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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:4E7V2WGH.xml</dcterms:identifier> <dcterms:creator identifier="GND:118503863">Archimedes</dcterms:creator> <dcterms:title xml:lang="la">Archimedis De iis quae ve huntur in aqua libri duo</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> </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/> <handwritten/> <pb file="0003" n="3"/> <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</head> <head xml:id="echoid-head2" xml:space="preserve">DE IIS QVAE VEHVNTVR <lb/>IN AQVA LIBRI DVO.</head> <head xml:id="echoid-head3" xml:space="preserve">A FEDERICO COMMANDINO <lb/>VRBINATE IN PRISTINVM <lb/>NITOREM RESTITVTI, ET <lb/>COMMENTARIIS ILLVSTRATI.</head> <figure> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0005-01"/> </figure> <figure> <image file="0005-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0005-02"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head4" xml:space="preserve">CVM PRIVILEGIO IN ANNOS X. <lb/>BONONIAE,</head> <p> <s xml:id="echoid-s1" xml:space="preserve">Ex Officina Alexandri Benacii.</s> <s xml:id="echoid-s2" xml:space="preserve"/> </p> <handwritten/> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head5" xml:space="preserve">M D LXV.</head> <figure> <image file="0005-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0005-03"/> </figure> <pb file="0006" n="6"/> <pb file="0007" n="7"/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head6" xml:space="preserve">RANVTIO FARNESIO <lb/>CARDINALI AMPLISSIMO <lb/>ET OPTIMO.</head> <p> <s xml:id="echoid-s3" xml:space="preserve"><emph style="sc">QVod</emph> tibi ſuperioribus diebus <lb/>pollicitus ſum, cum libellum <lb/>Ptolemæi de Analemmate in lu <lb/>cem proferrem, breui fore, vc<unsure/> <lb/>Archimedis etiam libri de ijs, <lb/>quæ in aqua vehuntur, & </s> <s xml:id="echoid-s4" xml:space="preserve">emen <lb/>datiores, & </s> <s xml:id="echoid-s5" xml:space="preserve">fortaſſe opera mea <lb/>illuſtriores ederentur: </s> <s xml:id="echoid-s6" xml:space="preserve">mihi non committendum eſ-<lb/>ſe duxi, vt iure optimo malum nomen, præſertim à<unsure/> <lb/>te, cui tantopere debeo, exiſtimari poſſem. </s> <s xml:id="echoid-s7" xml:space="preserve">quam-<lb/>uis cum mecum conſidero ſuſcepti negocij difficul <lb/>tates, quas multo plures, & </s> <s xml:id="echoid-s8" xml:space="preserve">multo grauiores, quàm <lb/>in libello de Analemmate deprehendi; </s> <s xml:id="echoid-s9" xml:space="preserve">vereor ne id <lb/>planè non aſſecutus ſim, quod ab initio ſpectaui, vt <lb/>mathematicarum diſciplinarum ſtudioſis hac in par-<lb/>te ſatisfacerem. </s> <s xml:id="echoid-s10" xml:space="preserve">cum enim græcus Archimedis co-<lb/>dex nondum in lucem venerit, non ſolum is, qui eum <lb/>latinitate donauit, multis in locis fœ de lapſus eſt, ve-<lb/>rum etiam codex ipſe, vt etiam interpres fatetur, ve-<lb/>tuſtate corruptus, & </s> <s xml:id="echoid-s11" xml:space="preserve">mancus eſt; </s> <s xml:id="echoid-s12" xml:space="preserve">duæq́; </s> <s xml:id="echoid-s13" xml:space="preserve">integræ <lb/>ἀποδείξεις, quas demonſtrationes dicimus, deperie-<lb/>runt. </s> <s xml:id="echoid-s14" xml:space="preserve">quæ iactura quantam vim habeat ad pertur-<lb/>bandum admirabilem illum ordinem, quo inter ſe <lb/>mathematicæ diſciplinæ quodãmodo connexæ ſunt, <pb file="0008" n="8"/> tibi, qui iam in iis multam operam, multumq́; </s> <s xml:id="echoid-s15" xml:space="preserve">ſtu-<lb/>dium poſuiſti, cogitandum relinquo. </s> <s xml:id="echoid-s16" xml:space="preserve">nonnulla præ-<lb/>terea Archimedes vt perſpicua in his tractandis po-<lb/>nere non dubitauit, quæ veteres mathematici, qui <lb/>de conicis conſcripſerunt, plurimis, & </s> <s xml:id="echoid-s17" xml:space="preserve">firmiſsimis <lb/>argumentis probauerũt. </s> <s xml:id="echoid-s18" xml:space="preserve">Hæc autem idcirco à nobis <lb/>omnino ignorantur; </s> <s xml:id="echoid-s19" xml:space="preserve">quòd poſtremi quatuor libri <lb/>conicorum Apollonii Pergæi adhuc in tenebris de-<lb/>liteſcunt. </s> <s xml:id="echoid-s20" xml:space="preserve">Qua quidem in re (vt mea fert opinio) <lb/>ſingulari fato fuerunt mathematicæ diſciplinæ, cum <lb/>tot ſcriptorum præclara monumenta interierint, per <lb/>quæ non ſolum in ſtudioſos homines, uerum etiam <lb/>in humanũ genus mirabiles utilitates importatæ fuiſ-<lb/>ſent. </s> <s xml:id="echoid-s21" xml:space="preserve">nam cum mecum conſidero quàm late pateant <lb/>hæ nobiliſsimæ ſcientiæ, quãtopere rebus publicis & </s> <s xml:id="echoid-s22" xml:space="preserve"><lb/>priuatis admirabili quadã ratione, atque ordine gu-<lb/>bernandis neceſſariæ ſint, dubitãdum non exiſtimo, <lb/>quin magna ſit habenda gratia huius diuini boni au-<lb/>ctoribus, & </s> <s xml:id="echoid-s23" xml:space="preserve">inuentoribus: </s> <s xml:id="echoid-s24" xml:space="preserve">ueterumq́; </s> <s xml:id="echoid-s25" xml:space="preserve">græcorum pru <lb/>dentiam ſatis admirari non poſſum, qui pueros cum <lb/>primum fari cœpiſſent, his diſciplinis imbuendos cu <lb/>rabant, ut à prima ætate multiplicis, ac ſubtilis ſcien-<lb/>tiæ contemplationi aſſueti nihil paruum, aut humile <lb/>cogitarent: </s> <s xml:id="echoid-s26" xml:space="preserve">ſed uel ſe totos ijs artibus traderent, qua-<lb/>rum ope ciuitatibus ſuis & </s> <s xml:id="echoid-s27" xml:space="preserve">præſidio, & </s> <s xml:id="echoid-s28" xml:space="preserve">ornamento <lb/>eſſe poſſent: </s> <s xml:id="echoid-s29" xml:space="preserve">uel humanis ſtudijs multam ſalutem di-<lb/>centes, diuinam philoſophiam toto animo amplexa-<lb/>rentur, cum ad eam per mathematicas diſciplinas fa- <pb file="0009" n="9"/> ciliorem ſibi aditum comparaſſent. </s> <s xml:id="echoid-s30" xml:space="preserve">quamobrem gra <lb/>uisſimum damnum factum eſt in tot præſtãtisſimis <lb/>uiris: </s> <s xml:id="echoid-s31" xml:space="preserve">quorũ ſcripta ſi in manus noſtras perueniſſent, <lb/>profecto multo præclarius cum rebus humanis age-<lb/>retur. </s> <s xml:id="echoid-s32" xml:space="preserve">complures enim, qui nunctot difficultatibus <lb/>ab his ſtudijs deterrentur, hac ratione priuatis & </s> <s xml:id="echoid-s33" xml:space="preserve">pu-<lb/>blicis rationibus optime conſuluiſſent. </s> <s xml:id="echoid-s34" xml:space="preserve">Cum hæc ita <lb/>eſſent, tamen nullum mihi laborem ſubterfugiendũ <lb/>eſſe iudicaui, quo ſtudioſis hominibus, qui in mathe <lb/>maticis diſciplinis toto animo incumbũt, facilior pa <lb/>teret aditus ad abſtruſa, & </s> <s xml:id="echoid-s35" xml:space="preserve">recondita ſenſa tanti ſcri-<lb/>ptoris intelligenda: </s> <s xml:id="echoid-s36" xml:space="preserve">nec à uetere meo in ſtituto diſce-<lb/>dere uolui; </s> <s xml:id="echoid-s37" xml:space="preserve">ſcis enim me multos abhinc annos hanc <lb/>eandem prouinciam, Archimedis quàm plurima ſcri <lb/>pta illuſtrandi ſuſcepiſſe. </s> <s xml:id="echoid-s38" xml:space="preserve">quod neque arrogãtia, nec <lb/>inanis gloriæ ſpe adductus ſum, ut facerẽ, ſed me ue-<lb/>hementer in hanc mentem impulit honeſtisſima cu-<lb/>piditas de ſtudioſis hominibus benemerẽdi: </s> <s xml:id="echoid-s39" xml:space="preserve">etenim <lb/>ſemper mea fuit ſentẽtia, mathematicum, qui libros <lb/>Archimedis accuratisſime non euoluerit, uix mathe-<lb/>maticum appellari debere: </s> <s xml:id="echoid-s40" xml:space="preserve">cum eū neceſſe ſit in mul <lb/>tarum rerum ignoratione uerſari, ſine quibus mathe <lb/>maticæ diſciplinæ imperfectæ quodammodo, atque <lb/>in choatæ ſunt habendæ. </s> <s xml:id="echoid-s41" xml:space="preserve">Dedi igitur operam, ut his <lb/>etiam Archimedis libris, quoad eius fieri poſſet, per <lb/>me aliqua lux afferretur. </s> <s xml:id="echoid-s42" xml:space="preserve">quos ut Archimedis eſſe nõ <lb/>dubitarem, duæ non contemnendæ cauſſæ fuerunt. <lb/></s> <s xml:id="echoid-s43" xml:space="preserve">una quòd in tanta obſcuritate ab interpretis inſcitia, <pb file="0010" n="10"/> & </s> <s xml:id="echoid-s44" xml:space="preserve">à uetuſtate profecta, neſcio quod ueſtigium illius <lb/>acuti, & </s> <s xml:id="echoid-s45" xml:space="preserve">perſpicacis ingenij, quo Archimedes excel-<lb/>luit, impreſſum apparet: </s> <s xml:id="echoid-s46" xml:space="preserve">altera quòd tum græci, tum <lb/>latini ſcriptores grauisſimi hos ut Archimedis libros <lb/>recognoſcũt. </s> <s xml:id="echoid-s47" xml:space="preserve">Strabo enim in primo libro hæc ad uer <lb/>bũ ſcribit. </s> <s xml:id="echoid-s48" xml:space="preserve">ὁδὲ οὕτος ἡδὺς ἐ<unsure/>στὶν, ὥστε η{αὶ} μὴ μαθηματιηὸςὤν, οὐδὲ <lb/>τὴν Αρχιμήδουςβεβαιοῖ δόξαν, ὅτιφησὶνἐη{εῖ} νος ἐν τοῖς περἱ τῶνὀχον-<lb/>μένων, παντὸς ὑγροῦ καθεστηηότος, καἱ μένοντος τὴνἑ πιφάν{ει}αν σφαιρι-<lb/>κὴν {εῖ}ν{αι}, σφ{αὶ} ρας ταυτὸ ηέντρον ἐφούσης τῆ γῆ. </s> <s xml:id="echoid-s49" xml:space="preserve">ταὺ την γάρ τὴν δοξαν <lb/>ἀποδέχονται πάντες οἱ μαθημάτων πῶς άψάμενοι. </s> <s xml:id="echoid-s50" xml:space="preserve">& </s> <s xml:id="echoid-s51" xml:space="preserve">Pappus Ale-<lb/>xandrinus in octauo mathematicarum collectionum <lb/>libro hæc ſcripta reliquit, ηαλοῦσι δὲ μηχανιηοὺςοἱ παλαιοὶ, <lb/>κ{αὶ} τοὺς θαυμασιουργοὺς, ὡνοἱ μὲν διὰ πνευμὰτων φιλοτεχνοῦσιν, ὡς <lb/>ἥρων πνευματιηοῖς, οἱ δὲ διὰ νευρίων καὶ σπάρτωνἐμψύχωνκινήσεις δο-<lb/>κοῦσι μιμ\~εισθαι, ὡςἥρων αὐτομάτοις, καὶ ζυγίοις: </s> <s xml:id="echoid-s52" xml:space="preserve">ἄλλοι δὲ διὰ τῶν ἐφ<unsure/> <lb/>ὕδατος ὁχουμένων, ὡςὰρχιμήδης ὀχουμένοις. </s> <s xml:id="echoid-s53" xml:space="preserve">Vitruuius etiam <lb/>in octauo libro de his eiſdem Archimedis libris me. <lb/></s> <s xml:id="echoid-s54" xml:space="preserve">minit. </s> <s xml:id="echoid-s55" xml:space="preserve">Fortaſſe, inquit, qui Archimedis libros legit, di <lb/>cet non poſſe fieri ueram ex aqua librationem: </s> <s xml:id="echoid-s56" xml:space="preserve">ſed ei <lb/>placet aquam non eſſe libratam, ſed ſphæroides habe <lb/>re ſchema: </s> <s xml:id="echoid-s57" xml:space="preserve">& </s> <s xml:id="echoid-s58" xml:space="preserve">ibi habere centrum, quo loci habet or-<lb/>bis terrarum. </s> <s xml:id="echoid-s59" xml:space="preserve">ut nemini dubium eſſe posſit, quin & </s> <s xml:id="echoid-s60" xml:space="preserve"><lb/>genere ſcriptionis, & </s> <s xml:id="echoid-s61" xml:space="preserve">tãtorum uirorum auctoritate, <lb/>ut germani Archimedis libri attente legendi, & </s> <s xml:id="echoid-s62" xml:space="preserve">per-<lb/>pendendi ſint: </s> <s xml:id="echoid-s63" xml:space="preserve">præſertim cum in ijs multa continean <lb/>tur cognitione dignisſirna, quæ nõ tam ad mathema <lb/>ticas diſciplinas, quàm ad naturæ obſcuritatem ſpe-<lb/>ctant. </s> <s xml:id="echoid-s64" xml:space="preserve">Quamobrem ego ne tanto, & </s> <s xml:id="echoid-s65" xml:space="preserve">tam fructuoſo <lb/>theſauro diutius ſtudioſi carerent, primum loca par- <pb file="0011" n="11"/> tim interpretis errore deprauata emendaui; </s> <s xml:id="echoid-s66" xml:space="preserve">partim <lb/>uetuſtate corrupta & </s> <s xml:id="echoid-s67" xml:space="preserve">conſumpta in priſtinam inte-<lb/>gritatem redegi, compluribus, quæ deſiderabantur, <lb/>meo, ut aiunt, marte ſuppletis. </s> <s xml:id="echoid-s68" xml:space="preserve">Deinde quoniam Ar-<lb/>chimedes, quemadmodum ſupra dixi, non nulla po-<lb/>nit, ut perſpicua, & </s> <s xml:id="echoid-s69" xml:space="preserve">quæ uel ipſe, uel ſuperiores ma-<lb/>thematici ἀποδείξεσι confirmauerunt, coactus ſum non <lb/>ſine maximo negotio ex ijs principijs conicæ diſcipli <lb/>næ Apollonij Pergæi, quæ in manus noſtras peruene-<lb/>rũt, nouas probationes adhibere, nequid eſſet, quod <lb/>diligentem lectorem in hac parte remorari poſſet. </s> <s xml:id="echoid-s70" xml:space="preserve">re <lb/>ſtabat, ut theorema illud, quod ſine cognitione cen-<lb/>tri grauitatis corporum ſolidorũ percipinon poteſt, <lb/>uidelicet, Centrum grauitatis in portionibus conoi-<lb/>dis rectanguli axem ita diuidere, ut pars, quæ ad uer-<lb/>ticem terminatur, reliquæ partis, quæ ad baſim ſit du <lb/>pla, certisſimis rationibus comprobarem. </s> <s xml:id="echoid-s71" xml:space="preserve">ſed huic <lb/>quoque rei prouiſum eſt à me: </s> <s xml:id="echoid-s72" xml:space="preserve">ſeorſumq́ ab his li-<lb/>bris de cẽtro grauitatis ſolidorũ uberrime cõſcripſi. <lb/></s> <s xml:id="echoid-s73" xml:space="preserve">denique nihil prætermiſi, quod ad Archimedem in <lb/>hac materia illuſtrandum attineret. </s> <s xml:id="echoid-s74" xml:space="preserve">quod ſi, ut ſpero, <lb/>aſſecutus ſum, ſatis magnum fructum mihi cepiſſe ui <lb/>debor laborum, & </s> <s xml:id="echoid-s75" xml:space="preserve">uigiliarum mearum: </s> <s xml:id="echoid-s76" xml:space="preserve">ſin ſecus acci <lb/>derit, hoc me tamen conſolabor, quòd omnes intelli <lb/>gent, honeſtisſimo meo conſilio, non tã ingenij mei <lb/>imbecillitatem, quàm rei obſcuritatem, & </s> <s xml:id="echoid-s77" xml:space="preserve">temporũ <lb/>iniurias obſtitiſſe. </s> <s xml:id="echoid-s78" xml:space="preserve">Hoc loco ſuperuacaneum eſſe arbi <lb/>tror pluribus uerbis exponere, cur tibi amplisſime <pb file="0012" n="12"/> Cardinalis, has lucubrationes meas dicare conſtitue-<lb/>rim. </s> <s xml:id="echoid-s79" xml:space="preserve">tantis enim beneficijs à te affectus, quanta fem-<lb/>per & </s> <s xml:id="echoid-s80" xml:space="preserve">meminero, & </s> <s xml:id="echoid-s81" xml:space="preserve">prædicabo; </s> <s xml:id="echoid-s82" xml:space="preserve">tanta liberalitate cõ-<lb/>plexus, quantam ne optare quidem unquam auſus eſ <lb/>ſem. </s> <s xml:id="echoid-s83" xml:space="preserve">cupio memorem, & </s> <s xml:id="echoid-s84" xml:space="preserve">erga te gratum animũ qua <lb/>ratione poſſum, oſtendere. </s> <s xml:id="echoid-s85" xml:space="preserve">quãuis ſi de te nihil aliud <lb/>præter auditum haberem, ſi amplitudini tuæ tanto-<lb/>pere deuinctus non eſſem; </s> <s xml:id="echoid-s86" xml:space="preserve">tua in omni genere diſci-<lb/>plinarum excellentia, tua grauitas, atque innocentia <lb/>me magnopere hortata eſſet, ut te potisſimum deli-<lb/>gerem, ſub cuius clarisſimi nominis ſplendore hi Ar-<lb/>chimedis libri ab obliuione hominum, atque à ſilen-<lb/>tio uindicarentur. </s> <s xml:id="echoid-s87" xml:space="preserve">uerecundius de te in præſentia di-<lb/>cerem, ne uiderer aſſentationi potius, quàm ueritati <lb/>ſeruire; </s> <s xml:id="echoid-s88" xml:space="preserve">niſi omnibus perſuaſisſimum eſſet, diuinas & </s> <s xml:id="echoid-s89" xml:space="preserve"><lb/>inauditas uirtutes tuas cum ſingulari eruditione con <lb/>iunctas in illo ſanctisſimo Reip. </s> <s xml:id="echoid-s90" xml:space="preserve">chriſtianæ conſilio <lb/>tanquam lumen aliquod elucere. </s> <s xml:id="echoid-s91" xml:space="preserve">quamobrem ea, <lb/>qua ſoles, benignitate, fidelisſimi clientis tui munus <lb/>accipies; </s> <s xml:id="echoid-s92" xml:space="preserve">quod tibi, qui & </s> <s xml:id="echoid-s93" xml:space="preserve">mathematicis diſciplinis, <lb/>& </s> <s xml:id="echoid-s94" xml:space="preserve">phiſiologiæ ſtudijs tantopere delectaris, non iniu-<lb/>cundum fore confido.</s> <s xml:id="echoid-s95" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head7" xml:space="preserve">Federicus Commandinus.</head> <pb o="1" file="0013" n="13"/> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head8" xml:space="preserve">ARCHIMEDIS DE IIS <lb/>QVAE VEHVNTVR IN AQVA <lb/>LIBER PRIMVS.</head> <head xml:id="echoid-head9" xml:space="preserve">CVM COMMENTARIIS FEDERICI <lb/>COMMANDINI VRBINATIS.</head> <head xml:id="echoid-head10" xml:space="preserve">POSITIO.</head> <p> <s xml:id="echoid-s96" xml:space="preserve">PONATVR humidi eam <lb/>eſſe naturam, vt partibus ip-<lb/>ſius æqualiter iacentibus, & </s> <s xml:id="echoid-s97" xml:space="preserve"><lb/>continuatis inter ſe ſe, minus <lb/>preſſa à magis preſſa expella <lb/>tur. </s> <s xml:id="echoid-s98" xml:space="preserve">Vnaquæque autem pars <lb/>eius premitur humido ſupra <lb/>ipſam exiſtente ad perpendiculum, ſi humidum <lb/>ſit deſcendens in aliquo, aut ab alio aliquo preſ-<lb/>ſum.</s> <s xml:id="echoid-s99" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div8" type="section" level="1" n="8"> <head xml:id="echoid-head11" xml:space="preserve">PROPOSITIO I.</head> <p> <s xml:id="echoid-s100" xml:space="preserve">SI ſuperficies aliqua plano ſecetur per idẽ ſem-<lb/>per punctum; </s> <s xml:id="echoid-s101" xml:space="preserve">ſitq́; </s> <s xml:id="echoid-s102" xml:space="preserve">ſectio circuli circunferen-<lb/>tia, centrum habens punctum illud, per quod pla <lb/>no ſecatur: </s> <s xml:id="echoid-s103" xml:space="preserve">ſphæræ ſuperficies erit.</s> <s xml:id="echoid-s104" xml:space="preserve"/> </p> <pb file="0014" n="14" rhead="ARCHIMEDIS"/> <p> <s xml:id="echoid-s105" xml:space="preserve">SECETVR ſuperficies aliqua plano per k punctum <lb/>ducto: </s> <s xml:id="echoid-s106" xml:space="preserve">& </s> <s xml:id="echoid-s107" xml:space="preserve">ſicſectio ſemper circuli circunferentia, centrum <lb/>habens punctum k. </s> <s xml:id="echoid-s108" xml:space="preserve">Dico eam ſphæræ ſuperficiem eſſe. </s> <s xml:id="echoid-s109" xml:space="preserve">Si <lb/>enim non eſt ſphæræ ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> cies; </s> <s xml:id="echoid-s110" xml:space="preserve">rectæ lineæ, quæ à pun-<lb/>cto k ad circunferentiam du-<lb/>cuntur non omnes æquales e-<lb/>runt. </s> <s xml:id="echoid-s111" xml:space="preserve">Itaque ſint a b puncta <lb/>in ſuperficie; </s> <s xml:id="echoid-s112" xml:space="preserve">& </s> <s xml:id="echoid-s113" xml:space="preserve">inæquales li-<lb/>neæ a k k b: </s> <s xml:id="echoid-s114" xml:space="preserve">per ipſas autem <lb/>a k k b planum ducatur, quod <lb/>ſectionem faciat in ſuperficie <lb/>lineam d a b c. </s> <s xml:id="echoid-s115" xml:space="preserve">ergo d a b c cir <lb/>culi circunferentia eſt, cuius <lb/>centrum k; </s> <s xml:id="echoid-s116" xml:space="preserve">quoniam ſuperficies eiuſmodi ponebatur: </s> <s xml:id="echoid-s117" xml:space="preserve">& </s> <s xml:id="echoid-s118" xml:space="preserve"><lb/>idcirco æquales inter ſe ſunt a k k b, ſed & </s> <s xml:id="echoid-s119" xml:space="preserve">inæquales; </s> <s xml:id="echoid-s120" xml:space="preserve">quod <lb/>fieri non poteſt. </s> <s xml:id="echoid-s121" xml:space="preserve">conſtat igitur ſuperficiem eam eſſe ſphæ-<lb/>ræ ſuperficiem.</s> <s xml:id="echoid-s122" xml:space="preserve"/> </p> <div xml:id="echoid-div8" 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/4E7V2WGH/figures/0014-01"/> </figure> </div> </div> <div xml:id="echoid-div10" type="section" level="1" n="9"> <head xml:id="echoid-head12" xml:space="preserve">PROPOSITIO II.</head> <p> <s xml:id="echoid-s123" xml:space="preserve"><emph style="sc">Omnis</emph> humidi conſiſtentis, atque manen-<lb/>tis ſuperficies ſphærica eſt; </s> <s xml:id="echoid-s124" xml:space="preserve">cuius ſphæræ centrũ <lb/>eſtidem, quod centrum terræ.</s> <s xml:id="echoid-s125" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s126" xml:space="preserve">INTELLIGATVR humidũ conſiſtens, manẽsq;</s> <s xml:id="echoid-s127" xml:space="preserve">: <lb/>& </s> <s xml:id="echoid-s128" xml:space="preserve">ſecetur ipſius ſuperficies plano per centrum terræ du-<lb/>cto. </s> <s xml:id="echoid-s129" xml:space="preserve">ſit autem terræ centrum k: </s> <s xml:id="echoid-s130" xml:space="preserve">& </s> <s xml:id="echoid-s131" xml:space="preserve">ſuperficieiſectio, linea <lb/>a b c d. </s> <s xml:id="echoid-s132" xml:space="preserve">Dico lineam a b c d circuli circunferentiam eſſe, cu <lb/>ius centrum k. </s> <s xml:id="echoid-s133" xml:space="preserve">Si enim non eſt, rectæ lineæ à puncto k ad <lb/>lineam a b c d ductæ non erunt æquales. </s> <s xml:id="echoid-s134" xml:space="preserve">Sumatur recta li <lb/>nea quibuſdam quidem à puncto k ad ipſam a b c d ductis <lb/>maior; </s> <s xml:id="echoid-s135" xml:space="preserve">quibuſdam uero minor; </s> <s xml:id="echoid-s136" xml:space="preserve">& </s> <s xml:id="echoid-s137" xml:space="preserve">ex centro k, interual- <pb o="2" file="0015" n="15" rhead="DE IIS QVAE VEH. IN AQVA."/> loq; </s> <s xml:id="echoid-s138" xml:space="preserve">lineæ ſumptæ circulus deſcribatur. </s> <s xml:id="echoid-s139" xml:space="preserve">cadet ergo ipſius <lb/>circunferentia partim <lb/> <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> extra lineam a b c d, par <lb/>tim intra; </s> <s xml:id="echoid-s140" xml:space="preserve">quoniam ea, <lb/>quæ ex centro quibuſ-<lb/>dam quidem à puncto <lb/>k ad ipſam ductis eſtma <lb/>ior; </s> <s xml:id="echoid-s141" xml:space="preserve">& </s> <s xml:id="echoid-s142" xml:space="preserve">quibuſdam mi-<lb/>nor. </s> <s xml:id="echoid-s143" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s144" xml:space="preserve">ſit circuli de-<lb/>ſcripti circunferentia <lb/>fb h: </s> <s xml:id="echoid-s145" xml:space="preserve">& </s> <s xml:id="echoid-s146" xml:space="preserve">ex b ad k ducta <lb/>linea, iungãtur fk k h e, <lb/>quæ angulos æquales faciant. </s> <s xml:id="echoid-s147" xml:space="preserve">deſcribatur autem & </s> <s xml:id="echoid-s148" xml:space="preserve">ex cen <lb/>tro k circunferentia quædam x o p in plano, & </s> <s xml:id="echoid-s149" xml:space="preserve">in humido. <lb/></s> <s xml:id="echoid-s150" xml:space="preserve">ergo partes humidi, quæ ſunt ad circunferentiam x o p æ-<lb/>qualiter iacent, ac continuatæ inter ſe ſe: </s> <s xml:id="echoid-s151" xml:space="preserve">& </s> <s xml:id="echoid-s152" xml:space="preserve">premũtur qui <lb/>dem partes, quæ ad x o circunferentiam, humido, quod lo <lb/>co a b continetur: </s> <s xml:id="echoid-s153" xml:space="preserve">quæ uero ad circunferentiam o p pre-<lb/>muntur humido, quod continetur b e. </s> <s xml:id="echoid-s154" xml:space="preserve">inæqualiter igitur <lb/>premuntur partes humidi ad cir cunferentiã x o, & </s> <s xml:id="echoid-s155" xml:space="preserve">ad o p. </s> <s xml:id="echoid-s156" xml:space="preserve"><lb/>quare minus preſſæ à magis presſis expellentur. </s> <s xml:id="echoid-s157" xml:space="preserve">non er-<lb/>go conſiſtet humidum. </s> <s xml:id="echoid-s158" xml:space="preserve">Atqui ponebatur conſiſtens, & </s> <s xml:id="echoid-s159" xml:space="preserve">ma <lb/>nens. </s> <s xml:id="echoid-s160" xml:space="preserve">neceſſarium eſt igitur lineam a b c d eſſe circuli cir <lb/>cunferentiam, cuius centrum k. </s> <s xml:id="echoid-s161" xml:space="preserve">Similiter autem demon-<lb/>ſtrabitur, & </s> <s xml:id="echoid-s162" xml:space="preserve">ſi quomodocunque aliter ſuperficies humidi <lb/>plano ſecta fuerit per centrum terræ ſectionem circuli cir <lb/>cunferentiam eſſe: </s> <s xml:id="echoid-s163" xml:space="preserve">& </s> <s xml:id="echoid-s164" xml:space="preserve">centrum ipſius eſſe, quod & </s> <s xml:id="echoid-s165" xml:space="preserve">terræ cen <lb/>trum. </s> <s xml:id="echoid-s166" xml:space="preserve">Ex quibus conſtat ſuperficiem humidi conſiſtentis, <lb/> <anchor type="note" xlink:label="note-0015-01a" xlink:href="note-0015-01"/> atque manentis ſphæricam eſſe: </s> <s xml:id="echoid-s167" xml:space="preserve">& </s> <s xml:id="echoid-s168" xml:space="preserve">eius ſphæræ centrum <lb/>idem, quod centrum terræ: </s> <s xml:id="echoid-s169" xml:space="preserve">quoniam eiuſmodi eſt, ut ſecta <lb/>per idem ſemper punctum ſectionem faciat circuli circun <lb/>ferentiam, centrum habentis punctum illud, per quod ipſa <lb/>plano ſecatur.</s> <s xml:id="echoid-s170" xml:space="preserve"/> </p> <div xml:id="echoid-div10" 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/4E7V2WGH/figures/0015-01"/> </figure> <note position="right" xlink:label="note-0015-01" xlink:href="note-0015-01a" xml:space="preserve">Prima hu <lb/>ius.</note> </div> <pb file="0016" n="16" rhead="ARCHIMEDIS"/> </div> <div xml:id="echoid-div12" type="section" level="1" n="10"> <head xml:id="echoid-head13" xml:space="preserve">PROPOSITIO III.</head> <p> <s xml:id="echoid-s171" xml:space="preserve"><emph style="sc">Solidarvm</emph> magnitudinum, quæ æqualẽ <lb/>molem habentes æque graues ſunt, atque humi-<lb/>dum; </s> <s xml:id="echoid-s172" xml:space="preserve">in humidum demiſſæ demergentur ita, vt <lb/>ex humidi ſuperficie nihil extet: </s> <s xml:id="echoid-s173" xml:space="preserve">non tamen ad <lb/>huc deorſum ferentur.</s> <s xml:id="echoid-s174" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s175" xml:space="preserve">SIT magnitudo aliqua æque grauis, atque humidum: <lb/></s> <s xml:id="echoid-s176" xml:space="preserve">& </s> <s xml:id="echoid-s177" xml:space="preserve">ſi fieri poteſt, in humidum demiſſa extet ex ſuperficie ip <lb/>ſius: </s> <s xml:id="echoid-s178" xml:space="preserve">conſiſtat autem humidum, maneatq;</s> <s xml:id="echoid-s179" xml:space="preserve">: & </s> <s xml:id="echoid-s180" xml:space="preserve">intelligatur <lb/>aliquod planum ductũ <lb/> <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> per cẽtrum terræ, & </s> <s xml:id="echoid-s181" xml:space="preserve">hu <lb/>midi, ac per ſolidam ma <lb/>gnitudinem, ut ſit ſuper <lb/>ficiei quidem humidi ſe <lb/>ctio a b c d; </s> <s xml:id="echoid-s182" xml:space="preserve">ſolidæ uero <lb/>magnitudinis inſiden-<lb/>tis e h t f; </s> <s xml:id="echoid-s183" xml:space="preserve">& </s> <s xml:id="echoid-s184" xml:space="preserve">terræ cen-<lb/>trum k: </s> <s xml:id="echoid-s185" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s186" xml:space="preserve">ſolidæ ma-<lb/>gnitudinis pars, quæ in <lb/>humido eſt, b h t c; </s> <s xml:id="echoid-s187" xml:space="preserve">& </s> <s xml:id="echoid-s188" xml:space="preserve"><lb/>quæ extra humidum b e f c. </s> <s xml:id="echoid-s189" xml:space="preserve">intelligatur etiam ſolida figu-<lb/>ra comprehenſa pyramide, baſim quidem habente paralle <lb/>logrammum, quod eſt in ſuperficie humidi; </s> <s xml:id="echoid-s190" xml:space="preserve">uerticem au-<lb/>tem centrum terræ: </s> <s xml:id="echoid-s191" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s192" xml:space="preserve">ſectio plani, in quo eſt a b c d cir-<lb/>cunferentia, & </s> <s xml:id="echoid-s193" xml:space="preserve">planorum pyramidis k l, k m: </s> <s xml:id="echoid-s194" xml:space="preserve">& </s> <s xml:id="echoid-s195" xml:space="preserve">deſcriba-<lb/>tur quædam alterius ſphæræ ſuperficies x o p circa centrũ <lb/>k, in humido ſub e f h t, ut ſit ipſa x o p ſectio facta à ſuper fi <lb/>cie plani. </s> <s xml:id="echoid-s196" xml:space="preserve">Sumatur præterea alia quædam pyramis æqua-<lb/>lis, & </s> <s xml:id="echoid-s197" xml:space="preserve">ſimilis comprehendenti ſolidam figuram, ipſi con- <pb o="3" file="0017" n="17" rhead="DE IIS QVAE VEH. IN AQVA."/> iuncta, & </s> <s xml:id="echoid-s198" xml:space="preserve">continuata: </s> <s xml:id="echoid-s199" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s200" xml:space="preserve">ſectio planorũ ipſius K m K n: <lb/></s> <s xml:id="echoid-s201" xml:space="preserve">& </s> <s xml:id="echoid-s202" xml:space="preserve">in humido intelligatur quædam magnitudo r s q y ex ip <lb/>ſo humido conſtans, æqualis, & </s> <s xml:id="echoid-s203" xml:space="preserve">ſimilis ſolidæ b h t c, quæ <lb/>quidem pars eſt ſolidæ magnitudinis in humido demerſa. </s> <s xml:id="echoid-s204" xml:space="preserve"><lb/>partes igitur humidi, quæ ſcilicet in prima pyramide ſuper <lb/>ficie x o continetur, & </s> <s xml:id="echoid-s205" xml:space="preserve">quæ in altera continetur p o, æquali <lb/>ter ſunt poſitæ, & </s> <s xml:id="echoid-s206" xml:space="preserve">continuatæ ſed non ſimiliter premun-<lb/>tur. </s> <s xml:id="echoid-s207" xml:space="preserve">nam contenta quidem x o, premitur ſolido e h t f, & </s> <s xml:id="echoid-s208" xml:space="preserve"><lb/>humido interiecto inter ſuperficies x o, l m, & </s> <s xml:id="echoid-s209" xml:space="preserve">plana pyra-<lb/>midis; </s> <s xml:id="echoid-s210" xml:space="preserve">contenta uero p o premitur ſolido r s q y, & </s> <s xml:id="echoid-s211" xml:space="preserve">humi-<lb/>do inter ſuperficies o p, m n, & </s> <s xml:id="echoid-s212" xml:space="preserve">pyramidis plana interiecto. </s> <s xml:id="echoid-s213" xml:space="preserve"><lb/>minor autem eſt grauitas humidi, quod eſt inter m n, o p, <lb/>quàm eius, quod inter l m, x o. </s> <s xml:id="echoid-s214" xml:space="preserve">ſolidum enim r s q y eſt mi <lb/>nus ſolido e h t f: </s> <s xml:id="echoid-s215" xml:space="preserve">cum ſit æquale ipſi b h t c; </s> <s xml:id="echoid-s216" xml:space="preserve">quia magnitu <lb/>dine æquale, & </s> <s xml:id="echoid-s217" xml:space="preserve">æque graue ponitur ſolidum, atque humi-<lb/>dum: </s> <s xml:id="echoid-s218" xml:space="preserve">reliquum autem reliquo inæquale eſt. </s> <s xml:id="echoid-s219" xml:space="preserve">conſtatigitur <lb/>partem contentã ſuperficie o p, expelli ab ea, quæ ipſa x o <lb/>continetur: </s> <s xml:id="echoid-s220" xml:space="preserve">& </s> <s xml:id="echoid-s221" xml:space="preserve">non conſiſtere humidum. </s> <s xml:id="echoid-s222" xml:space="preserve">ponebatur au-<lb/>tem conſiſtens, & </s> <s xml:id="echoid-s223" xml:space="preserve">manens: </s> <s xml:id="echoid-s224" xml:space="preserve">non ergo ex ſuperficie humidi <lb/>extat aliquid ſolidæ magnitudinis. </s> <s xml:id="echoid-s225" xml:space="preserve">ſed neque demerſum <lb/>ſolidum ad inferiora feretur. </s> <s xml:id="echoid-s226" xml:space="preserve">Similiter enim prementur <lb/>omnes partes humidi æqualiter poſitæ, cum ſolidum ſit æ-<lb/>que graue, atque humidum.</s> <s xml:id="echoid-s227" xml:space="preserve"/> </p> <div xml:id="echoid-div12" type="float" level="2" n="1"> <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/4E7V2WGH/figures/0016-01"/> </figure> </div> </div> <div xml:id="echoid-div14" type="section" level="1" n="11"> <head xml:id="echoid-head14" xml:space="preserve">PROPOSITIO IIII.</head> <p> <s xml:id="echoid-s228" xml:space="preserve"><emph style="sc">Solidarvm</emph> magnitudinum, quæcunque <lb/>leuior humido fuerit, demiſſa in humidum non <lb/>demergetur tota, ſed aliqua pars ipſius ex humi-<lb/>di ſuperficie extabit.</s> <s xml:id="echoid-s229" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s230" xml:space="preserve">SIT magnitudo ſolida humido leuior; </s> <s xml:id="echoid-s231" xml:space="preserve">& </s> <s xml:id="echoid-s232" xml:space="preserve">demiſſa in hu <lb/>midum demergatur tota, ſi fieri poteſt, ut nulla pars ipſius <pb file="0018" n="18" rhead="ARCHIMEDIS"/> extet ex humidi ſuperficie. </s> <s xml:id="echoid-s233" xml:space="preserve">conſiſtat autem humidum, ma <lb/>neatq;</s> <s xml:id="echoid-s234" xml:space="preserve">: & </s> <s xml:id="echoid-s235" xml:space="preserve">intelligatur aliquod planum ductum per centrũ <lb/>terræ, per humidum, & </s> <s xml:id="echoid-s236" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0018-01a" xlink:href="fig-0018-01"/> per magnitudinem ſoli-<lb/>dam: </s> <s xml:id="echoid-s237" xml:space="preserve">à quo ſuperficies <lb/>quidem humidi ſecetur <lb/>ſecundum circunferen-<lb/>tiam a b c; </s> <s xml:id="echoid-s238" xml:space="preserve">ſolida autem <lb/>magnitudo ſecundum fi <lb/>guram, in qua r: </s> <s xml:id="echoid-s239" xml:space="preserve">& </s> <s xml:id="echoid-s240" xml:space="preserve">cen-<lb/>trum terræ ſit K. </s> <s xml:id="echoid-s241" xml:space="preserve">Intelli <lb/>gatur etiam quædam py <lb/>ramis comprehendens <lb/>figuram r, ſicuti prius, quæ pũctum K pro uertice habeat: <lb/></s> <s xml:id="echoid-s242" xml:space="preserve">fecenturq; </s> <s xml:id="echoid-s243" xml:space="preserve">ipſius plana à ſuperficie plani a b c ſecundum <lb/>a K K b: </s> <s xml:id="echoid-s244" xml:space="preserve">& </s> <s xml:id="echoid-s245" xml:space="preserve">ſumatur pyramis alia æ qualis, & </s> <s xml:id="echoid-s246" xml:space="preserve">ſimilis ſuperio <lb/>ri, cuius plana ſecentur à plano a b c, ſecundum b K K c: </s> <s xml:id="echoid-s247" xml:space="preserve"><lb/>deinde alterius ſphæræ ſuperficies quædam deſcribatur in <lb/>humido circa centrum K, ſub ſolida magnitudine: </s> <s xml:id="echoid-s248" xml:space="preserve">& </s> <s xml:id="echoid-s249" xml:space="preserve">ſece-<lb/>tur ab eodem plano ſecundum x o p: </s> <s xml:id="echoid-s250" xml:space="preserve">poſtremo intelliga-<lb/>tur alia magnitudo h in poſteriori pyramide, quæ ex humi <lb/>do conſtet, & </s> <s xml:id="echoid-s251" xml:space="preserve">ſolidæ magnitudini r ſit æ qualis. </s> <s xml:id="echoid-s252" xml:space="preserve">partes igi-<lb/>tur humidi, & </s> <s xml:id="echoid-s253" xml:space="preserve">quæ in prima pyramide continetur ſuperfi-<lb/>cie x o; </s> <s xml:id="echoid-s254" xml:space="preserve">& </s> <s xml:id="echoid-s255" xml:space="preserve">quæ in ſecunda ſuperficie o p continetur, æquali <lb/>ter iacent, & </s> <s xml:id="echoid-s256" xml:space="preserve">continuatæ inter ſe ſe; </s> <s xml:id="echoid-s257" xml:space="preserve">non tamen ſimiliter <lb/>premuntur: </s> <s xml:id="echoid-s258" xml:space="preserve">nam quæ eſt in prima pyramide premitur ma <lb/>gnitudine ſolida r, & </s> <s xml:id="echoid-s259" xml:space="preserve">humido cõtinente ipſam, quod eſt in <lb/>loco pyramidis a b o x: </s> <s xml:id="echoid-s260" xml:space="preserve">quæ uero in altera pyramide pre-<lb/>mitur ſolida magnitudineh, & </s> <s xml:id="echoid-s261" xml:space="preserve">humido ipſam continente <lb/>in loco pyramidis p o b c. </s> <s xml:id="echoid-s262" xml:space="preserve">At grauitas ſolidæ magnitudi-<lb/>nis r, minor eſt grauitate humidi, in quo h: </s> <s xml:id="echoid-s263" xml:space="preserve">quoniam ma-<lb/>gnitudo ſolida mole quidem æqualis, & </s> <s xml:id="echoid-s264" xml:space="preserve">humido leuior po <lb/>nitur: </s> <s xml:id="echoid-s265" xml:space="preserve">grauitas autem humidi continentis magnitudines <lb/>r h eſt æqualis; </s> <s xml:id="echoid-s266" xml:space="preserve">cum pyramides æquales ſint. </s> <s xml:id="echoid-s267" xml:space="preserve">magis ergo <pb o="4" file="0019" n="19" rhead="DE IIS QVAE VEH. IN AQVA."/> premitur pars humidi, quæ eſt ſub ſuperficie o p. </s> <s xml:id="echoid-s268" xml:space="preserve">quare ex-<lb/>pellet partem minus preſſam, & </s> <s xml:id="echoid-s269" xml:space="preserve">non manebit humidum. <lb/></s> <s xml:id="echoid-s270" xml:space="preserve">ponebatur autem manens. </s> <s xml:id="echoid-s271" xml:space="preserve">non igitur demergetur tota, <lb/>ſed aliqua pars ipſius ex humidi ſuperficie extabit.</s> <s xml:id="echoid-s272" xml:space="preserve"/> </p> <div xml:id="echoid-div14" type="float" level="2" n="1"> <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a"> <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0018-01"/> </figure> </div> </div> <div xml:id="echoid-div16" type="section" level="1" n="12"> <head xml:id="echoid-head15" xml:space="preserve">PROPOSITIO V.</head> <p> <s xml:id="echoid-s273" xml:space="preserve"><emph style="sc">Solidarvm</emph> magnitudinum quæcunque le <lb/>uior humido fuerit, demiſſa in humidum vſque <lb/>eô demergetur, vt tanta moles humidi, quanta <lb/>eſt partis demerſæ, eandem, quam tota magnitu-<lb/>do, grauitatem habeat.</s> <s xml:id="echoid-s274" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s275" xml:space="preserve">DISPONANTVR eadem, quæſupra: </s> <s xml:id="echoid-s276" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s277" xml:space="preserve">humi-<lb/>dum manens: </s> <s xml:id="echoid-s278" xml:space="preserve">& </s> <s xml:id="echoid-s279" xml:space="preserve">magnitudo e h t f humido leuior. </s> <s xml:id="echoid-s280" xml:space="preserve">Si igitur <lb/>humidum manet, ſimiliter prementur eius partes, quæ æ-<lb/>qualiter iacent. </s> <s xml:id="echoid-s281" xml:space="preserve">ſimiliter ergo premetur humidum ſub ſu-<lb/>perficiebus x o o p. <lb/></s> <s xml:id="echoid-s282" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0019-01a" xlink:href="fig-0019-01"/> quare æ qualis eſt graui-<lb/>tas, qua premuntur. </s> <s xml:id="echoid-s283" xml:space="preserve">eſt <lb/>autem & </s> <s xml:id="echoid-s284" xml:space="preserve">grauitas humi <lb/>di, quod in prima pyra-<lb/>mide abſque ſolido b h <lb/>t c, æqualis grauitati hu <lb/>midi, quod in altera py-<lb/>ramide abſq; </s> <s xml:id="echoid-s285" xml:space="preserve">r s q y hu-<lb/>mido. </s> <s xml:id="echoid-s286" xml:space="preserve">perſpicuum eſt <lb/>igitur grauitatem ma-<lb/>gnitudinis e h t f grauitati humidi r s q y æqualem eſſe. </s> <s xml:id="echoid-s287" xml:space="preserve">ex <lb/>quibus conſtat, tantam humidi molem, quanta eſt pars de <lb/>merſa ſolidæ magnitudinis, eandem, quam tota magnitu-<lb/>do habere grauitatem.</s> <s xml:id="echoid-s288" xml:space="preserve"/> </p> <div xml:id="echoid-div16" type="float" level="2" n="1"> <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a"> <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0019-01"/> </figure> </div> <pb file="0020" n="20" rhead="ARCHIMEDIS"/> </div> <div xml:id="echoid-div18" type="section" level="1" n="13"> <head xml:id="echoid-head16" xml:space="preserve">PROPOSITIO VI.</head> <p> <s xml:id="echoid-s289" xml:space="preserve"><emph style="sc">Solidae</emph> magnitudines humido leuiores, in <lb/>humidum impulſæ ſurſum feruntur tanta ui, quã <lb/>to humidum molem habens magnitudini æqua-<lb/>lem, grauius eſt ipſa magnitudine.</s> <s xml:id="echoid-s290" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s291" xml:space="preserve">SIT enim magnitudo aleuior humido: </s> <s xml:id="echoid-s292" xml:space="preserve">& </s> <s xml:id="echoid-s293" xml:space="preserve">ſit magnitu <lb/>dinis quidem a grauitas b: </s> <s xml:id="echoid-s294" xml:space="preserve">humidi uero molem habentis <lb/>æqualem ipſi a, grauitas ſit b c. </s> <s xml:id="echoid-s295" xml:space="preserve">demonſtrandum eſt magni <lb/>tudinem a in humidum impulſam tanta ui ſurſum ferri, <lb/>quanta eſt grauitas c. </s> <s xml:id="echoid-s296" xml:space="preserve">accipiatur enim quædam magnitu-<lb/>do, in qua d habens grauitatem ipſi c æqualem. </s> <s xml:id="echoid-s297" xml:space="preserve">Itaque <lb/>magnitudo ex utriſque magnitudinibus conſtans, in qui-<lb/>bus a d, leuior eſt humido: </s> <s xml:id="echoid-s298" xml:space="preserve">nam magnitudinis quidem quæ <lb/>ex utriſque conſtat grauitas eſt b c; </s> <s xml:id="echoid-s299" xml:space="preserve">humidi uero habentis <lb/>molem ipſis æ qualem grauitas maior eſt, quàm b c: </s> <s xml:id="echoid-s300" xml:space="preserve">quo-<lb/>niam b c grauitas eſt humidi <lb/> <anchor type="figure" xlink:label="fig-0020-01a" xlink:href="fig-0020-01"/> molẽ habentis æqualem ipſia. <lb/></s> <s xml:id="echoid-s301" xml:space="preserve">Si ergo demittatur in humidũ <lb/>magnitudo ex utriſque a d con <lb/>ſtans; </s> <s xml:id="echoid-s302" xml:space="preserve">uſque eò demergetur, ut <lb/>tanta moles humidi, quanta eſt <lb/>pars magnitudinis demerſa eã <lb/>dem, quam tota magnitudo <lb/>grauitatem habeat. </s> <s xml:id="echoid-s303" xml:space="preserve">hoc enim <lb/>iam demonſtratum eſt. </s> <s xml:id="echoid-s304" xml:space="preserve">ſit autẽ <lb/>ſuperſicies humidi alicuius a b <lb/>c d circunferentia. </s> <s xml:id="echoid-s305" xml:space="preserve">Quoniam igitur tanta moles humidi, <lb/>quanta eſt magnitudo a grauitatem habet eandem, quam <lb/>magnitudines a d: </s> <s xml:id="echoid-s306" xml:space="preserve">perſpicuum eſt partem ipſius demer-<lb/>ſam eſſe magnitudinem a; </s> <s xml:id="echoid-s307" xml:space="preserve">reliquam uero d totam ex hu- <pb o="5" file="0021" n="21" rhead="DE IIS QVAE VEH. IN AQVA."/> midi ſuperficie extare. </s> <s xml:id="echoid-s308" xml:space="preserve">Quare conſtat magnitudinem a <lb/>tanta ui ſurſum ferri, quãta deorſum premitur ab eo, quod <lb/>eſt ſupra; </s> <s xml:id="echoid-s309" xml:space="preserve">uidelicet à d, cũ neutra ab altera expellatur, ſed <lb/>d fertur deorſum tanta grauitate, quanta eſt c: </s> <s xml:id="echoid-s310" xml:space="preserve">ponebatur <lb/>enim grauitas eius, in quo d ipſi c æqualis. </s> <s xml:id="echoid-s311" xml:space="preserve">patet igitur <lb/>illud quod demonſtrare oportebat.</s> <s xml:id="echoid-s312" xml:space="preserve"/> </p> <div xml:id="echoid-div18" type="float" level="2" n="1"> <figure xlink:label="fig-0020-01" xlink:href="fig-0020-01a"> <image file="0020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0020-01"/> </figure> </div> </div> <div xml:id="echoid-div20" type="section" level="1" n="14"> <head xml:id="echoid-head17" xml:space="preserve">PROPOSITIO VII.</head> <p> <s xml:id="echoid-s313" xml:space="preserve"><emph style="sc">Solidae</emph> magnitudines humido grauiores <lb/>demiſſæ in humidum ferentur deorſum, donec <lb/>deſcendant: </s> <s xml:id="echoid-s314" xml:space="preserve">& </s> <s xml:id="echoid-s315" xml:space="preserve">erunt in humido tanto leuiores, <lb/>quanta eſt grauitas humidi molem habentis ſoli-<lb/>dæ magnitudini æqualem.</s> <s xml:id="echoid-s316" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s317" xml:space="preserve">SOLIDAS magnitudines humido grauiores, in hu-<lb/>midum demiſſas deorſum quidam ferri, donec deſcẽdant, <lb/>manifeſtum eſt: </s> <s xml:id="echoid-s318" xml:space="preserve">partes enim humidi, quæ ſub eis ſunt, pre-<lb/>muntur magis, quàm partes æqualiter ipſis adiacentes; <lb/></s> <s xml:id="echoid-s319" xml:space="preserve">quoniam magnitudo ſolida humido grauior ponitur: </s> <s xml:id="echoid-s320" xml:space="preserve">le-<lb/>uiores autem eſſe uti dictum eſt, demonſtrabitur hoc mo-<lb/>do. </s> <s xml:id="echoid-s321" xml:space="preserve">Sit enim aliqua ma-<lb/> <anchor type="figure" xlink:label="fig-0021-01a" xlink:href="fig-0021-01"/> gnitudo a grauior hu-<lb/>mido: </s> <s xml:id="echoid-s322" xml:space="preserve">& </s> <s xml:id="echoid-s323" xml:space="preserve">fit magnitudi-<lb/>nis quidem a grauitas <lb/>b c: </s> <s xml:id="echoid-s324" xml:space="preserve">humidi uero molẽ <lb/>habentis æqualem ipſi a <lb/>grauitas ſit b. </s> <s xml:id="echoid-s325" xml:space="preserve">demon-<lb/>ſtrandum eſt magnitudi <lb/>nem a in humido exiſtē <lb/>tem habere grauitatem <lb/>æqualem ipſi c. </s> <s xml:id="echoid-s326" xml:space="preserve">Accipia <lb/>tur enim alia aliqua magnitudo, in qua d, leuior humido;</s> <s xml:id="echoid-s327" xml:space="preserve"> <pb file="0022" n="22" rhead="ARCHIMEDIS"/> cuius grauitas ſitipſi b æqualis: </s> <s xml:id="echoid-s328" xml:space="preserve">humidi ucro molem ha-<lb/>bentis æqualem magnitudini d, ſit grauitas æqualis b c. <lb/></s> <s xml:id="echoid-s329" xml:space="preserve">Itaque compoſitis magnitudinibus a d, magnitudo ex <lb/>utriſque conſtans æque grauis erit, atque ipſum humidũ: </s> <s xml:id="echoid-s330" xml:space="preserve"><lb/>grauitas enim utrarũque magnitudinum eſt æqualis utriſ-<lb/>que grauitatibus, uidelicet b c, & </s> <s xml:id="echoid-s331" xml:space="preserve">b: </s> <s xml:id="echoid-s332" xml:space="preserve">grauitas autem humi <lb/>di habentis molem æqualem utriſque magnitudinibus, eſt <lb/>eiſdem grauitatibus æqualis. </s> <s xml:id="echoid-s333" xml:space="preserve">Demisſis igitur magnitudini <lb/>bus, & </s> <s xml:id="echoid-s334" xml:space="preserve">in humidum proiectis æque graues erunt, atque hu <lb/>midum: </s> <s xml:id="echoid-s335" xml:space="preserve">neque ſurſum, neque deorſum ferentur: </s> <s xml:id="echoid-s336" xml:space="preserve">quoniam <lb/>magnitudo quidem a <lb/> <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> grauior humido feretur <lb/>deorſum; </s> <s xml:id="echoid-s337" xml:space="preserve">& </s> <s xml:id="echoid-s338" xml:space="preserve">eadem ui à <lb/>magnitudine d ſurſum <lb/>retrahetur: </s> <s xml:id="echoid-s339" xml:space="preserve">magnitudo <lb/>autem d humido leuior <lb/>feretur ſurſum tanta ui, <lb/>quanta eſt grauitas c: <lb/></s> <s xml:id="echoid-s340" xml:space="preserve">demõſtratũ enim eſt ma <lb/> <anchor type="note" xlink:label="note-0022-01a" xlink:href="note-0022-01"/> gnitudines ſolidas hu-<lb/>mido leuiores, impulſas <lb/>in humidum tanta uiretrahi ſurſum, quanto humidum ha <lb/>bens molem magnitudini æqualem grauius eſt ipſa magni <lb/>tudine. </s> <s xml:id="echoid-s341" xml:space="preserve">Athumidum molem habens æqualem d, grauius <lb/>eſt, quam d, ipſa c grauitate. </s> <s xml:id="echoid-s342" xml:space="preserve">Conſtatigitur magnitudinem <lb/>a deorſum ferri tanta grauitate, quanta eſt c. </s> <s xml:id="echoid-s343" xml:space="preserve">quod de-<lb/>monſtrare oportebat.</s> <s xml:id="echoid-s344" xml:space="preserve"/> </p> <div xml:id="echoid-div20" 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/4E7V2WGH/figures/0021-01"/> </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/4E7V2WGH/figures/0022-01"/> </figure> <note position="left" xlink:label="note-0022-01" xlink:href="note-0022-01a" xml:space="preserve">6. huius.</note> </div> </div> <div xml:id="echoid-div22" type="section" level="1" n="15"> <head xml:id="echoid-head18" xml:space="preserve">POSITIO II.</head> <p> <s xml:id="echoid-s345" xml:space="preserve"><emph style="sc">Ponatvr</emph> eorum, quæ in humido ſurſum <lb/>feruntur, vnumquodque ſurſum ferri ſecundum <lb/>perpendicularem, quæ per centrum grauitatis ip <lb/>ſorum ducitur.</s> <s xml:id="echoid-s346" xml:space="preserve"/> </p> <pb o="6" file="0023" n="23" rhead="DE IIS QVAE VEH. IN AQVA."/> </div> <div xml:id="echoid-div23" type="section" level="1" n="16"> <head xml:id="echoid-head19" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s347" xml:space="preserve">AT ucro ea, quæ feruntur deorſum, ſecundum perpendicula-<lb/>rem, quæ per centrum grauit atis ipſorum ducitur, ſimiliter ferri, <lb/>uel tanquam notum, uel ut ab alijs poſitum prætermiſit.</s> <s xml:id="echoid-s348" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div24" type="section" level="1" n="17"> <head xml:id="echoid-head20" xml:space="preserve">PROPOSITIO VIII.</head> <p> <s xml:id="echoid-s349" xml:space="preserve">SI aliqua magnitudo ſolida leuior humido, <lb/> <anchor type="note" xlink:label="note-0023-01a" xlink:href="note-0023-01"/> quæ figuram portionis ſphæræ habeat, in humi-<lb/> <anchor type="note" xlink:label="note-0023-02a" xlink:href="note-0023-02"/> dum demittatur, ita vt baſis portionis non tan-<lb/>gat humidum: </s> <s xml:id="echoid-s350" xml:space="preserve">figura inſidebit recta, ita vt axis <lb/>portionis ſit ſecundum perpendicularem. </s> <s xml:id="echoid-s351" xml:space="preserve">Et ſi <lb/>ab aliquo inclinetur figura, vt baſis portionis hu-<lb/>midum cõtingat; </s> <s xml:id="echoid-s352" xml:space="preserve">non manebit inclinata ſi demit <lb/>tatur, ſed recta reſtituetur.</s> <s xml:id="echoid-s353" xml:space="preserve"/> </p> <div xml:id="echoid-div24" type="float" level="2" n="1"> <note position="right" xlink:label="note-0023-01" xlink:href="note-0023-01a" xml:space="preserve">A</note> <note position="right" xlink:label="note-0023-02" xlink:href="note-0023-02a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s354" xml:space="preserve">[INTELLIGATVR quædam magnitudo, qualis <lb/> <anchor type="note" xlink:label="note-0023-03a" xlink:href="note-0023-03"/> dicta eſt, in humidum demiſſa: </s> <s xml:id="echoid-s355" xml:space="preserve">& </s> <s xml:id="echoid-s356" xml:space="preserve">ducatur planum per axẽ <lb/>portionis, & </s> <s xml:id="echoid-s357" xml:space="preserve">per terræ <lb/> <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> centrum, ut ſit ſuperfi-<lb/>ciei humidi ſectio circũ <lb/>ferentia a b c d: </s> <s xml:id="echoid-s358" xml:space="preserve">& </s> <s xml:id="echoid-s359" xml:space="preserve">figu-<lb/>ræ ſectio e f h circunfe-<lb/>rentia: </s> <s xml:id="echoid-s360" xml:space="preserve">ſit autem e h <lb/>recta linea; </s> <s xml:id="echoid-s361" xml:space="preserve">& </s> <s xml:id="echoid-s362" xml:space="preserve">f t axis <lb/>portionis. </s> <s xml:id="echoid-s363" xml:space="preserve">Si igitur in-<lb/>clinetur figura, ita ut a-<lb/>xis portionis f t non ſit <lb/>ſecundum perpendicu-<lb/>larem. </s> <s xml:id="echoid-s364" xml:space="preserve">demonſtrandum eſt, non manere ipſam figu-<lb/>ram; </s> <s xml:id="echoid-s365" xml:space="preserve">ſed in rectum reſtitui. </s> <s xml:id="echoid-s366" xml:space="preserve">Itaque centrum ſphæræ eſt <pb file="0024" n="24" rhead="ARCHIMEDIS"/> in linea ft. </s> <s xml:id="echoid-s367" xml:space="preserve">nam ſit primum figura maior dimidia ſphære: <lb/></s> <s xml:id="echoid-s368" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s369" xml:space="preserve">in dimidia ſphæra ſphæræ centrum t; </s> <s xml:id="echoid-s370" xml:space="preserve">in minori por-<lb/>tioneſit centrum p; </s> <s xml:id="echoid-s371" xml:space="preserve">& </s> <s xml:id="echoid-s372" xml:space="preserve">in maiori _k_: </s> <s xml:id="echoid-s373" xml:space="preserve">per _k_ uero, & </s> <s xml:id="echoid-s374" xml:space="preserve">terræ cen <lb/>trum l ducatur _k_ l ſecans circunferentiam e f h in pun-<lb/>cto n. </s> <s xml:id="echoid-s375" xml:space="preserve">Quoniam igitur unaquæque ſphæræportio axem <lb/> <anchor type="note" xlink:label="note-0024-01a" xlink:href="note-0024-01"/> habet in linea, quæ à cẽtro ſphæræ ad cius baſim perpen-<lb/>dicularis ducitur: </s> <s xml:id="echoid-s376" xml:space="preserve">habetq; </s> <s xml:id="echoid-s377" xml:space="preserve">in axe grauitatis centrum: <lb/></s> <s xml:id="echoid-s378" xml:space="preserve">portionis in humido demerſæ, quæ ex duabus ſphæræ <lb/>portionibus conſtat, axis erit in perpendiculari per _k_ du-<lb/>cta. </s> <s xml:id="echoid-s379" xml:space="preserve">& </s> <s xml:id="echoid-s380" xml:space="preserve">idcirco centrum grauitatis ipſius erit in linea n _k_, <lb/>quod ſit r. </s> <s xml:id="echoid-s381" xml:space="preserve">ſed totius portionis grauitatis centrum eſt in li <lb/> <anchor type="note" xlink:label="note-0024-02a" xlink:href="note-0024-02"/> nea f t inter _k_, & </s> <s xml:id="echoid-s382" xml:space="preserve">f, quod ſit x. </s> <s xml:id="echoid-s383" xml:space="preserve">reliquæ ergo figuræ, quæ eſt <lb/> <anchor type="note" xlink:label="note-0024-03a" xlink:href="note-0024-03"/> extra humidum, centrum erit in linea r x producta ad par <lb/>tes x; </s> <s xml:id="echoid-s384" xml:space="preserve">& </s> <s xml:id="echoid-s385" xml:space="preserve">aſſumpta ex ea, linea quadam, quæ ad r x eandem <lb/>proportionem habeat, quam grauitas portionis in humi-<lb/>do demerſæ habet ad grauitatem figuræ, quæ eſt extra hu-<lb/>midum. </s> <s xml:id="echoid-s386" xml:space="preserve">Sit autem s centrum dictæ figuræ: </s> <s xml:id="echoid-s387" xml:space="preserve">& </s> <s xml:id="echoid-s388" xml:space="preserve">per s duca-<lb/>tur perpendicularis l s. </s> <s xml:id="echoid-s389" xml:space="preserve">Feretur ergo grauitas figuræ qui-<lb/> <anchor type="note" xlink:label="note-0024-04a" xlink:href="note-0024-04"/> dem, quæ extra humidum per rectam s l deorſum; </s> <s xml:id="echoid-s390" xml:space="preserve">portio <lb/>nis autem, quæ in humido, ſurſum per rectam r l. </s> <s xml:id="echoid-s391" xml:space="preserve">quare <lb/>non manebit figura: </s> <s xml:id="echoid-s392" xml:space="preserve">ſed partes eius, quæ ſunt ad e, deor-<lb/>ſum; </s> <s xml:id="echoid-s393" xml:space="preserve">& </s> <s xml:id="echoid-s394" xml:space="preserve">quæ ad h ſurſum ſerẽtur: </s> <s xml:id="echoid-s395" xml:space="preserve">idq; </s> <s xml:id="echoid-s396" xml:space="preserve">cõtinenter fiet, quoad <lb/>ſ t ſit ſecundum perpendicularem. </s> <s xml:id="echoid-s397" xml:space="preserve">Eodem modo in aliis <lb/>portionibus idem demonſtrabitur.</s> <s xml:id="echoid-s398" xml:space="preserve">]</s> </p> <div xml:id="echoid-div25" type="float" level="2" n="2"> <note position="right" xlink:label="note-0023-03" xlink:href="note-0023-03a" xml:space="preserve">Suppleta <lb/>a Federi-<lb/>co Cõm.</note> <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0023-01"/> </figure> <note position="left" xlink:label="note-0024-01" xlink:href="note-0024-01a" xml:space="preserve">C</note> <note position="left" xlink:label="note-0024-02" xlink:href="note-0024-02a" xml:space="preserve">D</note> <note position="left" xlink:label="note-0024-03" xlink:href="note-0024-03a" xml:space="preserve">E</note> <note position="left" xlink:label="note-0024-04" xlink:href="note-0024-04a" xml:space="preserve">F</note> </div> <figure> <image file="0024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0024-01"/> </figure> <pb o="7" file="0025" n="25" rhead="DE IIS QVAE VEH. IN AQVA."/> </div> <div xml:id="echoid-div27" type="section" level="1" n="18"> <head xml:id="echoid-head21" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s399" xml:space="preserve">H_<emph style="sc">VIVS</emph>_ propoſitionis demonſtratio iniuria temporum deſidera-<lb/>tur, quam nos ita reſtituimus, ut ex figuris, quæ remanſerunt Archi <lb/>medem ſcripſiſſe colligi potuit: </s> <s xml:id="echoid-s400" xml:space="preserve">neque enim eas immutare uiſum est, <lb/>quæ uero ad declarationem, explicationémque addenda fuerant, in <lb/>commentarijs ſuppleuimus, id quod etiam præstitimus in ſecunda <lb/>propoſitione ſecundi libri.</s> <s xml:id="echoid-s401" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s402" xml:space="preserve">_SI aliqua magnitudo ſolida leuior humido.</s> <s xml:id="echoid-s403" xml:space="preserve">]_ Ea uerba, <lb/> <anchor type="note" xlink:label="note-0025-01a" xlink:href="note-0025-01"/> leuior bumido, nos addidimus, quæ in translatione non erant; </s> <s xml:id="echoid-s404" xml:space="preserve">quo-<lb/>niam de eiuſmodi magnitudinibus in bac propoſitione agitur.</s> <s xml:id="echoid-s405" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="1"> <note position="right" xlink:label="note-0025-01" xlink:href="note-0025-01a" xml:space="preserve">A</note> </div> <p> <s xml:id="echoid-s406" xml:space="preserve">In humidũ demittatur, ita ut baſis portionis nõ tangat hu <lb/> <anchor type="note" xlink:label="note-0025-02a" xlink:href="note-0025-02"/> midum.</s> <s xml:id="echoid-s407" xml:space="preserve">] _Hoc est in humidum ita demitt atur, ut baſis ſurſum ſpe_ <lb/>_ctet; </s> <s xml:id="echoid-s408" xml:space="preserve">uertex autem deorſum. </s> <s xml:id="echoid-s409" xml:space="preserve">quod quidem opponitur ei, quod in ſe-_ <lb/>_quenti dixit._ </s> <s xml:id="echoid-s410" xml:space="preserve">In humidum demittatur, ita ut baſis tota ſit in <lb/>humido. </s> <s xml:id="echoid-s411" xml:space="preserve">_His enim uerbis ſignificat portionem oppoſito modo in_ <lb/>_humidum demitti, ut ſcilicet uertex ſurſum; </s> <s xml:id="echoid-s412" xml:space="preserve">baſis autem deorſum_ <lb/>_uergat. </s> <s xml:id="echoid-s413" xml:space="preserve">eodem dicendi modo frequenter uſus est in ſecundo libro; </s> <s xml:id="echoid-s414" xml:space="preserve">in_ <lb/>_quo de portionibus conoidis rectangulitractatur._</s> <s xml:id="echoid-s415" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="2"> <note position="right" xlink:label="note-0025-02" xlink:href="note-0025-02a" xml:space="preserve">B</note> </div> <p style="it"> <s xml:id="echoid-s416" xml:space="preserve">_Quoniã igitur unaquæq; </s> <s xml:id="echoid-s417" xml:space="preserve">ſphæræ portio axẽ habet in linea,_ <lb/> <anchor type="note" xlink:label="note-0025-03a" xlink:href="note-0025-03"/> _quæ à cẽtro ſphæræ ad eius baſim perpẽdicularis ducitur.</s> <s xml:id="echoid-s418" xml:space="preserve">]_ <lb/>Iungatur enim b c, & </s> <s xml:id="echoid-s419" xml:space="preserve">k l ſecet circunferentiam a b c d in puncto g; <lb/></s> <s xml:id="echoid-s420" xml:space="preserve">lineam uero rectam b c in m. </s> <s xml:id="echoid-s421" xml:space="preserve">& </s> <s xml:id="echoid-s422" xml:space="preserve">quoniam duo circuli a b c d, e f b <lb/>ſecant ſe ſe in punctis b c; </s> <s xml:id="echoid-s423" xml:space="preserve">recta linea, quæ ipſorum centra coniun-<lb/>git, uidelicet k l lineam b c bifariam, & </s> <s xml:id="echoid-s424" xml:space="preserve">ad angulos rectos ſecat: </s> <s xml:id="echoid-s425" xml:space="preserve"><lb/>ut in commentarij s in Ptolemæi planiſpbærium oſtendimus. </s> <s xml:id="echoid-s426" xml:space="preserve">quare <lb/>portionis circuli b n c diameter eſt m n; </s> <s xml:id="echoid-s427" xml:space="preserve">& </s> <s xml:id="echoid-s428" xml:space="preserve">portionis b g c diame-<lb/> <anchor type="note" xlink:label="note-0025-04a" xlink:href="note-0025-04"/> ter m g: </s> <s xml:id="echoid-s429" xml:space="preserve">nam rectæ lineæ, quæ ipſi b c æquidistantes ex utraque <lb/>parte ducuntur, cum linea n g rectos angulos faciunt; </s> <s xml:id="echoid-s430" xml:space="preserve">& </s> <s xml:id="echoid-s431" xml:space="preserve">idcirco ab <lb/> <anchor type="note" xlink:label="note-0025-05a" xlink:href="note-0025-05"/> ipſa bifariam ſecantur. </s> <s xml:id="echoid-s432" xml:space="preserve">portionis igitur ſpbæræ b n c axis eſt n m; <lb/></s> <s xml:id="echoid-s433" xml:space="preserve">& </s> <s xml:id="echoid-s434" xml:space="preserve">portionis b g c axis m g. </s> <s xml:id="echoid-s435" xml:space="preserve">ex quo ſequitur, portionis in bumido <lb/>demerſæ axem eſſe in linea k l; </s> <s xml:id="echoid-s436" xml:space="preserve">ipſam ſcilicet n g. </s> <s xml:id="echoid-s437" xml:space="preserve">& </s> <s xml:id="echoid-s438" xml:space="preserve">cum grauita-<lb/>tis centrum cuius libet ſpbæræ portionis ſit in axe; </s> <s xml:id="echoid-s439" xml:space="preserve">quod nos in libro <pb file="0026" n="26" rhead="ARCHIMEDIS"/> @e centro grauitatis ſolidorum demonstrauimus: </s> <s xml:id="echoid-s440" xml:space="preserve">erit magnitudi-<lb/>nis ex utriſque portionibus b n c, b g c conſtantis; </s> <s xml:id="echoid-s441" xml:space="preserve">hoc eſt portionis <lb/>in humido demerſa grauitatis centrum in linea n g, quæ ipſarum <lb/>ſphæræ portionum centra graui-<lb/> <anchor type="figure" xlink:label="fig-0026-01a" xlink:href="fig-0026-01"/> tatis coniungit. </s> <s xml:id="echoid-s442" xml:space="preserve">ſi enim fieri po-<lb/>teſt, ſit extra lineam n g, ut in <lb/>q: </s> <s xml:id="echoid-s443" xml:space="preserve">sîtq; </s> <s xml:id="echoid-s444" xml:space="preserve">portionis b n c centrum <lb/>grauitatis u; </s> <s xml:id="echoid-s445" xml:space="preserve">& </s> <s xml:id="echoid-s446" xml:space="preserve">ducatur u q.</s> <s xml:id="echoid-s447" xml:space="preserve"/> </p> <div xml:id="echoid-div29" type="float" level="2" n="3"> <note position="right" xlink:label="note-0025-03" xlink:href="note-0025-03a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0025-04" xlink:href="note-0025-04a" xml:space="preserve">29. primi</note> <note position="right" xlink:label="note-0025-05" xlink:href="note-0025-05a" xml:space="preserve">3. tertii.</note> <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/4E7V2WGH/figures/0026-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s448" xml:space="preserve">Quoniam igitur à portione in bu-<lb/>mido demerſa aufertur ſphæræ <lb/>portio b n c, non habens idem cen <lb/>trum grauitatis: </s> <s xml:id="echoid-s449" xml:space="preserve">erit ex octaua <lb/>primi libri Archimcdis de centro <lb/>grauitatis planorum, reliquæ por <lb/>tionis b g c centrum in linea u q <lb/>producta. </s> <s xml:id="echoid-s450" xml:space="preserve">quod fieri non potest; </s> <s xml:id="echoid-s451" xml:space="preserve">eſt enim in axe ipſius mg. </s> <s xml:id="echoid-s452" xml:space="preserve">sequi-<lb/>tur ergo ut portionis in humido demerſæ centrum grauitatis ſit in li <lb/>nean k. </s> <s xml:id="echoid-s453" xml:space="preserve">quod oſtendendum propoſuimus.</s> <s xml:id="echoid-s454" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s455" xml:space="preserve">_Sed totius portionis grauitatis centrum eſt in linea ft, in-_ <lb/> <anchor type="note" xlink:label="note-0026-01a" xlink:href="note-0026-01"/> _ter_ k, _& </s> <s xml:id="echoid-s456" xml:space="preserve">f, quod ſit x.</s> <s xml:id="echoid-s457" xml:space="preserve">]_ Compleatur ſphæra, ut ſit portionis additæ <lb/>axis t y; </s> <s xml:id="echoid-s458" xml:space="preserve">& </s> <s xml:id="echoid-s459" xml:space="preserve">centrú grauitatis z. </s> <s xml:id="echoid-s460" xml:space="preserve">Itaque quoniá à tota ſphæra, cuius <lb/>grauitatis cétrum eſt k, ut etiam in eodem libro demóſtrauimus, au <lb/> <anchor type="note" xlink:label="note-0026-02a" xlink:href="note-0026-02"/> fertur portio e y h centrú grauitatis habens z: </s> <s xml:id="echoid-s461" xml:space="preserve">erit reliquæ portionis <lb/>e f h cétrú in linea z k producta. </s> <s xml:id="echoid-s462" xml:space="preserve">quare inter k. </s> <s xml:id="echoid-s463" xml:space="preserve">& </s> <s xml:id="echoid-s464" xml:space="preserve">f neceſſario cadet.</s> <s xml:id="echoid-s465" xml:space="preserve"/> </p> <div xml:id="echoid-div30" type="float" level="2" n="4"> <note position="left" xlink:label="note-0026-01" xlink:href="note-0026-01a" xml:space="preserve">D</note> <note position="left" xlink:label="note-0026-02" xlink:href="note-0026-02a" xml:space="preserve">8. primi <lb/>Archime <lb/>dis.</note> </div> <p> <s xml:id="echoid-s466" xml:space="preserve">Reliquæ ergo figuræ, quæ eſt extra humidum, centrum erit <lb/> <anchor type="note" xlink:label="note-0026-03a" xlink:href="note-0026-03"/> in linea r x producta.</s> <s xml:id="echoid-s467" xml:space="preserve">] _Ex eadem octaua primi libri Archime-_ <lb/>_dis de centro grauitatis planorum._</s> <s xml:id="echoid-s468" xml:space="preserve"/> </p> <div xml:id="echoid-div31" type="float" level="2" n="5"> <note position="left" xlink:label="note-0026-03" xlink:href="note-0026-03a" xml:space="preserve">E</note> </div> <p style="it"> <s xml:id="echoid-s469" xml:space="preserve">_Feretur ergo grauitas, figuræ quidem quæ extra humi_-<lb/> <anchor type="note" xlink:label="note-0026-04a" xlink:href="note-0026-04"/> _dum per rectam s l deorſum; </s> <s xml:id="echoid-s470" xml:space="preserve">portionis autem, quæ in_ <lb/>_humido ſurſum per rectam r l.</s> <s xml:id="echoid-s471" xml:space="preserve">]_ Ex antecedenti poſitio-<lb/>ne. </s> <s xml:id="echoid-s472" xml:space="preserve">magnitudo @enim, quæ in humido demerſa est, tanta ui per li-<lb/>neam r l ſurſum@fertur, quanta quæ extra humidum per li-<lb/>neam s l, deorſum: </s> <s xml:id="echoid-s473" xml:space="preserve">id quod ex propoſitione ſexta huius li-<lb/>briconſtare poteſt. </s> <s xml:id="echoid-s474" xml:space="preserve">& </s> <s xml:id="echoid-s475" xml:space="preserve">quoniam feruntur per alias, atque alias li- <pb o="8" file="0027" n="27" rhead="DE IIS QVAE VEH. IN AQVA."/> neas; </s> <s xml:id="echoid-s476" xml:space="preserve">neutra alteri obſistit, quo minus moueatur; </s> <s xml:id="echoid-s477" xml:space="preserve">ídq; </s> <s xml:id="echoid-s478" xml:space="preserve">continenter <lb/>fiat, dum portio in rectum fuerit conſtituta: </s> <s xml:id="echoid-s479" xml:space="preserve">tunc enim utrarumque <lb/>magnitudinum grauitatis centra in unam, eandémq; </s> <s xml:id="echoid-s480" xml:space="preserve">perpendicula-<lb/>rum conueniunt, uidelicet in axem portionis: </s> <s xml:id="echoid-s481" xml:space="preserve">& </s> <s xml:id="echoid-s482" xml:space="preserve">quanto conatu, im <lb/>petùue ea, quæ in humido eſt ſurſum, tanto quæ extra humidum de-<lb/>orſum per eandem lineam contendit. </s> <s xml:id="echoid-s483" xml:space="preserve">quare cum altera alteram non <lb/>ſuperet, non amplius mouebitur portio; </s> <s xml:id="echoid-s484" xml:space="preserve">ſed conſiſtet, manebítq; </s> <s xml:id="echoid-s485" xml:space="preserve">in <lb/>eodem ſemper ſitu; </s> <s xml:id="echoid-s486" xml:space="preserve">niſi forte aliqua cauſſa extrinſecus acceſſerit.</s> <s xml:id="echoid-s487" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="6"> <note position="left" xlink:label="note-0026-04" xlink:href="note-0026-04a" xml:space="preserve">F</note> </div> </div> <div xml:id="echoid-div34" type="section" level="1" n="19"> <head xml:id="echoid-head22" xml:space="preserve">PROPOSITIO IX.</head> <p> <s xml:id="echoid-s488" xml:space="preserve"><emph style="sc">Qvòd</emph> ſi figura humido leuior in humidum <lb/>demittatur, ita ut baſis tota ſit in humido; </s> <s xml:id="echoid-s489" xml:space="preserve">inſide <lb/>bit recta, ita ut axis ipſius ſecundum perpendicu <lb/>larem conſtituatur.</s> <s xml:id="echoid-s490" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s491" xml:space="preserve">INTELLIGATVR enim magnitudo aliqua, qua-<lb/>lis dicta eſt, in humidum demiſſa: </s> <s xml:id="echoid-s492" xml:space="preserve">& </s> <s xml:id="echoid-s493" xml:space="preserve">intelligatur planum <lb/>per axem portionis, & </s> <s xml:id="echoid-s494" xml:space="preserve">per centrum terræ ductum: </s> <s xml:id="echoid-s495" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s496" xml:space="preserve">ſu <lb/>perficiei quidem humidi ſectio a b c d circunferentia; </s> <s xml:id="echoid-s497" xml:space="preserve">figu <lb/>ræ autem ſectio circun ferentia e f h: </s> <s xml:id="echoid-s498" xml:space="preserve">& </s> <s xml:id="echoid-s499" xml:space="preserve">ſit e h recta linea: <lb/></s> <s xml:id="echoid-s500" xml:space="preserve">& </s> <s xml:id="echoid-s501" xml:space="preserve">axis portionis f t. </s> <s xml:id="echoid-s502" xml:space="preserve">Si igitur fieri poteſt, non ſit f t ſecun <lb/>dum perpendicularem.</s> <s xml:id="echoid-s503" xml:space="preserve"/> </p> <figure> <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0027-01"/> </figure> <p> <s xml:id="echoid-s504" xml:space="preserve">Demonſtrandum eſt non <lb/>manerefiguram; </s> <s xml:id="echoid-s505" xml:space="preserve">ſed in re <lb/>ctum reſtitui. </s> <s xml:id="echoid-s506" xml:space="preserve">eſt autem <lb/>centrum ſphæræ in linea <lb/>f t: </s> <s xml:id="echoid-s507" xml:space="preserve">rurſus enim ſit figu-<lb/>ra primo maior dimidia <lb/>ſphæra: </s> <s xml:id="echoid-s508" xml:space="preserve">& </s> <s xml:id="echoid-s509" xml:space="preserve">ſphæræ centrũ <lb/>in dimidia ſphæra ſit pun-<lb/>ctum t; </s> <s xml:id="echoid-s510" xml:space="preserve">in minore portione p; </s> <s xml:id="echoid-s511" xml:space="preserve">in maiori uero ſit _k_: </s> <s xml:id="echoid-s512" xml:space="preserve">& </s> <s xml:id="echoid-s513" xml:space="preserve">per <lb/>_k_, & </s> <s xml:id="echoid-s514" xml:space="preserve">terræ centrum l ducatur _k_ l. </s> <s xml:id="echoid-s515" xml:space="preserve">Itaque figura quæ eſt <lb/> <anchor type="note" xlink:label="note-0027-01a" xlink:href="note-0027-01"/> <pb file="0028" n="28" rhead="ARCHIMEDIS"/> extra humidi ſuperficiem, axem habetin perpendiculari <lb/>per _k_: </s> <s xml:id="echoid-s516" xml:space="preserve">& </s> <s xml:id="echoid-s517" xml:space="preserve">propter ea, quæ ſuperius dicta ſunt, centrum gra-<lb/>uitatis ipſius eſt in linea n _k_, quod ſitr; </s> <s xml:id="echoid-s518" xml:space="preserve">totius autem por-<lb/>tionis centrum grauitatis eſt in linea f t, inter _k_ & </s> <s xml:id="echoid-s519" xml:space="preserve">f, quod <lb/>ſit x. </s> <s xml:id="echoid-s520" xml:space="preserve">reliquæ ergo figuræ, eius ſcilicet, quæ eſt in humido, <lb/>centrum erit in rectalinea r x producta ad partes x; </s> <s xml:id="echoid-s521" xml:space="preserve">& </s> <s xml:id="echoid-s522" xml:space="preserve">aſ-<lb/> <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> ſumpta ex ea linea quadam, quæ ad x r eandem habeat pro <lb/>portionem, quam grauitas portionis, quæ eſt extra humi-<lb/>dum, ad grauitatem figuræ, quæ in humido. </s> <s xml:id="echoid-s523" xml:space="preserve">Sit autem o <lb/>centrum dictæ figuræ: </s> <s xml:id="echoid-s524" xml:space="preserve">& </s> <s xml:id="echoid-s525" xml:space="preserve">per o perpendicularis ducatur <lb/>l o. </s> <s xml:id="echoid-s526" xml:space="preserve">Feretur ergo grauitas portionis quidem, quæ eſt ex-<lb/>tra humidum, per rectam r l deorſum; </s> <s xml:id="echoid-s527" xml:space="preserve">figuræ autem, quæ <lb/>in humido, per rectam o l ſurſum. </s> <s xml:id="echoid-s528" xml:space="preserve">non manet igitur flgu-<lb/>ra; </s> <s xml:id="echoid-s529" xml:space="preserve">ſed partes eius, quæ ſuntad h, deorſum ferẽtur; </s> <s xml:id="echoid-s530" xml:space="preserve">& </s> <s xml:id="echoid-s531" xml:space="preserve">quæ <lb/>ad e ſurſum. </s> <s xml:id="echoid-s532" xml:space="preserve">atque hoc ſemper erit, donec f t ſecundum <lb/>perpendicularem fiat.</s> <s xml:id="echoid-s533" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="1"> <note position="right" xlink:label="note-0027-01" xlink:href="note-0027-01a" xml:space="preserve">A</note> <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/4E7V2WGH/figures/0028-01"/> </figure> </div> </div> <div xml:id="echoid-div36" type="section" level="1" n="20"> <head xml:id="echoid-head23" xml:space="preserve">COMMENTARIVS.</head> <p> <s xml:id="echoid-s534" xml:space="preserve">ITAQVE figura, quæ extra humidi ſuperficiem, <lb/> <anchor type="note" xlink:label="note-0028-01a" xlink:href="note-0028-01"/> axem habet in perpendiculari per _k_.</s> <s xml:id="echoid-s535" xml:space="preserve">]</s> </p> <div xml:id="echoid-div36" type="float" level="2" n="1"> <note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">A</note> </div> <p style="it"> <s xml:id="echoid-s536" xml:space="preserve"><emph style="sc">D_vcatvr_</emph> enim b c, quæ ſecet lineam n k in m: </s> <s xml:id="echoid-s537" xml:space="preserve">ipſa uero <lb/>n k circunferentiam a b c d ſecet in g. </s> <s xml:id="echoid-s538" xml:space="preserve">eodem modo, quo ſupra, de- <pb o="9" file="0029" n="29" rhead="DE IIS QVAE VEH. IN AQVA."/> monſtrabimus portionis ſphæræ b n c axem eſſe ipſam n m: </s> <s xml:id="echoid-s539" xml:space="preserve">& </s> <s xml:id="echoid-s540" xml:space="preserve"><lb/>portionis b g c axem g m. <lb/></s> <s xml:id="echoid-s541" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> quare centrum grauitatis utri <lb/>uſque, erit in linea n m. </s> <s xml:id="echoid-s542" xml:space="preserve">& </s> <s xml:id="echoid-s543" xml:space="preserve"><lb/>quoniam à portione b n c au-<lb/>fertur portio b g c, non ha-<lb/>bens@idem grauitatis centrú: <lb/></s> <s xml:id="echoid-s544" xml:space="preserve">reliquæ magnitudinis, quæ est <lb/>extra humidi ſuperficiem, cen-<lb/>trum grauitatis erit in linea <lb/>n k; </s> <s xml:id="echoid-s545" xml:space="preserve">quæ ſcilicet earum portionum centra grauitatis coniungit: </s> <s xml:id="echoid-s546" xml:space="preserve">ex <lb/>eadem octaua Archimedis.</s> <s xml:id="echoid-s547" xml:space="preserve"/> </p> <div xml:id="echoid-div37" type="float" level="2" n="2"> <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/4E7V2WGH/figures/0029-01"/> </figure> </div> <pb file="0030" n="30"/> </div> <div xml:id="echoid-div39" type="section" level="1" n="21"> <head xml:id="echoid-head24" xml:space="preserve">ARCHIMEDIS DE IIS <lb/>QVAE VEHVNTVR IN AQVA <lb/>LIBER SECVNDVS.</head> <head xml:id="echoid-head25" xml:space="preserve">CVM COMMENTARIIS FEDERICI <lb/>COMMANDINI VRBINATIS.</head> <head xml:id="echoid-head26" xml:space="preserve">PROPOSITIO I.</head> <p> <s xml:id="echoid-s548" xml:space="preserve">SI magnitudo aliqua humido <lb/>leuior demittatur in humi-<lb/>dum, eam in grauitate pro-<lb/>portionem habebit ad humi-<lb/>dum æqualis molis, quã pars <lb/>magnitudinis demerſa habet <lb/>ad totam magnitudinem.</s> <s xml:id="echoid-s549" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s550" xml:space="preserve">DEMITTATVR enim in humidum aliqua magni-<lb/>tudo ſolida, quæ ſit fa, leuior humido: </s> <s xml:id="echoid-s551" xml:space="preserve">& </s> <s xml:id="echoid-s552" xml:space="preserve">pars quidem ip-<lb/>ſius demerſa ſit a; </s> <s xml:id="echoid-s553" xml:space="preserve">quæ autem extra humidum f. </s> <s xml:id="echoid-s554" xml:space="preserve">demon-<lb/>ſtrandum eſt, ma <lb/> <anchor type="figure" xlink:label="fig-0030-01a" xlink:href="fig-0030-01"/> gnitudinem f a <lb/>ad humidum æ-<lb/>qualis molis eam <lb/>in grauitate pro-<lb/>portionem habe <lb/>re, quam habet <lb/>a ad fa. </s> <s xml:id="echoid-s555" xml:space="preserve">accipiatur enim aliqua humidi magnitudo n i <pb o="10" file="0031" n="31" rhead="DE IIS QVAE VEH. IN AQVA."/> æqualis magnitudini f a; </s> <s xml:id="echoid-s556" xml:space="preserve">ſitq, ipſi f æqualis n: </s> <s xml:id="echoid-s557" xml:space="preserve">& </s> <s xml:id="echoid-s558" xml:space="preserve">ipſi a æ-<lb/>qualis i. </s> <s xml:id="echoid-s559" xml:space="preserve">magnitudinis autem f a grauitas ſit b: </s> <s xml:id="echoid-s560" xml:space="preserve">& </s> <s xml:id="echoid-s561" xml:space="preserve">magni-<lb/>tudinis n i grauitas o r; </s> <s xml:id="echoid-s562" xml:space="preserve">& </s> <s xml:id="echoid-s563" xml:space="preserve">ipſius i ſit r. </s> <s xml:id="echoid-s564" xml:space="preserve">magnitudo igi-<lb/>tur f a ad n i eam proportionem habet, quam grauitas b <lb/>ad grauitatem or. </s> <s xml:id="echoid-s565" xml:space="preserve">Sed quoniam magnitudo f a in humi-<lb/>dum demiſſa leuior eſt humido; </s> <s xml:id="echoid-s566" xml:space="preserve">patet tantam humidi mo-<lb/>lem, quanta eſt pars magnitudin_i_s demerſa, eandem quam <lb/>magnitudo f a habere grauitatem. </s> <s xml:id="echoid-s567" xml:space="preserve">hoc enim ſuperius de-<lb/> <anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> monſtratum eſt. </s> <s xml:id="echoid-s568" xml:space="preserve">Atipſi a reſpondet humidum i, cuius qui <lb/>dem grauitas eſt r; </s> <s xml:id="echoid-s569" xml:space="preserve">& </s> <s xml:id="echoid-s570" xml:space="preserve">ipſius f a grauitas b. </s> <s xml:id="echoid-s571" xml:space="preserve">ergo b graui-<lb/>tas eius, quod habet molem æqualem toti magnitudini <lb/>f a, æqualis erit grauitati humidi i, uidelicetipſi r. </s> <s xml:id="echoid-s572" xml:space="preserve">Et quo <lb/>niam ut magnitudo f a ad humidum n i ſibi reſpondens, <lb/>ita eſt b ad o r: </s> <s xml:id="echoid-s573" xml:space="preserve">eſt autem b æqualis ipſi r: </s> <s xml:id="echoid-s574" xml:space="preserve">& </s> <s xml:id="echoid-s575" xml:space="preserve">utr ad o r, ita <lb/>i ad n i; </s> <s xml:id="echoid-s576" xml:space="preserve">& </s> <s xml:id="echoid-s577" xml:space="preserve">a ad f a. </s> <s xml:id="echoid-s578" xml:space="preserve">Sequitur ut f a ad humidum æqualis <lb/> <anchor type="note" xlink:label="note-0031-02a" xlink:href="note-0031-02"/> molis eam in grauitate proportionem habeat, quam ma-<lb/>gnitudo a habet ad f a. </s> <s xml:id="echoid-s579" xml:space="preserve">quod demonſtrare oportebat.</s> <s xml:id="echoid-s580" xml:space="preserve"/> </p> <div xml:id="echoid-div39" type="float" level="2" n="1"> <figure xlink:label="fig-0030-01" xlink:href="fig-0030-01a"> <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0030-01"/> </figure> <note position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">5. priml <lb/>huius.</note> <note position="right" xlink:label="note-0031-02" xlink:href="note-0031-02a" xml:space="preserve">11. quinta<unsure/></note> </div> </div> <div xml:id="echoid-div41" type="section" level="1" n="22"> <head xml:id="echoid-head27" xml:space="preserve">PROPOSITIO II.</head> <p> <s xml:id="echoid-s581" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/> <anchor type="note" xlink:label="note-0031-03a" xlink:href="note-0031-03"/> axem habuerit minorem, quam ſeſquialterum <lb/>eius, quæ uſque ad axem, quamcunque propor-<lb/>tionem habens ad humidum in grauitate; </s> <s xml:id="echoid-s582" xml:space="preserve">demiſ <lb/>ſa in humidum, ita ut baſis ipſius humidum non <lb/>contingat; </s> <s xml:id="echoid-s583" xml:space="preserve">& </s> <s xml:id="echoid-s584" xml:space="preserve">poſita inelinata, non manebit incli <lb/>nata; </s> <s xml:id="echoid-s585" xml:space="preserve">ſed recta reſtituetur. </s> <s xml:id="echoid-s586" xml:space="preserve">Rectam dico conſi-<lb/>ſtere talem portionem, quando planum quod ip <lb/>ſam ſecuit, ſuperficiei humidi fuerit æquidiſtans.</s> <s xml:id="echoid-s587" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="1"> <note position="right" xlink:label="note-0031-03" xlink:href="note-0031-03a" xml:space="preserve">A</note> </div> <p> <s xml:id="echoid-s588" xml:space="preserve">SIT portio rectanguli conoidis, qualis dicta eſt; </s> <s xml:id="echoid-s589" xml:space="preserve">& </s> <s xml:id="echoid-s590" xml:space="preserve">ia- <pb file="0032" n="32" rhead="ARCHIMEDIS"/> ceatinclinata. </s> <s xml:id="echoid-s591" xml:space="preserve">Demonſtrandum eſt non manere ipſam; </s> <s xml:id="echoid-s592" xml:space="preserve">ſed <lb/>rectam reſtitui. </s> <s xml:id="echoid-s593" xml:space="preserve">Itaque ſecta ipſa plano per axem, recto ad <lb/>planum, quod eſt in ſuperficie humidi, portionis ſectio ſit <lb/>a p o l rectanguli coni ſectio: </s> <s xml:id="echoid-s594" xml:space="preserve">axis portionis, & </s> <s xml:id="echoid-s595" xml:space="preserve">ſectionis <lb/>diameter n o: </s> <s xml:id="echoid-s596" xml:space="preserve">ſuperficiei autem humidi ſectio ſit i s. </s> <s xml:id="echoid-s597" xml:space="preserve">Si <lb/>igitur portio non eſt recta; </s> <s xml:id="echoid-s598" xml:space="preserve">non utique erit a l ipſi i s æ-<lb/>quidiſtans. </s> <s xml:id="echoid-s599" xml:space="preserve">quare n o cum i s non faciet angulos rectos. <lb/></s> <s xml:id="echoid-s600" xml:space="preserve">ducatur crgo k ω contingens ſectionem coni in p [quæ <lb/> <anchor type="note" xlink:label="note-0032-01a" xlink:href="note-0032-01"/> ipſi i s æquidiſtet: </s> <s xml:id="echoid-s601" xml:space="preserve">& </s> <s xml:id="echoid-s602" xml:space="preserve">à puncto p ad i s ducatur p f æquidi <lb/>ſtans ipſi o n, quæ erit ſectionis i p o s diameter, & </s> <s xml:id="echoid-s603" xml:space="preserve">axis por <lb/> <anchor type="note" xlink:label="note-0032-02a" xlink:href="note-0032-02"/> tionis in humido demerſæ. </s> <s xml:id="echoid-s604" xml:space="preserve">ſumantur deinde centra graui <lb/>tatum: </s> <s xml:id="echoid-s605" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s606" xml:space="preserve">ſolidæ magnitudinis a p o l grauitatis centrũ <lb/> <anchor type="note" xlink:label="note-0032-03a" xlink:href="note-0032-03"/> r; </s> <s xml:id="echoid-s607" xml:space="preserve">ipſius uero i p o s centrum ſit b: </s> <s xml:id="echoid-s608" xml:space="preserve">& </s> <s xml:id="echoid-s609" xml:space="preserve">iuncta b r produca-<lb/> <anchor type="note" xlink:label="note-0032-04a" xlink:href="note-0032-04"/> tur ad g, quod ſit centrum grauitatis reliquæ figuræ i s l a. <lb/></s> <s xml:id="echoid-s610" xml:space="preserve">Quoniam igitur n o ipſius quidem r o ſeſquialtera eſt; </s> <s xml:id="echoid-s611" xml:space="preserve"><lb/>eius autẽ, quæ uſque ad axẽ minor, quam ſeſquialtera; </s> <s xml:id="echoid-s612" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-0032-05a" xlink:href="note-0032-05"/> r o minor, quàm quæ uſque ad axem. </s> <s xml:id="echoid-s613" xml:space="preserve">Quare angulus r p ω <lb/> <anchor type="note" xlink:label="note-0032-06a" xlink:href="note-0032-06"/> acutus erit: </s> <s xml:id="echoid-s614" xml:space="preserve">cum enim linea, quæ uſque ad axem maior ſit <lb/>ipſa r o; </s> <s xml:id="echoid-s615" xml:space="preserve">quæ à puncto r ad k ω perpendicularis ducitur, <lb/>uidelicet r t, cũ <lb/> <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/> linea f p extra <lb/>ſectionem con <lb/>ueniet: </s> <s xml:id="echoid-s616" xml:space="preserve">& </s> <s xml:id="echoid-s617" xml:space="preserve">pro-<lb/>pterea inter p <lb/>& </s> <s xml:id="echoid-s618" xml:space="preserve">ω puncta ca-<lb/>datneceſſe eſt. <lb/></s> <s xml:id="echoid-s619" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s620" xml:space="preserve">ſi per b g <lb/>ducantur lineæ <lb/>ipſi r t æquidi-<lb/>ſtantes; </s> <s xml:id="echoid-s621" xml:space="preserve">angu-<lb/>los rectos cum <lb/>ſuperficie humidi continebunt: </s> <s xml:id="echoid-s622" xml:space="preserve">& </s> <s xml:id="echoid-s623" xml:space="preserve">quod in humido eſt ſur-<lb/> <anchor type="note" xlink:label="note-0032-07a" xlink:href="note-0032-07"/> ſum feretur ſecundum perpendicularem, quæ per b ducta <lb/>eſt, ipſi r t æquidiſtans: </s> <s xml:id="echoid-s624" xml:space="preserve">quod uero eſt extra humi dum ſe- <pb o="11" file="0033" n="33" rhead="DE IIS QVAE VEH. IN AQVA."/> cundum eam, quæ per g, deorſum ferctur; </s> <s xml:id="echoid-s625" xml:space="preserve">& </s> <s xml:id="echoid-s626" xml:space="preserve">non ita mane <lb/>bit ſolidum a p o l: </s> <s xml:id="echoid-s627" xml:space="preserve">nam quod eſt ad a feretur ſurſum; </s> <s xml:id="echoid-s628" xml:space="preserve">& </s> <s xml:id="echoid-s629" xml:space="preserve"><lb/>quod ad b deorſum, donec n o ſecundum perpendicu-<lb/>larem conſtituatur.</s> <s xml:id="echoid-s630" xml:space="preserve">]</s> </p> <div xml:id="echoid-div42" type="float" level="2" n="2"> <note position="left" xlink:label="note-0032-01" xlink:href="note-0032-01a" xml:space="preserve">Suppleta <lb/>a. Federi-<lb/>co Cõm.</note> <note position="left" xlink:label="note-0032-02" xlink:href="note-0032-02a" xml:space="preserve">B</note> <note position="left" xlink:label="note-0032-03" xlink:href="note-0032-03a" xml:space="preserve">C</note> <note position="left" xlink:label="note-0032-04" xlink:href="note-0032-04a" xml:space="preserve">D</note> <note position="left" xlink:label="note-0032-05" xlink:href="note-0032-05a" xml:space="preserve">E</note> <note position="left" xlink:label="note-0032-06" xlink:href="note-0032-06a" xml:space="preserve">F</note> <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a"> <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0032-01"/> </figure> <note position="left" xlink:label="note-0032-07" xlink:href="note-0032-07a" xml:space="preserve">G</note> </div> </div> <div xml:id="echoid-div44" type="section" level="1" n="23"> <head xml:id="echoid-head28" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s631" xml:space="preserve"><emph style="sc">D_esideratvr_</emph> propoſitionis huius demonstratio, quam nos <lb/>etiam ad Archimedis figuram appoſite restituimus, commentarijs-<lb/>que illustrauimus.</s> <s xml:id="echoid-s632" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s633" xml:space="preserve">_Recta portio conoidis rectanguli, quando axem habue_ <lb/> <anchor type="note" xlink:label="note-0033-01a" xlink:href="note-0033-01"/> _rit minorem, quàm ſeſquialterum eius, quæ uſque ad axẽ]_ <lb/>In tranſlatione mendoſe legebatur. </s> <s xml:id="echoid-s634" xml:space="preserve">maiorem quàm ſeſquialterum: <lb/></s> <s xml:id="echoid-s635" xml:space="preserve">& </s> <s xml:id="echoid-s636" xml:space="preserve">ita legebatur in ſequenti propoſitione. </s> <s xml:id="echoid-s637" xml:space="preserve">est autem recta portio co <lb/>noidis, quæ plano ad axem recto abſcinditur: </s> <s xml:id="echoid-s638" xml:space="preserve">eâmque rectam tunc <lb/>conſiſtere dicimus, quando planum abſcindens, uidelicet baſis pla-<lb/>num, ſuperficiei humidi æquidiſtans fuerit.</s> <s xml:id="echoid-s639" xml:space="preserve"/> </p> <div xml:id="echoid-div44" type="float" level="2" n="1"> <note position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">A</note> </div> <p> <s xml:id="echoid-s640" xml:space="preserve">Quæ erit ſectionis i p o s diameter, & </s> <s xml:id="echoid-s641" xml:space="preserve">axis portionis in <lb/> <anchor type="note" xlink:label="note-0033-02a" xlink:href="note-0033-02"/> humido demerſæ] _ex_ 46 _primi conicorum Apollonij: </s> <s xml:id="echoid-s642" xml:space="preserve">uel ex co-_ <lb/>_rollario_ 51 _eiuſdem_.</s> <s xml:id="echoid-s643" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="2"> <note position="right" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">B</note> </div> <p style="it"> <s xml:id="echoid-s644" xml:space="preserve">_Sitque ſolidæ magnitudinis a p o l grauitatis centrum r,_ <lb/> <anchor type="note" xlink:label="note-0033-03a" xlink:href="note-0033-03"/> _ipſius uero i p o s centrum ſit b.</s> <s xml:id="echoid-s645" xml:space="preserve">]_ Portionis enim conoidis <lb/>rectanguli centrum grauitatis eſt in axe, quem ita diuidit, ut pars <lb/>eius, quæ ad uerticem terminatur, reliquæ partis, quæ ad baſim, ſit <lb/>dupla: </s> <s xml:id="echoid-s646" xml:space="preserve">quod nos in libro de centro grauitatis ſolidorum propoſitio-<lb/>ne 29 demonstrauimus. </s> <s xml:id="echoid-s647" xml:space="preserve">Cum igitur portionis a p o l centrum gra-<lb/>uitatis ſit r, erit o r dupla r n: </s> <s xml:id="echoid-s648" xml:space="preserve">& </s> <s xml:id="echoid-s649" xml:space="preserve">propterea n o ipſius o r ſeſqui-<lb/>altera. </s> <s xml:id="echoid-s650" xml:space="preserve">Eadem ratione b centrum grauitatis portionis i p o s est in <lb/>axe p f, ita ut p b dupla ſit b f.</s> <s xml:id="echoid-s651" xml:space="preserve"/> </p> <div xml:id="echoid-div46" type="float" level="2" n="3"> <note position="right" xlink:label="note-0033-03" xlink:href="note-0033-03a" xml:space="preserve">C</note> </div> <p style="it"> <s xml:id="echoid-s652" xml:space="preserve">_Etiuncta b r producatur ad g, quod ſit centrum graui_ <lb/> <anchor type="note" xlink:label="note-0033-04a" xlink:href="note-0033-04"/> _tatis reliquæ figuræ i s l a]_ Si enim linea b r in g producta, ha <lb/>beat g r ad r b proportionem eam, quam conoidis portio i p o s ad <lb/>reliquam figuram, quæ ex humidi ſuperficie extat: </s> <s xml:id="echoid-s653" xml:space="preserve">erit punctum g <lb/>ipſius grauitatis centrum, ex octaua Archimedis.</s> <s xml:id="echoid-s654" xml:space="preserve"/> </p> <div xml:id="echoid-div47" type="float" level="2" n="4"> <note position="right" xlink:label="note-0033-04" xlink:href="note-0033-04a" xml:space="preserve">D</note> </div> <pb file="0034" n="34" rhead="ARCHIMEDIS"/> <p style="it"> <s xml:id="echoid-s655" xml:space="preserve">_Erit r o minor, quàm, quæ uſque ad axem]_ Ex decima <lb/> <anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/> propoſitione quinti libri elementorum. </s> <s xml:id="echoid-s656" xml:space="preserve">Linea, quæ uſque ad axem <lb/>apud Archimedem, eſt dimidia eius, iuxta quam poſſunt, quæ à ſe-<lb/>ctione ducuntur; </s> <s xml:id="echoid-s657" xml:space="preserve">ut ex quarta propoſitione libri de conoidibus, & </s> <s xml:id="echoid-s658" xml:space="preserve"><lb/>ſphæroidibus apparet. </s> <s xml:id="echoid-s659" xml:space="preserve">cur uero ita appellata ſit, nos in commentarijs <lb/>in eam editis tradidimus.</s> <s xml:id="echoid-s660" xml:space="preserve"/> </p> <div xml:id="echoid-div48" type="float" level="2" n="5"> <note position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">E</note> </div> <p style="it"> <s xml:id="echoid-s661" xml:space="preserve">_Quare angulus r p ω acutus erit]_ producatur linea n o ad <lb/> <anchor type="note" xlink:label="note-0034-02a" xlink:href="note-0034-02"/> h, ut ſit r h æqualis ei, quæ uſque ad axem. </s> <s xml:id="echoid-s662" xml:space="preserve">ſi igitur à puncto h du-<lb/>catur linea ad rectos angulos ipſi n h, conueniet cum f p extra ſe-<lb/>ctionem: </s> <s xml:id="echoid-s663" xml:space="preserve">ducta enim per o ipſi a l æquidiſtans, extra ſectionem ca <lb/>dit ex decima ſepti-<lb/> <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> ma primi libri coni-<lb/>corum. </s> <s xml:id="echoid-s664" xml:space="preserve">Itaque con-<lb/>ueniat in u. </s> <s xml:id="echoid-s665" xml:space="preserve">& </s> <s xml:id="echoid-s666" xml:space="preserve">quo <lb/>niam f p est æqui-<lb/>distans diametro; <lb/></s> <s xml:id="echoid-s667" xml:space="preserve">h u uero ad diame-<lb/>trum perpendicula-<lb/>ris; </s> <s xml:id="echoid-s668" xml:space="preserve">& </s> <s xml:id="echoid-s669" xml:space="preserve">r h æqualis <lb/>ei, quæ uſq; </s> <s xml:id="echoid-s670" xml:space="preserve">ad axẽ, <lb/>linea à puncto r ad <lb/>u ducta angulos re-<lb/>ctos faciet cum ea, quæ ſectionem in puncto p contingit, hoc eſt cum <lb/>k ω, ut mox demonstrabitur. </s> <s xml:id="echoid-s671" xml:space="preserve">quare perpendicularis r t inter p & </s> <s xml:id="echoid-s672" xml:space="preserve"><lb/>ω cadet; </s> <s xml:id="echoid-s673" xml:space="preserve">erítque r p ω angulus acutus.</s> <s xml:id="echoid-s674" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="6"> <note position="left" xlink:label="note-0034-02" xlink:href="note-0034-02a" xml:space="preserve">F</note> <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0034-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s675" xml:space="preserve">Sit rectanguli coni ſectio, ſeu parabole a b c, cuius <lb/>diameter b d: </s> <s xml:id="echoid-s676" xml:space="preserve">atque ipſam contingat linea e f in pun-<lb/>cto g: </s> <s xml:id="echoid-s677" xml:space="preserve">ſumatur autem in diametro b d linea h k æqua-<lb/>lis ei, quæ uſque ad axem: </s> <s xml:id="echoid-s678" xml:space="preserve">& </s> <s xml:id="echoid-s679" xml:space="preserve">per g ducta g l, diame-<lb/>tro æquidistante, à puncto _k_ ad rectos angulos ipſi b d <lb/>ducatur _k_ m, ſecans g l in m. </s> <s xml:id="echoid-s680" xml:space="preserve">Dico lineam ab h ad <pb o="12" file="0035" n="35" rhead="DE IIS QVAE VEH. IN AQVA."/> m productam per pendicularem eſſe ad ipſam e f, quam <lb/>quidem ſecet in n.</s> <s xml:id="echoid-s681" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s682" xml:space="preserve"><emph style="sc">D_vcatvr_</emph> enim à puncto g linea g o ad rectos angulos ipſi <lb/>e f, diametrum in o ſecans: </s> <s xml:id="echoid-s683" xml:space="preserve">& </s> <s xml:id="echoid-s684" xml:space="preserve">rurſus ab eodem puncto ducatur g p <lb/>ad diametrum perpendicularis: </s> <s xml:id="echoid-s685" xml:space="preserve">ſecet autem ipſa diameter producta <lb/>lineã e f in q. </s> <s xml:id="echoid-s686" xml:space="preserve">erit p b ipſi b q æqualis, ex trigeſimaquinta primi co <lb/>nicorum: </s> <s xml:id="echoid-s687" xml:space="preserve">& </s> <s xml:id="echoid-s688" xml:space="preserve">g p pro-<lb/> <anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> portionalis ĩter q p, p o <lb/>quare quadratũ g p re-<lb/> <anchor type="note" xlink:label="note-0035-02a" xlink:href="note-0035-02"/> ctangulo o p q æquale <lb/>erit: </s> <s xml:id="echoid-s689" xml:space="preserve">ſed etiã æquale est <lb/>rectangulo cõtento ipſa <lb/>p b, & </s> <s xml:id="echoid-s690" xml:space="preserve">linea, iuxta quã <lb/>poſſunt, quæ à ſectione <lb/>ad diametrũ ordinatim <lb/>ducuntur, ex undecima <lb/>primi conicorum. </s> <s xml:id="echoid-s691" xml:space="preserve">ergo <lb/> <anchor type="note" xlink:label="note-0035-03a" xlink:href="note-0035-03"/> quæ est proportio q p <lb/>ad p b eadem est lineæ, <lb/>iuxta quã poſſunt, quæ <lb/>à ſectione ducũtur ad ip <lb/>ſam p o: </s> <s xml:id="echoid-s692" xml:space="preserve">est autem q p <lb/>dupla p b: </s> <s xml:id="echoid-s693" xml:space="preserve">cũ ſint p b, <lb/>b q æquales, ut dictum <lb/>est. </s> <s xml:id="echoid-s694" xml:space="preserve">Linea igitur iuxta <lb/>quam poſſunt, quæ à ſe-<lb/>ctione ducuntur ipſi-<lb/>us p o dupla erit: </s> <s xml:id="echoid-s695" xml:space="preserve">& </s> <s xml:id="echoid-s696" xml:space="preserve"><lb/>propterea p o æqualis <lb/>ei, quæ uſque ad axem, <lb/>uidelicet ipſi k h: </s> <s xml:id="echoid-s697" xml:space="preserve">ſed eſt p g æqualis k m; </s> <s xml:id="echoid-s698" xml:space="preserve">& </s> <s xml:id="echoid-s699" xml:space="preserve">angulus o p g angu-<lb/> <anchor type="note" xlink:label="note-0035-04a" xlink:href="note-0035-04"/> lo h k m; </s> <s xml:id="echoid-s700" xml:space="preserve">quòd uterque rectus. </s> <s xml:id="echoid-s701" xml:space="preserve">quare & </s> <s xml:id="echoid-s702" xml:space="preserve">o g ipſi h m est œqualis: <lb/></s> <s xml:id="echoid-s703" xml:space="preserve"> <anchor type="note" xlink:label="note-0035-05a" xlink:href="note-0035-05"/> & </s> <s xml:id="echoid-s704" xml:space="preserve">angulus p o g angulo _k_ h m. </s> <s xml:id="echoid-s705" xml:space="preserve">æquidistantes igitur ſunt o g, h n: <lb/></s> <s xml:id="echoid-s706" xml:space="preserve"> <anchor type="note" xlink:label="note-0035-06a" xlink:href="note-0035-06"/> <pb file="0036" n="36" rhead="ARCHIMEDIS"/> angulus b n f œqualis angulo o g f: </s> <s xml:id="echoid-s707" xml:space="preserve">quòd cum ſit g o perpendi-<lb/> <anchor type="note" xlink:label="note-0036-01a" xlink:href="note-0036-01"/> cularis ad e f, & </s> <s xml:id="echoid-s708" xml:space="preserve">h n ad eandem perpendicularis erit. </s> <s xml:id="echoid-s709" xml:space="preserve">quod de-<lb/>monstrare oportebat.</s> <s xml:id="echoid-s710" xml:space="preserve"/> </p> <div xml:id="echoid-div50" type="float" level="2" n="7"> <note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">cor. 8. ſe-<lb/>xti.</note> <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/4E7V2WGH/figures/0035-01"/> </figure> <note position="right" xlink:label="note-0035-02" xlink:href="note-0035-02a" xml:space="preserve">17. ſextĩ.</note> <note position="right" xlink:label="note-0035-03" xlink:href="note-0035-03a" xml:space="preserve">14. ſexti.</note> <note position="right" xlink:label="note-0035-04" xlink:href="note-0035-04a" xml:space="preserve">32. primi</note> <note position="right" xlink:label="note-0035-05" xlink:href="note-0035-05a" xml:space="preserve">4. primi.</note> <note position="right" xlink:label="note-0035-06" xlink:href="note-0035-06a" xml:space="preserve">28</note> <note position="left" xlink:label="note-0036-01" xlink:href="note-0036-01a" xml:space="preserve">29, primi</note> </div> <p> <s xml:id="echoid-s711" xml:space="preserve">Et quod in humido eſt ſurſum ſeretur ſecundum per-<lb/> <anchor type="note" xlink:label="note-0036-02a" xlink:href="note-0036-02"/> pendicularem, quæ per b ducta eſtipſi rt æquidiſtans.</s> <s xml:id="echoid-s712" xml:space="preserve">] <lb/>_Cur hoc quidem ſurſum, illud uero deorſum per lineam perpen-_ <lb/>_dicularem feratur, diximus ſupra in octauam primi libri buius. </s> <s xml:id="echoid-s713" xml:space="preserve">qua_ <lb/>_re neque in hac, neque in alijs, quæ ſequuntur, eadem iterare neceſſa_ <lb/>_rium exiſtimauimus._</s> <s xml:id="echoid-s714" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="8"> <note position="left" xlink:label="note-0036-02" xlink:href="note-0036-02a" xml:space="preserve">G</note> </div> </div> <div xml:id="echoid-div53" type="section" level="1" n="24"> <head xml:id="echoid-head29" xml:space="preserve">PROPOSITIO III.</head> <p> <s xml:id="echoid-s715" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli quando <lb/>axem habuerit minorem, quam ſeſquialterum <lb/>eius, quæ uſque ad axem, quamcunque propor-<lb/>tionem habens ad humidum in grauitate; </s> <s xml:id="echoid-s716" xml:space="preserve">demiſ-<lb/>ſa in humidum, ita ut baſis ipſius tota ſit in humi <lb/>do; </s> <s xml:id="echoid-s717" xml:space="preserve">& </s> <s xml:id="echoid-s718" xml:space="preserve">poſita inclinata, non manebit inclinata, ſed <lb/>ita reſtituetur, ut axis ipſius ſecundum perpendi <lb/>cularem fiat.</s> <s xml:id="echoid-s719" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s720" xml:space="preserve">DEMITTATVR enim aliqua portio in humidum, <lb/>qualis dicca eſt: </s> <s xml:id="echoid-s721" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s722" xml:space="preserve">ipſius baſis in humido: </s> <s xml:id="echoid-s723" xml:space="preserve">& </s> <s xml:id="echoid-s724" xml:space="preserve">ſecta ipſa <lb/>plano per axẽ, recto ad ſuperficiẽ humidi, ſit ſectio a p ol <lb/>rectanguli coniſectio: </s> <s xml:id="echoid-s725" xml:space="preserve">axis portionis, & </s> <s xml:id="echoid-s726" xml:space="preserve">ſectionis diame-<lb/>ter p f: </s> <s xml:id="echoid-s727" xml:space="preserve">ſuperficiei autem humidi ſectio ſit is. </s> <s xml:id="echoid-s728" xml:space="preserve">Quòd ſi incli <lb/>nata iaceat portio, non erit axis ſecundum perpendicula-<lb/>rem. </s> <s xml:id="echoid-s729" xml:space="preserve">ergo p f cum is angulos rectos non faciet. </s> <s xml:id="echoid-s730" xml:space="preserve">Itaque <lb/>ducatur linea quædã k ω æquidiſtans ipſi is; </s> <s xml:id="echoid-s731" xml:space="preserve">contingensq; <lb/></s> <s xml:id="echoid-s732" xml:space="preserve">ſectionẽ ap ol in o: </s> <s xml:id="echoid-s733" xml:space="preserve">& </s> <s xml:id="echoid-s734" xml:space="preserve">ſolidæ quidẽ magnitudinis a p o l <lb/>ſit r grauitatis centrum: </s> <s xml:id="echoid-s735" xml:space="preserve">ipſius autem i p o s centrum ſit <pb o="13" file="0037" n="37" rhead="DE IIS QVAE VEH. IN AQVA."/> b: </s> <s xml:id="echoid-s736" xml:space="preserve">iunctaq; </s> <s xml:id="echoid-s737" xml:space="preserve">br producatur: </s> <s xml:id="echoid-s738" xml:space="preserve">& </s> <s xml:id="echoid-s739" xml:space="preserve">ſit g centrum grauitatis <lb/>reliquæ figuræ isla. </s> <s xml:id="echoid-s740" xml:space="preserve">ſimiliter demonſtrabitur angulum <lb/>rok acutu eſ-<lb/> <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> ſe: </s> <s xml:id="echoid-s741" xml:space="preserve">& </s> <s xml:id="echoid-s742" xml:space="preserve">perpendi <lb/>culare ab r ad <lb/>k ω ductam ca <lb/>dereinter k & </s> <s xml:id="echoid-s743" xml:space="preserve"><lb/>o, quæ ſit rt. <lb/></s> <s xml:id="echoid-s744" xml:space="preserve">ſi autem à pun <lb/>ctis g b ducan <lb/>tur ipſi r t æqui <lb/>diſtantes; </s> <s xml:id="echoid-s745" xml:space="preserve">pars <lb/>quidem ſolidæ <lb/>magnitudinis, <lb/>quæ in humido eſt, ſurſum feretur ſecundum perpendicu-<lb/>larem per g ductam: </s> <s xml:id="echoid-s746" xml:space="preserve">quæ autem extra humidum ſecundũ <lb/>perpendicularem per b deorſum feretur: </s> <s xml:id="echoid-s747" xml:space="preserve">& </s> <s xml:id="echoid-s748" xml:space="preserve">non manebit <lb/>ſolidum a p o l ſic habens in humido: </s> <s xml:id="echoid-s749" xml:space="preserve">ſed quod quidem <lb/>eſt ad a feretur ſurſum: </s> <s xml:id="echoid-s750" xml:space="preserve">quod autem ad l deorſum, donec <lb/>p f fiat ſecundum perpendicularem.</s> <s xml:id="echoid-s751" xml:space="preserve"/> </p> <div xml:id="echoid-div53" 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/4E7V2WGH/figures/0037-01"/> </figure> </div> </div> <div xml:id="echoid-div55" type="section" level="1" n="25"> <head xml:id="echoid-head30" xml:space="preserve">PROPOSITIO IIII.</head> <p> <s xml:id="echoid-s752" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>fuerit humido leuior, & </s> <s xml:id="echoid-s753" xml:space="preserve">axem habuerit maiorẽ, <lb/>qnàm ſeſquialterum eius, quæ uſque ad axem: </s> <s xml:id="echoid-s754" xml:space="preserve">ſi <lb/>in grauitate ad humidum æqualis molis non mi-<lb/>norem proportionem habeat ea, quàm quadra-<lb/>tũ, quod fit ab exceſſu, quo axis maior eſt, quàm <lb/>ſeſquialter eius, quæ uſque ad axẽ, habet ad qua-<lb/>dratum, quod ab axe; </s> <s xml:id="echoid-s755" xml:space="preserve">demiſſa in humidum, ita <pb file="0038" n="38" rhead="ARCHIMEDIS"/> ut baſis ipſius humidum non contingat; </s> <s xml:id="echoid-s756" xml:space="preserve">& </s> <s xml:id="echoid-s757" xml:space="preserve">poſi-<lb/>ta inclinata, non manebit inclinata, ſed recta re-<lb/>ſtituetur.</s> <s xml:id="echoid-s758" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s759" xml:space="preserve">SIT portio conoidis rectanguli, qualis dicta eſt: </s> <s xml:id="echoid-s760" xml:space="preserve">& </s> <s xml:id="echoid-s761" xml:space="preserve">de-<lb/>miſſa in humidum, ſi fieri poteſt, non ſitrecta; </s> <s xml:id="echoid-s762" xml:space="preserve">ſed inclina-<lb/>ta: </s> <s xml:id="echoid-s763" xml:space="preserve">ſecta autem ipſa plano per axem, recto ad ſuperficiem <lb/>humidi, portionis quidem ſectio ſit rectanguli coni ſectio <lb/>a p o l, axis portionis, & </s> <s xml:id="echoid-s764" xml:space="preserve">ſectionis diameter n o; </s> <s xml:id="echoid-s765" xml:space="preserve">& </s> <s xml:id="echoid-s766" xml:space="preserve">ſuper-<lb/>ficiei humidi ſectio ſit is. </s> <s xml:id="echoid-s767" xml:space="preserve">ſi igitur portio non eſt recta, nõ <lb/>faciet n o cum is angulos æquales. </s> <s xml:id="echoid-s768" xml:space="preserve">Ducatur k ω contin-<lb/>gens rectanguli coni ſectionem in p; </s> <s xml:id="echoid-s769" xml:space="preserve">æquidiſtanſq; </s> <s xml:id="echoid-s770" xml:space="preserve">ipſi <lb/>is: </s> <s xml:id="echoid-s771" xml:space="preserve">& </s> <s xml:id="echoid-s772" xml:space="preserve">à puncto p ipſi o n æquidiſtans ducatur p f. </s> <s xml:id="echoid-s773" xml:space="preserve">Itaque <lb/>ſumantur centra grauitatum: </s> <s xml:id="echoid-s774" xml:space="preserve">& </s> <s xml:id="echoid-s775" xml:space="preserve">ſolidi quidem a p o l cen <lb/>trum ſit r; </s> <s xml:id="echoid-s776" xml:space="preserve">eius autem, quod intra humidum, centrum b: <lb/></s> <s xml:id="echoid-s777" xml:space="preserve">iunctaq; </s> <s xml:id="echoid-s778" xml:space="preserve">b r pro-<lb/> <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> ducatur ad g, ut <lb/>g ſit centrũ graui <lb/>tatis ſolidi, quod <lb/>extra humidum. <lb/></s> <s xml:id="echoid-s779" xml:space="preserve">Quoniam igitur <lb/>n o ipſius quidem <lb/>r o ſeſquialtera ẽ; </s> <s xml:id="echoid-s780" xml:space="preserve"><lb/>eius autẽ, quæ uſ-<lb/>que ad axẽ maior, <lb/>quàm ſeſquialte-<lb/>ra: </s> <s xml:id="echoid-s781" xml:space="preserve">patet r o maio <lb/> <anchor type="note" xlink:label="note-0038-01a" xlink:href="note-0038-01"/> rẽ eſſe, quàm quæ <lb/>uſq; </s> <s xml:id="echoid-s782" xml:space="preserve">ad axẽ. </s> <s xml:id="echoid-s783" xml:space="preserve">Sit ei, <lb/> <anchor type="note" xlink:label="note-0038-02a" xlink:href="note-0038-02"/> quæ uſque ad axẽ <lb/>æqualis r h: </s> <s xml:id="echoid-s784" xml:space="preserve">& </s> <s xml:id="echoid-s785" xml:space="preserve">o h dupla ipſius h m. </s> <s xml:id="echoid-s786" xml:space="preserve">quòd cū n o ipſius r o <lb/> <anchor type="note" xlink:label="note-0038-03a" xlink:href="note-0038-03"/> ſeſquialtera ſit; </s> <s xml:id="echoid-s787" xml:space="preserve">item q; </s> <s xml:id="echoid-s788" xml:space="preserve">m o ipſius o h: </s> <s xml:id="echoid-s789" xml:space="preserve">& </s> <s xml:id="echoid-s790" xml:space="preserve">reliqua n m reli <lb/> <anchor type="note" xlink:label="note-0038-04a" xlink:href="note-0038-04"/> quæ r h ſeſquialtera erit. </s> <s xml:id="echoid-s791" xml:space="preserve">ergo axis tanto maior eſt, quàm <pb o="14" file="0039" n="39" rhead="DE IIS QVAE VEH. IN AQVA."/> ſeſquialter eius, quæ uſque ad axem, quanta eſt linea m o. <lb/></s> <s xml:id="echoid-s792" xml:space="preserve">Ponebatur autem portio ad humidum æqualis molis non <lb/>minorem in grauitate proportionem habere, quam qua-<lb/>dratum, quod fit ab exceſſu, quo axis eſt maior, quam ſeſ-<lb/>quialter eius, quæ uſque ad axem, ad quadratum, quod ab <lb/>axe. </s> <s xml:id="echoid-s793" xml:space="preserve">quare conſtat portionem ad humidum in grauitate <lb/>non minorem proportionem habere, quàm quadratum li <lb/>neæ m o ad quadratum ipſius n o. </s> <s xml:id="echoid-s794" xml:space="preserve">Sed quam proportio-<lb/>nem habet portio ad humidum in grauitate, eandem por-<lb/>tio ipſius demerla habet ad totam portionem: </s> <s xml:id="echoid-s795" xml:space="preserve">hoc enim <lb/> <anchor type="note" xlink:label="note-0039-01a" xlink:href="note-0039-01"/> ſupra demonſtratum eſt: </s> <s xml:id="echoid-s796" xml:space="preserve">& </s> <s xml:id="echoid-s797" xml:space="preserve">quam proportionem habet de <lb/> <anchor type="note" xlink:label="note-0039-02a" xlink:href="note-0039-02"/> merſa portio ad totam, eam quadratum p f habet ad n o <lb/>quadratum: </s> <s xml:id="echoid-s798" xml:space="preserve">cum demonſtratum ſit in iis, quæ de conoidi <lb/>bus, & </s> <s xml:id="echoid-s799" xml:space="preserve">ſphæroidibus, ſi à rectangulo conoide duæ portio-<lb/>nes planis quomodocunque ductis abſcindantur, portio-<lb/>nes inter ſe eandem habere proportionem, qnàm quadra-<lb/>ta, quæ ab ipſorum axibus conſtituuntur. </s> <s xml:id="echoid-s800" xml:space="preserve">non minorem <lb/>ergo proportionẽ habet quadratum pf ad quadratũ n o, <lb/>quàm quadratum m o ad idem n o quadratum. </s> <s xml:id="echoid-s801" xml:space="preserve">quare <lb/> <anchor type="note" xlink:label="note-0039-03a" xlink:href="note-0039-03"/> p f non eſt minor ipſa m o; </s> <s xml:id="echoid-s802" xml:space="preserve">nec b p item minor h o. </s> <s xml:id="echoid-s803" xml:space="preserve">Si <lb/> <anchor type="note" xlink:label="note-0039-04a" xlink:href="note-0039-04"/> igitur ab h ducatur linea ad rectos angulos ipſi n o, coi-<lb/> <anchor type="note" xlink:label="note-0039-05a" xlink:href="note-0039-05"/> bit cum b p, atque inter b, & </s> <s xml:id="echoid-s804" xml:space="preserve">p cadet. </s> <s xml:id="echoid-s805" xml:space="preserve">coeat in t. </s> <s xml:id="echoid-s806" xml:space="preserve">& </s> <s xml:id="echoid-s807" xml:space="preserve">quo <lb/> <anchor type="note" xlink:label="note-0039-06a" xlink:href="note-0039-06"/> niam p f quidem æquidiſtans eſt diametro, h t autem ad <lb/>diametrum perpendicularis; </s> <s xml:id="echoid-s808" xml:space="preserve">& </s> <s xml:id="echoid-s809" xml:space="preserve">r h æqualis ei, quæ uſque <lb/>ad axem: </s> <s xml:id="echoid-s810" xml:space="preserve">ducta linea ab r ad t & </s> <s xml:id="echoid-s811" xml:space="preserve">producta angulos rectos <lb/>faciet cum linea ſectionem in puncto p contingente. </s> <s xml:id="echoid-s812" xml:space="preserve">qua-<lb/>re & </s> <s xml:id="echoid-s813" xml:space="preserve">cum is, & </s> <s xml:id="echoid-s814" xml:space="preserve">cum humidi ſuperficie, quæ per is tran-<lb/>ſit. </s> <s xml:id="echoid-s815" xml:space="preserve">Itaque ſi per b g puncta lineæ ipſi r t æquidiſtantes du <lb/>cantur, angulos rectos facient cum ſuperficie humidi: </s> <s xml:id="echoid-s816" xml:space="preserve">& </s> <s xml:id="echoid-s817" xml:space="preserve"><lb/>quod quidem in humido eſt ſolidum conoidis feretur ſur-<lb/>ſum ſecundum eam, quæ per b ducta fuerit ipſi r t æquidi <lb/>ſtans: </s> <s xml:id="echoid-s818" xml:space="preserve">quod autem extra humidum, ſecundum eam, quæ <lb/>per g deorſum feretur. </s> <s xml:id="echoid-s819" xml:space="preserve">atque hoc tandiu fiet, quoad co-<lb/>noides rectum conſtituatur.</s> <s xml:id="echoid-s820" xml:space="preserve"/> </p> <div xml:id="echoid-div55" type="float" level="2" n="1"> <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/4E7V2WGH/figures/0038-01"/> </figure> <note position="left" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">10. quinti</note> <note position="left" xlink:label="note-0038-02" xlink:href="note-0038-02a" xml:space="preserve">A</note> <note position="left" xlink:label="note-0038-03" xlink:href="note-0038-03a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0038-04" xlink:href="note-0038-04a" xml:space="preserve">19. quinti</note> <note position="right" xlink:label="note-0039-01" xlink:href="note-0039-01a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0039-02" xlink:href="note-0039-02a" xml:space="preserve">D</note> <note position="right" xlink:label="note-0039-03" xlink:href="note-0039-03a" xml:space="preserve">E</note> <note position="right" xlink:label="note-0039-04" xlink:href="note-0039-04a" xml:space="preserve">F</note> <note position="right" xlink:label="note-0039-05" xlink:href="note-0039-05a" xml:space="preserve">G</note> <note position="right" xlink:label="note-0039-06" xlink:href="note-0039-06a" xml:space="preserve">H</note> </div> <pb file="0040" n="40" rhead="ARCHIMEDIS"/> </div> <div xml:id="echoid-div57" type="section" level="1" n="26"> <head xml:id="echoid-head31" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s821" xml:space="preserve">_Sit ei, quæ uſque ad axem æqualis r h.</s> <s xml:id="echoid-s822" xml:space="preserve">]_ Ita legendum eſt, <lb/> <anchor type="note" xlink:label="note-0040-01a" xlink:href="note-0040-01"/> non r m, ut tranſlatio habet, quod ex ijs, quæ ſequuntur, manifeſte <lb/>conſtare poteſt.</s> <s xml:id="echoid-s823" xml:space="preserve"/> </p> <div xml:id="echoid-div57" type="float" level="2" n="1"> <note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">A</note> </div> <p style="it"> <s xml:id="echoid-s824" xml:space="preserve">_Et oh dupla ipſius h m.</s> <s xml:id="echoid-s825" xml:space="preserve">]_ In tranſlatione mendoſe legeba-<lb/> <anchor type="note" xlink:label="note-0040-02a" xlink:href="note-0040-02"/> tur, on dupla ipſius rm.</s> <s xml:id="echoid-s826" xml:space="preserve"/> </p> <div xml:id="echoid-div58" type="float" level="2" n="2"> <note position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s827" xml:space="preserve">Hoc enim ſupra demonſtratum eſt.</s> <s xml:id="echoid-s828" xml:space="preserve">] _In prima huius_.</s> <s xml:id="echoid-s829" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">C</note> <p> <s xml:id="echoid-s830" xml:space="preserve">Et quam proportionem habet demerſa portio ad totã, <lb/> <anchor type="note" xlink:label="note-0040-04a" xlink:href="note-0040-04"/> eam quadratum p f habet ad n o quadratum.</s> <s xml:id="echoid-s831" xml:space="preserve">] _Hoc loco in_ <lb/>_tranſlatione non nulli deſider abantur, quænos reſtituimus. </s> <s xml:id="echoid-s832" xml:space="preserve">Illud au_ <lb/>_tem ab Archimede demonſtratum eſt in libro de conoidibus & </s> <s xml:id="echoid-s833" xml:space="preserve">ſphæ_ <lb/>_roidibus propoſitione_ 26.</s> <s xml:id="echoid-s834" xml:space="preserve"/> </p> <div xml:id="echoid-div59" type="float" level="2" n="3"> <note position="left" xlink:label="note-0040-04" xlink:href="note-0040-04a" xml:space="preserve">D</note> </div> <p style="it"> <s xml:id="echoid-s835" xml:space="preserve">_Quare p f non eſt minor ipſa m o.</s> <s xml:id="echoid-s836" xml:space="preserve">]_ Nam ex decima quinti <lb/> <anchor type="note" xlink:label="note-0040-05a" xlink:href="note-0040-05"/> ſequitur, quadratum p f non eſſe minus quadrato m o. </s> <s xml:id="echoid-s837" xml:space="preserve">quare neque <lb/>linea p f minor erit linea m o ex 22 ſexti.</s> <s xml:id="echoid-s838" xml:space="preserve"/> </p> <div xml:id="echoid-div60" type="float" level="2" n="4"> <note position="left" xlink:label="note-0040-05" xlink:href="note-0040-05a" xml:space="preserve">E</note> </div> <p style="it"> <s xml:id="echoid-s839" xml:space="preserve">_Nec b p item minor h o.</s> <s xml:id="echoid-s840" xml:space="preserve">]_ Eſt enim ut p f ad p b, ita m o, <lb/> <anchor type="note" xlink:label="note-0040-06a" xlink:href="note-0040-06"/> ad h o & </s> <s xml:id="echoid-s841" xml:space="preserve">permutando, ut p f ad mo, ita b p, ad b o. </s> <s xml:id="echoid-s842" xml:space="preserve">ſed p f non <lb/>est minor m o, ut oſtenſiim cst. </s> <s xml:id="echoid-s843" xml:space="preserve">ergo neque b p ipſa h o minor erit.</s> <s xml:id="echoid-s844" xml:space="preserve"/> </p> <div xml:id="echoid-div61" type="float" level="2" n="5"> <note position="left" xlink:label="note-0040-06" xlink:href="note-0040-06a" xml:space="preserve">F</note> </div> <note position="left" xml:space="preserve">14. quinti</note> <p> <s xml:id="echoid-s845" xml:space="preserve">Si igitur ab h <lb/> <anchor type="note" xlink:label="note-0040-08a" xlink:href="note-0040-08"/> <anchor type="figure" xlink:label="fig-0040-01a" xlink:href="fig-0040-01"/> ducatur linea ad <lb/>rectos angulos ip <lb/>ſi n o, coibit cum <lb/>b p, atque inter <lb/>b & </s> <s xml:id="echoid-s846" xml:space="preserve">p cadet.</s> <s xml:id="echoid-s847" xml:space="preserve">] <lb/>_Corruptus erat hic_ <lb/>_locus in tranſlatio-_ <lb/>_ne. </s> <s xml:id="echoid-s848" xml:space="preserve">Illud uero ita de-_ <lb/>_monſtr abitur. </s> <s xml:id="echoid-s849" xml:space="preserve">Quo-_ <lb/>_niam p f non eſt mi-_ <lb/>_nor o m, nec p b ip-_ <lb/>_ſa h o; </s> <s xml:id="echoid-s850" xml:space="preserve">ſi ponatur p f_ <lb/>_æqualis o m; </s> <s xml:id="echoid-s851" xml:space="preserve">& </s> <s xml:id="echoid-s852" xml:space="preserve">p b,_ <lb/>_ipſi h o æqualis erit._</s> <s xml:id="echoid-s853" xml:space="preserve"> <pb o="15" file="0041" n="41" rhead="DE IIS QVAE VEH. IN AQVA."/> _quare per o ductaipſi al æquidiſtans cadet extra ſectionem ex 17._ <lb/></s> <s xml:id="echoid-s854" xml:space="preserve">_primi conicorum: </s> <s xml:id="echoid-s855" xml:space="preserve">& </s> <s xml:id="echoid-s856" xml:space="preserve">cum b p producta coibit inſra p. </s> <s xml:id="echoid-s857" xml:space="preserve">ergò & </s> <s xml:id="echoid-s858" xml:space="preserve">per-_ <lb/>_pendicularis ducta per b cum eadem infra b coibit, at que inter b &_</s> <s xml:id="echoid-s859" xml:space="preserve"> <lb/>_p neceſſario cadet. </s> <s xml:id="echoid-s860" xml:space="preserve">multo autem magis illud idem ſequetur, ſi pona-_ <lb/>_mus pf ipſa om maiorem eſſe._</s> <s xml:id="echoid-s861" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="6"> <note position="left" xlink:label="note-0040-08" xlink:href="note-0040-08a" xml:space="preserve">G</note> <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/4E7V2WGH/figures/0040-01"/> </figure> </div> <p> <s xml:id="echoid-s862" xml:space="preserve">Et quoniam p f quidem æquidiſtans eſt diametro, htau <lb/> <anchor type="note" xlink:label="note-0041-01a" xlink:href="note-0041-01"/> tem ad diametrum perpendicularis; </s> <s xml:id="echoid-s863" xml:space="preserve">& </s> <s xml:id="echoid-s864" xml:space="preserve">rh æqualis ei, quæ <lb/>uſque ad axem, ducta linea ab r ad e, & </s> <s xml:id="echoid-s865" xml:space="preserve">producta angulos <lb/>rectos facere cum linea ſectionem in p contingente.</s> <s xml:id="echoid-s866" xml:space="preserve">] <lb/>_Hoc ſuperius à nobis demonſtratum eſt in ſecundam buin<unsure/>s._</s> <s xml:id="echoid-s867" xml:space="preserve"/> </p> <div xml:id="echoid-div63" type="float" level="2" n="7"> <note position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">H</note> </div> </div> <div xml:id="echoid-div65" type="section" level="1" n="27"> <head xml:id="echoid-head32" xml:space="preserve">PROPOSITIO V.</head> <p> <s xml:id="echoid-s868" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem, quàm <lb/>ſeſquialterum eius, quæ uſque ad axem; </s> <s xml:id="echoid-s869" xml:space="preserve">ſi ad hu-<lb/>midum in grauitate non maiorem proportionẽ <lb/>habeat, quàm exceſſus, quo quadratum quod ſit <lb/>ab axe maius eſt quadrato, quod ab exceſſu, quo <lb/>axis maior eſt, quàm ſeſquialter eius, quæ uſque <lb/>ad axem, ad quadratum, quod ab axe: </s> <s xml:id="echoid-s870" xml:space="preserve">demiſſa in <lb/>humidum, ita ut baſis ipſius tota ſit in humido; <lb/></s> <s xml:id="echoid-s871" xml:space="preserve">& </s> <s xml:id="echoid-s872" xml:space="preserve">poſita inclinata non manebit inclinata, ſed re-<lb/>ſtituetur ita, ut axis ipſius ſecundum perpendi-<lb/>cularem fiat.</s> <s xml:id="echoid-s873" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s874" xml:space="preserve">DEMITTATVR enim in humidum portio aliqna, <lb/>qualis dicta eſt: </s> <s xml:id="echoid-s875" xml:space="preserve">& </s> <s xml:id="echoid-s876" xml:space="preserve">ſit baſis ipſius tota in humido. </s> <s xml:id="echoid-s877" xml:space="preserve">Secta au-<lb/>tem ipſa plano per axem, recto ad ſuperſiciem humidi, erit <lb/>ſectio rectanguli coniſectio, quæ ſit apol: </s> <s xml:id="echoid-s878" xml:space="preserve">axis portionis, <pb file="0042" n="42" rhead="ARCHIMEDIS"/> & </s> <s xml:id="echoid-s879" xml:space="preserve">ſectionis diameter no: </s> <s xml:id="echoid-s880" xml:space="preserve">ſuperſiciei autem humidi ſectio <lb/>ſit is. </s> <s xml:id="echoid-s881" xml:space="preserve">Quoniam igitur axis non eſt ſecundum perpendicu <lb/>larem; </s> <s xml:id="echoid-s882" xml:space="preserve">ipſa no cum is non faciet angulos æquales. </s> <s xml:id="echoid-s883" xml:space="preserve">Du-<lb/>catur k ω contingens ſectionem apol in p; </s> <s xml:id="echoid-s884" xml:space="preserve">atque ipſi is <lb/>æquidiſtans: </s> <s xml:id="echoid-s885" xml:space="preserve">per p autem ducatur p f æquidiſtās ipſi n o: <lb/></s> <s xml:id="echoid-s886" xml:space="preserve">& </s> <s xml:id="echoid-s887" xml:space="preserve">ſumantur grauitatum centra: </s> <s xml:id="echoid-s888" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s889" xml:space="preserve">ipſius a p o l ſolidi <lb/>centrum r; </s> <s xml:id="echoid-s890" xml:space="preserve">eius quod extra humidum ſit b: </s> <s xml:id="echoid-s891" xml:space="preserve">& </s> <s xml:id="echoid-s892" xml:space="preserve">iuncta br <lb/>producatur adg, <lb/> <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> quodſit centrum <lb/>grauitatis ſolidi ĩ <lb/>humido demerſi: <lb/></s> <s xml:id="echoid-s893" xml:space="preserve">ſumatur præterea <lb/>r h æ qualis ei, quæ <lb/>uſque ad axẽ: </s> <s xml:id="echoid-s894" xml:space="preserve">o h <lb/>autem dupla ipſi-<lb/>us h m; </s> <s xml:id="echoid-s895" xml:space="preserve">& </s> <s xml:id="echoid-s896" xml:space="preserve">alia fiãt, <lb/>ſicuti ſuperius di-<lb/>ctum eſt. </s> <s xml:id="echoid-s897" xml:space="preserve">Itaque <lb/>cum portio ad hu <lb/>midum in grauita <lb/>te non maiorem <lb/>proportionem ha <lb/>bere ponatur, quã <lb/>exceſſus, quo quadratum n o excedit quadratum m o, ad <lb/>ipſum n o quadratum: </s> <s xml:id="echoid-s898" xml:space="preserve">& </s> <s xml:id="echoid-s899" xml:space="preserve">quam proportionem in grauita <lb/>te portio habet ad humidum æqualis molis, eandem ha-<lb/>beat magnitudo portionis demerſa ad totam portio-<lb/>nem, quod demonſtratum eſt in prima propoſitione: </s> <s xml:id="echoid-s900" xml:space="preserve"><lb/>magnitudo demerſa non maiorem proportionem ha-<lb/> <anchor type="note" xlink:label="note-0042-01a" xlink:href="note-0042-01"/> bebit ad totam portionem, quàm ſit dicta illa propor-<lb/>portio. </s> <s xml:id="echoid-s901" xml:space="preserve">quare non maiorem proportionem habet tota <lb/> <anchor type="note" xlink:label="note-0042-02a" xlink:href="note-0042-02"/> portio ad eam quæ eſt extra humidum, quàm quadratum <lb/>no ad quadratum m o. </s> <s xml:id="echoid-s902" xml:space="preserve">habet autem tota portio ad eam, <lb/> <anchor type="note" xlink:label="note-0042-03a" xlink:href="note-0042-03"/> quæ extra humidum proportionem eandem, quam qua- <pb o="16" file="0043" n="43" rhead="DE IIS QVAE VEH. IN AQVA."/> dratum n o ad quadratum p f. </s> <s xml:id="echoid-s903" xml:space="preserve">quadratum igitur n o ad <lb/>quadratum p f non maiorem proportionem habet, quàm <lb/>ad quadratum m o. </s> <s xml:id="echoid-s904" xml:space="preserve">ex quo eſſicitur, ut p f non ſit minor <lb/> <anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/> ipſa o m; </s> <s xml:id="echoid-s905" xml:space="preserve">neque p b ipſa o h. </s> <s xml:id="echoid-s906" xml:space="preserve">quæ ergo ab h ducitur ad <lb/> <anchor type="note" xlink:label="note-0043-02a" xlink:href="note-0043-02"/> rectos angulos ipſi n o, coibit cum b p inter p & </s> <s xml:id="echoid-s907" xml:space="preserve">b. </s> <s xml:id="echoid-s908" xml:space="preserve">co-<lb/>eatin t. </s> <s xml:id="echoid-s909" xml:space="preserve">& </s> <s xml:id="echoid-s910" xml:space="preserve">quoniam in rectanguli coniſectione p f eſt æqui <lb/>diſtans diametro n o; </s> <s xml:id="echoid-s911" xml:space="preserve">h t autem ad diametrum perpẽ-<lb/>dicularis: </s> <s xml:id="echoid-s912" xml:space="preserve">& </s> <s xml:id="echoid-s913" xml:space="preserve">r h æqualis ei, quæ uſque ad axem: </s> <s xml:id="echoid-s914" xml:space="preserve">conſtat r t <lb/>productam ſacere angulos rectos cum ipſa k p ω. </s> <s xml:id="echoid-s915" xml:space="preserve">quare <lb/>& </s> <s xml:id="echoid-s916" xml:space="preserve">cum is. </s> <s xml:id="echoid-s917" xml:space="preserve">ergo rt perpendicularis eſt ad ſuperſiciem hu <lb/>midi. </s> <s xml:id="echoid-s918" xml:space="preserve">et ſi per b g puncta ducantur æquidiſtantes ipſirt, <lb/>ad ſuperſiciem humidi perpendicular es erunt. </s> <s xml:id="echoid-s919" xml:space="preserve">portio igi <lb/>tur, qnæ eſt extra humidum, deorſum in humidum feretur <lb/>ſecundum perpendicularem per b ductam; </s> <s xml:id="echoid-s920" xml:space="preserve">quæ uero in-<lb/>tra humidum ſecundum perpendicularem per g ſurſum <lb/>feretur: </s> <s xml:id="echoid-s921" xml:space="preserve">& </s> <s xml:id="echoid-s922" xml:space="preserve">non manebit ſolida portio a p o l, ſedintra hu <lb/>midum mouebitur, donecutique ipſa n o ſecundum per-<lb/>pendicularem ſiat.</s> <s xml:id="echoid-s923" xml:space="preserve"/> </p> <div xml:id="echoid-div65" type="float" level="2" n="1"> <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a"> <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0042-01"/> </figure> <note position="left" xlink:label="note-0042-01" xlink:href="note-0042-01a" xml:space="preserve">11. quin-<lb/>ti.</note> <note position="left" xlink:label="note-0042-02" xlink:href="note-0042-02a" xml:space="preserve">A</note> <note position="left" xlink:label="note-0042-03" xlink:href="note-0042-03a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0043-02" xlink:href="note-0043-02a" xml:space="preserve">D</note> </div> </div> <div xml:id="echoid-div67" type="section" level="1" n="28"> <head xml:id="echoid-head33" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s924" xml:space="preserve">_Quare non maiorem proportionem habet tota portio_ <lb/> <anchor type="note" xlink:label="note-0043-03a" xlink:href="note-0043-03"/> _ad eam, quæ eſt extra humidum, quam quadratum n o ad_ <lb/>_quadratum m o]_ cum enim magnitudo portionis in bumidum <lb/>demerſa ad totam portionem non maiorem proportionem babeat, <lb/>quàm exceſſus, quo quadratum n o excedit quadratum m o, ad ip-<lb/>ſum no quadratum: </s> <s xml:id="echoid-s925" xml:space="preserve">conuertendo per uigeſimáſextam quinti ele-<lb/>mentorum ex traditione Campani, tota portio ad magnitudinem de <lb/>merſam non minorem proportionem babebit, quàm quadratum n o <lb/>ad exceſſum, quo ipſum quadratum no excedit quadratum m o. </s> <s xml:id="echoid-s926" xml:space="preserve">In <lb/>telligatur portio, quæ extra bumidum, magnitudo prima: </s> <s xml:id="echoid-s927" xml:space="preserve">quæ in bu <lb/>mido demerſa est, ſecunda: </s> <s xml:id="echoid-s928" xml:space="preserve">tertia autem magnitudo ſit quadratum <lb/>mo: </s> <s xml:id="echoid-s929" xml:space="preserve">& </s> <s xml:id="echoid-s930" xml:space="preserve">exceſſus, quo quadratum n o excedit quadratum m o ſit <lb/>quarta. </s> <s xml:id="echoid-s931" xml:space="preserve">ex his igitur magnitudinibus, primæ & </s> <s xml:id="echoid-s932" xml:space="preserve">ſecundæ ad ſecun- <pb file="0044" n="44" rhead="ARCHIME DIS"/> dam non minor est proportio, quàm tertiæ & </s> <s xml:id="echoid-s933" xml:space="preserve">quartæ ad quartam; <lb/></s> <s xml:id="echoid-s934" xml:space="preserve">est enim quadratum m o unà cum exceſſu, quo quadratum n o exce <lb/>dit quadratum m o æquale ipſi n o quadrato. </s> <s xml:id="echoid-s935" xml:space="preserve">quare per conuerſio <lb/>nem rationis ex 30 eiuſdem, primæ & </s> <s xml:id="echoid-s936" xml:space="preserve">ſecundæ ad primam non ma-<lb/>ior proportio erit, quàm tertiæ & </s> <s xml:id="echoid-s937" xml:space="preserve">quartæ ad tertiam: </s> <s xml:id="echoid-s938" xml:space="preserve">& </s> <s xml:id="echoid-s939" xml:space="preserve">idcirco to-<lb/>ta portio ad portionem eam, quæ est extra bumidum non maiorem <lb/>proportionem babebit, quàm quadratum n o ad quadratum mo. </s> <s xml:id="echoid-s940" xml:space="preserve"><lb/>quod demonstrandum proponebatur.</s> <s xml:id="echoid-s941" xml:space="preserve"/> </p> <div xml:id="echoid-div67" type="float" level="2" n="1"> <note position="right" xlink:label="note-0043-03" xlink:href="note-0043-03a" xml:space="preserve">A</note> </div> <p> <s xml:id="echoid-s942" xml:space="preserve">Habet autem tota portio ad eam, quæ extra humidum <lb/> <anchor type="note" xlink:label="note-0044-01a" xlink:href="note-0044-01"/> proportionem eandem, quam quadratum n o ad quadra <lb/>tum p f.</s> <s xml:id="echoid-s943" xml:space="preserve">] _Ex uigeſimaſexta libri de conoidibus, & </s> <s xml:id="echoid-s944" xml:space="preserve">ſpbæroi-_ <lb/>_dibus._</s> <s xml:id="echoid-s945" xml:space="preserve"/> </p> <div xml:id="echoid-div68" type="float" level="2" n="2"> <note position="left" xlink:label="note-0044-01" xlink:href="note-0044-01a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s946" xml:space="preserve">Ex quo eſſicitur, ut p ſ non ſit minor ipſa o m; </s> <s xml:id="echoid-s947" xml:space="preserve">neque <lb/> <anchor type="note" xlink:label="note-0044-02a" xlink:href="note-0044-02"/> pb ipſa o h.</s> <s xml:id="echoid-s948" xml:space="preserve">] _Sequitur illud ex decima & </s> <s xml:id="echoid-s949" xml:space="preserve">decimaquarta quinti,_ <lb/>_& </s> <s xml:id="echoid-s950" xml:space="preserve">ex uigeſimaſecunda ſexti elementorum, ut ſuperius dictum eſt._</s> <s xml:id="echoid-s951" xml:space="preserve"/> </p> <div xml:id="echoid-div69" type="float" level="2" n="3"> <note position="left" xlink:label="note-0044-02" xlink:href="note-0044-02a" xml:space="preserve">C</note> </div> <p> <s xml:id="echoid-s952" xml:space="preserve">Quæ ergo ab h ducitur ad rectos angulos ipſi n o coi-<lb/> <anchor type="note" xlink:label="note-0044-03a" xlink:href="note-0044-03"/> bit cum p b inter p & </s> <s xml:id="echoid-s953" xml:space="preserve">b.</s> <s xml:id="echoid-s954" xml:space="preserve">] _Cur boc ita contingat, nos proxi-_ <lb/>_me explicauimus._</s> <s xml:id="echoid-s955" xml:space="preserve"/> </p> <div xml:id="echoid-div70" type="float" level="2" n="4"> <note position="left" xlink:label="note-0044-03" xlink:href="note-0044-03a" xml:space="preserve">D</note> </div> </div> <div xml:id="echoid-div72" type="section" level="1" n="29"> <head xml:id="echoid-head34" xml:space="preserve">PROPOSITIO VI.</head> <p> <s xml:id="echoid-s956" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem quidem <lb/>quàm ſeſquialterum eius, quæ uſque ad axem, <lb/>minorem nero, quàm ut ad eam, quæ uſque ad <lb/>axem proportionem habeat, quam quindecim <lb/>ad quatuor; </s> <s xml:id="echoid-s957" xml:space="preserve">in humidum demiſſa adeo, ut baſis <lb/>ipſius contingat humidum, nunquam conſiſtet <lb/>inclinata ita, ut baſis in uno puncto humidum <lb/>contingat.</s> <s xml:id="echoid-s958" xml:space="preserve"/> </p> <pb o="17" file="0045" n="45" rhead="DE IIS QVAE VEH. IN AQVA."/> <p> <s xml:id="echoid-s959" xml:space="preserve">SIT portio, qualis dicta eſt, & </s> <s xml:id="echoid-s960" xml:space="preserve">in humidum demittatur, <lb/>ſicuti diximus, adeo ut baſis eius in uno puncto contingat <lb/>humidum. </s> <s xml:id="echoid-s961" xml:space="preserve">demonſtrandum eſtnon manere ipſam portio-<lb/>nem, ſed reuoluiita, ut baſis nullo modo humidi ſuperſicie <lb/> <anchor type="note" xlink:label="note-0045-01a" xlink:href="note-0045-01"/> contingat. </s> <s xml:id="echoid-s962" xml:space="preserve">Secta enim ipſa per axem, plano ad ſuper ſiciem <lb/>humidi recto, ſit ſectio ſuperſiciei portionis a p o l re-<lb/>ctãguli coni ſe <lb/> <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> ctio: </s> <s xml:id="echoid-s963" xml:space="preserve">ſuperſi-<lb/>ciei humidi ſe-<lb/>ctio ſit a s: </s> <s xml:id="echoid-s964" xml:space="preserve">axis <lb/>autem portio-<lb/>nis, ac ſectio-<lb/>nis diameter n <lb/>o: </s> <s xml:id="echoid-s965" xml:space="preserve">& </s> <s xml:id="echoid-s966" xml:space="preserve">ſccetur in <lb/>f quidẽ ita, ut <lb/>o f ſit dupla ip <lb/>ſius ſn; </s> <s xml:id="echoid-s967" xml:space="preserve">in ω ue <lb/>ro, ut n o ad <lb/>f ω eandem ha <lb/>beat proportionem, quam quindecim ad quatuor: </s> <s xml:id="echoid-s968" xml:space="preserve">& </s> <s xml:id="echoid-s969" xml:space="preserve">ipſi <lb/>n o ad rectos angulos ducatur ω k. </s> <s xml:id="echoid-s970" xml:space="preserve">Itaque quoniam n o <lb/> <anchor type="note" xlink:label="note-0045-02a" xlink:href="note-0045-02"/> ad f ω maiorem habet proportionem, quàm ad eam, quæ <lb/>uſque ad axem; </s> <s xml:id="echoid-s971" xml:space="preserve">ſit ei, quæ uſque ad axem æqualis f b: </s> <s xml:id="echoid-s972" xml:space="preserve">& </s> <s xml:id="echoid-s973" xml:space="preserve">du <lb/>catur p c quidem ipſi a s æquidiſtans, cõtingensq; </s> <s xml:id="echoid-s974" xml:space="preserve">ſectio-<lb/>nem a p o l in p; </s> <s xml:id="echoid-s975" xml:space="preserve">pi uero æquidiſtans ipſi n o: </s> <s xml:id="echoid-s976" xml:space="preserve">& </s> <s xml:id="echoid-s977" xml:space="preserve">primum <lb/>ſecet pi ipſam κ ω in h. </s> <s xml:id="echoid-s978" xml:space="preserve">Quoniã ergo in portione a p o l, <lb/> <anchor type="note" xlink:label="note-0045-03a" xlink:href="note-0045-03"/> quæ continetur recta linea, & </s> <s xml:id="echoid-s979" xml:space="preserve">rectanguli coni ſectione, κ ω <lb/>quidem æ quidiſtans eſtipſi a l; </s> <s xml:id="echoid-s980" xml:space="preserve">p i uero diametro æquidi-<lb/>ſtat: </s> <s xml:id="echoid-s981" xml:space="preserve">ſecaturq; </s> <s xml:id="echoid-s982" xml:space="preserve">ab ipſa κ ω in h: </s> <s xml:id="echoid-s983" xml:space="preserve">& </s> <s xml:id="echoid-s984" xml:space="preserve">a s æquidiſtat contingen-<lb/>ti in p: </s> <s xml:id="echoid-s985" xml:space="preserve">neceſſarium eſtipſam p i ad p h uel ean dem pro-<lb/>portionem habere, quam habet n ω ad ω o, uel maiorem: <lb/></s> <s xml:id="echoid-s986" xml:space="preserve">hocenim iam demonſtratum eſt. </s> <s xml:id="echoid-s987" xml:space="preserve">At uero n ω ſeſquialtera <lb/>eſt ipſius ω o. </s> <s xml:id="echoid-s988" xml:space="preserve">& </s> <s xml:id="echoid-s989" xml:space="preserve">pi igitur uel ſeſquialtera eſt ipſius h p; </s> <s xml:id="echoid-s990" xml:space="preserve"><lb/>uel maior, quàm ſeſquialtera. </s> <s xml:id="echoid-s991" xml:space="preserve">Quare ph ipſius h i aut du <lb/> <anchor type="note" xlink:label="note-0045-04a" xlink:href="note-0045-04"/> <pb file="0046" n="46" rhead="ARCHIMEDIS"/> pla eſt, aut minor, quàm dupla. </s> <s xml:id="echoid-s992" xml:space="preserve">Sit autem p t dupla t i. </s> <s xml:id="echoid-s993" xml:space="preserve">erit <lb/>centrum grauitatis eius, quod eſt in humido, punctum t. <lb/></s> <s xml:id="echoid-s994" xml:space="preserve">Itaque iuncta t f producatur; </s> <s xml:id="echoid-s995" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s996" xml:space="preserve">eius, quod extra humi <lb/>dum grauitatis centrum g: </s> <s xml:id="echoid-s997" xml:space="preserve">& </s> <s xml:id="echoid-s998" xml:space="preserve">à puncto b ad rectos angu-<lb/>los ipſi n o ducatur b r. </s> <s xml:id="echoid-s999" xml:space="preserve">Quòd cum p i quidem ſit æqui-<lb/>diſtans diametro n o: </s> <s xml:id="echoid-s1000" xml:space="preserve">br autem ad diametrum perpendi <lb/>cularis. </s> <s xml:id="echoid-s1001" xml:space="preserve">& </s> <s xml:id="echoid-s1002" xml:space="preserve">f b æqualis ei, quæ uſque ad axem: </s> <s xml:id="echoid-s1003" xml:space="preserve">perſpicuum <lb/>eſt f r productam æquales facere angulos cum ea, quæ ſe-<lb/>ctionem a p o l in puncto p contingit. </s> <s xml:id="echoid-s1004" xml:space="preserve">quare & </s> <s xml:id="echoid-s1005" xml:space="preserve">cum a s: </s> <s xml:id="echoid-s1006" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s1007" xml:space="preserve">cum ſuperficie humidi. </s> <s xml:id="echoid-s1008" xml:space="preserve">lineæ autem ductæ per tg æqui-<lb/>diſtantes ipſi f r, erunt & </s> <s xml:id="echoid-s1009" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> ad humidi ſuperficiẽ per-<lb/>pendiculares: </s> <s xml:id="echoid-s1010" xml:space="preserve">& </s> <s xml:id="echoid-s1011" xml:space="preserve">ſolidi <lb/>a p o l magnitudo, quæ ẽ <lb/>intra humidum ſurſum fe <lb/>retur ſecundum perpen-<lb/>dicularem per t ductam; <lb/></s> <s xml:id="echoid-s1012" xml:space="preserve">quæ uero extra humidum <lb/>ſecundum eam, quæ per g <lb/>deorſum feretur. </s> <s xml:id="echoid-s1013" xml:space="preserve">reuolue <lb/> <anchor type="note" xlink:label="note-0046-01a" xlink:href="note-0046-01"/> tur ergo ſolidum a p o l: <lb/></s> <s xml:id="echoid-s1014" xml:space="preserve">& </s> <s xml:id="echoid-s1015" xml:space="preserve">baſis ipſius nullo modo <lb/>humidi ſuperficiem con-<lb/>tinget. </s> <s xml:id="echoid-s1016" xml:space="preserve">At ſi pi lineam k ω <lb/>non ſecet, ut in ſecunda <lb/>figura; </s> <s xml:id="echoid-s1017" xml:space="preserve">manifeſtum eſt punctum t, quod eſt centrum gra-<lb/>uitatis demerſæ portionis, cadere inter p & </s> <s xml:id="echoid-s1018" xml:space="preserve">i: </s> <s xml:id="echoid-s1019" xml:space="preserve">& </s> <s xml:id="echoid-s1020" xml:space="preserve">reliqua <lb/>ſimiliter demonſtrabuntur.</s> <s xml:id="echoid-s1021" xml:space="preserve"/> </p> <div xml:id="echoid-div72" type="float" level="2" n="1"> <note position="right" xlink:label="note-0045-01" xlink:href="note-0045-01a" xml:space="preserve">A</note> <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/4E7V2WGH/figures/0045-01"/> </figure> <note position="right" xlink:label="note-0045-02" xlink:href="note-0045-02a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0045-03" xlink:href="note-0045-03a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0045-04" xlink:href="note-0045-04a" xml:space="preserve">D</note> <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/4E7V2WGH/figures/0046-01"/> </figure> <note position="left" xlink:label="note-0046-01" xlink:href="note-0046-01a" xml:space="preserve">E</note> </div> </div> <div xml:id="echoid-div74" type="section" level="1" n="30"> <head xml:id="echoid-head35" xml:space="preserve">COMMENTARIVS.</head> <p> <s xml:id="echoid-s1022" xml:space="preserve">Demonſtrandum eſt non manere ipſam portionem, ſed <lb/> <anchor type="note" xlink:label="note-0046-02a" xlink:href="note-0046-02"/> reuolui ita, ut baſis nullo modo ſuperficiem humidi con-<lb/>tingat.</s> <s xml:id="echoid-s1023" xml:space="preserve">] _Hæcnos addidimus tanquam ab interprete omiſſa_.</s> <s xml:id="echoid-s1024" xml:space="preserve"/> </p> <div xml:id="echoid-div74" type="float" level="2" n="1"> <note position="left" xlink:label="note-0046-02" xlink:href="note-0046-02a" xml:space="preserve">A</note> </div> <pb o="18" file="0047" n="47" rhead="DE IIS QVAE VEH. IN AQVA."/> <p> <s xml:id="echoid-s1025" xml:space="preserve">Itaque quoniam no ad f ω maiorem habetproportio-<lb/> <anchor type="note" xlink:label="note-0047-01a" xlink:href="note-0047-01"/> nem, quam ad eam, quæ uſque ad axem.</s> <s xml:id="echoid-s1026" xml:space="preserve">] _Habet enim diame-_ <lb/>_ter portioms n o ad f ω proportionem eandem, quam quindeeim ad_ <lb/>_quatuor; </s> <s xml:id="echoid-s1027" xml:space="preserve">ad eam uero, quæ uſque ad axem minorem proportionem_ <lb/>_habere ponitur, quàm quindecim ad quatuor. </s> <s xml:id="echoid-s1028" xml:space="preserve">quare n o ad f ω ma_ <lb/>_iorem habebit proportionem, quàm ad eam, quæ uſque ad axem: </s> <s xml:id="echoid-s1029" xml:space="preserve">&_</s> <s xml:id="echoid-s1030" xml:space="preserve"> <lb/>_propterea quæ uſque ad axem ipſa f ω maior erit_. <lb/></s> <s xml:id="echoid-s1031" xml:space="preserve"/> </p> <div xml:id="echoid-div75" type="float" level="2" n="2"> <note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">B</note> </div> <note position="right" xml:space="preserve">10. quinti</note> <p> <s xml:id="echoid-s1032" xml:space="preserve">Quoniam ergo in portione a p o l, quæ continetur re-<lb/>cta linea, & </s> <s xml:id="echoid-s1033" xml:space="preserve">rectanguli coni ſectione, _k_ ω quidem æ quidi-<lb/>ſtans eſt ipſi a l; </s> <s xml:id="echoid-s1034" xml:space="preserve">p i uero diametro æquidiſtat; </s> <s xml:id="echoid-s1035" xml:space="preserve">ſecaturq; <lb/></s> <s xml:id="echoid-s1036" xml:space="preserve">ab ipſa k ω in h: </s> <s xml:id="echoid-s1037" xml:space="preserve">& </s> <s xml:id="echoid-s1038" xml:space="preserve">a c æquidiſtat contingenti in p neceſ-<lb/>ſarium eſt ipſam p i ad p h uel eandem proportionem ha <lb/>bere, quam habet n ω ad ω o, uel maiorem. </s> <s xml:id="echoid-s1039" xml:space="preserve">hoc enim iam <lb/>demonſtratum eſt] _Vbi hoc demonſtratum ſit uel ab ipſo Ar-_ <lb/>_chimede, uel ab alio, numdum apparet, quocircanos demonstra-_ <lb/>_tionem afferemus, poſteaquam non nulla, quæ ad eam pertinent ex-_ <lb/>_plicauerimus_.</s> <s xml:id="echoid-s1040" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div77" type="section" level="1" n="31"> <head xml:id="echoid-head36" xml:space="preserve">LEMMAI.</head> <p style="it"> <s xml:id="echoid-s1041" xml:space="preserve">Sint lineæ a b, a c angulum b a c continentes: </s> <s xml:id="echoid-s1042" xml:space="preserve">& </s> <s xml:id="echoid-s1043" xml:space="preserve">à <lb/>puncto d, quod in linea a c ſumptum ſit, ducantur d e, <lb/>d f utcunque ad ipſam a b. </s> <s xml:id="echoid-s1044" xml:space="preserve">Sumptis uero in eadem li. <lb/></s> <s xml:id="echoid-s1045" xml:space="preserve">nea quotlibet punctis g l, ducantur g h, l m ipſi d e <lb/>æquidistantes; </s> <s xml:id="echoid-s1046" xml:space="preserve">& </s> <s xml:id="echoid-s1047" xml:space="preserve">g k, l n æquidiſtantes f d. </s> <s xml:id="echoid-s1048" xml:space="preserve">deinde à <lb/>punctis d, g uſque ad lineam m l ducantur, d o p qui <lb/>dem ſecans g h in o; </s> <s xml:id="echoid-s1049" xml:space="preserve">& </s> <s xml:id="echoid-s1050" xml:space="preserve">g q, quæ æquidistent ipſi b a. </s> <s xml:id="echoid-s1051" xml:space="preserve"><lb/>Dico lineas, quæ inter æquidiſtantes ipſi f d ad eas, quæ <lb/>inter æquidiſtantes d e interiiciuntur, uidelicet k n ad g q, <lb/>uel ad o p; </s> <s xml:id="echoid-s1052" xml:space="preserve">f k ad d o; </s> <s xml:id="echoid-s1053" xml:space="preserve">& </s> <s xml:id="echoid-s1054" xml:space="preserve">f n ad d p eandem inter ſe ſe <lb/>proportionem habere: </s> <s xml:id="echoid-s1055" xml:space="preserve">nempe eam, quã habet a f ad a e.</s> <s xml:id="echoid-s1056" xml:space="preserve"> <pb file="0048" n="48" rhead="ARCHIMEDIS"/> Quoniam enim triangula afd, akg, anl ſi-<lb/> <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/> milia ſunt; </s> <s xml:id="echoid-s1057" xml:space="preserve">itémq; </s> <s xml:id="echoid-s1058" xml:space="preserve">ſimilia efd, h k g, mnl: <lb/></s> <s xml:id="echoid-s1059" xml:space="preserve">erit ut af ad fd, ita ak ad kg; </s> <s xml:id="echoid-s1060" xml:space="preserve">ut autem fd <lb/> <anchor type="note" xlink:label="note-0048-01a" xlink:href="note-0048-01"/> ad fe, ita kg ad kh. </s> <s xml:id="echoid-s1061" xml:space="preserve">quare ex æquali ut af <lb/>ad fe, ita ak ad kh: </s> <s xml:id="echoid-s1062" xml:space="preserve">& </s> <s xml:id="echoid-s1063" xml:space="preserve">per conuerſionem ra-<lb/>tionis ut af ad ae, ita ak ad ah. </s> <s xml:id="echoid-s1064" xml:space="preserve">eodem <lb/>modo oſtendetur, ut af ad a e, ita an ad am. <lb/></s> <s xml:id="echoid-s1065" xml:space="preserve">cum igitur an ad am ſit, ut a k ad a h; </s> <s xml:id="echoid-s1066" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-0048-02a" xlink:href="note-0048-02"/> reliqua kn ad reliquam h m, hoc eſt ad g q, <lb/>uel o p, ut a n ad a m; </s> <s xml:id="echoid-s1067" xml:space="preserve">hoc estut a f ad a e. <lb/></s> <s xml:id="echoid-s1068" xml:space="preserve">rurſus a k ad a h est, ut a f ad a e. </s> <s xml:id="echoid-s1069" xml:space="preserve">er-<lb/>go reliqua f k ad e h reliquam, uidelicet <lb/>ad do, ut a f ad a e. </s> <s xml:id="echoid-s1070" xml:space="preserve">Similiter demonſtrabi-<lb/>mus ita eſſe fn ad d p. </s> <s xml:id="echoid-s1071" xml:space="preserve">quod quidem demonſtra <lb/>re oportebat.</s> <s xml:id="echoid-s1072" xml:space="preserve"/> </p> <div xml:id="echoid-div77" type="float" level="2" n="1"> <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a"> <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0048-01"/> </figure> <note position="left" xlink:label="note-0048-01" xlink:href="note-0048-01a" xml:space="preserve">4. ſexti.</note> <note position="left" xlink:label="note-0048-02" xlink:href="note-0048-02a" xml:space="preserve">19. quinti</note> </div> </div> <div xml:id="echoid-div79" type="section" level="1" n="32"> <head xml:id="echoid-head37" xml:space="preserve">LEMMA II.</head> <p style="it"> <s xml:id="echoid-s1073" xml:space="preserve">Sint in eadem linea a b puncta <lb/> <anchor type="figure" xlink:label="fig-0048-02a" xlink:href="fig-0048-02"/> duo r s ita diſpoſita, ut a s ad a r <lb/>eandem proportionem habeat, quam <lb/>a f ad ae: </s> <s xml:id="echoid-s1074" xml:space="preserve">& </s> <s xml:id="echoid-s1075" xml:space="preserve">per r ducatur rtipſi <lb/>e d æquidiſtans; </s> <s xml:id="echoid-s1076" xml:space="preserve">per s uero ducatur <lb/>s t æquidiſtans fd, ita ut cum r t in <lb/>t puncto conueniat. </s> <s xml:id="echoid-s1077" xml:space="preserve">Dico punctum t <lb/>cadere in lineam a c.</s> <s xml:id="echoid-s1078" xml:space="preserve"/> </p> <div xml:id="echoid-div79" type="float" level="2" n="1"> <figure xlink:label="fig-0048-02" xlink:href="fig-0048-02a"> <image file="0048-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0048-02"/> </figure> </div> <p style="it"> <s xml:id="echoid-s1079" xml:space="preserve">Si enim fieri potest, cadat citra: </s> <s xml:id="echoid-s1080" xml:space="preserve">& </s> <s xml:id="echoid-s1081" xml:space="preserve">produca <lb/>tur rt uſque ad ipſam a c in u. </s> <s xml:id="echoid-s1082" xml:space="preserve">deinde per u <lb/>ducatur u x ipſi f d æquidiſtans. </s> <s xml:id="echoid-s1083" xml:space="preserve">Itaque ex <lb/>ijs, quæ proxime demonstrauimus a x ad ar <pb o="19" file="0049" n="49" rhead="DE IIS QVAE VEH. IN AQVA."/> eam proportionem babebit, quam a f ad a e. </s> <s xml:id="echoid-s1084" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1085" xml:space="preserve">eandem habet <lb/>a s ad a r. </s> <s xml:id="echoid-s1086" xml:space="preserve">quare a s ipſi a x eſt æqualis, pars toti, quod fieri non <lb/> <anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/> poteſt. </s> <s xml:id="echoid-s1087" xml:space="preserve">Idem abſurdum ſequetur, ſi ponamus punctum t cadere ul-<lb/>tra lineam a c. </s> <s xml:id="echoid-s1088" xml:space="preserve">neceſſarium igitur est, ut in ipſam a c cadat. </s> <s xml:id="echoid-s1089" xml:space="preserve">quod <lb/>demonſtrandum propoſuimus.</s> <s xml:id="echoid-s1090" xml:space="preserve"/> </p> <div xml:id="echoid-div80" type="float" level="2" n="2"> <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">9. quinti</note> </div> </div> <div xml:id="echoid-div82" type="section" level="1" n="33"> <head xml:id="echoid-head38" xml:space="preserve">LEMMA III.</head> <p style="it"> <s xml:id="echoid-s1091" xml:space="preserve">Sit parabole, cuius diameter a b: </s> <s xml:id="echoid-s1092" xml:space="preserve">atque eam cŏtingen <lb/>tes rectæ lineæ a c, b d; </s> <s xml:id="echoid-s1093" xml:space="preserve">a c quidem in puncto c, b d ue <lb/>ro in b: </s> <s xml:id="echoid-s1094" xml:space="preserve">& </s> <s xml:id="echoid-s1095" xml:space="preserve">per c ductis duabus lineis; </s> <s xml:id="echoid-s1096" xml:space="preserve">quarum alter a c e <lb/>diametro æquidiſtet, alter a c f æquidiſtet ipſi b d: </s> <s xml:id="echoid-s1097" xml:space="preserve">ſuma <lb/>tur quod uis punctum g in diametro: </s> <s xml:id="echoid-s1098" xml:space="preserve">fiatque ut f b, ad <lb/>b g, ita b g ad b h: </s> <s xml:id="echoid-s1099" xml:space="preserve">& </s> <s xml:id="echoid-s1100" xml:space="preserve">per g h ducantur g k l, h e m, <lb/>æquidiſtantes b d: </s> <s xml:id="echoid-s1101" xml:space="preserve">per m uero ducatur m n o ipſi a c <lb/>æquidistans, quæ diametrum ſecet in o: </s> <s xml:id="echoid-s1102" xml:space="preserve">& </s> <s xml:id="echoid-s1103" xml:space="preserve">per n ducta <lb/>n p uſque ad diametrum, ipſi b d æquidistet. </s> <s xml:id="echoid-s1104" xml:space="preserve">Dico h o <lb/>ipſius g b duplam eſſe.</s> <s xml:id="echoid-s1105" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1106" xml:space="preserve">V_EL_ igitur linea m n o ſccat diametrum in g, uel in alijs pun-<lb/>ctis: </s> <s xml:id="echoid-s1107" xml:space="preserve">& </s> <s xml:id="echoid-s1108" xml:space="preserve">ſi quidem ſecat in g, unum at que idem punctum duabus li-<lb/>teris go notabitur. </s> <s xml:id="echoid-s1109" xml:space="preserve">Itaque quoniam f c, p n, h e m ſibiipſis æqui <lb/>distant: </s> <s xml:id="echoid-s1110" xml:space="preserve">& </s> <s xml:id="echoid-s1111" xml:space="preserve">ipſi a c æquidiſtat m n o: </s> <s xml:id="echoid-s1112" xml:space="preserve">fient triangula a f c, o p n, <lb/>o h m inter ſe ſimilia. </s> <s xml:id="echoid-s1113" xml:space="preserve">quare erit o h ad h m, ut a f ad fc: </s> <s xml:id="echoid-s1114" xml:space="preserve">& </s> <s xml:id="echoid-s1115" xml:space="preserve">per-<lb/> <anchor type="note" xlink:label="note-0049-02a" xlink:href="note-0049-02"/> mut ando o h ad a f, ut h m ad fc. </s> <s xml:id="echoid-s1116" xml:space="preserve">est autem quadratum h m ad <lb/>quadratum g l, ut linea h b ad lineam b g, ex uigeſima primi libri <lb/>conicorum: </s> <s xml:id="echoid-s1117" xml:space="preserve">& </s> <s xml:id="echoid-s1118" xml:space="preserve">quadratum g l ad quadratum fc, ut linea g b ad <lb/>ipſam b f: </s> <s xml:id="echoid-s1119" xml:space="preserve">ſuntq; </s> <s xml:id="echoid-s1120" xml:space="preserve">h b, b g, b f lineæ deinceps proportionales. </s> <s xml:id="echoid-s1121" xml:space="preserve">er-<lb/> <anchor type="note" xlink:label="note-0049-03a" xlink:href="note-0049-03"/> go & </s> <s xml:id="echoid-s1122" xml:space="preserve">quadrata h m, g l, f c, & </s> <s xml:id="echoid-s1123" xml:space="preserve">ipſorum latera proportionalia <lb/>erunt. </s> <s xml:id="echoid-s1124" xml:space="preserve">atque idcirco ut quadratum h m ad quadratum g l, ita li- <pb file="0050" n="50" rhead="ARCHIMEDIS"/> nea h m ad li-<lb/> <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> neam fc. </s> <s xml:id="echoid-s1125" xml:space="preserve">at uero <lb/>ut h m ad f c, ita <lb/>o h ad a f: </s> <s xml:id="echoid-s1126" xml:space="preserve">& </s> <s xml:id="echoid-s1127" xml:space="preserve">ut <lb/>quadratum h m <lb/>ad quadratú g l, <lb/>ita linea h b ad <lb/>b g; </s> <s xml:id="echoid-s1128" xml:space="preserve">hoc est b g <lb/>ad b f. </s> <s xml:id="echoid-s1129" xml:space="preserve">ex quibus <lb/>ſequitur o h ad <lb/>a f ita eſſe, ut b g <lb/>ad b f: </s> <s xml:id="echoid-s1130" xml:space="preserve">& </s> <s xml:id="echoid-s1131" xml:space="preserve">permu <lb/>tando oh ad b g, <lb/>ut a f ad f b. </s> <s xml:id="echoid-s1132" xml:space="preserve">ſed <lb/>eſt a f dupla ip-<lb/>ſius fb: </s> <s xml:id="echoid-s1133" xml:space="preserve">ſunt eni <lb/>a b, b f æquales <lb/>ex 35 primi libri <lb/>conicorum. </s> <s xml:id="echoid-s1134" xml:space="preserve">ergo <lb/>& </s> <s xml:id="echoid-s1135" xml:space="preserve">h o ipſius g b <lb/>eſt dupla. </s> <s xml:id="echoid-s1136" xml:space="preserve">quod demonſtrare oportebat.</s> <s xml:id="echoid-s1137" xml:space="preserve"/> </p> <div xml:id="echoid-div82" type="float" level="2" n="1"> <note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">22. ſexti. <lb/>cor. 20. ſe <lb/>xti.</note> <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a"> <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0050-01"/> </figure> </div> </div> <div xml:id="echoid-div84" type="section" level="1" n="34"> <head xml:id="echoid-head39" xml:space="preserve">LEMMA IIII.</head> <p style="it"> <s xml:id="echoid-s1138" xml:space="preserve">Iiſdem manentibus, & </s> <s xml:id="echoid-s1139" xml:space="preserve">à puncto m ducta m q uſque <lb/>ad diametrum, quæ ſectionem in puncto m conting at; <lb/></s> <s xml:id="echoid-s1140" xml:space="preserve">Dico h q ad q o eandem proportionem habere, quam <lb/>habet g h ad c n.</s> <s xml:id="echoid-s1141" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1142" xml:space="preserve">F_IAT_ enim h r æqualis g f. </s> <s xml:id="echoid-s1143" xml:space="preserve">& </s> <s xml:id="echoid-s1144" xml:space="preserve">cumtriangula a f c, o p n ſimi <lb/>lia ſint, & </s> <s xml:id="echoid-s1145" xml:space="preserve">p n ſit æqualis f c; </s> <s xml:id="echoid-s1146" xml:space="preserve">eodem modo demonſtrabimus p o, f a <lb/>inter ſe æquales eſſe. </s> <s xml:id="echoid-s1147" xml:space="preserve">quare p o ipſius f b dupla erit. </s> <s xml:id="echoid-s1148" xml:space="preserve">Sed eſt h o du <lb/>pla g b. </s> <s xml:id="echoid-s1149" xml:space="preserve">ergo & </s> <s xml:id="echoid-s1150" xml:space="preserve">reliqua p h reliquæ f g; </s> <s xml:id="echoid-s1151" xml:space="preserve">uidelicet ipſius r h eſt du- <pb o="20" file="0051" n="51" rhead="DE IIS QVAE VEH. IN AQVA."/> pla. </s> <s xml:id="echoid-s1152" xml:space="preserve">ex quo fit ut pr, rh, fg inter ſe ſint æquales; </s> <s xml:id="echoid-s1153" xml:space="preserve">itémq; </s> <s xml:id="echoid-s1154" xml:space="preserve">æquales <lb/>rg, pf. </s> <s xml:id="echoid-s1155" xml:space="preserve">eſt enim pg utrique r p, gf communis. </s> <s xml:id="echoid-s1156" xml:space="preserve">Quoniam igitur <lb/>hb ad bg est, ut <lb/> <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> gb ad bf; </s> <s xml:id="echoid-s1157" xml:space="preserve">per c<gap/> <lb/>uerſionem ratio-<lb/>mis erit b h ad <lb/>h g, ut b g ad gf. <lb/></s> <s xml:id="echoid-s1158" xml:space="preserve">eſt autem q h ad <lb/>h b, ut h o ad gb. </s> <s xml:id="echoid-s1159" xml:space="preserve"><lb/>nam ex 35 primi <lb/>libri conicorum, <lb/>cum linea qm có <lb/>tingat ſectionem <lb/>in punctom; </s> <s xml:id="echoid-s1160" xml:space="preserve">erút <lb/>h b, bq æquales; </s> <s xml:id="echoid-s1161" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s1162" xml:space="preserve">gh ipſius h b <lb/>dupla. </s> <s xml:id="echoid-s1163" xml:space="preserve">ergo ex æ-<lb/>quali q h ad hg, <lb/>ut ho ad g f; </s> <s xml:id="echoid-s1164" xml:space="preserve">hoc <lb/>eſt ad hr: </s> <s xml:id="echoid-s1165" xml:space="preserve">& </s> <s xml:id="echoid-s1166" xml:space="preserve">per <lb/>mutando q h ad <lb/>h o, ut g h ad h r. </s> <s xml:id="echoid-s1167" xml:space="preserve"><lb/>rurſus per conuerſionem rationis h q ad qo, ut h g ad g r; </s> <s xml:id="echoid-s1168" xml:space="preserve">hoc eſt <lb/>p f: </s> <s xml:id="echoid-s1169" xml:space="preserve">& </s> <s xml:id="echoid-s1170" xml:space="preserve">propterea ad ipſam cn, quod demonstrandum fuerat.</s> <s xml:id="echoid-s1171" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="1"> <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a"> <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0051-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s1172" xml:space="preserve">His igitur explicatis, iam adid, quod propoſitum fue <lb/>rat, accedamus. </s> <s xml:id="echoid-s1173" xml:space="preserve">Itaque dico primum nc ad c k eandem <lb/>proportionem babere, quam h g ad g b.</s> <s xml:id="echoid-s1174" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1175" xml:space="preserve">Quoniam enim h q ad qo eſt, ut h g ad c n, hoc eſt ad a o ipſi <lb/>cn æqualem; </s> <s xml:id="echoid-s1176" xml:space="preserve">erit reliqua gq ad reliquam q a, ut h q ad q o: </s> <s xml:id="echoid-s1177" xml:space="preserve">& </s> <s xml:id="echoid-s1178" xml:space="preserve"><lb/>ob eam cauſſam lineæ a c g l productæ ex ijs, quæ ſupra demonſtra <lb/>uimus in linea q m conueniunt. </s> <s xml:id="echoid-s1179" xml:space="preserve">Rurſus gq ad qa eſt, ut h q ad <pb file="0052" n="52" rhead="ARCHIMEDIS"/> q o; </s> <s xml:id="echoid-s1180" xml:space="preserve">uidelicet ut h g ad f p: </s> <s xml:id="echoid-s1181" xml:space="preserve">quod proxime demonſtr atum eſt. </s> <s xml:id="echoid-s1182" xml:space="preserve">At <lb/> <anchor type="note" xlink:label="note-0052-01a" xlink:href="note-0052-01"/> ueroipſi g q æquales ſunt duæ lineæ ſimul ſumptæ qb, hoc eſt h b, <lb/> <anchor type="note" xlink:label="note-0052-02a" xlink:href="note-0052-02"/> & </s> <s xml:id="echoid-s1183" xml:space="preserve">b g: </s> <s xml:id="echoid-s1184" xml:space="preserve">atque ipſi q a æqualis eſt h f. </s> <s xml:id="echoid-s1185" xml:space="preserve">Sienim ab æqualibus h b, <lb/>bq, æqualia fb, <lb/> <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> ba demantur, re <lb/>manentia æqua-<lb/>lia erunt. </s> <s xml:id="echoid-s1186" xml:space="preserve">ergo <lb/>dempta h g ex <lb/>duabus lineis h <lb/>b, h g, relinqui-<lb/>tur dupla ipſius <lb/>b g; </s> <s xml:id="echoid-s1187" xml:space="preserve">hoc eſt o h: <lb/></s> <s xml:id="echoid-s1188" xml:space="preserve">& </s> <s xml:id="echoid-s1189" xml:space="preserve">dempta p f ex <lb/>f h, reliqua est <lb/>b p. </s> <s xml:id="echoid-s1190" xml:space="preserve">quare o h <lb/> <anchor type="note" xlink:label="note-0052-03a" xlink:href="note-0052-03"/> ad h p, eſt ut g q <lb/>ad q a. </s> <s xml:id="echoid-s1191" xml:space="preserve">Sed ut <lb/>g q ad q a, ita <lb/>h q ad q o; </s> <s xml:id="echoid-s1192" xml:space="preserve">hoc <lb/>eſt h g ad n c: <lb/></s> <s xml:id="echoid-s1193" xml:space="preserve">& </s> <s xml:id="echoid-s1194" xml:space="preserve">ut o h ad h p, <lb/> <anchor type="note" xlink:label="note-0052-04a" xlink:href="note-0052-04"/> ita g b ad c k. </s> <s xml:id="echoid-s1195" xml:space="preserve">eſt <lb/>cnim o h dupla <lb/>g b, & </s> <s xml:id="echoid-s1196" xml:space="preserve">h p item <lb/>dupla gf; </s> <s xml:id="echoid-s1197" xml:space="preserve">hoc eſt <lb/>c k. </s> <s xml:id="echoid-s1198" xml:space="preserve">eandem igitur proportionem habet h g ad n c, qnam g b ad <lb/>c k: </s> <s xml:id="echoid-s1199" xml:space="preserve">& </s> <s xml:id="echoid-s1200" xml:space="preserve">permutando n c ad c k eandem habet, quam b g ad g b.</s> <s xml:id="echoid-s1201" xml:space="preserve"/> </p> <div xml:id="echoid-div85" type="float" level="2" n="2"> <note position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">2. lem:</note> <note position="left" xlink:label="note-0052-02" xlink:href="note-0052-02a" xml:space="preserve">4. lem.</note> <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a"> <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0052-01"/> </figure> <note position="left" xlink:label="note-0052-03" xlink:href="note-0052-03a" xml:space="preserve">19. quinti</note> <note position="left" xlink:label="note-0052-04" xlink:href="note-0052-04a" xml:space="preserve">15. quin-<lb/>ti.</note> </div> <p style="it"> <s xml:id="echoid-s1202" xml:space="preserve">Sumatur deinde aliud quod uis punctum in ſectum in ſectione, <lb/>quod ſit s: </s> <s xml:id="echoid-s1203" xml:space="preserve">& </s> <s xml:id="echoid-s1204" xml:space="preserve">per s duæ lineæ ducantur: </s> <s xml:id="echoid-s1205" xml:space="preserve">st quidem <lb/>æquidistans ipſi db, diametrumque in puncto t ſecans; <lb/></s> <s xml:id="echoid-s1206" xml:space="preserve">s u uero æquidistans ac, & </s> <s xml:id="echoid-s1207" xml:space="preserve">ſecans c e in u. </s> <s xml:id="echoid-s1208" xml:space="preserve">Dico u c <lb/>ad ck maiorem proportionem habere, quamtg ad gb.</s> <s xml:id="echoid-s1209" xml:space="preserve"/> </p> <pb o="21" file="0053" n="53" rhead="DE IIS QVAE VEH. IN AQVA."/> <p style="it"> <s xml:id="echoid-s1210" xml:space="preserve">Producatur enim u s ad lineam qm in x: </s> <s xml:id="echoid-s1211" xml:space="preserve">& </s> <s xml:id="echoid-s1212" xml:space="preserve">à puncto x duca <lb/>tur ad diametrum x y ipſi bd æquidistans. </s> <s xml:id="echoid-s1213" xml:space="preserve">erit gt minor quàm <lb/>gy, quoniam u s minor eſt quàm ux: </s> <s xml:id="echoid-s1214" xml:space="preserve">& </s> <s xml:id="echoid-s1215" xml:space="preserve">ex primo lemmate yg <lb/>ad uc erit, ut h g ad n c; </s> <s xml:id="echoid-s1216" xml:space="preserve">uidelicet ut g b ad c k, quod proxime de <lb/>monstrauimus: </s> <s xml:id="echoid-s1217" xml:space="preserve">& </s> <s xml:id="echoid-s1218" xml:space="preserve">permutando yg ad gb, ut uc ad c k. </s> <s xml:id="echoid-s1219" xml:space="preserve">Sed t g <lb/>cum ſit ipſa y g minor, habet ad g b proportionem minorem, quàm <lb/>y g ad eandem. </s> <s xml:id="echoid-s1220" xml:space="preserve">ergo u c ad c K maiorem proportioné habet, quàm <lb/>t g ad g b. </s> <s xml:id="echoid-s1221" xml:space="preserve">quod demonstraſſe oportuit. </s> <s xml:id="echoid-s1222" xml:space="preserve">Itaque poſitione data g K <lb/>unum duntaxat erit in ſectione punctum, uidelicet m, à quo ductis <lb/>duabus lineis m e h, mno, habeat n c ad c K proportionem ean-<lb/>dem, quam h g ad g b. </s> <s xml:id="echoid-s1223" xml:space="preserve">nam ſi ab alijs omnibus ducantur, ſemper <lb/>ea, quæ inter a c, & </s> <s xml:id="echoid-s1224" xml:space="preserve">lineam ipſi æquidistantem interijcitur, ad c K <lb/>proportionem maiorem habebit, quàm quæ inter g K atque ei æqui <lb/>diſtantem, ad ipſam g b. </s> <s xml:id="echoid-s1225" xml:space="preserve">Conſtat igitur id, quod ab Archimede di-<lb/>ctum est; </s> <s xml:id="echoid-s1226" xml:space="preserve">nempe lineam pi ad p h uel eandem, quam n ω ad ω o, <lb/>uel maiorem habere proportionem.</s> <s xml:id="echoid-s1227" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1228" xml:space="preserve">Quare p h ipſius h i aut dupla eſt, aut minor quàm du <lb/> <anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/> pla.</s> <s xml:id="echoid-s1229" xml:space="preserve">] _Si quidé_ <lb/> <anchor type="figure" xlink:label="fig-0053-01a" xlink:href="fig-0053-01"/> _minor, quàm du-_ <lb/>_pla, ſit pt dupl. </s> <s xml:id="echoid-s1230" xml:space="preserve">2_ <lb/>_ti. </s> <s xml:id="echoid-s1231" xml:space="preserve">erit centrum_ <lb/>_grauitatis eius,_ <lb/>_quod in humido_ <lb/>_est, punctumt. </s> <s xml:id="echoid-s1232" xml:space="preserve">ſi_ <lb/>_uero p h ſit ip-_ <lb/>_ſius h i dupla,_ <lb/>_erit h grauitatis_ <lb/>_centrum: </s> <s xml:id="echoid-s1233" xml:space="preserve">ductâq;_ <lb/></s> <s xml:id="echoid-s1234" xml:space="preserve">_h f, & </s> <s xml:id="echoid-s1235" xml:space="preserve">producta_ <lb/>_ad centrum eius,_ <lb/>_quod est extra humidum, uidelicct ad g, alia ſimiliter demonstra-_ <lb/>_buntur. </s> <s xml:id="echoid-s1236" xml:space="preserve">atque idem intelligendum est in propoſitione, quæ ſe_-<lb/>_quitur._</s> <s xml:id="echoid-s1237" xml:space="preserve"/> </p> <div xml:id="echoid-div86" type="float" level="2" n="3"> <note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">D</note> <figure xlink:label="fig-0053-01" xlink:href="fig-0053-01a"> <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0053-01"/> </figure> </div> <p> <s xml:id="echoid-s1238" xml:space="preserve">Reuoluetur ergo ſolidum a p o l, & </s> <s xml:id="echoid-s1239" xml:space="preserve">baſis ipſius nullo <lb/> <anchor type="note" xlink:label="note-0053-02a" xlink:href="note-0053-02"/> <pb file="0054" n="54" rhead="ARCHIMEDIS"/> modo hum idi ſuperficiem continget.</s> <s xml:id="echoid-s1240" xml:space="preserve">] _In translatione le-_ <lb/>_gebatur ut baſis ipſius non tangat ſuperficiem humidi ſecundum_ <lb/>_unum ſignum. </s> <s xml:id="echoid-s1241" xml:space="preserve">nos autem ita uertere maluimus, & </s> <s xml:id="echoid-s1242" xml:space="preserve">hic & </s> <s xml:id="echoid-s1243" xml:space="preserve">in ijs,_ <lb/>_quæ ſequuntur, quoniam græci οὐδὲ \~εις, οὐδὲ ἓν, pro οὐ δεὶς, &_</s> <s xml:id="echoid-s1244" xml:space="preserve"> <lb/>_οὐδε ν frequenter utútur. </s> <s xml:id="echoid-s1245" xml:space="preserve">ut οὐδ ε ςιν οὐ δεὶς, nullus eſt: </s> <s xml:id="echoid-s1246" xml:space="preserve">οὐ \’δ ὑφ<unsure/> ἑ νὸς,_ <lb/>_ànullo & </s> <s xml:id="echoid-s1247" xml:space="preserve">alia eiuſmodi._</s> <s xml:id="echoid-s1248" xml:space="preserve"/> </p> <div xml:id="echoid-div87" type="float" level="2" n="4"> <note position="right" xlink:label="note-0053-02" xlink:href="note-0053-02a" xml:space="preserve">E</note> </div> </div> <div xml:id="echoid-div89" type="section" level="1" n="35"> <head xml:id="echoid-head40" xml:space="preserve">PROPOSITIO VII.</head> <p> <s xml:id="echoid-s1249" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem quidem <lb/>quàm ſeſquialtérum eius, quæ uſque ad axem; <lb/></s> <s xml:id="echoid-s1250" xml:space="preserve">minorem uero, quàm ut ad eam, quæ uſque ad <lb/>axem proportionem habeat, quam quindecim <lb/>ad quatuor: </s> <s xml:id="echoid-s1251" xml:space="preserve">in humidum demiſſa, adeo ut baſis <lb/>ipſius tota ſitin humido; </s> <s xml:id="echoid-s1252" xml:space="preserve">nunquam conſiſtet ita, <lb/>ut baſis contingat humidi ſuperficiem: </s> <s xml:id="echoid-s1253" xml:space="preserve">ſed ut to-<lb/>ta in humido ſit, & </s> <s xml:id="echoid-s1254" xml:space="preserve">nullo modo eius ſuperficiem <lb/>contingat.</s> <s xml:id="echoid-s1255" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1256" xml:space="preserve">SIT portio qualis dicta eſt: </s> <s xml:id="echoid-s1257" xml:space="preserve">& </s> <s xml:id="echoid-s1258" xml:space="preserve">demittatur in humidũ, <lb/>ut diximus, adeo ut baſis ipſius in uno puncto contingat <lb/>humidi ſuperficiem. </s> <s xml:id="echoid-s1259" xml:space="preserve">Demonſtrandum eſt non manere ip-<lb/>ſam: </s> <s xml:id="echoid-s1260" xml:space="preserve">ſed reuolui ita ut baſis fuperficiem humidi nullo mo-<lb/>do contingat. </s> <s xml:id="echoid-s1261" xml:space="preserve">Secta enim ipſa plano per axem, recto ad ſu <lb/>perficiem humidi, ſectio ſit a p o l rectanguli coni ſectio: <lb/></s> <s xml:id="echoid-s1262" xml:space="preserve">ſuperficiei humidi ſectio ſit s 1: </s> <s xml:id="echoid-s1263" xml:space="preserve">axis portionis, & </s> <s xml:id="echoid-s1264" xml:space="preserve">ſectio-<lb/>nis diameter p f: </s> <s xml:id="echoid-s1265" xml:space="preserve">ſeceturq; </s> <s xml:id="echoid-s1266" xml:space="preserve">p f in r quidem ita ut r p ſit <lb/>dupla ipſius r f; </s> <s xml:id="echoid-s1267" xml:space="preserve">in ω autem ut p f ad r ω proportionem <lb/>habeat, quam quindecim ad quatuor: </s> <s xml:id="echoid-s1268" xml:space="preserve">& </s> <s xml:id="echoid-s1269" xml:space="preserve">ω k ipſi p f ad re-<lb/>ctos angulos ducatur erit r ω minor, quàm quæ uſque ad <lb/>axem. </s> <s xml:id="echoid-s1270" xml:space="preserve">Itaque accipiatur ei, quæ uſque ad axem æqualis rh:</s> <s xml:id="echoid-s1271" xml:space="preserve"> <pb o="22" file="0055" n="55" rhead="DE IIS QVAE VEH. IN AQVA."/> & </s> <s xml:id="echoid-s1272" xml:space="preserve">c o quidẽ <lb/> <anchor type="figure" xlink:label="fig-0055-01a" xlink:href="fig-0055-01"/> ducatur con <lb/>tingẽs ſectio <lb/>nẽin o, quæ <lb/>ipſi s l æqui-<lb/>diſtet; </s> <s xml:id="echoid-s1273" xml:space="preserve">n o au <lb/>tem æquidi-<lb/> <anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/> ſtet p f: </s> <s xml:id="echoid-s1274" xml:space="preserve">& </s> <s xml:id="echoid-s1275" xml:space="preserve">pri <lb/>mum ipſam <lb/>k ω ſecet, at-<lb/>quein pũcto <lb/>i ſimiliter ut <lb/>in ſuperiori-<lb/>bus demonſtrabitur no, uel ſeſquialtera ipſius oi, uel <lb/>maior, quàm ſeſquialtera. </s> <s xml:id="echoid-s1276" xml:space="preserve">Sit autem o i minor, quam du-<lb/>pla ipſius in: </s> <s xml:id="echoid-s1277" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s1278" xml:space="preserve">o b dupla b n: </s> <s xml:id="echoid-s1279" xml:space="preserve">& </s> <s xml:id="echoid-s1280" xml:space="preserve">diſponantur eadem, <lb/>quæſupra. </s> <s xml:id="echoid-s1281" xml:space="preserve">Similiter demonſtrabimus, ſi ducatur linea r t, <lb/>facere eam angulos rectos cum linea c o, & </s> <s xml:id="echoid-s1282" xml:space="preserve">cum ſuperficie <lb/>humidi. </s> <s xml:id="echoid-s1283" xml:space="preserve">quare à punctis b g lineæ ductæipſi r t æquidiſtã <lb/>tes, etiã ad humidi ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0055-02a" xlink:href="fig-0055-02"/> ciẽ perpẽdiculares erunt. <lb/></s> <s xml:id="echoid-s1284" xml:space="preserve">portio igitur quæ eſt extra <lb/>humidũ deorſum feretur <lb/>ſecundum eam perpendi-<lb/>cularem, quæ per b tran-<lb/>ſit; </s> <s xml:id="echoid-s1285" xml:space="preserve">quæ uero intra humi-<lb/>dum ſecundum eam, quæ <lb/>per g ſurſum ſeretur. </s> <s xml:id="echoid-s1286" xml:space="preserve">ex <lb/>quibus conſtat reuolui ſo-<lb/>lidum, ita ut baſis ipſius <lb/>nullo modo humidi ſuper <lb/>ficiem contingat: </s> <s xml:id="echoid-s1287" xml:space="preserve">quo-<lb/>niam nuncin uno puncto <lb/>contingens deorſum fer- <pb file="0056" n="56" rhead="ARCHIMEDIS"/> tur ex parte 1. </s> <s xml:id="echoid-s1288" xml:space="preserve">Quod ſi n o non ſecuerit ipſam ω k, <lb/>eadem nihilominus demonſtrabuntur.</s> <s xml:id="echoid-s1289" xml:space="preserve"/> </p> <div xml:id="echoid-div89" type="float" level="2" n="1"> <figure xlink:label="fig-0055-01" xlink:href="fig-0055-01a"> <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0055-01"/> </figure> <note position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">10. quinti</note> <figure xlink:label="fig-0055-02" xlink:href="fig-0055-02a"> <image file="0055-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0055-02"/> </figure> </div> </div> <div xml:id="echoid-div91" type="section" level="1" n="36"> <head xml:id="echoid-head41" xml:space="preserve">PROPOSITIO VIII.</head> <p> <s xml:id="echoid-s1290" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>axem habuerit maiorem quidem, quàm ſeſqui-<lb/>alterum eius, quæ uſque ad axem; </s> <s xml:id="echoid-s1291" xml:space="preserve">minorem ue-<lb/>ro, quàm ut ad eam, quæ uſque ad axem propor-<lb/>tionem habeat, quam quindecim ad quatuor: </s> <s xml:id="echoid-s1292" xml:space="preserve">ſi <lb/>in grauitate ad humidum habeat proportionem <lb/>minorem ea, quam quadratum, quod fit ab exceſ <lb/>ſu, quo axis maior eſt, quàm ſeſquialter eius, quæ <lb/>uſque ad axem, habet ad quadratum, quod ab <lb/>axe: </s> <s xml:id="echoid-s1293" xml:space="preserve">demiſſa in humidum, ita ut baſis ipſius humi <lb/>dum non contingat; </s> <s xml:id="echoid-s1294" xml:space="preserve">neque in rectum reſtitue-<lb/>tur, neque manebit inclinata, niſi quando axis <lb/>cum ſuperficie humidi angulum fecerit æqualẽ <lb/>ei, de quo infra dicetur.</s> <s xml:id="echoid-s1295" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1296" xml:space="preserve">SIT portio qualis dicta eſt; </s> <s xml:id="echoid-s1297" xml:space="preserve">ſitque b d æqualis axi: </s> <s xml:id="echoid-s1298" xml:space="preserve">& </s> <s xml:id="echoid-s1299" xml:space="preserve"><lb/>b k quidem dupla ipſius _K_ d: </s> <s xml:id="echoid-s1300" xml:space="preserve">r _K_ uero æqualis ei, quæ uſ-<lb/>que ad axem: </s> <s xml:id="echoid-s1301" xml:space="preserve">& </s> <s xml:id="echoid-s1302" xml:space="preserve">ſit c b ſeſquialtera b r. </s> <s xml:id="echoid-s1303" xml:space="preserve">erit & </s> <s xml:id="echoid-s1304" xml:space="preserve">c d ipſius <lb/>_k_ r ſeſquialtera. </s> <s xml:id="echoid-s1305" xml:space="preserve">Quam uero portionem habet portio ad <lb/> <anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/> humidum in grauitate, habeat quadratum f q ad quadra-<lb/>tum d b: </s> <s xml:id="echoid-s1306" xml:space="preserve">& </s> <s xml:id="echoid-s1307" xml:space="preserve">ſit f dupla ipſius q. </s> <s xml:id="echoid-s1308" xml:space="preserve">perſpicuum igitur eſt f q <lb/>ad d b proportionem minorem habere ea, quam habet <lb/>c b ad b d. </s> <s xml:id="echoid-s1309" xml:space="preserve">eſt enim c b exceſſus, quo axis maior eſt, quàm <lb/>ſeſquialter eins, quæ uſque ad axem: </s> <s xml:id="echoid-s1310" xml:space="preserve">quare f q minor eſt <lb/> <anchor type="note" xlink:label="note-0056-02a" xlink:href="note-0056-02"/> <pb o="23" file="0057" n="57" rhead="DE IIS QVAE VEH. IN AQVA."/> ipſa b c: </s> <s xml:id="echoid-s1311" xml:space="preserve">& </s> <s xml:id="echoid-s1312" xml:space="preserve">idcirco f minor ipſa b r. </s> <s xml:id="echoid-s1313" xml:space="preserve">ſit ipſi f æqualis r ψ: <lb/></s> <s xml:id="echoid-s1314" xml:space="preserve"> <anchor type="note" xlink:label="note-0057-01a" xlink:href="note-0057-01"/> ducaturq; </s> <s xml:id="echoid-s1315" xml:space="preserve">ad b d perpendicularis ψ e, quæ posſit dimidiũ <lb/>eius, quod lineis _k_ r, ψ b continetur: </s> <s xml:id="echoid-s1316" xml:space="preserve">& </s> <s xml:id="echoid-s1317" xml:space="preserve">iungatur b c. </s> <s xml:id="echoid-s1318" xml:space="preserve">De-<lb/>monſtrandum eſt portionem in humidum demiſſam, ſicu-<lb/>ti dictum eſt, conſiſtere inclinatam ita, ut axis cum ſuperſi-<lb/>cie humidi angulum faciat angulo c b ψ æqualem. </s> <s xml:id="echoid-s1319" xml:space="preserve">demit-<lb/>tatur enim aliqua portio in humidum, ut baſis ipſius hu-<lb/>midi ſuperficiem non contingat: </s> <s xml:id="echoid-s1320" xml:space="preserve">& </s> <s xml:id="echoid-s1321" xml:space="preserve">ſi fieri poteſt, axis cum <lb/>ſuperficie humidi non faciat angulum æqualem angulo <lb/>e b ψ; </s> <s xml:id="echoid-s1322" xml:space="preserve">ſed primo maiorem. </s> <s xml:id="echoid-s1323" xml:space="preserve">ſecta autẽ portione plano per <lb/>axem, recto ad ſu-<lb/> <anchor type="figure" xlink:label="fig-0057-01a" xlink:href="fig-0057-01"/> perficiem humi-<lb/>di, ſit ſectio a p o l <lb/>rectanguli coni ſe <lb/>ctio: </s> <s xml:id="echoid-s1324" xml:space="preserve">ſuperficiei <lb/>humidi ſectio x s: <lb/></s> <s xml:id="echoid-s1325" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s1326" xml:space="preserve">axis portio-<lb/>nis, & </s> <s xml:id="echoid-s1327" xml:space="preserve">ſectiõis dia <lb/>meter n o: </s> <s xml:id="echoid-s1328" xml:space="preserve">& </s> <s xml:id="echoid-s1329" xml:space="preserve">du-<lb/>catur p y quidem <lb/>ipſi x s æquidi-<lb/>ſtans, quæ ſectio-<lb/>nem a p o l contin <lb/>gat in p: </s> <s xml:id="echoid-s1330" xml:space="preserve">p m ue-<lb/>ro æquidiſtans ip-<lb/>ſi n o: </s> <s xml:id="echoid-s1331" xml:space="preserve">& </s> <s xml:id="echoid-s1332" xml:space="preserve">p i ad <lb/>n o perpendicularis. </s> <s xml:id="echoid-s1333" xml:space="preserve">ſit præterea b r æqualis o ω. </s> <s xml:id="echoid-s1334" xml:space="preserve">itemq; </s> <s xml:id="echoid-s1335" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0057-02a" xlink:href="note-0057-02"/> r k ipſi t _a_ & </s> <s xml:id="echoid-s1336" xml:space="preserve">ω h perpendicularis ad axem. </s> <s xml:id="echoid-s1337" xml:space="preserve">Itaque quo-<lb/>niam ponitut axis portionis cum ſuperficie humidi facere <lb/>angulum maiorem angulo b: </s> <s xml:id="echoid-s1338" xml:space="preserve">erit angulus p y i angulo b <lb/> <anchor type="note" xlink:label="note-0057-03a" xlink:href="note-0057-03"/> maior. </s> <s xml:id="echoid-s1339" xml:space="preserve">maiorem ergo proportionem habet quadratum <lb/>p i ad quadratum y i, quam quadratum e ψ ad ψ b qua-<lb/> <anchor type="note" xlink:label="note-0057-04a" xlink:href="note-0057-04"/> dratum. </s> <s xml:id="echoid-s1340" xml:space="preserve">Sed quam proportionem habet quadratum p i <lb/>ad quadratum i y, eandem linea k r habet ad lineam i y:</s> <s xml:id="echoid-s1341" xml:space="preserve"> <pb file="0058" n="58" rhead="ARCHIMEDIS"/> & </s> <s xml:id="echoid-s1342" xml:space="preserve">quam proportionem habet quadratum e ψ ad quadra-<lb/> <anchor type="note" xlink:label="note-0058-01a" xlink:href="note-0058-01"/> tum ψ b, eandem habet dimidium lineæ _k_ r ad lineã ψ b. <lb/></s> <s xml:id="echoid-s1343" xml:space="preserve">quare maiorem babet proportionem _k_ r ad i y, quàm di-<lb/> <anchor type="note" xlink:label="note-0058-02a" xlink:href="note-0058-02"/> midium k r ad ψ b: </s> <s xml:id="echoid-s1344" xml:space="preserve">& </s> <s xml:id="echoid-s1345" xml:space="preserve">idcirco i y minor eſt, quàm dupla <lb/> <anchor type="note" xlink:label="note-0058-03a" xlink:href="note-0058-03"/> ψ b. </s> <s xml:id="echoid-s1346" xml:space="preserve">eſt autem ipſius o i dupla. </s> <s xml:id="echoid-s1347" xml:space="preserve">ergo o i minor eſt, quàm <lb/>ψ b: </s> <s xml:id="echoid-s1348" xml:space="preserve">& </s> <s xml:id="echoid-s1349" xml:space="preserve">i ω maior, quàm ψ r. </s> <s xml:id="echoid-s1350" xml:space="preserve">ſed ψ r eſt æqualis ipſi f. </s> <s xml:id="echoid-s1351" xml:space="preserve">maior <lb/> <anchor type="note" xlink:label="note-0058-04a" xlink:href="note-0058-04"/> igitur eſt i ω, quàm f. </s> <s xml:id="echoid-s1352" xml:space="preserve">& </s> <s xml:id="echoid-s1353" xml:space="preserve">quoniam portio ad humidum in <lb/>grauitate eam ponitur habere proportionem, quam qua-<lb/>dratum f q ad quadratum b d: </s> <s xml:id="echoid-s1354" xml:space="preserve">quam uero proportionem <lb/>habet portio ad humidum in grauitate, eam habet pars ip <lb/>ſius demerſa ad totam portionem: </s> <s xml:id="echoid-s1355" xml:space="preserve">& </s> <s xml:id="echoid-s1356" xml:space="preserve">quam pars ipſius de-<lb/>merſa habet ad totam, eandem habet quadratum p m ad <lb/>quadratnm o n: </s> <s xml:id="echoid-s1357" xml:space="preserve">ſequitur quadratum p m ad quadratum <lb/>o n eam proportionem habere, quam quadratum f q ad <lb/>b d quadratum. <lb/></s> <s xml:id="echoid-s1358" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> atque ideo ſ q æ-<lb/> <anchor type="note" xlink:label="note-0058-05a" xlink:href="note-0058-05"/> qualis eſt ipſi p m. <lb/></s> <s xml:id="echoid-s1359" xml:space="preserve">demõſtrata eſt au <lb/> <anchor type="note" xlink:label="note-0058-06a" xlink:href="note-0058-06"/> tem p h maior, <lb/>quàm f. </s> <s xml:id="echoid-s1360" xml:space="preserve">cõſtat igi <lb/>tur p m minorem <lb/>eſſe, quàm ſeſqui-<lb/>alterã ipſius p h: <lb/></s> <s xml:id="echoid-s1361" xml:space="preserve">& </s> <s xml:id="echoid-s1362" xml:space="preserve">idcirco p h ma <lb/>iorem, quàm du-<lb/>plam h m. </s> <s xml:id="echoid-s1363" xml:space="preserve">Sit p z <lb/>ipſius z m dupla. </s> <s xml:id="echoid-s1364" xml:space="preserve"><lb/>erit t quidem cẽ-<lb/>trũ grauitatis to-<lb/>tius ſolidi: </s> <s xml:id="echoid-s1365" xml:space="preserve">centrũ <lb/>eius partis, quæ intra humidum, punctumz: </s> <s xml:id="echoid-s1366" xml:space="preserve">reliquæ uero <lb/>partis centrum erit in linea z t producta uſque ad g. </s> <s xml:id="echoid-s1367" xml:space="preserve">Eodẽ <lb/> <anchor type="note" xlink:label="note-0058-07a" xlink:href="note-0058-07"/> modo demonſtrabitur linea th perpendicularis ad ſuper-<lb/>ficiem humidi. </s> <s xml:id="echoid-s1368" xml:space="preserve">& </s> <s xml:id="echoid-s1369" xml:space="preserve">portio demerſa in humido ſeretur extra <pb o="24" file="0059" n="59" rhead="DE IIS QVAE VEH. IN AQVA."/> humidum ſecundum perpendicularem, quæ per z ad hu-<lb/>midi ſuperficiem ducta fuerit: </s> <s xml:id="echoid-s1370" xml:space="preserve">quæ autem eſt extra humi-<lb/>dum ſecundum eam, quæ per gintra humidum feretur. </s> <s xml:id="echoid-s1371" xml:space="preserve">nõ <lb/>ergo manebit portio ſic inclinata, ut ponitur: </s> <s xml:id="echoid-s1372" xml:space="preserve">ſed neque re <lb/>ſtituecur recta: </s> <s xml:id="echoid-s1373" xml:space="preserve">quoniam perpendicularium per z g ducta <lb/>rum, quæ quidem per z ducitur ad eas partes cadit, in qui <lb/>bus eſt l; </s> <s xml:id="echoid-s1374" xml:space="preserve">& </s> <s xml:id="echoid-s1375" xml:space="preserve">quæ per g ad eas, in quibus eſt a. </s> <s xml:id="echoid-s1376" xml:space="preserve">quare ſequi-<lb/>tur centrum z ſurſum ferri: </s> <s xml:id="echoid-s1377" xml:space="preserve">& </s> <s xml:id="echoid-s1378" xml:space="preserve">g deorſum. </s> <s xml:id="echoid-s1379" xml:space="preserve">ergo partes to <lb/>tius ſolidi, quæ ſunt ad a deorſum, quæ uero ad l ſurſum <lb/>ferentur. </s> <s xml:id="echoid-s1380" xml:space="preserve">Rurſus alia eadem ponantur: </s> <s xml:id="echoid-s1381" xml:space="preserve">axis autem <lb/>portionis cum ſuperficie humidi angulum faciat minorẽ <lb/>eo, qui eſt ad b. </s> <s xml:id="echoid-s1382" xml:space="preserve">minorem igitur proportionem habet qua <lb/> <anchor type="note" xlink:label="note-0059-01a" xlink:href="note-0059-01"/> dratum p i ad quadratum i y, quàm quadratum e ψ ad <lb/>ψ b quadratum: </s> <s xml:id="echoid-s1383" xml:space="preserve">quare k r ad i y minorem proportionẽ <lb/>habet, quàm dimidium k r ad ψ b: </s> <s xml:id="echoid-s1384" xml:space="preserve">& </s> <s xml:id="echoid-s1385" xml:space="preserve">propterea i y maior <lb/>eſt, quam dupla ψ b. </s> <s xml:id="echoid-s1386" xml:space="preserve">eſt autem ipſius o i dupla. </s> <s xml:id="echoid-s1387" xml:space="preserve">ergo o i <lb/>ipſa ψ b maior e-<lb/> <anchor type="figure" xlink:label="fig-0059-01a" xlink:href="fig-0059-01"/> rit. </s> <s xml:id="echoid-s1388" xml:space="preserve">ſed tota o ω eſt <lb/>æqualis ipſi r b: <lb/></s> <s xml:id="echoid-s1389" xml:space="preserve">& </s> <s xml:id="echoid-s1390" xml:space="preserve">reliqua ω i mi-<lb/>nor quàm ψ r. </s> <s xml:id="echoid-s1391" xml:space="preserve">qua <lb/>re & </s> <s xml:id="echoid-s1392" xml:space="preserve">p h minor e-<lb/>rit, quàm f. </s> <s xml:id="echoid-s1393" xml:space="preserve">Quòd <lb/>cum m p ipſi f q <lb/>ſit æqualis, cõſtat <lb/>p m maiorẽ eſſe, <lb/>quàm ſeſquialterã <lb/>ipſius p h: </s> <s xml:id="echoid-s1394" xml:space="preserve">& </s> <s xml:id="echoid-s1395" xml:space="preserve">p h <lb/>minorem, quam <lb/>duplam h m. </s> <s xml:id="echoid-s1396" xml:space="preserve">Sit <lb/>p z ipſius z m du <lb/>pla. </s> <s xml:id="echoid-s1397" xml:space="preserve">Rurſus to-<lb/>tius quidem ſolidi centrum grauitatis erit pũctum t; </s> <s xml:id="echoid-s1398" xml:space="preserve">eius <lb/>uero partis, quæ intra humidum z: </s> <s xml:id="echoid-s1399" xml:space="preserve">& </s> <s xml:id="echoid-s1400" xml:space="preserve">iuncta z t inuenia- <pb file="0060" n="60" rhead="ARCHIMEDIS"/> tur centrum grauitatis eius, quæ extra humidum in pro-<lb/>tracta, quod ſit g. </s> <s xml:id="echoid-s1401" xml:space="preserve">Itaque per z g ductis perpendiculari-<lb/> <anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/> bus ad humidi ſuperficiem, quæ ipſi t h æquidiſtent; </s> <s xml:id="echoid-s1402" xml:space="preserve">ſequi-<lb/>tur portionem ipſam non manere, fed reuolui adeo, ut a-<lb/>xis cum ſuperficie <lb/> <anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/> humidi angulum <lb/>faciat maiorẽ eo, <lb/>quem nunc facit.</s> <s xml:id="echoid-s1403" xml:space="preserve"/> </p> <div xml:id="echoid-div91" type="float" level="2" n="1"> <note position="left" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">A</note> <note position="left" xlink:label="note-0056-02" xlink:href="note-0056-02a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0057-01" xlink:href="note-0057-01a" xml:space="preserve">C</note> <figure xlink:label="fig-0057-01" xlink:href="fig-0057-01a"> <image file="0057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0057-01"/> </figure> <note position="right" xlink:label="note-0057-02" xlink:href="note-0057-02a" xml:space="preserve">D</note> <note position="right" xlink:label="note-0057-03" xlink:href="note-0057-03a" xml:space="preserve">E</note> <note position="right" xlink:label="note-0057-04" xlink:href="note-0057-04a" xml:space="preserve">F</note> <note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">G</note> <note position="left" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">13. quin-<lb/>ti.</note> <note position="left" xlink:label="note-0058-03" xlink:href="note-0058-03a" xml:space="preserve">H</note> <note position="left" xlink:label="note-0058-04" xlink:href="note-0058-04a" xml:space="preserve">K</note> <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a"> <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0058-01"/> </figure> <note position="left" xlink:label="note-0058-05" xlink:href="note-0058-05a" xml:space="preserve">L</note> <note position="left" xlink:label="note-0058-06" xlink:href="note-0058-06a" xml:space="preserve">M</note> <note position="left" xlink:label="note-0058-07" xlink:href="note-0058-07a" xml:space="preserve">N</note> <note position="right" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">O</note> <figure xlink:label="fig-0059-01" xlink:href="fig-0059-01a"> <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0059-01"/> </figure> <note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">P</note> <figure xlink:label="fig-0060-01" xlink:href="fig-0060-01a"> <image file="0060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0060-01"/> </figure> </div> <p> <s xml:id="echoid-s1404" xml:space="preserve">Et quoniam cũ <lb/>antea poſuiſſem´<unsure/> <lb/>facere angulũ ma <lb/>iorem angulo b, <lb/>portio neque tũc <lb/>cõſiſtebat; </s> <s xml:id="echoid-s1405" xml:space="preserve">perſpi <lb/>cuũ eſt ipſam con <lb/>ſiſtere, ſi angulum <lb/>fecerit angulo b <lb/>æqualem. </s> <s xml:id="echoid-s1406" xml:space="preserve">Sic e-<lb/> <anchor type="note" xlink:label="note-0060-02a" xlink:href="note-0060-02"/> nim eriti o æqua-<lb/>lis ψ b: </s> <s xml:id="echoid-s1407" xml:space="preserve">itemq; </s> <s xml:id="echoid-s1408" xml:space="preserve">ω i <lb/>æqualis ψ r: </s> <s xml:id="echoid-s1409" xml:space="preserve">& </s> <s xml:id="echoid-s1410" xml:space="preserve">p h ipſi f. </s> <s xml:id="echoid-s1411" xml:space="preserve">erit igitur m p ſeſquialtera p h; <lb/></s> <s xml:id="echoid-s1412" xml:space="preserve">& </s> <s xml:id="echoid-s1413" xml:space="preserve">p h dupla h m. </s> <s xml:id="echoid-s1414" xml:space="preserve">quare cum h ſit centrum grauitatis eius <lb/>partis, quæ eſt in humido, per eandem perpendicularem, <lb/>& </s> <s xml:id="echoid-s1415" xml:space="preserve">ipſa ſurſum, & </s> <s xml:id="echoid-s1416" xml:space="preserve">quæ extra eſt feretur deorſum. </s> <s xml:id="echoid-s1417" xml:space="preserve">mane-<lb/>bitigitur portio; </s> <s xml:id="echoid-s1418" xml:space="preserve">quoniam altera pars ab altera non re-<lb/>pelletur.</s> <s xml:id="echoid-s1419" xml:space="preserve"/> </p> <div xml:id="echoid-div92" type="float" level="2" n="2"> <note position="left" xlink:label="note-0060-02" xlink:href="note-0060-02a" xml:space="preserve">Q</note> </div> </div> <div xml:id="echoid-div94" type="section" level="1" n="37"> <head xml:id="echoid-head42" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s1420" xml:space="preserve">_E T ſit c b ſeſquialtera b r. </s> <s xml:id="echoid-s1421" xml:space="preserve">erit & </s> <s xml:id="echoid-s1422" xml:space="preserve">c d ipſius k r ſeſqui-_ <lb/> <anchor type="note" xlink:label="note-0060-03a" xlink:href="note-0060-03"/> _altera_.</s> <s xml:id="echoid-s1423" xml:space="preserve">] In translatione ita legebatur. </s> <s xml:id="echoid-s1424" xml:space="preserve">ſit autem & </s> <s xml:id="echoid-s1425" xml:space="preserve">c b quidem <lb/>hemiolia ipſius b r: </s> <s xml:id="echoid-s1426" xml:space="preserve">c d autem ipſius K r. </s> <s xml:id="echoid-s1427" xml:space="preserve">Sed nos quod postremo <lb/>loco legitur, idcirco corrigendum duximus, quoniam illud non po-<lb/>nitur ita eſſe, ſed ex ijs, quæ poſita ſunt, neceſſario colligitur. </s> <s xml:id="echoid-s1428" xml:space="preserve">ſi enim <pb o="25" file="0061" n="61" rhead="DE IIS QVAE VEH. IN AQVA."/> b ψ dupla ſit ψ d, erit d b ipſius b ψ ſeſquialtera. </s> <s xml:id="echoid-s1429" xml:space="preserve">& </s> <s xml:id="echoid-s1430" xml:space="preserve">quoniam e b ſeſ <lb/>quialtera est b r, ſequitur reliquam c d ipſius ψ r, boc est eius, quæ <lb/> <anchor type="note" xlink:label="note-0061-01a" xlink:href="note-0061-01"/> uſque ad axem ſeſquialteram eſſe. </s> <s xml:id="echoid-s1431" xml:space="preserve">quare b c erit exceſſus, quo axis <lb/>maior est, quàm ſeſquialter eius, quæ uſque ad axem.</s> <s xml:id="echoid-s1432" xml:space="preserve"/> </p> <div xml:id="echoid-div94" type="float" level="2" n="1"> <note position="left" xlink:label="note-0060-03" xlink:href="note-0060-03a" xml:space="preserve">A</note> <note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">12. quinti</note> </div> <p style="it"> <s xml:id="echoid-s1433" xml:space="preserve">_Quare f q minor éſtipſa b c.</s> <s xml:id="echoid-s1434" xml:space="preserve">]_ Nam cum portio ad bumi-<lb/> <anchor type="note" xlink:label="note-0061-02a" xlink:href="note-0061-02"/> dum in grauitate proportionem habeat eandem, quàm quadratum <lb/>f q ad quadratum d b: </s> <s xml:id="echoid-s1435" xml:space="preserve">habeatq, minorem proportionem, quàm qua <lb/>dratum factum ab exceſſu, quo axis maior eſt, quàm ſeſquialter eius, <lb/>quæ uſque ad axem, ad quadratum ab axe; </s> <s xml:id="echoid-s1436" xml:space="preserve">boc eſt minorem, quàm <lb/>quadratum c b ad quadratum b d: </s> <s xml:id="echoid-s1437" xml:space="preserve">ponitur enim linea b d æqualis <lb/>axi: </s> <s xml:id="echoid-s1438" xml:space="preserve">quadratum f q ad quadratum d b proportionem minorem ha-<lb/>bebit, quàm quadratum c b ad idem b d quadratum. </s> <s xml:id="echoid-s1439" xml:space="preserve">ergo quadra-<lb/> <anchor type="note" xlink:label="note-0061-03a" xlink:href="note-0061-03"/> tum f q minus erit quadrato c b: </s> <s xml:id="echoid-s1440" xml:space="preserve">& </s> <s xml:id="echoid-s1441" xml:space="preserve">propterea linea f q ipſa b c <lb/>minor.</s> <s xml:id="echoid-s1442" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="2"> <note position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0061-03" xlink:href="note-0061-03a" xml:space="preserve">8. quinti.</note> </div> <p style="it"> <s xml:id="echoid-s1443" xml:space="preserve">_Etidcirco f minor ipſa b r.</s> <s xml:id="echoid-s1444" xml:space="preserve">]_ Quoniam enim c b ſeſquial-<lb/> <anchor type="note" xlink:label="note-0061-04a" xlink:href="note-0061-04"/> tera eſt b r, & </s> <s xml:id="echoid-s1445" xml:space="preserve">f q ipſius f ſeſquialtera: </s> <s xml:id="echoid-s1446" xml:space="preserve">estq; </s> <s xml:id="echoid-s1447" xml:space="preserve">f q minor b c; </s> <s xml:id="echoid-s1448" xml:space="preserve">& </s> <s xml:id="echoid-s1449" xml:space="preserve">f <lb/> <anchor type="note" xlink:label="note-0061-05a" xlink:href="note-0061-05"/> ipſa b r minor erit.</s> <s xml:id="echoid-s1450" xml:space="preserve"/> </p> <div xml:id="echoid-div96" type="float" level="2" n="3"> <note position="right" xlink:label="note-0061-04" xlink:href="note-0061-04a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0061-05" xlink:href="note-0061-05a" xml:space="preserve">14. quin-<lb/>ti.</note> </div> <p style="it"> <s xml:id="echoid-s1451" xml:space="preserve">_Itaque quoniam ponitur axis portionis cum ſuperficie_ <lb/> <anchor type="note" xlink:label="note-0061-06a" xlink:href="note-0061-06"/> _humidi facere angulum maiorem angulo b: </s> <s xml:id="echoid-s1452" xml:space="preserve">erit angulus_ <lb/>_p y i angulo b maior.</s> <s xml:id="echoid-s1453" xml:space="preserve">]_ Nam cum linea p y ſuperficiei bumidi <lb/>æ quidistet; </s> <s xml:id="echoid-s1454" xml:space="preserve">uidelicet ipſi x s: </s> <s xml:id="echoid-s1455" xml:space="preserve">angulus p y i æqualis erit angulo, qui <lb/> <anchor type="note" xlink:label="note-0061-07a" xlink:href="note-0061-07"/> diametro portionis n o, & </s> <s xml:id="echoid-s1456" xml:space="preserve">linea x s continetur. </s> <s xml:id="echoid-s1457" xml:space="preserve">quare & </s> <s xml:id="echoid-s1458" xml:space="preserve">angulo <lb/>b maior erit.</s> <s xml:id="echoid-s1459" xml:space="preserve"/> </p> <div xml:id="echoid-div97" type="float" level="2" n="4"> <note position="right" xlink:label="note-0061-06" xlink:href="note-0061-06a" xml:space="preserve">D</note> <note position="right" xlink:label="note-0061-07" xlink:href="note-0061-07a" xml:space="preserve">29. primi</note> </div> <p style="it"> <s xml:id="echoid-s1460" xml:space="preserve">_Maiorem igitur proportionem habet quadratum p i ad_ <lb/> <anchor type="note" xlink:label="note-0061-08a" xlink:href="note-0061-08"/> _quadratum i y, quàm quadratum e ψ ad ψ b quadratu.</s> <s xml:id="echoid-s1461" xml:space="preserve">]_ <lb/>Deſcribantur ſeorſum triangula p i y, e ψ b. </s> <s xml:id="echoid-s1462" xml:space="preserve">& </s> <s xml:id="echoid-s1463" xml:space="preserve">cum angulus p y i <lb/>maior ſit angulo e b ψ, ad lineam i y, atque ad punctum y in ea da-<lb/>tum fiat angulus u y i æqualis angulo e b ψ. </s> <s xml:id="echoid-s1464" xml:space="preserve">est autem angulus ad <lb/>i rectus æqualis recto ad ψ. </s> <s xml:id="echoid-s1465" xml:space="preserve">reliquus igitur y u i reliquo b c ψ est <lb/>æqualis. </s> <s xml:id="echoid-s1466" xml:space="preserve">quare linea u i ad lineam i y eandem proportionem ha-<lb/> <anchor type="note" xlink:label="note-0061-09a" xlink:href="note-0061-09"/> bet, quam linea e ψ ad ψ b. </s> <s xml:id="echoid-s1467" xml:space="preserve">Sed linea p i, quæ maior est ipſa u i ad <lb/> <anchor type="note" xlink:label="note-0061-10a" xlink:href="note-0061-10"/> lineam in maiorem habet proportionem quam u i ad eandem. </s> <s xml:id="echoid-s1468" xml:space="preserve">ergo <lb/> <anchor type="note" xlink:label="note-0061-11a" xlink:href="note-0061-11"/> p i ad i y maiorem proportionem habebit, quàm e ψ ad ψ b: </s> <s xml:id="echoid-s1469" xml:space="preserve">& </s> <s xml:id="echoid-s1470" xml:space="preserve"><lb/>propterea quadratum p i ad quadratum i y maiorem habebit, quàm <pb file="0062" n="62" rhead="ARCHIMEDIS"/> quadratum e ψ ad quadr. </s> <s xml:id="echoid-s1471" xml:space="preserve">itum ψ b.</s> <s xml:id="echoid-s1472" xml:space="preserve"/> </p> <div xml:id="echoid-div98" type="float" level="2" n="5"> <note position="right" xlink:label="note-0061-08" xlink:href="note-0061-08a" xml:space="preserve">E</note> <note position="right" xlink:label="note-0061-09" xlink:href="note-0061-09a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-0061-10" xlink:href="note-0061-10a" xml:space="preserve">8. quinti.</note> <note position="right" xlink:label="note-0061-11" xlink:href="note-0061-11a" xml:space="preserve">13. quin-<lb/>ti.</note> </div> <p style="it"> <s xml:id="echoid-s1473" xml:space="preserve">_Sed quam proportionem habet qua-_ <lb/> <anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/> <anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/> _dratum p i ad quadratum i y, eandem li_ <lb/>_nea k r habet ad lineam i y.</s> <s xml:id="echoid-s1474" xml:space="preserve">]_ Est enim ex <lb/>undecima primi conicorum quadratum p i æqua <lb/>le rectangulo contento linea i o, & </s> <s xml:id="echoid-s1475" xml:space="preserve">ea, iuxta quam poſſunt quæ à <lb/>ſectione ad diametrum ducuntur, uidelicet duplaipſius k r. </s> <s xml:id="echoid-s1476" xml:space="preserve">atque <lb/>est i y dupla i o, extrigeſimatertia eiuſdem: </s> <s xml:id="echoid-s1477" xml:space="preserve">quare ex decimaſext a <lb/>ſexti elementorum, rectangulum, quod fit ex k r, & </s> <s xml:id="echoid-s1478" xml:space="preserve">i y æ quale eſt <lb/>rectangulo contento linea i o & </s> <s xml:id="echoid-s1479" xml:space="preserve">ea, iuxta quam poſſunt: </s> <s xml:id="echoid-s1480" xml:space="preserve">hoc eſt qua <lb/>drato p i. </s> <s xml:id="echoid-s1481" xml:space="preserve">Sed ut rectangulnm ex k r, & </s> <s xml:id="echoid-s1482" xml:space="preserve">i y ad quadratum i y, ita <lb/> <anchor type="note" xlink:label="note-0062-02a" xlink:href="note-0062-02"/> linea κ r ad ipſam i y. </s> <s xml:id="echoid-s1483" xml:space="preserve">ergo linea κ r ad i y eandem proportionem <lb/>habebit, quam rectangulum ex κ r & </s> <s xml:id="echoid-s1484" xml:space="preserve">i y, hoc eſt quadratum p i ad <lb/>quadratum i y.</s> <s xml:id="echoid-s1485" xml:space="preserve"/> </p> <div xml:id="echoid-div99" type="float" level="2" n="6"> <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a"> <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0062-01"/> </figure> <note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">F</note> <note position="left" xlink:label="note-0062-02" xlink:href="note-0062-02a" xml:space="preserve">lem. 22. <lb/>decimi.</note> </div> <p> <s xml:id="echoid-s1486" xml:space="preserve">Et quam proportionem habet quadratũ e ψ ad quadra <lb/> <anchor type="note" xlink:label="note-0062-03a" xlink:href="note-0062-03"/> tum ψ b, eandem habet dimidium lineæ K r ad lineã ψ b.</s> <s xml:id="echoid-s1487" xml:space="preserve">]</s> </p> <div xml:id="echoid-div100" type="float" level="2" n="7"> <note position="left" xlink:label="note-0062-03" xlink:href="note-0062-03a" xml:space="preserve">G</note> </div> <p style="it"> <s xml:id="echoid-s1488" xml:space="preserve">Nam cum quadratum e ψ poſitum ſit æquale dimidio rectanguli <lb/>contenti linea κ r, & </s> <s xml:id="echoid-s1489" xml:space="preserve">ψ b; </s> <s xml:id="echoid-s1490" xml:space="preserve">hoc est ei, quod dimidia ipſius κ r <lb/>& </s> <s xml:id="echoid-s1491" xml:space="preserve">linea ψ b continetur: </s> <s xml:id="echoid-s1492" xml:space="preserve">& </s> <s xml:id="echoid-s1493" xml:space="preserve">ut rectangulum ex dimidia κ r, & </s> <s xml:id="echoid-s1494" xml:space="preserve">ψ b <lb/> <anchor type="note" xlink:label="note-0062-04a" xlink:href="note-0062-04"/> ad quadratum ψ b, ita ſit dimidia κ r ad line am ψ b: </s> <s xml:id="echoid-s1495" xml:space="preserve">habebit dimi-<lb/>dia κ r ad ψ b proportionem eandem, quam quadratum e ψ ad qua-<lb/>dratum ψ b.</s> <s xml:id="echoid-s1496" xml:space="preserve"/> </p> <div xml:id="echoid-div101" type="float" level="2" n="8"> <note position="left" xlink:label="note-0062-04" xlink:href="note-0062-04a" xml:space="preserve">lem. 22. <lb/>decimi</note> </div> <p style="it"> <s xml:id="echoid-s1497" xml:space="preserve">_Etidcirco i y minor eſt, quàm dupla ψ b.</s> <s xml:id="echoid-s1498" xml:space="preserve">]_ Quam enim pro <lb/> <anchor type="note" xlink:label="note-0062-05a" xlink:href="note-0062-05"/> portionem habet dimidium κ r ad ψ b, habeat κ r ad aliam lineam. <lb/></s> <s xml:id="echoid-s1499" xml:space="preserve">erit ea maior, quàm i y; </s> <s xml:id="echoid-s1500" xml:space="preserve">nempe ad quam κ r minorem proportioné <lb/> <anchor type="note" xlink:label="note-0062-06a" xlink:href="note-0062-06"/> habet: </s> <s xml:id="echoid-s1501" xml:space="preserve">at que erit dupla ψ b. </s> <s xml:id="echoid-s1502" xml:space="preserve">ergo i y minor eſt, quam dupla ψ b.</s> <s xml:id="echoid-s1503" xml:space="preserve"/> </p> <div xml:id="echoid-div102" type="float" level="2" n="9"> <note position="left" xlink:label="note-0062-05" xlink:href="note-0062-05a" xml:space="preserve">H</note> <note position="left" xlink:label="note-0062-06" xlink:href="note-0062-06a" xml:space="preserve">10. quinti.</note> </div> <p style="it"> <s xml:id="echoid-s1504" xml:space="preserve">_Et i ω maior, quam ψ r.</s> <s xml:id="echoid-s1505" xml:space="preserve">]_ Cum enim o ω poſita ſit æ qualis b r <lb/> <anchor type="note" xlink:label="note-0062-07a" xlink:href="note-0062-07"/> ſi ex b r dematur ψ b, & </s> <s xml:id="echoid-s1506" xml:space="preserve">ex o ω dematur o i, quæ minor eſt ψ b: </s> <s xml:id="echoid-s1507" xml:space="preserve">erit <lb/>reliqua i ω maior reliqua ψ r.</s> <s xml:id="echoid-s1508" xml:space="preserve"/> </p> <div xml:id="echoid-div103" type="float" level="2" n="10"> <note position="left" xlink:label="note-0062-07" xlink:href="note-0062-07a" xml:space="preserve">K</note> </div> <p style="it"> <s xml:id="echoid-s1509" xml:space="preserve">_Atqueideo f q æqualis eſt ipſi p m.</s> <s xml:id="echoid-s1510" xml:space="preserve">]_ Ex decimaquarta <lb/> <anchor type="note" xlink:label="note-0062-08a" xlink:href="note-0062-08"/> quinti elementorum, nam linea o n ipſi b d eſt æ qualis.</s> <s xml:id="echoid-s1511" xml:space="preserve"/> </p> <div xml:id="echoid-div104" type="float" level="2" n="11"> <note position="left" xlink:label="note-0062-08" xlink:href="note-0062-08a" xml:space="preserve">L</note> </div> <p style="it"> <s xml:id="echoid-s1512" xml:space="preserve">_Demonſtrata eſt autem p h maior, quàm f.</s> <s xml:id="echoid-s1513" xml:space="preserve">]_ Etenim de-<lb/> <anchor type="note" xlink:label="note-0062-09a" xlink:href="note-0062-09"/> monstrata est i ω maior, quàm f; </s> <s xml:id="echoid-s1514" xml:space="preserve">atque est p h æqualis ipſi i ω.</s> <s xml:id="echoid-s1515" xml:space="preserve"/> </p> <div xml:id="echoid-div105" type="float" level="2" n="12"> <note position="left" xlink:label="note-0062-09" xlink:href="note-0062-09a" xml:space="preserve">M</note> </div> <p style="it"> <s xml:id="echoid-s1516" xml:space="preserve">_Eodem modo demonſtrabitur t h perpendicularis ad_ <lb/> <anchor type="note" xlink:label="note-0062-10a" xlink:href="note-0062-10"/> <pb o="26" file="0063" n="63" rhead="DE IIS QVAE VEH. IN AQVA."/> _humidi ſuperficiem.</s> <s xml:id="echoid-s1517" xml:space="preserve">]_ Est enim t ω æqualis κ r, hoc eſt ei, quæ <lb/>uſque ad axem. </s> <s xml:id="echoid-s1518" xml:space="preserve">quare ex ijs, quæ ſuperius demonſtrata ſunt, linea <lb/>t h ducta erit ad humidi ſuperficiem perpendicularis.</s> <s xml:id="echoid-s1519" xml:space="preserve"/> </p> <div xml:id="echoid-div106" type="float" level="2" n="13"> <note position="left" xlink:label="note-0062-10" xlink:href="note-0062-10a" xml:space="preserve">N</note> </div> <p style="it"> <s xml:id="echoid-s1520" xml:space="preserve">_Minorem igitur proportionem habet quadratum p i_ <lb/> <anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/> _ad quadratum i y, quàm quadratum e ψ ad ψ b quadratũ]_ <lb/>Hæc & </s> <s xml:id="echoid-s1521" xml:space="preserve">alia, quæ ſequuntur, tum in hac, tum in ſequenti propoſitio-<lb/>ne non alio, quàm quo ſupra modo demonstrabimus.</s> <s xml:id="echoid-s1522" xml:space="preserve"/> </p> <div xml:id="echoid-div107" type="float" level="2" n="14"> <note position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">O</note> </div> <p style="it"> <s xml:id="echoid-s1523" xml:space="preserve">_Itaque per z g ductis perpendicularibus ad humidi ſu-_ <lb/> <anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/> _perficiem, quæ i pſi t h æ quidiftent; </s> <s xml:id="echoid-s1524" xml:space="preserve">ſequitur portionem ip_ <lb/>_ſam non manere, ſed reuolui adeo, ut axis cum ſuperſicie_ <lb/>_humidi angulum faciat maiorem eo, quem nunc facit.</s> <s xml:id="echoid-s1525" xml:space="preserve">]_ <lb/>Nam cum perpendicularis, quæ per g, ducitur ad eas partes cadat, <lb/>in quibus eſt l; </s> <s xml:id="echoid-s1526" xml:space="preserve">quæ autem per Z ad eis in quibus a: </s> <s xml:id="echoid-s1527" xml:space="preserve">neceſſarium eſt <lb/>centrum g deorſum ferri, & </s> <s xml:id="echoid-s1528" xml:space="preserve">Z ſurſum. </s> <s xml:id="echoid-s1529" xml:space="preserve">quare partes ſolidi, quæ <lb/>ſunt ad l deorſum; </s> <s xml:id="echoid-s1530" xml:space="preserve">quæ uero ad a ſurſum ferentur, ut axis cum ſu-<lb/>perficie humidi maiorem angulum contineat.</s> <s xml:id="echoid-s1531" xml:space="preserve"/> </p> <div xml:id="echoid-div108" type="float" level="2" n="15"> <note position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">P</note> </div> <p> <s xml:id="echoid-s1532" xml:space="preserve">Sic enim erit i o æ qualis ψ b, itẽq; </s> <s xml:id="echoid-s1533" xml:space="preserve">ω i æ qualis ψ r, & </s> <s xml:id="echoid-s1534" xml:space="preserve">p h <lb/> <anchor type="note" xlink:label="note-0063-03a" xlink:href="note-0063-03"/> ipſi f.</s> <s xml:id="echoid-s1535" xml:space="preserve">] _Hoc in tertia figura, quam nos addidimus, perſpicue apparet_.</s> <s xml:id="echoid-s1536" xml:space="preserve"/> </p> <div xml:id="echoid-div109" type="float" level="2" n="16"> <note position="right" xlink:label="note-0063-03" xlink:href="note-0063-03a" xml:space="preserve">Q</note> </div> </div> <div xml:id="echoid-div111" type="section" level="1" n="38"> <head xml:id="echoid-head43" xml:space="preserve">PROPOSITIO IX.</head> <p> <s xml:id="echoid-s1537" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>axem habuerit maiorem quidem, quàm ſeſquial-<lb/>terum eius, quæ uſque ad axem; </s> <s xml:id="echoid-s1538" xml:space="preserve">minorem uero, <lb/>quàm ut ad eam, quæ uſque ad axem proportio-<lb/>nem habeat, quam quindecim ad quatuor; </s> <s xml:id="echoid-s1539" xml:space="preserve">& </s> <s xml:id="echoid-s1540" xml:space="preserve">in <lb/>grauitate ad humidum proportionem habeat ma <lb/>iorem, quàm exceſſus, quo quadratum, quod fit <lb/>ab axe maius eſt quadrato, quod ab exceſſu, quo <lb/>axis eſt maior, quàm ſeſquialter eius, quæ uſq; </s> <s xml:id="echoid-s1541" xml:space="preserve">ad <lb/>axem, habet ad quadratum, quod ab axe: </s> <s xml:id="echoid-s1542" xml:space="preserve">in hu- <pb file="0064" n="64" rhead="ARCHIMEDIS"/> midum demiſſa adeo, ut baſis ipſius tota ſit in hu <lb/>mido, & </s> <s xml:id="echoid-s1543" xml:space="preserve">poſita inclinata, nec conuertetur ita, ut <lb/>axis ipſius ſecũdum perpendicularem ſit, nec ma <lb/>nebit inclinata, niſi quãdo axis cum ſuperficie hu <lb/>midi angulum fecerit æqualem angulo ſimiliter <lb/>ut prius, aſſumpto.</s> <s xml:id="echoid-s1544" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1545" xml:space="preserve">SIT portio, qualis dicta eſt: </s> <s xml:id="echoid-s1546" xml:space="preserve">ponaturq; </s> <s xml:id="echoid-s1547" xml:space="preserve">d b æ qualis axi <lb/>portionis: </s> <s xml:id="echoid-s1548" xml:space="preserve">& </s> <s xml:id="echoid-s1549" xml:space="preserve">b _k_ quidem ſit dupla ipſius _k_ d; </s> <s xml:id="echoid-s1550" xml:space="preserve">k r autem <lb/>æqualis ei, quæ uſque ad axem: </s> <s xml:id="echoid-s1551" xml:space="preserve">& </s> <s xml:id="echoid-s1552" xml:space="preserve">c b ſeſquialtera b r. <lb/></s> <s xml:id="echoid-s1553" xml:space="preserve">Quam uero proportionem habet portio ad humidum in <lb/>grauitate, eam habeat exceſſus, quo quadratum b d exce-<lb/>dit quadratum f q, ad ipſum b d quadratum: </s> <s xml:id="echoid-s1554" xml:space="preserve">& </s> <s xml:id="echoid-s1555" xml:space="preserve">ſit f ipſius <lb/>q dupla. </s> <s xml:id="echoid-s1556" xml:space="preserve">conſtat igitur exceſſum, quo quadratum b d ex-<lb/>cedit quadratum <lb/> <anchor type="figure" xlink:label="fig-0064-01a" xlink:href="fig-0064-01"/> b c ad quadratum <lb/>b d, minorem ha-<lb/>bere proportio-<lb/>nem, quàm exceſ-<lb/>ſus, quo quadratũ <lb/>b d excedit qua-<lb/>dratum f q ad b d <lb/>quadratum. </s> <s xml:id="echoid-s1557" xml:space="preserve">eſt e-<lb/>nim b c exceſſus <lb/>quo axis portiõis <lb/>maior eſt, quã ſeſ-<lb/>quialter eius, quæ <lb/>uſque ad axem. <lb/></s> <s xml:id="echoid-s1558" xml:space="preserve">quare quadr atum <lb/> <anchor type="note" xlink:label="note-0064-01a" xlink:href="note-0064-01"/> b d magis excedit <lb/>quadratum f q, quàm b c quadratum: </s> <s xml:id="echoid-s1559" xml:space="preserve">& </s> <s xml:id="echoid-s1560" xml:space="preserve">idcirco linea f q <lb/>minor eſt, quàm b c itemq; </s> <s xml:id="echoid-s1561" xml:space="preserve">f minor, quàm br. </s> <s xml:id="echoid-s1562" xml:space="preserve">Sit ipſi ſ <pb o="27" file="0065" n="65" rhead="DE IIS QVAE VEH. IN AQVA."/> æqualis r ψ: </s> <s xml:id="echoid-s1563" xml:space="preserve">& </s> <s xml:id="echoid-s1564" xml:space="preserve">ducatur ψ r perpendicularis ad b d, quæ <lb/>posſit dimidium eius, quod ipſis k r, ψ b, continetur. </s> <s xml:id="echoid-s1565" xml:space="preserve">Dico <lb/>portionem in humidum demiſſam adeo, ut baſis ipſius to-<lb/>ta ſit in humido, ita conſiſtere, ut axis cum ſuperficie humi <lb/>di faciat angulum angulo b æqualem. </s> <s xml:id="echoid-s1566" xml:space="preserve">Demittatur enim <lb/>portio in humidum, ſicuti dictum eſt; </s> <s xml:id="echoid-s1567" xml:space="preserve">& </s> <s xml:id="echoid-s1568" xml:space="preserve">axis cum humidi <lb/>ſuperficie non faciat angulum æqualẽ ipſi b, ſed primo ma <lb/>iorem: </s> <s xml:id="echoid-s1569" xml:space="preserve">ſecta autem ipſa plano per axem, recto ad ſuperfi-<lb/>ciem humidi, ſectio portionis ſit a p o l rectanguli coni ſe-<lb/>ctio; </s> <s xml:id="echoid-s1570" xml:space="preserve">ſuperficiei humidi ſectio c i; </s> <s xml:id="echoid-s1571" xml:space="preserve">ſitq, axis portionis, & </s> <s xml:id="echoid-s1572" xml:space="preserve">ſe <lb/>ctionis diameter n o, quæ fecetur in punctis ω t, ut prius. </s> <s xml:id="echoid-s1573" xml:space="preserve">& </s> <s xml:id="echoid-s1574" xml:space="preserve"><lb/>ducantur y p quidem ipſi ci æquidiſtans, contingensq; </s> <s xml:id="echoid-s1575" xml:space="preserve">ſe <lb/>ctionem in p; </s> <s xml:id="echoid-s1576" xml:space="preserve">m p uero æquidiſtans n o: </s> <s xml:id="echoid-s1577" xml:space="preserve">& </s> <s xml:id="echoid-s1578" xml:space="preserve">p s ad axem <lb/>perpendicularis. </s> <s xml:id="echoid-s1579" xml:space="preserve">Quoniam igitur axis portionis cum ſu-<lb/>perficie humidi facit angulum maiorem angulo b; </s> <s xml:id="echoid-s1580" xml:space="preserve">erit & </s> <s xml:id="echoid-s1581" xml:space="preserve"><lb/>angulus s y p angulo b maior. </s> <s xml:id="echoid-s1582" xml:space="preserve">quare quadratum p s ad <lb/>quadratum s y maiorem habet proportionem, quàm qua <lb/>dratum ψ e ad quadratum ψ b: </s> <s xml:id="echoid-s1583" xml:space="preserve">& </s> <s xml:id="echoid-s1584" xml:space="preserve">propterea _K_ r ad s y ma <lb/> <anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/> iorem habet, quàm dimidium ipſius κ r ad ψ b. </s> <s xml:id="echoid-s1585" xml:space="preserve">ergo s y <lb/>minor eſt, quam dupla ψ b; </s> <s xml:id="echoid-s1586" xml:space="preserve">& </s> <s xml:id="echoid-s1587" xml:space="preserve">s o minor, quam ψ b. </s> <s xml:id="echoid-s1588" xml:space="preserve">quare <lb/> <anchor type="note" xlink:label="note-0065-02a" xlink:href="note-0065-02"/> s ω maior, quàm r ψ; </s> <s xml:id="echoid-s1589" xml:space="preserve">& </s> <s xml:id="echoid-s1590" xml:space="preserve">p h maior, quàm f. </s> <s xml:id="echoid-s1591" xml:space="preserve">Itaque quoniã <lb/>portio ad humidum in grauitate eam habet proportionẽ, <lb/> <anchor type="note" xlink:label="note-0065-03a" xlink:href="note-0065-03"/> quam exceſſus, quo quadratum b d excedit quadratum f q <lb/>ad quadratum b d: </s> <s xml:id="echoid-s1592" xml:space="preserve">quam uero proportionem habet por-<lb/>tio ad humidum in grauitate, eandem pars ipſius demerſa <lb/>habet ad totam portionẽ: </s> <s xml:id="echoid-s1593" xml:space="preserve">ſequitur partẽ demerſam ad to <lb/>tam portionem, eam proportionem habere, quã exceſſus, <lb/>quo quadratum b d excedit quadratũ f q, ad quadratū b d. <lb/></s> <s xml:id="echoid-s1594" xml:space="preserve">habebit ergo tota portio ad eam, quæ eſt extra humidum <lb/> <anchor type="note" xlink:label="note-0065-04a" xlink:href="note-0065-04"/> proportionem eandem, quam quadratum b d ad quadra-<lb/>tum f q. </s> <s xml:id="echoid-s1595" xml:space="preserve">Sed quam proportionem habet tota portio ad eã, <lb/>quæ eſt extra humidum, eandem habet quadratum n o ad <lb/>quadratum p m. </s> <s xml:id="echoid-s1596" xml:space="preserve">ergo p m ipſi f q æ qualis etit. </s> <s xml:id="echoid-s1597" xml:space="preserve">demonſtra <lb/>ta eſt autem p h maior, quàm f: </s> <s xml:id="echoid-s1598" xml:space="preserve">quare m h minor erit, <pb file="0066" n="66" rhead="ARCHIMEDIS"/> quàm q; </s> <s xml:id="echoid-s1599" xml:space="preserve">& </s> <s xml:id="echoid-s1600" xml:space="preserve">p h maior, quàm dupla h m. </s> <s xml:id="echoid-s1601" xml:space="preserve">Sit igitur <lb/>p z dupla ip-<lb/> <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> ſius z m: </s> <s xml:id="echoid-s1602" xml:space="preserve">& </s> <s xml:id="echoid-s1603" xml:space="preserve">iun <lb/>cta z t produca <lb/>tur ad g. </s> <s xml:id="echoid-s1604" xml:space="preserve">erit <lb/>totius quidem <lb/>portionis gra-<lb/>uitatis centrũ <lb/>t: </s> <s xml:id="echoid-s1605" xml:space="preserve">eius, quæ eſt <lb/>extra humidũ <lb/>z: </s> <s xml:id="echoid-s1606" xml:space="preserve">reliquæ uero <lb/>partis, quæ in <lb/>humido, cen-<lb/>trum erit in li-<lb/>nea z t produ-<lb/>cta; </s> <s xml:id="echoid-s1607" xml:space="preserve">quod ſit g. <lb/></s> <s xml:id="echoid-s1608" xml:space="preserve">demõſtrabitur <lb/>ſimiliter, ut <lb/>prius, th per-<lb/>pẽdicularis ad <lb/>ſuperficiem hu <lb/>midi: </s> <s xml:id="echoid-s1609" xml:space="preserve">& </s> <s xml:id="echoid-s1610" xml:space="preserve">quæ <lb/>per z, g ducun-<lb/>tur æquidiſtan-<lb/>tes ipſi th, ad <lb/>eandem perpẽ <lb/>diculares. </s> <s xml:id="echoid-s1611" xml:space="preserve">ergo <lb/>portio, quæ eſt <lb/>extra humidũ <lb/>deorſum fere-<lb/>tur ſecundum <lb/>eam quæ per z <lb/>tranſit; </s> <s xml:id="echoid-s1612" xml:space="preserve">quæ ue <lb/>to intra ſecun- <pb o="22" file="0067" n="67" rhead="DE IIS QVAE VEH. IN AQVA."/> dum eam, quæ per g ſurſum eleuabitur. </s> <s xml:id="echoid-s1613" xml:space="preserve">non igitur manebit <lb/>portio ſic inclinata, nec conuertetur ita, ut axis ad ſuperfi-<lb/>ciem humidi ſit perpendicularis: </s> <s xml:id="echoid-s1614" xml:space="preserve">quoniam quæ ex parte 1 <lb/> <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> deorſum; </s> <s xml:id="echoid-s1615" xml:space="preserve">quæ uero ex parte a ſurſum ferentur, ut ex iam de <lb/>monſtratis apparere poteſt. </s> <s xml:id="echoid-s1616" xml:space="preserve">Quòd ſi axis cum ſuperficie <lb/>humidi fecerit angulum minorem angulo b, ſimiliter de-<lb/> <anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/> monſtrabitur, nõ manere portionem, ſed inclinari, donec <lb/>utique axis cum ſuperficie humidi faciat angulum angulo <lb/>b æqualem.</s> <s xml:id="echoid-s1617" xml:space="preserve"/> </p> <div xml:id="echoid-div111" type="float" level="2" n="1"> <figure xlink:label="fig-0064-01" xlink:href="fig-0064-01a"> <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0064-01"/> </figure> <note position="left" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve">A</note> <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0065-02" xlink:href="note-0065-02a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0065-03" xlink:href="note-0065-03a" xml:space="preserve">D</note> <note position="right" xlink:label="note-0065-04" xlink:href="note-0065-04a" xml:space="preserve">E</note> <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a"> <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0066-01"/> </figure> <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">F</note> <note position="right" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">G</note> </div> </div> <div xml:id="echoid-div113" type="section" level="1" n="39"> <head xml:id="echoid-head44" xml:space="preserve">COMMENTARIVS.</head> <p> <s xml:id="echoid-s1618" xml:space="preserve">QVARE quadratum b d magis excedit quadratum <lb/> <anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/> f q, quàm b c quadratum: </s> <s xml:id="echoid-s1619" xml:space="preserve">& </s> <s xml:id="echoid-s1620" xml:space="preserve">idcirco linea f q minor eſt, <lb/>quàm b c: </s> <s xml:id="echoid-s1621" xml:space="preserve">itemq; </s> <s xml:id="echoid-s1622" xml:space="preserve">f minor quam b r.</s> <s xml:id="echoid-s1623" xml:space="preserve">] _Quoniani exceſſus, quo_ <lb/>_quadratum b d excedit quadratum b c ad quadratum b d minorem_ <lb/>_proportionem habet, quàm exceſſus, quo quadratum b d excedit qua_ <lb/>_dratum f q, ad idem quadratum: </s> <s xml:id="echoid-s1624" xml:space="preserve">erit ex octaua quinti exceſſus, quo_ <lb/>_quadratum b d excedit quadratum b c, minor quàm exceſſus, quo ex_ <lb/>_cedit quadratum f q. </s> <s xml:id="echoid-s1625" xml:space="preserve">ergo quadratum f q minus est quadrato b c: </s> <s xml:id="echoid-s1626" xml:space="preserve">&_</s> <s xml:id="echoid-s1627" xml:space="preserve"> <lb/>_propterea linea f q minor linea b c. </s> <s xml:id="echoid-s1628" xml:space="preserve">Sed f q ad f eandem proportionẽ_ <lb/>_habet, quam b c ad b r; </s> <s xml:id="echoid-s1629" xml:space="preserve">utraque enim utriuſque ſeſquialtera est. </s> <s xml:id="echoid-s1630" xml:space="preserve">cum_ <lb/> <anchor type="note" xlink:label="note-0067-04a" xlink:href="note-0067-04"/> _igitur f q ſit minor b c, & </s> <s xml:id="echoid-s1631" xml:space="preserve">f ipſa b r minor erit_.</s> <s xml:id="echoid-s1632" xml:space="preserve"/> </p> <div xml:id="echoid-div113" type="float" level="2" n="1"> <note position="right" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">A</note> <note position="right" xlink:label="note-0067-04" xlink:href="note-0067-04a" xml:space="preserve">14. quinti</note> </div> <p> <s xml:id="echoid-s1633" xml:space="preserve">Et propterea k r ad ſ y maiorem habet, quàm dimidium <lb/> <anchor type="note" xlink:label="note-0067-05a" xlink:href="note-0067-05"/> ipſius k r ad ψ b.</s> <s xml:id="echoid-s1634" xml:space="preserve">] _Est enim k r ad ſ y, ut quadratum p s ad qua_ <lb/>_dratum ſ y<unsure/>: </s> <s xml:id="echoid-s1635" xml:space="preserve">& </s> <s xml:id="echoid-s1636" xml:space="preserve">dimidium lineæ K r ad lineam ψ b, ut quadratum e ψ_ <lb/>_ad quadratum ψ b_.</s> <s xml:id="echoid-s1637" xml:space="preserve"/> </p> <div xml:id="echoid-div114" type="float" level="2" n="2"> <note position="right" xlink:label="note-0067-05" xlink:href="note-0067-05a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s1638" xml:space="preserve">Et s o minor quàm ψ b] _Est enim ſ y dupla ipſius ſ o._</s> <s xml:id="echoid-s1639" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">C</note> <p> <s xml:id="echoid-s1640" xml:space="preserve">Et p h maior, quàm f.</s> <s xml:id="echoid-s1641" xml:space="preserve">] _Nam p h eſt æqualis ſ ω, & </s> <s xml:id="echoid-s1642" xml:space="preserve">r ψ_ <lb/> <anchor type="note" xlink:label="note-0067-07a" xlink:href="note-0067-07"/> _ipſi f_.</s> <s xml:id="echoid-s1643" xml:space="preserve"/> </p> <div xml:id="echoid-div115" type="float" level="2" n="3"> <note position="right" xlink:label="note-0067-07" xlink:href="note-0067-07a" xml:space="preserve">D</note> </div> <p> <s xml:id="echoid-s1644" xml:space="preserve">Habebit ergo tota portio ad eam, quæ eſt extra humi-<lb/> <anchor type="note" xlink:label="note-0067-08a" xlink:href="note-0067-08"/> dum proportionem eandem, quam quadratum b d ad qua <lb/>dratum f q.</s> <s xml:id="echoid-s1645" xml:space="preserve">] _Cum pars demerſa ad totam portionem ita ſit, ut_ <lb/>_exceſſus, quo quadratum b d excedit quadratum f q ad b d quadratu<unsure/>:_</s> <s xml:id="echoid-s1646" xml:space="preserve"> <pb file="0068" n="68" rhead="ARCHIMEDIS"/> _erit conuertendo tota portio ad partem ipſius demerſam, ut quadrd-_ <lb/>_tum b d ad exceſſum, quo quadratum f q excedit. </s> <s xml:id="echoid-s1647" xml:space="preserve">quare per conuer-_ <lb/>_ſionem rationis tota portio ad eam, quæ extra humidum est ut_ <lb/>_qu.</s> <s xml:id="echoid-s1648" xml:space="preserve">idratum b d ad quadratum f q: </s> <s xml:id="echoid-s1649" xml:space="preserve">nam quadratum b d tanto maius_ <lb/>_est exceſſu, quo excedit quadratum f q, quantum est ipſum f q qua-_ <lb/>_dratum_.</s> <s xml:id="echoid-s1650" xml:space="preserve"/> </p> <div xml:id="echoid-div116" type="float" level="2" n="4"> <note position="right" xlink:label="note-0067-08" xlink:href="note-0067-08a" xml:space="preserve">E</note> </div> <p style="it"> <s xml:id="echoid-s1651" xml:space="preserve">_Quoniam quæ ex parte 1 deorſum, quæ uero ex parte a_ <lb/> <anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> _ſurſum ferentur.</s> <s xml:id="echoid-s1652" xml:space="preserve">]_ Hæc nos ita correximus, nam in translatione <lb/>mendoſe, ut opinor, legebatur, quoniam quæ ex parte l ad ſuperiora <lb/>ferentur, perpendicularis enim quæ tranſit per z ad partes l, & </s> <s xml:id="echoid-s1653" xml:space="preserve">quæ <lb/>per g ad partes a cadit. </s> <s xml:id="echoid-s1654" xml:space="preserve">quare centrum z unà cum p. </s> <s xml:id="echoid-s1655" xml:space="preserve">trtibus ijs, quæ <lb/>ſunt ad l deorſum feretur, centrum uero g unà cum partibus quæ ad <lb/>a ſurſum.</s> <s xml:id="echoid-s1656" xml:space="preserve"/> </p> <div xml:id="echoid-div117" type="float" level="2" n="5"> <note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">F</note> </div> <p> <s xml:id="echoid-s1657" xml:space="preserve">Similiter demonſtrabitur non manere portionem, ſed <lb/> <anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/> inclinari, donec utique axis cum ſuperficie humidi faciat <lb/>angulum angulo b æqualem.</s> <s xml:id="echoid-s1658" xml:space="preserve">] _Illud uero tum ex ijs, quæ in an_ <lb/>_tecedenti dicta ſunt, tum ex figuris, quas appoſuimus, facile demon-_ <lb/>_strari potest._</s> <s xml:id="echoid-s1659" xml:space="preserve"/> </p> <div xml:id="echoid-div118" type="float" level="2" n="6"> <note position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">G</note> </div> </div> <div xml:id="echoid-div120" type="section" level="1" n="40"> <head xml:id="echoid-head45" xml:space="preserve">PROPOSITIO X.</head> <p> <s xml:id="echoid-s1660" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem, quàm <lb/>ut ad eam, quæ uſque ad axem proportionem ha-<lb/>beat, quam quindecim ad quatuor: </s> <s xml:id="echoid-s1661" xml:space="preserve">in humidum <lb/>demiſſa, ita ut baſis ipſius non contingat humi-<lb/>dum: </s> <s xml:id="echoid-s1662" xml:space="preserve">non nunquam quidem recta conſiſtet; </s> <s xml:id="echoid-s1663" xml:space="preserve">non <lb/> <anchor type="note" xlink:label="note-0068-03a" xlink:href="note-0068-03"/> nunquam inclinata: </s> <s xml:id="echoid-s1664" xml:space="preserve">& </s> <s xml:id="echoid-s1665" xml:space="preserve">interdum adeo inclinata, <lb/> <anchor type="note" xlink:label="note-0068-04a" xlink:href="note-0068-04"/> ut baſis ipſius in uno puncto contingat ſuperfi-<lb/>ciem humidi: </s> <s xml:id="echoid-s1666" xml:space="preserve">idq; </s> <s xml:id="echoid-s1667" xml:space="preserve">in duabus diſpoſitionibus:</s> <s xml:id="echoid-s1668" xml:space="preserve"> <pb o="29" file="0069" n="69" rhead="DE IIS QVAE VEH. IN AQVA."/> interdũ quidem ita, ut baſis in humidum magis <lb/> <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> demergatur: </s> <s xml:id="echoid-s1669" xml:space="preserve">interdum uero ita, ut ſuperficiem <lb/> <anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/> humidi nullo modo contingat; </s> <s xml:id="echoid-s1670" xml:space="preserve">ſecundum pro-<lb/> <anchor type="note" xlink:label="note-0069-03a" xlink:href="note-0069-03"/> portionem, quam habet ad humidum in grauita-<lb/>te. </s> <s xml:id="echoid-s1671" xml:space="preserve">Eorum quæ dicta ſunt, ſingula inferius de-<lb/>monſtrabuntur.</s> <s xml:id="echoid-s1672" xml:space="preserve"/> </p> <div xml:id="echoid-div120" type="float" level="2" n="1"> <note position="left" xlink:label="note-0068-03" xlink:href="note-0068-03a" xml:space="preserve">A</note> <note position="left" xlink:label="note-0068-04" xlink:href="note-0068-04a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">C</note> <note position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">D</note> <note position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">E</note> </div> <p> <s xml:id="echoid-s1673" xml:space="preserve">SIT portio qualis dicta eſt: </s> <s xml:id="echoid-s1674" xml:space="preserve">& </s> <s xml:id="echoid-s1675" xml:space="preserve">ſecta ipſa plano per axẽ. <lb/></s> <s xml:id="echoid-s1676" xml:space="preserve">recto ad ſuperficiem humidi, ſectio ſit a p o l rectanguli co <lb/>ni ſeccio: </s> <s xml:id="echoid-s1677" xml:space="preserve">axis portionis, & </s> <s xml:id="echoid-s1678" xml:space="preserve">ſectionis diameter b d: </s> <s xml:id="echoid-s1679" xml:space="preserve">ſece-<lb/>turq; </s> <s xml:id="echoid-s1680" xml:space="preserve">b d in puncto quidem _k_ ita, ut b k dupla ſitipſius <lb/>_k_ d: </s> <s xml:id="echoid-s1681" xml:space="preserve">in c uero ita, ut b d ad <emph style="sc">K C</emph> proportionẽ habeat ean-<lb/>dem, quam quindecim ad quatuor. </s> <s xml:id="echoid-s1682" xml:space="preserve">conſtat igitur k c ma-<lb/> <anchor type="note" xlink:label="note-0069-04a" xlink:href="note-0069-04"/> iorem eſſe, quàm quæ uſque ad axem. </s> <s xml:id="echoid-s1683" xml:space="preserve">Sit ei quæ uſque ad <lb/> <anchor type="note" xlink:label="note-0069-05a" xlink:href="note-0069-05"/> axem æqualis k r: </s> <s xml:id="echoid-s1684" xml:space="preserve">& </s> <s xml:id="echoid-s1685" xml:space="preserve">ipſius k r ſeſquialtera d s. </s> <s xml:id="echoid-s1686" xml:space="preserve">Eſt autem <lb/> <anchor type="note" xlink:label="note-0069-06a" xlink:href="note-0069-06"/> & </s> <s xml:id="echoid-s1687" xml:space="preserve">s b ſeſquial-<lb/> <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> tera ipſius b r. <lb/></s> <s xml:id="echoid-s1688" xml:space="preserve">Itaque iũgatur <lb/>a b, & </s> <s xml:id="echoid-s1689" xml:space="preserve">per c du <lb/>catur c e per-<lb/>pẽdicularis ad <lb/>b d, quæ lineã <lb/>a b in puncto <lb/>e ſecet: </s> <s xml:id="echoid-s1690" xml:space="preserve">& </s> <s xml:id="echoid-s1691" xml:space="preserve">per <lb/>e ducatur e z <lb/>æquidiſtãs b d. </s> <s xml:id="echoid-s1692" xml:space="preserve"><lb/>Rurſus ipſa a b <lb/>bifariã in t di-<lb/>uiſa, ducatur t <lb/>h eidem b d æ-<lb/>quidiſtans: </s> <s xml:id="echoid-s1693" xml:space="preserve">& </s> <s xml:id="echoid-s1694" xml:space="preserve"><lb/>intelligantur rectanguli coni ſectiones deſcriptæ a e i qui- <pb file="0070" n="70" rhead="ARCHIMEDIS"/> dem circa e z diametrum; </s> <s xml:id="echoid-s1695" xml:space="preserve">a t d uero circa diametrum t h; <lb/></s> <s xml:id="echoid-s1696" xml:space="preserve"> <anchor type="note" xlink:label="note-0070-01a" xlink:href="note-0070-01"/> quæ ſimiles ſint portioni a b l. </s> <s xml:id="echoid-s1697" xml:space="preserve">tranſibit igitur a e i coni <lb/> <anchor type="note" xlink:label="note-0070-02a" xlink:href="note-0070-02"/> ſectio per _K_: </s> <s xml:id="echoid-s1698" xml:space="preserve">& </s> <s xml:id="echoid-s1699" xml:space="preserve">quæ ab r ducta eſt perpendicularis ad b d, <lb/>ipſam a e i ſecabit. </s> <s xml:id="echoid-s1700" xml:space="preserve">ſecet in punctis y g: </s> <s xml:id="echoid-s1701" xml:space="preserve">& </s> <s xml:id="echoid-s1702" xml:space="preserve">per y g ducan <lb/>tur ipſi b d æquidiſtantes p y q, o g n, quæ ſecent a t d in <lb/>f x. </s> <s xml:id="echoid-s1703" xml:space="preserve">ducantur poſtremo, & </s> <s xml:id="echoid-s1704" xml:space="preserve">p χ, o φ contingentes ſectionẽ <lb/>a p o l in punctis p o. </s> <s xml:id="echoid-s1705" xml:space="preserve">cũ<unsure/> ergo tres portiones ſint a p o l, <lb/> <anchor type="note" xlink:label="note-0070-03a" xlink:href="note-0070-03"/> a e i, a t d, contentæ rectis lineis, & </s> <s xml:id="echoid-s1706" xml:space="preserve">rectangulorum cono-<lb/>rum ſectionibus; </s> <s xml:id="echoid-s1707" xml:space="preserve">rectæq, ſimiles, & </s> <s xml:id="echoid-s1708" xml:space="preserve">inæquales, quæ contin <lb/>gunt ſe ſe ſuper unamquanque baſim: </s> <s xml:id="echoid-s1709" xml:space="preserve">à puncto autem n <lb/>ſurſum ducta ſit n x g o; </s> <s xml:id="echoid-s1710" xml:space="preserve">& </s> <s xml:id="echoid-s1711" xml:space="preserve">à q ipſa q fy p: </s> <s xml:id="echoid-s1712" xml:space="preserve">habebit o g ad <lb/>g x proportionem compoſitam ex proportione, quam ha <lb/>bet i l ad l a; </s> <s xml:id="echoid-s1713" xml:space="preserve">& </s> <s xml:id="echoid-s1714" xml:space="preserve">ex proportione, quam a d habet ad d i. <lb/></s> <s xml:id="echoid-s1715" xml:space="preserve">Sed i l ad l a <lb/> <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> habet eandem, <lb/>quam duo ad <lb/>quinque. </s> <s xml:id="echoid-s1716" xml:space="preserve">ete-<lb/>nim c b ad b d <lb/> <anchor type="note" xlink:label="note-0070-04a" xlink:href="note-0070-04"/> eſt, ut ſex ad <lb/>quĩdecim; </s> <s xml:id="echoid-s1717" xml:space="preserve">hoc <lb/>eſt ut duo ad <lb/>quinque: </s> <s xml:id="echoid-s1718" xml:space="preserve">& </s> <s xml:id="echoid-s1719" xml:space="preserve">ut <lb/> <anchor type="note" xlink:label="note-0070-05a" xlink:href="note-0070-05"/> c b ad b d, ita <lb/>e b ad b a: </s> <s xml:id="echoid-s1720" xml:space="preserve">& </s> <s xml:id="echoid-s1721" xml:space="preserve"><lb/>d z ad d a. </s> <s xml:id="echoid-s1722" xml:space="preserve">ha-<lb/>rum autẽ d z, <lb/> <anchor type="note" xlink:label="note-0070-06a" xlink:href="note-0070-06"/> d a duplæ ſunt <lb/>ipſæ l i, l a: </s> <s xml:id="echoid-s1723" xml:space="preserve">& </s> <s xml:id="echoid-s1724" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0070-07a" xlink:href="note-0070-07"/> a d ad d i eã pro <lb/>portionem habet, quam quinque ad unum. </s> <s xml:id="echoid-s1725" xml:space="preserve">ſed proportio <lb/>compoſita ex proportione, quam habet duo ad quinque; <lb/></s> <s xml:id="echoid-s1726" xml:space="preserve">& </s> <s xml:id="echoid-s1727" xml:space="preserve">ex proportione, quam quinque ad unum; </s> <s xml:id="echoid-s1728" xml:space="preserve">eſt eadem, <lb/>quam habent duo ad unum: </s> <s xml:id="echoid-s1729" xml:space="preserve">duo autem ad unum duplam <lb/>proportionem habent. </s> <s xml:id="echoid-s1730" xml:space="preserve">dupla eſt igitur g b ipſius g x: </s> <s xml:id="echoid-s1731" xml:space="preserve">&</s> <s xml:id="echoid-s1732" xml:space="preserve"> <pb o="30" file="0071" n="71" rhead="DE IIS QVAE VEH. IN AQVA."/> eadem ratione oſtẽdetur p y ipſius y f dupla. </s> <s xml:id="echoid-s1733" xml:space="preserve">Itaque quo <lb/>niam d s ſeſquialtera eſt ipſius _k_ r; </s> <s xml:id="echoid-s1734" xml:space="preserve">erit b s exceſſus, quo <lb/>axis eſt maior, quàm ſeſquialter eius, quæ uſque ad axem. <lb/></s> <s xml:id="echoid-s1735" xml:space="preserve">Si igitur portio ad humidũ in grauitate eã habet propor-<lb/>tionem, quam quadratum, quod fit à linea b ſ ad quadra-<lb/>tum, quod à b d, aut maiorem; </s> <s xml:id="echoid-s1736" xml:space="preserve">in hnmidum demiſſa, ita <lb/>ut baſis ipſius non contingat humidum, recta conſiſtet. </s> <s xml:id="echoid-s1737" xml:space="preserve">de <lb/>monſtratum eſt enim ſuperius, portionem, cuius axis eſt <lb/> <anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/> maior, quàm ſeſquiaiter eius, quæ uſque ad axem, ſi ad hu-<lb/>midum in grauitate non minorem proportionem habeat, <lb/>quàm quadratum, quod fit ab exceſſu, quo axis maior eſt, <lb/>quam ſeſquialter eius, quæ uſque ad axem, ad quadratum, <lb/>quod ab axe; </s> <s xml:id="echoid-s1738" xml:space="preserve">demiſſam in humidum, ita ut dictum eſt, re-<lb/>ctam conſiſtere.</s> <s xml:id="echoid-s1739" xml:space="preserve"/> </p> <div xml:id="echoid-div121" type="float" level="2" n="2"> <note position="right" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">F</note> <note position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">G</note> <note position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">H</note> <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a"> <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0069-01"/> </figure> <note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">K</note> <note position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">L</note> <note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">M</note> <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a"> <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0070-01"/> </figure> <note position="left" xlink:label="note-0070-04" xlink:href="note-0070-04a" xml:space="preserve">N</note> <note position="left" xlink:label="note-0070-05" xlink:href="note-0070-05a" xml:space="preserve">O</note> <note position="left" xlink:label="note-0070-06" xlink:href="note-0070-06a" xml:space="preserve">P</note> <note position="left" xlink:label="note-0070-07" xlink:href="note-0070-07a" xml:space="preserve">Q</note> <note position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">R</note> </div> </div> <div xml:id="echoid-div123" type="section" level="1" n="41"> <head xml:id="echoid-head46" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s1740" xml:space="preserve">_<emph style="sc">Qvae</emph>_ hac decima propoſitione continentur, Archimedes in <lb/>quinque partes diſſecuit, & </s> <s xml:id="echoid-s1741" xml:space="preserve">ſingulas ſeorſum demonſtrauit.</s> <s xml:id="echoid-s1742" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1743" xml:space="preserve">Nonnunquam quidem recta conſiſtat.</s> <s xml:id="echoid-s1744" xml:space="preserve">] _Hæc eſt prima_ <lb/> <anchor type="note" xlink:label="note-0071-02a" xlink:href="note-0071-02"/> _pars, cuius demonstr ationem ſtatim ſubiungit._</s> <s xml:id="echoid-s1745" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="1"> <note position="right" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">A</note> </div> <p> <s xml:id="echoid-s1746" xml:space="preserve">Etinterdum adeo inclinata, ut baſis ipſius in uno pun-<lb/> <anchor type="note" xlink:label="note-0071-03a" xlink:href="note-0071-03"/> cto contingat ſuperficiem humidi; </s> <s xml:id="echoid-s1747" xml:space="preserve">idq; </s> <s xml:id="echoid-s1748" xml:space="preserve">in duabus diſpoſi-<lb/>tionibus.</s> <s xml:id="echoid-s1749" xml:space="preserve">] _Demonſtratum eſt illud in tertia parte._</s> <s xml:id="echoid-s1750" xml:space="preserve"/> </p> <div xml:id="echoid-div124" type="float" level="2" n="2"> <note position="right" xlink:label="note-0071-03" xlink:href="note-0071-03a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s1751" xml:space="preserve">Interdum ita, ut baſis in humidum magis demergatur.</s> <s xml:id="echoid-s1752" xml:space="preserve">] <lb/> <anchor type="note" xlink:label="note-0071-04a" xlink:href="note-0071-04"/> _Pertinet id ad quartam partem._</s> <s xml:id="echoid-s1753" xml:space="preserve"/> </p> <div xml:id="echoid-div125" type="float" level="2" n="3"> <note position="right" xlink:label="note-0071-04" xlink:href="note-0071-04a" xml:space="preserve">C</note> </div> <p> <s xml:id="echoid-s1754" xml:space="preserve">Interdum uero ita, ut ſuperficiem humidi nullo modo <lb/> <anchor type="note" xlink:label="note-0071-05a" xlink:href="note-0071-05"/> contingat.</s> <s xml:id="echoid-s1755" xml:space="preserve">] _Hoc duobus item modis fit, quorum unus in ſecunda,_ <lb/>_alter in quarta parte explicatur._</s> <s xml:id="echoid-s1756" xml:space="preserve"/> </p> <div xml:id="echoid-div126" type="float" level="2" n="4"> <note position="right" xlink:label="note-0071-05" xlink:href="note-0071-05a" xml:space="preserve">D</note> </div> <p> <s xml:id="echoid-s1757" xml:space="preserve">Secundum proportionem, quam habet ad humidum in <lb/> <anchor type="note" xlink:label="note-0071-06a" xlink:href="note-0071-06"/> grauitate.</s> <s xml:id="echoid-s1758" xml:space="preserve">] _In translatione ita legebatur, quam autem proportio_ <lb/>_nem habet ad humidum in grauitate._</s> <s xml:id="echoid-s1759" xml:space="preserve"/> </p> <div xml:id="echoid-div127" type="float" level="2" n="5"> <note position="right" xlink:label="note-0071-06" xlink:href="note-0071-06a" xml:space="preserve">E</note> </div> <p style="it"> <s xml:id="echoid-s1760" xml:space="preserve">_Conſtat igitur k c maiorem eſſe, quàm quæ uſque ad_ <lb/> <anchor type="note" xlink:label="note-0071-07a" xlink:href="note-0071-07"/> _axem.</s> <s xml:id="echoid-s1761" xml:space="preserve">]_ Nam cum b d ad k c eandem habeat proportionem, quam <pb file="0072" n="72" rhead="ARCHIMEDIS"/> quindecim ad quatuor; </s> <s xml:id="echoid-s1762" xml:space="preserve">& </s> <s xml:id="echoid-s1763" xml:space="preserve">ad eam, quæ uſque ad axem maiorem pro <lb/>portionem habeat: </s> <s xml:id="echoid-s1764" xml:space="preserve">erit quæ uſ que ad axem minor ipſa k c.</s> <s xml:id="echoid-s1765" xml:space="preserve"/> </p> <div xml:id="echoid-div128" type="float" level="2" n="6"> <note position="right" xlink:label="note-0071-07" xlink:href="note-0071-07a" xml:space="preserve">F</note> </div> <note position="left" xml:space="preserve">10. quinti</note> <p> <s xml:id="echoid-s1766" xml:space="preserve">Sit ei, quæ uſque ad axem æ qualis k r.</s> <s xml:id="echoid-s1767" xml:space="preserve">] _Hac nos addidimus,_ <lb/> <anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/> _quæ in translatione non erant._</s> <s xml:id="echoid-s1768" xml:space="preserve"/> </p> <div xml:id="echoid-div129" type="float" level="2" n="7"> <note position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">G</note> </div> <p style="it"> <s xml:id="echoid-s1769" xml:space="preserve">_Eſt autem & </s> <s xml:id="echoid-s1770" xml:space="preserve">s b ſeſquialtera ipſius b r.</s> <s xml:id="echoid-s1771" xml:space="preserve">]_ Ponitur enim <lb/> <anchor type="note" xlink:label="note-0072-03a" xlink:href="note-0072-03"/> d b ſeſquialtera ipſius b k; </s> <s xml:id="echoid-s1772" xml:space="preserve">itémq; </s> <s xml:id="echoid-s1773" xml:space="preserve">d ſ ſeſquialtera k r. </s> <s xml:id="echoid-s1774" xml:space="preserve">quare ut to <lb/>ta d b ad totam b K, ita pars d s ad partem K r. </s> <s xml:id="echoid-s1775" xml:space="preserve">ergo & </s> <s xml:id="echoid-s1776" xml:space="preserve">reliqua <lb/> <anchor type="note" xlink:label="note-0072-04a" xlink:href="note-0072-04"/> s b ad reliquim b r, ut d b ad b k.</s> <s xml:id="echoid-s1777" xml:space="preserve"/> </p> <div xml:id="echoid-div130" type="float" level="2" n="8"> <note position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">H</note> <note position="left" xlink:label="note-0072-04" xlink:href="note-0072-04a" xml:space="preserve">19. quinti</note> </div> <p style="it"> <s xml:id="echoid-s1778" xml:space="preserve">_Quæ ſimiles ſint portioni a b l.</s> <s xml:id="echoid-s1779" xml:space="preserve">]_ Similes portiones coni ſe-<lb/> <anchor type="note" xlink:label="note-0072-05a" xlink:href="note-0072-05"/> ctionum Apollonius it. </s> <s xml:id="echoid-s1780" xml:space="preserve">i diffiniuit in ſexto libro conicorum, ut ſcri-<lb/>bit Eutocius, εν οἱς α χ θεισω<unsure/>νἐν ἑηάστω παραλλήλων τῆ βάσει, ἵσωι <lb/>τὸ πλῆθος, ὰι παρὰλληλοι, καὶ αἱ βάσ{ει}ς πρ<unsure/>ὸς τὰςἀποτεμνομένας <lb/>ἀπὸ τῶν διαμέ τρων ταῖς νορυφαῖς ἐν τοῖς αὐτοῖ ς λὄγοιςεἰσἰ, καὶἁι <lb/>ἀποτεμνόμεναι πρ<unsure/>ὸς τάς ἀποτεμνομένας; </s> <s xml:id="echoid-s1781" xml:space="preserve">hoc est. </s> <s xml:id="echoid-s1782" xml:space="preserve">in quibus ſi du-<lb/>cantnr lineæ æquidistantes baſi numero æquales: </s> <s xml:id="echoid-s1783" xml:space="preserve">æquidiſtantes atq; <lb/></s> <s xml:id="echoid-s1784" xml:space="preserve">baſes ad partes diametrorum, quæ ab ipſis ad uerticem abſcindũtur, <lb/>eandem proportionem babent: </s> <s xml:id="echoid-s1785" xml:space="preserve">it émq; </s> <s xml:id="echoid-s1786" xml:space="preserve">partes abſciſſæ ad abſciſſas. </s> <s xml:id="echoid-s1787" xml:space="preserve"><lb/>ducuntur autem lineæ baſi æquidistantes: </s> <s xml:id="echoid-s1788" xml:space="preserve">ut opinor, deſcripta in ſin <lb/>gulis plane rectilinea figura, quæ lateribus numero æqualibus conti <lb/> <anchor type="note" xlink:label="note-0072-06a" xlink:href="note-0072-06"/> neatur. </s> <s xml:id="echoid-s1789" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s1790" xml:space="preserve">portiones ſimiles à ſimilibus coni ſectionibus abſcindũ <lb/>tur: </s> <s xml:id="echoid-s1791" xml:space="preserve">& </s> <s xml:id="echoid-s1792" xml:space="preserve">earum diametri ſiue ad baſes rectæ, ſiue cum baſibus æ qua-<lb/>les angulos facientes, ad ipſas baſes eandem habent proportionem.</s> <s xml:id="echoid-s1793" xml:space="preserve"/> </p> <div xml:id="echoid-div131" type="float" level="2" n="9"> <note position="left" xlink:label="note-0072-05" xlink:href="note-0072-05a" xml:space="preserve">K</note> <note position="left" xlink:label="note-0072-06" xlink:href="note-0072-06a" xml:space="preserve">γνωρίμως</note> </div> <p style="it"> <s xml:id="echoid-s1794" xml:space="preserve">_Tranſibit igitur a e i coni ſectio per k.</s> <s xml:id="echoid-s1795" xml:space="preserve">]_ Sienim fieri po <lb/> <anchor type="note" xlink:label="note-0072-07a" xlink:href="note-0072-07"/> teſt non tranſeat per k, ſed per aliud punctum lineæ d b, ut per u. <lb/></s> <s xml:id="echoid-s1796" xml:space="preserve">Quoniam igitur in rectáguli coni ſectione a e i, cuius diameter e z, <lb/>ducta eſt a e, & </s> <s xml:id="echoid-s1797" xml:space="preserve">producta: </s> <s xml:id="echoid-s1798" xml:space="preserve">& </s> <s xml:id="echoid-s1799" xml:space="preserve">d b diametro æquidistans utraſque <lb/>a e, a i ſecat; </s> <s xml:id="echoid-s1800" xml:space="preserve">a e quidem in b, ai uero in d: </s> <s xml:id="echoid-s1801" xml:space="preserve">habebit d b ad b u <lb/>proportionem eandem, quam a z, ad z d, ex quarta propoſitione li <lb/>bri. </s> <s xml:id="echoid-s1802" xml:space="preserve">Archimedis de quadratura parabol<unsure/>æ. </s> <s xml:id="echoid-s1803" xml:space="preserve">Sed a z ſeſquialtera eſt <lb/>ipſius z d: </s> <s xml:id="echoid-s1804" xml:space="preserve">eſt enim ut tria ad duo, quod mox demonſtrabimus. </s> <s xml:id="echoid-s1805" xml:space="preserve">ergo <lb/>d b ſeſquialtera eſt ipſius b u. </s> <s xml:id="echoid-s1806" xml:space="preserve">eſt auté d b & </s> <s xml:id="echoid-s1807" xml:space="preserve">ipſius b k ſeſquialte <lb/>ra. </s> <s xml:id="echoid-s1808" xml:space="preserve">quare lineæ b u, b k inter ſe æ quales ſunt; </s> <s xml:id="echoid-s1809" xml:space="preserve">quod fieri non po-<lb/> <anchor type="note" xlink:label="note-0072-08a" xlink:href="note-0072-08"/> teſt. </s> <s xml:id="echoid-s1810" xml:space="preserve">restanguli igitur com ſectio a e i per punctum k tranſibit. <lb/></s> <s xml:id="echoid-s1811" xml:space="preserve">quod demonstrare uolebamus.</s> <s xml:id="echoid-s1812" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="10"> <note position="left" xlink:label="note-0072-07" xlink:href="note-0072-07a" xml:space="preserve">L</note> <note position="left" xlink:label="note-0072-08" xlink:href="note-0072-08a" xml:space="preserve">2. quinti.</note> </div> <pb o="37" file="0073" n="73" rhead="DE IIS QVAE VEH. IN AQVA."/> <p> <s xml:id="echoid-s1813" xml:space="preserve">Cum ergo tres portiones ſint a p o i, a ei, atd, con-<lb/> <anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/> tentæ rectis lineis, & </s> <s xml:id="echoid-s1814" xml:space="preserve">rectãgulorum conorum ſectionibus; <lb/></s> <s xml:id="echoid-s1815" xml:space="preserve">rectæq;</s> <s xml:id="echoid-s1816" xml:space="preserve">, ſimiles, & </s> <s xml:id="echoid-s1817" xml:space="preserve">inæquales, quæ contingunt ſe ſe ſuper <lb/>unam quamque baſim.</s> <s xml:id="echoid-s1818" xml:space="preserve">] _Poſt ea uerba, ſuper unamquanque_ <lb/>_baſim, in trans latione aliqua deſiderari uidentur. </s> <s xml:id="echoid-s1819" xml:space="preserve">Ad borum autem_ <lb/>_demonſtrationem non nulla præmittere oportet, quæ etiam ad alia,_ <lb/>_quæ ſequuntur, neceſſaria erunt._</s> <s xml:id="echoid-s1820" xml:space="preserve"/> </p> <div xml:id="echoid-div133" type="float" level="2" n="11"> <note position="right" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">M</note> </div> </div> <div xml:id="echoid-div135" type="section" level="1" n="42"> <head xml:id="echoid-head47" xml:space="preserve">LEMMA I.</head> <p style="it"> <s xml:id="echoid-s1821" xml:space="preserve">Sit recta linea a b, quam ſecent duæ lineæ inter ſeſe <lb/>æquidiſtantes a c, d e, ita ut quam proportionem ba-<lb/>bet a b ad b d, eandern haheat a c ad de. </s> <s xml:id="echoid-s1822" xml:space="preserve">Dico li-<lb/>neam, quæ c b puncta coniungit, etiam per ipſum e <lb/>tr anſire.</s> <s xml:id="echoid-s1823" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1824" xml:space="preserve">SI enim fieri poteſt, non tranſeat pere, ſed nel ſupra, uel infra. <lb/></s> <s xml:id="echoid-s1825" xml:space="preserve">tranſeat primum infra, ut per f. </s> <s xml:id="echoid-s1826" xml:space="preserve">erunt triangula a b c, d b f inter ſe <lb/>ſimilia. </s> <s xml:id="echoid-s1827" xml:space="preserve">quare ut a b ad b d, ita a c ad d f. </s> <s xml:id="echoid-s1828" xml:space="preserve">ſed ut a b ad bd, ita <lb/> <anchor type="note" xlink:label="note-0073-02a" xlink:href="note-0073-02"/> erat a c ad d e. </s> <s xml:id="echoid-s1829" xml:space="preserve">ergo d f ipſi d e æqualis erit, uidelicet pars to-<lb/> <anchor type="note" xlink:label="note-0073-03a" xlink:href="note-0073-03"/> ti, quod eſt <lb/> <anchor type="figure" xlink:label="fig-0073-01a" xlink:href="fig-0073-01"/> cbſurdum. <lb/></s> <s xml:id="echoid-s1830" xml:space="preserve">Idem ab-<lb/>ſurdum ſe <lb/>quetur, ſi <lb/>linea c b <lb/>ſupra e pú <lb/>ctum tran <lb/>ſire pona-<lb/>tur. </s> <s xml:id="echoid-s1831" xml:space="preserve">quare <lb/>c b etiam <lb/>per e ne-<lb/>ceſſario tranſibit. </s> <s xml:id="echoid-s1832" xml:space="preserve">quod oportebat demonſtrare.</s> <s xml:id="echoid-s1833" xml:space="preserve"/> </p> <div xml:id="echoid-div135" type="float" level="2" n="1"> <note position="right" xlink:label="note-0073-02" xlink:href="note-0073-02a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-0073-03" xlink:href="note-0073-03a" xml:space="preserve">9. quinti.</note> <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a"> <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0073-01"/> </figure> </div> <pb file="0074" n="74" rhead="ARCHIMEDIS"/> </div> <div xml:id="echoid-div137" type="section" level="1" n="43"> <head xml:id="echoid-head48" xml:space="preserve">LEMMA II.</head> <p style="it"> <s xml:id="echoid-s1834" xml:space="preserve">Sint duæ portionis ſimiles, contentæ rectis lineis, & </s> <s xml:id="echoid-s1835" xml:space="preserve"><lb/>rectangulorum conorum ſectionibus; </s> <s xml:id="echoid-s1836" xml:space="preserve">a b c quidem ma-<lb/>ior, cuius diameter b d; </s> <s xml:id="echoid-s1837" xml:space="preserve">e f c uero minor, cuius diameter <lb/>fg: </s> <s xml:id="echoid-s1838" xml:space="preserve">aptenturq; </s> <s xml:id="echoid-s1839" xml:space="preserve">inter ſeſe, ita ut maior minorem includat <lb/>& </s> <s xml:id="echoid-s1840" xml:space="preserve">ſint earum baſes a c, e c in eadem recta linea, ut idẽ <lb/>punctum c ſit utriuſque terminus: </s> <s xml:id="echoid-s1841" xml:space="preserve">ſumatur deinde in ſe <lb/>ctione a b c quodlibet punctum b: </s> <s xml:id="echoid-s1842" xml:space="preserve">& </s> <s xml:id="echoid-s1843" xml:space="preserve">iungatur h c. </s> <s xml:id="echoid-s1844" xml:space="preserve">Di <lb/>co lineam h c ad partem ſui ipſius, quæ inter c, & </s> <s xml:id="echoid-s1845" xml:space="preserve">ſe-<lb/>ctionem e f c interiicitur, eam proportionẽ habere, quam <lb/>habet a c ad c e.</s> <s xml:id="echoid-s1846" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1847" xml:space="preserve">_<emph style="sc">Dvcatvr</emph>_ b c, quæ tranſibit per f. </s> <s xml:id="echoid-s1848" xml:space="preserve">quoniam enim portiones <lb/>ſimiles ſunt, diametri cú baſibus æquales continent angulos. </s> <s xml:id="echoid-s1849" xml:space="preserve">quare <lb/>æquidiſtant inter ſe ſe b d, f g: </s> <s xml:id="echoid-s1850" xml:space="preserve">éſtq; </s> <s xml:id="echoid-s1851" xml:space="preserve">b d ad a c, ut f g ad e c: <lb/></s> <s xml:id="echoid-s1852" xml:space="preserve">& </s> <s xml:id="echoid-s1853" xml:space="preserve">permu-<lb/> <anchor type="figure" xlink:label="fig-0074-01a" xlink:href="fig-0074-01"/> tando b d ad <lb/>f g, ut a c ad <lb/>c e: </s> <s xml:id="echoid-s1854" xml:space="preserve">hoc eſt <lb/> <anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/> ut earum di-<lb/>midiæ d c ad <lb/>c g. </s> <s xml:id="echoid-s1855" xml:space="preserve">ergo ex <lb/>antecedēti lé <lb/>mate ſequi-<lb/>tur lineá b c <lb/>per punctum <lb/>f tranſire. <lb/></s> <s xml:id="echoid-s1856" xml:space="preserve">Ducatur præ <lb/>terea à puncto h ad diametrum b d linea h K, æquidiſtans baſi <lb/>a c: </s> <s xml:id="echoid-s1857" xml:space="preserve">& </s> <s xml:id="echoid-s1858" xml:space="preserve">iuncta k c, quæ diametrum f g ſecet in l; </s> <s xml:id="echoid-s1859" xml:space="preserve">per l ducatur <pb o="32" file="0075" n="75" rhead="DE IIS QVAE VEH. IN AQVA."/> ad ſectionem e f g ex parte e linea l m, eidem a c baſi æquidi-<lb/>stans. </s> <s xml:id="echoid-s1860" xml:space="preserve">Sit autem ſectionis a b c, linea b n iuxta quam poſſunt, quæ <lb/>à ſectione ducuntur: </s> <s xml:id="echoid-s1861" xml:space="preserve">& </s> <s xml:id="echoid-s1862" xml:space="preserve">ſectionis e f c ſit ipſa f o. </s> <s xml:id="echoid-s1863" xml:space="preserve">quoniam igi-<lb/>tur triangula c d b, c f g ſimilia ſunt, erit ut b c ad c f, ita d c <lb/> <anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/> ad c g; </s> <s xml:id="echoid-s1864" xml:space="preserve">& </s> <s xml:id="echoid-s1865" xml:space="preserve">b d ad f g. </s> <s xml:id="echoid-s1866" xml:space="preserve">rurſus quoniam triangula c k b, c l f etiã <lb/>inter ſe ſunt ſimilia, ut b c ad c f, boc eſt ut b d ad f g, ita erit k c <lb/>ad c l; </s> <s xml:id="echoid-s1867" xml:space="preserve">& </s> <s xml:id="echoid-s1868" xml:space="preserve">b K ad f l. </s> <s xml:id="echoid-s1869" xml:space="preserve">quare K c ad c l, & </s> <s xml:id="echoid-s1870" xml:space="preserve">b k ad f l ſunt ut d c <lb/>ad c g: </s> <s xml:id="echoid-s1871" xml:space="preserve">hoc eſt ut earum duplæ a c ad c e. </s> <s xml:id="echoid-s1872" xml:space="preserve">ſed ut b d ad f g, ita d c <lb/> <anchor type="note" xlink:label="note-0075-02a" xlink:href="note-0075-02"/> ad c g; </s> <s xml:id="echoid-s1873" xml:space="preserve">hoc ẽ a d ad e g: </s> <s xml:id="echoid-s1874" xml:space="preserve">& </s> <s xml:id="echoid-s1875" xml:space="preserve">permutãdo ut b d ad a d, ita f g ad e g. <lb/></s> <s xml:id="echoid-s1876" xml:space="preserve">quadratum autem a d æquale eſt rectangulo d b n ex undecima pri <lb/>mi conicorum. </s> <s xml:id="echoid-s1877" xml:space="preserve">ergo tres lineæ b d, a d, b n inter ſe ſunt proportio <lb/> <anchor type="note" xlink:label="note-0075-03a" xlink:href="note-0075-03"/> nales. </s> <s xml:id="echoid-s1878" xml:space="preserve">eadem quoque ratione cum quadratum e g æquale ſit rectan <lb/>gulo g f o, tres aliæ lineæ f g, e g, f o, deinceps proportionales <lb/>erũt. </s> <s xml:id="echoid-s1879" xml:space="preserve">& </s> <s xml:id="echoid-s1880" xml:space="preserve">ut b d ad, a d, ita f g ad e g. </s> <s xml:id="echoid-s1881" xml:space="preserve">quare ut a d ad b n, ita e g <lb/>ad f o. </s> <s xml:id="echoid-s1882" xml:space="preserve">ex æquali igitur, ut d b ad b n, ita g f ad f o: </s> <s xml:id="echoid-s1883" xml:space="preserve">& </s> <s xml:id="echoid-s1884" xml:space="preserve">permu-<lb/>tando ut d b ad g f, ita b n ad f o. </s> <s xml:id="echoid-s1885" xml:space="preserve">ut autem d b ad g f, ita b k <lb/>ad f l. </s> <s xml:id="echoid-s1886" xml:space="preserve">ergo b k ad f l, ut b n ad f o: </s> <s xml:id="echoid-s1887" xml:space="preserve">& </s> <s xml:id="echoid-s1888" xml:space="preserve">permutando, ut b k ad <lb/>bn, ita f l ad f o. </s> <s xml:id="echoid-s1889" xml:space="preserve">Rurſus quoniá quadratú h K æquale eſt rectan <lb/> <anchor type="note" xlink:label="note-0075-04a" xlink:href="note-0075-04"/> gulo k b n: </s> <s xml:id="echoid-s1890" xml:space="preserve">& </s> <s xml:id="echoid-s1891" xml:space="preserve">quadratum m l rectangulo l f o æquale: </s> <s xml:id="echoid-s1892" xml:space="preserve">erunt tres <lb/>lineæ b k, k h, b n proportionales: </s> <s xml:id="echoid-s1893" xml:space="preserve">itémq; </s> <s xml:id="echoid-s1894" xml:space="preserve">proportionales inter ſe <lb/>f l, l m, f o. </s> <s xml:id="echoid-s1895" xml:space="preserve">quare ut linea b K ad lineam b n, ita quadratum b K <lb/> <anchor type="note" xlink:label="note-0075-05a" xlink:href="note-0075-05"/> ad quadratum h k: </s> <s xml:id="echoid-s1896" xml:space="preserve">& </s> <s xml:id="echoid-s1897" xml:space="preserve">ut linea f l ad ipſam f o, ita quadratú f l <lb/>ad quadratum l m. </s> <s xml:id="echoid-s1898" xml:space="preserve">Itaque quoniam, ut b K ad b n, ita eſt f l ad <lb/>f o; </s> <s xml:id="echoid-s1899" xml:space="preserve">erit ut quadratum b K ad quadratum k h, ita quadratum f l <lb/>ad l m quadratum. </s> <s xml:id="echoid-s1900" xml:space="preserve">ergo ut linea b k, ad lineam K h, ita linea f l <lb/> <anchor type="note" xlink:label="note-0075-06a" xlink:href="note-0075-06"/> ad ipsã lm: </s> <s xml:id="echoid-s1901" xml:space="preserve">& </s> <s xml:id="echoid-s1902" xml:space="preserve">permutãdo ut b k ad f l, ita k h ad lm. </s> <s xml:id="echoid-s1903" xml:space="preserve">ſed b k ad <lb/>f l erat ut k c ad c l. </s> <s xml:id="echoid-s1904" xml:space="preserve">ergo k h ad lm, ut K c ad c l. </s> <s xml:id="echoid-s1905" xml:space="preserve">quare ex eo <lb/>dem lemmate patet lineam h c, & </s> <s xml:id="echoid-s1906" xml:space="preserve">per m punctum tranſire. </s> <s xml:id="echoid-s1907" xml:space="preserve">ut igi-<lb/>tur K c ad c l: </s> <s xml:id="echoid-s1908" xml:space="preserve">hoc eſt ut a c ad c e, ita h c ad c m; </s> <s xml:id="echoid-s1909" xml:space="preserve">hoc eſt ad eam <lb/>ipſius partem, quæ inter c, & </s> <s xml:id="echoid-s1910" xml:space="preserve">e g c ſectionem interyeitur. </s> <s xml:id="echoid-s1911" xml:space="preserve">ſimiliter <lb/>demonſtrabimus idem contingere in alijs lineis, quæ à puncto c ad <lb/>a b c ſectionem perducuntur. </s> <s xml:id="echoid-s1912" xml:space="preserve">At uero b c ad e f eandern propor-<lb/>tionem habere, liquido apparet; </s> <s xml:id="echoid-s1913" xml:space="preserve">nam b c ad c f, eſt ut d c ad c g; <lb/></s> <s xml:id="echoid-s1914" xml:space="preserve">uidelicet ut earum duplæ, a c ad c e.</s> <s xml:id="echoid-s1915" xml:space="preserve"/> </p> <div xml:id="echoid-div137" type="float" level="2" n="1"> <figure xlink:label="fig-0074-01" xlink:href="fig-0074-01a"> <image file="0074-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0074-01"/> </figure> <note position="left" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">15. quin-<lb/>ti.</note> <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">15. quin-<lb/>ti.</note> <note position="right" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">17. ſexti.</note> <note position="right" xlink:label="note-0075-04" xlink:href="note-0075-04a" xml:space="preserve">11. primi <lb/>conicorũ</note> <note position="right" xlink:label="note-0075-05" xlink:href="note-0075-05a" xml:space="preserve">cor. 20. ſe <lb/>xti.</note> <note position="right" xlink:label="note-0075-06" xlink:href="note-0075-06a" xml:space="preserve">22. ſexti</note> </div> <pb file="0076" n="76" rhead="ARCHIMEDIS"/> <p style="it"> <s xml:id="echoid-s1916" xml:space="preserve">Ex quibus perſpicuum eſt lineas omnes ſic ductas ab <lb/>ipſis ſectionibus in eandem proportionem ſecari. </s> <s xml:id="echoid-s1917" xml:space="preserve">eſt enim <lb/>diuidendo, conuertendoque cm ad mb, & </s> <s xml:id="echoid-s1918" xml:space="preserve">cf ad fb, ut <lb/>ce ad ea.</s> <s xml:id="echoid-s1919" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div139" type="section" level="1" n="44"> <head xml:id="echoid-head49" xml:space="preserve">LEMMA III.</head> <p style="it"> <s xml:id="echoid-s1920" xml:space="preserve">Sed & </s> <s xml:id="echoid-s1921" xml:space="preserve">illud constare potest; </s> <s xml:id="echoid-s1922" xml:space="preserve">lineas, quæ in portioni-<lb/>bus eiuſmodi ſimilibus ita ducuntur, ut cú baſibus æqua-<lb/>les angulos contineant, ab ipſis ſimiles quoque portiones <lb/>abſcindere: </s> <s xml:id="echoid-s1923" xml:space="preserve">hoc eſt, ut in propoſita figura, portiones h b c, <lb/>m f c, quas lineæ c h, c m abſcindunt, etiam inter ſe <lb/>ſimiles eſſe.</s> <s xml:id="echoid-s1924" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1925" xml:space="preserve"><emph style="sc">D_ividantvr_</emph> enim ch, cm bifariam in punctis p q: </s> <s xml:id="echoid-s1926" xml:space="preserve">& </s> <s xml:id="echoid-s1927" xml:space="preserve">per <lb/>ipſa ducantur lineæ r p s, t q u diametris æquidiſtantes. </s> <s xml:id="echoid-s1928" xml:space="preserve">erit portio-<lb/>nis b s c diameter p s, & </s> <s xml:id="echoid-s1929" xml:space="preserve">portionis m u c diameter q u. </s> <s xml:id="echoid-s1930" xml:space="preserve">Itaque fiat <lb/>ut quadratum c r ad quadratum c p, ita linea b n ad aliam lineam, <lb/>quæ ſit s x: </s> <s xml:id="echoid-s1931" xml:space="preserve">& </s> <s xml:id="echoid-s1932" xml:space="preserve">ut quadratum c t ad quadratum c q, ita fiat f o ad <lb/>u y. </s> <s xml:id="echoid-s1933" xml:space="preserve">iam exijs <lb/> <anchor type="figure" xlink:label="fig-0076-01a" xlink:href="fig-0076-01"/> quæ demóſtra <lb/>uimus in com-<lb/>mentarijs in <lb/>quartam pro-<lb/>poſitioné. </s> <s xml:id="echoid-s1934" xml:space="preserve">Ar-<lb/>chrmedis de co <lb/>noidibus, & </s> <s xml:id="echoid-s1935" xml:space="preserve"><lb/>ſphæroidibus, <lb/>patet quadra-<lb/>tum c p æqua-<lb/>le eſſe rectan-<lb/>gulo p s x:</s> <s xml:id="echoid-s1936" xml:space="preserve"> <pb o="25" file="0077" n="77" rhead="DE IIS QVAE VEH. IN AQVA."/> itêmq; </s> <s xml:id="echoid-s1937" xml:space="preserve">quadratum c q æquale rectangulo q u y, hoc eſt ſectionum <lb/>h s c, m u c lineas s x, u y, eas eſſe, iuxta quas poſſunt, quæ à ſectio-<lb/>ne ad diametrum ducuntur. </s> <s xml:id="echoid-s1938" xml:space="preserve">ſed cú triangula c p r, c q t ſimilia ſint, <lb/>habebit c r ad c p eandem proportionem, quam c t ad c q: </s> <s xml:id="echoid-s1939" xml:space="preserve">& </s> <s xml:id="echoid-s1940" xml:space="preserve">id-<lb/> <anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/> circo quadratum c r ad quadratum c p eandem habebit, quam <lb/>quadratum c t ad quadratum c q. </s> <s xml:id="echoid-s1941" xml:space="preserve">ergo & </s> <s xml:id="echoid-s1942" xml:space="preserve">linea b n, ad lineam <lb/>ſ x ita erit, ut linea fo ad ipſam u y. </s> <s xml:id="echoid-s1943" xml:space="preserve">erat autem b c ad c m, ut a c <lb/>ad c e. </s> <s xml:id="echoid-s1944" xml:space="preserve">quare & </s> <s xml:id="echoid-s1945" xml:space="preserve">earum dimidiæ c p ad c q, ut a d ad e g: </s> <s xml:id="echoid-s1946" xml:space="preserve">& </s> <s xml:id="echoid-s1947" xml:space="preserve"><lb/>permutando c p ad a d, ut c q ad e g. </s> <s xml:id="echoid-s1948" xml:space="preserve">Sed oſtenſum est a d ad b n <lb/>ita eſſe, ut e g ad f o: </s> <s xml:id="echoid-s1949" xml:space="preserve">& </s> <s xml:id="echoid-s1950" xml:space="preserve">b n ad s x, ut f o ad u y. </s> <s xml:id="echoid-s1951" xml:space="preserve">ergo ex <lb/>æquali c p ad ſ x erit, ut c q ad u y. </s> <s xml:id="echoid-s1952" xml:space="preserve">Quòd cum quadratú c p æqua <lb/>le ſit rectangulo p s x & </s> <s xml:id="echoid-s1953" xml:space="preserve">quadratum c q rectangulo q u y, erunt <lb/>tres lineæ ſ p, p c, ſ x proportionales; </s> <s xml:id="echoid-s1954" xml:space="preserve">itemq; </s> <s xml:id="echoid-s1955" xml:space="preserve">proportionales ip-<lb/>ſæ u q, q c, u y. </s> <s xml:id="echoid-s1956" xml:space="preserve">quare & </s> <s xml:id="echoid-s1957" xml:space="preserve">ſ p ad p c, ut u q ad q c: </s> <s xml:id="echoid-s1958" xml:space="preserve">& </s> <s xml:id="echoid-s1959" xml:space="preserve">ut p c ad <lb/>c h, ita q c ad c m. </s> <s xml:id="echoid-s1960" xml:space="preserve">ex æquali igitur ut portionis h ſ c diameter ſ p <lb/>ad eius baſim c h, ita portionis m u s diameter u q ad baſim c m. <lb/></s> <s xml:id="echoid-s1961" xml:space="preserve">& </s> <s xml:id="echoid-s1962" xml:space="preserve">anguli, quos diametri cum baſibus continent, ſunt æquales, quòd <lb/>lineæ ſ p, u q ſibi ipſis æquidiſtent, ergo & </s> <s xml:id="echoid-s1963" xml:space="preserve">portiones h ſ c, m u c <lb/>inter ſe ſimiles erunt. </s> <s xml:id="echoid-s1964" xml:space="preserve">id quod demonstrandum proponebatur.</s> <s xml:id="echoid-s1965" xml:space="preserve"/> </p> <div xml:id="echoid-div139" type="float" level="2" n="1"> <figure xlink:label="fig-0076-01" xlink:href="fig-0076-01a"> <image file="0076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0076-01"/> </figure> <note position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">22. fexti</note> </div> </div> <div xml:id="echoid-div141" type="section" level="1" n="45"> <head xml:id="echoid-head50" xml:space="preserve">LEMMA IIII.</head> <p style="it"> <s xml:id="echoid-s1966" xml:space="preserve">Sint duæ lineæ a b, c d, quæ ſecentur in punctis e f, <lb/>ita ut quam proportionem habet a e ad e b, habeat c f <lb/>ad f d: </s> <s xml:id="echoid-s1967" xml:space="preserve">rurſus ſecentur in aliis duobus punctis g h; </s> <s xml:id="echoid-s1968" xml:space="preserve">& </s> <s xml:id="echoid-s1969" xml:space="preserve"><lb/>habeat c h ad h d eandem proportionem, quam a g ad <lb/>g b. </s> <s xml:id="echoid-s1970" xml:space="preserve">Dico c f ad f h ita eſſe, ut a e ad e g.</s> <s xml:id="echoid-s1971" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1972" xml:space="preserve"><emph style="sc">Q_voniam_</emph> enim ut a e ad e b, ita c f ad f d, erit componen <lb/>do ut a b ad e b, ita c d ad f d. </s> <s xml:id="echoid-s1973" xml:space="preserve">Rurſus cum ſit ut a g ad g b, ita <lb/>c h ad h d; </s> <s xml:id="echoid-s1974" xml:space="preserve">componendo, conuertendoq; </s> <s xml:id="echoid-s1975" xml:space="preserve">ut g b ad a b, ita erit h d <lb/>ad c d. </s> <s xml:id="echoid-s1976" xml:space="preserve">ergo ex æquali, conuertendoq; </s> <s xml:id="echoid-s1977" xml:space="preserve">ut e b ad g b, ita f d ad h d:</s> <s xml:id="echoid-s1978" xml:space="preserve"> <pb file="0078" n="78" rhead="ARCHIMEDIS"/> & </s> <s xml:id="echoid-s1979" xml:space="preserve">per conuer-<lb/> <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> ſionem rationis <lb/>ut e b ad e g, <lb/>ita f d ad f h. <lb/></s> <s xml:id="echoid-s1980" xml:space="preserve">eſt autem ut a e <lb/>ad e b, ita c f <lb/>ad f d. </s> <s xml:id="echoid-s1981" xml:space="preserve">ex æqua <lb/>li igitur ut a e <lb/>ad e g, ita c f <lb/>ad f h.</s> <s xml:id="echoid-s1982" xml:space="preserve"/> </p> <div xml:id="echoid-div141" type="float" level="2" n="1"> <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a"> <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0078-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s1983" xml:space="preserve"><emph style="sc">A_liter_</emph>. </s> <s xml:id="echoid-s1984" xml:space="preserve">Aptentur lineæ a b, c d inter ſe ſe, ita ut ad partes <lb/>a c angulum faciant; </s> <s xml:id="echoid-s1985" xml:space="preserve">& </s> <s xml:id="echoid-s1986" xml:space="preserve">ſint a c in uno atque eodem puncto: </s> <s xml:id="echoid-s1987" xml:space="preserve">deinde <lb/>iungantur d b, h g, fe. </s> <s xml:id="echoid-s1988" xml:space="preserve">cum igitur ſit ut a e ad e b, ita c f, hoc eſt <lb/>a f ad f d; </s> <s xml:id="echoid-s1989" xml:space="preserve">æquidiſtabit fe ipſi d b: </s> <s xml:id="echoid-s1990" xml:space="preserve">& </s> <s xml:id="echoid-s1991" xml:space="preserve">ſimiliter h g eidem d b <lb/> <anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/> æquidiſtabit: </s> <s xml:id="echoid-s1992" xml:space="preserve">quoniam a h ad h d eſt, ut a g ad g b. </s> <s xml:id="echoid-s1993" xml:space="preserve">ergo f c, h g <lb/> <anchor type="note" xlink:label="note-0078-02a" xlink:href="note-0078-02"/> inter ſe ſe æquidiſtant: </s> <s xml:id="echoid-s1994" xml:space="preserve">& </s> <s xml:id="echoid-s1995" xml:space="preserve">idcirco ut a e ad e g, ita a f; </s> <s xml:id="echoid-s1996" xml:space="preserve">hoc eſt c f ad <lb/>fh. </s> <s xml:id="echoid-s1997" xml:space="preserve">quod demonſtrare oportebat.</s> <s xml:id="echoid-s1998" xml:space="preserve"/> </p> <div xml:id="echoid-div142" type="float" level="2" n="2"> <note position="left" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">2. ſexti:</note> <note position="left" xlink:label="note-0078-02" xlink:href="note-0078-02a" xml:space="preserve">30. primi</note> </div> </div> <div xml:id="echoid-div144" type="section" level="1" n="46"> <head xml:id="echoid-head51" xml:space="preserve">LEMMA V.</head> <p style="it"> <s xml:id="echoid-s1999" xml:space="preserve">Sint rurſus duæ portiones ſimiles, contentæ rectis li-<lb/>neis, & </s> <s xml:id="echoid-s2000" xml:space="preserve">rectangulorum conorum ſectionibus, ut in ſupe-<lb/>riori figura a b c, cuius diameter b d: </s> <s xml:id="echoid-s2001" xml:space="preserve">& </s> <s xml:id="echoid-s2002" xml:space="preserve">e f c, cuius <lb/>diameter f g: </s> <s xml:id="echoid-s2003" xml:space="preserve">ducaturque à puncto e linea e h, diame-<lb/>tris b d, f g æquidiſtans, quæ ſectionem a b c in _k_ ſe-<lb/>cet: </s> <s xml:id="echoid-s2004" xml:space="preserve">& </s> <s xml:id="echoid-s2005" xml:space="preserve">à puncto c ducatur c h contingens ſectionem <lb/>a b c in c conueniensque cumlinea e h in h, quæ ſectio <lb/>nem quoque e f c in eodem c puncto continget, ut demon <lb/>strabitur. </s> <s xml:id="echoid-s2006" xml:space="preserve">Dico lineam ductam ab ipſa c h uſque ad ſe-<lb/>ctionem e f c, ita ut lineæ e h æquidistet, in eandem pro <lb/>portionem diuidi à ſectione a b c; </s> <s xml:id="echoid-s2007" xml:space="preserve">in quam linea c a à <pb o="34" file="0079" n="79" rhead="DE IIS QVAE VEH. IN AQVA."/> ſectione e f c diuiditur: </s> <s xml:id="echoid-s2008" xml:space="preserve">pars uero lineæ c a, quæ eſt in-<lb/>ter duas ſectiones proportione reſpondebit parti lineæ <lb/>ductæ, quæ itidem inter eaſdem ſectiones interiicitur; </s> <s xml:id="echoid-s2009" xml:space="preserve">hoc <lb/>est ut in propoſita figura, ſi producatur d b uſque ad c h <lb/>in l, ut ſectioni e f c in puncto m occurrat; </s> <s xml:id="echoid-s2010" xml:space="preserve">lineam l <lb/>b ad b m eãdem proportionem habere, quàm c e ad e a.</s> <s xml:id="echoid-s2011" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2012" xml:space="preserve">Produc. </s> <s xml:id="echoid-s2013" xml:space="preserve">itur enim <lb/> <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/> q f ad eandem lineá <lb/>c h in n, ſecás a b c <lb/>ſectionem in o: </s> <s xml:id="echoid-s2014" xml:space="preserve">& </s> <s xml:id="echoid-s2015" xml:space="preserve"><lb/>iuncta b c, quæ tran <lb/>ſibit per f, ut oſten-<lb/>ſum eſt, erunt trian-<lb/>gula c g f, c d b ſi-<lb/>milia: </s> <s xml:id="echoid-s2016" xml:space="preserve">itémq; </s> <s xml:id="echoid-s2017" xml:space="preserve">ſimi-<lb/>lia íter ſe, c f n, c b l. <lb/></s> <s xml:id="echoid-s2018" xml:space="preserve">quare ut g f ad d b, <lb/> <anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/> ita erit c f ad c b: <lb/></s> <s xml:id="echoid-s2019" xml:space="preserve">& </s> <s xml:id="echoid-s2020" xml:space="preserve">ut c f ad c b, ita <lb/>f n ad b l. </s> <s xml:id="echoid-s2021" xml:space="preserve">ergo g f <lb/> <anchor type="note" xlink:label="note-0079-02a" xlink:href="note-0079-02"/> ad d b, ut f n ad b l: <lb/></s> <s xml:id="echoid-s2022" xml:space="preserve">& </s> <s xml:id="echoid-s2023" xml:space="preserve">permutando g f <lb/>ad f n, ut d b ad b l. </s> <s xml:id="echoid-s2024" xml:space="preserve"><lb/>eſt autem d b æqua <lb/>lis ipſi b l ex trigeſi <lb/>maquinta primi li-<lb/> <anchor type="note" xlink:label="note-0079-03a" xlink:href="note-0079-03"/> bri conicorum. </s> <s xml:id="echoid-s2025" xml:space="preserve">ergo <lb/>& </s> <s xml:id="echoid-s2026" xml:space="preserve">g f ipſi p i æqua <lb/>lis erit: </s> <s xml:id="echoid-s2027" xml:space="preserve">& </s> <s xml:id="echoid-s2028" xml:space="preserve">ex trige <lb/>ſimatertia eiuſdem <lb/>linea c h ſectionem <lb/>e f c in eodem pun- <pb file="0080" n="80" rhead="ARCHIMEDIS"/> cto continget. </s> <s xml:id="echoid-s2029" xml:space="preserve">Itaque iuncta cm producatur ad ſectionem a b c in p: <lb/></s> <s xml:id="echoid-s2030" xml:space="preserve">& </s> <s xml:id="echoid-s2031" xml:space="preserve">à p ad a c ducatur p q, quæ ipſi b d æquidiſtet. </s> <s xml:id="echoid-s2032" xml:space="preserve">quoniam igi-<lb/>tur linea c h contingit ſectionem e f c in c puncto; </s> <s xml:id="echoid-s2033" xml:space="preserve">habebit l m <lb/>ad m d proportionem eandem, quam c d ad d e, ex quinta propoſi-<lb/>tione Archimedis in libro de quadratura parabolæ. </s> <s xml:id="echoid-s2034" xml:space="preserve">& </s> <s xml:id="echoid-s2035" xml:space="preserve">propter triá <lb/>gulorum c m d, c p q <lb/> <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> ſimilitudinem, ut c m <lb/>ad c d, ita erit c p ad <lb/>c q: </s> <s xml:id="echoid-s2036" xml:space="preserve">permutandôq; <lb/></s> <s xml:id="echoid-s2037" xml:space="preserve">ut c m ad c p, ita c d <lb/>ad c q. </s> <s xml:id="echoid-s2038" xml:space="preserve">ut autem c m <lb/>ad c p, ſic c e ad c a: </s> <s xml:id="echoid-s2039" xml:space="preserve"><lb/>quod proxime demó-<lb/>ſtrauimus. </s> <s xml:id="echoid-s2040" xml:space="preserve">quare ut <lb/>c e ad c a, ſit c d ad <lb/>c q: </s> <s xml:id="echoid-s2041" xml:space="preserve">hoc eſt ut totum <lb/>ad totum, ſic pars ad <lb/>partem, reliquum igi <lb/>tur d e ad reliquum <lb/>q a eſt ut c e ad c a; </s> <s xml:id="echoid-s2042" xml:space="preserve"><lb/>uidelicet ut c d ad <lb/>c q: </s> <s xml:id="echoid-s2043" xml:space="preserve">& </s> <s xml:id="echoid-s2044" xml:space="preserve">permutando <lb/>c d ad d e, ut c q ad <lb/>q a. </s> <s xml:id="echoid-s2045" xml:space="preserve">êſtq; </s> <s xml:id="echoid-s2046" xml:space="preserve">l m ad m <lb/>d, ut c d ad d e. </s> <s xml:id="echoid-s2047" xml:space="preserve">ergo <lb/>l m ad m d, ut c q ad <lb/>q a. </s> <s xml:id="echoid-s2048" xml:space="preserve">ſed l b ad b d <lb/>ex quinta Archime-<lb/>dis, quam diximus; </s> <s xml:id="echoid-s2049" xml:space="preserve"><lb/>eſt ut c d ad d a. </s> <s xml:id="echoid-s2050" xml:space="preserve">con <lb/>ſtat igitur ex antece-<lb/>denti lemmate c d ad d q ita eſſe, ut l b ad b m. </s> <s xml:id="echoid-s2051" xml:space="preserve">ut autem c d ad d q, <lb/> <anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/> ita c m ad m p. </s> <s xml:id="echoid-s2052" xml:space="preserve">ergo l b ad b m, ut c m ad m p. </s> <s xml:id="echoid-s2053" xml:space="preserve">Quòd cum demon <lb/>ſtratum fuerit, c m ad m p, ut c e ad e a: </s> <s xml:id="echoid-s2054" xml:space="preserve">habebit l b ad b m eandé <pb o="35" file="0081" n="81" rhead="DE IIS QVAE VEH. IN AQVA."/> proportionem, quam c e ad e a. </s> <s xml:id="echoid-s2055" xml:space="preserve">ſimiliter demonſtrabitur eandem <lb/>babere n o ad o f: </s> <s xml:id="echoid-s2056" xml:space="preserve">& </s> <s xml:id="echoid-s2057" xml:space="preserve">reliquas eiuſmodi, at uero b K ad K e eam <lb/>habere proportionem, quam habet c e ad e a, ex eadem quinta. </s> <s xml:id="echoid-s2058" xml:space="preserve">Ar-<lb/>chimedis perſpicue apparet. </s> <s xml:id="echoid-s2059" xml:space="preserve">at que illud eſt, quod demonſtr andum <lb/>propoſuimus.</s> <s xml:id="echoid-s2060" xml:space="preserve"/> </p> <div xml:id="echoid-div144" type="float" level="2" n="1"> <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a"> <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0079-01"/> </figure> <note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">4. ſexti:</note> <note position="right" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">11. quinti</note> <note position="right" xlink:label="note-0079-03" xlink:href="note-0079-03a" xml:space="preserve">14. quinti</note> <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a"> <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0080-01"/> </figure> <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">2. ſexti.</note> </div> </div> <div xml:id="echoid-div146" type="section" level="1" n="47"> <head xml:id="echoid-head52" xml:space="preserve">LEMMA VI.</head> <p style="it"> <s xml:id="echoid-s2061" xml:space="preserve">Itaque maneant eadem, quæ ſupra: </s> <s xml:id="echoid-s2062" xml:space="preserve">& </s> <s xml:id="echoid-s2063" xml:space="preserve">itidem deſcri-<lb/>batur alia portio ſimilis contenta recta linea & </s> <s xml:id="echoid-s2064" xml:space="preserve">rectan-<lb/>guli coni ſectione d r c; </s> <s xml:id="echoid-s2065" xml:space="preserve">cuius diameter r s, ut ſecet li-<lb/>neam f g in t: </s> <s xml:id="echoid-s2066" xml:space="preserve">producaturque s r ad lineam c h in u; <lb/></s> <s xml:id="echoid-s2067" xml:space="preserve">cuiſectio a b c occurrat in x, & </s> <s xml:id="echoid-s2068" xml:space="preserve">e f c in y. </s> <s xml:id="echoid-s2069" xml:space="preserve">Dico b m <lb/>ad m d proportionem habere compoſitam ex propor-<lb/>tione, quam babet e a ad a c; </s> <s xml:id="echoid-s2070" xml:space="preserve">& </s> <s xml:id="echoid-s2071" xml:space="preserve">ex ea, quam c d ba-<lb/>bet ad de.</s> <s xml:id="echoid-s2072" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2073" xml:space="preserve"><emph style="sc">S_imiliter_</emph> enim ut ſupra, demonſtrabimus lineam c h con-<lb/>tingere ſectioné d r c in c puncto: </s> <s xml:id="echoid-s2074" xml:space="preserve">& </s> <s xml:id="echoid-s2075" xml:space="preserve">l m ad m d, itêmq; </s> <s xml:id="echoid-s2076" xml:space="preserve">n f ad f t; <lb/></s> <s xml:id="echoid-s2077" xml:space="preserve">& </s> <s xml:id="echoid-s2078" xml:space="preserve">u y ad y r ita eſſe, ut c d ad d e. </s> <s xml:id="echoid-s2079" xml:space="preserve">Quoniam igitur lb ad b m eſt, <lb/>ut c e ad e a; </s> <s xml:id="echoid-s2080" xml:space="preserve">erit componendo, conuertendôq; </s> <s xml:id="echoid-s2081" xml:space="preserve">bm ad lm, ut e a ad <lb/>a c: </s> <s xml:id="echoid-s2082" xml:space="preserve">& </s> <s xml:id="echoid-s2083" xml:space="preserve">ut lm ad m d, ita c d ad d e. </s> <s xml:id="echoid-s2084" xml:space="preserve">proportio autem b m ad m d <lb/>compoſita eſt ex proportione, quam habet b m ad l m, & </s> <s xml:id="echoid-s2085" xml:space="preserve">ex propor <lb/>tione, quam l m habet ad m d. </s> <s xml:id="echoid-s2086" xml:space="preserve">ergo proportio b m ad m d etiam com <lb/>poſita erit ex proportione, quam habet e a, ad a c; </s> <s xml:id="echoid-s2087" xml:space="preserve">& </s> <s xml:id="echoid-s2088" xml:space="preserve">ex ea, quam <lb/>c d habet ad d e. </s> <s xml:id="echoid-s2089" xml:space="preserve">Eadem ratione demonſtrabitur o f ad f t; </s> <s xml:id="echoid-s2090" xml:space="preserve">itêmq; </s> <s xml:id="echoid-s2091" xml:space="preserve"><lb/>x y ad y r proportionem habere ex eiſdem proportionibus compo-<lb/>ſitam: </s> <s xml:id="echoid-s2092" xml:space="preserve">& </s> <s xml:id="echoid-s2093" xml:space="preserve">ita in aijs. </s> <s xml:id="echoid-s2094" xml:space="preserve">quod demonſtrare oportebat.</s> <s xml:id="echoid-s2095" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2096" xml:space="preserve">Ex quibus apparet lineas ſic ductas, quæ inter ſectio <lb/>nes a b c, d r c interiiciuntur à ſectione e f c in eandem <lb/>proportionem diuidi.</s> <s xml:id="echoid-s2097" xml:space="preserve"/> </p> <pb file="0082" n="82" rhead="ARCHIMEDIS"/> <p style="it"> <s xml:id="echoid-s2098" xml:space="preserve">_Etenim c b ad b d eſt ut ſex ad quindecim.</s> <s xml:id="echoid-s2099" xml:space="preserve">]_ Poſuimus <lb/> <anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/> enim b K duplam eſſe ipſius K d. </s> <s xml:id="echoid-s2100" xml:space="preserve">quare componendo b d ad k d erit, <lb/>ut tria ad unum; </s> <s xml:id="echoid-s2101" xml:space="preserve">hoc eſt ut quindecim ad quinque. </s> <s xml:id="echoid-s2102" xml:space="preserve">ſed b d ad K c <lb/>erat ut quídecim <lb/> <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> ad quatuor. </s> <s xml:id="echoid-s2103" xml:space="preserve">ergo <lb/>b d ad d c, ut quin <lb/>decim ad nouem: <lb/></s> <s xml:id="echoid-s2104" xml:space="preserve">& </s> <s xml:id="echoid-s2105" xml:space="preserve">per conuerſio <lb/>nem rationis, con <lb/>uertendôq; </s> <s xml:id="echoid-s2106" xml:space="preserve">c b ad <lb/>b d, ut ſex ad quí <lb/>decim.</s> <s xml:id="echoid-s2107" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="1"> <note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">N</note> <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0082-01"/> </figure> </div> <p> <s xml:id="echoid-s2108" xml:space="preserve">Etut c b ad <lb/> <anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/> b d, ita e b ad <lb/>b a, & </s> <s xml:id="echoid-s2109" xml:space="preserve">d z ad <lb/>d a.</s> <s xml:id="echoid-s2110" xml:space="preserve">] _Nam cum_ <lb/>_triangula c b e,_ <lb/>_d b a ſint ſimilia,_ <lb/>_erit ut c b ad b e,_ <lb/>_ita d b, ad b a & </s> <s xml:id="echoid-s2111" xml:space="preserve">permutando, ut c b ad b d; </s> <s xml:id="echoid-s2112" xml:space="preserve">ita e b ad b a. </s> <s xml:id="echoid-s2113" xml:space="preserve">Rurſus_ <lb/>_ut b c ad c e, ita b d ad d a: </s> <s xml:id="echoid-s2114" xml:space="preserve">permutandôq; </s> <s xml:id="echoid-s2115" xml:space="preserve">ut c b ad b d, ita c e, hoc_ <lb/>_eſt d z ei æqualis ad d a._</s> <s xml:id="echoid-s2116" xml:space="preserve"/> </p> <div xml:id="echoid-div147" type="float" level="2" n="2"> <note position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">O</note> </div> <p> <s xml:id="echoid-s2117" xml:space="preserve">Harum autem d z d a duplæ ſuntipſæ l i, la.</s> <s xml:id="echoid-s2118" xml:space="preserve">] _Lineam_ <lb/> <anchor type="note" xlink:label="note-0082-03a" xlink:href="note-0082-03"/> _quidem l a duplam eſſe ipſius d a, cum b d ſit portionis diameter,_ <lb/>_manifeſte conſtat. </s> <s xml:id="echoid-s2119" xml:space="preserve">At uero l i ipſius d z dupla hoc pacto demon-_ <lb/>_ſtrabitur. </s> <s xml:id="echoid-s2120" xml:space="preserve">Quoniam enim z d ad d a eſt, ut duo ad quinque; </s> <s xml:id="echoid-s2121" xml:space="preserve">erit có_ <lb/>_uertendo, diuidendôq; </s> <s xml:id="echoid-s2122" xml:space="preserve">a z, hoc eſt i z ad z d, ut tria ad duo: </s> <s xml:id="echoid-s2123" xml:space="preserve">&_</s> <s xml:id="echoid-s2124" xml:space="preserve"> <lb/>_rurſus diuidendo i d ad d z, ut <gap/>m ad duo. </s> <s xml:id="echoid-s2125" xml:space="preserve">erat autem z d ad_ <lb/>_d a, hoc eſt ad d l, ut duo ad quinque. </s> <s xml:id="echoid-s2126" xml:space="preserve">ergo ex æquali, conuertendóq;_ <lb/></s> <s xml:id="echoid-s2127" xml:space="preserve">_l d ad d i, ut quinque ad unum: </s> <s xml:id="echoid-s2128" xml:space="preserve">& </s> <s xml:id="echoid-s2129" xml:space="preserve">per conuerſionem rationis d l ad_ <lb/>_li, ut quinque ad quatuor. </s> <s xml:id="echoid-s2130" xml:space="preserve">ſed d z ad d l erat, ut duo ad quinque._ </s> <s xml:id="echoid-s2131" xml:space="preserve"><lb/>_ergo rurſus ex æquali d z ad l i, ut duo ad quatuor. </s> <s xml:id="echoid-s2132" xml:space="preserve">dupla eſt igitur_ <lb/>_l i ipſius d z. </s> <s xml:id="echoid-s2133" xml:space="preserve">quod demonſtrandum fuerat._</s> <s xml:id="echoid-s2134" xml:space="preserve"/> </p> <div xml:id="echoid-div148" type="float" level="2" n="3"> <note position="left" xlink:label="note-0082-03" xlink:href="note-0082-03a" xml:space="preserve">P</note> </div> <p> <s xml:id="echoid-s2135" xml:space="preserve">Et a d ad d i eam proportionem habet, quã quinque <lb/> <anchor type="note" xlink:label="note-0082-04a" xlink:href="note-0082-04"/> <pb o="36" file="0083" n="83" rhead="DE IIS QVAE VEH. IN AQVA."/> ad unum.</s> <s xml:id="echoid-s2136" xml:space="preserve">] _Hoc nos proxime demonſtrauimus._</s> <s xml:id="echoid-s2137" xml:space="preserve"/> </p> <div xml:id="echoid-div149" type="float" level="2" n="4"> <note position="left" xlink:label="note-0082-04" xlink:href="note-0082-04a" xml:space="preserve">Q</note> </div> <p> <s xml:id="echoid-s2138" xml:space="preserve">Demonſtratum eſt enim ſuperius portionem cuius axis <lb/> <anchor type="note" xlink:label="note-0083-01a" xlink:href="note-0083-01"/> eſt maior, quàm ſeſquialter eius, quæ uſque ad axem, ſiad <lb/>humidum in grauitate non minorem proportionem ha-<lb/>beat &</s> <s xml:id="echoid-s2139" xml:space="preserve">c.</s> <s xml:id="echoid-s2140" xml:space="preserve">] _Illud uero demonſtrauit in quarta propoſitione buius_ <lb/>_libri_.</s> <s xml:id="echoid-s2141" xml:space="preserve"/> </p> <div xml:id="echoid-div150" type="float" level="2" n="5"> <note position="right" xlink:label="note-0083-01" xlink:href="note-0083-01a" xml:space="preserve">R</note> </div> </div> <div xml:id="echoid-div152" type="section" level="1" n="48"> <head xml:id="echoid-head53" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s2142" xml:space="preserve">Si portio ad humidum in grauitate minorem <lb/> <anchor type="note" xlink:label="note-0083-02a" xlink:href="note-0083-02"/> quidem proportionem habeat, quàm quadra-<lb/>tum ſ b ad quadratum b d; </s> <s xml:id="echoid-s2143" xml:space="preserve">maiorem uero, <lb/>quàm quadratum x o ad quadratum b d; </s> <s xml:id="echoid-s2144" xml:space="preserve">de-<lb/>miſſa in humidum, adeo inclinata, ut baſis ip-<lb/>ſius non contingat humidum, inclinata conſi-<lb/>ſtet; </s> <s xml:id="echoid-s2145" xml:space="preserve">ita ut baſis ſuperficiem humidi nullo modo <lb/>contingat; </s> <s xml:id="echoid-s2146" xml:space="preserve">& </s> <s xml:id="echoid-s2147" xml:space="preserve">axis cum humidi ſuperficie angu-<lb/>lum faciat maiorem angulo χ.</s> <s xml:id="echoid-s2148" xml:space="preserve"/> </p> <div xml:id="echoid-div152" type="float" level="2" n="1"> <note position="right" xlink:label="note-0083-02" xlink:href="note-0083-02a" xml:space="preserve">A</note> </div> </div> <div xml:id="echoid-div154" type="section" level="1" n="49"> <head xml:id="echoid-head54" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s2149" xml:space="preserve">Si portio ad humidum in grauitate, eam ha-<lb/>beat proportionem, quam quadratum x o ad <lb/>quadratum b d; </s> <s xml:id="echoid-s2150" xml:space="preserve">demiſſa in humidum inclinata <lb/>adeo, ut baſis ipſius non contingat humidum; <lb/></s> <s xml:id="echoid-s2151" xml:space="preserve">conſiſtet, & </s> <s xml:id="echoid-s2152" xml:space="preserve">manebit ita, ut baſis in uno pun-<lb/>cto humidi ſuperficiem contingat: </s> <s xml:id="echoid-s2153" xml:space="preserve">& </s> <s xml:id="echoid-s2154" xml:space="preserve">axis cum <lb/>ſuperficie humidi angulũ faciat angulo χ æqualẽ. </s> <s xml:id="echoid-s2155" xml:space="preserve"><lb/>Quòd ſi portio ad humidum in grauitate cam <lb/>proportionem habeat, quam quadratum p f ad <pb file="0084" n="84" rhead="ARCHIMEDIS"/> quadratum b d; </s> <s xml:id="echoid-s2156" xml:space="preserve">in humidum demiſſa, & </s> <s xml:id="echoid-s2157" xml:space="preserve">poſi-<lb/>ta inclinata adeo, ut baſis ipſius non contingat <lb/>humidum; </s> <s xml:id="echoid-s2158" xml:space="preserve">conſiſtet inclinata, ita ut baſis in uno <lb/>puncto humidi ſuperficiem contingat: </s> <s xml:id="echoid-s2159" xml:space="preserve">& </s> <s xml:id="echoid-s2160" xml:space="preserve">axis cũ <lb/>ea faciat angulum angulo φ æqualem.</s> <s xml:id="echoid-s2161" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div155" type="section" level="1" n="50"> <head xml:id="echoid-head55" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s2162" xml:space="preserve">Si portio ad humidum in grauitate maiorem <lb/> <anchor type="note" xlink:label="note-0084-01a" xlink:href="note-0084-01"/> quidem proportionem habeat, quàm quadra-<lb/>tum f p ad quadratum b d; </s> <s xml:id="echoid-s2163" xml:space="preserve">minorem uero, <lb/>quàm quadratum x o ad b d quadratum; </s> <s xml:id="echoid-s2164" xml:space="preserve">in hu-<lb/>midum demiſſa, & </s> <s xml:id="echoid-s2165" xml:space="preserve">inclinata adeo, ut baſis ipſius <lb/>non contingat humidum conſiſtet, & </s> <s xml:id="echoid-s2166" xml:space="preserve">manebit <lb/>ita, ut baſis in humidum magis demergatur.</s> <s xml:id="echoid-s2167" xml:space="preserve"/> </p> <div xml:id="echoid-div155" type="float" level="2" n="1"> <note position="left" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">B</note> </div> </div> <div xml:id="echoid-div157" type="section" level="1" n="51"> <head xml:id="echoid-head56" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s2168" xml:space="preserve">Si portio ad humidum in grauitate proportio <lb/>nem habeat minorem, quàm quadratum f p ad <lb/>quadratum b d: </s> <s xml:id="echoid-s2169" xml:space="preserve">demiſſa in humidum, & </s> <s xml:id="echoid-s2170" xml:space="preserve">poſita <lb/>inclinata adeo ut baſis ipſius non contingat humi <lb/>dum: </s> <s xml:id="echoid-s2171" xml:space="preserve">conſiſtet inclinata, ita ut axis ipſius cum <lb/>humidi ſuperficie angulum faciat minorem an-<lb/>gulo φ: </s> <s xml:id="echoid-s2172" xml:space="preserve">& </s> <s xml:id="echoid-s2173" xml:space="preserve">baſis nullo modo ſuperficiem humi-<lb/>di contingat. </s> <s xml:id="echoid-s2174" xml:space="preserve">Hæc autem omnia deinceps de-<lb/>monſtrabuntur.</s> <s xml:id="echoid-s2175" xml:space="preserve"/> </p> <pb o="37" file="0085" n="85" rhead="DE IIS QVAE VEH. IN AQVA."/> </div> <div xml:id="echoid-div158" type="section" level="1" n="52"> <head xml:id="echoid-head57" xml:space="preserve">DEMONSTRATIO SECVNDAE PARTIS.</head> <p> <s xml:id="echoid-s2176" xml:space="preserve">ITAQVE primum habeat portio ad humidum in <lb/>grauitate proportionem quidem maiorem, quàm qua dra <lb/>tum x o ad quadratum b d; </s> <s xml:id="echoid-s2177" xml:space="preserve">minorem uero, quàm quadra <lb/>tum, quod fit ab exceſſu, quo axis eſt maior, quàm ſeſquial-<lb/>ter eius, quæ uſque ad axem, ad quadratum b d: </s> <s xml:id="echoid-s2178" xml:space="preserve">& </s> <s xml:id="echoid-s2179" xml:space="preserve">quam <lb/>proportionem habet portio ad humidum in grauitate, eã <lb/>habeat quadratum, quod fit à linea ψ ad quadratum b d: <lb/></s> <s xml:id="echoid-s2180" xml:space="preserve">erit ψ maior quidem, quàm x o, minor uero, quàm exceſ-<lb/> <anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/> ſus, quo axis eſt maior, quàm ſeſquialter eius, quæ uſque ad <lb/>axem. </s> <s xml:id="echoid-s2181" xml:space="preserve">aptetur quædam recta linea m n conicis ſectioni-<lb/>bus a m q l, <lb/> <anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/> a x d interiecta, <lb/>ac media, quæ li <lb/>neæ ψ ſit æqua-<lb/>lis; </s> <s xml:id="echoid-s2182" xml:space="preserve">ſecetq; </s> <s xml:id="echoid-s2183" xml:space="preserve">reli-<lb/>quã coni ſectio <lb/>nem in pun cto <lb/>h; </s> <s xml:id="echoid-s2184" xml:space="preserve">& </s> <s xml:id="echoid-s2185" xml:space="preserve">rectam li-<lb/>neam r g in u. <lb/></s> <s xml:id="echoid-s2186" xml:space="preserve">demõſtrabitur <lb/> <anchor type="note" xlink:label="note-0085-02a" xlink:href="note-0085-02"/> m h dupla ip-<lb/>ſius h n, ſicuti <lb/>demonſtratum <lb/>eſt o g ipſius g x <lb/>duplam eſſe. </s> <s xml:id="echoid-s2187" xml:space="preserve">à <lb/>puncto autẽ m <lb/>ducatur m y contingens ſectionem a m q l in m: </s> <s xml:id="echoid-s2188" xml:space="preserve">& </s> <s xml:id="echoid-s2189" xml:space="preserve">m c a d <lb/>b d perpendicularis. </s> <s xml:id="echoid-s2190" xml:space="preserve">poſtea ducta a n, & </s> <s xml:id="echoid-s2191" xml:space="preserve">producta ad q li <lb/>neæ a n, n q inter ſe æquales erunt. </s> <s xml:id="echoid-s2192" xml:space="preserve">quoniã enim in ſimi-<lb/> <anchor type="note" xlink:label="note-0085-03a" xlink:href="note-0085-03"/> libus portionibus a m q l, a x d ductæ ſunt à baſibus ad <lb/>portiones lineæ a q, a n, quæ æquales angulos continent <lb/>cum ipſis baſibus, eandem proportionem habebit q a ad <lb/>an, quam la ad a d. </s> <s xml:id="echoid-s2193" xml:space="preserve">æqualis eſt ergo a n ipſi n q; </s> <s xml:id="echoid-s2194" xml:space="preserve">& </s> <s xml:id="echoid-s2195" xml:space="preserve">a q <lb/> <anchor type="note" xlink:label="note-0085-04a" xlink:href="note-0085-04"/> <pb file="0086" n="86" rhead="ARCHIMEDIS"/> ipſi my æquidiſtans. </s> <s xml:id="echoid-s2196" xml:space="preserve">Demonſtrandum eſt portionem in <lb/> <anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/> humidum demiſſam, inclinatamq; </s> <s xml:id="echoid-s2197" xml:space="preserve">adeo, ut baſis ipſius nõ <lb/>contingat humidum, inclinatam conſiſtere ita, ut baſis ſu-<lb/>perficiem humidi nullo modo contingat: </s> <s xml:id="echoid-s2198" xml:space="preserve">& </s> <s xml:id="echoid-s2199" xml:space="preserve">axis cum ea fa <lb/>ciat angulum angulo χ maiorem. </s> <s xml:id="echoid-s2200" xml:space="preserve">Demittatur enim in hu-<lb/>midum, conſiſtatq; </s> <s xml:id="echoid-s2201" xml:space="preserve">ita, ut baſis ipſius in uno puncto cõtin <lb/>gat humidi ſuperficiem: </s> <s xml:id="echoid-s2202" xml:space="preserve">& </s> <s xml:id="echoid-s2203" xml:space="preserve">ſecta ipſa portione per axem, <lb/>plano ad humidi ſuperficiem recto; </s> <s xml:id="echoid-s2204" xml:space="preserve">ſuperficiei quidẽ por-<lb/>tionis ſectio ſit a p o l rectanguli coni ſectio: </s> <s xml:id="echoid-s2205" xml:space="preserve">ſuperficiei <lb/>humidi ſectio ſit a o: </s> <s xml:id="echoid-s2206" xml:space="preserve">axis autem portionis, & </s> <s xml:id="echoid-s2207" xml:space="preserve">ſectionis dia <lb/>meter b d: </s> <s xml:id="echoid-s2208" xml:space="preserve">& </s> <s xml:id="echoid-s2209" xml:space="preserve">ſecetur b d in punctis k r, ut dictum eſt: </s> <s xml:id="echoid-s2210" xml:space="preserve">du-<lb/> <anchor type="note" xlink:label="note-0086-02a" xlink:href="note-0086-02"/> catur etiam p g æquidiſtans ipſi a o, quæ ſectionem a p o l <lb/>contingat in p: </s> <s xml:id="echoid-s2211" xml:space="preserve">atque ab eo puncto ducatur p t æquidiſtãs <lb/>ipſi b d; </s> <s xml:id="echoid-s2212" xml:space="preserve">& </s> <s xml:id="echoid-s2213" xml:space="preserve">p s ad b d perpendicularis. </s> <s xml:id="echoid-s2214" xml:space="preserve">Itaque quoniam <lb/>portio ad humidum in grauitate eam proportionem ha-<lb/>bet, quam qua-<lb/> <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> dratũ, quod fit <lb/>à linea χ ad qua <lb/>dratum b d: </s> <s xml:id="echoid-s2215" xml:space="preserve">quã <lb/>uero proportio <lb/>nem habet por-<lb/>tio ad humidũ, <lb/>eandem pars ip <lb/>ſius demerſa ha <lb/>bet ad totã por <lb/>tionẽ: </s> <s xml:id="echoid-s2216" xml:space="preserve">& </s> <s xml:id="echoid-s2217" xml:space="preserve">quam <lb/>pars demerſa ad <lb/>totam, eandem <lb/>habet quadra-<lb/>tum t p ad b d <lb/>quadratum: </s> <s xml:id="echoid-s2218" xml:space="preserve">erit <lb/>linea ψ æqualis <lb/>ipſi t p. </s> <s xml:id="echoid-s2219" xml:space="preserve">quare & </s> <s xml:id="echoid-s2220" xml:space="preserve">lineæ m n, p t; </s> <s xml:id="echoid-s2221" xml:space="preserve">itemq, portiones a m q, <lb/>a p o inter ſe ſunt æquales. </s> <s xml:id="echoid-s2222" xml:space="preserve">Quòd cumin portionibus <lb/> <anchor type="note" xlink:label="note-0086-03a" xlink:href="note-0086-03"/> <pb o="38" file="0087" n="87" rhead="DE IIS QVAE VEH. IN AQVA."/> æqualibus, & </s> <s xml:id="echoid-s2223" xml:space="preserve">ſimilibus, a p o l, a m q l ab extremitati-<lb/>bus baſium ductæ ſint a o, a q ita, ut portiones ablatæ <lb/>faciant cum diametris angulos æquales; </s> <s xml:id="echoid-s2224" xml:space="preserve">& </s> <s xml:id="echoid-s2225" xml:space="preserve">anguli, qui <lb/>ad y g: </s> <s xml:id="echoid-s2226" xml:space="preserve">& </s> <s xml:id="echoid-s2227" xml:space="preserve">lineæ y b, g b, & </s> <s xml:id="echoid-s2228" xml:space="preserve">b c, b s inter ſe æquales erunt. <lb/></s> <s xml:id="echoid-s2229" xml:space="preserve">quare & </s> <s xml:id="echoid-s2230" xml:space="preserve">ipſæ c r, s r: </s> <s xml:id="echoid-s2231" xml:space="preserve">& </s> <s xml:id="echoid-s2232" xml:space="preserve">m u, p z: </s> <s xml:id="echoid-s2233" xml:space="preserve">& </s> <s xml:id="echoid-s2234" xml:space="preserve">u n, z t. </s> <s xml:id="echoid-s2235" xml:space="preserve">Quo-<lb/> <anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/> niam igitur m u minor eſt, quàm dupla u n; </s> <s xml:id="echoid-s2236" xml:space="preserve">conſtat p z ip-<lb/>ſius z t minorem eſſe, quàm duplam. </s> <s xml:id="echoid-s2237" xml:space="preserve">Sit p α dupla ipſius <lb/>ω t: </s> <s xml:id="echoid-s2238" xml:space="preserve">& </s> <s xml:id="echoid-s2239" xml:space="preserve">iuncta α k ad e producatur. </s> <s xml:id="echoid-s2240" xml:space="preserve">ergo totius quidem por <lb/>tionis centrum grauitatis erit puntum κ; </s> <s xml:id="echoid-s2241" xml:space="preserve">partis eius, quæ <lb/>in humido eſt, centrum ω; </s> <s xml:id="echoid-s2242" xml:space="preserve">eius uero, quæ extra humidum <lb/>in linea k e, quod ſit e. </s> <s xml:id="echoid-s2243" xml:space="preserve">Sed linea k z perpendicularis erit <lb/>ad ſuperficiem humidi. </s> <s xml:id="echoid-s2244" xml:space="preserve">quare & </s> <s xml:id="echoid-s2245" xml:space="preserve">lineæ quæ per puncta e, <lb/>ω, æ quidiſtantes ipſi κ z ducuntur. </s> <s xml:id="echoid-s2246" xml:space="preserve">non ergo manebit por-<lb/> <anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/> tio, ſed reuoluetur ita, ut baſis ipſius ſuperficiem humidi <lb/>nullo modo contingat: </s> <s xml:id="echoid-s2247" xml:space="preserve">quoniã nuncin uno puncto contin <lb/>gens, ſurſum fertur ex parte a. </s> <s xml:id="echoid-s2248" xml:space="preserve">perſpicuum eſt igitur por-<lb/> <anchor type="note" xlink:label="note-0087-03a" xlink:href="note-0087-03"/> tionem conſiſt ere ita, ut axis cum ſuperſicie humidi faciat <lb/>angulum maiorem angulo χ.</s> <s xml:id="echoid-s2249" xml:space="preserve"/> </p> <div xml:id="echoid-div158" type="float" level="2" n="1"> <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">C</note> <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a"> <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0085-01"/> </figure> <note position="left" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">D</note> <note position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">E</note> <note position="right" xlink:label="note-0085-04" xlink:href="note-0085-04a" xml:space="preserve">F</note> <note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">G</note> <note position="left" xlink:label="note-0086-02" xlink:href="note-0086-02a" xml:space="preserve">H</note> <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a"> <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0086-01"/> </figure> <note position="left" xlink:label="note-0086-03" xlink:href="note-0086-03a" xml:space="preserve">K</note> <note position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">L</note> <note position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">M</note> <note position="right" xlink:label="note-0087-03" xlink:href="note-0087-03a" xml:space="preserve">N</note> </div> </div> <div xml:id="echoid-div160" type="section" level="1" n="53"> <head xml:id="echoid-head58" xml:space="preserve">COMMENTARIVS.</head> <p> <s xml:id="echoid-s2250" xml:space="preserve">Siportio ad humidum in grauitate minorẽ proportio-<lb/> <anchor type="note" xlink:label="note-0087-04a" xlink:href="note-0087-04"/> nem habeat; </s> <s xml:id="echoid-s2251" xml:space="preserve">quàm quadratum s b ad quadratum b d; </s> <s xml:id="echoid-s2252" xml:space="preserve">ma-<lb/>iorem uero, quàm quadratum x o ad b d quadratum.</s> <s xml:id="echoid-s2253" xml:space="preserve">] _Hæc_ <lb/>_eſt ſecunda pars propoſitionis, quam aliæ deinceps, postea ipſarum_ <lb/>_demonstrationes eodem ordine ſequuntur_.</s> <s xml:id="echoid-s2254" xml:space="preserve"/> </p> <div xml:id="echoid-div160" type="float" level="2" n="1"> <note position="right" xlink:label="note-0087-04" xlink:href="note-0087-04a" xml:space="preserve">A</note> </div> <p> <s xml:id="echoid-s2255" xml:space="preserve">SI portio ad humidum in grauitate maiorem quidem <lb/> <anchor type="note" xlink:label="note-0087-05a" xlink:href="note-0087-05"/> proportionẽ habeat, quàm quadratũ f p ad quadratũb d.</s> <s xml:id="echoid-s2256" xml:space="preserve">] <lb/>_Hãc quartá parté nos reſtituimus, quæ ĩ trãſlatione deſiderabatur_.</s> <s xml:id="echoid-s2257" xml:space="preserve"/> </p> <div xml:id="echoid-div161" type="float" level="2" n="2"> <note position="right" xlink:label="note-0087-05" xlink:href="note-0087-05a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s2258" xml:space="preserve">Erit ψ maior quidem, quàm x o, minor uero, quàm ex-<lb/> <anchor type="note" xlink:label="note-0087-06a" xlink:href="note-0087-06"/> ceſſus, quo axis eſt maior, quam ſeſquialter eius, quæ uſque <lb/>ad axem,] _Sequitur illud ex decima quinti libri elementornm_.</s> <s xml:id="echoid-s2259" xml:space="preserve"/> </p> <div xml:id="echoid-div162" type="float" level="2" n="3"> <note position="right" xlink:label="note-0087-06" xlink:href="note-0087-06a" xml:space="preserve">C</note> </div> <p> <s xml:id="echoid-s2260" xml:space="preserve">Demonſtrabitur m h duplaipſius h n, ſicuti demonſtra <lb/> <anchor type="note" xlink:label="note-0087-07a" xlink:href="note-0087-07"/> tũ eſt o gipſius g x duplam eſſe.</s> <s xml:id="echoid-s2261" xml:space="preserve">] _Vt in prima parte huius, &_</s> <s xml:id="echoid-s2262" xml:space="preserve"> <lb/>_exijs, quæ nos proxime in ipſam conſcripſimus_.</s> <s xml:id="echoid-s2263" xml:space="preserve"/> </p> <div xml:id="echoid-div163" type="float" level="2" n="4"> <note position="right" xlink:label="note-0087-07" xlink:href="note-0087-07a" xml:space="preserve">D</note> </div> <p> <s xml:id="echoid-s2264" xml:space="preserve">Quoniam enim in ſimilibus portionibus a p o l, a x d, <lb/> <anchor type="note" xlink:label="note-0087-08a" xlink:href="note-0087-08"/> <pb file="0088" n="88" rhead="ARCHIMEDIS"/> ductæ ſunt à baſibus ad portiones lineæ a n, a q, quæ angu <lb/>los æquales continent cum ipſis baſibus, eandem propor-<lb/>tionem habebit q a ad a n, quam l a ad a d.</s> <s xml:id="echoid-s2265" xml:space="preserve">] _Hoc nos ſu_-<lb/>_pra demonstrauimus_.</s> <s xml:id="echoid-s2266" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="5"> <note position="right" xlink:label="note-0087-08" xlink:href="note-0087-08a" xml:space="preserve">E</note> </div> <p> <s xml:id="echoid-s2267" xml:space="preserve">Aequalis eſt ergo a n ipſi n q.</s> <s xml:id="echoid-s2268" xml:space="preserve">] _Cum enim q a ad a n ſit_, <lb/> <anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/> _ut l a ad a d; </s> <s xml:id="echoid-s2269" xml:space="preserve">diuidendo, conuertendoq; </s> <s xml:id="echoid-s2270" xml:space="preserve">erit an ad n q, ut a d ad_ <lb/>_d l. </s> <s xml:id="echoid-s2271" xml:space="preserve">eſt autem a d æqualis ipſi d l, quoniam d b ponitur diameter_ <lb/>_portionis. </s> <s xml:id="echoid-s2272" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2273" xml:space="preserve">a n ipſi n q eſt æqualis_. <lb/></s> <s xml:id="echoid-s2274" xml:space="preserve"/> </p> <div xml:id="echoid-div165" type="float" level="2" n="6"> <note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">F</note> </div> <note position="left" xml:space="preserve">14. quinti</note> <p> <s xml:id="echoid-s2275" xml:space="preserve">Et a q ipſi m y æquidiſtans.</s> <s xml:id="echoid-s2276" xml:space="preserve">] _Ex quinta ſecundi libri coni_-<lb/> <anchor type="note" xlink:label="note-0088-03a" xlink:href="note-0088-03"/> _corum. </s> <s xml:id="echoid-s2277" xml:space="preserve">Apollonij_.</s> <s xml:id="echoid-s2278" xml:space="preserve"/> </p> <div xml:id="echoid-div166" type="float" level="2" n="7"> <note position="left" xlink:label="note-0088-03" xlink:href="note-0088-03a" xml:space="preserve">G</note> </div> <p> <s xml:id="echoid-s2279" xml:space="preserve">Etſecetur b d in punctis k r, ut dictum eſt.</s> <s xml:id="echoid-s2280" xml:space="preserve">] _In prima_ <lb/> <anchor type="note" xlink:label="note-0088-04a" xlink:href="note-0088-04"/> _parte huius propoſitionis. </s> <s xml:id="echoid-s2281" xml:space="preserve">ſecetur autem in K ita, ut b k ſit dupla ip_-<lb/>_ſius k d; </s> <s xml:id="echoid-s2282" xml:space="preserve">& </s> <s xml:id="echoid-s2283" xml:space="preserve">in r, ut K r ſit æqualis ei, quæ uſque ad axcm_.</s> <s xml:id="echoid-s2284" xml:space="preserve"/> </p> <div xml:id="echoid-div167" type="float" level="2" n="8"> <note position="left" xlink:label="note-0088-04" xlink:href="note-0088-04a" xml:space="preserve">H</note> </div> <p> <s xml:id="echoid-s2285" xml:space="preserve">Quòd cũ in portionibus æqualibus, & </s> <s xml:id="echoid-s2286" xml:space="preserve">ſimilibus, a p o l, <lb/> <anchor type="note" xlink:label="note-0088-05a" xlink:href="note-0088-05"/> a m q l ab extremitatibus baſium ductæ ſint a o, a q, ita ut <lb/>portiones ablatæ faciant cum diametris angulos æquales: <lb/></s> <s xml:id="echoid-s2287" xml:space="preserve">& </s> <s xml:id="echoid-s2288" xml:space="preserve">anguli, qui ad y g: </s> <s xml:id="echoid-s2289" xml:space="preserve">& </s> <s xml:id="echoid-s2290" xml:space="preserve">lineæ y b, g b inter ſe æquales erũt.</s> <s xml:id="echoid-s2291" xml:space="preserve">] <lb/>_Secet linea a q diametrum d b in θ, & </s> <s xml:id="echoid-s2292" xml:space="preserve">a o ſecet in η. </s> <s xml:id="echoid-s2293" xml:space="preserve">Itaque quo_-<lb/>_niam in portionibus æqualibus, & </s> <s xml:id="echoid-s2294" xml:space="preserve">ſimilibus a p o l, a m q l ab ex_-<lb/>_tremitatibus baſium_ <lb/> <anchor type="figure" xlink:label="fig-0088-01a" xlink:href="fig-0088-01"/> _ducũtur a o, a q, quæ_ <lb/>_æquales angulos con_ <lb/>_tinent cum ipſis baſi_ <lb/>_bus: </s> <s xml:id="echoid-s2295" xml:space="preserve">& </s> <s xml:id="echoid-s2296" xml:space="preserve">anguli ad d_ <lb/>_utrique ſunt recti_: <lb/></s> <s xml:id="echoid-s2297" xml:space="preserve">_erũt & </s> <s xml:id="echoid-s2298" xml:space="preserve">reliqui a η d_, <lb/>_a θ d inter ſe æqua_-<lb/>_les. </s> <s xml:id="echoid-s2299" xml:space="preserve">linea autem p g_ <lb/>_æquidiſtat lineæ a o_: </s> <s xml:id="echoid-s2300" xml:space="preserve"><lb/>_itémq; </s> <s xml:id="echoid-s2301" xml:space="preserve">m y ipſi a q_: </s> <s xml:id="echoid-s2302" xml:space="preserve"><lb/>_& </s> <s xml:id="echoid-s2303" xml:space="preserve">p s, m c ipſis a d_. </s> <s xml:id="echoid-s2304" xml:space="preserve"><lb/>_triágula igitur p g s_, <lb/>_m y c triãgulis a η d_ <lb/>_a θ d, atque inter ſe_ <lb/>_ſunt ſimilia: </s> <s xml:id="echoid-s2305" xml:space="preserve">& </s> <s xml:id="echoid-s2306" xml:space="preserve">ut a d ad a η, ita a d ad a θ: </s> <s xml:id="echoid-s2307" xml:space="preserve">& </s> <s xml:id="echoid-s2308" xml:space="preserve">permutando. </s> <s xml:id="echoid-s2309" xml:space="preserve">li_-<lb/> <anchor type="note" xlink:label="note-0088-06a" xlink:href="note-0088-06"/> <pb o="39" file="0089" n="89" rhead="DE IIS QVAE VEH. IN AQVA."/> _neæ autem a d inter ſe æquales ſunt. </s> <s xml:id="echoid-s2310" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2311" xml:space="preserve">ipſæ a η, a θ. </s> <s xml:id="echoid-s2312" xml:space="preserve">Sed ſunt_ <lb/>_æquales a o, a q: </s> <s xml:id="echoid-s2313" xml:space="preserve">& </s> <s xml:id="echoid-s2314" xml:space="preserve">earum dimidiæ a t a n. </s> <s xml:id="echoid-s2315" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2316" xml:space="preserve">reliquæ t η, n θ_ <lb/>_boc eſt p g, m y. </s> <s xml:id="echoid-s2317" xml:space="preserve">ut autem p g ad g h, ita m y ad y c; </s> <s xml:id="echoid-s2318" xml:space="preserve">& </s> <s xml:id="echoid-s2319" xml:space="preserve">permutan_ <lb/> <anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/> _do, ut p g ad m y, ita g s ad y c. </s> <s xml:id="echoid-s2320" xml:space="preserve">quare g s, y c æquales ſunt: </s> <s xml:id="echoid-s2321" xml:space="preserve">&_</s> <s xml:id="echoid-s2322" xml:space="preserve"> <lb/>_ipſarum dimidiæ b s, b c: </s> <s xml:id="echoid-s2323" xml:space="preserve">ex quibus ſequitur ut & </s> <s xml:id="echoid-s2324" xml:space="preserve">reliquæ s r, c r_: <lb/></s> <s xml:id="echoid-s2325" xml:space="preserve">_& </s> <s xml:id="echoid-s2326" xml:space="preserve">idcirco p z, m u & </s> <s xml:id="echoid-s2327" xml:space="preserve">u n, z t inter ſe ſunt æquales_.</s> <s xml:id="echoid-s2328" xml:space="preserve"/> </p> <div xml:id="echoid-div168" type="float" level="2" n="9"> <note position="left" xlink:label="note-0088-05" xlink:href="note-0088-05a" xml:space="preserve">K</note> <figure xlink:label="fig-0088-01" xlink:href="fig-0088-01a"> <image file="0088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0088-01"/> </figure> <note position="left" xlink:label="note-0088-06" xlink:href="note-0088-06a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">34. primi</note> </div> <p> <s xml:id="echoid-s2329" xml:space="preserve">Quoniam igitur m u minor eſt, quàm dupla u n.</s> <s xml:id="echoid-s2330" xml:space="preserve">] _Eſt_ <lb/> <anchor type="note" xlink:label="note-0089-02a" xlink:href="note-0089-02"/> _enim m h ipſius h n dupla, & </s> <s xml:id="echoid-s2331" xml:space="preserve">m u minor ipſa m h. </s> <s xml:id="echoid-s2332" xml:space="preserve">ergo m u minor_ <lb/>_eſt, quàm dupla h n; </s> <s xml:id="echoid-s2333" xml:space="preserve">& </s> <s xml:id="echoid-s2334" xml:space="preserve">multo minor, quàm dupla ipſius u n_.</s> <s xml:id="echoid-s2335" xml:space="preserve"/> </p> <div xml:id="echoid-div169" type="float" level="2" n="10"> <note position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">L</note> </div> <p> <s xml:id="echoid-s2336" xml:space="preserve">Non ergo manebit portio, ſed reuoluetur, ita ut baſis ip <lb/> <anchor type="note" xlink:label="note-0089-03a" xlink:href="note-0089-03"/> ſius humidi ſuperſiciem nullo modo contingat. </s> <s xml:id="echoid-s2337" xml:space="preserve">quoniam <lb/>nunc in uno puncto contingens ſurſum fertur ex parte a.</s> <s xml:id="echoid-s2338" xml:space="preserve">] <lb/>_Tranſlatio ſic habet. </s> <s xml:id="echoid-s2339" xml:space="preserve">non ergo manet portio ſed inclinabitur, ut ba_-<lb/>_ſis ipſius nec ſecundum unum tang at ſuperficiem humidi, quoniam_ <lb/>_nunc ſecundum unum tacta ipſa reclinatur. </s> <s xml:id="echoid-s2340" xml:space="preserve">Quæ nos ex alijs Ar_-<lb/>_chimedis locis, & </s> <s xml:id="echoid-s2341" xml:space="preserve">perſpicuitatis cauſſa in eum modum corrigenda_ <lb/>_duximus. </s> <s xml:id="echoid-s2342" xml:space="preserve">In ſexta enim propoſitione huius ita ſcribit, ut habetur in_ <lb/>_tranſlatione. </s> <s xml:id="echoid-s2343" xml:space="preserve">reuoluetur ergo ſolidum a p o l, & </s> <s xml:id="echoid-s2344" xml:space="preserve">baſis ipſius nó tan_ <lb/>_get ſuperficiem humidi ſecundum unum ſignum. </s> <s xml:id="echoid-s2345" xml:space="preserve">Rurſus in ſeptima_ <lb/>_propoſitione. </s> <s xml:id="echoid-s2346" xml:space="preserve">manifeſtum igitur, quòd reuoluetur ſolidum ita ut ba_-<lb/>_ſis ipſius nec ſecundum unum ſignum contingat ſuperficiem humidi_, <lb/>_quoniam nunc ſecundum unum tangens deorſum fertur ex parte l_. <lb/></s> <s xml:id="echoid-s2347" xml:space="preserve">_At uero portionem ſurſum ferri ex parte a manifeſte constat. </s> <s xml:id="echoid-s2348" xml:space="preserve">nam_ <lb/>_cumperpendicularis ad ſuperficiem humidi, quæ tranſit per ω ad_ <lb/>_partes a cadat, & </s> <s xml:id="echoid-s2349" xml:space="preserve">quæ per e ad partes l, neceſſe eſt ut centrum ω_ <lb/>_ſurſum, e uero deorſum feratur_.</s> <s xml:id="echoid-s2350" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="11"> <note position="right" xlink:label="note-0089-03" xlink:href="note-0089-03a" xml:space="preserve">M</note> </div> <p> <s xml:id="echoid-s2351" xml:space="preserve">Perſpicuum eſtigitur portionem conſiſtere ita, ut axis <lb/> <anchor type="note" xlink:label="note-0089-04a" xlink:href="note-0089-04"/> cum ſuperficie humidi faciat angulum maiorem angu-<lb/>10 χ.</s> <s xml:id="echoid-s2352" xml:space="preserve">] _Iuncta enim a x producatur, ut diametrum b d ſe_-<lb/>_cet in λ, & </s> <s xml:id="echoid-s2353" xml:space="preserve">ab o puncto ipſi æquidistans ducatur o χ. </s> <s xml:id="echoid-s2354" xml:space="preserve">con_-<lb/>_tinget eaſectionem in o, ut in prima figura: </s> <s xml:id="echoid-s2355" xml:space="preserve">atque erit angu_-<lb/> <anchor type="note" xlink:label="note-0089-05a" xlink:href="note-0089-05"/> _lus ad χ angulo ad λ æqualis. </s> <s xml:id="echoid-s2356" xml:space="preserve">Sed angulus ad y æqualis est_ <lb/>_angulo ad θ: </s> <s xml:id="echoid-s2357" xml:space="preserve">& </s> <s xml:id="echoid-s2358" xml:space="preserve">angulus a θ d maior angulo a λ d; </s> <s xml:id="echoid-s2359" xml:space="preserve">quod ex_-<lb/> <anchor type="note" xlink:label="note-0089-06a" xlink:href="note-0089-06"/> _traipſum cadat. </s> <s xml:id="echoid-s2360" xml:space="preserve">ergo angulus ad y eo, qui ad χ maior erit_.</s> <s xml:id="echoid-s2361" xml:space="preserve"> <pb file="0090" n="90" rhead="ARCHIMEDIS"/> _Quoniam igi_-<lb/> <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> _tur portio con_-<lb/>_uertitur, ita ut_ <lb/>_baſis humidum_ <lb/>_non contingat_, <lb/>_axis cum ſuper_ <lb/>_ficie eius faciet_ <lb/>_angulum maio_-<lb/>_rem angulo g;_ <lb/></s> <s xml:id="echoid-s2362" xml:space="preserve">_hoc est angulo_ <lb/>_y: </s> <s xml:id="echoid-s2363" xml:space="preserve">& </s> <s xml:id="echoid-s2364" xml:space="preserve">propter_-<lb/>_ea multo maio_-<lb/>_rem angulo χ_.</s> <s xml:id="echoid-s2365" xml:space="preserve"/> </p> <div xml:id="echoid-div171" type="float" level="2" n="12"> <note position="right" xlink:label="note-0089-04" xlink:href="note-0089-04a" xml:space="preserve">N</note> <note position="right" xlink:label="note-0089-05" xlink:href="note-0089-05a" xml:space="preserve">29. primi</note> <note position="right" xlink:label="note-0089-06" xlink:href="note-0089-06a" xml:space="preserve">16. primi</note> <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a"> <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0090-01"/> </figure> </div> </div> <div xml:id="echoid-div173" type="section" level="1" n="54"> <head xml:id="echoid-head59" xml:space="preserve">DEMONSTRATIO TERTIAE PARTIS.</head> <p> <s xml:id="echoid-s2366" xml:space="preserve">HABEAT deinde portio ad humidum eam in graui-<lb/>tate proportionem, quam quadratũ x o habet ad quadra-<lb/>tum b d: </s> <s xml:id="echoid-s2367" xml:space="preserve">& </s> <s xml:id="echoid-s2368" xml:space="preserve">in humidum demittatur adeo inclinata, ut ba-<lb/>ſis ipſius non con <lb/> <anchor type="figure" xlink:label="fig-0090-02a" xlink:href="fig-0090-02"/> tingat humidum. <lb/></s> <s xml:id="echoid-s2369" xml:space="preserve">Secta aũt ipſa per <lb/>axem plano ad hu <lb/>midi ſuperficiem <lb/>recto, ſolidi ſectio <lb/>ſit rectanguli co-<lb/>ni ſectio a p m l: </s> <s xml:id="echoid-s2370" xml:space="preserve">ſu <lb/>perficiei humidi <lb/>ſectio ſit i m: </s> <s xml:id="echoid-s2371" xml:space="preserve">axis <lb/>portionis, & </s> <s xml:id="echoid-s2372" xml:space="preserve">ſe-<lb/>ctionis diameter <lb/>b d: </s> <s xml:id="echoid-s2373" xml:space="preserve">ſeceturq; </s> <s xml:id="echoid-s2374" xml:space="preserve">b d <lb/>ſicuti prius: </s> <s xml:id="echoid-s2375" xml:space="preserve">& </s> <s xml:id="echoid-s2376" xml:space="preserve">du-<lb/>catur p n quidem <pb o="40" file="0091" n="91" rhead="DE IIS QVAE VEH. IN AQVA."/> ipſi i m æquidiſtãs, & </s> <s xml:id="echoid-s2377" xml:space="preserve">contingens ſectionem in p; </s> <s xml:id="echoid-s2378" xml:space="preserve">pt uero <lb/>æquidiſtans b d, & </s> <s xml:id="echoid-s2379" xml:space="preserve">p s ad ipſam b d perpendicularis. </s> <s xml:id="echoid-s2380" xml:space="preserve">Demõ <lb/>ſtrandũ eſt, portionẽ non cõſiſtereita, ſed inclinari, donec <lb/>baſis in uno puncto ſuperficiem humidi cõtingat. </s> <s xml:id="echoid-s2381" xml:space="preserve">Maneãt <lb/>enim eadem, quæ in ſuperiori figura: </s> <s xml:id="echoid-s2382" xml:space="preserve">ducaturq; </s> <s xml:id="echoid-s2383" xml:space="preserve">o c ad b d <lb/>perpendicularis: </s> <s xml:id="echoid-s2384" xml:space="preserve">& </s> <s xml:id="echoid-s2385" xml:space="preserve">iuncta a x ad q producatur. </s> <s xml:id="echoid-s2386" xml:space="preserve">erit a x <lb/>æqualis ipſi x q. </s> <s xml:id="echoid-s2387" xml:space="preserve">deĩde ducatur o χ ipſi a q æquidiſtãs. </s> <s xml:id="echoid-s2388" xml:space="preserve">Quo <lb/>niã igitur portio ad humidũ eã in grauitate proportione <lb/>habere ponitur, quam quadratum x o ad quadratum b d: <lb/></s> <s xml:id="echoid-s2389" xml:space="preserve">& </s> <s xml:id="echoid-s2390" xml:space="preserve">eandem proportionem habet pars ipſius demerſa ad to <lb/>tam; </s> <s xml:id="echoid-s2391" xml:space="preserve">hoc eſt quadratum t p ad quadratum b d: </s> <s xml:id="echoid-s2392" xml:space="preserve">æqualis uti <lb/>que erit t p ipſi x o: </s> <s xml:id="echoid-s2393" xml:space="preserve">cumq; </s> <s xml:id="echoid-s2394" xml:space="preserve">portionum i p m, a o q diame-<lb/>tri ſint æquales, & </s> <s xml:id="echoid-s2395" xml:space="preserve">portiones ipſæ æquales erunt. </s> <s xml:id="echoid-s2396" xml:space="preserve">Rurſus <lb/> <anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/> quoniam in por <lb/> <anchor type="note" xlink:label="note-0091-02a" xlink:href="note-0091-02"/> <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> tionibus æquali <lb/>bus, & </s> <s xml:id="echoid-s2397" xml:space="preserve">ſimilibus <lb/>a o q l, a p m l, <lb/>ductæ ſunt lineæ <lb/>a q, i m, quæ æ-<lb/>quales portio-<lb/>nes auferunt; </s> <s xml:id="echoid-s2398" xml:space="preserve">il-<lb/>la quidem ab ex <lb/>tremitate baſis, <lb/>hæc autem non <lb/>ab extremitate: <lb/></s> <s xml:id="echoid-s2399" xml:space="preserve">cõſtat eam, quæ <lb/>ab extremitate <lb/>baſis ducta eſt, <lb/>minorem facere <lb/>angulum acutũ <lb/>cum diametro totius portionis. </s> <s xml:id="echoid-s2400" xml:space="preserve">& </s> <s xml:id="echoid-s2401" xml:space="preserve">quoniam angulus, qui <lb/> <anchor type="note" xlink:label="note-0091-03a" xlink:href="note-0091-03"/> ad χ minor eſt angulo, qui ad n; </s> <s xml:id="echoid-s2402" xml:space="preserve">maior erit b c, quàm b s: <lb/></s> <s xml:id="echoid-s2403" xml:space="preserve">cr autem, quàm ſr minor. </s> <s xml:id="echoid-s2404" xml:space="preserve">quare & </s> <s xml:id="echoid-s2405" xml:space="preserve">o g minor, quàm p z: </s> <s xml:id="echoid-s2406" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s2407" xml:space="preserve">g x maior, quàm z t. </s> <s xml:id="echoid-s2408" xml:space="preserve">ergo p z maior eſt, quàm dupla z t;</s> <s xml:id="echoid-s2409" xml:space="preserve"> <pb file="0092" n="92" rhead="ARCHIMEDIS"/> quia o g ipſius g x eſt dupla. </s> <s xml:id="echoid-s2410" xml:space="preserve">Sit p h dupla h t: </s> <s xml:id="echoid-s2411" xml:space="preserve">& </s> <s xml:id="echoid-s2412" xml:space="preserve">iun-<lb/>cta h κ ad ω producatur. </s> <s xml:id="echoid-s2413" xml:space="preserve">erit totius quidem portionis cen <lb/>trum grauitatis k; </s> <s xml:id="echoid-s2414" xml:space="preserve">partis eius, quæ intra humidum h; </s> <s xml:id="echoid-s2415" xml:space="preserve">eius <lb/>uero, quæ extra humidum in linea κ ω, quod ſit ω. </s> <s xml:id="echoid-s2416" xml:space="preserve">Itaque <lb/>demonſtrabitur <lb/> <anchor type="figure" xlink:label="fig-0092-01a" xlink:href="fig-0092-01"/> ſimiliter & </s> <s xml:id="echoid-s2417" xml:space="preserve">k z ad <lb/>humidi ſuperſi-<lb/>ciem perpẽdicu-<lb/>laris, & </s> <s xml:id="echoid-s2418" xml:space="preserve">quæ per <lb/>puncta h ω æqui-<lb/>diſtantes ipſi κ z <lb/>ducuntur. </s> <s xml:id="echoid-s2419" xml:space="preserve">quare <lb/>nõ manebit por <lb/>tio, ſed inclinabi <lb/>tur, donec baſis <lb/>ipſius in uno pũ <lb/>cto contingat ſu <lb/>perficiem humi-<lb/>di: </s> <s xml:id="echoid-s2420" xml:space="preserve">atque ita con <lb/>ſiſtet. </s> <s xml:id="echoid-s2421" xml:space="preserve">nam in por <lb/>tionibus æquali-<lb/>bus a o q l, a p m l, ductæ erunt ab extremitatibus baſium <lb/>a q, a m, quæ æquales portiones abſcindunt: </s> <s xml:id="echoid-s2422" xml:space="preserve">etenim a o q <lb/>ipſi a p m, utin ſuperioribus æqualis demonſtrabitur. </s> <s xml:id="echoid-s2423" xml:space="preserve">ergo <lb/> <anchor type="note" xlink:label="note-0092-01a" xlink:href="note-0092-01"/> æquales faciunt acutos angulos a q, a m cum diametris ba <lb/>ſium: </s> <s xml:id="echoid-s2424" xml:space="preserve">quòd anguli ad χ & </s> <s xml:id="echoid-s2425" xml:space="preserve">n æquales ſint. </s> <s xml:id="echoid-s2426" xml:space="preserve">quare ſi ducta <lb/>h k ad ω producatur, erit totius portionis grauitatis cen-<lb/>trum k; </s> <s xml:id="echoid-s2427" xml:space="preserve">partis eius, quæ in humido h; </s> <s xml:id="echoid-s2428" xml:space="preserve">at eius, quæ extra <lb/>humidum in linea h κ; </s> <s xml:id="echoid-s2429" xml:space="preserve">quod ſit ω: </s> <s xml:id="echoid-s2430" xml:space="preserve">& </s> <s xml:id="echoid-s2431" xml:space="preserve">h k ad humidi ſuper-<lb/>ficiem perpendicularis. </s> <s xml:id="echoid-s2432" xml:space="preserve">per eaſdem igitur rectas lineas, <lb/>quod quidem in humido eſt, ſurſum, & </s> <s xml:id="echoid-s2433" xml:space="preserve">quod extra humi-<lb/>dum deorſum feretur. </s> <s xml:id="echoid-s2434" xml:space="preserve">quare manebit portio, cuius baſis <lb/>humidi ſuperficiem in uno puncto continget: </s> <s xml:id="echoid-s2435" xml:space="preserve">& </s> <s xml:id="echoid-s2436" xml:space="preserve">axis cum <lb/>ipſa angulum faciet æqualem angulo χ. </s> <s xml:id="echoid-s2437" xml:space="preserve">Similiter demon-<lb/> <anchor type="note" xlink:label="note-0092-02a" xlink:href="note-0092-02"/> <pb o="41" file="0093" n="93" rhead="DE IIS QVAE VEH. IN AQVA."/> ſtrabitur portionem, quæ ad humidum in grauitate eandẽ <lb/>proportionem habeat, quàm quadratum p f ad quadratũ <lb/>b d in humidum demiſſam, ita ut baſis ipſius nõ cõtingat <lb/>humidum, inclinatam conſiſtere adeo, ut baſis in uno pun <lb/>cto humidi ſuperficiem contingat. </s> <s xml:id="echoid-s2438" xml:space="preserve">& </s> <s xml:id="echoid-s2439" xml:space="preserve">axis cum ipſa faciat <lb/>angulum angulo φ æqualem.</s> <s xml:id="echoid-s2440" xml:space="preserve"/> </p> <div xml:id="echoid-div173" type="float" level="2" n="1"> <figure xlink:label="fig-0090-02" xlink:href="fig-0090-02a"> <image file="0090-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0090-02"/> </figure> <note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">B</note> <note position="right" xlink:label="note-0091-02" xlink:href="note-0091-02a" xml:space="preserve">C</note> <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a"> <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0091-01"/> </figure> <note position="right" xlink:label="note-0091-03" xlink:href="note-0091-03a" xml:space="preserve">D</note> <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a"> <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0092-01"/> </figure> <note position="left" xlink:label="note-0092-01" xlink:href="note-0092-01a" xml:space="preserve">E</note> <note position="left" xlink:label="note-0092-02" xlink:href="note-0092-02a" xml:space="preserve">F</note> </div> </div> <div xml:id="echoid-div175" type="section" level="1" n="55"> <head xml:id="echoid-head60" xml:space="preserve">COMMENTARIVS.</head> <p style="it"> <s xml:id="echoid-s2441" xml:space="preserve">_Hoc eſt quadratum t p ad quadratum b d.</s> <s xml:id="echoid-s2442" xml:space="preserve">]_ Ex uigeſima <lb/> <anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/> ſexta libri Archimedis de conoidibus, & </s> <s xml:id="echoid-s2443" xml:space="preserve">ſphæroidibus. </s> <s xml:id="echoid-s2444" xml:space="preserve">ergo ex no <lb/>na quinti erit quadratum t p æquale quadrato x o: </s> <s xml:id="echoid-s2445" xml:space="preserve">& </s> <s xml:id="echoid-s2446" xml:space="preserve">propterea li <lb/>nea t p lineæ x o æqualis.</s> <s xml:id="echoid-s2447" xml:space="preserve"/> </p> <div xml:id="echoid-div175" type="float" level="2" n="1"> <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">A</note> </div> <p style="it"> <s xml:id="echoid-s2448" xml:space="preserve">_Et portiones ipſæ æquales erunt.</s> <s xml:id="echoid-s2449" xml:space="preserve">]_ Ex uigeſimaquinta eiuſ-<lb/> <anchor type="note" xlink:label="note-0093-02a" xlink:href="note-0093-02"/> dem libri.</s> <s xml:id="echoid-s2450" xml:space="preserve"/> </p> <div xml:id="echoid-div176" type="float" level="2" n="2"> <note position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">B</note> </div> <p> <s xml:id="echoid-s2451" xml:space="preserve">Rurſus <lb/> <anchor type="note" xlink:label="note-0093-03a" xlink:href="note-0093-03"/> <anchor type="figure" xlink:label="fig-0093-01a" xlink:href="fig-0093-01"/> quoniam <lb/>in portio <lb/>nibus æ-<lb/>qualibus, <lb/>& </s> <s xml:id="echoid-s2452" xml:space="preserve">ſimili-<lb/>bus a o q <lb/>l, a p m l.</s> <s xml:id="echoid-s2453" xml:space="preserve">] <lb/>_In portio-_ <lb/>_ne enim a p_ <lb/>_m l deſcri-_ <lb/>_batur por-_ <lb/>_tio a o q æ-_ <lb/>_qualis por_ <lb/>_tioni i p m_, <lb/>_cadet pun-_ <lb/>_ctum q in-_ <lb/>_fram, alio-_ <lb/>_qui totum parti eſſet æquale. </s> <s xml:id="echoid-s2454" xml:space="preserve">Ducatur deinde i u æquidiſtans a q_, <pb file="0094" n="94" rhead="ARCHIMEDIS"/> _quæ diametrum ſecet in ψ; </s> <s xml:id="echoid-s2455" xml:space="preserve">ſecet autem i m eandem in σ: </s> <s xml:id="echoid-s2456" xml:space="preserve">& </s> <s xml:id="echoid-s2457" xml:space="preserve">a q in_ <lb/>_v. </s> <s xml:id="echoid-s2458" xml:space="preserve">Dico angulum a ν d angulo i σ d minoré eſſe. </s> <s xml:id="echoid-s2459" xml:space="preserve">angulus enimi ψ d_ <lb/>_æqualis est angulo a ν d. </s> <s xml:id="echoid-s2460" xml:space="preserve">ſed angulus interior i ψ d minor eſt exte-_ <lb/> <anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/> _riore i σ d. </s> <s xml:id="echoid-s2461" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2462" xml:space="preserve">a ν d ipſo i σ d minor erit_.</s> <s xml:id="echoid-s2463" xml:space="preserve"/> </p> <div xml:id="echoid-div177" type="float" level="2" n="3"> <note position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">C</note> <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a"> <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0093-01"/> </figure> <note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">29. primi</note> </div> <note position="left" xml:space="preserve">16. primi</note> <p style="it"> <s xml:id="echoid-s2464" xml:space="preserve">_Et quoniam angulus, qui ad χ minor eſt angulo, qui ad_ <lb/> <anchor type="note" xlink:label="note-0094-03a" xlink:href="note-0094-03"/> _n.</s> <s xml:id="echoid-s2465" xml:space="preserve">]_ Ducantur per o duæ lineæ, o c quidem ad diametrum b d per-<lb/>pendicularis: </s> <s xml:id="echoid-s2466" xml:space="preserve">& </s> <s xml:id="echoid-s2467" xml:space="preserve">o χ in puncto o ſectionem contingens, quæ diame <lb/>trum ſecet in χ. </s> <s xml:id="echoid-s2468" xml:space="preserve">æquidiſtabit o χ ipſi a q: </s> <s xml:id="echoid-s2469" xml:space="preserve">atque erit angulus ad <lb/> <anchor type="note" xlink:label="note-0094-04a" xlink:href="note-0094-04"/> χ æqualis ei, qui ad ν. </s> <s xml:id="echoid-s2470" xml:space="preserve">ergo angulus ad χ angulo ad σ, uidelicet eo, <lb/> <anchor type="note" xlink:label="note-0094-05a" xlink:href="note-0094-05"/> qui ad n minor erit: </s> <s xml:id="echoid-s2471" xml:space="preserve">& </s> <s xml:id="echoid-s2472" xml:space="preserve">propterea χ infra n cadet. </s> <s xml:id="echoid-s2473" xml:space="preserve">linea igitur χ b <lb/> <anchor type="note" xlink:label="note-0094-06a" xlink:href="note-0094-06"/> maior eſt, quàm n b. </s> <s xml:id="echoid-s2474" xml:space="preserve">Sed cum b c ſit æqualis χ b, & </s> <s xml:id="echoid-s2475" xml:space="preserve">b s ipſi n b: <lb/></s> <s xml:id="echoid-s2476" xml:space="preserve">erit b c ipſa b s maior.</s> <s xml:id="echoid-s2477" xml:space="preserve"/> </p> <div xml:id="echoid-div178" type="float" level="2" n="4"> <note position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">D</note> <note position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">5. ſecũdi <lb/>conicoiũ</note> <note position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">29. primi.</note> <note position="left" xlink:label="note-0094-06" xlink:href="note-0094-06a" xml:space="preserve">35. primi <lb/>conicorũ</note> </div> <p> <s xml:id="echoid-s2478" xml:space="preserve">Ergo æquales faciunt angulos a q, a m cum diametris <lb/> <anchor type="note" xlink:label="note-0094-07a" xlink:href="note-0094-07"/> portionum.</s> <s xml:id="echoid-s2479" xml:space="preserve">] _Hoc demonstrabimus ut in commentarijs in ſecun-_ <lb/>_dam partem_.</s> <s xml:id="echoid-s2480" xml:space="preserve"/> </p> <div xml:id="echoid-div179" type="float" level="2" n="5"> <note position="left" xlink:label="note-0094-07" xlink:href="note-0094-07a" xml:space="preserve">E</note> </div> <p style="it"> <s xml:id="echoid-s2481" xml:space="preserve">_Similiter demonſtrabitur, portionem, quæ ad humidũ_ <lb/> <anchor type="note" xlink:label="note-0094-08a" xlink:href="note-0094-08"/> _in grauitate ean-_ <lb/> <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> _dem proportio-_ <lb/>_nem habeat, quã_ <lb/>_quadratum p fad_ <lb/>_quadratũ b d; </s> <s xml:id="echoid-s2482" xml:space="preserve">in_ <lb/>_humidum demiſ-_ <lb/>_ſam, ita ut baſis ip_ <lb/>_ſius non cõtingat_ <lb/>_humidum, incli-_ <lb/>_natam conſiſtere_ <lb/>_adeo, ut baſis in_ <lb/>_uno pũcto humi-_ <lb/>_di ſuperficiem cõ_ <lb/>_tingat: </s> <s xml:id="echoid-s2483" xml:space="preserve">& </s> <s xml:id="echoid-s2484" xml:space="preserve">axis cũ_ <lb/>_ipſa faciat angulũ_ <lb/>_angulo φ æqualẽ]_ <lb/>Habeat portio ad humidum in grauitate proportionem eam, quam <lb/>p f quadratum ad quadratum b d: </s> <s xml:id="echoid-s2485" xml:space="preserve">& </s> <s xml:id="echoid-s2486" xml:space="preserve">demiſſa in humidum adeo in- <pb o="42" file="0095" n="95" rhead="DE IIS QVAE VEH. IN AQVA."/> clinata, ut baſis humidum non contingat, ſectur plano per axem, <lb/>recto ad ſuperficiem humidi, ut ſectio ſit a m o l rectanguli coni ſe-<lb/>ctio: </s> <s xml:id="echoid-s2487" xml:space="preserve">ſuperficiei humidi ſectio ſit i o: </s> <s xml:id="echoid-s2488" xml:space="preserve">axis portionis, & </s> <s xml:id="echoid-s2489" xml:space="preserve">ſectionis <lb/>diameter b d; </s> <s xml:id="echoid-s2490" xml:space="preserve">quæ in eaſdem, quas diximus, partes ſecetur: </s> <s xml:id="echoid-s2491" xml:space="preserve">duca-<lb/>turq; </s> <s xml:id="echoid-s2492" xml:space="preserve">m n quidem ipſi i o æquidiſtans, ut in puncto m ſectionem <lb/>cótingat: </s> <s xml:id="echoid-s2493" xml:space="preserve">mt uero æquidiſtans ipſi b d: </s> <s xml:id="echoid-s2494" xml:space="preserve">& </s> <s xml:id="echoid-s2495" xml:space="preserve">m s ad eandem perpen <lb/>dicularis. </s> <s xml:id="echoid-s2496" xml:space="preserve">Demonſtrandum eſt non manere portionem, ſed inclinari <lb/>ita, ut in uno puncto contingat ſuperficiem humidi. </s> <s xml:id="echoid-s2497" xml:space="preserve">ducatur enim p c <lb/>ad ipſam b d perpendicularis: </s> <s xml:id="echoid-s2498" xml:space="preserve">& </s> <s xml:id="echoid-s2499" xml:space="preserve">iuncta a f uſque ad ſectionem <lb/>producatur in q: </s> <s xml:id="echoid-s2500" xml:space="preserve">& </s> <s xml:id="echoid-s2501" xml:space="preserve">per p ducatur p φ ipſi a q æquidiſtans. </s> <s xml:id="echoid-s2502" xml:space="preserve">erunt <lb/>iam ex ijs, quæ demonſtrauimus a f, f q inter ſe ſe æquales. </s> <s xml:id="echoid-s2503" xml:space="preserve">& </s> <s xml:id="echoid-s2504" xml:space="preserve">cum <lb/>portio ad humi-<lb/> <anchor type="figure" xlink:label="fig-0095-01a" xlink:href="fig-0095-01"/> dum eam in gra-<lb/>uitate proportio <lb/>nem habeat, quá <lb/>quadratú p f ad <lb/>b d quadratum: <lb/></s> <s xml:id="echoid-s2505" xml:space="preserve">atque eandem ha <lb/>beat portio ipſi-<lb/>us demerſa ad to <lb/>tam portionem; </s> <s xml:id="echoid-s2506" xml:space="preserve"><lb/>hoc eſt quadratú <lb/>m t ad quadratú <lb/> <anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> b d: </s> <s xml:id="echoid-s2507" xml:space="preserve">erit quadra <lb/>tum m t quadra-<lb/>to p f æquale: </s> <s xml:id="echoid-s2508" xml:space="preserve">& </s> <s xml:id="echoid-s2509" xml:space="preserve"><lb/>idcirco linea m t <lb/>æqualis lmeæ p <lb/>f. </s> <s xml:id="echoid-s2510" xml:space="preserve">Itaque quoniam in portionibus æqualibus, & </s> <s xml:id="echoid-s2511" xml:space="preserve">ſimilibus a p q l, a <lb/>m o l ductæ ſunt lineæ a q, i o, quæ æquales portiones abſcindunt; <lb/></s> <s xml:id="echoid-s2512" xml:space="preserve">illa quidem ab extremitate baſis; </s> <s xml:id="echoid-s2513" xml:space="preserve">hæc uero non ab extremitate: </s> <s xml:id="echoid-s2514" xml:space="preserve">ſe-<lb/>quitur ut a q, quæ ab extremitate ducitur, minorem acutum angulú <lb/>contineat cum diametro portionis, quàm ipſa i o. </s> <s xml:id="echoid-s2515" xml:space="preserve">Sed linea p φ li-<lb/>neæ a q æquidiſtat, & </s> <s xml:id="echoid-s2516" xml:space="preserve">m n ipſi i o. </s> <s xml:id="echoid-s2517" xml:space="preserve">angulus igitur ad φ angulo ad n <pb file="0096" n="96" rhead="ARCHIMEDIS"/> minor erit: </s> <s xml:id="echoid-s2518" xml:space="preserve">linea uero b c maior, quàm b s: </s> <s xml:id="echoid-s2519" xml:space="preserve">& </s> <s xml:id="echoid-s2520" xml:space="preserve">s r; </s> <s xml:id="echoid-s2521" xml:space="preserve">hoc eſt m χ ma-<lb/>ior, quàm c r, hoc eſt, quàm p y: </s> <s xml:id="echoid-s2522" xml:space="preserve">& </s> <s xml:id="echoid-s2523" xml:space="preserve">propterea χ t minor, quàm y f. <lb/></s> <s xml:id="echoid-s2524" xml:space="preserve">quòd cum p y ſit dupla y f, erit m χ maior, quàm dupla y f; </s> <s xml:id="echoid-s2525" xml:space="preserve">& </s> <s xml:id="echoid-s2526" xml:space="preserve"><lb/>multo maior, quàm dupla χ t. </s> <s xml:id="echoid-s2527" xml:space="preserve">fiat m h dupla ipſius h t: </s> <s xml:id="echoid-s2528" xml:space="preserve">& </s> <s xml:id="echoid-s2529" xml:space="preserve">copu-<lb/>lata h k producatur. </s> <s xml:id="echoid-s2530" xml:space="preserve">I am grauitatis centrum totius portionis erit <lb/>punctum k: </s> <s xml:id="echoid-s2531" xml:space="preserve">eius, quæ in humido est, h: </s> <s xml:id="echoid-s2532" xml:space="preserve">at rel iquæ partis, quæ ex-<lb/>tra humidum in linea h k producta; </s> <s xml:id="echoid-s2533" xml:space="preserve">quod ſit ω. </s> <s xml:id="echoid-s2534" xml:space="preserve">eodem modo demon <lb/>strabitur, & </s> <s xml:id="echoid-s2535" xml:space="preserve">lineam k h, & </s> <s xml:id="echoid-s2536" xml:space="preserve">quæ per h ω puncta ipſi k h æquidi-<lb/>ſtantes ducuntur, ad humidi ſuperficiem perpendiculares eſſe. </s> <s xml:id="echoid-s2537" xml:space="preserve">non <lb/>igitur maneb it <lb/> <anchor type="figure" xlink:label="fig-0096-01a" xlink:href="fig-0096-01"/> portio, ſed cum <lb/>uſque eò inclina-<lb/>ta fuerit, ut in <lb/>uno puncto con-<lb/>tingat ſuperfi-<lb/>cié humidi, tunc <lb/>conſiſtet. </s> <s xml:id="echoid-s2538" xml:space="preserve">an-<lb/>gulus enim ad n <lb/>angulo ad φ æ-<lb/>qualis erit; </s> <s xml:id="echoid-s2539" xml:space="preserve">li-<lb/>neáq; </s> <s xml:id="echoid-s2540" xml:space="preserve">b s lineæ <lb/>b c; </s> <s xml:id="echoid-s2541" xml:space="preserve">& </s> <s xml:id="echoid-s2542" xml:space="preserve">s r ipſi <lb/>c r. </s> <s xml:id="echoid-s2543" xml:space="preserve">quare & </s> <s xml:id="echoid-s2544" xml:space="preserve">m h <lb/>ipſi p y eſt æqua <lb/>lis. </s> <s xml:id="echoid-s2545" xml:space="preserve">Itaque ducta <lb/>h k producatur. <lb/></s> <s xml:id="echoid-s2546" xml:space="preserve">erit totius portionis grauitatis centrum K; </s> <s xml:id="echoid-s2547" xml:space="preserve">eius, quæ in humido eſt <lb/>h; </s> <s xml:id="echoid-s2548" xml:space="preserve">& </s> <s xml:id="echoid-s2549" xml:space="preserve">reliquæ partis centrum in linea producta; </s> <s xml:id="echoid-s2550" xml:space="preserve">ſit autem ω. </s> <s xml:id="echoid-s2551" xml:space="preserve">per ean <lb/>dem igitur rectam lineam k h, quæ eſt ad humidi ſuperficiem perpen <lb/>dicularis, id quod in humido eſt ſurſum; </s> <s xml:id="echoid-s2552" xml:space="preserve">& </s> <s xml:id="echoid-s2553" xml:space="preserve">quod extra humidum de <lb/>orſum feretur. </s> <s xml:id="echoid-s2554" xml:space="preserve">atque ob hác cauſſam portio non amplius mouebitur; </s> <s xml:id="echoid-s2555" xml:space="preserve"><lb/>ſed conſiſtet, manebítq, ita, ut eius baſis ſuperficiem humidi in uno <lb/>punsto contingat; </s> <s xml:id="echoid-s2556" xml:space="preserve">& </s> <s xml:id="echoid-s2557" xml:space="preserve">axis, cum ipſa angulum faciat æqualem angulo <lb/>φ. </s> <s xml:id="echoid-s2558" xml:space="preserve">at que illud eſt, quod demonſtrare oportebat.</s> <s xml:id="echoid-s2559" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="6"> <note position="left" xlink:label="note-0094-08" xlink:href="note-0094-08a" xml:space="preserve">F</note> <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a"> <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0094-01"/> </figure> <figure xlink:label="fig-0095-01" xlink:href="fig-0095-01a"> <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0095-01"/> </figure> <note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">8. quinti.</note> <figure xlink:label="fig-0096-01" xlink:href="fig-0096-01a"> <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0096-01"/> </figure> </div> <pb o="43" file="0097" n="97" rhead="DE I _IS_ QVAE VEH. IN AQVA."/> </div> <div xml:id="echoid-div182" type="section" level="1" n="56"> <head xml:id="echoid-head61" xml:space="preserve">DEMONSTRATIO QVARTAE PARTIS.</head> <p> <s xml:id="echoid-s2560" xml:space="preserve">HABEAT rurſum portio ad humidum in grauitate <lb/>proportionem quidem maiorem, quàm quadratum f p ad <lb/>quadratum b d; </s> <s xml:id="echoid-s2561" xml:space="preserve">minorem uero, quàm quadratum x o ad <lb/>b d quadratum: </s> <s xml:id="echoid-s2562" xml:space="preserve">& </s> <s xml:id="echoid-s2563" xml:space="preserve">quam proportionem habet portio ad <lb/>humidum in grauitate, eandem habeat quadratum, quod <lb/>fit à linea ψ ad quadratum b d. </s> <s xml:id="echoid-s2564" xml:space="preserve">erit ψ maior, quàm f p, & </s> <s xml:id="echoid-s2565" xml:space="preserve">mi <lb/>nor, quàm x o. </s> <s xml:id="echoid-s2566" xml:space="preserve">aptetur ergo quæ dam rectalinea i u inter <lb/>portiones a u q l, a x d interiecta, quæ ſit æqualis ψ, & </s> <s xml:id="echoid-s2567" xml:space="preserve">ipſi <lb/>b d æquidiſtans: </s> <s xml:id="echoid-s2568" xml:space="preserve">occurratq; </s> <s xml:id="echoid-s2569" xml:space="preserve">reliquæ ſectioni in y. </s> <s xml:id="echoid-s2570" xml:space="preserve">rurſus <lb/>u y dupla ipſius y i demonſtrabitur, ſicuti demonſtrata eſt <lb/>o g ipſius g x dupla. </s> <s xml:id="echoid-s2571" xml:space="preserve">ducatur autem ab u linea u ο, quæ ſe <lb/>ctionem a u q l in u contingat: </s> <s xml:id="echoid-s2572" xml:space="preserve">& </s> <s xml:id="echoid-s2573" xml:space="preserve">iuncta a i ad q produca <lb/>tur. </s> <s xml:id="echoid-s2574" xml:space="preserve">eodem modo oſtendemus lineam a i ipſi i q æqualem <lb/>eſſe: </s> <s xml:id="echoid-s2575" xml:space="preserve">& </s> <s xml:id="echoid-s2576" xml:space="preserve">a q ipſi <lb/> <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/> u ω æquidiſtan-<lb/>tem. </s> <s xml:id="echoid-s2577" xml:space="preserve">Demon-<lb/>ſtrãdum eſt por <lb/>tionem in humi <lb/>dum demiſſam, <lb/>ĩclinatãq; </s> <s xml:id="echoid-s2578" xml:space="preserve">adeo, <lb/>ut baſis ipſius <lb/>non contingat <lb/>humidũ, ita con <lb/>ſiſtere, ut baſis <lb/>in humidũ ma-<lb/>gis demergatur <lb/>quam ut in uno <lb/>puncto eius ſu-<lb/>perficiem cõtin <lb/>gat. </s> <s xml:id="echoid-s2579" xml:space="preserve">Demittatur enim in humidum, ut dictum eſt; </s> <s xml:id="echoid-s2580" xml:space="preserve">& </s> <s xml:id="echoid-s2581" xml:space="preserve">iaceat <lb/>primo ſic inclinata, ut baſis nullo modo contingat ſuperfi-<lb/>ciem humidi. </s> <s xml:id="echoid-s2582" xml:space="preserve">ſecta autem ipſa plano per axem ad humidi <pb file="0098" n="98" rhead="ARCHIMEDIS"/> ſuperficiem recto, ſit portionis ſectio anzg; </s> <s xml:id="echoid-s2583" xml:space="preserve">ſuperficiei <lb/>humidi ez: </s> <s xml:id="echoid-s2584" xml:space="preserve">a-<lb/> <anchor type="figure" xlink:label="fig-0098-01a" xlink:href="fig-0098-01"/> xis portionis, <lb/>& </s> <s xml:id="echoid-s2585" xml:space="preserve">ſectionis dia-<lb/>meter b d: </s> <s xml:id="echoid-s2586" xml:space="preserve">ſece-<lb/>turq, b d in pũ-<lb/>ctis _K_r, ſicuti <lb/>prius; </s> <s xml:id="echoid-s2587" xml:space="preserve">& </s> <s xml:id="echoid-s2588" xml:space="preserve">duca-<lb/>tur n l quidem <lb/>ipſi e z æquidi-<lb/>ſtans, quæ con-<lb/>tingat ſectionẽ <lb/>a n z g in n; </s> <s xml:id="echoid-s2589" xml:space="preserve">& </s> <s xml:id="echoid-s2590" xml:space="preserve"><lb/>n t æquidiſtans <lb/>ipſi b d; </s> <s xml:id="echoid-s2591" xml:space="preserve">n s ue-<lb/>ro ad b d perpẽ <lb/>dicularis. </s> <s xml:id="echoid-s2592" xml:space="preserve">Itaq; <lb/></s> <s xml:id="echoid-s2593" xml:space="preserve">quoniam portio ad humidum in grauitate eam proportio <lb/>nem habet, quam quadratum, quod fit à linea ψ ad quadra <lb/>tum b d: </s> <s xml:id="echoid-s2594" xml:space="preserve">erit ψ ipſi n t æqualis: </s> <s xml:id="echoid-s2595" xml:space="preserve">quod ſimiliter demonſtrabi <lb/>tur, ut ſuperius. </s> <s xml:id="echoid-s2596" xml:space="preserve">quare & </s> <s xml:id="echoid-s2597" xml:space="preserve">n t eſt æqualis ipſi u i. </s> <s xml:id="echoid-s2598" xml:space="preserve">portiones <lb/>igitur a u q, e n z inter ſe ſunt æquales. </s> <s xml:id="echoid-s2599" xml:space="preserve">Et cum in æquali-<lb/>bus, & </s> <s xml:id="echoid-s2600" xml:space="preserve">ſimilibus portionibus a u q l, a n z g ductæ ſint a q <lb/>e z, quæ æquales portiones auferunt; </s> <s xml:id="echoid-s2601" xml:space="preserve">illa quidem ab extre <lb/>mitate baſis; </s> <s xml:id="echoid-s2602" xml:space="preserve">hæc autem non ab extremitate: </s> <s xml:id="echoid-s2603" xml:space="preserve">minorem fa-<lb/>ciet acutum angulum cum portionis diametro, quæ ab ex-<lb/>tremitate baſis ducitur. </s> <s xml:id="echoid-s2604" xml:space="preserve">At triangulorum n l s, u ω c angu <lb/>lus ad l angulo ad ω maior eſt. </s> <s xml:id="echoid-s2605" xml:space="preserve">ergo b s minor erit, quam <lb/>b c: </s> <s xml:id="echoid-s2606" xml:space="preserve">& </s> <s xml:id="echoid-s2607" xml:space="preserve">ſ r maior, quàm c r: </s> <s xml:id="echoid-s2608" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s2609" xml:space="preserve">n χ maior, quam u h; </s> <s xml:id="echoid-s2610" xml:space="preserve">& </s> <s xml:id="echoid-s2611" xml:space="preserve"><lb/>χ t minor, quàm h i. </s> <s xml:id="echoid-s2612" xml:space="preserve">Quoniam igitur u y dupla eſt ipſius <lb/>y i; </s> <s xml:id="echoid-s2613" xml:space="preserve">conſtat n χ maiorem eſſe, quàm duplã χ t. </s> <s xml:id="echoid-s2614" xml:space="preserve">Sit n m dupla <lb/>ipſius m t. </s> <s xml:id="echoid-s2615" xml:space="preserve">perſpicuũ eſt ex iis, quæ dicta ſunt, non manere <lb/>portionẽ; </s> <s xml:id="echoid-s2616" xml:space="preserve">ſed in clinari, donec eius baſis contingat ſuperfi-<lb/>ciem humidi: </s> <s xml:id="echoid-s2617" xml:space="preserve">contingat autem in puncto uno, ut patet in fi <pb o="44" file="0099" n="99" rhead="DE IIS QVAE VEH. IN AQVA."/> gura: </s> <s xml:id="echoid-s2618" xml:space="preserve">& </s> <s xml:id="echoid-s2619" xml:space="preserve">alia eadem diſponantur demonſtrabimus rurſum <lb/>n t æqualem eſſe ipſi u i: </s> <s xml:id="echoid-s2620" xml:space="preserve">& </s> <s xml:id="echoid-s2621" xml:space="preserve">portiones a u q, a n z inter <lb/>ſe ſe æquales. <lb/></s> <s xml:id="echoid-s2622" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> Itaque quoniã <lb/>ĩ portionibus <lb/>æqualibus, & </s> <s xml:id="echoid-s2623" xml:space="preserve">ſi <lb/>milibus a u q l, <lb/>a n z g ductæ <lb/>sũt a q, a z, por <lb/>tiones æqua-<lb/>les auferentes; <lb/></s> <s xml:id="echoid-s2624" xml:space="preserve">cum diametris <lb/>portionum æ-<lb/>quales angu-<lb/>los cõtinebũt. </s> <s xml:id="echoid-s2625" xml:space="preserve"><lb/>ergo triangulo <lb/>rum n l s, u ω c <lb/>anguli, qui cõ-<lb/>ſiſtũt ad l ω pũ-<lb/>cta, æquales ſunt: </s> <s xml:id="echoid-s2626" xml:space="preserve">& </s> <s xml:id="echoid-s2627" xml:space="preserve">b s recta linea æqualis ipſi b c: </s> <s xml:id="echoid-s2628" xml:space="preserve">ſ r ipſi cr, <lb/>n χ ipſi u h: </s> <s xml:id="echoid-s2629" xml:space="preserve">& </s> <s xml:id="echoid-s2630" xml:space="preserve">χ tipſi h i. </s> <s xml:id="echoid-s2631" xml:space="preserve">quòd cum u y dupla ſit ipſius y i, <lb/>erit n χ maior, quàm dupla χ t. </s> <s xml:id="echoid-s2632" xml:space="preserve">Sit igitur n m ipſius m t du <lb/>pla. </s> <s xml:id="echoid-s2633" xml:space="preserve">Rurſus ex his manifeſtum eſt, non manere ipſam por-<lb/>tionem; </s> <s xml:id="echoid-s2634" xml:space="preserve">ſed inclinari ex parte a: </s> <s xml:id="echoid-s2635" xml:space="preserve">ponebatur autem portio <lb/>humidi ſuperficiem in uno puncto contingere. </s> <s xml:id="echoid-s2636" xml:space="preserve">ergo ne-<lb/>ceſſe eſt, ut eius baſis in humidum magis demergatur.</s> <s xml:id="echoid-s2637" xml:space="preserve"/> </p> <div xml:id="echoid-div182" type="float" level="2" n="1"> <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a"> <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0097-01"/> </figure> <figure xlink:label="fig-0098-01" xlink:href="fig-0098-01a"> <image file="0098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0098-01"/> </figure> <figure xlink:label="fig-0099-01" xlink:href="fig-0099-01a"> <image file="0099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0099-01"/> </figure> </div> </div> <div xml:id="echoid-div184" type="section" level="1" n="57"> <head xml:id="echoid-head62" xml:space="preserve">DEMONSTRATIO QVINT AE PARTIS.</head> <p> <s xml:id="echoid-s2638" xml:space="preserve">HABEAT denique portio ad humidum in grauitate <lb/>minorem proportionem, quàm quadratum f p ad quadra-<lb/>tum b d: </s> <s xml:id="echoid-s2639" xml:space="preserve">& </s> <s xml:id="echoid-s2640" xml:space="preserve">quam proportionem habet portio ad humidũ <lb/>in grauitate, eandem quadratum, quod fit à linea ψ habeat <lb/>ad quadratum b d. </s> <s xml:id="echoid-s2641" xml:space="preserve">erit χ minor ipſa p f. </s> <s xml:id="echoid-s2642" xml:space="preserve">Rurſus aptetur <pb file="0100" n="100" rhead="ARCHIMEDIS"/> quædam recta linea g i, ſectionibus a g q l, a x d interiecta, <lb/>& </s> <s xml:id="echoid-s2643" xml:space="preserve">ipſi b d æquidiſtans; </s> <s xml:id="echoid-s2644" xml:space="preserve">quæ mediam coni ſectionem in pun <lb/>cto h, & </s> <s xml:id="echoid-s2645" xml:space="preserve">rectam <lb/> <anchor type="figure" xlink:label="fig-0100-01a" xlink:href="fig-0100-01"/> lineam r y in y <lb/>ſecet. </s> <s xml:id="echoid-s2646" xml:space="preserve">demonſtra <lb/>bitur g h dupla <lb/>h i, quemadmo-<lb/>dum demonſtra <lb/>ta eſt o g ipſius <lb/>g x dupla. </s> <s xml:id="echoid-s2647" xml:space="preserve">duca-<lb/>tur poſtea g ω cõ <lb/>tingens a g q l ſe <lb/>ctioneming: </s> <s xml:id="echoid-s2648" xml:space="preserve">& </s> <s xml:id="echoid-s2649" xml:space="preserve"><lb/>g c ad b d perpé <lb/>dicularis: </s> <s xml:id="echoid-s2650" xml:space="preserve">iun-<lb/>ctaq; </s> <s xml:id="echoid-s2651" xml:space="preserve">ai produ-<lb/>catur ad q. </s> <s xml:id="echoid-s2652" xml:space="preserve">erit <lb/>ergo a i æqualis <lb/>i q: </s> <s xml:id="echoid-s2653" xml:space="preserve">& </s> <s xml:id="echoid-s2654" xml:space="preserve">a q ipſi g ω <lb/>æquidiſtans. </s> <s xml:id="echoid-s2655" xml:space="preserve">Demonſtrandũ eſt portionẽ in humidũ demiſ <lb/>fam, inclinatamq; </s> <s xml:id="echoid-s2656" xml:space="preserve">adeo, ut baſis ipſius non cõtingat humi-<lb/>dũ, conſiſtere inclinatã ita, ut axis cum ſuperficie humidi <lb/>angulum faciat minorem angulo φ: </s> <s xml:id="echoid-s2657" xml:space="preserve">& </s> <s xml:id="echoid-s2658" xml:space="preserve">baſis humidi ſuper-<lb/>ficiem nullo modo contingat. </s> <s xml:id="echoid-s2659" xml:space="preserve">Demittatur enim in humi-<lb/>dum; </s> <s xml:id="echoid-s2660" xml:space="preserve">& </s> <s xml:id="echoid-s2661" xml:space="preserve">conſiſtat ita, ut baſis ipſius in uno puncto contin-<lb/>gat ſuperficiem humidi. </s> <s xml:id="echoid-s2662" xml:space="preserve">ſecta autem portione per axem, <lb/>plano ad humidi ſuperficiem recto, ſit portionis ſectio a n <lb/>z l rectanguli coni ſectio: </s> <s xml:id="echoid-s2663" xml:space="preserve">ſuperficiei humidi a z: </s> <s xml:id="echoid-s2664" xml:space="preserve">axis autẽ <lb/>portionis, & </s> <s xml:id="echoid-s2665" xml:space="preserve">ſectionis diameter b d: </s> <s xml:id="echoid-s2666" xml:space="preserve">ſeceturq; </s> <s xml:id="echoid-s2667" xml:space="preserve">b d in pun-<lb/>ctis _K_ r, ut ſuperius dictum eſt: </s> <s xml:id="echoid-s2668" xml:space="preserve">& </s> <s xml:id="echoid-s2669" xml:space="preserve">ducatur n f quidem ipſi <lb/>a z æquidiſtans, & </s> <s xml:id="echoid-s2670" xml:space="preserve">contingens coni ſectionem in pũcto n; <lb/></s> <s xml:id="echoid-s2671" xml:space="preserve">n t uero æquidiſtans ipſi b d: </s> <s xml:id="echoid-s2672" xml:space="preserve">& </s> <s xml:id="echoid-s2673" xml:space="preserve">n s ad eandem perpendi-<lb/>cularis. </s> <s xml:id="echoid-s2674" xml:space="preserve">Quoniam igitur portio ad humidum in grauitate, <lb/>cam habet proportionem, quam quadratum, quod fit à χ <pb o="43" file="0101" n="101" rhead="DEIIS QVAE VEH. IN AQVA."/> ad quadratum bd: </s> <s xml:id="echoid-s2675" xml:space="preserve">& </s> <s xml:id="echoid-s2676" xml:space="preserve">quam habet portio ad humidum in <lb/>grauitate, eandem quadratum nt habet ad bd quadratũ, <lb/>ex iis, quæ dicta ſunt: </s> <s xml:id="echoid-s2677" xml:space="preserve">conſtat n t lineæ ψ æqualem eſſe, <lb/>quare & </s> <s xml:id="echoid-s2678" xml:space="preserve">portio-<lb/> <anchor type="figure" xlink:label="fig-0101-01a" xlink:href="fig-0101-01"/> nes a n z, a g q <lb/>ſunt æquales. </s> <s xml:id="echoid-s2679" xml:space="preserve">Et <lb/>quoniam in por <lb/>tionibus æquali <lb/>bus, & </s> <s xml:id="echoid-s2680" xml:space="preserve">ſimilibus <lb/>a g q l, a n z l, ab <lb/>extremitatibus <lb/>baſiũ ductæ ſunt <lb/>a q, a z, quæ æ-<lb/>quales portiões <lb/>abſcindunt: </s> <s xml:id="echoid-s2681" xml:space="preserve">per <lb/>ſpicuum eſt an-<lb/>gulos facere æ-<lb/>quales cum por <lb/>tionum diame-<lb/>tris: </s> <s xml:id="echoid-s2682" xml:space="preserve">& </s> <s xml:id="echoid-s2683" xml:space="preserve">triangu-<lb/>lorum n fs, g ω c, angulos, qui ad f ω æquales eſſe: </s> <s xml:id="echoid-s2684" xml:space="preserve">itemque <lb/>æquales inter ſe, s b, c b; </s> <s xml:id="echoid-s2685" xml:space="preserve">& </s> <s xml:id="echoid-s2686" xml:space="preserve">s r, c r, quare & </s> <s xml:id="echoid-s2687" xml:space="preserve">n χ, g y æquales: <lb/></s> <s xml:id="echoid-s2688" xml:space="preserve">& </s> <s xml:id="echoid-s2689" xml:space="preserve">χ t y i. </s> <s xml:id="echoid-s2690" xml:space="preserve">cũq; </s> <s xml:id="echoid-s2691" xml:space="preserve">g h dupla ſit ipſius h i, erit n χ minor, quàm <lb/>duplaipſius χ t. </s> <s xml:id="echoid-s2692" xml:space="preserve">Sit igitur n m ipſius m t dupla: </s> <s xml:id="echoid-s2693" xml:space="preserve">& </s> <s xml:id="echoid-s2694" xml:space="preserve">iuncta <lb/>m K protrahatur ad e. </s> <s xml:id="echoid-s2695" xml:space="preserve">Itaque centrum grauitatis totius <lb/>erit punctum K: </s> <s xml:id="echoid-s2696" xml:space="preserve">partis eius, quæ eſt in humido, punctũ m: </s> <s xml:id="echoid-s2697" xml:space="preserve"><lb/>eius autem, quæ extra humidum in linea protracta, quod <lb/>ſit e. </s> <s xml:id="echoid-s2698" xml:space="preserve">ergo ex proxime demonſtratis patet, nõ manere por <lb/>tionem, ſed inclinari adeo, ut baſis nullo modo ſuperficiẽ <lb/>humidi contingat. </s> <s xml:id="echoid-s2699" xml:space="preserve">At uero portionem conſiſtere ita, uta-<lb/>xis cum ſuperficie humidi faciat angulum angulo φ mino-<lb/>rem, ſic demonſtrabitur. </s> <s xml:id="echoid-s2700" xml:space="preserve">conſiſtat enim, ſi fieri poteſt, ut <lb/>non faciat angulum minorem angulo φ: </s> <s xml:id="echoid-s2701" xml:space="preserve">& </s> <s xml:id="echoid-s2702" xml:space="preserve">alia eadem diſ-<lb/>ponantur; </s> <s xml:id="echoid-s2703" xml:space="preserve">ut in ſubiecta figura. </s> <s xml:id="echoid-s2704" xml:space="preserve">eodem modo demonſtra <pb file="0102" n="102" rhead="ARCHIMEDIS"/> bimus n t æqualem eſſe ψ, & </s> <s xml:id="echoid-s2705" xml:space="preserve">propterea ipſi gi. </s> <s xml:id="echoid-s2706" xml:space="preserve">& </s> <s xml:id="echoid-s2707" xml:space="preserve">quo-<lb/>niam triangulornm p φ c, n f s angulus f non eſt minor an <lb/>gulo φ, non erit b f maior, quam b c. </s> <s xml:id="echoid-s2708" xml:space="preserve">ergo neque s r mi-<lb/>nor, quàm c r: </s> <s xml:id="echoid-s2709" xml:space="preserve">neque n χ minor, quàm p y. </s> <s xml:id="echoid-s2710" xml:space="preserve">Sed cum p f ſit <lb/>maior, quàm n t: <lb/></s> <s xml:id="echoid-s2711" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/> ſitq; </s> <s xml:id="echoid-s2712" xml:space="preserve">p f ſeſquialte <lb/>ra p y: </s> <s xml:id="echoid-s2713" xml:space="preserve">erit n t mi-<lb/>nor, quàm ſeſquial <lb/>tera n χ: </s> <s xml:id="echoid-s2714" xml:space="preserve">& </s> <s xml:id="echoid-s2715" xml:space="preserve">idcir-<lb/>co n χ maior, quã <lb/>dupla χ t. </s> <s xml:id="echoid-s2716" xml:space="preserve">ſit autẽ <lb/>n m dupla m t: </s> <s xml:id="echoid-s2717" xml:space="preserve">& </s> <s xml:id="echoid-s2718" xml:space="preserve"><lb/>iuncta m K produ <lb/>catur. </s> <s xml:id="echoid-s2719" xml:space="preserve">conſtat igi-<lb/>tur ex iam dictis <lb/>non manere por-<lb/>tionem; </s> <s xml:id="echoid-s2720" xml:space="preserve">ſed reuol <lb/>ui ita, ut axis cum <lb/>ſuperficie humidi <lb/>faciat angulum an <lb/>gulo φ minorem.</s> <s xml:id="echoid-s2721" xml:space="preserve"/> </p> <div xml:id="echoid-div184" type="float" level="2" n="1"> <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a"> <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0100-01"/> </figure> <figure xlink:label="fig-0101-01" xlink:href="fig-0101-01a"> <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0101-01"/> </figure> <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a"> <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0102-01"/> </figure> </div> </div> <div xml:id="echoid-div186" type="section" level="1" n="58"> <head xml:id="echoid-head63" xml:space="preserve">FINIS LIBRORVM ARCHIMEDIS DE <lb/>IIS, QVAE IN AQVA VEHVNTVR.</head> <pb file="0103" n="103"/> <pb file="0104" n="104"/> <pb file="0105" n="105"/> </div> <div xml:id="echoid-div187" type="section" level="1" n="59"> <head xml:id="echoid-head64" xml:space="preserve">FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</head> <head xml:id="echoid-head65" xml:space="preserve">LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORV M.</head> <figure> <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0105-01"/> </figure> </div> <div xml:id="echoid-div188" type="section" level="1" n="60"> <head xml:id="echoid-head66" xml:space="preserve">CVM PRIVILEGIO IN ANNOS X. <lb/>BONONIAE, <lb/>Ex Officina Alexandri Benacii.</head> <head xml:id="echoid-head67" xml:space="preserve">M D LXV.</head> <pb file="0106" n="106"/> <pb file="0107" n="107"/> </div> <div xml:id="echoid-div189" type="section" level="1" n="61"> <head xml:id="echoid-head68" xml:space="preserve">ALEXANDRO FARNESIO <lb/>CARDINALI AMPLISSIMO <lb/>ET OPTIMO.</head> <p> <s xml:id="echoid-s2722" xml:space="preserve">CVM multæ res in mathematicis <lb/>diſciplinis nequaquam ſatis ad-<lb/>huc explicatæ ſint, tum perdif-<lb/>ficilis, & </s> <s xml:id="echoid-s2723" xml:space="preserve">perobſcura quæſtio <lb/>eſt de centro grauitatis corpo-<lb/>rum ſolidorum; </s> <s xml:id="echoid-s2724" xml:space="preserve">quæ, & </s> <s xml:id="echoid-s2725" xml:space="preserve">ad co-<lb/>gnoſcendum pulcherrima eſt, <lb/>& </s> <s xml:id="echoid-s2726" xml:space="preserve">ad multa, quæ à mathematicis proponuntur, præ-<lb/>clare intelligenda maximum affert adiumentum. </s> <s xml:id="echoid-s2727" xml:space="preserve">de <lb/>qua neminem ex mathematicis, neque noſtra, neque <lb/>patrum noſtrorum memoria ſcriptum reliquiſſe ſci-<lb/>mus. </s> <s xml:id="echoid-s2728" xml:space="preserve">& </s> <s xml:id="echoid-s2729" xml:space="preserve">quamuis in earum monumentis literarum nõ <lb/>nulla reperiantur, ex quibus in hanc ſententiam addu <lb/>ci poſſumus, vt exiſtimemus hanc rem ab ijſdẽ vber-<lb/>rime tractatam eſſe; </s> <s xml:id="echoid-s2730" xml:space="preserve">tamen neſcio quo fato adhuc <lb/>in eiuſmodi librorum ignoratione verſamur. </s> <s xml:id="echoid-s2731" xml:space="preserve">Archi-<lb/>medes quidem mathematicorũ princeps in libello, <lb/>cuius inſcriptio eſt, κέντρα βάρων ἐπιπέδων, de centro pla-<lb/>norum copioſiſsime, atque acutiſsime conſcripſit: </s> <s xml:id="echoid-s2732" xml:space="preserve">& </s> <s xml:id="echoid-s2733" xml:space="preserve"><lb/>in eo explicando ſummã ingenii, & </s> <s xml:id="echoid-s2734" xml:space="preserve">ſcientiæ gloriã eſt <lb/>cõſecutus. </s> <s xml:id="echoid-s2735" xml:space="preserve">Sed de cognitione cẽtri grauitatis corporũ <lb/>ſolidorũ nulla in eius libris litera inuenitur. </s> <s xml:id="echoid-s2736" xml:space="preserve">non mul <lb/>tos abhinc annos <emph style="sc">Marcellvs i</emph>I. </s> <s xml:id="echoid-s2737" xml:space="preserve"><emph style="sc">Pont</emph>. </s> <s xml:id="echoid-s2738" xml:space="preserve"><emph style="sc">Max</emph>.</s> <s xml:id="echoid-s2739" xml:space="preserve"> <pb file="0108" n="108"/> cum adhuc Cardinalis eſſet, mihi, quæ ſua erat hu-<lb/>manitas, libros eiuſdem Archimedis de ijs, quæ ve-<lb/>huntur in aqua, latine redditos dono dedit. </s> <s xml:id="echoid-s2740" xml:space="preserve">hos cum <lb/>ego, ut aliorum ſtudia incitarem, emendãdos, & </s> <s xml:id="echoid-s2741" xml:space="preserve">cõ-<lb/>mentariis illuſtrandos ſuſcepiſſem, animaduerti dubi <lb/>tari non poſſe, quin Archimedes vel de hac materia <lb/>ſcripſiſſet, vel aliorum mathematicorum ſcripta per-<lb/>legiſſet. </s> <s xml:id="echoid-s2742" xml:space="preserve">nam in iis tum alia nonnulla, tum maxime <lb/>illam propoſitionem, ut euidentem, & </s> <s xml:id="echoid-s2743" xml:space="preserve">aliàs proba-<lb/>tam aſſumit, Centrũ grauitatis in portionibus conoi <lb/>dis rectan guli axem ita diuidere, vt pars, quæ ad verti <lb/>cem terminatur, alterius partis, quæ ad baſim dupla <lb/>ſit. </s> <s xml:id="echoid-s2744" xml:space="preserve">Verum hæc ad eam partem mathematicarum <lb/>diſciplinarum præcipue refertur, in qua de centro <lb/>grauitatis corporum ſolidorum tractatur. </s> <s xml:id="echoid-s2745" xml:space="preserve">non eſt au <lb/>tem conſentaneum Archimedem illum admirabilem <lb/>virum hanc propoſitionem ſibi argumentis con-<lb/>firmandam exiſtimaturum non fuiſſe, niſi eam vel <lb/>aliis in locis probauiſſet, vel ab aliis probatam eſſe <lb/>comperiſſet. </s> <s xml:id="echoid-s2746" xml:space="preserve">quamobrem nequid in iis libris intel-<lb/>ligendis deſiderari poſſet, ſtatui hanc etiam partem <lb/>vel à veteribus prætermiſſam, vel tractatam quidem, <lb/>ſed in tenebris iacentem, non intactam relinquere; <lb/></s> <s xml:id="echoid-s2747" xml:space="preserve">atque ex aſsidua mathematicorum, præſertim Archi-<lb/>medis lectione, quæ mihi in mentem venerunt, ea in <lb/>medium afferre; </s> <s xml:id="echoid-s2748" xml:space="preserve">ut centri grauitatis corporum ſoli-<lb/>dorum, ſi non perfectam, at certe aliquam noti- <pb file="0109" n="109"/> tiam haberemus. </s> <s xml:id="echoid-s2749" xml:space="preserve">Q uem meum laborem nō mathe-<lb/>maticis ſolum, verum iis etiam, qui naturæ obſcuri-<lb/>tate delectantur, nó iniucundam fore ſperaui: </s> <s xml:id="echoid-s2750" xml:space="preserve">multa <lb/>enim προβλήματα cognitiòne digniſsima, quæ ad vtrã-<lb/>que ſcientiam attinent, ſeſe legentibus obtuliſſent. <lb/></s> <s xml:id="echoid-s2751" xml:space="preserve">neque id vlli mirandum videri debet. </s> <s xml:id="echoid-s2752" xml:space="preserve">vt enim in cor-<lb/>poribus noſtris omnia membra, ex quibus certa quæ <lb/>dam officia naſcuntur, diuino quodam ordine inter <lb/>ſe implicata, & </s> <s xml:id="echoid-s2753" xml:space="preserve">colligata ſunt: </s> <s xml:id="echoid-s2754" xml:space="preserve">in iisq́; </s> <s xml:id="echoid-s2755" xml:space="preserve">admirabilis il-<lb/>la conſpiratio, quam σύμπνοιαν græci vocant, eluceſcit, <lb/>ita tres illæ Philoſophiæ (ut Ariſtotelis verbo vtar) <lb/>quæ veritatem ſ<unsure/>olam propoſitam habent, licet qui-<lb/>buſdam quaſi finibus ſuis regantur: </s> <s xml:id="echoid-s2756" xml:space="preserve">tamen earũ vna-<lb/>quæque per ſe ipſam quodammodo imperfecta eſt: </s> <s xml:id="echoid-s2757" xml:space="preserve"><lb/>neque altera ſine alterius auxilio plene comprehen-<lb/>di poteſt. </s> <s xml:id="echoid-s2758" xml:space="preserve">complures præterea mathematicorum no-<lb/>di ante hac explicatu difficillimi nullo negotio expe <lb/>diti eſſent: </s> <s xml:id="echoid-s2759" xml:space="preserve">atque (ut vno, verbo complectar) niſi <lb/>mea valde amo, tractationem hanc meam ſtudioſis <lb/>non mediocrem vtilitatem, & </s> <s xml:id="echoid-s2760" xml:space="preserve">magnam volupta-<lb/>tem allaturam eſſe mihi perſuaſi. </s> <s xml:id="echoid-s2761" xml:space="preserve">cum autem ad hoc <lb/>ſcribendum aggreſſus eſsem, allatus eſt ad me liber <lb/>Franciſci Maurolici Meſſanenſis, in quo vir ille do-<lb/>ctiſsimus, & </s> <s xml:id="echoid-s2762" xml:space="preserve">in iis diſciplinis exercitatiſsimus af-<lb/>firmabat ſe de centro grauitatis corporum ſolido-<lb/>rum conſcripſiſſe. </s> <s xml:id="echoid-s2763" xml:space="preserve">cum hoc intellexiſſem, ſuſtinui <lb/>me pauliſper: </s> <s xml:id="echoid-s2764" xml:space="preserve">tacitusque expectaui, dum opus cla- <pb file="0110" n="110"/> risſimi uiri, quem ſemper honoris cauſſa nomino, <lb/>in lucem proferretur: </s> <s xml:id="echoid-s2765" xml:space="preserve">mihi enim exploratisſimum <lb/>erat: </s> <s xml:id="echoid-s2766" xml:space="preserve">Franciſcum Maurolicum multo doctius, & </s> <s xml:id="echoid-s2767" xml:space="preserve"><lb/>exquiſitius hoc diſciplinarum genus ſcriptis ſuis tra <lb/>diturum. </s> <s xml:id="echoid-s2768" xml:space="preserve">ſed cum id tardius fieret, hoc eſt, ut ego <lb/>interpretor, diligentius, mihi diutius hac ſcriptione <lb/>non ſuperſedendum eſſe duxi, præſertim cum iam li-<lb/>bri Archimedis de iis, quæ uehuntur in aqua, opera <lb/>mea illuſtrati typis excudẽdi eſſent. </s> <s xml:id="echoid-s2769" xml:space="preserve">nec me alia cauſ <lb/>ſa impuliſſet, ut de centro grauitatis corporum ſoli-<lb/>dorum ſcriberem, niſi ut hac etiam ratione lux eis <lb/>quâm maxime fieri poſſet afferretur. </s> <s xml:id="echoid-s2770" xml:space="preserve">atq; </s> <s xml:id="echoid-s2771" xml:space="preserve">id eò mihi <lb/>faciendum exiſtimaui, quòd in ſpem ueniebam fore, <lb/>ut cum ego ex omnibus mathematicis primus, hanc <lb/>materiam explicandam ſuſcepiſſem; </s> <s xml:id="echoid-s2772" xml:space="preserve">ſi quid errati for <lb/>te à me commiſſum eſſet, boni uiri potius id meæ de <lb/>ſtudioſis hominibus bene merẽdi cupiditati, quàm <lb/>arrogantiæ aſcriberent. </s> <s xml:id="echoid-s2773" xml:space="preserve">reſtabat ut conſiderarem, cui <lb/>potisſimum ex principibus uiris contemplationem <lb/>hanc, nunc primum memoriæ, ac literis proditam de <lb/>dicarem. </s> <s xml:id="echoid-s2774" xml:space="preserve">harum mearum cogitationum ſumma fa-<lb/>cta, exiſtimaui nemini conuenientius de centro graui <lb/>tatis corporum opus dicari oportere, quàm <emph style="sc">Ale-<lb/>xandro</emph> <emph style="sc">Farnesio</emph> grauisſimo, ac prudentisſi-<lb/>mo Cardinali, quo in uiro ſumma fortuna ſemper cũ <lb/>ſumma uirtute certauit. </s> <s xml:id="echoid-s2775" xml:space="preserve">quid enim maxime in te ad-<lb/>mirari debeant homines, obſcurum eſt; </s> <s xml:id="echoid-s2776" xml:space="preserve">uſuḿne re- <pb file="0111" n="111"/> rum, qui pueritiæ tempus extremum principium ha <lb/>buiſti, & </s> <s xml:id="echoid-s2777" xml:space="preserve">imperiorũ, & </s> <s xml:id="echoid-s2778" xml:space="preserve">ad Reges, & </s> <s xml:id="echoid-s2779" xml:space="preserve">Imperatores ho-<lb/>norificentiſsimarum legationum; </s> <s xml:id="echoid-s2780" xml:space="preserve">an excellentiam <lb/>in omni genere literarum, qui vix adoleſcẽtulus, quæ <lb/>homines iam confirmata ætate ſummo ſtudio, diu-<lb/>turnisq́; </s> <s xml:id="echoid-s2781" xml:space="preserve">laboribus didicerunt, ſcientia, & </s> <s xml:id="echoid-s2782" xml:space="preserve">cogaitione <lb/>comprehendiſti: </s> <s xml:id="echoid-s2783" xml:space="preserve">an conſilium, & </s> <s xml:id="echoid-s2784" xml:space="preserve">ſapientiam in re-<lb/>gendis, & </s> <s xml:id="echoid-s2785" xml:space="preserve">gubernãdis Ciuitatibus, cuius grauiísimæ <lb/>ſententiæ in ſanctiſsimo Reip. </s> <s xml:id="echoid-s2786" xml:space="preserve">Chrſtianæ conſilio di-<lb/>ctæ, potius diuina oracula, quàm ſententiæ habitæ <lb/>ſunt, & </s> <s xml:id="echoid-s2787" xml:space="preserve">habentur. </s> <s xml:id="echoid-s2788" xml:space="preserve">prætermitto liberalitatem, & </s> <s xml:id="echoid-s2789" xml:space="preserve">mu-<lb/>nificentiam tuam, quam in ſtudio ſiſsimo quoque ho <lb/>neſtando quotidie magis oſtendis, ne videar auribus <lb/>tuis potius, quàm veritati ſeruire. </s> <s xml:id="echoid-s2790" xml:space="preserve">quamuis à te in tot <lb/>præclaros viros tanta beneficia collata ſunt, & </s> <s xml:id="echoid-s2791" xml:space="preserve">confe-<lb/>rũtur, vt omnibus teſtatum ſit, nihil tibi eſſe charius, <lb/>nihil iucundius, quàm eximia tua liberalitate homi-<lb/>nes ad amplexandam virtutem, licet currentes incita-<lb/>re. </s> <s xml:id="echoid-s2792" xml:space="preserve">nihil dico de ceteris virtutibus tuis, quæ tantæ <lb/>ſunt, quantæ ne cogitatione quidem comprehendi <lb/>poſſunt. </s> <s xml:id="echoid-s2793" xml:space="preserve">Quamobrem hac præcipue de cauſſa te hu-<lb/>ius meæ lucubrationis patronum eſſe volui, quam ea, <lb/>qua ſoles, humanitate accipies. </s> <s xml:id="echoid-s2794" xml:space="preserve">te enim ſemper ob <lb/>diuinas virtutes tuas colui, & </s> <s xml:id="echoid-s2795" xml:space="preserve">obſeruaui: </s> <s xml:id="echoid-s2796" xml:space="preserve">nihilq́; </s> <s xml:id="echoid-s2797" xml:space="preserve">mi-<lb/>hi fuit optatius; </s> <s xml:id="echoid-s2798" xml:space="preserve">quàm tibi perſpectum eſſe meum <lb/>erga te animum; </s> <s xml:id="echoid-s2799" xml:space="preserve">ſingularemq́; </s> <s xml:id="echoid-s2800" xml:space="preserve">obſeruantiam. </s> <s xml:id="echoid-s2801" xml:space="preserve">cœ-<lb/>lum igitur digito attingam, ſi poſt grauiſsimas oc- <pb file="0112" n="112"/> cupationes tuas legendo Federici tui libro aliquid <lb/>impertiri temporis non grauaberis: </s> <s xml:id="echoid-s2802" xml:space="preserve">eumq́; </s> <s xml:id="echoid-s2803" xml:space="preserve">in iis, qui <lb/>tibi ſemper addicti erunt, numerare. </s> <s xml:id="echoid-s2804" xml:space="preserve">Vale.</s> <s xml:id="echoid-s2805" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2806" xml:space="preserve">Federicus Commandinus.</s> <s xml:id="echoid-s2807" xml:space="preserve"/> </p> <pb o="1" file="0113" n="113"/> </div> <div xml:id="echoid-div190" type="section" level="1" n="62"> <head xml:id="echoid-head69" xml:space="preserve">FEDERICI COMMANDINI <lb/>VRBINATIS LIBER DE CENTRO <lb/>GRAVITATIS SOLIDORVM. <lb/>DIFFINITIONES.</head> <p> <s xml:id="echoid-s2808" xml:space="preserve"><emph style="sc">CEntrvm</emph> grauitatis, Pappus <lb/>Alexandrinus in octauo ma-<lb/>thematicarum collectionum <lb/>libro ita diffiniuit.</s> <s xml:id="echoid-s2809" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2810" xml:space="preserve">λέγομεν δἐ κἐντρον βάρ<unsure/>ους ἑκά στου σἀ<unsure/> <lb/>ματος {εἶ}ναι σημ\~ειον τικείμενον ἑντὸς, ἀφ<unsure/> <lb/>οὖ κα\’τ ἐποίνιαν ἀρτηθέν τό βάρ<unsure/>ος ἡμερ{εἶ} <lb/>φερ<unsure/>όμενον, καὶ φυλὰσσει τήν ἐξἀρχ\~νςθἐ-<lb/>σιν, οὐ μὴ περιτρ<unsure/>επ ὸμενον ἐντῆ φορ<unsure/>ᾶ. </s> <s xml:id="echoid-s2811" xml:space="preserve">hoc eſt,</s> </p> <p> <s xml:id="echoid-s2812" xml:space="preserve">Dicimus autem centrum grauitatis uniuſcu-<lb/>inſque corporis punctum quoddam intra poſi-<lb/>cum, à quo ſi graue appenſum mente concipia-<lb/>tur, dum fertur quieſcit; </s> <s xml:id="echoid-s2813" xml:space="preserve">& </s> <s xml:id="echoid-s2814" xml:space="preserve">ſeruat eam, quam in <lb/>principio habebat poſitionem: </s> <s xml:id="echoid-s2815" xml:space="preserve">neque in ipſa la-<lb/>tione circumuertitur.</s> <s xml:id="echoid-s2816" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2817" xml:space="preserve">Poſſumus etiam hoc modo diffinire.</s> <s xml:id="echoid-s2818" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2819" xml:space="preserve">Centrum grauitatis uniuſcuiuſque ſolidæ figu <lb/>ræ eſt punctum illud intra poſitum, circa quod <lb/>undique partes æqualium momentorum conſi-<lb/>ſtunt. </s> <s xml:id="echoid-s2820" xml:space="preserve">ſi enim per tale centrum ducatur planum <lb/>figuram quomodocunque ſecans ſemper in par- <pb file="0114" n="114" rhead="FED. COMMANDINI"/> tes æqueponderantes ipſam diuidet.</s> <s xml:id="echoid-s2821" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2822" xml:space="preserve">2 Priſmatis, cylindri, & </s> <s xml:id="echoid-s2823" xml:space="preserve">portionis cylindri axem <lb/>appello rectam lineam, quæ oppoſitorum plano-<lb/>rum centra grauitatis coniungit.</s> <s xml:id="echoid-s2824" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2825" xml:space="preserve">3 Pyramidis, coni, & </s> <s xml:id="echoid-s2826" xml:space="preserve">portionis coni axem dico li <lb/>neam, quæ à uertice ad centrum grauitatis baſis <lb/>perducitur.</s> <s xml:id="echoid-s2827" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2828" xml:space="preserve">4 Si pyramis, conus, portio coni, uel conoidis ſe-<lb/>cetur plano baſi æquidiſtante, pars, quæ eſt ad ba-<lb/>ſim, fruſtum pyramidis, coni, portionis coni, uel <lb/>conoidis dicetur; </s> <s xml:id="echoid-s2829" xml:space="preserve">quorum plana æquidiſtantia, <lb/>quæ opponuntur ſimilia ſunt, & </s> <s xml:id="echoid-s2830" xml:space="preserve">inæqualia: </s> <s xml:id="echoid-s2831" xml:space="preserve">axes <lb/>uero ſunt axium figurarum partes, quæ in ipſis <lb/>comprehenduntur.</s> <s xml:id="echoid-s2832" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div191" type="section" level="1" n="63"> <head xml:id="echoid-head70" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s2833" xml:space="preserve">1 Solidarum figurarum ſimilium centra grauita-<lb/>tis ſimiliter ſunt poſita.</s> <s xml:id="echoid-s2834" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2835" xml:space="preserve">2 Solidis figuris ſimilibus, & </s> <s xml:id="echoid-s2836" xml:space="preserve">æqualibus inter ſe <lb/>aptatis, centra quoque grauitatis ipſarum inter ſe <lb/>aptata erunt.</s> <s xml:id="echoid-s2837" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div192" type="section" level="1" n="64"> <head xml:id="echoid-head71" xml:space="preserve">THEOREMA I. PROPOSITIO I.</head> <p> <s xml:id="echoid-s2838" xml:space="preserve">Omnis figuræ rectilineæ in circulo deſcriptæ, <lb/>quæ æqualibus lateribus, & </s> <s xml:id="echoid-s2839" xml:space="preserve">angulis contine- <pb o="2" file="0115" n="115" rhead="DE CENTRO GRAVIT. SOLID."/> tur, centrum grauitatis eſt idem, quod circuli cen <lb/>trum.</s> <s xml:id="echoid-s2840" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2841" xml:space="preserve">Sit primo triangulum æquilaterum a b c in circulo de-<lb/>ſcriptum: </s> <s xml:id="echoid-s2842" xml:space="preserve">& </s> <s xml:id="echoid-s2843" xml:space="preserve">diuiſa a c bifariam in d, ducatur b d. </s> <s xml:id="echoid-s2844" xml:space="preserve">erit in li-<lb/>nea b d centrum grauitatis triãguli a b c, ex tertia decima <lb/>primi libri Archimedis de centro grauitatis planorum. </s> <s xml:id="echoid-s2845" xml:space="preserve">Et <lb/>quoniam linea a b eſt æqualis <lb/> <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> lineæ b c; </s> <s xml:id="echoid-s2846" xml:space="preserve">& </s> <s xml:id="echoid-s2847" xml:space="preserve">a d ipſi d c; </s> <s xml:id="echoid-s2848" xml:space="preserve">eſtq́; <lb/></s> <s xml:id="echoid-s2849" xml:space="preserve">b d utrique communis: </s> <s xml:id="echoid-s2850" xml:space="preserve">trian-<lb/>gulum a b d æquale erit trian <lb/> <anchor type="note" xlink:label="note-0115-01a" xlink:href="note-0115-01"/> gulo c b d: </s> <s xml:id="echoid-s2851" xml:space="preserve">& </s> <s xml:id="echoid-s2852" xml:space="preserve">anguli angulis æ-<lb/>quales, qui æqualibus lateri-<lb/>bus ſubtenduntur. </s> <s xml:id="echoid-s2853" xml:space="preserve">ergo angu <lb/> <anchor type="note" xlink:label="note-0115-02a" xlink:href="note-0115-02"/> li ad d utriq; </s> <s xml:id="echoid-s2854" xml:space="preserve">recti ſunt. </s> <s xml:id="echoid-s2855" xml:space="preserve">quòd <lb/>cum linea b d ſecet a c biſa-<lb/>riam, & </s> <s xml:id="echoid-s2856" xml:space="preserve">ad angulos rectos; </s> <s xml:id="echoid-s2857" xml:space="preserve">in <lb/> <anchor type="note" xlink:label="note-0115-03a" xlink:href="note-0115-03"/> ipſa b d eſt centrum circuli. <lb/></s> <s xml:id="echoid-s2858" xml:space="preserve">quare in eadem b d linea erit <lb/>centrum grauitatis trianguli, & </s> <s xml:id="echoid-s2859" xml:space="preserve">circuli centrum. </s> <s xml:id="echoid-s2860" xml:space="preserve">Similiter <lb/>diuiſa a b bifariam in e, & </s> <s xml:id="echoid-s2861" xml:space="preserve">ducta c e, oſtendetur in ipſa utrũ <lb/>que centrum contineri. </s> <s xml:id="echoid-s2862" xml:space="preserve">ergo ea erunt in puncto, in quo li-<lb/>neæ b d, c e conueniunt. </s> <s xml:id="echoid-s2863" xml:space="preserve">trianguli igitur a b c centrum gra <lb/>uitatis eſt idem, quod circuli centrum.</s> <s xml:id="echoid-s2864" xml:space="preserve"/> </p> <div xml:id="echoid-div192" type="float" level="2" n="1"> <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a"> <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0115-01"/> </figure> <note position="right" xlink:label="note-0115-01" xlink:href="note-0115-01a" xml:space="preserve">8. primi.</note> <note position="right" xlink:label="note-0115-02" xlink:href="note-0115-02a" xml:space="preserve">13. primi.</note> <note position="right" xlink:label="note-0115-03" xlink:href="note-0115-03a" xml:space="preserve">corol. p@@ <lb/>mæ tertii</note> </div> <p> <s xml:id="echoid-s2865" xml:space="preserve">Sit quadratum a b c d in cir-<lb/> <anchor type="figure" xlink:label="fig-0115-02a" xlink:href="fig-0115-02"/> culo deſcriptum: </s> <s xml:id="echoid-s2866" xml:space="preserve">& </s> <s xml:id="echoid-s2867" xml:space="preserve">ducantur <lb/>a c, b d, quæ conueniant in e. </s> <s xml:id="echoid-s2868" xml:space="preserve">er-<lb/>go punctum e eſt centrum gra <lb/>uitatis quadrati, ex decima eiuſ <lb/>dem libri Archimedis. </s> <s xml:id="echoid-s2869" xml:space="preserve">Sed cum <lb/>omnes anguli ad a b c d recti <lb/>ſint; </s> <s xml:id="echoid-s2870" xml:space="preserve">erit a b c femicirculus: <lb/></s> <s xml:id="echoid-s2871" xml:space="preserve"> <anchor type="note" xlink:label="note-0115-04a" xlink:href="note-0115-04"/> itemq́; </s> <s xml:id="echoid-s2872" xml:space="preserve">b c d: </s> <s xml:id="echoid-s2873" xml:space="preserve">& </s> <s xml:id="echoid-s2874" xml:space="preserve">propterea li-<lb/>neæ a c, b d diametri circuli:</s> <s xml:id="echoid-s2875" xml:space="preserve"> <pb file="0116" n="116" rhead="FED. COMMANDINI"/> quæ quidem in centro conueniunt. </s> <s xml:id="echoid-s2876" xml:space="preserve">idem igitur eſt centrum <lb/>grauitatis quadrati, & </s> <s xml:id="echoid-s2877" xml:space="preserve">circuli centrum.</s> <s xml:id="echoid-s2878" xml:space="preserve"/> </p> <div xml:id="echoid-div193" type="float" level="2" n="2"> <figure xlink:label="fig-0115-02" xlink:href="fig-0115-02a"> <image file="0115-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0115-02"/> </figure> <note position="right" xlink:label="note-0115-04" xlink:href="note-0115-04a" xml:space="preserve">51. tortil.</note> </div> <p> <s xml:id="echoid-s2879" xml:space="preserve">Sit pentagonum æquilaterum, & </s> <s xml:id="echoid-s2880" xml:space="preserve">æquiangulum in circu-<lb/>lo deſcriptum a b c d e: </s> <s xml:id="echoid-s2881" xml:space="preserve">& </s> <s xml:id="echoid-s2882" xml:space="preserve">iun-<lb/> <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> cta b d, bifariamq́; </s> <s xml:id="echoid-s2883" xml:space="preserve">in ſ diuiſa, <lb/>ducatur c f, & </s> <s xml:id="echoid-s2884" xml:space="preserve">producatur ad <lb/>circuli circumferentiam in g; <lb/></s> <s xml:id="echoid-s2885" xml:space="preserve">quæ lineam a e in h ſecet: </s> <s xml:id="echoid-s2886" xml:space="preserve">de-<lb/>inde iungantur a c, c e. </s> <s xml:id="echoid-s2887" xml:space="preserve">Eodem <lb/>modo, quo ſupra demonſtra-<lb/>bimus angulum b c f æqualem <lb/>eſſe angulo d c f; </s> <s xml:id="echoid-s2888" xml:space="preserve">& </s> <s xml:id="echoid-s2889" xml:space="preserve">angulos <lb/>ad f utroſque rectos: </s> <s xml:id="echoid-s2890" xml:space="preserve">& </s> <s xml:id="echoid-s2891" xml:space="preserve">idcir-<lb/>colineam c f g per circuli cen <lb/>trum tranſire. </s> <s xml:id="echoid-s2892" xml:space="preserve">Quoniam igi-<lb/>tur latera c b, b a, & </s> <s xml:id="echoid-s2893" xml:space="preserve">c d, d e æqualia ſunt; </s> <s xml:id="echoid-s2894" xml:space="preserve">& </s> <s xml:id="echoid-s2895" xml:space="preserve">æquales anguli <lb/>c b a, c d e: </s> <s xml:id="echoid-s2896" xml:space="preserve">erit baſis c a baſi c e, & </s> <s xml:id="echoid-s2897" xml:space="preserve">angulus b c a angulo <lb/> <anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/> d c e æqualis. </s> <s xml:id="echoid-s2898" xml:space="preserve">ergo & </s> <s xml:id="echoid-s2899" xml:space="preserve">reliquus a c h, reliquo e c h. </s> <s xml:id="echoid-s2900" xml:space="preserve">eſt au-<lb/>tem c h utrique triangulo a c h, e c h communis. </s> <s xml:id="echoid-s2901" xml:space="preserve">quare <lb/>baſis a h æqualis eſt baſi h e: </s> <s xml:id="echoid-s2902" xml:space="preserve">& </s> <s xml:id="echoid-s2903" xml:space="preserve">anguli, quiad h recti: </s> <s xml:id="echoid-s2904" xml:space="preserve">ſuntq́; <lb/></s> <s xml:id="echoid-s2905" xml:space="preserve">recti, qui ad f. </s> <s xml:id="echoid-s2906" xml:space="preserve">ergo lineæ a e, b d inter ſe ſe æquidiſtant. </s> <s xml:id="echoid-s2907" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0116-02a" xlink:href="note-0116-02"/> Itaque cum trapezij a b d e latera b d, a e æquidiſtantia à li <lb/>nea fh bifariam diuidantur; </s> <s xml:id="echoid-s2908" xml:space="preserve">centrum grauitatis ipſius erit <lb/>in linea f h, ex ultima eiuſdem libri Archimedis. </s> <s xml:id="echoid-s2909" xml:space="preserve">Sed trian-<lb/> <anchor type="note" xlink:label="note-0116-03a" xlink:href="note-0116-03"/> guli b c d centrum grauitatis eſt in linea c f. </s> <s xml:id="echoid-s2910" xml:space="preserve">ergo in eadem <lb/>linea c h eſt centrum grauitatis trapezij a b d e, & </s> <s xml:id="echoid-s2911" xml:space="preserve">trian-<lb/>guli b c d: </s> <s xml:id="echoid-s2912" xml:space="preserve">hoc eſt pentagoni ipſius centrum & </s> <s xml:id="echoid-s2913" xml:space="preserve">centrum <lb/>circuli. </s> <s xml:id="echoid-s2914" xml:space="preserve">Rurſus ſi iuncta a d, bifariamq́; </s> <s xml:id="echoid-s2915" xml:space="preserve">ſecta in k, duca-<lb/>tur e k l: </s> <s xml:id="echoid-s2916" xml:space="preserve">demonſtrabimus in ipſa utrumque centrum in <lb/>eſſe. </s> <s xml:id="echoid-s2917" xml:space="preserve">Sequitur ergo, ut punctum, in quo lineæ c g, e l con-<lb/>ueniunt, idem ſit centrum circuli, & </s> <s xml:id="echoid-s2918" xml:space="preserve">centrum grauitatis <lb/>pentagoni.</s> <s xml:id="echoid-s2919" xml:space="preserve"/> </p> <div xml:id="echoid-div194" type="float" level="2" n="3"> <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a"> <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0116-01"/> </figure> <note position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">4. Primi.</note> <note position="left" xlink:label="note-0116-02" xlink:href="note-0116-02a" xml:space="preserve">08. primi.</note> <note position="left" xlink:label="note-0116-03" xlink:href="note-0116-03a" xml:space="preserve">13. Archi-<lb/>medis.</note> </div> <p> <s xml:id="echoid-s2920" xml:space="preserve">Sit hexagonum a b c d e f æquilaterum, & </s> <s xml:id="echoid-s2921" xml:space="preserve">æquiangulum <lb/>in circulo deſignatum: </s> <s xml:id="echoid-s2922" xml:space="preserve">iunganturq́; </s> <s xml:id="echoid-s2923" xml:space="preserve">b d, a c: </s> <s xml:id="echoid-s2924" xml:space="preserve">& </s> <s xml:id="echoid-s2925" xml:space="preserve">bifariam ſe- <pb o="3" file="0117" n="117" rhead="DE CENTRO GRAVIT. SOLID."/> cta b d in g puncto, ducatur c g; </s> <s xml:id="echoid-s2926" xml:space="preserve">& </s> <s xml:id="echoid-s2927" xml:space="preserve">protrahatur ad circuli <lb/>uſque circumferentiam; </s> <s xml:id="echoid-s2928" xml:space="preserve">quæ ſecet a e in h. </s> <s xml:id="echoid-s2929" xml:space="preserve">Similiter conclu <lb/>demus c g per centrum circuli tranſire: </s> <s xml:id="echoid-s2930" xml:space="preserve">& </s> <s xml:id="echoid-s2931" xml:space="preserve">bifariam ſecare <lb/>lineam a e; </s> <s xml:id="echoid-s2932" xml:space="preserve">itemq́; </s> <s xml:id="echoid-s2933" xml:space="preserve">lineas b d, a e inter ſe æquidiſtantes eſſe. <lb/></s> <s xml:id="echoid-s2934" xml:space="preserve">Cumigitur c g per centrum circuli tranſeat; </s> <s xml:id="echoid-s2935" xml:space="preserve">& </s> <s xml:id="echoid-s2936" xml:space="preserve">ad punctũ <lb/>f perueniat neceſſe eſt: </s> <s xml:id="echoid-s2937" xml:space="preserve">quòd c d e f ſit dimidium circumfe <lb/>rentiæ circuli. </s> <s xml:id="echoid-s2938" xml:space="preserve">Quare in eadem <lb/> <anchor type="figure" xlink:label="fig-0117-01a" xlink:href="fig-0117-01"/> diametro c f erunt centra gra <lb/> <anchor type="note" xlink:label="note-0117-01a" xlink:href="note-0117-01"/> uitatis triangulorum b c d, <lb/>a f e, & </s> <s xml:id="echoid-s2939" xml:space="preserve">quadrilateri a b d e, ex <lb/> <anchor type="note" xlink:label="note-0117-02a" xlink:href="note-0117-02"/> quibus conſtat hexagonum a b <lb/>c d e f. </s> <s xml:id="echoid-s2940" xml:space="preserve">perſpicuum eſt igitur in <lb/>ipſa c f eſſe circuli centrum, & </s> <s xml:id="echoid-s2941" xml:space="preserve"><lb/>centrum grauitatis hexagoni. <lb/></s> <s xml:id="echoid-s2942" xml:space="preserve">Rurſus ducta altera diametro <lb/>a d, eiſdem rationibus oſtende-<lb/>mus in ipſa utrumque cẽtrum <lb/>ineſſe. </s> <s xml:id="echoid-s2943" xml:space="preserve">Centrum ergo grauita-<lb/>tis hexagoni, & </s> <s xml:id="echoid-s2944" xml:space="preserve">centrum circuli idem erit.</s> <s xml:id="echoid-s2945" xml:space="preserve"/> </p> <div xml:id="echoid-div195" type="float" level="2" n="4"> <figure xlink:label="fig-0117-01" xlink:href="fig-0117-01a"> <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0117-01"/> </figure> <note position="right" xlink:label="note-0117-01" xlink:href="note-0117-01a" xml:space="preserve">13. Archi <lb/>medis.</note> <note position="right" xlink:label="note-0117-02" xlink:href="note-0117-02a" xml:space="preserve">9. @iuſdé.</note> </div> <p> <s xml:id="echoid-s2946" xml:space="preserve">Sit heptagonum a b c d e f g æquilaterum atque æquian <lb/>gulum in circulo deſcriptum: <lb/></s> <s xml:id="echoid-s2947" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0117-02a" xlink:href="fig-0117-02"/> & </s> <s xml:id="echoid-s2948" xml:space="preserve">iungantur c e, b f, a g: </s> <s xml:id="echoid-s2949" xml:space="preserve">di-<lb/>uiſa autem c e bifariam in pũ <lb/>cto h: </s> <s xml:id="echoid-s2950" xml:space="preserve">& </s> <s xml:id="echoid-s2951" xml:space="preserve">iuncta d h produca-<lb/>tur in k. </s> <s xml:id="echoid-s2952" xml:space="preserve">non aliter demon-<lb/>ſtrabimus in linea d k eſſe cen <lb/>trum circuli, & </s> <s xml:id="echoid-s2953" xml:space="preserve">centrum gra-<lb/>uitatis trianguli c d e, & </s> <s xml:id="echoid-s2954" xml:space="preserve">tra-<lb/>peziorum b c e f, a b f g, hoc <lb/>eſt centrum totius heptago-<lb/>ni: </s> <s xml:id="echoid-s2955" xml:space="preserve">& </s> <s xml:id="echoid-s2956" xml:space="preserve">rurſus eadem centra in <lb/>alia diametro cl ſimiliter du-<lb/>cta contineri. </s> <s xml:id="echoid-s2957" xml:space="preserve">Quare & </s> <s xml:id="echoid-s2958" xml:space="preserve">centrum grauitatis heptagoni, & </s> <s xml:id="echoid-s2959" xml:space="preserve"><lb/>centrum circuli in idem punctum conucniunt. </s> <s xml:id="echoid-s2960" xml:space="preserve">Eodem mo <pb file="0118" n="118" rhead="FED. COMMANDINI"/> do in reliquis figuris æquilateris, & </s> <s xml:id="echoid-s2961" xml:space="preserve">æquiangulis, quæ in cir-<lb/>culo deſcribuntur, probabimus cẽtrum grauitatis earum, <lb/>& </s> <s xml:id="echoid-s2962" xml:space="preserve">centrum circuli idem eſſe. </s> <s xml:id="echoid-s2963" xml:space="preserve">quod quidem demonſtrare <lb/>oportebat.</s> <s xml:id="echoid-s2964" xml:space="preserve"/> </p> <div xml:id="echoid-div196" type="float" level="2" n="5"> <figure xlink:label="fig-0117-02" xlink:href="fig-0117-02a"> <image file="0117-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0117-02"/> </figure> </div> <p> <s xml:id="echoid-s2965" xml:space="preserve">Ex quibus apparet cuiuslibet figuræ rectilineæ <lb/>in circulo plane deſcriptæ centrum grauitatis idẽ <lb/>eſſe, quod & </s> <s xml:id="echoid-s2966" xml:space="preserve">circuli centrum.</s> <s xml:id="echoid-s2967" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2968" xml:space="preserve">Figuram in circulo plane deſcriptam appella-<lb/> <anchor type="note" xlink:label="note-0118-01a" xlink:href="note-0118-01"/> mus, cuiuſmodi eſt ea, quæ in duodecimo elemen <lb/>torum libro, propoſitione ſecunda deſcribitur. <lb/></s> <s xml:id="echoid-s2969" xml:space="preserve">ex æqualibus enim lateribus, & </s> <s xml:id="echoid-s2970" xml:space="preserve">angulis conſtare <lb/>perſpicuum eſt.</s> <s xml:id="echoid-s2971" xml:space="preserve"/> </p> <div xml:id="echoid-div197" type="float" level="2" n="6"> <note position="left" xlink:label="note-0118-01" xlink:href="note-0118-01a" xml:space="preserve">γνωρ@ μω@</note> </div> </div> <div xml:id="echoid-div199" type="section" level="1" n="65"> <head xml:id="echoid-head72" xml:space="preserve">THEOREMA II. PROPOSITIO II.</head> <p> <s xml:id="echoid-s2972" xml:space="preserve">Omnis figuræ rectilineæ in ellipſi plane deſcri-<lb/>ptæ centrum grauitatis eſt idem, quod ellipſis <lb/>centrum.</s> <s xml:id="echoid-s2973" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2974" xml:space="preserve">Quo modo figura rectilinea in ellipſi plane deſcribatur, <lb/>docuimus in commentarijs in quintam propoſitionem li-<lb/>bri Archimedis de conoidibus, & </s> <s xml:id="echoid-s2975" xml:space="preserve">ſphæroidibus.</s> <s xml:id="echoid-s2976" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2977" xml:space="preserve">Sit ellipſis a b c d, cuius maior axis a c, minor b d: </s> <s xml:id="echoid-s2978" xml:space="preserve">iun-<lb/>ganturq́; </s> <s xml:id="echoid-s2979" xml:space="preserve">a b, b c, c d, d a: </s> <s xml:id="echoid-s2980" xml:space="preserve">& </s> <s xml:id="echoid-s2981" xml:space="preserve">bifariam diuidantur in pun-<lb/>ctis e f g h. </s> <s xml:id="echoid-s2982" xml:space="preserve">à centro autem, quod ſit k ductæ lineæ k e, k f, <lb/>k g, k h uſque ad ſectionem in puncta l m n o protrahan-<lb/>tur: </s> <s xml:id="echoid-s2983" xml:space="preserve">& </s> <s xml:id="echoid-s2984" xml:space="preserve">iungantur l m, m n, n o, o l, ita ut a c ſecet li-<lb/>neas l o, m n, in z φ punctis, & </s> <s xml:id="echoid-s2985" xml:space="preserve">b d ſecet l m, o n in χ ψ. <lb/></s> <s xml:id="echoid-s2986" xml:space="preserve">erunt l k, k n linea una, itemq́ue linea unaipſæ m k, k o: </s> <s xml:id="echoid-s2987" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s2988" xml:space="preserve">lineæ b a, c d æquidiſtabunt lineæ m o: </s> <s xml:id="echoid-s2989" xml:space="preserve">& </s> <s xml:id="echoid-s2990" xml:space="preserve">b c, a d ipſi <lb/>l n. </s> <s xml:id="echoid-s2991" xml:space="preserve">rurſus l o, m n axi b d æquidiſtabunt: </s> <s xml:id="echoid-s2992" xml:space="preserve">& </s> <s xml:id="echoid-s2993" xml:space="preserve">l m, <pb o="4" file="0119" n="119" rhead="DE CENTRO GRAVIT. SOLID."/> o n ipſi a c. </s> <s xml:id="echoid-s2994" xml:space="preserve">Quoniam enim triangulorum a b k, a d k, latus <lb/>b k eſt æquale lateri k d, & </s> <s xml:id="echoid-s2995" xml:space="preserve">a k utrique commune; </s> <s xml:id="echoid-s2996" xml:space="preserve">anguliq́; <lb/></s> <s xml:id="echoid-s2997" xml:space="preserve">ad k recti baſis a b baſi a d; </s> <s xml:id="echoid-s2998" xml:space="preserve">& </s> <s xml:id="echoid-s2999" xml:space="preserve">reliqui anguli reliquis an-<lb/> <anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/> gulis æquales erunt. </s> <s xml:id="echoid-s3000" xml:space="preserve">eadem quoqueratione oſtendetur b c <lb/>æqualis c d; </s> <s xml:id="echoid-s3001" xml:space="preserve">& </s> <s xml:id="echoid-s3002" xml:space="preserve">a b ipſi <lb/> <anchor type="figure" xlink:label="fig-0119-01a" xlink:href="fig-0119-01"/> b c. </s> <s xml:id="echoid-s3003" xml:space="preserve">quare omnes a b, <lb/>b c, c d, d a ſunt æqua-<lb/>les. </s> <s xml:id="echoid-s3004" xml:space="preserve">& </s> <s xml:id="echoid-s3005" xml:space="preserve">quoniam anguli <lb/>ad a æquales ſunt angu <lb/>lis ad c; </s> <s xml:id="echoid-s3006" xml:space="preserve">erunt anguli b <lb/>a c, a c d coalterni inter <lb/>ſe æquales; </s> <s xml:id="echoid-s3007" xml:space="preserve">itemq́; </s> <s xml:id="echoid-s3008" xml:space="preserve">d a c, <lb/>a c b. </s> <s xml:id="echoid-s3009" xml:space="preserve">ergo c d ipſi b a; <lb/></s> <s xml:id="echoid-s3010" xml:space="preserve">& </s> <s xml:id="echoid-s3011" xml:space="preserve">a d ipſi b c æquidi-<lb/>ſtat. </s> <s xml:id="echoid-s3012" xml:space="preserve">Atuero cum lineæ <lb/>a b, c d inter ſe æquidi-<lb/>ſtantes bifariam ſecen-<lb/>tur in punctis e g; </s> <s xml:id="echoid-s3013" xml:space="preserve">erit li <lb/>nea l e k g n diameter ſe <lb/>ctionis, & </s> <s xml:id="echoid-s3014" xml:space="preserve">linea una, ex <lb/>demonſtratis in uigeſi-<lb/>ma octaua ſecundi coni <lb/>corum. </s> <s xml:id="echoid-s3015" xml:space="preserve">Et eadem ratione linea una m f k h o. </s> <s xml:id="echoid-s3016" xml:space="preserve">Sunt autẽ a d, <lb/>b c inter ſe ſe æquales, & </s> <s xml:id="echoid-s3017" xml:space="preserve">æquidiſtantes. </s> <s xml:id="echoid-s3018" xml:space="preserve">quare & </s> <s xml:id="echoid-s3019" xml:space="preserve">earum di-<lb/>midiæ a h, b f; </s> <s xml:id="echoid-s3020" xml:space="preserve">itemq́; </s> <s xml:id="echoid-s3021" xml:space="preserve">h d, f e; </s> <s xml:id="echoid-s3022" xml:space="preserve">& </s> <s xml:id="echoid-s3023" xml:space="preserve">quæ ipſas coniunguntrectæ <lb/> <anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/> lineæ æquales, & </s> <s xml:id="echoid-s3024" xml:space="preserve">æquidiſtantes erunt. </s> <s xml:id="echoid-s3025" xml:space="preserve">æquidiſtãt igitur b a, <lb/>c d diametro m o: </s> <s xml:id="echoid-s3026" xml:space="preserve">& </s> <s xml:id="echoid-s3027" xml:space="preserve">pariter a d, b c ipſi l n æquidiſtare o-<lb/>ſtendemus. </s> <s xml:id="echoid-s3028" xml:space="preserve">Si igitur manẽte diametro a c intelligatur a b c <lb/>portio ellipſis ad portionem a d c moueri, cum primum b <lb/>applicuerit ad d, cõgruet tota portio toti portioni, lineaq́; <lb/></s> <s xml:id="echoid-s3029" xml:space="preserve">b a lineæ a d; </s> <s xml:id="echoid-s3030" xml:space="preserve">& </s> <s xml:id="echoid-s3031" xml:space="preserve">b c ipſi c d congruet: </s> <s xml:id="echoid-s3032" xml:space="preserve">punctum uero e ca-<lb/>det in h; </s> <s xml:id="echoid-s3033" xml:space="preserve">f in g: </s> <s xml:id="echoid-s3034" xml:space="preserve">& </s> <s xml:id="echoid-s3035" xml:space="preserve">linea k e in lineam k h: </s> <s xml:id="echoid-s3036" xml:space="preserve">& </s> <s xml:id="echoid-s3037" xml:space="preserve">k f in k g. </s> <s xml:id="echoid-s3038" xml:space="preserve">qua <lb/>re & </s> <s xml:id="echoid-s3039" xml:space="preserve">el in h o, et fm in g n. </s> <s xml:id="echoid-s3040" xml:space="preserve">Atipſa lz in z o; </s> <s xml:id="echoid-s3041" xml:space="preserve">et m φ in φ n <lb/>cadet. </s> <s xml:id="echoid-s3042" xml:space="preserve">congruet igitur triangulum l k z triangulo o k z: </s> <s xml:id="echoid-s3043" xml:space="preserve">et <pb file="0120" n="120" rhead="FED. COMMANDINI"/> triangulum m k φ triangulo n k φ. </s> <s xml:id="echoid-s3044" xml:space="preserve">ergo anguli l z k, o z k, <lb/>m φ k, n φ k æquales ſunt, ac recti. </s> <s xml:id="echoid-s3045" xml:space="preserve">quòd cum etiam recti <lb/>ſint, qui ad k; </s> <s xml:id="echoid-s3046" xml:space="preserve">æquidiſtabunt lineæ l o, m n axi b d. </s> <s xml:id="echoid-s3047" xml:space="preserve">& </s> <s xml:id="echoid-s3048" xml:space="preserve">ita. <lb/></s> <s xml:id="echoid-s3049" xml:space="preserve"> <anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/> demonſtrabuntur l m, o n ipſi a c æquidiſtare. </s> <s xml:id="echoid-s3050" xml:space="preserve">Rurſus ſi <lb/>iungantur a l, l b, b m, m c, c n, n d, d o, o a: </s> <s xml:id="echoid-s3051" xml:space="preserve">& </s> <s xml:id="echoid-s3052" xml:space="preserve">bifariam di <lb/>uidantur: </s> <s xml:id="echoid-s3053" xml:space="preserve">à centro autem k ad diuiſiones ductæ lineæ pro-<lb/>trahantur uſque ad ſectionem in puncta p q r s t u x y: </s> <s xml:id="echoid-s3054" xml:space="preserve">& </s> <s xml:id="echoid-s3055" xml:space="preserve">po <lb/>ſtremo p y, q x, r u, s t, q r, p s, y t, x u coniungantur. </s> <s xml:id="echoid-s3056" xml:space="preserve">Simili-<lb/>ter oſtendemus lineas <lb/> <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> p y, q x, r u, s t axi b d æ-<lb/>quidiſtantes eſſe: </s> <s xml:id="echoid-s3057" xml:space="preserve">& </s> <s xml:id="echoid-s3058" xml:space="preserve">q r, <lb/>p s, y t, x u æquidiſtan-<lb/>tesipſi a c. </s> <s xml:id="echoid-s3059" xml:space="preserve">Itaque dico <lb/>harum figurarum in el-<lb/>lipſi deſcriptarum cen-<lb/>trum grauitatis eſſe pũ-<lb/>ctum k, idem quod & </s> <s xml:id="echoid-s3060" xml:space="preserve">el <lb/>lipſis centrum. </s> <s xml:id="echoid-s3061" xml:space="preserve">quadri-<lb/>lateri enim a b c d cen-<lb/>trum eſt k, ex decima e-<lb/>iuſdem libri Archime-<lb/>dis, quippe cũ in eo om <lb/>nes diametri cõueniãt. <lb/></s> <s xml:id="echoid-s3062" xml:space="preserve">Sed in figura alb m c n <lb/> <anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/> d o, quoniam trianguli <lb/>alb centrum grauitatis <lb/> <anchor type="note" xlink:label="note-0120-03a" xlink:href="note-0120-03"/> eſt in linea l e: </s> <s xml:id="echoid-s3063" xml:space="preserve">trapezijq́; </s> <s xml:id="echoid-s3064" xml:space="preserve">a b m o centrum in linea e k: </s> <s xml:id="echoid-s3065" xml:space="preserve">trape <lb/>zij o m c d in k g: </s> <s xml:id="echoid-s3066" xml:space="preserve">& </s> <s xml:id="echoid-s3067" xml:space="preserve">trianguli c n d in ipſa g n: </s> <s xml:id="echoid-s3068" xml:space="preserve">erit magnitu <lb/>dinis ex his omnibus conſtantis, uidelicet totius figuræ cen <lb/>trum grauitatis in linea l n: </s> <s xml:id="echoid-s3069" xml:space="preserve">& </s> <s xml:id="echoid-s3070" xml:space="preserve">o b eandem cauſſam in linea <lb/>o m. </s> <s xml:id="echoid-s3071" xml:space="preserve">eſt enim trianguli a o d centrum in linea o h: </s> <s xml:id="echoid-s3072" xml:space="preserve">trapezij <lb/>a l n d in h k: </s> <s xml:id="echoid-s3073" xml:space="preserve">trapezij l b c n in k f: </s> <s xml:id="echoid-s3074" xml:space="preserve">& </s> <s xml:id="echoid-s3075" xml:space="preserve">trianguli b m c in fm. <lb/></s> <s xml:id="echoid-s3076" xml:space="preserve">cum ergo figuræ a l b m c n d o centrum grauitatis ſit in li-<lb/>nea l n, & </s> <s xml:id="echoid-s3077" xml:space="preserve">in linea o m; </s> <s xml:id="echoid-s3078" xml:space="preserve">erit centrum ipſius punctum k, in <pb o="5" file="0121" n="121" rhead="DE CENTRO GRAVIT. SOLID."/> quo ſcilicet ln, om conueniunt. </s> <s xml:id="echoid-s3079" xml:space="preserve">Poſtremo in figura <lb/>a p l q b r m s c t n u d x o y centrum grauitatis trian <lb/>guli pay, & </s> <s xml:id="echoid-s3080" xml:space="preserve">trapezii ploy eſtin linea a z: </s> <s xml:id="echoid-s3081" xml:space="preserve">trapeziorum <lb/>uero lqxo, q b d x centrum eſtin linea z k: </s> <s xml:id="echoid-s3082" xml:space="preserve">& </s> <s xml:id="echoid-s3083" xml:space="preserve">trapeziorũ <lb/>b r u d, r m n u in k φ: </s> <s xml:id="echoid-s3084" xml:space="preserve">& </s> <s xml:id="echoid-s3085" xml:space="preserve">denique trapezii m s t n; </s> <s xml:id="echoid-s3086" xml:space="preserve">& </s> <s xml:id="echoid-s3087" xml:space="preserve">triangu <lb/>li s c t in φ c. </s> <s xml:id="echoid-s3088" xml:space="preserve">quare magnitudinis ex his compoſitæ centrū <lb/>in linea a c conſiſtit. </s> <s xml:id="echoid-s3089" xml:space="preserve">Rurſus trianguli q b r, & </s> <s xml:id="echoid-s3090" xml:space="preserve">trapezii q l <lb/>m r centrum eſt in linea b χ: </s> <s xml:id="echoid-s3091" xml:space="preserve">trapeziorum l p s m, p a c s, <lb/>a y t c, y o n t in linea χ φ: </s> <s xml:id="echoid-s3092" xml:space="preserve">trapeziiq; </s> <s xml:id="echoid-s3093" xml:space="preserve">o x u n, & </s> <s xml:id="echoid-s3094" xml:space="preserve">trianguli <lb/>x d u centrum in ψ d. </s> <s xml:id="echoid-s3095" xml:space="preserve">totius ergo magnitudinis centrum <lb/>eſtin linea b d. </s> <s xml:id="echoid-s3096" xml:space="preserve">ex quo ſequitur, centrum grauitatis figuræ <lb/>a p l q b r m s c t n u d x o y eſſe punctū _K_, lineis ſcilicet a c, <lb/>b d commune, quæ omnia demonſtrare oportebat.</s> <s xml:id="echoid-s3097" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="1"> <note position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">8. primi</note> <figure xlink:label="fig-0119-01" xlink:href="fig-0119-01a"> <image file="0119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0119-01"/> </figure> <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">33. primit</note> <note position="left" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve">28. primi.</note> <figure xlink:label="fig-0120-01" xlink:href="fig-0120-01a"> <image file="0120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0120-01"/> </figure> <note position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">13. Archi <lb/>medis.</note> <note position="left" xlink:label="note-0120-03" xlink:href="note-0120-03a" xml:space="preserve">Vltima.</note> </div> </div> <div xml:id="echoid-div201" type="section" level="1" n="66"> <head xml:id="echoid-head73" xml:space="preserve">THE OREMA III. PROPOSITIO III.</head> <p> <s xml:id="echoid-s3098" xml:space="preserve">Cuiuslibet portio-<lb/> <anchor type="figure" xlink:label="fig-0121-01a" xlink:href="fig-0121-01"/> nis circuli, & </s> <s xml:id="echoid-s3099" xml:space="preserve">ellipſis, <lb/>quæ dimidia non ſit <lb/>maior, centrum graui <lb/>tatis in portionis dia-<lb/>metro conſiſtit.</s> <s xml:id="echoid-s3100" xml:space="preserve"/> </p> <div xml:id="echoid-div201" type="float" level="2" n="1"> <figure xlink:label="fig-0121-01" xlink:href="fig-0121-01a"> <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0121-01"/> </figure> </div> <p> <s xml:id="echoid-s3101" xml:space="preserve">HOC eodem prorſus <lb/>modo demonſtrabitur, <lb/>quo in libro de centro gra <lb/>uitatis planorum ab Ar-<lb/>chimede demonſtratũ eſt, <lb/>in portione cõtenta recta <lb/>linea, & </s> <s xml:id="echoid-s3102" xml:space="preserve">rectanguli coni ſe <lb/>ctione grauitatis cẽtrum <lb/>eſſe in diametro portio-<lb/>nis. </s> <s xml:id="echoid-s3103" xml:space="preserve">Etita demonſtrari po <lb/> <anchor type="handwritten" xlink:label="hd-0121-02a" xlink:href="hd-0121-02"/> <pb file="0122" n="122" rhead="FED. COMMANDINI"/> teſt in portione, quæ recta linea & </s> <s xml:id="echoid-s3104" xml:space="preserve">obtuſianguli coni ſe-<lb/>ctione, ſeu hyperbola continetur.</s> <s xml:id="echoid-s3105" xml:space="preserve"/> </p> <div xml:id="echoid-div202" type="float" level="2" n="2"> <handwritten xlink:label="hd-0121-02" xlink:href="hd-0121-02a"/> </div> </div> <div xml:id="echoid-div204" type="section" level="1" n="67"> <head xml:id="echoid-head74" xml:space="preserve">THE OREMA IIII. PROPOSITIO IIII.</head> <p> <s xml:id="echoid-s3106" xml:space="preserve"><emph style="sc">In</emph> circulo & </s> <s xml:id="echoid-s3107" xml:space="preserve">ellipſiidem eſt figuræ & </s> <s xml:id="echoid-s3108" xml:space="preserve">graui-<lb/>tatis centrum.</s> <s xml:id="echoid-s3109" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3110" xml:space="preserve">SIT circulus, uel ellipſis, cuius centrum a. </s> <s xml:id="echoid-s3111" xml:space="preserve">Dico a gra-<lb/>uitatis quoque centrum eſſe. </s> <s xml:id="echoid-s3112" xml:space="preserve">Si enim fieri poteſt, ſit b cen-<lb/>trum grauitatis: </s> <s xml:id="echoid-s3113" xml:space="preserve">& </s> <s xml:id="echoid-s3114" xml:space="preserve">iuncta a b extra figuram in c produca <lb/>tur: </s> <s xml:id="echoid-s3115" xml:space="preserve">quam uero proportionem habetlinea c a ad a b, ha-<lb/>beat circulus a ad alium circulum, in quo d; </s> <s xml:id="echoid-s3116" xml:space="preserve">uel ellipſis ad <lb/>aliam ellipſim: </s> <s xml:id="echoid-s3117" xml:space="preserve">& </s> <s xml:id="echoid-s3118" xml:space="preserve">in circulo, uel ellipſi ſigura rectilinea pla-<lb/>ne deſcribatur adeo, ut tandem relinquantur portiones <lb/>quædam minores circulo, uel ellipſid; </s> <s xml:id="echoid-s3119" xml:space="preserve">quæ figura ſit e f g <lb/>h _k_ l m n. </s> <s xml:id="echoid-s3120" xml:space="preserve">Illud uero in circulo fieri poſſe ex duodecimo <lb/>elementorum libro, propoſitione ſecunda manifeſte con-<lb/>ſtat; </s> <s xml:id="echoid-s3121" xml:space="preserve">at in ellipſi nos demonſtra-<lb/> <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> uinius in commentariis in quin-<lb/>tam propoſitionem Archimedis <lb/>de conoidibus, & </s> <s xml:id="echoid-s3122" xml:space="preserve">ſphæroidibus. <lb/></s> <s xml:id="echoid-s3123" xml:space="preserve">erit igitur a centrum grauitatis <lb/>ipſius figuræ, quod proxime oſtē <lb/>dimus. </s> <s xml:id="echoid-s3124" xml:space="preserve">Itaque quoniam circulus <lb/>a ad circulum d; </s> <s xml:id="echoid-s3125" xml:space="preserve">uel ellipſis a ad <lb/>ellipſim d eandem proportionē <lb/>habet, quam linea c a ad a b: </s> <s xml:id="echoid-s3126" xml:space="preserve"><lb/>portiones uero ſunt minores cir <lb/> <anchor type="note" xlink:label="note-0122-01a" xlink:href="note-0122-01"/> culo uel ellipſi d: </s> <s xml:id="echoid-s3127" xml:space="preserve">habebit circu-<lb/>lus, uel ellipſis ad portiones ma-<lb/>iorem proportionem, quàm c a <lb/> <anchor type="note" xlink:label="note-0122-02a" xlink:href="note-0122-02"/> ad a b: </s> <s xml:id="echoid-s3128" xml:space="preserve">& </s> <s xml:id="echoid-s3129" xml:space="preserve">diuidendo figura recti-<lb/>linea e f g h _k_ l m n ad portiones <pb o="6" file="0123" n="123" rhead="DE CENTRO GRAVIT. SOLID."/> habebit maiorem proportionẽ, <lb/> <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> quam c b ad b a. </s> <s xml:id="echoid-s3130" xml:space="preserve">fiat o b ad b a, <lb/>ut figura rectilinea ad portio-<lb/>nes. </s> <s xml:id="echoid-s3131" xml:space="preserve">cum igitur à circulo, uel el-<lb/>lipſi, cuius grauitatis centrum <lb/>eſt b, auferatur figura rectilinea <lb/>e f g h k l m n, cuius centrum a; <lb/></s> <s xml:id="echoid-s3132" xml:space="preserve">reliquæ magnitudinis ex portio <lb/> <anchor type="note" xlink:label="note-0123-01a" xlink:href="note-0123-01"/> nibus compoſitæ centrum graui <lb/>tatis erit in linea a b producta, <lb/>& </s> <s xml:id="echoid-s3133" xml:space="preserve">in puncto o, extra figuram po <lb/>ſito. </s> <s xml:id="echoid-s3134" xml:space="preserve">quod quidem fieri nullo mo <lb/>do poſſe perſpicuum eſt. </s> <s xml:id="echoid-s3135" xml:space="preserve">ſequi-<lb/>tur ergo, ut circuli & </s> <s xml:id="echoid-s3136" xml:space="preserve">ellipſis cen <lb/>trum grauitatis ſit punctum a, <lb/>idem quod figuræ centrum.</s> <s xml:id="echoid-s3137" xml:space="preserve"/> </p> <div xml:id="echoid-div204" type="float" level="2" n="1"> <figure xlink:label="fig-0122-01" xlink:href="fig-0122-01a"> <image file="0122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0122-01"/> </figure> <note position="left" xlink:label="note-0122-01" xlink:href="note-0122-01a" xml:space="preserve">8. quinti.</note> <note position="left" xlink:label="note-0122-02" xlink:href="note-0122-02a" xml:space="preserve">19. quinti <lb/>apud Cã <lb/>panum.</note> <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a"> <image file="0123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0123-01"/> </figure> <note position="right" xlink:label="note-0123-01" xlink:href="note-0123-01a" xml:space="preserve">8. Archi-<lb/>medis.</note> </div> </div> <div xml:id="echoid-div206" type="section" level="1" n="68"> <head xml:id="echoid-head75" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3138" xml:space="preserve">Sit circulus, uel ellipſis a b c d, <lb/>cuius diameter d b, & </s> <s xml:id="echoid-s3139" xml:space="preserve">centrum e: </s> <s xml:id="echoid-s3140" xml:space="preserve">ducaturq; </s> <s xml:id="echoid-s3141" xml:space="preserve">per e recta li <lb/>nea a c, ſecans ipſam d b adrectos angulos. </s> <s xml:id="echoid-s3142" xml:space="preserve">erunt a d c, <lb/>a b c circuli, uel ellipſis dimidiæ portiones. </s> <s xml:id="echoid-s3143" xml:space="preserve">Itaque quo-<lb/>niam por <lb/> <anchor type="figure" xlink:label="fig-0123-02a" xlink:href="fig-0123-02"/> tiõis a d c <lb/>cétrū gra-<lb/>uitatis eſt <lb/>in diame-<lb/>tro d e: </s> <s xml:id="echoid-s3144" xml:space="preserve">& </s> <s xml:id="echoid-s3145" xml:space="preserve"><lb/>portionis <lb/>a b c cen-<lb/>trum eſt ĩ <lb/>ipſa e b: </s> <s xml:id="echoid-s3146" xml:space="preserve">to <lb/>tius circu <lb/>li, uel ellipſis grauitatis centrum eritin diametro d b. <lb/></s> <s xml:id="echoid-s3147" xml:space="preserve">Sit autem portionis a d c cẽtrum grauitatis f: </s> <s xml:id="echoid-s3148" xml:space="preserve">& </s> <s xml:id="echoid-s3149" xml:space="preserve">ſumatur <pb file="0124" n="124" rhead="FED. COMMANDINI"/> in linea e b punctũ g, it aut ſit g e æqualis e f. </s> <s xml:id="echoid-s3150" xml:space="preserve">erit g por-<lb/>tionis a b c centrum. </s> <s xml:id="echoid-s3151" xml:space="preserve">nam ſi hæ portiones, quæ æquales <lb/>& </s> <s xml:id="echoid-s3152" xml:space="preserve">ſimiles ſunt, inter ſe ſe aptentur, ita ut b e cadat in d e, <lb/>& </s> <s xml:id="echoid-s3153" xml:space="preserve">punctum b in d cadet, & </s> <s xml:id="echoid-s3154" xml:space="preserve">g in f: </s> <s xml:id="echoid-s3155" xml:space="preserve">figuris autem æquali-<lb/>bus, & </s> <s xml:id="echoid-s3156" xml:space="preserve">ſimilibus inter ſe aptatis, centra quoque grauitatis <lb/>ipſarum inter ſe aptata erunt, ex quinta petitione Archi-<lb/>medis in libro de centro grauitatis planorum. </s> <s xml:id="echoid-s3157" xml:space="preserve">Quare cum <lb/>portionis a d c centrum grauitatis ſit ſ: </s> <s xml:id="echoid-s3158" xml:space="preserve">& </s> <s xml:id="echoid-s3159" xml:space="preserve">portionis <lb/>a b c centrum g: </s> <s xml:id="echoid-s3160" xml:space="preserve">magnitudinis; </s> <s xml:id="echoid-s3161" xml:space="preserve">quæ ex utriſque efficitur: <lb/></s> <s xml:id="echoid-s3162" xml:space="preserve">hoc eſt circuli uel ellipſis grauitatis centrum in medio li-<lb/>neæ f g, quod eſt e, conſiſtet, ex quarta propoſitione eiuſ-<lb/>dem libri Archimedis. </s> <s xml:id="echoid-s3163" xml:space="preserve">ergo circuli, uel ellipſis centrum <lb/>grauitatis eſt idem, quod figuræ centrum. </s> <s xml:id="echoid-s3164" xml:space="preserve">atque illud eſt, <lb/>quod demonſtrare oportebat.</s> <s xml:id="echoid-s3165" xml:space="preserve"/> </p> <div xml:id="echoid-div206" type="float" level="2" n="1"> <figure xlink:label="fig-0123-02" xlink:href="fig-0123-02a"> <image file="0123-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0123-02"/> </figure> </div> <p> <s xml:id="echoid-s3166" xml:space="preserve">Ex quibus ſequitur portionis circuli, uel ellip-<lb/>ſis, quæ dimidia maior ſit, centrum grauitatis in <lb/>diametro quoque ipſius conſiſtere.</s> <s xml:id="echoid-s3167" xml:space="preserve"/> </p> <figure> <image file="0124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0124-01"/> </figure> <p> <s xml:id="echoid-s3168" xml:space="preserve">Sit enim maior portio a b c, cu_i_us diameter b d, & </s> <s xml:id="echoid-s3169" xml:space="preserve">com-<lb/>pleatur circulus, uel ellipſis, ut portio reliqua ſit a e c, dia <pb o="7" file="0125" n="125" rhead="DE CENTRO GRAVIT. SOLID."/> metrum habens e d. </s> <s xml:id="echoid-s3170" xml:space="preserve">Quoniam igitur circuli uel ellipſis <lb/>a e c b grauitatis centrum eſt in diametro b e, & </s> <s xml:id="echoid-s3171" xml:space="preserve">portio-<lb/>nis a e c centrum in linea e d: </s> <s xml:id="echoid-s3172" xml:space="preserve">reliquæ portionis, uidelicet <lb/>a b c centrum grauitatis in ipſa b d conſiſtat neceſſe eſt, ex <lb/>octaua propoſitione eiuſdem.</s> <s xml:id="echoid-s3173" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div208" type="section" level="1" n="69"> <head xml:id="echoid-head76" xml:space="preserve">THEOREMA V. PROPOSITIO V.</head> <p> <s xml:id="echoid-s3174" xml:space="preserve">SI priſma ſecetur plano oppoſitis planis æqui <lb/>diſtante, ſectio erit figura æqualis & </s> <s xml:id="echoid-s3175" xml:space="preserve">ſimilis ei, <lb/>quæ eſt oppoſitorum planorum, centrum graui <lb/>tatis in axe habens.</s> <s xml:id="echoid-s3176" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3177" xml:space="preserve">Sit priſma, in quo plana oppoſita ſint triangula a b c, <lb/>d e f; </s> <s xml:id="echoid-s3178" xml:space="preserve">axis g h: </s> <s xml:id="echoid-s3179" xml:space="preserve">& </s> <s xml:id="echoid-s3180" xml:space="preserve">ſecetur plano iam dictis planis æquidiſtã <lb/>te; </s> <s xml:id="echoid-s3181" xml:space="preserve">quod faciat ſectionem <emph style="sc">K</emph> l m; </s> <s xml:id="echoid-s3182" xml:space="preserve">& </s> <s xml:id="echoid-s3183" xml:space="preserve">axi in pũcto n occurrat. <lb/></s> <s xml:id="echoid-s3184" xml:space="preserve">Dico _k_ l m triangulum æquale eſſe, & </s> <s xml:id="echoid-s3185" xml:space="preserve">ſimile triangulis a b c <lb/>d e f; </s> <s xml:id="echoid-s3186" xml:space="preserve">atque eius grauitatis centrum eſſe punctum n. </s> <s xml:id="echoid-s3187" xml:space="preserve">Quo-<lb/>niam enim plana a b c <lb/> <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> K l m æquidiſtantia ſecã <lb/> <anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/> tur a plano a e; </s> <s xml:id="echoid-s3188" xml:space="preserve">rectæ li-<lb/>neæ a b, K l, quæ ſunt ip <lb/>ſorum cõmunes ſectio-<lb/>nes inter ſe ſe æquidi-<lb/>ſtant. </s> <s xml:id="echoid-s3189" xml:space="preserve">Sed æquidiſtant <lb/>a d, b e; </s> <s xml:id="echoid-s3190" xml:space="preserve">cum a e ſit para <lb/>lelogrammum, ex priſ-<lb/>matis diffinitione. </s> <s xml:id="echoid-s3191" xml:space="preserve">ergo <lb/>& </s> <s xml:id="echoid-s3192" xml:space="preserve">al parallelogrammũ <lb/>erit; </s> <s xml:id="echoid-s3193" xml:space="preserve">& </s> <s xml:id="echoid-s3194" xml:space="preserve">propterea linea <lb/> <anchor type="note" xlink:label="note-0125-02a" xlink:href="note-0125-02"/> _k_l, ipſi a b æqualis. </s> <s xml:id="echoid-s3195" xml:space="preserve">Si-<lb/>militer demonſtrabitur <lb/>l m æquidiſtans, & </s> <s xml:id="echoid-s3196" xml:space="preserve">æqua <lb/>lis b c; </s> <s xml:id="echoid-s3197" xml:space="preserve">& </s> <s xml:id="echoid-s3198" xml:space="preserve">m <emph style="sc">K</emph> ipſi c a.</s> <s xml:id="echoid-s3199" xml:space="preserve"/> </p> <div xml:id="echoid-div208" type="float" level="2" n="1"> <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a"> <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0125-01"/> </figure> <note position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve">16. unde-<lb/>cimi.</note> <note position="right" xlink:label="note-0125-02" xlink:href="note-0125-02a" xml:space="preserve">34. prim@</note> </div> <pb file="0126" n="126" rhead="FED. COMMANDINI"/> <p> <s xml:id="echoid-s3200" xml:space="preserve">Itaque quoniam duæ lineæ K l, l m ſe ſe tangentes, duabus <lb/>lineis ſe ſe tangentibus a b, b c æquidiſtant; </s> <s xml:id="echoid-s3201" xml:space="preserve">nec ſunt in eo-<lb/>dem plano: </s> <s xml:id="echoid-s3202" xml:space="preserve">angulus <emph style="sc">K</emph> l m æqualis eſt angulo a b c: </s> <s xml:id="echoid-s3203" xml:space="preserve">& </s> <s xml:id="echoid-s3204" xml:space="preserve">ita an <lb/> <anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/> gulus l m <emph style="sc">K</emph>, angulo b c a, & </s> <s xml:id="echoid-s3205" xml:space="preserve">m <emph style="sc">K</emph> lipſi c a b æqualis prob abi <lb/>tur. </s> <s xml:id="echoid-s3206" xml:space="preserve">triangulum ergo <emph style="sc">K</emph> l m eſt æquale, & </s> <s xml:id="echoid-s3207" xml:space="preserve">ſimile triang ulo <lb/>a b c. </s> <s xml:id="echoid-s3208" xml:space="preserve">quare & </s> <s xml:id="echoid-s3209" xml:space="preserve">triangulo d e f. </s> <s xml:id="echoid-s3210" xml:space="preserve">Ducatur linea c g o, & </s> <s xml:id="echoid-s3211" xml:space="preserve">per ip <lb/>ſam, & </s> <s xml:id="echoid-s3212" xml:space="preserve">per c f ducatur planum ſecans priſma, cuius & </s> <s xml:id="echoid-s3213" xml:space="preserve">paral <lb/>lelogrammi a e communis ſectio ſit o p q. </s> <s xml:id="echoid-s3214" xml:space="preserve">tranſibit linea <lb/>f q per h, & </s> <s xml:id="echoid-s3215" xml:space="preserve">m p per n. </s> <s xml:id="echoid-s3216" xml:space="preserve">nam cum plana æquidiſtantia ſecen <lb/>tur à plano c q, communes eorum ſectiones c g o, m p, f q <lb/>ſibi ipſis æquidiſtabunt. </s> <s xml:id="echoid-s3217" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3218" xml:space="preserve">æquidiſtant a b, <emph style="sc">K</emph> l, d e. </s> <s xml:id="echoid-s3219" xml:space="preserve">an-<lb/>guli ergo a o c, <emph style="sc">K</emph> p m, d q f inter ſe æquales ſunt: </s> <s xml:id="echoid-s3220" xml:space="preserve">& </s> <s xml:id="echoid-s3221" xml:space="preserve">ſunt <lb/> <anchor type="note" xlink:label="note-0126-02a" xlink:href="note-0126-02"/> æquales qui ad puncta a k d conſtituuntur. </s> <s xml:id="echoid-s3222" xml:space="preserve">quare & </s> <s xml:id="echoid-s3223" xml:space="preserve">reliqui <lb/>reliquis æquales; </s> <s xml:id="echoid-s3224" xml:space="preserve">& </s> <s xml:id="echoid-s3225" xml:space="preserve">triangula a c o, _K_ m p, d f q inter ſe ſimi <lb/>lia erunt. </s> <s xml:id="echoid-s3226" xml:space="preserve">Vtigitur ca ad a o, ita fd ad d q: </s> <s xml:id="echoid-s3227" xml:space="preserve">& </s> <s xml:id="echoid-s3228" xml:space="preserve">permutando <lb/> <anchor type="note" xlink:label="note-0126-03a" xlink:href="note-0126-03"/> ut c a ad fd, ita a o ad d q. </s> <s xml:id="echoid-s3229" xml:space="preserve">eſt autem c a æqualis fd. </s> <s xml:id="echoid-s3230" xml:space="preserve">ergo & </s> <s xml:id="echoid-s3231" xml:space="preserve"><lb/>a o ipſi d q. </s> <s xml:id="echoid-s3232" xml:space="preserve">eadem quoque ratione & </s> <s xml:id="echoid-s3233" xml:space="preserve">a o ipſi _K_ p æqualis <lb/>demonſtrabitur. </s> <s xml:id="echoid-s3234" xml:space="preserve">Itaque ſi triangula, a b c, d e f æqualia & </s> <s xml:id="echoid-s3235" xml:space="preserve"><lb/>ſimilia inter ſe aptétur, <lb/> <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> cadet linea f q in lineam <lb/>c g o. </s> <s xml:id="echoid-s3236" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3237" xml:space="preserve">centrũ gra <lb/> <anchor type="note" xlink:label="note-0126-04a" xlink:href="note-0126-04"/> uitatis h in g centrũ ca-<lb/>det. </s> <s xml:id="echoid-s3238" xml:space="preserve">trãſibit igitur linea <lb/>f q per h: </s> <s xml:id="echoid-s3239" xml:space="preserve">& </s> <s xml:id="echoid-s3240" xml:space="preserve">planum per <lb/>c o & </s> <s xml:id="echoid-s3241" xml:space="preserve">c f ductũ per axẽ <lb/>g h ducetur: </s> <s xml:id="echoid-s3242" xml:space="preserve">idcircoq; </s> <s xml:id="echoid-s3243" xml:space="preserve">li <lb/>neam m p etiã per n trã <lb/>ſire neceſſe erit. </s> <s xml:id="echoid-s3244" xml:space="preserve">Quo-<lb/>niam ergo ſh, c g æqua-<lb/>les ſunt, & </s> <s xml:id="echoid-s3245" xml:space="preserve">æquidiſtãtes: <lb/></s> <s xml:id="echoid-s3246" xml:space="preserve">itemq; </s> <s xml:id="echoid-s3247" xml:space="preserve">h q, g o; </s> <s xml:id="echoid-s3248" xml:space="preserve">rectæ li-<lb/>neæ, quæ ipſas cónectũt <lb/>c m f, g n h, o p q æqua-<lb/>les & </s> <s xml:id="echoid-s3249" xml:space="preserve">æquidiſtãtes erũt.</s> <s xml:id="echoid-s3250" xml:space="preserve"> <pb o="8" file="0127" n="127" rhead="DE CENTRO GRAVIT. SOLID."/> æquidiſtant autem c g o, m n p. </s> <s xml:id="echoid-s3251" xml:space="preserve">ergo parallelogrãma ſunt <lb/>o n, g m, & </s> <s xml:id="echoid-s3252" xml:space="preserve">linea m n æqualis c g; </s> <s xml:id="echoid-s3253" xml:space="preserve">& </s> <s xml:id="echoid-s3254" xml:space="preserve">n p ipſi g o. </s> <s xml:id="echoid-s3255" xml:space="preserve">aptatis igi-<lb/>tur <emph style="sc">K</emph> l m, a b c triãgulis, quæ æqualia & </s> <s xml:id="echoid-s3256" xml:space="preserve">ſimilia sũt; </s> <s xml:id="echoid-s3257" xml:space="preserve">linea m p <lb/>in c o, & </s> <s xml:id="echoid-s3258" xml:space="preserve">punctum n in g cadet. </s> <s xml:id="echoid-s3259" xml:space="preserve">Quòd cũ g ſit centrum gra-<lb/>uitatis trianguli a b c, & </s> <s xml:id="echoid-s3260" xml:space="preserve">n trianguli <emph style="sc">K</emph> l m grauitatis cen-<lb/>trum erit id, quod demonſtrandum relinquebatur. </s> <s xml:id="echoid-s3261" xml:space="preserve">Simili <lb/>ratione idem contingere demonſtrabimus in aliis priſma-<lb/>tibus, ſiue quadrilatera, ſiue plurilatera habeant plana, <lb/>quæ opponuntur.</s> <s xml:id="echoid-s3262" xml:space="preserve"/> </p> <div xml:id="echoid-div209" type="float" level="2" n="2"> <note position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">10. unde <lb/>cimi</note> <note position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve">10. unde-<lb/>cimi</note> <note position="left" xlink:label="note-0126-03" xlink:href="note-0126-03a" xml:space="preserve">4. ſexti</note> <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0126-01"/> </figure> <note position="left" xlink:label="note-0126-04" xlink:href="note-0126-04a" xml:space="preserve">per 5. pe-<lb/>titionem <lb/>Archime <lb/>dis.</note> </div> </div> <div xml:id="echoid-div211" type="section" level="1" n="70"> <head xml:id="echoid-head77" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s3263" xml:space="preserve">Exiam demonſtratis perſpicue apparet, cuius <lb/>Iibet priſmatis axem, parallelogrammorum lat eri <lb/>bus, quæ ab oppoſitis planis ducũtur æquidiſtare.</s> <s xml:id="echoid-s3264" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div212" type="section" level="1" n="71"> <head xml:id="echoid-head78" xml:space="preserve">THEOREMA VI. PROPOSITIO VI.</head> <p> <s xml:id="echoid-s3265" xml:space="preserve">Cuiuslibet priſmatis centrum grauitatis eſt in <lb/>plano, quod oppoſitis planis æquidiſtans, reli-<lb/>quorum planorum latera bifariam diuidit.</s> <s xml:id="echoid-s3266" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3267" xml:space="preserve">Sit priſma, in quo plana, quæ opponuntur ſint trian-<lb/>gula a c e, b d f: </s> <s xml:id="echoid-s3268" xml:space="preserve">& </s> <s xml:id="echoid-s3269" xml:space="preserve">parallelogrammorum latera a b, c d, <lb/>e f bifariam diuidãtur in punctis g h _K_: </s> <s xml:id="echoid-s3270" xml:space="preserve">per diuiſiones au-<lb/>tem planum ducatur; </s> <s xml:id="echoid-s3271" xml:space="preserve">cuius ſectio figura g h _K_. </s> <s xml:id="echoid-s3272" xml:space="preserve">eritlinea <lb/> <anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/> g h æquidiſtans lineis a c, b d & </s> <s xml:id="echoid-s3273" xml:space="preserve">h k ipſis c e, d f. </s> <s xml:id="echoid-s3274" xml:space="preserve">quare ex <lb/>decimaquinta undecimi elementorum, planum illud pla <lb/>nis a c e, b d f æquidiſtabit, & </s> <s xml:id="echoid-s3275" xml:space="preserve">ſaciet ſectionem figu-<lb/> <anchor type="note" xlink:label="note-0127-02a" xlink:href="note-0127-02"/> ram ipſis æqualem, & </s> <s xml:id="echoid-s3276" xml:space="preserve">ſimilem, ut proxime demonſtra-<lb/>uimus. </s> <s xml:id="echoid-s3277" xml:space="preserve">Dico centrum grauitatis priſmatis eſſe in plano <lb/>g h <emph style="sc">K</emph>. </s> <s xml:id="echoid-s3278" xml:space="preserve">Si enim fieri poteſt, ſit eius centrum l: </s> <s xml:id="echoid-s3279" xml:space="preserve">& </s> <s xml:id="echoid-s3280" xml:space="preserve">ducatur <lb/>l m uſque ad planum g h <emph style="sc">K</emph>, quæ ipſi a b æquidiſtet.</s> <s xml:id="echoid-s3281" xml:space="preserve"> <pb file="0128" n="128" rhead="FED. COMMANDINI"/> ergo linea a g continenter in duas partes æquales diui-<lb/> <anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/> ſa, relinquetur tãdem pars aliqua n g, quæ minor eritl m. <lb/></s> <s xml:id="echoid-s3282" xml:space="preserve">Vtraque uero linearum a g, g b diuidatur in partes æqua-<lb/>les ipſi n g: </s> <s xml:id="echoid-s3283" xml:space="preserve">& </s> <s xml:id="echoid-s3284" xml:space="preserve">per puncta diuiſionum plana oppoſitis pla-<lb/> <anchor type="note" xlink:label="note-0128-02a" xlink:href="note-0128-02"/> nis æquidiſtantia ducantur. </s> <s xml:id="echoid-s3285" xml:space="preserve">erunt ſectiones figuræ æqua-<lb/>les, ac ſimiles ipſis a c e, b d f: </s> <s xml:id="echoid-s3286" xml:space="preserve">& </s> <s xml:id="echoid-s3287" xml:space="preserve">totum priſma diuiſum erit <lb/>in priſmata æqualia, & </s> <s xml:id="echoid-s3288" xml:space="preserve">ſimilia: </s> <s xml:id="echoid-s3289" xml:space="preserve">quæ cum inter ſe congruãt; <lb/></s> <s xml:id="echoid-s3290" xml:space="preserve">& </s> <s xml:id="echoid-s3291" xml:space="preserve">grauitatis centra ſibi ipſis congruentia, reſpondentiaq; </s> <s xml:id="echoid-s3292" xml:space="preserve"><lb/>habebunt. </s> <s xml:id="echoid-s3293" xml:space="preserve">Itaq: </s> <s xml:id="echoid-s3294" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0128-01a" xlink:href="fig-0128-01"/> ſunt magnitudi-<lb/>nes quædã æqua-<lb/>les ipſi n h, & </s> <s xml:id="echoid-s3295" xml:space="preserve">nu-<lb/>mero pares, qua-<lb/>rum centra gra-<lb/>uitatis in eadẽ re <lb/>cta linea conſti-<lb/>tuuntur: </s> <s xml:id="echoid-s3296" xml:space="preserve">duæ ue-<lb/>ro mediæ æqua-<lb/>les ſunt: </s> <s xml:id="echoid-s3297" xml:space="preserve">& </s> <s xml:id="echoid-s3298" xml:space="preserve">quæ ex <lb/>utraque parte i-<lb/>pſarum ſimili --<lb/>ter æquales: </s> <s xml:id="echoid-s3299" xml:space="preserve">& </s> <s xml:id="echoid-s3300" xml:space="preserve">æ-<lb/>quales rectæ li-<lb/>neæ, quæ inter <lb/>grauitatis centra <lb/>interiiciuntur. <lb/></s> <s xml:id="echoid-s3301" xml:space="preserve">quare ex corolla-<lb/>rio quintæ pro-<lb/>poſitionis primi <lb/>libri Archimedis <lb/>de centro graui-<lb/>tatis planorum; </s> <s xml:id="echoid-s3302" xml:space="preserve">magnitudinis ex his omnibus compoſitæ <lb/>centrum grauitatis eſt in medio lineæ, quæ magnitudi-<lb/>num mediarum centra coniungit. </s> <s xml:id="echoid-s3303" xml:space="preserve">at qui non ita res ha- <pb o="9" file="0129" n="129" rhead="DE CENTRO GRAVIT. SOLID."/> bet, ſi quidem 1 extra medias magnitudines poſitum eſt. <lb/></s> <s xml:id="echoid-s3304" xml:space="preserve">Conſtatigitur centrum grauitatis priſmatis eſſe in plano <lb/> <anchor type="figure" xlink:label="fig-0129-01a" xlink:href="fig-0129-01"/> g h k, quod nos demonſtrandum propoſuimus. </s> <s xml:id="echoid-s3305" xml:space="preserve">At ſi op-<lb/>poſita plana in priſmate ſint quadrilatera, uel plurilatera, <lb/>eadem erit in omnibus demonſtratio.</s> <s xml:id="echoid-s3306" xml:space="preserve"/> </p> <div xml:id="echoid-div212" type="float" level="2" n="1"> <note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">33. primi</note> <note position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">5. huius</note> <note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">1. decimi</note> <note position="left" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">5 huius</note> <figure xlink:label="fig-0128-01" xlink:href="fig-0128-01a"> <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0128-01"/> </figure> <figure xlink:label="fig-0129-01" xlink:href="fig-0129-01a"> <image file="0129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0129-01"/> </figure> </div> </div> <div xml:id="echoid-div214" type="section" level="1" n="72"> <head xml:id="echoid-head79" xml:space="preserve">THE OREMA VII. PROPOSITIO VII.</head> <p> <s xml:id="echoid-s3307" xml:space="preserve">Cuiuslibet cylindri, & </s> <s xml:id="echoid-s3308" xml:space="preserve">cuiuslibet cylindri por <lb/>tionis centrum grauitatis eſt in plano, quod baſi-<lb/>bus æquidiſtans, parallelogrammi per axem late-<lb/>ra bifariam ſecat.</s> <s xml:id="echoid-s3309" xml:space="preserve"/> </p> <pb file="0130" n="130" rhead="FED. COMMANDINI"/> <p> <s xml:id="echoid-s3310" xml:space="preserve">SIT cylindrus, uel cylindri po rtio a c: </s> <s xml:id="echoid-s3311" xml:space="preserve">& </s> <s xml:id="echoid-s3312" xml:space="preserve">plano per a-<lb/>xem ducto ſecetur; </s> <s xml:id="echoid-s3313" xml:space="preserve">cuius ſectio ſit parallelogrammum a b <lb/>c d: </s> <s xml:id="echoid-s3314" xml:space="preserve">& </s> <s xml:id="echoid-s3315" xml:space="preserve">bifariam diuiſis a d, b c parallelogrammi lateribus, <lb/>per diuiſionum puncta e f planum baſi æquidiſtans duca-<lb/>tur; </s> <s xml:id="echoid-s3316" xml:space="preserve">quod faciet ſectionem, in cy lindro quidem circulum <lb/>æqualem iis, qui ſunt in baſibus, ut demonſtrauit Serenus <lb/>in libro cylindricorum, propoſitione quinta: </s> <s xml:id="echoid-s3317" xml:space="preserve">in cylindri <lb/>uero portione ellipſim æqualem, & </s> <s xml:id="echoid-s3318" xml:space="preserve">ſimilem eis, quæ ſunt <lb/>in oppoſitis planis, quod nos <lb/> <anchor type="figure" xlink:label="fig-0130-01a" xlink:href="fig-0130-01"/> demonſtrauimus in commen <lb/>tariis in librum Archimedis <lb/>de conoidibus, & </s> <s xml:id="echoid-s3319" xml:space="preserve">ſphæroidi-<lb/>bus. </s> <s xml:id="echoid-s3320" xml:space="preserve">Dico centrum grauita-<lb/>tis cylindri, uel cylindri por-<lb/>tionis eſſe in plano e f. </s> <s xml:id="echoid-s3321" xml:space="preserve">Si enĩ <lb/>fieri poteſt, fit centrum g: </s> <s xml:id="echoid-s3322" xml:space="preserve">& </s> <s xml:id="echoid-s3323" xml:space="preserve"><lb/>ducatur g h ipſi a d æquidi-<lb/>ſtans, uſque ad e f planum. <lb/></s> <s xml:id="echoid-s3324" xml:space="preserve">Itaque linea a e continenter <lb/>diuiſa bifariam, erit tandem <lb/>pars aliqua ipſius k e, minor <lb/>g h. </s> <s xml:id="echoid-s3325" xml:space="preserve">Diuidantur ergo lineæ <lb/>a e, e d in partes æquales ipſi <lb/>k e: </s> <s xml:id="echoid-s3326" xml:space="preserve">& </s> <s xml:id="echoid-s3327" xml:space="preserve">per diuiſiones plana ba <lb/>ſibus æquidiſtantia ducãtur. </s> <s xml:id="echoid-s3328" xml:space="preserve"><lb/>erunt iam ſectiones, figuræ æ-<lb/>quales, & </s> <s xml:id="echoid-s3329" xml:space="preserve">ſimiles eis, quæ ſunt <lb/>in baſibus: </s> <s xml:id="echoid-s3330" xml:space="preserve">atque erit cylindrus in cylindros diuiſus: </s> <s xml:id="echoid-s3331" xml:space="preserve">& </s> <s xml:id="echoid-s3332" xml:space="preserve">cy <lb/>lindri portio in portiones æquales, & </s> <s xml:id="echoid-s3333" xml:space="preserve">ſimiles ipſi k f. </s> <s xml:id="echoid-s3334" xml:space="preserve">reli-<lb/>qua ſimiliter, ut ſuperius in priſmate concludentur.</s> <s xml:id="echoid-s3335" xml:space="preserve"/> </p> <div xml:id="echoid-div214" type="float" level="2" n="1"> <figure xlink:label="fig-0130-01" xlink:href="fig-0130-01a"> <image file="0130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0130-01"/> </figure> </div> <pb o="10" file="0131" n="131" rhead="DE CENTRO GRA VIT. SOLID."/> <figure> <image file="0131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0131-01"/> </figure> </div> <div xml:id="echoid-div216" type="section" level="1" n="73"> <head xml:id="echoid-head80" xml:space="preserve">THE OREMA VIII. PROPOSITIO VIII.</head> <p> <s xml:id="echoid-s3336" xml:space="preserve">Cuiuslibet priſmatis, & </s> <s xml:id="echoid-s3337" xml:space="preserve">cuiuslibet cylindri, uel <lb/>cylindri portionis grauitatis centrum in medio <lb/>ipſius axis conſiſtit.</s> <s xml:id="echoid-s3338" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3339" xml:space="preserve">Sit primum a f priſma æ quidiſtantibus planis contentũ, <lb/>quod ſolidum parallelepipedum appellatur: </s> <s xml:id="echoid-s3340" xml:space="preserve">& </s> <s xml:id="echoid-s3341" xml:space="preserve">oppoſito-<lb/>rum planorum c f, a h, d a, f g latera bifariam diuidantur in <lb/>punctis k l m n o p q r s t u x: </s> <s xml:id="echoid-s3342" xml:space="preserve">& </s> <s xml:id="echoid-s3343" xml:space="preserve">per diuiſiones ducantur <lb/>plana κ n, o r, s x. </s> <s xml:id="echoid-s3344" xml:space="preserve">communes autem eorum planorum ſe-<lb/>ctiones ſint lineæ y z, θ φ, χ ψ: </s> <s xml:id="echoid-s3345" xml:space="preserve">quæ in puncto ω conueniãt. <lb/></s> <s xml:id="echoid-s3346" xml:space="preserve">erit ex decima eiuſdem libri Archimedis parallelogrammi <lb/>c f centrum grauitatis punctum y; </s> <s xml:id="echoid-s3347" xml:space="preserve">parallelogrammi a h <pb file="0132" n="132" rhead="FED. COMMANDINI"/> centrum z: </s> <s xml:id="echoid-s3348" xml:space="preserve">parallelogram mi a d, θ: </s> <s xml:id="echoid-s3349" xml:space="preserve">parallelogrammi f g, φ: <lb/></s> <s xml:id="echoid-s3350" xml:space="preserve">parallelogrammi d h, χ: </s> <s xml:id="echoid-s3351" xml:space="preserve">& </s> <s xml:id="echoid-s3352" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> parallelogrammi c g centrũ <lb/>ψ: </s> <s xml:id="echoid-s3353" xml:space="preserve">atque erit ω punctum me <lb/>dium uniuſcuiuſque axis, ui <lb/>delicet eius lineæ, quæ oppo <lb/>ſitorum planorũ centra con <lb/>iungit. </s> <s xml:id="echoid-s3354" xml:space="preserve">Dico ω centrum effe <lb/>grauitatis ipſius ſolidi. </s> <s xml:id="echoid-s3355" xml:space="preserve">eſt <lb/>enim, ut demonſtrauimus, <lb/> <anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/> ſolidi a f centrum grauitatis <lb/>in plano K n; </s> <s xml:id="echoid-s3356" xml:space="preserve">quod oppoſi-<lb/>tis planis a d, g f æ quidiſtans <lb/>reliquorum planorum late-<lb/>ra biſariam diuidit: </s> <s xml:id="echoid-s3357" xml:space="preserve">& </s> <s xml:id="echoid-s3358" xml:space="preserve">fimili <lb/>rationeidem centrum eſt in plano o r, æ quidiſtante planis <lb/>a e, b f oppo ſitis. </s> <s xml:id="echoid-s3359" xml:space="preserve">ergo in communi ipſorum fectione: </s> <s xml:id="echoid-s3360" xml:space="preserve">ui-<lb/>delicet in linea y z. </s> <s xml:id="echoid-s3361" xml:space="preserve">Sed eſt etiam in plano t u, quod quidẽ <lb/>y z ſecat in ω. </s> <s xml:id="echoid-s3362" xml:space="preserve">Conſtat igitur centrum grauitatis ſolidi eſſe <lb/>punctum ω, medium ſcilicet axium, hoc eſt linearum, quæ <lb/>planorum oppoſitorum centra coniungunt.</s> <s xml:id="echoid-s3363" xml:space="preserve"/> </p> <div xml:id="echoid-div216" type="float" level="2" n="1"> <figure xlink:label="fig-0132-01" xlink:href="fig-0132-01a"> <image file="0132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0132-01"/> </figure> <note position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">6. huius</note> </div> <p> <s xml:id="echoid-s3364" xml:space="preserve">Sit aliud prima a f; </s> <s xml:id="echoid-s3365" xml:space="preserve">& </s> <s xml:id="echoid-s3366" xml:space="preserve">in eo plana, quæ opponuntur, tri-<lb/>angula a b c, d e f: </s> <s xml:id="echoid-s3367" xml:space="preserve">diuiſisq; </s> <s xml:id="echoid-s3368" xml:space="preserve">bifariam parallelogrammorum <lb/>lateribus a d, b e, c f in punctis g h κ, per diuiſiones planũ <lb/>ducatur, quod oppoſitis planis æ quidiſtans faciet ſe ctionẽ <lb/>triangulum g h k æ quale, & </s> <s xml:id="echoid-s3369" xml:space="preserve">ſimile ipſis a b c, d e f. </s> <s xml:id="echoid-s3370" xml:space="preserve">Rurſus <lb/>diuidatur a b bifariam in l: </s> <s xml:id="echoid-s3371" xml:space="preserve">& </s> <s xml:id="echoid-s3372" xml:space="preserve">iuncta c l per ipſam, & </s> <s xml:id="echoid-s3373" xml:space="preserve">per <lb/>c _K_ f planum ducatur priſma ſecans, cuius, & </s> <s xml:id="echoid-s3374" xml:space="preserve">parallelogrã <lb/>mi a e communis ſcctio ſit l m n. </s> <s xml:id="echoid-s3375" xml:space="preserve">diuidet pun ctum m li-<lb/>neam g h bifariam; </s> <s xml:id="echoid-s3376" xml:space="preserve">& </s> <s xml:id="echoid-s3377" xml:space="preserve">ita n diuidet lineam d e: </s> <s xml:id="echoid-s3378" xml:space="preserve">quoniam <lb/>triangula a c l, g k m, d f n æ qualia ſunt, & </s> <s xml:id="echoid-s3379" xml:space="preserve">ſimilia, ut ſu pra <lb/> <anchor type="note" xlink:label="note-0132-02a" xlink:href="note-0132-02"/> demonſtrauimus. </s> <s xml:id="echoid-s3380" xml:space="preserve">Iam ex iis, quæ tradita ſunt, conſtat cen <lb/>trum greuitatis priſmatis in plano g h k contineri. </s> <s xml:id="echoid-s3381" xml:space="preserve">Dico <lb/>ipſum eſſe in linea k m. </s> <s xml:id="echoid-s3382" xml:space="preserve">Si enim fieri poteſt, ſit o centrum;</s> <s xml:id="echoid-s3383" xml:space="preserve"> <pb o="11" file="0133" n="133" rhead="DE CENTRO GRA VIT. SOLID."/> & </s> <s xml:id="echoid-s3384" xml:space="preserve">per o ducatur o p ad k m ipſi h g æquidiſtans. </s> <s xml:id="echoid-s3385" xml:space="preserve">Itaque li <lb/>nea h m bifariã uſque eò diuidatur, quoad reliqua ſit pars <lb/>quædam q m, minor o p. </s> <s xml:id="echoid-s3386" xml:space="preserve">deinde h m, m g diuidantur in <lb/>partes æ quales ipſi m q: </s> <s xml:id="echoid-s3387" xml:space="preserve">& </s> <s xml:id="echoid-s3388" xml:space="preserve">per diuiſiones lineæ ipſi m K <lb/>æ quidiſtantes ducantur. </s> <s xml:id="echoid-s3389" xml:space="preserve">puncta uero, in quibus hæ trian-<lb/>gulorum latera ſecant, coniungantur ductis lineis r s, t u, <lb/> <anchor type="figure" xlink:label="fig-0133-01a" xlink:href="fig-0133-01"/> x y; </s> <s xml:id="echoid-s3390" xml:space="preserve">quæ baſi g h æquidiſtabunt. </s> <s xml:id="echoid-s3391" xml:space="preserve">Quoniam enim lineæ g z, <lb/>h α ſunt æ quales: </s> <s xml:id="echoid-s3392" xml:space="preserve">itemq; </s> <s xml:id="echoid-s3393" xml:space="preserve">æquales g m, m h: </s> <s xml:id="echoid-s3394" xml:space="preserve">ut m g ad g z, <lb/>ita erit m h, ad h α: </s> <s xml:id="echoid-s3395" xml:space="preserve">& </s> <s xml:id="echoid-s3396" xml:space="preserve">diuidendo, ut m z ad z g, ita m α ad <lb/>α h. </s> <s xml:id="echoid-s3397" xml:space="preserve">Sed ut m z ad z g, ita k r ad r g: </s> <s xml:id="echoid-s3398" xml:space="preserve">& </s> <s xml:id="echoid-s3399" xml:space="preserve">ut m α ad α h, ita k s <lb/> <anchor type="note" xlink:label="note-0133-01a" xlink:href="note-0133-01"/> ad s h. </s> <s xml:id="echoid-s3400" xml:space="preserve">quare ut κ r ad r g, ita k s ad s h. </s> <s xml:id="echoid-s3401" xml:space="preserve">æ quidiſtant igitur <lb/> <anchor type="note" xlink:label="note-0133-02a" xlink:href="note-0133-02"/> inter ſe ſe r s, g h. </s> <s xml:id="echoid-s3402" xml:space="preserve">eadem quoque ratione demonſtrabimus <lb/> <anchor type="note" xlink:label="note-0133-03a" xlink:href="note-0133-03"/> <pb file="0134" n="134" rhead="FED. COMMANDINI"/> t u, x y ipſi g h æquidiſtare. </s> <s xml:id="echoid-s3403" xml:space="preserve">Et quoniam triangula, quæ <lb/>fiunt à lineis K y, y u, u s, s h æqualia ſuntinter ſe, & </s> <s xml:id="echoid-s3404" xml:space="preserve">ſimilia <lb/>triangulo K m h: </s> <s xml:id="echoid-s3405" xml:space="preserve">habebit triangulum K m h ad triangulũ <lb/> <anchor type="note" xlink:label="note-0134-01a" xlink:href="note-0134-01"/> K δ y duplam proportionem eius, quæ eſt lineæ k h ad K y. <lb/></s> <s xml:id="echoid-s3406" xml:space="preserve">ſed _K_ h poſita eſt quadrupla ipſius k y. </s> <s xml:id="echoid-s3407" xml:space="preserve">ergo triangulum <lb/>κ m h ad triangulum _K_ δ y eãdem proportionem habebit, <lb/>quam ſexdecim ad unũ & </s> <s xml:id="echoid-s3408" xml:space="preserve">ad quatuor triangula k δ y, y u, <lb/>u s, s α h habebit eandem, quam fexdecim ad quatuor, hoc <lb/>eſt quam h K ad κ y: </s> <s xml:id="echoid-s3409" xml:space="preserve">& </s> <s xml:id="echoid-s3410" xml:space="preserve">ſimiliter eandem habere demonſtra <lb/>bitur trian-<lb/> <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> gulum κ m g <lb/>ad quatuor <lb/>triãgula K δ <lb/>x, x γ t, t β r, <lb/>r z g. </s> <s xml:id="echoid-s3411" xml:space="preserve">quare <lb/> <anchor type="note" xlink:label="note-0134-02a" xlink:href="note-0134-02"/> totum trian <lb/>gulum K g h <lb/>ad omnia tri <lb/>angula g z r, <lb/>r β t, t γ x, x δ <lb/>_K_, K δ y, y u, <lb/>u s, s α h ita <lb/>erit, ut h κ a d <lb/>k y, hoc eſt <lb/>ut h m ad m <lb/>q. </s> <s xml:id="echoid-s3412" xml:space="preserve">Si igitur in <lb/>triangulis a b c, d e f deſcribantur figuræ ſimiles ei, quæ de-<lb/>ſcripta eſt in g h K triangulo: </s> <s xml:id="echoid-s3413" xml:space="preserve">& </s> <s xml:id="echoid-s3414" xml:space="preserve">per lineas ſibi reſp onden-<lb/>tes plana ducantur: </s> <s xml:id="echoid-s3415" xml:space="preserve">totum priſma a f diuiſum eritin tria <lb/>ſolida parallelepipeda y γ, u β, s z, quorum baſes ſunt æ qua <lb/>les & </s> <s xml:id="echoid-s3416" xml:space="preserve">ſimiles ipſis parallelogrammis y γ, u β, s z: </s> <s xml:id="echoid-s3417" xml:space="preserve">& </s> <s xml:id="echoid-s3418" xml:space="preserve">in octo <lb/>priſmata g z r, r β t, t γ x, x δ K, κ δ y, y u, u s, s α h: </s> <s xml:id="echoid-s3419" xml:space="preserve">quorum <lb/>item baſes æquales, & </s> <s xml:id="echoid-s3420" xml:space="preserve">ſimiles ſunt dictis triangulis; </s> <s xml:id="echoid-s3421" xml:space="preserve">altitu-<lb/>do autem in omnibus, totius priſmatis altitudini æ qualis.</s> <s xml:id="echoid-s3422" xml:space="preserve"> <pb o="12" file="0135" n="135" rhead="DE CENTRO GRA VIT. SOLID."/> Itaque ſolidi parallelepipedi y γ centrum grauitatis eſt in <lb/>linea δ: </s> <s xml:id="echoid-s3423" xml:space="preserve">ſolidi u β centrum eſt in linea ε η: </s> <s xml:id="echoid-s3424" xml:space="preserve">& </s> <s xml:id="echoid-s3425" xml:space="preserve">ſolidi s z in li <lb/>nea η m, quæ quidem lineæ axes ſunt, cum planorum oppo <lb/>ſitorum centra coniungant. </s> <s xml:id="echoid-s3426" xml:space="preserve">ergo magnitudinis ex his ſoli <lb/>dis compoſitæ centrum grauitatis eſt in linea δ m, quod ſit <lb/>θ; </s> <s xml:id="echoid-s3427" xml:space="preserve">& </s> <s xml:id="echoid-s3428" xml:space="preserve">iuncta θ o producatur: </s> <s xml:id="echoid-s3429" xml:space="preserve">à puncto autem h ducatur h μ <lb/>ipſi m κ æquidiſtans, quæ cum θ o in μ conueniat. </s> <s xml:id="echoid-s3430" xml:space="preserve">triangu <lb/>lum igitur g h κ ad omnia triangula g z r, r β t, t γ x, x δ k, <lb/>κ δ y, y u, u s, s α h eandem habet proportionem, quam h m <lb/>ad m q; </s> <s xml:id="echoid-s3431" xml:space="preserve">hoc eſt, quam μ θ ad θ λ: </s> <s xml:id="echoid-s3432" xml:space="preserve">nam ſi h m, μ θ produci in <lb/>telligantur, quouſque coeant; </s> <s xml:id="echoid-s3433" xml:space="preserve">erit ob linearum q y, m k æ-<lb/>quidiſtantiam, ut h q ad q m, ita μ λ ad ad λ θ: </s> <s xml:id="echoid-s3434" xml:space="preserve">& </s> <s xml:id="echoid-s3435" xml:space="preserve">componen <lb/>do, ut h m ad m q, ita μ θ ad θ λ. </s> <s xml:id="echoid-s3436" xml:space="preserve">linea uero θ o maior eſt, <lb/>quàm θ λ: </s> <s xml:id="echoid-s3437" xml:space="preserve">habebit igitur μ θ ad θ λ maiorem proportio-<lb/> <anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/> nem, quàm ad θ o. </s> <s xml:id="echoid-s3438" xml:space="preserve">quare triangulum etiam g h k ad omnia <lb/>iam dicta triangula maiorem proportionẽ habebit, quàm <lb/>μ θ ad θ o. </s> <s xml:id="echoid-s3439" xml:space="preserve">ſed ut triangulũ g h k ad omnia triangula, ita to-<lb/>tũ priſma a f ad omnia priſmata g z r, r β t, t γ x, x δ k, k δ y, <lb/>y u, u s, s α h: </s> <s xml:id="echoid-s3440" xml:space="preserve">quoniam enim ſolida parallelepipeda æque al <lb/>ta, eandem inter ſe proportionem habent, quam baſes; </s> <s xml:id="echoid-s3441" xml:space="preserve">ut <lb/>ex trigeſimaſecunda undecimi elementorum conſtat. </s> <s xml:id="echoid-s3442" xml:space="preserve">ſunt <lb/> <anchor type="note" xlink:label="note-0135-02a" xlink:href="note-0135-02"/> autem ſolida parallelepipeda priſmatum triangulares ba-<lb/>ſes habentium dupla: </s> <s xml:id="echoid-s3443" xml:space="preserve">ſequitur, ut etiam huiuſmodi priſ-<lb/> <anchor type="note" xlink:label="note-0135-03a" xlink:href="note-0135-03"/> matainter ſe ſint, ſicut eorum baſes. </s> <s xml:id="echoid-s3444" xml:space="preserve">ergo totum priſma ad <lb/>omnia priſmata maiorem proportionem habet, quam μ θ <lb/>ad θ o: </s> <s xml:id="echoid-s3445" xml:space="preserve">& </s> <s xml:id="echoid-s3446" xml:space="preserve">diuidendo ſolida parallelepipeda y γ, u β, s z ad o-<lb/> <anchor type="note" xlink:label="note-0135-04a" xlink:href="note-0135-04"/> mnia prifmata proportionem habent maiorem, quàm μ o <lb/>ad o θ. </s> <s xml:id="echoid-s3447" xml:space="preserve">fiat @ o ad o θ, ut folida parallelepipeda y γ, u β, s z ad <lb/>omnia priſmata. </s> <s xml:id="echoid-s3448" xml:space="preserve">Itaque cum à priſmate a f, cuius cẽtrum <lb/>grauitatis eſt o, auferatur magnitudo ex ſolidis parallelepi <lb/>pedis y γ, u β, s z conſtans: </s> <s xml:id="echoid-s3449" xml:space="preserve">atque ipfius grauitatis centrum <lb/>ſit θ: </s> <s xml:id="echoid-s3450" xml:space="preserve">reliquæ magnitudinis, quæ ex omnibus priſmatibus <lb/>conſtat, grauitatis centrum erit in linea θ o producta: </s> <s xml:id="echoid-s3451" xml:space="preserve">& </s> <s xml:id="echoid-s3452" xml:space="preserve"><lb/>in puncto ν, ex o ctaua propoſitione eiuſdem libri Archi- <pb file="0136" n="136" rhead="FED. COMMANDINI"/> medis. </s> <s xml:id="echoid-s3453" xml:space="preserve">ergo punctum v extra p riſima a f poſitum, centrũ <lb/>erit magnitudinis cõpoſitæ e x omnibus priſmatibus g z r, <lb/>r β t, t γ x, x δ k, k δ y, y u, u s, s α h, quod fieri nullo modo po <lb/>teſt. </s> <s xml:id="echoid-s3454" xml:space="preserve">eſt enim ex diſſinitione centrum grauitatis ſolidæ figu <lb/>ræ intra ipſam poſitum, non extra. </s> <s xml:id="echoid-s3455" xml:space="preserve">quare relinquitur, ut cẽ <lb/>trum grauitatis priſmatis ſit in linea K m. </s> <s xml:id="echoid-s3456" xml:space="preserve">Rurſus b c bifa-<lb/>riam in ξ diuidatur: </s> <s xml:id="echoid-s3457" xml:space="preserve">& </s> <s xml:id="echoid-s3458" xml:space="preserve">ducta a ξ, per ipſam, & </s> <s xml:id="echoid-s3459" xml:space="preserve">per lineam <lb/>a g d plan um ducatur; </s> <s xml:id="echoid-s3460" xml:space="preserve">quod priſma ſecet: </s> <s xml:id="echoid-s3461" xml:space="preserve">faciatq; </s> <s xml:id="echoid-s3462" xml:space="preserve">in paral <lb/>lelogrammo b f ſectionem ξ π di uidet punctum π lineam <lb/>quoque c f bifariam: </s> <s xml:id="echoid-s3463" xml:space="preserve">& </s> <s xml:id="echoid-s3464" xml:space="preserve">erit p lani eius, & </s> <s xml:id="echoid-s3465" xml:space="preserve">trianguli g h K <lb/>communis ſectio g u; </s> <s xml:id="echoid-s3466" xml:space="preserve">quòd p ũctum u in inedio lineæ h K <lb/> <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> poſitum ſi t. </s> <s xml:id="echoid-s3467" xml:space="preserve">Similiter demonſtrabimus centrum grauita-<lb/>tis priſm atis in ipſa g u ineſſe. </s> <s xml:id="echoid-s3468" xml:space="preserve">ſit autem planorum c f n l, <lb/>a d π ξ communis ſectio linea ρ ο τ quæ quidem priſmatis <lb/>axis erit, cum tranſeat per centra grauitatis triangulorum <lb/>a b c, g h k, d e f, ex quartadecima eiuſdem. </s> <s xml:id="echoid-s3469" xml:space="preserve">ergo centrum <lb/>grauitatis pri ſmatis a f eſt punctum σ, centrum ſcilicet <pb o="13" file="0137" n="137" rhead="DE CENTRO GRAVIT. SOLID."/> trianguli g h K, & </s> <s xml:id="echoid-s3470" xml:space="preserve">ipſius ρ τ axis medium.</s> <s xml:id="echoid-s3471" xml:space="preserve"/> </p> <div xml:id="echoid-div217" type="float" level="2" n="2"> <note position="left" xlink:label="note-0132-02" xlink:href="note-0132-02a" xml:space="preserve">5. huius</note> <figure xlink:label="fig-0133-01" xlink:href="fig-0133-01a"> <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0133-01"/> </figure> <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">2. ſexti.</note> <note position="right" xlink:label="note-0133-02" xlink:href="note-0133-02a" xml:space="preserve">I1. quinti</note> <note position="right" xlink:label="note-0133-03" xlink:href="note-0133-03a" xml:space="preserve">2. ſexti.</note> <note position="left" xlink:label="note-0134-01" xlink:href="note-0134-01a" xml:space="preserve">19. ſexti</note> <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a"> <image file="0134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0134-01"/> </figure> <note position="left" xlink:label="note-0134-02" xlink:href="note-0134-02a" xml:space="preserve">2. uel 121 <lb/>quinti.</note> <note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">8. quinti.</note> <note position="right" xlink:label="note-0135-02" xlink:href="note-0135-02a" xml:space="preserve">28. unde <lb/>cimi</note> <note position="right" xlink:label="note-0135-03" xlink:href="note-0135-03a" xml:space="preserve">15. quinti</note> <note position="right" xlink:label="note-0135-04" xlink:href="note-0135-04a" xml:space="preserve">19. quinti <lb/>apud Cã <lb/>panum.</note> <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a"> <image file="0136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0136-01"/> </figure> </div> <p> <s xml:id="echoid-s3472" xml:space="preserve">Sit priſma a g, cuius oppoſita plana ſint quadrilatera <lb/>a b c d, e f g h: </s> <s xml:id="echoid-s3473" xml:space="preserve">ſecenturq; </s> <s xml:id="echoid-s3474" xml:space="preserve">a e, b f, c g, d h bifariam: </s> <s xml:id="echoid-s3475" xml:space="preserve">& </s> <s xml:id="echoid-s3476" xml:space="preserve">per di-<lb/>uiſiones planum ducatur; </s> <s xml:id="echoid-s3477" xml:space="preserve">quod ſectionem faciat quadrila-<lb/>terum _K_ l m n. </s> <s xml:id="echoid-s3478" xml:space="preserve">Deinde iuncta a c per lineas a c, a e ducatur <lb/>planum ſecãs priſma, quod ipſum diuidet in duo priſmata <lb/>triangulares baſes habentia a b c e f g, a d c e h g. </s> <s xml:id="echoid-s3479" xml:space="preserve">Sint autẽ <lb/>triangulorum a b c, e f g gra-<lb/> <anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/> uitatis centra o p: </s> <s xml:id="echoid-s3480" xml:space="preserve">& </s> <s xml:id="echoid-s3481" xml:space="preserve">triangu-<lb/>lorum a d c, e h g centra q r: <lb/></s> <s xml:id="echoid-s3482" xml:space="preserve">iunganturq; </s> <s xml:id="echoid-s3483" xml:space="preserve">o p, q r; </s> <s xml:id="echoid-s3484" xml:space="preserve">quæ pla-<lb/>no _k_ l m n occurrant in pun-<lb/>ctis s t. </s> <s xml:id="echoid-s3485" xml:space="preserve">erit ex iis, quæ demon <lb/>ſtrauimus, punctum s grauita <lb/>tis centrum trianguli k l m; </s> <s xml:id="echoid-s3486" xml:space="preserve">& </s> <s xml:id="echoid-s3487" xml:space="preserve"><lb/>ipſius priſmatis a b c e f g: </s> <s xml:id="echoid-s3488" xml:space="preserve">pun <lb/>ctum uero t centrum grauita <lb/>tis trianguli _K_ n m, & </s> <s xml:id="echoid-s3489" xml:space="preserve">priſma-<lb/>tis a d c, e h g. </s> <s xml:id="echoid-s3490" xml:space="preserve">iunctis igitur <lb/>o q, p r, s t, erit in linea o q cẽ <lb/>trum grauitatis quadrilateri <lb/>a b c d, quod ſit u: </s> <s xml:id="echoid-s3491" xml:space="preserve">& </s> <s xml:id="echoid-s3492" xml:space="preserve">in linea <lb/>p r cẽtrum quadrilateri e f g h <lb/>ſit autem x. </s> <s xml:id="echoid-s3493" xml:space="preserve">deniqueiungatur <lb/>u x, quæ ſecet lineam ſ t in y. </s> <s xml:id="echoid-s3494" xml:space="preserve">ſe <lb/>cabit enim cum ſint in eodem <lb/> <anchor type="note" xlink:label="note-0137-01a" xlink:href="note-0137-01"/> plano: </s> <s xml:id="echoid-s3495" xml:space="preserve">atq; </s> <s xml:id="echoid-s3496" xml:space="preserve">erit y grauitatis centrum quadril ateri _K_ lm n. <lb/></s> <s xml:id="echoid-s3497" xml:space="preserve">Dico idem punctum y centrum quoque gra uitatis eſſe to-<lb/>tius priſmatis. </s> <s xml:id="echoid-s3498" xml:space="preserve">Quoniam enim quadri lateri k lm n graui-<lb/>tatis centrum eſt y: </s> <s xml:id="echoid-s3499" xml:space="preserve">linea s y ad y t eandem proportionem <lb/>habebit, quam triangulum k n m ad triangulum k lm, ex 8 <lb/>Archimedis de centro grauitatis planorum. </s> <s xml:id="echoid-s3500" xml:space="preserve">Vtautem triã <lb/>gulum k n m ad ipſum k l m, hoc eſt ut triangulum a d c ad <lb/>triangulum a b c, æqualia enim ſunt, ita priſina a d c e h g <pb file="0138" n="138" rhead="FED. COMMANDINI"/> ad priſma a b c e f g. </s> <s xml:id="echoid-s3501" xml:space="preserve">quare linea s y ad y t eandem propor-<lb/>tionem habet, quam priſma a d c e h g ad priſma a b c e f g. <lb/></s> <s xml:id="echoid-s3502" xml:space="preserve">Sed priſmatis a b c e f g centrum grauitatis eſts: </s> <s xml:id="echoid-s3503" xml:space="preserve">& </s> <s xml:id="echoid-s3504" xml:space="preserve">priſma-<lb/>tis a d c e h g centrum t. </s> <s xml:id="echoid-s3505" xml:space="preserve">magnitudinis igitur ex his compo <lb/>ſitæ, hoc eſt totius priſmatis a g centrum grauitatis eſt pun <lb/>ctum y; </s> <s xml:id="echoid-s3506" xml:space="preserve">medium ſcilicet axis u x, qui oppoſitorum plano-<lb/>rum centra coniungit.</s> <s xml:id="echoid-s3507" xml:space="preserve"/> </p> <div xml:id="echoid-div218" type="float" level="2" n="3"> <figure xlink:label="fig-0137-01" xlink:href="fig-0137-01a"> <image file="0137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0137-01"/> </figure> <note position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">5. huius.</note> </div> <p> <s xml:id="echoid-s3508" xml:space="preserve">Rurſus ſit priſma baſim habens pentagonum a b c d e: <lb/></s> <s xml:id="echoid-s3509" xml:space="preserve">& </s> <s xml:id="echoid-s3510" xml:space="preserve">quod ei opponitur ſit f g h _K_ l: </s> <s xml:id="echoid-s3511" xml:space="preserve">ſec enturq; </s> <s xml:id="echoid-s3512" xml:space="preserve">a f, b g, c h, <lb/>d _k_, el bifariam: </s> <s xml:id="echoid-s3513" xml:space="preserve">& </s> <s xml:id="echoid-s3514" xml:space="preserve">per diuiſiones ducto plano, ſectio ſit pẽ <lb/>tagonũ m n o p q. </s> <s xml:id="echoid-s3515" xml:space="preserve">deinde iuncta e b per lineas le, e b aliud <lb/>planum ducatur, diuidẽs priſ <lb/> <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> ma a k in duo priſmata, in priſ <lb/>ma ſcilicet al, cuius plana op-<lb/>poſita ſint triangula a b e f g l: <lb/></s> <s xml:id="echoid-s3516" xml:space="preserve">& </s> <s xml:id="echoid-s3517" xml:space="preserve">in prima b _k_ cuius plana op <lb/>poſita ſint quadrilatera b c d e <lb/>g h _k_ l. </s> <s xml:id="echoid-s3518" xml:space="preserve">Sint autem triangulo-<lb/>rum a b e, f g l centra grauita <lb/>tis puncta r ſ: </s> <s xml:id="echoid-s3519" xml:space="preserve">& </s> <s xml:id="echoid-s3520" xml:space="preserve">b c d e, g h _k_ l <lb/>quadrilaterorum centra tu: </s> <s xml:id="echoid-s3521" xml:space="preserve"><lb/>iunganturq; </s> <s xml:id="echoid-s3522" xml:space="preserve">r s, t u o ccurren-<lb/>tes plano m n o p q in punctis <lb/>x y. </s> <s xml:id="echoid-s3523" xml:space="preserve">& </s> <s xml:id="echoid-s3524" xml:space="preserve">itidem iungãtur r t, ſu, <lb/>x y. </s> <s xml:id="echoid-s3525" xml:space="preserve">erit in linea r t cẽtrum gra <lb/>uitatis pentagoni a b c d e; </s> <s xml:id="echoid-s3526" xml:space="preserve"><lb/>quod ſit z: </s> <s xml:id="echoid-s3527" xml:space="preserve">& </s> <s xml:id="echoid-s3528" xml:space="preserve">in linea ſu cen-<lb/>trum pentagoni f g h k l: </s> <s xml:id="echoid-s3529" xml:space="preserve">ſit au <lb/>tem χ: </s> <s xml:id="echoid-s3530" xml:space="preserve">& </s> <s xml:id="echoid-s3531" xml:space="preserve">ducatur z χ, quæ di-<lb/>cto plano in χ occurrat. </s> <s xml:id="echoid-s3532" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s3533" xml:space="preserve"><lb/>punctum x eſt centrum graui <lb/>tatis trianguli m n q, ac priſ-<lb/>matis al: </s> <s xml:id="echoid-s3534" xml:space="preserve">& </s> <s xml:id="echoid-s3535" xml:space="preserve">y grauitatis centrum quadrilateri n o p q, ac <lb/>priſmatis b k. </s> <s xml:id="echoid-s3536" xml:space="preserve">quare y centrum erit pentagoni m n o p q. </s> <s xml:id="echoid-s3537" xml:space="preserve">&</s> <s xml:id="echoid-s3538" xml:space="preserve"> <pb o="14" file="0139" n="139" rhead="DE CENTRO GRAVIT. SOLID."/> ſimiliter demonſtrabitur totius priſmatis a _K_ grauitatis eſ <lb/>ſe centrum. </s> <s xml:id="echoid-s3539" xml:space="preserve">Simili ratione & </s> <s xml:id="echoid-s3540" xml:space="preserve">in aliis priſinatibus illud <lb/>idem ſacile demonſtrabitur. </s> <s xml:id="echoid-s3541" xml:space="preserve">Quo autem pacto in omni <lb/>figura rectilinea centrum grauitatis inueniatur, do cuimus <lb/>in commentariis in ſextam propoſitionem Archimedis de <lb/>quadratura parabolæ.</s> <s xml:id="echoid-s3542" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="4"> <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a"> <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0138-01"/> </figure> </div> <p> <s xml:id="echoid-s3543" xml:space="preserve">Sit cylindrus, uel cylindri portio c e cuius axis a b: </s> <s xml:id="echoid-s3544" xml:space="preserve">ſece-<lb/>turq, plano per axem ducto; </s> <s xml:id="echoid-s3545" xml:space="preserve">quod ſectionem faciat paral-<lb/>lelo grammum c d e f: </s> <s xml:id="echoid-s3546" xml:space="preserve">& </s> <s xml:id="echoid-s3547" xml:space="preserve">diuiſis c f, d e bifariam in punctis <lb/> <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> g h, per ea ducatur planum baſi æquidiſtans. </s> <s xml:id="echoid-s3548" xml:space="preserve">erit ſectio g h <lb/>circulus, uel ellipſis, centrum habens in axe; </s> <s xml:id="echoid-s3549" xml:space="preserve">quod ſit K: </s> <s xml:id="echoid-s3550" xml:space="preserve">at-<lb/> <anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> que erunt ex iis, quæ demonſtrauimus, centra grauitatis <lb/>planorum oppoſitorum puncta a b: </s> <s xml:id="echoid-s3551" xml:space="preserve">& </s> <s xml:id="echoid-s3552" xml:space="preserve">plani g h ipſum _k_. </s> <s xml:id="echoid-s3553" xml:space="preserve">in <lb/>quo quidem plano eſt centrum grauitatis cylindri, uel cy-<lb/>lindri portionis. </s> <s xml:id="echoid-s3554" xml:space="preserve">Dico punctum K cylindri quoque, uel cy <lb/>lindri portionis grauitatis centrum eſſe. </s> <s xml:id="echoid-s3555" xml:space="preserve">Si enim fieri po-<lb/>teſt, ſitl centrum: </s> <s xml:id="echoid-s3556" xml:space="preserve">ducaturq; </s> <s xml:id="echoid-s3557" xml:space="preserve">k l, & </s> <s xml:id="echoid-s3558" xml:space="preserve">extra figuram in m pro-<lb/>ducatur. </s> <s xml:id="echoid-s3559" xml:space="preserve">quam uero proportionem habet linea m K ad _k_ l <pb file="0140" n="140" rhead="FED. COMMANDINI"/> habeat circulus, uel ellipſis g h ad aliud ſpacium, in quo u: <lb/></s> <s xml:id="echoid-s3560" xml:space="preserve">& </s> <s xml:id="echoid-s3561" xml:space="preserve">in circulo, uel ellipſi plane deſcribatur rectilinea figura, <lb/>ita ut tãdem relinquãtur portiones minores ſpacio u, quæ <lb/>ſit o p g q r s h t: </s> <s xml:id="echoid-s3562" xml:space="preserve">deſcriptaq; </s> <s xml:id="echoid-s3563" xml:space="preserve">ſimili figura in oppoſitis pla-<lb/>nis c d, f e, per lineas ſibi ipſis reſpondentes plana ducãtur. </s> <s xml:id="echoid-s3564" xml:space="preserve"><lb/>Itaque cylindrus, uel cylindri portio diuiditur in priſma, <lb/>cuius quidem baſis eſt figura rectilinea iam dicta, centrum <lb/>que grauitatis punctum K: </s> <s xml:id="echoid-s3565" xml:space="preserve">& </s> <s xml:id="echoid-s3566" xml:space="preserve">in multa ſolida, quæ pro baſi <lb/>bus habent relictas portiones, quas nos ſolidas portiones <lb/>appellabimus. </s> <s xml:id="echoid-s3567" xml:space="preserve">cum igitur portiones ſint minores ſpacio <lb/>u, circulus, uel ellipſis g h ad portiones maiorem propor-<lb/>tionem habebit, quàm linea m k ad K l. </s> <s xml:id="echoid-s3568" xml:space="preserve">fiat n k ad K l, ut <lb/>circulus uel ellipſis g h ad ipſas portiones. </s> <s xml:id="echoid-s3569" xml:space="preserve">Sed ut circulus <lb/>uel ellipſis g h ad figuram rectilineam in ipſa deſcri-<lb/>ptam, ita eſt cylindrus uel cylindri portio c e ad priſma, <lb/>quod rectilineam figuram pro baſi habet, & </s> <s xml:id="echoid-s3570" xml:space="preserve">altitudinem <lb/>æqualem; </s> <s xml:id="echoid-s3571" xml:space="preserve">id, quod infra demonſtrabitur, ergo per conuer <lb/>ſionem rationis, ut circulus, uel ellipſis g h ad portiones re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad ſolidas por-<lb/>tiones, quare cylindrus uel cylindri portio ad ſolidas por-<lb/>tiones eandem proportionem habet, quam linea n k a d _k_ <lb/>& </s> <s xml:id="echoid-s3572" xml:space="preserve">diuidendo priſma, cuius baſis eſt rectilinea figura ad ſo-<lb/>lidas portiones eandem proportionem habet, quam n lad <lb/>1 _k_. </s> <s xml:id="echoid-s3573" xml:space="preserve">& </s> <s xml:id="echoid-s3574" xml:space="preserve">quoniam a cylindro uel cylindri portione, cuius gra-<lb/>uitatis centrum eſt l, aufertur priſma baſim habens rectili-<lb/>neam figurã, cuius centrũ grauitatis eſt _K_: </s> <s xml:id="echoid-s3575" xml:space="preserve">reſiduæ magnitu <lb/>dinis ex ſolidis portionibus cõpoſitæ grauitatis cẽtrũ erit <lb/>in linea k l protracta, & </s> <s xml:id="echoid-s3576" xml:space="preserve">in puncto n; </s> <s xml:id="echoid-s3577" xml:space="preserve">quod eſt abſurdū. </s> <s xml:id="echoid-s3578" xml:space="preserve">relin <lb/>quitur ergo, ut cẽtrum grauitatis cylindri; </s> <s xml:id="echoid-s3579" xml:space="preserve">uel cylin dri por <lb/>tionis ſit punctũ k. </s> <s xml:id="echoid-s3580" xml:space="preserve">quæ omnia demonſtrãda propoſuimus.</s> <s xml:id="echoid-s3581" xml:space="preserve"/> </p> <div xml:id="echoid-div220" type="float" level="2" n="5"> <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a"> <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0139-01"/> </figure> <note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">4. huius.</note> </div> <p> <s xml:id="echoid-s3582" xml:space="preserve">At uero cylindrum, uel cylindri portionẽ ce <lb/>ad priſma, cuius baſis eſt rectilinea figura in ſpa-<lb/>cio g h deſcripta, & </s> <s xml:id="echoid-s3583" xml:space="preserve">altitudo æqualis; </s> <s xml:id="echoid-s3584" xml:space="preserve">eandem ha- <pb o="15" file="0141" n="141" rhead="DE CENTRO GRAVIT. SOLID."/> bere proportionem, quam ſpacium g h ad dictã <lb/>figuram, hoc modo demonſtrabimus.</s> <s xml:id="echoid-s3585" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3586" xml:space="preserve">Intelligatur circulus, uel ellipſis x æqualis figuræ rectili-<lb/>neæ in g h ſpacio deſcriptæ: </s> <s xml:id="echoid-s3587" xml:space="preserve">& </s> <s xml:id="echoid-s3588" xml:space="preserve">ab x conſtituatur conus, uel <lb/> <anchor type="figure" xlink:label="fig-0141-01a" xlink:href="fig-0141-01"/> coni portio, altitudinẽ habens eandẽ, quã cylindrus uel cy <lb/>lindri portio c e. </s> <s xml:id="echoid-s3589" xml:space="preserve">Sit deinde rectilinea figura, in quay eade, <lb/>quæ in ſpacio g h deſcripta eſt: </s> <s xml:id="echoid-s3590" xml:space="preserve">& </s> <s xml:id="echoid-s3591" xml:space="preserve">ab hac pyramis æquealta <lb/>conſtituatur. </s> <s xml:id="echoid-s3592" xml:space="preserve">Dico conũ uel coni portionẽ x pyramidiy æ-<lb/>qualẽ eſſe. </s> <s xml:id="echoid-s3593" xml:space="preserve">niſi enim ſit æqualis, uel maior, uel minor erit.</s> <s xml:id="echoid-s3594" xml:space="preserve"/> </p> <div xml:id="echoid-div221" type="float" level="2" n="6"> <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a"> <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0141-01"/> </figure> </div> <p> <s xml:id="echoid-s3595" xml:space="preserve">Sit primum maior, et exuperet ſolido z. </s> <s xml:id="echoid-s3596" xml:space="preserve">Itaque in circu <lb/>lo, uel ellipſi x deſcribatur figura rectilinea; </s> <s xml:id="echoid-s3597" xml:space="preserve">& </s> <s xml:id="echoid-s3598" xml:space="preserve">in ea pyra-<lb/>mis eandem, quam conus, uel coni portio altitudinem ha-<lb/>bens, ita ut portiones relictæ minores ſint ſolido z, quem-<lb/>admodum docetur in duodecimo libro elementorum pro <lb/>poſitione undecima. </s> <s xml:id="echoid-s3599" xml:space="preserve">erit pyramis x adhuc pyramide y ma <lb/>ior. </s> <s xml:id="echoid-s3600" xml:space="preserve">& </s> <s xml:id="echoid-s3601" xml:space="preserve">quoniam piramides æque altæ inter ſe ſunt, ſicuti ba <lb/> <anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/> ſes; </s> <s xml:id="echoid-s3602" xml:space="preserve">pyramis x ad piramidem y eandem proportionem ha-<lb/>bet, quàm figura rectilinea x ad figuram y. </s> <s xml:id="echoid-s3603" xml:space="preserve">Sed ſigura recti <pb file="0142" n="142" rhead="FED. COMMANDINI"/> <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> linea x cum ſit minor circulo, uel ellipſi, eſt etiam minor fi-<lb/>gura rectilinea y. </s> <s xml:id="echoid-s3604" xml:space="preserve">ergo pyramis x pyramide y minor erit. <lb/></s> <s xml:id="echoid-s3605" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3606" xml:space="preserve">maior; </s> <s xml:id="echoid-s3607" xml:space="preserve">quod fieri nõ poteſt. </s> <s xml:id="echoid-s3608" xml:space="preserve">At ſi conus, uel coni por <lb/>tio x ponatur minor pyramide y: </s> <s xml:id="echoid-s3609" xml:space="preserve">ſit alter conus æque al-<lb/>tus, uel altera coni portio χ ipſi pyramidi y æqualis. </s> <s xml:id="echoid-s3610" xml:space="preserve">erit <lb/>eius baſis circulus, uel ellipſis maior circulo, uel ellipſi x, <lb/>quorum exceſſus ſit ſpacium ω. </s> <s xml:id="echoid-s3611" xml:space="preserve">Siigitur in circulo, uel elli-<lb/>pſi χ figura rectilinea deſcribatur, ita ut portiones relictæ <lb/>ſint ω ſpacio minores, eiuſinodi figura adhuc maior erit cir <lb/>culo, uel ellipſi x, hoc eſt figura rectilinea _y_. </s> <s xml:id="echoid-s3612" xml:space="preserve">& </s> <s xml:id="echoid-s3613" xml:space="preserve">p_y_ramis in <lb/>ea conſtituta minor cono, uel coni portione χ, hoc eſt mi-<lb/>nor p_y_ramide_y_. </s> <s xml:id="echoid-s3614" xml:space="preserve">eſt ergo ut χ figura rectilinea ad figuram <lb/>rectilineam _y_, ita pyramis χ ad pyramidem _y_. </s> <s xml:id="echoid-s3615" xml:space="preserve">quare cum <lb/>figura rectilinea χ ſit maior figura_y_: </s> <s xml:id="echoid-s3616" xml:space="preserve">erit & </s> <s xml:id="echoid-s3617" xml:space="preserve">p_y_ramis χ p_y_-<lb/>ramide_y_ maior. </s> <s xml:id="echoid-s3618" xml:space="preserve">ſed erat minor; </s> <s xml:id="echoid-s3619" xml:space="preserve">quod rurſus fieri non po-<lb/>teſt. </s> <s xml:id="echoid-s3620" xml:space="preserve">non eſt igitur conus, uel coni portio x neque maior, <lb/>neque minor p_y_ramide_y_. </s> <s xml:id="echoid-s3621" xml:space="preserve">ergo ipſi neceſſario eſt æqualis. </s> <s xml:id="echoid-s3622" xml:space="preserve"><lb/>Itaque quoniam ut conus ad conum, uel coni portio ad co <pb o="15" file="0143" n="143" rhead="DE CENTRO GRAVIT. SOLID."/> <anchor type="figure" xlink:label="fig-0143-01a" xlink:href="fig-0143-01"/> ni portionem, ita eſt c_y_lindrus ad c_y_lindrum, uel c_y_lin-<lb/>dri portio ad c_y_lindri portionem: </s> <s xml:id="echoid-s3623" xml:space="preserve">& </s> <s xml:id="echoid-s3624" xml:space="preserve">ut p_y_ramis ad p_y_ra-<lb/>midem, ita priſma ad priſma, cum eadem ſit baſis, & </s> <s xml:id="echoid-s3625" xml:space="preserve">æqua <lb/>lis altitudo; </s> <s xml:id="echoid-s3626" xml:space="preserve">erit c_y_lindrus uel c_y_lindri portio x priſma-<lb/>ti _y_ æqualis. </s> <s xml:id="echoid-s3627" xml:space="preserve">eftq; </s> <s xml:id="echoid-s3628" xml:space="preserve">ut ſpacium g h ad ſpacium x, ita c_y_lin-<lb/>drus, uel c_y_lindri portio c e ad c_y_lindrum, uel c_y_lindri por-<lb/>tionem x. </s> <s xml:id="echoid-s3629" xml:space="preserve">Conſtatigitur c_y_lindrum uel c_y_lindri portionẽ <lb/>c e, ad priſina_y_, quippe cuius baſis eſt figura rectilinea in <lb/> <anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/> ſpacio g h deſcripta, eandem proportionem habere, quam <lb/>ſpacium g h habet ad ſpacium x, hoc eſt ad dictam figuram. <lb/></s> <s xml:id="echoid-s3630" xml:space="preserve">quod demonſtrandum fuerat.</s> <s xml:id="echoid-s3631" xml:space="preserve"/> </p> <div xml:id="echoid-div222" type="float" level="2" n="7"> <note position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">6. duode-<lb/>cimi.</note> <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a"> <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0142-01"/> </figure> <figure xlink:label="fig-0143-01" xlink:href="fig-0143-01a"> <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0143-01"/> </figure> <note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">7. quinti</note> </div> </div> <div xml:id="echoid-div224" type="section" level="1" n="74"> <head xml:id="echoid-head81" xml:space="preserve">THE OREMA IX. PROPOSITIO IX.</head> <p> <s xml:id="echoid-s3632" xml:space="preserve">Si pyramis ſecetur plano baſi æquidiſtante; </s> <s xml:id="echoid-s3633" xml:space="preserve">ſe-<lb/>ctio erit figura ſimilis ei, quæ eſt baſis, centrum <lb/>grauitatis in axe habens.</s> <s xml:id="echoid-s3634" xml:space="preserve"/> </p> <pb o="16" file="0144" n="144" rhead="FED. COMMANDINI"/> <p> <s xml:id="echoid-s3635" xml:space="preserve">SIT pyramis, cuius baſis triangulum a b c; </s> <s xml:id="echoid-s3636" xml:space="preserve">axis d e: </s> <s xml:id="echoid-s3637" xml:space="preserve">& </s> <s xml:id="echoid-s3638" xml:space="preserve"><lb/>ſecetur plano baſi æquidiſtante; </s> <s xml:id="echoid-s3639" xml:space="preserve">quod ſectionẽ faciat f g h; <lb/></s> <s xml:id="echoid-s3640" xml:space="preserve">occurratq; </s> <s xml:id="echoid-s3641" xml:space="preserve">axi in puncto k. </s> <s xml:id="echoid-s3642" xml:space="preserve">Dico f g h triangulum eſſe, ipſi <lb/>a b c ſimile; </s> <s xml:id="echoid-s3643" xml:space="preserve">cuius grauitatis centrum eſt K. </s> <s xml:id="echoid-s3644" xml:space="preserve">Quoniã enim <lb/> <anchor type="note" xlink:label="note-0144-01a" xlink:href="note-0144-01"/> duo plana æquidiſtantia a b c, f g h ſecantur à plano a b d; <lb/></s> <s xml:id="echoid-s3645" xml:space="preserve">communes eorum ſectiones a b, f g æquidiſtantes erunt: </s> <s xml:id="echoid-s3646" xml:space="preserve">& </s> <s xml:id="echoid-s3647" xml:space="preserve"><lb/>eadem ratione æquidiſtantes ipſæ b c, g h: </s> <s xml:id="echoid-s3648" xml:space="preserve">& </s> <s xml:id="echoid-s3649" xml:space="preserve">c a, h f. </s> <s xml:id="echoid-s3650" xml:space="preserve">Quòd <lb/>cum duæ lineæ f g, g h, duabus a b, b c æquidiſtent, nec <lb/>ſintin eodem plano; </s> <s xml:id="echoid-s3651" xml:space="preserve">angulus ad g æqualis eſt angulo ad <lb/> <anchor type="note" xlink:label="note-0144-02a" xlink:href="note-0144-02"/> b: </s> <s xml:id="echoid-s3652" xml:space="preserve">& </s> <s xml:id="echoid-s3653" xml:space="preserve">ſimiliter angulus ad h angulo ad c: </s> <s xml:id="echoid-s3654" xml:space="preserve">angulusq; </s> <s xml:id="echoid-s3655" xml:space="preserve">ad f ei, <lb/>qui ad a eſt æqualis. </s> <s xml:id="echoid-s3656" xml:space="preserve">triangulum igitur f g h ſimile eſt tri-<lb/>angulo a b c. </s> <s xml:id="echoid-s3657" xml:space="preserve">At uero punctum k centrum eſſe grauita-<lb/>tis trianguli f g h hoc modo oſtendemus. </s> <s xml:id="echoid-s3658" xml:space="preserve">Ducantur pla-<lb/>na per axem, & </s> <s xml:id="echoid-s3659" xml:space="preserve">per lineas d a, d b, d c: </s> <s xml:id="echoid-s3660" xml:space="preserve">erunt communes ſe-<lb/> <anchor type="note" xlink:label="note-0144-03a" xlink:href="note-0144-03"/> ctiones f K, a e æquidiſtantes: </s> <s xml:id="echoid-s3661" xml:space="preserve">pariterq; </s> <s xml:id="echoid-s3662" xml:space="preserve">k g, e b; </s> <s xml:id="echoid-s3663" xml:space="preserve">& </s> <s xml:id="echoid-s3664" xml:space="preserve">k h, e c: <lb/></s> <s xml:id="echoid-s3665" xml:space="preserve">quare angulus k f h angulo e a c; </s> <s xml:id="echoid-s3666" xml:space="preserve">& </s> <s xml:id="echoid-s3667" xml:space="preserve">angulus k f g ipſi e a b <lb/> <anchor type="note" xlink:label="note-0144-04a" xlink:href="note-0144-04"/> eſt æqualis. </s> <s xml:id="echoid-s3668" xml:space="preserve">Eadem ratione <lb/> <anchor type="figure" xlink:label="fig-0144-01a" xlink:href="fig-0144-01"/> anguli ad g angulis ad b: </s> <s xml:id="echoid-s3669" xml:space="preserve">& </s> <s xml:id="echoid-s3670" xml:space="preserve"><lb/>anguli ad h iis, qui ad c æ-<lb/>quales erunt. </s> <s xml:id="echoid-s3671" xml:space="preserve">ergo puncta <lb/>e _K_ in triangulis a b c, f g h <lb/>ſimiliter ſunt poſita, per ſe-<lb/>xtam poſitionem Archime-<lb/>dis in libro de centro graui-<lb/>tatis planorum. </s> <s xml:id="echoid-s3672" xml:space="preserve">Sed cum e <lb/>ſit centrum grauitatis trian <lb/>guli a b c, erit ex undecíma <lb/>propoſitione eiuſdem libri, <lb/>& </s> <s xml:id="echoid-s3673" xml:space="preserve">K trianguli f g h grauita <lb/>tis centrum. </s> <s xml:id="echoid-s3674" xml:space="preserve">id quod demonſtrare oportebat. </s> <s xml:id="echoid-s3675" xml:space="preserve">Non aliter <lb/>in ceteris pyramidibus, quod propoſitum eſt demonſtra-<lb/>bitur.</s> <s xml:id="echoid-s3676" xml:space="preserve"/> </p> <div xml:id="echoid-div224" type="float" level="2" n="1"> <note position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">16. unde <lb/>cimi</note> <note position="left" xlink:label="note-0144-02" xlink:href="note-0144-02a" xml:space="preserve">10. undeci <lb/>mi.</note> <note position="left" xlink:label="note-0144-03" xlink:href="note-0144-03a" xml:space="preserve">16. unde-<lb/>cimi</note> <note position="left" xlink:label="note-0144-04" xlink:href="note-0144-04a" xml:space="preserve">10. unde-<lb/>cimi</note> <figure xlink:label="fig-0144-01" xlink:href="fig-0144-01a"> <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0144-01"/> </figure> </div> <pb o="17" file="0145" n="145" rhead="DE CENTRO GRAVIT. SOLID."/> </div> <div xml:id="echoid-div226" type="section" level="1" n="75"> <head xml:id="echoid-head82" xml:space="preserve">PROBLEMA I. PROPOSITIO X.</head> <p> <s xml:id="echoid-s3677" xml:space="preserve"><emph style="sc">Data</emph> qualibet pyramide, fieri poteſt, ut fi-<lb/>gura ſolida in ipſa in ſcribatur, & </s> <s xml:id="echoid-s3678" xml:space="preserve">altera circũſcri-<lb/>batur ex priſmatibus æqualem aItitudinem ha-<lb/>bẽtibus, ita ut cir cumſcripta inſcriptam excedat <lb/>magnitudine, quæ minor ſit quacũque ſolida ma <lb/>gnitudine propoſita.</s> <s xml:id="echoid-s3679" xml:space="preserve"/> </p> <figure> <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0145-01"/> </figure> <p> <s xml:id="echoid-s3680" xml:space="preserve">Sit pyramis, cuius baſis <lb/>triangulũ a b c; </s> <s xml:id="echoid-s3681" xml:space="preserve">axis d e. <lb/></s> <s xml:id="echoid-s3682" xml:space="preserve">Sitq; </s> <s xml:id="echoid-s3683" xml:space="preserve">priſma, quod eandẽ <lb/>baſim habeat, & </s> <s xml:id="echoid-s3684" xml:space="preserve">axem eun <lb/>dem. </s> <s xml:id="echoid-s3685" xml:space="preserve">Itaque hoc priſma-<lb/>te continenter ſecto bifa-<lb/>riam, plano baſi æquidiftã <lb/>te, relinquetur tãdem priſ <lb/>ma quoddam minus pro-<lb/>poſita magnitudine: </s> <s xml:id="echoid-s3686" xml:space="preserve">quod <lb/>quidem baſim eandem ha <lb/>beat, quam pyramis, & </s> <s xml:id="echoid-s3687" xml:space="preserve">a-<lb/>xem e f. </s> <s xml:id="echoid-s3688" xml:space="preserve">diuidatur d e in <lb/>partes æquales ipſi e f in <lb/>punctis g h k l m n: </s> <s xml:id="echoid-s3689" xml:space="preserve">& </s> <s xml:id="echoid-s3690" xml:space="preserve">per <lb/>diuiſiones plana ducãtur: </s> <s xml:id="echoid-s3691" xml:space="preserve"><lb/>quæ baſibus æquidiſtent, <lb/>erunt ſectiones, triangula <lb/>ipſi a b c ſimilia, ut proxi-<lb/>me oſtendimus. </s> <s xml:id="echoid-s3692" xml:space="preserve">ab uno <lb/>quoque autẽ horum trian <lb/>gulorum duo priſmata cõ <lb/>ſtruantur; </s> <s xml:id="echoid-s3693" xml:space="preserve">unum quidem <lb/>ad partes e; </s> <s xml:id="echoid-s3694" xml:space="preserve">alterum ad <pb file="0146" n="146" rhead="FED. COMMANDINI"/> partes d. </s> <s xml:id="echoid-s3695" xml:space="preserve">in pyramide igitur inſcripta erit quædam figura, <lb/>ex priſinatibus æqualem altitudinem habentibus cóſtans, <lb/>ad partes e: </s> <s xml:id="echoid-s3696" xml:space="preserve">& </s> <s xml:id="echoid-s3697" xml:space="preserve">altera circumſcripta ad partes d. </s> <s xml:id="echoid-s3698" xml:space="preserve">Sed unum-<lb/>quodque eorum priſmatum, quæ in figura inſcripta conti-<lb/>nentur, æquale eſt priſmati, quod ab eodem fit triangulo in <lb/>figura circumſcripta: </s> <s xml:id="echoid-s3699" xml:space="preserve">nam priſma p q priſmati p o eſt æ-<lb/>quale; </s> <s xml:id="echoid-s3700" xml:space="preserve">priſma s t æquale priſmati s r; </s> <s xml:id="echoid-s3701" xml:space="preserve">priſma x y priſmati <lb/>x u; </s> <s xml:id="echoid-s3702" xml:space="preserve">priſma η θ priſinati η z; </s> <s xml:id="echoid-s3703" xml:space="preserve">priſina μ ν priſmati μ λ; </s> <s xml:id="echoid-s3704" xml:space="preserve">priſ-<lb/>ma ρ σ priſmati ρ π; </s> <s xml:id="echoid-s3705" xml:space="preserve">& </s> <s xml:id="echoid-s3706" xml:space="preserve">priſma φ χ priſinati φ τ æquale. </s> <s xml:id="echoid-s3707" xml:space="preserve">re-<lb/>linquitur ergo, ut circumſcripta figura exuperet inſcriptã <lb/>priſmate, quod baſim habet a b c triangulum, & </s> <s xml:id="echoid-s3708" xml:space="preserve">axem e f. <lb/></s> <s xml:id="echoid-s3709" xml:space="preserve">Illud uero minus eſt ſolida magnitudine propoſita. </s> <s xml:id="echoid-s3710" xml:space="preserve">Eadȩ <lb/>ratione inſcribetur, & </s> <s xml:id="echoid-s3711" xml:space="preserve">circumſcribetur ſolida figura in py-<lb/>ramide, quæ quadrilateram, uel plurilaterã baſim habeat.</s> <s xml:id="echoid-s3712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div227" type="section" level="1" n="76"> <head xml:id="echoid-head83" xml:space="preserve">PROBLEMA II. PROPOSITIO XI.</head> <p> <s xml:id="echoid-s3713" xml:space="preserve"><emph style="sc">Dato</emph> cono, fieri poteſt, ut figura ſolida in-<lb/>ſcribatur, & </s> <s xml:id="echoid-s3714" xml:space="preserve">altera circumſcribatur ex cylindris <lb/>æqualem habentibus altitudinem, ita ut circum-<lb/>ſcripta ſuperet inſcriptam, magnitudine, quæ ſo-<lb/>lida magnitudine propoſita ſit minor.</s> <s xml:id="echoid-s3715" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3716" xml:space="preserve">SIT conus, cuius axis b d: </s> <s xml:id="echoid-s3717" xml:space="preserve">& </s> <s xml:id="echoid-s3718" xml:space="preserve">ſecetur plano per axem <lb/>ducto, ut ſectio ſit triangulum a b c: </s> <s xml:id="echoid-s3719" xml:space="preserve">intelligaturq; </s> <s xml:id="echoid-s3720" xml:space="preserve">cylin-<lb/>drus, qui baſim eandem, & </s> <s xml:id="echoid-s3721" xml:space="preserve">eundem axem habeat. </s> <s xml:id="echoid-s3722" xml:space="preserve">Hoc igi-<lb/>tur cylindro continenter bifariam ſecto, relinquetur cylin <lb/>drus minor ſolida magnitudine propoſita. </s> <s xml:id="echoid-s3723" xml:space="preserve">Sit autem is cy <lb/>lindrus, qui baſim habet circulum circa diametrum a c, & </s> <s xml:id="echoid-s3724" xml:space="preserve"><lb/>axem d e. </s> <s xml:id="echoid-s3725" xml:space="preserve">Itaque diuidatur b d in partes æquales ipſi d e <lb/>in punctis f g h _K_lm: </s> <s xml:id="echoid-s3726" xml:space="preserve">& </s> <s xml:id="echoid-s3727" xml:space="preserve">per ea ducantur plana conum ſe-<lb/>cantia; </s> <s xml:id="echoid-s3728" xml:space="preserve">quæ baſi æquidiſtent. </s> <s xml:id="echoid-s3729" xml:space="preserve">erunt ſectiones circuli, cen-<lb/>tra in axi habentes, ut in primo libro conicorum, propoſi- <pb o="18" file="0147" n="147" rhead="DE CENTRO GRAVIT. SOLID."/> tione quarta Apollonius demonſtrauit. </s> <s xml:id="echoid-s3730" xml:space="preserve">Si igitur à ſingu-<lb/>lis horum circulorum, duo cylindri fiant; </s> <s xml:id="echoid-s3731" xml:space="preserve">unus quidem ad <lb/>baſis partes; </s> <s xml:id="echoid-s3732" xml:space="preserve">alter ad partes uerticis: </s> <s xml:id="echoid-s3733" xml:space="preserve">inſcripta erit in co-<lb/>no ſolida quædam figura, & </s> <s xml:id="echoid-s3734" xml:space="preserve">altera circumſcripta ex cylin-<lb/>dris æqualem altitudinem habentibus conſtans; </s> <s xml:id="echoid-s3735" xml:space="preserve">quorum <lb/>unuſquiſque, qui in <lb/> <anchor type="figure" xlink:label="fig-0147-01a" xlink:href="fig-0147-01"/> figura inſcripta con-<lb/>tinetur æqualis eſt ei, <lb/>qui ab eodem fit cir-<lb/>culo in figura circũ-<lb/>ſcripta. </s> <s xml:id="echoid-s3736" xml:space="preserve">Itaque cylin <lb/>drus o p æqualis eſt <lb/>cylindro o n; </s> <s xml:id="echoid-s3737" xml:space="preserve">cylin-<lb/>drus r s cylĩdro r q; <lb/></s> <s xml:id="echoid-s3738" xml:space="preserve">cylindrus u x cylin-<lb/>dro u t cſt æqualis; </s> <s xml:id="echoid-s3739" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s3740" xml:space="preserve">alii aliis ſimiliter. </s> <s xml:id="echoid-s3741" xml:space="preserve"><lb/>quare conſtat circũ-<lb/>ſcriptam figuram ſu-<lb/>perare inſcriptam cy <lb/>lindro, cuius baſis eſt <lb/>circulus circa diametrum a c, & </s> <s xml:id="echoid-s3742" xml:space="preserve">axis d e. </s> <s xml:id="echoid-s3743" xml:space="preserve">atque hic eſtmi-<lb/>nor ſolida magnitudine propoſita.</s> <s xml:id="echoid-s3744" xml:space="preserve"/> </p> <div xml:id="echoid-div227" type="float" level="2" n="1"> <figure xlink:label="fig-0147-01" xlink:href="fig-0147-01a"> <image file="0147-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0147-01"/> </figure> </div> </div> <div xml:id="echoid-div229" type="section" level="1" n="77"> <head xml:id="echoid-head84" xml:space="preserve">PROBLEMA III. PROPOSITIO XII.</head> <p> <s xml:id="echoid-s3745" xml:space="preserve"><emph style="sc">Data</emph> coni portione, poteſt ſolida quædam <lb/>figura inſcribi, & </s> <s xml:id="echoid-s3746" xml:space="preserve">altera circumſcribi ex cylindri <lb/>portionibus æqualem altitudinem habentibus; <lb/></s> <s xml:id="echoid-s3747" xml:space="preserve">ita ut circumſcripta inſcriptam exuperet, magni <lb/>tudine, quæ minor ſit ſolida magnitudine pro-<lb/>poſita.</s> <s xml:id="echoid-s3748" xml:space="preserve"/> </p> <pb file="0148" n="148" rhead="FED. COMMANDINI"/> <p> <s xml:id="echoid-s3749" xml:space="preserve">Figuram ciuſmodi, & </s> <s xml:id="echoid-s3750" xml:space="preserve">inſcribemus, & </s> <s xml:id="echoid-s3751" xml:space="preserve">circũſcribemus, ita <lb/>ut in cono dictum eſt.</s> <s xml:id="echoid-s3752" xml:space="preserve"/> </p> <figure> <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0148-01"/> </figure> </div> <div xml:id="echoid-div230" type="section" level="1" n="78"> <head xml:id="echoid-head85" xml:space="preserve">PROBLEMA IIII. PROPOSITIO XIII.</head> <p> <s xml:id="echoid-s3753" xml:space="preserve"><emph style="sc">Data</emph> ſphæræ portione, quæ dimidia ſphæ-<lb/>ra maior non ſit, poteſt ſolida quædam portio in-<lb/>ſcribi & </s> <s xml:id="echoid-s3754" xml:space="preserve">altera circumſcribi ex cylindris æqualem <lb/>altitudinem habentibus, ita ut circumſcripta in-<lb/>ſcriptam excedat magnitudine, quæ ſolida ma-<lb/>gnitudine propoſita ſit minor.</s> <s xml:id="echoid-s3755" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3756" xml:space="preserve">HOC etiam eodem prorſus modo fiet: </s> <s xml:id="echoid-s3757" xml:space="preserve">atque ut ab <lb/>Archimede traditum eſt in conoidum, & </s> <s xml:id="echoid-s3758" xml:space="preserve">ſphæroidum por <lb/>tionibus, propofitione uigeſimaprima libri de conoidi-<lb/>bus, & </s> <s xml:id="echoid-s3759" xml:space="preserve">ſphæroidibus.</s> <s xml:id="echoid-s3760" xml:space="preserve"/> </p> <pb o="19" file="0149" n="149" rhead="DE CENTRO GRAVIT. SOLID."/> <figure> <image file="0149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0149-01"/> </figure> </div> <div xml:id="echoid-div231" type="section" level="1" n="79"> <head xml:id="echoid-head86" xml:space="preserve">THEOREMA X. PROPOSITIO XIIII.</head> <p> <s xml:id="echoid-s3761" xml:space="preserve">Cuiuslibet pyramidis, & </s> <s xml:id="echoid-s3762" xml:space="preserve">cuiuslibet coni, uel <lb/>coni portionis, centrum grauitatis in axe cõſiſtit.</s> <s xml:id="echoid-s3763" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3764" xml:space="preserve">SIT pyramis, cuius baſis triangulum a b c: </s> <s xml:id="echoid-s3765" xml:space="preserve">& </s> <s xml:id="echoid-s3766" xml:space="preserve">axis d e. <lb/></s> <s xml:id="echoid-s3767" xml:space="preserve">Dico in linea d e ipſius grauitatis centrum ineſſe. </s> <s xml:id="echoid-s3768" xml:space="preserve">Si enim <lb/>fieri poteſt, ſit centrum f: </s> <s xml:id="echoid-s3769" xml:space="preserve">& </s> <s xml:id="echoid-s3770" xml:space="preserve">ab f ducatur ad baſim pyrami <lb/>dis linea f g, axi æquidiſtans: </s> <s xml:id="echoid-s3771" xml:space="preserve">iunctaq; </s> <s xml:id="echoid-s3772" xml:space="preserve">e g ad latera trian-<lb/>guli a b c producatur in h. </s> <s xml:id="echoid-s3773" xml:space="preserve">quam uero proportionem ha-<lb/>bet linea h e ad e g, habeat pyramis ad aliud ſolidum, in <lb/>quo K: </s> <s xml:id="echoid-s3774" xml:space="preserve">inſcribaturq; </s> <s xml:id="echoid-s3775" xml:space="preserve">in pyramide ſolida figura, & </s> <s xml:id="echoid-s3776" xml:space="preserve">altera cir <lb/>cumſcribatur ex priſmatibus æqualem habentibus altitu-<lb/>dinem, ita ut circumſcripta inſcriptam exuperet magnitu-<lb/>dine, quæ ſolido _k_ ſit minor. </s> <s xml:id="echoid-s3777" xml:space="preserve">Et quoniam in pyramide pla <lb/>num baſi æquidiſtans ductum ſectionem facit figuram ſi-<lb/>milem ei, quæ eſt baſis; </s> <s xml:id="echoid-s3778" xml:space="preserve">centrumq; </s> <s xml:id="echoid-s3779" xml:space="preserve">grauitatis in axe haben <lb/>tem: </s> <s xml:id="echoid-s3780" xml:space="preserve">erit priſmatis s t grauitatis centrũ in linear q; </s> <s xml:id="echoid-s3781" xml:space="preserve">priſ-<lb/>matis u x centrum in linea q p; </s> <s xml:id="echoid-s3782" xml:space="preserve">priſmatis y z in linea p o; </s> <s xml:id="echoid-s3783" xml:space="preserve"><lb/>priſmatis η θ in l_i_nea o n; </s> <s xml:id="echoid-s3784" xml:space="preserve">priſmatis λ μ in linea n m; </s> <s xml:id="echoid-s3785" xml:space="preserve">priſ-<lb/>matis ν π in m l; </s> <s xml:id="echoid-s3786" xml:space="preserve">& </s> <s xml:id="echoid-s3787" xml:space="preserve">denique priſmatis ρ σ in l e. </s> <s xml:id="echoid-s3788" xml:space="preserve">quare to- <pb file="0150" n="150" rhead="FED. COMMANDINI"/> tius figuræ inſcriptæ centrum grauitatis eſt in linea r e: <lb/></s> <s xml:id="echoid-s3789" xml:space="preserve">quod ſit τ: </s> <s xml:id="echoid-s3790" xml:space="preserve">iũ-<lb/> <anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/> ctaque τ f, & </s> <s xml:id="echoid-s3791" xml:space="preserve"><lb/>producta, à <lb/>puncto h du-<lb/>catur linea a-<lb/>xi pyramidis <lb/>æquidiſtans, <lb/>quæ cũ linea <lb/>τ f conueniat <lb/>in φ. </s> <s xml:id="echoid-s3792" xml:space="preserve">habebit <lb/>φ τ ad τ f ean-<lb/>dem propor-<lb/>tionem, quã <lb/>h e ad e g. <lb/></s> <s xml:id="echoid-s3793" xml:space="preserve">Quoniam igi <lb/>tur exceſſus, <lb/>quo circũſcri <lb/>pta figura in-<lb/>ſcriptam ſupe <lb/>rat, minor eſt <lb/>ſolido <emph style="sc">K</emph>; </s> <s xml:id="echoid-s3794" xml:space="preserve">py-<lb/>ramis ad eun-<lb/>dẽ exceſsũ ma <lb/>ioré propor-<lb/>tionȩ habet, <lb/>quàm ad _K_ ſo <lb/>lidum: </s> <s xml:id="echoid-s3795" xml:space="preserve">uideli <lb/>cet maiorem, <lb/>quàm linea h <lb/>e ad e g; </s> <s xml:id="echoid-s3796" xml:space="preserve">hoc <lb/>eſt quam φ τ <lb/>ad τ f: </s> <s xml:id="echoid-s3797" xml:space="preserve">& </s> <s xml:id="echoid-s3798" xml:space="preserve">propterea multo maiorem habet ad partem ex-<lb/>ceſſus, quæ intra pyramidem comprehenditur. </s> <s xml:id="echoid-s3799" xml:space="preserve">Itaque ha- <pb o="20" file="0151" n="151" rhead="DE CENTRO GRAVIT. SOLID."/> beat eam, quam χ τ ad τ f. </s> <s xml:id="echoid-s3800" xml:space="preserve">erit diuidendo ut χ f ad f τ, ita fi <lb/>gura ſolida inſcripta ad partem exceſſus, quæ eſtintra pyra <lb/>midem. </s> <s xml:id="echoid-s3801" xml:space="preserve">Cum ergo à pyramide, cuius grauitatis cẽtrum eſt <lb/>punctum f, ſolida figura inſcripta auferatur, cuius centrũ <lb/>τ: </s> <s xml:id="echoid-s3802" xml:space="preserve">reliquæ magnitudinis conſtantis ex parte exceſſus, quæ <lb/>eſtintra pyramidem, centrum grauitatis erit in linea τ f <lb/>producta, & </s> <s xml:id="echoid-s3803" xml:space="preserve">in puncto χ. </s> <s xml:id="echoid-s3804" xml:space="preserve">quod fieri non poteſt. </s> <s xml:id="echoid-s3805" xml:space="preserve">Sequitur <lb/>igitur, ut centrum grauitatis pyramidis in linea d e; </s> <s xml:id="echoid-s3806" xml:space="preserve">hoc <lb/>eſt in eius axe conſiſtat.</s> <s xml:id="echoid-s3807" xml:space="preserve"/> </p> <div xml:id="echoid-div231" type="float" level="2" n="1"> <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a"> <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0150-01"/> </figure> </div> <p> <s xml:id="echoid-s3808" xml:space="preserve">Sit conus, uel coni portio, cuius axis b d: </s> <s xml:id="echoid-s3809" xml:space="preserve">& </s> <s xml:id="echoid-s3810" xml:space="preserve">ſecetur plano <lb/>per axem, ut ſectio ſit triangulum a b c. </s> <s xml:id="echoid-s3811" xml:space="preserve">Dico centrum gra <lb/>uitatis ipſius eſſe in linea b d. </s> <s xml:id="echoid-s3812" xml:space="preserve">Sit enim, ſi fieri poteſt, centrũ <lb/> <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> e: </s> <s xml:id="echoid-s3813" xml:space="preserve">perq; </s> <s xml:id="echoid-s3814" xml:space="preserve">e ducatur e f axi æquidiſtans: </s> <s xml:id="echoid-s3815" xml:space="preserve">& </s> <s xml:id="echoid-s3816" xml:space="preserve">quam propor-<lb/>tionem habet c d ad d f, habeat conus, uel coni portio ad <lb/>ſolidum g. </s> <s xml:id="echoid-s3817" xml:space="preserve">inſcribatur ergo in cono, uel coni portione ſoli <pb file="0152" n="152" rhead="FED. COMMANDINI"/> da figura, & </s> <s xml:id="echoid-s3818" xml:space="preserve">altera circumſcribatur ex cylindris, uel cylin-<lb/>dri portionibus, ſicuti dictum eſt, ita ut exceſſus, quo figu-<lb/>ra circumſcripta inſcriptam ſuperat, ſit ſolido g minor. <lb/></s> <s xml:id="echoid-s3819" xml:space="preserve">Itaque centrum grauitatis cylindri, uel cylindri portionis <lb/>q r eſt in linea p o; </s> <s xml:id="echoid-s3820" xml:space="preserve">cylindri, uel cylindri portionis st cen-<lb/>trum in linea on; </s> <s xml:id="echoid-s3821" xml:space="preserve">centrum u x in linea n m; </s> <s xml:id="echoid-s3822" xml:space="preserve">y z in m b; </s> <s xml:id="echoid-s3823" xml:space="preserve">η @ <lb/>in 1k; </s> <s xml:id="echoid-s3824" xml:space="preserve">λ μ in K h; </s> <s xml:id="echoid-s3825" xml:space="preserve">& </s> <s xml:id="echoid-s3826" xml:space="preserve">denique ν π centrum in h d. </s> <s xml:id="echoid-s3827" xml:space="preserve">ergo figu-<lb/> <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> ræ inſcriptæ centrum eſt in linea p d. </s> <s xml:id="echoid-s3828" xml:space="preserve">Sitautem ρ: </s> <s xml:id="echoid-s3829" xml:space="preserve">& </s> <s xml:id="echoid-s3830" xml:space="preserve">iun-<lb/>cta ρ e protendatur, ut cum linea, quæ à pũctoc ducta fue-<lb/>rit axi æquidiſtans, conueniat in σ. </s> <s xml:id="echoid-s3831" xml:space="preserve">erit σ ζ ad ρ e, ut c d <lb/>ad d f: </s> <s xml:id="echoid-s3832" xml:space="preserve">& </s> <s xml:id="echoid-s3833" xml:space="preserve">conus, ſeu coni portio ad exceſſum, quo circum-<lb/>ſcripta figura inſcriptam ſuperat, habebit maiorem pro-<lb/>portionem, quàm σ ζ ad ρ e. </s> <s xml:id="echoid-s3834" xml:space="preserve">ergo ad partem exceſſus, quæ <lb/>intra ipſius ſuperficiem comprehenditur, multo maiorem <lb/>proportionem habebit. </s> <s xml:id="echoid-s3835" xml:space="preserve">habeat eam, quam τ ρ ad ρ e. </s> <s xml:id="echoid-s3836" xml:space="preserve">erit <pb o="21" file="0153" n="153" rhead="DE CENTRO GRAVIT. SOLID."/> diuidendo figura ſolida inſcripta ad dictam exceſſus par-<lb/>tem, ut τ e ad e ρ. </s> <s xml:id="echoid-s3837" xml:space="preserve">& </s> <s xml:id="echoid-s3838" xml:space="preserve">quoniam à cono, ſeu coni portione, <lb/>cuius grauitatis centrum eſt e, aufertur figura inſcripta, <lb/>cuius centrum ρ: </s> <s xml:id="echoid-s3839" xml:space="preserve">reſiduæ magnitudinis compoſitæ ex par <lb/>te exceſſus, quæ intra coni, uel coni portionis ſuperficiem <lb/>continetur, centrum grauitatis erit in linea ζ e protracta, <lb/>atque in puncto τ. </s> <s xml:id="echoid-s3840" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s3841" xml:space="preserve">cõſtat ergo centrũ <lb/>grauitatis coni, uel coni portionis, eſſe in axe b d: </s> <s xml:id="echoid-s3842" xml:space="preserve">quod de <lb/>monſcrandum propoſuimus.</s> <s xml:id="echoid-s3843" xml:space="preserve"/> </p> <div xml:id="echoid-div232" type="float" level="2" n="2"> <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a"> <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0151-01"/> </figure> <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0152-01"/> </figure> </div> </div> <div xml:id="echoid-div234" type="section" level="1" n="80"> <head xml:id="echoid-head87" xml:space="preserve">THE OREMA XI. PROPOSITIO XV.</head> <p> <s xml:id="echoid-s3844" xml:space="preserve">Cuiuslibet portionis ſphæræ uel ſphæroidis, <lb/>quæ dimidia maior non ſit: </s> <s xml:id="echoid-s3845" xml:space="preserve">itemq́; </s> <s xml:id="echoid-s3846" xml:space="preserve">cuiuslibet por <lb/>tionis conoidis, uel abſciſſæ plano ad axem recto, <lb/>uel non recto, centrum grauitatis in axe con-<lb/>ſiſtit.</s> <s xml:id="echoid-s3847" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3848" xml:space="preserve">Demonſtratio ſimilis erit ei, quam ſupra in cono, uel co <lb/>ni portione attulimus, ne toties eadem fruſtra iterentur.</s> <s xml:id="echoid-s3849" xml:space="preserve"/> </p> <figure> <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0153-01"/> </figure> <pb file="0154" n="154" rhead="FED. COMMANDINI"/> </div> <div xml:id="echoid-div235" type="section" level="1" n="81"> <head xml:id="echoid-head88" xml:space="preserve">THE OREMA XII. PROPOSITIO XVI.</head> <p> <s xml:id="echoid-s3850" xml:space="preserve">In ſphæra, & </s> <s xml:id="echoid-s3851" xml:space="preserve">ſphæroide idem eſt grauitatis, & </s> <s xml:id="echoid-s3852" xml:space="preserve"><lb/>figuræ centrum.</s> <s xml:id="echoid-s3853" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3854" xml:space="preserve">Secetur ſphæra, uel ſphæroid<gap/>no per axem ducto; <lb/></s> <s xml:id="echoid-s3855" xml:space="preserve">quod ſectionem faciat circulum, <gap/>ellipſim a b c d, cuius <lb/>diameter, & </s> <s xml:id="echoid-s3856" xml:space="preserve">ſphæræ, uelſphæroidis axis d b; </s> <s xml:id="echoid-s3857" xml:space="preserve">& </s> <s xml:id="echoid-s3858" xml:space="preserve">centrume. </s> <s xml:id="echoid-s3859" xml:space="preserve"><lb/>Dico e grauitatis etiam centrum eſſe. </s> <s xml:id="echoid-s3860" xml:space="preserve">ſecetur enim altero <lb/>plano per e, ad planum ſecans recto, cuius fectio ſit circu-<lb/>lus circa diametrum a c. </s> <s xml:id="echoid-s3861" xml:space="preserve">erunt a d c, a b c dimidiæ portio-<lb/>nes ſphæræ, uel fphæroidis. </s> <s xml:id="echoid-s3862" xml:space="preserve">& </s> <s xml:id="echoid-s3863" xml:space="preserve">quoniam portionis a d c gra <lb/>uitatis centrum eſt in linea d, & </s> <s xml:id="echoid-s3864" xml:space="preserve">centrum portionis a b c in <lb/>ipſa b e; </s> <s xml:id="echoid-s3865" xml:space="preserve">totius ſphæræ, uel ſphæroidis grauitatis centrum <lb/>in axe d b conſiſtet. </s> <s xml:id="echoid-s3866" xml:space="preserve">Quòd ſi portionis a d c centrum graui <lb/>tatis ponatur eſſe f. </s> <s xml:id="echoid-s3867" xml:space="preserve">& </s> <s xml:id="echoid-s3868" xml:space="preserve">fiat ipſi f e æqualis e g: </s> <s xml:id="echoid-s3869" xml:space="preserve">punctũ g por <lb/> <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> tionis a b c centrum erit. </s> <s xml:id="echoid-s3870" xml:space="preserve">ſolidis enim figuris ſimilibus & </s> <s xml:id="echoid-s3871" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0154-01a" xlink:href="note-0154-01"/> æqualibus inter ſe aptatis, & </s> <s xml:id="echoid-s3872" xml:space="preserve">centra grauitatis ipſarum in-<lb/>ter fe aptentur neceſſe eſt. </s> <s xml:id="echoid-s3873" xml:space="preserve">ex quo fit, ut magnitudinis, quæ <lb/> <anchor type="note" xlink:label="note-0154-02a" xlink:href="note-0154-02"/> ex utriſque cõſtat, hoc eſt ipſius ſphæræ, uel ſphæroidis gra <lb/>uitatis centrum ſitin medio lineæ f g, uidelicet in e. </s> <s xml:id="echoid-s3874" xml:space="preserve">Sphæ-<lb/>ræ igitur, uel ſphæroidis grauitatis centrum eſtidem, quod <lb/>centrum figuræ.</s> <s xml:id="echoid-s3875" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="1"> <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0154-01"/> </figure> <note position="left" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve">per 2. pe-<lb/>titionem</note> <note position="left" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve">4 Arch-<lb/>medis.</note> </div> <pb o="22" file="0155" n="155" rhead="DE CENTRO GRAVIT. SOLID."/> <p> <s xml:id="echoid-s3876" xml:space="preserve">Ex demonſtratis perſpicue apparet, portioni <lb/>ſphæræ uel ſphæroidis, quæ dimidia maior eſt, cẽ <lb/>trum grauitatis in axe conſiſtere.</s> <s xml:id="echoid-s3877" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3878" xml:space="preserve">Data enim <lb/> <anchor type="figure" xlink:label="fig-0155-01a" xlink:href="fig-0155-01"/> qualibet maio <lb/>ri portiõe, quo <lb/>niã totius ſphæ <lb/>ræ, uel ſphæroi <lb/>dis grauitatis <lb/>centrum eſt in <lb/>axe; </s> <s xml:id="echoid-s3879" xml:space="preserve">eſt autem <lb/>& </s> <s xml:id="echoid-s3880" xml:space="preserve">in axe cen-<lb/>trum portio-<lb/>nis minoris: <lb/></s> <s xml:id="echoid-s3881" xml:space="preserve">reliquæ portionis uidelicet maioris centrum in axe neceſ-<lb/>ſario conſiſtet.</s> <s xml:id="echoid-s3882" xml:space="preserve"/> </p> <div xml:id="echoid-div236" type="float" level="2" n="2"> <figure xlink:label="fig-0155-01" xlink:href="fig-0155-01a"> <image file="0155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0155-01"/> </figure> </div> </div> <div xml:id="echoid-div238" type="section" level="1" n="82"> <head xml:id="echoid-head89" xml:space="preserve">THE OREMA XIII. PROPOSITIO XVII.</head> <p> <s xml:id="echoid-s3883" xml:space="preserve">Cuiuslibet pyramidis triã <lb/> <anchor type="figure" xlink:label="fig-0155-02a" xlink:href="fig-0155-02"/> gularem baſim habẽtis gra <lb/>uitatis centrum eſt in pun-<lb/>cto, in quo ipſius axes con-<lb/>ueniunt.</s> <s xml:id="echoid-s3884" xml:space="preserve"/> </p> <div xml:id="echoid-div238" type="float" level="2" n="1"> <figure xlink:label="fig-0155-02" xlink:href="fig-0155-02a"> <image file="0155-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0155-02"/> </figure> </div> <p> <s xml:id="echoid-s3885" xml:space="preserve">Sit pyramis, cuius baſis trian <lb/>gulum a b c, axis d e: </s> <s xml:id="echoid-s3886" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s3887" xml:space="preserve">trian <lb/>guli b d c grauitatis centrum f: <lb/></s> <s xml:id="echoid-s3888" xml:space="preserve">& </s> <s xml:id="echoid-s3889" xml:space="preserve">iungatur a f. </s> <s xml:id="echoid-s3890" xml:space="preserve">erit & </s> <s xml:id="echoid-s3891" xml:space="preserve">a faxis eiuſ <lb/>dem pyramidis ex tertia diffini-<lb/>tione huius. </s> <s xml:id="echoid-s3892" xml:space="preserve">Itaque quoniam centrum grauitatis eſt in <lb/>axe d e; </s> <s xml:id="echoid-s3893" xml:space="preserve">eſt autem & </s> <s xml:id="echoid-s3894" xml:space="preserve">in axe a f; </s> <s xml:id="echoid-s3895" xml:space="preserve">quod proxime demonſtraui <pb file="0156" n="156" rhead="FED. COMMANDINI"/> mus: </s> <s xml:id="echoid-s3896" xml:space="preserve">erit utique grauitatis centrum pyramidis punctum <lb/>g: </s> <s xml:id="echoid-s3897" xml:space="preserve">in quo ſcilicet ipſi axes conueniunt.</s> <s xml:id="echoid-s3898" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div240" type="section" level="1" n="83"> <head xml:id="echoid-head90" xml:space="preserve">THEOREMA XIIII. PROPOSITIO XVIII.</head> <p> <s xml:id="echoid-s3899" xml:space="preserve"><emph style="sc">Si</emph> ſolidum parallelepipedum ſecetur plano <lb/>baſibus æquidiſtante; </s> <s xml:id="echoid-s3900" xml:space="preserve">erit ſolidum ad ſolidum, <lb/>ſicut altitudo ad altitudinem, uel ſicut axisad <lb/>axem.</s> <s xml:id="echoid-s3901" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3902" xml:space="preserve">Sit ſolidum parallelepipe <lb/> <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> dum a b c d e f g h, cuius axis <lb/>k 1: </s> <s xml:id="echoid-s3903" xml:space="preserve">ſeceturq; </s> <s xml:id="echoid-s3904" xml:space="preserve">plano baſibus <lb/>æquidiſtante, quod faciat <lb/>fectionem m n o p; </s> <s xml:id="echoid-s3905" xml:space="preserve">& </s> <s xml:id="echoid-s3906" xml:space="preserve">axi in <lb/>puncto q occurrat. </s> <s xml:id="echoid-s3907" xml:space="preserve">Dico <lb/>ſolidum g m ad ſolidum m c <lb/>eam proportionem habere, <lb/>quam altitudo ſolidi g m ha-<lb/>betad ſolidi m c altitudi-<lb/>nem; </s> <s xml:id="echoid-s3908" xml:space="preserve">uel quam axis k q ad <lb/>axem q l. </s> <s xml:id="echoid-s3909" xml:space="preserve">Sienim axis K l ad <lb/>baſis planum ſit perpendicu <lb/>laris, & </s> <s xml:id="echoid-s3910" xml:space="preserve">linea g c, quæ ex quin <lb/>ta huius ipſi k l æquidiſtat, <lb/>perpendicularis erit ad idẽ <lb/>planum, & </s> <s xml:id="echoid-s3911" xml:space="preserve">ſolidi altitudi-<lb/>nem dimetietur. </s> <s xml:id="echoid-s3912" xml:space="preserve">Itaqueſo-<lb/> <anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/> lidum g m ad ſolidum m c <lb/>eam proportionem habet, <lb/>quam parallelogrammũ g n <lb/>ad parallelogrammum n c, <lb/>hoc eſt quam linea g o, quæ <lb/> <anchor type="note" xlink:label="note-0156-02a" xlink:href="note-0156-02"/> <pb o="23" file="0157" n="157" rhead="DE CENTRO GRAVIT. SOLID."/> eſtſolidi g m altitudo ad o e altitudinem ſolidi m c, uel quã <lb/>axis k q ad q l axem. </s> <s xml:id="echoid-s3913" xml:space="preserve">Si uero axis k l non ſit perpendicularis <lb/>ad planum baſis; </s> <s xml:id="echoid-s3914" xml:space="preserve">ducatur a puncto k ad idem planum per <lb/>pendicularis k r, occurrẽs plano m n o p in s. </s> <s xml:id="echoid-s3915" xml:space="preserve">ſimiliter de-<lb/>mõſtrabimus ſolidum g m ad ſoli<gap/> m c ita eſſe, ut axis k q <lb/>ad axem q l. </s> <s xml:id="echoid-s3916" xml:space="preserve">Sed ut K q ad q l, ita k s altitudo ad altitudi-<lb/>nem s r, nam lineæ K l, K r à planis æquidiſtantibus in eaſ-<lb/> <anchor type="note" xlink:label="note-0157-01a" xlink:href="note-0157-01"/> dem proportiones ſecantur. </s> <s xml:id="echoid-s3917" xml:space="preserve">ergo ſolidum g m ad ſolidum <lb/>m c eandẽ proportionem habet, quam altitudo ad altitu <lb/>dinẽ, uel quam axis ad axem. </s> <s xml:id="echoid-s3918" xml:space="preserve">quod demõſtrare oportebat.</s> <s xml:id="echoid-s3919" xml:space="preserve"/> </p> <div xml:id="echoid-div240" type="float" level="2" n="1"> <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a"> <image file="0156-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0156-01"/> </figure> <note position="left" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve">2. undeci <lb/>mi.</note> <note position="left" xlink:label="note-0156-02" xlink:href="note-0156-02a" xml:space="preserve">i. ſexti.</note> <note position="right" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve">17. unde-<lb/>cimi</note> </div> </div> <div xml:id="echoid-div242" type="section" level="1" n="84"> <head xml:id="echoid-head91" xml:space="preserve">THEOREMA XV. PROPOSITIO XIX.</head> <p> <s xml:id="echoid-s3920" xml:space="preserve">Solida parallelepipedain eadem baſi, uel in <lb/>æqualibus baſibus conſtituta eam inter ſe propor <lb/>tionem habent, quam altitudines: </s> <s xml:id="echoid-s3921" xml:space="preserve">& </s> <s xml:id="echoid-s3922" xml:space="preserve">ſi axes ipſo-<lb/>rum cum baſibus æquales angulos contineant, <lb/>eam quoque, quam axes proportionem habebũt.</s> <s xml:id="echoid-s3923" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3924" xml:space="preserve">Sint ſolida parallelepipeda in eadẽ baſi cõſtituta a b c d, <lb/>a b e f: </s> <s xml:id="echoid-s3925" xml:space="preserve">& </s> <s xml:id="echoid-s3926" xml:space="preserve">ſit ſolidi a b c d altitudo minor: </s> <s xml:id="echoid-s3927" xml:space="preserve">producatur au-<lb/>tem planum c d adeo, utſolidum a b e f ſecet; </s> <s xml:id="echoid-s3928" xml:space="preserve">cuius ſectio <lb/>ſit g h. </s> <s xml:id="echoid-s3929" xml:space="preserve">erũſoli <lb/> <anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/> <anchor type="note" xlink:label="note-0157-02a" xlink:href="note-0157-02"/> da a b c d, a b g h <lb/>in eadem baſi, <lb/>& </s> <s xml:id="echoid-s3930" xml:space="preserve">æquali altitu <lb/>dine inter ſe æ-<lb/>qualia. </s> <s xml:id="echoid-s3931" xml:space="preserve">Quoniã <lb/>igitur ſolidum <lb/>a b e f ſecatur <lb/>plano baſibus <lb/>æquidiſtãte, erit <lb/>ſolidum g h e f <lb/> <anchor type="note" xlink:label="note-0157-03a" xlink:href="note-0157-03"/> adipſum a b g h <pb file="0158" n="158" rhead="FED. COMMANDINI"/> ut altitudo ad altitudinem & </s> <s xml:id="echoid-s3932" xml:space="preserve">componendo conuertendo <lb/>que ſolidum a b g h, hoc eſt ſolidum a b c d ipſi æquale, ad <lb/> <anchor type="note" xlink:label="note-0158-01a" xlink:href="note-0158-01"/> ſolidum a b e f, ut altitudo ſolidi a b c d ad ſolidi a b e f al-<lb/>titudinem.</s> <s xml:id="echoid-s3933" xml:space="preserve"/> </p> <div xml:id="echoid-div242" type="float" level="2" n="1"> <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a"> <image file="0157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0157-01"/> </figure> <note position="right" xlink:label="note-0157-02" xlink:href="note-0157-02a" xml:space="preserve">29. unde-<lb/>cimi</note> <note position="right" xlink:label="note-0157-03" xlink:href="note-0157-03a" xml:space="preserve">18. huius</note> <note position="left" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s3934" xml:space="preserve">Sint ſolida parallelepipeda a b, c d in æqualibus baſibus <lb/>conſtituta: </s> <s xml:id="echoid-s3935" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s3936" xml:space="preserve">b e altitudo ſolidi a b: </s> <s xml:id="echoid-s3937" xml:space="preserve">& </s> <s xml:id="echoid-s3938" xml:space="preserve">ſolidi c d altitudo <lb/>d f; </s> <s xml:id="echoid-s3939" xml:space="preserve">quæ quidem maior ſit, quàm b e. </s> <s xml:id="echoid-s3940" xml:space="preserve">Dico ſolidum a b ad <lb/>ſolidum c d eandem habere proportionem, quam be ad <lb/>d f. </s> <s xml:id="echoid-s3941" xml:space="preserve">abſcindatur enim à linea d f æqualis ipſi b e, quæ ſit g f: <lb/></s> <s xml:id="echoid-s3942" xml:space="preserve">& </s> <s xml:id="echoid-s3943" xml:space="preserve">per g ducatur planum ſecans ſolidum c d; </s> <s xml:id="echoid-s3944" xml:space="preserve">quod baſibus <lb/>æquidiſtet, faciatq; </s> <s xml:id="echoid-s3945" xml:space="preserve">ſectionẽ h K. </s> <s xml:id="echoid-s3946" xml:space="preserve">erunt ſolida a b, c k æque <lb/> <anchor type="note" xlink:label="note-0158-02a" xlink:href="note-0158-02"/> alta inter <lb/> <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> ſe æqualia <lb/>cũ æqua-<lb/>les baſes <lb/>habeant. <lb/></s> <s xml:id="echoid-s3947" xml:space="preserve">Sed ſolidũ <lb/> <anchor type="note" xlink:label="note-0158-03a" xlink:href="note-0158-03"/> h d ad ſoli <lb/>dum c _K_ <lb/>eſt, ut alti <lb/>tudo d g <lb/>ad g f alti-<lb/>tudinẽ ſe <lb/>catur enim ſolidum c d plano baſi <lb/> <anchor type="figure" xlink:label="fig-0158-02a" xlink:href="fig-0158-02"/> bus æquidiſtante: </s> <s xml:id="echoid-s3948" xml:space="preserve">& </s> <s xml:id="echoid-s3949" xml:space="preserve">rurſus cõpo-<lb/>nendo, conuertendoq; </s> <s xml:id="echoid-s3950" xml:space="preserve">ſolidũ c _k_ <lb/>ad ſolidum c d, ut g f ad fd. </s> <s xml:id="echoid-s3951" xml:space="preserve">ergo <lb/> <anchor type="note" xlink:label="note-0158-04a" xlink:href="note-0158-04"/> ſolidum a b, quod eſt æquale ipſi <lb/>c k ad ſolidum c d eam proportio <lb/>nem habet, quam altitudo g f, hoc <lb/>eſt b e ad d f altitudinem.</s> <s xml:id="echoid-s3952" xml:space="preserve"/> </p> <div xml:id="echoid-div243" type="float" level="2" n="2"> <note position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">31. unde <lb/>cimi</note> <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a"> <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0158-01"/> </figure> <note position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">18. huius</note> <figure xlink:label="fig-0158-02" xlink:href="fig-0158-02a"> <image file="0158-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0158-02"/> </figure> <note position="left" xlink:label="note-0158-04" xlink:href="note-0158-04a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s3953" xml:space="preserve">Sint deinde ſolida parallelepipe <lb/>da a b, a c in eadem baſi; </s> <s xml:id="echoid-s3954" xml:space="preserve">quorum <lb/>axes d e, ſ e cum ipſa æquales angu <pb o="24" file="0159" n="159" rhead="DE CENTRO GRAVIT. SOLID."/> los contineant. </s> <s xml:id="echoid-s3955" xml:space="preserve">Dico ſolidum a b ad ſolidum a c eãdem ha <lb/>bere proportionem, quam axis d e ad axem e f. </s> <s xml:id="echoid-s3956" xml:space="preserve">Sienim <lb/>axes in eadem recta linea fuerint conſtituti, hæc duo ſoli-<lb/>da, in unum, atque i @m ſolidum conuenient. </s> <s xml:id="echoid-s3957" xml:space="preserve">quare ex <lb/>iis, quæ proxime tradita ſunt, habebit ſolidum a b ad ſo-<lb/>lidum a c eandem proportionem, quam axis d e ad e f <lb/>axem. </s> <s xml:id="echoid-s3958" xml:space="preserve">Siuero axes non ſint in eadem recta linea, demittan <lb/>tur a punctis d, f perpendiculares ad baſis planum, d g, fh: <lb/></s> <s xml:id="echoid-s3959" xml:space="preserve">& </s> <s xml:id="echoid-s3960" xml:space="preserve">iungantur e g, e h. </s> <s xml:id="echoid-s3961" xml:space="preserve">Quoniam igitur axes cum baſibus <lb/>æquales angulos eontinent, erit d e g angulus æqualis an-<lb/>gulo f e h: </s> <s xml:id="echoid-s3962" xml:space="preserve">& </s> <s xml:id="echoid-s3963" xml:space="preserve">ſunt <lb/> <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> anguli ad g h re-<lb/>cti, quare & </s> <s xml:id="echoid-s3964" xml:space="preserve">re-<lb/>liquus e d g æqua <lb/>lis erit reliquo <lb/>e fh: </s> <s xml:id="echoid-s3965" xml:space="preserve">& </s> <s xml:id="echoid-s3966" xml:space="preserve">triangu-<lb/>lum d e g triãgu-<lb/>lo f e h ſimile. </s> <s xml:id="echoid-s3967" xml:space="preserve">er-<lb/>go g d ad d e eſt, <lb/>ut h f ad f e: </s> <s xml:id="echoid-s3968" xml:space="preserve">& </s> <s xml:id="echoid-s3969" xml:space="preserve">per <lb/>mutando g d ad <lb/>h f, ut d e ad e f. <lb/></s> <s xml:id="echoid-s3970" xml:space="preserve">Sed ſolidum a b <lb/>ad ſolidum a c <lb/>eandem propor-<lb/>tionem habet, <lb/>quam d g altitu-<lb/>do ad altitudinẽ <lb/>f h. </s> <s xml:id="echoid-s3971" xml:space="preserve">ergo & </s> <s xml:id="echoid-s3972" xml:space="preserve">ean-<lb/>dẽ habebit, quã <lb/>axis d e a l e f axẽ</s> </p> <div xml:id="echoid-div244" type="float" level="2" n="3"> <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0159-01"/> </figure> </div> <p> <s xml:id="echoid-s3973" xml:space="preserve">Poſtremo ſint <lb/>ſolida parallelepi <lb/>peda a b, c d in <pb file="0160" n="160" rhead="FED. COMMANDINI"/> æqualibus baſibus, quorum axes cum baſibus æquales an <lb/>gulos faciant. </s> <s xml:id="echoid-s3974" xml:space="preserve">Dico ſolidum a b adſolidũ c d ita eſſe, ut axis <lb/>e f ad axem g h: </s> <s xml:id="echoid-s3975" xml:space="preserve">nam ſi axes ad planum baſis recti ſint, il-<lb/>lud perſpicue conſtat: </s> <s xml:id="echoid-s3976" xml:space="preserve">quoniam eadem linea, & </s> <s xml:id="echoid-s3977" xml:space="preserve">axem & </s> <s xml:id="echoid-s3978" xml:space="preserve">ſoli <lb/>di altitudinem determinabit. </s> <s xml:id="echoid-s3979" xml:space="preserve">Si uero ſintinclinati, à pun-<lb/>ctis e g ad ſubiectum planum perpendiculares ducantur <lb/>e k, g l: </s> <s xml:id="echoid-s3980" xml:space="preserve">& </s> <s xml:id="echoid-s3981" xml:space="preserve">iungantur f_k_, h l. </s> <s xml:id="echoid-s3982" xml:space="preserve">rurſus quoniam axes cum ba <lb/>ſibus æquales faciunt angulos, eodem modo demonſtrabi <lb/>tur, triangulum e f K triangulo g h l ſimile eſſe: </s> <s xml:id="echoid-s3983" xml:space="preserve">& </s> <s xml:id="echoid-s3984" xml:space="preserve">e k ad g l, <lb/>ut e f ad g h. </s> <s xml:id="echoid-s3985" xml:space="preserve">Solidum autem a b ad ſolidum c d eſt, ut <lb/>e K ad g l. </s> <s xml:id="echoid-s3986" xml:space="preserve">ergo & </s> <s xml:id="echoid-s3987" xml:space="preserve">ut axis e f ad axem g h. </s> <s xml:id="echoid-s3988" xml:space="preserve">quæ omnia de <lb/>monſtrare oportebat.</s> <s xml:id="echoid-s3989" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3990" xml:space="preserve">Ex iis quæ demonſtrata ſunt, facile conſtare <lb/>poteſt, priſmata omnia & </s> <s xml:id="echoid-s3991" xml:space="preserve">pyramides, quæ trian-<lb/>gulares baſes habent, ſiue in eiſdem, ſiue in æqua <lb/>libus baſibus conſtituantur, eandem proportio-<lb/> <anchor type="note" xlink:label="note-0160-01a" xlink:href="note-0160-01"/> nem habere, quam altitudines: </s> <s xml:id="echoid-s3992" xml:space="preserve">& </s> <s xml:id="echoid-s3993" xml:space="preserve">ſi axes cum ba <lb/>ſibus æquales angulos contineant, ſimiliter ean-<lb/>dem, quam axes, habere proportionem: </s> <s xml:id="echoid-s3994" xml:space="preserve">ſunt <lb/> <anchor type="note" xlink:label="note-0160-02a" xlink:href="note-0160-02"/> enim ſolida parallelepipeda priſmatum triangula <lb/>res baſes habentiũ dupla; </s> <s xml:id="echoid-s3995" xml:space="preserve">& </s> <s xml:id="echoid-s3996" xml:space="preserve">pyramidum ſextupla.</s> <s xml:id="echoid-s3997" xml:space="preserve"/> </p> <div xml:id="echoid-div245" type="float" level="2" n="4"> <note position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve">15. quinti</note> <note position="left" xlink:label="note-0160-02" xlink:href="note-0160-02a" xml:space="preserve">28. unde-<lb/>cimi.</note> </div> <note position="left" xml:space="preserve">7. duode-<lb/>cimi.</note> </div> <div xml:id="echoid-div247" type="section" level="1" n="85"> <head xml:id="echoid-head92" xml:space="preserve">THE OREMA XVI. PROPOSITIO XX.</head> <p> <s xml:id="echoid-s3998" xml:space="preserve">Priſmata omnia & </s> <s xml:id="echoid-s3999" xml:space="preserve">pyramides, quæ in eiſdem, <lb/>uel æqualibus baſibus conſtituuntur, eam inter <lb/>ſe proportionem habent, quam altitudines: </s> <s xml:id="echoid-s4000" xml:space="preserve">& </s> <s xml:id="echoid-s4001" xml:space="preserve">ſi <lb/>axes cum baſibus faciant angulos æquales, eam <lb/>etiam, quam axes habent proportionem.</s> <s xml:id="echoid-s4002" xml:space="preserve"/> </p> <pb o="25" file="0161" n="161" rhead="DE CENTRO GRAVIT. SOLID."/> <p> <s xml:id="echoid-s4003" xml:space="preserve">Sint duo priſmata a e, a f, quorum eadem baſis quadri-<lb/>latera a b c d: </s> <s xml:id="echoid-s4004" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s4005" xml:space="preserve">priſmatis a e altitudo e g; </s> <s xml:id="echoid-s4006" xml:space="preserve">& </s> <s xml:id="echoid-s4007" xml:space="preserve">priſmatis <lb/>a f altitudo f h. </s> <s xml:id="echoid-s4008" xml:space="preserve">Dico priſma a e ad priſma a f eam habere <lb/>proportionem, quam e g ad f h. </s> <s xml:id="echoid-s4009" xml:space="preserve">iungatur enim a c: </s> <s xml:id="echoid-s4010" xml:space="preserve">& </s> <s xml:id="echoid-s4011" xml:space="preserve">in <lb/>unoquoque priſmate duo priſmata intelligantur, quorum <lb/>baſes ſint triangu <lb/> <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> la a b c, a c d. </s> <s xml:id="echoid-s4012" xml:space="preserve">habe <lb/>bunt duo priſma-<lb/>te in eadem baſi <lb/>a b c conſtituta, <lb/>proportionem eã <lb/>dem, quam ipſo-<lb/>rum altitudines e <lb/>g, f h, exiam de-<lb/>monſtratis. </s> <s xml:id="echoid-s4013" xml:space="preserve">& </s> <s xml:id="echoid-s4014" xml:space="preserve">ſi-<lb/>militer alia duo, <lb/>quæ ſunt in baſi a <lb/>c d. </s> <s xml:id="echoid-s4015" xml:space="preserve">quare totum priſma a e ad priſma a f eandem propor <lb/> <anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/> tionem habebit, quam altitudo e g ad f h altitudinem. <lb/></s> <s xml:id="echoid-s4016" xml:space="preserve">Quòd cum priſmata ſint pyramidum tripla, & </s> <s xml:id="echoid-s4017" xml:space="preserve">ipſæ pyrami <lb/>des, quarum eadem eſt baſis quadrilatera, & </s> <s xml:id="echoid-s4018" xml:space="preserve">altitudo priſ-<lb/>matum altitudini æqualis, eam inter ſe proportionem ha-<lb/>bebunt, quam altitudines.</s> <s xml:id="echoid-s4019" xml:space="preserve"/> </p> <div xml:id="echoid-div247" type="float" level="2" n="1"> <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a"> <image file="0161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0161-01"/> </figure> <note position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve">12. quinti</note> </div> <p> <s xml:id="echoid-s4020" xml:space="preserve">Si uero priſmata baſes æquales habeant, nõ eaſdem, ſint <lb/>duo eiuſmodi priſmata a e, f l: </s> <s xml:id="echoid-s4021" xml:space="preserve">& </s> <s xml:id="echoid-s4022" xml:space="preserve">ſit baſis priſmatis a e qua <lb/>drilaterum a b c d; </s> <s xml:id="echoid-s4023" xml:space="preserve">& </s> <s xml:id="echoid-s4024" xml:space="preserve">priſmatis f l quadrilaterum f g h k. <lb/></s> <s xml:id="echoid-s4025" xml:space="preserve">Dico priſma a e ad priſma f l ita eſſe, ut altitudo illius ad <lb/>huius altitudinem. </s> <s xml:id="echoid-s4026" xml:space="preserve">nam ſi altitudo ſit eadem, intelligãtur <lb/>duæ pyramides a b c d e, f g h k l. </s> <s xml:id="echoid-s4027" xml:space="preserve">quæ ĩter ſe æquales erũt, <lb/> <anchor type="note" xlink:label="note-0161-02a" xlink:href="note-0161-02"/> cum æ quales baſes, & </s> <s xml:id="echoid-s4028" xml:space="preserve">altitudinem eandem habeant. </s> <s xml:id="echoid-s4029" xml:space="preserve">quare <lb/>& </s> <s xml:id="echoid-s4030" xml:space="preserve">priſmata a e, f l, quæ ſunt harù pyramidum tripla, æqua-<lb/> <anchor type="note" xlink:label="note-0161-03a" xlink:href="note-0161-03"/> lia ſint neceſſe eſt. </s> <s xml:id="echoid-s4031" xml:space="preserve">ex quibus perſpicue conſtat propoſitũ. <lb/></s> <s xml:id="echoid-s4032" xml:space="preserve">Si uero altitudo priſmatis f l ſit maior, à priſmate f l ab-<lb/>ſcindatur priſma fm, quod æque altum ſit, atq; </s> <s xml:id="echoid-s4033" xml:space="preserve">ipſum a e.</s> <s xml:id="echoid-s4034" xml:space="preserve"> <pb file="0162" n="162" rhead="FED. COMMANDINI"/> erunteadem ra-<lb/> <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> tione priſmata a <lb/>e, f m inter ſe æ-<lb/>qualia. </s> <s xml:id="echoid-s4035" xml:space="preserve">quare ſi-<lb/>militer demon-<lb/>ſtrabitur priſma <lb/>f m ad priſma f l <lb/>eandem habere <lb/>proportionem, <lb/>quam priſmatis <lb/>f m altitudo ad <lb/>altitudinem ip-<lb/>ſius f l. </s> <s xml:id="echoid-s4036" xml:space="preserve">ergo & </s> <s xml:id="echoid-s4037" xml:space="preserve">priſma a e ad priſma f l eandem propor-<lb/>tionem habebit, quam altitudo ad altitudinem. </s> <s xml:id="echoid-s4038" xml:space="preserve">ſequitur <lb/>igitur ut & </s> <s xml:id="echoid-s4039" xml:space="preserve">pyramides, quæ in æqualibus baſibus conſtituũ <lb/>tur, eandem inter ſe ſe, quam altitudines, proportionem <lb/>habeant.</s> <s xml:id="echoid-s4040" xml:space="preserve"/> </p> <div xml:id="echoid-div248" type="float" level="2" n="2"> <note position="right" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">6. duode <lb/>cimi</note> <note position="right" xlink:label="note-0161-03" xlink:href="note-0161-03a" xml:space="preserve">15. quintĩ</note> <figure xlink:label="fig-0162-01" xlink:href="fig-0162-01a"> <image file="0162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0162-01"/> </figure> </div> <figure> <image file="0162-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0162-02"/> </figure> <p> <s xml:id="echoid-s4041" xml:space="preserve">Sint deinde priſmata a e, a f in eadem baſi a b c d; </s> <s xml:id="echoid-s4042" xml:space="preserve">quorũ <lb/>axes cum baſibus æ quales angulos contineant: </s> <s xml:id="echoid-s4043" xml:space="preserve">& </s> <s xml:id="echoid-s4044" xml:space="preserve">ſit priſ- <pb o="26" file="0163" n="163" rhead="DE CENTRO GRAVIT. SOLID."/> matis a e axis g h; </s> <s xml:id="echoid-s4045" xml:space="preserve">& </s> <s xml:id="echoid-s4046" xml:space="preserve">priſmatis a f axis l h. </s> <s xml:id="echoid-s4047" xml:space="preserve">Dico priſma <lb/>a e ad priſma a f eam proportionem habere, quam g h ad <lb/>h l. </s> <s xml:id="echoid-s4048" xml:space="preserve">ducantur à punctis g l perpendiculares ad baſis pla-<lb/>num g K, l m: </s> <s xml:id="echoid-s4049" xml:space="preserve">& </s> <s xml:id="echoid-s4050" xml:space="preserve">iungantur k h, <lb/> <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> h m. </s> <s xml:id="echoid-s4051" xml:space="preserve">Itaque quoniam anguli g h <lb/>k, l h m ſunt æquales, ſimiliter ut <lb/>ſupra demonſtrabimus, triangu-<lb/>la g h K, l h m ſimilia eſſe; </s> <s xml:id="echoid-s4052" xml:space="preserve">& </s> <s xml:id="echoid-s4053" xml:space="preserve">ut g <lb/>K adlm, ita g h ad h l. </s> <s xml:id="echoid-s4054" xml:space="preserve">habet au <lb/>tem priſma a e ad priſma a f ean <lb/>dem proportionem, quam altitu <lb/>do g k ad altitudinem l m, ſicuti <lb/>demonſtratum eſt. </s> <s xml:id="echoid-s4055" xml:space="preserve">ergo & </s> <s xml:id="echoid-s4056" xml:space="preserve">ean-<lb/>dem habebit, quam g h, ad h l. </s> <s xml:id="echoid-s4057" xml:space="preserve">py <lb/>ramis igitur a b c d g ad pyrami-<lb/>dem a b c d l eandem proportio-<lb/>nem habebit, quam axis g h ad h l axem.</s> <s xml:id="echoid-s4058" xml:space="preserve"/> </p> <div xml:id="echoid-div249" type="float" level="2" n="3"> <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a"> <image file="0163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0163-01"/> </figure> </div> <figure> <image file="0163-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0163-02"/> </figure> <p> <s xml:id="echoid-s4059" xml:space="preserve">Denique ſint priſmata a e, k o in æqualibus baſibus a b <lb/>c d, k l m n conſtituta; </s> <s xml:id="echoid-s4060" xml:space="preserve">quorum axes cum baſibus æquales <lb/>faciant angulos: </s> <s xml:id="echoid-s4061" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s4062" xml:space="preserve">priſmatis a e axis f g, & </s> <s xml:id="echoid-s4063" xml:space="preserve">altitudo f h: <lb/></s> <s xml:id="echoid-s4064" xml:space="preserve">priſmatis autem k o axis p q, & </s> <s xml:id="echoid-s4065" xml:space="preserve">altitudo p r. </s> <s xml:id="echoid-s4066" xml:space="preserve">Dico priſma <lb/>a e ad priſma k o ita eſſe, ut f g ad p q. </s> <s xml:id="echoid-s4067" xml:space="preserve">iunctis enim g h, <pb file="0164" n="164" rhead="FED. COMMANDINI"/> qr, eodem, quo ſupra, modo oſtendemns f g ad p q, ut f h <lb/>ad p r. </s> <s xml:id="echoid-s4068" xml:space="preserve">ſed priſma a e ad ipſum k o eſt, ut f h ad p r. </s> <s xml:id="echoid-s4069" xml:space="preserve">ergo <lb/>& </s> <s xml:id="echoid-s4070" xml:space="preserve">ut f g axis ad axem p q. </s> <s xml:id="echoid-s4071" xml:space="preserve">ex quibus fit, ut pyramis a b c d f <lb/>ad pyrami-<lb/> <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> dẽ k l m n p <lb/>eandem-ha <lb/>beat pro-<lb/>portionẽ, <lb/>quãaxis ad <lb/>axẽ. </s> <s xml:id="echoid-s4072" xml:space="preserve">quod <lb/>demonſtrã <lb/>dũ fuerat.</s> <s xml:id="echoid-s4073" xml:space="preserve"/> </p> <div xml:id="echoid-div250" type="float" level="2" n="4"> <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a"> <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0164-01"/> </figure> </div> <p> <s xml:id="echoid-s4074" xml:space="preserve">Simili ra <lb/>tione in a-<lb/>liis priſma-<lb/>tibus & </s> <s xml:id="echoid-s4075" xml:space="preserve">py <lb/>ramidibus eadem demonſtrabuntur.</s> <s xml:id="echoid-s4076" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div252" type="section" level="1" n="86"> <head xml:id="echoid-head93" xml:space="preserve">THEOREMA XVII. PROPOSITIO XXI.</head> <p> <s xml:id="echoid-s4077" xml:space="preserve">Priſmata omnia, & </s> <s xml:id="echoid-s4078" xml:space="preserve">pyramides inter ſe propor <lb/>tionem habent compoſitam ex proportione ba-<lb/>ſium, & </s> <s xml:id="echoid-s4079" xml:space="preserve">proportione altitudinum.</s> <s xml:id="echoid-s4080" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4081" xml:space="preserve">Sint duo priſmata a e, g m: </s> <s xml:id="echoid-s4082" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s4083" xml:space="preserve">priſmatis a e baſis qua <lb/>drilaterum a b c d, & </s> <s xml:id="echoid-s4084" xml:space="preserve">altitudo e f: </s> <s xml:id="echoid-s4085" xml:space="preserve">priſmatis uero g m ba-<lb/>fis quadrilaterum g h K l, & </s> <s xml:id="echoid-s4086" xml:space="preserve">altitudo m n. </s> <s xml:id="echoid-s4087" xml:space="preserve">Dico priſma a e <lb/>ad priſma g m proportionem habere compoſitam ex pro <lb/>portione baſis a b c d ad baſim g h k l, & </s> <s xml:id="echoid-s4088" xml:space="preserve">ex proportione <lb/>altitudinis e f, ad altitudinem m n.</s> <s xml:id="echoid-s4089" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4090" xml:space="preserve">Sint enim primum e f, m n æquales: </s> <s xml:id="echoid-s4091" xml:space="preserve">& </s> <s xml:id="echoid-s4092" xml:space="preserve">ut baſis a b c d <lb/>ad baſim g h k l, ita fiat linea, in qua o ad lineam, in qua p: <lb/></s> <s xml:id="echoid-s4093" xml:space="preserve">ut autem e f ad m n, ita linea p ad lineam q. </s> <s xml:id="echoid-s4094" xml:space="preserve">erunt lineæ <lb/>p q inter ſe æquales. </s> <s xml:id="echoid-s4095" xml:space="preserve">Itaque priſma a e ad priſma g m eã <pb o="27" file="0165" n="165" rhead="DE CENTRO GRAVIT. SOLID."/> proportionem habet, quam baſis a b c d ad baſim g h k l: <lb/></s> <s xml:id="echoid-s4096" xml:space="preserve">ſi enim intelligantur duæ pyramides a b c d e, g h k l m, ha-<lb/>bebunt hæ inter ſe proportionem eandem, quam ipſarum <lb/>baſes ex ſexta duodecimi elementorum. </s> <s xml:id="echoid-s4097" xml:space="preserve">Sed ut baſis a b c d <lb/>ad g h K l baſim, ita linea o ad lineam p; </s> <s xml:id="echoid-s4098" xml:space="preserve">hoc eſt ad lineam q <lb/>ei æqualem. </s> <s xml:id="echoid-s4099" xml:space="preserve">ergo priſma a e ad priſma g m eſt, ut linea o <lb/>ad lineam q. </s> <s xml:id="echoid-s4100" xml:space="preserve">proportio autem o ad q cõpoſita eſt ex pro-<lb/>portione o ad p, & </s> <s xml:id="echoid-s4101" xml:space="preserve">ex proportione p ad q. </s> <s xml:id="echoid-s4102" xml:space="preserve">quare priſma <lb/>a e ad priſma g m, & </s> <s xml:id="echoid-s4103" xml:space="preserve">idcirco pyramis a b c d e, ad pyrami-<lb/>dem g h K l m proportionem habet ex eiſdem proportio-<lb/>nibus compoſitam, uidelicet ex proportione baſis a b c d <lb/>ad baſim g h _K_ l, & </s> <s xml:id="echoid-s4104" xml:space="preserve">ex proportione altitudinis e f ad m n al <lb/>titudinem. </s> <s xml:id="echoid-s4105" xml:space="preserve">Quòd ſi lineæ e f, m n inæquales ponantur, ſit <lb/>e f minor: </s> <s xml:id="echoid-s4106" xml:space="preserve">& </s> <s xml:id="echoid-s4107" xml:space="preserve">ut e f ad m n, ita fiat linea p ad lineam u: </s> <s xml:id="echoid-s4108" xml:space="preserve">de <lb/> <anchor type="figure" xlink:label="fig-0165-01a" xlink:href="fig-0165-01"/> inde ab ipſa m n abſcindatur r n æqualis e f: </s> <s xml:id="echoid-s4109" xml:space="preserve">& </s> <s xml:id="echoid-s4110" xml:space="preserve">per r duca-<lb/>tur planum, quod oppoſitis planis æquidiſtans faciat ſe-<lb/>ctionem s t. </s> <s xml:id="echoid-s4111" xml:space="preserve">erit priſma a e, ad priſma g t, ut baſis a b c d <lb/>ad baſim g h k l; </s> <s xml:id="echoid-s4112" xml:space="preserve">hoc eſt ut o ad p: </s> <s xml:id="echoid-s4113" xml:space="preserve">ut autem priſma g t ad <lb/>priſma g m, ita altitudo r n; </s> <s xml:id="echoid-s4114" xml:space="preserve">hoc eſt e f ad altitudinẽ m n; <lb/></s> <s xml:id="echoid-s4115" xml:space="preserve"> <anchor type="note" xlink:label="note-0165-01a" xlink:href="note-0165-01"/> uidelicet linea p ad lineam u. </s> <s xml:id="echoid-s4116" xml:space="preserve">ergo ex æquali priſma a e ad <lb/>priſma g m eſt, ut linea o ad ipſam u. </s> <s xml:id="echoid-s4117" xml:space="preserve">Sed proportio o ad <lb/>u cõpoſita eſt ex proportione o ad p, quæ eſt baſis a b c d <lb/>ad baſim g h k l; </s> <s xml:id="echoid-s4118" xml:space="preserve">& </s> <s xml:id="echoid-s4119" xml:space="preserve">ex proportione p ad u, quæ eſt altitudi-<lb/>nis e f ad altitudinem m n. </s> <s xml:id="echoid-s4120" xml:space="preserve">priſma igitur a e ad priſma g m <pb file="0166" n="166" rhead="FED. COMMANDINI"/> compoſitam proportionem habet ex proportione baſiũ, <lb/>& </s> <s xml:id="echoid-s4121" xml:space="preserve">proportione altitudinum. </s> <s xml:id="echoid-s4122" xml:space="preserve">Quare & </s> <s xml:id="echoid-s4123" xml:space="preserve">pyramis, cuius ba-<lb/>ſis eſt quadrilaterum a b c d, & </s> <s xml:id="echoid-s4124" xml:space="preserve">altitudo e f ad pyramidem, <lb/> <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> cuius baſis quadrilaterum g h K l, & </s> <s xml:id="echoid-s4125" xml:space="preserve">altitudo m n, compoſi <lb/>tam habet proportionem ex proportione baſium a b c d, <lb/>g h k l, & </s> <s xml:id="echoid-s4126" xml:space="preserve">ex proportione altitudinum e f, m n. </s> <s xml:id="echoid-s4127" xml:space="preserve">quod qui-<lb/>dem demonſtraſſe oportebat.</s> <s xml:id="echoid-s4128" xml:space="preserve"/> </p> <div xml:id="echoid-div252" type="float" level="2" n="1"> <figure xlink:label="fig-0165-01" xlink:href="fig-0165-01a"> <image file="0165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0165-01"/> </figure> <note position="right" xlink:label="note-0165-01" xlink:href="note-0165-01a" xml:space="preserve">20. huius</note> <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a"> <image file="0166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0166-01"/> </figure> </div> <p> <s xml:id="echoid-s4129" xml:space="preserve">Ex iam demonſtratis perſpicuum eſt, priſma <lb/>ta omnia, & </s> <s xml:id="echoid-s4130" xml:space="preserve">pyramides, in quibus axes cum baſi-<lb/>bus æquales angulos continent, proportionem <lb/>habere compoſitam ex baſium proportione, & </s> <s xml:id="echoid-s4131" xml:space="preserve"><lb/>proportione axium. </s> <s xml:id="echoid-s4132" xml:space="preserve">demonſttatum eſt enim, a-<lb/>xes inter ſe eandem proportionem habere, quam <lb/>ipſæ altitudines.</s> <s xml:id="echoid-s4133" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div254" type="section" level="1" n="87"> <head xml:id="echoid-head94" xml:space="preserve">THE OREMA XVIII. PROPOSITIO XXII.</head> <p> <s xml:id="echoid-s4134" xml:space="preserve"><emph style="sc">Cvivslibet</emph> pyramidis, & </s> <s xml:id="echoid-s4135" xml:space="preserve">cuiuslibet coni, <pb o="28" file="0167" n="167" rhead="DE CENTRO GRAVIT. SOLID."/> uel coni portionis axis à centro grauitatis ita diui <lb/>ditur, ut pars, quæ terminatur ad uerticem reli-<lb/>quæ partis, quæ ad baſim, ſit tripla.</s> <s xml:id="echoid-s4136" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4137" xml:space="preserve">Sit pyramis, cuius baſis triangulum a b c; </s> <s xml:id="echoid-s4138" xml:space="preserve">axis d e; </s> <s xml:id="echoid-s4139" xml:space="preserve">& </s> <s xml:id="echoid-s4140" xml:space="preserve">gra <lb/>uitatis centrum _K_. </s> <s xml:id="echoid-s4141" xml:space="preserve">Dico lineam d k ipſius _K_ e triplam eſſe. <lb/></s> <s xml:id="echoid-s4142" xml:space="preserve">trianguli enim b d c centrum grauitatis ſit punctum f; </s> <s xml:id="echoid-s4143" xml:space="preserve">triã <lb/>guli a d c centrũ g; </s> <s xml:id="echoid-s4144" xml:space="preserve">& </s> <s xml:id="echoid-s4145" xml:space="preserve">trianguli a d b ſit h: </s> <s xml:id="echoid-s4146" xml:space="preserve">& </s> <s xml:id="echoid-s4147" xml:space="preserve">iungantur a f, <lb/>b g, c h. </s> <s xml:id="echoid-s4148" xml:space="preserve">Quoniam igitur centrũ grauitatis pyramidis in axe <lb/>cõſiſtit: </s> <s xml:id="echoid-s4149" xml:space="preserve">ſuntq; </s> <s xml:id="echoid-s4150" xml:space="preserve">d e, a f, b g, c h eiuſdẽ pyramidis axes: </s> <s xml:id="echoid-s4151" xml:space="preserve">conue <lb/> <anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> nient omnes in idẽ punctũ _k_, quod eſt grauitatis centrum. <lb/></s> <s xml:id="echoid-s4152" xml:space="preserve">Itaque animo concipiamus hanc pyramidem diuiſam in <lb/>quatuor pyramides, quarum baſes ſint ipſa pyramidis <lb/>triangula; </s> <s xml:id="echoid-s4153" xml:space="preserve">& </s> <s xml:id="echoid-s4154" xml:space="preserve">axis pun-<lb/> <anchor type="handwritten" xlink:label="hd-0167-01a" xlink:href="hd-0167-01"/> <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> ctum k quæ quidem py-<lb/>ramides inter ſe æquales <lb/>ſunt, ut demõſtrabitur. <lb/></s> <s xml:id="echoid-s4155" xml:space="preserve">Ducatur enĩ per lineas <lb/>d c, d e planum ſecãs, ut <lb/>ſit ipſius, & </s> <s xml:id="echoid-s4156" xml:space="preserve">baſis a b c cõ <lb/>munis ſectio recta linea <lb/>c e l: </s> <s xml:id="echoid-s4157" xml:space="preserve">eiuſdẽ uero & </s> <s xml:id="echoid-s4158" xml:space="preserve">triã-<lb/>guli a d b ſitlinea d h l. </s> <s xml:id="echoid-s4159" xml:space="preserve"><lb/>erit linea a l æqualis ipſi <lb/>l b: </s> <s xml:id="echoid-s4160" xml:space="preserve">nam centrum graui-<lb/>tatis trianguli conſiſtit <lb/>in linea, quæ ab angulo <lb/>ad dimidiam baſim per-<lb/>ducitur, ex tertia deci-<lb/>ma Archimedis. </s> <s xml:id="echoid-s4161" xml:space="preserve">quare <lb/> <anchor type="note" xlink:label="note-0167-02a" xlink:href="note-0167-02"/> triangulum a c l æquale <lb/>eſt triangulo b c l: </s> <s xml:id="echoid-s4162" xml:space="preserve">& </s> <s xml:id="echoid-s4163" xml:space="preserve">propterea pyramis, cuius baſis trian-<lb/>gulum a c l, uertex d, eſt æqualis pyramidi, cuius baſis b c l <lb/>triangulum, & </s> <s xml:id="echoid-s4164" xml:space="preserve">idem uertex. </s> <s xml:id="echoid-s4165" xml:space="preserve">pyramides enim, quæ ab eodẽ <lb/> <anchor type="note" xlink:label="note-0167-03a" xlink:href="note-0167-03"/> <pb file="0168" n="168" rhead="FED. COMMANDINI"/> ſunt uertice, eandem proportionem habent, quam ipſarũ <lb/>baſes. </s> <s xml:id="echoid-s4166" xml:space="preserve">eadem ratione pyramis a c l k pyramidi b c l k: </s> <s xml:id="echoid-s4167" xml:space="preserve">& </s> <s xml:id="echoid-s4168" xml:space="preserve">py <lb/>ramis a d l k ipſi b d l k pyramidi æqualis erit. </s> <s xml:id="echoid-s4169" xml:space="preserve">Itaque ſi a py <lb/>ramide a c l d auferantur pyramides a clk, a d l k: </s> <s xml:id="echoid-s4170" xml:space="preserve">& </s> <s xml:id="echoid-s4171" xml:space="preserve">à pyra <lb/>mide b c l d auferãtur pyramides b c l k, d b l K: </s> <s xml:id="echoid-s4172" xml:space="preserve">quæ relin-<lb/>quuntur erunt æqualia. </s> <s xml:id="echoid-s4173" xml:space="preserve">æqualis igitur eſt pyramis a c d k <lb/>pyramidi b c d _K_. </s> <s xml:id="echoid-s4174" xml:space="preserve">Rurſus ſi per lineas a d, d e ducatur pla-<lb/>num quod pyramidem ſecet: </s> <s xml:id="echoid-s4175" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s4176" xml:space="preserve">eius & </s> <s xml:id="echoid-s4177" xml:space="preserve">baſis communis <lb/>ſectio a e m: </s> <s xml:id="echoid-s4178" xml:space="preserve">ſimiliter oſtendetur pyramis a b d K æqualis <lb/>pyramidi a c d <emph style="sc">K</emph>. </s> <s xml:id="echoid-s4179" xml:space="preserve">ducto denique alio piano per lineas c a, <lb/>a f: </s> <s xml:id="echoid-s4180" xml:space="preserve">ut eius, & </s> <s xml:id="echoid-s4181" xml:space="preserve">trianguli c d b communis ſectio ſit c fn, py-<lb/>ramis a b c k pyramidi a c d <emph style="sc">K</emph> æqualis demonſtrabitur. </s> <s xml:id="echoid-s4182" xml:space="preserve">cũ <lb/>ergo tres pyramides b c d _k_, a b d k, a b c k uni, & </s> <s xml:id="echoid-s4183" xml:space="preserve">eidem py <lb/>ramidia c d k ſint æquales, omnes inter ſe ſe æquales erũt. <lb/></s> <s xml:id="echoid-s4184" xml:space="preserve">Sed ut pyramis a b c d ad pyramidem a b c k, ita d e axis ad <lb/>axem k e, ex uigeſima propoſitione huius: </s> <s xml:id="echoid-s4185" xml:space="preserve">ſunt enim hæ <lb/>pyramides in eadem baſi, & </s> <s xml:id="echoid-s4186" xml:space="preserve">axes cum baſibus æquales con <lb/>tinent angulos, quòd in eadem recta linea conſtituantur. </s> <s xml:id="echoid-s4187" xml:space="preserve"><lb/>quare diuidendo, ut tres pyramides a c d k, b c d _K_, a b d _K_ <lb/>ad pyramidem a b c _K_, ita d _k_ ad _K_ e. </s> <s xml:id="echoid-s4188" xml:space="preserve">conſtat igitur lineam <lb/>d K ipſius _K_ e triplam eſſe. </s> <s xml:id="echoid-s4189" xml:space="preserve">ſed & </s> <s xml:id="echoid-s4190" xml:space="preserve">a k tripla eſt K f: </s> <s xml:id="echoid-s4191" xml:space="preserve">itemque <lb/>b K ipſius _K_ g: </s> <s xml:id="echoid-s4192" xml:space="preserve">& </s> <s xml:id="echoid-s4193" xml:space="preserve">c <emph style="sc">K</emph> ipſius <emph style="sc">K</emph> l tripla. </s> <s xml:id="echoid-s4194" xml:space="preserve">quod eodem modo <lb/>demonſtrabimus.</s> <s xml:id="echoid-s4195" xml:space="preserve"/> </p> <div xml:id="echoid-div254" type="float" level="2" n="1"> <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">17. huíus</note> <handwritten xlink:label="hd-0167-01" xlink:href="hd-0167-01a"/> <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a"> <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0167-01"/> </figure> <note position="right" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">1. ſexti.</note> <note position="right" xlink:label="note-0167-03" xlink:href="note-0167-03a" xml:space="preserve">5. duode-<lb/>cimi.</note> </div> <p> <s xml:id="echoid-s4196" xml:space="preserve">Sit pyramis, cuius baſis quadrilaterum a b c d; </s> <s xml:id="echoid-s4197" xml:space="preserve">axis e f: <lb/></s> <s xml:id="echoid-s4198" xml:space="preserve">& </s> <s xml:id="echoid-s4199" xml:space="preserve">diuidatur e fin g, ita ut e g ipſius g f ſit tripla. </s> <s xml:id="echoid-s4200" xml:space="preserve">Dico cen-<lb/>trum grauitatis pyramidis eſſe punctum g. </s> <s xml:id="echoid-s4201" xml:space="preserve">ducatur enim <lb/>linea b d diuidens baſim in duo triangula a b d, b c d: </s> <s xml:id="echoid-s4202" xml:space="preserve">ex <lb/>quibus intelligãtur cõſtitui duæ pyramides a b d e, b c d e: </s> <s xml:id="echoid-s4203" xml:space="preserve"><lb/>ſitque pyramidis a b d e axis e h; </s> <s xml:id="echoid-s4204" xml:space="preserve">& </s> <s xml:id="echoid-s4205" xml:space="preserve">pyramidis b c d e axis <lb/>e K: </s> <s xml:id="echoid-s4206" xml:space="preserve">& </s> <s xml:id="echoid-s4207" xml:space="preserve">iungatur h _K_, quæ per ftranſibit: </s> <s xml:id="echoid-s4208" xml:space="preserve">eſt enim in ipſa h K <lb/>centrum grauitatis magnitudinis compoſitæ ex triangulis <lb/>a b d, b c d, hoc eſt ipſius quadrilateri. </s> <s xml:id="echoid-s4209" xml:space="preserve">Itaque centrum gra <lb/>uitatis pyramidis a b d e ſit punctum l: </s> <s xml:id="echoid-s4210" xml:space="preserve">& </s> <s xml:id="echoid-s4211" xml:space="preserve">pyramidis b c d e <lb/>ſit m. </s> <s xml:id="echoid-s4212" xml:space="preserve">ductaigitur l m ipſi h m lineæ æquidiſtabit: </s> <s xml:id="echoid-s4213" xml:space="preserve">nam el ad <lb/> <anchor type="note" xlink:label="note-0168-01a" xlink:href="note-0168-01"/> <pb o="29" file="0169" n="169" rhead="DE CENTRO GRAVIT. SOLID."/> l h eandem habet proportionem, quam e m ad m k, uideli-<lb/>cet triplam. </s> <s xml:id="echoid-s4214" xml:space="preserve">quare linea l m ipſam e f ſecabit in puncto g: <lb/></s> <s xml:id="echoid-s4215" xml:space="preserve">etenim e g ad g f eſt, ut el ad l h. </s> <s xml:id="echoid-s4216" xml:space="preserve">præterea quoniam h k, l m <lb/>æquidiſtant, erunt triangula h e f, l e g ſimilia: </s> <s xml:id="echoid-s4217" xml:space="preserve">itemq; </s> <s xml:id="echoid-s4218" xml:space="preserve">inter <lb/>ſe ſimilia f e k, g e m: </s> <s xml:id="echoid-s4219" xml:space="preserve">& </s> <s xml:id="echoid-s4220" xml:space="preserve">ut e fad e g, ita h fad l g: </s> <s xml:id="echoid-s4221" xml:space="preserve">& </s> <s xml:id="echoid-s4222" xml:space="preserve">ita f _K_ ad <lb/>g m. </s> <s xml:id="echoid-s4223" xml:space="preserve">ergo uth fadlg, ita f k ad g m: </s> <s xml:id="echoid-s4224" xml:space="preserve">& </s> <s xml:id="echoid-s4225" xml:space="preserve">permutando uth f <lb/>ad f _K_, ita l g ad g m. </s> <s xml:id="echoid-s4226" xml:space="preserve">ſed cum h ſit centrum trianguli a b d; </s> <s xml:id="echoid-s4227" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s4228" xml:space="preserve">K triãguli b c d: </s> <s xml:id="echoid-s4229" xml:space="preserve">punctũ uero f totius quadrilateri a b c d <lb/>centrum: </s> <s xml:id="echoid-s4230" xml:space="preserve">erit ex 8. </s> <s xml:id="echoid-s4231" xml:space="preserve">Archimedis de centro grauitatis plano <lb/>rum h fad f <emph style="sc">K</emph>, ut triangulum b c d ad triangulum a b d: </s> <s xml:id="echoid-s4232" xml:space="preserve">ut <lb/>autem b c d triangulum ad triangulum a b d, ita pyramis <lb/>b c d e ad pyramidem a b d e. </s> <s xml:id="echoid-s4233" xml:space="preserve">ergo <lb/> <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> linea lg ad g m erit, ut pyramis <lb/>b c d e ad pyramidé a b d e. </s> <s xml:id="echoid-s4234" xml:space="preserve">ex quo <lb/>ſequitur, ut totius pyramidis <lb/>a b c d e punctum g ſit grauitatis <lb/>centrum. </s> <s xml:id="echoid-s4235" xml:space="preserve">Rurſus ſit pyramis ba-<lb/>ſim habens pentagonum a b c d e: <lb/></s> <s xml:id="echoid-s4236" xml:space="preserve">& </s> <s xml:id="echoid-s4237" xml:space="preserve">axem f g: </s> <s xml:id="echoid-s4238" xml:space="preserve">diuidaturq; </s> <s xml:id="echoid-s4239" xml:space="preserve">axis in pũ <lb/>cto h, ita ut fh ad h g triplam habe <lb/>at proportionem. </s> <s xml:id="echoid-s4240" xml:space="preserve">Dico h grauita-<lb/>tis centrũ eſſe pyramidis a b c d e f. </s> <s xml:id="echoid-s4241" xml:space="preserve"><lb/>iungatur enim e b: </s> <s xml:id="echoid-s4242" xml:space="preserve">intelligaturq; </s> <s xml:id="echoid-s4243" xml:space="preserve"><lb/>pyramis, cuius uertex f, & </s> <s xml:id="echoid-s4244" xml:space="preserve">baſis <lb/>triangulum a b e: </s> <s xml:id="echoid-s4245" xml:space="preserve">& </s> <s xml:id="echoid-s4246" xml:space="preserve">alia pyramis <lb/>intelligatur eundem uerticem ha-<lb/>bens, & </s> <s xml:id="echoid-s4247" xml:space="preserve">baſim b c d e quadrilaterũ: </s> <s xml:id="echoid-s4248" xml:space="preserve"><lb/>ſit autem pyramidis a b e faxis f <emph style="sc">K</emph>, <lb/>& </s> <s xml:id="echoid-s4249" xml:space="preserve">grauitatis centrum l: </s> <s xml:id="echoid-s4250" xml:space="preserve">& </s> <s xml:id="echoid-s4251" xml:space="preserve">pyrami <lb/>dis b c d e faxis f m, & </s> <s xml:id="echoid-s4252" xml:space="preserve">centrum gra <lb/>uitatis n: </s> <s xml:id="echoid-s4253" xml:space="preserve">iunganturq; </s> <s xml:id="echoid-s4254" xml:space="preserve"><emph style="sc">K</emph> m, l n; </s> <s xml:id="echoid-s4255" xml:space="preserve"><lb/>quæ per puncta g h tranſibunt. </s> <s xml:id="echoid-s4256" xml:space="preserve"><lb/>Rurſus eodem modo, quo ſup ra, <lb/>demonſtrabimus lineas K g m, l h n ſibiipſis æ quidiſtare <pb file="0170" n="170" rhead="FED. COMMANDINI"/> & </s> <s xml:id="echoid-s4257" xml:space="preserve">denique punctum h pyramidis a b c d e f grauitatis eſſe <lb/>centrum, & </s> <s xml:id="echoid-s4258" xml:space="preserve">ita in aliis.</s> <s xml:id="echoid-s4259" xml:space="preserve"/> </p> <div xml:id="echoid-div255" type="float" level="2" n="2"> <note position="right" xlink:label="note-0168-01" xlink:href="note-0168-01a" xml:space="preserve">2. ſexti.</note> <figure xlink:label="fig-0169-01" xlink:href="fig-0169-01a"> <image file="0169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0169-01"/> </figure> </div> <p> <s xml:id="echoid-s4260" xml:space="preserve">Sit conus, uel coni portio axem habens b d: </s> <s xml:id="echoid-s4261" xml:space="preserve">ſecetur que <lb/>plano per axem, quod ſectionem faciat triangulum a b c: <lb/></s> <s xml:id="echoid-s4262" xml:space="preserve">& </s> <s xml:id="echoid-s4263" xml:space="preserve">b d axis diuidatur in e, ita ut b e ipſius e d ſit tripla. </s> <s xml:id="echoid-s4264" xml:space="preserve"><lb/>Dico punctum e coni, uel coni portionis, grauitatis <lb/>eſſe centrum. </s> <s xml:id="echoid-s4265" xml:space="preserve">Sienim fieri poteſt, ſit centrum f: </s> <s xml:id="echoid-s4266" xml:space="preserve">& </s> <s xml:id="echoid-s4267" xml:space="preserve">pro-<lb/>ducatur e f extra figuram in g. </s> <s xml:id="echoid-s4268" xml:space="preserve">quam uero proportionem <lb/>habet g e ad e f, habeat baſis coni, uel coni portionis, hoc <lb/>eſt circulus, uel ellipſis circa diametrum a c ad aliud ſpa-<lb/>cium, in quo h. </s> <s xml:id="echoid-s4269" xml:space="preserve">Itaque in circulo, uel ellipſi plane deſcri-<lb/>batur rectilinea figura a k l m c n o p, ita ut quæ relinquũ-<lb/>tur portiones ſint minores ſpacio h: </s> <s xml:id="echoid-s4270" xml:space="preserve">& </s> <s xml:id="echoid-s4271" xml:space="preserve">intelligatur pyra-<lb/>mis baſim habens rectilineam figuram a K l m c n o p, & </s> <s xml:id="echoid-s4272" xml:space="preserve"><lb/>axem b d; </s> <s xml:id="echoid-s4273" xml:space="preserve">cuius quidem grauitatis centrum erit punctum <lb/>e, ut iam demonſtrauimus. </s> <s xml:id="echoid-s4274" xml:space="preserve">Et quoniam portiones ſunt <lb/>minores ſpacio h, circulus, uel ellipſis ad portiones ma-<lb/> <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> iorem proportionem habet, quam g e a d e f. </s> <s xml:id="echoid-s4275" xml:space="preserve">ſed ut circu-<lb/>lus, uel ellipſis ad figuram rectilineam ſibi inſcriptam, ita <lb/>conus, uel coni portio ad pyramidem, quæ figuram rectili-<lb/>neam pro baſi habet; </s> <s xml:id="echoid-s4276" xml:space="preserve">& </s> <s xml:id="echoid-s4277" xml:space="preserve">altitudinem æqualem: </s> <s xml:id="echoid-s4278" xml:space="preserve">etenim ſu- <pb o="30" file="0171" n="171" rhead="DE CENTRO GRAVIT. SOLID."/> pra demonſtratum eſt, ita eſſe cylindrum, uel cylindri por-<lb/> <anchor type="note" xlink:label="note-0171-01a" xlink:href="note-0171-01"/> tionem ad priſina, cuius baſis rectilinea figura, & </s> <s xml:id="echoid-s4279" xml:space="preserve">æqua-<lb/>lis altitudo. </s> <s xml:id="echoid-s4280" xml:space="preserve">ergo per conuerſionem rationis, ut circulus, <lb/>uel ellipſis ad portiones, ita conus, uel coni portio ad por-<lb/>tiones ſolidas. </s> <s xml:id="echoid-s4281" xml:space="preserve">quare conus uel coni portio ad portiones <lb/>ſolidas maiorem habet proportionem, quam g e ad e f: </s> <s xml:id="echoid-s4282" xml:space="preserve">& </s> <s xml:id="echoid-s4283" xml:space="preserve"><lb/>diuidendo, pyramis ad portiones ſolidas maiorem pro-<lb/>portionem habet, quam g f ad f e. </s> <s xml:id="echoid-s4284" xml:space="preserve">ſiat igitur q f ad f e <lb/>ut pyramis ad dictas portiones. </s> <s xml:id="echoid-s4285" xml:space="preserve">Itaque quoniam à cono <lb/>uel coni portione, cuius grauitatis centrum eſt f, aufer-<lb/>tur pyramis, cuius centrum e; </s> <s xml:id="echoid-s4286" xml:space="preserve">reliquæ magnitudinis, <lb/>quæ ex ſolidis portionibus conſtat, centrum grauitatis <lb/>erit in linea e f protracta, & </s> <s xml:id="echoid-s4287" xml:space="preserve">in puncto q. </s> <s xml:id="echoid-s4288" xml:space="preserve">quod fieri <lb/>non poteft: </s> <s xml:id="echoid-s4289" xml:space="preserve">eſt enim centrum grauitatis intra. </s> <s xml:id="echoid-s4290" xml:space="preserve">Conſtat <lb/>igitur coni, uel coni portionis grauitatis centrum eſſe pun <lb/>ctum e. </s> <s xml:id="echoid-s4291" xml:space="preserve">quæ omnia demonſtrare oportebat.</s> <s xml:id="echoid-s4292" xml:space="preserve"/> </p> <div xml:id="echoid-div256" type="float" level="2" n="3"> <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a"> <image file="0170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0170-01"/> </figure> <note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">8. huius</note> </div> </div> <div xml:id="echoid-div258" type="section" level="1" n="88"> <head xml:id="echoid-head95" xml:space="preserve">THEOREMA XIX. PROPOSITIO XXIII.</head> <p> <s xml:id="echoid-s4293" xml:space="preserve"><emph style="sc">Qvodlibet</emph> fruſtum à pyramide, quæ <lb/>triangularem baſim habeat, abſciſſum, diuiditur <lb/>in tres pyramides proportionales, in ea proportio <lb/>ne, quæ eſt lateris maioris baſis ad latus minoris <lb/>ipſi reſpondens.</s> <s xml:id="echoid-s4294" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4295" xml:space="preserve">Hoc demonſtrauit Leonardus Piſanus in libro, qui de-<lb/>praxi geometriæ inſcribitur. </s> <s xml:id="echoid-s4296" xml:space="preserve">Sed quoniam is adhucim-<lb/>preſſus non eſt, nos ipſius demonſtrationem breuíter <lb/>perſtringemus, rem ipſam ſecuti, non uerba. </s> <s xml:id="echoid-s4297" xml:space="preserve">Sit fru-<lb/>ſtum pyramidis a b c d e f, cuíus maior baſis triangulum <lb/>a b c, minor d e f: </s> <s xml:id="echoid-s4298" xml:space="preserve">& </s> <s xml:id="echoid-s4299" xml:space="preserve">iunctis a e, e c, c d, per line-<lb/>as a e, e c ducatur planum ſecans fruſtum: </s> <s xml:id="echoid-s4300" xml:space="preserve">itemque per <lb/>lineas e c, c d; </s> <s xml:id="echoid-s4301" xml:space="preserve">& </s> <s xml:id="echoid-s4302" xml:space="preserve">per c d, d a alia plana ducantur, quæ, <lb/>diuident fruſtum in tres pyramides a b c e, a d c e, d e f c.</s> <s xml:id="echoid-s4303" xml:space="preserve"> <pb file="0172" n="172" rhead="FED. COMMANDINI"/> Dico eas proportion ales eſſe in proportione, quæ eſt la-<lb/>teris a b adlatus d e, itaut earum maior ſit a b c e, me-<lb/>dia a d c e, & </s> <s xml:id="echoid-s4304" xml:space="preserve">minor d e f c. </s> <s xml:id="echoid-s4305" xml:space="preserve">Quoniam enim lineæ d e, <lb/>a b æquidiſtant; </s> <s xml:id="echoid-s4306" xml:space="preserve">& </s> <s xml:id="echoid-s4307" xml:space="preserve">interipſas ſunt triangula a b e, a d e; <lb/></s> <s xml:id="echoid-s4308" xml:space="preserve">erit triangulum a b e <lb/> <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/> ad triangulum a d e, <lb/>ut linea a b ad lineam <lb/>d e. </s> <s xml:id="echoid-s4309" xml:space="preserve">ut autem triangu <lb/>lum a b e ad triangu-<lb/>lum a d e, ita pyramis <lb/> <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/> a b e c ad pyramidem <lb/>a d e c: </s> <s xml:id="echoid-s4310" xml:space="preserve">habent enim <lb/>altitudinem eandem, <lb/>quæ eſt à puncto c ad <lb/>planum, in quo qua-<lb/>drilaterum a b e d. </s> <s xml:id="echoid-s4311" xml:space="preserve">er-<lb/> <anchor type="note" xlink:label="note-0172-03a" xlink:href="note-0172-03"/> go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c. <lb/></s> <s xml:id="echoid-s4312" xml:space="preserve">Rurſus quoniam æquidiſtantes ſunt a c, d f; </s> <s xml:id="echoid-s4313" xml:space="preserve">erit eadem <lb/>ratione pyramis a d c e ad pyramidem c d f e, ut a c ad <lb/> <anchor type="note" xlink:label="note-0172-04a" xlink:href="note-0172-04"/> d f. </s> <s xml:id="echoid-s4314" xml:space="preserve">Sed ut a c a l d f, ita a b ad d e, quoniam triangula <lb/>a b c, d e f ſimilia ſunt, ex nona huius. </s> <s xml:id="echoid-s4315" xml:space="preserve">quare ut pyramis <lb/>a b c e ad pyramidem a d c e, ita pyramis a d c e ad ipſam <lb/>d e f c. </s> <s xml:id="echoid-s4316" xml:space="preserve">fruſtum igitur a b c d e f diuiditur in tres pyramides <lb/>proportionales in ea proportione, quæ eſt lateris a b ad d e <lb/>latus, & </s> <s xml:id="echoid-s4317" xml:space="preserve">earum maior eſt c a b e, media a d c e, & </s> <s xml:id="echoid-s4318" xml:space="preserve">minor <lb/>d e f c. </s> <s xml:id="echoid-s4319" xml:space="preserve">quod demonſtrare oportebat.</s> <s xml:id="echoid-s4320" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="1"> <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a"> <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0172-01"/> </figure> <note position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve">1. ſextí.</note> <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">5. duodeci <lb/>mi.</note> <note position="left" xlink:label="note-0172-03" xlink:href="note-0172-03a" xml:space="preserve">11. quinti.</note> <note position="left" xlink:label="note-0172-04" xlink:href="note-0172-04a" xml:space="preserve">4 ſexti.</note> </div> </div> <div xml:id="echoid-div260" type="section" level="1" n="89"> <head xml:id="echoid-head96" xml:space="preserve">PROBLEMA V. PROPOSITIO XXIIII.</head> <p> <s xml:id="echoid-s4321" xml:space="preserve"><emph style="sc">Qvodlibet</emph> fruſtum pyramidis, uel coni, <lb/>uel coni portionis, plano baſi æquidiſtanti ita ſe-<lb/>care, ut ſectio ſit proportionalis inter maiorem, <lb/>& </s> <s xml:id="echoid-s4322" xml:space="preserve">minorem baſim.</s> <s xml:id="echoid-s4323" xml:space="preserve"/> </p> <pb o="31" file="0173" n="173" rhead="DE CENTRO GRAVIT. SOLID."/> <p> <s xml:id="echoid-s4324" xml:space="preserve">SIT fruſtum pyramidis a e, cuius maior baſis triangu-<lb/>lum a b c, minor d e f: </s> <s xml:id="echoid-s4325" xml:space="preserve">& </s> <s xml:id="echoid-s4326" xml:space="preserve">oporteat ipſum plano, quod baſi <lb/>æquidiſtet, ita ſecare, ut ſectio ſit proportionalis inter triã <lb/>gula a b c, d e f. </s> <s xml:id="echoid-s4327" xml:space="preserve">Inueniatur inter lineas a b, d e media pro-<lb/>portionalis, quæ ſit b g: </s> <s xml:id="echoid-s4328" xml:space="preserve">& </s> <s xml:id="echoid-s4329" xml:space="preserve">à puncto g erigatur g h æquidi-<lb/>ſtans b e, ſecansq; </s> <s xml:id="echoid-s4330" xml:space="preserve">a d in h: </s> <s xml:id="echoid-s4331" xml:space="preserve">deinde per h ducatur planum <lb/>baſibus æ quidiſtans, cuius ſectio ſit triangulum h _k_ 1. </s> <s xml:id="echoid-s4332" xml:space="preserve">Dico <lb/>triangulum h K l proportionale eſſe inter triangula a b c, <lb/>d e f, hoc eſt triangulum a b c ad <lb/> <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> triangulum h K l eandem habere <lb/>proportionem, quam triãgulum <lb/>h K l ad ipſum d e f. </s> <s xml:id="echoid-s4333" xml:space="preserve">Quoniã enim <lb/>lineæ a b, h K æquidiſtantium pla <lb/> <anchor type="note" xlink:label="note-0173-01a" xlink:href="note-0173-01"/> norum ſectiones inter ſe æquidi-<lb/>ſtant: </s> <s xml:id="echoid-s4334" xml:space="preserve">atque æquidiſtant b _k_, g h: <lb/></s> <s xml:id="echoid-s4335" xml:space="preserve">linea h _k_ ipſi g b eſt æqualis: </s> <s xml:id="echoid-s4336" xml:space="preserve">& </s> <s xml:id="echoid-s4337" xml:space="preserve">pro <lb/> <anchor type="note" xlink:label="note-0173-02a" xlink:href="note-0173-02"/> pterea proportionalis inter a b, <lb/>d e. </s> <s xml:id="echoid-s4338" xml:space="preserve">quare ut a b ad h K, ita eſt h <emph style="sc">K</emph> <lb/>ad d e. </s> <s xml:id="echoid-s4339" xml:space="preserve">fiat ut h k ad d e, ita d e <lb/>ad aliam lineam, in qua ſit m. </s> <s xml:id="echoid-s4340" xml:space="preserve">erit <lb/>ex æquali ut a b ad d e, ita h k ad <lb/>m. </s> <s xml:id="echoid-s4341" xml:space="preserve">Et quoniam triangula a b c, <lb/> <anchor type="note" xlink:label="note-0173-03a" xlink:href="note-0173-03"/> h K l, d e f ſimilia ſunt; </s> <s xml:id="echoid-s4342" xml:space="preserve">triangulū <lb/>a b c ad triangulum h k l eſt, ut li-<lb/> <anchor type="note" xlink:label="note-0173-04a" xlink:href="note-0173-04"/> nea a b ad lineam d e: </s> <s xml:id="echoid-s4343" xml:space="preserve">triangulũ <lb/>autem h k l ad ipſum d e f eſt, ut h _k_ ad m. </s> <s xml:id="echoid-s4344" xml:space="preserve">ergo tríangulum <lb/> <anchor type="note" xlink:label="note-0173-05a" xlink:href="note-0173-05"/> a b c ad triangulum h k l eandem proportionem habet, <lb/>quam triangulum h K l ad ipſum d e f. </s> <s xml:id="echoid-s4345" xml:space="preserve">Eodem modo in a-<lb/>liis fruſtis pyramidis idem demonſtrabitur.</s> <s xml:id="echoid-s4346" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="2" n="1"> <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a"> <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0173-01"/> </figure> <note position="right" xlink:label="note-0173-01" xlink:href="note-0173-01a" xml:space="preserve">16. unde <lb/>cimi</note> <note position="right" xlink:label="note-0173-02" xlink:href="note-0173-02a" xml:space="preserve">34. primi</note> <note position="right" xlink:label="note-0173-03" xlink:href="note-0173-03a" xml:space="preserve">9. huius <lb/>corol.</note> <note position="right" xlink:label="note-0173-04" xlink:href="note-0173-04a" xml:space="preserve">20. ſexti</note> <note position="right" xlink:label="note-0173-05" xlink:href="note-0173-05a" xml:space="preserve">11. quinti</note> </div> <p> <s xml:id="echoid-s4347" xml:space="preserve">Sit fruſtum coni, uel coni portionis a d: </s> <s xml:id="echoid-s4348" xml:space="preserve">& </s> <s xml:id="echoid-s4349" xml:space="preserve">ſecetur plano <lb/>per axem, cuius ſectio ſit a b c d, ita ut maior ipſius baſis ſit <lb/>circulus, uel ellipſis circa diametrum a b; </s> <s xml:id="echoid-s4350" xml:space="preserve">minor circa c d. <lb/></s> <s xml:id="echoid-s4351" xml:space="preserve">Rurſus inter lineas a b, c d inueniatur proportionalis b e: </s> <s xml:id="echoid-s4352" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s4353" xml:space="preserve">ab e ducta e ſ æquid_i_ſtante b d, quæ lineam c a in f ſecet, <pb file="0174" n="174" rhead="FED. COMMANDINI"/> per f planum baſibus æquidiſtans ducatur, ut ſit ſectio cir <lb/>culus, uel ellipſis circa diametrum f g. </s> <s xml:id="echoid-s4354" xml:space="preserve">Dico ſectionem a b <lb/>ad ſectionem f g eandem proportionem habere, quam f g <lb/>ad ipſam c d. </s> <s xml:id="echoid-s4355" xml:space="preserve">Simili enim ratione, qua ſupra, demonſtrabi-<lb/>tur quadratum a b ad quadratum f g ita eſſe, ut quadratũ <lb/>f g ad c d quadratum. </s> <s xml:id="echoid-s4356" xml:space="preserve">Sed circuli inter ſe eandem propor-<lb/> <anchor type="note" xlink:label="note-0174-01a" xlink:href="note-0174-01"/> tionem habent, quam diametrorum quadrata. </s> <s xml:id="echoid-s4357" xml:space="preserve">ellipſes au-<lb/>tem circa a b, f g, c d, quæ ſimiles ſunt, ut oſten dimus in cõ-<lb/>mentariis in principium libri Archimedis de conoidibus, <lb/>& </s> <s xml:id="echoid-s4358" xml:space="preserve">ſphæroidibus, eam habẽt proportionem, quam quadrar <lb/>ta diametrorum, quæ eiuſdem rationis ſunt, ex corollaio-<lb/>ſeptimæ propoſitionis eiuſdem li-<lb/> <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> bri. </s> <s xml:id="echoid-s4359" xml:space="preserve">ellipſes enim nunc appello ip-<lb/>ſa ſpacia ellipſibus contenta. </s> <s xml:id="echoid-s4360" xml:space="preserve">ergo <lb/>circulus, uel ellipſis a b ad circulũ, <lb/>uel ellipſim f g eam proportionem <lb/>habet, quam circulus, uel ellipſis <lb/>f g ad circulum uel ellipſim c d. <lb/></s> <s xml:id="echoid-s4361" xml:space="preserve">quod quidem facienduni propo-<lb/>ſuimus.</s> <s xml:id="echoid-s4362" xml:space="preserve"/> </p> <div xml:id="echoid-div261" type="float" level="2" n="2"> <note position="left" xlink:label="note-0174-01" xlink:href="note-0174-01a" xml:space="preserve">2. duode <lb/>cimi</note> <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a"> <image file="0174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0174-01"/> </figure> </div> </div> <div xml:id="echoid-div263" type="section" level="1" n="90"> <head xml:id="echoid-head97" xml:space="preserve">THEOREMA XX. PROPOSITIO XXV.</head> <p> <s xml:id="echoid-s4363" xml:space="preserve"><emph style="sc">Qvodlibet</emph> fruſtum pyramidis, uel coni, <lb/>uel coni portionis ad pyramidem, uel conum, uel <lb/>coni portionem, cuius baſis eadem eſt, & </s> <s xml:id="echoid-s4364" xml:space="preserve">æqualis <lb/>altitudo, eandem proportionẽ habet, quam utræ <lb/>que baſes, maior, & </s> <s xml:id="echoid-s4365" xml:space="preserve">minor ſimul ſumptæ vnà cũ <lb/>ea, quæ inter ipſas ſit proportionalis, ad baſim ma <lb/>iorem.</s> <s xml:id="echoid-s4366" xml:space="preserve"/> </p> <pb o="32" file="0175" n="175" rhead="DE CENTRO GRAVIT. SOLID."/> <p> <s xml:id="echoid-s4367" xml:space="preserve">SIT fruſtũ pyramidis, uel coni, uel coni portionis a d, <lb/>cuius maior baſis a b, minor c d. </s> <s xml:id="echoid-s4368" xml:space="preserve">& </s> <s xml:id="echoid-s4369" xml:space="preserve">ſecetur altero plano <lb/>baſi æquidiſtante, ita utſectio e f ſit proportionalis inter <lb/>baſes a b, c d. </s> <s xml:id="echoid-s4370" xml:space="preserve">conſtituatur autẽ pyramis, uel conus, uel co-<lb/>ni portio a g b, cuius baſis ſit eadem, quæ baſis maior fru-<lb/>ſti, & </s> <s xml:id="echoid-s4371" xml:space="preserve">altitudo æqualis. </s> <s xml:id="echoid-s4372" xml:space="preserve">Di-<lb/> <anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/> co fruſtum a d ad pyrami-<lb/>dem, uel conum, uel coni <lb/>portionem a g b eandem <lb/>proportionẽ habere, quã <lb/>utræque baſes, a b, c d unà <lb/>cum e f ad baſim a b. </s> <s xml:id="echoid-s4373" xml:space="preserve">eſt <lb/>enim fruſtum a d æquale <lb/>pyramidi, uel cono, uel co-<lb/>ni portioni, cuius baſis ex <lb/>tribus baſibus a b, e f, c d <lb/>conſtat; </s> <s xml:id="echoid-s4374" xml:space="preserve">& </s> <s xml:id="echoid-s4375" xml:space="preserve">altitudo ipſius <lb/>altitudini eſt æqualis: </s> <s xml:id="echoid-s4376" xml:space="preserve">quod mox oſtendemus. </s> <s xml:id="echoid-s4377" xml:space="preserve">Sed pyrami <lb/>des, coni, uel coni portiões, <lb/> <anchor type="figure" xlink:label="fig-0175-02a" xlink:href="fig-0175-02"/> quæ ſunt æquali altitudine, <lb/>eãdem inter ſe, quam baſes, <lb/>proportionem habent, ſicu-<lb/>ti demonſtratum eſt, partim <lb/>ab Euclide in duodecimo li-<lb/> <anchor type="note" xlink:label="note-0175-01a" xlink:href="note-0175-01"/> bro elementorum, partim à <lb/>nobis in cõmentariis in un-<lb/>decimam propoſitionẽ Ar-<lb/>chimedis de conoidibus, & </s> <s xml:id="echoid-s4378" xml:space="preserve"><lb/>ſphæroidibus. </s> <s xml:id="echoid-s4379" xml:space="preserve">quare pyra-<lb/>mis, uel conus, uel coni por-<lb/>tio, cuius baſis eſt tribus illis <lb/>baſibus æqualis ad a g b eam <lb/>habet proportionem, quam <lb/>baſes a b, e f, c d ad ab bafim. </s> <s xml:id="echoid-s4380" xml:space="preserve">Fruſtum igitur a d ad a g b <pb file="0176" n="176" rhead="FED. COMMANDINI"/> pyramidem, uel conum, uel coni portionem candem pro-<lb/>portionem habet, quam baſes ab, cd unà cum e ſ ad ba-<lb/>ſim a b. </s> <s xml:id="echoid-s4381" xml:space="preserve">quod demonſtrare uolebamus.</s> <s xml:id="echoid-s4382" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="1"> <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a"> <image file="0175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0175-01"/> </figure> <figure xlink:label="fig-0175-02" xlink:href="fig-0175-02a"> <image file="0175-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0175-02"/> </figure> <note position="right" xlink:label="note-0175-01" xlink:href="note-0175-01a" xml:space="preserve">6. 11. duo <lb/>decimi</note> </div> <p> <s xml:id="echoid-s4383" xml:space="preserve">Fruſtum uero a d æquale eſſe pyramidi, uel co <lb/>no, uel coni portioni, cuius baſis conſtat ex baſi-<lb/>bus a b, c d, e f, & </s> <s xml:id="echoid-s4384" xml:space="preserve">altitudo fruſti altitudini eſt æ-<lb/>qualis, hoc modo oſten demus.</s> <s xml:id="echoid-s4385" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4386" xml:space="preserve">Sit fruſtum pyramidis a b c d e f, cuius maior baſis trian-<lb/>gulum a b c; </s> <s xml:id="echoid-s4387" xml:space="preserve">minor d e f: </s> <s xml:id="echoid-s4388" xml:space="preserve">& </s> <s xml:id="echoid-s4389" xml:space="preserve">ſecetur plano baſibus æquidi-<lb/>ſtante, quod ſectionem faciat triangulum g h k inter trian-<lb/>gula a b c, d e f proportionale. </s> <s xml:id="echoid-s4390" xml:space="preserve">Iam ex iis, quæ demonſtrata <lb/>ſuntin 23. </s> <s xml:id="echoid-s4391" xml:space="preserve">huius, patet ſruſtum a b c d e f diuidi in tres pyra <lb/>mides proportionales; </s> <s xml:id="echoid-s4392" xml:space="preserve">& </s> <s xml:id="echoid-s4393" xml:space="preserve">earum maiorem eſſe pyramidẽ <lb/>a b c d minorẽ uero d e f b. </s> <s xml:id="echoid-s4394" xml:space="preserve">ergo pyramis à triangulo g h k <lb/>conſtituta, quæ altitudinem habeat ſruſti altitudini æqua-<lb/>lem, proportionalis eſtinter pyramides a b c d, d e f b: </s> <s xml:id="echoid-s4395" xml:space="preserve">& </s> <s xml:id="echoid-s4396" xml:space="preserve"><lb/>idcirco fruſtum a b c d e f tribus dictis pyramidibus æqua <lb/>le erit. </s> <s xml:id="echoid-s4397" xml:space="preserve">Itaque ſi intelligatur alia pyra-<lb/> <anchor type="figure" xlink:label="fig-0176-01a" xlink:href="fig-0176-01"/> mis æque alta, quæ baſim habeat ex tri <lb/>bus baſibus a b c, d e f, g h k conſtan-<lb/>tem; </s> <s xml:id="echoid-s4398" xml:space="preserve">perſpicuum eſtipſam eiſdem py-<lb/>ramidibus, & </s> <s xml:id="echoid-s4399" xml:space="preserve">propterea ipſi fruſto æ-<lb/>qualem eſſe.</s> <s xml:id="echoid-s4400" xml:space="preserve"/> </p> <div xml:id="echoid-div264" type="float" level="2" n="2"> <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a"> <image file="0176-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0176-01"/> </figure> </div> <p> <s xml:id="echoid-s4401" xml:space="preserve">Rurſus ſit ſruſtum pyramidis a g, cu <lb/>ius maior baſis quadrilaterum a b c d, <lb/>minor e f g h: </s> <s xml:id="echoid-s4402" xml:space="preserve">& </s> <s xml:id="echoid-s4403" xml:space="preserve">ſecetur plano baſi-<lb/>bus æquidiſtante, ita ut fiat ſectio qua-<lb/>drilaterum K lm n, quod ſit proportio <lb/>nale inter quadrilatera a b c d, e f g h. </s> <s xml:id="echoid-s4404" xml:space="preserve">Dico pyramidem, <lb/>cuius baſis ſit æqualis tribus quadrilateris a b c d, _k_ l m n, <lb/>e f g h, & </s> <s xml:id="echoid-s4405" xml:space="preserve">altitudo æqualis altitudini fruſti, ipſi fruſto a g <lb/>æqualem eſſe. </s> <s xml:id="echoid-s4406" xml:space="preserve">Ducatur enim planum per lineas f b, h d, <pb o="33" file="0177" n="177" rhead="DE CENTRO GRAVIT. SOLID."/> quod diuidat fruſtum in duo fruſta triangulares baſes ha-<lb/>bentia, uidelicet in fruſtum a b d e f h, & </s> <s xml:id="echoid-s4407" xml:space="preserve">in fruſtũ b c d f g h. <lb/></s> <s xml:id="echoid-s4408" xml:space="preserve">erit triangulum k l n proportionale inter triangula a b d, <lb/>e f h: </s> <s xml:id="echoid-s4409" xml:space="preserve">& </s> <s xml:id="echoid-s4410" xml:space="preserve">triangulum l m n proportionale inter b c d, f g h. </s> <s xml:id="echoid-s4411" xml:space="preserve"><lb/>ſed pyramis æque alta, cuius baſis conſtat ex tribus trian-<lb/>gulis a b d, k l n, e f h, demonſtrata <lb/> <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> eſt ſruſto a b d e f h æqualis. </s> <s xml:id="echoid-s4412" xml:space="preserve">& </s> <s xml:id="echoid-s4413" xml:space="preserve">ſi-<lb/>militer pyramis, cuius baſis con-<lb/>ſtat ex triangulis b c d, l m n, f g h <lb/>æqualis fruſto b c d f g h: </s> <s xml:id="echoid-s4414" xml:space="preserve">compo-<lb/>nuntur autem tria quadrilatera a <lb/>b c d, _k_ l m n, e f g h è ſex triangu-<lb/>lis iam dictis. </s> <s xml:id="echoid-s4415" xml:space="preserve">pyramis igitur ba-<lb/>ſim habens æqualem tribus qua-<lb/>drilateris, & </s> <s xml:id="echoid-s4416" xml:space="preserve">altitudinem eandem <lb/>ipſi fruſto a g eſt æqualis. </s> <s xml:id="echoid-s4417" xml:space="preserve">Eodem <lb/>modo illud demõſtrabitur in aliis <lb/>eiuſmodi fruſtis.</s> <s xml:id="echoid-s4418" xml:space="preserve"/> </p> <div xml:id="echoid-div265" type="float" level="2" n="3"> <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a"> <image file="0177-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0177-01"/> </figure> </div> <p> <s xml:id="echoid-s4419" xml:space="preserve">Sit fruſtum coni, uel coni, uel coni portionis a d; </s> <s xml:id="echoid-s4420" xml:space="preserve">cuius maior ba-<lb/>ſis circulus, uel ellipſis circa diametrum a b; </s> <s xml:id="echoid-s4421" xml:space="preserve">minor circa <lb/>c d: </s> <s xml:id="echoid-s4422" xml:space="preserve">& </s> <s xml:id="echoid-s4423" xml:space="preserve">ſecetur plano, quod baſibus æquidiſtet, faciatq; </s> <s xml:id="echoid-s4424" xml:space="preserve">ſe-<lb/>ctionem circulum, uel ellipſim circa diametrum e f, ita ut <lb/>inter circulos, uel ellipſes a b, c d ſit proportionalis. </s> <s xml:id="echoid-s4425" xml:space="preserve">Dico <lb/>conum, uel coni portionem, cuius baſis eſt æqualis tribus <lb/>circulis, uel tribus ellipſibus a b, e f, c d; </s> <s xml:id="echoid-s4426" xml:space="preserve">& </s> <s xml:id="echoid-s4427" xml:space="preserve">altitudo eadem, <lb/>quæ fruſti a d, ipſi fruſto æqualem eſſe. </s> <s xml:id="echoid-s4428" xml:space="preserve">producatur enim <lb/>fruſti ſuperficies quouſque coeat in unum punctum, quod <lb/>ſit g: </s> <s xml:id="echoid-s4429" xml:space="preserve">& </s> <s xml:id="echoid-s4430" xml:space="preserve">coni, uel coni portionis a g b axis ſit g h, occurrens <lb/>planis a b, e f, c d in punctis h _k_ l: </s> <s xml:id="echoid-s4431" xml:space="preserve">circa circulum uero de-<lb/>ſcribatur quadratum m n o p, & </s> <s xml:id="echoid-s4432" xml:space="preserve">circa ellipſim rectangulũ <lb/>m n o p, quod ex ipſius diametris conſtat: </s> <s xml:id="echoid-s4433" xml:space="preserve">iunctisq; </s> <s xml:id="echoid-s4434" xml:space="preserve">g m, <lb/>g n, g o, g p, ex eodem uertice intelligatur pyramis baſim <lb/>habens dictum quadratum, uel rectangulum: </s> <s xml:id="echoid-s4435" xml:space="preserve">& </s> <s xml:id="echoid-s4436" xml:space="preserve">plana in <lb/>quibus ſunt circuli, uel ellipſes e f, c d uſque ad eius latera <pb file="0178" n="178" rhead="FED. COMMANDINI"/> producantur. </s> <s xml:id="echoid-s4437" xml:space="preserve">Quoniam igitur pyramis ſecatur planis bafi <lb/>æquidiſtantibus, ſectiones ſimiles erunt: </s> <s xml:id="echoid-s4438" xml:space="preserve">atque erunt qua-<lb/> <anchor type="note" xlink:label="note-0178-01a" xlink:href="note-0178-01"/> drata, uel rectangula circa circulos, uel ellipſes deſcripta, <lb/>quemadmodum & </s> <s xml:id="echoid-s4439" xml:space="preserve">in ipſa baſi. </s> <s xml:id="echoid-s4440" xml:space="preserve">Sed cum circuli inter ſe eã <lb/>proportionem habeant, quam diametrorum quadrata: <lb/></s> <s xml:id="echoid-s4441" xml:space="preserve"> <anchor type="note" xlink:label="note-0178-02a" xlink:href="note-0178-02"/> itemq; </s> <s xml:id="echoid-s4442" xml:space="preserve">ellipſes eam quam rectangula ex ipſarum diametris <lb/>conſtantia: </s> <s xml:id="echoid-s4443" xml:space="preserve">& </s> <s xml:id="echoid-s4444" xml:space="preserve">ſit circulus, uel ellipſis circa diametrum e f <lb/> <anchor type="figure" xlink:label="fig-0178-01a" xlink:href="fig-0178-01"/> <anchor type="note" xlink:label="note-0178-03a" xlink:href="note-0178-03"/> proportionalis inter circulos, uel ellipſes a b, c d; </s> <s xml:id="echoid-s4445" xml:space="preserve">erit re-<lb/>ctangulum e f etiam inter rectangula a b, c d proportio-<lb/>nale: </s> <s xml:id="echoid-s4446" xml:space="preserve">per rectangulum enim nunc breuitatis cauſa etiã ip-<lb/>ſum quadratum intelligemus. </s> <s xml:id="echoid-s4447" xml:space="preserve">quare ex iis, quæ proxime <lb/>dicta ſunt, pyramis baſim habens æqualem dictis rectangu <lb/>lis, & </s> <s xml:id="echoid-s4448" xml:space="preserve">altitudinem eandem, quam fruſtum a d, ipſi fruſto à <lb/>pyramide abſciſſo æqualis probabitur. </s> <s xml:id="echoid-s4449" xml:space="preserve">ut autem rectangu <lb/>lum c d ad rectangulũ e f, ita circulus, uel ellipſis c d a d e f <lb/>circulum, uel ellipſim: </s> <s xml:id="echoid-s4450" xml:space="preserve">componendoq; </s> <s xml:id="echoid-s4451" xml:space="preserve">ut rectangula c d, <lb/>e f, ad e f rectangulum, ita circuli, uel ellipſes e d, e f, ad e f: <lb/></s> <s xml:id="echoid-s4452" xml:space="preserve">& </s> <s xml:id="echoid-s4453" xml:space="preserve">ut rectangulum e f ad rectangulum a b, ita cir culus, uel <lb/>cllipſis e f ad a b circulum, uel ellipſim. </s> <s xml:id="echoid-s4454" xml:space="preserve">ergo ex æquali, & </s> <s xml:id="echoid-s4455" xml:space="preserve"><lb/>componendo, utrectãgula c d, e f, a b ad ipſum a b, ita cir- <pb o="34" file="0179" n="179" rhead="DE CENTRO GRAVIT. SOLID."/> culi, uel ellipſes c d, e ſ a b ad circulum, uel ellipſim a b. </s> <s xml:id="echoid-s4456" xml:space="preserve">In-<lb/>telligatur pyramis q baſim habens æqualem tribus rectan <lb/>gulis a b, e f, c d; </s> <s xml:id="echoid-s4457" xml:space="preserve">& </s> <s xml:id="echoid-s4458" xml:space="preserve">altitudinem eãdem, quam fruſtum a d. <lb/></s> <s xml:id="echoid-s4459" xml:space="preserve">intelligatur etiam conus, uel coni portio q, eadem altitudi <lb/>ne, cuius baſis ſit tribus circulis, uel tribus ellipſibus a b, <lb/>e f, c d æqualis. </s> <s xml:id="echoid-s4460" xml:space="preserve">poſtremo intelligatur pyramis a l b, cuius <lb/>baſis ſit rectangulum m n o p, & </s> <s xml:id="echoid-s4461" xml:space="preserve">altitudo eadem, quæ fru-<lb/>ſti: </s> <s xml:id="echoid-s4462" xml:space="preserve">itemq, intelligatur conus, uel coni portio a l b, cuius <lb/>baſis circulus, uel ellipſis circa diametrum a b, & </s> <s xml:id="echoid-s4463" xml:space="preserve">eadem al <lb/>titudo. </s> <s xml:id="echoid-s4464" xml:space="preserve">ut igitur rectangula a b, e f, c d ad rectangulum a b, <lb/> <anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/> ita pyramis q ad pyramidem a l b; </s> <s xml:id="echoid-s4465" xml:space="preserve">& </s> <s xml:id="echoid-s4466" xml:space="preserve">ut circuli, uel ellip-<lb/>ſes a b, e f, c d ad a b circulum, uel ellipſim, ita conus, uel co <lb/>ni portio q ad conum, uel coni portionem a l b. </s> <s xml:id="echoid-s4467" xml:space="preserve">conus <lb/>igitur, uel coni portio q ad conum, uel coni portionem <lb/>a l b eſt, ut pyramis q ad pyramidem a l b. </s> <s xml:id="echoid-s4468" xml:space="preserve">ſed pyramis <lb/>a l b ad pyramidem a g b eſt, ut altitudo ad altitudinem, ex <lb/>20. </s> <s xml:id="echoid-s4469" xml:space="preserve">huius: </s> <s xml:id="echoid-s4470" xml:space="preserve">& </s> <s xml:id="echoid-s4471" xml:space="preserve">ita eſt conus, uel coni portio al b ad conum, <lb/>uel coni portionem a g b ex 14. </s> <s xml:id="echoid-s4472" xml:space="preserve">duodecimi elementorum, <lb/>& </s> <s xml:id="echoid-s4473" xml:space="preserve">ex iis, quæ nos demonſtrauimus in commentariis in un-<lb/>decimam de conoidibus, & </s> <s xml:id="echoid-s4474" xml:space="preserve">ſphæroidibus, propoſitione <lb/>quarta. </s> <s xml:id="echoid-s4475" xml:space="preserve">pyramis autem a g b ad pyramidem c g d propor-<lb/>tionem habet compoſitam ex proportione baſium & </s> <s xml:id="echoid-s4476" xml:space="preserve">pro <lb/>portione altitudinum, ex uigeſima prima huius: </s> <s xml:id="echoid-s4477" xml:space="preserve">& </s> <s xml:id="echoid-s4478" xml:space="preserve">ſimili-<lb/>ter conus, uel coni portio a g b a d conum, uel coni portio-<lb/>nem c g d proportionem habet compoſitã ex eiſdem pro-<lb/>portionibus, per ea, quæ in dictis commentariis demon-<lb/>ſtrauimus, propoſitione quinta, & </s> <s xml:id="echoid-s4479" xml:space="preserve">ſexta: </s> <s xml:id="echoid-s4480" xml:space="preserve">altitudo enim in <lb/>utriſque eadem eſt, & </s> <s xml:id="echoid-s4481" xml:space="preserve">baſes inter ſe ſe eandem habent pro-<lb/>portionem. </s> <s xml:id="echoid-s4482" xml:space="preserve">ergo ut pyramis a g b ad pyramidem c g d, ita <lb/>eſt conus, uel coni portio a g b ad a g d conum, uel coni <lb/>portionem: </s> <s xml:id="echoid-s4483" xml:space="preserve">& </s> <s xml:id="echoid-s4484" xml:space="preserve">per conuerſionẽ rationis, ut pyramis a g b <lb/>ad fruſtū à pyramide abſciſſum, ita conus uel coni portio <lb/>a g b ad fruſtum a d. </s> <s xml:id="echoid-s4485" xml:space="preserve">ex æquali igitur, ut pyramis q ad fru-<lb/>ſtum à pyramide abſciſſum, ita conus uel coni portio q ad <pb file="0180" n="180" rhead="FED. COMMANDINI"/> fruſtum a d. </s> <s xml:id="echoid-s4486" xml:space="preserve">Sed pyramis q æqualis eſt fruſto à pyramide <lb/>abſciſſo, ut dem onſtrauimus. </s> <s xml:id="echoid-s4487" xml:space="preserve">ergo & </s> <s xml:id="echoid-s4488" xml:space="preserve">conus, uel coni por-<lb/>tio q, cuius baſis ex tribus circulis, uel ellipſibus a b, e f, c d <lb/>conſtat, & </s> <s xml:id="echoid-s4489" xml:space="preserve">altitudo eadem, quæ fruſti: </s> <s xml:id="echoid-s4490" xml:space="preserve">ipſi fruſto a d eſt æ-<lb/>qualis. </s> <s xml:id="echoid-s4491" xml:space="preserve">atque illud eſt, quod demonſtrare oportebat.</s> <s xml:id="echoid-s4492" xml:space="preserve"/> </p> <div xml:id="echoid-div266" type="float" level="2" n="4"> <note position="left" xlink:label="note-0178-01" xlink:href="note-0178-01a" xml:space="preserve">9. huius</note> <note position="left" xlink:label="note-0178-02" xlink:href="note-0178-02a" xml:space="preserve">2. duode-<lb/>cimi.</note> <figure xlink:label="fig-0178-01" xlink:href="fig-0178-01a"> <image file="0178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0178-01"/> </figure> <note position="left" xlink:label="note-0178-03" xlink:href="note-0178-03a" xml:space="preserve">7. de co-<lb/>noidibus <lb/>& ſphæ-<lb/>roidibus</note> <note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">6. 11. duo <lb/>decimi</note> </div> </div> <div xml:id="echoid-div268" type="section" level="1" n="91"> <head xml:id="echoid-head98" xml:space="preserve">THEOREMA XXI. PROPOSITIO XXVI.</head> <p> <s xml:id="echoid-s4493" xml:space="preserve"><emph style="sc">Cvivslibet</emph> fruſti à pyramide, uel cono, <lb/>uel coni portione abſcisſi, centrum grauitatis eſt <lb/>in axe, ita ut eo primum in duas portiones diui-<lb/>ſo, portio ſuperior, quæ minorem baſim attingit <lb/>ad portionem reliquam eam habeat proportio-<lb/>nem, quam duplum lateris, uel diametri maioris <lb/>baſis, vnà cum latere, uel diametro minoris, ipſi <lb/>reſpondente, habet ad duplum lateris, uel diame-<lb/>tri minoris baſis vnà cũ latere, uel diametro ma-<lb/>ioris: </s> <s xml:id="echoid-s4494" xml:space="preserve">deinde à puncto diuiſionis quarta parte ſu <lb/>perioris portionis in ipſa ſumpta: </s> <s xml:id="echoid-s4495" xml:space="preserve">& </s> <s xml:id="echoid-s4496" xml:space="preserve">rurſus ab in-<lb/>ferioris portionis termino, qui eſt ad baſim maio <lb/>rem, ſumpta quarta parte totius axis: </s> <s xml:id="echoid-s4497" xml:space="preserve">centrum ſit <lb/>in linea, quæ his finibus continetur, atque in eo li <lb/>neæ puncto, quo ſic diuiditur, ut tota linea ad par <lb/>tem propinquiorem minori baſi, eãdem propor-<lb/>tionem habeat, quam fruſtum ad pyramidẽ, uel <lb/>conum, uel coni portionem, cuius baſis ſit ea-<lb/>dem, quæ baſis maior, & </s> <s xml:id="echoid-s4498" xml:space="preserve">altitudo fruſti altitudini <lb/>æqualis.</s> <s xml:id="echoid-s4499" xml:space="preserve"/> </p> <pb o="35" file="0181" n="181" rhead="DE CENTRO GRAVIT. SOLID."/> <p> <s xml:id="echoid-s4500" xml:space="preserve">Sit ſruſtum a e a pyramide, quæ triangularem baſim ha-<lb/>beat abſciſſum: </s> <s xml:id="echoid-s4501" xml:space="preserve">cuius maior baſis triangulum a b c, minor <lb/>d e f; </s> <s xml:id="echoid-s4502" xml:space="preserve">& </s> <s xml:id="echoid-s4503" xml:space="preserve">axis g h. </s> <s xml:id="echoid-s4504" xml:space="preserve">ducto autem plano per axem & </s> <s xml:id="echoid-s4505" xml:space="preserve">per lineã <lb/>d a, quod ſectionem faciat d a k l quadrilaterum; </s> <s xml:id="echoid-s4506" xml:space="preserve">puncta <lb/>K l lineas b c, e f bifariam ſecabunt. </s> <s xml:id="echoid-s4507" xml:space="preserve">nam cum g h ſit axis <lb/>ſruſti: </s> <s xml:id="echoid-s4508" xml:space="preserve">erit h centrum grauitatis trianguli a b c: </s> <s xml:id="echoid-s4509" xml:space="preserve">& </s> <s xml:id="echoid-s4510" xml:space="preserve">g <lb/>centrum trianguli d e f: </s> <s xml:id="echoid-s4511" xml:space="preserve">cen-<lb/> <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> <anchor type="note" xlink:label="note-0181-01a" xlink:href="note-0181-01"/> trum uero cuiuslibet triangu <lb/>li eſt in recta linea, quæ ab an-<lb/>gulo ipſius ad dimidiã baſim <lb/>ducitur ex decimatertia primi <lb/>libri Archimedis de cẽtro gra <lb/>uitatis planorum. </s> <s xml:id="echoid-s4512" xml:space="preserve">quare cen-<lb/> <anchor type="note" xlink:label="note-0181-02a" xlink:href="note-0181-02"/> trũ grauitatis trapezii b c f e <lb/>eſt in linea _K_ l, quod ſit m: </s> <s xml:id="echoid-s4513" xml:space="preserve">& </s> <s xml:id="echoid-s4514" xml:space="preserve">à <lb/>puncto m ad axem ducta m n <lb/>ipſi a k, uel d l æquidiſtante; <lb/></s> <s xml:id="echoid-s4515" xml:space="preserve">erit axis g h diuiſus in portio-<lb/>nes g n, n h, quas diximus: </s> <s xml:id="echoid-s4516" xml:space="preserve">ean <lb/>dem enim proportionem ha-<lb/>bet g n ad n h, quã l m ad m _k_. </s> <s xml:id="echoid-s4517" xml:space="preserve"><lb/>At l m ad m K habet eam, quã <lb/>duplum lateris maioris baſis <lb/>b c una cum latere minoris e f <lb/>ad duplum lateris e f unà cum <lb/>later b c, ex ultima eiuſdem <lb/>libri Archimedis. </s> <s xml:id="echoid-s4518" xml:space="preserve">Itaque à li-<lb/>nea n g abſcindatur, quarta <lb/>pars, quæ ſit n p: </s> <s xml:id="echoid-s4519" xml:space="preserve">& </s> <s xml:id="echoid-s4520" xml:space="preserve">ab axe h g abſcindatur itidem <lb/>quarta pars h o: </s> <s xml:id="echoid-s4521" xml:space="preserve">& </s> <s xml:id="echoid-s4522" xml:space="preserve">quam proportionem habet fruſtum ad <lb/>pyramidem, cuius maior baſis eſt triangulum a b c, & </s> <s xml:id="echoid-s4523" xml:space="preserve">alti-<lb/>tudo ipſi æqualis; </s> <s xml:id="echoid-s4524" xml:space="preserve">habeat o p ad p q. </s> <s xml:id="echoid-s4525" xml:space="preserve">Dico centrum graui-<lb/>tatis fruſti eſſe in linea p o, & </s> <s xml:id="echoid-s4526" xml:space="preserve">in puncto q. </s> <s xml:id="echoid-s4527" xml:space="preserve">namque ipſum <lb/>eſſe in linea g h manifeſte conſtat. </s> <s xml:id="echoid-s4528" xml:space="preserve">protractis enim fruſti pla <pb file="0182" n="182" rhead="FED. COMMANDINI"/> nis, quouſque in unum punctum r conueniant; </s> <s xml:id="echoid-s4529" xml:space="preserve">erit pyra-<lb/>midis a b c r, & </s> <s xml:id="echoid-s4530" xml:space="preserve">pyramidis d e f r grauitatis centrum in li-<lb/>nea r h. </s> <s xml:id="echoid-s4531" xml:space="preserve">ergo & </s> <s xml:id="echoid-s4532" xml:space="preserve">reliquæ magnitudinis, uidelicet fruſti cen-<lb/>trum in eadem linea neceſſario comperietur. </s> <s xml:id="echoid-s4533" xml:space="preserve">Iungantur <lb/>d b, d c, d h, d m: </s> <s xml:id="echoid-s4534" xml:space="preserve">& </s> <s xml:id="echoid-s4535" xml:space="preserve">per lineas d b, d c ducto altero plano <lb/>intelligatur fruſtum in duas pyramides diuiſum: </s> <s xml:id="echoid-s4536" xml:space="preserve">in pyra-<lb/>midem quidem, cuius baſis eſt triangulum a b c, uertex d: <lb/></s> <s xml:id="echoid-s4537" xml:space="preserve">& </s> <s xml:id="echoid-s4538" xml:space="preserve">in eam, cuius idem uertex, & </s> <s xml:id="echoid-s4539" xml:space="preserve">baſis trapezium b c f e. </s> <s xml:id="echoid-s4540" xml:space="preserve">erit <lb/>igitur pyramidis a b c d axis d h, & </s> <s xml:id="echoid-s4541" xml:space="preserve">pyramidis b c f e d axis <lb/>d m: </s> <s xml:id="echoid-s4542" xml:space="preserve">atque erunt tres axes g h, d h, d m in eodem plano <lb/>d a K l. </s> <s xml:id="echoid-s4543" xml:space="preserve">ducatur præterea per o linea ſt ip ſi a K æquidiſtãs, <lb/>quæ lineam d h in u ſecet: </s> <s xml:id="echoid-s4544" xml:space="preserve">per p uero ducatur x y æquidi-<lb/>ſtans eidem, ſecansque d m in <lb/> <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> z: </s> <s xml:id="echoid-s4545" xml:space="preserve">& </s> <s xml:id="echoid-s4546" xml:space="preserve">iungatur z u, quæ ſecet <lb/>g h in φ. </s> <s xml:id="echoid-s4547" xml:space="preserve">tranſibit ea per q: </s> <s xml:id="echoid-s4548" xml:space="preserve">& </s> <s xml:id="echoid-s4549" xml:space="preserve"><lb/>erunt φ q unum, atque idem <lb/>pun ctum; </s> <s xml:id="echoid-s4550" xml:space="preserve">ut inferius appare-<lb/>bit. </s> <s xml:id="echoid-s4551" xml:space="preserve">Quoniam igitur linea u o <lb/>æ quidiſtat ipſi d g, erit d u ad <lb/> <anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/> u h, ut g o ad o h. </s> <s xml:id="echoid-s4552" xml:space="preserve">Sed g o tri-<lb/>pla eſt o h. </s> <s xml:id="echoid-s4553" xml:space="preserve">quare & </s> <s xml:id="echoid-s4554" xml:space="preserve">d u ipſius <lb/>u h eſt tripla: </s> <s xml:id="echoid-s4555" xml:space="preserve">& </s> <s xml:id="echoid-s4556" xml:space="preserve">ideo pyrami-<lb/>dis a b c d centrum grauitatis <lb/>erit punctum 11. </s> <s xml:id="echoid-s4557" xml:space="preserve">Rurſus quo-<lb/>niam z y ipſi d l æquidiſtat, d z <lb/>a d z m eſt, utly ad y m: </s> <s xml:id="echoid-s4558" xml:space="preserve">eſtque <lb/>ly ad y m, ut g p ad p n. </s> <s xml:id="echoid-s4559" xml:space="preserve">ergo <lb/>d z ad z m eſt, ut g p ad p n. <lb/></s> <s xml:id="echoid-s4560" xml:space="preserve">Quòd cum g p ſit tripla p n; </s> <s xml:id="echoid-s4561" xml:space="preserve"><lb/>erit etiam d z ipſius z m tri-<lb/>pla. </s> <s xml:id="echoid-s4562" xml:space="preserve">atque ob eandem cauſ-<lb/>ſam punctum z eſt centrũ gra-<lb/>uitatis pyramidis b c f e d. </s> <s xml:id="echoid-s4563" xml:space="preserve">iun <lb/>ctaigitur z u, in ea erit cẽtrum <pb o="36" file="0183" n="183" rhead="DE CENTRO GRAVIT. SOLID."/> grauitatis magnitudinis, quæ ex utriſque pyramidibus cõ <lb/>ſtat; </s> <s xml:id="echoid-s4564" xml:space="preserve">hoc eſt ipſius fruſti. </s> <s xml:id="echoid-s4565" xml:space="preserve">Sed fruſti centrum eſt etiam in a-<lb/>xe g h. </s> <s xml:id="echoid-s4566" xml:space="preserve">ergo in puncto φ, in quo lineæ z u, g h conueniunt. <lb/></s> <s xml:id="echoid-s4567" xml:space="preserve">Itaque u φ ad φ z eam proportionem habet, quam pyramis <lb/> <anchor type="note" xlink:label="note-0183-01a" xlink:href="note-0183-01"/> b c f e d ad pyramidem a b c d. </s> <s xml:id="echoid-s4568" xml:space="preserve">& </s> <s xml:id="echoid-s4569" xml:space="preserve">componendo u z ad z φ <lb/>eam habet, quam fruſtum ad pyramidem a b c d. </s> <s xml:id="echoid-s4570" xml:space="preserve">Vtuero <lb/>u z ad z φ, ita o p ad p φ ob ſimilitudinem triangulorum, <lb/>u o φ, z p φ. </s> <s xml:id="echoid-s4571" xml:space="preserve">quare o p ad p φ eſt ut fruſtum ad pyramidem <lb/>a b c d. </s> <s xml:id="echoid-s4572" xml:space="preserve">ſed ita erat o p ad p q. </s> <s xml:id="echoid-s4573" xml:space="preserve">æquales igitur ſunt p φ, p q: </s> <s xml:id="echoid-s4574" xml:space="preserve">& </s> <s xml:id="echoid-s4575" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0183-02a" xlink:href="note-0183-02"/> q φ unum atque idem punctum. </s> <s xml:id="echoid-s4576" xml:space="preserve">ex quibus ſequitur lineam <lb/>z u ſecare o p in q: </s> <s xml:id="echoid-s4577" xml:space="preserve">& </s> <s xml:id="echoid-s4578" xml:space="preserve">propterea pũctum q ipſius fruſti gra-<lb/>uitatis centrum eſſe.</s> <s xml:id="echoid-s4579" xml:space="preserve"/> </p> <div xml:id="echoid-div268" type="float" level="2" n="1"> <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a"> <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0181-01"/> </figure> <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve">3. diffi. hu <lb/>ius.</note> <note position="right" xlink:label="note-0181-02" xlink:href="note-0181-02a" xml:space="preserve">Vltima e-<lb/>auſdẽ libri <lb/>Archime-<lb/>dis.</note> <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a"> <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0182-01"/> </figure> <note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">2. ſexti.</note> <note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">8. prim I <lb/>libri Ar-<lb/>chimedis <lb/>de cẽtro <lb/>grauita-<lb/>tis plano <lb/>runi</note> <note position="right" xlink:label="note-0183-02" xlink:href="note-0183-02a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s4580" xml:space="preserve">Sit fruſtum a g à pyramide, quæ quadrangularem baſim <lb/>habeat abſciſſum, cuius maior baſis a b c d, minor e f g h, <lb/>& </s> <s xml:id="echoid-s4581" xml:space="preserve">axis k l. </s> <s xml:id="echoid-s4582" xml:space="preserve">diuidatur autem primũ _k_ l, ita ut quam propor-<lb/>tionem habet duplum lateris a b unà cum latere e f ad du <lb/>plum lateris e f unà cum a b; </s> <s xml:id="echoid-s4583" xml:space="preserve">habeat k m ad m l. </s> <s xml:id="echoid-s4584" xml:space="preserve">deinde à <lb/>púcto m ad k ſumatur quarta pars ipſius m k, quæ ſit m n. <lb/></s> <s xml:id="echoid-s4585" xml:space="preserve">& </s> <s xml:id="echoid-s4586" xml:space="preserve">rurſus ab l ſumatur quarta pars totius axis l k, quæ ſit <lb/>l o. </s> <s xml:id="echoid-s4587" xml:space="preserve">poſtremo fiat o n ad n p, ut fruſtum a g ad pyramidẽ, <lb/>cuius baſis ſit eadem, quæ fruſti, & </s> <s xml:id="echoid-s4588" xml:space="preserve">altitudo æqualis. </s> <s xml:id="echoid-s4589" xml:space="preserve">Dico <lb/>punctum p fruſti a g grauitatis centrum eſſe. </s> <s xml:id="echoid-s4590" xml:space="preserve">ducantur <lb/>enim a c, e g: </s> <s xml:id="echoid-s4591" xml:space="preserve">& </s> <s xml:id="echoid-s4592" xml:space="preserve">intelligantur duo fruſta triangulares ba-<lb/>ſes habentia, quorum alterum l f ex baſibus a b c, e f g cõ-<lb/>ſtet; </s> <s xml:id="echoid-s4593" xml:space="preserve">alterum l h ex baſibus a c d, e g h. </s> <s xml:id="echoid-s4594" xml:space="preserve">Sitq; </s> <s xml:id="echoid-s4595" xml:space="preserve">fruſti l f axis <lb/>q r; </s> <s xml:id="echoid-s4596" xml:space="preserve">in quo grauitatis centrum s: </s> <s xml:id="echoid-s4597" xml:space="preserve">fruſti uero l h axis t u, & </s> <s xml:id="echoid-s4598" xml:space="preserve"><lb/>x grauitatis centrum: </s> <s xml:id="echoid-s4599" xml:space="preserve">deinde iungantur u r, t q, x s. </s> <s xml:id="echoid-s4600" xml:space="preserve">tranſi-<lb/>bit u r per l: </s> <s xml:id="echoid-s4601" xml:space="preserve">quoniam l eſt centrum grauitatis quadran-<lb/>guli a b c d: </s> <s xml:id="echoid-s4602" xml:space="preserve">& </s> <s xml:id="echoid-s4603" xml:space="preserve">puncta r u grauitatis centra triangulorum <lb/>a b c, a c d; </s> <s xml:id="echoid-s4604" xml:space="preserve">in quæ quadrangulum ipſum diuiditur. </s> <s xml:id="echoid-s4605" xml:space="preserve">eadem <lb/>quoque ratione t q per punctum _k_ tranſibit. </s> <s xml:id="echoid-s4606" xml:space="preserve">At uero pro <lb/>portiones, ex quibus fruſtorum grauitatis centra inquiri-<lb/>mus, eædem ſunt in toto ſruſto a g, & </s> <s xml:id="echoid-s4607" xml:space="preserve">in fruſtis l f, l h. </s> <s xml:id="echoid-s4608" xml:space="preserve">Sunt <lb/>enim per octauam huius quadrilatera a b c d, e f g h ſimilia:</s> <s xml:id="echoid-s4609" xml:space="preserve"> <pb file="0184" n="184" rhead="FED. COMMANDINI"/> itemq; </s> <s xml:id="echoid-s4610" xml:space="preserve">ſimilia triangula a b c, e f g; </s> <s xml:id="echoid-s4611" xml:space="preserve">& </s> <s xml:id="echoid-s4612" xml:space="preserve">a c d, e g h. </s> <s xml:id="echoid-s4613" xml:space="preserve">idcir-<lb/>coq; </s> <s xml:id="echoid-s4614" xml:space="preserve">latera ſibi ipſis reſpondentia eandem inter ſe ſe pro-<lb/>portionem ſeruant. </s> <s xml:id="echoid-s4615" xml:space="preserve">Vt igitur duplum lateris a b unà <lb/>cum latere e f ad duplum lateris e f unà cum a b, ita eſt <lb/>duplum a d late-<lb/> <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> ris unà cum late-<lb/>re e h ad duplum <lb/>e h unà cum a d: <lb/></s> <s xml:id="echoid-s4616" xml:space="preserve">& </s> <s xml:id="echoid-s4617" xml:space="preserve">ita in aliis. </s> <s xml:id="echoid-s4618" xml:space="preserve"><lb/>Rurſus fruſtum <lb/>a g ad pyramidẽ, <lb/>cuius eadem eſt <lb/>bafis, & </s> <s xml:id="echoid-s4619" xml:space="preserve">æqualis <lb/>altitudo eandem <lb/>proportionẽ ha <lb/>bet, quam fruſtũ <lb/>l f ad pyramidẽ, <lb/>quæ eſt eadẽ ba-<lb/>ſi, & </s> <s xml:id="echoid-s4620" xml:space="preserve">æquali alti-<lb/>tudine: </s> <s xml:id="echoid-s4621" xml:space="preserve">& </s> <s xml:id="echoid-s4622" xml:space="preserve">ſimili-<lb/>ter quam l h fru-<lb/>ſtum ad pyrami-<lb/>dem, quæ ex ea-<lb/>dẽ baſi, & </s> <s xml:id="echoid-s4623" xml:space="preserve">æquali <lb/>altitudine con-<lb/>ſtat. </s> <s xml:id="echoid-s4624" xml:space="preserve">nam ſi inter <lb/>ipſas baſes me-<lb/>diæ proportio-<lb/>nales conſtituan <lb/>tur, tres baſes ſimul ſumptæ ad maiorem baſim in om-<lb/>nibus eodem modo ſe habebunt. </s> <s xml:id="echoid-s4625" xml:space="preserve">Vnde fit, ut axes K l, <lb/>q r, t u à punctis p s x in eandem proportionem ſecen-<lb/>tur. </s> <s xml:id="echoid-s4626" xml:space="preserve">ergo linea x s per p tranſibit: </s> <s xml:id="echoid-s4627" xml:space="preserve">& </s> <s xml:id="echoid-s4628" xml:space="preserve">lineæ r u, s x, q t in-<lb/> <anchor type="note" xlink:label="note-0184-01a" xlink:href="note-0184-01"/> ter ſe æquidiſtantes erunt. </s> <s xml:id="echoid-s4629" xml:space="preserve">Itaque cum fruſti a g latera pro- <pb o="37" file="0185" n="185" rhead="DE CENTRO GRAVIT. SOLID."/> ducta fuerìnt, ira ut in unum punctum y coeant, erunt triã <lb/>gala u y l, x y p, t y _k_ inter ſe ſimilia: </s> <s xml:id="echoid-s4630" xml:space="preserve">& </s> <s xml:id="echoid-s4631" xml:space="preserve">ſimilia etiam triangu <lb/>la l y r, p y s, _k_ y q. </s> <s xml:id="echoid-s4632" xml:space="preserve">quare ut in 19 huius, demonſtrabitur <lb/>x p, ad p s: </s> <s xml:id="echoid-s4633" xml:space="preserve">itemq; </s> <s xml:id="echoid-s4634" xml:space="preserve">t k ad _k_ q èandem habere proportionẽ, <lb/>quam u l ad l r. </s> <s xml:id="echoid-s4635" xml:space="preserve">Sed ut u l ad l r, ita eſt triangulum a b c ad <lb/>triangulum a c d: </s> <s xml:id="echoid-s4636" xml:space="preserve">& </s> <s xml:id="echoid-s4637" xml:space="preserve">ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s> <s xml:id="echoid-s4638" xml:space="preserve">Vt autem triangulum a b c ad triangu-<lb/>lum a c d, ita pyramis a b c y ad pyramidem a c d y. </s> <s xml:id="echoid-s4639" xml:space="preserve">& </s> <s xml:id="echoid-s4640" xml:space="preserve">ut <lb/>triangulum e f g ad triangulum e g h, ita pyramis e f g y <lb/>ad pyramidem e g h y; </s> <s xml:id="echoid-s4641" xml:space="preserve">ergo ut pyramis a b c y ad pyramidẽ <lb/>a c d y, ita pyramis e f g y ad pyramidem e g h y. </s> <s xml:id="echoid-s4642" xml:space="preserve">reliquum <lb/> <anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/> igitur fruſtũ l f ad reliquum fruſtũ l h eſt ut pyramis a b c y <lb/>ad pyramidem a c d y, hoc eſt ut u l ad l r, & </s> <s xml:id="echoid-s4643" xml:space="preserve">ut x p ad p s. <lb/></s> <s xml:id="echoid-s4644" xml:space="preserve">Quòd cum fruſti l f centrum grauitatis ſit s: </s> <s xml:id="echoid-s4645" xml:space="preserve">& </s> <s xml:id="echoid-s4646" xml:space="preserve">fruſti l h ſit <lb/>centrum x: </s> <s xml:id="echoid-s4647" xml:space="preserve">conſtat punctum p totius fruſti a g grauitatis <lb/> <anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/> eſſe centrum. </s> <s xml:id="echoid-s4648" xml:space="preserve">Eodem modo fiet demonſtratio etiam in <lb/>aliis pyramidibus.</s> <s xml:id="echoid-s4649" xml:space="preserve"/> </p> <div xml:id="echoid-div269" type="float" level="2" n="2"> <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a"> <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0184-01"/> </figure> <note position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve">2. ſexti.</note> <note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">19. quinti</note> <note position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">8. Archi-<lb/>medis.</note> </div> <p> <s xml:id="echoid-s4650" xml:space="preserve">Sit fruſtum a d à cono, uel coni portione abſciſſum, cu-<lb/>ius maior baſis circulus, uel ellipſis circa diametrum a b; <lb/></s> <s xml:id="echoid-s4651" xml:space="preserve">minor circa diametrum c d: </s> <s xml:id="echoid-s4652" xml:space="preserve">& </s> <s xml:id="echoid-s4653" xml:space="preserve">axis e f. </s> <s xml:id="echoid-s4654" xml:space="preserve">diuidatur autẽ e f <lb/>in g, ita ut e g ad g f eandem proportionem habeat, quam <lb/>duplum diametri a b unà cum diametro c d ad duplum c d <lb/>unà cum a b. </s> <s xml:id="echoid-s4655" xml:space="preserve">Sitq; </s> <s xml:id="echoid-s4656" xml:space="preserve">g h quarta pars lineæ g e: </s> <s xml:id="echoid-s4657" xml:space="preserve">& </s> <s xml:id="echoid-s4658" xml:space="preserve">ſit ſ K item <lb/>quarta pars totius f e axis. </s> <s xml:id="echoid-s4659" xml:space="preserve">Rurfus quam proportionem <lb/>habet fruſtum a d ad conum, uel coni portionem, in eadẽ <lb/>baſi, & </s> <s xml:id="echoid-s4660" xml:space="preserve">æquali altitudine, habeat linea _k_ h ad h l. </s> <s xml:id="echoid-s4661" xml:space="preserve">Dico pun-<lb/>ctum l fruſti a d grauitatis centrum eſſe. </s> <s xml:id="echoid-s4662" xml:space="preserve">Si enim fieri po-<lb/>teſt, ſit m centrum: </s> <s xml:id="echoid-s4663" xml:space="preserve">producaturq; </s> <s xml:id="echoid-s4664" xml:space="preserve">l m extra fruſtum in n: </s> <s xml:id="echoid-s4665" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s4666" xml:space="preserve">ut n l ad l m, ita fiat circulus, uel ellipſis circa diametrũ <lb/>a b ad aliud ſpacium, in quo ſit o. </s> <s xml:id="echoid-s4667" xml:space="preserve">Itaque in circulo, uel <lb/>ellipſi circa diametrum a b rectilinea figura plane deſcri-<lb/>batur, ita ut quæ relinquuntur portiones ſint o ſpacio mi-<lb/>nores: </s> <s xml:id="echoid-s4668" xml:space="preserve">& </s> <s xml:id="echoid-s4669" xml:space="preserve">inteiligatur pyramis a p b, baſim habens rectili-<lb/>neam figuram in circulo, uel ellipſi a b deſcriptam: </s> <s xml:id="echoid-s4670" xml:space="preserve">à qua <pb file="0186" n="186" rhead="FED. COMMANDINI"/> fruſtum pyramidis ſit abſciſſum. </s> <s xml:id="echoid-s4671" xml:space="preserve">erit ex iis quæ proxime <lb/>tradidimus, fruſti pyramidis a d centrum grauitatis l. </s> <s xml:id="echoid-s4672" xml:space="preserve">Quo <lb/>niam igitur portiones ſpacio o minores ſunt; </s> <s xml:id="echoid-s4673" xml:space="preserve">habebit cir <lb/>culus, uel ellipſis a b ad <lb/> <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> portiones dictas maiorẽ <lb/>proportionem, quàm n l <lb/>ad lm. </s> <s xml:id="echoid-s4674" xml:space="preserve">ſed ut circulus, uel <lb/>ellipſis a b ad portiones, <lb/>ita a p b conus, uel coni <lb/>portio ad ſolidas portio-<lb/>nes, id quod ſupra demon <lb/>ſtratum eſt: </s> <s xml:id="echoid-s4675" xml:space="preserve">& </s> <s xml:id="echoid-s4676" xml:space="preserve">ut circulus <lb/>uel ellipſis c d ad portio-<lb/> <anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/> nes, quæ ipſi inſunt, ita co <lb/>nus, uel coni portio c p d <lb/>ad ſolidas ipſius portio-<lb/>nes. </s> <s xml:id="echoid-s4677" xml:space="preserve">Quòd cum figuræ in <lb/>circulis, uel ellipſibus a b <lb/>c d deſcriptæ ſimiles ſint, <lb/>erit proportio circuli, uel <lb/>ellipſis a b ad ſuas portio <lb/>nes, eadẽ, quæ circuli uel <lb/>ellipſis c d ad ſuas. </s> <s xml:id="echoid-s4678" xml:space="preserve">ergo <lb/>conus, uel coni portio a p <lb/>b ad portiones ſolidas eã-<lb/>dem habet proportionẽ, <lb/>quam conus, uel coni por <lb/>tio c p d ad ſolidas ipſius <lb/>portiones. </s> <s xml:id="echoid-s4679" xml:space="preserve">reliquum igi-<lb/> <anchor type="note" xlink:label="note-0186-02a" xlink:href="note-0186-02"/> tur coni, uel coni portionis fruſtũ, ſcilicet a d ad reliquas <lb/>portiones ſolidas in ipſo contentas eandem proportionẽ <lb/>habet, quam conus, uel coni portio a p b ad ſolidas portio <lb/>nes: </s> <s xml:id="echoid-s4680" xml:space="preserve">hoc eſt eandem, quam circulus, uel ellipſis a b ad por <lb/>tiones planas. </s> <s xml:id="echoid-s4681" xml:space="preserve">quare fruſtum coni, uel coni portionis a d <pb o="38" file="0187" n="187" rhead="DE CENTRO GRA VIT. SOLID."/> ad portiones ſolidas maiorem habet proportioné, quàm <lb/>n l ad l m: </s> <s xml:id="echoid-s4682" xml:space="preserve">& </s> <s xml:id="echoid-s4683" xml:space="preserve">diuidendo fruſtum pyramidis ad dictas por-<lb/>tiones maiorem proportionem habet, quàm n m ad m l. <lb/></s> <s xml:id="echoid-s4684" xml:space="preserve">fiat igitur ut fruſtum pyramidis ad portiones, ita q m ad <lb/>m l. </s> <s xml:id="echoid-s4685" xml:space="preserve">Itaque quoniam à fruſto coni, uel coni portionis a d, <lb/>cuius grauitatis centrum eſtm, aufertur fruſtum pyrami-<lb/>dis habens centruml; </s> <s xml:id="echoid-s4686" xml:space="preserve">erit reliquæ magnitudinis, quæ ex <lb/>portionibus ſolidis conſtat; </s> <s xml:id="echoid-s4687" xml:space="preserve">grauitatis cẽtrum in linea l m <lb/>producta, atque in puncto q, extra figuram poſito. </s> <s xml:id="echoid-s4688" xml:space="preserve">quod <lb/>fieri nullo modo poteſt. </s> <s xml:id="echoid-s4689" xml:space="preserve">relinquitur ergo, ut punctum l ſit <lb/>fruſti a d grauitatis centrum. </s> <s xml:id="echoid-s4690" xml:space="preserve">quæ omnia demonſtranda <lb/>proponebantur.</s> <s xml:id="echoid-s4691" xml:space="preserve"/> </p> <div xml:id="echoid-div270" type="float" level="2" n="3"> <figure xlink:label="fig-0186-01" xlink:href="fig-0186-01a"> <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0186-01"/> </figure> <note position="left" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve">22. huius</note> <note position="left" xlink:label="note-0186-02" xlink:href="note-0186-02a" xml:space="preserve">19. quinti</note> </div> </div> <div xml:id="echoid-div272" type="section" level="1" n="92"> <head xml:id="echoid-head99" xml:space="preserve">THEOREMA XXII. PROPOSITIO XXVII.</head> <p> <s xml:id="echoid-s4692" xml:space="preserve"><emph style="sc">Omnivm</emph> ſolidorum in ſphæra deſcripto-<lb/>rum, quæ æqualibus, & </s> <s xml:id="echoid-s4693" xml:space="preserve">ſimilibus baſibus conti-<lb/>nentur, centrum grauitatis eſt idem, quod ſphæ-<lb/>ræ centrum.</s> <s xml:id="echoid-s4694" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4695" xml:space="preserve">Solida eiuſmodi corpora regularia appellare ſolent, de <lb/>quibus agitur in tribus ultimis libris elementorum: </s> <s xml:id="echoid-s4696" xml:space="preserve">ſunt <lb/>autem numero quinque, tetrahedrum, uel pyramis, hexa-<lb/>hedrum, uel cubus, octahedrum, dodecahedrum, & </s> <s xml:id="echoid-s4697" xml:space="preserve">icoſa-<lb/>hedrum.</s> <s xml:id="echoid-s4698" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4699" xml:space="preserve">Sit primo a b c d pyramis ĩ ſphæra deſcripta, cuíus ſphæ <lb/>ræ centrum ſit e. </s> <s xml:id="echoid-s4700" xml:space="preserve">Dico e pyramidis a b c d grauitatis eſſe <lb/>centrum. </s> <s xml:id="echoid-s4701" xml:space="preserve">Si enim iuncta d e producatur ad baſim a b c in <lb/>f; </s> <s xml:id="echoid-s4702" xml:space="preserve">ex iis, quæ demonſtrauit Campanus in quartodecimo li <lb/>bro elementorum, propoſitione decima quinta, & </s> <s xml:id="echoid-s4703" xml:space="preserve">decima <lb/>ſeptima, erit f centrum circuli circa triangulum a b c de-<lb/>ſcripti: </s> <s xml:id="echoid-s4704" xml:space="preserve">atque erit e f ſexta pars ipſius ſphæræ axis. </s> <s xml:id="echoid-s4705" xml:space="preserve">quare <lb/>ex prima huius conſtat trianguli a b c grauitatis centrum <lb/>eſſe punctum f: </s> <s xml:id="echoid-s4706" xml:space="preserve">& </s> <s xml:id="echoid-s4707" xml:space="preserve">idcirco lineam d f eſſe pyramidis axem.</s> <s xml:id="echoid-s4708" xml:space="preserve"> <pb file="0188" n="188" rhead="FED. COMMANDINI"/> At cum e f ſit ſexta pars axis <lb/> <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> ſphæræ, crit d e tripla e f. </s> <s xml:id="echoid-s4709" xml:space="preserve">ergo <lb/>punctum e eſt grauitatis cen-<lb/>trum ipſius pyramidis: </s> <s xml:id="echoid-s4710" xml:space="preserve">quod <lb/>in uigeſima ſecunda huius de-<lb/>monſtratum fuit. </s> <s xml:id="echoid-s4711" xml:space="preserve">Sed e eſt cen <lb/>trum ſphæræ. </s> <s xml:id="echoid-s4712" xml:space="preserve">Sequitur igitur, <lb/>ut centrum grauitatis pyrami-<lb/>dis in ſphæra deſcriptæ idem <lb/>ſit, quod ipſius ſphæræ cen-<lb/>trum.</s> <s xml:id="echoid-s4713" xml:space="preserve"/> </p> <div xml:id="echoid-div272" type="float" level="2" n="1"> <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a"> <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0188-01"/> </figure> </div> <p> <s xml:id="echoid-s4714" xml:space="preserve">Sit cubus in ſphæra deſcriptus a b, & </s> <s xml:id="echoid-s4715" xml:space="preserve">oppoſitorum pla-<lb/>norum lateribus bifariam diuiſis, per puncta diuiſionum <lb/>plana ducantur, ut communis ipſorum ſectio ſit recta li-<lb/>nea c d. </s> <s xml:id="echoid-s4716" xml:space="preserve">Itaque ſi ducatur a b, ſolidi ſcilicet diameter, lineæ <lb/>a b, c d ex trigeſimanona undecimi ſeſe bifariam ſecabunt. <lb/></s> <s xml:id="echoid-s4717" xml:space="preserve">ſecent autem in puncto e. </s> <s xml:id="echoid-s4718" xml:space="preserve">erit <lb/> <anchor type="figure" xlink:label="fig-0188-02a" xlink:href="fig-0188-02"/> e centrũ grauitatis ſolidi a b, <lb/>id quod demonſtratum eſt in <lb/>octaua huius. </s> <s xml:id="echoid-s4719" xml:space="preserve">Sed quoniam ab <lb/>eſt ſphæræ diametro æqualis, <lb/>ut in decima quinta propoſi-<lb/>tione tertii decimi libri elemẽ <lb/>torum oſtenditur: </s> <s xml:id="echoid-s4720" xml:space="preserve">punctum e <lb/>ſphæræ quoque centrum erit. <lb/></s> <s xml:id="echoid-s4721" xml:space="preserve">Cubi igitur in ſphæra deſcri-<lb/>pti grauitatis centrum idem <lb/>eſt, quod centrum ipſius ſphæræ.</s> <s xml:id="echoid-s4722" xml:space="preserve"/> </p> <div xml:id="echoid-div273" type="float" level="2" n="2"> <figure xlink:label="fig-0188-02" xlink:href="fig-0188-02a"> <image file="0188-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0188-02"/> </figure> </div> <p> <s xml:id="echoid-s4723" xml:space="preserve">Sit octahedrum a b c d e f, in ſphæra deſcriptum, cuius <lb/>ſphæræ centrum ſit g. </s> <s xml:id="echoid-s4724" xml:space="preserve">Dico punctum g ipſius octahedri <lb/>grauitatis centrum eſſe. </s> <s xml:id="echoid-s4725" xml:space="preserve">Conſtat enim ex iis, quæ demon-<lb/>ſtrata ſunt à Campano in quinto decimo libro elemento-<lb/>rum, propoſitione ſextadecima eiuſimodi ſolidum diuidi <lb/>in duas pyramides æquales, & </s> <s xml:id="echoid-s4726" xml:space="preserve">ſimiles; </s> <s xml:id="echoid-s4727" xml:space="preserve">uidelicetin pyrami- <pb o="39" file="0189" n="189" rhead="DE CENTRO GRAVIT. SOLID."/> dem, cuius baſis eſt quadratum a b c d, & </s> <s xml:id="echoid-s4728" xml:space="preserve">altitudo e g: </s> <s xml:id="echoid-s4729" xml:space="preserve">& </s> <s xml:id="echoid-s4730" xml:space="preserve"><lb/>in pyramidem, cuius eadé baſis, altitudoq; </s> <s xml:id="echoid-s4731" xml:space="preserve">f g; </s> <s xml:id="echoid-s4732" xml:space="preserve">ut ſint e g, <lb/>g f ſemidiametri ſphæræ, & </s> <s xml:id="echoid-s4733" xml:space="preserve">linea una. </s> <s xml:id="echoid-s4734" xml:space="preserve">Cũigitur g ſit ſphæ-<lb/>ræ centrum, erit etiam centrum circuli, qui circa quadratũ <lb/>a b c d deſcribitur: </s> <s xml:id="echoid-s4735" xml:space="preserve">& </s> <s xml:id="echoid-s4736" xml:space="preserve">propterea eiuſdem quadrati grauita <lb/>tis centrum: </s> <s xml:id="echoid-s4737" xml:space="preserve">quod in prima propoſitione huius demon-<lb/>ſtratum eſt. </s> <s xml:id="echoid-s4738" xml:space="preserve">quare pyramidis a b c d e axis erit e g: </s> <s xml:id="echoid-s4739" xml:space="preserve">& </s> <s xml:id="echoid-s4740" xml:space="preserve">pyra <lb/>midis a b c d f axis f g. </s> <s xml:id="echoid-s4741" xml:space="preserve">Itaque ſit h centrum grauitatis py-<lb/>ramidis a b c d e, & </s> <s xml:id="echoid-s4742" xml:space="preserve">pyramidis a b c d f centrum ſit _K_: </s> <s xml:id="echoid-s4743" xml:space="preserve">per-<lb/>ſpicuum eſt ex uigeſima ſecunda propoſitione huius, lineã <lb/>e h triplam eſſe h g: </s> <s xml:id="echoid-s4744" xml:space="preserve">cõ <lb/> <anchor type="figure" xlink:label="fig-0189-01a" xlink:href="fig-0189-01"/> ponendoq; </s> <s xml:id="echoid-s4745" xml:space="preserve">e g ipſius g <lb/>h quadruplam. </s> <s xml:id="echoid-s4746" xml:space="preserve">& </s> <s xml:id="echoid-s4747" xml:space="preserve">eadẽ <lb/>ratione f g quadruplã <lb/>ipſius g k. </s> <s xml:id="echoid-s4748" xml:space="preserve">quod cum e <lb/>g, g f ſintæquales, & </s> <s xml:id="echoid-s4749" xml:space="preserve">h <lb/>g, g _k_ neceſſario æqua-<lb/>les erunt. </s> <s xml:id="echoid-s4750" xml:space="preserve">ergo ex quar <lb/>ta propoſitione primi <lb/>libri Archimedis de cẽ-<lb/>tro grauitatis planorũ, <lb/>totius octahedri, quod <lb/>ex dictis pyramidibus <lb/>conſtat, centrum graui <lb/>tatis erit punctum g idem, quodipſius ſphæræ centrum.</s> <s xml:id="echoid-s4751" xml:space="preserve"/> </p> <div xml:id="echoid-div274" type="float" level="2" n="3"> <figure xlink:label="fig-0189-01" xlink:href="fig-0189-01a"> <image file="0189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0189-01"/> </figure> </div> <p> <s xml:id="echoid-s4752" xml:space="preserve">Sit icoſahedrum a d deſcriptum in ſphæra, cuius centrū <lb/>ſit g. </s> <s xml:id="echoid-s4753" xml:space="preserve">Dico g ipſius icoſahedri grauitatis eſſe centrum. </s> <s xml:id="echoid-s4754" xml:space="preserve">Si <lb/>enim ab angnlo a per g ducatur rectalinea uſque ad ſphæ <lb/>ræ ſuperficiem; </s> <s xml:id="echoid-s4755" xml:space="preserve">conſtat ex ſexta decima propoſitione libri <lb/>tertii decimi elementorum, cadere eam in angulum ipſi a <lb/>oppoſitum. </s> <s xml:id="echoid-s4756" xml:space="preserve">cadat in d: </s> <s xml:id="echoid-s4757" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s4758" xml:space="preserve">una aliqua baſis icoſahedri tri-<lb/>angulum a b c: </s> <s xml:id="echoid-s4759" xml:space="preserve">& </s> <s xml:id="echoid-s4760" xml:space="preserve">iunctæ b g, c g producantur, & </s> <s xml:id="echoid-s4761" xml:space="preserve">cadant in <lb/>angulos e f, ipſis b c oppoſitos. </s> <s xml:id="echoid-s4762" xml:space="preserve">Itaque per triangula <lb/>a b c, d e f ducantur plana ſphæram ſecantia. </s> <s xml:id="echoid-s4763" xml:space="preserve">erunt hæ ſe- <pb file="0190" n="190" rhead="FED. COMMANDINI"/> ctiones circuli ex prima propofitione ſphæricorum Theo <lb/>doſii: </s> <s xml:id="echoid-s4764" xml:space="preserve">unus quidem circa triangulum a b c deſcriptus: </s> <s xml:id="echoid-s4765" xml:space="preserve">al-<lb/>ter uero circa d e f: </s> <s xml:id="echoid-s4766" xml:space="preserve">& </s> <s xml:id="echoid-s4767" xml:space="preserve">quoniam triangula a b c, d e f æqua-<lb/>lia ſunt, & </s> <s xml:id="echoid-s4768" xml:space="preserve">ſimilia; </s> <s xml:id="echoid-s4769" xml:space="preserve">erunt ex prima, & </s> <s xml:id="echoid-s4770" xml:space="preserve">ſecunda propoſitione <lb/>duodecimi libri elementorum, circuli quoque inter ſe ſe <lb/>æquales. </s> <s xml:id="echoid-s4771" xml:space="preserve">poſtremo a centro g ad circulum a b c perpendi <lb/>cularis ducatur g h; </s> <s xml:id="echoid-s4772" xml:space="preserve">& </s> <s xml:id="echoid-s4773" xml:space="preserve">alia perpendicularis ducatur ad cir <lb/>culum d e f, quæ ſit g _k_; </s> <s xml:id="echoid-s4774" xml:space="preserve">& </s> <s xml:id="echoid-s4775" xml:space="preserve">iungantur a h, d k. </s> <s xml:id="echoid-s4776" xml:space="preserve">perſpicuum <lb/>eſt ex corollario primæ ſphæricorum Theodoſii, punctum <lb/>h centrum eſſe circuli a b c, & </s> <s xml:id="echoid-s4777" xml:space="preserve">k centrum circuli d e f. </s> <s xml:id="echoid-s4778" xml:space="preserve">Quo <lb/>niam igitur triangulorum g a h, g d K latus a g eſt æquale la <lb/>teri g d; </s> <s xml:id="echoid-s4779" xml:space="preserve">ſunt enim à centro ſphæræ ad ſuperficiem: </s> <s xml:id="echoid-s4780" xml:space="preserve">atque <lb/>eſt a h æquale d k: </s> <s xml:id="echoid-s4781" xml:space="preserve">& </s> <s xml:id="echoid-s4782" xml:space="preserve">ex ſexta propoſitione libri primi ſphæ <lb/>ricorum Theodoſii g h ipſi g K: </s> <s xml:id="echoid-s4783" xml:space="preserve">triangulum g a h æquale <lb/>erit, & </s> <s xml:id="echoid-s4784" xml:space="preserve">ſimile g d k triangulo: </s> <s xml:id="echoid-s4785" xml:space="preserve">& </s> <s xml:id="echoid-s4786" xml:space="preserve">angulus a g h æqualis an-<lb/>gulo d g _K_. </s> <s xml:id="echoid-s4787" xml:space="preserve">ſed anguli a g h, h g d ſunt æquales duobus re-<lb/> <anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/> ctis. </s> <s xml:id="echoid-s4788" xml:space="preserve">ergo & </s> <s xml:id="echoid-s4789" xml:space="preserve">ipſi h g d, d g k duobus rectis æquales erunt. <lb/></s> <s xml:id="echoid-s4790" xml:space="preserve">& </s> <s xml:id="echoid-s4791" xml:space="preserve">idcirco h g, g _K_ una, atque eadem erit linea. </s> <s xml:id="echoid-s4792" xml:space="preserve">cum autem <lb/> <anchor type="note" xlink:label="note-0190-02a" xlink:href="note-0190-02"/> h ſit centrũ circuli, & </s> <s xml:id="echoid-s4793" xml:space="preserve">tri-<lb/> <anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/> anguli a b c grauitatis cen <lb/>trũ probabitur ex iis, quæ <lb/>in prima propoſitione hu <lb/>ius tradita funt. </s> <s xml:id="echoid-s4794" xml:space="preserve">quare g h <lb/>erit pyramidis a b c g axis. <lb/></s> <s xml:id="echoid-s4795" xml:space="preserve">& </s> <s xml:id="echoid-s4796" xml:space="preserve">ob eandem cauſſam g k <lb/>axis pyramidis d e f g. </s> <s xml:id="echoid-s4797" xml:space="preserve">Ita-<lb/>que centrum grauitatis py <lb/>ramidis a b c g ſit púctum <lb/>l, & </s> <s xml:id="echoid-s4798" xml:space="preserve">pyramidis d e f g ſit m. </s> <s xml:id="echoid-s4799" xml:space="preserve"><lb/>Similiter ut ſupra demon-<lb/>ſtrabimus m g, g linter ſe æquales eſſe, & </s> <s xml:id="echoid-s4800" xml:space="preserve">punctum g graui <lb/>tatis centrum magnitudinis, quæ ex utriſque pyramidibus <lb/>conſtat. </s> <s xml:id="echoid-s4801" xml:space="preserve">eodem modo demonſtrabitur, quarumcunque <lb/>duarum pyramidum, quæ opponuntur, grauitatis centrũ <pb o="40" file="0191" n="191" rhead="DE CENTRO GRAVIT. SOLID."/> eſſe pun ctum g. </s> <s xml:id="echoid-s4802" xml:space="preserve">Sequitur ergo uticoſahedri centrum gra <lb/>uitatis fit idem, quodipſius ſphæræ centrum.</s> <s xml:id="echoid-s4803" xml:space="preserve"/> </p> <div xml:id="echoid-div275" type="float" level="2" n="4"> <note position="left" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve">13. primi</note> <note position="left" xlink:label="note-0190-02" xlink:href="note-0190-02a" xml:space="preserve">14. primi</note> <figure xlink:label="fig-0190-01" xlink:href="fig-0190-01a"> <image file="0190-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0190-01"/> </figure> </div> <p> <s xml:id="echoid-s4804" xml:space="preserve">Sit dodecahedrum a ſin ſphæra deſignatum, ſitque ſphæ <lb/>ræ centrum m. </s> <s xml:id="echoid-s4805" xml:space="preserve">Dico m centrum eſſe grauitatis ipſius do-<lb/>decahedri. </s> <s xml:id="echoid-s4806" xml:space="preserve">Sit enim pentagonum a b c d e una ex duode-<lb/>cim baſibus ſolidi a f: </s> <s xml:id="echoid-s4807" xml:space="preserve">& </s> <s xml:id="echoid-s4808" xml:space="preserve">iuncta a m producatur ad ſphæræ <lb/>ſuperficiem. </s> <s xml:id="echoid-s4809" xml:space="preserve">cadetin angulum ipſi a oppoſitum; </s> <s xml:id="echoid-s4810" xml:space="preserve">quod col-<lb/>ligitur ex decima ſeptima propoſitione tertiidecimilibri <lb/>elementorum. </s> <s xml:id="echoid-s4811" xml:space="preserve">cadat in f. </s> <s xml:id="echoid-s4812" xml:space="preserve">at ſi ab aliis angulis b c d e per cẽ <lb/>trum itidem lineæ ducantur ad ſuperficiem ſphæræ in pun <lb/>cta g h k l; </s> <s xml:id="echoid-s4813" xml:space="preserve">cadent hæ in alios angulos baſis, quæ ipſi a b c d <lb/>baſi opponitur. </s> <s xml:id="echoid-s4814" xml:space="preserve">tranſeant ergo per pentagona a b c d e, <lb/>f g h K l plana ſphæram ſecantia, quæ facient ſectiones cir-<lb/>culos æquales inter ſe ſe poſtea ducantur ex centro ſphæræ <lb/>m perpen diculares ad pla-<lb/> <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> na dictorum circulorũ; </s> <s xml:id="echoid-s4815" xml:space="preserve">ad <lb/>circulum quidem a b c d e <lb/>perpendicularis m n: </s> <s xml:id="echoid-s4816" xml:space="preserve">& </s> <s xml:id="echoid-s4817" xml:space="preserve">ad <lb/>circulum f g h K l ipſa m o, <lb/> <anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/> erunt puncta n o circulorũ <lb/>centra: </s> <s xml:id="echoid-s4818" xml:space="preserve">& </s> <s xml:id="echoid-s4819" xml:space="preserve">lineæ m n, m o in <lb/>ter ſe æquales: </s> <s xml:id="echoid-s4820" xml:space="preserve">quòd circu-<lb/>li æquales ſint. </s> <s xml:id="echoid-s4821" xml:space="preserve">Eodem mo <lb/> <anchor type="note" xlink:label="note-0191-02a" xlink:href="note-0191-02"/> do, quo ſupra, demonſtrabi <lb/>mus lineas m n, m o in unã <lb/>atque eandem lineam con-<lb/>uenire. </s> <s xml:id="echoid-s4822" xml:space="preserve">ergo cum puncta n o ſint centra circulorum, con-<lb/>ſtat ex prima huius & </s> <s xml:id="echoid-s4823" xml:space="preserve">pentagonorũ grauitatis eſſe centra: <lb/></s> <s xml:id="echoid-s4824" xml:space="preserve">idcircoq; </s> <s xml:id="echoid-s4825" xml:space="preserve">m n, m o pyramidum a b c d e m, ſ g h _K_ l m axes. </s> <s xml:id="echoid-s4826" xml:space="preserve"><lb/>ponatur a b c d e m pyramidis grauitatis centrum p: </s> <s xml:id="echoid-s4827" xml:space="preserve">& </s> <s xml:id="echoid-s4828" xml:space="preserve">py <lb/>ramidis f g h <emph style="sc">K</emph> l m ipſum q centrum. </s> <s xml:id="echoid-s4829" xml:space="preserve">erunt p m, m q æqua-<lb/>les, & </s> <s xml:id="echoid-s4830" xml:space="preserve">punctum m grauitatis centrum magnitudinis, quæ <lb/>ex ipſis pyramidibus conſtat. </s> <s xml:id="echoid-s4831" xml:space="preserve">eodẽ modo probabitur qua-<lb/>rumlibet pyramidum, quæ è regione opponuntur, centrũ <pb file="0192" n="192" rhead="FED. COMMANDINI"/> grauitatis eſſe punctum m. </s> <s xml:id="echoid-s4832" xml:space="preserve">patetigitur totius dodecahe-<lb/>dri, centrum grauitatis idẽ eſſe, quod & </s> <s xml:id="echoid-s4833" xml:space="preserve">ſphæræ ipſum com <lb/>prehendentis centrum. </s> <s xml:id="echoid-s4834" xml:space="preserve">quæ quidem omnia demonſtraſſe <lb/>oportebat.</s> <s xml:id="echoid-s4835" xml:space="preserve"/> </p> <div xml:id="echoid-div276" type="float" level="2" n="5"> <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a"> <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0191-01"/> </figure> <note position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">corol. pri <lb/>mæ ſphæ <lb/>ricorum <lb/>Theod.</note> <note position="right" xlink:label="note-0191-02" xlink:href="note-0191-02a" xml:space="preserve">6. primi <lb/>phærico <lb/>rum.</note> </div> </div> <div xml:id="echoid-div278" type="section" level="1" n="93"> <head xml:id="echoid-head100" xml:space="preserve">PROBLEMA VI. PROPOSITIO XX VIII.</head> <p> <s xml:id="echoid-s4836" xml:space="preserve"><emph style="sc">Data</emph> qualibet portione conoidis rectangu <lb/>li, abſciſſa plano ad axem recto, uel non recto; </s> <s xml:id="echoid-s4837" xml:space="preserve">fie-<lb/>ri poteſt, ut portio ſolida inſcribatur, uel circum-<lb/>ſcribatur ex cylindris, uel cylindri portionibus, <lb/>æqualem habentibus altitudinem, ita ut recta li-<lb/>nea, quæ inter centrum grauitatis portionis, & </s> <s xml:id="echoid-s4838" xml:space="preserve"><lb/>figuræ inſcriptæ, uel circumſcriptæ interiicitur, <lb/>ſit minor qualibet recta linea propoſita.</s> <s xml:id="echoid-s4839" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4840" xml:space="preserve">Sit portio conoidis rectanguli a b c, cuius axis b d, gra-<lb/>uitatisq; </s> <s xml:id="echoid-s4841" xml:space="preserve">centrum e: </s> <s xml:id="echoid-s4842" xml:space="preserve">& </s> <s xml:id="echoid-s4843" xml:space="preserve">fit g recta linea propoſita. </s> <s xml:id="echoid-s4844" xml:space="preserve">quam ue <lb/>ro proportionem habet linea b e ad lineam g, eandem ha-<lb/>beat portio conoidis ad ſolidum h: </s> <s xml:id="echoid-s4845" xml:space="preserve">& </s> <s xml:id="echoid-s4846" xml:space="preserve">circumſcribatur por <lb/>tioni figura, ſicuti dictum eſt, ita ut portiones reliquæ ſint <lb/>ſolido h minores: </s> <s xml:id="echoid-s4847" xml:space="preserve">cuius quidem figuræ centrum grauitatis <lb/>ſit punctum <emph style="sc">K</emph>. </s> <s xml:id="echoid-s4848" xml:space="preserve">Dico lineã k e minorem eſſe linea g propo-<lb/>ſita. </s> <s xml:id="echoid-s4849" xml:space="preserve">niſi enim ſit minor, uel æqualis, uel maior erit. </s> <s xml:id="echoid-s4850" xml:space="preserve">& </s> <s xml:id="echoid-s4851" xml:space="preserve">quo-<lb/>niam figura circumſcripta ad reliquas portiones maiorem <lb/> <anchor type="note" xlink:label="note-0192-01a" xlink:href="note-0192-01"/> proportionem habet, quàm portio conoidis ad ſolidum h; <lb/></s> <s xml:id="echoid-s4852" xml:space="preserve">hoc eſt maiorem, quàm b c ad g: </s> <s xml:id="echoid-s4853" xml:space="preserve">& </s> <s xml:id="echoid-s4854" xml:space="preserve">b e ad g non minorem <lb/>habet proportionem, quàm ad _k_ e, propterea quod k e non <lb/>ponitur minor ipſa g: </s> <s xml:id="echoid-s4855" xml:space="preserve">habebit figura circumſcripta ad por <lb/>tiones reliquas maiorem proportionem quàm b e ad e k: </s> <s xml:id="echoid-s4856" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0192-02a" xlink:href="note-0192-02"/> & </s> <s xml:id="echoid-s4857" xml:space="preserve">diuidendo portio conoidis ad reliquas portiones habe-<lb/>bit maiorem, quàm b <emph style="sc">K</emph> ad K e. </s> <s xml:id="echoid-s4858" xml:space="preserve">quare ſi fiat ut portio co- <pb o="41" file="0193" n="193" rhead="DE CENTRO GRAVIT. SOLID."/> noidis ad portiones reliquas, ita alia linea, quæ ſit 1 <emph style="sc">K</emph> ad <lb/>k e: </s> <s xml:id="echoid-s4859" xml:space="preserve">erit 1k maior, quam b k: </s> <s xml:id="echoid-s4860" xml:space="preserve">& </s> <s xml:id="echoid-s4861" xml:space="preserve">ideo punctum l extra por-<lb/>tionem cadet. </s> <s xml:id="echoid-s4862" xml:space="preserve">Quoniã <lb/> <anchor type="figure" xlink:label="fig-0193-01a" xlink:href="fig-0193-01"/> igitur à figura circum-<lb/>ſcripta, cuius grauitatis <lb/>centrum eſt k, aufertur <lb/>portio conoidis, cuius <lb/>centrum e. </s> <s xml:id="echoid-s4863" xml:space="preserve">habetq; </s> <s xml:id="echoid-s4864" xml:space="preserve">l K <lb/>ad K e eam proportio-<lb/>nem, quam portio co-<lb/>noidis ad reliquas por-<lb/>tiones; </s> <s xml:id="echoid-s4865" xml:space="preserve">erit punctum l <lb/>extra portionem cadẽs, <lb/>centrum magnitudinis <lb/>ex reliquis portionibus compoſitæ. </s> <s xml:id="echoid-s4866" xml:space="preserve">illud autem fieri nullo <lb/>modo poteſt. </s> <s xml:id="echoid-s4867" xml:space="preserve">quare conſtat lineam k e ipſa g linea propoſi <lb/>ta minorem eſſe.</s> <s xml:id="echoid-s4868" xml:space="preserve"/> </p> <div xml:id="echoid-div278" type="float" level="2" n="1"> <note position="left" xlink:label="note-0192-01" xlink:href="note-0192-01a" xml:space="preserve">8. quĭnti.</note> <note position="left" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">29. quĭnti <lb/>ex tradi-<lb/>tione Cã-<lb/>ſàni.</note> <figure xlink:label="fig-0193-01" xlink:href="fig-0193-01a"> <image file="0193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0193-01"/> </figure> </div> <p> <s xml:id="echoid-s4869" xml:space="preserve">Rurfus inſcribatur portioni figura, uidelicet cylindr us <lb/>m n, ut ſit ipſius altitudo <lb/> <anchor type="figure" xlink:label="fig-0193-02a" xlink:href="fig-0193-02"/> æqualis dimidio axis b d: <lb/></s> <s xml:id="echoid-s4870" xml:space="preserve">& </s> <s xml:id="echoid-s4871" xml:space="preserve">quam proportionem <lb/>habet b e ad g, habeat m n <lb/>cylindrus ad ſolidum o. </s> <s xml:id="echoid-s4872" xml:space="preserve"><lb/>inſcrib itur deinde eidem <lb/>alia figura, ita ut portio-<lb/>nes reliquæ ſint ſolido o <lb/>minores: </s> <s xml:id="echoid-s4873" xml:space="preserve">& </s> <s xml:id="echoid-s4874" xml:space="preserve">centrum gra <lb/>uitatis figuræ ſit p. </s> <s xml:id="echoid-s4875" xml:space="preserve">Dico <lb/>lineam p e ipſa g minorẽ <lb/>eſſe. </s> <s xml:id="echoid-s4876" xml:space="preserve">ſi enim non ſit mi-<lb/>nor, eodem, quo ſupra modo demonſtrabimus figuram in <lb/>ſcriptam ad reliquas portiones maiorem proportionem <lb/>habere, quàm b e ad e p. </s> <s xml:id="echoid-s4877" xml:space="preserve">& </s> <s xml:id="echoid-s4878" xml:space="preserve">ſi fiat alia linea l e ad e p, ut eſt <lb/>figura inſcripta ad reliquas portiones, pũctum l extra por <pb file="0194" n="194" rhead="FED. COMMANDINI"/> tionem cadet: </s> <s xml:id="echoid-s4879" xml:space="preserve">Itaque cum à portione conoidis, cuius gra-<lb/>uitatis centrum e auferatur inſcripta figura, centrum ha-<lb/>bens p: </s> <s xml:id="echoid-s4880" xml:space="preserve">& </s> <s xml:id="echoid-s4881" xml:space="preserve">ſit l e ad e p, ut figura inſcripta ad portiones reli <lb/>quas: </s> <s xml:id="echoid-s4882" xml:space="preserve">erit magnitudinis, quæ ex reliquis portionibus con <lb/>ſtat, centrum grauitatis punctum l, extra portionem ca-<lb/>dens. </s> <s xml:id="echoid-s4883" xml:space="preserve">quod fieri nequit. </s> <s xml:id="echoid-s4884" xml:space="preserve">ergo linea p e minor eſt ip ſa g li-<lb/>nea propoſita.</s> <s xml:id="echoid-s4885" xml:space="preserve"/> </p> <div xml:id="echoid-div279" type="float" level="2" n="2"> <figure xlink:label="fig-0193-02" xlink:href="fig-0193-02a"> <image file="0193-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0193-02"/> </figure> </div> <p> <s xml:id="echoid-s4886" xml:space="preserve">Ex quibus perſpicuum eſt centrum grauitatis <lb/>figuræ inſcriptæ, & </s> <s xml:id="echoid-s4887" xml:space="preserve">circumſcriptæ eo magis acce <lb/>dere ad portionis centrum, quo pluribus cylin-<lb/>dris, uel cylindri portionibus conſtet: </s> <s xml:id="echoid-s4888" xml:space="preserve">fiatq́ figu <lb/>ra inſcripta maior, & </s> <s xml:id="echoid-s4889" xml:space="preserve">circumſcripta minor. </s> <s xml:id="echoid-s4890" xml:space="preserve">& </s> <s xml:id="echoid-s4891" xml:space="preserve"><lb/>quanquam continenter ad portionis centrū pro-<lb/>pius admoueatur nunquam tamen ad ipſum per <lb/>ueniet. </s> <s xml:id="echoid-s4892" xml:space="preserve">ſequeretur enim figuram inſcriptam, nó <lb/>ſolum portioni, ſed etiam circumſcriptæ figuræ <lb/>æqualem eſſe. </s> <s xml:id="echoid-s4893" xml:space="preserve">quod eſt abſurdum.</s> <s xml:id="echoid-s4894" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div281" type="section" level="1" n="94"> <head xml:id="echoid-head101" xml:space="preserve">THE OREMA XXIII. PROPOSITIO XXIX.</head> <p> <s xml:id="echoid-s4895" xml:space="preserve"><emph style="sc">Cvivslibet</emph> portionis conoidis rectangu-<lb/>li axis à cẽtro grauitatis ita diuiditur, ut pars quæ <lb/>terminatur ad uerticem, reliquæ partis, quæ ad ba <lb/>ſim ſit dupla.</s> <s xml:id="echoid-s4896" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4897" xml:space="preserve">SIT portio conoidis rectanguli uel abſciſſa plano ad <lb/>axem recto, uel non recto: </s> <s xml:id="echoid-s4898" xml:space="preserve">& </s> <s xml:id="echoid-s4899" xml:space="preserve">ſecta ipſa altero plano per axé <lb/>ſit ſuperſiciei ſe ctio a b c r ectanguli coni ſectio, uel parabo <lb/>le; </s> <s xml:id="echoid-s4900" xml:space="preserve">plani abſcindentis portionem ſectio ſit recta linea a c: <lb/></s> <s xml:id="echoid-s4901" xml:space="preserve">axis portionis, & </s> <s xml:id="echoid-s4902" xml:space="preserve">ſectionis diameter b d. </s> <s xml:id="echoid-s4903" xml:space="preserve">Sumatur autem <lb/>in linea b d punctum e, ita ut b e ſit ipſius e d dupla. </s> <s xml:id="echoid-s4904" xml:space="preserve">Dico <pb o="42" file="0195" n="195" rhead="DE CENTRO GRAVIT. SOLID."/> e portionis a b <lb/> <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> c grauitatis eſſe <lb/>centrum. </s> <s xml:id="echoid-s4905" xml:space="preserve">Diui-<lb/>datur enim b d <lb/>bifariam in m: <lb/></s> <s xml:id="echoid-s4906" xml:space="preserve">&</s> <s xml:id="echoid-s4907" xml:space="preserve">rurſus d m, m <lb/>b bifariam diui-<lb/>dantur in pun-<lb/>ctis n, o: </s> <s xml:id="echoid-s4908" xml:space="preserve">inſcri-<lb/>baturq; </s> <s xml:id="echoid-s4909" xml:space="preserve">portio-<lb/>ni figura ſolida, <lb/>& </s> <s xml:id="echoid-s4910" xml:space="preserve">altera circum <lb/>ſcribatur ex cy-<lb/>lindris æqualem <lb/>altitudinem ha-<lb/>bentibus, utſu-<lb/>perius dictũ eſt. </s> <s xml:id="echoid-s4911" xml:space="preserve"><lb/>Sit autem pri-<lb/>mum figura in-<lb/>ſcripta cylĩ drus <lb/>f g: </s> <s xml:id="echoid-s4912" xml:space="preserve">& </s> <s xml:id="echoid-s4913" xml:space="preserve">circũſcri-<lb/>pta ex cylindris <lb/>a h, K l conſtet. </s> <s xml:id="echoid-s4914" xml:space="preserve"><lb/>punctum n erit <lb/> <anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/> centrum graui-<lb/>tatis figuræ in-<lb/>fcriptæ, mediũ <lb/>ſcilicet ipſius d <lb/>m axis: </s> <s xml:id="echoid-s4915" xml:space="preserve">atq; </s> <s xml:id="echoid-s4916" xml:space="preserve">idẽ <lb/>erit centrum cy <lb/>lindri a h: </s> <s xml:id="echoid-s4917" xml:space="preserve">& </s> <s xml:id="echoid-s4918" xml:space="preserve">cy-<lb/>lindri <emph style="sc">K</emph> l centrũ <lb/>o, axis b m me-<lb/>dium, quare ſi li <pb file="0196" n="196" rhead="FED. COMMANDINI"/> neam on ita di <lb/> <anchor type="figure" xlink:label="fig-0196-01a" xlink:href="fig-0196-01"/> uiſerimus in p, <lb/>ut quã propor-<lb/>tionẽ habet cy-<lb/>lindrus a h ad <lb/>cylindrum <emph style="sc">K</emph> l, <lb/>habeat linea o p <lb/>ad p n: </s> <s xml:id="echoid-s4919" xml:space="preserve">centrum <lb/> <anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/> grauitatis toti-<lb/>us figuræ circũ-<lb/>ſcriptæ erit pun <lb/>ctum p. </s> <s xml:id="echoid-s4920" xml:space="preserve">Sed cy-<lb/> <anchor type="note" xlink:label="note-0196-02a" xlink:href="note-0196-02"/> lindri, qui ſunt <lb/>æquali altitudi-<lb/>ne, candem in-<lb/>ter ſe ſe, quam <lb/>baſes propor-<lb/>tionem habent: <lb/></s> <s xml:id="echoid-s4921" xml:space="preserve">eſtq; </s> <s xml:id="echoid-s4922" xml:space="preserve">ut linea d b <lb/>ad b m, ita qua-<lb/>dratũ lineæ a d <lb/>ad quadratũ ip-<lb/>ſius _K_ in, ex uige <lb/>ſima primi libri <lb/>conicorũ: </s> <s xml:id="echoid-s4923" xml:space="preserve">& </s> <s xml:id="echoid-s4924" xml:space="preserve">ita <lb/> <anchor type="note" xlink:label="note-0196-03a" xlink:href="note-0196-03"/> quadratum a c <lb/>ad quadratũ K <lb/>g: </s> <s xml:id="echoid-s4925" xml:space="preserve">hoc eſt circu-<lb/> <anchor type="note" xlink:label="note-0196-04a" xlink:href="note-0196-04"/> lus circa diame <lb/>trum a c ad cir-<lb/>culum circa dia <lb/>metrum k g. </s> <s xml:id="echoid-s4926" xml:space="preserve">du <lb/>pla eſt autem li-<lb/>nea d b lineæ <pb o="43" file="0197" n="197" rhead="DE CENTRO GRAVIT. SOLID."/> b m. </s> <s xml:id="echoid-s4927" xml:space="preserve">ergo circulus a c circuli _k_ g: </s> <s xml:id="echoid-s4928" xml:space="preserve">& </s> <s xml:id="echoid-s4929" xml:space="preserve">idcirco cylindrus <lb/>a h cylindri _k_ l duplus erit. </s> <s xml:id="echoid-s4930" xml:space="preserve">quare & </s> <s xml:id="echoid-s4931" xml:space="preserve">linea o p dupla <lb/>ipſius p n. </s> <s xml:id="echoid-s4932" xml:space="preserve">Deinde inſcripta & </s> <s xml:id="echoid-s4933" xml:space="preserve">circumſcripta portioni <lb/>alia figura, ita ut inſcripta conſtituatur ex tribus cylin-<lb/>dris q r, s g, tu: </s> <s xml:id="echoid-s4934" xml:space="preserve">circumſcripta uero ex quatuor a x, y z, <lb/>_K_ ν, θ λ: </s> <s xml:id="echoid-s4935" xml:space="preserve">diuidantur b o, o m, m n, n d bifariam in punctis <lb/>μ ν π ρ. </s> <s xml:id="echoid-s4936" xml:space="preserve">Itaque cylindri θ λ centrum grauitætis eſt punctum <lb/>μ: </s> <s xml:id="echoid-s4937" xml:space="preserve">& </s> <s xml:id="echoid-s4938" xml:space="preserve">cylindri <emph style="sc">K</emph> ν centrum ν. </s> <s xml:id="echoid-s4939" xml:space="preserve">ergo ſi linea μ ν diuidatur in σ, <lb/>ita ut μ σ ad σ ν proportionẽ eã habeat, quam cylindrus K ν <lb/>ad cylindrum θ λ, uidelicet quam quadratum <emph style="sc">K</emph> m ad qua-<lb/>dratum θ o, hoc eſt, quam linea m b ad b o: </s> <s xml:id="echoid-s4940" xml:space="preserve">erit σ centrum <lb/> <anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/> magnitudinis compoſitæ ex cylindris <emph style="sc">K</emph> ν, θ λ. </s> <s xml:id="echoid-s4941" xml:space="preserve">& </s> <s xml:id="echoid-s4942" xml:space="preserve">cum linea <lb/>m b ſit dupla b o, erit & </s> <s xml:id="echoid-s4943" xml:space="preserve">μ σ ipſius σ ν dupla. </s> <s xml:id="echoid-s4944" xml:space="preserve">præterea quo-<lb/>niam cylindri y z centrum grauitatis eſt π, linea σ π ita diui <lb/>ſa in τ, ut σ τ ad τ π eam habeat proportionem, quam cylin <lb/>drus y z ad duos cylindros K ν, θ λ: </s> <s xml:id="echoid-s4945" xml:space="preserve">erit τ centrum magnitu <lb/>dinis, quæ ex dictis tribus cylindris conſtat. </s> <s xml:id="echoid-s4946" xml:space="preserve">cylindrus au-<lb/>tẽ y z ad cylindrum θ λ eſt, ut linea n b ad b o, hoc eſt ut 3 <lb/>ad 1: </s> <s xml:id="echoid-s4947" xml:space="preserve">& </s> <s xml:id="echoid-s4948" xml:space="preserve">ad cylindrum k ν, ut n b ad b m, uidelicet ut 3 ad 2. <lb/></s> <s xml:id="echoid-s4949" xml:space="preserve">quare y z cylĩdrus duobus cylindris k ν, θ λ æqualis erit. </s> <s xml:id="echoid-s4950" xml:space="preserve">& </s> <s xml:id="echoid-s4951" xml:space="preserve"><lb/>propterea linea σ τ æqualis ipſi τ π. </s> <s xml:id="echoid-s4952" xml:space="preserve">denique cylindri a x <lb/>centrum grauitatis eſt punctum ρ. </s> <s xml:id="echoid-s4953" xml:space="preserve">& </s> <s xml:id="echoid-s4954" xml:space="preserve">cum τ ζ diuiſa fuerit <lb/>in eã proportionem, quam habet cylindrus a x ad tres cy-<lb/>lindros y z, _k_ ν, θ λ: </s> <s xml:id="echoid-s4955" xml:space="preserve">erit in eo puncto centrum grauitatis <lb/>totius figuræ circũſcriptæ. </s> <s xml:id="echoid-s4956" xml:space="preserve">Sed cylindrus a x ad ipſum y z <lb/>eſt ut linea d b ad b n: </s> <s xml:id="echoid-s4957" xml:space="preserve">hoc eſt ut 4 ad 3: </s> <s xml:id="echoid-s4958" xml:space="preserve">& </s> <s xml:id="echoid-s4959" xml:space="preserve">duo cylindri _k_ ν <lb/>θ λ cylindro y z ſunt æquales. </s> <s xml:id="echoid-s4960" xml:space="preserve">cylindrns igitur a x ad tres <lb/>iam dictos cylindros eſt ut 2 ad 3. </s> <s xml:id="echoid-s4961" xml:space="preserve">Sed quoniã μ σ eſt dua-<lb/>rum partium, & </s> <s xml:id="echoid-s4962" xml:space="preserve">σ ν unius, qualium μ π eſt ſex; </s> <s xml:id="echoid-s4963" xml:space="preserve">erit σ π par-<lb/>tium quatuor: </s> <s xml:id="echoid-s4964" xml:space="preserve">proptereaq; </s> <s xml:id="echoid-s4965" xml:space="preserve">τ π duarum, & </s> <s xml:id="echoid-s4966" xml:space="preserve">ν π, hoc eſt π ρ <lb/>trium. </s> <s xml:id="echoid-s4967" xml:space="preserve">quare ſequitur ut punctum π totius figuræ circum <lb/>ſcriptæ ſit centrum. </s> <s xml:id="echoid-s4968" xml:space="preserve">Itaque fiat ν υ ad υ π, ut μ σ ad σ ν. </s> <s xml:id="echoid-s4969" xml:space="preserve">& </s> <s xml:id="echoid-s4970" xml:space="preserve">υ ρ <lb/>bifariam diuidatur in φ. </s> <s xml:id="echoid-s4971" xml:space="preserve">Similiter ut in circumſcripta figu <lb/>ra oſtendetur centrum magnitudinis compoſitæ ex cylin- <pb file="0198" n="198" rhead="FED. COMMANDINI"/> dris s g, t u eſſe <lb/> <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> punctum υ & </s> <s xml:id="echoid-s4972" xml:space="preserve"><lb/>totius figuræ in <lb/>ſcriptæ, quæ cõ-<lb/>ſtat ex cylindris <lb/>q r, ſ g, t u eſſe φ <lb/>centrum. </s> <s xml:id="echoid-s4973" xml:space="preserve">Sunt <lb/>enim hi cylindri <lb/>æquales & </s> <s xml:id="echoid-s4974" xml:space="preserve">ſimi-<lb/>les cylindris y z, <lb/>K ν, θ λ, figuræ <lb/>circumſcriptæ. <lb/></s> <s xml:id="echoid-s4975" xml:space="preserve">Quoniã igitur <lb/>ut b e ad e d, ita <lb/>eſt o p ad p n; </s> <s xml:id="echoid-s4976" xml:space="preserve"><lb/>utraq; </s> <s xml:id="echoid-s4977" xml:space="preserve">enim u-<lb/>triuſque eſt du-<lb/>pla: </s> <s xml:id="echoid-s4978" xml:space="preserve">erit compo <lb/>nendo, ut b d ad <lb/>d e, ita o n ad n <lb/>p; </s> <s xml:id="echoid-s4979" xml:space="preserve">& </s> <s xml:id="echoid-s4980" xml:space="preserve">permutan <lb/>do, ut b d ad o <lb/>n, ita d e ad n p. </s> <s xml:id="echoid-s4981" xml:space="preserve"><lb/>Sed b d dupla <lb/>eſt o n. </s> <s xml:id="echoid-s4982" xml:space="preserve">ergo & </s> <s xml:id="echoid-s4983" xml:space="preserve"><lb/>e d ipſius n p du <lb/>pla erit. </s> <s xml:id="echoid-s4984" xml:space="preserve">quòd ſi <lb/>e d bifariam di-<lb/>uidatur ĩ χ, erit <lb/>χ d, uel e χ æ-<lb/>qualis n p: </s> <s xml:id="echoid-s4985" xml:space="preserve">& </s> <s xml:id="echoid-s4986" xml:space="preserve"><lb/>ſublata e n, quæ <lb/>eſt cõmunis u-<lb/>trique e χ, p n, <pb o="44" file="0199" n="199" rhead="DE CENTRO GRAVIT. SOLID."/> relinquetur p e ipſi n χ æqualis. </s> <s xml:id="echoid-s4987" xml:space="preserve">cum autem b e ſit dupla <lb/>e d, & </s> <s xml:id="echoid-s4988" xml:space="preserve">o p dupla p n, hoc eſt ipſius e χ, & </s> <s xml:id="echoid-s4989" xml:space="preserve">reliquum, uideli-<lb/>cet b o unà cum p e ipſius reliqui χ d duplnm erit. </s> <s xml:id="echoid-s4990" xml:space="preserve">eſtque <lb/> <anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/> b o dupla ζ d. </s> <s xml:id="echoid-s4991" xml:space="preserve">ergo p e, hoc eſt n χ ipſius χ ρ dupla. </s> <s xml:id="echoid-s4992" xml:space="preserve">ſed d n <lb/>dupla eſt n ζ. </s> <s xml:id="echoid-s4993" xml:space="preserve">reliqua igitur d χ dupla reliquæ χ n. </s> <s xml:id="echoid-s4994" xml:space="preserve">ſunt au-<lb/>tem d χ, p n inter ſe æquales: </s> <s xml:id="echoid-s4995" xml:space="preserve">itemq; </s> <s xml:id="echoid-s4996" xml:space="preserve">æquales χ n, p e. </s> <s xml:id="echoid-s4997" xml:space="preserve">qua-<lb/>re conſtat n p ipſius p e duplam eſſe. </s> <s xml:id="echoid-s4998" xml:space="preserve">& </s> <s xml:id="echoid-s4999" xml:space="preserve">idcirco p e ipſi e n <lb/>æqualem. </s> <s xml:id="echoid-s5000" xml:space="preserve">Rurſus cum ſit μ ν dupla o ν, & </s> <s xml:id="echoid-s5001" xml:space="preserve">μ σ dupla σ ν; </s> <s xml:id="echoid-s5002" xml:space="preserve">erit <lb/>etiam reliqua ν σ o dupla. </s> <s xml:id="echoid-s5003" xml:space="preserve">Eadem quoque ratione <lb/>cõcludetur π υ dupla υ m. </s> <s xml:id="echoid-s5004" xml:space="preserve">ergo ut ν σ ad σ O, ita π υ ad υ m: <lb/></s> <s xml:id="echoid-s5005" xml:space="preserve">componendoq;</s> <s xml:id="echoid-s5006" xml:space="preserve">, & </s> <s xml:id="echoid-s5007" xml:space="preserve">permutando, ut υ o ad π m, ita o σ ad <lb/>m υ & </s> <s xml:id="echoid-s5008" xml:space="preserve">ſunt æquales ν o, π m. </s> <s xml:id="echoid-s5009" xml:space="preserve">quare & </s> <s xml:id="echoid-s5010" xml:space="preserve">o σ, m υ æquales. </s> <s xml:id="echoid-s5011" xml:space="preserve">præ <lb/>terea σ π dupla eſt π τ, & </s> <s xml:id="echoid-s5012" xml:space="preserve">ν π ipſius π m. </s> <s xml:id="echoid-s5013" xml:space="preserve">reliqua igitur σ ν re <lb/>liquæ m τ dupla. </s> <s xml:id="echoid-s5014" xml:space="preserve">atque erat ν σ dupla σ o. </s> <s xml:id="echoid-s5015" xml:space="preserve">ergo m τ, σ o æ-<lb/>quales ſunt: </s> <s xml:id="echoid-s5016" xml:space="preserve">& </s> <s xml:id="echoid-s5017" xml:space="preserve">ita æquales m υ, n φ. </s> <s xml:id="echoid-s5018" xml:space="preserve">at o σ, eſt æqualis <lb/>m υ. </s> <s xml:id="echoid-s5019" xml:space="preserve">Sequitur igitur, ut omnes o σ, m τ, m υ, n φ in-<lb/>ter ſe ſint æquales. </s> <s xml:id="echoid-s5020" xml:space="preserve">Sed ut ρ π ad π τ, hoc eſt ut 3 ad 2, ita n d <lb/>ad d χ: </s> <s xml:id="echoid-s5021" xml:space="preserve">permutãdoq; </s> <s xml:id="echoid-s5022" xml:space="preserve">ut ρ π ad n d, ita π τ ad d χ. </s> <s xml:id="echoid-s5023" xml:space="preserve">& </s> <s xml:id="echoid-s5024" xml:space="preserve">ſũt æqua <lb/>les ζ π, n d. </s> <s xml:id="echoid-s5025" xml:space="preserve">ergo d χ, hoc eſt n p, & </s> <s xml:id="echoid-s5026" xml:space="preserve">π τ æquales. </s> <s xml:id="echoid-s5027" xml:space="preserve">Sed etiam æ-<lb/>quales n π, π m. </s> <s xml:id="echoid-s5028" xml:space="preserve">reliqua igitur π p reliquæ m τ, hoc eſt ipſi <lb/>n φ æqualis erit. </s> <s xml:id="echoid-s5029" xml:space="preserve">quare dempta p π ex p e, & </s> <s xml:id="echoid-s5030" xml:space="preserve">φ n dempta ex <lb/>n e, relinquitur p e æqualis e φ. </s> <s xml:id="echoid-s5031" xml:space="preserve">Itaque π, ρ centra figurarũ <lb/>ſecundo loco deſcriptarum a primis centris p n æquali in-<lb/>teruallo recedunt. </s> <s xml:id="echoid-s5032" xml:space="preserve">quòd ſi rurſus aliæ figuræ deſcribantur, <lb/>eodem modo demonſtrabimus earum centra æqualiter ab <lb/>his recedere, & </s> <s xml:id="echoid-s5033" xml:space="preserve">ad portionis conoidis centrum propius ad <lb/>moueri. </s> <s xml:id="echoid-s5034" xml:space="preserve">Ex quibus conſtat lineam π φ à centro grauitatis <lb/>portionis diuidi in partes æquales. </s> <s xml:id="echoid-s5035" xml:space="preserve">Si enim fieri poteſt, non <lb/>ſit centrum in puncto e, quod eſt lineæ π φ medium: </s> <s xml:id="echoid-s5036" xml:space="preserve">ſed in <lb/>ψ: </s> <s xml:id="echoid-s5037" xml:space="preserve">& </s> <s xml:id="echoid-s5038" xml:space="preserve">ipſi π ψ æqualis fiat φ ω. </s> <s xml:id="echoid-s5039" xml:space="preserve">Cum igitur in portione ſolida <lb/>quædam figura inſcribi posſit, ita ut linea, quæ inter cen-<lb/>trum grauitatis portionis, & </s> <s xml:id="echoid-s5040" xml:space="preserve">inſcriptæ figuræ interiicitur, <lb/>qualibet linea propoſita ſit minor, quod proxime demon-<lb/>ſtrauimus: </s> <s xml:id="echoid-s5041" xml:space="preserve">perueniet tandem φ centrum inſcriptæ figuræ <pb file="0200" n="200" rhead="FED. COMMANDINI"/> <anchor type="figure" xlink:label="fig-0200-01a" xlink:href="fig-0200-01"/> <pb o="45" file="0201" n="201" rhead="DE CENTRO GRAVIT. SOLID."/> ad punctum ω. </s> <s xml:id="echoid-s5042" xml:space="preserve">Sed quoniam π circum ſcripta itidem alia <lb/>figura æquali interuallo ad portionis centrum accedit, ubi <lb/>primum φ applieuerit ſe ad ω, & </s> <s xml:id="echoid-s5043" xml:space="preserve">π ad punctũ ψ, hoc eſt ad <lb/>portionis centrum ſe applicabit. </s> <s xml:id="echoid-s5044" xml:space="preserve">quod fieri nullo modo <lb/>poſſe perſpicuum eſt. </s> <s xml:id="echoid-s5045" xml:space="preserve">non aliter idem abſurdum ſequetur, <lb/>ſi ponamus centrum portionis recedere à medio ad par-<lb/>tes ω; </s> <s xml:id="echoid-s5046" xml:space="preserve">eſſet enim aliquando centrum figuræ inſcriptæ idem <lb/>quod portionis centrũ. </s> <s xml:id="echoid-s5047" xml:space="preserve">ergo punctum e centrum erit gra <lb/>uitatis portionis a b c. </s> <s xml:id="echoid-s5048" xml:space="preserve">quod demonſtrare oportebat.</s> <s xml:id="echoid-s5049" xml:space="preserve"/> </p> <div xml:id="echoid-div281" type="float" level="2" n="1"> <figure xlink:label="fig-0195-01" xlink:href="fig-0195-01a"> <image file="0195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0195-01"/> </figure> <note position="right" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">7. huius</note> <figure xlink:label="fig-0196-01" xlink:href="fig-0196-01a"> <image file="0196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0196-01"/> </figure> <note position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve">8. primi <lb/>libri Ar-<lb/>chimedis</note> <note position="left" xlink:label="note-0196-02" xlink:href="note-0196-02a" xml:space="preserve">11. duo-<lb/>decimi.</note> <note position="left" xlink:label="note-0196-03" xlink:href="note-0196-03a" xml:space="preserve">15. quinti</note> <note position="left" xlink:label="note-0196-04" xlink:href="note-0196-04a" xml:space="preserve">2. duode-<lb/>cimi.</note> <note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">20. primi <lb/>conicorũ</note> <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a"> <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0198-01"/> </figure> <note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">19. quinti</note> <figure xlink:label="fig-0200-01" xlink:href="fig-0200-01a"> <image file="0200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0200-01"/> </figure> </div> <p> <s xml:id="echoid-s5050" xml:space="preserve">Quod autem ſupra demõſtratum eſt in portione conoi-<lb/>dis recta per figuras, quæ ex cylindris æqualem altitudi-<lb/>dinem habentibus conſtant, idem ſimiliter demonſtrabi-<lb/>mus per figuras ex cylindri portionibus conſtantes in ea <lb/>portione, quæ plano non ad axem recto abſcinditur. </s> <s xml:id="echoid-s5051" xml:space="preserve">ut <lb/>enim tradidimus in commentariis in undecimam propoſi <lb/>tionem libri Archimedis de conoidibus & </s> <s xml:id="echoid-s5052" xml:space="preserve">ſphæroidibus. <lb/></s> <s xml:id="echoid-s5053" xml:space="preserve">portiones cylindri, quæ æquali ſunt altitudine eam inter ſe <lb/>ſe proportionem habent, quam ipſarum baſes; </s> <s xml:id="echoid-s5054" xml:space="preserve">baſes autẽ <lb/>quæ ſunt ellipſes ſimiles eandem proportionem habere, <lb/> <anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/> quam quadrata diametrorum eiuſdem rationis, ex corol-<lb/>lario ſeptimæ propoſitionis libri de conoidibus, & </s> <s xml:id="echoid-s5055" xml:space="preserve">ſphæ-<lb/>roidibus, manifeſte apparet.</s> <s xml:id="echoid-s5056" xml:space="preserve"/> </p> <div xml:id="echoid-div282" type="float" level="2" n="2"> <note position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">corol. 15 <lb/>deconoi-<lb/>dibus & <lb/>ſphæroi-<lb/>dibus.</note> </div> </div> <div xml:id="echoid-div284" type="section" level="1" n="95"> <head xml:id="echoid-head102" xml:space="preserve">THEOREMA XXIIII. PROPOSITIO XXX.</head> <p> <s xml:id="echoid-s5057" xml:space="preserve">SI à portione conoidis rectanguli alia portio <lb/>abſcindatur, plano baſi æquidiſtante; </s> <s xml:id="echoid-s5058" xml:space="preserve">habebit <lb/>portio tota ad eam, quæ abſciſſa eſt, duplam pro <lb/>portio nem eius, quæ eſt baſis maioris portionis <lb/>ad baſi m minoris, uel quæ axis maioris ad axem <lb/>minoris.</s> <s xml:id="echoid-s5059" xml:space="preserve"/> </p> <pb file="0202" n="202" rhead="FED. COMMANDINI"/> <p> <s xml:id="echoid-s5060" xml:space="preserve">ABSCINDATVR à portione conoidis rectanguli <lb/>a b c alia portio e b f, plano baſi æquidiſtante: </s> <s xml:id="echoid-s5061" xml:space="preserve">& </s> <s xml:id="echoid-s5062" xml:space="preserve">eadem <lb/>portio ſecetur alio plano per axem; </s> <s xml:id="echoid-s5063" xml:space="preserve">ut ſuperficiei ſectio ſit <lb/>parabole a b c: </s> <s xml:id="echoid-s5064" xml:space="preserve">planorũ portiones abſcindentium rectæ <lb/>lineæ a c, e f: </s> <s xml:id="echoid-s5065" xml:space="preserve">axis autem portionis, & </s> <s xml:id="echoid-s5066" xml:space="preserve">ſectionis diameter <lb/>b d; </s> <s xml:id="echoid-s5067" xml:space="preserve">quam linea e fin puncto g ſecet. </s> <s xml:id="echoid-s5068" xml:space="preserve">Dico portionem co-<lb/>noidis a b c ad portionem e b f duplam proportionem ha-<lb/>bere eius, quæ eſt baſis a c ad baſim e f; </s> <s xml:id="echoid-s5069" xml:space="preserve">uel axis d b ad b g <lb/>axem. </s> <s xml:id="echoid-s5070" xml:space="preserve">Intelligantur enim duo coni, ſeu coni portiones <lb/>a b c, e b f, eãdem baſim, quam portiones conoidis, & </s> <s xml:id="echoid-s5071" xml:space="preserve">æqua <lb/>lem habentes altitudinem. </s> <s xml:id="echoid-s5072" xml:space="preserve">& </s> <s xml:id="echoid-s5073" xml:space="preserve">quoniam a b c portio conoi <lb/>dis ſeſquialtera eſt coni, ſeu portionis coni a b c; </s> <s xml:id="echoid-s5074" xml:space="preserve">& </s> <s xml:id="echoid-s5075" xml:space="preserve">portio <lb/>e b f coniſeu portionis coni e b feſt ſeſquialtera, quod de-<lb/> <anchor type="figure" xlink:label="fig-0202-01a" xlink:href="fig-0202-01"/> monſtrauit Archimedes in propoſitionibus 23, & </s> <s xml:id="echoid-s5076" xml:space="preserve">24 libri <lb/>de conoidibus, & </s> <s xml:id="echoid-s5077" xml:space="preserve">ſphæroidibus: </s> <s xml:id="echoid-s5078" xml:space="preserve">erit conoidis portio ad <lb/>conoidis portionem, ut conus ad conum, uel ut coni por-<lb/>tio ad coni portionem. </s> <s xml:id="echoid-s5079" xml:space="preserve">Sed conus, uel coni portio a b c ad <lb/>conum, uel coni portionem e b f compoſitam proportio-<lb/>nem habet ex proportione baſis a c ad baſim e f, & </s> <s xml:id="echoid-s5080" xml:space="preserve">ex pro-<lb/>portione altitudinis coni, uel coni portionis a b c ad alti-<lb/>tudinem ipſius e b f, ut nos demonſtrauimus in com men-<lb/>tariis in undecimam propoſitionem eiuſdem libri A rchi-<lb/>medis: </s> <s xml:id="echoid-s5081" xml:space="preserve">altitudo autem ad altitudinem eſt, ut axis ad axem. <lb/></s> <s xml:id="echoid-s5082" xml:space="preserve">quod quidem in conis rectis perſpicuum eſt, in ſcalenis ue <pb o="46" file="0203" n="203" rhead="DE CENTRO GRAVIT. SOLID."/> ro ita demonſtrabitur. </s> <s xml:id="echoid-s5083" xml:space="preserve">Ducatur à puncto b ad planum ba-<lb/>ſis a c perpendicularis linea b h, quæ ipſam e fin K ſecet. <lb/></s> <s xml:id="echoid-s5084" xml:space="preserve">erit b h altitudo coni, uel coni portionis a b c: </s> <s xml:id="echoid-s5085" xml:space="preserve">& </s> <s xml:id="echoid-s5086" xml:space="preserve">b K altitu <lb/> <anchor type="note" xlink:label="note-0203-01a" xlink:href="note-0203-01"/> do e f g. </s> <s xml:id="echoid-s5087" xml:space="preserve">Quod cum lineæ a c, e f inter ſe æ quidiſtent, ſunt <lb/>enim planorum æ quidiſtantium ſectiones: </s> <s xml:id="echoid-s5088" xml:space="preserve">habebit d b ad <lb/> <anchor type="note" xlink:label="note-0203-02a" xlink:href="note-0203-02"/> b g proportionem ean dem, quam h b ad b k. </s> <s xml:id="echoid-s5089" xml:space="preserve">quare por-<lb/>tio conoidis a b c ad portionem e f g proportionem habet <lb/>compoſitam ex proportione baſis a c ad baſim e f; </s> <s xml:id="echoid-s5090" xml:space="preserve">& </s> <s xml:id="echoid-s5091" xml:space="preserve">ex <lb/>proportione d b axis ad axem b g. </s> <s xml:id="echoid-s5092" xml:space="preserve">Sed circulus, uel <lb/> <anchor type="note" xlink:label="note-0203-03a" xlink:href="note-0203-03"/> ellipſis circa diametrum a c ad circulum, uel ellipſim <lb/> <anchor type="note" xlink:label="note-0203-04a" xlink:href="note-0203-04"/> circa e f, eſt ut quadratum a c ad quadratum e f; </s> <s xml:id="echoid-s5093" xml:space="preserve">hoc eſt ut <lb/>quadratũ a d ad quadratũ e g. </s> <s xml:id="echoid-s5094" xml:space="preserve">& </s> <s xml:id="echoid-s5095" xml:space="preserve">quadratum a d ad quadra <lb/>tum e g eſt, ut linea d b ad lineam b g. </s> <s xml:id="echoid-s5096" xml:space="preserve">circulus igitur, uel el <lb/>lipſis circa diametrum a c ad circulũ, uel ellipſim circa e f, <lb/> <anchor type="note" xlink:label="note-0203-05a" xlink:href="note-0203-05"/> hoc eſt baſis ad baſim eandem proportionem habet, quã <lb/> <anchor type="note" xlink:label="note-0203-06a" xlink:href="note-0203-06"/> d b axis ad axem b g. </s> <s xml:id="echoid-s5097" xml:space="preserve">ex quibus ſequitur portionem a b c <lb/>ad portionem e b f habere proportionem duplam eius, <lb/>quæ eſt baſis a c ad bafim e f: </s> <s xml:id="echoid-s5098" xml:space="preserve">uel axis d b ad b g axem. </s> <s xml:id="echoid-s5099" xml:space="preserve">quod <lb/>demonſtrandum proponebatur.</s> <s xml:id="echoid-s5100" xml:space="preserve"/> </p> <div xml:id="echoid-div284" type="float" level="2" n="1"> <figure xlink:label="fig-0202-01" xlink:href="fig-0202-01a"> <image file="0202-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0202-01"/> </figure> <note position="right" xlink:label="note-0203-01" xlink:href="note-0203-01a" xml:space="preserve">16. unde-<lb/>cimi.</note> <note position="right" xlink:label="note-0203-02" xlink:href="note-0203-02a" xml:space="preserve">4 ſexti.</note> <note position="right" xlink:label="note-0203-03" xlink:href="note-0203-03a" xml:space="preserve">2. duode <lb/>cimi</note> <note position="right" xlink:label="note-0203-04" xlink:href="note-0203-04a" xml:space="preserve">7. de co-<lb/>noidibus <lb/>& ſphæ-<lb/>roidibus</note> <note position="right" xlink:label="note-0203-05" xlink:href="note-0203-05a" xml:space="preserve">15. quinti</note> <note position="right" xlink:label="note-0203-06" xlink:href="note-0203-06a" xml:space="preserve">20. primi <lb/>conicorũ</note> </div> </div> <div xml:id="echoid-div286" type="section" level="1" n="96"> <head xml:id="echoid-head103" xml:space="preserve">THEOREMA XXV. PROPOSITIO XXXI.</head> <p> <s xml:id="echoid-s5101" xml:space="preserve">Cuiuslibet fruſti à portione rectanguli conoi <lb/>dis abſcisſi, centrum grauitatis eſt in axe, ita ut <lb/>demptis primum à quadrato, quod fit ex diame-<lb/>tro maioris baſis, tertia ipſius parte, & </s> <s xml:id="echoid-s5102" xml:space="preserve">duabus <lb/>tertiis quadrati, quod fit ex diametro baſis mino-<lb/>ris: </s> <s xml:id="echoid-s5103" xml:space="preserve">deinde à tertia parte quadrati maioris baſis <lb/>rurſus dempta portione, ad quam reliquum qua <lb/>drati baſis maioris unà cum dicta portione duplã <lb/>proportionem habeat eius, quæ eſt quadrati ma- <pb file="0204" n="204" rhead="FED. COMMANDINI"/> ioris baſis ad quadratum minoris: </s> <s xml:id="echoid-s5104" xml:space="preserve">centrum ſit in <lb/>eo axis puncto, quo ita diuiditur ut pars, quæ mi <lb/>norem baſim attingit ad alteram partem eandem <lb/>proportionem habeat, quam dempto quadrato <lb/>minoris baſis à duabus tertiis quadrati maioris, <lb/>habet id, quod reliquum eſt unà cum portione à <lb/>tertia quadrati maioris parte dempta, ad reliquà <lb/>eiuſdem tertiæ portionem.</s> <s xml:id="echoid-s5105" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5106" xml:space="preserve">SIT fruſtum à portione rectanguli conoidis abſciſſum <lb/>a b c d, cuius maior baſis circulus, uel ellipſis circa diame-<lb/>trum b c, minor circa diametrum a d; </s> <s xml:id="echoid-s5107" xml:space="preserve">& </s> <s xml:id="echoid-s5108" xml:space="preserve">axis e f. </s> <s xml:id="echoid-s5109" xml:space="preserve">deſcriba-<lb/>tur autem portio conoidis, à quo illud abſciſſum eſt, & </s> <s xml:id="echoid-s5110" xml:space="preserve">pla-<lb/> <anchor type="figure" xlink:label="fig-0204-01a" xlink:href="fig-0204-01"/> no per axem ducto ſecetur; </s> <s xml:id="echoid-s5111" xml:space="preserve">ut ſuperficiei ſectio ſit parabo-<lb/>le b g c, cuius diameter, & </s> <s xml:id="echoid-s5112" xml:space="preserve">axis portionis g f: </s> <s xml:id="echoid-s5113" xml:space="preserve">deinde g f diui <lb/>datur in puncto h, ita ut g h ſit dupla h f: </s> <s xml:id="echoid-s5114" xml:space="preserve">& </s> <s xml:id="echoid-s5115" xml:space="preserve">rurſus g e in ean <lb/>dem proportionem diuidatur: </s> <s xml:id="echoid-s5116" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s5117" xml:space="preserve">g _k_ ipſius k e dupla. </s> <s xml:id="echoid-s5118" xml:space="preserve">Iã <lb/>ex iis, quæ proxime demonſtrauimus, conſtat centrum gra <lb/>uitatis portionis b g c eſſe h punctum: </s> <s xml:id="echoid-s5119" xml:space="preserve">& </s> <s xml:id="echoid-s5120" xml:space="preserve">portionis a g c <lb/>punctum k. </s> <s xml:id="echoid-s5121" xml:space="preserve">ſumpto igitur infra h punctol, ita ut k h ad h l <pb o="47" file="0205" n="205" rhead="DE CENTRO GRAVIT. SOLID."/> eani proportionem habeat, quam a b c d fruſtum ad por-<lb/>tionem a g d; </s> <s xml:id="echoid-s5122" xml:space="preserve">erit punctum l eius fruſti grauitatis cẽtrum: <lb/></s> <s xml:id="echoid-s5123" xml:space="preserve">habebitq; </s> <s xml:id="echoid-s5124" xml:space="preserve">componendo K l ad 1 h proportionem eandem, <lb/>quam portio conoidis b gc ad a g d portionem. </s> <s xml:id="echoid-s5125" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s5126" xml:space="preserve">quo <lb/> <anchor type="note" xlink:label="note-0205-01a" xlink:href="note-0205-01"/> niam quadratum b f ad quadratum a e, hoc eſt quadratum <lb/>b c ad quadratum a d eſt, ut linea f g ad g e: </s> <s xml:id="echoid-s5127" xml:space="preserve">erunt duæ ter-<lb/>tiæ quadrati b c ad duas tertias quadrati a d, ut h g ad g _k_: <lb/></s> <s xml:id="echoid-s5128" xml:space="preserve">& </s> <s xml:id="echoid-s5129" xml:space="preserve">ſi à duabus tertiis quadrati b c demptæ fuerint duæ ter-<lb/>tiæ quadrati a d: </s> <s xml:id="echoid-s5130" xml:space="preserve">erit diuidẽdo id, quod relinquitur ad duas <lb/>tertias quadrati a d, ut h k ad k g. </s> <s xml:id="echoid-s5131" xml:space="preserve">Rurſus duæ tertiæ quadra <lb/>ti a d ad duas tertias quadrati b c ſunt, ut _k_ g ad g h: </s> <s xml:id="echoid-s5132" xml:space="preserve">& </s> <s xml:id="echoid-s5133" xml:space="preserve">duæ <lb/>tertiæ quadrati b c ad tertiã partẽ ipſius, ut g h ad h f. </s> <s xml:id="echoid-s5134" xml:space="preserve">ergo <lb/>ex æ quali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ipſis quadrati a d duabus tertiis, ad tertiã <lb/>partem quadrati b c, ut _k_ h ad h f: </s> <s xml:id="echoid-s5135" xml:space="preserve">& </s> <s xml:id="echoid-s5136" xml:space="preserve">ad portionem eiuſdẽ <lb/>tertiæ partis, ad quam unà cum ipſa portione, duplam pro <lb/>portionem habeat eius, quæ eſt quadrati b c ad quadratũ <lb/>a d, ut K 1 ad 1 h. </s> <s xml:id="echoid-s5137" xml:space="preserve">habet enim _K_l ad 1 h ean dem proportio-<lb/>nem, quam conoidis portio b g c ad portionem a g d: </s> <s xml:id="echoid-s5138" xml:space="preserve">por-<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, quæ eſt baſis b c ad baſim a d: </s> <s xml:id="echoid-s5139" xml:space="preserve">hoc eſt quadrati <lb/>b c ad quadratum a d; </s> <s xml:id="echoid-s5140" xml:space="preserve">ut proxime demonſtratum eſt. </s> <s xml:id="echoid-s5141" xml:space="preserve">quare <lb/> <anchor type="note" xlink:label="note-0205-02a" xlink:href="note-0205-02"/> dempto a d quadrato à duabus tertiis quadrati b c, erit id, <lb/>quod relin quitur unà cum dicta portione tertiæ partis ad <lb/>reliquam eiuſdem portionem, ut el ad 1 f. </s> <s xml:id="echoid-s5142" xml:space="preserve">Cum igitur cen-<lb/>trum grauitatis fruſti a b c d ſit l, à quo axis e f in eam, quã <lb/>diximus, proportionem diuidatur; </s> <s xml:id="echoid-s5143" xml:space="preserve">conſtat uerũ eſſe illud, <lb/>quod demonſtrandum propoſuimus.</s> <s xml:id="echoid-s5144" xml:space="preserve"/> </p> <div xml:id="echoid-div286" type="float" level="2" n="1"> <figure xlink:label="fig-0204-01" xlink:href="fig-0204-01a"> <image file="0204-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0204-01"/> </figure> <note position="right" xlink:label="note-0205-01" xlink:href="note-0205-01a" xml:space="preserve">20. I. coni <lb/>corum.</note> <note position="right" xlink:label="note-0205-02" xlink:href="note-0205-02a" xml:space="preserve">30. huius</note> </div> </div> <div xml:id="echoid-div288" type="section" level="1" n="97"> <head xml:id="echoid-head104" xml:space="preserve">FINIS LIBRI DE CENTRO <lb/>GRAVITATIS SOLIDORVM.</head> <p> <s xml:id="echoid-s5145" xml:space="preserve">Impreſſ. </s> <s xml:id="echoid-s5146" xml:space="preserve">Bononiæ cum licentia Superiorum.</s> <s xml:id="echoid-s5147" xml:space="preserve"/> </p> <pb file="0206" n="206"/> <pb file="0207" n="207"/> <pb file="0208" n="208"/> <figure> <image file="0208-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0208-01"/> </figure> <pb file="0209" n="209"/> <pb file="0210" n="210"/> <pb file="0211" n="211"/> <pb file="0212" n="212"/> </div></text> </echo>