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Show 124. EVCLIDIS ELEMENTORVM TOTA fiipcrficics Pymmidis, VCi oéhhc'dri , infm (film-am, Cuius diameter rationalis othdcfcripti m cdialis ME Nam '1 iplierrcr diameter fit rariomliszcrit iii-um {oiidi lzrtus)pcr IGiVL'l ‘7‘.Iil~ri prmccdcnl‘is, PROPO'SITIONES. mcomprchenil, ('iiiiil‘uitiilfln latcris cubi. Patcthoccx 18.19. (St 4 o. vndccimi. rationale . anrc , par pixiecudcnrcm. v 11:1 indium {olidi tiledialis cit, Igitur&'ommum bufium congerics, hoc cfk Lora inpcrhcics {ohdi, medialis ctk, [icut propobimr . P Y R AMIDM‘ Isms dd pci‘p:ndicuiarcm , gm}: :‘1 Yel‘ticc ad bnfim hf, iccdchxjprjoncm ,5*_ dclnbirur, porcntin {quuialrcmm «IE. "minim ii'nriin qua ficut cit k 1. 1d ipf‘iim k f'. fic (E k f. ad ipfiim k 6. Sad k Lipfius kf‘. potcn tin {Etlliiiaitcruln‘L-G. Ergo k f: latus iblidi ad iplhm k c. Perpendiculnrcm Porcntia icilluiglrumm cfh @od CH: Propofitum .. A 5p H A: R A? ccnrro' ad bafim circumréripm' pyramidis 11'3th 5 pcrpédicularis cfi {exta pars {plmricx diamantNam in. dei‘criptionc 1 aprrzfcmi librLk Idizimcrer ad k cpcrpcndiCuhrcm c-i‘c ficut and 4. Ergo {Emiliamctcrml ipfiim k e. {icut 4.:1d 4. 1rd rxc-cfliis ipfius k c. i iiipcr ilmidinmcrrum ci‘ripfa 51 Ccntro ad baiim pa‘rpcmiicuhris. Ergoipfa pm‘pendicularis cf: ILmidiamrtri pars tertia. Qu/irc din: 6 7 mctur fem.' i S p H .f. R A {Emidiamcrcr ad'pcrpcndicnlarcm hcenrro 1d bafim oé‘tnhcdri circumil‘ripri,porvntiz tripium cflt Vndc iarus ip‘fius {Oiidi ad tandem Perpendicularcm potcmia (Exrupimnerit ., Nam per 17.131‘xtcriti, {cmidiamcrcr {191193112 ad Iamé ochhsdri po- atcnria eff, ficut 3. 1d 6. latus aurem ofclihtdriadftm-"diamctrumf, circuiiqui bafim oflahedri circum {cribenpsr 1 yummifli, CR: ficut 6. ad 2, in'porenrizi. Ergo, pcrmquam‘ prdpom'oncm , {cmidiamcter fphxra‘: ad‘ {amidmmctrum difh' circuii , cf'c {icur ;. ad 2 . Scd per Pcnultimmn primi , quadratum {cmidian ctri diéh circuii cum qua: dmto.,pcrpcndicul';1ris mqunm ef'c quadrato fcmidiametri {‘phmraz . Igirur quadratum femidiamcrri {phmrm ad quadratum perpendicu' laris triplum . @relarus 0&ahcdrifcxcuplum Potentialircr ad candcm, {icut prhpommr; 8 "125 PERPENDLQVL {a 1! IS éccnrro fphxm: ad [mfim cubi ab ipfa {phra- 3;. ‘ , D VA: perpendzcni arr»: , vna h ccritro {Phatrit 21d bafm oé‘rahedri: gin-fa a'r) codem centre :zri balm-3 min in cadcm fphmm ccmprchcn- 9 1orum {um {$9113 '68 . Nan: c5: th'll'iiil 8; 18.1v‘a'ccdcnris , lilia‘m: {cmid'iamrrcr porcmiglircr tzg‘nia «F: 3d p‘i‘pcndicu‘xurem culti. Et per 6".huius, cadcm (cmidi' :nctcr ITOJCEEiCifICI triiila cf} adpcrpcndicu- larcm oéhhcdri . Q3112 pmpuu‘iiculares il‘ils iiintinuiccm Equalcs. quod d'c Propofitum . vBAS is~ pyramidis ad bnfim odhhedri in cadcm fbhrzm comerc~ hcnfi cl‘t feiguitertia. Nam quadmmm iatmis {yin m‘idis ad quxdm(tum diarnctri fphxrm, c1} [icut Lad3.1m-16.pra:ccdcntis.&idco ficut 4.:1d 6.11‘hxricu2 aurcm diametri quadmtmn ad quadratum Ilreris oé‘cahedri,iicut 64d 3.17m17.pr9:ccjdenris.Quarc pcrmquam proper. tion-{nuqundratum latcris pyramidis ad qundi‘arum lateris oflahedri crit (icut 4.:1d 3. uarc Iic trinngulum ad triangulum per 18fexti. . Hinc ergo mamfeitum sibquéd to {a pvramidis fupcrficics ad totam oétahcdri {upcrficicm eff ficut [dad I. 4.videlicct filbfcf'quialtera . R A "r 1 o {excupla {lipcrparti‘ens trcs quartas, dupla ell ad rationcm , quam habct oé‘cahcdri tblidum ad Pyramidis lblidum in endcm {phgem exiflcntium . E duéris :‘iccn m) (phémw ad angulos {olidorum re€cis , fecetur oé‘mhedrum in 8. pyramidcs : Tctraiicdrum verc‘) {cu pyramis in quntuor. cruntt'luc 8.pyramidum ccliitudines ipfie Per. pendiCularcs 5. ccntro fphxrm ad baiés 0&ahcdri. Lhamor veto Pytg. midum celfitudincs ipfk: perpendicularcs abcodem ccntro ad bafes tetrahedri . Sit imquc pyramis arming bafis fit filperficiei oaahcdri xqualis: cclfitudo vcro :rqualis perpendiculari ofl‘ahedri . Sit item 13‘. pyramis , cuius bafis fuperficzci tetrahedri , cchirudo vero perpendiculari termhcdri fit mqualis . Eritc'lue per 6 5. vndecimi, pyramis a. pélahcdro: Pymmis vcro b.tctrahcdro xquah‘s. ‘lzibus fuppofitis, crit Per '7" huius , cclflrudo pyramidis a . ad 1' 5 R P b N b L c v L A R IS 5; centro {pimzrre ad bafim oéhhedri pqtshtialiter trinId cf]: 1d Dei‘psndicularam ab eodcm ccnrro ad ba~fim pyramidis in erdcm: {pitta locut'r. Nam , pcrprzmilfim pcrpcndiculnris ofhhcdri ad firmidiamermm (phat: porcntia CH: {icut ,'. ad a). RT ant: prfemillfim aurem, feiiiidinmetcr {phcene ad pctpcndicu‘;xrc;11 pyrmnidis , porun-riiilitertfl: , Hair 9. ad I. '3 Per squnm‘ ergo proportionrsm, pci‘m-ndicularis oftahedriad P6"? pendicrilarcm pymmi'dis , Pqtcntia. iicur tcrrium ad. Vnum , llcut 3‘ Bropommr. ; ,4 , .v ‘ PERPEN‘ i is~~..,._.g .w A»: ~ Cellitudinem py -amidis b.potcntialircr tripla. & idco ficur 2.7. ad 9.ba1is verb pyra- mldis and balim Pyramidis b. Per corolla- \f::,\v;.\ rium prseccdcntis :crit ibilptialtcra : & idC‘O potentialiterdicut 9. ad 4.ErgoFera'quam proportioncm,mtio pyramidis mad Pymmidem b. (qua cx mticnibus cclfitudinum 8c bafium componirurfiiuplicam crirgficut 2.7.ad 4. licut cnim limer (implam, fic duping duplam rationcm componunt. Igirur 8: cadcm ratio 061:1hedri ad tctrahcd:um,ficut proponitur dcmonfl‘randum. C V i. '5 vnx |