Mercurial > hg > mpdl-xml-content
view texts/XML/echo/la/Clavius_1586_YC97H42F.xml @ 31:edf6e8fcf323 default tip
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 |
line wrap: on
line source
<?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:YC97H42F.xml</dcterms:identifier> <dcterms:creator identifier="GND:118880632">Clavius, Christoph</dcterms:creator> <dcterms:title xml:lang="la">Theodosii Tripolitae Sphaericorum libri tres; Sinus, vel semisses rectarum in circulo subtensarum</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1586</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="001" n="1"/> <pb file="002" n="2"/> <pb file="003" n="3"/> <pb file="004" n="4"/> <pb file="005" n="5"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">THEODOSII <lb/>TRIPOLITAE <lb/>SPHAERICORVM <lb/>LIBRI III.</head> <head xml:id="echoid-head2" xml:space="preserve">_A CHRISTOPHORO CLAVIO BAMBER-_ <lb/>_genſi Societatis IESV_ <lb/>PERSPICVIS DEMONSTRATIONIBVS, <lb/>_ac ſcholijs illuſtrati._ <lb/>_Item Eiuſdem_</head> <head xml:id="echoid-head3" xml:space="preserve">CHRISTOPHORI CLAVII <lb/>SINVS. LINEAE TANGENTES. ETSECANTES. <lb/>TRIANGVLA RECTILINEA. ATQVE SPHAERICA.</head> <figure> <image file="005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/005-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head4" xml:space="preserve">ROMAE,</head> <head xml:id="echoid-head5" xml:space="preserve">Ex Typographia Dominici Baſæ. M. D. LXXXVI.</head> <head xml:id="echoid-head6" style="it" xml:space="preserve">PERMISSV SVPERIORVM.</head> <pb file="006" n="6"/> <handwritten/> <handwritten/> <handwritten/> <pb file="007" n="7"/> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head7" xml:space="preserve">ILLVSTRISS. ET EXCELL. <lb/>PRINCIPI,</head> <head xml:id="echoid-head8" xml:space="preserve">DOM. IACOBO BONCOMPAGNO, <lb/>Duci Soræ, & Marchioni <lb/>Vignolæ, &c.</head> <head xml:id="echoid-head9" style="it" xml:space="preserve">CHRISTOPHORVS CLAVIVS <lb/>è Societate IESV. S. P. D.</head> <figure> <image file="007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/007-01"/> </figure> <p> <s xml:id="echoid-s1" xml:space="preserve">QVANTVM ſit, Illuſtriſſime <lb/>Princeps, ad perfectam Mathe <lb/>maticorum ſcientiam in ſphę-<lb/>ricis Elementis, & </s> <s xml:id="echoid-s2" xml:space="preserve">in Sinuum, <lb/>linearumq́; </s> <s xml:id="echoid-s3" xml:space="preserve">Tangentium atque <lb/>Secantium, & </s> <s xml:id="echoid-s4" xml:space="preserve">in tota denique <lb/>Triangulorum diſciplina momenti, nemo me-<lb/>lius intelligit, mea quidem ſententia, quam qui <lb/>in rerum cęleſtium cognitionem neruos omnes <lb/>intendit. </s> <s xml:id="echoid-s5" xml:space="preserve">Nam nec Orbium globorumve cæle-<lb/>ſtium, nec ſiderum ſeu quæ vocantur errantia, <lb/>ſeu quæ infixa ſunt cælo, curſus ac magnitudo <lb/>(quæ vna omnium iucundiſſima, & </s> <s xml:id="echoid-s6" xml:space="preserve">ad decus a-<lb/>ptiſſima ſcientia eſt) ſine eorum ope ac benefi-<lb/>cio teneri poteſt. </s> <s xml:id="echoid-s7" xml:space="preserve">Locupletiſſimus teſtis eſt ex-<lb/>cellenti doctrina vir Ptolemæus eo libro, quem <lb/>Conſtructionem magnam inſcripſit: </s> <s xml:id="echoid-s8" xml:space="preserve">vbi quic- <pb file="008" n="8"/> quid ad cæleſtem doctrinam pertinet, id omne <lb/>vel ſphæricis ex elementis, vel ex triangulorum <lb/>doctrina, vel ex arcuum denique chordis (vnde <lb/>omnis Sinuum, omnis linearum Tangentium ac <lb/>Secantium diſciplina manauit) peracute demon <lb/>ſtrat: </s> <s xml:id="echoid-s9" xml:space="preserve">vt fruſtra laborem operamq́; </s> <s xml:id="echoid-s10" xml:space="preserve">ſuſcipiat, quiſ-<lb/>quis hoc ſine præſidio Demonſtrationes illas <lb/>conetur attingere. </s> <s xml:id="echoid-s11" xml:space="preserve">Mitto quanta in Gnomoni-<lb/>ca, quanta in Coſmographia, Geodæſia, & </s> <s xml:id="echoid-s12" xml:space="preserve">vni-<lb/>uerſa pene Geometria earum rerum ſit oppor-<lb/>tunitas: </s> <s xml:id="echoid-s13" xml:space="preserve">ne aut in re perſpicua ſim multus, aut <lb/>aliena diſputatio videatur. </s> <s xml:id="echoid-s14" xml:space="preserve">Quocirca fatebor, <lb/>quod res eſt, Sapientiſſime Princeps, multas me <lb/>annorum vigilias ac labores pro eo amore, quo <lb/>ad Mathematicas voluptates incendor, in ea-<lb/>rum, quas dixi, rerum planam ac ſolidam cogni-<lb/>tionem contuliſſe: </s> <s xml:id="echoid-s15" xml:space="preserve">& </s> <s xml:id="echoid-s16" xml:space="preserve">lucubrationes quaſdam <lb/>ſepoſuiſſe, meos duntaxat in vſus. </s> <s xml:id="echoid-s17" xml:space="preserve">Verum vt <lb/>eruditorum, & </s> <s xml:id="echoid-s18" xml:space="preserve">eorum, qui ſcripta noſtra tracta-<lb/>bant, Gnomonica præſertim, vbi Sinuum, Trian <lb/>gulorum, cæterarumq́; </s> <s xml:id="echoid-s19" xml:space="preserve">rerum frequens incidit <lb/>mentio, poſtulationi concederem; </s> <s xml:id="echoid-s20" xml:space="preserve">commode <lb/>me facturum exiſtimaui, ſi quæ priuatos in vſus <lb/>temere collegiſſem, ea in ordinem adducta, & </s> <s xml:id="echoid-s21" xml:space="preserve">in <lb/>vnum quaſi coacta corpus in commune confer-<lb/>rem: </s> <s xml:id="echoid-s22" xml:space="preserve">Et commentarios hoſce in tres Theodoſij <lb/>Tripolitæ libros de ſphæricis elemẽtis, vna cum <pb file="009" n="9"/> demonſtrationibus noſtris Sinuum, linearum <lb/>Tangentium atque Secantium, præcipueq́; </s> <s xml:id="echoid-s23" xml:space="preserve">Trian <lb/>gulorum tam Rectilineorum quam Sphærico-<lb/>rum diſciplina, paterer in apertum lucemq́; </s> <s xml:id="echoid-s24" xml:space="preserve">pro-<lb/>ferri. </s> <s xml:id="echoid-s25" xml:space="preserve">Quæ quidem peropportuna mihi viſa res <lb/>eſt ad declarandam noſtram in te voluntatem, <lb/>perpetuamq́; </s> <s xml:id="echoid-s26" xml:space="preserve">beneuolentiam, ſi cum tui nomi-<lb/>nis inſcriptione diuulgarem. </s> <s xml:id="echoid-s27" xml:space="preserve">Quibus enim ti-<lb/>tulis labores noſtros cohoneſtemus, niſi eorum <lb/>virorum, qui vel eiuſdem ſtudij delectatione <lb/>ducuntur, vel eas res, in quibus elaboramus, ſuo <lb/>pondere momentoq́; </s> <s xml:id="echoid-s28" xml:space="preserve">perpendunt? </s> <s xml:id="echoid-s29" xml:space="preserve">Scientiæ co-<lb/>mes iucunditas eſt, & </s> <s xml:id="echoid-s30" xml:space="preserve">ea demum ſunt pretia re-<lb/>rum, quæ ſtatuerit cuiuſque cognitio. </s> <s xml:id="echoid-s31" xml:space="preserve">Tu Ma-<lb/>thematicorum ſcientia præſtas, atq; </s> <s xml:id="echoid-s32" xml:space="preserve">idcirco mi-<lb/>rificas inde voluptates hauris. </s> <s xml:id="echoid-s33" xml:space="preserve">Apud te maxi-<lb/>mo ſemper in pretio atque in honore fuit, quia <lb/>quanti facienda eſſet, eius diuturnus vſus tracta-<lb/>tioq; </s> <s xml:id="echoid-s34" xml:space="preserve">te docuit. </s> <s xml:id="echoid-s35" xml:space="preserve">Accedit, quod cum Societati <lb/>noſtræ gens tua tot iam ſummæ liberalitatis of-<lb/>ficia tribuerit, vix videtur ſummam ingrati ani-<lb/>mi notam effugere poſſe, niſi cum ſe dedit occa-<lb/>ſio, aliquid aliquando retribuat. </s> <s xml:id="echoid-s36" xml:space="preserve">Extant Grego-<lb/>rianæ liberalitatis quam ampliſſima monumen <lb/>ta: </s> <s xml:id="echoid-s37" xml:space="preserve">extant eius de Societate noſtra iudicia perho-<lb/>norifica: </s> <s xml:id="echoid-s38" xml:space="preserve">Iura data atque amplificata. </s> <s xml:id="echoid-s39" xml:space="preserve">Conceſſæ <lb/>immunitates: </s> <s xml:id="echoid-s40" xml:space="preserve">Collegia familiæ noſtræ in vlti- <pb file="010" n="10"/> mis etiam terrarum partibus excitata. </s> <s xml:id="echoid-s41" xml:space="preserve">quæ om-<lb/>nia, quantum tuo ſimus obſtricti nomini, non <lb/>modo ijs, qui nunc ſunt, ſed omnibus etiam <lb/>poſteris teſtabuntur. </s> <s xml:id="echoid-s42" xml:space="preserve">Quare quî poteram in <lb/>quærendis honeſtis titulis operibus meis te, <lb/>Princeps Ampliſſime, præterire? </s> <s xml:id="echoid-s43" xml:space="preserve">Appareat igitur <lb/>in tuo nomine munuſculum hoc vigiliarum <lb/>mearum tenue illud quidem & </s> <s xml:id="echoid-s44" xml:space="preserve">perexiguum: </s> <s xml:id="echoid-s45" xml:space="preserve">ſed <lb/>tamen eiuſmodi, quod & </s> <s xml:id="echoid-s46" xml:space="preserve">noſtra in te ſtudia, & </s> <s xml:id="echoid-s47" xml:space="preserve"><lb/>tua in nos officia grata quadam ſignificatione <lb/>memoriæ cunctis gentibus patefaciat. </s> <s xml:id="echoid-s48" xml:space="preserve">Romæ <lb/>Octauo Id. </s> <s xml:id="echoid-s49" xml:space="preserve">Decemb. </s> <s xml:id="echoid-s50" xml:space="preserve">M D LXXXV.</s> <s xml:id="echoid-s51" xml:space="preserve"/> </p> <figure> <image file="010-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/010-01"/> </figure> <pb file="011" n="11"/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head10" xml:space="preserve">ERRATORVM CORRECTIO.</head> <note position="right" xml:space="preserve"> <lb/>Pag. # Lin. # Errata. # Correctiones. <lb/>29. # 22. # Quoniam igitur in # Quoniam in <lb/>36. # 30. # atque punctum E, # atque punctum F, <lb/>44. # 20. # Quod ſecundo loco # Quod primo loco <lb/>47. ### Secunda figura cum prima paginæ 48. locum permuter. <lb/>48. #### Prima figura cum ſecunda paginæ 47. locum commutet: & in tertio caſu de. \ṁonſtrationis adhibeatur ſecunda figura paginæ 48. <lb/>48. # 11. # ADC, B E. # ADC, G H. <lb/>54. # 42. # & arcus C H, # & arcus C N, <lb/>58. # 13. # incidant. # indicant. <lb/>61. # 6. # arcus ſingulorum # arcus ſingulos <lb/>64. # 4. # IO, I B. # IO, I P. <lb/>64. # 4. ## in margine pro [12. 1. huius] lege [12. huius] <lb/>64. ### linea antepenultima, in margine apponatur [22. huius.] <lb/>65. # 12. # ad minorem # ad maiorem <lb/>67. # 6. # maiorem eſſe # minorem eſſe <lb/>69. # 3. # per C, # per D, <lb/>70. # 35. # AFC, # AFG. <lb/>75. # 22. # FB, circuli minotis # FB, circuli maioris <lb/>77. # 5. # quorum # quarum <lb/>77. # ultima. # & K L, # & K X, <lb/>80. # 10. # Si igitur ſphæra # Si igitur in ſphæra <lb/>109. # 30. ## in margine apponatur [14. quinti.] immediate ſupra [34. primi] <lb/>112. ### prope finem in margine pro [33. primi.] reponatur [34. primi.] <lb/>127. ### infra lineam vltimam pro [DF,] ponatur [HL.] <lb/>170. # 7. # Sio. # Si 60. <lb/>172. # 43. # vel partem # vel per partem <lb/>176. # 21. # quæ rectæ F E, # quàm rectæ F E, <lb/>187. ### Infra vltimam lineam pro [TABV] reponatur [LINEÆ] <lb/>188. # 1. # LINÆ # LINEÆ <lb/>189. # 2. # CAC, # CAD, <lb/>197. # 28. # tangentem arcus # ad tangentem arcus <lb/> #### In tabulis tang entium, & ſecantium pag. 203. 209. 211. 221. 223. 225. 227. 233. 235. 237. \\ 247. 249. 253. 255. 263. in margine dextro pro [39] ſcribe [29] <lb/> #### In tabula ſecantium pag. 252. ſub gradu 14. è regione minuti 29. antepenultima litera, nem- \ṗe 1. verſus dexteram debet eſſe 2. <lb/>309. # 17. # ſit angens B C, # ſi tangens B C, <lb/>315. # 28. # differentia, hæcnempe # differentia hæc, nempe <lb/>322. # 7. # in E. # in D. <lb/>334. # 14. # ipſi D C, æqualis # ipſi D C, æquali <lb/>359. # 3. # BC, # quàm B C, <lb/>374. # 28. # propoſ. 45. # propoſ. 41. & 42. <lb/>375. # vltima. # propoſ. 45. # propoſ. 66. <lb/>377. # 11. # propoſ. 45. # propoſ. 66. <lb/>378. # antepen. # propoſ. 45. # propoſ. 67. <lb/>388. ## prima figura inuerſa eſt. <lb/>395. # 14. # triangulo ſphætico # triangulo ſphærico rectangulo. <lb/>409. # 1. # eſſe totum # eſſe ſinum totum <lb/>412. # 23. ## deleantur hæcverba [vt in propoſ. dictum eſt.] <lb/>415. # 16. ## ad ſinum complemẽti, ad finum arcus E F, hoc eſt, ad finum com \ṗlementi. <lb/>433. # 22. # theorema ſequens # problema ſequens. <lb/>446. # 28. # aicu B Q, # arcu B K, <lb/>448. # 30. # ſub A V, K V, # ſub A V, K Y, <lb/></note> <pb file="012" n="12"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head11" xml:space="preserve">ERRATORVM CORRECTIO.</head> <note position="right" xml:space="preserve"> <lb/>Pag. # Lin. # Errata. # Correctiones. <lb/>471. # 24. # arum B D, # arcum B D, <lb/>472. # penult. # angum # angulum <lb/>478. # 21. # BD, notum # AD, notum <lb/>486. # 3. & 10. ## in margine pro [propoſ. 1.] ponatur [propoſ. 1.] <lb/>492. # 14. # vt æqualis. # ſit æqualis. <lb/></note> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head12" style="it" xml:space="preserve">Errata leuiora, quæ ſtudio negleximus, prudens lector facilè emendabit.</head> <figure> <image file="012-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/012-01"/> </figure> <pb file="013" n="13"/> </div> <div xml:id="echoid-div8" type="section" level="1" n="8"> <head xml:id="echoid-head13" xml:space="preserve">THEODOSII</head> <head xml:id="echoid-head14" xml:space="preserve">TRIPOLITAE</head> <head xml:id="echoid-head15" xml:space="preserve">SPHAERICORVM</head> <head xml:id="echoid-head16" xml:space="preserve">LIBRI TRES.</head> <figure> <image file="013-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/013-01"/> </figure> </div> <div xml:id="echoid-div9" type="section" level="1" n="9"> <head xml:id="echoid-head17" xml:space="preserve">PRAEF ATIO.</head> <p style="it"> <s xml:id="echoid-s52" xml:space="preserve">CVM & </s> <s xml:id="echoid-s53" xml:space="preserve">in Phænicia, & </s> <s xml:id="echoid-s54" xml:space="preserve">in <lb/>Africa vrbs, cuinomen Tri-<lb/>polis, à Georgraphis, atque Hi-<lb/>Storicis deſcribatur, certo non <lb/>conſtat apudſcriptores, vtra <lb/>harum ciuitatum Theodoſit <lb/>patria fuerit. </s> <s xml:id="echoid-s55" xml:space="preserve">Quo item tempore floruerit, non <lb/>ſatis inter eoſdem conuenit: </s> <s xml:id="echoid-s56" xml:space="preserve">Non tamen leuis <lb/>coniectura eſt, eum circatempora Pompeij Ma-<lb/>gni vixiſſe: </s> <s xml:id="echoid-s57" xml:space="preserve">propterea quod eum ſimul cum Aſcle <lb/>piade medico (quitemporibus Pompeij Magni <lb/>floruit, ſi Plinio credimus) in Bithynia floruiſſe <lb/>ſcribit Strabo. </s> <s xml:id="echoid-s58" xml:space="preserve">Scripſit autem varia opuſcula <lb/>Mathematica, vt De Habitiationibus, De <lb/>Noctibus, & </s> <s xml:id="echoid-s59" xml:space="preserve">diebus, atque etiam tres hoſce ſphæ <pb o="2" file="014" n="14" rhead=""/> ricorũlibros ſumma eruditione refertos, in qui-<lb/>bus varias ſphæræproprietates demõſtrat, qua-<lb/>rum quidem cognitio magnopere eſt neceſſaria <lb/>adrerum cæleſtium doctrinam conſequendam. <lb/></s> <s xml:id="echoid-s60" xml:space="preserve">Ctenim ſine his Aſtronomia ſuã dignit atẽ tue-<lb/>rinullaratione potect: </s> <s xml:id="echoid-s61" xml:space="preserve">Gnomonice quoque, ſeu <lb/>ratio horologiorum Solarium deſcribendorũ ex <lb/>his maximè pendet. </s> <s xml:id="echoid-s62" xml:space="preserve">Adde quod & </s> <s xml:id="echoid-s63" xml:space="preserve">ad Geogra-<lb/>phiam, & </s> <s xml:id="echoid-s64" xml:space="preserve">ad Perſpectiuam rectè intelligendam <lb/>non parum momenti habeant, vt interim alias <lb/>vtilitates ſphæricorum elementorum taceamus.</s> <s xml:id="echoid-s65" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s66" xml:space="preserve">QVONIAM verò duplex verſio Sphæ-<lb/>ricorum Theodoſie circumfertur, germana alte <lb/>ra & </s> <s xml:id="echoid-s67" xml:space="preserve">propria loannis Penæ exemplarigræco ad <lb/>verbumre ſpondens, altera Franciſci Mauro-<lb/>lyci Abbatis Meſſanenſis extraditione Ara-<lb/>bum: </s> <s xml:id="echoid-s68" xml:space="preserve">nos priorem ſecuti ſumus, quæ nouem & </s> <s xml:id="echoid-s69" xml:space="preserve"><lb/>quinquagint a propoſitionibus abſoluitur, inſe-<lb/>ruimus{q́ue} varia ſcholia, quibus plurima theore-<lb/>mat a neceſſaria, & </s> <s xml:id="echoid-s70" xml:space="preserve">ſcitu iucunda, à T heodoſio <lb/>quidem omiſſa, ab Arabibus autem adiuncta, <lb/>demonstrauimus. </s> <s xml:id="echoid-s71" xml:space="preserve">In demonſtrationibus autem <lb/>non ſumus ſecuti verba codicis Graci, ſed ſen-<lb/>ſum, vt demonstrationes ipſæ clariores fierent:</s> <s xml:id="echoid-s72" xml:space="preserve"> <pb o="3" file="015" n="15" rhead=""/> adiecimus{q́ue} nonnunquam corollaria quædam, <lb/>& </s> <s xml:id="echoid-s73" xml:space="preserve">ſcholia, necnõ lemmata, vt illis vtipoſsimus, <lb/>quandores poſtulabit. </s> <s xml:id="echoid-s74" xml:space="preserve">In margine porro appo-<lb/>ſuimus numeros ſeriem propoſitionu iuxta ver-<lb/>ſionẽ Franciſci Maurolyci referentes; </s> <s xml:id="echoid-s75" xml:space="preserve">vt facile <lb/>à quouis propoſitiones T heodoſii, quas nonnulli <lb/>ſecundũ ordinẽ Arabũ citant, poßint inueniri. <lb/></s> <s xml:id="echoid-s76" xml:space="preserve">Figuras quoque, quæ in græco exemplari extãt, <lb/>plerunque negleximus, quòd illæ, quas Mauro-<lb/>licus pinxit, commodiores ſint, & </s> <s xml:id="echoid-s77" xml:space="preserve">adintelligen-<lb/>dasres ſphæricas multò faciliores. </s> <s xml:id="echoid-s78" xml:space="preserve">Poſtremo, ne <lb/>demonstrationum curſus interrumperetur, ci-<lb/>tauimus propoſitiones Euclidis, & </s> <s xml:id="echoid-s79" xml:space="preserve">horum libro-<lb/>rum in margine. </s> <s xml:id="echoid-s80" xml:space="preserve">Id quod & </s> <s xml:id="echoid-s81" xml:space="preserve">in ſequentibus ope-<lb/>ribus obſeruauimus. </s> <s xml:id="echoid-s82" xml:space="preserve">Citationes autem hoc mo-<lb/>do intelligendæ ſunt.</s> <s xml:id="echoid-s83" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>1. primi. # Prima propoſitio lib. 1. Euclid. <lb/>18. vndec. # Decimaoctaua propoſitio lib. 11. \\ Euclid. <lb/>Coroll. 16. \ṫertij. # Corollarium propoſitionis ſex - \ṫędecimæ lib. 3. Eucl. <lb/>Coroll. 2. \\ 33. ſexti. # Corollarium ſecundum propoſi \ṫionis trigeſimætertiæ ſexti \l̇ib. Eucl. <lb/>Schol. 1. 2. \ȯctaui. # Scholium primum propoſitio- \ṅis ſecundæ lib. 8. Euclid. <lb/>4. huius. # Propoſitio quarta huius libri. <lb/>12. 2. huius. # Propoſitio duodecima libri 2. \ḣuius operis. <lb/>Coroll. 10. \ḣuius. # Corollarium propoſitionis deci \ṁæ huius lib. <lb/>Coroll. 1. \\ 1. huius. # Corollarium propoſitionis pri- \ṁæ lib. 1. huius operis. <lb/>Schol. 15. \ḣuius. # Scholium propoſitionis quintæ \ḋecimæ huius lib. <lb/>Schol. 15. \\ 1. huius. # Scholium propoſitionis quintæ \ḋecimæ lib. 1. huius operis. <lb/>20. 1. Theod. # Propoſitio vigeſima lib. 1. Theod. <lb/>Coroll. 16. \\ 1. Theod. # Corollarium propoſitionis ſex- \ṫædecimæ lib. 1. Theodoſij. <lb/>Schol. 19. \\ 1. Theod. # Scholium propoſitionis decimę. \ṅonæ lib. 1. Theodoſij. <lb/></note> <p style="it"> <s xml:id="echoid-s84" xml:space="preserve">Ex his aliæ citationes facilè percipientur, <lb/>cum in omnibus eadem ſit ratio.</s> <s xml:id="echoid-s85" xml:space="preserve"/> </p> <pb file="016" n="16"/> </div> <div xml:id="echoid-div10" type="section" level="1" n="10"> <head xml:id="echoid-head18" xml:space="preserve">THEODOSII</head> <head xml:id="echoid-head19" xml:space="preserve">SPHAERICORVM</head> <head xml:id="echoid-head20" xml:space="preserve">LIBER PRIMVS.</head> <figure> <image file="016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/016-01"/> </figure> </div> <div xml:id="echoid-div11" type="section" level="1" n="11"> <head xml:id="echoid-head21" style="it" xml:space="preserve">DEFINIT IONES.</head> <head xml:id="echoid-head22" xml:space="preserve">I</head> <p> <s xml:id="echoid-s86" xml:space="preserve">SPHAERA eſt figura ſolida compre-<lb/>henſa vna ſuperficie, ad quam ab vno <lb/>eorum punctorum, quæ intra figuram <lb/>ſunt, omnes rectæ lineæ ductæ ſunt in-<lb/>ter ſe æquales.</s> <s xml:id="echoid-s87" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div12" type="section" level="1" n="12"> <head xml:id="echoid-head23" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s88" xml:space="preserve">Centrum autem Sphæræ, eſt eiuſmodi punctũ.</s> <s xml:id="echoid-s89" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div13" type="section" level="1" n="13"> <head xml:id="echoid-head24" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s90" xml:space="preserve">Axis verò Sphæræ, eſt recta quædã linea per cen <lb/>trũ ducta, & </s> <s xml:id="echoid-s91" xml:space="preserve">vtrin que terminata in ſphæræ ſuper-<lb/>ficie, circa quã quieſcentẽ circumuoluitur ſphęra.</s> <s xml:id="echoid-s92" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div14" type="section" level="1" n="14"> <head xml:id="echoid-head25" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s93" xml:space="preserve">Poli ſphæræ ſunt extrema puncta ipſius axis.</s> <s xml:id="echoid-s94" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div15" type="section" level="1" n="15"> <head xml:id="echoid-head26" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s95" xml:space="preserve">Polus Circuli in Sphæra, eſt punctum in ſuper-<lb/>ficie ſphæræ, à quo omnes rectæ lineæ ad Circuli <lb/>circumferentiam tendentes ſuntinter ſe æquales.</s> <s xml:id="echoid-s96" xml:space="preserve"/> </p> <pb o="5" file="017" n="17" rhead=""/> </div> <div xml:id="echoid-div16" type="section" level="1" n="16"> <head xml:id="echoid-head27" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s97" xml:space="preserve">_ADDITVR_ in exemplari græco alia adhuc definitio, qua explicatur, quid ſit <lb/>planum ad planum ſimiliter inclinari, atque alterum ad alterum. </s> <s xml:id="echoid-s98" xml:space="preserve">Sed quoniam in-<lb/>clinatio plani ad planum ab Euclide explicata eſt lib. </s> <s xml:id="echoid-s99" xml:space="preserve">11. </s> <s xml:id="echoid-s100" xml:space="preserve">defin. </s> <s xml:id="echoid-s101" xml:space="preserve">6. </s> <s xml:id="echoid-s102" xml:space="preserve">At vero, quan-<lb/>do planum ad planum ſimiliter inclinari dicitur, atque alterum ad alterum, eodem <lb/>lib defin. </s> <s xml:id="echoid-s103" xml:space="preserve">7. </s> <s xml:id="echoid-s104" xml:space="preserve">declaratum eſt, ſtatui eam omnino omittere hoc loco, & </s> <s xml:id="echoid-s105" xml:space="preserve">ſequentem ap-<lb/>ponere non dißimilem definitioni 4. </s> <s xml:id="echoid-s106" xml:space="preserve">lib. </s> <s xml:id="echoid-s107" xml:space="preserve">3. </s> <s xml:id="echoid-s108" xml:space="preserve">Euclidis, ita vt ſextum locum obtineat.</s> <s xml:id="echoid-s109" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div17" type="section" level="1" n="17"> <head xml:id="echoid-head28" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s110" xml:space="preserve">IN Sphæra æqualiter diſtare à centro ſphæræ <lb/>circuli dicuntur, cum perpendiculares, quæ à cen-<lb/>tro ſphærę in ipſorum plana ducuntur, ſunt æqua-<lb/>les. </s> <s xml:id="echoid-s111" xml:space="preserve">Longius autem abeſſe ille dicitur, in cuius pla-<lb/>num maior perpendicularis cadit.</s> <s xml:id="echoid-s112" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div18" type="section" level="1" n="18"> <head xml:id="echoid-head29" xml:space="preserve">THEOREMA 1. PROPOS. 1.</head> <note position="right" xml:space="preserve">1.</note> <p> <s xml:id="echoid-s113" xml:space="preserve">SI Sphærica ſuperficies plano aliquo ſece-<lb/>tur, linea quæ fit in ſphæræ ſuperficie, eſt <lb/>circumferentia circuli.</s> <s xml:id="echoid-s114" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s115" xml:space="preserve">SECETVR Sphærica ſuperficies A B C, cuius centrum D, plano ali-<lb/>quo ſaciente in ſuperficie ſphæræ lineam B E F C G. </s> <s xml:id="echoid-s116" xml:space="preserve">Dico B E F C G, cir-<lb/> <anchor type="figure" xlink:label="fig-017-01a" xlink:href="fig-017-01"/> cumferentiam eſ-<lb/>ſe circuli. </s> <s xml:id="echoid-s117" xml:space="preserve">Tran-<lb/>ſeat enim primò <lb/>planum ſecans per <lb/>centrũ ſphæræ D, <lb/>ita vt D, ſit in pla-<lb/>no ſecante, in quo <lb/>ex D, ad lineam fa <lb/>ctam B E F C G, du <lb/>cantur lineæ rectæ <lb/>quotcunque D E, <lb/>D F, D G. </s> <s xml:id="echoid-s118" xml:space="preserve">Quo-<lb/>niam igitur omnes <lb/>hæ lineæ ductæ, <lb/>quotcunque fuerint, cum ex centro ſphæræ ad eius ſuperficiem cadant, inter <lb/>ſe æquales ſunt, erit, per defin. </s> <s xml:id="echoid-s119" xml:space="preserve">15. </s> <s xml:id="echoid-s120" xml:space="preserve">lib. </s> <s xml:id="echoid-s121" xml:space="preserve">1 Eucl. </s> <s xml:id="echoid-s122" xml:space="preserve">linea B E F C G, circunferen-<lb/>tia circulia, cuius centrum D, idem quod ſphæræ.</s> <s xml:id="echoid-s123" xml:space="preserve"/> </p> <div xml:id="echoid-div18" type="float" level="2" n="1"> <figure xlink:label="fig-017-01" xlink:href="fig-017-01a"> <image file="017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/017-01"/> </figure> </div> <pb o="6" file="018" n="18" rhead=""/> <p> <s xml:id="echoid-s124" xml:space="preserve">TRANSEAT deinde planum ſecans non per centrum ſphæræ. </s> <s xml:id="echoid-s125" xml:space="preserve">Du-<lb/> <anchor type="note" xlink:label="note-018-01a" xlink:href="note-018-01"/> catur autem ex D, centro ſphæræ ad planum ſecans perpendicularis D H, <lb/> <anchor type="figure" xlink:label="fig-018-01a" xlink:href="fig-018-01"/> emittãturq; </s> <s xml:id="echoid-s126" xml:space="preserve">ex H, <lb/>rectę vtcunq; </s> <s xml:id="echoid-s127" xml:space="preserve">H E, <lb/>H F, ad lineam B E <lb/>F C G, & </s> <s xml:id="echoid-s128" xml:space="preserve">cõnectan <lb/>tur rectæ D E, D F. <lb/></s> <s xml:id="echoid-s129" xml:space="preserve">Quoniã igitur an-<lb/>guli D H E, D H F, <lb/>recti ſunt, ex defin. </s> <s xml:id="echoid-s130" xml:space="preserve"><lb/>3. </s> <s xml:id="echoid-s131" xml:space="preserve">lib. </s> <s xml:id="echoid-s132" xml:space="preserve">11. </s> <s xml:id="echoid-s133" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s134" xml:space="preserve"><lb/>erit tam quadratũ <lb/>ex D E, quadratis <lb/>ex D H, H E, quàm <lb/> <anchor type="note" xlink:label="note-018-02a" xlink:href="note-018-02"/> quadratũ ex D F, <lb/>quadratis ex D H, <lb/>H F, æ quale: </s> <s xml:id="echoid-s135" xml:space="preserve">Sunt autem quadrata ex D E, D F, inter ſe æqualia, quod & </s> <s xml:id="echoid-s136" xml:space="preserve"><lb/>rectæ D E, D F, ex centro ſphæræ in eius ſuperficiẽ cadentes inter ſe æqua-<lb/>les ſint. </s> <s xml:id="echoid-s137" xml:space="preserve">Quadrata igitur ex D H, H E, ſimul quadratis ex D H, H F, ſi-<lb/>mul æqualia erunt. </s> <s xml:id="echoid-s138" xml:space="preserve">Dempto igitur communi quadrato rectæ D H, reliquæ <lb/>quadrata rectarum H E, H F, inter ſe æqualia, & </s> <s xml:id="echoid-s139" xml:space="preserve">rectæ propterea H E, H F, <lb/>inter ſe æquales erunt. </s> <s xml:id="echoid-s140" xml:space="preserve">Eodem argumento oſtendemus, omnes lineas ex H, ad <lb/>lineam B E F C G, cadentes eſſe æquales & </s> <s xml:id="echoid-s141" xml:space="preserve">inter ſe, & </s> <s xml:id="echoid-s142" xml:space="preserve">dictis duabus H E, <lb/>H F. </s> <s xml:id="echoid-s143" xml:space="preserve">Linea ergo B E F C G, circum ſerentia erit circuli, ex defin. </s> <s xml:id="echoid-s144" xml:space="preserve">15. </s> <s xml:id="echoid-s145" xml:space="preserve">lib. </s> <s xml:id="echoid-s146" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s147" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s148" xml:space="preserve">cuius centrum eſt punctum H, in quod perpendicularis D H, cadit. </s> <s xml:id="echoid-s149" xml:space="preserve"><lb/>Quare ſi ſphærica ſuperficies Plano aliquo ſecetur, &</s> <s xml:id="echoid-s150" xml:space="preserve">c. </s> <s xml:id="echoid-s151" xml:space="preserve">Quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s152" xml:space="preserve"/> </p> <div xml:id="echoid-div19" type="float" level="2" n="2"> <note position="left" xlink:label="note-018-01" xlink:href="note-018-01a" xml:space="preserve">11. vndec.</note> <figure xlink:label="fig-018-01" xlink:href="fig-018-01a"> <image file="018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/018-01"/> </figure> <note position="left" xlink:label="note-018-02" xlink:href="note-018-02a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div21" type="section" level="1" n="19"> <head xml:id="echoid-head30" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s153" xml:space="preserve">ITAQVE ſi planum ſecans per centrum ſphæræ tranſierit’, efficietur circulus idem <lb/>centrum habens, quod ſphæra. </s> <s xml:id="echoid-s154" xml:space="preserve">Si verò non per centrum tranſierit, efficientur circulus aliud <lb/>habens centrum, quàm ſphæra, illud videlicet punctum, in quod cadit perpendicularis ex <lb/>centro ſphæræ ad planum ſecans deducta. </s> <s xml:id="echoid-s155" xml:space="preserve">Nam ſemper demonſtrabuntur lineæ rectæ caden <lb/>tes ex hoc puncto in circum ferentiam circuli eſſe æquales.</s> <s xml:id="echoid-s156" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div22" type="section" level="1" n="20"> <head xml:id="echoid-head31" xml:space="preserve">HOCEST.</head> <p> <s xml:id="echoid-s157" xml:space="preserve">IDEM eſt ſphæræ centrum, & </s> <s xml:id="echoid-s158" xml:space="preserve">circuli per ſphæræ centrum traiecti. </s> <s xml:id="echoid-s159" xml:space="preserve">Et perpendiculatis <lb/>ducta à centro ſphæræ in planum circuli per centrum ſphæræ non traiecti, cadit in centrum <lb/>circuli: </s> <s xml:id="echoid-s160" xml:space="preserve">quia punctum H, in quod perpendicularis D H, cadit, demonſtratum cſt centrum <lb/>eſſe circuli.</s> <s xml:id="echoid-s161" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div23" type="section" level="1" n="21"> <head xml:id="echoid-head32" xml:space="preserve">PROBL. 1. PROPOS. 2.</head> <note position="left" xml:space="preserve">2.</note> </div> <div xml:id="echoid-div24" type="section" level="1" n="22"> <head xml:id="echoid-head33" xml:space="preserve">DATAE Sphæræ centrum inuenire.</head> <p> <s xml:id="echoid-s162" xml:space="preserve">SIT centrum inueniendum Sphæræ A B C D. </s> <s xml:id="echoid-s163" xml:space="preserve">Secetur eius ſuperficies <lb/> <anchor type="note" xlink:label="note-018-04a" xlink:href="note-018-04"/> plano quopiam faciente in ipſa lineam B D E, quæ circuli circumferentia <lb/>crit. </s> <s xml:id="echoid-s164" xml:space="preserve">Sit huius circuli centrum F. </s> <s xml:id="echoid-s165" xml:space="preserve">Siigitur circulus B D E, per centrum ſphæ <lb/>ræ traijcitur, erit punctum F, centrum quoque ſphæræ. </s> <s xml:id="echoid-s166" xml:space="preserve">Si verò per centrum <lb/>ſphæræ non traijcitur, erigatur ex F, ad planum circuli B D E, perpendicu- <pb o="7" file="019" n="19" rhead=""/> laris F G, quæ vtrinque ad ſuperficiem ſphæ-<lb/> <anchor type="figure" xlink:label="fig-019-01a" xlink:href="fig-019-01"/> ræ educta ad puncta A, C, ſecetur bifariam in <lb/>G. </s> <s xml:id="echoid-s167" xml:space="preserve">Dico G, centrum eſſe ſphæræ. </s> <s xml:id="echoid-s168" xml:space="preserve">Si enim nõ <lb/>eſt, ſit, ſi fieri poteſt, centrum H, ſecans diame <lb/>tros omnes bifariã, quod quidem in linea A C, <lb/>nõ exiſtet, cũ ea in puncto G, ſolũ bifariã diui <lb/>datur, ſed extra illã. </s> <s xml:id="echoid-s169" xml:space="preserve">Demittatur ex H, centro <lb/>ſphæræ ad planum circuli B D E, perpendicu <lb/>laris H I, quæ æquidiſtans erit lineæ F G; </s> <s xml:id="echoid-s170" xml:space="preserve">ac <lb/>proinde in punctum F, non cadet: </s> <s xml:id="echoid-s171" xml:space="preserve">coirent em̃ <lb/>tunc duæ parallelæ H I, G F, in F, puncto, <lb/>quod fieri non poteſt. </s> <s xml:id="echoid-s172" xml:space="preserve">Quoniam verò perpen <lb/>dicularis ex centro ſphæræ in planũ circuli B D E, demiſſa cadit in eius cen-<lb/> <anchor type="note" xlink:label="note-019-01a" xlink:href="note-019-01"/> trum, erit I, centrum circuli B D E. </s> <s xml:id="echoid-s173" xml:space="preserve">Sed & </s> <s xml:id="echoid-s174" xml:space="preserve">F, ex conſtructione, centrum eſt <lb/>eiuſdem circuli. </s> <s xml:id="echoid-s175" xml:space="preserve">Quod abſurdum eſt. </s> <s xml:id="echoid-s176" xml:space="preserve">Idem enim circulus vnum tantum ha-<lb/>beat centrum neceſſe eſt. </s> <s xml:id="echoid-s177" xml:space="preserve">Non ergo aliud punctum præter G, centrum erit <lb/>ſphæræ. </s> <s xml:id="echoid-s178" xml:space="preserve">Quare datæ ſphæræ centrum inuenimus. </s> <s xml:id="echoid-s179" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s180" xml:space="preserve"/> </p> <div xml:id="echoid-div24" type="float" level="2" n="1"> <note position="left" xlink:label="note-018-04" xlink:href="note-018-04a" xml:space="preserve">1. huius. <lb/>1. tertij. <lb/>Coroll. 1. <lb/>huius.</note> <figure xlink:label="fig-019-01" xlink:href="fig-019-01a"> <image file="019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/019-01"/> </figure> <note position="right" xlink:label="note-019-01" xlink:href="note-019-01a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div26" type="section" level="1" n="23"> <head xml:id="echoid-head34" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s181" xml:space="preserve">HINC conſtat, ſi in ſphæra ſit circulus non per centrum ſphæræ traiectus, à cuius cen-<lb/>tro excitetur perpendicularis ad ipſius planum, in linea perpendiculari centrũ eſſe ſphærę. <lb/></s> <s xml:id="echoid-s182" xml:space="preserve">Oſtenſum enim eſt, punctum G, quod perpendicularẽ A C, bifariã diuidit, eſſe ſphærę centrũ.</s> <s xml:id="echoid-s183" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div27" type="section" level="1" n="24"> <head xml:id="echoid-head35" xml:space="preserve">THEOREMA 2. PROPOS. 3.</head> <p> <s xml:id="echoid-s184" xml:space="preserve">SPHAERA planum, à quo non ſecatur, non <lb/> <anchor type="note" xlink:label="note-019-02a" xlink:href="note-019-02"/> tangit in pluribus punctis vno.</s> <s xml:id="echoid-s185" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="1"> <note position="left" xlink:label="note-019-02" xlink:href="note-019-02a" xml:space="preserve">3.</note> </div> <p> <s xml:id="echoid-s186" xml:space="preserve">SI enim fieri poteſt, ſphæra planum, à quo non ſecatur, tangat in pluri-<lb/> <anchor type="note" xlink:label="note-019-03a" xlink:href="note-019-03"/> bus punctis vno, vt in A, & </s> <s xml:id="echoid-s187" xml:space="preserve">B. </s> <s xml:id="echoid-s188" xml:space="preserve">Inuento igitur C, centro ſphæræ, ducantur re <lb/> <anchor type="figure" xlink:label="fig-019-02a" xlink:href="fig-019-02"/> ctæ C A, C B: </s> <s xml:id="echoid-s189" xml:space="preserve">& </s> <s xml:id="echoid-s190" xml:space="preserve">per C A, C B, ducatur pla-<lb/>num faciens quidem in ſuperficie ſphæræ cir <lb/> <anchor type="note" xlink:label="note-019-04a" xlink:href="note-019-04"/> cumferentiam circuli A B D, in plano autẽ <lb/>ſecante rectam lineam E A B F. </s> <s xml:id="echoid-s191" xml:space="preserve">Quia igitur <lb/> <anchor type="note" xlink:label="note-019-05a" xlink:href="note-019-05"/> planũ tangens, in quo eſt recta E A B F, ſphæ <lb/>ram non ſecat, atque adeò neque circulum <lb/>A B D, in ſphęrę ſuperſicie exiſtentem, fit vt <lb/>neq; </s> <s xml:id="echoid-s192" xml:space="preserve">recta E A B F, circulũ A B D, ſecet. </s> <s xml:id="echoid-s193" xml:space="preserve">Cadet <lb/>ergo recta A B, tota extra circulũ. </s> <s xml:id="echoid-s194" xml:space="preserve">Quoniã <lb/>vero duo puncta ſumpta ſunt A, B, in circũfe <lb/>rentia circuli A B D, cadet eadem recta A B, à <lb/>pũcto A, in punctũ B, ducta tota in tra circulũ <lb/> <anchor type="note" xlink:label="note-019-06a" xlink:href="note-019-06"/> A B D. </s> <s xml:id="echoid-s195" xml:space="preserve">Quod eſt abſurdũ. </s> <s xml:id="echoid-s196" xml:space="preserve">Sphęra igit̃ planũ, <lb/>à quo nõ ſecatur, nõ tangit in pluribus pũctis vno. </s> <s xml:id="echoid-s197" xml:space="preserve">Quod erat demonſtrandũ.</s> <s xml:id="echoid-s198" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="2"> <note position="right" xlink:label="note-019-03" xlink:href="note-019-03a" xml:space="preserve">2. huius.</note> <figure xlink:label="fig-019-02" xlink:href="fig-019-02a"> <image file="019-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/019-02"/> </figure> <note position="right" xlink:label="note-019-04" xlink:href="note-019-04a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-019-05" xlink:href="note-019-05a" xml:space="preserve">3. vndec.</note> <note position="right" xlink:label="note-019-06" xlink:href="note-019-06a" xml:space="preserve">2. tertij.</note> </div> </div> <div xml:id="echoid-div30" type="section" level="1" n="25"> <head xml:id="echoid-head36" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s199" xml:space="preserve">HINC fit, ſi duo puncta ſignentur in ſuperficie ſphæræ, rectam, quæ illa connectit, intra <lb/>ſphæram cadere. </s> <s xml:id="echoid-s200" xml:space="preserve">quia videlicet cadit intra circulum, qui in ſphæræ ſuperficie circumferen <lb/> <anchor type="note" xlink:label="note-019-07a" xlink:href="note-019-07"/> tiam habet.</s> <s xml:id="echoid-s201" xml:space="preserve"/> </p> <div xml:id="echoid-div30" type="float" level="2" n="1"> <note position="right" xlink:label="note-019-07" xlink:href="note-019-07a" xml:space="preserve">2. tertij.</note> </div> <pb o="8" file="020" n="20" rhead=""/> <note position="left" xml:space="preserve">4.</note> </div> <div xml:id="echoid-div32" type="section" level="1" n="26"> <head xml:id="echoid-head37" xml:space="preserve">THEOREMA 3. PROPOS. 4.</head> <p> <s xml:id="echoid-s202" xml:space="preserve">SI Sphæra planum tangat, quod eam non ſe-<lb/>cet, recta linea ducta à centro ſphæræ ad conta-<lb/>ctum, perpendicularis erit ad planum.</s> <s xml:id="echoid-s203" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s204" xml:space="preserve">TANGAT Sphæra planum, quod ip <lb/> <anchor type="note" xlink:label="note-020-02a" xlink:href="note-020-02"/> <anchor type="figure" xlink:label="fig-020-01a" xlink:href="fig-020-01"/> ſam non ſecet, in puncto A: </s> <s xml:id="echoid-s205" xml:space="preserve">Et inuento B, <lb/>centro ſphæræ, ducatur ab eo recta B A, ad <lb/>punctum contactus A. </s> <s xml:id="echoid-s206" xml:space="preserve">Dico rectam B A, ad <lb/>dictum planum perpendicularem eſſe. </s> <s xml:id="echoid-s207" xml:space="preserve">Nam <lb/>per rectam A B, ducantur duo plana vtcun <lb/>que ſe mutuo ſecãtia, quæ in ſuperficie qui-<lb/> <anchor type="note" xlink:label="note-020-03a" xlink:href="note-020-03"/> dem ſphæræ faciant circulorum circumfe-<lb/>rentias A C D E, A F D G, in plano autẽ <lb/> <anchor type="note" xlink:label="note-020-04a" xlink:href="note-020-04"/> tangente rectas H A I, K A L. </s> <s xml:id="echoid-s208" xml:space="preserve">Quoniara <lb/>igitur vterque circulus A C D E, A F D G, <lb/>per centrum B, ſphæræ traijcitur, erit quo-<lb/> <anchor type="note" xlink:label="note-020-05a" xlink:href="note-020-05"/> que B, vtriuſque centrum. </s> <s xml:id="echoid-s209" xml:space="preserve">Rurſus quia planum tangens ſphęram non ſecat, <lb/>fit, vt neque rectæ H A I, K A L, in eo exiſtentes eandem ſecent; </s> <s xml:id="echoid-s210" xml:space="preserve">ac proinde <lb/>neque circulos A C D E, A F D G, in ſphæræ ſuperficie exiſtentes. </s> <s xml:id="echoid-s211" xml:space="preserve">Tanget <lb/>igitur recta H A I, circulum A C D E, in puncto A, & </s> <s xml:id="echoid-s212" xml:space="preserve">recta K A L, circulum <lb/> <anchor type="note" xlink:label="note-020-06a" xlink:href="note-020-06"/> A F D G, in eodem puncto A. </s> <s xml:id="echoid-s213" xml:space="preserve">Igitur recta B A, & </s> <s xml:id="echoid-s214" xml:space="preserve">ad rectam H A I, & </s> <s xml:id="echoid-s215" xml:space="preserve">ad re-<lb/>ctam K A L, perpendicularis eſt. </s> <s xml:id="echoid-s216" xml:space="preserve">Quare eadem recta B A, & </s> <s xml:id="echoid-s217" xml:space="preserve">ad planum tan-<lb/> <anchor type="note" xlink:label="note-020-07a" xlink:href="note-020-07"/> gens, quod per rectas H A I, K A L, ducitur, perpendicularis erit. </s> <s xml:id="echoid-s218" xml:space="preserve">Si ſphæra <lb/>ergo planum tangat, quod eam non ſecet, &</s> <s xml:id="echoid-s219" xml:space="preserve">c. </s> <s xml:id="echoid-s220" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s221" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="1"> <note position="left" xlink:label="note-020-02" xlink:href="note-020-02a" xml:space="preserve">2. huius.</note> <figure xlink:label="fig-020-01" xlink:href="fig-020-01a"> <image file="020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/020-01"/> </figure> <note position="left" xlink:label="note-020-03" xlink:href="note-020-03a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-020-04" xlink:href="note-020-04a" xml:space="preserve">3. vndec.</note> <note position="left" xlink:label="note-020-05" xlink:href="note-020-05a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="left" xlink:label="note-020-06" xlink:href="note-020-06a" xml:space="preserve">18. tertij.</note> <note position="left" xlink:label="note-020-07" xlink:href="note-020-07a" xml:space="preserve">4. vndec.</note> </div> </div> <div xml:id="echoid-div34" type="section" level="1" n="27"> <head xml:id="echoid-head38" xml:space="preserve">THEOREMA 4. PROPOS. 5.</head> <note position="left" xml:space="preserve">5.</note> <p> <s xml:id="echoid-s222" xml:space="preserve">SI Sphæra planum tangat, quod ipſam non ſe-<lb/>cet, à contactu autem excitetur recta linea ad an-<lb/>gulos rectos ipſi plano, in linea excitata erit cen-<lb/>trum ſphæræ.</s> <s xml:id="echoid-s223" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s224" xml:space="preserve">SPHAERA A B C D, tãgat in C, pun <lb/> <anchor type="figure" xlink:label="fig-020-02a" xlink:href="fig-020-02"/> cto planum E F, quod eam non ſecet, à pun <lb/>cto autem C, excitetur ad planum E F, per-<lb/> <anchor type="note" xlink:label="note-020-09a" xlink:href="note-020-09"/> pendicularis C A. </s> <s xml:id="echoid-s225" xml:space="preserve">Dico in A C, centrum eſ <lb/>ſe ſphæræ. </s> <s xml:id="echoid-s226" xml:space="preserve">Si enim non eſt, ſit G, centrum <lb/>ſphæræ extra rectam A C, ſi fieri poteſt, & </s> <s xml:id="echoid-s227" xml:space="preserve">à <lb/>G, ad C, recta ducatur G C, quę ad planum <lb/> <anchor type="note" xlink:label="note-020-10a" xlink:href="note-020-10"/> E F, perpendicularis erit: </s> <s xml:id="echoid-s228" xml:space="preserve">Erat autẽ & </s> <s xml:id="echoid-s229" xml:space="preserve">A C, <lb/>ad idem planum perpendicularis. </s> <s xml:id="echoid-s230" xml:space="preserve">Igitur ex <lb/>eodem puncto C, ad idem planum E F, duæ <lb/>perpendiculares ducuntur. </s> <s xml:id="echoid-s231" xml:space="preserve">Quod eſt abſur- <pb o="9" file="021" n="21" rhead=""/> dum. </s> <s xml:id="echoid-s232" xml:space="preserve">Dato enim plano, à puncto, quod in illo datum eſt, duæ rectæ lincæ ad <lb/> <anchor type="note" xlink:label="note-021-01a" xlink:href="note-021-01"/> rectos angulos non excitantur. </s> <s xml:id="echoid-s233" xml:space="preserve">Quare ſi ſphæra planum tangat, quod ipſam <lb/>non ſecet, &</s> <s xml:id="echoid-s234" xml:space="preserve">c. </s> <s xml:id="echoid-s235" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s236" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="1"> <figure xlink:label="fig-020-02" xlink:href="fig-020-02a"> <image file="020-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/020-02"/> </figure> <note position="left" xlink:label="note-020-09" xlink:href="note-020-09a" xml:space="preserve">12. vndec.</note> <note position="left" xlink:label="note-020-10" xlink:href="note-020-10a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-021-01" xlink:href="note-021-01a" xml:space="preserve">13. vndee.</note> </div> </div> <div xml:id="echoid-div36" type="section" level="1" n="28"> <head xml:id="echoid-head39" xml:space="preserve">THEOREMA 5. PROPOS. 6.</head> <note position="right" xml:space="preserve">6.</note> <note position="right" xml:space="preserve">7.</note> <p> <s xml:id="echoid-s237" xml:space="preserve">CIRCVLORVM, qui in ſphæra ſunt, ma-<lb/>ximi ſunt, qui per ſphærę centrum ducuntui: </s> <s xml:id="echoid-s238" xml:space="preserve">alio-<lb/>rum autem illi inter ſe æquales ſunt, qui æqualiter <lb/>à centro diſtát: </s> <s xml:id="echoid-s239" xml:space="preserve">qui vero longius à centro diſtant, <lb/>minores ſunt. </s> <s xml:id="echoid-s240" xml:space="preserve">Et circuli in ſphæra maximi per <lb/>ſphæræ centrum tranſeunt: </s> <s xml:id="echoid-s241" xml:space="preserve">aliorum autem æqua-<lb/>les à centro æqualiter diſtant: </s> <s xml:id="echoid-s242" xml:space="preserve">minores verò lon-<lb/>gius à centro diſtant.</s> <s xml:id="echoid-s243" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s244" xml:space="preserve">IN ſphæra A B C D E F, cuius centrum G, tranſeat circulus A D, per <lb/>centrum G, & </s> <s xml:id="echoid-s245" xml:space="preserve">alij B C, F E, non per centrum. </s> <s xml:id="echoid-s246" xml:space="preserve">Dico A D, circulum eſſe om-<lb/>nium maximum, &</s> <s xml:id="echoid-s247" xml:space="preserve">c. </s> <s xml:id="echoid-s248" xml:space="preserve">Ducantur ex centro G, ad plana circulorum B C, F E, <lb/> <anchor type="note" xlink:label="note-021-04a" xlink:href="note-021-04"/> perpendiculares G H, G I, quæ in ipſorum centra cadent; </s> <s xml:id="echoid-s249" xml:space="preserve">ita vt H, I, cen-<lb/>tra ſint circulorum B C, F E: </s> <s xml:id="echoid-s250" xml:space="preserve">Eſt autem G, centrũ ſphæræ, centrũ quoq; </s> <s xml:id="echoid-s251" xml:space="preserve">cir-<lb/> <anchor type="figure" xlink:label="fig-021-01a" xlink:href="fig-021-01"/> <anchor type="note" xlink:label="note-021-05a" xlink:href="note-021-05"/> culi A D, per centrum ſphæræ tra-<lb/>iecti. </s> <s xml:id="echoid-s252" xml:space="preserve">Si igitur ex G, H, I, ad ſuper-<lb/>ficiem ſphæræ rectæ ducantur G D, <lb/>H C, I E, erũt hæ ſemidiametri cir <lb/>culorum A D, B C, F E. </s> <s xml:id="echoid-s253" xml:space="preserve">Conne-<lb/>ctantur autem rectæ G C, G E. </s> <s xml:id="echoid-s254" xml:space="preserve">Quo <lb/>niam igitur in triangulo G H C, an <lb/>gulus H, rectus eſt, ex defin. </s> <s xml:id="echoid-s255" xml:space="preserve">3. </s> <s xml:id="echoid-s256" xml:space="preserve">lib. </s> <s xml:id="echoid-s257" xml:space="preserve">11 <lb/>Eucl. </s> <s xml:id="echoid-s258" xml:space="preserve">erit quadratum ex G C, æqua <lb/> <anchor type="note" xlink:label="note-021-06a" xlink:href="note-021-06"/> le quadratis ex G H, H C. </s> <s xml:id="echoid-s259" xml:space="preserve">Dempto <lb/>ergo quadrato rectæ G H, maius e-<lb/>rit quadratum ex G C, quadrato ex <lb/>H C; </s> <s xml:id="echoid-s260" xml:space="preserve">atque adeò & </s> <s xml:id="echoid-s261" xml:space="preserve">recta G C, hoc <lb/>eſt, ſibi æqualis G D, (ducuntur em̃ <lb/>G C, G D, ex centro ſphæræ ad ſu-<lb/>perficiem) maior erit, quàm recta <lb/>H C. </s> <s xml:id="echoid-s262" xml:space="preserve">Quare circulus A D, maiorẽ <lb/>habens ſemidiametrum, quàm circulus B C, maior erit circulo B C. </s> <s xml:id="echoid-s263" xml:space="preserve">Non ſe-<lb/>cus oſtendemus, circulum A D, quocunque alio, qui per centrum G, non <lb/>tranſeat, maiorem eſſe. </s> <s xml:id="echoid-s264" xml:space="preserve">Maximus eſt ergo circulus A D.</s> <s xml:id="echoid-s265" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="1"> <note position="right" xlink:label="note-021-04" xlink:href="note-021-04a" xml:space="preserve">11. vndec. <lb/>Coroll. 1. <lb/>huius.</note> <figure xlink:label="fig-021-01" xlink:href="fig-021-01a"> <image file="021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/021-01"/> </figure> <note position="right" xlink:label="note-021-05" xlink:href="note-021-05a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="right" xlink:label="note-021-06" xlink:href="note-021-06a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s266" xml:space="preserve">DISTENT iam circuli B C, F E, à centro G, æqualiter, hoc eſt, per-<lb/>pendiculares G H, G I, æquales ſint, ex deſin. </s> <s xml:id="echoid-s267" xml:space="preserve">6. </s> <s xml:id="echoid-s268" xml:space="preserve">huius libri. </s> <s xml:id="echoid-s269" xml:space="preserve">Dico circulos <lb/>B C, F E, æquales eſſe. </s> <s xml:id="echoid-s270" xml:space="preserve">Cum enim rectæ G C, G E, à centro ſphæræ in eius ſu <pb o="10" file="022" n="22" rhead=""/> perficiem cadentes ſint æquales, ac proinde & </s> <s xml:id="echoid-s271" xml:space="preserve">earum quadrata æqualia; </s> <s xml:id="echoid-s272" xml:space="preserve">ſit au <lb/>tem tam quadratum ex G C, quadratis ex G H, H C, quàm quadratum ex <lb/> <anchor type="note" xlink:label="note-022-01a" xlink:href="note-022-01"/> G E, quadratis ex G I, I E, æquale; </s> <s xml:id="echoid-s273" xml:space="preserve">erũt quadrata ex G H, H C, ſimul æqua-<lb/>lia quadratis ex G I, I E, ſimul. </s> <s xml:id="echoid-s274" xml:space="preserve">Demptis ergo æqualibus quadratis rectarum <lb/>G H, G I, (poſitæ enim ſunt hæ lineæ æquales) æqualia erunt reliqua quadra <lb/>ta rectarum H C, I E, ac proinde & </s> <s xml:id="echoid-s275" xml:space="preserve">rectæ H C, I E, æquales erunt: </s> <s xml:id="echoid-s276" xml:space="preserve">quæ cum <lb/>ſint ſemidiametri circulorum B C, F E, æquales erunt circuli ipſi B C, F E.</s> <s xml:id="echoid-s277" xml:space="preserve"/> </p> <div xml:id="echoid-div37" type="float" level="2" n="2"> <note position="left" xlink:label="note-022-01" xlink:href="note-022-01a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s278" xml:space="preserve">QVOD ſi alter horũ circulorũ, nempe B C, longius à centro G, ponatur <lb/>diſtare, quàm alter F E, hoc eſt, perpẽdicularis G H, maior ponatur perpen-<lb/>diculari G I, oſtendemus eodem fere modo, circulum B C, minorem eſſe cir-<lb/>culo F E. </s> <s xml:id="echoid-s279" xml:space="preserve">Cum enim quadrata ex G H, H C, æqualia ſint demonſtrata qua-<lb/>dratis ex G I, I E; </s> <s xml:id="echoid-s280" xml:space="preserve">ſi auferantur quadrata inæqualia rectarum inæqualium <lb/> <anchor type="figure" xlink:label="fig-022-01a" xlink:href="fig-022-01"/> G H, G I, quorum illud maius eſt, <lb/>(quòd & </s> <s xml:id="echoid-s281" xml:space="preserve">recta G H, maior ponatur <lb/>quàm recta G I,) erit reliquum qua <lb/>dratum rectæ H C, minus quadrato <lb/>reliquo rectæ I E; </s> <s xml:id="echoid-s282" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s283" xml:space="preserve">re <lb/>cta H C, minor erit, quàm recta I E. <lb/></s> <s xml:id="echoid-s284" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s285" xml:space="preserve">circulus B C, circulo F E, <lb/>minor erit.</s> <s xml:id="echoid-s286" xml:space="preserve"/> </p> <div xml:id="echoid-div38" type="float" level="2" n="3"> <figure xlink:label="fig-022-01" xlink:href="fig-022-01a"> <image file="022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/022-01"/> </figure> </div> <p> <s xml:id="echoid-s287" xml:space="preserve">SIT iam circulus omnium ma-<lb/>ximus A D. </s> <s xml:id="echoid-s288" xml:space="preserve">Dico eum per G, cen-<lb/>trum ſphæræ tranſire. </s> <s xml:id="echoid-s289" xml:space="preserve">Sienim non <lb/>tranſeat per centrũ, erit alius quiſ-<lb/>piam circulus per centrum G, tran <lb/>ſiens maior circulo A D, non per <lb/>centrũ tranſeũte, vt in hac propoſ, <lb/>demonſtratum eſt. </s> <s xml:id="echoid-s290" xml:space="preserve">Quare A D, non <lb/>eſt maximus circulus. </s> <s xml:id="echoid-s291" xml:space="preserve">Quod eſt ab-<lb/>ſurdum. </s> <s xml:id="echoid-s292" xml:space="preserve">Ponitur enim maximus. </s> <s xml:id="echoid-s293" xml:space="preserve">Tranſit ergo per G, centrum ſphæræ.</s> <s xml:id="echoid-s294" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s295" xml:space="preserve">DEINDE ſint æquales circuli B C, F E. </s> <s xml:id="echoid-s296" xml:space="preserve">Dico eos à centro G, æquali-<lb/>ter diſtare. </s> <s xml:id="echoid-s297" xml:space="preserve">Conſtructa enim figura, vt prius, erunt ſemidiametri H C, I E, æ-<lb/>quales. </s> <s xml:id="echoid-s298" xml:space="preserve">Et quoniam quadrata ex G H, H C, æqualia ſunt quadratis ex G I, <lb/> <anchor type="note" xlink:label="note-022-02a" xlink:href="note-022-02"/> I E, vt demonſtratum eſt; </s> <s xml:id="echoid-s299" xml:space="preserve">ablatis æqualibus quadratis linearum æqualium <lb/>H C, I E, erunt reliqua quadrata rectarum G H, G I, æqualia; </s> <s xml:id="echoid-s300" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s301" xml:space="preserve"><lb/>lineæ G H, G I, æquales erunt. </s> <s xml:id="echoid-s302" xml:space="preserve">Quæ cum perpendiculares ſint, ex conſtru-<lb/>ctione, ad plana circulorum B C, F E, æqualiter à centro G, diſtabunt cir-<lb/>culi B C, F E, ex defin. </s> <s xml:id="echoid-s303" xml:space="preserve">6. </s> <s xml:id="echoid-s304" xml:space="preserve">huius lib.</s> <s xml:id="echoid-s305" xml:space="preserve"/> </p> <div xml:id="echoid-div39" type="float" level="2" n="4"> <note position="left" xlink:label="note-022-02" xlink:href="note-022-02a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s306" xml:space="preserve">QVOD ſi alter circulorum B C, F E, nimirum circulus B C, minor po-<lb/>natur altero circulo F E, oſtendemus eodem ferè modo, perpendicularem <lb/>G H, maiorem eſſe perpendiculari G I. </s> <s xml:id="echoid-s307" xml:space="preserve">Cum enim quadrata ex G H, H C, <lb/>oſtenſa ſint æqualia quadratis ex G I, I E; </s> <s xml:id="echoid-s308" xml:space="preserve">ſit autem quadratum ex H C, mi-<lb/>nus quadrato ex I E; </s> <s xml:id="echoid-s309" xml:space="preserve">(quòd & </s> <s xml:id="echoid-s310" xml:space="preserve">ſemidiameter H C, circuli minoris minor ſit <lb/>ſemidiametro I E, circuli maioris) erit quadratum reliquum rectæ G H, reli <lb/>quo quadrato rectæ G I, maius; </s> <s xml:id="echoid-s311" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s312" xml:space="preserve">recta G H, maior erit, quàm <lb/>G I. </s> <s xml:id="echoid-s313" xml:space="preserve">Quare cum G H, G I, perpendiculares ſint, ex conſtructione, ad plana <lb/>circulorum, longius diſtabit, per defin. </s> <s xml:id="echoid-s314" xml:space="preserve">6. </s> <s xml:id="echoid-s315" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s316" xml:space="preserve">circulus B C, minor à cen <lb/>tro G, quàm circulus maior F E. </s> <s xml:id="echoid-s317" xml:space="preserve">Itaque circulorum, qui in ſphæra ſunt, <pb o="11" file="023" n="23" rhead=""/> maximi ſunt, qui per ſphæræ centrũ ducũtur, &</s> <s xml:id="echoid-s318" xml:space="preserve">c. </s> <s xml:id="echoid-s319" xml:space="preserve">Quod erat demonſtrandũ.</s> <s xml:id="echoid-s320" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div41" type="section" level="1" n="29"> <head xml:id="echoid-head40" xml:space="preserve">THEOREMA 6. PROPOS. 7.</head> <note position="right" xml:space="preserve">8.</note> <p> <s xml:id="echoid-s321" xml:space="preserve">SI in ſphæra ſit circulus, à centro autem ſphæ-<lb/>ræ ad centrum circuli connectatur recta linea, con <lb/>nexa linea ad circuli planum recta erit.</s> <s xml:id="echoid-s322" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s323" xml:space="preserve">IN ſphæra A B C, cuius centrum D, ſit circulus B F C G, cuius centrũ <lb/>E: </s> <s xml:id="echoid-s324" xml:space="preserve">Et recta D E, connectat duo centra D, E. </s> <s xml:id="echoid-s325" xml:space="preserve">Dico D E, rectam eſſe ad planũ <lb/>circuli B F C G. </s> <s xml:id="echoid-s326" xml:space="preserve">Ductis enim duabus diametris vtcunque B C, F G, in circu <lb/>lo, ducantur ab earum extremis ad D, centrum ſphæræ rectæ lineæ, B D, <lb/> <anchor type="figure" xlink:label="fig-023-01a" xlink:href="fig-023-01"/> C D, F D, G D, quæ omnes inter ſe æqua-<lb/>les erunt, cum à centro ſphæræ ad eius ſuper <lb/>ficiem cadant: </s> <s xml:id="echoid-s327" xml:space="preserve">Sunt autem & </s> <s xml:id="echoid-s328" xml:space="preserve">B E, C E, F E, <lb/>G E, ſemidiametri circuli B F C G, æquales. <lb/></s> <s xml:id="echoid-s329" xml:space="preserve">Igitur duo triangula D E B, D E C, duo la-<lb/>tera D E, E B, duobus lateribus D E, E C, <lb/>& </s> <s xml:id="echoid-s330" xml:space="preserve">baſim D B, baſi D C, æqualem habent; </s> <s xml:id="echoid-s331" xml:space="preserve">ex <lb/>quo fit, angulos D E B, D E C, æquales, at-<lb/> <anchor type="note" xlink:label="note-023-02a" xlink:href="note-023-02"/> que adeò rectos eſſe. </s> <s xml:id="echoid-s332" xml:space="preserve">Recta igitur D E, rectę <lb/>B C, ad rectos inſiſtet angulos. </s> <s xml:id="echoid-s333" xml:space="preserve">Non aliter <lb/>oſtendemus, rectam D E, rectæ F G, ad re-<lb/>ctos angulos inſiſtere. </s> <s xml:id="echoid-s334" xml:space="preserve">Quamobrem & </s> <s xml:id="echoid-s335" xml:space="preserve">pla-<lb/>no circuli B F C G, per rectas B C, F G, du-<lb/> <anchor type="note" xlink:label="note-023-03a" xlink:href="note-023-03"/> cto ad rectos angulos inſiſtet. </s> <s xml:id="echoid-s336" xml:space="preserve">Si igitur in ſphæra ſit circulus, &</s> <s xml:id="echoid-s337" xml:space="preserve">c. </s> <s xml:id="echoid-s338" xml:space="preserve">Quod oſten <lb/>dendum erat.</s> <s xml:id="echoid-s339" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="1"> <figure xlink:label="fig-023-01" xlink:href="fig-023-01a"> <image file="023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/023-01"/> </figure> <note position="right" xlink:label="note-023-02" xlink:href="note-023-02a" xml:space="preserve">8. primi.</note> <note position="right" xlink:label="note-023-03" xlink:href="note-023-03a" xml:space="preserve">4. vndec.</note> </div> </div> <div xml:id="echoid-div43" type="section" level="1" n="30"> <head xml:id="echoid-head41" xml:space="preserve">THEOREMA 7. PROPOS. 8.</head> <note position="right" xml:space="preserve">9.</note> <p> <s xml:id="echoid-s340" xml:space="preserve">SI ſit in ſphæra circulus, & </s> <s xml:id="echoid-s341" xml:space="preserve">à centro ſphæræ ad <lb/>circulũ ducatur perpendicularis, quæ ad vtramq; <lb/></s> <s xml:id="echoid-s342" xml:space="preserve">partẽ producatur, cadet ea in polos ipſius circuli.</s> <s xml:id="echoid-s343" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s344" xml:space="preserve">IN ſphæra A B C D, <lb/> <anchor type="figure" xlink:label="fig-023-02a" xlink:href="fig-023-02"/> cuius centrum E, ſit cir-<lb/>culus B G D H, in cuius <lb/>planum à centro ſphæræ <lb/> <anchor type="note" xlink:label="note-023-05a" xlink:href="note-023-05"/> E, per pendicularis dedu <lb/>cta ſit E F, quæ in vtram-<lb/>que partem protracta ca <lb/>dat in ſuperficiem ſphæ-<lb/>ræ ad puncta A, C. </s> <s xml:id="echoid-s345" xml:space="preserve">Dico <lb/>A, C, polos eſſe circuli <lb/>BGDH. </s> <s xml:id="echoid-s346" xml:space="preserve">Cadet em̃ per-<lb/>pendicularis E F, in cen- <pb o="12" file="024" n="24" rhead=""/> trum circuli B G D H, atque adeo F, centrum erit circuli. </s> <s xml:id="echoid-s347" xml:space="preserve">Quòd ſi circu-<lb/> <anchor type="note" xlink:label="note-024-01a" xlink:href="note-024-01"/> lus B G D H, per centrum ſphæræ ducatur, erit ipſum centrum ſphæræ E, <lb/>idem quod F, centrum circuli; </s> <s xml:id="echoid-s348" xml:space="preserve">ex quo ad planum circuli excitata ſit perpen-<lb/> <anchor type="note" xlink:label="note-024-02a" xlink:href="note-024-02"/> dicularis A C. </s> <s xml:id="echoid-s349" xml:space="preserve">Ductis igitur diametris B D, G H, vtcunque, ducantur ab ea <lb/>rum extremis rectæ ad puncta A, C. </s> <s xml:id="echoid-s350" xml:space="preserve">Et quia A F, perpendicularis eſt ad planũ <lb/> <anchor type="figure" xlink:label="fig-024-01a" xlink:href="fig-024-01"/> circuli B G D H, erunt <lb/>anguli omnes, quos ad F, <lb/>facit, recti, ex defin. </s> <s xml:id="echoid-s351" xml:space="preserve">3. </s> <s xml:id="echoid-s352" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s353" xml:space="preserve">11. </s> <s xml:id="echoid-s354" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s355" xml:space="preserve">quare duo <lb/>triangula A F B, A F H, <lb/>duo latera A F, F B, duo <lb/>bus lateribus A F, F H, <lb/>æqualia habent, quę qui <lb/>dem angulos comprehen <lb/>dunt æquales, nempe re-<lb/>ctos. </s> <s xml:id="echoid-s356" xml:space="preserve">Igitur baſes A B, <lb/>A H, æquales erunt. </s> <s xml:id="echoid-s357" xml:space="preserve">Eo-<lb/> <anchor type="note" xlink:label="note-024-03a" xlink:href="note-024-03"/> dem modo oſtẽ demus & </s> <s xml:id="echoid-s358" xml:space="preserve"><lb/>rectas A D, A G, & </s> <s xml:id="echoid-s359" xml:space="preserve">alias <lb/>quaſcunque ex A, ad circumferentiam circuli B G D H, ductas tam inter ſe, <lb/>quàm rectis A B, A H, æquales eſſe. </s> <s xml:id="echoid-s360" xml:space="preserve">Punctũ ergo A, polus eſt circuli B G D H, <lb/>ex defin. </s> <s xml:id="echoid-s361" xml:space="preserve">5. </s> <s xml:id="echoid-s362" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s363" xml:space="preserve">Non aliter demonſtrabimus, & </s> <s xml:id="echoid-s364" xml:space="preserve">C, punctum eiuſdem cir <lb/>culi polum eſſe. </s> <s xml:id="echoid-s365" xml:space="preserve">Si igitur ſit in ſphæra circulus, & </s> <s xml:id="echoid-s366" xml:space="preserve">à centro, &</s> <s xml:id="echoid-s367" xml:space="preserve">c. </s> <s xml:id="echoid-s368" xml:space="preserve">Quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s369" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="1"> <figure xlink:label="fig-023-02" xlink:href="fig-023-02a"> <image file="023-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/023-02"/> </figure> <note position="right" xlink:label="note-023-05" xlink:href="note-023-05a" xml:space="preserve">11. vndec.</note> <note position="left" xlink:label="note-024-01" xlink:href="note-024-01a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="left" xlink:label="note-024-02" xlink:href="note-024-02a" xml:space="preserve">12. vndec.</note> <figure xlink:label="fig-024-01" xlink:href="fig-024-01a"> <image file="024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/024-01"/> </figure> <note position="left" xlink:label="note-024-03" xlink:href="note-024-03a" xml:space="preserve">4. primi.</note> </div> </div> <div xml:id="echoid-div45" type="section" level="1" n="31"> <head xml:id="echoid-head42" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s370" xml:space="preserve">_IN_ verſione Maurolyci adduntur ſequentia duo theoremata, quæ Arabes adie-<lb/>cerunt.</s> <s xml:id="echoid-s371" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div46" type="section" level="1" n="32"> <head xml:id="echoid-head43" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s372" xml:space="preserve">SI ſit in ſphæra circulus, a cuius centro educatur perpendicu-<lb/> <anchor type="note" xlink:label="note-024-04a" xlink:href="note-024-04"/> laris ad circuli planum, quæ in vtramque partem producatur, cadet <lb/>hæc in vtrumque polum circuli.</s> <s xml:id="echoid-s373" xml:space="preserve"/> </p> <div xml:id="echoid-div46" type="float" level="2" n="1"> <note position="left" xlink:label="note-024-04" xlink:href="note-024-04a" xml:space="preserve">10.</note> </div> <p style="it"> <s xml:id="echoid-s374" xml:space="preserve">_IN_ eadem figura ex _F,_ centro circuli _B G D H,_ erigatur recta _F A,_ perpendi-<lb/> <anchor type="note" xlink:label="note-024-05a" xlink:href="note-024-05"/> cularis ad circuli planum, quæ occurr at ſuperficiei ſphæræ in punctis _A, C._ </s> <s xml:id="echoid-s375" xml:space="preserve">Dico <lb/>_A, C,_ eſſe polos circuli _B G D H._ </s> <s xml:id="echoid-s376" xml:space="preserve">Erunt enim rurſus ex definit. </s> <s xml:id="echoid-s377" xml:space="preserve">3. </s> <s xml:id="echoid-s378" xml:space="preserve">lib. </s> <s xml:id="echoid-s379" xml:space="preserve">11. </s> <s xml:id="echoid-s380" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s381" xml:space="preserve">om <lb/>nes anguli, quos ad _F,_ facit recta _A F,_ recti. </s> <s xml:id="echoid-s382" xml:space="preserve">Quare, vt prius, lineæ _A B, A D, A G,_ <lb/>_A H,_ &</s> <s xml:id="echoid-s383" xml:space="preserve">c. </s> <s xml:id="echoid-s384" xml:space="preserve">æquales inter ſe erunt, &</s> <s xml:id="echoid-s385" xml:space="preserve">c.</s> <s xml:id="echoid-s386" xml:space="preserve"/> </p> <div xml:id="echoid-div47" type="float" level="2" n="2"> <note position="left" xlink:label="note-024-05" xlink:href="note-024-05a" xml:space="preserve">12. vndec.</note> </div> <note position="left" xml:space="preserve">4. primi. <lb/>Coroll. 2. <lb/>huius.</note> <p style="it"> <s xml:id="echoid-s387" xml:space="preserve">_ALITER._ </s> <s xml:id="echoid-s388" xml:space="preserve">Quoniam perpendicularis _F A,_ tranſit per centrum ſphæræ _E;_ </s> <s xml:id="echoid-s389" xml:space="preserve">du <lb/>cta erit recta _E F,_ ex _E,_ centro ſphæræ ad planum circuli _B G D H,_ perpendicu-<lb/>laris. </s> <s xml:id="echoid-s390" xml:space="preserve">Quare vt demonſtratum eſt, cadet in polos eiuſdem circuli. </s> <s xml:id="echoid-s391" xml:space="preserve">Quod eſt pro-<lb/> <anchor type="note" xlink:label="note-024-07a" xlink:href="note-024-07"/> poſitum.</s> <s xml:id="echoid-s392" xml:space="preserve"/> </p> <div xml:id="echoid-div48" type="float" level="2" n="3"> <note position="left" xlink:label="note-024-07" xlink:href="note-024-07a" xml:space="preserve">3. huius.</note> </div> <pb o="13" file="025" n="25" rhead=""/> </div> <div xml:id="echoid-div50" type="section" level="1" n="33"> <head xml:id="echoid-head44" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s393" xml:space="preserve">SI ſit in ſphæra circulus, & </s> <s xml:id="echoid-s394" xml:space="preserve">ab altero polorum eius recta duca-<lb/> <anchor type="note" xlink:label="note-025-01a" xlink:href="note-025-01"/> tur per centrum illius, erit hęc ad planum circuli perpendicularis, <lb/>& </s> <s xml:id="echoid-s395" xml:space="preserve">producta cadet in reliquum polum.</s> <s xml:id="echoid-s396" xml:space="preserve"/> </p> <div xml:id="echoid-div50" type="float" level="2" n="1"> <note position="right" xlink:label="note-025-01" xlink:href="note-025-01a" xml:space="preserve">11.</note> </div> <p style="it"> <s xml:id="echoid-s397" xml:space="preserve">_IN_ eadem adbuc figura ex _A,_ polo circuli _B G D H,_ per centrum eius _F,_ demit <lb/>tatur linea recta _A F,_ occurrens ſuperficiei ſphæræ in _C._ </s> <s xml:id="echoid-s398" xml:space="preserve">Dico rectam _A F,_ perpen <lb/>dicularem eſſe ad planum circuli _B G D H,_ & </s> <s xml:id="echoid-s399" xml:space="preserve">_C,_ eſſe reliquum polum eiuſdem cir-<lb/>culi. </s> <s xml:id="echoid-s400" xml:space="preserve">Quoniam enim duo triangula _A F B, A F D,_ duo latera _A F, F B,_ duobus la-<lb/>teribus _A F, F D,_ & </s> <s xml:id="echoid-s401" xml:space="preserve">baſim _A B,_ baſi _A D,_ æqualem habent, ex defin. </s> <s xml:id="echoid-s402" xml:space="preserve">poli; </s> <s xml:id="echoid-s403" xml:space="preserve">habebunt <lb/>quoque duos angulos _A F B, A F D,_ æquales, atque adeo rectos. </s> <s xml:id="echoid-s404" xml:space="preserve">Igitur _A F,_ re-<lb/> <anchor type="note" xlink:label="note-025-02a" xlink:href="note-025-02"/> ctæ _B D,_ inſiſtit ad angulos rectos. </s> <s xml:id="echoid-s405" xml:space="preserve">Similiter oſtendemus, eandẽ _A F,_ ad angulos rectos <lb/>inſiſtere rectæ _G H._ </s> <s xml:id="echoid-s406" xml:space="preserve">Quare & </s> <s xml:id="echoid-s407" xml:space="preserve">plano circuli _B G D H,_ per rectas _B D, G H,_ ducto eadẽ <lb/> <anchor type="note" xlink:label="note-025-03a" xlink:href="note-025-03"/> recta _A F,_ ad rectos inſiſtet angulos. </s> <s xml:id="echoid-s408" xml:space="preserve">Quod eſt primò propoſitum. </s> <s xml:id="echoid-s409" xml:space="preserve">Quoniamigitur _A F,_ <lb/>ad rectos eſt angulos plano circuli _B G D H,_ ducta erit _F A,_ ex centro circuli _F,_ ad pla <lb/>num circuli perpendicularis. </s> <s xml:id="echoid-s410" xml:space="preserve">Quare, vt in hoc ſcholio proxime demonſtratum eſt, in <lb/>vtramque partem protracta in vtrumque polum circuli cadet, ac proinde _C,_ reli-<lb/>quus polus erit circuli _B G D H,_ quod eſt ſecundo loco propoſitum.</s> <s xml:id="echoid-s411" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="2"> <note position="right" xlink:label="note-025-02" xlink:href="note-025-02a" xml:space="preserve">8 primi.</note> <note position="right" xlink:label="note-025-03" xlink:href="note-025-03a" xml:space="preserve">4. vndec.</note> </div> </div> <div xml:id="echoid-div53" type="section" level="1" n="34"> <head xml:id="echoid-head45" xml:space="preserve">THEOR. 8. PROPOS. 9.</head> <note position="right" xml:space="preserve">12.</note> <p> <s xml:id="echoid-s412" xml:space="preserve">SI ſit in ſphæra circulus, & </s> <s xml:id="echoid-s413" xml:space="preserve">ab altero polorum <lb/>eius in ipſum ducatur perpẽdicularis recta linea, <lb/>cadet hæc in circuli centrum, & </s> <s xml:id="echoid-s414" xml:space="preserve">inde producta ca <lb/>det in reliquum polum ipſius circuli.</s> <s xml:id="echoid-s415" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s416" xml:space="preserve">IN Sphæra A B C D, ſit circulus B F D G, à cuius polo A, ad eius pla-<lb/>num perpendicularis ducatur A E, occurrens ſuperficiei ſphæræ in C. </s> <s xml:id="echoid-s417" xml:space="preserve">Dico <lb/> <anchor type="figure" xlink:label="fig-025-01a" xlink:href="fig-025-01"/> <anchor type="note" xlink:label="note-025-05a" xlink:href="note-025-05"/> E, centrum eſſe circuli B F D G, & </s> <s xml:id="echoid-s418" xml:space="preserve">C, reliquũ <lb/>polum. </s> <s xml:id="echoid-s419" xml:space="preserve">Ductis enim per E, duabus rectis vtcun <lb/>que B D, F G, connectantur earum extrema <lb/>cum polo A, rectis A B, A D, A F, A G, quæ <lb/>omnes inter ſe æquales erũt, ex definitione po <lb/>li. </s> <s xml:id="echoid-s420" xml:space="preserve">Omnes item anguli, quos recta A E, facit ad <lb/>E, recti, ex defin. </s> <s xml:id="echoid-s421" xml:space="preserve">3. </s> <s xml:id="echoid-s422" xml:space="preserve">lib. </s> <s xml:id="echoid-s423" xml:space="preserve">11. </s> <s xml:id="echoid-s424" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s425" xml:space="preserve">Erit igitur tam <lb/>quadratũ ex A B, quadratis ex A E, E B, quàm <lb/> <anchor type="note" xlink:label="note-025-06a" xlink:href="note-025-06"/> quadratum ex A G, quadratis ex A E, E G, æ-<lb/>quale; </s> <s xml:id="echoid-s426" xml:space="preserve">atq; </s> <s xml:id="echoid-s427" xml:space="preserve">adeò cum quadrata rectarum A B, <lb/>A G, æqualium æqualia ſint, erunt quadrata <lb/>ex A E, E B, ſimul quadratis ex A E, G E, ſi-<lb/>mul æqualia. </s> <s xml:id="echoid-s428" xml:space="preserve">Dempto ergo communi quadrato rectæ A E, reliqua quadrata <lb/>rectarum E B, E G, æqualia erunt, ac proinde & </s> <s xml:id="echoid-s429" xml:space="preserve">rectæ E B, E G, æquales. <lb/></s> <s xml:id="echoid-s430" xml:space="preserve">Eodem modo oſtendemus, rectas E G, E D, æquales eſſe. </s> <s xml:id="echoid-s431" xml:space="preserve">Quare E, centrum <lb/>eſt circuli BFDG; </s> <s xml:id="echoid-s432" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s433" xml:space="preserve">Quoniam igitur ex E, centro cir <lb/> <anchor type="note" xlink:label="note-025-07a" xlink:href="note-025-07"/> <pb o="14" file="026" n="26" rhead=""/> culi B F D G, ad ipſius planum educta eſt perpendicularis E A, tranſibit hęc <lb/> <anchor type="note" xlink:label="note-026-01a" xlink:href="note-026-01"/> per H, centrum ſphæræ, atq; </s> <s xml:id="echoid-s434" xml:space="preserve">adeo ex H, centro ſphæræ eadem H E, ducta <lb/>erit perpendicularis ad planum circuli B F D G. </s> <s xml:id="echoid-s435" xml:space="preserve">Quocirca H E, vtrinq; </s> <s xml:id="echoid-s436" xml:space="preserve">edu-<lb/> <anchor type="note" xlink:label="note-026-02a" xlink:href="note-026-02"/> cta cadetin polos eiuſdem circuli; </s> <s xml:id="echoid-s437" xml:space="preserve">ac proinde C, reliquus polus erit circuli <lb/>BFDG. </s> <s xml:id="echoid-s438" xml:space="preserve">Si igitur ſit in ſphæra circulus, & </s> <s xml:id="echoid-s439" xml:space="preserve">ab altero polorum eius, &</s> <s xml:id="echoid-s440" xml:space="preserve">c. </s> <s xml:id="echoid-s441" xml:space="preserve">Quod <lb/>oſtendendum erat.</s> <s xml:id="echoid-s442" xml:space="preserve"/> </p> <div xml:id="echoid-div53" type="float" level="2" n="1"> <figure xlink:label="fig-025-01" xlink:href="fig-025-01a"> <image file="025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/025-01"/> </figure> <note position="right" xlink:label="note-025-05" xlink:href="note-025-05a" xml:space="preserve">11. vndce.</note> <note position="right" xlink:label="note-025-06" xlink:href="note-025-06a" xml:space="preserve">47. primi.</note> <note position="right" xlink:label="note-025-07" xlink:href="note-025-07a" xml:space="preserve">9. tertij.</note> <note position="left" xlink:label="note-026-01" xlink:href="note-026-01a" xml:space="preserve">Coroll. 2. <lb/>huius.</note> <note position="left" xlink:label="note-026-02" xlink:href="note-026-02a" xml:space="preserve">8. huius.</note> </div> </div> <div xml:id="echoid-div55" type="section" level="1" n="35"> <head xml:id="echoid-head46" xml:space="preserve">THEOR. 9. PROPOS. 10.</head> <note position="left" xml:space="preserve">13.</note> <p> <s xml:id="echoid-s443" xml:space="preserve">SI ſit in ſphæra circulus, linea recta per eius po <lb/>los ducta, ad circulum recta eſt, tranſitq́ per cen-<lb/>trum circuli, & </s> <s xml:id="echoid-s444" xml:space="preserve">ſphæræ.</s> <s xml:id="echoid-s445" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s446" xml:space="preserve">IN ſphæra A B C D, ſit circulus B F D G, per cuius polos A, C, recta du <lb/>catur A C, occurrens plano circuli in E. </s> <s xml:id="echoid-s447" xml:space="preserve">Dico rectam A C, ad planum circu <lb/>li rectam eſſe, tranſireq́; </s> <s xml:id="echoid-s448" xml:space="preserve">per eius centrum, (hoc eſt, E, eſſe ipſius centrum) <lb/>nec non per centrũ ſphæræ. </s> <s xml:id="echoid-s449" xml:space="preserve">Ductis namq; </s> <s xml:id="echoid-s450" xml:space="preserve">per E, duabus rectis vtcunq; </s> <s xml:id="echoid-s451" xml:space="preserve">B D, <lb/>F G, quarum extrema cum polis A, C, iungantur rectis, vt in figura; </s> <s xml:id="echoid-s452" xml:space="preserve">erunt <lb/> <anchor type="figure" xlink:label="fig-026-01a" xlink:href="fig-026-01"/> tam A B, A G, A F, A D, inter ſe, quàm C B, <lb/>C G, C F, C D, inter ſe æquales, ex defin. </s> <s xml:id="echoid-s453" xml:space="preserve">poli. <lb/></s> <s xml:id="echoid-s454" xml:space="preserve">Igitur duo triangula A B C, A D C, duo late-<lb/>ra A B, A C, duobus lateribus A D, A C, & </s> <s xml:id="echoid-s455" xml:space="preserve">ba <lb/>ſim B C, baſi D C, æqualem habent. </s> <s xml:id="echoid-s456" xml:space="preserve">Quapro-<lb/>pter & </s> <s xml:id="echoid-s457" xml:space="preserve">angulos B A C, D A C, æquales habe-<lb/> <anchor type="note" xlink:label="note-026-04a" xlink:href="note-026-04"/> bunt. </s> <s xml:id="echoid-s458" xml:space="preserve">Quoniam igitur duo triangula A B E, <lb/>A D E, duo latera A B, A E, duobus lateribus <lb/>A D, A E; </s> <s xml:id="echoid-s459" xml:space="preserve">æqualia habent, anguloſq́; </s> <s xml:id="echoid-s460" xml:space="preserve">ſub ip-<lb/>ſis contentos B A E, D A E, æquales, vt pro-<lb/>xime demonſtratum eſt, erunt & </s> <s xml:id="echoid-s461" xml:space="preserve">anguli A E B, <lb/> <anchor type="note" xlink:label="note-026-05a" xlink:href="note-026-05"/> A E D, æquales, & </s> <s xml:id="echoid-s462" xml:space="preserve">ob id recti. </s> <s xml:id="echoid-s463" xml:space="preserve">Non aliter de-<lb/>monſtrabimus, rectos eſſe angu los A E G, A E F. </s> <s xml:id="echoid-s464" xml:space="preserve">Recta igitur A E, duabus re-<lb/>ctis B D, F G, ad rectos inſiſtit angulos. </s> <s xml:id="echoid-s465" xml:space="preserve">Quare perpendicularis erit ad planũ <lb/>circuli B F D G, per rectas B D, F G, ductum. </s> <s xml:id="echoid-s466" xml:space="preserve">Quod eſt primo loco propoſi-<lb/> <anchor type="note" xlink:label="note-026-06a" xlink:href="note-026-06"/> tum. </s> <s xml:id="echoid-s467" xml:space="preserve">Quoniam igitur ex A, polo circuli B F D G, ad eius planum perpendi-<lb/>cularis eſt ducta A E, cadet A E, in centrum ipſius. </s> <s xml:id="echoid-s468" xml:space="preserve">Eſt ergo E, centrum cir-<lb/> <anchor type="note" xlink:label="note-026-07a" xlink:href="note-026-07"/> culi B F D G. </s> <s xml:id="echoid-s469" xml:space="preserve">Rurſus quia ex E, centro circuli B F D G, educta eſt ad eius pla <lb/>num perpendicularis E A, tranſibit hæc per centrum quoq; </s> <s xml:id="echoid-s470" xml:space="preserve">ſphæræ. </s> <s xml:id="echoid-s471" xml:space="preserve">Quare <lb/> <anchor type="note" xlink:label="note-026-08a" xlink:href="note-026-08"/> recta A C, perpendicularis eſt ad planum circuli B F D G, tranſitq́ per eius <lb/>centrum, & </s> <s xml:id="echoid-s472" xml:space="preserve">ſphæræ. </s> <s xml:id="echoid-s473" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s474" xml:space="preserve">Si ſit igitur in ſphæra circulus, <lb/>linea recta per eius polos ducta, &</s> <s xml:id="echoid-s475" xml:space="preserve">c. </s> <s xml:id="echoid-s476" xml:space="preserve">Quod erat demonftrandum.</s> <s xml:id="echoid-s477" xml:space="preserve"/> </p> <div xml:id="echoid-div55" type="float" level="2" n="1"> <figure xlink:label="fig-026-01" xlink:href="fig-026-01a"> <image file="026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/026-01"/> </figure> <note position="left" xlink:label="note-026-04" xlink:href="note-026-04a" xml:space="preserve">8. primi.</note> <note position="left" xlink:label="note-026-05" xlink:href="note-026-05a" xml:space="preserve">4. primi.</note> <note position="left" xlink:label="note-026-06" xlink:href="note-026-06a" xml:space="preserve">4. vndec.</note> <note position="left" xlink:label="note-026-07" xlink:href="note-026-07a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-026-08" xlink:href="note-026-08a" xml:space="preserve">Coroll. 2. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div57" type="section" level="1" n="36"> <head xml:id="echoid-head47" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s478" xml:space="preserve">_ADDVNTVR_ hoc loco alia duo theoremata huiuſmodi.</s> <s xml:id="echoid-s479" xml:space="preserve"/> </p> <pb o="15" file="027" n="27" rhead=""/> </div> <div xml:id="echoid-div58" type="section" level="1" n="37"> <head xml:id="echoid-head48" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s480" xml:space="preserve">SI in ſphæra ſit circulus, & </s> <s xml:id="echoid-s481" xml:space="preserve">ab altero polorum eius per centrum <lb/> <anchor type="note" xlink:label="note-027-01a" xlink:href="note-027-01"/> ſphæræ recta linea ducatur, erit hæc ad planum circuli perpendi-<lb/>cularis, & </s> <s xml:id="echoid-s482" xml:space="preserve">producta cadet in centrum ipſius, & </s> <s xml:id="echoid-s483" xml:space="preserve">in reliquum polum.</s> <s xml:id="echoid-s484" xml:space="preserve"/> </p> <div xml:id="echoid-div58" type="float" level="2" n="1"> <note position="right" xlink:label="note-027-01" xlink:href="note-027-01a" xml:space="preserve">14.</note> </div> <p style="it"> <s xml:id="echoid-s485" xml:space="preserve">_IN_ ſphæra _A B C D,_ cuius centrum _E,_ ſit circulus _B G D H,_ a cuius polo _A,_ <lb/>per _E,_ centrum ſphæræ ducatur recta _A E,_ occurrens plano circuli in _F,_ & </s> <s xml:id="echoid-s486" xml:space="preserve">ſuperſi <lb/>ciei ſphæræ in _C. </s> <s xml:id="echoid-s487" xml:space="preserve">D_ico _A E,_ perpendicularem eſſe ad planum circuli, tranſireq́ per <lb/>eius centrum, & </s> <s xml:id="echoid-s488" xml:space="preserve">reliquum polum, hoc eſt, _F,_ eſſe eius centrum; </s> <s xml:id="echoid-s489" xml:space="preserve">& </s> <s xml:id="echoid-s490" xml:space="preserve">_C,_ reliquum <lb/> <anchor type="figure" xlink:label="fig-027-01a" xlink:href="fig-027-01"/> polum. </s> <s xml:id="echoid-s491" xml:space="preserve">_D_uctis enim per _F,_ duabus xectis vtcun-<lb/>que _B D, G H,_ iungantur extrema cum punctis <lb/>_A,_ & </s> <s xml:id="echoid-s492" xml:space="preserve">_E,_ vt in figura; </s> <s xml:id="echoid-s493" xml:space="preserve">eruntq́; </s> <s xml:id="echoid-s494" xml:space="preserve">_A B, A H, A D,_ <lb/>_A G,_ ex definitione poli, inter ſe æquales; </s> <s xml:id="echoid-s495" xml:space="preserve">nec <lb/>non & </s> <s xml:id="echoid-s496" xml:space="preserve">_E B, E H, E D, E G,_ ſemidiametri ſphæ-<lb/>ræinter ſe æquales. </s> <s xml:id="echoid-s497" xml:space="preserve">Quoniamigitur duo trian-<lb/>gula _A B E, A D E,_ duo latera _A B, A E,_ duo-<lb/>bus lateribus _A D, A E,_ & </s> <s xml:id="echoid-s498" xml:space="preserve">baſim _E B,_ baſi _E D,_ <lb/>habent æqualem; </s> <s xml:id="echoid-s499" xml:space="preserve">erunt anguli _B A E, D A E._ <lb/></s> <s xml:id="echoid-s500" xml:space="preserve"> <anchor type="note" xlink:label="note-027-02a" xlink:href="note-027-02"/> æquales. </s> <s xml:id="echoid-s501" xml:space="preserve">_I_taque duo triangula _A B F, A D F,_ <lb/>duo latera _A B, A F,_ duobus lateribus _A D, A F,_ <lb/>æqualia habent, anguloſq́ ſub ipſis contentos <lb/>_B A F, D A F,_ æquales, vt proxime oſtenſum eſt. <lb/></s> <s xml:id="echoid-s502" xml:space="preserve">Quare anguli _A F B, A F D,_ æquales erunt, at-<lb/> <anchor type="note" xlink:label="note-027-03a" xlink:href="note-027-03"/> que adeo recti. </s> <s xml:id="echoid-s503" xml:space="preserve">_E_odem modo demonſtrabimus re-<lb/>ctos eſſe angulos _A F H, A F G,._ </s> <s xml:id="echoid-s504" xml:space="preserve">_R_ecta igitur _A F,_ duabus rectis _B D, G H,_ inſiſtit <lb/>ad angulos rectos. </s> <s xml:id="echoid-s505" xml:space="preserve">Quare perpendicularis erit ad planum circuli _B G D H,_ per re-<lb/> <anchor type="note" xlink:label="note-027-04a" xlink:href="note-027-04"/> ctas _B D, G H,_ ductum. </s> <s xml:id="echoid-s506" xml:space="preserve">_I_taque producta cadet & </s> <s xml:id="echoid-s507" xml:space="preserve">in centrum circuli, & </s> <s xml:id="echoid-s508" xml:space="preserve">in reli-<lb/> <anchor type="note" xlink:label="note-027-05a" xlink:href="note-027-05"/> quum polum: </s> <s xml:id="echoid-s509" xml:space="preserve">ac proinde _F,_ centrum erit circuli, & </s> <s xml:id="echoid-s510" xml:space="preserve">_C,_ reliquus polus. </s> <s xml:id="echoid-s511" xml:space="preserve">Quod eſt <lb/>propoſitum. </s> <s xml:id="echoid-s512" xml:space="preserve">Si in ſphæra igitur ſit circulus, &</s> <s xml:id="echoid-s513" xml:space="preserve">c. </s> <s xml:id="echoid-s514" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s515" xml:space="preserve"/> </p> <div xml:id="echoid-div59" type="float" level="2" n="2"> <figure xlink:label="fig-027-01" xlink:href="fig-027-01a"> <image file="027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/027-01"/> </figure> <note position="right" xlink:label="note-027-02" xlink:href="note-027-02a" xml:space="preserve">8. primi.</note> <note position="right" xlink:label="note-027-03" xlink:href="note-027-03a" xml:space="preserve">4. primi.</note> <note position="right" xlink:label="note-027-04" xlink:href="note-027-04a" xml:space="preserve">4. vndec.</note> <note position="right" xlink:label="note-027-05" xlink:href="note-027-05a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div61" type="section" level="1" n="38"> <head xml:id="echoid-head49" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s516" xml:space="preserve">HINC fit, circulum maximum, qui per alterum polorum cuiuſlibet circuli in ſphæ-<lb/>ra tranſit, tranſire quoq; </s> <s xml:id="echoid-s517" xml:space="preserve">per polum reliquum. </s> <s xml:id="echoid-s518" xml:space="preserve">Nam ſi ex vno polo per centrum ſphæræ dia <lb/>meter ducatur circuli maximi, qui per illum polum tranſit, cadet hæc in alterum polum, <lb/>vt demonſtratum eſt. </s> <s xml:id="echoid-s519" xml:space="preserve">Idem ergo circulus maximus per reliquum polum tranſibit.</s> <s xml:id="echoid-s520" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s521" xml:space="preserve">Et quia diameter circuli maximi eſt quoq; </s> <s xml:id="echoid-s522" xml:space="preserve">diameter ſphæræ, manifeſtum eſt, duos po-<lb/>los circuli cuiuſlibet in ſphæra per diametrum eſſe oppoſitos: </s> <s xml:id="echoid-s523" xml:space="preserve">atq; </s> <s xml:id="echoid-s524" xml:space="preserve">adeò inter ipſos inter-<lb/>poſitum eſſe ſemicircuium maximi circuli.</s> <s xml:id="echoid-s525" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div62" type="section" level="1" n="39"> <head xml:id="echoid-head50" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s526" xml:space="preserve">SI in ſphæra ſit circulus, & </s> <s xml:id="echoid-s527" xml:space="preserve">à centro ſphæræ per centrum circu-<lb/> <anchor type="note" xlink:label="note-027-06a" xlink:href="note-027-06"/> lirecta linea ducatur, cadet hæc in vtrumque polum circuli.</s> <s xml:id="echoid-s528" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="1"> <note position="right" xlink:label="note-027-06" xlink:href="note-027-06a" xml:space="preserve">15.</note> </div> <p style="it"> <s xml:id="echoid-s529" xml:space="preserve">_IN_ eadem figura ducatur per _E,_ centrum ſphæræ, & </s> <s xml:id="echoid-s530" xml:space="preserve">_F,_ centrum circuli _B G D H,_ <pb o="16" file="028" n="28" rhead=""/> recta _E F,_ in vtramque partem. </s> <s xml:id="echoid-s531" xml:space="preserve">_D_ico _E F,_ cadere in vtrumque polum circulè <lb/>_BGDH;_ </s> <s xml:id="echoid-s532" xml:space="preserve">Quoniam enim recta _E F,_ centrum ſphæræ, & </s> <s xml:id="echoid-s533" xml:space="preserve">centrum circuli _B G D H,_ <lb/>connectens perpendicularis eſt ad planum eiuſdem circuli, cadet eadem _E F,_ vtrin <lb/> <anchor type="note" xlink:label="note-028-01a" xlink:href="note-028-01"/> que protracta in polum vtrumque eiuſdem circuli. </s> <s xml:id="echoid-s534" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s535" xml:space="preserve"/> </p> <div xml:id="echoid-div63" type="float" level="2" n="2"> <note position="left" xlink:label="note-028-01" xlink:href="note-028-01a" xml:space="preserve">7. huius.</note> </div> <note position="left" xml:space="preserve">8. huius.</note> </div> <div xml:id="echoid-div65" type="section" level="1" n="40"> <head xml:id="echoid-head51" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s536" xml:space="preserve">EX his omnibus conſtat, in ſphæra quatuor hæc puncta, nempe duos polos cuiuſq; </s> <s xml:id="echoid-s537" xml:space="preserve">cir-<lb/>culi, eiuſdem centrum, & </s> <s xml:id="echoid-s538" xml:space="preserve">centrum ſphæræ, perpetuo in vna ſinea recta, nempe diametro <lb/>ſphæræ, exiſtere, & </s> <s xml:id="echoid-s539" xml:space="preserve">ipſam quidem diametrum ad planum eiuſdem circuli eſſe perpendi-<lb/>cularem: </s> <s xml:id="echoid-s540" xml:space="preserve">Adeo vt recta pet quælibet duo puncta ex his ducta tranſeat per reliqua duo, ſitq́; <lb/></s> <s xml:id="echoid-s541" xml:space="preserve">ad planum circuli perpendicularis: </s> <s xml:id="echoid-s542" xml:space="preserve">Et recta pet vnum eorum ducta perpendicularis ad pla-<lb/>num circuli, tranſeat quoq; </s> <s xml:id="echoid-s543" xml:space="preserve">per tria puncta reliqua.</s> <s xml:id="echoid-s544" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div66" type="section" level="1" n="41"> <head xml:id="echoid-head52" xml:space="preserve">THEOR. 10. PROP. 11.</head> <note position="left" xml:space="preserve">16.</note> <p> <s xml:id="echoid-s545" xml:space="preserve">IN ſphęra maximi circuli ſe mutuo ſecant bi-<lb/>fariam.</s> <s xml:id="echoid-s546" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s547" xml:space="preserve">IN ſphæra A B C D, ſecent ſe mutuo duo circuli maximi A C, B D, in <lb/>punctis E, F. </s> <s xml:id="echoid-s548" xml:space="preserve">Dico ſe mutuo ſecare bifariam. </s> <s xml:id="echoid-s549" xml:space="preserve">Quoniam enim circuli maximi <lb/>in ſphæra per centrum ſphæræ tranſeunt, tranſibunt circuli A C, B D, per <lb/> <anchor type="note" xlink:label="note-028-04a" xlink:href="note-028-04"/> ſphæræ centrum, quod ſit G. </s> <s xml:id="echoid-s550" xml:space="preserve">Et quoniam idem eſt ſphæræ centrum, & </s> <s xml:id="echoid-s551" xml:space="preserve">circu-<lb/>li per ſphæræ centrum traiecti, erit punctum G, quod ſphæræ centrum poni-<lb/> <anchor type="note" xlink:label="note-028-05a" xlink:href="note-028-05"/> tur, centrum quoq; </s> <s xml:id="echoid-s552" xml:space="preserve">vtriuſq; </s> <s xml:id="echoid-s553" xml:space="preserve">circuli A C, B D, ita vt in vtroq; </s> <s xml:id="echoid-s554" xml:space="preserve">plano circu-<lb/>lorum A C, B D, exiſtat. </s> <s xml:id="echoid-s555" xml:space="preserve">Sunt autem & </s> <s xml:id="echoid-s556" xml:space="preserve">puncta E, F, in vtroq; </s> <s xml:id="echoid-s557" xml:space="preserve">eodem plano. <lb/></s> <s xml:id="echoid-s558" xml:space="preserve"> <anchor type="figure" xlink:label="fig-028-01a" xlink:href="fig-028-01"/> Tria igitur pũcta E, G, F, in vtroq; </s> <s xml:id="echoid-s559" xml:space="preserve">plano circulo <lb/>rũ A C, B D, exiſtunt; </s> <s xml:id="echoid-s560" xml:space="preserve">atq; </s> <s xml:id="echoid-s561" xml:space="preserve">adeo in cõmuni eorũ <lb/>ſectione erunt, cum ſolũ cõmunis eorum ſectio <lb/>ſit in vtroq; </s> <s xml:id="echoid-s562" xml:space="preserve">plano: </s> <s xml:id="echoid-s563" xml:space="preserve">Eſt autem communis eo-<lb/> <anchor type="note" xlink:label="note-028-06a" xlink:href="note-028-06"/> rum ſectio linea recta. </s> <s xml:id="echoid-s564" xml:space="preserve">Igitur tria puncta E, G, F, <lb/>in linea recta ex E, per G, ad F, ducta exiſtunt. <lb/></s> <s xml:id="echoid-s565" xml:space="preserve">quæ cum tranſeat per G, centrum vtriuſq; </s> <s xml:id="echoid-s566" xml:space="preserve">cir-<lb/>culi, & </s> <s xml:id="echoid-s567" xml:space="preserve">ſphæræ, vt oſtenſum eſt, diameter erit <lb/>& </s> <s xml:id="echoid-s568" xml:space="preserve">ſphæræ, & </s> <s xml:id="echoid-s569" xml:space="preserve">vtriuſq; </s> <s xml:id="echoid-s570" xml:space="preserve">circuli; </s> <s xml:id="echoid-s571" xml:space="preserve">atq; </s> <s xml:id="echoid-s572" xml:space="preserve">adeo vtrum-<lb/>que eorum bifariam ſecabit, ita vtſemicirculi <lb/>ſint E A F, F C E, E B F, F D E: </s> <s xml:id="echoid-s573" xml:space="preserve">In ſphæra er-<lb/>go maximi circuli ſe mutuo ſecant bifariam. </s> <s xml:id="echoid-s574" xml:space="preserve"><lb/>Quod erat demonſtrandum.</s> <s xml:id="echoid-s575" xml:space="preserve"/> </p> <div xml:id="echoid-div66" type="float" level="2" n="1"> <note position="left" xlink:label="note-028-04" xlink:href="note-028-04a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-028-05" xlink:href="note-028-05a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <figure xlink:label="fig-028-01" xlink:href="fig-028-01a"> <image file="028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/028-01"/> </figure> <note position="left" xlink:label="note-028-06" xlink:href="note-028-06a" xml:space="preserve">3. vndec.</note> </div> </div> <div xml:id="echoid-div68" type="section" level="1" n="42"> <head xml:id="echoid-head53" xml:space="preserve">THEOR. 11. PROP. 12.</head> <note position="left" xml:space="preserve">17.</note> <p> <s xml:id="echoid-s576" xml:space="preserve">IN ſphæra circuli, qui ſe mutuo bifariam ſe-<lb/>cant, ſunt maximi.</s> <s xml:id="echoid-s577" xml:space="preserve"/> </p> <pb o="17" file="029" n="29" rhead=""/> <p> <s xml:id="echoid-s578" xml:space="preserve">IN ſphæra A B C D, circuli A E, B D, ſe mutuo ſecent bifariam in pun-<lb/>ctis E, F. </s> <s xml:id="echoid-s579" xml:space="preserve">Dico circulos A C, B D, eſſe maximos. </s> <s xml:id="echoid-s580" xml:space="preserve">Cum enim ſe mutuo fecent <lb/>bifariam in E, F, erit ducta recta E F, vtriuſq; </s> <s xml:id="echoid-s581" xml:space="preserve">diameter, cum ſola diameter <lb/> <anchor type="figure" xlink:label="fig-029-01a" xlink:href="fig-029-01"/> circulũ quemcunq; </s> <s xml:id="echoid-s582" xml:space="preserve">bifariam diuidat; </s> <s xml:id="echoid-s583" xml:space="preserve">ac proin-<lb/>de diuiſa recta E F, bifariã in G, erit G, vtriuſq; <lb/></s> <s xml:id="echoid-s584" xml:space="preserve">circuli centrum: </s> <s xml:id="echoid-s585" xml:space="preserve">quod dico etiam eſſe ſphæræ <lb/>centrum, atq; </s> <s xml:id="echoid-s586" xml:space="preserve">adeo vtrumq; </s> <s xml:id="echoid-s587" xml:space="preserve">circulum per ſphę <lb/>ræ centrum duci. </s> <s xml:id="echoid-s588" xml:space="preserve">Sinamq; </s> <s xml:id="echoid-s589" xml:space="preserve">G, dicatur non eſſe <lb/>centrum ſphæræ, ac proinde circulos A C, B D, <lb/>non eſſe per ſphæræ centrum ductos; </s> <s xml:id="echoid-s590" xml:space="preserve">hoc ipſo <lb/>oſtendemus, G, eſſe centrum ſphæræ, atq; </s> <s xml:id="echoid-s591" xml:space="preserve">idcir <lb/>co vtrumq; </s> <s xml:id="echoid-s592" xml:space="preserve">circulum per ſphæræ centrum du-<lb/>ci. </s> <s xml:id="echoid-s593" xml:space="preserve">Erigatur enim ex G, ad planum circuli A C, <lb/> <anchor type="note" xlink:label="note-029-01a" xlink:href="note-029-01"/> perpendicularis G H: </s> <s xml:id="echoid-s594" xml:space="preserve">Item G I, perpendicula-<lb/>ris ad planum circuli B D. </s> <s xml:id="echoid-s595" xml:space="preserve">Quoniam igitur cir <lb/>culi A C, B D, ponuntur non tranſire per centrum ſphæræ, tranſibit vtraq; <lb/></s> <s xml:id="echoid-s596" xml:space="preserve">perpendicularis G H, G I, per centrum ſphæræ. </s> <s xml:id="echoid-s597" xml:space="preserve">Quare punctum G, in quo <lb/> <anchor type="note" xlink:label="note-029-02a" xlink:href="note-029-02"/> conueniunt, centrum erit ſphæræ, aliàs centrum non exiſteret in vtraque: <lb/></s> <s xml:id="echoid-s598" xml:space="preserve">ac proinde vterq; </s> <s xml:id="echoid-s599" xml:space="preserve">circulus per centrum ſphæræ traijcietur. </s> <s xml:id="echoid-s600" xml:space="preserve">Sunt ergo circu <lb/> <anchor type="note" xlink:label="note-029-03a" xlink:href="note-029-03"/> li A C, B D, per centrum ſphæræ traiecti, maximi. </s> <s xml:id="echoid-s601" xml:space="preserve">In ſphæra ergo circuli, qui <lb/>ſe mutuo bifariam ſecant, ſunt maximi. </s> <s xml:id="echoid-s602" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s603" xml:space="preserve"/> </p> <div xml:id="echoid-div68" type="float" level="2" n="1"> <figure xlink:label="fig-029-01" xlink:href="fig-029-01a"> <image file="029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/029-01"/> </figure> <note position="right" xlink:label="note-029-01" xlink:href="note-029-01a" xml:space="preserve">12. vndec.</note> <note position="right" xlink:label="note-029-02" xlink:href="note-029-02a" xml:space="preserve">Coroll. 2. <lb/>huius.</note> <note position="right" xlink:label="note-029-03" xlink:href="note-029-03a" xml:space="preserve">6. huius.</note> </div> </div> <div xml:id="echoid-div70" type="section" level="1" n="43"> <head xml:id="echoid-head54" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s604" xml:space="preserve">_HIC_ vides mirabilem ſane argumentandi modum. </s> <s xml:id="echoid-s605" xml:space="preserve">_N_am ex eo, quòd _G,_ dici-<lb/>tur non eſſe centrum ſphæræ, demonſtratum eſt demonſtratione affirmatiua, _G,_ eſ-<lb/>ſe centrum ſphæræ. </s> <s xml:id="echoid-s606" xml:space="preserve">quo modo argumentandi etiam vſus eſt _E_uclides lib. </s> <s xml:id="echoid-s607" xml:space="preserve">9. </s> <s xml:id="echoid-s608" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s609" xml:space="preserve">12. </s> <s xml:id="echoid-s610" xml:space="preserve">& </s> <s xml:id="echoid-s611" xml:space="preserve">_C_ardanus lib. </s> <s xml:id="echoid-s612" xml:space="preserve">5. </s> <s xml:id="echoid-s613" xml:space="preserve">de _P_roport. </s> <s xml:id="echoid-s614" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s615" xml:space="preserve">201. </s> <s xml:id="echoid-s616" xml:space="preserve">vt in ſcholio eiuſdẽ propoſ. </s> <s xml:id="echoid-s617" xml:space="preserve">monuimus.</s> <s xml:id="echoid-s618" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div71" type="section" level="1" n="44"> <head xml:id="echoid-head55" xml:space="preserve">THEOREMA 12. PROPOS. 13.</head> <note position="right" xml:space="preserve">18.</note> <p> <s xml:id="echoid-s619" xml:space="preserve">SI in ſphæra maximus circulus circulum quẽ-<lb/>piam ad rectos angulos ſecet; </s> <s xml:id="echoid-s620" xml:space="preserve">& </s> <s xml:id="echoid-s621" xml:space="preserve">bifariam eum ſe-<lb/>cat, & </s> <s xml:id="echoid-s622" xml:space="preserve">per polos.</s> <s xml:id="echoid-s623" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s624" xml:space="preserve">IN ſphæra maximus circulus A B C D, <lb/> <anchor type="figure" xlink:label="fig-029-02a" xlink:href="fig-029-02"/> ſecet circulũ B E D, in punctis B, D, ad an-<lb/>gulos rectos, hoc eſt, planũ circuli A B C D, <lb/>rectum ſit ad planum circuli B E D; </s> <s xml:id="echoid-s625" xml:space="preserve">ſitq́; </s> <s xml:id="echoid-s626" xml:space="preserve">cõ-<lb/>munis eorum ſectio recta B D. </s> <s xml:id="echoid-s627" xml:space="preserve">Dico circu-<lb/>lum A B C D, bifariam, & </s> <s xml:id="echoid-s628" xml:space="preserve">per polos ſecare <lb/>circulum B E D. </s> <s xml:id="echoid-s629" xml:space="preserve">Sumpto enim F, centro cir <lb/> <anchor type="note" xlink:label="note-029-05a" xlink:href="note-029-05"/> culi maximi A B C D, quod & </s> <s xml:id="echoid-s630" xml:space="preserve">centrũ ſphę-<lb/>ræ erit, (Nam cum circulus maximus duca-<lb/> <anchor type="note" xlink:label="note-029-06a" xlink:href="note-029-06"/> tur per centrum ſphæræ, erit eius centrum <lb/> <anchor type="note" xlink:label="note-029-07a" xlink:href="note-029-07"/> idem, quod ſphæræ.) </s> <s xml:id="echoid-s631" xml:space="preserve">ducatur ex F, ad planũ <lb/>circuli B E D, perpendicularis F G, quæ in <lb/> <anchor type="note" xlink:label="note-029-08a" xlink:href="note-029-08"/> <pb o="18" file="030" n="30" rhead=""/> B D, communem ſectionem cadet. </s> <s xml:id="echoid-s632" xml:space="preserve">Cadat autem in punctum G. </s> <s xml:id="echoid-s633" xml:space="preserve">Et quoniam <lb/> <anchor type="note" xlink:label="note-030-01a" xlink:href="note-030-01"/> eadem cadit quoq; </s> <s xml:id="echoid-s634" xml:space="preserve">in centrum circuli B E D, erit G, centrum circuli B E D; <lb/></s> <s xml:id="echoid-s635" xml:space="preserve"> <anchor type="figure" xlink:label="fig-030-01a" xlink:href="fig-030-01"/> atq; </s> <s xml:id="echoid-s636" xml:space="preserve">adeo B D, per G, ducta, diameter eiuſ-<lb/>dem: </s> <s xml:id="echoid-s637" xml:space="preserve">quæ cum diuidat eirculum B E D, bi-<lb/>fariam, diuidet quoq; </s> <s xml:id="echoid-s638" xml:space="preserve">eundem bifariam cir-<lb/>culus maximus A B C D, per rectam B D, <lb/>ductus. </s> <s xml:id="echoid-s639" xml:space="preserve">Quod eſt primo loco propoſitum. <lb/></s> <s xml:id="echoid-s640" xml:space="preserve">Quoniam verò recta F G, in plano eſt circu <lb/>li A B C D, cadet ea producta in circum-<lb/>ferentiam ad A, C, puncta, quæ in ſuperfi-<lb/>cie ſphæræ ſunt: </s> <s xml:id="echoid-s641" xml:space="preserve">cadit autem & </s> <s xml:id="echoid-s642" xml:space="preserve">in vtrumq; </s> <s xml:id="echoid-s643" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-030-02a" xlink:href="note-030-02"/> polum circuli B E D, quòd ex F, centro <lb/>ſphæræ ad circuli planum perpendicularis <lb/>ſit ducta. </s> <s xml:id="echoid-s644" xml:space="preserve">Igitur A, C, poli ſunt circuli B E D, <lb/>ac proinde circulus maximus A B C D, per <lb/>polos circuli B E D, tranſit. </s> <s xml:id="echoid-s645" xml:space="preserve">quod ſecundo loco proponebatur demonſtrã-<lb/>dum. </s> <s xml:id="echoid-s646" xml:space="preserve">Si igitur in ſphæra ma ximus circulus circulum quempiam, &</s> <s xml:id="echoid-s647" xml:space="preserve">c. </s> <s xml:id="echoid-s648" xml:space="preserve">Quod <lb/>oſtendendum erat.</s> <s xml:id="echoid-s649" xml:space="preserve"/> </p> <div xml:id="echoid-div71" type="float" level="2" n="1"> <figure xlink:label="fig-029-02" xlink:href="fig-029-02a"> <image file="029-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/029-02"/> </figure> <note position="right" xlink:label="note-029-05" xlink:href="note-029-05a" xml:space="preserve">1. tertij.</note> <note position="right" xlink:label="note-029-06" xlink:href="note-029-06a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-029-07" xlink:href="note-029-07a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="right" xlink:label="note-029-08" xlink:href="note-029-08a" xml:space="preserve">11. vndec.</note> <note position="left" xlink:label="note-030-01" xlink:href="note-030-01a" xml:space="preserve">38. vndec. <lb/>Coroll. 1. <lb/>huius.</note> <figure xlink:label="fig-030-01" xlink:href="fig-030-01a"> <image file="030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/030-01"/> </figure> <note position="left" xlink:label="note-030-02" xlink:href="note-030-02a" xml:space="preserve">8. huius.</note> </div> </div> <div xml:id="echoid-div73" type="section" level="1" n="45"> <head xml:id="echoid-head56" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s650" xml:space="preserve">_CAETERVM_ hæc propoſ. </s> <s xml:id="echoid-s651" xml:space="preserve">vnà cum 8. </s> <s xml:id="echoid-s652" xml:space="preserve">9. </s> <s xml:id="echoid-s653" xml:space="preserve">10. </s> <s xml:id="echoid-s654" xml:space="preserve">& </s> <s xml:id="echoid-s655" xml:space="preserve">earum ſcholijs intelligenda <lb/>etiam eſt, quando circulus _B D,_ maximus eſt, & </s> <s xml:id="echoid-s656" xml:space="preserve">per ſphæræ centrum tranſit. </s> <s xml:id="echoid-s657" xml:space="preserve">_E_adem <lb/>enim eſt ferè ſemper demonſtratio, vtperſpicuum eſt.</s> <s xml:id="echoid-s658" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div74" type="section" level="1" n="46"> <head xml:id="echoid-head57" xml:space="preserve">THEOR. 13. PROPOS. 14.</head> <note position="left" xml:space="preserve">19.</note> <p> <s xml:id="echoid-s659" xml:space="preserve">SI in ſphæra maximus circulus circulum non <lb/>maximum bifariam ſecet; </s> <s xml:id="echoid-s660" xml:space="preserve">ad angulos rectos eum <lb/>ſecat, & </s> <s xml:id="echoid-s661" xml:space="preserve">per polos.</s> <s xml:id="echoid-s662" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s663" xml:space="preserve">IN ſphęra maximus circulus A B C D, non maximum B E D, ſecet bifa-<lb/> <anchor type="figure" xlink:label="fig-030-02a" xlink:href="fig-030-02"/> riam in punctis B, D, ſitq́; </s> <s xml:id="echoid-s664" xml:space="preserve">communis eorum <lb/>ſectio recta B D. </s> <s xml:id="echoid-s665" xml:space="preserve">Dico circulum A B C D, <lb/>ſecare circulum B E D, ad angulos rectos, <lb/>& </s> <s xml:id="echoid-s666" xml:space="preserve">per polos. </s> <s xml:id="echoid-s667" xml:space="preserve">Quia enim circulus B E D, bi <lb/>fariam ſecatur in B, D, hoc eſt, in ſemicircu <lb/>los, erit B D, communis ſectio diameter eius. <lb/></s> <s xml:id="echoid-s668" xml:space="preserve">Diuiſa ergo B D, bifariam in F, erit F, cen-<lb/> <anchor type="note" xlink:label="note-030-04a" xlink:href="note-030-04"/> trum circuli B E D. </s> <s xml:id="echoid-s669" xml:space="preserve">Sumpto autem G, cen <lb/>tro ſphæræ, quod & </s> <s xml:id="echoid-s670" xml:space="preserve">centrũ erit maximi cir-<lb/>culi A B C D, ducatur ex G, ad F, recta F G, <lb/> <anchor type="note" xlink:label="note-030-05a" xlink:href="note-030-05"/> quæ perpendicularis erit ad planum circuli <lb/>B E D. </s> <s xml:id="echoid-s671" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s672" xml:space="preserve">planum circuli maximi <lb/> <anchor type="note" xlink:label="note-030-06a" xlink:href="note-030-06"/> A B C D, per rectã F G, ductum ad idẽ planũ circuli B E D, rectũ erit. </s> <s xml:id="echoid-s673" xml:space="preserve">Secat igitur <pb o="19" file="031" n="31" rhead=""/> circulus maximus A B C D, circulum B E D, non maximum ad angulos re-<lb/>ctos. </s> <s xml:id="echoid-s674" xml:space="preserve">Quod eſt primo loco propoſitum. </s> <s xml:id="echoid-s675" xml:space="preserve">Et quoniam oſtenſum eſt, rectã F G, <lb/>ex G, centro ſphæræ ductam ad planum circuli B E D, eſſe perpendicularẽ, <lb/>cadet F G, vtrinque producta in polos circuli B E D. </s> <s xml:id="echoid-s676" xml:space="preserve">Quare cum G F, in <lb/> <anchor type="note" xlink:label="note-031-01a" xlink:href="note-031-01"/> plano circuli A B C D, exiſtens, producta cadat in circunferentiam eius ad <lb/>puncta A, C, quæ etiam in ſuperficie ſphæræ funt, erunt A, C, poli circuli <lb/>B E D, atque adeo circulus maximus A B C D, circulũ non maximũ B E D, <lb/>per polos A, C, ſecabit. </s> <s xml:id="echoid-s677" xml:space="preserve">quod ſecundo loco propoſitũ fuit. </s> <s xml:id="echoid-s678" xml:space="preserve">Si igitur in ſphæ <lb/>ra maximus circulus circulum non maximum, &</s> <s xml:id="echoid-s679" xml:space="preserve">c. </s> <s xml:id="echoid-s680" xml:space="preserve">Quod erat oftendendum.</s> <s xml:id="echoid-s681" xml:space="preserve"/> </p> <div xml:id="echoid-div74" type="float" level="2" n="1"> <figure xlink:label="fig-030-02" xlink:href="fig-030-02a"> <image file="030-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/030-02"/> </figure> <note position="left" xlink:label="note-030-04" xlink:href="note-030-04a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-030-05" xlink:href="note-030-05a" xml:space="preserve">7. huius.</note> <note position="left" xlink:label="note-030-06" xlink:href="note-030-06a" xml:space="preserve">18. vndec.</note> <note position="right" xlink:label="note-031-01" xlink:href="note-031-01a" xml:space="preserve">8. huius.</note> </div> </div> <div xml:id="echoid-div76" type="section" level="1" n="47"> <head xml:id="echoid-head58" xml:space="preserve">THEOREMA 14. PROPOS. 15.</head> <note position="right" xml:space="preserve">20.</note> <p> <s xml:id="echoid-s682" xml:space="preserve">Si in ſphæra maximus circulus, eorum, qui in <lb/>ſphæra ſunt, circulorum aliquem per polos ſecet; <lb/></s> <s xml:id="echoid-s683" xml:space="preserve">bifariam, & </s> <s xml:id="echoid-s684" xml:space="preserve">ad angulos rectos eum ſecat.</s> <s xml:id="echoid-s685" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s686" xml:space="preserve">IN Sphæra maximus circulus A B C D, ſecet circulum B E D, per polos <lb/>A, C. </s> <s xml:id="echoid-s687" xml:space="preserve">Dico circulum A B C D, ſecare circulum B E D, fifariam, & </s> <s xml:id="echoid-s688" xml:space="preserve">ad angu <lb/> <anchor type="figure" xlink:label="fig-031-01a" xlink:href="fig-031-01"/> los rectos. </s> <s xml:id="echoid-s689" xml:space="preserve">Connectat enim recta A C, polos <lb/>A, C, occurrens plano circuli B E D, in F, <lb/>puhcto. </s> <s xml:id="echoid-s690" xml:space="preserve">Et quoniam recta A C, ad planũ cir <lb/>culi B E D, per pendicularis eſt, tranſitq́; </s> <s xml:id="echoid-s691" xml:space="preserve">per <lb/> <anchor type="note" xlink:label="note-031-03a" xlink:href="note-031-03"/> centrum ſphæræ, & </s> <s xml:id="echoid-s692" xml:space="preserve">circuli B E D; </s> <s xml:id="echoid-s693" xml:space="preserve">erit F, cen <lb/>trum circuli B E D. </s> <s xml:id="echoid-s694" xml:space="preserve">Cum ergo circulus ma <lb/>ximus A B C D, circulum B E D, ſecans tran <lb/>ſeat per rectam A C, ac proinde per centrũ <lb/>F, erit communis ſectio B F D, diameter cir <lb/>culi B E D. </s> <s xml:id="echoid-s695" xml:space="preserve">Bifariam ergo ſecatur circulus <lb/>B E D. </s> <s xml:id="echoid-s696" xml:space="preserve">Dico quod & </s> <s xml:id="echoid-s697" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s698" xml:space="preserve">Cum <lb/>enim recta A C, oſtenſa ſit perpendicularis <lb/>ad planum circuli B E D, erit quoque planũ <lb/>circuli maximi A B C D, per rectam A C, ductum ad idem planum circuli <lb/> <anchor type="note" xlink:label="note-031-04a" xlink:href="note-031-04"/> B E D, rectum. </s> <s xml:id="echoid-s699" xml:space="preserve">Igitur ſi in ſphęra maximus circulus, &</s> <s xml:id="echoid-s700" xml:space="preserve">c. </s> <s xml:id="echoid-s701" xml:space="preserve">Quod demonſtran <lb/>dum erat.</s> <s xml:id="echoid-s702" xml:space="preserve"/> </p> <div xml:id="echoid-div76" type="float" level="2" n="1"> <figure xlink:label="fig-031-01" xlink:href="fig-031-01a"> <image file="031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/031-01"/> </figure> <note position="right" xlink:label="note-031-03" xlink:href="note-031-03a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-031-04" xlink:href="note-031-04a" xml:space="preserve">18. vndes.</note> </div> </div> <div xml:id="echoid-div78" type="section" level="1" n="48"> <head xml:id="echoid-head59" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s703" xml:space="preserve">_QVATVOR_alia theoremata hoc loco addútur in alia verſione, hoc ordine.</s> <s xml:id="echoid-s704" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div79" type="section" level="1" n="49"> <head xml:id="echoid-head60" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s705" xml:space="preserve">SI in ſphæra maximus circulus per polos alterius cuiuſpiam ma <lb/> <anchor type="note" xlink:label="note-031-05a" xlink:href="note-031-05"/> ximi circuli tranſeat, tranſibit viciſſim hic per polos illius.</s> <s xml:id="echoid-s706" xml:space="preserve"/> </p> <div xml:id="echoid-div79" type="float" level="2" n="1"> <note position="right" xlink:label="note-031-05" xlink:href="note-031-05a" xml:space="preserve">21.</note> </div> <pb o="20" file="032" n="32" rhead=""/> <p style="it"> <s xml:id="echoid-s707" xml:space="preserve">_IN_ ſphæra tranſeat maximus circulus <lb/> <anchor type="figure" xlink:label="fig-032-01a" xlink:href="fig-032-01"/> _A B C D,_ per _A, C,_ polos circuli maximi _B D._ <lb/></s> <s xml:id="echoid-s708" xml:space="preserve">_Dico_ & </s> <s xml:id="echoid-s709" xml:space="preserve">maximum circulum _B D,_ per polos ma-<lb/>ximi circuli _A B C D,_ tranſire. </s> <s xml:id="echoid-s710" xml:space="preserve">Quoniam enim <lb/>circulus maximus _A B C D,_ circulum _B D,_ ſe-<lb/> <anchor type="note" xlink:label="note-032-01a" xlink:href="note-032-01"/> cat per polos, ſecabit ipſum ad angulos rectos. <lb/></s> <s xml:id="echoid-s711" xml:space="preserve">Quare vicißim maximus circulus _B D,_ circu-<lb/>lum _A B C D,_ ad angulos rectos ſccabit; </s> <s xml:id="echoid-s712" xml:space="preserve">at-<lb/>que adeo per ipſius polos eum ſecabit. </s> <s xml:id="echoid-s713" xml:space="preserve">Quod eſt <lb/> <anchor type="note" xlink:label="note-032-02a" xlink:href="note-032-02"/> pr opoſitum.</s> <s xml:id="echoid-s714" xml:space="preserve"/> </p> <div xml:id="echoid-div80" type="float" level="2" n="2"> <figure xlink:label="fig-032-01" xlink:href="fig-032-01a"> <image file="032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/032-01"/> </figure> <note position="left" xlink:label="note-032-01" xlink:href="note-032-01a" xml:space="preserve">15. huius.</note> <note position="left" xlink:label="note-032-02" xlink:href="note-032-02a" xml:space="preserve">13. huius.</note> </div> </div> <div xml:id="echoid-div82" type="section" level="1" n="50"> <head xml:id="echoid-head61" xml:space="preserve">II.</head> <note position="left" xml:space="preserve">22.</note> <p> <s xml:id="echoid-s715" xml:space="preserve">SI in ſphæra circulus circulum per polos ſecet, circulus maxi-<lb/>mus eſt, & </s> <s xml:id="echoid-s716" xml:space="preserve">bifariam eum ſecat, & </s> <s xml:id="echoid-s717" xml:space="preserve">ad angulos rectos.</s> <s xml:id="echoid-s718" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s719" xml:space="preserve">_IN_ ſphæra circulus _A B C D,_ ſecet circu-<lb/> <anchor type="figure" xlink:label="fig-032-02a" xlink:href="fig-032-02"/> lum _B D,_ per polos _A, C._ </s> <s xml:id="echoid-s720" xml:space="preserve">Dico ipſum eſſe circu-<lb/>culum maximum, ſecareq́; </s> <s xml:id="echoid-s721" xml:space="preserve">circulum _B D,_ bifa-<lb/>riam, & </s> <s xml:id="echoid-s722" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s723" xml:space="preserve">Coniungat enim re-<lb/>cta _A C,_ polos _A, C,_ quæ neceſſario in plano <lb/>circuli _A B C D,_ erit, quod circunferentia eius <lb/>per eoſdem polos _A, C,_ ponatur tranſire. </s> <s xml:id="echoid-s724" xml:space="preserve">Quo-<lb/>niam vero recta _A C,_ per _A, C,_ polos circuli <lb/> <anchor type="note" xlink:label="note-032-04a" xlink:href="note-032-04"/> _B D,_ ducta tranſit per centrum ſphæræ, tranſi-<lb/>bit quoque cir culus _A B C D,_ (cum per rectã <lb/>_A C,_ ducatur) per centrũ ſphæræ; </s> <s xml:id="echoid-s725" xml:space="preserve">atque adeo <lb/> <anchor type="note" xlink:label="note-032-05a" xlink:href="note-032-05"/> maximus erit. </s> <s xml:id="echoid-s726" xml:space="preserve">Quare cum per _A, C,_ polos cir <lb/> <anchor type="note" xlink:label="note-032-06a" xlink:href="note-032-06"/> culi _B D,_ ponatur tranſire, ſecabit eum bifariam, & </s> <s xml:id="echoid-s727" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s728" xml:space="preserve">Quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s729" xml:space="preserve"/> </p> <div xml:id="echoid-div82" type="float" level="2" n="1"> <figure xlink:label="fig-032-02" xlink:href="fig-032-02a"> <image file="032-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/032-02"/> </figure> <note position="left" xlink:label="note-032-04" xlink:href="note-032-04a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-032-05" xlink:href="note-032-05a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-032-06" xlink:href="note-032-06a" xml:space="preserve">15. huius.</note> </div> </div> <div xml:id="echoid-div84" type="section" level="1" n="51"> <head xml:id="echoid-head62" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s730" xml:space="preserve">SI in ſphæra circulus circulum bifariam, & </s> <s xml:id="echoid-s731" xml:space="preserve">ad angulos rectos <lb/> <anchor type="note" xlink:label="note-032-07a" xlink:href="note-032-07"/> ſecet, circulus maximus eſt, & </s> <s xml:id="echoid-s732" xml:space="preserve">per polos eum ſecat.</s> <s xml:id="echoid-s733" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="1"> <note position="left" xlink:label="note-032-07" xlink:href="note-032-07a" xml:space="preserve">23.</note> </div> <p style="it"> <s xml:id="echoid-s734" xml:space="preserve">_IN_ſphæra circulus _A B C D,_ ſecet circulum <lb/> <anchor type="figure" xlink:label="fig-032-03a" xlink:href="fig-032-03"/> _B D,_ bifariam, & </s> <s xml:id="echoid-s735" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s736" xml:space="preserve">_Dico_ ipſum <lb/>eſſe circulum maximum, tranſireq́; </s> <s xml:id="echoid-s737" xml:space="preserve">per polos cir-<lb/>culi _B D._ </s> <s xml:id="echoid-s738" xml:space="preserve">Sit recta _BD;_ </s> <s xml:id="echoid-s739" xml:space="preserve">communis circu lorum ſe-<lb/>etio. </s> <s xml:id="echoid-s740" xml:space="preserve">Quo niam igitur circulus _A B C D,_ circu-<lb/>lum _B D,_ ſecat bifariam, erit recta _B D,_ nempe <lb/>communis ſectio circulorũ, diameter circuli B D, <lb/>atque adeo diuiſa recta _B D,_ bifari am in E: </s> <s xml:id="echoid-s741" xml:space="preserve">erit <lb/>E, eiuſdem circuli centrum. </s> <s xml:id="echoid-s742" xml:space="preserve">Ducatur in plano cir <lb/>culi _A B C D,_ recta E A, perpendicularis ad re <lb/>etam _B D._ </s> <s xml:id="echoid-s743" xml:space="preserve">Et quoniam circulus _A B C D,_ circu <pb o="21" file="033" n="33" rhead=""/> lum _B D,_ ponitur ſecare ad angulos rectos, erit ex defin. </s> <s xml:id="echoid-s744" xml:space="preserve">4. </s> <s xml:id="echoid-s745" xml:space="preserve">lib. </s> <s xml:id="echoid-s746" xml:space="preserve">11. </s> <s xml:id="echoid-s747" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s748" xml:space="preserve">_E A,_ ad pla <lb/>num circuli _B D,_ recta; </s> <s xml:id="echoid-s749" xml:space="preserve">ac proinde cum ex E, centro ipſius educatur, in vtrunque <lb/>polum eiuſdem cadet. </s> <s xml:id="echoid-s750" xml:space="preserve">Cadit autem in circunferentiam circuli _A B C D,_ in ſuperficie <lb/> <anchor type="note" xlink:label="note-033-01a" xlink:href="note-033-01"/> ſphæræ exiſtem ad puncta _A, C._ </s> <s xml:id="echoid-s751" xml:space="preserve">Sunt ergo _A, C,_ poli circuli _BD;_ </s> <s xml:id="echoid-s752" xml:space="preserve">at que adeo cir <lb/>culus _A B C D,_ circulũ _B D,_ per polos _A, C,_ ſecat. </s> <s xml:id="echoid-s753" xml:space="preserve">Quare ex præcedenti theoremate, <lb/>maximus circulus eſt. </s> <s xml:id="echoid-s754" xml:space="preserve">Probatum autem eſt, quod & </s> <s xml:id="echoid-s755" xml:space="preserve">circulum _B D,_ per polos ſecat. <lb/></s> <s xml:id="echoid-s756" xml:space="preserve">Conſtat ergo propoſitum.</s> <s xml:id="echoid-s757" xml:space="preserve"/> </p> <div xml:id="echoid-div85" type="float" level="2" n="2"> <figure xlink:label="fig-032-03" xlink:href="fig-032-03a"> <image file="032-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/032-03"/> </figure> <note position="right" xlink:label="note-033-01" xlink:href="note-033-01a" xml:space="preserve">Schol. 8. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div87" type="section" level="1" n="52"> <head xml:id="echoid-head63" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s758" xml:space="preserve">SI in ſphæra ſit circulus, & </s> <s xml:id="echoid-s759" xml:space="preserve">ab altero polorum eius recta cadens <lb/> <anchor type="note" xlink:label="note-033-02a" xlink:href="note-033-02"/> in planum ipſius ad angulos rectos æqualis ſit ſemidiametro eius, <lb/>circulus maximus eſt.</s> <s xml:id="echoid-s760" xml:space="preserve"/> </p> <div xml:id="echoid-div87" type="float" level="2" n="1"> <note position="right" xlink:label="note-033-02" xlink:href="note-033-02a" xml:space="preserve">24.</note> </div> <p style="it"> <s xml:id="echoid-s761" xml:space="preserve">_IN_ſphæra ſit circulus _AB_, à cuius altero polorum _C,_ in planum eius cadens re <lb/>eta perpendicularis _C D,_ æqualis ſit ipſius ſemidiametro. </s> <s xml:id="echoid-s762" xml:space="preserve">_Dico A B,_ eſſe circulum ma <lb/>ximum. </s> <s xml:id="echoid-s763" xml:space="preserve">Cum enim _C D,_ perpendicularis ſit ad circulum _A B,_ cadet ipſa in circuli <lb/>centrum, & </s> <s xml:id="echoid-s764" xml:space="preserve">producta cadet in alterum polum, qui ſit E. </s> <s xml:id="echoid-s765" xml:space="preserve">Eſt ergo _D,_ centrum circu <lb/> <anchor type="figure" xlink:label="fig-033-01a" xlink:href="fig-033-01"/> <anchor type="note" xlink:label="note-033-03a" xlink:href="note-033-03"/> li _AB;_ </s> <s xml:id="echoid-s766" xml:space="preserve">atque adeo perpendicularis _C D,_ tran-<lb/>ſit per centrum ſphæræ. </s> <s xml:id="echoid-s767" xml:space="preserve">Ducatur per rectã _C E,_ <lb/> <anchor type="note" xlink:label="note-033-04a" xlink:href="note-033-04"/> in ſphæra planum vtcunque faciens in ſphæra <lb/>circulum _A E B C,_ qui cum tranſeat per centrũ, <lb/> <anchor type="note" xlink:label="note-033-05a" xlink:href="note-033-05"/> ſphæræ, maximus erit: </s> <s xml:id="echoid-s768" xml:space="preserve">qui circulum _A B,_ ſecet <lb/>in punctis _A, B,_ & </s> <s xml:id="echoid-s769" xml:space="preserve">iungatur ſemidiameter _D B,_ <lb/>cui ex hypotheſi æqualis eſt _G D._ </s> <s xml:id="echoid-s770" xml:space="preserve">Quoniam vero <lb/>_C D,_ perpendicularis ponitur ad circulum A B, <lb/>erit, ex deſin. </s> <s xml:id="echoid-s771" xml:space="preserve">3. </s> <s xml:id="echoid-s772" xml:space="preserve">lib. </s> <s xml:id="echoid-s773" xml:space="preserve">11. </s> <s xml:id="echoid-s774" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s775" xml:space="preserve">angulus _C D B,_ re-<lb/> <anchor type="note" xlink:label="note-033-06a" xlink:href="note-033-06"/> ctus. </s> <s xml:id="echoid-s776" xml:space="preserve">Quare _B D,_ media proportionalis eſt inter <lb/>_C D, D E,_ hoc eſt, erit, vt _C D,_ ad _B D,_ ita _B D,_ <lb/>ad _D E._ </s> <s xml:id="echoid-s777" xml:space="preserve">Eſt autem _C D,_ ipſi _B D,_ æqualis. </s> <s xml:id="echoid-s778" xml:space="preserve">Igi-<lb/>tur & </s> <s xml:id="echoid-s779" xml:space="preserve">_D E,_ eidem _B D,_ æqualis erit; </s> <s xml:id="echoid-s780" xml:space="preserve">atq; </s> <s xml:id="echoid-s781" xml:space="preserve">adeo <lb/>& </s> <s xml:id="echoid-s782" xml:space="preserve">_C D, D E,_ inter ſe æquales erunt. </s> <s xml:id="echoid-s783" xml:space="preserve">Cum ergo _C E,_ oſtenſa ſit tranſire per centrũ <lb/>ſphæræ, erit _D,_ centrum ſphæræ. </s> <s xml:id="echoid-s784" xml:space="preserve">Erat autem & </s> <s xml:id="echoid-s785" xml:space="preserve">centraum circuli _A B._ </s> <s xml:id="echoid-s786" xml:space="preserve">Idem ergo <lb/>eſt centrum ſphæræ. </s> <s xml:id="echoid-s787" xml:space="preserve">& </s> <s xml:id="echoid-s788" xml:space="preserve">circuli _A B,_ ac proinde circulus _A B,_ maximus eſt. </s> <s xml:id="echoid-s789" xml:space="preserve">Quod eſt <lb/> <anchor type="note" xlink:label="note-033-07a" xlink:href="note-033-07"/> propoſitum.</s> <s xml:id="echoid-s790" xml:space="preserve"/> </p> <div xml:id="echoid-div88" type="float" level="2" n="2"> <figure xlink:label="fig-033-01" xlink:href="fig-033-01a"> <image file="033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/033-01"/> </figure> <note position="right" xlink:label="note-033-03" xlink:href="note-033-03a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-033-04" xlink:href="note-033-04a" xml:space="preserve">Coroll. 2. <lb/>huius.</note> <note position="right" xlink:label="note-033-05" xlink:href="note-033-05a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-033-06" xlink:href="note-033-06a" xml:space="preserve">Schol. 13. <lb/>fextf.</note> <note position="right" xlink:label="note-033-07" xlink:href="note-033-07a" xml:space="preserve">6. huius.</note> </div> </div> <div xml:id="echoid-div90" type="section" level="1" n="53"> <head xml:id="echoid-head64" xml:space="preserve">THEOREMA 15. PROPOS. 16.</head> <note position="right" xml:space="preserve">25.</note> <p> <s xml:id="echoid-s791" xml:space="preserve">SI in ſphæra ſit maximus circulus, recta linea <lb/>ducta ab eiuſdem circuli polo ad circunferentiã <lb/>æqualis eſt lateri quadrati inſcripti in maximo cir-<lb/>culo.</s> <s xml:id="echoid-s792" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s793" xml:space="preserve">IN ſphæra ſit circulus maximus A B, à cuius polo C, ad eius circũferentiã <lb/>ducatur recta C B. </s> <s xml:id="echoid-s794" xml:space="preserve">Dico C B, æqualẽ eſſe lateri quadrati in circulo A B, vel <pb o="22" file="034" n="34" rhead=""/> quouis alio maximo inſcripti. </s> <s xml:id="echoid-s795" xml:space="preserve">Ducatur ex C, ad circulum A B, perpendicu <lb/> <anchor type="note" xlink:label="note-034-01a" xlink:href="note-034-01"/> laris C E, quæ in centrum ipſius cadet, quod ſit E, & </s> <s xml:id="echoid-s796" xml:space="preserve">producta in reliquum <lb/> <anchor type="figure" xlink:label="fig-034-01a" xlink:href="fig-034-01"/> <anchor type="note" xlink:label="note-034-02a" xlink:href="note-034-02"/> polum, qui ſit D, cadet. </s> <s xml:id="echoid-s797" xml:space="preserve">Iam per rectas C B, <lb/> <anchor type="note" xlink:label="note-034-03a" xlink:href="note-034-03"/> C D, planum ducatur faciens in ſphæra cir-<lb/>culum A D B C, qui cum per E, centrum <lb/>ſphæræ (Eſt enim E, centrum circuli maxi-<lb/> <anchor type="note" xlink:label="note-034-04a" xlink:href="note-034-04"/> mi A B, quòd per centrum ſphæræ tranſeat, <lb/> <anchor type="note" xlink:label="note-034-05a" xlink:href="note-034-05"/> idem, quod ſphæræ) tranſeat, maximus erit, <lb/>atq; </s> <s xml:id="echoid-s798" xml:space="preserve">adeo circulum maximum A B, bifariam <lb/> <anchor type="note" xlink:label="note-034-06a" xlink:href="note-034-06"/> ſecabit. </s> <s xml:id="echoid-s799" xml:space="preserve">Quod etiam inde patet, quòd per <lb/> <anchor type="note" xlink:label="note-034-07a" xlink:href="note-034-07"/> eius polos incedat. </s> <s xml:id="echoid-s800" xml:space="preserve">Hinc enim fit, vt ipſum <lb/> <anchor type="note" xlink:label="note-034-08a" xlink:href="note-034-08"/> bifariam diuidat. </s> <s xml:id="echoid-s801" xml:space="preserve">Sit ergo communis ſectio <lb/>diameter B E A. </s> <s xml:id="echoid-s802" xml:space="preserve">Et quoniam C E, perpendi <lb/>cularis ducta eſt ad circulum A B, erit eadé <lb/>perpendicularis ad rectam A B, ex defin. </s> <s xml:id="echoid-s803" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s804" xml:space="preserve">lib. </s> <s xml:id="echoid-s805" xml:space="preserve">11. </s> <s xml:id="echoid-s806" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s807" xml:space="preserve">Duæ ergo diametri A B, C D, in maximo circulo A D B C, ſeſe <lb/>mutuo ſecãt ad angulos rectos; </s> <s xml:id="echoid-s808" xml:space="preserve">ac propterea vt in lib. </s> <s xml:id="echoid-s809" xml:space="preserve">4. </s> <s xml:id="echoid-s810" xml:space="preserve">Euclidis demonſtra <lb/> <anchor type="note" xlink:label="note-034-09a" xlink:href="note-034-09"/> tum eſt, C B, latus eſt quadrati in circulo maximo A D B C, atq; </s> <s xml:id="echoid-s811" xml:space="preserve">adeò & </s> <s xml:id="echoid-s812" xml:space="preserve">in <lb/>maximo A B, deſcripti. </s> <s xml:id="echoid-s813" xml:space="preserve">Si igitur in ſphæra fit maximus circulus, recta linea <lb/>ducta, &</s> <s xml:id="echoid-s814" xml:space="preserve">c. </s> <s xml:id="echoid-s815" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s816" xml:space="preserve"/> </p> <div xml:id="echoid-div90" type="float" level="2" n="1"> <note position="left" xlink:label="note-034-01" xlink:href="note-034-01a" xml:space="preserve">11. vndee.</note> <figure xlink:label="fig-034-01" xlink:href="fig-034-01a"> <image file="034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/034-01"/> </figure> <note position="left" xlink:label="note-034-02" xlink:href="note-034-02a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-034-03" xlink:href="note-034-03a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-034-04" xlink:href="note-034-04a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-034-05" xlink:href="note-034-05a" xml:space="preserve">Coroll. 1. <lb/>huius.</note> <note position="left" xlink:label="note-034-06" xlink:href="note-034-06a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-034-07" xlink:href="note-034-07a" xml:space="preserve">11. huius.</note> <note position="left" xlink:label="note-034-08" xlink:href="note-034-08a" xml:space="preserve">15. huius.</note> <note position="left" xlink:label="note-034-09" xlink:href="note-034-09a" xml:space="preserve">6. quarti.</note> </div> </div> <div xml:id="echoid-div92" type="section" level="1" n="54"> <head xml:id="echoid-head65" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s817" xml:space="preserve">QVONIAM verò quatuor anguli recti ad centrum E, æquales ſunt, atq; </s> <s xml:id="echoid-s818" xml:space="preserve">adeò qua-<lb/>tuor arcus B C, C A, A D, D B, ſuper quos aſcendetunt, æquales, nem pe quadrantes, per-<lb/> <anchor type="note" xlink:label="note-034-10a" xlink:href="note-034-10"/> ſpicuum eſt, in ſphæra polum maximi citculi abeſſe à circunferentia maximi circuli, qua-<lb/>drante maximi circuli. </s> <s xml:id="echoid-s819" xml:space="preserve">Abeſt enim C, polus circuli maximi A B, ab eius circunferentia <lb/>quadrante C B, eademq́; </s> <s xml:id="echoid-s820" xml:space="preserve">ratio de ceteris habenda eſt. </s> <s xml:id="echoid-s821" xml:space="preserve">Semper enim recta ducta à circunfe-<lb/>rentia maximi circuli ad eiuſdem polum æqualis eſt lateri quadrati in maximo circulo <lb/> <anchor type="note" xlink:label="note-034-11a" xlink:href="note-034-11"/> inſcripti, arq; </s> <s xml:id="echoid-s822" xml:space="preserve">adeò quadrantem in maximo circulo ſubtendet.</s> <s xml:id="echoid-s823" xml:space="preserve"/> </p> <div xml:id="echoid-div92" type="float" level="2" n="1"> <note position="left" xlink:label="note-034-10" xlink:href="note-034-10a" xml:space="preserve">26. tertij.</note> <note position="left" xlink:label="note-034-11" xlink:href="note-034-11a" xml:space="preserve">16. huius.</note> </div> </div> <div xml:id="echoid-div94" type="section" level="1" n="55"> <head xml:id="echoid-head66" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s824" xml:space="preserve">_CONVERSVM_quoq; </s> <s xml:id="echoid-s825" xml:space="preserve">huius demonſtratur in alia verſione hoc theoremate.</s> <s xml:id="echoid-s826" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s827" xml:space="preserve">SI in ſphæra ſit circulus, & </s> <s xml:id="echoid-s828" xml:space="preserve">ab eius polo ad circunferentiam du <lb/> <anchor type="note" xlink:label="note-034-12a" xlink:href="note-034-12"/> cta recta æqualis ſit lateri quadtati in eo deſcripti, circulus ipſe <lb/>maximus eſt.</s> <s xml:id="echoid-s829" xml:space="preserve"/> </p> <div xml:id="echoid-div94" type="float" level="2" n="1"> <note position="left" xlink:label="note-034-12" xlink:href="note-034-12a" xml:space="preserve">26.</note> </div> <p style="it"> <s xml:id="echoid-s830" xml:space="preserve">_IN_ eadem figura ex _C,_ polo ad circunferentiã circuli _A B,_ ductarecta _C B,_ ſit <lb/>equalis lateri quadrati in circulo _A B,_ deſcripti. </s> <s xml:id="echoid-s831" xml:space="preserve">Dico _A B,_ circulum eſſe maxi-<lb/>mum. </s> <s xml:id="echoid-s832" xml:space="preserve">Ducatur enim ex _C,_ ad circulum _A B,_ perpendicularis _C E,_ quæ in eius <lb/> <anchor type="note" xlink:label="note-034-13a" xlink:href="note-034-13"/> centrum cadet, quod ſit _E._ </s> <s xml:id="echoid-s833" xml:space="preserve">Ducta autem ſemidiametro _E B,_ erit ex deſin. </s> <s xml:id="echoid-s834" xml:space="preserve">3. </s> <s xml:id="echoid-s835" xml:space="preserve">lib. </s> <s xml:id="echoid-s836" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s837" xml:space="preserve"> <anchor type="note" xlink:label="note-034-14a" xlink:href="note-034-14"/> Eucl. </s> <s xml:id="echoid-s838" xml:space="preserve">angulus _E,_ rectus. </s> <s xml:id="echoid-s839" xml:space="preserve">Igitur quadratum in circul _A B,_ deſcriptum, æquale eſt <lb/>quadratis ex _B E, C E:_ </s> <s xml:id="echoid-s840" xml:space="preserve">ſed quadratum ſemidiametri _B E,_ dimiaium eſt quadrati <lb/> <anchor type="note" xlink:label="note-034-15a" xlink:href="note-034-15"/> in circulo _A B,_ deſcripti, vt mox oſtendemus. </s> <s xml:id="echoid-s841" xml:space="preserve">I gitur & </s> <s xml:id="echoid-s842" xml:space="preserve">quadratum ex _C E,_ eiuſ-<lb/>dem quadrati in circulo _A B,_ deſcripti dimidium erit; </s> <s xml:id="echoid-s843" xml:space="preserve">atque adeo quadrata ex <lb/>_B E, C E,_ inter ſe æqualia, necnon & </s> <s xml:id="echoid-s844" xml:space="preserve">lineæ propterea _B E, C E._ </s> <s xml:id="echoid-s845" xml:space="preserve">aquales erunt. <lb/></s> <s xml:id="echoid-s846" xml:space="preserve">Quare cum _C E,_ ducta ſit ex C, polo circuli _A B,_ ad ipſum circulum perpendicu-<lb/>laris, oſtenſaq̀; </s> <s xml:id="echoid-s847" xml:space="preserve">ſit ſemidiametro _B E,_ aequalis; </s> <s xml:id="echoid-s848" xml:space="preserve">erit circulus _A B,_ maximus.</s> <s xml:id="echoid-s849" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="2"> <note position="left" xlink:label="note-034-13" xlink:href="note-034-13a" xml:space="preserve">11. vndec.</note> <note position="left" xlink:label="note-034-14" xlink:href="note-034-14a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-034-15" xlink:href="note-034-15a" xml:space="preserve">47. primi.</note> </div> <note position="left" xml:space="preserve">Schol. 15. <lb/>huius.</note> <pb o="23" file="035" n="35" rhead=""/> </div> <div xml:id="echoid-div97" type="section" level="1" n="56"> <head xml:id="echoid-head67" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s850" xml:space="preserve">IN omni circulo quadratum ſemidiametri dimidium eſt qua-<lb/>drati in ipſo circulo deſcripti.</s> <s xml:id="echoid-s851" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s852" xml:space="preserve">_IN_ circulo, cuius centrum E, ductæ ſint duæ diametri A C, B D, <lb/> <anchor type="figure" xlink:label="fig-035-01a" xlink:href="fig-035-01"/> ſeſe ad angulos rectos ſecantes in E, cen-<lb/>tro. </s> <s xml:id="echoid-s853" xml:space="preserve">lunctis igitur rectis A B, B C, C D, <lb/>_D A,_ quadratum erit A B C D, in circu <lb/>lo inſcriptum, vt conſtat ex propoſ. </s> <s xml:id="echoid-s854" xml:space="preserve">6. </s> <s xml:id="echoid-s855" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s856" xml:space="preserve">4. </s> <s xml:id="echoid-s857" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s858" xml:space="preserve">Quoniam vero quadrata ex ſemi-<lb/>diametris æqualibus E A, E B, æqualia <lb/>inter ſe, æqualia ſimul ſunt quadrato ex <lb/>A B; </s> <s xml:id="echoid-s859" xml:space="preserve">dimidium erit quadratum ſemidia <lb/> <anchor type="note" xlink:label="note-035-01a" xlink:href="note-035-01"/> metri E A, quadrati ex A B, quod in cir <lb/>culo deſcribitur. </s> <s xml:id="echoid-s860" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s861" xml:space="preserve">Ex <lb/>quo conſtat, in ſuperiorifigura, quadratum ſemidiametri B E, dimidium <lb/>eſſe quadrati ex C B, quod æquale ponitur ei, quod in circulo A B, in-<lb/>ſcribitur.</s> <s xml:id="echoid-s862" xml:space="preserve"/> </p> <div xml:id="echoid-div97" type="float" level="2" n="1"> <figure xlink:label="fig-035-01" xlink:href="fig-035-01a"> <image file="035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/035-01"/> </figure> <note position="right" xlink:label="note-035-01" xlink:href="note-035-01a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div99" type="section" level="1" n="57"> <head xml:id="echoid-head68" xml:space="preserve">THEOR. 16. PROPOS. 17.</head> <note position="right" xml:space="preserve">27.</note> <p> <s xml:id="echoid-s863" xml:space="preserve">SI in ſphæra ſit circulus, à cuius polo in ipſius <lb/>circunferentiam ducta recta linea æqualis ſit late-<lb/>ri quadrati in ſcripti in maximo circulo, ipſe circu <lb/>lus maximus erit.</s> <s xml:id="echoid-s864" xml:space="preserve"/> </p> <figure> <image file="035-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/035-02"/> </figure> <p> <s xml:id="echoid-s865" xml:space="preserve">IN ſphæra ſit circulus A B, à cuius polo <lb/>C, ad eius circunferentiam recta ducta C A, <lb/>æqualis ſit lateri quadrati in maximo circulo <lb/>ſphæræ deſcripti. </s> <s xml:id="echoid-s866" xml:space="preserve">Dico A B, circulum eſſe ma <lb/>ximum. </s> <s xml:id="echoid-s867" xml:space="preserve">Per rectam enim A C, & </s> <s xml:id="echoid-s868" xml:space="preserve">centrũ ſphæ <lb/>ræ planum ducatur, faciens in ſphæra circulũ <lb/> <anchor type="note" xlink:label="note-035-03a" xlink:href="note-035-03"/> A C B, qui maximus erit, cum per ſphæræ cen <lb/> <anchor type="note" xlink:label="note-035-04a" xlink:href="note-035-04"/> trum ducatur. </s> <s xml:id="echoid-s869" xml:space="preserve">Ducatur quoq; </s> <s xml:id="echoid-s870" xml:space="preserve">ex C, recta li-<lb/>nea C B, ad B, punctũ, in quo circulus maxi-<lb/>mus A C B, circulũ A B, ſecat; </s> <s xml:id="echoid-s871" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s872" xml:space="preserve">per deſi <lb/>nit. </s> <s xml:id="echoid-s873" xml:space="preserve">poli, recta C B, rectæ C A, æqualis. </s> <s xml:id="echoid-s874" xml:space="preserve">Cũ <lb/>ergo A C, ponatur latus quadrati in maximo circulo A C B, deſcripti, erit <lb/>quoque C B, latus eiuſdem quadrati; </s> <s xml:id="echoid-s875" xml:space="preserve">atque adeò duo arcus A C, C B, qua-<lb/>drantes erunt conſicientes ſemicirculũ A C B, quòd quatuor latera quadra-<lb/>ti æqualia ſubtendãt quatuor circuli arcus æquales. </s> <s xml:id="echoid-s876" xml:space="preserve">Recta igitur A B, com-<lb/> <anchor type="note" xlink:label="note-035-05a" xlink:href="note-035-05"/> <pb o="24" file="036" n="36" rhead=""/> munis fectio circulorum diameter erit circuli maximi A C B; </s> <s xml:id="echoid-s877" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s878" xml:space="preserve"><lb/>ſphæræ. </s> <s xml:id="echoid-s879" xml:space="preserve">Quoniamverò circulus maximus A C B, circulum A B, per polos ſe-<lb/>cans ſecat bifariam, erit quoq; </s> <s xml:id="echoid-s880" xml:space="preserve">A B, communis ſectio diameter circuli A B, <lb/> <anchor type="note" xlink:label="note-036-01a" xlink:href="note-036-01"/> ac proinde cum & </s> <s xml:id="echoid-s881" xml:space="preserve">ſphæræ diameter ſit, circulus maximus erit A B. </s> <s xml:id="echoid-s882" xml:space="preserve">Si in ſphæ <lb/>ra ergo ſit circulus, à cuius polo, &</s> <s xml:id="echoid-s883" xml:space="preserve">c. </s> <s xml:id="echoid-s884" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s885" xml:space="preserve"/> </p> <div xml:id="echoid-div99" type="float" level="2" n="1"> <note position="right" xlink:label="note-035-03" xlink:href="note-035-03a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-035-04" xlink:href="note-035-04a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-035-05" xlink:href="note-035-05a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-036-01" xlink:href="note-036-01a" xml:space="preserve">15. huius.</note> </div> </div> <div xml:id="echoid-div101" type="section" level="1" n="58"> <head xml:id="echoid-head69" xml:space="preserve">PROBL. 2. PROP. 18.</head> <note position="left" xml:space="preserve">28.</note> <p> <s xml:id="echoid-s886" xml:space="preserve">LINEAM rectam deſcribere æqualem dia-<lb/>metro circuli cuiuſlibetin ſphæra dati.</s> <s xml:id="echoid-s887" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s888" xml:space="preserve">IN ſphæra ſit datus circulus quilibet A B C D, cuius diametro rectam <lb/>æqualem oporteat deſcribere. </s> <s xml:id="echoid-s889" xml:space="preserve">Sumptis tribus punctis in circunferentia circu <lb/>li vtcunq; </s> <s xml:id="echoid-s890" xml:space="preserve">A, B, D, & </s> <s xml:id="echoid-s891" xml:space="preserve">iunctis rectis A B, A D, B D, conſtituatur triangulo <lb/>A B D, triangulum æquale E F G, ita vt latus E F, lateri A B, & </s> <s xml:id="echoid-s892" xml:space="preserve">E G, ipfi <lb/> <anchor type="figure" xlink:label="fig-036-01a" xlink:href="fig-036-01"/> <anchor type="note" xlink:label="note-036-03a" xlink:href="note-036-03"/> A D, & </s> <s xml:id="echoid-s893" xml:space="preserve">F G, ipſi B D, æqua-<lb/>le ſit. </s> <s xml:id="echoid-s894" xml:space="preserve">Deinde ex G, F, ducan-<lb/>tur ad rectas E F, E G, perpen <lb/>diculares F H, G H, coeuntes <lb/>in H, connectaturq́; </s> <s xml:id="echoid-s895" xml:space="preserve">recta E H. <lb/></s> <s xml:id="echoid-s896" xml:space="preserve">Dico E H, æqualem eſſe diame <lb/>tro circuli A B C D. </s> <s xml:id="echoid-s897" xml:space="preserve">Ducta enim <lb/>diam etro A C, iungatur recta <lb/>D C. </s> <s xml:id="echoid-s898" xml:space="preserve">Quoniam vero quatuor <lb/>anguli quadrilateri E F H G, <lb/>quatuor rectis æquales ſunt, <lb/> <anchor type="note" xlink:label="note-036-04a" xlink:href="note-036-04"/> ſuntq́; </s> <s xml:id="echoid-s899" xml:space="preserve">E F H, E G H, recti; <lb/></s> <s xml:id="echoid-s900" xml:space="preserve">erunt F E G, F H G, duobus re <lb/>ctis æquales; </s> <s xml:id="echoid-s901" xml:space="preserve">atq; </s> <s xml:id="echoid-s902" xml:space="preserve">adeo in quadrilatero E F H G, duo quilibet anguli ex ad-<lb/>uerſo duobus rectis æqua les erunt. </s> <s xml:id="echoid-s903" xml:space="preserve">Quare circa ipſum circulus deſcribi po-<lb/> <anchor type="note" xlink:label="note-036-05a" xlink:href="note-036-05"/> teſt: </s> <s xml:id="echoid-s904" xml:space="preserve">quo deſcripto erunt anguli E F G, E H G, eidem ſegmento, cuius chor <lb/>da E G, inſiſtentes, æquales. </s> <s xml:id="echoid-s905" xml:space="preserve">Eſt autem angulus E F G, angulo A B D, æqua-<lb/> <anchor type="note" xlink:label="note-036-06a" xlink:href="note-036-06"/> lis; </s> <s xml:id="echoid-s906" xml:space="preserve">quod duo latera E F, F G, duobus lateribus A B, B D, æqualia ſint, & </s> <s xml:id="echoid-s907" xml:space="preserve">ba-<lb/> <anchor type="note" xlink:label="note-036-07a" xlink:href="note-036-07"/> ſis E G, baſi A D, ex conſtructione: </s> <s xml:id="echoid-s908" xml:space="preserve">& </s> <s xml:id="echoid-s909" xml:space="preserve">angulus A B D, angulo A C D, æqua-<lb/> <anchor type="note" xlink:label="note-036-08a" xlink:href="note-036-08"/> lis eſt. </s> <s xml:id="echoid-s910" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s911" xml:space="preserve">angulus E H G, angulo A C D, æqualis erit. </s> <s xml:id="echoid-s912" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s913" xml:space="preserve">re-<lb/>ctus angulus E G H, angulo A D C, æqualis, quòd hic quoque rectus ſit in ſe-<lb/> <anchor type="note" xlink:label="note-036-09a" xlink:href="note-036-09"/> micirculo A D C, exiſtens. </s> <s xml:id="echoid-s914" xml:space="preserve">Igitur triangula E H G, A C D, duos angulos <lb/>duobus angulis æquales habent, necnon & </s> <s xml:id="echoid-s915" xml:space="preserve">latus E G, lateri A D, quod æqua-<lb/> <anchor type="note" xlink:label="note-036-10a" xlink:href="note-036-10"/> lium angulorum vni ſubtenditur, æquale. </s> <s xml:id="echoid-s916" xml:space="preserve">Quare & </s> <s xml:id="echoid-s917" xml:space="preserve">latus E H, lateri A C, <lb/>æquale erit. </s> <s xml:id="echoid-s918" xml:space="preserve">Lineam igitur rectam E H, deſcripſimus æqualem diametro A C, <lb/>circuli A B C D. </s> <s xml:id="echoid-s919" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s920" xml:space="preserve"/> </p> <div xml:id="echoid-div101" type="float" level="2" n="1"> <figure xlink:label="fig-036-01" xlink:href="fig-036-01a"> <image file="036-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/036-01"/> </figure> <note position="left" xlink:label="note-036-03" xlink:href="note-036-03a" xml:space="preserve">Schol 22. <lb/>primi.</note> <note position="left" xlink:label="note-036-04" xlink:href="note-036-04a" xml:space="preserve">Schol. 32. <lb/>primi.</note> <note position="left" xlink:label="note-036-05" xlink:href="note-036-05a" xml:space="preserve">Schol. 22. <lb/>tertij.</note> <note position="left" xlink:label="note-036-06" xlink:href="note-036-06a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-036-07" xlink:href="note-036-07a" xml:space="preserve">8. primi.</note> <note position="left" xlink:label="note-036-08" xlink:href="note-036-08a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-036-09" xlink:href="note-036-09a" xml:space="preserve">31. tertij.</note> <note position="left" xlink:label="note-036-10" xlink:href="note-036-10a" xml:space="preserve">26. primi.</note> </div> </div> <div xml:id="echoid-div103" type="section" level="1" n="59"> <head xml:id="echoid-head70" xml:space="preserve">PROBL. 3. PROPOS. 19.</head> <note position="left" xml:space="preserve">29.</note> <p> <s xml:id="echoid-s921" xml:space="preserve">LINEAM rectam deſcribere æqualem dia-<lb/>metro datæ ſphæræ.</s> <s xml:id="echoid-s922" xml:space="preserve"/> </p> <pb o="25" file="037" n="37" rhead=""/> <p> <s xml:id="echoid-s923" xml:space="preserve">IN ſphæra data ſumptis vtcunq́ue duobus punctis A, B, deſcribatur ex <lb/>A, polo, & </s> <s xml:id="echoid-s924" xml:space="preserve">interuallo A B, circulus B D, cuius diametro æqualis recta deſcri <lb/> <anchor type="note" xlink:label="note-037-01a" xlink:href="note-037-01"/> batur F G: </s> <s xml:id="echoid-s925" xml:space="preserve">& </s> <s xml:id="echoid-s926" xml:space="preserve">fiat ſupra F G, triangulum E F G, habens vtruque reliquorum <lb/> <anchor type="figure" xlink:label="fig-037-01a" xlink:href="fig-037-01"/> <anchor type="note" xlink:label="note-037-02a" xlink:href="note-037-02"/> laterum E F, E G, rectæ ducte <lb/> <anchor type="note" xlink:label="note-037-03a" xlink:href="note-037-03"/> A B, æquale. </s> <s xml:id="echoid-s927" xml:space="preserve">Deinde ex F, G, <lb/>ad E F, E G, perpendiculares <lb/>educantur F H, G H, coeun-<lb/>tes in H;</s> <s xml:id="echoid-s928" xml:space="preserve">iungaturq́; </s> <s xml:id="echoid-s929" xml:space="preserve">recta E H. <lb/></s> <s xml:id="echoid-s930" xml:space="preserve">Dico E H, æqualem eſſe dia-<lb/>metro datæ ſphæræ. </s> <s xml:id="echoid-s931" xml:space="preserve">Ducta em̃ <lb/>ſphæræ diametro A C, traijcia <lb/>tur per rectas A B, A C, pla-<lb/>num ſaciens in ſphæra circulũ <lb/> <anchor type="note" xlink:label="note-037-04a" xlink:href="note-037-04"/> A B C D, qui maximus erit, <lb/> <anchor type="note" xlink:label="note-037-05a" xlink:href="note-037-05"/> cum per diametrum ſphæræ, <lb/>atque adeo per centrum eiuſ-<lb/>dem ducatur. </s> <s xml:id="echoid-s932" xml:space="preserve">Quare idẽ per A, polũ circuli B D, ductus circulum B D, bifa-<lb/> <anchor type="note" xlink:label="note-037-06a" xlink:href="note-037-06"/> riam ſecabit; </s> <s xml:id="echoid-s933" xml:space="preserve">ac propterea communis ſectio B D, diameter erit circuli B D. <lb/></s> <s xml:id="echoid-s934" xml:space="preserve">Iunctis autem rectis A D, D C, erunt duo latera A B, B D, duobus lateribus <lb/>E F, F G, æqualia, nec non & </s> <s xml:id="echoid-s935" xml:space="preserve">baſes A D, E G, æquales. </s> <s xml:id="echoid-s936" xml:space="preserve">Eſt enium F G, diame-<lb/>tro B D, æqualis, ex conſtructione: </s> <s xml:id="echoid-s937" xml:space="preserve">& </s> <s xml:id="echoid-s938" xml:space="preserve">vtraque E F, E G, rectæ A B, vel A D. </s> <s xml:id="echoid-s939" xml:space="preserve"><lb/>Igitur & </s> <s xml:id="echoid-s940" xml:space="preserve">anguli A B D, E F G, æquales erunt. </s> <s xml:id="echoid-s941" xml:space="preserve">Eſt autem angulo A B D, an-<lb/> <anchor type="note" xlink:label="note-037-07a" xlink:href="note-037-07"/> gulus A C D, æqualis: </s> <s xml:id="echoid-s942" xml:space="preserve">& </s> <s xml:id="echoid-s943" xml:space="preserve">angulo E F G, angulus E H G, vt in præcedenti <lb/> <anchor type="note" xlink:label="note-037-08a" xlink:href="note-037-08"/> propoſ. </s> <s xml:id="echoid-s944" xml:space="preserve">demonſtratum eſt. </s> <s xml:id="echoid-s945" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s946" xml:space="preserve">anguli A C D, E H G, æquales erunt. <lb/></s> <s xml:id="echoid-s947" xml:space="preserve">Sunt autem & </s> <s xml:id="echoid-s948" xml:space="preserve">recti A D C, E G H, æquales, & </s> <s xml:id="echoid-s949" xml:space="preserve">latus A D, lateri E G, quod <lb/>vni æqualium angulorum obijcitur, æquale. </s> <s xml:id="echoid-s950" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s951" xml:space="preserve">recta E H, rectæ A C, <lb/> <anchor type="note" xlink:label="note-037-09a" xlink:href="note-037-09"/> æqualis erit. </s> <s xml:id="echoid-s952" xml:space="preserve">Lineam igitur rectam E H, deſcripſimus æqualẽ diametro A C, <lb/>datæ ſphæræ. </s> <s xml:id="echoid-s953" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s954" xml:space="preserve"/> </p> <div xml:id="echoid-div103" type="float" level="2" n="1"> <note position="right" xlink:label="note-037-01" xlink:href="note-037-01a" xml:space="preserve">18. huius.</note> <figure xlink:label="fig-037-01" xlink:href="fig-037-01a"> <image file="037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/037-01"/> </figure> <note position="right" xlink:label="note-037-02" xlink:href="note-037-02a" xml:space="preserve">Schol 22. <lb/>primi.</note> <note position="right" xlink:label="note-037-03" xlink:href="note-037-03a" xml:space="preserve">primi.</note> <note position="right" xlink:label="note-037-04" xlink:href="note-037-04a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-037-05" xlink:href="note-037-05a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-037-06" xlink:href="note-037-06a" xml:space="preserve">15. huius.</note> <note position="right" xlink:label="note-037-07" xlink:href="note-037-07a" xml:space="preserve">8. primi.</note> <note position="right" xlink:label="note-037-08" xlink:href="note-037-08a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-037-09" xlink:href="note-037-09a" xml:space="preserve">26. primi.</note> </div> </div> <div xml:id="echoid-div105" type="section" level="1" n="60"> <head xml:id="echoid-head71" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s955" xml:space="preserve">_ADDITVR_in alia verſione ſequens hoc Theorema.</s> <s xml:id="echoid-s956" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s957" xml:space="preserve">LINEA recta à polo cuiuſuis circuli in ſphæra ad ſuperficiem <lb/> <anchor type="note" xlink:label="note-037-10a" xlink:href="note-037-10"/> ſphæræ ducta, quæ ſit æqualis lineæ rectæ ab eodem polo ad circun-<lb/>ferentiam circuli ductæ, in circuli circunferentiam cadit.</s> <s xml:id="echoid-s958" xml:space="preserve"/> </p> <div xml:id="echoid-div105" type="float" level="2" n="1"> <note position="right" xlink:label="note-037-10" xlink:href="note-037-10a" xml:space="preserve">30.</note> </div> <figure> <image file="037-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/037-02"/> </figure> <p style="it"> <s xml:id="echoid-s959" xml:space="preserve">_IN_ſphæra ex _A,_ polo circuli _B C,_ recta du-<lb/>cta ſit vtcumque _A D,_ ad eius circunferentiã, <lb/>quæ minor erit diametro ſphæræ, atque adeo dia <lb/>metro circuli maximi in ſphæra, cum diameter <lb/>ſphæræ ſit omnium rectarum in ſphæra ductarũ <lb/>maxima Ducatur iam ex eodem polo A. </s> <s xml:id="echoid-s960" xml:space="preserve">ad ſu-<lb/>perficiem ſphæræ recta _A E,_ quæ ipſi _A D,_ æqua-<lb/>lis ſit. </s> <s xml:id="echoid-s961" xml:space="preserve">Dico rectam A E, caderein circunferen-<lb/>tiam circuli _B C._ </s> <s xml:id="echoid-s962" xml:space="preserve">Si enim ſi eri poteſt, non cadat <lb/>in eius circunferentiam. </s> <s xml:id="echoid-s963" xml:space="preserve">Et per rectam _A E,_ & </s> <s xml:id="echoid-s964" xml:space="preserve"><lb/>centrum ſphæræ ducatur planũ faciens in ſphæ-<lb/> <anchor type="note" xlink:label="note-037-11a" xlink:href="note-037-11"/> <pb o="26" file="038" n="38" rhead=""/> racirculum _A B C,_ qui maximus erit, cum per centrum ſphæræ tranſeat. </s> <s xml:id="echoid-s965" xml:space="preserve">Secet au <lb/> <anchor type="figure" xlink:label="fig-038-01a" xlink:href="fig-038-01"/> <anchor type="note" xlink:label="note-038-01a" xlink:href="note-038-01"/> tem circulus _A B C,_ circulum _B C,_ in punctis <lb/>_B, C._ </s> <s xml:id="echoid-s966" xml:space="preserve">Non cadet ergo recta _A E,_ in puncta _B, C._ <lb/></s> <s xml:id="echoid-s967" xml:space="preserve">cum ponatur non cadere in circunferentiam cir <lb/>culi _B C._ </s> <s xml:id="echoid-s968" xml:space="preserve">Ducta igitur recta _A B,_ erit hæc, ex <lb/>defin. </s> <s xml:id="echoid-s969" xml:space="preserve">poli, rectæ _A D,_ atque adeo rectæ _A E,_ <lb/>æqualis. </s> <s xml:id="echoid-s970" xml:space="preserve">Et quia vtraque _A B, A E,_ minor eſt <lb/>diametro maximi circuli _A B C,_ vt dictum eſt, <lb/>erunt areus _A B, A E,_ cum ſint ſegmenta ſemi-<lb/>circulo minora, æquales, pars & </s> <s xml:id="echoid-s971" xml:space="preserve">totum. </s> <s xml:id="echoid-s972" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-038-02a" xlink:href="note-038-02"/> Quod eſt abſurdum. </s> <s xml:id="echoid-s973" xml:space="preserve">Cadet ergo recta _A E,_ in <lb/>circunferentiam circuli _B C._ </s> <s xml:id="echoid-s974" xml:space="preserve">Quod eſt pros <lb/>poſitum.</s> <s xml:id="echoid-s975" xml:space="preserve"/> </p> <div xml:id="echoid-div106" type="float" level="2" n="2"> <note position="right" xlink:label="note-037-11" xlink:href="note-037-11a" xml:space="preserve">1. huius.</note> <figure xlink:label="fig-038-01" xlink:href="fig-038-01a"> <image file="038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/038-01"/> </figure> <note position="left" xlink:label="note-038-01" xlink:href="note-038-01a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-038-02" xlink:href="note-038-02a" xml:space="preserve">28. tertij.</note> </div> </div> <div xml:id="echoid-div108" type="section" level="1" n="61"> <head xml:id="echoid-head72" xml:space="preserve">PROBL. 4. PROP. 20.</head> <note position="left" xml:space="preserve">31.</note> <p> <s xml:id="echoid-s976" xml:space="preserve">PER duo puncta data in ſphærica ſuperficie <lb/>maximum circulum deſcribere.</s> <s xml:id="echoid-s977" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s978" xml:space="preserve">IN ſphærica ſuperficie data ſint duo pũcta A, B, per quæ deſcribere opor <lb/>teat circulum maximum. </s> <s xml:id="echoid-s979" xml:space="preserve">Si ergo puncta A, B, ſint oppoſita ex diametro <lb/>ſphęræ, certum eſt, inſinitos circulos maximos per ipſa duci poſſe, ductis ni-<lb/>mirum inſinitis planis per diametrum ſphæræ puncta illa connectentem. </s> <s xml:id="echoid-s980" xml:space="preserve">Si <lb/> <anchor type="figure" xlink:label="fig-038-02a" xlink:href="fig-038-02"/> autem puncta A, B, non ſint in ſphæræ dia-<lb/>metro, deſcribatur ex A, polo, & </s> <s xml:id="echoid-s981" xml:space="preserve">interual-<lb/>lo quod lateri quadrati in maximo circulo <lb/>deſcripti æquale ſit, circulus C D, qui ma-<lb/>ximus erit, cum recta ex A, polo ad eius cir <lb/> <anchor type="note" xlink:label="note-038-04a" xlink:href="note-038-04"/> cunferentiam ducta æqualis ſit lateri qua-<lb/>drati in circulo maximo deſcripti, propter <lb/>interuallum, quo circulus C D, deſeriptus <lb/>eſt. </s> <s xml:id="echoid-s982" xml:space="preserve">Similiter ex B, polo, & </s> <s xml:id="echoid-s983" xml:space="preserve">interuallo eodẽ, <lb/>quo prius, circulus deſcribatur E F, qui rur <lb/> <anchor type="note" xlink:label="note-038-05a" xlink:href="note-038-05"/> ſus erit maximus. </s> <s xml:id="echoid-s984" xml:space="preserve">Secet autem hic priorem <lb/>in puncto G, a quo ad polos A, B, rectæ du <lb/>cantur G A, G B; </s> <s xml:id="echoid-s985" xml:space="preserve">quarum vtraque, ex con <lb/>ſtructione, æqualis erit lateri quadrati in <lb/>maximo circulo deſcripti. </s> <s xml:id="echoid-s986" xml:space="preserve">Tanto enim interuallo ex polis A, B, circuli C D, <lb/>E F, deſcripti ſunt. </s> <s xml:id="echoid-s987" xml:space="preserve">Aequales ergo ſunt G A, G B. </s> <s xml:id="echoid-s988" xml:space="preserve">Iam ex G, polo, & </s> <s xml:id="echoid-s989" xml:space="preserve">inter-<lb/>uallo G A, circulus deſcribatur A E D F C B, qui maximus erit; </s> <s xml:id="echoid-s990" xml:space="preserve">cum recta <lb/> <anchor type="note" xlink:label="note-038-06a" xlink:href="note-038-06"/> G A, ex G, polo ad eius circunferentiam ducta æqualis ſit lateri quadrati in <lb/>maximo circulo inſcripti, vt demonſtratum eſt. </s> <s xml:id="echoid-s991" xml:space="preserve">Quoniam vero recta G B, æ-<lb/>qualis ipſi G A, ducta ad ſuperficiem ſphæræ cadit in circunferentiam circu-<lb/> <anchor type="note" xlink:label="note-038-07a" xlink:href="note-038-07"/> li A E D F C B, deſcriptus propterea erit circulus maximus A E D F C B, <lb/>per data duo puncta A, B, in ſuperficie ſphæræ. </s> <s xml:id="echoid-s992" xml:space="preserve">Per duo ergo puncta data in <lb/>ſphærica ſuperſicie maximum circulum deſcripſimus, Quod faciendum erat.</s> <s xml:id="echoid-s993" xml:space="preserve"/> </p> <div xml:id="echoid-div108" type="float" level="2" n="1"> <figure xlink:label="fig-038-02" xlink:href="fig-038-02a"> <image file="038-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/038-02"/> </figure> <note position="left" xlink:label="note-038-04" xlink:href="note-038-04a" xml:space="preserve">17. huius.</note> <note position="left" xlink:label="note-038-05" xlink:href="note-038-05a" xml:space="preserve">17. huius.</note> <note position="left" xlink:label="note-038-06" xlink:href="note-038-06a" xml:space="preserve">17. huius.</note> <note position="left" xlink:label="note-038-07" xlink:href="note-038-07a" xml:space="preserve">Schol. 19. <lb/>huius.</note> </div> <pb o="27" file="039" n="39" rhead=""/> </div> <div xml:id="echoid-div110" type="section" level="1" n="62"> <head xml:id="echoid-head73" xml:space="preserve">PROBL. 5. PROP. 21.</head> <note position="right" xml:space="preserve">32.</note> <p> <s xml:id="echoid-s994" xml:space="preserve">CVIVSLIBET circuli in ſphæra dati po-<lb/>lum inuenire.</s> <s xml:id="echoid-s995" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s996" xml:space="preserve">SIT inueniendus polus circuli A B, in ſphæra dati, ſitq́; </s> <s xml:id="echoid-s997" xml:space="preserve">primum circu-<lb/>lus A B, non maximus. </s> <s xml:id="echoid-s998" xml:space="preserve">Sumptis duobus punctis in circunferentia vtcumque <lb/>C, D, diuidatur vterque arcus C A D, C B D, bifariam in A, & </s> <s xml:id="echoid-s999" xml:space="preserve">B, punctis, per <lb/> <anchor type="note" xlink:label="note-039-02a" xlink:href="note-039-02"/> quæ deſcribatur maximus circulus A E B; </s> <s xml:id="echoid-s1000" xml:space="preserve">ſeceturq́; </s> <s xml:id="echoid-s1001" xml:space="preserve">arcus A E B, bifariam <lb/> <anchor type="note" xlink:label="note-039-03a" xlink:href="note-039-03"/> in E. </s> <s xml:id="echoid-s1002" xml:space="preserve">Dico E, polum eſſe circuli A B; </s> <s xml:id="echoid-s1003" xml:space="preserve">Quoniam enim arcus A C, A D, æqua-<lb/>les ſunt, necnon B C, B D, erunt toti arcus A C B, A D B, æquales. </s> <s xml:id="echoid-s1004" xml:space="preserve">Qua-<lb/> <anchor type="figure" xlink:label="fig-039-01a" xlink:href="fig-039-01"/> re maximus circulus <lb/>A E B, cum circulum <lb/>non maximum A B, <lb/>bifariam ſecet in A, <lb/>& </s> <s xml:id="echoid-s1005" xml:space="preserve">B, ſecabit eum per <lb/>polos. </s> <s xml:id="echoid-s1006" xml:space="preserve">Punctum ergo <lb/> <anchor type="note" xlink:label="note-039-04a" xlink:href="note-039-04"/> E, æqualiter diſtans <lb/>a circunferentia cir-<lb/>culi A B, polus eſt cir <lb/>culi A B. </s> <s xml:id="echoid-s1007" xml:space="preserve">Eodem mo-<lb/>do ſi reliquus arcus <lb/>A F B, ſecetur bifa-<lb/>riam in F, erit F, al-<lb/>ter polus circuli A B.</s> <s xml:id="echoid-s1008" xml:space="preserve"/> </p> <div xml:id="echoid-div110" type="float" level="2" n="1"> <note position="right" xlink:label="note-039-02" xlink:href="note-039-02a" xml:space="preserve">30. tertij.</note> <note position="right" xlink:label="note-039-03" xlink:href="note-039-03a" xml:space="preserve">20. huius.</note> <figure xlink:label="fig-039-01" xlink:href="fig-039-01a"> <image file="039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/039-01"/> </figure> <note position="right" xlink:label="note-039-04" xlink:href="note-039-04a" xml:space="preserve">14. huius.</note> </div> <p> <s xml:id="echoid-s1009" xml:space="preserve">SED ſit iam datus circulus A B, maximus. </s> <s xml:id="echoid-s1010" xml:space="preserve">Sumptis rurſus punctis C, D, <lb/>vtcumque, & </s> <s xml:id="echoid-s1011" xml:space="preserve">diuiſis arcubus C A D, C B D, bifariam in A, B, oſtendemus, <lb/> <anchor type="note" xlink:label="note-039-05a" xlink:href="note-039-05"/> vt prius, totos arcus A C B, A D B, eſſe æquales, ac propterea vtrumque eſ <lb/>ſe ſemicirculũ circuli maximi. </s> <s xml:id="echoid-s1012" xml:space="preserve">Diuiſo ergo altero ſemicirculo, nempe A C B, <lb/>bifariam in G, erit recta G A, ſubtendens quadrantem circuli, latus quadrati <lb/>in maximo circulo A B, deſcripti; </s> <s xml:id="echoid-s1013" xml:space="preserve">vt ex prop.</s> <s xml:id="echoid-s1014" xml:space="preserve">6.</s> <s xml:id="echoid-s1015" xml:space="preserve">lib.</s> <s xml:id="echoid-s1016" xml:space="preserve">4.</s> <s xml:id="echoid-s1017" xml:space="preserve">Eucl.</s> <s xml:id="echoid-s1018" xml:space="preserve">cõſtat. </s> <s xml:id="echoid-s1019" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s1020" xml:space="preserve">ex po <lb/>lo G, & </s> <s xml:id="echoid-s1021" xml:space="preserve">in teruallo G A, circulus deſcribatur A E B, qui maximus erit, cũ recta <lb/> <anchor type="note" xlink:label="note-039-06a" xlink:href="note-039-06"/> ex G, polo ad eius circunſerentiã ducta nimirũ ad punctũ A, ſit æqualis lateri <lb/>quadrati in circulo maximo A B, deſcripti, Diuidatur deniq; </s> <s xml:id="echoid-s1022" xml:space="preserve">arcus A E B, biſa <lb/>riam in E. </s> <s xml:id="echoid-s1023" xml:space="preserve">Dico E, polum eſſe circuli A B. </s> <s xml:id="echoid-s1024" xml:space="preserve">Cum enim maximus circulus A C B, <lb/>tranſeat per G, polum maximi circuli A E B, tranſibit viciſsim maximus cir <lb/> <anchor type="note" xlink:label="note-039-07a" xlink:href="note-039-07"/> culus A E B, per polos maximi circuli A C B. </s> <s xml:id="echoid-s1025" xml:space="preserve">Quare punctum E, æqualiter <lb/>remotum à circunferentia circuli A C B, polus eſt circuli A C B. </s> <s xml:id="echoid-s1026" xml:space="preserve">Eodem mo <lb/>do diuiſo arcu A F B, bifariam in F, erit F, alter polus circuli A C B. </s> <s xml:id="echoid-s1027" xml:space="preserve">Cuiuſli <lb/>bet ergo circuli in ſphæra dati polum inuenimus. </s> <s xml:id="echoid-s1028" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s1029" xml:space="preserve"/> </p> <div xml:id="echoid-div111" type="float" level="2" n="2"> <note position="right" xlink:label="note-039-05" xlink:href="note-039-05a" xml:space="preserve">30. tertij.</note> <note position="right" xlink:label="note-039-06" xlink:href="note-039-06a" xml:space="preserve">17. huius.</note> <note position="right" xlink:label="note-039-07" xlink:href="note-039-07a" xml:space="preserve">Schol. 15. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div113" type="section" level="1" n="63"> <head xml:id="echoid-head74" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1030" xml:space="preserve">_IN_ alia verſione demonſtrantur ſequentia duo theoremata.</s> <s xml:id="echoid-s1031" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div114" type="section" level="1" n="64"> <head xml:id="echoid-head75" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s1032" xml:space="preserve">SI in ſuperſicie ſphæræ acceptum fuerit punctum aliquod, & </s> <s xml:id="echoid-s1033" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-039-08a" xlink:href="note-039-08"/> <pb o="28" file="040" n="40" rhead=""/> ab eo puncto ad circunferentiam circuli cuiuſpiam in ſphæra dati <lb/>cadant plures, quàm duæ rectæ lineę æquales, acceptum punctum <lb/>polus eſt ipſius circuli.</s> <s xml:id="echoid-s1034" xml:space="preserve"/> </p> <div xml:id="echoid-div114" type="float" level="2" n="1"> <note position="right" xlink:label="note-039-08" xlink:href="note-039-08a" xml:space="preserve">33.</note> </div> <p style="it"> <s xml:id="echoid-s1035" xml:space="preserve">_IN_ ſuperficie ſphæræ _A B C,_ acceptum ſit punctum _A,_ a quo ad circunferentiã <lb/>circuli _B C,_ cadant plures, quàm duæ, rectæ linæ æquales _A D, A E, A F._ </s> <s xml:id="echoid-s1036" xml:space="preserve">Dico <lb/> <anchor type="figure" xlink:label="fig-040-01a" xlink:href="fig-040-01"/> _A,_ polum eſſe circuli _B C._ </s> <s xml:id="echoid-s1037" xml:space="preserve">Demittatur enim ex <lb/>_A,_ in planum circuli _B C,_ perpendicularis <lb/> <anchor type="note" xlink:label="note-040-01a" xlink:href="note-040-01"/> _A G,_ iungãturq́; </s> <s xml:id="echoid-s1038" xml:space="preserve">rectæ _D G, E G, F G,_ eruntq́; <lb/></s> <s xml:id="echoid-s1039" xml:space="preserve">ex 3. </s> <s xml:id="echoid-s1040" xml:space="preserve">defin. </s> <s xml:id="echoid-s1041" xml:space="preserve">lib. </s> <s xml:id="echoid-s1042" xml:space="preserve">I I. </s> <s xml:id="echoid-s1043" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1044" xml:space="preserve">omnes treas anguli ad <lb/>G, recti. </s> <s xml:id="echoid-s1045" xml:space="preserve">Quare tam quadratum ex _A D,_ qua-<lb/>dratis ex _A G, G D,_ quàm quadratum ex _A E,_ <lb/> <anchor type="note" xlink:label="note-040-02a" xlink:href="note-040-02"/> quadratis ex _A G, G E,_ & </s> <s xml:id="echoid-s1046" xml:space="preserve">quadratum ex _A F,_ <lb/>quadratis ex _A G, G F,_ æquale erit. </s> <s xml:id="echoid-s1047" xml:space="preserve">Cum er-<lb/>go quadrata rectarum æqualiũ _A D, A E, A F._ <lb/></s> <s xml:id="echoid-s1048" xml:space="preserve">æqualia ſint, erunt & </s> <s xml:id="echoid-s1049" xml:space="preserve">quadrata ex _A G, G D,_ <lb/>ſimul quadratis ex _A G, G E,_ ſimul, nec non <lb/>quadratis ex _A G, G F,_ ſimul æqualia; </s> <s xml:id="echoid-s1050" xml:space="preserve">dem-<lb/>ptoq́; </s> <s xml:id="echoid-s1051" xml:space="preserve">communi quadrato lineæ _A G,_ æqualia <lb/>erunt reliqua quadrata linearum _G D, G E,_ <lb/>_G F,_ at que adeo & </s> <s xml:id="echoid-s1052" xml:space="preserve">rectæ _G D, G E, G F,_ æquales erunt, Igitur _G,_ centrum erit <lb/> <anchor type="note" xlink:label="note-040-03a" xlink:href="note-040-03"/> circuli _BC;_ </s> <s xml:id="echoid-s1053" xml:space="preserve">ac proinde recta _G A,_ quæ ex centro _G,_ ad circulum _B C,_ perpendi-<lb/>cularis eſt ducta, in polum circuli _B C,_ cadet. </s> <s xml:id="echoid-s1054" xml:space="preserve">Punctum ergo _A,_ polus eſt circuli <lb/> <anchor type="note" xlink:label="note-040-04a" xlink:href="note-040-04"/> B C. </s> <s xml:id="echoid-s1055" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s1056" xml:space="preserve"/> </p> <div xml:id="echoid-div115" type="float" level="2" n="2"> <figure xlink:label="fig-040-01" xlink:href="fig-040-01a"> <image file="040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/040-01"/> </figure> <note position="left" xlink:label="note-040-01" xlink:href="note-040-01a" xml:space="preserve">11. valec.</note> <note position="left" xlink:label="note-040-02" xlink:href="note-040-02a" xml:space="preserve">47. primi.</note> <note position="left" xlink:label="note-040-03" xlink:href="note-040-03a" xml:space="preserve">9. tertij.</note> <note position="left" xlink:label="note-040-04" xlink:href="note-040-04a" xml:space="preserve">Schol. 8. hu <lb/>ius.</note> </div> </div> <div xml:id="echoid-div117" type="section" level="1" n="65"> <head xml:id="echoid-head76" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s1057" xml:space="preserve">IN ſphæra circuli, à quorum polis rectæ ad eorum circunferen <lb/> <anchor type="note" xlink:label="note-040-05a" xlink:href="note-040-05"/> tias ductæ ſunt æquales, inter ſe ęquales ſunt. </s> <s xml:id="echoid-s1058" xml:space="preserve">Et circulorum ęqua-<lb/>lium ęquales ſunt rectę ab eorum polis ad circunferentias ductæ.</s> <s xml:id="echoid-s1059" xml:space="preserve"/> </p> <div xml:id="echoid-div117" type="float" level="2" n="1"> <note position="left" xlink:label="note-040-05" xlink:href="note-040-05a" xml:space="preserve">34.</note> </div> <p style="it"> <s xml:id="echoid-s1060" xml:space="preserve">_IN_ ſphæra _A B C D E F,_ cuius centrum _G,_ ſint duo circuli _B F, C E,_ a quorum <lb/> <anchor type="figure" xlink:label="fig-040-02a" xlink:href="fig-040-02"/> polis _A, D,_ rectæ _A F, D E,_ ad eorum circunfe <lb/>rentias ductæ ſint æquales. </s> <s xml:id="echoid-s1061" xml:space="preserve">Dico circulos _B F,_ <lb/>_C E,_ æquales eſſe. </s> <s xml:id="echoid-s1062" xml:space="preserve">Ducantur ex polis _A, D,_ ad <lb/> <anchor type="note" xlink:label="note-040-06a" xlink:href="note-040-06"/> plana circulorum perpendiculares _A H, D I,_ quæ <lb/>cadent in eorum centra _H, I,_ & </s> <s xml:id="echoid-s1063" xml:space="preserve">inde productæ <lb/> <anchor type="note" xlink:label="note-040-07a" xlink:href="note-040-07"/> in reliquos polos; </s> <s xml:id="echoid-s1064" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s1065" xml:space="preserve">in _G,_ centrum <lb/> <anchor type="note" xlink:label="note-040-08a" xlink:href="note-040-08"/> ſphæræ. </s> <s xml:id="echoid-s1066" xml:space="preserve">Ductis igitur ſemidiametris ſphæræ _F G,_ <lb/>_E G,_ & </s> <s xml:id="echoid-s1067" xml:space="preserve">ſemidiametris circulorũ _F H, E I,_ cum <lb/>latera _A G, G F,_ lateribus _D G, G E,_ ſint æqua <lb/> <anchor type="note" xlink:label="note-040-09a" xlink:href="note-040-09"/> lia, & </s> <s xml:id="echoid-s1068" xml:space="preserve">baſis _A F,_ baſi _DE_ erunt anquli _A G F,_ <lb/>_D G E,_ æquales. </s> <s xml:id="echoid-s1069" xml:space="preserve">Sunt autem anguli _H, I,_ ex <lb/>defin. </s> <s xml:id="echoid-s1070" xml:space="preserve">3. </s> <s xml:id="echoid-s1071" xml:space="preserve">lib. </s> <s xml:id="echoid-s1072" xml:space="preserve">11. </s> <s xml:id="echoid-s1073" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1074" xml:space="preserve">recti. </s> <s xml:id="echoid-s1075" xml:space="preserve">Triangula igitur <lb/>_F G H, E G I,_ duos angulos duobus angulis æ-<lb/>quales habent: </s> <s xml:id="echoid-s1076" xml:space="preserve">habent autem & </s> <s xml:id="echoid-s1077" xml:space="preserve">latus _F G,_ lateri _E G,_ quod recto angulo opponi- <pb o="29" file="041" n="41" rhead=""/> tur æquale: </s> <s xml:id="echoid-s1078" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s1079" xml:space="preserve">ſemidiametri F H, E I, æquales erunt, atque adeo & </s> <s xml:id="echoid-s1080" xml:space="preserve">cir-<lb/> <anchor type="note" xlink:label="note-041-01a" xlink:href="note-041-01"/> culi B F, C E, æquales. </s> <s xml:id="echoid-s1081" xml:space="preserve">quod primo loco propoſitum eſt.</s> <s xml:id="echoid-s1082" xml:space="preserve"/> </p> <div xml:id="echoid-div118" type="float" level="2" n="2"> <figure xlink:label="fig-040-02" xlink:href="fig-040-02a"> <image file="040-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/040-02"/> </figure> <note position="left" xlink:label="note-040-06" xlink:href="note-040-06a" xml:space="preserve">21. vndec.</note> <note position="left" xlink:label="note-040-07" xlink:href="note-040-07a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-040-08" xlink:href="note-040-08a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-040-09" xlink:href="note-040-09a" xml:space="preserve">3. primi.</note> <note position="right" xlink:label="note-041-01" xlink:href="note-041-01a" xml:space="preserve">26. primi.</note> </div> <p style="it"> <s xml:id="echoid-s1083" xml:space="preserve">_SINT_ iam circuli B F, C E, æquales. </s> <s xml:id="echoid-s1084" xml:space="preserve">Dico & </s> <s xml:id="echoid-s1085" xml:space="preserve">rectas A F, D E, ab eorum po-<lb/>lis ad circunferentias ductas eſſe æquales. </s> <s xml:id="echoid-s1086" xml:space="preserve">Conſtructis enim eiſdem, erunt ſemidiame-<lb/>tri F H, E I, æquales, & </s> <s xml:id="echoid-s1087" xml:space="preserve">circuli ipſi æqualiter acentro ſphæræ diſtabunt. </s> <s xml:id="echoid-s1088" xml:space="preserve">Perpen-<lb/> <anchor type="note" xlink:label="note-041-02a" xlink:href="note-041-02"/> diculares ergo G H, G I, æquales erunt; </s> <s xml:id="echoid-s1089" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s1090" xml:space="preserve">reliquæ lineæ A H, D I, <lb/>erunt æquales. </s> <s xml:id="echoid-s1091" xml:space="preserve">Quoniam igitur latera A H, H F, lateribus D I, I E, æqualia <lb/>ſunt, continentq́; </s> <s xml:id="echoid-s1092" xml:space="preserve">angulos H, I, æquales, cum recti ſint ex defin. </s> <s xml:id="echoid-s1093" xml:space="preserve">3. </s> <s xml:id="echoid-s1094" xml:space="preserve">lib, 11. </s> <s xml:id="echoid-s1095" xml:space="preserve">Eucl, erũt <lb/> <anchor type="note" xlink:label="note-041-03a" xlink:href="note-041-03"/> baſes A F, D E, æquales. </s> <s xml:id="echoid-s1096" xml:space="preserve">Quod ſecundo loco propoſitum erat.</s> <s xml:id="echoid-s1097" xml:space="preserve"/> </p> <div xml:id="echoid-div119" type="float" level="2" n="3"> <note position="right" xlink:label="note-041-02" xlink:href="note-041-02a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-041-03" xlink:href="note-041-03a" xml:space="preserve">4. primi,</note> </div> </div> <div xml:id="echoid-div121" type="section" level="1" n="66"> <head xml:id="echoid-head77" xml:space="preserve">THEOR. 17. PROPOS. 22.</head> <note position="right" xml:space="preserve">0</note> <p> <s xml:id="echoid-s1098" xml:space="preserve">SI in ſphæra recta linea per centrum ducta re-<lb/>ctam aliquam lineam non per centrum ductam <lb/>bifariam ſecet, ad angulos rectos ipſam ſecabit. <lb/></s> <s xml:id="echoid-s1099" xml:space="preserve">Quod ſi ad angulos rectos eam ſecet, bifariam <lb/>quoqueipſam ſecabit.</s> <s xml:id="echoid-s1100" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1101" xml:space="preserve">IN ſphæra, cuius centrum A, recta A B, per centrum ducta rectam C D, <lb/> <anchor type="figure" xlink:label="fig-041-01a" xlink:href="fig-041-01"/> non per centrum ductam ſecet bifariam in <lb/>B. </s> <s xml:id="echoid-s1102" xml:space="preserve">Dico ipſam C D, ſecari ad angulos re-<lb/>ctos. </s> <s xml:id="echoid-s1103" xml:space="preserve">Ducto enim per rectas A B, C D, pla-<lb/> <anchor type="note" xlink:label="note-041-05a" xlink:href="note-041-05"/> no, quod circulum faciat C D, qui maxi-<lb/> <anchor type="note" xlink:label="note-041-06a" xlink:href="note-041-06"/> mus erit, cum per centrum ſphæræ tranſeat. <lb/></s> <s xml:id="echoid-s1104" xml:space="preserve">Quoniam igitur in circulo C D, recta A B, <lb/>per eius centrum A, tranſiens rectam C D, <lb/>non per centrum ductam ſecat bifariam in <lb/>B, ad angulos rectos ipſam ſecabit. </s> <s xml:id="echoid-s1105" xml:space="preserve">Et ſi ad <lb/> <anchor type="note" xlink:label="note-041-07a" xlink:href="note-041-07"/> angulos rectos ipſam ſecet, bifariam ipſam <lb/>ſecabit. </s> <s xml:id="echoid-s1106" xml:space="preserve">Si igitur in ſphæra recta linea, &</s> <s xml:id="echoid-s1107" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1108" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s1109" xml:space="preserve"/> </p> <div xml:id="echoid-div121" type="float" level="2" n="1"> <figure xlink:label="fig-041-01" xlink:href="fig-041-01a"> <image file="041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/041-01"/> </figure> <note position="right" xlink:label="note-041-05" xlink:href="note-041-05a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-041-06" xlink:href="note-041-06a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-041-07" xlink:href="note-041-07a" xml:space="preserve">3. tertij.</note> </div> </div> <div xml:id="echoid-div123" type="section" level="1" n="67"> <head xml:id="echoid-head78" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1110" xml:space="preserve">_ADDITVR_ hic in exemplari græco theorema aliud, quod id em prorſus eſt, <lb/>quod prop. </s> <s xml:id="echoid-s1111" xml:space="preserve">7. </s> <s xml:id="echoid-s1112" xml:space="preserve">demonſtratum eſt. </s> <s xml:id="echoid-s1113" xml:space="preserve">Vnde ſuperuacaneũ eſſe duximus, illud hic repetere.</s> <s xml:id="echoid-s1114" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div124" type="section" level="1" n="68"> <head xml:id="echoid-head79" xml:space="preserve">FINIS LIBRI PRIMI THEODOSII.</head> <figure> <image file="041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/041-02"/> </figure> <pb file="042" n="42"/> </div> <div xml:id="echoid-div125" type="section" level="1" n="69"> <head xml:id="echoid-head80" xml:space="preserve">THEODOSII</head> <head xml:id="echoid-head81" xml:space="preserve">SPHAE RICORVM <lb/>LIBER SECVNDVS.</head> <figure> <image file="042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/042-01"/> </figure> </div> <div xml:id="echoid-div126" type="section" level="1" n="70"> <head xml:id="echoid-head82" style="it" xml:space="preserve">DEFINITIO.</head> <p> <s xml:id="echoid-s1115" xml:space="preserve">IN ſphæra circuli ſe mutuo tangere di-<lb/>cuntur, cum communis ſectio plano-<lb/>rum vtrumque circulum tetigerit.</s> <s xml:id="echoid-s1116" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div127" type="section" level="1" n="71"> <head xml:id="echoid-head83" xml:space="preserve">THEOREMA 1. PROPOS. 1.</head> <note position="left" xml:space="preserve">1</note> <p> <s xml:id="echoid-s1117" xml:space="preserve">IN ſphæra paralleli circuli circa eoſdem po-<lb/>los ſunt.</s> <s xml:id="echoid-s1118" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1119" xml:space="preserve">IN ſphæra A B C D E F, paralleli circuli <lb/> <anchor type="figure" xlink:label="fig-042-02a" xlink:href="fig-042-02"/> ſint B F, C E. </s> <s xml:id="echoid-s1120" xml:space="preserve">Dico eos circa eoſdem polos <lb/>eſſe. </s> <s xml:id="echoid-s1121" xml:space="preserve">Sint enim A, D, poli circuli B, F, & </s> <s xml:id="echoid-s1122" xml:space="preserve">cõ-<lb/> <anchor type="note" xlink:label="note-042-02a" xlink:href="note-042-02"/> nectatur recta A D, quæ ad circulum B F, re-<lb/>cta erit, tranſibitq́; </s> <s xml:id="echoid-s1123" xml:space="preserve">per centrum ſphæræ. <lb/></s> <s xml:id="echoid-s1124" xml:space="preserve"> <anchor type="note" xlink:label="note-042-03a" xlink:href="note-042-03"/> Quoniam igitur recta A D, ad circulũ B F, <lb/>perpendicularis eſt, erit quoque ad circulũ <lb/>parallelum C E, perpendicularis. </s> <s xml:id="echoid-s1125" xml:space="preserve">Quare cũ <lb/> <anchor type="note" xlink:label="note-042-04a" xlink:href="note-042-04"/> tranſeat per centrum ſphæræ, vt oſtenſum <lb/>eſt, cadet in polos circuli C E. </s> <s xml:id="echoid-s1126" xml:space="preserve">Sunt ergo <lb/> <anchor type="note" xlink:label="note-042-05a" xlink:href="note-042-05"/> A, D, poli circuli C E: </s> <s xml:id="echoid-s1127" xml:space="preserve">ſunt autem & </s> <s xml:id="echoid-s1128" xml:space="preserve">poli <lb/>circuli B F. </s> <s xml:id="echoid-s1129" xml:space="preserve">In ſphæra igitur paralleli circu-<lb/>li B F, C E, circa eoſdem polos A, D, ſunt. </s> <s xml:id="echoid-s1130" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1131" xml:space="preserve"/> </p> <div xml:id="echoid-div127" type="float" level="2" n="1"> <figure xlink:label="fig-042-02" xlink:href="fig-042-02a"> <image file="042-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/042-02"/> </figure> <note position="left" xlink:label="note-042-02" xlink:href="note-042-02a" xml:space="preserve">21. 1. huius.</note> <note position="left" xlink:label="note-042-03" xlink:href="note-042-03a" xml:space="preserve">10. 1. huius.</note> <note position="left" xlink:label="note-042-04" xlink:href="note-042-04a" xml:space="preserve">Schol. 14. <lb/>vndec.</note> <note position="left" xlink:label="note-042-05" xlink:href="note-042-05a" xml:space="preserve">8. 1. huius.</note> </div> </div> <div xml:id="echoid-div129" type="section" level="1" n="72"> <head xml:id="echoid-head84" xml:space="preserve">THEOREMA 2. PROPOS. 2.</head> <note position="left" xml:space="preserve">2</note> <p> <s xml:id="echoid-s1132" xml:space="preserve">IN ſphæra circuli, qui ſunt circa eoſdem po-<lb/>los, ſunt paralleli.</s> <s xml:id="echoid-s1133" xml:space="preserve"/> </p> <pb o="31" file="043" n="43" rhead=""/> <p> <s xml:id="echoid-s1134" xml:space="preserve">IN eadem ſphæra A B C D E F, circa eoſdẽ polos A, D, ſint circuli B F, <lb/>C E. </s> <s xml:id="echoid-s1135" xml:space="preserve">Dico eos parallelos eſſe. </s> <s xml:id="echoid-s1136" xml:space="preserve">Connexa enim recta A D, erit hæc ad vtrunq; <lb/></s> <s xml:id="echoid-s1137" xml:space="preserve"> <anchor type="note" xlink:label="note-043-01a" xlink:href="note-043-01"/> circulum perpendicularis. </s> <s xml:id="echoid-s1138" xml:space="preserve">Quare plana circulorum B F, C E, parallela ſunt. <lb/></s> <s xml:id="echoid-s1139" xml:space="preserve"> <anchor type="note" xlink:label="note-043-02a" xlink:href="note-043-02"/> In ſphæra igitur circuli, qui ſunt circa eoſdem polos, ſunt paralleli. </s> <s xml:id="echoid-s1140" xml:space="preserve">Quod <lb/>oſtendendum erat.</s> <s xml:id="echoid-s1141" xml:space="preserve"/> </p> <div xml:id="echoid-div129" type="float" level="2" n="1"> <note position="right" xlink:label="note-043-01" xlink:href="note-043-01a" xml:space="preserve">10. 1. huius.</note> <note position="right" xlink:label="note-043-02" xlink:href="note-043-02a" xml:space="preserve">14. vndec.</note> </div> </div> <div xml:id="echoid-div131" type="section" level="1" n="73"> <head xml:id="echoid-head85" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1142" xml:space="preserve">_SED_ & </s> <s xml:id="echoid-s1143" xml:space="preserve">hoc theorema ſequens in alia verſione demonſtratur.</s> <s xml:id="echoid-s1144" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1145" xml:space="preserve">IN ſphæra non ſunt plures circuli æquales, & </s> <s xml:id="echoid-s1146" xml:space="preserve">paralleli, quàm <lb/> <anchor type="note" xlink:label="note-043-03a" xlink:href="note-043-03"/> duo.</s> <s xml:id="echoid-s1147" xml:space="preserve"/> </p> <div xml:id="echoid-div131" type="float" level="2" n="1"> <note position="right" xlink:label="note-043-03" xlink:href="note-043-03a" xml:space="preserve">3.</note> </div> <p style="it"> <s xml:id="echoid-s1148" xml:space="preserve">_IN_ ſphæra quacunque ſint, ſi fieri poteſt, plu <lb/> <anchor type="figure" xlink:label="fig-043-01a" xlink:href="fig-043-01"/> res quàm duo circuli æquales, & </s> <s xml:id="echoid-s1149" xml:space="preserve">paralleli, nem <lb/> <anchor type="note" xlink:label="note-043-04a" xlink:href="note-043-04"/> pe tres _A B, C D, E F,_ qui circa eoſdem polos <lb/>erunt. </s> <s xml:id="echoid-s1150" xml:space="preserve">Sint ergo eorum poli G, H, & </s> <s xml:id="echoid-s1151" xml:space="preserve">iungatur <lb/> <anchor type="note" xlink:label="note-043-05a" xlink:href="note-043-05"/> recta _G H,_ quæ tranſibit per I, centrum ſphæ-<lb/>ræ, & </s> <s xml:id="echoid-s1152" xml:space="preserve">per _K, L, M,_ centra circulorum; </s> <s xml:id="echoid-s1153" xml:space="preserve">perpen <lb/>dicularisq́; </s> <s xml:id="echoid-s1154" xml:space="preserve">erit ad circulos _A B, C D, E F._ </s> <s xml:id="echoid-s1155" xml:space="preserve">Quo <lb/>niam igitur circuli _A B, C D, E F,_ æquales <lb/> <anchor type="note" xlink:label="note-043-06a" xlink:href="note-043-06"/> ſunt, ipſi æqualiter diſtabunt à centro ſphæræ _I._ <lb/></s> <s xml:id="echoid-s1156" xml:space="preserve">Per defin. </s> <s xml:id="echoid-s1157" xml:space="preserve">ergo 6. </s> <s xml:id="echoid-s1158" xml:space="preserve">lib. </s> <s xml:id="echoid-s1159" xml:space="preserve">1. </s> <s xml:id="echoid-s1160" xml:space="preserve">huius, perpendiculares <lb/>_I K, I L, I M,_ æquales erunt, nempe pars _I L,_ <lb/>& </s> <s xml:id="echoid-s1161" xml:space="preserve">totum _I M._ </s> <s xml:id="echoid-s1162" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s1163" xml:space="preserve">In ſphæra <lb/>igitur non ſunt plures circuli æquales, & </s> <s xml:id="echoid-s1164" xml:space="preserve">paralleli, quàm duo. </s> <s xml:id="echoid-s1165" xml:space="preserve">Quod demonſtran-<lb/>dum erat.</s> <s xml:id="echoid-s1166" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="2"> <figure xlink:label="fig-043-01" xlink:href="fig-043-01a"> <image file="043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/043-01"/> </figure> <note position="right" xlink:label="note-043-04" xlink:href="note-043-04a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-043-05" xlink:href="note-043-05a" xml:space="preserve">10. 1. huius.</note> <note position="right" xlink:label="note-043-06" xlink:href="note-043-06a" xml:space="preserve">6. 1. huius.</note> </div> </div> <div xml:id="echoid-div134" type="section" level="1" n="74"> <head xml:id="echoid-head86" xml:space="preserve">THEOREMA 3. PROPOS. 3.</head> <note position="right" xml:space="preserve">4.</note> <p> <s xml:id="echoid-s1167" xml:space="preserve">SI in ſphæra duo circuli ſecent in eodem pun <lb/>cto circunferentiam illius maximi circuli, in quo <lb/>polos habent, ſe mutuo tangent illi circuli.</s> <s xml:id="echoid-s1168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1169" xml:space="preserve">IN ſphæra duo circuli A B, A C, ſecent <lb/> <anchor type="figure" xlink:label="fig-043-02a" xlink:href="fig-043-02"/> in puncto A, circunferentiam maximi circu-<lb/>li A B C, qui per illorum polos tranſeat. </s> <s xml:id="echoid-s1170" xml:space="preserve">Di <lb/>co circulos A B, A C, ſe mutuo tangere in <lb/>A. </s> <s xml:id="echoid-s1171" xml:space="preserve">Quoniam enim circulus maximus A B C, <lb/>ſecat circulos A B, A C, per polos, bifariam <lb/>ipſos ſecabit, & </s> <s xml:id="echoid-s1172" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s1173" xml:space="preserve">Commu-<lb/> <anchor type="note" xlink:label="note-043-08a" xlink:href="note-043-08"/> nes ergo ſectiones circuli A B C, & </s> <s xml:id="echoid-s1174" xml:space="preserve">circulo-<lb/>rum A B, A C, nempe rectæ A B, A C, dia-<lb/>metri ſunt circulorum A B, A C. </s> <s xml:id="echoid-s1175" xml:space="preserve">Sit quo-<lb/>que communis ſectio planorum, in quo <lb/>circuli A B, A C, exiſtunt, recta D E, quæ <lb/>per punctum A, tranſibit, propterea quod plana circulorum in A, ponan- <pb o="32" file="044" n="44" rhead=""/> <anchor type="figure" xlink:label="fig-044-01a" xlink:href="fig-044-01"/> tur ſecare circulum A B C. </s> <s xml:id="echoid-s1176" xml:space="preserve">Et quoniam pla <lb/>num circuli A B C, ad plana circulorũ A B, <lb/>A C, rectum eſt oſtenſum, erunt vicisſim pla <lb/>na circulorum A B, A C, ad planum circuli <lb/>A B C, recta; </s> <s xml:id="echoid-s1177" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s1178" xml:space="preserve">D E, communis <lb/> <anchor type="note" xlink:label="note-044-01a" xlink:href="note-044-01"/> ipſorum ſectio ad idem planũ circuli A B C, <lb/>perpendicularis erit. </s> <s xml:id="echoid-s1179" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s1180" xml:space="preserve">ad diametros <lb/>A B, A C, in eodem plano exiſtentes perpen <lb/>dicularis erit, ex defin. </s> <s xml:id="echoid-s1181" xml:space="preserve">3. </s> <s xml:id="echoid-s1182" xml:space="preserve">lib. </s> <s xml:id="echoid-s1183" xml:space="preserve">11. </s> <s xml:id="echoid-s1184" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1185" xml:space="preserve">Quare <lb/>D E, vtrumque circulum A B, A C, tanget <lb/> <anchor type="note" xlink:label="note-044-02a" xlink:href="note-044-02"/> in A; </s> <s xml:id="echoid-s1186" xml:space="preserve">ac proinde per deſin. </s> <s xml:id="echoid-s1187" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s1188" xml:space="preserve">circuli <lb/>A B, A C, ſe mutuo tangent in A, puncto. <lb/></s> <s xml:id="echoid-s1189" xml:space="preserve">Si igitur in ſphæra duo circuli ſecent, &</s> <s xml:id="echoid-s1190" xml:space="preserve">c. </s> <s xml:id="echoid-s1191" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1192" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="1"> <figure xlink:label="fig-043-02" xlink:href="fig-043-02a"> <image file="043-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/043-02"/> </figure> <note position="right" xlink:label="note-043-08" xlink:href="note-043-08a" xml:space="preserve">15. 1. huius.</note> <figure xlink:label="fig-044-01" xlink:href="fig-044-01a"> <image file="044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/044-01"/> </figure> <note position="left" xlink:label="note-044-01" xlink:href="note-044-01a" xml:space="preserve">19. vndec.</note> <note position="left" xlink:label="note-044-02" xlink:href="note-044-02a" xml:space="preserve">Coroll. 16. <lb/>tertij.</note> </div> </div> <div xml:id="echoid-div136" type="section" level="1" n="75"> <head xml:id="echoid-head87" xml:space="preserve">THEOREMA 4. PROPOS. 4.</head> <note position="left" xml:space="preserve">5.</note> <p> <s xml:id="echoid-s1193" xml:space="preserve">SI in ſphæra duo circuli ſe mutuo tangant, ma-<lb/>ximus circulus per eorum polos deſcriptus, per <lb/>eorum contactum tranſibit.</s> <s xml:id="echoid-s1194" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1195" xml:space="preserve">IN ſphæra tangant ſe mutuo circuli A B, C B, in B; </s> <s xml:id="echoid-s1196" xml:space="preserve">& </s> <s xml:id="echoid-s1197" xml:space="preserve">per D, polum cir-<lb/> <anchor type="note" xlink:label="note-044-04a" xlink:href="note-044-04"/> culi A B, & </s> <s xml:id="echoid-s1198" xml:space="preserve">E, polum circuli C B, deſcribatur circulus maximus D E. </s> <s xml:id="echoid-s1199" xml:space="preserve">Dico <lb/>circulum D E, per contactum B, tranſire. </s> <s xml:id="echoid-s1200" xml:space="preserve">Non tranſeat enim, ſi fieri poteſt, <lb/>per tactum B, ſed ſecet circunferentiam v. </s> <s xml:id="echoid-s1201" xml:space="preserve">g. </s> <s xml:id="echoid-s1202" xml:space="preserve">circuli C B, in F. </s> <s xml:id="echoid-s1203" xml:space="preserve">Polo igitur <lb/>D, & </s> <s xml:id="echoid-s1204" xml:space="preserve">interuallo D F, circulus deſcribatur F G, qui, cum ad maius interual-<lb/> <anchor type="figure" xlink:label="fig-044-02a" xlink:href="fig-044-02"/> lum deſcriptus ſit, quàm circu <lb/>lus A B, ſecabit circulũ C B, <lb/>in F; </s> <s xml:id="echoid-s1205" xml:space="preserve">quandoquidem circulus <lb/>A B, eundem tangit in B, pun <lb/>cto, vltra quod circulus G F, <lb/>ex polo D, deſcriptus eſt. </s> <s xml:id="echoid-s1206" xml:space="preserve">Quo <lb/>niam vero in ſphæra duo cir-<lb/>culi G F, C F, ſecant in eodẽ <lb/>puncto F, maximum circulum <lb/>D F E, per eorum polos de-<lb/>ſcriptum, tangent ſe mutuo in <lb/>F, duo circuli G F, C F: </s> <s xml:id="echoid-s1207" xml:space="preserve">Sed <lb/> <anchor type="note" xlink:label="note-044-05a" xlink:href="note-044-05"/> & </s> <s xml:id="echoid-s1208" xml:space="preserve">mutuo ſeſe ſecant in F, vt <lb/>dictum eſt. </s> <s xml:id="echoid-s1209" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s1210" xml:space="preserve">Non ergo circulus maximus D E, ſecat ali-<lb/>bi circulos A B, C B, quàm in B, contactu, atque adeo per eorum tactũ tran <lb/>ſibit. </s> <s xml:id="echoid-s1211" xml:space="preserve">Itaque ſi in ſphæra duo circuli ſe mutuo tangant, &</s> <s xml:id="echoid-s1212" xml:space="preserve">c. </s> <s xml:id="echoid-s1213" xml:space="preserve">Quod oſtenden-<lb/>dum erat.</s> <s xml:id="echoid-s1214" xml:space="preserve"/> </p> <div xml:id="echoid-div136" type="float" level="2" n="1"> <note position="left" xlink:label="note-044-04" xlink:href="note-044-04a" xml:space="preserve">20. 1. huius.</note> <figure xlink:label="fig-044-02" xlink:href="fig-044-02a"> <image file="044-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/044-02"/> </figure> <note position="left" xlink:label="note-044-05" xlink:href="note-044-05a" xml:space="preserve">3. huius.</note> </div> <note position="left" xml:space="preserve">6.</note> </div> <div xml:id="echoid-div138" type="section" level="1" n="76"> <head xml:id="echoid-head88" xml:space="preserve">THEOR. 5. PROPOS. 5.</head> <p> <s xml:id="echoid-s1215" xml:space="preserve">SI in ſphęra duo circuli ſe mutuo tangant, ma- <pb o="33" file="045" n="45" rhead=""/> ximus circulus deſcriptus per vnius polos, & </s> <s xml:id="echoid-s1216" xml:space="preserve">per <lb/>contactum amborum circulorũ, per reliqui quo-<lb/>que circuli polos tranſibit.</s> <s xml:id="echoid-s1217" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1218" xml:space="preserve">IN ſphæra duo circuli A B, C B, tangãt ſe mutuo in B, ſintq́ D, E, poli <lb/>ipſorum. </s> <s xml:id="echoid-s1219" xml:space="preserve">Dico maximum circulum per D, polum circuli A B, & </s> <s xml:id="echoid-s1220" xml:space="preserve">per conta-<lb/>ctum B, deſcriptum tranſire quoque per E, polum circuli C B. </s> <s xml:id="echoid-s1221" xml:space="preserve">Si enim fieri <lb/>poteſt, non tranſeat per E, ſed per aliud quoduis punctum F, cuiuſmodi eſt <lb/>circulus maximus D B F: </s> <s xml:id="echoid-s1222" xml:space="preserve">Et per polos D, E, maximus circulus deſcribatur <lb/> <anchor type="figure" xlink:label="fig-045-01a" xlink:href="fig-045-01"/> <anchor type="note" xlink:label="note-045-01a" xlink:href="note-045-01"/> D E, qui omnino per conta-<lb/> <anchor type="note" xlink:label="note-045-02a" xlink:href="note-045-02"/> ctum B, tranſibit; </s> <s xml:id="echoid-s1223" xml:space="preserve">atque adeo <lb/>duo circuli maximi D B F, <lb/>D B E, ſe mutuo ſecabuntin <lb/>D, & </s> <s xml:id="echoid-s1224" xml:space="preserve">B, ac proinde bifariam. <lb/></s> <s xml:id="echoid-s1225" xml:space="preserve"> <anchor type="note" xlink:label="note-045-03a" xlink:href="note-045-03"/> Semicirculus ergo erit vterq; <lb/></s> <s xml:id="echoid-s1226" xml:space="preserve">arcus D B. </s> <s xml:id="echoid-s1227" xml:space="preserve">Quoniam vero cir <lb/>culus maximus per alterũ po-<lb/>lorũ cuiuſlibet circuli in ſphæ <lb/>ra tranſiens, tranſit quoque <lb/> <anchor type="note" xlink:label="note-045-04a" xlink:href="note-045-04"/> per reliquum polum, eſtq́; </s> <s xml:id="echoid-s1228" xml:space="preserve">in-<lb/>ter duos polos eiuſdem circu-<lb/>li ſemicirculus circuli maximi <lb/>interpoſitus; </s> <s xml:id="echoid-s1229" xml:space="preserve">fit, vt exiſtente D, vno polorum circuli A B, punctum B, ſit al <lb/>ter polus. </s> <s xml:id="echoid-s1230" xml:space="preserve">Quod eſt abſurdũ. </s> <s xml:id="echoid-s1231" xml:space="preserve">Eſt enim B, in circunferentia circuli. </s> <s xml:id="echoid-s1232" xml:space="preserve">Tranſit <lb/>igitur circulus maximus D B, per E. </s> <s xml:id="echoid-s1233" xml:space="preserve">Quocirca, ſi in ſphæra duo circuliſe <lb/>mutuo tangant, &</s> <s xml:id="echoid-s1234" xml:space="preserve">c. </s> <s xml:id="echoid-s1235" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1236" xml:space="preserve"/> </p> <div xml:id="echoid-div138" type="float" level="2" n="1"> <figure xlink:label="fig-045-01" xlink:href="fig-045-01a"> <image file="045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/045-01"/> </figure> <note position="right" xlink:label="note-045-01" xlink:href="note-045-01a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-045-02" xlink:href="note-045-02a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-045-03" xlink:href="note-045-03a" xml:space="preserve">11. 1. huius.</note> <note position="right" xlink:label="note-045-04" xlink:href="note-045-04a" xml:space="preserve">Coroll. 10. <lb/>1. huius.</note> </div> </div> <div xml:id="echoid-div140" type="section" level="1" n="77"> <head xml:id="echoid-head89" xml:space="preserve">THEOREMA 6. PROPOS. 6.</head> <note position="right" xml:space="preserve">7.</note> <p> <s xml:id="echoid-s1237" xml:space="preserve">SI in ſphæra maximus circulus aliquem circu <lb/>lorum in ſphęrica ſuperficie deſc@iptorum tangat, <lb/>tanget & </s> <s xml:id="echoid-s1238" xml:space="preserve">alterum ei æqualem, & </s> <s xml:id="echoid-s1239" xml:space="preserve">parallelum.</s> <s xml:id="echoid-s1240" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1241" xml:space="preserve">IN ſphæra maximus circulus A B, tan-<lb/> <anchor type="figure" xlink:label="fig-045-02a" xlink:href="fig-045-02"/> gat circulum A C, in A. </s> <s xml:id="echoid-s1242" xml:space="preserve">Dico circulũ A B, <lb/>tangere quoque alterum circulum ipſi A C, <lb/>æqualem, & </s> <s xml:id="echoid-s1243" xml:space="preserve">parallelum. </s> <s xml:id="echoid-s1244" xml:space="preserve">Sit enim D, polus <lb/> <anchor type="note" xlink:label="note-045-06a" xlink:href="note-045-06"/> circuli A C: </s> <s xml:id="echoid-s1245" xml:space="preserve">ac per D, A, circulus maximus <lb/>deſcribatur D A: </s> <s xml:id="echoid-s1246" xml:space="preserve">qui, cum per D, polũ cir-<lb/>culi A C, & </s> <s xml:id="echoid-s1247" xml:space="preserve">per contactum A, tranſeat, tran <lb/>ſibit per polos quoque circuli A B. </s> <s xml:id="echoid-s1248" xml:space="preserve">Aſſum-<lb/> <anchor type="note" xlink:label="note-045-07a" xlink:href="note-045-07"/> pto autem E, reliquo polo circuli A C, du-<lb/>catur recta D E, quæ per centrum ſphæræ <lb/> <anchor type="note" xlink:label="note-045-08a" xlink:href="note-045-08"/> tranſibit, atque adeo ſphæræ diameter erit.</s> <s xml:id="echoid-s1249" xml:space="preserve"> <pb o="34" file="046" n="46" rhead=""/> Ex polo igitur E, & </s> <s xml:id="echoid-s1250" xml:space="preserve">ad interuallum E B, circulus deſcribatur B F. </s> <s xml:id="echoid-s1251" xml:space="preserve">Dico cir-<lb/>culum maximum A B, tangere quoque circulum B F, in B, & </s> <s xml:id="echoid-s1252" xml:space="preserve">circulum B F, <lb/>æqualem eſſe, ac parallelum circulo A C. </s> <s xml:id="echoid-s1253" xml:space="preserve">Quoniam enim recta D E, per po-<lb/>los circulorũ A C, B F, tranſiens perpendicularis eſt ad ipſos circulos, erunt <lb/> <anchor type="figure" xlink:label="fig-046-01a" xlink:href="fig-046-01"/> <anchor type="note" xlink:label="note-046-01a" xlink:href="note-046-01"/> circuli A C, B F, paralleli. </s> <s xml:id="echoid-s1254" xml:space="preserve">Rurſus quia cir <lb/> <anchor type="note" xlink:label="note-046-02a" xlink:href="note-046-02"/> culi maximi in ſphęra bifariam ſe ſecant, ſe-<lb/> <anchor type="note" xlink:label="note-046-03a" xlink:href="note-046-03"/> micirculus erit A C B; </s> <s xml:id="echoid-s1255" xml:space="preserve">atque adeo ſemicir-<lb/>culo D C E, æqualis. </s> <s xml:id="echoid-s1256" xml:space="preserve">Dempto ergo commu <lb/>ni arcu B D, æquales remanebũt arcus D A, <lb/>E B; </s> <s xml:id="echoid-s1257" xml:space="preserve">atque adeo rectæ D A, E B, à polis D, <lb/> <anchor type="note" xlink:label="note-046-04a" xlink:href="note-046-04"/> E, ad circunferentias circulorum A C, B F, <lb/>ductæ æquales. </s> <s xml:id="echoid-s1258" xml:space="preserve">Quare æquales ſunt circuli <lb/> <anchor type="note" xlink:label="note-046-05a" xlink:href="note-046-05"/> A C, B F. </s> <s xml:id="echoid-s1259" xml:space="preserve">Denique quia circuli A B, B F, in <lb/>eodem puncto B, ſecant maximum circulũ <lb/>A E B, in quo quidem polos habent, ſe <lb/>mutuo tangent in B, circuli A B, B F. </s> <s xml:id="echoid-s1260" xml:space="preserve">Qua-<lb/> <anchor type="note" xlink:label="note-046-06a" xlink:href="note-046-06"/> re circulus maximus A B, tangens in ſphæra <lb/>circulum A C, tangit quoque alterum circulum B F, ipſi A C, æqualem, & </s> <s xml:id="echoid-s1261" xml:space="preserve"><lb/>parallelũ. </s> <s xml:id="echoid-s1262" xml:space="preserve">Ac proinde ſi in ſphæra maximus circulus aliquem circulorum, &</s> <s xml:id="echoid-s1263" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1264" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1265" xml:space="preserve"/> </p> <div xml:id="echoid-div140" type="float" level="2" n="1"> <figure xlink:label="fig-045-02" xlink:href="fig-045-02a"> <image file="045-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/045-02"/> </figure> <note position="right" xlink:label="note-045-06" xlink:href="note-045-06a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-045-07" xlink:href="note-045-07a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-045-08" xlink:href="note-045-08a" xml:space="preserve">10. 1. huius.</note> <figure xlink:label="fig-046-01" xlink:href="fig-046-01a"> <image file="046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/046-01"/> </figure> <note position="left" xlink:label="note-046-01" xlink:href="note-046-01a" xml:space="preserve">10. i. huius.</note> <note position="left" xlink:label="note-046-02" xlink:href="note-046-02a" xml:space="preserve">14. vndec.</note> <note position="left" xlink:label="note-046-03" xlink:href="note-046-03a" xml:space="preserve">11. 1. huius.</note> <note position="left" xlink:label="note-046-04" xlink:href="note-046-04a" xml:space="preserve">29. tertij.</note> <note position="left" xlink:label="note-046-05" xlink:href="note-046-05a" xml:space="preserve">Schol. 21. <lb/>1. huius.</note> <note position="left" xlink:label="note-046-06" xlink:href="note-046-06a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div142" type="section" level="1" n="78"> <head xml:id="echoid-head90" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s1266" xml:space="preserve">HINC perſpicuum eſt, puncta contactuum A, B, per diametrum eſſe oppoſita. </s> <s xml:id="echoid-s1267" xml:space="preserve">Oſten-<lb/>ſum enim eſt, A C B, eſſe ſemicirculum, ac propterea rectam ex A, ad B, ductam eſſe dia-<lb/>metrum ſphæræ, ſen circuli maximi A C B, &</s> <s xml:id="echoid-s1268" xml:space="preserve">c.</s> <s xml:id="echoid-s1269" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">8.</note> </div> <div xml:id="echoid-div143" type="section" level="1" n="79"> <head xml:id="echoid-head91" xml:space="preserve">THEOREMA 7. PROPOS. 7.</head> <p> <s xml:id="echoid-s1270" xml:space="preserve">SI ſint in ſphæra duo æquales, & </s> <s xml:id="echoid-s1271" xml:space="preserve">paralleli cir-<lb/>culi, maximus circulus, qui eorum alterum tetige <lb/>rit, reliquum quoque tanget.</s> <s xml:id="echoid-s1272" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1273" xml:space="preserve">IN eadem figura ſint duo circuli æquales, & </s> <s xml:id="echoid-s1274" xml:space="preserve">paralleli A C, B F, & </s> <s xml:id="echoid-s1275" xml:space="preserve">maxi-<lb/>mus A B, tangat A C. </s> <s xml:id="echoid-s1276" xml:space="preserve">Dico eundem A B, tangere quoque B F. </s> <s xml:id="echoid-s1277" xml:space="preserve">Sienim A B, <lb/>non tangat ipſum B F, tanget vtique alterum ipſi A C, æqualem, & </s> <s xml:id="echoid-s1278" xml:space="preserve">paralle-<lb/> <anchor type="note" xlink:label="note-046-08a" xlink:href="note-046-08"/> lum. </s> <s xml:id="echoid-s1279" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s1280" xml:space="preserve">B F, eidem A C, æqualis ponatur, & </s> <s xml:id="echoid-s1281" xml:space="preserve">parellelus, erunt tres <lb/>circuli in ſphæra, nempe A C, B F, & </s> <s xml:id="echoid-s1282" xml:space="preserve">ille alius, quem A B, tangit, inter ſe <lb/>æquales, & </s> <s xml:id="echoid-s1283" xml:space="preserve">paralleli. </s> <s xml:id="echoid-s1284" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s1285" xml:space="preserve">Non enim plures circuli æquales <lb/> <anchor type="note" xlink:label="note-046-09a" xlink:href="note-046-09"/> ſunt, & </s> <s xml:id="echoid-s1286" xml:space="preserve">paralleli in ſphæra, quàm duo. </s> <s xml:id="echoid-s1287" xml:space="preserve">Tanget igitur circulus A B, circulũ <lb/>B F. </s> <s xml:id="echoid-s1288" xml:space="preserve">Quamobrẽ, ſi ſint in ſphæra duo æquales, & </s> <s xml:id="echoid-s1289" xml:space="preserve">paralleli circuli, &</s> <s xml:id="echoid-s1290" xml:space="preserve">c. </s> <s xml:id="echoid-s1291" xml:space="preserve">Quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s1292" xml:space="preserve"/> </p> <div xml:id="echoid-div143" type="float" level="2" n="1"> <note position="left" xlink:label="note-046-08" xlink:href="note-046-08a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-046-09" xlink:href="note-046-09a" xml:space="preserve">Schol. 2. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div145" type="section" level="1" n="80"> <head xml:id="echoid-head92" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1293" xml:space="preserve">_IN_ alia verſione demonſtratur & </s> <s xml:id="echoid-s1294" xml:space="preserve">ſequens theorema.</s> <s xml:id="echoid-s1295" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1296" xml:space="preserve">CIRCVLI in ſphæra paralleli, quos maximus aliquis circu-<lb/> <anchor type="note" xlink:label="note-046-10a" xlink:href="note-046-10"/> lus tangit, æquales inter ſe ſunt.</s> <s xml:id="echoid-s1297" xml:space="preserve"/> </p> <div xml:id="echoid-div145" type="float" level="2" n="1"> <note position="left" xlink:label="note-046-10" xlink:href="note-046-10a" xml:space="preserve">9.</note> </div> <pb o="35" file="047" n="47" rhead=""/> <p style="it"> <s xml:id="echoid-s1298" xml:space="preserve">IN eadem adhuc figura ſint duo circuli paralleli _A C, B F,_ quos circulus maxis <lb/>mus _A B,_ tangat in _A, B._ </s> <s xml:id="echoid-s1299" xml:space="preserve">Dico circulos _A C, B F,_ æquales inter ſe eſſe. </s> <s xml:id="echoid-s1300" xml:space="preserve">Quoniam <lb/>enim paralleli ponuntur circuli _A C, B F,_ ipſi circa eoſdem polos erunt, qui ſint _D,_ <lb/> <anchor type="note" xlink:label="note-047-01a" xlink:href="note-047-01"/> _E;_ </s> <s xml:id="echoid-s1301" xml:space="preserve">per quos, & </s> <s xml:id="echoid-s1302" xml:space="preserve">polos circuli _A B,_ circulus maximus deſcribatur _A F B,_ qui per con <lb/> <anchor type="note" xlink:label="note-047-02a" xlink:href="note-047-02"/> tactus _A, B,_ tranſibit. </s> <s xml:id="echoid-s1303" xml:space="preserve">Quoniam vero circuli maximi in ſphæra ſe mutuo ſecant bi <lb/> <anchor type="note" xlink:label="note-047-03a" xlink:href="note-047-03"/> fariam, ſemicirculus erit _A D B,_ atque adeo ſemicirculo _D B E,_ æqualis. </s> <s xml:id="echoid-s1304" xml:space="preserve">Dempto <lb/>ergo arcu communi _D B,_ æquales remanebunt arcus _D A, E B;_ </s> <s xml:id="echoid-s1305" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s1306" xml:space="preserve">rectæ <lb/>_D A, E B,_ ex polis _D, E,_ ad circunferentias circulorum _A C, B F,_ ductæ æquales. <lb/></s> <s xml:id="echoid-s1307" xml:space="preserve"> <anchor type="note" xlink:label="note-047-04a" xlink:href="note-047-04"/> Quare circuli _A C, B F,_ æquales erunt. </s> <s xml:id="echoid-s1308" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s1309" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="2"> <note position="right" xlink:label="note-047-01" xlink:href="note-047-01a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-047-02" xlink:href="note-047-02a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-047-03" xlink:href="note-047-03a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-047-04" xlink:href="note-047-04a" xml:space="preserve">29. tertij. <lb/>Schol. 21. 1. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div148" type="section" level="1" n="81"> <head xml:id="echoid-head93" xml:space="preserve">THEOR. 8. PROP. 8.</head> <note position="right" xml:space="preserve">10.</note> <p> <s xml:id="echoid-s1310" xml:space="preserve">SI in ſphæra maximus circulus ad aliquẽ ſphæ <lb/>ræ circulum obliquus ſit, tanget is duos circulos <lb/>æqualcs quidem inter ſe, parallelos autem prædi-<lb/>cto circulo, ad quem obliquus eſt.</s> <s xml:id="echoid-s1311" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1312" xml:space="preserve">IN ſphæra maximus circulus A B, ad circulum quemcunque C D, obli-<lb/>quus ſit. </s> <s xml:id="echoid-s1313" xml:space="preserve">Dico circulum A B, tangere duos circulos inter ſe quidem æquales, <lb/>parallelos autem ipſi C D. </s> <s xml:id="echoid-s1314" xml:space="preserve">Sint E,F, poli circuli C D, per quos, & </s> <s xml:id="echoid-s1315" xml:space="preserve">polos cir <lb/> <anchor type="figure" xlink:label="fig-047-01a" xlink:href="fig-047-01"/> <anchor type="note" xlink:label="note-047-06a" xlink:href="note-047-06"/> culi A B, circulus maximus deſcribatur <lb/> <anchor type="note" xlink:label="note-047-07a" xlink:href="note-047-07"/> E A B, ſecans A B, in A, & </s> <s xml:id="echoid-s1316" xml:space="preserve">B. </s> <s xml:id="echoid-s1317" xml:space="preserve">Ex polo dein <lb/>de E, & </s> <s xml:id="echoid-s1318" xml:space="preserve">interuallo E A, circulus deſcriba-<lb/>tur A G. </s> <s xml:id="echoid-s1319" xml:space="preserve">Et quoniam circuli A B, A G, in <lb/>eodem puncto A, ſecant maximum circulũ <lb/> <anchor type="note" xlink:label="note-047-08a" xlink:href="note-047-08"/> E A B, in quo polos habent, ipſi ſe mutuo <lb/>tangent in A. </s> <s xml:id="echoid-s1320" xml:space="preserve">Circulus igitur maximus A B, <lb/>tangens circulum A G, tanget alterum il-<lb/> <anchor type="note" xlink:label="note-047-09a" xlink:href="note-047-09"/> li æqualem, & </s> <s xml:id="echoid-s1321" xml:space="preserve">parallelum, qui ſit B H. </s> <s xml:id="echoid-s1322" xml:space="preserve">Quia <lb/>vero circuli paralleli A G, B H, circa eoſdẽ <lb/> <anchor type="note" xlink:label="note-047-10a" xlink:href="note-047-10"/> polos ſunt E, F: </s> <s xml:id="echoid-s1323" xml:space="preserve">Sunt autem E, F, poli etiã <lb/>circuli C D; </s> <s xml:id="echoid-s1324" xml:space="preserve">erunt tres circuli A G, C D, <lb/> <anchor type="note" xlink:label="note-047-11a" xlink:href="note-047-11"/> B H, circa eoſdem polos; </s> <s xml:id="echoid-s1325" xml:space="preserve">atque adeo paralle <lb/>li inter ſe erũt. </s> <s xml:id="echoid-s1326" xml:space="preserve">Tangit igitur maximus circu <lb/>lus A B, duos A G, B H, æquales quidem inter ſe, parallelos autem ipſi C D, <lb/>ad quem obliquus eſt. </s> <s xml:id="echoid-s1327" xml:space="preserve">Quocirca, ſi in ſphæra maximus circulus ad aliquem, <lb/>&</s> <s xml:id="echoid-s1328" xml:space="preserve">c. </s> <s xml:id="echoid-s1329" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s1330" xml:space="preserve"/> </p> <div xml:id="echoid-div148" type="float" level="2" n="1"> <figure xlink:label="fig-047-01" xlink:href="fig-047-01a"> <image file="047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/047-01"/> </figure> <note position="right" xlink:label="note-047-06" xlink:href="note-047-06a" xml:space="preserve">21. 1. huius.</note> <note position="right" xlink:label="note-047-07" xlink:href="note-047-07a" xml:space="preserve">20. 2. huius.</note> <note position="right" xlink:label="note-047-08" xlink:href="note-047-08a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-047-09" xlink:href="note-047-09a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-047-10" xlink:href="note-047-10a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-047-11" xlink:href="note-047-11a" xml:space="preserve">2. huius.</note> </div> </div> <div xml:id="echoid-div150" type="section" level="1" n="82"> <head xml:id="echoid-head94" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1331" xml:space="preserve">_ALIVD_ theorema hoc loco adijcitur in alia verſione, videlicet.</s> <s xml:id="echoid-s1332" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1333" xml:space="preserve">SI in ſphæra maximus circulus aliquem circulorum in ſphæri-<lb/> <anchor type="note" xlink:label="note-047-12a" xlink:href="note-047-12"/> <pb o="36" file="048" n="48" rhead=""/> ca ſuperficie tangat, obliquus erit ad alios circulos, quos ſecat, paral <lb/>lelos ei, quem tangit.</s> <s xml:id="echoid-s1334" xml:space="preserve"/> </p> <div xml:id="echoid-div150" type="float" level="2" n="1"> <note position="right" xlink:label="note-047-12" xlink:href="note-047-12a" xml:space="preserve">11.</note> </div> <p style="it"> <s xml:id="echoid-s1335" xml:space="preserve">_IN_ eadem figura maximus circulus _A B,_ tangat circulum _A G,_ ſecet autem circu <lb/> <anchor type="figure" xlink:label="fig-048-01a" xlink:href="fig-048-01"/> lum _C D,_ ipſi _A G,_ parallelum. </s> <s xml:id="echoid-s1336" xml:space="preserve">Dico circulum <lb/>_A B,_ obliquum eſſe ad circulum _C D._ </s> <s xml:id="echoid-s1337" xml:space="preserve">Quoniã <lb/>enim maximus circulus A B, tangens circulum <lb/>_A G,_ non tranſit per ipſius polos, (Si namque per <lb/> <anchor type="note" xlink:label="note-048-01a" xlink:href="note-048-01"/> ipſius polos duceretur, ſecaret ipſum bifariam, <lb/>non autem tangeret.) </s> <s xml:id="echoid-s1338" xml:space="preserve">atque adeo neque per po <lb/>los circuli _CD;_ </s> <s xml:id="echoid-s1339" xml:space="preserve">(habent enim paralleli circuli <lb/> <anchor type="note" xlink:label="note-048-02a" xlink:href="note-048-02"/> _A G, C D,_ eoſdem polos) non ſecabit maximus <lb/>circulus _A B,_ circulum _C D,_ ad angulos rectos: <lb/></s> <s xml:id="echoid-s1340" xml:space="preserve">Aliàs tranſiret per eius polos. </s> <s xml:id="echoid-s1341" xml:space="preserve">Igitur obliquus <lb/> <anchor type="note" xlink:label="note-048-03a" xlink:href="note-048-03"/> eſt ad circulum _C D._ </s> <s xml:id="echoid-s1342" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s1343" xml:space="preserve"/> </p> <div xml:id="echoid-div151" type="float" level="2" n="2"> <figure xlink:label="fig-048-01" xlink:href="fig-048-01a"> <image file="048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/048-01"/> </figure> <note position="left" xlink:label="note-048-01" xlink:href="note-048-01a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-048-02" xlink:href="note-048-02a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-048-03" xlink:href="note-048-03a" xml:space="preserve">13. 1. huius.</note> </div> </div> <div xml:id="echoid-div153" type="section" level="1" n="83"> <head xml:id="echoid-head95" xml:space="preserve">THEOR. 9. PROPOS. 9.</head> <note position="left" xml:space="preserve">12.</note> <p> <s xml:id="echoid-s1344" xml:space="preserve">SI in ſphæra duo circuli ſe mutuo ſecent, ma-<lb/>ximus circulus per eorum polos ductus ſecabit bi <lb/>fariam ſegmenta ipſorum circulorum.</s> <s xml:id="echoid-s1345" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1346" xml:space="preserve">IN ſphæra ſe mutuo ſecent duo circuli A B C D, E D F B, in punctis B, <lb/>D, & </s> <s xml:id="echoid-s1347" xml:space="preserve">per eorum polos deſcribatur maxim us circulus A F C E, ſecans circu-<lb/> <anchor type="note" xlink:label="note-048-05a" xlink:href="note-048-05"/> los dictos in punctis A, C, E, F. </s> <s xml:id="echoid-s1348" xml:space="preserve">Dico circulum A F C E, ſecare bifariã ſeg-<lb/> <anchor type="figure" xlink:label="fig-048-02a" xlink:href="fig-048-02"/> menta B A D, B C D, B E D, B F D. </s> <s xml:id="echoid-s1349" xml:space="preserve">Quo-<lb/>niam enim circulus maximus A F C E, cir-<lb/> <anchor type="note" xlink:label="note-048-06a" xlink:href="note-048-06"/> culos A B C D, E D F B, ſecat bifariam, & </s> <s xml:id="echoid-s1350" xml:space="preserve"><lb/>ad angulos rectos, quòd per eorum polos du <lb/>ctus ſit, erunt communes ſectiones A C, E F, <lb/>quas cum ipſis facit, diametri ipſorum ſecan <lb/>tes ſeſe in G. </s> <s xml:id="echoid-s1351" xml:space="preserve">Secabunt enim ſe mutuo rectę <lb/>A C, E F, cum in eodẽ plano circuli A F C E, <lb/>exiſtant, ſitq́; </s> <s xml:id="echoid-s1352" xml:space="preserve">punctum E, inter puncta A, <lb/>& </s> <s xml:id="echoid-s1353" xml:space="preserve">C; </s> <s xml:id="echoid-s1354" xml:space="preserve">atque punctum E, inter eadem pun-<lb/>cta. </s> <s xml:id="echoid-s1355" xml:space="preserve">Connectantur rectæ B G, D G: </s> <s xml:id="echoid-s1356" xml:space="preserve">Eruntq́; <lb/></s> <s xml:id="echoid-s1357" xml:space="preserve">tria puncta B, G, D, in vtroque plano circu <lb/>lorum A B C D, EDFB; </s> <s xml:id="echoid-s1358" xml:space="preserve">atque adeo in cõ <lb/>muni eorum ſectione: </s> <s xml:id="echoid-s1359" xml:space="preserve">Eſt autem communis <lb/>eorum ſectio linea recta. </s> <s xml:id="echoid-s1360" xml:space="preserve">Igitur recta erit B G D. </s> <s xml:id="echoid-s1361" xml:space="preserve">Et quoniam circulus A F C E, <lb/> <anchor type="note" xlink:label="note-048-07a" xlink:href="note-048-07"/> oſtenſus eſt ſecare ad angulos rectos vtrumque circulum A B C D, E D F B, <lb/>crit viciſsim vterque rectus ad circulum AFCE; </s> <s xml:id="echoid-s1362" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s1363" xml:space="preserve">B D, com-<lb/>munis eorum ſectio ad eundem perpendicularis erit. </s> <s xml:id="echoid-s1364" xml:space="preserve">Recti igitur erunt angu <lb/> <anchor type="note" xlink:label="note-048-08a" xlink:href="note-048-08"/> <pb o="37" file="049" n="49" rhead=""/> li B G A, D G A, B G C, D G C, ex definit. </s> <s xml:id="echoid-s1365" xml:space="preserve">3. </s> <s xml:id="echoid-s1366" xml:space="preserve">lib. </s> <s xml:id="echoid-s1367" xml:space="preserve">11. </s> <s xml:id="echoid-s1368" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1369" xml:space="preserve">Quare diameter <lb/>A C, cum per centrum circuli A B C D, tranſeat, ſecetq̃; </s> <s xml:id="echoid-s1370" xml:space="preserve">rectam B D, ad <lb/>angulos rectos, bifariam eam ſecabit. </s> <s xml:id="echoid-s1371" xml:space="preserve">Itaque cum latera A G, G B, æqualia <lb/> <anchor type="note" xlink:label="note-049-01a" xlink:href="note-049-01"/> ſint lateribus A G, G D, contineantq́; </s> <s xml:id="echoid-s1372" xml:space="preserve">angulos æquales, nempe rectos, erunt <lb/> <anchor type="note" xlink:label="note-049-02a" xlink:href="note-049-02"/> baſes A B, A D, ſubtendentes arcus A B, A D, inter ſe æquales, ac proinde <lb/> <anchor type="note" xlink:label="note-049-03a" xlink:href="note-049-03"/> & </s> <s xml:id="echoid-s1373" xml:space="preserve">arcus A B, A D, æquales erunt. </s> <s xml:id="echoid-s1374" xml:space="preserve">Eodem modo oſtendemus arcus C B, C D, <lb/>æquales eſſe; </s> <s xml:id="echoid-s1375" xml:space="preserve">nec non & </s> <s xml:id="echoid-s1376" xml:space="preserve">arcus E B, E D; </s> <s xml:id="echoid-s1377" xml:space="preserve">& </s> <s xml:id="echoid-s1378" xml:space="preserve">F B, F D. </s> <s xml:id="echoid-s1379" xml:space="preserve">Circulus igitur A F C E, <lb/>ſegmenta B A D, B C D, B E D, B F D, bifariam diuidit. </s> <s xml:id="echoid-s1380" xml:space="preserve">Quapropter ſi in <lb/>ſphæra duo circuli ſe mutuo ſecent, &</s> <s xml:id="echoid-s1381" xml:space="preserve">c. </s> <s xml:id="echoid-s1382" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s1383" xml:space="preserve"/> </p> <div xml:id="echoid-div153" type="float" level="2" n="1"> <note position="left" xlink:label="note-048-05" xlink:href="note-048-05a" xml:space="preserve">10. 1. huius.</note> <figure xlink:label="fig-048-02" xlink:href="fig-048-02a"> <image file="048-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/048-02"/> </figure> <note position="left" xlink:label="note-048-06" xlink:href="note-048-06a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-048-07" xlink:href="note-048-07a" xml:space="preserve">3. vndee.</note> <note position="left" xlink:label="note-048-08" xlink:href="note-048-08a" xml:space="preserve">19. vndec.</note> <note position="right" xlink:label="note-049-01" xlink:href="note-049-01a" xml:space="preserve">3. tertij.</note> <note position="right" xlink:label="note-049-02" xlink:href="note-049-02a" xml:space="preserve">4. primi.</note> <note position="right" xlink:label="note-049-03" xlink:href="note-049-03a" xml:space="preserve">28. terij.</note> </div> </div> <div xml:id="echoid-div155" type="section" level="1" n="84"> <head xml:id="echoid-head96" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1384" xml:space="preserve">_DVO_ alia theoremata in alia verſione hoc loco adduntur, hæc videlicet.</s> <s xml:id="echoid-s1385" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div156" type="section" level="1" n="85"> <head xml:id="echoid-head97" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s1386" xml:space="preserve">SI in ſphæra duo circuli ſe mutuo ſecent, circulus alius eorum <lb/> <anchor type="note" xlink:label="note-049-04a" xlink:href="note-049-04"/> ſegmenta bifariam ſecans, it per polos eorum, eſtq́; </s> <s xml:id="echoid-s1387" xml:space="preserve">circulus ma-<lb/>ximus.</s> <s xml:id="echoid-s1388" xml:space="preserve"/> </p> <div xml:id="echoid-div156" type="float" level="2" n="1"> <note position="right" xlink:label="note-049-04" xlink:href="note-049-04a" xml:space="preserve">13.</note> </div> <p style="it"> <s xml:id="echoid-s1389" xml:space="preserve">_IN_ eadẽ figura ſecent ſe mutuo duo circuli _A B C D, E D F B,_ in punctis _B, D,_ & </s> <s xml:id="echoid-s1390" xml:space="preserve"><lb/>alius quiſpiã circulus _A F C E,_ ſecet ſegmenta _B A D, B C D, B E D, B F D,_ bifariã. </s> <s xml:id="echoid-s1391" xml:space="preserve">_D_i <lb/>co ciculũ _A F C E,_ ire per polos ipſorũ, eſſeq́; </s> <s xml:id="echoid-s1392" xml:space="preserve">circulũ maximũ. </s> <s xml:id="echoid-s1393" xml:space="preserve">Quoniã enim arcus _A D,_ <lb/>_A B,_ æquales ſunt, nec nõ _C D, C B,_ erũt toti arcus _A D C, A B C,_ æquales, & </s> <s xml:id="echoid-s1394" xml:space="preserve">propte <lb/>rea ſemicirculi. </s> <s xml:id="echoid-s1395" xml:space="preserve">_E_odemq́; </s> <s xml:id="echoid-s1396" xml:space="preserve">modo ſemicirculi erũt _E D F, E B F. </s> <s xml:id="echoid-s1397" xml:space="preserve">C_irculus igitur _A F C E,_ <lb/>bifariam ſecat circulos _A B C D, E D F B,_ atque adeo communes ſectiones _A C, E F,_ <lb/>ſe interſecantes in _G,_ ipſorum diametri ſunt. </s> <s xml:id="echoid-s1398" xml:space="preserve">_Q_uòd ſi connectantur rectæ _B G, D G,_ <lb/>cum tria puncta _<emph style="sc">B</emph>, G, D,_ in vtroque plano circulorum _A B C D, <emph style="sc">EDFb</emph>_, ſint, at-<lb/>que adeo in communi ipſorum ſectione; </s> <s xml:id="echoid-s1399" xml:space="preserve">ſit autem communis eorum ſectio linea recta; <lb/></s> <s xml:id="echoid-s1400" xml:space="preserve"> <anchor type="note" xlink:label="note-049-05a" xlink:href="note-049-05"/> recta erit _<emph style="sc">B</emph> G D. </s> <s xml:id="echoid-s1401" xml:space="preserve">Q_uoniam vero ſubtenſæ rectæ _D A, D C,_ ſubtenſis rectis <emph style="sc">B</emph> _A_, <emph style="sc">B</emph> _C,_ <lb/> <anchor type="note" xlink:label="note-049-06a" xlink:href="note-049-06"/> ſingulæ ſingulis æquales ſunt, ob æquales arcus, anguloſq́; </s> <s xml:id="echoid-s1402" xml:space="preserve">continent æquales, nem-<lb/> <anchor type="note" xlink:label="note-049-07a" xlink:href="note-049-07"/> pe rectos in ſemicirculis exiſtentes; </s> <s xml:id="echoid-s1403" xml:space="preserve">æquales erunt anguli _D A C, <emph style="sc">B</emph>AC. </s> <s xml:id="echoid-s1404" xml:space="preserve">Q_uod etiã <lb/> <anchor type="note" xlink:label="note-049-08a" xlink:href="note-049-08"/> ita probari poterit. </s> <s xml:id="echoid-s1405" xml:space="preserve">_Q_uoniam latera _D A, A C,_ lateribus _<emph style="sc">B</emph>A, A C,_ æqualia ſunt, <lb/>baſiſq́; </s> <s xml:id="echoid-s1406" xml:space="preserve">_D C,_ baſi _<emph style="sc">B</emph>C,_ æqualis, erunt anguli _D A C, <emph style="sc">B</emph>AC,_ æquales. </s> <s xml:id="echoid-s1407" xml:space="preserve">Rurſus quia <lb/> <anchor type="note" xlink:label="note-049-09a" xlink:href="note-049-09"/> latera _A D, A G,_ lateribus _<emph style="sc">Ab</emph>, A G,_ æqualia ſunt, angulosq́; </s> <s xml:id="echoid-s1408" xml:space="preserve">continent æqua-<lb/> <anchor type="note" xlink:label="note-049-10a" xlink:href="note-049-10"/> les, vt demonſtratum eſt; </s> <s xml:id="echoid-s1409" xml:space="preserve">æquales erunt anguli _A G D, A G <emph style="sc">B</emph>,_ ac propterea recti. <lb/></s> <s xml:id="echoid-s1410" xml:space="preserve">Perpendicularis igitur eſt _<emph style="sc">B</emph>GD,_ ad rectam _A C, E_odem modo oſtendemus rectam <lb/>eandem _<emph style="sc">B</emph>GD,_ ad _E F,_ perpendicularem eſſe. </s> <s xml:id="echoid-s1411" xml:space="preserve">_Q_uare eadem _<emph style="sc">B</emph>GD,_ perpendicula-<lb/>ris erit ad planum circuli _A F C E,_ per rectas _A C, E F,_ ductum; </s> <s xml:id="echoid-s1412" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s1413" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-049-11a" xlink:href="note-049-11"/> vtrumque planum circulorum _<emph style="sc">Ab</emph>CD, <emph style="sc">EDFb</emph>,_ per rectam _<emph style="sc">B</emph>GD,_ ductum ad <lb/>idem planum circuli _A F C E,_ rectum erit: </s> <s xml:id="echoid-s1414" xml:space="preserve">& </s> <s xml:id="echoid-s1415" xml:space="preserve">vicißim circulus _A F C E,_ ad circu-<lb/> <anchor type="note" xlink:label="note-049-12a" xlink:href="note-049-12"/> los _<emph style="sc">Ab</emph>CD, <emph style="sc">EDFb</emph>,_ rectus erit. </s> <s xml:id="echoid-s1416" xml:space="preserve">_I_taque circulus _A F C E,_ circulos _A <emph style="sc">B</emph> C D,_ <lb/> <anchor type="note" xlink:label="note-049-13a" xlink:href="note-049-13"/> _E D F <emph style="sc">B</emph>,_ & </s> <s xml:id="echoid-s1417" xml:space="preserve">bifariam & </s> <s xml:id="echoid-s1418" xml:space="preserve">ad angulos rectos ſecat, _Q_uare maximus eſt, tranſitq́; </s> <s xml:id="echoid-s1419" xml:space="preserve">per <lb/>ipſorum polos. </s> <s xml:id="echoid-s1420" xml:space="preserve">_Q_uod eſt propoſitum.</s> <s xml:id="echoid-s1421" xml:space="preserve"/> </p> <div xml:id="echoid-div157" type="float" level="2" n="2"> <note position="right" xlink:label="note-049-05" xlink:href="note-049-05a" xml:space="preserve">3. vndec.</note> <note position="right" xlink:label="note-049-06" xlink:href="note-049-06a" xml:space="preserve">29. tertij.</note> <note position="right" xlink:label="note-049-07" xlink:href="note-049-07a" xml:space="preserve">31. tertij.</note> <note position="right" xlink:label="note-049-08" xlink:href="note-049-08a" xml:space="preserve">4. primi.</note> <note position="right" xlink:label="note-049-09" xlink:href="note-049-09a" xml:space="preserve">8. primi.</note> <note position="right" xlink:label="note-049-10" xlink:href="note-049-10a" xml:space="preserve">4. primi.</note> <note position="right" xlink:label="note-049-11" xlink:href="note-049-11a" xml:space="preserve">4. vndec.</note> <note position="right" xlink:label="note-049-12" xlink:href="note-049-12a" xml:space="preserve">18. vndec.</note> <note position="right" xlink:label="note-049-13" xlink:href="note-049-13a" xml:space="preserve">Schol. 15. 1. <lb/>huius.</note> </div> <pb o="38" file="050" n="50" rhead=""/> <p> <s xml:id="echoid-s1422" xml:space="preserve">SI in ſphæra duo circuli ſe mutuo ſecent, maximus circulus ſe-<lb/> <anchor type="note" xlink:label="note-050-01a" xlink:href="note-050-01"/> cans bifariam duo illorum ſegmenta quæcumque, habens tamen <lb/>arcum inter illa ſegmenta poſitum ſemicirculo inæqualem; </s> <s xml:id="echoid-s1423" xml:space="preserve">tranfit <lb/>per polos ipſorum, duoq́; </s> <s xml:id="echoid-s1424" xml:space="preserve">reliqua ſegmenta bifariam ſecat.</s> <s xml:id="echoid-s1425" xml:space="preserve"/> </p> <div xml:id="echoid-div158" type="float" level="2" n="3"> <note position="left" xlink:label="note-050-01" xlink:href="note-050-01a" xml:space="preserve">14.</note> </div> <p style="it"> <s xml:id="echoid-s1426" xml:space="preserve">_IN_ ſphæra duo circuli _<emph style="sc">Ab</emph>CD, <emph style="sc">Eb</emph>FD,_ ſe mutuo ſecent in punctis _<emph style="sc">B</emph>, D:_ </s> <s xml:id="echoid-s1427" xml:space="preserve">Et <lb/> <anchor type="figure" xlink:label="fig-050-01a" xlink:href="fig-050-01"/> maximus circulus _A F C E,_ ſecet <lb/>duo quæcumque illorum ſegmẽta, <lb/>nempe, _<emph style="sc">B</emph>AD, <emph style="sc">B</emph>ED,_ bifariam <lb/>in punctis _A, E,_ & </s> <s xml:id="echoid-s1428" xml:space="preserve">arcus _A F C E,_ <lb/>interceptus inter dicta ſegmenta <lb/>non ſit ſemicirculus. </s> <s xml:id="echoid-s1429" xml:space="preserve">Dico circu-<lb/>lum _A F C E,_ tranſire per polos <lb/>circulorum _<emph style="sc">Ab</emph>CD, <emph style="sc">Eb</emph>FD,_ <lb/>ſecareé; </s> <s xml:id="echoid-s1430" xml:space="preserve">reliqua ſegmenta _<emph style="sc">B</emph>CD,_ <lb/>_<emph style="sc">B</emph>FD,_ bifariam. </s> <s xml:id="echoid-s1431" xml:space="preserve">Si enim circulus <lb/>_A F C E,_ non tranſeat per ipſorũ <lb/>polos, deſcribatur, ſi fieri poteſt, <lb/>alius circulus maximus _A G E,_ per <lb/>eorum polos, qui ſegmenta ipſo-<lb/>rum bifariam ſecabit; </s> <s xml:id="echoid-s1432" xml:space="preserve">atque adeo per puncta _A, E,_ tranſibit. </s> <s xml:id="echoid-s1433" xml:space="preserve">Secabunt ſe igitur cir <lb/> <anchor type="note" xlink:label="note-050-02a" xlink:href="note-050-02"/> <anchor type="note" xlink:label="note-050-03a" xlink:href="note-050-03"/> culi maximi _A F C E, A G E,_ in _A, E,_ bifariam: </s> <s xml:id="echoid-s1434" xml:space="preserve">ac propterea ſemicirculus erit <lb/>_AFCE._ </s> <s xml:id="echoid-s1435" xml:space="preserve">Quod eſt contra hypotheſim. </s> <s xml:id="echoid-s1436" xml:space="preserve">Tranſit ergo circulus _A F C E,_ per polos cir-<lb/> <anchor type="note" xlink:label="note-050-04a" xlink:href="note-050-04"/> culorum _A B C D, <emph style="sc">Eb</emph>FD._ </s> <s xml:id="echoid-s1437" xml:space="preserve">Quare omnia ſegmenta ipſorum ſecabit bifariam. </s> <s xml:id="echoid-s1438" xml:space="preserve">Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s1439" xml:space="preserve"/> </p> <div xml:id="echoid-div159" type="float" level="2" n="4"> <figure xlink:label="fig-050-01" xlink:href="fig-050-01a"> <image file="050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/050-01"/> </figure> <note position="left" xlink:label="note-050-02" xlink:href="note-050-02a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-050-03" xlink:href="note-050-03a" xml:space="preserve">11. 1. huius.</note> <note position="left" xlink:label="note-050-04" xlink:href="note-050-04a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div161" type="section" level="1" n="86"> <head xml:id="echoid-head98" xml:space="preserve">THEOR, 10. PROP. 10.</head> <note position="left" xml:space="preserve">15.</note> <p> <s xml:id="echoid-s1440" xml:space="preserve">SI ſint in ſphæra paralleli circuli, per quo-<lb/>rum polos deſcribantur maximi circuli; </s> <s xml:id="echoid-s1441" xml:space="preserve">paralle-<lb/>lorum quidem circunferentiæ inter maximos cir-<lb/>culos interceptæ, ſimiles ſunt; </s> <s xml:id="echoid-s1442" xml:space="preserve">maximorum autem <lb/>circulorum circunferentiæ inter parallelos circu-<lb/>los interceptæ, ſuntæ quales.</s> <s xml:id="echoid-s1443" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1444" xml:space="preserve">SINT in ſphæra circuli paralleli A B C D, E F G H, quorũ polus I:</s> <s xml:id="echoid-s1445" xml:space="preserve">(ſunt <lb/>enim paralleli circuli in ſphæra circa eoſdem polos.) </s> <s xml:id="echoid-s1446" xml:space="preserve">Per I, autẽ circuli maxi <lb/> <anchor type="note" xlink:label="note-050-06a" xlink:href="note-050-06"/> mi deſcribantur vtcumque A E I G C, B F I H D. </s> <s xml:id="echoid-s1447" xml:space="preserve">Dico circunferentias paral-<lb/>lelorum A B, E F, ſimiles, nec non B C, F G; </s> <s xml:id="echoid-s1448" xml:space="preserve">Item C D, G H; </s> <s xml:id="echoid-s1449" xml:space="preserve">& </s> <s xml:id="echoid-s1450" xml:space="preserve">D A, H E:</s> <s xml:id="echoid-s1451" xml:space="preserve"> <pb o="39" file="051" n="51" rhead=""/> circunferentias vero maximorum circulorum inter parallelos, nempe A E, <lb/>B F, C G, D H, æquales eſſe. </s> <s xml:id="echoid-s1452" xml:space="preserve">Sintenim communes ſectiones circuli A I C, & </s> <s xml:id="echoid-s1453" xml:space="preserve"><lb/>parallelorum rectæ A C, E G, quæ parallelæ erunt: </s> <s xml:id="echoid-s1454" xml:space="preserve">communes vero ſectiones <lb/> <anchor type="note" xlink:label="note-051-01a" xlink:href="note-051-01"/> circuli B I D, & </s> <s xml:id="echoid-s1455" xml:space="preserve">parallelorum eorundem, rectæ B D, F H, quæ ſimiliter pa-<lb/> <anchor type="figure" xlink:label="fig-051-01a" xlink:href="fig-051-01"/> rallelæ erũt. </s> <s xml:id="echoid-s1456" xml:space="preserve">Et quia circuli maximi A I C, <lb/>B I D, per polos parallelorum deſcripti ſe-<lb/>cant parallelos bifariam; </s> <s xml:id="echoid-s1457" xml:space="preserve">erunt A C, B D, <lb/> <anchor type="note" xlink:label="note-051-02a" xlink:href="note-051-02"/> diametri circuli A B C D, & </s> <s xml:id="echoid-s1458" xml:space="preserve">punctum L, vbi <lb/>ſe interſecant, centrũ eiuſdem: </s> <s xml:id="echoid-s1459" xml:space="preserve">Item E G, <lb/>F H, diametri circuli E F G H, & </s> <s xml:id="echoid-s1460" xml:space="preserve">punctum <lb/>K, vbi ſe interſecant, centrum eiuſdẽ. </s> <s xml:id="echoid-s1461" xml:space="preserve">Quo <lb/>niam igitur rectæ E K, K F, rectis A L, L B, <lb/>parallelæ ſunt, ſuntq́; </s> <s xml:id="echoid-s1462" xml:space="preserve">in diuerſis planis, e-<lb/>runt anguli E K F, A L B, ad centra K, L, <lb/> <anchor type="note" xlink:label="note-051-03a" xlink:href="note-051-03"/> æquales. </s> <s xml:id="echoid-s1463" xml:space="preserve">Quare circunſerentiæ A B, E F, <lb/>per ea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s1464" xml:space="preserve">33. </s> <s xml:id="echoid-s1465" xml:space="preserve">lib 6. </s> <s xml:id="echoid-s1466" xml:space="preserve">Eu-<lb/>clid. </s> <s xml:id="echoid-s1467" xml:space="preserve">oſtendimus, ſimiles erunt. </s> <s xml:id="echoid-s1468" xml:space="preserve">Eodemq́; <lb/></s> <s xml:id="echoid-s1469" xml:space="preserve">modo ſimiles erunt B C, F G, & </s> <s xml:id="echoid-s1470" xml:space="preserve">C D, G H, nec non D A, H E.</s> <s xml:id="echoid-s1471" xml:space="preserve"/> </p> <div xml:id="echoid-div161" type="float" level="2" n="1"> <note position="left" xlink:label="note-050-06" xlink:href="note-050-06a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-051-01" xlink:href="note-051-01a" xml:space="preserve">16. vndec.</note> <figure xlink:label="fig-051-01" xlink:href="fig-051-01a"> <image file="051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/051-01"/> </figure> <note position="right" xlink:label="note-051-02" xlink:href="note-051-02a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-051-03" xlink:href="note-051-03a" xml:space="preserve">10. vndeo.</note> </div> <p> <s xml:id="echoid-s1472" xml:space="preserve">RVRSVS, quia rectæ ex polo I, ad puncta A, B, C, D, demiſſæ æquales <lb/>ſunt, ex defin. </s> <s xml:id="echoid-s1473" xml:space="preserve">poli, erunt quoque arcus I A, I B, I C, I D, æquales: </s> <s xml:id="echoid-s1474" xml:space="preserve">Et eo-<lb/> <anchor type="note" xlink:label="note-051-04a" xlink:href="note-051-04"/> dem modo æquales erunt arcus I E, I F, I G, I H. </s> <s xml:id="echoid-s1475" xml:space="preserve">Reliquæ igitur circunfe-<lb/>rentiæ A E, B F, C G, D H, æquales inter ſe erunt. </s> <s xml:id="echoid-s1476" xml:space="preserve">Quapropter, ſi ſint in <lb/>ſphæra paralleli circuli, &</s> <s xml:id="echoid-s1477" xml:space="preserve">c. </s> <s xml:id="echoid-s1478" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1479" xml:space="preserve"/> </p> <div xml:id="echoid-div162" type="float" level="2" n="2"> <note position="right" xlink:label="note-051-04" xlink:href="note-051-04a" xml:space="preserve">28. tertij.</note> </div> </div> <div xml:id="echoid-div164" type="section" level="1" n="87"> <head xml:id="echoid-head99" xml:space="preserve">THEOR. 11. PROP. 11</head> <note position="right" xml:space="preserve">16.</note> <p> <s xml:id="echoid-s1480" xml:space="preserve">SI in diametris circulorum æqualium æqua-<lb/>lia circulorum ſegmenta ad angulos rectos inſi-<lb/>ſtant, à quibus ſumantur æquales circunferentiæ, <lb/>quarum quælibet inchoata ab extremitate ſui ſe-<lb/>gmenti, ſit minor ſemiſſe circunferentiæ integri <lb/>ſegmenti, à punctis autem æquales circunferen-<lb/>tias terminantibus ducátur æquales rectæ lineę ad <lb/>circunferentias circulorum primo poſitorum; </s> <s xml:id="echoid-s1481" xml:space="preserve">ip-<lb/>ſæ circulorum primo poſitorum circunferentiæ <lb/>interceptæ inter illas rectas lineas, & </s> <s xml:id="echoid-s1482" xml:space="preserve">extremitates <lb/>diametrorum, erunt æquales.</s> <s xml:id="echoid-s1483" xml:space="preserve"/> </p> <pb o="40" file="052" n="52" rhead=""/> <p> <s xml:id="echoid-s1484" xml:space="preserve">IN diametris A C, D F, circulorum æqualium A B C, D E F, inſiſtant <lb/>ipſis circulis ad angulos rectos ſegmenta circulorũ æqualia A G C, D H F: <lb/></s> <s xml:id="echoid-s1485" xml:space="preserve">ſumanturq́ æquales arcus A G, D H, ita vt puncta G, H, ſecent ſegmenta <lb/>A G C, D H F, non bifariam. </s> <s xml:id="echoid-s1486" xml:space="preserve">Ex G, H, denique in circunferentias circulo-<lb/>rum A B C, D E F, cadant rectæ æquales G B, H E. </s> <s xml:id="echoid-s1487" xml:space="preserve">Dico circunferentias <lb/>A B, D E, eſſe æquales. </s> <s xml:id="echoid-s1488" xml:space="preserve">Demittantur ex G, H, rectæ G I, H K, ad plana cir-<lb/>culorum A B C, D E F, perpendiculares, quæ in communes ſectiones A C, <lb/> <anchor type="note" xlink:label="note-052-01a" xlink:href="note-052-01"/> D F, cadent in puncta I, K. </s> <s xml:id="echoid-s1489" xml:space="preserve">Sumptis quoque L, M, centris circulorũ A B C, <lb/> <anchor type="note" xlink:label="note-052-02a" xlink:href="note-052-02"/> D E F, ducantur rectæ L B, B I, A G; </s> <s xml:id="echoid-s1490" xml:space="preserve">M E, E K, D H: </s> <s xml:id="echoid-s1491" xml:space="preserve">cadantq́; </s> <s xml:id="echoid-s1492" xml:space="preserve">primum pun <lb/>cta I, K, in ſemidiametros A L, D M. </s> <s xml:id="echoid-s1493" xml:space="preserve">Quoniam igitur arcus A G C, D H F, <lb/>æquales ſunt, nec non & </s> <s xml:id="echoid-s1494" xml:space="preserve">arcus A G, D H; </s> <s xml:id="echoid-s1495" xml:space="preserve">æquales quoque erunt arcus, C G, <lb/>F H; </s> <s xml:id="echoid-s1496" xml:space="preserve">ac propterea anguli G A C, H D F, illis inſiſtentes æquales. </s> <s xml:id="echoid-s1497" xml:space="preserve">Sunt autem <lb/> <anchor type="note" xlink:label="note-052-03a" xlink:href="note-052-03"/> & </s> <s xml:id="echoid-s1498" xml:space="preserve">anguli A I G, D K H, æquales, quòd recti ſint ex defin. </s> <s xml:id="echoid-s1499" xml:space="preserve">3. </s> <s xml:id="echoid-s1500" xml:space="preserve">lib. </s> <s xml:id="echoid-s1501" xml:space="preserve">11. </s> <s xml:id="echoid-s1502" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1503" xml:space="preserve">Ita-<lb/>que duo triangula A I G, D K H, habent duos angulos G A I, A I G, duo-<lb/> <anchor type="figure" xlink:label="fig-052-01a" xlink:href="fig-052-01"/> bus angulis H D K, <lb/>D K H, æquales. </s> <s xml:id="echoid-s1504" xml:space="preserve">Ha-<lb/>bent autem & </s> <s xml:id="echoid-s1505" xml:space="preserve">latus <lb/>A G, lateri D H, ęqua <lb/> <anchor type="note" xlink:label="note-052-04a" xlink:href="note-052-04"/> le, (ob æqualitatẽ ar <lb/>cuum A G, D H.) <lb/></s> <s xml:id="echoid-s1506" xml:space="preserve">quod angulis æquali-<lb/>bus I, K, ſubtenditur. </s> <s xml:id="echoid-s1507" xml:space="preserve"><lb/>Igitur & </s> <s xml:id="echoid-s1508" xml:space="preserve">latus A I, la <lb/> <anchor type="note" xlink:label="note-052-05a" xlink:href="note-052-05"/> teri D K, & </s> <s xml:id="echoid-s1509" xml:space="preserve">latus G I, <lb/>lateri H K, æquale e-<lb/>rit. </s> <s xml:id="echoid-s1510" xml:space="preserve">Quoniam vero an <lb/>guli G I B, H K E, re <lb/>cti ſunt ex defin. </s> <s xml:id="echoid-s1511" xml:space="preserve">3. </s> <s xml:id="echoid-s1512" xml:space="preserve">lib. </s> <s xml:id="echoid-s1513" xml:space="preserve">11. </s> <s xml:id="echoid-s1514" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1515" xml:space="preserve">erunt quadrata ex G B, H E, quæ inter ſe æ-<lb/> <anchor type="note" xlink:label="note-052-06a" xlink:href="note-052-06"/> qualia ſunt, ob æqualitatem rectarum G B, H E, quadratis ex G I, I B, & </s> <s xml:id="echoid-s1516" xml:space="preserve">ex <lb/>H K, K E, æqualia, ac {pro}pterea quadrata ex G I, I B, quadratis ex H K, K E, <lb/>æqualia erunt. </s> <s xml:id="echoid-s1517" xml:space="preserve">Ablatis ergo quadratis æqualibus rectarum æqualiũ G I, H K, <lb/>remanebunt quadrata rectarũ I B, K E, æqualia; </s> <s xml:id="echoid-s1518" xml:space="preserve">& </s> <s xml:id="echoid-s1519" xml:space="preserve">idcirco & </s> <s xml:id="echoid-s1520" xml:space="preserve">rectæ I B, K E, <lb/>æquales. </s> <s xml:id="echoid-s1521" xml:space="preserve">Et quia A L, D M, ſemidiametri circulorum æqualiũ æquales ſunt; <lb/></s> <s xml:id="echoid-s1522" xml:space="preserve">oſtenſæ autem quoque ſunt æquales A I, D K, erunt & </s> <s xml:id="echoid-s1523" xml:space="preserve">reliquæ I L, K M, æ-<lb/>quales. </s> <s xml:id="echoid-s1524" xml:space="preserve">Quare latera I L, L B, lateribus K M, M E, æqualia erunt: </s> <s xml:id="echoid-s1525" xml:space="preserve">ſunt au <lb/>tem & </s> <s xml:id="echoid-s1526" xml:space="preserve">baſes I B, K E, oſtenſæ æquales. </s> <s xml:id="echoid-s1527" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s1528" xml:space="preserve">anguli L, M, ad centra æqua <lb/> <anchor type="note" xlink:label="note-052-07a" xlink:href="note-052-07"/> les erunt; </s> <s xml:id="echoid-s1529" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s1530" xml:space="preserve">arcus A B, D E, æquales erunt.</s> <s xml:id="echoid-s1531" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="1"> <note position="left" xlink:label="note-052-01" xlink:href="note-052-01a" xml:space="preserve">11. vndec.</note> <note position="left" xlink:label="note-052-02" xlink:href="note-052-02a" xml:space="preserve">33. vndec.</note> <note position="left" xlink:label="note-052-03" xlink:href="note-052-03a" xml:space="preserve">27. tertij.</note> <figure xlink:label="fig-052-01" xlink:href="fig-052-01a"> <image file="052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/052-01"/> </figure> <note position="left" xlink:label="note-052-04" xlink:href="note-052-04a" xml:space="preserve">29. tertij.</note> <note position="left" xlink:label="note-052-05" xlink:href="note-052-05a" xml:space="preserve">26. primi.</note> <note position="left" xlink:label="note-052-06" xlink:href="note-052-06a" xml:space="preserve">47. primi.</note> <note position="left" xlink:label="note-052-07" xlink:href="note-052-07a" xml:space="preserve">8. primi.</note> </div> <note position="left" xml:space="preserve">26. tertij.</note> <p> <s xml:id="echoid-s1532" xml:space="preserve">CADANT deinde puncta I, K, in ſemidiametros L A, M D, produ-<lb/>ctas ad A, & </s> <s xml:id="echoid-s1533" xml:space="preserve">D: </s> <s xml:id="echoid-s1534" xml:space="preserve">quod quidem contingere poteſt, quando ſegmenta A G C, <lb/>D H F, ſemicirculo ſunt maiora; </s> <s xml:id="echoid-s1535" xml:space="preserve">fiatq́; </s> <s xml:id="echoid-s1536" xml:space="preserve">eadem conſtructio, quæ prius. </s> <s xml:id="echoid-s1537" xml:space="preserve">Oſten <lb/>demus, vt prius, angulos G A C, H D F, eſſe æquales; </s> <s xml:id="echoid-s1538" xml:space="preserve">ac propterea cum tam <lb/> <anchor type="note" xlink:label="note-052-09a" xlink:href="note-052-09"/> G A C, G A I, quàm H D F, H D K, duobus ſint rectis æquales, erũt & </s> <s xml:id="echoid-s1539" xml:space="preserve">G A I, <lb/> <anchor type="note" xlink:label="note-052-10a" xlink:href="note-052-10"/> H D K, æquales. </s> <s xml:id="echoid-s1540" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s1541" xml:space="preserve">anguli I, K, æquales ſint, nempe recti, & </s> <s xml:id="echoid-s1542" xml:space="preserve">late <lb/>ra G A, H D, æqualia, ob æquales arcus A G, D H, erunt, vt prius, rectæ <lb/> <anchor type="note" xlink:label="note-052-11a" xlink:href="note-052-11"/> G I, I A, rectis H K, K D, æquales; </s> <s xml:id="echoid-s1543" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s1544" xml:space="preserve">totæ I L, K M, inter ſe <lb/> <anchor type="note" xlink:label="note-052-12a" xlink:href="note-052-12"/> æquales erunt. </s> <s xml:id="echoid-s1545" xml:space="preserve">Igitur, vt prius, oſtendemus rectam I B, rectæ K E, & </s> <s xml:id="echoid-s1546" xml:space="preserve">angu-<lb/>lum L, angulo M, æqualem eſſe: </s> <s xml:id="echoid-s1547" xml:space="preserve">ac denique arcum A B, arcui D E.</s> <s xml:id="echoid-s1548" xml:space="preserve"/> </p> <div xml:id="echoid-div165" type="float" level="2" n="2"> <note position="left" xlink:label="note-052-09" xlink:href="note-052-09a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-052-10" xlink:href="note-052-10a" xml:space="preserve">13. primi.</note> <note position="left" xlink:label="note-052-11" xlink:href="note-052-11a" xml:space="preserve">29. tertij.</note> <note position="left" xlink:label="note-052-12" xlink:href="note-052-12a" xml:space="preserve">26. primi.</note> </div> <note position="left" xml:space="preserve">47. primi.</note> <note position="left" xml:space="preserve">8. primi.</note> <p> <s xml:id="echoid-s1549" xml:space="preserve">CADANT tertio perpendiculares ex G, H, demiſſæ in plana circulo-<lb/> <anchor type="note" xlink:label="note-052-15a" xlink:href="note-052-15"/> <pb o="41" file="053" n="53" rhead=""/> <anchor type="figure" xlink:label="fig-053-01a" xlink:href="fig-053-01"/> rum A B C, D E F, in pun-<lb/>cta A, D: </s> <s xml:id="echoid-s1550" xml:space="preserve">quod etiam con-<lb/>tingere poteſt, quando ſe-<lb/>gmenta A G C, D H F, ſe-<lb/>micirculo ſunt maiora. </s> <s xml:id="echoid-s1551" xml:space="preserve">Du-<lb/>ctis igitur rectis A B, D E, <lb/>erũt anguli G A B, H D E, <lb/>recti, ex defin. </s> <s xml:id="echoid-s1552" xml:space="preserve">3. </s> <s xml:id="echoid-s1553" xml:space="preserve">lib. </s> <s xml:id="echoid-s1554" xml:space="preserve">11. </s> <s xml:id="echoid-s1555" xml:space="preserve">Eu-<lb/>clid. </s> <s xml:id="echoid-s1556" xml:space="preserve">Quare, vt prius, æqua <lb/> <anchor type="note" xlink:label="note-053-01a" xlink:href="note-053-01"/> lia erunt quadrata rectarũ <lb/>G A, A B, quadratis recta-<lb/>rum H D, D E: </s> <s xml:id="echoid-s1557" xml:space="preserve">Sunt autẽ <lb/>quadrata ex G A, H D, æ-<lb/>qualia, quòd & </s> <s xml:id="echoid-s1558" xml:space="preserve">rectæ G A, <lb/> <anchor type="note" xlink:label="note-053-02a" xlink:href="note-053-02"/> H D, æquales ſint, ob æqua <lb/>les arcus A G, D H. </s> <s xml:id="echoid-s1559" xml:space="preserve">Igitur <lb/>& </s> <s xml:id="echoid-s1560" xml:space="preserve">quadrata ex A B, D E, æ-<lb/>qualia erunt; </s> <s xml:id="echoid-s1561" xml:space="preserve">& </s> <s xml:id="echoid-s1562" xml:space="preserve">propterea <lb/>& </s> <s xml:id="echoid-s1563" xml:space="preserve">rectæ A B, D E, æquales. <lb/></s> <s xml:id="echoid-s1564" xml:space="preserve">Quare & </s> <s xml:id="echoid-s1565" xml:space="preserve">arcus A B, D E, <lb/>æquales erunt. </s> <s xml:id="echoid-s1566" xml:space="preserve">Quod eſt {pro}-<lb/>poſitum. </s> <s xml:id="echoid-s1567" xml:space="preserve">Itaque ſi in diametris circulorum æqualium æqualia circulorum <lb/>ſegmenta ad angulos rectos inſiſtant, &</s> <s xml:id="echoid-s1568" xml:space="preserve">c. </s> <s xml:id="echoid-s1569" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s1570" xml:space="preserve"/> </p> <div xml:id="echoid-div166" type="float" level="2" n="3"> <note position="left" xlink:label="note-052-15" xlink:href="note-052-15a" xml:space="preserve">26. tertij.</note> <figure xlink:label="fig-053-01" xlink:href="fig-053-01a"> <image file="053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/053-01"/> </figure> <note position="right" xlink:label="note-053-01" xlink:href="note-053-01a" xml:space="preserve">47. primi.</note> <note position="right" xlink:label="note-053-02" xlink:href="note-053-02a" xml:space="preserve">29. tertij.</note> </div> </div> <div xml:id="echoid-div168" type="section" level="1" n="88"> <head xml:id="echoid-head100" xml:space="preserve">THEOR. 12. PROPOS. 12.</head> <note position="right" xml:space="preserve">16.</note> <p> <s xml:id="echoid-s1571" xml:space="preserve">SI in diametris circulorum æqualium, æqua-<lb/>lia ſegmenta circulorum erigantur, & </s> <s xml:id="echoid-s1572" xml:space="preserve">ab ipſis ſe-<lb/>gmentis æquales circunferentiæ ad extremitates <lb/>ſegmentorum deſumantur minores dimidijs ip-<lb/>ſorum partibus, ab ipſis autem circulis æquales <lb/>circunferentiæ ſumantur ad eaſdem partes, quæ <lb/>ſunt ad extremitates diametrorum, rectæ lineæ <lb/>ductæ à punctis in circunferentijs ſegmentorum <lb/>ad puncta in circunferentijs circulorum, erunt <lb/>æquales.</s> <s xml:id="echoid-s1573" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1574" xml:space="preserve">REPETANTVR figuræ propoſition is præcedentis, cum eiſdẽ con-<lb/>ſtructionibus, ponanturq́; </s> <s xml:id="echoid-s1575" xml:space="preserve">arcus A B, D E, æquales. </s> <s xml:id="echoid-s1576" xml:space="preserve">Dico & </s> <s xml:id="echoid-s1577" xml:space="preserve">rectas G B, H E, <lb/> <anchor type="note" xlink:label="note-053-04a" xlink:href="note-053-04"/> æquales eſſe. </s> <s xml:id="echoid-s1578" xml:space="preserve">Quoniam enim, vt in præcedenti propoſ. </s> <s xml:id="echoid-s1579" xml:space="preserve">demonſtratum eſt, <lb/> <anchor type="note" xlink:label="note-053-05a" xlink:href="note-053-05"/> <anchor type="note" xlink:label="note-053-06a" xlink:href="note-053-06"/> <pb o="42" file="054" n="54" rhead=""/> rectæ A I, I G, rectis D K, K H, æquales ſunt; </s> <s xml:id="echoid-s1580" xml:space="preserve">& </s> <s xml:id="echoid-s1581" xml:space="preserve">reliquæ I L, K M, ex <lb/>ſemidiametris A L, D M, vt in prima figura, vbi puncta I, K, cadunt in ſe-<lb/>midiametros A L, D M, vel certe erunt & </s> <s xml:id="echoid-s1582" xml:space="preserve">totæ I L, K M, æquales, vt in ſe-<lb/>cunda figura, vbi puncta I, K, cadunt in ſemidiametros A L, D M, productas <lb/> <anchor type="figure" xlink:label="fig-054-01a" xlink:href="fig-054-01"/> ad A, & </s> <s xml:id="echoid-s1583" xml:space="preserve">D. </s> <s xml:id="echoid-s1584" xml:space="preserve">Quia igit̃ <lb/>I L, L B, rectis K M, <lb/>M E, æquales ſunt; </s> <s xml:id="echoid-s1585" xml:space="preserve">cõ <lb/>tinentq́ue angulos ad <lb/>L, M, æquales, ob æ-<lb/> <anchor type="note" xlink:label="note-054-01a" xlink:href="note-054-01"/> qualitatẽ arcuũ A B, <lb/>D E; </s> <s xml:id="echoid-s1586" xml:space="preserve">erunt & </s> <s xml:id="echoid-s1587" xml:space="preserve">baſes <lb/>I B, K E, æquales. <lb/></s> <s xml:id="echoid-s1588" xml:space="preserve"> <anchor type="note" xlink:label="note-054-02a" xlink:href="note-054-02"/> Quamobrem cum la-<lb/>tera G I, I B, lateri-<lb/>bus H K, K E, æqua-<lb/>lia ſint, contineantq́; <lb/></s> <s xml:id="echoid-s1589" xml:space="preserve">angulos G I B, H K E, <lb/>æquales, nimirum rectos, ex defin. </s> <s xml:id="echoid-s1590" xml:space="preserve">3. </s> <s xml:id="echoid-s1591" xml:space="preserve">lib. </s> <s xml:id="echoid-s1592" xml:space="preserve">11. </s> <s xml:id="echoid-s1593" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1594" xml:space="preserve">erunt & </s> <s xml:id="echoid-s1595" xml:space="preserve">baſes G B, H E, æ-<lb/> <anchor type="note" xlink:label="note-054-03a" xlink:href="note-054-03"/> quales. </s> <s xml:id="echoid-s1596" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s1597" xml:space="preserve">Facilius idem concludetur, ſi perpendiculares <lb/>ex G, H, in plana circulorum A B C, D E F, demiſſæ cadant in puncta A, D, <lb/>vt in tertia figura. </s> <s xml:id="echoid-s1598" xml:space="preserve">Nam quia rectæ G A, A B, rectis H D, D E, æquales ſunt, <lb/> <anchor type="note" xlink:label="note-054-04a" xlink:href="note-054-04"/> ob æquales arcus A G, D H, & </s> <s xml:id="echoid-s1599" xml:space="preserve">A B, D E, continentq́; </s> <s xml:id="echoid-s1600" xml:space="preserve">angulos æquales, vt-<lb/>pote rectos, ex defin. </s> <s xml:id="echoid-s1601" xml:space="preserve">3. </s> <s xml:id="echoid-s1602" xml:space="preserve">lib. </s> <s xml:id="echoid-s1603" xml:space="preserve">11. </s> <s xml:id="echoid-s1604" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1605" xml:space="preserve">erunt baſes G B, H E, æquales. </s> <s xml:id="echoid-s1606" xml:space="preserve">Si igitur <lb/> <anchor type="note" xlink:label="note-054-05a" xlink:href="note-054-05"/> in diametris circulorum æqualium, æqualia ſegmenta, &</s> <s xml:id="echoid-s1607" xml:space="preserve">c. </s> <s xml:id="echoid-s1608" xml:space="preserve">Quod erat oſten <lb/>dendum.</s> <s xml:id="echoid-s1609" xml:space="preserve"/> </p> <div xml:id="echoid-div168" type="float" level="2" n="1"> <note position="right" xlink:label="note-053-04" xlink:href="note-053-04a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-053-05" xlink:href="note-053-05a" xml:space="preserve">29. tertij.</note> <note position="right" xlink:label="note-053-06" xlink:href="note-053-06a" xml:space="preserve">26. primi.</note> <figure xlink:label="fig-054-01" xlink:href="fig-054-01a"> <image file="054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/054-01"/> </figure> <note position="left" xlink:label="note-054-01" xlink:href="note-054-01a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-054-02" xlink:href="note-054-02a" xml:space="preserve">4. primi.</note> <note position="left" xlink:label="note-054-03" xlink:href="note-054-03a" xml:space="preserve">4. primi.</note> <note position="left" xlink:label="note-054-04" xlink:href="note-054-04a" xml:space="preserve">29. tertij.</note> <note position="left" xlink:label="note-054-05" xlink:href="note-054-05a" xml:space="preserve">4. primi.</note> </div> </div> <div xml:id="echoid-div170" type="section" level="1" n="89"> <head xml:id="echoid-head101" xml:space="preserve">THEOREMA 13. PROPOS. 13.</head> <note position="left" xml:space="preserve">18.</note> <p> <s xml:id="echoid-s1610" xml:space="preserve">SI in ſphæra ſint paralleli circuli, & </s> <s xml:id="echoid-s1611" xml:space="preserve">deſcriban <lb/>tur maximi circuli, qui vnum quidem parallelo-<lb/>rum tangant, reliquos vero ſecent; </s> <s xml:id="echoid-s1612" xml:space="preserve">circunferentię <lb/>parallelorum interceptæ inter eos maximorum <lb/>circulorum ſemicirculos, qui non concurrunt, <lb/>ſimiles erunt; </s> <s xml:id="echoid-s1613" xml:space="preserve">maximorum vero circulorum cir-<lb/>cunferentiæ inter duos quoſcunque parallelos in-<lb/>terceptæ, erunt æquales.</s> <s xml:id="echoid-s1614" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1615" xml:space="preserve">SINT in ſphæra paralleli circuli A B, C D E, F G H, quieundem polũ <lb/> <anchor type="note" xlink:label="note-054-07a" xlink:href="note-054-07"/> habebunt, nempe I. </s> <s xml:id="echoid-s1616" xml:space="preserve">Circuli autem maximi A F K, B H K, tangant parallelũ <lb/>A B, in punctis A, B, & </s> <s xml:id="echoid-s1617" xml:space="preserve">reliquos ſecent in punctis F, C, L, M; </s> <s xml:id="echoid-s1618" xml:space="preserve">H, E, D, G: <lb/></s> <s xml:id="echoid-s1619" xml:space="preserve">ſeipſos aũt mutuo ſecent in K, N, vt ſint ſemicirculi K M N, N F K; </s> <s xml:id="echoid-s1620" xml:space="preserve">K G N, <lb/>N H K. </s> <s xml:id="echoid-s1621" xml:space="preserve">Maximi enim circuli ſe ſecant mutuo bifariam. </s> <s xml:id="echoid-s1622" xml:space="preserve">Sumatur quoque ar <lb/> <anchor type="note" xlink:label="note-054-08a" xlink:href="note-054-08"/> <pb o="43" file="055" n="55" rhead=""/> <anchor type="figure" xlink:label="fig-055-01a" xlink:href="fig-055-01"/> cus K O, arcui N A, & </s> <s xml:id="echoid-s1623" xml:space="preserve"><lb/>arcus K P, arcui N B, æ-<lb/>qualis, vt ſint quoque ſe <lb/>micirculi A M O, O F A; <lb/></s> <s xml:id="echoid-s1624" xml:space="preserve">B G P, P H B. </s> <s xml:id="echoid-s1625" xml:space="preserve">Eruntigi-<lb/>tur ſemicirculi A M O, <lb/>B H P, non coeuntes, cũ <lb/>ſe mutuo non ſecent. </s> <s xml:id="echoid-s1626" xml:space="preserve">Eo <lb/>dem modo nõ coeuntes <lb/>erunt ſemicirculi B G P, <lb/>A F O. </s> <s xml:id="echoid-s1627" xml:space="preserve">Dico arcus paral <lb/>lelorum A B, L E, M H, <lb/>interceptos inter ſemi-<lb/>circulos A M O, B H P, <lb/>non coeuntes ſimiles eſ-<lb/>ſe, necnon & </s> <s xml:id="echoid-s1628" xml:space="preserve">arcus A B, <lb/>C D, F G, interceptos in <lb/>ter ſemicirculos B G P, <lb/>A F O, non concurren-<lb/>tes ſimiles eſſe: </s> <s xml:id="echoid-s1629" xml:space="preserve">Arcus vero maximorum circulorum A C, A L, B D, B E, æ-<lb/>quales eſſe; </s> <s xml:id="echoid-s1630" xml:space="preserve">necnon & </s> <s xml:id="echoid-s1631" xml:space="preserve">arcus C F, L M, D G, E H: </s> <s xml:id="echoid-s1632" xml:space="preserve">quorum illi inter paralle-<lb/>los A B, C D E, hi vero inter parallelos C D E, F G H, interijciuntur: </s> <s xml:id="echoid-s1633" xml:space="preserve">Eo-<lb/>demq́; </s> <s xml:id="echoid-s1634" xml:space="preserve">pacto æquales eſſe arcus A F, A M, B G, B H, inter parallelos A B, <lb/> <anchor type="note" xlink:label="note-055-01a" xlink:href="note-055-01"/> F G H, interiectos. </s> <s xml:id="echoid-s1635" xml:space="preserve">Per polum enim I, & </s> <s xml:id="echoid-s1636" xml:space="preserve">puncta contactuum A, B, circuli <lb/>maximi deſcribantur Q A I R, S B I T, ſecantes parallelos in Q, S, V, X. <lb/></s> <s xml:id="echoid-s1637" xml:space="preserve">Tranſibunt hi circuli maximi per polos quoque circulorum A F K, B H K; </s> <s xml:id="echoid-s1638" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-055-02a" xlink:href="note-055-02"/> ac proinde bifariam ſecabunt ſegmenta C A L, D B E, C V L, D X E: </s> <s xml:id="echoid-s1639" xml:space="preserve">necnõ <lb/> <anchor type="note" xlink:label="note-055-03a" xlink:href="note-055-03"/> ſegmenta F A M, G B H, F Q M, G S H. </s> <s xml:id="echoid-s1640" xml:space="preserve">Præterea ijdem circuli ad angulos <lb/>rectos ſecabunt parallelos A B, C D E, F G H, & </s> <s xml:id="echoid-s1641" xml:space="preserve">maximos circulos A F K, <lb/> <anchor type="note" xlink:label="note-055-04a" xlink:href="note-055-04"/> B H K. </s> <s xml:id="echoid-s1642" xml:space="preserve">Quoniam igitur diametris circulorum æqualium A F K, B H K, inſi <lb/>ſtunt ad angulos rectos ſegmenta circulorum æqualia, nempe ſemicirculi in-<lb/>choati à punctis A, B, & </s> <s xml:id="echoid-s1643" xml:space="preserve">per I, tranſeũtes, donec iterũ ſecent circulos A F K, <lb/>B H K; </s> <s xml:id="echoid-s1644" xml:space="preserve">ſuntq́; </s> <s xml:id="echoid-s1645" xml:space="preserve">arcus æquales A I, B I, quòd ex defin. </s> <s xml:id="echoid-s1646" xml:space="preserve">poli recta I A, IB æqua <lb/> <anchor type="note" xlink:label="note-055-05a" xlink:href="note-055-05"/> les ſint; </s> <s xml:id="echoid-s1647" xml:space="preserve">qui quidem minores ſunt dimidijs ſemicirculorum partibus: </s> <s xml:id="echoid-s1648" xml:space="preserve">(cum <lb/>enim dimidij ſint arcuum A I R, B I T, quòd, ex defin. </s> <s xml:id="echoid-s1649" xml:space="preserve">poli, rectæ ex I, ad <lb/> <anchor type="note" xlink:label="note-055-06a" xlink:href="note-055-06"/> puncta A, B, R, T, atque adeo arcus quoque ſint æquales: </s> <s xml:id="echoid-s1650" xml:space="preserve">ſint autem arcus <lb/>A I R, B I T, ſemicirculo minores, quòd ſemicirculi tendant ex A, & </s> <s xml:id="echoid-s1651" xml:space="preserve">B, per <lb/>I, vſque ad circulos A F K, B H K; </s> <s xml:id="echoid-s1652" xml:space="preserve">erunt arcus A I, B I, minores dimidijs par <lb/>tibus illorum ſemicirculorum.) </s> <s xml:id="echoid-s1653" xml:space="preserve">ſunt quoque æquales rectæ I C, I E, ex po-<lb/>li defin. </s> <s xml:id="echoid-s1654" xml:space="preserve">erunt arcus A C, B E, æquales: </s> <s xml:id="echoid-s1655" xml:space="preserve">Eſt autem A C, ipſi A L, & </s> <s xml:id="echoid-s1656" xml:space="preserve">B E, ipſi <lb/> <anchor type="note" xlink:label="note-055-07a" xlink:href="note-055-07"/> B D, æqualis, propterea quòd arcus C A L, D B E, bifariam ſecantur, vt de-<lb/> <anchor type="note" xlink:label="note-055-08a" xlink:href="note-055-08"/> monſtratum eſt. </s> <s xml:id="echoid-s1657" xml:space="preserve">Quatuor ergo arcus A C, A L, B E, B D, æquales ſunt. </s> <s xml:id="echoid-s1658" xml:space="preserve">Eo-<lb/>dem modo oſtendemus, æquales eſſe quatuor arcus A F, A M, B H, B G; </s> <s xml:id="echoid-s1659" xml:space="preserve">ac <lb/>propterea & </s> <s xml:id="echoid-s1660" xml:space="preserve">reliquos C F, L M, E H, D G, qui quidem ſinguli inter binos <lb/>parallelos intercipiuntur. </s> <s xml:id="echoid-s1661" xml:space="preserve">Quodſecundo loco proponebatur demonſtrandũ.</s> <s xml:id="echoid-s1662" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="1"> <note position="left" xlink:label="note-054-07" xlink:href="note-054-07a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-054-08" xlink:href="note-054-08a" xml:space="preserve">11. 1. huius.</note> <figure xlink:label="fig-055-01" xlink:href="fig-055-01a"> <image file="055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/055-01"/> </figure> <note position="right" xlink:label="note-055-01" xlink:href="note-055-01a" xml:space="preserve">20 1. huius.</note> <note position="right" xlink:label="note-055-02" xlink:href="note-055-02a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-055-03" xlink:href="note-055-03a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-055-04" xlink:href="note-055-04a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-055-05" xlink:href="note-055-05a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-055-06" xlink:href="note-055-06a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-055-07" xlink:href="note-055-07a" xml:space="preserve">11. huius.</note> <note position="right" xlink:label="note-055-08" xlink:href="note-055-08a" xml:space="preserve">9. huius.</note> </div> <p> <s xml:id="echoid-s1663" xml:space="preserve">QVIA vero arcus toti C A L, D B E, æquales ſunt, quòd ipſorum dimi <lb/>dia æqualia ſint, vt demonſtratum eſt; </s> <s xml:id="echoid-s1664" xml:space="preserve">erunt & </s> <s xml:id="echoid-s1665" xml:space="preserve">rectæ ſubtenſæ C L, D E, æ-<lb/> <anchor type="note" xlink:label="note-055-09a" xlink:href="note-055-09"/> quales, quæ quidem arcubus quoque C V L, D X E, ſubtenduntur; </s> <s xml:id="echoid-s1666" xml:space="preserve">ac pro- <pb o="44" file="056" n="56" rhead=""/> <anchor type="figure" xlink:label="fig-056-01a" xlink:href="fig-056-01"/> pterea & </s> <s xml:id="echoid-s1667" xml:space="preserve">arcus parallelo <lb/> <anchor type="note" xlink:label="note-056-01a" xlink:href="note-056-01"/> rum C V L, D X E, æ-<lb/>quales erunt. </s> <s xml:id="echoid-s1668" xml:space="preserve">Cum ergo <lb/> <anchor type="note" xlink:label="note-056-02a" xlink:href="note-056-02"/> ſecẽtur bifariam in V, X, <lb/>vt dictum eſt, æquales e-<lb/>runt eorum medietates, <lb/>nimirum quatuor arcus <lb/>C V, V L, D X, X E. </s> <s xml:id="echoid-s1669" xml:space="preserve">Si <lb/>igitur arcubus ęqualibus <lb/>C V, D X, communis ar-<lb/>cus addatur V D, æqua-<lb/>les erunt arcus C D, V X: <lb/></s> <s xml:id="echoid-s1670" xml:space="preserve">Eſt autem arcus V X, ar <lb/> <anchor type="note" xlink:label="note-056-03a" xlink:href="note-056-03"/> cui A B, ſimilis. </s> <s xml:id="echoid-s1671" xml:space="preserve">Igitur <lb/>& </s> <s xml:id="echoid-s1672" xml:space="preserve">C D, eidem A B, ſimi <lb/>lis erit. </s> <s xml:id="echoid-s1673" xml:space="preserve">Non ſecus oſten <lb/>demus F G, eidem A B, <lb/>ſimilem eſſE; </s> <s xml:id="echoid-s1674" xml:space="preserve">nec non & </s> <s xml:id="echoid-s1675" xml:space="preserve"><lb/>arcus E L, H M, eidem <lb/>arcui A B, eſſe ſimiles. </s> <s xml:id="echoid-s1676" xml:space="preserve">Quod ſecundo loco proponebatur demonſtrandum. <lb/></s> <s xml:id="echoid-s1677" xml:space="preserve">Siergo in ſphæra ſint paralleli circuli, &</s> <s xml:id="echoid-s1678" xml:space="preserve">c. </s> <s xml:id="echoid-s1679" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s1680" xml:space="preserve"/> </p> <div xml:id="echoid-div171" type="float" level="2" n="2"> <note position="right" xlink:label="note-055-09" xlink:href="note-055-09a" xml:space="preserve">29. tertij.</note> <figure xlink:label="fig-056-01" xlink:href="fig-056-01a"> <image file="056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/056-01"/> </figure> <note position="left" xlink:label="note-056-01" xlink:href="note-056-01a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-056-02" xlink:href="note-056-02a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-056-03" xlink:href="note-056-03a" xml:space="preserve">10. huius.</note> </div> </div> <div xml:id="echoid-div173" type="section" level="1" n="90"> <head xml:id="echoid-head102" xml:space="preserve">PROBL. 1. PROP. 14.</head> <note position="left" xml:space="preserve">17.</note> <p> <s xml:id="echoid-s1681" xml:space="preserve">CIRCVLO in ſphæra dato, qui minor ſit <lb/>quàm circulus maximus, datoq́ aliquo puncto <lb/>in eius circunferentia, per illud punctum deſcri-<lb/>bere circulum maximum, qui tangat datum cir-<lb/>culum.</s> <s xml:id="echoid-s1682" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1683" xml:space="preserve">IN ſphæra datus circulus ſit non maximus A B, cuius polus C, opor-<lb/> <anchor type="figure" xlink:label="fig-056-02a" xlink:href="fig-056-02"/> teatq́; </s> <s xml:id="echoid-s1684" xml:space="preserve">per A, punctum in eius circũferentia <lb/>datum, deſcribere maximum circulum, qui <lb/>circulum A B, tangat. </s> <s xml:id="echoid-s1685" xml:space="preserve">Per polum C, & </s> <s xml:id="echoid-s1686" xml:space="preserve">pun <lb/> <anchor type="note" xlink:label="note-056-05a" xlink:href="note-056-05"/> ctum A, deſcribatur circulus maximus <lb/>C A D E B, in quo ſumatur quadrans A D, <lb/>& </s> <s xml:id="echoid-s1687" xml:space="preserve">polo D, interuallo D A, circulus deſcri-<lb/>batur A E, qui maximus erit, quòd recta <lb/> <anchor type="note" xlink:label="note-056-06a" xlink:href="note-056-06"/> ſubtenſa D A, latus ſit quadrati in maxi-<lb/>mo circulo deſcripti. </s> <s xml:id="echoid-s1688" xml:space="preserve">Dico circulum maxi-<lb/>mum A E, tangere circulum A B, in A. </s> <s xml:id="echoid-s1689" xml:space="preserve">Quo-<lb/>niam enim duo circuli A B, A E, eundem <lb/>circulum C A D, per eorum polos tranſeũ <pb o="45" file="057" n="57" rhead=""/> tem ſecant in eodem puncto A, ipſi ſe mutuo tangent in puncto A. </s> <s xml:id="echoid-s1690" xml:space="preserve">Circu-<lb/> <anchor type="note" xlink:label="note-057-01a" xlink:href="note-057-01"/> lo igitur in ſphæra dato, &</s> <s xml:id="echoid-s1691" xml:space="preserve">c. </s> <s xml:id="echoid-s1692" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s1693" xml:space="preserve"/> </p> <div xml:id="echoid-div173" type="float" level="2" n="1"> <figure xlink:label="fig-056-02" xlink:href="fig-056-02a"> <image file="056-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/056-02"/> </figure> <note position="left" xlink:label="note-056-05" xlink:href="note-056-05a" xml:space="preserve">20. i. huius.</note> <note position="left" xlink:label="note-056-06" xlink:href="note-056-06a" xml:space="preserve">17. 1. huius.</note> <note position="right" xlink:label="note-057-01" xlink:href="note-057-01a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div175" type="section" level="1" n="91"> <head xml:id="echoid-head103" xml:space="preserve">PROBL. 2. PROPOS. 15.</head> <note position="right" xml:space="preserve">19.</note> <p> <s xml:id="echoid-s1694" xml:space="preserve">CIRCVLO in ſphæra dato, qui minor ſit <lb/>maximo circulo, & </s> <s xml:id="echoid-s1695" xml:space="preserve">puncto aliquo dato in ſphæ-<lb/>ræ ſuperficie, quod ſit inter datum circulum, & </s> <s xml:id="echoid-s1696" xml:space="preserve"><lb/>alium eidem æqualem, & </s> <s xml:id="echoid-s1697" xml:space="preserve">parallelum, per pun-<lb/>ctum illud datum deſcribere maximum circu-<lb/>lum, qui tangat datum circulum maximo minorẽ.</s> <s xml:id="echoid-s1698" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1699" xml:space="preserve">SIT in ſphæra datus circulus non maximus A B, cui æqualis ſit & </s> <s xml:id="echoid-s1700" xml:space="preserve">paral <lb/>lelus C D, datumq́; </s> <s xml:id="echoid-s1701" xml:space="preserve">punctum ſit G, inter duos circulos A B, C D; </s> <s xml:id="echoid-s1702" xml:space="preserve">oporteatq́; <lb/></s> <s xml:id="echoid-s1703" xml:space="preserve">per G, circulum maximum deſcribere, qui tangat circulum A B. </s> <s xml:id="echoid-s1704" xml:space="preserve">Sint E, F, <lb/>poli parallelorum A B, C D, (habent enim paralleli eoſdem polos.) </s> <s xml:id="echoid-s1705" xml:space="preserve">& </s> <s xml:id="echoid-s1706" xml:space="preserve">per <lb/> <anchor type="note" xlink:label="note-057-03a" xlink:href="note-057-03"/> E, G, circulus maximus deſcribatur E A C, qui per reliquum polum F, tran <lb/> <anchor type="note" xlink:label="note-057-04a" xlink:href="note-057-04"/> ſibit, ex coroll. </s> <s xml:id="echoid-s1707" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s1708" xml:space="preserve">10. </s> <s xml:id="echoid-s1709" xml:space="preserve">lib. </s> <s xml:id="echoid-s1710" xml:space="preserve">1. </s> <s xml:id="echoid-s1711" xml:space="preserve">huius. </s> <s xml:id="echoid-s1712" xml:space="preserve">In hoc accipiatur quadrans <lb/> <anchor type="figure" xlink:label="fig-057-01a" xlink:href="fig-057-01"/> B H; </s> <s xml:id="echoid-s1713" xml:space="preserve">ca-<lb/>detq́; </s> <s xml:id="echoid-s1714" xml:space="preserve">pũ <lb/>ctum H, <lb/>vel ſupra <lb/>D, vel in <lb/>D, vel in <lb/>fra D: <lb/></s> <s xml:id="echoid-s1715" xml:space="preserve">Quodcũ <lb/>que autẽ <lb/>horũ cõ-<lb/>tingat, <lb/>ita rẽ exe <lb/>quemur. </s> <s xml:id="echoid-s1716" xml:space="preserve"><lb/>Ex polo E, ad interuallum E H, vel ex polo F, ad interuallum F H, cir-<lb/>culus deſcribatur H I, qui ipſis A B, C D, parallelus erit, exiſtetq́; </s> <s xml:id="echoid-s1717" xml:space="preserve">vel ſu-<lb/> <anchor type="note" xlink:label="note-057-05a" xlink:href="note-057-05"/> pra C D, vel idem erit qui C D, vel infra C D, ſitus erit, prout punctum <lb/>H, ſupra D, vel in D, vel infra D, poſitum fuerit. </s> <s xml:id="echoid-s1718" xml:space="preserve">Sumatur rurſum qua-<lb/>drans G K, eritq́; </s> <s xml:id="echoid-s1719" xml:space="preserve">punctum K, vltra H, cum G H, quadrante minor ſit. </s> <s xml:id="echoid-s1720" xml:space="preserve">Po-<lb/>lo deinde G, interuallo autem G K, circulus deſcribatur K L, qui maxi-<lb/>mus erit, quòd recta ſubtendens quadrantẽ G K, æqualis ſit lateri quadrati <lb/> <anchor type="note" xlink:label="note-057-06a" xlink:href="note-057-06"/> in maximo circulo deſcripti. </s> <s xml:id="echoid-s1721" xml:space="preserve">Secet autem K L, circulum H I, in L, & </s> <s xml:id="echoid-s1722" xml:space="preserve">per <lb/>L, F, circulus maximus deſcribatur F L, qui per reliquum polum E, tranſi-<lb/> <anchor type="note" xlink:label="note-057-07a" xlink:href="note-057-07"/> bit, ex coroll. </s> <s xml:id="echoid-s1723" xml:space="preserve">ſcholij propoſ.</s> <s xml:id="echoid-s1724" xml:space="preserve">10.</s> <s xml:id="echoid-s1725" xml:space="preserve">lib. </s> <s xml:id="echoid-s1726" xml:space="preserve">1. </s> <s xml:id="echoid-s1727" xml:space="preserve">huius. </s> <s xml:id="echoid-s1728" xml:space="preserve">Secet autem hic circulus F L E, <lb/>circulum A B, in M. </s> <s xml:id="echoid-s1729" xml:space="preserve">Eruntq́; </s> <s xml:id="echoid-s1730" xml:space="preserve">arcus M L, B H, circulorum maximorum per <pb o="46" file="058" n="58" rhead=""/> E, F, polos parallelorum tranſeuntium, intercepti inter parallelos A B, H I, <lb/> <anchor type="note" xlink:label="note-058-01a" xlink:href="note-058-01"/> æquales, ac propterea exiſtente B H, quadrante per conſtructionem, erit & </s> <s xml:id="echoid-s1731" xml:space="preserve"><lb/>L M, quadrans. </s> <s xml:id="echoid-s1732" xml:space="preserve">Polo igitur L, interuallo autem L M, circulus deſcribatur <lb/>M N, qui maximus erit, quòd recta ſubtendens quadrantem L M, æqualis <lb/> <anchor type="note" xlink:label="note-058-02a" xlink:href="note-058-02"/> ſit lateri quadrati in maximo circulo deſcripti. </s> <s xml:id="echoid-s1733" xml:space="preserve">Quoniam vero maximus cir <lb/>culus K L, tranſit per L, polum maximi circuli N M, tranſibit viciſsim ma-<lb/>ximus circulus N M, per G, polum circuli K L: </s> <s xml:id="echoid-s1734" xml:space="preserve">atque ita tranſit maximus <lb/> <anchor type="note" xlink:label="note-058-03a" xlink:href="note-058-03"/> circulus N M, per datum punctum G. </s> <s xml:id="echoid-s1735" xml:space="preserve">Dico iam eundem tangere circulum <lb/>A B, in M. </s> <s xml:id="echoid-s1736" xml:space="preserve">Quoniã enim circuli A B, G N, in eodem puncto M, ſecãt maximũ <lb/>circulum E F, in quo polos habent, ipſi ſe mutuo tangent in M. </s> <s xml:id="echoid-s1737" xml:space="preserve">Deſcriptus <lb/> <anchor type="note" xlink:label="note-058-04a" xlink:href="note-058-04"/> eſt ergo per G, circulus maximus G N, tangens circulum A B, in M. </s> <s xml:id="echoid-s1738" xml:space="preserve">Quare <lb/>circulo in ſphæra dato, &</s> <s xml:id="echoid-s1739" xml:space="preserve">c. </s> <s xml:id="echoid-s1740" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s1741" xml:space="preserve"/> </p> <div xml:id="echoid-div175" type="float" level="2" n="1"> <note position="right" xlink:label="note-057-03" xlink:href="note-057-03a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-057-04" xlink:href="note-057-04a" xml:space="preserve">20. 1. huius.</note> <figure xlink:label="fig-057-01" xlink:href="fig-057-01a"> <image file="057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/057-01"/> </figure> <note position="right" xlink:label="note-057-05" xlink:href="note-057-05a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-057-06" xlink:href="note-057-06a" xml:space="preserve">17. 1. huius.</note> <note position="right" xlink:label="note-057-07" xlink:href="note-057-07a" xml:space="preserve">20. 1. huius</note> <note position="left" xlink:label="note-058-01" xlink:href="note-058-01a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-058-02" xlink:href="note-058-02a" xml:space="preserve">17. i. huius.</note> <note position="left" xlink:label="note-058-03" xlink:href="note-058-03a" xml:space="preserve">Scho. 15. 1. <lb/>huius.</note> <note position="left" xlink:label="note-058-04" xlink:href="note-058-04a" xml:space="preserve">8. huius.</note> </div> </div> <div xml:id="echoid-div177" type="section" level="1" n="92"> <head xml:id="echoid-head104" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1742" xml:space="preserve">_QVOD_ ſi punctum G, datum ſit præciſe in medio arcus _B D,_ erit quadrans _G F._ <lb/></s> <s xml:id="echoid-s1743" xml:space="preserve">Polo igitur _G_, interualloq́; </s> <s xml:id="echoid-s1744" xml:space="preserve">_G F,_ circulus deſcriptus _F E,_ ſecabit _H I,_ in _L,_ puncto, <lb/>quod rurſum erit polus circuli tangentis, vt prius. </s> <s xml:id="echoid-s1745" xml:space="preserve">Si vero _G,_ punctum datum ſit <lb/>idem, quod _D,_ erit polus circuli tangentis in medio arcus _D C A,_ cum hic arcus ſe-<lb/>micirculus ſit. </s> <s xml:id="echoid-s1746" xml:space="preserve">Circulus aut em ex illo polo deſcriptus tanget _A B,_ in _A,_ & </s> <s xml:id="echoid-s1747" xml:space="preserve">_C D,_ <lb/> <anchor type="figure" xlink:label="fig-058-01a" xlink:href="fig-058-01"/> <anchor type="note" xlink:label="note-058-05a" xlink:href="note-058-05"/> in _D_, vt <lb/>patet: </s> <s xml:id="echoid-s1748" xml:space="preserve">quo <lb/>niam vi-<lb/>delicet cir <lb/>culus hic <lb/>maximus, <lb/>& </s> <s xml:id="echoid-s1749" xml:space="preserve">paral. <lb/></s> <s xml:id="echoid-s1750" xml:space="preserve">leli _A B,_ <lb/>_C D,_ ſe-<lb/>cãt in pũ-<lb/>ctis _A,_ _D,_ <lb/>circunfe-<lb/>rentiã ma <lb/>ximi circuli _A C D B,_ in quo polos habent.</s> <s xml:id="echoid-s1751" xml:space="preserve"/> </p> <div xml:id="echoid-div177" type="float" level="2" n="1"> <figure xlink:label="fig-058-01" xlink:href="fig-058-01a"> <image file="058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/058-01"/> </figure> <note position="left" xlink:label="note-058-05" xlink:href="note-058-05a" xml:space="preserve">1. huius.</note> </div> <p style="it"> <s xml:id="echoid-s1752" xml:space="preserve">_QVONIAM_ vero ſicut _L,_ polus eſt oſtenſus circuli maximi _G N,_ tangentis, <lb/>circulum _A B,_ ita quoque oſtendi poteſt, aliud punctũ, in quo maximus circulus _K L,_ <lb/>circulum _H I,_ ex altera parte ſecat, polum eſſe alterius cuiuſdam circuli maximi, <lb/>qui per _G,_ tranſeat, tangatq́; </s> <s xml:id="echoid-s1753" xml:space="preserve">circulum _A B,_ in alio puncto; </s> <s xml:id="echoid-s1754" xml:space="preserve">perſpicuum eſt per pũ-<lb/>ctum in ſphæra datum inter duos circulos æquales, & </s> <s xml:id="echoid-s1755" xml:space="preserve">parallelos deſcribi poſſe duos <lb/>circulos maximos, qui circulum _A B,_ tangant in duobus punctis.</s> <s xml:id="echoid-s1756" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div179" type="section" level="1" n="93"> <head xml:id="echoid-head105" xml:space="preserve">THEOR. 14. PROPOS. 16.</head> <note position="left" xml:space="preserve">20.</note> <p> <s xml:id="echoid-s1757" xml:space="preserve">MAXIMI circuli, qui ſimiles circũferentias <pb o="47" file="059" n="59" rhead=""/> parallelorum circulorum in ſphæra auferunt, aut <lb/>per parallelorum polos tranſeunt, aut eundem v-<lb/>num parallelum tangunt.</s> <s xml:id="echoid-s1758" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1759" xml:space="preserve">IN ſphæra maximi circuli A B C, D B E, auferant ex paralleli: </s> <s xml:id="echoid-s1760" xml:space="preserve">A D C, <lb/>F G, circunferentias ſimiles A D, F G. </s> <s xml:id="echoid-s1761" xml:space="preserve">Dico maximos circulos A B C, D B E, <lb/> <anchor type="figure" xlink:label="fig-059-01a" xlink:href="fig-059-01"/> aut tranſire per polos parallelorum A D C, <lb/>F G, aut vnum eundem parallelum tangere. <lb/></s> <s xml:id="echoid-s1762" xml:space="preserve">Aut enim alter illorum, nempe A B C, tran-<lb/>ſit per polos parallelorum, atque ita oſten-<lb/>demus, alterum per eoſdem tranſire, aut nõ <lb/>tranſit quidẽ per polos parallelorũ, ſed alte <lb/>rũ tamen illorũ tangit, atq; </s> <s xml:id="echoid-s1763" xml:space="preserve">ita demonſtra-<lb/>bimus, alterum cundem tangere; </s> <s xml:id="echoid-s1764" xml:space="preserve">aut deniq; </s> <s xml:id="echoid-s1765" xml:space="preserve"><lb/>neque per polos parallelorum incedit, neq; </s> <s xml:id="echoid-s1766" xml:space="preserve"><lb/>alterum illorum tangit: </s> <s xml:id="echoid-s1767" xml:space="preserve">quo poſito conclu <lb/>demus circulos maximos datos aliquẽ aliũ <lb/>parallelum tangere datis parallelis minorẽ. </s> <s xml:id="echoid-s1768" xml:space="preserve"><lb/>Tranſeat enim primum A B C, per polos pa-<lb/>rallelorum. </s> <s xml:id="echoid-s1769" xml:space="preserve">Dico & </s> <s xml:id="echoid-s1770" xml:space="preserve">D B E, per eoſdem trã <lb/>ſire, hoc eſt, pũctum B, in quo ſe ſecant maximi circuli A B C, D B E, polum <lb/>eſſe parallelorum A D C, F G. </s> <s xml:id="echoid-s1771" xml:space="preserve">Si namque B, non eſt eorum polus, ſit H, po-<lb/>lus ipſorum. </s> <s xml:id="echoid-s1772" xml:space="preserve">Et quia circulus A B C, ponitur tranſire per eorum polos, erit <lb/>H, in circunferentia A B C. </s> <s xml:id="echoid-s1773" xml:space="preserve">Per H, G, deſcribatur circulus maximus H G, <lb/> <anchor type="note" xlink:label="note-059-01a" xlink:href="note-059-01"/> <anchor type="note" xlink:label="note-059-02a" xlink:href="note-059-02"/> ſecans A D C. </s> <s xml:id="echoid-s1774" xml:space="preserve">in I. </s> <s xml:id="echoid-s1775" xml:space="preserve">Eruntq́; </s> <s xml:id="echoid-s1776" xml:space="preserve">arcus A I, F G, ſimiles, cum intercipiantur in-<lb/>ter maximos circulos A H, H I, per polum H, deſcriptos: </s> <s xml:id="echoid-s1777" xml:space="preserve">Ponitur autem <lb/>& </s> <s xml:id="echoid-s1778" xml:space="preserve">arcus A D, eidem arcui F G, ſimilis. </s> <s xml:id="echoid-s1779" xml:space="preserve">Similes ergo ſunt arcus A I, A D; <lb/></s> <s xml:id="echoid-s1780" xml:space="preserve">atque adeo cum ſint eiuſdem circuli, inter ſe æquales erunt, totum & </s> <s xml:id="echoid-s1781" xml:space="preserve">pars. </s> <s xml:id="echoid-s1782" xml:space="preserve"><lb/>Quod eſt abſurdum. </s> <s xml:id="echoid-s1783" xml:space="preserve">Non ergo aliud punctum, præter B, polus erit parallelo-<lb/>rum, ſi alter circulorum A B C, D B E, nempe A B C, per illorum polos du <lb/>citur, Quare vterque circulus maximus A B C, D B E, per polum B, paralle <lb/>lorum tranſit, ſi vnusipſorum tranſit.</s> <s xml:id="echoid-s1784" xml:space="preserve"/> </p> <div xml:id="echoid-div179" type="float" level="2" n="1"> <figure xlink:label="fig-059-01" xlink:href="fig-059-01a"> <image file="059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/059-01"/> </figure> <note position="right" xlink:label="note-059-01" xlink:href="note-059-01a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-059-02" xlink:href="note-059-02a" xml:space="preserve">10. huius.</note> </div> <figure> <image file="059-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/059-02"/> </figure> <p> <s xml:id="echoid-s1785" xml:space="preserve">SED iam duo maximi circuli A B C, D E F, <lb/>auferant rurſum ex parallelis A D C, B E, <lb/>circunferentias ſimiles A D, B E, & </s> <s xml:id="echoid-s1786" xml:space="preserve">neuter <lb/>illorum tranſeat per parallelorum polos, <lb/>ſed alter, nempe A B C, vnum eorum, puta <lb/>B E, tangat in B. </s> <s xml:id="echoid-s1787" xml:space="preserve">Dico & </s> <s xml:id="echoid-s1788" xml:space="preserve">circulum D E F, <lb/>eundem B E, tangere in E. </s> <s xml:id="echoid-s1789" xml:space="preserve">Si enim non tan <lb/>git, ſed ſecat, deſcribatur per E, punctũ in <lb/>parallelo B E, datũ maximus circulus G E H, <lb/> <anchor type="note" xlink:label="note-059-03a" xlink:href="note-059-03"/> tangens parallelum B E, in E; </s> <s xml:id="echoid-s1790" xml:space="preserve">eruntq́; </s> <s xml:id="echoid-s1791" xml:space="preserve">ſemi <lb/>circuli, quorum alter ex E, per G, ducitur, <lb/>alter vero ex B, per A, tranſit, non coeun-<lb/>tes, vt conſtat ex figura propoſ. </s> <s xml:id="echoid-s1792" xml:space="preserve">13. </s> <s xml:id="echoid-s1793" xml:space="preserve">huius libri, & </s> <s xml:id="echoid-s1794" xml:space="preserve">ex demonſtratis ibidem. <lb/></s> <s xml:id="echoid-s1795" xml:space="preserve"> <anchor type="note" xlink:label="note-059-04a" xlink:href="note-059-04"/> <pb o="48" file="060" n="60" rhead=""/> Igitur arcus B E, A G, ſimiles erunt: </s> <s xml:id="echoid-s1796" xml:space="preserve">Ponuntur autem & </s> <s xml:id="echoid-s1797" xml:space="preserve">ſimiles B E, A D. <lb/></s> <s xml:id="echoid-s1798" xml:space="preserve">Similes ergo ſunt inter ſe A G, A D; </s> <s xml:id="echoid-s1799" xml:space="preserve">ac proinde, cum ſint eiuſdem circuli, <lb/>inter ſe æquales erunt, totum, & </s> <s xml:id="echoid-s1800" xml:space="preserve">pars. </s> <s xml:id="echoid-s1801" xml:space="preserve">Quod eſt abſurdũ. </s> <s xml:id="echoid-s1802" xml:space="preserve">Nullus ergo alius <lb/>circulus maximus per E, ductus præter D E F, parallelum B E, tangit in E, <lb/>ſi A B C, eundem in B, tangit. </s> <s xml:id="echoid-s1803" xml:space="preserve">Quare ſi A B C, tangit B E, tanget & </s> <s xml:id="echoid-s1804" xml:space="preserve">D E F, <lb/>eundem B E.</s> <s xml:id="echoid-s1805" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="2"> <note position="right" xlink:label="note-059-03" xlink:href="note-059-03a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-059-04" xlink:href="note-059-04a" xml:space="preserve">13. huius.</note> </div> <p> <s xml:id="echoid-s1806" xml:space="preserve">POSTREMO maximi circuli A B C, D E F, auferant ex parallelis <lb/>A D C, G H, circunferentias ſimiles A D, G H; </s> <s xml:id="echoid-s1807" xml:space="preserve">& </s> <s xml:id="echoid-s1808" xml:space="preserve">neuter illorum per paral-<lb/>lelorum polos ducatur, aut alterum eorum tangat. </s> <s xml:id="echoid-s1809" xml:space="preserve">Dico circulos maximos <lb/> <anchor type="figure" xlink:label="fig-060-01a" xlink:href="fig-060-01"/> A B C, D E F, tangere alium quendam pa-<lb/>rallelum ipſis A D C, B E, minorẽ. </s> <s xml:id="echoid-s1810" xml:space="preserve">Quo-<lb/>niam enim circulus maximus A B C, neque <lb/>tranſit per polos parallelorum, neque alte <lb/>rum ipſorum tangit, erit circulus maximus <lb/>A B C, ad vtrumque parallelorum A D C, <lb/>G H, obliquus. </s> <s xml:id="echoid-s1811" xml:space="preserve">Sienim rectus eſſet, tranſi-<lb/>ret per ipſorum polos, quod non ponitur. <lb/></s> <s xml:id="echoid-s1812" xml:space="preserve"> <anchor type="note" xlink:label="note-060-01a" xlink:href="note-060-01"/> Tanget igitur A B C, duos circulos æqua-<lb/>les inter ſe, & </s> <s xml:id="echoid-s1813" xml:space="preserve">parallelos vtrique A D C, <lb/> <anchor type="note" xlink:label="note-060-02a" xlink:href="note-060-02"/> G H. </s> <s xml:id="echoid-s1814" xml:space="preserve">Tangat ergo parallelum B E, qui mi <lb/>nor erit vtroque A D C, G H; </s> <s xml:id="echoid-s1815" xml:space="preserve">(cum A B C, <lb/>ipſos ſecet) atque adeo & </s> <s xml:id="echoid-s1816" xml:space="preserve">alter ſibi æqua-<lb/>lis, & </s> <s xml:id="echoid-s1817" xml:space="preserve">parallelus minor erit vtroque A D C, G H: </s> <s xml:id="echoid-s1818" xml:space="preserve">ac {pro}inde paralleli A D C, <lb/>G H, poſiti erũt inter illos duos, quos circulus A B C, tangit. </s> <s xml:id="echoid-s1819" xml:space="preserve">Dico & </s> <s xml:id="echoid-s1820" xml:space="preserve">D E F, <lb/>eundem B E, tangere. </s> <s xml:id="echoid-s1821" xml:space="preserve">Si enim non tangit, deſcribatur per punctum H, quod <lb/>eſt inter circulum B E, & </s> <s xml:id="echoid-s1822" xml:space="preserve">ſibi æqualem, ac parallelum, ut oſtendimus, cir-<lb/> <anchor type="note" xlink:label="note-060-03a" xlink:href="note-060-03"/> culus maximus K H, tangens B E, in I; </s> <s xml:id="echoid-s1823" xml:space="preserve">eruntq́; </s> <s xml:id="echoid-s1824" xml:space="preserve">ſemicirculi, quorum alter ex <lb/>I, per H, alter vero ex B, per G, tranſit, non coeuntes, vt conſtat ex figura <lb/> <anchor type="figure" xlink:label="fig-060-02a" xlink:href="fig-060-02"/> propoſi. </s> <s xml:id="echoid-s1825" xml:space="preserve">13. </s> <s xml:id="echoid-s1826" xml:space="preserve">huius libri, & </s> <s xml:id="echoid-s1827" xml:space="preserve">ex demonſtra-<lb/> <anchor type="note" xlink:label="note-060-04a" xlink:href="note-060-04"/> tis ibidem. </s> <s xml:id="echoid-s1828" xml:space="preserve">Igitur arcus A K, G H, ſimi-<lb/>les erunt: </s> <s xml:id="echoid-s1829" xml:space="preserve">Ponuntur autem & </s> <s xml:id="echoid-s1830" xml:space="preserve">A D, G H, <lb/>ſimiles. </s> <s xml:id="echoid-s1831" xml:space="preserve">Similes igitur ſunt A K, A D; </s> <s xml:id="echoid-s1832" xml:space="preserve">atque <lb/>adeo, cum ſint eiuſdem circuli, inter ſe æ-<lb/>quales erunt, totum & </s> <s xml:id="echoid-s1833" xml:space="preserve">pars. </s> <s xml:id="echoid-s1834" xml:space="preserve">Quod eſt ab-<lb/>ſurdum. </s> <s xml:id="echoid-s1835" xml:space="preserve">Nullus ergo alius circulus maxi-<lb/>mus per H, deſcriptus, præter D E F, paral-<lb/>lelum B E, tangit, ſi A B C, eundem tangit <lb/>in B. </s> <s xml:id="echoid-s1836" xml:space="preserve">Quare ſi A B C, tangit circulum B E, <lb/>tanget & </s> <s xml:id="echoid-s1837" xml:space="preserve">D E F, eundem B E. </s> <s xml:id="echoid-s1838" xml:space="preserve">Quapropter <lb/>maximi circuli, qui ſimiles circunferentias, <lb/>&</s> <s xml:id="echoid-s1839" xml:space="preserve">c. </s> <s xml:id="echoid-s1840" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s1841" xml:space="preserve"/> </p> <div xml:id="echoid-div181" type="float" level="2" n="3"> <figure xlink:label="fig-060-01" xlink:href="fig-060-01a"> <image file="060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/060-01"/> </figure> <note position="left" xlink:label="note-060-01" xlink:href="note-060-01a" xml:space="preserve">13 .1. huius.</note> <note position="left" xlink:label="note-060-02" xlink:href="note-060-02a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-060-03" xlink:href="note-060-03a" xml:space="preserve">15. huius.</note> <figure xlink:label="fig-060-02" xlink:href="fig-060-02a"> <image file="060-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/060-02"/> </figure> <note position="left" xlink:label="note-060-04" xlink:href="note-060-04a" xml:space="preserve">13. huius.</note> </div> </div> <div xml:id="echoid-div183" type="section" level="1" n="94"> <head xml:id="echoid-head106" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s1842" xml:space="preserve">_MANIFESTVM_ autem eſt, circulos maximos _A B C, D E F,_ ita tangere <lb/>eundem parallelum _B E,_ vt ſemicirculi eorum à contactibus per arcus ſimiles proce-<lb/>dentes non coeant. </s> <s xml:id="echoid-s1843" xml:space="preserve">Alias non eſſent arcus ablati ſimiles, vt conſtat ex propoſ.</s> <s xml:id="echoid-s1844" xml:space="preserve">13. <lb/></s> <s xml:id="echoid-s1845" xml:space="preserve">buius libri.</s> <s xml:id="echoid-s1846" xml:space="preserve"/> </p> <pb o="49" file="061" n="61" rhead=""/> </div> <div xml:id="echoid-div184" type="section" level="1" n="95"> <head xml:id="echoid-head107" xml:space="preserve">THEOREMA 15. PROPOS. 17.</head> <note position="right" xml:space="preserve">21.</note> <p> <s xml:id="echoid-s1847" xml:space="preserve">IN ſphæra paralleli circuli, inter quos & </s> <s xml:id="echoid-s1848" xml:space="preserve">ma-<lb/>ximum parallelorum æquales circunferentiæ ma-<lb/>ximorum circulorum intercipiuntur, ſuntinter ſe <lb/>æquales: </s> <s xml:id="echoid-s1849" xml:space="preserve">Illi vero, inter quos, & </s> <s xml:id="echoid-s1850" xml:space="preserve">maximum paralle-<lb/>lorum maiores maximorum circulorum circun-<lb/>ferentiæ intercipiuntur, ſunt minores.</s> <s xml:id="echoid-s1851" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1852" xml:space="preserve">SINT in ſphæra paralleli circuli A B, C D, E F; </s> <s xml:id="echoid-s1853" xml:space="preserve">ſitque C D, maximus <lb/>parallelorum. </s> <s xml:id="echoid-s1854" xml:space="preserve">Inter circulum vero C D, & </s> <s xml:id="echoid-s1855" xml:space="preserve">vtrumq; </s> <s xml:id="echoid-s1856" xml:space="preserve">parallelorum A B, E F, <lb/>intercipiantur æquales circunferentiæ A C, C E, maximi alicuius circuli <lb/> <anchor type="figure" xlink:label="fig-061-01a" xlink:href="fig-061-01"/> ACEFDB. </s> <s xml:id="echoid-s1857" xml:space="preserve">Dico parallelos A B, E F, ęqua <lb/>lès eſſe. </s> <s xml:id="echoid-s1858" xml:space="preserve">Sint enim communes ſectiones paral-<lb/>lelorum, & </s> <s xml:id="echoid-s1859" xml:space="preserve">circuli A C E F D B, rectæ A B, <lb/>C D, E F, quæ parallelæ inter ſe erunt. </s> <s xml:id="echoid-s1860" xml:space="preserve">Tran <lb/> <anchor type="note" xlink:label="note-061-02a" xlink:href="note-061-02"/> ſeat autem primum circulus maximus ACE-<lb/>F D B, per polos parallelorum. </s> <s xml:id="echoid-s1861" xml:space="preserve">Quo poſito, <lb/>fecabit circulus A C E F D B, parallelos A B, <lb/>C D, E F, bifariam, & </s> <s xml:id="echoid-s1862" xml:space="preserve">ad angulos rectos; <lb/></s> <s xml:id="echoid-s1863" xml:space="preserve"> <anchor type="note" xlink:label="note-061-03a" xlink:href="note-061-03"/> atq; </s> <s xml:id="echoid-s1864" xml:space="preserve">adeo diametri erunt A B, C D, E F, pa-<lb/>rallelorum. </s> <s xml:id="echoid-s1865" xml:space="preserve">Quoniam vero arcus A C, B D, <lb/>æquales ſunt, nec non & </s> <s xml:id="echoid-s1866" xml:space="preserve">arcus C E, D F; </s> <s xml:id="echoid-s1867" xml:space="preserve">po-<lb/> <anchor type="note" xlink:label="note-061-04a" xlink:href="note-061-04"/> niturque A C, æqualis ipſi C E; </s> <s xml:id="echoid-s1868" xml:space="preserve">erunt A C, <lb/>B D, ſimul ipſis C E, D F, ſimul æquales: <lb/></s> <s xml:id="echoid-s1869" xml:space="preserve">Sunt autcm ſemicirculi æquales C A B D, <lb/>C E F D: </s> <s xml:id="echoid-s1870" xml:space="preserve">quia circuli maximi C D, A C E F D B, ſe mutuo bifariam diuidunt. </s> <s xml:id="echoid-s1871" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-061-05a" xlink:href="note-061-05"/> Igitur reliqui arcus A B, E F, æquales erunt; </s> <s xml:id="echoid-s1872" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s1873" xml:space="preserve">rectæ A B, E F, <lb/>hoc eſt, diametri circulorum A B, E F, æquales. </s> <s xml:id="echoid-s1874" xml:space="preserve">Circuli ergo A B, E F, <lb/> <anchor type="note" xlink:label="note-061-06a" xlink:href="note-061-06"/> æquales ſunt.</s> <s xml:id="echoid-s1875" xml:space="preserve"/> </p> <div xml:id="echoid-div184" type="float" level="2" n="1"> <figure xlink:label="fig-061-01" xlink:href="fig-061-01a"> <image file="061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/061-01"/> </figure> <note position="right" xlink:label="note-061-02" xlink:href="note-061-02a" xml:space="preserve">16. vndes.</note> <note position="right" xlink:label="note-061-03" xlink:href="note-061-03a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-061-04" xlink:href="note-061-04a" xml:space="preserve">10. 1. huius.</note> <note position="right" xlink:label="note-061-05" xlink:href="note-061-05a" xml:space="preserve">11. 1. huius.</note> <note position="right" xlink:label="note-061-06" xlink:href="note-061-06a" xml:space="preserve">29. tertij.</note> </div> <p> <s xml:id="echoid-s1876" xml:space="preserve">QVOD ſi arcus A C, maior ponatur arcu C E. </s> <s xml:id="echoid-s1877" xml:space="preserve">Dico circulum A B, mi-<lb/>norem eſſe circulo E F. </s> <s xml:id="echoid-s1878" xml:space="preserve">Poſita enim eadem conſtructione, & </s> <s xml:id="echoid-s1879" xml:space="preserve">demonſtratione, <lb/>erunt vt prius, arcus A C, B D, æquales, nec non C E, D F, cum ergo A C, ma <lb/> <anchor type="note" xlink:label="note-061-07a" xlink:href="note-061-07"/> ior ponatur quam C E, erunt duo arcus A C, B D, ſimul, maiores duobus ar-<lb/>cubus C E, D F, ſimul. </s> <s xml:id="echoid-s1880" xml:space="preserve">Reliquus igitur A B, ex ſemicirculo C A B D, minor <lb/>erit reliquo E F, ex ſemicirculo CEFD; </s> <s xml:id="echoid-s1881" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s1882" xml:space="preserve">recta A B, hoc eſt, <lb/>diameter circuli A B, minor erit, quàm recta E F, hoc eſt, quàm diameter cir-<lb/>culi E F, vt in ſcholio propoſ. </s> <s xml:id="echoid-s1883" xml:space="preserve">29. </s> <s xml:id="echoid-s1884" xml:space="preserve">lib. </s> <s xml:id="echoid-s1885" xml:space="preserve">3. </s> <s xml:id="echoid-s1886" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s1887" xml:space="preserve">à nobis eſt demonſtratum, cum <lb/>arcus A B, E F, ſemicirculo ſint minores. </s> <s xml:id="echoid-s1888" xml:space="preserve">Quare minor erit circulus A B, cir-<lb/>culo E F. </s> <s xml:id="echoid-s1889" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s1890" xml:space="preserve"/> </p> <div xml:id="echoid-div185" type="float" level="2" n="2"> <note position="right" xlink:label="note-061-07" xlink:href="note-061-07a" xml:space="preserve">1@. huius.</note> </div> <p> <s xml:id="echoid-s1891" xml:space="preserve">SED iam circulus maximus A C E F D B, non tranſeat per polos paral-<lb/>lelorum A B, C D, E F; </s> <s xml:id="echoid-s1892" xml:space="preserve">ſintque rurſus arcus A C, C E, æquales. </s> <s xml:id="echoid-s1893" xml:space="preserve">Dico adhuc <lb/>circulos A B, E F, eſſe æquales. </s> <s xml:id="echoid-s1894" xml:space="preserve">Sint enim G, H, poli parallelorum A B, C D, <lb/>E F, & </s> <s xml:id="echoid-s1895" xml:space="preserve">per G, H, ac polos circuli maximi A C E F D B, crrculus maximus de- <pb o="50" file="062" n="62" rhead=""/> ſcribatur G I H K, qui circulum A C E F D B, ſecabit duobus in punctis, vt in <lb/> <anchor type="note" xlink:label="note-062-01a" xlink:href="note-062-01"/> I, K, ad angulos rectos. </s> <s xml:id="echoid-s1896" xml:space="preserve">Quoniam igitur circulus maximus G I H K, per po <lb/> <anchor type="note" xlink:label="note-062-02a" xlink:href="note-062-02"/> los maximorum circulorum A C E F D B, C D, tranſit, ex conftructione, trã-<lb/>ſibunt hi viciſsim per illius polos. </s> <s xml:id="echoid-s1897" xml:space="preserve">Puncta igitur C, D, vbi ſe duo hi circuli <lb/> <anchor type="note" xlink:label="note-062-03a" xlink:href="note-062-03"/> interſecant, poli erunt circuli GIHK; </s> <s xml:id="echoid-s1898" xml:space="preserve">(alias non vterque circulus A C E F D, <lb/>C D, per polos circuli G I H K, tranſiret) ac proinde ductæ rectę C I, C K, ex <lb/> <anchor type="figure" xlink:label="fig-062-01a" xlink:href="fig-062-01"/> defin. </s> <s xml:id="echoid-s1899" xml:space="preserve">poli, æquales erunt, ac propterea & </s> <s xml:id="echoid-s1900" xml:space="preserve">ar <lb/>cus C I, C K, inter ſe erunt æquales. </s> <s xml:id="echoid-s1901" xml:space="preserve">Sunt <lb/> <anchor type="note" xlink:label="note-062-04a" xlink:href="note-062-04"/> autem & </s> <s xml:id="echoid-s1902" xml:space="preserve">arcus A C, C E, per hypotheſim, <lb/>æquales Reliqui igitur arcus A I, E K, æqua <lb/>les quoque erunt. </s> <s xml:id="echoid-s1903" xml:space="preserve">Rurſus quia ſemicirculus <lb/>I G K; </s> <s xml:id="echoid-s1904" xml:space="preserve">ſemicirculo G K H, æqualis eſt; </s> <s xml:id="echoid-s1905" xml:space="preserve">(Diui-<lb/> <anchor type="note" xlink:label="note-062-05a" xlink:href="note-062-05"/> dunt enim ſe mutuo circuli A C E F D B, & </s> <s xml:id="echoid-s1906" xml:space="preserve"><lb/>G I H K, bifariam; </s> <s xml:id="echoid-s1907" xml:space="preserve">ac proinde I G K, ſemicir <lb/>culus eft; </s> <s xml:id="echoid-s1908" xml:space="preserve">Arcus autem G K H, ſemicireulus <lb/>eſt propter G, H, polos parallelorum.) </s> <s xml:id="echoid-s1909" xml:space="preserve">dem <lb/>pto communi arcu G K, erunt reliqui arcus <lb/>G I, H K, æquales. </s> <s xml:id="echoid-s1910" xml:space="preserve">Quoniam igitur in dia-<lb/>metro circuli I C K D, ſegmenta circulorũ <lb/>æqualia I G K, K H I, quæ ſemicirculi ſunt, <lb/> <anchor type="note" xlink:label="note-062-06a" xlink:href="note-062-06"/> vt oſtendimus, inſiſtunt ad angulos rectos, ſuntque arcus I G, K H, æquales, <lb/>& </s> <s xml:id="echoid-s1911" xml:space="preserve">non ſunt ſegmentorum ſemiſſes, ſiue quadrantes, cum G, H, non ſint poli <lb/>circuli I C K D: </s> <s xml:id="echoid-s1912" xml:space="preserve">Item æquales ſunt arcus I A, K E, vt demonſtratum eſt; <lb/></s> <s xml:id="echoid-s1913" xml:space="preserve">erunt rectæ demiſſæ G A, H E, æquales. </s> <s xml:id="echoid-s1914" xml:space="preserve">Quarc circuli A B, E F, æquales in-<lb/> <anchor type="note" xlink:label="note-062-07a" xlink:href="note-062-07"/> ter ſe erunt.</s> <s xml:id="echoid-s1915" xml:space="preserve"/> </p> <div xml:id="echoid-div186" type="float" level="2" n="3"> <note position="left" xlink:label="note-062-01" xlink:href="note-062-01a" xml:space="preserve">10. 1. huius.</note> <note position="left" xlink:label="note-062-02" xlink:href="note-062-02a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-062-03" xlink:href="note-062-03a" xml:space="preserve">Schol. 15. 1. <lb/>huius.</note> <figure xlink:label="fig-062-01" xlink:href="fig-062-01a"> <image file="062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/062-01"/> </figure> <note position="left" xlink:label="note-062-04" xlink:href="note-062-04a" xml:space="preserve">23. tertij.</note> <note position="left" xlink:label="note-062-05" xlink:href="note-062-05a" xml:space="preserve">11. 1. huius.</note> <note position="left" xlink:label="note-062-06" xlink:href="note-062-06a" xml:space="preserve">@@ 1. huius.</note> <note position="left" xlink:label="note-062-07" xlink:href="note-062-07a" xml:space="preserve">12. huius.</note> </div> <note position="left" xml:space="preserve">Schol. 21. 1 <lb/>huius.</note> <p> <s xml:id="echoid-s1916" xml:space="preserve">QVOD ſi arcus A C, maiot ponatur arcu C E; </s> <s xml:id="echoid-s1917" xml:space="preserve">Dico circulum A B, mi-<lb/>norom eſſe circulo E F. </s> <s xml:id="echoid-s1918" xml:space="preserve">Sumpto enim arcu C L, quiæqualis ſit arcui C E, erit, <lb/>vt proxime demonſtratum eſt, parallelus per L, deſcriptus æqualis parallelo <lb/>E F: </s> <s xml:id="echoid-s1919" xml:space="preserve">ſed parallclus A B, minor eſt, quàm parallelus per L, deſcriptus, cum ille <lb/> <anchor type="note" xlink:label="note-062-09a" xlink:href="note-062-09"/> longius à maximo parallelorum, atque adeo à centro ſphæræ, abſit. </s> <s xml:id="echoid-s1920" xml:space="preserve">Minor igi <lb/>tur quoque eſt parallelus A B, quam E F. </s> <s xml:id="echoid-s1921" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s1922" xml:space="preserve">In ſphæra <lb/>ergo paralleli circuli, inter quos & </s> <s xml:id="echoid-s1923" xml:space="preserve">maximum parallelorum, &</s> <s xml:id="echoid-s1924" xml:space="preserve">c. </s> <s xml:id="echoid-s1925" xml:space="preserve">Quod erat. <lb/></s> <s xml:id="echoid-s1926" xml:space="preserve">demonſtrandum.</s> <s xml:id="echoid-s1927" xml:space="preserve"/> </p> <div xml:id="echoid-div187" type="float" level="2" n="4"> <note position="left" xlink:label="note-062-09" xlink:href="note-062-09a" xml:space="preserve">6. 1. huius</note> </div> </div> <div xml:id="echoid-div189" type="section" level="1" n="96"> <head xml:id="echoid-head108" xml:space="preserve">THEOR 16. PROPOS. 18.</head> <note position="left" xml:space="preserve">22.</note> <p> <s xml:id="echoid-s1928" xml:space="preserve">IN ſphæra circunferentiæ maximorum circu-<lb/>lorum interceptæ inter maximum parallelorum, <lb/>& </s> <s xml:id="echoid-s1929" xml:space="preserve">duos alios circulos æquales, & </s> <s xml:id="echoid-s1930" xml:space="preserve">parallelos, ſunt <lb/>æquales: </s> <s xml:id="echoid-s1931" xml:space="preserve">Illæ vero, quæ intercipiuntur inter maio-<lb/>rem parallelum, & </s> <s xml:id="echoid-s1932" xml:space="preserve">maximum, ſunt minores.</s> <s xml:id="echoid-s1933" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1934" xml:space="preserve">IN ſphæra ſint duo paralleli æquales A B, C D, & </s> <s xml:id="echoid-s1935" xml:space="preserve">maximus parallelorũ <lb/>fit E F: </s> <s xml:id="echoid-s1936" xml:space="preserve">Hos autem omnes parallelos ſecet maximus alius circulus A C D B. <lb/></s> <s xml:id="echoid-s1937" xml:space="preserve">Dico arcus A E, E C, nec non B F, F D, æquales eſſe. </s> <s xml:id="echoid-s1938" xml:space="preserve">Si enim non ſunt æqua <pb o="51" file="063" n="63" rhead=""/> les, ſit A E, maior. </s> <s xml:id="echoid-s1939" xml:space="preserve">Erit igitur circulus A B, minor circulo C D. </s> <s xml:id="echoid-s1940" xml:space="preserve">quod eſt con-<lb/> <anchor type="figure" xlink:label="fig-063-01a" xlink:href="fig-063-01"/> <anchor type="note" xlink:label="note-063-01a" xlink:href="note-063-01"/> tra hypotheſim. </s> <s xml:id="echoid-s1941" xml:space="preserve">Sunt ergo ęquales arcus A E, <lb/>E C, nec non B F, F D.</s> <s xml:id="echoid-s1942" xml:space="preserve"/> </p> <div xml:id="echoid-div189" type="float" level="2" n="1"> <figure xlink:label="fig-063-01" xlink:href="fig-063-01a"> <image file="063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/063-01"/> </figure> <note position="right" xlink:label="note-063-01" xlink:href="note-063-01a" xml:space="preserve">17. huius.</note> </div> <p> <s xml:id="echoid-s1943" xml:space="preserve">QVOD ſi circulus A B, maior po-<lb/>natur circulo C D; </s> <s xml:id="echoid-s1944" xml:space="preserve">Dico arcum A E, mino-<lb/>rem eſſe arcu E C. </s> <s xml:id="echoid-s1945" xml:space="preserve">Si enim non eſt minor, <lb/>erit vel æqualis, vel maior. </s> <s xml:id="echoid-s1946" xml:space="preserve">Si æqualis, erunt <lb/>circuli A B, C D, æquales: </s> <s xml:id="echoid-s1947" xml:space="preserve">ſi maior, erit cir-<lb/> <anchor type="note" xlink:label="note-063-02a" xlink:href="note-063-02"/> culus A B, minor circulo C D, quorum vtrũ-<lb/> <anchor type="note" xlink:label="note-063-03a" xlink:href="note-063-03"/> que eſt cõtra hypotheſim. </s> <s xml:id="echoid-s1948" xml:space="preserve">Minor ergo eſt ar-<lb/>cus A E, quam E C. </s> <s xml:id="echoid-s1949" xml:space="preserve">Quamobrem In ſphæra <lb/>circunferenriæ maximorum circulorum in-<lb/>terceptæ, &</s> <s xml:id="echoid-s1950" xml:space="preserve">c. </s> <s xml:id="echoid-s1951" xml:space="preserve">Quod oſtendendũ erat.</s> <s xml:id="echoid-s1952" xml:space="preserve"/> </p> <div xml:id="echoid-div190" type="float" level="2" n="2"> <note position="right" xlink:label="note-063-02" xlink:href="note-063-02a" xml:space="preserve">17. huius.</note> <note position="right" xlink:label="note-063-03" xlink:href="note-063-03a" xml:space="preserve">17. huius.</note> </div> </div> <div xml:id="echoid-div192" type="section" level="1" n="97"> <head xml:id="echoid-head109" xml:space="preserve">THEOR. 17. PROPOS. 19.</head> <note position="right" xml:space="preserve">23.</note> <p> <s xml:id="echoid-s1953" xml:space="preserve">SI in ſphæra maximus circulus parallelos ali-<lb/>quot circulos in ſphærica ſuperficie deſcriptos ſe-<lb/>cet quidẽ, non tamen per polos, in partes inæqua-<lb/>les eos ſecabit, excepto maximo parallelorum: </s> <s xml:id="echoid-s1954" xml:space="preserve">De <lb/>parallelorum autem ſegmentis in vno hemiſphæ-<lb/>riorum interceptis, ea quæ ſunt inter maximum <lb/>parallelorum, & </s> <s xml:id="echoid-s1955" xml:space="preserve">polum conſpicuum, ſunt maiora <lb/>ſemicirculo; </s> <s xml:id="echoid-s1956" xml:space="preserve">reliqua vero, quæ ſunt inter maximũ <lb/>parallelorum, & </s> <s xml:id="echoid-s1957" xml:space="preserve">polum occultum, ſunt ſemicircu <lb/>lo minora: </s> <s xml:id="echoid-s1958" xml:space="preserve">Æqualium denique ac parallelorum cir <lb/>culorum alterna ſegmenta ſunt inter ſe æqualia,</s> </p> <p> <s xml:id="echoid-s1959" xml:space="preserve">IN ſphęra maximus circulus A B C D, <lb/>parallelos E F, G H, I K, ſecet in L, M; </s> <s xml:id="echoid-s1960" xml:space="preserve">B, <lb/> <anchor type="figure" xlink:label="fig-063-02a" xlink:href="fig-063-02"/> D; </s> <s xml:id="echoid-s1961" xml:space="preserve">& </s> <s xml:id="echoid-s1962" xml:space="preserve">O, P, non per polos, qui ſint Q, R; </s> <s xml:id="echoid-s1963" xml:space="preserve">& </s> <s xml:id="echoid-s1964" xml:space="preserve"><lb/>fit G H, parallelorum maximus, & </s> <s xml:id="echoid-s1965" xml:space="preserve">Q, polus <lb/>conſpicuus, & </s> <s xml:id="echoid-s1966" xml:space="preserve">R, occultus in hemiſphęrio, <lb/>quod ſupra circulum maximum A B C D, ex <lb/>tat, & </s> <s xml:id="echoid-s1967" xml:space="preserve">ad partes F, vergit. </s> <s xml:id="echoid-s1968" xml:space="preserve">Dico circulum <lb/>A B C D, parallelos non bifariam ſecare, ex <lb/>cepto maximo G H; </s> <s xml:id="echoid-s1969" xml:space="preserve">hunc enim bifariam ſe-<lb/> <anchor type="note" xlink:label="note-063-05a" xlink:href="note-063-05"/> cat: </s> <s xml:id="echoid-s1970" xml:space="preserve">ſegmentum autem L F M, inter maximũ <lb/>parallelum, & </s> <s xml:id="echoid-s1971" xml:space="preserve">polum Q, conſpicuum ſemicir <lb/>culo eſſe maius, & </s> <s xml:id="echoid-s1972" xml:space="preserve">O K P, minus. </s> <s xml:id="echoid-s1973" xml:space="preserve">Si denique <lb/>paralleli E F, I K, ęquales ſint, alterna ſegmẽ <pb o="52" file="064" n="64" rhead=""/> ta L F M, O I P, ęqualia eſſe. </s> <s xml:id="echoid-s1974" xml:space="preserve">Per polum <lb/>enim Q, & </s> <s xml:id="echoid-s1975" xml:space="preserve">punctum B, circulus maximus <lb/> <anchor type="note" xlink:label="note-064-01a" xlink:href="note-064-01"/> <anchor type="figure" xlink:label="fig-064-01a" xlink:href="fig-064-01"/> deſcribatur QBRD; </s> <s xml:id="echoid-s1976" xml:space="preserve">qui per reliquum po-<lb/>lum R, tranſibit ex coroll. </s> <s xml:id="echoid-s1977" xml:space="preserve">ſcholij pro-<lb/>poſ. </s> <s xml:id="echoid-s1978" xml:space="preserve">10. </s> <s xml:id="echoid-s1979" xml:space="preserve">lib. </s> <s xml:id="echoid-s1980" xml:space="preserve">1. </s> <s xml:id="echoid-s1981" xml:space="preserve">huius; </s> <s xml:id="echoid-s1982" xml:space="preserve">nec non per pun-<lb/>ctum D, cum vtrumque circulum G B H D, <lb/> <anchor type="note" xlink:label="note-064-02a" xlink:href="note-064-02"/> A B C D, bifariam diuidat; </s> <s xml:id="echoid-s1983" xml:space="preserve">circuli au-<lb/>tem hi ſecentur bifariam in B, D. </s> <s xml:id="echoid-s1984" xml:space="preserve">Ex <lb/>quo fit, circulum Q B R D, paralle-<lb/>lum E F, ſecare ſupra circulum A B C D, <lb/>at parallelum I K, infra eundem; </s> <s xml:id="echoid-s1985" xml:space="preserve">vt in pun-<lb/>ctis S, T; </s> <s xml:id="echoid-s1986" xml:space="preserve">& </s> <s xml:id="echoid-s1987" xml:space="preserve">V, X. </s> <s xml:id="echoid-s1988" xml:space="preserve">Quoniam vero circulus <lb/>Q B R D, parallelos E F, I K, bifariam ſe-<lb/> <anchor type="note" xlink:label="note-064-03a" xlink:href="note-064-03"/> cat, erunt S F T, V K X, ſemicirculi; </s> <s xml:id="echoid-s1989" xml:space="preserve">ac <lb/>propterea arcus L F M, ſemicirculo maior, & </s> <s xml:id="echoid-s1990" xml:space="preserve">O K P, ſemicirculo minor erit. <lb/></s> <s xml:id="echoid-s1991" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s1992" xml:space="preserve"/> </p> <div xml:id="echoid-div192" type="float" level="2" n="1"> <figure xlink:label="fig-063-02" xlink:href="fig-063-02a"> <image file="063-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/063-02"/> </figure> <note position="right" xlink:label="note-063-05" xlink:href="note-063-05a" xml:space="preserve">11. 1. huius</note> <note position="left" xlink:label="note-064-01" xlink:href="note-064-01a" xml:space="preserve">20. 1. huius.</note> <figure xlink:label="fig-064-01" xlink:href="fig-064-01a"> <image file="064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/064-01"/> </figure> <note position="left" xlink:label="note-064-02" xlink:href="note-064-02a" xml:space="preserve">11. 1. huius.</note> <note position="left" xlink:label="note-064-03" xlink:href="note-064-03a" xml:space="preserve">15. 1. huius.</note> </div> <p> <s xml:id="echoid-s1993" xml:space="preserve">SINT iam paralleli E F, I K, ęquales. </s> <s xml:id="echoid-s1994" xml:space="preserve">Dico alterna ſegmenta L F M, O I P, <lb/>ęqualia inter ſe eſſe; </s> <s xml:id="echoid-s1995" xml:space="preserve">nec non ſegmenta alterna L E M, O K P. </s> <s xml:id="echoid-s1996" xml:space="preserve">Nam per polos <lb/>parallelorũ, & </s> <s xml:id="echoid-s1997" xml:space="preserve">polos circuli A B C D, deſcribatur circulus maximus A G C H, <lb/> <anchor type="note" xlink:label="note-064-04a" xlink:href="note-064-04"/> qui diuidet ſegmenta L A M, O C P, bifariam. </s> <s xml:id="echoid-s1998" xml:space="preserve">Aequales ergo ſunt arcus A L, <lb/> <anchor type="note" xlink:label="note-064-05a" xlink:href="note-064-05"/> A M, inter ſe, & </s> <s xml:id="echoid-s1999" xml:space="preserve">C O, C P, inter ſe. </s> <s xml:id="echoid-s2000" xml:space="preserve">Et quoniam circulus maximus A G C H, <lb/>tranſit per polos maximorum circulorum G H, A C; </s> <s xml:id="echoid-s2001" xml:space="preserve">tranſibunt viciſsim hi <lb/> <anchor type="note" xlink:label="note-064-06a" xlink:href="note-064-06"/> per illius polos. </s> <s xml:id="echoid-s2002" xml:space="preserve">Puncta igitur B, D, poli ſunt circuli AGCH; </s> <s xml:id="echoid-s2003" xml:space="preserve">ac propte-<lb/>rea rectę B A, B C, æquales erunt, ex defin. </s> <s xml:id="echoid-s2004" xml:space="preserve">poli; </s> <s xml:id="echoid-s2005" xml:space="preserve">atque idcirco & </s> <s xml:id="echoid-s2006" xml:space="preserve">arcus ipſi B A <lb/> <anchor type="note" xlink:label="note-064-07a" xlink:href="note-064-07"/> B C, æquales erunt: </s> <s xml:id="echoid-s2007" xml:space="preserve">Sunt autem & </s> <s xml:id="echoid-s2008" xml:space="preserve">arcus B L, B O, ęquales; </s> <s xml:id="echoid-s2009" xml:space="preserve">propterea quod <lb/> <anchor type="note" xlink:label="note-064-08a" xlink:href="note-064-08"/> æquales ponuntur paralleli E F, I K. </s> <s xml:id="echoid-s2010" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2011" xml:space="preserve">reliqui arcus A L, C O, ęqua-<lb/>les erunt: </s> <s xml:id="echoid-s2012" xml:space="preserve">Sunt autem arcus A L, C O, dimidij arcuum E A M, O C P; </s> <s xml:id="echoid-s2013" xml:space="preserve">pro-<lb/>pterea quòd A L, ipſi A M, & </s> <s xml:id="echoid-s2014" xml:space="preserve">C O, ipſi C P, oſtenſus eſt ęqualis. </s> <s xml:id="echoid-s2015" xml:space="preserve">Aequales ergo <lb/>ſunt quoque arcus L A M, O C P, ae proinde & </s> <s xml:id="echoid-s2016" xml:space="preserve">rectę ſubtenſę L M, O P, <lb/> <anchor type="note" xlink:label="note-064-09a" xlink:href="note-064-09"/> æquales erunt. </s> <s xml:id="echoid-s2017" xml:space="preserve">Quare ex circulis ęqualibus E F, I K, auferent æquales arcus, <lb/> <anchor type="note" xlink:label="note-064-10a" xlink:href="note-064-10"/> maiorem quidem L F M, maiori O I P, & </s> <s xml:id="echoid-s2018" xml:space="preserve">minorem L E M, minori O K P, <lb/>(hoc eſt alternum ſegmentum alterno ſegmento) ęqualem. </s> <s xml:id="echoid-s2019" xml:space="preserve">Quod eſt pro po-<lb/>ſitum. </s> <s xml:id="echoid-s2020" xml:space="preserve">Itaque ſi in ſphęra maximus circulus parallelos aliquot circulos in <lb/>ſphęrica ſuperficie deſcriptos ſecet quidem, &</s> <s xml:id="echoid-s2021" xml:space="preserve">c. </s> <s xml:id="echoid-s2022" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2023" xml:space="preserve"/> </p> <div xml:id="echoid-div193" type="float" level="2" n="2"> <note position="left" xlink:label="note-064-04" xlink:href="note-064-04a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-064-05" xlink:href="note-064-05a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-064-06" xlink:href="note-064-06a" xml:space="preserve">Schol. 15. 1. <lb/>huius.</note> <note position="left" xlink:label="note-064-07" xlink:href="note-064-07a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-064-08" xlink:href="note-064-08a" xml:space="preserve">18. huius.</note> <note position="left" xlink:label="note-064-09" xlink:href="note-064-09a" xml:space="preserve">29. tetij.</note> <note position="left" xlink:label="note-064-10" xlink:href="note-064-10a" xml:space="preserve">28. tertij.</note> </div> </div> <div xml:id="echoid-div195" type="section" level="1" n="98"> <head xml:id="echoid-head110" xml:space="preserve">THEOREMA 18. PROPOS. 20.</head> <note position="left" xml:space="preserve">24.</note> <p> <s xml:id="echoid-s2024" xml:space="preserve">SI in ſphæra maximus circulus parallelos ali-<lb/>quot circulos ſecet, non tamen per polos; </s> <s xml:id="echoid-s2025" xml:space="preserve">de paral <lb/>lelorum aſſumptis cirtumferentijs in vno hemi-<lb/>ſphærio, illæ quæ propius accedunt ad polũ con-<lb/>ſpicuum, erunt maiores, quàm vt ſimiles eſſe poſ-<lb/>ſint illis, quæ ab eodem conſpicuo polo longius <lb/>abſunt.</s> <s xml:id="echoid-s2026" xml:space="preserve"/> </p> <pb o="53" file="065" n="65" rhead=""/> <p> <s xml:id="echoid-s2027" xml:space="preserve">IN ſphæra parallelos A B, C D, E F, ſecet in H, O; </s> <s xml:id="echoid-s2028" xml:space="preserve">I, N; </s> <s xml:id="echoid-s2029" xml:space="preserve">K, M, non ta-<lb/>men per polos, circulus maximus GHIKLMNO, ſitque ſupra hemiſphæ-<lb/> <anchor type="figure" xlink:label="fig-065-01a" xlink:href="fig-065-01"/> rium G B L, polus conſpicuus P, occultus <lb/>autem Q Dico arcum O B H, maiorem eſſe, <lb/>quàm vt ſimilis ſit arcui N D I, & </s> <s xml:id="echoid-s2030" xml:space="preserve">N D I, ma <lb/>iorem, quàm vt ſimilis ſit arcui M F K. </s> <s xml:id="echoid-s2031" xml:space="preserve">Per <lb/>polum enim parallelorum P, & </s> <s xml:id="echoid-s2032" xml:space="preserve">puncta I, N, <lb/> <anchor type="note" xlink:label="note-065-01a" xlink:href="note-065-01"/> deſcribãtur duo circuli maximi P I, P N, ſe-<lb/>cantes parallelum A B, ſupra circulũ G I L N, <lb/>in R, S: </s> <s xml:id="echoid-s2033" xml:space="preserve">eritque arcus R B S, arcui I D N, ſi-<lb/> <anchor type="note" xlink:label="note-065-02a" xlink:href="note-065-02"/> milis. </s> <s xml:id="echoid-s2034" xml:space="preserve">Cum ergo arcus O B H, maior ſit ar-<lb/>cu R B S, maior quoque erit, quam vt ſimilis <lb/>ſit arcui N D I. </s> <s xml:id="echoid-s2035" xml:space="preserve">Eodem modo oſtendemus <lb/>arcum N D I, maiorem eſſe, quàm vt ſimilis <lb/>ſit arcui M F K, ſi nimirum per polum P, & </s> <s xml:id="echoid-s2036" xml:space="preserve"><lb/>puncta K, M, duo alij circuli maximi deſcri-<lb/>bantur. </s> <s xml:id="echoid-s2037" xml:space="preserve">Igitur ſi in ſphæra maximus circulus parallelos aliquot, &</s> <s xml:id="echoid-s2038" xml:space="preserve">c. </s> <s xml:id="echoid-s2039" xml:space="preserve">Quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s2040" xml:space="preserve"/> </p> <div xml:id="echoid-div195" type="float" level="2" n="1"> <figure xlink:label="fig-065-01" xlink:href="fig-065-01a"> <image file="065-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/065-01"/> </figure> <note position="right" xlink:label="note-065-01" xlink:href="note-065-01a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-065-02" xlink:href="note-065-02a" xml:space="preserve">10. huius.</note> </div> </div> <div xml:id="echoid-div197" type="section" level="1" n="99"> <head xml:id="echoid-head111" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s2041" xml:space="preserve">HINC fit, ſimpliciter arcum O B H, maiorem eſſe partem ſui paralleli A B, quàm ar-<lb/>cum N D I, ſui paralleli, &</s> <s xml:id="echoid-s2042" xml:space="preserve">c. </s> <s xml:id="echoid-s2043" xml:space="preserve">quandoquidem arcus R B S, tanta pars eſt ſui paralleli, quanta <lb/>eſt arcus I D N, ſui paralleli, cum hi arcus demonſtrati ſint eſſe ſimiles, &</s> <s xml:id="echoid-s2044" xml:space="preserve">c.</s> <s xml:id="echoid-s2045" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div198" type="section" level="1" n="100"> <head xml:id="echoid-head112" xml:space="preserve">THEOREMA 19. PROPOS. 21.</head> <note position="right" xml:space="preserve">25.</note> <p> <s xml:id="echoid-s2046" xml:space="preserve">SI in ſphæris æqualibus maximi circuli ad ma-<lb/>ximos circulos inclinentur, ille cuius polus ſubli-<lb/>mior ſupra planum ſubiectum eſt, inclinatior erit: <lb/></s> <s xml:id="echoid-s2047" xml:space="preserve">illi vero circuli, quorum poli æqualiter diſtant à ſu <lb/>biectis planis, æqualiter inclinantur.</s> <s xml:id="echoid-s2048" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2049" xml:space="preserve">IN ſphæris æqua-<lb/>libus A B C D, E F G H, <lb/> <anchor type="figure" xlink:label="fig-065-02a" xlink:href="fig-065-02"/> quarum centra I, K, <lb/>ad circulos maximos <lb/>A B C D, E F G H, quo <lb/>rum poli L, M, incli-<lb/>nẽtur duo circuli ma-<lb/>ximi B N D, F O H, quo-<lb/>rum poli, P, Q; </s> <s xml:id="echoid-s2050" xml:space="preserve">ſitque <lb/>primum polus P, ſubli-<lb/>mior ſupra planum cir <lb/>culi A B C D, quàm po <lb/>lus Q, ſupra planũ cir-<lb/>culi E F G H. </s> <s xml:id="echoid-s2051" xml:space="preserve">Dico cir- <pb o="54" file="066" n="66" rhead=""/> culum B N D, inclinatiorem eſſe ad circulum A B C D, quàm F O H, <lb/>ad E F G H. </s> <s xml:id="echoid-s2052" xml:space="preserve">Deſcribantur enim per L, P, polos, & </s> <s xml:id="echoid-s2053" xml:space="preserve">per polos, M, Q, cir-<lb/> <anchor type="note" xlink:label="note-066-01a" xlink:href="note-066-01"/> culi maximi A N C, E O G; </s> <s xml:id="echoid-s2054" xml:space="preserve">ſitque communis ſectio circulorum A B C D, <lb/>B N D, recta B D; </s> <s xml:id="echoid-s2055" xml:space="preserve">circulorum autem A B C D, A N C, recta A C; </s> <s xml:id="echoid-s2056" xml:space="preserve">& </s> <s xml:id="echoid-s2057" xml:space="preserve">circulo-<lb/>rum B N D, A N C, recta N I: </s> <s xml:id="echoid-s2058" xml:space="preserve">quæ omnes rectæ per centrum ſphæræ I, tran-<lb/>ſibunt, cum circuli maximi per idem centrum ſphæræ ducantur. </s> <s xml:id="echoid-s2059" xml:space="preserve">Eodem ordi-<lb/> <anchor type="note" xlink:label="note-066-02a" xlink:href="note-066-02"/> dine ſint in alia ſphæra communes ſectiones circulorum, vt recta F H, circu-<lb/>lorum E F G H, F O H; </s> <s xml:id="echoid-s2060" xml:space="preserve">recta vero E G, circulorum E F G H, E O G; </s> <s xml:id="echoid-s2061" xml:space="preserve">& </s> <s xml:id="echoid-s2062" xml:space="preserve">re-<lb/>cta O K, circulorum F O H, E O G: </s> <s xml:id="echoid-s2063" xml:space="preserve">quæ omnes rectæ ſimiliter per centrum <lb/>ſphæræ K, tranſibunt. </s> <s xml:id="echoid-s2064" xml:space="preserve">Et quoniam circulus A N C, per polos circulorum <lb/>A B C D, B N D, tranſiens, eos ſecat ad angulos rectos; </s> <s xml:id="echoid-s2065" xml:space="preserve">erit viciſsim vterque <lb/> <anchor type="note" xlink:label="note-066-03a" xlink:href="note-066-03"/> circulus A B C D, B N D, ad circulum A N C, rectus, atque adeo & </s> <s xml:id="echoid-s2066" xml:space="preserve">recta B D, <lb/>communis eorum ſectio, ad eundem circulum A N C, perpendicularis erit. <lb/></s> <s xml:id="echoid-s2067" xml:space="preserve"> <anchor type="note" xlink:label="note-066-04a" xlink:href="note-066-04"/> Quare anguli A I D, N I D, recti erunt, ex defin. </s> <s xml:id="echoid-s2068" xml:space="preserve">3. </s> <s xml:id="echoid-s2069" xml:space="preserve">lib. </s> <s xml:id="echoid-s2070" xml:space="preserve">11. </s> <s xml:id="echoid-s2071" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2072" xml:space="preserve">ac pro-<lb/> <anchor type="figure" xlink:label="fig-066-01a" xlink:href="fig-066-01"/> inde A I N, angulus <lb/>erit inclinationis cir-<lb/>culi B N D, ad circu-<lb/>lum A B C D, ex de-<lb/>fin. </s> <s xml:id="echoid-s2073" xml:space="preserve">6. </s> <s xml:id="echoid-s2074" xml:space="preserve">lib. </s> <s xml:id="echoid-s2075" xml:space="preserve">11. </s> <s xml:id="echoid-s2076" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2077" xml:space="preserve">Eo-<lb/>dem modo erit E K O, <lb/>angulus inclinationis <lb/>circuli F O H, ad cir-<lb/>circulũ E F G H. </s> <s xml:id="echoid-s2078" xml:space="preserve">Quo-<lb/>niam vero P, polus cir <lb/>culi B N D, ſublimior <lb/>ponitur ſupra circu-<lb/>lum A B C D, quàm <lb/>polus Q, circuli F O H, <lb/>ſupra circulum E F G H, erit maior arcus C P, arcu G Q. </s> <s xml:id="echoid-s2079" xml:space="preserve">Hi enim arcus, <lb/>cum ſint perpendiculares ad circulos A B C D, E F G H, altitudines polo-<lb/>rum P, Q, ſupra ipſos circulos metiuntur. </s> <s xml:id="echoid-s2080" xml:space="preserve">Sunt autem arcus P N, Q O, <lb/>æquales, cum ſint quadrantes. </s> <s xml:id="echoid-s2081" xml:space="preserve">Poli enim P, Q, à circulis maximis B N D, <lb/>F O H, per quadrantem abſunt. </s> <s xml:id="echoid-s2082" xml:space="preserve">Arcus ergo C N, maior erit arcu G O, ac pro <lb/> <anchor type="note" xlink:label="note-066-05a" xlink:href="note-066-05"/> rea A N, reliquus ex ſemicirculo A N C, minor erit reliquo E O, ex ſemicir <lb/>culo E O G. </s> <s xml:id="echoid-s2083" xml:space="preserve">Quare angulus A I N, angulo E K O, minor erit, ac proinde <lb/> <anchor type="note" xlink:label="note-066-06a" xlink:href="note-066-06"/> magis inclinatus erit circulus B N D, ad circulum A B C D, quam circulus <lb/>F O H, ad circulum E F G H, vt in explicatione definitionis 7. </s> <s xml:id="echoid-s2084" xml:space="preserve">lib. </s> <s xml:id="echoid-s2085" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s2086" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2087" xml:space="preserve">ſcripſimus.</s> <s xml:id="echoid-s2088" xml:space="preserve"/> </p> <div xml:id="echoid-div198" type="float" level="2" n="1"> <figure xlink:label="fig-065-02" xlink:href="fig-065-02a"> <image file="065-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/065-02"/> </figure> <note position="left" xlink:label="note-066-01" xlink:href="note-066-01a" xml:space="preserve">30. i. huius</note> <note position="left" xlink:label="note-066-02" xlink:href="note-066-02a" xml:space="preserve">6. 1. huius.</note> <note position="left" xlink:label="note-066-03" xlink:href="note-066-03a" xml:space="preserve">15. 1. huius</note> <note position="left" xlink:label="note-066-04" xlink:href="note-066-04a" xml:space="preserve">19. vndec.</note> <figure xlink:label="fig-066-01" xlink:href="fig-066-01a"> <image file="066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/066-01"/> </figure> <note position="left" xlink:label="note-066-05" xlink:href="note-066-05a" xml:space="preserve">Coroll. 16. <lb/>huius.</note> <note position="left" xlink:label="note-066-06" xlink:href="note-066-06a" xml:space="preserve">Schol. 27. <lb/>tertij.</note> </div> <p> <s xml:id="echoid-s2089" xml:space="preserve">SED ſintiam arcus C P, G Q, æquales, hoc eſt, poli B, Q, æqualiter di <lb/>ſtent à planis circulorum A B C D, E F G H. </s> <s xml:id="echoid-s2090" xml:space="preserve">Dico circulos B N D, F O H, <lb/>æqualiter inclinari ad circulos A B C D, E F G H. </s> <s xml:id="echoid-s2091" xml:space="preserve">Quoniam enim arcus C P, <lb/>G Q, æquales ſunt, ſi addantur quadrantes P N, Q O, erunt & </s> <s xml:id="echoid-s2092" xml:space="preserve">arcus C H, <lb/>G O, æquales; </s> <s xml:id="echoid-s2093" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s2094" xml:space="preserve">reliqui arcus A N, E O, ex ſęmicirculis æqua-<lb/>les erunt, Anguli igitur A I N, E K O, æquales erunt, ac propterea, ex defin. <lb/></s> <s xml:id="echoid-s2095" xml:space="preserve"> <anchor type="note" xlink:label="note-066-07a" xlink:href="note-066-07"/> 7. </s> <s xml:id="echoid-s2096" xml:space="preserve">lib. </s> <s xml:id="echoid-s2097" xml:space="preserve">11. </s> <s xml:id="echoid-s2098" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2099" xml:space="preserve">ſimiles, ſiue æquales erunt inclinationes circulorũ B N D, F O H, <lb/>ad circulos A B C D, E F G H. </s> <s xml:id="echoid-s2100" xml:space="preserve">Si igitur in ſphæris ęqualibus maximi circuli <lb/>ad maximos circulos, &</s> <s xml:id="echoid-s2101" xml:space="preserve">c. </s> <s xml:id="echoid-s2102" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s2103" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="2"> <note position="left" xlink:label="note-066-07" xlink:href="note-066-07a" xml:space="preserve">27. tertij.</note> </div> <pb o="55" file="067" n="67" rhead=""/> </div> <div xml:id="echoid-div201" type="section" level="1" n="101"> <head xml:id="echoid-head113" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s2104" xml:space="preserve">_HINC_ fit, ſi circulorum maximorũ ad alios inclinatorum poli equaliter diſtent <lb/>à polis maximorum, ad quos inclinantur, inclinationes eſſe equales: </s> <s xml:id="echoid-s2105" xml:space="preserve">cuius vero polus <lb/>vicinior ſit pola eius, ad queminclinantur, inclinationem eſſe maiorem. </s> <s xml:id="echoid-s2106" xml:space="preserve">Nam ſi arcus <lb/> <anchor type="note" xlink:label="note-067-01a" xlink:href="note-067-01"/> _L P, MQ_, ſint æquales, erunt & </s> <s xml:id="echoid-s2107" xml:space="preserve">_C P, G Q,_ æquales, cum quadrantes ſint _C L,_ <lb/>_GM;_ </s> <s xml:id="echoid-s2108" xml:space="preserve">atque adeo poli _P, Q,_ circulorum inclinatorum æqualiter diſtabunt à ſubie-<lb/>ctis planis circulorum _A B C D, E F G H._ </s> <s xml:id="echoid-s2109" xml:space="preserve">Quare, vt demonſtratum eſt in hac propoſ. <lb/></s> <s xml:id="echoid-s2110" xml:space="preserve">æqualeserunt inclinationes circulorum _B N D, F O H,_ ad circulos _A B C D, E F G H._ </s> <s xml:id="echoid-s2111" xml:space="preserve"><lb/>Si vero arcus _L P,_ minor ſit arcu _M Q,_ erit reliquus arcus _C P,_ ex quadrante <lb/>maior arcu _G Q,_ reliquo ex quadrante. </s> <s xml:id="echoid-s2112" xml:space="preserve">Igitur, vt oſtendimus in hac propeſ. </s> <s xml:id="echoid-s2113" xml:space="preserve">maior <lb/>erit inclinatio circuli _B N D,_ ad circulum _A B C D,_ quam circuli _F O H,_ ad cir-<lb/>culum _E F G H._</s> <s xml:id="echoid-s2114" xml:space="preserve"/> </p> <div xml:id="echoid-div201" type="float" level="2" n="1"> <note position="right" xlink:label="note-067-01" xlink:href="note-067-01a" xml:space="preserve">Coroll. 16. <lb/>1. huius.</note> </div> <p style="it"> <s xml:id="echoid-s2115" xml:space="preserve">_CONVERSVM_ quoque huius Theorematis, & </s> <s xml:id="echoid-s2116" xml:space="preserve">ſcholij demonſtrabimus in <lb/>bunc modum.</s> <s xml:id="echoid-s2117" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2118" xml:space="preserve">SI in ſphæris æqualibus maximi circuli ad maximos circulos <lb/>æqualiter inclinentur, erunt diſtantiæ polorum ipſorum à ſubiectis <lb/>planis æquales: </s> <s xml:id="echoid-s2119" xml:space="preserve">Illius verò, qui magis inclinatur, ſublimior erit po-<lb/>lus. </s> <s xml:id="echoid-s2120" xml:space="preserve">Item diſtantiæ polorum illorum circulorum, qui æqualiter incli <lb/>nantur, à polis circulorum, ad quos inclinantur, æquales erunt: </s> <s xml:id="echoid-s2121" xml:space="preserve">Di-<lb/>ſtantia vero poli illius circuli, qui magis inclinatur, à polo circuli, <lb/>ad quem inclinatur, minor erit.</s> <s xml:id="echoid-s2122" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2123" xml:space="preserve">_SI_ namque circuli _B N D, F O H,_ al circulos _A B C D, E F G H,_ æqualiter in-<lb/>clinentur, erunt anguli _A I N, E K O,_ æquales, ex defin 7 lib. </s> <s xml:id="echoid-s2124" xml:space="preserve">11. </s> <s xml:id="echoid-s2125" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2126" xml:space="preserve">ac propterea <lb/> <anchor type="note" xlink:label="note-067-02a" xlink:href="note-067-02"/> & </s> <s xml:id="echoid-s2127" xml:space="preserve">arcus _A N, E O,_ æquales erunt. </s> <s xml:id="echoid-s2128" xml:space="preserve">Additis igitur quadrantibus _N P, O Q,_ æqudo <lb/>les erunt arcus _A P, E Q;_ </s> <s xml:id="echoid-s2129" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s2130" xml:space="preserve">reliqui _C P, G Q,_ ex ſemicirculis <lb/>æquales erunt.</s> <s xml:id="echoid-s2131" xml:space="preserve"/> </p> <div xml:id="echoid-div202" type="float" level="2" n="2"> <note position="right" xlink:label="note-067-02" xlink:href="note-067-02a" xml:space="preserve">26. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s2132" xml:space="preserve">_SI_ verò circulus _B N D,_ ad circulum _A B C D,_ magis inclinetur, quam circulus <lb/>_F O H,_ ad circulum _E F G H,_ erit minor angulus _A I N,_ angulo _E K O,_ vt in defi-<lb/>nitionem 7. </s> <s xml:id="echoid-s2133" xml:space="preserve">lib. </s> <s xml:id="echoid-s2134" xml:space="preserve">11 Eucl. </s> <s xml:id="echoid-s2135" xml:space="preserve">ſeripſimus; </s> <s xml:id="echoid-s2136" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s2137" xml:space="preserve">arcus _A H,_ minor erit arcu _F O._ <lb/></s> <s xml:id="echoid-s2138" xml:space="preserve"> <anchor type="note" xlink:label="note-067-03a" xlink:href="note-067-03"/> Additis igitur quadrantibus _N P, O Q,_ minor erit arcus _A P,_ arcu _EQ;_ </s> <s xml:id="echoid-s2139" xml:space="preserve">ac proin-<lb/>de reliquus _C P,_ ex ſemicirculo _A N C,_ reliquo _G Q,_ ex ſemicirculo _F O G,_ maior erit.</s> <s xml:id="echoid-s2140" xml:space="preserve"/> </p> <div xml:id="echoid-div203" type="float" level="2" n="3"> <note position="right" xlink:label="note-067-03" xlink:href="note-067-03a" xml:space="preserve">Scho. 26. <lb/>tcrtij.</note> </div> <p style="it"> <s xml:id="echoid-s2141" xml:space="preserve">_RVRSVS,_ ſi circuli æqualiter inclinentur, erunt arcus _C P, G Q,_ vt pro-<lb/>xime oſtendimus, æquales. </s> <s xml:id="echoid-s2142" xml:space="preserve">Cum ergo quadrantes ſint _C L, G M;_ </s> <s xml:id="echoid-s2143" xml:space="preserve">erunt & </s> <s xml:id="echoid-s2144" xml:space="preserve">arcus <lb/> <anchor type="note" xlink:label="note-067-04a" xlink:href="note-067-04"/> _L P, M Q,_ æquales.</s> <s xml:id="echoid-s2145" xml:space="preserve"/> </p> <div xml:id="echoid-div204" type="float" level="2" n="4"> <note position="right" xlink:label="note-067-04" xlink:href="note-067-04a" xml:space="preserve">Coroll. 16. <lb/>1. huius.</note> </div> <p style="it"> <s xml:id="echoid-s2146" xml:space="preserve">_SI_ denique circulus _B N D,_ magis inclinetur, erit exproxime demoſtratis, ar@ <lb/>cus _C P,_ maior arcu _G Q._ </s> <s xml:id="echoid-s2147" xml:space="preserve">Reliquus igitur _L P,_ ex quadrante _C L,_ minor erit re-<lb/>lique _M Q,_ ex quadrante _G M,_ &</s> <s xml:id="echoid-s2148" xml:space="preserve">c.</s> <s xml:id="echoid-s2149" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2150" xml:space="preserve">_DVO_ quoque alia Theoremata in alia verſione hoc loco adiecta ſunt, vide-<lb/>licet.</s> <s xml:id="echoid-s2151" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div206" type="section" level="1" n="102"> <head xml:id="echoid-head114" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s2152" xml:space="preserve">CIRCVLI maximi tangentes eundem parallelum, æqualiter <lb/> <anchor type="note" xlink:label="note-067-05a" xlink:href="note-067-05"/> inclinantur ad maximum parallelorum: </s> <s xml:id="echoid-s2153" xml:space="preserve">qui vero maiorem paralle-<lb/>lum tangit, inclinatior eſt ad maximum parallelorum. </s> <s xml:id="echoid-s2154" xml:space="preserve">Et circuli <pb o="56" file="068" n="68" rhead=""/> æqualiter inclinati ad maximum parallelorum, tangunt eundem pa-<lb/>rallelum: </s> <s xml:id="echoid-s2155" xml:space="preserve">Qui vero inclinatior eſt ad maximum parallelorum, ma-<lb/>iorem parallelum tangit.</s> <s xml:id="echoid-s2156" xml:space="preserve"/> </p> <div xml:id="echoid-div206" type="float" level="2" n="1"> <note position="right" xlink:label="note-067-05" xlink:href="note-067-05a" xml:space="preserve">26.</note> </div> <p style="it"> <s xml:id="echoid-s2157" xml:space="preserve">_MAXIMI_ circuli _A B, C B,_ tan-<lb/>gant eundem parallelum _A C,_ ſitque <lb/> <anchor type="figure" xlink:label="fig-068-01a" xlink:href="fig-068-01"/> parallelorum maximus _D E._ </s> <s xml:id="echoid-s2158" xml:space="preserve">Dico cir-<lb/>culos _A B, C B,_ æqualiter inclinari ad <lb/>circulum _D E._ </s> <s xml:id="echoid-s2159" xml:space="preserve">Sit enim _F,_ polus pa-<lb/>rallelorum, & </s> <s xml:id="echoid-s2160" xml:space="preserve">per _F,_ & </s> <s xml:id="echoid-s2161" xml:space="preserve">contactus _A,_ <lb/> <anchor type="note" xlink:label="note-068-01a" xlink:href="note-068-01"/> _C,_ circuli maximi deſcribantur _F A D,_ <lb/>_F C E,_ qui per polos circulorum _A B,_ <lb/> <anchor type="note" xlink:label="note-068-02a" xlink:href="note-068-02"/> _C B,_ tranſibunt; </s> <s xml:id="echoid-s2162" xml:space="preserve">atque adeo ipſos ad <lb/> <anchor type="note" xlink:label="note-068-03a" xlink:href="note-068-03"/> angulos rectos ſecabunt. </s> <s xml:id="echoid-s2163" xml:space="preserve">Quare arcus <lb/>_A F, C F,_ metientur altitudinem po-<lb/>li _F,_ circuli _D E,_ ſupra circulos _A B,_ <lb/>_CB;_ </s> <s xml:id="echoid-s2164" xml:space="preserve">ac proinde cum arcus _A F, C F;_ <lb/></s> <s xml:id="echoid-s2165" xml:space="preserve"> <anchor type="note" xlink:label="note-068-04a" xlink:href="note-068-04"/> æquales ſint, propterea quòd rectæ ſub <lb/>tenſæ _F A, F C,_ æquales ſunt, ex defin. <lb/></s> <s xml:id="echoid-s2166" xml:space="preserve"> <anchor type="note" xlink:label="note-068-05a" xlink:href="note-068-05"/> poli, æqualiter inclinabitur circulus <lb/>_D E,_ ad circulos _A B, C B;_ </s> <s xml:id="echoid-s2167" xml:space="preserve">& </s> <s xml:id="echoid-s2168" xml:space="preserve">hi vi-<lb/>ciſsim ad illũ æqualiter inclinabuntur.</s> <s xml:id="echoid-s2169" xml:space="preserve"/> </p> <div xml:id="echoid-div207" type="float" level="2" n="2"> <figure xlink:label="fig-068-01" xlink:href="fig-068-01a"> <image file="068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/068-01"/> </figure> <note position="left" xlink:label="note-068-01" xlink:href="note-068-01a" xml:space="preserve">20. 1. huius</note> <note position="left" xlink:label="note-068-02" xlink:href="note-068-02a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-068-03" xlink:href="note-068-03a" xml:space="preserve">@5. 1. huius.</note> <note position="left" xlink:label="note-068-04" xlink:href="note-068-04a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-068-05" xlink:href="note-068-05a" xml:space="preserve">21. huius.</note> </div> <p style="it"> <s xml:id="echoid-s2170" xml:space="preserve">_TANGAT_ iam maximus cireulus _G H,_ maiorem parallelum _G I._ </s> <s xml:id="echoid-s2171" xml:space="preserve">Dico maio-<lb/>rem eſſe inclinationem circuli _G H,_ ad maximum parallelorum _D E,_ quàm circuli <lb/>_A B._ </s> <s xml:id="echoid-s2172" xml:space="preserve">Deſcripto enim per _F,_ & </s> <s xml:id="echoid-s2173" xml:space="preserve">contactum _G,_ circulo maximo _F G E,_ metietur eodem <lb/> <anchor type="note" xlink:label="note-068-06a" xlink:href="note-068-06"/> modo, vt proxime demonſtratum eſt, arcus _F G,_ altitudinem poli _F,_ circuli _D E,_ ſu-<lb/>pra circulum _G H._ </s> <s xml:id="echoid-s2174" xml:space="preserve">Eſt autem arcus _F G,_ maior arcu _F A,_ quòd circulus _G I,_ maior <lb/>pònatur circulo _A C,_ ac proinde à polo _F,_ remotior. </s> <s xml:id="echoid-s2175" xml:space="preserve">Igitur magis inclinabitur cir-<lb/>culus _D E,_ ad circulum _G H,_ quàm ad circulum _AB;_ </s> <s xml:id="echoid-s2176" xml:space="preserve">& </s> <s xml:id="echoid-s2177" xml:space="preserve">viciſsim _G H,_ magis ad <lb/> <anchor type="note" xlink:label="note-068-07a" xlink:href="note-068-07"/> _D E,_ inclinabitur, quàm _A B._</s> <s xml:id="echoid-s2178" xml:space="preserve"/> </p> <div xml:id="echoid-div208" type="float" level="2" n="3"> <note position="left" xlink:label="note-068-06" xlink:href="note-068-06a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-068-07" xlink:href="note-068-07a" xml:space="preserve">21. 1. huius.</note> </div> <p style="it"> <s xml:id="echoid-s2179" xml:space="preserve">_RVRSVS_ circuli maximi _A B, C B,_ æqualiter inclinentur ad circulum _D E,_ <lb/>maximnm parallelorum. </s> <s xml:id="echoid-s2180" xml:space="preserve">Dico illos eundem parallelum tangere. </s> <s xml:id="echoid-s2181" xml:space="preserve">Per F, enim polum pa-<lb/> <anchor type="note" xlink:label="note-068-08a" xlink:href="note-068-08"/> rallelorum, & </s> <s xml:id="echoid-s2182" xml:space="preserve">polos circulorum _A B, C B,_ circuli maximi deſcribantur _F A D,_ <lb/>_F C E,_ ſecantes circulos _A B, C B,_ in _A, C._ </s> <s xml:id="echoid-s2183" xml:space="preserve">Et quoniam cos ſecant ad angulos re-<lb/> <anchor type="note" xlink:label="note-068-09a" xlink:href="note-068-09"/> ctos; </s> <s xml:id="echoid-s2184" xml:space="preserve">metientur arcus _F A, F C,_ altitudinem poli _F,_ circuli _D E,_ ſupra circulos _A B,_ <lb/>_C B:_ </s> <s xml:id="echoid-s2185" xml:space="preserve">ſunt autem arcus _F A, F C,_ æquales, quòd circuli _<emph style="sc">Ab</emph>, C B,_ æqualiter ponantur <lb/> <anchor type="note" xlink:label="note-068-10a" xlink:href="note-068-10"/> inclinari ad circulum _D E,_ atque adeo & </s> <s xml:id="echoid-s2186" xml:space="preserve">hic viciſsim ad illos. </s> <s xml:id="echoid-s2187" xml:space="preserve">Si igitur ex polo _F,_ <lb/>interuallo _F A,_ vel _F C,_ circulus deſcribatur _A C,_ tanget hic circulos _<emph style="sc">Ab</emph>, C B;_ <lb/></s> <s xml:id="echoid-s2188" xml:space="preserve"> <anchor type="note" xlink:label="note-068-11a" xlink:href="note-068-11"/> propterea quod circulus _<emph style="sc">A</emph>C,_ & </s> <s xml:id="echoid-s2189" xml:space="preserve">circuli _A B, C B,_ in eiſaem punctis _A, C,_ ſecant <lb/>circulos maximos _F D, F E,_ qui per eorum polos tranſeunt.</s> <s xml:id="echoid-s2190" xml:space="preserve"/> </p> <div xml:id="echoid-div209" type="float" level="2" n="4"> <note position="left" xlink:label="note-068-08" xlink:href="note-068-08a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-068-09" xlink:href="note-068-09a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-068-10" xlink:href="note-068-10a" xml:space="preserve">Schol. 21. <lb/>huius.</note> <note position="left" xlink:label="note-068-11" xlink:href="note-068-11a" xml:space="preserve">3. huius.</note> </div> <p style="it"> <s xml:id="echoid-s2191" xml:space="preserve">_IAM_ vero circulus maximus _G H,_ magis inclinatus ſit ad circulum _D E._ <lb/></s> <s xml:id="echoid-s2192" xml:space="preserve">Dico illum tangere maiorem parallelum. </s> <s xml:id="echoid-s2193" xml:space="preserve">Deſcripto enim per _F,_ polum parallelo-<lb/> <anchor type="note" xlink:label="note-068-12a" xlink:href="note-068-12"/> rum, & </s> <s xml:id="echoid-s2194" xml:space="preserve">per polum circuli _G H,_ circulo maximo _F G,_ qui circulum _G H,_ ſeca-<lb/>bit adangulos rectos, nimirum in puncto _G;_ </s> <s xml:id="echoid-s2195" xml:space="preserve">metietur rurſus arcus _F G,_ altitu-<lb/> <anchor type="note" xlink:label="note-068-13a" xlink:href="note-068-13"/> dinem poli _F,_ circuli _D E,_ ſupra circulum _G H:_ </s> <s xml:id="echoid-s2196" xml:space="preserve">Eſt autem _F G,_ maior quàm _F A,_ quod <lb/> <anchor type="note" xlink:label="note-068-14a" xlink:href="note-068-14"/> magis inclinatus ponatur circulus _G H,_ quàm _<emph style="sc">A</emph>B._ </s> <s xml:id="echoid-s2197" xml:space="preserve">Igitur circulus ex polo _F,_ & </s> <s xml:id="echoid-s2198" xml:space="preserve">in-<lb/>teruallo _F G,_ deſcriptus maior erit circulo ex eodẽ polo _F,_ & </s> <s xml:id="echoid-s2199" xml:space="preserve">interuallo _F A,_ deſcripto.</s> <s xml:id="echoid-s2200" xml:space="preserve"> <pb o="57" file="069" n="69" rhead=""/> Cumergo _A B, A C,_ ſe mutuo tangant in _A,_ & </s> <s xml:id="echoid-s2201" xml:space="preserve">_G H, G I,_ ſe mutue quoq; </s> <s xml:id="echoid-s2202" xml:space="preserve">tangant <lb/> <anchor type="note" xlink:label="note-069-01a" xlink:href="note-069-01"/> in _G,_ conſtat propoſitum.</s> <s xml:id="echoid-s2203" xml:space="preserve"/> </p> <div xml:id="echoid-div210" type="float" level="2" n="5"> <note position="left" xlink:label="note-068-12" xlink:href="note-068-12a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-068-13" xlink:href="note-068-13a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-068-14" xlink:href="note-068-14a" xml:space="preserve">Schol. 21. <lb/>huius.</note> <note position="right" xlink:label="note-069-01" xlink:href="note-069-01a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div212" type="section" level="1" n="103"> <head xml:id="echoid-head115" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s2204" xml:space="preserve">CIRCVLI maximi ad maximum parallelorum æqualiter in-<lb/> <anchor type="note" xlink:label="note-069-02a" xlink:href="note-069-02"/> clinati, polos habent in circunferentia eiuſdem paralleli. </s> <s xml:id="echoid-s2205" xml:space="preserve">Et circuli <lb/>maximi, qui polos habent in circunferentia eiuſdem paralleli, ad ma-<lb/>ximum parallelorum æqualiter inclinantur.</s> <s xml:id="echoid-s2206" xml:space="preserve"/> </p> <div xml:id="echoid-div212" type="float" level="2" n="1"> <note position="right" xlink:label="note-069-02" xlink:href="note-069-02a" xml:space="preserve">27.</note> </div> <p style="it"> <s xml:id="echoid-s2207" xml:space="preserve">_CIRCVLI_ maximi _A B, C D,_ quorum poli _E, F,_ æqualiter ſint inclinati ad <lb/> <anchor type="figure" xlink:label="fig-069-01a" xlink:href="fig-069-01"/> _D B,_ maximum parallelorum. </s> <s xml:id="echoid-s2208" xml:space="preserve">_D_ico eo-<lb/>rum polos _E, F,_ eſſe in eodem parallelo. <lb/></s> <s xml:id="echoid-s2209" xml:space="preserve"> <anchor type="note" xlink:label="note-069-03a" xlink:href="note-069-03"/> Deſcriptis enim per _G,_ polum paralle-<lb/>lorum, & </s> <s xml:id="echoid-s2210" xml:space="preserve">per _E, F,_ polos circulorum <lb/>_A B, C D,_ maximis circulis _G E, G F,_ <lb/>qui recti erunt ad circulos _A B, C D;_ <lb/></s> <s xml:id="echoid-s2211" xml:space="preserve"> <anchor type="note" xlink:label="note-069-04a" xlink:href="note-069-04"/> erunt arcus _E G, F G,_ diſtantiæ polorũ <lb/>_E, F,_ à polo _G:_ </s> <s xml:id="echoid-s2212" xml:space="preserve">ſunt autem æquales, <lb/> <anchor type="note" xlink:label="note-069-05a" xlink:href="note-069-05"/> quòd circuli _A B, C D,_ ponantur æqua <lb/>liter inclinati ad circulum _D B._ </s> <s xml:id="echoid-s2213" xml:space="preserve">Igitur <lb/>circulus _E F,_ ex polo _G,_ & </s> <s xml:id="echoid-s2214" xml:space="preserve">interuallo <lb/>_G E,_ vel _G F,_ deſcriptus, parallelus <lb/>eſt circulo _DB;_ </s> <s xml:id="echoid-s2215" xml:space="preserve">in quo quidem paralle-<lb/> <anchor type="note" xlink:label="note-069-06a" xlink:href="note-069-06"/> lo _E F,_ circuli _A B, C D,_ polos _E, F_ <lb/>habent. </s> <s xml:id="echoid-s2216" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s2217" xml:space="preserve"/> </p> <div xml:id="echoid-div213" type="float" level="2" n="2"> <figure xlink:label="fig-069-01" xlink:href="fig-069-01a"> <image file="069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/069-01"/> </figure> <note position="right" xlink:label="note-069-03" xlink:href="note-069-03a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-069-04" xlink:href="note-069-04a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-069-05" xlink:href="note-069-05a" xml:space="preserve">Schol. 21. <lb/>huius.</note> <note position="right" xlink:label="note-069-06" xlink:href="note-069-06a" xml:space="preserve">2. huius.</note> </div> <p style="it"> <s xml:id="echoid-s2218" xml:space="preserve">_SED_ iam circuli maximi _A B, C D,_ <lb/>habeant polos _E, F,_ in parallelo, _E F._ <lb/></s> <s xml:id="echoid-s2219" xml:space="preserve">Dico eos æqualiter inclinari ad _D B,_ ma <lb/>ximum parallelorum. </s> <s xml:id="echoid-s2220" xml:space="preserve">Erunt enim ex defin. </s> <s xml:id="echoid-s2221" xml:space="preserve">poli, rectæ _G E, G F,_ æquales, atque obid <lb/>arcus _E G, F G,_ æquales quoque erunt. </s> <s xml:id="echoid-s2222" xml:space="preserve">Cum ergo ijdem arcus ſint diſtantiæpolorum <lb/> <anchor type="note" xlink:label="note-069-07a" xlink:href="note-069-07"/> _E, F,_ à _G,_ polo parallelorum; </s> <s xml:id="echoid-s2223" xml:space="preserve">æqualiter inclinati erunt circuli _A B, C D,_ ad _D B,_ <lb/>parallelorum maximum.</s> <s xml:id="echoid-s2224" xml:space="preserve"/> </p> <div xml:id="echoid-div214" type="float" level="2" n="3"> <note position="right" xlink:label="note-069-07" xlink:href="note-069-07a" xml:space="preserve">28. tertij. <lb/>Schol. 21. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s2225" xml:space="preserve">_SEQVITVR_ iam in codice græco prepoſitio 22. </s> <s xml:id="echoid-s2226" xml:space="preserve">cuius demon ſtratio longiſsi-<lb/>ma eſt. </s> <s xml:id="echoid-s2227" xml:space="preserve">Vnde quoniam in alia verſione multo breuius, dilucidiusque eadem demon-<lb/>ſtratur, viſum eſt hoc loco inſerere alia tria theoremata a lterius verſionis, vt faci-<lb/>lius deinde propoſitionem 22. </s> <s xml:id="echoid-s2228" xml:space="preserve">huius libri demonſtremus. </s> <s xml:id="echoid-s2229" xml:space="preserve">Eſt autem primum Theorema <lb/>ſecunda pars propoſ. </s> <s xml:id="echoid-s2230" xml:space="preserve">1. </s> <s xml:id="echoid-s2231" xml:space="preserve">lib. </s> <s xml:id="echoid-s2232" xml:space="preserve">3. </s> <s xml:id="echoid-s2233" xml:space="preserve">Theodoſii, quamuis magis vniuerſale ſit, vt hic proponi-<lb/>tur. </s> <s xml:id="echoid-s2234" xml:space="preserve">Primum ergo Theorema, quod ordine tertium eſt in hoc ſcholio, ita ſe habet.</s> <s xml:id="echoid-s2235" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div216" type="section" level="1" n="104"> <head xml:id="echoid-head116" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s2236" xml:space="preserve">SI ſuper diametro circuli conſtituatur rectum circuli ſegmen-<lb/> <anchor type="note" xlink:label="note-069-08a" xlink:href="note-069-08"/> tum, diuidatur autem ſegmenti inſiſtẽtis circunferentia in duas inæ-<lb/>quales partes, & </s> <s xml:id="echoid-s2237" xml:space="preserve">à puncto ſectionis ad circunferentiam circuli primi <lb/>plurimæ rectæ lineæ cadant; </s> <s xml:id="echoid-s2238" xml:space="preserve">erit recta ſubtendens minorem partem <lb/>inſiſtentis ſegmenti omnium minima: </s> <s xml:id="echoid-s2239" xml:space="preserve">quæ autem maiorem ſubten-<lb/>dit, omnium maxima. </s> <s xml:id="echoid-s2240" xml:space="preserve">Reliquarum vero propinquior maximæ remo <lb/>tiore ſem per maior eſt: </s> <s xml:id="echoid-s2241" xml:space="preserve">At propinquior minimæ remotiore ſemper <pb o="58" file="070" n="70" rhead=""/> minor eſt. </s> <s xml:id="echoid-s2242" xml:space="preserve">Duæ vero rectæ lineæ æquales ab eodem puncto in circun <lb/>ferentiam circuli cadunt, à maxima æqualiter diſtantes.</s> <s xml:id="echoid-s2243" xml:space="preserve"/> </p> <div xml:id="echoid-div216" type="float" level="2" n="1"> <note position="right" xlink:label="note-069-08" xlink:href="note-069-08a" xml:space="preserve">28.</note> </div> <p style="it"> <s xml:id="echoid-s2244" xml:space="preserve">_SVPER_ diametro _A D,_ circuli _A B C D E,_ conſtituatur rectum circuli ſegmen-<lb/>tum _A F D,_ quod ſecetur non bifariam in _F,_ ſitque minor pars _A F,_ & </s> <s xml:id="echoid-s2245" xml:space="preserve">maior _D F:_ <lb/></s> <s xml:id="echoid-s2246" xml:space="preserve">Cadant autem ex _F,_ plurimæ rectæ lineæ _F A, F I, F H, F B, F C, F D, F E._ </s> <s xml:id="echoid-s2247" xml:space="preserve">_D_ico <lb/>omnium minimam eſſe _FA;_ </s> <s xml:id="echoid-s2248" xml:space="preserve">maximam vero _F D:_ </s> <s xml:id="echoid-s2249" xml:space="preserve">At _F C,_ maiorem, quàm _F B,_ &</s> <s xml:id="echoid-s2250" xml:space="preserve">c. </s> <s xml:id="echoid-s2251" xml:space="preserve">Et <lb/>_F I,_ minorem, quàm _F H._ </s> <s xml:id="echoid-s2252" xml:space="preserve">&</s> <s xml:id="echoid-s2253" xml:space="preserve">c. </s> <s xml:id="echoid-s2254" xml:space="preserve">Denique duas _F E, F C,_ æquales eſſe, ſi æqualiter diſtent <lb/>à maxima _F D,_ hoc eſt, ſiarcus _D E, D C,_ æquales ſint. </s> <s xml:id="echoid-s2255" xml:space="preserve">Demittatur enim ex _F,_ in pla <lb/> <anchor type="note" xlink:label="note-070-01a" xlink:href="note-070-01"/> num circuli _A B C D E,_ perpendicularis _F G,_ quæ in _A D,_ communem ſectionem ca-<lb/> <anchor type="note" xlink:label="note-070-02a" xlink:href="note-070-02"/> det: </s> <s xml:id="echoid-s2256" xml:space="preserve">eritque punctum _G,_ vel inter puncta _A D,_ vt in prima figura; </s> <s xml:id="echoid-s2257" xml:space="preserve">(Id quod ſemper <lb/>continget, quando ſegmentum _A F D,_ ſemicirculo maius non eſt, quamuis idem accide-<lb/>re poſsit in ſegmento maiore.) </s> <s xml:id="echoid-s2258" xml:space="preserve">vel idem quod A; </s> <s xml:id="echoid-s2259" xml:space="preserve">vel extra circulum in diametro _D <emph style="sc">A</emph>,_ <lb/>protracta, vt poſteriores duæ figuræ indicant. </s> <s xml:id="echoid-s2260" xml:space="preserve">Id quod ſolumin ſegmento, quod ſemi-<lb/>circulo maius ſit, contingere poteſt. </s> <s xml:id="echoid-s2261" xml:space="preserve">In prima autem figura non erit _G,_ centrum cir-<lb/>culi _A B C D E,_ quod _G F,_ non diuidat bifariam ſegmentum _A F D:_ </s> <s xml:id="echoid-s2262" xml:space="preserve">Multò minus <lb/>in poſterioribus duabus figuris erit _G,_ centrum circuli _<emph style="sc">Ab</emph>CDE._ </s> <s xml:id="echoid-s2263" xml:space="preserve">Iungantur rectæ <lb/>_G I, G H, <emph style="sc">Gb</emph>, G C, G E;_ </s> <s xml:id="echoid-s2264" xml:space="preserve">eruntque omnes anguli ad _G,_ recti, ex defin. </s> <s xml:id="echoid-s2265" xml:space="preserve">3. </s> <s xml:id="echoid-s2266" xml:space="preserve">lib. </s> <s xml:id="echoid-s2267" xml:space="preserve">11. </s> <s xml:id="echoid-s2268" xml:space="preserve">Eucl. <lb/></s> <s xml:id="echoid-s2269" xml:space="preserve"> <anchor type="figure" xlink:label="fig-070-01a" xlink:href="fig-070-01"/> Quoniam vero rectarum ex _G,_ in circulum _<emph style="sc">Ab</emph>CDE,_ cadentium in prima figura, <lb/> <anchor type="note" xlink:label="note-070-03a" xlink:href="note-070-03"/> & </s> <s xml:id="echoid-s2270" xml:space="preserve">tertia minima eſt _GA;_ </s> <s xml:id="echoid-s2271" xml:space="preserve">In omnibus autem figuris maxima eſt _<emph style="sc">G</emph>D;_ </s> <s xml:id="echoid-s2272" xml:space="preserve">& </s> <s xml:id="echoid-s2273" xml:space="preserve">_<emph style="sc">G</emph>C,_ ma-<lb/>ior, quàm _<emph style="sc">GB</emph>_; </s> <s xml:id="echoid-s2274" xml:space="preserve">atque _<emph style="sc">G</emph>I,_ minor, quàm _<emph style="sc">G</emph>H;_ </s> <s xml:id="echoid-s2275" xml:space="preserve">duæ denique _<emph style="sc">G</emph>C:_ </s> <s xml:id="echoid-s2276" xml:space="preserve">_<emph style="sc">G</emph>E,_ æquales: </s> <s xml:id="echoid-s2277" xml:space="preserve">erunt <lb/> <anchor type="note" xlink:label="note-070-04a" xlink:href="note-070-04"/> propterea in prima, & </s> <s xml:id="echoid-s2278" xml:space="preserve">tertia figura duo quadrata rectarum _<emph style="sc">Ag</emph>, <emph style="sc">G</emph>F,_ minora duo-<lb/> <anchor type="note" xlink:label="note-070-05a" xlink:href="note-070-05"/> bus quadratis recfarum _<emph style="sc">Ig</emph>, <emph style="sc">G</emph>F:_ </s> <s xml:id="echoid-s2279" xml:space="preserve">quibus cum æqualia ſint quadrata rectarum _<emph style="sc">Fa</emph>,_ <lb/> <anchor type="note" xlink:label="note-070-06a" xlink:href="note-070-06"/> _FI;_ </s> <s xml:id="echoid-s2280" xml:space="preserve">minus quoque erit quadratum ex _F A,_ quadrato ex _FI;_ </s> <s xml:id="echoid-s2281" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s2282" xml:space="preserve">recta <lb/>_F A,_ minor erit quàm _F I._ </s> <s xml:id="echoid-s2283" xml:space="preserve">Eodem modo oſtendemus _F A,_ in eadem figura prima, & </s> <s xml:id="echoid-s2284" xml:space="preserve"><lb/>tertia minorem eſſe, quàm F H, &</s> <s xml:id="echoid-s2285" xml:space="preserve">c. </s> <s xml:id="echoid-s2286" xml:space="preserve">In ſecunda verà figura minor quoque eſt _F A,_ <lb/>quam _F I,_ vel _F H,_ &</s> <s xml:id="echoid-s2287" xml:space="preserve">c. </s> <s xml:id="echoid-s2288" xml:space="preserve">propterea quòd in triangulis _A I F, A H F,_ (in quibus an-<lb/> <anchor type="note" xlink:label="note-070-07a" xlink:href="note-070-07"/> gulus _A,_ rectus eſt, ex defin. </s> <s xml:id="echoid-s2289" xml:space="preserve">3. </s> <s xml:id="echoid-s2290" xml:space="preserve">lib. </s> <s xml:id="echoid-s2291" xml:space="preserve">11. </s> <s xml:id="echoid-s2292" xml:space="preserve">Eucl ac proinde alij acuti.) </s> <s xml:id="echoid-s2293" xml:space="preserve">recta _F A,_ ſub-<lb/>tendit angulum acutum _I,_ vel _H,_ at recta _F I,_ vel _F H,_ &</s> <s xml:id="echoid-s2294" xml:space="preserve">c. </s> <s xml:id="echoid-s2295" xml:space="preserve">angulum rectum _A._ <lb/></s> <s xml:id="echoid-s2296" xml:space="preserve">Minima ergo omnium èſt recta _F A._ </s> <s xml:id="echoid-s2297" xml:space="preserve">Rurſus in omnibus figuris erunt duo quadrata <lb/>ex _G D, G F,_ maiora duobus quadratis ex _G C, G F:_ </s> <s xml:id="echoid-s2298" xml:space="preserve">quibus cum æqualia ſint qua-<lb/> <anchor type="note" xlink:label="note-070-08a" xlink:href="note-070-08"/> dxata ex _F D, F C;_ </s> <s xml:id="echoid-s2299" xml:space="preserve">maius quoque erit quadratum ex _F D,_ quadrato ex _FC;_ </s> <s xml:id="echoid-s2300" xml:space="preserve">ac pro-<lb/>inde & </s> <s xml:id="echoid-s2301" xml:space="preserve">recta _F D,_ maior erit, quam recta _F C._ </s> <s xml:id="echoid-s2302" xml:space="preserve">Non aliter oſtendemus, rectam _F D,_ <pb o="59" file="071" n="71" rhead=""/> maiorem eſſe, quàm _F B,_ &</s> <s xml:id="echoid-s2303" xml:space="preserve">c. </s> <s xml:id="echoid-s2304" xml:space="preserve">Maxima ergo omnium eſt recta _F D._ </s> <s xml:id="echoid-s2305" xml:space="preserve">Præterea in om-<lb/>nibus figuris erunt duo quadrata ex _G C,_ _<emph style="sc">G</emph>F,_ maiora duobus quadratis ex _<emph style="sc">GB</emph>,_ <lb/>_<emph style="sc">G</emph>F:_ </s> <s xml:id="echoid-s2306" xml:space="preserve">quibus cum æqualia ſint quadrata ex _F C, <emph style="sc">Fb</emph>;_ </s> <s xml:id="echoid-s2307" xml:space="preserve">erit quoque quadratum ex _F C,_ <lb/> <anchor type="note" xlink:label="note-071-01a" xlink:href="note-071-01"/> maius quadrato ex _FB;_ </s> <s xml:id="echoid-s2308" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s2309" xml:space="preserve">recta _F C,_ maior erit, quàm _F B._ </s> <s xml:id="echoid-s2310" xml:space="preserve">Non ali-<lb/>ter oſtendemus, rectam _F C,_ quæ propinquior eſt maximæ _F D,_ maiorem eſſe quacun-<lb/>quealia remotiore, &</s> <s xml:id="echoid-s2311" xml:space="preserve">c. </s> <s xml:id="echoid-s2312" xml:space="preserve">Adhuc in omnibus figuris erunt duo quadrata ex _G I, <emph style="sc">G</emph>F,_ <lb/>minora duobus quadratis ex _<emph style="sc">G</emph>H, <emph style="sc">G</emph>F:_ </s> <s xml:id="echoid-s2313" xml:space="preserve">quibus cum æqualia ſint quadrata ex _F I,_ <lb/> <anchor type="note" xlink:label="note-071-02a" xlink:href="note-071-02"/> _FH;_ </s> <s xml:id="echoid-s2314" xml:space="preserve">erit quoque quadratum ex _F I,_ minus quadrato ex _FH;_ </s> <s xml:id="echoid-s2315" xml:space="preserve">proptereaq́ & </s> <s xml:id="echoid-s2316" xml:space="preserve">recta <lb/>_F I,_ minor, quàm _F H,_ erit. </s> <s xml:id="echoid-s2317" xml:space="preserve">Eodemq́; </s> <s xml:id="echoid-s2318" xml:space="preserve">modo demonſtrabimus, rectam _F I,_ quæ pro-<lb/>pinquior eſt minimæ _F A,_ minorem eſſe quacunque alia remotiore, &</s> <s xml:id="echoid-s2319" xml:space="preserve">c. </s> <s xml:id="echoid-s2320" xml:space="preserve">Poſtremo <lb/>erunt duo quadrata ex _<emph style="sc">G</emph>C, <emph style="sc">G</emph>F,_ æqualia duobus quadratis ex _<emph style="sc">G</emph>E, <emph style="sc">G</emph>F:_ </s> <s xml:id="echoid-s2321" xml:space="preserve">quibus <lb/>cum æqualia ſint quadrata ex _F C, F E,_ æqualia quoque erunt quadrata ex _F C,_ <lb/> <anchor type="note" xlink:label="note-071-03a" xlink:href="note-071-03"/> _FE;_ </s> <s xml:id="echoid-s2322" xml:space="preserve">atque adeò & </s> <s xml:id="echoid-s2323" xml:space="preserve">rectæ _F C, F E,_ æquales erunt. </s> <s xml:id="echoid-s2324" xml:space="preserve">Conſtat ergo id, quod proponitur. <lb/></s> <s xml:id="echoid-s2325" xml:space="preserve">Cæterum vt ex demonſtratione patet, eam rectam dicimus propinquiorem maximæ <lb/>_F D,_ quæ cadit in puctum vicinius pucto _D:_ </s> <s xml:id="echoid-s2326" xml:space="preserve">Illam verò propinquiorem minimæ _F A,_ <lb/>quæ cadit in puctum propinquius puncto _A._</s> <s xml:id="echoid-s2327" xml:space="preserve"/> </p> <div xml:id="echoid-div217" type="float" level="2" n="2"> <note position="left" xlink:label="note-070-01" xlink:href="note-070-01a" xml:space="preserve">11. vndec.</note> <note position="left" xlink:label="note-070-02" xlink:href="note-070-02a" xml:space="preserve">38. vndec.</note> <figure xlink:label="fig-070-01" xlink:href="fig-070-01a"> <image file="070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/070-01"/> </figure> <note position="left" xlink:label="note-070-03" xlink:href="note-070-03a" xml:space="preserve">7. vel 8. ten <lb/>tij.</note> <note position="left" xlink:label="note-070-04" xlink:href="note-070-04a" xml:space="preserve">7. vel 15. vol.</note> <note position="left" xlink:label="note-070-05" xlink:href="note-070-05a" xml:space="preserve">8. tertij.</note> <note position="left" xlink:label="note-070-06" xlink:href="note-070-06a" xml:space="preserve">47. primi.</note> <note position="left" xlink:label="note-070-07" xlink:href="note-070-07a" xml:space="preserve">19. paimi.</note> <note position="left" xlink:label="note-070-08" xlink:href="note-070-08a" xml:space="preserve">47. paimi.</note> <note position="right" xlink:label="note-071-01" xlink:href="note-071-01a" xml:space="preserve">47. primi.</note> <note position="right" xlink:label="note-071-02" xlink:href="note-071-02a" xml:space="preserve">47. primi.</note> <note position="right" xlink:label="note-071-03" xlink:href="note-071-03a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div219" type="section" level="1" n="105"> <head xml:id="echoid-head117" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s2328" xml:space="preserve">SI in ſphæræ ſuperficie intra circuli cuiuſque peripheriam pun-<lb/> <anchor type="note" xlink:label="note-071-04a" xlink:href="note-071-04"/> ctum ſignetur præter eius polum, ab eo autem ad circuli circunfe-<lb/>rétiam plurimi arcus circulorum maximorum ducantur ſemicircu <lb/>lo minores; </s> <s xml:id="echoid-s2329" xml:space="preserve">maximus eſt, qui per circuli polum ducitur; </s> <s xml:id="echoid-s2330" xml:space="preserve">minimus <lb/>autem, qui ei adiacet: </s> <s xml:id="echoid-s2331" xml:space="preserve">Reliquorum verò propinquior maximo, re-<lb/>motiore ſemper maior eſt: </s> <s xml:id="echoid-s2332" xml:space="preserve">Duo verò arcus ab eodem maximo, vel <lb/>minimo æqualiter remoti inter ſe æquales ſunt.</s> <s xml:id="echoid-s2333" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="1"> <note position="right" xlink:label="note-071-04" xlink:href="note-071-04a" xml:space="preserve">31.</note> </div> <p style="it"> <s xml:id="echoid-s2334" xml:space="preserve">_SIT_ in ſphæra circulus _A B C D E,_ cuius polus F, ſigneturq́; </s> <s xml:id="echoid-s2335" xml:space="preserve">in ſphæræ ſuperfi-<lb/>tie intra peripheriam circuli præter polum _F,_ punctum quodlibet _G,_ à quo plurimi <lb/> <anchor type="figure" xlink:label="fig-071-01a" xlink:href="fig-071-01"/> arcus maximorum circulorum ad circunferen-<lb/>tiam circuli _A B C D E,_ ducantur, quorum _G A,_ <lb/>in vtramque partem eductus tranſeat per polum <lb/>F; </s> <s xml:id="echoid-s2336" xml:space="preserve">arcus verò _G B,_ propinquior ſit ipſi _G A,_ quàm <lb/>_GC;_ </s> <s xml:id="echoid-s2337" xml:space="preserve">duo denique _G B, G E,_ æqualiter diſtent ab <lb/>eodem _G A,_ vel à _GD;_ </s> <s xml:id="echoid-s2338" xml:space="preserve">ſintque omnes hi arcus ſe-<lb/>micirculo minores: </s> <s xml:id="echoid-s2339" xml:space="preserve">quod tum demam erit, cum <lb/>ſe mutuo non interſecabunt in alio puncto, quàm <lb/>in _G._ </s> <s xml:id="echoid-s2340" xml:space="preserve">Cum enim circuli maximi ſe mutuo diui-<lb/> <anchor type="note" xlink:label="note-071-05a" xlink:href="note-071-05"/> dant bifariam, erunt arcus _G A, G E,_ ſemicircu-<lb/>lo minores, cum nondum ſe interſecent. </s> <s xml:id="echoid-s2341" xml:space="preserve">Eademq́; <lb/></s> <s xml:id="echoid-s2342" xml:space="preserve">ratione erunt alij arcus ex _G,_ exeuntes minores <lb/>ſemicirculo, ſi ſe mutuo non interſecent. </s> <s xml:id="echoid-s2343" xml:space="preserve">Quòd ſi vnus eorum, vt v. </s> <s xml:id="echoid-s2344" xml:space="preserve">g. </s> <s xml:id="echoid-s2345" xml:space="preserve">arcus _G A,_ <lb/>eſſet ſemicirculus, tranſirent omnes alij per punctum A, eſſentq́; </s> <s xml:id="echoid-s2346" xml:space="preserve">ſemicirculi quoque: </s> <s xml:id="echoid-s2347" xml:space="preserve"><lb/>Si vero _G A,_ eſſet ſemicirculo maior, ſecarent eum omnes alij, antequam ad circun-<lb/>ferentiam peruenirent, eſſentq́; </s> <s xml:id="echoid-s2348" xml:space="preserve">ſemicirculo maiores, vt patst. </s> <s xml:id="echoid-s2349" xml:space="preserve">Vnde nihil colligi <lb/>poſſet. </s> <s xml:id="echoid-s2350" xml:space="preserve">Dico arcum _G A,_ omnium eſſe maximum, & </s> <s xml:id="echoid-s2351" xml:space="preserve">_G D,_ minimum: </s> <s xml:id="echoid-s2352" xml:space="preserve">_G B,_ verò ma-<lb/>iorem eſſe arcu _<emph style="sc">G</emph>C;_ </s> <s xml:id="echoid-s2353" xml:space="preserve">duos denique _<emph style="sc">G</emph>B, G E,_ eſſe æquales. </s> <s xml:id="echoid-s2354" xml:space="preserve">Quoniam enim arcus _A D,_ <lb/>ſecat circulum _<emph style="sc">Ab</emph>C,_ bifariam, & </s> <s xml:id="echoid-s2355" xml:space="preserve">ad angulos rectos; </s> <s xml:id="echoid-s2356" xml:space="preserve">erit recta ſubtenſa _A D,_ dia-<lb/> <anchor type="note" xlink:label="note-071-06a" xlink:href="note-071-06"/> <pb o="60" file="072" n="72" rhead=""/> meter circuli _ABC;_ </s> <s xml:id="echoid-s2357" xml:space="preserve">& </s> <s xml:id="echoid-s2358" xml:space="preserve">ſuper ipſam rectum circuli ſegmentum _A G D,_ conſtitutum, <lb/>quod quidem inæqualiter ſecatur in _<emph style="sc">G</emph>,_ (Nam quia, ex defin. </s> <s xml:id="echoid-s2359" xml:space="preserve">poli, rectæ ſubtenſæ <lb/>_F A, F D,_ æquales ſunt, erunt quoque arcus _F A, F D,_ æquales; </s> <s xml:id="echoid-s2360" xml:space="preserve">ac proinde arcus <lb/> <anchor type="note" xlink:label="note-072-01a" xlink:href="note-072-01"/> _A D,_ ſectus erit bifariam in _F,_ at que ob id in _G,_ non bifariam ) maiorq́; </s> <s xml:id="echoid-s2361" xml:space="preserve">pars eſt _G A._ <lb/></s> <s xml:id="echoid-s2362" xml:space="preserve"> <anchor type="note" xlink:label="note-072-02a" xlink:href="note-072-02"/> & </s> <s xml:id="echoid-s2363" xml:space="preserve">minor _G D._ </s> <s xml:id="echoid-s2364" xml:space="preserve">Igitur rectarum ductarum ex _G,_ ad circunferentiam circuli _A B C,_ <lb/>maxima eſt _G A,_ & </s> <s xml:id="echoid-s2365" xml:space="preserve">minima _G D: </s> <s xml:id="echoid-s2366" xml:space="preserve">G B,_ verò maior quàm _GC;_ </s> <s xml:id="echoid-s2367" xml:space="preserve">& </s> <s xml:id="echoid-s2368" xml:space="preserve">_G B,_ _G E,_ æqua-<lb/>les. </s> <s xml:id="echoid-s2369" xml:space="preserve">Quare cum arcus, quibus ſúbtenduntur, ponantur ſemicirculo minores, erit <lb/> <anchor type="note" xlink:label="note-072-03a" xlink:href="note-072-03"/> & </s> <s xml:id="echoid-s2370" xml:space="preserve">arcus _G A,_ maximus, & </s> <s xml:id="echoid-s2371" xml:space="preserve">_G D,_ minimus: </s> <s xml:id="echoid-s2372" xml:space="preserve">_<emph style="sc">G</emph>B,_ verò maior, quàm _<emph style="sc">G</emph>C;_ </s> <s xml:id="echoid-s2373" xml:space="preserve">Arcus de-<lb/>nique _<emph style="sc">G</emph>B, <emph style="sc">G</emph>E,_ æquales.</s> <s xml:id="echoid-s2374" xml:space="preserve"/> </p> <div xml:id="echoid-div220" type="float" level="2" n="2"> <figure xlink:label="fig-071-01" xlink:href="fig-071-01a"> <image file="071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/071-01"/> </figure> <note position="right" xlink:label="note-071-05" xlink:href="note-071-05a" xml:space="preserve">11. 1. huius.</note> <note position="right" xlink:label="note-071-06" xlink:href="note-071-06a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-072-01" xlink:href="note-072-01a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-072-02" xlink:href="note-072-02a" xml:space="preserve">Schol. 21. <lb/>huius.</note> <note position="left" xlink:label="note-072-03" xlink:href="note-072-03a" xml:space="preserve">Schol. 28. <lb/>terti.j</note> </div> <note position="left" xml:space="preserve">28. tertij.</note> </div> <div xml:id="echoid-div222" type="section" level="1" n="106"> <head xml:id="echoid-head118" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s2375" xml:space="preserve">SI in ſphæræ ſuperficie extra circuli cuiuſque peripheriam pun-<lb/> <anchor type="note" xlink:label="note-072-05a" xlink:href="note-072-05"/> ctum ſignetur præter eius polum, ab eo autem ad circuli circunfe-<lb/>rentiam plurimi arcus circulorum maximorum ducantur ſemicir-<lb/>culo minores, ſecantesq́; </s> <s xml:id="echoid-s2376" xml:space="preserve">circunferentiam circuli; </s> <s xml:id="echoid-s2377" xml:space="preserve">maximus eſt, qui <lb/>per circuli polum ducitur; </s> <s xml:id="echoid-s2378" xml:space="preserve">Reliquorum verò maximo propinquior, <lb/>remotiore ſemper maior eſt: </s> <s xml:id="echoid-s2379" xml:space="preserve">Minimus autem eſt ille, qui inter pun-<lb/>ctum, & </s> <s xml:id="echoid-s2380" xml:space="preserve">circuli circunferentiam extra circulum interijcitur; </s> <s xml:id="echoid-s2381" xml:space="preserve">Reli-<lb/>quorum verò minimo propinquior, remotiore ſemper minor eſt: <lb/></s> <s xml:id="echoid-s2382" xml:space="preserve">Duo verò arcus ab eodem maximo, vel minimo æqualiter remoti in-<lb/>ter ſe æquales ſunt.</s> <s xml:id="echoid-s2383" xml:space="preserve"/> </p> <div xml:id="echoid-div222" type="float" level="2" n="1"> <note position="left" xlink:label="note-072-05" xlink:href="note-072-05a" xml:space="preserve">32.</note> </div> <p style="it"> <s xml:id="echoid-s2384" xml:space="preserve">_IN_ ſphara circulus ſit _A B C D E,_ cuius polus _F;_ </s> <s xml:id="echoid-s2385" xml:space="preserve">ſigneturq́; </s> <s xml:id="echoid-s2386" xml:space="preserve">in ſphæræ ſuperficie <lb/>extra peripheriam circuli, punctum quodvis _<emph style="sc">G</emph>,_ præter alterum polum circuli <lb/> <anchor type="figure" xlink:label="fig-072-01a" xlink:href="fig-072-01"/> _<emph style="sc">Ab</emph> C D E:_ </s> <s xml:id="echoid-s2387" xml:space="preserve">& </s> <s xml:id="echoid-s2388" xml:space="preserve">à _<emph style="sc">G</emph>,_ plurimi arcus maximorum <lb/>circulorum ducantur ad circunferentiam circu-<lb/>li _A B C D E,_ ipſam ſecantes; </s> <s xml:id="echoid-s2389" xml:space="preserve">quorum _<emph style="sc">G</emph> D F A,_ <lb/>per polum _F,_ tranſeat;</s> <s xml:id="echoid-s2390" xml:space="preserve">arcus verò _<emph style="sc">G</emph> H B,_ pro-<lb/>pinquior ſit ipſi _<emph style="sc">G</emph> D F A,_ quàm _<emph style="sc">G</emph> I C:_ </s> <s xml:id="echoid-s2391" xml:space="preserve">duo de-<lb/>nique _<emph style="sc">G</emph><emph style="sc">Hb</emph>, <emph style="sc">G</emph> K E,_ æqualiter diſtent ab eo-<lb/>dem _<emph style="sc">G</emph> D F A,_ vel à _<emph style="sc">G</emph> D,_ ſintque omnes hi ar-<lb/>cus ſemicirculo minores: </s> <s xml:id="echoid-s2392" xml:space="preserve">quod tum demum erit, <lb/>cum ſe mutuo non interſecabunt in alio puncto, <lb/>quàm in _<emph style="sc">G</emph>,_ veluti in antecedenti theoremate <lb/>eſt oſtenſum. </s> <s xml:id="echoid-s2393" xml:space="preserve">Dico arcum _<emph style="sc">G</emph>A,_ eſſe omnium <lb/>maximum; </s> <s xml:id="echoid-s2394" xml:space="preserve">& </s> <s xml:id="echoid-s2395" xml:space="preserve">_<emph style="sc">G</emph>B,_ maiorem quàm _<emph style="sc">G</emph>C:_ </s> <s xml:id="echoid-s2396" xml:space="preserve">Mini-<lb/>mum autem eſſe _<emph style="sc">G</emph>D;_ </s> <s xml:id="echoid-s2397" xml:space="preserve">& </s> <s xml:id="echoid-s2398" xml:space="preserve">_<emph style="sc">G</emph>H,_ minorem quàm _<emph style="sc">G</emph>I:_ </s> <s xml:id="echoid-s2399" xml:space="preserve">Denique duos arcus _<emph style="sc">G</emph>B, <emph style="sc">G</emph>E,_ <lb/>Item _<emph style="sc">G</emph>H,_ _<emph style="sc">G</emph>K,_ æquales eße. </s> <s xml:id="echoid-s2400" xml:space="preserve">Quoniam enim arcus _<emph style="sc">G</emph>A,_ ſecat circulum _A B C D E,_ <lb/> <anchor type="note" xlink:label="note-072-06a" xlink:href="note-072-06"/> bifariam, & </s> <s xml:id="echoid-s2401" xml:space="preserve">ad angulos rectos;</s> <s xml:id="echoid-s2402" xml:space="preserve">erit recta ſubten ſa _<emph style="sc">A</emph>D,_ diameter circuli _A B C D E,_ <lb/>& </s> <s xml:id="echoid-s2403" xml:space="preserve">ſuper ipſam rectum circuli ſegmentũ conſtitutum _<emph style="sc">Dg</emph>,_ quod initium ſumens à _D,_ <lb/>per _<emph style="sc">G</emph>,_ ducitur, donec in alio puncto _A,_ circulum _A B C D E,_ iterum ſecet: </s> <s xml:id="echoid-s2404" xml:space="preserve">quod qui-<lb/>dem non bifariam ſectum eſt in _<emph style="sc">G</emph>,_ (quòd _<emph style="sc">G</emph>,_ non ponatur polus circuli _A B C D E,_ <lb/>in quo dictum ſegmentum bifariam diuiditur, vtin præcedenti theoremate oſtenſum <lb/>eſt.) </s> <s xml:id="echoid-s2405" xml:space="preserve">maiorque pars eſt à puncto _<emph style="sc">G</emph>,_ vſque ad _A,_ cum in ea ſit reliquus polus, (alias ar <lb/>cus _<emph style="sc">G</emph>DA,_ per vtrumque polum duceretur.)</s> <s xml:id="echoid-s2406" xml:space="preserve">minor vero _<emph style="sc">Dg</emph>._ </s> <s xml:id="echoid-s2407" xml:space="preserve">Igitur rectarum ex _<emph style="sc">G</emph>,_ <lb/> <anchor type="note" xlink:label="note-072-07a" xlink:href="note-072-07"/> ad circunferentiam circuli _A B C D E,_ ductarum, maxima eſt _<emph style="sc">G</emph>A,_ & </s> <s xml:id="echoid-s2408" xml:space="preserve">minima _<emph style="sc">G</emph>D,_ <pb o="61" file="073" n="73" rhead=""/> _<emph style="sc">G</emph>B,_ verò maior quàm _<emph style="sc">G</emph>C;_ </s> <s xml:id="echoid-s2409" xml:space="preserve">& </s> <s xml:id="echoid-s2410" xml:space="preserve">_<emph style="sc">G</emph>B, <emph style="sc">G</emph>E,_ æquales Item _<emph style="sc">G</emph>H,_ minor quàm _<emph style="sc">G</emph>I;_ </s> <s xml:id="echoid-s2411" xml:space="preserve">& </s> <s xml:id="echoid-s2412" xml:space="preserve">_<emph style="sc">G</emph>H_ <lb/>_<emph style="sc">G</emph>K,_ æquales. </s> <s xml:id="echoid-s2413" xml:space="preserve">Quapropter cum arcubus ſemicirculo minoribus ſubtendantur, ex by-<lb/> <anchor type="note" xlink:label="note-073-01a" xlink:href="note-073-01"/> pothcſi, erit quoque arcus _<emph style="sc">G</emph>A,_ omnium maximus, & </s> <s xml:id="echoid-s2414" xml:space="preserve">_<emph style="sc">G</emph>D,_ minimus: </s> <s xml:id="echoid-s2415" xml:space="preserve">at _<emph style="sc">G</emph>B,_ mater, <lb/>quàm _<emph style="sc">G</emph>C;_ </s> <s xml:id="echoid-s2416" xml:space="preserve">& </s> <s xml:id="echoid-s2417" xml:space="preserve">_<emph style="sc">G</emph>H,_ minor quàm _<emph style="sc">G</emph>I:_ </s> <s xml:id="echoid-s2418" xml:space="preserve">Denique _<emph style="sc">G</emph>B, <emph style="sc">G</emph>E,_ nec non _<emph style="sc">G</emph>H, <emph style="sc">G</emph>K,_ æqua-<lb/> <anchor type="note" xlink:label="note-073-02a" xlink:href="note-073-02"/> les inter ſe. </s> <s xml:id="echoid-s2419" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s2420" xml:space="preserve"/> </p> <div xml:id="echoid-div223" type="float" level="2" n="2"> <figure xlink:label="fig-072-01" xlink:href="fig-072-01a"> <image file="072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/072-01"/> </figure> <note position="left" xlink:label="note-072-06" xlink:href="note-072-06a" xml:space="preserve">35. 1. huius.</note> <note position="left" xlink:label="note-072-07" xlink:href="note-072-07a" xml:space="preserve">Schol. 21. <lb/>huius.</note> <note position="right" xlink:label="note-073-01" xlink:href="note-073-01a" xml:space="preserve">Schol. 28. <lb/>tertij.</note> <note position="right" xlink:label="note-073-02" xlink:href="note-073-02a" xml:space="preserve">28. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s2421" xml:space="preserve">_PERSPICVVM_ autem eſt in proximis duobus theorematibus arcus ſingulorũ <lb/>ex <emph style="sc">G</emph>, ductos non debere eſſe maiores ſemicirculo: </s> <s xml:id="echoid-s2422" xml:space="preserve">alias non auferrent maiores lineæ <lb/>maiores arcus, & </s> <s xml:id="echoid-s2423" xml:space="preserve">contra, vt conſtat exſcholio propoſ. </s> <s xml:id="echoid-s2424" xml:space="preserve">28. </s> <s xml:id="echoid-s2425" xml:space="preserve">lib. </s> <s xml:id="echoid-s2426" xml:space="preserve">3. </s> <s xml:id="echoid-s2427" xml:space="preserve">Eucl.</s> <s xml:id="echoid-s2428" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div225" type="section" level="1" n="107"> <head xml:id="echoid-head119" xml:space="preserve">THEOREMA 20. PROPOS. 22.</head> <note position="right" xml:space="preserve">33.</note> <p> <s xml:id="echoid-s2429" xml:space="preserve">SI in ſphæra maximus circulus vnum quidem <lb/>circulum tangat, alium vero ei parallelum ſecet, <lb/>poſitum inter ſphæræ centrum, & </s> <s xml:id="echoid-s2430" xml:space="preserve">eum circulum, <lb/>quem tangit maximus circulus, polus autem maxi <lb/>mi circuli fuerit inter vtrumque parallelorum, de-<lb/>ſcribanturque maximi circuli tangentes duorum <lb/>parallelorum maiorem: </s> <s xml:id="echoid-s2431" xml:space="preserve">hi omnes erunt inclinati <lb/>ad maximum circulum, & </s> <s xml:id="echoid-s2432" xml:space="preserve">eorum rectiſſimus qui-<lb/>dem eritille, cuius contactus erit in eo puncto, in <lb/>quo maius ſegmentum paralleli maioris bifariam <lb/>diuiditur; </s> <s xml:id="echoid-s2433" xml:space="preserve">humillimus vero & </s> <s xml:id="echoid-s2434" xml:space="preserve">maxime inclina-<lb/>tus, cuius contactus eritin eo puncto, in quo mi-<lb/>nus ſegmentum bifariã diuiditur; </s> <s xml:id="echoid-s2435" xml:space="preserve">Reliquorum au-<lb/>tem illi quidem, quiæqualiter diſtant ab alterutro <lb/>eorum punctorum, in quibus fegmenta bifariam <lb/>ſecantur, ſunt ſimiliter inclinati: </s> <s xml:id="echoid-s2436" xml:space="preserve">qui vero conta-<lb/>ctum remotiorem habet à puncto, in quo maius <lb/>ſegmentum bifariam ſecatur, inclinatior perpetuo <lb/>eſt, quam qui contactum eidem puncto propio-<lb/>rem habet. </s> <s xml:id="echoid-s2437" xml:space="preserve">Poli denique maximorum circulorum <lb/>erunt in vno circulo, qui & </s> <s xml:id="echoid-s2438" xml:space="preserve">minor erit eo circulo, <pb o="62" file="074" n="74" rhead=""/> quem tangit maximus in principio circulus, & </s> <s xml:id="echoid-s2439" xml:space="preserve">ei-<lb/>dem parallelus erit.</s> <s xml:id="echoid-s2440" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2441" xml:space="preserve">IN ſphæra maximus circulus A B C D, cuius polus E, tangat circulum <lb/>A F, ſecet autem alium huic parallelum G B H D, poſitum inter ſphæræ cen-<lb/>trum, & </s> <s xml:id="echoid-s2442" xml:space="preserve">circulum A F, ita vt circulus G B H D, maior ſit, quam A F; </s> <s xml:id="echoid-s2443" xml:space="preserve">ſitque <lb/>E, polus circuli maximi A B C D, inter vtrumque circulum A F, G B H D. <lb/></s> <s xml:id="echoid-s2444" xml:space="preserve">Quoniam verò maximus circulus A B C D, ſecat circulum G B H D, non bi-<lb/>fariam, cum non tranfeat per eius polos, hoc eſt, per polos parallelorum, erit <lb/> <anchor type="figure" xlink:label="fig-074-01a" xlink:href="fig-074-01"/> ſegmentum B H D, ad po <lb/>lum conſpicuum, qui ſit I, <lb/>maiꝰ ſemicirculo, & </s> <s xml:id="echoid-s2445" xml:space="preserve">B G D, <lb/> <anchor type="note" xlink:label="note-074-01a" xlink:href="note-074-01"/> minus. </s> <s xml:id="echoid-s2446" xml:space="preserve">Ducatur per E, po-<lb/>lum circuli A B C D, & </s> <s xml:id="echoid-s2447" xml:space="preserve">I, <lb/> <anchor type="note" xlink:label="note-074-02a" xlink:href="note-074-02"/> polũ parallelorũ circulus <lb/>maximus G A C, qui ſeca-<lb/>bit ſegmenta B G D, B H D, <lb/> <anchor type="note" xlink:label="note-074-03a" xlink:href="note-074-03"/> bifariam: </s> <s xml:id="echoid-s2448" xml:space="preserve">puncta autem M, <lb/>N, æqualiter diſtent ab H; <lb/></s> <s xml:id="echoid-s2449" xml:space="preserve">& </s> <s xml:id="echoid-s2450" xml:space="preserve">O, magis diſtet ab H, <lb/>quàm N. </s> <s xml:id="echoid-s2451" xml:space="preserve">Tangant autem <lb/>parallelum G B H D, in <lb/> <anchor type="note" xlink:label="note-074-04a" xlink:href="note-074-04"/> punctis G, H, M, N, O, cir-<lb/>culi maximiGL, H K, M P, <lb/>N K, O L, qui quidem om-<lb/>nes inclinati erunt ad ma-<lb/>ximum circulum A B C D, <lb/>cum non tranſeant per E, <lb/>polum ipſius. </s> <s xml:id="echoid-s2452" xml:space="preserve">Cum enim E, <lb/>polus ponatur inter parallelos A F, G B H D, non poterunt circuli tangen-<lb/>tes circulum G B H D, per E, tranſire, alias ſecarent ipſum, cum alter po-<lb/>lus, per quem etiam neceſſario tranſeunt, ſit extra dictos parallelos, vt patet. <lb/></s> <s xml:id="echoid-s2453" xml:space="preserve"> <anchor type="note" xlink:label="note-074-05a" xlink:href="note-074-05"/> Dico circulum H K, eſſe rectiſsimum, hoc eſt, minime inclinatum, humilimum <lb/>autem, id eſt, maximè inclinatum eſſe G L; </s> <s xml:id="echoid-s2454" xml:space="preserve">At M P, N K, ſimiliter inclinari, <lb/>& </s> <s xml:id="echoid-s2455" xml:space="preserve">O L, magis quàm N K: </s> <s xml:id="echoid-s2456" xml:space="preserve">Polos denique horum circulorum tangentium eſſe <lb/>in vno eodemq́ue parallelo, qui minor ſit, quàm A F. </s> <s xml:id="echoid-s2457" xml:space="preserve">Quoniam enim E, polus <lb/>eſt circuli A B C D, erit E A, quadrans maximi circuli; </s> <s xml:id="echoid-s2458" xml:space="preserve">ſumatur ei æqualis <lb/> <anchor type="note" xlink:label="note-074-06a" xlink:href="note-074-06"/> arcus H Q; </s> <s xml:id="echoid-s2459" xml:space="preserve">eritque punctum Q, inter puncta A, & </s> <s xml:id="echoid-s2460" xml:space="preserve">I, cum arcus H A, ma-<lb/>ior ſit quadrante, (quòd E A, quadrans ſit oſtenſus.) </s> <s xml:id="echoid-s2461" xml:space="preserve">& </s> <s xml:id="echoid-s2462" xml:space="preserve">H I, quadrante mi-<lb/> <anchor type="note" xlink:label="note-074-07a" xlink:href="note-074-07"/> nor, propterea quòd arcus ex I, polo per H, vſque ad maximum parallelorum <lb/>porrectus ſit quadrans. </s> <s xml:id="echoid-s2463" xml:space="preserve">Si igitur ex polo I, ad interuallum I Q, circulus de-<lb/> <anchor type="note" xlink:label="note-074-08a" xlink:href="note-074-08"/> ſcribatur Q T R, erit is ipſi A F, parallelus, & </s> <s xml:id="echoid-s2464" xml:space="preserve">eo minor. </s> <s xml:id="echoid-s2465" xml:space="preserve">In hoc ergo paral-<lb/>lelo Q T R, dico eſſe polos omnium circulorum parallelum G B H D, tan-<lb/> <anchor type="note" xlink:label="note-074-09a" xlink:href="note-074-09"/> gentium. </s> <s xml:id="echoid-s2466" xml:space="preserve">Per polum enim I, & </s> <s xml:id="echoid-s2467" xml:space="preserve">puncta contactuum deſcribantur circuli maxi-<lb/>mi M I S, N I T, O I V; </s> <s xml:id="echoid-s2468" xml:space="preserve">qui tranſibunt quoque per polos tangentium. </s> <s xml:id="echoid-s2469" xml:space="preserve">Quia <lb/> <anchor type="note" xlink:label="note-074-10a" xlink:href="note-074-10"/> vero arcus H I, M I, N I, O I, G I, æquales ſunt, quòd ex definitione poli, <lb/>rectæ illis arcubus ſubtenſæ æquales ſint, eademq́ue ratione & </s> <s xml:id="echoid-s2470" xml:space="preserve">arcus I Q, I S, <lb/>I T, I V, I R, æquales ſunt;</s> <s xml:id="echoid-s2471" xml:space="preserve">erunt toti arcus H Q, M S, N T, O V, G R, <pb o="63" file="075" n="75" rhead=""/> æquales; </s> <s xml:id="echoid-s2472" xml:space="preserve">atque adeò cum H Q, ſit quadrans, omnes illi arcus quadrantes <lb/>erunt. </s> <s xml:id="echoid-s2473" xml:space="preserve">Quare cum demonſtratum ſit eos tranſire per polos tangentium, erunt <lb/>puncta Q, S, T, V, R, poli circulorum tangentium, quæ quidem omnia <lb/> <anchor type="note" xlink:label="note-075-01a" xlink:href="note-075-01"/> ſunt in parallelo Q T R, quod vltimo loco proponebatur demonſtrandum. <lb/></s> <s xml:id="echoid-s2474" xml:space="preserve">Iam vero quia arcus circulorum maximorũ ex E, polo circuli maximi A B C D, <lb/>ad Q, S, T, V, R, polos tangentium ducti metiuntur diſtantias poli E, à <lb/>polis tangentium; </s> <s xml:id="echoid-s2475" xml:space="preserve">eſtq́ue omnium maximus E Q; </s> <s xml:id="echoid-s2476" xml:space="preserve">minimus autem E R; </s> <s xml:id="echoid-s2477" xml:space="preserve">æqua <lb/> <anchor type="note" xlink:label="note-075-02a" xlink:href="note-075-02"/> les verò E S, E T; </s> <s xml:id="echoid-s2478" xml:space="preserve">& </s> <s xml:id="echoid-s2479" xml:space="preserve">denique E T, maior, quàm E V, quòd omnes hi arcùs <lb/>ſint ſemicirculo minores; </s> <s xml:id="echoid-s2480" xml:space="preserve">(eſt enim E Q, quadrante E A, minor; </s> <s xml:id="echoid-s2481" xml:space="preserve">atque adeo <lb/>reliqui eum non ſecabunt citra punctum Q, ideoque ſemicirculo minores <lb/>erunt.) </s> <s xml:id="echoid-s2482" xml:space="preserve">erit circulus H K, minimè inclinatus ad circulum maximum A B C D; <lb/></s> <s xml:id="echoid-s2483" xml:space="preserve"> <anchor type="note" xlink:label="note-075-03a" xlink:href="note-075-03"/> & </s> <s xml:id="echoid-s2484" xml:space="preserve">G L, maximè; </s> <s xml:id="echoid-s2485" xml:space="preserve">at M P, N K, æqualiter, ſeu ſimiliter; </s> <s xml:id="echoid-s2486" xml:space="preserve">& </s> <s xml:id="echoid-s2487" xml:space="preserve">O L, magis quàm <lb/>N K, quod primo loco demonſtrandum proponebatur. </s> <s xml:id="echoid-s2488" xml:space="preserve">Quocirca ſi in ſphæ-<lb/>ra maximus circulus. </s> <s xml:id="echoid-s2489" xml:space="preserve">&</s> <s xml:id="echoid-s2490" xml:space="preserve">c. </s> <s xml:id="echoid-s2491" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2492" xml:space="preserve"/> </p> <div xml:id="echoid-div225" type="float" level="2" n="1"> <figure xlink:label="fig-074-01" xlink:href="fig-074-01a"> <image file="074-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/074-01"/> </figure> <note position="left" xlink:label="note-074-01" xlink:href="note-074-01a" xml:space="preserve">19. huius.</note> <note position="left" xlink:label="note-074-02" xlink:href="note-074-02a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-074-03" xlink:href="note-074-03a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-074-04" xlink:href="note-074-04a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-074-05" xlink:href="note-074-05a" xml:space="preserve">Coroll. 10. <lb/>1. huius.</note> <note position="left" xlink:label="note-074-06" xlink:href="note-074-06a" xml:space="preserve">Coroll. 16. <lb/>1. huius.</note> <note position="left" xlink:label="note-074-07" xlink:href="note-074-07a" xml:space="preserve">Coroll. 16. <lb/>1. huius.</note> <note position="left" xlink:label="note-074-08" xlink:href="note-074-08a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-074-09" xlink:href="note-074-09a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-074-10" xlink:href="note-074-10a" xml:space="preserve">5. huius. <lb/>28. tertij.</note> <note position="right" xlink:label="note-075-01" xlink:href="note-075-01a" xml:space="preserve">Coroll. 16. <lb/>1. huius.</note> <note position="right" xlink:label="note-075-02" xlink:href="note-075-02a" xml:space="preserve">Schol. 21. <lb/>huius.</note> <note position="right" xlink:label="note-075-03" xlink:href="note-075-03a" xml:space="preserve">Schol. 21. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div227" type="section" level="1" n="108"> <head xml:id="echoid-head120" xml:space="preserve">THEOR. 21. PROPOS. 23.</head> <note position="right" xml:space="preserve">34.</note> <p> <s xml:id="echoid-s2493" xml:space="preserve">IISDEM poſitis, ſi circunferétiæ circulorum <lb/>tangentium à contactibus ad nodos ſint æqua-<lb/>les;</s> <s xml:id="echoid-s2494" xml:space="preserve">prædicti circuli maximi ſimiliter inclinati erút.</s> <s xml:id="echoid-s2495" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2496" xml:space="preserve">RVRSVS in ſphæra maximus circulus A B C D, cuius polus E, tangat <lb/>circulum A F, ſecet autem alium huic parallelum G B H D, poſitum inter <lb/>ſphæræ centrum, & </s> <s xml:id="echoid-s2497" xml:space="preserve">circulum A F, ita vt G B H D, maior ſit, quàm A F; </s> <s xml:id="echoid-s2498" xml:space="preserve">ſit-<lb/>que E, polus maximi circuli A B C D, inter vtrumque circulum A F, G B H D: <lb/></s> <s xml:id="echoid-s2499" xml:space="preserve"> <anchor type="figure" xlink:label="fig-075-01a" xlink:href="fig-075-01"/> Tangãt deinde in punctis <lb/>M, N, circuli maximi <lb/>M O, N P, circulũ G B H D, <lb/>ſecantes A B C D, in O, <lb/>P, nodis, ſintq́ue arcus <lb/>M O, N P, æquales. </s> <s xml:id="echoid-s2500" xml:space="preserve">Di-<lb/>co circulos M O, N P, <lb/>ſimiliter inclinari ad ma-<lb/>ximum circulum A B C D. <lb/></s> <s xml:id="echoid-s2501" xml:space="preserve">Ducatur enim per E,po-<lb/> <anchor type="note" xlink:label="note-075-05a" xlink:href="note-075-05"/> lum circuli A B C D, & </s> <s xml:id="echoid-s2502" xml:space="preserve">I, <lb/>polum parallelorum cir-<lb/>culus maximus G A C: </s> <s xml:id="echoid-s2503" xml:space="preserve">Itẽ <lb/>per I, polum parallelorũ, <lb/>& </s> <s xml:id="echoid-s2504" xml:space="preserve">puncta contactuum cir <lb/>culi maximi I M, I N, qui <lb/>per polos quoque circu-<lb/> <anchor type="note" xlink:label="note-075-06a" xlink:href="note-075-06"/> lorum tangentium tran-<lb/>ſibũt;</s> <s xml:id="echoid-s2505" xml:space="preserve">atque adeo ipſos ad <lb/>angulos rectos ſecabunt. <lb/></s> <s xml:id="echoid-s2506" xml:space="preserve"> <anchor type="note" xlink:label="note-075-07a" xlink:href="note-075-07"/> Quoniam igitur ſegmenta circulorum æqualia, nempe ſemicirculi, qui ten-<lb/>dunt ex M, & </s> <s xml:id="echoid-s2507" xml:space="preserve">N, per I, donec iterum ſecent circulos tangentes M O, N P, <lb/>inſiſtunt diametris circulorum M O, N P, (eſt enim communis ſectio circu- <pb o="64" file="076" n="76" rhead=""/> lorum maximorum I M, M O, diameter vtriuſque, cum ſe mutuo ſecent bifa-<lb/> <anchor type="note" xlink:label="note-076-01a" xlink:href="note-076-01"/> riam) ad angulos rectos, & </s> <s xml:id="echoid-s2508" xml:space="preserve">diuiduntur non bifariam in I, quod I, polus paral-<lb/>lelorum non ſit polus tangentium; </s> <s xml:id="echoid-s2509" xml:space="preserve">ponunturque arcus M O, N P, æquales; <lb/></s> <s xml:id="echoid-s2510" xml:space="preserve">erunt ductæ rectæ I O, I B, æquales. </s> <s xml:id="echoid-s2511" xml:space="preserve">Si igitur ex I, polo parallelus deſcriba-<lb/> <anchor type="note" xlink:label="note-076-02a" xlink:href="note-076-02"/> tur O K, ad interuallum I O, tranſibit is quoque per P. </s> <s xml:id="echoid-s2512" xml:space="preserve">Et quia circulus <lb/>maximus I M, tranſiens per polos circulorum M O, O Q, ſe ſecantium in <lb/>O, Q, ſecat eorum ſegmenta bifariam, æquales erunt arcus M O, M Q, & </s> <s xml:id="echoid-s2513" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-076-03a" xlink:href="note-076-03"/> S O, S Q; </s> <s xml:id="echoid-s2514" xml:space="preserve">Eodemque argumento æquales erunt arcus N P, N R, & </s> <s xml:id="echoid-s2515" xml:space="preserve">T P, <lb/>T R; </s> <s xml:id="echoid-s2516" xml:space="preserve">nec non K O, K P, & </s> <s xml:id="echoid-s2517" xml:space="preserve">C O, C P; </s> <s xml:id="echoid-s2518" xml:space="preserve">propterea quòd circulus maximus IkC, <lb/>tranſiens per polos circulorum O K P, O C P, ſecat eorum ſegmenta bifa-<lb/> <anchor type="note" xlink:label="note-076-04a" xlink:href="note-076-04"/> riam in K, & </s> <s xml:id="echoid-s2519" xml:space="preserve">C. </s> <s xml:id="echoid-s2520" xml:space="preserve">Cum ergo arcus M O, N P, ponantur æquales, erunt & </s> <s xml:id="echoid-s2521" xml:space="preserve">toti <lb/> <anchor type="figure" xlink:label="fig-076-01a" xlink:href="fig-076-01"/> O M Q, P N R, quorum <lb/>ipſi dimidij ſunt, æqua-<lb/>les; </s> <s xml:id="echoid-s2522" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s2523" xml:space="preserve">rectæ <lb/> <anchor type="note" xlink:label="note-076-05a" xlink:href="note-076-05"/> ſubtenſę O Q, P R, æqua <lb/>les erunt. </s> <s xml:id="echoid-s2524" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2525" xml:space="preserve">arcus <lb/> <anchor type="note" xlink:label="note-076-06a" xlink:href="note-076-06"/> O S Q, P T R, ęquales <lb/>erunt; </s> <s xml:id="echoid-s2526" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s2527" xml:space="preserve">eo-<lb/>rum dimidij O S, P T, æ-<lb/>quales erunt. </s> <s xml:id="echoid-s2528" xml:space="preserve">Sunt autem <lb/>& </s> <s xml:id="echoid-s2529" xml:space="preserve">toti K O, K P, oſtenſi <lb/>ęquales. </s> <s xml:id="echoid-s2530" xml:space="preserve">Reliqui ergo K S, <lb/>K T, æquales erunt; </s> <s xml:id="echoid-s2531" xml:space="preserve">atque <lb/>adeo, cum ſint vnius eiuſ-<lb/>demq́ue circuli, ſimiles in <lb/>ter ſe erunt. </s> <s xml:id="echoid-s2532" xml:space="preserve">Quia verò ar-<lb/> <anchor type="note" xlink:label="note-076-07a" xlink:href="note-076-07"/> cubus K S, K T, ſimiles <lb/>ſunt arcus H M, H N; </s> <s xml:id="echoid-s2533" xml:space="preserve">erũt <lb/>quoq;</s> <s xml:id="echoid-s2534" xml:space="preserve">æquales arcus H M, <lb/>H N. </s> <s xml:id="echoid-s2535" xml:space="preserve">Itaque cum ſeg-<lb/>mentum B H D, bifariam <lb/> <anchor type="note" xlink:label="note-076-08a" xlink:href="note-076-08"/> ſeceturin H, fintque equales arcus H M, H N; </s> <s xml:id="echoid-s2536" xml:space="preserve">erunt circuli M O, N P, ſimili-<lb/>ter inclinati ad circulum A B C D. </s> <s xml:id="echoid-s2537" xml:space="preserve">Quare ijſdem poſitis, ſi circunferentiæ à <lb/>contactibus, &</s> <s xml:id="echoid-s2538" xml:space="preserve">c. </s> <s xml:id="echoid-s2539" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2540" xml:space="preserve"/> </p> <div xml:id="echoid-div227" type="float" level="2" n="1"> <figure xlink:label="fig-075-01" xlink:href="fig-075-01a"> <image file="075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/075-01"/> </figure> <note position="right" xlink:label="note-075-05" xlink:href="note-075-05a" xml:space="preserve">20. 1. huius.</note> <note position="right" xlink:label="note-075-06" xlink:href="note-075-06a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-075-07" xlink:href="note-075-07a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-076-01" xlink:href="note-076-01a" xml:space="preserve">@1. 1. huius.</note> <note position="left" xlink:label="note-076-02" xlink:href="note-076-02a" xml:space="preserve">12. 1. huius.</note> <note position="left" xlink:label="note-076-03" xlink:href="note-076-03a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-076-04" xlink:href="note-076-04a" xml:space="preserve">9. huius.</note> <figure xlink:label="fig-076-01" xlink:href="fig-076-01a"> <image file="076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/076-01"/> </figure> <note position="left" xlink:label="note-076-05" xlink:href="note-076-05a" xml:space="preserve">29. tertij.</note> <note position="left" xlink:label="note-076-06" xlink:href="note-076-06a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-076-07" xlink:href="note-076-07a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-076-08" xlink:href="note-076-08a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div229" type="section" level="1" n="109"> <head xml:id="echoid-head121" xml:space="preserve">FINIS LIBRI I I. THEODOSII.</head> <pb o="65" file="077" n="77"/> </div> <div xml:id="echoid-div230" type="section" level="1" n="110"> <head xml:id="echoid-head122" xml:space="preserve">THEODOSII</head> <head xml:id="echoid-head123" xml:space="preserve">SPHAERICORVM</head> <head xml:id="echoid-head124" xml:space="preserve">LIBER TERTIVS.</head> <figure> <image file="077-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/077-01"/> </figure> </div> <div xml:id="echoid-div231" type="section" level="1" n="111"> <head xml:id="echoid-head125" xml:space="preserve">THEOREMA 1. PROPOS. 1.</head> <p> <s xml:id="echoid-s2541" xml:space="preserve">SI recta linea circulum in partes inæ-<lb/>quales ſecet, ſuper qua conſtituatur re <lb/>ctum circuli ſegmentum, quod non <lb/>ſit maius ſemicirculo; </s> <s xml:id="echoid-s2542" xml:space="preserve">diuidatur au-<lb/>tem ſegmenti inſiſtentis circunferentia in duas in <lb/>æquales partes: </s> <s xml:id="echoid-s2543" xml:space="preserve">Recta linea ſubtendens earum mi-<lb/>norem, minima eſt linearum rectarum ductarum <lb/>ab eodem puncto ad minorem partem circunfe-<lb/>rentiæ primi circuli: </s> <s xml:id="echoid-s2544" xml:space="preserve">Rectarum verò ductarum ab <lb/>eo ipſo puncto ad circunferentiam interceptam <lb/>inter illam minimam rectam, & </s> <s xml:id="echoid-s2545" xml:space="preserve">diametrum, in <lb/>quam cadit perpendicularis deducta ab illo pun-<lb/>cto ſemper minimæ propior remotiore minor eſt. <lb/></s> <s xml:id="echoid-s2546" xml:space="preserve">Omnium autem maxima eſt ea, quæ ab illo eodẽ <lb/>puncto ducitur ad extremitatem eiuſdem diame-<lb/>tri: </s> <s xml:id="echoid-s2547" xml:space="preserve">Item recta ſubtendens maiorem circunferen-<lb/>tiam ſegmenti inſiſtentis, minima eſt earum, quæ <lb/>cadunt in circunferentiam interceptam inter ip-<lb/>ſam, & </s> <s xml:id="echoid-s2548" xml:space="preserve">diametrum, ſemperque huic propior remo <pb o="66" file="078" n="78" rhead=""/> tiore minor eſt. </s> <s xml:id="echoid-s2549" xml:space="preserve">Si verò recta linea ſubiectum circu <lb/>lum ſecans ſit eius diameter, & </s> <s xml:id="echoid-s2550" xml:space="preserve">reliqua omnia ea-<lb/>dem ſint, vt ſupra; </s> <s xml:id="echoid-s2551" xml:space="preserve">recta linea ſubtendens mino-<lb/>rem partem circunferentiæ ſegmenti inſiſtentis, <lb/>minima eſt rectarum ductarum ab illo eodem <lb/>puncto ad primi, & </s> <s xml:id="echoid-s2552" xml:space="preserve">ſubiecti circuli circunferen-<lb/>tiam; </s> <s xml:id="echoid-s2553" xml:space="preserve">ea verò, quæ maiorem partem circunferen-<lb/>tiæ ſegmenti inſiſtentis ſubtendit, maxima eſt.</s> <s xml:id="echoid-s2554" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2555" xml:space="preserve">RECTA linea A B, ſecet circulum A C B D, cuius centrum E, in partes <lb/> <anchor type="figure" xlink:label="fig-078-01a" xlink:href="fig-078-01"/> inæquales, quarum maior ſit A C B: </s> <s xml:id="echoid-s2556" xml:space="preserve">Inſiſtat <lb/>autem ipſi A B, rectum circuli ſegmentum <lb/>A F B, ſemicirculonó maius, quod in partes in-<lb/>æquales diuidatur in F; </s> <s xml:id="echoid-s2557" xml:space="preserve">ſitque minor pars B F: <lb/></s> <s xml:id="echoid-s2558" xml:space="preserve">Ex F, demittatur in circulum A C B D, per-<lb/> <anchor type="note" xlink:label="note-078-01a" xlink:href="note-078-01"/> pendicularis F L, quæ in A B, communem ſe-<lb/> <anchor type="note" xlink:label="note-078-02a" xlink:href="note-078-02"/> ctionem cadet: </s> <s xml:id="echoid-s2559" xml:space="preserve">Per E, autem, & </s> <s xml:id="echoid-s2560" xml:space="preserve">L, diameter <lb/>agatur C D; </s> <s xml:id="echoid-s2561" xml:space="preserve">& </s> <s xml:id="echoid-s2562" xml:space="preserve">ex F, in circunferentiam A C B, <lb/>maioris ſegmenti circuli A C B D, plurimę <lb/>rectæ cadãt F B, F G, F H, F C, F A, F I, F K. <lb/></s> <s xml:id="echoid-s2563" xml:space="preserve">Dico omnium minimam eſſe F B, & </s> <s xml:id="echoid-s2564" xml:space="preserve">F G, mino-<lb/>rem, quàm F H; </s> <s xml:id="echoid-s2565" xml:space="preserve">Omnium autem maximam eſ <lb/>ſe F C. </s> <s xml:id="echoid-s2566" xml:space="preserve">Item F A, eſſe omnium minimam, quæ <lb/>ex F, in portioncm A C, cadent; </s> <s xml:id="echoid-s2567" xml:space="preserve">& </s> <s xml:id="echoid-s2568" xml:space="preserve">F I, minorem, quàm F K. </s> <s xml:id="echoid-s2569" xml:space="preserve">Ducantur ex L, <lb/>lineæ rectę L G, L H, L I, L K; </s> <s xml:id="echoid-s2570" xml:space="preserve">eruntq́ue ex defin. </s> <s xml:id="echoid-s2571" xml:space="preserve">3. </s> <s xml:id="echoid-s2572" xml:space="preserve">lib. </s> <s xml:id="echoid-s2573" xml:space="preserve">11. </s> <s xml:id="echoid-s2574" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2575" xml:space="preserve">omnes an-<lb/>guli ad L, quos recta F L, facit, recti. </s> <s xml:id="echoid-s2576" xml:space="preserve">Quoniam igitur recta L D, eſt omnium <lb/>rectarum ex L, cadentium minima, & </s> <s xml:id="echoid-s2577" xml:space="preserve">L B, minor, quàm L G, L H, L C, L K, <lb/> <anchor type="note" xlink:label="note-078-03a" xlink:href="note-078-03"/> L I, L A; </s> <s xml:id="echoid-s2578" xml:space="preserve">erunt quadrata ex F L, L B, minora quadratis ex F L, L G: </s> <s xml:id="echoid-s2579" xml:space="preserve">Eſt au-<lb/>tem tam quadratum ex F B, quadratis ex F L, L B, quàm quadratum ex F G, <lb/> <anchor type="note" xlink:label="note-078-04a" xlink:href="note-078-04"/> quadratis ex F L, L G,æquale. </s> <s xml:id="echoid-s2580" xml:space="preserve">Igitur erit quoq; </s> <s xml:id="echoid-s2581" xml:space="preserve">quadratum ex F B, minus qua-<lb/>drato ex F G; </s> <s xml:id="echoid-s2582" xml:space="preserve">atq; </s> <s xml:id="echoid-s2583" xml:space="preserve">adeo & </s> <s xml:id="echoid-s2584" xml:space="preserve">recta F B, minor erit quàm F G. </s> <s xml:id="echoid-s2585" xml:space="preserve">Nõ aliter oſtẽdemus, <lb/>rectá F B, minoré eſſe, quàm F H, F C, F K, F I, F A. </s> <s xml:id="echoid-s2586" xml:space="preserve">Quare F B, omniũ minima eſt.</s> <s xml:id="echoid-s2587" xml:space="preserve"/> </p> <div xml:id="echoid-div231" type="float" level="2" n="1"> <figure xlink:label="fig-078-01" xlink:href="fig-078-01a"> <image file="078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/078-01"/> </figure> <note position="left" xlink:label="note-078-01" xlink:href="note-078-01a" xml:space="preserve">11. vndec.</note> <note position="left" xlink:label="note-078-02" xlink:href="note-078-02a" xml:space="preserve">38. vndec.</note> <note position="left" xlink:label="note-078-03" xlink:href="note-078-03a" xml:space="preserve">7. tertij.</note> <note position="left" xlink:label="note-078-04" xlink:href="note-078-04a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s2588" xml:space="preserve">RVRSVS quia L G, minor eſt, quàm L H, erunt quadrata ex F L, L G, <lb/> <anchor type="note" xlink:label="note-078-05a" xlink:href="note-078-05"/> minora quadratis ex F L, L H: </s> <s xml:id="echoid-s2589" xml:space="preserve">Eſt autem tam quadratum ex F G, quadratis <lb/>ex F L, L G, quàm quadratum ex F H, quadratis ex F L, L H, æquale. </s> <s xml:id="echoid-s2590" xml:space="preserve">Igitur <lb/> <anchor type="note" xlink:label="note-078-06a" xlink:href="note-078-06"/> & </s> <s xml:id="echoid-s2591" xml:space="preserve">quadratum ex F G, quadrato ex F H, minus erit; </s> <s xml:id="echoid-s2592" xml:space="preserve">atq; </s> <s xml:id="echoid-s2593" xml:space="preserve">adeo & </s> <s xml:id="echoid-s2594" xml:space="preserve">recta F G, mi-<lb/>nor erit, quàm recta F H.</s> <s xml:id="echoid-s2595" xml:space="preserve"/> </p> <div xml:id="echoid-div232" type="float" level="2" n="2"> <note position="left" xlink:label="note-078-05" xlink:href="note-078-05a" xml:space="preserve">7. tertij.</note> <note position="left" xlink:label="note-078-06" xlink:href="note-078-06a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s2596" xml:space="preserve">AMPLIVS quia L C, omnium ex L, cadentium maxima eſt;</s> <s xml:id="echoid-s2597" xml:space="preserve">erunt qua-<lb/> <anchor type="note" xlink:label="note-078-07a" xlink:href="note-078-07"/> drata ex F L, L C, maiora quadratis ex F L, L K: </s> <s xml:id="echoid-s2598" xml:space="preserve">Eſt autem tam quadratum <lb/>ex F C, quadratis ex F L, L C, quàm quadratum ex Fk, quadratis ex F L, Lk, <lb/> <anchor type="note" xlink:label="note-078-08a" xlink:href="note-078-08"/> æquale. </s> <s xml:id="echoid-s2599" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2600" xml:space="preserve">qua dratum ex F C, maius erit quadrato ex F K;</s> <s xml:id="echoid-s2601" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s2602" xml:space="preserve"><lb/>recta F C, maior erit, quàm recta F K. </s> <s xml:id="echoid-s2603" xml:space="preserve">Non aliter demonſtrabimus, rectam F C, <lb/>maiorem eſſe, quàm F I, & </s> <s xml:id="echoid-s2604" xml:space="preserve">F A. </s> <s xml:id="echoid-s2605" xml:space="preserve">Eſt ergo recta F C, omnium maxima.</s> <s xml:id="echoid-s2606" xml:space="preserve"/> </p> <div xml:id="echoid-div233" type="float" level="2" n="3"> <note position="left" xlink:label="note-078-07" xlink:href="note-078-07a" xml:space="preserve">7. tertij.</note> <note position="left" xlink:label="note-078-08" xlink:href="note-078-08a" xml:space="preserve">47. primi.</note> </div> <pb o="67" file="079" n="79" rhead=""/> <p> <s xml:id="echoid-s2607" xml:space="preserve">ITEM quia L A, minor eſt, quàm L I, Lk, L C; </s> <s xml:id="echoid-s2608" xml:space="preserve">erunt quadrata ex F L, <lb/> <anchor type="note" xlink:label="note-079-01a" xlink:href="note-079-01"/> L A, minora quadratis ex F L, L I: </s> <s xml:id="echoid-s2609" xml:space="preserve">Eſt autem tam quadratum ex F A, qua-<lb/>dratis ex F L, L A, quam quadratum ex F I, quadratis ex F L, L I, æqua-<lb/> <anchor type="note" xlink:label="note-079-02a" xlink:href="note-079-02"/> le. </s> <s xml:id="echoid-s2610" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2611" xml:space="preserve">quadratum ex F A, minus erit quadrato ex F I; </s> <s xml:id="echoid-s2612" xml:space="preserve">atque ob id re-<lb/>cta quoque F A, minor erit quàm recta F I. </s> <s xml:id="echoid-s2613" xml:space="preserve">Eodem modo oſtendemus, rectam <lb/>F A, maiorem eſſe, quàm F K, F C. </s> <s xml:id="echoid-s2614" xml:space="preserve">Eſt ergo F A, omnium rectarum ex F, in <lb/>arcum A C, cadentium minima.</s> <s xml:id="echoid-s2615" xml:space="preserve"/> </p> <div xml:id="echoid-div234" type="float" level="2" n="4"> <note position="right" xlink:label="note-079-01" xlink:href="note-079-01a" xml:space="preserve">7. tertij.</note> <note position="right" xlink:label="note-079-02" xlink:href="note-079-02a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s2616" xml:space="preserve">DENIQVE quia L I, minor eſt, quàm L K; </s> <s xml:id="echoid-s2617" xml:space="preserve">erunt quadrata ex F L, L I, <lb/> <anchor type="note" xlink:label="note-079-03a" xlink:href="note-079-03"/> minora quadratis ex F L, L K: </s> <s xml:id="echoid-s2618" xml:space="preserve">Eſtautem tam quadratum ex F I, quadra-<lb/>tis ex F L, L I, quàm quadratum ex F K, quadratis ex F L, L K, æquale. </s> <s xml:id="echoid-s2619" xml:space="preserve">Igi-<lb/> <anchor type="note" xlink:label="note-079-04a" xlink:href="note-079-04"/> tur & </s> <s xml:id="echoid-s2620" xml:space="preserve">quadratum ex F I, minus erit quadrato ex F K, ideoque & </s> <s xml:id="echoid-s2621" xml:space="preserve">recta F I, mi-<lb/>nor erit, quàm recta F K.</s> <s xml:id="echoid-s2622" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="5"> <note position="right" xlink:label="note-079-03" xlink:href="note-079-03a" xml:space="preserve">7. tertij.</note> <note position="right" xlink:label="note-079-04" xlink:href="note-079-04a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s2623" xml:space="preserve">QVOD ſi recta A B, ſecet circulum A C B D, bifariam, ita vt ſit eius <lb/>diameter, demonſtratum à nobis iam eſt theoremate tertio ſcholij propoſ.</s> <s xml:id="echoid-s2624" xml:space="preserve">21. <lb/></s> <s xml:id="echoid-s2625" xml:space="preserve">præcedentis libri, rectam F B, minimam eſſe, & </s> <s xml:id="echoid-s2626" xml:space="preserve">F A, maximam. </s> <s xml:id="echoid-s2627" xml:space="preserve">Vnde non eſt <lb/>neceſſe, idem hoc loco demonſtrare. </s> <s xml:id="echoid-s2628" xml:space="preserve">Immo plura ibi ſunt demonſtrata, quàm <lb/>hic proponuntur. </s> <s xml:id="echoid-s2629" xml:space="preserve">Sirecta igitur linea circulum in partes inæquales ſecet, &</s> <s xml:id="echoid-s2630" xml:space="preserve">c. </s> <s xml:id="echoid-s2631" xml:space="preserve"><lb/>Quod oſtendendum erat.</s> <s xml:id="echoid-s2632" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div237" type="section" level="1" n="112"> <head xml:id="echoid-head126" xml:space="preserve">THEOREMA 2. PROPOS. 2.</head> <note position="right" xml:space="preserve">30. Secundi <lb/>huius.</note> <p> <s xml:id="echoid-s2633" xml:space="preserve">SI recta linea ſecans circulum ſegmentum au-<lb/>ferat, quod ſemicirculo minus non ſit, ſuper ipſa <lb/>autem recta linea ſtatuatur aliud circuli ſegmen-<lb/>tum, quod & </s> <s xml:id="echoid-s2634" xml:space="preserve">ſemicirculo maius non ſit, & </s> <s xml:id="echoid-s2635" xml:space="preserve">incli-<lb/>natum ſit ad alterum ſegmentum, quod ſemicircu <lb/>lo maius non eſt; </s> <s xml:id="echoid-s2636" xml:space="preserve">diuidatur vero inſiſtentis ſeg-<lb/>menti circunferentia in partes inæquales: </s> <s xml:id="echoid-s2637" xml:space="preserve">Recta <lb/>linea ſubtendens minorem circunferentiæ partem <lb/>minima eſtrectarum omnium ductarum ab illo <lb/>puncto, à quo ipſa ducitur, ad ſubiecti circuli cir-<lb/>cunferentiam illam, quæ ſemicirculo minor non <lb/>eſt: </s> <s xml:id="echoid-s2638" xml:space="preserve">& </s> <s xml:id="echoid-s2639" xml:space="preserve">reliqua omnia, quæ in præcedẽti, ſequuntur.</s> <s xml:id="echoid-s2640" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2641" xml:space="preserve">RECTA linea A B, à circulo A C B D, cuius centrum E, auferat ſeg-<lb/>mentum A C B, ſemicirculo non minus, ſed vel ſemicirculo æquale, vt in pri-<lb/>ma figura, vel maius, vt in alijs figuris; </s> <s xml:id="echoid-s2642" xml:space="preserve">& </s> <s xml:id="echoid-s2643" xml:space="preserve">ſuper recta A B, ſtatuatur ſegmen-<lb/>tum aliud circuli A F B, ſemicirculo non maius, ſed vel ſemicirculo æquale, <lb/>vt in poſtrema trium figurarum, vel minus, vt in primis duabus figuris, & </s> <s xml:id="echoid-s2644" xml:space="preserve">in-<lb/>clinatum ad ſegmentum alterum A D B, quod ſemicirculo maius non eſt, cum <lb/>A C B, vel ſemicirculo æquale, vel maius ponatur. </s> <s xml:id="echoid-s2645" xml:space="preserve">Diuidatur quoque cir- <pb o="68" file="080" n="80" rhead=""/> cunferentia A F B, in F, in partes inæquales, & </s> <s xml:id="echoid-s2646" xml:space="preserve">ſit F B, minor. </s> <s xml:id="echoid-s2647" xml:space="preserve">Ex F, demitta-<lb/>tur in planum circuli A C B D, perpendicularis F L, quæ ad partes ſegmenti <lb/>A D B, cadet, propterea quod ſegmentum A F B, ad ſegmentum A D C, eſt <lb/>inclinatum, ita vt punctum L, ſit vel intra ſegmentum A D B, vel extra, vel <lb/>certe in ipſa circunferentia A D B. </s> <s xml:id="echoid-s2648" xml:space="preserve">Per centrum autem E, & </s> <s xml:id="echoid-s2649" xml:space="preserve">punctum L, dia-<lb/>meter agatur C D, & </s> <s xml:id="echoid-s2650" xml:space="preserve">ex F, in circunferentiam A C B, plurimæ rectæ cadant <lb/>F B, F G, &</s> <s xml:id="echoid-s2651" xml:space="preserve">c. </s> <s xml:id="echoid-s2652" xml:space="preserve">Dico omnium minimam eſſe F B; </s> <s xml:id="echoid-s2653" xml:space="preserve">& </s> <s xml:id="echoid-s2654" xml:space="preserve">F G, minorem quàm F H: <lb/></s> <s xml:id="echoid-s2655" xml:space="preserve">omnium autem maximam eſſe F C: </s> <s xml:id="echoid-s2656" xml:space="preserve">Item F A, eſſe omnium minimam, quæ ex <lb/>F, in circunferentiam A C, cadunt; </s> <s xml:id="echoid-s2657" xml:space="preserve">& </s> <s xml:id="echoid-s2658" xml:space="preserve">F I, minorem quàm F K. </s> <s xml:id="echoid-s2659" xml:space="preserve">Ducantur ex <lb/>L, rectæ lineæ L B, L G, L H, L A, L I, L K, eruntque omnes anguli ad L, <lb/>quos facit perpendicularis F L, recti, ex defin. </s> <s xml:id="echoid-s2660" xml:space="preserve">3. </s> <s xml:id="echoid-s2661" xml:space="preserve">lib. </s> <s xml:id="echoid-s2662" xml:space="preserve">11. </s> <s xml:id="echoid-s2663" xml:space="preserve">Eucl.</s> <s xml:id="echoid-s2664" xml:space="preserve"/> </p> <figure> <image file="080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/080-01"/> </figure> <p> <s xml:id="echoid-s2665" xml:space="preserve">Quoniam igitur recta L D, eſt omnium minima, (hæc autem linea nihil eſt om <lb/> <anchor type="note" xlink:label="note-080-01a" xlink:href="note-080-01"/> nino in ea figura, vbi punctum L, cadit in D.) </s> <s xml:id="echoid-s2666" xml:space="preserve">& </s> <s xml:id="echoid-s2667" xml:space="preserve">L B, minor, quàm L G, L H, <lb/>L C, L K, L I, L A, & </s> <s xml:id="echoid-s2668" xml:space="preserve">omnium maxima L C, &</s> <s xml:id="echoid-s2669" xml:space="preserve">c. </s> <s xml:id="echoid-s2670" xml:space="preserve">demonſtrabimus, vt in præ-<lb/> <anchor type="note" xlink:label="note-080-02a" xlink:href="note-080-02"/> cedenti, rectam F B, eſſe omnium minimam, & </s> <s xml:id="echoid-s2671" xml:space="preserve">F G, minorem quàm F H: </s> <s xml:id="echoid-s2672" xml:space="preserve">Item <lb/>F C, omnium maximam, & </s> <s xml:id="echoid-s2673" xml:space="preserve">F A, minimam omnium ex F, in circunferentiam <lb/>A C, cadentium; </s> <s xml:id="echoid-s2674" xml:space="preserve">& </s> <s xml:id="echoid-s2675" xml:space="preserve">F I, minorem quàm F K. </s> <s xml:id="echoid-s2676" xml:space="preserve">Si igitur recta linea ſecans circu-<lb/>lum, &</s> <s xml:id="echoid-s2677" xml:space="preserve">c. </s> <s xml:id="echoid-s2678" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s2679" xml:space="preserve"/> </p> <div xml:id="echoid-div237" type="float" level="2" n="1"> <note position="left" xlink:label="note-080-01" xlink:href="note-080-01a" xml:space="preserve">7. vel 8. vel <lb/>15. tertil.</note> <note position="left" xlink:label="note-080-02" xlink:href="note-080-02a" xml:space="preserve">7. vel 8. vel <lb/>15. tertij. & <lb/>47. primi.</note> </div> </div> <div xml:id="echoid-div239" type="section" level="1" n="113"> <head xml:id="echoid-head127" xml:space="preserve">THEOREMA 3. PROPOS. 3.</head> <p> <s xml:id="echoid-s2680" xml:space="preserve">SI in ſphæra duo circuli maximi ſe mutuo ſe-<lb/>cent, ab eorum verò vtroque æquales circunfe-<lb/>rentiæ ſumantur vtrinque à puncto, in quo ſe ſe-<lb/>cant: </s> <s xml:id="echoid-s2681" xml:space="preserve">Rectæ lineæ, quæ extrema puncta circunfe-<lb/>rentiarum connectunt ad eaſdem partes, æquales <lb/>inter ſe ſunt.</s> <s xml:id="echoid-s2682" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2683" xml:space="preserve">IN ſphæra duo circuli maximi A B C, D B E, ſe mutuo ſecent in B, & </s> <s xml:id="echoid-s2684" xml:space="preserve">in <lb/>vno quoque vtrinque à B, ſumantur duo arcus æquales B A, B C, & </s> <s xml:id="echoid-s2685" xml:space="preserve">B D, B E, <pb o="69" file="081" n="81" rhead=""/> ſunganturq́ue rectæ A D, C E. </s> <s xml:id="echoid-s2686" xml:space="preserve">Dico rectas A D, C E, æquales eſſe. </s> <s xml:id="echoid-s2687" xml:space="preserve">Polo enim <lb/>B, & </s> <s xml:id="echoid-s2688" xml:space="preserve">interuallo B A, circulus deſcribatur, qui etiam per C, tranſibit, ob æqua <lb/>litatem arcuum B A, B C. </s> <s xml:id="echoid-s2689" xml:space="preserve">Aut igitur idem circulus tranſit etiam per C, atque <lb/> <anchor type="figure" xlink:label="fig-081-01a" xlink:href="fig-081-01"/> adeo & </s> <s xml:id="echoid-s2690" xml:space="preserve">per E, ob æquali-<lb/>tatem arcuum B D, B E, <lb/>aut non. </s> <s xml:id="echoid-s2691" xml:space="preserve">Tranſeat primũ <lb/>per D, & </s> <s xml:id="echoid-s2692" xml:space="preserve">E, vt in priori <lb/>figura; </s> <s xml:id="echoid-s2693" xml:space="preserve">ſintq́ue communes <lb/>ſectiones circulorum ma-<lb/>ximorũ, & </s> <s xml:id="echoid-s2694" xml:space="preserve">circuli A D C E, <lb/>rectæ A C, D E. </s> <s xml:id="echoid-s2695" xml:space="preserve">Et quo-<lb/>niã circuli maximi A B C, <lb/>D B E, per B, polum cir-<lb/>culi A D C E, tranſeun-<lb/>tes ſecant ipſum bifariã, <lb/> <anchor type="note" xlink:label="note-081-01a" xlink:href="note-081-01"/> erunt A C, D E, diametri circuli A D C E, & </s> <s xml:id="echoid-s2696" xml:space="preserve">F, centrum; </s> <s xml:id="echoid-s2697" xml:space="preserve">ac proinde rectæ <lb/>F A, F D, rectis F C, F E, æquales. </s> <s xml:id="echoid-s2698" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s2699" xml:space="preserve">angulos æquales compre-<lb/> <anchor type="note" xlink:label="note-081-02a" xlink:href="note-081-02"/> hendant ad verticem F; </s> <s xml:id="echoid-s2700" xml:space="preserve">erunt & </s> <s xml:id="echoid-s2701" xml:space="preserve">rectæ A D, C E, æquales.</s> <s xml:id="echoid-s2702" xml:space="preserve"/> </p> <div xml:id="echoid-div239" type="float" level="2" n="1"> <figure xlink:label="fig-081-01" xlink:href="fig-081-01a"> <image file="081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/081-01"/> </figure> <note position="right" xlink:label="note-081-01" xlink:href="note-081-01a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-081-02" xlink:href="note-081-02a" xml:space="preserve">15. primi.</note> </div> <note position="right" xml:space="preserve">4. primi.</note> <p> <s xml:id="echoid-s2703" xml:space="preserve">SED non tranſeat iam circulus ex B, polo deſcriptus ad interuallum B A, <lb/>per D, ſed vltra punctum D, atque adeò & </s> <s xml:id="echoid-s2704" xml:space="preserve">vltra punctum E, excurrat. </s> <s xml:id="echoid-s2705" xml:space="preserve">Produ-<lb/> <anchor type="note" xlink:label="note-081-04a" xlink:href="note-081-04"/> cantur arcus B D, B E, ad G, H. </s> <s xml:id="echoid-s2706" xml:space="preserve">Quoniam igitur arcus B G, B H, æquales <lb/>ſunt, quòd ex defin. </s> <s xml:id="echoid-s2707" xml:space="preserve">poli, rectæ ſubtenſæ B G, B H, æquales ſint: </s> <s xml:id="echoid-s2708" xml:space="preserve">Sunt autem <lb/>& </s> <s xml:id="echoid-s2709" xml:space="preserve">B D, B E, ex hypotheſi, æquales; </s> <s xml:id="echoid-s2710" xml:space="preserve">erunt & </s> <s xml:id="echoid-s2711" xml:space="preserve">reliqui D G, E H, æquales. </s> <s xml:id="echoid-s2712" xml:space="preserve">Et <lb/>quoniam rectæ ductæ A G, C H, æquales ſunt, vt proxime demonſtratum eſt <lb/>in prima parte huius propoſ. </s> <s xml:id="echoid-s2713" xml:space="preserve">erunt & </s> <s xml:id="echoid-s2714" xml:space="preserve">arcus A G, C H, æquales. </s> <s xml:id="echoid-s2715" xml:space="preserve">Quia igitur <lb/> <anchor type="note" xlink:label="note-081-05a" xlink:href="note-081-05"/> circulus maximus G B H, per polum B, ductus ſecat circulum A G C H, bifa-<lb/> <anchor type="note" xlink:label="note-081-06a" xlink:href="note-081-06"/> riam, & </s> <s xml:id="echoid-s2716" xml:space="preserve">ad angulos rectos, inſiſtet ſegmentum G H, rectum diametro circuli <lb/>AGCH. </s> <s xml:id="echoid-s2717" xml:space="preserve">Cum ergo arcus D G, E H, æquales ſint, & </s> <s xml:id="echoid-s2718" xml:space="preserve">minores dimidio arcu <lb/>G D H; </s> <s xml:id="echoid-s2719" xml:space="preserve">ſintq́ue arcus G A, H C, oſtenſi quoque æquales; </s> <s xml:id="echoid-s2720" xml:space="preserve">erunt rectę D A, <lb/>E C, inter ſe æquales. </s> <s xml:id="echoid-s2721" xml:space="preserve">Si igitur in ſphæra duo maximi circuliſe mutuo ſecent, <lb/> <anchor type="note" xlink:label="note-081-07a" xlink:href="note-081-07"/> &</s> <s xml:id="echoid-s2722" xml:space="preserve">c. </s> <s xml:id="echoid-s2723" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2724" xml:space="preserve"/> </p> <div xml:id="echoid-div240" type="float" level="2" n="2"> <note position="right" xlink:label="note-081-04" xlink:href="note-081-04a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-081-05" xlink:href="note-081-05a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-081-06" xlink:href="note-081-06a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-081-07" xlink:href="note-081-07a" xml:space="preserve">12. 2. huius.</note> </div> </div> <div xml:id="echoid-div242" type="section" level="1" n="114"> <head xml:id="echoid-head128" xml:space="preserve">THEOREMA 4. PROPOS. 4.</head> <note position="right" xml:space="preserve">2.</note> <p> <s xml:id="echoid-s2725" xml:space="preserve">SI in ſphæra duo maximi circuli ſe mutuo ſe-<lb/>cent, ab eorumque altero æquales circunferen-<lb/>tiæ ſumantur vtrinque à puncto, in quo ſeinterſe-<lb/>cant, & </s> <s xml:id="echoid-s2726" xml:space="preserve">per puncta terminantia æquales circunfe-<lb/>rentias ducantur duo plana parallela, quorum alte <lb/>rum conueniat cum communi ſectione ipſorum <lb/>circulorum extra ſphæram verſus prædictum pun <lb/>ctum; </s> <s xml:id="echoid-s2727" xml:space="preserve">ſit vero vna illarum æqualium circunferen-<lb/>tiarum maior vtralibet circunferentiarum in alte- <pb o="70" file="082" n="82" rhead=""/> ro maximo circulo interceptarum inter prædictũ <lb/>punctum, & </s> <s xml:id="echoid-s2728" xml:space="preserve">vtrumque planorum parallelorum: <lb/></s> <s xml:id="echoid-s2729" xml:space="preserve">Ea circunferentia, quæ eſt inter illud punctum, & </s> <s xml:id="echoid-s2730" xml:space="preserve"><lb/>planum, quod non conuenit cum communi ſe-<lb/>ctione ipſorum circulorum, maior eſt, quam ea <lb/>eiuſdem circuli circunferentia, quæ eſt inter idem <lb/>punctum, & </s> <s xml:id="echoid-s2731" xml:space="preserve">planum, quod conuenit cum com-<lb/>muni ſectione circulorum.</s> <s xml:id="echoid-s2732" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2733" xml:space="preserve">IN ſphæra duo maximi circuli A B C, D B E, ſe mutuo ſecent in B, & </s> <s xml:id="echoid-s2734" xml:space="preserve">in <lb/>A B C, ſumantur arcus B A, B C, æquales, & </s> <s xml:id="echoid-s2735" xml:space="preserve">per A, C, puncta duo plana pa-<lb/>rallela inter ſe ducantur facientia in ſuperficie ſphæræ circunferentias circu <lb/> <anchor type="note" xlink:label="note-082-01a" xlink:href="note-082-01"/> lorum A F G, C H I, quæ ſecent circunferentiam D B E, in punctis F, H; </s> <s xml:id="echoid-s2736" xml:space="preserve">ſit <lb/>verò arcus B A, vel B C, maior vtralibet circunferentiarum B F, B H, inter <lb/>punctum B, & </s> <s xml:id="echoid-s2737" xml:space="preserve">plana parallela interceptarum. </s> <s xml:id="echoid-s2738" xml:space="preserve">Ex polo deinde B, & </s> <s xml:id="echoid-s2739" xml:space="preserve">interual-<lb/>lo B A, vel B C, circulus deſcribatur A D C E, qui puncta F, H, tranſcen-<lb/>det, propterea quòd arcus B F, B H, minores ponuntur arcubus B A, B C. <lb/></s> <s xml:id="echoid-s2740" xml:space="preserve">Producantur arcus B F, B H, vſque ad circunferentiam circuli A D C E, <lb/>ad puncta D, E; </s> <s xml:id="echoid-s2741" xml:space="preserve">ſintq́ue communes ſectiones circuli A D C E, & </s> <s xml:id="echoid-s2742" xml:space="preserve">circulorum <lb/>A F G, C H I, rectæ A G, C I; </s> <s xml:id="echoid-s2743" xml:space="preserve">communes autem ſectiones circulorum ma-<lb/> <anchor type="figure" xlink:label="fig-082-01a" xlink:href="fig-082-01"/> ximorum, & </s> <s xml:id="echoid-s2744" xml:space="preserve">circuli A D C E, rectæ A C, <lb/>D E; </s> <s xml:id="echoid-s2745" xml:space="preserve">quæ ipſius diametri erunt, atque adeo <lb/>eiuſdem centrum K, cum circuli maximi ip-<lb/>ſum per B, polum bifariam ſecent: </s> <s xml:id="echoid-s2746" xml:space="preserve">Secet au-<lb/> <anchor type="note" xlink:label="note-082-02a" xlink:href="note-082-02"/> tem recta D E, rectas A G, C I, in M, N. <lb/></s> <s xml:id="echoid-s2747" xml:space="preserve">Sit quoque maximorum circulorum commu <lb/>nis ſectio K B, recta, cum qua producta ad par <lb/>tes B, conueniat planum A F G, productum <lb/>extra ſphæram in puncto L. </s> <s xml:id="echoid-s2748" xml:space="preserve">Quo poſito, non <lb/>conueniet alterum planum C H I, cum re-<lb/>cta K B, ad partes B, necum ſibi parallelo <lb/>plano A F G, conueniat. </s> <s xml:id="echoid-s2749" xml:space="preserve">Dico arcum B H, <lb/>maiorem eſſe arcu B F. </s> <s xml:id="echoid-s2750" xml:space="preserve">Sint enim rectæ F M, <lb/>H N, communes ſectiones circuli D B E, & </s> <s xml:id="echoid-s2751" xml:space="preserve"><lb/>circulorum A F G, C H I. </s> <s xml:id="echoid-s2752" xml:space="preserve">Et quoniam pla-<lb/>num A F C, conuenit productum cum recta <lb/>K B, producta in L, erit L, punctum tam in <lb/>plano D B E, quàm in plano A F G; </s> <s xml:id="echoid-s2753" xml:space="preserve">atque <lb/>adeo in cõmuni eorum ſectione, nempe in recta M F. </s> <s xml:id="echoid-s2754" xml:space="preserve">Producta ergo M F, coi-<lb/>bit cum K B, producta in L. </s> <s xml:id="echoid-s2755" xml:space="preserve">Quoniam verò planum D B E, ſecat plana pa-<lb/>rallela A F G, C H I, erunt ſectiones factæ M F, N H, parallelæ. </s> <s xml:id="echoid-s2756" xml:space="preserve">Rurſus quia <lb/> <anchor type="note" xlink:label="note-082-03a" xlink:href="note-082-03"/> planum A D C E, eadem plana parallela ſecat, erunt quoque ſectiones factæ <lb/> <anchor type="note" xlink:label="note-082-04a" xlink:href="note-082-04"/> A G, C I, parallelæ. </s> <s xml:id="echoid-s2757" xml:space="preserve">Anguli ergo alterni K A M, K C N, æquales ſunt: <lb/></s> <s xml:id="echoid-s2758" xml:space="preserve"> <anchor type="note" xlink:label="note-082-05a" xlink:href="note-082-05"/> <pb o="71" file="083" n="83" rhead=""/> ſunt autem & </s> <s xml:id="echoid-s2759" xml:space="preserve">anguli A K M, C K N, ad verticem æquales, & </s> <s xml:id="echoid-s2760" xml:space="preserve">latera K A, K C, <lb/> <anchor type="note" xlink:label="note-083-01a" xlink:href="note-083-01"/> æqualia, cum ſint ſemidiametri circuli A D C E. </s> <s xml:id="echoid-s2761" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2762" xml:space="preserve">latera K M, K N, <lb/> <anchor type="note" xlink:label="note-083-02a" xlink:href="note-083-02"/> æqualia erunt:</s> <s xml:id="echoid-s2763" xml:space="preserve">ſunt autem & </s> <s xml:id="echoid-s2764" xml:space="preserve">ſemidiametri K D, K E, æquales. </s> <s xml:id="echoid-s2765" xml:space="preserve">Reliquæ ergo <lb/>rectæ D M, E N, æquales erunt. </s> <s xml:id="echoid-s2766" xml:space="preserve">Rurſus quoniam recta B K, ex B, polo circuli <lb/>A D C E, ad eiuſdem centrum K, ducta, recta eſt ad planum circuli, erit an-<lb/> <anchor type="note" xlink:label="note-083-03a" xlink:href="note-083-03"/> gulus M K L, in triangulo K L M, rectus, ex defin. </s> <s xml:id="echoid-s2767" xml:space="preserve">3. </s> <s xml:id="echoid-s2768" xml:space="preserve">lib. </s> <s xml:id="echoid-s2769" xml:space="preserve">11. </s> <s xml:id="echoid-s2770" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s2771" xml:space="preserve">Angulus igi-<lb/> <anchor type="note" xlink:label="note-083-04a" xlink:href="note-083-04"/> tur K M L, acutus erit. </s> <s xml:id="echoid-s2772" xml:space="preserve">Cum ergo duo anguli F M N, H N M, duobus ſint <lb/> <anchor type="note" xlink:label="note-083-05a" xlink:href="note-083-05"/> rectis æquales; </s> <s xml:id="echoid-s2773" xml:space="preserve">erit angulus H N M, obtuſus. </s> <s xml:id="echoid-s2774" xml:space="preserve">Quare, vt mox, lemmate ſequen <lb/>ti oſtendemus, arcus E H, minor erit, arcu D F; </s> <s xml:id="echoid-s2775" xml:space="preserve">atque adeo, cum æquales <lb/>ſint arcus B D, B E, quòd rectæ ſubtenſæ B D, B E, ex defin. </s> <s xml:id="echoid-s2776" xml:space="preserve">poli, ſint æqua-<lb/> <anchor type="note" xlink:label="note-083-06a" xlink:href="note-083-06"/> les, maior erit arcus B H, arcu B F. </s> <s xml:id="echoid-s2777" xml:space="preserve">Si igitur in ſphæra duo maximi circuli ſe <lb/>mutuo ſecent, &</s> <s xml:id="echoid-s2778" xml:space="preserve">c. </s> <s xml:id="echoid-s2779" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s2780" xml:space="preserve"/> </p> <div xml:id="echoid-div242" type="float" level="2" n="1"> <note position="left" xlink:label="note-082-01" xlink:href="note-082-01a" xml:space="preserve">L. 1. huius.</note> <figure xlink:label="fig-082-01" xlink:href="fig-082-01a"> <image file="082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/082-01"/> </figure> <note position="left" xlink:label="note-082-02" xlink:href="note-082-02a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-082-03" xlink:href="note-082-03a" xml:space="preserve">16. vndee.</note> <note position="left" xlink:label="note-082-04" xlink:href="note-082-04a" xml:space="preserve">16. vndec.</note> <note position="left" xlink:label="note-082-05" xlink:href="note-082-05a" xml:space="preserve">29. primi.</note> <note position="right" xlink:label="note-083-01" xlink:href="note-083-01a" xml:space="preserve">15. primi.</note> <note position="right" xlink:label="note-083-02" xlink:href="note-083-02a" xml:space="preserve">26. primi.</note> <note position="right" xlink:label="note-083-03" xlink:href="note-083-03a" xml:space="preserve">Schol. 8. 1. <lb/>huius.</note> <note position="right" xlink:label="note-083-04" xlink:href="note-083-04a" xml:space="preserve">17. primi.</note> <note position="right" xlink:label="note-083-05" xlink:href="note-083-05a" xml:space="preserve">29. primi.</note> <note position="right" xlink:label="note-083-06" xlink:href="note-083-06a" xml:space="preserve">28. tertij.</note> </div> </div> <div xml:id="echoid-div244" type="section" level="1" n="115"> <head xml:id="echoid-head129" xml:space="preserve">LEMMA.</head> <p style="it"> <s xml:id="echoid-s2781" xml:space="preserve">_QVOD_ autem arcus _E H,_ arcw _D F,_ minor ſit, facile demonſtrabimus, hoc propoſi-<lb/>to theoremate prius demonſtrato.</s> <s xml:id="echoid-s2782" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2783" xml:space="preserve">SI arcui circuli recta ſubtendatur, ad quam ex arcu duæ perpen-<lb/>diculares demittantur auferentes verſus terminos arcus duos arcus <lb/>æquales; </s> <s xml:id="echoid-s2784" xml:space="preserve">auferent eædem duas rectas ex recta ſubtenſa æquales. </s> <s xml:id="echoid-s2785" xml:space="preserve">Et ſi <lb/>duæ perpendiculares ad rectam ſubtenſam ducantur auferẽtes duas <lb/>rectas æquales; </s> <s xml:id="echoid-s2786" xml:space="preserve">auferent eædem duos arcus æquales.</s> <s xml:id="echoid-s2787" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2788" xml:space="preserve">ARCVI circuli A B C D, ſubtendatur recta A D, ad quam ex <lb/>arcu demittantur duæ perpendiculares B E, C F, auferentes duos arcus <lb/> <anchor type="figure" xlink:label="fig-083-01a" xlink:href="fig-083-01"/> A B, D C, æquales. </s> <s xml:id="echoid-s2789" xml:space="preserve">Dico eaſdem auferre <lb/>æquales rectas A E, D F. </s> <s xml:id="echoid-s2790" xml:space="preserve">Ducta enim <lb/> <anchor type="note" xlink:label="note-083-07a" xlink:href="note-083-07"/> recta B C, erunt A D, B C, parallelæ, <lb/>ob æqualitatem arcuum A B, D C: </s> <s xml:id="echoid-s2791" xml:space="preserve">ſunt <lb/>autem & </s> <s xml:id="echoid-s2792" xml:space="preserve">B E, C F, parallelæ. </s> <s xml:id="echoid-s2793" xml:space="preserve">Parallelo-<lb/> <anchor type="note" xlink:label="note-083-08a" xlink:href="note-083-08"/> grammum igitur eſt B E F C, atque adeò <lb/>& </s> <s xml:id="echoid-s2794" xml:space="preserve">rectæ B E, C F, æquales. </s> <s xml:id="echoid-s2795" xml:space="preserve">Et quoniam æqualibus arcubus A B, D C, re-<lb/> <anchor type="note" xlink:label="note-083-09a" xlink:href="note-083-09"/> ctæ ſubtenſæ A B, D C, æquales ſunt; </s> <s xml:id="echoid-s2796" xml:space="preserve">erunt quadrata ex A B, D C, æqua-<lb/> <anchor type="note" xlink:label="note-083-10a" xlink:href="note-083-10"/> lia. </s> <s xml:id="echoid-s2797" xml:space="preserve">Cum ergo tam illud æquale ſit quadratis ex A E, B E, quàm boc qua-<lb/> <anchor type="note" xlink:label="note-083-11a" xlink:href="note-083-11"/> dratis ex D F, C F; </s> <s xml:id="echoid-s2798" xml:space="preserve">ſi auferantur æqualia quadrata rectarum B E, C F, <lb/>æqualia erunt quadrata rectarum A E, D F; </s> <s xml:id="echoid-s2799" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s2800" xml:space="preserve">rectæ A E, <lb/>D F, æquales erunt. </s> <s xml:id="echoid-s2801" xml:space="preserve">quod primo loco proponebatur.</s> <s xml:id="echoid-s2802" xml:space="preserve"/> </p> <div xml:id="echoid-div244" type="float" level="2" n="1"> <figure xlink:label="fig-083-01" xlink:href="fig-083-01a"> <image file="083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/083-01"/> </figure> <note position="right" xlink:label="note-083-07" xlink:href="note-083-07a" xml:space="preserve">Schol. 27. <lb/>tertij.</note> <note position="right" xlink:label="note-083-08" xlink:href="note-083-08a" xml:space="preserve">28. primi.</note> <note position="right" xlink:label="note-083-09" xlink:href="note-083-09a" xml:space="preserve">34. primi.</note> <note position="right" xlink:label="note-083-10" xlink:href="note-083-10a" xml:space="preserve">29. tertij.</note> <note position="right" xlink:label="note-083-11" xlink:href="note-083-11a" xml:space="preserve">47. primi</note> </div> <p style="it"> <s xml:id="echoid-s2803" xml:space="preserve">SED iam perpendiculares B E, C F, auferant æquales rectas A E, <lb/>D F. </s> <s xml:id="echoid-s2804" xml:space="preserve">Dico eaſdem auferre æquales arcus A B, D C. </s> <s xml:id="echoid-s2805" xml:space="preserve">Si enim non ſunt <lb/>æquales, ſit, ſi fieri potest, maior arcus A B, à quo æqualis abſcindatur <lb/>A G, & </s> <s xml:id="echoid-s2806" xml:space="preserve">ex G, ad A D, perpendicularis ducatur G H. </s> <s xml:id="echoid-s2807" xml:space="preserve">Erit igitur, vt <lb/>proxime demonſtr atum eſt, recta A H, rectæ D F, æqualis, atque adeò <lb/>& </s> <s xml:id="echoid-s2808" xml:space="preserve">rectæ A E, pars toti: </s> <s xml:id="echoid-s2809" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s2810" xml:space="preserve">Non eſt ergo arcus A B, <lb/>maior arcu D C: </s> <s xml:id="echoid-s2811" xml:space="preserve">eademque ratione neque minor erit. </s> <s xml:id="echoid-s2812" xml:space="preserve">Aequalis ergo eſt.</s> <s xml:id="echoid-s2813" xml:space="preserve"> <pb o="72" file="084" n="84" rhead=""/> quod eſt propoſitum. </s> <s xml:id="echoid-s2814" xml:space="preserve">Ex his constat, arcum H E, in figura propoſitionis <lb/>minorem eſſe arcu D F. </s> <s xml:id="echoid-s2815" xml:space="preserve">Nam cum angulus F M K, acutus ſit, & </s> <s xml:id="echoid-s2816" xml:space="preserve">H N K, <lb/>ebtuſus, ſi ex M, N, ad D E, perpẽdiculares ducerentur, caderent hæ in ar <lb/>cus D F, B H, auferrentque, vt in proximo lemmatc oſtendimus, arcus <lb/>æquales. </s> <s xml:id="echoid-s2817" xml:space="preserve">Quare arcus H E, minor est arcu D F.</s> <s xml:id="echoid-s2818" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div246" type="section" level="1" n="116"> <head xml:id="echoid-head130" xml:space="preserve">THEOR. 5. PROPOS. 5.</head> <p> <s xml:id="echoid-s2819" xml:space="preserve">SI in circunferentia maximi circuli ſit polus <lb/>parallelorum, huncque maximum circulum ſecẽt <lb/>ad angulos rectos duo alij maximi circuli, quorú <lb/>alter ſit vnus parallelorum, alter verò obliquus ſit <lb/>ad parallelos; </s> <s xml:id="echoid-s2820" xml:space="preserve">ab hoc autem obliquo circulo æqua <lb/>les circunferentiæ ſumantur deinceps ad eandem <lb/>partem maximi parallelorum, perque illa puncta <lb/>terminantia æquales circunferentias deſcriban-<lb/>tur paralleli circuli: </s> <s xml:id="echoid-s2821" xml:space="preserve">Circunferentiæ maximi illius <lb/>circuli primo poſiti inter parallelos interceptæ in-<lb/>æquales erunt, ſemperque ea, quæ propior fuerit <lb/>maximo parallelorum, remotiore maior erit.</s> <s xml:id="echoid-s2822" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2823" xml:space="preserve">IN circunferentia maximi circuli A B C D, ſit A, polus parallelorum, <lb/>cumq́ue fecent duo maximi circuli B D, E C, ad angulos rectos, quorum B D, <lb/> <anchor type="figure" xlink:label="fig-084-01a" xlink:href="fig-084-01"/> ſit maximus parallelorum, & </s> <s xml:id="echoid-s2824" xml:space="preserve">E C, ad paralle <lb/>los obliquus: </s> <s xml:id="echoid-s2825" xml:space="preserve">& </s> <s xml:id="echoid-s2826" xml:space="preserve">per F, G, H, puncta, quæ ex <lb/>obliquo circulo arcus æquales auferunt F G, <lb/>G H, deſcribantur paralleli I K, L M, N O, ex <lb/>polo A. </s> <s xml:id="echoid-s2827" xml:space="preserve">Dico arcum I L, maiorẽ eſſe arcu L N. <lb/></s> <s xml:id="echoid-s2828" xml:space="preserve"> <anchor type="note" xlink:label="note-084-01a" xlink:href="note-084-01"/> Per polum enim A, & </s> <s xml:id="echoid-s2829" xml:space="preserve">punctum G, circulus <lb/>maximus deſcribatur A P, ſecans parallelos in <lb/>P, Q. </s> <s xml:id="echoid-s2830" xml:space="preserve">Quoniam igitur in ſphæræ ſuperficie <lb/>intra periphæriam circuli I K, punctum G, ſi-<lb/>gnatum eſt præter polum A, & </s> <s xml:id="echoid-s2831" xml:space="preserve">ex G, duo ar-<lb/>cus G P, G F, circulorum maximorum ca-<lb/>dunt in circunferentiam circuli I K; </s> <s xml:id="echoid-s2832" xml:space="preserve">erit ar-<lb/> <anchor type="note" xlink:label="note-084-02a" xlink:href="note-084-02"/> cus G P, omnium minimus; </s> <s xml:id="echoid-s2833" xml:space="preserve">atque adeo minor <lb/>quam G F: </s> <s xml:id="echoid-s2834" xml:space="preserve">quod arcus G P, G F, minores ſint ſemicirculo, cum ſe non inter-<lb/>ſecent, antequam parallelum I K, diuidunt. </s> <s xml:id="echoid-s2835" xml:space="preserve">Rurſus quia in ſuperficie ſphæræ <lb/>extra periphæriam circuli N O, punctum G, ſignatum eſt præter eius polum;</s> <s xml:id="echoid-s2836" xml:space="preserve"> <pb o="73" file="085" n="85" rhead=""/> erit & </s> <s xml:id="echoid-s2837" xml:space="preserve">arcus G Q, omnium ex G, cadentium minimus, hoc eſt, minor, quàm <lb/> <anchor type="note" xlink:label="note-085-01a" xlink:href="note-085-01"/> G H: </s> <s xml:id="echoid-s2838" xml:space="preserve">quod arcus G Q, G H, minores ſint ſemicirculo, cum ſe non inter-<lb/>ſecent, antequàm parallelo N O, occurrant. </s> <s xml:id="echoid-s2839" xml:space="preserve">Vterque igitur arcus F G, <lb/>G H, vtroque G P, G Q, maior eſt. </s> <s xml:id="echoid-s2840" xml:space="preserve">Et quoniam recta per G, & </s> <s xml:id="echoid-s2841" xml:space="preserve">centrum <lb/>ſphæræ ducta, id eſt, communis ſectio circulorum maximorum A P, E C, ſe-<lb/>cant paralleli I K, planum intra ſphæram; </s> <s xml:id="echoid-s2842" xml:space="preserve">(non enim recta illa ad centrum <lb/>ſphæræ perueniet, hoc eſt, ad centrum maximi circuli B D, niſi prius planum <lb/>circuli I K, ſecet; </s> <s xml:id="echoid-s2843" xml:space="preserve">quòd parallelus I K, poſitus ſit inter maximum parallelo-<lb/>rum, & </s> <s xml:id="echoid-s2844" xml:space="preserve">punctum G.) </s> <s xml:id="echoid-s2845" xml:space="preserve">ſecabit eadem recta planum paralleli N O, extra ſphæ-<lb/>ram, ſirecta illa, & </s> <s xml:id="echoid-s2846" xml:space="preserve">planum circuli ad partes G, producantur: </s> <s xml:id="echoid-s2847" xml:space="preserve">propterea <lb/>quòd punctum G, poſitum eſt inter maximum parallelorum, & </s> <s xml:id="echoid-s2848" xml:space="preserve">parallelum <lb/>N O. </s> <s xml:id="echoid-s2849" xml:space="preserve">Quoniam igitur duo circuli maximi A P, E C, ſe mutuo ſecant in G, <lb/>puncto, & </s> <s xml:id="echoid-s2850" xml:space="preserve">à circulo E C, vtrinque à puncto G, duo arcus æquales ſumpti <lb/>ſunt G F, G H, & </s> <s xml:id="echoid-s2851" xml:space="preserve">per F, H, plana parallela circulorum I K, N O, ducta, <lb/>quorum N O, occurrit commnni ſectioni circulorum maximorum A P, <lb/>E C, extra ſphæram, vt oſtenſum eſt, eſtq́ue vterque arcuum G F, G H, ma-<lb/>ior vtroque arcuum G P, G Q, erit arcus G P, maior arcu G Q. </s> <s xml:id="echoid-s2852" xml:space="preserve">Eſt au-<lb/> <anchor type="note" xlink:label="note-085-02a" xlink:href="note-085-02"/> tem arcus G P, arcui I L, & </s> <s xml:id="echoid-s2853" xml:space="preserve">arcus G Q, arcui L N, æqualis. </s> <s xml:id="echoid-s2854" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2855" xml:space="preserve">arcus <lb/> <anchor type="note" xlink:label="note-085-03a" xlink:href="note-085-03"/> I L, arcu L N, maior erit. </s> <s xml:id="echoid-s2856" xml:space="preserve">Quare ſi in circunferentia maximi circuli ſit po-<lb/>lus, &</s> <s xml:id="echoid-s2857" xml:space="preserve">c. </s> <s xml:id="echoid-s2858" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s2859" xml:space="preserve"/> </p> <div xml:id="echoid-div246" type="float" level="2" n="1"> <figure xlink:label="fig-084-01" xlink:href="fig-084-01a"> <image file="084-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/084-01"/> </figure> <note position="left" xlink:label="note-084-01" xlink:href="note-084-01a" xml:space="preserve">20. 1. huius</note> <note position="left" xlink:label="note-084-02" xlink:href="note-084-02a" xml:space="preserve">Schol. 11. <lb/>@. huius.</note> <note position="right" xlink:label="note-085-01" xlink:href="note-085-01a" xml:space="preserve">Schol. 21. 2 <lb/>huius.</note> <note position="right" xlink:label="note-085-02" xlink:href="note-085-02a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-085-03" xlink:href="note-085-03a" xml:space="preserve">10. 2. huius.</note> </div> </div> <div xml:id="echoid-div248" type="section" level="1" n="117"> <head xml:id="echoid-head131" xml:space="preserve">THEOREMA 6. PROPOS. 6.</head> <p> <s xml:id="echoid-s2860" xml:space="preserve">SI in circunferentia maximi circuli ſit polus <lb/>parallelorum, huncq́; </s> <s xml:id="echoid-s2861" xml:space="preserve">maximum circulum ad an-<lb/>gulos rectos ſecentduo alij circuli maximi, quo-<lb/>rum alter ſit vnus parallelorũ, alter verò obliquus <lb/>ſit ad parallelos; </s> <s xml:id="echoid-s2862" xml:space="preserve">ſumantur autem ab obliquo circu <lb/>lo æquales circunferentiæ deinceps ad eaſdem par <lb/>tes maximi illius paralleli, & </s> <s xml:id="echoid-s2863" xml:space="preserve">per puncta terminan-<lb/>tia æquales circũferentias, perq́; </s> <s xml:id="echoid-s2864" xml:space="preserve">polum, deſcriban-<lb/>tur maximi circuli: </s> <s xml:id="echoid-s2865" xml:space="preserve">Hi circunferentias inæquales <lb/>intercipient de maximo parallelorum, quarum <lb/>propior maximo circulo primo poſito ſemper erit <lb/>remotiore maior.</s> <s xml:id="echoid-s2866" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2867" xml:space="preserve">IN circunferentia maximi circuli A B C D, ſit A, polus parallelorum, <lb/>eumq́ue ſecent duo maximi circuli B D, E C, adangulos rectos, quorum B D, <lb/>ſit parallelorum maximus, at E C, ad parallelos obliquus, ex quo ſumantur <pb o="74" file="086" n="86" rhead=""/> arcus æqùales F G, G H; </s> <s xml:id="echoid-s2868" xml:space="preserve">& </s> <s xml:id="echoid-s2869" xml:space="preserve">per puncta F, G, H, perq́ue polum A, circuli ma-<lb/>ximi deſeribantur A I, A K, A L, ſecantes B D, in I, K, L. </s> <s xml:id="echoid-s2870" xml:space="preserve">Dico arcum K L, <lb/>maiorem eſſe arcu I K. </s> <s xml:id="echoid-s2871" xml:space="preserve">Deſcribantur enim per eadem puncta F, G, H, paral-<lb/> <anchor type="figure" xlink:label="fig-086-01a" xlink:href="fig-086-01"/> <anchor type="note" xlink:label="note-086-01a" xlink:href="note-086-01"/> leli M N, O P, Q R, ſecantes A K, <lb/>in V, X. </s> <s xml:id="echoid-s2872" xml:space="preserve">Erit igitur arcus M O, <lb/> <anchor type="note" xlink:label="note-086-02a" xlink:href="note-086-02"/> maior arcu O Q; </s> <s xml:id="echoid-s2873" xml:space="preserve">atque adeo, cũ <lb/> <anchor type="note" xlink:label="note-086-03a" xlink:href="note-086-03"/> arcui M O, arcus V G, & </s> <s xml:id="echoid-s2874" xml:space="preserve">arcui O Q, <lb/>arcus G X, ſit æqualis; </s> <s xml:id="echoid-s2875" xml:space="preserve">erit & </s> <s xml:id="echoid-s2876" xml:space="preserve">V G, <lb/>maior, quàm G X. </s> <s xml:id="echoid-s2877" xml:space="preserve">Sumatur arcus <lb/>G Y, ipſi G X, æqualis, & </s> <s xml:id="echoid-s2878" xml:space="preserve">per Y, <lb/>parallelus deſcribatur S T, ſecans <lb/>circulum A I, in Z. </s> <s xml:id="echoid-s2879" xml:space="preserve">Quoniam igi-<lb/>tur arcus G Y, G X, æquales ſunt, <lb/>nec non G F, G H, erunt ductæ re-<lb/>ctæ H X, Y F, æquales. </s> <s xml:id="echoid-s2880" xml:space="preserve">Et quia cir-<lb/> <anchor type="note" xlink:label="note-086-04a" xlink:href="note-086-04"/> culus maximus A I, per polum A, <lb/>ſecat cir culum S T, ad angulos re <lb/> <anchor type="note" xlink:label="note-086-05a" xlink:href="note-086-05"/> ctos, & </s> <s xml:id="echoid-s2881" xml:space="preserve">bifariam, erit communis <lb/>ſectio, nempe recta ex Z, ad alte-<lb/>ram ſectionem ducta diameter circuli S T, ſuper quam inſiſtit ſemicirculus <lb/>rectus ad circulum A I, nempe ſemicirculus à puncto Z, incipiens, & </s> <s xml:id="echoid-s2882" xml:space="preserve">per S, vſq; <lb/></s> <s xml:id="echoid-s2883" xml:space="preserve">ad alteram ſectionem progrediens, (hoc eſt, ſegmentum circuli, quod ſemicir-<lb/>culo maius non eſt.) </s> <s xml:id="echoid-s2884" xml:space="preserve">aufertque recta illa ex circulo A I, ſegmentum ſemicir-<lb/>culo maius, quod nimirum à puucto Z, per I, vſque ad alteram ſectionem cum <lb/>cireulo S T, ducitur, atque eſt Y Z, arcus inſiſtentis ſemicirculi quadrante <lb/>minor, (propterea quòd arcus Ik, qui illi eſt ſimilis, quadrante quoque mi-<lb/> <anchor type="note" xlink:label="note-086-06a" xlink:href="note-086-06"/> nor eſt. </s> <s xml:id="echoid-s2885" xml:space="preserve">quod ita oſtendi poteſt. </s> <s xml:id="echoid-s2886" xml:space="preserve">Quoniam circuli maximi B D, E C, recti ſunt <lb/>ad maximum circulum A B C D, erit hic viciſsim ad illos rectos, ac proinde <lb/> <anchor type="note" xlink:label="note-086-07a" xlink:href="note-086-07"/> per illorum polos tranſibit. </s> <s xml:id="echoid-s2887" xml:space="preserve">Quare eorum ſegmenta, quæ ſemicirculi ſunt, bi-<lb/> <anchor type="note" xlink:label="note-086-08a" xlink:href="note-086-08"/> fariam ſecabit, id eſt, in quadrantes. </s> <s xml:id="echoid-s2888" xml:space="preserve">Quadrans ergo eſt arcus circuli B D, po-<lb/>ſitus inter B, & </s> <s xml:id="echoid-s2889" xml:space="preserve">illud punctum, vbiſe mutuo ſecant circuli B D, E C, ideoque <lb/>I K, quadrante minor. </s> <s xml:id="echoid-s2890" xml:space="preserve">Nam circulus Ak, cadit inter puncta B, I, cum circu-<lb/>lum A B C D, ſecet in altero polo.) </s> <s xml:id="echoid-s2891" xml:space="preserve">atque adeo reliquus arcus ex ſemicirculo <lb/>inſiſtente interceptus inter Y, & </s> <s xml:id="echoid-s2892" xml:space="preserve">altetam ſectionem cum circulo A I, quadran-<lb/>te maior; </s> <s xml:id="echoid-s2893" xml:space="preserve">erit recta Y Z, omnium rectarum ex Y, cadẽtium in circunferentiam <lb/> <anchor type="note" xlink:label="note-086-09a" xlink:href="note-086-09"/> Z P, minima; </s> <s xml:id="echoid-s2894" xml:space="preserve">atq; </s> <s xml:id="echoid-s2895" xml:space="preserve">adeò minor quàm Y F, hoceſt, quàm H X, quam ęqualẽ oſten <lb/>ndimus eſſe rectæ Y F. </s> <s xml:id="echoid-s2896" xml:space="preserve">Quocirca cum eirculus Q R, minor ſit circulo S T, au-<lb/>feret recta H X, maior maiorem arcum ex ſuo circulo, quàm recta Y Z, minor <lb/>ex ſuo, vt mox oſtendemus. </s> <s xml:id="echoid-s2897" xml:space="preserve">Maior igitur eſt arcus H X, quàm vt ſimilis eſſe <lb/>poſsit arcui Y Z: </s> <s xml:id="echoid-s2898" xml:space="preserve">Eſt autem arcui H X, arcus kL, & </s> <s xml:id="echoid-s2899" xml:space="preserve">arcui Y Z, arcus Ik, ſimilis. <lb/></s> <s xml:id="echoid-s2900" xml:space="preserve"> <anchor type="note" xlink:label="note-086-10a" xlink:href="note-086-10"/> Igitur & </s> <s xml:id="echoid-s2901" xml:space="preserve">kL, maior eſt, quàm vt ſimilis fit ipſi Ik; </s> <s xml:id="echoid-s2902" xml:space="preserve">ac proinde, cum ſint in eo-<lb/>dem circulo, maior erit arcus kL, quàm Ik. </s> <s xml:id="echoid-s2903" xml:space="preserve">Quamobrem, ſi in circumferentia <lb/>maximi circuli ſit polus parallelorum, &</s> <s xml:id="echoid-s2904" xml:space="preserve">c. </s> <s xml:id="echoid-s2905" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s2906" xml:space="preserve"/> </p> <div xml:id="echoid-div248" type="float" level="2" n="1"> <figure xlink:label="fig-086-01" xlink:href="fig-086-01a"> <image file="086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/086-01"/> </figure> <note position="left" xlink:label="note-086-01" xlink:href="note-086-01a" xml:space="preserve">20. 1. huius</note> <note position="left" xlink:label="note-086-02" xlink:href="note-086-02a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-086-03" xlink:href="note-086-03a" xml:space="preserve">10. 2. huius</note> <note position="left" xlink:label="note-086-04" xlink:href="note-086-04a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-086-05" xlink:href="note-086-05a" xml:space="preserve">15. 1. huius.</note> <note position="left" xlink:label="note-086-06" xlink:href="note-086-06a" xml:space="preserve">10. 2. huius</note> <note position="left" xlink:label="note-086-07" xlink:href="note-086-07a" xml:space="preserve">13. 1 huius.</note> <note position="left" xlink:label="note-086-08" xlink:href="note-086-08a" xml:space="preserve">9. 2. huius.</note> <note position="left" xlink:label="note-086-09" xlink:href="note-086-09a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-086-10" xlink:href="note-086-10a" xml:space="preserve">10. 2. huius.</note> </div> </div> <div xml:id="echoid-div250" type="section" level="1" n="118"> <head xml:id="echoid-head132" xml:space="preserve">LEMMA.</head> <p style="it"> <s xml:id="echoid-s2907" xml:space="preserve">_QVOD_ autem recta _H X,_ maiorem arcum auferatex ſuo circulo quàm recta <lb/>Y Z, ex ſuo, perſpicuum fiet, ſi prius theorema, quod ſequitur, demonſtretur.</s> <s xml:id="echoid-s2908" xml:space="preserve"/> </p> <pb o="75" file="087" n="87" rhead=""/> <p> <s xml:id="echoid-s2909" xml:space="preserve">ÆQVALES rectæ lineæ ex circulis inæqualibus auferunt ar-<lb/>cus inæquales, maiorq́ue eſt arcus minoris circuli, quàm vt ſimilis <lb/>ſit arcui maioris circuli.</s> <s xml:id="echoid-s2910" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2911" xml:space="preserve">_SINT_ circuli inæquales A B, C D, circa idem centrum E, deſcripti: <lb/></s> <s xml:id="echoid-s2912" xml:space="preserve">ducantur autem ex E, duærectę vtcunque E A, E B, ſecantes circulum <lb/>C D, in punctis C, D: </s> <s xml:id="echoid-s2913" xml:space="preserve">erunt{quam} arcus A B, C D, ſimiles, cum illis idem an-<lb/> <anchor type="note" xlink:label="note-087-01a" xlink:href="note-087-01"/> gulus E, inſiſtat ad centrum. </s> <s xml:id="echoid-s2914" xml:space="preserve">Et quoniam rectæ E A, E B, proportiona-<lb/> <anchor type="figure" xlink:label="fig-087-01a" xlink:href="fig-087-01"/> liter ſunt ſectæ in punctis C, D, quòd E A, <lb/>E B, æquales ſint, nec non E C, E D; </s> <s xml:id="echoid-s2915" xml:space="preserve">erunt re-<lb/> <anchor type="note" xlink:label="note-087-02a" xlink:href="note-087-02"/> etæ ductæ A B, C D, parallelæ; </s> <s xml:id="echoid-s2916" xml:space="preserve">atque adeo <lb/>triangula E A B, E C D, ſimilia, habentia <lb/> <anchor type="note" xlink:label="note-087-03a" xlink:href="note-087-03"/> angulos E A B, E C D, inter ſe æquales, nec <lb/>non & </s> <s xml:id="echoid-s2917" xml:space="preserve">angulos E B A, E D C, & </s> <s xml:id="echoid-s2918" xml:space="preserve">angulũ E, <lb/> <anchor type="note" xlink:label="note-087-04a" xlink:href="note-087-04"/> communem. </s> <s xml:id="echoid-s2919" xml:space="preserve">Quare erit, vt E A, ad A B, <lb/>ita E C, ad C D: </s> <s xml:id="echoid-s2920" xml:space="preserve">Eſt autem E A, maior quam <lb/>E C. </s> <s xml:id="echoid-s2921" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s2922" xml:space="preserve">A B, maior erit, quàm C D. <lb/></s> <s xml:id="echoid-s2923" xml:space="preserve"> <anchor type="note" xlink:label="note-087-05a" xlink:href="note-087-05"/> Accommodetur igitur ipſi C D, in circulo <lb/> <anchor type="note" xlink:label="note-087-06a" xlink:href="note-087-06"/> A B, æqualis B F; </s> <s xml:id="echoid-s2924" xml:space="preserve">erit{quam} arcus A B, maior, quàm F B. </s> <s xml:id="echoid-s2925" xml:space="preserve">Quare cum ar-<lb/> <anchor type="note" xlink:label="note-087-07a" xlink:href="note-087-07"/> cus C D, arcui A B, ſit ſimilis; </s> <s xml:id="echoid-s2926" xml:space="preserve">erit arcus C D, maior, quàm vt ſimilis ſit <lb/>ipſi F B. </s> <s xml:id="echoid-s2927" xml:space="preserve">Aequales igitur rectæ F B, C D, ex circulis inæqualibus A B, <lb/>C D, inæquales arcus auferunt, maior{quam} eſt arcus C D, circuli minoris, <lb/>quàm vt ſimilis ſit arcui F B, circuli minoris. </s> <s xml:id="echoid-s2928" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s2929" xml:space="preserve"/> </p> <div xml:id="echoid-div250" type="float" level="2" n="1"> <note position="right" xlink:label="note-087-01" xlink:href="note-087-01a" xml:space="preserve">ſchol. 33. <lb/>ſexti.</note> <figure xlink:label="fig-087-01" xlink:href="fig-087-01a"> <image file="087-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/087-01"/> </figure> <note position="right" xlink:label="note-087-02" xlink:href="note-087-02a" xml:space="preserve">2. ſexti.</note> <note position="right" xlink:label="note-087-03" xlink:href="note-087-03a" xml:space="preserve">Coroll. 4. <lb/>ſexti.</note> <note position="right" xlink:label="note-087-04" xlink:href="note-087-04a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-087-05" xlink:href="note-087-05a" xml:space="preserve">14. quinti.</note> <note position="right" xlink:label="note-087-06" xlink:href="note-087-06a" xml:space="preserve">1. quarti.</note> <note position="right" xlink:label="note-087-07" xlink:href="note-087-07a" xml:space="preserve">Schol. 28. <lb/>tertij.</note> </div> <p style="it"> <s xml:id="echoid-s2930" xml:space="preserve">HINC perſpicuum eſt, multo magis maiorem lineam ex circulo mi-<lb/>nore auferre arcum maiorem, quàm vt ſimilis ſit ei, quem ex circulo ma-<lb/>iore aufert linea minor. </s> <s xml:id="echoid-s2931" xml:space="preserve">Cum enim recta C D, æqualis ipſi F B, auferat arcũ <lb/>C D, maiorem, quàm vt ſimilis ſit arcui F B; </s> <s xml:id="echoid-s2932" xml:space="preserve">multo magis linea maior quàm <lb/>C D, auferet maiorem arcum, quàm vt ſimilis ſit arcui F B; </s> <s xml:id="echoid-s2933" xml:space="preserve">cum illa maior <lb/> <anchor type="note" xlink:label="note-087-08a" xlink:href="note-087-08"/> maiorem arcum abſcindat, quam C D. </s> <s xml:id="echoid-s2934" xml:space="preserve">Quare in propoſ. </s> <s xml:id="echoid-s2935" xml:space="preserve">hac ſexta etiam <lb/>recta H X, maior exiſtens, quàm recta Y Z, auferet ex circulo minore <lb/>Q R, arcum H X, maiorem, quàm vt ſimilis ſit arcui Y Z, quem recta <lb/>Y Z, aufert ex S T, circulo maiore.</s> <s xml:id="echoid-s2936" xml:space="preserve"/> </p> <div xml:id="echoid-div251" type="float" level="2" n="2"> <note position="right" xlink:label="note-087-08" xlink:href="note-087-08a" xml:space="preserve">Schol. 28. <lb/>tertij.</note> </div> <p style="it"> <s xml:id="echoid-s2937" xml:space="preserve">HOC autem lemmate demonſtrato, facile etiam oſtendemus, æquales <lb/>rectas lineas ex circulis inæqualibus auferre arcus inæquales ſimpliciter, <lb/>ita vt arcus minoris circuli ſimpliciter maior ſit arcu circuli maioris, & </s> <s xml:id="echoid-s2938" xml:space="preserve"><lb/>nθn ſolũ maior, quàm vt ſimilis ſit. </s> <s xml:id="echoid-s2939" xml:space="preserve">Sint enim rectæ lineæ C D, B F, æquales, <lb/>auferat{quam} C D, arcum minoris circuli C E D, & </s> <s xml:id="echoid-s2940" xml:space="preserve">F B, arcum circuli maioris <lb/>F G B. </s> <s xml:id="echoid-s2941" xml:space="preserve">Dico ſimpliciter arcum C E D, maiorem eſſe arcu F G B. </s> <s xml:id="echoid-s2942" xml:space="preserve">Congruente <lb/>enim recta C D, rectæ F B, cadet neceſſario arcus C E D, extra arcũ F G B; <lb/></s> <s xml:id="echoid-s2943" xml:space="preserve">at que adeo arcus C E D, maior erit arcu F G B, cum ille hunc totum intra ſe <pb o="76" file="088" n="88" rhead=""/> contineat, ſint{quam} ambo arcus in eandẽ partem caui, at{quam} eadem extrema pun <lb/>cta habeant, vt vult Archimedes in ſuppoſitionibus ante lib. </s> <s xml:id="echoid-s2944" xml:space="preserve">1. </s> <s xml:id="echoid-s2945" xml:space="preserve">deſphæra <lb/>& </s> <s xml:id="echoid-s2946" xml:space="preserve">cylindro. </s> <s xml:id="echoid-s2947" xml:space="preserve">Neque vero arcus C E D, arcui F G B, congruet, aut imra ip-<lb/> <anchor type="figure" xlink:label="fig-088-01a" xlink:href="fig-088-01"/> ſum cadet. </s> <s xml:id="echoid-s2948" xml:space="preserve">Nam ſi dicatur congruere, <lb/>congruet etiam tota circumferentia <lb/>circuli C E D, toti circumferentiæ cir-<lb/>culi F G B, atque adeo æquales erunt <lb/>circuli. </s> <s xml:id="echoid-s2949" xml:space="preserve">quod eſt abſurdum, cum inæquæ <lb/>les ponantur: </s> <s xml:id="echoid-s2950" xml:space="preserve">Si vero arcus C E D, di-<lb/>catur cadere intra arcum F G B, cu-<lb/>inſmodi eſt arcus C A D, quoniam vt <lb/>paulo ante in hoc lemmate ostenſum <lb/>eſt, arcus C E D, id eſt, C A D, maior <lb/>eſt, quàm vt ſimilis ſit arcui F G B, ſu-<lb/>matur arcus H F B, arcui C A D, ſimilis, atque adeo maior arcu F G B: <lb/></s> <s xml:id="echoid-s2951" xml:space="preserve">Aſſumpto autem in arcu C A D, puncto A, vtcunque, ducantur rectæ <lb/>A F, A B; </s> <s xml:id="echoid-s2952" xml:space="preserve">productaque recta F A, donec arcum F G B, ſecet in G, <lb/>ducantur rectæ G H, G B. </s> <s xml:id="echoid-s2953" xml:space="preserve">Itaque quoniam arcus C A D, H F B, ſimiles <lb/>ſunt, erunt anguli C A D, H G B, inillis fegmentis exiſtentes, æquales. </s> <s xml:id="echoid-s2954" xml:space="preserve">Quia <lb/>vero angulus C A D, angulo C G B, maior eſt, externus interno; </s> <s xml:id="echoid-s2955" xml:space="preserve">& </s> <s xml:id="echoid-s2956" xml:space="preserve">angulus <lb/> <anchor type="note" xlink:label="note-088-01a" xlink:href="note-088-01"/> C G B, angulo H G B, maior quoque, totum parte; </s> <s xml:id="echoid-s2957" xml:space="preserve">erit multò maior angulus <lb/>C A D, angulo H G B. </s> <s xml:id="echoid-s2958" xml:space="preserve">quod eſt abſurdũ. </s> <s xml:id="echoid-s2959" xml:space="preserve">Oſtenſus enim eſt æqualis. </s> <s xml:id="echoid-s2960" xml:space="preserve">Non ergo <lb/>arcus C E D, cadet intra arcũ F G B: </s> <s xml:id="echoid-s2961" xml:space="preserve">ſed neq; </s> <s xml:id="echoid-s2962" xml:space="preserve">ei congruit, vt demõſtratũ eſt. <lb/></s> <s xml:id="echoid-s2963" xml:space="preserve">Cadet ergo extra, atq; </s> <s xml:id="echoid-s2964" xml:space="preserve">adeo maior erit arcus C E D, arcu F G B, vt dictũ eſ.</s> <s xml:id="echoid-s2965" xml:space="preserve"/> </p> <div xml:id="echoid-div252" type="float" level="2" n="3"> <figure xlink:label="fig-088-01" xlink:href="fig-088-01a"> <image file="088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/088-01"/> </figure> <note position="left" xlink:label="note-088-01" xlink:href="note-088-01a" xml:space="preserve">16. primi.</note> </div> <p style="it"> <s xml:id="echoid-s2966" xml:space="preserve">_HINC_ etiam liquido constat, multo magis maiorem lineam ex cir-<lb/>culo minore auferre arcum maiorem ſimpliciter eo, quem minor linea ex <lb/>circulo maiore aufert.</s> <s xml:id="echoid-s2967" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div254" type="section" level="1" n="119"> <head xml:id="echoid-head133" xml:space="preserve">THEOR. 7. PROPOS. 7.</head> <note position="left" xml:space="preserve">5.</note> <p> <s xml:id="echoid-s2968" xml:space="preserve">SI in ſphæra maximus circulus tãgat aliquem <lb/>ſphæræ circulum, alius autem maximus circulus <lb/>ad parallelos obliquus ſit, tangatq́; </s> <s xml:id="echoid-s2969" xml:space="preserve">circulos maio-<lb/>res illis, quos tangit maximus circulus primo poſi-<lb/>tus, fuerintq́; </s> <s xml:id="echoid-s2970" xml:space="preserve">eorum contactus in maximo circu-<lb/>lo primo poſito, & </s> <s xml:id="echoid-s2971" xml:space="preserve">ſumanturà circulo obliquo cir <pb o="77" file="089" n="89" rhead=""/> cunferentiæ æquales, & </s> <s xml:id="echoid-s2972" xml:space="preserve">continuæ ad eaſdem par-<lb/>tes maximi parallelorum; </s> <s xml:id="echoid-s2973" xml:space="preserve">per puncta autem termi-<lb/>nantia æquales circunferentias deſcribantur paral <lb/>leli circuli: </s> <s xml:id="echoid-s2974" xml:space="preserve">Hi circumferentias inæquales interci-<lb/>pient de maximo circulo primo poſito, quorum <lb/>ea, quæ propior erit maximo parallelorum, erit <lb/>maior remotiore.</s> <s xml:id="echoid-s2975" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2976" xml:space="preserve">IN ſphæra maximus circulus A B C D, tangat circulum A E, in puncto <lb/>A; </s> <s xml:id="echoid-s2977" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s2978" xml:space="preserve">alium C F, illi æqualem: </s> <s xml:id="echoid-s2979" xml:space="preserve">Alius autem circulus maximus G H, <lb/> <anchor type="note" xlink:label="note-089-01a" xlink:href="note-089-01"/> ad parallelos obliquus tangat alios duos circulos maiores illis, quos A B C D, <lb/>tangit, ſintq́ue puncta contactuum G, H, in maximo circulo ABCD; </s> <s xml:id="echoid-s2980" xml:space="preserve">ſitq́; <lb/></s> <s xml:id="echoid-s2981" xml:space="preserve">B D, maximus parallelorum: </s> <s xml:id="echoid-s2982" xml:space="preserve">Ex obliquo denique circulo G H, ſumantur <lb/>arcus æquales Ik, K L, & </s> <s xml:id="echoid-s2983" xml:space="preserve">per puncta I, k, L, paralleli deſcribantur M N, O P, <lb/>Q R. </s> <s xml:id="echoid-s2984" xml:space="preserve">Dico arcum M O, maiorem eſſe arcu O Q. </s> <s xml:id="echoid-s2985" xml:space="preserve">Nam per k, & </s> <s xml:id="echoid-s2986" xml:space="preserve">S, polum pa-<lb/>rallelorum circulus maximus dcfcribatur Sk, ſecans parallelos in punctis T, <lb/> <anchor type="note" xlink:label="note-089-02a" xlink:href="note-089-02"/> <anchor type="figure" xlink:label="fig-089-01a" xlink:href="fig-089-01"/> V. </s> <s xml:id="echoid-s2987" xml:space="preserve">Item per k, deſcribatur ma-<lb/>ximus circulus kE, tangens <lb/>parallelum A E, in E, ſecansq́; <lb/></s> <s xml:id="echoid-s2988" xml:space="preserve"> <anchor type="note" xlink:label="note-089-03a" xlink:href="note-089-03"/> parallelos alios in X, Y; </s> <s xml:id="echoid-s2989" xml:space="preserve">ita ta-<lb/>men, vt hæc puncta X, Y, ſint <lb/>inter puncta L, T, & </s> <s xml:id="echoid-s2990" xml:space="preserve">V, I. </s> <s xml:id="echoid-s2991" xml:space="preserve">quod <lb/>ita fiet. </s> <s xml:id="echoid-s2992" xml:space="preserve">Quoniam per k, duo <lb/> <anchor type="note" xlink:label="note-089-04a" xlink:href="note-089-04"/> circuli deſcribi poſſunt tágen-<lb/>ntes circulum A E, quorum <lb/>vnus inter arcus kG, kS, ca-<lb/>dit, alter vero extra ipſos; </s> <s xml:id="echoid-s2993" xml:space="preserve">(Nã <lb/>ſi ambo ex eadem parte circu-<lb/>lum A E, tangerent, ſecarent <lb/>ſeſe mutuo prope puncta con-<lb/>tactuum, quòd alter alteri oc-<lb/>curreret. </s> <s xml:id="echoid-s2994" xml:space="preserve">quod eſt abſurdum; <lb/></s> <s xml:id="echoid-s2995" xml:space="preserve">cum ſe interſecent in puncto, <lb/>quod ipſi K, opponitur inter <lb/>alterum polum, & </s> <s xml:id="echoid-s2996" xml:space="preserve">maximum <lb/>parallelorũ.) </s> <s xml:id="echoid-s2997" xml:space="preserve">ſi prior ſumatur, <lb/>cadẽt puncta X, Y, inter puncta L, T, & </s> <s xml:id="echoid-s2998" xml:space="preserve">V, I, vt patet. </s> <s xml:id="echoid-s2999" xml:space="preserve">Igitur quoniã in ſpheræ <lb/>ſuperficie intra peripheriam circuli M N, punctum k, ſignatum eſt præter po-<lb/>lum S, & </s> <s xml:id="echoid-s3000" xml:space="preserve">ex k, tres arcus cadunt in eius circunferentiam kV, kY, kI; </s> <s xml:id="echoid-s3001" xml:space="preserve">erit <lb/>kV, omnium minimus, & </s> <s xml:id="echoid-s3002" xml:space="preserve">K Y, minor, quàm kI. </s> <s xml:id="echoid-s3003" xml:space="preserve">Rurſus quia in ſuperficie <lb/> <anchor type="note" xlink:label="note-089-05a" xlink:href="note-089-05"/> ſphæræ extra peripheriam circuli Q R, ſignatum eſt punctum K, præter eius <lb/>polum, & </s> <s xml:id="echoid-s3004" xml:space="preserve">ex K, in eius circunferentiam cadunt tres arcus K T, K X, K L, <lb/> <anchor type="note" xlink:label="note-089-06a" xlink:href="note-089-06"/> erit K T, omnium minimus, & </s> <s xml:id="echoid-s3005" xml:space="preserve">kL, minor quam K L. </s> <s xml:id="echoid-s3006" xml:space="preserve">Vterque igitur arcus <pb o="78" file="090" n="90" rhead=""/> K I, K L, vtroque K Y, K X, maior eſt. </s> <s xml:id="echoid-s3007" xml:space="preserve">Et quoniam recta per K, & </s> <s xml:id="echoid-s3008" xml:space="preserve">ſphęræ centrũ <lb/>ducta, id eſt, communis ſectio maximorum circulorum G H, E Y, ſecat pla-<lb/>num paralleli Q R, extra ſphæram, ſi recta illa, & </s> <s xml:id="echoid-s3009" xml:space="preserve">planum circuli Q R, <lb/>producantur ad partes K, vt in demonſtratione propoſ. </s> <s xml:id="echoid-s3010" xml:space="preserve">5. </s> <s xml:id="echoid-s3011" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s3012" xml:space="preserve">dictum <lb/>eſt; </s> <s xml:id="echoid-s3013" xml:space="preserve">erit arcus K Y, maior arcu K X: </s> <s xml:id="echoid-s3014" xml:space="preserve">Sed arcui K Y, arcus M O, & </s> <s xml:id="echoid-s3015" xml:space="preserve">arcui K X, <lb/> <anchor type="note" xlink:label="note-090-01a" xlink:href="note-090-01"/> arcus O Q, æqualis eſt; </s> <s xml:id="echoid-s3016" xml:space="preserve">Sunt enim ſemicirculi, quorum vnus ex A, per B, al-<lb/> <anchor type="note" xlink:label="note-090-02a" xlink:href="note-090-02"/> ter vero ex E, per K, ducitur, non conuenientes, vt ex ijs, quæ in demonſtra-<lb/>tione propoſ. </s> <s xml:id="echoid-s3017" xml:space="preserve">13. </s> <s xml:id="echoid-s3018" xml:space="preserve">ſecundi lib. </s> <s xml:id="echoid-s3019" xml:space="preserve">diximus, perſpicuum eſt. </s> <s xml:id="echoid-s3020" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3021" xml:space="preserve">arcus M O, ma-<lb/>ior erit arcu O Q. </s> <s xml:id="echoid-s3022" xml:space="preserve">Siergo in ſphæra maximus circulus tangat, &</s> <s xml:id="echoid-s3023" xml:space="preserve">c. </s> <s xml:id="echoid-s3024" xml:space="preserve">Quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s3025" xml:space="preserve"/> </p> <div xml:id="echoid-div254" type="float" level="2" n="1"> <note position="right" xlink:label="note-089-01" xlink:href="note-089-01a" xml:space="preserve">6. 3. huius.</note> <note position="right" xlink:label="note-089-02" xlink:href="note-089-02a" xml:space="preserve">20. 1. huius</note> <figure xlink:label="fig-089-01" xlink:href="fig-089-01a"> <image file="089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/089-01"/> </figure> <note position="right" xlink:label="note-089-03" xlink:href="note-089-03a" xml:space="preserve">15. 1. huius</note> <note position="right" xlink:label="note-089-04" xlink:href="note-089-04a" xml:space="preserve">ſchol 15. 2. <lb/>huius.</note> <note position="right" xlink:label="note-089-05" xlink:href="note-089-05a" xml:space="preserve">Schol. 21. 2 <lb/>huius.</note> <note position="right" xlink:label="note-089-06" xlink:href="note-089-06a" xml:space="preserve">Schol. 21. 2 <lb/>huius.</note> <note position="left" xlink:label="note-090-01" xlink:href="note-090-01a" xml:space="preserve">4. huius.</note> <note position="left" xlink:label="note-090-02" xlink:href="note-090-02a" xml:space="preserve">13. 2. huius</note> </div> </div> <div xml:id="echoid-div256" type="section" level="1" n="120"> <head xml:id="echoid-head134" xml:space="preserve">THEOREMA 8. PROPOS. 8.</head> <note position="left" xml:space="preserve">6.</note> <p> <s xml:id="echoid-s3026" xml:space="preserve">SI in ſphæra maximus circulus aliquem ſphæ-<lb/>ræ circulum tangat, aliquis autem alius maximus <lb/>circulus obliquus ad parallelos tangat circulos ma <lb/>iores illis, quos tangebat maximus circulus primo <lb/>poſitus, fuerintque eorum contactus in maximo <lb/>circulo primo poſito; </s> <s xml:id="echoid-s3027" xml:space="preserve">ſumantur autem de obliquo <lb/>circulo æquales circunferentiæ continuæ ad eaſ-<lb/>dem partes maximi parallelorum, perque puncta <lb/>terminantia æquales circunferentias deſcribantur <lb/>maximi circuli, qui & </s> <s xml:id="echoid-s3028" xml:space="preserve">tangant eundem circulum, <lb/>quem tangebat maximus circulus primo poſitus, <lb/>& </s> <s xml:id="echoid-s3029" xml:space="preserve">ſimiles parallelorú circunferentias intercipiant, <lb/>habeantque eos ſemicirculos, qui tendunt à pun-<lb/>ctis contactuum ad puncta terminantia æquales <lb/>obliqui circuli circunferentias, per quæ deſcribun-<lb/>tur, eiuſmodi, vt minime conueniant cum illo cir <lb/>culi maximi primo poſiti ſemicirculo, in quo eſt <lb/>contactus obliqui circuli inter apparentem po-<lb/>lum, & </s> <s xml:id="echoid-s3030" xml:space="preserve">maximum parallelorum: </s> <s xml:id="echoid-s3031" xml:space="preserve">Inæquales inter- <pb o="79" file="091" n="91" rhead=""/> cipient circunferentias de maximo parallelorum, <lb/>quarum propior circulo maximo primò poſito <lb/>ſemper erit maior remotiore.</s> <s xml:id="echoid-s3032" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3033" xml:space="preserve">IN ſphæra maximus circulus A B, tangat circulum A C, in A; </s> <s xml:id="echoid-s3034" xml:space="preserve">atque adeo <lb/> <anchor type="note" xlink:label="note-091-01a" xlink:href="note-091-01"/> alium illi æqualem, & </s> <s xml:id="echoid-s3035" xml:space="preserve">parallelum: </s> <s xml:id="echoid-s3036" xml:space="preserve">& </s> <s xml:id="echoid-s3037" xml:space="preserve">alius circulus maximus D E, ad paralle-<lb/>los obliquus tangat alios parallelos maiores, ſintq́ue cõtactus in circulo A B, <lb/>cuiuſmodi eſt punctum D; </s> <s xml:id="echoid-s3038" xml:space="preserve">& </s> <s xml:id="echoid-s3039" xml:space="preserve">ſit B E, parallelorum maximus: </s> <s xml:id="echoid-s3040" xml:space="preserve">Ex obliquo au-<lb/>tem circulo D E, ſumantur arcus æquales F G, G H; </s> <s xml:id="echoid-s3041" xml:space="preserve">& </s> <s xml:id="echoid-s3042" xml:space="preserve">per puncta F, G, H, <lb/>circuli maximi deſeribantur C I, K L, M N, tangentes parallelum A C, in <lb/>C, K, M, ſecantesq́ue B E, maximum parallelorum in I, L, N, ita vt ſimiles <lb/> <anchor type="figure" xlink:label="fig-091-01a" xlink:href="fig-091-01"/> arcus parallelorum interci-<lb/>piant, eorumque ſemicirculi <lb/>à punctis C, K, M, incipien-<lb/>tes, & </s> <s xml:id="echoid-s3043" xml:space="preserve">per F, G, H, tranſeun-<lb/>tes non conueniant cum ſe-<lb/>micirculo circuli A B, ab A, <lb/>incipiente, & </s> <s xml:id="echoid-s3044" xml:space="preserve">per B, tran-<lb/>ſeunte. </s> <s xml:id="echoid-s3045" xml:space="preserve">Dico arcum I L, ma-<lb/>iorem eſſe arcu L N. </s> <s xml:id="echoid-s3046" xml:space="preserve">Deſcri-<lb/>bantur enim per F, G, H, pa-<lb/>ralleli P F, Q G, R H, ſecan-<lb/>tes circulum K L, in O, S. </s> <s xml:id="echoid-s3047" xml:space="preserve">Erit <lb/> <anchor type="note" xlink:label="note-091-02a" xlink:href="note-091-02"/> ergo arcus P Q, maior arcu <lb/>Q R; </s> <s xml:id="echoid-s3048" xml:space="preserve">quibus cum ſint æqua-<lb/> <anchor type="note" xlink:label="note-091-03a" xlink:href="note-091-03"/> les arcus G O, G S, erit & </s> <s xml:id="echoid-s3049" xml:space="preserve">G O, <lb/>maior, quàm G S. </s> <s xml:id="echoid-s3050" xml:space="preserve">Fiat G T, <lb/>ipſi G S, æqualis, & </s> <s xml:id="echoid-s3051" xml:space="preserve">per T, pa <lb/>rallelus deſeribatur V T, ſe-<lb/>cans circulum M N, in X. </s> <s xml:id="echoid-s3052" xml:space="preserve">Et <lb/>quoniam eommunis ſectio cir <lb/>culorum M N, V X, hoc eſt, <lb/>recta ab X, ſectione, ad alte-<lb/>ram ſectionem ducta auſert ſegmentum, quod incipit ab X, & </s> <s xml:id="echoid-s3053" xml:space="preserve">tranſit per V, <lb/>vſq; </s> <s xml:id="echoid-s3054" xml:space="preserve">ad alteram ſectionem, ſemicirculo minus; </s> <s xml:id="echoid-s3055" xml:space="preserve">(Nam circulus maximus M N, <lb/>ſecans parallelum V X, non per polos auſert ſegmentum maius ſemicirculo, <lb/> <anchor type="note" xlink:label="note-091-04a" xlink:href="note-091-04"/> quod nimirum eſt inter maximum parallelorum, & </s> <s xml:id="echoid-s3056" xml:space="preserve">polum conſpicuum, quale <lb/>eſt ſegmentum incipiens ab X, & </s> <s xml:id="echoid-s3057" xml:space="preserve">tranſiens per α, vſque ad alteram ſectio-<lb/>nem cum circulo M N.) </s> <s xml:id="echoid-s3058" xml:space="preserve">aufertq́ue ex maximo circulo M N, ſegmentum maius <lb/>ſemicirculo, quod nimirum ab X, incipiens per N, ad alteram ſectionem tran-<lb/>ſit; </s> <s xml:id="echoid-s3059" xml:space="preserve">eſtq́ue ſegmentum X V, ad ſegmentum X M, inclinatum verſus partes R. <lb/></s> <s xml:id="echoid-s3060" xml:space="preserve">Nam ſi per N, & </s> <s xml:id="echoid-s3061" xml:space="preserve">Y, polum parallelorum circulus maximus deſcribatur Y N, <lb/>erit hic rectus ad B E. </s> <s xml:id="echoid-s3062" xml:space="preserve">Ergo M N, qui inter hos duos eſt poſitus, (Quoniam <lb/> <anchor type="note" xlink:label="note-091-05a" xlink:href="note-091-05"/> enim ex puncto F, duo circuli tangentes parallelum A C, duci poſſunt, vnus <lb/>ad ſiniſtram circuli maximi Y N, & </s> <s xml:id="echoid-s3063" xml:space="preserve">ad dexteram alter, nos priorem eligimus, <lb/>vt nimirum ponatur inter maxim os circulos Y N, B E.) </s> <s xml:id="echoid-s3064" xml:space="preserve">ad eundem B E, in- <pb o="80" file="092" n="92" rhead=""/> clinatus eſt ad partes R, & </s> <s xml:id="echoid-s3065" xml:space="preserve">viciſsim B E, atque adeò & </s> <s xml:id="echoid-s3066" xml:space="preserve">ſibi parallelus V X, ad <lb/>M N, ad eaſdem partes R, erit inclinatus. </s> <s xml:id="echoid-s3067" xml:space="preserve">Item ſegmentum incipiens ab X, & </s> <s xml:id="echoid-s3068" xml:space="preserve"><lb/>per V, vſque ad alteram ſectionem tranſiens ſectum eſt inæqualiter in T, eſt-<lb/>q́ue minor pars T X, vt mox oſtcndemus. </s> <s xml:id="echoid-s3069" xml:space="preserve">Igitur recta T X, minor eſt, quàm <lb/> <anchor type="note" xlink:label="note-092-01a" xlink:href="note-092-01"/> recta T F: </s> <s xml:id="echoid-s3070" xml:space="preserve">Sed recta T F, æqualis eſt rectæ H S. </s> <s xml:id="echoid-s3071" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3072" xml:space="preserve">recta T X, minor erit <lb/> <anchor type="note" xlink:label="note-092-02a" xlink:href="note-092-02"/> quàm recta H S; </s> <s xml:id="echoid-s3073" xml:space="preserve">atquo adeo, vt in lemmate propoſ. </s> <s xml:id="echoid-s3074" xml:space="preserve">6. </s> <s xml:id="echoid-s3075" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s3076" xml:space="preserve">demonſtratum <lb/>eſt, maior erit arcus H S, quàm vt ſimilis eſſe poſsit arcui T X. </s> <s xml:id="echoid-s3077" xml:space="preserve">Cum ergo ar-<lb/>cus I L, arcui H S, & </s> <s xml:id="echoid-s3078" xml:space="preserve">arcus L N, arcui T X, ſit ſimilis, maior erit quoque ar-<lb/> <anchor type="note" xlink:label="note-092-03a" xlink:href="note-092-03"/> cus I L, quàm vt ſimilis ſit arcui L N; </s> <s xml:id="echoid-s3079" xml:space="preserve">atque adeo, cum in eodem circulo ſint, <lb/>erit I L, maior, quam L N. </s> <s xml:id="echoid-s3080" xml:space="preserve">Si igitur ſphæra maximus circulus aliquem ſphæræ <lb/>circulum tangat, &</s> <s xml:id="echoid-s3081" xml:space="preserve">c. </s> <s xml:id="echoid-s3082" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s3083" xml:space="preserve"/> </p> <div xml:id="echoid-div256" type="float" level="2" n="1"> <note position="right" xlink:label="note-091-01" xlink:href="note-091-01a" xml:space="preserve">6. 2. huius.</note> <figure xlink:label="fig-091-01" xlink:href="fig-091-01a"> <image file="091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/091-01"/> </figure> <note position="right" xlink:label="note-091-02" xlink:href="note-091-02a" xml:space="preserve">7 huius.</note> <note position="right" xlink:label="note-091-03" xlink:href="note-091-03a" xml:space="preserve">13. 2. huius.</note> <note position="right" xlink:label="note-091-04" xlink:href="note-091-04a" xml:space="preserve">19. 2. huius.</note> <note position="right" xlink:label="note-091-05" xlink:href="note-091-05a" xml:space="preserve">15. 1. huius <lb/>School. 15. 2. <lb/>huius.</note> <note position="left" xlink:label="note-092-01" xlink:href="note-092-01a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-092-02" xlink:href="note-092-02a" xml:space="preserve">3. huius.</note> <note position="left" xlink:label="note-092-03" xlink:href="note-092-03a" xml:space="preserve">13. 2. huius.</note> </div> </div> <div xml:id="echoid-div258" type="section" level="1" n="121"> <head xml:id="echoid-head135" xml:space="preserve">LEMMA. I.</head> <p style="it"> <s xml:id="echoid-s3084" xml:space="preserve">QVOD autem arcus T X, minor ſit ſemiſſe ſegmenti, quod ab X, inci <lb/>pit, et per V, vſque ad alteram ſectionẽ protenditur, it a demonſtrabimus. <lb/></s> <s xml:id="echoid-s3085" xml:space="preserve">Per E, ducatur circulus maximus E Z, tangens parallelum A C, in Z, pun <lb/>cto, quod ſit ad dexteram cir culimaximi N Y: </s> <s xml:id="echoid-s3086" xml:space="preserve">cùmex E, duo circuli tan-<lb/> <anchor type="figure" xlink:label="fig-092-01a" xlink:href="fig-092-01"/> gẽtes A C, deſcribi poſſint, <lb/>vnus ad ſinistram circuli <lb/> <anchor type="note" xlink:label="note-092-04a" xlink:href="note-092-04"/> N Y, et ad dexteram alter. <lb/></s> <s xml:id="echoid-s3087" xml:space="preserve">Eritq́ E Z, quadrans. </s> <s xml:id="echoid-s3088" xml:space="preserve">Nam <lb/>circulus maximus Z Y, per <lb/>Y, polum circuli A C, & </s> <s xml:id="echoid-s3089" xml:space="preserve"><lb/>per Z, cõtactum deſcriptus <lb/>trãſit quoq; </s> <s xml:id="echoid-s3090" xml:space="preserve">per polum cir-<lb/> <anchor type="note" xlink:label="note-092-05a" xlink:href="note-092-05"/> culi tangentis E Z. </s> <s xml:id="echoid-s3091" xml:space="preserve">Quare <lb/>idem circulus Y Z, ſecabit <lb/>ſegmenta circulorum B E, <lb/> <anchor type="note" xlink:label="note-092-06a" xlink:href="note-092-06"/> E Z, bifariam. </s> <s xml:id="echoid-s3092" xml:space="preserve">Cum ergo hi <lb/> <anchor type="note" xlink:label="note-092-07a" xlink:href="note-092-07"/> maximi cir culi ſe bifariam <lb/>ſecent, ſecabitur ſegmẽtum <lb/>à puncto E, per Z, vſque ad <lb/>alter am ſectionem, in duos <lb/>quadrantes in puncto Z; <lb/></s> <s xml:id="echoid-s3093" xml:space="preserve">atque adeo E Z, quadrans <lb/>erit. </s> <s xml:id="echoid-s3094" xml:space="preserve">Eodem modo quadrans erit E D, ſi per polum Y, & </s> <s xml:id="echoid-s3095" xml:space="preserve">contactum D, <lb/>circulus maximus Y D, deſcribatur. </s> <s xml:id="echoid-s3096" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s3097" xml:space="preserve">arcus cir culi maximi <lb/>inter E, & </s> <s xml:id="echoid-s3098" xml:space="preserve">Y, polum, quadrans. </s> <s xml:id="echoid-s3099" xml:space="preserve">lgitur cir culus maximus ex E, tanquam <lb/> <anchor type="note" xlink:label="note-092-08a" xlink:href="note-092-08"/> polo, & </s> <s xml:id="echoid-s3100" xml:space="preserve">interuallo E Z, deſcriptus tranſibit per puncta Y, D. </s> <s xml:id="echoid-s3101" xml:space="preserve">Non aliter <lb/>oſtendemus N M, eſſe quadrantem; </s> <s xml:id="echoid-s3102" xml:space="preserve">atque adeo circulum maximum ex <lb/>N, polo, & </s> <s xml:id="echoid-s3103" xml:space="preserve">interuallo N M, deſcriptum tranſire per Y, polum paralle-<lb/>lorum, qualis eſt M Y, atque adeo ſecare arcum B D, vltra punctum D, <pb o="81" file="093" n="93" rhead=""/> & </s> <s xml:id="echoid-s3104" xml:space="preserve">arcum N B, vltra arcum D B, ideoque & </s> <s xml:id="echoid-s3105" xml:space="preserve">arcum X V, vltra cundem <lb/>arcum D B: </s> <s xml:id="echoid-s3106" xml:space="preserve">propterea quòd maximi circuli Z Y D, M Y, ſe mutuo ſe-<lb/>cant in Y, polo, & </s> <s xml:id="echoid-s3107" xml:space="preserve">punctum M, e§t inter D, & </s> <s xml:id="echoid-s3108" xml:space="preserve">Z. </s> <s xml:id="echoid-s3109" xml:space="preserve">Quoniam verò circu-<lb/>lus maximus M Y, ductus per Y, polum paralleli A C, & </s> <s xml:id="echoid-s3110" xml:space="preserve">per cont actum <lb/>M, tranſit etiam per polum circulitangentis N M; </s> <s xml:id="echoid-s3111" xml:space="preserve">tranſibit per polos <lb/> <anchor type="note" xlink:label="note-093-01a" xlink:href="note-093-01"/> circulorum X V, & </s> <s xml:id="echoid-s3112" xml:space="preserve">N M, ſe mutuo ſecantium in X. </s> <s xml:id="echoid-s3113" xml:space="preserve">Quare bifariam ſe-<lb/>cabit ipſorum ſegmenta. </s> <s xml:id="echoid-s3114" xml:space="preserve">Cum ergo vltra punctum V, ſecet ſegmentum ab <lb/> <anchor type="note" xlink:label="note-093-02a" xlink:href="note-093-02"/> X, per V, vſque ad aliud punctum, vbi ſe mutuo ſecãt circuli X V, N M, <lb/>vt proxime eſt ostenſum; </s> <s xml:id="echoid-s3115" xml:space="preserve">erit X V, arcus minor ſemiſſe ſegmentiab X, per <lb/>V, vſque ad alteram ſectionem; </s> <s xml:id="echoid-s3116" xml:space="preserve">ac proinde multo minor ſemiſſe eiuſdem <lb/>ſegmenti erit T X. </s> <s xml:id="echoid-s3117" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s3118" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="1"> <figure xlink:label="fig-092-01" xlink:href="fig-092-01a"> <image file="092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/092-01"/> </figure> <note position="left" xlink:label="note-092-04" xlink:href="note-092-04a" xml:space="preserve">schol. 13. 3 <lb/>huius.</note> <note position="left" xlink:label="note-092-05" xlink:href="note-092-05a" xml:space="preserve">5. 2. huius.</note> <note position="left" xlink:label="note-092-06" xlink:href="note-092-06a" xml:space="preserve">5. 2. huius.</note> <note position="left" xlink:label="note-092-07" xlink:href="note-092-07a" xml:space="preserve">13. 1. huius</note> <note position="left" xlink:label="note-092-08" xlink:href="note-092-08a" xml:space="preserve">Cotol. 16.1 <lb/>huius.</note> <note position="right" xlink:label="note-093-01" xlink:href="note-093-01a" xml:space="preserve">5. 2. huius</note> <note position="right" xlink:label="note-093-02" xlink:href="note-093-02a" xml:space="preserve">9. 2. huius.</note> </div> </div> <div xml:id="echoid-div260" type="section" level="1" n="122"> <head xml:id="echoid-head136" xml:space="preserve">LEMMA. I I.</head> <p> <s xml:id="echoid-s3119" xml:space="preserve">PROPOSITIS duabus magnitudinibus inæqualibus, repe-<lb/>rire aliam mediam, quæ datæ cuicunque magnitudini commen-<lb/>ſurabilis ſit.</s> <s xml:id="echoid-s3120" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3121" xml:space="preserve">SINT propoſitæ duæ magnitudines inæquales A B, A C, & </s> <s xml:id="echoid-s3122" xml:space="preserve">data <lb/>alia quæcunque D G: </s> <s xml:id="echoid-s3123" xml:space="preserve">oporteat{q́ue} inuenire aliam mediam, boc eſt, quæ <lb/>maior quidem ſit, quàm A C, minor vero, quàm A B, & </s> <s xml:id="echoid-s3124" xml:space="preserve">ipſi D G, com-<lb/>menſurabilis. </s> <s xml:id="echoid-s3125" xml:space="preserve">Sit primum D G, minor, quàm B C, exceſſus inter magnitu-<lb/> <anchor type="figure" xlink:label="fig-093-01a" xlink:href="fig-093-01"/> dines A B, A C; </s> <s xml:id="echoid-s3126" xml:space="preserve">& </s> <s xml:id="echoid-s3127" xml:space="preserve">E, multiplex ipſius <lb/>D G, proxime maior quàm A C. </s> <s xml:id="echoid-s3128" xml:space="preserve">Quo po-<lb/>ſito, erit E, minor, quàm A B. </s> <s xml:id="echoid-s3129" xml:space="preserve">Si enim <lb/>æqualis eſſet, ſi detraheretur ex E, vna <lb/>magnitudo ipſi D G, æqualis (quę quidem <lb/>minor ponitur, quàm B C,) maneret adhuc <lb/>reliqua multiplex ipſius D G, maior quàm <lb/>A C. </s> <s xml:id="echoid-s3130" xml:space="preserve">Non ergo E, eſſet multiplex ipſius D G, proxime maior, quàm A C. <lb/></s> <s xml:id="echoid-s3131" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s3132" xml:space="preserve">Non ergo æqualis eſt E, ipſi A B; </s> <s xml:id="echoid-s3133" xml:space="preserve">atque adeo multo <lb/>magis neque maior erit. </s> <s xml:id="echoid-s3134" xml:space="preserve">Minor igitur eſt, quàm A B; </s> <s xml:id="echoid-s3135" xml:space="preserve">atque adeo cum ma-<lb/>ior quoque ſit quà A C, & </s> <s xml:id="echoid-s3136" xml:space="preserve">ipſi D G, commenſurabilis, quòd eius mul-<lb/>tiplex ſit, conſtat propoſitum.</s> <s xml:id="echoid-s3137" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="2" n="1"> <figure xlink:label="fig-093-01" xlink:href="fig-093-01a"> <image file="093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/093-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s3138" xml:space="preserve">SED iam data magnitudo D G, non minor ſit, quàm B C. </s> <s xml:id="echoid-s3139" xml:space="preserve">Diuiſa igi-<lb/>tur D G, bifariam, & </s> <s xml:id="echoid-s3140" xml:space="preserve">dimidia parte rurſus bifariam, & </s> <s xml:id="echoid-s3141" xml:space="preserve">ſic deinceps, do-<lb/>nec relinquatur pars D F, minor quàm B C; </s> <s xml:id="echoid-s3142" xml:space="preserve">ſit E, ipſius D F, multiplex <lb/> <anchor type="note" xlink:label="note-093-03a" xlink:href="note-093-03"/> proximè maior, quàm A C; </s> <s xml:id="echoid-s3143" xml:space="preserve">erit{q́ue} E, ipſi D F, commenſurabilis; </s> <s xml:id="echoid-s3144" xml:space="preserve">atque <lb/>adeo & </s> <s xml:id="echoid-s3145" xml:space="preserve">ipſi D G: </s> <s xml:id="echoid-s3146" xml:space="preserve">propterea quod vtraque E, & </s> <s xml:id="echoid-s3147" xml:space="preserve">D G, ipſi D F, commen <lb/> <anchor type="note" xlink:label="note-093-04a" xlink:href="note-093-04"/> ſurabilis eſt. </s> <s xml:id="echoid-s3148" xml:space="preserve">Rurſus eodem pacto, vt paulo ante demonſtrauimus, erit E, <lb/>minor, quam A B. </s> <s xml:id="echoid-s3149" xml:space="preserve">Cum ergo maior quoque ſit, quàm A C, & </s> <s xml:id="echoid-s3150" xml:space="preserve">ipſi D G, <lb/>commenſurabilis; </s> <s xml:id="echoid-s3151" xml:space="preserve">constat propoſitum.</s> <s xml:id="echoid-s3152" xml:space="preserve"/> </p> <div xml:id="echoid-div261" type="float" level="2" n="2"> <note position="right" xlink:label="note-093-03" xlink:href="note-093-03a" xml:space="preserve">1. decimi.</note> <note position="right" xlink:label="note-093-04" xlink:href="note-093-04a" xml:space="preserve">12. decimi.</note> </div> <pb o="82" file="094" n="94" rhead=""/> </div> <div xml:id="echoid-div263" type="section" level="1" n="123"> <head xml:id="echoid-head137" xml:space="preserve">THEOREMA 9. PROPOS. 9.</head> <note position="left" xml:space="preserve">8.</note> <p> <s xml:id="echoid-s3153" xml:space="preserve">SI polus parallelorum ſit in circunſerentia ma-<lb/>ximi circuli, quem duo alij maximi circuli ad an -<lb/>gulos rectos ſecent, quorum circulorum alter ſit <lb/>vnus parallelorũ, alter verò ad parallelos obliquus <lb/>ſit: </s> <s xml:id="echoid-s3154" xml:space="preserve">& </s> <s xml:id="echoid-s3155" xml:space="preserve">ab hoc obliquo circulo ſumantur æquales <lb/>circunferentiæ, quæ continuæ quidem non ſint, <lb/>ſed tamen ſint ad eaſdem partes maximi illius pa-<lb/>ralleli; </s> <s xml:id="echoid-s3156" xml:space="preserve">per polum autem, & </s> <s xml:id="echoid-s3157" xml:space="preserve">ſingula puncta æqua-<lb/>les circunferentias terminantia deſcribantur ma-<lb/>ximi circuli: </s> <s xml:id="echoid-s3158" xml:space="preserve">Inæquales circunferentias de maxi-<lb/>mo parallelo intercipient, quarum ea, quæ pro-<lb/>pior erit maximo circulo primo poſito, ſemper <lb/>erit maior remotiore.</s> <s xml:id="echoid-s3159" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3160" xml:space="preserve">IN circunſerentia maximi circuli A B, ſit A, polus parallelorum, eum-<lb/>que ſecent duo maximi circuli B C, D C, ad angulos rectos, quorum B C, <lb/>ſit maximus parallelorum, & </s> <s xml:id="echoid-s3161" xml:space="preserve">D C, ad parallelos obliquus; </s> <s xml:id="echoid-s3162" xml:space="preserve">ex quo ſuman-<lb/>tur arcus æquales non continui E F, G H: </s> <s xml:id="echoid-s3163" xml:space="preserve">& </s> <s xml:id="echoid-s3164" xml:space="preserve">per puncta E, F, G, H, & </s> <s xml:id="echoid-s3165" xml:space="preserve">polum <lb/>A, deſcribantur maximi circuli A E I, A F K, A G L, A H M. </s> <s xml:id="echoid-s3166" xml:space="preserve">Dico arcum M L, <lb/> <anchor type="note" xlink:label="note-094-02a" xlink:href="note-094-02"/> maiorem eſſe arcu K I. </s> <s xml:id="echoid-s3167" xml:space="preserve">Autenim intermedius arcus F G, vtrique æqualium <lb/> <anchor type="figure" xlink:label="fig-094-01a" xlink:href="fig-094-01"/> E F, G H, commenſurabilis eſt, aut incommen <lb/>ſurabilis. </s> <s xml:id="echoid-s3168" xml:space="preserve">Sit primum commenſurabilis. </s> <s xml:id="echoid-s3169" xml:space="preserve">In-<lb/>uenta autem maxima communi menſura X, <lb/> <anchor type="note" xlink:label="note-094-03a" xlink:href="note-094-03"/> diuidantur tres arcus E F, F G, G H, in par-<lb/>tes ipſi X, æquales, vt in prima figura appa-<lb/>ret; </s> <s xml:id="echoid-s3170" xml:space="preserve">& </s> <s xml:id="echoid-s3171" xml:space="preserve">per puncta diuiſionum, & </s> <s xml:id="echoid-s3172" xml:space="preserve">polum A, <lb/>circuli maximi ducantur. </s> <s xml:id="echoid-s3173" xml:space="preserve">Quoniam igitur ar-<lb/> <anchor type="note" xlink:label="note-094-04a" xlink:href="note-094-04"/> cus E Q, Q F, F P, &</s> <s xml:id="echoid-s3174" xml:space="preserve">c. </s> <s xml:id="echoid-s3175" xml:space="preserve">æquales ſunt, ma-<lb/>ior erit arcus M R, arcu R L, & </s> <s xml:id="echoid-s3176" xml:space="preserve">R L, maior, <lb/> <anchor type="note" xlink:label="note-094-05a" xlink:href="note-094-05"/> quàm L, S, &</s> <s xml:id="echoid-s3177" xml:space="preserve">c. </s> <s xml:id="echoid-s3178" xml:space="preserve">Igitur cum M R, maior ſit <lb/>quàm K V, & </s> <s xml:id="echoid-s3179" xml:space="preserve">R L, maior quàm V I, erit & </s> <s xml:id="echoid-s3180" xml:space="preserve"><lb/>totus M L, maior toto K I. </s> <s xml:id="echoid-s3181" xml:space="preserve">quod eſt propo-<lb/>ſitum.</s> <s xml:id="echoid-s3182" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="1"> <note position="left" xlink:label="note-094-02" xlink:href="note-094-02a" xml:space="preserve">20. 1. huius</note> <figure xlink:label="fig-094-01" xlink:href="fig-094-01a"> <image file="094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/094-01"/> </figure> <note position="left" xlink:label="note-094-03" xlink:href="note-094-03a" xml:space="preserve">4. decimi.</note> <note position="left" xlink:label="note-094-04" xlink:href="note-094-04a" xml:space="preserve">20. 1. huius.</note> <note position="left" xlink:label="note-094-05" xlink:href="note-094-05a" xml:space="preserve">6. huius.</note> </div> <p> <s xml:id="echoid-s3183" xml:space="preserve">SED iam ſit arcus intermedius F G, in <lb/>commenſurabilis vtrique arcuum æqualium E F, G H. </s> <s xml:id="echoid-s3184" xml:space="preserve">Dico Rurſus arcum <lb/>M L, maiorem eſſe arcu K I. </s> <s xml:id="echoid-s3185" xml:space="preserve">Si enim maior non eſt, erit vel minor, vel æqua-<lb/>lis. </s> <s xml:id="echoid-s3186" xml:space="preserve">Sit primum, ſi ſieri poteſt, M L, minor quàm K I, vt in ſecunda figura;</s> <s xml:id="echoid-s3187" xml:space="preserve"> <pb o="83" file="095" n="95" rhead=""/> & </s> <s xml:id="echoid-s3188" xml:space="preserve">ex K I, ſumatur K N, ipſi M L, æqualis; </s> <s xml:id="echoid-s3189" xml:space="preserve">& </s> <s xml:id="echoid-s3190" xml:space="preserve">per N, & </s> <s xml:id="echoid-s3191" xml:space="preserve">A, circulus maximus <lb/>deſeribatur A O N, ſecans circulum C D, & </s> <s xml:id="echoid-s3192" xml:space="preserve">in O. </s> <s xml:id="echoid-s3193" xml:space="preserve">Deinde per lemma 2. </s> <s xml:id="echoid-s3194" xml:space="preserve">præ-<lb/> <anchor type="note" xlink:label="note-095-01a" xlink:href="note-095-01"/> cedentis propoſ. </s> <s xml:id="echoid-s3195" xml:space="preserve">inueniatur arcus F P, maior quidem, quàm F O, minor ve-<lb/> <anchor type="figure" xlink:label="fig-095-01a" xlink:href="fig-095-01"/> rò quàm F E, & </s> <s xml:id="echoid-s3196" xml:space="preserve">ipſi F G, commenſurabilis: <lb/></s> <s xml:id="echoid-s3197" xml:space="preserve">ſitque G Q, ipſi F P, (qui minor eſt, quàm <lb/>E F, atque adeo minor etiam quàm G H, ip-<lb/>ſi E F, æqualis.) </s> <s xml:id="echoid-s3198" xml:space="preserve">æqualis: </s> <s xml:id="echoid-s3199" xml:space="preserve">& </s> <s xml:id="echoid-s3200" xml:space="preserve">per P, Q, & </s> <s xml:id="echoid-s3201" xml:space="preserve">A, <lb/>circuli maximi deſcribantur A P R, A Q S. </s> <s xml:id="echoid-s3202" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-095-02a" xlink:href="note-095-02"/> Quoniam igitur arcus P F, G Q, æquales <lb/>ſunt non continui, eſtq́ue vtrique illorum <lb/>commenſurabilis arcus intermedius F G; </s> <s xml:id="echoid-s3203" xml:space="preserve">erit, <lb/>vt demon ſtratum iam eſt in prima ſigura, ar-<lb/>cus S L, maior arcu K R. </s> <s xml:id="echoid-s3204" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3205" xml:space="preserve">multo <lb/>maior erit, quàm K N; </s> <s xml:id="echoid-s3206" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s3207" xml:space="preserve">M L, mul <lb/>to maior erit, quàm K N: </s> <s xml:id="echoid-s3208" xml:space="preserve">Sed & </s> <s xml:id="echoid-s3209" xml:space="preserve">K N, ipſi <lb/>M L, æqualis poſitus eſt. </s> <s xml:id="echoid-s3210" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s3211" xml:space="preserve">Non ergo M L, minor eſt <lb/>quàm K I.</s> <s xml:id="echoid-s3212" xml:space="preserve"/> </p> <div xml:id="echoid-div264" type="float" level="2" n="2"> <note position="right" xlink:label="note-095-01" xlink:href="note-095-01a" xml:space="preserve">10. 1. huius.</note> <figure xlink:label="fig-095-01" xlink:href="fig-095-01a"> <image file="095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/095-01"/> </figure> <note position="right" xlink:label="note-095-02" xlink:href="note-095-02a" xml:space="preserve">20. 1. huius</note> </div> <p> <s xml:id="echoid-s3213" xml:space="preserve">SIT deinde, ſi fieri poteſt, arcus M L, æqualis arcui K I, vt in tertia figura. <lb/></s> <s xml:id="echoid-s3214" xml:space="preserve">Diuiſis autẽ arcubus E F, G H, bifariã in N, O, deſcribantur per N, O, & </s> <s xml:id="echoid-s3215" xml:space="preserve">A, cir <lb/> <anchor type="note" xlink:label="note-095-03a" xlink:href="note-095-03"/> culi maximi A N P, A O Q. </s> <s xml:id="echoid-s3216" xml:space="preserve">Erit igitur arcus M Q, maior arcu Q L, & </s> <s xml:id="echoid-s3217" xml:space="preserve">K P, <lb/> <anchor type="figure" xlink:label="fig-095-02a" xlink:href="fig-095-02"/> <anchor type="note" xlink:label="note-095-04a" xlink:href="note-095-04"/> maior quàm P I. </s> <s xml:id="echoid-s3218" xml:space="preserve">Quare Q L, minor erit, quàm <lb/>dimidiũ ipſius M L; </s> <s xml:id="echoid-s3219" xml:space="preserve">& </s> <s xml:id="echoid-s3220" xml:space="preserve">K P, maior, quàm dimi-<lb/>dium ipſius K I. </s> <s xml:id="echoid-s3221" xml:space="preserve">Cum ergo M L, K I, ponãtur <lb/>æquales; </s> <s xml:id="echoid-s3222" xml:space="preserve">erit Q L, minor, quàm K P, quod eſt <lb/>abſurdum. </s> <s xml:id="echoid-s3223" xml:space="preserve">Quoniam enim arcus F N, G O, <lb/>dimidij æqualium arcuum E F, G H, æquales <lb/>ſunt non continui, non poterit Q L, minor <lb/>eſſe, quàm K B; </s> <s xml:id="echoid-s3224" xml:space="preserve">vt proximè in ſecunda figura <lb/>demonſtratum eſt. </s> <s xml:id="echoid-s3225" xml:space="preserve">Non ergo arcus M L, ar-<lb/>cui K I, æqualis eſt: </s> <s xml:id="echoid-s3226" xml:space="preserve">ſed neque minor eſt oſten <lb/>ſus. </s> <s xml:id="echoid-s3227" xml:space="preserve">Maior ergo eſt. </s> <s xml:id="echoid-s3228" xml:space="preserve">Si igitur polus paralle-<lb/>lorum ſit in circunferentia, &</s> <s xml:id="echoid-s3229" xml:space="preserve">c. </s> <s xml:id="echoid-s3230" xml:space="preserve">Quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s3231" xml:space="preserve"/> </p> <div xml:id="echoid-div265" type="float" level="2" n="3"> <note position="right" xlink:label="note-095-03" xlink:href="note-095-03a" xml:space="preserve">20. 1. huius</note> <figure xlink:label="fig-095-02" xlink:href="fig-095-02a"> <image file="095-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/095-02"/> </figure> <note position="right" xlink:label="note-095-04" xlink:href="note-095-04a" xml:space="preserve">6. huius.</note> </div> </div> <div xml:id="echoid-div267" type="section" level="1" n="124"> <head xml:id="echoid-head138" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s3232" xml:space="preserve">_SICVT_ Theodoſius in hac propoſitione 9. </s> <s xml:id="echoid-s3233" xml:space="preserve">idem demonſtrauit de arcubus non <lb/>continuis, quod de continuis propoſ. </s> <s xml:id="echoid-s3234" xml:space="preserve">6. </s> <s xml:id="echoid-s3235" xml:space="preserve">docuit, ita in alia verſione demonſtrantur tris <lb/>bus Theorematibus eadem de arcubus non continuis, quæ Theodoſius de continuis de-<lb/>monſtrauit propoſ. </s> <s xml:id="echoid-s3236" xml:space="preserve">5. </s> <s xml:id="echoid-s3237" xml:space="preserve">7. </s> <s xml:id="echoid-s3238" xml:space="preserve">& </s> <s xml:id="echoid-s3239" xml:space="preserve">8. </s> <s xml:id="echoid-s3240" xml:space="preserve">Primum autem theorema eiuſmodi eſt.</s> <s xml:id="echoid-s3241" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div268" type="section" level="1" n="125"> <head xml:id="echoid-head139" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s3242" xml:space="preserve">SI polus parallelorum ſit in circunferentia maximi circuli, quem <lb/> <anchor type="note" xlink:label="note-095-05a" xlink:href="note-095-05"/> duo alij maximi circuli ad angulos rectos ſecẽt, quorum circulorum <lb/>alter ſit vnus parallelorum, alter verò ad parallelos obliquus ſit, & </s> <s xml:id="echoid-s3243" xml:space="preserve">ab <lb/>hoc obliquo circulo ſumantur æquales circunferentię, quę continuę <lb/>quidem non ſint, ſed tamen ſint ad eaſdem partes maximi illius pa- <pb o="84" file="096" n="96" rhead=""/> ralleli; </s> <s xml:id="echoid-s3244" xml:space="preserve">per ſingula autem puncta æquales circun ſerentias terminan-<lb/>tia, deſcribãtur paralleli circuli. </s> <s xml:id="echoid-s3245" xml:space="preserve">Circunferentię maximi illins circuli <lb/>primo poſiti inter parallelos interceptæ, inæquales erunt, ſemperq́; <lb/></s> <s xml:id="echoid-s3246" xml:space="preserve">ea, quæ propior fuerit maximo parallelorum, remotiore maior erit.</s> <s xml:id="echoid-s3247" xml:space="preserve"/> </p> <div xml:id="echoid-div268" type="float" level="2" n="1"> <note position="right" xlink:label="note-095-05" xlink:href="note-095-05a" xml:space="preserve">7.</note> </div> <p> <s xml:id="echoid-s3248" xml:space="preserve">IN _circunferentia maximi circuli_ A B, _ſit polus parallelorum, quem alij duo_ <lb/>_maximi_ B C, A C, _ſecent ad angulos rectos, ſitque_ B C, _parall lorum maximus, &_</s> <s xml:id="echoid-s3249" xml:space="preserve"> <lb/>A C, _ad parallelos obliquus. </s> <s xml:id="echoid-s3250" xml:space="preserve">Sumantur arcus non continui æquales_ D E, F G; </s> <s xml:id="echoid-s3251" xml:space="preserve">_ac per_ <lb/>D, E, F, G, _paralleli ducantur_ D H, E I, F K, G L. </s> <s xml:id="echoid-s3252" xml:space="preserve">_Dico arcum_ H I, _maiorem eſſe arcu_ <lb/>K L. </s> <s xml:id="echoid-s3253" xml:space="preserve">_Aut enim arcus intermedius_ E F, _vtrique æqualium_ D E, F G, _commenſurabilis_ <lb/>_eſt, aut incommenſurabilis. </s> <s xml:id="echoid-s3254" xml:space="preserve">Sit primum commenſurabilis. </s> <s xml:id="echoid-s3255" xml:space="preserve">Inuenta autem maxima_ <lb/> <anchor type="note" xlink:label="note-096-01a" xlink:href="note-096-01"/> _menſura V, ſecentur tres arcus_ D E, E F, F G, _in partes ipſi_ V, _æquales, & </s> <s xml:id="echoid-s3256" xml:space="preserve">per pun=_ <lb/>_cta diuiſionum paralleli deſcribantur, vt in prima figura apparet. </s> <s xml:id="echoid-s3257" xml:space="preserve">Quoniam igitur_ <lb/> <anchor type="figure" xlink:label="fig-096-01a" xlink:href="fig-096-01"/> _arcus continui_ D P, P E, E O, _&</s> <s xml:id="echoid-s3258" xml:space="preserve">c. </s> <s xml:id="echoid-s3259" xml:space="preserve">æquales ſunt; </s> <s xml:id="echoid-s3260" xml:space="preserve">erit arcus_ H T, _maior arcu_ T I, <lb/> <anchor type="note" xlink:label="note-096-02a" xlink:href="note-096-02"/> _&_</s> <s xml:id="echoid-s3261" xml:space="preserve"> T I, _maior, quàm_ I S, _&</s> <s xml:id="echoid-s3262" xml:space="preserve">c. </s> <s xml:id="echoid-s3263" xml:space="preserve">Quare cum_ H T, _maior ſit, quàm_ K Q, _&_</s> <s xml:id="echoid-s3264" xml:space="preserve"> T I, _ma=_ <lb/>_ior quam_ Q L; </s> <s xml:id="echoid-s3265" xml:space="preserve">_erit totus_ H I, _maior toto_ K L. </s> <s xml:id="echoid-s3266" xml:space="preserve">_Quod eſt propoſitum._</s> <s xml:id="echoid-s3267" xml:space="preserve"/> </p> <div xml:id="echoid-div269" type="float" level="2" n="2"> <note position="left" xlink:label="note-096-01" xlink:href="note-096-01a" xml:space="preserve">4. decim.</note> <figure xlink:label="fig-096-01" xlink:href="fig-096-01a"> <image file="096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/096-01"/> </figure> <note position="left" xlink:label="note-096-02" xlink:href="note-096-02a" xml:space="preserve">5. huius.</note> </div> <p> <s xml:id="echoid-s3268" xml:space="preserve">SED _iam_ E F, _incommenſurabilis ſit vtrique_ D E, F G. </s> <s xml:id="echoid-s3269" xml:space="preserve">_Dico adhucarcum_ H I, <lb/>_maiorem eſſe arcu_ K L. </s> <s xml:id="echoid-s3270" xml:space="preserve">_Sienim maior non eſt, erit vel minor, vel æqualis. </s> <s xml:id="echoid-s3271" xml:space="preserve">Sit pri=_ <lb/>_mum minor; </s> <s xml:id="echoid-s3272" xml:space="preserve">& </s> <s xml:id="echoid-s3273" xml:space="preserve">ex_ K L, _(vt in ſecunda ſigura) auferatur ipſi_ H I, _æqualis_ K M; </s> <s xml:id="echoid-s3274" xml:space="preserve">_&_</s> <s xml:id="echoid-s3275" xml:space="preserve"> <lb/>_per_ M, _parallelus ducatur_ M N. </s> <s xml:id="echoid-s3276" xml:space="preserve">_Deinde per Lemma 2. </s> <s xml:id="echoid-s3277" xml:space="preserve">Propoſ._ </s> <s xml:id="echoid-s3278" xml:space="preserve">8. </s> <s xml:id="echoid-s3279" xml:space="preserve">_huius lib. </s> <s xml:id="echoid-s3280" xml:space="preserve">reperia=_ <lb/>_tur arcus_ F O, _maior quidem, quàm_ F N, _minor verò quam_ F G, _& </s> <s xml:id="echoid-s3281" xml:space="preserve">commenſurabi=_ <lb/>_lis intermedio arcui_ E F: </s> <s xml:id="echoid-s3282" xml:space="preserve">_Sitque_ E P, _ipſi_ F O, _(qui minor eſt, quàm_ F G, _atque_ <lb/>_adeo minor etiam, quam_ D E, _ipſi_ F G, _æqualis) æqualis, ac per_ O, P, _paralleli de=_ <lb/>_ſcribantur_ O R, P Q. </s> <s xml:id="echoid-s3283" xml:space="preserve">_Quoniam igitur arcus non continui_ P E, F O, _æquales ſunt,_ <lb/>_eſtque vtrique illorum commenſurabilis arcus intermedius_ E F; </s> <s xml:id="echoid-s3284" xml:space="preserve">_erit, vt iam eſt de=_ <lb/>_monſtratum in prima figura, arcus_ Q I, _maior arcu_ K R. </s> <s xml:id="echoid-s3285" xml:space="preserve">_Ergo & </s> <s xml:id="echoid-s3286" xml:space="preserve">multo maior erit,_ <lb/>_quaàm_ K M; </s> <s xml:id="echoid-s3287" xml:space="preserve">_ac proinde multo magis arcus_ H I, _maior erit quàm_ K M: </s> <s xml:id="echoid-s3288" xml:space="preserve">_Sed &_</s> <s xml:id="echoid-s3289" xml:space="preserve"> H I, <lb/>_equalis ponitur ipſi_ K M. </s> <s xml:id="echoid-s3290" xml:space="preserve">_Quod eſt abſurdum. </s> <s xml:id="echoid-s3291" xml:space="preserve">Non ergo_ H I, _minor eſt, quàm_ K L.</s> <s xml:id="echoid-s3292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3293" xml:space="preserve">SIT _deinde, ſi fieripoteſt, arcus_ H I, _arcui_ K L, _æqualis, vt in tertia figura._ <lb/></s> <s xml:id="echoid-s3294" xml:space="preserve">_Diniſis autem arcubus_ D E, F G, _bifariam in_ M, N, _ducantur per_ M, N, _paralleli_ M O, <lb/>N P. </s> <s xml:id="echoid-s3295" xml:space="preserve">_Erit igitur arcus_ H O, _maior, quàm_ O I; </s> <s xml:id="echoid-s3296" xml:space="preserve">_&_</s> <s xml:id="echoid-s3297" xml:space="preserve"> K P, _maior, quàm_ P L. </s> <s xml:id="echoid-s3298" xml:space="preserve">_Quare_ <lb/> <anchor type="note" xlink:label="note-096-03a" xlink:href="note-096-03"/> O I, _minor erit, quam dimidium ipſius_ H I, _&_</s> <s xml:id="echoid-s3299" xml:space="preserve"> K P, _maior dimidio ipſius_ K L. </s> <s xml:id="echoid-s3300" xml:space="preserve">_Cum_ <pb o="85" file="097" n="97" rhead=""/> _ergo_ H I, K L, _ponantur æquales, minor erit_ O I, _quàm_ K P. </s> <s xml:id="echoid-s3301" xml:space="preserve">_Quod eſt abſurdum._ <lb/></s> <s xml:id="echoid-s3302" xml:space="preserve">_Quia enim arcus_ E M, F N, _dimidij æqualium_ D E, F G, _æquales ſunt, & </s> <s xml:id="echoid-s3303" xml:space="preserve">non con=_ <lb/>_tinui, non poterit_ O I, _minor eſſe, quam_ K P, _vt in ſecunda figura demonſtratum_ <lb/>_eſt. </s> <s xml:id="echoid-s3304" xml:space="preserve">Nonergo arcus_ H I, _arcui_ K L, _æqualis eſt: </s> <s xml:id="echoid-s3305" xml:space="preserve">Sed nequeminor eſt oſtenſus. </s> <s xml:id="echoid-s3306" xml:space="preserve">Maior_ <lb/>_igitur eſt. </s> <s xml:id="echoid-s3307" xml:space="preserve">Quod eſt propoſitum._</s> <s xml:id="echoid-s3308" xml:space="preserve"/> </p> <div xml:id="echoid-div270" type="float" level="2" n="3"> <note position="left" xlink:label="note-096-03" xlink:href="note-096-03a" xml:space="preserve">5. huius.</note> </div> </div> <div xml:id="echoid-div272" type="section" level="1" n="126"> <head xml:id="echoid-head140" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s3309" xml:space="preserve">SI in ſphæra maximus circulus tangat aliquem ſphæræ circu-<lb/> <anchor type="note" xlink:label="note-097-01a" xlink:href="note-097-01"/> lum, alius autem maximus circulus ad parallelos obliquus ſit, tan-<lb/>gatque circulos maiores illis, quos tangit maximus circulus primò <lb/>poſitus, fuerintque eorum contactus in maximo circulo primo poſi-<lb/>to; </s> <s xml:id="echoid-s3310" xml:space="preserve">& </s> <s xml:id="echoid-s3311" xml:space="preserve">ſumantur à circulo obliquo circun ferentiæ æquales, quæ con-<lb/>tinuæ quidem non ſint, ſed tamen ſintad eaſdem partes maximi pa-<lb/>rallelorum; </s> <s xml:id="echoid-s3312" xml:space="preserve">per puncta autem terminantia æquales circunferentias <lb/>deſcribantur paralleli circuli: </s> <s xml:id="echoid-s3313" xml:space="preserve">HI circunferentias inæ quales interci-<lb/>pient de maximo circulo primò poſito, quarum ea, quæ propior erit <lb/>maximo parallelorum, maior erit remotiore.</s> <s xml:id="echoid-s3314" xml:space="preserve"/> </p> <div xml:id="echoid-div272" type="float" level="2" n="1"> <note position="right" xlink:label="note-097-01" xlink:href="note-097-01a" xml:space="preserve">9.</note> </div> <p> <s xml:id="echoid-s3315" xml:space="preserve">HOC _Theorema demonſtrabitur ex propoſ. </s> <s xml:id="echoid-s3316" xml:space="preserve">7. </s> <s xml:id="echoid-s3317" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s3318" xml:space="preserve">quemadmodum præce=_ <lb/>_dens Theorema ex propoſ. </s> <s xml:id="echoid-s3319" xml:space="preserve">5. </s> <s xml:id="echoid-s3320" xml:space="preserve">demonſtratũ ſuit: </s> <s xml:id="echoid-s3321" xml:space="preserve">dummodo duo circuli maximi_ A B, A C, <lb/>_præcedentis Theorematis tangant duos parallelos, vt in propoſ. </s> <s xml:id="echoid-s3322" xml:space="preserve">7. </s> <s xml:id="echoid-s3323" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s3324" xml:space="preserve">dictum_ <lb/>_eſt. </s> <s xml:id="echoid-s3325" xml:space="preserve">Reliqua conſtructio figuræ à conſtructione præcedentis Theorematis non dif=_ <lb/>_fert, &</s> <s xml:id="echoid-s3326" xml:space="preserve">c._</s> <s xml:id="echoid-s3327" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div274" type="section" level="1" n="127"> <head xml:id="echoid-head141" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s3328" xml:space="preserve">SI in ſphæra maximus circulus aliquem ſphæræ circulum tan-<lb/> <anchor type="note" xlink:label="note-097-02a" xlink:href="note-097-02"/> gat, aliquis autem alius maximus circulus obliquus ad parallelos tan-<lb/>gat circulos maiores illis, quos tangebat maximus circulus primo <lb/>poſitus, fuerintque eorum contactus in maximo circulo primo poſi-<lb/>to; </s> <s xml:id="echoid-s3329" xml:space="preserve">ſumantur autem de obliquo circulo æquales circunferentiæ, quæ <lb/>continuæ quidem non ſint, ſed tamen ſint ad eaſdem partes maximi <lb/>parallelorum, per que puncta terminantia æquales circunferentias <lb/>deſcribantur maximi circuli, qui & </s> <s xml:id="echoid-s3330" xml:space="preserve">tangant eundem circulum, quem <lb/>tangebat maximus circulus primo poſitus, & </s> <s xml:id="echoid-s3331" xml:space="preserve">ſimiles parallelorum <lb/>circunferentias intercipiant, habeantque eos ſemicirculos, qui ten-<lb/>dunt à punctis contactuum ad puncta terminantia æquales obliqui <lb/>circuli circunferentias, per quæ deſcribuntur, eiuſmodi, vt minimè <lb/>cõueniant cum illo circuli maximi primò poſiti ſemicirculo, in quo <lb/>eſt contactus obliqui circuli inter apparentem polum, & </s> <s xml:id="echoid-s3332" xml:space="preserve">maximum <lb/>parallelorum: </s> <s xml:id="echoid-s3333" xml:space="preserve">Inæquales intercipient circunferentias de maximo <lb/>parallelorum, quarum propior circulo maximo primò poſito, ſem-<lb/>per erit maior remotiore.</s> <s xml:id="echoid-s3334" xml:space="preserve"/> </p> <div xml:id="echoid-div274" type="float" level="2" n="1"> <note position="right" xlink:label="note-097-02" xlink:href="note-097-02a" xml:space="preserve">10.</note> </div> <pb o="86" file="098" n="98" rhead=""/> <p> <s xml:id="echoid-s3335" xml:space="preserve">HOC _etiam Theorema demonſtrabitur ex propoſ. </s> <s xml:id="echoid-s3336" xml:space="preserve">8. </s> <s xml:id="echoid-s3337" xml:space="preserve">buius lib. </s> <s xml:id="echoid-s3338" xml:space="preserve">quemadmodum_ <lb/>_propoſitio 9. </s> <s xml:id="echoid-s3339" xml:space="preserve">ex propoſ. </s> <s xml:id="echoid-s3340" xml:space="preserve">6. </s> <s xml:id="echoid-s3341" xml:space="preserve">fuit oſtenſa, dummodo maximi circuli propoſ. </s> <s xml:id="echoid-s3342" xml:space="preserve">9. </s> <s xml:id="echoid-s3343" xml:space="preserve">ex A,_ <lb/>_prodeuntes tangant eundem circulum minoremillo, quem_ D C, _tangere debet, &</s> <s xml:id="echoid-s3344" xml:space="preserve">c._</s> <s xml:id="echoid-s3345" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div276" type="section" level="1" n="128"> <head xml:id="echoid-head142" xml:space="preserve">THEOREMA 10. PROPOS. 10.</head> <note position="left" xml:space="preserve">11.</note> <p> <s xml:id="echoid-s3346" xml:space="preserve">SI polus parallelorum ſit in circunferentia ma <lb/>ximi circuli, quem duo alij maximi circuli ad angu <lb/>los rectos ſecent, quorum alter ſit vnus parallelo-<lb/>rum, alter verò ſit obliquus ad parallelos; </s> <s xml:id="echoid-s3347" xml:space="preserve">in hoc <lb/>autein obliquo circulo ſumãtur duo quælibet pun <lb/>cta ad eaſdem partes maximi illius paralleli, perq́; <lb/></s> <s xml:id="echoid-s3348" xml:space="preserve">polum parallelorum, & </s> <s xml:id="echoid-s3349" xml:space="preserve">per vtium que illorum pun <lb/>ctorum deſcribantur maximi circuli: </s> <s xml:id="echoid-s3350" xml:space="preserve">Erit, vt cir-<lb/>cunferentia maximi parallelorum intercepta inter <lb/>maximum circulum primò poſitum, & </s> <s xml:id="echoid-s3351" xml:space="preserve">proximum <lb/>maximum circulum per polum, & </s> <s xml:id="echoid-s3352" xml:space="preserve">per vnum pun-<lb/>ctorum deſcriptum, ad circunferentiam obliqui <lb/>circuli inter eoſdem circulos interceptam, ita cir-<lb/>cunferentia maximi parallelorum intercepta inter <lb/>duos magnos circulos per polum, perque vtrum-<lb/>que punctorum deſcriptos, ad circunferentiam <lb/>aliquam, quæ ſit minor, quam circunferentia obli-<lb/>qui circuli inter vtrum que punctum intercepta.</s> <s xml:id="echoid-s3353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3354" xml:space="preserve">SIT polus A, parallelorum in circunferentia maximi circuli A B, quem <lb/>duo alij maximi circuli B D, C D, ſecent ad angulos rectos, & </s> <s xml:id="echoid-s3355" xml:space="preserve">ſit B D, paral-<lb/>lelorum maximus, & </s> <s xml:id="echoid-s3356" xml:space="preserve">C D, ad parallelos obliquus; </s> <s xml:id="echoid-s3357" xml:space="preserve">in quo ſumptis duobus <lb/>punctis vtcunque E, F, deſcribantur per A, polum, & </s> <s xml:id="echoid-s3358" xml:space="preserve">per E, F, circuli ma-<lb/> <anchor type="note" xlink:label="note-098-02a" xlink:href="note-098-02"/> ximi A E G, A F H. </s> <s xml:id="echoid-s3359" xml:space="preserve">Dico, vt eſt arcus B H, ad arcum C F, ita eſſe arcum H G, <lb/>ad arcum minorem arcu F E. </s> <s xml:id="echoid-s3360" xml:space="preserve">Aut enim arcus C F, F E, commenſurabiles <lb/>ſunt, aut incommenſurabiles. </s> <s xml:id="echoid-s3361" xml:space="preserve">Sint primum commenſurabiles, vt in prima fi-<lb/>gura; </s> <s xml:id="echoid-s3362" xml:space="preserve">& </s> <s xml:id="echoid-s3363" xml:space="preserve">inuenta eorum maxima menſura P, diuidantur arcus C F, F E, in ar-<lb/> <anchor type="note" xlink:label="note-098-03a" xlink:href="note-098-03"/> cus maximæ menſuræ æquales, perque puncta diuiſionum, & </s> <s xml:id="echoid-s3364" xml:space="preserve">polum A, circu-<lb/> <anchor type="note" xlink:label="note-098-04a" xlink:href="note-098-04"/> li maximi ducantur I M, K N, L O. </s> <s xml:id="echoid-s3365" xml:space="preserve">Quoniam igitur arcus continui C L, L K, <pb o="87" file="099" n="99" rhead=""/> K F, F I, I E, æquales ſunt, erit arcus B O, maior quàm O N, & </s> <s xml:id="echoid-s3366" xml:space="preserve">O N, maior <lb/> <anchor type="note" xlink:label="note-099-01a" xlink:href="note-099-01"/> quàm N H, &</s> <s xml:id="echoid-s3367" xml:space="preserve">c. </s> <s xml:id="echoid-s3368" xml:space="preserve">Igitur maior erit proportio B O, ad C L, quàm O N, ad <lb/> <anchor type="note" xlink:label="note-099-02a" xlink:href="note-099-02"/> L K; </s> <s xml:id="echoid-s3369" xml:space="preserve">& </s> <s xml:id="echoid-s3370" xml:space="preserve">maior proportio O N, ad L K, quàm N H, ad K F, &</s> <s xml:id="echoid-s3371" xml:space="preserve">c. </s> <s xml:id="echoid-s3372" xml:space="preserve">Quare, cum <lb/>ſint quotcunque magnitudines B O, O N, N H, & </s> <s xml:id="echoid-s3373" xml:space="preserve">totidem numero C L, L K, <lb/> <anchor type="figure" xlink:label="fig-099-01a" xlink:href="fig-099-01"/> K F, ſitq́ue maior proportio primæ B O, ad primã C L, quàm ſecundæ O N, <lb/>ad ſecundam L K; </s> <s xml:id="echoid-s3374" xml:space="preserve">& </s> <s xml:id="echoid-s3375" xml:space="preserve">maior ſecundæ O N, ad ſecundam L K, quàm tertiæ <lb/>N H, ad tertiam K F; </s> <s xml:id="echoid-s3376" xml:space="preserve">maior erit proportio B H, ad C F, quàm N H, ad K F: <lb/></s> <s xml:id="echoid-s3377" xml:space="preserve"> <anchor type="note" xlink:label="note-099-03a" xlink:href="note-099-03"/> Sed proportio N H, ad K F, maior adhuc eſt proportione H M, ad F I, vt <lb/> <anchor type="note" xlink:label="note-099-04a" xlink:href="note-099-04"/> oſtenſum eſt. </s> <s xml:id="echoid-s3378" xml:space="preserve">Multo ergo maior eſt proportio B H, ad C F, quàm H M, ad <lb/>F I: </s> <s xml:id="echoid-s3379" xml:space="preserve">Sed adhuc maior eſt proportio H M, ad F I, quàm H G, ad F E; </s> <s xml:id="echoid-s3380" xml:space="preserve">propte-<lb/> <anchor type="note" xlink:label="note-099-05a" xlink:href="note-099-05"/> rea quòd arcus H M, M G, multitudine æquales ſunt arcubus F I, I E; </s> <s xml:id="echoid-s3381" xml:space="preserve">eſtq́ue <lb/>maior proportio primæ H M, ad primam F I, quæ ſecundæ M G, ad ſecundam <lb/> <anchor type="note" xlink:label="note-099-06a" xlink:href="note-099-06"/> I E, vt dictum eſt. </s> <s xml:id="echoid-s3382" xml:space="preserve">Multo igitur maior eſt proportio B H, ad C F, quàm H G, <lb/>ad F E. </s> <s xml:id="echoid-s3383" xml:space="preserve">Sit vt B H, ad C F, ita H G, ad P. </s> <s xml:id="echoid-s3384" xml:space="preserve">Erit ergo maior proportio quoque <lb/>H G, ad P, quam H G, ad F E; </s> <s xml:id="echoid-s3385" xml:space="preserve">ac proinde P, arcus minor erit arcu F E. </s> <s xml:id="echoid-s3386" xml:space="preserve">Qua-<lb/> <anchor type="note" xlink:label="note-099-07a" xlink:href="note-099-07"/> re eſt, vt arcus B H, ad arcum C F, ita arcus H G, ad arcum P, arcu F E, mi-<lb/>norem. </s> <s xml:id="echoid-s3387" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s3388" xml:space="preserve"/> </p> <div xml:id="echoid-div276" type="float" level="2" n="1"> <note position="left" xlink:label="note-098-02" xlink:href="note-098-02a" xml:space="preserve">20. 1. huius</note> <note position="left" xlink:label="note-098-03" xlink:href="note-098-03a" xml:space="preserve">3. decimi.</note> <note position="left" xlink:label="note-098-04" xlink:href="note-098-04a" xml:space="preserve">20. 1. huius</note> <note position="right" xlink:label="note-099-01" xlink:href="note-099-01a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-099-02" xlink:href="note-099-02a" xml:space="preserve">8. quinti.</note> <figure xlink:label="fig-099-01" xlink:href="fig-099-01a"> <image file="099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/099-01"/> </figure> <note position="right" xlink:label="note-099-03" xlink:href="note-099-03a" xml:space="preserve">34. quinti.</note> <note position="right" xlink:label="note-099-04" xlink:href="note-099-04a" xml:space="preserve">8. quinti.</note> <note position="right" xlink:label="note-099-05" xlink:href="note-099-05a" xml:space="preserve">34. quinti.</note> <note position="right" xlink:label="note-099-06" xlink:href="note-099-06a" xml:space="preserve">8. quinti.</note> <note position="right" xlink:label="note-099-07" xlink:href="note-099-07a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s3389" xml:space="preserve">SED iam ſint arcus C F, F E, incommenſurabiles, vt in ſecunda figura. <lb/></s> <s xml:id="echoid-s3390" xml:space="preserve">Dico adhuc, vt eſt arcus B H, ad arcum C F, ita eſſe arcum H G, ad arcum ar-<lb/>cu F E, minorem. </s> <s xml:id="echoid-s3391" xml:space="preserve">Si enim non ita ſit, erit, vt B H, ad C F, ita H G, vel ad arcũ <lb/>arcu F E, maiorem, vel ad ipſummet F E. </s> <s xml:id="echoid-s3392" xml:space="preserve">Sit primum, ſi fieri poteſt, vt B H, ad <lb/>C F, ita H G, ad arcum F I, arcu F E, maiorem. </s> <s xml:id="echoid-s3393" xml:space="preserve">Inueniatur per lemma 2. </s> <s xml:id="echoid-s3394" xml:space="preserve">pro-<lb/>poſ. </s> <s xml:id="echoid-s3395" xml:space="preserve">8. </s> <s xml:id="echoid-s3396" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s3397" xml:space="preserve">arcus F K, maior quidem quàm F E, minor autem quàm F I, <lb/>& </s> <s xml:id="echoid-s3398" xml:space="preserve">ipſi C F, commenſurabilis, ducaturq́ue per K, & </s> <s xml:id="echoid-s3399" xml:space="preserve">A, polum circulus maxi-<lb/> <anchor type="note" xlink:label="note-099-08a" xlink:href="note-099-08"/> mus K L. </s> <s xml:id="echoid-s3400" xml:space="preserve">Quoniam igitur commenſurabiles ſunt arcus C F, F K, erit, vt de-<lb/>monſtratum iam eſt in prima figura, vt B H, ad C F, ita H L, ad arcum arcu <lb/>F K, minorem: </s> <s xml:id="echoid-s3401" xml:space="preserve">Sed vt B H, ad C F, ita ponebatur H G, ad F I. </s> <s xml:id="echoid-s3402" xml:space="preserve">Igitur erit quo-<lb/>que, vt H G, ad F I, ita H L, ad arcum arcu F K, minorem: </s> <s xml:id="echoid-s3403" xml:space="preserve">& </s> <s xml:id="echoid-s3404" xml:space="preserve">permutando, vt <lb/>H G, ad H L, ita F I, ad arcum arcu F K, minorem: </s> <s xml:id="echoid-s3405" xml:space="preserve">Sed H G, arcus minor eſt <lb/>arcu H L. </s> <s xml:id="echoid-s3406" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3407" xml:space="preserve">arcus F I, minor erit, quàm arcus arcu F K, minor, totum <lb/>quàm pars. </s> <s xml:id="echoid-s3408" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s3409" xml:space="preserve">Non ergo eſt, vt B H, ad C F, ita H G, ad ar-<lb/>cum arcu F E, maiorem.</s> <s xml:id="echoid-s3410" xml:space="preserve"/> </p> <div xml:id="echoid-div277" type="float" level="2" n="2"> <note position="right" xlink:label="note-099-08" xlink:href="note-099-08a" xml:space="preserve">20. 1. huius.</note> </div> <p> <s xml:id="echoid-s3411" xml:space="preserve">SIT deinde, ſi ſieri poteſt, vt B H, ad C F, ita H G, ad F E, vt in tertia <lb/>figura. </s> <s xml:id="echoid-s3412" xml:space="preserve">Diuiſo arcu F E, bifariam in I, deſcribatur per I, & </s> <s xml:id="echoid-s3413" xml:space="preserve">per A, polum cir-<lb/> <anchor type="note" xlink:label="note-099-09a" xlink:href="note-099-09"/> <pb o="88" file="100" n="100" rhead=""/> culus maximus I K. </s> <s xml:id="echoid-s3414" xml:space="preserve">Quoniam igitur arcus continui F I, I E, æquales ſunt, erit <lb/>H K, maior quàm K G; </s> <s xml:id="echoid-s3415" xml:space="preserve">atque adeo H K, maior erit dimidio ipſius H G, Qua-<lb/> <anchor type="note" xlink:label="note-100-01a" xlink:href="note-100-01"/> re maior erit proportio H K, ad F I, quàm arcus dimidij ipſius H G, ad F I: <lb/></s> <s xml:id="echoid-s3416" xml:space="preserve"> <anchor type="note" xlink:label="note-100-02a" xlink:href="note-100-02"/> Sed vt dimidium arcus H G, ad F I, dimidium arcus F E, ita eſt totus arcus <lb/> <anchor type="note" xlink:label="note-100-03a" xlink:href="note-100-03"/> H G, ad totum arcum F E, Igitur maior erit proportio H K, ad F I, quam H G, <lb/>ad F E: </s> <s xml:id="echoid-s3417" xml:space="preserve">Ponitur autem, vt H G, ad F E, ita B H, ad C F. </s> <s xml:id="echoid-s3418" xml:space="preserve">Igitur maior erit quo-<lb/>que proportio H K, ad F I, quàm B H, ad C F; </s> <s xml:id="echoid-s3419" xml:space="preserve">atque adeo arcus H K, ad ar-<lb/>cum arcu F I, maiorem erit, vt B H, ad C F. </s> <s xml:id="echoid-s3420" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s3421" xml:space="preserve">Demonſtra-<lb/> <anchor type="note" xlink:label="note-100-04a" xlink:href="note-100-04"/> tum enim proxime ſuit in ſecunda figura, non poſſe eſſe, vt eſt arcus B H, ad <lb/>C F, ita arcum H K, ad arcum arcu F I, maiorem. </s> <s xml:id="echoid-s3422" xml:space="preserve">Non ergo eſt, vt B H, ad <lb/>C F, ita H G, ad F E: </s> <s xml:id="echoid-s3423" xml:space="preserve">ſed neque, vt B H, ad C F, ita eſt H G, ad arcum arcu <lb/>F E, maiorem, vt demonſtratum eſt. </s> <s xml:id="echoid-s3424" xml:space="preserve">Igitur erit, vt B H, ad C F, ita H G, ad <lb/>arcum arcu F E, minorem. </s> <s xml:id="echoid-s3425" xml:space="preserve">Quare ſi polus parallelorum ſit in circunferentia, <lb/>&</s> <s xml:id="echoid-s3426" xml:space="preserve">c. </s> <s xml:id="echoid-s3427" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s3428" xml:space="preserve"/> </p> <div xml:id="echoid-div278" type="float" level="2" n="3"> <note position="right" xlink:label="note-099-09" xlink:href="note-099-09a" xml:space="preserve">20. 1. huius</note> <note position="left" xlink:label="note-100-01" xlink:href="note-100-01a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-100-02" xlink:href="note-100-02a" xml:space="preserve">8. quinti.</note> <note position="left" xlink:label="note-100-03" xlink:href="note-100-03a" xml:space="preserve">15. quinti.</note> <note position="left" xlink:label="note-100-04" xlink:href="note-100-04a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div280" type="section" level="1" n="129"> <head xml:id="echoid-head143" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s3429" xml:space="preserve">HINC ſit, maiorem eſſe proportionem arcus B H, ad atcum C F, quàm arcus H G, ad <lb/>arcum F E. </s> <s xml:id="echoid-s3430" xml:space="preserve">Cum enim ſit, vt B H, ad C F, ita H G, ad atcum arcu F E, minorem: </s> <s xml:id="echoid-s3431" xml:space="preserve">Sit autem <lb/> <anchor type="note" xlink:label="note-100-05a" xlink:href="note-100-05"/> maior proportio arcus H G, ad arcum arcu F E, minorem, quàm ad F E; </s> <s xml:id="echoid-s3432" xml:space="preserve">erit quoque maior <lb/> <anchor type="note" xlink:label="note-100-06a" xlink:href="note-100-06"/> proportio B H, ad C F, quàm H G, ad F E.</s> <s xml:id="echoid-s3433" xml:space="preserve"/> </p> <div xml:id="echoid-div280" type="float" level="2" n="1"> <note position="left" xlink:label="note-100-05" xlink:href="note-100-05a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-100-06" xlink:href="note-100-06a" xml:space="preserve">8. quinti.</note> </div> </div> <div xml:id="echoid-div282" type="section" level="1" n="130"> <head xml:id="echoid-head144" xml:space="preserve">THEOR. 11. PROPOS. 11.</head> <p> <s xml:id="echoid-s3434" xml:space="preserve">SI polus parallelorum ſit in circunferentia ma <lb/>ximi circuli, quem duo alij maximi circuli ad an-<lb/>gulos rectos ſecent, quorum alter ſit vnus paralle-<lb/>lorum, alter vero ſit obliquus ad parallelos; </s> <s xml:id="echoid-s3435" xml:space="preserve">alius <lb/>autem maximus circulus per polos parallelorum <lb/>tranſiens obliquum circulũ ſecet inter maximum <lb/>parallelorum, & </s> <s xml:id="echoid-s3436" xml:space="preserve">eum, quem obliquus circulus tan <lb/>git: </s> <s xml:id="echoid-s3437" xml:space="preserve">Diameter ſphæræ ad diametrum eius circuli, <lb/>quem tãgit obliquus circulus, maiorem rationem <lb/>habet, quàm circunferentia maximi parallelorum <lb/>intercepta inter maximum circulum primo poſi-<lb/>tum, & </s> <s xml:id="echoid-s3438" xml:space="preserve">maximum circulum per polos parallelo-<lb/>rum tranſeuntem, ad circunferentiam obliqui cir-<lb/>culi inter eoſdem circulos interceptam.</s> <s xml:id="echoid-s3439" xml:space="preserve"/> </p> <pb o="89" file="101" n="101" rhead=""/> <p> <s xml:id="echoid-s3440" xml:space="preserve">IN circunferentia maximi circuli AB, ſit parallelorum polus A, eumque <lb/>duo alij circuli maximi BC, DE, ad angulos rectos ſecent, quorum BC, ſit <lb/>maximus parallelorum, & </s> <s xml:id="echoid-s3441" xml:space="preserve">DE, ad parallelos obliquus tãgens parallelum DF. <lb/></s> <s xml:id="echoid-s3442" xml:space="preserve">Per polum quoq; </s> <s xml:id="echoid-s3443" xml:space="preserve">A, alius <lb/> <anchor type="figure" xlink:label="fig-101-01a" xlink:href="fig-101-01"/> circulus maximus deſcri-<lb/>batur AE, ſecans obliquũ <lb/>DE, in puncto E, inter ma <lb/>ximũ parallelorum BC, & </s> <s xml:id="echoid-s3444" xml:space="preserve"><lb/>parallelum DF, quem ob-<lb/>liquus tangit, poſito. </s> <s xml:id="echoid-s3445" xml:space="preserve">Di-<lb/>co diametrum ſphæræ ad <lb/>diametrum paralleli DF, <lb/>maiorem habere rationé, <lb/>quàm circunferétiam BC, <lb/>ad circunferentiam DE. <lb/></s> <s xml:id="echoid-s3446" xml:space="preserve">Sit AG, recta communis <lb/>ſectio circulorũ AB, AE; </s> <s xml:id="echoid-s3447" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s3448" xml:space="preserve">BG, communis ſectio <lb/>circulorũ AB, BC; </s> <s xml:id="echoid-s3449" xml:space="preserve">eruntq; </s> <s xml:id="echoid-s3450" xml:space="preserve"><lb/>AG, BG, ſemidiametri <lb/>ipſorum, (cum ſe mutuo <lb/>ſecent bifariã circuli ma-<lb/> <anchor type="note" xlink:label="note-101-01a" xlink:href="note-101-01"/> ximi in ſphæra) atque adeo & </s> <s xml:id="echoid-s3451" xml:space="preserve">ſphæræ, ſecantes ſe ſe in G, centro ſphæræ, & </s> <s xml:id="echoid-s3452" xml:space="preserve"><lb/>circulorum maximorum. </s> <s xml:id="echoid-s3453" xml:space="preserve">Sit quoque DL, communis ſectio circulorum AB, <lb/>DE, quæ quoque diameter ſphæræ erit tranſiens per centrum G. </s> <s xml:id="echoid-s3454" xml:space="preserve">Rurſus <lb/>DM, ſit communis ſectio circulorum AB, DF; </s> <s xml:id="echoid-s3455" xml:space="preserve">eritque DM, diameter cir-<lb/>culi DF, propterea quòd circulus AB, parallelum DF, ſecet bifariam per <lb/> <anchor type="note" xlink:label="note-101-02a" xlink:href="note-101-02"/> polos. </s> <s xml:id="echoid-s3456" xml:space="preserve">Item FN, CG, ſint communes ſectiones circulorum DF, BC, <lb/>cum circulo AE. </s> <s xml:id="echoid-s3457" xml:space="preserve">Ex polo A, interuallo vero AE, parallelus deſcribatur <lb/>OE, fintq́ue OH, EH, communes eius ſectiones cum circulis AB, AE; <lb/></s> <s xml:id="echoid-s3458" xml:space="preserve">Eruntq́ue & </s> <s xml:id="echoid-s3459" xml:space="preserve">FN, EH, CG, ſemidiametri circulorum DF, OE, BC, quòd <lb/>ipſos bifariam ſecet circulus maximus AE, per polos; </s> <s xml:id="echoid-s3460" xml:space="preserve">atque adeo communes <lb/> <anchor type="note" xlink:label="note-101-03a" xlink:href="note-101-03"/> ſectiones diametri ſint occurrentes diametris DM, OH, BG, in centris N, <lb/>H, G. </s> <s xml:id="echoid-s3461" xml:space="preserve">Eſt enim & </s> <s xml:id="echoid-s3462" xml:space="preserve">OH, diameter circuli OE, cum eum circulus AB, per po-<lb/> <anchor type="note" xlink:label="note-101-04a" xlink:href="note-101-04"/> lum A, bifariam ſecet. </s> <s xml:id="echoid-s3463" xml:space="preserve">Sit rurſum EG, communis ſectio circulorum maximo-<lb/>rum AE, DE, quæ etiam diameter erit tran ſiens per G, centrum ſphæræ. <lb/></s> <s xml:id="echoid-s3464" xml:space="preserve">Denique EI, communis ſit ſectio circulorum DE, OE. </s> <s xml:id="echoid-s3465" xml:space="preserve">Et quoniam re-<lb/>cta AG, ducta per polos paralleli OE, recta eſt ad planum paralleli, ca-<lb/> <anchor type="note" xlink:label="note-101-05a" xlink:href="note-101-05"/> ditq́ue in eius centrum H; </s> <s xml:id="echoid-s3466" xml:space="preserve">erit angulus OHG, ex defin. </s> <s xml:id="echoid-s3467" xml:space="preserve">3. </s> <s xml:id="echoid-s3468" xml:space="preserve">lib. </s> <s xml:id="echoid-s3469" xml:space="preserve">11. </s> <s xml:id="echoid-s3470" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s3471" xml:space="preserve">in <lb/>triangulo GHI, rectus; </s> <s xml:id="echoid-s3472" xml:space="preserve">atque adeo angulus HGI, acutus. </s> <s xml:id="echoid-s3473" xml:space="preserve">Latus igitur GI, <lb/>maius erit latere HI. </s> <s xml:id="echoid-s3474" xml:space="preserve">Auferatur recta IK, rectæ IH, æqualis, iungaturq́ue <lb/> <anchor type="note" xlink:label="note-101-06a" xlink:href="note-101-06"/> recta EK. </s> <s xml:id="echoid-s3475" xml:space="preserve">Rurſus quia vterq; </s> <s xml:id="echoid-s3476" xml:space="preserve">circulus DE, OE, rectus eſt ad circulum AB; <lb/></s> <s xml:id="echoid-s3477" xml:space="preserve">erit & </s> <s xml:id="echoid-s3478" xml:space="preserve">EI, communis eorum ſectio ad eundem perpendicularis: </s> <s xml:id="echoid-s3479" xml:space="preserve">ac proinde, <lb/> <anchor type="note" xlink:label="note-101-07a" xlink:href="note-101-07"/> ex defin. </s> <s xml:id="echoid-s3480" xml:space="preserve">3. </s> <s xml:id="echoid-s3481" xml:space="preserve">lib. </s> <s xml:id="echoid-s3482" xml:space="preserve">11. </s> <s xml:id="echoid-s3483" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s3484" xml:space="preserve">vterque angulus EIH, EIK, rectus. </s> <s xml:id="echoid-s3485" xml:space="preserve">Quoniam igi-<lb/>tur duo latera EI, IH, trianguli EIH, duobus lateribus EI, IK, trianguli <lb/>EIK, ęqualia ſunt, angulosq́; </s> <s xml:id="echoid-s3486" xml:space="preserve">continent æquales, népe rectos, vt oſtendimus, <lb/>crunt anguli quoq; </s> <s xml:id="echoid-s3487" xml:space="preserve">IHE, IKE, æquales. </s> <s xml:id="echoid-s3488" xml:space="preserve">Quia verò maior eſt proportio re-<lb/> <anchor type="note" xlink:label="note-101-08a" xlink:href="note-101-08"/> ctæ GI, ad rectam I k, quàm anguli I k E, hoc eſt anguli OHE, ſibi æqualis, <pb o="90" file="102" n="102" rhead=""/> ad angulũ IGE, vt mox demonſtrabimus: </s> <s xml:id="echoid-s3489" xml:space="preserve">Eſt autem angulus OHE, angulo <lb/> <anchor type="note" xlink:label="note-102-01a" xlink:href="note-102-01"/> BGC, æqualis; </s> <s xml:id="echoid-s3490" xml:space="preserve">(ſunt enim rectæ OH, BG, communes ſectiones planorum pa-<lb/>rallelorũ OE, BC, factæ à plano AB, parallelæ; </s> <s xml:id="echoid-s3491" xml:space="preserve">necnõ & </s> <s xml:id="echoid-s3492" xml:space="preserve">rectæ EH, CG, com-<lb/> <anchor type="note" xlink:label="note-102-02a" xlink:href="note-102-02"/> munes ſectiones eorundem planorum factæ à plano AE.) </s> <s xml:id="echoid-s3493" xml:space="preserve">erit quoque maior <lb/>proportio rectę GI, ad rectam IK, hoc eſt, ad rectam ſibi æqualem IH, quàm <lb/>anguli BGC, ad angulum DGE: </s> <s xml:id="echoid-s3494" xml:space="preserve">Vt autem angulus BGC, ad angulum DGE, <lb/>ita eſt arcus BC, ad arcum DE. </s> <s xml:id="echoid-s3495" xml:space="preserve">Maior igitur proportio quoq; </s> <s xml:id="echoid-s3496" xml:space="preserve">erit rectæ GI, <lb/> <anchor type="note" xlink:label="note-102-03a" xlink:href="note-102-03"/> ad rectam IH, quàm arcus BC, ad arcum DE: </s> <s xml:id="echoid-s3497" xml:space="preserve">Eſt autem, vt GI, ad IH, ita <lb/> <anchor type="note" xlink:label="note-102-04a" xlink:href="note-102-04"/> GD, ad DN, hoc eſt, ita tota diameter DL, ad totam diametrum DM. <lb/></s> <s xml:id="echoid-s3498" xml:space="preserve"> <anchor type="note" xlink:label="note-102-05a" xlink:href="note-102-05"/> (ſunt enim DN, OH, communes ſectiones planorum parallelorum DF, <lb/>OE, factæ à plano AB, parallelæ.) </s> <s xml:id="echoid-s3499" xml:space="preserve">Igitur maior quoque proportio erit DL, <lb/> <anchor type="note" xlink:label="note-102-06a" xlink:href="note-102-06"/> diametri ſphæræ ad DM, diametrum paralleli DF, quàm arcus BC, ad arcum <lb/>DE. </s> <s xml:id="echoid-s3500" xml:space="preserve">Quapropter, ſi polus parallelorum ſit in circunferentia maximi circu-<lb/>li, &</s> <s xml:id="echoid-s3501" xml:space="preserve">c. </s> <s xml:id="echoid-s3502" xml:space="preserve">Quod demon ſtrandum erat.</s> <s xml:id="echoid-s3503" xml:space="preserve"/> </p> <div xml:id="echoid-div282" type="float" level="2" n="1"> <figure xlink:label="fig-101-01" xlink:href="fig-101-01a"> <image file="101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/101-01"/> </figure> <note position="right" xlink:label="note-101-01" xlink:href="note-101-01a" xml:space="preserve">11.1.huius.</note> <note position="right" xlink:label="note-101-02" xlink:href="note-101-02a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-101-03" xlink:href="note-101-03a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-101-04" xlink:href="note-101-04a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-101-05" xlink:href="note-101-05a" xml:space="preserve">10. 1. huius.</note> <note position="right" xlink:label="note-101-06" xlink:href="note-101-06a" xml:space="preserve">19. primi.</note> <note position="right" xlink:label="note-101-07" xlink:href="note-101-07a" xml:space="preserve">19. vndec.</note> <note position="right" xlink:label="note-101-08" xlink:href="note-101-08a" xml:space="preserve">4. primi.</note> <note position="left" xlink:label="note-102-01" xlink:href="note-102-01a" xml:space="preserve">10. vndec.</note> <note position="left" xlink:label="note-102-02" xlink:href="note-102-02a" xml:space="preserve">15. vndec.</note> <note position="left" xlink:label="note-102-03" xlink:href="note-102-03a" xml:space="preserve">33. ſexti.</note> <note position="left" xlink:label="note-102-04" xlink:href="note-102-04a" xml:space="preserve">4. ſexti.</note> <note position="left" xlink:label="note-102-05" xlink:href="note-102-05a" xml:space="preserve">15. quinti.</note> <note position="left" xlink:label="note-102-06" xlink:href="note-102-06a" xml:space="preserve">16. vndec.</note> </div> </div> <div xml:id="echoid-div284" type="section" level="1" n="131"> <head xml:id="echoid-head145" xml:space="preserve">LEMMA.</head> <p style="it"> <s xml:id="echoid-s3504" xml:space="preserve">_QVOD_ autem maior ſit proportio rectæ _GI,_ ad rectam _IK,_ quàm anguli _IKE,_ <lb/>ad angulum _<emph style="sc">Ig</emph>E,_ hoc theoremate propoſito demonſtrabimus.</s> <s xml:id="echoid-s3505" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3506" xml:space="preserve">IN omni triangulo rectangulo, ſi ab vno acutorum angulorum <lb/>vtcunque ad latus oppoſitum linea recta ducatur; </s> <s xml:id="echoid-s3507" xml:space="preserve">erit maior propor-<lb/>tio huius lateris ad eius ſegmentum, quod prope angulum rectum <lb/>exiſtit, quàm anguli acuti, quem linea ducta cum prædicto latere, ef-<lb/>fecit, ad reliquum angulum acutum trianguli.</s> <s xml:id="echoid-s3508" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3509" xml:space="preserve">SIT triangulum rectangulum EGI, habens angulum I, rectum, <lb/>ducaturque ab angulo acuto IEG, ad latus oppoſitum GI, recta li-<lb/>nea EK, vtcunque. </s> <s xml:id="echoid-s3510" xml:space="preserve">Dico maiorem eſſe proportionem rectæ GI, ad IK, <lb/>quàm anguli acuti IKE, ad angulum acutum IGE. </s> <s xml:id="echoid-s3511" xml:space="preserve">Ducatur enim <lb/>per G, recta GA, ipſi EK, parallela, occurrens rectæ IE, protractæ <lb/> <anchor type="note" xlink:label="note-102-07a" xlink:href="note-102-07"/> <anchor type="figure" xlink:label="fig-102-01a" xlink:href="fig-102-01"/> in A. </s> <s xml:id="echoid-s3512" xml:space="preserve">Et quoniam angulus I, rectus eſt, <lb/>erit angulus IEG, acutus, & </s> <s xml:id="echoid-s3513" xml:space="preserve">propte-<lb/>rea AEG, obtuſus. </s> <s xml:id="echoid-s3514" xml:space="preserve">Latus igitur EG, <lb/>in triangulo GEI, maius eſt latere GI; <lb/></s> <s xml:id="echoid-s3515" xml:space="preserve"> <anchor type="note" xlink:label="note-102-08a" xlink:href="note-102-08"/> in triangulo verò AEG, minus latere <lb/>AG. </s> <s xml:id="echoid-s3516" xml:space="preserve">Quare arcus circuli ex centro G, <lb/>ad interuallum GE, deſcriptus ſecabit <lb/>rectam GI, productam vltra I, nempe <lb/>in B, rectam vero GA, citra A, vt <lb/>in C. </s> <s xml:id="echoid-s3517" xml:space="preserve">Quoniam igitur triangulum GAE, <lb/>maius eſt ſectore GCE, maior erit proportio trianguli GAE, ad <lb/>triangulum GEI, quàm ſectoris GCE, ad triangulum GEI: </s> <s xml:id="echoid-s3518" xml:space="preserve">Eſt <lb/> <anchor type="note" xlink:label="note-102-09a" xlink:href="note-102-09"/> autcm maior adhuc proportio ſectoris GCE, ad triangulum GEI, quàm <lb/> <anchor type="note" xlink:label="note-102-10a" xlink:href="note-102-10"/> <pb o="91" file="103" n="103" rhead=""/> ad ſectorem GEB; </s> <s xml:id="echoid-s3519" xml:space="preserve">quòd triangulum GEI, minus ſit ſectore GEB. </s> <s xml:id="echoid-s3520" xml:space="preserve">Mul-<lb/>to igitur maior erit proportio trianguli GAE, ad triangulum GEI, <lb/>quàm ſectoris GCE, ad ſectorem GEB: </s> <s xml:id="echoid-s3521" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s3522" xml:space="preserve">componendo <lb/>maior erit proportio trianguli GAI, ad triangulum GEI, quàm ſe-<lb/>ctoris GCB, ad ſectorem GEB: </s> <s xml:id="echoid-s3523" xml:space="preserve">Eſt autem vt triangulum GAI, ad <lb/> <anchor type="note" xlink:label="note-103-01a" xlink:href="note-103-01"/> triangulum GEI, itarecta AI, ad rectam IE; </s> <s xml:id="echoid-s3524" xml:space="preserve">& </s> <s xml:id="echoid-s3525" xml:space="preserve">vt ſector GCB, <lb/> <anchor type="note" xlink:label="note-103-02a" xlink:href="note-103-02"/> ad ſectorem GEB, ita angulus BGC, ad angulum BGE. </s> <s xml:id="echoid-s3526" xml:space="preserve">Maior igitur <lb/> <anchor type="note" xlink:label="note-103-03a" xlink:href="note-103-03"/> erit quoque proportio AI, ad IE, quàm anguli BGA, hoc eſt, quàm an-<lb/>guli ſibi æqualis IKE, ad angulum IGE: </s> <s xml:id="echoid-s3527" xml:space="preserve">Vt autem AI, ad IE, ita eſt <lb/> <anchor type="note" xlink:label="note-103-04a" xlink:href="note-103-04"/> GI, ad IK. </s> <s xml:id="echoid-s3528" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3529" xml:space="preserve">maior erit proportio rectæ GI, adrectam IK, <lb/> <anchor type="note" xlink:label="note-103-05a" xlink:href="note-103-05"/> quàm anguli IKE, ad angulum IGE. </s> <s xml:id="echoid-s3530" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s3531" xml:space="preserve"/> </p> <div xml:id="echoid-div284" type="float" level="2" n="1"> <note position="left" xlink:label="note-102-07" xlink:href="note-102-07a" xml:space="preserve">31.primi.</note> <figure xlink:label="fig-102-01" xlink:href="fig-102-01a"> <image file="102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/102-01"/> </figure> <note position="left" xlink:label="note-102-08" xlink:href="note-102-08a" xml:space="preserve">19.primi.</note> <note position="left" xlink:label="note-102-09" xlink:href="note-102-09a" xml:space="preserve">3.quinti.</note> <note position="left" xlink:label="note-102-10" xlink:href="note-102-10a" xml:space="preserve">3.quinti.</note> <note position="right" xlink:label="note-103-01" xlink:href="note-103-01a" xml:space="preserve">28. quinti.</note> <note position="right" xlink:label="note-103-02" xlink:href="note-103-02a" xml:space="preserve">1.ſexti.</note> <note position="right" xlink:label="note-103-03" xlink:href="note-103-03a" xml:space="preserve">Corol. 1. 33 <lb/>ſexti.</note> <note position="right" xlink:label="note-103-04" xlink:href="note-103-04a" xml:space="preserve">29. primi.</note> <note position="right" xlink:label="note-103-05" xlink:href="note-103-05a" xml:space="preserve">2. vel 4. ſex <lb/>ti.</note> </div> </div> <div xml:id="echoid-div286" type="section" level="1" n="132"> <head xml:id="echoid-head146" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s3532" xml:space="preserve">_ADDITVR_ in alia verſione hoc loco ſequens Theorema.</s> <s xml:id="echoid-s3533" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3534" xml:space="preserve">IISDEM poſitis, Diameter ſphæræ ad diametrum paralleli per <lb/> <anchor type="note" xlink:label="note-103-06a" xlink:href="note-103-06"/> punctum obliqui circuli, per quod maximus circulus è polo tranſit, <lb/>deſcripti, minorem rationem habet quàm circunferentia maximi pa <lb/>rallelorum intercepta inter maximum circulum primo poſitum, & </s> <s xml:id="echoid-s3535" xml:space="preserve"><lb/>maxmum circulum per polos parallelorum tranſeuntem, ad circun-<lb/>ferentiam obliqui circuli inter eoſdem circulos interceptam.</s> <s xml:id="echoid-s3536" xml:space="preserve"/> </p> <div xml:id="echoid-div286" type="float" level="2" n="1"> <note position="right" xlink:label="note-103-06" xlink:href="note-103-06a" xml:space="preserve">13.</note> </div> <p style="it"> <s xml:id="echoid-s3537" xml:space="preserve">_SINT_ deſcripti circuli, vt in præcedenti propoſ. </s> <s xml:id="echoid-s3538" xml:space="preserve">Dico minorem eſſe proportionem <lb/>diametri ſphæræ ad diametrum paralleli _GE,_ quàm circunferentiæ _BC,_ ad circun-<lb/>ferentiam _DE._ </s> <s xml:id="echoid-s3539" xml:space="preserve">Sint _GH, BI,_ communes ſectiones circulorum _GE, BC,_ cum circule <lb/>_<emph style="sc">Ab</emph>,_ quæ diametri illorum <lb/>erunt, cum _AB,_ per eorum po-<lb/> <anchor type="figure" xlink:label="fig-103-01a" xlink:href="fig-103-01"/> los ductus ipſos ſecet bifariã, <lb/> <anchor type="note" xlink:label="note-103-07a" xlink:href="note-103-07"/> & </s> <s xml:id="echoid-s3540" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s3541" xml:space="preserve">Erit er <lb/>go _BI,_ diameter etiã ſphæræ. <lb/></s> <s xml:id="echoid-s3542" xml:space="preserve">Et quoniã circulus _DE,_ po-<lb/>nitur rectus ad _AB,_ tranſi-<lb/>bit _DE,_ per polos ipſius _AB._ </s> <s xml:id="echoid-s3543" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-103-08a" xlink:href="note-103-08"/> Eodem modo _<emph style="sc">B</emph>C,_ per polos <lb/>eiuſdem _AB,_ tanſibit, cum re-<lb/>ctus ad ipſum ponatur. </s> <s xml:id="echoid-s3544" xml:space="preserve">Qua-<lb/>re M, punctum, vbi ſe mutuo <lb/>ſecant, polus erit circuli _AB;_ <lb/></s> <s xml:id="echoid-s3545" xml:space="preserve">ac propterea ſegmẽtum _DEL,_ <lb/>quod rectum eſt ad circulum <lb/>_AB,_ inæqualiter diuidetur in <lb/>E, puncto, vbi circuli _DE, GE,_ <lb/>ſe interſecant, minorq́ pars <lb/>erit _ED:_ </s> <s xml:id="echoid-s3546" xml:space="preserve">quandoquidem ar-<lb/>cus _MD, ML,_ æquales ſunt, quod rectæ illis ſubtenſæ, ex defin. </s> <s xml:id="echoid-s3547" xml:space="preserve">poli, æquales ſint. </s> <s xml:id="echoid-s3548" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-103-09a" xlink:href="note-103-09"/> <pb o="92" file="104" n="104" rhead=""/> Recta igitur ducta _ED,_ minor erit, quàm recta _EG;_ </s> <s xml:id="echoid-s3549" xml:space="preserve">ac proinde cum circulus _GE,_ <lb/> <anchor type="note" xlink:label="note-104-01a" xlink:href="note-104-01"/> minor ſit circulo _DE,_ mator erit circunferentia _<emph style="sc">Eg</emph>,_ quàm circunferentia _DE._ <lb/></s> <s xml:id="echoid-s3550" xml:space="preserve">Sienim recta rectæ _ED,_ æqualis aufert ex circulo _GE,_ maiorem arcum, quàm <lb/> <anchor type="note" xlink:label="note-104-02a" xlink:href="note-104-02"/> recta _DE,_ ex circulo _DE;_ </s> <s xml:id="echoid-s3551" xml:space="preserve">multo magis recta _<emph style="sc">Eg</emph>,_ quæ maior eſt, quàm recta <lb/>_ED,_ vt oſtendimus, maiorem arcum auferet, &</s> <s xml:id="echoid-s3552" xml:space="preserve">c. </s> <s xml:id="echoid-s3553" xml:space="preserve">Quare minor erit propor-<lb/>tio arcus _<emph style="sc">B</emph>C,_ ad arcum _GE,_ quàm ad arcum _DE._ </s> <s xml:id="echoid-s3554" xml:space="preserve">Quoniam vero eſt, vt arcus _BC,_ <lb/> <anchor type="note" xlink:label="note-104-03a" xlink:href="note-104-03"/> <anchor type="figure" xlink:label="fig-104-01a" xlink:href="fig-104-01"/> ad totam circunferentiam cir-<lb/>culi _BC,_ ita arcus _<emph style="sc">G</emph>E,_ ad to-<lb/>tam circũferentiã circuli _GE,_ <lb/>propter ſimilitudinem arcuum <lb/>_BC, GE;_ </s> <s xml:id="echoid-s3555" xml:space="preserve">(In hoc enim conſiſtit <lb/>ſimilitudo arcuum, vt ad ſuo-<lb/>rum circulorum circunferen-<lb/>tias integras eandem habeant <lb/>proportionem, vt in ſcholio pro <lb/>poſ 33. </s> <s xml:id="echoid-s3556" xml:space="preserve">li. </s> <s xml:id="echoid-s3557" xml:space="preserve">6. </s> <s xml:id="echoid-s3558" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s3559" xml:space="preserve">tradidimus) <lb/>atque adeo permutando, vt ar-<lb/>cus _BC,_ ad arcũ _<emph style="sc">G</emph>E,_ itateta <lb/>circunferentia circuli _BC,_ ad <lb/>totam circunferentiam circuli <lb/>_<emph style="sc">G</emph>E;_ </s> <s xml:id="echoid-s3560" xml:space="preserve">erit quoque minor propor <lb/>tio circunferentiæ circuli _BC,_ <lb/>ad circunferentiã circuli _<emph style="sc">G</emph>E,_ <lb/>quàm arcus _<emph style="sc">B</emph>C,_ ad arcum <lb/>_DE:_ </s> <s xml:id="echoid-s3561" xml:space="preserve">Vt autem circunferentia <lb/>circuli _<emph style="sc">B</emph>C,_ ad circunferentiam circuli _GE,_ ita eſt diameter _BI,_ (quæ ſphæræ etiam <lb/>diameter eſt.) </s> <s xml:id="echoid-s3562" xml:space="preserve">ad diametrum _GH,_ vt Pappus demonſtrauit, & </s> <s xml:id="echoid-s3563" xml:space="preserve">nos in libello Archi-<lb/>medis de dimenſione circuli oſtendimus. </s> <s xml:id="echoid-s3564" xml:space="preserve">Igitur minor quoque erit proportio diame-<lb/>tri ſphæræ _<emph style="sc">B</emph>I,_ ad _<emph style="sc">G</emph>H,_ diametrum paralleli _<emph style="sc">G</emph>E,_ quàm arcus _<emph style="sc">B</emph>C,_ ad circunferen-<lb/>tiam _DE._ </s> <s xml:id="echoid-s3565" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s3566" xml:space="preserve"/> </p> <div xml:id="echoid-div287" type="float" level="2" n="2"> <figure xlink:label="fig-103-01" xlink:href="fig-103-01a"> <image file="103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/103-01"/> </figure> <note position="right" xlink:label="note-103-07" xlink:href="note-103-07a" xml:space="preserve">15. 1. huius.</note> <note position="right" xlink:label="note-103-08" xlink:href="note-103-08a" xml:space="preserve">13. 1. huius.</note> <note position="right" xlink:label="note-103-09" xlink:href="note-103-09a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-104-01" xlink:href="note-104-01a" xml:space="preserve">ſchol. 21. 2. <lb/>huius.</note> <note position="left" xlink:label="note-104-02" xlink:href="note-104-02a" xml:space="preserve">lemma 6. <lb/>huius.</note> <note position="left" xlink:label="note-104-03" xlink:href="note-104-03a" xml:space="preserve">8.quinti.</note> <figure xlink:label="fig-104-01" xlink:href="fig-104-01a"> <image file="104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/104-01"/> </figure> </div> </div> <div xml:id="echoid-div289" type="section" level="1" n="133"> <head xml:id="echoid-head147" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s3567" xml:space="preserve">HINC ſit, ijſdem poſitis, maiorem eſſe rationem circunferentiæ BC, maximi parallelo-<lb/>rum interceptæ inter maximum circulum AB, primo poſitum, & </s> <s xml:id="echoid-s3568" xml:space="preserve">maximum circulum AC, <lb/>per polos parallelorum tranſeuntem, ad circunferentiam DE, obliqui circuli inter coſdem <lb/>circulos interceptam, quàm ſinus totius ad ſinum circunferentiæ AE, maximi circuli per <lb/>polos parallelorum tranſeuntis; </s> <s xml:id="echoid-s3569" xml:space="preserve">minorem vero, quàm ſinus totius ad ſinum circunferenciæ <lb/>AD, maximi circuli primò poſiti inter polos parall elorum, & </s> <s xml:id="echoid-s3570" xml:space="preserve">obliquum circulum inter-<lb/>ceptæ. </s> <s xml:id="echoid-s3571" xml:space="preserve">Quoniam enim hoc Theoremate oſtenſum eſt, maiorem eſſe rationem arcus BC, ad <lb/>arcum DE, quàm diametri ſphæræ ad diametrum paralleli GE: </s> <s xml:id="echoid-s3572" xml:space="preserve">vt autem diameter BI, <lb/>ſphæræ ad GH, diametrum circuli GE, ita eſt BK, ſemidiameter, hoc eſt, ſinus rotus, ad <lb/> <anchor type="note" xlink:label="note-104-04a" xlink:href="note-104-04"/> GN, ſemidiametrum, hoc eſt, ad ſinum arcus AE. </s> <s xml:id="echoid-s3573" xml:space="preserve">(Cum enim arcus AG, AE, æquales ſint, <lb/> <anchor type="note" xlink:label="note-104-05a" xlink:href="note-104-05"/> ſitque GN, ſinus arcus AG; </s> <s xml:id="echoid-s3574" xml:space="preserve">erit quoque GN, ſinus arcus AE.) </s> <s xml:id="echoid-s3575" xml:space="preserve">Maiorigitur erit quoque <lb/>tatio arcus BC, ad arcum DE, quàm ſinus totius BK, ad GN, ſinum arcus AE.</s> <s xml:id="echoid-s3576" xml:space="preserve"/> </p> <div xml:id="echoid-div289" type="float" level="2" n="1"> <note position="left" xlink:label="note-104-04" xlink:href="note-104-04a" xml:space="preserve">15. quinti.</note> <note position="left" xlink:label="note-104-05" xlink:href="note-104-05a" xml:space="preserve">10.2.huius.</note> </div> <p> <s xml:id="echoid-s3577" xml:space="preserve">RVRSVS, quoniã oſtenſum eſt, minorem eſſe rationem ascus BC, ad arcum DE, quàm <lb/> <anchor type="note" xlink:label="note-104-06a" xlink:href="note-104-06"/> diametri ſphæræ ad diametrum paralleli DF: </s> <s xml:id="echoid-s3578" xml:space="preserve">Vt autem diameter ſphæræ BI, ad DF, diame <lb/>trum paralleli DF, ita eſt BK, ſinus totus ad DO, ſinum artus AD. </s> <s xml:id="echoid-s3579" xml:space="preserve">Minor igitur quoque <lb/> <anchor type="note" xlink:label="note-104-07a" xlink:href="note-104-07"/> eſt proportio arcus BC, ad arcum DE, q̃ ſinus totius ad ſinũ arcus AD. </s> <s xml:id="echoid-s3580" xml:space="preserve">Quod eſt propoſitũ.</s> <s xml:id="echoid-s3581" xml:space="preserve"/> </p> <div xml:id="echoid-div290" type="float" level="2" n="2"> <note position="left" xlink:label="note-104-06" xlink:href="note-104-06a" xml:space="preserve">11. huius.</note> <note position="left" xlink:label="note-104-07" xlink:href="note-104-07a" xml:space="preserve">15. quinti.</note> </div> <p> <s xml:id="echoid-s3582" xml:space="preserve">CÆTERVM quid ſit ſinus, ex ſequenti tractatione intelligetur.</s> <s xml:id="echoid-s3583" xml:space="preserve"/> </p> <pb o="93" file="105" n="105" rhead=""/> </div> <div xml:id="echoid-div292" type="section" level="1" n="134"> <head xml:id="echoid-head148" xml:space="preserve">THEOREMA 12. PROPOS 12.</head> <note position="right" xml:space="preserve">14.</note> <p> <s xml:id="echoid-s3584" xml:space="preserve">Slin ſphæra maximi circuli tangant vnum, eun <lb/>demq́; </s> <s xml:id="echoid-s3585" xml:space="preserve">parallelorum, intercipiantq́; </s> <s xml:id="echoid-s3586" xml:space="preserve">ſimiles paralle-<lb/>lorum circunferentias inter vtrũque maximorum <lb/>circulorum interiectas; </s> <s xml:id="echoid-s3587" xml:space="preserve">alius autem maximus cir-<lb/>culus ad parallelos obliquus circulos tangat ma-<lb/>iores illis, quos tangunt maximi circuli primò po-<lb/>ſiti, ſecetq́; </s> <s xml:id="echoid-s3588" xml:space="preserve">obliquus idem circulus eoſdem maxi-<lb/>mos circulos primò poſitos in punctis poſitis in-<lb/>ter maximum parallelorum, & </s> <s xml:id="echoid-s3589" xml:space="preserve">circulum, quem tan <lb/>gunt circuli maximi primo poſiti: </s> <s xml:id="echoid-s3590" xml:space="preserve">Diameter ſphæ <lb/>ræ ad diametrum circuli, quem tangit obliquus <lb/>circulus, maiorem rationem habet, quàm circun-<lb/>ferentia maximi paralleli intercepta inter circulos <lb/>primo poſitos, eundemq́; </s> <s xml:id="echoid-s3591" xml:space="preserve">circulum tangentes ad <lb/>circunferentiam obliqui circuli inter eoſdem cir-<lb/>culos interceptam.</s> <s xml:id="echoid-s3592" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3593" xml:space="preserve">IN ſphæra duo maximi circuli AB, CD, tangant eundem parallelum <lb/>AC, intercipiantq́; </s> <s xml:id="echoid-s3594" xml:space="preserve">ſimiles paralle-<lb/> <anchor type="figure" xlink:label="fig-105-01a" xlink:href="fig-105-01"/> lorum circunferentias inter ipſos in-<lb/>teriectas; </s> <s xml:id="echoid-s3595" xml:space="preserve">alius autem circulus maxi-<lb/>mus EF, tangat parallelum EG, ma-<lb/>iorem parallelo AC, in E, ſitque o-<lb/>bliquus ad parallelos, & </s> <s xml:id="echoid-s3596" xml:space="preserve">ſecet duos <lb/>priores AB, CD, inter maximum pa <lb/>rallelorum HF, & </s> <s xml:id="echoid-s3597" xml:space="preserve">parallelum AC, <lb/>in punctis I, K. </s> <s xml:id="echoid-s3598" xml:space="preserve">Dico maiorem eſſe <lb/>rationem diametri ſphæræ ad diame-<lb/>trum paralleli EG, quàm circunfe-<lb/>rentiæ BD, ad circunferentiam IK. <lb/></s> <s xml:id="echoid-s3599" xml:space="preserve">Per L, enim polum parallelorum, & </s> <s xml:id="echoid-s3600" xml:space="preserve"><lb/>puncta E, I, K, maximi circuli deſcri-<lb/> <anchor type="note" xlink:label="note-105-02a" xlink:href="note-105-02"/> bantur LH, LM, LN; </s> <s xml:id="echoid-s3601" xml:space="preserve">ac per K, pa-<lb/>rallelus KO, ſecanscirculum AB, in P. </s> <s xml:id="echoid-s3602" xml:space="preserve">Quoniam igitur maior eſt ratio dia-<lb/>metri ſphæræ ad diametrum circuli EG, quàm arcus HM, ad arcum EI; </s> <s xml:id="echoid-s3603" xml:space="preserve">ra-<lb/> <anchor type="note" xlink:label="note-105-03a" xlink:href="note-105-03"/> <pb o="94" file="106" n="106" rhead=""/> tio autem arcus HM, ad arcum EI, maior eſt, quàm arcus MN, ad arcum <lb/> <anchor type="note" xlink:label="note-106-01a" xlink:href="note-106-01"/> IK; </s> <s xml:id="echoid-s3604" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s3605" xml:space="preserve">maior ratio diametri ſphæræ ad diametrum circuli EG, quàm <lb/>arcus MN, ad arcum IK. </s> <s xml:id="echoid-s3606" xml:space="preserve">Et quia arcus PK, ſimilis eſt arcui BD, ex hy-<lb/>potheſi, & </s> <s xml:id="echoid-s3607" xml:space="preserve">arcus OK, ſimilis arcui MN; </s> <s xml:id="echoid-s3608" xml:space="preserve">eſtq́ue arcus PK, minor arcu OK; </s> <s xml:id="echoid-s3609" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-106-02a" xlink:href="note-106-02"/> quoque arcus BD, minor arcu MN; </s> <s xml:id="echoid-s3610" xml:space="preserve">ac proinde minor erit ratio arcus BD, <lb/>ad arcum IK, quàm arcus MN, ad eundẽ arcum IK. </s> <s xml:id="echoid-s3611" xml:space="preserve">Cum ergo oſtenſum ſit, ra <lb/> <anchor type="note" xlink:label="note-106-03a" xlink:href="note-106-03"/> tionem diametri ſphæræ ad diametrum circuli EG, maiorem eſſe, quàm arcus <lb/>MN, ad arcum IK; </s> <s xml:id="echoid-s3612" xml:space="preserve">Multo maior erit ratio diametri ſphæræ ad diametrum <lb/>cireuli EG, quàm arcus BD, ad arcum IK. </s> <s xml:id="echoid-s3613" xml:space="preserve">Si igitur in ſphæra maximi cir-<lb/>culi tangant vnum, &</s> <s xml:id="echoid-s3614" xml:space="preserve">c. </s> <s xml:id="echoid-s3615" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s3616" xml:space="preserve"/> </p> <div xml:id="echoid-div292" type="float" level="2" n="1"> <figure xlink:label="fig-105-01" xlink:href="fig-105-01a"> <image file="105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/105-01"/> </figure> <note position="right" xlink:label="note-105-02" xlink:href="note-105-02a" xml:space="preserve">20.1.huius.</note> <note position="right" xlink:label="note-105-03" xlink:href="note-105-03a" xml:space="preserve">11.huius.</note> <note position="left" xlink:label="note-106-01" xlink:href="note-106-01a" xml:space="preserve">coroll. 10. <lb/>huius.</note> <note position="left" xlink:label="note-106-02" xlink:href="note-106-02a" xml:space="preserve">10. 2. huius.</note> <note position="left" xlink:label="note-106-03" xlink:href="note-106-03a" xml:space="preserve">8. quinti.</note> </div> </div> <div xml:id="echoid-div294" type="section" level="1" n="135"> <head xml:id="echoid-head149" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s3617" xml:space="preserve">_IN_ exemplari græco habetur, maiorem eſſe rationem duplæ diametri ſphæræ ad <lb/>diametrum circuli _<emph style="sc">Eg</emph>,_ quàm arcus _<emph style="sc">B</emph>D,_ ad arcum _IK_ Quod quidem ex noſtra de-<lb/>monſtratione liquidò conſtat. </s> <s xml:id="echoid-s3618" xml:space="preserve">Cum enim diameter ſphæræ maiorem habeat rationem <lb/>ad diametrum circuli _EG,_ quàm arcus _BD,_ ad arcum _IK;_ </s> <s xml:id="echoid-s3619" xml:space="preserve">multo maiorem rationem <lb/>habebit dupla diametri ſphæræ ad diametrum circuli _<emph style="sc">Eg</emph>,_ quàm arcus _<emph style="sc">B</emph>D,_ ad ar-<lb/>cum _IK;_ </s> <s xml:id="echoid-s3620" xml:space="preserve">propterea quòd dupla diametri ſphæræ ad diametrum circuli _<emph style="sc">Eg</emph>,_ maiorem <lb/> <anchor type="note" xlink:label="note-106-04a" xlink:href="note-106-04"/> rationem habet, quàm diameter ſphæræ ad eandem diametrum circuli EG.</s> <s xml:id="echoid-s3621" xml:space="preserve"/> </p> <div xml:id="echoid-div294" type="float" level="2" n="1"> <note position="left" xlink:label="note-106-04" xlink:href="note-106-04a" xml:space="preserve">8.quinti.</note> </div> </div> <div xml:id="echoid-div296" type="section" level="1" n="136"> <head xml:id="echoid-head150" xml:space="preserve">THEOR. 13. PROPOS. 13.</head> <note position="left" xml:space="preserve">15.</note> <p> <s xml:id="echoid-s3622" xml:space="preserve">SI in ſphæra paralleli circuli intercipiant cir-<lb/>cunferentias maximi alicuius circuli vtrinq; </s> <s xml:id="echoid-s3623" xml:space="preserve">æqua <lb/>les ab illo puncto, in quo ipſe maximus circulus <lb/>ſecat maximum parallelorum; </s> <s xml:id="echoid-s3624" xml:space="preserve">per puncta autem <lb/>terminantia æquales circunferentias, & </s> <s xml:id="echoid-s3625" xml:space="preserve">per paral-<lb/>lelorum polos deſcribantur maximi circuli, aut ſi <lb/>deſcribantur maximi circuli, qui vnum eundem-<lb/>que parallelorum tangant: </s> <s xml:id="echoid-s3626" xml:space="preserve">æquales intercipient cir <lb/>cunferentias de maximo parallelorum.</s> <s xml:id="echoid-s3627" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3628" xml:space="preserve">IN ſphæra AB, paralleli circuli CD, EF, auferant de maximo circulo <lb/> <anchor type="figure" xlink:label="fig-106-01a" xlink:href="fig-106-01"/> AF, duas circunferentias <lb/>æquales GC, GF, vtrin-<lb/>que à puncto G, in quo <lb/>circulus AF, ſecat maxi-<lb/>mum parallelorum BG; <lb/></s> <s xml:id="echoid-s3629" xml:space="preserve">& </s> <s xml:id="echoid-s3630" xml:space="preserve">per puncta C, G, F, du <lb/>cãtur maximi circuli ſi-<lb/>ue per polos parallelo-<lb/>rum, vt in priori figura, <lb/>ſiue tangẽtes vnum eun-<lb/>demque parallelũ, vt in figura poſteriori, ſecantes maximum parallelorum in <pb o="95" file="107" n="107" rhead=""/> H,I. </s> <s xml:id="echoid-s3631" xml:space="preserve">Dico arcus GH, GI, æquales eſſe. </s> <s xml:id="echoid-s3632" xml:space="preserve">Quoniam enim arcus GC, GF, æqua-<lb/>les ponuntur, erunt paralleli CD, EF, æquales. </s> <s xml:id="echoid-s3633" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3634" xml:space="preserve">arcus GK, GL, <lb/> <anchor type="note" xlink:label="note-107-01a" xlink:href="note-107-01"/> æquales erunt. </s> <s xml:id="echoid-s3635" xml:space="preserve">Quare rectæ ductæ CK, FL, æquales erunt; </s> <s xml:id="echoid-s3636" xml:space="preserve">ac proinde in cir-<lb/> <anchor type="note" xlink:label="note-107-02a" xlink:href="note-107-02"/> culis æqualibus CD, EF, arcus æquales auferent CK, FL; </s> <s xml:id="echoid-s3637" xml:space="preserve">& </s> <s xml:id="echoid-s3638" xml:space="preserve">idcirco inter <lb/> <anchor type="note" xlink:label="note-107-03a" xlink:href="note-107-03"/> ſe ſimiles erunt arcus Ck, FL: </s> <s xml:id="echoid-s3639" xml:space="preserve">Eſt autem arcus GH, arcui CK, & </s> <s xml:id="echoid-s3640" xml:space="preserve">arcus <lb/> <anchor type="note" xlink:label="note-107-04a" xlink:href="note-107-04"/> GI, arcui FL, ſimilis. </s> <s xml:id="echoid-s3641" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3642" xml:space="preserve">arcus GH, GI, ſimiles inter ſe erunt, ac <lb/> <anchor type="note" xlink:label="note-107-05a" xlink:href="note-107-05"/> proinde, cum ſint eiuſdem circuli, æquales interſe. </s> <s xml:id="echoid-s3643" xml:space="preserve">Siigitur in ſphæra ma-<lb/> <anchor type="note" xlink:label="note-107-06a" xlink:href="note-107-06"/> ximus circulus, &</s> <s xml:id="echoid-s3644" xml:space="preserve">c. </s> <s xml:id="echoid-s3645" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s3646" xml:space="preserve"/> </p> <div xml:id="echoid-div296" type="float" level="2" n="1"> <figure xlink:label="fig-106-01" xlink:href="fig-106-01a"> <image file="106-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/106-01"/> </figure> <note position="right" xlink:label="note-107-01" xlink:href="note-107-01a" xml:space="preserve">17. 2. huius</note> <note position="right" xlink:label="note-107-02" xlink:href="note-107-02a" xml:space="preserve">18. 2. huius</note> <note position="right" xlink:label="note-107-03" xlink:href="note-107-03a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-107-04" xlink:href="note-107-04a" xml:space="preserve">28 tertij.</note> <note position="right" xlink:label="note-107-05" xlink:href="note-107-05a" xml:space="preserve">10. vel. 13.</note> <note position="right" xlink:label="note-107-06" xlink:href="note-107-06a" xml:space="preserve">2. huius.</note> </div> </div> <div xml:id="echoid-div298" type="section" level="1" n="137"> <head xml:id="echoid-head151" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s3647" xml:space="preserve">_HINC_ etiam conſtat, ijsdem poſitis, omnes arcus maximorum circuloruminter <lb/>parallelos interceptos inter ſe æquales eſſe, quales ſunt _CH, HE, KG, GL, DI,_ <lb/>_IF._ </s> <s xml:id="echoid-s3648" xml:space="preserve">Cum enim arcus _<emph style="sc">G</emph>C, GH,_ arcubus _<emph style="sc">G</emph>F, <emph style="sc">GI</emph>,_ æquales ſint, erunt & </s> <s xml:id="echoid-s3649" xml:space="preserve">rectæ _CH_, <lb/> <anchor type="note" xlink:label="note-107-07a" xlink:href="note-107-07"/> _FI,_ æquales; </s> <s xml:id="echoid-s3650" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s3651" xml:space="preserve">arcus _CH, FI,_ æquales erunt: </s> <s xml:id="echoid-s3652" xml:space="preserve">Sunt autem arcui _CH_, <lb/> <anchor type="note" xlink:label="note-107-08a" xlink:href="note-107-08"/> arcus _<emph style="sc">Kg</emph>, DI,_ & </s> <s xml:id="echoid-s3653" xml:space="preserve">arcui _FI_, arcus _LG, EH,_ æquales. </s> <s xml:id="echoid-s3654" xml:space="preserve">Igitur omnes illi ſex ars <lb/> <anchor type="note" xlink:label="note-107-09a" xlink:href="note-107-09"/> cus æquales erunt.</s> <s xml:id="echoid-s3655" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="2" n="1"> <note position="right" xlink:label="note-107-07" xlink:href="note-107-07a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-107-08" xlink:href="note-107-08a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-107-09" xlink:href="note-107-09a" xml:space="preserve">10. vel 13.</note> </div> <note position="right" xml:space="preserve">2. huius.</note> </div> <div xml:id="echoid-div300" type="section" level="1" n="138"> <head xml:id="echoid-head152" xml:space="preserve">THEOREMA 14. PROPOS. 14.</head> <note position="right" xml:space="preserve">16.</note> <p> <s xml:id="echoid-s3656" xml:space="preserve">SI in ſphæra maximus circulus aliquem circu-<lb/>lumtangat, alius autem maximus circulus obli-<lb/>quus ad parallelos tangat circulos maiores illis, <lb/>quos tangebat maximus circulus primo poſitus: <lb/></s> <s xml:id="echoid-s3657" xml:space="preserve">inæquales intercipient circunferẽtias parallelorũ <lb/>circulorum, quarum propiores vtriuis polorum <lb/>maiores erunt, quàm vt ſimiles ſint remotioribus.</s> <s xml:id="echoid-s3658" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3659" xml:space="preserve">IN ſphæra maximus circulus AB, tangat circulum AC; </s> <s xml:id="echoid-s3660" xml:space="preserve">& </s> <s xml:id="echoid-s3661" xml:space="preserve">alius maximus <lb/>DE, tãgat alium maiorẽ DF, ſecet-<lb/> <anchor type="figure" xlink:label="fig-107-01a" xlink:href="fig-107-01"/> q́ue duos parallelos quoſcũq; </s> <s xml:id="echoid-s3662" xml:space="preserve">GH, <lb/>BI, in k, E. </s> <s xml:id="echoid-s3663" xml:space="preserve">Dico arcus k H, EI, in-<lb/>æquales eſſe, maioremque eſſe k H, <lb/>polo conſpicuo propiorem, quàm <lb/>vt ſimilis ſit arcui EI, remotiori: <lb/></s> <s xml:id="echoid-s3664" xml:space="preserve">vel ipſum EB, polo occulto pro-<lb/>piorẽ eſſe maiorem, quam vt arcui <lb/>KG, remotiori ſimilis ſit. </s> <s xml:id="echoid-s3665" xml:space="preserve">Per pun-<lb/>cta enim E, K, deſcribantur maximi <lb/> <anchor type="note" xlink:label="note-107-12a" xlink:href="note-107-12"/> circuli LE, CN, tangentes circu-<lb/>lum AC, ita vt ſemicirculià C, per <lb/>N, & </s> <s xml:id="echoid-s3666" xml:space="preserve">ab A, per B, procedentes non <lb/>conueniant: </s> <s xml:id="echoid-s3667" xml:space="preserve">item ſemicirculi ab L, <lb/>per E, & </s> <s xml:id="echoid-s3668" xml:space="preserve">ab A, per I, tendentes non <lb/>coeant. </s> <s xml:id="echoid-s3669" xml:space="preserve">Erunt igitur arcus MH, <lb/> <anchor type="note" xlink:label="note-107-13a" xlink:href="note-107-13"/> EI, ſimiles. </s> <s xml:id="echoid-s3670" xml:space="preserve">Quare k H, maior eſt, quàm vt arcui EI, ſimilis ſit. </s> <s xml:id="echoid-s3671" xml:space="preserve">Eodem <pb o="96" file="108" n="108" rhead=""/> modo, quoniam ſimiles ſunt arcus BN, Gk, erit BE, alteri polo propior ma-<lb/>ior, quàm vt ſimilis ſit arcui Gk, ab eodem polo remotiori. </s> <s xml:id="echoid-s3672" xml:space="preserve">Itaque ſi in ſphæ-<lb/>ra maximus circulus aliquẽ circulum tangat, &</s> <s xml:id="echoid-s3673" xml:space="preserve">c. </s> <s xml:id="echoid-s3674" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s3675" xml:space="preserve"/> </p> <div xml:id="echoid-div300" type="float" level="2" n="1"> <figure xlink:label="fig-107-01" xlink:href="fig-107-01a"> <image file="107-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/107-01"/> </figure> <note position="right" xlink:label="note-107-12" xlink:href="note-107-12a" xml:space="preserve">15. 2. huius.</note> <note position="right" xlink:label="note-107-13" xlink:href="note-107-13a" xml:space="preserve">13. 2. huius.</note> </div> </div> <div xml:id="echoid-div302" type="section" level="1" n="139"> <head xml:id="echoid-head153" xml:space="preserve">FINIS LIBRI III. THEODOSII.</head> <head xml:id="echoid-head154" xml:space="preserve">AD LECTOREM.</head> <p style="it"> <s xml:id="echoid-s3676" xml:space="preserve">POTERVNT, ſi placet, hæ duæ figuræ tribus illis propoſitionis <lb/>ſecundæ lib. </s> <s xml:id="echoid-s3677" xml:space="preserve">3. </s> <s xml:id="echoid-s3678" xml:space="preserve">adiungi, vt omnes caſus lineæ perpendicularis FL, per-<lb/>ſpiciantur. </s> <s xml:id="echoid-s3679" xml:space="preserve">In prima namque harum figurarum ſegmentum inſiſtens <lb/> <anchor type="figure" xlink:label="fig-108-01a" xlink:href="fig-108-01"/> AFB, eſt ſemicircu-<lb/>lus, caditq́, perpendicu <lb/>laris FL, intra ſegmen <lb/>tum ADB: </s> <s xml:id="echoid-s3680" xml:space="preserve">In poſte-<lb/>riore autẽ eadem FL, <lb/>in ipſam circunferen-<lb/>tiam ADB, cadit, <lb/>exiſtente eodem ſegmen <lb/>to inſiſtente AFB, ſe-<lb/>micirculo; </s> <s xml:id="echoid-s3681" xml:space="preserve">quemadmo-<lb/>dum & </s> <s xml:id="echoid-s3682" xml:space="preserve">in tertia figura dictæ propoſitionis idem ſegmentum inſiſtens <lb/>AFB, ſemicirculus eſt, lineá perpendicularis FL, extra ſegmentum <lb/>ADB, cadit. </s> <s xml:id="echoid-s3683" xml:space="preserve">Hoc, benigne lector, te latere noluimus.</s> <s xml:id="echoid-s3684" xml:space="preserve"/> </p> <div xml:id="echoid-div302" type="float" level="2" n="1"> <figure xlink:label="fig-108-01" xlink:href="fig-108-01a"> <image file="108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/108-01"/> </figure> </div> <pb file="109" n="109"/> </div> <div xml:id="echoid-div304" type="section" level="1" n="140"> <head xml:id="echoid-head155" xml:space="preserve">CHRISTOPHORI</head> <head xml:id="echoid-head156" xml:space="preserve">CLAVII BAMBERGENSIS <lb/>E SOCIETATE IESV</head> <head xml:id="echoid-head157" xml:space="preserve">SINVS, VEL SEMISSES RECTARVM <lb/>IN CIRCVLO SVBTENSARVM:</head> <head xml:id="echoid-head158" style="it" xml:space="preserve">LINEAE TANGENTES: ATQVE <lb/>SECANTES.</head> <figure> <image file="109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/109-01"/> </figure> <pb file="110" n="110"/> <pb o="99" file="111" n="111"/> </div> <div xml:id="echoid-div305" type="section" level="1" n="141"> <head xml:id="echoid-head159" xml:space="preserve">CHRISTOPHORI CLAV II</head> <head xml:id="echoid-head160" xml:space="preserve">BAMBERGENSIS</head> <head xml:id="echoid-head161" xml:space="preserve">E SOCIETATE IESV</head> <head xml:id="echoid-head162" style="it" xml:space="preserve">SINVS, VEL SEMISSES RECTARVM</head> <head xml:id="echoid-head163" style="it" xml:space="preserve">in circulo ſubtenſarum:</head> <head xml:id="echoid-head164" xml:space="preserve">LINEÆ TANGENTES, ATQVE</head> <head xml:id="echoid-head165" xml:space="preserve">SECANTES.</head> <head xml:id="echoid-head166" xml:space="preserve">PRÆFATIO.</head> <p style="it"> <s xml:id="echoid-s3685" xml:space="preserve">DICI vix poteſt, quantam in <lb/> <anchor type="note" xlink:label="note-111-01a" xlink:href="note-111-01"/> rebustã Aſtronomicis, quàm <lb/>Geometricis, vtilitatẽ babeat <lb/>Sinuũ cognitio: </s> <s xml:id="echoid-s3686" xml:space="preserve">cũ innumera-<lb/>bilia peneproblemata Aſtro-<lb/>nomica, & </s> <s xml:id="echoid-s3687" xml:space="preserve">Geometrica ad <lb/>vſumper calculum & </s> <s xml:id="echoid-s3688" xml:space="preserve">rationem Sinuũreuocen <lb/>tur, vt tumex noſtris triangulis rectilineis, ac <lb/>sphæricis, tum ex Almageſto Ptolemæi, ex noſtra <lb/>Gnomonica, & </s> <s xml:id="echoid-s3689" xml:space="preserve">exalijs variorum Aſtronomo-<lb/>rum libris manifeſtumeſt. </s> <s xml:id="echoid-s3690" xml:space="preserve">Quare, cum à paucis <lb/>admodum Sinuum demonſtrationes ſint expli-<lb/>catæ, operæpretiũme facturum arbitror, ſi, quan <lb/>ta potero breuitate, ac perſpicuitate, ex varijs <lb/>auctoribus, præſertim ex Ptolemæo, Purbachio, <pb o="100" file="112" n="112" rhead=""/> atque Iohanne Regiomontano, demonſtrationes <lb/>colligam, quibus omnium arcuum ſinus & </s> <s xml:id="echoid-s3691" xml:space="preserve">chor-<lb/>das cognitas babeamus, vt & </s> <s xml:id="echoid-s3692" xml:space="preserve">tabulas Sinuum, <lb/>ac chordarumiam à multis ſcriptoribus ſuppu-<lb/>tatas examinare, (facile enim error in numero-<lb/>rum impreßione committitur) & </s> <s xml:id="echoid-s3693" xml:space="preserve">nouas alias, <lb/>quandores tulerit (poſito Sinu toto vel diame-<lb/>tro quotcunq; </s> <s xml:id="echoid-s3694" xml:space="preserve">particularum) condere poßimus.</s> <s xml:id="echoid-s3695" xml:space="preserve"/> </p> <div xml:id="echoid-div305" type="float" level="2" n="1"> <note position="right" xlink:label="note-111-01" xlink:href="note-111-01a" xml:space="preserve">Sinuũ vti-<lb/>litas.</note> </div> <p style="it"> <s xml:id="echoid-s3696" xml:space="preserve">QVONIAM vero Recentiores ſumma <lb/>felicitate ex ſinubus alias lineas collegerũt, nimi-<lb/>rum Tangentes, atque Secantes, vt facilius <lb/>quædam, ac breuius demonſtrarent; </s> <s xml:id="echoid-s3697" xml:space="preserve">de biſce e-<lb/>tiam lineis agemus. </s> <s xml:id="echoid-s3698" xml:space="preserve">Habent enim lineæ hæ egre-<lb/>gium vſum in rebus Aſtronomicis & </s> <s xml:id="echoid-s3699" xml:space="preserve">Geome-<lb/>tricis, vt ex noſtris triangulis planis, ac sphæri-<lb/>cis fiet per ſpicuum. </s> <s xml:id="echoid-s3700" xml:space="preserve">Initium autem ſumemus à <lb/>definitionibus.</s> <s xml:id="echoid-s3701" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div307" type="section" level="1" n="142"> <head xml:id="echoid-head167" xml:space="preserve">DEFINITIONES.</head> <head xml:id="echoid-head168" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s3702" xml:space="preserve">COMPLEMENTVM arcus alicuius, eſt <lb/> <anchor type="note" xlink:label="note-112-01a" xlink:href="note-112-01"/> exceſſus, quo quadrans eum ſuperat, ſi arcus mi-<lb/>nor eſt quadrante, vel ab eo ſuperatur, ſi eſt qua-<lb/>drante maior.</s> <s xml:id="echoid-s3703" xml:space="preserve"/> </p> <div xml:id="echoid-div307" type="float" level="2" n="1"> <note position="left" xlink:label="note-112-01" xlink:href="note-112-01a" xml:space="preserve">Comple-<lb/>mentũ at-<lb/>ous quid.</note> </div> <pb o="101" file="113" n="113" rhead=""/> </div> <div xml:id="echoid-div309" type="section" level="1" n="143"> <head xml:id="echoid-head169" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s3704" xml:space="preserve">CHORDA eſt linea recta arcum quemcun-<lb/> <anchor type="note" xlink:label="note-113-01a" xlink:href="note-113-01"/> quein circulo ſubtendens.</s> <s xml:id="echoid-s3705" xml:space="preserve"/> </p> <div xml:id="echoid-div309" type="float" level="2" n="1"> <note position="right" xlink:label="note-113-01" xlink:href="note-113-01a" xml:space="preserve">Chotda <lb/>quid.</note> </div> </div> <div xml:id="echoid-div311" type="section" level="1" n="144"> <head xml:id="echoid-head170" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s3706" xml:space="preserve">SINVS rectus eſt dimidium chordæ ſubten-<lb/> <anchor type="note" xlink:label="note-113-02a" xlink:href="note-113-02"/> dentis duplũ eius arcus, cuius dicitur ſinus rectus.</s> <s xml:id="echoid-s3707" xml:space="preserve"/> </p> <div xml:id="echoid-div311" type="float" level="2" n="1"> <note position="right" xlink:label="note-113-02" xlink:href="note-113-02a" xml:space="preserve">Sinus re-<lb/>ctus quid.</note> </div> </div> <div xml:id="echoid-div313" type="section" level="1" n="145"> <head xml:id="echoid-head171" xml:space="preserve">Vel aliter.</head> <p> <s xml:id="echoid-s3708" xml:space="preserve">SINVS rectus eſt linea perpendicularis cadens <lb/>ab vno extremo arcus, cuius dicitur ſinus rectus, in <lb/>diametrum circuli ab altero extremo eiuſdem ar-<lb/>cus ductam.</s> <s xml:id="echoid-s3709" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div314" type="section" level="1" n="146"> <head xml:id="echoid-head172" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s3710" xml:space="preserve">SINVS verſus eſt pars diametri circuli inter <lb/> <anchor type="note" xlink:label="note-113-03a" xlink:href="note-113-03"/> extremum dati arcus, cuius dicitur ſinus verſus, & </s> <s xml:id="echoid-s3711" xml:space="preserve"><lb/>ſinum rectum eiuſdem arcus intercepta.</s> <s xml:id="echoid-s3712" xml:space="preserve"/> </p> <div xml:id="echoid-div314" type="float" level="2" n="1"> <note position="right" xlink:label="note-113-03" xlink:href="note-113-03a" xml:space="preserve">Sinus vet-<lb/>ſur quid.</note> </div> </div> <div xml:id="echoid-div316" type="section" level="1" n="147"> <head xml:id="echoid-head173" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s3713" xml:space="preserve">SINVS complementi alicuius arcus eſt ſinus <lb/> <anchor type="note" xlink:label="note-113-04a" xlink:href="note-113-04"/> rectus alterius arcus, qui complementum eſt illius <lb/>arcus, cuius dicitur ſinus complementi.</s> <s xml:id="echoid-s3714" xml:space="preserve"/> </p> <div xml:id="echoid-div316" type="float" level="2" n="1"> <note position="right" xlink:label="note-113-04" xlink:href="note-113-04a" xml:space="preserve">Sinus com <lb/>plementi <lb/>quid.</note> </div> </div> <div xml:id="echoid-div318" type="section" level="1" n="148"> <head xml:id="echoid-head174" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s3715" xml:space="preserve">SINVS totus eſt ſemidiameter circuli, hoc <lb/> <anchor type="note" xlink:label="note-113-05a" xlink:href="note-113-05"/> eſt, ſinus rectus, vel verſus quadrantis circuli.</s> <s xml:id="echoid-s3716" xml:space="preserve"/> </p> <div xml:id="echoid-div318" type="float" level="2" n="1"> <note position="right" xlink:label="note-113-05" xlink:href="note-113-05a" xml:space="preserve">Sinus totus <lb/>quid.</note> </div> </div> <div xml:id="echoid-div320" type="section" level="1" n="149"> <head xml:id="echoid-head175" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s3717" xml:space="preserve">SINVS tam rectus, & </s> <s xml:id="echoid-s3718" xml:space="preserve">verſus, quàm comple-<lb/> <anchor type="note" xlink:label="note-113-06a" xlink:href="note-113-06"/> menti alicuius anguli rectilinei eſt ſinus illius ar-<lb/>cus, qui in circulo deſcripto ex angulo inter duas <lb/>rectas angulum conſtituentes interijcitur.</s> <s xml:id="echoid-s3719" xml:space="preserve"/> </p> <div xml:id="echoid-div320" type="float" level="2" n="1"> <note position="right" xlink:label="note-113-06" xlink:href="note-113-06a" xml:space="preserve">Sinus angu <lb/>li rectilinei <lb/>quid.</note> </div> <pb o="102" file="114" n="114" rhead=""/> <p style="it"> <s xml:id="echoid-s3720" xml:space="preserve">_EXPONATVR_ circulus _ABCD,_ cuius centrum _E,_ per quod ducantur duæ <lb/>diametri _AC, BD,_ ſeſe ad rectos angulos ſecantes & </s> <s xml:id="echoid-s3721" xml:space="preserve">totum circulum in quatuor <lb/>quadrantes equales diuidentes, vtpote qui angulis rectis æqualibus in centro ſub <lb/> <anchor type="note" xlink:label="note-114-01a" xlink:href="note-114-01"/> tenduntur: </s> <s xml:id="echoid-s3722" xml:space="preserve">ſumanturque arcus æquales _AF, AG:_ </s> <s xml:id="echoid-s3723" xml:space="preserve">Item _BF, BI;_ </s> <s xml:id="echoid-s3724" xml:space="preserve">& </s> <s xml:id="echoid-s3725" xml:space="preserve">rectæ ducantur <lb/>_FG, FI,_ ſecantes diametros in _H,_ & </s> <s xml:id="echoid-s3726" xml:space="preserve">_K._ </s> <s xml:id="echoid-s3727" xml:space="preserve">Arcusigitur _FB,_ dicitur complementum <lb/> <anchor type="note" xlink:label="note-114-02a" xlink:href="note-114-02"/> arcus _FA;_ </s> <s xml:id="echoid-s3728" xml:space="preserve">quia quadrans _AB,_ arcum _FA,_ ſuperat arcu _FB._ </s> <s xml:id="echoid-s3729" xml:space="preserve">Eadem ratione arcus <lb/>_FA,_ complementum nominatur arcus _FB._ </s> <s xml:id="echoid-s3730" xml:space="preserve">Item arcus _BI,_ complementum appellatur <lb/>arcus _AI;_ </s> <s xml:id="echoid-s3731" xml:space="preserve">quiaarcu _BI,_ ſuperatur quadrans _AB,_ ab arcu _AI._</s> <s xml:id="echoid-s3732" xml:space="preserve"/> </p> <div xml:id="echoid-div321" type="float" level="2" n="2"> <note position="left" xlink:label="note-114-01" xlink:href="note-114-01a" xml:space="preserve">26. tettij.</note> <note position="left" xlink:label="note-114-02" xlink:href="note-114-02a" xml:space="preserve">Exẽpla deſi <lb/>nitionum.</note> </div> <figure> <image file="114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/114-01"/> </figure> <p style="it"> <s xml:id="echoid-s3733" xml:space="preserve">_RECTA_ deinde _FG,_ <lb/>chorda dicitur arcus _FAG;_ <lb/></s> <s xml:id="echoid-s3734" xml:space="preserve">& </s> <s xml:id="echoid-s3735" xml:space="preserve">recta _FI,_ chorda arcus <lb/>_FBI._ </s> <s xml:id="echoid-s3736" xml:space="preserve">Et quia diameter _AC,_ <lb/>ſecãs arcũ _FAG,_ bifariã ſecat <lb/>quoq; </s> <s xml:id="echoid-s3737" xml:space="preserve">rectã _FG,_ bifariã, vt ex <lb/>coroll, 1. </s> <s xml:id="echoid-s3738" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s3739" xml:space="preserve">_IO._ </s> <s xml:id="echoid-s3740" xml:space="preserve">lib. </s> <s xml:id="echoid-s3741" xml:space="preserve">13. </s> <s xml:id="echoid-s3742" xml:space="preserve"><lb/>Euil. </s> <s xml:id="echoid-s3743" xml:space="preserve">cõſtat, (quod tamen in <lb/>lẽmate ſequẽti breuius oſtẽde <lb/>mus.) </s> <s xml:id="echoid-s3744" xml:space="preserve">erit recta _FH,_ ſinus re-<lb/>ctus arcus _FA,_ iuxtapriorẽ de <lb/>fin. </s> <s xml:id="echoid-s3745" xml:space="preserve">ſinus recti: </s> <s xml:id="echoid-s3746" xml:space="preserve">quia eſt dimi-<lb/>diũ chordæ _FG,_ ſubtendentis <lb/>arcũ _FAG,_ duplum arcus _FA,_ <lb/>cuius _FH,_ dicitur ſinus. </s> <s xml:id="echoid-s3747" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s3748" xml:space="preserve"><lb/>ſi quemlibet arcũ, eiusq́; </s> <s xml:id="echoid-s3749" xml:space="preserve">chor <lb/>dam bifariã ſecemus, dimidiũ <lb/>chordæ dicetur ſinus rectus <lb/>dimidiati arcus. </s> <s xml:id="echoid-s3750" xml:space="preserve">Hinc factum <lb/>eſt, vt Sinus recti à plerisque <lb/> <anchor type="note" xlink:label="note-114-03a" xlink:href="note-114-03"/> dicantur Semiſſes rectarum <lb/>in circulo ſubtenſarum. </s> <s xml:id="echoid-s3751" xml:space="preserve">Eadem quoque recta _FH,_ erit ſinus rectus eiuſdem arcus <lb/>_FA,_ ſecundum poſteriorem defin. </s> <s xml:id="echoid-s3752" xml:space="preserve">ſinus recti: </s> <s xml:id="echoid-s3753" xml:space="preserve">quoniam perpendicularis eſt, ducta ab _F,_ <lb/>extremo dicti arcus ad diametrum _AC,_ ab altero extremo _A,_ eiuſdem arcus ductam; <lb/></s> <s xml:id="echoid-s3754" xml:space="preserve">propterea quod recta _EA,_ ſecans rectã _FG,_ bifariam, ſecat eandẽ ad angulos rectos.</s> <s xml:id="echoid-s3755" xml:space="preserve"/> </p> <div xml:id="echoid-div322" type="float" level="2" n="3"> <note position="left" xlink:label="note-114-03" xlink:href="note-114-03a" xml:space="preserve">Sinus recti <lb/>curdicãtur <lb/>ſemiſſes re-<lb/>ctarum in <lb/>circulo ſub <lb/>tenſarum.</note> </div> <note position="left" xml:space="preserve">3. tertij.</note> <p style="it"> <s xml:id="echoid-s3756" xml:space="preserve">_RECTA_ vero _AH,_ ſinus verſus eſt eiuſdem arcus _FA;_ </s> <s xml:id="echoid-s3757" xml:space="preserve">cum ſit pars diametri <lb/>_AC,_ inter _A,_ extremum dicti arcus, & </s> <s xml:id="echoid-s3758" xml:space="preserve">ſinum eius rectum _FH,_ intercepta. </s> <s xml:id="echoid-s3759" xml:space="preserve">Dicitur <lb/>autem verſus Inc Sinus, quia verſomodo collocatur, ſi cum ſinu recto conferatur. <lb/></s> <s xml:id="echoid-s3760" xml:space="preserve"> <anchor type="note" xlink:label="note-114-05a" xlink:href="note-114-05"/> Hunc nonnulli dicunt ſagittam, quoniam inſtar ſagittæ eſt in arcu _FAG,_ à chorda <lb/>_FG,_ excuſſæ.</s> <s xml:id="echoid-s3761" xml:space="preserve"/> </p> <div xml:id="echoid-div323" type="float" level="2" n="4"> <note position="left" xlink:label="note-114-05" xlink:href="note-114-05a" xml:space="preserve">Sinus vet-<lb/>ſus cur di-<lb/>catur ſagit <lb/>ta.</note> </div> <p style="it"> <s xml:id="echoid-s3762" xml:space="preserve">_RECTA_ porrò _FK,_ est ſinus complementi arcus _FA;_ </s> <s xml:id="echoid-s3763" xml:space="preserve">quia eſt ſinus rectus arcus <lb/> <anchor type="note" xlink:label="note-114-06a" xlink:href="note-114-06"/> FB, qui complementum eſt arcus FA.</s> <s xml:id="echoid-s3764" xml:space="preserve"/> </p> <div xml:id="echoid-div324" type="float" level="2" n="5"> <note position="left" xlink:label="note-114-06" xlink:href="note-114-06a" xml:space="preserve">Sinꝰ verſus <lb/>cõplemẽti <lb/>alicuius ar <lb/>cus quid.</note> </div> <p style="it"> <s xml:id="echoid-s3765" xml:space="preserve">_RECTA_ autem _KB,_ eſt ſinus verſus complementi arcus _FA,_ hoc eſt, ſinus ber-<lb/>ſus arcus _FB,_ qui complementum eſt arcus _FA._</s> <s xml:id="echoid-s3766" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Sinus rectꝰ <lb/>primꝰ ac ſe <lb/>cundus: Itẽ <lb/>ſinus pri-<lb/>mus & ſinꝰ <lb/>ſecundus <lb/>apud quos <lb/>dam quid.</note> <p style="it"> <s xml:id="echoid-s3767" xml:space="preserve">_NONNVLLI_ porro Aſtronomi Sinum, quem nos rectum diximus, appellant <lb/>Sinum rectum primum; </s> <s xml:id="echoid-s3768" xml:space="preserve">Sinum vero, quem Sinum complementi diximus, bocant Sinum <lb/>rectum ſecundum. </s> <s xml:id="echoid-s3769" xml:space="preserve">Vt rectam _FH,_ appellant ſinum rectum primum arcus _FA;_ </s> <s xml:id="echoid-s3770" xml:space="preserve">re-<lb/>ctam bero _FK,_ ſinum rectum ſecundum eiuſdem arcus. </s> <s xml:id="echoid-s3771" xml:space="preserve">Sunt alij etiam, qui ſinum <lb/>rectum ſimpliciter appellent ſinum primum, verſum autem dicantſinum ſecundum.</s> <s xml:id="echoid-s3772" xml:space="preserve"> <pb o="103" file="115" n="115" rhead=""/> Quod dixerim, btintelligas auctores, qui varie de ſinubus ſunt locuti Nos communem <lb/> <anchor type="note" xlink:label="note-115-01a" xlink:href="note-115-01"/> modum loquendi retinuimus. </s> <s xml:id="echoid-s3773" xml:space="preserve">Caterum cum ſcriptores de ſinu alique loquuntur, ſem-<lb/>per intelligunt ſinum rectum: </s> <s xml:id="echoid-s3774" xml:space="preserve">niſi illum bocent ſinum complementi, aut verſum.</s> <s xml:id="echoid-s3775" xml:space="preserve"/> </p> <div xml:id="echoid-div325" type="float" level="2" n="6"> <note position="right" xlink:label="note-115-01" xlink:href="note-115-01a" xml:space="preserve">Quandofit <lb/>mentio ali <lb/>cuius ſinus <lb/>abſolure, <lb/>intelligit ſi <lb/>nus rectus.</note> </div> <p style="it"> <s xml:id="echoid-s3776" xml:space="preserve">_SEMIDIAMETER_ deinde _AE,_ ſinus eſt tam rectus, quàm verſus quadran-<lb/>tis _AB:_ </s> <s xml:id="echoid-s3777" xml:space="preserve">qui totus dicitur, ſiue maximus, propterea quò d maximus ſit omnium ſinuum <lb/>tam rectorum, quàm complemẽtorum; </s> <s xml:id="echoid-s3778" xml:space="preserve">immo vero & </s> <s xml:id="echoid-s3779" xml:space="preserve">maior omnibus ſinubus verſis il-<lb/> <anchor type="note" xlink:label="note-115-02a" xlink:href="note-115-02"/> lorum arcuum, qui quadrante minores ſunt: </s> <s xml:id="echoid-s3780" xml:space="preserve">Solum minor eſt ſinubus verſis illorum <lb/>arcuum, qui quadrante ſunt maiores, vt infradicemus, qui quidem rarius in vſum <lb/>veniunt, quàm alij. </s> <s xml:id="echoid-s3781" xml:space="preserve">Vel certe dicitur totus, ſiue maximus, quia in tabula Sinuum, in <lb/>qua Sinus recti tantummodo ponuntur, omnium Sinuum maximus eſt ille, qui qua-<lb/>dranti, ſeu gradibus 90. </s> <s xml:id="echoid-s3782" xml:space="preserve">reſpondet, vt ex tabula Sinuum, quam infra ponemus, perſpi-<lb/>cuumerit.</s> <s xml:id="echoid-s3783" xml:space="preserve"/> </p> <div xml:id="echoid-div326" type="float" level="2" n="7"> <note position="right" xlink:label="note-115-02" xlink:href="note-115-02a" xml:space="preserve">Sinus totus <lb/>vel maxi-<lb/>mus cur ſic <lb/>dicatur.</note> </div> <p style="it"> <s xml:id="echoid-s3784" xml:space="preserve">_POSTREMO,_ ducta recta _EF,_ erit recta _FH,_ ſinus rectus anguli _FEH;_ </s> <s xml:id="echoid-s3785" xml:space="preserve">re-<lb/>cta autem _FK,_ ſinus complementi eiusdem anguli; </s> <s xml:id="echoid-s3786" xml:space="preserve">& </s> <s xml:id="echoid-s3787" xml:space="preserve">recta _AH,_ eiusdem anguli ſi-<lb/> <anchor type="note" xlink:label="note-115-03a" xlink:href="note-115-03"/> nus verſus: </s> <s xml:id="echoid-s3788" xml:space="preserve">quoniam recta _FH,_ eſt ſinus rectus arcus _FA,_ in circulo deſcripto ex an-<lb/>gulo _FEH,_ interceptus inter rectas _EF, EA,_ angulum dictum conſt: </s> <s xml:id="echoid-s3789" xml:space="preserve">tuentes: </s> <s xml:id="echoid-s3790" xml:space="preserve">recta <lb/>autem _FK,_ eſt ſinus complementi eiusdem arcus; </s> <s xml:id="echoid-s3791" xml:space="preserve">& </s> <s xml:id="echoid-s3792" xml:space="preserve">recta _AH,_ ſinus verſus.</s> <s xml:id="echoid-s3793" xml:space="preserve"/> </p> <div xml:id="echoid-div327" type="float" level="2" n="8"> <note position="right" xlink:label="note-115-03" xlink:href="note-115-03a" xml:space="preserve">Duo arcus <lb/>ſemicircu-<lb/>lum conſi-<lb/>cientes eũ-<lb/>dem habẽt <lb/>ſinum, quẽ <lb/>admodum <lb/>& duo arc <lb/>circulũ con <lb/>ficiẽtes, eã-<lb/>dẽ chordã: <lb/>ſinꝰ tń ver-<lb/>ſos habent <lb/>differentes, <lb/>conficiẽtes <lb/>totã diame <lb/>trum.</note> </div> <p style="it"> <s xml:id="echoid-s3794" xml:space="preserve">_CAETERVM_ duo arcus ſemicirculum conſtituentes eundem prorſus habent ſi-<lb/>num, tam rectum, quàm complementi; </s> <s xml:id="echoid-s3795" xml:space="preserve">quemadmodũ & </s> <s xml:id="echoid-s3796" xml:space="preserve">duo arcus circulum conficien-<lb/>tes vnam eandemq́; </s> <s xml:id="echoid-s3797" xml:space="preserve">chordam habent: </s> <s xml:id="echoid-s3798" xml:space="preserve">ſinus tamen verſi eorum differunt, conficiuntq́; <lb/></s> <s xml:id="echoid-s3799" xml:space="preserve">totam circuli diametrum. </s> <s xml:id="echoid-s3800" xml:space="preserve">Vt duo arcus _FA, FC,_ conficientes ſemicirculum _ABC,_ <lb/>cundem habent ſinum rectum _FH,_ quemadmodum & </s> <s xml:id="echoid-s3801" xml:space="preserve">duo arcus _FAG, FCG,_ eorum <lb/>dupli, circulum conficientes, eandem habent chordam _<emph style="sc">Fg</emph>,_ cuius dimidium eſt ſinus <lb/>rectus _FH,_ vt vult prior deſinitio ſinus recti; </s> <s xml:id="echoid-s3802" xml:space="preserve">qui quidem ſinus rectus _FH,_ linea <lb/>perpendi cularis eſt, ducta à communi extremo _F,_ vtriuſque arcus _FA, FC,_ ad dia-<lb/>metrum _AC,_ ab extremis reliquis _A, C,_ eorundem arcuum ductam, vt vult poſte-<lb/>rior ſinus recti definitio. </s> <s xml:id="echoid-s3803" xml:space="preserve">Iidem duo arcus _FA, FC,_ eundem ſinum complementi ha-<lb/> <anchor type="note" xlink:label="note-115-04a" xlink:href="note-115-04"/> bent _FK;_ </s> <s xml:id="echoid-s3804" xml:space="preserve">propterea quòd arcus _FB,_ cuius ſinus rectus eſt _FK,_ eſt complementum <lb/>vtriuſque arcus. </s> <s xml:id="echoid-s3805" xml:space="preserve">Sinus tamen ve ſi ijdem non ſunt, ſed _AH,_ eſt ſinus verſus arcus <lb/>_FA;_ </s> <s xml:id="echoid-s3806" xml:space="preserve">& </s> <s xml:id="echoid-s3807" xml:space="preserve">_CH,_ ſt ſinus verſus arcus _FC:_ </s> <s xml:id="echoid-s3808" xml:space="preserve">qui quidem duo ſinus ve ſi diametrum _AC,_ <lb/>conſtituũt. </s> <s xml:id="echoid-s3809" xml:space="preserve">V bi vides ſinum verſum _CH,_ arcus _FC,_ quadrantem ſuperantis maiorem <lb/> <anchor type="note" xlink:label="note-115-05a" xlink:href="note-115-05"/> eſſe ſemidiametro, ſeu ſinu toto _CE._</s> <s xml:id="echoid-s3810" xml:space="preserve"/> </p> <div xml:id="echoid-div328" type="float" level="2" n="9"> <note position="right" xlink:label="note-115-04" xlink:href="note-115-04a" xml:space="preserve">Sinꝰ verſus <lb/>arcus qua-<lb/>drãte maio <lb/>ris maior ẽ <lb/>ſinu toto.</note> <note position="right" xlink:label="note-115-05" xlink:href="note-115-05a" xml:space="preserve">Duo angu-<lb/>li duobꝰ re <lb/>ctis ęquales <lb/>eundẽ ſinũ <lb/>habent, ſed <lb/>ſinus ver-<lb/>ſos differẽ-<lb/>tes, vtpote <lb/>ꝗ totã dia-<lb/>metrũ cõfi <lb/>ciant.</note> </div> <p style="it"> <s xml:id="echoid-s3811" xml:space="preserve">_SIC_ etiam duo anguli duobus rectis æquales eundem ſinum babent tam rectum, <lb/>quam complementi. </s> <s xml:id="echoid-s3812" xml:space="preserve">Vt patet in angulis _AEF, FEC,_ quorum vtriuſque ſinus re-<lb/>ctus eſt _FH;_ </s> <s xml:id="echoid-s3813" xml:space="preserve">ſinus autem complementi _FK:_ </s> <s xml:id="echoid-s3814" xml:space="preserve">propterea quòd arcubus _AF, FC,_ inſi-<lb/>ſtunt, quorum vtriuſque ſinus rectus eſt _FH,_ complementi autem ſinus _FK,_ vt dictum <lb/>eſt. </s> <s xml:id="echoid-s3815" xml:space="preserve">Sinus tamen verſi eorundem angulorum ijdem non ſunt, ſed _AH,_ ſinus verſus eſt <lb/>anguli _AEF,_ nempe arcus _AF; </s> <s xml:id="echoid-s3816" xml:space="preserve">& </s> <s xml:id="echoid-s3817" xml:space="preserve">HC,_ eſt ſinus verſus anguli _FEC,_ puta arcus <lb/>_FC._ </s> <s xml:id="echoid-s3818" xml:space="preserve">Conficiunt autem ambo ſinus verſi totam diametrum _AC._</s> <s xml:id="echoid-s3819" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3820" xml:space="preserve">_RVRSVM_ ſinus rectus cuiuſuis arcus æqualis eſt ſegmento diametri inter cen-<lb/> <anchor type="note" xlink:label="note-115-06a" xlink:href="note-115-06"/> trum, & </s> <s xml:id="echoid-s3821" xml:space="preserve">ſinum rectum complementi eiusdem arcus interiecto: </s> <s xml:id="echoid-s3822" xml:space="preserve">Sinus autem comple-<lb/>menti cuiuslibet arcus æqualis eſt ſegmento diametri inter centrum, & </s> <s xml:id="echoid-s3823" xml:space="preserve">ſinum rectum <lb/>eiuſdem arcus poſito. </s> <s xml:id="echoid-s3824" xml:space="preserve">Vt _FH,_ ſinus rectus arcus _FA,_ æqualis eſt ſegmento diamet ri <lb/>_EK: </s> <s xml:id="echoid-s3825" xml:space="preserve">& </s> <s xml:id="echoid-s3826" xml:space="preserve">FK,_ ſinus complementi eiuſdem arcus _FA,_ æqualis eſt ſegmento diametri _EH;_ <lb/></s> <s xml:id="echoid-s3827" xml:space="preserve">ob parallelogrammum HK: </s> <s xml:id="echoid-s3828" xml:space="preserve">ſunt enim tam rectæ _HF, EK,_ quàm rectæ _KF, EH,_ <lb/>parallelæ, propter rectos angulos _H, E, K, F._ </s> <s xml:id="echoid-s3829" xml:space="preserve">Hinc fit, ſi ſint duo arcus, quorum vnus <lb/>alterius ſit complementum, vtriuſuis ſinum rectum æqualem eſſe complemento ſinus <pb o="104" file="116" n="116" rhead=""/> verſi alterius arcus. </s> <s xml:id="echoid-s3830" xml:space="preserve">Vocamus autẽ cõplementum ſinus verſi ſe gmentum diametri, quo <lb/> <anchor type="note" xlink:label="note-116-01a" xlink:href="note-116-01"/> ipſe ſinus verſus à ſemidiametro ſuperatur, ſi eius arcus quadrante minor eſt, vel ſe-<lb/>midiametrum ſuperat, ſi eius arcus maior eſt quadrante. </s> <s xml:id="echoid-s3831" xml:space="preserve">Vt _HE,_ dicimus comple-<lb/>mentum tam ſinus verſi _AH,_ arcui _FA,_ reſpondentis, quàm ſinus verſi _CH,_ arcui <lb/>_FC,_ reſpondentis. </s> <s xml:id="echoid-s3832" xml:space="preserve">Vides igitur, duorum arcuum _FA, FB,_ quorum vnus alterius <lb/>eſt complementum, ſinum rectum _FK,_ arcus _FB,_ æqualem eſſe ipſi _HE,_ complemente <lb/>ſinus verſi _AH,_ alterius arcus _FA:_ </s> <s xml:id="echoid-s3833" xml:space="preserve">Et ſinum rectum _FH,_ arcus _FA,_ æqualem eſſe <lb/>ipſi _EK,_ complemento ſinus verſi _BK,_ alterius arcus _FB._ </s> <s xml:id="echoid-s3834" xml:space="preserve">Eadem ratione, quoniam <lb/>arcus _FB,_ complementum eſt arcus _FC,_ vides ſinum rectum _FH,_ arcus _FC,_ æqualem <lb/>eſſe ipſi _EK,_ complemente ſinus verſi _BK,_ alterius arcus _FB:_ </s> <s xml:id="echoid-s3835" xml:space="preserve">Et ſinum rectum _FK,_ <lb/> <anchor type="figure" xlink:label="fig-116-01a" xlink:href="fig-116-01"/> arcus _FB,_ æqualem eſſe ipſi <lb/> <anchor type="note" xlink:label="note-116-02a" xlink:href="note-116-02"/> _EH,_ complemento ſinus verſi <lb/>_CH,_ alterius arcus _FC._ </s> <s xml:id="echoid-s3836" xml:space="preserve">Sic <lb/>etiam ſinus complemẽtiarcus <lb/>cuiuſuis æqualis eſt complemẽ <lb/>to ſinus verſi eiuſdem arcus. <lb/></s> <s xml:id="echoid-s3837" xml:space="preserve">Vt _FK,_ ſinus complementi are <lb/>cus _AF,_ vel _FC,_ æqualis eſt <lb/> <anchor type="note" xlink:label="note-116-03a" xlink:href="note-116-03"/> ipſi _HE,_ complemento ſinus <lb/>verſi _AH,_ vel _CH,_ arcus <lb/>_AF,_ vel _FC._</s> <s xml:id="echoid-s3838" xml:space="preserve"/> </p> <div xml:id="echoid-div329" type="float" level="2" n="10"> <note position="right" xlink:label="note-115-06" xlink:href="note-115-06a" xml:space="preserve">Sinus tam <lb/>rectus, ꝗ̃ cõ <lb/>plementi. <lb/>cui ſegmen <lb/>todiametri <lb/>ſit e qualis. <lb/>34. primi. <lb/>28. primi.</note> <note position="left" xlink:label="note-116-01" xlink:href="note-116-01a" xml:space="preserve">Duotumat <lb/>euum, quo <lb/>rum vnus <lb/>alterius eſt <lb/>complemẽ <lb/>tum, ſinus <lb/>rectꝰ vtriuf <lb/>libet æqua <lb/>lis eſt com-<lb/>plemento <lb/>ſinus verſi <lb/>alterius at <lb/>eus.</note> <figure xlink:label="fig-116-01" xlink:href="fig-116-01a"> <image file="116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/116-01"/> </figure> <note position="left" xlink:label="note-116-02" xlink:href="note-116-02a" xml:space="preserve">Comple-<lb/>mentum ſi <lb/>nus verſi <lb/>quid.</note> <note position="left" xlink:label="note-116-03" xlink:href="note-116-03a" xml:space="preserve">14. primi.</note> </div> <p style="it"> <s xml:id="echoid-s3839" xml:space="preserve">_PARI_ ratione in eodem <lb/>circulo, vel in circulis æqua-<lb/> <anchor type="note" xlink:label="note-116-04a" xlink:href="note-116-04"/> libus, ſinus tam recti, quàm <lb/>verſi, aut ſinus complemento-<lb/>rum arcuum æqualium, & </s> <s xml:id="echoid-s3840" xml:space="preserve"><lb/>quadrante minorum, æquales <lb/>ſunt: </s> <s xml:id="echoid-s3841" xml:space="preserve">Et contra, æqualium ſi-<lb/>nuũ tam rectorum, quàm ver <lb/>ſorum, aut ſinuum complemen <lb/>torum, arcus quadrãte mino-<lb/>res æquales ſunt. </s> <s xml:id="echoid-s3842" xml:space="preserve">Arcuũ verò <lb/>inæqualium, & </s> <s xml:id="echoid-s3843" xml:space="preserve">quadrante minorum, ſinus inæquales ſunt, ſinus quidem tam rectus <lb/>quàm verſus maioris maior, minoris verò minor; </s> <s xml:id="echoid-s3844" xml:space="preserve">ſinus autem compleẽti maioris <lb/>arcus minor, & </s> <s xml:id="echoid-s3845" xml:space="preserve">minoris maior; </s> <s xml:id="echoid-s3846" xml:space="preserve">Et contra, inæqualium ſinuum tam rectorum, <lb/>quam verſorum, aut ſinuum complementorum, inæquales arcus ſunt, maioris qui-<lb/>dem ſinus tam recti quàm verſi maior arcus, & </s> <s xml:id="echoid-s3847" xml:space="preserve">minoris minor; </s> <s xml:id="echoid-s3848" xml:space="preserve">maioris autem <lb/>ſinus complementi arcus minor, & </s> <s xml:id="echoid-s3849" xml:space="preserve">minoris maior. </s> <s xml:id="echoid-s3850" xml:space="preserve">Sint enim arcus æquales _BF,_ <lb/>_DG._ </s> <s xml:id="echoid-s3851" xml:space="preserve">Dico eorum ſinus rectos _FK, GL,_ æquales eſſe; </s> <s xml:id="echoid-s3852" xml:space="preserve">Item ſinus verſos _KB, LD;_ <lb/></s> <s xml:id="echoid-s3853" xml:space="preserve">nec non ſinus complementorũ _EK, EL._ </s> <s xml:id="echoid-s3854" xml:space="preserve">Cum enim arcus _BF, DG,_ æquales ſunt, <lb/>erunt quoque anguli _BEF, DEG,_ æquales. </s> <s xml:id="echoid-s3855" xml:space="preserve">Sunt autẽ & </s> <s xml:id="echoid-s3856" xml:space="preserve">recti anguli _K, L,_ æquales, <lb/> <anchor type="note" xlink:label="note-116-05a" xlink:href="note-116-05"/> nec non & </s> <s xml:id="echoid-s3857" xml:space="preserve">latera _EF, EG,_ æqualia, vtpote ſemidiametri. </s> <s xml:id="echoid-s3858" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3859" xml:space="preserve">tam latera _FK,_ <lb/>_GL,_ quàm latera _EK, EL,_ inter ſe æqualia erunt, nempe ſinus recti inter ſe, & </s> <s xml:id="echoid-s3860" xml:space="preserve">ſi-<lb/> <anchor type="note" xlink:label="note-116-06a" xlink:href="note-116-06"/> nus complementorum inter ſe. </s> <s xml:id="echoid-s3861" xml:space="preserve">Detractis autem _EK, EL,_ æqualibus ex ſemidiametru <lb/>_EB, ED,_ reliqui erunt ſinus verſi _KB, LD,_ æquales. </s> <s xml:id="echoid-s3862" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s3863" xml:space="preserve">Sint iam <lb/>ſinus æquales ſiue recti _FK, GL,_ ſiue verſi _KB, LD,_ ſiue ſinus complementorum _EK,_ <lb/>_EL._ </s> <s xml:id="echoid-s3864" xml:space="preserve">Dico arcus _BF, DG,_ eſſe æquales, Nam ſi _FK, GL,_ ſint æquales, erunt eorum <pb o="105" file="117" n="117" rhead=""/> quadrata æqualia. </s> <s xml:id="echoid-s3865" xml:space="preserve">Cum ergo quadrata rectarum _EF, EG,_ æqualia quoque ſint, & </s> <s xml:id="echoid-s3866" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-117-01a" xlink:href="note-117-01"/> illi quidem æqualia ſint quadrata ex _FK, KE,_ huic vero quadrata ex _GL, LE;_ </s> <s xml:id="echoid-s3867" xml:space="preserve">ac <lb/>proinde duo quadrata ex _FK, KE,_ duobus quadratis ex _GL, LE,_ æqualia: </s> <s xml:id="echoid-s3868" xml:space="preserve">ſiau-<lb/>ferantur duo æqualia quadrata rectarum _FK, GL,_ æqualia remanebunt quadra-<lb/>ta ex _EK, EL;_ </s> <s xml:id="echoid-s3869" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s3870" xml:space="preserve">rectæ _EK, EL,_ æquales erunt Quare cum latera _EF,_ <lb/>_EK,_ lateribus _EG, EL,_ æqualia ſint, & </s> <s xml:id="echoid-s3871" xml:space="preserve">baſis _FK,_ baſi _GL,_ æqualis; </s> <s xml:id="echoid-s3872" xml:space="preserve">erunt anguli <lb/> <anchor type="note" xlink:label="note-117-02a" xlink:href="note-117-02"/> _FEB, GED,_ æquales; </s> <s xml:id="echoid-s3873" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s3874" xml:space="preserve">arcus _BF, DG,_ æquales erunt. </s> <s xml:id="echoid-s3875" xml:space="preserve">Quod ſi ſinus <lb/> <anchor type="note" xlink:label="note-117-03a" xlink:href="note-117-03"/> complementorum _EK, EL,_ ſint æquales, oſtendemus eodem modo, rectas _FK, GL,_ <lb/>æquales eſſe. </s> <s xml:id="echoid-s3876" xml:space="preserve">Quare vt prius, erunt arcus _BF, DG,_ æquales. </s> <s xml:id="echoid-s3877" xml:space="preserve">Si tandem ſinus verſi _KB,_ <lb/>_LD,_ ponantur æquales; </s> <s xml:id="echoid-s3878" xml:space="preserve">ijs ablatis ex ſemidiametris _EB, ED,_ relinquentur ſinus <lb/>complementorum _EK, EL,_ æquales. </s> <s xml:id="echoid-s3879" xml:space="preserve">Quare rurſus oſtendemus, vt prius, arcus _BF,_ <lb/>_DG,_ æquales eſſe. </s> <s xml:id="echoid-s3880" xml:space="preserve">quoderat oſtendendum. </s> <s xml:id="echoid-s3881" xml:space="preserve">Iam vero ſit arcus _BF,_ maior arcu _DM,_ <lb/>& </s> <s xml:id="echoid-s3882" xml:space="preserve">ducatur ſinus _MO._ </s> <s xml:id="echoid-s3883" xml:space="preserve">Dico ſinum rectum _FK,_ maiorẽ eſſe ſinu recto _MO:_ </s> <s xml:id="echoid-s3884" xml:space="preserve">Item ſinũ <lb/>verſum _KB,_ maiorẽ ſinu verſo _OD:_ </s> <s xml:id="echoid-s3885" xml:space="preserve">ſinũ vero cõplementi _EK,_ minorẽ ſinu cõplementi <lb/>_EO._ </s> <s xml:id="echoid-s3886" xml:space="preserve">Poſito enim arcu _DG,_ æquali arcui _BF,_ erunt, vt demonſtr auimus, tam ſinus recti <lb/>_FK, <emph style="sc">G</emph>L,_ quàm _OD,_ erit quoque ſinus verſus _KB,_ ſinu verſo _OD,_ maior: </s> <s xml:id="echoid-s3887" xml:space="preserve">Item cũ _EL,_ <lb/>maior ſit, quàm _OD,_ erit quoque ſinus verſus _KB,_ ſinu verſo _OD,_ maior: </s> <s xml:id="echoid-s3888" xml:space="preserve">Item cũ _EL,_ <lb/>minor ſit, quàm _EO,_ erit quoque ſinus cõplementi _EK,_ minor ſinu cõplementi _EO._ <lb/></s> <s xml:id="echoid-s3889" xml:space="preserve">Ducatur _MN,_ ad _GL,_ perpendicularis, eritque _NL,_ ipſi _MO,_ æqualis. </s> <s xml:id="echoid-s3890" xml:space="preserve">Cum ergo _GL,_ <lb/> <anchor type="note" xlink:label="note-117-04a" xlink:href="note-117-04"/> maior ſit, quàm _NL,_ hoc eſt, quàm _MO,_ erit quoq; </s> <s xml:id="echoid-s3891" xml:space="preserve">ſinus rectus _FK,_ maior ſinu recto <lb/>_MO._ </s> <s xml:id="echoid-s3892" xml:space="preserve">quod demonſt randum er at. </s> <s xml:id="echoid-s3893" xml:space="preserve">Sit denique tam ſinus rectus _FK,_ maior ſinu recto <lb/>_MO,_ quàm ſinus verſus _KB,_ ſinu verſo _OD;_ </s> <s xml:id="echoid-s3894" xml:space="preserve">& </s> <s xml:id="echoid-s3895" xml:space="preserve">ſinus complementi _EO,_ maior ſinu <lb/>complementi _EK._ </s> <s xml:id="echoid-s3896" xml:space="preserve">Dico ſinui maiori tam recto, quàm verſo reſpondentem arcum _BF,_ <lb/>maiorem eſſe arcu _DM,_ qui minori ſinui tam recto, quàm verſo reſpondet. </s> <s xml:id="echoid-s3897" xml:space="preserve">At maio-<lb/>ri ſinui complementi arcum reſpondentem _DM,_ minorem eſſe arcu _BF,_ qui minori <lb/>ſinui complementi reſpondet. </s> <s xml:id="echoid-s3898" xml:space="preserve">Nam ſi _FK,_ maior ſit, quàm _MO,_ auferatur _KP,_ ipſi <lb/>_MO,_ æqualis, & </s> <s xml:id="echoid-s3899" xml:space="preserve">ducatur _PQ,_ ad _FK,_ perpendicularis, ducaturque _QR,_ ad _BE,_ <lb/>perpendicularis, quæipſi _PK,_ hoc eſt, ipſi _MO,_ æqualis erit; </s> <s xml:id="echoid-s3900" xml:space="preserve">ac proinde, vt paulò ante <lb/> <anchor type="note" xlink:label="note-117-05a" xlink:href="note-117-05"/> oſtenſum eſt, erunt arcus _BQ, DM,_ æquales, propter æqualitatem ſinuum rectorum <lb/>_QR, MO._ </s> <s xml:id="echoid-s3901" xml:space="preserve">Cum ergoarcus _BF,_ arcu _BQ,_ maior ſit, erit idem arcus _BF,_ arcu _DM,_ <lb/>maior. </s> <s xml:id="echoid-s3902" xml:space="preserve">Quòd ſi _KB,_ maior ſit, quàm _OD,_ abſcindatur _BR,_ ipſi _DO,_ æqualis, duca-<lb/>turque _RQ,_ ad _BE,_ perpendicularis: </s> <s xml:id="echoid-s3903" xml:space="preserve">eruntq́ arcus _BQ, DM,_ vt paulo antemon-<lb/>ſtrauimus, æquales, ob æqualitatem ſinuum verſorum _RB, OD._ </s> <s xml:id="echoid-s3904" xml:space="preserve">Quare cũ arcus _BF,_ <lb/>maior ſit arcu _<emph style="sc">B</emph>Q,_ eritidem arcus _<emph style="sc">B</emph>F,_ arcu _DM,_ maior. </s> <s xml:id="echoid-s3905" xml:space="preserve">Si tandem maior ſit _EO,_ <lb/>quàm _EK,_ detrahatur _EL,_ ipsi _<emph style="sc">E</emph>K,_ æqualis, ducaturque ad _<emph style="sc">E</emph>D,_ perpendicularis <lb/> <anchor type="note" xlink:label="note-117-06a" xlink:href="note-117-06"/> _LG:_ </s> <s xml:id="echoid-s3906" xml:space="preserve">Eruntq́; </s> <s xml:id="echoid-s3907" xml:space="preserve">arcus _BF, <emph style="sc">Dg</emph>,_ ob æqualitatem sinuum complementorum _<emph style="sc">E</emph>K, <emph style="sc">E</emph>L,_ <lb/>æquales, vt paulo ante fuit oſtenſum. </s> <s xml:id="echoid-s3908" xml:space="preserve">Quam ob rem cum arcus _DM,_ arcu _DG,_ sit <lb/>minor, erit idem arcus _DM,_ arcn _BF,_ minor. </s> <s xml:id="echoid-s3909" xml:space="preserve">Quod eſt propositum. <lb/></s> <s xml:id="echoid-s3910" xml:space="preserve"/> </p> <div xml:id="echoid-div330" type="float" level="2" n="11"> <note position="left" xlink:label="note-116-04" xlink:href="note-116-04a" xml:space="preserve">In eodem <lb/>circulo, aut <lb/>æqualibus, <lb/>arcuum æ-<lb/>qualium <lb/>finus æqua <lb/>les ſunt; & <lb/>contra. At <lb/>arcuũ in æ-<lb/>qualium ſi <lb/>nus inęqua <lb/>les ſunt; & <lb/>contra.</note> <note position="left" xlink:label="note-116-05" xlink:href="note-116-05a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-116-06" xlink:href="note-116-06a" xml:space="preserve">26. primi.</note> <note position="right" xlink:label="note-117-01" xlink:href="note-117-01a" xml:space="preserve">47. primi.</note> <note position="right" xlink:label="note-117-02" xlink:href="note-117-02a" xml:space="preserve">8. primi.</note> <note position="right" xlink:label="note-117-03" xlink:href="note-117-03a" xml:space="preserve">26. tertij.</note> <note position="right" xlink:label="note-117-04" xlink:href="note-117-04a" xml:space="preserve">34. primi.</note> <note position="right" xlink:label="note-117-05" xlink:href="note-117-05a" xml:space="preserve">34. ptimi.</note> <note position="right" xlink:label="note-117-06" xlink:href="note-117-06a" xml:space="preserve">Anguli æ-<lb/>quales ha-<lb/>bent ſinus <lb/>ęquales, &c.</note> </div> <p style="it"> <s xml:id="echoid-s3911" xml:space="preserve">_<emph style="sc">IDe</emph>M_ prorſus dicendum eſt de sinubus angulorum. </s> <s xml:id="echoid-s3912" xml:space="preserve">Nam & </s> <s xml:id="echoid-s3913" xml:space="preserve">anguli æquales ha-<lb/> <anchor type="note" xlink:label="note-117-07a" xlink:href="note-117-07"/> bent sinus æquales tam rectos, quam complemẽtorum, & </s> <s xml:id="echoid-s3914" xml:space="preserve">verſos, &</s> <s xml:id="echoid-s3915" xml:space="preserve">c. </s> <s xml:id="echoid-s3916" xml:space="preserve">propterea quod <lb/>æquales anguli insiſtunt in centro æqualibus arcubus, &</s> <s xml:id="echoid-s3917" xml:space="preserve">c.</s> <s xml:id="echoid-s3918" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="2" n="12"> <note position="right" xlink:label="note-117-07" xlink:href="note-117-07a" xml:space="preserve">Si in trian <lb/>gulo rectã <lb/>gulo latus <lb/>recto angu <lb/>lo oppoſitũ <lb/>ſit ſinus to <lb/>tus, erit <lb/>vtrumuis <lb/>laterum re <lb/>liquorum <lb/>ſinus rectꝰ <lb/>anguli acu <lb/>ti oppoſiti.</note> </div> <p style="it"> <s xml:id="echoid-s3919" xml:space="preserve">_<emph style="sc">POSTRe</emph>MO_ in omni triangulo rectangulo, si latus recto angulo oppositum <lb/>ponatur sinus totus, reliqua duo latera ſunt sinus recti reliquorum angulorum acu-<lb/>torum, quibus opponuntur. </s> <s xml:id="echoid-s3920" xml:space="preserve">Vt in triangulo rectangulo _EKF,_ in quo _EF,_ eſt sinus to-<lb/>tus, vtpote ſemidiameter circuli ex F, deſcripti, latus _<emph style="sc">F</emph>K,_ eſtsinus rectus anguli <lb/>_<emph style="sc">FE</emph>K,_ ex deſin. </s> <s xml:id="echoid-s3921" xml:space="preserve">6. </s> <s xml:id="echoid-s3922" xml:space="preserve">Sic quoque si idem circulus ex _<emph style="sc">F</emph>,_ deſcriberetur, eſſet latus _<emph style="sc">E</emph>K,_ si-<lb/>nus reclus anguli _<emph style="sc">EF</emph>K,_ ex eadem deſin. </s> <s xml:id="echoid-s3923" xml:space="preserve">6. </s> <s xml:id="echoid-s3924" xml:space="preserve">Quod etiam hinc patet, quèd angulus _<emph style="sc">EF</emph>K,_ <pb o="106" file="118" n="118" rhead=""/> æqualis eſt angulo alterno _<emph style="sc">Aef</emph>;_ </s> <s xml:id="echoid-s3925" xml:space="preserve">cuius sinus rectus est, ex defin. </s> <s xml:id="echoid-s3926" xml:space="preserve">6. </s> <s xml:id="echoid-s3927" xml:space="preserve">recta _<emph style="sc">F</emph>H,_ quæ <lb/> <anchor type="note" xlink:label="note-118-01a" xlink:href="note-118-01"/> quidem æqualis eſt lateri _<emph style="sc">E</emph>K._ </s> <s xml:id="echoid-s3928" xml:space="preserve">Eodem pacto vtrumuis reliquorum laterum in trian-<lb/> <anchor type="note" xlink:label="note-118-02a" xlink:href="note-118-02"/> gulo rectangulo eſt sinus complementi anguli acuti sibi adiacentis, nempe sinus com-<lb/>plementi illius arcus, cuius alterum latus est sinus rectus. </s> <s xml:id="echoid-s3929" xml:space="preserve">Vt in eodem triangulo re-<lb/> <anchor type="note" xlink:label="note-118-03a" xlink:href="note-118-03"/> etangulo _<emph style="sc">E</emph><emph style="sc">Kf</emph>,_ latus _<emph style="sc">F</emph>K,_ eſt sinus complementi anguli _<emph style="sc">EF</emph>K,_ siue arcus _<emph style="sc">F</emph>A,_ cuius <lb/>sinui recto _<emph style="sc">F</emph>H,_ alterũ latus _<emph style="sc">E</emph>K,_ æquale eſt. </s> <s xml:id="echoid-s3930" xml:space="preserve">Item latus _<emph style="sc">E</emph>K,_ æquale estipsi _<emph style="sc">F</emph>H,_ si-<lb/>nui complementi anguli _<emph style="sc">FE</emph>K,_ siue arcus _<emph style="sc">FB</emph>,_ cuius alterum latus _<emph style="sc">F</emph>K,_ sinus rectus <lb/>eſt. </s> <s xml:id="echoid-s3931" xml:space="preserve">Sed iam lemma, cuius ſupra fecimus mentionem, demon ſtremus.</s> <s xml:id="echoid-s3932" xml:space="preserve"/> </p> <div xml:id="echoid-div332" type="float" level="2" n="13"> <note position="left" xlink:label="note-118-01" xlink:href="note-118-01a" xml:space="preserve">29. primi.</note> <note position="left" xlink:label="note-118-02" xlink:href="note-118-02a" xml:space="preserve">34 primi.</note> <note position="left" xlink:label="note-118-03" xlink:href="note-118-03a" xml:space="preserve">Si in trian-<lb/>gulo rectã-<lb/>gulo latus <lb/>recto angu <lb/>lo oppoſi-<lb/>tũ ſit ſinus <lb/>totus, erit <lb/>vttumui re <lb/>liquorum <lb/>alterum la <lb/>tus ſinus re <lb/>ctus eſt.</note> </div> </div> <div xml:id="echoid-div334" type="section" level="1" n="150"> <head xml:id="echoid-head176" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s3933" xml:space="preserve">SI in circulo recta linea è centro ducta aliam rectam non per cen <lb/>trum ductam bifariam ſecet, ſecabit eadem & </s> <s xml:id="echoid-s3934" xml:space="preserve">arcum, cui illa recta <lb/> <anchor type="note" xlink:label="note-118-04a" xlink:href="note-118-04"/> ſubtenditur, bifariam: </s> <s xml:id="echoid-s3935" xml:space="preserve">Et ſi arcum ſe cet bifariam, ſecabit quoque re-<lb/>ctam ei ſubtenſam bifariam.</s> <s xml:id="echoid-s3936" xml:space="preserve"/> </p> <div xml:id="echoid-div334" type="float" level="2" n="1"> <note position="left" xlink:label="note-118-04" xlink:href="note-118-04a" xml:space="preserve">Recta linea <lb/>è cẽtro du-<lb/>cta ſecans <lb/>aliam rectã <lb/>bifariã ſe-<lb/>cat quoque <lb/>arcum, cui <lb/>illa, ſubten <lb/>ditur, bifa-<lb/>tiam: Et cõ <lb/>tra.</note> </div> <p style="it"> <s xml:id="echoid-s3937" xml:space="preserve">SECET in eadem figura recta EA, rectam FG, bifariam in H, <lb/> <anchor type="figure" xlink:label="fig-118-01a" xlink:href="fig-118-01"/> Dico eandem ſecare quo <lb/>que arcum FG, bifariã <lb/>in A, & </s> <s xml:id="echoid-s3938" xml:space="preserve">contra. </s> <s xml:id="echoid-s3939" xml:space="preserve">Ducta <lb/>enim recta EG; </s> <s xml:id="echoid-s3940" xml:space="preserve">quoniã <lb/>duo latera EF, EH, triã <lb/>guli EFH, æqualia ſunt <lb/>duobus lateribus EG, <lb/>EH, trianguli EGH, <lb/>vtrumq; </s> <s xml:id="echoid-s3941" xml:space="preserve">vtrique; </s> <s xml:id="echoid-s3942" xml:space="preserve">baſisq́, <lb/>HF, baſi HG, ponitur <lb/>æqualis; </s> <s xml:id="echoid-s3943" xml:space="preserve">erit angulus <lb/> <anchor type="note" xlink:label="note-118-05a" xlink:href="note-118-05"/> FEH, angulo GEH, <lb/> <anchor type="note" xlink:label="note-118-06a" xlink:href="note-118-06"/> æqualis. </s> <s xml:id="echoid-s3944" xml:space="preserve">Igitur arcus <lb/>AF, arcui AG, æqualis <lb/>erit. </s> <s xml:id="echoid-s3945" xml:space="preserve">Quod eſt propoſitũ.</s> <s xml:id="echoid-s3946" xml:space="preserve"/> </p> <div xml:id="echoid-div335" type="float" level="2" n="2"> <figure xlink:label="fig-118-01" xlink:href="fig-118-01a"> <image file="118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/118-01"/> </figure> <note position="left" xlink:label="note-118-05" xlink:href="note-118-05a" xml:space="preserve">2. primi.</note> <note position="left" xlink:label="note-118-06" xlink:href="note-118-06a" xml:space="preserve">25. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s3947" xml:space="preserve">VERVM ſecet iã <lb/>recta EA, arcum FG, <lb/>bifariam in A. </s> <s xml:id="echoid-s3948" xml:space="preserve">Dico eandem ſecare quoque rectam FG, bifariam in H. <lb/></s> <s xml:id="echoid-s3949" xml:space="preserve">Quoniam enim arcus AF, AG, æquales ſunt, erunt quoq; </s> <s xml:id="echoid-s3950" xml:space="preserve">anguli FEH, <lb/> <anchor type="note" xlink:label="note-118-07a" xlink:href="note-118-07"/> GEH, æquales. </s> <s xml:id="echoid-s3951" xml:space="preserve">Igitur cum & </s> <s xml:id="echoid-s3952" xml:space="preserve">duo latera EF, EH, trianguli EFH, <lb/>duobus lateribus EG, EH, trianguli EGH, æqualia ſint; </s> <s xml:id="echoid-s3953" xml:space="preserve">erunt & </s> <s xml:id="echoid-s3954" xml:space="preserve">ba-<lb/>ſes HF, HG, æquales. </s> <s xml:id="echoid-s3955" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s3956" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="3"> <note position="left" xlink:label="note-118-07" xlink:href="note-118-07a" xml:space="preserve">27. tertij.</note> </div> <note position="left" xml:space="preserve">4. primi.</note> <p style="it"> <s xml:id="echoid-s3957" xml:space="preserve">EX boc ſequitur, rectam EA, quæ arcum FG, bifariam ſecat in A, <lb/>ſecare quoque rectam FG, bifariam in H; </s> <s xml:id="echoid-s3958" xml:space="preserve">vt ſupra poſuimus.</s> <s xml:id="echoid-s3959" xml:space="preserve"/> </p> <pb o="107" file="119" n="119" rhead=""/> <p style="it"> <s xml:id="echoid-s3960" xml:space="preserve">_QVONIAM_ vero Sinus totus (hoc eſt, ſemidiameter cuiusuis circuli) intelligi-<lb/>tur ab Aſtronomis diuiſus in aliquot partes æquales, vt ratione barum partium om-<lb/> <anchor type="note" xlink:label="note-119-01a" xlink:href="note-119-01"/> nes alios ſinus metiantur, proportiones ve omnium ſinuum ad ſinum totum, ſiue ad ſe-<lb/>midiametrum in numeris exprimant; </s> <s xml:id="echoid-s3961" xml:space="preserve">ex plicãdum paucis erit, in quot partes ſemidia-<lb/>metrum diſtribuerint: </s> <s xml:id="echoid-s3962" xml:space="preserve">Neque enim omnes eodem modo eam ſunt partiti. </s> <s xml:id="echoid-s3963" xml:space="preserve">Ptolemæus <lb/>namque ſemidiametrum ſecatin _60._ </s> <s xml:id="echoid-s3964" xml:space="preserve">partes æquales, totam vero diametrum in 120. <lb/></s> <s xml:id="echoid-s3965" xml:space="preserve">Quamlibet deinde partem concipit diuiſam eſſe in _60._ </s> <s xml:id="echoid-s3966" xml:space="preserve">Minuta, & </s> <s xml:id="echoid-s3967" xml:space="preserve">quoduis Minutum <lb/>in 60. </s> <s xml:id="echoid-s3968" xml:space="preserve">Secunda. </s> <s xml:id="echoid-s3969" xml:space="preserve">Hanc diuiſionem omnes fermè antiqui, & </s> <s xml:id="echoid-s3970" xml:space="preserve">nonnulli ex recentioribus, <lb/>inter quoseſt Orontius, ſecuti ſunt. </s> <s xml:id="echoid-s3971" xml:space="preserve">Supputauit autem Ptolemæus lib. </s> <s xml:id="echoid-s3972" xml:space="preserve">I. </s> <s xml:id="echoid-s3973" xml:space="preserve">Almageſti ta-<lb/> <anchor type="note" xlink:label="note-119-02a" xlink:href="note-119-02"/> bulam omnium chordarum, quæarcubus ſemicirculi dimidiato gradu ſeſe ordine ſu-<lb/>perantibus, initio facto ab arcu _30._ </s> <s xml:id="echoid-s3974" xml:space="preserve">Minutorum, reſpondẽt, in partibus, quarum 120. <lb/></s> <s xml:id="echoid-s3975" xml:space="preserve">tota diameter continet. </s> <s xml:id="echoid-s3976" xml:space="preserve">Orontius vero tabulam condidit omnium ſinuum, qui arcu-<lb/>bus quadrantis vno Minuto ſeſe ordine ſuperantibus, initio facto ab arcu I. </s> <s xml:id="echoid-s3977" xml:space="preserve">Minuti, <lb/>reſpondent, in partibus, quarum _60._ </s> <s xml:id="echoid-s3978" xml:space="preserve">ſemidiameter, ſeu ſinus totus continet. </s> <s xml:id="echoid-s3979" xml:space="preserve">Ar Zahel <lb/>vero Arabs conſtituit ſemidiametrum partium 150. </s> <s xml:id="echoid-s3980" xml:space="preserve">ac proinde totam diametrum <lb/>partium 300. </s> <s xml:id="echoid-s3981" xml:space="preserve">quas quidem partes rurſus diſtribuit in Minuta, & </s> <s xml:id="echoid-s3982" xml:space="preserve">Secunda, vt Pto-<lb/>lemæus. </s> <s xml:id="echoid-s3983" xml:space="preserve">Sed vtrouis modo ſemidiameter, siue diameter diuidatur, permoleſtum eſt, <lb/>alios omnes ſinus ſiue chordas in eiusmodi partibus inueſtigare, cum ſemper multi-<lb/>plicatio, diuiſio, extractioq́; </s> <s xml:id="echoid-s3984" xml:space="preserve">radicum per fractiones Aſtronomicas in ſtituenda ſit; </s> <s xml:id="echoid-s3985" xml:space="preserve">Vel <lb/>certe Partes in Minuta, ac Secunda conuertendæ, & </s> <s xml:id="echoid-s3986" xml:space="preserve">contra, Minuta ac Secunda in <lb/>Partes: </s> <s xml:id="echoid-s3987" xml:space="preserve">quæres valde laborioſa eſt non ſolum parum exercitatis in Arithmeticis, ve-<lb/>rum etiam peritißimis.</s> <s xml:id="echoid-s3988" xml:space="preserve"/> </p> <div xml:id="echoid-div337" type="float" level="2" n="4"> <note position="right" xlink:label="note-119-01" xlink:href="note-119-01a" xml:space="preserve">Omnes ſi-<lb/>nus expri-<lb/>muntur in <lb/>partibus, <lb/>in quas ſi <lb/>nus totus <lb/>concipitut <lb/>eſſe diui-<lb/>ſus.</note> <note position="right" xlink:label="note-119-02" xlink:href="note-119-02a" xml:space="preserve">Semidiame <lb/>ter circult <lb/>in quot par <lb/>tes ſecetur <lb/>à Ptolemęo <lb/>& Arzahe-<lb/>le.</note> </div> <p style="it"> <s xml:id="echoid-s3989" xml:space="preserve">_QVAMOBREM_ alij Aſtronomi, inter quos eſt Georgius Purbachius, Ioan-<lb/>nes Regiomontanus. </s> <s xml:id="echoid-s3990" xml:space="preserve">Petrus Appianus, ſemidiametrum, hoc eſt, sinum totum, in multo <lb/> <anchor type="note" xlink:label="note-119-03a" xlink:href="note-119-03"/> plures particulas æquales partiti ſunt, vtpote in partes _10000000._ </s> <s xml:id="echoid-s3991" xml:space="preserve">vel _100000._ <lb/></s> <s xml:id="echoid-s3992" xml:space="preserve">Ita enim opus non erit partes has in Minuta, ac Secunda diſtribuere; </s> <s xml:id="echoid-s3993" xml:space="preserve">cum vna ha-<lb/>rum partium sit vel multo minor, quàm vnum Secũdum ſemidiametri in partes _60._ </s> <s xml:id="echoid-s3994" xml:space="preserve"><lb/>ſecundum Ptolemæum diuiſæ, vel certe non multo maior. </s> <s xml:id="echoid-s3995" xml:space="preserve">Nam vnum Secundum eſt <lb/>{1/216000} totius ſemidiametri diuiſæ in 60. </s> <s xml:id="echoid-s3996" xml:space="preserve">partes, cum 60. </s> <s xml:id="echoid-s3997" xml:space="preserve">partes contineant <lb/>_216000._ </s> <s xml:id="echoid-s3998" xml:space="preserve">Secunda: </s> <s xml:id="echoid-s3999" xml:space="preserve">At vna particula ſemidiametridiuiſæ in partes _10000000._ </s> <s xml:id="echoid-s4000" xml:space="preserve">vel <lb/>_100000._ </s> <s xml:id="echoid-s4001" xml:space="preserve">eſt {1/10000000}. </s> <s xml:id="echoid-s4002" xml:space="preserve">vel {1/100000}. </s> <s xml:id="echoid-s4003" xml:space="preserve">totius ſemidiametri. </s> <s xml:id="echoid-s4004" xml:space="preserve">Con <lb/>ſtat autẽ minutiã hãc {1/10000000}. </s> <s xml:id="echoid-s4005" xml:space="preserve">eſſe multo minorẽilla {1/216000}. </s> <s xml:id="echoid-s4006" xml:space="preserve"><lb/>hanc vero {1/100000}. </s> <s xml:id="echoid-s4007" xml:space="preserve">non eſſe multo maiorẽ illa eadem {1/216000}. </s> <s xml:id="echoid-s4008" xml:space="preserve">Ita-<lb/>que etiã si in sinu aliquo negligatur interdum vna ferme particula ex _10000000._ </s> <s xml:id="echoid-s4009" xml:space="preserve"><lb/>particulis sinus totius, multò tamen minor error committetur, quàm si negligatur <lb/>vnum ferè ſecundum ex _216000._ </s> <s xml:id="echoid-s4010" xml:space="preserve">Secundis, in quæ sinus totus intelligitur eſſe diui-<lb/>ſus: </s> <s xml:id="echoid-s4011" xml:space="preserve">Si vero negligatur vna ferè particula ex _100000._ </s> <s xml:id="echoid-s4012" xml:space="preserve">particulis sinus totius, non <lb/>multò maior error cõmittetur, quàm ſinegligatur vnum fere ſecundum ex _216000._ </s> <s xml:id="echoid-s4013" xml:space="preserve"><lb/>ſecundis, in quæ sinus totus diſtribuitur.</s> <s xml:id="echoid-s4014" xml:space="preserve"/> </p> <div xml:id="echoid-div338" type="float" level="2" n="5"> <note position="right" xlink:label="note-119-03" xlink:href="note-119-03a" xml:space="preserve">Cõmodiot <lb/>eſt diuiſio <lb/>ſemidiame <lb/>tr i, vel ſin <lb/>totius in <lb/>particulas <lb/>10000000. <lb/>vel _100000._ <lb/>quàm in <lb/>60.</note> </div> <note position="right" xml:space="preserve">Quantue <lb/>ſit ſinus to-<lb/>tus ſecun-<lb/>dum com-<lb/>munem v-<lb/>ſum, & quã <lb/>tus ab au-<lb/>ctore ſta -<lb/>tuatur.</note> <p style="it"> <s xml:id="echoid-s4015" xml:space="preserve">_SVNT_ etiam, qui diſtribuant sinum totum in partes _6000000._ </s> <s xml:id="echoid-s4016" xml:space="preserve">vel _60000._ <lb/></s> <s xml:id="echoid-s4017" xml:space="preserve">extantq́ tabulæ sinuũ à Ioan. </s> <s xml:id="echoid-s4018" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s4019" xml:space="preserve">compositæ, in quibus sinus totus tot particu-<lb/>las ponitur continere: </s> <s xml:id="echoid-s4020" xml:space="preserve">ſed magis in vſu eſt apud Aſtronomos diuisio sinus totius in <lb/>particulas _10000000._ </s> <s xml:id="echoid-s4021" xml:space="preserve">vel _100000._ </s> <s xml:id="echoid-s4022" xml:space="preserve">Immo communis fere vſus omnium obtinuit, vt <lb/>in ſupputationibus, quæ ex sinubus depromuntur, sinus totus ſtatuatur particula-<lb/>rum _100000._ </s> <s xml:id="echoid-s4023" xml:space="preserve">qualem & </s> <s xml:id="echoid-s4024" xml:space="preserve">nos tam in ſphæra, quàm in Gnomonica alijsque operibus <lb/>conſtituimus: </s> <s xml:id="echoid-s4025" xml:space="preserve">quamuis, quo maior fuerit sinus totus, eo etiam accuratior ſupputatio <lb/> <anchor type="note" xlink:label="note-119-05a" xlink:href="note-119-05"/> atque calculus reddatur. </s> <s xml:id="echoid-s4026" xml:space="preserve">Conſtruxit porrò Ioan. </s> <s xml:id="echoid-s4027" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s4028" xml:space="preserve">tabulam omnium Sinuum, <pb o="108" file="120" n="120" rhead=""/> qui arcubus quadrantis vno Minuto ſeſe ordine ſuperantibus reſpondent, in parti-<lb/> <anchor type="note" xlink:label="note-120-01a" xlink:href="note-120-01"/> bus sinus totius in partes _10000000_ diuisi, initio facto ab arcu 1. </s> <s xml:id="echoid-s4029" xml:space="preserve">Minuti: </s> <s xml:id="echoid-s4030" xml:space="preserve">quam nos <lb/>ſumma cura, ac diligentia exam inauimus, & </s> <s xml:id="echoid-s4031" xml:space="preserve">in quibuſdam locis correximus; </s> <s xml:id="echoid-s4032" xml:space="preserve">quo-<lb/>niam propter typographorum incuriam mendis omnino non carebat. </s> <s xml:id="echoid-s4033" xml:space="preserve">Hanc tabulam <lb/>emendatam infra ſubijciemus, si prius demonſtrationes ex Ioanne Regiomontano po-<lb/>tißimum decerpt as exponamus, qu bus omnium arcuum sinus numeris exprimi poßint <lb/>in partibus sinus totius in quotuis partes diſtributi. </s> <s xml:id="echoid-s4034" xml:space="preserve">Poſt tabulæ vero vſumſubijcie <lb/>mus quoque Ptolemæi & </s> <s xml:id="echoid-s4035" xml:space="preserve">aliorum demonſtrationes, quibus omnium arcuum chordæ <lb/>numeris exprimantur, ex quibus rurſus facili negotio tabula sinuũ conſtrui poteſt.</s> <s xml:id="echoid-s4036" xml:space="preserve"/> </p> <div xml:id="echoid-div339" type="float" level="2" n="6"> <note position="right" xlink:label="note-119-05" xlink:href="note-119-05a" xml:space="preserve">Item eo fie <lb/>ri calculũ</note> <note position="left" xlink:label="note-120-01" xlink:href="note-120-01a" xml:space="preserve">accuratio <lb/>tem, quo <lb/>maior fue-<lb/>eit ſinus to <lb/>tus.</note> </div> <p style="it"> <s xml:id="echoid-s4037" xml:space="preserve">_ATQVE_ in primis, si in plano aliquo Quadrans tantæ magnitudinis conſtrue-<lb/>retur, vt eius arcus commode in _90._ </s> <s xml:id="echoid-s4038" xml:space="preserve">gradus, & </s> <s xml:id="echoid-s4039" xml:space="preserve">singuli gradus in _60._ </s> <s xml:id="echoid-s4040" xml:space="preserve">Minuta; </s> <s xml:id="echoid-s4041" xml:space="preserve">item-<lb/>que vtraque eius ſemidiameter, siue sinus totus, in _10000000._ </s> <s xml:id="echoid-s4042" xml:space="preserve">partes æquales, vel <lb/>etiam in plures, pauciores ve diuidi poſſet, facili negotio sine vlla ſupputationis mole-<lb/>ſtia, aut labore, omniũ ſinuũ magnitudines cognoſcerẽtur, si ex singulis arcus Minu-<lb/>tis rectæ ad vtramque ſemidiametrũ perpendiculares ducerentur. </s> <s xml:id="echoid-s4043" xml:space="preserve">Vt in quadrãte hoc <lb/>_ABC,_ si arcus _<emph style="sc">B</emph>C,_ in Gradus, ac Minuta ſecetur; </s> <s xml:id="echoid-s4044" xml:space="preserve">(Nos ob ſpatij anguſtias eum in <lb/> <anchor type="note" xlink:label="note-120-02a" xlink:href="note-120-02"/> <anchor type="figure" xlink:label="fig-120-01a" xlink:href="fig-120-01"/> nouem partes ſecuimus, vt sin <lb/>gulæ denos complectãtur gra <lb/>dus.) </s> <s xml:id="echoid-s4045" xml:space="preserve">Item vtraque ſemidia-<lb/>meter in _10000000._ </s> <s xml:id="echoid-s4046" xml:space="preserve">particu-<lb/>las, vel in plures, pauciores ve <lb/>diſtribuatur, atque ad vtram <lb/>que ſemidiametrum perpendi-<lb/>culares ducantur: </s> <s xml:id="echoid-s4047" xml:space="preserve">erunt per-<lb/>pendiculares ad ſemidiame-<lb/>trum _AB,_ ductæ, sinus recti <lb/>arcuum quadrantis à puncto <lb/>B, incipientium; </s> <s xml:id="echoid-s4048" xml:space="preserve">quibus æqua <lb/> <anchor type="note" xlink:label="note-120-03a" xlink:href="note-120-03"/> les ſunt portiones ſemidiame <lb/>tri _AC,_ inter punctum _A,_ & </s> <s xml:id="echoid-s4049" xml:space="preserve"><lb/>perpẽdiculares ad ſemidiame <lb/>trum _AC,_ ductas. </s> <s xml:id="echoid-s4050" xml:space="preserve">Quot ergo <lb/>particulas cõtinebunt hæ por <lb/>tiones ex illis _10000000._ </s> <s xml:id="echoid-s4051" xml:space="preserve">tot <lb/>particularũ erunt sinus recti <lb/>arcũ quadrantis. </s> <s xml:id="echoid-s4052" xml:space="preserve">Eodem modo <lb/>tam sinus complementorum arcuum eorundem, quàm sinus versi cognoſcentur. </s> <s xml:id="echoid-s4053" xml:space="preserve">Per-<lb/>pendiculares enim ad ſemidiametrum _AC,_ demiſſæ ſunt sinus complementorum, qui-<lb/>bus æquales ſunt portiones ſemidiametri _AB,_ inter punctum _A,_ & </s> <s xml:id="echoid-s4054" xml:space="preserve">perpendiculares <lb/> <anchor type="note" xlink:label="note-120-04a" xlink:href="note-120-04"/> ad ſemidiametrum _AB,_ ductas: </s> <s xml:id="echoid-s4055" xml:space="preserve">Portiones vero eiuſdem ſemidiam etri _AB,_ inter pun-<lb/>ctum _B,_ & </s> <s xml:id="echoid-s4056" xml:space="preserve">dictas perpendiculares ſunt sinus versi eorundem arcuum à puncto _B,_ <lb/>incipientium. </s> <s xml:id="echoid-s4057" xml:space="preserve">Sed quoniam fieri non poteſt, vt Quadrans tantæ magnitudinis repe-<lb/>riatur, qui commode tot diuisiones recipiat, inueſtigabimus sinuum magnitudines <lb/>per demonſtr ationes Geometricas, posito sinu to to quotcunq; </s> <s xml:id="echoid-s4058" xml:space="preserve">particularum, ſequen-<lb/>tibus propoſitionibus. </s> <s xml:id="echoid-s4059" xml:space="preserve">Satis autem erit, ſinus rectos omnium arcuum inquiramus: </s> <s xml:id="echoid-s4060" xml:space="preserve">ex <lb/>his enim cognitis & </s> <s xml:id="echoid-s4061" xml:space="preserve">ſinus complementorum, & </s> <s xml:id="echoid-s4062" xml:space="preserve">verſi eorundem arcuum pateſient, <lb/>@@ in vſutabulæ Sinuum exponemus.</s> <s xml:id="echoid-s4063" xml:space="preserve"/> </p> <div xml:id="echoid-div340" type="float" level="2" n="7"> <note position="left" xlink:label="note-120-02" xlink:href="note-120-02a" xml:space="preserve">Quo pacto <lb/>omnes ſi-<lb/>nus poſsint <lb/>cognoſci in <lb/>maximo a-<lb/>liquo qua-<lb/>drante, ſine <lb/>vllo ſuppu-<lb/>tationis la-<lb/>bore, aut <lb/>moleſtia.</note> <figure xlink:label="fig-120-01" xlink:href="fig-120-01a"> <image file="120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/120-01"/> </figure> <note position="left" xlink:label="note-120-03" xlink:href="note-120-03a" xml:space="preserve">34. primi.</note> <note position="left" xlink:label="note-120-04" xlink:href="note-120-04a" xml:space="preserve">34. primi.</note> </div> <pb o="109" file="121" n="121" rhead=""/> </div> <div xml:id="echoid-div342" type="section" level="1" n="151"> <head xml:id="echoid-head177" xml:space="preserve">THEOR. 1. PROPOS. 1.</head> <p> <s xml:id="echoid-s4064" xml:space="preserve">IN Quadrante circuli ſumptis arcubus æqua-<lb/> <anchor type="note" xlink:label="note-121-01a" xlink:href="note-121-01"/> libus, ſi ab eorum terminis ad alterutram ſemidia-<lb/>metrorum, vel ad rectam ſemidiametro paralle-<lb/>lam, perpendiculares ducantur; </s> <s xml:id="echoid-s4065" xml:space="preserve">erunt ſegmenta <lb/>ſemidiametri, velillius parallelæ interillas perpen-<lb/>diculares intercepta, inæqualia, maiusq́ erit illud, <lb/>quod alteri ſemidiametro propinquius elt.</s> <s xml:id="echoid-s4066" xml:space="preserve"/> </p> <div xml:id="echoid-div342" type="float" level="2" n="1"> <note position="right" xlink:label="note-121-01" xlink:href="note-121-01a" xml:space="preserve">Perpĕdicu-<lb/>lares ex ar-<lb/>cubus qua-<lb/>drãtis ęqua <lb/>libus ad al-<lb/>terutrá ſe-<lb/>midiame -<lb/>trorum, vel <lb/>ad rectam <lb/>ſemidiame <lb/>tro paral -<lb/>lelam du-<lb/>ctę auferũt <lb/>fegm enta <lb/>inæqualia, <lb/>maiufq́ eſt <lb/>illud, qd al <lb/>teri femi-<lb/>diametro <lb/>{pro}pinquius <lb/>eſt.</note> </div> <p> <s xml:id="echoid-s4067" xml:space="preserve">SIT Quadrans ABC, in quo arcus æquales ſint DE, EF, à quorum ter-<lb/>minis ad ſemidiametrum AC, vel ad rectam RS, ipſi AC, parallelam per-<lb/>pendiculares ducantur DKG, ELH, FMI. </s> <s xml:id="echoid-s4068" xml:space="preserve">Dico ſegmenta GH, HI, vel <lb/>KL, LM, inæqualia eſſe, maiusq́ue eſſe GH, <lb/> <anchor type="figure" xlink:label="fig-121-01a" xlink:href="fig-121-01"/> quàm HI, vel KL, maius, quàm LM. </s> <s xml:id="echoid-s4069" xml:space="preserve">Com-<lb/>pleto enim ſemicirculo BCN, producantur <lb/>rectæ DG, EH, FI, vſque ad O, P, Q. </s> <s xml:id="echoid-s4070" xml:space="preserve">Du-<lb/>ctis quoque rectis ET, FV, ad DO, EP, per-<lb/>pendicularibus, iungantur rectæ EO, FP. <lb/></s> <s xml:id="echoid-s4071" xml:space="preserve">Et quoniam arcus DE, EF, æquales ſunt, <lb/> <anchor type="note" xlink:label="note-121-02a" xlink:href="note-121-02"/> erunt anguli quoque DOE, EPF, illis inſi-<lb/>ſtentes, æquales: </s> <s xml:id="echoid-s4072" xml:space="preserve">Sunt autem & </s> <s xml:id="echoid-s4073" xml:space="preserve">recti anguli <lb/>T, V, æquales. </s> <s xml:id="echoid-s4074" xml:space="preserve">Igitur cum tres anguli trian-<lb/>guli EOT, tribus angulis trianguli FPV, <lb/>ſint æquales; </s> <s xml:id="echoid-s4075" xml:space="preserve">quòd tam illi, quàm hiduobus <lb/> <anchor type="note" xlink:label="note-121-03a" xlink:href="note-121-03"/> rectis ſint æquales; </s> <s xml:id="echoid-s4076" xml:space="preserve">erit & </s> <s xml:id="echoid-s4077" xml:space="preserve">reliquus angulus <lb/>TEO, reliquo angulo VFP, æqualis: </s> <s xml:id="echoid-s4078" xml:space="preserve">ac <lb/>propterea æquiangula erũt triãgula EOT, <lb/> <anchor type="note" xlink:label="note-121-04a" xlink:href="note-121-04"/> FPV. </s> <s xml:id="echoid-s4079" xml:space="preserve">Quare erit vt OE, ad ET, ita PF, ad <lb/> <anchor type="note" xlink:label="note-121-05a" xlink:href="note-121-05"/> EV: </s> <s xml:id="echoid-s4080" xml:space="preserve">Eſt auté recta OE, maior, quàm recta <lb/>PF; </s> <s xml:id="echoid-s4081" xml:space="preserve">quod illa centro propinquior ſit, quàm <lb/>hęc. </s> <s xml:id="echoid-s4082" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s4083" xml:space="preserve">recta ET, maior eſt, quàm re-<lb/>cta FV. </s> <s xml:id="echoid-s4084" xml:space="preserve">Cum ergo recta ET, æqualis ſit <lb/> <anchor type="note" xlink:label="note-121-06a" xlink:href="note-121-06"/> ſegmentis GH, KL, ob parallelogramma <lb/>TH, TL; </s> <s xml:id="echoid-s4085" xml:space="preserve">& </s> <s xml:id="echoid-s4086" xml:space="preserve">recta FV, ſegmentis HI, LM, <lb/>ob parallelogramma VI, VM; </s> <s xml:id="echoid-s4087" xml:space="preserve">erit quoque ſegmentum GH, maius ſegmen-<lb/>to HI, & </s> <s xml:id="echoid-s4088" xml:space="preserve">ſegmentum KL, ſegmento LM. </s> <s xml:id="echoid-s4089" xml:space="preserve">In quadrante ergo circuli ſumptis <lb/>arcubus æqualibus, &</s> <s xml:id="echoid-s4090" xml:space="preserve">c. </s> <s xml:id="echoid-s4091" xml:space="preserve">Quod erat den. </s> <s xml:id="echoid-s4092" xml:space="preserve">onſtrandum.</s> <s xml:id="echoid-s4093" xml:space="preserve"/> </p> <div xml:id="echoid-div343" type="float" level="2" n="2"> <figure xlink:label="fig-121-01" xlink:href="fig-121-01a"> <image file="121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/121-01"/> </figure> <note position="right" xlink:label="note-121-02" xlink:href="note-121-02a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-121-03" xlink:href="note-121-03a" xml:space="preserve">32. primi.</note> <note position="right" xlink:label="note-121-04" xlink:href="note-121-04a" xml:space="preserve">4. @exti.</note> <note position="right" xlink:label="note-121-05" xlink:href="note-121-05a" xml:space="preserve">15.tertij.</note> <note position="right" xlink:label="note-121-06" xlink:href="note-121-06a" xml:space="preserve">34. primi.</note> </div> <p> <s xml:id="echoid-s4094" xml:space="preserve">BREVIVS. </s> <s xml:id="echoid-s4095" xml:space="preserve">Ducatur recta DF, ſecans ſemidiametrum ductam AE, in <lb/>Z, & </s> <s xml:id="echoid-s4096" xml:space="preserve">rectam EH, in a, producaturq́ue recta FV, vſque ad b. </s> <s xml:id="echoid-s4097" xml:space="preserve">Quoniam igi-<lb/>tur arcus DF, ſectus eſt biſariam in E, ſecta quoque erit recta DF, biſariam <lb/>in Z, ex lemmate in definitionibus poſito, ac proinde Da, maior erit quàm <pb o="110" file="122" n="122" rhead=""/> a F. </s> <s xml:id="echoid-s4098" xml:space="preserve">Cum ergo ſit, vt Da, ad aF, ita b V, ad VF, erit quoque b V, maior, <lb/> <anchor type="note" xlink:label="note-122-01a" xlink:href="note-122-01"/> quàm VF, hoc eſt, GH, maior, quàm HI; </s> <s xml:id="echoid-s4099" xml:space="preserve">& </s> <s xml:id="echoid-s4100" xml:space="preserve">KL, maior quàm LM.</s> <s xml:id="echoid-s4101" xml:space="preserve"/> </p> <div xml:id="echoid-div344" type="float" level="2" n="3"> <note position="left" xlink:label="note-122-01" xlink:href="note-122-01a" xml:space="preserve">@. fexti.</note> </div> </div> <div xml:id="echoid-div346" type="section" level="1" n="152"> <head xml:id="echoid-head178" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s4102" xml:space="preserve">CONSTAT ex hac propoſitione, ſi quotcunque arcus quadrãtis à ſemidiametro eadé <lb/> <anchor type="note" xlink:label="note-122-02a" xlink:href="note-122-02"/> incipientes habeant æquales differentias, exceſſusve; </s> <s xml:id="echoid-s4103" xml:space="preserve">ſinus rectos minorum arcuum habe-<lb/>re maiores differentias, quàm ſinus arcuum maiorum; </s> <s xml:id="echoid-s4104" xml:space="preserve">adeo vt differentiæ ſinuum à prin-<lb/>cipio quadrantis ad finem vſque ſemper decreſcant. </s> <s xml:id="echoid-s4105" xml:space="preserve">Nam ſi in eadem figura huius propof. <lb/></s> <s xml:id="echoid-s4106" xml:space="preserve">a cci piatur arcus FX, arcubus DE, EF, æ qualis, ducaturq́ue recta XY, ad ſemidiametrum <lb/>AC, perpendicularis, habebunt quatuor arcus BX, BF, BE, BD, æquales exceſlus, cum <lb/>BX, ipſum BF, ſuperetarcu FX; </s> <s xml:id="echoid-s4107" xml:space="preserve">& </s> <s xml:id="echoid-s4108" xml:space="preserve">BF, ipſum BE, arcu EF, qui arcui FX, poſitus eſt <lb/>æqualis; </s> <s xml:id="echoid-s4109" xml:space="preserve">& </s> <s xml:id="echoid-s4110" xml:space="preserve">arcus BE, arcum BD, arcu DE, qui arcui EF, æqualis eſt. </s> <s xml:id="echoid-s4111" xml:space="preserve">Sinus autem recti <lb/>eorum arcuum ſunt AY, AI, AH, AG, vt ſupra in ex poſitione definitionum docuimus, <lb/>cum ſint partes ſemidiametri AC, inter centrum A, & </s> <s xml:id="echoid-s4112" xml:space="preserve">ſinus complementorum interiectæ, <lb/>vt patet. </s> <s xml:id="echoid-s4113" xml:space="preserve">Et quoniam in hac propof. </s> <s xml:id="echoid-s4114" xml:space="preserve">demonftrauimus, rectam GH, maiorem eſſe, quàm <lb/>HI, & </s> <s xml:id="echoid-s4115" xml:space="preserve">HI, maiorem, quàm IY; </s> <s xml:id="echoid-s4116" xml:space="preserve">liquet, exceſſum GH, inter ſinus arcuum minorum BE, <lb/>BD, maiorem eſſe exceſſu HI, inter ſinus arcuum maiorum BF, BE: </s> <s xml:id="echoid-s4117" xml:space="preserve">Item exceſſum HI, <lb/>inter ſinus arcuum minorum BF, BE, maiorem eſſe exceſſu IY, inter ſinus maiorum ar-<lb/>cuum BX, BF. </s> <s xml:id="echoid-s4118" xml:space="preserve">Eademq́ue ratio eſt de cæteris. </s> <s xml:id="echoid-s4119" xml:space="preserve">Conſtat igitur, differentias ſinuum rectorum <lb/>ſenſim decreſcere à principio quadrantis v ſque ad eius finem: </s> <s xml:id="echoid-s4120" xml:space="preserve">Id quod perſpicuè ex ſinuum <lb/>tabula apparet.</s> <s xml:id="echoid-s4121" xml:space="preserve"/> </p> <div xml:id="echoid-div346" type="float" level="2" n="1"> <note position="left" xlink:label="note-122-02" xlink:href="note-122-02a" xml:space="preserve">Differentię <lb/>ſinuũ recto <lb/>rũ à princi <lb/>pio quadrã <lb/>tis vſq; ad <lb/>eius finem <lb/>ſenſim de-<lb/>creſcunt:a-<lb/>deo vi ſinꝰ <lb/>minorú ar <lb/>cuũ maio-<lb/>res habeát <lb/>differétias, <lb/>q̇<unsure/> ſinꝰ arcuũ <lb/>maiorũ; dũ <lb/>modo arcꝰ <lb/>habeát dif-<lb/>ferentias ę <lb/>quales.</note> </div> </div> <div xml:id="echoid-div348" type="section" level="1" n="153"> <head xml:id="echoid-head179" xml:space="preserve">PROBL. 1. PROPOS. 2.</head> <p> <s xml:id="echoid-s4122" xml:space="preserve">LATERA Decagoni, & </s> <s xml:id="echoid-s4123" xml:space="preserve">Pentagoni æquila-<lb/> <anchor type="note" xlink:label="note-122-03a" xlink:href="note-122-03"/> teri in vno eodemq́; </s> <s xml:id="echoid-s4124" xml:space="preserve">circulo inueſtigare.</s> <s xml:id="echoid-s4125" xml:space="preserve"/> </p> <div xml:id="echoid-div348" type="float" level="2" n="1"> <note position="left" xlink:label="note-122-03" xlink:href="note-122-03a" xml:space="preserve">Latera De. <lb/>cagoni, & <lb/>Penta goni <lb/>in vno eo-<lb/>demq; cir-<lb/>culo quo <lb/>pacto inue-<lb/>niantur.</note> </div> <p> <s xml:id="echoid-s4126" xml:space="preserve">QVAMVIS hæc latera inueniantur per ea, quæ ab Euclide lib. </s> <s xml:id="echoid-s4127" xml:space="preserve">4. </s> <s xml:id="echoid-s4128" xml:space="preserve">ſunt <lb/>demonſtrata: </s> <s xml:id="echoid-s4129" xml:space="preserve">nihilominus eadem à Ptolemæo lib. </s> <s xml:id="echoid-s4130" xml:space="preserve">1. </s> <s xml:id="echoid-s4131" xml:space="preserve">Almageſti cap. </s> <s xml:id="echoid-s4132" xml:space="preserve">9. </s> <s xml:id="echoid-s4133" xml:space="preserve">inueſti-<lb/>gantur ratione alia, quæ ad plurimorum ſinuum inuentionem multum con-<lb/>ducit. </s> <s xml:id="echoid-s4134" xml:space="preserve">Eſt autem hæc ratio. </s> <s xml:id="echoid-s4135" xml:space="preserve">Sit circulus, vel (quod ſatis eſt) ſemicirculus ABC, <lb/> <anchor type="figure" xlink:label="fig-122-01a" xlink:href="fig-122-01"/> ad cuius diametrum AC, ex D, centro <lb/>educatur perpendicularis DB. </s> <s xml:id="echoid-s4136" xml:space="preserve">Diuiſa <lb/>quoque ſemidiametro CD, bifariam <lb/>in E, ducatur recta EB, cui ęqualis ab-<lb/>ſcindatur EF, iungaturq́ue recta FB. <lb/></s> <s xml:id="echoid-s4137" xml:space="preserve">Dico rectam BF, eſſe latus Penta-<lb/>goni, & </s> <s xml:id="echoid-s4138" xml:space="preserve">DF, latus Decagoni in cir-<lb/>culo ABC. </s> <s xml:id="echoid-s4139" xml:space="preserve">Cum enim recta CD, <lb/>ſecta ſit bifariam in E, eique addi-<lb/>ta DF; </s> <s xml:id="echoid-s4140" xml:space="preserve">erit rectangulum ſub CF, DF, <lb/> <anchor type="note" xlink:label="note-122-04a" xlink:href="note-122-04"/> vna cum quadrato rectæ DE, æquale <lb/>quadrato rectæ EF, ideoq́ quadrato rectæ EB, quæ ipſi EF, ęqualis <lb/>eſt: </s> <s xml:id="echoid-s4141" xml:space="preserve">Eſt autem quadratum rectæ EB, æquale quadratis rectarum BD, DE. <lb/></s> <s xml:id="echoid-s4142" xml:space="preserve"> <anchor type="note" xlink:label="note-122-05a" xlink:href="note-122-05"/> Igitur rectangulum ſub CF, DF, vnà cum quadrato rectæ DE, æquale eſt <lb/>quadratis rectarum BD, DE: </s> <s xml:id="echoid-s4143" xml:space="preserve">Ac proinde, dempto communi quadrato rectæ <lb/><emph style="sc">De</emph>, relinquetur rectangulum ſub CF, DF, æquale quadrato rectæ BD, <lb/>hoc eſt, quadrato rectæ CD. </s> <s xml:id="echoid-s4144" xml:space="preserve">Quamobrem erit, vt CF, ad CD, ita CD, ad <lb/> <anchor type="note" xlink:label="note-122-06a" xlink:href="note-122-06"/> DF; </s> <s xml:id="echoid-s4145" xml:space="preserve">proptereaq́ue recta CF, diuiſa erit in D, extrema ac media ratione. </s> <s xml:id="echoid-s4146" xml:space="preserve">Cum <lb/>igitur maius ſegmentum CD, ſit latus Hexagoni in circulo ABC, ex co- <pb o="111" file="123" n="123" rhead=""/> coll. </s> <s xml:id="echoid-s4147" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s4148" xml:space="preserve">15. </s> <s xml:id="echoid-s4149" xml:space="preserve">lib. </s> <s xml:id="echoid-s4150" xml:space="preserve">4. </s> <s xml:id="echoid-s4151" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s4152" xml:space="preserve">erit minus ſegmentum DF, latus Decagoni in eo-<lb/>dem circulo, vt ad propoſ. </s> <s xml:id="echoid-s4153" xml:space="preserve">9. </s> <s xml:id="echoid-s4154" xml:space="preserve">lib. </s> <s xml:id="echoid-s4155" xml:space="preserve">13. </s> <s xml:id="echoid-s4156" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s4157" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s4158" xml:space="preserve">Rurſus quoniam <lb/>quadrato lateris Hexagoni BD, vna cum quadrato lateris Decagoni DF, <lb/> <anchor type="note" xlink:label="note-123-01a" xlink:href="note-123-01"/> æquale eſt quadratum lateris Pentagoni in eodem circulo: </s> <s xml:id="echoid-s4159" xml:space="preserve">Eſt autem eiſdem <lb/>quadratis rectarum BD, DF, æquale quadratum rectæ BF; </s> <s xml:id="echoid-s4160" xml:space="preserve">erit quadratum <lb/> <anchor type="note" xlink:label="note-123-02a" xlink:href="note-123-02"/> lateris Pentagoni æquale quadrato rectæ BF; </s> <s xml:id="echoid-s4161" xml:space="preserve">ac propterea recta BF, lateri <lb/>Pentagoni æqualis. </s> <s xml:id="echoid-s4162" xml:space="preserve">Latera igitur Decagoni, & </s> <s xml:id="echoid-s4163" xml:space="preserve">Pentagoni æquilateri in vno <lb/>@odemq́ue circulo inueſtigauimus. </s> <s xml:id="echoid-s4164" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s4165" xml:space="preserve"/> </p> <div xml:id="echoid-div349" type="float" level="2" n="2"> <figure xlink:label="fig-122-01" xlink:href="fig-122-01a"> <image file="122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/122-01"/> </figure> <note position="left" xlink:label="note-122-04" xlink:href="note-122-04a" xml:space="preserve">6.ſecundi.</note> <note position="left" xlink:label="note-122-05" xlink:href="note-122-05a" xml:space="preserve">47.primi.</note> <note position="left" xlink:label="note-122-06" xlink:href="note-122-06a" xml:space="preserve">17.fexti.</note> <note position="right" xlink:label="note-123-01" xlink:href="note-123-01a" xml:space="preserve">10. tertij-<lb/>dec.</note> <note position="right" xlink:label="note-123-02" xlink:href="note-123-02a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div351" type="section" level="1" n="154"> <head xml:id="echoid-head180" xml:space="preserve">PROBL. 2. PROPOS. 3.</head> <note position="right" xml:space="preserve">Ex ſinu re-<lb/>ctocuiuſuis <lb/>arcus quo <lb/>pacto ſinus <lb/>com plemé <lb/>ti eiuſdem <lb/>arcus, & ex <lb/>chorda cu-<lb/>iuſuis ar -<lb/>cus qua ra-<lb/>tione chor-<lb/>da reliqui <lb/>arcus ſemi-<lb/>circuli co-<lb/>gnoſcatur. <lb/>47. primi.</note> <p> <s xml:id="echoid-s4166" xml:space="preserve">EX ſinu recto cuiuſuis arcus quadrante mi-<lb/>noris cognito, ſinum complementi eiuſdem ar-<lb/>cus; </s> <s xml:id="echoid-s4167" xml:space="preserve">lté ex chorda cuiuſuis arcus ſemicirculo mino <lb/>ris, chordam rehqui arcus ſemicircuh cognoſcere.</s> <s xml:id="echoid-s4168" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4169" xml:space="preserve">SIT primo cognitus ſinus rectus DE, arcus BD, cuius arcus complemen <lb/>ti ſinus ſit DF, quem cognoſcere debemus. </s> <s xml:id="echoid-s4170" xml:space="preserve">Ducta recta DA, erit quadratum <lb/>rectæ DA, æquale quadratis rectarum DE, EA. </s> <s xml:id="echoid-s4171" xml:space="preserve">Si igitur ex quadrato ſinus <lb/>totius DA, noti (Ponitur enim ſinus totus particularum certo numero com-<lb/>prehenſarum) detrahatur <lb/> <anchor type="figure" xlink:label="fig-123-01a" xlink:href="fig-123-01"/> quadratũ ſinus recti DE, <lb/>cogniti in partibus ſinus <lb/>totius DA, relinquetur <lb/>quadratum rectæ EA, no <lb/>tum; </s> <s xml:id="echoid-s4172" xml:space="preserve">ac proinde per radi-<lb/>cem quadratam recta EA, <lb/>in eiſdem partibus nota <lb/>erit. </s> <s xml:id="echoid-s4173" xml:space="preserve">Cum ergo recta EA, <lb/>æqualis ſit ſinui comple-<lb/>menti arcus BD, hoc eſt, <lb/> <anchor type="note" xlink:label="note-123-04a" xlink:href="note-123-04"/> rectæ DF, cognitus erit <lb/>DF, ſinus complementi arcus BD, cuius ſinus rectus DE, notus eſt poſitus.</s> <s xml:id="echoid-s4174" xml:space="preserve"/> </p> <div xml:id="echoid-div351" type="float" level="2" n="1"> <figure xlink:label="fig-123-01" xlink:href="fig-123-01a"> <image file="123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/123-01"/> </figure> <note position="right" xlink:label="note-123-04" xlink:href="note-123-04a" xml:space="preserve">34. primi.</note> </div> <p> <s xml:id="echoid-s4175" xml:space="preserve">SIT deinde cognita chorda AB, arcus AB, & </s> <s xml:id="echoid-s4176" xml:space="preserve">chorda BC, ſubtendens re-<lb/>liquum arcum BC, ſemicirculi, quam iubemur inueſtigare. </s> <s xml:id="echoid-s4177" xml:space="preserve">Quoniam angulus <lb/> <anchor type="note" xlink:label="note-123-05a" xlink:href="note-123-05"/> B, rectus eſt in ſemicirculo, erit quadratum diametri AC, æquale quadratis <lb/> <anchor type="note" xlink:label="note-123-06a" xlink:href="note-123-06"/> chordarum AB, BC. </s> <s xml:id="echoid-s4178" xml:space="preserve">Si igitur ex quadrato diametri AC, notæ (Ponitur enim <lb/>diameter diuiſa in particulas certo numero comprehenſas) dematur quadra-<lb/>tum chordæ AB, notæ in partibus diametri AC, notum relinquetur quadra-<lb/>tum chordæ BC; </s> <s xml:id="echoid-s4179" xml:space="preserve">ac proinde per radicem quadratam chorda BC, in eiſdem <lb/>partibus nota efficietur. </s> <s xml:id="echoid-s4180" xml:space="preserve">Ex ſinu igitur recto cuiuſuis arcus, &</s> <s xml:id="echoid-s4181" xml:space="preserve">c. </s> <s xml:id="echoid-s4182" xml:space="preserve">cognouimus. <lb/></s> <s xml:id="echoid-s4183" xml:space="preserve">Quod ſaciendum erat.</s> <s xml:id="echoid-s4184" xml:space="preserve"/> </p> <div xml:id="echoid-div352" type="float" level="2" n="2"> <note position="right" xlink:label="note-123-05" xlink:href="note-123-05a" xml:space="preserve">31. tertij.</note> <note position="right" xlink:label="note-123-06" xlink:href="note-123-06a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div354" type="section" level="1" n="155"> <head xml:id="echoid-head181" xml:space="preserve">COROLLARIVM.</head> <note position="right" xml:space="preserve">Sinus vet-<lb/>ſus cogno-<lb/>ſcitur ex co <lb/>gnito ſinu <lb/>recto,.</note> <p> <s xml:id="echoid-s4185" xml:space="preserve">HINC efficitur, ſinum verſum cuiuſuis arcus cognoſci quoque ex cognito ſinu recto. <lb/></s> <s xml:id="echoid-s4186" xml:space="preserve">Quoniam enim ex ſinu recto DE, cognoſcitur ſinus complementi DP, hoc eſt, AE; </s> <s xml:id="echoid-s4187" xml:space="preserve">ſi ſinus <pb o="112" file="124" n="124" rhead=""/> complementi dati arcus BD, auferatur ex ſinu toto AB, notus relinquetur ſinus verſus EB, <lb/>dati arcus BD. </s> <s xml:id="echoid-s4188" xml:space="preserve">Pari ratione, ſi ſinus rectus DE, hoc eſt, AF, dati arcus BD, dematur ex ſinu <lb/>toto AC, notus relinquetur ſinus verſus CF, complementi dati arcus BD.</s> <s xml:id="echoid-s4189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div355" type="section" level="1" n="156"> <head xml:id="echoid-head182" xml:space="preserve">THEOR. 2. PROPOS. 4.</head> <p> <s xml:id="echoid-s4190" xml:space="preserve">SINVS rectus cuiuſlibet arcus quadrante mi <lb/> <anchor type="note" xlink:label="note-124-01a" xlink:href="note-124-01"/> noris medio loco proportionalis eſt inter ſemiſ-<lb/>ſem ſemidiametri, ſeu ſinus totius, & </s> <s xml:id="echoid-s4191" xml:space="preserve">ſinum ver-<lb/>ſum arcus alterius, qui prioris arcus duplus eſt, & </s> <s xml:id="echoid-s4192" xml:space="preserve"><lb/>quadrante quoque minor.</s> <s xml:id="echoid-s4193" xml:space="preserve"/> </p> <div xml:id="echoid-div355" type="float" level="2" n="1"> <note position="left" xlink:label="note-124-01" xlink:href="note-124-01a" xml:space="preserve">Cuiuſuisar <lb/>cꝰ quadrá-<lb/>te minoris <lb/>finꝰ rectus <lb/>medioloco <lb/>{pro} portiona <lb/>lis eſt inter <lb/>ſemiſsé ſi-<lb/>nus totius, <lb/>& ſinũ ver-<lb/>ſum alteriꝰ <lb/>arcꝰ, ꝗ prio <lb/>ris duplus <lb/>eſt, & qua-<lb/>drante quo <lb/>queminor.</note> </div> <p> <s xml:id="echoid-s4194" xml:space="preserve">SIT arcus quicunque CE, quadrante minor, cuius dimidium ſit CD. </s> <s xml:id="echoid-s4195" xml:space="preserve">Di-<lb/>uiſa autem ſemidiametro AC, bifariã in G, ducatur ex E, ad AC, perpendi-<lb/>cularis EF, iungaturque recta AD, quæ ductã chordam CE, ſecabit in H, bifa-<lb/>riam, ex lemmate à nobis ad definitiones ſupra demonſtrato, atque adeo & </s> <s xml:id="echoid-s4196" xml:space="preserve">ad <lb/>angulos rectos. </s> <s xml:id="echoid-s4197" xml:space="preserve">Erit igitur CH, ſinus rectus arcus CD, & </s> <s xml:id="echoid-s4198" xml:space="preserve">CF, ſinus verſus <lb/> <anchor type="note" xlink:label="note-124-02a" xlink:href="note-124-02"/> <anchor type="figure" xlink:label="fig-124-01a" xlink:href="fig-124-01"/> arcus CE, qui duplus eſt arcus CD, cum EF, ſit <lb/>eiuſdem arcus CE, ſinus rectus: </s> <s xml:id="echoid-s4199" xml:space="preserve">vt ex definitionibus <lb/>conſtat. </s> <s xml:id="echoid-s4200" xml:space="preserve">Dico CH, ſinum rectum arcus CD, medio <lb/>loco eſſe proportionalẽ inter CG, dimidiũ ſinus to-<lb/>tius, & </s> <s xml:id="echoid-s4201" xml:space="preserve">CF, ſinum verſum arcus CE, qui arcus CD, <lb/>duplus eſt. </s> <s xml:id="echoid-s4202" xml:space="preserve">Quoniã enim duo anguli ACH, AHC, <lb/>trianguli ACH, æquales ſunt duobus angulis ECF, <lb/>EFC, trianguli ECF, quod angulus C, vtrique <lb/>triangulo ſit communis, & </s> <s xml:id="echoid-s4203" xml:space="preserve">anguli H, F, recti; </s> <s xml:id="echoid-s4204" xml:space="preserve">ęquian <lb/>gula erunt triangula ACH, ECF. </s> <s xml:id="echoid-s4205" xml:space="preserve">Igitur erit, vt <lb/> <anchor type="note" xlink:label="note-124-03a" xlink:href="note-124-03"/> AC, ad CH, ita EC, ad CF: </s> <s xml:id="echoid-s4206" xml:space="preserve">Et permutando, vt AC, ad CE, ita CH, ad <lb/> <anchor type="note" xlink:label="note-124-04a" xlink:href="note-124-04"/> CF. </s> <s xml:id="echoid-s4207" xml:space="preserve">Vt autem AC, ad CE, ita eſt CG, dimidium ipſius AC, ad CH, dimi-<lb/> <anchor type="note" xlink:label="note-124-05a" xlink:href="note-124-05"/> dium ipſius CE. </s> <s xml:id="echoid-s4208" xml:space="preserve">Igitur erit quoque vt CG, ad CH, ita CH, ad CF; </s> <s xml:id="echoid-s4209" xml:space="preserve">ac pro-<lb/>pterea CH, ſinus rectus arcus CD, medio loco proportionalis eſt inter CG, <lb/>ſemiſſem ſinus totius, & </s> <s xml:id="echoid-s4210" xml:space="preserve">CF, ſinum verſum arcus CE, qui arcus CD, duplus <lb/> <anchor type="note" xlink:label="note-124-06a" xlink:href="note-124-06"/> eſt. </s> <s xml:id="echoid-s4211" xml:space="preserve">Igitur ſinus rectus cuiuſlibet arcus quadrante minoris, &</s> <s xml:id="echoid-s4212" xml:space="preserve">c. </s> <s xml:id="echoid-s4213" xml:space="preserve">Quod de-<lb/>monſtrandum erat.</s> <s xml:id="echoid-s4214" xml:space="preserve"/> </p> <div xml:id="echoid-div356" type="float" level="2" n="2"> <note position="left" xlink:label="note-124-02" xlink:href="note-124-02a" xml:space="preserve">3. tertij.</note> <figure xlink:label="fig-124-01" xlink:href="fig-124-01a"> <image file="124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/124-01"/> </figure> <note position="left" xlink:label="note-124-03" xlink:href="note-124-03a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-124-04" xlink:href="note-124-04a" xml:space="preserve">4.ſexti.</note> <note position="left" xlink:label="note-124-05" xlink:href="note-124-05a" xml:space="preserve">15. quinti.</note> <note position="left" xlink:label="note-124-06" xlink:href="note-124-06a" xml:space="preserve">Ex ſinu re-<lb/>cto cuiuſ-<lb/>uis arcus <lb/>cognito no <lb/>tus fit ſinus <lb/>rectus alte-<lb/>rius arcus, <lb/>qui illius <lb/>dimidiũ ſit <lb/>33. primi. <lb/>3. huius, <lb/>27. ſexti.</note> </div> </div> <div xml:id="echoid-div358" type="section" level="1" n="157"> <head xml:id="echoid-head183" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s4215" xml:space="preserve">COLLIGITVR hinc, ſi ſinus rectus alicuius arcus cognitus ſit, notum etiam ſieri <lb/>ſinum rectum alterius arcus, qui illius di midium ſit: </s> <s xml:id="echoid-s4216" xml:space="preserve">ita vt ex EF, ſinu recto arcus CE, <lb/>cognito cognoſcatur etiam CH, ſinus rectus arcus CD, qui dimidium eſt arcus CE. </s> <s xml:id="echoid-s4217" xml:space="preserve">Nam <lb/>ex noto ſinu recto EF, notus fiet ſinus EI, complementi: </s> <s xml:id="echoid-s4218" xml:space="preserve">quo ablato ex ſinu toto AC, <lb/>(æqualis enim eſt ſinus EI, rectæ AF.) </s> <s xml:id="echoid-s4219" xml:space="preserve">notus relinquetur ſinus verſus CF, arcus CE, vt in <lb/>coroll. </s> <s xml:id="echoid-s4220" xml:space="preserve">præcedentis propoſ. </s> <s xml:id="echoid-s4221" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s4222" xml:space="preserve">Cum ergo ſinus CH, ſit medio loco proporti onalis <lb/>inter medietatem ſinus totius, & </s> <s xml:id="echoid-s4223" xml:space="preserve">ſinum verſum CF, vt oſtendimus; </s> <s xml:id="echoid-s4224" xml:space="preserve">erit rectangulum ſub di <lb/>midio ſinus totius, & </s> <s xml:id="echoid-s4225" xml:space="preserve">ſinu verſo CF, contentum æquale quadrato ſinus CH. </s> <s xml:id="echoid-s4226" xml:space="preserve">Quare ſi <lb/>multiplicetur medietas ſinus totius in ſinum verſum CF, producetur quadratus numerus <pb o="113" file="125" n="125" rhead=""/> ſinus CH, cuius radix quadrata notum dabit ſinum rectum CH. </s> <s xml:id="echoid-s4227" xml:space="preserve">Eademq́ue ratio eſt <lb/>de cæteris.</s> <s xml:id="echoid-s4228" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4229" xml:space="preserve">IDEM hac etiam ratione oſtendi poteſt. </s> <s xml:id="echoid-s4230" xml:space="preserve">Quoniam enim EF, ſinus rectus arcus <lb/>CE, notus ponitur, cognoſcetur & </s> <s xml:id="echoid-s4231" xml:space="preserve">EI, ſinus complementi ei uſd em arcus, hoc eſt, re@ <lb/> <anchor type="note" xlink:label="note-125-01a" xlink:href="note-125-01"/> cta AF, illi æqualis. </s> <s xml:id="echoid-s4232" xml:space="preserve">Detracta igitur recta AF, hoc eſt, ſinu complementi arcus CE, <lb/> <anchor type="note" xlink:label="note-125-02a" xlink:href="note-125-02"/> ex ſinu toto AC, cognitus erit ſinus verſus FC, arcus eiuſdem CE, vt etiam in coroll. <lb/></s> <s xml:id="echoid-s4233" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s4234" xml:space="preserve">3. </s> <s xml:id="echoid-s4235" xml:space="preserve">oſtendinaus. </s> <s xml:id="echoid-s4236" xml:space="preserve">Quia vero quadratum rectæ CE, æquale eſt quadratis rectarum EF, <lb/>FC; </s> <s xml:id="echoid-s4237" xml:space="preserve">fit, vt quadrata rectarum EF, FC, nota rum in vnam ſummam collecta efficiant <lb/>quadratum rectæ CE: </s> <s xml:id="echoid-s4238" xml:space="preserve">cuius radix quadrata ipſam rectam CE, reddet notam; </s> <s xml:id="echoid-s4239" xml:space="preserve">ac proinde <lb/>huius radicis dimidium dabit CH, ſinum rectum arcus CD, qui dimidium eſt dati arcus <lb/>CE, notum.</s> <s xml:id="echoid-s4240" xml:space="preserve"/> </p> <div xml:id="echoid-div358" type="float" level="2" n="1"> <note position="right" xlink:label="note-125-01" xlink:href="note-125-01a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-125-02" xlink:href="note-125-02a" xml:space="preserve">34. primi.</note> </div> <p> <s xml:id="echoid-s4241" xml:space="preserve">VICISSIM ex hac eadem propof. </s> <s xml:id="echoid-s4242" xml:space="preserve">4. </s> <s xml:id="echoid-s4243" xml:space="preserve">colligitur, ſi ſinus rectus alicuius arcus cognitus <lb/> <anchor type="note" xlink:label="note-125-03a" xlink:href="note-125-03"/> ſit, notum etiam fieri ſinum rectum alrerius arcus, qui illius duplus ſit, dummodo quadran-<lb/>te ſit minor: </s> <s xml:id="echoid-s4244" xml:space="preserve">ita vt ex CH, ſinu recto arcus CD, cognito cognoſcatur etiam EF, ſinus rectus <lb/>arcus CE, qui arcus CD, eſt duplus. </s> <s xml:id="echoid-s4245" xml:space="preserve">Cum enim ſin us CH, ſit medio loco proportionalis in-<lb/>ter medictatem ſinus totius, & </s> <s xml:id="echoid-s4246" xml:space="preserve">ſinum verſum FC, vt oſtendimus; </s> <s xml:id="echoid-s4247" xml:space="preserve">erit rectangulum ſub di-<lb/>midio ſinus totius, & </s> <s xml:id="echoid-s4248" xml:space="preserve">ſinu verſo FC, contentum æquale quadrato ſinus recti CH. </s> <s xml:id="echoid-s4249" xml:space="preserve">Quare <lb/>quadratum ſinus CH, noti erit illud rectangulum; </s> <s xml:id="echoid-s4250" xml:space="preserve">quo diuiſo per dimidium ſinus totius, <lb/>notus euadet ſinus verſus FC. </s> <s xml:id="echoid-s4251" xml:space="preserve">Quia vero recta CE, cum ſit dupla ſinus CH, noti nota eſt, <lb/>erit & </s> <s xml:id="echoid-s4252" xml:space="preserve">eius quadratum notum: </s> <s xml:id="echoid-s4253" xml:space="preserve">à quo ſi auferatur quadratum ſinus verſi FC, noti, relinque-<lb/>tur etiam quadratum rectæ EF, notum; </s> <s xml:id="echoid-s4254" xml:space="preserve">(cum quadratũ rectæ CE, quadratis rectarum CF, <lb/>FE, ſit æquale.) </s> <s xml:id="echoid-s4255" xml:space="preserve">ac proinde radix quadrata illius notum dabit ſinum rectum EF.</s> <s xml:id="echoid-s4256" xml:space="preserve"/> </p> <div xml:id="echoid-div359" type="float" level="2" n="2"> <note position="right" xlink:label="note-125-03" xlink:href="note-125-03a" xml:space="preserve">Ex ſinu re-<lb/>cto cuiuſ-<lb/>uis arcꝰ co-<lb/>gnito notꝰ <lb/>ſit ſinus re <lb/>ctus alteriꝰ <lb/>arcꝰ, qui il-<lb/>liꝰ ſit duplꝰ <lb/>dummodo <lb/>quadrante <lb/>minor ſit.</note> </div> </div> <div xml:id="echoid-div361" type="section" level="1" n="158"> <head xml:id="echoid-head184" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s4257" xml:space="preserve">QVOD _ſi quando perpendicularis_ <emph style="sc">Ef</emph>, _ſemidiametrum_ AC, _ſecet bifariam, vt_ <lb/>_in hac figura contingit, erit adhuc_ CH, _ſinus arcus_ CD, <lb/> <anchor type="figure" xlink:label="fig-125-01a" xlink:href="fig-125-01"/> _medio loco proporlionalis inter_ <emph style="sc">Cf</emph>, _ſemiſſem ſinus totius,_ <lb/>& </s> <s xml:id="echoid-s4258" xml:space="preserve"><emph style="sc">Cf</emph>, _ſinũ verſum arcus_ CE, _qui arcus_ CD, _duplus eſt. </s> <s xml:id="echoid-s4259" xml:space="preserve"><emph style="sc">E</emph>rũt_ <lb/>_enim rurſum triangula_ ACH, <emph style="sc">E</emph><emph style="sc">Cf</emph>, _æquiangula; </s> <s xml:id="echoid-s4260" xml:space="preserve">ac_ <lb/>_proinde, vt_ AC, _ad_ CH, _ita_ EC, _ad_ <emph style="sc">Cf</emph>: </s> <s xml:id="echoid-s4261" xml:space="preserve">_Et permutan-_ <lb/> <anchor type="note" xlink:label="note-125-04a" xlink:href="note-125-04"/> _do, vt_ AC, _ad_ CE, _ita_ CH, _ad_ <emph style="sc">Cf</emph>. </s> <s xml:id="echoid-s4262" xml:space="preserve">_Cum ergo ſit, vt_ <lb/>AC, _ad_ <emph style="sc">Cf</emph>, _dimidium ipſius_ AC, _ad_ CH, _dimi-_ <lb/> <anchor type="note" xlink:label="note-125-05a" xlink:href="note-125-05"/> _dium ipſius_ <emph style="sc">Ce</emph>; </s> <s xml:id="echoid-s4263" xml:space="preserve">_erit quoq; </s> <s xml:id="echoid-s4264" xml:space="preserve">vt_ <emph style="sc">Cf</emph>, _ad_ CH, _ita_ CH, _ad_ <emph style="sc">Cf</emph>: <lb/></s> <s xml:id="echoid-s4265" xml:space="preserve">_proptereaq́;_ </s> <s xml:id="echoid-s4266" xml:space="preserve">CH, _ſinus rectus arcus_ CD, _medio loco propor-_ <lb/>_tionalis eſt inter_ <emph style="sc">Cf</emph>, _ſemiſſem ſinus totius,_ & </s> <s xml:id="echoid-s4267" xml:space="preserve"><emph style="sc">Cf</emph>, _ſinum ver_ <lb/>_ſum arcus_ CG, _qui duplus eſt arcus_ CD.</s> <s xml:id="echoid-s4268" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="1"> <figure xlink:label="fig-125-01" xlink:href="fig-125-01a"> <image file="125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/125-01"/> </figure> <note position="right" xlink:label="note-125-04" xlink:href="note-125-04a" xml:space="preserve">4. fexti.</note> <note position="right" xlink:label="note-125-05" xlink:href="note-125-05a" xml:space="preserve">15. quinti.</note> </div> <p> <s xml:id="echoid-s4269" xml:space="preserve">HINC _fit, ſiperpendicularis_ EF, _ſemidiametrum_ AC, _ſecet bifariam, rectam_ <lb/>CH, _æqualem eſſe rectæ_ <emph style="sc">Cf</emph>. </s> <s xml:id="echoid-s4270" xml:space="preserve">_Si enim maior eſſet, aut minor, non poßet eſſe, vt_ <emph style="sc">Cf</emph>, <lb/>_ad_ CH, _ita_ CH, _ad_ <emph style="sc">Cf</emph>: </s> <s xml:id="echoid-s4271" xml:space="preserve">_cum vna proportio eſſet maioris inæqualitatis, & </s> <s xml:id="echoid-s4272" xml:space="preserve">altera_ <lb/>_minoris inæqualitatis._</s> <s xml:id="echoid-s4273" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Sinus rectꝰ <lb/>grad. 54. æ-<lb/>qualis eſt <lb/>ſemiſsi ſinꝰ <lb/>totiꝰ, & ſi-<lb/>nui gra. 18. <lb/>ſimul. Sinꝰ <lb/>aũt verſus <lb/>grad. 72 æ-<lb/>qualiseſt ſe <lb/>miſſi ſinus <lb/>totius, & ſi-<lb/>nui verſo <lb/>grad. 36. ſi-<lb/>mul.</note> </div> <div xml:id="echoid-div363" type="section" level="1" n="159"> <head xml:id="echoid-head185" xml:space="preserve">THEOR 3. PROPOS. 5.</head> <p> <s xml:id="echoid-s4274" xml:space="preserve">SINVS rectus arcus graduum 54. </s> <s xml:id="echoid-s4275" xml:space="preserve">componi-<lb/>tur ex ſemiſſe ſinus totius, & </s> <s xml:id="echoid-s4276" xml:space="preserve">ſinu recto arcus grad <lb/>18. </s> <s xml:id="echoid-s4277" xml:space="preserve">Sinus autem verſus arcus grad. </s> <s xml:id="echoid-s4278" xml:space="preserve">72. </s> <s xml:id="echoid-s4279" xml:space="preserve">componitur <lb/>ex ſemiſſe ſinus totius, & </s> <s xml:id="echoid-s4280" xml:space="preserve">ſinu verſo arcus grad. </s> <s xml:id="echoid-s4281" xml:space="preserve">36.</s> <s xml:id="echoid-s4282" xml:space="preserve"/> </p> <pb o="114" file="126" n="126" rhead=""/> <p> <s xml:id="echoid-s4283" xml:space="preserve">IN quadrante ABC, ſit BD, arcus grad. </s> <s xml:id="echoid-s4284" xml:space="preserve">54. </s> <s xml:id="echoid-s4285" xml:space="preserve">ac proinde eius cõplementum <lb/>CD, grad. </s> <s xml:id="echoid-s4286" xml:space="preserve">36. </s> <s xml:id="echoid-s4287" xml:space="preserve">quod diuidatur bifariam in H, vt vterq; </s> <s xml:id="echoid-s4288" xml:space="preserve">arcuũ CH, HD, habeat <lb/>grad. </s> <s xml:id="echoid-s4289" xml:space="preserve">18. </s> <s xml:id="echoid-s4290" xml:space="preserve">Ducatur DM, ad AB, perpendicularis pro ſinu arcus grad. </s> <s xml:id="echoid-s4291" xml:space="preserve">54. </s> <s xml:id="echoid-s4292" xml:space="preserve">& </s> <s xml:id="echoid-s4293" xml:space="preserve">DE, <lb/> <anchor type="figure" xlink:label="fig-126-01a" xlink:href="fig-126-01"/> ad AC, perpédicularis pro ſinu arcus grad. </s> <s xml:id="echoid-s4294" xml:space="preserve">36. </s> <s xml:id="echoid-s4295" xml:space="preserve">Iunga <lb/>tur quoq; </s> <s xml:id="echoid-s4296" xml:space="preserve">recta AH, quæ per lẽma in definitionibus <lb/>demonſtratũ ſecabit rectã CD, in I, bifariam, ac pro <lb/>inde & </s> <s xml:id="echoid-s4297" xml:space="preserve">ad angulos rectos: </s> <s xml:id="echoid-s4298" xml:space="preserve">eritq́ propterea CI, ſinus <lb/> <anchor type="note" xlink:label="note-126-01a" xlink:href="note-126-01"/> rectus arcus CH, grad. </s> <s xml:id="echoid-s4299" xml:space="preserve">18. </s> <s xml:id="echoid-s4300" xml:space="preserve">Sũpta tandẽ recta EF, ipſi <lb/>EC, æquali, diuidantur AC, AF, bifariã in G, K, & </s> <s xml:id="echoid-s4301" xml:space="preserve"><lb/>ex K, ad AC, perpendicularis ducatur KL. </s> <s xml:id="echoid-s4302" xml:space="preserve">Dico ſi-<lb/>num rectũ DM, arcus grad. </s> <s xml:id="echoid-s4303" xml:space="preserve">54. </s> <s xml:id="echoid-s4304" xml:space="preserve">hoc eſt, rectam AE, <lb/> <anchor type="note" xlink:label="note-126-02a" xlink:href="note-126-02"/> illi ęqualẽ, componi ex AG, dimidio ſinus totius, & </s> <s xml:id="echoid-s4305" xml:space="preserve"><lb/>ex CI, ſinu recto arcus grad. </s> <s xml:id="echoid-s4306" xml:space="preserve">18. </s> <s xml:id="echoid-s4307" xml:space="preserve">hoc eſt, rectam GE, <lb/>(quæ cũ AG, conſtituit totam rectã AE,) ęqualẽ eſ <lb/>ſe ſinui recto CI. </s> <s xml:id="echoid-s4308" xml:space="preserve">Item ſinũ verſum arcus grad. </s> <s xml:id="echoid-s4309" xml:space="preserve">72. </s> <s xml:id="echoid-s4310" xml:space="preserve">componi ex dimidio ſinus to <lb/>tius, & </s> <s xml:id="echoid-s4311" xml:space="preserve">ex CE, ſinu verſo arcus CD, grad. </s> <s xml:id="echoid-s4312" xml:space="preserve">36. </s> <s xml:id="echoid-s4313" xml:space="preserve">hoc eſt, rectam EK, (quæ cum <lb/>ſinu verſo CE, rectam CK, componit) æqualem eſſe dimidio ſinus totius, ip-<lb/>ſam vero CK, eſſe ſinum verſum arcus grad. </s> <s xml:id="echoid-s4314" xml:space="preserve">72. </s> <s xml:id="echoid-s4315" xml:space="preserve">hoc eſt, arcum CL, (cuius ſi-<lb/>nus verſus eſt CK,) eſſe grad. </s> <s xml:id="echoid-s4316" xml:space="preserve">72. </s> <s xml:id="echoid-s4317" xml:space="preserve">Ducta enim recta LN, ad AB, perpendicu-<lb/>lari, pro ſinu arcus BL, iungantur rectæ AD, DF. </s> <s xml:id="echoid-s4318" xml:space="preserve">Quoniam igitur arcus <lb/>CH, grad. </s> <s xml:id="echoid-s4319" xml:space="preserve">18. </s> <s xml:id="echoid-s4320" xml:space="preserve">continet {1/5}. </s> <s xml:id="echoid-s4321" xml:space="preserve">quadrantis BC, (quòd quinquies 18. </s> <s xml:id="echoid-s4322" xml:space="preserve">faciant 90.) <lb/></s> <s xml:id="echoid-s4323" xml:space="preserve">continebit arcus CD, {2/5}. </s> <s xml:id="echoid-s4324" xml:space="preserve">eiuſdem quadrantis, ac proinde proportio arcus <lb/>CD, ad arcum BC, erit vt 2. </s> <s xml:id="echoid-s4325" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s4326" xml:space="preserve">Eſt autem, vt arcus CD, ad arcum BC, ita <lb/> <anchor type="note" xlink:label="note-126-03a" xlink:href="note-126-03"/> angulus CAD, ad rectum angulum BAC. </s> <s xml:id="echoid-s4327" xml:space="preserve">Igitur proportio anguli CAD, <lb/>ad angulum rectum BAC, erit quoque, vt 2. </s> <s xml:id="echoid-s4328" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s4329" xml:space="preserve">ac proinde angulus CAD, <lb/>continebit {2/5}. </s> <s xml:id="echoid-s4330" xml:space="preserve">vnius anguli recti. </s> <s xml:id="echoid-s4331" xml:space="preserve">Cum ergo tres anguli trianguli CAD, con-<lb/>tineant {10/5}. </s> <s xml:id="echoid-s4332" xml:space="preserve">vnius recti, hoc eſt, æquales ſint duobus rectis, ſintq́ue inter ſe <lb/> <anchor type="note" xlink:label="note-126-04a" xlink:href="note-126-04"/> æquales duo anguli ACD, ADC; </s> <s xml:id="echoid-s4333" xml:space="preserve">continebit vterque eorum {4/5}. </s> <s xml:id="echoid-s4334" xml:space="preserve">vnius recti. <lb/></s> <s xml:id="echoid-s4335" xml:space="preserve"> <anchor type="note" xlink:label="note-126-05a" xlink:href="note-126-05"/> Et quoniam angulus DFC, angulo DCF, eſt æqualis, quòd & </s> <s xml:id="echoid-s4336" xml:space="preserve">rectę DF, DC, <lb/> <anchor type="note" xlink:label="note-126-06a" xlink:href="note-126-06"/> æquales ſint; </s> <s xml:id="echoid-s4337" xml:space="preserve">(cum enim DE, EF, latera trianguli DEF, æqualia ſint lateri-<lb/>bus DE, EC, trianguli DEC, angulosq́ue ad E, contineant æquales, vtpo-<lb/>te rectos; </s> <s xml:id="echoid-s4338" xml:space="preserve">æquales erunt baſes DF, DC,) continebit quoque angulus DFC, <lb/> <anchor type="note" xlink:label="note-126-07a" xlink:href="note-126-07"/> {4/5}. </s> <s xml:id="echoid-s4339" xml:space="preserve">vnius recti; </s> <s xml:id="echoid-s4340" xml:space="preserve">ac proinde reliquus angulus DFA, ex duobus rectis, hoc eſt, ex <lb/>{10/5}. </s> <s xml:id="echoid-s4341" xml:space="preserve">vnius recti, continebit {6/5}. </s> <s xml:id="echoid-s4342" xml:space="preserve">vnius recti. </s> <s xml:id="echoid-s4343" xml:space="preserve">Cum ergo angulus DAF, oſtenſus <lb/>ſit continere {2/5}. </s> <s xml:id="echoid-s4344" xml:space="preserve">vnius recti, & </s> <s xml:id="echoid-s4345" xml:space="preserve">omnes tres anguli in triangulo AFD, conti-<lb/> <anchor type="note" xlink:label="note-126-08a" xlink:href="note-126-08"/> neant {10/5}. </s> <s xml:id="echoid-s4346" xml:space="preserve">vnius recti, continebit angulus ADF, {2/5}. </s> <s xml:id="echoid-s4347" xml:space="preserve">vnius recti, propte-<lb/>reaq́ue angulo DAF, æqualis erit. </s> <s xml:id="echoid-s4348" xml:space="preserve">Quare æqualia erunt latera DF, AF. <lb/></s> <s xml:id="echoid-s4349" xml:space="preserve"> <anchor type="note" xlink:label="note-126-09a" xlink:href="note-126-09"/> Cum ergo recta DF, rectæ DC, oſtenſa ſit æqualis, erit & </s> <s xml:id="echoid-s4350" xml:space="preserve">recta AF, rectæ DC, <lb/>æqualis: </s> <s xml:id="echoid-s4351" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s4352" xml:space="preserve">k F, medietas ipſius AF, ipſi CI, medietati ipſius DC, <lb/>æqualis erit.</s> <s xml:id="echoid-s4353" xml:space="preserve"/> </p> <div xml:id="echoid-div363" type="float" level="2" n="1"> <figure xlink:label="fig-126-01" xlink:href="fig-126-01a"> <image file="126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/126-01"/> </figure> <note position="left" xlink:label="note-126-01" xlink:href="note-126-01a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-126-02" xlink:href="note-126-02a" xml:space="preserve">34. primi.</note> <note position="left" xlink:label="note-126-03" xlink:href="note-126-03a" xml:space="preserve">33. fexti.</note> <note position="left" xlink:label="note-126-04" xlink:href="note-126-04a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-126-05" xlink:href="note-126-05a" xml:space="preserve">5.primi.</note> <note position="left" xlink:label="note-126-06" xlink:href="note-126-06a" xml:space="preserve">5.primi.</note> <note position="left" xlink:label="note-126-07" xlink:href="note-126-07a" xml:space="preserve">4. primi.</note> <note position="left" xlink:label="note-126-08" xlink:href="note-126-08a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-126-09" xlink:href="note-126-09a" xml:space="preserve">6. primi.</note> </div> <p> <s xml:id="echoid-s4354" xml:space="preserve">RVRSVS quoniam AK, KF, æquales ſunt; </s> <s xml:id="echoid-s4355" xml:space="preserve">additis æqualibus EC, FE, <lb/>erit recta compoſita ex Ak, EC, æqualis rectæ KE: </s> <s xml:id="echoid-s4356" xml:space="preserve">ac proinde KE, medie-<lb/>tas erit ſemidiametri AC; </s> <s xml:id="echoid-s4357" xml:space="preserve">quandoquidem AC, diuiſa eſt in duas partes æqua <lb/>les, quarum vna eſt KE, altera vero, recta ex AK, EC, compoſita. </s> <s xml:id="echoid-s4358" xml:space="preserve">Eſt igi-<lb/>tur KE, æqualis ipſi CG. </s> <s xml:id="echoid-s4359" xml:space="preserve">Ablata ergo communi recta GE, remanebunt <lb/>æquales GK, EC. </s> <s xml:id="echoid-s4360" xml:space="preserve">Eſt autem EC, ſumpta ipſi EF, æqualis. </s> <s xml:id="echoid-s4361" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s4362" xml:space="preserve"><lb/>GK, ipſi EF, æqualis erit; </s> <s xml:id="echoid-s4363" xml:space="preserve">additaque communi recta FG, erit EG, ipſi FK, <lb/>æqualis, hoc eſt, ipſi CI, cui oſtendimus ſupra rectam k F, eſſe æqualem. </s> <s xml:id="echoid-s4364" xml:space="preserve">Com- <pb o="115" file="127" n="127" rhead=""/> ponitur ergo AE, (quæ ſinui DM, arcus grad. </s> <s xml:id="echoid-s4365" xml:space="preserve">54. </s> <s xml:id="echoid-s4366" xml:space="preserve">æqualis eſt.) </s> <s xml:id="echoid-s4367" xml:space="preserve">ex AG, me-<lb/>dietate ſinus totius, & </s> <s xml:id="echoid-s4368" xml:space="preserve">GE, quæ æqualis eſt oſtenſa ſinui CI, arcus grad. </s> <s xml:id="echoid-s4369" xml:space="preserve">18. <lb/></s> <s xml:id="echoid-s4370" xml:space="preserve">Quod eſt primum.</s> <s xml:id="echoid-s4371" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4372" xml:space="preserve">IAM vero, quoniam KF, ipſi EG; </s> <s xml:id="echoid-s4373" xml:space="preserve">& </s> <s xml:id="echoid-s4374" xml:space="preserve">EG, ipſi CI, oſtenſa eſt æqualis: <lb/></s> <s xml:id="echoid-s4375" xml:space="preserve">erit, KF, ſinui recto CI, æqualis: </s> <s xml:id="echoid-s4376" xml:space="preserve">Eſt autem KF, ipſi AK, æqualis. </s> <s xml:id="echoid-s4377" xml:space="preserve">Igitur <lb/>erit quoque AK, ipſi CI, æqualis. </s> <s xml:id="echoid-s4378" xml:space="preserve">Cum ergo AK, ſinui LN, ſit æqualis, erit <lb/> <anchor type="note" xlink:label="note-127-01a" xlink:href="note-127-01"/> etiam ſinus LN, ſinui CI, æqualis. </s> <s xml:id="echoid-s4379" xml:space="preserve">Eſt autem CI, ſinus arcus grad. </s> <s xml:id="echoid-s4380" xml:space="preserve">18. </s> <s xml:id="echoid-s4381" xml:space="preserve">Igitur <lb/>& </s> <s xml:id="echoid-s4382" xml:space="preserve">LN, ſinus erit arcus grad. </s> <s xml:id="echoid-s4383" xml:space="preserve">18. </s> <s xml:id="echoid-s4384" xml:space="preserve">ac proinde arcus BL, cuius ſinus eſt LN, <lb/>continebit grad. </s> <s xml:id="echoid-s4385" xml:space="preserve">18. </s> <s xml:id="echoid-s4386" xml:space="preserve">ideoq́ue eius complementum CL; </s> <s xml:id="echoid-s4387" xml:space="preserve">continebit grad. </s> <s xml:id="echoid-s4388" xml:space="preserve">72. <lb/></s> <s xml:id="echoid-s4389" xml:space="preserve">cuius ſinus verſus KC, cõponitur ex CG, medietate ſinus totius, & </s> <s xml:id="echoid-s4390" xml:space="preserve">ex GK, <lb/>quæ ſinui verſo EC, arcus CD, grad. </s> <s xml:id="echoid-s4391" xml:space="preserve">36. </s> <s xml:id="echoid-s4392" xml:space="preserve">oſtenſa eſt æqualis. </s> <s xml:id="echoid-s4393" xml:space="preserve">Quod eſt ſecun-<lb/>dum. </s> <s xml:id="echoid-s4394" xml:space="preserve">Itaque Sinus rectus arcus graduum 54. </s> <s xml:id="echoid-s4395" xml:space="preserve">componitur, &</s> <s xml:id="echoid-s4396" xml:space="preserve">c. </s> <s xml:id="echoid-s4397" xml:space="preserve">Quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s4398" xml:space="preserve"/> </p> <div xml:id="echoid-div364" type="float" level="2" n="2"> <note position="right" xlink:label="note-127-01" xlink:href="note-127-01a" xml:space="preserve">34. primi.</note> </div> </div> <div xml:id="echoid-div366" type="section" level="1" n="160"> <head xml:id="echoid-head186" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s4399" xml:space="preserve">CONSTAT cx his, triangulum ACD, cuius baſis CD, ſubtẽdit gradus 36. </s> <s xml:id="echoid-s4400" xml:space="preserve">verticemq́; <lb/></s> <s xml:id="echoid-s4401" xml:space="preserve">habet in centro, eſſe Iſoſceles, cuius vterque æqualium angulorum C, D, reliqui anguli ad <lb/>centrum duplus eſt. </s> <s xml:id="echoid-s4402" xml:space="preserve">Nam angulus CAD, oſtenſus eſt continere {2/5}. </s> <s xml:id="echoid-s4403" xml:space="preserve">vnius recti, vtrumque <lb/>vero C, & </s> <s xml:id="echoid-s4404" xml:space="preserve">D, {4/5}.</s> <s xml:id="echoid-s4405" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div367" type="section" level="1" n="161"> <head xml:id="echoid-head187" xml:space="preserve">THEOR. 4. PROPOS. 6.</head> <p> <s xml:id="echoid-s4406" xml:space="preserve">DIFFERENTIA chordarũ duorum arcuũ <lb/> <anchor type="note" xlink:label="note-127-02a" xlink:href="note-127-02"/> ſemicirculi, quorum alter tãto minor ſit arcu grad. <lb/></s> <s xml:id="echoid-s4407" xml:space="preserve">120. </s> <s xml:id="echoid-s4408" xml:space="preserve">quanto alter maior eſt, æqualis eſt chordæ ar-<lb/>cus, quo alteruter dictorum arcuum ab arcu grad. </s> <s xml:id="echoid-s4409" xml:space="preserve"><lb/>120. </s> <s xml:id="echoid-s4410" xml:space="preserve">differt.</s> <s xml:id="echoid-s4411" xml:space="preserve"/> </p> <div xml:id="echoid-div367" type="float" level="2" n="1"> <note position="right" xlink:label="note-127-02" xlink:href="note-127-02a" xml:space="preserve">Differentia <lb/>inter chor-<lb/>das duorũ <lb/>arcuũ, quo-<lb/>rũ alter tá <lb/>to ſit mi-<lb/>nor arcu <lb/>grad. 120. <lb/>quáto alter <lb/>maior eſt, <lb/>ęquat chot <lb/>dæ arcus, <lb/>quo alteru-<lb/>ter dictorũ <lb/>arcuũ dif-<lb/>fert ab arcu <lb/>grad. 120.</note> </div> <p> <s xml:id="echoid-s4412" xml:space="preserve">IN ſemicirculo ABC, ſit arcus BA, grad. </s> <s xml:id="echoid-s4413" xml:space="preserve">120. </s> <s xml:id="echoid-s4414" xml:space="preserve">arcus vero BD, eo tan-<lb/>to minor, quanto arcus BE, maior eſt; </s> <s xml:id="echoid-s4415" xml:space="preserve">quorum chordæ BD, BE: </s> <s xml:id="echoid-s4416" xml:space="preserve">abſcin-<lb/>daturq́ue BF, ipſi BD, æqualis, & </s> <s xml:id="echoid-s4417" xml:space="preserve">iungantur rectæ AD, AE, AF. </s> <s xml:id="echoid-s4418" xml:space="preserve">Dico EF, <lb/>differentiã duarũ chor <lb/> <anchor type="figure" xlink:label="fig-127-01a" xlink:href="fig-127-01"/> darum BD, BE, æqua-<lb/>lem eſſe chordæ AE, <lb/>vel AD. </s> <s xml:id="echoid-s4419" xml:space="preserve">Cõpleto enim <lb/>circulo, & </s> <s xml:id="echoid-s4420" xml:space="preserve">inſcriptotriã <lb/>gulo æquilatero ABG, <lb/>cuius vnum latus eſt <lb/>AB, chorda arcus grad. <lb/></s> <s xml:id="echoid-s4421" xml:space="preserve">120. </s> <s xml:id="echoid-s4422" xml:space="preserve">cum ſubtendat ter <lb/>tiã circunferentiæ par-<lb/>tem; </s> <s xml:id="echoid-s4423" xml:space="preserve">erit angulus AGB, <lb/>tertia pars duorum re-<lb/> <anchor type="note" xlink:label="note-127-03a" xlink:href="note-127-03"/> ctorum. </s> <s xml:id="echoid-s4424" xml:space="preserve">Cum ergo ei æqualis ſit angulus AEB, in eodem cum illo exiſtens <pb o="116" file="128" n="128" rhead=""/> ſegmento AGB; </s> <s xml:id="echoid-s4425" xml:space="preserve">erit & </s> <s xml:id="echoid-s4426" xml:space="preserve">AEB, tertia pars duorum rectorum. </s> <s xml:id="echoid-s4427" xml:space="preserve">Deinde, quo-<lb/>niam latera DB, BA, trianguli DBA, lateribus FB, BA, trianguli FBA, <lb/>æqualia ſunt, angulosq́ue continent æquales; </s> <s xml:id="echoid-s4428" xml:space="preserve">erunt baſes AD, AF, inter ſe <lb/> <anchor type="note" xlink:label="note-128-01a" xlink:href="note-128-01"/> æquales. </s> <s xml:id="echoid-s4429" xml:space="preserve">Cum ergo AD, ipſi AE, æqualis ſit, propter æquales arcus AD, <lb/> <anchor type="note" xlink:label="note-128-02a" xlink:href="note-128-02"/> AE; </s> <s xml:id="echoid-s4430" xml:space="preserve">erit & </s> <s xml:id="echoid-s4431" xml:space="preserve">AF, eidem AE, æqualis; </s> <s xml:id="echoid-s4432" xml:space="preserve">ac propterea anguli AEF, AFE, æqua <lb/> <anchor type="note" xlink:label="note-128-03a" xlink:href="note-128-03"/> les inter ſe erunt: </s> <s xml:id="echoid-s4433" xml:space="preserve">Eſt autem AEF, vt oſtendimus, tertia pars duorum recto-<lb/>rum. </s> <s xml:id="echoid-s4434" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s4435" xml:space="preserve">AFE, tertia pars erit duorum rectorum; </s> <s xml:id="echoid-s4436" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s4437" xml:space="preserve">reliquus <lb/>EAF, tertia pars erit duorum rectorum. </s> <s xml:id="echoid-s4438" xml:space="preserve">Quare triangulum AEF, æquilate-<lb/> <anchor type="note" xlink:label="note-128-04a" xlink:href="note-128-04"/> rum erit, ex coroll. </s> <s xml:id="echoid-s4439" xml:space="preserve">propof. </s> <s xml:id="echoid-s4440" xml:space="preserve">6. </s> <s xml:id="echoid-s4441" xml:space="preserve">lib. </s> <s xml:id="echoid-s4442" xml:space="preserve">1. </s> <s xml:id="echoid-s4443" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s4444" xml:space="preserve">ideoque recta EF, differentia chorda-<lb/>rum BD, BE, chordæ AE, vel AD, æqualis erit. </s> <s xml:id="echoid-s4445" xml:space="preserve">Differentia ergo chorda-<lb/>rum duorum arcuum ſemicirculi, &</s> <s xml:id="echoid-s4446" xml:space="preserve">c. </s> <s xml:id="echoid-s4447" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s4448" xml:space="preserve"/> </p> <div xml:id="echoid-div368" type="float" level="2" n="2"> <figure xlink:label="fig-127-01" xlink:href="fig-127-01a"> <image file="127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/127-01"/> </figure> <note position="right" xlink:label="note-127-03" xlink:href="note-127-03a" xml:space="preserve">32. primi. <lb/>21. tertij.</note> <note position="left" xlink:label="note-128-01" xlink:href="note-128-01a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-128-02" xlink:href="note-128-02a" xml:space="preserve">29.tertij.</note> <note position="left" xlink:label="note-128-03" xlink:href="note-128-03a" xml:space="preserve">5.primi.</note> <note position="left" xlink:label="note-128-04" xlink:href="note-128-04a" xml:space="preserve">32. primi.</note> </div> </div> <div xml:id="echoid-div370" type="section" level="1" n="162"> <head xml:id="echoid-head188" xml:space="preserve">COROLLARIVM.</head> <note position="left" xml:space="preserve">Duæ chor-<lb/>dę duorum <lb/>arcuũ cõſi-<lb/>ciẽtiũ gra. <lb/>120. ſimul <lb/>ęquales sũt <lb/>chordęarcꝰ <lb/>cõpoſiti ex <lb/>arcu grad. <lb/>120. & arcu <lb/>minore il-<lb/>lorum duo <lb/>rum.</note> <p> <s xml:id="echoid-s4449" xml:space="preserve">SEQVITVR hinc, ſi duorum arcuum, qui ſimul grad. </s> <s xml:id="echoid-s4450" xml:space="preserve">120. </s> <s xml:id="echoid-s4451" xml:space="preserve">conficiant, chordæ ſimul <lb/>iungantur, effici chordam arcus compoſiti ex arcu grad. </s> <s xml:id="echoid-s4452" xml:space="preserve">120, & </s> <s xml:id="echoid-s4453" xml:space="preserve">arcu minore illorum duo-<lb/>rum, ſi in æquales ſint. </s> <s xml:id="echoid-s4454" xml:space="preserve">Ita namque vides chordas BD, DA, arcuum BD, DA, conſicientium <lb/>grad. </s> <s xml:id="echoid-s4455" xml:space="preserve">120. </s> <s xml:id="echoid-s4456" xml:space="preserve">ſimul ſumptas æquari chordæ BE, arcus BAE, compoſiti ex arcu BA, grad. </s> <s xml:id="echoid-s4457" xml:space="preserve">120. <lb/></s> <s xml:id="echoid-s4458" xml:space="preserve">& </s> <s xml:id="echoid-s4459" xml:space="preserve">arcu AE, qui minori AD, æqualis eſt: </s> <s xml:id="echoid-s4460" xml:space="preserve">propterea quòd vt demonſtratum eſt, differentia <lb/>EF, inter choidas BD, BE, æqualis eſt chordæ AD.</s> <s xml:id="echoid-s4461" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div371" type="section" level="1" n="163"> <head xml:id="echoid-head189" xml:space="preserve">THEOR. 5. PROPOS. 7.</head> <p> <s xml:id="echoid-s4462" xml:space="preserve">SI quantitas quantitatem excedat, ſemiſsis il-<lb/> <anchor type="note" xlink:label="note-128-06a" xlink:href="note-128-06"/> lius ſemiſsem huius ſuperabit exceſſus ſemiſſe.</s> <s xml:id="echoid-s4463" xml:space="preserve"/> </p> <div xml:id="echoid-div371" type="float" level="2" n="1"> <note position="left" xlink:label="note-128-06" xlink:href="note-128-06a" xml:space="preserve">Si quãtitas <lb/>ſuꝑ & quã-<lb/>titaté ſemiſ <lb/>ſis ſemiſs ẽ <lb/>ſuperabit <lb/>exceſſus ſe <lb/>miſſe.</note> </div> <p> <s xml:id="echoid-s4464" xml:space="preserve">SVPERET quantitas AB, quantitatẽ C, exceſſu DB, qui bifariã ſecetur in <lb/> <anchor type="figure" xlink:label="fig-128-01a" xlink:href="fig-128-01"/> E, & </s> <s xml:id="echoid-s4465" xml:space="preserve">ipſi EB, æqualis pona <lb/>tur AF. </s> <s xml:id="echoid-s4466" xml:space="preserve">Quoniã igitur AF, <lb/>EB, toti exceſſui DB, æ-<lb/>quales ſunt, erit reliqua <lb/>FE, ipſi C, æqualis. </s> <s xml:id="echoid-s4467" xml:space="preserve">Sece-<lb/>tur FE, bifaria in G. </s> <s xml:id="echoid-s4468" xml:space="preserve">Quia <lb/>ergo GE, GF, æquales <lb/>sũt; </s> <s xml:id="echoid-s4469" xml:space="preserve">additis æqualibus EB, <lb/>FA, æquales quoque erũt <lb/>GB, GA; </s> <s xml:id="echoid-s4470" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s4471" xml:space="preserve">AB, <lb/>in G, ſecta erit bifariã. </s> <s xml:id="echoid-s4472" xml:space="preserve">Se-<lb/>miſsis igitur BG, ipſius <lb/>AB, ſuperat GE, ſemiſſem <lb/>ipſius FE, hoc eſt, ipſius C, exceſſu EB, qui ſem iſsis eſt exceſſus DB. </s> <s xml:id="echoid-s4473" xml:space="preserve">Si quan-<lb/>titas ergo quantitatem excedat, &</s> <s xml:id="echoid-s4474" xml:space="preserve">c. </s> <s xml:id="echoid-s4475" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s4476" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="2" n="2"> <figure xlink:label="fig-128-01" xlink:href="fig-128-01a"> <image file="128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/128-01"/> </figure> </div> </div> <div xml:id="echoid-div374" type="section" level="1" n="164"> <head xml:id="echoid-head190" xml:space="preserve">THEOR. 6. PROPOS. 8.</head> <p> <s xml:id="echoid-s4477" xml:space="preserve">DIFFERENTIA ſinuum duorum arcuũ <pb o="117" file="129" n="129" rhead=""/> quadrantis, quorum alter tanto minor ſit arcu <lb/> <anchor type="note" xlink:label="note-129-01a" xlink:href="note-129-01"/> grad. </s> <s xml:id="echoid-s4478" xml:space="preserve">60. </s> <s xml:id="echoid-s4479" xml:space="preserve">quanto alter maior eſt, æqualis eſt ſinui <lb/>arcus, quo alteruter dictorũ arcuum ab arcu grad. <lb/></s> <s xml:id="echoid-s4480" xml:space="preserve">60. </s> <s xml:id="echoid-s4481" xml:space="preserve">differt.</s> <s xml:id="echoid-s4482" xml:space="preserve"/> </p> <div xml:id="echoid-div374" type="float" level="2" n="1"> <note position="right" xlink:label="note-129-01" xlink:href="note-129-01a" xml:space="preserve">Differentia <lb/>inter ſinus <lb/>duorũ ar-<lb/>cuum, quo-<lb/>rum alter <lb/>tanto ſit <lb/>minorarcu <lb/>grad. 60. <lb/>quãto alter <lb/>maior eſt, <lb/>ęqua@ ſinui <lb/>arcus, quo <lb/>alteruter di <lb/>ctorũar@@ũ <lb/>differt ab <lb/>a@cu grad. <lb/>60.</note> </div> <p> <s xml:id="echoid-s4483" xml:space="preserve">IN quadrante ABC, ſit arcus CD, grad. </s> <s xml:id="echoid-s4484" xml:space="preserve">60. </s> <s xml:id="echoid-s4485" xml:space="preserve">arcus vero CE, eo tanto <lb/>minor, quanto arcus CF, maior eſt: </s> <s xml:id="echoid-s4486" xml:space="preserve">quorum ſinus recti EG, FH. </s> <s xml:id="echoid-s4487" xml:space="preserve">Dico horum <lb/>ſinuum differẽtiam æqualem eſſe ſinui arcus DE, vel DF. </s> <s xml:id="echoid-s4488" xml:space="preserve">Producto enim qua-<lb/>drante, vna cum ſinubus EG, FH, ad I, K; </s> <s xml:id="echoid-s4489" xml:space="preserve">ſumatur arcus CL, arcui CD, <lb/>æqualis, ita vt totus arcus <lb/> <anchor type="figure" xlink:label="fig-129-01a" xlink:href="fig-129-01"/> DCL, cõtineat grad. </s> <s xml:id="echoid-s4490" xml:space="preserve">120. <lb/></s> <s xml:id="echoid-s4491" xml:space="preserve">Et quia & </s> <s xml:id="echoid-s4492" xml:space="preserve">arcus CI, CK, <lb/>æquales ſunt arcubus CE, <lb/>CF; </s> <s xml:id="echoid-s4493" xml:space="preserve">quòd per lemma in <lb/>definitionibus poſitum re-<lb/>cta BC, ſecet arcus ECI, <lb/>FCK, bifariam, cum & </s> <s xml:id="echoid-s4494" xml:space="preserve">re-<lb/>ctas EI, FK, bifariam ſe-<lb/> <anchor type="note" xlink:label="note-129-02a" xlink:href="note-129-02"/> cet: </s> <s xml:id="echoid-s4495" xml:space="preserve">erunt quoque reliqui <lb/>arcus LI, LK, reliquis ar <lb/>cubus DE, DF, æquales. <lb/></s> <s xml:id="echoid-s4496" xml:space="preserve">Sumptis quoque arcubus <lb/>EM, FN, qui arcubus DE, <lb/>DF, æquales ſint, ducan-<lb/>tur chordæ LM, LN. </s> <s xml:id="echoid-s4497" xml:space="preserve">Et <lb/>quoniam arcus FN, arcui <lb/>LK, & </s> <s xml:id="echoid-s4498" xml:space="preserve">arcus EM, arcui <lb/>LI, ęqualis eſt; </s> <s xml:id="echoid-s4499" xml:space="preserve">additis com <lb/>munibus FL, MI, erit tam arcus NL, arcui FK, quàm arcus ML, arcui EI, <lb/>æqualis: </s> <s xml:id="echoid-s4500" xml:space="preserve">ac proinde tan chorda NL, chordæ FK, quam chorda ML, chordæ <lb/> <anchor type="note" xlink:label="note-129-03a" xlink:href="note-129-03"/> EI, æqualis. </s> <s xml:id="echoid-s4501" xml:space="preserve">Quoniam igitur arcus LM, tanto minor eſt arcu LD, grad. </s> <s xml:id="echoid-s4502" xml:space="preserve">120. <lb/></s> <s xml:id="echoid-s4503" xml:space="preserve">quanto arcus LN, eodem maior eſt; </s> <s xml:id="echoid-s4504" xml:space="preserve">erit per propoſ. </s> <s xml:id="echoid-s4505" xml:space="preserve">6. </s> <s xml:id="echoid-s4506" xml:space="preserve">differentia chor-<lb/>darum LN, LM, chordæ DN, vel DM, æqualis; </s> <s xml:id="echoid-s4507" xml:space="preserve">hoc eſt, recta k F, rectam <lb/>IE, ſuperabit chorda DN, vel DM. </s> <s xml:id="echoid-s4508" xml:space="preserve">Quare per antecedentem propoſ. </s> <s xml:id="echoid-s4509" xml:space="preserve">ſemiſ-<lb/>ſis HF, hoc eſt, ſinus arcus CF, ſuperabit ſemiſſem GE, id eſt, ſinum arcus CE, <lb/>ſemiſſe chordæ DN, vel DM, hoc eſt, ſinu arcus DF, vel DE. </s> <s xml:id="echoid-s4510" xml:space="preserve">Quod de-<lb/>monſtrandum erat.</s> <s xml:id="echoid-s4511" xml:space="preserve"/> </p> <div xml:id="echoid-div375" type="float" level="2" n="2"> <figure xlink:label="fig-129-01" xlink:href="fig-129-01a"> <image file="129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/129-01"/> </figure> <note position="right" xlink:label="note-129-02" xlink:href="note-129-02a" xml:space="preserve">3. tertij.</note> <note position="right" xlink:label="note-129-03" xlink:href="note-129-03a" xml:space="preserve">29. tertij.</note> </div> <p> <s xml:id="echoid-s4512" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s4513" xml:space="preserve">In quadrante ABD, arcus BE, ſit grad. </s> <s xml:id="echoid-s4514" xml:space="preserve">60. </s> <s xml:id="echoid-s4515" xml:space="preserve">& </s> <s xml:id="echoid-s4516" xml:space="preserve">arcus EF, <lb/>EG, æquales, ac proinde arcus BF, tanto minor arcu BE, quanto arcus BG, <lb/>eodem arcu BE, maior eſt: </s> <s xml:id="echoid-s4517" xml:space="preserve">ducanturq́ue FH, GI, ad BD, perpendiculares, <lb/>quæ ſinus erunt arcuum BF, BG. </s> <s xml:id="echoid-s4518" xml:space="preserve">Ducta autem chorda FG, ſecet eam ſemi-<lb/>diameter ducta DE, in L. </s> <s xml:id="echoid-s4519" xml:space="preserve">Et quoniam arcus FG, bifariam ſectus eſt in E, erit <lb/>quoque recta FG, bifariam ſecta in L, ex lemmate in definitionibus demon-<lb/>ſtrato; </s> <s xml:id="echoid-s4520" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s4521" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s4522" xml:space="preserve">Eſt ergo FL, ſinus arcus EF; </s> <s xml:id="echoid-s4523" xml:space="preserve">& </s> <s xml:id="echoid-s4524" xml:space="preserve">GL, <lb/> <anchor type="note" xlink:label="note-129-04a" xlink:href="note-129-04"/> ſinus arcus EG. </s> <s xml:id="echoid-s4525" xml:space="preserve">Ducta quoque recta FK, ad GI, perpendiculari, erit IK, re- <pb o="118" file="130" n="130" rhead=""/> ctæ FH, æqualis, ob parallelogrammum FI. </s> <s xml:id="echoid-s4526" xml:space="preserve">Quare GK, differentia erit ſi-<lb/> <anchor type="note" xlink:label="note-130-01a" xlink:href="note-130-01"/> nuum FH, GI. </s> <s xml:id="echoid-s4527" xml:space="preserve">Dico hanc differentiam GK, æqualem eſſe ſinui FL, vel GL. <lb/></s> <s xml:id="echoid-s4528" xml:space="preserve">Ducta enim recta BE, quæ latus hexagoni eſt, ac propterea, ex coroll propoſ. </s> <s xml:id="echoid-s4529" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-130-01a" xlink:href="fig-130-01"/> 15. </s> <s xml:id="echoid-s4530" xml:space="preserve">lib. </s> <s xml:id="echoid-s4531" xml:space="preserve">4. </s> <s xml:id="echoid-s4532" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s4533" xml:space="preserve">ſemidiametro DE, æqualis; </s> <s xml:id="echoid-s4534" xml:space="preserve">ſecetur <lb/>BD, bifariã in M, iungaturq́ recta EM. </s> <s xml:id="echoid-s4535" xml:space="preserve">Quoniã igi-<lb/>tur latera DM, ME, lateribus BM, ME, æqualia <lb/>ſunt, & </s> <s xml:id="echoid-s4536" xml:space="preserve">baſis DE, baſi BE, æqualis, erunt anguli ad <lb/> <anchor type="note" xlink:label="note-130-02a" xlink:href="note-130-02"/> M, ęquales, atque adeo recti. </s> <s xml:id="echoid-s4537" xml:space="preserve">Completo autem ſe-<lb/>micirculo ABC, & </s> <s xml:id="echoid-s4538" xml:space="preserve">productis rectis GI, EM, ad <lb/>N, O, erit arcus NO, arcui GE, hoc eſt, arcui EF, <lb/>æqualis, ex ſcholio propoſ. </s> <s xml:id="echoid-s4539" xml:space="preserve">27. </s> <s xml:id="echoid-s4540" xml:space="preserve">li b. </s> <s xml:id="echoid-s4541" xml:space="preserve">3. </s> <s xml:id="echoid-s4542" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s4543" xml:space="preserve">propterea <lb/>quod rectæ GN, EO, parallelæ ſunt, ob rectos angu <lb/> <anchor type="note" xlink:label="note-130-03a" xlink:href="note-130-03"/> los I, M. </s> <s xml:id="echoid-s4544" xml:space="preserve">Addito ergo communi arcu FO, erit ar-<lb/>cus FN, arcui EO, æqualis: </s> <s xml:id="echoid-s4545" xml:space="preserve">Sed arcus EO, duplus <lb/>eſt arcus BE. </s> <s xml:id="echoid-s4546" xml:space="preserve">(Nam recta DB, rectam EO, ſecans ad <lb/> <anchor type="note" xlink:label="note-130-04a" xlink:href="note-130-04"/> angulos rectos ſecat eandem bifariam: </s> <s xml:id="echoid-s4547" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s4548" xml:space="preserve"><lb/>arcum EO, bifariam, ex ſcholio in definitionibus <lb/>poſito) Igitur & </s> <s xml:id="echoid-s4549" xml:space="preserve">arcus FN, eiuſdem arcus BE, du-<lb/>plus erit. </s> <s xml:id="echoid-s4550" xml:space="preserve">Quare ductis rectis DF, DN, erit quoque <lb/>angulus FDN, anguli EDB, duplus: </s> <s xml:id="echoid-s4551" xml:space="preserve">Eſt autem idem angulus FDN, in cen <lb/> <anchor type="note" xlink:label="note-130-05a" xlink:href="note-130-05"/> tro anguli FGN, in circunferentia duplus. </s> <s xml:id="echoid-s4552" xml:space="preserve">Igitur æquales ſunt anguli EDM, <lb/> <anchor type="note" xlink:label="note-130-06a" xlink:href="note-130-06"/> FGK: </s> <s xml:id="echoid-s4553" xml:space="preserve">Suntautem & </s> <s xml:id="echoid-s4554" xml:space="preserve">recti M, K, æquales. </s> <s xml:id="echoid-s4555" xml:space="preserve">Aequiangula ergo ſunt triangula <lb/>EDM, <emph style="sc">F</emph>GK: </s> <s xml:id="echoid-s4556" xml:space="preserve">atque idcirco erit vt ED, ad DM, ita FG, ad GK. </s> <s xml:id="echoid-s4557" xml:space="preserve">Cum ergo <lb/> <anchor type="note" xlink:label="note-130-07a" xlink:href="note-130-07"/> ED, dupla ſit ipſius DM, (ſecta enim eſt DB, ipſi DE, æqualis, bifariam in <lb/>M.) </s> <s xml:id="echoid-s4558" xml:space="preserve">erit & </s> <s xml:id="echoid-s4559" xml:space="preserve">FG, ipſius GK, dupla: </s> <s xml:id="echoid-s4560" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s4561" xml:space="preserve">FG, ipſius FL, vel GL, du-<lb/>pla. </s> <s xml:id="echoid-s4562" xml:space="preserve">Igitur recta GK, differentia ſinuum FH, GI, æqualis eſt rectæ FL, ſi-<lb/>nui arcus EF, vel rectæ GL, ſinui arcus EG. </s> <s xml:id="echoid-s4563" xml:space="preserve">Differentia ergo ſinuum duo-<lb/>rum arcuum quadrantis, &</s> <s xml:id="echoid-s4564" xml:space="preserve">c. </s> <s xml:id="echoid-s4565" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s4566" xml:space="preserve"/> </p> <div xml:id="echoid-div376" type="float" level="2" n="3"> <note position="right" xlink:label="note-129-04" xlink:href="note-129-04a" xml:space="preserve">3. terttj.</note> <note position="left" xlink:label="note-130-01" xlink:href="note-130-01a" xml:space="preserve">34. primi.</note> <figure xlink:label="fig-130-01" xlink:href="fig-130-01a"> <image file="130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/130-01"/> </figure> <note position="left" xlink:label="note-130-02" xlink:href="note-130-02a" xml:space="preserve">8. primi.</note> <note position="left" xlink:label="note-130-03" xlink:href="note-130-03a" xml:space="preserve">28. primi.</note> <note position="left" xlink:label="note-130-04" xlink:href="note-130-04a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-130-05" xlink:href="note-130-05a" xml:space="preserve">33. ſexti.</note> <note position="left" xlink:label="note-130-06" xlink:href="note-130-06a" xml:space="preserve">20.tertij.</note> <note position="left" xlink:label="note-130-07" xlink:href="note-130-07a" xml:space="preserve">4. ſexti.</note> </div> </div> <div xml:id="echoid-div378" type="section" level="1" n="165"> <head xml:id="echoid-head191" xml:space="preserve">COROLLARIVM.</head> <note position="left" xml:space="preserve">Duo ſinus <lb/>duorum ar <lb/>cuũ confi-<lb/>cientium <lb/>grad. 60. ſi-<lb/>mul æqua-<lb/>les ſunt ſi-<lb/>nui arcus <lb/>compoſiti <lb/>ex arcu <lb/>grad. 60. & <lb/>arcu mino <lb/>re illorum <lb/>duorum.</note> <p> <s xml:id="echoid-s4567" xml:space="preserve">HINC ſequitur, ſi duorum arcuum conficientium grad. </s> <s xml:id="echoid-s4568" xml:space="preserve">60. </s> <s xml:id="echoid-s4569" xml:space="preserve">ſinus ſimul componantur, <lb/>effici ſinum arcus cõpoſiti ex arcu grad. </s> <s xml:id="echoid-s4570" xml:space="preserve">60. </s> <s xml:id="echoid-s4571" xml:space="preserve">& </s> <s xml:id="echoid-s4572" xml:space="preserve">arcu minore illorũ duorũ, ſi inæquales ſunt. <lb/></s> <s xml:id="echoid-s4573" xml:space="preserve">Ita enim vides in ſigura poſterioris demonſtrationis huius propoſ. </s> <s xml:id="echoid-s4574" xml:space="preserve">ſinus rectos FH, FL, ar-<lb/>cuum BF, FE, conſicientium grad. </s> <s xml:id="echoid-s4575" xml:space="preserve">60. </s> <s xml:id="echoid-s4576" xml:space="preserve">ſimul ſumptos æquari ſinui recto GI, arcus BEG, <lb/>compoſiti ex arcu BE, grad. </s> <s xml:id="echoid-s4577" xml:space="preserve">60. </s> <s xml:id="echoid-s4578" xml:space="preserve">& </s> <s xml:id="echoid-s4579" xml:space="preserve">arcu EG, qui minori EF, æqualis eſt: </s> <s xml:id="echoid-s4580" xml:space="preserve">propterea quòd, <lb/>vt demonſtratum eſt, differentia GK, inter ſinus FH, GI, æqualis eſt ſinui FL.</s> <s xml:id="echoid-s4581" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div379" type="section" level="1" n="166"> <head xml:id="echoid-head192" xml:space="preserve">PROBL. 3. PROP. 9.</head> <p> <s xml:id="echoid-s4582" xml:space="preserve">SINVS rectos omnium arcuum quadrantis <lb/> <anchor type="note" xlink:label="note-130-09a" xlink:href="note-130-09"/> ſeſe ordine ſuperantium vno Minuto, in partibus <lb/>Sinus totius in quotcunque particulas diſſributi, <lb/>ſupputare.</s> <s xml:id="echoid-s4583" xml:space="preserve"/> </p> <div xml:id="echoid-div379" type="float" level="2" n="1"> <note position="left" xlink:label="note-130-09" xlink:href="note-130-09a" xml:space="preserve">Qua rõne <lb/>omniũ ar-<lb/>cuum ſinus <lb/>recti ſuppu <lb/>tentur.</note> </div> <p> <s xml:id="echoid-s4584" xml:space="preserve">PRIMVM omnium ſupputabimus ſinus rectos arcuum ſeſe 15. </s> <s xml:id="echoid-s4585" xml:space="preserve">gradibus <pb o="119" file="131" n="131" rhead=""/> ſuperantium, reſpectu Sinus totius particularum 100000. </s> <s xml:id="echoid-s4586" xml:space="preserve">quotnimirum com <lb/>muniter ab omnibus, & </s> <s xml:id="echoid-s4587" xml:space="preserve">ànobis etiam conſtituitur, vt ſupra diximus. </s> <s xml:id="echoid-s4588" xml:space="preserve">Vt auté <lb/>accuratior ſiat ſupputatio, ponemus in hiſce ſupputationibus Sinum totum <lb/> <anchor type="note" xlink:label="note-131-01a" xlink:href="note-131-01"/> partium 10000000. </s> <s xml:id="echoid-s4589" xml:space="preserve">Ita enim fiet, vt abiectis duabus primis figuris ad dexte-<lb/>rá ex ſingulis ſinubus inuentis, (addita tamen vnitate, ſi duæ figuræ abiectæ nu <lb/>merũ 50. </s> <s xml:id="echoid-s4590" xml:space="preserve">ſuperent) relinquãtur ſinus magis exquiſiti reſpectu ſinus totius par <lb/>tium 100000. </s> <s xml:id="echoid-s4591" xml:space="preserve">Quòd ſi quis ſinus deſideret plurium particularum, poſito ni-<lb/>mirũ ſinu toto partiũ 10000000. </s> <s xml:id="echoid-s4592" xml:space="preserve">quot eum Ioan. </s> <s xml:id="echoid-s4593" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s4594" xml:space="preserve">poſuit, & </s> <s xml:id="echoid-s4595" xml:space="preserve">nos in ſe <lb/>quẽti tabula ſtatuemus, conſtituendus erit Sinus totus partiũ 1000000000. <lb/></s> <s xml:id="echoid-s4596" xml:space="preserve">Nam hac ratione, abiectis duabus primis figuris ad dexteram ex ſingulis ſinu-<lb/>bus inuentis, vt dictum eſt, remanebunt Sinus magis exquiſiti reſpectu Sinus <lb/>totius partium 10000000. </s> <s xml:id="echoid-s4597" xml:space="preserve">Ratio huius rei eſt, quòd in ſinuum inueſtigatio-<lb/>ne error ſolum contingere poteſt in vna aut altera figura ad dextram: </s> <s xml:id="echoid-s4598" xml:space="preserve">quare, <lb/>abiectis duabus figuris ad dextram, relinquentur ſinus reſpectu ſinus totius <lb/>minoris exquiſrtiſsimi. </s> <s xml:id="echoid-s4599" xml:space="preserve">Id quod in ſupputationibus ſinuum quiuis facile ex pe-<lb/>rietur, & </s> <s xml:id="echoid-s4600" xml:space="preserve">nos infra demonſtrabimus. </s> <s xml:id="echoid-s4601" xml:space="preserve">Pari ratione, ſi inueſtigandi ſint ſinus re-<lb/>ſpectu alterius Sinus totius, qui plures, aut pauciores particulas contineat, <lb/>quàm 100000. </s> <s xml:id="echoid-s4602" xml:space="preserve">vel 10000000. </s> <s xml:id="echoid-s4603" xml:space="preserve">conſtituendus erit in ſupputatione Sinus to-<lb/>tus, qui ad dexteram illum ſuperet duabus figuris his, 00; </s> <s xml:id="echoid-s4604" xml:space="preserve">adeo vt illius ſit <lb/>centuplus, quemadmodum & </s> <s xml:id="echoid-s4605" xml:space="preserve">hic 10000000. </s> <s xml:id="echoid-s4606" xml:space="preserve">quem in calculo aſſumimus, cen-<lb/>tuplus eſt illius 100000. </s> <s xml:id="echoid-s4607" xml:space="preserve">quem nos cum alijs Aſtronomis in vſu recipimus.</s> <s xml:id="echoid-s4608" xml:space="preserve"/> </p> <div xml:id="echoid-div380" type="float" level="2" n="2"> <note position="right" xlink:label="note-131-01" xlink:href="note-131-01a" xml:space="preserve">Sinus totꝰ <lb/>in cuiꝰ par <lb/>tibꝰ alij ſi-<lb/>nus ompu <lb/>tátur, quoe <lb/>particula <lb/>rú ponédus <lb/>ſit, vt alij <lb/>ſinus inue-<lb/>niátur ma-<lb/>gis exquiſi <lb/>ti reſp ectu <lb/>Sinus to-<lb/>tius pattiũ <lb/>100000. vel <lb/>pauciotum <lb/>pluriúve.</note> </div> <p> <s xml:id="echoid-s4609" xml:space="preserve">SIT igitur in quadrante ABC, arcus CD, grad. </s> <s xml:id="echoid-s4610" xml:space="preserve">15. </s> <s xml:id="echoid-s4611" xml:space="preserve">CE, 30. </s> <s xml:id="echoid-s4612" xml:space="preserve">CF, 45. </s> <s xml:id="echoid-s4613" xml:space="preserve">ac <lb/>proinde EB, grad. </s> <s xml:id="echoid-s4614" xml:space="preserve">60. </s> <s xml:id="echoid-s4615" xml:space="preserve">& </s> <s xml:id="echoid-s4616" xml:space="preserve">DB, 75. </s> <s xml:id="echoid-s4617" xml:space="preserve">vtpote complementa arcuum grad. </s> <s xml:id="echoid-s4618" xml:space="preserve">30. </s> <s xml:id="echoid-s4619" xml:space="preserve">& </s> <s xml:id="echoid-s4620" xml:space="preserve">15. <lb/></s> <s xml:id="echoid-s4621" xml:space="preserve">Horum ergo arcuum ſinus rectos ita ſupputabimus. </s> <s xml:id="echoid-s4622" xml:space="preserve">Ducantur EH, DI, ad AB, <lb/>perpendiculares, quæ ſinus recti erunt arcuum grad. </s> <s xml:id="echoid-s4623" xml:space="preserve">60. </s> <s xml:id="echoid-s4624" xml:space="preserve">& </s> <s xml:id="echoid-s4625" xml:space="preserve">grad. </s> <s xml:id="echoid-s4626" xml:space="preserve">75. </s> <s xml:id="echoid-s4627" xml:space="preserve">Ductam <lb/>autem chordam BC, ſecet recta AF, in L, bi-<lb/> <anchor type="figure" xlink:label="fig-131-01a" xlink:href="fig-131-01"/> fariam, ex lemmate in definitionibus demon-<lb/>ſtrato; </s> <s xml:id="echoid-s4628" xml:space="preserve">ac proinde ad angulos rectos: </s> <s xml:id="echoid-s4629" xml:space="preserve">eritq̀ue <lb/> <anchor type="note" xlink:label="note-131-02a" xlink:href="note-131-02"/> BL, ſinus rectus grad. </s> <s xml:id="echoid-s4630" xml:space="preserve">45. </s> <s xml:id="echoid-s4631" xml:space="preserve">hoc eſt, arcus BF. <lb/></s> <s xml:id="echoid-s4632" xml:space="preserve">Ducatur rurſus EG, ad AC, perpendicula-<lb/>ris pro ſinu grad. </s> <s xml:id="echoid-s4633" xml:space="preserve">30. </s> <s xml:id="echoid-s4634" xml:space="preserve">Item ductam chordã CE, <lb/> <anchor type="note" xlink:label="note-131-03a" xlink:href="note-131-03"/> ſecet recta AD, in K, bifariam, ex dicto lem-<lb/>mate; </s> <s xml:id="echoid-s4635" xml:space="preserve">ac propterea ad angulos rectos; </s> <s xml:id="echoid-s4636" xml:space="preserve">eritq́ue <lb/>CK, ſinus rectus grad. </s> <s xml:id="echoid-s4637" xml:space="preserve">15. </s> <s xml:id="echoid-s4638" xml:space="preserve">Denique rectæ iun-<lb/>gantur AE, EB. </s> <s xml:id="echoid-s4639" xml:space="preserve">Quoniam igitur arcus BE, <lb/>grad. </s> <s xml:id="echoid-s4640" xml:space="preserve">60. </s> <s xml:id="echoid-s4641" xml:space="preserve">ſexta pars eſt totius circunferentiæ <lb/>circuli, cum ſexies 60. </s> <s xml:id="echoid-s4642" xml:space="preserve">ſaciant 360. </s> <s xml:id="echoid-s4643" xml:space="preserve">grad. </s> <s xml:id="echoid-s4644" xml:space="preserve">erit <lb/>recta BE, latus Hexagoni; </s> <s xml:id="echoid-s4645" xml:space="preserve">atque adeo, ex <lb/>coroll. </s> <s xml:id="echoid-s4646" xml:space="preserve">propof. </s> <s xml:id="echoid-s4647" xml:space="preserve">15. </s> <s xml:id="echoid-s4648" xml:space="preserve">lib. </s> <s xml:id="echoid-s4649" xml:space="preserve">4. </s> <s xml:id="echoid-s4650" xml:space="preserve">Euel. </s> <s xml:id="echoid-s4651" xml:space="preserve">ſemidiametro AE, æqualis. </s> <s xml:id="echoid-s4652" xml:space="preserve">Anguliergo EAB, <lb/>EBA, æquales erunt: </s> <s xml:id="echoid-s4653" xml:space="preserve">Sunt autem & </s> <s xml:id="echoid-s4654" xml:space="preserve">anguli ad H, æquales, vtpote recti. </s> <s xml:id="echoid-s4655" xml:space="preserve">Igi-<lb/> <anchor type="note" xlink:label="note-131-04a" xlink:href="note-131-04"/> tur cum duo anguli EAH, EHA, trianguli AEH, æquales ſint duobus an-<lb/>gulis EBH, EHB, trianguli BEH, latusq́ue AE, lateri BE, æquale; </s> <s xml:id="echoid-s4656" xml:space="preserve">erit la-<lb/>tus AH, lateri BH, æquale; </s> <s xml:id="echoid-s4657" xml:space="preserve">ac proinde AH, medietas erit ſemidia metri AB. <lb/></s> <s xml:id="echoid-s4658" xml:space="preserve"> <anchor type="note" xlink:label="note-131-05a" xlink:href="note-131-05"/> Quare cum EG, ſinus rectus grad. </s> <s xml:id="echoid-s4659" xml:space="preserve">30. </s> <s xml:id="echoid-s4660" xml:space="preserve">ſit ipſi AH, æqualis, erit ſinus rectus <lb/> <anchor type="note" xlink:label="note-131-06a" xlink:href="note-131-06"/> grad. </s> <s xml:id="echoid-s4661" xml:space="preserve">30. </s> <s xml:id="echoid-s4662" xml:space="preserve">medietati ſemidiametri, ſiue ſinus totius, æqualis. </s> <s xml:id="echoid-s4663" xml:space="preserve">Cum ergo ſinus <lb/> <anchor type="note" xlink:label="note-131-07a" xlink:href="note-131-07"/> totus ponatur 10000000. </s> <s xml:id="echoid-s4664" xml:space="preserve">erit EG, ſinus grad. </s> <s xml:id="echoid-s4665" xml:space="preserve">30. </s> <s xml:id="echoid-s4666" xml:space="preserve">talium particularum <lb/>5000000. </s> <s xml:id="echoid-s4667" xml:space="preserve">nempe medietas ſinus totius.</s> <s xml:id="echoid-s4668" xml:space="preserve"/> </p> <div xml:id="echoid-div381" type="float" level="2" n="3"> <figure xlink:label="fig-131-01" xlink:href="fig-131-01a"> <image file="131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/131-01"/> </figure> <note position="right" xlink:label="note-131-02" xlink:href="note-131-02a" xml:space="preserve">3. tertij.</note> <note position="right" xlink:label="note-131-03" xlink:href="note-131-03a" xml:space="preserve">Supputatio <lb/>ſinuum ar <lb/>cuũ gradi-<lb/>bus 15. ſeſe <lb/>ſuperantiũ <lb/>quales ſunt <lb/>arcus grad. <lb/>15. 30. 45. <lb/>60. 75. & <lb/>90.</note> <note position="right" xlink:label="note-131-04" xlink:href="note-131-04a" xml:space="preserve">5. primi.</note> <note position="right" xlink:label="note-131-05" xlink:href="note-131-05a" xml:space="preserve">26. primi.</note> <note position="right" xlink:label="note-131-06" xlink:href="note-131-06a" xml:space="preserve">34. primi.</note> <note position="right" xlink:label="note-131-07" xlink:href="note-131-07a" xml:space="preserve">Sinus rectꝰ <lb/>grad. 30. æ-<lb/>qualis eſt <lb/>medietati <lb/>ſinus totiꝰ;</note> </div> <p> <s xml:id="echoid-s4669" xml:space="preserve">EX hoc ſinu, per propof. </s> <s xml:id="echoid-s4670" xml:space="preserve">3. </s> <s xml:id="echoid-s4671" xml:space="preserve">cognoſcetur ſinus complementi arcus grad. </s> <s xml:id="echoid-s4672" xml:space="preserve">30.</s> <s xml:id="echoid-s4673" xml:space="preserve"> <pb o="120" file="132" n="132" rhead=""/> nempe ſinus EH, grad. </s> <s xml:id="echoid-s4674" xml:space="preserve">60. </s> <s xml:id="echoid-s4675" xml:space="preserve">ſi nimirum quadratũ ſinus 5000000. </s> <s xml:id="echoid-s4676" xml:space="preserve">ex quadrato <lb/>ſinus totius 10000000. </s> <s xml:id="echoid-s4677" xml:space="preserve">auferatur, & </s> <s xml:id="echoid-s4678" xml:space="preserve">reliqui numeri radix quadrata accipia-<lb/>tur, quę eſt 8660254. </s> <s xml:id="echoid-s4679" xml:space="preserve">fere.</s> <s xml:id="echoid-s4680" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4681" xml:space="preserve">DEINDE, quoniam recta AF, fecans <lb/> <anchor type="figure" xlink:label="fig-132-01a" xlink:href="fig-132-01"/> arcum BC, bifariam, fecat quoque, ex lem-<lb/>mate definitionum, rectam BC, bifariam, at-<lb/>que adeo & </s> <s xml:id="echoid-s4682" xml:space="preserve">ad angulos rectos; </s> <s xml:id="echoid-s4683" xml:space="preserve">erit CL, ſinus <lb/> <anchor type="note" xlink:label="note-132-01a" xlink:href="note-132-01"/> arcus CF, grad. </s> <s xml:id="echoid-s4684" xml:space="preserve">45. </s> <s xml:id="echoid-s4685" xml:space="preserve">quem ita inueniemus. <lb/></s> <s xml:id="echoid-s4686" xml:space="preserve">Cum in triangulo CAL, angulus L, rectus <lb/>ſit, & </s> <s xml:id="echoid-s4687" xml:space="preserve">angulus CAL, ſemirectus, erit & </s> <s xml:id="echoid-s4688" xml:space="preserve">angu-<lb/> <anchor type="note" xlink:label="note-132-02a" xlink:href="note-132-02"/> lus ACL, ſemirectus, atque adeo angulo <lb/>CAL, æqualis. </s> <s xml:id="echoid-s4689" xml:space="preserve">Igitur rectæ AL, CL, æqua-<lb/> <anchor type="note" xlink:label="note-132-03a" xlink:href="note-132-03"/> les erunt. </s> <s xml:id="echoid-s4690" xml:space="preserve">Cum ergo quadratum rectæ AC, <lb/>æquale ſit quadratis rectarum AL, CL, ſi-<lb/> <anchor type="note" xlink:label="note-132-04a" xlink:href="note-132-04"/> mul; </s> <s xml:id="echoid-s4691" xml:space="preserve">erit quadratum ſinus totius AC, du-<lb/>plum quadrati ſinus CL, grad. </s> <s xml:id="echoid-s4692" xml:space="preserve">45. </s> <s xml:id="echoid-s4693" xml:space="preserve">Medietas <lb/>igitur quadrati ſinus totius erit quadratũ rectæ CL, cuius radix quadrata da-<lb/>bit ſinum CL, 7071068. </s> <s xml:id="echoid-s4694" xml:space="preserve">fere pro arcu grad. </s> <s xml:id="echoid-s4695" xml:space="preserve">45. </s> <s xml:id="echoid-s4696" xml:space="preserve">Qui etiam hoc modo re-<lb/>perietur. </s> <s xml:id="echoid-s4697" xml:space="preserve">Quoniam quadratum rectæ BC, æquale eſt quadratis rectarum AB, <lb/> <anchor type="note" xlink:label="note-132-05a" xlink:href="note-132-05"/> AC; </s> <s xml:id="echoid-s4698" xml:space="preserve">atque adeo duplum quadrati ſinus totius AC, ſi quadratum ſinus to-<lb/>tius duplicetur, habebitur quadratum rectæ BC, cuius quadrati radix dabit re <lb/>ctam BC, partium 14142136. </s> <s xml:id="echoid-s4699" xml:space="preserve">fere, & </s> <s xml:id="echoid-s4700" xml:space="preserve">huius dimidium 7071068. </s> <s xml:id="echoid-s4701" xml:space="preserve">dabit ſinum <lb/>CL, grad. </s> <s xml:id="echoid-s4702" xml:space="preserve">45.</s> <s xml:id="echoid-s4703" xml:space="preserve"/> </p> <div xml:id="echoid-div382" type="float" level="2" n="4"> <figure xlink:label="fig-132-01" xlink:href="fig-132-01a"> <image file="132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/132-01"/> </figure> <note position="left" xlink:label="note-132-01" xlink:href="note-132-01a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-132-02" xlink:href="note-132-02a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-132-03" xlink:href="note-132-03a" xml:space="preserve">6 primi.</note> <note position="left" xlink:label="note-132-04" xlink:href="note-132-04a" xml:space="preserve">47. primi</note> <note position="left" xlink:label="note-132-05" xlink:href="note-132-05a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s4704" xml:space="preserve">RVRSVS, quia recta AD, ſecans arcum CE, bifariam, ſecat quoque <lb/>rectam CE, bifariam in K, ex lemmate definitionum, atque adeo & </s> <s xml:id="echoid-s4705" xml:space="preserve">ad angu-<lb/> <anchor type="note" xlink:label="note-132-06a" xlink:href="note-132-06"/> los rectos; </s> <s xml:id="echoid-s4706" xml:space="preserve">erit CK, ſinus arcus CD, grad. </s> <s xml:id="echoid-s4707" xml:space="preserve">15. </s> <s xml:id="echoid-s4708" xml:space="preserve">quem ſic inueniemus. </s> <s xml:id="echoid-s4709" xml:space="preserve">Quoniã <lb/>ex propoſ. </s> <s xml:id="echoid-s4710" xml:space="preserve">4. </s> <s xml:id="echoid-s4711" xml:space="preserve">ſinus CK, medio loco proportionalis eſt inter medietatem ſi-<lb/>nus totius, & </s> <s xml:id="echoid-s4712" xml:space="preserve">finum verfum CG; </s> <s xml:id="echoid-s4713" xml:space="preserve">(qui quidem habetur, ſi EH, ſinus grad. </s> <s xml:id="echoid-s4714" xml:space="preserve">60. <lb/></s> <s xml:id="echoid-s4715" xml:space="preserve">ex ſinu toto AC, detrahatur, vt in coroll. </s> <s xml:id="echoid-s4716" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s4717" xml:space="preserve">3. </s> <s xml:id="echoid-s4718" xml:space="preserve">diximus) ſit vt, per co-<lb/>roll. </s> <s xml:id="echoid-s4719" xml:space="preserve">propof. </s> <s xml:id="echoid-s4720" xml:space="preserve">4. </s> <s xml:id="echoid-s4721" xml:space="preserve">notus ſiat ſinus CK, arcus CD, qui dimidium eſt arcus CE. </s> <s xml:id="echoid-s4722" xml:space="preserve"><lb/>Nam ſi medietas ſinus totius multiplicetur in ſinum verſum CG, cognitum, <lb/>producetur quadratum rectæ CK; </s> <s xml:id="echoid-s4723" xml:space="preserve">quòd rectangulum ſub medietate ſinus to-<lb/>tius, & </s> <s xml:id="echoid-s4724" xml:space="preserve">ſinu verſo CG, contentum, æquale ſit quadrato mediæ proportiona-<lb/> <anchor type="note" xlink:label="note-132-07a" xlink:href="note-132-07"/> lis Ck. </s> <s xml:id="echoid-s4725" xml:space="preserve">Si igitur quadrati rectæ CK, radix eruatur, habebitur ſinus CK, par-<lb/>tium 2588190. </s> <s xml:id="echoid-s4726" xml:space="preserve">vt in dicto coroll. </s> <s xml:id="echoid-s4727" xml:space="preserve">propof. </s> <s xml:id="echoid-s4728" xml:space="preserve">4. </s> <s xml:id="echoid-s4729" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s4730" xml:space="preserve">Quem ſinum hoc etiam <lb/> <anchor type="note" xlink:label="note-132-08a" xlink:href="note-132-08"/> modo inueſtigabimus. </s> <s xml:id="echoid-s4731" xml:space="preserve">Quoniam quadratis rectarum notarum EG, GC, æqua <lb/> <anchor type="note" xlink:label="note-132-09a" xlink:href="note-132-09"/> le eſt quadratum rectæ EC; </s> <s xml:id="echoid-s4732" xml:space="preserve">fiet notum quadratum <lb/>rectæ EC; </s> <s xml:id="echoid-s4733" xml:space="preserve">cuius quadrati radix quadrata dabit re-<lb/>ctam EC, notam, & </s> <s xml:id="echoid-s4734" xml:space="preserve">huius medietas erit ſinus CK, <lb/>cognitus.</s> <s xml:id="echoid-s4735" xml:space="preserve"/> </p> <div xml:id="echoid-div383" type="float" level="2" n="5"> <note position="left" xlink:label="note-132-06" xlink:href="note-132-06a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-132-07" xlink:href="note-132-07a" xml:space="preserve">17. ſexti.</note> <note position="left" xlink:label="note-132-08" xlink:href="note-132-08a" xml:space="preserve">47. primi.</note> <note position="right" xlink:label="note-132-09" xlink:href="note-132-09a" xml:space="preserve"> <lb/>Arcus. # Sinus <lb/>G. <lb/>15 # 2588190 <lb/>30 # 5000000 <lb/>45 # 7071068 <lb/>60 # 8660254 <lb/>75 # 9659258 <lb/>90 # 10000000 <lb/></note> </div> <p> <s xml:id="echoid-s4736" xml:space="preserve">EX ſinu autem grad. </s> <s xml:id="echoid-s4737" xml:space="preserve">15. </s> <s xml:id="echoid-s4738" xml:space="preserve">cognito cognofcetur <lb/>quoque, per propof. </s> <s xml:id="echoid-s4739" xml:space="preserve">3. </s> <s xml:id="echoid-s4740" xml:space="preserve">ſinus DI, complementi arcus <lb/>CD, hoc eſt, ſinus arcus BD, grad. </s> <s xml:id="echoid-s4741" xml:space="preserve">75. </s> <s xml:id="echoid-s4742" xml:space="preserve">qui quidem <lb/>deprehendetur partium 9659258. </s> <s xml:id="echoid-s4743" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s4744" xml:space="preserve">inuenti ſunt <lb/>hactenus ſinus recti arcuum continentium grad. <lb/></s> <s xml:id="echoid-s4745" xml:space="preserve">15. </s> <s xml:id="echoid-s4746" xml:space="preserve">30. </s> <s xml:id="echoid-s4747" xml:space="preserve">45. </s> <s xml:id="echoid-s4748" xml:space="preserve">60. </s> <s xml:id="echoid-s4749" xml:space="preserve">75. </s> <s xml:id="echoid-s4750" xml:space="preserve">& </s> <s xml:id="echoid-s4751" xml:space="preserve">90. </s> <s xml:id="echoid-s4752" xml:space="preserve">vt in hac formula apparet.</s> <s xml:id="echoid-s4753" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4754" xml:space="preserve">HORVM autem ſinuũ beneficio ad aliorum <lb/>inueſtigationem ita progrediemur. </s> <s xml:id="echoid-s4755" xml:space="preserve">Quoniam per <pb o="121" file="133" n="133" rhead=""/> coroll. </s> <s xml:id="echoid-s4756" xml:space="preserve">propof. </s> <s xml:id="echoid-s4757" xml:space="preserve">4. </s> <s xml:id="echoid-s4758" xml:space="preserve">ex ſinu recto cuiuſuis arcus noto cognoſcitur quoque ſinus <lb/>rectus dimidij illius arous; </s> <s xml:id="echoid-s4759" xml:space="preserve">cognofcemus ex ſinu a rcus grad. </s> <s xml:id="echoid-s4760" xml:space="preserve">15. </s> <s xml:id="echoid-s4761" xml:space="preserve">ſinum arcus <lb/>grad. </s> <s xml:id="echoid-s4762" xml:space="preserve">7. </s> <s xml:id="echoid-s4763" xml:space="preserve">Min. </s> <s xml:id="echoid-s4764" xml:space="preserve">30. </s> <s xml:id="echoid-s4765" xml:space="preserve">Atque ex hoc ſinum arcus grad. </s> <s xml:id="echoid-s4766" xml:space="preserve">3. </s> <s xml:id="echoid-s4767" xml:space="preserve">Min. </s> <s xml:id="echoid-s4768" xml:space="preserve">45. </s> <s xml:id="echoid-s4769" xml:space="preserve">qui arcus amplius <lb/>bifariam ſecari non poteſt ſine Secundis, (continet enim eius medietas grad. </s> <s xml:id="echoid-s4770" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s4771" xml:space="preserve"> <anchor type="note" xlink:label="note-133-01a" xlink:href="note-133-01"/> Min. </s> <s xml:id="echoid-s4772" xml:space="preserve">52. </s> <s xml:id="echoid-s4773" xml:space="preserve">Sec. </s> <s xml:id="echoid-s4774" xml:space="preserve">30.) </s> <s xml:id="echoid-s4775" xml:space="preserve">quæ in Sinuum tractatione negliguntur. </s> <s xml:id="echoid-s4776" xml:space="preserve">Deinde quia per pro <lb/>poſ. </s> <s xml:id="echoid-s4777" xml:space="preserve">3. </s> <s xml:id="echoid-s4778" xml:space="preserve">ex ſinu recto cuiuſlibet arcus cognito notus quoque efficitur ſinus <lb/>complementi illius arcus, cognoſcemus ex ſinu arcus grad. </s> <s xml:id="echoid-s4779" xml:space="preserve">7. </s> <s xml:id="echoid-s4780" xml:space="preserve">Min. </s> <s xml:id="echoid-s4781" xml:space="preserve">30. </s> <s xml:id="echoid-s4782" xml:space="preserve">ſi-<lb/>num arcus grad. </s> <s xml:id="echoid-s4783" xml:space="preserve">82. </s> <s xml:id="echoid-s4784" xml:space="preserve">Min. </s> <s xml:id="echoid-s4785" xml:space="preserve">30. </s> <s xml:id="echoid-s4786" xml:space="preserve">Ex hoc autem, per coroll. </s> <s xml:id="echoid-s4787" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s4788" xml:space="preserve">4. </s> <s xml:id="echoid-s4789" xml:space="preserve">ſinum ar-<lb/>cus grad. </s> <s xml:id="echoid-s4790" xml:space="preserve">41. </s> <s xml:id="echoid-s4791" xml:space="preserve">Min. </s> <s xml:id="echoid-s4792" xml:space="preserve">15. </s> <s xml:id="echoid-s4793" xml:space="preserve">Atque ex hoc, per propof. </s> <s xml:id="echoid-s4794" xml:space="preserve">3. </s> <s xml:id="echoid-s4795" xml:space="preserve">ſinum arcus grad. </s> <s xml:id="echoid-s4796" xml:space="preserve">48. </s> <s xml:id="echoid-s4797" xml:space="preserve">Min. <lb/></s> <s xml:id="echoid-s4798" xml:space="preserve">45. </s> <s xml:id="echoid-s4799" xml:space="preserve">Item ex ſinu arcus gra. </s> <s xml:id="echoid-s4800" xml:space="preserve">3. </s> <s xml:id="echoid-s4801" xml:space="preserve">Min. </s> <s xml:id="echoid-s4802" xml:space="preserve">45. </s> <s xml:id="echoid-s4803" xml:space="preserve">cognoſcemus, per propoſ. </s> <s xml:id="echoid-s4804" xml:space="preserve">3. </s> <s xml:id="echoid-s4805" xml:space="preserve">ſinum arcus <lb/>grad. </s> <s xml:id="echoid-s4806" xml:space="preserve">86. </s> <s xml:id="echoid-s4807" xml:space="preserve">Min. </s> <s xml:id="echoid-s4808" xml:space="preserve">15. </s> <s xml:id="echoid-s4809" xml:space="preserve">Quòd ſi alij arcus, quorum ſinus in uenti funt, bifariam quo <lb/>que fecentur, & </s> <s xml:id="echoid-s4810" xml:space="preserve">eorum medietates rurſus bifariam, & </s> <s xml:id="echoid-s4811" xml:space="preserve">ita deinceps, donec ad <lb/>Minuta numero imparia, quæ amplius bifariã diuidi ſine Secundis nequeunt, <lb/>peruentum ſit; </s> <s xml:id="echoid-s4812" xml:space="preserve">Itemque harum medietatum complementa accipiantur, quæ <lb/>rurſus, eodem modo continue bifariam ſecẽtur, donec ad Minuta numero im-<lb/>paria ſit peruentum; </s> <s xml:id="echoid-s4813" xml:space="preserve">& </s> <s xml:id="echoid-s4814" xml:space="preserve">medietatum complementa ſumantur, &</s> <s xml:id="echoid-s4815" xml:space="preserve">c. </s> <s xml:id="echoid-s4816" xml:space="preserve">cognoſce-<lb/>mus, per coroll. </s> <s xml:id="echoid-s4817" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s4818" xml:space="preserve">4. </s> <s xml:id="echoid-s4819" xml:space="preserve">& </s> <s xml:id="echoid-s4820" xml:space="preserve">per propof. </s> <s xml:id="echoid-s4821" xml:space="preserve">3. </s> <s xml:id="echoid-s4822" xml:space="preserve">ſinus omnium harum medieta-<lb/>tum, & </s> <s xml:id="echoid-s4823" xml:space="preserve">complementorum. </s> <s xml:id="echoid-s4824" xml:space="preserve">Qui ſinus cum illis ſex primò inuentis conſtituent <lb/>ſinus 24. </s> <s xml:id="echoid-s4825" xml:space="preserve">arcuum fefe ordine ſuperantium gradibus 3. </s> <s xml:id="echoid-s4826" xml:space="preserve">Min. </s> <s xml:id="echoid-s4827" xml:space="preserve">45. </s> <s xml:id="echoid-s4828" xml:space="preserve">vt in hac ta-<lb/>bula vides.</s> <s xml:id="echoid-s4829" xml:space="preserve"/> </p> <div xml:id="echoid-div384" type="float" level="2" n="6"> <note position="right" xlink:label="note-133-01" xlink:href="note-133-01a" xml:space="preserve">Supputatia <lb/>ſinuum ac-<lb/>cuum gta-<lb/>dibus 3. <lb/>Min. 45. ſe-<lb/>ſe ſupetan-<lb/>tium.</note> </div> <note position="right" xml:space="preserve"> <lb/> ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus <lb/>G # M # # G # M # # G # M # # G # M <lb/>3 # 45 # 654031 # 26 # 15 # 4422887 # 48 # 45 # 7518398 # 71 # 15 # 9469301 <lb/>7 # 30 # 1305262 # 30 # 0 # 5000000 # 52 # 30 # 7933533 # 75 # 0 # 9659258 <lb/>11 # 15 # 1950903 # 33 # 45 # 5555702 # 56 # 15 # 8314696 # 78 # 45 # 9807853 <lb/>15 # 0 # 2588190 # 37 # 30 # 6087614 # 60 # 0 # 8660254 # 82 # 30 # 9914449 <lb/>18 # 45 # 3214395 # 41 # 15 # 6593458 # 63 # 45 # 8968727 # 86 # 15 # 9978589 <lb/>22 # 30 # 3826834 # 45 # 0 # 7071068 # 67 # 30 # 9238795 # 90 # 0 # 10000000 <lb/></note> <p> <s xml:id="echoid-s4830" xml:space="preserve">POST hæc, per ea, quæ propof. </s> <s xml:id="echoid-s4831" xml:space="preserve">2. </s> <s xml:id="echoid-s4832" xml:space="preserve">de inuentione lateris Decagoni, & </s> <s xml:id="echoid-s4833" xml:space="preserve">pen-<lb/> <anchor type="note" xlink:label="note-133-03a" xlink:href="note-133-03"/> tagoniæquilateri in eodem circulo demon ſtrauimus, inquiremus ſinum arcus <lb/>grad. </s> <s xml:id="echoid-s4834" xml:space="preserve">36. </s> <s xml:id="echoid-s4835" xml:space="preserve">hac ratione. </s> <s xml:id="echoid-s4836" xml:space="preserve">Repetatur figura propoſitionis 2. </s> <s xml:id="echoid-s4837" xml:space="preserve">vbi demonſtratum eſt, <lb/>DF, eſſe latus Decagoni, & </s> <s xml:id="echoid-s4838" xml:space="preserve">BF, latus <lb/> <anchor type="figure" xlink:label="fig-133-01a" xlink:href="fig-133-01"/> Pentagoni ęquilateri. </s> <s xml:id="echoid-s4839" xml:space="preserve">Et quoniam qua-<lb/>drata rectarum BD, DE, notarum (Eſt <lb/>enim BD, ſinus totus, & </s> <s xml:id="echoid-s4840" xml:space="preserve">DE, eius me-<lb/>dietas.) </s> <s xml:id="echoid-s4841" xml:space="preserve">nota ſunt, & </s> <s xml:id="echoid-s4842" xml:space="preserve">æqualia quadrato <lb/> <anchor type="note" xlink:label="note-133-04a" xlink:href="note-133-04"/> rectæ EB; </s> <s xml:id="echoid-s4843" xml:space="preserve">notum erit quadratum rectæ <lb/>EB; </s> <s xml:id="echoid-s4844" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s4845" xml:space="preserve">ipſa recta EB, hoc <lb/>eſt, recta EF, illi æqualis, nota erit, nem <lb/>pe partium fere 11180339. </s> <s xml:id="echoid-s4846" xml:space="preserve">Ex qua ſi <lb/>detrahatur recta ED, medietas ſinus to <lb/>tius partiũ 5000000. </s> <s xml:id="echoid-s4847" xml:space="preserve">nota fiet recta DF, <lb/>partium 6180339. </s> <s xml:id="echoid-s4848" xml:space="preserve">cuius quadratum ſi addatur quadrato ſinus totius BD, no <lb/>tum fiet aggregatum quadratorum ex rectis DF, BD, deſcriptorum; </s> <s xml:id="echoid-s4849" xml:space="preserve">atque <pb o="122" file="134" n="134" rhead=""/> adeo & </s> <s xml:id="echoid-s4850" xml:space="preserve">quadratum rectæ BF, quod illis duobus æquale eſt, cognitum erit; </s> <s xml:id="echoid-s4851" xml:space="preserve">cu-<lb/> <anchor type="note" xlink:label="note-134-01a" xlink:href="note-134-01"/> ius radix quadrata notam reddet rectam BF: </s> <s xml:id="echoid-s4852" xml:space="preserve">quæ cũ fubtendat grad. </s> <s xml:id="echoid-s4853" xml:space="preserve">72. </s> <s xml:id="echoid-s4854" xml:space="preserve">in cir <lb/>culo ABC, vtpote quintam partem circunferentiæ, quòd ſit latus pentago-<lb/>ni, nota erit chorda arcus grad. </s> <s xml:id="echoid-s4855" xml:space="preserve">72. </s> <s xml:id="echoid-s4856" xml:space="preserve">cuius medietas partium 5877852. </s> <s xml:id="echoid-s4857" xml:space="preserve">dabit ſi-<lb/>num rectum arcus grad. </s> <s xml:id="echoid-s4858" xml:space="preserve">36. </s> <s xml:id="echoid-s4859" xml:space="preserve">qui illius dimidium eſt.</s> <s xml:id="echoid-s4860" xml:space="preserve"/> </p> <div xml:id="echoid-div385" type="float" level="2" n="7"> <note position="right" xlink:label="note-133-03" xlink:href="note-133-03a" xml:space="preserve">Supputatio <lb/>ſinus arcus <lb/>grad. 36.</note> <figure xlink:label="fig-133-01" xlink:href="fig-133-01a"> <image file="133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/133-01"/> </figure> <note position="right" xlink:label="note-133-04" xlink:href="note-133-04a" xml:space="preserve">47. primi.</note> <note position="left" xlink:label="note-134-01" xlink:href="note-134-01a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s4861" xml:space="preserve">PORRO ex hoc ſinu arcus grad. </s> <s xml:id="echoid-s4862" xml:space="preserve">36. </s> <s xml:id="echoid-s4863" xml:space="preserve">inueniemus, per propof. </s> <s xml:id="echoid-s4864" xml:space="preserve">3. </s> <s xml:id="echoid-s4865" xml:space="preserve">ſinum eius <lb/> <anchor type="note" xlink:label="note-134-02a" xlink:href="note-134-02"/> complementi, hoc eſt, ſinum arcus grad. </s> <s xml:id="echoid-s4866" xml:space="preserve">54. </s> <s xml:id="echoid-s4867" xml:space="preserve">Quos duos arcus ſi biſariam fece-<lb/>mus, eorumq́ue medietates rurſus bifariam, & </s> <s xml:id="echoid-s4868" xml:space="preserve">ita deinceps, donec ad minuta <lb/>numero imparia veniamus, quæ amplius diuidi nequeunt, reperiemus quoque <lb/>per coroll. </s> <s xml:id="echoid-s4869" xml:space="preserve">propof. </s> <s xml:id="echoid-s4870" xml:space="preserve">4. </s> <s xml:id="echoid-s4871" xml:space="preserve">harum medietatum ſinus, nec non, per propoſ. </s> <s xml:id="echoid-s4872" xml:space="preserve">3. </s> <s xml:id="echoid-s4873" xml:space="preserve">ſinus ea-<lb/>rum complementorum: </s> <s xml:id="echoid-s4874" xml:space="preserve">quæ complementa ſi rurſus bifariam ſecemus, quoad <lb/>poſſumus, & </s> <s xml:id="echoid-s4875" xml:space="preserve">harum medietatum complementa accipiamus, quæ rurſus ſece-<lb/>mus bifariam, &</s> <s xml:id="echoid-s4876" xml:space="preserve">c. </s> <s xml:id="echoid-s4877" xml:space="preserve">inueniemus eodem modo ſinus omnium ar cuum in hac ta-<lb/>bella contentorum. </s> <s xml:id="echoid-s4878" xml:space="preserve">Quos ſinus ſi cum præcedentibus hactenus inuentis in or-<lb/> <anchor type="note" xlink:label="note-134-03a" xlink:href="note-134-03"/> dinem redigamus, habebimus ſinus rectos omnium arcuum ſeſe ordine ſupe-<lb/>rantium gradibus 2. </s> <s xml:id="echoid-s4879" xml:space="preserve">& </s> <s xml:id="echoid-s4880" xml:space="preserve">Min. </s> <s xml:id="echoid-s4881" xml:space="preserve">15. </s> <s xml:id="echoid-s4882" xml:space="preserve">initio facto ab arcu grad. </s> <s xml:id="echoid-s4883" xml:space="preserve">2. </s> <s xml:id="echoid-s4884" xml:space="preserve">Min. </s> <s xml:id="echoid-s4885" xml:space="preserve">15. </s> <s xml:id="echoid-s4886" xml:space="preserve">vſque ad <lb/>arcum grad. </s> <s xml:id="echoid-s4887" xml:space="preserve">87. </s> <s xml:id="echoid-s4888" xml:space="preserve">Min. </s> <s xml:id="echoid-s4889" xml:space="preserve">45. </s> <s xml:id="echoid-s4890" xml:space="preserve">incluſiue; </s> <s xml:id="echoid-s4891" xml:space="preserve">& </s> <s xml:id="echoid-s4892" xml:space="preserve">inſuper ſinus aliorum 17. </s> <s xml:id="echoid-s4893" xml:space="preserve">arcuum, qui <lb/>non ſeruant huiuſmodi incrementum. </s> <s xml:id="echoid-s4894" xml:space="preserve">Vt in hac tabella appparet. </s> <s xml:id="echoid-s4895" xml:space="preserve">Vbi manife <lb/> <anchor type="note" xlink:label="note-134-04a" xlink:href="note-134-04"/> <pb o="123" file="135" n="135" rhead=""/> <anchor type="note" xlink:label="note-135-01a" xlink:href="note-135-01"/> ſtum eſt, ſinum grad. </s> <s xml:id="echoid-s4896" xml:space="preserve">54. </s> <s xml:id="echoid-s4897" xml:space="preserve">hoc eſt. </s> <s xml:id="echoid-s4898" xml:space="preserve">8090170. </s> <s xml:id="echoid-s4899" xml:space="preserve">componi præciſe ex ſinubus grad. </s> <s xml:id="echoid-s4900" xml:space="preserve">18. <lb/></s> <s xml:id="echoid-s4901" xml:space="preserve">& </s> <s xml:id="echoid-s4902" xml:space="preserve">30. </s> <s xml:id="echoid-s4903" xml:space="preserve">hoc eſt, ex 3090170. </s> <s xml:id="echoid-s4904" xml:space="preserve">& </s> <s xml:id="echoid-s4905" xml:space="preserve">5000000. </s> <s xml:id="echoid-s4906" xml:space="preserve">Et quia ſinus grad. </s> <s xml:id="echoid-s4907" xml:space="preserve">30. </s> <s xml:id="echoid-s4908" xml:space="preserve">æqualis eſt ſemiſ-<lb/>ſi ſinus totius, vt ſupra oſtenſum eſt, liquet ſinum grad. </s> <s xml:id="echoid-s4909" xml:space="preserve">54. </s> <s xml:id="echoid-s4910" xml:space="preserve">componi ad vngué <lb/>ex ſemiſſe ſinus totius, & </s> <s xml:id="echoid-s4911" xml:space="preserve">ſinu recto grad. </s> <s xml:id="echoid-s4912" xml:space="preserve">18. </s> <s xml:id="echoid-s4913" xml:space="preserve">Id quod propoſ. </s> <s xml:id="echoid-s4914" xml:space="preserve">6. </s> <s xml:id="echoid-s4915" xml:space="preserve">à nobis eſt de-<lb/>monſtratum.</s> <s xml:id="echoid-s4916" xml:space="preserve"/> </p> <div xml:id="echoid-div386" type="float" level="2" n="8"> <note position="left" xlink:label="note-134-02" xlink:href="note-134-02a" xml:space="preserve">Supputatio <lb/>ſinuum at-<lb/>cuum gra-<lb/>dibus 2. <lb/>Min. 15. ſe-<lb/>ſe ſuperan <lb/>tiũ, & alio-<lb/>rum quorũ <lb/>dam.</note> <note position="right" xlink:label="note-134-03" xlink:href="note-134-03a" xml:space="preserve"> <lb/> ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus <lb/>G # M # # G # M # # G # M # # G # M <lb/>36 # 0 # 5877852 # 12 # 15 # 392598 # 40 # 30 # 6494480 # 15 # 45 # 2714405 <lb/>54 # 0 # 8090170 # 87 # 45 # 9992290 # 49 # 30 # 7604060 # 74 # 15 # 9624552 <lb/>18 # 0 # 3090170 # 27 # 0 # 4539905 # 20 # 15 # 3461171 # 38 # 15 # 6190940 <lb/>72 # 0 # 9510565 # 63 # 0 # 8910065 # 69 # 45 # 9381913 # 51 # 45 # 7853169 <lb/>9 # 0 # 1564345 # 13 # 30 # 2334454 # 42 # 45 # 6788007 # 24 # 45 # 4186597 <lb/>81 # 0 # 9876883 # 76 # 30 # 9723699 # 47 # 15 # 7343225 # 65 # 15 # 9081432 <lb/>4 # 30 # 784591 # 6 # 45 # 1175374 # 31 # 30 # 5224986 # 29 # 15 # 4886212 <lb/>85 # 30 # 9969173 # 83 # 15 # 9930665 # 58 # 30 # 8526402 # 60 # 45 # 8724960 <lb/></note> <note position="right" xlink:label="note-134-04" xlink:href="note-134-04a" xml:space="preserve"> <lb/> ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus <lb/>G # M # # G # M # # G # M # # G # M <lb/>2 # 15 # 392598 # 33 # 45 # 5555702 # 65 # 15 # 9081432 # 18 # 45 # 3214395 <lb/>4 # 30 # 784591 # 36 # 0 # 5877852 # 67 # 30 # 9238795 # 26 # 15 # 4422887 <lb/>6 # 45 # 1175374 # 38 # 15 # 6190940 # 69 # 45 # 9381913 # 30 # 0 # 5000000 <lb/>9 # 0 # 1564345 # 40 # 30 # 6494480 # 72 # 0 # 9510565 # 37 # 30 # 6087614 <lb/>11 # 15 # 1950903 # 42 # 45 # 6788007 # 74 # 15 # 9624552 # 41 # 15 # 6593458 <lb/>13 # 30 # 2334454 # 45 # 0 # 7071068 # 76 # 30 # 9723699 # 48 # 45 # 7518398 <lb/>15 # 45 # 2714405 # 47 # 15 # 7343225 # 78 # 45 # 9807853 # 52 # 30 # 7933533 <lb/>18 # 0 # 3090170 # 49 # 30 # 7604060 # 81 # 0 # 9876883 # 60 # 0 # 8660254 <lb/>20 # 15 # 3461171 # 51 # 45 # 7853169 # 83 # 15 # 9930685 # 63 # 45 # 8968727 <lb/>22 # 30 # 3826834 # 54 # 0 # 8090170 # 85 # 30 # 9969173 # 71 # 15 # 9469301 <lb/></note> <note position="right" xlink:label="note-135-01" xlink:href="note-135-01a" xml:space="preserve"> <lb/>24 # 45 # 4186597 # 56 # 15 # 8314696 # 87 # 45 # 9992290 # 75 # 0 # 965925 <lb/>27 # 0 # 4539905 # 58 # 30 # 8526402 # 3 # 45 # 654031 # 82 # 30 # 9914449 <lb/>9 # 15 # 4886212 # 60 # 45 # 8724960 # 7 # 30 # 1305262 # 86 # 15 # 9978589 <lb/>31 # 30 # 5224986 # 63 # 0 # 8910065 # 15 # 0 # 2588190 # 90 # 0 # 10000000 <lb/></note> </div> <p> <s xml:id="echoid-s4917" xml:space="preserve">IAM vero ſinum arcus grad. </s> <s xml:id="echoid-s4918" xml:space="preserve">12. </s> <s xml:id="echoid-s4919" xml:space="preserve">beneſicio ſinuum grad. </s> <s xml:id="echoid-s4920" xml:space="preserve">30. </s> <s xml:id="echoid-s4921" xml:space="preserve">& </s> <s xml:id="echoid-s4922" xml:space="preserve">54. </s> <s xml:id="echoid-s4923" xml:space="preserve">qui iam <lb/> <anchor type="note" xlink:label="note-135-02a" xlink:href="note-135-02"/> noti facti ſunt, ita inueſtigabimus. </s> <s xml:id="echoid-s4924" xml:space="preserve">Sit in quadrā <lb/> <anchor type="figure" xlink:label="fig-135-01a" xlink:href="fig-135-01"/> te ABC, arcus BD, grad. </s> <s xml:id="echoid-s4925" xml:space="preserve">30. </s> <s xml:id="echoid-s4926" xml:space="preserve">& </s> <s xml:id="echoid-s4927" xml:space="preserve">BE, grad. </s> <s xml:id="echoid-s4928" xml:space="preserve">54. </s> <s xml:id="echoid-s4929" xml:space="preserve">at-<lb/>que adeò arcus DE, eorum differentia, grad. </s> <s xml:id="echoid-s4930" xml:space="preserve">24. <lb/></s> <s xml:id="echoid-s4931" xml:space="preserve">ducanturque rectæ DF, EG, ad AB, perpendi-<lb/>culares, quæ ſinus recti erunt arcuum BD, BE; </s> <s xml:id="echoid-s4932" xml:space="preserve"><lb/>Item rectæ DH, EI, ad AC, perpendiculares, <lb/>quæ finus recti erunt arcuum CD, CE, grad. </s> <s xml:id="echoid-s4933" xml:space="preserve"><lb/>60. </s> <s xml:id="echoid-s4934" xml:space="preserve">& </s> <s xml:id="echoid-s4935" xml:space="preserve">36. </s> <s xml:id="echoid-s4936" xml:space="preserve">qui complementa ſunt arcuum BD, <lb/>BE. </s> <s xml:id="echoid-s4937" xml:space="preserve">Secet autem recta DH, rectam EG, in K. </s> <s xml:id="echoid-s4938" xml:space="preserve"><lb/>Et quoniam ſinus DF, hoc eſt, KG, illi æqua-<lb/> <anchor type="note" xlink:label="note-135-03a" xlink:href="note-135-03"/> lis, & </s> <s xml:id="echoid-s4939" xml:space="preserve">EG, noti ſunt; </s> <s xml:id="echoid-s4940" xml:space="preserve">erit quoque eorum diffe-<lb/>rentia EK, nota. </s> <s xml:id="echoid-s4941" xml:space="preserve">Similiter quia ſinus EI, hoc <lb/>eſt, KH, illi æqualis, & </s> <s xml:id="echoid-s4942" xml:space="preserve">DH, noti quoque ſunt, per propof. </s> <s xml:id="echoid-s4943" xml:space="preserve">3. </s> <s xml:id="echoid-s4944" xml:space="preserve">cum ſint ſinus <lb/> <anchor type="note" xlink:label="note-135-04a" xlink:href="note-135-04"/> complementorum arcuum BE, BD; </s> <s xml:id="echoid-s4945" xml:space="preserve">erit etiam eorum differentia DK, nota; <lb/></s> <s xml:id="echoid-s4946" xml:space="preserve">ac proinde duo quadrata rectarum notarum Ek, DK, nota erunt: </s> <s xml:id="echoid-s4947" xml:space="preserve">quæ cum <lb/>æqualia ſint quadrato rectæ DE, notum quoque erit quadratum rectæ DE, <lb/> <anchor type="note" xlink:label="note-135-05a" xlink:href="note-135-05"/> proptereaque & </s> <s xml:id="echoid-s4948" xml:space="preserve">ipſarecta DE, nota erit, nem pe chorda arcus grad. </s> <s xml:id="echoid-s4949" xml:space="preserve">24. </s> <s xml:id="echoid-s4950" xml:space="preserve">Hinc <lb/>& </s> <s xml:id="echoid-s4951" xml:space="preserve">eius medietas nota erit, nimirum ſinus rectus arcus grad. </s> <s xml:id="echoid-s4952" xml:space="preserve">12. </s> <s xml:id="echoid-s4953" xml:space="preserve">continebitque <lb/> <anchor type="note" xlink:label="note-135-06a" xlink:href="note-135-06"/> particulas 2079117. </s> <s xml:id="echoid-s4954" xml:space="preserve">Hac eadem arte; </s> <s xml:id="echoid-s4955" xml:space="preserve">ſi duorum arcuum quorumcumque ſinus <lb/>cogniti ſint, cognoſcetur etiam ſinus medietatis differentiæ illorum arcuum. <lb/></s> <s xml:id="echoid-s4956" xml:space="preserve">Vt ſi ſinus arcuum BD, BE, quorumcumque etiam ſi non contineant grad. </s> <s xml:id="echoid-s4957" xml:space="preserve">30. </s> <s xml:id="echoid-s4958" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s4959" xml:space="preserve">54. </s> <s xml:id="echoid-s4960" xml:space="preserve">noti ſint, erunt & </s> <s xml:id="echoid-s4961" xml:space="preserve">ſinus complementorum CD, CE, per propof. </s> <s xml:id="echoid-s4962" xml:space="preserve">3. </s> <s xml:id="echoid-s4963" xml:space="preserve">noti. </s> <s xml:id="echoid-s4964" xml:space="preserve"><lb/>Quare, vt modo demon ſtrauimus, nota ſiet chorda DE; </s> <s xml:id="echoid-s4965" xml:space="preserve">ac proinde eius me-<lb/>dietas nota quoque erit, nempe ſinus medietatis arcus DE.</s> <s xml:id="echoid-s4966" xml:space="preserve"/> </p> <div xml:id="echoid-div387" type="float" level="2" n="9"> <note position="right" xlink:label="note-135-02" xlink:href="note-135-02a" xml:space="preserve">Supputatio <lb/>ſinus arcus <lb/>grad. 12.</note> <figure xlink:label="fig-135-01" xlink:href="fig-135-01a"> <image file="135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/135-01"/> </figure> <note position="right" xlink:label="note-135-03" xlink:href="note-135-03a" xml:space="preserve">34. primi.</note> <note position="right" xlink:label="note-135-04" xlink:href="note-135-04a" xml:space="preserve">34. primi.</note> <note position="right" xlink:label="note-135-05" xlink:href="note-135-05a" xml:space="preserve">47. primi</note> <note position="right" xlink:label="note-135-06" xlink:href="note-135-06a" xml:space="preserve">Ex duorũ <lb/>arcuũ ſinu <lb/>bus notis <lb/>notus quo-<lb/>que ſit ſinꝰ <lb/>medietatis <lb/>differentiæ <lb/>arcuum.</note> </div> <p> <s xml:id="echoid-s4967" xml:space="preserve"><emph style="sc">Cae</emph><emph style="sc">Tervm</emph> ex ſinu arcus grad. </s> <s xml:id="echoid-s4968" xml:space="preserve">12. </s> <s xml:id="echoid-s4969" xml:space="preserve">(ſi hunc arcum bifariam ſecemus <lb/> <anchor type="note" xlink:label="note-135-07a" xlink:href="note-135-07"/> continue, quoad poſſumus, medietatumq́; </s> <s xml:id="echoid-s4970" xml:space="preserve">complementa accipiamus, vnà cum <lb/>ſinu complementi arcus grad. </s> <s xml:id="echoid-s4971" xml:space="preserve">12. </s> <s xml:id="echoid-s4972" xml:space="preserve">quæ rurſus ſecemus bifariam continue, &</s> <s xml:id="echoid-s4973" xml:space="preserve">c.) <lb/></s> <s xml:id="echoid-s4974" xml:space="preserve">reperiemus per coroll. </s> <s xml:id="echoid-s4975" xml:space="preserve">propof. </s> <s xml:id="echoid-s4976" xml:space="preserve">4. </s> <s xml:id="echoid-s4977" xml:space="preserve">& </s> <s xml:id="echoid-s4978" xml:space="preserve">per propoſ. </s> <s xml:id="echoid-s4979" xml:space="preserve">3. </s> <s xml:id="echoid-s4980" xml:space="preserve">ſinus omnium hor um arcuũ, <lb/>qui in hac tabella continentur. </s> <s xml:id="echoid-s4981" xml:space="preserve">Quos ſinus ſi cum ſinubus proxime antece-<lb/>dentis tabellæ in ordinem redigamus, habebimus ſinus arcuum numero 120. </s> <s xml:id="echoid-s4982" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-135-08a" xlink:href="note-135-08"/> <pb o="124" file="136" n="136" rhead=""/> <anchor type="note" xlink:label="note-136-01a" xlink:href="note-136-01"/> qui ſe ordine continuo ſuperant 45. </s> <s xml:id="echoid-s4983" xml:space="preserve">Minutis, quorum primus eſt arcus grad. <lb/></s> <s xml:id="echoid-s4984" xml:space="preserve">o. </s> <s xml:id="echoid-s4985" xml:space="preserve">Min. </s> <s xml:id="echoid-s4986" xml:space="preserve">45. </s> <s xml:id="echoid-s4987" xml:space="preserve">vltimus vero totus quadrans grad. </s> <s xml:id="echoid-s4988" xml:space="preserve">90. </s> <s xml:id="echoid-s4989" xml:space="preserve">cuiuſm odi ſunt arcus ſequen <lb/>tis tabellæ. </s> <s xml:id="echoid-s4990" xml:space="preserve">Omnium enim illorum arcuum ſinus inuentos eſſe comperies in <lb/>proximis duabus tabellis antecedentibus, licet non eo ordine ſint collocati.</s> <s xml:id="echoid-s4991" xml:space="preserve"/> </p> <div xml:id="echoid-div388" type="float" level="2" n="10"> <note position="right" xlink:label="note-135-07" xlink:href="note-135-07a" xml:space="preserve">Supputatio <lb/>ſinuu ar-<lb/>cuum Mi-<lb/>nutis 45. ſe <lb/>ſe ſupeian-<lb/>tium.</note> <note position="right" xlink:label="note-135-08" xlink:href="note-135-08a" xml:space="preserve"> <lb/> ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus # ## Arcus # Sinus <lb/>G # M # # G # M # # G # M # # G # M <lb/>12 # 0 # 2079117 # 42 # 0 # 6691306 # 12 # 45 # 2206974 # 33 # 0 # 5446390 <lb/>78 # 0 # 9781476 # 48 # 0 # 7431448 # 77 # 15 # 9753423 # 57 # 0 # 8386706 <lb/></note> <note position="right" xlink:label="note-136-01" xlink:href="note-136-01a" xml:space="preserve"> <lb/>6 # 0 # 1045285 # 21 # 0 # 3583679 # 35 # 15 # 5771452 # 16 # 30 # 2840153 <lb/>84 # 0 # 9945219 # 69 # 0 # 9335804 # 54 # 45 # 8166416 # 73 # 30 # 9588197 <lb/>3 # 0 # 523360 # 10 # 30 # 1822355 # 24 # 0 # 4067366 # 8 # 15 # 1434926 <lb/>87 # 0 # 9986295 # 79 # 30 # 9832549 # 66 # 0 # 9135455 # 81 # 45 # 9896514 <lb/>1 # 30 # 261769 # 5 # 15 # 915016 # 34 # 30 # 5664062 # 27 # 45 # 4656145 <lb/>88 # 30 # 9996573 # 84 # 45 # 9958049 # 55 # 30 # 8241262 # 62 # 15 # 8849876 <lb/>0 # 45 # 130896 # 43 # 30 # 6883546 # 17 # 15 # 2965416 # 28 # 30 # 4771588 <lb/>89 # 15 # 9999143 # 46 # 30 # 7253744 # 72 # 45 # 9550199 # 61 # 30 # 8788171 <lb/>39 # 0 # 6293204 # 21 # 45 # 3705574 # 39 # 45 # 6394390 # 14 # 15 # 2461533 <lb/>51 # 0 # 7771460 # 68 # 15 # 9288096 # 50 # 15 # 7688418 # 75 # 45 # 9692309 <lb/>19 # 30 # 3338069 # 44 # 15 # 6977905 # 23 # 15 # 3947439 # 36 # 45 # 5983246 <lb/>70 # 30 # 9426415 # 45 # 45 # 7163019 # 66 # 45 # 9187912 # 53 # 15 # 8012538 <lb/>9 # 45 # 1693495 # 25 # 30 # 4305111 # 32 # 15 # 5336145 # 30 # 45 # 5112931 <lb/>80 # 15 # 9855561 # 64 # 30 # 9025853 # 57 # 45 # 8457278 # 59 # 15 # 8594064 <lb/></note> </div> <note position="right" xml:space="preserve"> <lb/> ## Arcus # ## Arcus # ## Arcus # ## Arcus # ## Arcus # ## Arcus # ## Arcus # ## Arcus # ## Arcus # ## Arcus <lb/>G # M # G # M # G # M # G # M # G # M # G # M # G # M # G # M # G # M # G # M <lb/>0 # 45 # 9 # 45 # 18 # 45 # 27 # 45 # 36 # 45 # 45 # 45 # 54 # 45 # 63 # 45 # 72 # 45 # 81 # 45 <lb/>1 # 30 # 10 # 30 # 19 # 30 # 28 # 30 # 37 # 30 # 46 # 30 # 55 # 30 # 64 # 30 # 73 # 30 # 82 # 30 <lb/>2 # 15 # 11 # 15 # 20 # 15 # 29 # 15 # 38 # 15 # 47 # 15 # 56 # 15 # 65 # 15 # 74 # 15 # 83 # 15 <lb/>3 # 0 # 12 # 0 # 21 # 0 # 30 # 0 # 39 # 0 # 48 # 0 # 57 # 0 # 66 # 0 # 75 # 0 # 84 # 0 <lb/>3 # 45 # 12 # 45 # 21 # 45 # 30 # 45 # 39 # 45 # 48 # 45 # 57 # 45 # 66 # 45 # 75 # 45 # 84 # 45 <lb/>4 # 30 # 13 # 30 # 22 # 30 # 31 # 30 # 40 # 30 # 49 # 0 # 58 # 30 # 67 # 30 # 76 # 30 # 85 # 30 <lb/>5 # 15 # 14 # 15 # 23 # 15 # 32 # 15 # 41 # 15 # 50 # 15 # 59 # 15 # 68 # 15 # 77 # 15 # 86 # 15 <lb/>6 # 0 # 15 # 0 # 24 # 0 # 33 # 0 # 42 # 0 # 51 # 0 # 60 # 0 # 69 # 0 # 78 # 0 # 87 # 0 <lb/>6 # 45 # 15 # 45 # 24 # 45 # 33 # 45 # 42 # 45 # 51 # 45 # 60 # 45 # 69 # 45 # 78 # 45 # 87 # 45 <lb/>7 # 30 # 16 # 30 # 25 # 30 # 34 # 30 # 43 # 30 # 52 # 30 # 61 # 30 # 70 # 30 # 79 # 30 # 88 # 30 <lb/>8 # 15 # 17 # 15 # 26 # 15 # 35 # 15 # 44 # 15 # 53 # 15 # 62 # 15 # 71 # 15 # 80 # 15 # 89 # 15 <lb/>9 # 0 # 18 # 0 # 27 # 0 # 36 # 0 # 45 # 0 # 54 # 0 # 63 # 0 # 72 # 0 # 81 # 0 # 90 # 0 <lb/></note> <p> <s xml:id="echoid-s4992" xml:space="preserve">POSTREMO aliorum arcuum ſinus ita inueniemus. </s> <s xml:id="echoid-s4993" xml:space="preserve">Ponatur in qua-<lb/>drante ABC, arcus BD, grad. </s> <s xml:id="echoid-s4994" xml:space="preserve">1. </s> <s xml:id="echoid-s4995" xml:space="preserve">& </s> <s xml:id="echoid-s4996" xml:space="preserve">arcus BE, grad. </s> <s xml:id="echoid-s4997" xml:space="preserve">1. </s> <s xml:id="echoid-s4998" xml:space="preserve">Min. </s> <s xml:id="echoid-s4999" xml:space="preserve">30. </s> <s xml:id="echoid-s5000" xml:space="preserve">& </s> <s xml:id="echoid-s5001" xml:space="preserve">arcus BH, <lb/>grad. </s> <s xml:id="echoid-s5002" xml:space="preserve">0. </s> <s xml:id="echoid-s5003" xml:space="preserve">Min. </s> <s xml:id="echoid-s5004" xml:space="preserve">45. </s> <s xml:id="echoid-s5005" xml:space="preserve">Ducta autem recta EK, ad AB, perpendiculari, diuiſoque ar-<lb/>cu BH, in tres partes æquales BF, FG, GH, & </s> <s xml:id="echoid-s5006" xml:space="preserve">arcu DE, in duas DI, IE, <lb/>vt ſinguli arcus contineant 15. </s> <s xml:id="echoid-s5007" xml:space="preserve">Min. </s> <s xml:id="echoid-s5008" xml:space="preserve">ducantur ad EK, perpendiculares FL, <lb/>GM, HN, DO, IP. </s> <s xml:id="echoid-s5009" xml:space="preserve">Eritque EK, ſinus rectus arcus BE, & </s> <s xml:id="echoid-s5010" xml:space="preserve">OK, ſinus re-<lb/>ctus arcus BD, cum æqualis ſit rectæ ex D, ductæ ad AB, perpendiculari, quæ <lb/> <anchor type="note" xlink:label="note-136-03a" xlink:href="note-136-03"/> <pb o="125" file="137" n="137" rhead=""/> quidem ſinus eſt arcus BD. </s> <s xml:id="echoid-s5011" xml:space="preserve">Eademq́ue ratione erit NK, ſinus rectus arcus <lb/> <anchor type="note" xlink:label="note-137-01a" xlink:href="note-137-01"/> BH. </s> <s xml:id="echoid-s5012" xml:space="preserve">Sit ergo propoſitum inuenire ſinum OK, arcus BD, grad. </s> <s xml:id="echoid-s5013" xml:space="preserve">1. </s> <s xml:id="echoid-s5014" xml:space="preserve">Quoniam <lb/>NK, ſinus grad. </s> <s xml:id="echoid-s5015" xml:space="preserve">0. </s> <s xml:id="echoid-s5016" xml:space="preserve">Min. </s> <s xml:id="echoid-s5017" xml:space="preserve">45. </s> <s xml:id="echoid-s5018" xml:space="preserve">eſt inuentus 130896. </s> <s xml:id="echoid-s5019" xml:space="preserve">erit eius tertia pars, nempe <lb/>43632. </s> <s xml:id="echoid-s5020" xml:space="preserve">maior, quàm MN: </s> <s xml:id="echoid-s5021" xml:space="preserve">propterea quod, per propof. </s> <s xml:id="echoid-s5022" xml:space="preserve">1. </s> <s xml:id="echoid-s5023" xml:space="preserve">maior eſt KL, quàm <lb/>LM, & </s> <s xml:id="echoid-s5024" xml:space="preserve">LM, maior, quàm MN, adeo vt MN, mi-<lb/> <anchor type="figure" xlink:label="fig-137-01a" xlink:href="fig-137-01"/> nor ſit, quàm tertia pars ipſius NK, hoc eſt, minor, <lb/>quàm 43632. </s> <s xml:id="echoid-s5025" xml:space="preserve">Multo igitur maior erit eadem tertia <lb/>pars rectæ NK, nempe 43632. </s> <s xml:id="echoid-s5026" xml:space="preserve">quàm NO: </s> <s xml:id="echoid-s5027" xml:space="preserve">quod, per <lb/>eandẽ propof. </s> <s xml:id="echoid-s5028" xml:space="preserve">1. </s> <s xml:id="echoid-s5029" xml:space="preserve">maior quoque ſit MN, quàm NO. <lb/></s> <s xml:id="echoid-s5030" xml:space="preserve">Quare ſi addamus 43632. </s> <s xml:id="echoid-s5031" xml:space="preserve">ad 130896. </s> <s xml:id="echoid-s5032" xml:space="preserve">id eſt, ad NK, <lb/>ſinum Minutorum 45. </s> <s xml:id="echoid-s5033" xml:space="preserve">efficiemus numerum 174528. </s> <s xml:id="echoid-s5034" xml:space="preserve"><lb/>qui maior erit, quàm OK, ſinus rectus grad. </s> <s xml:id="echoid-s5035" xml:space="preserve">1. </s> <s xml:id="echoid-s5036" xml:space="preserve">quan <lb/>doquidem ad NK, plus addimus, quàm rectã NO, <lb/>vt dictum eſt. </s> <s xml:id="echoid-s5037" xml:space="preserve">Rurſus quia EK, ſinus grad. </s> <s xml:id="echoid-s5038" xml:space="preserve">1. </s> <s xml:id="echoid-s5039" xml:space="preserve">Min. </s> <s xml:id="echoid-s5040" xml:space="preserve"><lb/>30. </s> <s xml:id="echoid-s5041" xml:space="preserve">inuentus eſt 261769. </s> <s xml:id="echoid-s5042" xml:space="preserve">ſi ex eo detrahamus ſinum NK, Minutorum 45. </s> <s xml:id="echoid-s5043" xml:space="preserve">nem <lb/>pe 130896. </s> <s xml:id="echoid-s5044" xml:space="preserve">relinquetur recta EN, partium 130873. </s> <s xml:id="echoid-s5045" xml:space="preserve">cuius tertia pars, nempe <lb/>43624. </s> <s xml:id="echoid-s5046" xml:space="preserve">fere, minor erit, quàm NO: </s> <s xml:id="echoid-s5047" xml:space="preserve">propterea quod, per propof. </s> <s xml:id="echoid-s5048" xml:space="preserve">1. </s> <s xml:id="echoid-s5049" xml:space="preserve">maior eſt <lb/>NO, quàm OP, & </s> <s xml:id="echoid-s5050" xml:space="preserve">OP, maior, quàm PE, adeo vt NO, maior ſit, quàm ter-<lb/>tia parsipſius EN, hoc eſt, maior, quàm 43624. </s> <s xml:id="echoid-s5051" xml:space="preserve">Quare ſi addamus 43624. </s> <s xml:id="echoid-s5052" xml:space="preserve">ad <lb/>130896. </s> <s xml:id="echoid-s5053" xml:space="preserve">id eſt, ad NK, ſinum Min. </s> <s xml:id="echoid-s5054" xml:space="preserve">45. </s> <s xml:id="echoid-s5055" xml:space="preserve">efficiemus numerum 174520. </s> <s xml:id="echoid-s5056" xml:space="preserve">qui minor <lb/>erit, quàm OK, ſinus re ctus grad. </s> <s xml:id="echoid-s5057" xml:space="preserve">1. </s> <s xml:id="echoid-s5058" xml:space="preserve">quandoquidem ad NK, minus addimus, <lb/>quàm rectam NO. </s> <s xml:id="echoid-s5059" xml:space="preserve">Conſtatigitur, ſinum rectum grad. </s> <s xml:id="echoid-s5060" xml:space="preserve">1. </s> <s xml:id="echoid-s5061" xml:space="preserve">conſiſtere inter hos <lb/>duos numeros, 174528. </s> <s xml:id="echoid-s5062" xml:space="preserve">174520. </s> <s xml:id="echoid-s5063" xml:space="preserve">cum ille maior ſit, hic auté minor. </s> <s xml:id="echoid-s5064" xml:space="preserve">Statuamus <lb/>ergo eum eſſe, 174524. </s> <s xml:id="echoid-s5065" xml:space="preserve">inter illos numeros omnino medium. </s> <s xml:id="echoid-s5066" xml:space="preserve">Ita enim non dif-<lb/>feret ſenſibiliter hic numerus à vero ſinu grad. </s> <s xml:id="echoid-s5067" xml:space="preserve">1.</s> <s xml:id="echoid-s5068" xml:space="preserve"/> </p> <div xml:id="echoid-div389" type="float" level="2" n="11"> <note position="left" xlink:label="note-136-03" xlink:href="note-136-03a" xml:space="preserve">34. primi.</note> <note position="right" xlink:label="note-137-01" xlink:href="note-137-01a" xml:space="preserve">Supputatie <lb/>ſinus grad. <lb/>1.</note> <figure xlink:label="fig-137-01" xlink:href="fig-137-01a"> <image file="137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/137-01"/> </figure> </div> <p> <s xml:id="echoid-s5069" xml:space="preserve">EX hoc ſinu grad. </s> <s xml:id="echoid-s5070" xml:space="preserve">1. </s> <s xml:id="echoid-s5071" xml:space="preserve">inueniemus, per propoſ. </s> <s xml:id="echoid-s5072" xml:space="preserve">3. </s> <s xml:id="echoid-s5073" xml:space="preserve">ſinum eius cõplementi, hoc <lb/> <anchor type="note" xlink:label="note-137-02a" xlink:href="note-137-02"/> eſt, ſinum arcus grad. </s> <s xml:id="echoid-s5074" xml:space="preserve">89. </s> <s xml:id="echoid-s5075" xml:space="preserve">eſſe 9998477. </s> <s xml:id="echoid-s5076" xml:space="preserve">Quem hoc etiã modo reperiemus. </s> <s xml:id="echoid-s5077" xml:space="preserve">Quo-<lb/>niam numerus 174528. </s> <s xml:id="echoid-s5078" xml:space="preserve">maior eſt, quam ſinus grad. </s> <s xml:id="echoid-s5079" xml:space="preserve">1. </s> <s xml:id="echoid-s5080" xml:space="preserve">vt diximus: </s> <s xml:id="echoid-s5081" xml:space="preserve">ſit, vt per eum <lb/>inueſtigatus, iuxta doctrinam propof. </s> <s xml:id="echoid-s5082" xml:space="preserve">3. </s> <s xml:id="echoid-s5083" xml:space="preserve">ſinus cóplementi grad. </s> <s xml:id="echoid-s5084" xml:space="preserve">1. </s> <s xml:id="echoid-s5085" xml:space="preserve">hoc eſt, ſinus <lb/>grad. </s> <s xml:id="echoid-s5086" xml:space="preserve">89. </s> <s xml:id="echoid-s5087" xml:space="preserve">ſit numerus 9998476. </s> <s xml:id="echoid-s5088" xml:space="preserve">& </s> <s xml:id="echoid-s5089" xml:space="preserve">paulo amplius, qui minor erit neceſſario, <lb/>quàm verus ſinus arcus grad. </s> <s xml:id="echoid-s5090" xml:space="preserve">89. </s> <s xml:id="echoid-s5091" xml:space="preserve">quandoquidem pro ſinu grad. </s> <s xml:id="echoid-s5092" xml:space="preserve">1. </s> <s xml:id="echoid-s5093" xml:space="preserve">numerum <lb/>ſumpſimus vero ſinu maiorem. </s> <s xml:id="echoid-s5094" xml:space="preserve">Rurſus quia numerus 174520. </s> <s xml:id="echoid-s5095" xml:space="preserve">minor eſt, quàm <lb/>ſinus grad. </s> <s xml:id="echoid-s5096" xml:space="preserve">1. </s> <s xml:id="echoid-s5097" xml:space="preserve">vt diximus; </s> <s xml:id="echoid-s5098" xml:space="preserve">fit, vt per eum inueſtigatus, iuxta doctrinam propof. <lb/></s> <s xml:id="echoid-s5099" xml:space="preserve">3. </s> <s xml:id="echoid-s5100" xml:space="preserve">ſinus eius complementi, nimirum ſinus grad. </s> <s xml:id="echoid-s5101" xml:space="preserve">89. </s> <s xml:id="echoid-s5102" xml:space="preserve">ſit numerus 9998477. </s> <s xml:id="echoid-s5103" xml:space="preserve">& </s> <s xml:id="echoid-s5104" xml:space="preserve">pau <lb/>lo amplius, qui maior erit neceſſario, quàm verus ſinus grad. </s> <s xml:id="echoid-s5105" xml:space="preserve">89. </s> <s xml:id="echoid-s5106" xml:space="preserve">quandoqui-<lb/>dem pro ſinu grad. </s> <s xml:id="echoid-s5107" xml:space="preserve">1. </s> <s xml:id="echoid-s5108" xml:space="preserve">numerum accepimus vero ſinu minorem. </s> <s xml:id="echoid-s5109" xml:space="preserve">Conſtitutus er-<lb/>go erit ſinus grad. </s> <s xml:id="echoid-s5110" xml:space="preserve">89. </s> <s xml:id="echoid-s5111" xml:space="preserve">inter duos hos numeros 9998476. </s> <s xml:id="echoid-s5112" xml:space="preserve">& </s> <s xml:id="echoid-s5113" xml:space="preserve">paulo amplius, at-<lb/>que 9998477. </s> <s xml:id="echoid-s5114" xml:space="preserve">& </s> <s xml:id="echoid-s5115" xml:space="preserve">paulo amplius: </s> <s xml:id="echoid-s5116" xml:space="preserve">ac proinde ſine notabili errore eum ſtatuemus <lb/>eſſe 9998477. </s> <s xml:id="echoid-s5117" xml:space="preserve">quantum videlicet eundem inuenimus ex ſinugrad. </s> <s xml:id="echoid-s5118" xml:space="preserve">1. </s> <s xml:id="echoid-s5119" xml:space="preserve">Ex quo <lb/>conſtat, recte conſtitutum eſſe ſinum grad. </s> <s xml:id="echoid-s5120" xml:space="preserve">1. </s> <s xml:id="echoid-s5121" xml:space="preserve">partium 174524. </s> <s xml:id="echoid-s5122" xml:space="preserve">cum ex eo, per <lb/>propoſ. </s> <s xml:id="echoid-s5123" xml:space="preserve">3. </s> <s xml:id="echoid-s5124" xml:space="preserve">repertus ſit ſinus complementiipſius tot particularum, quot vere ac <lb/>reipſa continere debet.</s> <s xml:id="echoid-s5125" xml:space="preserve"/> </p> <div xml:id="echoid-div390" type="float" level="2" n="12"> <note position="right" xlink:label="note-137-02" xlink:href="note-137-02a" xml:space="preserve">Supputatio <lb/>ſinus grad. <lb/>89.</note> </div> <p> <s xml:id="echoid-s5126" xml:space="preserve">PRAETEREA ex finu arcus grad. </s> <s xml:id="echoid-s5127" xml:space="preserve">1. </s> <s xml:id="echoid-s5128" xml:space="preserve">reperiemus, per coroll. </s> <s xml:id="echoid-s5129" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s5130" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s5131" xml:space="preserve"> <anchor type="note" xlink:label="note-137-03a" xlink:href="note-137-03"/> ſinum rectum arcus Min. </s> <s xml:id="echoid-s5132" xml:space="preserve">30. </s> <s xml:id="echoid-s5133" xml:space="preserve">& </s> <s xml:id="echoid-s5134" xml:space="preserve">ex hoc, per propoſ. </s> <s xml:id="echoid-s5135" xml:space="preserve">3. </s> <s xml:id="echoid-s5136" xml:space="preserve">ſinum complementi, nem-<lb/>pe ſinum grad. </s> <s xml:id="echoid-s5137" xml:space="preserve">89. </s> <s xml:id="echoid-s5138" xml:space="preserve">Min. </s> <s xml:id="echoid-s5139" xml:space="preserve">30. </s> <s xml:id="echoid-s5140" xml:space="preserve">Item ex ſinu arcus Min. </s> <s xml:id="echoid-s5141" xml:space="preserve">30. </s> <s xml:id="echoid-s5142" xml:space="preserve">inueniemus, per coroll. <lb/></s> <s xml:id="echoid-s5143" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s5144" xml:space="preserve">4. </s> <s xml:id="echoid-s5145" xml:space="preserve">ſinum arcus Min. </s> <s xml:id="echoid-s5146" xml:space="preserve">15. </s> <s xml:id="echoid-s5147" xml:space="preserve">atque ex hoe ſinum complementi, hoc eſt, ſi-<lb/>num grad. </s> <s xml:id="echoid-s5148" xml:space="preserve">89. </s> <s xml:id="echoid-s5149" xml:space="preserve">Min. </s> <s xml:id="echoid-s5150" xml:space="preserve">45.</s> <s xml:id="echoid-s5151" xml:space="preserve"/> </p> <div xml:id="echoid-div391" type="float" level="2" n="13"> <note position="right" xlink:label="note-137-03" xlink:href="note-137-03a" xml:space="preserve">Supputatie <lb/>ſinus Min. <lb/>30. & ſinus <lb/>grad. 89. <lb/>Min. 30. It@ <lb/>ſinus Min. <lb/>15. & ſinus <lb/>grad. 89. <lb/>Min. 45.</note> </div> <p> <s xml:id="echoid-s5152" xml:space="preserve">QVEMADMODVM autem ſupra ex ſinubus rectis grad. </s> <s xml:id="echoid-s5153" xml:space="preserve">30. </s> <s xml:id="echoid-s5154" xml:space="preserve">& </s> <s xml:id="echoid-s5155" xml:space="preserve">54.</s> <s xml:id="echoid-s5156" xml:space="preserve"> <pb o="126" file="138" n="138" rhead=""/> indagauimus ſinum rectum grad. </s> <s xml:id="echoid-s5157" xml:space="preserve">12. </s> <s xml:id="echoid-s5158" xml:space="preserve">qui dimidium eſt chordæ arcus grad. </s> <s xml:id="echoid-s5159" xml:space="preserve">24. <lb/></s> <s xml:id="echoid-s5160" xml:space="preserve">quo dicti arcus grad. </s> <s xml:id="echoid-s5161" xml:space="preserve">30. </s> <s xml:id="echoid-s5162" xml:space="preserve">& </s> <s xml:id="echoid-s5163" xml:space="preserve">54. </s> <s xml:id="echoid-s5164" xml:space="preserve">inter ſe differunt: </s> <s xml:id="echoid-s5165" xml:space="preserve">ita quoque ex ſinubus rectis <lb/> <anchor type="note" xlink:label="note-138-01a" xlink:href="note-138-01"/> arcus Min. </s> <s xml:id="echoid-s5166" xml:space="preserve">30. </s> <s xml:id="echoid-s5167" xml:space="preserve">& </s> <s xml:id="echoid-s5168" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5169" xml:space="preserve">52. </s> <s xml:id="echoid-s5170" xml:space="preserve">Min. </s> <s xml:id="echoid-s5171" xml:space="preserve">30. </s> <s xml:id="echoid-s5172" xml:space="preserve">cognitis cognoſeemus ſinum rectum <lb/>grad. </s> <s xml:id="echoid-s5173" xml:space="preserve">26. </s> <s xml:id="echoid-s5174" xml:space="preserve">qui dimidium eſt chordæ arcus grad. </s> <s xml:id="echoid-s5175" xml:space="preserve">52. </s> <s xml:id="echoid-s5176" xml:space="preserve">quo dicti arcus Min. </s> <s xml:id="echoid-s5177" xml:space="preserve">30. </s> <s xml:id="echoid-s5178" xml:space="preserve">& </s> <s xml:id="echoid-s5179" xml:space="preserve"><lb/>grad. </s> <s xml:id="echoid-s5180" xml:space="preserve">52. </s> <s xml:id="echoid-s5181" xml:space="preserve">Min. </s> <s xml:id="echoid-s5182" xml:space="preserve">30. </s> <s xml:id="echoid-s5183" xml:space="preserve">inter ſe differunt, vt ſupra oſtendimus; </s> <s xml:id="echoid-s5184" xml:space="preserve">atque ex ſinu grad. </s> <s xml:id="echoid-s5185" xml:space="preserve">26. <lb/></s> <s xml:id="echoid-s5186" xml:space="preserve">ſiet, per propof. </s> <s xml:id="echoid-s5187" xml:space="preserve">3. </s> <s xml:id="echoid-s5188" xml:space="preserve">notus quoque ſinus complementi, nimirum ſinus grad. </s> <s xml:id="echoid-s5189" xml:space="preserve">64.</s> <s xml:id="echoid-s5190" xml:space="preserve"/> </p> <div xml:id="echoid-div392" type="float" level="2" n="14"> <note position="left" xlink:label="note-138-01" xlink:href="note-138-01a" xml:space="preserve">Supputatio <lb/>ſinus grad. <lb/>26. & grad. <lb/>64.</note> </div> <p> <s xml:id="echoid-s5191" xml:space="preserve">QVOD ſi arcus grad. </s> <s xml:id="echoid-s5192" xml:space="preserve">26. </s> <s xml:id="echoid-s5193" xml:space="preserve">& </s> <s xml:id="echoid-s5194" xml:space="preserve">grad. </s> <s xml:id="echoid-s5195" xml:space="preserve">64. </s> <s xml:id="echoid-s5196" xml:space="preserve">& </s> <s xml:id="echoid-s5197" xml:space="preserve">grad 89. </s> <s xml:id="echoid-s5198" xml:space="preserve">& </s> <s xml:id="echoid-s5199" xml:space="preserve">grad. </s> <s xml:id="echoid-s5200" xml:space="preserve">89. </s> <s xml:id="echoid-s5201" xml:space="preserve">Min. </s> <s xml:id="echoid-s5202" xml:space="preserve">30. </s> <s xml:id="echoid-s5203" xml:space="preserve">quo-<lb/>rum ſinus inuenti ſunt, ſecemus continue bifariam, quoad poſſumus, accipia-<lb/> <anchor type="note" xlink:label="note-138-02a" xlink:href="note-138-02"/> musque medictatum complementa, quę rurſus continue biſariam diuidamus, <lb/>&</s> <s xml:id="echoid-s5204" xml:space="preserve">c. </s> <s xml:id="echoid-s5205" xml:space="preserve">adhibeamusq́ue doctrinam illam, qua ex duorum arcuum ſinubus notis <lb/>cognoſcitur ſinus medietatis differentiæ illorum arcuum, reperiemus ſin us ar <lb/>cuum numero 240. </s> <s xml:id="echoid-s5206" xml:space="preserve">qui in ordinem redacti cum ſinubus arcuum numero 120. <lb/></s> <s xml:id="echoid-s5207" xml:space="preserve">prius inuentis, conſtituent ſinus arcuum numero 360. </s> <s xml:id="echoid-s5208" xml:space="preserve">ſeſe ordine ſuperantium <lb/>Minutis 15.</s> <s xml:id="echoid-s5209" xml:space="preserve"/> </p> <div xml:id="echoid-div393" type="float" level="2" n="15"> <note position="left" xlink:label="note-138-02" xlink:href="note-138-02a" xml:space="preserve">Supputatio <lb/>ſinuum ar <lb/>cuũ Miau-<lb/>tis 15. ſeſe <lb/>ſuperãuũ.</note> </div> <p> <s xml:id="echoid-s5210" xml:space="preserve">VERVM quia laborioſum eſt, atque moleſtum tot ſinus ea ratione in-<lb/> <anchor type="note" xlink:label="note-138-03a" xlink:href="note-138-03"/> dagare, ſatis erit, tanta difficultate inueniſſe ſinus illos arcuum ſuperiorum <lb/>numero 120. </s> <s xml:id="echoid-s5211" xml:space="preserve">ſeſe ordine ſuperantium Minutis 45. </s> <s xml:id="echoid-s5212" xml:space="preserve">Ex illis enim ſacile per re-<lb/>gulam proportionum reperiemus ſinus arcuum ſe ordine ſuperantium Minu-<lb/>tis 15. </s> <s xml:id="echoid-s5213" xml:space="preserve">Deinde ex his ſinus arcuum, qui ordine ſe ſuperant Minutis 5. </s> <s xml:id="echoid-s5214" xml:space="preserve">Acdeni-<lb/>que ex iſtis ſinus arcuum per ſingula Minuta extenſorum; </s> <s xml:id="echoid-s5215" xml:space="preserve">neque vnquàm in <lb/>hac ſupputatione error ſenſibilis continget. </s> <s xml:id="echoid-s5216" xml:space="preserve">Cum enim ſinum totum poſue-<lb/>rimus centies maiorem, quàm 100000. </s> <s xml:id="echoid-s5217" xml:space="preserve">ſit vt abiectis primis duabus figuris ad <lb/>dexteram, vt ſupra dictum eſt, exquiſitiſsimi relinquantur ſinus reſpectu ſinus <lb/>totius 100000. </s> <s xml:id="echoid-s5218" xml:space="preserve">quod totus error, qui in hac ſupputatione contingere poteſt, <lb/>conſiſtat in prima figura ad dexterá, vel ad ſummum in duabus primis. </s> <s xml:id="echoid-s5219" xml:space="preserve">Qua-<lb/>re abiectis duabus primis figuris, remanebunt omnic ijdem ſinus, qui inuen-<lb/>ti eſſent priori illo modo Geometrico reſpectu eiuſdem ſinus totius 100000. <lb/></s> <s xml:id="echoid-s5220" xml:space="preserve">ſi in ſupputatione poneretur ſinus totus centies etiam maior, nempe partium <lb/>10000000. </s> <s xml:id="echoid-s5221" xml:space="preserve">& </s> <s xml:id="echoid-s5222" xml:space="preserve">ex inuentis ſinubus duæ figuræ reijcerentur: </s> <s xml:id="echoid-s5223" xml:space="preserve">Id quod experien-<lb/>tia docebit. </s> <s xml:id="echoid-s5224" xml:space="preserve">Ita autem rem exequemur. </s> <s xml:id="echoid-s5225" xml:space="preserve">Statuátur ordine illi arcus cum ſinu-<lb/>bus, & </s> <s xml:id="echoid-s5226" xml:space="preserve">ad dexteram cuiuſque ſinus aſcribatur differentia, qua à præcedenti ſi-<lb/> <anchor type="note" xlink:label="note-138-04a" xlink:href="note-138-04"/> nu differt; </s> <s xml:id="echoid-s5227" xml:space="preserve">vt hic factum eſſe vides in quin <lb/>que arcubus. </s> <s xml:id="echoid-s5228" xml:space="preserve">Deinde dic. </s> <s xml:id="echoid-s5229" xml:space="preserve">Si Minuta 45. </s> <s xml:id="echoid-s5230" xml:space="preserve">re-<lb/>quirunt differentiam 130896. </s> <s xml:id="echoid-s5231" xml:space="preserve">addendam ad <lb/>ſinum Minuti. </s> <s xml:id="echoid-s5232" xml:space="preserve">0. </s> <s xml:id="echoid-s5233" xml:space="preserve">vt ſiat ſinus Minutorum <lb/>45. </s> <s xml:id="echoid-s5234" xml:space="preserve">Minuta 15. </s> <s xml:id="echoid-s5235" xml:space="preserve">quantam poſtulant diffe-<lb/>rentiam addendam eidem ſinui Minur. </s> <s xml:id="echoid-s5236" xml:space="preserve">0. <lb/></s> <s xml:id="echoid-s5237" xml:space="preserve">vt fiat ſinus Minutorum 15? </s> <s xml:id="echoid-s5238" xml:space="preserve">Inuenies enim <lb/>requiri differentiã 43632. </s> <s xml:id="echoid-s5239" xml:space="preserve">quæ addita ſinui <lb/>Minuti. </s> <s xml:id="echoid-s5240" xml:space="preserve">0. </s> <s xml:id="echoid-s5241" xml:space="preserve">conſtituet ſinũ Minutorum 15. </s> <s xml:id="echoid-s5242" xml:space="preserve"><lb/>partium 43632. </s> <s xml:id="echoid-s5243" xml:space="preserve">Rurſus dic. </s> <s xml:id="echoid-s5244" xml:space="preserve">Poſitis ijſdem, <lb/>quantam differẽtiam exigune Minuta 30. </s> <s xml:id="echoid-s5245" xml:space="preserve"><lb/>addendam eidem ſinui Minuti 0. </s> <s xml:id="echoid-s5246" xml:space="preserve">vt ſiat ſi-<lb/>nus Minutorum 30? </s> <s xml:id="echoid-s5247" xml:space="preserve">Reperies enim differentiam 87264. </s> <s xml:id="echoid-s5248" xml:space="preserve">quæ addita ſinui Minu <lb/>tio. </s> <s xml:id="echoid-s5249" xml:space="preserve">faciet 87264. </s> <s xml:id="echoid-s5250" xml:space="preserve">ſinum Min 30. </s> <s xml:id="echoid-s5251" xml:space="preserve">Item dic. </s> <s xml:id="echoid-s5252" xml:space="preserve">Si Minuta 45. </s> <s xml:id="echoid-s5253" xml:space="preserve">quibus arcus Min. </s> <s xml:id="echoid-s5254" xml:space="preserve">45. </s> <s xml:id="echoid-s5255" xml:space="preserve"><lb/>ab arcu ſequenti grad. </s> <s xml:id="echoid-s5256" xml:space="preserve">1. </s> <s xml:id="echoid-s5257" xml:space="preserve">Min. </s> <s xml:id="echoid-s5258" xml:space="preserve">30. </s> <s xml:id="echoid-s5259" xml:space="preserve">differt, requirunt differentiam 130873. </s> <s xml:id="echoid-s5260" xml:space="preserve">ad-<lb/>dendam ſinui Min. </s> <s xml:id="echoid-s5261" xml:space="preserve">45. </s> <s xml:id="echoid-s5262" xml:space="preserve">vt ſiat ſinus grad. </s> <s xml:id="echoid-s5263" xml:space="preserve">1. </s> <s xml:id="echoid-s5264" xml:space="preserve">Min. </s> <s xml:id="echoid-s5265" xml:space="preserve">30. </s> <s xml:id="echoid-s5266" xml:space="preserve">quantam differentiam po-<lb/>ſtulant Minuta 15. </s> <s xml:id="echoid-s5267" xml:space="preserve">addédam eidem ſinui Min. </s> <s xml:id="echoid-s5268" xml:space="preserve">45. </s> <s xml:id="echoid-s5269" xml:space="preserve">vt fiat ſinus Min. </s> <s xml:id="echoid-s5270" xml:space="preserve">60. </s> <s xml:id="echoid-s5271" xml:space="preserve">hoc eſt, <pb o="127" file="139" n="139" rhead=""/> ſinus grad. </s> <s xml:id="echoid-s5272" xml:space="preserve">1? </s> <s xml:id="echoid-s5273" xml:space="preserve">Inuenies enim differentiã 43624. </s> <s xml:id="echoid-s5274" xml:space="preserve">quæ addita ad 130896. </s> <s xml:id="echoid-s5275" xml:space="preserve">ſinũ Min. <lb/></s> <s xml:id="echoid-s5276" xml:space="preserve">45. </s> <s xml:id="echoid-s5277" xml:space="preserve">faciat 174520. </s> <s xml:id="echoid-s5278" xml:space="preserve">ſinũ grad. </s> <s xml:id="echoid-s5279" xml:space="preserve">1. </s> <s xml:id="echoid-s5280" xml:space="preserve">qui licet minor aliquanto ſit illo, quem alio mo <lb/>do inuenimus, abiectis tamẽ duabus primis ſiguris 20. </s> <s xml:id="echoid-s5281" xml:space="preserve">relinquetur ſinus 1745. </s> <s xml:id="echoid-s5282" xml:space="preserve"><lb/>exquiſitiſsimus grad. </s> <s xml:id="echoid-s5283" xml:space="preserve">1. </s> <s xml:id="echoid-s5284" xml:space="preserve">reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5285" xml:space="preserve">Dic pręterea. </s> <s xml:id="echoid-s5286" xml:space="preserve">Iiſdẽ poſi-<lb/>tis, quantam differentiam poſcunt Minuta 30. </s> <s xml:id="echoid-s5287" xml:space="preserve">addendam eidẽ ſinui Min. </s> <s xml:id="echoid-s5288" xml:space="preserve">45. </s> <s xml:id="echoid-s5289" xml:space="preserve"><lb/>vt ſiat ſinus Min. </s> <s xml:id="echoid-s5290" xml:space="preserve">75. </s> <s xml:id="echoid-s5291" xml:space="preserve">hoc eſt, ſinus grad. </s> <s xml:id="echoid-s5292" xml:space="preserve">1. </s> <s xml:id="echoid-s5293" xml:space="preserve">Min. </s> <s xml:id="echoid-s5294" xml:space="preserve">15? </s> <s xml:id="echoid-s5295" xml:space="preserve">Inuenies enim differentiam <lb/>87249. </s> <s xml:id="echoid-s5296" xml:space="preserve">quæ addita ad 130896. </s> <s xml:id="echoid-s5297" xml:space="preserve">ſinum Min. </s> <s xml:id="echoid-s5298" xml:space="preserve">45. </s> <s xml:id="echoid-s5299" xml:space="preserve">faciet 218145. </s> <s xml:id="echoid-s5300" xml:space="preserve">ſinum grad. </s> <s xml:id="echoid-s5301" xml:space="preserve">1. </s> <s xml:id="echoid-s5302" xml:space="preserve">Min. </s> <s xml:id="echoid-s5303" xml:space="preserve"><lb/>15. </s> <s xml:id="echoid-s5304" xml:space="preserve">qui licet ſit aliquanto minor illo, quem prior ille modus exhibet, tamen <lb/>abiectis duabus primis figuris 45. </s> <s xml:id="echoid-s5305" xml:space="preserve">remanebit ſinus 2181. </s> <s xml:id="echoid-s5306" xml:space="preserve">ex quiſitiſsimus grad. </s> <s xml:id="echoid-s5307" xml:space="preserve">1. </s> <s xml:id="echoid-s5308" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s5309" xml:space="preserve">15. </s> <s xml:id="echoid-s5310" xml:space="preserve">reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5311" xml:space="preserve">Item dic. </s> <s xml:id="echoid-s5312" xml:space="preserve">Si Minuta 45. </s> <s xml:id="echoid-s5313" xml:space="preserve">quibus arcus <lb/>grad. </s> <s xml:id="echoid-s5314" xml:space="preserve">1. </s> <s xml:id="echoid-s5315" xml:space="preserve">Min. </s> <s xml:id="echoid-s5316" xml:space="preserve">30. </s> <s xml:id="echoid-s5317" xml:space="preserve">differt ab arcu ſequentigrad 2. </s> <s xml:id="echoid-s5318" xml:space="preserve">Min. </s> <s xml:id="echoid-s5319" xml:space="preserve">15. </s> <s xml:id="echoid-s5320" xml:space="preserve">requirunt differẽtiam <lb/>130829. </s> <s xml:id="echoid-s5321" xml:space="preserve">addendã ſinuigrad. </s> <s xml:id="echoid-s5322" xml:space="preserve">1. </s> <s xml:id="echoid-s5323" xml:space="preserve">Min. </s> <s xml:id="echoid-s5324" xml:space="preserve">30 vt ſiat ſinus grad. </s> <s xml:id="echoid-s5325" xml:space="preserve">2. </s> <s xml:id="echoid-s5326" xml:space="preserve">Min. </s> <s xml:id="echoid-s5327" xml:space="preserve">15. </s> <s xml:id="echoid-s5328" xml:space="preserve">quantã diffe <lb/>rentiam poſcunt Minuta 15. </s> <s xml:id="echoid-s5329" xml:space="preserve">addendam eidẽ ſinui grad. </s> <s xml:id="echoid-s5330" xml:space="preserve">1. </s> <s xml:id="echoid-s5331" xml:space="preserve">Min. </s> <s xml:id="echoid-s5332" xml:space="preserve">30. </s> <s xml:id="echoid-s5333" xml:space="preserve">vt ſiat ſinus <lb/>grad. </s> <s xml:id="echoid-s5334" xml:space="preserve">1. </s> <s xml:id="echoid-s5335" xml:space="preserve">Min. </s> <s xml:id="echoid-s5336" xml:space="preserve">45? </s> <s xml:id="echoid-s5337" xml:space="preserve">quantam præterea differentiã ſlagitant Minuta 30. </s> <s xml:id="echoid-s5338" xml:space="preserve">addendam <lb/>eidem ſinui grad. </s> <s xml:id="echoid-s5339" xml:space="preserve">1. </s> <s xml:id="echoid-s5340" xml:space="preserve">Min. </s> <s xml:id="echoid-s5341" xml:space="preserve">30. </s> <s xml:id="echoid-s5342" xml:space="preserve">vt fiat ſinus grad. </s> <s xml:id="echoid-s5343" xml:space="preserve">2? </s> <s xml:id="echoid-s5344" xml:space="preserve">Reperies enim priorem diffe-<lb/>rentiã eſſe 43610. </s> <s xml:id="echoid-s5345" xml:space="preserve">quę addita ad 261769. </s> <s xml:id="echoid-s5346" xml:space="preserve">ſinum grad. </s> <s xml:id="echoid-s5347" xml:space="preserve">1. </s> <s xml:id="echoid-s5348" xml:space="preserve">Min. </s> <s xml:id="echoid-s5349" xml:space="preserve">30. </s> <s xml:id="echoid-s5350" xml:space="preserve">efficiet 305379. </s> <s xml:id="echoid-s5351" xml:space="preserve"><lb/>ſinum grad. </s> <s xml:id="echoid-s5352" xml:space="preserve">1. </s> <s xml:id="echoid-s5353" xml:space="preserve">Min. </s> <s xml:id="echoid-s5354" xml:space="preserve">45. </s> <s xml:id="echoid-s5355" xml:space="preserve">Poſteriorem vero differentiam inuenies eſſe 87219. </s> <s xml:id="echoid-s5356" xml:space="preserve">quæ <lb/>addita ad 261769. </s> <s xml:id="echoid-s5357" xml:space="preserve">ſinum eundem grad. </s> <s xml:id="echoid-s5358" xml:space="preserve">1. </s> <s xml:id="echoid-s5359" xml:space="preserve">Min. </s> <s xml:id="echoid-s5360" xml:space="preserve">30. </s> <s xml:id="echoid-s5361" xml:space="preserve">componet 348988. </s> <s xml:id="echoid-s5362" xml:space="preserve">ſinũ grad. </s> <s xml:id="echoid-s5363" xml:space="preserve"><lb/>2. </s> <s xml:id="echoid-s5364" xml:space="preserve">qui duo ſinus quamuis minores ſint, quàm illi, quos prior ille modus exhi-<lb/>bet, reiectis tamen duabus figuris primis ex vtroque, reliquierunt ſinus 3053. </s> <s xml:id="echoid-s5365" xml:space="preserve"><lb/>3489. </s> <s xml:id="echoid-s5366" xml:space="preserve">exquiſitiſsimi reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5367" xml:space="preserve">Hac eadem arte progredien <lb/>dum erit in cæteris, donec inuenti ſint ſinus omnium arcuum per 15. </s> <s xml:id="echoid-s5368" xml:space="preserve">Minuta <lb/> <anchor type="note" xlink:label="note-139-01a" xlink:href="note-139-01"/> extenſorum vſque ad arcum grad. </s> <s xml:id="echoid-s5369" xml:space="preserve">45. </s> <s xml:id="echoid-s5370" xml:space="preserve">Vltra hunc etenim progredi hac via nõ <lb/>expedit, cum magis exquiſite ſinus complementorum arcuũ illorum, per pro-<lb/>poſ. </s> <s xml:id="echoid-s5371" xml:space="preserve">3. </s> <s xml:id="echoid-s5372" xml:space="preserve">inueſtigari poſsint, reſpectu ſinus totius 10000000. </s> <s xml:id="echoid-s5373" xml:space="preserve">quàm per regulam <lb/>proportionum; </s> <s xml:id="echoid-s5374" xml:space="preserve">propterea quòd, vt ſupra oſtendimus in coroll. </s> <s xml:id="echoid-s5375" xml:space="preserve">propof. </s> <s xml:id="echoid-s5376" xml:space="preserve">1. </s> <s xml:id="echoid-s5377" xml:space="preserve">dif-<lb/>ferentiæ ſinuum verſus ſinem quadrantis minores ſunt, quàm prope initium <lb/>quadrantis; </s> <s xml:id="echoid-s5378" xml:space="preserve">ac proinde fæpius differentiæ ſinuum mutantur prope ſinem qua-<lb/>drantis, quàm iuxta principium. </s> <s xml:id="echoid-s5379" xml:space="preserve">Ex quo ſit rectius prope quadrantis initium <lb/>reperiri ſinus per regulam proportionum, quàm prope ſinem: </s> <s xml:id="echoid-s5380" xml:space="preserve">quandoquidem <lb/>ibi vna eademq́ue differentia pluribus ſinubus deſeruit, quàm hic, vt dictũ eſt. <lb/></s> <s xml:id="echoid-s5381" xml:space="preserve">Id quod experientia a pertiſsime quoque demonſtrat.</s> <s xml:id="echoid-s5382" xml:space="preserve"/> </p> <div xml:id="echoid-div394" type="float" level="2" n="16"> <note position="left" xlink:label="note-138-03" xlink:href="note-138-03a" xml:space="preserve">Alia ſuppu <lb/>tatio ſinuũ <lb/>arcuũ ſeſe <lb/>Minutis 15 <lb/>ſuperantiũ</note> <note position="right" xlink:label="note-138-04" xlink:href="note-138-04a" xml:space="preserve"> <lb/> ## Arcus # Sinus # Differentiæ <lb/>G # M <lb/>0 # 0 # 000000 <lb/>0 # 45 # 130896 # 130896 <lb/>1 # 30 # 261769 # 130873 <lb/>2 # 15 # 392598 # 130829 <lb/>3 # 0 # 523360 # 130762 <lb/>3 # 45 # 654031 # 130671 <lb/></note> <note position="right" xlink:label="note-139-01" xlink:href="note-139-01a" xml:space="preserve">Sinus eut <lb/>magis ex-<lb/>quiſite per <lb/>regulã pro-<lb/>portionum <lb/>inueniátur <lb/>prope qua-<lb/>drantis ini <lb/>tiũ quàm <lb/>prope finé.</note> </div> <p> <s xml:id="echoid-s5383" xml:space="preserve">QVOD ſi omnes ſinus arcuum per 15. </s> <s xml:id="echoid-s5384" xml:space="preserve">Minuta progredientium, initio fa-<lb/> <anchor type="note" xlink:label="note-139-02a" xlink:href="note-139-02"/> cto ab arcu Min. </s> <s xml:id="echoid-s5385" xml:space="preserve">0. </s> <s xml:id="echoid-s5386" xml:space="preserve">in ordinem redigantur, vna cum eorum differentijs; </s> <s xml:id="echoid-s5387" xml:space="preserve">repe-<lb/>riemus eodem artificio ſinus omnium arcuum per 5. </s> <s xml:id="echoid-s5388" xml:space="preserve">minuta extenſorum: </s> <s xml:id="echoid-s5389" xml:space="preserve">Ex <lb/>quibus demum eadem ratione ſinus omnium arcuum per ſingula minuta pro-<lb/>gredientium explorabimus; </s> <s xml:id="echoid-s5390" xml:space="preserve">ac proinde tabulam ſinuum conficiem us vſque ad <lb/>arcum gra. </s> <s xml:id="echoid-s5391" xml:space="preserve">45. </s> <s xml:id="echoid-s5392" xml:space="preserve">Per ſinus autem horũ arcuũ eliciemus, per propof. </s> <s xml:id="echoid-s5393" xml:space="preserve">3. </s> <s xml:id="echoid-s5394" xml:space="preserve">ſinus com-<lb/>plementorum arcuum eorundem. </s> <s xml:id="echoid-s5395" xml:space="preserve">Quare tota ſinuum tabula conſecta erit: </s> <s xml:id="echoid-s5396" xml:space="preserve">ac <lb/>proinde ſinus rectos omnium arcuum Quadrantis ſeſe ordine ſuperantium <lb/>vno Minuto, in partibus Sinus totius in quotcunque particulas diſtributi, <lb/>ſupputauimus. </s> <s xml:id="echoid-s5397" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s5398" xml:space="preserve"/> </p> <div xml:id="echoid-div395" type="float" level="2" n="17"> <note position="right" xlink:label="note-139-02" xlink:href="note-139-02a" xml:space="preserve">Supputatio <lb/>ſinuum ar-<lb/>cuum per <lb/>5. Minuta, <lb/>& perſingu <lb/>la Minura <lb/>extenſorũ.</note> </div> </div> <div xml:id="echoid-div397" type="section" level="1" n="167"> <head xml:id="echoid-head193" xml:space="preserve">SCHOLIVM.</head> <note position="right" xml:space="preserve">Demonſtr@ <lb/>tio ſupputa <lb/>tiõis ſinuũ <lb/>per regulã <lb/>proportio-<lb/>num.</note> <p style="it"> <s xml:id="echoid-s5399" xml:space="preserve">_CAETERVM_ vt rationem ſupputationis ſinuum per proportionum regulam <lb/>videas, ſint in Quadrante _ABC_, arcuum _BD_, _BE_, ſinus _DF_, _EG_, noti, ex quibus <lb/>propoſitum ſit elicere ſinum _HI_, arcus _BH_, inter duos illos arcus poſiti Ducta recta <lb/>_DK_, perpendiculari ad _EG_; </s> <s xml:id="echoid-s5400" xml:space="preserve">erunt rectæ _LI_, _KG_, ſinui _DF_, æquales s atque adeo <pb o="128" file="140" n="140" rhead=""/> _HL_, _EK_, differentiæ inter ſinus _HI_, _EG_, & </s> <s xml:id="echoid-s5401" xml:space="preserve">ſinum _DT_. </s> <s xml:id="echoid-s5402" xml:space="preserve">Quoniam vero arcus _DE_, <lb/>ſi e xiguus fuerit, qualem nos ponimus, nempe Minutorum _45_. </s> <s xml:id="echoid-s5403" xml:space="preserve">vel _15_. </s> <s xml:id="echoid-s5404" xml:space="preserve">vel _5_. </s> <s xml:id="echoid-s5405" xml:space="preserve">(Ex his <lb/>enim ſolis intermediorum arcuum ſinus inquirimus) à recta linea, quoad ſenſum, non <lb/> <anchor type="figure" xlink:label="fig-140-01a" xlink:href="fig-140-01"/> differt, ac proinde multo minus arcus _DH_; </s> <s xml:id="echoid-s5406" xml:space="preserve">erunt trianæ <lb/>gula _DEK_, _DHL_, quodammodo rectilinea. </s> <s xml:id="echoid-s5407" xml:space="preserve">& </s> <s xml:id="echoid-s5408" xml:space="preserve">æquian-<lb/>gula inter ſe. </s> <s xml:id="echoid-s5409" xml:space="preserve">Quamobrem erit, vt _DE_, differentia inter <lb/> <anchor type="note" xlink:label="note-140-01a" xlink:href="note-140-01"/> arcum _BD_, & </s> <s xml:id="echoid-s5410" xml:space="preserve">arcum _BE_, ad _EK_, differentiam inter ſi-<lb/>num _DF_, & </s> <s xml:id="echoid-s5411" xml:space="preserve">ſinum _EG_, ita _DH_, differentia inter ar-<lb/>cum _BD_, & </s> <s xml:id="echoid-s5412" xml:space="preserve">arcum _BH_, ad _HL_, differentiam inter ſi-<lb/>num _DF_, & </s> <s xml:id="echoid-s5413" xml:space="preserve">ſinum _HI_. </s> <s xml:id="echoid-s5414" xml:space="preserve">Cum ergo priores tres magnitu-<lb/>dines ſint cognitæ, (nempe arcus _DE_, quo arcus _BE_, arcũ <lb/>_BD_, ſuperat; </s> <s xml:id="echoid-s5415" xml:space="preserve">& </s> <s xml:id="echoid-s5416" xml:space="preserve">recta _EK_, qua ſinus _EG_, ſinum _DF_, <lb/>excedit; </s> <s xml:id="echoid-s5417" xml:space="preserve">nec non arcus _DH_, quo arcus _BH_, cuius ſinus <lb/>queritur, ſuperat arcum _BD_.) </s> <s xml:id="echoid-s5418" xml:space="preserve">cognita etiam erit quarta magnitudo, id eſt, recta _HL_, <lb/>quæ addenda eſt ſinui _DF_, vt fiat ſinus _HI_. </s> <s xml:id="echoid-s5419" xml:space="preserve">Conſtat igitur, ſinus per regulam pro-<lb/>portionum inuentos ſenſibiliter non differre à veris ſinubus, præſertim quando arcus <lb/>_DE_, quo arcus cognitorum ſinuuminter ſe differunt, valde exiguus eſt, ita vt à recta <lb/>linea vix differat.</s> <s xml:id="echoid-s5420" xml:space="preserve"/> </p> <div xml:id="echoid-div397" type="float" level="2" n="1"> <figure xlink:label="fig-140-01" xlink:href="fig-140-01a"> <image file="140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/140-01"/> </figure> <note position="left" xlink:label="note-140-01" xlink:href="note-140-01a" xml:space="preserve">4. fexti.</note> </div> <p style="it"> <s xml:id="echoid-s5421" xml:space="preserve">_MAGNVM_ quoque compendium in hoc negotio nobis afferet propoſitio octaua. <lb/></s> <s xml:id="echoid-s5422" xml:space="preserve"> <anchor type="note" xlink:label="note-140-02a" xlink:href="note-140-02"/> Ex ea enim plurimos ſinus ex alijs inuentis per ſolam additionem, ſubtractionemue <lb/>nanciſcemur. </s> <s xml:id="echoid-s5423" xml:space="preserve">_N_am ſi ſinum cuiuſuis arcus, qui maior non ſit, quàm grad. </s> <s xml:id="echoid-s5424" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5425" xml:space="preserve">addamus <lb/>ſinui arcus, qui ab arcugrad. </s> <s xml:id="echoid-s5426" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5427" xml:space="preserve">ſuperatur ſumpto illo arcu, componemus ſinum arcus <lb/>qui eodemillo arcu aſſumpto arcũgrad. </s> <s xml:id="echoid-s5428" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5429" xml:space="preserve">ſuperat: </s> <s xml:id="echoid-s5430" xml:space="preserve">propterea quod differentia inter <lb/>ſinus duorum horum arcuum maiorum æqualis eſt ſinui arcus illius aſſumpti, qui ma <lb/>ior nõ ponitur, quàm grad. </s> <s xml:id="echoid-s5431" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5432" xml:space="preserve">vt ibi demonſtrauimus. </s> <s xml:id="echoid-s5433" xml:space="preserve">Vt ſi _3461171_. </s> <s xml:id="echoid-s5434" xml:space="preserve">ſinũ arcus grad. <lb/></s> <s xml:id="echoid-s5435" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5436" xml:space="preserve">Min. </s> <s xml:id="echoid-s5437" xml:space="preserve">_15_. </s> <s xml:id="echoid-s5438" xml:space="preserve">adijciamus ad _6394390_. </s> <s xml:id="echoid-s5439" xml:space="preserve">ſinum arcus grad. </s> <s xml:id="echoid-s5440" xml:space="preserve">_39_. </s> <s xml:id="echoid-s5441" xml:space="preserve">Min. </s> <s xml:id="echoid-s5442" xml:space="preserve">_45._</s> <s xml:id="echoid-s5443" xml:space="preserve">. </s> <s xml:id="echoid-s5444" xml:space="preserve">qui abarcu grad. </s> <s xml:id="echoid-s5445" xml:space="preserve"><lb/>_60_. </s> <s xml:id="echoid-s5446" xml:space="preserve">ſuperatur dicto arcu grad. </s> <s xml:id="echoid-s5447" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5448" xml:space="preserve">Min. </s> <s xml:id="echoid-s5449" xml:space="preserve">_15_. </s> <s xml:id="echoid-s5450" xml:space="preserve">conficiemus _9855561_. </s> <s xml:id="echoid-s5451" xml:space="preserve">ſinum arcus grad. </s> <s xml:id="echoid-s5452" xml:space="preserve"><lb/>_80_. </s> <s xml:id="echoid-s5453" xml:space="preserve">Min. </s> <s xml:id="echoid-s5454" xml:space="preserve">_15_. </s> <s xml:id="echoid-s5455" xml:space="preserve">qui arcum gra. </s> <s xml:id="echoid-s5456" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5457" xml:space="preserve">eodem arcu grad. </s> <s xml:id="echoid-s5458" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5459" xml:space="preserve">Min _15_. </s> <s xml:id="echoid-s5460" xml:space="preserve">ſuperat. </s> <s xml:id="echoid-s5461" xml:space="preserve">Sic ſi _5000000_. </s> <s xml:id="echoid-s5462" xml:space="preserve"><lb/>ſinum arcus grad. </s> <s xml:id="echoid-s5463" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5464" xml:space="preserve">addamus ſinui _5000000_. </s> <s xml:id="echoid-s5465" xml:space="preserve">arcus grad _30_. </s> <s xml:id="echoid-s5466" xml:space="preserve">quem arcus grad. </s> <s xml:id="echoid-s5467" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5468" xml:space="preserve"><lb/>fuperat dicto illo arcu grad. </s> <s xml:id="echoid-s5469" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5470" xml:space="preserve">componemus _10000000_. </s> <s xml:id="echoid-s5471" xml:space="preserve">ſinum arcus grad. </s> <s xml:id="echoid-s5472" xml:space="preserve">_90_. </s> <s xml:id="echoid-s5473" xml:space="preserve">qui <lb/>arcum grad. </s> <s xml:id="echoid-s5474" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5475" xml:space="preserve">eodem illo arcu aſſumpto grad. </s> <s xml:id="echoid-s5476" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5477" xml:space="preserve">ſuperat.</s> <s xml:id="echoid-s5478" xml:space="preserve"/> </p> <div xml:id="echoid-div398" type="float" level="2" n="2"> <note position="left" xlink:label="note-140-02" xlink:href="note-140-02a" xml:space="preserve">Compédiũ <lb/>miri ficum <lb/>pro inuen-<lb/>rione pluti <lb/>morum ſi-<lb/>nuum.</note> </div> <p style="it"> <s xml:id="echoid-s5479" xml:space="preserve">_ITEM_ ſi ſinum cuiuslibet arcus, qui arcu grad. </s> <s xml:id="echoid-s5480" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5481" xml:space="preserve">maior non ſit, ſubducamus ex <lb/>ſinu arcus, qui arcum grad _60_. </s> <s xml:id="echoid-s5482" xml:space="preserve">ſumpto illo arcu ſuperat, relinquetur ſinus arcus, qui <lb/>eodem illo arcu aſſumpto ab arcu grad. </s> <s xml:id="echoid-s5483" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5484" xml:space="preserve">ſuperatur. </s> <s xml:id="echoid-s5485" xml:space="preserve">Vt ſi _3502075_. </s> <s xml:id="echoid-s5486" xml:space="preserve">ſinum arcus <lb/>grad. </s> <s xml:id="echoid-s5487" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5488" xml:space="preserve">Min. </s> <s xml:id="echoid-s5489" xml:space="preserve">_30_ detrahamus ex _9862856_. </s> <s xml:id="echoid-s5490" xml:space="preserve">ſinu arcus grad _80_. </s> <s xml:id="echoid-s5491" xml:space="preserve">Min. </s> <s xml:id="echoid-s5492" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5493" xml:space="preserve">qui arcum <lb/>grad. </s> <s xml:id="echoid-s5494" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5495" xml:space="preserve">dicto arcu grad. </s> <s xml:id="echoid-s5496" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5497" xml:space="preserve">Min. </s> <s xml:id="echoid-s5498" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5499" xml:space="preserve">ſuperat, reliquus erit _6360781_. </s> <s xml:id="echoid-s5500" xml:space="preserve">ſinus arcus gra. <lb/></s> <s xml:id="echoid-s5501" xml:space="preserve">_39_. </s> <s xml:id="echoid-s5502" xml:space="preserve">Min. </s> <s xml:id="echoid-s5503" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5504" xml:space="preserve">quem arcus grad. </s> <s xml:id="echoid-s5505" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5506" xml:space="preserve">eodemillo arcu grad. </s> <s xml:id="echoid-s5507" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5508" xml:space="preserve">Min. </s> <s xml:id="echoid-s5509" xml:space="preserve">_30_ ſuperat.</s> <s xml:id="echoid-s5510" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5511" xml:space="preserve">_RVRSVS_, ſi ex ſinu cuiuſuis arcus, qui maior ſit arcu grad. </s> <s xml:id="echoid-s5512" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5513" xml:space="preserve">detrahatur ſinus <lb/>arcus, quitanto minor ſit arcu grad. </s> <s xml:id="echoid-s5514" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5515" xml:space="preserve">quanto ille maior eſt, relinquetur ſinus ar-<lb/>cus, quo vteruis illorũ ab arcu grad. </s> <s xml:id="echoid-s5516" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5517" xml:space="preserve">differt. </s> <s xml:id="echoid-s5518" xml:space="preserve">Vt ſi ex _9781476_. </s> <s xml:id="echoid-s5519" xml:space="preserve">ſinu arcus gra. </s> <s xml:id="echoid-s5520" xml:space="preserve">_78_. <lb/></s> <s xml:id="echoid-s5521" xml:space="preserve">auferatur _6691306_. </s> <s xml:id="echoid-s5522" xml:space="preserve">ſinus arcus gra. </s> <s xml:id="echoid-s5523" xml:space="preserve">_42_. </s> <s xml:id="echoid-s5524" xml:space="preserve">reliquus erit _3090170_. </s> <s xml:id="echoid-s5525" xml:space="preserve">ſinus arcus grad. </s> <s xml:id="echoid-s5526" xml:space="preserve">_18_. </s> <s xml:id="echoid-s5527" xml:space="preserve"><lb/>quo vterq; </s> <s xml:id="echoid-s5528" xml:space="preserve">illorũ ab arcu grad. </s> <s xml:id="echoid-s5529" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5530" xml:space="preserve">differt. </s> <s xml:id="echoid-s5531" xml:space="preserve">Quæ omnia ex dicta propeſ. </s> <s xml:id="echoid-s5532" xml:space="preserve">_8_. </s> <s xml:id="echoid-s5533" xml:space="preserve">colliguntur.</s> <s xml:id="echoid-s5534" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5535" xml:space="preserve">_ITAQVE_ ſatis eſt, vt inueniantur per regulam proportionum ſinus ommum ar-<lb/>cuum à principio quadrantis vſque ad arcum grad. </s> <s xml:id="echoid-s5536" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5537" xml:space="preserve">Si enim ex his eliciantur ſinus <lb/>complementorum, & </s> <s xml:id="echoid-s5538" xml:space="preserve">ex his inuentis detrahãtur priores illi ſinus, reli qui erunt ſinus <lb/>omnium arcuum inter arcum grad. </s> <s xml:id="echoid-s5539" xml:space="preserve">_30_ & </s> <s xml:id="echoid-s5540" xml:space="preserve">arcum grad. </s> <s xml:id="echoid-s5541" xml:space="preserve">_90_. </s> <s xml:id="echoid-s5542" xml:space="preserve">Item ſicogniti eſſent ſinus <lb/>arcuũ omniũ ab arcu grad. </s> <s xml:id="echoid-s5543" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5544" xml:space="preserve">vſq; </s> <s xml:id="echoid-s5545" xml:space="preserve">ad finem quadrantis, & </s> <s xml:id="echoid-s5546" xml:space="preserve">ſinus omniũ arcuum, qui <pb o="129" file="141" n="141" rhead=""/> minores ſint arcu grad. </s> <s xml:id="echoid-s5547" xml:space="preserve">60. </s> <s xml:id="echoid-s5548" xml:space="preserve">detraherentur ex ſinubus omnium arcuum maiorũ, quàm <lb/>grad. </s> <s xml:id="echoid-s5549" xml:space="preserve">60. </s> <s xml:id="echoid-s5550" xml:space="preserve">remanerent ſinus omnium arcuum à principio quadrantis vſque ad arcum <lb/>grad. </s> <s xml:id="echoid-s5551" xml:space="preserve">30. </s> <s xml:id="echoid-s5552" xml:space="preserve">Denique ſi ſinus omnium arcuum à principio quadrantis vſque ad arcum <lb/>grad. </s> <s xml:id="echoid-s5553" xml:space="preserve">60. </s> <s xml:id="echoid-s5554" xml:space="preserve">inuenti eſſent, & </s> <s xml:id="echoid-s5555" xml:space="preserve">ſinus omnium arcuum minorum, quàm grad. </s> <s xml:id="echoid-s5556" xml:space="preserve">30. </s> <s xml:id="echoid-s5557" xml:space="preserve">ſinubus <lb/>omnium arcuum maiorum, quàm grad. </s> <s xml:id="echoid-s5558" xml:space="preserve">30. </s> <s xml:id="echoid-s5559" xml:space="preserve">adijcerentur, componerentur ſinus omnium <lb/>arcuum maiorum, quàm grad. </s> <s xml:id="echoid-s5560" xml:space="preserve">60.</s> <s xml:id="echoid-s5561" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s5562" xml:space="preserve">_SED_ iam ſubijciamus tabulam ſinuum omnium arcuum quadrantis per ſingula <lb/>Minuta extenſorumà Ioanne Regiom. </s> <s xml:id="echoid-s5563" xml:space="preserve">ſupputatam, quam tamen pleriſque in locis ab <lb/>erroribus, qui incuria typographorum irrepſerant, purgauimus. </s> <s xml:id="echoid-s5564" xml:space="preserve">Cõtinet autem in hac <lb/>tabula ſinus totus particulas 10000000. </s> <s xml:id="echoid-s5565" xml:space="preserve">ratione cuius omnes alij ſinus inuenti ſunt. <lb/></s> <s xml:id="echoid-s5566" xml:space="preserve">Quòd ſi à ſingulis reijciantur primæ duæ figuræ ad dexteram, (addita tamen vnita-<lb/>te, ſi duæ figuræ abiectæ numerum maiorem conſtituant, quàm 50.) </s> <s xml:id="echoid-s5567" xml:space="preserve">relinquetur ſinus <lb/>reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5568" xml:space="preserve">Si tamen quis eandem tabulam per præcepta tradita <lb/> <anchor type="note" xlink:label="note-141-01a" xlink:href="note-141-01"/> proprio Marte conſtrnere velit, poſito ſinu toto partium 10000000. </s> <s xml:id="echoid-s5569" xml:space="preserve">conſtituendus <lb/>ei erit ſinus totus in ſupputatione partium 1000000000. </s> <s xml:id="echoid-s5570" xml:space="preserve">_N_am ſi ex ſingulis ſinubus <lb/>inuentis abijciantur duæ primæ figuræ ad dexteram, vt diximus, reliqui erunt ſinus <lb/>reſpectu ſinus totius 10000000. </s> <s xml:id="echoid-s5571" xml:space="preserve">ijdem omnino, qui in tabula Ioannis Regiom. </s> <s xml:id="echoid-s5572" xml:space="preserve">de-<lb/>ſcripti ſunt. </s> <s xml:id="echoid-s5573" xml:space="preserve">_H_oc idcirco dico, ne mireris, non omnes ſinus per regulam proportionũ <lb/>inuentos ex ſinubus arcuum Minutis 45. </s> <s xml:id="echoid-s5574" xml:space="preserve">ſe ordine ſuperantium, poſito ſinu toto <lb/>10000000. </s> <s xml:id="echoid-s5575" xml:space="preserve">ad vnguem reſpondere ſinubus huius tabulæ. </s> <s xml:id="echoid-s5576" xml:space="preserve">Vt enim ſinus exquiſitere-<lb/>periatur reſpectu alicuius ſinus totius, conſtituendus eſt in ſupputatione ſinus totus <lb/>centies maior, vt ſupra dictum eſt.</s> <s xml:id="echoid-s5577" xml:space="preserve"/> </p> <div xml:id="echoid-div399" type="float" level="2" n="3"> <note position="right" xlink:label="note-141-01" xlink:href="note-141-01a" xml:space="preserve">Quid ageu <lb/>dum, vt ſe-<lb/>quens tabu <lb/>la ſinuum <lb/>exquiſite có <lb/>ſtruat, poſi-<lb/>to ſinu to-<lb/>to partium <lb/>10000000.</note> </div> <p style="it"> <s xml:id="echoid-s5578" xml:space="preserve">_QVOD_ autẽ, abiectis duabus primis figuris ad dexterã ex ſingulis ſinubus, rema <lb/> <anchor type="note" xlink:label="note-141-02a" xlink:href="note-141-02"/> neant ſinus exquiſiti reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5579" xml:space="preserve">quàmuis priores illi nõ ſint exqui <lb/>ſite inuenti, manifeſtũ eſt. </s> <s xml:id="echoid-s5580" xml:space="preserve">Quoniã enim ſinus totus, ſiue ſemidiameter ad ſinum rectũ <lb/>quemcumq; </s> <s xml:id="echoid-s5581" xml:space="preserve">determinatã quandã proportionẽ habet; </s> <s xml:id="echoid-s5582" xml:space="preserve">fit, vt omnes partes illius, quot-<lb/>cunq; </s> <s xml:id="echoid-s5583" xml:space="preserve">illæ ſint, ad partes huius inuẽtas reſpectu illarũ partiũ ſinus totius eandẽ habeãt <lb/>proportionem, quam omnes partes eiusdẽ ſinus totius pauciores, quàm illæ priores, ha <lb/>bent ad partes eiuſdem ſinus reſpectu illarum partium ſinus totius pauciorum : </s> <s xml:id="echoid-s5584" xml:space="preserve">alio-<lb/>quin ſinus totus non haberet ſemper ad eundem ſinum eandem proportionem, ſed ali-<lb/>quando eſſet maioris quantitatis reſpectu illius, & </s> <s xml:id="echoid-s5585" xml:space="preserve">ali quando minoris : </s> <s xml:id="echoid-s5586" xml:space="preserve">quod eſt ab-<lb/>ſurdum. </s> <s xml:id="echoid-s5587" xml:space="preserve">Quocirca ſi ſinus cuiuslibet arcus, vt v. </s> <s xml:id="echoid-s5588" xml:space="preserve">g illius, qui continet grad. </s> <s xml:id="echoid-s5589" xml:space="preserve">28. </s> <s xml:id="echoid-s5590" xml:space="preserve">inuen-<lb/>tus ſit reſpectu ſinus totius in quotuis particulas diſtributi, facile per regulam pro-<lb/>portionum inueniemus eundem ſinũ reſpectu eiuſdem ſinus totius in pauciores partes <lb/>diuiſi, ſi ita dicamus. </s> <s xml:id="echoid-s5591" xml:space="preserve">Si ſinus totus partium 10000000. </s> <s xml:id="echoid-s5592" xml:space="preserve">dat ſinum arcus grad. </s> <s xml:id="echoid-s5593" xml:space="preserve">28. </s> <s xml:id="echoid-s5594" xml:space="preserve">par <lb/>tium 4694716. </s> <s xml:id="echoid-s5595" xml:space="preserve">Idem ſinus totus partium 100000. </s> <s xml:id="echoid-s5596" xml:space="preserve">quot partium dabit ſinam eiuſdẽ <lb/>arcus grad. </s> <s xml:id="echoid-s5597" xml:space="preserve">28? </s> <s xml:id="echoid-s5598" xml:space="preserve">Inueniemus enim ſinum partium 46947 {1600000/10000000}. </s> <s xml:id="echoid-s5599" xml:space="preserve">Omiſ-<lb/>ſa autem hac fractione, (quòd minor ſit, quàm vnitas.) </s> <s xml:id="echoid-s5600" xml:space="preserve">continebit idem ſinus partes <lb/>duntaxat 46947. </s> <s xml:id="echoid-s5601" xml:space="preserve">quemadmodum in tabulis ſinuum ponitur, in quibus ſinus totus <lb/>continet partes 100000. </s> <s xml:id="echoid-s5602" xml:space="preserve">Vbi vides, ſinum hunc relinqui, ſi ex illo duæ primæ figuræ <lb/>ad dexteram abijciantur. </s> <s xml:id="echoid-s5603" xml:space="preserve">Ratio huius abiectionis eſt; </s> <s xml:id="echoid-s5604" xml:space="preserve">quia vt ſinus ille grad. </s> <s xml:id="echoid-s5605" xml:space="preserve">28. </s> <s xml:id="echoid-s5606" xml:space="preserve">mul-<lb/>tiplicatus per ſinum totum 100000 (quæ multiplicatio fit per appoſitionem quinque <lb/>cifrarũ, hoc modo; </s> <s xml:id="echoid-s5607" xml:space="preserve">469471600000. </s> <s xml:id="echoid-s5608" xml:space="preserve">vt in cap. </s> <s xml:id="echoid-s5609" xml:space="preserve">4. </s> <s xml:id="echoid-s5610" xml:space="preserve">Arithmeticæ diximus.) </s> <s xml:id="echoid-s5611" xml:space="preserve">diuidatur per <lb/>ſinum totum 10000000. </s> <s xml:id="echoid-s5612" xml:space="preserve">vt regula proportionnm præcipit; </s> <s xml:id="echoid-s5613" xml:space="preserve">ſatis eſt, ſi ex numero pro-<lb/>ducto reijciantur ſeptem figuræ, quotnimirum cifræ ſunt in diuiſore 10000000. <lb/></s> <s xml:id="echoid-s5614" xml:space="preserve">vt in cap. </s> <s xml:id="echoid-s5615" xml:space="preserve">5. </s> <s xml:id="echoid-s5616" xml:space="preserve">Arithmeticæ docuimus. </s> <s xml:id="echoid-s5617" xml:space="preserve">Quare reijciendæ ſunt quing; </s> <s xml:id="echoid-s5618" xml:space="preserve">illæ cifræ appoſitæ, <lb/>& </s> <s xml:id="echoid-s5619" xml:space="preserve">præterea duæ figuræ primæ, nẽpe hic numerus 1600000. </s> <s xml:id="echoid-s5620" xml:space="preserve">qui cum diuiſore hanc mi <pb o="130" file="142" n="142" rhead=""/> nutiam {1600000/10000000}. </s> <s xml:id="echoid-s5621" xml:space="preserve">conſtituit, quæ vnitate minor eſt; </s> <s xml:id="echoid-s5622" xml:space="preserve">proptered quòd n@ <lb/>merator cotinet ſeptem figuras, denominator autem octo. </s> <s xml:id="echoid-s5623" xml:space="preserve">Eademq́; </s> <s xml:id="echoid-s5624" xml:space="preserve">ratio eſt in omnibus <lb/>alijs ſinubus. </s> <s xml:id="echoid-s5625" xml:space="preserve">Hinc fit, ſinum relictum poſt abiectionem duarum primarum figurarum <lb/>ſatis exquiſitum eſſe reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5626" xml:space="preserve">etiamſi ille, a quo duæ figuræ abij <lb/>ciuntur, reſpectu ſinus totius 10000000. </s> <s xml:id="echoid-s5627" xml:space="preserve">non eſſet exquiſite inuentus. </s> <s xml:id="echoid-s5628" xml:space="preserve">Cum enim to-<lb/>tus error, qui in ſupputatione contingere poteſt, (quando nimirum conſtituendus eſt <lb/>ſinus inter duos numeros, quorum vnus vero ſinu maior eſt, & </s> <s xml:id="echoid-s5629" xml:space="preserve">alter minor; </s> <s xml:id="echoid-s5630" xml:space="preserve">Vel quan-<lb/>do per regulam proportionum ſinus inquiritur: </s> <s xml:id="echoid-s5631" xml:space="preserve">hic enim maius periculum errandi eſ-<lb/>ſe poteſt. </s> <s xml:id="echoid-s5632" xml:space="preserve">Nam quando ſinus inuenitur per extractionem radicis quadratæ, error <lb/>vnitatem non excedit.) </s> <s xml:id="echoid-s5633" xml:space="preserve">conſiſtat vel in prima ſola figura ad dexteram, vel in duabus <lb/>primis, ita vt ad ſummum error ſit in 99. </s> <s xml:id="echoid-s5634" xml:space="preserve">vnitatibus, quibus ſinus inuentus verum ſi-<lb/>num excedat, vel ab eo deficiat; </s> <s xml:id="echoid-s5635" xml:space="preserve">(quis enim pluribus vnitatibus à ſcopo aberret, niſi <lb/>plane rerum Geometricarum, at que Arithmeticarum ſit ignarus?) </s> <s xml:id="echoid-s5636" xml:space="preserve">Duæ vero primæ <lb/>figuræ cum quinque cifris appoſitis conſtituant numeratorem fractionis, quam diui-<lb/>ſio exhibet, minorem denominatore, ita vt fractio minor ſit, quàm vnitas; </s> <s xml:id="echoid-s5637" xml:space="preserve">liquet, <lb/>ſatis exquiſitum ſinum relinqui.</s> <s xml:id="echoid-s5638" xml:space="preserve"/> </p> <div xml:id="echoid-div400" type="float" level="2" n="4"> <note position="right" xlink:label="note-141-02" xlink:href="note-141-02a" xml:space="preserve">Ratio, <lb/>cur abtectis <lb/>duabus pri <lb/>mis figuris <lb/>ex ſinu <lb/>quocũque <lb/>reſpectu ſi-<lb/>nus totius <lb/>10000000. <lb/>relinqua -<lb/>tut idé ſi-<lb/>nus reſpe-<lb/>ctu ſinꝰ to-<lb/>tiꝰ 100000. <lb/>licet prior <lb/>ille Ron ſit <lb/>exquiſitein <lb/>uentus.</note> </div> <note position="left" xml:space="preserve">Siad ſinũ <lb/>quemcúq; <lb/>reſpectu ſi-<lb/>nus totius <lb/>100000. <lb/>adijciãtut <lb/>duæ cifræ <lb/>ad dexterã, <lb/>fitidẽ ſinꝰ <lb/>reſpectu ſi-<lb/>nus totius <lb/>10000000.</note> <p style="it"> <s xml:id="echoid-s5639" xml:space="preserve">_E A D E M_ ratione, ſicuilibet ſinui reſpectu ſinus totius partium 100000. </s> <s xml:id="echoid-s5640" xml:space="preserve">inuen-<lb/>to apponãtur duæ cifræ ad dexteram, habebitur idem ſinus reſpectu ſinus totius par-<lb/>tium 10000000. </s> <s xml:id="echoid-s5641" xml:space="preserve">Nam ſi dicamus verbi gratia; </s> <s xml:id="echoid-s5642" xml:space="preserve">Sinus totus partium 100000. </s> <s xml:id="echoid-s5643" xml:space="preserve">dat <lb/>ſinum arcus grad. </s> <s xml:id="echoid-s5644" xml:space="preserve">28. </s> <s xml:id="echoid-s5645" xml:space="preserve">partium 46947: </s> <s xml:id="echoid-s5646" xml:space="preserve">Sinus ergo totus partiũ 10000000. </s> <s xml:id="echoid-s5647" xml:space="preserve">quot par-<lb/>tium dabit eundẽ ſinũ arcus grad. </s> <s xml:id="echoid-s5648" xml:space="preserve">28? </s> <s xml:id="echoid-s5649" xml:space="preserve">reperiemus ſinũ partium 4694700. </s> <s xml:id="echoid-s5650" xml:space="preserve">Vbi vides, <lb/>ſinum hunc procreari, ſi illi duæ ciſræ ad dexteram adijciantur. </s> <s xml:id="echoid-s5651" xml:space="preserve">Ratio huius adiectio-<lb/>nis eſt; </s> <s xml:id="echoid-s5652" xml:space="preserve">quia vt ſinus ille grad. </s> <s xml:id="echoid-s5653" xml:space="preserve">28. </s> <s xml:id="echoid-s5654" xml:space="preserve">multiplicatus per ſinum totum 10000000. </s> <s xml:id="echoid-s5655" xml:space="preserve">(quæ <lb/>multiplicatio fit per appoſitionem ſeptem cifrarum, vt in cap 4. </s> <s xml:id="echoid-s5656" xml:space="preserve">Arithmeticæ tradidi-<lb/>mus) diuidatur per ſinum totum 100000. </s> <s xml:id="echoid-s5657" xml:space="preserve">vt regula proportionum præcipit, ſatis eſt, <lb/>ſi ex numero producto 469470000000. </s> <s xml:id="echoid-s5658" xml:space="preserve">auferantur quinque cifræ, vt ex cap. </s> <s xml:id="echoid-s5659" xml:space="preserve">5. </s> <s xml:id="echoid-s5660" xml:space="preserve">no <lb/>ſtræ Arithmeticæ conſtat. </s> <s xml:id="echoid-s5661" xml:space="preserve">Quocirca relinquetur prior ſinus cum duabus cifris ad dex-<lb/>tram appoſitis. </s> <s xml:id="echoid-s5662" xml:space="preserve">Quòd autẽ ex ſinu 46947. </s> <s xml:id="echoid-s5663" xml:space="preserve">non sit inuentus sinus 4694716. </s> <s xml:id="echoid-s5664" xml:space="preserve">ille idẽ, ex <lb/>quo prius illum elicuimus, ſed ſolum hic 4694700. </s> <s xml:id="echoid-s5665" xml:space="preserve">cauſa eſt, quòd sinus 46947. </s> <s xml:id="echoid-s5666" xml:space="preserve">non eſt <lb/>omnino exquisitus reſpectu sinus totius 100000. </s> <s xml:id="echoid-s5667" xml:space="preserve">Deberet namq; </s> <s xml:id="echoid-s5668" xml:space="preserve">eſſe 46947. </s> <s xml:id="echoid-s5669" xml:space="preserve">& </s> <s xml:id="echoid-s5670" xml:space="preserve">inſu-<lb/>per {1600000/10000000}. </s> <s xml:id="echoid-s5671" xml:space="preserve">vt ex dictis patet, ex quo præciſe inuenietur sinus ille <lb/>4694716. </s> <s xml:id="echoid-s5672" xml:space="preserve">Sed licet hæ 16. </s> <s xml:id="echoid-s5673" xml:space="preserve">vnitates negligantur, accipiaturq́; </s> <s xml:id="echoid-s5674" xml:space="preserve">sinus 4694700. </s> <s xml:id="echoid-s5675" xml:space="preserve">qua-<lb/>lem inuenimus, non tamen fit error notabilis, cum 16. </s> <s xml:id="echoid-s5676" xml:space="preserve">vnitates reſpectu sinus totius <lb/>sint {16/10000000}. </s> <s xml:id="echoid-s5677" xml:space="preserve">quæ minutia multo minor eſt, quàm {1/216000}. <lb/></s> <s xml:id="echoid-s5678" xml:space="preserve">hoc eſt, quàm vnũ ſecundũ reſpectu sinus totius 60. </s> <s xml:id="echoid-s5679" xml:space="preserve">vt merito negligi poßit. </s> <s xml:id="echoid-s5680" xml:space="preserve">Ad ſum <lb/>mũ poterit aliquãdo cõtingere error, quàmuis valde raro, in {99/10000000}. </s> <s xml:id="echoid-s5681" xml:space="preserve"><lb/>quæ minutia licet sit ali quanto maior, quàm {1/216000}. </s> <s xml:id="echoid-s5682" xml:space="preserve">hoc eſt, quàm vnum <lb/>Secundum reſpectu sinus totius 60 eſt tamen multo minor, quàm {1/3600}. </s> <s xml:id="echoid-s5683" xml:space="preserve">hoc eſt, <lb/>quàm vnum Minutum reſpectu sinus totius 60. </s> <s xml:id="echoid-s5684" xml:space="preserve">Id vero, quod de sinubus totis partiũ <lb/>10000000 & </s> <s xml:id="echoid-s5685" xml:space="preserve">100000. </s> <s xml:id="echoid-s5686" xml:space="preserve">diximus, intelligendum quoq; </s> <s xml:id="echoid-s5687" xml:space="preserve">eſt de alijs sinubus totis quot-<lb/> <anchor type="note" xlink:label="note-142-02a" xlink:href="note-142-02"/> cunq; </s> <s xml:id="echoid-s5688" xml:space="preserve">partium siue plurium, siue pauciorum. </s> <s xml:id="echoid-s5689" xml:space="preserve">Semper enim ex sinubus re ſpectu sinus <lb/>totius maioris inuentis abijciendæ ſunt tot figuræ, vt relinquantur sinus reſpectu si-<lb/>nus totius minoris, quot ci fris sinus totus maior sinum totum minorem ſuperat: </s> <s xml:id="echoid-s5690" xml:space="preserve">Item <lb/>sinubus reſpectu sinus totius minoris inuentis adijciẽ dæ ſunt tot cifræ, vt fiãt sinus re <lb/>ſpectu sinus totius maioris, quot cifris minor sinus totus à sinu toto maiore ſupera-<lb/>tur. </s> <s xml:id="echoid-s5691" xml:space="preserve">Quod eodem modo demonſtrabitur. </s> <s xml:id="echoid-s5692" xml:space="preserve">Vt si ſupputentur sinus reſpectu sinus totius <lb/>100000000. </s> <s xml:id="echoid-s5693" xml:space="preserve">& </s> <s xml:id="echoid-s5694" xml:space="preserve">ex singulis abijciantur tres primæ figuræ ad dexteram, reliqui erũt <pb o="131" file="143" n="143" rhead=""/> sinus reſpectu sinus totius 100000. </s> <s xml:id="echoid-s5695" xml:space="preserve">& </s> <s xml:id="echoid-s5696" xml:space="preserve">quidem multò exquisitiores, quàm si sinus <lb/>ſupputentur reſpectu sinus totius 10000000. </s> <s xml:id="echoid-s5697" xml:space="preserve">& </s> <s xml:id="echoid-s5698" xml:space="preserve">ex singulis duæ figuræ abijciantur. <lb/></s> <s xml:id="echoid-s5699" xml:space="preserve">Quòd si sinubus reſpectu sinus totius 100000. </s> <s xml:id="echoid-s5700" xml:space="preserve">inuentis adijciantur tres cifræ, fient <lb/>sinus reſpectu sinus totius 100000000. </s> <s xml:id="echoid-s5701" xml:space="preserve">at que ita de cæteris.</s> <s xml:id="echoid-s5702" xml:space="preserve"/> </p> <div xml:id="echoid-div401" type="float" level="2" n="5"> <note position="left" xlink:label="note-142-02" xlink:href="note-142-02a" xml:space="preserve">Quo pacto <lb/>ex ſinubus <lb/>maioribus <lb/>ſiant mino <lb/>res, & con-<lb/>tra, quotcũ <lb/>que parti-<lb/>cularum ſi <lb/>nus totus <lb/>@atuatur.</note> </div> <note position="right" xml:space="preserve">Cognitis <lb/>duabus li-<lb/>neis rectis <lb/>reſpectu ali <lb/>cuius men-<lb/>ſuræ, dein-<lb/>de vero v-<lb/>na earum <lb/>reſpectu al <lb/>terius men <lb/>ſuræ cogni <lb/>ta, quo pa-<lb/>cto alterare <lb/>ſpectu hu-<lb/>ius alteri <lb/>mẽſurę co. <lb/>gnoſcatur. <lb/>Id qd Aſtro <lb/>nomis eſt <lb/>familiatiſ-<lb/>ſimum.</note> <p style="it"> <s xml:id="echoid-s5703" xml:space="preserve">_E X_ his patet ratio illius operationis, qua frequenter & </s> <s xml:id="echoid-s5704" xml:space="preserve">in mea Gnomonica, & </s> <s xml:id="echoid-s5705" xml:space="preserve"><lb/>alijs in locis vſus ſum; </s> <s xml:id="echoid-s5706" xml:space="preserve">cum duabus lineis cognitis reſpectu alicuius lineæ rectæ, tan-<lb/>quam sinus totius, deinde vero vna earum iterum cognita reſpectu alterius lineæ <lb/>rectæ maioris vel minoris veluti sinus totius, vel reſpectu alterius cuiuſpiam menſu-<lb/>ræ, alteram reſpectu huius alterius sinus totius, vel reſpectu alterius huius menſuræ <lb/>inueſtigo. </s> <s xml:id="echoid-s5707" xml:space="preserve">Id quod & </s> <s xml:id="echoid-s5708" xml:space="preserve">Ptolemæus, & </s> <s xml:id="echoid-s5709" xml:space="preserve">alij Aſtronomi non raro etiam faciunt. </s> <s xml:id="echoid-s5710" xml:space="preserve">Exempli <lb/>gratia; </s> <s xml:id="echoid-s5711" xml:space="preserve">cum duabus lineis rectis A, B, co-<lb/> <anchor type="figure" xlink:label="fig-143-01a" xlink:href="fig-143-01"/> gnitis reſpectu lineæ rectæ C, tanquã sinus <lb/>totius cõ tinentis particulas 100000. </s> <s xml:id="echoid-s5712" xml:space="preserve">linea <lb/>quidem A, partium 91354. </s> <s xml:id="echoid-s5713" xml:space="preserve">linea vero <lb/>B, partium 40673. </s> <s xml:id="echoid-s5714" xml:space="preserve">Deinde vero recta A, <lb/>reſpectu alterius lineæ maioris D, veluti <lb/>sinus totius cõplectentis quoque particulas <lb/>100000. </s> <s xml:id="echoid-s5715" xml:space="preserve">deprehenſa iterum sit partium <lb/>80901. </s> <s xml:id="echoid-s5716" xml:space="preserve">vel palmorum. </s> <s xml:id="echoid-s5717" xml:space="preserve">4 reſpectu menſuræ <lb/>E. </s> <s xml:id="echoid-s5718" xml:space="preserve">quæ palmo ſit æqualis, vel reſpectu men <lb/>ſuræ F, quæ plures palmos, nempe quinque, cõtineat, inquiro, quot partes, aut palmos <lb/>linea B, contineat reſpectu poſterioris sinus totius, aut reſpectu dictæ illius menſuræ <lb/>E, vel F. </s> <s xml:id="echoid-s5719" xml:space="preserve">Quod quidem expedio per regulam proportionum hoc modo. </s> <s xml:id="echoid-s5720" xml:space="preserve">Si linea A, partiũ <lb/>91354. </s> <s xml:id="echoid-s5721" xml:space="preserve">dat lineam B, partium 40673. </s> <s xml:id="echoid-s5722" xml:space="preserve">Eadem linea A, partium 80901. </s> <s xml:id="echoid-s5723" xml:space="preserve">vel palmo-<lb/>r@m 4. </s> <s xml:id="echoid-s5724" xml:space="preserve">quot partium, aut palmorũ dabit eandem lineã B ? </s> <s xml:id="echoid-s5725" xml:space="preserve">Inuenietur namque lineæ <lb/>B, partium 36019. </s> <s xml:id="echoid-s5726" xml:space="preserve">& </s> <s xml:id="echoid-s5727" xml:space="preserve">paulo amplius, vel palmorũ 1 {35669/45677}. </s> <s xml:id="echoid-s5728" xml:space="preserve">Cuius operationis <lb/>ratio à ſuperiori non differt, cum recta A, ad rectam B, habeat ſemper vnam & </s> <s xml:id="echoid-s5729" xml:space="preserve">ean <lb/>dem, determinatamq́; </s> <s xml:id="echoid-s5730" xml:space="preserve">proportionem.</s> <s xml:id="echoid-s5731" xml:space="preserve"/> </p> <div xml:id="echoid-div402" type="float" level="2" n="6"> <figure xlink:label="fig-143-01" xlink:href="fig-143-01a"> <image file="143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/143-01"/> </figure> </div> </div> <div xml:id="echoid-div404" type="section" level="1" n="168"> <head xml:id="echoid-head194" xml:space="preserve">SEQVITVR TABVLA SINVVM RECTORVM <lb/>per ſingula Quadrantis Minuta extenſa, & à Ioan. Regio-<lb/>montano quondam ſupputata, nunc autem per me <lb/>examinata, & pleriſque in locis caſtigata, <lb/>atque correcta.</head> <pb o="132" file="144" n="144" rhead=""/> </div> <div xml:id="echoid-div405" type="section" level="1" n="169"> <head xml:id="echoid-head195" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 0 # 1 # 2 # 3 # 4 <lb/>0 # 0000 # 174524 # 348995 # 523360 # 697565 # 60 <lb/>1 # 2909 # 177433 # 351902 # 526265 # 700467 # 59 <lb/>2 # 5818 # 180341 # 354809 # 529170 # 703369 # 58 <lb/>3 # 8727 # 183250 # 357716 # 532075 # 706270 # 57 <lb/>4 # 11636 # 186158 # 360623 # 534980 # 709172 # 56 <lb/>5 # 14544 # 189066 # 363530 # 537884 # 712073 # 55 <lb/>6 # 17453 # 191975 # 366437 # 540789 # 714975 # 54 <lb/>7 # 20362 # 194883 # 369344 # 543694 # 717876 # 53 <lb/>8 # 23271 # 197792 # 372251 # 546598 # 720777 # 52 <lb/>9 # 26180 # 200700 # 375158 # 549503 # 723678 # 51 <lb/>10 # 29088 # 203608 # 378064 # 552407 # 726579 # 50 <lb/>11 # 31997 # 206517 # 380971 # 555312 # 729480 # 49 <lb/>12 # 34906 # 209425 # 383878 # 558216 # 732381 # 48 <lb/>13 # 37815 # 212333 # 386785 # 561120 # 735282 # 47 <lb/>14 # 40724 # 215241 # 389692 # 564024 # 738183 # 46 <lb/>15 # 43632 # 218149 # 392598 # 566928 # 741084 # 45 <lb/>16 # 46541 # 221057 # 395505 # 569832 # 743985 # 44 <lb/>17 # 49450 # 223965 # 398412 # 572736 # 746886 # 43 <lb/>18 # 52359 # 226873 # 401318 # 575640 # 749787 # 42 <lb/>19 # 55268 # 229781 # 404225 # 578544 # 752688 # 41 <lb/>20 # 58177 # 232689 # 407131 # 581448 # 755588 # 40 <lb/>21 # 61086 # 235597 # 410038 # 584352 # 758489 # 39 <lb/>22 # 63995 # 238505 # 412944 # 587256 # 761389 # 38 <lb/>23 # 66904 # 241413 # 415851 # 590160 # 764290 # 37 <lb/>24 # 59813 # 244321 # 418757 # 593064 # 767190 # 36 <lb/>25 # 72721 # 247229 # 421663 # 595967 # 770090 # 35 <lb/>26 # 75630 # 250137 # 424570 # 598871 # 772991 # 34 <lb/>27 # 78539 # 253045 # 427476 # 601775 # 775891 # 33 <lb/>28 # 81448 # 255953 # 430382 # 604678 # 778791 # 32 <lb/>29 # 84357 # 258861 # 433288 # 607582 # 781691 # 31 <lb/>30 # 87265 # 261769 # 436194 # 610485 # 784591 # 30 <lb/> # 89 # 88 # 87 # 86 # 85 <lb/></note> </div> <div xml:id="echoid-div406" type="section" level="1" n="170"> <head xml:id="echoid-head196" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="133" file="145" n="145" rhead=""/> </div> <div xml:id="echoid-div407" type="section" level="1" n="171"> <head xml:id="echoid-head197" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 0 # 1 # 2 # 3 # 4 <lb/>30 # 87265 # 261769 # 436194 # 610485 # 784591 # 30 <lb/>31 # 90174 # 264677 # 439100 # 613389 # 787491 # 29 <lb/>32 # 93083 # 267585 # 442006 # 616292 # 790391 # 28 <lb/>33 # 95992 # 270493 # 444912 # 619196 # 793291 # 27 <lb/>34 # 98901 # 273401 # 447818 # 622099 # 796191 # 26 <lb/>35 # 101809 # 276308 # 450724 # 625002 # 799090 # 25 <lb/>36 # 104718 # 279216 # 453630 # 627905 # 801990 # 24 <lb/>37 # 107627 # 282124 # 456536 # 630808 # 804889 # 23 <lb/>38 # 110536 # 285032 # 459442 # 633711 # 807789 # 22 <lb/>39 # 113445 # 287940 # 462348 # 636614 # 810688 # 21 <lb/>40 # 116353 # 290847 # 465253 # 639517 # 813587 # 20 <lb/>41 # 119262 # 293755 # 468159 # 642420 # 816486 # 19 <lb/>42 # 122171 # 296663 # 471065 # 645323 # 819385 # 18 <lb/>43 # 125079 # 299570 # 473970 # 648226 # 822284 # 17 <lb/>44 # 127988 # 302478 # 476876 # 651129 # 825183 # 16 <lb/>45 # 130896 # 305385 # 479781 # 654031 # 828082 # 15 <lb/>46 # 133805 # 308293 # 482687 # 656934 # 830981 # 14 <lb/>47 # 136714 # 311200 # 485592 # 659837 # 833880 # 13 <lb/>48 # 139622 # 314108 # 488498 # 662739 # 836778 # 12 <lb/>49 # 142531 # 317015 # 491403 # 665642 # 839677 # 11 <lb/>50 # 145439 # 319922 # 494308 # 668544 # 842576 # 10 <lb/>51 # 148348 # 322830 # 497214 # 671447 # 845474 # 9 <lb/>52 # 151257 # 325737 # 500119 # 674349 # 848372 # 8 <lb/>53 # 154165 # 328645 # 503024 # 677251 # 851271 # 7 <lb/>54 # 157074 # 331552 # 505929 # 680153 # 854169 # 6 <lb/>55 # 159982 # 334459 # 508834 # 683055 # 857067 # 5 <lb/>56 # 162891 # 337367 # 511740 # 685957 # 859965 # 4 <lb/>57 # 165799 # 340274 # 514645 # 688859 # 862863 # 3 <lb/>58 # 168708 # 343181 # 517550 # 691761 # 865761 # 2 <lb/>59 # 171616 # 346088 # 520455 # 694663 # 868659 # 1 <lb/>60 # 174524 # 348995 # 523360 # 667565 # 871557 # 0 <lb/> # 89 # 88 # 87 # 86 # 85 <lb/></note> </div> <div xml:id="echoid-div408" type="section" level="1" n="172"> <head xml:id="echoid-head198" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="134" file="146" n="146" rhead=""/> </div> <div xml:id="echoid-div409" type="section" level="1" n="173"> <head xml:id="echoid-head199" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 5 # 6 # 7 # 8 # 9 <lb/>0 # 871557 # 1045285 # 1218693 # 1391731 # 1564345 # 60 <lb/>1 # 874455 # 1048178 # 1221580 # 1394612 # 1567218 # 59 <lb/>2 # 877353 # 1051071 # 1224467 # 1397492 # 1570091 # 58 <lb/>3 # 880250 # 1053964 # 1227354 # 1400373 # 1572964 # 57 <lb/>4 # 883148 # 1056857 # 1230241 # 1403253 # 1575837 # 56 <lb/>5 # 886045 # 1059749 # 1233128 # 1406133 # 1578709 # 55 <lb/>6 # 888943 # 1062642 # 1236015 # 1409013 # 1581581 # 54 <lb/>7 # 891840 # 1065534 # 1238901 # 1411893 # 1584453 # 53 <lb/>8 # 894737 # 1068426 # 1241788 # 1414772 # 1587325 # 52 <lb/>9 # 897634 # 1071318 # 1244674 # 1417652 # 1590197 # 51 <lb/>10 # 900531 # 1074210 # 1247560 # 1420531 # 1593069 # 50 <lb/>11 # 903428 # 1077102 # 1250446 # 1423410 # 1595941 # 49 <lb/>12 # 906325 # 1079994 # 1253332 # 1426289 # 1598812 # 48 <lb/>13 # 909222 # 1082886 # 1256218 # 1429168 # 1601684 # 47 <lb/>14 # 912119 # 1085778 # 1259104 # 1432047 # 1604555 # 46 <lb/>15 # 915016 # 1088669 # 1261990 # 1434926 # 1607426 # 45 <lb/>16 # 917913 # 1091561 # 1264876 # 1437805 # 1610297 # 44 <lb/>17 # 920809 # 1094452 # 1267761 # 1440684 # 1613168 # 43 <lb/>18 # 923706 # 1097344 # 1270647 # 1443562 # 1616038 # 42 <lb/>19 # 926602 # 1100235 # 1273532 # 1446441 # 1618909 # 41 <lb/>20 # 929498 # 1103126 # 1276417 # 1449319 # 1621779 # 40 <lb/>21 # 932395 # 1106017 # 1279302 # 1452197 # 1624649 # 39 <lb/>22 # 935291 # 1108908 # 1282187 # 1455075 # 1627519 # 38 <lb/>23 # 938187 # 1111799 # 1285072 # 1457953 # 1630389 # 37 <lb/>24 # 941083 # 1114690 # 1287957 # 1460831 # 1633259 # 36 <lb/>25 # 943979 # 1117580 # 1290841 # 1463708 # 1636129 # 35 <lb/>26 # 946875 # 1120471 # 1293726 # 1466586 # 1638999 # 34 <lb/>27 # 949771 # 1123361 # 1296610 # 1469463 # 1641868 # 33 <lb/>28 # 952667 # 1126252 # 1299495 # 1472340 # 1644738 # 32 <lb/>29 # 955563 # 1129142 # 1302378 # 1475217 # 1647607 # 31 <lb/>30 # 958458 # 1132032 # 1305262 # 1478094 # 1650476 # 30 <lb/> # 84 # 83 # 82 # 81 # 80 <lb/></note> </div> <div xml:id="echoid-div410" type="section" level="1" n="174"> <head xml:id="echoid-head200" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="135" file="147" n="147" rhead=""/> </div> <div xml:id="echoid-div411" type="section" level="1" n="175"> <head xml:id="echoid-head201" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 5 # 6 # 7 # 8 # 9 <lb/>30 # 958458 # 1132032 # 1305262 # 1478094 # 1650476 # 30 <lb/>31 # 961354 # 1134922 # 1308146 # 1480971 # 1653345 # 29 <lb/>32 # 964249 # 1137812 # 1311030 # 1483848 # 1656214 # 28 <lb/>33 # 967144 # 1140702 # 1313914 # 1486724 # 1659082 # 27 <lb/>34 # 970039 # 1143592 # 1316798 # 1489601 # 1661951 # 26 <lb/>35 # 972934 # 1146482 # 1319681 # 1492477 # 1664819 # 25 <lb/>36 # 975829 # 1149372 # 1322564 # 1495353 # 1667687 # 24 <lb/>37 # 978724 # 1152261 # 1325447 # 1498229 # 1670555 # 23 <lb/>38 # 981619 # 1155151 # 1328330 # 1501105 # 1673423 # 22 <lb/>39 # 984514 # 1158040 # 1331213 # 1503981 # 1676291 # 21 <lb/>40 # 987408 # 1160929 # 1334096 # 1506857 # 1679159 # 20 <lb/>41 # 990303 # 1163818 # 1336979 # 1509733 # 1682027 # 19 <lb/>42 # 993198 # 1166707 # 1339862 # 1512608 # 1684894 # 18 <lb/>43 # 996092 # 1169596 # 1342744 # 1515484 # 1687761 # 17 <lb/>44 # 998987 # 1172485 # 1345627 # 1518359 # 1690628 # 16 <lb/>45 # 1001881 # 1175374 # 1348509 # 1521234 # 1693495 # 15 <lb/>46 # 1004775 # 1178263 # 1351392 # 1524109 # 1696362 # 14 <lb/>47 # 1007669 # 1181151 # 1354274 # 1526984 # 1699229 # 13 <lb/>48 # 1010563 # 1184040 # 1357156 # 1529859 # 1702095 # 12 <lb/>49 # 1013457 # 1186928 # 1360038 # 1532734 # 1704962 # 11 <lb/>50 # 1016351 # 1189816 # 1362920 # 1535608 # 1707828 # 10 <lb/>51 # 1019245 # 1192704 # 1365802 # 1538482 # 1710694 # 9 <lb/>52 # 1022139 # 1195592 # 1368683 # 1541356 # 1713560 # 8 <lb/>53 # 1025032 # 1198480 # 1371564 # 1544230 # 1716426 # 7 <lb/>54 # 1027926 # 1201368 # 1374446 # 1547104 # 1719292 # 6 <lb/>55 # 1030819 # 1204255 # 1377327 # 1549978 # 1722157 # 5 <lb/>56 # 1033713 # 1207143 # 1380208 # 1552852 # 1725022 # 4 <lb/>57 # 1036606 # 1210031 # 1383089 # 1555725 # 1727887 # 3 <lb/>58 # 1039499 # 1212918 # 1385970 # 1558599 # 1730752 # 2 <lb/>59 # 1042392 # 1215806 # 1388851 # 1561472 # 1733617 # 1 <lb/>60 # 1045285 # 1218693 # 1391731 # 1564345 # 1736482 # 0 <lb/> # 84 # 83 # 82 # 81 # 80 <lb/></note> </div> <div xml:id="echoid-div412" type="section" level="1" n="176"> <head xml:id="echoid-head202" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="136" file="148" n="148" rhead=""/> </div> <div xml:id="echoid-div413" type="section" level="1" n="177"> <head xml:id="echoid-head203" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 10 # 11 # 12 # 13 # 14 <lb/>0 # 1736482 # 1908090 # 2079117 # 2249511 # 2419219 # 60 <lb/>1 # 1139347 # 1910945 # 2081962 # 2252345 # 2422041 # 59 <lb/>2 # 1742211 # 1913800 # 2084807 # 2255179 # 2424863 # 58 <lb/>3 # 1745075 # 1916655 # 2087652 # 2258013 # 2427685 # 57 <lb/>4 # 1747939 # 1919510 # 2090497 # 2260847 # 2430507 # 56 <lb/>5 # 1750803 # 1922365 # 2093342 # 2263680 # 2433329 # 55 <lb/>6 # 1753667 # 1925220 # 2096186 # 2266513 # 2436150 # 54 <lb/>7 # 1756531 # 1928074 # 2099030 # 2269346 # 2438971 # 53 <lb/>8 # 1759394 # 1930928 # 2101874 # 2272179 # 2441792 # 52 <lb/>9 # 1762258 # 1933782 # 2104718 # 2275012 # 2444613 # 51 <lb/>10 # 1765121 # 1936636 # 2107562 # 2277844 # 2447434 # 50 <lb/>11 # 1767984 # 1939490 # 2110405 # 2280676 # 2450254 # 49 <lb/>12 # 1770847 # 1942344 # 2113248 # 2283508 # 2453074 # 48 <lb/>13 # 1773710 # 1945197 # 2116091 # 2286340 # 2455894 # 47 <lb/>14 # 1776573 # 1948050 # 2118934 # 2289172 # 2458714 # 46 <lb/>15 # 1779435 # 1950903 # 2121777 # 2292004 # 2461533 # 45 <lb/>16 # 1782298 # 1953756 # 2124620 # 2294835 # 2464352 # 44 <lb/>17 # 1785160 # 1956609 # 2127462 # 2297666 # 2467171 # 43 <lb/>18 # 1788022 # 1959462 # 2130304 # 2300497 # 2469990 # 42 <lb/>19 # 1790884 # 1962314 # 2133146 # 2303328 # 2472809 # 41 <lb/>20 # 1793746 # 1965166 # 2135988 # 2306159 # 2475628 # 40 <lb/>21 # 1796608 # 1968018 # 2138830 # 2308989 # 2478446 # 39 <lb/>22 # 1799469 # 1970870 # 2141671 # 2311819 # 2481264 # 38 <lb/>23 # 1802331 # 1973722 # 2144512 # 2314649 # 2484082 # 37 <lb/>24 # 1805192 # 1976574 # 2147353 # 2317479 # 2486900 # 36 <lb/>25 # 1808053 # 1979425 # 2150194 # 2320309 # 2489717 # 35 <lb/>26 # 1810914 # 1982276 # 2153035 # 2323138 # 2492534 # 34 <lb/>27 # 1813774 # 1985127 # 2155876 # 2325967 # 2495351 # 33 <lb/>28 # 1816634 # 1987978 # 2158716 # 2328796 # 2498168 # 32 <lb/>29 # 1819495 # 1990829 # 2161556 # 2331625 # 2500984 # 31 <lb/>30 # 1822355 # 1993679 # 2164396 # 2334454 # 2503800 # 30 <lb/> # 79 # 78 # 77 # 76 # 75 <lb/></note> </div> <div xml:id="echoid-div414" type="section" level="1" n="178"> <head xml:id="echoid-head204" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="137" file="149" n="149" rhead=""/> </div> <div xml:id="echoid-div415" type="section" level="1" n="179"> <head xml:id="echoid-head205" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 10 # 11 # 12 # 13 # 14 <lb/>30 # 1822355 # 1993679 # 2164396 # 2334454 # 2503800 # 30 <lb/>31 # 1825215 # 1996530 # 2167236 # 2337282 # 2506616 # 29 <lb/>32 # 1828075 # 1999380 # 2170076 # 2340110 # 2509432 # 28 <lb/>33 # 1830935 # 2002230 # 2172916 # 2342938 # 2512248 # 27 <lb/>34 # 1833795 # 2005080 # 2175755 # 2345766 # 2515064 # 26 <lb/>35 # 1836654 # 2007930 # 2178594 # 2348594 # 2517879 # 25 <lb/>36 # 1839513 # 2010780 # 2181433 # 2351421 # 2520694 # 24 <lb/>37 # 1842372 # 2013629 # 2184272 # 2354248 # 2523509 # 23 <lb/>38 # 1845231 # 2016478 # 2187111 # 2357075 # 2526324 # 22 <lb/>39 # 1848090 # 2019327 # 2189949 # 2359902 # 2529138 # 21 <lb/>40 # 1850949 # 2022176 # 2192787 # 2362729 # 2531952 # 20 <lb/>41 # 1853808 # 2025025 # 2195625 # 2365555 # 2534766 # 19 <lb/>42 # 1856666 # 2027874 # 2198463 # 2368381 # 2537580 # 18 <lb/>43 # 1859524 # 2030722 # 2201300 # 2371207 # 2540393 # 17 <lb/>44 # 1862382 # 2033570 # 2204137 # 2374033 # 2543206 # 16 <lb/>45 # 1865240 # 2036418 # 2206974 # 2376859 # 2546019 # 15 <lb/>46 # 1868098 # 2039266 # 2209811 # 2379684 # 2548832 # 14 <lb/>47 # 1870956 # 2042114 # 2212648 # 2382509 # 2551645 # 13 <lb/>48 # 1873813 # 2044962 # 2215485 # 2385334 # 2554458 # 12 <lb/>49 # 1876670 # 2047809 # 2218322 # 2388159 # 2557270 # 11 <lb/>50 # 1879527 # 2050656 # 2221158 # 2390983 # 2560082 # 10 <lb/>51 # 1882384 # 2053503 # 2223994 # 2393808 # 2562894 # 9 <lb/>52 # 1885241 # 2056350 # 2226830 # 2396632 # 2565706 # 8 <lb/>53 # 1888098 # 2059197 # 2229666 # 2399456 # 2568517 # 7 <lb/>54 # 1890954 # 2062043 # 2232502 # 2402280 # 2571328 # 6 <lb/>55 # 1893810 # 2064889 # 2235337 # 2405104 # 2574139 # 5 <lb/>56 # 1896666 # 2067735 # 2238172 # 2407927 # 2576950 # 4 <lb/>57 # 1899522 # 2070581 # 2241007 # 2410750 # 2579760 # 3 <lb/>58 # 1902378 # 2073427 # 2243842 # 2413573 # 2582570 # 2 <lb/>59 # 1905234 # 2076272 # 2246677 # 2416396 # 2585380 # 1 <lb/>60 # 1908090 # 2079117 # 2249511 # 2419219 # 2588190 # 0 <lb/> # 79 # 78 # 77 # 76 # 75 <lb/></note> </div> <div xml:id="echoid-div416" type="section" level="1" n="180"> <head xml:id="echoid-head206" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="138" file="150" n="150" rhead=""/> </div> <div xml:id="echoid-div417" type="section" level="1" n="181"> <head xml:id="echoid-head207" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 15 # 16 # 17 # 18 # 19 <lb/>0 # 2588190 # 2756373 # 2923717 # 3090170 # 3255682 # 60 <lb/>1 # 2591000 # 2759169 # 2926499 # 3092936 # 3258532 # 59 <lb/>2 # 2593809 # 2761965 # 2929280 # 3095702 # 3261182 # 58 <lb/>3 # 2596618 # 2764761 # 2932061 # 3098468 # 3263931 # 57 <lb/>4 # 2599427 # 2767556 # 2934842 # 3101234 # 3266681 # 56 <lb/>5 # 2602236 # 2770351 # 2937623 # 3103999 # 3269430 # 55 <lb/>6 # 2605045 # 2773146 # 2940403 # 3106764 # 3272179 # 54 <lb/>7 # 2607853 # 2775941 # 2943183 # 3109529 # 3274927 # 53 <lb/>8 # 2610661 # 2778735 # 2945963 # 3112294 # 3277675 # 52 <lb/>9 # 2613469 # 2781529 # 2948743 # 3115058 # 3280423 # 51 <lb/>10 # 2616277 # 2784323 # 2951523 # 3117822 # 3283171 # 50 <lb/>11 # 2619084 # 2787117 # 2954302 # 3120586 # 3285918 # 49 <lb/>12 # 2621891 # 2789911 # 2957081 # 3123349 # 3288665 # 48 <lb/>13 # 2624698 # 2792704 # 2959860 # 3126112 # 3291412 # 47 <lb/>14 # 2627505 # 2795497 # 2962638 # 3128875 # 3294159 # 46 <lb/>15 # 2630312 # 2798290 # 2965416 # 3131638 # 3296906 # 45 <lb/>16 # 2633118 # 2801082 # 2968194 # 3134400 # 3299652 # 44 <lb/>17 # 2635924 # 2803874 # 2970972 # 3137162 # 3302398 # 43 <lb/>18 # 2638730 # 2806666 # 2973750 # 3139924 # 3305144 # 42 <lb/>19 # 2641536 # 2809458 # 2976527 # 3142686 # 3307889 # 41 <lb/>20 # 2644342 # 2812250 # 2979304 # 3145448 # 3310634 # 40 <lb/>21 # 2647147 # 2815041 # 2982081 # 3148209 # 3313379 # 39 <lb/>22 # 2649952 # 2817832 # 2984857 # 3150970 # 3316123 # 38 <lb/>23 # 2652757 # 2820623 # 2987633 # 3153731 # 3318867 # 37 <lb/>24 # 2655562 # 2823414 # 2990409 # 3156491 # 3321611 # 36 <lb/>25 # 2658366 # 2826204 # 2993185 # 3159251 # 3324355 # 35 <lb/>26 # 2661170 # 2828994 # 2995960 # 3162011 # 3327098 # 34 <lb/>27 # 2663974 # 2831784 # 2998735 # 3164770 # 3329841 # 33 <lb/>28 # 2666777 # 2834574 # 3001510 # 3167529 # 3332585 # 32 <lb/>29 # 2669580 # 2837364 # 3004284 # 3170288 # 3335327 # 31 <lb/>30 # 2672383 # 2840153 # 3007058 # 3173047 # 3338069 # 30 <lb/> # 74 # 73 # 72 # 71 # 70 <lb/></note> </div> <div xml:id="echoid-div418" type="section" level="1" n="182"> <head xml:id="echoid-head208" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="139" file="151" n="151" rhead=""/> </div> <div xml:id="echoid-div419" type="section" level="1" n="183"> <head xml:id="echoid-head209" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 15 # 16 # 17 # 18 # 19 <lb/>30 # 2672383 # 2840153 # 3007058 # 3173047 # 3338069 # 30 <lb/>31 # 2675186 # 2842942 # 3009832 # 3175805 # 3340811 # 29 <lb/>32 # 2677989 # 2845731 # 3012606 # 3178563 # 3343553 # 28 <lb/>33 # 2680792 # 2848520 # 3015380 # 3181321 # 3346294 # 27 <lb/>34 # 2683595 # 2851308 # 3018153 # 3184079 # 3349035 # 26 <lb/>35 # 2686397 # 2854096 # 3020926 # 3186837 # 3351776 # 25 <lb/>36 # 2689199 # 2856884 # 3023699 # 3189594 # 3354516 # 24 <lb/>37 # 2692001 # 2859672 # 3026472 # 3192351 # 3357256 # 23 <lb/>38 # 2694802 # 2862459 # 3029244 # 3195108 # 3359996 # 22 <lb/>39 # 2697603 # 2865246 # 3032016 # 3197864 # 3362736 # 21 <lb/>40 # 2700404 # 2868033 # 3034788 # 3200620 # 3365475 # 20 <lb/>41 # 2703205 # 2870819 # 3037559 # 3203375 # 3368214 # 19 <lb/>42 # 2706005 # 2873605 # 3040330 # 3206130 # 3370953 # 18 <lb/>43 # 2708805 # 2876391 # 3043101 # 3208885 # 3373691 # 17 <lb/>44 # 2711605 # 2879177 # 3045872 # 3211640 # 3376429 # 16 <lb/>45 # 2714405 # 2881963 # 3048643 # 3214395 # 3379167 # 15 <lb/>46 # 2717204 # 2884748 # 3051413 # 3217150 # 3381905 # 14 <lb/>47 # 2720003 # 2887533 # 3054183 # 3219904 # 3384642 # 13 <lb/>48 # 2722802 # 2890318 # 3056953 # 3222658 # 3387379 # 12 <lb/>49 # 2725601 # 2893103 # 3059723 # 3225412 # 3390116 # 11 <lb/>50 # 2728400 # 2895888 # 3062492 # 3228165 # 3392852 # 10 <lb/>51 # 2731198 # 2898672 # 3065261 # 3230918 # 3395588 # 9 <lb/>52 # 2733996 # 2901456 # 3068030 # 3233671 # 3398324 # 8 <lb/>53 # 2736794 # 2904240 # 3070798 # 3236423 # 3401060 # 7 <lb/>54 # 2739592 # 2907023 # 3073566 # 3239175 # 3403795 # 6 <lb/>55 # 2742389 # 2909806 # 3076334 # 3241927 # 3406530 # 5 <lb/>56 # 2745186 # 2912589 # 3079102 # 3244679 # 3409265 # 4 <lb/>57 # 2747983 # 2915371 # 3081869 # 3247430 # 3411999 # 3 <lb/>58 # 2750780 # 2918153 # 3084636 # 3250181 # 3414733 # 2 <lb/>59 # 2753577 # 2920935 # 3087403 # 3252932 # 3417467 # 1 <lb/>60 # 2756373 # 2923717 # 3090170 # 3255682 # 3420201 # 0 <lb/> # 74 # 73 # 72 # 71 # 70 <lb/></note> </div> <div xml:id="echoid-div420" type="section" level="1" n="184"> <head xml:id="echoid-head210" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="140" file="152" n="152" rhead=""/> </div> <div xml:id="echoid-div421" type="section" level="1" n="185"> <head xml:id="echoid-head211" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 20 # 21 # 22 # 23 # 24 <lb/>0 # 3420201 # 3583679 # 3746066 # 3907311 # 4067366 # 60 <lb/>1 # 3422934 # 3586395 # 3748763 # 3909989 # 4070023 # 59 <lb/>2 # 3425667 # 3589110 # 3751460 # 3912666 # 4072680 # 58 <lb/>3 # 3428400 # 3591825 # 3754156 # 3915343 # 4075337 # 57 <lb/>4 # 3431133 # 3594540 # 3756852 # 3918020 # 4077993 # 56 <lb/>5 # 3433865 # 3597254 # 3759548 # 3920696 # 4080649 # 55 <lb/>6 # 3436597 # 3599968 # 3762243 # 3923372 # 4083305 # 54 <lb/>7 # 3439329 # 3602682 # 3764938 # 3926048 # 4085960 # 53 <lb/>8 # 3442060 # 3605395 # 3767633 # 3928723 # 4088615 # 52 <lb/>9 # 3444791 # 3608108 # 3770327 # 3931398 # 4091269 # 51 <lb/>10 # 3447522 # 3610821 # 3773021 # 3934072 # 4093923 # 50 <lb/>11 # 3450253 # 3613533 # 3775715 # 3936746 # 4096577 # 49 <lb/>12 # 3452983 # 3616245 # 3778408 # 3939420 # 4099231 # 48 <lb/>13 # 3455713 # 3618957 # 3781101 # 3942093 # 4101884 # 47 <lb/>14 # 3458442 # 3621669 # 3783794 # 3944766 # 4104537 # 46 <lb/>15 # 3461171 # 3624380 # 3786486 # 3947439 # 4107189 # 45 <lb/>16 # 3463900 # 3627091 # 3789178 # 3950112 # 4109841 # 44 <lb/>17 # 3466629 # 3629802 # 3791870 # 3952784 # 4112493 # 43 <lb/>18 # 3469357 # 3632512 # 3794562 # 3955456 # 4115144 # 42 <lb/>19 # 3472085 # 3635222 # 3797253 # 3958128 # 4117795 # 41 <lb/>20 # 3474813 # 3637932 # 3799944 # 3960799 # 4120446 # 40 <lb/>21 # 3477540 # 3640642 # 3802635 # 3963470 # 4123096 # 39 <lb/>22 # 3480267 # 3643351 # 3805325 # 3966140 # 4125746 # 38 <lb/>23 # 3482994 # 3646060 # 3808015 # 3968810 # 4128395 # 37 <lb/>24 # 3485721 # 3648768 # 3810704 # 3971480 # 4131044 # 36 <lb/>25 # 3488447 # 3651476 # 3813393 # 3974149 # 4133693 # 35 <lb/>26 # 3491173 # 3654184 # 3816082 # 3976818 # 4136341 # 34 <lb/>27 # 3493899 # 3656892 # 3818771 # 3979487 # 4138989 # 33 <lb/>28 # 3496624 # 3659599 # 3821459 # 3982155 # 4141637 # 32 <lb/>29 # 3499349 # 3662306 # 3824147 # 3984823 # 4144285 # 31 <lb/>30 # 3502075 # 3665012 # 3826834 # 3987491 # 4146932 # 30 <lb/> # 69 # 68 # 67 # 66 # 65 <lb/></note> </div> <div xml:id="echoid-div422" type="section" level="1" n="186"> <head xml:id="echoid-head212" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="141" file="153" n="153" rhead=""/> </div> <div xml:id="echoid-div423" type="section" level="1" n="187"> <head xml:id="echoid-head213" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 20 # 21 # 22 # 23 # 24 <lb/>30 # 3502075 # 3665012 # 3826834 # 3987491 # 4146932 # 30 <lb/>31 # 3504799 # 3667718 # 3829521 # 3990159 # 4149579 # 29 <lb/>32 # 3507523 # 3670424 # 3832208 # 3992826 # 4152226 # 28 <lb/>33 # 3510247 # 3673130 # 3834895 # 3995493 # 4154872 # 27 <lb/>34 # 3512971 # 3675835 # 3837581 # 3998159 # 4157518 # 26 <lb/>35 # 3515694 # 3678541 # 3840267 # 4000825 # 4160163 # 25 <lb/>36 # 3518417 # 3681246 # 3842953 # 4003491 # 4162808 # 24 <lb/>37 # 3521140 # 3683951 # 3845638 # 4006156 # 4165453 # 23 <lb/>38 # 3523862 # 3686655 # 3848323 # 4008821 # 4168097 # 22 <lb/>39 # 3526584 # 3689359 # 3851008 # 4011486 # 4170741 # 21 <lb/>40 # 3529306 # 3692062 # 3853692 # 4014150 # 4173385 # 20 <lb/>41 # 3532027 # 3694765 # 3856376 # 4016814 # 4176028 # 19 <lb/>42 # 3534748 # 3697468 # 3859060 # 4019478 # 4178671 # 18 <lb/>43 # 3537469 # 3700170 # 3861743 # 4022141 # 4181313 # 17 <lb/>44 # 3540190 # 2702872 # 3864426 # 4024804 # 4183955 # 16 <lb/>45 # 3542910 # 3705574 # 3867109 # 4027467 # 4186597 # 15 <lb/>46 # 3545630 # 3708276 # 3869791 # 4030130 # 4189239 # 14 <lb/>47 # 3548350 # 3710977 # 3872473 # 4032792 # 4191880 # 13 <lb/>48 # 3551070 # 3713678 # 3875155 # 4035454 # 4194521 # 12 <lb/>49 # 3553789 # 3716379 # 3877837 # 4038115 # 4197162 # 11 <lb/>50 # 3556508 # 3719080 # 3880518 # 4040776 # 4199802 # 10 <lb/>51 # 3559227 # 3721780 # 3883199 # 4043437 # 4202442 # 9 <lb/>52 # 3561945 # 3724480 # 3885880 # 4046097 # 4205081 # 8 <lb/>53 # 3564663 # 3727179 # 3888560 # 4048757 # 4207720 # 7 <lb/>54 # 3567380 # 3729878 # 3891240 # 4051416 # 4210359 # 6 <lb/>55 # 3570097 # 3732577 # 3893919 # 4054075 # 4212997 # 5 <lb/>56 # 3572814 # 3735275 # 3896598 # 4056734 # 4215635 # 4 <lb/>57 # 3575531 # 3737973 # 3899277 # 4059392 # 4218273 # 3 <lb/>58 # 3578247 # 3740671 # 3901955 # 4062050 # 4220910 # 2 <lb/>59 # 3580963 # 3743369 # 3904633 # 4064708 # 4223547 # 1 <lb/>60 # 3583679 # 3746066 # 3907311 # 4067366 # 4226183 # 0 <lb/> # 69 # 68 # 67 # 66 # 65 <lb/></note> </div> <div xml:id="echoid-div424" type="section" level="1" n="188"> <head xml:id="echoid-head214" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="142" file="154" n="154" rhead=""/> </div> <div xml:id="echoid-div425" type="section" level="1" n="189"> <head xml:id="echoid-head215" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 25 # 26 # 27 # 28 # 29 <lb/>0 # 4226183 # 4383712 # 4539905 # 4694716 # 4848096 # 60 <lb/>1 # 4228819 # 4386326 # 4542497 # 4697284 # 4850640 # 59 <lb/>2 # 4231455 # 4388940 # 4545088 # 4699852 # 4853184 # 58 <lb/>3 # 4234090 # 4391554 # 4547679 # 4702419 # 4855727 # 57 <lb/>4 # 4236725 # 4394167 # 4550270 # 4704986 # 4858270 # 56 <lb/>5 # 4239360 # 4396780 # 4552860 # 4707553 # 4860812 # 55 <lb/>6 # 4241994 # 4399392 # 4555450 # 4710119 # 4863354 # 54 <lb/>7 # 4244628 # 4402004 # 4558039 # 4712685 # 4865895 # 53 <lb/>8 # 4247262 # 4404616 # 4560628 # 4715250 # 4868436 # 52 <lb/>9 # 4249895 # 4407227 # 4563216 # 4717815 # 4870977 # 51 <lb/>10 # 4252528 # 4409838 # 4565804 # 4720380 # 4873517 # 50 <lb/>11 # 4255161 # 4412449 # 4568392 # 4722944 # 4876057 # 49 <lb/>12 # 4257793 # 4415059 # 4570979 # 4725508 # 4878596 # 48 <lb/>13 # 4260425 # 4417669 # 4573566 # 4728071 # 4881135 # 47 <lb/>14 # 4263056 # 4420278 # 4576153 # 4730634 # 4883674 # 46 <lb/>15 # 4265687 # 4422887 # 4578739 # 4733197 # 4886212 # 45 <lb/>16 # 4268318 # 4425496 # 4581325 # 4735759 # 4888750 # 44 <lb/>17 # 4270949 # 4428104 # 4583911 # 4738321 # 4891287 # 43 <lb/>18 # 4273579 # 4430712 # 4586496 # 4740882 # 4893824 # 42 <lb/>19 # 4276209 # 4433320 # 4589081 # 4743443 # 4896361 # 41 <lb/>20 # 4278838 # 4435927 # 4591665 # 4746004 # 4898897 # 40 <lb/>21 # 4281467 # 4438534 # 4594249 # 4748564 # 4901433 # 39 <lb/>22 # 4284096 # 4441140 # 4596833 # 4751124 # 4903968 # 38 <lb/>23 # 4286724 # 4443746 # 4599416 # 4753683 # 4906503 # 37 <lb/>24 # 4289352 # 4446352 # 4601999 # 4756242 # 4909037 # 36 <lb/>25 # 4291979 # 4448957 # 4604581 # 4758801 # 4911571 # 35 <lb/>26 # 4294606 # 4451562 # 4607163 # 4761359 # 4914105 # 34 <lb/>27 # 4297233 # 4454167 # 4609744 # 4763917 # 4916638 # 33 <lb/>28 # 4299859 # 4456771 # 4612325 # 4766474 # 4919171 # 32 <lb/>29 # 4302485 # 4459375 # 4614906 # 4769031 # 4921703 # 31 <lb/>30 # 4305111 # 4461978 # 4617486 # 4771588 # 4924235 # 30 <lb/> # 64 # 63 # 62 # 61 # 60 <lb/></note> </div> <div xml:id="echoid-div426" type="section" level="1" n="190"> <head xml:id="echoid-head216" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="143" file="155" n="155" rhead=""/> </div> <div xml:id="echoid-div427" type="section" level="1" n="191"> <head xml:id="echoid-head217" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 25 # 26 # 27 # 28 # 29 <lb/>30 # 4305111 # 4461978 # 4617486 # 4771588 # 4924235 # 30 <lb/>31 # 4307736 # 4464581 # 4620066 # 4774144 # 4926767 # 29 <lb/>32 # 4310361 # 4467184 # 4622646 # 4776700 # 4929298 # 28 <lb/>33 # 4312986 # 4469786 # 4625225 # 4779255 # 4931829 # 27 <lb/>34 # 4315610 # 4472388 # 4627804 # 4781810 # 4934359 # 26 <lb/>35 # 4318234 # 4474990 # 4630382 # 4784365 # 4936889 # 25 <lb/>36 # 4320858 # 4477591 # 4632960 # 4786919 # 4939418 # 24 <lb/>37 # 4323481 # 4480192 # 4635538 # 4789473 # 4941947 # 23 <lb/>38 # 4326104 # 4482792 # 4638115 # 4792026 # 4944476 # 22 <lb/>39 # 4328726 # 4485392 # 4640692 # 4794579 # 4947004 # 21 <lb/>40 # 4331348 # 4487992 # 4643268 # 4797132 # 4949532 # 20 <lb/>41 # 4333970 # 4490591 # 4645844 # 4799684 # 4952059 # 19 <lb/>42 # 4336591 # 4493190 # 4648420 # 4802236 # 4954586 # 18 <lb/>43 # 4339212 # 4495788 # 4650995 # 4804787 # 4957113 # 17 <lb/>44 # 4341833 # 4498386 # 4653570 # 4807338 # 5959639 # 16 <lb/>45 # 4344453 # 4500984 # 4656145 # 4809888 # 4962165 # 15 <lb/>46 # 4347073 # 4503582 # 4658719 # 4812438 # 4964690 # 14 <lb/>47 # 4349693 # 4506179 # 4661293 # 4814988 # 4967215 # 13 <lb/>48 # 4352312 # 4508776 # 4663866 # 4817537 # 4969740 # 12 <lb/>49 # 4354931 # 4511372 # 4666439 # 4820086 # 4972264 # 11 <lb/>50 # 4357549 # 4513968 # 4669012 # 4822635 # 4974788 # 10 <lb/>51 # 4360167 # 4516563 # 4671584 # 4825183 # 4977311 # 9 <lb/>52 # 4362785 # 4519158 # 4674156 # 4827731 # 4979834 # 8 <lb/>53 # 4365402 # 4521753 # 4676727 # 4830278 # 4982356 # 7 <lb/>54 # 4368019 # 4524347 # 4679298 # 4832825 # 4984878 # 6 <lb/>55 # 4370635 # 4526941 # 4681869 # 4835371 # 4987399 # 5 <lb/>56 # 4373251 # 4529535 # 4684439 # 4837917 # 4989920 # 4 <lb/>57 # 4375867 # 4532128 # 4687009 # 4840462 # 4992441 # 3 <lb/>58 # 4378482 # 4534721 # 4689578 # 4843007 # 4994961 # 2 <lb/>59 # 4381097 # 4537313 # 4692147 # 4845552 # 4997481 # 1 <lb/>60 # 4383712 # 4539905 # 4694716 # 4848096 # 5000000 # 0 <lb/> # 64 # 63 # 62 # 61 # 60 <lb/></note> </div> <div xml:id="echoid-div428" type="section" level="1" n="192"> <head xml:id="echoid-head218" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="144" file="156" n="156" rhead=""/> </div> <div xml:id="echoid-div429" type="section" level="1" n="193"> <head xml:id="echoid-head219" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 30 # 31 # 32 # 33 # 34 <lb/>0 # 5000000 # 5150381 # 5299192 # 5446390 # 5591929 # 60 <lb/>1 # 5002519 # 5152874 # 5301659 # 5448829 # 5594340 # 59 <lb/>2 # 5005038 # 5155367 # 5304125 # 5451268 # 5596751 # 58 <lb/>3 # 5007556 # 5157859 # 5306591 # 5453707 # 5599161 # 57 <lb/>4 # 5010074 # 5160351 # 5309056 # 5456145 # 5601571 # 56 <lb/>5 # 5012591 # 5162843 # 5311521 # 5458583 # 5603981 # 55 <lb/>6 # 5015108 # 5165334 # 5313985 # 5461020 # 5606390 # 54 <lb/>7 # 5017624 # 5167825 # 5316449 # 5463456 # 5608798 # 53 <lb/>8 # 5020140 # 5170315 # 5318913 # 5465892 # 5611206 # 52 <lb/>9 # 5022656 # 5172805 # 5321376 # 5468328 # 5613614 # 51 <lb/>10 # 5025171 # 5175294 # 5323839 # 5470763 # 5616021 # 50 <lb/>11 # 5027686 # 5177783 # 5326301 # 5473198 # 5618427 # 49 <lb/>12 # 5030200 # 5180271 # 5328763 # 5475632 # 5620833 # 48 <lb/>13 # 5032714 # 5182759 # 5331224 # 5478066 # 5623239 # 47 <lb/>14 # 5035227 # 5185246 # 5333685 # 5480499 # 5625644 # 46 <lb/>15 # 5037740 # 5187733 # 5336145 # 5482932 # 5628049 # 45 <lb/>16 # 5040253 # 5190220 # 5338605 # 5485364 # 5630453 # 44 <lb/>17 # 5042765 # 5192706 # 5341065 # 5487796 # 5632857 # 43 <lb/>18 # 5045277 # 5195192 # 5343524 # 5490228 # 5635260 # 42 <lb/>19 # 5047788 # 5197677 # 5345983 # 5492659 # 5637663 # 41 <lb/>20 # 5050299 # 5200162 # 5348441 # 5495090 # 5640066 # 40 <lb/>21 # 5052809 # 5202646 # 5350898 # 5497520 # 5642468 # 39 <lb/>22 # 5055319 # 5205130 # 5353355 # 5499950 # 5644869 # 38 <lb/>23 # 5057829 # 5207614 # 5355812 # 5502379 # 5647270 # 37 <lb/>24 # 5060338 # 5210097 # 5358268 # 5504808 # 5649670 # 36 <lb/>25 # 5062847 # 5212580 # 5360724 # 5507236 # 5652070 # 35 <lb/>26 # 5065355 # 5215062 # 5363179 # 5509664 # 5654469 # 34 <lb/>27 # 5067863 # 5217544 # 5365634 # 5512091 # 5656868 # 33 <lb/>28 # 5070370 # 5220025 # 5368088 # 5514518 # 5659266 # 32 <lb/>29 # 5072877 # 5222506 # 5370542 # 5516944 # 5661664 # 31 <lb/>30 # 5075384 # 5224986 # 5372996 # 5519370 # 5664062 # 30 <lb/> # 59 # 58 # 57 # 56 # 55 <lb/></note> </div> <div xml:id="echoid-div430" type="section" level="1" n="194"> <head xml:id="echoid-head220" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="145" file="157" n="157" rhead=""/> </div> <div xml:id="echoid-div431" type="section" level="1" n="195"> <head xml:id="echoid-head221" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 30 # 31 # 32 # 33 # 34 <lb/>30 # 5075384 # 5224986 # 5372996 # 5519370 # 5664062 # 30 <lb/>31 # 5077890 # 5227466 # 5375449 # 5521795 # 5666459 # 29 <lb/>32 # 5080396 # 5229949 # 5377902 # 5524220 # 5668856 # 28 <lb/>33 # 5082901 # 5232425 # 5380354 # 5526645 # 5671252 # 27 <lb/>34 # 5085406 # 5234904 # 5382806 # 5529069 # 5673648 # 26 <lb/>35 # 5087911 # 5237382 # 5385258 # 5531493 # 5676043 # 25 <lb/>36 # 5090415 # 5239860 # 5387709 # 5533916 # 5678438 # 24 <lb/>37 # 5092619 # 5242337 # 5390159 # 5536338 # 5680832 # 23 <lb/>38 # 5095422 # 5244814 # 5392609 # 5538760 # 5683226 # 22 <lb/>39 # 5097925 # 5247290 # 5395058 # 5541182 # 5685619 # 21 <lb/>40 # 5100427 # 5249766 # 5397507 # 5543603 # 5688012 # 20 <lb/>41 # 5102929 # 5252241 # 5399955 # 5546024 # 5690404 # 19 <lb/>42 # 5105430 # 5254716 # 5402403 # 5548444 # 5692796 # 18 <lb/>43 # 5107931 # 5257191 # 5404851 # 5550864 # 5695187 # 17 <lb/>44 # 5110431 # 5259665 # 5407298 # 5553283 # 5697578 # 16 <lb/>45 # 5112931 # 5262139 # 5409745 # 5555702 # 5699968 # 15 <lb/>46 # 5115431 # 5264612 # 5412191 # 5558120 # 5702358 # 14 <lb/>47 # 5117930 # 5267085 # 5414637 # 5560538 # 5704747 # 13 <lb/>48 # 5120429 # 5269557 # 5417082 # 5562956 # 5707136 # 12 <lb/>49 # 5122927 # 5272029 # 5419527 # 5565373 # 5709524 # 11 <lb/>50 # 5125425 # 5274501 # 5421972 # 5567790 # 5711912 # 10 <lb/>51 # 5127922 # 5276972 # 5424416 # 5570206 # 5714299 # 9 <lb/>52 # 5130419 # 5279443 # 5426859 # 5572622 # 5716686 # 8 <lb/>53 # 5132916 # 5281913 # 5429302 # 5575037 # 5719072 # 7 <lb/>54 # 5135412 # 5284383 # 5431745 # 5577452 # 5721458 # 6 <lb/>55 # 5137908 # 5286852 # 5434187 # 5579866 # 5723844 # 5 <lb/>56 # 5140403 # 5289321 # 5436629 # 5582280 # 5726229 # 4 <lb/>57 # 5141898 # 5291789 # 5439070 # 5584693 # 5728613 # 3 <lb/>58 # 5145393 # 5294257 # 5441510 # 5587106 # 5730997 # 2 <lb/>59 # 5147887 # 5296725 # 5443950 # 5589518 # 5733381 # 1 <lb/>60 # 5150381 # 5299192 # 5446390 # 5591929 # 5735764 # 0 <lb/> # 59 # 58 # 57 # 56 # 55 <lb/></note> </div> <div xml:id="echoid-div432" type="section" level="1" n="196"> <head xml:id="echoid-head222" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="146" file="158" n="158" rhead=""/> </div> <div xml:id="echoid-div433" type="section" level="1" n="197"> <head xml:id="echoid-head223" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 35 # 36 # 37 # 38 # 39 <lb/>0 # 5735764 # 5877852 # 6018150 # 6156615 # 6293204 # 60 <lb/>1 # 5738147 # 5880205 # 6020473 # 6158907 # 6295464 # 59 <lb/>2 # 5740529 # 5882558 # 6022796 # 6161198 # 6297724 # 58 <lb/>3 # 5742911 # 5884910 # 6025118 # 6163489 # 6299983 # 57 <lb/>4 # 5745292 # 5887262 # 6027439 # 6165780 # 6302242 # 56 <lb/>5 # 5747672 # 5889613 # 6029760 # 6168070 # 6304501 # 55 <lb/>6 # 5750052 # 5891964 # 6032080 # 6170359 # 6306759 # 54 <lb/>7 # 5752432 # 5894314 # 6034400 # 6172648 # 6309016 # 53 <lb/>8 # 5754811 # 5896664 # 6036719 # 6174936 # 6311273 # 52 <lb/>9 # 5757190 # 5899013 # 6039038 # 6177224 # 6313529 # 51 <lb/>10 # 5759568 # 5901361 # 6041357 # 6179512 # 6315784 # 50 <lb/>11 # 5761946 # 5903709 # 6043675 # 6181799 # 6318039 # 49 <lb/>12 # 5764323 # 5906056 # 6045992 # 6184085 # 6320293 # 48 <lb/>13 # 5766700 # 5908403 # 6048309 # 6186371 # 6322547 # 47 <lb/>14 # 5769076 # 5910750 # 6050625 # 6188656 # 6324800 # 46 <lb/>15 # 5771452 # 5913096 # 6052940 # 6190940 # 6327053 # 45 <lb/>16 # 5773827 # 5915442 # 6055255 # 6193224 # 6329305 # 44 <lb/>17 # 5776202 # 5917787 # 6057570 # 6195508 # 6331557 # 43 <lb/>18 # 5778576 # 5920132 # 6059884 # 6197781 # 6333808 # 42 <lb/>19 # 5780950 # 5922476 # 6062198 # 6200074 # 6336059 # 41 <lb/>20 # 5783324 # 5924820 # 6064511 # 6202356 # 6338310 # 40 <lb/>21 # 5785697 # 5927163 # 6066824 # 6204638 # 6340560 # 39 <lb/>22 # 5788069 # 5929505 # 6069136 # 6206919 # 6342809 # 38 <lb/>23 # 5790441 # 5931847 # 6071448 # 6209199 # 6345058 # 37 <lb/>24 # 5792812 # 5934189 # 6073759 # 6211479 # 6347306 # 36 <lb/>25 # 5795183 # 5936530 # 6076069 # 6213758 # 6349553 # 35 <lb/>26 # 5797553 # 5938871 # 6078379 # 6216037 # 6351800 # 34 <lb/>27 # 5799923 # 5941211 # 6080688 # 6218315 # 6354046 # 33 <lb/>28 # 5802292 # 5943551 # 6082997 # 6220593 # 6356292 # 32 <lb/>29 # 5804661 # 5945890 # 6085306 # 6222870 # 6358537 # 31 <lb/>30 # 5807030 # 5948228 # 6087614 # 6225146 # 6360782 # 30 <lb/> # 54 # 53 # 52 # 51 # 50 <lb/></note> </div> <div xml:id="echoid-div434" type="section" level="1" n="198"> <head xml:id="echoid-head224" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="147" file="159" n="159" rhead=""/> </div> <div xml:id="echoid-div435" type="section" level="1" n="199"> <head xml:id="echoid-head225" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 35 # 36 # 37 # 38 # 39 <lb/>30 # 5807030 # 5948228 # 6087614 # 6225146 # 6360782 # 30 <lb/>31 # 5809398 # 5950566 # 6089922 # 6227422 # 6363026 # 29 <lb/>32 # 5811766 # 5952904 # 6092229 # 6229698 # 6365270 # 28 <lb/>33 # 5814133 # 5955241 # 6094536 # 6231973 # 6367513 # 27 <lb/>34 # 5816499 # 5957578 # 6096842 # 6234248 # 6369756 # 26 <lb/>35 # 5818865 # 5959914 # 6099147 # 6236522 # 6371999 # 25 <lb/>36 # 6821230 # 5962250 # 6101452 # 6238796 # 6374241 # 24 <lb/>37 # 5823595 # 5964585 # 6103756 # 6241069 # 6376482 # 23 <lb/>38 # 5825959 # 5966919 # 6106060 # 6243342 # 6378722 # 22 <lb/>39 # 5828323 # 5969253 # 6108364 # 6245614 # 6380962 # 21 <lb/>40 # 5830687 # 5971586 # 6110667 # 6247885 # 6383201 # 20 <lb/>41 # 5833050 # 5973919 # 6112970 # 6250156 # 6385440 # 19 <lb/>42 # 5835412 # 5976251 # 6115272 # 6252426 # 6387678 # 18 <lb/>43 # 5837774 # 5978583 # 6117573 # 6254696 # 6389916 # 17 <lb/>44 # 5840136 # 5980915 # 6119873 # 6256966 # 6392153 # 16 <lb/>45 # 5842497 # 5983246 # 6122173 # 6259235 # 6394390 # 15 <lb/>46 # 5844858 # 5985577 # 6124473 # 6261503 # 6396626 # 14 <lb/>47 # 5847218 # 5987907 # 6126772 # 6263771 # 6398862 # 13 <lb/>48 # 5849578 # 5990237 # 6129071 # 6266038 # 6401097 # 12 <lb/>49 # 5851937 # 5992566 # 6131369 # 6268305 # 6403332 # 11 <lb/>50 # 5854295 # 5994894 # 6133667 # 6270572 # 6405566 # 10 <lb/>51 # 5856653 # 5997222 # 6135964 # 6272838 # 6407799 # 9 <lb/>52 # 5859010 # 5999549 # 6138261 # 6275103 # 6410032 # 8 <lb/>53 # 5861367 # 6001876 # 6140557 # 6277368 # 6412264 # 7 <lb/>54 # 5863724 # 6004202 # 6142853 # 6279632 # 6414496 # 6 <lb/>55 # 5866080 # 6006528 # 6145148 # 6281895 # 6416728 # 5 <lb/>56 # 5868436 # 6008853 # 6147442 # 6284158 # 6418959 # 4 <lb/>57 # 5870791 # 6011178 # 6149736 # 6286420 # 6421189 # 3 <lb/>58 # 5873145 # 6013502 # 6152030 # 6288682 # 6423419 # 2 <lb/>59 # 5875499 # 6015826 # 6154323 # 6290943 # 6425648 # 1 <lb/>60 # 5877852 # 6018150 # 6156615 # 6293204 # 6427876 # 0 <lb/> # 54 # 53 # 52 # 51 # 50 <lb/></note> </div> <div xml:id="echoid-div436" type="section" level="1" n="200"> <head xml:id="echoid-head226" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="148" file="160" n="160" rhead=""/> </div> <div xml:id="echoid-div437" type="section" level="1" n="201"> <head xml:id="echoid-head227" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 40 # 41 # 42 # 43 # 44 <lb/>0 # 6427876 # 6560590 # 6691306 # 6819984 # 6946584 # 60 <lb/>1 # 6430104 # 6562785 # 6693468 # 6822111 # 6948676 # 59 <lb/>2 # 6432331 # 6564979 # 6695629 # 6824237 # 6950767 # 58 <lb/>3 # 6434558 # 6567173 # 6697789 # 6826363 # 6952858 # 57 <lb/>4 # 6436785 # 6569367 # 6699949 # 6828489 # 6954949 # 56 <lb/>5 # 6439011 # 6571560 # 6702108 # 6830614 # 6957039 # 55 <lb/>6 # 6441236 # 6573753 # 6704267 # 6832738 # 6959128 # 54 <lb/>7 # 6443461 # 6575945 # 6706425 # 6834861 # 6961216 # 53 <lb/>8 # 6445685 # 6578136 # 6708582 # 6836984 # 6963304 # 52 <lb/>9 # 6447909 # 6580326 # 6710739 # 6839107 # 6965392 # 51 <lb/>10 # 6450132 # 6582516 # 6712895 # 6841229 # 6967479 # 50 <lb/>11 # 6452355 # 6584705 # 6715051 # 6843350 # 6969565 # 49 <lb/>12 # 6454577 # 6586894 # 6717206 # 6845471 # 6971651 # 48 <lb/>13 # 6456799 # 6589082 # 6719361 # 6847591 # 6973736 # 47 <lb/>14 # 6459020 # 6591270 # 6721515 # 6849711 # 6975821 # 46 <lb/>15 # 6461240 # 6593458 # 6723668 # 6851830 # 6977905 # 45 <lb/>16 # 6463460 # 6595645 # 6725821 # 6853949 # 6979988 # 44 <lb/>17 # 6465679 # 6597831 # 6727973 # 6856067 # 6982071 # 43 <lb/>18 # 6467898 # 6600016 # 6730125 # 6858184 # 6984153 # 42 <lb/>19 # 6470116 # 6602201 # 6732276 # 6860301 # 6986235 # 41 <lb/>20 # 6472333 # 6604386 # 6734427 # 6862417 # 6988319 # 40 <lb/>21 # 6474550 # 6606570 # 6736577 # 6864533 # 6990396 # 39 <lb/>22 # 6476766 # 6608753 # 6738726 # 6866648 # 6992476 # 38 <lb/>23 # 6478982 # 6610936 # 6740875 # 6868762 # 6994555 # 37 <lb/>24 # 6481198 # 6613118 # 6743024 # 6870876 # 6996634 # 36 <lb/>25 # 6483413 # 6615300 # 6745172 # 6872989 # 6998712 # 35 <lb/>26 # 6485628 # 6617481 # 6747319 # 6875102 # 7000789 # 34 <lb/>27 # 6487842 # 6619661 # 6749465 # 6877214 # 7002866 # 33 <lb/>28 # 6490055 # 6621841 # 6751611 # 6879325 # 7004942 # 32 <lb/>29 # 6492268 # 6624021 # 6753757 # 6881436 # 7007018 # 31 <lb/>30 # 6494480 # 6626200 # 6755902 # 6883546 # 7009093 # 30 <lb/> # 49 # 48 # 47 # 46 # 45 <lb/></note> </div> <div xml:id="echoid-div438" type="section" level="1" n="202"> <head xml:id="echoid-head228" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="149" file="161" n="161" rhead=""/> </div> <div xml:id="echoid-div439" type="section" level="1" n="203"> <head xml:id="echoid-head229" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 40 # 41 # 42 # 43 # 44 <lb/>30 # 6494480 # 6626200 # 6755902 # 6883546 # 7009093 # 30 <lb/>31 # 6496692 # 6628379 # 6758047 # 6885656 # 7011167 # 29 <lb/>32 # 6498903 # 6630557 # 6760191 # 6887765 # 7013241 # 28 <lb/>33 # 6501114 # 6632734 # 6762334 # 6889874 # 7015314 # 27 <lb/>34 # 6503324 # 6634911 # 6764477 # 6891982 # 7017387 # 26 <lb/>35 # 6505533 # 6637087 # 6766619 # 6894089 # 7019459 # 25 <lb/>36 # 6507742 # 6639263 # 6768760 # 6896196 # 7021530 # 24 <lb/>37 # 6509950 # 6641438 # 6770901 # 6898302 # 7023601 # 23 <lb/>38 # 6512158 # 6643612 # 6773041 # 6900408 # 7025671 # 22 <lb/>39 # 6514365 # 6645786 # 6775181 # 6902513 # 7027741 # 21 <lb/>40 # 6516572 # 6647959 # 6777320 # 6904617 # 7029810 # 20 <lb/>41 # 6518778 # 6650132 # 6779459 # 6906721 # 7031879 # 19 <lb/>42 # 6520984 # 6652304 # 6781597 # 6908824 # 7033947 # 18 <lb/>43 # 6523189 # 6654476 # 6783734 # 6910927 # 7036014 # 17 <lb/>44 # 6525394 # 6656647 # 6785871 # 6913029 # 7038081 # 16 <lb/>45 # 6527598 # 6658817 # 6788007 # 6915131 # 7040147 # 15 <lb/>46 # 6529801 # 6660987 # 6790143 # 6917232 # 7042213 # 14 <lb/>47 # 6532004 # 6663156 # 6792278 # 6919332 # 7044278 # 13 <lb/>48 # 6534206 # 6665325 # 6794413 # 6921432 # 7046342 # 12 <lb/>49 # 6536408 # 6667493 # 6796547 # 6923531 # 7048406 # 11 <lb/>50 # 6538609 # 6669661 # 6798681 # 6925630 # 7050469 # 10 <lb/>51 # 6540809 # 6671828 # 6800814 # 6927728 # 7052432 # 9 <lb/>52 # 6543009 # 6673994 # 6802946 # 6929725 # 7054594 # 8 <lb/>53 # 6545208 # 6676160 # 6805078 # 6931922 # 7056655 # 7 <lb/>54 # 6547407 # 6678326 # 6807209 # 6934018 # 7058716 # 6 <lb/>55 # 6549606 # 6680491 # 6809340 # 6936114 # 7060776 # 5 <lb/>56 # 6551804 # 6682655 # 6811470 # 6938209 # 7062836 # 4 <lb/>57 # 6554001 # 6684818 # 6813599 # 6940303 # 7064895 # 3 <lb/>58 # 6556198 # 6686981 # 6815728 # 6942397 # 7066953 # 2 <lb/>59 # 6558394 # 6689144 # 6817856 # 6944491 # 7069011 # 1 <lb/>60 # 6560590 # 6691306 # 6819984 # 6946584 # 7071063 # 0 <lb/> # 49 # 48 # 47 # 46 # 45 <lb/></note> </div> <div xml:id="echoid-div440" type="section" level="1" n="204"> <head xml:id="echoid-head230" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="150" file="162" n="162" rhead=""/> </div> <div xml:id="echoid-div441" type="section" level="1" n="205"> <head xml:id="echoid-head231" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 45 # 46 # 47 # 48 # 49 <lb/>0 # 7071068 # 7193398 # 7313537 # 7431448 # 7547096 # 60 <lb/>1 # 7073125 # 7195418 # 7315521 # 7433394 # 7549004 # 59 <lb/>2 # 7075181 # 7197438 # 7317504 # 7435339 # 7550911 # 58 <lb/>3 # 7077236 # 7199457 # 7319486 # 7437284 # 7552818 # 57 <lb/>4 # 7079291 # 7201476 # 7321468 # 7439229 # 7554724 # 56 <lb/>5 # 7081345 # 7203494 # 7323449 # 7441173 # 7556630 # 55 <lb/>6 # 7083399 # 7205511 # 7325429 # 7443116 # 7558535 # 54 <lb/>7 # 7085452 # 7207527 # 7327409 # 7445058 # 75@@439 # 53 <lb/>8 # 7087504 # 3209543 # 7329388 # 7447000 # 7562343 # 52 <lb/>9 # 7089556 # 7211559 # 7331367 # 7448941 # 7564246 # 51 <lb/>10 # 7091607 # 7213574 # 7333345 # 7450882 # 7566148 # 50 <lb/>11 # 7093658 # 7215588 # 7335322 # 7452822 # 7568050 # 49 <lb/>12 # 7095708 # 7217601 # 7337298 # 7454761 # 7569951 # 48 <lb/>13 # 7097757 # 7219614 # 7339274 # 7456699 # 7571851 # 47 <lb/>14 # 7099806 # 7221627 # 7341250 # 7458637 # 7573751 # 46 <lb/>15 # 7101854 # 7223639 # 7343225 # 7460574 # 7575650 # 45 <lb/>16 # 7103902 # 7225651 # 7345199 # 7462511 # 7577548 # 44 <lb/>17 # 7105949 # 7227662 # 7347173 # 7464447 # 7579446 # 43 <lb/>18 # 7107995 # 7229672 # 7349146 # 7466382 # 7581343 # 42 <lb/>19 # 7110041 # 7231681 # 7351118 # 7468317 # 7583240 # 41 <lb/>20 # 7112086 # 7233689 # 7353090 # 7470251 # 7585136 # 40 <lb/>21 # 7114131 # 7235697 # 7355061 # 7472184 # 7587031 # 39 <lb/>22 # 7116175 # 7237704 # 7357031 # 7474117 # 7588925 # 38 <lb/>23 # 7118218 # 7239711 # 7359001 # 7476049 # 7590819 # 37 <lb/>24 # 7120261 # 7241718 # 7360970 # 7477981 # 7592713 # 36 <lb/>25 # 7122303 # 7243724 # 7362939 # 7479912 # 7594606 # 35 <lb/>26 # 7124344 # 7245729 # 7364907 # 7481842 # 7596498 # 34 <lb/>27 # 7126385 # 7247733 # 7366874 # 7483771 # 7598389 # 33 <lb/>28 # 7128425 # 7249737 # 7368841 # 7485700 # 7600280 # 32 <lb/>29 # 7130465 # 7251741 # 7370807 # 7487629 # 7501170 # 31 <lb/>30 # 7132504 # 7253744 # 7372773 # 7489557 # 7604060 # 30 <lb/> # 44 # 43 # 42 # 41 # 40 <lb/></note> </div> <div xml:id="echoid-div442" type="section" level="1" n="206"> <head xml:id="echoid-head232" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="151" file="163" n="163" rhead=""/> </div> <div xml:id="echoid-div443" type="section" level="1" n="207"> <head xml:id="echoid-head233" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 45 # 46 # 47 # 48 # 49 <lb/>30 # 7132504 # 7253744 # 7372773 # 7489557 # 7604960 # 30 <lb/>31 # 7134543 # 7255746 # 7374738 # 7491484 # 7605949 # 29 <lb/>32 # 7136581 # 7257747 # 7376702 # 7493410 # 7607837 # 28 <lb/>33 # 7138618 # 7259748 # 7378666 # 7495336 # 7609725 # 27 <lb/>34 # 7140655 # 7261749 # 7380629 # 7497262 # 7611612 # 26 <lb/>35 # 7142691 # 7263749 # 7382592 # 7499187 # 7613498 # 25 <lb/>36 # 7144727 # 7265748 # 7384554 # 7501111 # 7645384 # 24 <lb/>37 # 7146762 # 7267746 # 7386515 # 7503034 # 7617269 # 23 <lb/>38 # 7148796 # 7269744 # 7388475 # 7504957 # 7619153 # 22 <lb/>39 # 7150830 # 7271741 # 7390435 # 7506879 # 7621037 # 21 <lb/>40 # 7152863 # 7273737 # 7392394 # 7508801 # 7622920 # 20 <lb/>41 # 7154895 # 7275733 # 7394353 # 7510722 # 7624802 # 19 <lb/>42 # 7156927 # 7277728 # 7396311 # 7512642 # 7626683 # 18 <lb/>43 # 7158958 # 7279722 # 7398268 # 7514561 # 7628564 # 17 <lb/>44 # 7160989 # 7281716 # 7400225 # 7516480 # 7630445 # 16 <lb/>45 # 7163019 # 7283710 # 7402181 # 7518398 # 7632325 # 15 <lb/>46 # 7165049 # 7285703 # 7404137 # 7520316 # 7634204 # 14 <lb/>47 # 7167078 # 7287695 # 7406092 # 7522233 # 7636082 # 13 <lb/>48 # 7169106 # 7289687 # 7408046 # 7524149 # 7637960 # 12 <lb/>49 # 7171134 # 7291678 # 7410000 # 7526065 # 7639838 # 11 <lb/>50 # 7173161 # 7293668 # 7411953 # 7527980 # 7641715 # 10 <lb/>51 # 7175187 # 7295658 # 7413905 # 7529894 # 7643591 # 9 <lb/>52 # 7177213 # 7297647 # 7415856 # 7531808 # 7645466 # 8 <lb/>53 # 7179238 # 7299635 # 7417807 # 7533721 # 7647341 # 7 <lb/>54 # 7181263 # 7301623 # 7419758 # 7535634 # 7649215 # 6 <lb/>55 # 7183287 # 7303610 # 7421708 # 7537546 # 7651088 # 5 <lb/>56 # 7185310 # 7305597 # 7423657 # 7539457 # 7652961 # 4 <lb/>57 # 7187333 # 7307583 # 7425605 # 7541367 # 7654833 # 3 <lb/>58 # 7189355 # 7309568 # 7427552 # 7543277 # 7656704 # 2 <lb/>59 # 7191377 # 7311553 # 7429501 # 7545187 # 7658575 # 1 <lb/>60 # 7193398 # 7313537 # 7431448 # 7547096 # 7660445 # 0 <lb/> # 44 # 43 # 42 # 41 # 40 <lb/></note> </div> <div xml:id="echoid-div444" type="section" level="1" n="208"> <head xml:id="echoid-head234" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="152" file="164" n="164" rhead=""/> </div> <div xml:id="echoid-div445" type="section" level="1" n="209"> <head xml:id="echoid-head235" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 50 # 51 # 52 # 53 # 54 <lb/>0 # 7660445 # 7771460 # 7880108 # 7986355 # 8090170 # 60 <lb/>1 # 7662314 # 7773290 # 7881898 # 7988105 # 8091879 # 59 <lb/>2 # 7664183 # 7775120 # 7883688 # 7989855 # 8093588 # 58 <lb/>3 # 7666051 # 7776949 # 7885477 # 7991604 # 8095296 # 57 <lb/>4 # 7667919 # 7778777 # 7887266 # 7993352 # 8097004 # 56 <lb/>5 # 7669786 # 7780605 # 7889054 # 7995100 # 8098711 # 55 <lb/>6 # 7671652 # 7782432 # 7890841 # 7996847 # 8100417 # 54 <lb/>7 # 7673517 # 7784258 # 7892927 # 7998593 # 8102122 # 53 <lb/>8 # 7675382 # 7786084 # 7894413 # 8000339 # 8103827 # 52 <lb/>9 # 7677246 # 7787909 # 7896198 # 8002084 # 8105531 # 51 <lb/>10 # 7679110 # 7789833 # 7897983 # 8003828 # 8107234 # 50 <lb/>11 # 7680973 # 7791557 # 7899767 # 8005571 # 8108936 # 49 <lb/>12 # 7682835 # 7793380 # 7901550 # 8007314 # 8110638 # 48 <lb/>13 # 7684697 # 7795202 # 7903332 # 8009056 # 8112339 # 47 <lb/>14 # 7686558 # 7797024 # 7905114 # 8010797 # 8114040 # 46 <lb/>15 # 7688418 # 7798845 # 7906895 # 8012538 # 8115740 # 45 <lb/>16 # 7690278 # 7800665 # 7908676 # 8014278 # 8117439 # 44 <lb/>17 # 7692137 # 7802485 # 7910456 # 8016017 # 8119137 # 43 <lb/>18 # 7693995 # 7804304 # 7912235 # 8017756 # 8120835 # 42 <lb/>19 # 7695853 # 7806123 # 7914014 # 8019494 # 8122532 # 41 <lb/>20 # 7697710 # 7807941 # 7915792 # 8021232 # 8124229 # 40 <lb/>21 # 7699566 # 7809758 # 7917569 # 8022969 # 8125925 # 39 <lb/>22 # 7701422 # 7811574 # 7919345 # 8024705 # 8127620 # 38 <lb/>23 # 7703277 # 7813390 # 7921121 # 8026440 # 8129314 # 37 <lb/>24 # 7705132 # 7815205 # 7922896 # 8028175 # 8131008 # 36 <lb/>25 # 7706986 # 7817020 # 7924671 # 8029909 # 8132701 # 35 <lb/>26 # 7708839 # 7818834 # 7926445 # 8031642 # 8134393 # 34 <lb/>27 # 7710692 # 7820647 # 7928218 # 8033375 # 8136084 # 33 <lb/>28 # 7712544 # 7822459 # 7929990 # 8035107 # 8137775 # 32 <lb/>29 # 7714395 # 7824271 # 7931762 # 8036838 # 8139465 # 31 <lb/>30 # 7716246 # 7826082 # 7933533 # 8038569 # 8141155 # 30 <lb/> # 39 # 38 # 37 # 36 # 35 <lb/></note> </div> <div xml:id="echoid-div446" type="section" level="1" n="210"> <head xml:id="echoid-head236" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="153" file="165" n="165" rhead=""/> </div> <div xml:id="echoid-div447" type="section" level="1" n="211"> <head xml:id="echoid-head237" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 50 # 51 # 52 # 53 # 54 <lb/>30 # 7716246 # 7826082 # 7933533 # 8038569 # 8141155 # 30 <lb/>31 # 7718096 # 7827892 # 7935303 # 8040299 # 8142844 # 29 <lb/>32 # 7719945 # 7829702 # 7937073 # 8042028 # 8144532 # 28 <lb/>33 # 7721794 # 7831511 # 7938842 # 8043757 # 8146220 # 27 <lb/>34 # 7723642 # 7833320 # 7940611 # 8045485 # 8147907 # 26 <lb/>35 # 7725490 # 7835128 # 7942379 # 8047212 # 8149593 # 25 <lb/>36 # 7727337 # 7836935 # 7944146 # 8048938 # 8151278 # 24 <lb/>37 # 7729183 # 7838741 # 7945912 # 8050664 # 8152963 # 23 <lb/>38 # 7731028 # 7840547 # 7947678 # 8052389 # 8154647 # 22 <lb/>39 # 7732872 # 7842352 # 7949443 # 8054114 # 8156330 # 21 <lb/>40 # 7734716 # 7844157 # 7951208 # 8055838 # 8158013 # 20 <lb/>41 # 7736559 # 7845961 # 7952972 # 8057561 # 8159695 # 19 <lb/>42 # 7738402 # 7847764 # 7954735 # 8059283 # 8161376 # 18 <lb/>43 # 7740244 # 7849566 # 7956497 # 8061005 # 8163057 # 17 <lb/>44 # 7742085 # 7851368 # 7958259 # 8062726 # 8164737 # 16 <lb/>45 # 7743926 # 7853169 # 7960020 # 8064446 # 8166416 # 15 <lb/>46 # 7745766 # 7854970 # 7961780 # 8066166 # 8168094 # 14 <lb/>47 # 7747606 # 7856770 # 7963540 # 8067885 # 8169772 # 13 <lb/>48 # 7749445 # 7858569 # 7965299 # 8069603 # 8171449 # 12 <lb/>49 # 7751283 # 7860368 # 7967057 # 8071321 # 8173126 # 11 <lb/>50 # 7753121 # 7862166 # 7968815 # 8073038 # 8174802 # 10 <lb/>51 # 7754958 # 7863963 # 7970572 # 8074754 # 8176477 # 9 <lb/>52 # 7756794 # 7865759 # 7972328 # 8076470 # 8178151 # 8 <lb/>53 # 7758630 # 7867555 # 7974084 # 8078185 # 8179825 # 7 <lb/>54 # 7760465 # 7869350 # 7975839 # 8079899 # 8181498 # 6 <lb/>55 # 7762299 # 7871145 # 7977593 # 8081613 # 8183170 # 5 <lb/>56 # 7764132 # 7872939 # 7979347 # 8083326 # 8184841 # 4 <lb/>57 # 7765965 # 7874732 # 7981100 # 8085038 # 8186512 # 3 <lb/>58 # 7767797 # 7876525 # 7982852 # 8086749 # 8188182 # 2 <lb/>59 # 7769629 # 7878317 # 7984604 # 8088460 # 8189851 # 1 <lb/>60 # 7771460 # 7880108 # 7986355 # 8090170 # 8191520 # 0 <lb/> # 39 # 38 # 37 # 36 # 35 <lb/></note> </div> <div xml:id="echoid-div448" type="section" level="1" n="212"> <head xml:id="echoid-head238" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="154" file="166" n="166" rhead=""/> </div> <div xml:id="echoid-div449" type="section" level="1" n="213"> <head xml:id="echoid-head239" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 55 # 56 # 57 # 58 # 59 <lb/>0 # 8191520 # 8290376 # 8386706 # 8480481 # 8571673 # 60 <lb/>1 # 8193188 # 8292002 # 8388290 # 8482022 # 8573171 # 59 <lb/>2 # 8194855 # 8293628 # 8389873 # 8483562 # 8574668 # 58 <lb/>3 # 8196522 # 8295253 # 8391456 # 8485102 # 8576164 # 57 <lb/>4 # 8198188 # 8296877 # 8393038 # 8486641 # 8577660 # 56 <lb/>5 # 8199854 # 8298501 # 8394619 # 8488180 # 8579155 # 55 <lb/>6 # 8201519 # 8300124 # 8396199 # 8489718 # 8580649 # 54 <lb/>7 # 8203183 # 8301746 # 8397778 # 8491255 # 8582142 # 53 <lb/>8 # 8204846 # 8303367 # 8399357 # 8492791 # 8583635 # 52 <lb/>9 # 8206508 # 8304987 # 8400935 # 8494326 # 8585127 # 51 <lb/>10 # 8208170 # 8306607 # 8402513 # 8495860 # 8586619 # 50 <lb/>11 # 8209831 # 8308226 # 8404090 # 8497394 # 8588110 # 49 <lb/>12 # 8211491 # 8309844 # 8405666 # 8498927 # 8589600 # 48 <lb/>13 # 8213151 # 8311462 # 8407241 # 8500459 # 8591089 # 47 <lb/>14 # 8214810 # 8313079 # 8408816 # 8501991 # 8592577 # 46 <lb/>15 # 8216469 # 8314696 # 8410390 # 8503522 # 8594064 # 45 <lb/>16 # 8218127 # 8316312 # 8411963 # 8505052 # 8595551 # 44 <lb/>17 # 8219784 # 8317927 # 8413536 # 8506582 # 8597037 # 43 <lb/>18 # 8221440 # 8319541 # 8415108 # 8508111 # 8598523 # 42 <lb/>19 # 8223096 # 8321155 # 8416679 # 8509639 # 8600008 # 41 <lb/>20 # 8224751 # 8322768 # 8418250 # 8511167 # 8601492 # 40 <lb/>21 # 8226405 # 8324380 # 8419820 # 8512694 # 8602975 # 39 <lb/>22 # 8228058 # 8325991 # 8421389 # 8514220 # 8604457 # 38 <lb/>23 # 8229711 # 8327602 # 8422957 # 8515745 # 8605939 # 37 <lb/>24 # 8231363 # 8329212 # 8424525 # 8517270 # 8607420 # 36 <lb/>25 # 8233015 # 8330822 # 8426092 # 8518794 # 8608901 # 35 <lb/>26 # 8234666 # 8332431 # 8427658 # 8520317 # 8610381 # 34 <lb/>27 # 8236316 # 8334039 # 8429223 # 8521839 # 8611860 # 33 <lb/>28 # 8237965 # 8335646 # 8430788 # 8523361 # 8613338 # 32 <lb/>29 # 8239614 # 8337252 # 8432352 # 8524882 # 8614815 # 31 <lb/>30 # 8241262 # 8338858 # 8433915 # 8526402 # 8616292 # 30 <lb/> # 34 # 33 # 32 # 31 # 30 <lb/></note> </div> <div xml:id="echoid-div450" type="section" level="1" n="214"> <head xml:id="echoid-head240" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="155" file="167" n="167" rhead=""/> </div> <div xml:id="echoid-div451" type="section" level="1" n="215"> <head xml:id="echoid-head241" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 55 # 56 # 57 # 58 # 59 <lb/>30 # 8241262 # 8338858 # 8433915 # 8526402 # 8616292 # 30 <lb/>31 # 8242909 # 8340463 # 8435477 # 8527921 # 8617768 # 29 <lb/>32 # 8244556 # 8342067 # 8437039 # 8529440 # 8619243 # 28 <lb/>33 # 8246202 # 8343671 # 8438600 # 8530958 # 8620718 # 27 <lb/>34 # 8247847 # 8345247 # 8440161 # 8532476 # 8622192 # 26 <lb/>35 # 8249492 # 8346877 # 8441721 # 8533993 # 8623665 # 25 <lb/>36 # 8251136 # 8348479 # 8443280 # 8535509 # 8625137 # 24 <lb/>37 # 8252779 # 8350080 # 8444838 # 8537024 # 8626608 # 23 <lb/>38 # 8254421 # 8251680 # 8446396 # 8538538 # 8628079 # 22 <lb/>39 # 8256062 # 8353279 # 8447953 # 8540052 # 8629549 # 21 <lb/>40 # 8257703 # 8354878 # 8449509 # 8541565 # 8631019 # 20 <lb/>41 # 8259343 # 8356476 # 8451064 # 8543077 # 8632488 # 19 <lb/>42 # 8260982 # 8358073 # 8452618 # 8544588 # 8633956 # 18 <lb/>43 # 8262621 # 8359670 # 8454172 # 8546099 # 8635423 # 17 <lb/>44 # 8264259 # 8361266 # 8455725 # 8547609 # 8636889 # 16 <lb/>45 # 8265897 # 8362862 # 8457278 # 8549119 # 8638355 # 15 <lb/>46 # 8267534 # 8364457 # 8458830 # 8550628 # 8639820 # 14 <lb/>47 # 8269170 # 8366051 # 8460381 # 8552136 # 8641284 # 13 <lb/>48 # 8270806 # 8367644 # 8461932 # 8553643 # 8642748 # 12 <lb/>49 # 8272441 # 8369236 # 8463482 # 8555149 # 8644211 # 11 <lb/>50 # 8274075 # 8370828 # 8465031 # 8556655 # 8645673 # 10 <lb/>51 # 8275708 # 8372419 # 8466579 # 8558160 # 8647134 # 9 <lb/>52 # 8277340 # 8374009 # 8468126 # 8559664 # 8648595 # 8 <lb/>53 # 8278972 # 8375599 # 8469673 # 8561168 # 8650055 # 7 <lb/>54 # 8280603 # 8377188 # 8471219 # 8562671 # 8651514 # 6 <lb/>55 # 8282234 # 8378776 # 8472765 # 8564173 # 8652973 # 5 <lb/>56 # 8283864 # 8380363 # 8474310 # 8565675 # 8654431 # 4 <lb/>57 # 8285493 # 8381950 # 8475854 # 8567176 # 8655888 # 3 <lb/>58 # 8287121 # 8383536 # 8477397 # 8568676 # 8657344 # 2 <lb/>59 # 8288749 # 8385121 # 8478939 # 8570175 # 8658799 # 1 <lb/>60 # 8290376 # 8386706 # 8480481 # 8571673 # 8660254 # 0 <lb/> # 34 # 33 # 32 # 31 # 30 <lb/></note> <pb o="156" file="168" n="168" rhead=""/> </div> <div xml:id="echoid-div452" type="section" level="1" n="216"> <head xml:id="echoid-head242" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 60 # 61 # 62 # 63 # 64 <lb/>0 # 8660254 # 8746197 # 8829476 # 8910065 # 8987940 # 60 <lb/>1 # 8661708 # 8747607 # 8830841 # 8911385 # 8989215 # 59 <lb/>2 # 8663162 # 8749016 # 8832205 # 8912704 # 8990489 # 58 <lb/>3 # 8664615 # 8750425 # 8833569 # 8914023 # 8991762 # 57 <lb/>4 # 8666067 # 8751833 # 8834932 # 8915341 # 8993035 # 56 <lb/>5 # 8667518 # 8753240 # 8836295 # 8916659 # 8994307 # 55 <lb/>6 # 8668968 # 8754646 # 8837657 # 8917976 # 8995578 # 54 <lb/>7 # 8670417 # 8756051 # 8839018 # 8919292 # 8996848 # 53 <lb/>8 # 8671866 # 8757456 # 8840378 # 8920607 # 8998117 # 52 <lb/>9 # 8673314 # 8758860 # 8841737 # 8921921 # 8999386 # 51 <lb/>10 # 8674762 # 8760263 # 8843095 # 8923234 # 9000654 # 50 <lb/>11 # 8676209 # 8761665 # 8844452 # 8924546 # 9001921 # 49 <lb/>12 # 8677655 # 8763067 # 8845809 # 8925858 # 9003187 # 48 <lb/>13 # 8679100 # 8764468 # 8847165 # 8927169 # 9004453 # 47 <lb/>14 # 8680544 # 8765868 # 8848521 # 8928479 # 9005718 # 46 <lb/>15 # 8681988 # 8767268 # 8849876 # 8929789 # 9006982 # 45 <lb/>16 # 8683431 # 8768667 # 8851230 # 8931098 # 9008245 # 44 <lb/>17 # 8684874 # 8770065 # 8852583 # 8932406 # 9009508 # 43 <lb/>18 # 8686316 # 8771462 # 8853936 # 8933714 # 9010770 # 42 <lb/>19 # 8687757 # 8772859 # 8855288 # 8935021 # 9012031 # 41 <lb/>20 # 8689197 # 8774255 # 8856639 # 8936327 # 9013292 # 40 <lb/>21 # 8690636 # 8775650 # 8857989 # 8937632 # 9014552 # 39 <lb/>22 # 8692074 # 8777044 # 8859338 # 8938936 # 9015811 # 38 <lb/>23 # 8693512 # 8778437 # 8860687 # 8940240 # 9017069 # 37 <lb/>24 # 8694949 # 8779830 # 8862035 # 8941543 # 9018326 # 36 <lb/>25 # 8696386 # 8781222 # 8863383 # 8942845 # 9019582 # 35 <lb/>26 # 8697822 # 8782613 # 8864730 # 8944146 # 9020838 # 34 <lb/>27 # 8699257 # 8784003 # 8866076 # 8945446 # 9022093 # 33 <lb/>28 # 8700691 # 8785393 # 8867421 # 8946746 # 9023347 # 32 <lb/>29 # 8702124 # 8786782 # 8868765 # 8948045 # 9024600 # 31 <lb/>30 # 8703557 # 8788171 # 8870108 # 8949344 # 9025853 # 30 <lb/> # 29 # 28 # 27 # 26 # 25 <lb/></note> </div> <div xml:id="echoid-div453" type="section" level="1" n="217"> <head xml:id="echoid-head243" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="157" file="169" n="169" rhead=""/> </div> <div xml:id="echoid-div454" type="section" level="1" n="218"> <head xml:id="echoid-head244" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 60 # 61 # 62 # 63 # 64 <lb/>30 # 8703557 # 8788171 # 8870108 # 8949344 # 9025853 # 30 <lb/>31 # 8704989 # 8789559 # 8871451 # 8950642 # 9027105 # 29 <lb/>32 # 8706420 # 8790946 # 8872793 # 8951939 # 9028356 # 28 <lb/>33 # 8707851 # 8792332 # 8874134 # 8953235 # 9029606 # 27 <lb/>34 # 8709281 # 8793717 # 8875475 # 8954530 # 9030856 # 26 <lb/>35 # 8710710 # 8795102 # 8876815 # 8955824 # 9032105 # 25 <lb/>36 # 8712138 # 8796486 # 8878154 # 8957117 # 9033353 # 24 <lb/>37 # 8713565 # 8797869 # 8879492 # 8958410 # 9034600 # 23 <lb/>38 # 8714992 # 8799251 # 8880830 # 8959702 # 9035847 # 22 <lb/>39 # 8716418 # 8800633 # 8882167 # 8960994 # 9037093 # 21 <lb/>40 # 8717844 # 8802014 # 8883503 # 8962285 # 9038338 # 20 <lb/>41 # 8719269 # 8803394 # 8884838 # 8963575 # 9039582 # 19 <lb/>42 # 8720693 # 8804773 # 8886172 # 8964864 # 9040825 # 18 <lb/>43 # 8722116 # 8806152 # 8887506 # 8966152 # 9042068 # 17 <lb/>44 # 8723538 # 8807530 # 8888839 # 8967440 # 9043310 # 16 <lb/>45 # 8724960 # 8808907 # 8890171 # 8968727 # 9044551 # 15 <lb/>46 # 8726381 # 8810283 # 8891502 # 8970013 # 9045791 # 14 <lb/>47 # 8727801 # 8811659 # 8892833 # 8971299 # 9047031 # 13 <lb/>48 # 8729221 # 8813034 # 8894163 # 8972584 # 9048270 # 12 <lb/>49 # 8730640 # 8814408 # 8895492 # 8973868 # 9049508 # 11 <lb/>50 # 8732058 # 8815782 # 8896821 # 8975151 # 9050746 # 10 <lb/>51 # 8733475 # 8817155 # 8898149 # 8976433 # 9051983 # 9 <lb/>52 # 8734891 # 8818527 # 8899476 # 8977715 # 9053219 # 8 <lb/>53 # 8736307 # 8819898 # 8900802 # 8978996 # 9054454 # 7 <lb/>54 # 8737722 # 8821268 # 8902127 # 8980276 # 9055688 # 6 <lb/>55 # 8739137 # 8822638 # 8903452 # 8981555 # 9056922 # 5 <lb/>56 # 8740551 # 8824007 # 8904776 # 9882833 # 9058155 # 4 <lb/>57 # 8741964 # 8825375 # 8906099 # 8984111 # 9059387 # 3 <lb/>58 # 8743376 # 8826743 # 8907422 # 8985388 # 9060618 # 2 <lb/>59 # 8744787 # 8828110 # 8908744 # 8986664 # 9061848 # 1 <lb/>60 # 8746197 # 8829476 # 8910065 # 8987940 # 9063078 # 0 <lb/> # 25 # 26 # 27 # 28 # 29 <lb/></note> </div> <div xml:id="echoid-div455" type="section" level="1" n="219"> <head xml:id="echoid-head245" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="158" file="170" n="170" rhead=""/> </div> <div xml:id="echoid-div456" type="section" level="1" n="220"> <head xml:id="echoid-head246" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 65 # 66 # 67 # 68 # 69 <lb/>0 # 9063078 # 9135455 # 9205049 # 9271839 # 9335804 # 60 <lb/>1 # 9064307 # 9136638 # 9206185 # 9272928 # 9336846 # 59 <lb/>2 # 9065535 # 9137820 # 9207321 # 9274017 # 9337887 # 58 <lb/>3 # 9066763 # 9139001 # 9208456 # 9275105 # 9338928 # 57 <lb/>4 # 9067990 # 9140181 # 9209590 # 9276192 # 9339968 # 56 <lb/>5 # 9069216 # 9141361 # 9210723 # 9277278 # 9341007 # 55 <lb/>6 # 9070441 # 9142540 # 9211855 # 9278363 # 9342045 # 54 <lb/>7 # 9071665 # 9143718 # 9212986 # 9279448 # 9343082 # 53 <lb/>8 # 9072889 # 9144895 # 9214117 # 9280532 # 9344119 # 52 <lb/>9 # 9074112 # 9146072 # 9215247 # 9281615 # 9345155 # 51 <lb/>10 # 9075334 # 9147248 # 9216376 # 9282697 # 9346190 # 50 <lb/>11 # 9076555 # 9148423 # 9217504 # 9283778 # 9347224 # 49 <lb/>12 # 9077775 # 9149597 # 9218631 # 9284859 # 9348257 # 48 <lb/>13 # 9078995 # 9150770 # 9219758 # 9285939 # 9349289 # 47 <lb/>14 # 9080214 # 9151943 # 9220884 # 9287018 # 9350321 # 46 <lb/>15 # 9081432 # 9153115 # 9222010 # 9288096 # 9351352 # 45 <lb/>16 # 9082649 # 9154286 # 9223135 # 9289173 # 9352382 # 44 <lb/>17 # 9083866 # 9155457 # 9224259 # 9290250 # 9353411 # 43 <lb/>18 # 9085082 # 9156627 # 9225382 # 9291326 # 9354440 # 42 <lb/>19 # 9086297 # 9157796 # 9226504 # 9292401 # 9355468 # 41 <lb/>20 # 9087512 # 9158964 # 9227625 # 9293476 # 9356495 # 40 <lb/>21 # 9088726 # 9160131 # 9228746 # 9294550 # 9357521 # 39 <lb/>22 # 9089939 # 9161297 # 9229866 # 9295623 # 9358546 # 38 <lb/>23 # 9091151 # 9162463 # 9230985 # 9296695 # 9359571 # 37 <lb/>24 # 9092362 # 9163628 # 9232103 # 9297766 # 9360595 # 36 <lb/>25 # 9093572 # 9164792 # 9233220 # 9298836 # 9361618 # 35 <lb/>26 # 9094781 # 9165955 # 9234337 # 9299905 # 9362640 # 34 <lb/>27 # 9095990 # 9167117 # 9235453 # 9300974 # 9363662 # 33 <lb/>28 # 9097198 # 9168279 # 9236568 # 9302042 # 9364683 # 32 <lb/>29 # 9098406 # 9169440 # 9237682 # 9303109 # 9365703 # 31 <lb/>30 # 9099613 # 9170601 # 9238795 # 9304176 # 9366722 # 30 <lb/> # 24 # 23 # 22 # 21 # 20 <lb/></note> </div> <div xml:id="echoid-div457" type="section" level="1" n="221"> <head xml:id="echoid-head247" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="159" file="171" n="171" rhead=""/> </div> <div xml:id="echoid-div458" type="section" level="1" n="222"> <head xml:id="echoid-head248" xml:space="preserve">rectis arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 65 # 66 # 67 # 68 # 69 <lb/>30 # 9099613 # 9170601 # 9238795 # 9304176 # 9366722 # 30 <lb/>31 # 9100819 # 9171761 # 9239908 # 9305242 # 9367740 # 29 <lb/>32 # 9102024 # 9172920 # 9241020 # 9306307 # 9368758 # 28 <lb/>33 # 9103228 # 9174078 # 9242131 # 9307371 # 9369775 # 27 <lb/>34 # 9104432 # 9175235 # 9243242 # 9308434 # 9370791 # 26 <lb/>35 # 9105635 # 9176391 # 9244352 # 9309497 # 9371806 # 25 <lb/>36 # 9106837 # 9177547 # 9245461 # 9310559 # 9372820 # 24 <lb/>37 # 9108038 # 9178702 # 9246569 # 9311620 # 9373834 # 23 <lb/>38 # 9109238 # 9179856 # 9247676 # 9312680 # 9374847 # 22 <lb/>39 # 9110438 # 9181009 # 9248782 # 9313739 # 9375859 # 21 <lb/>40 # 9111637 # 9182161 # 9249888 # 9314798 # 9376870 # 20 <lb/>41 # 9112835 # 9183313 # 9250993 # 9315856 # 9377880 # 19 <lb/>42 # 9114032 # 9184464 # 9252097 # 9316913 # 9378889 # 18 <lb/>43 # 9115229 # 9185614 # 9253200 # 9317969 # 9379898 # 17 <lb/>44 # 9116425 # 9186763 # 9254303 # 9319024 # 9380906 # 16 <lb/>45 # 9117620 # 9187912 # 9255405 # 9320079 # 9381913 # 15 <lb/>46 # 9118814 # 9189060 # 9256506 # 9321133 # 9382919 # 14 <lb/>47 # 9120007 # 9190207 # 9257606 # 9322186 # 9383925 # 13 <lb/>48 # 9121200 # 9191353 # 9258706 # 9323238 # 9384930 # 12 <lb/>49 # 9122392 # 9192499 # 9259805 # 9324290 # 9385934 # 11 <lb/>50 # 9123584 # 9193644 # 9260903 # 9325341 # 9386937 # 10 <lb/>51 # 9124775 # 9194788 # 9262000 # 9326391 # 9387939 # 9 <lb/>52 # 9125965 # 9195931 # 9263096 # 9327440 # 9388941 # 8 <lb/>53 # 9127154 # 9197073 # 9264192 # 9328488 # 9389942 # 7 <lb/>54 # 9128342 # 9198215 # 9265287 # 9329535 # 9390942 # 6 <lb/>55 # 9129529 # 9199356 # 9266381 # 9330582 # 9391941 # 5 <lb/>56 # 9130716 # 9200496 # 9267474 # 9331628 # 9392940 # 4 <lb/>57 # 9131902 # 9201635 # 9268566 # 9332673 # 9393938 # 3 <lb/>58 # 9133087 # 9202774 # 9269658 # 9333717 # 9394935 # 2 <lb/>59 # 9134271 # 9203912 # 9270749 # 9334761 # 9395931 # 1 <lb/>60 # 9135455 # 9205049 # 9271839 # 9335804 # 9396926 # 0 <lb/> # 24 # 23 # 22 # 21 # 20 <lb/></note> </div> <div xml:id="echoid-div459" type="section" level="1" n="223"> <head xml:id="echoid-head249" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="160" file="172" n="172" rhead=""/> </div> <div xml:id="echoid-div460" type="section" level="1" n="224"> <head xml:id="echoid-head250" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 70 # 71 # 72 # 73 # 74 <lb/>0 # 9396926 # 9455186 # 9510565 # 9563048 # 9612617 # 60 <lb/>1 # 9397921 # 9456133 # 9511464 # 9563898 # 9613418 # 59 <lb/>2 # 9398915 # 9457079 # 9512362 # 9564747 # 9614219 # 58 <lb/>3 # 9399908 # 9458024 # 9513259 # 9565596 # 9615019 # 57 <lb/>4 # 9400900 # 9458968 # 9514155 # 9566444 # 9615818 # 56 <lb/>5 # 9401891 # 9459911 # 9515050 # 9567291 # 9616616 # 55 <lb/>6 # 9402882 # 9460854 # 9515944 # 9568137 # 9617413 # 54 <lb/>7 # 9403872 # 9461796 # 9516838 # 9568982 # 9618209 # 53 <lb/>8 # 9404861 # 9462737 # 9517731 # 9569826 # 9619005 # 52 <lb/>9 # 9405849 # 9463677 # 9518623 # 9570670 # 9619800 # 51 <lb/>10 # 9406836 # 9464616 # 9519514 # 9571513 # 9620594 # 50 <lb/>11 # 9407822 # 9465555 # 9520404 # 9572355 # 9621387 # 49 <lb/>12 # 9408808 # 9466493 # 9521294 # 9573196 # 9622179 # 48 <lb/>13 # 9409793 # 9467430 # 9522183 # 9574036 # 9622971 # 47 <lb/>14 # 9410777 # 9468366 # 9523071 # 9574875 # 9623762 # 46 <lb/>15 # 9411760 # 9469301 # 9523958 # 9575714 # 9624552 # 45 <lb/>16 # 9412742 # 9470236 # 9524844 # 9576552 # 9625341 # 44 <lb/>17 # 9413724 # 9471170 # 9525730 # 9577389 # 9626129 # 43 <lb/>18 # 9414705 # 9472103 # 9526615 # 9578225 # 9626917 # 42 <lb/>19 # 9415685 # 9473035 # 9527499 # 9579061 # 9627704 # 41 <lb/>20 # 9416665 # 9473967 # 9528382 # 9579896 # 9628490 # 40 <lb/>21 # 9417644 # 9474898 # 9529264 # 9580730 # 9629275 # 39 <lb/>22 # 9418622 # 9475828 # 9530146 # 9581563 # 9630059 # 38 <lb/>23 # 9419599 # 9476757 # 9531027 # 9582395 # 9630843 # 37 <lb/>24 # 9420575 # 9477685 # 9531907 # 9583226 # 9631626 # 36 <lb/>25 # 9421550 # 9478612 # 9532786 # 9584057 # 9632408 # 35 <lb/>26 # 9422525 # 9479539 # 9533664 # 9584887 # 9633189 # 34 <lb/>27 # 9423499 # 9480465 # 9534541 # 9585716 # 9633969 # 33 <lb/>28 # 9425472 # 9481390 # 9535418 # 9586544 # 9634748 # 32 <lb/>29 # 9425444 # 9482314 # 9536294 # 9587371 # 9635527 # 31 <lb/>30 # 9426415 # 9483237 # 9537169 # 9588197 # 9636305 # 30 <lb/> # 19 # 18 # 17 # 16 # 15 <lb/></note> </div> <div xml:id="echoid-div461" type="section" level="1" n="225"> <head xml:id="echoid-head251" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="161" file="173" n="173" rhead=""/> </div> <div xml:id="echoid-div462" type="section" level="1" n="226"> <head xml:id="echoid-head252" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 70 # 71 # 72 # 73 # 74 <lb/>30 # 9426415 # 9483237 # 9537169 # 9588197 # 9636305 # 30 <lb/>31 # 9427386 # 9484160 # 9538043 # 9589023 # 9637082 # 29 <lb/>32 # 9428356 # 9485082 # 9538917 # 9589848 # 9637858 # 28 <lb/>33 # 9429325 # 9486003 # 9539790 # 9590672 # 9638633 # 27 <lb/>34 # 9430293 # 9486923 # 9540662 # 9591495 # 9639408 # 26 <lb/>35 # 9431260 # 9487842 # 9541533 # 9592318 # 9640182 # 25 <lb/>36 # 9432227 # 9488761 # 9542403 # 9593140 # 9640955 # 24 <lb/>37 # 9433193 # 9489679 # 9543272 # 9593961 # 9641727 # 23 <lb/>38 # 9434158 # 9490596 # 9544141 # 9594781 # 9642498 # 22 <lb/>39 # 9435122 # 9491512 # 9545009 # 9595600 # 9643268 # 21 <lb/>40 # 9436085 # 9492427 # 9545876 # 9596419 # 9644038 # 20 <lb/>41 # 9437048 # 9493341 # 9546742 # 9597237 # 9644807 # 19 <lb/>42 # 9438010 # 9494255 # 9547607 # 9598054 # 9645575 # 18 <lb/>43 # 9438971 # 9495168 # 9548472 # 9598870 # 9646342 # 17 <lb/>44 # 9439931 # 9496080 # 9549336 # 9599685 # 9647108 # 16 <lb/>45 # 9440890 # 9496991 # 9550199 # 9600499 # 9647873 # 15 <lb/>46 # 9441849 # 9497902 # 9551061 # 9601313 # 9648638 # 14 <lb/>47 # 9442807 # 9498812 # 9551922 # 9602126 # 9649402 # 13 <lb/>48 # 9443764 # 9499721 # 9552783 # 9602938 # 9650165 # 12 <lb/>49 # 9444720 # 9500629 # 9553643 # 9603749 # 9650927 # 11 <lb/>50 # 9445676 # 9501536 # 9554502 # 9604559 # 9651689 # 10 <lb/>51 # 9446631 # 9502443 # 9555360 # 9605368 # 9652450 # 9 <lb/>52 # 9447585 # 9503349 # 9556217 # 9606177 # 9653210 # 8 <lb/>53 # 9448538 # 9504254 # 9557074 # 9606985 # 9653969 # 7 <lb/>54 # 9449490 # 9505158 # 9557930 # 9607792 # 9654727 # 6 <lb/>55 # 9450441 # 9506061 # 9558785 # 9608598 # 9655484 # 5 <lb/>56 # 9451392 # 9506963 # 9559639 # 9609403 # 9656240 # 4 <lb/>57 # 9452342 # 9507865 # 9560492 # 9610208 # 9656996 # 3 <lb/>58 # 9453291 # 9508766 # 9561345 # 9611012 # 9657751 # 2 <lb/>59 # 9454239 # 9509666 # 9562197 # 9611815 # 9658505 # 1 <lb/>60 # 9455186 # 9510565 # 9563048 # 9612617 # 9659258 # 0 <lb/> # 19 # 18 # 17 # 16 # 15 <lb/></note> </div> <div xml:id="echoid-div463" type="section" level="1" n="227"> <head xml:id="echoid-head253" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="162" file="174" n="174" rhead=""/> </div> <div xml:id="echoid-div464" type="section" level="1" n="228"> <head xml:id="echoid-head254" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 75 # 76 # 77 # 78 # 79 <lb/>0 # 9659258 # 9702957 # 9743600 # 9781476 # 9816272 # 60 <lb/>1 # 9660011 # 9703660 # 9744355 # 9782080 # 9816827 # 59 <lb/>2 # 9660163 # 9704363 # 9745008 # 9782684 # 9817381 # 58 <lb/>3 # 9661514 # 9705065 # 9745660 # 9783281 # 9817934 # 57 <lb/>4 # 9662264 # 9705766 # 9746312 # 9783889 # 9818486 # 56 <lb/>5 # 9663013 # 9706466 # 9746963 # 9784490 # 9819037 # 55 <lb/>6 # 9663761 # 9707165 # 9747613 # 9785090 # 9819587 # 54 <lb/>7 # 9664508 # 9707863 # 9748262 # 9785689 # 9820137 # 53 <lb/>8 # 9665255 # 9708561 # 9748910 # 9786288 # 9820686 # 52 <lb/>9 # 9666001 # 9709258 # 9749557 # 9786886 # 9821234 # 51 <lb/>10 # 9666746 # 9709954 # 9750203 # 9787483 # 9821781 # 50 <lb/>11 # 9667490 # 9710649 # 9750849 # 9788079 # 9822327 # 49 <lb/>12 # 9668233 # 9711343 # 9751494 # 9788674 # 9822872 # 48 <lb/>13 # 9668976 # 9712036 # 9752138 # 9789268 # 9823417 # 47 <lb/>14 # 9669718 # 9712729 # 9752781 # 9789862 # 9823961 # 46 <lb/>15 # 9670459 # 9713421 # 9753423 # 9790455 # 9314504 # 45 <lb/>16 # 9671199 # 9714112 # 9754065 # 9791047 # 9825046 # 44 <lb/>17 # 9671938 # 9714802 # 9754706 # 9791638 # 9825587 # 43 <lb/>18 # 9672677 # 9715491 # 9755346 # 9792228 # 9826128 # 42 <lb/>19 # 9673415 # 9716180 # 9755985 # 9792818 # 9826668 # 41 <lb/>20 # 9674152 # 9716868 # 9756623 # 9793407 # 9827207 # 40 <lb/>21 # 9674888 # 9717555 # 9757260 # 9793995 # 9827745 # 39 <lb/>22 # 9675623 # 9718241 # 9757897 # 9794582 # 9828282 # 38 <lb/>23 # 9676357 # 9718926 # 9758533 # 9795168 # 9828818 # 37 <lb/>24 # 9677091 # 9719610 # 9759168 # 9795753 # 9829354 # 36 <lb/>25 # 9677824 # 9720294 # 9759802 # 9796337 # 9829889 # 35 <lb/>26 # 9678556 # 9720977 # 9760435 # 9796921 # 9830423 # 34 <lb/>27 # 9679287 # 9721659 # 9761067 # 9797504 # 9830956 # 33 <lb/>28 # 9680017 # 9722340 # 9761699 # 9798086 # 9831488 # 32 <lb/>29 # 9680747 # 9723020 # 9762330 # 9798667 # 9832019 # 31 <lb/>30 # 9681476 # 9723699 # 9762960 # 9799247 # 9832549 # 30 <lb/> # 14 # 13 # 12 # 11 # 10 <lb/></note> </div> <div xml:id="echoid-div465" type="section" level="1" n="229"> <head xml:id="echoid-head255" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="163" file="175" n="175" rhead=""/> </div> <div xml:id="echoid-div466" type="section" level="1" n="230"> <head xml:id="echoid-head256" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 75 # 76 # 77 # 78 # 79 <lb/>30 # 9681476 # 9723699 # 9762960 # 9799247 # 9832549 # 30 <lb/>31 # 9682804 # 9724378 # 9763589 # 9799827 # 9833079 # 29 <lb/>32 # 9682931 # 9725056 # 9764217 # 9800406 # 9833608 # 28 <lb/>33 # 9683657 # 9725733 # 9764845 # 9800984 # 9834136 # 27 <lb/>34 # 9684383 # 9726409 # 9765472 # 9801561 # 9834663 # 26 <lb/>35 # 9685108 # 9727085 # 9766098 # 9802137 # 9835189 # 25 <lb/>36 # 9685832 # 9727760 # 9766723 # 9802712 # 9835714 # 24 <lb/>37 # 9686555 # 9728434 # 9767347 # 9803287 # 9836239 # 23 <lb/>38 # 9687277 # 9729107 # 9767970 # 9803861 # 9836763 # 22 <lb/>39 # 9687998 # 9729779 # 9768593 # 9804434 # 9837286 # 21 <lb/>40 # 9688719 # 9730450 # 9769215 # 9805006 # 9837808 # 20 <lb/>41 # 9689439 # 9731120 # 9769836 # 9805577 # 9838329 # 19 <lb/>42 # 9690158 # 9731789 # 9770456 # 9806147 # 9838850 # 18 <lb/>43 # 9690879 # 9732458 # 9771075 # 9806716 # 9839370 # 17 <lb/>44 # 9691593 # 9733126 # 9771693 # 9807285 # 9839889 # 16 <lb/>45 # 9692309 # 9733793 # 9772311 # 9807853 # 9840407 # 15 <lb/>46 # 9693025 # 9734459 # 9772928 # 9808420 # 9840924 # 14 <lb/>47 # 9693740 # 9735124 # 9773544 # 9808986 # 9841440 # 13 <lb/>48 # 9694454 # 9735789 # 9774159 # 9809551 # 9841956 # 12 <lb/>49 # 9695167 # 9736453 # 9774773 # 9810116 # 9842471 # 11 <lb/>50 # 9695879 # 9737116 # 9775387 # 9810680 # 9842985 # 10 <lb/>51 # 9696590 # 9737778 # 9776000 # 9811243 # 9843498 # 9 <lb/>52 # 9697301 # 9738439 # 9776612 # 9811805 # 9844010 # 8 <lb/>53 # 9698011 # 9739099 # 9777223 # 9812366 # 9844521 # 7 <lb/>54 # 9698720 # 9739759 # 9777833 # 9812926 # 9845032 # 6 <lb/>55 # 9699428 # 9740418 # 9778442 # 9813486 # 9845542 # 5 <lb/>56 # 9700135 # 9741076 # 9779050 # 9814045 # 9846051 # 4 <lb/>57 # 9700842 # 9741733 # 9779658 # 9814603 # 9846559 # 3 <lb/>58 # 9701548 # 9742389 # 9780265 # 9815160 # 9847066 # 2 <lb/>59 # 9702253 # 9743045 # 9780871 # 9815716 # 9847572 # 1 <lb/>60 # 9702957 # 9743700 # 9781476 # 9816272 # 9848078 # 0 <lb/> # 14 # 13 # 12 # 11 # 10 <lb/></note> </div> <div xml:id="echoid-div467" type="section" level="1" n="231"> <head xml:id="echoid-head257" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="164" file="176" n="176" rhead=""/> </div> <div xml:id="echoid-div468" type="section" level="1" n="232"> <head xml:id="echoid-head258" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 80 # 81 # 82 # 83 # 84 <lb/>0 # 9848078 # 9876883 # 9902681 # 9925461 # 9945219 # 60 <lb/>1 # 9848583 # 9877338 # 9903085 # 9925816 # 9945523 # 59 <lb/>2 # 9849087 # 9877792 # 9903489 # 9926169 # 9945826 # 58 <lb/>3 # 9849590 # 9878245 # 9903892 # 9926521 # 9946128 # 57 <lb/>4 # 9850092 # 9878697 # 9904294 # 9926873 # 9946429 # 56 <lb/>5 # 9850593 # 9879148 # 9904695 # 9927224 # 9946729 # 55 <lb/>6 # 9851093 # 9879598 # 9905095 # 9927574 # 9947028 # 54 <lb/>7 # 9851593 # 9880048 # 9905494 # 9927923 # 9947327 # 53 <lb/>8 # 9852092 # 9880497 # 9905893 # 9928271 # 9947625 # 52 <lb/>9 # 9852590 # 9880945 # 9906291 # 9928618 # 9947922 # 51 <lb/>10 # 9853087 # 9881392 # 9906688 # 9928965 # 9948218 # 50 <lb/>11 # 9853583 # 9881838 # 9907084 # 9929311 # 9948513 # 49 <lb/>12 # 9854079 # 9882283 # 9907479 # 9929656 # 9948807 # 48 <lb/>13 # 9854574 # 9882728 # 9907873 # 9930000 # 9949100 # 47 <lb/>14 # 9855068 # 9883172 # 9908266 # 9930343 # 9949393 # 46 <lb/>15 # 9855561 # 9883615 # 9908659 # 9930685 # 9949685 # 45 <lb/>16 # 9856053 # 9884057 # 9909051 # 9931026 # 9949976 # 44 <lb/>17 # 9856544 # 9884498 # 9909442 # 9931367 # 9950266 # 43 <lb/>18 # 9857035 # 9884938 # 9909832 # 9931707 # 9950555 # 42 <lb/>19 # 9857525 # 9885378 # 9910221 # 9932046 # 9950844 # 41 <lb/>20 # 9858014 # 9885817 # 9910610 # 9932384 # 9951132 # 40 <lb/>21 # 9858502 # 9886255 # 9910998 # 9932721 # 9951419 # 39 <lb/>22 # 9858989 # 9886692 # 9911385 # 9933057 # 9951705 # 38 <lb/>23 # 9859475 # 9887128 # 9911771 # 9933393 # 9951990 # 37 <lb/>24 # 9859961 # 9887564 # 9912156 # 9933728 # 9952274 # 36 <lb/>25 # 9860446 # 9887999 # 9912540 # 9934062 # 9952557 # 35 <lb/>26 # 9860930 # 9888433 # 9912923 # 9934395 # 9952840 # 34 <lb/>27 # 9861413 # 9888866 # 9913306 # 9934727 # 9953122 # 33 <lb/>28 # 9861895 # 9889298 # 9913688 # 9935058 # 9953403 # 32 <lb/>29 # 9862376 # 9889729 # 9914069 # 9935389 # 9953683 # 31 <lb/>30 # 9862856 # 9890159 # 9914449 # 9935719 # 9953962 # 30 <lb/> # 9 # 8 # 7 # 6 # 5 <lb/></note> </div> <div xml:id="echoid-div469" type="section" level="1" n="233"> <head xml:id="echoid-head259" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="165" file="177" n="177" rhead=""/> </div> <div xml:id="echoid-div470" type="section" level="1" n="234"> <head xml:id="echoid-head260" xml:space="preserve">rectis arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 80 # 81 # 82 # 83 # 84 <lb/>30 # 9862856 # 9890159 # 9914449 # 9935719 # 9953962 # 30 <lb/>31 # 9863336 # 9890588 # 9914828 # 9936048 # 9954240 # 29 <lb/>32 # 9863815 # 9891017 # 9915206 # 9936376 # 9954518 # 28 <lb/>33 # 9864293 # 9891445 # 9915584 # 9936703 # 9954795 # 27 <lb/>34 # 9864770 # 9891872 # 9915961 # 9937029 # 9955071 # 26 <lb/>35 # 9865246 # 9892298 # 9916337 # 9937355 # 9955346 # 25 <lb/>36 # 9865722 # 9892723 # 9916712 # 9937680 # 9955620 # 24 <lb/>37 # 9866197 # 9893147 # 9917086 # 9938004 # 9955893 # 23 <lb/>38 # 9866671 # 9893571 # 9917459 # 9938327 # 9956165 # 22 <lb/>39 # 9867144 # 9893994 # 9917832 # 9938649 # 9956437 # 21 <lb/>40 # 9867616 # 9894416 # 9918204 # 9938970 # 9956708 # 20 <lb/>41 # 9868087 # 9894837 # 9918575 # 9939290 # 9956978 # 19 <lb/>42 # 9868557 # 9895257 # 9918945 # 9939609 # 9957247 # 18 <lb/>43 # 9869027 # 9895677 # 9919314 # 9939928 # 9957515 # 17 <lb/>44 # 9869496 # 9896096 # 9919682 # 9940246 # 9957782 # 16 <lb/>45 # 9869964 # 9896514 # 9920049 # 9940563 # 9958049 # 15 <lb/>46 # 9870431 # 9896931 # 9920416 # 9940879 # 9958315 # 14 <lb/>47 # 9870897 # 9897347 # 9920782 # 9941194 # 9958580 # 13 <lb/>48 # 9871362 # 9897762 # 9921147 # 9941509 # 9958844 # 12 <lb/>49 # 9871827 # 9898177 # 9921511 # 9941823 # 9959307 # 11 <lb/>50 # 9872291 # 9898591 # 9921874 # 9942136 # 9959370 # 10 <lb/>51 # 9872754 # 9899004 # 9922236 # 9942448 # 9959632 # 9 <lb/>52 # 9873216 # 9899416 # 9922598 # 9942759 # 9959893 # 8 <lb/>53 # 9873677 # 9899827 # 9922959 # 9943069 # 9960153 # 7 <lb/>54 # 9874137 # 9900237 # 9923319 # 9943379 # 9960412 # 6 <lb/>55 # 9874597 # 9900646 # 9923678 # 9943688 # 9960670 # 5 <lb/>56 # 9875056 # 9901055 # 9924036 # 9943996 # 9960927 # 4 <lb/>57 # 9875514 # 9901463 # 9924393 # 9944303 # 9961183 # 3 <lb/>58 # 9875971 # 9901870 # 9924750 # 9944609 # 9961438 # 2 <lb/>59 # 9876427 # 9902276 # 9925106 # 9944914 # 9961693 # 1 <lb/>60 # 9876883 # 9902681 # 9925461 # 9945219 # 9961947 # 0 <lb/> # 9 # 8 # 7 # 6 # 5 # <lb/></note> </div> <div xml:id="echoid-div471" type="section" level="1" n="235"> <head xml:id="echoid-head261" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="166" file="178" n="178" rhead=""/> </div> <div xml:id="echoid-div472" type="section" level="1" n="236"> <head xml:id="echoid-head262" xml:space="preserve">Gradus Quadrantis pro ſinubus</head> <note position="right" xml:space="preserve"> <lb/> # 85 # 86 # 87 # 88 # 89 <lb/>0 # 9961947 # 9975640 # 9986295 # 9993908 # 9998477 # 60 <lb/>1 # 9962200 # 9975843 # 9986447 # 9994009 # 9998527 # 59 <lb/>2 # 9962452 # 9976045 # 9986598 # 9994109 # 9998576 # 58 <lb/>3 # 9962703 # 9976246 # 9986748 # 9994208 # 9998625 # 57 <lb/>4 # 9962954 # 9976446 # 9986897 # 9994307 # 9998673 # 56 <lb/>5 # 9963204 # 9976645 # 9987045 # 9994405 # 9998720 # 55 <lb/>6 # 9963453 # 9976843 # 9987193 # 9994502 # 9998766 # 54 <lb/>7 # 9963701 # 9977040 # 9987340 # 9994598 # 9998811 # 53 <lb/>8 # 9963948 # 9977237 # 9987486 # 9994693 # 9998855 # 52 <lb/>9 # 9964194 # 9977433 # 9987631 # 9994787 # 9998899 # 51 <lb/>10 # 9964440 # 9977628 # 9987775 # 9994881 # 9998942 # 50 <lb/>11 # 9964685 # 9977822 # 9987918 # 9994974 # 9998984 # 49 <lb/>12 # 9964929 # 9978015 # 9988061 # 9995066 # 9999025 # 48 <lb/>13 # 9965172 # 9978207 # 9988203 # 9995157 # 9999065 # 47 <lb/>14 # 9965414 # 9978398 # 9988344 # 9995247 # 9999104 # 46 <lb/>15 # 9965655 # 9978589 # 9988484 # 9995336 # 9999143 # 45 <lb/>16 # 9965895 # 9978779 # 9988623 # 9995424 # 9999181 # 44 <lb/>17 # 9966135 # 9978968 # 9988761 # 9995512 # 9999218 # 43 <lb/>18 # 9966374 # 9979156 # 9988899 # 9995599 # 9999254 # 42 <lb/>19 # 9966612 # 9979343 # 9989036 # 9995685 # 9999289 # 41 <lb/>20 # 9966849 # 9979530 # 9989172 # 9995770 # 9999323 # 40 <lb/>21 # 9967085 # 9979716 # 9989307 # 9995854 # 9999356 # 39 <lb/>22 # 9967320 # 9979901 # 9989441 # 9995937 # 9999389 # 38 <lb/>23 # 9967555 # 9980085 # 9989574 # 9996019 # 9999421 # 37 <lb/>24 # 9967789 # 9980268 # 9989706 # 9996101 # 9999452 # 36 <lb/>25 # 9968022 # 9980450 # 9989837 # 9996182 # 9999482 # 35 <lb/>26 # 9968254 # 9980631 # 9989968 # 9996262 # 9999511 # 34 <lb/>27 # 9968485 # 9980811 # 9990098 # 9996341 # 9999539 # 33 <lb/>28 # 9968715 # 9980991 # 9990227 # 9996419 # 9999566 # 32 <lb/>29 # 9968944 # 9981170 # 9990355 # 9996496 # 9999593 # 31 <lb/>30 # 9969173 # 9981348 # 9990482 # 9996573 # 9999619 # 30 <lb/> # 4 # 3 # 2 # 1 # 0 <lb/></note> </div> <div xml:id="echoid-div473" type="section" level="1" n="237"> <head xml:id="echoid-head263" xml:space="preserve">Gradus Quadrantis pro ſinubus rectis</head> <pb o="167" file="179" n="179" rhead=""/> </div> <div xml:id="echoid-div474" type="section" level="1" n="238"> <head xml:id="echoid-head264" xml:space="preserve">rectis arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 85 # 86 # 87 # 88 # 89 <lb/>30 # 9969173 # 9981348 # 9990482 # 9996573 # 9999616 # 30 <lb/>31 # 9969401 # 9981525 # 9990608 # 9996649 # 9999644 # 29 <lb/>32 # 9969628 # 9981701 # 9990734 # 9996724 # 9999668 # 28 <lb/>33 # 9969854 # 9981877 # 9990859 # 9996798 # 9999691 # 27 <lb/>34 # 9970079 # 9982052 # 9990983 # 9996871 # 9999713 # 26 <lb/>35 # 9970304 # 9982226 # 9991106 # 9996943 # 9999735 # 25 <lb/>36 # 9970528 # 9982399 # 9991228 # 9997014 # 9999756 # 24 <lb/>37 # 9970751 # 9982571 # 9991349 # 9997085 # 9999776 # 23 <lb/>38 # 9970973 # 9982742 # 9991470 # 9997155 # 9999795 # 22 <lb/>39 # 9971194 # 9982912 # 9991590 # 9997224 # 9999813 # 21 <lb/>40 # 9971414 # 9983082 # 9991709 # 9997292 # 9999830 # 20 <lb/>41 # 9971633 # 9983251 # 9991827 # 9997359 # 9999846 # 19 <lb/>42 # 9971851 # 9983419 # 9991944 # 9997425 # 9999862 # 18 <lb/>43 # 9972069 # 9983586 # 9992060 # 9997491 # 9999877 # 17 <lb/>44 # 9972286 # 9983752 # 9992175 # 9997556 # 9999891 # 16 <lb/>45 # 9972502 # 9983917 # 9992290 # 9997620 # 9999904 # 15 <lb/>46 # 9972717 # 9984081 # 9992404 # 997683 # 99999916 # 14 <lb/>47 # 9972931 # 9984245 # 9992517 # 9997745 # 9999927 # 13 <lb/>48 # 9973145 # 9984408 # 9992629 # 9997806 # 9999938 # 12 <lb/>49 # 9973358 # 9984570 # 9992740 # 9997867 # 9999948 # 11 <lb/>50 # 9973570 # 9984731 # 9992850 # 9997927 # 9999957 # 10 <lb/>51 # 9973781 # 9984891 # 9992960 # 9997986 # 9999965 # 9 <lb/>52 # 9973991 # 9985050 # 9993069 # 9998044 # 9999972 # 8 <lb/>53 # 9974200 # 9985209 # 9993177 # 9998101 # 9999978 # 7 <lb/>54 # 9974408 # 9985367 # 9993284 # 9998157 # 9999984 # 6 <lb/>55 # 9974615 # 9985524 # 9993390 # 9998212 # 9999989 # 5 <lb/>56 # 9974822 # 9985680 # 9993495 # 9998267 # 9999993 # 4 <lb/>57 # 9975028 # 9985835 # 9993599 # 9998321 # 9999996 # 3 <lb/>58 # 9975233 # 9985989 # 9993703 # 9998374 # 9999998 # 2 <lb/>59 # 9975437 # 9986143 # 9993806 # 9998426 # 9999999 # 1 <lb/>60 # 9975640 # 9986295 # 9993908 # 9998477 # 10000000 # 0 <lb/> # 4 # 3 # 2 # 1 # 0 <lb/></note> </div> <div xml:id="echoid-div475" type="section" level="1" n="239"> <head xml:id="echoid-head265" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="168" file="180" n="180" rhead=""/> </div> <div xml:id="echoid-div476" type="section" level="1" n="240"> <head xml:id="echoid-head266" xml:space="preserve">EXPLICATIO, ATQVE VSVS TABVLAE</head> <head xml:id="echoid-head267" xml:space="preserve">præcedentis Sinuum rectorum.</head> <p style="it"> <s xml:id="echoid-s5732" xml:space="preserve">_IN_vertice præcedentis tabulæ ordine deſcripti ſunt 90. </s> <s xml:id="echoid-s5733" xml:space="preserve">gradus Quadrantis, & </s> <s xml:id="echoid-s5734" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-180-01a" xlink:href="note-180-01"/> ad ſiniſtram deorſum verſus, 60. </s> <s xml:id="echoid-s5735" xml:space="preserve">Minuta. </s> <s xml:id="echoid-s5736" xml:space="preserve">In infimo deinde latere ijdem 90. </s> <s xml:id="echoid-s5737" xml:space="preserve">gradus <lb/>Quadrantis repoſiti ſunt ordine retrogrado, & </s> <s xml:id="echoid-s5738" xml:space="preserve">ad dexteram ſurſum verſus, 60. </s> <s xml:id="echoid-s5739" xml:space="preserve">Mi-<lb/>nuta. </s> <s xml:id="echoid-s5740" xml:space="preserve">Quod ideofactum est à nobis, vt illico cuiuslibet arcus complementum cognoſca-<lb/>tur. </s> <s xml:id="echoid-s5741" xml:space="preserve">Nam quilibet gradus in vertice tabulæ poſitus cum quouis Minuto ad ſiniſtram <lb/> <anchor type="note" xlink:label="note-180-02a" xlink:href="note-180-02"/> collocato, habet pro complemento gradum in infimo latere gradui accepto in vertice <lb/>reſpondentem cum Minuto, quod ad dextram Minuto ad ſiniſtram accepto reſpondet. <lb/></s> <s xml:id="echoid-s5742" xml:space="preserve">Vt quomam gradui 46. </s> <s xml:id="echoid-s5743" xml:space="preserve">in vertice, & </s> <s xml:id="echoid-s5744" xml:space="preserve">Minuto O ad ſiniſtram poſito, reſpondet in infi-<lb/>mo latere gradus 43. </s> <s xml:id="echoid-s5745" xml:space="preserve">& </s> <s xml:id="echoid-s5746" xml:space="preserve">Minutum 60. </s> <s xml:id="echoid-s5747" xml:space="preserve">ad dexteram collocatum; </s> <s xml:id="echoid-s5748" xml:space="preserve">erit arcus grad. </s> <s xml:id="echoid-s5749" xml:space="preserve">43. </s> <s xml:id="echoid-s5750" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s5751" xml:space="preserve">60. </s> <s xml:id="echoid-s5752" xml:space="preserve">hoceſt, arcus grad. </s> <s xml:id="echoid-s5753" xml:space="preserve">44. </s> <s xml:id="echoid-s5754" xml:space="preserve">Min. </s> <s xml:id="echoid-s5755" xml:space="preserve">o. </s> <s xml:id="echoid-s5756" xml:space="preserve">complementum arcus grad. </s> <s xml:id="echoid-s5757" xml:space="preserve">46. </s> <s xml:id="echoid-s5758" xml:space="preserve">Min. </s> <s xml:id="echoid-s5759" xml:space="preserve">o. </s> <s xml:id="echoid-s5760" xml:space="preserve">Sic <lb/>quoque arcus grad. </s> <s xml:id="echoid-s5761" xml:space="preserve">43. </s> <s xml:id="echoid-s5762" xml:space="preserve">Min. </s> <s xml:id="echoid-s5763" xml:space="preserve">13. </s> <s xml:id="echoid-s5764" xml:space="preserve">complementum habebit arcum grad. </s> <s xml:id="echoid-s5765" xml:space="preserve">46 Min. </s> <s xml:id="echoid-s5766" xml:space="preserve">47. </s> <s xml:id="echoid-s5767" xml:space="preserve">Ea-<lb/>dem ratione quilibet gradus in infimo latere poſitus cum quouis Minuto ad dexteram <lb/>collocato, habet pro complemento gradum in vertice gradui acceptoin infimo latere <lb/>reſpondentem cum Minuto, quod ad ſiniſtram Minuto ad dextram accepto reſpondet. </s> <s xml:id="echoid-s5768" xml:space="preserve"><lb/>Poſtremo ſub gradibus in vertice tabulæ deſcriptis poſiti ſunt ſinus recti omnium ar-<lb/>cuũ per ſingula Quadrãtis Minuta progredientiũ, quatenus ſinus totus eſt 10000000. </s> <s xml:id="echoid-s5769" xml:space="preserve"><lb/>Quòd ſi ex ſingulis ſinubus binæ priores figuræ ad dexter am abijciãtur, (addita tamen <lb/>vnitate, ſi duæ figuræ abiectæ numerum 50. </s> <s xml:id="echoid-s5770" xml:space="preserve">excedunt) reliqui erunt ſinus eorundem <lb/>arcuum, quatenus ſinus totus eſt 100000. </s> <s xml:id="echoid-s5771" xml:space="preserve">vt ſupra diximus. </s> <s xml:id="echoid-s5772" xml:space="preserve">Vnde quicquid in vſu hu-<lb/>ius tabulæ præcipiemus de ſinubus reſpectu ſinus totius 10000000. </s> <s xml:id="echoid-s5773" xml:space="preserve">intelligendum quo-<lb/>que erit de ſinubus reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s5774" xml:space="preserve">abiectis nimirum duabus primis fi-<lb/> <anchor type="note" xlink:label="note-180-03a" xlink:href="note-180-03"/> guris ad dexteram, vt diximus.</s> <s xml:id="echoid-s5775" xml:space="preserve"/> </p> <div xml:id="echoid-div476" type="float" level="2" n="1"> <note position="left" xlink:label="note-180-01" xlink:href="note-180-01a" xml:space="preserve">Expoſitio <lb/>partium ta <lb/>bulęSinuũ.</note> <note position="left" xlink:label="note-180-02" xlink:href="note-180-02a" xml:space="preserve">Comple <lb/>mẽtum cu <lb/>iuſuis arcꝰ <lb/>quo pacto <lb/>ex hac ta-<lb/>bula elicia <lb/>tur.</note> <note position="left" xlink:label="note-180-03" xlink:href="note-180-03a" xml:space="preserve">Vſus tabu-<lb/>bulę ſinuũ <lb/>duplex.</note> </div> <p style="it"> <s xml:id="echoid-s5776" xml:space="preserve">_HVIVS_tabulæ vſus duplex eſt. </s> <s xml:id="echoid-s5777" xml:space="preserve">Nam in ea vel cuiuslibet arcus inquiritur ſi-<lb/>nus, vel cuiuſuis ſinus cogniti arcus inueſtigatur. </s> <s xml:id="echoid-s5778" xml:space="preserve">Quando ergo dati arcus Quadrante <lb/> <anchor type="note" xlink:label="note-180-04a" xlink:href="note-180-04"/> minoris ſinum rectum quæris, ſume gradus illius in vertice tabulæ, Minuta vero ad ſini <lb/>ſtram. </s> <s xml:id="echoid-s5779" xml:space="preserve">In communi enim angulo, procedendo nimirum à Minutis dextram verſus, do-<lb/>nec adlocum ſub gradibus acceptis peruenias, illico ſinum rectum inuenies. </s> <s xml:id="echoid-s5780" xml:space="preserve">Ita ſinum <lb/>rectum arcus grad 25. </s> <s xml:id="echoid-s5781" xml:space="preserve">ſub grad 25. </s> <s xml:id="echoid-s5782" xml:space="preserve">è regione Minuti o ad ſinistram collocati repe-<lb/>ries 4226183. </s> <s xml:id="echoid-s5783" xml:space="preserve">Sinum vero rectum arcus grad. </s> <s xml:id="echoid-s5784" xml:space="preserve">25. </s> <s xml:id="echoid-s5785" xml:space="preserve">Min. </s> <s xml:id="echoid-s5786" xml:space="preserve">19. </s> <s xml:id="echoid-s5787" xml:space="preserve">inuenies 4276209. </s> <s xml:id="echoid-s5788" xml:space="preserve">Sinum <lb/>denique rectum arcus grad. </s> <s xml:id="echoid-s5789" xml:space="preserve">25. </s> <s xml:id="echoid-s5790" xml:space="preserve">Min. </s> <s xml:id="echoid-s5791" xml:space="preserve">50. </s> <s xml:id="echoid-s5792" xml:space="preserve">offendes 4357549. </s> <s xml:id="echoid-s5793" xml:space="preserve">Si vero ſinum rectum ar-<lb/> <anchor type="note" xlink:label="note-180-05a" xlink:href="note-180-05"/> cus Quadrante maioris, ſed ſemicirculo minoris deſideras, detrabe arcum datum ex <lb/>ſemicirculo, & </s> <s xml:id="echoid-s5794" xml:space="preserve">reſidui arcus ſinum rectum cape, vt prius. </s> <s xml:id="echoid-s5795" xml:space="preserve">Hic enim ſinus erit etiam <lb/>ſinus rectus arcus quadrante maioris: </s> <s xml:id="echoid-s5796" xml:space="preserve">propterea quòd duo arcus ſemicirculum con-<lb/>ficientes eundem ſinũ rectum habent, vtin definitionũ expoſitione diximus. </s> <s xml:id="echoid-s5797" xml:space="preserve">Vt ſi da-<lb/>tus ſit arcus grad. </s> <s xml:id="echoid-s5798" xml:space="preserve">138. </s> <s xml:id="echoid-s5799" xml:space="preserve">Min. </s> <s xml:id="echoid-s5800" xml:space="preserve">47. </s> <s xml:id="echoid-s5801" xml:space="preserve">detrabe eum ex ſemicirculo, hoc eſt, ex grad. </s> <s xml:id="echoid-s5802" xml:space="preserve">180. <lb/></s> <s xml:id="echoid-s5803" xml:space="preserve">Sinus namq; </s> <s xml:id="echoid-s5804" xml:space="preserve">rectus 6589082. </s> <s xml:id="echoid-s5805" xml:space="preserve">reſidui arcus grad. </s> <s xml:id="echoid-s5806" xml:space="preserve">41. </s> <s xml:id="echoid-s5807" xml:space="preserve">Min. </s> <s xml:id="echoid-s5808" xml:space="preserve">13. </s> <s xml:id="echoid-s5809" xml:space="preserve">eſt quoq; </s> <s xml:id="echoid-s5810" xml:space="preserve">ſinus rectus <lb/>arcus grad. </s> <s xml:id="echoid-s5811" xml:space="preserve">138. </s> <s xml:id="echoid-s5812" xml:space="preserve">Min. </s> <s xml:id="echoid-s5813" xml:space="preserve">47. </s> <s xml:id="echoid-s5814" xml:space="preserve">cum illo arcu grad. </s> <s xml:id="echoid-s5815" xml:space="preserve">41. </s> <s xml:id="echoid-s5816" xml:space="preserve">Min. </s> <s xml:id="echoid-s5817" xml:space="preserve">13. </s> <s xml:id="echoid-s5818" xml:space="preserve">ſemicirculũ coficientis.</s> <s xml:id="echoid-s5819" xml:space="preserve"/> </p> <div xml:id="echoid-div477" type="float" level="2" n="2"> <note position="left" xlink:label="note-180-04" xlink:href="note-180-04a" xml:space="preserve">Sinus tectꝰ <lb/>cuiuſuis ar <lb/>cus quadrã <lb/>te minoris <lb/>quo pacto í <lb/>tabula repe <lb/>riatur.</note> <note position="left" xlink:label="note-180-05" xlink:href="note-180-05a" xml:space="preserve">Sinꝰ rectus <lb/>cuiuſuis ar <lb/>cus quadrã <lb/>te maioris, <lb/>qua rõne <lb/>inueniatur</note> </div> <p style="it"> <s xml:id="echoid-s5820" xml:space="preserve">_QVOD_ſiarcus datus præter gradus, ac minuta habeat etiam Secunda, inquiren-<lb/> <anchor type="note" xlink:label="note-180-06a" xlink:href="note-180-06"/> da erit pars proportionalis, hoc modo. </s> <s xml:id="echoid-s5821" xml:space="preserve">Accipe differentiam inter ſinum rectum arcus <lb/>proxime minoris, & </s> <s xml:id="echoid-s5822" xml:space="preserve">ſinũ arcus proxime maioris, & </s> <s xml:id="echoid-s5823" xml:space="preserve">dic. </s> <s xml:id="echoid-s5824" xml:space="preserve">Si 60. </s> <s xml:id="echoid-s5825" xml:space="preserve">ſecunda (qui-<lb/>bus ſinguli arcus proximiin hae tabula inter ſe differunt) requirũt tota eam differen <lb/>tiam addendam ſinui arcus proxime minoris, vt componatur ſinus arcus proxime <lb/>maioris, quantam differentiam requirunt propoſita ſecunda addendam eidem ſinui <pb o="169" file="181" n="181" rhead=""/> areus proxime minoris, vt fiat ſinus propositi arcus? </s> <s xml:id="echoid-s5826" xml:space="preserve">_N_am differentia inuenta erit <lb/> <anchor type="note" xlink:label="note-181-01a" xlink:href="note-181-01"/> pars proportionalis, quæ ſi addatur ſinui arcus proxime minoris, efficietur ſinus re-<lb/>ctus arcus propoſiti. </s> <s xml:id="echoid-s5827" xml:space="preserve">_V_t ſi propoſitus ſit arcus grad. </s> <s xml:id="echoid-s5828" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5829" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5830" xml:space="preserve">_43_. </s> <s xml:id="echoid-s5831" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s5832" xml:space="preserve">_20._ </s> <s xml:id="echoid-s5833" xml:space="preserve">_A_ccipe dif-<lb/>ferentiam _2721_. </s> <s xml:id="echoid-s5834" xml:space="preserve">inter ſinum _3537469_. </s> <s xml:id="echoid-s5835" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5836" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5837" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5838" xml:space="preserve">_43_. </s> <s xml:id="echoid-s5839" xml:space="preserve">proxime minoris, <lb/>& </s> <s xml:id="echoid-s5840" xml:space="preserve">ſinum _3540190_. </s> <s xml:id="echoid-s5841" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5842" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5843" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5844" xml:space="preserve">_44_. </s> <s xml:id="echoid-s5845" xml:space="preserve">proxime maioris, & </s> <s xml:id="echoid-s5846" xml:space="preserve">dic. </s> <s xml:id="echoid-s5847" xml:space="preserve">_S_i _60._ </s> <s xml:id="echoid-s5848" xml:space="preserve">ſecunda <lb/>vequirunt differentiam _2721_. </s> <s xml:id="echoid-s5849" xml:space="preserve">addendam ſinui _3537469_. </s> <s xml:id="echoid-s5850" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5851" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5852" xml:space="preserve">min _43_. <lb/></s> <s xml:id="echoid-s5853" xml:space="preserve">vt efficiatur ſinus _3540190_. </s> <s xml:id="echoid-s5854" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5855" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5856" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5857" xml:space="preserve">_44_. </s> <s xml:id="echoid-s5858" xml:space="preserve">quantam differentiam poſtu-<lb/>lant _20_. </s> <s xml:id="echoid-s5859" xml:space="preserve">ſecunda addendam eidem ſinui _3537469_. </s> <s xml:id="echoid-s5860" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5861" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5862" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5863" xml:space="preserve">_43_. </s> <s xml:id="echoid-s5864" xml:space="preserve">vt fiat <lb/>ſinus arcus grad. </s> <s xml:id="echoid-s5865" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5866" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5867" xml:space="preserve">_43_. </s> <s xml:id="echoid-s5868" xml:space="preserve">ſec. </s> <s xml:id="echoid-s5869" xml:space="preserve">_20_? </s> <s xml:id="echoid-s5870" xml:space="preserve">_I_nuenies enim differentiam, ſiue partem pro-<lb/>portionalem _907_. </s> <s xml:id="echoid-s5871" xml:space="preserve">quæ addita ſinui _3537469_. </s> <s xml:id="echoid-s5872" xml:space="preserve">efficiet _3538376_. </s> <s xml:id="echoid-s5873" xml:space="preserve">ſinum arcus grad. </s> <s xml:id="echoid-s5874" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5875" xml:space="preserve"><lb/>_M_in. </s> <s xml:id="echoid-s5876" xml:space="preserve">_43_ ſec. </s> <s xml:id="echoid-s5877" xml:space="preserve">_20_. </s> <s xml:id="echoid-s5878" xml:space="preserve">_C_ommuniter tamen ab _A_ſtronomis negliguntur in hoc negotio _S_e-<lb/> <anchor type="note" xlink:label="note-181-02a" xlink:href="note-181-02"/> cunda, ſi pauciora ſunt, quàm _30._ </s> <s xml:id="echoid-s5879" xml:space="preserve">_S_i vero pl@ra, addunt proillis vnum minutum <lb/>alijs _M_inutis. </s> <s xml:id="echoid-s5880" xml:space="preserve">_N_ullus enim error, qui alicuius momenti ſit, inde oritur. </s> <s xml:id="echoid-s5881" xml:space="preserve">_I_taq; </s> <s xml:id="echoid-s5882" xml:space="preserve">pro <lb/>dato arcugrad. </s> <s xml:id="echoid-s5883" xml:space="preserve">_20._ </s> <s xml:id="echoid-s5884" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5885" xml:space="preserve">_43._ </s> <s xml:id="echoid-s5886" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s5887" xml:space="preserve">_20._ </s> <s xml:id="echoid-s5888" xml:space="preserve">accipiunt ſinum arcus grad. </s> <s xml:id="echoid-s5889" xml:space="preserve">_20._ </s> <s xml:id="echoid-s5890" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5891" xml:space="preserve">_43._ </s> <s xml:id="echoid-s5892" xml:space="preserve">ne-<lb/>glectis illis _20_ ſec. </s> <s xml:id="echoid-s5893" xml:space="preserve">_P_ro arcu vero grad. </s> <s xml:id="echoid-s5894" xml:space="preserve">_20._ </s> <s xml:id="echoid-s5895" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5896" xml:space="preserve">_43_ ſec. </s> <s xml:id="echoid-s5897" xml:space="preserve">_48._ </s> <s xml:id="echoid-s5898" xml:space="preserve">ſumunt ſinum arcus grad. <lb/></s> <s xml:id="echoid-s5899" xml:space="preserve">_20._ </s> <s xml:id="echoid-s5900" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5901" xml:space="preserve">_44._ </s> <s xml:id="echoid-s5902" xml:space="preserve">computatis illis _48._ </s> <s xml:id="echoid-s5903" xml:space="preserve">ſec. </s> <s xml:id="echoid-s5904" xml:space="preserve">pro vno _M_inuto.</s> <s xml:id="echoid-s5905" xml:space="preserve"/> </p> <div xml:id="echoid-div478" type="float" level="2" n="3"> <note position="left" xlink:label="note-180-06" xlink:href="note-180-06a" xml:space="preserve">Sinꝰ rectus <lb/>cuiuſuis ar <lb/>cus habẽtis <lb/>Secũda, prę <lb/>ter Minuta <lb/>quomodo <lb/>eliciat̃ per</note> <note position="right" xlink:label="note-181-01" xlink:href="note-181-01a" xml:space="preserve">partem pre <lb/>pottionalẽ</note> <note position="right" xlink:label="note-181-02" xlink:href="note-181-02a" xml:space="preserve">Secũda cõl <lb/>ter ĩttacta <lb/>tione ſinuũ <lb/>negligũ tut <lb/>ab Aſtrone <lb/>mis.</note> </div> <p style="it"> <s xml:id="echoid-s5906" xml:space="preserve">_SEMICIRCVLI_porro, atq; </s> <s xml:id="echoid-s5907" xml:space="preserve">arcus ſemicirculo maioris, non eſt quòd ſinus rectus <lb/> <anchor type="note" xlink:label="note-181-03a" xlink:href="note-181-03"/> inueſtigetur, cùm nunquam in ſupputationibus _A_ſtronomicis buiuſmodi arcuum ſi-<lb/>nus adbibeantur, quod ijs perſptcuum eſt, qui in triangulis rectilineis, ac ſphæri-<lb/>cis, in quibus tota _S_inuum ſcientia verſatur, ſunt exercitati. </s> <s xml:id="echoid-s5908" xml:space="preserve">_I_mmo ſemicirculus <lb/>nullum habet ſinum, vt ex vtraq; </s> <s xml:id="echoid-s5909" xml:space="preserve">defin. </s> <s xml:id="echoid-s5910" xml:space="preserve">ſinus recti patet. </s> <s xml:id="echoid-s5911" xml:space="preserve">quod etiam de arcu ma-<lb/>iore dici poteſt, niſi quis in prima figura definitionum rectam _FK,_ ſinum rectum velit <lb/>appellare arcus _DBF_, ſecundum poſteriorem defin. </s> <s xml:id="echoid-s5912" xml:space="preserve">ſinus recti; </s> <s xml:id="echoid-s5913" xml:space="preserve">& </s> <s xml:id="echoid-s5914" xml:space="preserve">rectam _FH_, ſinum <lb/>rectum arcus _<emph style="sc">ACf</emph>_. </s> <s xml:id="echoid-s5915" xml:space="preserve">quod non videturproprie dici, cum huiuſmodi lineis prior defi-<lb/>nitio ſinus recti nullo modo conuenire poſsit, vt ex eadem figura perſpicuum eſt. <lb/></s> <s xml:id="echoid-s5916" xml:space="preserve"> <anchor type="note" xlink:label="note-181-04a" xlink:href="note-181-04"/> </s> </p> <div xml:id="echoid-div479" type="float" level="2" n="4"> <note position="right" xlink:label="note-181-03" xlink:href="note-181-03a" xml:space="preserve">Semicircu-<lb/>li, & arcꝰ ſe <lb/>micirculo <lb/>maioris, nõ <lb/>eſt ſinus re <lb/>ctus inqui-<lb/>rendus; im-<lb/>mo nullus <lb/>eft eorũ ſi-<lb/>nus rectus.</note> <note position="right" xlink:label="note-181-04" xlink:href="note-181-04a" xml:space="preserve">Sinus com <lb/>plemẽti ar-<lb/>cus quadrã <lb/>te minoris <lb/>quomodo <lb/>i tabula re <lb/>periatur.</note> </div> <p style="it"> <s xml:id="echoid-s5917" xml:space="preserve">_QVANDO_ autemdati arcus _Q_uadrante minoris ſinum complementi quæris, cape <lb/>gradus illius in in feriori parte tabulæ, _M_inuta vero ad dexteram. </s> <s xml:id="echoid-s5918" xml:space="preserve">_I_n communi enim <lb/>angulo continuò ſinum complementireperies, hoc eſt, ſinum rectum illius arcus, qui <lb/>dati arcus complementum eſt. </s> <s xml:id="echoid-s5919" xml:space="preserve">_N_am ſinus ille rectus debetur arcui, cuius gradus in <lb/>vertice tabulæ, & </s> <s xml:id="echoid-s5920" xml:space="preserve">minuta in ſiniſtro latere collocantur, qui quidem dati arcus com-<lb/>plementum eſt, vt ſupra diximus. </s> <s xml:id="echoid-s5921" xml:space="preserve">_I_ta ſinum complementi arcus grad. </s> <s xml:id="echoid-s5922" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5923" xml:space="preserve">ſupra <lb/>grad. </s> <s xml:id="echoid-s5924" xml:space="preserve">_30._ </s> <s xml:id="echoid-s5925" xml:space="preserve">infimi lateris tabulæ è regione _M_inuti _0._ </s> <s xml:id="echoid-s5926" xml:space="preserve">ad dextram collocati inuenies <lb/>_8660254_. </s> <s xml:id="echoid-s5927" xml:space="preserve">qui quidem ſinus rectus eſt arcus grad. </s> <s xml:id="echoid-s5928" xml:space="preserve">_59_. </s> <s xml:id="echoid-s5929" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5930" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5931" xml:space="preserve">hoc eſt, arcus grad. <lb/></s> <s xml:id="echoid-s5932" xml:space="preserve">_60_. </s> <s xml:id="echoid-s5933" xml:space="preserve">qui complementum eſt arcus grad. </s> <s xml:id="echoid-s5934" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5935" xml:space="preserve">_I_tem ſinum complementi arcus grad. </s> <s xml:id="echoid-s5936" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5937" xml:space="preserve"><lb/>_M_in. </s> <s xml:id="echoid-s5938" xml:space="preserve">_49_. </s> <s xml:id="echoid-s5939" xml:space="preserve">reperies _8588110_. </s> <s xml:id="echoid-s5940" xml:space="preserve">qui quidem sinus rectus eſt arcus grad. </s> <s xml:id="echoid-s5941" xml:space="preserve">_59._ </s> <s xml:id="echoid-s5942" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5943" xml:space="preserve">_11._ </s> <s xml:id="echoid-s5944" xml:space="preserve">qui <lb/>complementum eſt arcus grad. </s> <s xml:id="echoid-s5945" xml:space="preserve">_30_. </s> <s xml:id="echoid-s5946" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5947" xml:space="preserve">_49_. </s> <s xml:id="echoid-s5948" xml:space="preserve">_S_i vero offeratur arcus _Q_uadrante <lb/> <anchor type="note" xlink:label="note-181-05a" xlink:href="note-181-05"/> maior, ſed ſemicirculo minor, ita ſinum complementi ipſius reperies. </s> <s xml:id="echoid-s5949" xml:space="preserve">_D_etrabe ex eo <lb/>quadrantem, & </s> <s xml:id="echoid-s5950" xml:space="preserve">reſidui arcus ſinum rectum cape. </s> <s xml:id="echoid-s5951" xml:space="preserve">_C_um enim reliquus hic arcus ſit <lb/>complementum dati arcus, vt in definitionibus dictum eſt, erit eius ſinus rectus, ſinus <lb/>complementi dati arcus. </s> <s xml:id="echoid-s5952" xml:space="preserve">_V_t ſi oblatus ſit arcus grad. </s> <s xml:id="echoid-s5953" xml:space="preserve">_127_. </s> <s xml:id="echoid-s5954" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5955" xml:space="preserve">_30._ </s> <s xml:id="echoid-s5956" xml:space="preserve">_D_etrabe ex eo <lb/>quadrantem, hoc eſt, _90._ </s> <s xml:id="echoid-s5957" xml:space="preserve">gradus. </s> <s xml:id="echoid-s5958" xml:space="preserve">_S_inus namque rectus _6087614._ </s> <s xml:id="echoid-s5959" xml:space="preserve">reliqui arcus grad. <lb/></s> <s xml:id="echoid-s5960" xml:space="preserve">_37._ </s> <s xml:id="echoid-s5961" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5962" xml:space="preserve">_30._ </s> <s xml:id="echoid-s5963" xml:space="preserve">eſt ſinus complementi dati arcus grad. </s> <s xml:id="echoid-s5964" xml:space="preserve">_127._ </s> <s xml:id="echoid-s5965" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5966" xml:space="preserve">_30._ </s> <s xml:id="echoid-s5967" xml:space="preserve">cumille arcus ſit <lb/>buius complementum. </s> <s xml:id="echoid-s5968" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-181-06a" xlink:href="note-181-06"/> </s> </p> <div xml:id="echoid-div480" type="float" level="2" n="5"> <note position="right" xlink:label="note-181-05" xlink:href="note-181-05a" xml:space="preserve">Sinus cõple <lb/>mẽti cuiuſ <lb/>uis arcꝰ qua <lb/>drãte maio <lb/>ris, qua arte <lb/>deprehen -<lb/>datur.</note> <note position="right" xlink:label="note-181-06" xlink:href="note-181-06a" xml:space="preserve">Sinꝰ cõple-<lb/>mẽti cuiuſ <lb/>uis arcꝰ ha <lb/>bẽtis Secũ-<lb/>da, p̃ter Mi <lb/>nuta, quo</note> </div> <p style="it"> <s xml:id="echoid-s5969" xml:space="preserve">_QVOD_ ſi datus arcus præter gradus, ac _M_inuta habeat etiã _S_ecunda; </s> <s xml:id="echoid-s5970" xml:space="preserve">ſi quidem <lb/>_Q_uadrante minor ſit, inueſtigabis eius ſinum complementi per regulam proportio-<lb/>num, quemadmodum ſupra de ſinu recto diximus, niſi quòd hic differentia inuenta, <lb/>ſiue pars proportionalis ſubtrahenda eſt à ſinu arcus proxime minoris, _S_i vero ar- <pb o="170" file="182" n="182" rhead=""/> cus datus ſit _Q_uadrante maior, ſed ſemicirculo minor; </s> <s xml:id="echoid-s5971" xml:space="preserve">detracto _Q_uadrante, inqui-<lb/> <anchor type="note" xlink:label="note-182-01a" xlink:href="note-182-01"/> res reſidui arcus ſinum rectum per eandem regulam proportionum, eo modo, quem <lb/>ſupra de ſinu recto tradidimus. </s> <s xml:id="echoid-s5972" xml:space="preserve">_Q_uamuis _S_ecunda negligi poſsint, vt ſupra docui-<lb/>mus, in hoc ſinuum negotio _E_xemplum. </s> <s xml:id="echoid-s5973" xml:space="preserve">_S_it datus arcus grad. </s> <s xml:id="echoid-s5974" xml:space="preserve">_69_. </s> <s xml:id="echoid-s5975" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5976" xml:space="preserve">_16._ </s> <s xml:id="echoid-s5977" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s5978" xml:space="preserve">_40._ <lb/></s> <s xml:id="echoid-s5979" xml:space="preserve">_A_ccipedifferentiam _2721_. </s> <s xml:id="echoid-s5980" xml:space="preserve">inter ſinum _3540190._ </s> <s xml:id="echoid-s5981" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5982" xml:space="preserve">_69_. </s> <s xml:id="echoid-s5983" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5984" xml:space="preserve">_16_. </s> <s xml:id="echoid-s5985" xml:space="preserve">proxime <lb/>minoris in parte tabulæ inferiori deſcripti, & </s> <s xml:id="echoid-s5986" xml:space="preserve">ſinum _3537469_. </s> <s xml:id="echoid-s5987" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5988" xml:space="preserve">_69_. </s> <s xml:id="echoid-s5989" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s5990" xml:space="preserve"><lb/>_17._ </s> <s xml:id="echoid-s5991" xml:space="preserve">proxime maioris in eadem parteinferiori tabulæ poſiti; </s> <s xml:id="echoid-s5992" xml:space="preserve">& </s> <s xml:id="echoid-s5993" xml:space="preserve">dic. </s> <s xml:id="echoid-s5994" xml:space="preserve">_S_i _40._ </s> <s xml:id="echoid-s5995" xml:space="preserve">_S_ecundæ <lb/>requirunt differentiam _2721_. </s> <s xml:id="echoid-s5996" xml:space="preserve">ſubtrahendam à ſinu _3540190_. </s> <s xml:id="echoid-s5997" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s5998" xml:space="preserve">_69_. </s> <s xml:id="echoid-s5999" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6000" xml:space="preserve"><lb/>_16._ </s> <s xml:id="echoid-s6001" xml:space="preserve">in i feriori parte tabulæ poſiti, vt relinquatur ſinus _3537469_. </s> <s xml:id="echoid-s6002" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s6003" xml:space="preserve">_69_. </s> <s xml:id="echoid-s6004" xml:space="preserve"><lb/>_M_in. </s> <s xml:id="echoid-s6005" xml:space="preserve">_17_. </s> <s xml:id="echoid-s6006" xml:space="preserve">in eadem parte inferiori tabulæ deſcripti: </s> <s xml:id="echoid-s6007" xml:space="preserve">quantam differentiam poſtulãt <lb/>_40_. </s> <s xml:id="echoid-s6008" xml:space="preserve">_S_ecunda ſubtrahendam ab eodem ſinu _3540190_. </s> <s xml:id="echoid-s6009" xml:space="preserve">arcus grad _69_. </s> <s xml:id="echoid-s6010" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6011" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6012" xml:space="preserve">poſiti <lb/>in part in feriori tabulæ, vt relinquatur ſinus arcus grad. </s> <s xml:id="echoid-s6013" xml:space="preserve">_69_. </s> <s xml:id="echoid-s6014" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6015" xml:space="preserve">_16._ </s> <s xml:id="echoid-s6016" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6017" xml:space="preserve">_40_ in <lb/>eadẽ parte inferiori tabulæ contenti? </s> <s xml:id="echoid-s6018" xml:space="preserve">Inuenies enim differentiã ſiue partẽ proportio-<lb/>nalẽ _1814_. </s> <s xml:id="echoid-s6019" xml:space="preserve">quæ ſubtracta ex sinu _3540190_. </s> <s xml:id="echoid-s6020" xml:space="preserve">relinquet sinum _3538376_. </s> <s xml:id="echoid-s6021" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s6022" xml:space="preserve"><lb/>_69_. </s> <s xml:id="echoid-s6023" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6024" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6025" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6026" xml:space="preserve">_40_. </s> <s xml:id="echoid-s6027" xml:space="preserve">in inferiori parte tabulæ collocati: </s> <s xml:id="echoid-s6028" xml:space="preserve">qui quidem eſt sinus rectus <lb/>arcus grad. </s> <s xml:id="echoid-s6029" xml:space="preserve">_20_ _M_in. </s> <s xml:id="echoid-s6030" xml:space="preserve">_43_ _S_ec. </s> <s xml:id="echoid-s6031" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6032" xml:space="preserve">hoc eſt, complementi arcus datigrad. </s> <s xml:id="echoid-s6033" xml:space="preserve">_69._ </s> <s xml:id="echoid-s6034" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6035" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6036" xml:space="preserve"><lb/>_S_ec. </s> <s xml:id="echoid-s6037" xml:space="preserve">_40_. </s> <s xml:id="echoid-s6038" xml:space="preserve">_S_it rurſum datus arcus grad. </s> <s xml:id="echoid-s6039" xml:space="preserve">_110_. </s> <s xml:id="echoid-s6040" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6041" xml:space="preserve">_43_. </s> <s xml:id="echoid-s6042" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6043" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6044" xml:space="preserve">_D_etrahe _Q_uadrantem, <lb/>id eſt, grad. </s> <s xml:id="echoid-s6045" xml:space="preserve">_90_ & </s> <s xml:id="echoid-s6046" xml:space="preserve">residui arcus grad. </s> <s xml:id="echoid-s6047" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6048" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6049" xml:space="preserve">_43_. </s> <s xml:id="echoid-s6050" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6051" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6052" xml:space="preserve">quære sinum rectum, vt <lb/>ſupra tradidimus. </s> <s xml:id="echoid-s6053" xml:space="preserve">_I_nuenies enim per partem proportionalem, sinum _3538376_. </s> <s xml:id="echoid-s6054" xml:space="preserve">qui <lb/>eſt sinus complementi arcus propositi.</s> <s xml:id="echoid-s6055" xml:space="preserve"/> </p> <div xml:id="echoid-div481" type="float" level="2" n="6"> <note position="left" xlink:label="note-182-01" xlink:href="note-182-01a" xml:space="preserve">modo eli-<lb/>ciat̃ per par <lb/>tẽ propor-<lb/>tionalem.</note> </div> <p style="it"> <s xml:id="echoid-s6056" xml:space="preserve">_HOC_ etiam modo sinum complemẽti arcus propositi quadrante minoris reperies. <lb/></s> <s xml:id="echoid-s6057" xml:space="preserve"> <anchor type="note" xlink:label="note-182-02a" xlink:href="note-182-02"/> _S_ubtrahe propositũ arcum ex quadrante, vt habeas eius complementum _S_inus enim <lb/>rectus eius complementi inuentus, vt de sinu recto diximus, eſt is, qui quæritur. </s> <s xml:id="echoid-s6058" xml:space="preserve">_V_t <lb/>si quæratur sinus complementi arcus grad. </s> <s xml:id="echoid-s6059" xml:space="preserve">_69_. </s> <s xml:id="echoid-s6060" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6061" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6062" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6063" xml:space="preserve">_40_. </s> <s xml:id="echoid-s6064" xml:space="preserve">detrahe hunc ar-<lb/>cum ex grad. </s> <s xml:id="echoid-s6065" xml:space="preserve">_90_. </s> <s xml:id="echoid-s6066" xml:space="preserve">& </s> <s xml:id="echoid-s6067" xml:space="preserve">residui arcus grad. </s> <s xml:id="echoid-s6068" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6069" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6070" xml:space="preserve">_43_. </s> <s xml:id="echoid-s6071" xml:space="preserve">ſec. </s> <s xml:id="echoid-s6072" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6073" xml:space="preserve">(qui complementum <lb/>eſt dati arcus grad. </s> <s xml:id="echoid-s6074" xml:space="preserve">_69_. </s> <s xml:id="echoid-s6075" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6076" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6077" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6078" xml:space="preserve">_40_) sinum rectum quære, quem inuenies eſſe <lb/>_3538376_. </s> <s xml:id="echoid-s6079" xml:space="preserve">atque hic eſt sinus complementi dati arcus grad. </s> <s xml:id="echoid-s6080" xml:space="preserve">_69_. </s> <s xml:id="echoid-s6081" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6082" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6083" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6084" xml:space="preserve">_40_. <lb/></s> <s xml:id="echoid-s6085" xml:space="preserve"> <anchor type="note" xlink:label="note-182-03a" xlink:href="note-182-03"/> </s> </p> <div xml:id="echoid-div482" type="float" level="2" n="7"> <note position="left" xlink:label="note-182-02" xlink:href="note-182-02a" xml:space="preserve">@lia rõ in. <lb/>ueſtigãdi ſi <lb/>nũ comple <lb/>menti arcꝰ <lb/>quad rante <lb/>minoris.</note> <note position="left" xlink:label="note-182-03" xlink:href="note-182-03a" xml:space="preserve">Sinꝰ cõple <lb/>mẽti ſemi-<lb/>circuli, aut <lb/>arcus maio <lb/>ris, q̃rendus <lb/>non eſt.</note> </div> <p style="it"> <s xml:id="echoid-s6086" xml:space="preserve">_HIC_ quoque ſemicirculi, atque arcus ſemicirculo matoris sinus complementi <lb/>quærendus non eſt, ob rationem ſupra dictam.</s> <s xml:id="echoid-s6087" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6088" xml:space="preserve">_QVANDO_ deniq; </s> <s xml:id="echoid-s6089" xml:space="preserve">propositi arcus sinum verſum desideras; </s> <s xml:id="echoid-s6090" xml:space="preserve">si quidem _Q_uadrante <lb/>minor eſt, detrahe eius sinum complementi ex sinu toto: </s> <s xml:id="echoid-s6091" xml:space="preserve">si vero _Q_uadrante eſt ma-<lb/>ior, ſed ſemicirculo minor, adde eius sinum complementi sinui toti. </s> <s xml:id="echoid-s6092" xml:space="preserve">_N_umerus enim <lb/>reliquus, vel compositus, erit sinus verſus dati arcus: </s> <s xml:id="echoid-s6093" xml:space="preserve">propterea quod sinus comple-<lb/> <anchor type="note" xlink:label="note-182-04a" xlink:href="note-182-04"/> menti cuiusuis arcus æqualis est complemento sinus versi eiuſdem arcus, vt ſupra <lb/>in definitionibus oſtendimus. </s> <s xml:id="echoid-s6094" xml:space="preserve"><emph style="sc">E</emph>x quo fit, vt sinus complementi arcus cuiusuis abla-<lb/>tus ex sinu toto, vel ad eum adiectus relinquat, vel componat eiuſdem arcus sinum <lb/>verſum: </s> <s xml:id="echoid-s6095" xml:space="preserve">_I_d quod perſpicuum eſt ex prima figura, quam in expositione definitionum <lb/>poſuimus. </s> <s xml:id="echoid-s6096" xml:space="preserve">_E_xemplum. </s> <s xml:id="echoid-s6097" xml:space="preserve">_S_it quærendus sinus verſus arcus grad. </s> <s xml:id="echoid-s6098" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6099" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6100" xml:space="preserve">_57_. </s> <s xml:id="echoid-s6101" xml:space="preserve">_H_uius <lb/>sinuscomplementi eſt _9338928_ qui detractus ex sinu toto _10000000_. </s> <s xml:id="echoid-s6102" xml:space="preserve">relinquet <lb/>sinum verſum _661072_. </s> <s xml:id="echoid-s6103" xml:space="preserve">dati arcus grad. </s> <s xml:id="echoid-s6104" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6105" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6106" xml:space="preserve">_57_. </s> <s xml:id="echoid-s6107" xml:space="preserve">_R_urſus sit inueſtigandus sinus <lb/>verſus arcus grad. </s> <s xml:id="echoid-s6108" xml:space="preserve">_138_. </s> <s xml:id="echoid-s6109" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6110" xml:space="preserve">_31_. </s> <s xml:id="echoid-s6111" xml:space="preserve">_H_uius sinus complementi eſt _7491484_. </s> <s xml:id="echoid-s6112" xml:space="preserve">qui additus <lb/>sinui toti _10000000_. </s> <s xml:id="echoid-s6113" xml:space="preserve">efficiet sinum verſum _17491484_. </s> <s xml:id="echoid-s6114" xml:space="preserve">arcus propositi grad. </s> <s xml:id="echoid-s6115" xml:space="preserve">_138_. <lb/></s> <s xml:id="echoid-s6116" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6117" xml:space="preserve">_31_. </s> <s xml:id="echoid-s6118" xml:space="preserve">_P_oſtremo sit inueniendus sinus verſus arcus grad. </s> <s xml:id="echoid-s6119" xml:space="preserve">_69_. </s> <s xml:id="echoid-s6120" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6121" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6122" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6123" xml:space="preserve">_40_. </s> <s xml:id="echoid-s6124" xml:space="preserve"><lb/>_H_uius complementi sinus eſt _3538376_. </s> <s xml:id="echoid-s6125" xml:space="preserve">inuentus per partem proportionalem; </s> <s xml:id="echoid-s6126" xml:space="preserve">qui <lb/>ſubtractus ex ſinu toto _10000000_. </s> <s xml:id="echoid-s6127" xml:space="preserve">relinquet ſinũ verſum _6461624_. </s> <s xml:id="echoid-s6128" xml:space="preserve">dati arcus grad. </s> <s xml:id="echoid-s6129" xml:space="preserve"><lb/>_69_ _M_in. </s> <s xml:id="echoid-s6130" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6131" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6132" xml:space="preserve">_40_. </s> <s xml:id="echoid-s6133" xml:space="preserve">_S_ic quoque ſinus verſus arcus grad. </s> <s xml:id="echoid-s6134" xml:space="preserve">_159_. </s> <s xml:id="echoid-s6135" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6136" xml:space="preserve">_16_. </s> <s xml:id="echoid-s6137" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6138" xml:space="preserve">_40_. </s> <s xml:id="echoid-s6139" xml:space="preserve">reperie-<lb/>tur _13538376_. </s> <s xml:id="echoid-s6140" xml:space="preserve">per partem proportionalem.</s> <s xml:id="echoid-s6141" xml:space="preserve"/> </p> <div xml:id="echoid-div483" type="float" level="2" n="8"> <note position="left" xlink:label="note-182-04" xlink:href="note-182-04a" xml:space="preserve">Sinꝰ verſus <lb/>cuiuſuis ar <lb/>cꝰ ſiue qua <lb/>d@ãte mino <lb/>ris ſiue ma <lb/>ioris, qu<unsure/>o <lb/>pacto colli <lb/>gatur.</note> </div> <pb o="171" file="183" n="183" rhead=""/> <p style="it"> <s xml:id="echoid-s6142" xml:space="preserve">_CAETERVM_ _Q_uadrantist tam ſinus rectus, quàm verſus eſt ſinus totus; </s> <s xml:id="echoid-s6143" xml:space="preserve">ſinus <lb/> <anchor type="note" xlink:label="note-183-01a" xlink:href="note-183-01"/> vero complementi nihil eſt, vt manifeſtum eſt ex prima figura in definitionibus poſita.</s> <s xml:id="echoid-s6144" xml:space="preserve"/> </p> <div xml:id="echoid-div484" type="float" level="2" n="9"> <note position="right" xlink:label="note-183-01" xlink:href="note-183-01a" xml:space="preserve">Sinus tã re <lb/>ctus, q̃ ver-<lb/>ſus Quadiã <lb/>tis eſt ſinus <lb/>totus, ſinus <lb/>vero cõple-<lb/>menti ni-<lb/>hil eſt.</note> </div> <p style="it"> <s xml:id="echoid-s6145" xml:space="preserve">_IAM_ vero ex cognito ſinu recto ita arcum inuenies. </s> <s xml:id="echoid-s6146" xml:space="preserve">_Q_uære ſinum rectum propo-<lb/>ſitum inter ſinus tabulæ; </s> <s xml:id="echoid-s6147" xml:space="preserve">vel ſi eum non inueneris, ſume proxime maiorem, vel mino-<lb/>rem, qui nimirum paucioribus vnitatibus à propoſito ſinu diſtat. </s> <s xml:id="echoid-s6148" xml:space="preserve">_N_am in vertice ta-<lb/>bulæ reperies gradus, & </s> <s xml:id="echoid-s6149" xml:space="preserve">ad ſini ſtram è regione ſinus accepti, _M_inuta illius arcus, qui <lb/>propoſito ſinut reſpondet. </s> <s xml:id="echoid-s6150" xml:space="preserve">_V_t ſi cognitus ſit ſinus _7510767_. </s> <s xml:id="echoid-s6151" xml:space="preserve">_I_nuenio ſinum _7510722_. <lb/></s> <s xml:id="echoid-s6152" xml:space="preserve">proxime minorem, qui paucioribus vmtatibus à ſinu cognito diſtat, quàm ſinus <lb/> <anchor type="note" xlink:label="note-183-02a" xlink:href="note-183-02"/> _7512642_. </s> <s xml:id="echoid-s6153" xml:space="preserve">proxime maior: </s> <s xml:id="echoid-s6154" xml:space="preserve">_C_ui ſinui proxime minori reſpõdent in vertice tabulæ grad. <lb/></s> <s xml:id="echoid-s6155" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6156" xml:space="preserve">& </s> <s xml:id="echoid-s6157" xml:space="preserve">ad ſiniſtram _M_inuta _41_. </s> <s xml:id="echoid-s6158" xml:space="preserve">_A_rcum ergo grad. </s> <s xml:id="echoid-s6159" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6160" xml:space="preserve">& </s> <s xml:id="echoid-s6161" xml:space="preserve">_M_in _41_ dico deberi ſinui pro-<lb/>poſito. </s> <s xml:id="echoid-s6162" xml:space="preserve">_N_am vnitates illæ, quibus ſinus propoſitus à ſinu dicti arcus differt, non indu-<lb/>cunt errorem notabilem. </s> <s xml:id="echoid-s6163" xml:space="preserve">_S_i tamen arcum cupis præciſiorem, inueſtiganda erit pars <lb/>proportionalis, hac arte. </s> <s xml:id="echoid-s6164" xml:space="preserve">_C_ape differentiam inter ſinum proxime minorem, & </s> <s xml:id="echoid-s6165" xml:space="preserve">ſinum <lb/>proxime maiorem: </s> <s xml:id="echoid-s6166" xml:space="preserve">_I_tem differentiam inter ſinum propoſitum, & </s> <s xml:id="echoid-s6167" xml:space="preserve">illum in tabula re-<lb/>pertũ, à quo minus differt; </s> <s xml:id="echoid-s6168" xml:space="preserve">& </s> <s xml:id="echoid-s6169" xml:space="preserve">dic. </s> <s xml:id="echoid-s6170" xml:space="preserve">_S_i differentia inter duos ſinus in tabula repertos dat <lb/> <anchor type="note" xlink:label="note-183-03a" xlink:href="note-183-03"/> _60_. </s> <s xml:id="echoid-s6171" xml:space="preserve">_S_ecunda addenda arcui ſinus proxime minoris, vel auferenda ab arcu ſinus proxi-<lb/>me maioris, (prout videlicet ſumpta fuerit differentia inter ſinum propoſitum, & </s> <s xml:id="echoid-s6172" xml:space="preserve"><lb/>ſinum proxime minorem, vel proxime maiorem) vt habeatur arcus ſinus proxime ma-<lb/>ioris, vel proxime minoris; </s> <s xml:id="echoid-s6173" xml:space="preserve">quot _S_ecunda poſtulat differentia inter ſinum propoſitum <lb/>& </s> <s xml:id="echoid-s6174" xml:space="preserve">ſinum proxime minorem, vel proxime maiorem, addenda arcui ſinus proxime mi-<lb/>noris, vel auferenda ab arcu ſinus proxime maioris, vt habeatur arcus propoſiti ſinus? <lb/></s> <s xml:id="echoid-s6175" xml:space="preserve">_N_am hæc ſecunda inuenta addita arcui ſinus proxime minoris, vel ablata ab arcu <lb/>ſinus proxime maioris, dabunt arcum ſinus propoſiti. </s> <s xml:id="echoid-s6176" xml:space="preserve">_V_t in dato exemplo, ſi dicas. </s> <s xml:id="echoid-s6177" xml:space="preserve"><lb/>_D_ifferẽtia _1920_. </s> <s xml:id="echoid-s6178" xml:space="preserve">inter ſinum _7510722_. </s> <s xml:id="echoid-s6179" xml:space="preserve">proxime minorem, & </s> <s xml:id="echoid-s6180" xml:space="preserve">ſinum _17512642_. </s> <s xml:id="echoid-s6181" xml:space="preserve">pro-<lb/>xime maiorẽ, dat _60_. </s> <s xml:id="echoid-s6182" xml:space="preserve">_S_ecunda addenda arcui grad. </s> <s xml:id="echoid-s6183" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6184" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6185" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6186" xml:space="preserve">qui ſinui proxime mi-<lb/>nori reſpõdet, (quoniã propoſitus ſinus minus differt à ſinu proxime minori quàm à si <lb/>nu proxime maiori) vt habeatur arcus grad. </s> <s xml:id="echoid-s6187" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6188" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6189" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6190" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6191" xml:space="preserve">_60_. </s> <s xml:id="echoid-s6192" xml:space="preserve">hoc eſt, grad. </s> <s xml:id="echoid-s6193" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6194" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6195" xml:space="preserve"><lb/>_42_. </s> <s xml:id="echoid-s6196" xml:space="preserve">reſpõdens ſinui proxime maiori. </s> <s xml:id="echoid-s6197" xml:space="preserve">_Q_uot ergo Secũda poſtulat differẽtia _45_. </s> <s xml:id="echoid-s6198" xml:space="preserve">inter ſinũ <lb/>propoſitũ, & </s> <s xml:id="echoid-s6199" xml:space="preserve">ſinũ proxime minorẽ, addenda eidẽ arcui ſinus proxime minoris, vt fiat <lb/>arcus dati ſinus? </s> <s xml:id="echoid-s6200" xml:space="preserve">_I_nuenies enim _S_ecundũ _1._ </s> <s xml:id="echoid-s6201" xml:space="preserve">& </s> <s xml:id="echoid-s6202" xml:space="preserve">paulo amplius addendũ arcui grad. </s> <s xml:id="echoid-s6203" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6204" xml:space="preserve"><lb/>_M_in. </s> <s xml:id="echoid-s6205" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6206" xml:space="preserve">ita vt ſinui propoſito _7510767_. </s> <s xml:id="echoid-s6207" xml:space="preserve">reſpondeat arcus grad. </s> <s xml:id="echoid-s6208" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6209" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6210" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6211" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6212" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6213" xml:space="preserve">& </s> <s xml:id="echoid-s6214" xml:space="preserve"><lb/>paulo amplius. </s> <s xml:id="echoid-s6215" xml:space="preserve">_R_u-ſus ſit datus ſinus _455630_. </s> <s xml:id="echoid-s6216" xml:space="preserve">quem in tabula quæſitum non inuenio. </s> <s xml:id="echoid-s6217" xml:space="preserve"><lb/>_A_ccipio ergo proxime maiorem _456536_. </s> <s xml:id="echoid-s6218" xml:space="preserve">(_A_b hoc enim minus diſtat, quàm à proxime <lb/>minori _453630_.) </s> <s xml:id="echoid-s6219" xml:space="preserve">cui reſpondet arcus grad. </s> <s xml:id="echoid-s6220" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6221" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6222" xml:space="preserve">_37_. </s> <s xml:id="echoid-s6223" xml:space="preserve">_Q_uòd ſi magis præciſum arcum <lb/>deſiderem, inquiram partem proportionalem, hoc modo. </s> <s xml:id="echoid-s6224" xml:space="preserve">_D_ifferentia _2906_. </s> <s xml:id="echoid-s6225" xml:space="preserve">inter ſinũ <lb/>_456536_. </s> <s xml:id="echoid-s6226" xml:space="preserve">proxime maiorem, & </s> <s xml:id="echoid-s6227" xml:space="preserve">ſinum _453630_. </s> <s xml:id="echoid-s6228" xml:space="preserve">proxime minorem, dat _60_. </s> <s xml:id="echoid-s6229" xml:space="preserve">_S_ecũda au-<lb/>ferenda ab arcu grad. </s> <s xml:id="echoid-s6230" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6231" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6232" xml:space="preserve">_37_. </s> <s xml:id="echoid-s6233" xml:space="preserve">qui ſinui proxime maiori reſpondet, vt reliquus sit <lb/>arcus grad. </s> <s xml:id="echoid-s6234" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6235" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6236" xml:space="preserve">_36_. </s> <s xml:id="echoid-s6237" xml:space="preserve">reſpondens sinui proxime minori: </s> <s xml:id="echoid-s6238" xml:space="preserve">_Q_uot ergo _S_ecunda poſtulat <lb/>differentia _906_. </s> <s xml:id="echoid-s6239" xml:space="preserve">inter sinum propositum, & </s> <s xml:id="echoid-s6240" xml:space="preserve">sinum proxime maiorem, auferenda ab <lb/>eodem arcu ſinus proxime maioris, vt fiat arcus dati sinus? </s> <s xml:id="echoid-s6241" xml:space="preserve">_I_nuenio enim _S_ecunda _19_. </s> <s xml:id="echoid-s6242" xml:space="preserve"><lb/>fere, quæ ablata ex arcu grad. </s> <s xml:id="echoid-s6243" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6244" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6245" xml:space="preserve">_37_. </s> <s xml:id="echoid-s6246" xml:space="preserve">relinquunt arcum grad. </s> <s xml:id="echoid-s6247" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6248" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6249" xml:space="preserve">_36_. </s> <s xml:id="echoid-s6250" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6251" xml:space="preserve"><lb/>_41_. </s> <s xml:id="echoid-s6252" xml:space="preserve">sinui dato _455630_. </s> <s xml:id="echoid-s6253" xml:space="preserve">debitum. </s> <s xml:id="echoid-s6254" xml:space="preserve">_Q_uoniam veroidem sinus rectus reſpondet duobus <lb/> <anchor type="note" xlink:label="note-183-04a" xlink:href="note-183-04"/> arcubus ſemicirculum conficientibus, vt ſupra diximus, si arcus dicta arte ex sinu <lb/>recto inuentus ſubducatur ex ſemicirculo, ideſt, ex grad. </s> <s xml:id="echoid-s6255" xml:space="preserve">_180_. </s> <s xml:id="echoid-s6256" xml:space="preserve">reliquus erit alter ar-<lb/>cus quadrante maior, qui dicto etiam ſinui debetur. </s> <s xml:id="echoid-s6257" xml:space="preserve">_V_t ſi arcus grad. </s> <s xml:id="echoid-s6258" xml:space="preserve">_48._ </s> <s xml:id="echoid-s6259" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6260" xml:space="preserve">_41_. <lb/></s> <s xml:id="echoid-s6261" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6262" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6263" xml:space="preserve">inuentus dematur ex grad. </s> <s xml:id="echoid-s6264" xml:space="preserve">_180_. </s> <s xml:id="echoid-s6265" xml:space="preserve">remanebit arcus grad _131_. </s> <s xml:id="echoid-s6266" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6267" xml:space="preserve">_18_. </s> <s xml:id="echoid-s6268" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6269" xml:space="preserve">_59_. </s> <s xml:id="echoid-s6270" xml:space="preserve">eidẽ <lb/>sinui recto _7510767_. </s> <s xml:id="echoid-s6271" xml:space="preserve">debitus. </s> <s xml:id="echoid-s6272" xml:space="preserve">_P_ulchre autem operatio in triangulis tam rectilineis, <pb o="172" file="184" n="184" rhead=""/> quàm ſphæricis, docebit, num accipiendus sit arcus quadrante maior proposito ſinui <lb/>veſpondens, an vero minor, vt proprijs locis apparebit.</s> <s xml:id="echoid-s6273" xml:space="preserve"/> </p> <div xml:id="echoid-div485" type="float" level="2" n="10"> <note position="right" xlink:label="note-183-02" xlink:href="note-183-02a" xml:space="preserve">Arcus qua-<lb/>drante mi-<lb/>not quo pà <lb/>cto ex ſinu <lb/>recto cogni <lb/>to eliciat̃.</note> <note position="right" xlink:label="note-183-03" xlink:href="note-183-03a" xml:space="preserve">Arcus qua-<lb/>drante mi-<lb/>nor magis <lb/>p̃ciſus, quo <lb/>pacto p par <lb/>tẽ propor-<lb/>tionalẽ ex <lb/>dato ſinu <lb/>eliciatur.</note> <note position="right" xlink:label="note-183-04" xlink:href="note-183-04a" xml:space="preserve">Atcus qu@-<lb/>drante ma <lb/>tor qua ar-<lb/>te ex ſinu <lb/>recto etua-<lb/>tur.</note> </div> <p style="it"> <s xml:id="echoid-s6274" xml:space="preserve">_SI_ vero sinus cognitus, eſt sinus complementi arcus quæsiti, ſumendierunt gra-<lb/> <anchor type="note" xlink:label="note-184-01a" xlink:href="note-184-01"/> dus in parte inferiori tabulæ, & </s> <s xml:id="echoid-s6275" xml:space="preserve">_M_inuta ad dextram. </s> <s xml:id="echoid-s6276" xml:space="preserve">_I_ta enim habebitur arcus quæ-<lb/>situs. </s> <s xml:id="echoid-s6277" xml:space="preserve">_V_elcerte inueniendus erit arcus, vt prius diximus, ſinui dato, tanquam recto, <lb/>reſpondens, is que ex quadrante demendus, vt arcus quæsitus relinquatur. </s> <s xml:id="echoid-s6278" xml:space="preserve">_V_t si co-<lb/>gnitus sit _7510767_. </s> <s xml:id="echoid-s6279" xml:space="preserve">sinus complementi alicuius arcus. </s> <s xml:id="echoid-s6280" xml:space="preserve">_I_nuenio sinum _7510722_. </s> <s xml:id="echoid-s6281" xml:space="preserve">pro-<lb/>xime minorem; </s> <s xml:id="echoid-s6282" xml:space="preserve">quoniam paucioribus hic vnitatibus à sinu cognito diſtat, quàm sinus <lb/>_7512642_. </s> <s xml:id="echoid-s6283" xml:space="preserve">proxime maior in tabula sinuum: </s> <s xml:id="echoid-s6284" xml:space="preserve">_C_ui sinui proxime minori reſpondent in <lb/>ima ſede tabulæ grad. </s> <s xml:id="echoid-s6285" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6286" xml:space="preserve">& </s> <s xml:id="echoid-s6287" xml:space="preserve">ad dexteram _M_inuta _19_. </s> <s xml:id="echoid-s6288" xml:space="preserve">_A_rcus igitur grad. </s> <s xml:id="echoid-s6289" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6290" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6291" xml:space="preserve">_19_. <lb/></s> <s xml:id="echoid-s6292" xml:space="preserve">eſt is, qui quæritur. </s> <s xml:id="echoid-s6293" xml:space="preserve">_H_uius enim complementum eſt arcus grad. </s> <s xml:id="echoid-s6294" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6295" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6296" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6297" xml:space="preserve">cui sinus <lb/>datus debetur _I_dem arcus grad. </s> <s xml:id="echoid-s6298" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6299" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6300" xml:space="preserve">_19_. </s> <s xml:id="echoid-s6301" xml:space="preserve">reperietur, si arcus grad. </s> <s xml:id="echoid-s6302" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6303" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6304" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6305" xml:space="preserve"><lb/>sinui dato in vertice tabulæ, & </s> <s xml:id="echoid-s6306" xml:space="preserve">ad siniſtram reſpondens ex quadrante ſubducatur. </s> <s xml:id="echoid-s6307" xml:space="preserve"><lb/>_Q_uòd si partem proportionalem ſupra inuentam, nimirum _S_ec. </s> <s xml:id="echoid-s6308" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6309" xml:space="preserve">detrahas ex arcure <lb/> <anchor type="note" xlink:label="note-184-02a" xlink:href="note-184-02"/> perto grad. </s> <s xml:id="echoid-s6310" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6311" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6312" xml:space="preserve">_19_. </s> <s xml:id="echoid-s6313" xml:space="preserve">(quia maiorem arcum, quàm par eſt, dato sinui _7510767_. </s> <s xml:id="echoid-s6314" xml:space="preserve">tri-<lb/>buimus.) </s> <s xml:id="echoid-s6315" xml:space="preserve">inuenietur arcus magis præciſus grad. </s> <s xml:id="echoid-s6316" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6317" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6318" xml:space="preserve">_18_. </s> <s xml:id="echoid-s6319" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6320" xml:space="preserve">_59_. </s> <s xml:id="echoid-s6321" xml:space="preserve">_Q_ui etiam repe-<lb/>rietur, si arcum grad. </s> <s xml:id="echoid-s6322" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6323" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6324" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6325" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6326" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6327" xml:space="preserve">eidem ſinui, tanquam recto, debitum, & </s> <s xml:id="echoid-s6328" xml:space="preserve">ſe-<lb/>cundum partem propertionalem inuentum, detrahas ex quadrante. </s> <s xml:id="echoid-s6329" xml:space="preserve">_R_urſus detur <lb/>_455630_. </s> <s xml:id="echoid-s6330" xml:space="preserve">sinus complementi alicuius arcus, quem in tabula quæsitum non inuenio. <lb/></s> <s xml:id="echoid-s6331" xml:space="preserve">_A_ccipio ergo proxime maiorem _456536_. </s> <s xml:id="echoid-s6332" xml:space="preserve">(quoniam ab hoc minus diſtat, quàm à pro-<lb/>xime minore _453630_.) </s> <s xml:id="echoid-s6333" xml:space="preserve">cui in parte inferiori tabulæ reſpondet arcus grad. </s> <s xml:id="echoid-s6334" xml:space="preserve">_87_. </s> <s xml:id="echoid-s6335" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6336" xml:space="preserve"><lb/>_23_. </s> <s xml:id="echoid-s6337" xml:space="preserve">quæſitus; </s> <s xml:id="echoid-s6338" xml:space="preserve">cum huius complementum ſit arcus grad. </s> <s xml:id="echoid-s6339" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6340" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6341" xml:space="preserve">_37_. </s> <s xml:id="echoid-s6342" xml:space="preserve">sinui dato, tanquã <lb/>recto, debitus. </s> <s xml:id="echoid-s6343" xml:space="preserve">_Q_uòd si partem proportionalem ſupra inuentam, nimirum _S_ec. </s> <s xml:id="echoid-s6344" xml:space="preserve">_19_. </s> <s xml:id="echoid-s6345" xml:space="preserve">ad-<lb/>das arcui inuento grad. </s> <s xml:id="echoid-s6346" xml:space="preserve">_87_. </s> <s xml:id="echoid-s6347" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6348" xml:space="preserve">_23_. </s> <s xml:id="echoid-s6349" xml:space="preserve">(quia minorem arcum, quàm par eſt, dato sinui <lb/>_455630_. </s> <s xml:id="echoid-s6350" xml:space="preserve">tribuimus.) </s> <s xml:id="echoid-s6351" xml:space="preserve">inuenietur arcus magis præciſus grad. </s> <s xml:id="echoid-s6352" xml:space="preserve">_87_. </s> <s xml:id="echoid-s6353" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6354" xml:space="preserve">_23_. </s> <s xml:id="echoid-s6355" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6356" xml:space="preserve">_19_. </s> <s xml:id="echoid-s6357" xml:space="preserve">_Q_uẽ <lb/>etiam reperies, si arcum grad. </s> <s xml:id="echoid-s6358" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6359" xml:space="preserve">_M_in _36_. </s> <s xml:id="echoid-s6360" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6361" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6362" xml:space="preserve">eidem sinui, tanquam recto, debitũ, <lb/>& </s> <s xml:id="echoid-s6363" xml:space="preserve">ſecundum partem proportionalem inuentum, ex quadrante ſubducas _I_am vero si <lb/>sinus propositus, eſt sinus complementi arcus quadrante maioris, (_Q_uod quãdo fiat, <lb/> <anchor type="note" xlink:label="note-184-03a" xlink:href="note-184-03"/> pulchre operatio in triãgulis ſiue rectilineis, ſiue ſphæricis docebit) ſumẽdus erit arcus <lb/>ei in vertice tabulæ, tanquàm ſinui recto reſpondens, & </s> <s xml:id="echoid-s6364" xml:space="preserve">quadrãti adijciendus, vt ar-<lb/>cus quæſitus conficiatur. </s> <s xml:id="echoid-s6365" xml:space="preserve">_V_t sisinus complementi alicuius arcus quadrante maioris <lb/>cognitus sit _7510722_. </s> <s xml:id="echoid-s6366" xml:space="preserve">ſumendus erit arcus ei reſpondens in vertice tabulæ vna cum <lb/>parte proportionali, grad. </s> <s xml:id="echoid-s6367" xml:space="preserve">_48_. </s> <s xml:id="echoid-s6368" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6369" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6370" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6371" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6372" xml:space="preserve">& </s> <s xml:id="echoid-s6373" xml:space="preserve">quadranti adijciendus. </s> <s xml:id="echoid-s6374" xml:space="preserve">_C_omponetur <lb/>enim arcus grad. </s> <s xml:id="echoid-s6375" xml:space="preserve">_138_. </s> <s xml:id="echoid-s6376" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6377" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6378" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6379" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6380" xml:space="preserve">qui quæritur, cuius nimirum complement@ <lb/>datus sinus debetur.</s> <s xml:id="echoid-s6381" xml:space="preserve"/> </p> <div xml:id="echoid-div486" type="float" level="2" n="11"> <note position="left" xlink:label="note-184-01" xlink:href="note-184-01a" xml:space="preserve">Arcus qua-<lb/>dranre mi-<lb/>nor quo pa <lb/>cto ex ſinu <lb/>complemẽ <lb/>ti cruatur.</note> <note position="left" xlink:label="note-184-02" xlink:href="note-184-02a" xml:space="preserve">Arcus qua-<lb/>drante mi-<lb/>nor magis <lb/>pręcisꝰ, qua <lb/>via ex ſinu <lb/>cõplementi <lb/>@ognoſcat̃.</note> <note position="left" xlink:label="note-184-03" xlink:href="note-184-03a" xml:space="preserve">Arcus qua-<lb/>drante ma-<lb/>ior quomo <lb/>do ex ſinu <lb/>cõplementi <lb/>inueſtiget̃.</note> </div> <p style="it"> <s xml:id="echoid-s6382" xml:space="preserve">_<emph style="sc">DENIQVe</emph>_ ex sinu verſo cognito ita arcum inquires. </s> <s xml:id="echoid-s6383" xml:space="preserve">_S_i datus sinus verſus <lb/> <anchor type="note" xlink:label="note-184-04a" xlink:href="note-184-04"/> minor eſt, quàm sinus totus, detrahe eum ex sinu toto. </s> <s xml:id="echoid-s6384" xml:space="preserve">_R_eliquus enim eritsinus com-<lb/>plementi arcus quæsiti. </s> <s xml:id="echoid-s6385" xml:space="preserve">_Q_uare ex hoc, vt proxime docuimus, arcum quæsitum inue-<lb/>nies. </s> <s xml:id="echoid-s6386" xml:space="preserve">_S_i vero datus sinus verſus sinum totum ſuperat, ſubtrahe ex eo sinum totum. <lb/></s> <s xml:id="echoid-s6387" xml:space="preserve">_R_emanebit enim sinus rectus arcus, qui quadranti adiectus arcum quæsitum conficiet-<lb/>_E_xemplum. </s> <s xml:id="echoid-s6388" xml:space="preserve">_D_etur ſinus verſus _9544370_. </s> <s xml:id="echoid-s6389" xml:space="preserve">_H_unc detraho ex sinu toto _10000000_. </s> <s xml:id="echoid-s6390" xml:space="preserve">re-<lb/>manebitq; </s> <s xml:id="echoid-s6391" xml:space="preserve">_455630_. </s> <s xml:id="echoid-s6392" xml:space="preserve">sinus complementi arcus quæsiti, ex quo inuenietur arcus grad. </s> <s xml:id="echoid-s6393" xml:space="preserve"><lb/>_87_. </s> <s xml:id="echoid-s6394" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6395" xml:space="preserve">_23_. </s> <s xml:id="echoid-s6396" xml:space="preserve">_V_el partem proportionalem magis præciſus, grad. </s> <s xml:id="echoid-s6397" xml:space="preserve">_87_. </s> <s xml:id="echoid-s6398" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6399" xml:space="preserve">_23_. </s> <s xml:id="echoid-s6400" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6401" xml:space="preserve">_19_. </s> <s xml:id="echoid-s6402" xml:space="preserve"><lb/>_I_tem sit datus sinus verſus _10455630_. </s> <s xml:id="echoid-s6403" xml:space="preserve"><emph style="sc">E</emph>xhoc ſubduco sinum totum _10000000_. </s> <s xml:id="echoid-s6404" xml:space="preserve">re-<lb/>linqueturq́; </s> <s xml:id="echoid-s6405" xml:space="preserve">_455630_. </s> <s xml:id="echoid-s6406" xml:space="preserve">sinus rectus, cuius arcus grad. </s> <s xml:id="echoid-s6407" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6408" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6409" xml:space="preserve">_37_. </s> <s xml:id="echoid-s6410" xml:space="preserve">vel magis præciſus per <lb/>partem proportionalem inuentus, grad. </s> <s xml:id="echoid-s6411" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6412" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6413" xml:space="preserve">_36_. </s> <s xml:id="echoid-s6414" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6415" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6416" xml:space="preserve">adiectus quadranti efficiet <lb/>arcum quæsitum grad. </s> <s xml:id="echoid-s6417" xml:space="preserve">_92_. </s> <s xml:id="echoid-s6418" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6419" xml:space="preserve">_37_. </s> <s xml:id="echoid-s6420" xml:space="preserve">vel magis præciſum, grad. </s> <s xml:id="echoid-s6421" xml:space="preserve">_2_. </s> <s xml:id="echoid-s6422" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6423" xml:space="preserve">_36_. </s> <s xml:id="echoid-s6424" xml:space="preserve">_S_ec. </s> <s xml:id="echoid-s6425" xml:space="preserve">_41_. </s> <s xml:id="echoid-s6426" xml:space="preserve">_H_uius <pb o="173" file="185" n="185" rhead=""/> operationis ratio perſpicua eſt ex prima figura in expoſitione definitionũ poſita. </s> <s xml:id="echoid-s6427" xml:space="preserve">_I_n ea <lb/>enim ſinus verſus _AH,_ ex ſinu toto _<emph style="sc">Ae</emph>_, ſublatus relinquit _HE_, vel _<emph style="sc">F</emph>K_, ſinum comple <lb/>menti arcus _AF_, qui dicto ſinui verſo _AH_, debetur. </s> <s xml:id="echoid-s6428" xml:space="preserve">_I_tem ex ſinu verſo _HC_, ſubdu-<lb/>ctus ſinus totus _<emph style="sc">E</emph>C_, relinquit _<emph style="sc">E</emph>H_, vel _<emph style="sc">Kf</emph>_, ſinum rectum arcus _<emph style="sc">F</emph>B_, qui quadranti <lb/>_BC_, adiectus componit arcum _<emph style="sc">F</emph>C_, dicto ſinui verſo _HC_, reſpondentem.</s> <s xml:id="echoid-s6429" xml:space="preserve"/> </p> <div xml:id="echoid-div487" type="float" level="2" n="12"> <note position="left" xlink:label="note-184-04" xlink:href="note-184-04a" xml:space="preserve">Arcus quo <lb/>pacto ex ſi-<lb/>nu verſo <lb/>cruatur.</note> </div> <p style="it"> <s xml:id="echoid-s6430" xml:space="preserve">_EX_ eadem tabula ſinuum rectorum indagabimus quoq; </s> <s xml:id="echoid-s6431" xml:space="preserve">cuiusq; </s> <s xml:id="echoid-s6432" xml:space="preserve">arcus chordam; <lb/></s> <s xml:id="echoid-s6433" xml:space="preserve"> <anchor type="note" xlink:label="note-185-01a" xlink:href="note-185-01"/> & </s> <s xml:id="echoid-s6434" xml:space="preserve">contra datæ cuiusq; </s> <s xml:id="echoid-s6435" xml:space="preserve">chordæ arcum reperiemus. </s> <s xml:id="echoid-s6436" xml:space="preserve">_N_am ſi dimidij arcus propoſiti ſi-<lb/>num rectũ accipiamus, eumq; </s> <s xml:id="echoid-s6437" xml:space="preserve">duplicemus, conflabimus dicti arcus chordam. </s> <s xml:id="echoid-s6438" xml:space="preserve">_I_tem ſi <lb/>datæ chordæ dimidium, tanquàm sinum rectum ſumamus, eiuſq́; </s> <s xml:id="echoid-s6439" xml:space="preserve">arcum eliciamus, da <lb/>bit hic arcus duplicatus arcum datæ chordæ reſpondentem. </s> <s xml:id="echoid-s6440" xml:space="preserve">_I_d quod ex eadem figura <lb/>prima, quã in definitionum explicatione deſcripsimus, manifeſtum eſt. </s> <s xml:id="echoid-s6441" xml:space="preserve">_N_ã in ea _<emph style="sc">F</emph>H_, <lb/>ſinus rectus arcus _<emph style="sc">AF</emph>,_ vel _<emph style="sc">F</emph>C_, eſt ſemißis chordæ _<emph style="sc">F</emph>G_, arcus _<emph style="sc">F</emph>AG_, vel _<emph style="sc">F</emph>CG_, &</s> <s xml:id="echoid-s6442" xml:space="preserve">c.</s> <s xml:id="echoid-s6443" xml:space="preserve"/> </p> <div xml:id="echoid-div488" type="float" level="2" n="13"> <note position="right" xlink:label="note-185-01" xlink:href="note-185-01a" xml:space="preserve">Chorda cu-<lb/>iuſq; arcꝰ; <lb/>& cõtra, ar-<lb/>cus chordæ <lb/>cuiuſq; qua <lb/>rõne ex ta-<lb/>bula ſinuũ <lb/>eliciatur.</note> </div> <p style="it"> <s xml:id="echoid-s6444" xml:space="preserve">_<emph style="sc">Ve</emph>RVM_ quia tabulam ſinuum non ſemper in promptu habemus, non iniucun-<lb/>dum ſtudioſis fore ſum arbitratus, ſi breuiter hoc loco doceam, antequam ad alia <lb/>progrediar, qua ratione ſinubus _G_eometrice, ſine auxilio numerorum, vti poßimus in <lb/>theorematibus, atq; </s> <s xml:id="echoid-s6445" xml:space="preserve">problematibus _A_ſtronomorum ac _G_eometrarũ explicandis: </s> <s xml:id="echoid-s6446" xml:space="preserve">ita vt <lb/>ſolo circini beneficio omnia illa conſequamur, quæ longis multiplicationibus, diuiſioni <lb/>busq́; </s> <s xml:id="echoid-s6447" xml:space="preserve">numerorum in ſinuum tabula contentorũ inquiri ſolẽt. </s> <s xml:id="echoid-s6448" xml:space="preserve">_H_ac enim reijs præſer <lb/>tim conſultum erit, qui vel magnã moleſtiam in numerorũ ſupputationibus ſentiũt, <lb/>vel non admodum in ijs ſeſe exercuerunt. </s> <s xml:id="echoid-s6449" xml:space="preserve">_Q_uod vt commodius exequamur, rem totam <lb/>vno aut altero exemplo exponemus. </s> <s xml:id="echoid-s6450" xml:space="preserve">_S_it ergo, exempli cauſa, inueſtiganda deslinatio <lb/>cuiuſuis puncti _E_clipticæ, vt grad. </s> <s xml:id="echoid-s6451" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6452" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6453" xml:space="preserve">_D_eſcribatur circulus _<emph style="sc">Ab</emph>CD_, vnà cum <lb/> <anchor type="note" xlink:label="note-185-02a" xlink:href="note-185-02"/> duabus diame-<lb/> <anchor type="figure" xlink:label="fig-185-01a" xlink:href="fig-185-01"/> tris _AC, BD_, <lb/>ſeſe in centro <lb/>_<emph style="sc">E</emph>_, ad angulos <lb/>rectos ſecanti-<lb/>bus. </s> <s xml:id="echoid-s6454" xml:space="preserve">_<emph style="sc">E</emph>_tquoniã, <lb/>vt in coroll. <lb/></s> <s xml:id="echoid-s6455" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s6456" xml:space="preserve">_1._ </s> <s xml:id="echoid-s6457" xml:space="preserve">lib. </s> <s xml:id="echoid-s6458" xml:space="preserve"><lb/>_1._ </s> <s xml:id="echoid-s6459" xml:space="preserve">noſtræ _G_no-<lb/>monices oſten-<lb/>dimus, ea eſt <lb/>proportio ſi-<lb/>nus totius ad <lb/>ſinum maximæ declinationis, quæ ſinus illius arcus, quo datum punctum à viciniori <lb/>puncto æquinoctij diſtat, ad ſinum declinationis eiuſdem puncti; </s> <s xml:id="echoid-s6460" xml:space="preserve">ſumatur arcus maxi-<lb/>mæ declinationis _<emph style="sc">Df</emph>_, (quod quidem facile fiet, ſi adſit quadrãs æneus, aut ligneus ac-<lb/>curate in _90_. </s> <s xml:id="echoid-s6461" xml:space="preserve">gradus diuiſus, de quo in initio noſtræ _G_nomonices ſcripſimus. </s> <s xml:id="echoid-s6462" xml:space="preserve">_S_ine hoc <lb/>enim quadrãte non eſſet operæ pretium velle ſinubus vti ſine numeris.) </s> <s xml:id="echoid-s6463" xml:space="preserve">& </s> <s xml:id="echoid-s6464" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s6465" xml:space="preserve"><lb/>_50._ </s> <s xml:id="echoid-s6466" xml:space="preserve">_DG_, quo nimirum datus gradus _20._ </s> <s xml:id="echoid-s6467" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6468" xml:space="preserve">à principio γ. </s> <s xml:id="echoid-s6469" xml:space="preserve">abeſt: </s> <s xml:id="echoid-s6470" xml:space="preserve">atq; </s> <s xml:id="echoid-s6471" xml:space="preserve">ex _<emph style="sc">F</emph>, G_, ad _<emph style="sc">De</emph>_, <lb/>perpendiculares demittantur _<emph style="sc">F</emph>I, GH_; </s> <s xml:id="echoid-s6472" xml:space="preserve">quod facile fiet, ſi arcubus _DF, <emph style="sc">Dg</emph>,_ ſuman-<lb/>tur arcus _DK, DL_, æquales: </s> <s xml:id="echoid-s6473" xml:space="preserve">_R_ectæ enim puncta _<emph style="sc">G</emph>, L,_ & </s> <s xml:id="echoid-s6474" xml:space="preserve">_<emph style="sc">F</emph>, K_, iungentes erunt ad <lb/>_<emph style="sc">De</emph>_, perpendiculares, cum per ſcholium in definitionibus poſitum ſecentur in _H,_ <lb/>_I_, bifariam, ac propterea ad angulos rectos: </s> <s xml:id="echoid-s6475" xml:space="preserve">_<emph style="sc">E</emph>_runt autem _FI, <emph style="sc">G</emph>H_, ſinus re-<lb/> <anchor type="note" xlink:label="note-185-03a" xlink:href="note-185-03"/> cti arcuum _<emph style="sc">Df</emph>, <emph style="sc">Dg</emph>._ </s> <s xml:id="echoid-s6476" xml:space="preserve">_I_gitur ſi tribus rectis _ED_, ſinui toti, & </s> <s xml:id="echoid-s6477" xml:space="preserve">_<emph style="sc">F</emph>I_, ſinui ma-<lb/>ximæ declinationis, & </s> <s xml:id="echoid-s6478" xml:space="preserve">_GH_, ſinui arcus, quo datum punctum ab æquinoctio <pb o="174" file="186" n="186" rhead=""/> diſtat, inveniatur quarta proportionalis, inuentus erit ſinus rectus declinationis <lb/> <anchor type="note" xlink:label="note-186-01a" xlink:href="note-186-01"/> quæſitæ. </s> <s xml:id="echoid-s6479" xml:space="preserve">_I_ta autem ſine magno labore quartam proportionalem reperiemus cum _E_u-<lb/>clide. </s> <s xml:id="echoid-s6480" xml:space="preserve">_D_uctis duabus rectis _AB, AC_, angulum quemcunq; </s> <s xml:id="echoid-s6481" xml:space="preserve">facientibus in _A_, ſumatur <lb/>in earum altera, recta _AD_, primæ lineæ, boc eſt, ſinui toti _ED_, æqualis; </s> <s xml:id="echoid-s6482" xml:space="preserve">& </s> <s xml:id="echoid-s6483" xml:space="preserve">_DB_, <lb/>æqualis ſecundæ lineæ, vt ſinui maximæ declinationis _FI:_ </s> <s xml:id="echoid-s6484" xml:space="preserve">_I_n altera vero, recta _<emph style="sc">Ae</emph>_, <lb/>tertiæ lineæ, nempe ſinui _GH_, æqualis. </s> <s xml:id="echoid-s6485" xml:space="preserve">_D_einde ducta recta _<emph style="sc">De</emph>_, ægatur illi per _B_, <lb/>parallela _BC_. </s> <s xml:id="echoid-s6486" xml:space="preserve">_N_am _<emph style="sc">E</emph>C_, erit quarta proportionalis, boc eſt, ſinus rectus declina-<lb/>tionis grad. </s> <s xml:id="echoid-s6487" xml:space="preserve">_20._ </s> <s xml:id="echoid-s6488" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6489" xml:space="preserve">_C_uius arcum ita inquiremus. </s> <s xml:id="echoid-s6490" xml:space="preserve">_R_ectæ _EC_, inuentæ abſcindemus <lb/>ex ſemidiametro _<emph style="sc">E</emph>A_, æqualem _EM_, & </s> <s xml:id="echoid-s6491" xml:space="preserve">per _M_, rectæ _EB_, parallelam ducemus _MN_. <lb/></s> <s xml:id="echoid-s6492" xml:space="preserve">_A_rcus namq; </s> <s xml:id="echoid-s6493" xml:space="preserve">_BN_, erit arcus declinationis quæſitæ, cum reſpondeat ſinui recto _<emph style="sc">E</emph>M_, <lb/>ſiue _EC:_ </s> <s xml:id="echoid-s6494" xml:space="preserve">propterea quod, ducto ſinu recto _NO,_ arcus _NB_, inter ſe æquales ſintre-<lb/> <anchor type="note" xlink:label="note-186-02a" xlink:href="note-186-02"/> ctæ _<emph style="sc">E</emph>M, ON_, ob parallelogrammum _MO_. </s> <s xml:id="echoid-s6495" xml:space="preserve">_E_undem tamen arcumita quoq; </s> <s xml:id="echoid-s6496" xml:space="preserve">obtine-<lb/> <anchor type="figure" xlink:label="fig-186-01a" xlink:href="fig-186-01"/> bimus. </s> <s xml:id="echoid-s6497" xml:space="preserve">_R_ectæ inæ <lb/>uentæ _<emph style="sc">E</emph>C_, æquæ <lb/>lem abſcindemus <lb/>_<emph style="sc">Cf</emph>_, vt _<emph style="sc">E</emph>F_, <lb/>ipſius _EC_, ſit du <lb/>pla, hoc eſt, chor <lb/>da illius arcus, <lb/>qui duplus e§t <lb/>arcus, cuius ſis <lb/>nus rectus e§t <lb/>_<emph style="sc">E</emph>C_. </s> <s xml:id="echoid-s6498" xml:space="preserve">_N_am ſire-<lb/>ctæ _<emph style="sc">E</emph>F_, æqua-<lb/>lem chordã _PQ,_ <lb/>in circulo accommodemus, & </s> <s xml:id="echoid-s6499" xml:space="preserve">cius arcum _PQ_, bifariam ſecemus in _R_, erit quoq; </s> <s xml:id="echoid-s6500" xml:space="preserve">_QR_, <lb/>vel _<emph style="sc">Pr</emph>_, arcus declinationis quæſitæ reſpondens ſinui _<emph style="sc">E</emph>C_, hoc eſt, dimidiatæ chordæ <lb/>_PQ_, vt conſtat ex definitione ſinus recti. </s> <s xml:id="echoid-s6501" xml:space="preserve">_Q_uadrans porroin _90._ </s> <s xml:id="echoid-s6502" xml:space="preserve">gradus diuiſus mon-<lb/>ſtrabit, (ſihoc etiam ſcire lubeat) quot gradus ac _M_inuta in _BN_, vel _PR,_ arcu declina <lb/>tionis contineantur: </s> <s xml:id="echoid-s6503" xml:space="preserve">quamuis _M_inuta ſecundũ exiſtimationẽ accipienda ſint; </s> <s xml:id="echoid-s6504" xml:space="preserve">proptereæ <lb/>quòd gradus quadrantis, niſi admodum magnus eſſet, in _M_inuta diuidi non poßit.</s> <s xml:id="echoid-s6505" xml:space="preserve"/> </p> <div xml:id="echoid-div489" type="float" level="2" n="14"> <note position="right" xlink:label="note-185-02" xlink:href="note-185-02a" xml:space="preserve">Quo pacto <lb/>ſinubus vtẽ <lb/>dũ ſit Geo-<lb/>metricè ſi-<lb/>ne tabu la <lb/>finuum.</note> <figure xlink:label="fig-185-01" xlink:href="fig-185-01a"> <image file="185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/185-01"/> </figure> <note position="right" xlink:label="note-185-03" xlink:href="note-185-03a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-186-01" xlink:href="note-186-01a" xml:space="preserve">12. fexti.</note> <note position="left" xlink:label="note-186-02" xlink:href="note-186-02a" xml:space="preserve">@4. primi.</note> <figure xlink:label="fig-186-01" xlink:href="fig-186-01a"> <image file="186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/186-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s6506" xml:space="preserve">_RVRSVS_ inueſtiganda ſit aſcenſio recta grad. </s> <s xml:id="echoid-s6507" xml:space="preserve">_20._ </s> <s xml:id="echoid-s6508" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6509" xml:space="preserve">_Q_uoniamigitur, vtin ſch@ <lb/>lio propoſ. </s> <s xml:id="echoid-s6510" xml:space="preserve">_9_. </s> <s xml:id="echoid-s6511" xml:space="preserve">lib. </s> <s xml:id="echoid-s6512" xml:space="preserve">_2._ </s> <s xml:id="echoid-s6513" xml:space="preserve">_G_nomonices demonſtrauimus, eadem eſt proportio ſinus complemen <lb/>ti declinationis puncti propoſiti ad ſinum complementi arcus, quo datũ punctũ à vicia <lb/>niori puncto æquinoctij abeſt, quæ ſinus totius ad ſinum complemẽti aſcenſionis rectæ: <lb/></s> <s xml:id="echoid-s6514" xml:space="preserve">ſumatur in eadem figura arcus _<emph style="sc">B</emph>N_, declinationis grad. </s> <s xml:id="echoid-s6515" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6516" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6517" xml:space="preserve">quæ in tabula declina-<lb/>tionis continet grad. </s> <s xml:id="echoid-s6518" xml:space="preserve">_17_. </s> <s xml:id="echoid-s6519" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s6520" xml:space="preserve">_47_. </s> <s xml:id="echoid-s6521" xml:space="preserve">ducaturq́; </s> <s xml:id="echoid-s6522" xml:space="preserve">_NM_, ad _<emph style="sc">E</emph>A_, perpendicularis, quæ ſinus erit <lb/>complementi dictæ declinationis. </s> <s xml:id="echoid-s6523" xml:space="preserve">_C_apiatur quoq; </s> <s xml:id="echoid-s6524" xml:space="preserve">arcus _DG_, grad. </s> <s xml:id="echoid-s6525" xml:space="preserve">_50_. </s> <s xml:id="echoid-s6526" xml:space="preserve">quo datum pun <lb/>ctum ab æquinoctio verno abeſt, & </s> <s xml:id="echoid-s6527" xml:space="preserve">ad _<emph style="sc">E</emph>A_, perpendicularis ducatur _GS_, nempe ſinus <lb/>complementi dicti arcus _DG_. </s> <s xml:id="echoid-s6528" xml:space="preserve">_P_oſt hæc tribus rectis _NM_, ſinui complemẽti declinationis <lb/>dati puncti, & </s> <s xml:id="echoid-s6529" xml:space="preserve">_GS_, ſinui complementi arcus _DG_, quo datũ punctum ab æquinoctij pun <lb/>cto diſtat, et _<emph style="sc">E</emph>D_, ſinui toti, quarta proportionalis inueniatur _LI_, vtin lineis _GH, GI_, <lb/>ſeſein _G_, ſecantibus factũ eſt: </s> <s xml:id="echoid-s6530" xml:space="preserve">_S_umpta enim ibi eſt _<emph style="sc">G</emph>K_, ipſi _NM_, & </s> <s xml:id="echoid-s6531" xml:space="preserve">_KH,_ ipſi _GS_, & </s> <s xml:id="echoid-s6532" xml:space="preserve">_GL_, <lb/>ſinui toti _ED_, æqualis, &</s> <s xml:id="echoid-s6533" xml:space="preserve">c. </s> <s xml:id="echoid-s6534" xml:space="preserve">_N_am _LI_, inuenta erit ſinus complementi aſcenſionis rectæ <lb/>dati grad. </s> <s xml:id="echoid-s6535" xml:space="preserve">_20_. </s> <s xml:id="echoid-s6536" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6537" xml:space="preserve">_Q_uare ſi ipſi _LI_, ex ſemidiametro _EC_, abſcindatur æqualis recta _<emph style="sc">E</emph>T_, <lb/>ducaturq; </s> <s xml:id="echoid-s6538" xml:space="preserve">_TV_, ipſi _EB_, parallela, & </s> <s xml:id="echoid-s6539" xml:space="preserve">_VX_, ipſi _<emph style="sc">E</emph>C_, parallela, erunt æquales rectæ _<emph style="sc">E</emph>T,_ <lb/> <anchor type="note" xlink:label="note-186-03a" xlink:href="note-186-03"/> _VX. </s> <s xml:id="echoid-s6540" xml:space="preserve">C_um ergo ſinui _VX_, reſpondeat arcus _BV_, erit huius complementũ _VC_, aſcenſi@ <pb o="175" file="187" n="187" rhead=""/> recta gradus _20._ </s> <s xml:id="echoid-s6541" xml:space="preserve"><de:unknown code="022"/>. </s> <s xml:id="echoid-s6542" xml:space="preserve">_Q_uot autem gradus complectatur arcus _VC_, indicabit quadrans <lb/>in _90_. </s> <s xml:id="echoid-s6543" xml:space="preserve">gradus diuiſus.</s> <s xml:id="echoid-s6544" xml:space="preserve"/> </p> <div xml:id="echoid-div490" type="float" level="2" n="15"> <note position="left" xlink:label="note-186-03" xlink:href="note-186-03a" xml:space="preserve">34. primi.</note> </div> <p style="it"> <s xml:id="echoid-s6545" xml:space="preserve">_EODEM_ pacto omnia alia problemata _G_eometrice per ſinus abſoluemus, etiamſi <lb/>ſinubus verſis vti oportuerit aliquando, qui quidem eadem facilitate exdatis arcu-<lb/>bus inueniuntur, & </s> <s xml:id="echoid-s6546" xml:space="preserve">exipſis arcus, qua ſinus rectos, & </s> <s xml:id="echoid-s6547" xml:space="preserve">ſinus complemẽtorum reperi-<lb/>mus. </s> <s xml:id="echoid-s6548" xml:space="preserve">_S_i enim arcus datus minor eſt quadrante, vt _AG;_ </s> <s xml:id="echoid-s6549" xml:space="preserve">ducta _GS_. </s> <s xml:id="echoid-s6550" xml:space="preserve">ad _AE_, perpendi-<lb/>culari, erit _AS_ ſinus verſus arcus _AG_. </s> <s xml:id="echoid-s6551" xml:space="preserve">_S_i vero datus arcus quadrante maior eſt, vt <lb/>_AV;_ </s> <s xml:id="echoid-s6552" xml:space="preserve">ducta _VT_, ad _EC_, perpendiculari, erit _AT_, ſinus verſus arcus _AV_, vt ex <lb/>definitione manifeſtum eſt.</s> <s xml:id="echoid-s6553" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6554" xml:space="preserve">_SED_ iam ad inquiſitionem chordarum _G_eometricam aggrediamur, ex quibus <lb/>rurſum ſinuum tabulam facili negotio componemus.</s> <s xml:id="echoid-s6555" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div492" type="section" level="1" n="241"> <head xml:id="echoid-head268" xml:space="preserve">THEOR. 7. PROPOS. 10.</head> <p> <s xml:id="echoid-s6556" xml:space="preserve">IN circulo ſumptis duobus arcubus inæqua-<lb/> <anchor type="note" xlink:label="note-187-01a" xlink:href="note-187-01"/> libus, quorum maioris chorda maior ſit, quam <lb/>chorda minoris; </s> <s xml:id="echoid-s6557" xml:space="preserve">maior eſt proportio arcus maio-<lb/>ris ad minorem, quam chordæ arcus maioris ad <lb/>chordam minoris arcus.</s> <s xml:id="echoid-s6558" xml:space="preserve"/> </p> <div xml:id="echoid-div492" type="float" level="2" n="1"> <note position="right" xlink:label="note-187-01" xlink:href="note-187-01a" xml:space="preserve">Maior eſt <lb/>proportio <lb/>maioris ar-<lb/>cꝰ in circu-<lb/>lo ad arcũ <lb/>minorẽ, q̃ <lb/>chordæ ma <lb/>ioris arcus <lb/>ad chordã <lb/>minoris.</note> </div> <p> <s xml:id="echoid-s6559" xml:space="preserve">IN circulo ABCD, ſint inęquales arcus AB, <emph style="sc">Bc</emph>; </s> <s xml:id="echoid-s6560" xml:space="preserve">ille maior, hic vero minor: <lb/></s> <s xml:id="echoid-s6561" xml:space="preserve">quorum chordæ AB, BC; </s> <s xml:id="echoid-s6562" xml:space="preserve">illa maior, hæc vero minor. </s> <s xml:id="echoid-s6563" xml:space="preserve">Dico maiorem eſſe <lb/>proportionem arcus AB, ad arcum BC, quam chordæ AB, ad chordam BC. </s> <s xml:id="echoid-s6564" xml:space="preserve"><lb/>Contineant enim chordæ AB, BC, angulum ABC, ita vt arcus ſint conti-<lb/>nuati, minoreſq; </s> <s xml:id="echoid-s6565" xml:space="preserve">ſint tota circunferentia. </s> <s xml:id="echoid-s6566" xml:space="preserve">Nam ſi toti circunferentiæ forent <lb/>æquales, eſſet eadem chorda vtriuſq; </s> <s xml:id="echoid-s6567" xml:space="preserve">arcus: </s> <s xml:id="echoid-s6568" xml:space="preserve">ſi vero totam circunferentiam <lb/>excederent, eſſet chorda arcus minoris maior, quam maioris, vt patet in ſe-<lb/> <anchor type="figure" xlink:label="fig-187-01a" xlink:href="fig-187-01"/> cunda figura, ſi minor arcus foret BAI. </s> <s xml:id="echoid-s6569" xml:space="preserve">Angulus porrò ABC, bifariam ſe-<lb/> <anchor type="note" xlink:label="note-187-02a" xlink:href="note-187-02"/> cetur recta BD, connectãturq́; </s> <s xml:id="echoid-s6570" xml:space="preserve">rectæ AC, AD, CD, quarum AC, rectam BD, <lb/>ſecet in E. </s> <s xml:id="echoid-s6571" xml:space="preserve">Erunt autem rectæ AD, CD, æquales, propter arcus AD, CD, <lb/> <anchor type="note" xlink:label="note-187-03a" xlink:href="note-187-03"/> qui ſubten ſiangulis ABD, CBD, ex conſtructione æqualibus æqualesinter <lb/> <anchor type="note" xlink:label="note-187-04a" xlink:href="note-187-04"/> ſe ſunt. </s> <s xml:id="echoid-s6572" xml:space="preserve">Et quoniam in triangulo ABC, recta BE, angulum ABC, bifariam <lb/>ſecat; </s> <s xml:id="echoid-s6573" xml:space="preserve">erit, vt AB, ad BC, ita AE, ad EC: </s> <s xml:id="echoid-s6574" xml:space="preserve">Eſt autem recta AB, maior, quam <lb/> <anchor type="note" xlink:label="note-187-05a" xlink:href="note-187-05"/> <pb o="176" file="188" n="188" rhead=""/> recta BC, ex hypotheſi. </s> <s xml:id="echoid-s6575" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s6576" xml:space="preserve">AE, maior erit, quã EC. </s> <s xml:id="echoid-s6577" xml:space="preserve">Diuiſa ergo AC, bi-<lb/>fariã in F, erit punctũ F, in maiori ſegmento AE. </s> <s xml:id="echoid-s6578" xml:space="preserve">Ducta autẽ recta DF; </s> <s xml:id="echoid-s6579" xml:space="preserve">quoniã <lb/>latera AF, FD, trianguli AFD, lateribus CF, FD, trianguli CFD, baſisq́; <lb/></s> <s xml:id="echoid-s6580" xml:space="preserve">AD, baſi CD, oſtenſa eſt æqualis; </s> <s xml:id="echoid-s6581" xml:space="preserve">erit angulus AFD, angulo CFD, æqua-<lb/> <anchor type="note" xlink:label="note-188-01a" xlink:href="note-188-01"/> lis; </s> <s xml:id="echoid-s6582" xml:space="preserve">ac proinde vterq; </s> <s xml:id="echoid-s6583" xml:space="preserve">rectus erit. </s> <s xml:id="echoid-s6584" xml:space="preserve">Cum ergo in triangulo DEF, duo anguli <lb/>E, F, duobus rectis ſint minores, erit angulus E, acutus, ac proinde reliquus <lb/> <anchor type="note" xlink:label="note-188-02a" xlink:href="note-188-02"/> CED, obtuſus. </s> <s xml:id="echoid-s6585" xml:space="preserve">Quare cum in triangulo CDE, duo anguli C, E, ſint duo-<lb/>bus rectis minores, erit angulus C, acutus. </s> <s xml:id="echoid-s6586" xml:space="preserve">Eſt igitur in triangulo DEF, la-<lb/>tus DE, maius latere DF, & </s> <s xml:id="echoid-s6587" xml:space="preserve">in triangulo CDE, minus latere CD. </s> <s xml:id="echoid-s6588" xml:space="preserve">Quocir-<lb/> <anchor type="note" xlink:label="note-188-03a" xlink:href="note-188-03"/> ca arcus circuli ex D, centro per E, deſcriptus ſecabit rectam DF, productam <lb/>in H, rectam autem CD, infra punctum C, in G. </s> <s xml:id="echoid-s6589" xml:space="preserve">Quoniam vero ſector DHE, <lb/>ad triangulum DEC, maiorem proportionem habet, quam triangulũ DFE, <lb/> <anchor type="note" xlink:label="note-188-04a" xlink:href="note-188-04"/> <anchor type="figure" xlink:label="fig-188-01a" xlink:href="fig-188-01"/> adidem triangulum DEC: </s> <s xml:id="echoid-s6590" xml:space="preserve">Item ſector idem DHE, ad ſectorem DEG, ma-<lb/>iorem proportionem habet, quam ad triangulum DEC; </s> <s xml:id="echoid-s6591" xml:space="preserve">habebit multò ma-<lb/>iorem proportionem ſector DHE, ad ſectorem DEG, quam triangulum <lb/>DFE, ad triangulum DEC. </s> <s xml:id="echoid-s6592" xml:space="preserve">Eſt autem, vt ſector DHE, ad ſectorem DEG, <lb/> <anchor type="note" xlink:label="note-188-05a" xlink:href="note-188-05"/> ita angulus HDE, ad angulum EDG. </s> <s xml:id="echoid-s6593" xml:space="preserve">Maior ergo quoq; </s> <s xml:id="echoid-s6594" xml:space="preserve">erit proportio an-<lb/>guli HDE, ad angulum EDG, quam trianguli DFE, ad triangulum DEC: <lb/></s> <s xml:id="echoid-s6595" xml:space="preserve"> <anchor type="note" xlink:label="note-188-06a" xlink:href="note-188-06"/> Sed vt triangulum DFE, ad triangulum DEC, ita eſt recta FE, ad rectam <lb/>EC. </s> <s xml:id="echoid-s6596" xml:space="preserve">Eſt igitur maior quoq; </s> <s xml:id="echoid-s6597" xml:space="preserve">proportio anguli HDE, ad angulum EDG, <lb/>quæ rectę FE, ad rectam EC. </s> <s xml:id="echoid-s6598" xml:space="preserve">Etcomponendo, maior etiam erit proportio <lb/> <anchor type="note" xlink:label="note-188-07a" xlink:href="note-188-07"/> anguli HDG, ad angulum EDG, quam rectæ FC, ad rectam EC. </s> <s xml:id="echoid-s6599" xml:space="preserve">Quia igi-<lb/> <anchor type="note" xlink:label="note-188-08a" xlink:href="note-188-08"/> tur eſt, vt angulus ADC, ad angulum HDG, ita <lb/>recta AC, ad rectã FC: </s> <s xml:id="echoid-s6600" xml:space="preserve">(vtrobiq; </s> <s xml:id="echoid-s6601" xml:space="preserve">enim eſt proportio <lb/>dupla) Angulus autem HDG, ad angulum EDG, <lb/>maiorem habet proportionem, quam recta FC, ad <lb/>rectam EC, vt oſtendimus; </s> <s xml:id="echoid-s6602" xml:space="preserve">erit ex æquo maior quo-<lb/> <anchor type="note" xlink:label="note-188-09a" xlink:href="note-188-09"/> que proportio anguli ADC, ad angulum EDG, <lb/>quam rectæ AC, ad rectam EC, vt in hac formula <lb/>apparet. </s> <s xml:id="echoid-s6603" xml:space="preserve">Diuidendo ergo erit quoq; </s> <s xml:id="echoid-s6604" xml:space="preserve">maior proportio anguli ADE, ad angu-<lb/> <anchor type="note" xlink:label="note-188-10a" xlink:href="note-188-10"/> lum EDG, quam rectæ AE, ad rectam EC. </s> <s xml:id="echoid-s6605" xml:space="preserve">Atqui vt angulus ADE, ad an-<lb/>gulum EDG, ita eſt arcus AB, ad arcum BC; </s> <s xml:id="echoid-s6606" xml:space="preserve">Et vt recta AE, ad rectam EC, <lb/> <anchor type="note" xlink:label="note-188-11a" xlink:href="note-188-11"/> ita eſt chorda AB, ad chordam BC. </s> <s xml:id="echoid-s6607" xml:space="preserve">Igitur maior erit etiam proportio arcus <lb/> <anchor type="note" xlink:label="note-188-12a" xlink:href="note-188-12"/> AB, ad arcum BC, quam chordæ AB, ad chordam BC. </s> <s xml:id="echoid-s6608" xml:space="preserve">In circulo ergo ſum-<lb/>ptis duobus arcubus inæqualibus, &</s> <s xml:id="echoid-s6609" xml:space="preserve">c. </s> <s xml:id="echoid-s6610" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s6611" xml:space="preserve"/> </p> <div xml:id="echoid-div493" type="float" level="2" n="2"> <figure xlink:label="fig-187-01" xlink:href="fig-187-01a"> <image file="187-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/187-01"/> </figure> <note position="right" xlink:label="note-187-02" xlink:href="note-187-02a" xml:space="preserve">9. primi.</note> <note position="right" xlink:label="note-187-03" xlink:href="note-187-03a" xml:space="preserve">29. tertij.</note> <note position="right" xlink:label="note-187-04" xlink:href="note-187-04a" xml:space="preserve">26. tertij.</note> <note position="right" xlink:label="note-187-05" xlink:href="note-187-05a" xml:space="preserve">3. fextia</note> <note position="left" xlink:label="note-188-01" xlink:href="note-188-01a" xml:space="preserve">8. primi.</note> <note position="left" xlink:label="note-188-02" xlink:href="note-188-02a" xml:space="preserve">17. primi.</note> <note position="left" xlink:label="note-188-03" xlink:href="note-188-03a" xml:space="preserve">19. primi.</note> <note position="left" xlink:label="note-188-04" xlink:href="note-188-04a" xml:space="preserve">8, quinti.</note> <figure xlink:label="fig-188-01" xlink:href="fig-188-01a"> <image file="188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/188-01"/> </figure> <note position="left" xlink:label="note-188-05" xlink:href="note-188-05a" xml:space="preserve">Coroll. 1. <lb/>propoſ. 33. <lb/>lib. 6.</note> <note position="left" xlink:label="note-188-06" xlink:href="note-188-06a" xml:space="preserve">1. ſexti.</note> <note position="left" xlink:label="note-188-07" xlink:href="note-188-07a" xml:space="preserve">28. quinti.</note> <note position="right" xlink:label="note-188-08" xlink:href="note-188-08a" xml:space="preserve"> <lb/>Anguli # Rectæ. <lb/>ADC. # AC. <lb/>HDG. # FC. <lb/>EDG. # EC. <lb/></note> <note position="left" xlink:label="note-188-09" xlink:href="note-188-09a" xml:space="preserve">31. quinti.</note> <note position="left" xlink:label="note-188-10" xlink:href="note-188-10a" xml:space="preserve">29. quinti.</note> <note position="left" xlink:label="note-188-11" xlink:href="note-188-11a" xml:space="preserve">33. ſexti.</note> <note position="left" xlink:label="note-188-12" xlink:href="note-188-12a" xml:space="preserve">3. ſexti.</note> </div> <pb o="177" file="189" n="189" rhead=""/> </div> <div xml:id="echoid-div495" type="section" level="1" n="242"> <head xml:id="echoid-head269" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s6612" xml:space="preserve">_QVAMVIS_ autem Theorema hoc proponatur ſolum de arcubus illis inæqualin <lb/>bus, quorũ maiori maior chorda ſubtenditur, quam minori: </s> <s xml:id="echoid-s6613" xml:space="preserve">_I_dem tamen locum etiam <lb/>habet in illis arcubus inæqualibus, quorum maioris chorda minor eſt, quam chordæ <lb/>minoris. </s> <s xml:id="echoid-s6614" xml:space="preserve">_N_am quia tunc arcus maior ad minorem habet proportionẽ maioris inæqua-<lb/>litatis, chorda vero maioris arcus ad chordam minoris arcus proportionem habet mi-<lb/>noris inæqualitatis, maior erit proportio maioris arcus ad minorem, quam chordæ <lb/>arcus maioris ad chordam minoris arcus.</s> <s xml:id="echoid-s6615" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div496" type="section" level="1" n="243"> <head xml:id="echoid-head270" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s6616" xml:space="preserve">SEQVITVR ex hac propoſitione, minorem eſſe proportionem minoris arcus ad ma-<lb/>iorem, quam chordæ minoris arcus ad chordam maioris. </s> <s xml:id="echoid-s6617" xml:space="preserve">Cum enim maior arcus ad mi-<lb/>norem habeat maiorem proportionem, quam chorda maioris arcus ad chordam minoris, <lb/>vt demonſtratum eſt; </s> <s xml:id="echoid-s6618" xml:space="preserve">habebit conuertendo minor arcus ad maiorem, minorem propor-<lb/> <anchor type="note" xlink:label="note-189-01a" xlink:href="note-189-01"/> tionem, quam chorda arcus minoris ad chordam maioris.</s> <s xml:id="echoid-s6619" xml:space="preserve"/> </p> <div xml:id="echoid-div496" type="float" level="2" n="1"> <note position="right" xlink:label="note-189-01" xlink:href="note-189-01a" xml:space="preserve">26. quinti.</note> </div> </div> <div xml:id="echoid-div498" type="section" level="1" n="244"> <head xml:id="echoid-head271" xml:space="preserve">THEOR. 8. PROPOS. II.</head> <p> <s xml:id="echoid-s6620" xml:space="preserve">SI in circulo quadrilaterum deſcribatur cum <lb/> <anchor type="note" xlink:label="note-189-02a" xlink:href="note-189-02"/> ſuis diametris; </s> <s xml:id="echoid-s6621" xml:space="preserve">eritrectãgulum ſub diametris com-<lb/>prehenſum æquale duobus rectãgulis ſimul, quæ <lb/>ſub lateribus oppoſitis continentur.</s> <s xml:id="echoid-s6622" xml:space="preserve"/> </p> <div xml:id="echoid-div498" type="float" level="2" n="1"> <note position="right" xlink:label="note-189-02" xlink:href="note-189-02a" xml:space="preserve">Rectangu-<lb/>lũ ſub dia-<lb/>metris qua <lb/>drilateri in <lb/>circulo de-<lb/>ſcripti con <lb/>tentũ æqua <lb/>le eſt duo-<lb/>bus rectan-<lb/>gulis ſub <lb/>oppoſitis la <lb/>teribus con <lb/>tentis.</note> </div> <p> <s xml:id="echoid-s6623" xml:space="preserve">IN circulo ABCD, ſit quadrilaterum ABCD, cuius diametri AC, BD. <lb/></s> <s xml:id="echoid-s6624" xml:space="preserve">Dico rectangulum ſub AC, BD, comprehenſum æquale eſſe rectangulis ſi-<lb/>mul ſub AD, BC, & </s> <s xml:id="echoid-s6625" xml:space="preserve">ſub AB, DC, contentis. </s> <s xml:id="echoid-s6626" xml:space="preserve">Fiat angulo DAC, æqualis <lb/>angulus BAE; </s> <s xml:id="echoid-s6627" xml:space="preserve">cadetq́ recta AE, vel in ipſam rectam AC; </s> <s xml:id="echoid-s6628" xml:space="preserve">vel inter AC, <lb/>rectam, & </s> <s xml:id="echoid-s6629" xml:space="preserve">punctum B; </s> <s xml:id="echoid-s6630" xml:space="preserve">vel deniq; </s> <s xml:id="echoid-s6631" xml:space="preserve">inter rectam AC, & </s> <s xml:id="echoid-s6632" xml:space="preserve">punctum D: </s> <s xml:id="echoid-s6633" xml:space="preserve">atq; </s> <s xml:id="echoid-s6634" xml:space="preserve">erit <lb/>in primo caſu angulus BAC, angulo DAE; </s> <s xml:id="echoid-s6635" xml:space="preserve">& </s> <s xml:id="echoid-s6636" xml:space="preserve">in ſecundo caſu totus angu-<lb/>lus BAC, <lb/> <anchor type="figure" xlink:label="fig-189-01a" xlink:href="fig-189-01"/> toti angulo <lb/>DAE, pro-<lb/>pter cõ mu-<lb/>nem angu-<lb/>lum EAC, <lb/>additum; </s> <s xml:id="echoid-s6637" xml:space="preserve">& </s> <s xml:id="echoid-s6638" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s6639" xml:space="preserve">in tertio <lb/>caſu reli-<lb/>quus angu-<lb/>lus BAC, reliquo angulo DAE, ob communem angulum EAC, ablatum <lb/>æqualis. </s> <s xml:id="echoid-s6640" xml:space="preserve">Et quoniam angulus quoq; </s> <s xml:id="echoid-s6641" xml:space="preserve">ACB, angulo ADB, æqualis eſt; </s> <s xml:id="echoid-s6642" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-189-03a" xlink:href="note-189-03"/> reliquus etiam angulus ABC, in triangulo ABC, reliquo angulo AED, in <lb/> <anchor type="note" xlink:label="note-189-04a" xlink:href="note-189-04"/> triangulo AED, æqualis. </s> <s xml:id="echoid-s6643" xml:space="preserve">Erit igitur vt AC, ad CB, ita AD, ad DE. </s> <s xml:id="echoid-s6644" xml:space="preserve">Qua-<lb/> <anchor type="note" xlink:label="note-189-05a" xlink:href="note-189-05"/> re rectangulum ſub AC, DE, æquale eſt rectangulo ſub CB, AD. </s> <s xml:id="echoid-s6645" xml:space="preserve">Rurſus <lb/> <anchor type="note" xlink:label="note-189-06a" xlink:href="note-189-06"/> quia angulus BAE, angulo DAC, ex conſtructione æqualis eſt; </s> <s xml:id="echoid-s6646" xml:space="preserve">& </s> <s xml:id="echoid-s6647" xml:space="preserve">angulus <lb/>ABD, angulo ACD: </s> <s xml:id="echoid-s6648" xml:space="preserve">erit & </s> <s xml:id="echoid-s6649" xml:space="preserve">reliquus angulus AEB, in triangulo AEB, re-<lb/> <anchor type="note" xlink:label="note-189-07a" xlink:href="note-189-07"/> <pb o="178" file="190" n="190" rhead=""/> liquo angulo ADC, in triangulo ADC, æqualis. </s> <s xml:id="echoid-s6650" xml:space="preserve">Erit igitur, vt AC, ad CD, <lb/> <anchor type="note" xlink:label="note-190-01a" xlink:href="note-190-01"/> ita AB, ad BE. </s> <s xml:id="echoid-s6651" xml:space="preserve">Quare rectangulum ſub AC, BE, æquale eſt rectangulo ſub <lb/> <anchor type="note" xlink:label="note-190-02a" xlink:href="note-190-02"/> CD, AB. </s> <s xml:id="echoid-s6652" xml:space="preserve">Quoniam igitur rectangulum ſub AC, DE, rectangulo ſub CB, <lb/> <anchor type="figure" xlink:label="fig-190-01a" xlink:href="fig-190-01"/> AD, oſten-<lb/>ſum eſt æqua <lb/>le; </s> <s xml:id="echoid-s6653" xml:space="preserve">& </s> <s xml:id="echoid-s6654" xml:space="preserve">rectãgu <lb/>lum ſub AC, <lb/>BE, rectãgu <lb/>lo ſub CD, <lb/>AB: </s> <s xml:id="echoid-s6655" xml:space="preserve">Suntau <lb/>tem rectangu <lb/>la ſub AC, <lb/>DE, & </s> <s xml:id="echoid-s6656" xml:space="preserve">ſub <lb/>AC, BE, ſimul rectangulo ſub AC, BD, æqualia; </s> <s xml:id="echoid-s6657" xml:space="preserve">erit rectangulum ſub AC, <lb/> <anchor type="note" xlink:label="note-190-03a" xlink:href="note-190-03"/> BD, rectangulis ſub BC, AD, & </s> <s xml:id="echoid-s6658" xml:space="preserve">ſub CD, AB, contentis æquale. </s> <s xml:id="echoid-s6659" xml:space="preserve">Siergo <lb/>in circulo quadrilaterum deſcribatur, &</s> <s xml:id="echoid-s6660" xml:space="preserve">c. </s> <s xml:id="echoid-s6661" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s6662" xml:space="preserve"/> </p> <div xml:id="echoid-div499" type="float" level="2" n="2"> <figure xlink:label="fig-189-01" xlink:href="fig-189-01a"> <image file="189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/189-01"/> </figure> <note position="right" xlink:label="note-189-03" xlink:href="note-189-03a" xml:space="preserve">21. tertij.</note> <note position="right" xlink:label="note-189-04" xlink:href="note-189-04a" xml:space="preserve">32. primi.</note> <note position="right" xlink:label="note-189-05" xlink:href="note-189-05a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-189-06" xlink:href="note-189-06a" xml:space="preserve">16. ſexti.</note> <note position="right" xlink:label="note-189-07" xlink:href="note-189-07a" xml:space="preserve">21. tertij.</note> <note position="left" xlink:label="note-190-01" xlink:href="note-190-01a" xml:space="preserve">4. ſexti.</note> <note position="left" xlink:label="note-190-02" xlink:href="note-190-02a" xml:space="preserve">16. ſexti.</note> <figure xlink:label="fig-190-01" xlink:href="fig-190-01a"> <image file="190-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/190-01"/> </figure> <note position="left" xlink:label="note-190-03" xlink:href="note-190-03a" xml:space="preserve">1. ſecundi.</note> </div> </div> <div xml:id="echoid-div501" type="section" level="1" n="245"> <head xml:id="echoid-head272" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s6663" xml:space="preserve">_QVANDO_ figura in circulo deſcripta eſt quadratum, vt in prima figura, fa-<lb/>cilius demonſtrabitur theorema, hoc modo. </s> <s xml:id="echoid-s6664" xml:space="preserve">_Q_uoniam rectangulum ſub _AC, BD,_ <lb/> <anchor type="note" xlink:label="note-190-04a" xlink:href="note-190-04"/> hoc eſt, quadratum ex _AC,_ (ſunt enim diametri in quadrato æquales) æquale eſt <lb/>quadratis ex _AD, DC,_ hoc eſt, rectangulis ſub _AD, <emph style="sc">B</emph>C,_ & </s> <s xml:id="echoid-s6665" xml:space="preserve">ſub _AB, DC,_ conten-<lb/> <anchor type="note" xlink:label="note-190-05a" xlink:href="note-190-05"/> tis, propter æqualitatem rectarum _AD, BC,_ & </s> <s xml:id="echoid-s6666" xml:space="preserve">_AB, DC;_ </s> <s xml:id="echoid-s6667" xml:space="preserve">liquido conſtatid, quod <lb/>proponitur.</s> <s xml:id="echoid-s6668" xml:space="preserve"/> </p> <div xml:id="echoid-div501" type="float" level="2" n="1"> <note position="left" xlink:label="note-190-04" xlink:href="note-190-04a" xml:space="preserve">Schol. 34. <lb/>lib. 1.</note> <note position="left" xlink:label="note-190-05" xlink:href="note-190-05a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div503" type="section" level="1" n="246"> <head xml:id="echoid-head273" xml:space="preserve">PROBL. 4. PROP. 12.</head> <note position="left" xml:space="preserve">Ex data dia <lb/>metro cir-<lb/>culi quo pa <lb/>cto latera <lb/>trianguli ę-<lb/>quilateri, <lb/>quadrati, <lb/>hexagoni, <lb/>pentagoni, <lb/>& dccago-<lb/>ni eiuſdem <lb/>circuli in-<lb/>ueſtigent́.</note> <p> <s xml:id="echoid-s6669" xml:space="preserve">EX data circuli diametro quotlibet particula-<lb/>rum, latera trianguli æquilateri, quadrati, hexago-<lb/>ni, pentagoni, & </s> <s xml:id="echoid-s6670" xml:space="preserve">decagoni in eodem circulo de-<lb/>ſcriptorum, in eiſdem partibus inueſtigare.</s> <s xml:id="echoid-s6671" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6672" xml:space="preserve">PONATVR diameter partium 20000000. </s> <s xml:id="echoid-s6673" xml:space="preserve">Quoniam igitur latus he-<lb/>xagoni ſemidia metro circuli eſt æquale, ipſum notum fiet partium 10000000.</s> <s xml:id="echoid-s6674" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Coroll. 15. <lb/>quarti.</note> <p> <s xml:id="echoid-s6675" xml:space="preserve">RVRSVS, quia quadratum à diametro quadrati cuiuſuis deſcriptum, <lb/>duplum eſt quadrati eiuſdem; </s> <s xml:id="echoid-s6676" xml:space="preserve">Eſt autem diameter quadrati in circulo deſcri-<lb/> <anchor type="note" xlink:label="note-190-08a" xlink:href="note-190-08"/> pti eadem, quæ circuli diameter: </s> <s xml:id="echoid-s6677" xml:space="preserve">ſi accipiatur quadratum à diametro circuli <lb/>deſcriptũ, népe 400000000000000. </s> <s xml:id="echoid-s6678" xml:space="preserve">erit dimidiũ eius, puta 200000000000000. <lb/></s> <s xml:id="echoid-s6679" xml:space="preserve">quadratum lateris quadrati, cuius radix quadrata 14142136. </s> <s xml:id="echoid-s6680" xml:space="preserve">dabit latus qua-<lb/>drati. </s> <s xml:id="echoid-s6681" xml:space="preserve">Quod hoc etiam modo reperietur. </s> <s xml:id="echoid-s6682" xml:space="preserve">Quoniam quadratum in circulo de-<lb/>ſcriptum duplum eſt quadrati à ſemidiametro deſcriptum, vt patet in trian-<lb/> <anchor type="note" xlink:label="note-190-09a" xlink:href="note-190-09"/> gulo rectangulo ADE, primę figuræ præcedẽtis propoſ. </s> <s xml:id="echoid-s6683" xml:space="preserve">ſi 100000000000000. <lb/></s> <s xml:id="echoid-s6684" xml:space="preserve">quadratum ſemidiametri duplicetur, fiet quadratum in circulo deſcriptum <lb/>partium 200000000000000. </s> <s xml:id="echoid-s6685" xml:space="preserve">cuius radix quadrata 14142136. </s> <s xml:id="echoid-s6686" xml:space="preserve">rurſus dabit la-<lb/>tus quad rati.</s> <s xml:id="echoid-s6687" xml:space="preserve"/> </p> <div xml:id="echoid-div503" type="float" level="2" n="1"> <note position="left" xlink:label="note-190-08" xlink:href="note-190-08a" xml:space="preserve">Schol. 47. <lb/>primi.</note> <note position="left" xlink:label="note-190-09" xlink:href="note-190-09a" xml:space="preserve">47. ptimi.</note> </div> <pb o="179" file="191" n="191" rhead=""/> <p> <s xml:id="echoid-s6688" xml:space="preserve">PRAETEREA, cum latus trianguli æquilateri in circulo deſcripti ſit <lb/> <anchor type="note" xlink:label="note-191-01a" xlink:href="note-191-01"/> potentia triplum ſemidiametri eiuſdem circuli, efficitur, vt quadratum ſe-<lb/>midiametri triplicatum det quadratum lateris triãguli 300000000000000. </s> <s xml:id="echoid-s6689" xml:space="preserve">cu-<lb/>ius radix quadrata idem latus exhibebit partium 17320508.</s> <s xml:id="echoid-s6690" xml:space="preserve"/> </p> <div xml:id="echoid-div504" type="float" level="2" n="2"> <note position="right" xlink:label="note-191-01" xlink:href="note-191-01a" xml:space="preserve">12. tertij-<lb/>dec.</note> </div> <p> <s xml:id="echoid-s6691" xml:space="preserve">SIT inſuper AB, ſemidiameter circuli cuiuſuis, qua diuiſa ſecundum ex-<lb/> <anchor type="note" xlink:label="note-191-02a" xlink:href="note-191-02"/> tremam ac mediam rationem <lb/> <anchor type="figure" xlink:label="fig-191-01a" xlink:href="fig-191-01"/> in C, vt maius ſegmentum ſit <lb/>BC; </s> <s xml:id="echoid-s6692" xml:space="preserve">producta autem AB, & </s> <s xml:id="echoid-s6693" xml:space="preserve"><lb/>abſciſſa BD, quæ maiori ſeg-<lb/>mento BC, ſit æqualis; </s> <s xml:id="echoid-s6694" xml:space="preserve">erit <lb/>quoq; </s> <s xml:id="echoid-s6695" xml:space="preserve">AD, in B, diuiſa ſecundum extremam ac mediam rationem, maiusq́; <lb/></s> <s xml:id="echoid-s6696" xml:space="preserve"> <anchor type="note" xlink:label="note-191-03a" xlink:href="note-191-03"/> ſegmentum erit AB: </s> <s xml:id="echoid-s6697" xml:space="preserve">quod cum ſit latus hexagoni in circulo, cuius ſemidia-<lb/>meter AB; </s> <s xml:id="echoid-s6698" xml:space="preserve">erit BD, latus decagoni in eodem circulo. </s> <s xml:id="echoid-s6699" xml:space="preserve">Quod hac ratione <lb/>notum efficietur. </s> <s xml:id="echoid-s6700" xml:space="preserve">Secta AB, bifariam in E, erit quadratum rectæ DE, com-<lb/>poſitæ ex minori ſegmento DB, & </s> <s xml:id="echoid-s6701" xml:space="preserve">dimidio BE, maioris ſegmenti BA, quin-<lb/> <anchor type="note" xlink:label="note-191-04a" xlink:href="note-191-04"/> tuplum quadrati rectæ BE, quæ cognita eſt, cum ſit ſemiſsis ſemidiametri <lb/>AB, ac proinde partium 5000000. </s> <s xml:id="echoid-s6702" xml:space="preserve">Quare ſi quadratum rectæ BE, quincupli-<lb/>cetur, fiet quadratum rectæ DE, 125000000000000. </s> <s xml:id="echoid-s6703" xml:space="preserve">cuius radix quadrata <lb/>dabit rectam DE, partium 11180340. </s> <s xml:id="echoid-s6704" xml:space="preserve">ex qua ſi dematur recta BE, partium <lb/>5000000. </s> <s xml:id="echoid-s6705" xml:space="preserve">reliquum erit BD, latus decagoni partium 6180340.</s> <s xml:id="echoid-s6706" xml:space="preserve"/> </p> <div xml:id="echoid-div505" type="float" level="2" n="3"> <note position="right" xlink:label="note-191-02" xlink:href="note-191-02a" xml:space="preserve">30.ſexti.</note> <figure xlink:label="fig-191-01" xlink:href="fig-191-01a"> <image file="191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/191-01"/> </figure> <note position="right" xlink:label="note-191-03" xlink:href="note-191-03a" xml:space="preserve">5. terrijdec. <lb/>coroll. 15.6 <lb/>Schol. 9. 13</note> <note position="right" xlink:label="note-191-04" xlink:href="note-191-04a" xml:space="preserve">3. tertijdec.</note> </div> <p> <s xml:id="echoid-s6707" xml:space="preserve">POSTREMO, quoniam pentagoni latus poteſt & </s> <s xml:id="echoid-s6708" xml:space="preserve">latus hexagoni, & </s> <s xml:id="echoid-s6709" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-191-05a" xlink:href="note-191-05"/> latus decagoni; </s> <s xml:id="echoid-s6710" xml:space="preserve">ſi quadratum lateris hexagoni 100000000000000. </s> <s xml:id="echoid-s6711" xml:space="preserve">& </s> <s xml:id="echoid-s6712" xml:space="preserve">quadra-<lb/>tum lateris decagoni 38196602515600. </s> <s xml:id="echoid-s6713" xml:space="preserve">ſimul componantur, fiet quadratum <lb/>lateris pentagoni 138196602515600. </s> <s xml:id="echoid-s6714" xml:space="preserve">cuius radix quadrata dabit latus pen-<lb/>tagoni partium 11755705. </s> <s xml:id="echoid-s6715" xml:space="preserve">Atq; </s> <s xml:id="echoid-s6716" xml:space="preserve">ita latera trianguli æquilateri, quadrati, pen <lb/>tagoni, hexagoni, & </s> <s xml:id="echoid-s6717" xml:space="preserve">decagoni nota facta ſunt in partibus diametri circuli, in <lb/>quo deſcribuntur. </s> <s xml:id="echoid-s6718" xml:space="preserve">Ex data igitur circuli diametro quotlibet particularum, <lb/>latera trianguli æquilateri, quadrati, &</s> <s xml:id="echoid-s6719" xml:space="preserve">c. </s> <s xml:id="echoid-s6720" xml:space="preserve">inueſtigauimus. </s> <s xml:id="echoid-s6721" xml:space="preserve">Quod erat fa-<lb/>ciendum.</s> <s xml:id="echoid-s6722" xml:space="preserve"/> </p> <div xml:id="echoid-div506" type="float" level="2" n="4"> <note position="right" xlink:label="note-191-05" xlink:href="note-191-05a" xml:space="preserve">10. tertij-<lb/>dec.</note> </div> </div> <div xml:id="echoid-div508" type="section" level="1" n="247"> <head xml:id="echoid-head274" xml:space="preserve">PROBL. 5. PROP. 13.</head> <note position="right" xml:space="preserve">Qua ratio-<lb/>ne ex dua-<lb/>buschordis <lb/>cognitis in <lb/>ueſtigetur <lb/>chorda dif-<lb/>ferentiæ, <lb/>qua arcus <lb/>chordarũ <lb/>datarũ in-<lb/>ter ſe diffe <lb/>runt.</note> <p> <s xml:id="echoid-s6723" xml:space="preserve">EX datis chordis duorum arcuũ inæqualium <lb/>chordam arcus, quo maior arcus minorem ſupe-<lb/>rat, inquirere.</s> <s xml:id="echoid-s6724" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6725" xml:space="preserve">IN ſemicirculo ABCD, ſint datæ chordæ AB, AC, & </s> <s xml:id="echoid-s6726" xml:space="preserve">BC, ſit chorda <lb/>arcus BC, quo maior arcus AC, minorem AB, ſuperat: </s> <s xml:id="echoid-s6727" xml:space="preserve">oporteatq́; </s> <s xml:id="echoid-s6728" xml:space="preserve">inqui-<lb/>rere chordã BC. </s> <s xml:id="echoid-s6729" xml:space="preserve">Ductis rectis BD, CD; </s> <s xml:id="echoid-s6730" xml:space="preserve">quoniam <lb/> <anchor type="note" xlink:label="note-191-07a" xlink:href="note-191-07"/> <anchor type="figure" xlink:label="fig-191-02a" xlink:href="fig-191-02"/> chordæ AB, AC, ponuntur notæ, notæ quoque <lb/>erunt chordæ BD, CD. </s> <s xml:id="echoid-s6731" xml:space="preserve">Rectangulum ergo ſub <lb/>datis rectis AB, CD, comprehenſum, notum erit: <lb/></s> <s xml:id="echoid-s6732" xml:space="preserve">Itemrectangulum ſub datis rectis AC, BD. </s> <s xml:id="echoid-s6733" xml:space="preserve">Eſt <lb/>autem rectangulum ſub rectis, AC, BD, æquale <lb/> <anchor type="note" xlink:label="note-191-08a" xlink:href="note-191-08"/> duobus rectangulis ſub AB, CD, & </s> <s xml:id="echoid-s6734" xml:space="preserve">ſub BC, AD. <lb/></s> <s xml:id="echoid-s6735" xml:space="preserve">Ablato ergo rectangulo noto ſub AB, CD, ex <lb/>rectangulo ſub AC, BD, notum ſiet reliquum rectangulum ſub BC, AD.</s> <s xml:id="echoid-s6736" xml:space="preserve"> <pb o="180" file="192" n="192" rhead=""/> quod diuiſum per diametrum AD, notam, cognitam faciet chordam BC. </s> <s xml:id="echoid-s6737" xml:space="preserve">Ex <lb/>datis ergo chordis duorum arcuum inæqualium chordam arcus, quo maior <lb/>arcus minorem ſuperat, inquiſiuimus. </s> <s xml:id="echoid-s6738" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s6739" xml:space="preserve"/> </p> <div xml:id="echoid-div508" type="float" level="2" n="1"> <note position="right" xlink:label="note-191-07" xlink:href="note-191-07a" xml:space="preserve">3. huius.</note> <figure xlink:label="fig-191-02" xlink:href="fig-191-02a"> <image file="191-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/191-02"/> </figure> <note position="right" xlink:label="note-191-08" xlink:href="note-191-08a" xml:space="preserve">11.huius.</note> </div> </div> <div xml:id="echoid-div510" type="section" level="1" n="248"> <head xml:id="echoid-head275" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s6740" xml:space="preserve">ITAQVE datis chordis duorum arcuum inæqualium, ſi maioris chorda multiplice-<lb/> <anchor type="note" xlink:label="note-192-01a" xlink:href="note-192-01"/> tur in chordam arcus, qui cum minori arcu ſemicirculum conficit, quæ quidem per 3. </s> <s xml:id="echoid-s6741" xml:space="preserve">pro-<lb/>poſ. </s> <s xml:id="echoid-s6742" xml:space="preserve">datur; </s> <s xml:id="echoid-s6743" xml:space="preserve">& </s> <s xml:id="echoid-s6744" xml:space="preserve">ex producto ſubtrahatur numerus procreatus ex minoris arcus chorda in chor <lb/>dam arcus, qui cum arcu maiori ſemicirculum complet, quæ per eandem propoſ. </s> <s xml:id="echoid-s6745" xml:space="preserve">3. </s> <s xml:id="echoid-s6746" xml:space="preserve">dat@r; <lb/></s> <s xml:id="echoid-s6747" xml:space="preserve">reliquus autem numerus per diametrum diuidatur, reddetur chorda illius arcus, quo ma-<lb/>ior arcus minorem ſuperat, nota: </s> <s xml:id="echoid-s6748" xml:space="preserve">vt ex figura & </s> <s xml:id="echoid-s6749" xml:space="preserve">demonſtratione huius propoſ, manife-<lb/>ſtum eſt.</s> <s xml:id="echoid-s6750" xml:space="preserve"/> </p> <div xml:id="echoid-div510" type="float" level="2" n="1"> <note position="left" xlink:label="note-192-01" xlink:href="note-192-01a" xml:space="preserve">Praxis.</note> </div> </div> <div xml:id="echoid-div512" type="section" level="1" n="249"> <head xml:id="echoid-head276" xml:space="preserve">PROBL. 6. PROPOS. 14.</head> <note position="left" xml:space="preserve">Qua arte ex <lb/>datis dua-<lb/>bus chordis <lb/>cognoſcaf <lb/>chorda ar-<lb/>cus cõpoſiti <lb/>ex duobus <lb/>arcubus da <lb/>tarũ duarũ <lb/>chordarũ.</note> <p> <s xml:id="echoid-s6751" xml:space="preserve">EX datis chordis duorum arcuum chordam <lb/>arcus, qui ex duobus illis arcubus componitur, in-<lb/>ueſtigare.</s> <s xml:id="echoid-s6752" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6753" xml:space="preserve">IN circulo ABCDE, cuius centrum F, datæ ſint duæ chordæ AB, BC: <lb/></s> <s xml:id="echoid-s6754" xml:space="preserve">oporteatq́; </s> <s xml:id="echoid-s6755" xml:space="preserve">inueſtigare chordam AC, arcus ABC, ex duobus arcubus AB, <lb/>BC, compoſiti. </s> <s xml:id="echoid-s6756" xml:space="preserve">Ductis duabus diametris AD, BE, & </s> <s xml:id="echoid-s6757" xml:space="preserve">rectis BD, CE, CD, <lb/>DE; </s> <s xml:id="echoid-s6758" xml:space="preserve">quoniam data eſt chorda AB, dabitur quoq; </s> <s xml:id="echoid-s6759" xml:space="preserve">chorda BD, arcus BCD, <lb/> <anchor type="note" xlink:label="note-192-03a" xlink:href="note-192-03"/> <anchor type="figure" xlink:label="fig-192-01a" xlink:href="fig-192-01"/> reliqui in ſemicirculo ABD. </s> <s xml:id="echoid-s6760" xml:space="preserve">Pariratione, quia <lb/>data eſt chorda BC, dabitur quoq; </s> <s xml:id="echoid-s6761" xml:space="preserve">chorda CE, <lb/>arcus CDE, reliqui in ſemicirculo BCE. </s> <s xml:id="echoid-s6762" xml:space="preserve">Et quia <lb/>anguli AFB, DFE, æquales ſunt, lateraq́; </s> <s xml:id="echoid-s6763" xml:space="preserve">FA, <lb/> <anchor type="note" xlink:label="note-192-04a" xlink:href="note-192-04"/> FB, lateribus FD, FE, æqualia, æquales quoque <lb/>erunt baſes AB, DE; </s> <s xml:id="echoid-s6764" xml:space="preserve">ac proinde cum AB, data <lb/> <anchor type="note" xlink:label="note-192-05a" xlink:href="note-192-05"/> ſit, data quoq; </s> <s xml:id="echoid-s6765" xml:space="preserve">erit DE. </s> <s xml:id="echoid-s6766" xml:space="preserve">Quoniam igitur rectan-<lb/>gulum ſub datis rectis BD, CE, æquale eſt rectan <lb/> <anchor type="note" xlink:label="note-192-06a" xlink:href="note-192-06"/> gulis ſub datis rectis BC, DE, & </s> <s xml:id="echoid-s6767" xml:space="preserve">ſub diametro <lb/>BE, ac recta CD: </s> <s xml:id="echoid-s6768" xml:space="preserve">ſi rectangulum ſub BC, DE, <lb/>datum auferatur ex rectangulo dato ſub BD, CE; </s> <s xml:id="echoid-s6769" xml:space="preserve">notum relinquetur rectan <lb/>gulum ſub BE, CD. </s> <s xml:id="echoid-s6770" xml:space="preserve">quo diuiſo per diametrum notam BE, nota fiet chor-<lb/>da CD; </s> <s xml:id="echoid-s6771" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s6772" xml:space="preserve">chorda AC, reliqui arcus ABC, in ſemicirculo ACD, <lb/> <anchor type="note" xlink:label="note-192-07a" xlink:href="note-192-07"/> nota erit.</s> <s xml:id="echoid-s6773" xml:space="preserve"/> </p> <div xml:id="echoid-div512" type="float" level="2" n="1"> <note position="left" xlink:label="note-192-03" xlink:href="note-192-03a" xml:space="preserve">3.huius.</note> <figure xlink:label="fig-192-01" xlink:href="fig-192-01a"> <image file="192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/192-01"/> </figure> <note position="left" xlink:label="note-192-04" xlink:href="note-192-04a" xml:space="preserve">15. primi.</note> <note position="left" xlink:label="note-192-05" xlink:href="note-192-05a" xml:space="preserve">4.primi.</note> <note position="left" xlink:label="note-192-06" xlink:href="note-192-06a" xml:space="preserve">11.huius.</note> <note position="left" xlink:label="note-192-07" xlink:href="note-192-07a" xml:space="preserve">3. huius.</note> </div> <p> <s xml:id="echoid-s6774" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s6775" xml:space="preserve">Quoniam data eſt chorda AB, dabitur etiam BD, reliqui ar-<lb/> <anchor type="note" xlink:label="note-192-08a" xlink:href="note-192-08"/> cus BCD, in ſemicirculo ABD: </s> <s xml:id="echoid-s6776" xml:space="preserve">Data eſt autem & </s> <s xml:id="echoid-s6777" xml:space="preserve">BC. </s> <s xml:id="echoid-s6778" xml:space="preserve">Igitur cum chordæ <lb/>BD, BC, datæ ſint, dabitur quoq; </s> <s xml:id="echoid-s6779" xml:space="preserve">chorda CD, arcus CD, quo maior arcus <lb/> <anchor type="note" xlink:label="note-192-09a" xlink:href="note-192-09"/> BD, minorem arcum BC, ſuperat; </s> <s xml:id="echoid-s6780" xml:space="preserve">ac proinde rurſus chorda AC, reliqui <lb/> <anchor type="note" xlink:label="note-192-10a" xlink:href="note-192-10"/> arcus ABC, in ſemicirculo ACD, dabitur. </s> <s xml:id="echoid-s6781" xml:space="preserve">Ex datis ergo chordis duorum ar <lb/>cuum chordam arcus, qui ex duobus illis arcubus componitur, inueſtigaui-<lb/>mus. </s> <s xml:id="echoid-s6782" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s6783" xml:space="preserve"/> </p> <div xml:id="echoid-div513" type="float" level="2" n="2"> <note position="left" xlink:label="note-192-08" xlink:href="note-192-08a" xml:space="preserve">3.huius.</note> <note position="left" xlink:label="note-192-09" xlink:href="note-192-09a" xml:space="preserve">13. huius.</note> <note position="left" xlink:label="note-192-10" xlink:href="note-192-10a" xml:space="preserve">3.huius.</note> </div> </div> <div xml:id="echoid-div515" type="section" level="1" n="250"> <head xml:id="echoid-head277" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s6784" xml:space="preserve">ITAQVE datis chordis duorum arcuum, ſi chordæ arcuum duorum, qui cum illis ſe-<lb/> <anchor type="note" xlink:label="note-192-11a" xlink:href="note-192-11"/> micirculos complent, quæ quidem per propoſ. </s> <s xml:id="echoid-s6785" xml:space="preserve">3. </s> <s xml:id="echoid-s6786" xml:space="preserve">dantur, inter ſe multiplicentur; </s> <s xml:id="echoid-s6787" xml:space="preserve">& </s> <s xml:id="echoid-s6788" xml:space="preserve">ex pro- <pb o="181" file="193" n="193" rhead=""/> ducto auferatur numerus procreatus ex multiplicatione duarum chordarum datarum in-<lb/>ter ſe; </s> <s xml:id="echoid-s6789" xml:space="preserve">reliquus autem numerus per diametrum diuidatur, relinquetur chorda, ex qua ſi per <lb/>propoſ. </s> <s xml:id="echoid-s6790" xml:space="preserve">3. </s> <s xml:id="echoid-s6791" xml:space="preserve">inueſtigetur chorda arcus, qui cum relictæ chordæ arcu ſemicirculum conficit, <lb/>erit hæc inuenta ſubtendens arcum compoſitum ex duobus arcubus duarum chordarum <lb/>datarum. </s> <s xml:id="echoid-s6792" xml:space="preserve">Operatio hæc perſpicua eſt ex figura, & </s> <s xml:id="echoid-s6793" xml:space="preserve">demonſtrarione priori huius propoſ.</s> <s xml:id="echoid-s6794" xml:space="preserve"/> </p> <div xml:id="echoid-div515" type="float" level="2" n="1"> <note position="left" xlink:label="note-192-11" xlink:href="note-192-11a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s6795" xml:space="preserve">EADEM hæc operatio colligi poteſt ex poſteriori demonſtratione, vt manifeſtum eſt.</s> <s xml:id="echoid-s6796" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div517" type="section" level="1" n="251"> <head xml:id="echoid-head278" xml:space="preserve">PROBL. 7. PROPOS. 15.</head> <note position="right" xml:space="preserve">Quo pa-<lb/>cto ex da-<lb/>ta chorda <lb/>reperiatur <lb/>chorda ſe-<lb/>miſſis arcus <lb/>datæ chor-<lb/>dæ.</note> <p> <s xml:id="echoid-s6797" xml:space="preserve">EX data chorda cuiuſuis arcus chordã ſemiſ-<lb/>ſis illius arcus inuenire.</s> <s xml:id="echoid-s6798" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6799" xml:space="preserve">IN circulo ABC, cuius centrum E, data ſit chorda BC, arcus BDC, cu-<lb/>ius ſemiſsis ſit arcus BD, eiusq́; </s> <s xml:id="echoid-s6800" xml:space="preserve">chorda BD, quam inuenire oporteat. </s> <s xml:id="echoid-s6801" xml:space="preserve">Ducta <lb/>diametro DG, ſecabitea, per lemma in definitionibus poſitum, rectam BC, <lb/>bifariam, ac proinde ad angulos rectos. </s> <s xml:id="echoid-s6802" xml:space="preserve">Iunctis autem rectis BA, BG; </s> <s xml:id="echoid-s6803" xml:space="preserve">erunt <lb/> <anchor type="note" xlink:label="note-193-02a" xlink:href="note-193-02"/> duo triangula ABC, EFC, æquiangula, cum angulus EFC, oſtenſus ſitre-<lb/>ctus, & </s> <s xml:id="echoid-s6804" xml:space="preserve">angulus ABC, ſit quoq; </s> <s xml:id="echoid-s6805" xml:space="preserve">rectus in ſemicirculo, at angulus C, commu <lb/> <anchor type="note" xlink:label="note-193-03a" xlink:href="note-193-03"/> nis. </s> <s xml:id="echoid-s6806" xml:space="preserve">Igitur erit, vt CF, ad FE, ita CB, ad BA: </s> <s xml:id="echoid-s6807" xml:space="preserve">& </s> <s xml:id="echoid-s6808" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-193-04a" xlink:href="note-193-04"/> <anchor type="figure" xlink:label="fig-193-01a" xlink:href="fig-193-01"/> permutando, vt CF, ad CB, ita FE, ad BA. </s> <s xml:id="echoid-s6809" xml:space="preserve">Cum <lb/>ergo CF, dimidium ſit ipſius CB, vt oſtendimus, <lb/>erit & </s> <s xml:id="echoid-s6810" xml:space="preserve">EF, dimidium ipſius AB: </s> <s xml:id="echoid-s6811" xml:space="preserve">ac propterea cum <lb/>AB, data ſit ex data BC, data quoq; </s> <s xml:id="echoid-s6812" xml:space="preserve">erit EF; </s> <s xml:id="echoid-s6813" xml:space="preserve">qua <lb/> <anchor type="note" xlink:label="note-193-05a" xlink:href="note-193-05"/> dempta ex ſemidiametro ED, nota, data erit quoq; <lb/></s> <s xml:id="echoid-s6814" xml:space="preserve">reliqua FD. </s> <s xml:id="echoid-s6815" xml:space="preserve">Quoniam vero in triangulo GBD, an-<lb/>gulus B, rectus eſt, à quo demiſſa eſt BF, ad baſim <lb/> <anchor type="note" xlink:label="note-193-06a" xlink:href="note-193-06"/> GD, perpendicularis; </s> <s xml:id="echoid-s6816" xml:space="preserve">erit recta DB, media propor <lb/> <anchor type="note" xlink:label="note-193-07a" xlink:href="note-193-07"/> tionalis inter GD, & </s> <s xml:id="echoid-s6817" xml:space="preserve">FD: </s> <s xml:id="echoid-s6818" xml:space="preserve">atq; </s> <s xml:id="echoid-s6819" xml:space="preserve">adeo rectangulum <lb/>ſub GD, FD, notis quadrato rectæ DB, æquale. <lb/></s> <s xml:id="echoid-s6820" xml:space="preserve"> <anchor type="note" xlink:label="note-193-08a" xlink:href="note-193-08"/> Notum ergo erit quadratum rectæ DB; </s> <s xml:id="echoid-s6821" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s6822" xml:space="preserve">radix eius quadrata re-<lb/>ctam DB, notam exhibebit. </s> <s xml:id="echoid-s6823" xml:space="preserve">Quam etiam ita cognoſcemus. </s> <s xml:id="echoid-s6824" xml:space="preserve">Quoniam FD, <lb/>nota facta eſt, erunt quadrata rectarum FD, FB, nota: </s> <s xml:id="echoid-s6825" xml:space="preserve">quæ cum æqualia ſint <lb/> <anchor type="note" xlink:label="note-193-09a" xlink:href="note-193-09"/> quadrato rectæ BD; </s> <s xml:id="echoid-s6826" xml:space="preserve">erit & </s> <s xml:id="echoid-s6827" xml:space="preserve">hoc quadratum notum, cuius radix quadrata ite-<lb/>rum rectam BD, efficiet notam. </s> <s xml:id="echoid-s6828" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s6829" xml:space="preserve"/> </p> <div xml:id="echoid-div517" type="float" level="2" n="1"> <note position="right" xlink:label="note-193-02" xlink:href="note-193-02a" xml:space="preserve">3. tertij.</note> <note position="right" xlink:label="note-193-03" xlink:href="note-193-03a" xml:space="preserve">31. tertij.</note> <note position="right" xlink:label="note-193-04" xlink:href="note-193-04a" xml:space="preserve">4. ſexti.</note> <figure xlink:label="fig-193-01" xlink:href="fig-193-01a"> <image file="193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/193-01"/> </figure> <note position="right" xlink:label="note-193-05" xlink:href="note-193-05a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-193-06" xlink:href="note-193-06a" xml:space="preserve">31. tertij.</note> <note position="right" xlink:label="note-193-07" xlink:href="note-193-07a" xml:space="preserve">Coroll.8.6.</note> <note position="right" xlink:label="note-193-08" xlink:href="note-193-08a" xml:space="preserve">17. ſexti.</note> <note position="right" xlink:label="note-193-09" xlink:href="note-193-09a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s6830" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s6831" xml:space="preserve">SIT rurſus in ſemicirculo ABC, data chorda BC, arcus <lb/>BDC, cuius ſemiſsis ſit arcus DC, ciusq́; </s> <s xml:id="echoid-s6832" xml:space="preserve">chorda DC, quam oporteat dari. <lb/></s> <s xml:id="echoid-s6833" xml:space="preserve">Ducta chorda AB, abſcindatur ei æqualis AE, iunganturq; </s> <s xml:id="echoid-s6834" xml:space="preserve">rectæ BD, DE; </s> <s xml:id="echoid-s6835" xml:space="preserve"><lb/>Diuiſa quoq; </s> <s xml:id="echoid-s6836" xml:space="preserve">EC, bifariam in F, demittatur recta <lb/> <anchor type="figure" xlink:label="fig-193-02a" xlink:href="fig-193-02"/> DF. </s> <s xml:id="echoid-s6837" xml:space="preserve">Quoniam igitur duo latera BA, AD, equa-<lb/>lia ſunt duobus lateribus EA, AD, anguloſq; <lb/></s> <s xml:id="echoid-s6838" xml:space="preserve">comprehendunt æquales, ob arcus æquales BD, <lb/> <anchor type="note" xlink:label="note-193-10a" xlink:href="note-193-10"/> DC; </s> <s xml:id="echoid-s6839" xml:space="preserve">erunt baſes BD, DE, æquales. </s> <s xml:id="echoid-s6840" xml:space="preserve">Eſt autem <lb/> <anchor type="note" xlink:label="note-193-11a" xlink:href="note-193-11"/> BD, recta rectæ DC, æqualis. </s> <s xml:id="echoid-s6841" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s6842" xml:space="preserve">recta DE, <lb/> <anchor type="note" xlink:label="note-193-12a" xlink:href="note-193-12"/> eidem DC, æqualis erit. </s> <s xml:id="echoid-s6843" xml:space="preserve">Quare cum duo latera <lb/>EF, FD, duobus lateribus CF, FD, æqualia ſint, <lb/>baſisq́; </s> <s xml:id="echoid-s6844" xml:space="preserve">DE, baſi DC, æqualis; </s> <s xml:id="echoid-s6845" xml:space="preserve">erunt anguli ad F, <lb/> <anchor type="note" xlink:label="note-193-13a" xlink:href="note-193-13"/> æquales, ideoq́; </s> <s xml:id="echoid-s6846" xml:space="preserve">recti. </s> <s xml:id="echoid-s6847" xml:space="preserve">Quoniam vero chorda AB, nota eſt ex data chorda BC; <lb/></s> <s xml:id="echoid-s6848" xml:space="preserve"> <anchor type="note" xlink:label="note-193-14a" xlink:href="note-193-14"/> erit quoq; </s> <s xml:id="echoid-s6849" xml:space="preserve">AE, ipſi AB, æqualis, nota: </s> <s xml:id="echoid-s6850" xml:space="preserve">qua ablata ex diametro AC, nota re-<lb/>linquetur EC; </s> <s xml:id="echoid-s6851" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s6852" xml:space="preserve">huius medietas FC. </s> <s xml:id="echoid-s6853" xml:space="preserve">Iam vero, quia CD, media <lb/> <anchor type="note" xlink:label="note-193-15a" xlink:href="note-193-15"/> <pb o="182" file="194" n="194" rhead=""/> proportionalis eſt inter AC, FC; </s> <s xml:id="echoid-s6854" xml:space="preserve">erit rectangulum ſub AC, FC, æquale <lb/> <anchor type="note" xlink:label="note-194-01a" xlink:href="note-194-01"/> quadrato rectæ CD. </s> <s xml:id="echoid-s6855" xml:space="preserve">Cum ergo illud notũ ſit, ob notas AC, FC; </s> <s xml:id="echoid-s6856" xml:space="preserve">erit & </s> <s xml:id="echoid-s6857" xml:space="preserve">qua-<lb/>dratum ex DC, notum: </s> <s xml:id="echoid-s6858" xml:space="preserve">atq; </s> <s xml:id="echoid-s6859" xml:space="preserve">adeo radix eius quadrata rectam DC, efficiet <lb/>notam. </s> <s xml:id="echoid-s6860" xml:space="preserve">Quare ex data chorda cuiuſuis arcus chordam ſemiſsis illius arcus in-<lb/>uenimus. </s> <s xml:id="echoid-s6861" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s6862" xml:space="preserve"/> </p> <div xml:id="echoid-div518" type="float" level="2" n="2"> <figure xlink:label="fig-193-02" xlink:href="fig-193-02a"> <image file="193-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/193-02"/> </figure> <note position="right" xlink:label="note-193-10" xlink:href="note-193-10a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-193-11" xlink:href="note-193-11a" xml:space="preserve">4. primi.</note> <note position="right" xlink:label="note-193-12" xlink:href="note-193-12a" xml:space="preserve">29. tertij.</note> <note position="right" xlink:label="note-193-13" xlink:href="note-193-13a" xml:space="preserve">8.primi.</note> <note position="right" xlink:label="note-193-14" xlink:href="note-193-14a" xml:space="preserve">3. huius.</note> <note position="right" xlink:label="note-193-15" xlink:href="note-193-15a" xml:space="preserve">Coroll. 8.6.</note> <note position="left" xlink:label="note-194-01" xlink:href="note-194-01a" xml:space="preserve">15.ſexti.</note> </div> </div> <div xml:id="echoid-div520" type="section" level="1" n="252"> <head xml:id="echoid-head279" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s6863" xml:space="preserve">ITAQVE, ſi per propoſ. </s> <s xml:id="echoid-s6864" xml:space="preserve">3. </s> <s xml:id="echoid-s6865" xml:space="preserve">inueniatur chorda arcus, qui cum arcu datæ chordæ ſemicir-<lb/> <anchor type="note" xlink:label="note-194-02a" xlink:href="note-194-02"/> culum conſicit; </s> <s xml:id="echoid-s6866" xml:space="preserve">inuentæ autem huius chordæ dimidium ex ſemidiametro detrahatur, & </s> <s xml:id="echoid-s6867" xml:space="preserve"><lb/>reliquus numerus in diametrum multiplicetur, dabit radix quadrata huius producti chor-<lb/>dam ſemiſsis illius arcus, cuius chorda data eſt. </s> <s xml:id="echoid-s6868" xml:space="preserve">Vel ſi reliqui illius numeri quadratum iun <lb/>gatur quadrato ſemiſsis chordæ datæ, componetur numerus, cuius radix quadrata chordam <lb/>quæſitam exhibebit cognitam. </s> <s xml:id="echoid-s6869" xml:space="preserve">Quæ quidem operatio facile colligitur ex figura, & </s> <s xml:id="echoid-s6870" xml:space="preserve">priori de-<lb/>mon ſtratione huius propoſ.</s> <s xml:id="echoid-s6871" xml:space="preserve"/> </p> <div xml:id="echoid-div520" type="float" level="2" n="1"> <note position="left" xlink:label="note-194-02" xlink:href="note-194-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s6872" xml:space="preserve">ITEM ſi per propoſ. </s> <s xml:id="echoid-s6873" xml:space="preserve">3. </s> <s xml:id="echoid-s6874" xml:space="preserve">reperiatur chorda arcus, qui cum arcu datæ chordæ ſemicircu-<lb/>lum conſicit; </s> <s xml:id="echoid-s6875" xml:space="preserve">inuenta autem hæc chorda ex dia metro detrabatur, & </s> <s xml:id="echoid-s6876" xml:space="preserve">reliqui numeri dimi-<lb/>dium in diametrum multiplicetur, dabit radix quadrata huius producti chordam ſemiſſis <lb/>illius arcus, cuius chorda data eſt. </s> <s xml:id="echoid-s6877" xml:space="preserve">Vt perſpicuum eſt ex figura, & </s> <s xml:id="echoid-s6878" xml:space="preserve">poſteriori demonſtra-<lb/>tione huius propoſ.</s> <s xml:id="echoid-s6879" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div522" type="section" level="1" n="253"> <head xml:id="echoid-head280" xml:space="preserve">PROBL. 8. PROPOS. 16.</head> <p> <s xml:id="echoid-s6880" xml:space="preserve">CHORDAS omnium arcuum ſemicirculi <lb/> <anchor type="note" xlink:label="note-194-03a" xlink:href="note-194-03"/> ſeſe ordine ſuperantium vno Minuto, in partibus <lb/>diametri in quotius particulas diſtributæ, ſuppu-<lb/>tare.</s> <s xml:id="echoid-s6881" xml:space="preserve"/> </p> <div xml:id="echoid-div522" type="float" level="2" n="1"> <note position="left" xlink:label="note-194-03" xlink:href="note-194-03a" xml:space="preserve">Qua ratio-<lb/>tione om-<lb/>niu arcuũ <lb/>chordæ ſup <lb/>putentur.</note> </div> <p> <s xml:id="echoid-s6882" xml:space="preserve">STATVAMVS, chordas omnium arcuum ſupputandas eſſe reſpectu <lb/>diametri in partes 200000. </s> <s xml:id="echoid-s6883" xml:space="preserve">diſtributæ. </s> <s xml:id="echoid-s6884" xml:space="preserve">Quod vt fiat accuratius, ponenda erit <lb/>in ſupputationibus diameter partium 20000000. </s> <s xml:id="echoid-s6885" xml:space="preserve">Ita enim fiet, vt abiectis dua-<lb/>bus primis figuris ad dexteram ex ſingulis chordis inuentis, relinquãtur chor-<lb/>dæ magis exquiſitę reſpectu diametri partium 200000. </s> <s xml:id="echoid-s6886" xml:space="preserve">quemadmodum ad ini-<lb/>tium propoſ. </s> <s xml:id="echoid-s6887" xml:space="preserve">9. </s> <s xml:id="echoid-s6888" xml:space="preserve">de Sinubus docuimus, vbi etiam addidimus, quot particula-<lb/>rum ſinus totus aſſumi debeat in ſupputatione, ſi ſinus totus in tabula plu-<lb/>rium particularum deſideretur. </s> <s xml:id="echoid-s6889" xml:space="preserve">Quod etiam de tota diametro hic intelligi <lb/>debet, ſtatuendo ſemper diametrum duplo plurium particularum in ſuppu-<lb/>titione, quam ſinum totum ibi conſtituimus.</s> <s xml:id="echoid-s6890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6891" xml:space="preserve">PRIMVM ergo omnium inuentæ ſunt in propoſ. </s> <s xml:id="echoid-s6892" xml:space="preserve">12. </s> <s xml:id="echoid-s6893" xml:space="preserve">chordæ arcuum <lb/>grad. </s> <s xml:id="echoid-s6894" xml:space="preserve">36. </s> <s xml:id="echoid-s6895" xml:space="preserve">grad. </s> <s xml:id="echoid-s6896" xml:space="preserve">60. </s> <s xml:id="echoid-s6897" xml:space="preserve">grad. </s> <s xml:id="echoid-s6898" xml:space="preserve">72. </s> <s xml:id="echoid-s6899" xml:space="preserve">grad. </s> <s xml:id="echoid-s6900" xml:space="preserve">90. </s> <s xml:id="echoid-s6901" xml:space="preserve">& </s> <s xml:id="echoid-s6902" xml:space="preserve">grad. </s> <s xml:id="echoid-s6903" xml:space="preserve">120. </s> <s xml:id="echoid-s6904" xml:space="preserve">nempe latera decagoni, he-<lb/>xagoni, pentagoni, quadrati, & </s> <s xml:id="echoid-s6905" xml:space="preserve">trianguli æquilateri, partium 6180340. <lb/></s> <s xml:id="echoid-s6906" xml:space="preserve">10000000. </s> <s xml:id="echoid-s6907" xml:space="preserve">11755705. </s> <s xml:id="echoid-s6908" xml:space="preserve">14142136. </s> <s xml:id="echoid-s6909" xml:space="preserve">17320508. </s> <s xml:id="echoid-s6910" xml:space="preserve">qualium 20000000. </s> <s xml:id="echoid-s6911" xml:space="preserve">tota diame-<lb/>ter ſtatuitur. </s> <s xml:id="echoid-s6912" xml:space="preserve">Ex chordis autem 6180340. </s> <s xml:id="echoid-s6913" xml:space="preserve">11755705. </s> <s xml:id="echoid-s6914" xml:space="preserve">arcuum grad. </s> <s xml:id="echoid-s6915" xml:space="preserve">36. </s> <s xml:id="echoid-s6916" xml:space="preserve">& </s> <s xml:id="echoid-s6917" xml:space="preserve">grad. </s> <s xml:id="echoid-s6918" xml:space="preserve"><lb/>72. </s> <s xml:id="echoid-s6919" xml:space="preserve">inuenientur chordæ arcuum reliquorum in ſemicirculo, vt grad. </s> <s xml:id="echoid-s6920" xml:space="preserve">144. </s> <s xml:id="echoid-s6921" xml:space="preserve">& </s> <s xml:id="echoid-s6922" xml:space="preserve"><lb/>grad. </s> <s xml:id="echoid-s6923" xml:space="preserve">108. </s> <s xml:id="echoid-s6924" xml:space="preserve">partium 19021130. </s> <s xml:id="echoid-s6925" xml:space="preserve">16180340. </s> <s xml:id="echoid-s6926" xml:space="preserve">vt in propoſ. </s> <s xml:id="echoid-s6927" xml:space="preserve">3. </s> <s xml:id="echoid-s6928" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s6929" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6930" xml:space="preserve">DEINDE per propoſ. </s> <s xml:id="echoid-s6931" xml:space="preserve">13. </s> <s xml:id="echoid-s6932" xml:space="preserve">eiusq́ corollarium reperientur chordæ om-<lb/>nium arcuum, qui differentiæ ſint duorum quorumlibet arcuum, quorũ chor-<lb/>dæ ſint notæ. </s> <s xml:id="echoid-s6933" xml:space="preserve">Vt ex chorda arcus grad. </s> <s xml:id="echoid-s6934" xml:space="preserve">120. </s> <s xml:id="echoid-s6935" xml:space="preserve">& </s> <s xml:id="echoid-s6936" xml:space="preserve">ex chorda arcus grad. </s> <s xml:id="echoid-s6937" xml:space="preserve">36. </s> <s xml:id="echoid-s6938" xml:space="preserve">inue-<lb/>niemus chordam arcus grad. </s> <s xml:id="echoid-s6939" xml:space="preserve">84. </s> <s xml:id="echoid-s6940" xml:space="preserve">qui illorum differentia eſt. </s> <s xml:id="echoid-s6941" xml:space="preserve">Item ex chorda <pb o="183" file="195" n="195" rhead=""/> arcus grad. </s> <s xml:id="echoid-s6942" xml:space="preserve">90. </s> <s xml:id="echoid-s6943" xml:space="preserve">& </s> <s xml:id="echoid-s6944" xml:space="preserve">ex chorda arcus grad. </s> <s xml:id="echoid-s6945" xml:space="preserve">60. </s> <s xml:id="echoid-s6946" xml:space="preserve">cognoſcemus chordam arcus grad. <lb/></s> <s xml:id="echoid-s6947" xml:space="preserve">30. </s> <s xml:id="echoid-s6948" xml:space="preserve">quo duo illi inter ſe differunt. </s> <s xml:id="echoid-s6949" xml:space="preserve">Eodemq́; </s> <s xml:id="echoid-s6950" xml:space="preserve">modo plurimorum arcuum chor-<lb/>das inueſtigabimus.</s> <s xml:id="echoid-s6951" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6952" xml:space="preserve">RVRSVS perea, quæ propoſ. </s> <s xml:id="echoid-s6953" xml:space="preserve">14. </s> <s xml:id="echoid-s6954" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s6955" xml:space="preserve">corollario demonſtrauimus, <lb/>reperiemus chordam cuiuſq; </s> <s xml:id="echoid-s6956" xml:space="preserve">arcus compoſiti ex duobus, quorum chordæ no-<lb/>tæ ſint. </s> <s xml:id="echoid-s6957" xml:space="preserve">Vt ex chorda arcus grad. </s> <s xml:id="echoid-s6958" xml:space="preserve">60. </s> <s xml:id="echoid-s6959" xml:space="preserve">& </s> <s xml:id="echoid-s6960" xml:space="preserve">ex chorda arcus grad. </s> <s xml:id="echoid-s6961" xml:space="preserve">90. </s> <s xml:id="echoid-s6962" xml:space="preserve">nota red-<lb/>detur chorda arcus grad. </s> <s xml:id="echoid-s6963" xml:space="preserve">150. </s> <s xml:id="echoid-s6964" xml:space="preserve">ex illis duobus compoſiti. </s> <s xml:id="echoid-s6965" xml:space="preserve">Sic etiam ex chor-<lb/>da arcus grad. </s> <s xml:id="echoid-s6966" xml:space="preserve">90. </s> <s xml:id="echoid-s6967" xml:space="preserve">& </s> <s xml:id="echoid-s6968" xml:space="preserve">ex chorda arcus grad. </s> <s xml:id="echoid-s6969" xml:space="preserve">36. </s> <s xml:id="echoid-s6970" xml:space="preserve">cognoſcetur chorda arcus <lb/>grad. </s> <s xml:id="echoid-s6971" xml:space="preserve">126. </s> <s xml:id="echoid-s6972" xml:space="preserve">&</s> <s xml:id="echoid-s6973" xml:space="preserve">c.</s> <s xml:id="echoid-s6974" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6975" xml:space="preserve">PRAETEREA per doctrinam propoſ. </s> <s xml:id="echoid-s6976" xml:space="preserve">15. </s> <s xml:id="echoid-s6977" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s6978" xml:space="preserve">coroll. </s> <s xml:id="echoid-s6979" xml:space="preserve">cognita chor-<lb/>da cuiuſuis arcus, cognoſcemus & </s> <s xml:id="echoid-s6980" xml:space="preserve">chordam dimidiati arcus. </s> <s xml:id="echoid-s6981" xml:space="preserve">Vt ex chorda ar-<lb/>cus grad. </s> <s xml:id="echoid-s6982" xml:space="preserve">60. </s> <s xml:id="echoid-s6983" xml:space="preserve">notam efficiemus chordam arcus grad. </s> <s xml:id="echoid-s6984" xml:space="preserve">30. </s> <s xml:id="echoid-s6985" xml:space="preserve">Ex hac chordam ar-<lb/>cus grad. </s> <s xml:id="echoid-s6986" xml:space="preserve">15. </s> <s xml:id="echoid-s6987" xml:space="preserve">Ex hac vero chordam arcus grad. </s> <s xml:id="echoid-s6988" xml:space="preserve">7. </s> <s xml:id="echoid-s6989" xml:space="preserve">Min. </s> <s xml:id="echoid-s6990" xml:space="preserve">30. </s> <s xml:id="echoid-s6991" xml:space="preserve">& </s> <s xml:id="echoid-s6992" xml:space="preserve">ex hac chordam <lb/>arcus grad. </s> <s xml:id="echoid-s6993" xml:space="preserve">3. </s> <s xml:id="echoid-s6994" xml:space="preserve">Min. </s> <s xml:id="echoid-s6995" xml:space="preserve">45. </s> <s xml:id="echoid-s6996" xml:space="preserve">Item ex chorda arcus grad. </s> <s xml:id="echoid-s6997" xml:space="preserve">72. </s> <s xml:id="echoid-s6998" xml:space="preserve">explorabimus chordam <lb/>arcus grad. </s> <s xml:id="echoid-s6999" xml:space="preserve">36. </s> <s xml:id="echoid-s7000" xml:space="preserve">Et ex hac chordã arcus grad. </s> <s xml:id="echoid-s7001" xml:space="preserve">18. </s> <s xml:id="echoid-s7002" xml:space="preserve">Et ex hac chordã arcus grad. </s> <s xml:id="echoid-s7003" xml:space="preserve">9. <lb/></s> <s xml:id="echoid-s7004" xml:space="preserve">Et ex hac chordã arcus grad. </s> <s xml:id="echoid-s7005" xml:space="preserve">4. </s> <s xml:id="echoid-s7006" xml:space="preserve">Min. </s> <s xml:id="echoid-s7007" xml:space="preserve">30. </s> <s xml:id="echoid-s7008" xml:space="preserve">Et ex hac chordam arcus grad. </s> <s xml:id="echoid-s7009" xml:space="preserve">2. </s> <s xml:id="echoid-s7010" xml:space="preserve">Min. </s> <s xml:id="echoid-s7011" xml:space="preserve"><lb/>15. </s> <s xml:id="echoid-s7012" xml:space="preserve">Sic etiam, quoniam per propoſ. </s> <s xml:id="echoid-s7013" xml:space="preserve">13. </s> <s xml:id="echoid-s7014" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s7015" xml:space="preserve">coroll. </s> <s xml:id="echoid-s7016" xml:space="preserve">ex chordis arcuum <lb/>grad. </s> <s xml:id="echoid-s7017" xml:space="preserve">36. </s> <s xml:id="echoid-s7018" xml:space="preserve">& </s> <s xml:id="echoid-s7019" xml:space="preserve">grad. </s> <s xml:id="echoid-s7020" xml:space="preserve">48. </s> <s xml:id="echoid-s7021" xml:space="preserve">cognoſcitur chorda arcus grad. </s> <s xml:id="echoid-s7022" xml:space="preserve">12. </s> <s xml:id="echoid-s7023" xml:space="preserve">quo illi duo inter ſe <lb/>differunt, cognoſcemus ex chorda arcus grad. </s> <s xml:id="echoid-s7024" xml:space="preserve">12. </s> <s xml:id="echoid-s7025" xml:space="preserve">chordam arcus grad. </s> <s xml:id="echoid-s7026" xml:space="preserve">6. </s> <s xml:id="echoid-s7027" xml:space="preserve">Et ex <lb/>hac chordam arcus grad. </s> <s xml:id="echoid-s7028" xml:space="preserve">3. </s> <s xml:id="echoid-s7029" xml:space="preserve">Ex hac chordam arcus grad. </s> <s xml:id="echoid-s7030" xml:space="preserve">1. </s> <s xml:id="echoid-s7031" xml:space="preserve">Min. </s> <s xml:id="echoid-s7032" xml:space="preserve">30. </s> <s xml:id="echoid-s7033" xml:space="preserve">Et ex hac <lb/>chordam arcus grad. </s> <s xml:id="echoid-s7034" xml:space="preserve">0. </s> <s xml:id="echoid-s7035" xml:space="preserve">Min. </s> <s xml:id="echoid-s7036" xml:space="preserve">45. </s> <s xml:id="echoid-s7037" xml:space="preserve">Deinde ſi per ea, quæ demonſtrata ſunt, ar-<lb/> <anchor type="note" xlink:label="note-195-01a" xlink:href="note-195-01"/> cuum cæterorum chordas diligẽter ex inuentis inquiramus, inueniemus chor <lb/>das omnium arcuum, qui ſe ordine continuo ſuperant Minutis 45. </s> <s xml:id="echoid-s7038" xml:space="preserve">ita vt pri-<lb/>mus arcus contineat grad. </s> <s xml:id="echoid-s7039" xml:space="preserve">0. </s> <s xml:id="echoid-s7040" xml:space="preserve">Min. </s> <s xml:id="echoid-s7041" xml:space="preserve">45. </s> <s xml:id="echoid-s7042" xml:space="preserve">ſecundus grad. </s> <s xml:id="echoid-s7043" xml:space="preserve">1. </s> <s xml:id="echoid-s7044" xml:space="preserve">Min. </s> <s xml:id="echoid-s7045" xml:space="preserve">30. </s> <s xml:id="echoid-s7046" xml:space="preserve">tertius grad. </s> <s xml:id="echoid-s7047" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s7048" xml:space="preserve">Min. </s> <s xml:id="echoid-s7049" xml:space="preserve">15. </s> <s xml:id="echoid-s7050" xml:space="preserve">vltimus deniq; </s> <s xml:id="echoid-s7051" xml:space="preserve">ſit totus ſemicirculus grad. </s> <s xml:id="echoid-s7052" xml:space="preserve">180. </s> <s xml:id="echoid-s7053" xml:space="preserve">Immo vero, ſi, vt <lb/>proxime docuimus, inuenta fuerit chorda arcus grad. </s> <s xml:id="echoid-s7054" xml:space="preserve">0. </s> <s xml:id="echoid-s7055" xml:space="preserve">Min. </s> <s xml:id="echoid-s7056" xml:space="preserve">45. </s> <s xml:id="echoid-s7057" xml:space="preserve">inueniemus <lb/>ex hac, per doctrinam propoſ. </s> <s xml:id="echoid-s7058" xml:space="preserve">14. </s> <s xml:id="echoid-s7059" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s7060" xml:space="preserve">coroll. </s> <s xml:id="echoid-s7061" xml:space="preserve">chordas omnium arcuum ſe-<lb/>ſe continue Minutis 45. </s> <s xml:id="echoid-s7062" xml:space="preserve">ſuperantium, ſi primo inueſtigemus chordam arcus <lb/>grad. </s> <s xml:id="echoid-s7063" xml:space="preserve">1. </s> <s xml:id="echoid-s7064" xml:space="preserve">Min. </s> <s xml:id="echoid-s7065" xml:space="preserve">30. </s> <s xml:id="echoid-s7066" xml:space="preserve">ex duobus arcubus Min. </s> <s xml:id="echoid-s7067" xml:space="preserve">45. </s> <s xml:id="echoid-s7068" xml:space="preserve">& </s> <s xml:id="echoid-s7069" xml:space="preserve">Min. </s> <s xml:id="echoid-s7070" xml:space="preserve">45. </s> <s xml:id="echoid-s7071" xml:space="preserve">compoſiti: </s> <s xml:id="echoid-s7072" xml:space="preserve">Deinde ve <lb/>ro chordam arcus grad. </s> <s xml:id="echoid-s7073" xml:space="preserve">2. </s> <s xml:id="echoid-s7074" xml:space="preserve">Min. </s> <s xml:id="echoid-s7075" xml:space="preserve">15. </s> <s xml:id="echoid-s7076" xml:space="preserve">qui ex duobus arcubus grad. </s> <s xml:id="echoid-s7077" xml:space="preserve">1. </s> <s xml:id="echoid-s7078" xml:space="preserve">Min. </s> <s xml:id="echoid-s7079" xml:space="preserve">30. </s> <s xml:id="echoid-s7080" xml:space="preserve">& </s> <s xml:id="echoid-s7081" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s7082" xml:space="preserve">45. </s> <s xml:id="echoid-s7083" xml:space="preserve">componitur: </s> <s xml:id="echoid-s7084" xml:space="preserve">Et poſtea chordam arcus grad. </s> <s xml:id="echoid-s7085" xml:space="preserve">3. </s> <s xml:id="echoid-s7086" xml:space="preserve">qui componitur ex <lb/>arcu grad. </s> <s xml:id="echoid-s7087" xml:space="preserve">2. </s> <s xml:id="echoid-s7088" xml:space="preserve">Min. </s> <s xml:id="echoid-s7089" xml:space="preserve">15. </s> <s xml:id="echoid-s7090" xml:space="preserve">& </s> <s xml:id="echoid-s7091" xml:space="preserve">ex arcu Min. </s> <s xml:id="echoid-s7092" xml:space="preserve">45. </s> <s xml:id="echoid-s7093" xml:space="preserve">atq; </s> <s xml:id="echoid-s7094" xml:space="preserve">ita deinceps, apponendo ſemper <lb/>arcui antecedenti arcum Min. </s> <s xml:id="echoid-s7095" xml:space="preserve">45.</s> <s xml:id="echoid-s7096" xml:space="preserve"/> </p> <div xml:id="echoid-div523" type="float" level="2" n="2"> <note position="right" xlink:label="note-195-01" xlink:href="note-195-01a" xml:space="preserve">Supputa-<lb/>tio chorda-<lb/>rũ arcuum <lb/>Min. 45. ſe-<lb/>ſe ſnperan-<lb/>tium.</note> </div> <p> <s xml:id="echoid-s7097" xml:space="preserve">POSTREMO aliorum arcuum chordas inueſtigabimus hac arte. </s> <s xml:id="echoid-s7098" xml:space="preserve">Sit in <lb/> <anchor type="note" xlink:label="note-195-02a" xlink:href="note-195-02"/> ſemicirculo ABCDE, chorda AB, arcus Min. <lb/></s> <s xml:id="echoid-s7099" xml:space="preserve"> <anchor type="figure" xlink:label="fig-195-01a" xlink:href="fig-195-01"/> 45. </s> <s xml:id="echoid-s7100" xml:space="preserve">& </s> <s xml:id="echoid-s7101" xml:space="preserve">AD, chorda arcus grad. </s> <s xml:id="echoid-s7102" xml:space="preserve">1. </s> <s xml:id="echoid-s7103" xml:space="preserve">Min. </s> <s xml:id="echoid-s7104" xml:space="preserve">30. </s> <s xml:id="echoid-s7105" xml:space="preserve">at AC, <lb/>chorda arcus grad. </s> <s xml:id="echoid-s7106" xml:space="preserve">1. </s> <s xml:id="echoid-s7107" xml:space="preserve">quæ inueſtiganda propo-<lb/>natur. </s> <s xml:id="echoid-s7108" xml:space="preserve">Quoniam igitur maior eſt proportio arcus <lb/>AC, ad arcum AB, quam chordæ AC, ad chor-<lb/> <anchor type="note" xlink:label="note-195-03a" xlink:href="note-195-03"/> dam AB: </s> <s xml:id="echoid-s7109" xml:space="preserve">Habet autem arcus AC, ad arcum AB, <lb/>proportionẽ ſesquitertiam; </s> <s xml:id="echoid-s7110" xml:space="preserve">habebit chorda AC, <lb/>ad chordam AB, proportionem minorem, quàm <lb/>ſeſquitertiam. </s> <s xml:id="echoid-s7111" xml:space="preserve">Cum ergo chorda AB, arcus Min. <lb/></s> <s xml:id="echoid-s7112" xml:space="preserve">45. </s> <s xml:id="echoid-s7113" xml:space="preserve">ex præcedentibus inuenta ſit partium ferè 130899. </s> <s xml:id="echoid-s7114" xml:space="preserve">erit chorda AC, ar-<lb/>cus grad. </s> <s xml:id="echoid-s7115" xml:space="preserve">1 (quæ nimirum ad chordam AB, hoc eſt, ad 130899. </s> <s xml:id="echoid-s7116" xml:space="preserve">minorem pro-<lb/>portionem habet, quam ſeſquitertiam.) </s> <s xml:id="echoid-s7117" xml:space="preserve">minor, quàm 174532. </s> <s xml:id="echoid-s7118" xml:space="preserve">cum hic nume <lb/> <anchor type="note" xlink:label="note-195-04a" xlink:href="note-195-04"/> rus ad illum proportionem habeat ſeſquitertiam. </s> <s xml:id="echoid-s7119" xml:space="preserve">Rurſus quia maior eſt pro-<lb/>portio arcus AD, ad arcum AC, quam chordæ AD, ad chordam AC: </s> <s xml:id="echoid-s7120" xml:space="preserve">Habet <lb/> <anchor type="note" xlink:label="note-195-05a" xlink:href="note-195-05"/> <pb o="184" file="196" n="196" rhead=""/> autem arcus AD, ad arcum AC, proportionem ſeſquialteram; </s> <s xml:id="echoid-s7121" xml:space="preserve">habebit chor <lb/>da AD, ad chordam BC, minorem proportionem, quam ſeſquialteram. </s> <s xml:id="echoid-s7122" xml:space="preserve">Cum <lb/>ergo chorda AD, arcus grad. </s> <s xml:id="echoid-s7123" xml:space="preserve">1. </s> <s xml:id="echoid-s7124" xml:space="preserve">Min. </s> <s xml:id="echoid-s7125" xml:space="preserve">30. </s> <s xml:id="echoid-s7126" xml:space="preserve">ex præcedentibus inuẽta ſit partium <lb/>ferè 261792. </s> <s xml:id="echoid-s7127" xml:space="preserve">erit chorda AC, arcus grad. </s> <s xml:id="echoid-s7128" xml:space="preserve">1. </s> <s xml:id="echoid-s7129" xml:space="preserve">(ad quam nimirum chorda AD, <lb/>hoc eſt, numerus 261792. </s> <s xml:id="echoid-s7130" xml:space="preserve">minorem proportionem habet, quam ſeſquialterã.) <lb/></s> <s xml:id="echoid-s7131" xml:space="preserve">maior, quam 174528. </s> <s xml:id="echoid-s7132" xml:space="preserve">cum ad hunc numerum numerus 261792. </s> <s xml:id="echoid-s7133" xml:space="preserve">proportio-<lb/> <anchor type="note" xlink:label="note-196-01a" xlink:href="note-196-01"/> nem ſeſquialteram habeat. </s> <s xml:id="echoid-s7134" xml:space="preserve">Conſtat igitur, chordam arcus grad. </s> <s xml:id="echoid-s7135" xml:space="preserve">1. </s> <s xml:id="echoid-s7136" xml:space="preserve">conſiſtere <lb/>inter duos hos numeros, 174532. </s> <s xml:id="echoid-s7137" xml:space="preserve">174528. </s> <s xml:id="echoid-s7138" xml:space="preserve">cum ille maior ſit, hic vero minor. <lb/></s> <s xml:id="echoid-s7139" xml:space="preserve">Statuamus ergo eam eſſe 174530. </s> <s xml:id="echoid-s7140" xml:space="preserve">inter numeros illos omnino mediam. </s> <s xml:id="echoid-s7141" xml:space="preserve">Ita <lb/>enim ſenſibiliter non differet à vera chorda arcus grad. </s> <s xml:id="echoid-s7142" xml:space="preserve">1.</s> <s xml:id="echoid-s7143" xml:space="preserve"/> </p> <div xml:id="echoid-div524" type="float" level="2" n="3"> <note position="right" xlink:label="note-195-02" xlink:href="note-195-02a" xml:space="preserve">Supputatio <lb/>chordæ ar-<lb/>cus grad. 1.</note> <figure xlink:label="fig-195-01" xlink:href="fig-195-01a"> <image file="195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/195-01"/> </figure> <note position="right" xlink:label="note-195-03" xlink:href="note-195-03a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-195-04" xlink:href="note-195-04a" xml:space="preserve">10. quinti.</note> <note position="right" xlink:label="note-195-05" xlink:href="note-195-05a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-196-01" xlink:href="note-196-01a" xml:space="preserve">10.quinti.</note> </div> <p> <s xml:id="echoid-s7144" xml:space="preserve">NON putes autem, eadem hac arte inueſtigari poſſe chordam arcus cu-<lb/>iuſuis plurium graduum ex duabus chordis notis duorum arcuum circunſtan <lb/>tium. </s> <s xml:id="echoid-s7145" xml:space="preserve">Nam cum in maioribus arcubus magna ſit differentia inter arcus, & </s> <s xml:id="echoid-s7146" xml:space="preserve">chor <lb/>das, ægre iudicari poterit, quinam numerus ex intermedijs inter duos inuẽtos <lb/>conſtitui debeat chorda arcus propoſiti. </s> <s xml:id="echoid-s7147" xml:space="preserve">Quod hoc exemplo faciemus perſpi-<lb/>cuum. </s> <s xml:id="echoid-s7148" xml:space="preserve">Sint cognitæ chordæ arcuum grad. </s> <s xml:id="echoid-s7149" xml:space="preserve">59. </s> <s xml:id="echoid-s7150" xml:space="preserve">Min. </s> <s xml:id="echoid-s7151" xml:space="preserve">46. </s> <s xml:id="echoid-s7152" xml:space="preserve">& </s> <s xml:id="echoid-s7153" xml:space="preserve">grad. </s> <s xml:id="echoid-s7154" xml:space="preserve">60. </s> <s xml:id="echoid-s7155" xml:space="preserve">Min. </s> <s xml:id="echoid-s7156" xml:space="preserve">30. <lb/></s> <s xml:id="echoid-s7157" xml:space="preserve">partium 9964712. </s> <s xml:id="echoid-s7158" xml:space="preserve">& </s> <s xml:id="echoid-s7159" xml:space="preserve">10075480. </s> <s xml:id="echoid-s7160" xml:space="preserve">Si quis igitur ex his eruere vellet chordam ar-<lb/>cus grad. </s> <s xml:id="echoid-s7161" xml:space="preserve">60. </s> <s xml:id="echoid-s7162" xml:space="preserve">ita eſſet ei progrediendum. </s> <s xml:id="echoid-s7163" xml:space="preserve">Quoniam minor eſt proportio arcus <lb/> <anchor type="note" xlink:label="note-196-02a" xlink:href="note-196-02"/> grad. </s> <s xml:id="echoid-s7164" xml:space="preserve">59. </s> <s xml:id="echoid-s7165" xml:space="preserve">Min. </s> <s xml:id="echoid-s7166" xml:space="preserve">46. </s> <s xml:id="echoid-s7167" xml:space="preserve">ad arcum grad. </s> <s xml:id="echoid-s7168" xml:space="preserve">60. </s> <s xml:id="echoid-s7169" xml:space="preserve">quam chordæ ad chordam: </s> <s xml:id="echoid-s7170" xml:space="preserve">Habet autem <lb/>9964712. </s> <s xml:id="echoid-s7171" xml:space="preserve">chorda arcus grad. </s> <s xml:id="echoid-s7172" xml:space="preserve">59. </s> <s xml:id="echoid-s7173" xml:space="preserve">Min. </s> <s xml:id="echoid-s7174" xml:space="preserve">46. </s> <s xml:id="echoid-s7175" xml:space="preserve">ad 10003614 {1698/1793}. </s> <s xml:id="echoid-s7176" xml:space="preserve">proportio-<lb/>nem eandem, quam arcus grad. </s> <s xml:id="echoid-s7177" xml:space="preserve">59. </s> <s xml:id="echoid-s7178" xml:space="preserve">Min. </s> <s xml:id="echoid-s7179" xml:space="preserve">46. </s> <s xml:id="echoid-s7180" xml:space="preserve">ad arcum grad. </s> <s xml:id="echoid-s7181" xml:space="preserve">60. </s> <s xml:id="echoid-s7182" xml:space="preserve">erit chorda ar-<lb/>cus grad. </s> <s xml:id="echoid-s7183" xml:space="preserve">60. </s> <s xml:id="echoid-s7184" xml:space="preserve">minor, quam 10003614 {1698/1793}. </s> <s xml:id="echoid-s7185" xml:space="preserve">vtpote ad quam chorda <lb/> <anchor type="note" xlink:label="note-196-03a" xlink:href="note-196-03"/> 9964712. </s> <s xml:id="echoid-s7186" xml:space="preserve">maiorem proportionem habeat, quam ad 10003614 {1698/1793}. </s> <s xml:id="echoid-s7187" xml:space="preserve">Item <lb/>quia maior eſt proportio arcus grad. </s> <s xml:id="echoid-s7188" xml:space="preserve">60. </s> <s xml:id="echoid-s7189" xml:space="preserve">Min. </s> <s xml:id="echoid-s7190" xml:space="preserve">30. </s> <s xml:id="echoid-s7191" xml:space="preserve">ad arcum grad. </s> <s xml:id="echoid-s7192" xml:space="preserve">60. </s> <s xml:id="echoid-s7193" xml:space="preserve">quam chor <lb/> <anchor type="note" xlink:label="note-196-04a" xlink:href="note-196-04"/> dæ ad chordam: </s> <s xml:id="echoid-s7194" xml:space="preserve">Habet autem 10075480. </s> <s xml:id="echoid-s7195" xml:space="preserve">chorda arcus grad. </s> <s xml:id="echoid-s7196" xml:space="preserve">60. </s> <s xml:id="echoid-s7197" xml:space="preserve">Min. </s> <s xml:id="echoid-s7198" xml:space="preserve">30. </s> <s xml:id="echoid-s7199" xml:space="preserve">ad <lb/>9992211 {69/121}. </s> <s xml:id="echoid-s7200" xml:space="preserve">eandem proportionem, quam arcus grad. </s> <s xml:id="echoid-s7201" xml:space="preserve">60. </s> <s xml:id="echoid-s7202" xml:space="preserve">Min. </s> <s xml:id="echoid-s7203" xml:space="preserve">30. </s> <s xml:id="echoid-s7204" xml:space="preserve">ad <lb/>arcum grad. </s> <s xml:id="echoid-s7205" xml:space="preserve">60. </s> <s xml:id="echoid-s7206" xml:space="preserve">erit chorda arcus grad. </s> <s xml:id="echoid-s7207" xml:space="preserve">60. </s> <s xml:id="echoid-s7208" xml:space="preserve">maior, quam 9992211 {69/121}. </s> <s xml:id="echoid-s7209" xml:space="preserve">vt-<lb/> <anchor type="note" xlink:label="note-196-05a" xlink:href="note-196-05"/> pote ad quam chorda 10075480. </s> <s xml:id="echoid-s7210" xml:space="preserve">proportionem habeat minorem, quam ad <lb/>9992211 {69/121}. </s> <s xml:id="echoid-s7211" xml:space="preserve">Conſtituenda igitur eſſet chorda arcus grad. </s> <s xml:id="echoid-s7212" xml:space="preserve">60. </s> <s xml:id="echoid-s7213" xml:space="preserve">inter hos <lb/>duos numeros 10003614 {1698/1793}. </s> <s xml:id="echoid-s7214" xml:space="preserve">9992211 {69/121}. </s> <s xml:id="echoid-s7215" xml:space="preserve">qui cum valde inter ſe <lb/>differant (eſt enim eorum differentia ferme 11403.) </s> <s xml:id="echoid-s7216" xml:space="preserve">ambiguum erit, quanta ea <lb/>ſit aſſumenda. </s> <s xml:id="echoid-s7217" xml:space="preserve">Quæ ambiguitas in perueſtigatione chordæ arcus grad. </s> <s xml:id="echoid-s7218" xml:space="preserve">1. </s> <s xml:id="echoid-s7219" xml:space="preserve">lo-<lb/>cum non habet, cum in tam paruis arcubus chordę parũ ab arcubus differant.</s> <s xml:id="echoid-s7220" xml:space="preserve"/> </p> <div xml:id="echoid-div525" type="float" level="2" n="4"> <note position="left" xlink:label="note-196-02" xlink:href="note-196-02a" xml:space="preserve">Coroll. 10. <lb/>huius.</note> <note position="left" xlink:label="note-196-03" xlink:href="note-196-03a" xml:space="preserve">10. quinti.</note> <note position="left" xlink:label="note-196-04" xlink:href="note-196-04a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-196-05" xlink:href="note-196-05a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s7221" xml:space="preserve">IAM vero inuenta chorda arcus grad. </s> <s xml:id="echoid-s7222" xml:space="preserve">1. </s> <s xml:id="echoid-s7223" xml:space="preserve">reperiemus quoq;</s> <s xml:id="echoid-s7224" xml:space="preserve">, per propoſ. <lb/></s> <s xml:id="echoid-s7225" xml:space="preserve"> <anchor type="note" xlink:label="note-196-06a" xlink:href="note-196-06"/> 15. </s> <s xml:id="echoid-s7226" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s7227" xml:space="preserve">coroll. </s> <s xml:id="echoid-s7228" xml:space="preserve">chordam arcus Min. </s> <s xml:id="echoid-s7229" xml:space="preserve">30. </s> <s xml:id="echoid-s7230" xml:space="preserve">Et ex hac chordam arcus Min. </s> <s xml:id="echoid-s7231" xml:space="preserve">15. <lb/></s> <s xml:id="echoid-s7232" xml:space="preserve">Sed hanc poſtremam etiam inueniemus per propoſ. </s> <s xml:id="echoid-s7233" xml:space="preserve">13. </s> <s xml:id="echoid-s7234" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s7235" xml:space="preserve">coroll. </s> <s xml:id="echoid-s7236" xml:space="preserve">cum ar-<lb/>cus Min. </s> <s xml:id="echoid-s7237" xml:space="preserve">15. </s> <s xml:id="echoid-s7238" xml:space="preserve">ſit differentia inter arcum grad. </s> <s xml:id="echoid-s7239" xml:space="preserve">1. </s> <s xml:id="echoid-s7240" xml:space="preserve">& </s> <s xml:id="echoid-s7241" xml:space="preserve">arcum Min. </s> <s xml:id="echoid-s7242" xml:space="preserve">45. </s> <s xml:id="echoid-s7243" xml:space="preserve">quorũ chordæ <lb/>iam ſunt cognitæ. </s> <s xml:id="echoid-s7244" xml:space="preserve">Per chordã autem arcus Min. </s> <s xml:id="echoid-s7245" xml:space="preserve">15. </s> <s xml:id="echoid-s7246" xml:space="preserve">cognoſcemus per propoſ. </s> <s xml:id="echoid-s7247" xml:space="preserve"><lb/>14. </s> <s xml:id="echoid-s7248" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s7249" xml:space="preserve">coroll. </s> <s xml:id="echoid-s7250" xml:space="preserve">chordã arcus Min. </s> <s xml:id="echoid-s7251" xml:space="preserve">30. </s> <s xml:id="echoid-s7252" xml:space="preserve">qui ex arcu Min. </s> <s xml:id="echoid-s7253" xml:space="preserve">15. </s> <s xml:id="echoid-s7254" xml:space="preserve">& </s> <s xml:id="echoid-s7255" xml:space="preserve">ex arcu Min. </s> <s xml:id="echoid-s7256" xml:space="preserve">15. </s> <s xml:id="echoid-s7257" xml:space="preserve"><lb/>componitur. </s> <s xml:id="echoid-s7258" xml:space="preserve">Item chordam arcus Min. </s> <s xml:id="echoid-s7259" xml:space="preserve">45. </s> <s xml:id="echoid-s7260" xml:space="preserve">ex arcubus Min. </s> <s xml:id="echoid-s7261" xml:space="preserve">30. </s> <s xml:id="echoid-s7262" xml:space="preserve">& </s> <s xml:id="echoid-s7263" xml:space="preserve">Min. </s> <s xml:id="echoid-s7264" xml:space="preserve">15. </s> <s xml:id="echoid-s7265" xml:space="preserve">com <lb/>poſiti. </s> <s xml:id="echoid-s7266" xml:space="preserve">Item chordam arcus grad. </s> <s xml:id="echoid-s7267" xml:space="preserve">1. </s> <s xml:id="echoid-s7268" xml:space="preserve">ex arcubus Min. </s> <s xml:id="echoid-s7269" xml:space="preserve">45. </s> <s xml:id="echoid-s7270" xml:space="preserve">& </s> <s xml:id="echoid-s7271" xml:space="preserve">Min. </s> <s xml:id="echoid-s7272" xml:space="preserve">15. </s> <s xml:id="echoid-s7273" xml:space="preserve">conflati. </s> <s xml:id="echoid-s7274" xml:space="preserve"><lb/>Et chordam arcus grad. </s> <s xml:id="echoid-s7275" xml:space="preserve">1. </s> <s xml:id="echoid-s7276" xml:space="preserve">Min. </s> <s xml:id="echoid-s7277" xml:space="preserve">15. </s> <s xml:id="echoid-s7278" xml:space="preserve">Et chordam arcus grad. </s> <s xml:id="echoid-s7279" xml:space="preserve">1. </s> <s xml:id="echoid-s7280" xml:space="preserve">Min. </s> <s xml:id="echoid-s7281" xml:space="preserve">30. </s> <s xml:id="echoid-s7282" xml:space="preserve">quam-<lb/>uis omnes hæ chordæ iam factæ ſint alia ratione notæ. </s> <s xml:id="echoid-s7283" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s7284" xml:space="preserve">hac via reperie-<lb/>mus chordas omnium arcuum ſeſe ordine Minutis 15. </s> <s xml:id="echoid-s7285" xml:space="preserve">ſuperantium: </s> <s xml:id="echoid-s7286" xml:space="preserve">quamuis <lb/>multas illarum alijs rationibus inueſtigare poſsimus, vt ex propoſ. </s> <s xml:id="echoid-s7287" xml:space="preserve">13. </s> <s xml:id="echoid-s7288" xml:space="preserve">14. </s> <s xml:id="echoid-s7289" xml:space="preserve">15. </s> <s xml:id="echoid-s7290" xml:space="preserve"><lb/>earumq́; </s> <s xml:id="echoid-s7291" xml:space="preserve">corollarijs manifeſtum eſt.</s> <s xml:id="echoid-s7292" xml:space="preserve"/> </p> <div xml:id="echoid-div526" type="float" level="2" n="5"> <note position="left" xlink:label="note-196-06" xlink:href="note-196-06a" xml:space="preserve">Supputatio <lb/>chordarũ <lb/>arcuum per <lb/>Min.15. ex-<lb/>tenſorum.</note> </div> <p> <s xml:id="echoid-s7293" xml:space="preserve">QVOD ſi ſtatuantur ordine omnes arcus ſeſe Minutis 15. </s> <s xml:id="echoid-s7294" xml:space="preserve">ſuperantes, <lb/> <anchor type="note" xlink:label="note-196-07a" xlink:href="note-196-07"/> vnà cum eorum chordis; </s> <s xml:id="echoid-s7295" xml:space="preserve">& </s> <s xml:id="echoid-s7296" xml:space="preserve">ad dexteram cuiuſuis chordæ aſcribatur differen- <pb o="185" file="197" n="197" rhead=""/> tia, qua à præcedenti chorda differt, inueniemus per regulam proportionum <lb/> <anchor type="note" xlink:label="note-197-01a" xlink:href="note-197-01"/> chordas aliorum arcuum intermediorum per quina minuta extenſorum: </s> <s xml:id="echoid-s7297" xml:space="preserve">& </s> <s xml:id="echoid-s7298" xml:space="preserve">ex <lb/>his chordas omnium arcuum per ſingula minuta progredientium; </s> <s xml:id="echoid-s7299" xml:space="preserve">quemadmo-<lb/>dum ſupra de inuẽtione ſinuum diximus. </s> <s xml:id="echoid-s7300" xml:space="preserve">Quod vt facilius intelligatur, propo <lb/>nemus hoc vnum exemplum. </s> <s xml:id="echoid-s7301" xml:space="preserve">Sit inquirenda chorda arcus Min. </s> <s xml:id="echoid-s7302" xml:space="preserve">20. </s> <s xml:id="echoid-s7303" xml:space="preserve">Quoniam <lb/>igitur differentia inter 43632. </s> <s xml:id="echoid-s7304" xml:space="preserve">chordam Min. </s> <s xml:id="echoid-s7305" xml:space="preserve">15. </s> <s xml:id="echoid-s7306" xml:space="preserve">& </s> <s xml:id="echoid-s7307" xml:space="preserve">87264. </s> <s xml:id="echoid-s7308" xml:space="preserve">chordam Min. </s> <s xml:id="echoid-s7309" xml:space="preserve">30. <lb/></s> <s xml:id="echoid-s7310" xml:space="preserve">eſt 43632. </s> <s xml:id="echoid-s7311" xml:space="preserve">Dic. </s> <s xml:id="echoid-s7312" xml:space="preserve">Si Min. </s> <s xml:id="echoid-s7313" xml:space="preserve">15. </s> <s xml:id="echoid-s7314" xml:space="preserve">quibus arcus Min. </s> <s xml:id="echoid-s7315" xml:space="preserve">15. </s> <s xml:id="echoid-s7316" xml:space="preserve">ab arcu Min. </s> <s xml:id="echoid-s7317" xml:space="preserve">30. </s> <s xml:id="echoid-s7318" xml:space="preserve">differt, @ re-<lb/>quirunt differentiam 43632. </s> <s xml:id="echoid-s7319" xml:space="preserve">adijciendam ad chordam arcus Min. </s> <s xml:id="echoid-s7320" xml:space="preserve">15. </s> <s xml:id="echoid-s7321" xml:space="preserve">vt fiat <lb/>chorda arcus Min. </s> <s xml:id="echoid-s7322" xml:space="preserve">30. </s> <s xml:id="echoid-s7323" xml:space="preserve">quantã poſtulant differentiam Minuta 5. </s> <s xml:id="echoid-s7324" xml:space="preserve">quibus arcus <lb/>Min. </s> <s xml:id="echoid-s7325" xml:space="preserve">15. </s> <s xml:id="echoid-s7326" xml:space="preserve">ab arcu Min. </s> <s xml:id="echoid-s7327" xml:space="preserve">20. </s> <s xml:id="echoid-s7328" xml:space="preserve">differt, addendam ad eandem chordam arcus Min. </s> <s xml:id="echoid-s7329" xml:space="preserve">15. </s> <s xml:id="echoid-s7330" xml:space="preserve"><lb/>vt componatur chorda arcus Min. </s> <s xml:id="echoid-s7331" xml:space="preserve">20? </s> <s xml:id="echoid-s7332" xml:space="preserve">Inuenies enim requiri differentiam <lb/>14544. </s> <s xml:id="echoid-s7333" xml:space="preserve">quæ additaad 43632. </s> <s xml:id="echoid-s7334" xml:space="preserve">chordam arcus Min. </s> <s xml:id="echoid-s7335" xml:space="preserve">15. </s> <s xml:id="echoid-s7336" xml:space="preserve">conſtituet 58176. </s> <s xml:id="echoid-s7337" xml:space="preserve">chor-<lb/>dam arcus Min. </s> <s xml:id="echoid-s7338" xml:space="preserve">20. </s> <s xml:id="echoid-s7339" xml:space="preserve">Eademq́; </s> <s xml:id="echoid-s7340" xml:space="preserve">ratio eſt de cæteris.</s> <s xml:id="echoid-s7341" xml:space="preserve"/> </p> <div xml:id="echoid-div527" type="float" level="2" n="6"> <note position="left" xlink:label="note-196-07" xlink:href="note-196-07a" xml:space="preserve">Supputatio <lb/>chordarũ</note> <note position="right" xlink:label="note-197-01" xlink:href="note-197-01a" xml:space="preserve">arcuum per <lb/>ſingula Mi <lb/>nuta exten <lb/>ſorum.</note> </div> </div> <div xml:id="echoid-div529" type="section" level="1" n="254"> <head xml:id="echoid-head281" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s7342" xml:space="preserve">SED magnum compendium nobis in hac re afferet propoſitio ſexta. </s> <s xml:id="echoid-s7343" xml:space="preserve">Nam ex ea <lb/> <anchor type="note" xlink:label="note-197-02a" xlink:href="note-197-02"/> plurimas chordas ex alijs inuentis per ſolam additionem, ſubtractionemue cõficiemus. <lb/></s> <s xml:id="echoid-s7344" xml:space="preserve">Si namq; </s> <s xml:id="echoid-s7345" xml:space="preserve">chordam cuiuſuis arcus, qui maior non ſit, quàm grad. </s> <s xml:id="echoid-s7346" xml:space="preserve">60. </s> <s xml:id="echoid-s7347" xml:space="preserve">addamus chordæ <lb/>arcus, quem arcus grad. </s> <s xml:id="echoid-s7348" xml:space="preserve">120. </s> <s xml:id="echoid-s7349" xml:space="preserve">ſumpto illo arcu ſuperat, componemus chordam arcus, <lb/>qui eodem illo arcu aſſumpto arcum grad. </s> <s xml:id="echoid-s7350" xml:space="preserve">120. </s> <s xml:id="echoid-s7351" xml:space="preserve">excedit: </s> <s xml:id="echoid-s7352" xml:space="preserve">propterea quòd differentia <lb/>inter chordas duorum horum arcuum maiorum æqualis eſt chordæ arcus illius aſſum-<lb/>pti, qui maior non ponitur, quàm grad. </s> <s xml:id="echoid-s7353" xml:space="preserve">60. </s> <s xml:id="echoid-s7354" xml:space="preserve">vtibi oſtendimus. </s> <s xml:id="echoid-s7355" xml:space="preserve">Vt ſi 3472964. </s> <s xml:id="echoid-s7356" xml:space="preserve">chor-<lb/>dam arcus grad. </s> <s xml:id="echoid-s7357" xml:space="preserve">20. </s> <s xml:id="echoid-s7358" xml:space="preserve">adijciamus ad 15320890. </s> <s xml:id="echoid-s7359" xml:space="preserve">chordam arcus grad. </s> <s xml:id="echoid-s7360" xml:space="preserve">100. </s> <s xml:id="echoid-s7361" xml:space="preserve">quem arcus <lb/>grad. </s> <s xml:id="echoid-s7362" xml:space="preserve">120. </s> <s xml:id="echoid-s7363" xml:space="preserve">ſuperat dicto arcugrad. </s> <s xml:id="echoid-s7364" xml:space="preserve">20. </s> <s xml:id="echoid-s7365" xml:space="preserve">cõponemus 18793854. </s> <s xml:id="echoid-s7366" xml:space="preserve">chordã arcus grad. </s> <s xml:id="echoid-s7367" xml:space="preserve">140. </s> <s xml:id="echoid-s7368" xml:space="preserve"><lb/>qui arcum grad. </s> <s xml:id="echoid-s7369" xml:space="preserve">120 codem arcu grad. </s> <s xml:id="echoid-s7370" xml:space="preserve">20. </s> <s xml:id="echoid-s7371" xml:space="preserve">excedit. </s> <s xml:id="echoid-s7372" xml:space="preserve">Ita quoque, ſi 10000000. </s> <s xml:id="echoid-s7373" xml:space="preserve">chor-<lb/>dam arcus grad. </s> <s xml:id="echoid-s7374" xml:space="preserve">60. </s> <s xml:id="echoid-s7375" xml:space="preserve">addamus chordæ 10000000. </s> <s xml:id="echoid-s7376" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s7377" xml:space="preserve">60. </s> <s xml:id="echoid-s7378" xml:space="preserve">quem arcus grad. </s> <s xml:id="echoid-s7379" xml:space="preserve"><lb/>120. </s> <s xml:id="echoid-s7380" xml:space="preserve">dicto illo arcu grad. </s> <s xml:id="echoid-s7381" xml:space="preserve">60. </s> <s xml:id="echoid-s7382" xml:space="preserve">ſuperat, conficiemus 20000000. </s> <s xml:id="echoid-s7383" xml:space="preserve">chordam arcus grad. </s> <s xml:id="echoid-s7384" xml:space="preserve"><lb/>180. </s> <s xml:id="echoid-s7385" xml:space="preserve">qui arcum grad. </s> <s xml:id="echoid-s7386" xml:space="preserve">120. </s> <s xml:id="echoid-s7387" xml:space="preserve">eodem illo arcu grad. </s> <s xml:id="echoid-s7388" xml:space="preserve">60. </s> <s xml:id="echoid-s7389" xml:space="preserve">ſuperat.</s> <s xml:id="echoid-s7390" xml:space="preserve"/> </p> <div xml:id="echoid-div529" type="float" level="2" n="1"> <note position="right" xlink:label="note-197-02" xlink:href="note-197-02a" xml:space="preserve">Compen-<lb/>dium miri <lb/>ficũ pro in <lb/>uentione <lb/>plurim arũ <lb/>chordarũ.</note> </div> <p style="it"> <s xml:id="echoid-s7391" xml:space="preserve">ITEM ſichordam cuiuslibet arcus, qui arcu grad. </s> <s xml:id="echoid-s7392" xml:space="preserve">60. </s> <s xml:id="echoid-s7393" xml:space="preserve">maior non ſit, ſubtraha-<lb/>mus ex chorda arcus, qui arcum grad. </s> <s xml:id="echoid-s7394" xml:space="preserve">120 ſumpto illo arcu ſuperat, relinquetur chor <lb/>da arcus, quem arcus grad. </s> <s xml:id="echoid-s7395" xml:space="preserve">120. </s> <s xml:id="echoid-s7396" xml:space="preserve">eodem illo arcu aſſumpto excedit. </s> <s xml:id="echoid-s7397" xml:space="preserve">Vt ſi 3472964. <lb/></s> <s xml:id="echoid-s7398" xml:space="preserve">chordam arcus grad. </s> <s xml:id="echoid-s7399" xml:space="preserve">20. </s> <s xml:id="echoid-s7400" xml:space="preserve">detrahamus ex 18793854. </s> <s xml:id="echoid-s7401" xml:space="preserve">chorda arcus grad. </s> <s xml:id="echoid-s7402" xml:space="preserve">140. </s> <s xml:id="echoid-s7403" xml:space="preserve">qui ar-<lb/>cum grad. </s> <s xml:id="echoid-s7404" xml:space="preserve">120. </s> <s xml:id="echoid-s7405" xml:space="preserve">ſuperat arcu illo ſumpto grad. </s> <s xml:id="echoid-s7406" xml:space="preserve">20 remanebit 15320890. </s> <s xml:id="echoid-s7407" xml:space="preserve">chorda arcus <lb/>grad. </s> <s xml:id="echoid-s7408" xml:space="preserve">100. </s> <s xml:id="echoid-s7409" xml:space="preserve">qui eodem illo arcu grad. </s> <s xml:id="echoid-s7410" xml:space="preserve">20. </s> <s xml:id="echoid-s7411" xml:space="preserve">abarcu grad. </s> <s xml:id="echoid-s7412" xml:space="preserve">120. </s> <s xml:id="echoid-s7413" xml:space="preserve">ſuperatur.</s> <s xml:id="echoid-s7414" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7415" xml:space="preserve">RVRSVS ſiex chorda cuiuſuis arcus, qui maior ſit arcu grad. </s> <s xml:id="echoid-s7416" xml:space="preserve">120 ſubducatur <lb/>chorda arcus, qui tanto minor ſit arcu grad. </s> <s xml:id="echoid-s7417" xml:space="preserve">120. </s> <s xml:id="echoid-s7418" xml:space="preserve">quanto ille maior eſt, reliqua erit <lb/>chorda arcus, quo vteruis illorum ab arcu grad. </s> <s xml:id="echoid-s7419" xml:space="preserve">120. </s> <s xml:id="echoid-s7420" xml:space="preserve">differt. </s> <s xml:id="echoid-s7421" xml:space="preserve">Vt ſi ex 18793854. <lb/></s> <s xml:id="echoid-s7422" xml:space="preserve">chorda arcus grad. </s> <s xml:id="echoid-s7423" xml:space="preserve">140. </s> <s xml:id="echoid-s7424" xml:space="preserve">auferamus 15320890. </s> <s xml:id="echoid-s7425" xml:space="preserve">chordam arcus grad. </s> <s xml:id="echoid-s7426" xml:space="preserve">100. </s> <s xml:id="echoid-s7427" xml:space="preserve">relinque-<lb/>tur 3472964. </s> <s xml:id="echoid-s7428" xml:space="preserve">chorda arcus grad. </s> <s xml:id="echoid-s7429" xml:space="preserve">20. </s> <s xml:id="echoid-s7430" xml:space="preserve">quo vterq; </s> <s xml:id="echoid-s7431" xml:space="preserve">illorum ab arcu grad. </s> <s xml:id="echoid-s7432" xml:space="preserve">120. </s> <s xml:id="echoid-s7433" xml:space="preserve">differt. </s> <s xml:id="echoid-s7434" xml:space="preserve"><lb/>Quæ omnia ex dicta propoſ. </s> <s xml:id="echoid-s7435" xml:space="preserve">6. </s> <s xml:id="echoid-s7436" xml:space="preserve">colliguntur.</s> <s xml:id="echoid-s7437" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7438" xml:space="preserve">SATIS ergo eſt, vt per regulam proportionum inueſtigentur chordæ omnium ar <lb/>cuum à principio ſemicirculi vſque ad arcum grad. </s> <s xml:id="echoid-s7439" xml:space="preserve">60. </s> <s xml:id="echoid-s7440" xml:space="preserve">Si enim ex his reperiantur <lb/>chordæ arcuum, qui cum illis ſemicirculum conficiant, & </s> <s xml:id="echoid-s7441" xml:space="preserve">ex his repertis ſubducantur <lb/>priores illæ inuentæ, remanebunt chordæ omnium arcuum inter arcum grad. </s> <s xml:id="echoid-s7442" xml:space="preserve">60. </s> <s xml:id="echoid-s7443" xml:space="preserve">& </s> <s xml:id="echoid-s7444" xml:space="preserve"><lb/>arcum grad. </s> <s xml:id="echoid-s7445" xml:space="preserve">120. </s> <s xml:id="echoid-s7446" xml:space="preserve">Item ſinotæ eſſent chordæ omnium arcuum ab arcu grad. </s> <s xml:id="echoid-s7447" xml:space="preserve">60. </s> <s xml:id="echoid-s7448" xml:space="preserve">vſq; <lb/></s> <s xml:id="echoid-s7449" xml:space="preserve">ad finem ſemicirculi, & </s> <s xml:id="echoid-s7450" xml:space="preserve">chordæ omnium arcuũ, qui minores ſint arcu grad 120. </s> <s xml:id="echoid-s7451" xml:space="preserve">au- <pb o="186" file="198" n="198" rhead=""/> ferrentur ex chordis omnium arcuũ maiorũ, quàm grad. </s> <s xml:id="echoid-s7452" xml:space="preserve">120. </s> <s xml:id="echoid-s7453" xml:space="preserve">reliquæ fierent chordæ <lb/>mnium arcuum à principio ſemicirculi vſque ad arcum grad. </s> <s xml:id="echoid-s7454" xml:space="preserve">60. </s> <s xml:id="echoid-s7455" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s7456" xml:space="preserve">ſi chordæ om <lb/>nium arcuum à principio ſemicirculi vſque ad arcum grad. </s> <s xml:id="echoid-s7457" xml:space="preserve">120. </s> <s xml:id="echoid-s7458" xml:space="preserve">inuentæ eſſent, & </s> <s xml:id="echoid-s7459" xml:space="preserve"><lb/>chordæ omnium arcuum minorum, quàm 60. </s> <s xml:id="echoid-s7460" xml:space="preserve">grad. </s> <s xml:id="echoid-s7461" xml:space="preserve">chordis omnium arcuum maiorum <lb/>quàm grad. </s> <s xml:id="echoid-s7462" xml:space="preserve">60. </s> <s xml:id="echoid-s7463" xml:space="preserve">adijcerentur, componerentur chordæ omnium arcuum maiorum, <lb/>quàm grad. </s> <s xml:id="echoid-s7464" xml:space="preserve">120.</s> <s xml:id="echoid-s7465" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7466" xml:space="preserve">PORRO ſi chordæ omnium arcuum ſemicir culi in tabulam redigantur, & </s> <s xml:id="echoid-s7467" xml:space="preserve">chor <lb/> <anchor type="note" xlink:label="note-198-01a" xlink:href="note-198-01"/> dæ omnium arcuum, qui ſecontinuè Minutis 2. </s> <s xml:id="echoid-s7468" xml:space="preserve">excedunt, ſecentur bifariam, inuenti <lb/>erunt ſinus omnium arcuum per ſingula Minuta progredientiũ, vt ex defin. </s> <s xml:id="echoid-s7469" xml:space="preserve">3. </s> <s xml:id="echoid-s7470" xml:space="preserve">conſtat.</s> <s xml:id="echoid-s7471" xml:space="preserve"/> </p> <div xml:id="echoid-div530" type="float" level="2" n="2"> <note position="left" xlink:label="note-198-01" xlink:href="note-198-01a" xml:space="preserve">Qua rõne <lb/>ſinus oium <lb/>a@cuum ex <lb/>cho@dis in-<lb/>ueniantur.</note> </div> <p style="it"> <s xml:id="echoid-s7472" xml:space="preserve">HINC conſtat, rectius feciſſe recentiores, qui ſinuum tabulã confecerunt, quàm <lb/>Ptolemæum & </s> <s xml:id="echoid-s7473" xml:space="preserve">veteres, qui chordas in tabulam redegerunt. </s> <s xml:id="echoid-s7474" xml:space="preserve">Pro ſinubus enim ſatis eſt, <lb/> <anchor type="note" xlink:label="note-198-02a" xlink:href="note-198-02"/> ſi Quadrans circuli per ſingula Minuta extendatur: </s> <s xml:id="echoid-s7475" xml:space="preserve">at pro chordis neceſſe eſt totum <lb/>ſemicir culum per Minuia ſingula extendere: </s> <s xml:id="echoid-s7476" xml:space="preserve">ita vt chordarum tabula ſit duplo maior <lb/>quàm tabula ſinuum. </s> <s xml:id="echoid-s7477" xml:space="preserve">Taceo, multo expeditiorem, breuiorem, fac@ lioremq́; </s> <s xml:id="echoid-s7478" xml:space="preserve">eſſe ſinuum, <lb/>vjum in rebus Aſtronomicis, & </s> <s xml:id="echoid-s7479" xml:space="preserve">Geometricis, quàm chordarum: </s> <s xml:id="echoid-s7480" xml:space="preserve">vt ijs eſt manifeſtum, <lb/>qui ſeſe in bu@@ ſmodi vſu exercue; </s> <s xml:id="echoid-s7481" xml:space="preserve">unt aliquando.</s> <s xml:id="echoid-s7482" xml:space="preserve"/> </p> <div xml:id="echoid-div531" type="float" level="2" n="3"> <note position="left" xlink:label="note-198-02" xlink:href="note-198-02a" xml:space="preserve">Rectiꝰ face <lb/>re eos, ꝗ @a <lb/>bulã ſinuũ <lb/>cõſtruunt, <lb/>quàm qu@ <lb/>chordarũ.</note> </div> <p style="it"> <s xml:id="echoid-s7483" xml:space="preserve">QVEMADMODVM autem ſinuũ differentiæ ſenſim decreſcunt à principie <lb/> <anchor type="note" xlink:label="note-198-03a" xlink:href="note-198-03"/> quad@antis vſque ad eius finem; </s> <s xml:id="echoid-s7484" xml:space="preserve">ita vt, poſitis pluribus arcubus æqualiter ſeſe exc@-<lb/>dentibus, minorum ſinus habeani maiores differentias, quàm ſinus maiorum, vt in co-<lb/>roll. </s> <s xml:id="echoid-s7485" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7486" xml:space="preserve">1 oſtendimus@ ita quoque chordarum differentiæ paulatim decreſcunt à <lb/>principio ſemicirculiad eius finem vſq. </s> <s xml:id="echoid-s7487" xml:space="preserve">Nam poſitis pluribus arcubus, quorum æqua-<lb/>les ſint differentiæ, minorum chordæ maiores habent differentias, quàm chordæ ma-<lb/>iorum. </s> <s xml:id="echoid-s7488" xml:space="preserve">quod ita demonſtrabimus. </s> <s xml:id="echoid-s7489" xml:space="preserve">Sint in ſemicirculo <emph style="sc">ABCDe</emph>, arcus AB, AC, <lb/>AD, quorum differentiæ <emph style="sc">B</emph>C, CD, æquales ſint, <lb/> <anchor type="figure" xlink:label="fig-198-01a" xlink:href="fig-198-01"/> chordæ autem eorundem AB, AC, AD: </s> <s xml:id="echoid-s7490" xml:space="preserve">abſeinda-<lb/>turq; </s> <s xml:id="echoid-s7491" xml:space="preserve">recta <emph style="sc">Af</emph>, chordæ AB, æqualis; </s> <s xml:id="echoid-s7492" xml:space="preserve">& </s> <s xml:id="echoid-s7493" xml:space="preserve">recta AG, <lb/>chordæ AC, æqualis. </s> <s xml:id="echoid-s7494" xml:space="preserve">Dico FC, differentiam inter <lb/>chordas AB, AC, maiorem eſſe, quàm GD, diffe-<lb/>rentiam inter chordas AC, AD. </s> <s xml:id="echoid-s7495" xml:space="preserve">Abſeiſſa enim re-<lb/>cta AH, æquali ipſi AF, vel ipſi AB, iunctiſque re-<lb/>ctis BC, CD, CG, CH: </s> <s xml:id="echoid-s7496" xml:space="preserve">quoniam latera BA, AC, <lb/>lateribus HA, AC, æqualia ſunt, angulosq́; </s> <s xml:id="echoid-s7497" xml:space="preserve">con-<lb/>tinent æquales ad A, propter æquales arcus BC, CD; </s> <s xml:id="echoid-s7498" xml:space="preserve">erunt baſes BC, CD; </s> <s xml:id="echoid-s7499" xml:space="preserve">æqua-<lb/> <anchor type="note" xlink:label="note-198-04a" xlink:href="note-198-04"/> les. </s> <s xml:id="echoid-s7500" xml:space="preserve">Eſt autem recta BC, rectæ CD, æqualis, ob æquales arcus eoſdem BC, CD. </s> <s xml:id="echoid-s7501" xml:space="preserve">Igi-<lb/> <anchor type="note" xlink:label="note-198-05a" xlink:href="note-198-05"/> tur & </s> <s xml:id="echoid-s7502" xml:space="preserve">recta CH, rectæ CD, æqualis erit. </s> <s xml:id="echoid-s7503" xml:space="preserve">Anguli ergo CHD, CDH, æquales quo-<lb/> <anchor type="note" xlink:label="note-198-06a" xlink:href="note-198-06"/> que erunt: </s> <s xml:id="echoid-s7504" xml:space="preserve">qui cum ſint duobus rectis minores; </s> <s xml:id="echoid-s7505" xml:space="preserve">erit vterque eorum acutus. </s> <s xml:id="echoid-s7506" xml:space="preserve">Eodem pa-<lb/> <anchor type="note" xlink:label="note-198-07a" xlink:href="note-198-07"/> cto erit vterque angulorum ACG, <emph style="sc">AGc</emph>, acutus: </s> <s xml:id="echoid-s7507" xml:space="preserve">propterea quod inter ſe etiam <lb/>æqu@les ſunt ob æquales rectas AC, AG. </s> <s xml:id="echoid-s7508" xml:space="preserve">Quia igitur in triangulo CGH, anguli <lb/> <anchor type="note" xlink:label="note-198-08a" xlink:href="note-198-08"/> ad G, H, acuti ſunt; </s> <s xml:id="echoid-s7509" xml:space="preserve">fit, vt CI, ducta ad DH, perpendicularis cadat intra triangu-<lb/>lum in r@ctam GH, vt in ſeholio propoſ. </s> <s xml:id="echoid-s7510" xml:space="preserve">13. </s> <s xml:id="echoid-s7511" xml:space="preserve">lib. </s> <s xml:id="echoid-s7512" xml:space="preserve">2. </s> <s xml:id="echoid-s7513" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7514" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s7515" xml:space="preserve">Itaque quia <lb/>duo anguli CHI, CIH, trianguli CHI, duobus angulis CDI, CID, æquales <lb/>ſunt, ſuntq; </s> <s xml:id="echoid-s7516" xml:space="preserve">duo latera CH, CD, æqualibus rectis angulis oppoſita æqualia, vel <lb/>certe latus CI, commune eſt; </s> <s xml:id="echoid-s7517" xml:space="preserve">erunt quoque latera IH, ID, æqualia. </s> <s xml:id="echoid-s7518" xml:space="preserve">Cum ergo ID, <lb/> <anchor type="note" xlink:label="note-198-09a" xlink:href="note-198-09"/> maior ſit, quàm GD, erit quoq; </s> <s xml:id="echoid-s7519" xml:space="preserve">HI, & </s> <s xml:id="echoid-s7520" xml:space="preserve">à fortiori <emph style="sc">Hg</emph>, maior, quam GD. </s> <s xml:id="echoid-s7521" xml:space="preserve">Eſt au-<lb/>tem HG, ipſi FC, æqualis, propterea quòd & </s> <s xml:id="echoid-s7522" xml:space="preserve">rectæ AC, AG, inter ſe, & </s> <s xml:id="echoid-s7523" xml:space="preserve">rectæ <emph style="sc">Af</emph>, <lb/>AH, inter ſe æquales ſunt. </s> <s xml:id="echoid-s7524" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s7525" xml:space="preserve">FC, differentia chordarum AB, AC, maior <lb/>erit, quàm GD, differentia chordarum AC, AD. </s> <s xml:id="echoid-s7526" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s7527" xml:space="preserve"/> </p> <div xml:id="echoid-div532" type="float" level="2" n="4"> <note position="left" xlink:label="note-198-03" xlink:href="note-198-03a" xml:space="preserve">Differẽ@iæ <lb/>chordarũ à <lb/>prin c@pio <lb/>ſemici@culi <lb/>vſq; ad e@@s <lb/>finé ſen ſim <lb/>decreſcunt@ <lb/>ita vt chor-<lb/>dæ minorũ <lb/>arcuũmaio <lb/>res habeant <lb/>differentias <lb/>quàm chor <lb/>dæ arcuum <lb/>maiorũ dũ <lb/>modo arcꝰ <lb/>habeãt dif-<lb/>ferentias æ <lb/>quales.</note> <figure xlink:label="fig-198-01" xlink:href="fig-198-01a"> <image file="198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/198-01"/> </figure> <note position="left" xlink:label="note-198-04" xlink:href="note-198-04a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-198-05" xlink:href="note-198-05a" xml:space="preserve">29.tertij.</note> <note position="left" xlink:label="note-198-06" xlink:href="note-198-06a" xml:space="preserve">5. primi.</note> <note position="left" xlink:label="note-198-07" xlink:href="note-198-07a" xml:space="preserve">17.primi.</note> <note position="left" xlink:label="note-198-08" xlink:href="note-198-08a" xml:space="preserve">5.primi.</note> <note position="left" xlink:label="note-198-09" xlink:href="note-198-09a" xml:space="preserve">26.primi.</note> </div> <pb o="187" file="199" n="199" rhead=""/> <p style="it"> <s xml:id="echoid-s7528" xml:space="preserve">NEC vero prætereundum eſt, ſi, poſita diametro par <lb/> <anchor type="note" xlink:label="note-199-01a" xlink:href="note-199-01"/> tium 120 chordas arcuum inquiramus in partibus, Mi-<lb/>nutis, & </s> <s xml:id="echoid-s7529" xml:space="preserve">Secundis, vt Ptolemæus fecit, chordas arcuum <lb/>minorum, quàm grad. </s> <s xml:id="echoid-s7530" xml:space="preserve">60. </s> <s xml:id="echoid-s7531" xml:space="preserve">habere plures Partes, Minu@a, <lb/>ac Secunda: </s> <s xml:id="echoid-s7532" xml:space="preserve">quam arcus, quorum ſunt chordæ: </s> <s xml:id="echoid-s7533" xml:space="preserve">at ve-<lb/>ro chordas arcuum maiorum, quàm grad 60. </s> <s xml:id="echoid-s7534" xml:space="preserve">eſſe pau-<lb/>ciorum Partium, Minutorum, ac Secundorum, quàm ar-<lb/>cus illis reſpondentes. </s> <s xml:id="echoid-s7535" xml:space="preserve">Vt in tabula hic appoſita manife-<lb/>ſtum eſt, quam ex Ptolemæi tabula exerceptam huc tran-<lb/>ſtuli. </s> <s xml:id="echoid-s7536" xml:space="preserve">In qua cernis chordas arcuum minorum, quàm <lb/>grad. </s> <s xml:id="echoid-s7537" xml:space="preserve">60. </s> <s xml:id="echoid-s7538" xml:space="preserve">maiores eſſe, quoad numerum partium, Mi-<lb/>nutorum, & </s> <s xml:id="echoid-s7539" xml:space="preserve">ſecundorum, quàm arcus reſpondentes, <lb/>quoad gradus, ac Minuta; </s> <s xml:id="echoid-s7540" xml:space="preserve">chordas vero arcuum ma-<lb/>iorum, quàm grad. </s> <s xml:id="echoid-s7541" xml:space="preserve">60. </s> <s xml:id="echoid-s7542" xml:space="preserve">eſſe minores arcubus reſponden-<lb/>tibus, quoad eoſdem numeros. </s> <s xml:id="echoid-s7543" xml:space="preserve">Cuius quidem rei hæc eſt <lb/>demonſtratio.</s> <s xml:id="echoid-s7544" xml:space="preserve"/> </p> <div xml:id="echoid-div533" type="float" level="2" n="5"> <note position="right" xlink:label="note-199-01" xlink:href="note-199-01a" xml:space="preserve"> <lb/> ## Arcus ### Chordæ <lb/>G # M # Par. # M # Sec. <lb/>0 # 30 # 0 # 31 # 25 <lb/>10 # 0 # 10 # 27 # 32 <lb/>20 # 0 # 20 # 50 # 16 <lb/>45 # 0 # 45 # 55 # 19 <lb/>50 # 0 # 50 # 42 # 51 <lb/>60 # 0 # 60 # 0 # 0 <lb/>66 # 30 # 65 # 47 # 43 <lb/>80 # 30 # 77 # 32 # 6 <lb/>90 # 0 # 84 # 51 # 10 <lb/>120 # 0 # 103 # 55 # 23 <lb/>170 # 0 # 119 # 32 # 37 <lb/></note> </div> <p> <s xml:id="echoid-s7545" xml:space="preserve">IN ſemicirculo ABCDE, ſit arcus AC, grad. </s> <s xml:id="echoid-s7546" xml:space="preserve">60. <lb/></s> <s xml:id="echoid-s7547" xml:space="preserve">& </s> <s xml:id="echoid-s7548" xml:space="preserve"><emph style="sc">Ab</emph>, minor, nempe grad. </s> <s xml:id="echoid-s7549" xml:space="preserve">45. </s> <s xml:id="echoid-s7550" xml:space="preserve">at AD, maior, puta grad. </s> <s xml:id="echoid-s7551" xml:space="preserve"><lb/>80. </s> <s xml:id="echoid-s7552" xml:space="preserve">Min. </s> <s xml:id="echoid-s7553" xml:space="preserve">30. </s> <s xml:id="echoid-s7554" xml:space="preserve">ducanturque chordæ AB, <emph style="sc">Ac</emph>, AD. </s> <s xml:id="echoid-s7555" xml:space="preserve">Dico chordam AB, maiorem eſſe, <lb/>quàm Par. </s> <s xml:id="echoid-s7556" xml:space="preserve">45. </s> <s xml:id="echoid-s7557" xml:space="preserve">at chordam AD, minorem, quàm Partiũ <lb/> <anchor type="figure" xlink:label="fig-199-01a" xlink:href="fig-199-01"/> 80. </s> <s xml:id="echoid-s7558" xml:space="preserve">Min. </s> <s xml:id="echoid-s7559" xml:space="preserve">30. </s> <s xml:id="echoid-s7560" xml:space="preserve">Cumenim maior ſit proportio arcus AC, <lb/> <anchor type="note" xlink:label="note-199-02a" xlink:href="note-199-02"/> ad arcum AB, quam chordæ AC, ad chordam <emph style="sc">Ab</emph>: </s> <s xml:id="echoid-s7561" xml:space="preserve">ſit <lb/>autem proportio arcus AC, ad arcum AB, eadem, quæ <lb/>60. </s> <s xml:id="echoid-s7562" xml:space="preserve">ad 45. </s> <s xml:id="echoid-s7563" xml:space="preserve">erit proportio chordæ AC, ad chordam <emph style="sc">Ab</emph>, <lb/>minor, quàm 60. </s> <s xml:id="echoid-s7564" xml:space="preserve">ad 45. </s> <s xml:id="echoid-s7565" xml:space="preserve">Quare cum chorda <emph style="sc">A</emph>C, ſit <lb/>Partium 60. </s> <s xml:id="echoid-s7566" xml:space="preserve">vtpote quæ ſemidiametro æqualis eſt, per <lb/>coroll. </s> <s xml:id="echoid-s7567" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7568" xml:space="preserve">15. </s> <s xml:id="echoid-s7569" xml:space="preserve">lib. </s> <s xml:id="echoid-s7570" xml:space="preserve">4. </s> <s xml:id="echoid-s7571" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7572" xml:space="preserve">erit chorda <emph style="sc">Ab</emph>, maior, <lb/>quàm Partium 45. </s> <s xml:id="echoid-s7573" xml:space="preserve">propterea quòd 60. </s> <s xml:id="echoid-s7574" xml:space="preserve">ad numerum, quimaior ſit, quàm 45. </s> <s xml:id="echoid-s7575" xml:space="preserve">mi-<lb/>norem proportionem habet, quàm ad 45. </s> <s xml:id="echoid-s7576" xml:space="preserve">Atq; </s> <s xml:id="echoid-s7577" xml:space="preserve">ita in tabella ſuperiori vides chordam <lb/> <anchor type="note" xlink:label="note-199-03a" xlink:href="note-199-03"/> arcus grad. </s> <s xml:id="echoid-s7578" xml:space="preserve">45. </s> <s xml:id="echoid-s7579" xml:space="preserve">eſſe Partium 45. </s> <s xml:id="echoid-s7580" xml:space="preserve">Min. </s> <s xml:id="echoid-s7581" xml:space="preserve">55. </s> <s xml:id="echoid-s7582" xml:space="preserve">Sec. </s> <s xml:id="echoid-s7583" xml:space="preserve">19. </s> <s xml:id="echoid-s7584" xml:space="preserve">Rurſus quia maior eſt proportio <lb/>arcus <emph style="sc">A</emph>D, ad arcum AC, quàm chordæ AD, ad chordam AC: </s> <s xml:id="echoid-s7585" xml:space="preserve">Eſt autem proportio <lb/> <anchor type="note" xlink:label="note-199-04a" xlink:href="note-199-04"/> arcus AD, ad arcum <emph style="sc">A</emph>C, eadem, quæ grad. </s> <s xml:id="echoid-s7586" xml:space="preserve">80. </s> <s xml:id="echoid-s7587" xml:space="preserve">Min. </s> <s xml:id="echoid-s7588" xml:space="preserve">30. </s> <s xml:id="echoid-s7589" xml:space="preserve">ad 60. </s> <s xml:id="echoid-s7590" xml:space="preserve">erit chordæ AD, <lb/>ad chordam AC, minor proportio, quàm Par. </s> <s xml:id="echoid-s7591" xml:space="preserve">80. </s> <s xml:id="echoid-s7592" xml:space="preserve">Min. </s> <s xml:id="echoid-s7593" xml:space="preserve">30. </s> <s xml:id="echoid-s7594" xml:space="preserve">ad 60. </s> <s xml:id="echoid-s7595" xml:space="preserve">Cum ergo chorda <lb/>AC, ſit Partium 60. </s> <s xml:id="echoid-s7596" xml:space="preserve">erit chorda AD, minor, quàm Par. </s> <s xml:id="echoid-s7597" xml:space="preserve">80. </s> <s xml:id="echoid-s7598" xml:space="preserve">Min. </s> <s xml:id="echoid-s7599" xml:space="preserve">30. </s> <s xml:id="echoid-s7600" xml:space="preserve">propterea quòd <lb/>numerus, qui minor ſit, quàm Par. </s> <s xml:id="echoid-s7601" xml:space="preserve">80. </s> <s xml:id="echoid-s7602" xml:space="preserve">Min. </s> <s xml:id="echoid-s7603" xml:space="preserve">30. </s> <s xml:id="echoid-s7604" xml:space="preserve">ad 60. </s> <s xml:id="echoid-s7605" xml:space="preserve">minorem proportionẽ habet, <lb/> <anchor type="note" xlink:label="note-199-05a" xlink:href="note-199-05"/> quàm Par. </s> <s xml:id="echoid-s7606" xml:space="preserve">80. </s> <s xml:id="echoid-s7607" xml:space="preserve">Min. </s> <s xml:id="echoid-s7608" xml:space="preserve">30. </s> <s xml:id="echoid-s7609" xml:space="preserve">ad eundem numerum 60. </s> <s xml:id="echoid-s7610" xml:space="preserve">Atqueita cernis in ſuperiori tabella <lb/>chordã arcus grad. </s> <s xml:id="echoid-s7611" xml:space="preserve">80. </s> <s xml:id="echoid-s7612" xml:space="preserve">Min. </s> <s xml:id="echoid-s7613" xml:space="preserve">30. </s> <s xml:id="echoid-s7614" xml:space="preserve">cõtinere partes duntaxat 77. </s> <s xml:id="echoid-s7615" xml:space="preserve">Min. </s> <s xml:id="echoid-s7616" xml:space="preserve">32. </s> <s xml:id="echoid-s7617" xml:space="preserve">Sec. </s> <s xml:id="echoid-s7618" xml:space="preserve">6. </s> <s xml:id="echoid-s7619" xml:space="preserve">Eademq́; <lb/></s> <s xml:id="echoid-s7620" xml:space="preserve">ratio eſt dealij schordis minoribus, & </s> <s xml:id="echoid-s7621" xml:space="preserve">maioribus, quàm <emph style="sc">A</emph>C, hoc eſt, quàm chorda ar-<lb/>cus grad. </s> <s xml:id="echoid-s7622" xml:space="preserve">60.</s> <s xml:id="echoid-s7623" xml:space="preserve"/> </p> <div xml:id="echoid-div534" type="float" level="2" n="6"> <figure xlink:label="fig-199-01" xlink:href="fig-199-01a"> <image file="199-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/199-01"/> </figure> <note position="right" xlink:label="note-199-02" xlink:href="note-199-02a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-199-03" xlink:href="note-199-03a" xml:space="preserve">8. quinti.</note> <note position="right" xlink:label="note-199-04" xlink:href="note-199-04a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-199-05" xlink:href="note-199-05a" xml:space="preserve">8. quinti.</note> </div> <pb o="188" file="200" n="200" rhead=""/> </div> <div xml:id="echoid-div536" type="section" level="1" n="255"> <head xml:id="echoid-head282" xml:space="preserve">LINAE TANGENTES, <lb/>atque Secantes.</head> <p style="it"> <s xml:id="echoid-s7624" xml:space="preserve">QVANQVAM Aſtronomi omnia <lb/>ſua problemata, atque theoremata per ſolos ſi-<lb/>nus explicare poßint, vt communiter ab omni-<lb/>bus fieri ſolet, quia tamen multa facilius, ac bre-<lb/>uius expediuntur, ſi vnà cum ſinubus lineætan-<lb/>gentes, ſecantesque adhibeantur, vt ex doctri-<lb/>na triangulorum erit manifestum; </s> <s xml:id="echoid-s7625" xml:space="preserve">quas qui-<lb/>demlineas vtili ſane conſilio Recentiores exco-<lb/>gitarunt, atque in tabulas redegerunt: </s> <s xml:id="echoid-s7626" xml:space="preserve">viſum <lb/>est has etiam lineas paucis exponere, vt doctri-<lb/>na noſtrorum triangulorum perfectior euadat. <lb/></s> <s xml:id="echoid-s7627" xml:space="preserve">Vniuerſa ſiquidem triangulorum doctrina in <lb/> <anchor type="note" xlink:label="note-200-01a" xlink:href="note-200-01"/> tribus hiſce line arum generibus, nempe in ſinu-<lb/>bus, lineis tangentibus, & </s> <s xml:id="echoid-s7628" xml:space="preserve">ſecantibus, potißi-<lb/>mum conſiſtere videtur. </s> <s xml:id="echoid-s7629" xml:space="preserve">Primum autem expli-<lb/>candum eſt, quid ſit linea tangens, & </s> <s xml:id="echoid-s7630" xml:space="preserve">quid ſe-<lb/>cans propoſiti cuiuſuis arcus.</s> <s xml:id="echoid-s7631" xml:space="preserve"/> </p> <div xml:id="echoid-div536" type="float" level="2" n="1"> <note position="left" xlink:label="note-200-01" xlink:href="note-200-01a" xml:space="preserve">Doctrina <lb/>triangulo -<lb/>rũ in quo <lb/>conſiſtat.</note> </div> <p style="it"> <s xml:id="echoid-s7632" xml:space="preserve"><emph style="sc">CVm</emph> ergoab altero extremo cuius libet arcus, qui quadrante minor ſit, ſemi-<lb/> <anchor type="note" xlink:label="note-200-02a" xlink:href="note-200-02"/> diameter ducta fuerit, in cuius extremitate recta linea circulum tangat, & </s> <s xml:id="echoid-s7633" xml:space="preserve">per <lb/>alierum extremum eiuſdem arcus extendatur alia recta linea ex centro ad tangen-<lb/>tem lineam vſque: </s> <s xml:id="echoid-s7634" xml:space="preserve">appellatur portio lineæ tangentis inter duas rectas è centro egre-<lb/>dientes, Linea tangens illius arcus, quem eædẽ duæ rectæ e centro eductæ includunt: <lb/></s> <s xml:id="echoid-s7635" xml:space="preserve">Recta vero altera puncto contactus oppoſita inter centrum, & </s> <s xml:id="echoid-s7636" xml:space="preserve">lineam tangentem, di-<lb/>citur Linea ſecans eiuſdem arcus. </s> <s xml:id="echoid-s7637" xml:space="preserve">Vt ſi in circulo AB, cuius centrum C, ſumatur ar-<lb/>cus AB, quadrante minor, & </s> <s xml:id="echoid-s7638" xml:space="preserve">in extremitate ſemidiametri <emph style="sc">Ac</emph>, ab extremitate A, <pb o="189" file="201" n="201" rhead=""/> ducta recta AD, circulum tangat, recta autem CD, circulum ſecet, conueniens cum <lb/>AD, in D, (conueniet enim neceſſario, propterea quòd duo anguli <emph style="sc">Ca</emph>C, DCA, <lb/>duobus rectis ſunt minores; </s> <s xml:id="echoid-s7639" xml:space="preserve">cum ille rectus ſit, hic autem <lb/>recto minor, propter arcum <emph style="sc">A</emph>B, quadrante minorem.) <lb/></s> <s xml:id="echoid-s7640" xml:space="preserve">dicetur AD, Tangens arcus <emph style="sc">A</emph>B, at CD, Secans eiuſdẽ <lb/>arcus. </s> <s xml:id="echoid-s7641" xml:space="preserve">Tangentem vocant nonnulli Adſcriptam, quòd <lb/> <anchor type="note" xlink:label="note-201-01a" xlink:href="note-201-01"/> circulo quodãmodo adſcribatur; </s> <s xml:id="echoid-s7642" xml:space="preserve">Secantem vero, Hypo-<lb/>tenuſam, propterea quòd in triangulo rectãgulo ACD, <lb/>(angulus enim <emph style="sc">A</emph>, apud contactum rectus eſt) angulum <lb/>rectum ſubtendit: </s> <s xml:id="echoid-s7643" xml:space="preserve">Semidiametrum denique <emph style="sc">A</emph>C, ſiue ſi-<lb/> <anchor type="note" xlink:label="note-201-02a" xlink:href="note-201-02"/> num totum, dicunt baſem eiuſdem trianguli.</s> <s xml:id="echoid-s7644" xml:space="preserve"/> </p> <div xml:id="echoid-div537" type="float" level="2" n="2"> <note position="left" xlink:label="note-200-02" xlink:href="note-200-02a" xml:space="preserve">Linea tan-<lb/>gens, & ſe-<lb/>cans quid.</note> <note position="right" xlink:label="note-201-01" xlink:href="note-201-01a" xml:space="preserve">Linea ad-<lb/>ſcripta, & <lb/>Hypotenu-<lb/>ſa quid.</note> <note position="right" xlink:label="note-201-02" xlink:href="note-201-02a" xml:space="preserve">18. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s7645" xml:space="preserve"><emph style="sc">QVem</emph><emph style="sc">ADm</emph><emph style="sc">ODVm</emph> autemin omni triangulo <lb/> <anchor type="note" xlink:label="note-201-03a" xlink:href="note-201-03"/> rectangulo, ſilatus recto angulo oppoſitum ponatur ſinus <lb/>totus, reliqua duo latera ſunt ſinus recti reliquorum angulorum acutorum, quibus <lb/>opponuntur; </s> <s xml:id="echoid-s7646" xml:space="preserve">Item vtrumuis reliquorum laterum eſt ſinus complementi anguli ſibi <lb/>adiacentis, vt in definitionibus ſinuum traditum eſt: </s> <s xml:id="echoid-s7647" xml:space="preserve">ita quoque ſi alterutrum late-<lb/>rum circa angulum rectum ſtatuatur ſinus totus, erit alterum latus circa angulum <lb/>rectum Tangens anguli acuti ſibi oppoſiti, latus vero angulo recto oppoſitum Secans <lb/>eiuſdem anguli. </s> <s xml:id="echoid-s7648" xml:space="preserve">Vt in triangulo rectangulo ACD, latus CA, eſt ſinus totus, nempe <lb/>ſemidiameter circuli AB: </s> <s xml:id="echoid-s7649" xml:space="preserve">at <emph style="sc">A</emph>D, tangens anguli C, vel arcus <emph style="sc">Ab</emph>, & </s> <s xml:id="echoid-s7650" xml:space="preserve">CD, eiuſdem <lb/>ſecans. </s> <s xml:id="echoid-s7651" xml:space="preserve">Eodem pacto, ſt DA, ſtatuatur ſinus totus, erit AC, tangens anguli D, & </s> <s xml:id="echoid-s7652" xml:space="preserve"><lb/>DC, eiuſdem ſecans.</s> <s xml:id="echoid-s7653" xml:space="preserve"/> </p> <div xml:id="echoid-div538" type="float" level="2" n="3"> <note position="right" xlink:label="note-201-03" xlink:href="note-201-03a" xml:space="preserve">Si in trian-<lb/>gu’o rectan <lb/>gulo alteru <lb/>trum late-<lb/>rum circa <lb/>angulũ re-<lb/>ctum pona <lb/>tur ſinus to <lb/>tus, erit al-<lb/>terum latus <lb/>circa angu-<lb/>lum rectú <lb/>tangens an <lb/>gulĩ acutiſi <lb/>bi oppoſiti, <lb/>& latus re-<lb/>cto angulo <lb/>oppoſitum <lb/>eiuſdem ſe <lb/>cans.</note> </div> <p style="it"> <s xml:id="echoid-s7654" xml:space="preserve">ETSI autem diximus, tangentem, & </s> <s xml:id="echoid-s7655" xml:space="preserve">ſecantem ſumi reſpectu arcus quadrante <lb/>minoris, tamen eadem tangens, & </s> <s xml:id="echoid-s7656" xml:space="preserve">ſecans referri ſolet ad arcum etiam, qui cum illo <lb/>ſemicirculum complet: </s> <s xml:id="echoid-s7657" xml:space="preserve">adeo vt duo arcus ſemicirculum conficientes, vel duo anguli <lb/>duobus rectis æquales, vnam eandemq; </s> <s xml:id="echoid-s7658" xml:space="preserve">tangentem, atq; </s> <s xml:id="echoid-s7659" xml:space="preserve">ſ@cantem habeant: </s> <s xml:id="echoid-s7660" xml:space="preserve">quemad-<lb/>modum & </s> <s xml:id="echoid-s7661" xml:space="preserve">eundem ſinum rectum habent, vt in tractatione ſinuum tradidimus: </s> <s xml:id="echoid-s7662" xml:space="preserve">adeo <lb/> <anchor type="note" xlink:label="note-201-04a" xlink:href="note-201-04"/> vt ſi quæratur tangens & </s> <s xml:id="echoid-s7663" xml:space="preserve">ſecans alicuius arcus quadrante maioris, ſumenda ſit tan <lb/>gens, & </s> <s xml:id="echoid-s7664" xml:space="preserve">ſecans arcus quadrante minoris, qui cum illo ſemicireulum complet.</s> <s xml:id="echoid-s7665" xml:space="preserve"/> </p> <div xml:id="echoid-div539" type="float" level="2" n="4"> <note position="right" xlink:label="note-201-04" xlink:href="note-201-04a" xml:space="preserve">Duo arcus <lb/>ſem icircu-<lb/>lú cóficiéte <lb/>vel duo än <lb/>guli duobꝰ <lb/>rectis æqua <lb/>les habent <lb/>eandé tan-<lb/>gentem & <lb/>ſecantem.</note> </div> <p style="it"> <s xml:id="echoid-s7666" xml:space="preserve">PORRO qua ratione Tangentes, & </s> <s xml:id="echoid-s7667" xml:space="preserve">Secantes omnium arcuum quadrantis <lb/>reddantur cognitæ in partibus ſinus totius, ac proinde qua via tabula Tangentium, <lb/>tabula item Secantium componenda ſit, ſequentibus propoſitionibus, quæ ad line{as} <lb/>Tangentes, ac Secantes ſpectant, planum fiet.</s> <s xml:id="echoid-s7668" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div541" type="section" level="1" n="256"> <head xml:id="echoid-head283" xml:space="preserve">THEOR. .9. PROPOS. 17.</head> <p> <s xml:id="echoid-s7669" xml:space="preserve">TANGENS dimidij quadrantis ſinui toti <lb/> <anchor type="note" xlink:label="note-201-05a" xlink:href="note-201-05"/> æqualis eſt: </s> <s xml:id="echoid-s7670" xml:space="preserve">Tangens autem arcus maioris dimidi<unsure/>o <lb/>quadrantis maior eſt ſinu toto: </s> <s xml:id="echoid-s7671" xml:space="preserve">Et Tangens mino-<lb/>ris arcus minor eſt. </s> <s xml:id="echoid-s7672" xml:space="preserve">Secans denique dimidij qua-<lb/>drantis dupla eſt ſinus recti eiuſdem dimidij.</s> <s xml:id="echoid-s7673" xml:space="preserve"/> </p> <div xml:id="echoid-div541" type="float" level="2" n="1"> <note position="right" xlink:label="note-201-05" xlink:href="note-201-05a" xml:space="preserve">Tangentes <lb/>quomodo <lb/>ſe habeant <lb/>cũ ſinu to-<lb/>to com pa-<lb/>ratæ.</note> </div> <p> <s xml:id="echoid-s7674" xml:space="preserve">IN quadrante ABC, ſit arcus CD, ſemiſsis ipſius; </s> <s xml:id="echoid-s7675" xml:space="preserve">CE, ſemiſſe maior, &</s> <s xml:id="echoid-s7676" xml:space="preserve"> <pb o="190" file="202" n="202" rhead=""/> CF, minor. </s> <s xml:id="echoid-s7677" xml:space="preserve">Ducta autem recta CH, ad AC, perpendiculari, quæ circulum <lb/>tanget in C, ducantur rectæ AG, AH, AI, per puncta D, E, F. </s> <s xml:id="echoid-s7678" xml:space="preserve">Item DK, ad <lb/> <anchor type="note" xlink:label="note-202-01a" xlink:href="note-202-01"/> AC, perpendicularis. </s> <s xml:id="echoid-s7679" xml:space="preserve">Eritq; </s> <s xml:id="echoid-s7680" xml:space="preserve">CG, tangens arcus CD; </s> <s xml:id="echoid-s7681" xml:space="preserve">& </s> <s xml:id="echoid-s7682" xml:space="preserve">CH, tangens arcus <lb/>CE; </s> <s xml:id="echoid-s7683" xml:space="preserve">& </s> <s xml:id="echoid-s7684" xml:space="preserve">CI, tangens arcus CF. </s> <s xml:id="echoid-s7685" xml:space="preserve">at DK, ſinus rectus <lb/> <anchor type="figure" xlink:label="fig-202-01a" xlink:href="fig-202-01"/> arcus CD, & </s> <s xml:id="echoid-s7686" xml:space="preserve">AG, eiuſdem ſecans. </s> <s xml:id="echoid-s7687" xml:space="preserve">Dico CG, æqua-<lb/>lem eſſe ſinui toti AC; </s> <s xml:id="echoid-s7688" xml:space="preserve">at CH, maiorem, & </s> <s xml:id="echoid-s7689" xml:space="preserve">CI, mi-<lb/>norem. </s> <s xml:id="echoid-s7690" xml:space="preserve">Item AG, duplam ſinus DK. </s> <s xml:id="echoid-s7691" xml:space="preserve">Quoniam <lb/> <anchor type="note" xlink:label="note-202-02a" xlink:href="note-202-02"/> enim anguli CAG, GAB, æquales ſunt, ob arcus <lb/>æquales CD, DB; </s> <s xml:id="echoid-s7692" xml:space="preserve">eſtq́; </s> <s xml:id="echoid-s7693" xml:space="preserve">angulus BAC, rectus, erit <lb/>vterq; </s> <s xml:id="echoid-s7694" xml:space="preserve">illorum ſemirectus. </s> <s xml:id="echoid-s7695" xml:space="preserve">Quare & </s> <s xml:id="echoid-s7696" xml:space="preserve">reliquus angu-<lb/> <anchor type="note" xlink:label="note-202-03a" xlink:href="note-202-03"/> lus CGA, in triangulo ACG, ſemirectus erit; <lb/></s> <s xml:id="echoid-s7697" xml:space="preserve">propterea quòd angulus C, rectus eſt. </s> <s xml:id="echoid-s7698" xml:space="preserve">Igitur recta <lb/> <anchor type="note" xlink:label="note-202-04a" xlink:href="note-202-04"/> CG, tangens arcus CD, qui ſemiſsis eſt quadran-<lb/>tis, ſinui toti AC, æqualis erit. </s> <s xml:id="echoid-s7699" xml:space="preserve">Ex quo ſequi-<lb/>tur, CH, tangentem arcus CE, qui ſemiſſe quadrantis maior eſt, ſinu toto <lb/>AC, maiorem eſſe; </s> <s xml:id="echoid-s7700" xml:space="preserve">& </s> <s xml:id="echoid-s7701" xml:space="preserve">CI, tangentem arcus CF, qui ſemiſſe quadrantis mi-<lb/>nor eſt, minorem: </s> <s xml:id="echoid-s7702" xml:space="preserve">cum punctum H, neceſſario cadat ſupra G, & </s> <s xml:id="echoid-s7703" xml:space="preserve">punctum I, <lb/>infra.</s> <s xml:id="echoid-s7704" xml:space="preserve"/> </p> <div xml:id="echoid-div542" type="float" level="2" n="2"> <note position="left" xlink:label="note-202-01" xlink:href="note-202-01a" xml:space="preserve">coroll. 16 3</note> <figure xlink:label="fig-202-01" xlink:href="fig-202-01a"> <image file="202-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/202-01"/> </figure> <note position="left" xlink:label="note-202-02" xlink:href="note-202-02a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-202-03" xlink:href="note-202-03a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-202-04" xlink:href="note-202-04a" xml:space="preserve">5. primi.</note> </div> <p> <s xml:id="echoid-s7705" xml:space="preserve">QVOD tamen ſeorſum ita quoq; </s> <s xml:id="echoid-s7706" xml:space="preserve">oſtendi poteſt, nulla habita ratione tan <lb/>gentis CG, cuius arcus eſt ſemiſsis quadrantis. </s> <s xml:id="echoid-s7707" xml:space="preserve">Quoniam arcus CE, ſemiſſe <lb/>quadrantis maior eſt, erit arcus BE, ſemiſſe quadrantis minor. </s> <s xml:id="echoid-s7708" xml:space="preserve">Igitur angulus <lb/>CAH, angulo HAB, maior eſt, ac proinde maior ſemirecto. </s> <s xml:id="echoid-s7709" xml:space="preserve">Cum ergo an-<lb/> <anchor type="note" xlink:label="note-202-05a" xlink:href="note-202-05"/> gulus C, rectus ſit, erit reliquus AHC, in triangulo ACH, ſemirecto minor. <lb/></s> <s xml:id="echoid-s7710" xml:space="preserve"> <anchor type="note" xlink:label="note-202-06a" xlink:href="note-202-06"/> Quare recta CH, tangens arcus CE, qui ſemiſſe quadrantis maior eſt, ma-<lb/> <anchor type="note" xlink:label="note-202-07a" xlink:href="note-202-07"/> ior eſt ſinu toto AC.</s> <s xml:id="echoid-s7711" xml:space="preserve"/> </p> <div xml:id="echoid-div543" type="float" level="2" n="3"> <note position="left" xlink:label="note-202-05" xlink:href="note-202-05a" xml:space="preserve">Schol. 27. 3.</note> <note position="left" xlink:label="note-202-06" xlink:href="note-202-06a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-202-07" xlink:href="note-202-07a" xml:space="preserve">19. primi.</note> </div> <p> <s xml:id="echoid-s7712" xml:space="preserve">RVRSVS quia arcus CF, ſemiſſe quadrantis minor eſt, ac proinde BF, <lb/>maior, erit angulus CAI, angulo IAB, minor; </s> <s xml:id="echoid-s7713" xml:space="preserve">atque adeo minor ſemirecto. <lb/></s> <s xml:id="echoid-s7714" xml:space="preserve"> <anchor type="note" xlink:label="note-202-08a" xlink:href="note-202-08"/> Cum ergo angulus C, ſit rectus, erit reliquus AIC, in triangulo ACI, maior <lb/> <anchor type="note" xlink:label="note-202-09a" xlink:href="note-202-09"/> ſemirecto: </s> <s xml:id="echoid-s7715" xml:space="preserve">ac propterea recta CI, tangens arcus CF, qui quadrantis ſemiſſe <lb/>minor eſt, minor erit ſinu toto AC.</s> <s xml:id="echoid-s7716" xml:space="preserve"/> </p> <div xml:id="echoid-div544" type="float" level="2" n="4"> <note position="left" xlink:label="note-202-08" xlink:href="note-202-08a" xml:space="preserve">Schol. 27. 3.</note> <note position="left" xlink:label="note-202-09" xlink:href="note-202-09a" xml:space="preserve">32. primi.</note> </div> <note position="left" xml:space="preserve">19. primi.</note> <p> <s xml:id="echoid-s7717" xml:space="preserve">PRAETEREA quoniam angulus K, rectus eſt, & </s> <s xml:id="echoid-s7718" xml:space="preserve">DAK, ſemirectus, vt <lb/>oſtenſum eſt; </s> <s xml:id="echoid-s7719" xml:space="preserve">erit & </s> <s xml:id="echoid-s7720" xml:space="preserve">AD k, ſemirectus; </s> <s xml:id="echoid-s7721" xml:space="preserve">ac proinde AK, ſinui DK, æqualis <lb/> <anchor type="note" xlink:label="note-202-11a" xlink:href="note-202-11"/> erit. </s> <s xml:id="echoid-s7722" xml:space="preserve">Quia vero triangula GAC, DAK, æquiangula ſunt, erit vt AG, ad AC, <lb/> <anchor type="note" xlink:label="note-202-12a" xlink:href="note-202-12"/> ita AD, hoc eſt, AC, ad AK; </s> <s xml:id="echoid-s7723" xml:space="preserve">ac proinde tres rectæ GA, ſecans; </s> <s xml:id="echoid-s7724" xml:space="preserve">AC, ſinus <lb/>totus, & </s> <s xml:id="echoid-s7725" xml:space="preserve">AK, ſinus dimidij quadrantis, continue proportionales erunt. </s> <s xml:id="echoid-s7726" xml:space="preserve">Ita <lb/>ergo erit quadratum ex AG, ad quadratum ex AC, vt recta AG, ad rectam <lb/> <anchor type="note" xlink:label="note-202-13a" xlink:href="note-202-13"/> AK. </s> <s xml:id="echoid-s7727" xml:space="preserve">Eſt autem quadratum ex AG, quadrati ex AC, duplum propterea quòd <lb/> <anchor type="note" xlink:label="note-202-14a" xlink:href="note-202-14"/> æquale eſt quadratis ex AC, CG, æqualibus. </s> <s xml:id="echoid-s7728" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s7729" xml:space="preserve">AG, ſecans dimidij <lb/>quadrátis dupla eſt ſinus AK, vel DK, eiuſdem dimidij. </s> <s xml:id="echoid-s7730" xml:space="preserve">Quocirca tangens di-<lb/>midij quadrantis ſinui toti æqualis eſt, &</s> <s xml:id="echoid-s7731" xml:space="preserve">c. </s> <s xml:id="echoid-s7732" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s7733" xml:space="preserve"/> </p> <div xml:id="echoid-div545" type="float" level="2" n="5"> <note position="left" xlink:label="note-202-11" xlink:href="note-202-11a" xml:space="preserve">6. primi.</note> <note position="left" xlink:label="note-202-12" xlink:href="note-202-12a" xml:space="preserve">4. ſexti.</note> <note position="left" xlink:label="note-202-13" xlink:href="note-202-13a" xml:space="preserve">Corol. 20. 6</note> <note position="left" xlink:label="note-202-14" xlink:href="note-202-14a" xml:space="preserve">47 primi.</note> </div> <note position="left" xml:space="preserve">Q uæ tan-<lb/>gentes in ta <lb/>bula tágen-<lb/>tiũ mino -<lb/>res<unsure/>ſint ſinu <lb/>toto, & quę <lb/>maiores. <lb/>Ité cur ſe-<lb/>cás gt. 45. <lb/>dupla ſit ſi-<lb/>nus gt. 45.</note> </div> <div xml:id="echoid-div547" type="section" level="1" n="257"> <head xml:id="echoid-head284" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s7734" xml:space="preserve">E X hac propoſ. </s> <s xml:id="echoid-s7735" xml:space="preserve">aperte cauſa colligitur, cur in tabula Tangentium omnes tangen <lb/>tes arcuum minorum, quàm grad. </s> <s xml:id="echoid-s7736" xml:space="preserve">45. </s> <s xml:id="echoid-s7737" xml:space="preserve">minores ſint ſinu toto: </s> <s xml:id="echoid-s7738" xml:space="preserve">Tangens vero arcus gra. <lb/></s> <s xml:id="echoid-s7739" xml:space="preserve">45. </s> <s xml:id="echoid-s7740" xml:space="preserve">ſinui totiæqualis: </s> <s xml:id="echoid-s7741" xml:space="preserve">Tangentes denique omnes arcuum maiorum, quàm grad. </s> <s xml:id="echoid-s7742" xml:space="preserve">45. </s> <s xml:id="echoid-s7743" xml:space="preserve"><lb/>ſinu toto maiores. </s> <s xml:id="echoid-s7744" xml:space="preserve">Item cur in tabula Secantium ſecans arcus grad. </s> <s xml:id="echoid-s7745" xml:space="preserve">45. </s> <s xml:id="echoid-s7746" xml:space="preserve">dupla ſit ſinus <lb/>arcus eiuſdem grad. </s> <s xml:id="echoid-s7747" xml:space="preserve">45.</s> <s xml:id="echoid-s7748" xml:space="preserve"/> </p> <pb o="191" file="203" n="203" rhead=""/> </div> <div xml:id="echoid-div548" type="section" level="1" n="258"> <head xml:id="echoid-head285" xml:space="preserve">THEOR. 10. PROPOS. 18.</head> <p> <s xml:id="echoid-s7749" xml:space="preserve">QVAM proportionem habet ſinus comple <lb/> <anchor type="note" xlink:label="note-203-01a" xlink:href="note-203-01"/> menti arcus cuiuſuis ad ſinum rectum eiuſdem ar <lb/>cus, eam habet ſinus totus ad tangentem eiuſdem <lb/>arcus: </s> <s xml:id="echoid-s7750" xml:space="preserve">Item quam proportionem habet ſinus re-<lb/>ctus cuiuſlibet arcus ad ſinum complementi eiuſ-<lb/>dem arcus, eam habet ſinus totus ad tangentem <lb/>eiuſdem complementi. </s> <s xml:id="echoid-s7751" xml:space="preserve">Sinus autem totus medio <lb/>loco proportionalis eſt inter ſinum complementi <lb/>cuiuſuis arcus, & </s> <s xml:id="echoid-s7752" xml:space="preserve">ſecantem eiuſdem aicus: </s> <s xml:id="echoid-s7753" xml:space="preserve">Item <lb/>inter ſinum rectum cuiuſlibet arcus, & </s> <s xml:id="echoid-s7754" xml:space="preserve">ſecantem <lb/>complementi eiuſdem arcus. </s> <s xml:id="echoid-s7755" xml:space="preserve">Sinus denique idem <lb/>totus medius proportionalis eſt inter tangentem <lb/>arcus cuiuſuis, & </s> <s xml:id="echoid-s7756" xml:space="preserve">tangentem complementi eiuſ-<lb/>dem arcus.</s> <s xml:id="echoid-s7757" xml:space="preserve"/> </p> <div xml:id="echoid-div548" type="float" level="2" n="1"> <note position="right" xlink:label="note-203-01" xlink:href="note-203-01a" xml:space="preserve">Quam pro <lb/>portionem <lb/>habeat ſinꝰ <lb/>totus ad :5 <lb/>genté, & ſe <lb/>cantem cu-<lb/>iuſuis arcꝰ.</note> </div> <p> <s xml:id="echoid-s7758" xml:space="preserve">IN quadrante ABC, ſit DF, ſinus arcus CD; </s> <s xml:id="echoid-s7759" xml:space="preserve">& </s> <s xml:id="echoid-s7760" xml:space="preserve">CE, eiuſdem tangens <lb/>inter ſemidiametrum AC, & </s> <s xml:id="echoid-s7761" xml:space="preserve">ſecantem AE: </s> <s xml:id="echoid-s7762" xml:space="preserve">Item BG, tangens arcus BD, qui <lb/>complementum eſt arcus CD. </s> <s xml:id="echoid-s7763" xml:space="preserve">Dico ita eſſe ſinum complementi arcus CD, ad <lb/>ſinum rectum eiuſdé arcus, vt eſt ſinus totus AC, <lb/> <anchor type="figure" xlink:label="fig-203-01a" xlink:href="fig-203-01"/> ad tangentem CE, &</s> <s xml:id="echoid-s7764" xml:space="preserve">c. </s> <s xml:id="echoid-s7765" xml:space="preserve">Quoniam enim, vt in ex-<lb/>poſitione definitionum dictum eſt, AF, æqualis <lb/>eſt ſinui complementi arcus CD, cum ſit æqualis <lb/>ſinui recto arcus BD, qui complementum eſt arcus <lb/>CD; </s> <s xml:id="echoid-s7766" xml:space="preserve">ſuntq́; </s> <s xml:id="echoid-s7767" xml:space="preserve">triangula AFD, ACE, æquiangula, <lb/>ob rectos angulos F, C, & </s> <s xml:id="echoid-s7768" xml:space="preserve">communem angulum <lb/>A: </s> <s xml:id="echoid-s7769" xml:space="preserve">erit vt AF, ſinus complementi arcus CD, ad <lb/> <anchor type="note" xlink:label="note-203-02a" xlink:href="note-203-02"/> FD, ſinum rectum eiuſdem arcus CD, ita AC, ſi-<lb/>nus totus ad CE, tangentem eiuſdem arcus. </s> <s xml:id="echoid-s7770" xml:space="preserve">quod <lb/>eſt primum.</s> <s xml:id="echoid-s7771" xml:space="preserve"/> </p> <div xml:id="echoid-div549" type="float" level="2" n="2"> <figure xlink:label="fig-203-01" xlink:href="fig-203-01a"> <image file="203-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/203-01"/> </figure> <note position="right" xlink:label="note-203-02" xlink:href="note-203-02a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s7772" xml:space="preserve">DEINDE eadem ratione erit, vt AF, ſinus <lb/>rectus arcus BD, ad FD, ſinũ complementi eiuſdem arcus BD, ita AC, ſinus <lb/>totus ad CE, tangentem eiuſdem complementi arcus BD. </s> <s xml:id="echoid-s7773" xml:space="preserve">quod eſt ſecundum.</s> <s xml:id="echoid-s7774" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7775" xml:space="preserve">TERTIO in eiſdem triangulis erit, vt AF, ſinus complementi arcus <lb/>CD, ad AD, ſinum totum, ita AC, ſinus totus ad AE, ſecantem eiuſdem <pb o="192" file="204" n="204" rhead=""/> arcus CD. </s> <s xml:id="echoid-s7776" xml:space="preserve">Item vt AF, ſinus rectus arcus BD, ad AD, ſinum totum, ita AC, <lb/>ſinus totus ad AE, ſecantem arcus CD, qui complementum eſt eiuſdem arcus <lb/>BD. </s> <s xml:id="echoid-s7777" xml:space="preserve">Quare ſinus totus medius proportionalis eſt inter AF, ſinũ cõ plementi <lb/>arcus CD, & </s> <s xml:id="echoid-s7778" xml:space="preserve">AE, ſecantem eiuſdem arcus CD: <lb/></s> <s xml:id="echoid-s7779" xml:space="preserve"> <anchor type="figure" xlink:label="fig-204-01a" xlink:href="fig-204-01"/> Item inter AF, ſinum rectum arcus BD, & </s> <s xml:id="echoid-s7780" xml:space="preserve">AE, <lb/>ſecantem complementi eiuſdem arcus BD. </s> <s xml:id="echoid-s7781" xml:space="preserve">quod <lb/>eſt tertium.</s> <s xml:id="echoid-s7782" xml:space="preserve"/> </p> <div xml:id="echoid-div550" type="float" level="2" n="3"> <figure xlink:label="fig-204-01" xlink:href="fig-204-01a"> <image file="204-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/204-01"/> </figure> </div> <p> <s xml:id="echoid-s7783" xml:space="preserve">POSTREMO, quia triangula ACE; <lb/></s> <s xml:id="echoid-s7784" xml:space="preserve">GBA, æquiangula ſunt, quòd anguli C, B, ſint <lb/>recti; </s> <s xml:id="echoid-s7785" xml:space="preserve">& </s> <s xml:id="echoid-s7786" xml:space="preserve">alterni CAE, BGA, nec non alterni <lb/> <anchor type="note" xlink:label="note-204-01a" xlink:href="note-204-01"/> AEC, GAB, æquales: </s> <s xml:id="echoid-s7787" xml:space="preserve">erit vt CE, tangens ar-<lb/>cus CD, ad AC, ſinum totum, ita AB, ſinus to-<lb/>tus ad BG, tangentem complementi eiuſdem ar-<lb/>cus CD. </s> <s xml:id="echoid-s7788" xml:space="preserve">Pari ratione erit, vt BG, tangens arcus <lb/>BD, ad AB, ſinum totum, ita AC, ſinus totus <lb/>ad CE, tangentem complementi eiuſdem arcus BD. </s> <s xml:id="echoid-s7789" xml:space="preserve">quod eſt quartum. <lb/></s> <s xml:id="echoid-s7790" xml:space="preserve">Quã ergo proportionem habet ſinus complementi arcus cuiuſuis, &</s> <s xml:id="echoid-s7791" xml:space="preserve">c. </s> <s xml:id="echoid-s7792" xml:space="preserve">Quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s7793" xml:space="preserve"/> </p> <div xml:id="echoid-div551" type="float" level="2" n="4"> <note position="left" xlink:label="note-204-01" xlink:href="note-204-01a" xml:space="preserve">29. primi. <lb/>4. ſexti.</note> </div> </div> <div xml:id="echoid-div553" type="section" level="1" n="259"> <head xml:id="echoid-head286" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s7794" xml:space="preserve">ITAQVE per ea, quæ primo loco demonſtrata ſunt in hac propoſ. </s> <s xml:id="echoid-s7795" xml:space="preserve">ſi fiat, vt ſin{us} com <lb/> <anchor type="note" xlink:label="note-204-02a" xlink:href="note-204-02"/> plementi cuiuſuis arc{us} ad ſinũ rectũ eiuſdẽ arc{us}, ita ſin{us} tot{us} ad aliud; </s> <s xml:id="echoid-s7796" xml:space="preserve">hoc eſt, ſi ſi-<lb/>n{us} arc{us} cui{us}libet in ſinum totũ multiplicetur, (quod fiet facile, ſi ei ad dextrã præ-<lb/>ponas tot ci fras, quot in ſinu toto continentur, nempe ſeptem, ſi ſin{us} tot{us} fuerit <lb/>10000000. </s> <s xml:id="echoid-s7797" xml:space="preserve">vel quinque, ſi ſinus totus fuerit 100000.) </s> <s xml:id="echoid-s7798" xml:space="preserve">productusq́; </s> <s xml:id="echoid-s7799" xml:space="preserve">numerus per ſi-<lb/>num complementi eiuſdem arc{us} diuidatur: </s> <s xml:id="echoid-s7800" xml:space="preserve">inuenietur Tangens illi{us} arc{us}, cui{us} ſi-<lb/>num complementi accepiſti, vel cuius ſinum rectum in ſinum totum multiplicaſti. </s> <s xml:id="echoid-s7801" xml:space="preserve">Vt <lb/>ſi tangens arc{us} gra. </s> <s xml:id="echoid-s7802" xml:space="preserve">30. </s> <s xml:id="echoid-s7803" xml:space="preserve">quæratur, adiungemus eius ſinui recto 5000000 ſeptem ci-<lb/>fras, hoc modo. </s> <s xml:id="echoid-s7804" xml:space="preserve">50000000000000. </s> <s xml:id="echoid-s7805" xml:space="preserve">& </s> <s xml:id="echoid-s7806" xml:space="preserve">hunc numerum per 8660254 ſinumcomple-<lb/>menti eiuſdem arcus grad. </s> <s xml:id="echoid-s7807" xml:space="preserve">30. </s> <s xml:id="echoid-s7808" xml:space="preserve">partiemur. </s> <s xml:id="echoid-s7809" xml:space="preserve">Nam quotiens numerus 5773503. </s> <s xml:id="echoid-s7810" xml:space="preserve">dabit <lb/>tangentem arcus grad. </s> <s xml:id="echoid-s7811" xml:space="preserve">30. </s> <s xml:id="echoid-s7812" xml:space="preserve">quatenus ſinus totus eſt 10000000. </s> <s xml:id="echoid-s7813" xml:space="preserve">Hinc fit, tangentes <lb/> <anchor type="note" xlink:label="note-204-03a" xlink:href="note-204-03"/> emnes per ſolam diuiſionem inueniri. </s> <s xml:id="echoid-s7814" xml:space="preserve">Nam ſiomnium arcuum ſinubus, initio facto à <lb/>principio quadr antis, ſeptem ci fr{as} appon{as}, & </s> <s xml:id="echoid-s7815" xml:space="preserve">compoſitos numeros per ſinus comple <lb/>mentorum eorundem arcuum, quemlibet per ſuum correſpondentem, diuid{as}, prodi-<lb/>bunt omnium arcuum tangentes, vt ex demonſtratis liquido cõſtat. </s> <s xml:id="echoid-s7816" xml:space="preserve">Quamobrem per-<lb/>facilis eſt conſtructio tabulæ Tangentium.</s> <s xml:id="echoid-s7817" xml:space="preserve"/> </p> <div xml:id="echoid-div553" type="float" level="2" n="1"> <note position="left" xlink:label="note-204-02" xlink:href="note-204-02a" xml:space="preserve">Quo pacto <lb/>tangentes <lb/>omniú ar-<lb/>cuum repe-<lb/>tiantur.</note> <note position="left" xlink:label="note-204-03" xlink:href="note-204-03a" xml:space="preserve">Sola diui-<lb/>ſione om <lb/>nes tangen <lb/>tes eliciun-<lb/>tur.</note> </div> <p style="it"> <s xml:id="echoid-s7818" xml:space="preserve">RVRSVS per ea, quæ tertio loco in hac propoſ. </s> <s xml:id="echoid-s7819" xml:space="preserve">ſunt demonſtrata, ſi fiat, vt ſi-<lb/> <anchor type="note" xlink:label="note-204-04a" xlink:href="note-204-04"/> nus complementi cuiuſuis arcus ad ſinum totum, ita ſinus totus ad aliud: </s> <s xml:id="echoid-s7820" xml:space="preserve">hoc eſt, ſi ſi-<lb/>nus totus in ſeipſum multiplicetur, (quod facile fiet, ſi ei ad dextram præpon{as} tot <lb/>cifras, quot ſunt in ſinu toto, puta ſeptem, vel quinque, prout ſinus totus habuerit <lb/>ſeptem, aut quinque cifras.) </s> <s xml:id="echoid-s7821" xml:space="preserve">numerusq́; </s> <s xml:id="echoid-s7822" xml:space="preserve">productus per ſinum complementi cuiuſuis ar-<lb/>cus diuidatur: </s> <s xml:id="echoid-s7823" xml:space="preserve">reperietur Secans illius arcus, per cuius ſinum complementi diuiſio fa <lb/>cta eſt. </s> <s xml:id="echoid-s7824" xml:space="preserve">Vt ſi ſecans arcus grad. </s> <s xml:id="echoid-s7825" xml:space="preserve">30. </s> <s xml:id="echoid-s7826" xml:space="preserve">deſideretur, diuidemus 100000000000000. </s> <s xml:id="echoid-s7827" xml:space="preserve">(pro-<lb/>ductum ſcilicet numerum ex ſinu toto in ſeipſum) per 8660254. </s> <s xml:id="echoid-s7828" xml:space="preserve">ſinum complementi <lb/>arcus grad. </s> <s xml:id="echoid-s7829" xml:space="preserve">30. </s> <s xml:id="echoid-s7830" xml:space="preserve">Numerus enim Quotiens 11547005. </s> <s xml:id="echoid-s7831" xml:space="preserve">dabit ſecantem arcus grad. </s> <s xml:id="echoid-s7832" xml:space="preserve">30. <lb/></s> <s xml:id="echoid-s7833" xml:space="preserve">quatenus ſinus totus est 10000000. </s> <s xml:id="echoid-s7834" xml:space="preserve">Sola ergo diuiſ@one huius ſemper numeri <pb o="193" file="205" n="205" rhead=""/> 100000000000000. </s> <s xml:id="echoid-s7835" xml:space="preserve">per sinus complementorum omnium arcuum, initio facto à <lb/> <anchor type="note" xlink:label="note-205-01a" xlink:href="note-205-01"/> principto Quadrantis, omnium arcuum ſecantes eruuntur, vt ex demonſtratis liquet. <lb/></s> <s xml:id="echoid-s7836" xml:space="preserve">Ex quo facilima erit conſtructio tabulæ ſecantium.</s> <s xml:id="echoid-s7837" xml:space="preserve"/> </p> <div xml:id="echoid-div554" type="float" level="2" n="2"> <note position="left" xlink:label="note-204-04" xlink:href="note-204-04a" xml:space="preserve">Qua tõne <lb/>ſecátes om-<lb/>niú arcuũ <lb/>inueſtigen-<lb/>tur.</note> <note position="right" xlink:label="note-205-01" xlink:href="note-205-01a" xml:space="preserve">Sola diui-<lb/>ſione eiuſ-<lb/>dé ſemper <lb/>numeri p ſi <lb/>nus omnes <lb/>ſecantes in <lb/>ueniuntur.</note> </div> </div> <div xml:id="echoid-div556" type="section" level="1" n="260"> <head xml:id="echoid-head287" xml:space="preserve">THEOR 11. PROPOS. 19.</head> <p> <s xml:id="echoid-s7838" xml:space="preserve">TANGENS cuiuſuis arcus, qui ſemiſſe qua-<lb/> <anchor type="note" xlink:label="note-205-02a" xlink:href="note-205-02"/> drantis maior ſit, æqualis eſt tangenti & </s> <s xml:id="echoid-s7839" xml:space="preserve">ſecanti ſi-<lb/>mul arcus, qui duplus ſit exceſſus, quo datus ar-<lb/>cus ſemiſſem quadrantis ſuperat.</s> <s xml:id="echoid-s7840" xml:space="preserve"/> </p> <div xml:id="echoid-div556" type="float" level="2" n="1"> <note position="right" xlink:label="note-205-02" xlink:href="note-205-02a" xml:space="preserve">Tangens ar <lb/>cus maioris <lb/>ſemiſle qua <lb/>diantis, cui <lb/>tangenti, & <lb/>ſecanti ſi-<lb/>mul ſit æ-<lb/>qualis.</note> </div> <p> <s xml:id="echoid-s7841" xml:space="preserve">IN quadrante ABC, ſit CG, tangens arcus CF, qui ſemiſſe quadrantis <lb/>maior ſit, inter ſemidiametrum AC, & </s> <s xml:id="echoid-s7842" xml:space="preserve">ſecantem AG, eiuſdem arcus CF, com <lb/>prehenſa. </s> <s xml:id="echoid-s7843" xml:space="preserve">Dico CG, æqualem eſſe tangenti, & </s> <s xml:id="echoid-s7844" xml:space="preserve">ſecanti ſimul arcus, qui duplus <lb/>ſit exceſſus, quo arcus CF, ſemiſſem quadrantis <lb/> <anchor type="figure" xlink:label="fig-205-01a" xlink:href="fig-205-01"/> ſuperat. </s> <s xml:id="echoid-s7845" xml:space="preserve">Sumpto enim arcu FD, ipſi FB, æquali, <lb/>ducatur recta AD, extendaturq́; </s> <s xml:id="echoid-s7846" xml:space="preserve">vſque ad E. </s> <s xml:id="echoid-s7847" xml:space="preserve">Et <lb/>quoniam anguli BAF, FAE, ob æquales arcus <lb/> <anchor type="note" xlink:label="note-205-03a" xlink:href="note-205-03"/> BF, DF, æquales ſunt: </s> <s xml:id="echoid-s7848" xml:space="preserve">Et angulo BAF, æqualis <lb/>eſt alternus angulus G; </s> <s xml:id="echoid-s7849" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s7850" xml:space="preserve">idem angulus <lb/> <anchor type="note" xlink:label="note-205-04a" xlink:href="note-205-04"/> G, angulo GAE, æqualis. </s> <s xml:id="echoid-s7851" xml:space="preserve">Quare rectæ EG, EA, <lb/> <anchor type="note" xlink:label="note-205-05a" xlink:href="note-205-05"/> æquales ſunt: </s> <s xml:id="echoid-s7852" xml:space="preserve">ac propterea, addita communi CE, <lb/>erit CG, tota tangens arcus CF, duabus CE, & </s> <s xml:id="echoid-s7853" xml:space="preserve"><lb/>AE, hoc eſt, tangenti, & </s> <s xml:id="echoid-s7854" xml:space="preserve">ſecanti arcus CD, ſimul <lb/>æqualis. </s> <s xml:id="echoid-s7855" xml:space="preserve">Dico iam arcum CD, duplum eſſe exceſ-<lb/>ſus quo arcus CF, propoſitus ſemiſſem quadrãtis ſuperat. </s> <s xml:id="echoid-s7856" xml:space="preserve">Producto enim ar-<lb/>cu quadrantis ad partes B, ſumptoq́; </s> <s xml:id="echoid-s7857" xml:space="preserve">arcu BH, æquali ipſi CD, cum & </s> <s xml:id="echoid-s7858" xml:space="preserve">arcus <lb/>FB, arcui FD, ſit æqualis, erit totus arcus FH, toti arcui CF, æqualis, ac pro-<lb/>inde arcus CH, duplus erit arcus CF. </s> <s xml:id="echoid-s7859" xml:space="preserve">Quoniam vero arcus CH, quadrantem <lb/>CB, ſuperat arcu BH, hoc eſt, arcu CD; </s> <s xml:id="echoid-s7860" xml:space="preserve">ſuperabit CF, ſemiſsis arcus CH, ſe <lb/> <anchor type="note" xlink:label="note-205-06a" xlink:href="note-205-06"/> miſsem quadrantis CB, ſemiſse exceſſus CD. </s> <s xml:id="echoid-s7861" xml:space="preserve">Arcus igitur CD, duplus eſt ex-<lb/>ceſſus, quo datus arcus CF, ſemiſſem quadrantis ſuperat. </s> <s xml:id="echoid-s7862" xml:space="preserve">Eſt auté oſtenſum, <lb/>CG, tangentem arcus CF, æqualem eſſe tangenti CE, & </s> <s xml:id="echoid-s7863" xml:space="preserve">ſecanti AE, ſimul ar-<lb/>cus CD. </s> <s xml:id="echoid-s7864" xml:space="preserve">Igitur tangens cuiuſuis arcus, qui ſemiſſe quadrantis maior ſit, æqua <lb/>lis eſt tangenti, & </s> <s xml:id="echoid-s7865" xml:space="preserve">ſecanti ſimul arcus, qui duplus ſit exceſſus, quo datus ar-<lb/>cus ſemiſſem quadrantis ſuperat. </s> <s xml:id="echoid-s7866" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s7867" xml:space="preserve"/> </p> <div xml:id="echoid-div557" type="float" level="2" n="2"> <figure xlink:label="fig-205-01" xlink:href="fig-205-01a"> <image file="205-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/205-01"/> </figure> <note position="right" xlink:label="note-205-03" xlink:href="note-205-03a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-205-04" xlink:href="note-205-04a" xml:space="preserve">29. primi.</note> <note position="right" xlink:label="note-205-05" xlink:href="note-205-05a" xml:space="preserve">6. primi.</note> <note position="right" xlink:label="note-205-06" xlink:href="note-205-06a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div559" type="section" level="1" n="261"> <head xml:id="echoid-head288" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s7868" xml:space="preserve">HANC propoſitionem nonnulli ita proponunt.</s> <s xml:id="echoid-s7869" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7870" xml:space="preserve">SECANS cuiuſuis arcus vna cum tangente eiuſdem æqualis eſt <lb/> <anchor type="note" xlink:label="note-205-07a" xlink:href="note-205-07"/> tangenti arcus compoſiti ex dato arcu, & </s> <s xml:id="echoid-s7871" xml:space="preserve">ſemiſſe complementi <lb/>eiuſdem.</s> <s xml:id="echoid-s7872" xml:space="preserve"/> </p> <div xml:id="echoid-div559" type="float" level="2" n="1"> <note position="right" xlink:label="note-205-07" xlink:href="note-205-07a" xml:space="preserve">Secás @@ tá-<lb/>gens eiuſ-<lb/>dem arcus <lb/>cui tágenti <lb/>ſimul ęqua <lb/>les ſint.</note> </div> <p style="it"> <s xml:id="echoid-s7873" xml:space="preserve">N A M in eadem figura ſit AE, ſecans, & </s> <s xml:id="echoid-s7874" xml:space="preserve"><emph style="sc">Ce</emph>, tangens eiuſdem arcus CD. </s> <s xml:id="echoid-s7875" xml:space="preserve">Secto <pb o="194" file="206" n="206" rhead=""/> autem arcu BD, nempe complemente arcus C D, bifariam in F, ducatur ex centre <lb/>A, per <emph style="sc">F</emph>, recta A G, ſecans tangentem C <emph style="sc">E</emph>, productam in <emph style="sc">G</emph>. </s> <s xml:id="echoid-s7876" xml:space="preserve">Eritq́; </s> <s xml:id="echoid-s7877" xml:space="preserve">C <emph style="sc">G</emph>, tangens <lb/>arcus <emph style="sc">Cf</emph>, compoſiti ex dato arcu CD, & </s> <s xml:id="echoid-s7878" xml:space="preserve"><emph style="sc">Df</emph>, ſemiſſe complementi DB. </s> <s xml:id="echoid-s7879" xml:space="preserve">Dico ſe-<lb/>cantem <emph style="sc">Ae</emph>, & </s> <s xml:id="echoid-s7880" xml:space="preserve">tangentẽ <emph style="sc">Ce</emph>. </s> <s xml:id="echoid-s7881" xml:space="preserve">ſimul æquales eſſe tangenti <emph style="sc">Cg</emph>. </s> <s xml:id="echoid-s7882" xml:space="preserve">Quia enim anguli BAF, <lb/> <anchor type="note" xlink:label="note-206-01a" xlink:href="note-206-01"/> FAE, æquales ſunt, propter æquales arcus <emph style="sc">Bf</emph>, <emph style="sc">F</emph>D; </s> <s xml:id="echoid-s7883" xml:space="preserve">& </s> <s xml:id="echoid-s7884" xml:space="preserve">angulo BAF, alternus <lb/> <anchor type="note" xlink:label="note-206-02a" xlink:href="note-206-02"/> angulus G, æqualis eſt; </s> <s xml:id="echoid-s7885" xml:space="preserve">erit quoque angulus idem G, angulo <emph style="sc">GAe</emph>, æqualis Quare <lb/> <anchor type="note" xlink:label="note-206-03a" xlink:href="note-206-03"/> aquales ſunt rectæ <emph style="sc">E</emph>A, <emph style="sc">E</emph>G: </s> <s xml:id="echoid-s7886" xml:space="preserve">at que adeo, addita communi <emph style="sc">E</emph>C, duæ A E, <emph style="sc">E</emph>C, ſimul <lb/>toti CG, æquales erunt.</s> <s xml:id="echoid-s7887" xml:space="preserve"/> </p> <div xml:id="echoid-div560" type="float" level="2" n="2"> <note position="left" xlink:label="note-206-01" xlink:href="note-206-01a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-206-02" xlink:href="note-206-02a" xml:space="preserve">29. primi.</note> <note position="left" xlink:label="note-206-03" xlink:href="note-206-03a" xml:space="preserve">6. primi.</note> </div> </div> <div xml:id="echoid-div562" type="section" level="1" n="262"> <head xml:id="echoid-head289" xml:space="preserve">THEOR. 12. PROPOS. 20.</head> <p> <s xml:id="echoid-s7888" xml:space="preserve">SECANS cuiuſuis arcus æqualis eſt tangen-<lb/> <anchor type="note" xlink:label="note-206-04a" xlink:href="note-206-04"/> ti eiuſdem, vna cum tangente ſemiſſis comple-<lb/>menti arcus eiuſdem.</s> <s xml:id="echoid-s7889" xml:space="preserve"/> </p> <div xml:id="echoid-div562" type="float" level="2" n="1"> <note position="left" xlink:label="note-206-04" xlink:href="note-206-04a" xml:space="preserve">Secans cu-<lb/>iuſuis arcꝰ <lb/>quorũ duo <lb/>rú arcuum <lb/>tangentibꝰ <lb/>ſit æqualis.</note> </div> <p> <s xml:id="echoid-s7890" xml:space="preserve">IN quadrante ABC, ſit AD, ſecans, & </s> <s xml:id="echoid-s7891" xml:space="preserve">CD, tangens arcus CE, cuius <lb/>complementi EB, ſemiſsis ſit EF, vel FB, & </s> <s xml:id="echoid-s7892" xml:space="preserve">huic ſemiſsi æqualis ſit arcus <lb/>CG. </s> <s xml:id="echoid-s7893" xml:space="preserve">Ducta autem recta AG, & </s> <s xml:id="echoid-s7894" xml:space="preserve">producta, donec cum DC, protracta coeat in <lb/>H, erit CH, tangens arcus CG, qui ſemiſsis eſt complementi arcus CE. <lb/></s> <s xml:id="echoid-s7895" xml:space="preserve">Dico ſecantem AD, æqualem eſſe tangenti <lb/> <anchor type="figure" xlink:label="fig-206-01a" xlink:href="fig-206-01"/> CD, & </s> <s xml:id="echoid-s7896" xml:space="preserve">tangenti CH, ſimul, hoc eſt, toti li-<lb/>neæ DH. </s> <s xml:id="echoid-s7897" xml:space="preserve">Quoniam enim anguli EAF, CAG, <lb/> <anchor type="note" xlink:label="note-206-05a" xlink:href="note-206-05"/> æquales ſunt, ob æquales arcus EF, CG; <lb/></s> <s xml:id="echoid-s7898" xml:space="preserve">addito communi angulo EAC, erunt toti<unsure/> <lb/>anguli FAC, EAH, æquales. </s> <s xml:id="echoid-s7899" xml:space="preserve">Rurſus quia <lb/>in triangulo rectangulo ACH, duo anguli <lb/> <anchor type="note" xlink:label="note-206-06a" xlink:href="note-206-06"/> A, H, vni recto, nimirũ angulo BAC, æqua-<lb/>les ſunt; </s> <s xml:id="echoid-s7900" xml:space="preserve">ablatis angulis BAF, CAH, qui <lb/>propter æquales arcus BF, CG, æquales ſunt, <lb/> <anchor type="note" xlink:label="note-206-07a" xlink:href="note-206-07"/> erunt reliqui anguli FAC, & </s> <s xml:id="echoid-s7901" xml:space="preserve">H, æquales. </s> <s xml:id="echoid-s7902" xml:space="preserve">Eſt <lb/>autem angulus FAC, oſtenſus æqualis angu <lb/>lo EAH. </s> <s xml:id="echoid-s7903" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s7904" xml:space="preserve">angulus H, eidem angulo EAH, æqualis erit: </s> <s xml:id="echoid-s7905" xml:space="preserve">ac propte-<lb/>rea rectæ AD, DH, æquales erunt, hoc eſt, ſecans AD, tangentibus DC, CH, <lb/> <anchor type="note" xlink:label="note-206-08a" xlink:href="note-206-08"/> æqualis erit. </s> <s xml:id="echoid-s7906" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s7907" xml:space="preserve"/> </p> <div xml:id="echoid-div563" type="float" level="2" n="2"> <figure xlink:label="fig-206-01" xlink:href="fig-206-01a"> <image file="206-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/206-01"/> </figure> <note position="left" xlink:label="note-206-05" xlink:href="note-206-05a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-206-06" xlink:href="note-206-06a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-206-07" xlink:href="note-206-07a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-206-08" xlink:href="note-206-08a" xml:space="preserve">6. primi.</note> </div> <p> <s xml:id="echoid-s7908" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s7909" xml:space="preserve">Sit rurſus AD, ſecans, & </s> <s xml:id="echoid-s7910" xml:space="preserve">CD, tangens arcus CE. </s> <s xml:id="echoid-s7911" xml:space="preserve">Dico ſecan-<lb/>tem AD, æqualem eſſe tangenti CD, vnà cum tangente ſemiſsis complemen <lb/>ti arcus EC, ſeu anguli DAC, hoc eſt, vna cum tangente ſemiſsis anguli D, <lb/>qui complementum eſt anguli DAC, cum ambo in triangulo rectágulo ACD, <lb/>vni recto ſint æquales. </s> <s xml:id="echoid-s7912" xml:space="preserve">Centro namque D, & </s> <s xml:id="echoid-s7913" xml:space="preserve">interuallo DA, arcus circuli de-<lb/> <anchor type="note" xlink:label="note-206-09a" xlink:href="note-206-09"/> ſcribatur AHI, ſecans DC, productam in H, & </s> <s xml:id="echoid-s7914" xml:space="preserve">AC, productam in I, ducan-<lb/>turq́; </s> <s xml:id="echoid-s7915" xml:space="preserve">rectæ AH, HI. </s> <s xml:id="echoid-s7916" xml:space="preserve">Quia igitur recta DC, ex centro D, circuli AHI, edu-<lb/>cta ſecans rectam AI, ad angulos rectos, ſecat eam bifariam; </s> <s xml:id="echoid-s7917" xml:space="preserve">ſecabit eadem <lb/> <anchor type="note" xlink:label="note-206-10a" xlink:href="note-206-10"/> DCH, & </s> <s xml:id="echoid-s7918" xml:space="preserve">arcum AHI, bifariam, ex lemmate in definitionibus demonſtrato. <lb/></s> <s xml:id="echoid-s7919" xml:space="preserve">Quare anguli CAH, & </s> <s xml:id="echoid-s7920" xml:space="preserve">I, æquales ſunt. </s> <s xml:id="echoid-s7921" xml:space="preserve">Quoniam autem, cum anguli D, & </s> <s xml:id="echoid-s7922" xml:space="preserve">I, <lb/> <anchor type="note" xlink:label="note-206-11a" xlink:href="note-206-11"/> candem habeant baſim arcum AH, & </s> <s xml:id="echoid-s7923" xml:space="preserve">ille ſit ad centrum D, hic vero ad cir- <pb o="195" file="207" n="207" rhead=""/> cunferentiam, angulus D, anguli I, duplus eſt; </s> <s xml:id="echoid-s7924" xml:space="preserve">erit quoque idem angulus D, <lb/> <anchor type="note" xlink:label="note-207-01a" xlink:href="note-207-01"/> anguli CAH, duplus: </s> <s xml:id="echoid-s7925" xml:space="preserve">ac proinde angulus CAH, ſemiſsis erit anguli D, qui <lb/>complementum eſt anguli DAC. </s> <s xml:id="echoid-s7926" xml:space="preserve">Cum ergo CH, tangens ſit anguli CAH, <lb/>ſitq́; </s> <s xml:id="echoid-s7927" xml:space="preserve">DA, recta rectæ DH, ex defin. </s> <s xml:id="echoid-s7928" xml:space="preserve">circuli, æqualis: </s> <s xml:id="echoid-s7929" xml:space="preserve">liquido conſtat, ſecan-<lb/>tem AD, arcus CE, æqualem eſſe tangenti CD, eiuſdem arcus, vnà cum CH, <lb/>tangente ſemiſsis complementi arcus CE, ſeu anguli CAD. </s> <s xml:id="echoid-s7930" xml:space="preserve">Quapropter Se-<lb/>cans cuiuſuis arcus æqualis eſt tangenti eiuſdẽ, &</s> <s xml:id="echoid-s7931" xml:space="preserve">c. </s> <s xml:id="echoid-s7932" xml:space="preserve">Quod demonſtrandũ erat.</s> <s xml:id="echoid-s7933" xml:space="preserve"/> </p> <div xml:id="echoid-div564" type="float" level="2" n="3"> <note position="left" xlink:label="note-206-09" xlink:href="note-206-09a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-206-10" xlink:href="note-206-10a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-206-11" xlink:href="note-206-11a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-207-01" xlink:href="note-207-01a" xml:space="preserve">20. tertij.</note> </div> </div> <div xml:id="echoid-div566" type="section" level="1" n="263"> <head xml:id="echoid-head290" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s7934" xml:space="preserve">EX preximis duobus theorematibus mirificum nobis compendium ſuppeditatur ad <lb/> <anchor type="note" xlink:label="note-207-02a" xlink:href="note-207-02"/> tabulam tam Tangentium, quàm Secantium conſtruendam. </s> <s xml:id="echoid-s7935" xml:space="preserve">Nam ſiper ea, quæ pro-<lb/>poſ. </s> <s xml:id="echoid-s7936" xml:space="preserve">18. </s> <s xml:id="echoid-s7937" xml:space="preserve">eiuſq́; </s> <s xml:id="echoid-s7938" xml:space="preserve">ſcholio præcepim{us}, tangentes omnium arcuum per ſingula minuta ex-<lb/>tenſorum vſque ad ſemiſſem quadrantis, ſecantes vero omnium arcuum totius qua-<lb/>drantis inquiramus; </s> <s xml:id="echoid-s7939" xml:space="preserve">inueniemus earum beneficio per ſolam additionem tangentes <lb/>aliorum arcuum, vſque ad arcum grad 67. </s> <s xml:id="echoid-s7940" xml:space="preserve">Min. </s> <s xml:id="echoid-s7941" xml:space="preserve">30 ſi nimirum tangentem, & </s> <s xml:id="echoid-s7942" xml:space="preserve">ſecan-<lb/>tem cuiuſq; </s> <s xml:id="echoid-s7943" xml:space="preserve">arcus minoris ſemiſſe quadrantis, qui minuta numero paria contineat, in <lb/>vnam ſummá colligamus: </s> <s xml:id="echoid-s7944" xml:space="preserve">propterea quòd tangens cuiuſuis arc{us} maioris ſemiſſe qua-<lb/> <anchor type="note" xlink:label="note-207-03a" xlink:href="note-207-03"/> drantis æqualis eſt tangenti, ac ſecanti arcus, qui duplus ſit exceſſus, quo datus arcus <lb/>ſemiſſem quadrantis ſuperat; </s> <s xml:id="echoid-s7945" xml:space="preserve">quales ſunt omnes arcus minutorũ parium vſq; </s> <s xml:id="echoid-s7946" xml:space="preserve">ad gra. <lb/></s> <s xml:id="echoid-s7947" xml:space="preserve">45. </s> <s xml:id="echoid-s7948" xml:space="preserve">vt arcus Min. </s> <s xml:id="echoid-s7949" xml:space="preserve">2. </s> <s xml:id="echoid-s7950" xml:space="preserve">Min. </s> <s xml:id="echoid-s7951" xml:space="preserve">4. </s> <s xml:id="echoid-s7952" xml:space="preserve">Min. </s> <s xml:id="echoid-s7953" xml:space="preserve">6. </s> <s xml:id="echoid-s7954" xml:space="preserve">&</s> <s xml:id="echoid-s7955" xml:space="preserve">c. </s> <s xml:id="echoid-s7956" xml:space="preserve">Exempli cauſa. </s> <s xml:id="echoid-s7957" xml:space="preserve">ſi deſideretur tangens arcus <lb/>gra. </s> <s xml:id="echoid-s7958" xml:space="preserve">45. </s> <s xml:id="echoid-s7959" xml:space="preserve">Min. </s> <s xml:id="echoid-s7960" xml:space="preserve">1 colligemus in vnam ſummam tangentem 5818. </s> <s xml:id="echoid-s7961" xml:space="preserve">& </s> <s xml:id="echoid-s7962" xml:space="preserve">ſecantẽ 10000002. </s> <s xml:id="echoid-s7963" xml:space="preserve"><lb/>arcus Min. </s> <s xml:id="echoid-s7964" xml:space="preserve">2 qui duplus eſt arcus Min. </s> <s xml:id="echoid-s7965" xml:space="preserve">1. </s> <s xml:id="echoid-s7966" xml:space="preserve">quo datus arcus grad. </s> <s xml:id="echoid-s7967" xml:space="preserve">45 Min. </s> <s xml:id="echoid-s7968" xml:space="preserve">1. </s> <s xml:id="echoid-s7969" xml:space="preserve">ſemißem quæ <lb/>drantis, hoc eſt, arcũ grad. </s> <s xml:id="echoid-s7970" xml:space="preserve">45 ſuperat. </s> <s xml:id="echoid-s7971" xml:space="preserve">Numerus enim cõſlatus 10005820. </s> <s xml:id="echoid-s7972" xml:space="preserve">dabit tã-<lb/>gentẽ propoſiti arcus grad. </s> <s xml:id="echoid-s7973" xml:space="preserve">45. </s> <s xml:id="echoid-s7974" xml:space="preserve">Min. </s> <s xml:id="echoid-s7975" xml:space="preserve">1. </s> <s xml:id="echoid-s7976" xml:space="preserve">qui ſemiſſem quadrantis ſuperat ſemiſſe arcus <lb/>Min. </s> <s xml:id="echoid-s7977" xml:space="preserve">2. </s> <s xml:id="echoid-s7978" xml:space="preserve">vt propoſ. </s> <s xml:id="echoid-s7979" xml:space="preserve">19. </s> <s xml:id="echoid-s7980" xml:space="preserve">oſtenſum eſt. </s> <s xml:id="echoid-s7981" xml:space="preserve">Idẽ arcus grad. </s> <s xml:id="echoid-s7982" xml:space="preserve">45. </s> <s xml:id="echoid-s7983" xml:space="preserve">Min. </s> <s xml:id="echoid-s7984" xml:space="preserve">1. </s> <s xml:id="echoid-s7985" xml:space="preserve">cõponitur ex arcu Min. </s> <s xml:id="echoid-s7986" xml:space="preserve"><lb/>2. </s> <s xml:id="echoid-s7987" xml:space="preserve">& </s> <s xml:id="echoid-s7988" xml:space="preserve">ſemiſſe cõplemẽti eiuſdẽ arcus, quod cõplectitur grad. </s> <s xml:id="echoid-s7989" xml:space="preserve">89. </s> <s xml:id="echoid-s7990" xml:space="preserve">Min. </s> <s xml:id="echoid-s7991" xml:space="preserve">58. </s> <s xml:id="echoid-s7992" xml:space="preserve">hoc eſt, ex ar-<lb/>cu Min. </s> <s xml:id="echoid-s7993" xml:space="preserve">2. </s> <s xml:id="echoid-s7994" xml:space="preserve">& </s> <s xml:id="echoid-s7995" xml:space="preserve">arcu grad. </s> <s xml:id="echoid-s7996" xml:space="preserve">44. </s> <s xml:id="echoid-s7997" xml:space="preserve">Min. </s> <s xml:id="echoid-s7998" xml:space="preserve">59 ac proinde, vt in ſcholio propoſ. </s> <s xml:id="echoid-s7999" xml:space="preserve">19. </s> <s xml:id="echoid-s8000" xml:space="preserve">demonſtraui-<lb/>mus, numerus 10005820. </s> <s xml:id="echoid-s8001" xml:space="preserve">conſlatus ex tangente, & </s> <s xml:id="echoid-s8002" xml:space="preserve">ſecante arcus Min. </s> <s xml:id="echoid-s8003" xml:space="preserve">2. </s> <s xml:id="echoid-s8004" xml:space="preserve">dabit tan-<lb/>gentem dicti arcus compoſiti ex arcu Min. </s> <s xml:id="echoid-s8005" xml:space="preserve">2. </s> <s xml:id="echoid-s8006" xml:space="preserve">& </s> <s xml:id="echoid-s8007" xml:space="preserve">ſemiſſe complementi eiuſdem.</s> <s xml:id="echoid-s8008" xml:space="preserve"/> </p> <div xml:id="echoid-div566" type="float" level="2" n="1"> <note position="right" xlink:label="note-207-02" xlink:href="note-207-02a" xml:space="preserve">Compédl @ <lb/>mirificum <lb/>pro cóſtru-<lb/>ctione tabu <lb/>læ tam tan <lb/>gentium, q̃ <lb/>ſecantium.</note> <note position="right" xlink:label="note-207-03" xlink:href="note-207-03a" xml:space="preserve">19. huius.</note> </div> <p style="it"> <s xml:id="echoid-s8009" xml:space="preserve">EADEM ratione tangens cuiuſuis arcus mauris ſemiſſe quadrantis compone-<lb/>tur ex tangẽte, & </s> <s xml:id="echoid-s8010" xml:space="preserve">ſecante arcus, qui duplus ſit exceſſus, quo arcus ille ſemiſſem qua-<lb/>drantis ſuperat. </s> <s xml:id="echoid-s8011" xml:space="preserve">Item ſi in vnam ſummã colligantur tangens, & </s> <s xml:id="echoid-s8012" xml:space="preserve">ſecans cuiuſuis ar-<lb/>cus minoris ſemiſſe quadrátis, conſlabitur tàgens arcus cõpoſiti ex illo arcu, & </s> <s xml:id="echoid-s8013" xml:space="preserve">ſemiſ <lb/>ſe complementi eiuſdem. </s> <s xml:id="echoid-s8014" xml:space="preserve">Vt tangens 14352451. </s> <s xml:id="echoid-s8015" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s8016" xml:space="preserve">55 Min. </s> <s xml:id="echoid-s8017" xml:space="preserve">8. </s> <s xml:id="echoid-s8018" xml:space="preserve">compoſita eſt <lb/>ex tangente 3692500. </s> <s xml:id="echoid-s8019" xml:space="preserve">& </s> <s xml:id="echoid-s8020" xml:space="preserve">ſecante 10659951. </s> <s xml:id="echoid-s8021" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s8022" xml:space="preserve">20 Min. </s> <s xml:id="echoid-s8023" xml:space="preserve">16. </s> <s xml:id="echoid-s8024" xml:space="preserve">qui duplus eſt <lb/>arcus grad. </s> <s xml:id="echoid-s8025" xml:space="preserve">10. </s> <s xml:id="echoid-s8026" xml:space="preserve">Min. </s> <s xml:id="echoid-s8027" xml:space="preserve">8. </s> <s xml:id="echoid-s8028" xml:space="preserve">quo datus arcus grad. </s> <s xml:id="echoid-s8029" xml:space="preserve">55 Min. </s> <s xml:id="echoid-s8030" xml:space="preserve">8. </s> <s xml:id="echoid-s8031" xml:space="preserve">quadrantis ſemiſſem ſuperat. <lb/></s> <s xml:id="echoid-s8032" xml:space="preserve">Item ſi tangens 3692500. </s> <s xml:id="echoid-s8033" xml:space="preserve">& </s> <s xml:id="echoid-s8034" xml:space="preserve">ſecans 10659951. </s> <s xml:id="echoid-s8035" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s8036" xml:space="preserve">20. </s> <s xml:id="echoid-s8037" xml:space="preserve">Min. </s> <s xml:id="echoid-s8038" xml:space="preserve">16. </s> <s xml:id="echoid-s8039" xml:space="preserve">in vnam <lb/>ſummam colligantur, conſlabitur tangens 14352451. </s> <s xml:id="echoid-s8040" xml:space="preserve">arcus grad 55. </s> <s xml:id="echoid-s8041" xml:space="preserve">Min. </s> <s xml:id="echoid-s8042" xml:space="preserve">8. </s> <s xml:id="echoid-s8043" xml:space="preserve">comſoſi-<lb/>ti ex illo arcu grad. </s> <s xml:id="echoid-s8044" xml:space="preserve">20. </s> <s xml:id="echoid-s8045" xml:space="preserve">Min. </s> <s xml:id="echoid-s8046" xml:space="preserve">16. </s> <s xml:id="echoid-s8047" xml:space="preserve">& </s> <s xml:id="echoid-s8048" xml:space="preserve">ſemiſſe complementi eiuſdem, nempe ex arcu grad. </s> <s xml:id="echoid-s8049" xml:space="preserve"><lb/>34. </s> <s xml:id="echoid-s8050" xml:space="preserve">Min. </s> <s xml:id="echoid-s8051" xml:space="preserve">52. </s> <s xml:id="echoid-s8052" xml:space="preserve">cum complementũ arcus grad. </s> <s xml:id="echoid-s8053" xml:space="preserve">20. </s> <s xml:id="echoid-s8054" xml:space="preserve">Min. </s> <s xml:id="echoid-s8055" xml:space="preserve">16. </s> <s xml:id="echoid-s8056" xml:space="preserve">cõprehendat grad. </s> <s xml:id="echoid-s8057" xml:space="preserve">69. </s> <s xml:id="echoid-s8058" xml:space="preserve">Min. </s> <s xml:id="echoid-s8059" xml:space="preserve">44.</s> <s xml:id="echoid-s8060" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8061" xml:space="preserve"><emph style="sc">ITAQVe</emph> propoſito arcu quocunque, qui maior ſit quadrantis dimidio, ſi ex <lb/>eo detrahatur ſemiſsis quadrantis, id eſt, arcus grad. </s> <s xml:id="echoid-s8062" xml:space="preserve">45. </s> <s xml:id="echoid-s8063" xml:space="preserve">& </s> <s xml:id="echoid-s8064" xml:space="preserve">reliqui arcus ſumatur <lb/>duplus, component tangens & </s> <s xml:id="echoid-s8065" xml:space="preserve">ſecans huius dupli arcus ſumpti tangentem propoſiti <lb/>illius arcus. </s> <s xml:id="echoid-s8066" xml:space="preserve">Vt ſi quæratur tangens arcus grad. </s> <s xml:id="echoid-s8067" xml:space="preserve">55. </s> <s xml:id="echoid-s8068" xml:space="preserve">Min. </s> <s xml:id="echoid-s8069" xml:space="preserve">8. </s> <s xml:id="echoid-s8070" xml:space="preserve">detrahendi erunt grad. <lb/></s> <s xml:id="echoid-s8071" xml:space="preserve">45. </s> <s xml:id="echoid-s8072" xml:space="preserve">ex eo, & </s> <s xml:id="echoid-s8073" xml:space="preserve">reliqui arcus grad. </s> <s xml:id="echoid-s8074" xml:space="preserve">10. </s> <s xml:id="echoid-s8075" xml:space="preserve">Min. </s> <s xml:id="echoid-s8076" xml:space="preserve">8. </s> <s xml:id="echoid-s8077" xml:space="preserve">ſumendus duplus arcus grad. </s> <s xml:id="echoid-s8078" xml:space="preserve">20. </s> <s xml:id="echoid-s8079" xml:space="preserve">Min, <pb o="196" file="208" n="208" rhead=""/> 16. </s> <s xml:id="echoid-s8080" xml:space="preserve">Huius enim tangens, & </s> <s xml:id="echoid-s8081" xml:space="preserve">ſecant component illius tangentem, vt demonſtratum eſt.</s> <s xml:id="echoid-s8082" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">19. huius.</note> <p style="it"> <s xml:id="echoid-s8083" xml:space="preserve">INVENTIS autem hoc modo tangentibus arcuum ſemiſſe quadrantis maiorũ <lb/>vſque ad arcum grad. </s> <s xml:id="echoid-s8084" xml:space="preserve">67. </s> <s xml:id="echoid-s8085" xml:space="preserve">Min. </s> <s xml:id="echoid-s8086" xml:space="preserve">30. </s> <s xml:id="echoid-s8087" xml:space="preserve">incluſiue; </s> <s xml:id="echoid-s8088" xml:space="preserve">ſi rurſus tangentem, ac ſecantem cuiuſq; <lb/></s> <s xml:id="echoid-s8089" xml:space="preserve">horum arcuum, qui minuta numero paria complectatur, (quales ſunt arcus grad. </s> <s xml:id="echoid-s8090" xml:space="preserve">45. </s> <s xml:id="echoid-s8091" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s8092" xml:space="preserve">2. </s> <s xml:id="echoid-s8093" xml:space="preserve">& </s> <s xml:id="echoid-s8094" xml:space="preserve">grad. </s> <s xml:id="echoid-s8095" xml:space="preserve">45. </s> <s xml:id="echoid-s8096" xml:space="preserve">Min. </s> <s xml:id="echoid-s8097" xml:space="preserve">4. </s> <s xml:id="echoid-s8098" xml:space="preserve">&</s> <s xml:id="echoid-s8099" xml:space="preserve">c.) </s> <s xml:id="echoid-s8100" xml:space="preserve">in vnam ſummam colligamus, reperiemus tangentes <lb/>maiorum adhuc arcuum, nempe grad. </s> <s xml:id="echoid-s8101" xml:space="preserve">67. </s> <s xml:id="echoid-s8102" xml:space="preserve">Min. </s> <s xml:id="echoid-s8103" xml:space="preserve">31. </s> <s xml:id="echoid-s8104" xml:space="preserve">& </s> <s xml:id="echoid-s8105" xml:space="preserve">grad. </s> <s xml:id="echoid-s8106" xml:space="preserve">67. </s> <s xml:id="echoid-s8107" xml:space="preserve">Min. </s> <s xml:id="echoid-s8108" xml:space="preserve">32. </s> <s xml:id="echoid-s8109" xml:space="preserve">&</s> <s xml:id="echoid-s8110" xml:space="preserve">c. </s> <s xml:id="echoid-s8111" xml:space="preserve">vſque ad <lb/>arcũ gra. </s> <s xml:id="echoid-s8112" xml:space="preserve">78. </s> <s xml:id="echoid-s8113" xml:space="preserve">Min. </s> <s xml:id="echoid-s8114" xml:space="preserve">45. </s> <s xml:id="echoid-s8115" xml:space="preserve">incluſiue. </s> <s xml:id="echoid-s8116" xml:space="preserve">Nam huius arcus grad. </s> <s xml:id="echoid-s8117" xml:space="preserve">33. </s> <s xml:id="echoid-s8118" xml:space="preserve">Min. </s> <s xml:id="echoid-s8119" xml:space="preserve">45. </s> <s xml:id="echoid-s8120" xml:space="preserve">quo arcus grad. </s> <s xml:id="echoid-s8121" xml:space="preserve"><lb/>78 Min 45 ſemiſſem quadrantis excedit, duplus eſt arcus grad. </s> <s xml:id="echoid-s8122" xml:space="preserve">67. </s> <s xml:id="echoid-s8123" xml:space="preserve">Min. </s> <s xml:id="echoid-s8124" xml:space="preserve">30. </s> <s xml:id="echoid-s8125" xml:space="preserve">cu-<lb/>ius tangens vltimo loco inuenta ſuit. </s> <s xml:id="echoid-s8126" xml:space="preserve">Item ex his tangentibus arcuum maiorum, <lb/>quàm grad. </s> <s xml:id="echoid-s8127" xml:space="preserve">67. </s> <s xml:id="echoid-s8128" xml:space="preserve">Min. </s> <s xml:id="echoid-s8129" xml:space="preserve">30. </s> <s xml:id="echoid-s8130" xml:space="preserve">@ſque ad arcum grad. </s> <s xml:id="echoid-s8131" xml:space="preserve">78. </s> <s xml:id="echoid-s8132" xml:space="preserve">Min. </s> <s xml:id="echoid-s8133" xml:space="preserve">45. </s> <s xml:id="echoid-s8134" xml:space="preserve">incluſiue inuentis; </s> <s xml:id="echoid-s8135" xml:space="preserve">ſi <lb/>rurſus tangentem, & </s> <s xml:id="echoid-s8136" xml:space="preserve">ſecantem cuiuſq; </s> <s xml:id="echoid-s8137" xml:space="preserve">illorum, qui minuta numero paria compre-<lb/>hendat, in vnam colligamus ſummá, inueniemus tágentes maiorum adhuc arcuum, <lb/>cuiuſmodi ſunt arcus grad. </s> <s xml:id="echoid-s8138" xml:space="preserve">78. </s> <s xml:id="echoid-s8139" xml:space="preserve">Min. </s> <s xml:id="echoid-s8140" xml:space="preserve">46. </s> <s xml:id="echoid-s8141" xml:space="preserve">& </s> <s xml:id="echoid-s8142" xml:space="preserve">grad. </s> <s xml:id="echoid-s8143" xml:space="preserve">78. </s> <s xml:id="echoid-s8144" xml:space="preserve">Min. </s> <s xml:id="echoid-s8145" xml:space="preserve">47. </s> <s xml:id="echoid-s8146" xml:space="preserve">&</s> <s xml:id="echoid-s8147" xml:space="preserve">c. </s> <s xml:id="echoid-s8148" xml:space="preserve">vſque ad arcum <lb/>grad. </s> <s xml:id="echoid-s8149" xml:space="preserve">84. </s> <s xml:id="echoid-s8150" xml:space="preserve">Min. </s> <s xml:id="echoid-s8151" xml:space="preserve">22. </s> <s xml:id="echoid-s8152" xml:space="preserve">incluſiue. </s> <s xml:id="echoid-s8153" xml:space="preserve">Nam huius arcus grad. </s> <s xml:id="echoid-s8154" xml:space="preserve">39. </s> <s xml:id="echoid-s8155" xml:space="preserve">Min. </s> <s xml:id="echoid-s8156" xml:space="preserve">22. </s> <s xml:id="echoid-s8157" xml:space="preserve">quo arcus grad. </s> <s xml:id="echoid-s8158" xml:space="preserve"><lb/>84. </s> <s xml:id="echoid-s8159" xml:space="preserve">Min. </s> <s xml:id="echoid-s8160" xml:space="preserve">22. </s> <s xml:id="echoid-s8161" xml:space="preserve">ſemißem quadrantis ſuperat, duplus eſt arcus grad. </s> <s xml:id="echoid-s8162" xml:space="preserve">78. </s> <s xml:id="echoid-s8163" xml:space="preserve">Min. </s> <s xml:id="echoid-s8164" xml:space="preserve">44. </s> <s xml:id="echoid-s8165" xml:space="preserve">qui maxi-<lb/>mus eſt eorum, qui minuta numero paria habent, & </s> <s xml:id="echoid-s8166" xml:space="preserve">quorum tangentes iam inuentæ <lb/>ſunt. </s> <s xml:id="echoid-s8167" xml:space="preserve">Sic etiam ex his inuentis reperiemus tangentes maiorum adhuc arcuum, quàm <lb/>grad. </s> <s xml:id="echoid-s8168" xml:space="preserve">84. </s> <s xml:id="echoid-s8169" xml:space="preserve">Min. </s> <s xml:id="echoid-s8170" xml:space="preserve">22. </s> <s xml:id="echoid-s8171" xml:space="preserve">vſque ad arcum grad. </s> <s xml:id="echoid-s8172" xml:space="preserve">87. </s> <s xml:id="echoid-s8173" xml:space="preserve">Min. </s> <s xml:id="echoid-s8174" xml:space="preserve">11. </s> <s xml:id="echoid-s8175" xml:space="preserve">Quia huius arcus grad 42. </s> <s xml:id="echoid-s8176" xml:space="preserve">Min. </s> <s xml:id="echoid-s8177" xml:space="preserve"><lb/>11 quo arcus grad 87 Min 11. </s> <s xml:id="echoid-s8178" xml:space="preserve">dimidium quadrantis excedit, duplus eſt arcus grad. </s> <s xml:id="echoid-s8179" xml:space="preserve"><lb/>84. </s> <s xml:id="echoid-s8180" xml:space="preserve">Min 22. </s> <s xml:id="echoid-s8181" xml:space="preserve">cuius tangens vltimo loco fuit inuenta. </s> <s xml:id="echoid-s8182" xml:space="preserve">Ex his vero repertis conficiemus <lb/>tangentes ſequentium arcuum, vſque ad arcum grad. </s> <s xml:id="echoid-s8183" xml:space="preserve">88. </s> <s xml:id="echoid-s8184" xml:space="preserve">Min. </s> <s xml:id="echoid-s8185" xml:space="preserve">35. </s> <s xml:id="echoid-s8186" xml:space="preserve">propterea quòd hu-<lb/>ius arcus grad. </s> <s xml:id="echoid-s8187" xml:space="preserve">43 Min. </s> <s xml:id="echoid-s8188" xml:space="preserve">35. </s> <s xml:id="echoid-s8189" xml:space="preserve">quo arcus grad. </s> <s xml:id="echoid-s8190" xml:space="preserve">88. </s> <s xml:id="echoid-s8191" xml:space="preserve">Min. </s> <s xml:id="echoid-s8192" xml:space="preserve">35. </s> <s xml:id="echoid-s8193" xml:space="preserve">quadrátis dimidiũ excedit, d@ <lb/>plus eſt arcus gra. </s> <s xml:id="echoid-s8194" xml:space="preserve">87. </s> <s xml:id="echoid-s8195" xml:space="preserve">Min. </s> <s xml:id="echoid-s8196" xml:space="preserve">10. </s> <s xml:id="echoid-s8197" xml:space="preserve">qui maximus eſt eorũ, qui minuta habent numero paria, <lb/>& </s> <s xml:id="echoid-s8198" xml:space="preserve">quorum tangentes proxime inuent æ ſunt. </s> <s xml:id="echoid-s8199" xml:space="preserve">Per has quoque reperiemus aliorum ar-<lb/>cuum tangentes, vſque ad arcum grad. </s> <s xml:id="echoid-s8200" xml:space="preserve">89. </s> <s xml:id="echoid-s8201" xml:space="preserve">Min. </s> <s xml:id="echoid-s8202" xml:space="preserve">17. </s> <s xml:id="echoid-s8203" xml:space="preserve">incluſiue; </s> <s xml:id="echoid-s8204" xml:space="preserve">cum huius arcus grad. </s> <s xml:id="echoid-s8205" xml:space="preserve"><lb/>44. </s> <s xml:id="echoid-s8206" xml:space="preserve">Min. </s> <s xml:id="echoid-s8207" xml:space="preserve">17. </s> <s xml:id="echoid-s8208" xml:space="preserve">quo arcus grad. </s> <s xml:id="echoid-s8209" xml:space="preserve">89 Min. </s> <s xml:id="echoid-s8210" xml:space="preserve">17. </s> <s xml:id="echoid-s8211" xml:space="preserve">dimidiatum quadrantem excedit, duplus ſit <lb/>arcus grad. </s> <s xml:id="echoid-s8212" xml:space="preserve">88. </s> <s xml:id="echoid-s8213" xml:space="preserve">Min. </s> <s xml:id="echoid-s8214" xml:space="preserve">34. </s> <s xml:id="echoid-s8215" xml:space="preserve">vtpote maximus eorum, qui minuta numero paria continent, <lb/>& </s> <s xml:id="echoid-s8216" xml:space="preserve">quorum iam tangentes ſunt cognitæ. </s> <s xml:id="echoid-s8217" xml:space="preserve">Beneficio deinde harum tangentium inuen-<lb/>tarum eliciemus tangentes aliorum arcuum, vſque ad arcum grad. </s> <s xml:id="echoid-s8218" xml:space="preserve">89. </s> <s xml:id="echoid-s8219" xml:space="preserve">Min. </s> <s xml:id="echoid-s8220" xml:space="preserve">38. </s> <s xml:id="echoid-s8221" xml:space="preserve">in-<lb/>cluſiue; </s> <s xml:id="echoid-s8222" xml:space="preserve">eo quod huius arcus grad. </s> <s xml:id="echoid-s8223" xml:space="preserve">44. </s> <s xml:id="echoid-s8224" xml:space="preserve">Min 38. </s> <s xml:id="echoid-s8225" xml:space="preserve">quo arcus grad. </s> <s xml:id="echoid-s8226" xml:space="preserve">89. </s> <s xml:id="echoid-s8227" xml:space="preserve">Min. </s> <s xml:id="echoid-s8228" xml:space="preserve">38. </s> <s xml:id="echoid-s8229" xml:space="preserve">ſemiſſem <lb/>quadrantis ſuperat, duplus eſt arcus grad. </s> <s xml:id="echoid-s8230" xml:space="preserve">89. </s> <s xml:id="echoid-s8231" xml:space="preserve">Min. </s> <s xml:id="echoid-s8232" xml:space="preserve">16. </s> <s xml:id="echoid-s8233" xml:space="preserve">qui maximus eſt eorum, qui <lb/>minuta numero habent paria, & </s> <s xml:id="echoid-s8234" xml:space="preserve">quorũ tangentes iam factæ ſunt notæ; </s> <s xml:id="echoid-s8235" xml:space="preserve">Hinc aliorum <lb/>arcuũ tãgentes inquiremus, vſq; </s> <s xml:id="echoid-s8236" xml:space="preserve">ad arcũ gra. </s> <s xml:id="echoid-s8237" xml:space="preserve">89. </s> <s xml:id="echoid-s8238" xml:space="preserve">Min. </s> <s xml:id="echoid-s8239" xml:space="preserve">49. </s> <s xml:id="echoid-s8240" xml:space="preserve">quippe qui ſuperet quadrã <lb/>tis dimidiũ drcugrad. </s> <s xml:id="echoid-s8241" xml:space="preserve">44. </s> <s xml:id="echoid-s8242" xml:space="preserve">Min. </s> <s xml:id="echoid-s8243" xml:space="preserve">49 cuius duplus eſt arcus grad. </s> <s xml:id="echoid-s8244" xml:space="preserve">89. </s> <s xml:id="echoid-s8245" xml:space="preserve">Min. </s> <s xml:id="echoid-s8246" xml:space="preserve">38. </s> <s xml:id="echoid-s8247" xml:space="preserve">ad quẽ pro-<lb/>xime peruenimus. </s> <s xml:id="echoid-s8248" xml:space="preserve">At ex his inueſtigabimus tangentes ſequentium arcuum vſq; </s> <s xml:id="echoid-s8249" xml:space="preserve">ad ar-<lb/>cum grad. </s> <s xml:id="echoid-s8250" xml:space="preserve">89. </s> <s xml:id="echoid-s8251" xml:space="preserve">Min 54 quippe qui quadran<unsure/>tis medietatẽ ſuperet arcu grad. </s> <s xml:id="echoid-s8252" xml:space="preserve">44. </s> <s xml:id="echoid-s8253" xml:space="preserve">Min. </s> <s xml:id="echoid-s8254" xml:space="preserve"><lb/>54. </s> <s xml:id="echoid-s8255" xml:space="preserve">cuius duplus ſt arcus grad. </s> <s xml:id="echoid-s8256" xml:space="preserve">89. </s> <s xml:id="echoid-s8257" xml:space="preserve">Min. </s> <s xml:id="echoid-s8258" xml:space="preserve">48. </s> <s xml:id="echoid-s8259" xml:space="preserve">qui maximus eſt eorum, qui minuta ha-<lb/>bent numero paria, & </s> <s xml:id="echoid-s8260" xml:space="preserve">quorum tangentes iam ſunt inuentæ. </s> <s xml:id="echoid-s8261" xml:space="preserve">Eadẽ ratione ex his in-<lb/>ueniemus tangentes ſequentium arcuum vſq@ ad arcum grad. </s> <s xml:id="echoid-s8262" xml:space="preserve">89. </s> <s xml:id="echoid-s8263" xml:space="preserve">Min. </s> <s xml:id="echoid-s8264" xml:space="preserve">57 Quia hu-<lb/>ius arcus grad. </s> <s xml:id="echoid-s8265" xml:space="preserve">44. </s> <s xml:id="echoid-s8266" xml:space="preserve">Min. </s> <s xml:id="echoid-s8267" xml:space="preserve">57. </s> <s xml:id="echoid-s8268" xml:space="preserve">quo arcus grad. </s> <s xml:id="echoid-s8269" xml:space="preserve">89. </s> <s xml:id="echoid-s8270" xml:space="preserve">Min 57 quadrantis dimidium ſupe-<lb/>rat, duplus eſt arcus grad. </s> <s xml:id="echoid-s8271" xml:space="preserve">89. </s> <s xml:id="echoid-s8272" xml:space="preserve">Min. </s> <s xml:id="echoid-s8273" xml:space="preserve">54. </s> <s xml:id="echoid-s8274" xml:space="preserve">ad quem proxime peruentum fuit. </s> <s xml:id="echoid-s8275" xml:space="preserve">Denique ex <lb/>tangente, & </s> <s xml:id="echoid-s8276" xml:space="preserve">ſecante arcus grad. </s> <s xml:id="echoid-s8277" xml:space="preserve">89. </s> <s xml:id="echoid-s8278" xml:space="preserve">Min. </s> <s xml:id="echoid-s8279" xml:space="preserve">56. </s> <s xml:id="echoid-s8280" xml:space="preserve">conficiemus tangentem arcus grad. </s> <s xml:id="echoid-s8281" xml:space="preserve">89. </s> <s xml:id="echoid-s8282" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s8283" xml:space="preserve">58. </s> <s xml:id="echoid-s8284" xml:space="preserve">Et hinc tangentem explor abimus arcus grad. </s> <s xml:id="echoid-s8285" xml:space="preserve">89. </s> <s xml:id="echoid-s8286" xml:space="preserve">Min. </s> <s xml:id="echoid-s8287" xml:space="preserve">59. </s> <s xml:id="echoid-s8288" xml:space="preserve">Atq; </s> <s xml:id="echoid-s8289" xml:space="preserve">ita, vt vides, <lb/>ex tangentibus arcuum vſque ad grad. </s> <s xml:id="echoid-s8290" xml:space="preserve">45. </s> <s xml:id="echoid-s8291" xml:space="preserve">& </s> <s xml:id="echoid-s8292" xml:space="preserve">ex ſecantibus omnium arcuum qua-<lb/>drantis perficitur integra tabula tangentium.</s> <s xml:id="echoid-s8293" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8294" xml:space="preserve">QVOD ſi ſecantem cuiuſcunq; </s> <s xml:id="echoid-s8295" xml:space="preserve">arcus ſubducas ex tangente alterius arcus, qui <lb/>ex priore illo, ac ſemiſſe complementi eiuſdem componitur, reliquam facies tangen-<lb/> <lb/> <pb o="197" file="209" n="209" rhead=""/> tem eiuſdem prioris illius arcus, cuius ſecantem ſubduxiſti Item ſi tangentem cuiuſ@ <lb/>libet arcus ex eiuſdem ſecante detrahas, remanebit tangens ſemiſsis complementi <lb/>arcus eiuſdem. </s> <s xml:id="echoid-s8296" xml:space="preserve">Primum conſtat ex ſcholio propoſ. </s> <s xml:id="echoid-s8297" xml:space="preserve">19. </s> <s xml:id="echoid-s8298" xml:space="preserve">vbi oſtenſum eſt, ſecantem, & </s> <s xml:id="echoid-s8299" xml:space="preserve"><lb/>tangentem cuiuſuis arcus ſimul æquales eſſe tangenti arcus compoſiti ex illo, & </s> <s xml:id="echoid-s8300" xml:space="preserve">ex <lb/>ſemiſſe complementi eiuſdem. </s> <s xml:id="echoid-s8301" xml:space="preserve">H@nc enim fit, vt ſecans ex cõpoſita hac tangente ablata <lb/>relinquat alteram illá tangentem. </s> <s xml:id="echoid-s8302" xml:space="preserve">Secundum vero liquet ex propoſ. </s> <s xml:id="echoid-s8303" xml:space="preserve">hac 20. </s> <s xml:id="echoid-s8304" xml:space="preserve">vbi de-<lb/>monſtrauimus, ſeca ntem cuiusuis arcus æqualem eſſe tangenti eiuſdem, vna cum tan-<lb/>gente ſemiſsis complementi arcus eiuſdem. </s> <s xml:id="echoid-s8305" xml:space="preserve">Quare huius ſemiſsis tangens reliqua fiet <lb/>poſt ſubtractionem alterius illius tangentis ex ſecante. </s> <s xml:id="echoid-s8306" xml:space="preserve">V. </s> <s xml:id="echoid-s8307" xml:space="preserve">g. </s> <s xml:id="echoid-s8308" xml:space="preserve">ſi ſecantem arcus grad. <lb/></s> <s xml:id="echoid-s8309" xml:space="preserve">20. </s> <s xml:id="echoid-s8310" xml:space="preserve">quæ eſt 10641777. </s> <s xml:id="echoid-s8311" xml:space="preserve">detrahamus ex 14281480. </s> <s xml:id="echoid-s8312" xml:space="preserve">tangente arcus grad. </s> <s xml:id="echoid-s8313" xml:space="preserve">55. </s> <s xml:id="echoid-s8314" xml:space="preserve"><lb/>compoſiti ex arcu grad. </s> <s xml:id="echoid-s8315" xml:space="preserve">20. </s> <s xml:id="echoid-s8316" xml:space="preserve">& </s> <s xml:id="echoid-s8317" xml:space="preserve">ſemi<unsure/>ſſe complementi eiuſdem, relinquetur tangens <lb/>3639703. </s> <s xml:id="echoid-s8318" xml:space="preserve">arcus eiuſdem grad. </s> <s xml:id="echoid-s8319" xml:space="preserve">20. </s> <s xml:id="echoid-s8320" xml:space="preserve">Item ſi 4244748. </s> <s xml:id="echoid-s8321" xml:space="preserve">tangentem arcus grad. </s> <s xml:id="echoid-s8322" xml:space="preserve">23. </s> <s xml:id="echoid-s8323" xml:space="preserve"><lb/>ex 10863603 ſecante eiuſdem arcus ſubducamus, remanebit tangens 6618855. </s> <s xml:id="echoid-s8324" xml:space="preserve">ar-<lb/>cus grad 33. </s> <s xml:id="echoid-s8325" xml:space="preserve">Min. </s> <s xml:id="echoid-s8326" xml:space="preserve">30. </s> <s xml:id="echoid-s8327" xml:space="preserve">hoc eſt, ſem ſsis complementi dati arcus grad. </s> <s xml:id="echoid-s8328" xml:space="preserve">23. </s> <s xml:id="echoid-s8329" xml:space="preserve">Rurſus <lb/>ſi 11547004. </s> <s xml:id="echoid-s8330" xml:space="preserve">ſecantem arcus grad. </s> <s xml:id="echoid-s8331" xml:space="preserve">30 ex 17320508. </s> <s xml:id="echoid-s8332" xml:space="preserve">tangente arcus grad. </s> <s xml:id="echoid-s8333" xml:space="preserve">60. </s> <s xml:id="echoid-s8334" xml:space="preserve"><lb/>qui ex arcu grad. </s> <s xml:id="echoid-s8335" xml:space="preserve">30. </s> <s xml:id="echoid-s8336" xml:space="preserve">& </s> <s xml:id="echoid-s8337" xml:space="preserve">ſemiſſe complementi eiuſdem componitur, auferamus, @elin <lb/>quetur tangens 5773504. </s> <s xml:id="echoid-s8338" xml:space="preserve">arcus grad. </s> <s xml:id="echoid-s8339" xml:space="preserve">30. </s> <s xml:id="echoid-s8340" xml:space="preserve">Et ſi 1763268. </s> <s xml:id="echoid-s8341" xml:space="preserve">tangentem arcus grad. </s> <s xml:id="echoid-s8342" xml:space="preserve">10. </s> <s xml:id="echoid-s8343" xml:space="preserve"><lb/>demamus ex 10154264. </s> <s xml:id="echoid-s8344" xml:space="preserve">ſecante eiuſdem arcus gra. </s> <s xml:id="echoid-s8345" xml:space="preserve">10. </s> <s xml:id="echoid-s8346" xml:space="preserve">remanebit tangẽs 8390996. </s> <s xml:id="echoid-s8347" xml:space="preserve"><lb/>arcus grad. </s> <s xml:id="echoid-s8348" xml:space="preserve">40. </s> <s xml:id="echoid-s8349" xml:space="preserve">qui ſemiſsis eſt complementi dicti arcus grad. </s> <s xml:id="echoid-s8350" xml:space="preserve">10.</s> <s xml:id="echoid-s8351" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8352" xml:space="preserve">IAM vero ſi per ea, quæ propoſ. </s> <s xml:id="echoid-s8353" xml:space="preserve">18. </s> <s xml:id="echoid-s8354" xml:space="preserve">eiusq́; </s> <s xml:id="echoid-s8355" xml:space="preserve">ſcholio tradidimus, tangentes om-<lb/>nium arcuum quadrantis per ſingula Minuta extenſorum inueſtigemus; </s> <s xml:id="echoid-s8356" xml:space="preserve">reperiemus <lb/>earum beneficio per ſolam additionem ſecantes omnium arcuum per bina minuta pro-<lb/>gredientium, ſi mmirum tangentem cuiuſuis arcus minuta numero paria habentis <lb/>addamus ad tangentem ſemiſsis complementi arcus eiuſdem: </s> <s xml:id="echoid-s8357" xml:space="preserve">propterea quòd Secans <lb/>cuiuſuis arcus æqualis eſt tangenti eiuſdem, vna cum tangente ſemiſsis complementi <lb/> <anchor type="note" xlink:label="note-209-01a" xlink:href="note-209-01"/> eiuſdem; </s> <s xml:id="echoid-s8358" xml:space="preserve">conſtat autem omnium arcuum minuta numero paria habentium comple-<lb/>menta ſemiſſes habere, Exempli cauſa, ſi deſideretur ſecans arcus Min. </s> <s xml:id="echoid-s8359" xml:space="preserve">2. </s> <s xml:id="echoid-s8360" xml:space="preserve">addemus eius <lb/>tangentem 5818. </s> <s xml:id="echoid-s8361" xml:space="preserve">ad 9994184. </s> <s xml:id="echoid-s8362" xml:space="preserve">tangentem arcus grad. </s> <s xml:id="echoid-s8363" xml:space="preserve">44. </s> <s xml:id="echoid-s8364" xml:space="preserve">Min. </s> <s xml:id="echoid-s8365" xml:space="preserve">59. </s> <s xml:id="echoid-s8366" xml:space="preserve">qui ſemiſsis eſt com <lb/>plementi arcus dati Min. </s> <s xml:id="echoid-s8367" xml:space="preserve">2. </s> <s xml:id="echoid-s8368" xml:space="preserve">Numerus enim compoſitus 10000002. </s> <s xml:id="echoid-s8369" xml:space="preserve">erit ſecans arcus <lb/>Min. </s> <s xml:id="echoid-s8370" xml:space="preserve">2. </s> <s xml:id="echoid-s8371" xml:space="preserve">sic etiam ſi quæratur ſecans arcus grad. </s> <s xml:id="echoid-s8372" xml:space="preserve">89. </s> <s xml:id="echoid-s8373" xml:space="preserve">Min. </s> <s xml:id="echoid-s8374" xml:space="preserve">58. </s> <s xml:id="echoid-s8375" xml:space="preserve">addemus eius tangentem <lb/>17188033689. </s> <s xml:id="echoid-s8376" xml:space="preserve">ad 2909 tangentem arcus Min. </s> <s xml:id="echoid-s8377" xml:space="preserve">1. </s> <s xml:id="echoid-s8378" xml:space="preserve">qui ſemiſsis eſt complementi arcus <lb/>dati grad. </s> <s xml:id="echoid-s8379" xml:space="preserve">89. </s> <s xml:id="echoid-s8380" xml:space="preserve">Min. </s> <s xml:id="echoid-s8381" xml:space="preserve">58. </s> <s xml:id="echoid-s8382" xml:space="preserve">Ná numerus conflatus 17188036598. </s> <s xml:id="echoid-s8383" xml:space="preserve">erit ſecans arcus grad. <lb/></s> <s xml:id="echoid-s8384" xml:space="preserve">89 Min. </s> <s xml:id="echoid-s8385" xml:space="preserve">58. </s> <s xml:id="echoid-s8386" xml:space="preserve">Hac ratione conficietur dimidiata pars tabulæ Tangentium: </s> <s xml:id="echoid-s8387" xml:space="preserve">at Tangen-<lb/>tes arcuum minuta numero imparia habentium, quoniam eorum complementa ſemiſ-<lb/>ſes non habent, niſi ad Secunda venire velimus, inueſtigandæ erunt, vt propoſ. </s> <s xml:id="echoid-s8388" xml:space="preserve">18. </s> <s xml:id="echoid-s8389" xml:space="preserve"><lb/>eiuſq; </s> <s xml:id="echoid-s8390" xml:space="preserve">ſcholio præcepimus.</s> <s xml:id="echoid-s8391" xml:space="preserve"/> </p> <div xml:id="echoid-div567" type="float" level="2" n="2"> <note position="right" xlink:label="note-209-01" xlink:href="note-209-01a" xml:space="preserve">20. huius.</note> </div> <p style="it"> <s xml:id="echoid-s8392" xml:space="preserve">RVRSVS ſecantem cuiuſuis arcus inueniemus, ſi eius tangentem demamus ex <lb/>tangente arcus compoſiti ex ar cu illo, & </s> <s xml:id="echoid-s8393" xml:space="preserve">ſemiſſe complementi eiuſdem arcus. </s> <s xml:id="echoid-s8394" xml:space="preserve">Nam cũ, <lb/>vt demonſtrauimu, ſecans cuiuſuis arcus, vnà cum tangente eiuſdem æqualis ſit tan-<lb/> <anchor type="note" xlink:label="note-209-02a" xlink:href="note-209-02"/> genti arcus compoſiti ex dato arcu, & </s> <s xml:id="echoid-s8395" xml:space="preserve">ſemiſſe complementi eiuſdem; </s> <s xml:id="echoid-s8396" xml:space="preserve">efficitur, vt tan <lb/>gens dati arcus ex tangente arcus ex eo, & </s> <s xml:id="echoid-s8397" xml:space="preserve">ſemiſſe complementi compoſiti ablata re-<lb/>linquat ſecantem eiuſdem dati arcus. </s> <s xml:id="echoid-s8398" xml:space="preserve">Vt ſi cupiamus ſecantem arcus Min. </s> <s xml:id="echoid-s8399" xml:space="preserve">2. </s> <s xml:id="echoid-s8400" xml:space="preserve">aufere-<lb/>mus 5818. </s> <s xml:id="echoid-s8401" xml:space="preserve">tangentem ipſius ex 10005820. </s> <s xml:id="echoid-s8402" xml:space="preserve">tangente arcus grad. </s> <s xml:id="echoid-s8403" xml:space="preserve">45. </s> <s xml:id="echoid-s8404" xml:space="preserve">Min. </s> <s xml:id="echoid-s8405" xml:space="preserve">1 cõpoſiti ex <lb/>arcu Min. </s> <s xml:id="echoid-s8406" xml:space="preserve">2. </s> <s xml:id="echoid-s8407" xml:space="preserve">& </s> <s xml:id="echoid-s8408" xml:space="preserve">ex arcu grad. </s> <s xml:id="echoid-s8409" xml:space="preserve">44. </s> <s xml:id="echoid-s8410" xml:space="preserve">Min. </s> <s xml:id="echoid-s8411" xml:space="preserve">59. </s> <s xml:id="echoid-s8412" xml:space="preserve">qui ſemiſsis eſt complementi arcus dati <lb/>Min. </s> <s xml:id="echoid-s8413" xml:space="preserve">2 Relictus namque numerus 10000002 erit ſecans arcus dati Min. </s> <s xml:id="echoid-s8414" xml:space="preserve">2. </s> <s xml:id="echoid-s8415" xml:space="preserve">Ita quoq; <lb/></s> <s xml:id="echoid-s8416" xml:space="preserve">ſi velimus habere ſecantem arcus grad. </s> <s xml:id="echoid-s8417" xml:space="preserve">60. </s> <s xml:id="echoid-s8418" xml:space="preserve">ſubducemus 17320508. </s> <s xml:id="echoid-s8419" xml:space="preserve">eius tangentem <lb/>ex 37320514. </s> <s xml:id="echoid-s8420" xml:space="preserve">tangente arcus grad. </s> <s xml:id="echoid-s8421" xml:space="preserve">75. </s> <s xml:id="echoid-s8422" xml:space="preserve">compoſiti ex dato arcu grad. </s> <s xml:id="echoid-s8423" xml:space="preserve">60. </s> <s xml:id="echoid-s8424" xml:space="preserve">& </s> <s xml:id="echoid-s8425" xml:space="preserve">ex arcu <pb o="198" file="210" n="210" rhead=""/> grad. </s> <s xml:id="echoid-s8426" xml:space="preserve">15. </s> <s xml:id="echoid-s8427" xml:space="preserve">qui ſemiſsis eſt complementi dati arcus grad. </s> <s xml:id="echoid-s8428" xml:space="preserve">60. </s> <s xml:id="echoid-s8429" xml:space="preserve">Remanebit enim numeras <lb/>20000006 pro ſecante dati arcus grad. </s> <s xml:id="echoid-s8430" xml:space="preserve">60.</s> <s xml:id="echoid-s8431" xml:space="preserve"/> </p> <div xml:id="echoid-div568" type="float" level="2" n="3"> <note position="right" xlink:label="note-209-02" xlink:href="note-209-02a" xml:space="preserve">Schol. 19. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s8432" xml:space="preserve">HAEC, quæ hoc ſcholio tradita à nobis ſunt, vera ſunt, ſi ſinus exquiſite inuen@ <lb/> <anchor type="note" xlink:label="note-210-01a" xlink:href="note-210-01"/> ti fuerint: </s> <s xml:id="echoid-s8433" xml:space="preserve">ſed quia non omnes ſinus accurate ſunt cogniti, maxime ſinus arcus grad. <lb/></s> <s xml:id="echoid-s8434" xml:space="preserve">1. </s> <s xml:id="echoid-s8435" xml:space="preserve">& </s> <s xml:id="echoid-s8436" xml:space="preserve">alij ex hoc dependentes, quales ſunt ſinus arcuum per ſingula minuta extenſo-<lb/>rum; </s> <s xml:id="echoid-s8437" xml:space="preserve">fit vt neq; </s> <s xml:id="echoid-s8438" xml:space="preserve">tangentes, neq; </s> <s xml:id="echoid-s8439" xml:space="preserve">ſecantes inuẽtæ per hoſce ſinus ſint admodũ accuratæ. </s> <s xml:id="echoid-s8440" xml:space="preserve"><lb/>Quare ſi ex inuentis quibuſdam aliæ per ſolam additionem, ſubtractionem ve inqui-<lb/>rantur vt hoc ſcholio docuimus, non parum different ab eiſdem, ſi per ſinus inueſtiga <lb/>rentur. </s> <s xml:id="echoid-s8441" xml:space="preserve">Nam tangentes & </s> <s xml:id="echoid-s8442" xml:space="preserve">ſecantes per ſinus inuentæ ex vno ſolo principio non omni <lb/>ex parte vero, nempe ex ſinubus, gignuntur: </s> <s xml:id="echoid-s8443" xml:space="preserve">at eædem per ſolam additionem, ſubtra, <lb/>ctionem ve procreatæ oriuntur ex pluribus falſis principijs, nimirum ex ſinubus pr <lb/>mum, deinde vero etiá ex tangentibus, & </s> <s xml:id="echoid-s8444" xml:space="preserve">ſecátibus per ſinus inuentis, quæ accuratæ <lb/>eſſe<unsure/> non poſſunt, vt diximus. </s> <s xml:id="echoid-s8445" xml:space="preserve">Magis exquiſite ergo cognoſcentur huiuſmodi lineæ per <lb/>ſinus, vt propoſ. </s> <s xml:id="echoid-s8446" xml:space="preserve">18. </s> <s xml:id="echoid-s8447" xml:space="preserve">eiuſq́; </s> <s xml:id="echoid-s8448" xml:space="preserve">ſcholio traditum eſt. </s> <s xml:id="echoid-s8449" xml:space="preserve">Hac ratione & </s> <s xml:id="echoid-s8450" xml:space="preserve">tabulam Tangentium, <lb/>& </s> <s xml:id="echoid-s8451" xml:space="preserve">tabulam Secantium breui ſupputabimus. </s> <s xml:id="echoid-s8452" xml:space="preserve">Non paruos enim errores in aliorum ta-<lb/>bulis deprehendimus; </s> <s xml:id="echoid-s8453" xml:space="preserve">vt tutò illis fid ere non poſsimus; </s> <s xml:id="echoid-s8454" xml:space="preserve">propterea quòd multas tangen-<lb/>tes, & </s> <s xml:id="echoid-s8455" xml:space="preserve">ſecátes vel per partem proportionalem, vel per ſolam additionem aut ſubtra-<lb/>ctionem inueſtigarunt, non autem omnes per ſinus. </s> <s xml:id="echoid-s8456" xml:space="preserve">Subiungemus tamen paulo infra <lb/>aliorum tabulas, donec per tempus nouas conſtruere licebit.</s> <s xml:id="echoid-s8457" xml:space="preserve"/> </p> <div xml:id="echoid-div569" type="float" level="2" n="4"> <note position="left" xlink:label="note-210-01" xlink:href="note-210-01a" xml:space="preserve">Tangétes, & <lb/>Secátes ma <lb/>gis eſſe ac <lb/>curatas, per <lb/>ſinus inué <lb/>tas, q̃ per <lb/>additioné, <lb/>ſubtractio -<lb/>néue, vt in <lb/>hoc ſcholio <lb/>traditú eſt.</note> </div> </div> <div xml:id="echoid-div571" type="section" level="1" n="264"> <head xml:id="echoid-head291" xml:space="preserve">THEOR. 13. PROPOS. 21.</head> <p> <s xml:id="echoid-s8458" xml:space="preserve">TANGENS cuiuſuis arcus eſt ad tangen-<lb/> <anchor type="note" xlink:label="note-210-02a" xlink:href="note-210-02"/> tem alterius arcus cuiuſlibet, vt tangens comple-<lb/>menti poſterioris arcus ad tangétem complemen-<lb/>ti prioris.</s> <s xml:id="echoid-s8459" xml:space="preserve"/> </p> <div xml:id="echoid-div571" type="float" level="2" n="1"> <note position="left" xlink:label="note-210-02" xlink:href="note-210-02a" xml:space="preserve">Tangentes <lb/>duorum at <lb/>cuú quotú-<lb/>libet sút re <lb/>ciprocè {pro}-<lb/>portionales <lb/>cũ tangen-<lb/>tibꝰ comple <lb/>métorú ar-<lb/>cuú eoiun-<lb/>dem.</note> </div> <p> <s xml:id="echoid-s8460" xml:space="preserve">IN quadrante ABC, arcus CD, tangens ſit CE, & </s> <s xml:id="echoid-s8461" xml:space="preserve">ſecans AE: </s> <s xml:id="echoid-s8462" xml:space="preserve">Item ar-<lb/>cus CF, tangens ſit CG, & </s> <s xml:id="echoid-s8463" xml:space="preserve">ſecans AG: </s> <s xml:id="echoid-s8464" xml:space="preserve">Ducta autem recta BH, circulum tan <lb/>gente, & </s> <s xml:id="echoid-s8465" xml:space="preserve">vtrique ſecanti AE, AG, occurrente in I, H; </s> <s xml:id="echoid-s8466" xml:space="preserve">erit BI, tangens com-<lb/>plementi arcus CD; </s> <s xml:id="echoid-s8467" xml:space="preserve">& </s> <s xml:id="echoid-s8468" xml:space="preserve">BH, tangens comple-<lb/> <anchor type="figure" xlink:label="fig-210-01a" xlink:href="fig-210-01"/> menti arcus CF. </s> <s xml:id="echoid-s8469" xml:space="preserve">Dico ita eſſe CE, tangentem <lb/>arcus CD, ad CG, tangentem arcus CF, vt eſt <lb/>BH, tangens complementi poſterioris arcus <lb/>CF, ad BI, tangentem complementi arcus prio <lb/>ris CD. </s> <s xml:id="echoid-s8470" xml:space="preserve">Cum enim ſinus totus ſit medius pro-<lb/> <anchor type="note" xlink:label="note-210-03a" xlink:href="note-210-03"/> portionalis tam inter CE, tangenté arcus CD, <lb/>& </s> <s xml:id="echoid-s8471" xml:space="preserve">BI, tangentem complementi arcus eiuſdem <lb/>CD, quàm inter CG, tangentem arcus CF, & </s> <s xml:id="echoid-s8472" xml:space="preserve"><lb/>BH, tangentem complementi arcus eiuſdem <lb/>CF; </s> <s xml:id="echoid-s8473" xml:space="preserve">erit tam rectangulum ſub CE, BI, quam re-<lb/>ctangulum ſub CG, BH, quadrato ſinus totius æquale: </s> <s xml:id="echoid-s8474" xml:space="preserve">ac proinde rectangu-<lb/> <anchor type="note" xlink:label="note-210-04a" xlink:href="note-210-04"/> lum ſub CE, BI, rectangulo ſub CG, BH, æquale erit. </s> <s xml:id="echoid-s8475" xml:space="preserve">Quare erit, vt CE, <lb/>prima ad CG, ſecundam, ita BH, tertia ad BI, quartam; </s> <s xml:id="echoid-s8476" xml:space="preserve">nempe vt CE, tan-<lb/> <anchor type="note" xlink:label="note-210-05a" xlink:href="note-210-05"/> <pb o="199" file="211" n="211" rhead=""/> gens arcus CD, ad CG, tangentem arcus CF, ita BH, tangens complemen-<lb/>ti arcus poſterioris CF, ad BI, tangentem complementi prioris arcus CD. <lb/></s> <s xml:id="echoid-s8477" xml:space="preserve">Tangens igitur cuiuſuis arcus eſt ad tangentem alterius, &</s> <s xml:id="echoid-s8478" xml:space="preserve">c. </s> <s xml:id="echoid-s8479" xml:space="preserve">Quod oſtenden-<lb/>dum erat.</s> <s xml:id="echoid-s8480" xml:space="preserve"/> </p> <div xml:id="echoid-div572" type="float" level="2" n="2"> <figure xlink:label="fig-210-01" xlink:href="fig-210-01a"> <image file="210-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/210-01"/> </figure> <note position="left" xlink:label="note-210-03" xlink:href="note-210-03a" xml:space="preserve">18. huius</note> <note position="left" xlink:label="note-210-04" xlink:href="note-210-04a" xml:space="preserve">17. ſexti.</note> <note position="left" xlink:label="note-210-05" xlink:href="note-210-05a" xml:space="preserve">16. ſexti.</note> </div> </div> <div xml:id="echoid-div574" type="section" level="1" n="265"> <head xml:id="echoid-head292" xml:space="preserve">THEOR. 14. PROPOS. 22.</head> <p> <s xml:id="echoid-s8481" xml:space="preserve">SECANS cuiuſuis arcus eſt ad Secantem al-<lb/> <anchor type="note" xlink:label="note-211-01a" xlink:href="note-211-01"/> terius arcus cuiuſlibet, vt ſinus complementi po <lb/>ſterioris arcus ad ſinum complementi prioris.</s> <s xml:id="echoid-s8482" xml:space="preserve"/> </p> <div xml:id="echoid-div574" type="float" level="2" n="1"> <note position="right" xlink:label="note-211-01" xlink:href="note-211-01a" xml:space="preserve">Secãtes du@ <lb/>rũ arcuum <lb/>quorũlibet <lb/>sũt recipro-<lb/>ce pportio <lb/>nales cũ fi-<lb/>nubus com <lb/>plemẽto rũ <lb/>arcuum eo <lb/>rundem.</note> </div> <p> <s xml:id="echoid-s8483" xml:space="preserve">IN quadrante ABC, ſit AD, ſecans arcus CE, & </s> <s xml:id="echoid-s8484" xml:space="preserve">AF, ſecans arcus CG: </s> <s xml:id="echoid-s8485" xml:space="preserve">& </s> <s xml:id="echoid-s8486" xml:space="preserve"><lb/>EH, ſinus complementi arcus CE, at GI, ſinus complementi arcus CG. </s> <s xml:id="echoid-s8487" xml:space="preserve">Di-<lb/>coita eſſe ſecantem AD, arcus CE, ad AF, ſecantem arcus CG, vt eſt GI, <lb/>ſinus complementi poſterioris arcus CG, ad EH, <lb/> <anchor type="figure" xlink:label="fig-211-01a" xlink:href="fig-211-01"/> ſinum complementi arcus prioris CE. </s> <s xml:id="echoid-s8488" xml:space="preserve">Quoniam <lb/>enim ſinus totus eſt medius proportionalis tã in-<lb/> <anchor type="note" xlink:label="note-211-02a" xlink:href="note-211-02"/> ter ſecantem AD, arcus CE, & </s> <s xml:id="echoid-s8489" xml:space="preserve">EH, ſinum com-<lb/>plementi eiuſdem arcus CE, quàm inter AF, ſe-<lb/>cantem arcus CG, & </s> <s xml:id="echoid-s8490" xml:space="preserve">GI, ſinum complementi eiu-<lb/>ſdem arcus CG; </s> <s xml:id="echoid-s8491" xml:space="preserve">erit tam rectangulum ſub AD, <lb/>EH, quàm rectangulum ſub AF, GI, quadrato <lb/> <anchor type="note" xlink:label="note-211-03a" xlink:href="note-211-03"/> ſinus totius æquale: </s> <s xml:id="echoid-s8492" xml:space="preserve">ac proinde rectangulum illud <lb/>huic æquale. </s> <s xml:id="echoid-s8493" xml:space="preserve">Quare erit vt AD, prima ad AF, ſe-<lb/> <anchor type="note" xlink:label="note-211-04a" xlink:href="note-211-04"/> cundam, ita GI, tertia ad EH, quartam; </s> <s xml:id="echoid-s8494" xml:space="preserve">hoc eſt, <lb/>vt AD, ſecans arcus CE, ad AF, ſecantem arcus CG, ita GI, ſinus comple-<lb/>menti arcus poſterioris CG, ad EH, ſinum complementi arcus prioris CE. <lb/></s> <s xml:id="echoid-s8495" xml:space="preserve">Secans igitur cuiuſuis arcus eſt ad ſecantem alterius arcus, &</s> <s xml:id="echoid-s8496" xml:space="preserve">c. </s> <s xml:id="echoid-s8497" xml:space="preserve">Quod demon-<lb/>ſtrandum erat.</s> <s xml:id="echoid-s8498" xml:space="preserve"/> </p> <div xml:id="echoid-div575" type="float" level="2" n="2"> <figure xlink:label="fig-211-01" xlink:href="fig-211-01a"> <image file="211-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/211-01"/> </figure> <note position="right" xlink:label="note-211-02" xlink:href="note-211-02a" xml:space="preserve">18. huius.</note> <note position="right" xlink:label="note-211-03" xlink:href="note-211-03a" xml:space="preserve">17. fexti.</note> <note position="right" xlink:label="note-211-04" xlink:href="note-211-04a" xml:space="preserve">16. fexti.</note> </div> </div> <div xml:id="echoid-div577" type="section" level="1" n="266"> <head xml:id="echoid-head293" xml:space="preserve">THEOR. 15. PROPOS. 23.</head> <p> <s xml:id="echoid-s8499" xml:space="preserve">SI plures ſint arcus æquali exceſſu progredien <lb/> <anchor type="note" xlink:label="note-211-05a" xlink:href="note-211-05"/> tes, habebunt tam tangentes, quàm Secantes ma-<lb/>iorum arcuum maiorem differentiam, quàm mi-<lb/>norum:</s> <s xml:id="echoid-s8500" xml:space="preserve">ita vt in tabula differentiæ tam tangentiũ, <lb/>quàm ſecantium ſemper creſcant vſque ad finem <lb/>quadrantis.</s> <s xml:id="echoid-s8501" xml:space="preserve"/> </p> <div xml:id="echoid-div577" type="float" level="2" n="1"> <note position="right" xlink:label="note-211-05" xlink:href="note-211-05a" xml:space="preserve">Tangentes <lb/>& ſecantes <lb/>arcuũ ęqna <lb/>liter creſcẽ <lb/>tiũ augent <lb/>femper dif-<lb/>ferentias.</note> </div> <p> <s xml:id="echoid-s8502" xml:space="preserve">IN quadrante ABC, ſintarcus CD, CE, CF, quorum differentiæ DE, <lb/>EF, æquales ſint, & </s> <s xml:id="echoid-s8503" xml:space="preserve">eorumdem tangentes ſint CG, CH, CI; </s> <s xml:id="echoid-s8504" xml:space="preserve">ſecantes autem <pb o="200" file="212" n="212" rhead=""/> AG, AH, AI.</s> <s xml:id="echoid-s8505" xml:space="preserve">Et quia in triangulo ACH, angulus C, rectus eſt; </s> <s xml:id="echoid-s8506" xml:space="preserve">erit AHC, <lb/>recto minor, cum ambo ſint duobus rectis minores. </s> <s xml:id="echoid-s8507" xml:space="preserve">Cum ergo duo anguli <lb/> <anchor type="note" xlink:label="note-212-01a" xlink:href="note-212-01"/> ad H, ſint duobus rectis æquales, erit AHI, maior recto, ac proinde angu-<lb/> <anchor type="note" xlink:label="note-212-02a" xlink:href="note-212-02"/> lus I, in triangulo AHI, recto minor. </s> <s xml:id="echoid-s8508" xml:space="preserve">Quare maior erit ſecans AI, ſecante <lb/> <anchor type="note" xlink:label="note-212-03a" xlink:href="note-212-03"/> <anchor type="figure" xlink:label="fig-212-01a" xlink:href="fig-212-01"/> AH. </s> <s xml:id="echoid-s8509" xml:space="preserve">Eadẽ ratione maior erit quam AG: </s> <s xml:id="echoid-s8510" xml:space="preserve">Item <lb/> <anchor type="note" xlink:label="note-212-04a" xlink:href="note-212-04"/> AH, maior, quã AG. </s> <s xml:id="echoid-s8511" xml:space="preserve">Abſcindatur ergo. </s> <s xml:id="echoid-s8512" xml:space="preserve">AK, ipſi <lb/>AH, & </s> <s xml:id="echoid-s8513" xml:space="preserve">AL, ipſi AG, æqualis. </s> <s xml:id="echoid-s8514" xml:space="preserve">Dico IH, differẽ <lb/>tiã tangentiũ CI, CH, arcuũ maiorũ CF, CE, <lb/>maiorem eſſe differentia HG, tãgentium CH, <lb/>CG, minorum arcuum CE, CD: </s> <s xml:id="echoid-s8515" xml:space="preserve">Item KI, <lb/>differentiam ſecantium AI, AH, arcuum ma-<lb/>iorum CF, CE, maiorem eſſe differentia LH, <lb/>ſecantium AH, AG, minorum arcuum CE, <lb/>CD. </s> <s xml:id="echoid-s8516" xml:space="preserve">Cum enim arcus DE, EF, æquales ſint, <lb/> <anchor type="note" xlink:label="note-212-05a" xlink:href="note-212-05"/> erunt & </s> <s xml:id="echoid-s8517" xml:space="preserve">anguli DAE, EAF, æquales: </s> <s xml:id="echoid-s8518" xml:space="preserve">ac pro-<lb/>inde angulus IAG, ſectus erit bifariam per re-<lb/> <anchor type="note" xlink:label="note-212-06a" xlink:href="note-212-06"/> ctam AH. </s> <s xml:id="echoid-s8519" xml:space="preserve">Igitur erit, vt IA, ad AG, ita IH, <lb/>ad HG: </s> <s xml:id="echoid-s8520" xml:space="preserve">Eſt autem AI, maior, quàm AG, vt oſtenſum eſt. </s> <s xml:id="echoid-s8521" xml:space="preserve">Recta ergo IH, <lb/>maior quoque erit, quàm HG. </s> <s xml:id="echoid-s8522" xml:space="preserve">quod eſt primum.</s> <s xml:id="echoid-s8523" xml:space="preserve"/> </p> <div xml:id="echoid-div578" type="float" level="2" n="2"> <note position="left" xlink:label="note-212-01" xlink:href="note-212-01a" xml:space="preserve">17. primi.</note> <note position="left" xlink:label="note-212-02" xlink:href="note-212-02a" xml:space="preserve">13. primi.</note> <note position="left" xlink:label="note-212-03" xlink:href="note-212-03a" xml:space="preserve">17. primi.</note> <figure xlink:label="fig-212-01" xlink:href="fig-212-01a"> <image file="212-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/212-01"/> </figure> <note position="left" xlink:label="note-212-04" xlink:href="note-212-04a" xml:space="preserve">19. primi.</note> <note position="left" xlink:label="note-212-05" xlink:href="note-212-05a" xml:space="preserve">27. tertij.</note> <note position="left" xlink:label="note-212-06" xlink:href="note-212-06a" xml:space="preserve">3. fexti.</note> </div> <p> <s xml:id="echoid-s8524" xml:space="preserve">DVCTIS iam FM, EN, DO, ad AC, perpendicularibus, nempe fi-<lb/>nubus rectis arcuum CF, CE, CD; </s> <s xml:id="echoid-s8525" xml:space="preserve">erit AM, ſinus complementi arcus CF; <lb/></s> <s xml:id="echoid-s8526" xml:space="preserve">& </s> <s xml:id="echoid-s8527" xml:space="preserve">AN, ſinus complementi arcus CE; </s> <s xml:id="echoid-s8528" xml:space="preserve">& </s> <s xml:id="echoid-s8529" xml:space="preserve">AO, ſinus complementi arcus CD, <lb/>vt in expoſitione definitionum dictum eſt. </s> <s xml:id="echoid-s8530" xml:space="preserve">Quoniam vero recta MN, maior <lb/> <anchor type="note" xlink:label="note-212-07a" xlink:href="note-212-07"/> eſt, quam NO; </s> <s xml:id="echoid-s8531" xml:space="preserve">maior erit proportio AN, ad NO, quàm ad MN: </s> <s xml:id="echoid-s8532" xml:space="preserve">Eſt autem <lb/> <anchor type="note" xlink:label="note-212-08a" xlink:href="note-212-08"/> adhuc maior proportio AO, ad NO, quàm AN, ad eandem NO. </s> <s xml:id="echoid-s8533" xml:space="preserve">Igitur <lb/>multo maior erit proportio AO, ad NO, quàm AN, ad MN. </s> <s xml:id="echoid-s8534" xml:space="preserve">Et per con-<lb/>uerſionem rationis, minor proportio AO, ad AN, quàm AN, ad AM: </s> <s xml:id="echoid-s8535" xml:space="preserve">hoc <lb/> <anchor type="note" xlink:label="note-212-09a" xlink:href="note-212-09"/> eſt, maior proportio AN, ad AM, quàm AO, ad AN. </s> <s xml:id="echoid-s8536" xml:space="preserve">Cum ergo ſit, vt <lb/>AN, ad AM, ita AI, ad AH; </s> <s xml:id="echoid-s8537" xml:space="preserve">Et vt AO, ad AN, ita AH, ad AG: </s> <s xml:id="echoid-s8538" xml:space="preserve">maior <lb/> <anchor type="note" xlink:label="note-212-10a" xlink:href="note-212-10"/> quoque erit proportio AI, ad AH, hoc eſt, ad AK, quàm AH, ad AG, <lb/>hoc eſt, ad AL. </s> <s xml:id="echoid-s8539" xml:space="preserve">Diuidendo ergo maior etiam proportio erit IK, ad AK, hoc <lb/> <anchor type="note" xlink:label="note-212-11a" xlink:href="note-212-11"/> eſt, ad AH, quàm HL, ad AL: </s> <s xml:id="echoid-s8540" xml:space="preserve">Et conuertendo minor erit proportio AH, <lb/> <anchor type="note" xlink:label="note-212-12a" xlink:href="note-212-12"/> ad KI, quàm AL, ad LH: </s> <s xml:id="echoid-s8541" xml:space="preserve">hoc eſt, maior proportio erit AL, ad LH, quàm <lb/>AH, ad KI. </s> <s xml:id="echoid-s8542" xml:space="preserve">Quare cum maior adhuc ſit proportio AH, ad LH, quàm AL, <lb/> <anchor type="note" xlink:label="note-212-13a" xlink:href="note-212-13"/> ad eandem LH: </s> <s xml:id="echoid-s8543" xml:space="preserve">multo maior proportio erit AH, ad LH, quàm eiuſdem <lb/>AH, ad KI; </s> <s xml:id="echoid-s8544" xml:space="preserve">ac propterea recta LH, minor erit, quàm kI. </s> <s xml:id="echoid-s8545" xml:space="preserve">quod eſt ſecun-<lb/> <anchor type="note" xlink:label="note-212-14a" xlink:href="note-212-14"/> dum. </s> <s xml:id="echoid-s8546" xml:space="preserve">Ex quo fit, differentias tam tangentium, quàm ſecantium in tabula ſem <lb/>per augeri ad finem vſq; </s> <s xml:id="echoid-s8547" xml:space="preserve">quadrantis: </s> <s xml:id="echoid-s8548" xml:space="preserve">cuius quidem contrarium in ſinubus ac-<lb/>cidit, vt ſupra demonſtratum eſt. </s> <s xml:id="echoid-s8549" xml:space="preserve">Quamobrem ſi plures ſint arcus æquali <lb/>exceſſu porgredientes, &</s> <s xml:id="echoid-s8550" xml:space="preserve">c. </s> <s xml:id="echoid-s8551" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s8552" xml:space="preserve"/> </p> <div xml:id="echoid-div579" type="float" level="2" n="3"> <note position="left" xlink:label="note-212-07" xlink:href="note-212-07a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-212-08" xlink:href="note-212-08a" xml:space="preserve">8. quinti.</note> <note position="left" xlink:label="note-212-09" xlink:href="note-212-09a" xml:space="preserve">30. quinti.</note> <note position="left" xlink:label="note-212-10" xlink:href="note-212-10a" xml:space="preserve">22. huius.</note> <note position="left" xlink:label="note-212-11" xlink:href="note-212-11a" xml:space="preserve">29. quinti.</note> <note position="left" xlink:label="note-212-12" xlink:href="note-212-12a" xml:space="preserve">26. quinti.</note> <note position="left" xlink:label="note-212-13" xlink:href="note-212-13a" xml:space="preserve">8. quinti.</note> <note position="left" xlink:label="note-212-14" xlink:href="note-212-14a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div581" type="section" level="1" n="267"> <head xml:id="echoid-head294" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s8553" xml:space="preserve">SEQVITVR hinc, ſi quotlibet arcuum tangentes æqualiter ſeſe excedant, arcus <lb/> <anchor type="note" xlink:label="note-212-15a" xlink:href="note-212-15"/> earum inæqualiter ſeſe excedere, exceſſusq; </s> <s xml:id="echoid-s8554" xml:space="preserve">ma orum arcuum eſſe minores: </s> <s xml:id="echoid-s8555" xml:space="preserve">quàm ma-<lb/>iorum Omnium item ſecantium ſegmenta extra quadrantem eſſe inæqualia, minoraq; </s> <s xml:id="echoid-s8556" xml:space="preserve">eſſe <lb/>illa, quæ principio quadrantis ſunt propinquiora. </s> <s xml:id="echoid-s8557" xml:space="preserve">Quoniam enim poſi is arcubus DE, <lb/>EF, æqualibus, oſtenſum fuit, rectam IH, maiorem eſſe quam HG; </s> <s xml:id="echoid-s8558" xml:space="preserve">liquido conſtat, ſi ex <lb/>HI, auferatur recta ipſi HG, æqualis. </s> <s xml:id="echoid-s8559" xml:space="preserve">ſecantem inter duas AI, AH, ductam diuidere ar- <pb o="201" file="213" n="213" rhead=""/> eum EF, atq; </s> <s xml:id="echoid-s8560" xml:space="preserve">adeo abſcindere arcum minorem arcu DE, nempe partem arcus EF. </s> <s xml:id="echoid-s8561" xml:space="preserve">Eademq́; <lb/></s> <s xml:id="echoid-s8562" xml:space="preserve">ratio eſt de alijs.</s> <s xml:id="echoid-s8563" xml:space="preserve"/> </p> <div xml:id="echoid-div581" type="float" level="2" n="1"> <note position="left" xlink:label="note-212-15" xlink:href="note-212-15a" xml:space="preserve">Arcus tan-<lb/>gentium æ <lb/>quales ex-<lb/>ceſſus habẽ <lb/>tium inæ-<lb/>quales ha</note> </div> <p> <s xml:id="echoid-s8564" xml:space="preserve">RVRSVS quia demonſtratum eſt, ſecantem AG, minorem eſſe, quàm AH; </s> <s xml:id="echoid-s8565" xml:space="preserve">fit, vt <lb/>ablatis femidiametris æqualibus AD, AE, ſegmentum DG, ſeliquum minus ſit ſegmen-<lb/>to reliquo EH, &</s> <s xml:id="echoid-s8566" xml:space="preserve">c.</s> <s xml:id="echoid-s8567" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div583" type="section" level="1" n="268"> <head xml:id="echoid-head295" xml:space="preserve">THEOR. 16. PROPOS. 24.</head> <p> <s xml:id="echoid-s8568" xml:space="preserve">TANGENS arcus maioris ad tangentem <lb/> <anchor type="note" xlink:label="note-213-01a" xlink:href="note-213-01"/> minoris arcus maioré proportionem habet, quá <lb/>ſecans maioris eiuſdem arcus ad ſecantem eiuſdé <lb/>minoris.</s> <s xml:id="echoid-s8569" xml:space="preserve"/> </p> <div xml:id="echoid-div583" type="float" level="2" n="1"> <note position="right" xlink:label="note-213-01" xlink:href="note-213-01a" xml:space="preserve">Arcuu inę-<lb/>qualiú tan-<lb/>gens maio-<lb/>ris ad tan-<lb/>gentem mi <lb/>noris pro-<lb/>portionem <lb/>habet maio <lb/>rem, quam <lb/>fecans ma-<lb/>ioris ad ſe-<lb/>cantem mi <lb/>noris.</note> </div> <p> <s xml:id="echoid-s8570" xml:space="preserve">REPETATVR figura pręcedentis propoſ. <lb/></s> <s xml:id="echoid-s8571" xml:space="preserve"> <anchor type="figure" xlink:label="fig-213-01a" xlink:href="fig-213-01"/> Dico maiorem eſſe proportionem tágentis CI, <lb/>ad tangentem CH, quàm ſecantis AI, ad ſecan <lb/>tem AH. </s> <s xml:id="echoid-s8572" xml:space="preserve">Quoniam enim eſt, vt AF, ad FM, <lb/> <anchor type="note" xlink:label="note-213-02a" xlink:href="note-213-02"/> ita AI, ad IC: </s> <s xml:id="echoid-s8573" xml:space="preserve">Item vt AE, ad EN, ita AH, <lb/>ad HC. </s> <s xml:id="echoid-s8574" xml:space="preserve">Eſt autem minor proportio ſemidia-<lb/> <anchor type="note" xlink:label="note-213-03a" xlink:href="note-213-03"/> metri AF, ad FM, quàm ſemidiametri AE ad, <lb/>EN; </s> <s xml:id="echoid-s8575" xml:space="preserve">quòd ſinus FM, maioris arcus CF, ma-<lb/>ior ſit ſinu EN, minoris arcus CE, vt in ex-<lb/>poſitione definitionum dictum eſt. </s> <s xml:id="echoid-s8576" xml:space="preserve">Igitur minor <lb/>quoq; </s> <s xml:id="echoid-s8577" xml:space="preserve">erit proportio AI, ad IC, quàm AH, <lb/>ad HC: </s> <s xml:id="echoid-s8578" xml:space="preserve">Et permutando, minor etiam propor-<lb/> <anchor type="note" xlink:label="note-213-04a" xlink:href="note-213-04"/> tio AI, ad AH, quàm IC, ad HC; </s> <s xml:id="echoid-s8579" xml:space="preserve">hoc eſt, tan <lb/>gens CI, ad tangentem CH, habebit maiorem <lb/>proportionem, quàm ſecans AI, ad ſecantem AH. </s> <s xml:id="echoid-s8580" xml:space="preserve">Quocirca Tangensar-<lb/>cus maioris ad tangentem minoris arcus, &</s> <s xml:id="echoid-s8581" xml:space="preserve">c. </s> <s xml:id="echoid-s8582" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s8583" xml:space="preserve"/> </p> <div xml:id="echoid-div584" type="float" level="2" n="2"> <figure xlink:label="fig-213-01" xlink:href="fig-213-01a"> <image file="213-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/213-01"/> </figure> <note position="right" xlink:label="note-213-02" xlink:href="note-213-02a" xml:space="preserve">4. fexti.</note> <note position="right" xlink:label="note-213-03" xlink:href="note-213-03a" xml:space="preserve">8. quinti.</note> <note position="right" xlink:label="note-213-04" xlink:href="note-213-04a" xml:space="preserve">ſchol. 27. 5.</note> </div> </div> <div xml:id="echoid-div586" type="section" level="1" n="269"> <head xml:id="echoid-head296" style="it" xml:space="preserve">SEQVVNTVR TABVLAE TANGEN-<lb/>tium atque ſecantium.</head> <pb o="202" file="214" n="214" rhead=""/> </div> <div xml:id="echoid-div587" type="section" level="1" n="270"> <head xml:id="echoid-head297" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>0 # 1 # 2 # 3 # 4 <lb/>0 # 0000 # 174550 # 349207 # 524078 # 699269 # 60 <lb/>1 # 2909 # 177459 # 352120 # 526995 # 702193 # 59 <lb/>2 # 5818 # 180369 # 355033 # 529911 # 705116 # 58 <lb/>3 # 8127 # 183279 # 357945 # 532828 # 708039 # 57 <lb/>4 # 11636 # 186189 # 360858 # 535745 # 710962 # 56 <lb/>5 # 14544 # 189100 # 363770 # 538663 # 713886 # 55 <lb/>6 # 17452 # 192010 # 366683 # 541580 # 716809 # 54 <lb/>7 # 20361 # 194920 # 369596 # 544498 # 719733 # 53 <lb/>8 # 23270 # 197830 # 372508 # 547415 # 722657 # 52 <lb/>9 # 26179 # 200740 # 375421 # 550333 # 725580 # 51 <lb/>10 # 29088 # 203650 # 378334 # 553251 # 728504 # 50 <lb/>11 # 31996 # 206561 # 381247 # 556169 # 731428 # 49 <lb/>12 # 34905 # 209471 # 384160 # 559087 # 734353 # 48 <lb/>13 # 37814 # 212381 # 387073 # 562005 # 737277 # 47 <lb/>14 # 40723 # 215291 # 389987 # 564923 # 740202 # 46 <lb/>15 # 43632 # 218201 # 392900 # 567841 # 743127 # 45 <lb/>16 # 46541 # 221111 # 395814 # 570759 # 746052 # 44 <lb/>17 # 49450 # 224022 # 398727 # 573678 # 748978 # 43 <lb/>18 # 52359 # 226932 # 401641 # 576596 # 751903 # 42 <lb/>19 # 55268 # 229842 # 404554 # 579514 # 754829 # 41 <lb/>20 # 58177 # 232752 # 407468 # 582433 # 757754 # 40 <lb/>21 # 61086 # 235663 # 410382 # 585352 # 760680 # 39 <lb/>22 # 63995 # 238574 # 413295 # 588270 # 763606 # 38 <lb/>23 # 66904 # 241485 # 416209 # 591189 # 766532 # 37 <lb/>24 # 69813 # 244395 # 419123 # 594108 # 769459 # 36 <lb/>25 # 72722 # 247306 # 422037 # 597028 # 772385 # 35 <lb/>26 # 75631 # 250217 # 424951 # 599947 # 775311 # 34 <lb/>27 # 78540 # 253128 # 427866 # 602866 # 778238 # 33 <lb/>28 # 81450 # 256038 # 430780 # 605786 # 781164 # 32 <lb/>29 # 84359 # 258949 # 433694 # 608705 # 784091 # 31 <lb/>30 # 87268 # 261859 # 436609 # 611625 # 787017 # 30 <lb/> # 89 # 88 # 87 # 86 # 85 <lb/></note> </div> <div xml:id="echoid-div588" type="section" level="1" n="271"> <head xml:id="echoid-head298" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="203" file="215" n="215" rhead=""/> </div> <div xml:id="echoid-div589" type="section" level="1" n="272"> <head xml:id="echoid-head299" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 0 # 1 # 2 # 3 # 4 <lb/>30 # 87268 # 261859 # 436609 # 611625 # 787017 # 30 <lb/>31 # 90177 # 264770 # 439523 # 614544 # 789944 # 39 <lb/>32 # 93086 # 267681 # 442438 # 617464 # 792871 # 28 <lb/>33 # 95995 # 270592 # 445353 # 620384 # 795799 # 27 <lb/>34 # 98904 # 273503 # 448267 # 623304 # 798726 # 26 <lb/>35 # 101814 # 276414 # 451182 # 626225 # 801653 # 25 <lb/>36 # 104723 # 279325 # 454097 # 629145 # 804581 # 24 <lb/>37 # 107632 # 282237 # 457012 # 632066 # 807509 # 23 <lb/>38 # 110541 # 285148 # 459927 # 634986 # 810437 # 22 <lb/>39 # 113450 # 288059 # 462842 # 637907 # 813365 # 21 <lb/>40 # 116360 # 290970 # 465757 # 640828 # 816293 # 20 <lb/>41 # 119269 # 293882 # 468672 # 643749 # 819221 # 19 <lb/>42 # 122178 # 296794 # 471588 # 646671 # 822150 # 18 <lb/>43 # 125088 # 299705 # 474503 # 649592 # 825079 # 17 <lb/>44 # 127997 # 302617 # 477419 # 652514 # 828008 # 16 <lb/>45 # 130906 # 305528 # 480335 # 655435 # 830937 # 15 <lb/>46 # 133816 # 308439 # 483251 # 658357 # 833866 # 14 <lb/>47 # 136725 # 311351 # 486166 # 661278 # 836795 # 13 <lb/>48 # 139635 # 314262 # 489082 # 664200 # 839724 # 12 <lb/>49 # 142544 # 317174 # 491997 # 667121 # 842653 # 11 <lb/>50 # 145454 # 320085 # 494913 # 670043 # 845583 # 10 <lb/>51 # 148363 # 322997 # 497829 # 672965 # 848513 # 9 <lb/>52 # 151273 # 325909 # 500745 # 675888 # 851443 # 8 <lb/>53 # 154182 # 328821 # 503662 # 678810 # 854374 # 7 <lb/>54 # 159092 # 331733 # 506578 # 681733 # 857304 # 6 <lb/>55 # 160001 # 334645 # 509495 # 684656 # 860234 # 5 <lb/>56 # 162911 # 337558 # 512411 # 687578 # 863164 # 4 <lb/>57 # 165820 # 340470 # 515328 # 690501 # 866095 # 3 <lb/>58 # 168730 # 343382 # 518244 # 693423 # 869025 # 2 <lb/>59 # 171640 # 346295 # 521161 # 696346 # 871956 # 1 <lb/>60 # 174550 # 349207 # 524078 # 699269 # 874886 # 0 <lb/> # 89 # 88 # 87 # 86 # 85 <lb/></note> </div> <div xml:id="echoid-div590" type="section" level="1" n="273"> <head xml:id="echoid-head300" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="204" file="216" n="216" rhead=""/> </div> <div xml:id="echoid-div591" type="section" level="1" n="274"> <head xml:id="echoid-head301" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 5 # 6 # 7 # 8 # 9 <lb/>0 # 874886 # 1051042 # 1227846 # 14008 # 1583844 # 60 <lb/>1 # 877817 # 1053983 # 1230798 # 1408374 # 1586826 # 59 <lb/>2 # 880748 # 1056924 # 1233751 # 1411341 # 1589808 # 58 <lb/>3 # 883680 # 1059866 # 1236704 # 1414308 # 1592791 # 57 <lb/>4 # 886611 # 1062808 # 1239658 # 1417275 # 1595774 # 56 <lb/>5 # 889543 # 1065750 # 1242612 # 1420242 # 1598757 # 55 <lb/>6 # 892475 # 1068692 # 1245566 # 1423210 # 1601740 # 54 <lb/>7 # 895407 # 1071634 # 1248520 # 1426178 # 1604723 # 53 <lb/>8 # 898339 # 1074576 # 1251474 # 1429146 # 1607707 # 52 <lb/>9 # 901271 # 1077518 # 1254428 # 1432115 # 1610691 # 51 <lb/>10 # 904204 # 1080461 # 1257383 # 1435084 # 1613675 # 50 <lb/>11 # 907137 # 1083404 # 1260338 # 1438053 # 1616660 # 49 <lb/>12 # 910070 # 1086347 # 1263293 # 1441022 # 1619645 # 48 <lb/>13 # 913003 # 1089291 # 1266249 # 1443992 # 1622630 # 47 <lb/>14 # 915936 # 1092234 # 1269205 # 1446961 # 1625615 # 46 <lb/>15 # 918870 # 1095178 # 1272161 # 1449931 # 1628601 # 45 <lb/>16 # 921804 # 1098122 # 1275117 # 1452901 # 1631587 # 44 <lb/>17 # 924738 # 1101066 # 1278073 # 1455871 # 1634573 # 43 <lb/>18 # 927771 # 1104010 # 1281029 # 1458842 # 1637560 # 42 <lb/>19 # 930605 # 1106954 # 1283986 # 1461813 # 1640547 # 41 <lb/>20 # 933539 # 1109899 # 1286943 # 1464784 # 1643534 # 40 <lb/>21 # 936473 # 1112844 # 1289900 # 1467755 # 1646522 # 39 <lb/>22 # 939407 # 1115789 # 1292857 # 1470727 # 1649510 # 38 <lb/>23 # 942342 # 1118734 # 1295815 # 1473699 # 1652499 # 37 <lb/>24 # 945277 # 1121680 # 1298773 # 1476671 # 1655488 # 36 <lb/>25 # 948212 # 1124625 # 1301731 # 1479644 # 1658477 # 35 <lb/>26 # 951147 # 1127571 # 1304689 # 1482617 # 1661466 # 34 <lb/>27 # 954083 # 1130517 # 1307648 # 1485590 # 1664456 # 33 <lb/>28 # 957019 # 1133463 # 1310607 # 1488563 # 1667446 # 32 <lb/>29 # 959954 # 1136409 # 1313566 # 1491536 # 1670436 # 31 <lb/>30 # 962890 # 1139355 # 1316525 # 1494510 # 1673426 # 30 <lb/> # 84 # 83 # 82 # 81 # 80 <lb/></note> </div> <div xml:id="echoid-div592" type="section" level="1" n="275"> <head xml:id="echoid-head302" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="205" file="217" n="217" rhead=""/> </div> <div xml:id="echoid-div593" type="section" level="1" n="276"> <head xml:id="echoid-head303" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 5 # 6 # 7 # 8 # 9 <lb/>30 # 962890 # 1139355 # 1316525 # 1494510 # 1673426 # 30 <lb/>31 # 965826 # 1142302 # 1319485 # 1497484 # 1676417 # 29 <lb/>32 # 968763 # 1145249 # 1322445 # 1500458 # 1679408 # 28 <lb/>33 # 971699 # 1148196 # 1325405 # 1503433 # 1682399 # 27 <lb/>34 # 974636 # 1151114 # 1328365 # 1506408 # 1685390 # 26 <lb/>35 # 977573 # 1154092 # 1331325 # 1509383 # 1688382 # 25 <lb/>36 # 980509 # 1157040 # 1334285 # 1512358 # 1691374 # 24 <lb/>37 # 983446 # 1159988 # 1337246 # 1515334 # 1694366 # 23 <lb/>38 # 986383 # 1162936 # 1340207 # 1518310 # 1697358 # 22 <lb/>39 # 989320 # 1165884 # 1343168 # 1521286 # 1700351 # 21 <lb/>40 # 992257 # 1168822 # 1346129 # 1524262 # 1703344 # 20 <lb/>41 # 995195 # 1171781 # 1349091 # 1527239 # 1706337 # 19 <lb/>42 # 998133 # 1174730 # 1352053 # 1530216 # 1709331 # 18 <lb/>43 # 1001072 # 1177679 # 1355015 # 1533193 # 1712325 # 17 <lb/>44 # 1004010 # 1180628 # 1357977 # 1536170 # 1715319 # 16 <lb/>45 # 1006949 # 1183577 # 1360940 # 1539148 # 1718313 # 15 <lb/>46 # 1009887 # 1186527 # 1363903 # 1542126 # 1721308 # 14 <lb/>47 # 1012825 # 1189477 # 1366866 # 1545104 # 1724304 # 13 <lb/>48 # 1015763 # 1192427 # 1369830 # 1548082 # 1727300 # 12 <lb/>49 # 1018702 # 1195377 # 1372793 # 1551061 # 1730296 # 11 <lb/>50 # 1021641 # 1198328 # 1375757 # 1554040 # 1733292 # 10 <lb/>51 # 1024580 # 1201279 # 1378721 # 1557019 # 1736287 # 9 <lb/>52 # 1027519 # 1204230 # 1381686 # 1559999 # 1739284 # 8 <lb/>53 # 1030459 # 1207181 # 1384650 # 1562979 # 1742281 # 7 <lb/>54 # 1033399 # 1210132 # 1387615 # 1565959 # 1745278 # 6 <lb/>55 # 1036339 # 1213084 # 1390580 # 1568939 # 1748275 # 5 <lb/>56 # 1039279 # 1216036 # 1393545 # 1571920 # 1751273 # 4 <lb/>57 # 1042219 # 1218988 # 1396510 # 1574901 # 1754271 # 3 <lb/>58 # 1045160 # 1221940 # 1399476 # 1577882 # 1757270 # 2 <lb/>59 # 1048101 # 1224892 # 1402442 # 1580863 # 1760269 # 1 <lb/>60 # 1051042 # 1227845 # 1405408 # 1583844 # 1763268 # 0 <lb/> # 84 # 83 # 82 # 81 # 80 <lb/></note> </div> <div xml:id="echoid-div594" type="section" level="1" n="277"> <head xml:id="echoid-head304" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="206" file="218" n="218" rhead=""/> </div> <div xml:id="echoid-div595" type="section" level="1" n="278"> <head xml:id="echoid-head305" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 10 # 11 # 12 # 13 # 14 <lb/>0 # 1763268 # 1943803 # 2125565 # 2308682 # 2493280 # 60 <lb/>1 # 1766268 # 1946822 # 2128605 # 2311746 # 2496370 # 59 <lb/>2 # 1769268 # 1949841 # 2131646 # 2314810 # 2499411 # 58 <lb/>3 # 1772268 # 1952861 # 2134687 # 2317875 # 2502552 # 57 <lb/>4 # 1775269 # 1955881 # 2137729 # 2320940 # 2505643 # 56 <lb/>5 # 1778270 # 1958901 # 2140771 # 2324006 # 2508735 # 55 <lb/>6 # 1781271 # 1961922 # 2143814 # 2327072 # 2511827 # 54 <lb/>7 # 1784272 # 1964943 # 2146857 # 2330139 # 2514920 # 53 <lb/>8 # 1787274 # 1967964 # 2149900 # 2333206 # 2518013 # 52 <lb/>9 # 1790276 # 1970985 # 2152944 # 2336273 # 2521106 # 51 <lb/>10 # 1793278 # 1974007 # 2155988 # 2339341 # 2524200 # 50 <lb/>11 # 1796281 # 1977029 # 2159032 # 2342419 # 2527294 # 49 <lb/>12 # 1799284 # 1980052 # 2162077 # 2345478 # 2530389 # 48 <lb/>13 # 1802287 # 1983075 # 2165122 # 2348547 # 2533484 # 47 <lb/>14 # 1805291 # 1986098 # 2168167 # 2351616 # 2536580 # 46 <lb/>15 # 1808295 # 1989122 # 2171213 # 2354686 # 2539676 # 45 <lb/>16 # 1811299 # 1992146 # 2174259 # 2357757 # 2542773 # 44 <lb/>17 # 1814303 # 1995171 # 2177306 # 2360828 # 2545870 # 43 <lb/>18 # 1817308 # 1998196 # 2180352 # 2363899 # 2548968 # 42 <lb/>19 # 1820313 # 2001221 # 2183400 # 2366971 # 2552066 # 41 <lb/>20 # 1823318 # 2004247 # 2186448 # 2370043 # 2555165 # 40 <lb/>21 # 1826324 # 2007273 # 2189496 # 2373116 # 2558264 # 39 <lb/>22 # 1829329 # 2010299 # 2192544 # 2376189 # 2561364 # 38 <lb/>23 # 1832335 # 2013326 # 2192544 # 2379263 # 2564464 # 37 <lb/>24 # 1835342 # 2016353 # 2195593 # 2382337 # 2567564 # 36 <lb/>25 # 1838349 # 2019380 # 2201692 # 2385411 # 2570665 # 35 <lb/>26 # 1841357 # 2022408 # 2204742 # 2388486 # 2573766 # 34 <lb/>27 # 1844365 # 2025436 # 2207792 # 2391561 # 2576868 # 33 <lb/>28 # 1847373 # 2028464 # 2210843 # 2394636 # 2579970 # 32 <lb/>29 # 1850382 # 2031493 # 2213894 # 2397712 # 2583073 # 31 <lb/>30 # 1853391 # 2034522 # 2216946 # 2400788 # 8586176 # 30 <lb/> # 79 # 78 # 77 # 76 # 75 <lb/></note> </div> <div xml:id="echoid-div596" type="section" level="1" n="279"> <head xml:id="echoid-head306" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="207" file="219" n="219" rhead=""/> </div> <div xml:id="echoid-div597" type="section" level="1" n="280"> <head xml:id="echoid-head307" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 10 # 11 # 12 # 13 # 14 <lb/>30 # 1853391 # 2034522 # 2216946 # 2400788 # 2586176 # 30 <lb/>31 # 1856400 # 2037552 # 2219998 # 2403865 # 2589280 # 29 <lb/>32 # 1859409 # 2040582 # 2223051 # 2406942 # 2592384 # 28 <lb/>33 # 1862419 # 2043612 # 2226104 # 2410020 # 2595489 # 27 <lb/>34 # 1865429 # 2046643 # 2229157 # 2413098 # 2598594 # 26 <lb/>35 # 1868439 # 2049674 # 2232211 # 2416176 # 2601700 # 25 <lb/>36 # 1871449 # 2052705 # 2235265 # 2419255 # 2604806 # 24 <lb/>37 # 1874460 # 2055737 # 2238319 # 2422334 # 2607912 # 23 <lb/>38 # 1877471 # 2058769 # 2241374 # 2425414 # 2611019 # 22 <lb/>39 # 1880482 # 2061801 # 2244429 # 2428494 # 2614126 # 21 <lb/>40 # 1883494 # 2064834 # 2247485 # 2431574 # 2617234 # 20 <lb/>41 # 1886506 # 2067867 # 2250541 # 2434655 # 2620342 # 19 <lb/>42 # 1889518 # 2070900 # 2253597 # 2437736 # 2623451 # 18 <lb/>43 # 1892531 # 2073934 # 2256654 # 2440818 # 2626560 # 17 <lb/>44 # 1895544 # 2076968 # 2259711 # 2443900 # 2629670 # 16 <lb/>45 # 1898558 # 2080002 # 2262769 # 2446983 # 2632780 # 15 <lb/>46 # 1901572 # 2083037 # 2265827 # 2450066 # 2635891 # 14 <lb/>47 # 1904586 # 2086073 # 2268885 # 2453150 # 2639002 # 13 <lb/>48 # 1907601 # 2089109 # 2271944 # 2456234 # 2642114 # 12 <lb/>49 # 1910616 # 2092145 # 2275003 # 2419319 # 2645226 # 11 <lb/>50 # 1913632 # 2095182 # 2278063 # 2462404 # 2648339 # 10 <lb/>51 # 1916648 # 2098219 # 2281123 # 2465490 # 2651452 # 9 <lb/>52 # 1999664 # 2101256 # 2284183 # 2468576 # 2654566 # 8 <lb/>53 # 1922680 # 2104293 # 2287244 # 2471662 # 2657680 # 7 <lb/>54 # 1925697 # 2107331 # 2290305 # 2474749 # 2660795 # 6 <lb/>55 # 1928714 # 2110369 # 2293367 # 2477836 # 2663910 # 5 <lb/>56 # 1931731 # 2113407 # 2296429 # 2480924 # 2667026 # 4 <lb/>57 # 1934749 # 2116446 # 2299492 # 2484012 # 2670142 # 3 <lb/>58 # 1937767 # 2119485 # 2302555 # 2487101 # 2673258 # 2 <lb/>59 # 1940785 # 2122525 # 2305618 # 2490191 # 2676375 # 1 <lb/>60 # 1943803 # 2125565 # 2308682 # 2493280 # 2679492 # 0 <lb/> # 79 # 78 # 77 # 76 # 75 <lb/></note> </div> <div xml:id="echoid-div598" type="section" level="1" n="281"> <head xml:id="echoid-head308" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="208" file="220" n="220" rhead=""/> </div> <div xml:id="echoid-div599" type="section" level="1" n="282"> <head xml:id="echoid-head309" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 15 # 16 # 17 # 18 # 19 <lb/>0 # 2679492 # 2867453 # 3057307 # 3249197 # 3443276 # 60 <lb/>1 # 2682610 # 2870601 # 3060487 # 3252413 # 3446530 # 59 <lb/>2 # 2685728 # 2873749 # 3063669 # 3255630 # 3449785 # 58 <lb/>3 # 2688847 # 2876898 # 3066851 # 3258848 # 3453040 # 57 <lb/>4 # 2691966 # 2880048 # 3070034 # 3262066 # 3456296 # 56 <lb/>5 # 2695086 # 2883198 # 3073218 # 3265285 # 3459553 # 55 <lb/>6 # 2698206 # 2886349 # 3076402 # 3268504 # 3462810 # 54 <lb/>7 # 2701327 # 2889501 # 3079587 # 3271724 # 3466068 # 53 <lb/>8 # 2704448 # 2892653 # 3082772 # 3274944 # 3469326 # 52 <lb/>9 # 2707570 # 2895806 # 3085958 # 3278165 # 3472585 # 51 <lb/>10 # 2710693 # 2898960 # 3085144 # 3281387 # 3475845 # 50 <lb/>11 # 2713816 # 2902114 # 3092331 # 3284609 # 3479105 # 49 <lb/>12 # 2716940 # 2905268 # 3095518 # 3287832 # 3482366 # 48 <lb/>13 # 2720064 # 2908423 # 3198706 # 3291055 # 3485628 # 47 <lb/>14 # 2723189 # 2911578 # 3101895 # 3294280 # 3488891 # 46 <lb/>15 # 2726314 # 2914734 # 3105084 # 3297505 # 3492154 # 45 <lb/>16 # 2729439 # 2917890 # 3108274 # 3300731 # 3495418 # 44 <lb/>17 # 2732565 # 2921047 # 3111464 # 3303957 # 3498683 # 43 <lb/>18 # 2735691 # 2924204 # 3114655 # 3307184 # 3501949 # 42 <lb/>19 # 2738818 # 2927362 # 3117846 # 3110411 # 3505215 # 41 <lb/>20 # 2741945 # 2930520 # 3121038 # 3313639 # 3508482 # 40 <lb/>21 # 2745073 # 2933679 # 3124230 # 3316868 # 3511749 # 39 <lb/>22 # 2748201 # 2936839 # 3127423 # 3320097 # 3515017 # 38 <lb/>23 # 2751330 # 2939999 # 3130617 # 3333327 # 3518286 # 37 <lb/>24 # 2754459 # 2943160 # 3133811 # 3326558 # 3521555 # 36 <lb/>25 # 2757589 # 2946321 # 3137006 # 3329789 # 3524825 # 35 <lb/>26 # 2760729 # 2946483 # 3140201 # 3333020 # 3528096 # 34 <lb/>27 # 2763850 # 2952645 # 3143397 # 3336252 # 3531368 # 33 <lb/>28 # 2766981 # 2955808 # 3146594 # 3339485 # 3534640 # 32 <lb/>29 # 2770113 # 2958971 # 3149791 # 3342719 # 3537913 # 31 <lb/>30 # 2773245 # 2962135 # 3152989 # 3345953 # 3541186 # 30 <lb/> # 74 # 73 # 72 # 71 # 70 <lb/></note> </div> <div xml:id="echoid-div600" type="section" level="1" n="283"> <head xml:id="echoid-head310" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="209" file="221" n="221" rhead=""/> </div> <div xml:id="echoid-div601" type="section" level="1" n="284"> <head xml:id="echoid-head311" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/>15 # 16 # 17 # 18 # 19 <lb/>30 # 2773245 # 2962135 # 3152989 # 3345953 # 3541186 # 30 <lb/>31 # 2776378 # 2965299 # 3156187 # 3349188 # 3544460 # 39 <lb/>32 # 2779511 # 2968464 # 3159386 # 3352423 # 3547735 # 28 <lb/>33 # 2782645 # 2971629 # 3162585 # 3355659 # 3551010 # 27 <lb/>34 # 2785779 # 2974795 # 3165785 # 3358896 # 3554286 # 26 <lb/>35 # 2788914 # 2977962 # 3168986 # 3362133 # 3557563 # 25 <lb/>36 # 2792050 # 2981129 # 3172187 # 3365371 # 3560840 # 24 <lb/>37 # 2795186 # 2984297 # 3175389 # 3368610 # 3564118 # 23 <lb/>38 # 2798323 # 2987465 # 3178591 # 3371850 # 3567397 # 22 <lb/>39 # 2801460 # 2990634 # 3181794 # 3375090 # 3570676 # 21 <lb/>40 # 2804597 # 2993804 # 3184998 # 3378331 # 3573956 # 20 <lb/>41 # 2807735 # 2996973 # 3188202 # 3381572 # 3577237 # 19 <lb/>42 # 2810873 # 3000143 # 3191407 # 3384814 # 3580519 # 18 <lb/>43 # 2814012 # 3003314 # 3194613 # 3388057 # 3583801 # 17 <lb/>44 # 2817151 # 3006486 # 3197819 # 3391300 # 3587084 # 16 <lb/>45 # 2820291 # 3009658 # 3201026 # 3394544 # 3590367 # 15 <lb/>46 # 2823432 # 3012831 # 3204233 # 3397798 # 3593651 # 14 <lb/>47 # 2826573 # 3016004 # 3207441 # 3401033 # 3596936 # 13 <lb/>48 # 2829714 # 3019178 # 3210649 # 3404279 # 3600221 # 12 <lb/>49 # 2832856 # 3022353 # 3213858 # 3407525 # 3603507 # 11 <lb/>50 # 2835999 # 3025528 # 3217067 # 3410772 # 3606794 # 10 <lb/>51 # 2839142 # 3028703 # 3220277 # 3414020 # 3610082 # 9 <lb/>52 # 2842286 # 3031879 # 3223488 # 3417268 # 3613370 # 8 <lb/>53 # 2845430 # 3035055 # 3226699 # 3420517 # 3616659 # 7 <lb/>54 # 2848575 # 3038232 # 3229911 # 3423766 # 3619949 # 6 <lb/>55 # 2851720 # 3041410 # 3233124 # 3427016 # 3623239 # 5 <lb/>56 # 2854866 # 3044588 # 3236337 # 3430267 # 3626530 # 4 <lb/>57 # 2858012 # 3047767 # 3239551 # 3433518 # 3629822 # 3 <lb/>58 # 2861159 # 3050946 # 3242766 # 3436770 # 3633115 # 2 <lb/>59 # 2864306 # 3054126 # 3245981 # 3440023 # 3636408 # 1 <lb/>60 # 2867453 # 3057307 # 3249197 # 3443276 # 3639702 # 0 <lb/>74 # 73 # 72 # 71 # 70 <lb/></note> </div> <div xml:id="echoid-div602" type="section" level="1" n="285"> <head xml:id="echoid-head312" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="210" file="222" n="222" rhead=""/> </div> <div xml:id="echoid-div603" type="section" level="1" n="286"> <head xml:id="echoid-head313" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>20 # 21 # 22 # 23 # 24 <lb/>0 # 3639702 # 3838640 # 4040262 # 4244748 # 4452286 # 60 <lb/>1 # 3642997 # 3841978 # 4043647 # 4248182 # 4455772 # 59 <lb/>2 # 3646293 # 3845316 # 4047031 # 4251617 # 4459259 # 58 <lb/>3 # 3649589 # 3848655 # 4050416 # 4255052 # 4462747 # 57 <lb/>4 # 3652886 # 3851995 # 4053802 # 4258488 # 4466236 # 56 <lb/>5 # 3656183 # 3855336 # 4057189 # 4261925 # 4469726 # 55 <lb/>6 # 3659481 # 3858678 # 4060577 # 4265363 # 4473216 # 54 <lb/>7 # 3662780 # 3862020 # 4063966 # 4268801 # 4476707 # 53 <lb/>8 # 3666079 # 3865363 # 4067356 # 4265363 # 4480199 # 52 <lb/>9 # 3669379 # 3868707 # 4070747 # 4268801 # 4483692 # 51 <lb/>10 # 3672680 # 3872052 # 4074139 # 4272240 # 4487186 # 50 <lb/>11 # 3675982 # 3875397 # 4077531 # 4275680 # 4490681 # 49 <lb/>12 # 3679284 # 3878743 # 4080924 # 4279121 # 4494177 # 48 <lb/>13 # 3682587 # 3882090 # 4084318 # 4282563 # 4497674 # 47 <lb/>14 # 3685891 # 3885438 # 4087713 # 4286006 # 4501172 # 46 <lb/>15 # 3689195 # 3888787 # 4091109 # 4289450 # 4504671 # 45 <lb/>16 # 3692500 # 3892136 # 4094506 # 4292895 # 4508171 # 44 <lb/>17 # 3695806 # 3895486 # 4097903 # 4296340 # 4511672 # 43 <lb/>18 # 3699113 # 3898837 # 4101301 # 4299786 # 4515173 # 42 <lb/>19 # 3702420 # 3902188 # 4104699 # 4303233 # 4518675 # 41 <lb/>20 # 3705728 # 3905540 # 4108097 # 4306681 # 4522178 # 40 <lb/>21 # 3709037 # 3908893 # 4111497 # 4310130 # 4525682 # 39 <lb/>22 # 3712347 # 3912247 # 4114898 # 4313580 # 4529187 # 38 <lb/>23 # 3715657 # 3915601 # 4118300 # 4317031 # 4532693 # 37 <lb/>24 # 3718968 # 3918956 # 4121703 # 4327387 # 4536200 # 36 <lb/>25 # 3722279 # 3922312 # 4125107 # 4330841 # 4539708 # 35 <lb/>26 # 3725591 # 3925669 # 4128511 # 4334296 # 4543217 # 34 <lb/>27 # 3728904 # 3929027 # 4131916 # 4337752 # 4546727 # 33 <lb/>28 # 3732218 # 3932385 # 4135322 # 4341209 # 4550238 # 32 <lb/>29 # 3735533 # 3935744 # 4138728 # 4344666 # 4553750 # 31 <lb/>30 # 3738848 # 3939104 # 4142135 # 4348124 # 4557264 # 30 <lb/>69 # 68 # 67 # 66 # 65 <lb/></note> </div> <div xml:id="echoid-div604" type="section" level="1" n="287"> <head xml:id="echoid-head314" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="211" file="223" n="223" rhead=""/> </div> <div xml:id="echoid-div605" type="section" level="1" n="288"> <head xml:id="echoid-head315" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/>20 # 21 # 22 # 23 # 24 <lb/>30 # 3738848 # 3939104 # 4142135 # 4348124 # 4557264 # 30 <lb/>31 # 3742164 # 3942465 # 4145544 # 4351583 # 4560778 # 39 <lb/>32 # 3745480 # 3945826 # 4148953 # 4355043 # 4564293 # 28 <lb/>33 # 3748797 # 3949188 # 4152363 # 4358504 # 4567809 # 27 <lb/>34 # 3752115 # 3952551 # 4155773 # 4361966 # 4571326 # 26 <lb/>35 # 3755434 # 3955915 # 4159184 # 4365429 # 4574843 # 25 <lb/>36 # 3758753 # 3959280 # 4162596 # 4368893 # 4578361 # 24 <lb/>37 # 3762073 # 3962646 # 4166009 # 4372357 # 4581880 # 23 <lb/>38 # 3765394 # 3966012 # 4169423 # 4375822 # 4585400 # 22 <lb/>39 # 3768716 # 3969379 # 4172838 # 4379288 # 4588921 # 21 <lb/>40 # 3772038 # 3972746 # 4176255 # 4382755 # 4592443 # 20 <lb/>41 # 3775361 # 3976114 # 4179672 # 4386223 # 4595966 # 19 <lb/>42 # 3778685 # 3979483 # 4183090 # 4389692 # 4599490 # 18 <lb/>43 # 3782010 # 3982853 # 4186509 # 4393162 # 4603015 # 17 <lb/>44 # 3785335 # 3986224 # 4189928 # 4396633 # 4606541 # 16 <lb/>45 # 3788661 # 3989596 # 4193348 # 4400105 # 4610068 # 15 <lb/>46 # 3791988 # 3992969 # 4196769 # 4403578 # 4613596 # 14 <lb/>47 # 3795315 # 3996342 # 4200191 # 4407051 # 4617125 # 13 <lb/>48 # 3798643 # 3999716 # 4203613 # 4410525 # 4620654 # 12 <lb/>49 # 3801972 # 4003090 # 4207036 # 4414000 # 4624184 # 11 <lb/>50 # 3805302 # 4006465 # 4210460 # 4417476 # 4627715 # 10 <lb/>51 # 3808632 # 4009841 # 4213885 # 4420953 # 4631247 # 9 <lb/>52 # 3811963 # 4013217 # 4217311 # 4424431 # 4634780 # 8 <lb/>53 # 3815295 # 4016594 # 4220738 # 4427910 # 4638314 # 7 <lb/>54 # 3818628 # 4019972 # 4224165 # 4431390 # 4641849 # 6 <lb/>55 # 3821961 # 4023351 # 4227593 # 4434871 # 4645385 # 5 <lb/>56 # 3825295 # 4026731 # 4231022 # 4438352 # 4648922 # 4 <lb/>57 # 3828630 # 4030112 # 4234452 # 4441834 # 4652460 # 3 <lb/>58 # 3831966 # 4033494 # 4237883 # 4445317 # 4655999 # 2 <lb/>59 # 3835303 # 4036877 # 4241315 # 4448801 # 4659540 # 1 <lb/>60 # 3838640 # 4040262 # 4244748 # 4452286 # 4663081 # 0 <lb/>69 # 68 # 67 # 66 # 65 <lb/></note> </div> <div xml:id="echoid-div606" type="section" level="1" n="289"> <head xml:id="echoid-head316" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="212" file="224" n="224" rhead=""/> </div> <div xml:id="echoid-div607" type="section" level="1" n="290"> <head xml:id="echoid-head317" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>25 # 26 # 27 # 28 # 29 <lb/>0 # 4663081 # 4877328 # 5095254 # 5317094 # 5543090 # 60 <lb/>1 # 4666623 # 4880930 # 5098919 # 5320826 # 5546893 # 59 <lb/>2 # 4670166 # 4884533 # 5102585 # 5324559 # 5550697 # 58 <lb/>3 # 4673710 # 4888137 # 5106252 # 5328293 # 5554503 # 57 <lb/>4 # 4677255 # 4891742 # 5109920 # 5332028 # 5558310 # 56 <lb/>5 # 4680801 # 4895347 # 5113589 # 5335765 # 5562118 # 55 <lb/>6 # 4684348 # 4898953 # 5117259 # 5339503 # 5565927 # 54 <lb/>7 # 4687896 # 4902560 # 5120930 # 5343242 # 5569738 # 53 <lb/>8 # 4691444 # 4906168 # 5124602 # 5346982 # 5573550 # 52 <lb/>9 # 4694993 # 4909777 # 5128275 # 5350723 # 5577363 # 51 <lb/>10 # 4698543 # 4913387 # 5131949 # 5354465 # 5581177 # 50 <lb/>11 # 4702094 # 4916998 # 5135625 # 5358209 # 5584993 # 49 <lb/>12 # 4704646 # 4920610 # 5139302 # 5361954 # 5588810 # 48 <lb/>13 # 4709199 # 4924223 # 5142980 # 5365700 # 5592628 # 47 <lb/>14 # 4712753 # 4927838 # 5146659 # 5369447 # 5596447 # 46 <lb/>15 # 4716308 # 4931454 # 5150339 # 5373195 # 5600268 # 45 <lb/>16 # 4719864 # 4935071 # 5154020 # 5376944 # 5604090 # 44 <lb/>17 # 4723422 # 4938689 # 5157702 # 5380694 # 5607913 # 43 <lb/>18 # 4726981 # 4942308 # 5161385 # 5384445 # 5611737 # 42 <lb/>19 # 4730541 # 4945928 # 5165069 # 5388198 # 5615562 # 41 <lb/>20 # 4734102 # 4949549 # 5168755 # 5391952 # 5619388 # 40 <lb/>21 # 4737664 # 4953171 # 5172442 # 5395707 # 5623216 # 39 <lb/>22 # 4741227 # 4956794 # 5176130 # 5399463 # 5627045 # 38 <lb/>23 # 4744790 # 4960418 # 5179819 # 5403221 # 5630875 # 37 <lb/>24 # 4748354 # 4964043 # 5183509 # 5406980 # 5634707 # 36 <lb/>25 # 4751919 # 4967669 # 5187200 # 5410740 # 5638540 # 35 <lb/>26 # 4755485 # 4971296 # 5190892 # 5414501 # 5642374 # 34 <lb/>27 # 4759052 # 4974924 # 5194585 # 5418263 # 5646210 # 33 <lb/>28 # 4762620 # 4978553 # 5198279 # 5422026 # 5650047 # 32 <lb/>29 # 4766189 # 4982184 # 5201974 # 5425791 # 5653885 # 31 <lb/>30 # 4769759 # 4985816 # 5205670 # 5429557 # 5657725 # 30 <lb/>64 # 63 # 62 # 61 # 60 <lb/></note> </div> <div xml:id="echoid-div608" type="section" level="1" n="291"> <head xml:id="echoid-head318" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="213" file="225" n="225" rhead=""/> </div> <div xml:id="echoid-div609" type="section" level="1" n="292"> <head xml:id="echoid-head319" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/>25 # 26 # 27 # 28 # 29 <lb/>30 # 4769759 # 4985816 # 5205670 # 5429557 # 5657725 # 30 <lb/>31 # 4773330 # 4989448 # 5209368 # 5433324 # 5661566 # 29 <lb/>32 # 4776902 # 4993081 # 5213067 # 5437092 # 5665408 # 28 <lb/>33 # 4780475 # 4996716 # 5216767 # 5440861 # 5669251 # 27 <lb/>34 # 4784049 # 5000352 # 5220468 # 5444632 # 5673096 # 26 <lb/>35 # 4787624 # 5003989 # 5224170 # 5448404 # 5676942 # 25 <lb/>36 # 4791200 # 5007627 # 5227873 # 5452177 # 5680789 # 24 <lb/>37 # 4794777 # 5011266 # 5231577 # 5455951 # 5684637 # 23 <lb/>38 # 4798355 # 5014906 # 5235283 # 5459726 # 5688486 # 22 <lb/>39 # 4801934 # 5018547 # 5238990 # 5463503 # 5692337 # 21 <lb/>40 # 4805515 # 5022189 # 5242698 # 5467281 # 5696189 # 20 <lb/>41 # 4809096 # 5025832 # 5246407 # 5471060 # 5700043 # 19 <lb/>42 # 4812678 # 5029476 # 5250117 # 5474840 # 5703898 # 18 <lb/>43 # 4816261 # 5033121 # 5253828 # 5478621 # 5707754 # 17 <lb/>44 # 4819845 # 5036767 # 5257540 # 5482404 # 5711611 # 16 <lb/>45 # 4823430 # 5040414 # 5261254 # 5486188 # 5715469 # 15 <lb/>46 # 4827016 # 5044062 # 5264969 # 5489973 # 5719329 # 14 <lb/>47 # 4830603 # 5047712 # 5268685 # 5493759 # 5723190 # 13 <lb/>48 # 4834191 # 5051363 # 5272402 # 5497546 # 5727052 # 12 <lb/>49 # 4837780 # 5055015 # 5276120 # 5501335 # 5730916 # 11 <lb/>50 # 4841371 # 5058668 # 5279839 # 5505125 # 5734781 # 10 <lb/>51 # 4844962 # 5062322 # 5283959 # 5508916 # 5738647 # 9 <lb/>52 # 4848554 # 5065977 # 5287280 # 5512708 # 5742515 # 8 <lb/>53 # 4852147 # 5069633 # 5291003 # 5516501 # 5746384 # 7 <lb/>54 # 4855741 # 5073290 # 5294727 # 5520296 # 5750254 # 6 <lb/>55 # 4859336 # 5076948 # 5298452 # 5524092 # 5754125 # 5 <lb/>56 # 4862932 # 5080607 # 5302178 # 5527889 # 5757998 # 4 <lb/>57 # 4866529 # 5084267 # 5305905 # 5531687 # 5761872 # 3 <lb/>58 # 4870127 # 5087928 # 5309633 # 5535487 # 5765747 # 2 <lb/>59 # 4873727 # 5091590 # 5313363 # 5539288 # 5769624 # 1 <lb/>60 # 4877328 # 5095254 # 5317094 # 5543090 # 5773502 # 0 <lb/>64 # 63 # 62 # 61 # 60 <lb/></note> </div> <div xml:id="echoid-div610" type="section" level="1" n="293"> <head xml:id="echoid-head320" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="214" file="226" n="226" rhead=""/> </div> <div xml:id="echoid-div611" type="section" level="1" n="294"> <head xml:id="echoid-head321" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>30 # 31 # 32 # 33 # 34 <lb/>0 # 5773502 # 6008606 # 6248693 # 6494076 # 6745085 # 60 <lb/>1 # 5777381 # 6012566 # 6252738 # 6498212 # 6749318 # 59 <lb/>2 # 5781262 # 6016528 # 6256785 # 6502350 # 6753553 # 58 <lb/>3 # 5785144 # 6020491 # 6260834 # 6506489 # 6757789 # 57 <lb/>4 # 5789027 # 6024455 # 6264884 # 6510630 # 6762027 # 56 <lb/>5 # 5792911 # 6028420 # 6268935 # 6514773 # 6766267 # 55 <lb/>6 # 5796797 # 6032387 # 6272988 # 6518917 # 6770508 # 54 <lb/>7 # 5800684 # 6036355 # 6277042 # 6523063 # 6774751 # 53 <lb/>8 # 5804572 # 6040324 # 6281098 # 6527200 # 6778996 # 52 <lb/>9 # 5808462 # 6044295 # 6285155 # 6531359 # 6783243 # 51 <lb/>10 # 5812353 # 6048267 # 6289214 # 6535510 # 6787491 # 50 <lb/>11 # 5816245 # 6052241 # 6293274 # 6539662 # 6791741 # 49 <lb/>12 # 5820139 # 6056216 # 6297336 # 6543816 # 6795993 # 48 <lb/>13 # 5824034 # 6060193 # 6301399 # 6547971 # 6800246 # 47 <lb/>14 # 5827930 # 6064171 # 6305464 # 6552128 # 6804501 # 46 <lb/>15 # 5831828 # 6068150 # 6309530 # 6556287 # 6808758 # 45 <lb/>16 # 5835727 # 6072131 # 6313598 # 6560447 # 6813016 # 44 <lb/>17 # 5839627 # 6076113 # 6317667 # 6564609 # 6217276 # 43 <lb/>18 # 5843528 # 6080096 # 6321738 # 6568772 # 6821538 # 42 <lb/>19 # 5847431 # 6084081 # 6325810 # 6572937 # 6825801 # 41 <lb/>20 # 5851335 # 6088067 # 6329883 # 6577103 # 6830066 # 40 <lb/>21 # 5855241 # 6092055 # 6333958 # 6581271 # 6834333 # 39 <lb/>22 # 5859148 # 6096044 # 6338034 # 6585440 # 6838602 # 38 <lb/>23 # 5863056 # 6100035 # 6342112 # 6589611 # 6842872 # 37 <lb/>24 # 5866966 # 6104027 # 6346191 # 6593784 # 6847144 # 36 <lb/>25 # 5870877 # 6108020 # 6350272 # 6597958 # 6851417 # 35 <lb/>26 # 5874489 # 6112015 # 6354355 # 6602134 # 6855692 # 34 <lb/>27 # 5878702 # 6116011 # 6358439 # 6606312 # 6859969 # 33 <lb/>28 # 5882617 # 6120009 # 6362525 # 6610491 # 6864247 # 32 <lb/>29 # 5886533 # 6124008 # 6366613 # 6614672 # 6868527 # 31 <lb/>30 # 5890450 # 6128008 # 6370702 # 6618855 # 6872809 # 30 <lb/>59 # 58 # 57 # 56 # 55 <lb/></note> </div> <div xml:id="echoid-div612" type="section" level="1" n="295"> <head xml:id="echoid-head322" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="215" file="227" n="227" rhead=""/> </div> <div xml:id="echoid-div613" type="section" level="1" n="296"> <head xml:id="echoid-head323" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/>30 # 31 # 32 # 33 # 34 <lb/>30 # 5890450 # 6128008 # 6370702 # 6618855 # 6872809 # 30 <lb/>31 # 5894369 # 6132010 # 6374792 # 6623039 # 6877093 # 29 <lb/>32 # 5898289 # 6136013 # 6378884 # 6627225 # 6881379 # 28 <lb/>33 # 5902211 # 6140018 # 6382977 # 6631413 # 6885666 # 27 <lb/>34 # 5906134 # 6144024 # 6387072 # 6635603 # 6889955 # 26 <lb/>35 # 5910058 # 6148032 # 6391169 # 6639792 # 6894246 # 25 <lb/>36 # 5913984 # 6152041 # 6395267 # 6643984 # 6898539 # 24 <lb/>37 # 5917911 # 6156052 # 6399366 # 6648178 # 6902833 # 23 <lb/>38 # 5921839 # 6160064 # 6403467 # 6952373 # 6907129 # 22 <lb/>39 # 5925769 # 6164077 # 6407569 # 6656570 # 6911426 # 21 <lb/>40 # 5929700 # 6168092 # 6411673 # 6660768 # 6915725 # 20 <lb/>41 # 5933633 # 6172108 # 6415779 # 6664968 # 6920026 # 19 <lb/>42 # 5937567 # 6176126 # 6419886 # 6669170 # 6924329 # 18 <lb/>43 # 5941502 # 6180147 # 6423995 # 6673373 # 6928634 # 17 <lb/>44 # 5945438 # 6184168 # 6428105 # 6677578 # 6932940 # 16 <lb/>45 # 5949376 # 6188190 # 6432216 # 6681785 # 6937248 # 15 <lb/>46 # 5955315 # 6192213 # 6436329 # 6685994 # 6941558 # 14 <lb/>47 # 5957255 # 6196237 # 6440444 # 6690204 # 6945869 # 13 <lb/>48 # 5961197 # 6200263 # 6444560 # 6694416 # 6950182 # 12 <lb/>49 # 5965140 # 6204290 # 6458678 # 6698630 # 6954497 # 11 <lb/>50 # 5969084 # 6208319 # 6452798 # 6702845 # 6958813 # 10 <lb/>51 # 5973030 # 6212350 # 6456919 # 6707062 # 6963131 # 9 <lb/>52 # 5976776 # 6216382 # 6461042 # 6711281 # 6967451 # 8 <lb/>53 # 5980926 # 6220416 # 6465166 # 6715501 # 6971773 # 7 <lb/>54 # 5984876 # 6224451 # 6469292 # 6719723 # 6976097 # 6 <lb/>55 # 5988827 # 6228488 # 6473419 # 6723946 # 6980423 # 5 <lb/>56 # 5992780 # 6232526 # 6477548 # 6728171 # 6984750 # 4 <lb/>57 # 5996734 # 6246566 # 6481678 # 6732397 # 6989079 # 3 <lb/>58 # 6000690 # 6240607 # 6485809 # 6736625 # 6993409 # 2 <lb/>59 # 6004647 # 6244649 # 6489942 # 6740854 # 6997741 # 1 <lb/>60 # 6008606 # 6248693 # 6494076 # 6745085 # 7002075 # 0 <lb/>59 # 58 # 57 # 56 # 55 <lb/></note> </div> <div xml:id="echoid-div614" type="section" level="1" n="297"> <head xml:id="echoid-head324" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="216" file="228" n="228" rhead=""/> </div> <div xml:id="echoid-div615" type="section" level="1" n="298"> <head xml:id="echoid-head325" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>35 # 36 # 37 # 38 # 39 <lb/>0 # 7002075 # 7265424 # 7535541 # 7812856 # 8097840 # 60 <lb/>1 # 7006411 # 7269869 # 7540103 # 7817542 # 8102658 # 59 <lb/>2 # 7010749 # 7274316 # 7544667 # 7822230 # 8107478 # 58 <lb/>3 # 7015088 # 7278765 # 7549233 # 7826920 # 8112300 # 57 <lb/>4 # 7019429 # 7283216 # 7553801 # 7831612 # 8117124 # 56 <lb/>5 # 7023772 # 7287669 # 7558371 # 7836306 # 8121951 # 55 <lb/>6 # 7028117 # 7292124 # 7562943 # 7841002 # 8126780 # 54 <lb/>7 # 7032463 # 7296581 # 7567517 # 7845700 # 8131611 # 53 <lb/>8 # 7036811 # 7301040 # 7572093 # 7850400 # 8136444 # 52 <lb/>9 # 7041161 # 7305501 # 7576670 # 7855102 # 8141280 # 51 <lb/>10 # 7045513 # 7309963 # 7581249 # 7859807 # 8146118 # 50 <lb/>11 # 7049867 # 7314427 # 7585830 # 7864514 # 8150958 # 49 <lb/>12 # 7054223 # 7318893 # 7590413 # 7869223 # 8155801 # 48 <lb/>13 # 7058581 # 7323361 # 7594999 # 7873934 # 8160646 # 47 <lb/>14 # 7062940 # 7327831 # 7599587 # 7878647 # 8165493 # 46 <lb/>15 # 7067301 # 7332303 # 7604177 # 7883363 # 8170343 # 45 <lb/>16 # 7071664 # 7336777 # 7608769 # 7888081 # 8175195 # 44 <lb/>17 # 7076029 # 7341253 # 7613363 # 7892801 # 8180049 # 43 <lb/>18 # 7070395 # 7345731 # 7617959 # 7897523 # 8184905 # 42 <lb/>19 # 7084763 # 7350210 # 7622557 # 7902247 # 8189764 # 41 <lb/>20 # 7089133 # 7354691 # 7627157 # 7906973 # 8194625 # 40 <lb/>21 # 7093505 # 7359174 # 7631759 # 7911702 # 8199488 # 39 <lb/>22 # 7097879 # 7363659 # 7636363 # 7916433 # 8204354 # 38 <lb/>23 # 7102254 # 7368146 # 7640969 # 7921166 # 8209222 # 37 <lb/>24 # 7106631 # 7372635 # 7645577 # 7925901 # 8214092 # 36 <lb/>25 # 7111010 # 7377126 # 7650187 # 7930638 # 8218965 # 35 <lb/>26 # 7115391 # 7381619 # 7654799 # 7935378 # 8223840 # 34 <lb/>27 # 7119773 # 7386114 # 7659413 # 7940120 # 8228717 # 33 <lb/>28 # 7124167 # 7390611 # 7664030 # 7944864 # 8233597 # 32 <lb/>29 # 7128543 # 7395110 # 7668649 # 7949610 # 8238479 # 31 <lb/>30 # 7132931 # 739961 # 7663270 # 7954358 # 8243363 # 30 <lb/>54 # 53 # 52 # 51 # 50 <lb/></note> </div> <div xml:id="echoid-div616" type="section" level="1" n="299"> <head xml:id="echoid-head326" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="217" file="229" n="229" rhead=""/> </div> <div xml:id="echoid-div617" type="section" level="1" n="300"> <head xml:id="echoid-head327" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/>35 # 36 # 37 # 38 # 39 <lb/>30 # 7132931 # 7399610 # 7673270 # 7954358 # 8243363 # 30 <lb/>31 # 7137321 # 7404112 # 7677893 # 7959109 # 8248250 # 29 <lb/>32 # 7141713 # 7408616 # 7682518 # 7963862 # 8253139 # 28 <lb/>33 # 7146106 # 7413122 # 7687145 # 7968617 # 8258031 # 27 <lb/>34 # 7150501 # 7417630 # 7691774 # 7973374 # 8262925 # 26 <lb/>35 # 7154898 # 7422140 # 7696405 # 7978133 # 8267821 # 25 <lb/>36 # 7159297 # 7426652 # 7701038 # 7982895 # 8272720 # 24 <lb/>37 # 7163698 # 7431167 # 7705673 # 7987659 # 8277621 # 23 <lb/>38 # 7168100 # 7435684 # 7710310 # 7992425 # 8282524 # 22 <lb/>39 # 7172504 # 7440203 # 7714949 # 7997193 # 8287429 # 21 <lb/>40 # 7176910 # 7444724 # 7719590 # 8001963 # 8292337 # 20 <lb/>41 # 7181318 # 7449246 # 7724233 # 8006736 # 8297247 # 19 <lb/>42 # 7185728 # 7453770 # 7728878 # 8011511 # 8302160 # 18 <lb/>43 # 7190140 # 7458296 # 7733525 # 8016288 # 8307075 # 17 <lb/>44 # 7194554 # 7462824 # 7738175 # 8021067 # 8311992 # 16 <lb/>45 # 7198970 # 7476354 # 7742827 # 8025849 # 8316912 # 15 <lb/>46 # 7203387 # 7471886 # 7747481 # 8030633 # 8321834 # 14 <lb/>47 # 7207806 # 7476420 # 7752137 # 8035419 # 8326759 # 13 <lb/>48 # 7212227 # 7480956 # 7756795 # 8040207 # 8331686 # 12 <lb/>49 # 7216650 # 7485494 # 7761455 # 8044997 # 8336615 # 11 <lb/>50 # 7221075 # 7490033 # 7766117 # 8049790 # 8341547 # 10 <lb/>51 # 7225502 # 7494574 # 7770781 # 8054585 # 8346481 # 9 <lb/>52 # 7229931 # 7499117 # 7775447 # 8059382 # 8351418 # 8 <lb/>53 # 7234362 # 7503663 # 7780116 # 8064181 # 8356357 # 7 <lb/>54 # 7238794 # 7508211 # 7784787 # 8068983 # 8361298 # 6 <lb/>55 # 7243228 # 7512761 # 7789460 # 8073787 # 8366242 # 5 <lb/>56 # 7247664 # 7517313 # 7794135 # 8078593 # 8371188 # 4 <lb/>57 # 7252102 # 7521867 # 7798812 # 8083401 # 8376136 # 3 <lb/>58 # 7256541 # 7526423 # 7803491 # 8088212 # 8381087 # 2 <lb/>59 # 7260982 # 7530981 # 7808172 # 8093025 # 8386040 # 1 <lb/>60 # 7265424 # 7535541 # 7812856 # 8097840 # 8390996 # 0 <lb/>54 # 53 # 52 # 51 # 50 <lb/></note> </div> <div xml:id="echoid-div618" type="section" level="1" n="301"> <head xml:id="echoid-head328" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="218" file="230" n="230" rhead=""/> </div> <div xml:id="echoid-div619" type="section" level="1" n="302"> <head xml:id="echoid-head329" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>40 # 41 # 42 # 43 # 44 <lb/>0 # 8390996 # 8692867 # 9004040 # 9325151 # 9656888 # 60 <lb/>1 # 8395954 # 8697975 # 9009308 # 9330591 # 9662511 # 59 <lb/>2 # 8400915 # 8703085 # 9014579 # 9336034 # 9668137 # 58 <lb/>3 # 8405878 # 8708198 # 9019853 # 9341480 # 9673766 # 57 <lb/>4 # 8410844 # 8713344 # 9025130 # 9346929 # 9679398 # 56 <lb/>5 # 8415812 # 8718433 # 9030410 # 9352381 # 9685034 # 55 <lb/>6 # 8420782 # 8723555 # 9035693 # 9357835 # 9690674 # 54 <lb/>7 # 8425754 # 8728679 # 9040978 # 9363292 # 9696315 # 53 <lb/>8 # 8430729 # 8733806 # 9046266 # 9368752 # 9701960 # 52 <lb/>9 # 8435706 # 8738935 # 9051557 # 9374215 # 9707609 # 51 <lb/>10 # 8440686 # 8744067 # 9056850 # 9379682 # 9713261 # 50 <lb/>11 # 8445668 # 8749201 # 9062146 # 9385152 # 9718916 # 49 <lb/>12 # 8450653 # 8754338 # 9067445 # 9390625 # 9724574 # 48 <lb/>13 # 8455640 # 8759478 # 9072747 # 9396101 # 9730235 # 47 <lb/>14 # 8460630 # 8764620 # 9078052 # 9401580 # 9735900 # 46 <lb/>15 # 8465622 # 8769764 # 9083360 # 9407062 # 9741568 # 45 <lb/>16 # 8470617 # 8774911 # 9088670 # 9412547 # 9747239 # 44 <lb/>17 # 8475614 # 8780061 # 9093983 # 9418034 # 9752913 # 43 <lb/>18 # 8480614 # 8785214 # 9099299 # 9423524 # 9758591 # 42 <lb/>19 # 8485617 # 8790369 # 9104618 # 9429017 # 9764272 # 41 <lb/>20 # 8490622 # 8795527 # 9109940 # 9434513 # 9769956 # 40 <lb/>21 # 8495629 # 8800688 # 9115265 # 9440012 # 9775643 # 39 <lb/>22 # 8500639 # 8805851 # 9120593 # 9445514 # 9781334 # 38 <lb/>23 # 8505651 # 8811017 # 9125923 # 9451019 # 9787028 # 37 <lb/>24 # 8510666 # 8816186 # 9131256 # 9456528 # 9792725 # 36 <lb/>25 # 8515683 # 8821357 # 9136592 # 9462040 # 9798425 # 35 <lb/>26 # 8520703 # 8826531 # 9141930 # 9467555 # 9804128 # 34 <lb/>27 # 8525725 # 8831708 # 9147271 # 9473073 # 9809835 # 33 <lb/>28 # 8530750 # 8836887 # 9152615 # 9478594 # 9815545 # 32 <lb/>29 # 8535777 # 8842069 # 9157962 # 9484118 # 9821258 # 31 <lb/>30 # 8540806 # 8847253 # 9163312 # 9489645 # 9826974 # 30 <lb/>49 # 48 # 47 # 46 # 45 <lb/></note> </div> <div xml:id="echoid-div620" type="section" level="1" n="303"> <head xml:id="echoid-head330" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="219" file="231" n="231" rhead=""/> </div> <div xml:id="echoid-div621" type="section" level="1" n="304"> <head xml:id="echoid-head331" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/>40 # 41 # 42 # 43 # 44 <lb/>30 # 8540806 # 8847253 # 9163312 # 9489645 # 9826974 # 30 <lb/>31 # 8545838 # 8852440 # 9168665 # 9495175 # 9832694 # 29 <lb/>32 # 8550872 # 8857630 # 9174021 # 9400708 # 9838417 # 28 <lb/>33 # 8555909 # 8862822 # 9179380 # 9506244 # 9844143 # 27 <lb/>34 # 8560949 # 8868017 # 9184741 # 9511783 # 9849872 # 26 <lb/>35 # 8565991 # 8873015 # 9190105 # 9517325 # 9855605 # 25 <lb/>36 # 8571036 # 8878415 # 9195472 # 9522870 # 9861341 # 24 <lb/>37 # 8576083 # 8883628 # 9200842 # 9528419 # 9867180 # 23 <lb/>38 # 8581133 # 8888824 # 9206215 # 9533971 # 9872922 # 22 <lb/>39 # 8586185 # 8899033 # 9211590 # 9539526 # 9878668 # 21 <lb/>40 # 8591239 # 8899244 # 9216968 # 9545084 # 9884317 # 20 <lb/>41 # 8596296 # 8904458 # 9222349 # 9550645 # 9890070 # 19 <lb/>42 # 8601355 # 8909675 # 9227733 # 9556209 # 9895826 # 18 <lb/>43 # 8606417 # 8914894 # 9233120 # 9561776 # 9901585 # 17 <lb/>44 # 8611482 # 8920116 # 9238510 # 9567346 # 9907347 # 16 <lb/>45 # 8616549 # 8925341 # 9243903 # 9572919 # 9913113 # 15 <lb/>46 # 8621619 # 8930568 # 9249299 # 9578495 # 9918882 # 14 <lb/>47 # 8626692 # 8935798 # 9254698 # 9584074 # 9924654 # 13 <lb/>48 # 8631767 # 8941031 # 9260100 # 9589656 # 9930430 # 12 <lb/>49 # 8636845 # 8946267 # 9265505 # 9595241 # 9936209 # 11 <lb/>50 # 8641926 # 8951506 # 9270913 # 9600830 # 9941991 # 10 <lb/>51 # 8647009 # 8956747 # 9276324 # 9606422 # 9947777 # 9 <lb/>52 # 8652095 # 8961991 # 9281738 # 9612017 # 9953566 # 8 <lb/>53 # 8657183 # 8967238 # 9287155 # 9617615 # 9959359 # 7 <lb/>54 # 8662273 # 8972487 # 9292574 # 9623216 # 9965155 # 6 <lb/>55 # 8667366 # 8977739 # 9297996 # 9628820 # 9970954 # 5 <lb/>56 # 8672461 # 8982994 # 9303421 # 9634427 # 9976756 # 4 <lb/>57 # 8677559 # 8988252 # 9308849 # 9640037 # 9982562 # 3 <lb/>58 # 8682659 # 8993512 # 9314280 # 9645651 # 9988371 # 2 <lb/>59 # 8687762 # 8998775 # 9319714 # 9651268 # 9994184 # 1 <lb/>60 # 8692867 # 9004040 # 9325151 # 9656888 # 10000000 # 0 <lb/>49 # 48 # 47 # 46 # 45 <lb/></note> </div> <div xml:id="echoid-div622" type="section" level="1" n="305"> <head xml:id="echoid-head332" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="220" file="232" n="232" rhead=""/> </div> <div xml:id="echoid-div623" type="section" level="1" n="306"> <head xml:id="echoid-head333" xml:space="preserve">Gradus Qudrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>45 # 46 # 47 # 48 <lb/>0 # 10000000 # 10355302 # 10723686 # 11106124 # 60 <lb/>1 # 10005820 # 10361332 # 10729942 # 11112623 # 59 <lb/>2 # 10011643 # 10367365 # 10736202 # 11119126 # 58 <lb/>3 # 10017469 # 10373402 # 10742466 # 11125634 # 57 <lb/>4 # 10023299 # 10379443 # 10748734 # 11132146 # 56 <lb/>5 # 10029132 # 10385487 # 10755006 # 11138662 # 55 <lb/>6 # 10034968 # 10391535 # 10761282 # 11145182 # 54 <lb/>7 # 10040808 # 10397587 # 10767562 # 11151706 # 53 <lb/>8 # 10046651 # 10403643 # 10773845 # 11158235 # 52 <lb/>9 # 10052497 # 10409702 # 10780132 # 11164768 # 51 <lb/>10 # 10058347 # 10415765 # 10786423 # 11171305 # 50 <lb/>11 # 10064201 # 10421832 # 10792718 # 11177846 # 49 <lb/>12 # 10070058 # 10427902 # 10799017 # 11184392 # 48 <lb/>13 # 10075918 # 10433976 # 10805320 # 11190942 # 47 <lb/>14 # 10081782 # 10340054 # 10811627 # 11197496 # 46 <lb/>15 # 10087649 # 10446135 # 10817938 # 11204054 # 45 <lb/>16 # 10093520 # 10452220 # 10824253 # 11210617 # 44 <lb/>17 # 10099394 # 10458309 # 10830572 # 11217184 # 43 <lb/>18 # 10105272 # 10464401 # 10836895 # 11223755 # 42 <lb/>19 # 10111153 # 10470497 # 10843222 # 11230330 # 41 <lb/>20 # 10117038 # 10476597 # 10849554 # 11236910 # 40 <lb/>21 # 10122926 # 10482701 # 10855889 # 11243494 # 39 <lb/>22 # 10128818 # 10488808 # 10862228 # 11250082 # 38 <lb/>23 # 10134713 # 10494919 # 10868571 # 11256675 # 37 <lb/>24 # 10140611 # 10501034 # 10874918 # 11263272 # 36 <lb/>25 # 10146513 # 10507153 # 10881269 # 11269873 # 35 <lb/>26 # 10152418 # 10513275 # 10887624 # 11276478 # 34 <lb/>27 # 10158327 # 10519401 # 10893983 # 11283088 # 33 <lb/>28 # 10164239 # 10525531 # 10900346 # 11289702 # 32 <lb/>29 # 10170154 # 10531664 # 10906713 # 11296321 # 31 <lb/>30 # 10176073 # 10537801 # 10913084 # 11302944 # 30 <lb/>44 # 43 # 42 # 41 <lb/></note> </div> <div xml:id="echoid-div624" type="section" level="1" n="307"> <head xml:id="echoid-head334" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="221" file="233" n="233" rhead=""/> </div> <div xml:id="echoid-div625" type="section" level="1" n="308"> <head xml:id="echoid-head335" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/>45 # 46 # 47 # 48 <lb/>30 # 10176073 # 10537801 # 10913084 # 11302944 # 30 <lb/>31 # 10181996 # 10543942 # 10919459 # 11309571 # 39 <lb/>32 # 10187922 # 10550087 # 10925838 # 11316203 # 28 <lb/>33 # 10193852 # 10556235 # 10932221 # 11322899 # 27 <lb/>34 # 10199785 # 10562387 # 10938608 # 11329480 # 26 <lb/>35 # 10205722 # 10568543 # 10945000 # 11336125 # 25 <lb/>36 # 10211663 # 10574703 # 10951396 # 11342774 # 24 <lb/>37 # 10217607 # 10580867 # 10957796 # 11349428 # 23 <lb/>38 # 10223555 # 10587034 # 10964200 # 11356086 # 22 <lb/>39 # 10229506 # 10593205 # 10970608 # 11362748 # 21 <lb/>40 # 10235460 # 10599280 # 10977020 # 11369415 # 20 <lb/>41 # 10241418 # 10605559 # 10983436 # 11376086 # 19 <lb/>42 # 10247380 # 10611742 # 10989856 # 11382762 # 18 <lb/>43 # 10253345 # 10617929 # 10996280 # 11389442 # 17 <lb/>44 # 10259314 # 10624119 # 11002708 # 11396126 # 16 <lb/>45 # 10265286 # 10630313 # 11009140 # 11402815 # 15 <lb/>46 # 10271262 # 10636511 # 11015577 # 11409508 # 14 <lb/>47 # 10277242 # 10642713 # 11022028 # 11416206 # 13 <lb/>48 # 10283225 # 10648919 # 11028463 # 11422908 # 12 <lb/>49 # 10289212 # 10655128 # 11034912 # 11429615 # 11 <lb/>50 # 10295202 # 10661341 # 11041365 # 11436326 # 10 <lb/>51 # 10301196 # 10667558 # 11047822 # 11443042 # 9 <lb/>52 # 10307193 # 10673779 # 11054283 # 11449762 # 8 <lb/>53 # 10313194 # 10680004 # 11060748 # 11456487 # 7 <lb/>54 # 10319199 # 10686233 # 11067218 # 11463216 # 6 <lb/>55 # 10325207 # 10692466 # 11073692 # 11469950 # 5 <lb/>56 # 10331219 # 10698702 # 11080170 # 11476688 # 4 <lb/>57 # 10337234 # 10704942 # 11086652 # 11483431 # 3 <lb/>58 # 10343253 # 10711186 # 11093138 # 11490178 # 2 <lb/>59 # 10349276 # 10717434 # 11099629 # 11496929 # 1 <lb/>60 # 10355302 # 10723686 # 11106124 # 11503684 # 0 <lb/>44 # 43 # 42 # 41 <lb/></note> </div> <div xml:id="echoid-div626" type="section" level="1" n="309"> <head xml:id="echoid-head336" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="222" file="234" n="234" rhead=""/> </div> <div xml:id="echoid-div627" type="section" level="1" n="310"> <head xml:id="echoid-head337" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>49 # 50 # 51 # 52 <lb/>0 # 11503684 # 11917537 # 12348972 # 12799416 # 60 <lb/>1 # 11510444 # 11924580 # 12356320 # 12807093 # 59 <lb/>2 # 11517208 # 11931628 # 12363673 # 12814776 # 58 <lb/>3 # 11523977 # 11938680 # 12371031 # 12822465 # 57 <lb/>4 # 11530751 # 11945737 # 12378394 # 12830159 # 56 <lb/>5 # 11537529 # 11952799 # 12385762 # 12837859 # 55 <lb/>6 # 11544312 # 11959866 # 12393136 # 12845565 # 54 <lb/>7 # 11551100 # 11966938 # 12400515 # 12853277 # 53 <lb/>8 # 11557893 # 11974015 # 12407999 # 12860994 # 52 <lb/>9 # 11564691 # 11981097 # 12415288 # 12868717 # 51 <lb/>10 # 11571494 # 11988183 # 12422683 # 12876445 # 50 <lb/>11 # 11578301 # 11995274 # 12430083 # 12884179 # 49 <lb/>12 # 11585112 # 12002370 # 12437489 # 12891919 # 48 <lb/>13 # 11591928 # 12009471 # 12444900 # 12899665 # 47 <lb/>14 # 11598748 # 12016578 # 12452317 # 12907417 # 46 <lb/>15 # 11605572 # 12023690 # 12459739 # 12915175 # 45 <lb/>16 # 11612401 # 12030807 # 12467167 # 12922939 # 44 <lb/>17 # 11619234 # 12037929 # 12474600 # 12930709 # 43 <lb/>18 # 11626072 # 12045056 # 12482039 # 12938485 # 42 <lb/>19 # 11632915 # 12052188 # 12489484 # 12946267 # 41 <lb/>20 # 11639763 # 12059325 # 12496934 # 12954055 # 40 <lb/>21 # 11646615 # 12066467 # 12504389 # 12961843 # 39 <lb/>22 # 11653472 # 12073614 # 12511850 # 12969647 # 38 <lb/>23 # 11660334 # 12080766 # 12519316 # 12977457 # 37 <lb/>24 # 11667200 # 12087923 # 12526787 # 12985263 # 36 <lb/>25 # 11674071 # 12095085 # 12534264 # 12993080 # 35 <lb/>26 # 11680947 # 12102252 # 12541746 # 13000903 # 34 <lb/>27 # 11687827 # 12109424 # 12549233 # 13008732 # 33 <lb/>28 # 11694712 # 12116601 # 12556725 # 13016567 # 32 <lb/>29 # 11701602 # 12123783 # 12564222 # 13024407 # 31 <lb/>30 # 11708497 # 12130970 # 12571724 # 13032253 # 30 <lb/>40 # 39 # 38 # 37 <lb/></note> </div> <div xml:id="echoid-div628" type="section" level="1" n="311"> <head xml:id="echoid-head338" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="223" file="235" n="235" rhead=""/> </div> <div xml:id="echoid-div629" type="section" level="1" n="312"> <head xml:id="echoid-head339" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/>49 # 50 # 51 # 52 <lb/>30 # 11708497 # 12130970 # 12571724 # 13032253 # 30 <lb/>31 # 11715396 # 12138162 # 12579232 # 13040105 # 39 <lb/>32 # 11722300 # 12145359 # 12586746 # 13047963 # 28 <lb/>33 # 11729208 # 12152561 # 12594265 # 13055827 # 27 <lb/>34 # 11736121 # 12159768 # 12601790 # 13063697 # 26 <lb/>35 # 11743039 # 12166981 # 12609321 # 13071573 # 25 <lb/>36 # 11749962 # 12174199 # 12616858 # 13079455 # 24 <lb/>37 # 11756989 # 12181422 # 12624400 # 13087343 # 23 <lb/>38 # 11763821 # 12188650 # 12631948 # 13095237 # 22 <lb/>39 # 11770758 # 12195883 # 12639501 # 13103138 # 21 <lb/>40 # 11777700 # 12203121 # 12647060 # 13111045 # 20 <lb/>41 # 11784646 # 12210364 # 12654624 # 13118958 # 19 <lb/>42 # 11791597 # 12217613 # 12662194 # 13126877 # 18 <lb/>43 # 11798553 # 12224867 # 12669769 # 13134802 # 17 <lb/>44 # 11805514 # 12232126 # 12677350 # 13142732 # 16 <lb/>45 # 11812479 # 12239390 # 12684937 # 13150668 # 15 <lb/>46 # 11819449 # 12246659 # 12692530 # 13158610 # 14 <lb/>47 # 11826424 # 12253933 # 12700128 # 13166558 # 13 <lb/>48 # 11833404 # 12261212 # 12707732 # 13174512 # 12 <lb/>49 # 11840388 # 12268496 # 12715341 # 13182472 # 11 <lb/>50 # 11847377 # 12275786 # 12722956 # 13190438 # 10 <lb/>51 # 11854371 # 12283081 # 12730577 # 13198411 # 9 <lb/>52 # 11861370 # 12290381 # 12738203 # 13206390 # 8 <lb/>53 # 11868374 # 12297687 # 12745835 # 13214375 # 7 <lb/>54 # 11875383 # 12304998 # 12753473 # 13222367 # 6 <lb/>55 # 11882397 # 12312314 # 12761116 # 13230365 # 5 <lb/>56 # 11889417 # 12319635 # 12768765 # 13238369 # 4 <lb/>57 # 11896438 # 12326961 # 12776420 # 13246379 # 3 <lb/>58 # 11903466 # 12334293 # 12784080 # 13254396 # 2 <lb/>59 # 11910499 # 12341630 # 12791745 # 13262419 # 1 <lb/>60 # 11917537 # 12348972 # 12799416 # 13270448 # 0 <lb/>40 # 39 # 38 # 37 <lb/></note> </div> <div xml:id="echoid-div630" type="section" level="1" n="313"> <head xml:id="echoid-head340" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="224" file="236" n="236" rhead=""/> </div> <div xml:id="echoid-div631" type="section" level="1" n="314"> <head xml:id="echoid-head341" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>53 # 54 # 55 # 56 <lb/>0 # 13270448 # 13763820 # 14281480 # 14825610 # 60 <lb/>1 # 13278483 # 13772243 # 14290325 # 14834916 # 59 <lb/>2 # 13286524 # 13780673 # 14299177 # 14844230 # 58 <lb/>3 # 13294571 # 13789109 # 14308037 # 14853553 # 57 <lb/>4 # 13302624 # 13797552 # 14316905 # 14862884 # 56 <lb/>5 # 13310683 # 13806002 # 14325780 # 14872223 # 55 <lb/>6 # 13318749 # 13814459 # 14334662 # 14881570 # 54 <lb/>7 # 13326821 # 13822922 # 14343552 # 14890925 # 53 <lb/>8 # 13334899 # 13831392 # 14352451 # 14909288 # 52 <lb/>9 # 13342984 # 13839869 # 14361354 # 14909659 # 51 <lb/>10 # 13351075 # 13848352 # 14370266 # 14919038 # 50 <lb/>11 # 13359172 # 13856842 # 14379186 # 14928426 # 49 <lb/>12 # 13367276 # 13865339 # 14388113 # 14937822 # 48 <lb/>13 # 13375386 # 13873843 # 14397048 # 14947226 # 47 <lb/>14 # 13383502 # 13882354 # 14405990 # 14956638 # 46 <lb/>15 # 13391624 # 13890872 # 14414939 # 14966058 # 45 <lb/>16 # 13399753 # 13899397 # 14423896 # 14975486 # 44 <lb/>17 # 13407888 # 13907930 # 14432861 # 14984923 # 43 <lb/>18 # 13416029 # 13916470 # 14441833 # 14994368 # 42 <lb/>19 # 13424177 # 13925017 # 14450812 # 15003821 # 41 <lb/>20 # 13432331 # 13933571 # 14459799 # 15013283 # 40 <lb/>21 # 13440492 # 13942131 # 14468794 # 15022753 # 39 <lb/>22 # 13448659 # 13950698 # 14477797 # 15032231 # 38 <lb/>23 # 13456832 # 13959272 # 14486807 # 15041717 # 37 <lb/>24 # 13465011 # 13967853 # 14495825 # 15051211 # 36 <lb/>25 # 13473197 # 13976441 # 14504850 # 15060714 # 35 <lb/>26 # 13481390 # 13985035 # 14513883 # 15070225 # 34 <lb/>27 # 13489589 # 13993636 # 14522924 # 15079744 # 33 <lb/>28 # 13497794 # 14002244 # 14531972 # 15089271 # 32 <lb/>29 # 13506006 # 14010859 # 14541028 # 15078807 # 31 <lb/>30 # 13514224 # 14019481 # 14550091 # 15108351 # 30 <lb/>36 # 35 # 34 # 33 <lb/></note> </div> <div xml:id="echoid-div632" type="section" level="1" n="315"> <head xml:id="echoid-head342" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="225" file="237" n="237" rhead=""/> </div> <div xml:id="echoid-div633" type="section" level="1" n="316"> <head xml:id="echoid-head343" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/>53 # 54 # 55 # 56 <lb/>30 # 13514224 # 14019481 # 14550091 # 15108351 # 30 <lb/>31 # 13522449 # 14028110 # 14559162 # 15117903 # 39 <lb/>32 # 13530680 # 14036746 # 14568241 # 15127464 # 28 <lb/>33 # 13538918 # 14045389 # 14577327 # 15137034 # 27 <lb/>34 # 13547162 # 14054040 # 14586421 # 15146612 # 26 <lb/>35 # 13555413 # 14062698 # 14595523 # 15156199 # 25 <lb/>36 # 13563670 # 14071363 # 14604633 # 15165794 # 24 <lb/>37 # 13571834 # 14080035 # 14613750 # 15175398 # 23 <lb/>38 # 13580104 # 14088715 # 14622875 # 15185011 # 22 <lb/>39 # 13588381 # 14097402 # 14632007 # 15194632 # 21 <lb/>40 # 13596764 # 14106097 # 14641146 # 15204261 # 20 <lb/>41 # 13605054 # 14114798 # 14650293 # 15213899 # 19 <lb/>42 # 13613350 # 14123506 # 14659449 # 15223545 # 18 <lb/>43 # 13621653 # 14132221 # 14668613 # 15233200 # 17 <lb/>44 # 13629963 # 14140923 # 14677785 # 15242863 # 16 <lb/>45 # 13638279 # 14149672 # 14686965 # 15252535 # 15 <lb/>46 # 13646602 # 14158409 # 14696153 # 15262216 # 14 <lb/>47 # 13654932 # 14167153 # 14705349 # 15271905 # 13 <lb/>48 # 13663268 # 14175904 # 14714553 # 15281603 # 12 <lb/>49 # 13671610 # 14184663 # 14723765 # 15291309 # 11 <lb/>50 # 13679959 # 14193429 # 14732985 # 15301024 # 10 <lb/>51 # 13688315 # 14202202 # 14742212 # 15310748 # 9 <lb/>52 # 13696677 # 14210982 # 14751447 # 15320481 # 8 <lb/>53 # 13705046 # 14219769 # 14760690 # 15330222 # 7 <lb/>54 # 13713422 # 14228563 # 14769941 # 15339972 # 6 <lb/>55 # 13721805 # 14237365 # 14779200 # 15349730 # 5 <lb/>56 # 13730194 # 14246174 # 14788466 # 15359497 # 4 <lb/>57 # 13738590 # 14254990 # 14797740 # 15369273 # 3 <lb/>58 # 13746993 # 14263813 # 14807022 # 15379057 # 2 <lb/>59 # 13755403 # 14272643 # 14816312 # 15388850 # 1 <lb/>60 # 13763820 # 14281480 # 14825610 # 15398651 # 0 <lb/>36 # 35 # 34 # 33 <lb/></note> </div> <div xml:id="echoid-div634" type="section" level="1" n="317"> <head xml:id="echoid-head344" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="226" file="238" n="238" rhead=""/> </div> <div xml:id="echoid-div635" type="section" level="1" n="318"> <head xml:id="echoid-head345" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>57 # 58 # 59 # 60 <lb/>0 # 15398651 # 16003347 # 16642794 # 17320508 # 60 <lb/>1 # 15408461 # 16013710 # 16653766 # 17332150 # 59 <lb/>2 # 15418280 # 16024083 # 16664749 # 17343804 # 58 <lb/>3 # 15428108 # 16034466 # 16675742 # 17355469 # 57 <lb/>4 # 15437945 # 16044859 # 16686746 # 17367146 # 56 <lb/>5 # 15447791 # 16055261 # 16697760 # 17378834 # 55 <lb/>6 # 15457646 # 16065673 # 16708785 # 17390534 # 54 <lb/>7 # 15467510 # 16076095 # 16719820 # 17402246 # 53 <lb/>8 # 15477382 # 16086527 # 16730866 # 17413969 # 52 <lb/>9 # 15487263 # 16096968 # 16741922 # 17425704 # 51 <lb/>10 # 15497153 # 16107419 # 16752989 # 17437451 # 50 <lb/>11 # 15507052 # 16117880 # 16764067 # 17449210 # 49 <lb/>12 # 15516960 # 16128351 # 16775156 # 17460981 # 48 <lb/>13 # 15526877 # 16138832 # 16786256 # 17472764 # 47 <lb/>14 # 15536803 # 16149322 # 16797367 # 17484559 # 46 <lb/>15 # 15546738 # 16159822 # 16808489 # 17496366 # 45 <lb/>16 # 15556682 # 16170332 # 16819621 # 17508185 # 44 <lb/>17 # 15566636 # 16180852 # 16830764 # 17520026 # 43 <lb/>18 # 15576599 # 16191381 # 16841918 # 17531869 # 42 <lb/>19 # 15586571 # 16201920 # 16853083 # 17543724 # 41 <lb/>20 # 15596552 # 16212469 # 16864259 # 17555591 # 40 <lb/>21 # 15606542 # 16223028 # 16875446 # 17567470 # 39 <lb/>22 # 15616541 # 16233597 # 16886644 # 17579362 # 38 <lb/>23 # 15626549 # 16244176 # 16897853 # 17591266 # 37 <lb/>24 # 15636566 # 16254766 # 16909074 # 17603182 # 36 <lb/>25 # 15646592 # 16265366 # 16920306 # 17615111 # 35 <lb/>26 # 15656627 # 16275976 # 16931549 # 17627052 # 34 <lb/>27 # 15666671 # 16286596 # 16942803 # 17639006 # 33 <lb/>28 # 15676724 # 16297226 # 16954068 # 17650972 # 32 <lb/>29 # 15686786 # 16307866 # 16965344 # 17662951 # 31 <lb/>30 # 15696857 # 16318516 # 16976631 # 17674941 # 30 <lb/>32 # 31 # 30 # 29 <lb/></note> </div> <div xml:id="echoid-div636" type="section" level="1" n="319"> <head xml:id="echoid-head346" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="227" file="239" n="239" rhead=""/> </div> <div xml:id="echoid-div637" type="section" level="1" n="320"> <head xml:id="echoid-head347" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/>57 # 58 # 59 # 60 <lb/>30 # 15696857 # 16318516 # 16976631 # 17674942 # 30 <lb/>31 # 15706938 # 16329176 # 16987929 # 17686945 # 39 <lb/>32 # 15717028 # 16339847 # 16999239 # 17698960 # 28 <lb/>33 # 15727127 # 16350528 # 17010560 # 17710987 # 27 <lb/>34 # 15737235 # 16361219 # 17021892 # 17723027 # 26 <lb/>35 # 15747353 # 16371920 # 17033236 # 17735079 # 25 <lb/>36 # 15757480 # 16382631 # 17044591 # 17747143 # 24 <lb/>37 # 15767616 # 16393352 # 17055957 # 17759220 # 23 <lb/>38 # 15777761 # 16404083 # 17067325 # 17771309 # 22 <lb/>39 # 15787915 # 16414824 # 17078714 # 17783410 # 21 <lb/>40 # 15798078 # 16425575 # 17090115 # 17795524 # 20 <lb/>41 # 15808251 # 16436337 # 17101527 # 17808651 # 19 <lb/>42 # 15818433 # 16447109 # 17112950 # 17819790 # 18 <lb/>43 # 15828625 # 16457892 # 17124384 # 17831942 # 17 <lb/>44 # 15838827 # 16468685 # 17135829 # 17844107 # 16 <lb/>45 # 15849038 # 16479488 # 17147285 # 17856285 # 15 <lb/>46 # 15859259 # 16490302 # 17158752 # 17868475 # 14 <lb/>47 # 15869489 # 16501126 # 17170231 # 17880678 # 13 <lb/>48 # 15879729 # 16511960 # 17181721 # 17892894 # 12 <lb/>49 # 15889979 # 16522805 # 17193222 # 17905123 # 11 <lb/>50 # 15900238 # 16533660 # 17204734 # 17917364 # 10 <lb/>51 # 15910507 # 16544526 # 17216258 # 17929618 # 9 <lb/>52 # 15920785 # 16555402 # 17227794 # 17941885 # 8 <lb/>53 # 15931073 # 16566289 # 17239342 # 17954164 # 7 <lb/>54 # 15941370 # 16577186 # 17250902 # 17966456 # 6 <lb/>55 # 15951676 # 16588094 # 17262473 # 17978761 # 5 <lb/>56 # 15961992 # 16599013 # 17274056 # 17991079 # 4 <lb/>57 # 15972317 # 16609942 # 17285651 # 18003410 # 3 <lb/>58 # 15982651 # 16620882 # 17297258 # 18015753 # 2 <lb/>59 # 15992994 # 16631833 # 17308877 # 18028109 # 1 <lb/>60 # 16003347 # 16642794 # 17320508 # 18040478 # 0 <lb/>32 # 31 # 30 # 29 <lb/></note> </div> <div xml:id="echoid-div638" type="section" level="1" n="321"> <head xml:id="echoid-head348" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="228" file="240" n="240" rhead=""/> </div> <div xml:id="echoid-div639" type="section" level="1" n="322"> <head xml:id="echoid-head349" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/>61 # 62 # 63 # 64 <lb/>0 # 18040478 # 18807265 # 19626104 # 20503034 # 60 <lb/>1 # 18052860 # 18820471 # 19640225 # 20518180 # 59 <lb/>2 # 18065255 # 18833691 # 19654362 # 20533344 # 58 <lb/>3 # 18077663 # 18846925 # 19668516 # 20548526 # 57 <lb/>4 # 18090084 # 18860174 # 19682686 # 20563726 # 56 <lb/>5 # 18102518 # 18873437 # 19696872 # 20578945 # 55 <lb/>6 # 18114966 # 18886715 # 19711074 # 20594182 # 54 <lb/>7 # 18127427 # 18900007 # 19725293 # 20609437 # 53 <lb/>8 # 18139901 # 18913314 # 19739528 # 20624711 # 52 <lb/>9 # 18152388 # 18926636 # 19753780 # 20640003 # 51 <lb/>10 # 18164889 # 18939972 # 19768048 # 20655313 # 50 <lb/>11 # 18177403 # 18953323 # 19782333 # 20670642 # 49 <lb/>12 # 18189930 # 18966689 # 19796634 # 20685989 # 48 <lb/>13 # 18202470 # 18980070 # 19810951 # 20701355 # 47 <lb/>14 # 18215024 # 18993466 # 19825285 # 20716739 # 46 <lb/>15 # 18227591 # 19006876 # 19839635 # 20732142 # 45 <lb/>16 # 18240171 # 19020301 # 19854002 # 20747564 # 44 <lb/>17 # 18252765 # 19033741 # 19868386 # 20763004 # 43 <lb/>18 # 18265372 # 19047196 # 19882786 # 20778463 # 42 <lb/>19 # 18277992 # 19060665 # 19897203 # 20793941 # 41 <lb/>20 # 18290626 # 19074149 # 19911637 # 20809438 # 40 <lb/>21 # 18303273 # 19087648 # 19926088 # 20824953 # 39 <lb/>22 # 18315934 # 19101162 # 19940555 # 20840487 # 38 <lb/>23 # 18328608 # 19114691 # 19955039 # 20856040 # 37 <lb/>24 # 18341296 # 19128235 # 19669540 # 20871612 # 36 <lb/>25 # 18353997 # 19141795 # 19984057 # 20887202 # 35 <lb/>26 # 18366712 # 19155370 # 19998591 # 20902811 # 34 <lb/>27 # 18379440 # 19168960 # 20013142 # 20918439 # 33 <lb/>28 # 18392182 # 19182565 # 20027709 # 20934086 # 32 <lb/>29 # 18404938 # 19196185 # 20042297 # 20949752 # 31 <lb/>30 # 18417707 # 19209821 # 20056898 # 20965436 # 30 <lb/>28 # 27 # 26 # 25 <lb/></note> </div> <div xml:id="echoid-div640" type="section" level="1" n="323"> <head xml:id="echoid-head350" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="229" file="241" n="241" rhead=""/> </div> <div xml:id="echoid-div641" type="section" level="1" n="324"> <head xml:id="echoid-head351" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 61 # 62 # 63 # 64 <lb/>30 # 18417707 # 19209821 # 20056898 # 20965436 # 30 <lb/>31 # 18430490 # 19223472 # 20071516 # 20981140 # 29 <lb/>32 # 18443287 # 19237138 # 20086152 # 20996863 # 28 <lb/>33 # 18456098 # 19250819 # 20100805 # 21012605 # 27 <lb/>34 # 18468922 # 19264516 # 20115475 # 21028367 # 26 <lb/>35 # 18481760 # 19278228 # 20130163 # 21044148 # 25 <lb/>36 # 18494612 # 19291955 # 20144868 # 21059949 # 24 <lb/>37 # 18507478 # 19305698 # 20159@90 # 21075769 # 23 <lb/>38 # 18520357 # 193194@6 # 20174329 # 21091609 # 22 <lb/>39 # 18533250 # 19333230 # 20189086 # 21107468 # 21 <lb/>40 # 18546157 # 19347019 # 20203860 # 21123347 # 20 <lb/>41 # 18559078 # 19360824 # 20218651 # 21139246 # 19 <lb/>42 # 18572013 # 19374644 # 20233460 # 21155164 # 18 <lb/>43 # 18584962 # 19388480 # 20248286 # 21171102 # 17 <lb/>44 # 18597925 # 19402331 # 20263130 # 21187059 # 16 <lb/>45 # 18610902 # 19416198 # 20277991 # 21203036 # 15 <lb/>46 # 18623894 # 19430081 # 20292870 # 21219032 # 14 <lb/>47 # 18636900 # 19443980 # 20307767 # 21235048 # 13 <lb/>48 # 18649920 # 19457894 # 20322681 # 21251083 # 12 <lb/>49 # 18662954 # 19471824 # 20337613 # 21267138 # 11 <lb/>50 # 18676002 # 19485770 # 20352563 # 21283213 # 10 <lb/>51 # 18689064 # 19499732 # 20367531 # 21299308 # 9 <lb/>52 # 18702140 # 19513710 # 20382516 # 21315423 # 8 <lb/>53 # 18715231 # 19527704 # 20397519 # 21331558 # 7 <lb/>54 # 18728335 # 19541714 # 20412539 # 21347713 # 6 <lb/>55 # 18741454 # 19555739 # 20427577 # 21363888 # 5 <lb/>56 # 18754587 # 19569780 # 20442633 # 21380083 # 4 <lb/>57 # 18767735 # 19583837 # 20457706 # 21396298 # 3 <lb/>58 # 18780897 # 19597910 # 20472797 # 21412534 # 2 <lb/>59 # 18794074 # 19611999 # 20487906 # 21428790 # 1 <lb/>60 # 18807265 # 19626104 # 20503034 # 21445067 # 0 <lb/> # 28 # 27 # 26 # 25 <lb/></note> </div> <div xml:id="echoid-div642" type="section" level="1" n="325"> <head xml:id="echoid-head352" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="230" file="242" n="242" rhead=""/> </div> <div xml:id="echoid-div643" type="section" level="1" n="326"> <head xml:id="echoid-head353" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 65 # 66 # 67 # 68 <lb/>0 # 21445067 # 22460371 # 23558529 # 24750869 # 60 <lb/>1 # 21461364 # 22477965 # 23577595 # 24771613 # 59 <lb/>2 # 21477681 # 22495582 # 23596687 # 24792387 # 58 <lb/>3 # 21494019 # 22513222 # 23615805 # 24813191 # 57 <lb/>4 # 21510377 # 22530885 # 23634950 # 24834024 # 56 <lb/>5 # 21526756 # 22548571 # 23654121 # 24854887 # 55 <lb/>6 # 21543155 # 22566281 # 23673318 # 24875780 # 54 <lb/>7 # 21559575 # 22584014 # 23692542 # 24896704 # 53 <lb/>8 # 21576015 # 22601771 # 23711793 # 24917659 # 52 <lb/>9 # 21592475 # 22619551 # 23731071 # 24938644 # 51 <lb/>10 # 21608956 # 22637355 # 23750375 # 24959659 # 50 <lb/>11 # 21625458 # 22655183 # 23769706 # 24980705 # 49 <lb/>12 # 21641981 # 22673034 # 23789064 # 25001782 # 48 <lb/>13 # 21658525 # 22690909 # 23808448 # 25022890 # 47 <lb/>14 # 21675090 # 22708808 # 23827859 # 25044029 # 46 <lb/>15 # 21691676 # 22726730 # 23847297 # 25065198 # 45 <lb/>16 # 21708283 # 22744676 # 23866762 # 25086398 # 44 <lb/>17 # 21724911 # 22762646 # 23886254 # 25107629 # 43 <lb/>18 # 21741559 # 22780639 # 23905773 # 25128991 # 42 <lb/>19 # 21758228 # 22798656 # 23925320 # 25150183 # 41 <lb/>20 # 21774918 # 22816696 # 23944895 # 25171506 # 40 <lb/>21 # 21791629 # 22834760 # 23964496 # 25192861 # 39 <lb/>22 # 21808362 # 22852848 # 23984124 # 25214248 # 38 <lb/>23 # 21825116 # 22870960 # 24003779 # 25235666 # 37 <lb/>24 # 21841892 # 22889096 # 24023462 # 25257116 # 36 <lb/>25 # 21858689 # 22907256 # 24043172 # 25278597 # 35 <lb/>26 # 21875508 # 22925441 # 24062910 # 25300110 # 34 <lb/>27 # 21892348 # 22943650 # 24082675 # 25321655 # 33 <lb/>28 # 21909210 # 22961883 # 24102468 # 25343232 # 32 <lb/>29 # 21926094 # 22980141 # 24122289 # 25364841 # 31 <lb/>30 # 21943000 # 22998424 # 24142137 # 25386482 # 30 <lb/> # 24 # 23 # 22 # 21 <lb/></note> </div> <div xml:id="echoid-div644" type="section" level="1" n="327"> <head xml:id="echoid-head354" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="231" file="243" n="243" rhead=""/> </div> <div xml:id="echoid-div645" type="section" level="1" n="328"> <head xml:id="echoid-head355" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 65 # 66 # 67 # 68 <lb/>30 # 21943000 # 22998424 # 24142137 # 25386482 # 30 <lb/>31 # 21959926 # 23016731 # 24162013 # 25408154 # 29 <lb/>32 # 21976874 # 23035062 # 24181917 # 25429858 # 28 <lb/>33 # 21993843 # 23053418 # 24201849 # 25451594 # 27 <lb/>34 # 22010834 # 23071798 # 24221809 # 25473362 # 26 <lb/>35 # 22027846 # 23090203 # 24241798 # 25495162 # 25 <lb/>36 # 22044879 # 23108632 # 24261815 # 25516995 # 24 <lb/>37 # 22061934 # 23127086 # 24281860 # 25538860 # 23 <lb/>38 # 22079011 # 23145565 # 24301934 # 25560758 # 22 <lb/>39 # 22096109 # 23164068 # 24322037 # 25582688 # 21 <lb/>40 # 22113229 # 23182597 # 24342169 # 25604651 # 20 <lb/>41 # 22130372 # 23201151 # 24362329 # 25626647 # 19 <lb/>42 # 22147537 # 23219730 # 24382518 # 25648675 # 18 <lb/>43 # 22164725 # 23238335 # 24402735 # 25670736 # 17 <lb/>44 # 22181935 # 23256965 # 24422981 # 25692830 # 16 <lb/>45 # 22199168 # 23275621 # 24443256 # 25714957 # 15 <lb/>46 # 22216424 # 23294302 # 24463559 # 25737118 # 14 <lb/>47 # 22233703 # 23313008 # 24483891 # 25759312 # 13 <lb/>48 # 22251004 # 23331740 # 24504252 # 25781540 # 12 <lb/>49 # 22268328 # 23350498 # 24524642 # 25803801 # 11 <lb/>50 # 22285675 # 23369282 # 24545061 # 25826096 # 10 <lb/>51 # 22303044 # 23388092 # 24565509 # 25848424 # 9 <lb/>52 # 22320435 # 23406927 # 24585986 # 25870786 # 8 <lb/>53 # 22337848 # 23425788 # 24606492 # 25893181 # 7 <lb/>54 # 22355284 # 23444674 # 24627028 # 25915610 # 6 <lb/>55 # 22372742 # 23463586 # 24647594 # 25938073 # 5 <lb/>56 # 22390223 # 23482523 # 24668189 # 25960569 # 4 <lb/>57 # 22407726 # 23501486 # 24688814 # 25983099 # 3 <lb/>58 # 22425252 # 23520475 # 24709469 # 26005663 # 2 <lb/>59 # 22442800 # 23539489 # 24730154 # 26028261 # 1 <lb/>60 # 22460371 # 23558529 # 24750869 # 26050893 # 0 <lb/> # 24 # 23 # 22 # 21 <lb/></note> </div> <div xml:id="echoid-div646" type="section" level="1" n="329"> <head xml:id="echoid-head356" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="232" file="244" n="244" rhead=""/> </div> <div xml:id="echoid-div647" type="section" level="1" n="330"> <head xml:id="echoid-head357" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 69 # 70 # 71 # 72 <lb/>0 # 26050893 # 27474777 # 29042105 # 30776834 # 60 <lb/>1 # 26073559 # 27499665 # 29069569 # 30807323 # 59 <lb/>2 # 26096260 # 27524592 # 29097080 # 30837866 # 58 <lb/>3 # 26118996 # 27549559 # 29124638 # 30868465 # 57 <lb/>4 # 26141766 # 27574565 # 29152243 # 30899119 # 56 <lb/>5 # 26164571 # 27599612 # 29179895 # 30929828 # 55 <lb/>6 # 26187411 # 27624699 # 29207595 # 30960593 # 54 <lb/>7 # 26210286 # 27649827 # 29235343 # 30991413 # 53 <lb/>8 # 26233196 # 27674995 # 29263139 # 31022289 # 52 <lb/>9 # 26256141 # 27700204 # 29290382 # 31053221 # 51 <lb/>10 # 26279120 # 27725453 # 29318873 # 31084208 # 50 <lb/>11 # 26302135 # 27750742 # 29346811 # 31115252 # 49 <lb/>12 # 26325185 # 27776072 # 29374797 # 31146352 # 48 <lb/>13 # 26348270 # 27801443 # 29402831 # 31177508 # 47 <lb/>14 # 26371390 # 27826855 # 29430913 # 31208720 # 46 <lb/>15 # 26394546 # 27852308 # 29459043 # 31239989 # 45 <lb/>16 # 26417738 # 27877803 # 29487221 # 31271315 # 44 <lb/>17 # 26440966 # 27903339 # 29515446 # 31302698 # 43 <lb/>18 # 26464229 # 27928917 # 29543719 # 31334138 # 42 <lb/>19 # 26487528 # 27954536 # 29572041 # 31365636 # 41 <lb/>20 # 26510863 # 27980196 # 29600411 # 31397191 # 40 <lb/>21 # 26534234 # 28005898 # 29628831 # 31428805 # 39 <lb/>22 # 26557641 # 28031642 # 29657301 # 31460476 # 38 <lb/>23 # 26581084 # 28057429 # 29685820 # 31492205 # 37 <lb/>24 # 26604563 # 28083258 # 29714388 # 31523992 # 36 <lb/>25 # 26628079 # 28109129 # 29743006 # 31555838 # 35 <lb/>26 # 26651631 # 28135043 # 29771674 # 31587742 # 34 <lb/>27 # 26675220 # 28160999 # 29800392 # 31619705 # 33 <lb/>28 # 26698845 # 28186998 # 29829160 # 31651727 # 32 <lb/>29 # 26722507 # 28213040 # 29857978 # 31683807 # 31 <lb/>30 # 26746206 # 28239125 # 29886847 # 31715946 # 30 <lb/> # 20 # 19 # 18 # 17 <lb/></note> </div> <div xml:id="echoid-div648" type="section" level="1" n="331"> <head xml:id="echoid-head358" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="233" file="245" n="245" rhead=""/> </div> <div xml:id="echoid-div649" type="section" level="1" n="332"> <head xml:id="echoid-head359" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 69 # 70 # 71 # 72 <lb/>30 # 26746206 # 28239125 # 29886847 # 31715946 # 30 <lb/>31 # 26769942 # 28265253 # 29915765 # 31748144 # 39 <lb/>32 # 26793716 # 28291424 # 29944734 # 31780401 # 28 <lb/>33 # 26816527 # 28317638 # 29973753 # 31812717 # 27 <lb/>34 # 26841375 # 28343895 # 30002823 # 31845093 # 26 <lb/>35 # 26865260 # 28379195 # 30031943 # 31877528 # 25 <lb/>36 # 26889183 # 28396539 # 30061113 # 31910024 # 24 <lb/>37 # 26913143 # 28422926 # 30090334 # 31942580 # 23 <lb/>38 # 26937141 # 28449357 # 30119605 # 31975197 # 22 <lb/>39 # 26961177 # 28475832 # 30148927 # 32007875 # 21 <lb/>40 # 26985251 # 28502350 # 30178299 # 32040613 # 20 <lb/>41 # 27009362 # 28528913 # 30207723 # 32073413 # 19 <lb/>42 # 27033511 # 28555520 # 30237200 # 32106275 # 18 <lb/>43 # 27057698 # 28582172 # 30266730 # 32139200 # 17 <lb/>44 # 27081922 # 28608868 # 30296312 # 32172187 # 16 <lb/>45 # 27106184 # 28635608 # 30325947 # 32205237 # 15 <lb/>46 # 27130484 # 28662393 # 30355635 # 32238349 # 14 <lb/>47 # 27154823 # 28689222 # 30385375 # 32271524 # 13 <lb/>48 # 27179200 # 28716096 # 30415169 # 32304762 # 12 <lb/>49 # 27203616 # 28743015 # 30445015 # 32338064 # 11 <lb/>50 # 27228070 # 28769979 # 30474915 # 32371430 # 10 <lb/>51 # 27252563 # 28796987 # 30504867 # 32404858 # 9 <lb/>52 # 27277095 # 28824040 # 30534872 # 32438348 # 8 <lb/>53 # 27301667 # 28851139 # 30564930 # 32471901 # 7 <lb/>54 # 27326278 # 28878283 # 30595041 # 32505517 # 6 <lb/>55 # 27350929 # 28905472 # 30625205 # 32539196 # 5 <lb/>56 # 27375620 # 28932707 # 30655423 # 32572937 # 4 <lb/>57 # 27400350 # 28959988 # 30685695 # 32606741 # 3 <lb/>58 # 27425120 # 28987315 # 30716020 # 32640907 # 2 <lb/>59 # 27449929 # 29014687 # 30746400 # 32674536 # 1 <lb/>60 # 27474777 # 29042105 # 30776834 # 32708528 # 0 <lb/> # 20 # 19 # 18 # 17 <lb/></note> </div> <div xml:id="echoid-div650" type="section" level="1" n="333"> <head xml:id="echoid-head360" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="234" file="246" n="246" rhead=""/> </div> <div xml:id="echoid-div651" type="section" level="1" n="334"> <head xml:id="echoid-head361" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 73 # 74 # 75 # 76 <lb/>0 # 32708528 # 34874151 # 37320517 # 40107808 # 60 <lb/>1 # 32745286 # 34912477 # 37363987 # 40157569 # 59 <lb/>2 # 32776709 # 34950881 # 37407551 # 40207446 # 58 <lb/>3 # 32810898 # 34989364 # 37451210 # 40257440 # 57 <lb/>4 # 32845153 # 35027925 # 37494964 # 40307552 # 56 <lb/>5 # 32879747 # 35066565 # 37538814 # 40357781 # 55 <lb/>6 # 32913862 # 35105283 # 37582760 # 40408129 # 54 <lb/>7 # 32948317 # 35144080 # 37626803 # 40458596 # 53 <lb/>8 # 32982839 # 35182956 # 37670943 # 40509183 # 52 <lb/>9 # 33017427 # 35221911 # 37715180 # 40559890 # 51 <lb/>10 # 33052082 # 35260945 # 37759515 # 40610718 # 50 <lb/>11 # 33086802 # 35300059 # 37803948 # 40661665 # 49 <lb/>12 # 33121588 # 35339253 # 37848479 # 40712731 # 48 <lb/>13 # 33156441 # 35378528 # 37893109 # 40763917 # 47 <lb/>14 # 33191362 # 35417883 # 37937838 # 40815224 # 46 <lb/>15 # 33226351 # 35457320 # 37982666 # 40866652 # 45 <lb/>16 # 33261408 # 35496838 # 38027592 # 40918201 # 44 <lb/>17 # 33296534 # 35536438 # 38072616 # 40969871 # 43 <lb/>18 # 33331728 # 35576121 # 38117740 # 41021663 # 42 <lb/>19 # 33366990 # 35615888 # 38162963 # 41073577 # 41 <lb/>20 # 33402321 # 35655739 # 38208285 # 41125614 # 40 <lb/>21 # 33437720 # 35695672 # 38253708 # 41177775 # 39 <lb/>22 # 33473188 # 35735689 # 38299232 # 41230062 # 38 <lb/>23 # 33508725 # 35775789 # 38344857 # 41282475 # 37 <lb/>24 # 33544330 # 35815973 # 38390584 # 41335015 # 36 <lb/>25 # 33580005 # 35856241 # 38436414 # 41387683 # 35 <lb/>26 # 33615750 # 35896593 # 38482347 # 41440480 # 34 <lb/>27 # 33651566 # 35937029 # 38528384 # 41493407 # 33 <lb/>28 # 33687453 # 35977550 # 38574525 # 41546464 # 32 <lb/>29 # 33723410 # 36018156 # 38620772 # 41599653 # 31 <lb/>30 # 33759438 # 36058848 # 38667125 # 41652974 # 30 <lb/> # 16 # 15 # 14 # 13 <lb/></note> </div> <div xml:id="echoid-div652" type="section" level="1" n="335"> <head xml:id="echoid-head362" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="235" file="247" n="247" rhead=""/> </div> <div xml:id="echoid-div653" type="section" level="1" n="336"> <head xml:id="echoid-head363" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 73 # 74 # 75 # 76 <lb/>30 # 33759438 # 36058848 # 38667125 # 41652974 # 30 <lb/>31 # 33795535 # 36099623 # 38713580 # 41706424 # 39 <lb/>32 # 33831703 # 36140483 # 38760139 # 41760003 # 28 <lb/>33 # 33867942 # 36181427 # 38806801 # 41813712 # 27 <lb/>34 # 33904252 # 36222456 # 38853567 # 41867550 # 26 <lb/>35 # 33940634 # 36263570 # 38900438 # 41921518 # 25 <lb/>36 # 33977088 # 36304771 # 38947416 # 41975617 # 24 <lb/>37 # 34013615 # 36346060 # 38994501 # 42029848 # 23 <lb/>38 # 34050215 # 36387437 # 39041695 # 42084211 # 22 <lb/>39 # 34086888 # 36428903 # 39088998 # 42138706 # 21 <lb/>40 # 34123634 # 36470459 # 39136409 # 42193334 # 20 <lb/>41 # 34160453 # 36512103 # 39183929 # 42248096 # 19 <lb/>42 # 34197345 # 36553836 # 39231557 # 42302993 # 18 <lb/>43 # 34234310 # 36595659 # 39279294 # 42358025 # 17 <lb/>44 # 34271348 # 36637572 # 39327139 # 42413193 # 16 <lb/>45 # 34308459 # 36679574 # 39375094 # 42468497 # 15 <lb/>46 # 34345644 # 36721666 # 39423158 # 42523937 # 14 <lb/>47 # 34382903 # 36763849 # 39471331 # 42579514 # 13 <lb/>48 # 34420237 # 36806121 # 39519614 # 42635228 # 12 <lb/>49 # 34457647 # 36848483 # 39568006 # 42691080 # 11 <lb/>50 # 34495132 # 36890936 # 39616509 # 42747070 # 10 <lb/>51 # 34532692 # 36933479 # 39665124 # 42803199 # 9 <lb/>52 # 34570327 # 36976114 # 39713852 # 42859468 # 8 <lb/>53 # 34608038 # 37018840 # 39762695 # 42915878 # 7 <lb/>54 # 34645824 # 37061659 # 39811654 # 42972429 # 6 <lb/>55 # 34683686 # 37104570 # 39860729 # 43029122 # 5 <lb/>56 # 34721625 # 37147574 # 39909917 # 43085958 # 4 <lb/>57 # 34759640 # 37190670 # 39959218 # 43142937 # 3 <lb/>58 # 34797733 # 37233859 # 40008633 # 43200060 # 2 <lb/>59 # 34835903 # 37277141 # 40058103 # 43257328 # 1 <lb/>60 # 34874151 # 37320517 # 40107808 # 43314742 # 0 <lb/> # 16 # 15 # 14 # 13 <lb/></note> </div> <div xml:id="echoid-div654" type="section" level="1" n="337"> <head xml:id="echoid-head364" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="236" file="248" n="248" rhead=""/> </div> <div xml:id="echoid-div655" type="section" level="1" n="338"> <head xml:id="echoid-head365" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 77 # 78 # 79 # 80 <lb/>0 # 43314742 # 47046295 # 51445543 # 56712854 # 60 <lb/>1 # 43372301 # 47113680 # 51525561 # 56809480 # 59 <lb/>2 # 43430006 # 47181249 # 51605820 # 56906425 # 58 <lb/>3 # 43487857 # 47249003 # 51686321 # 57003690 # 57 <lb/>4 # 43545855 # 47316942 # 51767065 # 57101277 # 56 <lb/>5 # 43604000 # 47385067 # 51848053 # 57199188 # 55 <lb/>6 # 43662293 # 47453380 # 51929285 # 57297425 # 54 <lb/>7 # 43720733 # 47521882 # 52010762 # 57395990 # 53 <lb/>8 # 43779321 # 47590575 # 52092485 # 57494885 # 52 <lb/>9 # 43838057 # 47659460 # 52174455 # 57594111 # 51 <lb/>10 # 43896942 # 47728538 # 52256673 # 57693670 # 50 <lb/>11 # 43955977 # 47797809 # 52339140 # 57793564 # 49 <lb/>12 # 44015163 # 47867274 # 52421857 # 57893795 # 48 <lb/>13 # 44074501 # 47936934 # 52504826 # 57994366 # 47 <lb/>14 # 44133992 # 48006790 # 52588048 # 58095279 # 46 <lb/>15 # 44193637 # 48076841 # 42671525 # 58196536 # 45 <lb/>16 # 44253435 # 48147088 # 52755259 # 58298138 # 44 <lb/>17 # 44313387 # 48217531 # 52839251 # 58400087 # 43 <lb/>18 # 44373494 # 48288171 # 52923503 # 58502385 # 42 <lb/>19 # 44433756 # 48359008 # 53008016 # 58605034 # 41 <lb/>20 # 44494174 # 48430043 # 53092792 # 58708035 # 40 <lb/>21 # 44554749 # 48501278 # 53177831 # 58811388 # 39 <lb/>22 # 44615481 # 48572714 # 53263134 # 58915095 # 38 <lb/>23 # 44676371 # 48644352 # 53348702 # 59019157 # 37 <lb/>24 # 44737419 # 48716193 # 53434536 # 59123576 # 36 <lb/>25 # 44798626 # 48788238 # 53520637 # 59228353 # 35 <lb/>26 # 44859993 # 48860488 # 53607006 # 59333490 # 34 <lb/>27 # 44921521 # 48932945 # 53693644 # 59438989 # 33 <lb/>28 # 44983211 # 49005610 # 53780552 # 59544852 # 32 <lb/>29 # 45045065 # 49078483 # 53867731 # 59651081 # 31 <lb/>30 # 45107083 # 49151565 # 53955183 # 59757678 # 30 <lb/> # 12 # 11 # 10 # 9 <lb/></note> </div> <div xml:id="echoid-div656" type="section" level="1" n="339"> <head xml:id="echoid-head366" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="237" file="249" n="249" rhead=""/> </div> <div xml:id="echoid-div657" type="section" level="1" n="340"> <head xml:id="echoid-head367" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 77 # 78 # 79 # 80 <lb/>30 # 45107083 # 49151565 # 53955183 # 59757678 # 30 <lb/>31 # 45169263 # 49224856 # 54042909 # 59864646 # 39 <lb/>32 # 45231607 # 49298357 # 54130911 # 59971987 # 28 <lb/>33 # 45294114 # 49372069 # 54219190 # 60079703 # 27 <lb/>34 # 45356785 # 49445993 # 54307748 # 60187796 # 26 <lb/>35 # 45419621 # 49520130 # 54396586 # 60296268 # 25 <lb/>36 # 45482623 # 49594481 # 54485705 # 60405121 # 24 <lb/>37 # 45545790 # 49669047 # 54575107 # 60514358 # 23 <lb/>38 # 45609123 # 49743829 # 54664793 # 60623981 # 22 <lb/>39 # 45672623 # 49818827 # 54754764 # 60733992 # 21 <lb/>40 # 45736291 # 49894042 # 55845022 # 60844392 # 20 <lb/>41 # 45800128 # 49969475 # 54935569 # 60955184 # 19 <lb/>42 # 45864135 # 50045127 # 55029406 # 61066370 # 18 <lb/>43 # 45928314 # 50120999 # 55117535 # 61177952 # 17 <lb/>44 # 45992666 # 50197092 # 55208958 # 61289930 # 16 <lb/>45 # 46057192 # 50273407 # 55300676 # 61402307 # 15 <lb/>46 # 46121892 # 50349935 # 55392692 # 61515085 # 14 <lb/>47 # 46186767 # 50246707 # 55485007 # 61628267 # 13 <lb/>48 # 46251817 # 50503695 # 55577622 # 61741856 # 12 <lb/>49 # 46318043 # 50580910 # 55670539 # 61855854 # 11 <lb/>50 # 46382445 # 50658353 # 55763759 # 61970263 # 10 <lb/>51 # 46448023 # 50736025 # 55857283 # 62085085 # 9 <lb/>52 # 46513778 # 50813927 # 55951112 # 62200323 # 8 <lb/>53 # 46579711 # 50892060 # 56045247 # 62315979 # 7 <lb/>54 # 46645823 # 50970425 # 56139689 # 62432056 # 6 <lb/>55 # 46712115 # 51049023 # 56234439 # 62548556 # 5 <lb/>56 # 46778587 # 51127855 # 56329498 # 62665481 # 4 <lb/>57 # 46845240 # 51206922 # 56424868 # 62782833 # 3 <lb/>58 # 46912075 # 51286225 # 56520550 # 62900615 # 2 <lb/>59 # 46979093 # 51365765 # 56616545 # 63018829 # 1 <lb/>60 # 47046295 # 51445543 # 56712854 # 63137478 # 0 <lb/> # 12 # 11 # 10 # 9 <lb/></note> </div> <div xml:id="echoid-div658" type="section" level="1" n="341"> <head xml:id="echoid-head368" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="238" file="250" n="250" rhead=""/> </div> <div xml:id="echoid-div659" type="section" level="1" n="342"> <head xml:id="echoid-head369" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 81 # 82 # 83 # 84 <lb/>0 # 63137478 # 71153707 # 81443502 # 95143611 # 60 <lb/>1 # 63256564 # 71304198 # 81639821 # 95410585 # 59 <lb/>2 # 63376089 # 71455313 # 81837074 # 95679034 # 58 <lb/>3 # 63496056 # 71607058 # 82035268 # 95948971 # 57 <lb/>4 # 63616468 # 71759440 # 82234410 # 96220411 # 56 <lb/>5 # 63737327 # 71912459 # 82434508 # 96493467 # 55 <lb/>6 # 63858635 # 72066117 # 82635570 # 96767939 # 54 <lb/>7 # 63980394 # 72220422 # 82837603 # 97044063 # 53 <lb/>8 # 64102607 # 72375376 # 83040614 # 97321646 # 52 <lb/>9 # 64225276 # 72530983 # 83244610 # 97600890 # 51 <lb/>10 # 64348404 # 72687247 # 83449598 # 97881716 # 50 <lb/>11 # 64471994 # 72844173 # 83655585 # 98164135 # 49 <lb/>12 # 64596049 # 73001766 # 83862572 # 98448162 # 48 <lb/>13 # 64720571 # 73160031 # 84070565 # 98733810 # 47 <lb/>14 # 64845563 # 73318972 # 84279571 # 99021104 # 46 <lb/>15 # 64971028 # 73478593 # 84489598 # 99310047 # 45 <lb/>16 # 65096969 # 73638898 # 84700687 # 99600655 # 44 <lb/>17 # 65223388 # 73799892 # 84912817 # 99893042 # 43 <lb/>18 # 65350287 # 73961579 # 85125995 # 100187022 # 42 <lb/>19 # 65477669 # 74123964 # 85340229 # 100482822 # 41 <lb/>20 # 65601537 # 74287052 # 85555525 # 100780346 # 40 <lb/>21 # 65733894 # 74450847 # 85771891 # 101079507 # 39 <lb/>22 # 65862743 # 74615354 # 85989335 # 101380525 # 38 <lb/>23 # 65992087 # 74780577 # 86207866 # 101683314 # 37 <lb/>24 # 66121928 # 74946521 # 86427493 # 101987889 # 36 <lb/>25 # 66252268 # 75113189 # 86648225 # 102294266 # 35 <lb/>26 # 66383110 # 75280586 # 86870072 # 102602473 # 34 <lb/>27 # 66514457 # 75448716 # 87093043 # 102912514 # 33 <lb/>28 # 66646313 # 75617584 # 87317150 # 103224405 # 32 <lb/>29 # 66778681 # 75787195 # 87542404 # 103538166 # 31 <lb/>30 # 66911564 # 75957554 # 87768816 # 103853919 # 30 <lb/> # 8 # 7 # 6 # 5 <lb/></note> </div> <div xml:id="echoid-div660" type="section" level="1" n="343"> <head xml:id="echoid-head370" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="239" file="251" n="251" rhead=""/> </div> <div xml:id="echoid-div661" type="section" level="1" n="344"> <head xml:id="echoid-head371" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 81 # 82 # 83 # 84 <lb/>30 # 66911564 # 75957554 # 87768816 # 103853919 # 30 <lb/>31 # 67044965 # 76128666 # 87996394 # 104171468 # 29 <lb/>32 # 67178887 # 76300536 # 88225146 # 104491055 # 28 <lb/>33 # 67313334 # 76473170 # 88455079 # 104812581 # 27 <lb/>34 # 67448309 # 76646573 # 88686196 # 105136063 # 26 <lb/>35 # 67583815 # 76820751 # 88918508 # 105461519 # 25 <lb/>36 # 67719855 # 76995710 # 89152021 # 105788969 # 24 <lb/>37 # 67856423 # 77171455 # 89386745 # 106118428 # 23 <lb/>38 # 67993549 # 77347991 # 89622688 # 106449917 # 22 <lb/>39 # 68131209 # 77525324 # 89859858 # 106783466 # 21 <lb/>40 # 68269416 # 77703459 # 90098268 # 107119198 # 20 <lb/>41 # 68408173 # 77882402 # 90337927 # 107456902 # 19 <lb/>42 # 68547438 # 78062159 # 90578848 # 107796712 # 18 <lb/>43 # 68687350 # 78242737 # 90821043 # 108138767 # 17 <lb/>44 # 68827777 # 78424142 # 91064526 # 108482852 # 16 <lb/>45 # 68968768 # 78606379 # 91309309 # 108829233 # 15 <lb/>46 # 69110326 # 78789454 # 91555401 # 109177805 # 14 <lb/>47 # 69252455 # 78973371 # 91802810 # 109528589 # 13 <lb/>48 # 69395158 # 79158136 # 92051546 # 109881598 # 12 <lb/>49 # 69538439 # 76343754 # 92301618 # 110236864 # 11 <lb/>50 # 69682302 # 79530231 # 92553036 # 110594415 # 10 <lb/>51 # 69826751 # 79717572 # 92805759 # 110954264 # 9 <lb/>52 # 69971789 # 79905783 # 93059875 # 111316432 # 8 <lb/>53 # 70117419 # 80094869 # 93315361 # 111680940 # 7 <lb/>54 # 70263645 # 80284835 # 93572238 # 112047814 # 6 <lb/>55 # 70410470 # 80475688 # 93830595 # 112417202 # 5 <lb/>56 # 70557898 # 80667435 # 94090270 # 112788838 # 4 <lb/>57 # 70705932 # 80860083 # 94351448 # 113163656 # 3 <lb/>58 # 70854576 # 81053639 # 94614055 # 113539681 # 2 <lb/>59 # 71003833 # 81248110 # 94878103 # 113918875 # 1 <lb/>60 # 71153706 # 81443502 # 95143611 # 114300579 # 0 <lb/>9 # 8 # 7 # 6 # 5 <lb/></note> </div> <div xml:id="echoid-div662" type="section" level="1" n="345"> <head xml:id="echoid-head372" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="240" file="252" n="252" rhead=""/> </div> <div xml:id="echoid-div663" type="section" level="1" n="346"> <head xml:id="echoid-head373" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 85 # 86 # 87 <lb/>0 # 114300579 # 143006601 # 190811200 # 60 <lb/>1 # 114684819 # 143606943 # 191879163 # 59 <lb/>2 # 115071619 # 144212307 # 192959095 # 58 <lb/>3 # 115461005 # 144822757 # 194051200 # 57 <lb/>4 # 115853017 # 145438358 # 195155685 # 56 <lb/>5 # 116247668 # 146059175 # 196273146 # 55 <lb/>6 # 116644985 # 146685275 # 197403054 # 54 <lb/>7 # 117044995 # 147316726 # 198545993 # 53 <lb/>8 # 117447864 # 147953611 # 199702191 # 52 <lb/>9 # 117853346 # 148595987 # 200871878 # 51 <lb/>10 # 118261757 # 149244148 # 202055705 # 50 <lb/>11 # 118672834 # 149897753 # 203253093 # 49 <lb/>12 # 119086890 # 150557233 # 204464726 # 48 <lb/>13 # 119503669 # 151222301 # 205691260 # 47 <lb/>14 # 119923488 # 151893462 # 206932111 # 46 <lb/>15 # 120346233 # 152570581 # 208188402 # 45 <lb/>16 # 120771937 # 153253487 # 209459545 # 44 <lb/>17 # 121200643 # 153942729 # 210746693 # 43 <lb/>18 # 121632370 # 154638158 # 212049271 # 42 <lb/>19 # 122067151 # 155339855 # 213368214 # 41 <lb/>20 # 122505017 # 156047923 # 214704085 # 40 <lb/>21 # 122946003 # 156762433 # 216056022 # 39 <lb/>22 # 123390142 # 157483474 # 217425507 # 38 <lb/>23 # 123837634 # 158211136 # 218812405 # 37 <lb/>24 # 124288195 # 158945509 # 220217049 # 36 <lb/>25 # 124742169 # 159686753 # 221639784 # 35 <lb/>26 # 125199280 # 160434770 # 223080983 # 34 <lb/>27 # 125659878 # 161189849 # 224540987 # 33 <lb/>28 # 126123842 # 161952305 # 226020167 # 32 <lb/>29 # 126591211 # 162721698 # 227518902 # 31 <lb/>30 # 127062036 # 163498660 # 229037584 # 30 <lb/>4 # 3 # 2 <lb/></note> </div> <div xml:id="echoid-div664" type="section" level="1" n="347"> <head xml:id="echoid-head374" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="241" file="253" n="253" rhead=""/> </div> <div xml:id="echoid-div665" type="section" level="1" n="348"> <head xml:id="echoid-head375" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 85 # 86 # 87 <lb/>30 # 127062036 # 163498660 # 229037584 # 30 <lb/>31 # 127536341 # 164282764 # 230576614 # 29 <lb/>32 # 128014165 # 165074651 # 232136427 # 28 <lb/>33 # 128495548 # 165873906 # 233717425 # 27 <lb/>34 # 128980531 # 166681172 # 235320041 # 26 <lb/>35 # 129469305 # 167496287 # 236945285 # 25 <lb/>36 # 129961652 # 168319085 # 238592501 # 24 <lb/>37 # 130457692 # 169150247 # 240262714 # 23 <lb/>38 # 130957670 # 169989613 # 241957021 # 22 <lb/>39 # 131461286 # 170837304 # 243674732 # 21 <lb/>40 # 131968930 # 171693461 # 245417543 # 20 <lb/>41 # 132480297 # 172558198 # 247184785 # 19 <lb/>42 # 132995769 # 173431641 # 248978216 # 18 <lb/>43 # 133515636 # 174313925 # 250797165 # 17 <lb/>44 # 134038804 # 175205183 # 252643455 # 16 <lb/>45 # 134566419 # 176105555 # 254517088 # 15 <lb/>46 # 135098153 # 177015180 # 256417991 # 14 <lb/>47 # 135634096 # 177934219 # 258348100 # 13 <lb/>48 # 136174272 # 178862806 # 260307416 # 12 <lb/>49 # 136718731 # 179801085 # 262296605 # 11 <lb/>50 # 137267523 # 180749537 # 264316358 # 10 <lb/>51 # 137820702 # 181707670 # 266366704 # 9 <lb/>52 # 138378319 # 182676299 # 268449755 # 8 <lb/>53 # 138940429 # 183654941 # 270565570 # 7 <lb/>54 # 139507087 # 184644417 # 272714927 # 6 <lb/>55 # 140078545 # 185644562 # 274898633 # 5 <lb/>56 # 140654481 # 186655202 # 277117516 # 4 <lb/>57 # 141235334 # 187677207 # 279372435 # 3 <lb/>58 # 141820765 # 188710414 # 281664304 # 2 <lb/>59 # 142411234 # 189755028 # 283994009 # 1 <lb/>60 # 143006601 # 190811200 # 286362498 # 0 <lb/> # 4 # 3 # 2 <lb/></note> </div> <div xml:id="echoid-div666" type="section" level="1" n="349"> <head xml:id="echoid-head376" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="242" file="254" n="254" rhead=""/> </div> <div xml:id="echoid-div667" type="section" level="1" n="350"> <head xml:id="echoid-head377" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <note position="right" xml:space="preserve"> <lb/> # 88 # 89 <lb/>0 # 286362498 # 572899830 # 60 <lb/>1 # 288770746 # 582610421 # 59 <lb/>2 # 291219764 # 592655713 # 58 <lb/>3 # 293710598 # 603057015 # 57 <lb/>4 # 296244357 # 613825994 # 56 <lb/>5 # 298823024 # 624990311 # 55 <lb/>6 # 301445987 # 636564040 # 54 <lb/>7 # 304115322 # 648591509 # 53 <lb/>8 # 306833212 # 661050728 # 52 <lb/>9 # 309599077 # 674016435 # 51 <lb/>10 # 312416191 # 687500725 # 50 <lb/>11 # 315283945 # 701531474 # 49 <lb/>12 # 318204757 # 716149676 # 48 <lb/>13 # 321181137 # 731385593 # 47 <lb/>14 # 324212583 # 747289264 # 46 <lb/>15 # 327302782 # 763899813 # 45 <lb/>16 # 330451272 # 781259259 # 44 <lb/>17 # 333661982 # 799432199 # 43 <lb/>18 # 336934467 # 818463792 # 42 <lb/>19 # 340272744 # 838430438 # 41 <lb/>20 # 343677949 # 859395374 # 40 <lb/>21 # 347150587 # 881427652 # 39 <lb/>22 # 350695255 # 904627361 # 38 <lb/>23 # 354312962 # 929081086 # 37 <lb/>24 # 358006024 # 954893332 # 36 <lb/>25 # 361776788 # 982180553 # 35 <lb/>26 # 365626388 # 1011062679 # 34 <lb/>27 # 369560062 # 1041705454 # 33 <lb/>28 # 373579199 # 1074263399 # 32 <lb/>29 # 377686614 # 1108922084 # 31 <lb/>30 # 381885288 # 1145891136 # 30 <lb/>1 # 0 <lb/></note> </div> <div xml:id="echoid-div668" type="section" level="1" n="351"> <head xml:id="echoid-head378" xml:space="preserve">Gradus Quadrantis pro tangentibus</head> <pb o="243" file="255" n="255" rhead=""/> </div> <div xml:id="echoid-div669" type="section" level="1" n="352"> <head xml:id="echoid-head379" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 88 # 89 <lb/>30 # 381885288 # 1145891136 # 30 <lb/>31 # 386178258 # 1185395877 # 29 <lb/>32 # 390568737 # 1227736470 # 28 <lb/>33 # 395060088 # 1273213435 # 27 <lb/>34 # 399655828 # 1322188681 # 26 <lb/>35 # 404359642 # 1375082163 # 25 <lb/>36 # 409175388 # 1432363027 # 24 <lb/>37 # 414111295 # 1494645462 # 23 <lb/>38 # 419159137 # 1562590046 # 22 <lb/>39 # 424335793 # 1637005697 # 21 <lb/>40 # 429641796 # 1718863124 # 20 <lb/>41 # 435082056 # 1809337410 # 19 <lb/>42 # 440661780 # 1909864971 # 18 <lb/>43 # 446386310 # 2022219818 # 17 <lb/>44 # 452261453 # 2148619711 # 16 <lb/>45 # 458293185 # 2291873854 # 15 <lb/>46 # 464487853 # 2455533838 # 14 <lb/>47 # 470852152 # 2644433955 # 13 <lb/>48 # 477393195 # 2864819229 # 12 <lb/>49 # 484118353 # 3125276745 # 11 <lb/>50 # 491038024 # 3437829002 # 10 <lb/>51 # 498155754 # 3819696333 # 9 <lb/>52 # 505482730 # 4297181900 # 8 <lb/>53 # 513030946 # 4911098124 # 7 <lb/>54 # 520805157 # 5729633839 # 6 <lb/>55 # 528821258 # 6875680006 # 5 <lb/>56 # 537085003 # 8594003953 # 4 <lb/>57 # 545610968 # 11457529506 # 3 <lb/>58 # 554414914 # 17188033688 # 2 <lb/>59 # 563504309 # 34376070815 # 1 <lb/>60 # 572899830 # infinita # 0 <lb/> # 1 # 0 <lb/></note> </div> <div xml:id="echoid-div670" type="section" level="1" n="353"> <head xml:id="echoid-head380" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb file="256" n="256"/> <pb file="257" n="257" rhead=""/> </div> <div xml:id="echoid-div671" type="section" level="1" n="354"> <head xml:id="echoid-head381" xml:space="preserve">LINEARVM SECANTIVM, <lb/>SIVE <lb/>BENEFICA.</head> <pb o="246" file="258" n="258" rhead=""/> </div> <div xml:id="echoid-div672" type="section" level="1" n="355"> <head xml:id="echoid-head382" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 0 # 1 # 2 # 3 <lb/>0 # 10000000 # 10001524 # 10006095 # 10013723 # 60 <lb/>1 # 10000001 # 10001574 # 10006198 # 10013875 # 59 <lb/>2 # 10000002 # 10001626 # 10006301 # 10014029 # 58 <lb/>3 # 10000004 # 10001679 # 10006405 # 10014184 # 57 <lb/>4 # 10000008 # 10001733 # 10006509 # 10014339 # 56 <lb/>5 # 10000010 # 10001788 # 10006615 # 10014495 # 55 <lb/>6 # 10000014 # 10001844 # 10006721 # 10014653 # 54 <lb/>7 # 10000020 # 10001900 # 10006828 # 10014811 # 53 <lb/>8 # 10000027 # 10001957 # 10006936 # 10014970 # 52 <lb/>9 # 10000034 # 10002015 # 10007045 # 10015130 # 51 <lb/>10 # 10000042 # 10002074 # 10007155 # 10015291 # 50 <lb/>11 # 10000051 # 10002134 # 10007265 # 10015453 # 49 <lb/>12 # 10000060 # 10002195 # 10007376 # 10015615 # 48 <lb/>13 # 10000071 # 10002256 # 10007488 # 10015778 # 47 <lb/>14 # 10000083 # 10002318 # 10007601 # 10015942 # 46 <lb/>15 # 10000095 # 10002381 # 10007716 # 10016107 # 45 <lb/>16 # 10000108 # 10002445 # 10007831 # 10016273 # 44 <lb/>17 # 10000122 # 10002510 # 10007946 # 10016440 # 43 <lb/>18 # 10000137 # 10002576 # 10008062 # 10016608 # 42 <lb/>19 # 10000152 # 10002642 # 10008179 # 10016777 # 41 <lb/>20 # 10000168 # 10002709 # 10008298 # 10016946 # 40 <lb/>21 # 10000186 # 10002777 # 10008417 # 10017116 # 39 <lb/>22 # 10000204 # 10002846 # 10008537 # 10017287 # 38 <lb/>23 # 10000223 # 10002916 # 10008658 # 10017459 # 37 <lb/>24 # 10000243 # 10002987 # 10008779 # 10017632 # 36 <lb/>25 # 10000264 # 10003058 # 10008902 # 10017806 # 35 <lb/>26 # 10000285 # 10003130 # 10009025 # 10017981 # 34 <lb/>27 # 10000308 # 10003203 # 10009149 # 10018157 # 33 <lb/>28 # 10000332 # 10003277 # 10009274 # 10018333 # 32 <lb/>29 # 10000357 # 10003352 # 10009400 # 10018510 # 31 <lb/>30 # 10000381 # 10003428 # 10009527 # 10018687 # 30 <lb/>89 # 88 # 87 # 86 <lb/></note> </div> <div xml:id="echoid-div673" type="section" level="1" n="356"> <head xml:id="echoid-head383" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="247" file="259" n="259" rhead=""/> </div> <div xml:id="echoid-div674" type="section" level="1" n="357"> <head xml:id="echoid-head384" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 0 # 1 # 2 # 3 <lb/>30 # 10000381 # 10003428 # 10009527 # 10018687 # 30 <lb/>31 # 10000407 # 10003505 # 10009655 # 10018865 # 39 <lb/>32 # 10000433 # 10003582 # 10009783 # 10019044 # 28 <lb/>33 # 10000461 # 10003660 # 10009912 # 10019224 # 27 <lb/>34 # 10000489 # 10003739 # 10010043 # 10019405 # 26 <lb/>35 # 10000518 # 10003819 # 10010174 # 10019587 # 25 <lb/>36 # 10000548 # 10003900 # 10010306 # 10019770 # 24 <lb/>37 # 10000579 # 10003982 # 10010439 # 10019954 # 23 <lb/>38 # 10000611 # 10004060 # 10010572 # 10020138 # 22 <lb/>39 # 10000643 # 10004148 # 10010706 # 10020324 # 21 <lb/>40 # 10000677 # 10004232 # 10010841 # 10020510 # 20 <lb/>41 # 10000711 # 10004317 # 10010977 # 10020698 # 19 <lb/>42 # 10000746 # 10004403 # 10011114 # 10020886 # 18 <lb/>43 # 10000782 # 10004490 # 10011252 # 10021086 # 17 <lb/>44 # 10000819 # 10004578 # 10011390 # 10021266 # 16 <lb/>45 # 10000857 # 10004666 # 10011529 # 10021456 # 15 <lb/>46 # 10000895 # 10004755 # 10011670 # 10021649 # 14 <lb/>47 # 10000934 # 10004845 # 10011811 # 10021842 # 13 <lb/>48 # 10000975 # 10004936 # 10011952 # 10022035 # 12 <lb/>49 # 10001016 # 10005028 # 10012098 # 10022239 # 11 <lb/>50 # 10001058 # 10005122 # 10012238 # 10022424 # 10 <lb/>51 # 10001100 # 10005216 # 10012383 # 10022620 # 9 <lb/>52 # 10001144 # 10005310 # 10012528 # 10022817 # 8 <lb/>53 # 10001188 # 10005405 # 10012674 # 10023015 # 7 <lb/>54 # 10001233 # 10005501 # 10012822 # 10023213 # 6 <lb/>55 # 10001280 # 10005598 # 10012970 # 10023412 # 5 <lb/>56 # 10001327 # 10005696 # 10013119 # 10023612 # 4 <lb/>57 # 10001375 # 10005795 # 10013269 # 10023813 # 3 <lb/>58 # 10001423 # 10005894 # 10013419 # 10024014 # 2 <lb/>59 # 10001473 # 10005994 # 10013570 # 10024217 # 1 <lb/>60 # 10001524 # 10006095 # 10013723 # 10024420 # 0 <lb/>89 # 88 # 87 # 86 <lb/></note> </div> <div xml:id="echoid-div675" type="section" level="1" n="358"> <head xml:id="echoid-head385" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="248" file="260" n="260" rhead=""/> </div> <div xml:id="echoid-div676" type="section" level="1" n="359"> <head xml:id="echoid-head386" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 4 # 5 # 6 # 7 <lb/>0 # 10024420 # 10038198 # 10055082 # 10075098 # 60 <lb/>1 # 10024625 # 10038454 # 10055390 # 10075459 # 59 <lb/>2 # 10024830 # 10038710 # 10055699 # 10075820 # 58 <lb/>3 # 10025036 # 10038968 # 10056009 # 10076182 # 57 <lb/>4 # 10025242 # 10039226 # 10056320 # 10076545 # 56 <lb/>5 # 10025450 # 10039486 # 10056632 # 10076909 # 55 <lb/>6 # 10025658 # 10039746 # 10056944 # 10077274 # 54 <lb/>7 # 10025868 # 10040008 # 10057256 # 10077639 # 53 <lb/>8 # 10026078 # 10040269 # 10057570 # 10078005 # 52 <lb/>9 # 10026289 # 10040532 # 10057884 # 10078372 # 51 <lb/>10 # 10026500 # 10040796 # 10058200 # 10078740 # 50 <lb/>11 # 10026713 # 10041061 # 10058517 # 10079009 # 49 <lb/>12 # 10026927 # 10041326 # 10058834 # 10079479 # 48 <lb/>13 # 10027141 # 10041592 # 10059153 # 10079850 # 47 <lb/>14 # 10027357 # 10041859 # 10059472 # 10080222 # 46 <lb/>15 # 10027573 # 10042128 # 10059792 # 10080595 # 45 <lb/>16 # 10027790 # 10042397 # 10060113 # 10080968 # 44 <lb/>17 # 10028009 # 10042667 # 10060435 # 10081332 # 43 <lb/>18 # 10028227 # 10042936 # 10060757 # 10081717 # 42 <lb/>19 # 10028447 # 10043207 # 10061080 # 10082093 # 41 <lb/>20 # 10028667 # 10043479 # 10061405 # 10082470 # 40 <lb/>21 # 10028889 # 10043752 # 10061730 # 10082848 # 39 <lb/>22 # 10029111 # 10044025 # 10062056 # 10083226 # 38 <lb/>23 # 10029334 # 10044300 # 10062383 # 10063606 # 37 <lb/>24 # 10029559 # 10044576 # 10062711 # 10083987 # 36 <lb/>25 # 10029784 # 10044853 # 10063039 # 10084368 # 35 <lb/>26 # 10030009 # 10045130 # 10063369 # 10084750 # 34 <lb/>27 # 10030236 # 10045409 # 10063700 # 00085134 # 33 <lb/>28 # 10030463 # 10045689 # 10064031 # 10085518 # 32 <lb/>29 # 10030692 # 10045969 # 10064364 # 10085903 # 31 <lb/>30 # 10030920 # 10046250 # 10064696 # 10086289 # 30 <lb/>85 # 84 # 83 # 82 <lb/></note> </div> <div xml:id="echoid-div677" type="section" level="1" n="360"> <head xml:id="echoid-head387" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="249" file="261" n="261" rhead=""/> </div> <div xml:id="echoid-div678" type="section" level="1" n="361"> <head xml:id="echoid-head388" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 4 # 5 # 6 # 7 <lb/>30 # 10030920 # 10046250 # 10064696 # 10086287 # 30 <lb/>31 # 10031150 # 10046532 # 10065035 # 10086677 # 39 <lb/>32 # 10031381 # 10046815 # 10065365 # 10087065 # 28 <lb/>33 # 10031614 # 10047098 # 10065701 # 10087454 # 27 <lb/>34 # 10031846 # 10047383 # 10066038 # 10087843 # 26 <lb/>35 # 10032079 # 10047669 # 10066376 # 10088243 # 25 <lb/>36 # 10032314 # 10047954 # 10066715 # 10088623 # 24 <lb/>37 # 10032550 # 10048241 # 10067054 # 10089015 # 23 <lb/>38 # 10032786 # 10048529 # 10067394 # 10089408 # 22 <lb/>39 # 10033023 # 10048818 # 10067735 # 10089802 # 21 <lb/>40 # 10033261 # 10049107 # 10068076 # 10090196 # 20 <lb/>41 # 10033500 # 10049398 # 10068419 # 10090592 # 19 <lb/>42 # 10033740 # 10049690 # 10068763 # 10090988 # 18 <lb/>43 # 10033981 # 10049983 # 10069107 # 10091385 # 17 <lb/>44 # 10034223 # 10050276 # 10069452 # 10091783 # 16 <lb/>45 # 10034465 # 10050571 # 10069808 # 10092182 # 15 <lb/>46 # 10034708 # 10050865 # 10070155 # 10092582 # 14 <lb/>47 # 10034952 # 10051160 # 10070493 # 10092983 # 13 <lb/>48 # 10035196 # 10051456 # 10070842 # 10093385 # 12 <lb/>49 # 10035441 # 10051753 # 10071192 # 10093787 # 11 <lb/>50 # 10035688 # 10052051 # 10071543 # 10094190 # 10 <lb/>51 # 10035936 # 10052350 # 10071895 # 10094624 # 9 <lb/>52 # 10036184 # 10052649 # 10072247 # 10095030 # 8 <lb/>53 # 10036434 # 10052951 # 10072600 # 10095406 # 7 <lb/>54 # 10036684 # 10053252 # 10072954 # 10095813 # 6 <lb/>55 # 10036934 # 10053555 # 10073310 # 10096221 # 5 <lb/>56 # 10037185 # 10053858 # 10073666 # 10096630 # 4 <lb/>57 # 10037438 # 10054162 # 10074023 # 10097040 # 3 <lb/>58 # 10037690 # 10054468 # 10074380 # 10097451 # 2 <lb/>59 # 10037944 # 10054775 # 10074737 # 10097863 # 1 <lb/>60 # 10038198 # 10055082 # 10075098 # 10098275 # 0 <lb/> # 85 # 84 # 83 # 82 <lb/></note> </div> <div xml:id="echoid-div679" type="section" level="1" n="362"> <head xml:id="echoid-head389" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="250" file="262" n="262" rhead=""/> </div> <div xml:id="echoid-div680" type="section" level="1" n="363"> <head xml:id="echoid-head390" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 8 # 9 # 10 # 11 <lb/>0 # 10098275 # 10124650 # 10154264 # 10187166 # 60 <lb/>1 # 10098698 # 10125117 # 10154786 # 10187743 # 59 <lb/>2 # 10099103 # 10125585 # 10155308 # 10188320 # 58 <lb/>3 # 10099518 # 10126054 # 10155831 # 10188899 # 57 <lb/>4 # 10099934 # 10126524 # 10156356 # 10189478 # 56 <lb/>5 # 10100351 # 10126994 # 10156881 # 10190058 # 55 <lb/>6 # 10100769 # 10127465 # 10157407 # 10190639 # 54 <lb/>7 # 10101188 # 10127947 # 10157934 # 10191221 # 53 <lb/>8 # 10101607 # 10128410 # 10158462 # 10191804 # 52 <lb/>9 # 10102028 # 10128684 # 10158991 # 10192387 # 51 <lb/>10 # 10102450 # 10129358 # 10159520 # 10192972 # 50 <lb/>11 # 10102872 # 10129634 # 10160051 # 10193557 # 49 <lb/>12 # 10103295 # 10130311 # 10160582 # 10194144 # 48 <lb/>13 # 10103720 # 10130788 # 10161114 # 10194732 # 47 <lb/>14 # 10104144 # 10131266 # 10161648 # 10195320 # 46 <lb/>15 # 10104570 # 10131746 # 10162182 # 10195910 # 45 <lb/>16 # 10104996 # 10132226 # 10162707 # 10196500 # 44 <lb/>17 # 10105423 # 10132707 # 10163252 # 10197092 # 43 <lb/>18 # 10105851 # 10133189 # 10163789 # 10197684 # 42 <lb/>19 # 10106286 # 10133672 # 10164327 # 10198277 # 41 <lb/>20 # 10106710 # 10134156 # 10165865 # 10198872 # 40 <lb/>21 # 10107140 # 10134641 # 10165495 # 10199467 # 39 <lb/>22 # 10107572 # 10135127 # 10165944 # 10200063 # 38 <lb/>23 # 10108005 # 10135614 # 10166485 # 10200660 # 37 <lb/>24 # 10108438 # 10136102 # 10167018 # 10201258 # 36 <lb/>25 # 10108873 # 10136591 # 10167571 # 10201857 # 35 <lb/>26 # 10109309 # 10137080 # 10168116 # 10202457 # 34 <lb/>27 # 10109755 # 10137571 # 10168661 # 10203058 # 33 <lb/>28 # 10110182 # 10138163 # 10169207 # 10203659 # 32 <lb/>29 # 10110620 # 10138555 # 10169765 # 10204262 # 31 <lb/>30 # 101@1059 # 10139048 # 10170303 # 10204867 # 30 <lb/> # 81 # 80 # 79 # 78 <lb/></note> </div> <div xml:id="echoid-div681" type="section" level="1" n="364"> <head xml:id="echoid-head391" xml:space="preserve">Gradus Quadratis pro ſecantibus</head> <pb o="251" file="263" n="263" rhead=""/> </div> <div xml:id="echoid-div682" type="section" level="1" n="365"> <head xml:id="echoid-head392" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 8 # 9 # 10 # 11 <lb/>30 # 10111059 # 10139048 # 10170303 # 10204867 # 30 <lb/>31 # 10111509 # 10139543 # 10170852 # 10205470 # 29 <lb/>32 # 10111940 # 10140038 # 10171401 # 10206075 # 28 <lb/>33 # 10112482 # 10140534 # 10171952 # 10206681 # 27 <lb/>34 # 10112825 # 10141036 # 10172504 # 10207289 # 26 <lb/>35 # 10113279 # 10141528 # 10173056 # 10207897 # 25 <lb/>36 # 10113713 # 10142027 # 10173609 # 10208506 # 24 <lb/>37 # 10114159 # 10142526 # 10174163 # 10209116 # 23 <lb/>38 # 10114606 # 10143026 # 10174718 # 10209727 # 22 <lb/>39 # 10115053 # 10143528 # 10175274 # 10210339 # 21 <lb/>40 # 10115501 # 10144030 # 10175831 # 10210952 # 20 <lb/>41 # 10115951 # 10144533 # 10176389 # 10211566 # 19 <lb/>42 # 10116401 # 10145037 # 10176947 # 10211180 # 18 <lb/>43 # 10116852 # 10145542 # 10177507 # 10212796 # 17 <lb/>44 # 10117303 # 10146048 # 10178068 # 10213412 # 16 <lb/>45 # 10117754 # 10146554 # 10178630 # 10214030 # 15 <lb/>46 # 10118209 # 10147062 # 10179193 # 10214668 # 14 <lb/>47 # 10118663 # 10147572 # 10179756 # 10215268 # 13 <lb/>48 # 10119118 # 10148082 # 10180321 # 10215889 # 12 <lb/>49 # 10119574 # 10348593 # 10180886 # 10216510 # 11 <lb/>50 # 10120031 # 10149104 # 10181453 # 10217113 # 10 <lb/>51 # 10120489 # 10149615 # 10182021 # 10217756 # 9 <lb/>52 # 10120948 # 10150128 # 10182589 # 10218380 # 8 <lb/>53 # 10121408 # 10150642 # 10183158 # 10219015 # 7 <lb/>54 # 10121868 # 10151156 # 10183728 # 10219631 # 6 <lb/>55 # 10122330 # 10151672 # 10184299 # 10220258 # 5 <lb/>56 # 10122792 # 10152188 # 10184870 # 10220885 # 4 <lb/>57 # 10123256 # 10152705 # 10185443 # 10221514 # 3 <lb/>58 # 10123720 # 10153224 # 10186017 # 10222143 # 2 <lb/>59 # 10124275 # 10153744 # 10186591 # 10222774 # 1 <lb/>60 # 10124650 # 10154264 # 10187166 # 10223405 # 0 <lb/> # 81 # 80 # 79 # 78 <lb/></note> </div> <div xml:id="echoid-div683" type="section" level="1" n="366"> <head xml:id="echoid-head393" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="252" file="264" n="264" rhead=""/> </div> <div xml:id="echoid-div684" type="section" level="1" n="367"> <head xml:id="echoid-head394" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 12 # 13 # 14 # 15 <lb/>0 # 10223405 # 10263040 # 10306136 # 10352762 # 60 <lb/>1 # 10224037 # 10263730 # 10306884 # 10353569 # 59 <lb/>2 # 10224671 # 10264420 # 10307633 # 10354377 # 58 <lb/>3 # 10225305 # 10265112 # 10308383 # 10355186 # 57 <lb/>4 # 10225941 # 10265804 # 10309134 # 10355996 # 56 <lb/>5 # 10226577 # 10266498 # 10309886 # 10356807 # 55 <lb/>6 # 10227215 # 10267192 # 10310639 # 10357619 # 54 <lb/>7 # 10227854 # 10267888 # 10311393 # 10358433 # 53 <lb/>8 # 10228493 # 10268584 # 10312148 # 10359247 # 52 <lb/>9 # 10229134 # 10269281 # 10312903 # 10360063 # 51 <lb/>10 # 10229775 # 10269979 # 10313660 # 10360880 # 50 <lb/>11 # 10230417 # 10270688 # 10314417 # 10361698 # 49 <lb/>12 # 10231060 # 10271379 # 10315176 # 10362517 # 48 <lb/>13 # 10231644 # 10272080 # 10315935 # 10363337 # 47 <lb/>14 # 10232288 # 10272782 # 10316696 # 10364158 # 46 <lb/>15 # 10232994 # 10273485 # 10317457 # 10364980 # 45 <lb/>16 # 10233641 # 10274190 # 10318220 # 10365802 # 44 <lb/>17 # 10234289 # 10274895 # 10318984 # 10366626 # 43 <lb/>18 # 10234938 # 10275601 # 10319749 # 10367450 # 42 <lb/>19 # 10235587 # 10276318 # 10320525 # 10368276 # 41 <lb/>20 # 10236238 # 10277016 # 10321282 # 10369102 # 40 <lb/>21 # 10236889 # 10277726 # 10322050 # 10369930 # 39 <lb/>22 # 10237541 # 10278436 # 10322819 # 10370758 # 38 <lb/>23 # 10238195 # 10279148 # 10323589 # 10371588 # 37 <lb/>24 # 10238849 # 10279860 # 10324359 # 10372418 # 36 <lb/>25 # 10239505 # 10280573 # 10325131 # 10373250 # 35 <lb/>26 # 10240161 # 10281287 # 10325903 # 10374092 # 34 <lb/>27 # 10240818 # 10282002 # 10326677 # 10374916 # 33 <lb/>28 # 10241476 # 10282717 # 10327451 # 10375750 # 32 <lb/>29 # 10242135 # 10283434 # 10328127 # 10376586 # 31 <lb/>30 # 10242795 # 10284151 # 10329003 # 10377422 # 30 <lb/> # 77 # 76 # 75 # 74 <lb/></note> </div> <div xml:id="echoid-div685" type="section" level="1" n="368"> <head xml:id="echoid-head395" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="253" file="265" n="265" rhead=""/> </div> <div xml:id="echoid-div686" type="section" level="1" n="369"> <head xml:id="echoid-head396" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 12 # 13 # 14 # 15 <lb/>30 # 10242795 # 10284151 # 10329003 # 10377422 # 30 <lb/>31 # 10243456 # 10284870 # 10329781 # 10378260 # 39 <lb/>32 # 10244118 # 10285589 # 10330559 # 10379098 # 28 <lb/>33 # 10245782 # 10286310 # 10331339 # 10379938 # 27 <lb/>34 # 10245445 # 10287032 # 10332119 # 10380778 # 26 <lb/>35 # 10246110 # 10287754 # 10332902 # 10381620 # 25 <lb/>36 # 10246776 # 10288478 # 10333684 # 10382463 # 24 <lb/>37 # 10247442 # 10289202 # 10334467 # 10383307 # 23 <lb/>38 # 10248110 # 10289928 # 10335252 # 10384153 # 22 <lb/>39 # 10248778 # 10290654 # 10336037 # 10384999 # 21 <lb/>40 # 10249448 # 10291381 # 10336824 # 10385846 # 20 <lb/>41 # 10250119 # 10292119 # 10337612 # 10386694 # 19 <lb/>42 # 10250790 # 10292838 # 10338400 # 10387543 # 18 <lb/>43 # 10251461 # 10293569 # 10339189 # 10388393 # 17 <lb/>44 # 10252136 # 10294300 # 10339980 # 10389244 # 16 <lb/>45 # 10252811 # 10295043 # 10340771 # 10390096 # 15 <lb/>46 # 10253482 # 10295766 # 10341564 # 10390949 # 14 <lb/>47 # 10254162 # 10296501 # 10342347 # 10391803 # 13 <lb/>48 # 10254839 # 10297237 # 10343152 # 10392657 # 12 <lb/>49 # 10255517 # 10297973 # 10343947 # 10393513 # 11 <lb/>50 # 10256196 # 10298710 # 10344743 # 10394370 # 10 <lb/>51 # 10256876 # 10299449 # 10345541 # 10395228 # 9 <lb/>52 # 10257557 # 10300188 # 10346340 # 10396087 # 8 <lb/>53 # 10258239 # 10300928 # 10347139 # 10396947 # 7 <lb/>54 # 10258922 # 10301669 # 10347940 # 10397808 # 6 <lb/>55 # 10259606 # 10302411 # 10348741 # 10398670 # 5 <lb/>56 # 10260291 # 10303154 # 10349544 # 10399533 # 4 <lb/>57 # 10260977 # 10303898 # 10350347 # 10400397 # 3 <lb/>58 # 10261661 # 10304643 # 10351151 # 10401262 # 2 <lb/>59 # 10262351 # 10305390 # 10351956 # 10402128 # 1 <lb/>60 # 10263040 # 10306136 # 10352762 # 10402994 # 0 <lb/> # 77 # 76 # 75 # 74 <lb/></note> </div> <div xml:id="echoid-div687" type="section" level="1" n="370"> <head xml:id="echoid-head397" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="254" file="266" n="266" rhead=""/> </div> <div xml:id="echoid-div688" type="section" level="1" n="371"> <head xml:id="echoid-head398" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 16 # 17 # 18 # 19 <lb/>0 # 10402994 # 10456917 # 10514621 # 10576207 # 60 <lb/>1 # 10403862 # 10457847 # 10515616 # 10577267 # 59 <lb/>2 # 10404730 # 10458779 # 10516612 # 10578328 # 58 <lb/>3 # 10405590 # 10459711 # 10517609 # 10579400 # 57 <lb/>4 # 10406471 # 10460645 # 10518607 # 10580463 # 56 <lb/>5 # 10407343 # 10461580 # 10519606 # 10581518 # 55 <lb/>6 # 10408216 # 10462516 # 10520606 # 10582583 # 54 <lb/>7 # 10409091 # 10463453 # 10521607 # 10583650 # 53 <lb/>8 # 10409966 # 10464391 # 10522608 # 10584717 # 52 <lb/>9 # 10410843 # 10465330 # 10523611 # 10585795 # 51 <lb/>10 # 10411721 # 10466270 # 10524615 # 10586855 # 50 <lb/>11 # 10412600 # 10467211 # 10525620 # 10587925 # 49 <lb/>12 # 10413479 # 10468153 # 10526626 # 10588997 # 48 <lb/>13 # 10414360 # 10469096 # 10527633 # 10590070 # 47 <lb/>14 # 10415241 # 10470041 # 10528642 # 10591145 # 46 <lb/>15 # 10416124 # 10470986 # 10529651 # 10592220 # 45 <lb/>16 # 10417007 # 10471933 # 10530662 # 10593297 # 44 <lb/>17 # 10417892 # 10472880 # 10531673 # 10594375 # 43 <lb/>18 # 10418778 # 10473829 # 10532686 # 10595455 # 42 <lb/>19 # 10419665 # 10474778 # 10533699 # 10596534 # 41 <lb/>20 # 10420553 # 10475729 # 10534714 # 10597615 # 40 <lb/>21 # 10421442 # 10476680 # 10535730 # 10598697 # 39 <lb/>22 # 10422333 # 10477633 # 10536747 # 10599780 # 38 <lb/>23 # 10423224 # 10478587 # 10537765 # 10600865 # 37 <lb/>24 # 10424116 # 10479542 # 10538785 # 10601950 # 36 <lb/>25 # 10425009 # 10480498 # 10539805 # 10603037 # 35 <lb/>26 # 10425903 # 10481454 # 10540826 # 10604125 # 34 <lb/>27 # 10426798 # 10482412 # 10541848 # 10605214 # 33 <lb/>28 # 10427694 # 10483371 # 10542872 # 10606304 # 32 <lb/>29 # 10428591 # 10484331 # 10543897 # 10607395 # 31 <lb/>30 # 10429489 # 10485292 # 10544923 # 10608487 # 30 <lb/> # 73 # 72 # 71 # 70 <lb/></note> </div> <div xml:id="echoid-div689" type="section" level="1" n="372"> <head xml:id="echoid-head399" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="255" file="267" n="267" rhead=""/> </div> <div xml:id="echoid-div690" type="section" level="1" n="373"> <head xml:id="echoid-head400" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 16 # 17 # 18 # 19 <lb/>30 # 10429489 # 10485292 # 10544923 # 10608487 # 30 <lb/>31 # 10430388 # 10486254 # 10545950 # 10609580 # 39 <lb/>32 # 10431288 # 10487217 # 10546977 # 10610675 # 28 <lb/>33 # 10432189 # 10488181 # 10548006 # 10611770 # 27 <lb/>34 # 10433091 # 10489146 # 10549036 # 10612867 # 26 <lb/>35 # 10433995 # 10490113 # 10550067 # 10613964 # 25 <lb/>36 # 10434899 # 10491080 # 10551099 # 10615063 # 24 <lb/>37 # 10435805 # 10492049 # 10552133 # 10616163 # 23 <lb/>38 # 10436711 # 10493018 # 10553168 # 10617264 # 22 <lb/>39 # 10437619 # 10493989 # 10554204 # 10618366 # 21 <lb/>40 # 10438528 # 10494961 # 10555241 # 10619469 # 20 <lb/>41 # 10439436 # 10494934 # 10556279 # 10620574 # 19 <lb/>42 # 10440346 # 10496908 # 10557318 # 10621680 # 18 <lb/>43 # 10441257 # 10497883 # 10558359 # 10622787 # 17 <lb/>44 # 10442170 # 10498059 # 10559400 # 10623895 # 16 <lb/>45 # 10443083 # 10499836 # 10560443 # 10625004 # 15 <lb/>46 # 10443998 # 10500814 # 10561496 # 10626114 # 14 <lb/>47 # 10444913 # 10501793 # 10562531 # 10627226 # 13 <lb/>48 # 10445830 # 10502773 # 10563577 # 10628338 # 12 <lb/>49 # 10446749 # 10503754 # 10564623 # 10629451 # 11 <lb/>50 # 10447668 # 10504736 # 10565670 # 10630566 # 10 <lb/>51 # 10448588 # 10505719 # 10566719 # 10631682 # 9 <lb/>52 # 10449509 # 10506704 # 10567769 # 10632799 # 8 <lb/>53 # 10450431 # 10507689 # 10568820 # 10633917 # 7 <lb/>54 # 10451354 # 10508676 # 10569872 # 10635037 # 6 <lb/>55 # 10452279 # 10509664 # 10570925 # 10636157 # 5 <lb/>56 # 10453204 # 10510653 # 10571980 # 10637279 # 4 <lb/>57 # 10454131 # 10511643 # 10573034 # 10638402 # 3 <lb/>58 # 10455058 # 10512635 # 10574091 # 10639526 # 2 <lb/>59 # 10455987 # 10513627 # 10575149 # 10640651 # 1 <lb/>60 # 10456917 # 10514621 # 10576207 # 10641777 # 0 <lb/> # 73 # 72 # 71 # 70 <lb/></note> </div> <div xml:id="echoid-div691" type="section" level="1" n="374"> <head xml:id="echoid-head401" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="256" file="268" n="268" rhead=""/> </div> <div xml:id="echoid-div692" type="section" level="1" n="375"> <head xml:id="echoid-head402" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 20 # 21 # 22 # 23 <lb/>0 # 10641777 # 10711449 # 10785347 # 10863603 # 60 <lb/>1 # 10642905 # 10712646 # 10786616 # 10864945 # 59 <lb/>2 # 10644034 # 10713888 # 10787885 # 10866289 # 58 <lb/>3 # 10645164 # 10715042 # 10789155 # 10867633 # 57 <lb/>4 # 10646295 # 10716242 # 10790427 # 10868979 # 56 <lb/>5 # 10647427 # 10717444 # 10791700 # 10870326 # 55 <lb/>6 # 10648560 # 10718647 # 10792974 # 10871675 # 54 <lb/>7 # 10649694 # 10719850 # 10794250 # 10873024 # 53 <lb/>8 # 10650829 # 10721056 # 10795527 # 10874374 # 52 <lb/>9 # 10651965 # 10722261 # 10796805 # 10875626 # 51 <lb/>10 # 10653103 # 10723469 # 10798085 # 10877079 # 50 <lb/>11 # 10654242 # 10724677 # 10799365 # 10878434 # 49 <lb/>12 # 10655381 # 10725887 # 10800647 # 10879790 # 48 <lb/>13 # 10656522 # 10727098 # 10801930 # 10881147 # 47 <lb/>14 # 10657664 # 10728310 # 10803214 # 10882506 # 46 <lb/>15 # 10658807 # 10729524 # 10804500 # 10883865 # 45 <lb/>16 # 10659951 # 10730738 # 10805787 # 10885226 # 44 <lb/>17 # 10661097 # 10731953 # 10807074 # 10886588 # 43 <lb/>18 # 10662244 # 10733170 # 10808363 # 10887952 # 42 <lb/>19 # 10663392 # 10734387 # 10809652 # 10889317 # 41 <lb/>20 # 10664541 # 10735606 # 10810942 # 10890683 # 40 <lb/>21 # 10665692 # 10736826 # 10812234 # 10892051 # 39 <lb/>22 # 10666844 # 10738048 # 10813528 # 10893417 # 38 <lb/>23 # 10667996 # 10739270 # 10814823 # 10894788 # 37 <lb/>24 # 10669150 # 10740494 # 10816119 # 10896159 # 36 <lb/>25 # 10670304 # 10741719 # 10817417 # 10897531 # 35 <lb/>26 # 10671460 # 10742945 # 10818715 # 10898905 # 34 <lb/>27 # 10672617 # 10744173 # 10820015 # 10900280 # 33 <lb/>28 # 10673776 # 10745401 # 10821316 # 10901656 # 32 <lb/>29 # 10674936 # 10746631 # 10822617 # 10903033 # 31 <lb/>30 # 10676096 # 10747864 # 10823920 # 10904413 # 30 <lb/> # 69 # 68 # 67 # 66 <lb/></note> </div> <div xml:id="echoid-div693" type="section" level="1" n="376"> <head xml:id="echoid-head403" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="257" file="269" n="269" rhead=""/> </div> <div xml:id="echoid-div694" type="section" level="1" n="377"> <head xml:id="echoid-head404" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 20 # 21 # 22 # 23 <lb/>30 # 10676096 # 10747864 # 10823920 # 10904413 # 30 <lb/>31 # 10677258 # 10749094 # 10825225 # 10905790 # 29 <lb/>32 # 10678420 # 10750327 # 10826531 # 10907171 # 28 <lb/>33 # 10679584 # 10751561 # 10827838 # 10908553 # 27 <lb/>34 # 10680749 # 10752797 # 10829146 # 10909936 # 26 <lb/>35 # 10681915 # 10754034 # 10830455 # 10911322 # 25 <lb/>36 # 10683082 # 10755273 # 10831766 # 10912709 # 24 <lb/>37 # 10684250 # 10756513 # 10833078 # 10914096 # 23 <lb/>38 # 10685420 # 10757753 # 10834391 # 10915484 # 22 <lb/>39 # 10686591 # 10758995 # 10835706 # 10916874 # 21 <lb/>40 # 10687763 # 10760237 # 10837023 # 10918265 # 20 <lb/>41 # 10688936 # 10761481 # 10838341 # 10919657 # 19 <lb/>42 # 10690111 # 10762726 # 10839660 # 10921051 # 18 <lb/>43 # 10691287 # 10763972 # 10840980 # 10922436 # 17 <lb/>44 # 10692464 # 10765220 # 10842301 # 10923833 # 16 <lb/>45 # 10693642 # 10766469 # 10843623 # 10925241 # 15 <lb/>46 # 10694821 # 10767720 # 10844947 # 10926641 # 14 <lb/>47 # 10696001 # 10768971 # 10846272 # 10928041 # 13 <lb/>48 # 10697182 # 10770224 # 10847597 # 10929442 # 12 <lb/>49 # 10698364 # 10771477 # 10848924 # 10930846 # 11 <lb/>50 # 10699548 # 10772732 # 10850252 # 10932249 # 10 <lb/>51 # 10700732 # 10773988 # 10851583 # 10933654 # 9 <lb/>52 # 10701918 # 10775244 # 10852914 # 10935061 # 8 <lb/>53 # 10703105 # 10776502 # 10854246 # 10936469 # 7 <lb/>54 # 10704294 # 10777761 # 10855578 # 10937879 # 6 <lb/>55 # 10705483 # 10779022 # 10856912 # 10939290 # 5 <lb/>56 # 10706674 # 10780284 # 10858247 # 10940702 # 4 <lb/>57 # 10707866 # 10781547 # 10859584 # 10942115 # 3 <lb/>58 # 10709059 # 10782802 # 10860922 # 10943527 # 2 <lb/>59 # 10710254 # 10784078 # 10862262 # 10944945 # 1 <lb/>60 # 10711449 # 10785347 # 10863603 # 10946362 # 0 <lb/> # 69 # 68 # 67 # 66 <lb/></note> </div> <div xml:id="echoid-div695" type="section" level="1" n="378"> <head xml:id="echoid-head405" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="258" file="270" n="270" rhead=""/> </div> <div xml:id="echoid-div696" type="section" level="1" n="379"> <head xml:id="echoid-head406" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 24 # 25 # 26 # 27 <lb/>0 # 10946362 # 11033783 # 11126021 # 11223262 # 60 <lb/>1 # 10947781 # 11035280 # 11127601 # 11224927 # 59 <lb/>2 # 10949201 # 11036779 # 11129182 # 11226593 # 58 <lb/>3 # 10950622 # 11038279 # 11130765 # 11228260 # 57 <lb/>4 # 10952045 # 11039780 # 11132349 # 11229929 # 56 <lb/>5 # 10953469 # 11041283 # 11133933 # 11231599 # 55 <lb/>6 # 10954898 # 11042787 # 11135519 # 11233270 # 54 <lb/>7 # 10956320 # 11044293 # 11137106 # 11234943 # 53 <lb/>8 # 10957747 # 11045799 # 11138694 # 11236617 # 52 <lb/>9 # 10959175 # 11047306 # 11140284 # 11238292 # 51 <lb/>10 # 10960605 # 11048815 # 11141875 # 11239969 # 50 <lb/>11 # 10962036 # 11050325 # 11143467 # 11241648 # 49 <lb/>12 # 10963469 # 11051837 # 11145061 # 11243329 # 48 <lb/>13 # 10964903 # 11053350 # 11146656 # 11245011 # 47 <lb/>14 # 10966338 # 11054865 # 11148254 # 11246694 # 46 <lb/>15 # 10967775 # 11056381 # 11149853 # 11248378 # 45 <lb/>16 # 10969213 # 11057898 # 11151453 # 11250064 # 44 <lb/>17 # 10970652 # 11059420 # 11153055 # 11251751 # 43 <lb/>18 # 10972092 # 11060939 # 11154658 # 11253440 # 42 <lb/>19 # 10973533 # 11062461 # 11156262 # 11255130 # 41 <lb/>20 # 10974976 # 11063985 # 11157868 # 11256822 # 40 <lb/>21 # 10976420 # 11065510 # 11159475 # 11258516 # 39 <lb/>22 # 10977865 # 11067037 # 11161084 # 11260211 # 38 <lb/>23 # 10979312 # 11068564 # 11162694 # 11261907 # 37 <lb/>24 # 10980760 # 11070092 # 11164306 # 11263605 # 36 <lb/>25 # 10982210 # 11071621 # 11165919 # 11265304 # 35 <lb/>26 # 10983661 # 11073152 # 11167533 # 11267005 # 34 <lb/>27 # 10985113 # 11074684 # 11169149 # 11268707 # 33 <lb/>28 # 10986567 # 11076218 # 11170766 # 11270410 # 32 <lb/>29 # 10988022 # 11077753 # 11172385 # 11272114 # 31 <lb/>30 # 10989480 # 11079289 # 11174006 # 11273820 # 30 <lb/> # 65 # 64 # 63 # 62 <lb/></note> </div> <div xml:id="echoid-div697" type="section" level="1" n="380"> <head xml:id="echoid-head407" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="259" file="271" n="271" rhead=""/> </div> <div xml:id="echoid-div698" type="section" level="1" n="381"> <head xml:id="echoid-head408" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 24 # 25 # 26 # 27 <lb/>30 # 10989480 # 11079289 # 11174006 # 11273820 # 30 <lb/>31 # 10990938 # 11080827 # 11175627 # 11275528 # 29 <lb/>32 # 10992398 # 11082366 # 11177249 # 11277238 # 28 <lb/>33 # 10993859 # 1103906 # 11178873 # 11278949 # 27 <lb/>34 # 10995321 # 11085448 # 11180499 # 11280661 # 26 <lb/>35 # 10996783 # 11086990 # 11182125 # 11282374 # 25 <lb/>36 # 10998247 # 11088536 # 11183753 # 11284089 # 24 <lb/>37 # 10999712 # 11090082 # 11185383 # 11285805 # 23 <lb/>38 # 11001179 # 11091629 # 11187014 # 11287524 # 22 <lb/>39 # 11002647 # 11093178 # 11188647 # 11289244 # 21 <lb/>40 # 11004116 # 11094729 # 11190281 # 11290965 # 20 <lb/>41 # 11005587 # 11096280 # 11191916 # 11292688 # 19 <lb/>42 # 11007059 # 11097833 # 11193553 # 11294412 # 18 <lb/>43 # 11008533 # 11099387 # 11195191 # 11296132 # 17 <lb/>44 # 11 0008 # 11100943 # 11196831 # 11297864 # 16 <lb/>45 # 11011484 # 11102500 # 11198472 # 11299593 # 15 <lb/>46 # 11019262 # 11104058 # 11200114 # 11301324 # 14 <lb/>47 # 11014441 # 11105618 # 11201758 # 11303056 # 13 <lb/>48 # 11015921 # 11107179 # 11203404 # 11304789 # 12 <lb/>49 # 11017402 # 11108741 # 11205051 # 11306523 # 11 <lb/>50 # 11018884 # 11110306 # 11206700 # 11308259 # 10 <lb/>51 # 11020367 # 11111871 # 11208350 # 11309996 # 9 <lb/>52 # 11021852 # 11113438 # 11210001 # 11311735 # 8 <lb/>53 # 11023338 # 11115006 # 11211654 # 11313476 # 7 <lb/>54 # 11024826 # 11116575 # 11213308 # 11315218 # 6 <lb/>55 # 11026315 # 11118145 # 11214963 # 11316961 # 5 <lb/>56 # 11027806 # 11119717 # 11216620 # 11318706 # 4 <lb/>57 # 11029298 # 11121290 # 11218278 # 11319452 # 3 <lb/>58 # 11030791 # 11122865 # 11219938 # 11322199 # 2 <lb/>59 # 11032287 # 11124442 # 11221599 # 11323949 # 1 <lb/>60 # 11033783 # 11126021 # 11223262 # 11325700 # 0 <lb/> # 65 # 64 # 63 # 62 <lb/></note> </div> <div xml:id="echoid-div699" type="section" level="1" n="382"> <head xml:id="echoid-head409" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="260" file="272" n="272" rhead=""/> </div> <div xml:id="echoid-div700" type="section" level="1" n="383"> <head xml:id="echoid-head410" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 28 # 29 # 30 # 31 <lb/>0 # 11325700 # 11433540 # 11547004 # 11666331 # 60 <lb/>1 # 11327452 # 11435384 # 11548944 # 11668371 # 59 <lb/>2 # 11329206 # 11437230 # 11550886 # 11670413 # 58 <lb/>3 # 11330961 # 11439078 # 11552829 # 11672457 # 57 <lb/>4 # 11332718 # 11440927 # 11554774 # 11674502 # 56 <lb/>5 # 11334479 # 11442777 # 11556720 # 11676548 # 55 <lb/>6 # 11336237 # 11444629 # 11558669 # 11678597 # 54 <lb/>7 # 11337999 # 11446483 # 11560619 # 11680647 # 53 <lb/>8 # 11339762 # 11448339 # 11562570 # 11682698 # 52 <lb/>9 # 11341526 # 11450196 # 11564523 # 11684752 # 51 <lb/>10 # 11343292 # 11452054 # 11566480 # 11686807 # 50 <lb/>11 # 11345060 # 11453915 # 11568434 # 11688864 # 49 <lb/>12 # 11346830 # 11455776 # 11570393 # 11690923 # 48 <lb/>13 # 11348601 # 11457639 # 11572353 # 11692984 # 47 <lb/>14 # 11350373 # 11459503 # 11574314 # 11695046 # 46 <lb/>15 # 11352149 # 11461370 # 11576277 # 11697110 # 45 <lb/>16 # 11353923 # 11463238 # 11578242 # 11699176 # 44 <lb/>17 # 11355698 # 11465107 # 11580208 # 11701243 # 43 <lb/>18 # 11357475 # 11466978 # 11582175 # 11703312 # 42 <lb/>19 # 11359255 # 11468850 # 11584145 # 11705383 # 41 <lb/>20 # 11361036 # 11470723 # 11586116 # 11707455 # 40 <lb/>21 # 11362819 # 11472599 # 11588089 # 11709530 # 39 <lb/>22 # 11364603 # 11474483 # 11590064 # 11711606 # 38 <lb/>23 # 11366389 # 11476354 # 11592040 # 11713684 # 37 <lb/>24 # 11368177 # 11478235 # 11594018 # 11715764 # 36 <lb/>25 # 11369966 # 11480117 # 11595998 # 11717845 # 35 <lb/>26 # 11371756 # 11482001 # 11597979 # 11719928 # 34 <lb/>27 # 11373548 # 11483887 # 11599961 # 11722012 # 33 <lb/>28 # 11375341 # 11485774 # 11601946 # 11724099 # 32 <lb/>29 # 11377136 # 11487662 # 11603932 # 11726187 # 31 <lb/>30 # 11378933 # 11489353 # 11605919 # 11728276 # 30 <lb/> # 61 # 60 # 59 # 58 <lb/></note> </div> <div xml:id="echoid-div701" type="section" level="1" n="384"> <head xml:id="echoid-head411" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="261" file="273" n="273" rhead=""/> </div> <div xml:id="echoid-div702" type="section" level="1" n="385"> <head xml:id="echoid-head412" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 28 # 29 # 30 # 31 <lb/>30 # 11378933 # 11489353 # 11605919 # 11728276 # 30 <lb/>31 # 11380731 # 11491445 # 11607909 # 11730367 # 29 <lb/>32 # 11382530 # 11493338 # 11609900 # 11732460 # 28 <lb/>33 # 11384331 # 11495233 # 11611893 # 11734555 # 27 <lb/>34 # 11386134 # 11497140 # 11613888 # 11736652 # 26 <lb/>35 # 11387938 # 11499028 # 11615876 # 11738751 # 25 <lb/>36 # 11389744 # 11500928 # 11617882 # 11740851 # 24 <lb/>37 # 11391551 # 11502829 # 11619881 # 11742953 # 23 <lb/>38 # 11393359 # 11504731 # 11621882 # 11745057 # 22 <lb/>39 # 11395169 # 11506626 # 11623885 # 11747162 # 21 <lb/>40 # 11396981 # 11508532 # 11625889 # 11749269 # 20 <lb/>41 # 11398794 # 11510450 # 11627996 # 11751378 # 19 <lb/>42 # 11400609 # 11512360 # 11629904 # 11753489 # 18 <lb/>43 # 11402425 # 11514271 # 11631913 # 11755603 # 17 <lb/>44 # 11404243 # 11516183 # 11633924 # 11757718 # 16 <lb/>45 # 11406063 # 11518097 # 11635937 # 11759834 # 15 <lb/>46 # 11407884 # 11520013 # 11637952 # 11761951 # 14 <lb/>47 # 11409706 # 11521930 # 11639968 # 11764069 # 13 <lb/>48 # 11411530 # 11523849 # 11641986 # 11766190 # 12 <lb/>49 # 11413356 # 11525770 # 11644005 # 11768312 # 11 <lb/>50 # 11415183 # 11527692 # 11646026 # 11770437 # 10 <lb/>51 # 11417012 # 11529616 # 11648049 # 11772564 # 9 <lb/>52 # 11418842 # 11531542 # 11650075 # 11774696 # 8 <lb/>53 # 11420673 # 11533469 # 11652099 # 11776822 # 7 <lb/>54 # 11422507 # 11535398 # 11654127 # 11778954 # 6 <lb/>55 # 11424342 # 11537328 # 11656156 # 11781088 # 5 <lb/>56 # 11426178 # 11539260 # 11658188 # 11783223 # 4 <lb/>57 # 11428016 # 11541193 # 11660221 # 11785361 # 3 <lb/>58 # 11429856 # 11543128 # 11662256 # 11787500 # 2 <lb/>59 # 11431689 # 11545065 # 11664292 # 11789640 # 1 <lb/>60 # 11433540 # 11547004 # 11666331 # 11791783 # 0 <lb/> # 61 # 60 # 59 # 58 <lb/></note> </div> <div xml:id="echoid-div703" type="section" level="1" n="386"> <head xml:id="echoid-head413" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="262" file="274" n="274" rhead=""/> </div> <div xml:id="echoid-div704" type="section" level="1" n="387"> <head xml:id="echoid-head414" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 32 # 33 # 34 # 35 <lb/>0 # 11791783 # 11923633 # 12062179 # 12207745 # 60 <lb/>1 # 11793927 # 11925886 # 12064546 # 12210233 # 59 <lb/>2 # 11796073 # 11928141 # 12066916 # 12212723 # 58 <lb/>3 # 11798221 # 11930397 # 12069286 # 12215214 # 57 <lb/>4 # 11800371 # 11932656 # 12071660 # 12217708 # 56 <lb/>5 # 11802522 # 11934917 # 12074036 # 12220204 # 55 <lb/>6 # 11804675 # 11937180 # 12076413 # 12222702 # 54 <lb/>7 # 11806830 # 11939445 # 12078792 # 12225201 # 53 <lb/>8 # 11808987 # 11941701 # 12081174 # 12227703 # 52 <lb/>9 # 11811145 # 11943979 # 12083558 # 12230207 # 51 <lb/>10 # 11813306 # 11946250 # 12085943 # 12232713 # 50 <lb/>11 # 11815468 # 11948522 # 12088330 # 12235221 # 49 <lb/>12 # 11817632 # 11950796 # 12090720 # 12237732 # 48 <lb/>13 # 11819797 # 11953071 # 12093111 # 12240245 # 47 <lb/>14 # 11821965 # 11955349 # 12095504 # 12242759 # 46 <lb/>15 # 11824134 # 11957629 # 12097899 # 12245275 # 45 <lb/>16 # 11826306 # 11959910 # 12100296 # 12247794 # 44 <lb/>17 # 11828479 # 11962194 # 12102696 # 12250315 # 43 <lb/>18 # 11830654 # 11964479 # 12105097 # 12252837 # 42 <lb/>19 # 11832830 # 11966766 # 12107500 # 12255361 # 41 <lb/>20 # 11835008 # 11969055 # 12109905 # 12257888 # 40 <lb/>21 # 11837188 # 11971346 # 12112312 # 12260417 # 39 <lb/>22 # 11839369 # 11973638 # 12114722 # 12262948 # 38 <lb/>23 # 11841552 # 11975932 # 12117133 # 12265481 # 37 <lb/>24 # 11843737 # 11978229 # 12119546 # 12268016 # 36 <lb/>25 # 11845924 # 11980527 # 12121960 # 12270553 # 35 <lb/>26 # 11848114 # 11982828 # 12124377 # 12273093 # 34 <lb/>27 # 11850305 # 11985131 # 12126796 # 12275634 # 33 <lb/>28 # 11852498 # 11987435 # 12129216 # 12278187 # 32 <lb/>29 # 11854693 # 11989741 # 12131638 # 12280722 # 31 <lb/>30 # 11856890 # 11992050 # 12134063 # 12283270 # 30 <lb/> # 57 # 56 # 55 # 54 <lb/></note> </div> <div xml:id="echoid-div705" type="section" level="1" n="388"> <head xml:id="echoid-head415" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="263" file="275" n="275" rhead=""/> </div> <div xml:id="echoid-div706" type="section" level="1" n="389"> <head xml:id="echoid-head416" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 32 # 33 # 34 # 35 <lb/>30 # 11856890 # 11992050 # 12134063 # 12283270 # 30 <lb/>31 # 11859088 # 11994360 # 12136490 # 12285820 # 39 <lb/>32 # 11861288 # 11996672 # 12138919 # 12288372 # 28 <lb/>33 # 11863489 # 11998986 # 12141350 # 12290925 # 27 <lb/>34 # 11865693 # 12001303 # 12143783 # 12293481 # 26 <lb/>35 # 11867899 # 12003619 # 12146218 # 12296039 # 25 <lb/>36 # 11870107 # 12005938 # 12148656 # 12298599 # 24 <lb/>37 # 11872316 # 12008259 # 12150095 # 12301161 # 23 <lb/>38 # 11874527 # 12010582 # 12153536 # 12303725 # 22 <lb/>39 # 11876739 # 12012907 # 12155978 # 12306291 # 21 <lb/>40 # 11878954 # 12015233 # 12158423 # 12308859 # 20 <lb/>41 # 11881171 # 12017562 # 12160870 # 12311430 # 19 <lb/>42 # 11883389 # 12019893 # 12163319 # 12314003 # 18 <lb/>43 # 11885609 # 12022226 # 12165770 # 12316578 # 17 <lb/>44 # 11887831 # 12024560 # 12168223 # 12319156 # 16 <lb/>45 # 11890054 # 12026897 # 12170677 # 12321736 # 15 <lb/>46 # 11892280 # 12029236 # 12173135 # 12324317 # 14 <lb/>47 # 11894508 # 12031576 # 12175594 # 12326900 # 13 <lb/>48 # 11896737 # 12033919 # 12178055 # 12329486 # 12 <lb/>49 # 11898968 # 12036264 # 12180518 # 12332074 # 11 <lb/>50 # 11901202 # 12038610 # 12182983 # 12334664 # 10 <lb/>51 # 11903437 # 12040958 # 12185450 # 12337256 # 9 <lb/>52 # 11905674 # 12043309 # 12187919 # 12339851 # 8 <lb/>53 # 11907912 # 12045661 # 12190390 # 12342448 # 7 <lb/>54 # 11910153 # 12048016 # 12192864 # 12345046 # 6 <lb/>55 # 11912395 # 12050372 # 12195340 # 12347646 # 5 <lb/>56 # 11914640 # 12052730 # 12197817 # 12350249 # 4 <lb/>57 # 11916886 # 12055089 # 12200296 # 12352854 # 3 <lb/>58 # 11919133 # 12057451 # 12202777 # 12355460 # 2 <lb/>59 # 11921382 # 12059814 # 12205260 # 12358068 # 1 <lb/>60 # 11923633 # 12063179 # 12207745 # 12360678 # 0 <lb/> # 57 # 56 # 55 # 54 <lb/></note> </div> <div xml:id="echoid-div707" type="section" level="1" n="390"> <head xml:id="echoid-head417" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="264" file="276" n="276" rhead=""/> </div> <div xml:id="echoid-div708" type="section" level="1" n="391"> <head xml:id="echoid-head418" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 36 # 37 # 38 # 39 <lb/>0 # 12360678 # 12521357 # 12690184 # 12867599 # 60 <lb/>1 # 12363290 # 12524103 # 12693070 # 12870632 # 59 <lb/>2 # 12365906 # 12526851 # 12695957 # 12873667 # 58 <lb/>3 # 12368524 # 12529601 # 12698847 # 12876704 # 57 <lb/>4 # 12371144 # 12532354 # 12701739 # 12879744 # 56 <lb/>5 # 12373766 # 12535110 # 12704634 # 12882787 # 55 <lb/>6 # 12376391 # 12537867 # 12707531 # 12885832 # 54 <lb/>7 # 12379018 # 12540627 # 12710430 # 12888879 # 53 <lb/>8 # 12381647 # 12543389 # 12713332 # 12891929 # 52 <lb/>9 # 12384278 # 12546152 # 12716236 # 12894982 # 51 <lb/>10 # 12386911 # 12548918 # 12719143 # 12898037 # 50 <lb/>11 # 12389546 # 12551686 # 1272@052 # 12901094 # 49 <lb/>12 # 12392183 # 12554456 # 12724964 # 12904155 # 48 <lb/>13 # 12394822 # 12557229 # 12727878 # 12907218 # 47 <lb/>14 # 12397464 # 12560005 # 12730794 # 12910283 # 46 <lb/>15 # 12400108 # 12562783 # 12733713 # 12913351 # 45 <lb/>16 # 12402754 # 12565563 # 12736635 # 12916422 # 44 <lb/>17 # 12405402 # 12568345 # 12739559 # 12919494 # 43 <lb/>18 # 12408053 # 12571130 # 12742485 # 12922569 # 42 <lb/>19 # 12410705 # 12573917 # 12745413 # 12925647 # 41 <lb/>20 # 12413359 # 12576706 # 12748344 # 12928727 # 40 <lb/>21 # 12416015 # 12579597 # 12751277 # 12931809 # 39 <lb/>22 # 12418674 # 12582912 # 12754213 # 12934895 # 38 <lb/>23 # 12421335 # 12585087 # 12757151 # 12937983 # 37 <lb/>24 # 12423998 # 12587885 # 12760092 # 12941073 # 36 <lb/>25 # 12426663 # 12590685 # 12763035 # 12944166 # 35 <lb/>26 # 12429331 # 12593488 # 12765981 # 12947262 # 34 <lb/>27 # 12432001 # 12596293 # 12768929 # 12950360 # 33 <lb/>28 # 12434673 # 12599101 # 12771880 # 12953461 # 32 <lb/>29 # 12437348 # 12601911 # 12774833 # 12956565 # 31 <lb/>30 # 12440024 # 12604724 # 12777788 # 12959671 # 30 <lb/> # 53 # 52 # 51 # 50 <lb/></note> </div> <div xml:id="echoid-div709" type="section" level="1" n="392"> <head xml:id="echoid-head419" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="265" file="277" n="277" rhead=""/> </div> <div xml:id="echoid-div710" type="section" level="1" n="393"> <head xml:id="echoid-head420" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 36 # 37 # 38 # 39 <lb/>30 # 12440024 # 12604724 # 12777788 # 12959671 # 30 <lb/>31 # 12442702 # 12607539 # 12780746 # 12962780 # 29 <lb/>32 # 12445383 # 12610356 # 12783707 # 12965892 # 28 <lb/>33 # 12448066 # 12613175 # 12786670 # 12969007 # 27 <lb/>34 # 12450751 # 12615997 # 12789635 # 12972124 # 26 <lb/>35 # 12453438 # 12618821 # 12792602 # 12975243 # 25 <lb/>36 # 12450128 # 12621648 # 12795573 # 12978366 # 24 <lb/>37 # 12458821 # 12624477 # 12798546 # 12981491 # 23 <lb/>38 # 12461516 # 12627308 # 12801521 # 12984618 # 22 <lb/>39 # 12464213 # 12630141 # 12804498 # 12987747 # 21 <lb/>40 # 12466913 # 12632977 # 12807478 # 12990880 # 20 <lb/>41 # 12469614 # 12635815 # 12810460 # 12994015 # 19 <lb/>42 # 12472317 # 12638655 # 12813445 # 12997153 # 18 <lb/>43 # 12475022 # 12641597 # 12816432 # 13000293 # 17 <lb/>44 # 12477730 # 12644343 # 12819422 # 13003436 # 16 <lb/>45 # 12480440 # 12646191 # 12822415 # 13006582 # 15 <lb/>46 # 12483152 # 12650041 # 12825410 # 13009730 # 14 <lb/>47 # 12485866 # 12652893 # 12828407 # 13012881 # 13 <lb/>48 # 12488583 # 12655748 # 12831407 # 13016034 # 12 <lb/>49 # 12491302 # 12658605 # 12834409 # 13019189 # 11 <lb/>50 # 12494022 # 12661464 # 12837414 # 13022348 # 10 <lb/>51 # 12496744 # 12664325 # 12840421 # 13025509 # 9 <lb/>52 # 12499469 # 12667189 # 12843431 # 13028673 # 8 <lb/>53 # 12502197 # 12670055 # 12846443 # 13031839 # 7 <lb/>54 # 12504927 # 12672924 # 12849458 # 13035008 # 6 <lb/>55 # 12507659 # 12675795 # 12852475 # 13038180 # 5 <lb/>56 # 12510394 # 12678668 # 12855495 # 13041354 # 4 <lb/>57 # 12513132 # 12681543 # 12858517 # 13044530 # 3 <lb/>58 # 12515871 # 12684421 # 12861542 # 13047710 # 2 <lb/>59 # 12518613 # 12687301 # 12864569 # 13050892 # 1 <lb/>60 # 12521357 # 12690184 # 12867599 # 13054077 # 0 <lb/> # 53 # 52 # 51 # 50 <lb/></note> </div> <div xml:id="echoid-div711" type="section" level="1" n="394"> <head xml:id="echoid-head421" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="266" file="278" n="278" rhead=""/> </div> <div xml:id="echoid-div712" type="section" level="1" n="395"> <head xml:id="echoid-head422" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 40 # 41 # 42 # 43 <lb/>0 # 13054077 # 13250131 # 13456326 # 13673275 # 60 <lb/>1 # 13057264 # 13253482 # 13459851 # 13676986 # 59 <lb/>2 # 13060455 # 13256835 # 13463380 # 13680700 # 58 <lb/>3 # 13063646 # 13260192 # 13466912 # 13684417 # 57 <lb/>4 # 13066843 # 13263582 # 13470447 # 13688138 # 56 <lb/>5 # 13070041 # 13266915 # 13473985 # 13691861 # 55 <lb/>6 # 13073242 # 13270282 # 13477527 # 13695587 # 54 <lb/>7 # 13076445 # 13273651 # 13481071 # 13699316 # 53 <lb/>8 # 13079651 # 13277023 # 13484618 # 13703048 # 52 <lb/>9 # 13082859 # 13280397 # 13488168 # 13706783 # 51 <lb/>10 # 13086071 # 13283775 # 13491721 # 13710523 # 50 <lb/>11 # 13089285 # 13287155 # 13495276 # 13714266 # 49 <lb/>12 # 13092502 # 13290538 # 13498835 # 13718012 # 48 <lb/>13 # 13095721 # 13293924 # 13502397 # 13721761 # 47 <lb/>14 # 13098944 # 13297313 # 13505962 # 13725514 # 46 <lb/>15 # 13102169 # 13300704 # 13509530 # 13729270 # 45 <lb/>16 # 13105397 # 13304098 # 13513101 # 13733029 # 44 <lb/>17 # 13108627 # 13307495 # 13516675 # 13736790 # 43 <lb/>18 # 13111861 # 13310896 # 13520252 # 13740555 # 42 <lb/>19 # 13114098 # 13314299 # 13523832 # 13744322 # 41 <lb/>20 # 13118337 # 13317705 # 13527416 # 13748092 # 40 <lb/>21 # 13121578 # 13321114 # 13531003 # 13751867 # 39 <lb/>22 # 13124823 # 13324526 # 13534593 # 13755644 # 38 <lb/>23 # 13128070 # 13327941 # 13538185 # 13759424 # 37 <lb/>24 # 13131320 # 13331359 # 13541781 # 13763209 # 36 <lb/>25 # 13134572 # 13334779 # 13545380 # 13766997 # 35 <lb/>26 # 13137828 # 13338203 # 13548981 # 13770788 # 34 <lb/>27 # 13141085 # 13341629 # 13552585 # 13774582 # 33 <lb/>28 # 13144346 # 13345058 # 13556193 # 13778380 # 32 <lb/>29 # 13147509 # 13348490 # 13559803 # 13782181 # 31 <lb/>30 # 13150874 # 13351924 # 13563417 # 13785985 # 30 <lb/> # 49 # 48 # 47 # 46 <lb/></note> </div> <div xml:id="echoid-div713" type="section" level="1" n="396"> <head xml:id="echoid-head423" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="267" file="279" n="279" rhead=""/> </div> <div xml:id="echoid-div714" type="section" level="1" n="397"> <head xml:id="echoid-head424" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 40 # 41 # 42 # 43 <lb/>30 # 13150874 # 13351924 # 13563417 # 13785985 # 30 <lb/>31 # 13154142 # 13355361 # 13567034 # 13789792 # 29 <lb/>32 # 13157413 # 13358802 # 13570654 # 13793603 # 28 <lb/>33 # 13160687 # 13362245 # 13574277 # 13797416 # 27 <lb/>34 # 13163964 # 13365691 # 13577903 # 13801233 # 26 <lb/>35 # 13167243 # 13369140 # 13581532 # 13805053 # 25 <lb/>36 # 13170526 # 13372592 # 13585164 # 13808876 # 24 <lb/>37 # 13173811 # 13376057 # 13588799 # 13812703 # 23 <lb/>38 # 13177099 # 13379505 # 13592438 # 13816534 # 22 <lb/>39 # 13180389 # 13382966 # 13596079 # 13820368 # 21 <lb/>40 # 13183682 # 13386430 # 13599723 # 13824205 # 20 <lb/>41 # 13186978 # 13389897 # 13603370 # 13828045 # 19 <lb/>42 # 13190276 # 13393367 # 13607021 # 13831889 # 18 <lb/>43 # 13193577 # 13396839 # 13610975 # 13835736 # 17 <lb/>44 # 13196882 # 13400315 # 13614332 # 13839586 # 16 <lb/>45 # 13200189 # 13403794 # 13617992 # 13843439 # 15 <lb/>46 # 13203499 # 13407275 # 13621656 # 13847296 # 14 <lb/>47 # 13206812 # 13410759 # 13625323 # 13851156 # 13 <lb/>48 # 13210128 # 13414247 # 13628993 # 13855019 # 12 <lb/>49 # 13213447 # 13417738 # 13632666 # 13858885 # 11 <lb/>50 # 13216769 # 13421232 # 13636342 # 13862755 # 10 <lb/>51 # 13220093 # 13424728 # 13640021 # 13866628 # 9 <lb/>52 # 13223421 # 13428227 # 13643704 # 13870505 # 8 <lb/>53 # 13226750 # 13431729 # 13647390 # 13874385 # 7 <lb/>54 # 13230082 # 13435234 # 13651078 # 13878268 # 6 <lb/>55 # 13233417 # 13438742 # 13654769 # 13882154 # 5 <lb/>56 # 13236754 # 13442253 # 13658464 # 13886044 # 4 <lb/>57 # 13240094 # 13445767 # 13662162 # 13889636 # 3 <lb/>58 # 13243437 # 13449284 # 13665863 # 13893833 # 2 <lb/>59 # 13246783 # 13452804 # 13669567 # 13897733 # 1 <lb/>60 # 13250131 # 13456326 # 13673275 # 13901636 # 0 <lb/> # 49 # 48 # 47 # 46 <lb/></note> </div> <div xml:id="echoid-div715" type="section" level="1" n="398"> <head xml:id="echoid-head425" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="268" file="280" n="280" rhead=""/> </div> <div xml:id="echoid-div716" type="section" level="1" n="399"> <head xml:id="echoid-head426" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 44 # 45 # 46 # 47 <lb/>0 # 13901636 # 14142135 # 14395564 # 14662790 # 60 <lb/>1 # 13905542 # 14146251 # 14399901 # 14667366 # 59 <lb/>2 # 13909452 # 14150371 # 14404242 # 14671946 # 58 <lb/>3 # 13913365 # 14154494 # 14408587 # 14676530 # 57 <lb/>4 # 13917281 # 14158621 # 14412937 # 14681119 # 56 <lb/>5 # 13921201 # 14162751 # 14417290 # 14685712 # 55 <lb/>6 # 13925126 # 14166884 # 14421647 # 14690309 # 54 <lb/>7 # 13929052 # 14171021 # 14426008 # 14694910 # 53 <lb/>8 # 13932982 # 14175162 # 14430374 # 14699514 # 52 <lb/>9 # 13936916 # 14179306 # 14434743 # 14704122 # 51 <lb/>10 # 13940854 # 14183454 # 14439116 # 14708735 # 50 <lb/>11 # 13944795 # 14187606 # 14443493 # 14713352 # 49 <lb/>12 # 13948739 # 14191761 # 14447874 # 14717973 # 48 <lb/>13 # 13952686 # 14195919 # 14452259 # 14722598 # 47 <lb/>14 # 13956638 # 14200082 # 14456648 # 14727228 # 46 <lb/>15 # 13960592 # 14204248 # 14461040 # 14731862 # 45 <lb/>16 # 13964550 # 14208418 # 14465437 # 14736500 # 44 <lb/>17 # 13968511 # 14212591 # 14469838 # 14741142 # 43 <lb/>18 # 13972476 # 14216769 # 14474242 # 14745788 # 42 <lb/>19 # 13976444 # 14220950 # 14478650 # 14750438 # 41 <lb/>20 # 13980416 # 14225135 # 14483062 # 14755094 # 40 <lb/>21 # 13984391 # 14229324 # 14487478 # 14759753 # 39 <lb/>22 # 13988370 # 14233517 # 14491898 # 14764416 # 38 <lb/>23 # 13992352 # 14237713 # 14496322 # 14769083 # 37 <lb/>24 # 13996338 # 14241912 # 14500750 # 14773755 # 36 <lb/>25 # 14000327 # 14246115 # 14505182 # 14778430 # 35 <lb/>26 # 14004319 # 14250321 # 14509617 # 14783110 # 34 <lb/>27 # 14008315 # 14254531 # 14514056 # 14787794 # 33 <lb/>28 # 14012314 # 14258745 # 14518500 # 14792482 # 32 <lb/>29 # 14016316 # 14262961 # 14522946 # 14797174 # 31 <lb/>30 # 14020322 # 14267182 # 14527397 # 14801871 # 30 <lb/> # 45 # 44 # 43 # 42 <lb/></note> </div> <div xml:id="echoid-div717" type="section" level="1" n="400"> <head xml:id="echoid-head427" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="269" file="281" n="281" rhead=""/> </div> <div xml:id="echoid-div718" type="section" level="1" n="401"> <head xml:id="echoid-head428" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 44 # 45 # 46 # 47 <lb/>30 # 14020322 # 14267182 # 14527397 # 14801871 # 30 <lb/>31 # 14024332 # 14271407 # 14531852 # 14806571 # 29 <lb/>32 # 14028345 # 14275635 # 14536311 # 14811276 # 28 <lb/>33 # 14032361 # 14279867 # 14540773 # 14815985 # 27 <lb/>34 # 14036381 # 14284103 # 14545240 # 14820698 # 26 <lb/>35 # 14040404 # 14288343 # 14549711 # 14825416 # 25 <lb/>36 # 14044431 # 14292587 # 14554186 # 14830139 # 24 <lb/>37 # 14048461 # 14296834 # 14558665 # 14834866 # 23 <lb/>38 # 14052494 # 14301086 # 14563148 # 14839597 # 22 <lb/>39 # 14056531 # 14305331 # 14567635 # 14844332 # 21 <lb/>40 # 14060572 # 14309599 # 14572126 # 14849072 # 20 <lb/>41 # 14064616 # 14313861 # 14576621 # 14853815 # 19 <lb/>42 # 14068664 # 14318127 # 14581120 # 14858563 # 18 <lb/>43 # 14072715 # 14322396 # 14585624 # 14863315 # 17 <lb/>44 # 14076770 # 14326670 # 14590131 # 14868071 # 16 <lb/>45 # 14080829 # 14330947 # 14594642 # 14872831 # 15 <lb/>46 # 14084891 # 14335228 # 14599157 # 14877597 # 14 <lb/>47 # 14088956 # 14339513 # 14603676 # 14882377 # 13 <lb/>48 # 14093026 # 14343802 # 14608199 # 14887141 # 12 <lb/>49 # 14097099 # 14348095 # 14612725 # 14891919 # 11 <lb/>50 # 14101175 # 14352391 # 14617256 # 14896701 # 10 <lb/>51 # 14105255 # 14356691 # 14621791 # 14901487 # 9 <lb/>52 # 14109339 # 14360995 # 14626330 # 14906278 # 8 <lb/>53 # 14113427 # 14365303 # 14630873 # 14911073 # 7 <lb/>54 # 14117518 # 14369615 # 14635421 # 14915873 # 6 <lb/>55 # 14121612 # 14373930 # 14639973 # 14920677 # 5 <lb/>56 # 14125709 # 14378350 # 14644528 # 14925486 # 4 <lb/>57 # 14129810 # 14382573 # 14649087 # 14930299 # 3 <lb/>58 # 14133915 # 14386900 # 14653651 # 14935116 # 2 <lb/>59 # 14138023 # 14391230 # 14658218 # 14939938 # 1 <lb/>60 # 14142135 # 14395564 # 14662790 # 14944764 # 0 <lb/> # 45 # 44 # 43 # 42 <lb/></note> </div> <div xml:id="echoid-div719" type="section" level="1" n="402"> <head xml:id="echoid-head429" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="270" file="282" n="282" rhead=""/> </div> <div xml:id="echoid-div720" type="section" level="1" n="403"> <head xml:id="echoid-head430" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 48 # 49 # 50 # 51 <lb/>0 # 14944764 # 15242532 # 15557239 # 15890158 # 60 <lb/>1 # 14949594 # 15247634 # 15562635 # 15895869 # 59 <lb/>2 # 14954429 # 15252741 # 15568036 # 15901586 # 58 <lb/>3 # 14959268 # 15257852 # 15573441 # 15907307 # 57 <lb/>4 # 14964112 # 15262969 # 15578852 # 15913034 # 56 <lb/>5 # 14968960 # 15268090 # 15584267 # 15918766 # 55 <lb/>6 # 14973812 # 15273216 # 15589688 # 15924504 # 54 <lb/>7 # 14978668 # 15278347 # 15595114 # 15930247 # 53 <lb/>8 # 14983530 # 15283484 # 15600545 # 15936095 # 52 <lb/>9 # 14988396 # 15288626 # 15605981 # 15941748 # 51 <lb/>10 # 14993266 # 15293773 # 15611422 # 15947508 # 50 <lb/>11 # 14998104 # 15298924 # 15616868 # 15953273 # 49 <lb/>12 # 15003020 # 15304080 # 15622319 # 15959044 # 48 <lb/>13 # 15007903 # 15309240 # 15627775 # 15964820 # 47 <lb/>14 # 15012791 # 15314405 # 15633237 # 15970603 # 46 <lb/>15 # 15017683 # 15319574 # 15639704 # 15976390 # 45 <lb/>16 # 15022580 # 15324748 # 15644177 # 15982184 # 44 <lb/>17 # 15027481 # 15329926 # 15649655 # 15987983 # 43 <lb/>18 # 15032387 # 15335109 # 15655138 # 15993788 # 42 <lb/>19 # 15037297 # 15340297 # 15660626 # 15999599 # 41 <lb/>20 # 15042212 # 15345491 # 15666119 # 16005416 # 40 <lb/>21 # 15047131 # 15350689 # 15671617 # 16011237 # 39 <lb/>22 # 15052054 # 15355892 # 15677121 # 16017065 # 38 <lb/>23 # 15056982 # 15361100 # 15682630 # 16022898 # 37 <lb/>24 # 15061915 # 15366313 # 15688144 # 16028736 # 36 <lb/>25 # 15066852 # 15371530 # 15693663 # 16034579 # 35 <lb/>26 # 15071791 # 15376753 # 15699188 # 16040429 # 34 <lb/>27 # 15076739 # 15381980 # 15704717 # 16046283 # 33 <lb/>28 # 15081690 # 15387212 # 15710252 # 16052143 # 32 <lb/>29 # 15086645 # 15392449 # 15715792 # 16058008 # 31 <lb/>30 # 15091605 # 15397692 # 15721337 # 16063878 # 30 <lb/> # 41 # 40 # 39 # 38 <lb/></note> </div> <div xml:id="echoid-div721" type="section" level="1" n="404"> <head xml:id="echoid-head431" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="271" file="283" n="283" rhead=""/> </div> <div xml:id="echoid-div722" type="section" level="1" n="405"> <head xml:id="echoid-head432" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 48 # 49 # 50 # 51 <lb/>30 # 15091605 # 15397692 # 15721337 # 16063878 # 30 <lb/>31 # 15096569 # 15402939 # 15726887 # 16069754 # 29 <lb/>32 # 15101538 # 15408191 # 15732443 # 16075637 # 28 <lb/>33 # 15106571 # 15413447 # 15738003 # 16081524 # 27 <lb/>34 # 15111490 # 15418708 # 15743569 # 16087418 # 26 <lb/>35 # 15116472 # 15423974 # 15749141 # 16093318 # 25 <lb/>36 # 15121459 # 15429246 # 15754718 # 16099224 # 24 <lb/>37 # 15126451 # 15434522 # 15760300 # 16105135 # 23 <lb/>38 # 15131447 # 15439803 # 15765887 # 16111053 # 22 <lb/>39 # 15136447 # 15445089 # 15771479 # 16116976 # 21 <lb/>40 # 15141453 # 15450380 # 15777077 # 16122905 # 20 <lb/>41 # 15146463 # 15455675 # 15782680 # 16128839 # 19 <lb/>42 # 15151478 # 15460976 # 15788289 # 16134779 # 18 <lb/>43 # 15156497 # 15466282 # 15793903 # 16140724 # 17 <lb/>44 # 15161520 # 15471593 # 15799523 # 16146676 # 16 <lb/>45 # 15166548 # 15476908 # 15805147 # 16152634 # 15 <lb/>46 # 15171581 # 15482229 # 15810777 # 16158598 # 14 <lb/>47 # 15176619 # 15487554 # 15816412 # 16164567 # 13 <lb/>48 # 15181661 # 15492885 # 15822052 # 16170542 # 12 <lb/>49 # 15186708 # 15498220 # 15827697 # 16176522 # 11 <lb/>50 # 15191760 # 15503560 # 15833349 # 16182509 # 10 <lb/>51 # 15196816 # 15508905 # 15839005 # 16188501 # 9 <lb/>52 # 15201877 # 15514256 # 15844667 # 16194499 # 8 <lb/>53 # 15206943 # 15519611 # 15850335 # 16200503 # 7 <lb/>54 # 15212013 # 15524972 # 15856008 # 16206513 # 6 <lb/>55 # 15217088 # 15530338 # 15861676 # 16212528 # 5 <lb/>56 # 15222168 # 15535710 # 15867370 # 16218550 # 4 <lb/>57 # 15227253 # 15541083 # 15873058 # 16224577 # 3 <lb/>58 # 15232342 # 15546463 # 15878753 # 16230610 # 2 <lb/>59 # 15237435 # 15551848 # 15884453 # 16236648 # 1 <lb/>60 # 15242532 # 15557239 # 15890158 # 16242692 # 0 <lb/> # 41 # 40 # 39 # 38 <lb/></note> </div> <div xml:id="echoid-div723" type="section" level="1" n="406"> <head xml:id="echoid-head433" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis</head> <pb o="272" file="284" n="284" rhead=""/> </div> <div xml:id="echoid-div724" type="section" level="1" n="407"> <head xml:id="echoid-head434" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 52 # 53 # 54 # 55 <lb/>0 # 16242692 # 16616401 # 17013017 # 17434469 # 60 <lb/>1 # 16248742 # 16622819 # 17019832 # 17441715 # 59 <lb/>2 # 16254799 # 16629243 # 17026654 # 17448968 # 58 <lb/>3 # 16260861 # 16635673 # 17033482 # 17456229 # 57 <lb/>4 # 16266929 # 16642109 # 17040318 # 17463499 # 56 <lb/>5 # 16273003 # 16648551 # 17047160 # 17470775 # 55 <lb/>6 # 16279083 # 16655001 # 17054010 # 17478059 # 54 <lb/>7 # 16285169 # 16661457 # 17060866 # 17485351 # 53 <lb/>8 # 16291261 # 16667919 # 17067729 # 17492650 # 52 <lb/>9 # 16297358 # 16674408 # 17074599 # 17499957 # 51 <lb/>10 # 16303461 # 16680864 # 17081476 # 17507272 # 50 <lb/>11 # 16309570 # 16687345 # 17088359 # 17514594 # 49 <lb/>12 # 16315685 # 16693834 # 17095250 # 17521924 # 48 <lb/>13 # 16321806 # 16700328 # 17102148 # 17529262 # 47 <lb/>14 # 16327934 # 16706829 # 17109053 # 17536607 # 46 <lb/>15 # 16334067 # 16713336 # 17115965 # 17543959 # 45 <lb/>16 # 16340197 # 16719850 # 17122885 # 17551319 # 44 <lb/>17 # 16346353 # 16726362 # 17129812 # 17558687 # 43 <lb/>18 # 16352505 # 16732877 # 17136747 # 17566063 # 42 <lb/>19 # 16358663 # 16739430 # 17143689 # 17573446 # 41 <lb/>20 # 16364827 # 16745970 # 17150638 # 17580837 # 40 <lb/>21 # 16370996 # 16752517 # 17157593 # 17588236 # 39 <lb/>22 # 16377172 # 16759070 # 17164556 # 17595643 # 38 <lb/>23 # 16383359 # 16765629 # 17171525 # 17603057 # 37 <lb/>24 # 16389542 # 16772195 # 17178502 # 17610480 # 36 <lb/>25 # 16395736 # 16778767 # 17185485 # 17617909 # 35 <lb/>26 # 16401936 # 16785347 # 17192476 # 17625347 # 34 <lb/>27 # 16408152 # 16791933 # 17199472 # 17632793 # 33 <lb/>28 # 16414365 # 16798525 # 17206477 # 17640246 # 32 <lb/>29 # 16420573 # 16805124 # 17213488 # 17647707 # 31 <lb/>30 # 16426798 # 16811729 # 17220507 # 17655174 # 30 <lb/> # 37 # 36 # 35 # 34 <lb/></note> </div> <div xml:id="echoid-div725" type="section" level="1" n="408"> <head xml:id="echoid-head435" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="273" file="285" n="285" rhead=""/> </div> <div xml:id="echoid-div726" type="section" level="1" n="409"> <head xml:id="echoid-head436" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 52 # 53 # 54 # 55 <lb/>30 # 16426798 # 16811729 # 17220507 # 17655175 # 30 <lb/>31 # 16433027 # 16818341 # 17227532 # 17662651 # 29 <lb/>32 # 16439263 # 16824960 # 17234565 # 17670136 # 28 <lb/>33 # 16445505 # 16831585 # 17241605 # 17677627 # 27 <lb/>34 # 16451754 # 16838217 # 17248653 # 17685127 # 26 <lb/>35 # 16458008 # 16844856 # 17255708 # 17692635 # 25 <lb/>36 # 16464269 # 16851502 # 17262770 # 17700151 # 24 <lb/>37 # 16470536 # 16858154 # 17269839 # 17707674 # 23 <lb/>38 # 16476809 # 16864813 # 17276917 # 17715206 # 22 <lb/>39 # 16483089 # 16871479 # 17284002 # 17722744 # 21 <lb/>40 # 16489385 # 16878151 # 17291095 # 17730290 # 20 <lb/>41 # 16495668 # 16884830 # 17298194 # 17737844 # 19 <lb/>42 # 16501967 # 16891515 # 17305300 # 17745407 # 18 <lb/>43 # 16508272 # 16898207 # 17312413 # 17752978 # 17 <lb/>44 # 16514582 # 16904907 # 17319514 # 17760555 # 16 <lb/>45 # 16520898 # 16911613 # 17326662 # 17768142 # 15 <lb/>46 # 16527220 # 16918326 # 17333798 # 17775740 # 14 <lb/>47 # 16533548 # 16925046 # 17340941 # 17783343 # 13 <lb/>48 # 16539883 # 16931772 # 17348091 # 17790955 # 12 <lb/>49 # 16546224 # 16948504 # 17355249 # 17798575 # 11 <lb/>50 # 16552571 # 16945244 # 17362415 # 17806203 # 10 <lb/>51 # 16558925 # 16951990 # 17369587 # 17813838 # 9 <lb/>52 # 16565286 # 16958743 # 17376767 # 17821481 # 8 <lb/>53 # 16571642 # 16965495 # 17383954 # 17829132 # 7 <lb/>54 # 16578026 # 16972270 # 17391148 # 17836792 # 6 <lb/>55 # 16584406 # 16979044 # 17398350 # 17844460 # 5 <lb/>56 # 16590792 # 16985824 # 17405560 # 17852135 # 4 <lb/>57 # 16597184 # 16992611 # 17412776 # 17859818 # 3 <lb/>58 # 16603584 # 16999406 # 17420000 # 17867509 # 2 <lb/>59 # 16609989 # 17006208 # 17427231 # 17875209 # 1 <lb/>60 # 16616401 # 17013017 # 17434469 # 17882917 # 0 <lb/> # 37 # 36 # 35 # 34 <lb/></note> </div> <div xml:id="echoid-div727" type="section" level="1" n="410"> <head xml:id="echoid-head437" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="274" file="286" n="286" rhead=""/> </div> <div xml:id="echoid-div728" type="section" level="1" n="411"> <head xml:id="echoid-head438" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 56 # 57 # 58 # 59 <lb/>0 # 17882917 # 18360816 # 18870800 # 19416039 # 60 <lb/>1 # 17890632 # 18369014 # 18879589 # 19425445 # 59 <lb/>2 # 17898356 # 18377251 # 18888389 # 19434862 # 58 <lb/>3 # 17906089 # 18385497 # 18897196 # 19444290 # 57 <lb/>4 # 17913830 # 18393753 # 18906018 # 19453727 # 56 <lb/>5 # 17921579 # 18402017 # 18914846 # 19463175 # 55 <lb/>6 # 17929337 # 18410291 # 18923685 # 19472635 # 54 <lb/>7 # 17937102 # 18418574 # 18932534 # 19482114 # 53 <lb/>8 # 17944876 # 18426865 # 18941393 # 19491595 # 52 <lb/>9 # 17952658 # 18435165 # 18950261 # 19501076 # 51 <lb/>10 # 17960448 # 18443454 # 18959139 # 19510578 # 50 <lb/>11 # 17968247 # 18451792 # 18968027 # 19520091 # 49 <lb/>12 # 17976054 # 18460120 # 18976926 # 19529615 # 48 <lb/>13 # 17983869 # 18468456 # 18985834 # 19539150 # 47 <lb/>14 # 17991693 # 18476802 # 18994752 # 19548697 # 46 <lb/>15 # 17999525 # 18485157 # 19003680 # 19558254 # 45 <lb/>16 # 18007365 # 18493521 # 19012618 # 19567822 # 44 <lb/>17 # 18015214 # 18501895 # 19021516 # 19577401 # 43 <lb/>18 # 18023071 # 18510278 # 19030523 # 19586991 # 42 <lb/>19 # 18030936 # 18518670 # 19039491 # 19596592 # 41 <lb/>20 # 18038811 # 18527072 # 19048468 # 19606204 # 40 <lb/>21 # 18046693 # 18535483 # 19057455 # 19615827 # 39 <lb/>22 # 18054584 # 18543903 # 19066453 # 19625462 # 38 <lb/>23 # 18062482 # 18552332 # 19075461 # 19635107 # 37 <lb/>24 # 18070389 # 18560770 # 19084480 # 19644765 # 36 <lb/>25 # 18078305 # 18569217 # 19093509 # 19654434 # 35 <lb/>26 # 18086129 # 18577674 # 19102549 # 19664114 # 34 <lb/>27 # 18094161 # 18586139 # 19111598 # 19673805 # 33 <lb/>28 # 18102102 # 18594614 # 19120658 # 19683507 # 32 <lb/>29 # 18110051 # 18603098 # 19129727 # 19693220 # 31 <lb/>30 # 18118009 # 18611591 # 19138807 # 19702945 # 30 <lb/> # 33 # 32 # 31 # 30 <lb/></note> </div> <div xml:id="echoid-div729" type="section" level="1" n="412"> <head xml:id="echoid-head439" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="275" file="287" n="287" rhead=""/> </div> <div xml:id="echoid-div730" type="section" level="1" n="413"> <head xml:id="echoid-head440" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 56 # 57 # 58 # 59 <lb/>30 # 18118009 # 18611591 # 19138807 # 19702945 # 30 <lb/>31 # 18125975 # 18620094 # 19147897 # 19712680 # 29 <lb/>32 # 18133950 # 18629606 # 19156998 # 19722428 # 28 <lb/>33 # 18141934 # 18637127 # 19166109 # 19732186 # 27 <lb/>34 # 18149926 # 18645658 # 19175231 # 19741956 # 26 <lb/>35 # 18157927 # 18654198 # 19184362 # 19751738 # 25 <lb/>36 # 18165937 # 18662748 # 19193504 # 19761531 # 24 <lb/>37 # 18173956 # 18671307 # 19202656 # 19771335 # 23 <lb/>38 # 18181984 # 18679875 # 19211818 # 19781141 # 22 <lb/>39 # 18190021 # 18688452 # 19220990 # 19790968 # 21 <lb/>40 # 18198065 # 18697038 # 19230172 # 19800808 # 20 <lb/>41 # 18206118 # 18705634 # 19239365 # 19810658 # 19 <lb/>42 # 18214179 # 18714239 # 19248569 # 19820320 # 18 <lb/>43 # 18222249 # 18722854 # 19257783 # 19830393 # 17 <lb/>44 # 18230328 # 18731480 # 19267008 # 19840277 # 16 <lb/>45 # 18238416 # 18740115 # 19276242 # 19850172 # 15 <lb/>46 # 18246513 # 18748760 # 19285488 # 19860079 # 14 <lb/>47 # 18254618 # 18757414 # 19294744 # 19869997 # 13 <lb/>48 # 18262732 # 18766078 # 19304010 # 19879927 # 12 <lb/>49 # 18270854 # 18774752 # 19313287 # 19889868 # 11 <lb/>50 # 18278986 # 18783436 # 19322574 # 19899820 # 10 <lb/>51 # 18287126 # 18792130 # 19331872 # 19909784 # 9 <lb/>52 # 18295276 # 18800833 # 19341181 # 19919760 # 8 <lb/>53 # 18303434 # 18809546 # 19350501 # 19929748 # 7 <lb/>54 # 18311601 # 18818268 # 19359831 # 19939749 # 6 <lb/>55 # 18319776 # 18826999 # 19369172 # 19949760 # 5 <lb/>56 # 18327961 # 18835741 # 19378524 # 19959784 # 4 <lb/>57 # 18337154 # 18844492 # 19387886 # 19966820 # 3 <lb/>58 # 18344356 # 18853252 # 19397260 # 19979868 # 2 <lb/>59 # 18352567 # 18862021 # 19406644 # 19989928 # 1 <lb/>60 # 18360816 # 18870800 # 19416039 # 20000000 # 0 <lb/> # 33 # 32 # 31 # 30 <lb/></note> </div> <div xml:id="echoid-div731" type="section" level="1" n="414"> <head xml:id="echoid-head441" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="276" file="288" n="288" rhead=""/> </div> <div xml:id="echoid-div732" type="section" level="1" n="415"> <head xml:id="echoid-head442" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 60 # 61 # 62 # 63 <lb/>0 # 20000000 # 20626654 # 21300545 # 22026892 # 60 <lb/>1 # 20010083 # 20637484 # 21312206 # 22039475 # 59 <lb/>2 # 20020179 # 20648338 # 21323882 # 22052074 # 58 <lb/>3 # 20030285 # 20659184 # 21335570 # 22064690 # 57 <lb/>4 # 20040404 # 20670054 # 21347275 # 22077322 # 56 <lb/>5 # 20050534 # 20680937 # 21358993 # 22089970 # 55 <lb/>6 # 20060676 # 20691834 # 21370727 # 22102635 # 54 <lb/>7 # 20070832 # 20702744 # 21382475 # 22115316 # 53 <lb/>8 # 20080995 # 20713667 # 21394238 # 22128014 # 52 <lb/>9 # 20091172 # 20724603 # 21407016 # 22140728 # 51 <lb/>10 # 20101361 # 20735554 # 21417808 # 22153459 # 50 <lb/>11 # 20111562 # 20746517 # 21429615 # 22166204 # 49 <lb/>12 # 20121776 # 20757494 # 21441438 # 22178971 # 48 <lb/>13 # 20132001 # 20768484 # 21453275 # 22191751 # 47 <lb/>14 # 20142239 # 20779488 # 21465128 # 22204548 # 46 <lb/>15 # 20152489 # 20790505 # 21476995 # 22217361 # 45 <lb/>16 # 20162751 # 20801535 # 21488877 # 22230191 # 44 <lb/>17 # 20173035 # 20812579 # 21500774 # 22243038 # 43 <lb/>18 # 20183321 # 20823636 # 21512686 # 22255902 # 42 <lb/>19 # 20193619 # 20834706 # 21524612 # 22268782 # 41 <lb/>20 # 20203930 # 20845791 # 21536553 # 22281680 # 40 <lb/>21 # 20214252 # 20856888 # 21548509 # 22294595 # 39 <lb/>22 # 20224588 # 20868000 # 21560481 # 22307526 # 38 <lb/>23 # 20234936 # 20879125 # 21572467 # 22320474 # 37 <lb/>24 # 20245296 # 20890264 # 21584469 # 22333439 # 36 <lb/>25 # 20255669 # 20901416 # 21596487 # 22346420 # 35 <lb/>26 # 20266054 # 20912582 # 21608520 # 22359419 # 34 <lb/>27 # 20276452 # 20923761 # 21620568 # 22372434 # 33 <lb/>28 # 20286863 # 20934955 # 21632631 # 22385466 # 32 <lb/>29 # 20297286 # 20946162 # 21644710 # 22398418 # 31 <lb/>30 # 20307721 # 20957383 # 21656804 # 22411584 # 30 <lb/> # 29 # 28 # 27 # 26 <lb/></note> </div> <div xml:id="echoid-div733" type="section" level="1" n="416"> <head xml:id="echoid-head443" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="277" file="289" n="289" rhead=""/> </div> <div xml:id="echoid-div734" type="section" level="1" n="417"> <head xml:id="echoid-head444" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 60 # 61 # 62 # 63 <lb/>30 # 20307721 # 20957383 # 21656804 # 22411584 # 30 <lb/>31 # 20318170 # 20968618 # 21668913 # 22424667 # 29 <lb/>32 # 20328630 # 20979867 # 21681038 # 22437768 # 28 <lb/>33 # 20339102 # 20991130 # 21693178 # 22450886 # 27 <lb/>34 # 20349587 # 21002406 # 21705334 # 22464022 # 26 <lb/>35 # 20360084 # 21013696 # 21717505 # 22477175 # 25 <lb/>36 # 20370594 # 21025001 # 21729691 # 22490346 # 24 <lb/>37 # 20381116 # 21036319 # 21741893 # 22503543 # 23 <lb/>38 # 20391751 # 21047651 # 21754111 # 22516748 # 22 <lb/>39 # 20402198 # 21058997 # 21766344 # 22529965 # 21 <lb/>40 # 20412758 # 21070357 # 21778593 # 22543201 # 20 <lb/>41 # 20423331 # 21081731 # 21790858 # 22556358 # 19 <lb/>42 # 20433916 # 21093119 # 21803138 # 22569723 # 18 <lb/>43 # 20444514 # 21104522 # 21815434 # 22583025 # 17 <lb/>44 # 20455126 # 21115938 # 21827745 # 22596336 # 16 <lb/>45 # 20465750 # 21127368 # 21840072 # 22609663 # 15 <lb/>46 # 20476387 # 21138814 # 21852415 # 22623009 # 14 <lb/>47 # 20487037 # 21150273 # 21864774 # 22636372 # 13 <lb/>48 # 20497700 # 21161747 # 21877149 # 22649754 # 12 <lb/>49 # 20508376 # 21173235 # 21889539 # 22663152 # 11 <lb/>50 # 20519064 # 21184737 # 21901946 # 22676569 # 10 <lb/>51 # 20529765 # 21196253 # 21914369 # 22690004 # 9 <lb/>52 # 20540479 # 21207783 # 21926808 # 22703456 # 8 <lb/>53 # 20551205 # 21219328 # 21939263 # 22716924 # 7 <lb/>54 # 20561945 # 21230887 # 21951734 # 22730414 # 6 <lb/>55 # 20572697 # 21242460 # 21964220 # 22743919 # 5 <lb/>56 # 20583463 # 21254048 # 21976722 # 22757443 # 4 <lb/>57 # 20594242 # 21265650 # 21989240 # 22770984 # 3 <lb/>58 # 20605033 # 21277267 # 22001775 # 22784543 # 2 <lb/>59 # 20615837 # 21288899 # 22014325 # 22798120 # 1 <lb/>60 # 20626654 # 21300545 # 22026892 # 22811726 # 0 <lb/> # 29 # 28 # 27 # 26 <lb/></note> </div> <div xml:id="echoid-div735" type="section" level="1" n="418"> <head xml:id="echoid-head445" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="278" file="290" n="290" rhead=""/> </div> <div xml:id="echoid-div736" type="section" level="1" n="419"> <head xml:id="echoid-head446" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 64 # 65 # 66 # 67 <lb/>0 # 22811726 # 23662013 # 24585936 # 25593051 # 60 <lb/>1 # 22825329 # 23676784 # 24602010 # 25610602 # 59 <lb/>2 # 22838962 # 23691575 # 24618107 # 25628180 # 58 <lb/>3 # 22852612 # 23706387 # 24634227 # 25645783 # 57 <lb/>4 # 22866281 # 23721220 # 24650370 # 25663414 # 56 <lb/>5 # 22879968 # 23736073 # 24666536 # 25681071 # 55 <lb/>6 # 22893674 # 23750947 # 24682727 # 25698754 # 54 <lb/>7 # 22907387 # 23765842 # 24698940 # 25716464 # 53 <lb/>8 # 22921140 # 23780757 # 24715178 # 25734201 # 52 <lb/>9 # 22934901 # 23795692 # 24731439 # 25751965 # 51 <lb/>10 # 22948680 # 23810648 # 24747724 # 25769755 # 50 <lb/>11 # 22962478 # 23825625 # 24764033 # 25787582 # 49 <lb/>12 # 22976294 # 23840623 # 24780365 # 25805417 # 48 <lb/>13 # 22990129 # 23855642 # 24796721 # 25823287 # 47 <lb/>14 # 23003983 # 23870683 # 24813101 # 25841185 # 46 <lb/>15 # 23017855 # 23885844 # 24829504 # 25859104 # 45 <lb/>16 # 23031747 # 23900827 # 24845932 # 25877061 # 44 <lb/>17 # 23045657 # 23915931 # 24862383 # 25895040 # 43 <lb/>18 # 23059586 # 23931055 # 24879958 # 25913046 # 42 <lb/>19 # 23073534 # 23946200 # 24895356 # 25931080 # 41 <lb/>20 # 23087501 # 23961366 # 24911878 # 25949142 # 40 <lb/>21 # 23101486 # 23976553 # 24928423 # 25967230 # 39 <lb/>22 # 23115490 # 23991762 # 24944993 # 25985345 # 38 <lb/>23 # 23129513 # 24006992 # 24961587 # 26003487 # 37 <lb/>24 # 23143556 # 24022245 # 24978205 # 26021658 # 36 <lb/>25 # 23157616 # 24037518 # 24994847 # 26039855 # 35 <lb/>26 # 23171696 # 24052814 # 25011514 # 26058081 # 34 <lb/>27 # 23185795 # 24068130 # 25028205 # 26076333 # 33 <lb/>28 # 23199913 # 24083469 # 25044920 # 26094614 # 32 <lb/>29 # 23214050 # 24098830 # 25061660 # 26112923 # 31 <lb/>30 # 23228205 # 24114213 # 25078426 # 26131259 # 30 <lb/> # 25 # 24 # 23 # 22 <lb/></note> </div> <div xml:id="echoid-div737" type="section" level="1" n="420"> <head xml:id="echoid-head447" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="279" file="291" n="291" rhead=""/> </div> <div xml:id="echoid-div738" type="section" level="1" n="421"> <head xml:id="echoid-head448" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 64 # 65 # 66 # 67 <lb/>30 # 23228205 # 24114213 # 25078426 # 26131259 # 30 <lb/>31 # 23242380 # 24129616 # 25095216 # 26149623 # 29 <lb/>32 # 23256574 # 24145041 # 25112030 # 26168015 # 28 <lb/>33 # 23270797 # 24160487 # 25128869 # 26186436 # 27 <lb/>34 # 23285021 # 24175956 # 25145732 # 26204884 # 26 <lb/>35 # 23299273 # 24191445 # 25162620 # 26223361 # 25 <lb/>36 # 23313546 # 24206956 # 25179532 # 26241867 # 24 <lb/>37 # 23327838 # 24222488 # 25196469 # 26260400 # 23 <lb/>38 # 23342150 # 24238043 # 25213432 # 26278963 # 22 <lb/>39 # 23356481 # 24253619 # 25230418 # 26297555 # 21 <lb/>40 # 23370832 # 24269217 # 25247431 # 26316176 # 20 <lb/>41 # 23385203 # 24284838 # 25264468 # 26334825 # 19 <lb/>42 # 23399593 # 24300481 # 25281531 # 26353503 # 18 <lb/>43 # 23414003 # 24316147 # 25298620 # 26372209 # 17 <lb/>44 # 23428433 # 24331835 # 25315734 # 26390945 # 16 <lb/>45 # 23442882 # 24347546 # 25332874 # 26409709 # 15 <lb/>46 # 23457351 # 24363281 # 25350039 # 26428502 # 14 <lb/>47 # 23471840 # 24379038 # 25367229 # 26447323 # 13 <lb/>48 # 23486348 # 24394818 # 25384445 # 26466174 # 12 <lb/>49 # 23500876 # 24410620 # 25401687 # 26485053 # 11 <lb/>50 # 23515424 # 24426446 # 25418956 # 26503962 # 10 <lb/>51 # 23529992 # 24442294 # 25436250 # 26522890 # 9 <lb/>52 # 23544580 # 24458164 # 25453570 # 26541867 # 8 <lb/>53 # 23559188 # 24474056 # 25470915 # 26560863 # 7 <lb/>54 # 23573817 # 24489973 # 25488286 # 26579889 # 6 <lb/>55 # 23588565 # 24505908 # 25505683 # 26598945 # 5 <lb/>56 # 23603134 # 24521869 # 25523005 # 26618030 # 4 <lb/>57 # 23617822 # 24537851 # 25540553 # 26637145 # 3 <lb/>58 # 23632532 # 24553857 # 25558027 # 26656291 # 2 <lb/>59 # 23647262 # 24569885 # 25575526 # 26675466 # 1 <lb/>60 # 23662013 # 24585936 # 25593051 # 26694672 # 0 <lb/> # 25 # 24 # 23 # 22 <lb/></note> </div> <div xml:id="echoid-div739" type="section" level="1" n="422"> <head xml:id="echoid-head449" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="280" file="292" n="292" rhead=""/> </div> <div xml:id="echoid-div740" type="section" level="1" n="423"> <head xml:id="echoid-head450" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 68 # 69 # 70 # 71 <lb/>0 # 26694672 # 27904284 # 29238045 # 30715531 # 60 <lb/>1 # 26713907 # 27925445 # 29261433 # 30741500 # 59 <lb/>2 # 26733172 # 27946642 # 29284861 # 30767516 # 58 <lb/>3 # 26752467 # 27967873 # 29308328 # 30793579 # 57 <lb/>4 # 26771791 # 27989139 # 29331835 # 30819689 # 56 <lb/>5 # 26791145 # 28010440 # 29355382 # 30845846 # 55 <lb/>6 # 26810529 # 28031776 # 29378970 # 30872051 # 54 <lb/>7 # 26829942 # 28053147 # 29402599 # 30898304 # 53 <lb/>8 # 26849390 # 28074553 # 29426268 # 30924605 # 52 <lb/>9 # 26868867 # 28095994 # 29449978 # 30950953 # 51 <lb/>10 # 26888373 # 28117469 # 29473728 # 30977350 # 50 <lb/>11 # 26907910 # 28138980 # 29497519 # 31003793 # 49 <lb/>12 # 26927479 # 28160527 # 29521350 # 31030285 # 48 <lb/>13 # 26947078 # 28182108 # 29545222 # 31056824 # 47 <lb/>14 # 26966709 # 28203725 # 29569136 # 31083412 # 46 <lb/>15 # 26986370 # 28225378 # 29593090 # 31110047 # 45 <lb/>16 # 27006062 # 28247067 # 29617087 # 31136731 # 44 <lb/>17 # 27025785 # 28268793 # 29641124 # 31163462 # 43 <lb/>18 # 27045539 # 28290553 # 29665204 # 31190241 # 42 <lb/>19 # 27065323 # 28312349 # 29689326 # 31217019 # 41 <lb/>20 # 27085138 # 28334181 # 29713488 # 31243945 # 40 <lb/>21 # 27104985 # 28356049 # 29737692 # 31270871 # 39 <lb/>22 # 27124864 # 28377954 # 29761938 # 31297848 # 38 <lb/>23 # 27144774 # 28399894 # 29786227 # 31324873 # 37 <lb/>24 # 27164717 # 28421871 # 29810558 # 31351948 # 36 <lb/>25 # 27184690 # 28443884 # 29834931 # 31379072 # 35 <lb/>26 # 27204686 # 28465934 # 29859347 # 31406247 # 34 <lb/>27 # 27224734 # 28488021 # 29883705 # 31433472 # 33 <lb/>28 # 27244804 # 28510144 # 29908306 # 31460747 # 32 <lb/>29 # 27264906 # 28532304 # 29932850 # 31488072 # 31 <lb/>30 # 27285040 # 28554501 # 29957438 # 31515448 # 30 <lb/> # 21 # 20 # 19 # 18 <lb/></note> </div> <div xml:id="echoid-div741" type="section" level="1" n="424"> <head xml:id="echoid-head451" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="281" file="293" n="293" rhead=""/> </div> <div xml:id="echoid-div742" type="section" level="1" n="425"> <head xml:id="echoid-head452" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 68 # 69 # 70 # 71 <lb/>30 # 27285040 # 28554501 # 29957438 # 31515448 # 30 <lb/>31 # 27305205 # 28576735 # 29982069 # 31542873 # 29 <lb/>32 # 27325402 # 28599007 # 30006743 # 31570349 # 28 <lb/>33 # 27345631 # 28621316 # 30031460 # 31597875 # 27 <lb/>34 # 27365893 # 28643662 # 30056220 # 31625453 # 26 <lb/>35 # 27386186 # 28666045 # 30081023 # 31653080 # 25 <lb/>36 # 27406513 # 28688467 # 30105870 # 31680758 # 24 <lb/>37 # 27426872 # 28710925 # 30130760 # 31708486 # 23 <lb/>38 # 27447264 # 28733422 # 30155714 # 31736265 # 22 <lb/>39 # 27467688 # 28755956 # 30180672 # 31764094 # 21 <lb/>40 # 27488145 # 28778549 # 30205694 # 31791974 # 20 <lb/>41 # 27508635 # 28801139 # 30230760 # 31819906 # 19 <lb/>42 # 27529157 # 28823787 # 30255871 # 31847891 # 18 <lb/>43 # 27549722 # 28846473 # 30281026 # 31875929 # 17 <lb/>44 # 27570301 # 28869196 # 30306226 # 31904019 # 16 <lb/>45 # 27590922 # 28891957 # 30331460 # 31932164 # 15 <lb/>46 # 27611578 # 28914756 # 30356759 # 31960358 # 14 <lb/>47 # 27632266 # 28937594 # 30382092 # 31988606 # 13 <lb/>48 # 27652989 # 28960471 # 30407470 # 32016909 # 12 <lb/>49 # 27673745 # 28983386 # 30432893 # 32045263 # 11 <lb/>50 # 27694535 # 29006340 # 30458361 # 32073672 # 10 <lb/>51 # 27715358 # 29029332 # 30483873 # 32102132 # 9 <lb/>52 # 27736215 # 29052363 # 30509430 # 32130649 # 8 <lb/>53 # 27757105 # 29075435 # 30535033 # 32159212 # 7 <lb/>54 # 27778029 # 29098546 # 30560682 # 32187832 # 6 <lb/>55 # 27798987 # 29121697 # 30586375 # 32216504 # 5 <lb/>56 # 27819978 # 29144888 # 30612115 # 32245231 # 4 <lb/>57 # 27841003 # 29168118 # 30637890 # 32274012 # 3 <lb/>58 # 27862060 # 29191388 # 30663732 # 32302846 # 2 <lb/>59 # 27883156 # 29214697 # 30689608 # 32331735 # 1 <lb/>60 # 27904284 # 29238045 # 30715531 # 32360678 # 0 <lb/> # 21 # 20 # 19 # 18 <lb/></note> </div> <div xml:id="echoid-div743" type="section" level="1" n="426"> <head xml:id="echoid-head453" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="282" file="294" n="294" rhead=""/> </div> <div xml:id="echoid-div744" type="section" level="1" n="427"> <head xml:id="echoid-head454" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 72 # 73 # 74 # 75 <lb/>0 # 32360678 # 34203038 # 36279559 # 38637042 # 60 <lb/>1 # 32389676 # 34235609 # 36316402 # 38679033 # 59 <lb/>2 # 32418726 # 34268245 # 36353333 # 38721117 # 58 <lb/>3 # 32447837 # 34300947 # 36390323 # 38763296 # 57 <lb/>4 # 32477001 # 34333716 # 36427401 # 38805571 # 56 <lb/>5 # 32506219 # 34366553 # 36464558 # 38847941 # 55 <lb/>6 # 32535494 # 34399452 # 36501793 # 38890408 # 54 <lb/>7 # 32564823 # 34432420 # 36539107 # 38932971 # 53 <lb/>8 # 32594209 # 34465456 # 36570511 # 38975632 # 52 <lb/>9 # 32623651 # 34498557 # 36613973 # 39018390 # 51 <lb/>10 # 32653148 # 34531726 # 36651525 # 39061246 # 50 <lb/>11 # 32682701 # 34564959 # 36689156 # 39104200 # 49 <lb/>12 # 32712311 # 34598259 # 36726868 # 39147252 # 48 <lb/>13 # 32741977 # 34631626 # 36764660 # 39190423 # 47 <lb/>14 # 32771699 # 34665061 # 36802533 # 39233653 # 46 <lb/>15 # 32801478 # 34698564 # 36840488 # 39277002 # 45 <lb/>16 # 32831314 # 34732135 # 36878524 # 39320449 # 44 <lb/>17 # 32861207 # 34765775 # 36916641 # 39363994 # 43 <lb/>18 # 32891157 # 34799483 # 36954842 # 39407640 # 42 <lb/>19 # 32921165 # 34833259 # 36993127 # 39451384 # 41 <lb/>20 # 32951231 # 34867105 # 37031496 # 39495228 # 40 <lb/>21 # 32981355 # 34901024 # 37069947 # 39539172 # 39 <lb/>22 # 33011537 # 34935005 # 37108482 # 39583218 # 38 <lb/>23 # 33041776 # 34966052 # 37147101 # 39627364 # 37 <lb/>24 # 33072074 # 35003172 # 37185803 # 39671613 # 36 <lb/>25 # 33102431 # 35037361 # 37224589 # 39715965 # 35 <lb/>26 # 33131846 # 35071621 # 37263459 # 39760420 # 34 <lb/>27 # 33163320 # 35105952 # 37302413 # 39804979 # 33 <lb/>28 # 33193853 # 35140354 # 37341453 # 39849642 # 32 <lb/>29 # 33224444 # 35174826 # 37380577 # 39894411 # 31 <lb/>30 # 33255094 # 35209369 # 37419788 # 39939286 # 30 <lb/> # 17 # 16 # 15 # 14 <lb/></note> </div> <div xml:id="echoid-div745" type="section" level="1" n="428"> <head xml:id="echoid-head455" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="283" file="295" n="295" rhead=""/> </div> <div xml:id="echoid-div746" type="section" level="1" n="429"> <head xml:id="echoid-head456" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 72 # 73 # 74 # 75 <lb/>30 # 33255094 # 35209369 # 37419788 # 39939286 # 30 <lb/>31 # 33285803 # 35243981 # 37459081 # 39984263 # 29 <lb/>32 # 33316571 # 35278664 # 37498460 # 40029344 # 28 <lb/>33 # 33347398 # 35313418 # 37537923 # 40074528 # 27 <lb/>34 # 33378286 # 35348244 # 37577471 # 40119816 # 26 <lb/>35 # 33409132 # 35383140 # 37617104 # 40165289 # 25 <lb/>36 # 33440240 # 35418110 # 37656824 # 40210709 # 24 <lb/>37 # 33471307 # 35453152 # 37696632 # 40256316 # 23 <lb/>38 # 33502436 # 35488268 # 37736518 # 40302033 # 22 <lb/>39 # 33533625 # 35523456 # 37776513 # 40347858 # 21 <lb/>40 # 33564875 # 35558718 # 37816588 # 40393792 # 20 <lb/>41 # 33596187 # 35594052 # 37856751 # 40439834 # 19 <lb/>42 # 33627561 # 35629460 # 37897004 # 40485985 # 18 <lb/>43 # 33658998 # 35664940 # 37937146 # 40532245 # 17 <lb/>44 # 33690497 # 35700494 # 37977779 # 40578613 # 16 <lb/>45 # 33722059 # 35736121 # 38018300 # 40625091 # 15 <lb/>46 # 33753683 # 35771822 # 38058912 # 40671678 # 14 <lb/>47 # 33785370 # 35807597 # 38099614 # 40718374 # 13 <lb/>48 # 33817120 # 35843447 # 38140406 # 40765180 # 12 <lb/>49 # 33848934 # 35879373 # 38181288 # 40812093 # 11 <lb/>50 # 33880813 # 35915374 # 38222261 # 40859121 # 10 <lb/>51 # 33912753 # 35951451 # 38263324 # 40906259 # 9 <lb/>52 # 33944756 # 35987602 # 38304479 # 40953510 # 8 <lb/>53 # 33976821 # 36023829 # 38345725 # 41004876 # 7 <lb/>54 # 34008950 # 36060132 # 38387064 # 41048358 # 6 <lb/>55 # 34041141 # 36096510 # 38428495 # 41095957 # 5 <lb/>56 # 34073395 # 36132966 # 38470019 # 41143668 # 4 <lb/>57 # 34105712 # 36169497 # 38511635 # 41191492 # 3 <lb/>58 # 34138091 # 36206107 # 38553344 # 41239431 # 2 <lb/>59 # 34170523 # 36242794 # 38595146 # 41287425 # 1 <lb/>60 # 34203038 # 36279559 # 38637042 # 41335654 # 0 <lb/> # 17 # 16 # 15 # 14 <lb/></note> </div> <div xml:id="echoid-div747" type="section" level="1" n="430"> <head xml:id="echoid-head457" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="284" file="296" n="296" rhead=""/> </div> <div xml:id="echoid-div748" type="section" level="1" n="431"> <head xml:id="echoid-head458" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 76 # 77 # 78 # 79 <lb/>0 # 41335654 # 44454097 # 48097335 # 52408433 # 60 <lb/>1 # 41383937 # 44510183 # 48163151 # 52486983 # 59 <lb/>2 # 41432338 # 44566415 # 48229350 # 52565774 # 58 <lb/>3 # 41480856 # 44622793 # 48295633 # 52644807 # 57 <lb/>4 # 41529492 # 44679318 # 48362102 # 52724084 # 56 <lb/>5 # 41578245 # 44735990 # 48428756 # 52803604 # 55 <lb/>6 # 41627117 # 44792810 # 48495599 # 52883368 # 54 <lb/>7 # 41676108 # 44849777 # 48562631 # 52963377 # 53 <lb/>8 # 41725219 # 44906892 # 48629854 # 53043632 # 52 <lb/>9 # 41774450 # 44964155 # 48697269 # 53124134 # 51 <lb/>10 # 41823802 # 45021567 # 48764877 # 53204885 # 50 <lb/>11 # 41873273 # 45079129 # 48832678 # 53285884 # 49 <lb/>12 # 41922863 # 45136843 # 48900673 # 53367134 # 48 <lb/>13 # 41972573 # 45194707 # 48968853 # 53448635 # 47 <lb/>14 # 42022405 # 45252726 # 49037249 # 53530390 # 46 <lb/>15 # 42072357 # 45310898 # 49105830 # 53612399 # 45 <lb/>16 # 42122431 # 45369224 # 49174607 # 53694666 # 44 <lb/>17 # 42172625 # 45427703 # 49243590 # 53777191 # 43 <lb/>18 # 42222942 # 45486338 # 49312751 # 53859976 # 42 <lb/>19 # 42273380 # 45545127 # 49382118 # 53943022 # 41 <lb/>20 # 42323942 # 45604073 # 49451684 # 54026331 # 40 <lb/>21 # 42374627 # 45663175 # 49521449 # 54109903 # 39 <lb/>22 # 42425439 # 45722435 # 49591416 # 54193739 # 38 <lb/>23 # 42476377 # 45781853 # 49661584 # 54277840 # 37 <lb/>24 # 42527442 # 45841429 # 49731956 # 54362207 # 36 <lb/>25 # 42578635 # 45901164 # 49802532 # 54446842 # 35 <lb/>26 # 42629957 # 45961059 # 49873313 # 54531744 # 34 <lb/>27 # 42681409 # 46021115 # 49944301 # 54616915 # 33 <lb/>28 # 42732991 # 46081333 # 50015497 # 54702356 # 32 <lb/>29 # 42784705 # 46141715 # 50086901 # 54788068 # 31 <lb/>30 # 42836551 # 46202261 # 50158514 # 54874053 # 30 <lb/> # 13 # 12 # 11 # 10 <lb/></note> </div> <div xml:id="echoid-div749" type="section" level="1" n="432"> <head xml:id="echoid-head459" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="285" file="297" n="297" rhead=""/> </div> <div xml:id="echoid-div750" type="section" level="1" n="433"> <head xml:id="echoid-head460" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 76 # 77 # 78 # 79 <lb/>30 # 42836551 # 46202261 # 50158514 # 54874053 # 30 <lb/>31 # 42888527 # 46262969 # 50230335 # 54960312 # 29 <lb/>32 # 42940631 # 46323841 # 50302367 # 55046847 # 28 <lb/>33 # 42992865 # 46384877 # 50374610 # 55133659 # 27 <lb/>34 # 43045229 # 46446076 # 50447065 # 55220751 # 26 <lb/>35 # 43097722 # 46507440 # 50519732 # 55308122 # 25 <lb/>36 # 43150347 # 46568970 # 50592614 # 55395775 # 24 <lb/>37 # 43203103 # 46630665 # 50665711 # 55483710 # 23 <lb/>38 # 43255992 # 46692527 # 50739024 # 55571930 # 22 <lb/>39 # 43309012 # 46754555 # 50812553 # 55660434 # 21 <lb/>40 # 43362166 # 46816752 # 50886299 # 55749226 # 20 <lb/>41 # 43415454 # 46879117 # 50960263 # 55838300 # 19 <lb/>42 # 43468877 # 46941653 # 51034447 # 55927677 # 18 <lb/>43 # 43522435 # 47004361 # 51108850 # 56017340 # 17 <lb/>44 # 43576129 # 47067242 # 51183475 # 56107297 # 16 <lb/>45 # 43629959 # 47130297 # 51258321 # 56197549 # 15 <lb/>46 # 43683925 # 47193526 # 51333391 # 56288099 # 14 <lb/>47 # 43738728 # 47256930 # 51408684 # 56378948 # 13 <lb/>48 # 43792268 # 47320509 # 51484204 # 56470097 # 12 <lb/>49 # 43846646 # 47384264 # 51559951 # 56561548 # 11 <lb/>50 # 43901162 # 47448195 # 51635936 # 56653302 # 10 <lb/>51 # 43955817 # 47512302 # 51712129 # 56745360 # 9 <lb/>52 # 44000612 # 47576586 # 51788563 # 56837723 # 8 <lb/>53 # 44065548 # 47641048 # 51865227 # 56930392 # 7 <lb/>54 # 44120625 # 47705689 # 51942124 # 57023369 # 6 <lb/>55 # 44175844 # 47770510 # 52019254 # 57116653 # 5 <lb/>56 # 44231207 # 47835511 # 52096618 # 57210246 # 4 <lb/>57 # 44286712 # 47900693 # 52174216 # 57304150 # 3 <lb/>58 # 44342362 # 47966058 # 52252051 # 57398367 # 2 <lb/>59 # 44398156 # 48031605 # 52330123 # 57492896 # 1 <lb/>60 # 44454097 # 48097335 # 52408433 # 57587740 # 0 <lb/> # 13 # 12 # 11 # 10 <lb/></note> </div> <div xml:id="echoid-div751" type="section" level="1" n="434"> <head xml:id="echoid-head461" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="286" file="298" n="298" rhead=""/> </div> <div xml:id="echoid-div752" type="section" level="1" n="435"> <head xml:id="echoid-head462" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 80 # 81 # 82 # 83 <lb/>0 # 57587740 # 63924495 # 71852975 # 82055127 # 60 <lb/>1 # 57682901 # 64042118 # 72002006 # 82249986 # 59 <lb/>2 # 57778381 # 64160180 # 72151659 # 82445779 # 58 <lb/>3 # 57874180 # 64278683 # 72301942 # 82642513 # 57 <lb/>4 # 57970302 # 64397632 # 72452863 # 82840196 # 56 <lb/>5 # 58066748 # 64517028 # 72604421 # 83038833 # 55 <lb/>6 # 58163520 # 64636873 # 72756618 # 83238436 # 54 <lb/>7 # 58260619 # 64757168 # 72909461 # 83439009 # 53 <lb/>8 # 58358049 # 64877918 # 73062954 # 83640561 # 52 <lb/>9 # 58455810 # 64999124 # 73217100 # 83843097 # 51 <lb/>10 # 58553904 # 65120789 # 73371903 # 84046626 # 50 <lb/>11 # 58652333 # 65242916 # 73527367 # 84251153 # 49 <lb/>12 # 58751099 # 65365508 # 73683499 # 84456680 # 48 <lb/>13 # 58850205 # 65488566 # 73840302 # 84663213 # 47 <lb/>14 # 58949653 # 65612095 # 73997782 # 84870760 # 46 <lb/>15 # 59049444 # 65736097 # 74155942 # 85079327 # 45 <lb/>16 # 59149581 # 65859675 # 74314786 # 85288957 # 44 <lb/>17 # 59250065 # 65985531 # 74474318 # 85499628 # 43 <lb/>18 # 59350898 # 66110967 # 74634544 # 85711347 # 42 <lb/>19 # 59452082 # 66246886 # 74795468 # 85924121 # 41 <lb/>20 # 59553618 # 66363291 # 74957095 # 86137958 # 40 <lb/>21 # 59655506 # 66490185 # 75119429 # 86352864 # 39 <lb/>22 # 59757728 # 66617572 # 75282475 # 86568849 # 38 <lb/>23 # 59860346 # 66745453 # 75446238 # 86785921 # 37 <lb/>24 # 59963291 # 66873831 # 75610721 # 87004089 # 36 <lb/>25 # 60066612 # 67002708 # 75775928 # 87223362 # 35 <lb/>26 # 60170285 # 67132088 # 75941864 # 87443750 # 34 <lb/>27 # 60274319 # 67261972 # 76108533 # 87665261 # 33 <lb/>28 # 60378718 # 67392365 # 76275941 # 87887909 # 32 <lb/>29 # 60483482 # 67523270 # 76444091 # 88111704 # 31 <lb/>30 # 60588615 # 67654691 # 76612989 # 88336657 # 30 <lb/> # 9 # 8 # 7 # 6 <lb/></note> </div> <div xml:id="echoid-div753" type="section" level="1" n="436"> <head xml:id="echoid-head463" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="287" file="299" n="299" rhead=""/> </div> <div xml:id="echoid-div754" type="section" level="1" n="437"> <head xml:id="echoid-head464" xml:space="preserve">arcuum eiuſdem Quadrantis.</head> <note position="right" xml:space="preserve"> <lb/> # 80 # 81 # 82 # 83 <lb/>30 # 60588615 # 67654691 # 76612989 # 88336657 # 30 <lb/>31 # 60694118 # 67786629 # 76782641 # 88562776 # 29 <lb/>32 # 60799995 # 67919089 # 76953050 # 88790069 # 28 <lb/>33 # 60906246 # 68052073 # 77124223 # 89018543 # 27 <lb/>34 # 61012875 # 68185585 # 77296165 # 89248201 # 26 <lb/>35 # 61119882 # 68319630 # 77468882 # 89479054 # 25 <lb/>36 # 61227271 # 68454208 # 77642381 # 89711108 # 24 <lb/>37 # 61335043 # 68589313 # 77816665 # 89944373 # 23 <lb/>38 # 61443202 # 68724977 # 77991740 # 90178856 # 22 <lb/>39 # 61551749 # 68861175 # 78167612 # 90414568 # 21 <lb/>40 # 61660686 # 68997920 # 78344287 # 90651519 # 20 <lb/>41 # 61770013 # 69135315 # 78521769 # 90889717 # 19 <lb/>42 # 61879735 # 69273018 # 78700066 # 91129181 # 18 <lb/>43 # 61989853 # 69411469 # 78879183 # 91369917 # 17 <lb/>44 # 62100367 # 69550434 # 79059128 # 91611941 # 16 <lb/>45 # 62211280 # 69689963 # 79239905 # 91855265 # 15 <lb/>46 # 62322594 # 69830059 # 79421520 # 92099899 # 14 <lb/>47 # 62434312 # 69970726 # 79603976 # 92345849 # 13 <lb/>48 # 62546437 # 70111967 # 79787381 # 92593126 # 12 <lb/>49 # 62658971 # 70253786 # 79971439 # 92841739 # 11 <lb/>50 # 62771918 # 70396188 # 80156456 # 93091699 # 10 <lb/>51 # 62885274 # 70539174 # 80342336 # 93342963 # 9 <lb/>52 # 62999049 # 70682751 # 80529087 # 93595620 # 8 <lb/>53 # 63113241 # 70826919 # 80716713 # 93849647 # 7 <lb/>54 # 63227855 # 70971684 # 80905219 # 94105066 # 6 <lb/>55 # 63342890 # 71117047 # 81094612 # 94361964 # 5 <lb/>56 # 63458352 # 71263014 # 81284899 # 94620181 # 4 <lb/>57 # 63574240 # 71409586 # 81476087 # 94879901 # 3 <lb/>58 # 63690559 # 71556760 # 81668183 # 95141050 # 2 <lb/>59 # 63807309 # 71704564 # 81861195 # 95403639 # 1 <lb/>60 # 63924495 # 71852975 # 82055127 # 95667689 # 0 <lb/> # 9 # 8 # 7 # 6 <lb/></note> </div> <div xml:id="echoid-div755" type="section" level="1" n="438"> <head xml:id="echoid-head465" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="288" file="300" n="300" rhead=""/> </div> <div xml:id="echoid-div756" type="section" level="1" n="439"> <head xml:id="echoid-head466" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 84 # 85 # 86 <lb/>0 # 95667689 # 114737188 # 143355808 # 60 <lb/>1 # 95933204 # 115119970 # 143954694 # 59 <lb/>2 # 96200195 # 115505313 # 144558602 # 58 <lb/>3 # 96468673 # 115893242 # 145167595 # 57 <lb/>4 # 96738655 # 116283797 # 145781740 # 56 <lb/>5 # 97010253 # 116676991 # 146401101 # 55 <lb/>6 # 97283267 # 117072851 # 147025745 # 54 <lb/>7 # 97557932 # 117471403 # 147655740 # 53 <lb/>8 # 97834057 # 117872815 # 148291169 # 52 <lb/>9 # 98111843 # 118276840 # 148932108 # 51 <lb/>10 # 98391211 # 118683794 # 149578791 # 50 <lb/>11 # 98672171 # 119093414 # 150230942 # 49 <lb/>12 # 98954738 # 119506013 # 150888966 # 48 <lb/>13 # 99236930 # 119921335 # 151552578 # 47 <lb/>14 # 99524766 # 120339695 # 152222283 # 46 <lb/>15 # 99812250 # 120760985 # 152897946 # 45 <lb/>16 # 100101400 # 121185232 # 153579394 # 44 <lb/>17 # 100392329 # 121612482 # 154267179 # 43 <lb/>18 # 100684851 # 122042752 # 154961155 # 42 <lb/>19 # 100979193 # 122476076 # 155661396 # 41 <lb/>20 # 101275259 # 122912485 # 156368008 # 40 <lb/>21 # 101572962 # 123352014 # 157081063 # 39 <lb/>22 # 101872522 # 123794696 # 157800648 # 38 <lb/>23 # 102173854 # 124240732 # 158526854 # 37 <lb/>24 # 102476971 # 124689836 # 159259771 # 36 <lb/>25 # 102781890 # 125142353 # 159999560 # 35 <lb/>26 # 103088639 # 125598007 # 160746121 # 34 <lb/>27 # 103397202 # 126057149 # 161499724 # 33 <lb/>28 # 103707656 # 126519656 # 162260744 # 32 <lb/>29 # 104019959 # 126985568 # 163028671 # 31 <lb/>30 # 104334254 # 127454936 # 163804188 # 30 <lb/> # 5 # 4 # 3 <lb/></note> </div> <div xml:id="echoid-div757" type="section" level="1" n="440"> <head xml:id="echoid-head467" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="289" file="301" n="301" rhead=""/> </div> <div xml:id="echoid-div758" type="section" level="1" n="441"> <head xml:id="echoid-head468" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 84 # 85 # 86 <lb/>30 # 104334254 # 127454936 # 163804188 # 30 <lb/>31 # 104650345 # 127927785 # 164586836 # 29 <lb/>32 # 104968474 # 128404152 # 165377268 # 28 <lb/>33 # 105288542 # 128884078 # 166175067 # 27 <lb/>34 # 105610566 # 129367604 # 166980877 # 26 <lb/>35 # 105934564 # 129854921 # 167794536 # 25 <lb/>36 # 106260557 # 130345812 # 168615879 # 24 <lb/>37 # 106588558 # 130840395 # 169445585 # 23 <lb/>38 # 106918589 # 131338917 # 170283495 # 22 <lb/>39 # 107250680 # 131841076 # 171129820 # 21 <lb/>40 # 107584955 # 132347264 # 171984431 # 20 <lb/>41 # 107921201 # 132857174 # 172847712 # 19 <lb/>42 # 108259554 # 133371390 # 173719700 # 18 <lb/>43 # 108600151 # 133889600 # 174600528 # 17 <lb/>44 # 108942779 # 134411312 # 175490331 # 16 <lb/>45 # 109287702 # 134937471 # 176389247 # 15 <lb/>46 # 109634817 # 135467749 # 177297417 # 14 <lb/>47 # 109984143 # 136002235 # 178215000 # 13 <lb/>48 # 110335695 # 136540955 # 179142131 # 12 <lb/>49 # 110689503 # 137083887 # 180078954 # 11 <lb/>50 # 111045597 # 137631223 # 181025951 # 10 <lb/>51 # 111403988 # 138183016 # 181982628 # 9 <lb/>52 # 111764699 # 138739177 # 182949802 # 8 <lb/>53 # 112127750 # 139299830 # 183926988 # 7 <lb/>54 # 112493167 # 139865032 # 184915009 # 6 <lb/>55 # 112861097 # 140435034 # 185913698 # 5 <lb/>56 # 113231316 # 141009514 # 186922883 # 4 <lb/>57 # 113604036 # 141588910 # 187943432 # 3 <lb/>58 # 113979204 # 142172885 # 188975184 # 2 <lb/>59 # 114356941 # 142761897 # 190018342 # 1 <lb/>60 # 114737188 # 143355808 # 191073059 # 0 <lb/> # 5 # 4 # 3 <lb/></note> </div> <div xml:id="echoid-div759" type="section" level="1" n="442"> <head xml:id="echoid-head469" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="290" file="302" n="302" rhead=""/> </div> <div xml:id="echoid-div760" type="section" level="1" n="443"> <head xml:id="echoid-head470" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <note position="right" xml:space="preserve"> <lb/> # 87 # 88 # 89 <lb/>0 # 191073059 # 286537048 # 572987098 # 60 <lb/>1 # 192139567 # 288943841 # 582696234 # 59 <lb/>2 # 193218044 # 291391404 # 592740072 # 58 <lb/>3 # 194308693 # 293880683 # 603139919 # 57 <lb/>4 # 195411723 # 296413087 # 613907444 # 56 <lb/>5 # 196527729 # 298990299 # 625070305 # 55 <lb/>6 # 197656182 # 301611807 # 636642580 # 54 <lb/>7 # 198797665 # 304279687 # 648655621 # 53 <lb/>8 # 199952408 # 306996123 # 661126359 # 52 <lb/>9 # 201120639 # 309760533 # 674090521 # 51 <lb/>10 # 202303011 # 312576192 # 687573461 # 50 <lb/>11 # 203498943 # 315442491 # 701612741 # 49 <lb/>12 # 204709121 # 318361849 # 716229489 # 48 <lb/>13 # 205934200 # 321336774 # 731453951 # 47 <lb/>14 # 207173596 # 324366765 # 747356168 # 46 <lb/>15 # 208428431 # 327455509 # 763965262 # 45 <lb/>16 # 209698119 # 330602545 # 781323254 # 44 <lb/>17 # 210983811 # 333811800 # 799494739 # 43 <lb/>18 # 212284914 # 337082830 # 818524878 # 42 <lb/>19 # 213602421 # 340419652 # 838490069 # 41 <lb/>20 # 214936837 # 343823403 # 859453551 # 40 <lb/>21 # 216287319 # 347294586 # 881484374 # 39 <lb/>22 # 217655350 # 350837799 # 904682629 # 38 <lb/>23 # 219040792 # 354454051 # 929134899 # 37 <lb/>24 # 220443981 # 358145679 # 954945691 # 36 <lb/>25 # 221865261 # 361914968 # 982231457 # 35 <lb/>26 # 223305005 # 365763113 # 1011112129 # 34 <lb/>27 # 224763453 # 369695332 # 1041753449 # 33 <lb/>28 # 226241278 # 373713015 # 1074309940 # 32 <lb/>29 # 227738558 # 377818975 # 1108967170 # 31 <lb/>30 # 229255785 # 382016194 # 1145934768 # 30 <lb/> # 2 # 1 # 0 <lb/></note> </div> <div xml:id="echoid-div761" type="section" level="1" n="444"> <head xml:id="echoid-head471" xml:space="preserve">Gradus Quadrantis pro ſecantibus</head> <pb o="291" file="303" n="303" rhead=""/> </div> <div xml:id="echoid-div762" type="section" level="1" n="445"> <head xml:id="echoid-head472" xml:space="preserve">arcuum eiuſdem Quadrantis</head> <note position="right" xml:space="preserve"> <lb/> # 87 # 88 # 89 <lb/>30 # 229255785 # 382016194 # 1145934768 # 30 <lb/>31 # 230793360 # 386307709 # 1185438054 # 29 <lb/>32 # 232351718 # 390696734 # 1227777193 # 28 <lb/>33 # 233931261 # 395186630 # 1273252703 # 27 <lb/>34 # 235532422 # 399780916 # 1322226495 # 26 <lb/>35 # 237156211 # 404483275 # 1375118522 # 25 <lb/>36 # 238801972 # 409397566 # 1432397932 # 24 <lb/>37 # 240470730 # 414227875 # 1494678912 # 23 <lb/>38 # 242163582 # 419278406 # 1562622042 # 22 <lb/>39 # 243879838 # 424453607 # 1637036239 # 21 <lb/>40 # 245621193 # 429758156 # 1718892212 # 20 <lb/>41 # 247386980 # 435196961 # 1809365043 # 19 <lb/>42 # 249178956 # 440775230 # 1909891150 # 18 <lb/>43 # 250996450 # 446498305 # 2022234532 # 17 <lb/>44 # 252841285 # 452371994 # 2148642981 # 16 <lb/>45 # 254713463 # 458402271 # 2291895669 # 15 <lb/>46 # 256612911 # 464595485 # 2455554199 # 14 <lb/>47 # 258541565 # 470958329 # 2644450861 # 13 <lb/>48 # 260499426 # 477497828 # 2864894681 # 12 <lb/>49 # 262487160 # 484221619 # 3125282743 # 11 <lb/>50 # 264505458 # 491139838 # 3437843546 # 10 <lb/>51 # 266554348 # 498256113 # 3819709423 # 9 <lb/>52 # 268635944 # 505581634 # 4297193536 # 8 <lb/>53 # 270750304 # 513128395 # 4911255640 # 7 <lb/>54 # 272898206 # 520901152 # 5729642566 # 6 <lb/>55 # 275080457 # 528915798 # 6875687278 # 5 <lb/>56 # 277297985 # 537178089 # 8594018365 # 4 <lb/>57 # 279551349 # 545702599 # 11458691197 # 3 <lb/>58 # 281841763 # 554505091 # 17188036598 # 2 <lb/>59 # 284170013 # 563593031 # 34376072269 # 1 <lb/>60 # 286537048 # 572987098 # Infinita. # 0 <lb/> # 2 # 1 # 0 <lb/></note> </div> <div xml:id="echoid-div763" type="section" level="1" n="446"> <head xml:id="echoid-head473" xml:space="preserve">complementorum arcuum eiuſdem Quadrantis.</head> <pb o="292" file="304" n="304" rhead=""/> </div> <div xml:id="echoid-div764" type="section" level="1" n="447"> <head xml:id="echoid-head474" xml:space="preserve">VSVS TABVLÆ TAM TANGEN-</head> <head xml:id="echoid-head475" xml:space="preserve">tium, quam ſecantium.</head> <p style="it"> <s xml:id="echoid-s8584" xml:space="preserve">EX vtraq; </s> <s xml:id="echoid-s8585" xml:space="preserve">tabula non aliter tangentes, ac ſecantes arcuum, vel complemento@ <lb/> <anchor type="note" xlink:label="note-304-01a" xlink:href="note-304-01"/> rum arcuum inueſtigabimus, ac ſupra ſinus rectos, & </s> <s xml:id="echoid-s8586" xml:space="preserve">ſinus complementorum arcuum <lb/>ex ſinuum tabula eruere docuimus. </s> <s xml:id="echoid-s8587" xml:space="preserve">Vt ſi quæratur tam tangens, quam ſecans arcus <lb/>grad. </s> <s xml:id="echoid-s8588" xml:space="preserve">50. </s> <s xml:id="echoid-s8589" xml:space="preserve">Min. </s> <s xml:id="echoid-s8590" xml:space="preserve">24. </s> <s xml:id="echoid-s8591" xml:space="preserve">inuenietur in tabula tangentium ſub grad. </s> <s xml:id="echoid-s8592" xml:space="preserve">50. </s> <s xml:id="echoid-s8593" xml:space="preserve">in vertice tabu-<lb/>læ poſitis, è regione Min. </s> <s xml:id="echoid-s8594" xml:space="preserve">24. </s> <s xml:id="echoid-s8595" xml:space="preserve">ad ſiniſtram collocatorum tangens particularum <lb/>1 2087923. </s> <s xml:id="echoid-s8596" xml:space="preserve">qualium ſinus totus ponitur 10000000. </s> <s xml:id="echoid-s8597" xml:space="preserve">In tabula vero ſecantium repe-<lb/>rietur ſub grad. </s> <s xml:id="echoid-s8598" xml:space="preserve">50. </s> <s xml:id="echoid-s8599" xml:space="preserve">è regione Min. </s> <s xml:id="echoid-s8600" xml:space="preserve">24. </s> <s xml:id="echoid-s8601" xml:space="preserve">ſecans earundem particularum. </s> <s xml:id="echoid-s8602" xml:space="preserve">15688144. <lb/></s> <s xml:id="echoid-s8603" xml:space="preserve">Quod ſi quæratur tam tangens, quam ſecans complementi arcus 39. </s> <s xml:id="echoid-s8604" xml:space="preserve">Min. </s> <s xml:id="echoid-s8605" xml:space="preserve">36. </s> <s xml:id="echoid-s8606" xml:space="preserve">repe-<lb/>rietur in priori quidem tabula ſupra grad. </s> <s xml:id="echoid-s8607" xml:space="preserve">39. </s> <s xml:id="echoid-s8608" xml:space="preserve">in ima ſede poſitos, è regione Min 36. </s> <s xml:id="echoid-s8609" xml:space="preserve"><lb/>ad dextram collocatorum tangens eadem, quæ prius, 12087923. </s> <s xml:id="echoid-s8610" xml:space="preserve">In poſteriori vero ta-<lb/>bula eadem ſecans 15688144. </s> <s xml:id="echoid-s8611" xml:space="preserve">propterea quòd complementum arcus grad. </s> <s xml:id="echoid-s8612" xml:space="preserve">39. </s> <s xml:id="echoid-s8613" xml:space="preserve">Min. </s> <s xml:id="echoid-s8614" xml:space="preserve"><lb/>36. </s> <s xml:id="echoid-s8615" xml:space="preserve">complectitur grad. </s> <s xml:id="echoid-s8616" xml:space="preserve">50. </s> <s xml:id="echoid-s8617" xml:space="preserve">Min. </s> <s xml:id="echoid-s8618" xml:space="preserve">24. </s> <s xml:id="echoid-s8619" xml:space="preserve">cui arcui dicta tangens, ac ſecans debetur, vt <lb/>patet.</s> <s xml:id="echoid-s8620" xml:space="preserve"/> </p> <div xml:id="echoid-div764" type="float" level="2" n="1"> <note position="left" xlink:label="note-304-01" xlink:href="note-304-01a" xml:space="preserve">Vſus tabu-<lb/>læ tam tan <lb/>gentium, <lb/>quam ſecá-<lb/>tium.</note> </div> <p style="it"> <s xml:id="echoid-s8621" xml:space="preserve">I AM verò ſi ſinus totus aſſumatur particularum tantummodo 100000. </s> <s xml:id="echoid-s8622" xml:space="preserve">abiectis <lb/>duabus cifris ex ſinu tote 10000000. </s> <s xml:id="echoid-s8623" xml:space="preserve">abijciendæ quoq; </s> <s xml:id="echoid-s8624" xml:space="preserve">erunt ex ſingulis tangentibus, <lb/>ac ſecantibus duæ priores figuræ ad dextram: </s> <s xml:id="echoid-s8625" xml:space="preserve">quemadmodum de ſinubus diximus.</s> <s xml:id="echoid-s8626" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div766" type="section" level="1" n="448"> <head xml:id="echoid-head476" xml:space="preserve">SINVVM, TANGENTIVM, <lb/>ET SECANTIVM FINIS.</head> <pb file="305" n="305" rhead=""/> </div> <div xml:id="echoid-div767" type="section" level="1" n="449"> <head xml:id="echoid-head477" xml:space="preserve">CLAVII BAMBER GENSIS <lb/>E SOCIETATE IESV</head> <head xml:id="echoid-head478" xml:space="preserve">TRIANGVLA <lb/>RECTILINEA.</head> <pb file="306" n="306"/> <pb o="295" file="307" n="307" rhead=""/> </div> <div xml:id="echoid-div768" type="section" level="1" n="450"> <head xml:id="echoid-head479" xml:space="preserve">BAMBER GENSIS E</head> <head xml:id="echoid-head480" xml:space="preserve">SOCIETATE IESV</head> <head xml:id="echoid-head481" xml:space="preserve">TRIANGVLA RECTILINEA.</head> <head xml:id="echoid-head482" xml:space="preserve">PRÆFATIO.</head> <p style="it"> <s xml:id="echoid-s8627" xml:space="preserve">SINVVM, linearum tan-<lb/> <anchor type="note" xlink:label="note-307-01a" xlink:href="note-307-01"/> gentium, & </s> <s xml:id="echoid-s8628" xml:space="preserve">ſecantium vſus <lb/>potißimum in doctrina trian-<lb/>gulorum tam rectilineorum, <lb/>quàm ſphæricorum conſiſtit. <lb/></s> <s xml:id="echoid-s8629" xml:space="preserve">Omnes enim Aſtronomi in mo <lb/>tibus cæleſtibus vel inueſtig andis, vel explican-<lb/>dis explorant in triangulis beneficio ſinuum, li-<lb/>nearum tang entium, & </s> <s xml:id="echoid-s8630" xml:space="preserve">ſecantium tumlatera <lb/>ex angulis notis, tum etiam angulos ex lateribus <lb/>cognitis. </s> <s xml:id="echoid-s8631" xml:space="preserve">Id quod ex Epitoma Ioan. </s> <s xml:id="echoid-s8632" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s8633" xml:space="preserve">in <lb/>Almageſtum, ſiue magnam cõſtructionem Pto-<lb/>lomæi, ex opere Copernici dereuolutionibus cæle-<lb/>ſtibus, & </s> <s xml:id="echoid-s8634" xml:space="preserve">ex aliorum Aſtronomorũ ſcriptis per-<lb/>ſpicuè conſtare potest. </s> <s xml:id="echoid-s8635" xml:space="preserve">Quam ob rem cum iarn <lb/>tractationem ſinuum, linearum{q́ue} tangentium, <lb/>ac ſecantium abſoluerimus, ordo poſtulat, vt <lb/>ſciẽtiam hanc triang ulorum à Foanne Regiom.</s> <s xml:id="echoid-s8636" xml:space="preserve"> <pb o="296" file="308" n="308" rhead=""/> quin{que} libris diffusè explicatam, & </s> <s xml:id="echoid-s8637" xml:space="preserve">à Gebro Hi-<lb/>ſpalenſi Arabe, necnon à Nicolao Copernico bre-<lb/>uiter quidem, ſed paulò obſcurius traditam, pro <lb/>virili etiam exponamus, cum incredibilis ſit eo-<lb/>rum vtilit as cum in rebus omnibus Mathema-<lb/>ticis, tum præſertim in cæleſtibus motibus, & </s> <s xml:id="echoid-s8638" xml:space="preserve">in <lb/>ijs rebus, quæ ex illis pendent, rectè intelligendis, <lb/>velinueſtig ãdis, vt dictum est, & </s> <s xml:id="echoid-s8639" xml:space="preserve">partim etiam <lb/>non obſcure ex noſtra Gnomonica colligi poteſt, <lb/>vbi permulta ad horologia pertinentia ex trian-<lb/>gulis à nobis ſunt demonſtrata. </s> <s xml:id="echoid-s8640" xml:space="preserve">Exordiemur <lb/>autem à triangulis rectilineis, tanquam facilio-<lb/>ribus, de quibus eaſolum demonſtr abimus, quæ <lb/>ad res Aſtronomicas, & </s> <s xml:id="echoid-s8641" xml:space="preserve">Geometric as recte per-<lb/>cipiend as neceſſaria eſſe iudicamus: </s> <s xml:id="echoid-s8642" xml:space="preserve">Id quod e-<lb/>tiam in ſphæricis triang ulis obſeruauimus. </s> <s xml:id="echoid-s8643" xml:space="preserve">Qui <lb/>plur a deſider at, leg at Menelaum, & </s> <s xml:id="echoid-s8644" xml:space="preserve">Mauro-<lb/>lycum de sphæricis triangulis, de rectilineis ve-<lb/>ro Ioannem Regiomontanum. </s> <s xml:id="echoid-s8645" xml:space="preserve">Ante omnia au-<lb/>tem explicandum erit, penes quid angulorum <lb/>rectilineorum quantitas ſumenda ſit.</s> <s xml:id="echoid-s8646" xml:space="preserve"/> </p> <div xml:id="echoid-div768" type="float" level="2" n="1"> <note position="right" xlink:label="note-307-01" xlink:href="note-307-01a" xml:space="preserve">vſus ſinu@, <lb/>linearũ tan <lb/>gentium, & <lb/>ſecantium <lb/>in doctrina <lb/>triangulo-<lb/>rum potiſ-<lb/>ſimum con <lb/>ſiſtit.</note> </div> <p> <s xml:id="echoid-s8647" xml:space="preserve">PENES QVID ANGVLI rectilinei magnitudo ſumatur.</s> <s xml:id="echoid-s8648" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Angulorũ <lb/>rectilineo-<lb/>rũ magni <lb/>tudo penes <lb/>quid ſuma <lb/>tur.</note> <p style="it"> <s xml:id="echoid-s8649" xml:space="preserve">ANGVLI cuiuſuis rectilinei magnitudo ſumitur penes arcum circuli ex ipſo <lb/>angulo, vt centro, deſcripti ad quodcunq; </s> <s xml:id="echoid-s8650" xml:space="preserve">interuallum, inter rectas lineas angulum <lb/>comprehendentes interceptum. </s> <s xml:id="echoid-s8651" xml:space="preserve">Nam quilibet angulus rectilineus tantus eſſe dicitur, <lb/>quantus eſt arcus circuli, cuius centrum eſt inipſo angulo, inter duas lineas rectas, <pb o="297" file="309" n="309" rhead=""/> quæ angulum continent, interiectus: </s> <s xml:id="echoid-s8652" xml:space="preserve">ita vt quot graduum fuerit ille arcus, totidem <lb/> <anchor type="note" xlink:label="note-309-01a" xlink:href="note-309-01"/> partium ſit & </s> <s xml:id="echoid-s8653" xml:space="preserve">angulus, qualium quatuor recti ſunt 360. </s> <s xml:id="echoid-s8654" xml:space="preserve">aut vnus rectus 90. </s> <s xml:id="echoid-s8655" xml:space="preserve">Ex <lb/>quo fit, indifferenter ſinum anguli rectilinei pro ſinù arcus accipi poſſe, & </s> <s xml:id="echoid-s8656" xml:space="preserve">contra; <lb/></s> <s xml:id="echoid-s8657" xml:space="preserve">quod etiam de tangente, & </s> <s xml:id="echoid-s8658" xml:space="preserve">ſecante intelligatur: </s> <s xml:id="echoid-s8659" xml:space="preserve">quandoquidem arcus, & </s> <s xml:id="echoid-s8660" xml:space="preserve">angulus il-<lb/>li in centro inſiſtens eundem habent partium numerum, licet diuerſi generis, cum par <lb/>tes arcus ſint arcus, partes vero anguli ſint anguli: </s> <s xml:id="echoid-s8661" xml:space="preserve">quamuis & </s> <s xml:id="echoid-s8662" xml:space="preserve">partes anguli dici <lb/>poſsint arcus, ita vt angulus dicatur habere tot gradus, quot in arcu, cui inſiſtit, <lb/>comprehenduntur.</s> <s xml:id="echoid-s8663" xml:space="preserve"/> </p> <div xml:id="echoid-div769" type="float" level="2" n="2"> <note position="right" xlink:label="note-309-01" xlink:href="note-309-01a" xml:space="preserve">Angulus r@ <lb/>ctilineus eſt <lb/>tot partiũ, <lb/>quot gra-<lb/>duũ eſt ar-<lb/>cus circuli, <lb/>cui in cen-<lb/>tro inſiſtit.</note> </div> <p style="it"> <s xml:id="echoid-s8664" xml:space="preserve">QVANDOCVNQVE ergo arcus angulum rectilineum metiens eſt quadrãs, <lb/>id eſt, quarta pars totius circunferentiæ, angulus ei inſiſtens in centro rectus erit, <lb/>nempe quarta pars quatuor rectorum, quibus ſpatium, quod circumſtat centrum <lb/> <anchor type="note" xlink:label="note-309-02a" xlink:href="note-309-02"/> circuli æqualiter omnes partes circunferentiæ reſpiciens, æquale eſt; </s> <s xml:id="echoid-s8665" xml:space="preserve">quando autem <lb/>arcus idem eſt quadrante minor, angulus quoq; </s> <s xml:id="echoid-s8666" xml:space="preserve">minor erit recto, nempe acutus: </s> <s xml:id="echoid-s8667" xml:space="preserve">quan <lb/>do deniq; </s> <s xml:id="echoid-s8668" xml:space="preserve">arcus eſt maior quadrante, angulus etiam recto maior erit, nimirum obtu-<lb/>ſus. </s> <s xml:id="echoid-s8669" xml:space="preserve"><emph style="sc">E</emph>t contra, quando angulus eſt rectus, erit arcus illum metiens quadrans: </s> <s xml:id="echoid-s8670" xml:space="preserve">quan-<lb/>do acutus, quadrante minor: </s> <s xml:id="echoid-s8671" xml:space="preserve">quando deniq; </s> <s xml:id="echoid-s8672" xml:space="preserve">obtuſus, maior quadrante. </s> <s xml:id="echoid-s8673" xml:space="preserve">Quæ omnia <lb/>ex lemmate ſequenti erunt perſpicua.</s> <s xml:id="echoid-s8674" xml:space="preserve"/> </p> <div xml:id="echoid-div770" type="float" level="2" n="3"> <note position="right" xlink:label="note-309-02" xlink:href="note-309-02a" xml:space="preserve">Coroll. 2. <lb/>15. primi.</note> </div> </div> <div xml:id="echoid-div772" type="section" level="1" n="451"> <head xml:id="echoid-head483" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s8675" xml:space="preserve">RECTÆ lineæ angulum rectum comprehendentes abſcindũt <lb/> <anchor type="note" xlink:label="note-309-03a" xlink:href="note-309-03"/> quadrantem ex circulo, qui ex ipſo angulo, vt centro, ad quodcunq; <lb/></s> <s xml:id="echoid-s8676" xml:space="preserve">interuallum deſcribitur: </s> <s xml:id="echoid-s8677" xml:space="preserve">lineæ vero rectæ angulum acutum conti-<lb/>nentes auferunt arcum quadrante minorem: </s> <s xml:id="echoid-s8678" xml:space="preserve">lineæ deniq; </s> <s xml:id="echoid-s8679" xml:space="preserve">rectę con-<lb/>ſtituentes angulum obtuſum intercipiunt in eodem circulo arcum <lb/>maiorem quadrante. </s> <s xml:id="echoid-s8680" xml:space="preserve">Et contra, rectæ lineæ ex centro circuli egre-<lb/>dientes, quadrantemq́; </s> <s xml:id="echoid-s8681" xml:space="preserve">intercipientes conſtituunt angulum rectum: </s> <s xml:id="echoid-s8682" xml:space="preserve"><lb/>lineæ vero arcum quadrante minorem abſcindentes angulum acu-<lb/>tum continent: </s> <s xml:id="echoid-s8683" xml:space="preserve">rectæ deniq; </s> <s xml:id="echoid-s8684" xml:space="preserve">lineæ auſerentes arcum maiorem qua-<lb/>drante obtuſum angulum comprehendunt.</s> <s xml:id="echoid-s8685" xml:space="preserve"/> </p> <div xml:id="echoid-div772" type="float" level="2" n="1"> <note position="right" xlink:label="note-309-03" xlink:href="note-309-03a" xml:space="preserve">Quomodo <lb/>ſe habeant <lb/>anguli recti <lb/>linei ad ar-<lb/>cus circulo-<lb/>rú ex ipſis, <lb/>vt centris, <lb/>deſcriptorũ, <lb/>& contra.</note> </div> <p style="it"> <s xml:id="echoid-s8686" xml:space="preserve">RECTAE lineæ AB, CB, angulum rectum contineant ABC, & </s> <s xml:id="echoid-s8687" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-309-01a" xlink:href="fig-309-01"/> ex B, circulus deſcribatur ACDE. </s> <s xml:id="echoid-s8688" xml:space="preserve">Dico <lb/>arcum AC, quadrantem eſſe, &</s> <s xml:id="echoid-s8689" xml:space="preserve">c. </s> <s xml:id="echoid-s8690" xml:space="preserve">Quoniam <lb/>enim eſt, vt angulus ABC, in centro ad qua-<lb/> <anchor type="note" xlink:label="note-309-04a" xlink:href="note-309-04"/> tuor rectos, ita arcus AC, ad totam circun. <lb/></s> <s xml:id="echoid-s8691" xml:space="preserve">ferentiam; </s> <s xml:id="echoid-s8692" xml:space="preserve">eſt autem angulus ABC, cum re-<lb/>ctus ſit, quarta pars quatuor rectorum: </s> <s xml:id="echoid-s8693" xml:space="preserve">erit <lb/>quoq; </s> <s xml:id="echoid-s8694" xml:space="preserve">arcus AC, totius circunferentiæ quar <lb/>t a pars, id eſt, quadrans. </s> <s xml:id="echoid-s8695" xml:space="preserve">Quoniam vero re-<lb/>cta linea conſtituens cum recta AB, in pun-<lb/>cto B, angulum acutum cadit in arcum AC, <lb/>recta vero linea cum eadem AB, conſtituens angulum obtuſum in puncto <lb/>B, cadit in arcum CD; </s> <s xml:id="echoid-s8696" xml:space="preserve">liquido conſtat, rectas lineas angulum acutum in <pb o="298" file="310" n="310" rhead=""/> centro B, conſtituentes inter cipere arcum quadrante AC, minorem, li-<lb/>neas vero rectas continẽtes angulum obtuſum abſcindere arcum quadran <lb/>te AC, maiorem.</s> <s xml:id="echoid-s8697" xml:space="preserve"/> </p> <div xml:id="echoid-div773" type="float" level="2" n="2"> <figure xlink:label="fig-309-01" xlink:href="fig-309-01a"> <image file="309-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/309-01"/> </figure> <note position="right" xlink:label="note-309-04" xlink:href="note-309-04a" xml:space="preserve">Coroll. 2. <lb/>33. ſexti.</note> </div> <p style="it"> <s xml:id="echoid-s8698" xml:space="preserve">SED auferant iam rectæ BA, BC, ex centro B, egredientes qua-<lb/>àrantem AC. </s> <s xml:id="echoid-s8699" xml:space="preserve">Dico angulum ABC, eſſe rectum, &</s> <s xml:id="echoid-s8700" xml:space="preserve">c. </s> <s xml:id="echoid-s8701" xml:space="preserve">Quoniam enim <lb/>eſt, vt arcus AC, ad totam circunferentiam, ita angulus ABC, in cen-<lb/> <anchor type="note" xlink:label="note-310-01a" xlink:href="note-310-01"/> tro ad quatuor rectos; </s> <s xml:id="echoid-s8702" xml:space="preserve">eſt autem arcus AC, quadrans, id eſt, quarta pars <lb/>circunferentiæ totius: </s> <s xml:id="echoid-s8703" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s8704" xml:space="preserve">angulus ABC, quarta pars quatuor <lb/>rectorum, atq; </s> <s xml:id="echoid-s8705" xml:space="preserve">adeo rectus. </s> <s xml:id="echoid-s8706" xml:space="preserve">Quia vero rectæ ex centro B, emiſſæ, atque <lb/>arcum quadrante AC, minorem auferentes angulum conſtituunt mino-<lb/>rem angulo recto ABC, auferentes vero arcum quadrantc AC, maio-<lb/>rem conſtituunt angulum recto angulo ABC, maiorem; </s> <s xml:id="echoid-s8707" xml:space="preserve">perſpicuum eſt, <lb/>rectas lineas arcum quadrante AC, minorem intercipientes conſtituere <lb/>in centro B, angulum acutum, lineas vero rectas arcum quadrante AC, <lb/>maiorem includentes continere in centro B, angulum obtuſum. </s> <s xml:id="echoid-s8708" xml:space="preserve">Quod est <lb/>propoſitum.</s> <s xml:id="echoid-s8709" xml:space="preserve"/> </p> <div xml:id="echoid-div774" type="float" level="2" n="3"> <note position="left" xlink:label="note-310-01" xlink:href="note-310-01a" xml:space="preserve">Coroll. 2. <lb/>33. ſexti.</note> </div> <p style="it"> <s xml:id="echoid-s8710" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s8711" xml:space="preserve">Contineant rurſum rectæ AB, CB, angulum rectum <lb/> <anchor type="figure" xlink:label="fig-310-01a" xlink:href="fig-310-01"/> ABC, et ex B, circulus deſcribatur ACDE. <lb/></s> <s xml:id="echoid-s8712" xml:space="preserve">Dico arcum AC, eſſe quadrantem, &</s> <s xml:id="echoid-s8713" xml:space="preserve">c. </s> <s xml:id="echoid-s8714" xml:space="preserve"><lb/>Productis enim rectis AB, CB, ad D, E, <lb/>erunt & </s> <s xml:id="echoid-s8715" xml:space="preserve">anguli ABE, CBD, cum ſint an-<lb/>gulo ABC, deinceps, recti, ex definitione; </s> <s xml:id="echoid-s8716" xml:space="preserve"><lb/>necnõ & </s> <s xml:id="echoid-s8717" xml:space="preserve">angulus DBE, quòd angulo ABC, <lb/> <anchor type="note" xlink:label="note-310-02a" xlink:href="note-310-02"/> ſit ad verticem æqualis, rectus. </s> <s xml:id="echoid-s8718" xml:space="preserve">Quare cum <lb/>omnes quatuor anguli ad B, centrum ſint re-<lb/> <anchor type="note" xlink:label="note-310-03a" xlink:href="note-310-03"/> cti, id eſt, æquales, æquales quoq; </s> <s xml:id="echoid-s8719" xml:space="preserve">erunt qua-<lb/>tuor arcus AC, CD, DE, EA; </s> <s xml:id="echoid-s8720" xml:space="preserve">atq; </s> <s xml:id="echoid-s8721" xml:space="preserve">adeo <lb/>quilibet eorum quadrans erit. </s> <s xml:id="echoid-s8722" xml:space="preserve">Reliqua demonſtrabuntur, vt prius.</s> <s xml:id="echoid-s8723" xml:space="preserve"/> </p> <div xml:id="echoid-div775" type="float" level="2" n="4"> <figure xlink:label="fig-310-01" xlink:href="fig-310-01a"> <image file="310-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/310-01"/> </figure> <note position="left" xlink:label="note-310-02" xlink:href="note-310-02a" xml:space="preserve">23. primi.</note> <note position="left" xlink:label="note-310-03" xlink:href="note-310-03a" xml:space="preserve">26. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s8724" xml:space="preserve">VERVM rectæ BA, BC, ex centro B, emiſſæ auferant iam qua-<lb/>drantem AC. </s> <s xml:id="echoid-s8725" xml:space="preserve">Dico angulum ABC, rectum eſſe, &</s> <s xml:id="echoid-s8726" xml:space="preserve">c. </s> <s xml:id="echoid-s8727" xml:space="preserve">Pro tuctis enim <lb/>rectis AB, CB, ad D, E, cum angulus DBE, angulo ABC, ad verti-<lb/> <anchor type="note" xlink:label="note-310-04a" xlink:href="note-310-04"/> cem ſit æqualis, erit & </s> <s xml:id="echoid-s8728" xml:space="preserve">arcus DE, arcui AC, æqualis, & </s> <s xml:id="echoid-s8729" xml:space="preserve">proinde qua-<lb/> <anchor type="note" xlink:label="note-310-05a" xlink:href="note-310-05"/> drans. </s> <s xml:id="echoid-s8730" xml:space="preserve">Semicir culum ergo conficiunt duo quadrantes AC, DE; </s> <s xml:id="echoid-s8731" xml:space="preserve">atque <lb/>adeo reliqui duo arcus AE, DC, alterum ſemicir culum conſtituent. </s> <s xml:id="echoid-s8732" xml:space="preserve">Cum <lb/> <anchor type="note" xlink:label="note-310-06a" xlink:href="note-310-06"/> ergo duo arcus AE, DC, æquales ſint, quòd anguli ABE, CBD, ad <lb/>verticem ſint æquales; </s> <s xml:id="echoid-s8733" xml:space="preserve">erit vterq; </s> <s xml:id="echoid-s8734" xml:space="preserve">eorum quadrans: </s> <s xml:id="echoid-s8735" xml:space="preserve">ac propterea qua-<lb/> <anchor type="note" xlink:label="note-310-07a" xlink:href="note-310-07"/> tuor arcus AC, CD, DE, EA, cum ſint quadrantes, æquales erunt. <lb/></s> <s xml:id="echoid-s8736" xml:space="preserve">Quatuor ergo anguli ad centrum B, æquales quoq; </s> <s xml:id="echoid-s8737" xml:space="preserve">erunt; </s> <s xml:id="echoid-s8738" xml:space="preserve">atq; </s> <s xml:id="echoid-s8739" xml:space="preserve">adeo eorum <lb/> <anchor type="note" xlink:label="note-310-08a" xlink:href="note-310-08"/> quilibet erit rectus. </s> <s xml:id="echoid-s8740" xml:space="preserve">Vel breuius. </s> <s xml:id="echoid-s8741" xml:space="preserve">Cum AC, ſit quadrans, erit quoque <pb o="299" file="311" n="311" rhead=""/> tam in ſemicirculo CAE, reliquus arcus AE, quam in ſemicirculo <lb/>ACD, reliquus arcus CD, quadrans: </s> <s xml:id="echoid-s8742" xml:space="preserve">Eodemq́, modo in ſemicirculo <lb/>AED, vel CDE, reliquus arcus DE, quadrans erit; </s> <s xml:id="echoid-s8743" xml:space="preserve">& </s> <s xml:id="echoid-s8744" xml:space="preserve">proinde qua-<lb/>tuor anguliad B, quatuor quadrantibus æqualibus inſiſtentes erunt æqua-<lb/> <anchor type="note" xlink:label="note-311-01a" xlink:href="note-311-01"/> les, & </s> <s xml:id="echoid-s8745" xml:space="preserve">recti. </s> <s xml:id="echoid-s8746" xml:space="preserve">Reliqua, vt prius, oſtendentur.</s> <s xml:id="echoid-s8747" xml:space="preserve"/> </p> <div xml:id="echoid-div776" type="float" level="2" n="5"> <note position="left" xlink:label="note-310-04" xlink:href="note-310-04a" xml:space="preserve">23. primi.</note> <note position="left" xlink:label="note-310-05" xlink:href="note-310-05a" xml:space="preserve">26. tertij.</note> <note position="left" xlink:label="note-310-06" xlink:href="note-310-06a" xml:space="preserve">26. tertij.</note> <note position="left" xlink:label="note-310-07" xlink:href="note-310-07a" xml:space="preserve">15. primi.</note> <note position="left" xlink:label="note-310-08" xlink:href="note-310-08a" xml:space="preserve">27. tertij.</note> <note position="right" xlink:label="note-311-01" xlink:href="note-311-01a" xml:space="preserve">27. tertij.</note> </div> </div> <div xml:id="echoid-div778" type="section" level="1" n="452"> <head xml:id="echoid-head484" xml:space="preserve">SCHOLIVM.</head> <note position="right" xml:space="preserve">Datis duo-<lb/>bus angulis <lb/>tirãguli re-<lb/>ctilinei, da-<lb/>tus etiã erit <lb/>tertius. Itẽ <lb/>in triãgulo <lb/>rectangulo, <lb/>ſi detur vnꝰ <lb/>angulus a-<lb/>cutus, datus <lb/>quoq; erit <lb/>acutus reli-<lb/>quus.</note> <p style="it"> <s xml:id="echoid-s8748" xml:space="preserve">_IN_ materia porro triangulorum rectilineorum, cum dantur duo anguli noti, ter <lb/>tius illico notus quoq; </s> <s xml:id="echoid-s8749" xml:space="preserve">erit, cum ſit complementum duorum rectorum: </s> <s xml:id="echoid-s8750" xml:space="preserve">Item cum in <lb/>triangulo rectãgulo datur vnus acutus angulus, notus etiã erit reliquus acutus, quod <lb/>ſit complementum vnius recti. </s> <s xml:id="echoid-s8751" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s8752" xml:space="preserve">detractis duobus angulis notis ſimul ex grad. </s> <s xml:id="echoid-s8753" xml:space="preserve">180. <lb/></s> <s xml:id="echoid-s8754" xml:space="preserve">reliquus erit tertius notus. </s> <s xml:id="echoid-s8755" xml:space="preserve">Item in triangulo rectangulo, ſi detrahatur acutus no@ <lb/>tus ex grad. </s> <s xml:id="echoid-s8756" xml:space="preserve">90. </s> <s xml:id="echoid-s8757" xml:space="preserve">remanebit alter acutus notus. </s> <s xml:id="echoid-s8758" xml:space="preserve">Quod ſemel monuiſſe ſatis ſit.</s> <s xml:id="echoid-s8759" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div779" type="section" level="1" n="453"> <head xml:id="echoid-head485" xml:space="preserve">THEOR. I. PROPOS. I.</head> <p> <s xml:id="echoid-s8760" xml:space="preserve">IN omni triangulo rectilineo latera quæuis <lb/> <anchor type="note" xlink:label="note-311-03a" xlink:href="note-311-03"/> duo eandem proportionem habent, quam ſinus <lb/>angulorum illis oppoſitorum.</s> <s xml:id="echoid-s8761" xml:space="preserve"/> </p> <div xml:id="echoid-div779" type="float" level="2" n="1"> <note position="right" xlink:label="note-311-03" xlink:href="note-311-03a" xml:space="preserve">Latera triã-<lb/>guli rectili-<lb/>nei ſunt ſi-<lb/>nubus an-<lb/>gulotũ op-<lb/>poſitorum <lb/>proportio-<lb/>nalia.</note> </div> <p> <s xml:id="echoid-s8762" xml:space="preserve">SIT primum triangulum rectangulum ABC, cuius angulus rectus B. <lb/></s> <s xml:id="echoid-s8763" xml:space="preserve">Dico eſſe AB, ad AC, vt eſt ſinus anguli C, ad ſinum anguli B. </s> <s xml:id="echoid-s8764" xml:space="preserve">Item <lb/>AB, ad BC, vt eſt ſinus anguli C, ad ſinum angu-<lb/> <anchor type="figure" xlink:label="fig-311-01a" xlink:href="fig-311-01"/> li A, &</s> <s xml:id="echoid-s8765" xml:space="preserve">c. </s> <s xml:id="echoid-s8766" xml:space="preserve">Quoniam enim, vt in definitionibus ſi-<lb/>nuum oſtendimus, ſi AC, ponatur ſinus totus, latus <lb/>AB, eſt ſinus anguli C; </s> <s xml:id="echoid-s8767" xml:space="preserve">& </s> <s xml:id="echoid-s8768" xml:space="preserve">BC, ſinus anguli A: </s> <s xml:id="echoid-s8769" xml:space="preserve">liquido <lb/>conſtat, ita eſſe latus AB, ad latus AC, vt eſt AB, ſinus <lb/>anguli C, ad AC, ſinum totum anguli recti B: </s> <s xml:id="echoid-s8770" xml:space="preserve">Vel ita <lb/>eſſe latus AC, ad latus AB, vt eſt AC, ſinus totus re-<lb/>cti anguli B, ad AB, ſinum anguli C; </s> <s xml:id="echoid-s8771" xml:space="preserve">cum ipſa latera <lb/>ſint ſinus angulorum oppoſitorum, ac proinde vtro-<lb/>biq; </s> <s xml:id="echoid-s8772" xml:space="preserve">ſit identitatis proportio. </s> <s xml:id="echoid-s8773" xml:space="preserve">Eadem ratione erit, vt latus AC, ad latus BC, <lb/>ita AC, ſinus totus anguli recti B, ad BC, ſinum anguli A: </s> <s xml:id="echoid-s8774" xml:space="preserve">Vel vt latus BC, <lb/>ad latus AC, ita BC, ſinus anguli A, ad AC, ſinum totum recti anguli B. <lb/></s> <s xml:id="echoid-s8775" xml:space="preserve">Item vt latus AB, ad latus BC, ita AB, ſinus anguli C, ad BC, ſinum an-<lb/>guli A: </s> <s xml:id="echoid-s8776" xml:space="preserve">Vel vt latus BC, ad latus AB, ita BC, ſinus anguli A, ad AB, ſi-<lb/>num anguli C.</s> <s xml:id="echoid-s8777" xml:space="preserve"/> </p> <div xml:id="echoid-div780" type="float" level="2" n="2"> <figure xlink:label="fig-311-01" xlink:href="fig-311-01a"> <image file="311-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/311-01"/> </figure> </div> <p> <s xml:id="echoid-s8778" xml:space="preserve">SIT deinde triangulum ABC, non rectangulum. </s> <s xml:id="echoid-s8779" xml:space="preserve">Dico rurſus eſſe la-<lb/>tus AB, ad latus AC, vt eſt ſinus anguli C, ad ſinum anguli B, &</s> <s xml:id="echoid-s8780" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s8781" xml:space="preserve">Aut enim latera aſſumpta AB, AC, æqualia ſunt, autinæqualia. </s> <s xml:id="echoid-s8782" xml:space="preserve">Siæqua-<lb/>lia, erunt quoq; </s> <s xml:id="echoid-s8783" xml:space="preserve">anguli C, B, æquales; </s> <s xml:id="echoid-s8784" xml:space="preserve">ac proinde, vt in definitionibus ſi-<lb/> <anchor type="note" xlink:label="note-311-04a" xlink:href="note-311-04"/> nuum docuimus, eorum ſinus æquales. </s> <s xml:id="echoid-s8785" xml:space="preserve">Quare erit, vt latus AB, ad latus <lb/>AC, ita ſinus anguli C, ad ſinum anguli B: </s> <s xml:id="echoid-s8786" xml:space="preserve">Vel vt latus AC, ad latus AB, ita <lb/>ſinus anguli B, ad ſinum anguli C; </s> <s xml:id="echoid-s8787" xml:space="preserve">cum ſemper ſit proportio æqualitatis. </s> <s xml:id="echoid-s8788" xml:space="preserve">Si <pb o="300" file="312" n="312" rhead=""/> vero latera AB, AC, ſunt inæqualia, ſit AC, maius, ex quo abſcindatur re-<lb/>cta CE, minori lateri AB, æqualis, & </s> <s xml:id="echoid-s8789" xml:space="preserve">ex A, E, ad tertium latus BC, perpen <lb/> <anchor type="figure" xlink:label="fig-312-01a" xlink:href="fig-312-01"/> diculares demittantur AD, EF, quarum vtraq; <lb/></s> <s xml:id="echoid-s8790" xml:space="preserve">cadet intra triangulum, quando angulus B, ma-<lb/>iori lateri AC, oppoſitus acutus eſt. </s> <s xml:id="echoid-s8791" xml:space="preserve">Erit enim <lb/>& </s> <s xml:id="echoid-s8792" xml:space="preserve">tunc angulus quoq; </s> <s xml:id="echoid-s8793" xml:space="preserve">C, acutus, cum minor <lb/>ſit, quam B. </s> <s xml:id="echoid-s8794" xml:space="preserve">Quare perpendicularis AD, intra <lb/> <anchor type="note" xlink:label="note-312-01a" xlink:href="note-312-01"/> triangulum cadet, ac proinde & </s> <s xml:id="echoid-s8795" xml:space="preserve">perpendi@ula-<lb/>ris EF. </s> <s xml:id="echoid-s8796" xml:space="preserve">Quando vero angulus B, obtuſus eſt, <lb/>cadet quidem AD, ſemper extra triangulum, <lb/>at EF, cadere poteſt vel extra etiam, vel in pun <lb/>ctum B, vel intra triangulum. </s> <s xml:id="echoid-s8797" xml:space="preserve">Quomodocunq; <lb/></s> <s xml:id="echoid-s8798" xml:space="preserve">autem cadant dictæ perpendiculares, ſemper ea-<lb/> <anchor type="note" xlink:label="note-312-02a" xlink:href="note-312-02"/> dem erit demonſtratio. </s> <s xml:id="echoid-s8799" xml:space="preserve">Nam cum AD, EF, ſint <lb/>parallelæ, erunt triangula CEF, CAD, ſimi-<lb/>lia. </s> <s xml:id="echoid-s8800" xml:space="preserve">Quamobrem erit, vt CE, ad EF, ita CA, ad AD. </s> <s xml:id="echoid-s8801" xml:space="preserve">Cum ergo ex ijs, quæ <lb/> <anchor type="note" xlink:label="note-312-03a" xlink:href="note-312-03"/> in definitionibus ſinuum tradidimus, poſito ſinu toto CE, recta EF, ſit ſinus <lb/>anguli C; </s> <s xml:id="echoid-s8802" xml:space="preserve">poſito item ſinu toto AB, recta AD, ſit ſinus anguli ABD; </s> <s xml:id="echoid-s8803" xml:space="preserve">ſintq; <lb/></s> <s xml:id="echoid-s8804" xml:space="preserve">ſinus toti CE, AB, reſpectu quorum illi ſunt ſinus, æquales; </s> <s xml:id="echoid-s8805" xml:space="preserve">liquet eſſe, vt <lb/>CE, hoc eſt, latus AB, ad EF, ſinum anguli C, ita latus CA, ad AD, ſinum <lb/>anguli ABD: </s> <s xml:id="echoid-s8806" xml:space="preserve">Et permutando, vt latus AB, ad latus AC, ita EF, ſinum <lb/>anguli C, ad AD, ſinum anguli ABD, hoc eſt, in poſteriori triangulo, ad <lb/>ſinum anguli ABC, cum duo anguli ad B, æquales ſint duobus rectis, & </s> <s xml:id="echoid-s8807" xml:space="preserve">pro-<lb/>inde eundem ſinum habeant, vt in definitionibus ſinuum docuimus. </s> <s xml:id="echoid-s8808" xml:space="preserve">Ex quo <lb/>conſtat, ita eſſe minus latus AB, ad maius AC, vt eſt EF, ſinus anguli C, mi-<lb/>nori lateri oppoſiti ad AD, ſinum anguli ABC, maiori lateri oppoſiti: </s> <s xml:id="echoid-s8809" xml:space="preserve">Et <lb/>conuertendo, ita eſſe maius latus AC, ad minus AB, vt eſt AD, ſinus angu-<lb/>li ABC, maiori lateri oppoſiti ad EF, ſinum anguli C, minori lateri oppo-<lb/>ſiti. </s> <s xml:id="echoid-s8810" xml:space="preserve">Non aliter oſtendemus eſſe, vt latus AB, ad latus BC, ita ſinum anguli <lb/>C, ad ſinum anguli A: </s> <s xml:id="echoid-s8811" xml:space="preserve">Vel vt latus BC, ad latus AB, ita ſinum anguli A, ad <lb/>ſinum anguli C. </s> <s xml:id="echoid-s8812" xml:space="preserve">&</s> <s xml:id="echoid-s8813" xml:space="preserve">c. </s> <s xml:id="echoid-s8814" xml:space="preserve">dummodo ex puncto, vbi conueniunt latera aſſumpta <lb/>inæqualia, (ſi forte æqualia non ſunt) ducas ad latus oppoſitum lineam per-<lb/>pendicularem, & </s> <s xml:id="echoid-s8815" xml:space="preserve">minori lateri ex maiore rectam æqualem abſcindas, initio <lb/>facto ab altero puncto extremo maioris lateris, vbi cum tertio latere coniun-<lb/>gitur, vt à nobis factum eſt, &</s> <s xml:id="echoid-s8816" xml:space="preserve">c.</s> <s xml:id="echoid-s8817" xml:space="preserve"/> </p> <div xml:id="echoid-div781" type="float" level="2" n="3"> <note position="right" xlink:label="note-311-04" xlink:href="note-311-04a" xml:space="preserve">5. primi.</note> <figure xlink:label="fig-312-01" xlink:href="fig-312-01a"> <image file="312-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/312-01"/> </figure> <note position="left" xlink:label="note-312-01" xlink:href="note-312-01a" xml:space="preserve">18. primi. <lb/>Schol. 13. <lb/>ſecundi. <lb/>Schol. 12. <lb/>ſecundi.</note> <note position="left" xlink:label="note-312-02" xlink:href="note-312-02a" xml:space="preserve">18. primi. <lb/>Coroll. 4. <lb/>ſexti.</note> <note position="left" xlink:label="note-312-03" xlink:href="note-312-03a" xml:space="preserve">4. ſexti.</note> </div> <figure> <image file="312-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/312-02"/> </figure> <p> <s xml:id="echoid-s8818" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s8819" xml:space="preserve">Sit rurſus triangulum non <lb/>rectangulum ABC: </s> <s xml:id="echoid-s8820" xml:space="preserve">de rectangulo enim in <lb/>principio huius demonſtrationis iam eſt de-<lb/>monſtratum. </s> <s xml:id="echoid-s8821" xml:space="preserve">Dico eſſe, vt latus AB, ad latus <lb/>AC, ita ſinum anguli C, ad ſinum anguli B: <lb/></s> <s xml:id="echoid-s8822" xml:space="preserve">Vel vt latus AC, ad latus AB, ita ſinum an-<lb/>guli B, ad ſinum anguli C, &</s> <s xml:id="echoid-s8823" xml:space="preserve">c. </s> <s xml:id="echoid-s8824" xml:space="preserve">Ducta enim ex <lb/>A, vbi duo late-<lb/> <anchor type="note" xlink:label="note-312-04a" xlink:href="note-312-04"/> ra aſſumpta co-<lb/>eunt, ad tertiũ <lb/>latus BC, per-<lb/>pẽdiculari AD, <lb/>quæ vel i@tra triangulum cadet, vel extra, prout anguli B, & </s> <s xml:id="echoid-s8825" xml:space="preserve">C, acuti fue- <pb o="301" file="313" n="313" rhead=""/> rint, vel alter eorũ obtuſus: </s> <s xml:id="echoid-s8826" xml:space="preserve">erit in triangulo rectangulo ABD, vt latus AB, <lb/>ad latus AD, ita ſinus anguli recti D, ad ſinum anguli B, vt ſupra eſt demon-<lb/>ſtratum: </s> <s xml:id="echoid-s8827" xml:space="preserve">Item in triangulo rectangulo ADC, vt latus AD, ad latus AC, ita <lb/>ſinus anguli C, ad ſinum anguli recti D. </s> <s xml:id="echoid-s8828" xml:space="preserve">Ex æqualitate ergo, & </s> <s xml:id="echoid-s8829" xml:space="preserve">perturbata <lb/>proportione erit, vt latus AB, ad latus AC, ita ſinus anguli C, (Habẽt enim <lb/>duo anguli ad C, in obtuſangulo triangulo eundem ſinum, vt in tractatione <lb/>ſinuum oſtendimus. </s> <s xml:id="echoid-s8830" xml:space="preserve">ad ſinum anguli B, vt in formula ſuprapoſita apparet: </s> <s xml:id="echoid-s8831" xml:space="preserve">Et <lb/>conuertendo quoq;</s> <s xml:id="echoid-s8832" xml:space="preserve">, vt latus AC, ad latus AB, ita ſinus anguli B, ad ſinum <lb/>anguli C. </s> <s xml:id="echoid-s8833" xml:space="preserve">Eodem modo concludemus eſſe, vt latus AB, ad latus BC, ita ſi-<lb/>num anguli C, ad ſinum anguli BAC: </s> <s xml:id="echoid-s8834" xml:space="preserve">Vel vt latus BC, ad latus AB, ita ſi-<lb/>num anguli BAC, ad ſinum anguli C, &</s> <s xml:id="echoid-s8835" xml:space="preserve">c. </s> <s xml:id="echoid-s8836" xml:space="preserve">Quocirca in omni triangulo re-<lb/>ctilineo latera quæuis duo eandem proportionem habent, quam ſinus angu-<lb/>lorum illis oppoſitorum. </s> <s xml:id="echoid-s8837" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s8838" xml:space="preserve"/> </p> <div xml:id="echoid-div782" type="float" level="2" n="4"> <note position="right" xlink:label="note-312-04" xlink:href="note-312-04a" xml:space="preserve"> <lb/>latus AB. # ſin. ang. C. <lb/>latus AD. # ſin. ang. D. <lb/>latus AC. # ſin. ang. B. <lb/></note> </div> </div> <div xml:id="echoid-div784" type="section" level="1" n="454"> <head xml:id="echoid-head486" xml:space="preserve">SCHOLIVM.</head> <note position="right" xml:space="preserve">Qua rŏne <lb/>ex @@ibꝰ vel <lb/>duobusan-<lb/>gulis notis <lb/>cuiuſuis <lb/>triãguli co-<lb/>gnoſcantur <lb/>propottio-<lb/>nes laterũ.</note> <p style="it"> <s xml:id="echoid-s8839" xml:space="preserve">_EX_ hac propoſ facile colligemus proportiones laterum cuiuſuis trianguli recti-<lb/>linei, cuius omnes anguli cogniti ſint, vel duo tantum Sint entm omnes anguli in <lb/>triangulo _ABC,_ noti. </s> <s xml:id="echoid-s8840" xml:space="preserve">Dico proportiones laterum notas eſſe. </s> <s xml:id="echoid-s8841" xml:space="preserve">Cum enim eadem <lb/>ſit proportio lateris _AB,_ ad latus <lb/> <anchor type="figure" xlink:label="fig-313-01a" xlink:href="fig-313-01"/> _AC,_ quæ ſinus anguli _C,_ ad ſinum <lb/>anguli _B;_ </s> <s xml:id="echoid-s8842" xml:space="preserve">ſint autem ſinus angulorũ <lb/> <anchor type="note" xlink:label="note-313-02a" xlink:href="note-313-02"/> _C, B,_ notorum cogniti ex tabula ſi-<lb/>nuum; </s> <s xml:id="echoid-s8843" xml:space="preserve">nota erit proportio lateris _AB,_ <lb/>ad latus _AC,_ &</s> <s xml:id="echoid-s8844" xml:space="preserve">c. </s> <s xml:id="echoid-s8845" xml:space="preserve">Exempli cauſa, <lb/>ponatur in primo triangulo angulus <lb/>_C,_ grad. </s> <s xml:id="echoid-s8846" xml:space="preserve">_60. </s> <s xml:id="echoid-s8847" xml:space="preserve">B,_ grad. </s> <s xml:id="echoid-s8848" xml:space="preserve">_50. </s> <s xml:id="echoid-s8849" xml:space="preserve">& </s> <s xml:id="echoid-s8850" xml:space="preserve">A,_ <lb/>grad. </s> <s xml:id="echoid-s8851" xml:space="preserve">_70._ </s> <s xml:id="echoid-s8852" xml:space="preserve">Horum ſinus ſunt _86602._ <lb/></s> <s xml:id="echoid-s8853" xml:space="preserve">_76604. </s> <s xml:id="echoid-s8854" xml:space="preserve">93969._ </s> <s xml:id="echoid-s8855" xml:space="preserve">Eſt ergo proportio <lb/>_AB,_ ad _AC,_ eadem, quæ _86602._ </s> <s xml:id="echoid-s8856" xml:space="preserve">ad _76604._ </s> <s xml:id="echoid-s8857" xml:space="preserve">& </s> <s xml:id="echoid-s8858" xml:space="preserve">_AB,_ ad _BC,_ eadem, quæ _86602._ </s> <s xml:id="echoid-s8859" xml:space="preserve"><lb/>ad _93969._ </s> <s xml:id="echoid-s8860" xml:space="preserve">& </s> <s xml:id="echoid-s8861" xml:space="preserve">_AC,_ ad _BC,_ eadem, quæ _76604._ </s> <s xml:id="echoid-s8862" xml:space="preserve">ad _93969._ </s> <s xml:id="echoid-s8863" xml:space="preserve">In triangulo vero ſecun <lb/>do ponatur angulus _B,_ rectus, ac proinde grad. </s> <s xml:id="echoid-s8864" xml:space="preserve">_90. </s> <s xml:id="echoid-s8865" xml:space="preserve">C,_ grad. </s> <s xml:id="echoid-s8866" xml:space="preserve">_50._ </s> <s xml:id="echoid-s8867" xml:space="preserve">& </s> <s xml:id="echoid-s8868" xml:space="preserve">_A,_ grad _40._ </s> <s xml:id="echoid-s8869" xml:space="preserve"><lb/>Horum ſinus ſunt _100000. </s> <s xml:id="echoid-s8870" xml:space="preserve">76604. </s> <s xml:id="echoid-s8871" xml:space="preserve">64278._ </s> <s xml:id="echoid-s8872" xml:space="preserve">Eritigitur _AB,_ ad _AC,_ vt _76604._ </s> <s xml:id="echoid-s8873" xml:space="preserve">ad <lb/>_100000._ </s> <s xml:id="echoid-s8874" xml:space="preserve">& </s> <s xml:id="echoid-s8875" xml:space="preserve">_AB,_ ad _BC,_ vt _76604._ </s> <s xml:id="echoid-s8876" xml:space="preserve">ad _64278._ </s> <s xml:id="echoid-s8877" xml:space="preserve">& </s> <s xml:id="echoid-s8878" xml:space="preserve">_AC,_ ad _BC,_ vt _100000._ </s> <s xml:id="echoid-s8879" xml:space="preserve">ad <lb/>_64278._ </s> <s xml:id="echoid-s8880" xml:space="preserve">In triangulo deniq; </s> <s xml:id="echoid-s8881" xml:space="preserve">tertio ſtatuatur angulus _B,_ obtuſus & </s> <s xml:id="echoid-s8882" xml:space="preserve">grad. </s> <s xml:id="echoid-s8883" xml:space="preserve">_124 C,_ <lb/>grad. </s> <s xml:id="echoid-s8884" xml:space="preserve">_30_ & </s> <s xml:id="echoid-s8885" xml:space="preserve">_A,_ grad _26_ Horum ſinus ſunt (ſi pro ſinu anguli obtuſi accipiatur ſi-<lb/>nus complementi ipſius vſq; </s> <s xml:id="echoid-s8886" xml:space="preserve">ad grad. </s> <s xml:id="echoid-s8887" xml:space="preserve">_180_ nempe ſinus grad _56._) </s> <s xml:id="echoid-s8888" xml:space="preserve">_82903. </s> <s xml:id="echoid-s8889" xml:space="preserve">50000._ </s> <s xml:id="echoid-s8890" xml:space="preserve"><lb/>_43837_ Quare erit _AB,_ ad _AC,_ vt _50000._ </s> <s xml:id="echoid-s8891" xml:space="preserve">ad _82903._ </s> <s xml:id="echoid-s8892" xml:space="preserve">& </s> <s xml:id="echoid-s8893" xml:space="preserve">_AB,_ ad _BC,_ vt _50000._ </s> <s xml:id="echoid-s8894" xml:space="preserve"><lb/>ad _43837._ </s> <s xml:id="echoid-s8895" xml:space="preserve">& </s> <s xml:id="echoid-s8896" xml:space="preserve">_AC,_ ad _BC,_ vt _82903._ </s> <s xml:id="echoid-s8897" xml:space="preserve">ad _43837._</s> <s xml:id="echoid-s8898" xml:space="preserve"/> </p> <div xml:id="echoid-div784" type="float" level="2" n="1"> <figure xlink:label="fig-313-01" xlink:href="fig-313-01a"> <image file="313-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/313-01"/> </figure> <note position="right" xlink:label="note-313-02" xlink:href="note-313-02a" xml:space="preserve">@.huius.</note> </div> <p style="it"> <s xml:id="echoid-s8899" xml:space="preserve">IT AQVE vt facile pro@ortiones laterum habeantur, ſatis eſt, ſila-<lb/> <anchor type="note" xlink:label="note-313-03a" xlink:href="note-313-03"/> teribus ſinus angulorum oppoſitorum aſcribantur: </s> <s xml:id="echoid-s8900" xml:space="preserve">propterea quòd late-<lb/>ra eandem proportionem habent, quam oppoſitorum angulorum ſinus, vt <lb/>demonſtratum eſt.</s> <s xml:id="echoid-s8901" xml:space="preserve"/> </p> <div xml:id="echoid-div785" type="float" level="2" n="2"> <note position="right" xlink:label="note-313-03" xlink:href="note-313-03a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s8902" xml:space="preserve">_QVOD_ ſi duo tantum anguli cogniti ſint, erit reliquus tertius quoq; </s> <s xml:id="echoid-s8903" xml:space="preserve">notus. <lb/></s> <s xml:id="echoid-s8904" xml:space="preserve">Quare, vt prius, laterum proportiones cognoſcentur.</s> <s xml:id="echoid-s8905" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8906" xml:space="preserve">_POSSVMVS_ eaſdem proportiones laterum cognoſcere ex angulis datis, ſine <lb/>auxilio antecedentis propoſ. </s> <s xml:id="echoid-s8907" xml:space="preserve">hoc modo. </s> <s xml:id="echoid-s8908" xml:space="preserve">Circa datum triangulum _ABC,_ deſcriba-<lb/> <anchor type="note" xlink:label="note-313-04a" xlink:href="note-313-04"/> <pb o="302" file="314" n="314" rhead=""/> tur circulus, cuius centrum _D,_ quod cadet vel intra triangulum, vel in vnum latus, <lb/> <anchor type="note" xlink:label="note-314-01a" xlink:href="note-314-01"/> vel extra triangulum, prout triangulum fuerit vel acutangulum, vel rectangulũ, <lb/>aut obtuſangulum. </s> <s xml:id="echoid-s8909" xml:space="preserve">Ductis deinde ex centro _D,_ ad omnes angulos rectis _DA, DB, DC,_ <lb/>(In rectangulo triangulo ſatis eſt, ſi ducatur _DA,_ quòd _DB, DC,_ partes ſint late-<lb/>ris _BC._) </s> <s xml:id="echoid-s8910" xml:space="preserve">ſecentur ſingula latera, & </s> <s xml:id="echoid-s8911" xml:space="preserve">arcus, quos ſubtendunt, bifariam in punctis _F,_ <lb/>_H, K,_ & </s> <s xml:id="echoid-s8912" xml:space="preserve">_E, G, I._ </s> <s xml:id="echoid-s8913" xml:space="preserve">In rectangulo tamen triangulo arcus _BC,_ cui rectus angulus inſiſtit, <lb/>non eſt diuidendus bifariã: </s> <s xml:id="echoid-s8914" xml:space="preserve">In obtusangulo autem ſecãdus eſt bifariam arcus _BAC,_ <lb/>in quo exiſtit obtuſus angulus, non autem arcus _BC,_ cui inſiſtit. </s> <s xml:id="echoid-s8915" xml:space="preserve">Erunt autem me-<lb/> <anchor type="figure" xlink:label="fig-314-01a" xlink:href="fig-314-01"/> dietates laterum ſinus recti medietatum arcuum, ex defin. </s> <s xml:id="echoid-s8916" xml:space="preserve">ſinus recti. </s> <s xml:id="echoid-s8917" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s8918" xml:space="preserve">quo-<lb/>niamtam angulus _ADB,_ anguli _ACB,_ quàm angulus _ADC,_ anguli _ABC,_ & </s> <s xml:id="echoid-s8919" xml:space="preserve">in <lb/> <anchor type="note" xlink:label="note-314-02a" xlink:href="note-314-02"/> triangulo acutangulo etiam angulus _BDC,_ anguli _BAC,_ duplus eſt: </s> <s xml:id="echoid-s8920" xml:space="preserve">ponuntur <lb/>autem anguli triangulorum noti; </s> <s xml:id="echoid-s8921" xml:space="preserve">erunt quoq; </s> <s xml:id="echoid-s8922" xml:space="preserve">eorum dupli in centro cogniti. </s> <s xml:id="echoid-s8923" xml:space="preserve">Qua-<lb/>re & </s> <s xml:id="echoid-s8924" xml:space="preserve">eorũ arcus _AB, AC,_ necnon & </s> <s xml:id="echoid-s8925" xml:space="preserve">arcus _BC,_ in primo circulo noti erunt; </s> <s xml:id="echoid-s8926" xml:space="preserve">ac proin <lb/>de & </s> <s xml:id="echoid-s8927" xml:space="preserve">ſemiſſes eorundem. </s> <s xml:id="echoid-s8928" xml:space="preserve">Igitur, ex tabula ſinuum, dabuntur ſinus harum ſemiſsium, <lb/>hoc eſt, ſemiſſes laterum _AB, AC,_ & </s> <s xml:id="echoid-s8929" xml:space="preserve">in triangulo acutangulo ſemiſsis quoq; </s> <s xml:id="echoid-s8930" xml:space="preserve">lateris <lb/>_BC;_ </s> <s xml:id="echoid-s8931" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s8932" xml:space="preserve">& </s> <s xml:id="echoid-s8933" xml:space="preserve">tota latera _AB, AC,_ vnà cum latere _BC,_ in triangulo acno <lb/>tangulo, cognita fient in partibus ſinus totius _AD._ </s> <s xml:id="echoid-s8934" xml:space="preserve">In triangulo porrò rectangulo <lb/>latus _BC,_ recto angulo oppoſitum duplum eſt ſinus totius, ac proinde notum in eiſdem <lb/>partibus ſinus totius _AD_ : </s> <s xml:id="echoid-s8935" xml:space="preserve">In triangulo vero obtuſangulo latus _BC,_ angulo obtuſo <lb/>oppoſitum ita dabitur. </s> <s xml:id="echoid-s8936" xml:space="preserve">Quoniam arcus _AB, AC,_ dati ſunt, datus etiamerit totus <lb/>arcus _BAC,_ ex ipſis conflatus. </s> <s xml:id="echoid-s8937" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s8938" xml:space="preserve">eius ſemiſsis _BE,_ & </s> <s xml:id="echoid-s8939" xml:space="preserve">proinde & </s> <s xml:id="echoid-s8940" xml:space="preserve">huius <lb/>ſemiſsis ſinus rectus _BF,_ dabitur; </s> <s xml:id="echoid-s8941" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s8942" xml:space="preserve">& </s> <s xml:id="echoid-s8943" xml:space="preserve">totum latus _BC._ </s> <s xml:id="echoid-s8944" xml:space="preserve">Cognita ergo <lb/>erunt hacratione omnia laterain partibus ſemidiametri circuli triangulo circun-<lb/>ſcripti; </s> <s xml:id="echoid-s8945" xml:space="preserve">& </s> <s xml:id="echoid-s8946" xml:space="preserve">proinde eorum proportiones notæ.</s> <s xml:id="echoid-s8947" xml:space="preserve"/> </p> <div xml:id="echoid-div786" type="float" level="2" n="3"> <note position="right" xlink:label="note-313-04" xlink:href="note-313-04a" xml:space="preserve">5. quarti.</note> <note position="left" xlink:label="note-314-01" xlink:href="note-314-01a" xml:space="preserve">Coroll. 5. <lb/>quarti.</note> <figure xlink:label="fig-314-01" xlink:href="fig-314-01a"> <image file="314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/314-01"/> </figure> <note position="left" xlink:label="note-314-02" xlink:href="note-314-02a" xml:space="preserve">20. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s8948" xml:space="preserve">ITA autem ſine longa circuitione latera cognoſces in partibus dictæ <lb/> <anchor type="note" xlink:label="note-314-03a" xlink:href="note-314-03"/> ſemidiametri. </s> <s xml:id="echoid-s8949" xml:space="preserve">Sinus rectus cuiuſuis anguli acuti duplicetur, & </s> <s xml:id="echoid-s8950" xml:space="preserve">habebi-<lb/>tur latus illi angulo oppoſitum in partibus dictæ ſemidiametri. </s> <s xml:id="echoid-s8951" xml:space="preserve">quod fa-<lb/>cile ex demonſtratis intelligi poteſt. </s> <s xml:id="echoid-s8952" xml:space="preserve">Nam quilibet angulus acutus con-<lb/>tinet tot gradus, quot ſunt in ſemiſſe arcus, cui inſiſtit; </s> <s xml:id="echoid-s8953" xml:space="preserve">Vt angulus ACB, <lb/>continet tot gradus, quot ſunt in arcu AG, ſemiſſe arcus AB, cui inſi-<lb/>ſtit, propterea quòd angulus ADB, cui totus arcus AB, debetur, du-<lb/> <anchor type="note" xlink:label="note-314-04a" xlink:href="note-314-04"/> plus eſt anguli ACB. </s> <s xml:id="echoid-s8954" xml:space="preserve">Quare cum AH, ſemiſſis lateris AB, ſit ſinus <lb/>arcus AG, erit eadem AH, ſinus anguli ACB: </s> <s xml:id="echoid-s8955" xml:space="preserve">atq; </s> <s xml:id="echoid-s8956" xml:space="preserve">adeo ſinus an- <pb o="303" file="315" n="315" rhead=""/> guli ACB, duplicatus dabit latus AB, in partibus ſemidiametri AD, <lb/>&</s> <s xml:id="echoid-s8957" xml:space="preserve">c. </s> <s xml:id="echoid-s8958" xml:space="preserve">Latus autem recto angulo oppoſitum perpetuò eſt diameter circuli <lb/>circunſcripti triangulo. </s> <s xml:id="echoid-s8959" xml:space="preserve">quare ſi ſemidiameter, ſinus ve totus duplicetur, <lb/>cognitum fiet ipſum latus. </s> <s xml:id="echoid-s8960" xml:space="preserve">Latus deniq; </s> <s xml:id="echoid-s8961" xml:space="preserve">obtuſo angulo oppoſitum habe-<lb/>bitur, ſi vterq; </s> <s xml:id="echoid-s8962" xml:space="preserve">angulorum acutorum duplicetur, & </s> <s xml:id="echoid-s8963" xml:space="preserve">duplicatorum ſemiſ-<lb/>ſis accipiatur. </s> <s xml:id="echoid-s8964" xml:space="preserve">Nam ſinus huius ſemiſsis duplicatus illico latus oſtendet <lb/> <anchor type="note" xlink:label="note-315-01a" xlink:href="note-315-01"/> notum. </s> <s xml:id="echoid-s8965" xml:space="preserve">Anguli namq; </s> <s xml:id="echoid-s8966" xml:space="preserve">ADB, ADC, dupli ſunt acutorum angulorum <lb/>ACB, ABC: </s> <s xml:id="echoid-s8967" xml:space="preserve">quibus quidem duplis angulis totus arcus BAC, de-<lb/>betur, &</s> <s xml:id="echoid-s8968" xml:space="preserve">c.</s> <s xml:id="echoid-s8969" xml:space="preserve"/> </p> <div xml:id="echoid-div787" type="float" level="2" n="4"> <note position="left" xlink:label="note-314-03" xlink:href="note-314-03a" xml:space="preserve">@raxis.</note> <note position="left" xlink:label="note-314-04" xlink:href="note-314-04a" xml:space="preserve">20.tertij.</note> <note position="right" xlink:label="note-315-01" xlink:href="note-315-01a" xml:space="preserve">20. tertlj.</note> </div> <p style="it"> <s xml:id="echoid-s8970" xml:space="preserve">_IMMO_ vero ſinon dentur anguli, ſed eorum tantum proportiones, cogneſce-<lb/> <anchor type="note" xlink:label="note-315-02a" xlink:href="note-315-02"/> mus nihilominus proportiones laterum, ſi prius ex angulorum proportionibus datis <lb/>eorundem magnitudines inueſtigemus, hocmodo. </s> <s xml:id="echoid-s8971" xml:space="preserve">In primo triangulo prioris figur æ <lb/>huius ſcholij ponatur proportio anguli _C,_ ad angulum _B,_ eadem, quæ _12._ </s> <s xml:id="echoid-s8972" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s8973" xml:space="preserve">& </s> <s xml:id="echoid-s8974" xml:space="preserve"><lb/>anguli _B,_ ad angulum _A,_ quæ _20._ </s> <s xml:id="echoid-s8975" xml:space="preserve">ad _28._ </s> <s xml:id="echoid-s8976" xml:space="preserve">quæ duæ proportiones notæ ſatis ſunt, e-<lb/>tiamſi proportio anguli _A,_ ad angulum _C,_ ignota ſit. </s> <s xml:id="echoid-s8977" xml:space="preserve">Inuentis autem minimis nu-<lb/>meris _6. </s> <s xml:id="echoid-s8978" xml:space="preserve">5._ </s> <s xml:id="echoid-s8979" xml:space="preserve">qui eandem proportio-<lb/> <anchor type="figure" xlink:label="fig-315-01a" xlink:href="fig-315-01"/> <anchor type="note" xlink:label="note-315-03a" xlink:href="note-315-03"/> nem habeant, quam anguli _C, B,_ hoc <lb/>eſt, quam numeri _12. </s> <s xml:id="echoid-s8980" xml:space="preserve">10._ </s> <s xml:id="echoid-s8981" xml:space="preserve">ſihi mini-<lb/>mi non ſint; </s> <s xml:id="echoid-s8982" xml:space="preserve">Item minimis _5. </s> <s xml:id="echoid-s8983" xml:space="preserve">7._ </s> <s xml:id="echoid-s8984" xml:space="preserve">ean-<lb/>dem proportionem habentibus, quam <lb/>anguli _B, A,_ ſiue numeri _20. </s> <s xml:id="echoid-s8985" xml:space="preserve">28._ </s> <s xml:id="echoid-s8986" xml:space="preserve">ſu-<lb/>memus tres hoſce numeros deinceps <lb/>minimos _6. </s> <s xml:id="echoid-s8987" xml:space="preserve">5. </s> <s xml:id="echoid-s8988" xml:space="preserve">7._ </s> <s xml:id="echoid-s8989" xml:space="preserve">in proportionibus <lb/>numerorum minimorum _6. </s> <s xml:id="echoid-s8990" xml:space="preserve">5._ </s> <s xml:id="echoid-s8991" xml:space="preserve">& </s> <s xml:id="echoid-s8992" xml:space="preserve">_5 7._ <lb/></s> <s xml:id="echoid-s8993" xml:space="preserve">qui ſi non eſſent deinceps minimi, in-<lb/>quirendi eſſent tres minimi, per ea, quæ ab Euclide demonſtrata ſunt lib. </s> <s xml:id="echoid-s8994" xml:space="preserve">_8._ </s> <s xml:id="echoid-s8995" xml:space="preserve">Erit <lb/> <anchor type="note" xlink:label="note-315-04a" xlink:href="note-315-04"/> ergo angulus _C,_ vt _6. </s> <s xml:id="echoid-s8996" xml:space="preserve">B,_ vt _5._ </s> <s xml:id="echoid-s8997" xml:space="preserve">& </s> <s xml:id="echoid-s8998" xml:space="preserve">_A,_ vt _7._ </s> <s xml:id="echoid-s8999" xml:space="preserve">quos in gradibus per regulam Societatum <lb/>ita notos efficiemus. </s> <s xml:id="echoid-s9000" xml:space="preserve">Collectis numeris _6. </s> <s xml:id="echoid-s9001" xml:space="preserve">5. </s> <s xml:id="echoid-s9002" xml:space="preserve">7._ </s> <s xml:id="echoid-s9003" xml:space="preserve">in vnam ſummam _18._ </s> <s xml:id="echoid-s9004" xml:space="preserve">dicemus per <lb/>auream regulam. </s> <s xml:id="echoid-s9005" xml:space="preserve">Si _18._ </s> <s xml:id="echoid-s9006" xml:space="preserve">dant grad. </s> <s xml:id="echoid-s9007" xml:space="preserve">_180._ </s> <s xml:id="echoid-s9008" xml:space="preserve">(tot enim gradibus omnes tres anguli, hoc <lb/>eſt, duo recti, æquiualent.) </s> <s xml:id="echoid-s9009" xml:space="preserve">quid dabunt 6? </s> <s xml:id="echoid-s9010" xml:space="preserve">quid 5? </s> <s xml:id="echoid-s9011" xml:space="preserve">& </s> <s xml:id="echoid-s9012" xml:space="preserve">quid 7? </s> <s xml:id="echoid-s9013" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s9014" xml:space="preserve"/> </p> <div xml:id="echoid-div788" type="float" level="2" n="5"> <note position="right" xlink:label="note-315-02" xlink:href="note-315-02a" xml:space="preserve">Quomodo <lb/>ex datis {pro}-<lb/>portionibꝰ <lb/>omniũ an-<lb/>gulotũ triã <lb/>guli cogno <lb/>ſcantur ipſi <lb/>anguli.</note> <figure xlink:label="fig-315-01" xlink:href="fig-315-01a"> <image file="315-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/315-01"/> </figure> <note position="right" xlink:label="note-315-03" xlink:href="note-315-03a" xml:space="preserve">35. ſeptimi</note> <note position="right" xlink:label="note-315-04" xlink:href="note-315-04a" xml:space="preserve">4. octaui.</note> </div> <p style="it"> <s xml:id="echoid-s9015" xml:space="preserve">18. </s> <s xml:id="echoid-s9016" xml:space="preserve">180. </s> <s xml:id="echoid-s9017" xml:space="preserve">grad. </s> <s xml:id="echoid-s9018" xml:space="preserve">{6? </s> <s xml:id="echoid-s9019" xml:space="preserve">\\ 5? </s> <s xml:id="echoid-s9020" xml:space="preserve">\\ 7?</s> <s xml:id="echoid-s9021" xml:space="preserve">} fiunt {60. </s> <s xml:id="echoid-s9022" xml:space="preserve">gr. </s> <s xml:id="echoid-s9023" xml:space="preserve">\\ 50. </s> <s xml:id="echoid-s9024" xml:space="preserve">gr. </s> <s xml:id="echoid-s9025" xml:space="preserve">\\ 70. </s> <s xml:id="echoid-s9026" xml:space="preserve">gr.</s> <s xml:id="echoid-s9027" xml:space="preserve">} pro angulo {C. </s> <s xml:id="echoid-s9028" xml:space="preserve">\\ B. </s> <s xml:id="echoid-s9029" xml:space="preserve">\\ A. <lb/></s> <s xml:id="echoid-s9030" xml:space="preserve">Inueniemusq́; </s> <s xml:id="echoid-s9031" xml:space="preserve">angulum C, grad. </s> <s xml:id="echoid-s9032" xml:space="preserve">60. </s> <s xml:id="echoid-s9033" xml:space="preserve">_<emph style="sc">B</emph>,_ grad. </s> <s xml:id="echoid-s9034" xml:space="preserve">_50._ </s> <s xml:id="echoid-s9035" xml:space="preserve">& </s> <s xml:id="echoid-s9036" xml:space="preserve">_A,_ grad. </s> <s xml:id="echoid-s9037" xml:space="preserve">_70._ </s> <s xml:id="echoid-s9038" xml:space="preserve">Quòd ſi duæ no-<lb/> <anchor type="note" xlink:label="note-315-05a" xlink:href="note-315-05"/> tæ proportiones angulorum non ſint continuatæ, vt in dato exemplo, continuandæ <lb/>erunt. </s> <s xml:id="echoid-s9039" xml:space="preserve">Vt ſi dicat quis. </s> <s xml:id="echoid-s9040" xml:space="preserve">Proportio anguli _C,_ ad angulum _B,_ eſt vt _12._ </s> <s xml:id="echoid-s9041" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s9042" xml:space="preserve">& </s> <s xml:id="echoid-s9043" xml:space="preserve">pro-<lb/>portio anguli _A,_ ad angulum _<emph style="sc">B</emph>,_ vt _28._ </s> <s xml:id="echoid-s9044" xml:space="preserve">ad _20._ </s> <s xml:id="echoid-s9045" xml:space="preserve">vbi vides, eundem angulum _C,_ in <lb/>vtraq; </s> <s xml:id="echoid-s9046" xml:space="preserve">proportione eſſe conſequens: </s> <s xml:id="echoid-s9047" xml:space="preserve">continuabimus illas, ſi dicamus, proportionem <lb/>_C,_ ad _<emph style="sc">B</emph>,_ eſſe vt _12._ </s> <s xml:id="echoid-s9048" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s9049" xml:space="preserve">& </s> <s xml:id="echoid-s9050" xml:space="preserve">_<emph style="sc">B</emph>,_ ad _A,_ vt _20._ </s> <s xml:id="echoid-s9051" xml:space="preserve">ad _28._ </s> <s xml:id="echoid-s9052" xml:space="preserve">Aut ſi quis dicat. </s> <s xml:id="echoid-s9053" xml:space="preserve">Proportio an-<lb/>guli _C,_ ad _<emph style="sc">B</emph>,_ eſt vt _12._ </s> <s xml:id="echoid-s9054" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s9055" xml:space="preserve">proportio autem _A,_ ad _C,_ eſt vt _28._ </s> <s xml:id="echoid-s9056" xml:space="preserve">ad _24._ </s> <s xml:id="echoid-s9057" xml:space="preserve">continua-<lb/>bimus eas, ponendo proportionem _A,_ ad _C,_ vt _28._ </s> <s xml:id="echoid-s9058" xml:space="preserve">ad _24._ </s> <s xml:id="echoid-s9059" xml:space="preserve">& </s> <s xml:id="echoid-s9060" xml:space="preserve">_C,_ ad _<emph style="sc">B</emph>,_ vt _12._ </s> <s xml:id="echoid-s9061" xml:space="preserve">ad _10._ <lb/></s> <s xml:id="echoid-s9062" xml:space="preserve">Aut deniq; </s> <s xml:id="echoid-s9063" xml:space="preserve">ſi quis dicat. </s> <s xml:id="echoid-s9064" xml:space="preserve">Proportio _C,_ ad _<emph style="sc">B</emph>,_ eſt vt _12._ </s> <s xml:id="echoid-s9065" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s9066" xml:space="preserve">& </s> <s xml:id="echoid-s9067" xml:space="preserve">_C,_ ad _A,_ vt _24._ </s> <s xml:id="echoid-s9068" xml:space="preserve">ad <pb o="304" file="316" n="316" rhead=""/> _28._ </s> <s xml:id="echoid-s9069" xml:space="preserve">continuabimus eas, ponendo _<emph style="sc">B</emph>,_ ad _C,_ vt _10._ </s> <s xml:id="echoid-s9070" xml:space="preserve">ad _12._ </s> <s xml:id="echoid-s9071" xml:space="preserve">& </s> <s xml:id="echoid-s9072" xml:space="preserve">_C,_ ad _A,_ vt _24._ </s> <s xml:id="echoid-s9073" xml:space="preserve">ad _28._ </s> <s xml:id="echoid-s9074" xml:space="preserve">&</s> <s xml:id="echoid-s9075" xml:space="preserve">c.</s> <s xml:id="echoid-s9076" xml:space="preserve"/> </p> <div xml:id="echoid-div789" type="float" level="2" n="6"> <note position="right" xlink:label="note-315-05" xlink:href="note-315-05a" xml:space="preserve">Quãdo pro <lb/>poitiones <lb/>angulorũ <lb/>notæ non <lb/>ſunt conti-<lb/>nuatę, quid <lb/>agendum.</note> </div> <p style="it"> <s xml:id="echoid-s9077" xml:space="preserve">_SED_ demus aliud exemplum in tertio triangulo eiuſdem figuræ, in quo ſit pro-<lb/>portio anguli _<emph style="sc">B</emph>,_ ad angulum _C,_ vt _62._ </s> <s xml:id="echoid-s9078" xml:space="preserve">ad _15._ </s> <s xml:id="echoid-s9079" xml:space="preserve">& </s> <s xml:id="echoid-s9080" xml:space="preserve">proportio anguli _<emph style="sc">B</emph>,_ ad angulum <lb/>_A,_ vt _248._ </s> <s xml:id="echoid-s9081" xml:space="preserve">ad _52._ </s> <s xml:id="echoid-s9082" xml:space="preserve">Quoniam angulus _B,_ bis fuit antecedens, hoc eſt, proportiones <lb/>datæ non ſunt continuatæ, eas continuabimus, ſtatuendo proportionem _A,_ ad _B,_ vt _52._ <lb/></s> <s xml:id="echoid-s9083" xml:space="preserve">ad _248._ </s> <s xml:id="echoid-s9084" xml:space="preserve">& </s> <s xml:id="echoid-s9085" xml:space="preserve">_<emph style="sc">B</emph>,_ ad _C,_ vt _62._ </s> <s xml:id="echoid-s9086" xml:space="preserve">ad _15._ </s> <s xml:id="echoid-s9087" xml:space="preserve">Inuentis autem minimis numeris _13. </s> <s xml:id="echoid-s9088" xml:space="preserve">62._ </s> <s xml:id="echoid-s9089" xml:space="preserve">eandem <lb/>proportionem habentibus, quam anguli _A, B,_ ſiue numeri _52. </s> <s xml:id="echoid-s9090" xml:space="preserve">248._ </s> <s xml:id="echoid-s9091" xml:space="preserve">erunt duæ datæ <lb/>proportiones continuatæ in his tribus numeris minimis _13. </s> <s xml:id="echoid-s9092" xml:space="preserve">62. </s> <s xml:id="echoid-s9093" xml:space="preserve">15._ </s> <s xml:id="echoid-s9094" xml:space="preserve">vt conſtat. </s> <s xml:id="echoid-s9095" xml:space="preserve">Col-<lb/>lectis ergo ipſis in vnam ſummam _90._ </s> <s xml:id="echoid-s9096" xml:space="preserve">inueniemus per regulam Societatum angulos in <lb/>gradibus, vt hic apparet.</s> <s xml:id="echoid-s9097" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9098" xml:space="preserve">90. </s> <s xml:id="echoid-s9099" xml:space="preserve">180. </s> <s xml:id="echoid-s9100" xml:space="preserve">grad. </s> <s xml:id="echoid-s9101" xml:space="preserve">{13? </s> <s xml:id="echoid-s9102" xml:space="preserve">\\ 62? </s> <s xml:id="echoid-s9103" xml:space="preserve">\\ 15?</s> <s xml:id="echoid-s9104" xml:space="preserve">} fiunt {26. </s> <s xml:id="echoid-s9105" xml:space="preserve">gr. </s> <s xml:id="echoid-s9106" xml:space="preserve">\\ 124. </s> <s xml:id="echoid-s9107" xml:space="preserve">gr. </s> <s xml:id="echoid-s9108" xml:space="preserve">\\ 30. </s> <s xml:id="echoid-s9109" xml:space="preserve">gr.</s> <s xml:id="echoid-s9110" xml:space="preserve">} pro angulo {A. </s> <s xml:id="echoid-s9111" xml:space="preserve">\\ B. </s> <s xml:id="echoid-s9112" xml:space="preserve">\\ C. <lb/></s> <s xml:id="echoid-s9113" xml:space="preserve">Inuentis hac ratione angulis, reperientur laterum proportiones, vt prius.</s> <s xml:id="echoid-s9114" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9115" xml:space="preserve">_PORRO_ in triangulo rectangulo ſatis eſt, ſi duorũ angulorum proportio detur. <lb/></s> <s xml:id="echoid-s9116" xml:space="preserve"> <anchor type="note" xlink:label="note-316-01a" xlink:href="note-316-01"/> Sit enim in ſecundo triangulo eiuſdem figuræ proportio anguli _A,_ ad angulum _<emph style="sc">B</emph>,_ re-<lb/>ctum, vt _8._ </s> <s xml:id="echoid-s9117" xml:space="preserve">ad _18._ </s> <s xml:id="echoid-s9118" xml:space="preserve">Quoniam ergo rectus angulus _B,_ eſt grad. </s> <s xml:id="echoid-s9119" xml:space="preserve">_90._ </s> <s xml:id="echoid-s9120" xml:space="preserve">inueniemus per re-<lb/>gulam auream angulum _A,_ eſſe grad. </s> <s xml:id="echoid-s9121" xml:space="preserve">_40._ </s> <s xml:id="echoid-s9122" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s9123" xml:space="preserve"/> </p> <div xml:id="echoid-div790" type="float" level="2" n="7"> <note position="left" xlink:label="note-316-01" xlink:href="note-316-01a" xml:space="preserve">Quo pacto <lb/>ex propor-<lb/>tione duo-<lb/>tũ tantum <lb/>angulorũ <lb/>in triangu-<lb/>lo rectangu <lb/>lo propot <lb/>tiones late-<lb/>rum cogno <lb/>ſeantur.</note> </div> <p style="it"> <s xml:id="echoid-s9124" xml:space="preserve">18. </s> <s xml:id="echoid-s9125" xml:space="preserve">90. </s> <s xml:id="echoid-s9126" xml:space="preserve">grad. </s> <s xml:id="echoid-s9127" xml:space="preserve">8? </s> <s xml:id="echoid-s9128" xml:space="preserve">fiunt 40. </s> <s xml:id="echoid-s9129" xml:space="preserve">gr. </s> <s xml:id="echoid-s9130" xml:space="preserve">pro angulo A. <lb/></s> <s xml:id="echoid-s9131" xml:space="preserve">Reliquus ergo angulus _C,_ complectetur grad. </s> <s xml:id="echoid-s9132" xml:space="preserve">_50._ </s> <s xml:id="echoid-s9133" xml:space="preserve">&</s> <s xml:id="echoid-s9134" xml:space="preserve">c. </s> <s xml:id="echoid-s9135" xml:space="preserve">Sit rurſum proportio aculi <lb/>anguli _A,_ ad angulum acutum _C,_ vt _16._ </s> <s xml:id="echoid-s9136" xml:space="preserve">ad _20._ </s> <s xml:id="echoid-s9137" xml:space="preserve">Quoniam ergo duo anguli _A, C,_ vni <lb/>recto ſunt æquales, hoc eſt, continẽt grad. </s> <s xml:id="echoid-s9138" xml:space="preserve">_90._ </s> <s xml:id="echoid-s9139" xml:space="preserve">Collectis numeris _16._ </s> <s xml:id="echoid-s9140" xml:space="preserve">& </s> <s xml:id="echoid-s9141" xml:space="preserve">_20._ </s> <s xml:id="echoid-s9142" xml:space="preserve">in vnam <lb/>ſummam _36._ </s> <s xml:id="echoid-s9143" xml:space="preserve">reperiemus per regulam Societatum vtrumq; </s> <s xml:id="echoid-s9144" xml:space="preserve">angulum in gradibus, vt <lb/>hic cernis.</s> <s xml:id="echoid-s9145" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9146" xml:space="preserve">36. </s> <s xml:id="echoid-s9147" xml:space="preserve">90. </s> <s xml:id="echoid-s9148" xml:space="preserve">grad. </s> <s xml:id="echoid-s9149" xml:space="preserve">{16? </s> <s xml:id="echoid-s9150" xml:space="preserve">\\ 20?</s> <s xml:id="echoid-s9151" xml:space="preserve">} fiunt {40. </s> <s xml:id="echoid-s9152" xml:space="preserve">gr. </s> <s xml:id="echoid-s9153" xml:space="preserve">\\ 50. </s> <s xml:id="echoid-s9154" xml:space="preserve">gr.</s> <s xml:id="echoid-s9155" xml:space="preserve">} pro angulo {A. </s> <s xml:id="echoid-s9156" xml:space="preserve">\\ C. <lb/></s> <s xml:id="echoid-s9157" xml:space="preserve">Iuuentis autem angulis hac ratione, notæ fient laterum proportiones, vt prius.</s> <s xml:id="echoid-s9158" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quo pacto <lb/>ex propor-<lb/>tione vtriuſ <lb/>uis angulo <lb/>rum æqua-<lb/>lium ad ter <lb/>tium angu <lb/>lum in triã <lb/>gulo Iſoſce <lb/>le inueniã-<lb/>cur laterũ <lb/>proportio-<lb/>nes.</note> <p style="it"> <s xml:id="echoid-s9159" xml:space="preserve">_EODEM_ modo in triangulo Iſoſcele ſatis eſt, ſi proportio vtriuslibet æqualium <lb/>angulorum ad tertium angulum cognoſcatur, aut tertij anguli ad vtrumlibet angu-<lb/> <anchor type="figure" xlink:label="fig-316-01a" xlink:href="fig-316-01"/> lorum æqualium. </s> <s xml:id="echoid-s9160" xml:space="preserve">Nam ſi in triangulo Iſoſcele _<emph style="sc">Ab</emph>C,_ cu-<lb/>ius duo latera _AB, AC,_ æqualia ſunt, cognita ſit propor-<lb/>tio anguli _<emph style="sc">B</emph>,_ ad angulum _A,_ nempe eadem, quæ _10._ </s> <s xml:id="echoid-s9161" xml:space="preserve">ad _16._ <lb/></s> <s xml:id="echoid-s9162" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s9163" xml:space="preserve">proportio anguli _C,_ @ad angulum _A,_ vt _10._ </s> <s xml:id="echoid-s9164" xml:space="preserve">ad _16._ </s> <s xml:id="echoid-s9165" xml:space="preserve"><lb/>Quare duæ proportiones notæ erunt, quas continuabimus, <lb/>ſi dicamus proportionem _<emph style="sc">A</emph>,_ ad _<emph style="sc">B</emph>,_ eſſe, vt _16._ </s> <s xml:id="echoid-s9166" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s9167" xml:space="preserve">& </s> <s xml:id="echoid-s9168" xml:space="preserve">_B,_ <lb/>ad _C,_ vt _10._ </s> <s xml:id="echoid-s9169" xml:space="preserve">ad _10._ </s> <s xml:id="echoid-s9170" xml:space="preserve">Ex quibus inuenietur angulus _A,_ grad. </s> <s xml:id="echoid-s9171" xml:space="preserve"><lb/>_80._ </s> <s xml:id="echoid-s9172" xml:space="preserve">& </s> <s xml:id="echoid-s9173" xml:space="preserve">vterq; </s> <s xml:id="echoid-s9174" xml:space="preserve">_<emph style="sc">B</emph>, C,_ grad. </s> <s xml:id="echoid-s9175" xml:space="preserve">_50._ </s> <s xml:id="echoid-s9176" xml:space="preserve">per ea, quæ iam demonſtra-<lb/>ta ſunt.</s> <s xml:id="echoid-s9177" xml:space="preserve"/> </p> <div xml:id="echoid-div791" type="float" level="2" n="8"> <figure xlink:label="fig-316-01" xlink:href="fig-316-01a"> <image file="316-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/316-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s9178" xml:space="preserve">_<emph style="sc">De</emph>_ æquilatero triangulo non eſt, quòd quicquam præcipiamus, cum in eo late-<lb/>@a babeant æqualitatis proportionem.</s> <s xml:id="echoid-s9179" xml:space="preserve"/> </p> <pb o="305" file="317" n="317" rhead=""/> <p style="it"> <s xml:id="echoid-s9180" xml:space="preserve">_SED_ iam ad inuentionem laterum, atq; </s> <s xml:id="echoid-s9181" xml:space="preserve">angulorum in triangulis rectilineis ex <lb/>quibuſdam datis ac cognitis accedamus; </s> <s xml:id="echoid-s9182" xml:space="preserve">qua in re, vt certum ordinem, ac methodum <lb/>ſeruemus, agemus primo loco de triangulis rectangulis, deinde vero de non rectangu-<lb/>lis, cum in illis minor, quàm in his, difficultas reperiatur.</s> <s xml:id="echoid-s9183" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div793" type="section" level="1" n="455"> <head xml:id="echoid-head487" xml:space="preserve">PROBL. 1. PROPOSITIO 2.</head> <p> <s xml:id="echoid-s9184" xml:space="preserve">DATO vno latere, cum vno angulo acuto <lb/> <anchor type="note" xlink:label="note-317-01a" xlink:href="note-317-01"/> trianguli rectáguli, vel cum proportione duorum <lb/>angulorum quorumcunq;</s> <s xml:id="echoid-s9185" xml:space="preserve">; reliqua duo latera co-<lb/>gnoſcere, & </s> <s xml:id="echoid-s9186" xml:space="preserve">quorumlibet duorum laterum pro-<lb/>portionem efficere notam.</s> <s xml:id="echoid-s9187" xml:space="preserve"/> </p> <div xml:id="echoid-div793" type="float" level="2" n="1"> <note position="right" xlink:label="note-317-01" xlink:href="note-317-01a" xml:space="preserve">In triangu <lb/>lo rectangu <lb/>lo ex vno la <lb/>tere dato, <lb/>vna cũ an-<lb/>gulo acuto <lb/>reliqua in-<lb/>ueſtigãtur.</note> </div> <p> <s xml:id="echoid-s9188" xml:space="preserve">SIT triangulum ABC, cuius angulus B, rectus, ſitq; </s> <s xml:id="echoid-s9189" xml:space="preserve">primò latus AC, <lb/> <anchor type="note" xlink:label="note-317-02a" xlink:href="note-317-02"/> recto angulo oppoſitũ datum 13. </s> <s xml:id="echoid-s9190" xml:space="preserve">palmorum, vna cum angulo acuto C, grad. <lb/></s> <s xml:id="echoid-s9191" xml:space="preserve">22. </s> <s xml:id="echoid-s9192" xml:space="preserve">Min. </s> <s xml:id="echoid-s9193" xml:space="preserve">37. </s> <s xml:id="echoid-s9194" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s9195" xml:space="preserve">cum angulo acuto A, grad. </s> <s xml:id="echoid-s9196" xml:space="preserve">67. </s> <s xml:id="echoid-s9197" xml:space="preserve">Min. </s> <s xml:id="echoid-s9198" xml:space="preserve">23. </s> <s xml:id="echoid-s9199" xml:space="preserve">Oportet <lb/>ex his indagare reliqua latera. </s> <s xml:id="echoid-s9200" xml:space="preserve">Quoniam, per ea, quæ in defin. </s> <s xml:id="echoid-s9201" xml:space="preserve">Sinuum tra-<lb/>didimus, poſito ſinu toto AC, latera AB, BC, ſunt ſinus oppoſitorum angulo <lb/>rum: </s> <s xml:id="echoid-s9202" xml:space="preserve">ſunt autem anguli dati; </s> <s xml:id="echoid-s9203" xml:space="preserve">noti erunt ſinus di-<lb/> <anchor type="figure" xlink:label="fig-317-01a" xlink:href="fig-317-01"/> ctorum angulorum; </s> <s xml:id="echoid-s9204" xml:space="preserve">AB, quidem 38456. </s> <s xml:id="echoid-s9205" xml:space="preserve">at BC, <lb/>92310. </s> <s xml:id="echoid-s9206" xml:space="preserve">Per regulam ergo auream dicemus. </s> <s xml:id="echoid-s9207" xml:space="preserve">Si AC, <lb/>partium 100000. </s> <s xml:id="echoid-s9208" xml:space="preserve">nempe quatenus ſinus totus, <lb/>dat 13. </s> <s xml:id="echoid-s9209" xml:space="preserve">palmos, quid dabit AB, ſinus partium <lb/>38456. </s> <s xml:id="echoid-s9210" xml:space="preserve">& </s> <s xml:id="echoid-s9211" xml:space="preserve">quid ſinus BC, partium 92310? </s> <s xml:id="echoid-s9212" xml:space="preserve">vt hic <lb/>vides. </s> <s xml:id="echoid-s9213" xml:space="preserve">Inueniemusq́ue latus AB, palm. </s> <s xml:id="echoid-s9214" xml:space="preserve">5. </s> <s xml:id="echoid-s9215" xml:space="preserve">& </s> <s xml:id="echoid-s9216" xml:space="preserve">BC, <lb/>palm. </s> <s xml:id="echoid-s9217" xml:space="preserve">12. </s> <s xml:id="echoid-s9218" xml:space="preserve">ferè.</s> <s xml:id="echoid-s9219" xml:space="preserve"/> </p> <div xml:id="echoid-div794" type="float" level="2" n="2"> <note position="right" xlink:label="note-317-02" xlink:href="note-317-02a" xml:space="preserve">Quando la <lb/>tꝰ recto an-<lb/>gulo oppo-<lb/>ſitũ datur, <lb/>cum acuto <lb/>angulo.</note> <figure xlink:label="fig-317-01" xlink:href="fig-317-01a"> <image file="317-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/317-01"/> </figure> </div> <note position="right" xml:space="preserve"> <lb/>AC. # AC. # {AB.} # # {AB. <lb/>100000. # 13. # 38456? # funt # 5 <lb/> # # BC. # # BC. <lb/> # # 92310? # # 12 <lb/></note> <p style="it"> <s xml:id="echoid-s9220" xml:space="preserve">IT AQVE quando latus recto angulo oppoſitum datur cum angu-<lb/> <anchor type="note" xlink:label="note-317-04a" xlink:href="note-317-04"/> lo vno acuto, ac proinde cum altero etiam acuto: </s> <s xml:id="echoid-s9221" xml:space="preserve">Sifiat, vt ſinus totus ad <lb/>latus datum recto angulo oppoſitum, ita ſinus vtriusque anguli acuti <lb/>ſeorſum ad aliud, reperientur latera eiſdem angulis oppoſita in partibus <lb/>menſuræ, ſecundum quam datum est latus angulo recto oppoſitum.</s> <s xml:id="echoid-s9222" xml:space="preserve"/> </p> <div xml:id="echoid-div795" type="float" level="2" n="3"> <note position="right" xlink:label="note-317-04" xlink:href="note-317-04a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9223" xml:space="preserve"><emph style="sc">DEINDe</emph> datum ſit vnum ex lateribus circa angulum rectum, vt BC, <lb/> <anchor type="note" xlink:label="note-317-05a" xlink:href="note-317-05"/> palmorum 12. </s> <s xml:id="echoid-s9224" xml:space="preserve">cum angulo C, grad. </s> <s xml:id="echoid-s9225" xml:space="preserve">22. </s> <s xml:id="echoid-s9226" xml:space="preserve">Min. </s> <s xml:id="echoid-s9227" xml:space="preserve">37. </s> <s xml:id="echoid-s9228" xml:space="preserve">& </s> <s xml:id="echoid-s9229" xml:space="preserve">proinde cum angulo etiam <lb/>A, grad. </s> <s xml:id="echoid-s9230" xml:space="preserve">67. </s> <s xml:id="echoid-s9231" xml:space="preserve">Min. </s> <s xml:id="echoid-s9232" xml:space="preserve">23. </s> <s xml:id="echoid-s9233" xml:space="preserve">Oportet ex his reliqua latera inueſtigare. </s> <s xml:id="echoid-s9234" xml:space="preserve">Quoniam per <lb/>ea, quæ in lineis tangentibus, atque ſecantibus ad initium oſtendimus, poſito <lb/>ſinu toto BC, latus AB, eſt tangens anguli C, & </s> <s xml:id="echoid-s9235" xml:space="preserve">latus AC, ciuſdem ſecans, <lb/>dabitur tangẽs AB, partium 41660. </s> <s xml:id="echoid-s9236" xml:space="preserve">& </s> <s xml:id="echoid-s9237" xml:space="preserve">ſecans AC, partium 108331. </s> <s xml:id="echoid-s9238" xml:space="preserve">Quare per <pb o="306" file="318" n="318" rhead=""/> regulam auream dicemus. </s> <s xml:id="echoid-s9239" xml:space="preserve">Si BC, quatenus ſinus totus partium 100000. </s> <s xml:id="echoid-s9240" xml:space="preserve">dat <lb/>12. </s> <s xml:id="echoid-s9241" xml:space="preserve">palmos, quid dabit tangens AB, inuenta partium 41660. </s> <s xml:id="echoid-s9242" xml:space="preserve">& </s> <s xml:id="echoid-s9243" xml:space="preserve">quid ſecans <lb/>AC, inuenta partium 108331? </s> <s xml:id="echoid-s9244" xml:space="preserve">vt hic cernis. </s> <s xml:id="echoid-s9245" xml:space="preserve">Inueniemusq́ue latus AB, palm. <lb/></s> <s xml:id="echoid-s9246" xml:space="preserve">5. </s> <s xml:id="echoid-s9247" xml:space="preserve">& </s> <s xml:id="echoid-s9248" xml:space="preserve">AC, palm. </s> <s xml:id="echoid-s9249" xml:space="preserve">13. </s> <s xml:id="echoid-s9250" xml:space="preserve">ferè.</s> <s xml:id="echoid-s9251" xml:space="preserve"/> </p> <div xml:id="echoid-div796" type="float" level="2" n="4"> <note position="right" xlink:label="note-317-05" xlink:href="note-317-05a" xml:space="preserve">Quãdo la-<lb/>tus vnum <lb/>circa angu <lb/>lum rectũ <lb/>datur, cum <lb/>acuto an-<lb/>gulo.</note> </div> <note position="right" xml:space="preserve"> <lb/>BC. # BC. # {AB.} # # {AB. <lb/>100000. # 12. # 41660? # fiunt # 5. <lb/> # # AC. # # AC. <lb/> # # 108331? # # 13. <lb/></note> <p style="it"> <s xml:id="echoid-s9252" xml:space="preserve">IT AQVE cum datur vnum latus circa angulum rectum, cum vno <lb/> <anchor type="note" xlink:label="note-318-02a" xlink:href="note-318-02"/> angulo acuto, ac proinde cum altero etiam acuto: </s> <s xml:id="echoid-s9253" xml:space="preserve">Sifiat, vt ſinus totus <lb/>ad datum latus circa angulum rectum, ita tam tangens anguli acuti dato <lb/>lateri adiacentis, quàm ſecans eiuſdem anguli, ad aliud, prodibit tam la-<lb/>tus, quod fuit tangens, quàm latus, quod fuit ſecans, notum in partibus <lb/>menſuræ, ſecundum quam latus circa angulum rectum fuit datum.</s> <s xml:id="echoid-s9254" xml:space="preserve"/> </p> <div xml:id="echoid-div797" type="float" level="2" n="5"> <note position="left" xlink:label="note-318-02" xlink:href="note-318-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9255" xml:space="preserve">PER ſolos autem ſinus, cum datur vnum latus circa rectum angulum, cũ <lb/> <anchor type="note" xlink:label="note-318-03a" xlink:href="note-318-03"/> vno acuto angulo, & </s> <s xml:id="echoid-s9256" xml:space="preserve">proinde etiam cum altero acuto, ita reliqua latera ex-<lb/>quiremus. </s> <s xml:id="echoid-s9257" xml:space="preserve">Sit rurſus datum latus BC, palm. </s> <s xml:id="echoid-s9258" xml:space="preserve">12. </s> <s xml:id="echoid-s9259" xml:space="preserve">& </s> <s xml:id="echoid-s9260" xml:space="preserve">angulus C, grad. </s> <s xml:id="echoid-s9261" xml:space="preserve">22. </s> <s xml:id="echoid-s9262" xml:space="preserve">Min. </s> <s xml:id="echoid-s9263" xml:space="preserve">37. <lb/></s> <s xml:id="echoid-s9264" xml:space="preserve">ac proinde angulus A, grad. </s> <s xml:id="echoid-s9265" xml:space="preserve">67. </s> <s xml:id="echoid-s9266" xml:space="preserve">Min. </s> <s xml:id="echoid-s9267" xml:space="preserve">23. </s> <s xml:id="echoid-s9268" xml:space="preserve">Quoniam igitur, vt in defin. </s> <s xml:id="echoid-s9269" xml:space="preserve">ſinuum <lb/>diximus, poſito ſinu toto AC, latus AB, eſt ſinus anguli C, & </s> <s xml:id="echoid-s9270" xml:space="preserve">BC, ſinus an-<lb/>guli A: </s> <s xml:id="echoid-s9271" xml:space="preserve">ſunt autem anguli dati; </s> <s xml:id="echoid-s9272" xml:space="preserve">noti erunt dicti ſinus, vt AB, 38456. </s> <s xml:id="echoid-s9273" xml:space="preserve">& </s> <s xml:id="echoid-s9274" xml:space="preserve">BC, <lb/>92310. </s> <s xml:id="echoid-s9275" xml:space="preserve">Per auream igitur regulam dicemus. </s> <s xml:id="echoid-s9276" xml:space="preserve">Si BC, ſinus partium 92310. </s> <s xml:id="echoid-s9277" xml:space="preserve">dat <lb/>palm. </s> <s xml:id="echoid-s9278" xml:space="preserve">12. </s> <s xml:id="echoid-s9279" xml:space="preserve">quid dabit ſinus AB, partium 38456. </s> <s xml:id="echoid-s9280" xml:space="preserve">& </s> <s xml:id="echoid-s9281" xml:space="preserve">quid ſinus totus AC, par-<lb/>tium 100000? </s> <s xml:id="echoid-s9282" xml:space="preserve">&</s> <s xml:id="echoid-s9283" xml:space="preserve">c. </s> <s xml:id="echoid-s9284" xml:space="preserve">vt hic apparet. </s> <s xml:id="echoid-s9285" xml:space="preserve">Inuenietur enim latus AB, palm. </s> <s xml:id="echoid-s9286" xml:space="preserve">5. </s> <s xml:id="echoid-s9287" xml:space="preserve">& </s> <s xml:id="echoid-s9288" xml:space="preserve"><lb/>AC, palm. </s> <s xml:id="echoid-s9289" xml:space="preserve">13. </s> <s xml:id="echoid-s9290" xml:space="preserve">ferè.</s> <s xml:id="echoid-s9291" xml:space="preserve"/> </p> <div xml:id="echoid-div798" type="float" level="2" n="6"> <note position="left" xlink:label="note-318-03" xlink:href="note-318-03a" xml:space="preserve">Aliter per <lb/>ſolos ſinus.</note> </div> <note position="right" xml:space="preserve"> <lb/>BC. # BC. # {AB.} # # {AB. <lb/>92310. # 12. # 38456? # fiunt # 5. <lb/> # # AC. # # AC. <lb/> # # 100000? # # 13. <lb/></note> <p style="it"> <s xml:id="echoid-s9292" xml:space="preserve">QV ANDO ergo vnum latus datur circa angulum rectũ, & </s> <s xml:id="echoid-s9293" xml:space="preserve">vnus <lb/> <anchor type="note" xlink:label="note-318-05a" xlink:href="note-318-05"/> acutus angulus, ac proinde & </s> <s xml:id="echoid-s9294" xml:space="preserve">alter acutus: </s> <s xml:id="echoid-s9295" xml:space="preserve">Si fiat, vt ſinus anguli acuti <lb/>dato lateri circa angulum rectum oppoſiti ad latus datum, ita tam ſinus <lb/>alterius anguli acuti, quam ſinus totus, ad aliud, prodibit tam latus alte-<lb/>rum circa angulum rectum, quàm latus recto angulo oppoſitum, notum in <lb/>partibus menſuræ, ſecundum quam latus circa angulum rectum fuit datũ. <lb/></s> <s xml:id="echoid-s9296" xml:space="preserve">Sed expeditior eſt via per lineas tangentes, & </s> <s xml:id="echoid-s9297" xml:space="preserve">ſecantes, cum ibiſinus to-<lb/>tus in regula aurea primum locũ obtineat, & </s> <s xml:id="echoid-s9298" xml:space="preserve">proinde diuiſio fiat facilior.</s> <s xml:id="echoid-s9299" xml:space="preserve"/> </p> <div xml:id="echoid-div799" type="float" level="2" n="7"> <note position="left" xlink:label="note-318-05" xlink:href="note-318-05a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9300" xml:space="preserve">QVOD ſidetur vnum latus, vnà cum proportione duorum angulorum, <lb/> <anchor type="note" xlink:label="note-318-06a" xlink:href="note-318-06"/> ita problema abſoluemus. </s> <s xml:id="echoid-s9301" xml:space="preserve">Ex proportione angulorum reperiemus acutorum <lb/>angulorum magnitudines, vt in ſcholio propoſ. </s> <s xml:id="echoid-s9302" xml:space="preserve">1. </s> <s xml:id="echoid-s9303" xml:space="preserve">oſtendimus. </s> <s xml:id="echoid-s9304" xml:space="preserve">Quam ob rem <lb/>inueniemus ex angulis notis reliqua latera, vt prius.</s> <s xml:id="echoid-s9305" xml:space="preserve"/> </p> <div xml:id="echoid-div800" type="float" level="2" n="8"> <note position="left" xlink:label="note-318-06" xlink:href="note-318-06a" xml:space="preserve">Quando la-<lb/>tus vnũ da-<lb/>tur, & pro-<lb/>portio duo-<lb/>rum angulo <lb/>rum quorũ <lb/>libet.</note> </div> <p> <s xml:id="echoid-s9306" xml:space="preserve">INVENTIS autem lateribus, manifeſtum eſt, proportionem quorum-<lb/>libet duorum dari in numeris, in quibus inuenta ſunt. </s> <s xml:id="echoid-s9307" xml:space="preserve">Erit enim proportio <pb o="307" file="319" n="319" rhead=""/> AB, ad AC, ut 5. </s> <s xml:id="echoid-s9308" xml:space="preserve">ad 13. </s> <s xml:id="echoid-s9309" xml:space="preserve">Vel ut 38456. </s> <s xml:id="echoid-s9310" xml:space="preserve">ad 100000. </s> <s xml:id="echoid-s9311" xml:space="preserve">Vel ut 41660. </s> <s xml:id="echoid-s9312" xml:space="preserve">ad 108331. <lb/></s> <s xml:id="echoid-s9313" xml:space="preserve">&</s> <s xml:id="echoid-s9314" xml:space="preserve">c. </s> <s xml:id="echoid-s9315" xml:space="preserve">In his enim omnibus numeris dicta latera inuenta ſunt. </s> <s xml:id="echoid-s9316" xml:space="preserve">Dato ergo uno la-<lb/>tere, cum uno angulo acuto trianguli rectanguli, &</s> <s xml:id="echoid-s9317" xml:space="preserve">c. </s> <s xml:id="echoid-s9318" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s9319" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div802" type="section" level="1" n="456"> <head xml:id="echoid-head488" xml:space="preserve">PROBL. 2. PROPOS. 3.</head> <p> <s xml:id="echoid-s9320" xml:space="preserve">DATIS duobus lateribus trianguli rectangu <lb/> <anchor type="note" xlink:label="note-319-01a" xlink:href="note-319-01"/> li, duos angulos acutos efficere notos, vna cum <lb/>tertio latere. </s> <s xml:id="echoid-s9321" xml:space="preserve">Item data proportione duorum late-<lb/>rum, & </s> <s xml:id="echoid-s9322" xml:space="preserve">inſuper vno latere dato quocunque, duos <lb/>angulos acutos, vna cum reliquis duobus lateri-<lb/>bus cognoſcere.</s> <s xml:id="echoid-s9323" xml:space="preserve"/> </p> <div xml:id="echoid-div802" type="float" level="2" n="1"> <note position="right" xlink:label="note-319-01" xlink:href="note-319-01a" xml:space="preserve">In triã gulo <lb/>rectãgulo ex <lb/>duobus late <lb/>ribus notis, <lb/>vel ex eorũ <lb/>proportione <lb/>nota, vna cũ <lb/>vno latere <lb/>quocunque, <lb/>reliqua in-<lb/>quiruntur.</note> </div> <p> <s xml:id="echoid-s9324" xml:space="preserve">IN triangulo ABC, cuius angulus B, rectus, ſit primum latus AC, recto <lb/> <anchor type="note" xlink:label="note-319-02a" xlink:href="note-319-02"/> angulo oppoſitum, & </s> <s xml:id="echoid-s9325" xml:space="preserve">inſuper latus AB, circa angulum rectum datum, nem-<lb/>pe AC, palm. </s> <s xml:id="echoid-s9326" xml:space="preserve">13. </s> <s xml:id="echoid-s9327" xml:space="preserve">& </s> <s xml:id="echoid-s9328" xml:space="preserve">AB, palm. </s> <s xml:id="echoid-s9329" xml:space="preserve">5. </s> <s xml:id="echoid-s9330" xml:space="preserve">Oportet ex his & </s> <s xml:id="echoid-s9331" xml:space="preserve">angulos A, C, & </s> <s xml:id="echoid-s9332" xml:space="preserve">latus ter-<lb/>tium BC, explorare. </s> <s xml:id="echoid-s9333" xml:space="preserve">Quoniam, poſito ſinu toto AC, latus AB, eſt ſinus an-<lb/>guli C, dicemus. </s> <s xml:id="echoid-s9334" xml:space="preserve">Si AC, palm. </s> <s xml:id="echoid-s9335" xml:space="preserve">13. </s> <s xml:id="echoid-s9336" xml:space="preserve">dat AC, ſinum totum partiũ 100000. </s> <s xml:id="echoid-s9337" xml:space="preserve">quid <lb/>dabit AB, palm. </s> <s xml:id="echoid-s9338" xml:space="preserve">5? </s> <s xml:id="echoid-s9339" xml:space="preserve">inueniemusq́ue <lb/> <anchor type="figure" xlink:label="fig-319-01a" xlink:href="fig-319-01"/> ſinum AB, partium 38461. </s> <s xml:id="echoid-s9340" xml:space="preserve">ut hic <lb/>uides.</s> <s xml:id="echoid-s9341" xml:space="preserve"/> </p> <div xml:id="echoid-div803" type="float" level="2" n="2"> <note position="right" xlink:label="note-319-02" xlink:href="note-319-02a" xml:space="preserve">Quando la-<lb/>tus angulo <lb/>recto oppoſi <lb/>tum, cũ vno <lb/>latere circa <lb/>angulum r@ <lb/>ctum datur.</note> <figure xlink:label="fig-319-01" xlink:href="fig-319-01a"> <image file="319-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/319-01"/> </figure> </div> <note position="right" xml:space="preserve"> <lb/>AC. # AC. # AB. # # AB. <lb/>13. # 100000. # 5? # fit # 38461. <lb/></note> <p> <s xml:id="echoid-s9342" xml:space="preserve">Ex tabula ergo ſinuum dabitur an-<lb/>gulus C, grad. </s> <s xml:id="echoid-s9343" xml:space="preserve">22. </s> <s xml:id="echoid-s9344" xml:space="preserve">Min. </s> <s xml:id="echoid-s9345" xml:space="preserve">37. </s> <s xml:id="echoid-s9346" xml:space="preserve">ac pro-<lb/>inde reliquus angulus A, grad. </s> <s xml:id="echoid-s9347" xml:space="preserve">67. <lb/></s> <s xml:id="echoid-s9348" xml:space="preserve">Min. </s> <s xml:id="echoid-s9349" xml:space="preserve">23. </s> <s xml:id="echoid-s9350" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s9351" xml:space="preserve">huius anguli A, <lb/>ſinus, nempe BC, dabitur partium <lb/>92310. </s> <s xml:id="echoid-s9352" xml:space="preserve">ex eadem tabula ſinuum. </s> <s xml:id="echoid-s9353" xml:space="preserve">Dicemus ergo rurſum. </s> <s xml:id="echoid-s9354" xml:space="preserve">Si ſinus totus AC, par <lb/>tium 100000. </s> <s xml:id="echoid-s9355" xml:space="preserve">dat AC, palm. </s> <s xml:id="echoid-s9356" xml:space="preserve">13. </s> <s xml:id="echoid-s9357" xml:space="preserve">Vel ſi ſinus AB, inuentus partium 38461. </s> <s xml:id="echoid-s9358" xml:space="preserve"><lb/>dat AB, palm. </s> <s xml:id="echoid-s9359" xml:space="preserve">5. </s> <s xml:id="echoid-s9360" xml:space="preserve">quid dabit ſinus BC, partium 92310? </s> <s xml:id="echoid-s9361" xml:space="preserve">reperiemusq́ue BC, eſ-<lb/>ſe palm. </s> <s xml:id="echoid-s9362" xml:space="preserve">12. </s> <s xml:id="echoid-s9363" xml:space="preserve">fermè, ut hic apparet.</s> <s xml:id="echoid-s9364" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AC. # AC. <lb/>100000. # 13. <lb/>AB. # AB.} # BC. # # BC. <lb/>38461. # 13. # 5. # 92310? # fit # 12. <lb/></note> <p style="it"> <s xml:id="echoid-s9365" xml:space="preserve">CVM ergo datur latus angulo recto oppoſitum, cum vno latere cir-<lb/> <anchor type="note" xlink:label="note-319-05a" xlink:href="note-319-05"/> ca eundem angulum rectum; </s> <s xml:id="echoid-s9366" xml:space="preserve">Si fiat, vt datum latus recto angulo oppoſi-<lb/>tum ad ſinum totum, ita alterum latus datum ad aliud, prodibit ſinus acu <lb/>ti anguli, qui lateri dato circarectum angulum opponitur. </s> <s xml:id="echoid-s9367" xml:space="preserve">Inuento autem, <lb/>beneficio huius ſinus inuenti, vtro angulo acuto; </s> <s xml:id="echoid-s9368" xml:space="preserve">Si iterum fiat, vt ſinus <pb o="308" file="320" n="320" rhead=""/> totus ad datum latus recto angulo oppoſitum; </s> <s xml:id="echoid-s9369" xml:space="preserve">vel vt ſinus anguli acuti <lb/>dato lateri circa rectum angulum oppoſiti ad datum latus circa angulum <lb/>rectum, ita ſinus alterius anguli acuti ad aliud, cognoſcetur tertium latus <lb/>in partibus menſuræ, ſec@ndum quam duo latera ſunt data.</s> <s xml:id="echoid-s9370" xml:space="preserve"/> </p> <div xml:id="echoid-div804" type="float" level="2" n="3"> <note position="right" xlink:label="note-319-05" xlink:href="note-319-05a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9371" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s9372" xml:space="preserve">Sit rurſus AC, palm. </s> <s xml:id="echoid-s9373" xml:space="preserve">13. </s> <s xml:id="echoid-s9374" xml:space="preserve">& </s> <s xml:id="echoid-s9375" xml:space="preserve">AB, palm. </s> <s xml:id="echoid-s9376" xml:space="preserve">5. </s> <s xml:id="echoid-s9377" xml:space="preserve">Quia igitur, ut ad <lb/> <anchor type="note" xlink:label="note-320-01a" xlink:href="note-320-01"/> initium lincarum tangentium, ac ſecantium oſtendimus, poſito AB, ſinu toto <lb/>latus AC, ſecans eſt anguli A, & </s> <s xml:id="echoid-s9378" xml:space="preserve">BC, tãgens eiuſdem; </s> <s xml:id="echoid-s9379" xml:space="preserve">dicemus. </s> <s xml:id="echoid-s9380" xml:space="preserve">Si AB, palm. <lb/></s> <s xml:id="echoid-s9381" xml:space="preserve">5. </s> <s xml:id="echoid-s9382" xml:space="preserve">dat AB, ſinum totum partium 100000 quid dabit AC, palm. </s> <s xml:id="echoid-s9383" xml:space="preserve">13? </s> <s xml:id="echoid-s9384" xml:space="preserve">inuenie-<lb/>musq́; </s> <s xml:id="echoid-s9385" xml:space="preserve">ſecantem AC, partium 260000. </s> <s xml:id="echoid-s9386" xml:space="preserve">ut hic patet.</s> <s xml:id="echoid-s9387" xml:space="preserve"/> </p> <div xml:id="echoid-div805" type="float" level="2" n="4"> <note position="left" xlink:label="note-320-01" xlink:href="note-320-01a" xml:space="preserve">Aliter per li <lb/>neas tangẽ-<lb/>tes & ſecan-<lb/>tes.</note> </div> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # AC. # # AC. <lb/>5. # 100000. # 13? # fit # 260000. <lb/></note> <p> <s xml:id="echoid-s9388" xml:space="preserve">Ex tabula ergo Secantium erit angulus A, grad 67. </s> <s xml:id="echoid-s9389" xml:space="preserve">Min. </s> <s xml:id="echoid-s9390" xml:space="preserve">23. </s> <s xml:id="echoid-s9391" xml:space="preserve">& </s> <s xml:id="echoid-s9392" xml:space="preserve">proinde reli-<lb/>quus angulus C, grad. </s> <s xml:id="echoid-s9393" xml:space="preserve">22. </s> <s xml:id="echoid-s9394" xml:space="preserve">Min. </s> <s xml:id="echoid-s9395" xml:space="preserve">37. </s> <s xml:id="echoid-s9396" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s9397" xml:space="preserve">tangens anguli A, nempe BC, da-<lb/>bitur partium 240038. </s> <s xml:id="echoid-s9398" xml:space="preserve">ex tangentium tabula. </s> <s xml:id="echoid-s9399" xml:space="preserve">Quare rurſum dicemus. </s> <s xml:id="echoid-s9400" xml:space="preserve">Si ſinus <lb/>totus AB, partium 100000. </s> <s xml:id="echoid-s9401" xml:space="preserve">dat AB, palm. </s> <s xml:id="echoid-s9402" xml:space="preserve">5. </s> <s xml:id="echoid-s9403" xml:space="preserve">Vel ſi ſecans AC, inuenta par-<lb/>tium 260000. </s> <s xml:id="echoid-s9404" xml:space="preserve">dat AC, palm. </s> <s xml:id="echoid-s9405" xml:space="preserve">13. </s> <s xml:id="echoid-s9406" xml:space="preserve">quid dabit tangens BC, partium 240038? </s> <s xml:id="echoid-s9407" xml:space="preserve">in-<lb/>ueniemusque iterum BC, eſſe ferme palm. </s> <s xml:id="echoid-s9408" xml:space="preserve">12. </s> <s xml:id="echoid-s9409" xml:space="preserve">ut hic conſtat.</s> <s xml:id="echoid-s9410" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AB. # AB. <lb/>100000. # 5. <lb/>AC. # AC.} # BC. # # BC. <lb/>260000. # 13. # 240038? # fit. # 12. <lb/></note> <p style="it"> <s xml:id="echoid-s9411" xml:space="preserve">IGITVR quando latus recto angulo oppoſitum datur, cum vno <lb/> <anchor type="note" xlink:label="note-320-04a" xlink:href="note-320-04"/> latere circa angulum rectum; </s> <s xml:id="echoid-s9412" xml:space="preserve">Si fiat, vt datum latus circa angulum re-<lb/>ctum ad ſinum totum, ita datum latus angulo recto oppoſitum ad aliud, <lb/>prodibit ſecans anguli acuti ſub datis lateribus comprehenſi. </s> <s xml:id="echoid-s9413" xml:space="preserve">Inuento er-<lb/>go, beneficio huius ſecantis repertæ, vtroque angulo acuto, & </s> <s xml:id="echoid-s9414" xml:space="preserve">tangente acu <lb/>ti anguli ſub datis lateribus comprehenſi, ex tarigentiũ tabula; </s> <s xml:id="echoid-s9415" xml:space="preserve">S@ iterum <lb/>fiat, vt ſinus totus ad datum latus circa angulum rectum; </s> <s xml:id="echoid-s9416" xml:space="preserve">Vel vt ſecans <lb/>acuti anguli ſub datis lateribus comprehenſi ad latus datum recto angu-<lb/>lo oppoſitum, ita tangens acuti anguli ſub lateribus datis comprehenſi ad <lb/>aliud, notum fiet tertium latus in partibus menſuræ, ſecundum quam ſunt <lb/>data duo latera. </s> <s xml:id="echoid-s9417" xml:space="preserve">Verum ſatius eſt per ſolos ſinus operari, cum tangentes <lb/>lineæ, at que ſecantes nihil compendij afferant, ſintque per ſinus inuentæ.</s> <s xml:id="echoid-s9418" xml:space="preserve"/> </p> <div xml:id="echoid-div806" type="float" level="2" n="5"> <note position="left" xlink:label="note-320-04" xlink:href="note-320-04a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9419" xml:space="preserve">ADHVC aliter. </s> <s xml:id="echoid-s9420" xml:space="preserve">Ponatur rurſum AC, palm. </s> <s xml:id="echoid-s9421" xml:space="preserve">13. </s> <s xml:id="echoid-s9422" xml:space="preserve">& </s> <s xml:id="echoid-s9423" xml:space="preserve">AB, palm. </s> <s xml:id="echoid-s9424" xml:space="preserve">5. </s> <s xml:id="echoid-s9425" xml:space="preserve">Quoniã <lb/>ergo quadratum rectæ AC, duobus quadratis rectarum AB, BC, æquale <lb/> <anchor type="note" xlink:label="note-320-05a" xlink:href="note-320-05"/> eſi; </s> <s xml:id="echoid-s9426" xml:space="preserve">ſi auferatur quadratum lateris AB, quod eſt 25. </s> <s xml:id="echoid-s9427" xml:space="preserve">ex quadrato lateris AC, <lb/>quod eſt 169 relinquetur quadratũ lateris BC, nempe 144. </s> <s xml:id="echoid-s9428" xml:space="preserve">cuius radix qua-<lb/>diata 12. </s> <s xml:id="echoid-s9429" xml:space="preserve">dabit latus BC, palm.</s> <s xml:id="echoid-s9430" xml:space="preserve">12. </s> <s xml:id="echoid-s9431" xml:space="preserve">Et quia, poſito AC, ſinu toto, latera AB, <lb/>BC, ſunt ſinus angulorum oppoſitorum, ut in defin. </s> <s xml:id="echoid-s9432" xml:space="preserve">ſinuum explicauimus: <lb/></s> <s xml:id="echoid-s9433" xml:space="preserve">Sifiat, ut latus AC, angulo recto oppoſitum palmorum 13. </s> <s xml:id="echoid-s9434" xml:space="preserve">ad AC, ſinum <lb/>totum partium 100000. </s> <s xml:id="echoid-s9435" xml:space="preserve">ita alterutrum laterum circa angulum rectum, nempe <lb/>BC, palm. </s> <s xml:id="echoid-s9436" xml:space="preserve">12. </s> <s xml:id="echoid-s9437" xml:space="preserve">ad aliud, prodibit ſinus anguli acuti A, ſumpto lateri oppoſiti <lb/>partium 92308. </s> <s xml:id="echoid-s9438" xml:space="preserve">Ex ſinuum ergo tabula dabitur angulus A, grad. </s> <s xml:id="echoid-s9439" xml:space="preserve">67. </s> <s xml:id="echoid-s9440" xml:space="preserve">Min. </s> <s xml:id="echoid-s9441" xml:space="preserve">23.</s> <s xml:id="echoid-s9442" xml:space="preserve"> <pb o="309" file="321" n="321" rhead=""/> atque adeo reliquus C, grad. </s> <s xml:id="echoid-s9443" xml:space="preserve">22. </s> <s xml:id="echoid-s9444" xml:space="preserve">Min. </s> <s xml:id="echoid-s9445" xml:space="preserve">37. </s> <s xml:id="echoid-s9446" xml:space="preserve">Hoc modo primo loco inuenitur <lb/>tertium latus, deinde vero anguli:</s> <s xml:id="echoid-s9447" xml:space="preserve">cũ alijs vijs inuenti ſint prius anguli, quam <lb/>tertium latus.</s> <s xml:id="echoid-s9448" xml:space="preserve"/> </p> <div xml:id="echoid-div807" type="float" level="2" n="6"> <note position="left" xlink:label="note-320-05" xlink:href="note-320-05a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s9449" xml:space="preserve">SINT iam duo latera AB, BC, circa rectum angulum data, vt AB, palm. <lb/></s> <s xml:id="echoid-s9450" xml:space="preserve"> <anchor type="note" xlink:label="note-321-01a" xlink:href="note-321-01"/> 5. </s> <s xml:id="echoid-s9451" xml:space="preserve">& </s> <s xml:id="echoid-s9452" xml:space="preserve">BC, palm. </s> <s xml:id="echoid-s9453" xml:space="preserve">12. </s> <s xml:id="echoid-s9454" xml:space="preserve">Oportet ex his tertium latus AC, & </s> <s xml:id="echoid-s9455" xml:space="preserve">acutos angulos in-<lb/>uenire. </s> <s xml:id="echoid-s9456" xml:space="preserve">Quoniam, ex demonſtratis in principio linearum tangentium, ſecan-<lb/>tiumq́ue, poſito AB, ſinu toto, latus BC, tangens eſt anguli A, & </s> <s xml:id="echoid-s9457" xml:space="preserve">latus AC, <lb/>eiuſdem ſecans; </s> <s xml:id="echoid-s9458" xml:space="preserve">dicemus. </s> <s xml:id="echoid-s9459" xml:space="preserve">Si AB, palm. </s> <s xml:id="echoid-s9460" xml:space="preserve">5. </s> <s xml:id="echoid-s9461" xml:space="preserve">dat AB, <lb/> <anchor type="figure" xlink:label="fig-321-01a" xlink:href="fig-321-01"/> ſinum totum partium 100000. </s> <s xml:id="echoid-s9462" xml:space="preserve">quid dabit BC, <lb/>palm. </s> <s xml:id="echoid-s9463" xml:space="preserve">12? </s> <s xml:id="echoid-s9464" xml:space="preserve">reperiemusq́ue tangentem BC, par-<lb/>tium 240000. </s> <s xml:id="echoid-s9465" xml:space="preserve">vt hic manifeſtum eſt.</s> <s xml:id="echoid-s9466" xml:space="preserve"/> </p> <div xml:id="echoid-div808" type="float" level="2" n="7"> <note position="right" xlink:label="note-321-01" xlink:href="note-321-01a" xml:space="preserve">Quando <lb/>duo latera <lb/>circa angu <lb/>lum rectú <lb/>data ſunt,</note> <figure xlink:label="fig-321-01" xlink:href="fig-321-01a"> <image file="321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/321-01"/> </figure> </div> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # BC. # # BC. <lb/>5. # 100000. # 12? # ſit. # 240000. <lb/></note> <p> <s xml:id="echoid-s9467" xml:space="preserve">Ex tabula ergo tangentium dabitur angulus A, <lb/>grad.</s> <s xml:id="echoid-s9468" xml:space="preserve">67. </s> <s xml:id="echoid-s9469" xml:space="preserve">Min.</s> <s xml:id="echoid-s9470" xml:space="preserve">23. </s> <s xml:id="echoid-s9471" xml:space="preserve">ac proinde reliquus angulus C, grad. </s> <s xml:id="echoid-s9472" xml:space="preserve">22. </s> <s xml:id="echoid-s9473" xml:space="preserve">Min. </s> <s xml:id="echoid-s9474" xml:space="preserve">37. </s> <s xml:id="echoid-s9475" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s9476" xml:space="preserve"><lb/>AC, ſecans anguli A, dabitur ex tabula ſecantium, partium 260035. </s> <s xml:id="echoid-s9477" xml:space="preserve">Rurſus <lb/>ergo dicemus. </s> <s xml:id="echoid-s9478" xml:space="preserve">Si AB, ſinus totus partium 100000. </s> <s xml:id="echoid-s9479" xml:space="preserve">dat AB, palm. </s> <s xml:id="echoid-s9480" xml:space="preserve">5. </s> <s xml:id="echoid-s9481" xml:space="preserve">Vel, ſit an-<lb/>gens BC, inuenta partium 240000. </s> <s xml:id="echoid-s9482" xml:space="preserve">dat BC, palm. </s> <s xml:id="echoid-s9483" xml:space="preserve">12. </s> <s xml:id="echoid-s9484" xml:space="preserve">quid dabit AC, ſecans <lb/>partium 260035? </s> <s xml:id="echoid-s9485" xml:space="preserve">inueniemusq́ue AC, palm. </s> <s xml:id="echoid-s9486" xml:space="preserve">13. </s> <s xml:id="echoid-s9487" xml:space="preserve">ferè vt hic vides.</s> <s xml:id="echoid-s9488" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AB. # AB. <lb/>100000. # 5. <lb/>BC. # BC.} # AC. # # AC. <lb/>240000. # 12. # 260035? # ſit. # 13. <lb/></note> <p style="it"> <s xml:id="echoid-s9489" xml:space="preserve">ITAQVE ſidentur duo latera circa angulum rectum: </s> <s xml:id="echoid-s9490" xml:space="preserve">Si fiat, vt <lb/> <anchor type="note" xlink:label="note-321-04a" xlink:href="note-321-04"/> alterutrum datorum laterum ad ſinum totum, ita alterum latus datum <lb/>ad aliud, proueniet tangẽs acuti anguli buic alteri dato lateri oppoſiti. </s> <s xml:id="echoid-s9491" xml:space="preserve">In-<lb/>uento ergo, beneficio huius tangentis inuentæ, vtroq; </s> <s xml:id="echoid-s9492" xml:space="preserve">angulo acuto, in tabu-<lb/>la tangentium; </s> <s xml:id="echoid-s9493" xml:space="preserve">& </s> <s xml:id="echoid-s9494" xml:space="preserve">extabula ſecantium, ſecante anguli acuti, qui alteri <lb/>buic dato lateri opponitur: </s> <s xml:id="echoid-s9495" xml:space="preserve">Sirurſum fiat, vt ſinus totus ad primum la-<lb/>tus datum; </s> <s xml:id="echoid-s9496" xml:space="preserve">Vel vt tangens inuenta ad ſecundum latus datum, ita ſecans <lb/>accepta ex tabula ſecantium, ad aliud, notum fiet latus tertium recto an-<lb/>gulo oppoſitum in ijſdem partibus, in quibus duo latera circa angulum <lb/> <anchor type="note" xlink:label="note-321-05a" xlink:href="note-321-05"/> rectum data ſunt.</s> <s xml:id="echoid-s9497" xml:space="preserve"/> </p> <div xml:id="echoid-div809" type="float" level="2" n="8"> <note position="right" xlink:label="note-321-04" xlink:href="note-321-04a" xml:space="preserve">Praxis.</note> <note position="right" xlink:label="note-321-05" xlink:href="note-321-05a" xml:space="preserve">Aliter ſine <lb/>tãgentibus <lb/>& Secanti-<lb/>bus.</note> </div> <p> <s xml:id="echoid-s9498" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s9499" xml:space="preserve">Sit rurſum AB, palm. </s> <s xml:id="echoid-s9500" xml:space="preserve">5. </s> <s xml:id="echoid-s9501" xml:space="preserve">& </s> <s xml:id="echoid-s9502" xml:space="preserve">BC, palm. </s> <s xml:id="echoid-s9503" xml:space="preserve">12. </s> <s xml:id="echoid-s9504" xml:space="preserve">Et quoniam qua-<lb/>drata laterum AB, BC, ſimul æqualia ſunt quadrato lateris AC; </s> <s xml:id="echoid-s9505" xml:space="preserve">erit qua-<lb/> <anchor type="note" xlink:label="note-321-06a" xlink:href="note-321-06"/> dratum lateris AC, palm. </s> <s xml:id="echoid-s9506" xml:space="preserve">169. </s> <s xml:id="echoid-s9507" xml:space="preserve">cuius radix quadrata dabit latus AC, palm. <lb/></s> <s xml:id="echoid-s9508" xml:space="preserve">13. </s> <s xml:id="echoid-s9509" xml:space="preserve">Quia vero, vt in deſin. </s> <s xml:id="echoid-s9510" xml:space="preserve">ſinuum traditum eſt, poſito AC, ſinu toto, latera <lb/>AB, BC, ſunt ſinus oppoſitorum angulorum: </s> <s xml:id="echoid-s9511" xml:space="preserve">Si ſiat, vt latus AC, quod an-<lb/>gulo recto opponitur, inuẽtum palm. </s> <s xml:id="echoid-s9512" xml:space="preserve">13. </s> <s xml:id="echoid-s9513" xml:space="preserve">ad AC, ſinum totum partiũ 100000. </s> <s xml:id="echoid-s9514" xml:space="preserve"><lb/>ita alterutrum laterum circa angulum rectum, nempe AB, palm. </s> <s xml:id="echoid-s9515" xml:space="preserve">5. </s> <s xml:id="echoid-s9516" xml:space="preserve">ad aliud, <lb/>reperietur ſinus anguli acuti C, qui accepto lateri opponitur, partiũ 38461. </s> <s xml:id="echoid-s9517" xml:space="preserve"><lb/>Ex tabula ergo ſinuum dabitur angulus C, grad. </s> <s xml:id="echoid-s9518" xml:space="preserve">22. </s> <s xml:id="echoid-s9519" xml:space="preserve">Min. </s> <s xml:id="echoid-s9520" xml:space="preserve">37. </s> <s xml:id="echoid-s9521" xml:space="preserve">ac propterea re-<lb/>liquus A, grad. </s> <s xml:id="echoid-s9522" xml:space="preserve">67. </s> <s xml:id="echoid-s9523" xml:space="preserve">Min. </s> <s xml:id="echoid-s9524" xml:space="preserve">23. </s> <s xml:id="echoid-s9525" xml:space="preserve">Hac via primo loco reperitur tertium latus, de-<lb/>inde vero duo anguli: </s> <s xml:id="echoid-s9526" xml:space="preserve">cum tamen alio modo anguli prius inuenti ſint, quam <lb/>tertium latus.</s> <s xml:id="echoid-s9527" xml:space="preserve"/> </p> <div xml:id="echoid-div810" type="float" level="2" n="9"> <note position="right" xlink:label="note-321-06" xlink:href="note-321-06a" xml:space="preserve">47.primi.</note> </div> <pb o="310" file="322" n="322" rhead=""/> <p> <s xml:id="echoid-s9528" xml:space="preserve">IAM vero ſi detur duorum laterũ quorumlibet proportio, & </s> <s xml:id="echoid-s9529" xml:space="preserve">vnum latus, <lb/> <anchor type="note" xlink:label="note-322-01a" xlink:href="note-322-01"/> quodcũque illud ſit, ſumemus numeros proportionis notæ, ac ſi eſſent partes <lb/>alicuius menſurę, in quibus duo illa latera dentur; </s> <s xml:id="echoid-s9530" xml:space="preserve">atq; </s> <s xml:id="echoid-s9531" xml:space="preserve">ex his, vt demonſtra-<lb/>uimus in hac propoſ. </s> <s xml:id="echoid-s9532" xml:space="preserve">angulos inueniemus, ac tertium latus in eiſdẽ partibus. <lb/></s> <s xml:id="echoid-s9533" xml:space="preserve">Deinde, ſi ſiat, vt numerus illius lateris, quod datum eſt, ad ipſum latus datũ, <lb/>ita numeri aliorum laterum ſigillatim ad aliud, reperientur alia latera in par-<lb/>tibus menſuræ, ſecundum quam illud alterum latus eſt datum. </s> <s xml:id="echoid-s9534" xml:space="preserve">Vt ſi propor-<lb/>tio AB, ad AC, ſit, vt 15. </s> <s xml:id="echoid-s9535" xml:space="preserve">ad 39. </s> <s xml:id="echoid-s9536" xml:space="preserve">& </s> <s xml:id="echoid-s9537" xml:space="preserve">latus BC, palm. </s> <s xml:id="echoid-s9538" xml:space="preserve">12. </s> <s xml:id="echoid-s9539" xml:space="preserve">reperietur, ex demon-<lb/>ſtratis, angulus A, grad. </s> <s xml:id="echoid-s9540" xml:space="preserve">67. </s> <s xml:id="echoid-s9541" xml:space="preserve">Min. </s> <s xml:id="echoid-s9542" xml:space="preserve">23. </s> <s xml:id="echoid-s9543" xml:space="preserve">& </s> <s xml:id="echoid-s9544" xml:space="preserve">angulus C, grad. </s> <s xml:id="echoid-s9545" xml:space="preserve">22. </s> <s xml:id="echoid-s9546" xml:space="preserve">Min. </s> <s xml:id="echoid-s9547" xml:space="preserve">37. </s> <s xml:id="echoid-s9548" xml:space="preserve">latus vero <lb/>BC, partium 36. </s> <s xml:id="echoid-s9549" xml:space="preserve">qualium AB, eſt 15. </s> <s xml:id="echoid-s9550" xml:space="preserve">& </s> <s xml:id="echoid-s9551" xml:space="preserve">AC, 39. </s> <s xml:id="echoid-s9552" xml:space="preserve">Quare ſi fiat, vt latus BC, <lb/>inuentum partium 36. </s> <s xml:id="echoid-s9553" xml:space="preserve">ad idem BC, datum palm. </s> <s xml:id="echoid-s9554" xml:space="preserve">12. </s> <s xml:id="echoid-s9555" xml:space="preserve">ita tam AB, partium 15. </s> <s xml:id="echoid-s9556" xml:space="preserve"><lb/>quàm AC, partium 39. </s> <s xml:id="echoid-s9557" xml:space="preserve">ad aliud, inuenietur AB, palm. </s> <s xml:id="echoid-s9558" xml:space="preserve">5. </s> <s xml:id="echoid-s9559" xml:space="preserve">& </s> <s xml:id="echoid-s9560" xml:space="preserve">AC, palm. </s> <s xml:id="echoid-s9561" xml:space="preserve">13. </s> <s xml:id="echoid-s9562" xml:space="preserve"><lb/>Datis ergo duobus lateribus trianguli rectanguli, duos angulos acutos effeci-<lb/>mus notos, &</s> <s xml:id="echoid-s9563" xml:space="preserve">c. </s> <s xml:id="echoid-s9564" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s9565" xml:space="preserve"/> </p> <div xml:id="echoid-div811" type="float" level="2" n="10"> <note position="left" xlink:label="note-322-01" xlink:href="note-322-01a" xml:space="preserve">Quãdo {pro}-<lb/>portio duo <lb/>rum laterũ <lb/>datur, & v-<lb/>nũlatus.</note> </div> </div> <div xml:id="echoid-div813" type="section" level="1" n="457"> <head xml:id="echoid-head489" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s9566" xml:space="preserve">_ABSOLVTVS_ iam eſt rectangulorum triangulorum calculus, ſequitur de <lb/>triangulis non rectangulis. </s> <s xml:id="echoid-s9567" xml:space="preserve">Sed prius quædam ad hanc rem neceſſaria demonſtranda <lb/>ſunt, quorum nonnulla plurimum etiam triangulis ſphæricis conducent.</s> <s xml:id="echoid-s9568" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div814" type="section" level="1" n="458"> <head xml:id="echoid-head490" xml:space="preserve">THEOR. 2. PROPOS. 4.</head> <p> <s xml:id="echoid-s9569" xml:space="preserve">SI diameter circuli chordam quamlibet, eiusq́; <lb/></s> <s xml:id="echoid-s9570" xml:space="preserve"> <anchor type="note" xlink:label="note-322-02a" xlink:href="note-322-02"/> arcum ſecet in duas partes; </s> <s xml:id="echoid-s9571" xml:space="preserve">habebunt ſegmenta <lb/>chordæ eandem proportionem, quam ſinus ſeg-<lb/>mentorum arcus reſpondentium.</s> <s xml:id="echoid-s9572" xml:space="preserve"/> </p> <div xml:id="echoid-div814" type="float" level="2" n="1"> <note position="left" xlink:label="note-322-02" xlink:href="note-322-02a" xml:space="preserve">Quam pro <lb/>portionem <lb/>habeãt duo <lb/>ſegmenta <lb/>cuiuſque <lb/>chordæ.</note> </div> <p> <s xml:id="echoid-s9573" xml:space="preserve">IN circulo ABCD, diameter AC, ſecet chordam BD, in E, eiuſq́ue ar-<lb/>cum BAD, in A, uel BCD, in C: </s> <s xml:id="echoid-s9574" xml:space="preserve">ducanturq́ue BF, DG, ad diametrum <lb/>AC, perpendiculares; </s> <s xml:id="echoid-s9575" xml:space="preserve">quarum BF, ſinus eſt arcus BA, uel BC: </s> <s xml:id="echoid-s9576" xml:space="preserve">& </s> <s xml:id="echoid-s9577" xml:space="preserve">DG, ſi-<lb/> <anchor type="figure" xlink:label="fig-322-01a" xlink:href="fig-322-01"/> nus arcus AD, uel CD. </s> <s xml:id="echoid-s9578" xml:space="preserve">Dico ita eſſe BE, ad ED, <lb/>ut BF, ad DG. </s> <s xml:id="echoid-s9579" xml:space="preserve">Quoniam enim in triangulis BE F, <lb/>DEG, anguli F, G, æquales ſunt, utpote recti: </s> <s xml:id="echoid-s9580" xml:space="preserve">Itẽ <lb/> <anchor type="note" xlink:label="note-322-03a" xlink:href="note-322-03"/> anguli E, ad uerticem æquales; </s> <s xml:id="echoid-s9581" xml:space="preserve">æquiangula erunt <lb/> <anchor type="note" xlink:label="note-322-04a" xlink:href="note-322-04"/> triangula BEF, DEG. </s> <s xml:id="echoid-s9582" xml:space="preserve">Quare erit, ut BE, ad BF, <lb/> <anchor type="note" xlink:label="note-322-05a" xlink:href="note-322-05"/> ita ED, ad DG: </s> <s xml:id="echoid-s9583" xml:space="preserve">Et permutando, ut BE, ad ED, <lb/>ita BF, ad DG. </s> <s xml:id="echoid-s9584" xml:space="preserve">Si ergo diameter circuli chordam <lb/>quamlibet, eiusq́; </s> <s xml:id="echoid-s9585" xml:space="preserve">arcum ſecet in duas partes, &</s> <s xml:id="echoid-s9586" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s9587" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s9588" xml:space="preserve"/> </p> <div xml:id="echoid-div815" type="float" level="2" n="2"> <figure xlink:label="fig-322-01" xlink:href="fig-322-01a"> <image file="322-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/322-01"/> </figure> <note position="left" xlink:label="note-322-03" xlink:href="note-322-03a" xml:space="preserve">15. primi.</note> <note position="left" xlink:label="note-322-04" xlink:href="note-322-04a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-322-05" xlink:href="note-322-05a" xml:space="preserve">4.ſexti.</note> </div> </div> <div xml:id="echoid-div817" type="section" level="1" n="459"> <head xml:id="echoid-head491" xml:space="preserve">THEOR. 3. PROPOS. 5.</head> <p> <s xml:id="echoid-s9589" xml:space="preserve">SI in circulo chorda cuiuſlibet arcus ad vnam <lb/> <anchor type="note" xlink:label="note-322-06a" xlink:href="note-322-06"/> partem producatur, conueniatq́; </s> <s xml:id="echoid-s9590" xml:space="preserve">cum diametro <pb o="311" file="323" n="323" rhead=""/> quauis ad eandem partem producta; </s> <s xml:id="echoid-s9591" xml:space="preserve">erit eadem <lb/> <anchor type="note" xlink:label="note-323-01a" xlink:href="note-323-01"/> proportio totius chordæ productæ ad ſegmẽtum <lb/>exterius, quæ ſinus arcus inter punctú, per quod <lb/>diameter producta eſt, & </s> <s xml:id="echoid-s9592" xml:space="preserve">remotius punctum ex-<lb/>tremum dictæ chordæ, ad ſinum arcus inter idem <lb/>punctum diametri, & </s> <s xml:id="echoid-s9593" xml:space="preserve">propinquius punctum ex-<lb/>tremum eiuſdem chordæ.</s> <s xml:id="echoid-s9594" xml:space="preserve"/> </p> <div xml:id="echoid-div817" type="float" level="2" n="1"> <note position="left" xlink:label="note-322-06" xlink:href="note-322-06a" xml:space="preserve">Quã {pro}por <lb/>tionem ha <lb/>beat chor-<lb/>da circuli</note> <note position="right" xlink:label="note-323-01" xlink:href="note-323-01a" xml:space="preserve">producta, <lb/>& cũ dia-<lb/>metro pro-<lb/>ducta con-<lb/>ueniens, ad <lb/>ſegmẽtum <lb/>exterius.</note> </div> <p> <s xml:id="echoid-s9595" xml:space="preserve">IN circulo ABCD, chorda AD, arcus AD, ad partes D, producta con-<lb/>ueniat cum diametro BC, ad eaſdem partes producta in puncto E; </s> <s xml:id="echoid-s9596" xml:space="preserve">demittan-<lb/>turq́ue AF, DG, ad diametrum BC, perpendiculares; </s> <s xml:id="echoid-s9597" xml:space="preserve">quarum AF, ſinus eſt <lb/>arcus AC: </s> <s xml:id="echoid-s9598" xml:space="preserve">& </s> <s xml:id="echoid-s9599" xml:space="preserve">DG, ſinus arcus CD. </s> <s xml:id="echoid-s9600" xml:space="preserve">Dico ita eſ-<lb/> <anchor type="figure" xlink:label="fig-323-01a" xlink:href="fig-323-01"/> ſe AE, ad DE, ut AF, ad DG. </s> <s xml:id="echoid-s9601" xml:space="preserve">Quoniam enim <lb/>AF, DG, parallelæ ſunt obangulos rectos F, <lb/> <anchor type="note" xlink:label="note-323-02a" xlink:href="note-323-02"/> G; </s> <s xml:id="echoid-s9602" xml:space="preserve">ſimilia erunt triangula AEF, DEG. </s> <s xml:id="echoid-s9603" xml:space="preserve">Qua-<lb/>re erit, ut AE, ad AF, ita DE, ad DG: </s> <s xml:id="echoid-s9604" xml:space="preserve">Et per-<lb/>mutando, ut AE, ad DE, ita AF, ad DG. </s> <s xml:id="echoid-s9605" xml:space="preserve">Si <lb/>igitur in circulo chorda cuiuſlibet arcus ad <lb/>unam partem producatur, conueniatq́ue cum <lb/>diametro quauis, &</s> <s xml:id="echoid-s9606" xml:space="preserve">c. </s> <s xml:id="echoid-s9607" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s9608" xml:space="preserve"/> </p> <div xml:id="echoid-div818" type="float" level="2" n="2"> <figure xlink:label="fig-323-01" xlink:href="fig-323-01a"> <image file="323-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/323-01"/> </figure> <note position="right" xlink:label="note-323-02" xlink:href="note-323-02a" xml:space="preserve">28. primi. <lb/>Coroll. 4. <lb/>ſexti.</note> </div> </div> <div xml:id="echoid-div820" type="section" level="1" n="460"> <head xml:id="echoid-head492" xml:space="preserve">PROBL. 3. PROPOS. 6.</head> <note position="right" xml:space="preserve">Ex ſumma <lb/>data duorũ <lb/>arcuũ, quo <lb/>rum quili-<lb/>bet ſemicir <lb/>culo minor <lb/>ſit, vel duo-<lb/>rum angu-<lb/>lotum, vna <lb/>cũ propor-<lb/>ticne, quã <lb/>eorũ ſinus <lb/>habẽ@, vter-<lb/>que cogno <lb/>ſcitur.</note> <p> <s xml:id="echoid-s9609" xml:space="preserve">DATO aggregato duorum arcuum, quo-<lb/>rum ſinguli ſemicirculo ſint minores, vel duorum <lb/>angulorum rectilineorum, ſiue minus illud ſit, ſi <lb/>ue maius, quàm grad. </s> <s xml:id="echoid-s9610" xml:space="preserve">180. </s> <s xml:id="echoid-s9611" xml:space="preserve">vnà cum proportione, <lb/>quam eorum ſinus habent: </s> <s xml:id="echoid-s9612" xml:space="preserve">vtrum queillorum ſi <lb/>gillatim exhibere notum.</s> <s xml:id="echoid-s9613" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9614" xml:space="preserve">IN circulo ABCD, cuius centrum E, datum ſit primo aggregatum ar-<lb/> <anchor type="note" xlink:label="note-323-04a" xlink:href="note-323-04"/> cuum BF, FD, quorum ſinguli ſint ſemicirculo minores, uel angulorũ BEF, <lb/>FED, & </s> <s xml:id="echoid-s9615" xml:space="preserve">aggregatum tam arcuum, quàm angulorum minus, quàm grad. </s> <s xml:id="echoid-s9616" xml:space="preserve">180. <lb/></s> <s xml:id="echoid-s9617" xml:space="preserve">nimirum datum ſit grad. </s> <s xml:id="echoid-s9618" xml:space="preserve">130. </s> <s xml:id="echoid-s9619" xml:space="preserve">Data quoque ſit proportio ſinus arcus BF, vel <lb/>anguli BEF, ad ſinum arcus FD, uel anguli FED, eadẽ, quæ 10.</s> <s xml:id="echoid-s9620" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9621" xml:space="preserve">Opor-<lb/>tet ex his utrumque arcum BF, FD, uel utrumque angulum BEF, FED, <lb/>notum efficere. </s> <s xml:id="echoid-s9622" xml:space="preserve">Ducta chorda BD, ducatur ex puncto F, ubi dati arcus con-<lb/>iunguntur, diameter FC, ſecans chordam BD, in G. </s> <s xml:id="echoid-s9623" xml:space="preserve">Diuiſo quoque toto ar- <pb o="312" file="324" n="324" rhead=""/> cu BAD, bifariam in A, ſecabit ſemidiameter ducta EA, chordam BD, bi-<lb/> <anchor type="note" xlink:label="note-324-01a" xlink:href="note-324-01"/> fariam in H, ex lemmate ad defin. </s> <s xml:id="echoid-s9624" xml:space="preserve">ſinuum demonſtrato; </s> <s xml:id="echoid-s9625" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s9626" xml:space="preserve">ad an-<lb/> <anchor type="figure" xlink:label="fig-324-01a" xlink:href="fig-324-01"/> gulos rectos. </s> <s xml:id="echoid-s9627" xml:space="preserve">Quoniam vero proportio ſinus <lb/>arcus BF, ad ſinum arcus FD, ponitur, vt 10. <lb/></s> <s xml:id="echoid-s9628" xml:space="preserve"> <anchor type="note" xlink:label="note-324-02a" xlink:href="note-324-02"/> ad 5. </s> <s xml:id="echoid-s9629" xml:space="preserve">eſtq́ue vt ſinus arcus BF, ad ſinum ar-<lb/>cus FD, ita BG, ad GD; </s> <s xml:id="echoid-s9630" xml:space="preserve">erit quoque BG, <lb/>ad GD, vt 10. </s> <s xml:id="echoid-s9631" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9632" xml:space="preserve">Poſita igitur recta BG, <lb/>10. </s> <s xml:id="echoid-s9633" xml:space="preserve">erit GD, 5. </s> <s xml:id="echoid-s9634" xml:space="preserve">ac proinde tota BD, 15. <lb/></s> <s xml:id="echoid-s9635" xml:space="preserve">vtraque vero ſemiſsis BH, HD, 7 {1/2}. </s> <s xml:id="echoid-s9636" xml:space="preserve">& </s> <s xml:id="echoid-s9637" xml:space="preserve">de-<lb/>nique HG, differentia inter ſemiſſem BH, <lb/>& </s> <s xml:id="echoid-s9638" xml:space="preserve">maius ſegmentum BG, vel inter ſemiſſem <lb/>HD, & </s> <s xml:id="echoid-s9639" xml:space="preserve">minus ſegmentum GD, erit 2 {1/2}. </s> <s xml:id="echoid-s9640" xml:space="preserve"><lb/>Rurſus quia totus arcus BAD, ponitur grad. </s> <s xml:id="echoid-s9641" xml:space="preserve"><lb/>130. </s> <s xml:id="echoid-s9642" xml:space="preserve">erit vtraque ſemiſsis BA, AD, grad. </s> <s xml:id="echoid-s9643" xml:space="preserve"><lb/>65. </s> <s xml:id="echoid-s9644" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s9645" xml:space="preserve">vterque angulus BEA, <lb/>AED, graduum quoque 65. </s> <s xml:id="echoid-s9646" xml:space="preserve">Et quoniam ex ijs, quæ ad initium tangentium, <lb/>ſecantiumq́ue tradidimus, poſito ſinu toto EH, recta BH, tangens eſt angu-<lb/>li BEH, & </s> <s xml:id="echoid-s9647" xml:space="preserve">HG, tangens anguli HEG; </s> <s xml:id="echoid-s9648" xml:space="preserve">dabitur, ex tangentium tabula, tan-<lb/>gens grad. </s> <s xml:id="echoid-s9649" xml:space="preserve">65. </s> <s xml:id="echoid-s9650" xml:space="preserve">nempe BH, partium 214451. </s> <s xml:id="echoid-s9651" xml:space="preserve">Quare, vt tangentem anguli AEF, <lb/>nimirum HG, inueniamus, dicemus per auream regulam. </s> <s xml:id="echoid-s9652" xml:space="preserve">Si BH, ſemiſsis ag-<lb/>gregati terminorum proportionis datæ, nempe 7 {1/2}. </s> <s xml:id="echoid-s9653" xml:space="preserve">dat BH, tangentem ſe-<lb/>miſsis aggregati angulorum BEF, FED, vel arcuũ BF, FD, partium 214451. </s> <s xml:id="echoid-s9654" xml:space="preserve"><lb/>quid dabit HG, differentia inter ſemiſſem aggregati terminorum datæ pro-<lb/>portionis, & </s> <s xml:id="echoid-s9655" xml:space="preserve">vtrumlibet terminorum eiuſdem proportionis, nimirum 2 {1/2}? </s> <s xml:id="echoid-s9656" xml:space="preserve">in-<lb/>ueniemusq́; </s> <s xml:id="echoid-s9657" xml:space="preserve">tangentem HG, partium 71484. </s> <s xml:id="echoid-s9658" xml:space="preserve">vt hic factum vides.</s> <s xml:id="echoid-s9659" xml:space="preserve"/> </p> <div xml:id="echoid-div820" type="float" level="2" n="1"> <note position="right" xlink:label="note-323-04" xlink:href="note-323-04a" xml:space="preserve">Quãdo ag <lb/>gregatũ ar-<lb/>cuum, vel <lb/>angulorũ <lb/>minus eſt, <lb/>quã grad. <lb/>180.</note> <note position="left" xlink:label="note-324-01" xlink:href="note-324-01a" xml:space="preserve">3. tertij.</note> <figure xlink:label="fig-324-01" xlink:href="fig-324-01a"> <image file="324-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/324-01"/> </figure> <note position="left" xlink:label="note-324-02" xlink:href="note-324-02a" xml:space="preserve">4.huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>BH. # BH. # HG. # # HG. <lb/>7 {1/2}. # 214451. # 2 {1/2}? # ſit # 71484. <lb/></note> <p> <s xml:id="echoid-s9660" xml:space="preserve">Ex tabula ergo tangentium elicietur angulus HEG, hoc eſt, arcus AF, grad. <lb/></s> <s xml:id="echoid-s9661" xml:space="preserve">35. </s> <s xml:id="echoid-s9662" xml:space="preserve">Min. </s> <s xml:id="echoid-s9663" xml:space="preserve">34. </s> <s xml:id="echoid-s9664" xml:space="preserve">qui additus ſemiſsi AB, grad. </s> <s xml:id="echoid-s9665" xml:space="preserve">65. </s> <s xml:id="echoid-s9666" xml:space="preserve">componet arcum BF, maiorem, <lb/>a@q; </s> <s xml:id="echoid-s9667" xml:space="preserve">adeo & </s> <s xml:id="echoid-s9668" xml:space="preserve">angulum BEF, grad. </s> <s xml:id="echoid-s9669" xml:space="preserve">100. </s> <s xml:id="echoid-s9670" xml:space="preserve">Min. </s> <s xml:id="echoid-s9671" xml:space="preserve">34. </s> <s xml:id="echoid-s9672" xml:space="preserve">ablatus vero ex ſemiſſe AD, <lb/>relinquet arcum minorem FD, & </s> <s xml:id="echoid-s9673" xml:space="preserve">proinde & </s> <s xml:id="echoid-s9674" xml:space="preserve">angulum FED, grad.</s> <s xml:id="echoid-s9675" xml:space="preserve">29. </s> <s xml:id="echoid-s9676" xml:space="preserve">Min.</s> <s xml:id="echoid-s9677" xml:space="preserve">26.</s> <s xml:id="echoid-s9678" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9679" xml:space="preserve">ITAQVE, quando duo arcus ſimul minores ſunt, quàm ſemi@ir-<lb/> <anchor type="note" xlink:label="note-324-04a" xlink:href="note-324-04"/> culus, vel duo anguli ſimul duobus rectis minores: </s> <s xml:id="echoid-s9680" xml:space="preserve">Si fiat, vt ſemiſſis ag-<lb/>gregatiterminorum proportionis datæ ad tangentem ſemiſſis aggregati ar <lb/>cuum, vel angulorum (quærendo tangentem per partem proportionalem <lb/>reſpondentem 30. </s> <s xml:id="echoid-s9681" xml:space="preserve">ſecundis, ſi forte aggregatum bifariam diuidi nequeat <lb/>ſine Secundis.) </s> <s xml:id="echoid-s9682" xml:space="preserve">ita differentia inter ſemiſſem aggregati terminorum datæ <lb/>proportionis, & </s> <s xml:id="echoid-s9683" xml:space="preserve">alterutrum terminorum, ad aliud, reperietur tangens ar <lb/>cus, vel anguli, quo vterque arcus, angulusve quæſitus à ſemiſſe aggregati <lb/>eorundem differt. </s> <s xml:id="echoid-s9684" xml:space="preserve">Additus igitur arcus, vel angulus buius inuentæ tan-<lb/>gentis ad ſemiſſem dabit maiorem arcum, vel angulum; </s> <s xml:id="echoid-s9685" xml:space="preserve">ablat us vero ex <lb/>eadem ſemiſſe relinquet arcum, vel angulum minorem.</s> <s xml:id="echoid-s9686" xml:space="preserve"/> </p> <div xml:id="echoid-div821" type="float" level="2" n="2"> <note position="left" xlink:label="note-324-04" xlink:href="note-324-04a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9687" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s9688" xml:space="preserve">Producta ſemidiametro BE, ad K, & </s> <s xml:id="echoid-s9689" xml:space="preserve">diametro CF, produ-<lb/> <anchor type="note" xlink:label="note-324-05a" xlink:href="note-324-05"/> cta, donec in L, conueniat cum recta KDL, ex K, per D, ducta, agatur EI, ad <lb/> <anchor type="note" xlink:label="note-324-06a" xlink:href="note-324-06"/> DK, perpendicularis, quæ ipſam DK, bifariam ſecabit; </s> <s xml:id="echoid-s9690" xml:space="preserve">ac proinde cum late- <pb o="313" file="325" n="325" rhead=""/> ra DE, EI, lateribus KE, EI, æqualia ſint, & </s> <s xml:id="echoid-s9691" xml:space="preserve">baſis DI, baſi KI; </s> <s xml:id="echoid-s9692" xml:space="preserve">angulus DEI, <lb/>angulo KEI, æqualis erit. </s> <s xml:id="echoid-s9693" xml:space="preserve">Quia ergo arcus BFD, datus eſt grad. </s> <s xml:id="echoid-s9694" xml:space="preserve">130. </s> <s xml:id="echoid-s9695" xml:space="preserve">dabi-<lb/> <anchor type="note" xlink:label="note-325-01a" xlink:href="note-325-01"/> tur reliquus DK, de ſemicirculo grad. </s> <s xml:id="echoid-s9696" xml:space="preserve">50. </s> <s xml:id="echoid-s9697" xml:space="preserve">& </s> <s xml:id="echoid-s9698" xml:space="preserve">eius ſemiſsis, id eſt, angulus DEI, <lb/>grad. </s> <s xml:id="echoid-s9699" xml:space="preserve">25. </s> <s xml:id="echoid-s9700" xml:space="preserve">Rurſus quia finus arcus BF, ad ſinum arcus FD, ponitur, ut 10. </s> <s xml:id="echoid-s9701" xml:space="preserve">ad <lb/>5. </s> <s xml:id="echoid-s9702" xml:space="preserve">eſtq́ue idem ſinus arcus KF, qui arcus BF, vt in defin. </s> <s xml:id="echoid-s9703" xml:space="preserve">ſinuum oſtenſum eſt: <lb/></s> <s xml:id="echoid-s9704" xml:space="preserve">erit quoque ſinus arcus KF, ad ſinum arcus DF, vt 10.</s> <s xml:id="echoid-s9705" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9706" xml:space="preserve">Cum ergo ſit, vt <lb/>ſinus arcus KF, ad ſinum arcus DF, ita KL, ad DL; </s> <s xml:id="echoid-s9707" xml:space="preserve">erit etiam KL, ad DL, vt <lb/> <anchor type="note" xlink:label="note-325-02a" xlink:href="note-325-02"/> 10.</s> <s xml:id="echoid-s9708" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9709" xml:space="preserve">Poſita igitur KL, 10. </s> <s xml:id="echoid-s9710" xml:space="preserve">erit DL, 5: </s> <s xml:id="echoid-s9711" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s9712" xml:space="preserve">reliqua KD, 5. </s> <s xml:id="echoid-s9713" xml:space="preserve">& </s> <s xml:id="echoid-s9714" xml:space="preserve">eius <lb/>ſemiſsis ID, 2 {1/2}. </s> <s xml:id="echoid-s9715" xml:space="preserve">At quoniam, vt ad initium tangentium & </s> <s xml:id="echoid-s9716" xml:space="preserve">ſecantiũ diximus, <lb/>poſito ſinu toto EI, recta ID, tangens eſt anguli DEI, hoc eſt, grad. </s> <s xml:id="echoid-s9717" xml:space="preserve">25. </s> <s xml:id="echoid-s9718" xml:space="preserve">da-<lb/>bitur ID, ex tabula tangẽtium, partium 46631. </s> <s xml:id="echoid-s9719" xml:space="preserve">Dicemus ergo per auream re <lb/>gulam. </s> <s xml:id="echoid-s9720" xml:space="preserve">Si ID, ſemiſsis differentiæ inter terminos proportionis datæ, nempe <lb/>2 {1/2}. </s> <s xml:id="echoid-s9721" xml:space="preserve">dat ID, tangentem ſemiſsis differentiæ inter aggregatum datum, & </s> <s xml:id="echoid-s9722" xml:space="preserve">ſemi-<lb/>circulum, partium 46631. </s> <s xml:id="echoid-s9723" xml:space="preserve">quid dabit IL, compoſita ex ſemiſſe differentiæ in-<lb/>ter terminos datæ proportionis, & </s> <s xml:id="echoid-s9724" xml:space="preserve">conſequente eiuſdem proportionis, nimi-<lb/>rum 7 {1/2}? </s> <s xml:id="echoid-s9725" xml:space="preserve">reperiemusq́ue IL, partium 139893. </s> <s xml:id="echoid-s9726" xml:space="preserve">qualium ID, eſt 46631. </s> <s xml:id="echoid-s9727" xml:space="preserve">vel <lb/>EI, 100000. </s> <s xml:id="echoid-s9728" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s9729" xml:space="preserve"/> </p> <div xml:id="echoid-div822" type="float" level="2" n="3"> <note position="left" xlink:label="note-324-05" xlink:href="note-324-05a" xml:space="preserve">Alia demõ. <lb/>ſtratio.</note> <note position="left" xlink:label="note-324-06" xlink:href="note-324-06a" xml:space="preserve">3.tertij.</note> <note position="right" xlink:label="note-325-01" xlink:href="note-325-01a" xml:space="preserve">8.primi.</note> <note position="right" xlink:label="note-325-02" xlink:href="note-325-02a" xml:space="preserve">5.huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>ID. # ID. # IL. # # IL. <lb/>2 {1/2}. # 46631. # 7 {1/2}? # ſit # 139893. <lb/></note> <p> <s xml:id="echoid-s9730" xml:space="preserve">Cum ergo IL, ſit tangens anguli IEL, poſito ſinu toto EI, vt in tractatione <lb/>tangentium ac ſecantium tradidimus; </s> <s xml:id="echoid-s9731" xml:space="preserve">dabitur, ex tangentium tabula, angulus <lb/>IEF, grad. </s> <s xml:id="echoid-s9732" xml:space="preserve">54. </s> <s xml:id="echoid-s9733" xml:space="preserve">Min. </s> <s xml:id="echoid-s9734" xml:space="preserve">26. </s> <s xml:id="echoid-s9735" xml:space="preserve">Ablato ergo angulo DEI, grad.</s> <s xml:id="echoid-s9736" xml:space="preserve">25. </s> <s xml:id="echoid-s9737" xml:space="preserve">nimirum ſemiſſe <lb/>differentiæ inter datum aggregatum, & </s> <s xml:id="echoid-s9738" xml:space="preserve">ſemicirculum, reliquus erit angulus <lb/>FED, ac propterea & </s> <s xml:id="echoid-s9739" xml:space="preserve">arcus FD, minor, grad.</s> <s xml:id="echoid-s9740" xml:space="preserve">29. </s> <s xml:id="echoid-s9741" xml:space="preserve">Min. </s> <s xml:id="echoid-s9742" xml:space="preserve">26. </s> <s xml:id="echoid-s9743" xml:space="preserve">qui ſubtractus ex da-<lb/>to aggregato grad. </s> <s xml:id="echoid-s9744" xml:space="preserve">130. </s> <s xml:id="echoid-s9745" xml:space="preserve">relinquet angulum BEF, & </s> <s xml:id="echoid-s9746" xml:space="preserve">proinde & </s> <s xml:id="echoid-s9747" xml:space="preserve">arcum BF, <lb/>maiorem, grad.</s> <s xml:id="echoid-s9748" xml:space="preserve">100. </s> <s xml:id="echoid-s9749" xml:space="preserve">Min.</s> <s xml:id="echoid-s9750" xml:space="preserve">34. </s> <s xml:id="echoid-s9751" xml:space="preserve">vt prius.</s> <s xml:id="echoid-s9752" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9753" xml:space="preserve">IGITVR ſi fiat, vt ſemiſſis differentiæ inter terminos proportio-<lb/> <anchor type="note" xlink:label="note-325-04a" xlink:href="note-325-04"/> nis datæ ad tangentem ſemiſſis diff rentiæ inter aggregatum datum, & </s> <s xml:id="echoid-s9754" xml:space="preserve"><lb/>ſemicirculum, ita aggregatum ex ſemiſſe differentiæ inter terminos datæ <lb/>proportionis, & </s> <s xml:id="echoid-s9755" xml:space="preserve">conſequente eiuſdem proportionis, ad aliud, inuenietur <lb/>tangens anguli, à quo ſi dematur ſemiſſis differentiæ inter datum aggrega-<lb/>tum, & </s> <s xml:id="echoid-s9756" xml:space="preserve">ſemicir culum, reliquus erit angulus, ſeu arcus minor quæſitus: </s> <s xml:id="echoid-s9757" xml:space="preserve">qui <lb/>detractus ex aggregato dato, relinquet maiorẽ angulum, ſine arcũ quęſitũ.</s> <s xml:id="echoid-s9758" xml:space="preserve"/> </p> <div xml:id="echoid-div823" type="float" level="2" n="4"> <note position="right" xlink:label="note-325-04" xlink:href="note-325-04a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9759" xml:space="preserve">ALITER adhuc per ſolos ſinus ſine tangentibus. </s> <s xml:id="echoid-s9760" xml:space="preserve">Ijsdem poſitis, quo-<lb/> <anchor type="note" xlink:label="note-325-05a" xlink:href="note-325-05"/> niam vt in ſinubus declarauimus, poſito ſinu toto EB, recta BH, eſt ſinus an-<lb/>guli BEH, nempe ſemiſsis aggregati angulorum, vel arcuũ dati, nempe grad. <lb/></s> <s xml:id="echoid-s9761" xml:space="preserve">65. </s> <s xml:id="echoid-s9762" xml:space="preserve">& </s> <s xml:id="echoid-s9763" xml:space="preserve">HE, ſinus anguli EBH, grad. </s> <s xml:id="echoid-s9764" xml:space="preserve">25. </s> <s xml:id="echoid-s9765" xml:space="preserve">vtpote complementi anguli BEH; </s> <s xml:id="echoid-s9766" xml:space="preserve"><lb/>dabitur ex tabula ſinuum, BH, partium 90631. </s> <s xml:id="echoid-s9767" xml:space="preserve">at HE, partiũ 42262. </s> <s xml:id="echoid-s9768" xml:space="preserve">Quòd <lb/>ſi dicamus. </s> <s xml:id="echoid-s9769" xml:space="preserve">Si BH, ſemiſsis aggregati terminorum proportionis datæ, nempe <lb/>7 {1/2}.</s> <s xml:id="echoid-s9770" xml:space="preserve">dat BH, ſinum ſemiſsis aggregati angulorum, vel arcuum, partiũ 90631. </s> <s xml:id="echoid-s9771" xml:space="preserve"><lb/>quid dabit HG, differentia inter ſemiſſem aggregati terminorum proportio-<lb/>nis datæ, & </s> <s xml:id="echoid-s9772" xml:space="preserve">alterutrum terminorum, nimirum 2 {1/2}? </s> <s xml:id="echoid-s9773" xml:space="preserve">reperiemus HG, partium <lb/>30210. </s> <s xml:id="echoid-s9774" xml:space="preserve">qualium ſinus totus EB, eſt 100000. </s> <s xml:id="echoid-s9775" xml:space="preserve">vel EH, 42262. </s> <s xml:id="echoid-s9776" xml:space="preserve">vt hic patet.</s> <s xml:id="echoid-s9777" xml:space="preserve"/> </p> <div xml:id="echoid-div824" type="float" level="2" n="5"> <note position="right" xlink:label="note-325-05" xlink:href="note-325-05a" xml:space="preserve">Aliter abs-<lb/>que tangẽ-<lb/>tibus.</note> </div> <note position="right" xml:space="preserve"> <lb/>BH. # BH. # HG. # # HG. <lb/>7 {1/2}. # 90631. # 2 {1/2}? # ſit # 30210. <lb/></note> <p> <s xml:id="echoid-s9778" xml:space="preserve">Quia vero quadrata rectarum HE, HG, nempe 1786076644. </s> <s xml:id="echoid-s9779" xml:space="preserve">912644100. <lb/></s> <s xml:id="echoid-s9780" xml:space="preserve"> <anchor type="note" xlink:label="note-325-07a" xlink:href="note-325-07"/> <pb o="314" file="326" n="326" rhead=""/> quadrato rectæ EG, æqualia ſunt, ſi ea in vnam ſummam colligamus, fiet qua-<lb/>dratum rectæ EG, 2698720744. </s> <s xml:id="echoid-s9781" xml:space="preserve">cuius radix quadrata dabit EG, partium <lb/>51949. </s> <s xml:id="echoid-s9782" xml:space="preserve">Cum autem, poſito ſinu toto EG, recta HG, ſinus ſit anguli HEG, <lb/>vt in ſinuum defin. </s> <s xml:id="echoid-s9783" xml:space="preserve">diximus, dicemus rurſum. </s> <s xml:id="echoid-s9784" xml:space="preserve">Si EG, inuenta partium 51949. <lb/></s> <s xml:id="echoid-s9785" xml:space="preserve">dat EG, ſinum totum partium 100000. </s> <s xml:id="echoid-s9786" xml:space="preserve">quid dabit HG, inuenta partium <lb/>30210? </s> <s xml:id="echoid-s9787" xml:space="preserve">inueniemusque HG, ſinum anguli HEG, partiũ 58153. </s> <s xml:id="echoid-s9788" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s9789" xml:space="preserve"/> </p> <div xml:id="echoid-div825" type="float" level="2" n="6"> <note position="right" xlink:label="note-325-07" xlink:href="note-325-07a" xml:space="preserve">47. primi.</note> </div> <note position="right" xml:space="preserve"> <lb/>EG. # EG. # HG. # # HG. <lb/>51949. # 100000. # 30210? # ſit. # 58153. <lb/></note> <p> <s xml:id="echoid-s9790" xml:space="preserve">Ex ſinuum ergo tabula dabitur angulus HEG, ſiue arcus AF, grad. </s> <s xml:id="echoid-s9791" xml:space="preserve">35. </s> <s xml:id="echoid-s9792" xml:space="preserve">Min. <lb/></s> <s xml:id="echoid-s9793" xml:space="preserve">34. </s> <s xml:id="echoid-s9794" xml:space="preserve">qui additus ſemiſsi AB, grad. </s> <s xml:id="echoid-s9795" xml:space="preserve">65. </s> <s xml:id="echoid-s9796" xml:space="preserve">exhibebit maiorem arcum BF, ideoq́ue <lb/>& </s> <s xml:id="echoid-s9797" xml:space="preserve">angulum BEF, grad. </s> <s xml:id="echoid-s9798" xml:space="preserve">100. </s> <s xml:id="echoid-s9799" xml:space="preserve">Min. </s> <s xml:id="echoid-s9800" xml:space="preserve">34. </s> <s xml:id="echoid-s9801" xml:space="preserve">ablatus vero ex ſemiſſe AD, reliquum <lb/>faciet arcum minorem FD, atque adeo & </s> <s xml:id="echoid-s9802" xml:space="preserve">angulum FED, grad. </s> <s xml:id="echoid-s9803" xml:space="preserve">29. </s> <s xml:id="echoid-s9804" xml:space="preserve">Min.</s> <s xml:id="echoid-s9805" xml:space="preserve">26. </s> <s xml:id="echoid-s9806" xml:space="preserve"><lb/>vt prius.</s> <s xml:id="echoid-s9807" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9808" xml:space="preserve">QVOCIRCA, quando aggregatum duorum arcuum, vel angulo-<lb/> <anchor type="note" xlink:label="note-326-02a" xlink:href="note-326-02"/> rum minus eſt, quam grad. </s> <s xml:id="echoid-s9809" xml:space="preserve">180. </s> <s xml:id="echoid-s9810" xml:space="preserve">Si fiat, vt ſemiſſis aggregati terminorum <lb/>proportionis datæ ad ſinum ſemiſſis aggregatiarcuum, angulorumve, ita <lb/>differentia inter ſemiſſem aggregati terminorum datæ proportionis, & </s> <s xml:id="echoid-s9811" xml:space="preserve"><lb/>alterutrum terminorum, ad aliud, inuenietur numerus; </s> <s xml:id="echoid-s9812" xml:space="preserve">cuius quadr atum <lb/>ſi adiungatur quadrato ſinus complementi ſemiſſis aggregati arcuum, ſeu <lb/>angulorum; </s> <s xml:id="echoid-s9813" xml:space="preserve">Et tandem fiat, vt compoſiti buius numeri radix quadrata <lb/>ad ſinum totum, it a numerus per auream regulam nuper inuẽtus ad aliud, <lb/>producetur ſinus anguli, ſiue arcus, quo vter angulus, arcusve quæſitus <lb/>ab eorundem aggregati ſemiſſe differt. </s> <s xml:id="echoid-s9814" xml:space="preserve">Additus ergo arcus, ſiue angulus <lb/>buius ſinus inuenti ad ſemiſſem aggregati dati, dabit maiorem arcum, vel <lb/>angulum; </s> <s xml:id="echoid-s9815" xml:space="preserve">ablatus vero ex eadem ſemiſſe relinquet minorem arcum, ſiu@ <lb/>angulum. </s> <s xml:id="echoid-s9816" xml:space="preserve">Sed priores duæ viæ longe ſunt expeditiores; </s> <s xml:id="echoid-s9817" xml:space="preserve">vt perſpicuum eſt</s> </p> <div xml:id="echoid-div826" type="float" level="2" n="7"> <note position="left" xlink:label="note-326-02" xlink:href="note-326-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9818" xml:space="preserve">DETVR deinde aggregatum arcuum BC, CD, quorum ſinguli ſemicit <lb/> <anchor type="note" xlink:label="note-326-03a" xlink:href="note-326-03"/> culo quoq; </s> <s xml:id="echoid-s9819" xml:space="preserve">ſint minores, vel angulorum BEC, CED, at aggregatum tam <lb/>arcuum, quam angulorum ſuperet grad. </s> <s xml:id="echoid-s9820" xml:space="preserve">180. </s> <s xml:id="echoid-s9821" xml:space="preserve">nempe detur grad. </s> <s xml:id="echoid-s9822" xml:space="preserve">230. </s> <s xml:id="echoid-s9823" xml:space="preserve">Detut <lb/>item proportio ſinus arcus BC, vel anguli BEC, ad ſinum arcus CD, vel an-<lb/>guli CED, eadem, quæ 10. </s> <s xml:id="echoid-s9824" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9825" xml:space="preserve">Oportet ex his vtrumq; </s> <s xml:id="echoid-s9826" xml:space="preserve">arcum BC, CD, vel <lb/>vtrumq; </s> <s xml:id="echoid-s9827" xml:space="preserve">angulum BEC, CED, elicere. </s> <s xml:id="echoid-s9828" xml:space="preserve">Ducta diametro CF, & </s> <s xml:id="echoid-s9829" xml:space="preserve">detracto da-<lb/>to aggregato ex integro circulo, hoc eſt, ex grad. </s> <s xml:id="echoid-s9830" xml:space="preserve">360. </s> <s xml:id="echoid-s9831" xml:space="preserve">reliquum erit aggrega-<lb/>tum arcuum BF, FD, vel angulorum BEF, FED, grad. </s> <s xml:id="echoid-s9832" xml:space="preserve">130. </s> <s xml:id="echoid-s9833" xml:space="preserve">minus, quam <lb/>grad. </s> <s xml:id="echoid-s9834" xml:space="preserve">180. </s> <s xml:id="echoid-s9835" xml:space="preserve">Et quoniam arcus BC, BF, eundem ſinuum habent, necnon & </s> <s xml:id="echoid-s9836" xml:space="preserve">ar-<lb/>cus CD, FD, vt in deſin. </s> <s xml:id="echoid-s9837" xml:space="preserve">ſinuum diximus, data quoq; </s> <s xml:id="echoid-s9838" xml:space="preserve">erit proportio ſinuum <lb/>arcuum BF, FD, vel angulorum BEF, FED, eadem, quæ 10. </s> <s xml:id="echoid-s9839" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9840" xml:space="preserve">Quam <lb/>ob rem, vt iam demonſtratum eſt, inueniemus arcus BF, FD, vel angulos <lb/>BEF, FED, grad. </s> <s xml:id="echoid-s9841" xml:space="preserve">100. </s> <s xml:id="echoid-s9842" xml:space="preserve">Min. </s> <s xml:id="echoid-s9843" xml:space="preserve">34. </s> <s xml:id="echoid-s9844" xml:space="preserve">& </s> <s xml:id="echoid-s9845" xml:space="preserve">grad. </s> <s xml:id="echoid-s9846" xml:space="preserve">29. </s> <s xml:id="echoid-s9847" xml:space="preserve">Min. </s> <s xml:id="echoid-s9848" xml:space="preserve">26. </s> <s xml:id="echoid-s9849" xml:space="preserve">qui ex ſemicirculo, hoc <lb/>eſt, ex grad. </s> <s xml:id="echoid-s9850" xml:space="preserve">180. </s> <s xml:id="echoid-s9851" xml:space="preserve">ſublati relinquent arcus BC, CD, vel angulos BEC, CED, <lb/>grad. </s> <s xml:id="echoid-s9852" xml:space="preserve">79. </s> <s xml:id="echoid-s9853" xml:space="preserve">Min. </s> <s xml:id="echoid-s9854" xml:space="preserve">26. </s> <s xml:id="echoid-s9855" xml:space="preserve">& </s> <s xml:id="echoid-s9856" xml:space="preserve">grad. </s> <s xml:id="echoid-s9857" xml:space="preserve">150. </s> <s xml:id="echoid-s9858" xml:space="preserve">Min. </s> <s xml:id="echoid-s9859" xml:space="preserve">34.</s> <s xml:id="echoid-s9860" xml:space="preserve"/> </p> <div xml:id="echoid-div827" type="float" level="2" n="8"> <note position="left" xlink:label="note-326-03" xlink:href="note-326-03a" xml:space="preserve">Quádo ag-<lb/>gregatũ ar <lb/>cuũ, vel an <lb/>gulorũ ma <lb/>ius eſt, quá <lb/>grad.180.</note> </div> <p style="it"> <s xml:id="echoid-s9861" xml:space="preserve">QVANDO ergo aggregatum duorum arcuum, ſeu angulorum ma-<lb/> <anchor type="note" xlink:label="note-326-04a" xlink:href="note-326-04"/> ius eſt, quam grad. </s> <s xml:id="echoid-s9862" xml:space="preserve">180. </s> <s xml:id="echoid-s9863" xml:space="preserve">Si illud ex grad. </s> <s xml:id="echoid-s9864" xml:space="preserve">360. </s> <s xml:id="echoid-s9865" xml:space="preserve">auferamus, remanebit <lb/>aggregatum aliorum duorum arcuum, vel angulorum minus, quam grad.</s> <s xml:id="echoid-s9866" xml:space="preserve"> <pb o="315" file="327" n="327" rhead=""/> _180._ </s> <s xml:id="echoid-s9867" xml:space="preserve">Quare ſi, vtiam demonſtratum eſt, beneficio buius aggregatimino-<lb/>ris, & </s> <s xml:id="echoid-s9868" xml:space="preserve">proportionis datæ, vtrum arcum, vel angulum inquir amus, & </s> <s xml:id="echoid-s9869" xml:space="preserve"><lb/>vtrumq, inuentum ſigillatim ex grad. </s> <s xml:id="echoid-s9870" xml:space="preserve">_180._ </s> <s xml:id="echoid-s9871" xml:space="preserve">demamus, noti relinquentur <lb/>arcus, vel anguli quæſiti.</s> <s xml:id="echoid-s9872" xml:space="preserve"/> </p> <div xml:id="echoid-div828" type="float" level="2" n="9"> <note position="left" xlink:label="note-326-04" xlink:href="note-326-04a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s9873" xml:space="preserve">QVOD ſi quando proportio ſinuum data ſit proportio æqualitatis, hoc <lb/> <anchor type="note" xlink:label="note-327-01a" xlink:href="note-327-01"/> eſt, ſinus ſint æquales, erunt quoq; </s> <s xml:id="echoid-s9874" xml:space="preserve">tam duo arcus, quam duo anguli æqua-<lb/>les, vt in deſin. </s> <s xml:id="echoid-s9875" xml:space="preserve">ſinuum oſtendimus. </s> <s xml:id="echoid-s9876" xml:space="preserve">Quapropter ſemiſsis dati aggregati dabit <lb/>vtrumq; </s> <s xml:id="echoid-s9877" xml:space="preserve">arcum, ſiue angulum cognitum. </s> <s xml:id="echoid-s9878" xml:space="preserve">Dato igitur aggregato duorum ar-<lb/>cuum, quorum ſinguli ſemicirculo ſint minores, vel duorum angulorum re-<lb/>ctilineorum, &</s> <s xml:id="echoid-s9879" xml:space="preserve">c. </s> <s xml:id="echoid-s9880" xml:space="preserve">vtrumq; </s> <s xml:id="echoid-s9881" xml:space="preserve">illorum ſigillatim exhibuimus notum. </s> <s xml:id="echoid-s9882" xml:space="preserve">Quod fa-<lb/>ciendum erat.</s> <s xml:id="echoid-s9883" xml:space="preserve"/> </p> <div xml:id="echoid-div829" type="float" level="2" n="10"> <note position="right" xlink:label="note-327-01" xlink:href="note-327-01a" xml:space="preserve">Quando fi-<lb/>nuum pro-<lb/>portio da-<lb/>ta eſt {pro}por <lb/>tio æquali-<lb/>tatis.</note> </div> </div> <div xml:id="echoid-div831" type="section" level="1" n="461"> <head xml:id="echoid-head493" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s9884" xml:space="preserve">_SI_ aggregatum duorum arcuum, vel angulorum fuerit præcisè grad. </s> <s xml:id="echoid-s9885" xml:space="preserve">_180_ non <lb/> <anchor type="note" xlink:label="note-327-02a" xlink:href="note-327-02"/> poterunt arcus illi, vel anguli cognoſci, etiam ſi proportio, quam eorum ſinus habent, <lb/>data ſit. </s> <s xml:id="echoid-s9886" xml:space="preserve">Nam quomodocunq; </s> <s xml:id="echoid-s9887" xml:space="preserve">ſemicirculus in duos arcus ſecetur, habebunt ſemper <lb/>eorum ſinus proportionem æqualitatis, cum vnus, & </s> <s xml:id="echoid-s9888" xml:space="preserve">idem ſinus ſit vtriuſq; </s> <s xml:id="echoid-s9889" xml:space="preserve">arcus, vt <lb/>ad defin. </s> <s xml:id="echoid-s9890" xml:space="preserve">ſinuum demonſtrauimus. </s> <s xml:id="echoid-s9891" xml:space="preserve">_Ne_ceſſe eſt ergo, aggregatum datum vel minus eſ-<lb/>ſe, vel maius, quam grad. </s> <s xml:id="echoid-s9892" xml:space="preserve">_180._ </s> <s xml:id="echoid-s9893" xml:space="preserve">vt in propoſitione expreſſum eſt.</s> <s xml:id="echoid-s9894" xml:space="preserve"/> </p> <div xml:id="echoid-div831" type="float" level="2" n="1"> <note position="right" xlink:label="note-327-02" xlink:href="note-327-02a" xml:space="preserve">Quãdo ag-<lb/>gregatũ da <lb/>tũ continet <lb/>grad. 180. <lb/>problema <lb/>ſolui non <lb/>poteſt.</note> </div> </div> <div xml:id="echoid-div833" type="section" level="1" n="462"> <head xml:id="echoid-head494" xml:space="preserve">PROBL. 4. PROPOS. 7.</head> <note position="right" xml:space="preserve">Ex diffe-<lb/>rentia data <lb/>duorum, ar <lb/>cuũ, quorũ <lb/>quilibet ſe-<lb/>micirculo <lb/>minor ſit, <lb/>vel duorũ <lb/>angulorũ, <lb/>vna cũ pro <lb/>portione, <lb/>quã eorum <lb/>ſinus ha-<lb/>bent, vter <lb/>cognoſcit.</note> <p> <s xml:id="echoid-s9895" xml:space="preserve">DATA differentia duorum arcuum, quorum <lb/>ſinguli ſemicirculo ſint minores, vel duorum an-<lb/>gulorum rectilineorum, vna cum proportione, <lb/>quam eorum ſinus habent: </s> <s xml:id="echoid-s9896" xml:space="preserve">vtrumq; </s> <s xml:id="echoid-s9897" xml:space="preserve">illorum ſigil-<lb/>latim notum efficere.</s> <s xml:id="echoid-s9898" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9899" xml:space="preserve">IN circulo ABCD, cuius centrum E, ſuperet arcus BF, ſemicirculo <lb/> <anchor type="figure" xlink:label="fig-327-01a" xlink:href="fig-327-01"/> <anchor type="note" xlink:label="note-327-04a" xlink:href="note-327-04"/> minor arcum DF, arcu BD, vel angu-<lb/>lus BEF, angulum DEF, angulo BED, <lb/>ſitque differentia hæc, nempe arcus BD, vel <lb/>angulus BED, data grad. </s> <s xml:id="echoid-s9900" xml:space="preserve">60. </s> <s xml:id="echoid-s9901" xml:space="preserve">Proportio <lb/>quoque ſinus maioris arcus BF, vel angu-<lb/>li BEF, ad ſinum arcus minoris DF, vel <lb/>anguli DEF, ſit primo maioris inæqualita-<lb/>tis data, eadem, quæ 11. </s> <s xml:id="echoid-s9902" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9903" xml:space="preserve">quod quidem <lb/>contingit, quando duo arcus BF, DF, ſemi-<lb/>circulo ſunt minores ſimul ſumpti. </s> <s xml:id="echoid-s9904" xml:space="preserve">Oportet <lb/>ex his vtrumque arcum BF, DF, ſiue vtrum-<lb/>que angulum BEF, DEF, cognitum facere. <lb/></s> <s xml:id="echoid-s9905" xml:space="preserve">Ducta chorda BD, & </s> <s xml:id="echoid-s9906" xml:space="preserve">diametro CF, conue- <pb o="316" file="328" n="328" rhead=""/> nient hæ lineæ productæ ad partes D, F, vt in puncto G. </s> <s xml:id="echoid-s9907" xml:space="preserve">Cum enim ſinuum <lb/>proportio ſit data maioris inæqualitatis, maior erit ſinus arcus BF, hoc eſt, <lb/>perpendicularis ex B, ad CF, demiſſa, ſinu arcus DF, hoc eſt, perpendiculari <lb/>ex D, ad CF, demiſſa. </s> <s xml:id="echoid-s9908" xml:space="preserve">Quare minus diſtabit punctum D, à recta CF, quàm pun-<lb/>ctum B; </s> <s xml:id="echoid-s9909" xml:space="preserve">atque adeo tandem coibunt BD, CF, productæ ad partes D, F. </s> <s xml:id="echoid-s9910" xml:space="preserve">Quod <lb/>etiam ita probabitur. </s> <s xml:id="echoid-s9911" xml:space="preserve">Si ambo ſinus, hoc eſt, perpendiculares ex B, D, ad CF, <lb/> <anchor type="note" xlink:label="note-328-01a" xlink:href="note-328-01"/> <anchor type="figure" xlink:label="fig-328-01a" xlink:href="fig-328-01"/> demiſſæ eſſent æquales, cum ipſæ ſint paral-<lb/>lelæ, eſſent quoque BD, CF, parallelæ. </s> <s xml:id="echoid-s9912" xml:space="preserve">Cũ <lb/> <anchor type="note" xlink:label="note-328-02a" xlink:href="note-328-02"/> ergo perpendicularis ex D, demiſſa minor <lb/>ſit, eſſicitur, vt conueniant, &</s> <s xml:id="echoid-s9913" xml:space="preserve">c. </s> <s xml:id="echoid-s9914" xml:space="preserve">Diuiſo dein-<lb/>de arcu BD, bifariam in A, ſecabit ſemidia-<lb/>meter ducta EA, chordã BD, quoq; </s> <s xml:id="echoid-s9915" xml:space="preserve">bifariã <lb/>in H, ex lemmate ad defin. </s> <s xml:id="echoid-s9916" xml:space="preserve">ſinuum demon-<lb/> <anchor type="note" xlink:label="note-328-03a" xlink:href="note-328-03"/> ſtrato; </s> <s xml:id="echoid-s9917" xml:space="preserve">& </s> <s xml:id="echoid-s9918" xml:space="preserve">proinde & </s> <s xml:id="echoid-s9919" xml:space="preserve">ad angulos rectos. </s> <s xml:id="echoid-s9920" xml:space="preserve">Quo-<lb/>niam vero proportio ſinus arcus BF, ad ſi-<lb/>num arcus DF, eſt, ex hypotheſi, vt 11. </s> <s xml:id="echoid-s9921" xml:space="preserve">ad <lb/>5. </s> <s xml:id="echoid-s9922" xml:space="preserve">eſtq́ue vt ſinus arcus BF, ad ſinum arcus <lb/> <anchor type="note" xlink:label="note-328-04a" xlink:href="note-328-04"/> DF, ita BG, ad DG; </s> <s xml:id="echoid-s9923" xml:space="preserve">erit quoque BG, ad <lb/>DG, vt 11. </s> <s xml:id="echoid-s9924" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s9925" xml:space="preserve">Poſita igitur recta BG, 11. <lb/></s> <s xml:id="echoid-s9926" xml:space="preserve">erit DG, 5. </s> <s xml:id="echoid-s9927" xml:space="preserve">ac proinde reliqua BD, 6. </s> <s xml:id="echoid-s9928" xml:space="preserve">vtraque uero ſemiſsis BH, HD, 3.</s> <s xml:id="echoid-s9929" xml:space="preserve">ac <lb/>denique HG, 8. </s> <s xml:id="echoid-s9930" xml:space="preserve">Rurſus quia arcus BD, ponitur grad. </s> <s xml:id="echoid-s9931" xml:space="preserve">60. </s> <s xml:id="echoid-s9932" xml:space="preserve">erit utraque ſemiſ-<lb/>ſis BA, AD, grad. </s> <s xml:id="echoid-s9933" xml:space="preserve">30. </s> <s xml:id="echoid-s9934" xml:space="preserve">proptereaq́ue & </s> <s xml:id="echoid-s9935" xml:space="preserve">uterque angulus BEA, AED, gra-<lb/>duum quoque 30. </s> <s xml:id="echoid-s9936" xml:space="preserve">Et quia, poſito ſinu toto EH, recta HD, tangens eſt angu-<lb/>li DEH, & </s> <s xml:id="echoid-s9937" xml:space="preserve">HG, tangens anguli HEG, ut ad initium tangentium, atque <lb/>ſecãtium monuimus; </s> <s xml:id="echoid-s9938" xml:space="preserve">dabitur ex tangentium tabula, tangens grad. </s> <s xml:id="echoid-s9939" xml:space="preserve">30. </s> <s xml:id="echoid-s9940" xml:space="preserve">hoc eſt, <lb/>HD, partium 57735. </s> <s xml:id="echoid-s9941" xml:space="preserve">Quapropter, ut tangentem HG, anguli HEG, cognoſ-<lb/>camus, dicemus per auream regulam. </s> <s xml:id="echoid-s9942" xml:space="preserve">Si HD, ſemiſsis differentiæ terminorum <lb/>proportionis datæ, nempe 3. </s> <s xml:id="echoid-s9943" xml:space="preserve">dat HD, tangentem ſemiſsis differentiæ datæ <lb/>arcuum BF, FD, uel angulorum BEF, DEF, partium 57735. </s> <s xml:id="echoid-s9944" xml:space="preserve">quid dabit <lb/>HG, aggregatum ex ſemiſſe differentiæ terminorum datæ proportionis, & </s> <s xml:id="echoid-s9945" xml:space="preserve"><lb/>conſequente eiuſdem proportionis, nimirum 8? </s> <s xml:id="echoid-s9946" xml:space="preserve">prouenietq́ue HG, tangens <lb/>partium 153960. </s> <s xml:id="echoid-s9947" xml:space="preserve">ut hic apparet.</s> <s xml:id="echoid-s9948" xml:space="preserve"/> </p> <div xml:id="echoid-div833" type="float" level="2" n="1"> <figure xlink:label="fig-327-01" xlink:href="fig-327-01a"> <image file="327-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/327-01"/> </figure> <note position="right" xlink:label="note-327-04" xlink:href="note-327-04a" xml:space="preserve">Quãdo ſi-<lb/>nꝰ maioris <lb/>arcus, vel <lb/>anguli ad <lb/>ſinum mi-<lb/>noris ha-<lb/>bet propor <lb/>tionẽ maio <lb/>ris inęqua-<lb/>litatis.</note> <note position="left" xlink:label="note-328-01" xlink:href="note-328-01a" xml:space="preserve">28.primi.</note> <figure xlink:label="fig-328-01" xlink:href="fig-328-01a"> <image file="328-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/328-01"/> </figure> <note position="left" xlink:label="note-328-02" xlink:href="note-328-02a" xml:space="preserve">33.primi.</note> <note position="left" xlink:label="note-328-03" xlink:href="note-328-03a" xml:space="preserve">3.tertij.</note> <note position="left" xlink:label="note-328-04" xlink:href="note-328-04a" xml:space="preserve">5.huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>HD. # HD. # HG. # # HG. <lb/>3. # 57735? # 8? # ſit. # 153960. <lb/></note> <p> <s xml:id="echoid-s9949" xml:space="preserve">In tangentium autem tabula hæc tangens inuenta offert angulum AEF, ſiue <lb/>arcum AF, grad. </s> <s xml:id="echoid-s9950" xml:space="preserve">57. </s> <s xml:id="echoid-s9951" xml:space="preserve">cui ſi addatur ſemiſsis AB, grad. </s> <s xml:id="echoid-s9952" xml:space="preserve">30. </s> <s xml:id="echoid-s9953" xml:space="preserve">dabitur maior arcus <lb/>BF, ſiue angulus BEF, grad. </s> <s xml:id="echoid-s9954" xml:space="preserve">87. </s> <s xml:id="echoid-s9955" xml:space="preserve">ſi uero ab eodem ſubtrahatur ſemiſsis AD, <lb/>grad.</s> <s xml:id="echoid-s9956" xml:space="preserve">30. </s> <s xml:id="echoid-s9957" xml:space="preserve">remanebit minor areus DF, uel angulus DEF, grad.</s> <s xml:id="echoid-s9958" xml:space="preserve">27.</s> <s xml:id="echoid-s9959" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s9960" xml:space="preserve">IGITVR quando proportio ſinus maioris arcus, vel anguli, ad ſi-<lb/> <anchor type="note" xlink:label="note-328-06a" xlink:href="note-328-06"/> nũ minoris eſt maioris inæqualitatis: </s> <s xml:id="echoid-s9961" xml:space="preserve">Si ſiat, vt ſemiſſis differentiæ termi-<lb/>norum proportionis datæ ad tangentem ſemiſſis differẽtiæ arcuum, vel an-<lb/>gulorum datæ, ita aggregatum ex ſemiſſe differentiæ terminorum propor-<lb/>tionis, & </s> <s xml:id="echoid-s9962" xml:space="preserve">conſequente proportionis ad aliud, producetur tangens arcus, <lb/>vel anguli, qui ſemiſſi differentiæ arcuum, vel angulorum datæ additus <lb/>componit maiorem arcum, ſeu angulum; </s> <s xml:id="echoid-s9963" xml:space="preserve">& </s> <s xml:id="echoid-s9964" xml:space="preserve">ſi ab eodem ſemiſſis dicta <lb/>ſubducatur, remanet arcus, vel angulus minor.</s> <s xml:id="echoid-s9965" xml:space="preserve"/> </p> <div xml:id="echoid-div834" type="float" level="2" n="2"> <note position="left" xlink:label="note-328-06" xlink:href="note-328-06a" xml:space="preserve">Praxis.</note> </div> <pb o="317" file="329" n="329" rhead=""/> <p> <s xml:id="echoid-s9966" xml:space="preserve">ALITER ſine tangentibus per ſolos ſinus. </s> <s xml:id="echoid-s9967" xml:space="preserve">Iisdem poſitis, quoniam, per <lb/> <anchor type="note" xlink:label="note-329-01a" xlink:href="note-329-01"/> ea, quæ in ſinuum deſin. </s> <s xml:id="echoid-s9968" xml:space="preserve">oſtendimus, poſito ſinu toto ED, recta HD, ſinus <lb/>eſt anguli HED, nimirum ſ@miſsis differentiæ arcuum, vel angulorum datæ, <lb/>hoc eſt, grad. </s> <s xml:id="echoid-s9969" xml:space="preserve">30. </s> <s xml:id="echoid-s9970" xml:space="preserve">& </s> <s xml:id="echoid-s9971" xml:space="preserve">HE, ſinus anguli HDE, grad. </s> <s xml:id="echoid-s9972" xml:space="preserve">60. </s> <s xml:id="echoid-s9973" xml:space="preserve">vt pote complementi <lb/>anguli HED; </s> <s xml:id="echoid-s9974" xml:space="preserve">dabitur ex ſinuum tabula, HD, partium 50000. </s> <s xml:id="echoid-s9975" xml:space="preserve">& </s> <s xml:id="echoid-s9976" xml:space="preserve">EH, par-<lb/>tium 86603. </s> <s xml:id="echoid-s9977" xml:space="preserve">Iam vero ſi dicamus. </s> <s xml:id="echoid-s9978" xml:space="preserve">Si HD, ſemiſsis differentiæ terminorum <lb/>proportionis datæ, nimirum 3. </s> <s xml:id="echoid-s9979" xml:space="preserve">dat HD, ſinum @0000. </s> <s xml:id="echoid-s9980" xml:space="preserve">vtpote ſinum ſemiſsis <lb/>differentiæ arcuũ, angulorum ve datæ, quid dabit HG, aggregatũ ex ſemiſſe <lb/>differentiæ terminorum proportionis, & </s> <s xml:id="echoid-s9981" xml:space="preserve">conſequente eiuſdem proportionis, <lb/>nempe 8? </s> <s xml:id="echoid-s9982" xml:space="preserve">inueniemus HG, eſſe 133333. </s> <s xml:id="echoid-s9983" xml:space="preserve">reſpectu ſinus totius ED, vt hic vides.</s> <s xml:id="echoid-s9984" xml:space="preserve"/> </p> <div xml:id="echoid-div835" type="float" level="2" n="3"> <note position="right" xlink:label="note-329-01" xlink:href="note-329-01a" xml:space="preserve">Aliter ſine <lb/>tãgẽtibus.</note> </div> <note position="right" xml:space="preserve"> <lb/>HD. # HD. # HG. # # HG. <lb/>3. # 50000. # 8? # ſit. # 133333. <lb/></note> <p> <s xml:id="echoid-s9985" xml:space="preserve">Igitur, cum quadrata rectarum EH, HG, nempe 7499906404. </s> <s xml:id="echoid-s9986" xml:space="preserve">17777688889. <lb/></s> <s xml:id="echoid-s9987" xml:space="preserve">æqualia ſint quadrato rectæ EG, fiet quadratum rectæ EG, 25277595293. </s> <s xml:id="echoid-s9988" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-329-03a" xlink:href="note-329-03"/> cuius radix quadrata indicabit rectam EG, eſſe 158989. </s> <s xml:id="echoid-s9989" xml:space="preserve">reſpectu ſinus totius <lb/>ED. </s> <s xml:id="echoid-s9990" xml:space="preserve">Cum autem, vt in noſtris ſinubus diximus, poſito ſinutoto EG, recta <lb/>HG, ſit ſinus anguli HEG; </s> <s xml:id="echoid-s9991" xml:space="preserve">dicemus rurſum. </s> <s xml:id="echoid-s9992" xml:space="preserve">Si EG, inuenta partiũ 158989. <lb/></s> <s xml:id="echoid-s9993" xml:space="preserve">dat EG, ſinum totum partium 100000. </s> <s xml:id="echoid-s9994" xml:space="preserve">quid dabit HG, inuenta partium <lb/>133333? </s> <s xml:id="echoid-s9995" xml:space="preserve">reperiemusq́; </s> <s xml:id="echoid-s9996" xml:space="preserve">HG, ſinum anguli HEG, partiũ 83863. </s> <s xml:id="echoid-s9997" xml:space="preserve">vt hic apparet.</s> <s xml:id="echoid-s9998" xml:space="preserve"/> </p> <div xml:id="echoid-div836" type="float" level="2" n="4"> <note position="right" xlink:label="note-329-03" xlink:href="note-329-03a" xml:space="preserve">47.primi.</note> </div> <note position="right" xml:space="preserve"> <lb/>EG. # EG. # HG. # # HG. <lb/>158989. # 100000. # 133333? # ſit. # 83863. <lb/></note> <p> <s xml:id="echoid-s9999" xml:space="preserve">Hic ſinus in tabula ſinuum monſtratarcum grad. </s> <s xml:id="echoid-s10000" xml:space="preserve">57. </s> <s xml:id="echoid-s10001" xml:space="preserve">Tantus eſt ergo angulus <lb/>AEF, ſiue arcus AF; </s> <s xml:id="echoid-s10002" xml:space="preserve">cui ſi adijciatur ſemiſsis AB, grad. </s> <s xml:id="echoid-s10003" xml:space="preserve">30. </s> <s xml:id="echoid-s10004" xml:space="preserve">fiet arcus maior <lb/>BF, & </s> <s xml:id="echoid-s10005" xml:space="preserve">angulus BEF, grad. </s> <s xml:id="echoid-s10006" xml:space="preserve">87. </s> <s xml:id="echoid-s10007" xml:space="preserve">Sivero ab eodem minuatur ſemiſsis AD, <lb/>grad. </s> <s xml:id="echoid-s10008" xml:space="preserve">30. </s> <s xml:id="echoid-s10009" xml:space="preserve">reliquus erit minor arcus DF, & </s> <s xml:id="echoid-s10010" xml:space="preserve">angulus DEF, grad. </s> <s xml:id="echoid-s10011" xml:space="preserve">27. </s> <s xml:id="echoid-s10012" xml:space="preserve">vt prius.</s> <s xml:id="echoid-s10013" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10014" xml:space="preserve">SI igitur (quando proportio ſinuum data eſt maioris inæqualitatis) <lb/> <anchor type="note" xlink:label="note-329-05a" xlink:href="note-329-05"/> fiat, vt ſemiſſis differentiæ terminorum proportionis datæ ad ſinum ſemiſ-<lb/>ſis differentiæ arcuum, vel angulorum datæ, ita aggregatum ex ſemiſſe <lb/>differentiæ terminorum proportionis, & </s> <s xml:id="echoid-s10015" xml:space="preserve">conſequente eiuſdem proportio-<lb/>nis, ad aliud, inuenietur numerus; </s> <s xml:id="echoid-s10016" xml:space="preserve">cuius quadratum ſi adijciatur quadra-<lb/>to ſinus compl@menti ſemiſſis differentiæ arcuum, vel angulorum datæ: </s> <s xml:id="echoid-s10017" xml:space="preserve">Et <lb/>tandem fiat, vt compoſiti buius numeriradix quadrata ad ſinum totum, <lb/>itanumerus per regulam auream nuper inuentus ad aliud, inuenietur ſi-<lb/>nus anguli, ſiue arcus, cui ſi addatur ſemiſſis differentiæ arcuum, vel an-<lb/>gulorum datæ, notus fict maior arcus, ſiue angulus: </s> <s xml:id="echoid-s10018" xml:space="preserve">Ab eodem vero ſi <lb/>eadem ſemiſſis detrahatur, remanebit minor arcus, angulusve cognitus. <lb/></s> <s xml:id="echoid-s10019" xml:space="preserve">Sed prior ratio breuior eſt, vt conſtat.</s> <s xml:id="echoid-s10020" xml:space="preserve"/> </p> <div xml:id="echoid-div837" type="float" level="2" n="5"> <note position="right" xlink:label="note-329-05" xlink:href="note-329-05a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s10021" xml:space="preserve">ADHVC aliter tam per tangentes, quam per ſinus. </s> <s xml:id="echoid-s10022" xml:space="preserve">Ijſdem poſitis, & </s> <s xml:id="echoid-s10023" xml:space="preserve"><lb/>extenſa recta BE, vſq; </s> <s xml:id="echoid-s10024" xml:space="preserve">ad I: </s> <s xml:id="echoid-s10025" xml:space="preserve">Quoniam arcus BD, hoc eſt, differentia arcuum <lb/>BF, DF, datur grad. </s> <s xml:id="echoid-s10026" xml:space="preserve">60. </s> <s xml:id="echoid-s10027" xml:space="preserve">dabitur reliquus ſemicirculiarcus DI, nimirum ag-<lb/>gregatum arcuum DF, FI, grad. </s> <s xml:id="echoid-s10028" xml:space="preserve">120. </s> <s xml:id="echoid-s10029" xml:space="preserve">Datur autem & </s> <s xml:id="echoid-s10030" xml:space="preserve">proportio ſinus arcus <lb/>BF, hoc eſt, arcus FI, (cum arcus BF, FI, eundem ſinum habeant, vt in de-<lb/>fin. </s> <s xml:id="echoid-s10031" xml:space="preserve">ſinuum diximus.) </s> <s xml:id="echoid-s10032" xml:space="preserve">ad ſinum arcus DF, eadem, quæ 11. </s> <s xml:id="echoid-s10033" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s10034" xml:space="preserve">Quare, vt de-<lb/>monſtratum eſt, vterq; </s> <s xml:id="echoid-s10035" xml:space="preserve">arcus IF, DF, cognoſcetur, quorum DF, eſt minor <lb/> <anchor type="note" xlink:label="note-329-06a" xlink:href="note-329-06"/> propoſitorum arcuum: </s> <s xml:id="echoid-s10036" xml:space="preserve">at FI, complementum maioris BF, vſq; </s> <s xml:id="echoid-s10037" xml:space="preserve">ad ſemicir- <pb o="318" file="330" n="330" rhead=""/> culum, ac proinde ex ſemicirculo ſublatus maiorẽ BF, notum relinquet. </s> <s xml:id="echoid-s10038" xml:space="preserve">Erit <lb/> <anchor type="note" xlink:label="note-330-01a" xlink:href="note-330-01"/> autẽ ſemper arcus IF, maior, quam DF, propterea quod ęqualis eſt arcus IF, <lb/>arcui BC, qui maior eſt arcu DF, quòd illius ſinus maior ponatur ſinu huius.</s> <s xml:id="echoid-s10039" xml:space="preserve"/> </p> <div xml:id="echoid-div838" type="float" level="2" n="6"> <note position="right" xlink:label="note-329-06" xlink:href="note-329-06a" xml:space="preserve">6.huius.</note> <note position="left" xlink:label="note-330-01" xlink:href="note-330-01a" xml:space="preserve">26. tertij.</note> </div> <p style="it"> <s xml:id="echoid-s10040" xml:space="preserve">QVOCIRCA ijſdem poſitis: </s> <s xml:id="echoid-s10041" xml:space="preserve">Siex data proportione ſinuum ma-<lb/> <anchor type="note" xlink:label="note-330-02a" xlink:href="note-330-02"/> ioris inæqualitatis, & </s> <s xml:id="echoid-s10042" xml:space="preserve">ex arcu, quirelinquitur poſt detractionem differen <lb/>tiæ datæ ex ſemicirculo, tanquam aggregato duorum arcuum, inquiran-<lb/>tur duo arcus buius aggregati, vt in antecedente propoſ. </s> <s xml:id="echoid-s10043" xml:space="preserve">oſtenſum eſt; </s> <s xml:id="echoid-s10044" xml:space="preserve">da-<lb/>bit maior inuentus, ſi ex ſemicirculo auferatur, maiorem arcum, atque <lb/>adeo & </s> <s xml:id="echoid-s10045" xml:space="preserve">angulum propoſitum; </s> <s xml:id="echoid-s10046" xml:space="preserve">minor vero inuẽtus erit minor propoſitus.</s> <s xml:id="echoid-s10047" xml:space="preserve"/> </p> <div xml:id="echoid-div839" type="float" level="2" n="7"> <note position="left" xlink:label="note-330-02" xlink:href="note-330-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s10048" xml:space="preserve">DEINDE ſuperet arcus DBC, ſemicirculo minor arcũ BC, arcu DB, vel <lb/> <anchor type="note" xlink:label="note-330-03a" xlink:href="note-330-03"/> angulus DEC, angulũ BEC, angulo DEB; </s> <s xml:id="echoid-s10049" xml:space="preserve">ſitq́; </s> <s xml:id="echoid-s10050" xml:space="preserve">differẽtia hęc, nẽpe arcus DB, <lb/>vel angulus DEB, data grad.</s> <s xml:id="echoid-s10051" xml:space="preserve">60. </s> <s xml:id="echoid-s10052" xml:space="preserve">Proportio quoq; </s> <s xml:id="echoid-s10053" xml:space="preserve">ſinus arcus maioris DBC, <lb/>vel anguli DEC, ad ſinũ arcus minoris BC, vel anguli BEC, ſit data, & </s> <s xml:id="echoid-s10054" xml:space="preserve">mi-<lb/>noris inæqualitatis, eadem, quæ 5. </s> <s xml:id="echoid-s10055" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10056" xml:space="preserve">quod quidem accidit, quando duo <lb/>arcus ſimul ſumpti DBC, BC, ſemicirculũ excedũt. </s> <s xml:id="echoid-s10057" xml:space="preserve">Oportet ex his vtrumq; <lb/></s> <s xml:id="echoid-s10058" xml:space="preserve">arcum DBC, BC, vel vtrumq; </s> <s xml:id="echoid-s10059" xml:space="preserve">angulum DEC, BEC, notum fieri. </s> <s xml:id="echoid-s10060" xml:space="preserve">Ijſdem <lb/>conſtructis, quæ prius, non conueniet chorda DB, cum diametro CF, ad par <lb/>tes B, C, minoris arcus producta, ſed ad partes D, F, vt ex ijs, quæ ad initium <lb/>huius propoſ. </s> <s xml:id="echoid-s10061" xml:space="preserve">oſtendimus, manifeſtum eſt. </s> <s xml:id="echoid-s10062" xml:space="preserve">Quia vero tam arcus DBC, DF, <lb/> <anchor type="figure" xlink:label="fig-330-01a" xlink:href="fig-330-01"/> eundem ſinum habent, quam arcus BC, BF, erit <lb/>quoq; </s> <s xml:id="echoid-s10063" xml:space="preserve">proportio ſinus arcus DF, ad ſinum ar-<lb/>cus BF, data, eadem, quæ 5. </s> <s xml:id="echoid-s10064" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10065" xml:space="preserve">& </s> <s xml:id="echoid-s10066" xml:space="preserve">proinde <lb/>proportio ſinus arcus BF, ad ſinum arcus DF, <lb/>vt 11. </s> <s xml:id="echoid-s10067" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s10068" xml:space="preserve">nempe maioris inæqualitatis. </s> <s xml:id="echoid-s10069" xml:space="preserve">Qua-<lb/>recum arcus BF, DF, eandem differentiam <lb/>habeant BD, datam, reperiemus vtrumq; </s> <s xml:id="echoid-s10070" xml:space="preserve">ar-<lb/>cum BF, DF, atq; </s> <s xml:id="echoid-s10071" xml:space="preserve">adeo & </s> <s xml:id="echoid-s10072" xml:space="preserve">vtrumq; </s> <s xml:id="echoid-s10073" xml:space="preserve">angulum <lb/>BEF, DEF, vt ante demonſtrauimus, illum ni-<lb/>mirum grad. </s> <s xml:id="echoid-s10074" xml:space="preserve">87. </s> <s xml:id="echoid-s10075" xml:space="preserve">hunc vero grad. </s> <s xml:id="echoid-s10076" xml:space="preserve">27. </s> <s xml:id="echoid-s10077" xml:space="preserve">qui ex ſe-<lb/>micirculo ſigillatim detracti relinquent mino-<lb/>rem arcum propoſitum BC, grad. </s> <s xml:id="echoid-s10078" xml:space="preserve">93. </s> <s xml:id="echoid-s10079" xml:space="preserve">maiorem <lb/>vero DBC, grad. </s> <s xml:id="echoid-s10080" xml:space="preserve">153.</s> <s xml:id="echoid-s10081" xml:space="preserve"/> </p> <div xml:id="echoid-div840" type="float" level="2" n="8"> <note position="left" xlink:label="note-330-03" xlink:href="note-330-03a" xml:space="preserve">Quãdo ſi-<lb/>nus maio-<lb/>ris arcus, <lb/>aut anguli <lb/>ad ſinum <lb/>minoris {pro} <lb/>portionem <lb/>minoris <lb/>inęqualita <lb/>tis habet.</note> <figure xlink:label="fig-330-01" xlink:href="fig-330-01a"> <image file="330-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/330-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s10082" xml:space="preserve">ITAQVE, quando proportio ſinuum data eſt minoris inæqualita-<lb/> <anchor type="note" xlink:label="note-330-04a" xlink:href="note-330-04"/> tis: </s> <s xml:id="echoid-s10083" xml:space="preserve">Si inuertatur, vt fiat maioris inęqualitatis proportio, & </s> <s xml:id="echoid-s10084" xml:space="preserve">ex hac, et dif <lb/>ferentia data inquirantur duo arcus, dabit maior inuentus, ſi ex ſemicir <lb/>culo dematur, minorem arcum, atq; </s> <s xml:id="echoid-s10085" xml:space="preserve">angulum propoſitum, minor vero, ſi <lb/>ex eodem ſemicirculo auſeratur, maiorem.</s> <s xml:id="echoid-s10086" xml:space="preserve"/> </p> <div xml:id="echoid-div841" type="float" level="2" n="9"> <note position="left" xlink:label="note-330-04" xlink:href="note-330-04a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s10087" xml:space="preserve">IAM vero ſi quãdo ſinuum data proportio fuerit æqualitatis; </s> <s xml:id="echoid-s10088" xml:space="preserve">quod qui-<lb/> <anchor type="note" xlink:label="note-330-05a" xlink:href="note-330-05"/> dem euenit, quando duo arcus propoſiti BF, DF, ſemicirculo æquantur, <lb/>erunt arcus DF, BC, æquales, vt in ſinuum defin. </s> <s xml:id="echoid-s10089" xml:space="preserve">demonſtrauimus, ob æqua-<lb/>litatem ſinuum.</s> <s xml:id="echoid-s10090" xml:space="preserve"/> </p> <div xml:id="echoid-div842" type="float" level="2" n="10"> <note position="left" xlink:label="note-330-05" xlink:href="note-330-05a" xml:space="preserve">Quandò <lb/>proportio <lb/>ſinuũ eſt ę-<lb/>qualitatis.</note> </div> <p style="it"> <s xml:id="echoid-s10091" xml:space="preserve">QVARE ſitunc data differentia grad. </s> <s xml:id="echoid-s10092" xml:space="preserve">60. </s> <s xml:id="echoid-s10093" xml:space="preserve">nempe arcus BD, ex ſe-<lb/> <anchor type="note" xlink:label="note-330-06a" xlink:href="note-330-06"/> micirculo detrabatur, & </s> <s xml:id="echoid-s10094" xml:space="preserve">reſidui arcus grad. </s> <s xml:id="echoid-s10095" xml:space="preserve">120. </s> <s xml:id="echoid-s10096" xml:space="preserve">ſemiſſis, nempe grad. <lb/></s> <s xml:id="echoid-s10097" xml:space="preserve">60. </s> <s xml:id="echoid-s10098" xml:space="preserve">ad differentiam addatur, componetur maior arcus BF, ſiue angulus <pb o="319" file="331" n="331" rhead=""/> BEF, grad. </s> <s xml:id="echoid-s10099" xml:space="preserve">_120._ </s> <s xml:id="echoid-s10100" xml:space="preserve">Ipſa vero ſemiſſis erit arcus minor DF, vel angulus <lb/>DEF, grad. </s> <s xml:id="echoid-s10101" xml:space="preserve">_60._ </s> <s xml:id="echoid-s10102" xml:space="preserve">Ita ſi differentia data complectatur grad. </s> <s xml:id="echoid-s10103" xml:space="preserve">_104._ </s> <s xml:id="echoid-s10104" xml:space="preserve">Min. <lb/></s> <s xml:id="echoid-s10105" xml:space="preserve">_20._ </s> <s xml:id="echoid-s10106" xml:space="preserve">detrahemus eam ex grad. </s> <s xml:id="echoid-s10107" xml:space="preserve">_180._ </s> <s xml:id="echoid-s10108" xml:space="preserve">& </s> <s xml:id="echoid-s10109" xml:space="preserve">reliqui arcus grad. </s> <s xml:id="echoid-s10110" xml:space="preserve">_75._ </s> <s xml:id="echoid-s10111" xml:space="preserve">Min. </s> <s xml:id="echoid-s10112" xml:space="preserve">_40._ </s> <s xml:id="echoid-s10113" xml:space="preserve"><lb/>ſemiſſem, nempe grad. </s> <s xml:id="echoid-s10114" xml:space="preserve">_37._ </s> <s xml:id="echoid-s10115" xml:space="preserve">Min. </s> <s xml:id="echoid-s10116" xml:space="preserve">_50._ </s> <s xml:id="echoid-s10117" xml:space="preserve">datæ differentiæ addemus, vt com-<lb/>ponatur maior arcus, ſeu angulus propoſitus, grad. </s> <s xml:id="echoid-s10118" xml:space="preserve">_142._ </s> <s xml:id="echoid-s10119" xml:space="preserve">Min. </s> <s xml:id="echoid-s10120" xml:space="preserve">_10._ </s> <s xml:id="echoid-s10121" xml:space="preserve">Minor <lb/>enim erit ioſa differentiæ ſemiſſis grad. </s> <s xml:id="echoid-s10122" xml:space="preserve">_37._ </s> <s xml:id="echoid-s10123" xml:space="preserve">Min. </s> <s xml:id="echoid-s10124" xml:space="preserve">_50._</s> <s xml:id="echoid-s10125" xml:space="preserve"/> </p> <div xml:id="echoid-div843" type="float" level="2" n="11"> <note position="left" xlink:label="note-330-06" xlink:href="note-330-06a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s10126" xml:space="preserve">QVOCIRCA, data diſſerentia duorum arcuum, quorum ſinguli ſe-<lb/>micirculo ſint minores, vel duorum angulorum rectilineorum, &</s> <s xml:id="echoid-s10127" xml:space="preserve">c. </s> <s xml:id="echoid-s10128" xml:space="preserve">vtrumq; <lb/></s> <s xml:id="echoid-s10129" xml:space="preserve">illorum ſigillatim notum effecimus. </s> <s xml:id="echoid-s10130" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s10131" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div845" type="section" level="1" n="463"> <head xml:id="echoid-head495" xml:space="preserve">THEOR. 4. PROPOS. 8.</head> <p> <s xml:id="echoid-s10132" xml:space="preserve">SI ab angulo trianguli cuiuſuis duobus lateri-<lb/> <anchor type="note" xlink:label="note-331-01a" xlink:href="note-331-01"/> bus inæqualibus comprehenſo linea perpendicu-<lb/>laris ad baſim ducatur, ſi quidem intra triangu-<lb/>lum cadit, erit quadratum maioris laterum dictum <lb/>angulum ambientium maius, quam quadratum <lb/>minoris, rectangulo ſub baſe, & </s> <s xml:id="echoid-s10133" xml:space="preserve">differentia ſeg-<lb/>mentorum à perpẽdiculari factorum comprehen-<lb/>ſo: </s> <s xml:id="echoid-s10134" xml:space="preserve">ſi vero extra cadit, erit quadratum maioris la-<lb/>teris maius, quam quadratum minoris, rectangu-<lb/>lo ſub baſe, & </s> <s xml:id="echoid-s10135" xml:space="preserve">recta linea, quæ ex baſe, & </s> <s xml:id="echoid-s10136" xml:space="preserve">duplo ex-<lb/>terioris lineæ inter perpendicularem, & </s> <s xml:id="echoid-s10137" xml:space="preserve">angulum <lb/>trianguli componitur, comprehenſo.</s> <s xml:id="echoid-s10138" xml:space="preserve"/> </p> <div xml:id="echoid-div845" type="float" level="2" n="1"> <note position="right" xlink:label="note-331-01" xlink:href="note-331-01a" xml:space="preserve">Quãto ma <lb/>ius ſit qua-<lb/>dratũ ma-<lb/>ioris late-<lb/>ris. quàm <lb/>minoris in <lb/>quouis triã <lb/>gulo.</note> </div> <p> <s xml:id="echoid-s10139" xml:space="preserve">IN triangulo ABC, cuius duo latera AB, <lb/> <anchor type="figure" xlink:label="fig-331-01a" xlink:href="fig-331-01"/> AC, inæqualia ſint, AC, maius, & </s> <s xml:id="echoid-s10140" xml:space="preserve">AB, minus, <lb/>ducatur ex angulo A, ad baſim BC, perpendicu-<lb/>laris AD, cadens primùm intra triangulum, vt <lb/>in priori figura. </s> <s xml:id="echoid-s10141" xml:space="preserve">Et quoniam tam quadrata re-<lb/>ctarum AD, DB, quadrato rectæ AB, quàm <lb/> <anchor type="note" xlink:label="note-331-02a" xlink:href="note-331-02"/> quadrata rectarum AD, DC, quadrato rectæ <lb/>AC, æqualia ſunt; </s> <s xml:id="echoid-s10142" xml:space="preserve">eſt autem quadratum rectæ <lb/>AB, minus quadrato rectæ AC, quòd minor po <lb/>natur recta AB, quam AC: </s> <s xml:id="echoid-s10143" xml:space="preserve">erunt quoque duo <lb/>quadrata rectarum AD, DB, ſimul; </s> <s xml:id="echoid-s10144" xml:space="preserve">minora <lb/>duobus quadratis AD, DC, ſimul; </s> <s xml:id="echoid-s10145" xml:space="preserve">ablato-<lb/>que propterea communi quadrato rectæ AD, <lb/>quadratum rectæ BD, quadrato rectæ DC, <pb o="320" file="332" n="332" rhead=""/> minus erit, & </s> <s xml:id="echoid-s10146" xml:space="preserve">proinde & </s> <s xml:id="echoid-s10147" xml:space="preserve">recta: </s> <s xml:id="echoid-s10148" xml:space="preserve">BD, minor, quam recta DC. </s> <s xml:id="echoid-s10149" xml:space="preserve">Abſciſſa erge <lb/>recta DE, ipſi BD, æquali, erit EC, differentia inter ſegmenta BD, <lb/>DC. </s> <s xml:id="echoid-s10150" xml:space="preserve">Dico quadratum lateris AC, ſuperare quadratum lateris AB, rectan-<lb/>gulo ſub BC, EC, comprehenſo. </s> <s xml:id="echoid-s10151" xml:space="preserve">Quia enim recta BE, ſecta eſt bifariam in <lb/>D, eiq́; </s> <s xml:id="echoid-s10152" xml:space="preserve">addita in continuum recta EC, erit rectangulum ſub BC, EC, <lb/> <anchor type="figure" xlink:label="fig-332-01a" xlink:href="fig-332-01"/> contentum vna cum quadrato rectæ DE, qua-<lb/>drato rectæ DC, æquale. </s> <s xml:id="echoid-s10153" xml:space="preserve">Addito ergo quadrato <lb/> <anchor type="note" xlink:label="note-332-01a" xlink:href="note-332-01"/> communi rectæ AD, erit rectangulum ſub BC, <lb/>EC, vnà cum quadratis rectarum DE, AD, hoc <lb/>eſt, rectarum BD, AD, hoc eſt, cum quadrato <lb/>rectæ AB, æquale quadratis rectarum DC, AD, <lb/>hoc eſt, quadrato rectæ AC. </s> <s xml:id="echoid-s10154" xml:space="preserve">Maius ergo eſt qua-<lb/>dratum lateris AC, quam quadratum lateris <lb/>AB, rectangulo ſub BC, EC, comprehenſo. <lb/></s> <s xml:id="echoid-s10155" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s10156" xml:space="preserve"/> </p> <div xml:id="echoid-div846" type="float" level="2" n="2"> <figure xlink:label="fig-331-01" xlink:href="fig-331-01a"> <image file="331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/331-01"/> </figure> <note position="right" xlink:label="note-331-02" xlink:href="note-331-02a" xml:space="preserve">47. primi.</note> <figure xlink:label="fig-332-01" xlink:href="fig-332-01a"> <image file="332-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/332-01"/> </figure> <note position="left" xlink:label="note-332-01" xlink:href="note-332-01a" xml:space="preserve">@. ſecundi.</note> </div> <p> <s xml:id="echoid-s10157" xml:space="preserve">CADAT deinde perpendicularis AD, ex-<lb/>tra triangulum in baſim CB, productam, vt in <lb/>figura poſteriori. </s> <s xml:id="echoid-s10158" xml:space="preserve">Abſciſſa recta DE, ipſi DB, <lb/>æquali, erit recta EC, compoſita ex baſe BC, & </s> <s xml:id="echoid-s10159" xml:space="preserve"><lb/>EB, quæ dupla eſt lineæ DB, inter perpendicu-<lb/>larem, & </s> <s xml:id="echoid-s10160" xml:space="preserve">angulum B. </s> <s xml:id="echoid-s10161" xml:space="preserve">Dico rurſus, quadratum <lb/>lateris AC, ſuperare quadratum lateris AB, rectangulo ſub BC, EC, com-<lb/> <anchor type="note" xlink:label="note-332-02a" xlink:href="note-332-02"/> prehenſo. </s> <s xml:id="echoid-s10162" xml:space="preserve">Eritenim rurſus rectangulum ſub BC, EC, vnà cum quadrato re-<lb/>ctæ DB, quadrato rectæ DC, æquale. </s> <s xml:id="echoid-s10163" xml:space="preserve">Addito ergo quadrato communi rectę <lb/>AD, erit rectangulum ſub BC, EC, vnà cum quadratis rectarum DB, AD, <lb/>hoc eſt, cum quadrato rectę AB, æquale quadratis rectarum DC, AD, hoc <lb/>eſt, quadrato rectæ AC. </s> <s xml:id="echoid-s10164" xml:space="preserve">Excedit igitur quadratum lateris AC, quadratum <lb/>lateris AB, rectangulo contento ſub BC, EC.</s> <s xml:id="echoid-s10165" xml:space="preserve"/> </p> <div xml:id="echoid-div847" type="float" level="2" n="3"> <note position="left" xlink:label="note-332-02" xlink:href="note-332-02a" xml:space="preserve">@. fecundi.</note> </div> <p> <s xml:id="echoid-s10166" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s10167" xml:space="preserve">Quoniã quadratis ex AD, DC, quadratum ex AC; </s> <s xml:id="echoid-s10168" xml:space="preserve">& </s> <s xml:id="echoid-s10169" xml:space="preserve">qua-<lb/> <anchor type="note" xlink:label="note-332-03a" xlink:href="note-332-03"/> dratis ex AD, DB, quadratum ex AB, æquale eſt: </s> <s xml:id="echoid-s10170" xml:space="preserve">idem erit exceſſus qua-<lb/>drati ex AC, ſupra quadratum ex AB, qui quadratorum ex AD, DC, ſu-<lb/>pra quadrata ex AD, DB: </s> <s xml:id="echoid-s10171" xml:space="preserve">Et, ablato communi quadrato ex AD, idem, qui <lb/>quadratiex DC, ſupra quadratum ex DB, per pronunciatum 17. </s> <s xml:id="echoid-s10172" xml:space="preserve">lib 1. </s> <s xml:id="echoid-s10173" xml:space="preserve">Eucl. <lb/></s> <s xml:id="echoid-s10174" xml:space="preserve">Sed quadratum ex DC, ſuperat quadratum ex DB, rectangulo ſub BC, CE, <lb/>comprehenſo; </s> <s xml:id="echoid-s10175" xml:space="preserve">propterea quòd quadratum ex DC, æquale eſt quadrato ex <lb/> <anchor type="note" xlink:label="note-332-04a" xlink:href="note-332-04"/> D B, vel ex DE, in prima ſigura, vnà cum rectangulo ſub BC, CE, contento. <lb/></s> <s xml:id="echoid-s10176" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s10177" xml:space="preserve">quadratum ex AC, ſuperat quadratum ex AB, rectangulo com-<lb/>prehenſo ſũb BC, CE. </s> <s xml:id="echoid-s10178" xml:space="preserve">Quocirca, Si ab angulo trianguli cuiuſuis duobus <lb/>lateribus inæqualibus com prehenſo linea perpendicularis ad baſim ducatur, <lb/>&</s> <s xml:id="echoid-s10179" xml:space="preserve">c. </s> <s xml:id="echoid-s10180" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s10181" xml:space="preserve"/> </p> <div xml:id="echoid-div848" type="float" level="2" n="4"> <note position="left" xlink:label="note-332-03" xlink:href="note-332-03a" xml:space="preserve">47 primi.</note> <note position="left" xlink:label="note-332-04" xlink:href="note-332-04a" xml:space="preserve">@. ſecundi.</note> </div> </div> <div xml:id="echoid-div850" type="section" level="1" n="464"> <head xml:id="echoid-head496" xml:space="preserve">COROLLARIVM.</head> <note position="left" xml:space="preserve">Perpẽdicu <lb/>laris in lſo <lb/>ſcele ſecat <lb/>baſim bifa <lb/>riam.</note> <p> <s xml:id="echoid-s10182" xml:space="preserve">EX demonſtratis conſtat, In Iſoſcele perpendicularem ſecare baſim bifariam. </s> <s xml:id="echoid-s10183" xml:space="preserve">Nam ſi in <lb/>priore triangulo latera AB, AC, ponantur æqualia, erunt eorum@uadrata quoque æqualia. <lb/></s> <s xml:id="echoid-s10184" xml:space="preserve">Quare cum quadratum ex AB, æquale ſit quadratis ex AD, BD; </s> <s xml:id="echoid-s10185" xml:space="preserve">& </s> <s xml:id="echoid-s10186" xml:space="preserve">quadratum ex AC, <lb/> <anchor type="note" xlink:label="note-332-06a" xlink:href="note-332-06"/> quadratis ex AD, CD: </s> <s xml:id="echoid-s10187" xml:space="preserve">erunt quoque quadrata ex AD, BD, quadratis ex AD, CD, æqua-<lb/>lia: </s> <s xml:id="echoid-s10188" xml:space="preserve">Ablatoque communi quadrato rectæ AD, reliqua erunt quadrata ex BD, CD; </s> <s xml:id="echoid-s10189" xml:space="preserve">æqua-<lb/>lia, & </s> <s xml:id="echoid-s10190" xml:space="preserve">proinde rectæ BD, CD, æquales.</s> <s xml:id="echoid-s10191" xml:space="preserve"/> </p> <div xml:id="echoid-div850" type="float" level="2" n="1"> <note position="left" xlink:label="note-332-06" xlink:href="note-332-06a" xml:space="preserve">47. primi.</note> </div> <pb o="321" file="333" n="333" rhead=""/> </div> <div xml:id="echoid-div852" type="section" level="1" n="465"> <head xml:id="echoid-head497" xml:space="preserve">PROBL. 5. PROPOS. 9.</head> <p> <s xml:id="echoid-s10192" xml:space="preserve">SI ab vno angulo trianguli cuiuſuis notorum <lb/> <anchor type="note" xlink:label="note-333-01a" xlink:href="note-333-01"/> laterum ad oppoſitũ latus perpendicularis demit-<lb/>tatur: </s> <s xml:id="echoid-s10193" xml:space="preserve">quanta ſit recta inter perpendicularem, & </s> <s xml:id="echoid-s10194" xml:space="preserve"><lb/>vtrumuis angulorum reliquorum comprehenſa, <lb/>cognoſcere.</s> <s xml:id="echoid-s10195" xml:space="preserve"/> </p> <div xml:id="echoid-div852" type="float" level="2" n="1"> <note position="right" xlink:label="note-333-01" xlink:href="note-333-01a" xml:space="preserve">Cognltis la <lb/>teribus triã <lb/>guli, cogno <lb/>fcútur ſeg <lb/>menta ba-<lb/>ſis inter p-<lb/>pendicula-<lb/>rẽ, & vtrũ-<lb/>que angu-<lb/>lum com-<lb/>prehenſa.</note> </div> <p> <s xml:id="echoid-s10196" xml:space="preserve">REPETANTVR duo triangula præcedẽtis propoſ. </s> <s xml:id="echoid-s10197" xml:space="preserve">ſitq́; </s> <s xml:id="echoid-s10198" xml:space="preserve">latus AC, <lb/>20. </s> <s xml:id="echoid-s10199" xml:space="preserve">AB, 13. </s> <s xml:id="echoid-s10200" xml:space="preserve">& </s> <s xml:id="echoid-s10201" xml:space="preserve">in priori quidem triangulo, BC, 21. </s> <s xml:id="echoid-s10202" xml:space="preserve">in poſteriori vero, 11. <lb/></s> <s xml:id="echoid-s10203" xml:space="preserve">Oporteatq; </s> <s xml:id="echoid-s10204" xml:space="preserve">cognoſcere, quanta ſit tam recta BD, quàm CD. </s> <s xml:id="echoid-s10205" xml:space="preserve">Quoniam qua-<lb/>dratum ex AC, ſuperat quadratum ex AB, re-<lb/> <anchor type="note" xlink:label="note-333-02a" xlink:href="note-333-02"/> <anchor type="figure" xlink:label="fig-333-01a" xlink:href="fig-333-01"/> ctangulo ſub BC, CE, contento; </s> <s xml:id="echoid-s10206" xml:space="preserve">ſi quadratum <lb/>rectæ AB, hoc eſt, 169. </s> <s xml:id="echoid-s10207" xml:space="preserve">detrahatur ex 400. </s> <s xml:id="echoid-s10208" xml:space="preserve">qua-<lb/>drato rectæ AC, reliquum erit rectangulum ſub <lb/>BC, CE, contentum 231. </s> <s xml:id="echoid-s10209" xml:space="preserve">quo diuiſo per latus <lb/>BC, hoc eſt, per 21. </s> <s xml:id="echoid-s10210" xml:space="preserve">in priori triangulo, prodibit <lb/>recta CE, 11. </s> <s xml:id="echoid-s10211" xml:space="preserve">quæ ablata ex latere BC, id eſt, <lb/>ex 21. </s> <s xml:id="echoid-s10212" xml:space="preserve">relinquet BE, 10. </s> <s xml:id="echoid-s10213" xml:space="preserve">Huius ergo dimidium <lb/>5. </s> <s xml:id="echoid-s10214" xml:space="preserve">dabit rectam BD: </s> <s xml:id="echoid-s10215" xml:space="preserve">ac proinde reliqua CD, <lb/>erit 16. </s> <s xml:id="echoid-s10216" xml:space="preserve">nempe reſiduum lateris BC. </s> <s xml:id="echoid-s10217" xml:space="preserve">In poſte-<lb/>riori vero triangulo, diuiſo eodem rectangulo <lb/>231. </s> <s xml:id="echoid-s10218" xml:space="preserve">per 11. </s> <s xml:id="echoid-s10219" xml:space="preserve">nimirum per latus BC, inuenietur <lb/>C E, 21. </s> <s xml:id="echoid-s10220" xml:space="preserve">à qua ſi latus BC, hoc eſt, 11. </s> <s xml:id="echoid-s10221" xml:space="preserve">aufera-<lb/>tur, remanebit BE, 10. </s> <s xml:id="echoid-s10222" xml:space="preserve">cuius ſemiſsis dabit BD, <lb/>5. </s> <s xml:id="echoid-s10223" xml:space="preserve">ac proinde CD, erit 16. </s> <s xml:id="echoid-s10224" xml:space="preserve">nempe compoſitum <lb/>ex latere BC, ac BD.</s> <s xml:id="echoid-s10225" xml:space="preserve"/> </p> <div xml:id="echoid-div853" type="float" level="2" n="2"> <note position="right" xlink:label="note-333-02" xlink:href="note-333-02a" xml:space="preserve">8. huius.</note> <figure xlink:label="fig-333-01" xlink:href="fig-333-01a"> <image file="333-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/333-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s10226" xml:space="preserve">ITAQVE, Sidifferentia inter duo quadrata laterum ambien-<lb/> <anchor type="note" xlink:label="note-333-03a" xlink:href="note-333-03"/> tium angulum, à quo porpendicularis ducta est, diuidatur per tertium <lb/>latus, in quod perpendicularis eſt demiſſa, producetur numerus, qui <lb/>ſi minor tertio latere fuerit, indicabit perpendicularem intra triangu-<lb/> <anchor type="note" xlink:label="note-333-04a" xlink:href="note-333-04"/> lum cecidiſſe, idemque ex tertio latere ſubductus relinquet numerum, <lb/>cuius ſemiſſis dabit minus ſegmentum baſis: </s> <s xml:id="echoid-s10227" xml:space="preserve">hoc autem ex tertio latere <lb/>ſubtractum exhibebit ſegmentum maius. </s> <s xml:id="echoid-s10228" xml:space="preserve">Si vero numerus ille ex diui-<lb/>ſione productus fuerit tertio latere maior, argumento eſt, perpendicula-<lb/>rem extra triangulum cecidiſſe. </s> <s xml:id="echoid-s10229" xml:space="preserve">Quare ſi ex eo tertium latus detraha-<lb/>tur, reliquus erit numerus, cuius ſemiſſis dabit rectam extra triangu-<lb/>l um inter perpendicularem, & </s> <s xml:id="echoid-s10230" xml:space="preserve">angulum obtuſum; </s> <s xml:id="echoid-s10231" xml:space="preserve">eadem vero ſemiſſis <lb/>tertio lateri addita exhibebit alteram rectam inter perpendicularem, & </s> <s xml:id="echoid-s10232" xml:space="preserve"><lb/>angulum acutum.</s> <s xml:id="echoid-s10233" xml:space="preserve"/> </p> <div xml:id="echoid-div854" type="float" level="2" n="3"> <note position="right" xlink:label="note-333-03" xlink:href="note-333-03a" xml:space="preserve">Praxis.</note> <note position="right" xlink:label="note-333-04" xlink:href="note-333-04a" xml:space="preserve">Quo pacto <lb/>ex opera-<lb/>tione intel <lb/>ligat̃, num <lb/>perpẽdicu-<lb/>laris intra <lb/>triãgulum <lb/>@adat, an <lb/>extra.</note> </div> <pb o="322" file="334" n="334" rhead=""/> <p> <s xml:id="echoid-s10234" xml:space="preserve">ALITER & </s> <s xml:id="echoid-s10235" xml:space="preserve">facilius. </s> <s xml:id="echoid-s10236" xml:space="preserve">Ex A, ad interuallum minoris lateris AB, circulus <lb/> <anchor type="note" xlink:label="note-334-01a" xlink:href="note-334-01"/> deſcribatur ſecans maius latus AC, in F, idemq́. </s> <s xml:id="echoid-s10237" xml:space="preserve">productum in G, & </s> <s xml:id="echoid-s10238" xml:space="preserve">latus BC, <lb/> <anchor type="figure" xlink:label="fig-334-01a" xlink:href="fig-334-01"/> ſi perpen dicularis intra trian <lb/>gulum cadit, vel certe, ſi ex-<lb/>tra cadit, ipſum productum in <lb/>E: </s> <s xml:id="echoid-s10239" xml:space="preserve">ſecabiturq́; </s> <s xml:id="echoid-s10240" xml:space="preserve">recta BE, bifa-<lb/> <anchor type="note" xlink:label="note-334-02a" xlink:href="note-334-02"/> riam in E. </s> <s xml:id="echoid-s10241" xml:space="preserve">Quia vero rectan-<lb/>gulum ſub BC, CE, rectan-<lb/> <anchor type="note" xlink:label="note-334-03a" xlink:href="note-334-03"/> gulo ſub GC, CF, æquale eſt; <lb/></s> <s xml:id="echoid-s10242" xml:space="preserve">erit, vt BC, latus, in quod <lb/> <anchor type="note" xlink:label="note-334-04a" xlink:href="note-334-04"/> perpendicularis ducitur, nem-<lb/>pe vt 21. </s> <s xml:id="echoid-s10243" xml:space="preserve">in priori triangulo, <lb/>vel vt 11. </s> <s xml:id="echoid-s10244" xml:space="preserve">in poſteriori, ad GC, <lb/>ſummam reliquorũ duorum <lb/>laterũ AC, AB, (quod AG, <lb/>ipſi AB, ſit æqualis) hoc eſt, <lb/>ad 33. </s> <s xml:id="echoid-s10245" xml:space="preserve">ita CF, differentia in-<lb/>ter eadem duo latera, id eſt, <lb/>ita 7. </s> <s xml:id="echoid-s10246" xml:space="preserve">ad CE. </s> <s xml:id="echoid-s10247" xml:space="preserve">Quare per <lb/>regulam auream inuenietur <lb/>C E, partium 11. </s> <s xml:id="echoid-s10248" xml:space="preserve">in triangulo <lb/>priori, in poſteriori autem 21. <lb/></s> <s xml:id="echoid-s10249" xml:space="preserve">vt hic perſpicuum eſt.</s> <s xml:id="echoid-s10250" xml:space="preserve"/> </p> <div xml:id="echoid-div855" type="float" level="2" n="4"> <note position="left" xlink:label="note-334-01" xlink:href="note-334-01a" xml:space="preserve">Alia inuẽ-<lb/>tio ſegmẽ-<lb/>torum ba-<lb/>ſis. & fac@-<lb/>lior.</note> <figure xlink:label="fig-334-01" xlink:href="fig-334-01a"> <image file="334-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/334-01"/> </figure> <note position="left" xlink:label="note-334-02" xlink:href="note-334-02a" xml:space="preserve">3. tertij.</note> <note position="left" xlink:label="note-334-03" xlink:href="note-334-03a" xml:space="preserve">corol. 1. 36. <lb/>tertij.</note> <note position="left" xlink:label="note-334-04" xlink:href="note-334-04a" xml:space="preserve">16. ſexti.</note> </div> <note position="right" xml:space="preserve"> <lb/>BC. # GC. # # CF. # # CE. <lb/>21. # 33. # # 7? # fit. # 11. <lb/> # # Item <lb/>11. # 33. # # 7? # fit. # 21. <lb/></note> <p> <s xml:id="echoid-s10251" xml:space="preserve">Quòd ſi EC, inuenta partium 11. </s> <s xml:id="echoid-s10252" xml:space="preserve">in priori triangulo auferatur ex latere <lb/>BC, nempe ex 21. </s> <s xml:id="echoid-s10253" xml:space="preserve">remanebit BE, 10. </s> <s xml:id="echoid-s10254" xml:space="preserve">cuius ſemiſsis 5. </s> <s xml:id="echoid-s10255" xml:space="preserve">erit ſegmentum BD, ac <lb/>proinde alterum CD, erit 16. </s> <s xml:id="echoid-s10256" xml:space="preserve">In poſteriori vero triangulo, ſi ex EC, inuenta <lb/>partium 21. </s> <s xml:id="echoid-s10257" xml:space="preserve">dematur latus BC, partium 11. </s> <s xml:id="echoid-s10258" xml:space="preserve">relinquetur rurſus BE, 10. </s> <s xml:id="echoid-s10259" xml:space="preserve">Qua-<lb/>re eius dimidium 5. </s> <s xml:id="echoid-s10260" xml:space="preserve">dabit rectam BD, extra triangulum inter perpendicula-<lb/>rem, & </s> <s xml:id="echoid-s10261" xml:space="preserve">angulum obtuſum; </s> <s xml:id="echoid-s10262" xml:space="preserve">ac proinde tota CD, compoſita ex latere BC, & </s> <s xml:id="echoid-s10263" xml:space="preserve"><lb/>dicto dimidio BD, erit 16.</s> <s xml:id="echoid-s10264" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10265" xml:space="preserve">SI igitur fiat, vt latus, in quod perpendicularis ducta eſt, ad ſum-<lb/> <anchor type="note" xlink:label="note-334-06a" xlink:href="note-334-06"/> mam aliorum duorũ laterum, ita diff@rentia eorundem laterum ad aliud, <lb/> <anchor type="note" xlink:label="note-334-07a" xlink:href="note-334-07"/> reperietur numerus, qui ſi minor fuerit tertio latere, indicabit perpendi-<lb/>cularem intra triangulum cecidiſſe, idemq́, extertio latere ablatus relin-<lb/>quet numerum, cuius dimidium erit minus ſegmentum baſis, hoc autem <lb/>ex tertio latere demptum reliquum faciet maius ſegmentum. </s> <s xml:id="echoid-s10266" xml:space="preserve">Si vero nu-<lb/>merus per auream regulam inuentus tertium latus ſuperet, argumento <lb/>eſt, perpenticularem cecidiſſe extra triangulum. </s> <s xml:id="echoid-s10267" xml:space="preserve">Quare ſi ex eolatus <lb/>tertium detrahatur, dabit ſemiſſis reliqui numeri rectam extra triangu-<lb/>lum inter perpendicularem, & </s> <s xml:id="echoid-s10268" xml:space="preserve">angulum obtuſum: </s> <s xml:id="echoid-s10269" xml:space="preserve">Eadem vero ſemiſſis <pb o="323" file="335" n="335" rhead=""/> tertio lateri adiuncta offeret alteram rectam inter perpendicularem, & </s> <s xml:id="echoid-s10270" xml:space="preserve"><lb/>angulum acutum.</s> <s xml:id="echoid-s10271" xml:space="preserve"/> </p> <div xml:id="echoid-div856" type="float" level="2" n="5"> <note position="left" xlink:label="note-334-06" xlink:href="note-334-06a" xml:space="preserve">Praxis.</note> <note position="left" xlink:label="note-334-07" xlink:href="note-334-07a" xml:space="preserve">Quo pacto <lb/>ex ipſa ope <lb/>ratione co <lb/>gnoſcatur, <lb/>an perpen-<lb/>dicularis <lb/>cadat intra <lb/>triangulũ, <lb/>an extra.</note> </div> <p> <s xml:id="echoid-s10272" xml:space="preserve">SI ergo ab vno angulo trianguli cuiuſuis notorum laterum, &</s> <s xml:id="echoid-s10273" xml:space="preserve">c. </s> <s xml:id="echoid-s10274" xml:space="preserve">Quod fa-<lb/>ciendum erat.</s> <s xml:id="echoid-s10275" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div858" type="section" level="1" n="466"> <head xml:id="echoid-head498" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s10276" xml:space="preserve">_<emph style="sc">VIDe</emph>S_ igitur, in vtraque praxi calculum ipſum monſtrare, num perpendicula <lb/>ris intra triangulum cadat, an vero extra.</s> <s xml:id="echoid-s10277" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10278" xml:space="preserve">_<emph style="sc">IDEm</emph>_ hoc problema abſolui poteſt per propoſ. </s> <s xml:id="echoid-s10279" xml:space="preserve">_13._ </s> <s xml:id="echoid-s10280" xml:space="preserve">aut _12._ </s> <s xml:id="echoid-s10281" xml:space="preserve">lib. </s> <s xml:id="echoid-s10282" xml:space="preserve">_2._ </s> <s xml:id="echoid-s10283" xml:space="preserve"><emph style="sc">E</emph>ucl. </s> <s xml:id="echoid-s10284" xml:space="preserve">prout per@ <lb/>pendicularis intra triangulum cadit, vel extra. </s> <s xml:id="echoid-s10285" xml:space="preserve">Cadat enim primũ perpendicularis <lb/>_AD,_ intra, ſintq́; </s> <s xml:id="echoid-s10286" xml:space="preserve">latera, vt prius; </s> <s xml:id="echoid-s10287" xml:space="preserve">_<emph style="sc">Ab</emph>, 13. </s> <s xml:id="echoid-s10288" xml:space="preserve">AC, 20 BC._ </s> <s xml:id="echoid-s10289" xml:space="preserve">_21._ </s> <s xml:id="echoid-s10290" xml:space="preserve">_<emph style="sc">E</emph>_ritq; </s> <s xml:id="echoid-s10291" xml:space="preserve">vterq; </s> <s xml:id="echoid-s10292" xml:space="preserve">angulus _B, C,_ <lb/>acutus, propter rectos angulos ad _D._ </s> <s xml:id="echoid-s10293" xml:space="preserve">cũ tam duo anguli _B,_ & </s> <s xml:id="echoid-s10294" xml:space="preserve">ADB, quàm duo _C,_ & </s> <s xml:id="echoid-s10295" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-335-01a" xlink:href="note-335-01"/> _<emph style="sc">A</emph>DC,_ ſint duobus rectis minores. </s> <s xml:id="echoid-s10296" xml:space="preserve">Quoniam igitur quadratũ ex _AC,_ minus eſt, quàm <lb/>duo quadrata ex _AB, <emph style="sc">B</emph>C,_ rectangulo bis cõprehenſo ſub _<emph style="sc">Cb, b</emph>D;_ </s> <s xml:id="echoid-s10297" xml:space="preserve">ſi quadratũ lateris <lb/> <anchor type="note" xlink:label="note-335-02a" xlink:href="note-335-02"/> _AC,_ nempe 400. </s> <s xml:id="echoid-s10298" xml:space="preserve">auferatur ex ſumma quadratorum laterum _AB, <emph style="sc">B</emph>C,_ nimirum ex <lb/>_610._ </s> <s xml:id="echoid-s10299" xml:space="preserve">reliquum erit rectangulũ ſub _<emph style="sc">Cb</emph>, BD,_ bis com-<lb/> <anchor type="figure" xlink:label="fig-335-01a" xlink:href="fig-335-01"/> prehẽſum _210._ </s> <s xml:id="echoid-s10300" xml:space="preserve">Semiſsis ergo huius, vtpote _105._ </s> <s xml:id="echoid-s10301" xml:space="preserve">erit re-<lb/>ctangulum ſub _<emph style="sc">Cb, b</emph>D,_ comprehenſum: </s> <s xml:id="echoid-s10302" xml:space="preserve">quo diuiſ@ <lb/>per latus _BC,_ hoc eſt, per _21._ </s> <s xml:id="echoid-s10303" xml:space="preserve">exibit ſegmentum _<emph style="sc">B</emph>D,_ <lb/>_5._ </s> <s xml:id="echoid-s10304" xml:space="preserve">Reliquum ergo _CD,_ erit _16._ </s> <s xml:id="echoid-s10305" xml:space="preserve">Quod tamen eodem mo <lb/>do reperirs poteſt, ſi quadratum lateris _AB,_ ex qua-<lb/>dratis laterum _AC, <emph style="sc">C</emph>B,_ ſubducatur, &</s> <s xml:id="echoid-s10306" xml:space="preserve">c. </s> <s xml:id="echoid-s10307" xml:space="preserve">_<emph style="sc">E</emph>_ſt enim <lb/> <anchor type="note" xlink:label="note-335-03a" xlink:href="note-335-03"/> & </s> <s xml:id="echoid-s10308" xml:space="preserve">illud minus, quàm hæc duo, rectangulo ſub _<emph style="sc">Bc</emph>,_ <lb/>_<emph style="sc">C</emph>D,_ bis comprehenſo.</s> <s xml:id="echoid-s10309" xml:space="preserve"/> </p> <div xml:id="echoid-div858" type="float" level="2" n="1"> <note position="right" xlink:label="note-335-01" xlink:href="note-335-01a" xml:space="preserve">17. primi.</note> <note position="right" xlink:label="note-335-02" xlink:href="note-335-02a" xml:space="preserve">13. ſecundi.</note> <figure xlink:label="fig-335-01" xlink:href="fig-335-01a"> <image file="335-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/335-01"/> </figure> <note position="right" xlink:label="note-335-03" xlink:href="note-335-03a" xml:space="preserve">13. ſecundi.</note> </div> <p style="it"> <s xml:id="echoid-s10310" xml:space="preserve">_<emph style="sc">CA</emph>DAT_ deinde perpendicularis _AD,_ extra, ſintq́; <lb/></s> <s xml:id="echoid-s10311" xml:space="preserve">latera, vt prius; </s> <s xml:id="echoid-s10312" xml:space="preserve">_AB, 13. </s> <s xml:id="echoid-s10313" xml:space="preserve"><emph style="sc">Ac</emph>, 20. </s> <s xml:id="echoid-s10314" xml:space="preserve"><emph style="sc">Bc</emph>, 11._ </s> <s xml:id="echoid-s10315" xml:space="preserve">_E_ritq́; </s> <s xml:id="echoid-s10316" xml:space="preserve"><lb/>angulus _<emph style="sc">A</emph><emph style="sc">Bc</emph>,_ obtuſus: </s> <s xml:id="echoid-s10317" xml:space="preserve">propterea quod duo _D,_ & </s> <s xml:id="echoid-s10318" xml:space="preserve"><lb/>_ABD,_ ſunt duobus rectis minores; </s> <s xml:id="echoid-s10319" xml:space="preserve">ac proinde _<emph style="sc">AB</emph>D,_ <lb/> <anchor type="note" xlink:label="note-335-04a" xlink:href="note-335-04"/> acutus, cum _D,_ rectus ſit. </s> <s xml:id="echoid-s10320" xml:space="preserve">Quia ergo quadratum ex <lb/>_<emph style="sc">AC</emph>,_ maius eſt, quàm duo quadrataex _<emph style="sc">A</emph>B, <emph style="sc">BC</emph>,_ re-<lb/> <anchor type="note" xlink:label="note-335-05a" xlink:href="note-335-05"/> ctangulo bis comprehenſo ſub _<emph style="sc">C</emph>B, BD;_ </s> <s xml:id="echoid-s10321" xml:space="preserve">ſi ſumma quadratorum laterum _<emph style="sc">Ab, bc</emph>,_ <lb/>id eſt, _290._ </s> <s xml:id="echoid-s10322" xml:space="preserve">auferatur ex _400._ </s> <s xml:id="echoid-s10323" xml:space="preserve">quadrato lateris _<emph style="sc">Ac</emph>,_ relinquetur rectangulum bis <lb/>comprehenſum ſub _<emph style="sc">CB, B</emph>D, 110._ </s> <s xml:id="echoid-s10324" xml:space="preserve">Semiſsis ergo huius, nimirum _55._ </s> <s xml:id="echoid-s10325" xml:space="preserve">erit rectangulum <lb/>ſub _<emph style="sc">CB, B</emph>D,_ contentum: </s> <s xml:id="echoid-s10326" xml:space="preserve">quo diuiſo per 11. </s> <s xml:id="echoid-s10327" xml:space="preserve">latus _<emph style="sc">BC</emph>,_ prodibit recta _<emph style="sc">B</emph>D,_ extra <lb/>triangulum, _5._ </s> <s xml:id="echoid-s10328" xml:space="preserve">quæcum _11._ </s> <s xml:id="echoid-s10329" xml:space="preserve">latere _<emph style="sc">BC</emph>,_ componet rectam _<emph style="sc">C</emph>D, 16._</s> <s xml:id="echoid-s10330" xml:space="preserve"/> </p> <div xml:id="echoid-div859" type="float" level="2" n="2"> <note position="right" xlink:label="note-335-04" xlink:href="note-335-04a" xml:space="preserve">17. primi</note> <note position="right" xlink:label="note-335-05" xlink:href="note-335-05a" xml:space="preserve">12. ſecundi.</note> </div> <p style="it"> <s xml:id="echoid-s10331" xml:space="preserve">ITAQVE, cadente perpendiculari intra triangulum; </s> <s xml:id="echoid-s10332" xml:space="preserve">Siſemiſſis <lb/> <anchor type="note" xlink:label="note-335-06a" xlink:href="note-335-06"/> differentiæinter quadratum vtriusuis laterum ambientium angulum, à <lb/>quo perpendicularis demiſſa eſt, & </s> <s xml:id="echoid-s10333" xml:space="preserve">ſummam quadratorum ex reliquis <lb/>duobus lateribus deſcriptorum, diuidatur per latus, in quod perpendicu-<lb/>laris cadit, producetur ſegmentum baſis prope angulum, quem continent <lb/>latera, quorum ſumma quadratorum accepta fuit: </s> <s xml:id="echoid-s10334" xml:space="preserve">Hoc autem ſegmentum <lb/>ex eadem-baſe det ractum relinquet alterum ſegmentum.</s> <s xml:id="echoid-s10335" xml:space="preserve"/> </p> <div xml:id="echoid-div860" type="float" level="2" n="3"> <note position="right" xlink:label="note-335-06" xlink:href="note-335-06a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s10336" xml:space="preserve">CADENTE ver o perpendiculari extra triangulum; </s> <s xml:id="echoid-s10337" xml:space="preserve">Si ſe miſſis <lb/>diffcrentiæinter quadratum lateris angulo obtuſo oppoſiti, & </s> <s xml:id="echoid-s10338" xml:space="preserve">ſummam <pb o="324" file="336" n="336" rhead=""/> quadratorum ex reliquis duobus lateribus deſcriptorum, diuidatur per <lb/>latus, in quod productum perpendicularis cadit, procreabitur linea ex-<lb/>tra triangulum inter perpendicularem, & </s> <s xml:id="echoid-s10339" xml:space="preserve">angulum obtuſum: </s> <s xml:id="echoid-s10340" xml:space="preserve">Hæc vero <lb/>addita baſi conſtituet alteram rectam inter perpendicularem, & </s> <s xml:id="echoid-s10341" xml:space="preserve">angu-<lb/>lum acutum baſis.</s> <s xml:id="echoid-s10342" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div862" type="section" level="1" n="467"> <head xml:id="echoid-head499" xml:space="preserve">PROBL. 6. PROPOS. 10.</head> <p> <s xml:id="echoid-s10343" xml:space="preserve">DATIS omnibus angulis trianguli non re-<lb/> <anchor type="note" xlink:label="note-336-01a" xlink:href="note-336-01"/> ctanguli, vel datis eorum proportionibus, vna cũ <lb/>vno latere; </s> <s xml:id="echoid-s10344" xml:space="preserve">reliqua duo latera cognoſcere, & </s> <s xml:id="echoid-s10345" xml:space="preserve">quo-<lb/>rumlibet duorum proportionem facere notam.</s> <s xml:id="echoid-s10346" xml:space="preserve"/> </p> <div xml:id="echoid-div862" type="float" level="2" n="1"> <note position="left" xlink:label="note-336-01" xlink:href="note-336-01a" xml:space="preserve">In triang@ <lb/>lo non re-<lb/>@tãgulo ex <lb/>angulis no <lb/>tis, vel ex <lb/>proportio-<lb/>nibꝰ angu <lb/>lorum no-<lb/>tis, vna cũ <lb/>vno latere, <lb/>reliqua in <lb/>@eſtigãtur.</note> </div> <p> <s xml:id="echoid-s10347" xml:space="preserve">IN triangulo ABC, ſint primum omnes anguli acuti, & </s> <s xml:id="echoid-s10348" xml:space="preserve">dati; </s> <s xml:id="echoid-s10349" xml:space="preserve">A, grad. <lb/></s> <s xml:id="echoid-s10350" xml:space="preserve">75. </s> <s xml:id="echoid-s10351" xml:space="preserve">Min. </s> <s xml:id="echoid-s10352" xml:space="preserve">45. </s> <s xml:id="echoid-s10353" xml:space="preserve">B, grad. </s> <s xml:id="echoid-s10354" xml:space="preserve">67. </s> <s xml:id="echoid-s10355" xml:space="preserve">Min. </s> <s xml:id="echoid-s10356" xml:space="preserve">23. </s> <s xml:id="echoid-s10357" xml:space="preserve">C, grad. </s> <s xml:id="echoid-s10358" xml:space="preserve">36. </s> <s xml:id="echoid-s10359" xml:space="preserve">Min. </s> <s xml:id="echoid-s10360" xml:space="preserve">52. </s> <s xml:id="echoid-s10361" xml:space="preserve">Datum quoq; </s> <s xml:id="echoid-s10362" xml:space="preserve">ſit la-<lb/> <anchor type="figure" xlink:label="fig-336-01a" xlink:href="fig-336-01"/> tus AB, 13. </s> <s xml:id="echoid-s10363" xml:space="preserve">Oportet ex his <lb/>reliqua duo latera inueni-<lb/>re. </s> <s xml:id="echoid-s10364" xml:space="preserve">Quoniam eſt, vt ſinus <lb/> <anchor type="note" xlink:label="note-336-02a" xlink:href="note-336-02"/> anguli C, ad ſinum anguli <lb/>A, ita latus AB, datum ad <lb/>latus BC: </s> <s xml:id="echoid-s10365" xml:space="preserve">ſi fiat, vt ſinus <lb/>anguli C, nempe 59996. </s> <s xml:id="echoid-s10366" xml:space="preserve">ad <lb/>96923. </s> <s xml:id="echoid-s10367" xml:space="preserve">ſinum anguli A, ita <lb/>latus AB, datũ 13. </s> <s xml:id="echoid-s10368" xml:space="preserve">ad aliud, <lb/>inuenietur latus BC, 21. <lb/></s> <s xml:id="echoid-s10369" xml:space="preserve">ferè, vt hic apparet.</s> <s xml:id="echoid-s10370" xml:space="preserve"/> </p> <div xml:id="echoid-div863" type="float" level="2" n="2"> <figure xlink:label="fig-336-01" xlink:href="fig-336-01a"> <image file="336-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/336-01"/> </figure> <note position="left" xlink:label="note-336-02" xlink:href="note-336-02a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>C. # A. # AB. # # BC. <lb/>59996. # 96923. # 13? # fit # 21. # ferè. <lb/></note> <p> <s xml:id="echoid-s10371" xml:space="preserve">Item quia eſt, vt ſinus anguli C, ad ſinum anguli B, ita latus AB, datum <lb/>ad latus AC: </s> <s xml:id="echoid-s10372" xml:space="preserve">ſi fiat, vt 59996. </s> <s xml:id="echoid-s10373" xml:space="preserve">ſinus anguli C, ad 92310. </s> <s xml:id="echoid-s10374" xml:space="preserve">ſinum anguli B, ita <lb/>latus AB, datum 13. </s> <s xml:id="echoid-s10375" xml:space="preserve">ad aliud, inuenietur latus AC, fere 20. </s> <s xml:id="echoid-s10376" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s10377" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>C. # A. # AB. # # BC. <lb/>59996. # 92310. # 13? # fit # 20. # ferè. <lb/></note> <p> <s xml:id="echoid-s10378" xml:space="preserve">Velquia eſt, vt ſinus anguli A, ad ſinum anguli B, ita latus BC, inuentum <lb/> <anchor type="note" xlink:label="note-336-05a" xlink:href="note-336-05"/> ad latus AC: </s> <s xml:id="echoid-s10379" xml:space="preserve">ſi fiat, vt 96923. </s> <s xml:id="echoid-s10380" xml:space="preserve">ſinus anguli A, ad 92310. </s> <s xml:id="echoid-s10381" xml:space="preserve">ſinum anguli B, ita <lb/>latus BC, inuentum 21. </s> <s xml:id="echoid-s10382" xml:space="preserve">ad aliud, reperietur latus AC, fere 20. </s> <s xml:id="echoid-s10383" xml:space="preserve">vt nic cernis.</s> <s xml:id="echoid-s10384" xml:space="preserve"/> </p> <div xml:id="echoid-div864" type="float" level="2" n="3"> <note position="left" xlink:label="note-336-05" xlink:href="note-336-05a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>A. # B. # BC. # # AC. <lb/>96923. # 92310. # 21? # fit # 20. # ferè. <lb/></note> <p> <s xml:id="echoid-s10385" xml:space="preserve">SIT deinde angulus B, obtuſus grad. </s> <s xml:id="echoid-s10386" xml:space="preserve">112. </s> <s xml:id="echoid-s10387" xml:space="preserve">Min. </s> <s xml:id="echoid-s10388" xml:space="preserve">37. </s> <s xml:id="echoid-s10389" xml:space="preserve">A, grad. </s> <s xml:id="echoid-s10390" xml:space="preserve">30. </s> <s xml:id="echoid-s10391" xml:space="preserve">Min. </s> <s xml:id="echoid-s10392" xml:space="preserve">31. <lb/></s> <s xml:id="echoid-s10393" xml:space="preserve">C, grad. </s> <s xml:id="echoid-s10394" xml:space="preserve">36. </s> <s xml:id="echoid-s10395" xml:space="preserve">Min 52. </s> <s xml:id="echoid-s10396" xml:space="preserve">& </s> <s xml:id="echoid-s10397" xml:space="preserve">rurſum latus AB, 13. </s> <s xml:id="echoid-s10398" xml:space="preserve">Quoniam idem eſt ſinus anguli <lb/>B, qui eius complementigrad. </s> <s xml:id="echoid-s10399" xml:space="preserve">67. </s> <s xml:id="echoid-s10400" xml:space="preserve">Min. </s> <s xml:id="echoid-s10401" xml:space="preserve">23. </s> <s xml:id="echoid-s10402" xml:space="preserve">vt in tractatione ſinuum demou-<lb/>ſtrauimus, inuenietur ratione iam expoſita latus BC, 11. </s> <s xml:id="echoid-s10403" xml:space="preserve">& </s> <s xml:id="echoid-s10404" xml:space="preserve">AC, 20. </s> <s xml:id="echoid-s10405" xml:space="preserve">fermè. </s> <s xml:id="echoid-s10406" xml:space="preserve"><lb/>vt hic liquido conſtat.</s> <s xml:id="echoid-s10407" xml:space="preserve"/> </p> <pb o="325" file="337" n="337" rhead=""/> <note position="right" xml:space="preserve"> <lb/>C. # A. # # AB. # # BC. <lb/>59996. # 50779. # # 13? # fit # 11. # ferè. <lb/> # # Item. <lb/>C. # B. # # AB. # # AC. <lb/>59996. # 92310. # # 13? # fit # 20. # ferè. <lb/> # # Vel. <lb/>A. # B. # # BC. # # AC. <lb/>50779. # 92310. # # 11? # fit # 20. # ferè. <lb/></note> <p style="it"> <s xml:id="echoid-s10408" xml:space="preserve">ITAQVE, datis angulis omnibus, cum vno latere; </s> <s xml:id="echoid-s10409" xml:space="preserve">Si fiat, vt ſi-<lb/> <anchor type="note" xlink:label="note-337-02a" xlink:href="note-337-02"/> nus anguli lateri dato oppoſiti ad ſinum vtriuſuis reliquorum angulorum, <lb/>ita latus datum ad aliud, inuenietur latus angulo illi, cuius ſinus acce-<lb/>ptus eſt, oppoſitum: </s> <s xml:id="echoid-s10410" xml:space="preserve">Et ſi rurſus fiat, vt ſinus anguli lateri dato oppo-<lb/>ſiti ad ſinum tertij anguli, ita datum latus ad aliud, reperietur latus ter <lb/>tio angulo oppoſitum, &</s> <s xml:id="echoid-s10411" xml:space="preserve">c.</s> <s xml:id="echoid-s10412" xml:space="preserve"/> </p> <div xml:id="echoid-div865" type="float" level="2" n="4"> <note position="right" xlink:label="note-337-02" xlink:href="note-337-02a" xml:space="preserve">Praxis.</note> </div> <note position="right" xml:space="preserve">Quãdo triã <lb/>gulũ eſt Iſo <lb/>ſceles, vel ę-<lb/>quilaterũ: <lb/>aut quãdo <lb/>in ſcaleno <lb/>dãtur duo <lb/>latera cum <lb/>angulis.</note> <p> <s xml:id="echoid-s10413" xml:space="preserve">QVOD ſi triangulum fuerit Iſoſceles, & </s> <s xml:id="echoid-s10414" xml:space="preserve">dentur anguli: </s> <s xml:id="echoid-s10415" xml:space="preserve">Vel ſi fuerit ſca-<lb/>lenum, & </s> <s xml:id="echoid-s10416" xml:space="preserve">duo dentur latera cum angulis, vnius tantum lateris inuentione <lb/>opus eſt, vt patet. </s> <s xml:id="echoid-s10417" xml:space="preserve">In æquilatero autem triangulo, ſi detur vnum latus, erunt <lb/>& </s> <s xml:id="echoid-s10418" xml:space="preserve">reliqua data, vtpoteilli æqualia.</s> <s xml:id="echoid-s10419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10420" xml:space="preserve">IAM verò ſi datæ ſint proportiones angulorum, cum vno latere, inue-<lb/>ſtigandæ erunt ex illis proportionibus magnitudines angulorum, vt in ſcho-<lb/>lio propoſ. </s> <s xml:id="echoid-s10421" xml:space="preserve">1. </s> <s xml:id="echoid-s10422" xml:space="preserve">demonſtrauimus: </s> <s xml:id="echoid-s10423" xml:space="preserve">Deinde vero reliqua latera exploranda, vt <lb/> <anchor type="note" xlink:label="note-337-04a" xlink:href="note-337-04"/> hic oſtenſum eſt.</s> <s xml:id="echoid-s10424" xml:space="preserve"/> </p> <div xml:id="echoid-div866" type="float" level="2" n="5"> <note position="right" xlink:label="note-337-04" xlink:href="note-337-04a" xml:space="preserve">Quãdo dã-<lb/>rur propor <lb/>tiones an-<lb/>gulorũ, cũ <lb/>vno latere.</note> </div> <p> <s xml:id="echoid-s10425" xml:space="preserve">INVENTIS autem lateribus, liquido conſtat, eorum proportiones <lb/>eſſe notas in numeris, in quibus ipſa cognita ſunt. </s> <s xml:id="echoid-s10426" xml:space="preserve">Datis ergo omnibus angu-<lb/>lis trianguli non rectanguli, &</s> <s xml:id="echoid-s10427" xml:space="preserve">c. </s> <s xml:id="echoid-s10428" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s10429" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div868" type="section" level="1" n="468"> <head xml:id="echoid-head500" xml:space="preserve">PROBL. 7. PROPOS. 11.</head> <p> <s xml:id="echoid-s10430" xml:space="preserve">DATIS omnibus trianguli non rectanguli <lb/> <anchor type="note" xlink:label="note-337-05a" xlink:href="note-337-05"/> lateribus, vel eorum proportionibus, omnes an-<lb/>gulos notos efficere.</s> <s xml:id="echoid-s10431" xml:space="preserve"/> </p> <div xml:id="echoid-div868" type="float" level="2" n="1"> <note position="right" xlink:label="note-337-05" xlink:href="note-337-05a" xml:space="preserve">In triãgulo <lb/>ex notis la-<lb/>teribus, vel <lb/>ex eorũ pro <lb/>portionibꝰ <lb/>notis, angu <lb/>li inueniũ-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s10432" xml:space="preserve">IN triangulo priori ABC, ſint data omnia <lb/> <anchor type="figure" xlink:label="fig-337-01a" xlink:href="fig-337-01"/> latera; </s> <s xml:id="echoid-s10433" xml:space="preserve">AB, 13. </s> <s xml:id="echoid-s10434" xml:space="preserve">AC, 20. </s> <s xml:id="echoid-s10435" xml:space="preserve">& </s> <s xml:id="echoid-s10436" xml:space="preserve">BC, 21. </s> <s xml:id="echoid-s10437" xml:space="preserve">Oporter ex <lb/>his inueſtigare angulos. </s> <s xml:id="echoid-s10438" xml:space="preserve">In maximum latus BC, <lb/>ex oppoſito angulo A, ducatur perpendicularis <lb/>AD, quę neceſlario intra triangulum cadet. </s> <s xml:id="echoid-s10439" xml:space="preserve">Cum <lb/>enim latus BC, ſit maximum, erit & </s> <s xml:id="echoid-s10440" xml:space="preserve">angulus A, <lb/> <anchor type="note" xlink:label="note-337-06a" xlink:href="note-337-06"/> ipſi oppoſitus, maximus: </s> <s xml:id="echoid-s10441" xml:space="preserve">ac propterea vterq; </s> <s xml:id="echoid-s10442" xml:space="preserve">B, <lb/> <anchor type="note" xlink:label="note-337-07a" xlink:href="note-337-07"/> C, acutus. </s> <s xml:id="echoid-s10443" xml:space="preserve">Ex quo fit, perpendicularem AD, in-<lb/> <anchor type="note" xlink:label="note-337-08a" xlink:href="note-337-08"/> tra triangulum cadere. </s> <s xml:id="echoid-s10444" xml:space="preserve">Primùm itaq; </s> <s xml:id="echoid-s10445" xml:space="preserve">inquiran-<lb/> <anchor type="note" xlink:label="note-337-09a" xlink:href="note-337-09"/> tur rectæ BD, CD. </s> <s xml:id="echoid-s10446" xml:space="preserve">Inuenietur BD, 5. </s> <s xml:id="echoid-s10447" xml:space="preserve">& </s> <s xml:id="echoid-s10448" xml:space="preserve">CD, <lb/> <anchor type="note" xlink:label="note-337-10a" xlink:href="note-337-10"/> 16. </s> <s xml:id="echoid-s10449" xml:space="preserve">Quia ergo, poſito ſinu toto AB, recta BD, <lb/>ſinus eſt anguli BAD, vt in tractatione ſinuum <lb/>oſtendimus, dicemus per auream regulam. </s> <s xml:id="echoid-s10450" xml:space="preserve">Si AB, <pb o="326" file="338" n="338" rhead=""/> 13. </s> <s xml:id="echoid-s10451" xml:space="preserve">dat AB, ſinum totum partium 100000. </s> <s xml:id="echoid-s10452" xml:space="preserve">quid dabit BD, 5? </s> <s xml:id="echoid-s10453" xml:space="preserve">inueniemusq́ <lb/>ſinum BD, 38461. </s> <s xml:id="echoid-s10454" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s10455" xml:space="preserve"/> </p> <div xml:id="echoid-div869" type="float" level="2" n="2"> <figure xlink:label="fig-337-01" xlink:href="fig-337-01a"> <image file="337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/337-01"/> </figure> <note position="right" xlink:label="note-337-06" xlink:href="note-337-06a" xml:space="preserve">19. primi.</note> <note position="right" xlink:label="note-337-07" xlink:href="note-337-07a" xml:space="preserve">17. primi.</note> <note position="right" xlink:label="note-337-08" xlink:href="note-337-08a" xml:space="preserve">Scholium</note> <note position="right" xlink:label="note-337-09" xlink:href="note-337-09a" xml:space="preserve">13. ſecundi.</note> <note position="right" xlink:label="note-337-10" xlink:href="note-337-10a" xml:space="preserve">9. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # BD. # # BD. <lb/>13. # 100000. # 5? # fit. # 38461. <lb/></note> <p> <s xml:id="echoid-s10456" xml:space="preserve">Ex tabula ergo ſinuum dabitur angulus BAD, <lb/> <anchor type="figure" xlink:label="fig-338-01a" xlink:href="fig-338-01"/> grad. </s> <s xml:id="echoid-s10457" xml:space="preserve">22. </s> <s xml:id="echoid-s10458" xml:space="preserve">Min. </s> <s xml:id="echoid-s10459" xml:space="preserve">37. </s> <s xml:id="echoid-s10460" xml:space="preserve">atq; </s> <s xml:id="echoid-s10461" xml:space="preserve">adeo eius complementum <lb/>B, grad. </s> <s xml:id="echoid-s10462" xml:space="preserve">67. </s> <s xml:id="echoid-s10463" xml:space="preserve">Min. </s> <s xml:id="echoid-s10464" xml:space="preserve">23. </s> <s xml:id="echoid-s10465" xml:space="preserve">qui eſt vnus angulorum quę-<lb/>ſitorum. </s> <s xml:id="echoid-s10466" xml:space="preserve">Rurſus, quia poſito AC, ſinu toto, CD, <lb/>ſinus eſt anguli CAD, dicemus iterum per regu-<lb/>lam auream. </s> <s xml:id="echoid-s10467" xml:space="preserve">Si AC, 20. </s> <s xml:id="echoid-s10468" xml:space="preserve">dat AC, 100000. </s> <s xml:id="echoid-s10469" xml:space="preserve">ſinum <lb/>totum, quid dabit CD, 16? </s> <s xml:id="echoid-s10470" xml:space="preserve">Inueniemuſq; </s> <s xml:id="echoid-s10471" xml:space="preserve">ſinum <lb/>CD, 80000. </s> <s xml:id="echoid-s10472" xml:space="preserve">Vt hic patet.</s> <s xml:id="echoid-s10473" xml:space="preserve"/> </p> <div xml:id="echoid-div870" type="float" level="2" n="3"> <figure xlink:label="fig-338-01" xlink:href="fig-338-01a"> <image file="338-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/338-01"/> </figure> </div> <note position="right" xml:space="preserve"> <lb/>AC. # AC. # CD. # # CD. <lb/>20. # 100000. # 16? # fit # 80000. <lb/></note> <p> <s xml:id="echoid-s10474" xml:space="preserve">Qui ſinus in tabula ſinuum monſtrat angulum <lb/>CAD, grad. </s> <s xml:id="echoid-s10475" xml:space="preserve">53. </s> <s xml:id="echoid-s10476" xml:space="preserve">Min. </s> <s xml:id="echoid-s10477" xml:space="preserve">8. </s> <s xml:id="echoid-s10478" xml:space="preserve">ac proinde eius comple-<lb/>mentum C, erit grad. </s> <s xml:id="echoid-s10479" xml:space="preserve">36. </s> <s xml:id="echoid-s10480" xml:space="preserve">Min. </s> <s xml:id="echoid-s10481" xml:space="preserve">52. </s> <s xml:id="echoid-s10482" xml:space="preserve">qui eſt vnus <lb/>etiam angulorum quæſitorum. </s> <s xml:id="echoid-s10483" xml:space="preserve">Quòd ſi duo an-<lb/>guli duorum ſinuum inuentorum, nempe grad. </s> <s xml:id="echoid-s10484" xml:space="preserve">22. </s> <s xml:id="echoid-s10485" xml:space="preserve">Min. </s> <s xml:id="echoid-s10486" xml:space="preserve">37. </s> <s xml:id="echoid-s10487" xml:space="preserve">& </s> <s xml:id="echoid-s10488" xml:space="preserve">grad. </s> <s xml:id="echoid-s10489" xml:space="preserve">53. </s> <s xml:id="echoid-s10490" xml:space="preserve">Min. <lb/></s> <s xml:id="echoid-s10491" xml:space="preserve"> <anchor type="note" xlink:label="note-338-03a" xlink:href="note-338-03"/> 8. </s> <s xml:id="echoid-s10492" xml:space="preserve">ſimul componantur, fiet tertius angulus BAC, grad. </s> <s xml:id="echoid-s10493" xml:space="preserve">75. </s> <s xml:id="echoid-s10494" xml:space="preserve">Min. </s> <s xml:id="echoid-s10495" xml:space="preserve">45. </s> <s xml:id="echoid-s10496" xml:space="preserve">Vel cer-<lb/>te ſi ſumma duorum angulorum B, C, inuentorum ex grad. </s> <s xml:id="echoid-s10497" xml:space="preserve">180. </s> <s xml:id="echoid-s10498" xml:space="preserve">auferatur, <lb/>reliquus fiet tertius angulus BAC, grad. </s> <s xml:id="echoid-s10499" xml:space="preserve">75. </s> <s xml:id="echoid-s10500" xml:space="preserve">Min. </s> <s xml:id="echoid-s10501" xml:space="preserve">45.</s> <s xml:id="echoid-s10502" xml:space="preserve"/> </p> <div xml:id="echoid-div871" type="float" level="2" n="4"> <note position="left" xlink:label="note-338-03" xlink:href="note-338-03a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s10503" xml:space="preserve">ITAQVE, (vt totam praxim complectamur) ſi fiat, vt maximum <lb/>latus (in quod perpendicularis ducta eſt) ad ſummam aliorum duorum, <lb/>ita differentia eorundem duorum ad aliud, reperietur numerus, qui ex <lb/>maximo latere ſubductus relinquet numerum, cuius ſemiſſis dabit minus <lb/>ſegmentum baſis, hoc autem ex baſi detractum relinquet maius ſegmen-<lb/>tum, vt conſtat ex ſecunda praxi propoſ. </s> <s xml:id="echoid-s10504" xml:space="preserve">9. </s> <s xml:id="echoid-s10505" xml:space="preserve">Quòd ſi rurſum fiat, vt mi-<lb/>nimum latus ad ſinum totum, ita ſegmentum baſis minus ad aliud, inue-<lb/>nietur ſinus, cuius arcus complementum dabit angulum ſupra baſim me-<lb/>dio lateri oppoſitum. </s> <s xml:id="echoid-s10506" xml:space="preserve">Deinde ſi rurſum fiat, vt medium latus ad ſinum <lb/>totum, ita maius ſegmentum baſis ad aliud, reperietur ſinus, cuius ar-<lb/>cus complementum dabit angulum ſupra baſim minimo lateri oppoſitum. <lb/></s> <s xml:id="echoid-s10507" xml:space="preserve">Tertius vero angulus maximo lateri oppoſitus conflabitur ex duobus il-<lb/>lis arcubus duorum ſinuum inuentorum: </s> <s xml:id="echoid-s10508" xml:space="preserve">vel certè relinquetur poſt de-<lb/>tractionem duorum angulorum inuentorum ex duobus rectis.</s> <s xml:id="echoid-s10509" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10510" xml:space="preserve">RVRSVS in poſteriori triangulo datum ſit latus AB, 11. </s> <s xml:id="echoid-s10511" xml:space="preserve">AC, 13. </s> <s xml:id="echoid-s10512" xml:space="preserve">& </s> <s xml:id="echoid-s10513" xml:space="preserve"><lb/>BC, 20. </s> <s xml:id="echoid-s10514" xml:space="preserve">Demiſſa in maximum latus BC, perpendiculari AD, inuenietur <lb/>ſegmentum BD, 8 {4/5}. </s> <s xml:id="echoid-s10515" xml:space="preserve">at CD, 11{@/5}. </s> <s xml:id="echoid-s10516" xml:space="preserve">vt hic apparet ſecundum praxim poſte-<lb/>riorem propoſ. </s> <s xml:id="echoid-s10517" xml:space="preserve">9.</s> <s xml:id="echoid-s10518" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>BC. # AB. AB. # Differ. inter # AB, AC. <lb/>20. # 24. # 2? # fit # 2 {2/5}. <lb/></note> <p> <s xml:id="echoid-s10519" xml:space="preserve">Hic numerus ex baſe 20. </s> <s xml:id="echoid-s10520" xml:space="preserve">ablatus relinquit 17 {3/5}. </s> <s xml:id="echoid-s10521" xml:space="preserve">cuius ſemiſsis 8 {4/5}. </s> <s xml:id="echoid-s10522" xml:space="preserve">dat ſeg-<lb/>mentum minus BD. </s> <s xml:id="echoid-s10523" xml:space="preserve">Ergo maius CD, erit 11 {1/5}. </s> <s xml:id="echoid-s10524" xml:space="preserve">Hinc inuenietur angulus <pb o="327" file="339" n="339" rhead=""/> B, grad. </s> <s xml:id="echoid-s10525" xml:space="preserve">36. </s> <s xml:id="echoid-s10526" xml:space="preserve">Min. </s> <s xml:id="echoid-s10527" xml:space="preserve">52. </s> <s xml:id="echoid-s10528" xml:space="preserve">C, grad. </s> <s xml:id="echoid-s10529" xml:space="preserve">30. </s> <s xml:id="echoid-s10530" xml:space="preserve">Min. </s> <s xml:id="echoid-s10531" xml:space="preserve">31. </s> <s xml:id="echoid-s10532" xml:space="preserve">& </s> <s xml:id="echoid-s10533" xml:space="preserve">BAC, grad. </s> <s xml:id="echoid-s10534" xml:space="preserve">112. </s> <s xml:id="echoid-s10535" xml:space="preserve">Min. </s> <s xml:id="echoid-s10536" xml:space="preserve">37. </s> <s xml:id="echoid-s10537" xml:space="preserve">vt <lb/>hic vides.</s> <s xml:id="echoid-s10538" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # # BD. # # BD. <lb/>11. # 100000. # # 8 {4/5}? # fit # 80000. <lb/> # # Item. <lb/>AC. # AC. # # CD. # # CD. <lb/>13. # 100000. # # 11 {1/5}? # fit # 86154. <lb/></note> <p> <s xml:id="echoid-s10539" xml:space="preserve">Complemétum arcus, quem prior ſinus inuentus offert, dat angulum B, grad. <lb/></s> <s xml:id="echoid-s10540" xml:space="preserve">36. </s> <s xml:id="echoid-s10541" xml:space="preserve">Min. </s> <s xml:id="echoid-s10542" xml:space="preserve">52. </s> <s xml:id="echoid-s10543" xml:space="preserve">At complementum arcus poſterioris ſinus inuenti dat angulum <lb/>C, grad. </s> <s xml:id="echoid-s10544" xml:space="preserve">30. </s> <s xml:id="echoid-s10545" xml:space="preserve">Min. </s> <s xml:id="echoid-s10546" xml:space="preserve">31. </s> <s xml:id="echoid-s10547" xml:space="preserve">&</s> <s xml:id="echoid-s10548" xml:space="preserve">c. </s> <s xml:id="echoid-s10549" xml:space="preserve">Eſt ergo doctrina huius propoſitionis generalis, ſi-<lb/> <anchor type="note" xlink:label="note-339-02a" xlink:href="note-339-02"/> ue angulus maximus A, acutus ſit, vt in priori triangulo, ſiue obtuſus, vt in <lb/>poſteriori, ſiue deniq; </s> <s xml:id="echoid-s10550" xml:space="preserve">rectus ſit; </s> <s xml:id="echoid-s10551" xml:space="preserve">quamuis in rectangulo triangulo iam ſupra <lb/>traditum ſit propoſ. </s> <s xml:id="echoid-s10552" xml:space="preserve">3. </s> <s xml:id="echoid-s10553" xml:space="preserve">quo pacto ex duobus lateribus cognitis facilius an-<lb/>guli duo acuti inueniantur.</s> <s xml:id="echoid-s10554" xml:space="preserve"/> </p> <div xml:id="echoid-div872" type="float" level="2" n="5"> <note position="right" xlink:label="note-339-02" xlink:href="note-339-02a" xml:space="preserve">Generalitas <lb/>huius pro-<lb/>poſ.</note> </div> <note position="right" xml:space="preserve">Quando la <lb/>terum pro-<lb/>portiones <lb/>datæ ſunt.</note> <p> <s xml:id="echoid-s10555" xml:space="preserve">IAM ſi dentur laterum proportiones, ſaltem duæ, continuabimus eas in <lb/>tribus minimis numeris, ſi proportionum numeri minimi non ſint, vt Eucl. <lb/></s> <s xml:id="echoid-s10556" xml:space="preserve">docuit propoſ. </s> <s xml:id="echoid-s10557" xml:space="preserve">4. </s> <s xml:id="echoid-s10558" xml:space="preserve">lib. </s> <s xml:id="echoid-s10559" xml:space="preserve">8. </s> <s xml:id="echoid-s10560" xml:space="preserve">eosq́; </s> <s xml:id="echoid-s10561" xml:space="preserve">numeros lateribus aſcribemus, perinde ac ſi in <lb/> <anchor type="note" xlink:label="note-339-04a" xlink:href="note-339-04"/> illis numeris darentur. </s> <s xml:id="echoid-s10562" xml:space="preserve">Vt ſi in priori triangulo proportio AB, ad BC, ſit, <lb/>quæ 26. </s> <s xml:id="echoid-s10563" xml:space="preserve">ad 42. </s> <s xml:id="echoid-s10564" xml:space="preserve">At AB, ad AC, quæ 39. </s> <s xml:id="echoid-s10565" xml:space="preserve">ad 60. </s> <s xml:id="echoid-s10566" xml:space="preserve">reuocabuntur hæ proportiones <lb/>ad minimos hoſce numeros 13. </s> <s xml:id="echoid-s10567" xml:space="preserve">21. </s> <s xml:id="echoid-s10568" xml:space="preserve">& </s> <s xml:id="echoid-s10569" xml:space="preserve">13. </s> <s xml:id="echoid-s10570" xml:space="preserve">20. </s> <s xml:id="echoid-s10571" xml:space="preserve">Dabitur ergo AB, 13. </s> <s xml:id="echoid-s10572" xml:space="preserve">AC, 20. <lb/></s> <s xml:id="echoid-s10573" xml:space="preserve"> <anchor type="note" xlink:label="note-339-05a" xlink:href="note-339-05"/> & </s> <s xml:id="echoid-s10574" xml:space="preserve">BC, 21. </s> <s xml:id="echoid-s10575" xml:space="preserve">Ex quibus angulos eruemus, vt prius.</s> <s xml:id="echoid-s10576" xml:space="preserve"/> </p> <div xml:id="echoid-div873" type="float" level="2" n="6"> <note position="right" xlink:label="note-339-04" xlink:href="note-339-04a" xml:space="preserve">35. ſeptimi.</note> <note position="right" xlink:label="note-339-05" xlink:href="note-339-05a" xml:space="preserve">Quãdo triã <lb/>gulũ eſt Iſo <lb/>ſceles. <lb/>coroll. 8. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s10577" xml:space="preserve">PORRO in Iſoſcele datorum laterum ducenda eſt perpendicularis ad <lb/>baſim, ſiue ea ſit maximum latus, ſiue minimum: </s> <s xml:id="echoid-s10578" xml:space="preserve">quæ diuidet baſim bifariam. <lb/></s> <s xml:id="echoid-s10579" xml:space="preserve">Quare ſi fiat, vt vnum æqualium laterum ad ſinũ totum, ita dimidium baſis <lb/>ad aliud, inuenietur ſinus cuiuſdam arcus, cuius complementum dabit vnum <lb/>æ qualium angulorum ſupra baſim, vt ex demonſtratis liquet. </s> <s xml:id="echoid-s10580" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s10581" xml:space="preserve">alter da-<lb/>bitur: </s> <s xml:id="echoid-s10582" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s10583" xml:space="preserve">tertius baſi oppoſitus, vtpote reliquus duorũ rectorum.</s> <s xml:id="echoid-s10584" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10585" xml:space="preserve">IN æquilatero deniq; </s> <s xml:id="echoid-s10586" xml:space="preserve">triangulo dabuntur anguli, etiamſi latera non den-<lb/> <anchor type="note" xlink:label="note-339-06a" xlink:href="note-339-06"/> tur, cum quilibet ſit tertia pars duorum rectorum, hoc eſt, contineat grad. <lb/></s> <s xml:id="echoid-s10587" xml:space="preserve">60. </s> <s xml:id="echoid-s10588" xml:space="preserve">Datis igitur omnibus trianguli non rectanguli lateribus, &</s> <s xml:id="echoid-s10589" xml:space="preserve">c. </s> <s xml:id="echoid-s10590" xml:space="preserve">Quod fa-<lb/>ciendum erat.</s> <s xml:id="echoid-s10591" xml:space="preserve"/> </p> <div xml:id="echoid-div874" type="float" level="2" n="7"> <note position="right" xlink:label="note-339-06" xlink:href="note-339-06a" xml:space="preserve">Quãdo triã <lb/>gulũ eſt æ-<lb/>quilaterũ.</note> </div> </div> <div xml:id="echoid-div876" type="section" level="1" n="469"> <head xml:id="echoid-head501" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s10592" xml:space="preserve">_ETSI_ in hac propoſ. </s> <s xml:id="echoid-s10593" xml:space="preserve">præcepimus, perpendicularem ad maximum latus eſſe du-<lb/>cendam ex angulo oppoſito, vt intra triangulum cadat, fiatq; </s> <s xml:id="echoid-s10594" xml:space="preserve">calculus facilior: </s> <s xml:id="echoid-s10595" xml:space="preserve">ta-<lb/>men eadem fere via problema abſoluemus, ſi in triangulo obtuſangulo perpendicula-<lb/>ris non ducatur ab obtſo angulo in maximum latus, ſed ab alterutro acutorum an-<lb/> <anchor type="note" xlink:label="note-339-07a" xlink:href="note-339-07"/> gulorum in latus oppoſitum protractum, ita vt cadat extra <lb/> <anchor type="figure" xlink:label="fig-339-01a" xlink:href="fig-339-01"/> triangulum, vt in hoc triangulo _ABC,_ manifeſtum eſt, in <lb/>quo latus _AB,_ datur _22. </s> <s xml:id="echoid-s10596" xml:space="preserve">AC, 31._ </s> <s xml:id="echoid-s10597" xml:space="preserve">& </s> <s xml:id="echoid-s10598" xml:space="preserve">_BC, 14._ </s> <s xml:id="echoid-s10599" xml:space="preserve">_N_am ſi fiat, <lb/>vt _BC, 14._ </s> <s xml:id="echoid-s10600" xml:space="preserve">(in quod latus perpendicularis eſt ducta) ad <lb/>_53._ </s> <s xml:id="echoid-s10601" xml:space="preserve">ſummam aliorum laterum _AB, AC,_ ita _9._ </s> <s xml:id="echoid-s10602" xml:space="preserve">differentia <lb/>eorundem laterum ad aliud, reperietur numerus _34 {1/14}._ </s> <s xml:id="echoid-s10603" xml:space="preserve">à <lb/>quo ſi ſubducatur latus _<emph style="sc">B</emph>C,_ remanebit numerus _20 {1/14}._ <lb/></s> <s xml:id="echoid-s10604" xml:space="preserve">cuius ſemiſsis _10 {1/28}._ </s> <s xml:id="echoid-s10605" xml:space="preserve">erit recta _BD,_ ac proinde _CD,_ <lb/>_24 {1/28}._ </s> <s xml:id="echoid-s10606" xml:space="preserve">Quam obrem ſi iam fiat, vt _<emph style="sc">Ab</emph>,_ _22._ </s> <s xml:id="echoid-s10607" xml:space="preserve">ad _<emph style="sc">Ab</emph>,_ ſi-<lb/>num totum 100000. </s> <s xml:id="echoid-s10608" xml:space="preserve">ita _BD, 10 {1/28}._ </s> <s xml:id="echoid-s10609" xml:space="preserve">ad aliud, innenietur _<emph style="sc">B</emph>D,_ ſinus _45617._</s> <s xml:id="echoid-s10610" xml:space="preserve"> <pb o="328" file="340" n="340" rhead=""/> cuius arcus complementum exhibebit angulum _ABD,_ grad. </s> <s xml:id="echoid-s10611" xml:space="preserve">_62._ </s> <s xml:id="echoid-s10612" xml:space="preserve">Min. </s> <s xml:id="echoid-s10613" xml:space="preserve">_52._ </s> <s xml:id="echoid-s10614" xml:space="preserve">ac propte-<lb/>rea reliquum duorum rectorum _ABC,_ grad. </s> <s xml:id="echoid-s10615" xml:space="preserve">_117._ </s> <s xml:id="echoid-s10616" xml:space="preserve">Min. </s> <s xml:id="echoid-s10617" xml:space="preserve">_8._ </s> <s xml:id="echoid-s10618" xml:space="preserve">_I_tem ſi fiat, vt _<emph style="sc">Ac</emph>, 31._ <lb/></s> <s xml:id="echoid-s10619" xml:space="preserve">ad _AC,_ _100000._ </s> <s xml:id="echoid-s10620" xml:space="preserve">ſinum totum, ita _CD, 24 {1/28}._ </s> <s xml:id="echoid-s10621" xml:space="preserve">ad aliud, reperietur ſinus _77534._ </s> <s xml:id="echoid-s10622" xml:space="preserve"><lb/>cuius arcus complementum offeret angulum _C,_ grad. </s> <s xml:id="echoid-s10623" xml:space="preserve">_39._ </s> <s xml:id="echoid-s10624" xml:space="preserve">_M_in. </s> <s xml:id="echoid-s10625" xml:space="preserve">_10._ </s> <s xml:id="echoid-s10626" xml:space="preserve">Quòd ſi duo an-<lb/>guli _<emph style="sc">Ab</emph>C,_ & </s> <s xml:id="echoid-s10627" xml:space="preserve">_<emph style="sc">C</emph>,_ ex grad. </s> <s xml:id="echoid-s10628" xml:space="preserve">_180._ </s> <s xml:id="echoid-s10629" xml:space="preserve">demantur, relinquetur angulus _<emph style="sc">B</emph><emph style="sc">Ac</emph>,_ grad. </s> <s xml:id="echoid-s10630" xml:space="preserve">_23._ </s> <s xml:id="echoid-s10631" xml:space="preserve"><lb/>_M_in. </s> <s xml:id="echoid-s10632" xml:space="preserve">_42._ </s> <s xml:id="echoid-s10633" xml:space="preserve">_R_atio huius operationis colligitur ex tractatione ſinuum, vbi oſtendimus, ſi <lb/>_<emph style="sc">Ab</emph>,_ ſtatuatur ſinus totus, _BD,_ eſſe ſinum anguli _<emph style="sc">B</emph>AD:_ </s> <s xml:id="echoid-s10634" xml:space="preserve">_I_tem ſi _<emph style="sc">Ac</emph>,_ ponatur ſinus <lb/>totus, _CD,_ eſſe ſinum anguli _<emph style="sc">C</emph>AD,_ &</s> <s xml:id="echoid-s10635" xml:space="preserve">c.</s> <s xml:id="echoid-s10636" xml:space="preserve"/> </p> <div xml:id="echoid-div876" type="float" level="2" n="1"> <note position="right" xlink:label="note-339-07" xlink:href="note-339-07a" xml:space="preserve">Quádo per <lb/>pendicũla-<lb/>ris in obtu-<lb/>sãgulo triã <lb/>gulo cadit <lb/>extta trian <lb/>gulum.</note> <figure xlink:label="fig-339-01" xlink:href="fig-339-01a"> <image file="339-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/339-01"/> </figure> </div> </div> <div xml:id="echoid-div878" type="section" level="1" n="470"> <head xml:id="echoid-head502" xml:space="preserve">PROBL. 8. PROPOS. 12.</head> <p> <s xml:id="echoid-s10637" xml:space="preserve">DATIS duobus lateribus trianguli non re-<lb/> <anchor type="note" xlink:label="note-340-01a" xlink:href="note-340-01"/> ctanguli, cum angulo ab ipſis comprehenſo; </s> <s xml:id="echoid-s10638" xml:space="preserve">vel <lb/>data proportione duorum laterum angulum da-<lb/>tum continentium: </s> <s xml:id="echoid-s10639" xml:space="preserve">tertium latus, & </s> <s xml:id="echoid-s10640" xml:space="preserve">reliquos an-<lb/>gulos inuenire.</s> <s xml:id="echoid-s10641" xml:space="preserve"/> </p> <div xml:id="echoid-div878" type="float" level="2" n="1"> <note position="left" xlink:label="note-340-01" xlink:href="note-340-01a" xml:space="preserve">In triágulo <lb/>nõ rectágu <lb/>lo ex duo-<lb/>bus lateri-<lb/>bus notis, <lb/>velex eorũ <lb/>proportio-<lb/>ne nota, cũ <lb/>angulo ab <lb/>ipſis com-<lb/>prehenſo, <lb/>tertium la-<lb/>tus, & reli-<lb/>qui anguli <lb/>exquirũt̃.</note> </div> <p> <s xml:id="echoid-s10642" xml:space="preserve">IN triangulo ABC, data ſint primum duo latera AB, AC, illud 3. </s> <s xml:id="echoid-s10643" xml:space="preserve">hoc <lb/>5. </s> <s xml:id="echoid-s10644" xml:space="preserve">ambientia angulum A, obtuſum, qui datus etiam ſit gnad. </s> <s xml:id="echoid-s10645" xml:space="preserve">93. </s> <s xml:id="echoid-s10646" xml:space="preserve">Min. </s> <s xml:id="echoid-s10647" xml:space="preserve">50. <lb/></s> <s xml:id="echoid-s10648" xml:space="preserve">Oportet ex his tertium latus BC, & </s> <s xml:id="echoid-s10649" xml:space="preserve">reliquos angulos B, C, inueſtigare. </s> <s xml:id="echoid-s10650" xml:space="preserve">Quo-<lb/>niam datur angulus A, grad. </s> <s xml:id="echoid-s10651" xml:space="preserve">93. </s> <s xml:id="echoid-s10652" xml:space="preserve">Min. </s> <s xml:id="echoid-s10653" xml:space="preserve">50. </s> <s xml:id="echoid-s10654" xml:space="preserve">ſi detrahatur ex grad. </s> <s xml:id="echoid-s10655" xml:space="preserve">180. </s> <s xml:id="echoid-s10656" xml:space="preserve">hoc eſt, <lb/>ex duobus rectis, reliquum erit aggregatum duorum angulorum B, C, grad. </s> <s xml:id="echoid-s10657" xml:space="preserve"><lb/>86. </s> <s xml:id="echoid-s10658" xml:space="preserve">Min. </s> <s xml:id="echoid-s10659" xml:space="preserve">10. </s> <s xml:id="echoid-s10660" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s10661" xml:space="preserve">proportio ſinuum angulorum C, B, data, nempe ea-<lb/>dem, quæ lateris AB, 3. </s> <s xml:id="echoid-s10662" xml:space="preserve">ad latus AC, 5. </s> <s xml:id="echoid-s10663" xml:space="preserve">propterea quod eſt, vt latus AB, ad <lb/>latus AC, ita ſinus anguli C, ad ſinum anguli B. </s> <s xml:id="echoid-s10664" xml:space="preserve">Quare vterq; </s> <s xml:id="echoid-s10665" xml:space="preserve">angulus C, <lb/> <anchor type="note" xlink:label="note-340-02a" xlink:href="note-340-02"/> B, ſigillatim cognitus erit, ille grad. </s> <s xml:id="echoid-s10666" xml:space="preserve">29. </s> <s xml:id="echoid-s10667" xml:space="preserve">Min. </s> <s xml:id="echoid-s10668" xml:space="preserve">55. </s> <s xml:id="echoid-s10669" xml:space="preserve">hic vero grad. </s> <s xml:id="echoid-s10670" xml:space="preserve">56. </s> <s xml:id="echoid-s10671" xml:space="preserve">Min. </s> <s xml:id="echoid-s10672" xml:space="preserve">15. <lb/></s> <s xml:id="echoid-s10673" xml:space="preserve"> <anchor type="note" xlink:label="note-340-03a" xlink:href="note-340-03"/> Quia vero latera ſunt ſinubus angulorum op-<lb/> <anchor type="figure" xlink:label="fig-340-01a" xlink:href="fig-340-01"/> <anchor type="note" xlink:label="note-340-04a" xlink:href="note-340-04"/> poſitorum proportionalia, erit, vt ſinus anguli <lb/>C, ad ſinum anguli A, ita latus AB, ad latus <lb/>BC: </s> <s xml:id="echoid-s10674" xml:space="preserve">vel vt ſinus anguli B, ad ſinum anguli A, <lb/>ita latus AC, ad latus BC. </s> <s xml:id="echoid-s10675" xml:space="preserve">Per auream ergo <lb/>regulam inuenietur latus BC, ferme 6. </s> <s xml:id="echoid-s10676" xml:space="preserve">vt hic <lb/>apparet.</s> <s xml:id="echoid-s10677" xml:space="preserve"/> </p> <div xml:id="echoid-div879" type="float" level="2" n="2"> <note position="left" xlink:label="note-340-02" xlink:href="note-340-02a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-340-03" xlink:href="note-340-03a" xml:space="preserve">6. huius.</note> <figure xlink:label="fig-340-01" xlink:href="fig-340-01a"> <image file="340-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/340-01"/> </figure> <note position="left" xlink:label="note-340-04" xlink:href="note-340-04a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>C. # A. # # AB. # # BC. <lb/>49874. # 99776. # # 3? # fit # 6. ferè <lb/> # # Vel. <lb/>B. # A. # # AC. # # BC. <lb/>83147. # 99776. # # 5? # fit # 6. ferè. <lb/></note> <p style="it"> <s xml:id="echoid-s10678" xml:space="preserve">VT ergo totam praxim complectamur: </s> <s xml:id="echoid-s10679" xml:space="preserve">Si (ablato angulo dato ex <lb/> <anchor type="note" xlink:label="note-340-06a" xlink:href="note-340-06"/> grad. </s> <s xml:id="echoid-s10680" xml:space="preserve">180. </s> <s xml:id="echoid-s10681" xml:space="preserve">vt ſumma aliorum duorum habeatur) fiat, vt ſemiſſis ſummæ <lb/>duorum lat erum datorum, nempe ſemiſſis ſummæ terminorum proportio-<lb/>nis ſinuum angulorum reliquorum, ad tangentem ſemiſſis ſummæ reliquo-<lb/>rum angulorum, ita differentia inter ſemiſſem ſummæ datorum laterum, <pb o="329" file="341" n="341" rhead=""/> hoc eſt, terminorum proportionis ſinuum angulorum reliquorum, & </s> <s xml:id="echoid-s10682" xml:space="preserve">al-<lb/>terum laterum, ſiue terminorum proportionis, ad aliud, reperietur tan-<lb/>gens anguli, qui cum ſemiſſe ſummæ reliquorum angulorum componet ma-<lb/>iorem angulum, qui nimirum maiori lateri dato opponitur: </s> <s xml:id="echoid-s10683" xml:space="preserve">idem verò ex <lb/>eadem ſemiſſe dictæ ſummæ detractus relinquet angulum minorem mino-<lb/>ri lateri dato oppoſitum; </s> <s xml:id="echoid-s10684" xml:space="preserve">vt perſpicuum eſt ex praxi priore propoſ. </s> <s xml:id="echoid-s10685" xml:space="preserve">6. <lb/></s> <s xml:id="echoid-s10686" xml:space="preserve">Quòd ſi rurſus fiat, vt ſinus vtriuſuis angulorum inuentorum ad ſinum <lb/>anguli in principio dati, ita latus inuento angulo, qui acceptus fuerit in <lb/>regula aurea, oppoſitum ad aliud, inuenietur tertium latus. </s> <s xml:id="echoid-s10687" xml:space="preserve">Et ad maio-<lb/>rem perſpicuit atem proponemus aliud exemplam.</s> <s xml:id="echoid-s10688" xml:space="preserve"/> </p> <div xml:id="echoid-div880" type="float" level="2" n="3"> <note position="left" xlink:label="note-340-06" xlink:href="note-340-06a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s10689" xml:space="preserve">SINT in triangulo ABC, data duo latera AB, AC, 22. </s> <s xml:id="echoid-s10690" xml:space="preserve">& </s> <s xml:id="echoid-s10691" xml:space="preserve">31. </s> <s xml:id="echoid-s10692" xml:space="preserve">vnà cum <lb/> <anchor type="figure" xlink:label="fig-341-01a" xlink:href="fig-341-01"/> angulo A, grad. </s> <s xml:id="echoid-s10693" xml:space="preserve">23. </s> <s xml:id="echoid-s10694" xml:space="preserve">Min. </s> <s xml:id="echoid-s10695" xml:space="preserve">42. </s> <s xml:id="echoid-s10696" xml:space="preserve">Hicangulus ablatus ex <lb/>grad. </s> <s xml:id="echoid-s10697" xml:space="preserve">180. </s> <s xml:id="echoid-s10698" xml:space="preserve">reliqua fiet ſumma angulorum B, C, grad. <lb/></s> <s xml:id="echoid-s10699" xml:space="preserve">156. </s> <s xml:id="echoid-s10700" xml:space="preserve">Min. </s> <s xml:id="echoid-s10701" xml:space="preserve">18. </s> <s xml:id="echoid-s10702" xml:space="preserve">Si ergo fiat, vt 26 {1/2}. </s> <s xml:id="echoid-s10703" xml:space="preserve">ſemiſsis laterum <lb/>AB, AC, id eſt, terminorum proportionis ſinuum <lb/>angulorum B, C, ad 476595. </s> <s xml:id="echoid-s10704" xml:space="preserve">tangentem ſemiſsis ſum-<lb/>mæ angulorum B, C, hoc eſt, ad tangentem grad. </s> <s xml:id="echoid-s10705" xml:space="preserve">78. </s> <s xml:id="echoid-s10706" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s10707" xml:space="preserve">9. </s> <s xml:id="echoid-s10708" xml:space="preserve">ita 4 {1/2}. </s> <s xml:id="echoid-s10709" xml:space="preserve">differentia inter ſemiſſem ſummæ la-<lb/>terũ AB, AC, vel terminorũ laterum, ſeu terminorum, <lb/>ad aliud, reperietur tangens 80931. </s> <s xml:id="echoid-s10710" xml:space="preserve">cuius arcus grad. </s> <s xml:id="echoid-s10711" xml:space="preserve">38. </s> <s xml:id="echoid-s10712" xml:space="preserve">Min. </s> <s xml:id="echoid-s10713" xml:space="preserve">59. </s> <s xml:id="echoid-s10714" xml:space="preserve">additus ad <lb/>grad. </s> <s xml:id="echoid-s10715" xml:space="preserve">78. </s> <s xml:id="echoid-s10716" xml:space="preserve">Min. </s> <s xml:id="echoid-s10717" xml:space="preserve">9. </s> <s xml:id="echoid-s10718" xml:space="preserve">nempe ad ſemiſſem ſummæ angulorum B, C, conſtituet an-<lb/>gulum maiorem B, maiori lateri dato AC, oppoſitum grad. </s> <s xml:id="echoid-s10719" xml:space="preserve">117. </s> <s xml:id="echoid-s10720" xml:space="preserve">Min. </s> <s xml:id="echoid-s10721" xml:space="preserve">8. </s> <s xml:id="echoid-s10722" xml:space="preserve">Idem <lb/>vero arcus grad. </s> <s xml:id="echoid-s10723" xml:space="preserve">38. </s> <s xml:id="echoid-s10724" xml:space="preserve">Min. </s> <s xml:id="echoid-s10725" xml:space="preserve">59. </s> <s xml:id="echoid-s10726" xml:space="preserve">ex eadem ſemiſſe ſummæ angulorum B, C, id <lb/>eſt, ex grad. </s> <s xml:id="echoid-s10727" xml:space="preserve">78. </s> <s xml:id="echoid-s10728" xml:space="preserve">Min. </s> <s xml:id="echoid-s10729" xml:space="preserve">9. </s> <s xml:id="echoid-s10730" xml:space="preserve">detractus relinquet minorem angulum C, minori la-<lb/>teri dato AB, oppoſitum grad. </s> <s xml:id="echoid-s10731" xml:space="preserve">39. </s> <s xml:id="echoid-s10732" xml:space="preserve">Min. </s> <s xml:id="echoid-s10733" xml:space="preserve">10. </s> <s xml:id="echoid-s10734" xml:space="preserve">Operationem aureæ regulæ <lb/>hic vides.</s> <s xml:id="echoid-s10735" xml:space="preserve"/> </p> <div xml:id="echoid-div881" type="float" level="2" n="4"> <figure xlink:label="fig-341-01" xlink:href="fig-341-01a"> <image file="341-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/341-01"/> </figure> </div> <note position="right" xml:space="preserve"> <lb/>26 {1/2}. # 476595. # 4 {1/2}? # fit # 80931. <lb/></note> <p> <s xml:id="echoid-s10736" xml:space="preserve">Quòd ſi iam fiat, vt 63158. </s> <s xml:id="echoid-s10737" xml:space="preserve">ſinus anguli C, ad 40195. </s> <s xml:id="echoid-s10738" xml:space="preserve">ſinum anguli A, ita la-<lb/>tus AB, 22. </s> <s xml:id="echoid-s10739" xml:space="preserve">ad aliud: </s> <s xml:id="echoid-s10740" xml:space="preserve">Vel vt 88995. </s> <s xml:id="echoid-s10741" xml:space="preserve">ſinus anguli B, ad 40195. </s> <s xml:id="echoid-s10742" xml:space="preserve">ſinum anguli <lb/>A, ita latus AC, 31. </s> <s xml:id="echoid-s10743" xml:space="preserve">ad aliud, inuenietur latus BC, 14. </s> <s xml:id="echoid-s10744" xml:space="preserve">ferè. </s> <s xml:id="echoid-s10745" xml:space="preserve">vt hic cernis.</s> <s xml:id="echoid-s10746" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>C. # A. # # AB. # # BC. <lb/>63158. # 40195. # # 22? # ſit # 14 ferè <lb/> # # Vel <lb/>B. # A. # # AC. # # BC. <lb/>88995. # 40195. # # 31? # fit # 14. ferè. <lb/></note> <p> <s xml:id="echoid-s10747" xml:space="preserve">SI vti nolis tangentibus, vſurpanda erit poſterior praxis propoſ. </s> <s xml:id="echoid-s10748" xml:space="preserve">6. </s> <s xml:id="echoid-s10749" xml:space="preserve">in in-<lb/>uentione angulorum B, C.</s> <s xml:id="echoid-s10750" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10751" xml:space="preserve">IAM ſi proportio laterum AB, AC, data ſit, vna cum angulo A, ab ipſis <lb/> <anchor type="note" xlink:label="note-341-03a" xlink:href="note-341-03"/> comprehenſo, aſcribemus dictis lateribus numeros proportionis, ac ſi in ipſis <lb/>data eſſent, problemaq́ abſoluemus, vt prius.</s> <s xml:id="echoid-s10752" xml:space="preserve"/> </p> <div xml:id="echoid-div882" type="float" level="2" n="5"> <note position="right" xlink:label="note-341-03" xlink:href="note-341-03a" xml:space="preserve">Quando la <lb/>terum pro-<lb/>portio data <lb/>eſt.</note> </div> <p> <s xml:id="echoid-s10753" xml:space="preserve">HOC etiam problema abſolui poteſt ſine auxilio propoſ. </s> <s xml:id="echoid-s10754" xml:space="preserve">6. </s> <s xml:id="echoid-s10755" xml:space="preserve">hoc modo. <lb/></s> <s xml:id="echoid-s10756" xml:space="preserve"> <anchor type="note" xlink:label="note-341-04a" xlink:href="note-341-04"/> In triangulo ABC, detur latus AB, 29. </s> <s xml:id="echoid-s10757" xml:space="preserve">AC, 24. </s> <s xml:id="echoid-s10758" xml:space="preserve">& </s> <s xml:id="echoid-s10759" xml:space="preserve">angulus A, primùm acutus <lb/>grad. </s> <s xml:id="echoid-s10760" xml:space="preserve">38. </s> <s xml:id="echoid-s10761" xml:space="preserve">Min. </s> <s xml:id="echoid-s10762" xml:space="preserve">16. </s> <s xml:id="echoid-s10763" xml:space="preserve">Ducatur ad maius latus A B, ab angulo oppoſito C, per-<lb/>pendicularis CD: </s> <s xml:id="echoid-s10764" xml:space="preserve">quæ neceſſario intra triangulum cadet. </s> <s xml:id="echoid-s10765" xml:space="preserve">Cum enim latus <pb o="330" file="342" n="342" rhead=""/> AB, maius ſit latere AC, erit & </s> <s xml:id="echoid-s10766" xml:space="preserve">angulus C, angulo B, maior. </s> <s xml:id="echoid-s10767" xml:space="preserve">Quare B, acu-<lb/> <anchor type="note" xlink:label="note-342-01a" xlink:href="note-342-01"/> <anchor type="figure" xlink:label="fig-342-01a" xlink:href="fig-342-01"/> tus erit. </s> <s xml:id="echoid-s10768" xml:space="preserve">Nam ſi rectus eſſet, aut maior, eſſet C, etiam <lb/>maior recto. </s> <s xml:id="echoid-s10769" xml:space="preserve">quod eſt abſurdum; </s> <s xml:id="echoid-s10770" xml:space="preserve">quòd B, C, ſint mino-<lb/> <anchor type="note" xlink:label="note-342-02a" xlink:href="note-342-02"/> res duobus rectis. </s> <s xml:id="echoid-s10771" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s10772" xml:space="preserve">A, ponatur acutus, ca-<lb/>det perpendicularis CD, intra triangulum. </s> <s xml:id="echoid-s10773" xml:space="preserve">In trian-<lb/> <anchor type="note" xlink:label="note-342-03a" xlink:href="note-342-03"/> gulo igitur rectangulo ACD, cum angulus A, ſit grad. <lb/></s> <s xml:id="echoid-s10774" xml:space="preserve">38. </s> <s xml:id="echoid-s10775" xml:space="preserve">Min. </s> <s xml:id="echoid-s10776" xml:space="preserve">16. </s> <s xml:id="echoid-s10777" xml:space="preserve">erit eius complementum ACD, grad. </s> <s xml:id="echoid-s10778" xml:space="preserve">51. </s> <s xml:id="echoid-s10779" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s10780" xml:space="preserve">44. </s> <s xml:id="echoid-s10781" xml:space="preserve">Quare cum ſit, vt ſinus totus anguli recti D, <lb/> <anchor type="note" xlink:label="note-342-04a" xlink:href="note-342-04"/> ad ſinum anguli A, ita latus AC, 24. </s> <s xml:id="echoid-s10782" xml:space="preserve">ad latus CD: <lb/></s> <s xml:id="echoid-s10783" xml:space="preserve">inuenietur latus CD, per regulã auream, 14 {2699/3125}. </s> <s xml:id="echoid-s10784" xml:space="preserve"><lb/>Vt hic apparet.</s> <s xml:id="echoid-s10785" xml:space="preserve"/> </p> <div xml:id="echoid-div883" type="float" level="2" n="6"> <note position="right" xlink:label="note-341-04" xlink:href="note-341-04a" xml:space="preserve">Alia demõ <lb/>ſtratio pro-<lb/>blematis ſi-<lb/>ne ppoſ.6.</note> <note position="left" xlink:label="note-342-01" xlink:href="note-342-01a" xml:space="preserve">19. primi.</note> <figure xlink:label="fig-342-01" xlink:href="fig-342-01a"> <image file="342-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/342-01"/> </figure> <note position="left" xlink:label="note-342-02" xlink:href="note-342-02a" xml:space="preserve">17. primi.</note> <note position="left" xlink:label="note-342-03" xlink:href="note-342-03a" xml:space="preserve">Schol. 13. <lb/>ſecundi.</note> <note position="left" xlink:label="note-342-04" xlink:href="note-342-04a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>D. # A. # AC. # # CD. <lb/>100000. # 61932. # 24? # fit # 14 {2699/3125}. <lb/></note> <p> <s xml:id="echoid-s10786" xml:space="preserve">Eadem ratione, cum ſit, vt ſinus totus anguli recti D, ad ſinum anguli ACD, <lb/> <anchor type="note" xlink:label="note-342-06a" xlink:href="note-342-06"/> ita latus AC, 24. </s> <s xml:id="echoid-s10787" xml:space="preserve">ad latus AD: </s> <s xml:id="echoid-s10788" xml:space="preserve">reperietur latus AD, 18 {5271/6250}. </s> <s xml:id="echoid-s10789" xml:space="preserve">vt hic <lb/>vides.</s> <s xml:id="echoid-s10790" xml:space="preserve"/> </p> <div xml:id="echoid-div884" type="float" level="2" n="7"> <note position="left" xlink:label="note-342-06" xlink:href="note-342-06a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>D. # ACD. # AC. # # AD. <lb/>100000. # 78514. # 24? # fit # 18 {5271/6250}. <lb/></note> <p> <s xml:id="echoid-s10791" xml:space="preserve">Ablata autem AD, inuenta ex AB, data 29. </s> <s xml:id="echoid-s10792" xml:space="preserve">relinquetur BD, 10 {979/6250}. <lb/></s> <s xml:id="echoid-s10793" xml:space="preserve">Et quia, vt in tractatu tangentium oſtendimus, poſito ſinu toto BD, recta <lb/>CD, tangens eſt anguli B, inuenietur CD, tangens 146344. </s> <s xml:id="echoid-s10794" xml:space="preserve">vt hic manife-<lb/>ſtum eſt.</s> <s xml:id="echoid-s10795" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>BD. # BD. # CD. # # CD. <lb/>10 {979/6250}. # 100000. # 14 {2699/3125}? # fit # 146344. <lb/></note> <p> <s xml:id="echoid-s10796" xml:space="preserve">Tangens autem 146344. </s> <s xml:id="echoid-s10797" xml:space="preserve">monſtrat in tabula tangentium angulum B, grad. <lb/></s> <s xml:id="echoid-s10798" xml:space="preserve">55. </s> <s xml:id="echoid-s10799" xml:space="preserve">Min. </s> <s xml:id="echoid-s10800" xml:space="preserve">40. </s> <s xml:id="echoid-s10801" xml:space="preserve">ac proinde duo anguli A, B, grad. </s> <s xml:id="echoid-s10802" xml:space="preserve">38. </s> <s xml:id="echoid-s10803" xml:space="preserve">Min. </s> <s xml:id="echoid-s10804" xml:space="preserve">16. </s> <s xml:id="echoid-s10805" xml:space="preserve">& </s> <s xml:id="echoid-s10806" xml:space="preserve">grad. </s> <s xml:id="echoid-s10807" xml:space="preserve">55. </s> <s xml:id="echoid-s10808" xml:space="preserve">Min. </s> <s xml:id="echoid-s10809" xml:space="preserve"><lb/>40. </s> <s xml:id="echoid-s10810" xml:space="preserve">ex grad 180. </s> <s xml:id="echoid-s10811" xml:space="preserve">ſubducti reliquum facient angulum ACB, grad. </s> <s xml:id="echoid-s10812" xml:space="preserve">86. </s> <s xml:id="echoid-s10813" xml:space="preserve">Min. </s> <s xml:id="echoid-s10814" xml:space="preserve">4. </s> <s xml:id="echoid-s10815" xml:space="preserve"><lb/>Quoniam autem eſt, vt ſinus anguli B, noti ad ſinum anguli A, dati, ita latus <lb/> <anchor type="note" xlink:label="note-342-09a" xlink:href="note-342-09"/> datum AC, 24. </s> <s xml:id="echoid-s10816" xml:space="preserve">adlatus BC, reperietur latus BC, 18. </s> <s xml:id="echoid-s10817" xml:space="preserve">ferè. </s> <s xml:id="echoid-s10818" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s10819" xml:space="preserve"/> </p> <div xml:id="echoid-div885" type="float" level="2" n="8"> <note position="left" xlink:label="note-342-09" xlink:href="note-342-09a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>B. # A. # AC. # # BC. <lb/>82577. # 61932. # 24? # fit # 18. ferè. <lb/></note> <p> <s xml:id="echoid-s10820" xml:space="preserve">RVRSVS in triangulo ABC, datum ſit latus AB, 13. </s> <s xml:id="echoid-s10821" xml:space="preserve">AC, 11. </s> <s xml:id="echoid-s10822" xml:space="preserve">& </s> <s xml:id="echoid-s10823" xml:space="preserve">an-<lb/>gulus A, ab ipſis comprehenſus obtuſus, & </s> <s xml:id="echoid-s10824" xml:space="preserve">datus grad. </s> <s xml:id="echoid-s10825" xml:space="preserve">112. </s> <s xml:id="echoid-s10826" xml:space="preserve">Min. </s> <s xml:id="echoid-s10827" xml:space="preserve">37. </s> <s xml:id="echoid-s10828" xml:space="preserve">Duca-<lb/> <anchor type="figure" xlink:label="fig-342-02a" xlink:href="fig-342-02"/> tur ex alterutro angulorum acutorum, vt ex B, <lb/>ad oppoſitum latus CA, perpendicularis: </s> <s xml:id="echoid-s10829" xml:space="preserve">quæ <lb/> <anchor type="note" xlink:label="note-342-11a" xlink:href="note-342-11"/> extra triangulum cadet. </s> <s xml:id="echoid-s10830" xml:space="preserve">Quia igitur angulus <lb/>BAC, ponitur grad. </s> <s xml:id="echoid-s10831" xml:space="preserve">112. </s> <s xml:id="echoid-s10832" xml:space="preserve">Min. </s> <s xml:id="echoid-s10833" xml:space="preserve">37. </s> <s xml:id="echoid-s10834" xml:space="preserve">erit BAD, <lb/>grad. </s> <s xml:id="echoid-s10835" xml:space="preserve">67. </s> <s xml:id="echoid-s10836" xml:space="preserve">Min. </s> <s xml:id="echoid-s10837" xml:space="preserve">23. </s> <s xml:id="echoid-s10838" xml:space="preserve">eiusq́ complementum pro-<lb/>pterea ABD, neceſſariò grad. </s> <s xml:id="echoid-s10839" xml:space="preserve">22. </s> <s xml:id="echoid-s10840" xml:space="preserve">Min. </s> <s xml:id="echoid-s10841" xml:space="preserve">37.</s> <s xml:id="echoid-s10842" xml:space="preserve"/> </p> <div xml:id="echoid-div886" type="float" level="2" n="9"> <figure xlink:label="fig-342-02" xlink:href="fig-342-02a"> <image file="342-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/342-02"/> </figure> <note position="left" xlink:label="note-342-11" xlink:href="note-342-11a" xml:space="preserve">Schol. 12. <lb/>ſecundi.</note> </div> <p> <s xml:id="echoid-s10843" xml:space="preserve">Quare cum ſit, vt ſinus totus anguli recti D, ad <lb/> <anchor type="note" xlink:label="note-342-12a" xlink:href="note-342-12"/> ſinum anguli BAD, ita latus AB, datum 13. </s> <s xml:id="echoid-s10844" xml:space="preserve">ad latus BD: </s> <s xml:id="echoid-s10845" xml:space="preserve">Item vt ſinus to-<lb/>tus anguli recti D, ad ſinum anguli ABD, ita latus datum AB, 13. </s> <s xml:id="echoid-s10846" xml:space="preserve">ad latus <lb/>AD; </s> <s xml:id="echoid-s10847" xml:space="preserve">in uenietur BD, quidem 12. </s> <s xml:id="echoid-s10848" xml:space="preserve">at AD, 5. </s> <s xml:id="echoid-s10849" xml:space="preserve">ferè. </s> <s xml:id="echoid-s10850" xml:space="preserve">vt hic apparet.</s> <s xml:id="echoid-s10851" xml:space="preserve"/> </p> <div xml:id="echoid-div887" type="float" level="2" n="10"> <note position="left" xlink:label="note-342-12" xlink:href="note-342-12a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>D. # BAD. # # AB. # # BD. <lb/>100000. # 92310. # # 13? # fit # 12. ferè. <lb/> # # Item. <lb/>D. # ABD. # # AB. # # AD. <lb/>100000. # 38456. # # 13? # fit # 5. ferè. <lb/></note> <pb o="331" file="343" n="343" rhead=""/> <p> <s xml:id="echoid-s10852" xml:space="preserve">Addita autem AD, inuenta 5. </s> <s xml:id="echoid-s10853" xml:space="preserve">ad latus AC, datum 11. </s> <s xml:id="echoid-s10854" xml:space="preserve">fiet tota CD, 16. </s> <s xml:id="echoid-s10855" xml:space="preserve">Cum <lb/>ergo, poſito ſinu toto CD, recta BD, ſit tangens anguli C; </s> <s xml:id="echoid-s10856" xml:space="preserve">reperietur tan-<lb/>gens BD, 75000. </s> <s xml:id="echoid-s10857" xml:space="preserve">vt hic cernis.</s> <s xml:id="echoid-s10858" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>CD. # CD. # BD. # # BD. <lb/>16. # 100000. # 12? # fit # 75000. <lb/></note> <p> <s xml:id="echoid-s10859" xml:space="preserve">Quæ tangens offeret in tangentium tabula angulum C, grad. </s> <s xml:id="echoid-s10860" xml:space="preserve">36. </s> <s xml:id="echoid-s10861" xml:space="preserve">Min. </s> <s xml:id="echoid-s10862" xml:space="preserve">52. </s> <s xml:id="echoid-s10863" xml:space="preserve">& </s> <s xml:id="echoid-s10864" xml:space="preserve"><lb/>proinde duo anguli A, C, grad. </s> <s xml:id="echoid-s10865" xml:space="preserve">112. </s> <s xml:id="echoid-s10866" xml:space="preserve">Min. </s> <s xml:id="echoid-s10867" xml:space="preserve">36. </s> <s xml:id="echoid-s10868" xml:space="preserve">& </s> <s xml:id="echoid-s10869" xml:space="preserve">grad. </s> <s xml:id="echoid-s10870" xml:space="preserve">36. </s> <s xml:id="echoid-s10871" xml:space="preserve">Min. </s> <s xml:id="echoid-s10872" xml:space="preserve">52. </s> <s xml:id="echoid-s10873" xml:space="preserve">ex grad. <lb/></s> <s xml:id="echoid-s10874" xml:space="preserve">180. </s> <s xml:id="echoid-s10875" xml:space="preserve">ablatirelinquent angulum ABC, grad. </s> <s xml:id="echoid-s10876" xml:space="preserve">30. </s> <s xml:id="echoid-s10877" xml:space="preserve">Min. </s> <s xml:id="echoid-s10878" xml:space="preserve">31. </s> <s xml:id="echoid-s10879" xml:space="preserve">Quoniam tandem <lb/>eſt, vt ſinus anguli C, cogniti ad ſinum anguli BAC, dati, ita latus AB, 13. </s> <s xml:id="echoid-s10880" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-343-02a" xlink:href="note-343-02"/> datum ad BC, inuenietur latus BC, 20. </s> <s xml:id="echoid-s10881" xml:space="preserve">ferè. </s> <s xml:id="echoid-s10882" xml:space="preserve">vt hic apparet.</s> <s xml:id="echoid-s10883" xml:space="preserve"/> </p> <div xml:id="echoid-div888" type="float" level="2" n="11"> <note position="right" xlink:label="note-343-02" xlink:href="note-343-02a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>C. # BAC. # AB. # # BC. <lb/>59996. # 92310. # 13? # fit # 20. ferè. <lb/></note> <p> <s xml:id="echoid-s10884" xml:space="preserve">Verum, vt vides, prior ratio multò eſt breuior, & </s> <s xml:id="echoid-s10885" xml:space="preserve">ex peditior.</s> <s xml:id="echoid-s10886" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10887" xml:space="preserve">QVOD ſi quando data duo latera datum angulum ambientia fuerint <lb/> <anchor type="note" xlink:label="note-343-04a" xlink:href="note-343-04"/> æqualia, facilius erit problema. </s> <s xml:id="echoid-s10888" xml:space="preserve">Sint namq; </s> <s xml:id="echoid-s10889" xml:space="preserve">in triangulo ABC, duo latera <lb/> <anchor type="figure" xlink:label="fig-343-01a" xlink:href="fig-343-01"/> AB, AC, æqualia, quodlibet 9. </s> <s xml:id="echoid-s10890" xml:space="preserve">& </s> <s xml:id="echoid-s10891" xml:space="preserve">angulus A, com-<lb/>prehenſus grad. </s> <s xml:id="echoid-s10892" xml:space="preserve">80. </s> <s xml:id="echoid-s10893" xml:space="preserve">Ablato hoc angulo ex grad. <lb/></s> <s xml:id="echoid-s10894" xml:space="preserve">180. </s> <s xml:id="echoid-s10895" xml:space="preserve">dabit ſemiſsis reſidui, quod eſt grad. </s> <s xml:id="echoid-s10896" xml:space="preserve">100. </s> <s xml:id="echoid-s10897" xml:space="preserve">vtrum-<lb/>que angulorum B, C, grad. </s> <s xml:id="echoid-s10898" xml:space="preserve">50. </s> <s xml:id="echoid-s10899" xml:space="preserve">Si autem fiat, vt ſinus <lb/>anguli B, vel C, ad ſinum anguli A, ita latus AC, <lb/>vel AB, ad aliud, prodibit latus BC, 11 {43685/76604}. </s> <s xml:id="echoid-s10900" xml:space="preserve"><lb/>vt hic vides.</s> <s xml:id="echoid-s10901" xml:space="preserve"/> </p> <div xml:id="echoid-div889" type="float" level="2" n="12"> <note position="right" xlink:label="note-343-04" xlink:href="note-343-04a" xml:space="preserve">Quado da <lb/>ta latera sũt <lb/>æqualia.</note> <figure xlink:label="fig-343-01" xlink:href="fig-343-01a"> <image file="343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/343-01"/> </figure> </div> <note position="right" xml:space="preserve"> <lb/>B, vel C. # A. # AB, vel AC. # # BC. <lb/>76604. # 98481. # 9? # fit # 11 {43685/76604}. <lb/></note> <p> <s xml:id="echoid-s10902" xml:space="preserve">Datis ergo duobus lateribus trianguli non rectanguli, cum angulo ab ipſis <lb/>comprehenſo, &</s> <s xml:id="echoid-s10903" xml:space="preserve">c. </s> <s xml:id="echoid-s10904" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s10905" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div891" type="section" level="1" n="471"> <head xml:id="echoid-head503" xml:space="preserve">PROBL. 9. PROPOS. 13.</head> <note position="right" xml:space="preserve">In triãgule <lb/>non rectan <lb/>gulo ex da-<lb/>tis duobus <lb/>lateribꝰ da <lb/>tis, vel ex eo <lb/>rũ propor-<lb/>cũ vno an-<lb/>gulo nõ ab <lb/>ipſis cõpre-<lb/>henſo, ter-<lb/>tium latus, <lb/>& reliqui <lb/>anguli in-<lb/>ueſtigãtur.</note> <p> <s xml:id="echoid-s10906" xml:space="preserve">DATIS duobus lateribus trianguli non re-<lb/>ctanguli, vel eorum proportione data, vnà cum <lb/>angulo, qui alteri datorum laterum opponitur: <lb/></s> <s xml:id="echoid-s10907" xml:space="preserve">reliquos angulos, & </s> <s xml:id="echoid-s10908" xml:space="preserve">tertium latus in quirere. </s> <s xml:id="echoid-s10909" xml:space="preserve">Opor <lb/>tet autem conſtare, num alter angulus reliquo la-<lb/>teri dato oppoſitus ſit acutus, an obtuſus, ſi datus <lb/>angulus acutus eſt.</s> <s xml:id="echoid-s10910" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10911" xml:space="preserve">SINT in triangulo ABC, data duo latera AB, AC. </s> <s xml:id="echoid-s10912" xml:space="preserve">13. </s> <s xml:id="echoid-s10913" xml:space="preserve">& </s> <s xml:id="echoid-s10914" xml:space="preserve">20. </s> <s xml:id="echoid-s10915" xml:space="preserve">datusq́; </s> <s xml:id="echoid-s10916" xml:space="preserve">ſit <lb/>acutus angulus C, grad. </s> <s xml:id="echoid-s10917" xml:space="preserve">36. </s> <s xml:id="echoid-s10918" xml:space="preserve">Min. </s> <s xml:id="echoid-s10919" xml:space="preserve">52. </s> <s xml:id="echoid-s10920" xml:space="preserve">lateri dato AB, oppoſitus, conſtetq́ue <lb/>de angulo B, qui alteri dato lateri AC, opponitur, num acutus ſit, an obtu-<lb/>ſus: </s> <s xml:id="echoid-s10921" xml:space="preserve">alias enim nihil certi colligi poſſet, vt ex ſequenti ſcholio patebit. </s> <s xml:id="echoid-s10922" xml:space="preserve">Quo-<lb/>niam ergo eſt, vt latus AB, ad latus AC, ita ſinus anguli C, ad ſinum anguli <lb/> <anchor type="note" xlink:label="note-343-07a" xlink:href="note-343-07"/> B; </s> <s xml:id="echoid-s10923" xml:space="preserve">ſuntque tria primadata, inuenietur per auream regulam quartum, hoc eſt, <lb/>ſinus anguli B, 92302. </s> <s xml:id="echoid-s10924" xml:space="preserve">vt hic liquet.</s> <s xml:id="echoid-s10925" xml:space="preserve"/> </p> <div xml:id="echoid-div891" type="float" level="2" n="1"> <note position="right" xlink:label="note-343-07" xlink:href="note-343-07a" xml:space="preserve">1. huiun</note> </div> <pb o="332" file="344" n="344" rhead=""/> <note position="right" xml:space="preserve"> <lb/>AB. # AC. # C. # # B. <lb/>13. # 20. # 59996? # fit # 92302. <lb/></note> <p> <s xml:id="echoid-s10926" xml:space="preserve">Qui ſinus in tabula ſinuum exhibet angulum B, ferè grad. </s> <s xml:id="echoid-s10927" xml:space="preserve">67. </s> <s xml:id="echoid-s10928" xml:space="preserve">Min. </s> <s xml:id="echoid-s10929" xml:space="preserve">23. </s> <s xml:id="echoid-s10930" xml:space="preserve">ſi acu <lb/>tus fuerit, vt in priori triangulo: </s> <s xml:id="echoid-s10931" xml:space="preserve">ſi autem obtuſus, vt in triangulo poſterio-<lb/>ri, grad. </s> <s xml:id="echoid-s10932" xml:space="preserve">112. </s> <s xml:id="echoid-s10933" xml:space="preserve">Min. </s> <s xml:id="echoid-s10934" xml:space="preserve">37. </s> <s xml:id="echoid-s10935" xml:space="preserve">vtpote qui cum illo ſemicirculum compleat; </s> <s xml:id="echoid-s10936" xml:space="preserve">cum ob-<lb/>tuſus, & </s> <s xml:id="echoid-s10937" xml:space="preserve">acutus conficientes grad. </s> <s xml:id="echoid-s10938" xml:space="preserve">180. </s> <s xml:id="echoid-s10939" xml:space="preserve">eundem ſinum habeant, vt ad deſini-<lb/> <anchor type="figure" xlink:label="fig-344-01a" xlink:href="fig-344-01"/> tiones ſinuum demõſtraui-<lb/>mus. </s> <s xml:id="echoid-s10940" xml:space="preserve">Ablatis autem duo-<lb/>bus angulis C, B, ex grad. <lb/></s> <s xml:id="echoid-s10941" xml:space="preserve">180. </s> <s xml:id="echoid-s10942" xml:space="preserve">relinquetur in priori <lb/>triangulo angulus A, grad. </s> <s xml:id="echoid-s10943" xml:space="preserve"><lb/>75. </s> <s xml:id="echoid-s10944" xml:space="preserve">Min. </s> <s xml:id="echoid-s10945" xml:space="preserve">45. </s> <s xml:id="echoid-s10946" xml:space="preserve">In poſteriori <lb/>vero grad. </s> <s xml:id="echoid-s10947" xml:space="preserve">30. </s> <s xml:id="echoid-s10948" xml:space="preserve">Min. </s> <s xml:id="echoid-s10949" xml:space="preserve">31. </s> <s xml:id="echoid-s10950" xml:space="preserve">La-<lb/>tus autem BC, inuenietur, <lb/>per auream regulam, par-<lb/>tium ferè 21. </s> <s xml:id="echoid-s10951" xml:space="preserve">in priori trian <lb/>gulo; </s> <s xml:id="echoid-s10952" xml:space="preserve">in poſteriori autem <lb/>ferè 11. </s> <s xml:id="echoid-s10953" xml:space="preserve">quòd ſit, vt ſinus anguli C, ad ſinum anguli A, ita latus AB, ad la-<lb/> <anchor type="note" xlink:label="note-344-02a" xlink:href="note-344-02"/> tus BC. </s> <s xml:id="echoid-s10954" xml:space="preserve">vt in hac operatione apparet.</s> <s xml:id="echoid-s10955" xml:space="preserve"/> </p> <div xml:id="echoid-div892" type="float" level="2" n="2"> <figure xlink:label="fig-344-01" xlink:href="fig-344-01a"> <image file="344-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/344-01"/> </figure> <note position="left" xlink:label="note-344-02" xlink:href="note-344-02a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>C. # A. # # AB. # # BC. <lb/>59996. # 96923. # # 13? # fit # 21. ferè. <lb/> # # Item <lb/>C. # A. # # AB. # # BC. <lb/>59996. # 50779. # # 13? # fit # 11. ferè. <lb/></note> <p style="it"> <s xml:id="echoid-s10956" xml:space="preserve">ITAQVE, ſi fiat, vt latus datum dato angulo oppoſitum ad al-<lb/> <anchor type="note" xlink:label="note-344-04a" xlink:href="note-344-04"/> terum latus datum, ita ſinus dati anguli ad aliud, reperietur ſinus, cuius <lb/>arcus dabit angulum alteri lateri dato oppoſitum, ſi acutus fuerit, (quod <lb/>quidem ſemper contingit, quando datus angulus eſt obtuſus) aut certe ex <lb/>ſemicirculo ſublatus dabit illum angulum, ſi obtuſus fuerit: </s> <s xml:id="echoid-s10957" xml:space="preserve">Summa ve-<lb/>ro ex dato angulo, & </s> <s xml:id="echoid-s10958" xml:space="preserve">inuento angulo conflata ex grad. </s> <s xml:id="echoid-s10959" xml:space="preserve">180. </s> <s xml:id="echoid-s10960" xml:space="preserve">ſubducta <lb/>exhibebit tertium angulum. </s> <s xml:id="echoid-s10961" xml:space="preserve">Si deni fiat, vt ſinus anguli dati ad ſinum <lb/>tertij anguli à datis lateribus comprehenſi, qui vltimo inuentus eſt, ita <lb/>latus datum dato angulo oppoſitum ad aliud, producetur tertium latus.</s> <s xml:id="echoid-s10962" xml:space="preserve"/> </p> <div xml:id="echoid-div893" type="float" level="2" n="3"> <note position="left" xlink:label="note-344-04" xlink:href="note-344-04a" xml:space="preserve">Praxis.</note> </div> <figure> <image file="344-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/344-02"/> </figure> <p> <s xml:id="echoid-s10963" xml:space="preserve">QVOD ſi detur duorum laterum proportio, <lb/> <anchor type="note" xlink:label="note-344-05a" xlink:href="note-344-05"/> ſi lateribus illis numeri proportionis aſcribantur, <lb/>eodem modo problema exequemur.</s> <s xml:id="echoid-s10964" xml:space="preserve"/> </p> <div xml:id="echoid-div894" type="float" level="2" n="4"> <note position="left" xlink:label="note-344-05" xlink:href="note-344-05a" xml:space="preserve">Quãdo pro <lb/>portio duo <lb/>rum laterũ <lb/>datur.</note> </div> <p> <s xml:id="echoid-s10965" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s10966" xml:space="preserve">Dentur rurſum duo latera AB, 13. <lb/></s> <s xml:id="echoid-s10967" xml:space="preserve">& </s> <s xml:id="echoid-s10968" xml:space="preserve">AC, 20. </s> <s xml:id="echoid-s10969" xml:space="preserve">vnà cum angulo C, acuto grad. </s> <s xml:id="echoid-s10970" xml:space="preserve">36 Min. </s> <s xml:id="echoid-s10971" xml:space="preserve"><lb/>52. </s> <s xml:id="echoid-s10972" xml:space="preserve">ſitq́; </s> <s xml:id="echoid-s10973" xml:space="preserve">primum alter angulus B, acutus etiam, vt <lb/>in priori triangulo. </s> <s xml:id="echoid-s10974" xml:space="preserve">Ducta ex A, ad BC, perpendi-<lb/>culari AD, quæ intra triangulum cadet: </s> <s xml:id="echoid-s10975" xml:space="preserve">quoniam <lb/> <anchor type="note" xlink:label="note-344-06a" xlink:href="note-344-06"/> in triãgulo rectangulo ACD, poſito ſinu toto AC, <lb/>recta AD, ſinus eſt anguli dati C, vt in tractatu ſi-<lb/>nuum docuimus; </s> <s xml:id="echoid-s10976" xml:space="preserve">ſi fiat, vt AC, ſinus totus ad AC, <lb/>latus datum, ita AD, ſinus dati anguli C, ad aliud, <lb/>reperietur AD, 12. </s> <s xml:id="echoid-s10977" xml:space="preserve">ferè, vt hic vides.</s> <s xml:id="echoid-s10978" xml:space="preserve"/> </p> <div xml:id="echoid-div895" type="float" level="2" n="5"> <note position="left" xlink:label="note-344-06" xlink:href="note-344-06a" xml:space="preserve">Schol. 13. <lb/>ſecundi.</note> </div> <pb o="333" file="345" n="345" rhead=""/> <note position="right" xml:space="preserve"> <lb/>AC. # AC. # AD. # # AD. <lb/>100000. # 20. # 59996? # fit # 12. ferè. <lb/></note> <p> <s xml:id="echoid-s10979" xml:space="preserve">Rurſus, quia poſito ſinu toto AB, recta AD, eſt ſinus anguli B, vt in ſinubus <lb/>traditum eſt; </s> <s xml:id="echoid-s10980" xml:space="preserve">ſi ſiat, vt AB, latus datum ad ſinum totum, ita AD, iam in-<lb/>uenta ad aliud, inuenietur ſinus AD, 92308. </s> <s xml:id="echoid-s10981" xml:space="preserve">vt hic apparet.</s> <s xml:id="echoid-s10982" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # AD. # fit # AD. <lb/>13. # 100000. # 12? # # 92308. <lb/></note> <p> <s xml:id="echoid-s10983" xml:space="preserve">Ex tabula ergo ſinuum dabitur angulus B, grad. </s> <s xml:id="echoid-s10984" xml:space="preserve">67. </s> <s xml:id="echoid-s10985" xml:space="preserve">Min. </s> <s xml:id="echoid-s10986" xml:space="preserve">23. </s> <s xml:id="echoid-s10987" xml:space="preserve">ac proinde <lb/>BAC, reliquus duorum rectorum, grad. </s> <s xml:id="echoid-s10988" xml:space="preserve">75. </s> <s xml:id="echoid-s10989" xml:space="preserve">Min. </s> <s xml:id="echoid-s10990" xml:space="preserve">45. </s> <s xml:id="echoid-s10991" xml:space="preserve">Quoniam vero eſt, vt ſi-<lb/>nus anguli dati C, ad ſinum anguli A, inuenti, ita latus AB, ad latus BC, in-<lb/> <anchor type="note" xlink:label="note-345-03a" xlink:href="note-345-03"/> uenietur latus BC, 21. </s> <s xml:id="echoid-s10992" xml:space="preserve">ferme: </s> <s xml:id="echoid-s10993" xml:space="preserve">vt hic cernis.</s> <s xml:id="echoid-s10994" xml:space="preserve"/> </p> <div xml:id="echoid-div896" type="float" level="2" n="6"> <note position="right" xlink:label="note-345-03" xlink:href="note-345-03a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>C. # BAC. # AB. # # BC. <lb/>59996. # 96923. # 13? # fit # 21. ferè. <lb/></note> <p> <s xml:id="echoid-s10995" xml:space="preserve">DEINDE, ijſdem poſitis, ſit alter angulus B, obtuſus, vt in poſterio-<lb/> <anchor type="note" xlink:label="note-345-05a" xlink:href="note-345-05"/> ri triangulo. </s> <s xml:id="echoid-s10996" xml:space="preserve">Ducta ex A, ad BC, perpendiculari AD, quæ extra triangu-<lb/>lum cadet: </s> <s xml:id="echoid-s10997" xml:space="preserve">quoniam in triangulo rectangulo ACD, poſito ſinu toto AC, <lb/>recta AD, ſinus eſt anguli C, vt dictum eſt in tractatione ſinuum; </s> <s xml:id="echoid-s10998" xml:space="preserve">ſi fiat, vt <lb/>finus totus ad @@tum latus AC, ita ſinus dati anguli C, ad aliud, inuenietur <lb/>AD, ferme 12. </s> <s xml:id="echoid-s10999" xml:space="preserve">vt hic apparet.</s> <s xml:id="echoid-s11000" xml:space="preserve"/> </p> <div xml:id="echoid-div897" type="float" level="2" n="7"> <note position="right" xlink:label="note-345-05" xlink:href="note-345-05a" xml:space="preserve">Schol. 12. <lb/>ſecundi</note> </div> <note position="right" xml:space="preserve"> <lb/>AC. # AC. # AD. # # AD. <lb/>100000. # 20. # 59996? # fit # 12. ferè. <lb/></note> <p> <s xml:id="echoid-s11001" xml:space="preserve">Rurſus, quia poſito ſinu toto AB, recta AD, ſinus eſt anguli ABD, vt in <lb/>defin. </s> <s xml:id="echoid-s11002" xml:space="preserve">ſinuum explicauimus; </s> <s xml:id="echoid-s11003" xml:space="preserve">Si fiat, vt latus datum ad ſinum totum, ita AD, <lb/>proxime inuenta ad aliud, reperietur ſinus AD, 92308. </s> <s xml:id="echoid-s11004" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s11005" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # AD. # # AD. <lb/>13. # 100000. # 12? # fit # 92308. <lb/></note> <p> <s xml:id="echoid-s11006" xml:space="preserve">Qui ſinus exhibet in tabula ſinuum angulũ ABD, grad. </s> <s xml:id="echoid-s11007" xml:space="preserve">67. </s> <s xml:id="echoid-s11008" xml:space="preserve">Min. </s> <s xml:id="echoid-s11009" xml:space="preserve">23. </s> <s xml:id="echoid-s11010" xml:space="preserve">ac proin-<lb/>de reliquum duorum rectorum ABC, grad. </s> <s xml:id="echoid-s11011" xml:space="preserve">11. </s> <s xml:id="echoid-s11012" xml:space="preserve">Min. </s> <s xml:id="echoid-s11013" xml:space="preserve">37. </s> <s xml:id="echoid-s11014" xml:space="preserve">Ablatis autem duo-<lb/>bus angulis C, & </s> <s xml:id="echoid-s11015" xml:space="preserve">ABC, notis ex grad. </s> <s xml:id="echoid-s11016" xml:space="preserve">180. </s> <s xml:id="echoid-s11017" xml:space="preserve">remanebit angulus BAC, grad. </s> <s xml:id="echoid-s11018" xml:space="preserve">30. <lb/></s> <s xml:id="echoid-s11019" xml:space="preserve">Min. </s> <s xml:id="echoid-s11020" xml:space="preserve">31. </s> <s xml:id="echoid-s11021" xml:space="preserve">Hinc, quoniam eſt, vt ſinus anguli C, dati ad ſinum anguli A, in-<lb/> <anchor type="note" xlink:label="note-345-08a" xlink:href="note-345-08"/> uenti, ita latus AB, datum ad latus BC, inuenietur latus BC, fere 11. </s> <s xml:id="echoid-s11022" xml:space="preserve">vt <lb/>hic manifeſtum eſt.</s> <s xml:id="echoid-s11023" xml:space="preserve"/> </p> <div xml:id="echoid-div898" type="float" level="2" n="8"> <note position="right" xlink:label="note-345-08" xlink:href="note-345-08a" xml:space="preserve">1. huius.</note> </div> <note position="right" xml:space="preserve"> <lb/>C. # A. # AB. # # BC. <lb/>59996. # 50779. # 13? # fit # 11. ferè. <lb/></note> <p> <s xml:id="echoid-s11024" xml:space="preserve">POSTREMO, datis ijſdem lateribus, detur angulus obtuſus ABC, <lb/>grad. </s> <s xml:id="echoid-s11025" xml:space="preserve">112. </s> <s xml:id="echoid-s11026" xml:space="preserve">Min. </s> <s xml:id="echoid-s11027" xml:space="preserve">37. </s> <s xml:id="echoid-s11028" xml:space="preserve">vt in poſteriori triangulo, factaq́ue eadem conſtructione; <lb/></s> <s xml:id="echoid-s11029" xml:space="preserve">quoniam poſito ſinu toto AB, recta AD, ſinus eſt anguli ABD, hoc eſt, an-<lb/>guli dati ABC, inuenietur rurſum AD, 12. </s> <s xml:id="echoid-s11030" xml:space="preserve">ferè, vt hic cernis.</s> <s xml:id="echoid-s11031" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>AB. # AB. # AD. # # AD. <lb/>100000. # 13. # 92308? # fit # 12. ferè. <lb/></note> <p> <s xml:id="echoid-s11032" xml:space="preserve">Rurſus, quia poſito ſinu toto AC, recta AD, ſinus eſt anguli C, reperietur <lb/>ſinus AD, 60000. </s> <s xml:id="echoid-s11033" xml:space="preserve">Vt hic apparet.</s> <s xml:id="echoid-s11034" xml:space="preserve"/> </p> <pb o="334" file="346" n="346" rhead=""/> <note position="right" xml:space="preserve"> <lb/>AC. # AC. # AD. # # AD. <lb/>20. # 100000. # 12? # fit # 60000. <lb/></note> <p> <s xml:id="echoid-s11035" xml:space="preserve">Eſt ergo angulus C, grad. </s> <s xml:id="echoid-s11036" xml:space="preserve">36. </s> <s xml:id="echoid-s11037" xml:space="preserve">Min. </s> <s xml:id="echoid-s11038" xml:space="preserve">52. </s> <s xml:id="echoid-s11039" xml:space="preserve">& </s> <s xml:id="echoid-s11040" xml:space="preserve">proinde reliquus duorum rectorum <lb/>BAC, grad. </s> <s xml:id="echoid-s11041" xml:space="preserve">30. </s> <s xml:id="echoid-s11042" xml:space="preserve">Min. </s> <s xml:id="echoid-s11043" xml:space="preserve">31. </s> <s xml:id="echoid-s11044" xml:space="preserve">Latus BC, inuenietur 11. </s> <s xml:id="echoid-s11045" xml:space="preserve">vt prius. </s> <s xml:id="echoid-s11046" xml:space="preserve">Igitur, Datis <lb/>duobus lateribus trianguli non rectanguli, &</s> <s xml:id="echoid-s11047" xml:space="preserve">c. </s> <s xml:id="echoid-s11048" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s11049" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div900" type="section" level="1" n="472"> <head xml:id="echoid-head504" xml:space="preserve">SCHOLIVM.</head> <note position="left" xml:space="preserve">Quãdo da-<lb/>tus angu-<lb/>lus eſt acu-<lb/>tus, cur cõ-<lb/>ſtare de-<lb/>beat, num <lb/>alter angu <lb/>lus ſit acu-<lb/>tus, vel ob-<lb/>tuſus.</note> <p> <s xml:id="echoid-s11050" xml:space="preserve">QVOD _autem nihil certi colligi poſsit, quando datus angulus vni datorum la-_ <lb/>_terum oppoſitus acutus eſt, niſi prius cognitum ſit, num angulus alteri dato la-_ <lb/>_teri oppoſitus ſit acutus, obtuſusue, vt in propoſ. </s> <s xml:id="echoid-s11051" xml:space="preserve">diximus, ita per ſpicuum faciemus._ <lb/></s> <s xml:id="echoid-s11052" xml:space="preserve">_Sint in triangulo_ <emph style="sc">Abc</emph>, _data latera_ <emph style="sc">Ab</emph>, <emph style="sc">A</emph>C, _vna cum angulo acuto_ <emph style="sc">B</emph>. </s> <s xml:id="echoid-s11053" xml:space="preserve">_Du-_ <lb/>_cta ex_ A, _ad_ <emph style="sc">B</emph>C, _perpendiculari_ AD, _ignorabitur, num ea cadat extra trian-_ <lb/> <anchor type="figure" xlink:label="fig-346-01a" xlink:href="fig-346-01"/> _gulum, an intra, niſi ſciatur, angulum alterum_ ACB, _eſſe_ <lb/>_obtuſum, acutumue. </s> <s xml:id="echoid-s11054" xml:space="preserve">Sumpta quoque recta_ DE, _ex altera_ <lb/>_parte perpendicularis_ <emph style="sc">A</emph>D, _ipſi_ DC, _æqualis, ductaq́; </s> <s xml:id="echoid-s11055" xml:space="preserve">recta_ <lb/> <anchor type="note" xlink:label="note-346-03a" xlink:href="note-346-03"/> <emph style="sc">AE</emph>; </s> <s xml:id="echoid-s11056" xml:space="preserve">_erit_ <emph style="sc">Ae</emph>, _ipſi_ <emph style="sc">A</emph>C, _æqualis: </s> <s xml:id="echoid-s11057" xml:space="preserve">propterea quòd latera_ <lb/>DC, <emph style="sc">Da</emph>, _lateribus_ <emph style="sc">De</emph>, <emph style="sc">Da</emph>, _æqualia ſunt, angulosq́; </s> <s xml:id="echoid-s11058" xml:space="preserve">com_ <lb/>_prehendunt æquales, vtpote rectos. </s> <s xml:id="echoid-s11059" xml:space="preserve">Itaque etiamſinota ſint_ <lb/>_latera_ <emph style="sc">Ab</emph>, AC, _vel_ AB, AE, _& </s> <s xml:id="echoid-s11060" xml:space="preserve">angulus acutus_ B, _non_ <lb/>_tamen idcirco reliquum latus notum erit, aut reliqui an-_ <lb/>_guli; </s> <s xml:id="echoid-s11061" xml:space="preserve">cum reliquum latus poſsit eſſe vel_ <emph style="sc">B</emph>C, _vel_ <emph style="sc">B</emph>E; </s> <s xml:id="echoid-s11062" xml:space="preserve">_ac_ <lb/>_propterea reliqui anguli vel_ BCA, <emph style="sc">B</emph>AC, _vel_ <emph style="sc">B</emph>EA, <lb/><emph style="sc">B</emph>AE: </s> <s xml:id="echoid-s11063" xml:space="preserve">_niſi prius conſtet, angulum_ <emph style="sc">B</emph>CA, _obtuſum eſſe, vel acutum._ </s> <s xml:id="echoid-s11064" xml:space="preserve">H_oc enim co-_ <lb/>_gnito, ſciemus, quando perpendicularis_ AD, _extra triangulum cadit, & </s> <s xml:id="echoid-s11065" xml:space="preserve">quando_ <lb/>_intra; </s> <s xml:id="echoid-s11066" xml:space="preserve">&</s> <s xml:id="echoid-s11067" xml:space="preserve">c._</s> <s xml:id="echoid-s11068" xml:space="preserve"/> </p> <div xml:id="echoid-div900" type="float" level="2" n="1"> <figure xlink:label="fig-346-01" xlink:href="fig-346-01a"> <image file="346-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/346-01"/> </figure> <note position="left" xlink:label="note-346-03" xlink:href="note-346-03a" xml:space="preserve">4. primi.</note> </div> <p> <s xml:id="echoid-s11069" xml:space="preserve">EX _quibus conſtat, Nicolaum Copernicum, alioquin diligentißimum, hallucina-_ <lb/> <anchor type="note" xlink:label="note-346-04a" xlink:href="note-346-04"/> _tum fuiſſe lb._ </s> <s xml:id="echoid-s11070" xml:space="preserve">1. </s> <s xml:id="echoid-s11071" xml:space="preserve">_Reuo lutionum propoſ._ </s> <s xml:id="echoid-s11072" xml:space="preserve">6. </s> <s xml:id="echoid-s11073" xml:space="preserve">_triangulorum rectilineorum, dum ſimplici-_ <lb/>_ter proponit: </s> <s xml:id="echoid-s11074" xml:space="preserve">Si duo latera trianguli data ſint, & </s> <s xml:id="echoid-s11075" xml:space="preserve">angulus vni eorum oppoſitus datus_ <lb/>_etiam, reliquum latus, & </s> <s xml:id="echoid-s11076" xml:space="preserve">reliquos angulos dari poſſe. </s> <s xml:id="echoid-s11077" xml:space="preserve">Hoc etenim fieri non poteſt, vt_ <lb/>_demonſtrauimus, quando datus angulus eſt acutus, niſi conſtet, num angulus alteri_ <lb/>_laterum daterum oppoſitus ſit acutus, an obtuſus._</s> <s xml:id="echoid-s11078" xml:space="preserve"/> </p> <div xml:id="echoid-div901" type="float" level="2" n="2"> <note position="left" xlink:label="note-346-04" xlink:href="note-346-04a" xml:space="preserve">Error Nico <lb/>lai Coper-<lb/>nici,</note> </div> </div> <div xml:id="echoid-div903" type="section" level="1" n="473"> <head xml:id="echoid-head505" xml:space="preserve">FINIS TRIANGVLORVM <lb/>RECTILINEORVM.</head> <figure> <image file="346-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/346-02"/> </figure> <pb file="347" n="347"/> <pb file="348" n="348"/> <pb file="349" n="349"/> </div> <div xml:id="echoid-div904" type="section" level="1" n="474"> <head xml:id="echoid-head506" xml:space="preserve">CHRISTOPHORI <lb/>CLAVII BAMBERGENSIS <lb/>ESOCIETATE <lb/>IESV</head> <head xml:id="echoid-head507" style="it" xml:space="preserve">TRIANGVLA <lb/>SPHÆRICA.</head> <figure> <image file="349-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/349-01"/> </figure> <pb file="350" n="350"/> <pb o="339" file="351" n="351"/> </div> <div xml:id="echoid-div905" type="section" level="1" n="475"> <head xml:id="echoid-head508" xml:space="preserve">CHRISTOPHORI CLAVII <lb/>BAMBERGENSIS E <lb/>SOCIETATE IESV</head> <head xml:id="echoid-head509" xml:space="preserve">TRIANGVLA SPHÆRICA. <lb/>PRÆFATIO.</head> <figure> <image file="351-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/351-01"/> </figure> <p style="it"> <s xml:id="echoid-s11079" xml:space="preserve">XPLICATIS ijs, quæ ad <lb/>triangulorum rectilineorum <lb/>ſcientiam, qua ex angulis no-<lb/>tis latera, & </s> <s xml:id="echoid-s11080" xml:space="preserve">vicißim ex notis <lb/>lateribus anguli cognoſcũtur, <lb/>neceſſaria eſſe duximus; </s> <s xml:id="echoid-s11081" xml:space="preserve">reli-<lb/>quum eſt, vt sphæricorum etiam triangulorum <lb/>doctrinam, qua arcus ex cognitis angulis, et con <lb/>tra, anguli ex arcubus notis inquiruntur, tra-<lb/>damus. </s> <s xml:id="echoid-s11082" xml:space="preserve">Quamuis enim Menelaus, qui etiam <lb/>Mileus, nobilis ſcriptor, qui temporibus Traia-<lb/>ni Imperatoris floruit, vt auctor eſt Ptolemæus, <lb/> <anchor type="note" xlink:label="note-351-01a" xlink:href="note-351-01"/> acutißimos tres libros de triãgulis ſphæricis com <lb/>poſuerit, non tamen eius ordinem nos in hiſce no <lb/>ſtris sphæricis triangulis ſequemur; </s> <s xml:id="echoid-s11083" xml:space="preserve">propterea <lb/>quòd & </s> <s xml:id="echoid-s11084" xml:space="preserve">plurimas eius propoſitiones, licet iucun-<lb/>dißimæ ſint, mira{q́ue} eruditione refertæ, tãquam <pb o="340" file="352" n="352" rhead=""/> non neceſſarias reiecimus, & </s> <s xml:id="echoid-s11085" xml:space="preserve">alias non paucas <lb/>ab eo omiſſas ex Gebro Hiſpalenſi, Ioanne Re-<lb/>giom. </s> <s xml:id="echoid-s11086" xml:space="preserve">Franciſco Maurolyco, & </s> <s xml:id="echoid-s11087" xml:space="preserve">ex alijs adie-<lb/>cimus, quas omnino neceßarias eſſe iudic aui-<lb/>mus ad res Aſtronomicas intelligendas. </s> <s xml:id="echoid-s11088" xml:space="preserve">Ple-<lb/>run{que} etiam nouas demonſtrationes, eas{q́ue} bre-<lb/>uiores, ac faciliores adhibuimus, nonnullas item <lb/>eodem modo demõſtrauimus, quoeædem de an-<lb/>gulis, & </s> <s xml:id="echoid-s11089" xml:space="preserve">triangulis rectilineis demonſtratæ ſunt <lb/>ab Euclide, vt planior fieret earum demonſtra-<lb/>tio: </s> <s xml:id="echoid-s11090" xml:space="preserve">ex quarum numero ſunt propoſ. </s> <s xml:id="echoid-s11091" xml:space="preserve">5. </s> <s xml:id="echoid-s11092" xml:space="preserve">6. </s> <s xml:id="echoid-s11093" xml:space="preserve">7. </s> <s xml:id="echoid-s11094" xml:space="preserve">8. <lb/></s> <s xml:id="echoid-s11095" xml:space="preserve">& </s> <s xml:id="echoid-s11096" xml:space="preserve">9. </s> <s xml:id="echoid-s11097" xml:space="preserve">Non parumtamen operæin eo poſuimus, <lb/>vt omnes propoſitiones triangulorum Sphærico-<lb/>rum it a in ordinem redigeremus, vt poſteriores <lb/>ex prioribus penderẽt, quemadmodum res Ma-<lb/>thematicæ poſtulant, & </s> <s xml:id="echoid-s11098" xml:space="preserve">in omnibus elementis <lb/>Geometricis fieri conſueuit Sed iam ad rem ve-<lb/>niamus, exordio ſumpto à definitionibus.</s> <s xml:id="echoid-s11099" xml:space="preserve"/> </p> <div xml:id="echoid-div905" type="float" level="2" n="1"> <note position="right" xlink:label="note-351-01" xlink:href="note-351-01a" xml:space="preserve">Lib. 7. Al-<lb/>mag. cap. 1.</note> </div> </div> <div xml:id="echoid-div907" type="section" level="1" n="476"> <head xml:id="echoid-head510" xml:space="preserve">DEFINITIONES. <lb/>I.</head> <p> <s xml:id="echoid-s11100" xml:space="preserve">ANGVLVS ſphæricus eſt, quem in ſphærę <lb/> <anchor type="note" xlink:label="note-352-01a" xlink:href="note-352-01"/> ſuperficie duo arcus circulorum maximorum ſe-<lb/>ſe mutuo ſecantes continent.</s> <s xml:id="echoid-s11101" xml:space="preserve"/> </p> <div xml:id="echoid-div907" type="float" level="2" n="1"> <note position="left" xlink:label="note-352-01" xlink:href="note-352-01a" xml:space="preserve">Angulus <lb/>fphæricus <lb/>quid.</note> </div> <p style="it"> <s xml:id="echoid-s11102" xml:space="preserve">_QVONIAM_ angulus ſphæricus, qui à Geometris in ſphærica ſuperficie conſia <lb/>deratur, ab arcubus maximorum circulorum tantummodo conſtituitur, omnes au@ <pb o="341" file="353" n="353" rhead=""/> tem circuli maximi in ſphæra ſe mutuo ſecantbifariam, fit, vtduo arcus angulum <lb/> <anchor type="note" xlink:label="note-353-01a" xlink:href="note-353-01"/> ſphæricum in ſuperficie ſphæræ continẽtes, ſi producantur, ſe mutuo ſecent, non autem <lb/>ſe mutuo contingant: </s> <s xml:id="echoid-s11103" xml:space="preserve">Ita vt omnis angulus ſphæricus fiat ex duobus arcubus ſeſe in-<lb/>terſecantibus in ſuperficie ſphæræ, non autem ex arcubus ſe mutuo tangentibus. </s> <s xml:id="echoid-s11104" xml:space="preserve">_I_d <lb/>quod de angulis in plana ſuperficie exiſtentibus dici non poi ſt. </s> <s xml:id="echoid-s11105" xml:space="preserve">_In_ hac enim non ſo-<lb/>lum duæ lineæ rectæ, vel curuæ, vel quarum vna recta eſt, & </s> <s xml:id="echoid-s11106" xml:space="preserve">altera curua, ſe mutuo <lb/>ſecantes. </s> <s xml:id="echoid-s11107" xml:space="preserve">ſi producantur, angulum planum conſtituunt, verum etiam duæ lineæ, qua-<lb/> <anchor type="note" xlink:label="note-353-02a" xlink:href="note-353-02"/> rum vtraque vel curua eſt, vel vna curua, & </s> <s xml:id="echoid-s11108" xml:space="preserve">altera recta, ſeſe tangentes tantum-<lb/>modo, angulum planum curuilineum, vel mixtum (qui quidem angulus contactus, vel <lb/>contingentiæ à Geometris vocatur,) conſtituere poſsunt, vt ex propoſ. </s> <s xml:id="echoid-s11109" xml:space="preserve">16. </s> <s xml:id="echoid-s11110" xml:space="preserve">lib. </s> <s xml:id="echoid-s11111" xml:space="preserve">3. </s> <s xml:id="echoid-s11112" xml:space="preserve">Eucl. <lb/></s> <s xml:id="echoid-s11113" xml:space="preserve">perſpicuum eſt: </s> <s xml:id="echoid-s11114" xml:space="preserve">licet Iacobus Peletarius neget eum eſſe angulum, coneturq́; </s> <s xml:id="echoid-s11115" xml:space="preserve">illud multis <lb/>rationibus confirmare, quæ omnes ſophiſticæ ſunt, & </s> <s xml:id="echoid-s11116" xml:space="preserve">friuolæ, vt ex ſolutionibus il-<lb/>larum, quas in ſcholio dictæ propoſ. </s> <s xml:id="echoid-s11117" xml:space="preserve">adduximus, perſpicuũ eſt. </s> <s xml:id="echoid-s11118" xml:space="preserve">Neque vero Peletarius <lb/>in Apologia de contactu linearũ, quam anno 1579. </s> <s xml:id="echoid-s11119" xml:space="preserve">in me conſcripſit, (quàm modeſte, <lb/>& </s> <s xml:id="echoid-s11120" xml:space="preserve">ſyncere, ipſe viderit.) </s> <s xml:id="echoid-s11121" xml:space="preserve">auſus eſt ſolutiones meas impugnare, aut opinionẽ ſuam, no <lb/>uamillam quidem, & </s> <s xml:id="echoid-s11122" xml:space="preserve">inauditam, nouis rationibus (ſcilicet nullas habebat) confirma-<lb/>re: </s> <s xml:id="echoid-s11123" xml:space="preserve">ſed verbis duntaxat, & </s> <s xml:id="echoid-s11124" xml:space="preserve">conuitijs ſe defendere conatur, vt facile ij, qui eam per-<lb/>legerint, iudicabunt.</s> <s xml:id="echoid-s11125" xml:space="preserve"/> </p> <div xml:id="echoid-div908" type="float" level="2" n="2"> <note position="right" xlink:label="note-353-01" xlink:href="note-353-01a" xml:space="preserve">11.1. Theod., <lb/>Angulus <lb/>ſphæricus <lb/>neceſſario <lb/>fit ex duo-<lb/>bus arcubꝰ <lb/>ſe mutuo <lb/>ſecantibus.</note> <note position="right" xlink:label="note-353-02" xlink:href="note-353-02a" xml:space="preserve">Angulus <lb/>planus fie-<lb/>ri etiam po <lb/>teſt ex dua <lb/>bus lineis ſe <lb/>non ſecan-<lb/>tibus, ſed tã <lb/>gentibus ſe <lb/>mutuo dũ-<lb/>taxat.</note> </div> <p> <s xml:id="echoid-s11126" xml:space="preserve">HVIC ego Apologiæ iam pridem non tam mei purgandi, quam ve-<lb/> <anchor type="note" xlink:label="note-353-03a" xlink:href="note-353-03"/> ritatis tuendę gratia reſpondiſſem, niſi me ab hoc conſilio grauiſsima eru-<lb/>ditiſsimorum hominum auctoritas, qui eam reſponſo omnino indignam <lb/>iudicabant, reuocaſſet. </s> <s xml:id="echoid-s11127" xml:space="preserve">Nunc vero quoniam de re ipſa, eiusq; </s> <s xml:id="echoid-s11128" xml:space="preserve">Apolo-<lb/>gia neceſſario mentio facta eſt, non alienum eſſe duxi, breuiter calumnias, <lb/>atq; </s> <s xml:id="echoid-s11129" xml:space="preserve">iniurias, quibus frequenter me in ea Apologia afficit, quanta pote-<lb/>ro modeſtia, depellere, vt benignus lector intelligat, ſine cauſa eum tan-<lb/>to animi dolore, & </s> <s xml:id="echoid-s11130" xml:space="preserve">iracundia, quantam præ ſe fert, contra me exarſiſſe, <lb/>falsoq; </s> <s xml:id="echoid-s11131" xml:space="preserve">mihi impoſuiſſe multa: </s> <s xml:id="echoid-s11132" xml:space="preserve">me autemè contrario nullum verbum in-<lb/>iurioſum in illum effudiſſe, aut conuitium, quod fruſtra in epiſtola nuncu-<lb/>patoria criminatur, vbi me à conuitijs non abſtinere, aperte teſtatur. </s> <s xml:id="echoid-s11133" xml:space="preserve">Atq; <lb/></s> <s xml:id="echoid-s11134" xml:space="preserve">in illa Apologia nihil adeo me offendit, quàm quòd me Peletarius non <lb/>ſyncere, ſed animoſe, atq; </s> <s xml:id="echoid-s11135" xml:space="preserve">adeo inuidioſe feciſſe inſimulat, vt eum in meis <lb/>commentarijs vel reprehenderem, vel laudarem: </s> <s xml:id="echoid-s11136" xml:space="preserve">quæ ſane vitia puſilli <lb/>ſemper animi eſſe duxi, & </s> <s xml:id="echoid-s11137" xml:space="preserve">ab homine liberaliter, chriſtianeq; </s> <s xml:id="echoid-s11138" xml:space="preserve">educato a-<lb/>lieniſsima. </s> <s xml:id="echoid-s11139" xml:space="preserve">Verum ea quàm longe abſint tum à noſtræ Societatis diſcipli-<lb/>na, tum à mea conſuetudine, nemo omnino, qui nos ac noſtra norit, igno-<lb/>rat. </s> <s xml:id="echoid-s11140" xml:space="preserve">His vero, qui noſtra minime norunt, liber ipſe fidem faciet, ſyncere <lb/>omnia dici, nihil inuidioſe, nihil animoſe: </s> <s xml:id="echoid-s11141" xml:space="preserve">planè vt veritatem quæſitam, <lb/>non cuiuſq; </s> <s xml:id="echoid-s11142" xml:space="preserve">auctoritatem contemptam eſſe appareat. </s> <s xml:id="echoid-s11143" xml:space="preserve">Neq; </s> <s xml:id="echoid-s11144" xml:space="preserve">enim mihi tan <lb/>tum derogo, (etſi nihil arrogo) vt mihi vni interdictum putem, ne, ſi quid <lb/>in alienis ſcriptis falſum videatur, occaſione oblata, cur id mhi minus pro-<lb/>betur, oſtendam; </s> <s xml:id="echoid-s11145" xml:space="preserve">modo (quod pudentium, ac bene moratorum hominum <lb/>conſuetudo poſtulat) id ſine conuitio, atq; </s> <s xml:id="echoid-s11146" xml:space="preserve">irriſione faciam: </s> <s xml:id="echoid-s11147" xml:space="preserve">Hoc autem <lb/>liber ipſe, qui in medio eſt, ita à me factum eſſe clamat. </s> <s xml:id="echoid-s11148" xml:space="preserve">Etenim vt totum <lb/>illum librum peruolutes, ne verbum quidem vnum reperias, quod vel <lb/>ſpeciem maledicti habeat, atq; </s> <s xml:id="echoid-s11149" xml:space="preserve">conuitij. </s> <s xml:id="echoid-s11150" xml:space="preserve">Nam in ſcholio de angulo con- <pb o="342" file="354" n="354" rhead=""/> tactus iurare liquido poſſem, nihil me minus cogitaſſe, quàm vt Peleta-<lb/>rium obtrectandi animo oppugnarem; </s> <s xml:id="echoid-s11151" xml:space="preserve">ſed illud habuiſſe propoſitum, (ſi <lb/>modo conſequi poſſem) vt ſuus eſſet veritati locus. </s> <s xml:id="echoid-s11152" xml:space="preserve">Quod quidem eò li-<lb/>berius feci, quòd Peletario ipſo non modo inuito, ſed etiam libenti exi-<lb/>ſtimaui me eſſe facturum, quod vel ipſo auctore facerem: </s> <s xml:id="echoid-s11153" xml:space="preserve">qui non à Car-<lb/>dano ſolum, atque Campano, ſed etiam à Proclo, Theone, Apollonio, <lb/>Eratoſthene, Pappo, Ptolemæo, Hippocrate Chio, Geometriæ lumini-<lb/>bus, ab ipſo deniq; </s> <s xml:id="echoid-s11154" xml:space="preserve">omnium magiſtro Euclide diſſentire non dubitauit-<lb/>Nimirum quia, vt ab eodem in Apologia vere dictum eſt, in omni doctri-<lb/>na, præſertim verò in Geometria, non auctoritas eſt ſpectanda, ſed veri-<lb/>tas: </s> <s xml:id="echoid-s11155" xml:space="preserve">quanquam non video, qui amicus veritatis ſit is, apud quem veritas <lb/>odium parit; </s> <s xml:id="echoid-s11156" xml:space="preserve">niſi forte aut decipi ſe non poſſe arbitratur, qui columina <lb/>illa Geometriæ erraſſe interdum prædicat, aut veritatem in alienis rebus <lb/>amat, ac quærit potius, quàm in ſuis. </s> <s xml:id="echoid-s11157" xml:space="preserve">Equidem ſi quis me in re quapiam <lb/>(quod pro humani ingenij imbecillitate fieri poſſe video) erraſſe oſten-<lb/>derit, næ ego maximam illi gratiam habuero, qui errantem in viam veri-<lb/>tatis reduxerit. </s> <s xml:id="echoid-s11158" xml:space="preserve">At enim probat ſtudium veritatis Peletarius, conuitia <lb/>ferre non poteſt: </s> <s xml:id="echoid-s11159" xml:space="preserve">quæ tandem conuitia? </s> <s xml:id="echoid-s11160" xml:space="preserve">rogas? </s> <s xml:id="echoid-s11161" xml:space="preserve">Demonſtrationes meas <lb/>appellas ſophiſmata. </s> <s xml:id="echoid-s11162" xml:space="preserve">Nunc demum, quæ conuitia dicat, intelligo. </s> <s xml:id="echoid-s11163" xml:space="preserve">Nam <lb/>alia nulla in meo libro eſſe certò ſcio: </s> <s xml:id="echoid-s11164" xml:space="preserve">ab his (ſi conuitia ſunt) fateor me <lb/>non abſtinere. </s> <s xml:id="echoid-s11165" xml:space="preserve">At ego homo ſimplex, & </s> <s xml:id="echoid-s11166" xml:space="preserve">ignarus verborum, conuitia eſſe <lb/>nunquam duxi, cum vere dicerentur. </s> <s xml:id="echoid-s11167" xml:space="preserve">Neq; </s> <s xml:id="echoid-s11168" xml:space="preserve">enim, quo alio vocabulo de-<lb/>monſtrationes plane fallaces, & </s> <s xml:id="echoid-s11169" xml:space="preserve">adulterinas appellarem, habebam: </s> <s xml:id="echoid-s11170" xml:space="preserve">neq; <lb/></s> <s xml:id="echoid-s11171" xml:space="preserve">vero philoſophorum, ac Mathematicorum conſuetudo loquendi magis <lb/>appoſitum mihi verbum ſnppeditabat. </s> <s xml:id="echoid-s11172" xml:space="preserve">Accedit, quòd cum à Peletario, <lb/>homine in loquendo conſideratiſsimo, germanas Campani, Cardanique <lb/>demonſtrationes paralogiſmos appellari viderem, exiſtimaui in falſis eius <lb/>demonſtrationibus refellendis impunius perſimili me vocabulo vſurum. </s> <s xml:id="echoid-s11173" xml:space="preserve"><lb/>Quòd ſi ſophiſma contumelioſius verbum eſt, quam paralogiſmus, in Gal <lb/>lia, ignoſcat conſuetudinis eius ignaro, atq; </s> <s xml:id="echoid-s11174" xml:space="preserve">exiſtimet, me paralogiſmos <lb/>dicere voluiſſe. </s> <s xml:id="echoid-s11175" xml:space="preserve">Atq; </s> <s xml:id="echoid-s11176" xml:space="preserve">vt plane intelligat Peletarius, me non contradicen <lb/>di ſtudio illa ſcripſiſſe, mecum vnà conſideret, quantam mihi materiam <lb/>ſui refellendi dederit, ſi hominem refellere potius, quam rem, quæ tum <lb/>agebatur, explanare in animo fuiſſet: </s> <s xml:id="echoid-s11177" xml:space="preserve">quanquam occaſionẽ eius reprehen <lb/>dendi in ſinum delatam ſæpius omiſi, ne illum mihi delegiſſe viderer, in <lb/>quem potiſsimum incurrerem. </s> <s xml:id="echoid-s11178" xml:space="preserve">Quàm præclara enim occaſio fuit in pro-<lb/>poſ. </s> <s xml:id="echoid-s11179" xml:space="preserve">4. </s> <s xml:id="echoid-s11180" xml:space="preserve">& </s> <s xml:id="echoid-s11181" xml:space="preserve">8. </s> <s xml:id="echoid-s11182" xml:space="preserve">lib. </s> <s xml:id="echoid-s11183" xml:space="preserve">1. </s> <s xml:id="echoid-s11184" xml:space="preserve">atq; </s> <s xml:id="echoid-s11185" xml:space="preserve">in propoſ. </s> <s xml:id="echoid-s11186" xml:space="preserve">24. </s> <s xml:id="echoid-s11187" xml:space="preserve">lib. </s> <s xml:id="echoid-s11188" xml:space="preserve">3. </s> <s xml:id="echoid-s11189" xml:space="preserve">quam ipſe 23. </s> <s xml:id="echoid-s11190" xml:space="preserve">facit? </s> <s xml:id="echoid-s11191" xml:space="preserve">In his <lb/>enim omnibus reijcit demonſtrationes antiquiſsimas Euclidis, tanquam <lb/>non Geometricas; </s> <s xml:id="echoid-s11192" xml:space="preserve">quippe in quibus figuram vnam alteri ſuperponi conci-<lb/>pere animo oporteat: </s> <s xml:id="echoid-s11193" xml:space="preserve">quod ipſe a Geometriæ dignitate putat eſſe alie-<lb/>num, hac ſolum inductus ratione, quòd ſuperpoſitionem illam mechani-<lb/>cum quid eſſe arbitretur, & </s> <s xml:id="echoid-s11194" xml:space="preserve">quòd omnes fere propoſitiones hoc modo, <lb/>vt ait, poſsint demonſtrari, etiam problemata, in quibus aliquid propo-<lb/>nitur conſtruendum: </s> <s xml:id="echoid-s11195" xml:space="preserve">atq; </s> <s xml:id="echoid-s11196" xml:space="preserve">in huius rei exemplum adducit propof. </s> <s xml:id="echoid-s11197" xml:space="preserve">2. </s> <s xml:id="echoid-s11198" xml:space="preserve">& </s> <s xml:id="echoid-s11199" xml:space="preserve">3.</s> <s xml:id="echoid-s11200" xml:space="preserve"> <pb o="343" file="355" n="355" rhead=""/> lib. </s> <s xml:id="echoid-s11201" xml:space="preserve">1. </s> <s xml:id="echoid-s11202" xml:space="preserve">quæ problemata ſunt. </s> <s xml:id="echoid-s11203" xml:space="preserve">Hic certe Peletarium iure carpere potuiſ-<lb/>ſem, ſi id mihi fuiſſet propoſitum, vt falſo criminatur; </s> <s xml:id="echoid-s11204" xml:space="preserve">maxime in eo, quòd <lb/>eadem ratione vſui fore exiſtimauit ſuperpoſitionem in demoſtrãdis pro-<lb/>blematibus, ac theorematibus. </s> <s xml:id="echoid-s11205" xml:space="preserve">Nam non ſatis intellexiſſe videtur, quo <lb/>pacto Geometræ ſuperpoſitionem illam vſurpent. </s> <s xml:id="echoid-s11206" xml:space="preserve">Neq; </s> <s xml:id="echoid-s11207" xml:space="preserve">enim volunt, re <lb/>ipſa faciendam eſſe figurarum ſuperpoſitionem, (hoc enim mechanicum <lb/> <anchor type="note" xlink:label="note-355-01a" xlink:href="note-355-01"/> quid eſſet) ſed cogitatione tantum, ac mente, quod opus eſt rationis atq; <lb/></s> <s xml:id="echoid-s11208" xml:space="preserve">intellectus. </s> <s xml:id="echoid-s11209" xml:space="preserve">Itaque in theorematibus quidem locum habebit genus hoc <lb/>argumentaudi, in problematibus vero non. </s> <s xml:id="echoid-s11210" xml:space="preserve">Namq; </s> <s xml:id="echoid-s11211" xml:space="preserve">in theorematibus, <lb/>propter magnitudinum æqualitatem, inæqualitatemve, quæ, vt nota, po-<lb/>nitur, facile intellectus cuiuſuis ſine vlla hęſitatione comprehendit, vnam <lb/>vel non excedere alteram, vel excedere, ſi animo concipiatur vna alteri <lb/>eſſe ſuperpoſita, quamuis re ipſa non fiat illa ſuperpoſitio, vt in propoſ. </s> <s xml:id="echoid-s11212" xml:space="preserve">4. </s> <s xml:id="echoid-s11213" xml:space="preserve"><lb/>lib. </s> <s xml:id="echoid-s11214" xml:space="preserve">1. </s> <s xml:id="echoid-s11215" xml:space="preserve">factum eſt: </s> <s xml:id="echoid-s11216" xml:space="preserve">At in problematibus, in quibus magnitudinẽ quis alte-<lb/>ri æqualem conſtruere iubetur, licet mente cogitet magnitudinem propo <lb/>ſitam ttansferri in alium locum, non tamen propterea quicquam efficiet, <lb/>cum reipſa tranſlatio nulla facta ſit: </s> <s xml:id="echoid-s11217" xml:space="preserve">Vt mirum ſit, Peletarium ſibi perſua-<lb/>dere potuiſſe, propoſ. </s> <s xml:id="echoid-s11218" xml:space="preserve">2. </s> <s xml:id="echoid-s11219" xml:space="preserve">& </s> <s xml:id="echoid-s11220" xml:space="preserve">3. </s> <s xml:id="echoid-s11221" xml:space="preserve">lib. </s> <s xml:id="echoid-s11222" xml:space="preserve">1. </s> <s xml:id="echoid-s11223" xml:space="preserve">& </s> <s xml:id="echoid-s11224" xml:space="preserve">alias pene omnes per ſuperpoſitio-<lb/>nem, ſiue tranſlationem linearum, figurarumve poſſe demonſtrari, ſi hoc <lb/>modo argumentandi in Geometria vti liceret. </s> <s xml:id="echoid-s11225" xml:space="preserve">Et certe hac in re non ſo-<lb/>lum Euclidem in crimen vocat Peletarius, verum etiam Archimedem, <lb/>quo, omnium iudicio, acutior in demonſtrando, & </s> <s xml:id="echoid-s11226" xml:space="preserve">ſubtilior fuit nemo, <lb/>eiuſque commentatorem grauiſsimum, eumque doctiſsimum Eutocium <lb/>Aſcalonitam, qui eodem argumentandi genere vtuntur in æqueponderan <lb/>tibus, immo vero & </s> <s xml:id="echoid-s11227" xml:space="preserve">omnes Geometras redarguat neceſſe eſt, qui non ra-<lb/>ro hoc argumenti genus adhibent. </s> <s xml:id="echoid-s11228" xml:space="preserve">Sed videamus, quò tandem egregius <lb/>hic noſter Geometra, qui omnes alios Geometras reprehendit, ſit deuo-<lb/>lutus. </s> <s xml:id="echoid-s11229" xml:space="preserve">Viderat Peletarius, (neq; </s> <s xml:id="echoid-s11230" xml:space="preserve">enim rem adeo manifeſtam videre non po <lb/>terat) ſi hunc modum argumentandi è medio tollat, vniuerſam ſe Geo-<lb/>metriam funditus euertere, cum plurimæ, eæque præcipuæ propoſitiones <lb/>in Geometria demonſtrentur ex propoſ. </s> <s xml:id="echoid-s11231" xml:space="preserve">4. </s> <s xml:id="echoid-s11232" xml:space="preserve">& </s> <s xml:id="echoid-s11233" xml:space="preserve">8. </s> <s xml:id="echoid-s11234" xml:space="preserve">lib. </s> <s xml:id="echoid-s11235" xml:space="preserve">1. </s> <s xml:id="echoid-s11236" xml:space="preserve">& </s> <s xml:id="echoid-s11237" xml:space="preserve">ex 24. </s> <s xml:id="echoid-s11238" xml:space="preserve">lib. </s> <s xml:id="echoid-s11239" xml:space="preserve">3. </s> <s xml:id="echoid-s11240" xml:space="preserve"><lb/>quæ quidem alio modo demonſtrari nequeunt, quam per dictam figura-<lb/>rum ſuperpoſitionem, non quidem re ipſa exiſtentem, ſed cogitatione <lb/>duntaxat, vt dixi, comprehenſam. </s> <s xml:id="echoid-s11241" xml:space="preserve">Quò igitur ſe verteret? </s> <s xml:id="echoid-s11242" xml:space="preserve">quid ageret? </s> <s xml:id="echoid-s11243" xml:space="preserve"><lb/>Excogitauit ſane rem magis à Geometria alienam, quam eſt ſuperpoſitio <lb/> <anchor type="note" xlink:label="note-355-02a" xlink:href="note-355-02"/> illa figurarum. </s> <s xml:id="echoid-s11244" xml:space="preserve">Coactus enim eſt aſſerere, propoſitionem 4. </s> <s xml:id="echoid-s11245" xml:space="preserve">lib. </s> <s xml:id="echoid-s11246" xml:space="preserve">1. </s> <s xml:id="echoid-s11247" xml:space="preserve">eſſe de-<lb/>finitionem angulorum ęqualium, (& </s> <s xml:id="echoid-s11248" xml:space="preserve">quis vnquam talem audiuit definitio-<lb/>nem?) </s> <s xml:id="echoid-s11249" xml:space="preserve">atq; </s> <s xml:id="echoid-s11250" xml:space="preserve">adeo concedendam eam eſſe ſine demonſtratione: </s> <s xml:id="echoid-s11251" xml:space="preserve">propoſi-<lb/>tionem vero 8. </s> <s xml:id="echoid-s11252" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s11253" xml:space="preserve">principium eſſe per ſe quoq; </s> <s xml:id="echoid-s11254" xml:space="preserve">notum. </s> <s xml:id="echoid-s11255" xml:space="preserve">Quod <lb/>vt credibile magis efficiat, ita ſcribit in propoſitionem 4. </s> <s xml:id="echoid-s11256" xml:space="preserve">lib. </s> <s xml:id="echoid-s11257" xml:space="preserve">1. </s> <s xml:id="echoid-s11258" xml:space="preserve">[_Etenim_ <lb/>_nulla euidentiori Specie æqualitas figurarum dignoſcitur, quam ex laterum_ <lb/>_æqualitate._</s> <s xml:id="echoid-s11259" xml:space="preserve">] Idemque quaſi confirmat, & </s> <s xml:id="echoid-s11260" xml:space="preserve">repetit in propoſitionem 8. </s> <s xml:id="echoid-s11261" xml:space="preserve">eiuſ-<lb/>dem lib. </s> <s xml:id="echoid-s11262" xml:space="preserve">dum ita loquitur. </s> <s xml:id="echoid-s11263" xml:space="preserve">[_Quis enim negauerit, duas ſuperficies eſſe æqua-_ <lb/>_les, quarum latera & </s> <s xml:id="echoid-s11264" xml:space="preserve">quantitate, & </s> <s xml:id="echoid-s11265" xml:space="preserve">numero ſunt æqualia?_</s> <s xml:id="echoid-s11266" xml:space="preserve">] Hæc Peleta- <pb o="344" file="356" n="356" rhead=""/> rius, vt dictę propoſitiones Euclidis ſine demonſtratione admittantur, cõ-<lb/>mẽtatus eſt, ſed quę omnino falſa ſunt: </s> <s xml:id="echoid-s11267" xml:space="preserve">vt magnopere mirandũ ſit, potuiſſe <lb/>eũ propoſitiones a Geometria prorſus alienas tam incõſiderate proferre. <lb/></s> <s xml:id="echoid-s11268" xml:space="preserve">Scilicet verum eſt, quod philoſophi aſſerunt; </s> <s xml:id="echoid-s11269" xml:space="preserve">Dato vno abſurdo, cætera <lb/>conſequũtur. </s> <s xml:id="echoid-s11270" xml:space="preserve">Aſſumpſerat enim Peletarius propoſ. </s> <s xml:id="echoid-s11271" xml:space="preserve">4. </s> <s xml:id="echoid-s11272" xml:space="preserve">& </s> <s xml:id="echoid-s11273" xml:space="preserve">8. </s> <s xml:id="echoid-s11274" xml:space="preserve">lib. </s> <s xml:id="echoid-s11275" xml:space="preserve">1. </s> <s xml:id="echoid-s11276" xml:space="preserve">pro prin <lb/>cipijs: </s> <s xml:id="echoid-s11277" xml:space="preserve">quod quidem falſum eſt, atq; </s> <s xml:id="echoid-s11278" xml:space="preserve">abſurdum. </s> <s xml:id="echoid-s11279" xml:space="preserve">Vnde ad eas abſurditates <lb/>neceſſario deuenit, quas etiam illi, qui vix adhuc principia Geometriæ <lb/>attigerunt, vel facile vitare potuiſſent. </s> <s xml:id="echoid-s11280" xml:space="preserve">Nam quis non videt, Rhombum, <lb/>& </s> <s xml:id="echoid-s11281" xml:space="preserve">Quadratum, etiamſi latera habeant & </s> <s xml:id="echoid-s11282" xml:space="preserve">quantitate, & </s> <s xml:id="echoid-s11283" xml:space="preserve">numero æqualia, <lb/>poſſe tamen inter ſe valde eſſe inæqualia? </s> <s xml:id="echoid-s11284" xml:space="preserve">Id quod in Pentagonis quoque <lb/>æquilateris, & </s> <s xml:id="echoid-s11285" xml:space="preserve">in alijs figuris pluriũ laterum æqualium cerni poteſt: </s> <s xml:id="echoid-s11286" xml:space="preserve">quod <lb/>non eſt huius loci pluribus verbis explicare. </s> <s xml:id="echoid-s11287" xml:space="preserve">Cum ergo in omnibus figu-<lb/>ris multilateris inæqualitas reperiatur, licet latera habeant & </s> <s xml:id="echoid-s11288" xml:space="preserve">quantitate, <lb/>& </s> <s xml:id="echoid-s11289" xml:space="preserve">numero æqualia, demonſtrandum fuit neceſſario Euclidi, æqualitatem <lb/>triangulorum colligi ex laterum ęqualitate, quandoquidem in alijs figuris <lb/>ea non colligitur. </s> <s xml:id="echoid-s11290" xml:space="preserve">Quare neq; </s> <s xml:id="echoid-s11291" xml:space="preserve">propoſitio 4. </s> <s xml:id="echoid-s11292" xml:space="preserve">Definitio, neq; </s> <s xml:id="echoid-s11293" xml:space="preserve">propoſitio 8. </s> <s xml:id="echoid-s11294" xml:space="preserve"><lb/>principium erit; </s> <s xml:id="echoid-s11295" xml:space="preserve">ac proinde omnes propoſitiones, quæ illis nituntur, quæ <lb/>innumerabiles propemodum ſunt, corruant neceſſe eſt, niſi demonſtra-<lb/>tiones Euclidis recipiãtur in illis propoſitionibus, cum alio modo demon <lb/> <anchor type="note" xlink:label="note-356-01a" xlink:href="note-356-01"/> ſtrari non poſsint. </s> <s xml:id="echoid-s11296" xml:space="preserve">Demonſtratio enim noua propof. </s> <s xml:id="echoid-s11297" xml:space="preserve">4. </s> <s xml:id="echoid-s11298" xml:space="preserve">quam Peletarius <lb/>confinxit, nihil aliud eſt, quam (vt cum Logicis loquamur) petitio prin-<lb/>cipij. </s> <s xml:id="echoid-s11299" xml:space="preserve">Id quod perſpicuum erit cuilibet, qui eam diligentius conſiderare <lb/>voluerit. </s> <s xml:id="echoid-s11300" xml:space="preserve">Nam in ea ſolum cõſtruitur vnum triangulum poſteriori ex duo-<lb/>bus datis æquale, immo idem, atq; </s> <s xml:id="echoid-s11301" xml:space="preserve">hoc ipſum quidem ineptiſsime, cum <lb/>ad id præſtandum circulos deſcribat Peletarius, quibus tamen in demon-<lb/>ſtratione non vtitur, quod vitioſum omnino eſt in Geometria: </s> <s xml:id="echoid-s11302" xml:space="preserve">Deinde <lb/>infert, triangulum hoc conſtructum, quod a poſteriori ex duobus propo-<lb/>ſitis non differt, priori eſſe æquale, ſine vlla demonſtratione; </s> <s xml:id="echoid-s11303" xml:space="preserve">certum au-<lb/>tem eſt, hoc ab initio propoſitum fuiſſe, vt demonſtretur. </s> <s xml:id="echoid-s11304" xml:space="preserve">Quocirca ma-<lb/>nifeſte principium petit, cum eadem facilitate ſtatim in principio conclu <lb/>dere potuiſſet, etiamſi nullam adhibuiſſet conſtructionem, triangula pro-<lb/>poſita eſſe æqualia; </s> <s xml:id="echoid-s11305" xml:space="preserve">quippe cum conſtructio illa ad rem non faciat. </s> <s xml:id="echoid-s11306" xml:space="preserve">Idem <lb/>dico de demonſtratione propof. </s> <s xml:id="echoid-s11307" xml:space="preserve">24. </s> <s xml:id="echoid-s11308" xml:space="preserve">lib. </s> <s xml:id="echoid-s11309" xml:space="preserve">3. </s> <s xml:id="echoid-s11310" xml:space="preserve">quam etiam nouam confinxit: <lb/></s> <s xml:id="echoid-s11311" xml:space="preserve">quod eorum iudicio, ad quorum manus eius commentarij peruenerunt, re-<lb/>linquo. </s> <s xml:id="echoid-s11312" xml:space="preserve">Prætereo alia loca innumerabilia, in quibus abutitur propoſitio-<lb/>nibus Euclidis in demonſtrando, vt quòd plerunq; </s> <s xml:id="echoid-s11313" xml:space="preserve">ſecundam propoſ. </s> <s xml:id="echoid-s11314" xml:space="preserve">lib. </s> <s xml:id="echoid-s11315" xml:space="preserve"><lb/>1. </s> <s xml:id="echoid-s11316" xml:space="preserve">inſcite pro tertia aſſumat, &</s> <s xml:id="echoid-s11317" xml:space="preserve">c. </s> <s xml:id="echoid-s11318" xml:space="preserve">Neq; </s> <s xml:id="echoid-s11319" xml:space="preserve">enim mihi in animo nunc eſt, eius <lb/>commentarios examinare, ſed ſolum calumnias, quas frequentes in ſua <lb/>Apologia adhibuit, a me depellere. </s> <s xml:id="echoid-s11320" xml:space="preserve">Quæ cum ita ſint, quod ille falsò de <lb/>me, verè ego de illo dicere poſſem, rubere me, (vt eius verbis vtar) Eu-<lb/>clidi interpretem contigiſſe, quinõ iam Theonem, aut Campanũ emẽdet, <lb/>ſed ipſum Euclidem ſine cauſa reprehendat; </s> <s xml:id="echoid-s11321" xml:space="preserve">quippe cum ego Euclidem <lb/>(vti par eſt) a calumnijs ipſius defendam, omneſque inſidias, ac fallacias, <lb/>quas contra eum inſtruxerat, detegam ac refellam. </s> <s xml:id="echoid-s11322" xml:space="preserve">Liquet igitur, me ea <pb o="345" file="357" n="357" rhead=""/> mente non fuiſſe, vt Peletarium redarguerem, cum tot ac tantos errores <lb/>diſsimulauerim: </s> <s xml:id="echoid-s11323" xml:space="preserve">quos ego ne nunc quidem in lucem protuliſſem, niſi vel-<lb/>lem omnes & </s> <s xml:id="echoid-s11324" xml:space="preserve">intelligere, quantum Peletarius a me, de quo tam acerbe <lb/>queritur, tum beneficium acceperit, & </s> <s xml:id="echoid-s11325" xml:space="preserve">ex breui hac diſputatione fructus <lb/>aliquid, vtilitatiſque percipere. </s> <s xml:id="echoid-s11326" xml:space="preserve">Nunc vt, quam diſpari ille animo in me <lb/>fuerit, appareat, eius calumnias breuiter exponam, atq; </s> <s xml:id="echoid-s11327" xml:space="preserve">ita refellam ac <lb/>diluam, vt omnes oculis videant, eas eſſe calumnias: </s> <s xml:id="echoid-s11328" xml:space="preserve">In quo tamen eiuſ-<lb/>modi a me moderatio adhibebitur, vt modeſtiæ, quæ hominem religio-<lb/>ſum decet, minime obliuiſcar. </s> <s xml:id="echoid-s11329" xml:space="preserve">Neq; </s> <s xml:id="echoid-s11330" xml:space="preserve">enim illi, vti prouocauit, reſpõdebo.</s> <s xml:id="echoid-s11331" xml:space="preserve"/> </p> <div xml:id="echoid-div909" type="float" level="2" n="3"> <note position="right" xlink:label="note-353-03" xlink:href="note-353-03a" xml:space="preserve">Digreſſio <lb/>cõtta Apo <lb/>logiam Pe <lb/>letarij in <lb/>auctotem <lb/>ſcriptam.</note> <note position="right" xlink:label="note-355-01" xlink:href="note-355-01a" xml:space="preserve">Superpoſi-<lb/>tio figura-<lb/>rum apud <lb/>Geometras <lb/>quo modo <lb/>intelliga--<lb/>tur, & cur <lb/>ea locum <lb/>habeat in <lb/>theorema-<lb/>tibus, non <lb/>autem in <lb/>problema-<lb/>tibus.</note> <note position="right" xlink:label="note-355-02" xlink:href="note-355-02a" xml:space="preserve">Abſurda sẽ <lb/>tentia Pele <lb/>tarij de pro <lb/>pof. 4. & 8. <lb/>lib. 1. Eucl.</note> <note position="left" xlink:label="note-356-01" xlink:href="note-356-01a" xml:space="preserve">Petitut <lb/>principiũ à <lb/>Peletario <lb/>in propoſ. <lb/>4. lib. 1. <lb/>Eucl.</note> </div> <p> <s xml:id="echoid-s11332" xml:space="preserve">PRIMVM itaq; </s> <s xml:id="echoid-s11333" xml:space="preserve">mihi obijcit Peletarius, quòd in eius demonſtra-<lb/>tionibus citandis ita me geſſerim, vt ſi quo modo nomen ipſius ſupprime-<lb/>re potuiſſem, id me oſtendam libenter fuiſſe facturum. </s> <s xml:id="echoid-s11334" xml:space="preserve">Quod quam ſit fal-<lb/>ſum, facile iudicabuntij, qui meos commentarios legerint; </s> <s xml:id="echoid-s11335" xml:space="preserve">cum vbiq; </s> <s xml:id="echoid-s11336" xml:space="preserve">eum <lb/>honorifice appellem, eique plurimas demonſtrationes aſcribam, tanquam <lb/>proprias, quas tamen aliter, quam ipſe, & </s> <s xml:id="echoid-s11337" xml:space="preserve">multo breuius demonſtro, & </s> <s xml:id="echoid-s11338" xml:space="preserve"><lb/>interdum etiã (quod maius eſt) vniuerſalius, vt liquido conſtat ex ijs, quæ <lb/>tum ad propof. </s> <s xml:id="echoid-s11339" xml:space="preserve">38. </s> <s xml:id="echoid-s11340" xml:space="preserve">tum ad propof. </s> <s xml:id="echoid-s11341" xml:space="preserve">45. </s> <s xml:id="echoid-s11342" xml:space="preserve">lib. </s> <s xml:id="echoid-s11343" xml:space="preserve">1. </s> <s xml:id="echoid-s11344" xml:space="preserve">ex Peletario demonſtraui, vt <lb/>alia interim taceam: </s> <s xml:id="echoid-s11345" xml:space="preserve">quæ non iniuria mihi vendicare potuiſſem: </s> <s xml:id="echoid-s11346" xml:space="preserve">vt mirer, <lb/>quid illi in mentem venerit, id a me parum ſyncere, atq; </s> <s xml:id="echoid-s11347" xml:space="preserve">adeo inuidioſe <lb/>factum exiſtimare, quod ego verebar, ne nimis ambitioſe factũ quiſpiam <lb/>iudicaret. </s> <s xml:id="echoid-s11348" xml:space="preserve">Quòd vero propoſ. </s> <s xml:id="echoid-s11349" xml:space="preserve">16. </s> <s xml:id="echoid-s11350" xml:space="preserve">lib. </s> <s xml:id="echoid-s11351" xml:space="preserve">3. </s> <s xml:id="echoid-s11352" xml:space="preserve">& </s> <s xml:id="echoid-s11353" xml:space="preserve">in prioribus duabus definitio-<lb/>nibus lib. </s> <s xml:id="echoid-s11354" xml:space="preserve">5. </s> <s xml:id="echoid-s11355" xml:space="preserve">vt ipſe obijcit, animoſe, vt ego fateor, libere, quid de eius de-<lb/>monſtrationibus ſentirẽ, expoſui, id feci, vt iam ante dixi, non cuiuſquam <lb/>lædendi cauſa, ſed quærendæ veritatis. </s> <s xml:id="echoid-s11356" xml:space="preserve">Ea enim eſt natura, & </s> <s xml:id="echoid-s11357" xml:space="preserve">conditio <lb/>eorum, qui liberalibus artibus dant operam, vt etiamſi alter alterius in-<lb/>terdum ſententiam impugnet, non tamen idcirco odijs potius, quam in-<lb/>genijs inter ſe certare videantur. </s> <s xml:id="echoid-s11358" xml:space="preserve">Qui ſit aliorum ſenſus ignoro, equidem, <lb/>vt ſupra dixi, ita ſum animo, vt ſi quis me alicuius erroris in demonſtran-<lb/>do commiſsi admoneret, ei quam maximas gratias haberem: </s> <s xml:id="echoid-s11359" xml:space="preserve">atq; </s> <s xml:id="echoid-s11360" xml:space="preserve">vt libe-<lb/>rius id facerent, enixe rogaui non paucos, & </s> <s xml:id="echoid-s11361" xml:space="preserve">nunc iterum eoſdem, atque <lb/>etiam alios amicè oratos volo. </s> <s xml:id="echoid-s11362" xml:space="preserve">Scio enim quam facile poſsit in ſuis quisq; <lb/></s> <s xml:id="echoid-s11363" xml:space="preserve">inuentis hallucinari; </s> <s xml:id="echoid-s11364" xml:space="preserve">video (quod ipſe quoq; </s> <s xml:id="echoid-s11365" xml:space="preserve">Peletarius in Apologia ſa-<lb/>pienter aſſeruit) omnibus hominibus commune eſſe, vt peccent. </s> <s xml:id="echoid-s11366" xml:space="preserve">Deinde <lb/>quòd in additionibus ad propof. </s> <s xml:id="echoid-s11367" xml:space="preserve">47. </s> <s xml:id="echoid-s11368" xml:space="preserve">lib. </s> <s xml:id="echoid-s11369" xml:space="preserve">1. </s> <s xml:id="echoid-s11370" xml:space="preserve">eius mentionem non fecerim, <lb/>non eſt, quod ægre ferat, cum illæ propoſitiones non ſint ab ipſo inuentæ. </s> <s xml:id="echoid-s11371" xml:space="preserve"><lb/>Quædam enim multo tempore ante ipſum demonſtratæ ſunt vel a Cam-<lb/>pano, vel a Proclo, aut Theone: </s> <s xml:id="echoid-s11372" xml:space="preserve">quaſdam vero demonſtraui egomet, ante-<lb/>quam ipſius demonſtrationes vidiſſem; </s> <s xml:id="echoid-s11373" xml:space="preserve">quòd adeo manifeſtæ ſint, & </s> <s xml:id="echoid-s11374" xml:space="preserve">faci-<lb/>les, vt nulla probatione egeant, ſed ſint inſtar corollariorum propoſ. </s> <s xml:id="echoid-s11375" xml:space="preserve">47. </s> <s xml:id="echoid-s11376" xml:space="preserve"><lb/>Vt nulla prorſus laus, aut gloria illi acceſſura videretur, ſi maxime eas ab <lb/>ipſo inuentas eſſe (quod tamen verum non eſt) prædicaſſem; </s> <s xml:id="echoid-s11377" xml:space="preserve">cum eas qui-<lb/>libet, modo primoribus labris ſtudia Mathematica deguſtarit, nullo <lb/>negotio ex illa propoſ. </s> <s xml:id="echoid-s11378" xml:space="preserve">47. </s> <s xml:id="echoid-s11379" xml:space="preserve">colligere poſsit: </s> <s xml:id="echoid-s11380" xml:space="preserve">Vt non videam, cur tandem <lb/>eas propoſitiones tanti ponderis eſſe dicat, cum ſint omnium iudicio leuiſ <pb o="346" file="358" n="358" rhead=""/> ſimæ; </s> <s xml:id="echoid-s11381" xml:space="preserve">adeo vt in pleriſque earum nec ipſe Peletarius demonſtrationem vl <lb/>lam, propter earum euidentiam, adducat, ſed eas nulla probatione egere <lb/>fateatur. </s> <s xml:id="echoid-s11382" xml:space="preserve">Denique non eſt, quod tantopere mihi ſuccenſeat idcirco, quod <lb/>conſtructionem Pentagoni æquilateri, & </s> <s xml:id="echoid-s11383" xml:space="preserve">æquianguli ſupra datam rectam <lb/>lineam finitam ei non tribuerim: </s> <s xml:id="echoid-s11384" xml:space="preserve">quoniam in ea conſtructione nihil prorſus <lb/>ab eo ſum mutuatus: </s> <s xml:id="echoid-s11385" xml:space="preserve">quod ijs dijudicandum relinquo, qui meam cum illius <lb/>conſtructione contulerint. </s> <s xml:id="echoid-s11386" xml:space="preserve">Nam & </s> <s xml:id="echoid-s11387" xml:space="preserve">mea omnino diuerſa eſt, & </s> <s xml:id="echoid-s11388" xml:space="preserve">ille in ſua <lb/>mirifice (vt alia peccata taceam) abutitur propoſitione 9. </s> <s xml:id="echoid-s11389" xml:space="preserve">lib. </s> <s xml:id="echoid-s11390" xml:space="preserve">3. </s> <s xml:id="echoid-s11391" xml:space="preserve">cum ex ea <lb/>probet, punctum quoddam eſſe centrũ circuli, qui nondum eſt deſcriptus. <lb/></s> <s xml:id="echoid-s11392" xml:space="preserve">Geometra ſanè dixiſſet, punctum illud eſſe eiuſmodi, vt circulus ex eo <lb/> <anchor type="note" xlink:label="note-358-01a" xlink:href="note-358-01"/> deſcriptus ad interuallum cuiuſlibet lineæ rectæ ex illis tribus, quæ ibi <lb/>oſtenſæ ſunt æquales, tranſeat per extremitates reliquarum duarum linea-<lb/>rum æqualium. </s> <s xml:id="echoid-s11393" xml:space="preserve">Nam propoſitio 9. </s> <s xml:id="echoid-s11394" xml:space="preserve">lib. </s> <s xml:id="echoid-s11395" xml:space="preserve">3. </s> <s xml:id="echoid-s11396" xml:space="preserve">nihil eo loco ad rem facit, cum <lb/>propoſitum ex ipſa cõſtructione poſsit cõcludi, & </s> <s xml:id="echoid-s11397" xml:space="preserve">ex demõſtratis, vt pro-<lb/>xime dixi, etiamſi propoſitio illa vera non eſſet, aut nuſquam demonſtra-<lb/>ta. </s> <s xml:id="echoid-s11398" xml:space="preserve">Idem peccatum committit Peletarius in omnibus propoſitionibus lib. <lb/></s> <s xml:id="echoid-s11399" xml:space="preserve">4. </s> <s xml:id="echoid-s11400" xml:space="preserve">in quibus vel intra ſiguram rectilineam, vel circa eandẽ circulus deſcri-<lb/>bẽdus eſt. </s> <s xml:id="echoid-s11401" xml:space="preserve">Quòd ſi ideo ſum reprehẽdendus, quòd propoſitionem vnã, mul <lb/>to aliter a me, & </s> <s xml:id="echoid-s11402" xml:space="preserve">breuius demonſtratam, ei non aſcripſerim, non video, <lb/>quo pacto in idem ipſe vitium non incurrat, cum problema hoc [_Propoſi-_ <lb/>_tis duabus lineis inæqualibus, potentiam maioris ſupra mincrem cognoſce-_ <lb/>_re._</s> <s xml:id="echoid-s11403" xml:space="preserve">] multis ſeculis ante ipſum a Theone demonſtratum ſibi arroget, hac <lb/>ſolum de cauſa, vt arbitror, quòd illud alia ratione, longiore tamen, de-<lb/>monſtrauerit. </s> <s xml:id="echoid-s11404" xml:space="preserve">Mitto hoc aliud problema, [_Dato angulo rectilineo æqua-_ <lb/>_lem angulum curuilineum constituere._</s> <s xml:id="echoid-s11405" xml:space="preserve">] quod in Apologia ſuum proprium <lb/>appellat, idemque hactenus deſideratum eſſe gloriatur; </s> <s xml:id="echoid-s11406" xml:space="preserve">cum tamen illud <lb/>ipſum ego ex Proclo, qui multis ante eum ſeculis floruit, in defin. </s> <s xml:id="echoid-s11407" xml:space="preserve">5. </s> <s xml:id="echoid-s11408" xml:space="preserve">lib. </s> <s xml:id="echoid-s11409" xml:space="preserve">5. </s> <s xml:id="echoid-s11410" xml:space="preserve"><lb/>multo breuius, & </s> <s xml:id="echoid-s11411" xml:space="preserve">clarius demonſtrauerim. </s> <s xml:id="echoid-s11412" xml:space="preserve">Nam, vt eo in loco oſtendi, ſi <lb/>rectæ lineæ datum angulum rectilineum continentes ponantur æquales, & </s> <s xml:id="echoid-s11413" xml:space="preserve"><lb/>circa ipſas duo ſemicirculi (qui æquales erunt) verſus eaſdem partes de-<lb/>ſcribantur, illico conſtitutus eritangulus curuilineus dato angulo rectili-<lb/>neo æqualis: </s> <s xml:id="echoid-s11414" xml:space="preserve">Neque opus eſt tot ambagibus vti, quot Peletarius ad eam <lb/>rem demonſtrandam adhibet; </s> <s xml:id="echoid-s11415" xml:space="preserve">quamuis robur demonſtrationis ipſius idem <lb/>ſit, quod meæ. </s> <s xml:id="echoid-s11416" xml:space="preserve">Et quod magis mirandum eſt, fatetur Peletarius, ſe meam <lb/>demonſtrationem vidiſſe, & </s> <s xml:id="echoid-s11417" xml:space="preserve">eam nihilominus ſibi audet, tanquam pro-<lb/>priam arrogare. </s> <s xml:id="echoid-s11418" xml:space="preserve">En cur Peletarius clamet, me non paucas demonſtratio-<lb/>nes parum honeſte, vt mihi vendicem, ſibi ſubducere conatum. </s> <s xml:id="echoid-s11419" xml:space="preserve">Quis au-<lb/>tem non videt, id eum in altero vituperare, quod ipſe ſibi glorioſum pu-<lb/>tat? </s> <s xml:id="echoid-s11420" xml:space="preserve">Itaque multo verius, ac iuſtius eodem illum crimine ego, quàm ille <lb/>me, condemnare poſſum; </s> <s xml:id="echoid-s11421" xml:space="preserve">cum nunquam propoſitionum illarum inuento-<lb/>rem me appellauerim, vtipſe, ſed ſolum eius nomen, obrationes a me ex-<lb/>poſitas, reticuerim.</s> <s xml:id="echoid-s11422" xml:space="preserve"/> </p> <div xml:id="echoid-div910" type="float" level="2" n="4"> <note position="left" xlink:label="note-358-01" xlink:href="note-358-01a" xml:space="preserve">Improprie <lb/>tates Pele-<lb/>tarij in de-<lb/>monſtran-<lb/>do.</note> </div> <p> <s xml:id="echoid-s11423" xml:space="preserve">DEINDE angulum contactus, & </s> <s xml:id="echoid-s11424" xml:space="preserve">acutum rectilineum eiuſdem ge-<lb/>neris eſſe, contra me pluribus verbis conatur oſtendere. </s> <s xml:id="echoid-s11425" xml:space="preserve">Sed neſcio quo <pb o="347" file="359" n="359" rhead=""/> modo aberrat, quod dicitur, a ſcopo. </s> <s xml:id="echoid-s11426" xml:space="preserve">Solum enim probat, vtrumque angu-<lb/>lum eodem genere quantitatis contineri, hoc eſt, vtrumque angulum pla-<lb/>num eſſe; </s> <s xml:id="echoid-s11427" xml:space="preserve">quod acutus angulus rectilineus, vel etiam rectus conſtare poſ-<lb/>ſit ex angulo contactus, & </s> <s xml:id="echoid-s11428" xml:space="preserve">alio angulo mixto: </s> <s xml:id="echoid-s11429" xml:space="preserve">quod neque ego, neque vllus <lb/>vnquam Geometra negauit. </s> <s xml:id="echoid-s11430" xml:space="preserve">Ego angulos illos eiuſdem eſſe generis nega-<lb/> <anchor type="note" xlink:label="note-359-01a" xlink:href="note-359-01"/> ui hac ſolum de cauſa, quòd angulus contactus quantumuis multiplicatus <lb/>angulum acutum rectilineum ſuperare nequeat, vt in ſcholio propof. </s> <s xml:id="echoid-s11431" xml:space="preserve">6. <lb/></s> <s xml:id="echoid-s11432" xml:space="preserve">lib. </s> <s xml:id="echoid-s11433" xml:space="preserve">3. </s> <s xml:id="echoid-s11434" xml:space="preserve">euidenter oſtendi. </s> <s xml:id="echoid-s11435" xml:space="preserve">Hinc enim fit, vt alter ad alterum proportionem <lb/>non habeat, atque adeo quodammodo diuerſi generis ſint: </s> <s xml:id="echoid-s11436" xml:space="preserve">quemadmo-<lb/>dum eadẽ de cauſa linea recta finita, & </s> <s xml:id="echoid-s11437" xml:space="preserve">infinita non cenſentur eſſe eiuſdem <lb/>generis, cum altera ad alteram proportionem non habeat; </s> <s xml:id="echoid-s11438" xml:space="preserve">quamuis ſub <lb/>eodem genere magnitudinis; </s> <s xml:id="echoid-s11439" xml:space="preserve">nimirum ſub linea recta, comprehendantur. </s> <s xml:id="echoid-s11440" xml:space="preserve"><lb/>Hoc itaque feriat, vt collimaſſe videatur: </s> <s xml:id="echoid-s11441" xml:space="preserve">quamquam vt omnia faciat, col-<lb/>limabit nunquam; </s> <s xml:id="echoid-s11442" xml:space="preserve">ita longè abeſt, quod eſt propoſitum. </s> <s xml:id="echoid-s11443" xml:space="preserve">Magnitudines au-<lb/>tem, quarum altera multiplicata alteram ſuperare nequit, non cenſeri eiu-<lb/>ſdem generis, (quod ad proportionem attinet) licet ſub eodẽ genere quan <lb/>titatis, hoc eſt, ſub longitudine, aut latitudine, aut profunditate, aut nume-<lb/>ro, collocentur, liquido conſtat ex defin. </s> <s xml:id="echoid-s11444" xml:space="preserve">5. </s> <s xml:id="echoid-s11445" xml:space="preserve">lib. </s> <s xml:id="echoid-s11446" xml:space="preserve">5. </s> <s xml:id="echoid-s11447" xml:space="preserve">vbi Euclides ſatis perſpi-<lb/>cue explicat, cuiuſmodi debeant eſſe magnitudines eiuſdem generis, inter <lb/>quas proportio reperitur. </s> <s xml:id="echoid-s11448" xml:space="preserve">Quare viderint alij, Peletarius homo conſide <lb/>ratus quam cogitatè me incogitantem hominem appellarit; </s> <s xml:id="echoid-s11449" xml:space="preserve">quaſi non re-<lb/>cte intellexerim, quæ magnitudines ſint eiuſdẽ generis, quæ non ſint. </s> <s xml:id="echoid-s11450" xml:space="preserve">Nun-<lb/>quam enim dixi (id quod mihi affinxit, vt carperet) duarum magnitudi-<lb/>num, quæ ſub diuerſis quãtitatis generibus collocantur, quales ſunt linea, <lb/>ſuperficies, corpus, ac numerus, alterutram ita poſſe multiplicari, vt alte-<lb/>ram ſuperet: </s> <s xml:id="echoid-s11451" xml:space="preserve">In quo, nẽmine reluctante, fruſtra ſeſe fatigat, vt doceat, id <lb/>fieri non poſſe; </s> <s xml:id="echoid-s11452" xml:space="preserve">ſed de illis duntaxat magnitudinibus ſum locutus, quæ cum <lb/>in eodem genere quantitatis verſentur, diuerſi tamen generis cenſeri poſ-<lb/>ſunt: </s> <s xml:id="echoid-s11453" xml:space="preserve">quales ſunt ſuperficies rectilinea & </s> <s xml:id="echoid-s11454" xml:space="preserve">curuilinea, ſiue mixta: </s> <s xml:id="echoid-s11455" xml:space="preserve">Itemque li <lb/>nea recta, & </s> <s xml:id="echoid-s11456" xml:space="preserve">curua. </s> <s xml:id="echoid-s11457" xml:space="preserve">Hæ etenim ita differre inter ſe videntur, vt Ariſtote-<lb/>les liquido affirmarit, vnam alteri æqualem eſſe non poſſe: </s> <s xml:id="echoid-s11458" xml:space="preserve">quod tamen (pa <lb/>ce Ariftotelis dictum ſit) verum vſquequaque non eſt; </s> <s xml:id="echoid-s11459" xml:space="preserve">cum Archimedes <lb/>in lib. </s> <s xml:id="echoid-s11460" xml:space="preserve">de lineis ſpiralibus demonſtrauerit, quænam linea recta æqualis poſ <lb/>ſit eſſe circunferentiæ cuiuſuis circuli dati. </s> <s xml:id="echoid-s11461" xml:space="preserve">Non igitur negare poterit Pele <lb/>tarius, aut quiſquam alius, ab Euclide defin. </s> <s xml:id="echoid-s11462" xml:space="preserve">5. </s> <s xml:id="echoid-s11463" xml:space="preserve">lib. </s> <s xml:id="echoid-s11464" xml:space="preserve">5. </s> <s xml:id="echoid-s11465" xml:space="preserve">aliquas quantitates <lb/>a proportionis definitione excludi, diuerſique propterea eſſe quodammo-<lb/>do generis, quod ad proportionem attinet, licet in eodem magnitudinis <lb/>genere ponantur: </s> <s xml:id="echoid-s11466" xml:space="preserve">quales ſunt angulus contactus, & </s> <s xml:id="echoid-s11467" xml:space="preserve">angulus rectilineus; </s> <s xml:id="echoid-s11468" xml:space="preserve">Li <lb/>nea item recta finita, & </s> <s xml:id="echoid-s11469" xml:space="preserve">infinita: </s> <s xml:id="echoid-s11470" xml:space="preserve">Multas item magnitudines comprehendi <lb/>in eadem definitione proportionis, quas plerique excludebant; </s> <s xml:id="echoid-s11471" xml:space="preserve">cuiuſmo-<lb/>di ſunt curuilinea ſuperficies, & </s> <s xml:id="echoid-s11472" xml:space="preserve">rectilinea; </s> <s xml:id="echoid-s11473" xml:space="preserve">necnon linea circularis, & </s> <s xml:id="echoid-s11474" xml:space="preserve">re-<lb/>cta, vt paulo ante diximus, latiuſque in defin. </s> <s xml:id="echoid-s11475" xml:space="preserve">5. </s> <s xml:id="echoid-s11476" xml:space="preserve">lib. </s> <s xml:id="echoid-s11477" xml:space="preserve">5. </s> <s xml:id="echoid-s11478" xml:space="preserve">expoſuimus. </s> <s xml:id="echoid-s11479" xml:space="preserve">Verum <lb/>Peletarius, ne opinionem illam ſuam, quam de angulo contactus ſemel im-<lb/>biberat, deſerere cogeretur, noluit hanc expoſitionem quintæ defin. </s> <s xml:id="echoid-s11480" xml:space="preserve">lib. </s> <s xml:id="echoid-s11481" xml:space="preserve">5.</s> <s xml:id="echoid-s11482" xml:space="preserve"> <pb o="348" file="360" n="360" rhead=""/> recipere; </s> <s xml:id="echoid-s11483" xml:space="preserve">immo eã vt oppugnet, omnes videtur in Apologia intẽdiſſe ner-<lb/>uos, oblitus ſui, ꝗ fere eodẽ modo illam defin. </s> <s xml:id="echoid-s11484" xml:space="preserve">in quinto lib. </s> <s xml:id="echoid-s11485" xml:space="preserve">olim expoſue-<lb/>rat; </s> <s xml:id="echoid-s11486" xml:space="preserve">niſi quod nõ recte inde colligit, angulum cõtactus non eſſe quantitatẽ, <lb/>propterea quod multiplicatus nullam magnitudinem, vt dicit, poſsit exce-<lb/>dere. </s> <s xml:id="echoid-s11487" xml:space="preserve">Hoc enim (pace eius dixerim) falſum eſt. </s> <s xml:id="echoid-s11488" xml:space="preserve">Nam licet angulus conta-<lb/>ctus multiplicatus angulum rectiiineum non poſsit excedere, excedet ta-<lb/>men alium angulum contactus. </s> <s xml:id="echoid-s11489" xml:space="preserve">Quare ex illa defin. </s> <s xml:id="echoid-s11490" xml:space="preserve">ſolum recte colligitur, <lb/>angulum contactus ad angulum rectilineum non habere proportionem <lb/>vllam; </s> <s xml:id="echoid-s11491" xml:space="preserve">ad angulum vero alium contactus quemcunque proportionem ha-<lb/>bere. </s> <s xml:id="echoid-s11492" xml:space="preserve">Sed ſiue ita intellexerit eam deſin. </s> <s xml:id="echoid-s11493" xml:space="preserve">vt ex commentarijs eius in lib. </s> <s xml:id="echoid-s11494" xml:space="preserve">5. <lb/></s> <s xml:id="echoid-s11495" xml:space="preserve">colligi poteſt, ſiue ſecus, vt in Apologia indicare videtur, non multum la-<lb/>boro: </s> <s xml:id="echoid-s11496" xml:space="preserve">Certè ita illam eſſe intelligendam, vt expoſui, nemo, qui verba Eu-<lb/>clidis diligenter expenderit, negabit. </s> <s xml:id="echoid-s11497" xml:space="preserve">Verum enim verò, ſi mihi fidem ha-<lb/>bere non vult Peletarius, habebit certe, (niſi arrogans haberi volet) aut <lb/>Proclo grauiſsimo ſcriptori, qui lib. </s> <s xml:id="echoid-s11498" xml:space="preserve">2. </s> <s xml:id="echoid-s11499" xml:space="preserve">in lib. </s> <s xml:id="echoid-s11500" xml:space="preserve">1. </s> <s xml:id="echoid-s11501" xml:space="preserve">Eucl, ad definitionem an-<lb/>guli plani eodem modo definitionem illam intellexit, aut Petro Nonio <lb/>Luſitano, quem tanti facit, (& </s> <s xml:id="echoid-s11502" xml:space="preserve">merito id quidẽ: </s> <s xml:id="echoid-s11503" xml:space="preserve">fuit enim acerrimo vir in-<lb/>genio, & </s> <s xml:id="echoid-s11504" xml:space="preserve">nullo hac noſtra ætate in Mathematicis inferior) vt eum vnum <lb/>pro multis millibus teſtem citet, & </s> <s xml:id="echoid-s11505" xml:space="preserve">ſuarũ demõſtrationum approbatorem, <lb/>qui diſertiſsimis verbis tum in libro de Erratis Orontij, tum in Algebra <lb/>ſua, illam definitionem explicat, vt a me eſt expoſita: </s> <s xml:id="echoid-s11506" xml:space="preserve">quinetiam ibidem <lb/>aſſerit, ex ea defin. </s> <s xml:id="echoid-s11507" xml:space="preserve">colligi, angulum contactus ad angulum rectilineum, & </s> <s xml:id="echoid-s11508" xml:space="preserve"><lb/>lineam finitam ad infinitam nullam habere proportionem; </s> <s xml:id="echoid-s11509" xml:space="preserve">vt Petrus No. </s> <s xml:id="echoid-s11510" xml:space="preserve"><lb/>nius, quem teſtem produxerat pro ſe Peletarius, iam pro me teſtimonium <lb/>dicat. </s> <s xml:id="echoid-s11511" xml:space="preserve">Atque ex hiſce duobus locis Petri Nonij facile quiuis intelliget, <lb/>quam ſine ratione, quanto contradicendi ſtudio mihi inſultet Peletarius, <lb/>cum ſemel atque iterum odioſe percontatur, vndenam potuerim illi lineã <lb/>inſinitam deportare. </s> <s xml:id="echoid-s11512" xml:space="preserve">In idem enim crimen (ſi crimen eſt, lineam infinitam <lb/>exempli cauſa nominare) vocat etiam Petrum Nonium teſtem ſuum, at-<lb/>que adeo omnes philoſophos, quorum eſt illa vox nemini inaudita, præ-<lb/>terquam Peletario, finiti ad infinitum nullam eſſe proportionem. </s> <s xml:id="echoid-s11513" xml:space="preserve">Deſi-<lb/>nat igitur a me ſciſcitari, vnde lineam infinitam deportauerim: </s> <s xml:id="echoid-s11514" xml:space="preserve">Inde enim <lb/>reſpondebo, vnde eam Petrus Nonius, vnde philoſophi omnes deporta-<lb/>runt. </s> <s xml:id="echoid-s11515" xml:space="preserve">Quid? </s> <s xml:id="echoid-s11516" xml:space="preserve">nonne ſophiſma illud Peletarij, ſemper in hoc erro, demon-<lb/>ſtratio illa, volui dicere, & </s> <s xml:id="echoid-s11517" xml:space="preserve">quidem palmaris, qua conatur oſtendere, pro-<lb/>poſitionem 1. </s> <s xml:id="echoid-s11518" xml:space="preserve">lib. </s> <s xml:id="echoid-s11519" xml:space="preserve">10. </s> <s xml:id="echoid-s11520" xml:space="preserve">cum propoſ. </s> <s xml:id="echoid-s11521" xml:space="preserve">16. </s> <s xml:id="echoid-s11522" xml:space="preserve">lib. </s> <s xml:id="echoid-s11523" xml:space="preserve">3. </s> <s xml:id="echoid-s11524" xml:space="preserve">ſtare non poſſe, ſi angulus con-<lb/>tactus concedatur eſſe quantitas, a Petro Nonio Peletarij cognitore ea-<lb/>dem prorſus ratione, qua a me ipſo, confutatur? </s> <s xml:id="echoid-s11525" xml:space="preserve">Quæ ſi germana demon-<lb/>ftratio eſt, miror quid ſit, cur eã Nonius Geometriæ ſcientiſsimus, idemq; </s> <s xml:id="echoid-s11526" xml:space="preserve"><lb/>Peletarij approbator, minus probarit: </s> <s xml:id="echoid-s11527" xml:space="preserve">Cur nihilo magis demouſtrationes <lb/>eiuſdem, quibus planum ſacit, (vt putat) angulum contactus quantitatem <lb/>non eſſe, eundem illum Nonium nihil admodum mouerint? </s> <s xml:id="echoid-s11528" xml:space="preserve">Id enim (niſi fal <lb/>lor) illa Nonij verba [_Si quis ſententiam Peletarij de angulo contactus am-_ <lb/>_plecti velit._</s> <s xml:id="echoid-s11529" xml:space="preserve">] declarant. </s> <s xml:id="echoid-s11530" xml:space="preserve">Nam ſi demonſtrationes exiſtimaſſet, profecto Pe- <pb o="349" file="361" n="361" rhead=""/> letarij doctrinä in eo retinendam eſſe dixiſſet, Geometricæ enim demon-<lb/>ſtrationes eiuſmodi ſunt, vt aſſenſum extorqueant, ac dubitationem om-<lb/>nem excludant, nulloq; </s> <s xml:id="echoid-s11531" xml:space="preserve">modo quempiam ſinant ancipiti opinione diſtra-<lb/>hi ſic, vt tum aſſentiatur, ſi velit, tum, ſi nolit, diſſentiat. </s> <s xml:id="echoid-s11532" xml:space="preserve">En cur Peletarius <lb/>Nonij teſtimonio aliorũ iudicia cõtemnat, en pręclarum teſtimoniũ, quod <lb/>Petrus Nonius eius demonſtrationibus dedit: </s> <s xml:id="echoid-s11533" xml:space="preserve">quò æquiore animo ferat, <lb/>eas a me nihilo magis, quã ab illo ſuo approbatore, demõſtrationes putari.</s> <s xml:id="echoid-s11534" xml:space="preserve"/> </p> <div xml:id="echoid-div911" type="float" level="2" n="5"> <note position="right" xlink:label="note-359-01" xlink:href="note-359-01a" xml:space="preserve">Angulus <lb/>contactus, <lb/>& rectili-<lb/>neus curdi <lb/>cantur eſſe <lb/>diuerſi g@-<lb/>neris.</note> </div> <p> <s xml:id="echoid-s11535" xml:space="preserve"><emph style="sc">Tertio</emph>, quòd exiſtimare dixi Peletarium, angulum contingen-<lb/>tiæ nihil eſſe, falſum eſſe, clamat: </s> <s xml:id="echoid-s11536" xml:space="preserve">Nuſquam enim dixiſſe ſe, nihil eſſe, ſed <lb/>quantitatem non eſſe. </s> <s xml:id="echoid-s11537" xml:space="preserve">Ita ne vero? </s> <s xml:id="echoid-s11538" xml:space="preserve">at in prædicamento Quantitatis, quod <lb/>neque eſt punctum, (quis enim inclinationem illam punctum eſſe dixerit?) <lb/></s> <s xml:id="echoid-s11539" xml:space="preserve">neque quantitas, quo alio nomine vocetur, quam Nihil? </s> <s xml:id="echoid-s11540" xml:space="preserve">Sed vt vt dixit, <lb/>profecto non modo mirabile eſt, ſed monſtri in Geometria fimile, putare <lb/>angulum contactus non eſſe quantitatem, qui poſtea additus alijs angulis <lb/>efficiat curuilineum angulum rectilineo æqualem. </s> <s xml:id="echoid-s11541" xml:space="preserve">Quis enim vnquã Geo-<lb/>metrarum id, quod quantitas non eſt, magnitudini adiunxit, vt æqualem <lb/>eam alteri efficeret? </s> <s xml:id="echoid-s11542" xml:space="preserve">Prætereo, quod figura trilatera curuilinea intra tres <lb/>circulos ſe mutuo tangentes concluſa nullum haberet angulũ ex Peleta-<lb/>rij ſententia; </s> <s xml:id="echoid-s11543" xml:space="preserve">quia tres illi contactus, anguli non ſunt: </s> <s xml:id="echoid-s11544" xml:space="preserve">cum tamen tribus di-<lb/>uerſis lineis contineatur; </s> <s xml:id="echoid-s11545" xml:space="preserve">quod omnino nouum eſt, & </s> <s xml:id="echoid-s11546" xml:space="preserve">inauditũ apud Geo-<lb/>metras. </s> <s xml:id="echoid-s11547" xml:space="preserve">Itemque, ſi quatuor, aut plures circuli ſe mutuo tangerent, vt fie-<lb/>ret figura curuilinea quadrilatera, vel plurium laterum, illa nullum angu-<lb/>lum haberet. </s> <s xml:id="echoid-s11548" xml:space="preserve">Atque etiam, ſi duæ lineæ rectæ angulum continentes, vnum <lb/>eundemque circulum tangerent, trilatera illa figura habens tertium latus <lb/>curuum, vnicum tantum haberet angulum. </s> <s xml:id="echoid-s11549" xml:space="preserve">Quæ omnia ſi ſunt abſurda, <lb/>conſentanea non eſt opinio Peletarij. </s> <s xml:id="echoid-s11550" xml:space="preserve">Sed nimis fortaſſe multa ad Nihil <lb/>illud Peletarij euertendum, ad quod tuendum ille nihil afferat. </s> <s xml:id="echoid-s11551" xml:space="preserve">Quoniam <lb/>vero, ne pro Nihilo ſuo nihil agere videatur, quando res non poteſt, mea <lb/>verba carpit, verba defendam: </s> <s xml:id="echoid-s11552" xml:space="preserve">quæ quidẽ ille neſcio quibus præſtigijs ita <lb/>deprauat, vt dicere videar, nihil eſſe minus quocũq; </s> <s xml:id="echoid-s11553" xml:space="preserve">angulo: </s> <s xml:id="echoid-s11554" xml:space="preserve">atq; </s> <s xml:id="echoid-s11555" xml:space="preserve">(vt ſim-<lb/>plicem, credo, hominẽ irretiat) quærit ex me, quod tãdem genus ſermonis <lb/>ſit illud. </s> <s xml:id="echoid-s11556" xml:space="preserve">Viderit is, cuius ex officina prodijt. </s> <s xml:id="echoid-s11557" xml:space="preserve">Neque enim ego eiuſmodi <lb/>ſermonem agnoſco, qui, nihil eſſe minus quocunque angulo, nuſquamdi-<lb/>xerim, niſi ex ſententia Peletarij. </s> <s xml:id="echoid-s11558" xml:space="preserve">Sed videlicet homo vehemens, vt ſuum <lb/>illud Nihil vlciſceretur, aliud mihi nihil affinxit, quo cum impune pugna-<lb/>ret: </s> <s xml:id="echoid-s11559" xml:space="preserve">At quam palæſtrice pugnat? </s> <s xml:id="echoid-s11560" xml:space="preserve">quam ſibi placet hoc loco, dum meum il-<lb/>lud argumentum, quo petitus fuerat, in me ipſum mira venuſtate conuer-<lb/>tit? </s> <s xml:id="echoid-s11561" xml:space="preserve">Sic enim argumentatur. </s> <s xml:id="echoid-s11562" xml:space="preserve">[_Angulus contactus nihil eſt: </s> <s xml:id="echoid-s11563" xml:space="preserve">Angulus conta-_ <lb/>_ctus angulo contactus maior eſt. </s> <s xml:id="echoid-s11564" xml:space="preserve">Angulus igitur maior nihilo eſt: </s> <s xml:id="echoid-s11565" xml:space="preserve">Atqui Cla_ <lb/>_uius eundẽ ponit minorẽ nihilo. </s> <s xml:id="echoid-s11566" xml:space="preserve">Eſt igitur angulus contactus nihilo maior, et_ <lb/>_idẽ nihilo minor_] Mox quaſi Nihil illud ab ſe effictũ iugulaſſet, exclamat. </s> <s xml:id="echoid-s11567" xml:space="preserve"><lb/>[_En Clauy argumenta, quæ vtrum tandem Peletary ſophiſmata ſunt, an_ <lb/>_Clauij potius ſigmenta, cum ipſe ſuum angulum contactus nihil eſſe dicat,_ <lb/>_non ego?_</s> <s xml:id="echoid-s11568" xml:space="preserve">] Verum vt hominem fæneum, atque adumbratum nequicquam <pb o="350" file="362" n="362" rhead=""/> petere deſinat, virum oſtendam, qui cum, ſi velit, certare cum laude poſ-<lb/>ſit. </s> <s xml:id="echoid-s11569" xml:space="preserve">Ego vt oſtenderem, angulum contactus, ex Euclidis ſententia, verè eſ-<lb/>ſe angulum, & </s> <s xml:id="echoid-s11570" xml:space="preserve">angulum ſemicirculi angulo recto rectilineo minorem, ita <lb/>ſum argumentatus. </s> <s xml:id="echoid-s11571" xml:space="preserve">Si Euclides ſenſiſſet, angulum contactus nihil prorſus <lb/>eſſe, (hoc eſt, vt Peletarius intelligit, non eſſe angulum, vel non eſſe quan-<lb/>titatem) & </s> <s xml:id="echoid-s11572" xml:space="preserve">angulum ſemicirculi æqualem recto rectilineo; </s> <s xml:id="echoid-s11573" xml:space="preserve">quid, obſecro, <lb/>tantopere deſudaſſet, vt demonſtraret, angulum contactus eſſe minorem <lb/>omni acuto rectilineo, angulum vero ſemicirculi maiorem? </s> <s xml:id="echoid-s11574" xml:space="preserve">Quid enim cla <lb/>rius, quàm nihil, cuiuſmodi eſt angulus contactus, ex Peletarij ſententia, <lb/>hoc eſt, quàm id, quod quantitas non eſt, minus eſſe quocunque angulo? <lb/></s> <s xml:id="echoid-s11575" xml:space="preserve">Quid rurſus magis perſpicuum, quàm angulum rectum, qualem ponit Pe-<lb/>letarius angulum ſemicirculi, maiorem eſſe quolibet acuto? </s> <s xml:id="echoid-s11576" xml:space="preserve">Agnoſcat <lb/>itaque Peletarius, Nihil illud ſuum male à nobis acceptum, idque ita <lb/>vlciſcatur, vt meum hoc argumentum refellat: </s> <s xml:id="echoid-s11577" xml:space="preserve">in quo ego ſi angulum <lb/>contactus dixi eſſe nihil, & </s> <s xml:id="echoid-s11578" xml:space="preserve">non potius eum nihil eſſe aſſerui ex ſenten-<lb/>tia Peletarij, libenter manus dabo. </s> <s xml:id="echoid-s11579" xml:space="preserve">Videtur Peletarius aut non intelle-<lb/>xiſſe meum argumentum, aut intelligere noluiſſe: </s> <s xml:id="echoid-s11580" xml:space="preserve">niſi eum quis dicat, de-<lb/>dita opera verba mea voluiſſe cauillari; </s> <s xml:id="echoid-s11581" xml:space="preserve">quod & </s> <s xml:id="echoid-s11582" xml:space="preserve">plerisque alijs in locis <lb/>facere videtur. </s> <s xml:id="echoid-s11583" xml:space="preserve">Nunquam enim dixi, angulum contactus minorem eſſe, aut <lb/>maiorem nihilo: </s> <s xml:id="echoid-s11584" xml:space="preserve">Solum affirmaui, angulum contactus quemcunque mino-<lb/>rem eſſe, aut maiorem aliquo alio angulo contactus, quem non ego dixi <lb/>nihil eſſe, ſed Peletarius, eundemque Euclides minorem quolibet acuto <lb/>rectilineo recte demonſtrauit. </s> <s xml:id="echoid-s11585" xml:space="preserve">Vt autem intelligat Peletarius, me, quod <lb/>ipſe negat, didiciſſe Dialecticam, illum ipſum tam lepidum, atque acu-<lb/>tum ſyllogiſmum, quo Nihil illud ab ſe cõfictum mira venuſtate confixit, <lb/>pauliſper conſiderabimus; </s> <s xml:id="echoid-s11586" xml:space="preserve">vt quàm ſuo iure Dialecticæ ignaros alios vo-<lb/>cet, appareat. </s> <s xml:id="echoid-s11587" xml:space="preserve">Nam mihi quidem male tornatus ille ipſe ſyllogiſmus vide-<lb/> <anchor type="note" xlink:label="note-362-01a" xlink:href="note-362-01"/> tur, incudique reddendus. </s> <s xml:id="echoid-s11588" xml:space="preserve">Etenim cum verſetur in tertia figura, in eo ma-<lb/>ior extremitas, (vt Dialectici loquuntur) quæ eſt, Nihil, de minori, quę eſt, <lb/>angulo contactus maior, in recto prædicari deberet, hoc pacto. </s> <s xml:id="echoid-s11589" xml:space="preserve">Angulus <lb/>contactus nihil eſt: </s> <s xml:id="echoid-s11590" xml:space="preserve">Angulus contactus angulo contactus maior eſt. </s> <s xml:id="echoid-s11591" xml:space="preserve">Igitur <lb/>aliquid, quod angulo contactus maius eſt, nihil eſt. </s> <s xml:id="echoid-s11592" xml:space="preserve">Quæ quidem con-<lb/>cluſio recte ſequitur ex præmiſsis, quarum prior Peletarij eſt, non mea, <lb/>poſterior autem mea, & </s> <s xml:id="echoid-s11593" xml:space="preserve">Procli, immo & </s> <s xml:id="echoid-s11594" xml:space="preserve">Euclidis. </s> <s xml:id="echoid-s11595" xml:space="preserve">Concluſio autem illa <lb/>Peletarij, Angulus igitur maior nihilo eſt, nulla ratione ex præmiſsis in-<lb/>ferri poteſt. </s> <s xml:id="echoid-s11596" xml:space="preserve">Nam ſi, angulum cum dicit, intelligit Peletarius angulum con <lb/>tactus, aſſumitur medius terminus, qui in vtraq; </s> <s xml:id="echoid-s11597" xml:space="preserve">præmiſſa ſubijcitur: </s> <s xml:id="echoid-s11598" xml:space="preserve">quod <lb/>nefas eſſe, Ariſtoteles in prioribus Anal. </s> <s xml:id="echoid-s11599" xml:space="preserve">& </s> <s xml:id="echoid-s11600" xml:space="preserve">Dialectici omnes clamant. </s> <s xml:id="echoid-s11601" xml:space="preserve">Si <lb/>autem alium angulum intelligit, aſſumitur in concluſione terminus, cuius <lb/>nulla facta eſt mentio in præmiſsis: </s> <s xml:id="echoid-s11602" xml:space="preserve">quod nihilo magis licere, nemo eſt tam <lb/>plumbeus in Dialecticis, qui neſciat. </s> <s xml:id="echoid-s11603" xml:space="preserve">Neque contendat Peletarius, men-<lb/>tionem factam eſſe anguli in minore extremitate, vbi dictum eſt, angulum <lb/>contactus angulo contactus maiorem eſſe. </s> <s xml:id="echoid-s11604" xml:space="preserve">Nam angulus in minore extre-<lb/>mitate poſitus eſt in obliquo, qui in concluſione ſubijcitur in recto: </s> <s xml:id="echoid-s11605" xml:space="preserve">quod, <pb o="351" file="363" n="363" rhead=""/> vt auctore Ariſtotele docent omnes Logici, ſine peccato fieri non poteſt. <lb/></s> <s xml:id="echoid-s11606" xml:space="preserve">Quod vt planum fiat, vtemur ea palæſtra, quam ab illo didicimus. </s> <s xml:id="echoid-s11607" xml:space="preserve">Si quiſ-<lb/>piam ita argumentetur; </s> <s xml:id="echoid-s11608" xml:space="preserve">Angulus in ſemicirculo rectus eſt: </s> <s xml:id="echoid-s11609" xml:space="preserve">Angulus in ſe-<lb/>micirculo angulo acuto maior eſt. </s> <s xml:id="echoid-s11610" xml:space="preserve">Angulus igitur acutus maior recto eſt; </s> <s xml:id="echoid-s11611" xml:space="preserve"><lb/>quis, modo ſit imbutus Dialecticis, eiuſmodi argumentationem probet, <lb/>cum præmiſſæ veræ ſint, concluſio autem falſa? </s> <s xml:id="echoid-s11612" xml:space="preserve">Talis ille ſyllogiſmus <lb/>eſt Peletarij, qui apud imperitam multitudinem alter Chryſippus videri <lb/>voluit. </s> <s xml:id="echoid-s11613" xml:space="preserve">Concluſio, quæ recte ex præmiſsis inferretur, hæc eſſet. </s> <s xml:id="echoid-s11614" xml:space="preserve">Igitur ali-<lb/>quis angulus, qui acuto maior eſt, rectus eſt. </s> <s xml:id="echoid-s11615" xml:space="preserve">Sed tamen ei veniam dandam <lb/>puto, quòd ſe Geometricum Dialecticum, ex alio quodam Dialecticorum <lb/>genere, profitetur, cuius ego me Dialecticæ, ſi ab Ariſtotelica abhorret, <lb/>planè fateor ignarum. </s> <s xml:id="echoid-s11616" xml:space="preserve">Fatetur deinde Peletarius, ſe non intelligere, quo <lb/>pacto dicere poſsim, angulum rectilineum minimum dari non poſſe, & </s> <s xml:id="echoid-s11617" xml:space="preserve">ta-<lb/>men angulum contactus eſſe omni acuto rectilineo minorem, (ipſe, vt ali-<lb/>quid addat de ſuo, dicit, omni minimo acuto rectilineo minorem; </s> <s xml:id="echoid-s11618" xml:space="preserve">cum ta-<lb/>men verbum illud, minimo, ego non addiderim) cupitque ſcire, quid aliud <lb/>ſit, angulum contactus minorem eſſe omni rectilineo acuto, quàm angu-<lb/>lum contactus eſſe acutorum rectilineorum minimum. </s> <s xml:id="echoid-s11619" xml:space="preserve">Qua in re morem <lb/>geram homini non grauate, etſi è ſcholio ad propoſ. </s> <s xml:id="echoid-s11620" xml:space="preserve">16. </s> <s xml:id="echoid-s11621" xml:space="preserve">lib. </s> <s xml:id="echoid-s11622" xml:space="preserve">3. </s> <s xml:id="echoid-s11623" xml:space="preserve">potuit id, <lb/>quod cupit, cognoſcere. </s> <s xml:id="echoid-s11624" xml:space="preserve">Nempe ea ratione me illud potuiſſe dicere, qua <lb/>dicimus, angulum obtuſum rectilineum minimum dari non poſſe, & </s> <s xml:id="echoid-s11625" xml:space="preserve">tamen <lb/>angulum rectilineum acutum eſſe omni obtuſo rectilineo minorem. </s> <s xml:id="echoid-s11626" xml:space="preserve">Item <lb/>quemadmodum aliud eſt, angulum rectilineum acutum minorem eſſe om-<lb/>ni rectilineo obtuſo, quàm angulum rectilineum acutum eſſe obtuſorum <lb/>rectilineorum minimum: </s> <s xml:id="echoid-s11627" xml:space="preserve">propterea quod angulus acutus non eſt obtuſus, <lb/>ſicut nec angulus contactus rectilineus eſt, aut acutus. </s> <s xml:id="echoid-s11628" xml:space="preserve">Id quod etiam cla-<lb/>riſsime docet Proclus lib. </s> <s xml:id="echoid-s11629" xml:space="preserve">2. </s> <s xml:id="echoid-s11630" xml:space="preserve">in primum Eucl. </s> <s xml:id="echoid-s11631" xml:space="preserve">ad defin. </s> <s xml:id="echoid-s11632" xml:space="preserve">anguli recti, obtuſi, <lb/>& </s> <s xml:id="echoid-s11633" xml:space="preserve">acuti. </s> <s xml:id="echoid-s11634" xml:space="preserve">Sed hæc puerilia ſunt, & </s> <s xml:id="echoid-s11635" xml:space="preserve">quæ magis ad Grammaticos ſpectent, <lb/>quàm ad Geometras. </s> <s xml:id="echoid-s11636" xml:space="preserve">Quòd etiam, ne librum meum parum ſpiſſum vide-<lb/>rer feciſſe, ſuas demonſtrationes ad verbum me recitaſſe queritur, id in me <lb/>reprehendit, quod ego in ipſo deſidero. </s> <s xml:id="echoid-s11637" xml:space="preserve">Id enim eo a me conſilio factum <lb/>eſt, vt omnes plane viderent, ſyncere me, ac fideliter eius opinionem retu-<lb/>liſſe, nullumq; </s> <s xml:id="echoid-s11638" xml:space="preserve">omnino verbũ immutaſſe. </s> <s xml:id="echoid-s11639" xml:space="preserve">Quod vtinam in meis verbis reci <lb/>tãdis ipſe facere in animum induxiſſet. </s> <s xml:id="echoid-s11640" xml:space="preserve">Multo enim minus ſpiſſam Apolo-<lb/>giam ſuam facere potuiſſet. </s> <s xml:id="echoid-s11641" xml:space="preserve">Nam ego, quid erat, cur laborarem meum li-<lb/>brum Peletarij verbis magis ſpiſſum efficere? </s> <s xml:id="echoid-s11642" xml:space="preserve">Qui enim parum ſpiſſum iu-<lb/>dicarem librum eum, qui nec raras, nec inanes in libros omnes Euclidis <lb/>commentationes contineret, cum Peletarius ſuum librum, qui ſex prio-<lb/>rum duntaxat librorum demonſtrationes complectitur, ſatis ſpiſſum ſit ar-<lb/>bitratus? </s> <s xml:id="echoid-s11643" xml:space="preserve">Sed eo ſum æquior Peletario, quòd ex ſe alios iudicat. </s> <s xml:id="echoid-s11644" xml:space="preserve">Nam in <lb/>Apologia ſua, ne inanis rerum videretur, tres demonſtrationes nihil pe-<lb/>nitus ad eam pertinentes inferſit: </s> <s xml:id="echoid-s11645" xml:space="preserve">quarum priorem immeritò ſuam pro-<lb/>priam facit, vt ſupra dixi: </s> <s xml:id="echoid-s11646" xml:space="preserve">poſteriorem vero, quam mirum in modum glo-<lb/>riatur ſe clariorem feciſſe, ego & </s> <s xml:id="echoid-s11647" xml:space="preserve">longè breuius, & </s> <s xml:id="echoid-s11648" xml:space="preserve">dilucidius (niſi meorum <pb o="352" file="364" n="364" rhead=""/> me amor fallat) iam pridem demonſtraui, vt mox, Deo adiuuante, ex libel-<lb/>lo meo de dimenſionibus magnitu dinum apparebit. </s> <s xml:id="echoid-s11649" xml:space="preserve">Sed licuerit Peletario <lb/>ſuæ A pologię, ne incomitata prodiret, nouo more comites ac pediſſequas <lb/>adiungere: </s> <s xml:id="echoid-s11650" xml:space="preserve">mihi cur non liceat, quod omnibus ſemper licuit, aliorum ſen-<lb/>tentias totas meis ſcriptis intexere? </s> <s xml:id="echoid-s11651" xml:space="preserve">Autigitur omnes reprehendat, atque <lb/>in primis Petrum Nonium laudatorem ſuum, qui idem fecit in refellen-<lb/>dis paralogiſmis Orontij, aut ſine cauſa id ſe mihi vitio dediſſe fateatur. <lb/></s> <s xml:id="echoid-s11652" xml:space="preserve">Quòd ſi, poſtquam tam fideliter eius verba propoſui, Peletarius crimi-<lb/>natur, me eius ſententiam perperam eſſe interpretatum, quid facturus fuiſ-<lb/>ſet, ſi alienis verbis eius opinionem in medium adduxiſſem? </s> <s xml:id="echoid-s11653" xml:space="preserve">Equidem fa-<lb/>cile ſibi perſuadebit quis, nullum eum verbum relicturum fuiſſe, quod <lb/>non reprehendiſſet.</s> <s xml:id="echoid-s11654" xml:space="preserve"/> </p> <div xml:id="echoid-div912" type="float" level="2" n="6"> <note position="left" xlink:label="note-362-01" xlink:href="note-362-01a" xml:space="preserve">Paralogif-<lb/>mus Pele-<lb/>tarij inſi-<lb/>gnis.</note> </div> <p> <s xml:id="echoid-s11655" xml:space="preserve">QVARTO vt leuiora hæc omittat, illud putat palmare, quòd me <lb/>laborare oſtendit, vt probem, angulos cõtactus alios alijs eſſe inæquales: <lb/></s> <s xml:id="echoid-s11656" xml:space="preserve">propterea quòd ſcripſi, æqualitatem angulorum eiuſdem generis require-<lb/>re eandem inclinationem linearum, ita vt lineæ vnius conueniant omni-<lb/>no lineis alterius, ſi alter alteri ſuperponatur, iuxta octauum pronuncia-<lb/>tum. </s> <s xml:id="echoid-s11657" xml:space="preserve">Qua in re dupliciter me peccare ait. </s> <s xml:id="echoid-s11658" xml:space="preserve">Primum quod dicam, ad æqua-<lb/>litatem angulorum eiuſdem generis requiri eandem linearum inclinatio-<lb/>nem; </s> <s xml:id="echoid-s11659" xml:space="preserve">cum tamen angulus rectilineus oſtenſus ſit a me æqualis circuilineo, <lb/>atque adeo eiuſdem generis cum illo, licet non ſit in vtroq; </s> <s xml:id="echoid-s11660" xml:space="preserve">eadem linea-<lb/>rum inclinatio. </s> <s xml:id="echoid-s11661" xml:space="preserve">Deinde quod putem angùlos contactus ideo inter ſe in-<lb/>æquales eſſe, quòd ſibi mutuo non congruant. </s> <s xml:id="echoid-s11662" xml:space="preserve">Equidem ſi quid in eo a me <lb/>peccatum eſſe intelligerem, & </s> <s xml:id="echoid-s11663" xml:space="preserve">peccatum (quod eſt ingenuo, & </s> <s xml:id="echoid-s11664" xml:space="preserve">liberali-<lb/>ter educato homine dignum) agnoſcerem, & </s> <s xml:id="echoid-s11665" xml:space="preserve">Peletario correctori, & </s> <s xml:id="echoid-s11666" xml:space="preserve"><lb/>emendatori meo (quo cunque id animo fecerit) gratias agerem. </s> <s xml:id="echoid-s11667" xml:space="preserve">Nunc ve-<lb/>ro, cum, totare etiam atque etiam conſiderata, nihil omnino vitij ineſſe <lb/>videam, ita, quæ obijciuntur, diluam, vt tamen gratiam habeam Peletario, <lb/>qui occaſionem dedit eius loci diligentius explicandi. </s> <s xml:id="echoid-s11668" xml:space="preserve">Ego igitur eo loco <lb/>intellexi angulos eiuſdem generis illos, qui vnam lineam habent rectam, <lb/>& </s> <s xml:id="echoid-s11669" xml:space="preserve">alteram circularem, quales ſunt anguli contactus, & </s> <s xml:id="echoid-s11670" xml:space="preserve">ſemicirculorum, <lb/>de quibus tunc agebamus. </s> <s xml:id="echoid-s11671" xml:space="preserve">Quare cum linea recta vnius congruat lineæ <lb/>rectæ alterius, circularis vero circulari non itẽ, niſi circuli ponantur æqua-<lb/>les, efficitur, angulos illos eſſe inæquales inter ſe, quippe cum alter alte-<lb/>rum excedat. </s> <s xml:id="echoid-s11672" xml:space="preserve">Eadem ratione, ſi dentur duo anguli curuilinei æqualium <lb/>circulorum æquales, neceſſe eſt, lineas vnius lineis alterius congruere, ſi <lb/>alter alteri ſuperponatur. </s> <s xml:id="echoid-s11673" xml:space="preserve">Quòd ſi Peletarius hanc doctrinam oppugnat, <lb/>ſciat, ſe iam bellum mouere non mihi, ſed Proclo, qui lib. </s> <s xml:id="echoid-s11674" xml:space="preserve">3. </s> <s xml:id="echoid-s11675" xml:space="preserve">in primum <lb/>Eucl. </s> <s xml:id="echoid-s11676" xml:space="preserve">ad propoſ. </s> <s xml:id="echoid-s11677" xml:space="preserve">4. </s> <s xml:id="echoid-s11678" xml:space="preserve">idem prorſus docet, quod ego. </s> <s xml:id="echoid-s11679" xml:space="preserve">Ait enim [_Angulorum_ <lb/>_autem æqualitatem ſumemus iuxta conuenientiam laterum in rectilineis,_ <lb/>_in cæterisque omnibus, qui eiuſdem ſunt speciei, vt in Lunaribus, in Syſtroi-_ <lb/>_dibus, atque in vtrinque conuexis, &</s> <s xml:id="echoid-s11680" xml:space="preserve">c._</s> <s xml:id="echoid-s11681" xml:space="preserve">] Et infra. </s> <s xml:id="echoid-s11682" xml:space="preserve">[_Quæ æqualia data ſunt,_ <lb/>_ſibi inuicem congruunt. </s> <s xml:id="echoid-s11683" xml:space="preserve">Hoc autem non in omnibus veru eſt, ſed in ys, quæ_ <lb/>_ſpecie ſimilia ſunt. </s> <s xml:id="echoid-s11684" xml:space="preserve">Specie autem ſimilia hæc dico, vt recta linea rectæ lineæ,_ <pb o="353" file="365" n="365" rhead=""/> _& </s> <s xml:id="echoid-s11685" xml:space="preserve">circunferentia circunferentiæ circuli eiuſdem, & </s> <s xml:id="echoid-s11686" xml:space="preserve">anguli, qui à ſimili-_ <lb/>_bus ſimiliter iacentibus lineis comprehenſi ſunt. </s> <s xml:id="echoid-s11687" xml:space="preserve">Horum autem dico, quòd_ <lb/>_quæ æqualia data fuerint, ſibi inuicem congruũt._</s> <s xml:id="echoid-s11688" xml:space="preserve">] Nonne luce clarius ex his <lb/>colligitur, Proclum illos ſolum angulos contactus concedere æquales, <lb/>quorum rectæ lineæ, & </s> <s xml:id="echoid-s11689" xml:space="preserve">curuæ ſibi mutuò congruunt? </s> <s xml:id="echoid-s11690" xml:space="preserve">Temere igitur Pele-<lb/>tarius mihi obijcit angulum rectilineum & </s> <s xml:id="echoid-s11691" xml:space="preserve">circuilineum, triangulum & </s> <s xml:id="echoid-s11692" xml:space="preserve"><lb/>quadratum, atque alia huiuſmodi, de quibus eo loco ſermo non erat; </s> <s xml:id="echoid-s11693" xml:space="preserve">quip-<lb/>pe quæ non ſint eiuſdem ſpeciei, atque adeo æqualitatem tueantur, etiamſi <lb/>alterum alteri non congruat. </s> <s xml:id="echoid-s11694" xml:space="preserve">Vtiam vereri incipiam, ne Peletarius noſter <lb/>contentionis ſit cupidior, quàm veritatis.</s> <s xml:id="echoid-s11695" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11696" xml:space="preserve">POSTREMO, vt nihil intactum relinquat, me non modo Geo-<lb/>metriæ ignarum vocat, ſed etiam Logices: </s> <s xml:id="echoid-s11697" xml:space="preserve">propterea quòd lib. </s> <s xml:id="echoid-s11698" xml:space="preserve">5. </s> <s xml:id="echoid-s11699" xml:space="preserve">dixi, non <lb/>recte à quibuſdam diuidi Proportionem rationalem in proportionẽ æqua-<lb/>litatis, atque inæqualitatis? </s> <s xml:id="echoid-s11700" xml:space="preserve">quòd multæ proportiones inæqualitatis ſint <lb/>etiam irrationales. </s> <s xml:id="echoid-s11701" xml:space="preserve">Ego vero (etſi non is ſum, qui mihi quicquã vllo in ge-<lb/>nere arrogem) tamen in hiſce ſtudijs, in quibus mediocriter verſatus ſum, <lb/>planè rudem non eſſe, præ me ſemper tuli. </s> <s xml:id="echoid-s11702" xml:space="preserve">Quantulum autem ſit id, quod <lb/>in vtroq; </s> <s xml:id="echoid-s11703" xml:space="preserve">poſſim, cæteri melius, qui vacant amore, & </s> <s xml:id="echoid-s11704" xml:space="preserve">odio, iudicabunt; </s> <s xml:id="echoid-s11705" xml:space="preserve">Pe-<lb/>letario quidem ipſi ita me adhuc reſpõdiſſe arbitror, vt iam minus fortaſſe <lb/>ignarus Geometriæ, ac Dialecticæ videar, quàm putarat. </s> <s xml:id="echoid-s11706" xml:space="preserve">Nunc, vt perſpi-<lb/>ciat, neq; </s> <s xml:id="echoid-s11707" xml:space="preserve">me pertinacem eſſe, neq; </s> <s xml:id="echoid-s11708" xml:space="preserve">illa, quæ exagitat, à Dialecticorũ præ-<lb/>ceptis abhorrere, libẽter ei concedo, diuiſionẽ illam, quam à me reprehen <lb/>ſam criminatur, probã eſſe, ita tamen, ſi in quolibet diuiſionis membro Di-<lb/>uiſum intelligatur: </s> <s xml:id="echoid-s11709" xml:space="preserve">neq; </s> <s xml:id="echoid-s11710" xml:space="preserve">vero hoc vnquã negaui, cum alibi ſimiles diuiſio-<lb/>nes vſurpem. </s> <s xml:id="echoid-s11711" xml:space="preserve">Solũ id eo loci contẽdi, rectius meo iudicio, diuidi Propor-<lb/>tionem in vniuerſum duplici diuiſione, priori quidem in proportionem ra <lb/>tionalem, & </s> <s xml:id="echoid-s11712" xml:space="preserve">irrationalem; </s> <s xml:id="echoid-s11713" xml:space="preserve">poſteriori vero in proportionẽ æqualitatis, atq; <lb/></s> <s xml:id="echoid-s11714" xml:space="preserve">inæqualitatis, (quod veriſsimum eſſe, neminem negaturum cenſeo, qui <lb/>rem diligentius expenderit) cum tam priora duo membra diuidẽtia, quam <lb/>poſteriora totum Diuiſum (vt Logici loquuntur) exhauriant: </s> <s xml:id="echoid-s11715" xml:space="preserve">quam ſi <lb/>prius membrum prioris diuiſionis, hoc eſt, proportio rationalis, ſecetur in <lb/>proportionem æqualitatis, & </s> <s xml:id="echoid-s11716" xml:space="preserve">inæqualitatis, cum hæc membra diuidentia <lb/>latius pateant, quam Diuiſum, niſi in illis Diuiſum intelligatur. </s> <s xml:id="echoid-s11717" xml:space="preserve">Atque eò <lb/>magis duplex illa diuiſio mihi probatur, quòd non deſint, qui primum par-<lb/>tiantur Proportionem in proportioneẽ æqualitatis, & </s> <s xml:id="echoid-s11718" xml:space="preserve">inæqualitatis; </s> <s xml:id="echoid-s11719" xml:space="preserve">poſte-<lb/>riorem deinde hanc in proportionem rationalem, & </s> <s xml:id="echoid-s11720" xml:space="preserve">irrationalem: </s> <s xml:id="echoid-s11721" xml:space="preserve">contra-<lb/>rio ſcilicet modo, quàm priores. </s> <s xml:id="echoid-s11722" xml:space="preserve">Vt igitur hanc controuerſiam dirimerem, <lb/>ac dubitationem, vtri rectius faciant, priores ne an poſteriores, tollerem, <lb/>ſt atui duabus diuiſionibus ſecandam eſſe Proportionem, quarum vtraque <lb/>abſolutiſsima eſt, ac perfectiſsima. </s> <s xml:id="echoid-s11723" xml:space="preserve">Non aliter arbitror, omnes magis eſſe <lb/>probaturos, ſi corpus duplici diuiſione ſecetur, primum quidem in viuens, <lb/>& </s> <s xml:id="echoid-s11724" xml:space="preserve">non viuens; </s> <s xml:id="echoid-s11725" xml:space="preserve">deinde vero in album, nigrum, ac mixto colore affectum: </s> <s xml:id="echoid-s11726" xml:space="preserve"><lb/>quam ſi corpus viuens diuidatur in album, nigrum, ac mixto colore affe-<lb/>ctum; </s> <s xml:id="echoid-s11727" xml:space="preserve">ob cauſam iam dictam: </s> <s xml:id="echoid-s11728" xml:space="preserve">licet hęc ſubdiuiſio bona ſit, ſi Diuiſum ſem- <pb o="354" file="366" n="366" rhead=""/> per intelligatur. </s> <s xml:id="echoid-s11729" xml:space="preserve">Huiuſmodi diuiſiones ſexcentas adducere poſſem: </s> <s xml:id="echoid-s11730" xml:space="preserve">ſed ſa-<lb/>tis eſt, me prudenti lectori inſtitutum meum in diuiſione Proportionis ex-<lb/>poſuiſſe, & </s> <s xml:id="echoid-s11731" xml:space="preserve">cur duplicem illam diuiſionem ſubdiuiſioni aliorum prætule-<lb/>rim. </s> <s xml:id="echoid-s11732" xml:space="preserve">Quòd ſi tam acres, & </s> <s xml:id="echoid-s11733" xml:space="preserve">ſeueri iudices ſingulorum verborum aut impro-<lb/>prietatum, quæ per incogitantiam interdum excidunt, eſſe velimus, næ <lb/>ſcriptorum nullus aliquo vitio carebit, neque ipſe quidem Peletarius, vt <lb/>partim ex ijs, quæ dicta ſunt, conſtat, partim etiam ex alijs eius demonſtra-<lb/>tionibus apparere poteſt: </s> <s xml:id="echoid-s11734" xml:space="preserve">quas ſi liberet ad certam illam Dialecticorum <lb/>normam exquirere, profecto reprehendendi materia non deeſſet. </s> <s xml:id="echoid-s11735" xml:space="preserve">Verum <lb/>non eſt hoc noſtri conſilij, refellendi ſtudio vitia aliena ſcrutari, ſed vbi ſe-<lb/>ſe occaſio obtulerit, meam (qualiſcunque eſt) de aliorum ſententijs ſenten-<lb/>tiam exponere: </s> <s xml:id="echoid-s11736" xml:space="preserve">Solum ab eo peto, (quoniam ſe tam acutum Dialecticum <lb/>iactat, vt alios contemnere videatur; </s> <s xml:id="echoid-s11737" xml:space="preserve">quanquàm ex ſuperiore ſyllogiſmo, <lb/>quem in me conuertit, liquido conſtat, quam ſit Dialecticæ peritus) ex qua <lb/>Logica hanc argumentationem hauſerit; </s> <s xml:id="echoid-s11738" xml:space="preserve">Omnes anguli contactus ſunt mi-<lb/> <anchor type="note" xlink:label="note-366-01a" xlink:href="note-366-01"/> nores quolibet angulo acuto rectilineo: </s> <s xml:id="echoid-s11739" xml:space="preserve">ergo omnes inter ſe ſunt æquales. <lb/></s> <s xml:id="echoid-s11740" xml:space="preserve">Itemque hanc; </s> <s xml:id="echoid-s11741" xml:space="preserve">Anguli ſemicirculorum, quò a maioribus circulis fiunt, eò <lb/>ſunt maiores: </s> <s xml:id="echoid-s11742" xml:space="preserve">igitur tandem ad aliquem perueniemus, qui recto rectilineo <lb/>maior ſit; </s> <s xml:id="echoid-s11743" xml:space="preserve">in qua quidem ad Cardanum ſcribit, nullum eſſe paralogiſmum. </s> <s xml:id="echoid-s11744" xml:space="preserve"><lb/>Ego ſane vehementer miror, qua ratione in tam apertas hallucinationes, <lb/>& </s> <s xml:id="echoid-s11745" xml:space="preserve">viro Geometra omnino indignas, incidere potuerit. </s> <s xml:id="echoid-s11746" xml:space="preserve">Sed argumentatio-<lb/>nes eiuſmodi ſatis ſuperque in ſcholio propoſ. </s> <s xml:id="echoid-s11747" xml:space="preserve">16. </s> <s xml:id="echoid-s11748" xml:space="preserve">lib. </s> <s xml:id="echoid-s11749" xml:space="preserve">3. </s> <s xml:id="echoid-s11750" xml:space="preserve">a me ſunt confu-<lb/>tatæ, adductis contra ipſas euidentiſsimis inſtantijs. </s> <s xml:id="echoid-s11751" xml:space="preserve">Deinde quòd me per-<lb/>ſtringit, quaſi parũ intellexerim, quæ ſit proportio rationalis, & </s> <s xml:id="echoid-s11752" xml:space="preserve">quæ irra-<lb/>tionalis, non multum laboro. </s> <s xml:id="echoid-s11753" xml:space="preserve">Conſtat enim eum ſtudio mihi detrahendi <lb/>id dixiſſe; </s> <s xml:id="echoid-s11754" xml:space="preserve">cum has proportiones vbique ex ſententia grauiſsimorum ſcri-<lb/>ptorum definierim: </s> <s xml:id="echoid-s11755" xml:space="preserve">neque vero ipſe, vllum peccatũ a me ea in re eſſe com-<lb/>miſſum, poterit oſtendere. </s> <s xml:id="echoid-s11756" xml:space="preserve">Certe commentarius meus in lib. </s> <s xml:id="echoid-s11757" xml:space="preserve">10. </s> <s xml:id="echoid-s11758" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s11759" xml:space="preserve">abun-<lb/>de declarat, numillas intellexerim, nec ne. </s> <s xml:id="echoid-s11760" xml:space="preserve">Denique quòd criminatur, me <lb/>in deſinitionibus lib. </s> <s xml:id="echoid-s11761" xml:space="preserve">5. </s> <s xml:id="echoid-s11762" xml:space="preserve">proportionis nomen confundere cum Rationis no-<lb/>mine, nullo modo verum eſt. </s> <s xml:id="echoid-s11763" xml:space="preserve">Perſpicuis enim verbis docui in defin. </s> <s xml:id="echoid-s11764" xml:space="preserve">4. </s> <s xml:id="echoid-s11765" xml:space="preserve">lib. </s> <s xml:id="echoid-s11766" xml:space="preserve"><lb/>5. </s> <s xml:id="echoid-s11767" xml:space="preserve">me in commentario comparationem duarum quantitatum Proportio-<lb/>nem cum pluribus Geometris appellaturum, habitudinem autem propor-<lb/>tionum, Proportionalitatem; </s> <s xml:id="echoid-s11768" xml:space="preserve">licet in textu cum interprete illam dicam Ra <lb/>tionem, hanc vero, Proportionem. </s> <s xml:id="echoid-s11769" xml:space="preserve">Neque enim quicquam in textu Eucli-<lb/>dis volui immutare. </s> <s xml:id="echoid-s11770" xml:space="preserve">Itaque nulla in meis verbis poteſt eſſe ambiguitas.</s> <s xml:id="echoid-s11771" xml:space="preserve"/> </p> <div xml:id="echoid-div913" type="float" level="2" n="7"> <note position="left" xlink:label="note-366-01" xlink:href="note-366-01a" xml:space="preserve">Argumẽta <lb/>tiones Pele <lb/>tarij ſophj <lb/>ſticæ.</note> </div> <p> <s xml:id="echoid-s11772" xml:space="preserve">EX HIS, quæ diximus, ſatis (vt opinior) apparet, doctiſsimos illos vi <lb/>ros, de quibus initio memini, non ſine cauſa Apologiam Peletarij inanem, <lb/>ac reſponſionis indignam iudicaſſe. </s> <s xml:id="echoid-s11773" xml:space="preserve">Ego tamen, ne contemnere hominem <lb/>viderer, quem ſemper laudandum eſſe duxi, occaſione inuitatus reſponden <lb/>dum amice putaui. </s> <s xml:id="echoid-s11774" xml:space="preserve">Exiſtimet ille, angulum contactus quantitatem non eſ-<lb/>ſe, atque adeo angulum ſemicirculi recto rectilineo eſſe æqualem, ego cer-<lb/>te contrariam ſententiam tuebor, donec aliud mihi demonſtratum ab ali-<lb/>quo fuerit; </s> <s xml:id="echoid-s11775" xml:space="preserve">rationes enim Peletarij fallaces ſunt, nihilque continent in ſe <pb o="355" file="367" n="367" rhead=""/> Probabilitatis, vt in ſcholio propoſ. </s> <s xml:id="echoid-s11776" xml:space="preserve">16. </s> <s xml:id="echoid-s11777" xml:space="preserve">lib. </s> <s xml:id="echoid-s11778" xml:space="preserve">3. </s> <s xml:id="echoid-s11779" xml:space="preserve">oſtendi, vbi omnes diſſolui: <lb/></s> <s xml:id="echoid-s11780" xml:space="preserve">neque meis ipſe ſolutionibus vel vnum verbum (exceptis ijs, quæ ſupra <lb/>ex lib. </s> <s xml:id="echoid-s11781" xml:space="preserve">10. </s> <s xml:id="echoid-s11782" xml:space="preserve">adduxi) reſpondit; </s> <s xml:id="echoid-s11783" xml:space="preserve">quod tamen maxime ad Apologiam pertine-<lb/>bat: </s> <s xml:id="echoid-s11784" xml:space="preserve">Vt non ſine cauſa permulti exiſtimauermt, eum non veritatis ſtudio <lb/>eam Apologiã ſcripſiſſe, ſed ne veritati ceſsiſſe videretur. </s> <s xml:id="echoid-s11785" xml:space="preserve">Nec vero quiſ-<lb/>quam putet, me vnum exiſtimare, angulum contactus vere eſſe angulum, <lb/>& </s> <s xml:id="echoid-s11786" xml:space="preserve">angulum ſemicirculi recto rectilineo minorem. </s> <s xml:id="echoid-s11787" xml:space="preserve">Multos enim eius rei <lb/> <anchor type="note" xlink:label="note-367-01a" xlink:href="note-367-01"/> auctores, eoſque grauiſsimos laudare poſſum, Theonem, Campanum, Pe-<lb/>trum Nonium, & </s> <s xml:id="echoid-s11788" xml:space="preserve">(vt Nonius refert) Archimedem, atque Iordanum: </s> <s xml:id="echoid-s11789" xml:space="preserve">quin <lb/>etiam (quod plurimi facio) Euclidem ipſum, eiuſque commentatorem ce-<lb/>leberrimum Proclum; </s> <s xml:id="echoid-s11790" xml:space="preserve">vt taceam ex Gallis præſtantiſsimos, atque eru-<lb/>ditiſsimos viros non paucos, è quorum numero in primis eſt Franciſcus <lb/>Candalla ex illuſtriſsima Fluſſatum familia oriundus, qui inſigne volumen <lb/>in elementa Geometrica Euclidis edidit, vbi ad propoſ. </s> <s xml:id="echoid-s11791" xml:space="preserve">16. </s> <s xml:id="echoid-s11792" xml:space="preserve">lib. </s> <s xml:id="echoid-s11793" xml:space="preserve">3. </s> <s xml:id="echoid-s11794" xml:space="preserve">apertiſ-<lb/>ſime docet, angulos contingentiæ verè eſſe angulos, ex definitione anguli <lb/>plani, aliosq; </s> <s xml:id="echoid-s11795" xml:space="preserve">alijs eſſe maiores, æquales, ac minores: </s> <s xml:id="echoid-s11796" xml:space="preserve">Eos autem, qui aliter <lb/>ſentiũt, (Peletariũ proculdubio intelligit. </s> <s xml:id="echoid-s11797" xml:space="preserve">Præter eum enim ad hunc diem <lb/>nemo hac de re ſcripſit) abſurde multa ex falſis ſuppoſitis concludere af-<lb/>firmat. </s> <s xml:id="echoid-s11798" xml:space="preserve">Huc accedat etiam Henricus Monantholius Mathematicarum ar-<lb/>tium profeſſor regius qui, cum Apologiam Peletarij in me conſcriptam <lb/>vidiſſet, opuſculum eruditum aduerſus Peletarium de angulo contactus <lb/>edidit. </s> <s xml:id="echoid-s11799" xml:space="preserve">Vt autem ſtudioſus lector videat, quid in hoc negotio ſentiat Pro-<lb/>clus, afferam in mediũ pauca quædam ex eius commentarijs in lib. </s> <s xml:id="echoid-s11800" xml:space="preserve">1. </s> <s xml:id="echoid-s11801" xml:space="preserve">Eucl. <lb/></s> <s xml:id="echoid-s11802" xml:space="preserve">quæ obiter notaui, & </s> <s xml:id="echoid-s11803" xml:space="preserve">ex quibus liquido conſtabit, eius ſententiam eſſe Pe-<lb/>letarij commento prorſus contrariam. </s> <s xml:id="echoid-s11804" xml:space="preserve">Primum itaque ita ſcribit lib. </s> <s xml:id="echoid-s11805" xml:space="preserve">2. </s> <s xml:id="echoid-s11806" xml:space="preserve">in <lb/> <anchor type="note" xlink:label="note-367-02a" xlink:href="note-367-02"/> primum Eucl. </s> <s xml:id="echoid-s11807" xml:space="preserve">ad definitionem anguli plani. </s> <s xml:id="echoid-s11808" xml:space="preserve">[_Duænamque circunferentiæ_ <lb/>_ſe inuicem ſecando, vel ſeſe contingendo, angulos efficiunt. </s> <s xml:id="echoid-s11809" xml:space="preserve">Quinetiam àre-_ <lb/>_cta linea, & </s> <s xml:id="echoid-s11810" xml:space="preserve">conuexa circunferentia angulus continetur, vt Cornicularis._</s> <s xml:id="echoid-s11811" xml:space="preserve">] <lb/>Intelligit autem nomine Cornicularis anguli angulum contactus mixtum. <lb/></s> <s xml:id="echoid-s11812" xml:space="preserve">Paulo enim ante dixerat, angulum Cornicularem eſſe omni rectilineo mi-<lb/>norem: </s> <s xml:id="echoid-s11813" xml:space="preserve">quod ſolius anguli contactus proprium eſt. </s> <s xml:id="echoid-s11814" xml:space="preserve">Deinde in eodem lib. </s> <s xml:id="echoid-s11815" xml:space="preserve"><lb/>ad definitionem anguli recti, obtuſi, & </s> <s xml:id="echoid-s11816" xml:space="preserve">acuti ita habet. </s> <s xml:id="echoid-s11817" xml:space="preserve">[_Cornicularis nam-_ <lb/>_que angulus omni recto eſt minor, quandoquidem & </s> <s xml:id="echoid-s11818" xml:space="preserve">acuto, nec tamen acu-_ <lb/>_tus eſt: </s> <s xml:id="echoid-s11819" xml:space="preserve">Semicir cularis itidem quocunque recto est minor, acutus tamen non_ <lb/>_eſt._</s> <s xml:id="echoid-s11820" xml:space="preserve">] Quid clarius, quam Proclum hic aſſerere, angulum ſemicirculi mino-<lb/>norem eſſe recto? </s> <s xml:id="echoid-s11821" xml:space="preserve">Rurſus lib. </s> <s xml:id="echoid-s11822" xml:space="preserve">3. </s> <s xml:id="echoid-s11823" xml:space="preserve">ad propoſ. </s> <s xml:id="echoid-s11824" xml:space="preserve">4. </s> <s xml:id="echoid-s11825" xml:space="preserve">lib. </s> <s xml:id="echoid-s11826" xml:space="preserve">1. </s> <s xml:id="echoid-s11827" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s11828" xml:space="preserve">ita ſcribit. </s> <s xml:id="echoid-s11829" xml:space="preserve">[_Addi-_ <lb/>_ſcemus enim, quòd angulus Cornicularis acuto ſemper inæqualis eſt, & </s> <s xml:id="echoid-s11830" xml:space="preserve">nun-_ <lb/>_quàm æqualis: </s> <s xml:id="echoid-s11831" xml:space="preserve">Et ſemicir cularis ſimiliter, tranſitusque à maiori ad minus_ <lb/>_non omnino per æquale fit._</s> <s xml:id="echoid-s11832" xml:space="preserve">] En quam aperte docet, angulum ſemicirculi <lb/>æqualem eſſe non poſſe angulo rectilineo, tranſitumque propterea fieri a <lb/>maiori ad minus non per æquale: </s> <s xml:id="echoid-s11833" xml:space="preserve">quorum vtrumque Peletarius negat, au-<lb/>detque poſterius appellare paralogiſmũ. </s> <s xml:id="echoid-s11834" xml:space="preserve">Denique in eodem lib. </s> <s xml:id="echoid-s11835" xml:space="preserve">3. </s> <s xml:id="echoid-s11836" xml:space="preserve">ad pro-<lb/>poſ. </s> <s xml:id="echoid-s11837" xml:space="preserve">23. </s> <s xml:id="echoid-s11838" xml:space="preserve">hæc verba habentur. </s> <s xml:id="echoid-s11839" xml:space="preserve">[_Cum autem nullus angulus mixtus rectilineo_ <lb/>_æqualis eſſe poſsit, &</s> <s xml:id="echoid-s11840" xml:space="preserve">c._</s> <s xml:id="echoid-s11841" xml:space="preserve">] Et Peletarius tamen non dubitat angulum ſemi- <pb o="356" file="368" n="368" rhead=""/> circuli, qui mixtus eſt, angulo recto rectilineo facere æqualem, Cõtra Pro-<lb/>cli ſententiam, Ex his liquere arbitror, vt de cæteris taceam, idem ſentire <lb/>Proclum de angulo contactus, & </s> <s xml:id="echoid-s11842" xml:space="preserve">ſemicirculi, quod ego contra Peletarium <lb/>ſcripſi: </s> <s xml:id="echoid-s11843" xml:space="preserve">quis autem neget, maiorem eſſe auctoritatem, meliora argumenta <lb/>Procli, quam Peletarij?</s> <s xml:id="echoid-s11844" xml:space="preserve"/> </p> <div xml:id="echoid-div914" type="float" level="2" n="8"> <note position="right" xlink:label="note-367-01" xlink:href="note-367-01a" xml:space="preserve">Varij au-<lb/>ctores, qui <lb/>ſenſerunt, <lb/>angulũ cõ-<lb/>tactus vere <lb/>eſſe angu-<lb/>lum, & an-<lb/>gulũ ſemi-<lb/>circuli an-<lb/>gulo recto <lb/>rectilineo <lb/>minorem.</note> <note position="right" xlink:label="note-367-02" xlink:href="note-367-02a" xml:space="preserve">Poeli ſen-<lb/>tentia de <lb/>angulo eõ-<lb/>tactus, & ſe <lb/>micirculi.</note> </div> <p> <s xml:id="echoid-s11845" xml:space="preserve">OBITER quoque hoc loco monendum lectorem cenſeo, id, quod <lb/> <anchor type="note" xlink:label="note-368-01a" xlink:href="note-368-01"/> de angulo contactus, qui fit in circulis, ex ſententia Euclidis, & </s> <s xml:id="echoid-s11846" xml:space="preserve">Procli do-<lb/>cui, verum etiam eſſe de angulo cõtactus, qui in conicis ſectionibus effici-<lb/>tur, nimirum in Parabola, Hyperbola, & </s> <s xml:id="echoid-s11847" xml:space="preserve">Ellipſi. </s> <s xml:id="echoid-s11848" xml:space="preserve">Vt enim Apollonius Per-<lb/>gæus demonſtrat lib. </s> <s xml:id="echoid-s11849" xml:space="preserve">1. </s> <s xml:id="echoid-s11850" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s11851" xml:space="preserve">32. </s> <s xml:id="echoid-s11852" xml:space="preserve">in locum, qui inter coni ſectionem, & </s> <s xml:id="echoid-s11853" xml:space="preserve"><lb/>rectam lineam tangentem interijcitur, altera recta linea non cadit; </s> <s xml:id="echoid-s11854" xml:space="preserve">atque <lb/>adeo angulus ille contactus minor etiam eſt omni acuto rectilineo, & </s> <s xml:id="echoid-s11855" xml:space="preserve">re-<lb/>liquus angulus ex recto (ſi nimirum ex puncto contactus ad lineam tan-<lb/>gentem excitetur perpendicularis) omni acuto rectilineo maior. </s> <s xml:id="echoid-s11856" xml:space="preserve">Si igitur, <lb/>vt opinatur Peletarius, angulus contactus quantitas non eſt, (eadem enim <lb/>hic eſt ratio, quæ in circulo) erunt omnes anguli contactus inter ſe æqua-<lb/>les, hoc eſt, vt ipſevult, non inæquales, & </s> <s xml:id="echoid-s11857" xml:space="preserve">reliquorum angulorum ſingu-<lb/>li recto rectilineo æquales. </s> <s xml:id="echoid-s11858" xml:space="preserve">Vbi ſanè maior abſurditas apparet, quo ad ſen <lb/>ſum, in Ellipſi, quæ perexiguam habeat latitudinem, & </s> <s xml:id="echoid-s11859" xml:space="preserve">in Hypeibola, quæ <lb/>ferè linea recta eſſe videatur. </s> <s xml:id="echoid-s11860" xml:space="preserve">Valde enim inæquales cernuntur anguli ad <lb/>verticem Ellipſis, & </s> <s xml:id="echoid-s11861" xml:space="preserve">Hyperbolæ conſtituti; </s> <s xml:id="echoid-s11862" xml:space="preserve">vt incredibile omnino ſit, ni-<lb/>ſi firma ratione demonſtretur, angulos illos contactus ad vertices ſectio-<lb/>num conſtitutos inter ſe, & </s> <s xml:id="echoid-s11863" xml:space="preserve">reliquos ex rectis inter ſe quoque eſſe æqua-<lb/>les; </s> <s xml:id="echoid-s11864" xml:space="preserve">propterea quod in ea Ellipſi linea tangens magis recedere perſpicia-<lb/>tur a circunferentia Ellipſis, quam in circulo; </s> <s xml:id="echoid-s11865" xml:space="preserve">in illa vero Hyperbola mi-<lb/>nus. </s> <s xml:id="echoid-s11866" xml:space="preserve">Sed hæc alio tempore examinanda relinquamus: </s> <s xml:id="echoid-s11867" xml:space="preserve">nunc ad interruptam <lb/>expoſitionem definitionum reuertamur.</s> <s xml:id="echoid-s11868" xml:space="preserve"/> </p> <div xml:id="echoid-div915" type="float" level="2" n="9"> <note position="left" xlink:label="note-368-01" xlink:href="note-368-01a" xml:space="preserve">Idem dicẽ-<lb/>dum eſt de <lb/>angulo cõ-<lb/>tactus, qui <lb/>in conicis <lb/>fectionibꝰ <lb/>fit, quod de <lb/>illo Eucli-<lb/>@is dicitur.</note> </div> </div> <div xml:id="echoid-div917" type="section" level="1" n="477"> <head xml:id="echoid-head511" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s11869" xml:space="preserve">ANGVLVS ſphæricus rectus eſt, quem in <lb/> <anchor type="note" xlink:label="note-368-02a" xlink:href="note-368-02"/> ſphærę ſuperficie duo arcus circulorum maximo-<lb/>rum ſeſe ad angulos rectos ſecantium, id eſt, quo-<lb/>rum alter ad alterum rectus eſt, continent.</s> <s xml:id="echoid-s11870" xml:space="preserve"/> </p> <div xml:id="echoid-div917" type="float" level="2" n="1"> <note position="left" xlink:label="note-368-02" xlink:href="note-368-02a" xml:space="preserve">Angulus <lb/>ſphæricus <lb/>rectꝰ quid.</note> </div> </div> <div xml:id="echoid-div919" type="section" level="1" n="478"> <head xml:id="echoid-head512" xml:space="preserve">III.</head> <note position="left" xml:space="preserve">Anguius <lb/>ſphæricus <lb/>obcu ſus <lb/>quid.</note> <p> <s xml:id="echoid-s11871" xml:space="preserve">ANGVLVS ſphæricus obtuſus eſt, qui re-<lb/>cto maior eſt.</s> <s xml:id="echoid-s11872" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div920" type="section" level="1" n="479"> <head xml:id="echoid-head513" xml:space="preserve">IIII.</head> <note position="left" xml:space="preserve">Angulus <lb/>ſphæricus <lb/>acutus qd.</note> <p> <s xml:id="echoid-s11873" xml:space="preserve">ACVTVS verò, qui minor eſt recto.</s> <s xml:id="echoid-s11874" xml:space="preserve"/> </p> <pb o="357" file="369" n="369" rhead=""/> <p style="it"> <s xml:id="echoid-s11875" xml:space="preserve">_CONSTITVITVR_ angulus ſphæricus rectus ad punctum datum in dato ar-<lb/> <anchor type="note" xlink:label="note-369-01a" xlink:href="note-369-01"/> tu circuli maximi ſuperficie in ſphæræ, ſi per illud punctũ & </s> <s xml:id="echoid-s11876" xml:space="preserve">per polum dati arcus (qui <lb/>per propoſ 21. </s> <s xml:id="echoid-s11877" xml:space="preserve">lib 1. </s> <s xml:id="echoid-s11878" xml:space="preserve">Theod. </s> <s xml:id="echoid-s11879" xml:space="preserve">inuenitur) circulus maximus deſcribatur. </s> <s xml:id="echoid-s11880" xml:space="preserve">Huius enim cir-<lb/>culi circunferentia cum arcu dato angulum rectum conſtituet; </s> <s xml:id="echoid-s11881" xml:space="preserve">cum circulus hic ad <lb/>circulum illius arcus ſit rectus. </s> <s xml:id="echoid-s11882" xml:space="preserve">Si vero per datum punctum deſcribatur arcus circuli <lb/> <anchor type="note" xlink:label="note-369-02a" xlink:href="note-369-02"/> maximi non per polos dati arcus, conſtituet circunferentiæ huius circuli cum date ar <lb/>cu angulos inæquales, obtuſum vnum, & </s> <s xml:id="echoid-s11883" xml:space="preserve">alterum acutum.</s> <s xml:id="echoid-s11884" xml:space="preserve"/> </p> <div xml:id="echoid-div920" type="float" level="2" n="1"> <note position="right" xlink:label="note-369-01" xlink:href="note-369-01a" xml:space="preserve">Cõſtructi@ <lb/>anguli ſphę <lb/>ralis recti, <lb/>obtuſi & <lb/>acuti.</note> <note position="right" xlink:label="note-369-02" xlink:href="note-369-02a" xml:space="preserve">15. 1. Theo.</note> </div> </div> <div xml:id="echoid-div922" type="section" level="1" n="480"> <head xml:id="echoid-head514" xml:space="preserve">V.</head> <note position="right" xml:space="preserve">Triangulũ <lb/>ſphęricum <lb/>quid.</note> <p> <s xml:id="echoid-s11885" xml:space="preserve">TRIANGVLVM ſphæricum eſt, quod tri <lb/> <anchor type="note" xlink:label="note-369-04a" xlink:href="note-369-04"/> bus arcubus circulorum maximorum in ſphæræ <lb/>ſuperficie continetur.</s> <s xml:id="echoid-s11886" xml:space="preserve"/> </p> <div xml:id="echoid-div922" type="float" level="2" n="1"> <note position="right" xlink:label="note-369-04" xlink:href="note-369-04a" xml:space="preserve">Triangulũ <lb/>ſphęri cum <lb/>diuiduur <lb/>vt rectili-<lb/>neum ab <lb/>Euclide.</note> </div> <p style="it"> <s xml:id="echoid-s11887" xml:space="preserve">_HOC_ autem eſt vel æquilaterum, ſi omnes arcus æquales fuerint; </s> <s xml:id="echoid-s11888" xml:space="preserve">vel Iſoſceles, <lb/>ſi duo arcus tantum fuerint æquales; </s> <s xml:id="echoid-s11889" xml:space="preserve">vel denique Scalenum, ſi omnes arcus inæqua-<lb/> <anchor type="note" xlink:label="note-369-05a" xlink:href="note-369-05"/> les inter ſe fuerint. </s> <s xml:id="echoid-s11890" xml:space="preserve">Itemq́; </s> <s xml:id="echoid-s11891" xml:space="preserve">vel rectangulum, ſi aliquem angulum habuerit rectum; <lb/></s> <s xml:id="echoid-s11892" xml:space="preserve">vel obtuſangulum, in quo angulus aliquis fuerit obtuſus; </s> <s xml:id="echoid-s11893" xml:space="preserve">vel denique acutangulum, <lb/>ſi omnes anguli fuerint acuti: </s> <s xml:id="echoid-s11894" xml:space="preserve">quemadmodum de rectilineo triangulo dixit Euclides. </s> <s xml:id="echoid-s11895" xml:space="preserve"><lb/>_Hoc_ tamen diſcrimen reperitur inter triangulum rectangulum, obtuſangulumque <lb/>rectilineum, & </s> <s xml:id="echoid-s11896" xml:space="preserve">ſphæricum, quòd in rectilineo reliqui duo anguli neceſſario ſint acu-<lb/>ti, propterea quòd duo anguli quomodolibet ſumpti minores sũt duobus rectissin ſphœ <lb/> <anchor type="note" xlink:label="note-369-06a" xlink:href="note-369-06"/> rico autem ſi vnus angulus fuerit rectus, vel obtuſus, poſſunt alij duo etiam eſſe recti, <lb/>vel obtuſi, vel alter ſaltem, vt ex demonſtrationibus ſequentibus perſpicuum fiet.</s> <s xml:id="echoid-s11897" xml:space="preserve"/> </p> <div xml:id="echoid-div923" type="float" level="2" n="2"> <note position="right" xlink:label="note-369-05" xlink:href="note-369-05a" xml:space="preserve">Diſcrimen <lb/>inter trian <lb/>gulũ rectã-<lb/>gulum, ob-<lb/>tuſangulũ-<lb/>que rectili-<lb/>neum, ac <lb/>ſphæticũ.</note> <note position="right" xlink:label="note-369-06" xlink:href="note-369-06a" xml:space="preserve">17. primi.</note> </div> </div> <div xml:id="echoid-div925" type="section" level="1" n="481"> <head xml:id="echoid-head515" xml:space="preserve">VI.</head> <note position="right" xml:space="preserve">Arcꝰ angu. <lb/>li ſphæri@ <lb/>quid.</note> <p> <s xml:id="echoid-s11898" xml:space="preserve">ARCVS anguli ſphærici eſt arcus circuli ma <lb/>ximi, cuius polus eſt in ipſo angulo, inter duos ar-<lb/>cus angulum ſphæricum comprehendentes inter-<lb/>ceptus.</s> <s xml:id="echoid-s11899" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s11900" xml:space="preserve">_QVIA_ vero polus circuli maximi quadrante maximi circuli ab eo abeſt, fit, vt <lb/> <anchor type="note" xlink:label="note-369-08a" xlink:href="note-369-08"/> vterque arcuum angulum comprehendentium inter angulum, & </s> <s xml:id="echoid-s11901" xml:space="preserve">arcum anguli poſi-<lb/> <anchor type="note" xlink:label="note-369-09a" xlink:href="note-369-09"/> torum ſu quadrans. </s> <s xml:id="echoid-s11902" xml:space="preserve">Quare ſi angulus ſuerit rectus, arcus anguli erit quadrans; </s> <s xml:id="echoid-s11903" xml:space="preserve">ſi <lb/>acutus, quadrante minor; </s> <s xml:id="echoid-s11904" xml:space="preserve">ſi denique obtuſus, maior quadrante: </s> <s xml:id="echoid-s11905" xml:space="preserve">& </s> <s xml:id="echoid-s11906" xml:space="preserve">contra. </s> <s xml:id="echoid-s11907" xml:space="preserve">Vt propoſ. <lb/></s> <s xml:id="echoid-s11908" xml:space="preserve">26, demonſtrabimus.</s> <s xml:id="echoid-s11909" xml:space="preserve"/> </p> <div xml:id="echoid-div925" type="float" level="2" n="1"> <note position="right" xlink:label="note-369-08" xlink:href="note-369-08a" xml:space="preserve">Coroll. 16.</note> <note position="right" xlink:label="note-369-09" xlink:href="note-369-09a" xml:space="preserve">1. Theod.</note> </div> </div> <div xml:id="echoid-div927" type="section" level="1" n="482"> <head xml:id="echoid-head516" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s11910" xml:space="preserve">COMPLEMENTVM arcus eſt exceſſus, <lb/> <anchor type="note" xlink:label="note-369-10a" xlink:href="note-369-10"/> quo quadrans eum ſuperat, ſi arcus minor eſt qua- <pb o="358" file="370" n="370" rhead=""/> drante, vel ab eo ſuperatur, ſi eſt quadrante maior.</s> <s xml:id="echoid-s11911" xml:space="preserve"/> </p> <div xml:id="echoid-div927" type="float" level="2" n="1"> <note position="right" xlink:label="note-369-10" xlink:href="note-369-10a" xml:space="preserve">Complem@ <lb/>tum arcus <lb/>quid.</note> </div> </div> <div xml:id="echoid-div929" type="section" level="1" n="483"> <head xml:id="echoid-head517" xml:space="preserve">VIII.</head> <p> <s xml:id="echoid-s11912" xml:space="preserve">COMPLEMENTVM anguli ſphærici di <lb/> <anchor type="note" xlink:label="note-370-01a" xlink:href="note-370-01"/> citur exceſſus, quo quadrans arcum ipſius anguli <lb/>ſuperat, vel ab eo ſuperatur.</s> <s xml:id="echoid-s11913" xml:space="preserve"/> </p> <div xml:id="echoid-div929" type="float" level="2" n="1"> <note position="left" xlink:label="note-370-01" xlink:href="note-370-01a" xml:space="preserve">Complem@ <lb/>tũ anguli <lb/>ſphærici <lb/>quid.</note> </div> </div> <div xml:id="echoid-div931" type="section" level="1" n="484"> <head xml:id="echoid-head518" xml:space="preserve">IX.</head> <p> <s xml:id="echoid-s11914" xml:space="preserve">SINVS, Tangens, & </s> <s xml:id="echoid-s11915" xml:space="preserve">Secans alicuius anguli <lb/>ſphærici eſt ſinus, tangens, & </s> <s xml:id="echoid-s11916" xml:space="preserve">ſecans illius arcus, <lb/>qui arcus anguli dicitur.</s> <s xml:id="echoid-s11917" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div932" type="section" level="1" n="485"> <head xml:id="echoid-head519" xml:space="preserve">PROBLEMA I. PROPOSITIO I.</head> <p> <s xml:id="echoid-s11918" xml:space="preserve">DATIS duobus arcubus circulorum ma-<lb/>ximorum in ſuperficie ſphæræ inæquali-<lb/>bus, quorũ neuter ſemicirculo maior ſit, <lb/>de maiore æqualem minori arcum detrahere.</s> <s xml:id="echoid-s11919" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11920" xml:space="preserve">SINT duo arcus circulorum maximorum inæquales AB, CD, quorum <lb/> <anchor type="figure" xlink:label="fig-370-01a" xlink:href="fig-370-01"/> neuter ſemicirculo maior ſit, & </s> <s xml:id="echoid-s11921" xml:space="preserve"><lb/>maior ſit CD; </s> <s xml:id="echoid-s11922" xml:space="preserve">oporteatq́ue ex ma <lb/>iori CD, minori AB, æqualem de-<lb/>trahere. </s> <s xml:id="echoid-s11923" xml:space="preserve">Ducta recta AB, applice-<lb/> <anchor type="note" xlink:label="note-370-02a" xlink:href="note-370-02"/> tur ei æqualis CE, in arcu CD. <lb/></s> <s xml:id="echoid-s11924" xml:space="preserve">Dico arcum ablatum CE, æqua-<lb/>lem eſſe arcui minori AB. </s> <s xml:id="echoid-s11925" xml:space="preserve">Cum <lb/>enim circuli arcuum AB, CD, <lb/>maximi ſint, & </s> <s xml:id="echoid-s11926" xml:space="preserve">propterea æqua-<lb/>les; </s> <s xml:id="echoid-s11927" xml:space="preserve">auferent rectæ æquales AB, <lb/>CE, arcus æquales AB, CE: </s> <s xml:id="echoid-s11928" xml:space="preserve">quòd <lb/> <anchor type="note" xlink:label="note-370-03a" xlink:href="note-370-03"/> vterque arcus ſemicirculo minor ponatur. </s> <s xml:id="echoid-s11929" xml:space="preserve">Datis igitur duobus arcubus cir-<lb/>culorum, &</s> <s xml:id="echoid-s11930" xml:space="preserve">c. </s> <s xml:id="echoid-s11931" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s11932" xml:space="preserve"/> </p> <div xml:id="echoid-div932" type="float" level="2" n="1"> <figure xlink:label="fig-370-01" xlink:href="fig-370-01a"> <image file="370-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/370-01"/> </figure> <note position="left" xlink:label="note-370-02" xlink:href="note-370-02a" xml:space="preserve">1. quarti.</note> <note position="left" xlink:label="note-370-03" xlink:href="note-370-03a" xml:space="preserve">28. tertij.</note> </div> </div> <div xml:id="echoid-div934" type="section" level="1" n="486"> <head xml:id="echoid-head520" xml:space="preserve">THEOR. 1. PROPOS. 2.</head> <p> <s xml:id="echoid-s11933" xml:space="preserve">IN omni triangulo ſphærico, latus quodcun-<lb/>que minus eſt ſemicirculo.</s> <s xml:id="echoid-s11934" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11935" xml:space="preserve">SIT triangulum ſphæricum ABC. </s> <s xml:id="echoid-s11936" xml:space="preserve">Dico quodcunque latus ſemicir-<lb/>culo eſſe minus. </s> <s xml:id="echoid-s11937" xml:space="preserve">Productis enim arcubus BA, BC, donec conueniant in D, vl <pb o="359" file="371" n="371" rhead=""/> tra A, & </s> <s xml:id="echoid-s11938" xml:space="preserve">C, erunt arcus BAD, BCD, ſemicirculi; </s> <s xml:id="echoid-s11939" xml:space="preserve">cum circuli maximi ſe mu-<lb/> <anchor type="note" xlink:label="note-371-01a" xlink:href="note-371-01"/> tuo bifariam ſecent. </s> <s xml:id="echoid-s11940" xml:space="preserve">Quare tam arcus BA, <lb/> <anchor type="figure" xlink:label="fig-371-01a" xlink:href="fig-371-01"/> BC, ſemicirculo minor eſt. </s> <s xml:id="echoid-s11941" xml:space="preserve">Eodem modo, <lb/>productis arcubus AB, AC, oſtendemus ar <lb/>cum AC, ſemicirculo eſſe minorem. </s> <s xml:id="echoid-s11942" xml:space="preserve">Con-<lb/>uenient autem arcus BA, BC, producti vl-<lb/>tra puncta A, & </s> <s xml:id="echoid-s11943" xml:space="preserve">C, propterea quòd ſphæ-<lb/>ricos angulos faciunt cũ arcu AC, ſuntq̀; <lb/></s> <s xml:id="echoid-s11944" xml:space="preserve">omnes tres arcus portiones circulorum ma <lb/>ximorum, qui ſe mutuo ſecant in punctis <lb/>A,B, C, non autem tangunt. </s> <s xml:id="echoid-s11945" xml:space="preserve">Hinc enim <lb/>fit, vt vterque arcus BA, BC, productus arcum AC, productum ſecet in pun-<lb/>ctis A, C, vt ex defin. </s> <s xml:id="echoid-s11946" xml:space="preserve">conſtat; </s> <s xml:id="echoid-s11947" xml:space="preserve">ac proinde inter ſe coeant vltra puncta A, C. </s> <s xml:id="echoid-s11948" xml:space="preserve">In <lb/>omniergo triangulo ſphærico, &</s> <s xml:id="echoid-s11949" xml:space="preserve">c. </s> <s xml:id="echoid-s11950" xml:space="preserve">Quod erat demon ſtrandum.</s> <s xml:id="echoid-s11951" xml:space="preserve"/> </p> <div xml:id="echoid-div934" type="float" level="2" n="1"> <note position="right" xlink:label="note-371-01" xlink:href="note-371-01a" xml:space="preserve">11. 1 Theod.</note> <figure xlink:label="fig-371-01" xlink:href="fig-371-01a"> <image file="371-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/371-01"/> </figure> </div> </div> <div xml:id="echoid-div936" type="section" level="1" n="487"> <head xml:id="echoid-head521" xml:space="preserve">THEOR. 2. PROPOS. 3.</head> <head xml:id="echoid-head522" xml:space="preserve">IN omni triangulo ſphærico, duo latera reli-<lb/>quo ſunt maiora, quomodocunque aſſumpta.</head> <p> <s xml:id="echoid-s11952" xml:space="preserve">SIT triangulum ſphæricum ABC. </s> <s xml:id="echoid-s11953" xml:space="preserve">Dico duo quælibet latera, vt AB, AC, <lb/>maiora eſſe latere BC. </s> <s xml:id="echoid-s11954" xml:space="preserve">Si enim triangulum eſt æquilaterum, manifeſtum eſt <lb/>duo ſimul dupla eſſe reliqui, atque adeo maiora. </s> <s xml:id="echoid-s11955" xml:space="preserve">Quod ſi alterum laterũ AB, <lb/>AC, æquale ſit lateri BC, vel maius, <lb/> <anchor type="figure" xlink:label="fig-371-02a" xlink:href="fig-371-02"/> vel etiã vtrumq; </s> <s xml:id="echoid-s11956" xml:space="preserve">maius, perſpicuum <lb/>quoque eſt, duo latera AB, AC, ma-<lb/>iora eſſe reliquo BC. </s> <s xml:id="echoid-s11957" xml:space="preserve">Si vero vtrum-<lb/>que latus AB, AC, aſſumptum late-<lb/>re tertio BC, minus ſit, demonſtrabi-<lb/>mus, latera AB, AC, ſimul maiora eſ-<lb/>ſe latere BC, hac ratione. </s> <s xml:id="echoid-s11958" xml:space="preserve">Perficia tur <lb/>circulus arcus tertij BC. </s> <s xml:id="echoid-s11959" xml:space="preserve">Deinde ex <lb/>polo B, nempe ex altero extremo ma <lb/>ioris lateris BC, ad interuallũ vtriuſ-<lb/>uis arcuum minorum, nimirum ad in-<lb/>teruallũ arcus BA, in ſuperficie ſphæ <lb/>ræ circulus deſcribatun AD, ſecans ar <lb/>cum BC, qui maior ponitur arcu BA, <lb/>in D, puncto inter B, & </s> <s xml:id="echoid-s11960" xml:space="preserve">C. </s> <s xml:id="echoid-s11961" xml:space="preserve">Et quoniam <lb/>circulus BC, tranſit quoque per reliquum polum circuli AD; </s> <s xml:id="echoid-s11962" xml:space="preserve">ſit alter po-<lb/> <anchor type="note" xlink:label="note-371-02a" xlink:href="note-371-02"/> lus E, qui per ſemicirculũ remotus erit à polo B; </s> <s xml:id="echoid-s11963" xml:space="preserve">ita vt ſemicirculus ſit BCE. <lb/></s> <s xml:id="echoid-s11964" xml:space="preserve"> <anchor type="note" xlink:label="note-371-03a" xlink:href="note-371-03"/> Cum ergo arcus BC, ſemicirculo minor ſit, exiſtet polus E, vltra punctum C: <lb/></s> <s xml:id="echoid-s11965" xml:space="preserve"> <anchor type="note" xlink:label="note-371-04a" xlink:href="note-371-04"/> Eſt autem punctum D, inter B, & </s> <s xml:id="echoid-s11966" xml:space="preserve">C, vt dictum eſt. </s> <s xml:id="echoid-s11967" xml:space="preserve">Punctum igitur C, inter <lb/>puncta D, E, cadet. </s> <s xml:id="echoid-s11968" xml:space="preserve">Quare cum ex puncto C, quod extra peripheriam circuli <lb/>AD, eſt, & </s> <s xml:id="echoid-s11969" xml:space="preserve">præter eiuſdem polum E, ſignatur, ducantur duo arcus maximorum <lb/>circulorum CB, CA, ſemicirculo minores (quòd latera ſint trianguli ſphæ-<lb/> <anchor type="note" xlink:label="note-371-05a" xlink:href="note-371-05"/> rici ABC.) </s> <s xml:id="echoid-s11970" xml:space="preserve">ad peripheriam AD, erit arcus CD, per polum B, tranſiens, mi-<lb/> <anchor type="note" xlink:label="note-371-06a" xlink:href="note-371-06"/> nor arcu CA. </s> <s xml:id="echoid-s11971" xml:space="preserve">Additis ergo æqualibus arcubus DB, AB; </s> <s xml:id="echoid-s11972" xml:space="preserve">(ſunt autem æqua-<lb/> <anchor type="note" xlink:label="note-371-07a" xlink:href="note-371-07"/> <pb o="360" file="372" n="372" rhead=""/> les, proptèrea quòd rectæ eos ſubten dentes æquales ſunt, per defin. </s> <s xml:id="echoid-s11973" xml:space="preserve">poli.) </s> <s xml:id="echoid-s11974" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-372-01a" xlink:href="note-372-01"/> totus arcus BC, minor duobus arcubus AB, AC; </s> <s xml:id="echoid-s11975" xml:space="preserve">hoc eſt, duo latera AB, <lb/>AC, maiora erunt latere BC. </s> <s xml:id="echoid-s11976" xml:space="preserve">Eodemque modo quælibet alia duo latera re-<lb/>liquo maiora demonſtrabuntur. </s> <s xml:id="echoid-s11977" xml:space="preserve">In omni ergo triangulo, &</s> <s xml:id="echoid-s11978" xml:space="preserve">c. </s> <s xml:id="echoid-s11979" xml:space="preserve">Quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s11980" xml:space="preserve"/> </p> <div xml:id="echoid-div936" type="float" level="2" n="1"> <figure xlink:label="fig-371-02" xlink:href="fig-371-02a"> <image file="371-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/371-02"/> </figure> <note position="right" xlink:label="note-371-02" xlink:href="note-371-02a" xml:space="preserve">ſchol. 10. 1.</note> <note position="right" xlink:label="note-371-03" xlink:href="note-371-03a" xml:space="preserve">Theod.</note> <note position="right" xlink:label="note-371-04" xlink:href="note-371-04a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-371-05" xlink:href="note-371-05a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-371-06" xlink:href="note-371-06a" xml:space="preserve">ſchol. 21.</note> <note position="right" xlink:label="note-371-07" xlink:href="note-371-07a" xml:space="preserve">2 Theod.</note> <note position="left" xlink:label="note-372-01" xlink:href="note-372-01a" xml:space="preserve">1@. tertij.</note> </div> </div> <div xml:id="echoid-div938" type="section" level="1" n="488"> <head xml:id="echoid-head523" xml:space="preserve">THEOR. 3. PROPOS. 4.</head> <p> <s xml:id="echoid-s11981" xml:space="preserve">IN omni triangulo ſphærico, tria latera ſimul <lb/>minora ſunt integro circulo maximo.</s> <s xml:id="echoid-s11982" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11983" xml:space="preserve">SIT triangulum ſphæricum ABC. </s> <s xml:id="echoid-s11984" xml:space="preserve">Dico tria latera ſimul minora eſſe in-<lb/>tegro circulo maximo. </s> <s xml:id="echoid-s11985" xml:space="preserve">Productis enim duobus arcubus quibuſlibet BA, BC, <lb/>donec coeant in D, puncto, (Coibunt autem neceſſario vltra A, C, quod cir-<lb/> <anchor type="figure" xlink:label="fig-372-01a" xlink:href="fig-372-01"/> culum maximum AC, ſecent in punctis A, <lb/>C. </s> <s xml:id="echoid-s11986" xml:space="preserve">vel propterea quòd vterque arcus BA, <lb/>BC, ſemicirculo minor eſt.) </s> <s xml:id="echoid-s11987" xml:space="preserve">erunt duo ar-<lb/>us BAD, BCD, ſemicirculi; </s> <s xml:id="echoid-s11988" xml:space="preserve">propte-<lb/>rea quòd circuli maximi ſeſe bifariam di-<lb/> <anchor type="note" xlink:label="note-372-02a" xlink:href="note-372-02"/> uidunt. </s> <s xml:id="echoid-s11989" xml:space="preserve">quoniam verò in triãgulo DAC, <lb/>latera DA, DC, maiora ſunt latere AC; </s> <s xml:id="echoid-s11990" xml:space="preserve">ſi <lb/> <anchor type="note" xlink:label="note-372-03a" xlink:href="note-372-03"/> addantur communes arcus AB, CB, hoc <lb/>eſt, aggregatum ex arcubus AB, CB, fient <lb/>quoque arcus BAD, BCD, maiores tribus arcubus AC, AB, BC; </s> <s xml:id="echoid-s11991" xml:space="preserve">hoc eſt, <lb/>tria latera AC, AB, BC, minora erunt duobus ſemicirculis BAD, BCD, <lb/>hoc eſt, integro circulo maximo. </s> <s xml:id="echoid-s11992" xml:space="preserve">In omni ergo triangulo ſphærico. </s> <s xml:id="echoid-s11993" xml:space="preserve">&</s> <s xml:id="echoid-s11994" xml:space="preserve">c. </s> <s xml:id="echoid-s11995" xml:space="preserve">Quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s11996" xml:space="preserve"/> </p> <div xml:id="echoid-div938" type="float" level="2" n="1"> <figure xlink:label="fig-372-01" xlink:href="fig-372-01a"> <image file="372-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/372-01"/> </figure> <note position="left" xlink:label="note-372-02" xlink:href="note-372-02a" xml:space="preserve">11. 1 Theod.</note> <note position="left" xlink:label="note-372-03" xlink:href="note-372-03a" xml:space="preserve">3. huius.</note> </div> </div> <div xml:id="echoid-div940" type="section" level="1" n="489"> <head xml:id="echoid-head524" xml:space="preserve">THEOR. 4. PROPOS. 5.</head> <p> <s xml:id="echoid-s11997" xml:space="preserve">CVM arcus circuli maximi in ſphęra ſuper ar <lb/>cum circuli maximi conſiſtens angulos facit; </s> <s xml:id="echoid-s11998" xml:space="preserve">aut <lb/>duos rectos, aut duobus rectis æquales efficier.</s> <s xml:id="echoid-s11999" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12000" xml:space="preserve">ARCVS circuli maximi AB, cõſiſtens ſuper arcum circuli maximi CD, <lb/>faciat duos angulos ſphærices ABC, ABD. </s> <s xml:id="echoid-s12001" xml:space="preserve">Si igitur circulus arcus AB, per <lb/>polum circuli arcus CD, tranſit, ſecabit, omnino arcum CD, ad angulos re-<lb/> <anchor type="note" xlink:label="note-372-04a" xlink:href="note-372-04"/> <anchor type="figure" xlink:label="fig-372-02a" xlink:href="fig-372-02"/> ctos; </s> <s xml:id="echoid-s12002" xml:space="preserve">atque idcirco anguli ABC, ABD, <lb/>recti erunt. </s> <s xml:id="echoid-s12003" xml:space="preserve">Si verò arcus AB, per polos ar-<lb/>cus CD, non tranſit, faciet vnum quidem <lb/>angulũ obtuſum, alterũ verò acutum. </s> <s xml:id="echoid-s12004" xml:space="preserve">Di-<lb/>coigitur ipſos duobus eſſe rectis æquales. <lb/></s> <s xml:id="echoid-s12005" xml:space="preserve">Ducatur enim arcus circuli maximi EB, per <lb/> <anchor type="note" xlink:label="note-372-05a" xlink:href="note-372-05"/> punctum B, & </s> <s xml:id="echoid-s12006" xml:space="preserve">polum arcus CD; </s> <s xml:id="echoid-s12007" xml:space="preserve">eruntque <lb/>duo anguli EBC, EBD, recti. </s> <s xml:id="echoid-s12008" xml:space="preserve">Quoniam <lb/> <anchor type="note" xlink:label="note-372-06a" xlink:href="note-372-06"/> verò angulus rectus EBD, æqualis eſt duo-<lb/>bus angulis DBA, ABE; </s> <s xml:id="echoid-s12009" xml:space="preserve">appoſito communi angulo recto EBC, erunt duo <lb/>recti EBD, EBC, tribus angulis DBA, ABE, EBC, æquales. </s> <s xml:id="echoid-s12010" xml:space="preserve">Rurſus quia an- <pb o="361" file="373" n="373" rhead=""/> gulus ABC, duobus angulis ABE, EBC, æqualis eſt, appoſito communi an-<lb/>gulo ABD, erunt duo anguli ABC, ABD, tribus angulis DBA, ABE, <lb/>EBC, æquales. </s> <s xml:id="echoid-s12011" xml:space="preserve">Sed eiſdem his tribus oſtenſum fuit eſſe etiam æquales duos <lb/>rectos EBD, EBC; </s> <s xml:id="echoid-s12012" xml:space="preserve">quæ autem eidem æqualia, inter ſe ſunt æqualia. </s> <s xml:id="echoid-s12013" xml:space="preserve">Duo <lb/>igitur anguli ABC, ABD, æquales ſunt duobus rectis EBD, EBC. </s> <s xml:id="echoid-s12014" xml:space="preserve">Cum <lb/>ergo arcus circuli maximi in ſphæra, &</s> <s xml:id="echoid-s12015" xml:space="preserve">c. </s> <s xml:id="echoid-s12016" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s12017" xml:space="preserve"/> </p> <div xml:id="echoid-div940" type="float" level="2" n="1"> <note position="left" xlink:label="note-372-04" xlink:href="note-372-04a" xml:space="preserve">15. 1. Theo.</note> <figure xlink:label="fig-372-02" xlink:href="fig-372-02a"> <image file="372-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/372-02"/> </figure> <note position="left" xlink:label="note-372-05" xlink:href="note-372-05a" xml:space="preserve">30. 1. Theo.</note> <note position="left" xlink:label="note-372-06" xlink:href="note-372-06a" xml:space="preserve">15. 1. Theo.</note> </div> </div> <div xml:id="echoid-div942" type="section" level="1" n="490"> <head xml:id="echoid-head525" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s12018" xml:space="preserve">SEQVITVR ex his, duos arcus duorum angulorum, qui <lb/> <anchor type="figure" xlink:label="fig-373-01a" xlink:href="fig-373-01"/> duobus rectis angulis ſunt æquales, hoc eſt, qui ab ateu circuli <lb/>maximi arcui alterius cireuli maximi inſiſtente efficiuntur, qua <lb/>les ſunt duo anguli ABC, ABD, ſemicireulum conſtituere. </s> <s xml:id="echoid-s12019" xml:space="preserve">Nam <lb/>ſi ex polo B, circulus maximus deſeribatur CAD, erunt, ex <lb/>defin. </s> <s xml:id="echoid-s12020" xml:space="preserve">6. </s> <s xml:id="echoid-s12021" xml:space="preserve">CA, AD, arcus angulorum ABC, ABD, Perſpicuum <lb/>autem eſt, arcus CA, AD, ſemicirculum conſicere; </s> <s xml:id="echoid-s12022" xml:space="preserve">cum circuli <lb/>maximi CBD, CAD, ſe mutuo ſecent bifariam in C, D.</s> <s xml:id="echoid-s12023" xml:space="preserve"/> </p> <div xml:id="echoid-div942" type="float" level="2" n="1"> <figure xlink:label="fig-373-01" xlink:href="fig-373-01a"> <image file="373-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/373-01"/> </figure> </div> <note position="right" xml:space="preserve">11. 1. Theod.</note> </div> <div xml:id="echoid-div944" type="section" level="1" n="491"> <head xml:id="echoid-head526" xml:space="preserve">THEOR. 5. PROPOS. 6.</head> <p> <s xml:id="echoid-s12024" xml:space="preserve">SI duo arcus circulorum maximorũ in ſphæ-<lb/>ra ſe mutuo ſecuerint, angulos ad verticem æqua-<lb/>les inter ſe efficient.</s> <s xml:id="echoid-s12025" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12026" xml:space="preserve">SECENT ſe duo arcus AB, CD, circulorum maximorum in ſphæra <lb/>in E, vtcunque. </s> <s xml:id="echoid-s12027" xml:space="preserve">Dico angulos, quos faciunt ad verticem E, inter ſe eſſe æqua-<lb/>les; </s> <s xml:id="echoid-s12028" xml:space="preserve">angulum videlicet AED, angulo BEC, <lb/> <anchor type="figure" xlink:label="fig-373-02a" xlink:href="fig-373-02"/> & </s> <s xml:id="echoid-s12029" xml:space="preserve">angulum AEC, angulo BED. </s> <s xml:id="echoid-s12030" xml:space="preserve">Quoniam <lb/>enim tam anguli AED, DEB, quàm angu-<lb/> <anchor type="note" xlink:label="note-373-02a" xlink:href="note-373-02"/> li DEB, BEC, duobus ſunt rectis æquales, <lb/>erunt illi duo his duobus æquales: </s> <s xml:id="echoid-s12031" xml:space="preserve">ablato <lb/>ergo communi angulo DEB, remanebit <lb/>angulus AED, angulo BEC, æqualis. </s> <s xml:id="echoid-s12032" xml:space="preserve">Ea-<lb/>demque ratione conſirmabimus, angulum <lb/>AEC, angulo BED, æqualem eſſe. </s> <s xml:id="echoid-s12033" xml:space="preserve">Si duo <lb/>ergo arcus circulorum maximorum, &</s> <s xml:id="echoid-s12034" xml:space="preserve">c. </s> <s xml:id="echoid-s12035" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s12036" xml:space="preserve"/> </p> <div xml:id="echoid-div944" type="float" level="2" n="1"> <figure xlink:label="fig-373-02" xlink:href="fig-373-02a"> <image file="373-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/373-02"/> </figure> <note position="right" xlink:label="note-373-02" xlink:href="note-373-02a" xml:space="preserve">5. huius.</note> </div> </div> <div xml:id="echoid-div946" type="section" level="1" n="492"> <head xml:id="echoid-head527" xml:space="preserve">THEOR. 6. PROPOS. 7.</head> <p> <s xml:id="echoid-s12037" xml:space="preserve">SI duo triangula ſphærica duo latera duobus <lb/>lateribus æqualia habeant, vtrumque vtrique; </s> <s xml:id="echoid-s12038" xml:space="preserve">ha-<lb/>beant verò & </s> <s xml:id="echoid-s12039" xml:space="preserve">angulum angulo æqualẽ ſub æqua-<lb/>libus arcubus contentũ: </s> <s xml:id="echoid-s12040" xml:space="preserve">Et baſim baſi æqualem ha <lb/>bebunt; </s> <s xml:id="echoid-s12041" xml:space="preserve">eritque triangulũ triangulo æquale, ac re-<lb/>liqui anguli reliquis angulis æquales erunt, vterq; <lb/></s> <s xml:id="echoid-s12042" xml:space="preserve">vtrique, ſub quibus æqualia latera ſubtenduntur.</s> <s xml:id="echoid-s12043" xml:space="preserve"/> </p> <pb o="362" file="374" n="374" rhead=""/> <p> <s xml:id="echoid-s12044" xml:space="preserve">SINT duo triangula ſphærica ABC, DEF, habentia duo latera AB, <lb/>AC, duobus lateribus DE, DF, æqualia, vtrumq; </s> <s xml:id="echoid-s12045" xml:space="preserve">vtriq;</s> <s xml:id="echoid-s12046" xml:space="preserve">, & </s> <s xml:id="echoid-s12047" xml:space="preserve">angulum A, an-<lb/> <anchor type="figure" xlink:label="fig-374-01a" xlink:href="fig-374-01"/> gulo D, æqualem. </s> <s xml:id="echoid-s12048" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12049" xml:space="preserve">baſem BC, ba-<lb/>ſi EF, æqualem eſſe, & </s> <s xml:id="echoid-s12050" xml:space="preserve">triangulum ABC, <lb/>triangulo DEF, & </s> <s xml:id="echoid-s12051" xml:space="preserve">reliquos angulos B, <lb/>C, reliquis angulis E, F, vtrumq; </s> <s xml:id="echoid-s12052" xml:space="preserve">vtriq;</s> <s xml:id="echoid-s12053" xml:space="preserve">. <lb/>Quoniam enim arcus AB, arcui DE, æ-<lb/>qualis ponitur, fit, vt ſi alter alteri intel-<lb/>ligatur ſuperponi in ſuperficie ſphæræ, <lb/>collocato puncto A, in puncto D, & </s> <s xml:id="echoid-s12054" xml:space="preserve">pun <lb/>cto B, in puncto E, plana circulorum AB, <lb/>DE, ſibi mutuo congruant, & </s> <s xml:id="echoid-s12055" xml:space="preserve">proinde ar <lb/>cus AB, arcui DE, congruat. </s> <s xml:id="echoid-s12056" xml:space="preserve">Alias ſe <lb/>mutuo ſecarent bifariam circuli illorum arcuum in A, & </s> <s xml:id="echoid-s12057" xml:space="preserve">B, atq; </s> <s xml:id="echoid-s12058" xml:space="preserve">adeo ſemicir <lb/> <anchor type="note" xlink:label="note-374-01a" xlink:href="note-374-01"/> culi eſſent AB, DE. </s> <s xml:id="echoid-s12059" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s12060" xml:space="preserve">Eſt enim ſemicirculo vterq; </s> <s xml:id="echoid-s12061" xml:space="preserve">mi-<lb/> <anchor type="note" xlink:label="note-374-02a" xlink:href="note-374-02"/> nor. </s> <s xml:id="echoid-s12062" xml:space="preserve">Cum ergo angulus A, angulo D, ponatur æqualis, congruet quoq; </s> <s xml:id="echoid-s12063" xml:space="preserve">ar-<lb/>cus AC, arcui DF, punctumq; </s> <s xml:id="echoid-s12064" xml:space="preserve">C, in punctum F, cadet, ob æqualitatem ar-<lb/>cuum AC, DF. </s> <s xml:id="echoid-s12065" xml:space="preserve">Baſis igitur BC, baſi EF, congruet quoq;</s> <s xml:id="echoid-s12066" xml:space="preserve">: alias, ſi ſupra ca <lb/>deret, aut infra, cuiuſmodi eſt arcus EGF, eſſent arcus EF, EGF, vel BC, <lb/>ſe mutuo ſecantes in E, F, ſemicitculi; </s> <s xml:id="echoid-s12067" xml:space="preserve">cum circuli maximi ſe mutuo ſecent <lb/> <anchor type="note" xlink:label="note-374-03a" xlink:href="note-374-03"/> bifariam. </s> <s xml:id="echoid-s12068" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s12069" xml:space="preserve">Singuli enim ſemicirculo minores ſunt. </s> <s xml:id="echoid-s12070" xml:space="preserve">Quo-<lb/> <anchor type="note" xlink:label="note-374-04a" xlink:href="note-374-04"/> circa baſis BC, baſi EF, æqualis erit, cum neutra alteram excedat; </s> <s xml:id="echoid-s12071" xml:space="preserve">& </s> <s xml:id="echoid-s12072" xml:space="preserve">trian <lb/>gulum ABC, triangulo DEF; </s> <s xml:id="echoid-s12073" xml:space="preserve">& </s> <s xml:id="echoid-s12074" xml:space="preserve">anguli B, C, angulis E, F, vterq; </s> <s xml:id="echoid-s12075" xml:space="preserve">vtrique, <lb/>æquales erunt, ob eandem cauſam. </s> <s xml:id="echoid-s12076" xml:space="preserve">Quare ſi duo trangula ſphærica, &</s> <s xml:id="echoid-s12077" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s12078" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s12079" xml:space="preserve"/> </p> <div xml:id="echoid-div946" type="float" level="2" n="1"> <figure xlink:label="fig-374-01" xlink:href="fig-374-01a"> <image file="374-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/374-01"/> </figure> <note position="left" xlink:label="note-374-01" xlink:href="note-374-01a" xml:space="preserve">11. 1. Theod.</note> <note position="left" xlink:label="note-374-02" xlink:href="note-374-02a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-374-03" xlink:href="note-374-03a" xml:space="preserve">11. 1. Theod.</note> <note position="left" xlink:label="note-374-04" xlink:href="note-374-04a" xml:space="preserve">a. huius.</note> </div> </div> <div xml:id="echoid-div948" type="section" level="1" n="493"> <head xml:id="echoid-head528" xml:space="preserve">THEOR. 7. PROPOS. 8.</head> <p> <s xml:id="echoid-s12080" xml:space="preserve">ISOSCELIVM triangulorum ſphærico-<lb/>rum, qui ad baſim ſunt, anguli inter ſe ſunt æqua-<lb/>les: </s> <s xml:id="echoid-s12081" xml:space="preserve">Et productis æqualibus arcubus, qui ſub baſi <lb/>ſunt, anguli inter ſe æquales erunt.</s> <s xml:id="echoid-s12082" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12083" xml:space="preserve">SIT triangulum ſphæricum Iſoſceles ABC, cuius duo latera AB, AC, <lb/>æqualia ſint. </s> <s xml:id="echoid-s12084" xml:space="preserve">Dico angulos B, C, ſupra baſim BC, æquales eſſe: </s> <s xml:id="echoid-s12085" xml:space="preserve">Item ſi pro-<lb/>ducantur arcus æquales AB, AC, infra baſim BC, quantumlibet, angulos <lb/>quoque B, C, ſub baſi BC, æquales eſſe. </s> <s xml:id="echoid-s12086" xml:space="preserve">Quoniam enim arcus AB, ſemicir-<lb/> <anchor type="note" xlink:label="note-374-05a" xlink:href="note-374-05"/> culo minor eſt, poterit in eo producto accipi adhuc arcus minor ſemicircu-<lb/>lo. </s> <s xml:id="echoid-s12087" xml:space="preserve">Sit igitur arcus AD, ſemicirculo minor; </s> <s xml:id="echoid-s12088" xml:space="preserve">& </s> <s xml:id="echoid-s12089" xml:space="preserve">ex arcu AE, quantumcunq; <lb/></s> <s xml:id="echoid-s12090" xml:space="preserve"> <anchor type="note" xlink:label="note-374-06a" xlink:href="note-374-06"/> producto abſcindatur arcus AF, æqualis arcui AD; </s> <s xml:id="echoid-s12091" xml:space="preserve">& </s> <s xml:id="echoid-s12092" xml:space="preserve">per duo puncta B, F, <lb/>nec non per C, D, ducantur duo arcus maximorum circulorum BF, CD. <lb/></s> <s xml:id="echoid-s12093" xml:space="preserve"> <anchor type="note" xlink:label="note-374-07a" xlink:href="note-374-07"/> Quia ergo duo latera AB, AF, trianguli ABF, æqualia ſunt duobus late-<lb/>ribus AC, AD, trianguli ACD, vtrumque vtrique, continentq́; </s> <s xml:id="echoid-s12094" xml:space="preserve">angulum <lb/>communem A; </s> <s xml:id="echoid-s12095" xml:space="preserve">erit baſis BF, baſi CD, æqualis, & </s> <s xml:id="echoid-s12096" xml:space="preserve">anguli ABF, & </s> <s xml:id="echoid-s12097" xml:space="preserve">F, angu-<lb/> <anchor type="note" xlink:label="note-374-08a" xlink:href="note-374-08"/> lis ACD, & </s> <s xml:id="echoid-s12098" xml:space="preserve">D. </s> <s xml:id="echoid-s12099" xml:space="preserve">Rurſus, quoniam arcus AD, AF, æquales ſunt; </s> <s xml:id="echoid-s12100" xml:space="preserve">ſi deman- <pb o="363" file="375" n="375" rhead=""/> tur æquales AB, AC, erunt & </s> <s xml:id="echoid-s12101" xml:space="preserve">BD, CF, æquales. </s> <s xml:id="echoid-s12102" xml:space="preserve">Quare duo latera DB, <lb/>DC, trianguli DBC, æqualia ſunt duobus <lb/> <anchor type="figure" xlink:label="fig-375-01a" xlink:href="fig-375-01"/> lateribus FC, FB, trianguli FCB: </s> <s xml:id="echoid-s12103" xml:space="preserve">quæ cum <lb/>contineant angulos æquales D, F, vt oſten-<lb/>dimus, erunt & </s> <s xml:id="echoid-s12104" xml:space="preserve">anguli DBC, DCB, angu-<lb/> <anchor type="note" xlink:label="note-375-01a" xlink:href="note-375-01"/> lis FCB, FBC, æquales. </s> <s xml:id="echoid-s12105" xml:space="preserve">Quòd ſi ex angu-<lb/>lis ABF, ACD, quos oſtendimus æquales <lb/>eſſe, auferantur anguli FBC, DCB, quos <lb/>etiam æquales eſſe demonſtrauimus, rema-<lb/>nebunt anguli ABC, ACB, ſupra baſim <lb/>BC, æquales: </s> <s xml:id="echoid-s12106" xml:space="preserve">Oſtenſum eſt autem & </s> <s xml:id="echoid-s12107" xml:space="preserve">angu-<lb/>los DBC, FCB, infra eandem baſim BC, <lb/>eſſe æquales. </s> <s xml:id="echoid-s12108" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12109" xml:space="preserve">anguli ſupra baſim in-<lb/>ter ſe, & </s> <s xml:id="echoid-s12110" xml:space="preserve">anguli infra eandem inter ſe æquales ſunt. </s> <s xml:id="echoid-s12111" xml:space="preserve">Quam ob rem Iſoſcelium <lb/>triangulorum ſphæricorum, &</s> <s xml:id="echoid-s12112" xml:space="preserve">c. </s> <s xml:id="echoid-s12113" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s12114" xml:space="preserve"/> </p> <div xml:id="echoid-div948" type="float" level="2" n="1"> <note position="left" xlink:label="note-374-05" xlink:href="note-374-05a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-374-06" xlink:href="note-374-06a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-374-07" xlink:href="note-374-07a" xml:space="preserve">20. 1. Theo.</note> <note position="left" xlink:label="note-374-08" xlink:href="note-374-08a" xml:space="preserve">7. huius.</note> <figure xlink:label="fig-375-01" xlink:href="fig-375-01a"> <image file="375-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/375-01"/> </figure> <note position="right" xlink:label="note-375-01" xlink:href="note-375-01a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div950" type="section" level="1" n="494"> <head xml:id="echoid-head529" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s12115" xml:space="preserve">HING manifeſtum eſt, omne triangulum ſphæricum æquilaterum, eſſe quoque <gap/> <lb/>@uiangulum.</s> <s xml:id="echoid-s12116" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div951" type="section" level="1" n="495"> <head xml:id="echoid-head530" xml:space="preserve">THEOR. 8. PROPOS. 9.</head> <p> <s xml:id="echoid-s12117" xml:space="preserve">SI trianguli ſphærici duo anguli æquales inter <lb/>ſe fuerint: </s> <s xml:id="echoid-s12118" xml:space="preserve">Et ſub æqualibus angulis ſubtenſa late-<lb/>ra æqualia inter ſe erunt.</s> <s xml:id="echoid-s12119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12120" xml:space="preserve">IN triangulo ABC, ſint duo anguli B, C, ſupra latus BC, æquales. </s> <s xml:id="echoid-s12121" xml:space="preserve">Di-<lb/>co latera quoque AB, AC, illis ſubtenſa eſſe æqualia. </s> <s xml:id="echoid-s12122" xml:space="preserve">Si enim non ſunt æ-<lb/>qualia, ſit, ſi fieri poteſt AB, maius. </s> <s xml:id="echoid-s12123" xml:space="preserve">Et quo-<lb/> <anchor type="figure" xlink:label="fig-375-02a" xlink:href="fig-375-02"/> niam arcus AC, minor eſt ſemicirculo, abſcin-<lb/> <anchor type="note" xlink:label="note-375-02a" xlink:href="note-375-02"/> datur ex arcu maiore AB, arcus BD, arcui mi-<lb/> <anchor type="note" xlink:label="note-375-03a" xlink:href="note-375-03"/> nori AC, æqualis; </s> <s xml:id="echoid-s12124" xml:space="preserve">& </s> <s xml:id="echoid-s12125" xml:space="preserve">per puncta C, D, arcus cir <lb/> <anchor type="note" xlink:label="note-375-04a" xlink:href="note-375-04"/> culi maximi ducatur CD. </s> <s xml:id="echoid-s12126" xml:space="preserve">Quoniam ergo duo <lb/>latera AC, CB, trianguli ACB, æqualia ſunt <lb/>duobus lateribus DB, BC, trianguli DBC, <lb/>continentq́; </s> <s xml:id="echoid-s12127" xml:space="preserve">angulos æquales ACB, DBC; <lb/></s> <s xml:id="echoid-s12128" xml:space="preserve">erunt triangula ACB, DBC, æqualia, totum <lb/> <anchor type="note" xlink:label="note-375-05a" xlink:href="note-375-05"/> & </s> <s xml:id="echoid-s12129" xml:space="preserve">pars. </s> <s xml:id="echoid-s12130" xml:space="preserve">Quod fieri non poteſt. </s> <s xml:id="echoid-s12131" xml:space="preserve">Non ergo inæ-<lb/>qualia ſunt latera AB, AC, ſed æqualia. </s> <s xml:id="echoid-s12132" xml:space="preserve">Si trian <lb/>guli igitur ſphærici duo anguli, &</s> <s xml:id="echoid-s12133" xml:space="preserve">c. </s> <s xml:id="echoid-s12134" xml:space="preserve">Quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s12135" xml:space="preserve"/> </p> <div xml:id="echoid-div951" type="float" level="2" n="1"> <figure xlink:label="fig-375-02" xlink:href="fig-375-02a"> <image file="375-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/375-02"/> </figure> <note position="right" xlink:label="note-375-02" xlink:href="note-375-02a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-375-03" xlink:href="note-375-03a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-375-04" xlink:href="note-375-04a" xml:space="preserve">20 1. Theo.</note> <note position="right" xlink:label="note-375-05" xlink:href="note-375-05a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div953" type="section" level="1" n="496"> <head xml:id="echoid-head531" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s12136" xml:space="preserve">SEQVITVR hinc, omne triangulum ſphætricum æquiangulum, eſſe quoque æqui-<lb/>laterum.</s> <s xml:id="echoid-s12137" xml:space="preserve"/> </p> <pb o="364" file="376" n="376" rhead=""/> </div> <div xml:id="echoid-div954" type="section" level="1" n="497"> <head xml:id="echoid-head532" xml:space="preserve">PROBL. 2. PROPOS. 10.</head> <p> <s xml:id="echoid-s12138" xml:space="preserve">AD datum arcum circuli maximi in ſphæra, <lb/>datumq́; </s> <s xml:id="echoid-s12139" xml:space="preserve">in eo punctum, dato angulo ſphærico æ-<lb/>qualem angulum ſphæricum conſtituere.</s> <s xml:id="echoid-s12140" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12141" xml:space="preserve">SIT datus arcus maximi circuli in ſphæra AB, datumq; </s> <s xml:id="echoid-s12142" xml:space="preserve">in eo punctum <lb/>C, oporteatq́; </s> <s xml:id="echoid-s12143" xml:space="preserve">dato angulo ſphærico D, ad punctum C, æqualem angulum <lb/>ſphæricum conſtituere. </s> <s xml:id="echoid-s12144" xml:space="preserve">Productis arcubus DE, DF, angulum D, continen-<lb/> <anchor type="note" xlink:label="note-376-01a" xlink:href="note-376-01"/> tibus quantumlibet, ſumatur quadrans DG; </s> <s xml:id="echoid-s12145" xml:space="preserve">atq; </s> <s xml:id="echoid-s12146" xml:space="preserve">per G, & </s> <s xml:id="echoid-s12147" xml:space="preserve">polum circuli <lb/> <anchor type="figure" xlink:label="fig-376-01a" xlink:href="fig-376-01"/> DE, arcus circuli ma-<lb/>ximi ducatur GH, ſe-<lb/>cans arcum DF, in H. <lb/></s> <s xml:id="echoid-s12148" xml:space="preserve">Erit igitur angulus G, <lb/> <anchor type="note" xlink:label="note-376-02a" xlink:href="note-376-02"/> rectus. </s> <s xml:id="echoid-s12149" xml:space="preserve">Deinde ſumpto <lb/>quoque quadrante CI, <lb/>ducatur per I, & </s> <s xml:id="echoid-s12150" xml:space="preserve">polum <lb/> <anchor type="note" xlink:label="note-376-03a" xlink:href="note-376-03"/> circuli AB, arcus ma-<lb/>ximi circuli IK. </s> <s xml:id="echoid-s12151" xml:space="preserve">Erit <lb/>igitur & </s> <s xml:id="echoid-s12152" xml:space="preserve">angulus 1, re-<lb/> <anchor type="note" xlink:label="note-376-04a" xlink:href="note-376-04"/> ctus. </s> <s xml:id="echoid-s12153" xml:space="preserve">Poſtremo, quia ar-<lb/>cus GH, ſemicirculo <lb/> <anchor type="note" xlink:label="note-376-05a" xlink:href="note-376-05"/> minor eſt, abſcindatur <lb/> <anchor type="note" xlink:label="note-376-06a" xlink:href="note-376-06"/> ei arcus IK, æqualis, ducaturque per C, K, arcus circuli maximi CK. </s> <s xml:id="echoid-s12154" xml:space="preserve">Dico <lb/> <anchor type="note" xlink:label="note-376-07a" xlink:href="note-376-07"/> angulum C, æqualem eſſe angulo D. </s> <s xml:id="echoid-s12155" xml:space="preserve">Cum enim latera DG, GH, æqualia <lb/>ſint lateribus CI, IK, contineantq́ue angulos æquales, vt pote rectos; </s> <s xml:id="echoid-s12156" xml:space="preserve">æqua-<lb/>les erunt anguli D, & </s> <s xml:id="echoid-s12157" xml:space="preserve">C. </s> <s xml:id="echoid-s12158" xml:space="preserve">Ad datum ergo arcum circuli maximi, &</s> <s xml:id="echoid-s12159" xml:space="preserve">c. </s> <s xml:id="echoid-s12160" xml:space="preserve">Quod fa-<lb/> <anchor type="note" xlink:label="note-376-08a" xlink:href="note-376-08"/> ciendum erat.</s> <s xml:id="echoid-s12161" xml:space="preserve"/> </p> <div xml:id="echoid-div954" type="float" level="2" n="1"> <note position="right" xlink:label="note-376-01" xlink:href="note-376-01a" xml:space="preserve">20. 1. Theo.</note> <figure xlink:label="fig-376-01" xlink:href="fig-376-01a"> <image file="376-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/376-01"/> </figure> <note position="left" xlink:label="note-376-02" xlink:href="note-376-02a" xml:space="preserve">25. 1. Theo.</note> <note position="left" xlink:label="note-376-03" xlink:href="note-376-03a" xml:space="preserve">20. 1. Theo.</note> <note position="left" xlink:label="note-376-04" xlink:href="note-376-04a" xml:space="preserve">15. 1. Theo.</note> <note position="left" xlink:label="note-376-05" xlink:href="note-376-05a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-376-06" xlink:href="note-376-06a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-376-07" xlink:href="note-376-07a" xml:space="preserve">20. 1. Theo.</note> <note position="left" xlink:label="note-376-08" xlink:href="note-376-08a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div956" type="section" level="1" n="498"> <head xml:id="echoid-head533" xml:space="preserve">THEOR. 9. PROPOS. 11.</head> <p> <s xml:id="echoid-s12162" xml:space="preserve">OMNIS trianguli ſphærici maior angulus <lb/>maiori lateriſubtenditur. </s> <s xml:id="echoid-s12163" xml:space="preserve">Et maius latus maiorem <lb/>angulum ſubtendit.</s> <s xml:id="echoid-s12164" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12165" xml:space="preserve">IN triangulo ſphærico ABC, ſit angulus ACB, angulo A, maior. </s> <s xml:id="echoid-s12166" xml:space="preserve">Dico <lb/> <anchor type="figure" xlink:label="fig-376-02a" xlink:href="fig-376-02"/> latus AB, maius eſſe latere BC. </s> <s xml:id="echoid-s12167" xml:space="preserve">Quoniam angulus <lb/> <anchor type="note" xlink:label="note-376-09a" xlink:href="note-376-09"/> ACB, maior ponitur angulo A, fiat angulus ACD, <lb/>angulo A, æqualis, ſecetq́ue arcus CD, arcum AB, <lb/>in D. </s> <s xml:id="echoid-s12168" xml:space="preserve">Quoniam igitur in triangulo ADC, anguli <lb/>A, & </s> <s xml:id="echoid-s12169" xml:space="preserve">ACD, æquales ſunt; </s> <s xml:id="echoid-s12170" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12171" xml:space="preserve">latera AD, CD, <lb/> <anchor type="note" xlink:label="note-376-10a" xlink:href="note-376-10"/> æqualia. </s> <s xml:id="echoid-s12172" xml:space="preserve">Addito ergo communiarcu DB, erunt ar-<lb/>cus BD, DC, æquales arcui AB: </s> <s xml:id="echoid-s12173" xml:space="preserve">Sed arcus BD, <lb/>DC, ſimul maiores ſunt arcu BC. </s> <s xml:id="echoid-s12174" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12175" xml:space="preserve">arcus <lb/> <anchor type="note" xlink:label="note-376-11a" xlink:href="note-376-11"/> AB, eodẽ arcu BC, maior erit. </s> <s xml:id="echoid-s12176" xml:space="preserve">Quod eſt propoſitũ.</s> <s xml:id="echoid-s12177" xml:space="preserve"/> </p> <div xml:id="echoid-div956" type="float" level="2" n="1"> <figure xlink:label="fig-376-02" xlink:href="fig-376-02a"> <image file="376-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/376-02"/> </figure> <note position="left" xlink:label="note-376-09" xlink:href="note-376-09a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-376-10" xlink:href="note-376-10a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-376-11" xlink:href="note-376-11a" xml:space="preserve">3. huius.</note> </div> <pb o="365" file="377" n="377" rhead=""/> <p> <s xml:id="echoid-s12178" xml:space="preserve">SED iam in triangulo ſphærico ABC, latus AB, maius ſit latere BC. <lb/></s> <s xml:id="echoid-s12179" xml:space="preserve">Dico angulum C, maiorem eſſe angulo A. </s> <s xml:id="echoid-s12180" xml:space="preserve">Si enim angu lus C, maior non eſt <lb/>angulo A, erit vel ei æqualis, vel minor. </s> <s xml:id="echoid-s12181" xml:space="preserve">Si eſt æqualis, erunt latera AB, CB, <lb/> <anchor type="note" xlink:label="note-377-01a" xlink:href="note-377-01"/> æqualia. </s> <s xml:id="echoid-s12182" xml:space="preserve">Quod eſt abſurdum, cum AB, ponatur ma-<lb/> <anchor type="figure" xlink:label="fig-377-01a" xlink:href="fig-377-01"/> ius, quàm CB: </s> <s xml:id="echoid-s12183" xml:space="preserve">Si vero minor eſt angulus C, angu-<lb/>lo A, erit latus BC, latere AB, maius, vt iã oſten-<lb/>ſum eſt. </s> <s xml:id="echoid-s12184" xml:space="preserve">Quod etiam abſurdum eſt. </s> <s xml:id="echoid-s12185" xml:space="preserve">ponitur enim <lb/>AB, maius, quàm BC. </s> <s xml:id="echoid-s12186" xml:space="preserve">Cum ergo angulus C, æqua-<lb/>lis non ſit, neque minor angulo A, erit vtique ma-<lb/>ior. </s> <s xml:id="echoid-s12187" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s12188" xml:space="preserve">Omnis ergo trianguli <lb/>ſphęrici maior angulus, &</s> <s xml:id="echoid-s12189" xml:space="preserve">c. </s> <s xml:id="echoid-s12190" xml:space="preserve">Quod erat oſtendendũ.</s> <s xml:id="echoid-s12191" xml:space="preserve"/> </p> <div xml:id="echoid-div957" type="float" level="2" n="2"> <note position="right" xlink:label="note-377-01" xlink:href="note-377-01a" xml:space="preserve">9. huius.</note> <figure xlink:label="fig-377-01" xlink:href="fig-377-01a"> <image file="377-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/377-01"/> </figure> </div> </div> <div xml:id="echoid-div959" type="section" level="1" n="499"> <head xml:id="echoid-head534" xml:space="preserve">THEOR. 10. PROPOS. 12.</head> <p> <s xml:id="echoid-s12192" xml:space="preserve">SI duo triangula ſphærica duo latera duobus <lb/>lateribus æqualia habuerint, vtrumque vtrique, <lb/>angulum verò angulo maiorem ſub æqualibus ar-<lb/>cubus contentum: </s> <s xml:id="echoid-s12193" xml:space="preserve">Et baſim baſi maiorem habe-<lb/>bunt. </s> <s xml:id="echoid-s12194" xml:space="preserve">Quòd ſi baſim baſi maiorem habuerint: </s> <s xml:id="echoid-s12195" xml:space="preserve">Et <lb/>angulum ſub æqualibus arcubus contentum an-<lb/>gulo maiorem habebunt.</s> <s xml:id="echoid-s12196" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12197" xml:space="preserve">SINT duo latera AB, AC, trianguli ABC, æqualia duobus lateri-<lb/>bus DE, DF, trianguli DEF, ſed angulus EDF, maior ſit angulo A. </s> <s xml:id="echoid-s12198" xml:space="preserve">Dico <lb/>baſim EF, maiorem quoque eſſe baſi BC. </s> <s xml:id="echoid-s12199" xml:space="preserve">Sint enim primum triangula hæc <lb/>ſphærica Iſoſcelia, & </s> <s xml:id="echoid-s12200" xml:space="preserve">ex D, polo per puncta <lb/> <anchor type="figure" xlink:label="fig-377-02a" xlink:href="fig-377-02"/> E, F, arcus circuli deſcribatur in ſuperficie <lb/>ſphæræ EGF, qui circulus, ſi maximus fue-<lb/>rit, idem erit omnino, qui EF: </s> <s xml:id="echoid-s12201" xml:space="preserve">alias, cum ma-<lb/>ximi circuli ſe bifariam ſecent, eſſet EF, ſe-<lb/> <anchor type="note" xlink:label="note-377-02a" xlink:href="note-377-02"/> micirculus. </s> <s xml:id="echoid-s12202" xml:space="preserve">quod eſt abſurdum, cum ſit ſe-<lb/>micirculo minor. </s> <s xml:id="echoid-s12203" xml:space="preserve">Tunc autem circulus arcus <lb/> <anchor type="note" xlink:label="note-377-03a" xlink:href="note-377-03"/> EGF, maximus erit, cum arcus DE, DF, <lb/>quadrantes fuerint; </s> <s xml:id="echoid-s12204" xml:space="preserve">quòd maximus circulus <lb/>quadrante abſit à ſuo polo. </s> <s xml:id="echoid-s12205" xml:space="preserve">Sit ergo iam ar-<lb/> <anchor type="note" xlink:label="note-377-04a" xlink:href="note-377-04"/> cus EGF, maximi circuli, & </s> <s xml:id="echoid-s12206" xml:space="preserve">idem, qui EF, <lb/> <anchor type="note" xlink:label="note-377-05a" xlink:href="note-377-05"/> fiatq́ue angulus FDG, angulo A, æqualis. <lb/></s> <s xml:id="echoid-s12207" xml:space="preserve"> <anchor type="note" xlink:label="note-377-06a" xlink:href="note-377-06"/> Erit arcus DG, arcui DE, atque adeo & </s> <s xml:id="echoid-s12208" xml:space="preserve">arcui AB, æqualis: </s> <s xml:id="echoid-s12209" xml:space="preserve">propterea quòd <lb/> <anchor type="note" xlink:label="note-377-07a" xlink:href="note-377-07"/> rectæ ſubtendentes DE, DG, ex defin. </s> <s xml:id="echoid-s12210" xml:space="preserve">poli, æqua les ſunt. </s> <s xml:id="echoid-s12211" xml:space="preserve">Quia igitur latera <lb/>AB, AC, æqualia ſunt lateribus DG, DF, angulosq̀ue continent æquales; <lb/></s> <s xml:id="echoid-s12212" xml:space="preserve">æquales reunt baſes BC, GF. </s> <s xml:id="echoid-s12213" xml:space="preserve">Cum ergo arcus EF, maior ſit arcu GF, ma-<lb/> <anchor type="note" xlink:label="note-377-08a" xlink:href="note-377-08"/> ior quoque erit arcus EF, arcu BC. </s> <s xml:id="echoid-s12214" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12215" xml:space="preserve"/> </p> <div xml:id="echoid-div959" type="float" level="2" n="1"> <figure xlink:label="fig-377-02" xlink:href="fig-377-02a"> <image file="377-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/377-02"/> </figure> <note position="right" xlink:label="note-377-02" xlink:href="note-377-02a" xml:space="preserve">11. 1. Theo.</note> <note position="right" xlink:label="note-377-03" xlink:href="note-377-03a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-377-04" xlink:href="note-377-04a" xml:space="preserve">Coroll. 16.</note> <note position="right" xlink:label="note-377-05" xlink:href="note-377-05a" xml:space="preserve">1. Theod.</note> <note position="right" xlink:label="note-377-06" xlink:href="note-377-06a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-377-07" xlink:href="note-377-07a" xml:space="preserve">28. ter@ij.</note> <note position="right" xlink:label="note-377-08" xlink:href="note-377-08a" xml:space="preserve">7. huius.</note> </div> <p> <s xml:id="echoid-s12216" xml:space="preserve">QVOD ſi circulus ex polo D, per puncta E, F, deſcriptus non fuerit ma- <pb o="366" file="378" n="378" rhead=""/> ximus, atque adeò idem non ſit, qui EF; </s> <s xml:id="echoid-s12217" xml:space="preserve">ſed vel cadat infra arcum EF, ſiue <lb/>ſupra, (nihil enim intereſt, quocunque cadat.) </s> <s xml:id="echoid-s12218" xml:space="preserve">fiat nihilominus angulus <lb/> <anchor type="note" xlink:label="note-378-01a" xlink:href="note-378-01"/> FDH, angulo A, æqualis: </s> <s xml:id="echoid-s12219" xml:space="preserve">eritq̀ue rurſus arcus DH, arcui DE, hoc eſt, ar-<lb/> <anchor type="note" xlink:label="note-378-02a" xlink:href="note-378-02"/> cui AB, æqualis; </s> <s xml:id="echoid-s12220" xml:space="preserve">eo quòd rectæ ſubtendentes DE, DH, æquales ſint, ex de-<lb/>fin. </s> <s xml:id="echoid-s12221" xml:space="preserve">poli. </s> <s xml:id="echoid-s12222" xml:space="preserve">Ducto igitur per puncta F, H, arcu circuli maximi FH; </s> <s xml:id="echoid-s12223" xml:space="preserve">cum latera <lb/> <anchor type="note" xlink:label="note-378-03a" xlink:href="note-378-03"/> AB, AC, lateribus DH, DF, æqualia ſint, angulosq̀ue contineant æquales; <lb/></s> <s xml:id="echoid-s12224" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12225" xml:space="preserve">baſes BC, HF, æquales. </s> <s xml:id="echoid-s12226" xml:space="preserve">Quoniam verò circulus maximus DF, per <lb/> <anchor type="note" xlink:label="note-378-04a" xlink:href="note-378-04"/> D, polum circuli EHF, tranſiens eum bifariam ſecat; </s> <s xml:id="echoid-s12227" xml:space="preserve">erit arcus EHF, ſemi-<lb/> <anchor type="note" xlink:label="note-378-05a" xlink:href="note-378-05"/> circulo minor; </s> <s xml:id="echoid-s12228" xml:space="preserve">(quia arcus à puncto F, per E, vſque ad illud punctum, in quo, <lb/>ſi protractus eſſet vltra E, ſecaretur ab arcu FD, ad partes D, producto, eſt <lb/>ſemicirculus: </s> <s xml:id="echoid-s12229" xml:space="preserve">quandoquidem circulus arcus EHF, bifariam ſecatur a circulo <lb/>arcus FD, vt dictum eſt.) </s> <s xml:id="echoid-s12230" xml:space="preserve">atque adeo recta FE, maior, quàm recta FH, in eo-<lb/> <anchor type="note" xlink:label="note-378-06a" xlink:href="note-378-06"/> dem circulo: </s> <s xml:id="echoid-s12231" xml:space="preserve">quia illa propin quior eſt centro circuli EHF, hoc eſt, diame-<lb/>tro, quàm hæc. </s> <s xml:id="echoid-s12232" xml:space="preserve">Cum ergo circuli arcuum EF, HF, maximi ſint, ideoq̀ æqua-<lb/>les; </s> <s xml:id="echoid-s12233" xml:space="preserve">ſit autem vterque arcus EF, HF, ſemicirculo minor; </s> <s xml:id="echoid-s12234" xml:space="preserve">erit arcus EF, ma-<lb/> <anchor type="note" xlink:label="note-378-07a" xlink:href="note-378-07"/> ior arcu HF: </s> <s xml:id="echoid-s12235" xml:space="preserve">Oſtenſus autem eſt arcus HF, æqualis arcui BC. </s> <s xml:id="echoid-s12236" xml:space="preserve">Maior igitur <lb/> <anchor type="note" xlink:label="note-378-08a" xlink:href="note-378-08"/> erit quoque arcus EF, arcu BC. </s> <s xml:id="echoid-s12237" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12238" xml:space="preserve"/> </p> <div xml:id="echoid-div960" type="float" level="2" n="2"> <note position="left" xlink:label="note-378-01" xlink:href="note-378-01a" xml:space="preserve">10. huius.</note> <note position="left" xlink:label="note-378-02" xlink:href="note-378-02a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-378-03" xlink:href="note-378-03a" xml:space="preserve">10. 1. Theo.</note> <note position="left" xlink:label="note-378-04" xlink:href="note-378-04a" xml:space="preserve">7. huius.</note> <note position="left" xlink:label="note-378-05" xlink:href="note-378-05a" xml:space="preserve">15. 1. Theo.</note> <note position="left" xlink:label="note-378-06" xlink:href="note-378-06a" xml:space="preserve">25. tertij.</note> <note position="left" xlink:label="note-378-07" xlink:href="note-378-07a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-378-08" xlink:href="note-378-08a" xml:space="preserve">ſchol. 28.</note> </div> <note position="left" xml:space="preserve">tertij.</note> <p> <s xml:id="echoid-s12239" xml:space="preserve">SINT deinde triangula propoſita non Iſoſcelia, ſed latus AB, maius <lb/>ſit latere AC, ac proinde & </s> <s xml:id="echoid-s12240" xml:space="preserve">latus DE, maius latere DF. </s> <s xml:id="echoid-s12241" xml:space="preserve">Producto ergo arcu <lb/>DF, ad partes F, abſciſſoq́ue arcu DG, æquali ipſi DE, qui minor eſt ſe-<lb/> <anchor type="note" xlink:label="note-378-10a" xlink:href="note-378-10"/> <anchor type="figure" xlink:label="fig-378-01a" xlink:href="fig-378-01"/> micirculo, deſcribatur ex polo D, per pun <lb/> <anchor type="note" xlink:label="note-378-11a" xlink:href="note-378-11"/> cta E, G, arcus circuli EHG, ſiue maxi-<lb/>ximus is ſit, ſiue non maximus. </s> <s xml:id="echoid-s12242" xml:space="preserve">Fiat rur-<lb/>ſus angulus FDH, angulo A, æqualis; <lb/></s> <s xml:id="echoid-s12243" xml:space="preserve"> <anchor type="note" xlink:label="note-378-12a" xlink:href="note-378-12"/> eritq́ue arcus DH, arcui DE, hoc eſt, ar-<lb/> <anchor type="note" xlink:label="note-378-13a" xlink:href="note-378-13"/> cui AB, æqualis; </s> <s xml:id="echoid-s12244" xml:space="preserve">eò quòd rectæ ſubtenſæ <lb/>DH, DE, æquales ſint, ex defin. </s> <s xml:id="echoid-s12245" xml:space="preserve">poli. </s> <s xml:id="echoid-s12246" xml:space="preserve">Du-<lb/>cto igitur per puncta H, F, arcu circuli <lb/>maximi HF, erit, vt prius, arcus BC, ar-<lb/>cui HF, æqualis. </s> <s xml:id="echoid-s12247" xml:space="preserve">Quoniam verò circulus <lb/>maximus DG, per D, polum circuli EG, <lb/> <anchor type="note" xlink:label="note-378-14a" xlink:href="note-378-14"/> ducitur, eſtq́ue punctum F, intra periphe-<lb/>riam circuli EG, (nempe inter circulum, & </s> <s xml:id="echoid-s12248" xml:space="preserve">polum D.) </s> <s xml:id="echoid-s12249" xml:space="preserve">& </s> <s xml:id="echoid-s12250" xml:space="preserve">præter eius polum; <lb/></s> <s xml:id="echoid-s12251" xml:space="preserve">erit arcus FE, maior arcu FH, cum ille propin quior ſit arcui FD, per po-<lb/> <anchor type="note" xlink:label="note-378-15a" xlink:href="note-378-15"/> lum D, tranſeunti, & </s> <s xml:id="echoid-s12252" xml:space="preserve">vterque arcus FE, FH, ſemicirculo ſit minor: </s> <s xml:id="echoid-s12253" xml:space="preserve">propte-<lb/> <anchor type="note" xlink:label="note-378-16a" xlink:href="note-378-16"/> rea quòd non ſe interſecant, niſi in puncto F: </s> <s xml:id="echoid-s12254" xml:space="preserve">Oſten ſus eſt autem arcus HF, <lb/>arcui BC, æqualis. </s> <s xml:id="echoid-s12255" xml:space="preserve">Maior ergo erit quoque arcus EF, arcu BC. </s> <s xml:id="echoid-s12256" xml:space="preserve">Quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s12257" xml:space="preserve"/> </p> <div xml:id="echoid-div961" type="float" level="2" n="3"> <note position="left" xlink:label="note-378-10" xlink:href="note-378-10a" xml:space="preserve">1. huius.</note> <figure xlink:label="fig-378-01" xlink:href="fig-378-01a"> <image file="378-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/378-01"/> </figure> <note position="left" xlink:label="note-378-11" xlink:href="note-378-11a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-378-12" xlink:href="note-378-12a" xml:space="preserve">20. huius.</note> <note position="left" xlink:label="note-378-13" xlink:href="note-378-13a" xml:space="preserve">28. tertij.</note> <note position="left" xlink:label="note-378-14" xlink:href="note-378-14a" xml:space="preserve">7. huius.</note> <note position="left" xlink:label="note-378-15" xlink:href="note-378-15a" xml:space="preserve">Schol. 21.</note> <note position="left" xlink:label="note-378-16" xlink:href="note-378-16a" xml:space="preserve">2. Theod.</note> </div> <figure> <image file="378-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/378-02"/> </figure> <p> <s xml:id="echoid-s12258" xml:space="preserve">SED iam baſis EF, maior ſit baſi BC. <lb/></s> <s xml:id="echoid-s12259" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12260" xml:space="preserve">angulum D, maiorem eſſe angulo <lb/>A. </s> <s xml:id="echoid-s12261" xml:space="preserve">Si enim angulus D, maior non eſt angu-<lb/>lo A, erit vel æqualis, vel minor. </s> <s xml:id="echoid-s12262" xml:space="preserve">Si æqualis <lb/>dicatur eſſe, erit arcus EF, æqualis arcui <lb/> <anchor type="note" xlink:label="note-378-17a" xlink:href="note-378-17"/> B C. </s> <s xml:id="echoid-s12263" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s12264" xml:space="preserve">Ponitur enim arcus <lb/>EF, maior arcu BC. </s> <s xml:id="echoid-s12265" xml:space="preserve">Si verò minor dicatur <lb/>eſſe angulus D, angulo A, erit, vt iam oſten <lb/>ſum eſt, arcus BC, maior arcu EF. </s> <s xml:id="echoid-s12266" xml:space="preserve">quod <lb/>etiam abſurdum eſt, cum EF, maior pona- <pb o="367" file="379" n="379" rhead=""/> tur, quàm BC. </s> <s xml:id="echoid-s12267" xml:space="preserve">Cum ergo angulus D, neque æqualis ſit angulo A, neque mi-<lb/>nor, erit vtique maior. </s> <s xml:id="echoid-s12268" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s12269" xml:space="preserve">Itaque ſi duo triangula ſphæ-<lb/>rica, &</s> <s xml:id="echoid-s12270" xml:space="preserve">c. </s> <s xml:id="echoid-s12271" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s12272" xml:space="preserve"/> </p> <div xml:id="echoid-div962" type="float" level="2" n="4"> <note position="left" xlink:label="note-378-17" xlink:href="note-378-17a" xml:space="preserve">9. huius.</note> </div> </div> <div xml:id="echoid-div964" type="section" level="1" n="500"> <head xml:id="echoid-head535" xml:space="preserve">THEOR. 11. PROPOS. 13.</head> <p> <s xml:id="echoid-s12273" xml:space="preserve">DVO ſemicirculi maximorum circulorum <lb/>ſe mutuo ſecantes continent duos angulos inter <lb/>ſe æquales.</s> <s xml:id="echoid-s12274" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12275" xml:space="preserve">DVO ſemicirculi maximorum circulorum ABC, ADC, ſe mutuo ſe-<lb/>cent in A, C. </s> <s xml:id="echoid-s12276" xml:space="preserve">Dico angulos A, & </s> <s xml:id="echoid-s12277" xml:space="preserve">C, æqua-<lb/> <anchor type="figure" xlink:label="fig-379-01a" xlink:href="fig-379-01"/> les eſſe. </s> <s xml:id="echoid-s12278" xml:space="preserve">Diuiſo enim ſemicirculo ABC, in <lb/>B, bifariam, vt AB, BC, quadrantes ſint, <lb/>ducatur per B, & </s> <s xml:id="echoid-s12279" xml:space="preserve">polum circuli ABC, ar-<lb/> <anchor type="note" xlink:label="note-379-01a" xlink:href="note-379-01"/> cus circuli maximi BD, ſecans arcũ ADC, <lb/>in D; </s> <s xml:id="echoid-s12280" xml:space="preserve">eritq̀ angulus B, ex vtraque parte <lb/> <anchor type="note" xlink:label="note-379-02a" xlink:href="note-379-02"/> rectus. </s> <s xml:id="echoid-s12281" xml:space="preserve">Quia igitur duo latera AB, BD, <lb/>duobus lateribus CB, BD, æqualia, ſunt, <lb/>cõtinentq̀ angulos æquales, vtpote rectos; <lb/></s> <s xml:id="echoid-s12282" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12283" xml:space="preserve">anguli A, & </s> <s xml:id="echoid-s12284" xml:space="preserve">C, æquales. </s> <s xml:id="echoid-s12285" xml:space="preserve">Quare duo <lb/> <anchor type="note" xlink:label="note-379-03a" xlink:href="note-379-03"/> ſemicirculi maximorum circulorum, &</s> <s xml:id="echoid-s12286" xml:space="preserve">c. </s> <s xml:id="echoid-s12287" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s12288" xml:space="preserve"/> </p> <div xml:id="echoid-div964" type="float" level="2" n="1"> <figure xlink:label="fig-379-01" xlink:href="fig-379-01a"> <image file="379-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/379-01"/> </figure> <note position="right" xlink:label="note-379-01" xlink:href="note-379-01a" xml:space="preserve">20. 1. Theo.</note> <note position="right" xlink:label="note-379-02" xlink:href="note-379-02a" xml:space="preserve">15. 1. Theo.</note> <note position="right" xlink:label="note-379-03" xlink:href="note-379-03a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div966" type="section" level="1" n="501"> <head xml:id="echoid-head536" xml:space="preserve">THEOR 12. PROPOS. 14.</head> <p> <s xml:id="echoid-s12289" xml:space="preserve">CVIVSCVNQVE trianguli ſphærici vno <lb/>latere producto, ſi reliqua latera ſimul ęqualia ſint <lb/>ſemicirculo, erit angulus externus æqualis angu-<lb/>lo interno oppoſito ſupra arcum productum: </s> <s xml:id="echoid-s12290" xml:space="preserve">Si <lb/>verò minora ſint ſemicirculo, erit angulus exter-<lb/>nus eodem interno oppoſito maior: </s> <s xml:id="echoid-s12291" xml:space="preserve">ſi denique <lb/>maiora ſint ſemicirculo, idem angulus externus <lb/>dicto angulo interno oppoſito minor erit.</s> <s xml:id="echoid-s12292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12293" xml:space="preserve">IN triangulo ſphærico ABC, produca-<lb/> <anchor type="figure" xlink:label="fig-379-02a" xlink:href="fig-379-02"/> tur latus BC, ad D, & </s> <s xml:id="echoid-s12294" xml:space="preserve">ſint primum reliqua <lb/>duo latera AB, AC, ſimul ſemicirculo æqua-<lb/>lia. </s> <s xml:id="echoid-s12295" xml:space="preserve">Dico angulum externum ACD, æqualem <lb/>eſſe interno oppoſito B, ſupra arcum produ-<lb/>ctum BC, &</s> <s xml:id="echoid-s12296" xml:space="preserve">c. </s> <s xml:id="echoid-s12297" xml:space="preserve">Coeat enim arcus BA, produ-<lb/>ctus cum arcu BC, producto in D; </s> <s xml:id="echoid-s12298" xml:space="preserve">eritq̀ue <lb/>BAD, ſemicirculus. </s> <s xml:id="echoid-s12299" xml:space="preserve">Quia vero arcus BA, <lb/> <anchor type="note" xlink:label="note-379-04a" xlink:href="note-379-04"/> <pb o="368" file="380" n="380" rhead=""/> AC, æquales ponuntur ſemicirculo BAD; </s> <s xml:id="echoid-s12300" xml:space="preserve">dempto communi arcu BA, erunt <lb/>reliqui arcus AC, AD, æquales. </s> <s xml:id="echoid-s12301" xml:space="preserve">Quare & </s> <s xml:id="echoid-s12302" xml:space="preserve">angulus ACD, angulo D, æqua-<lb/> <anchor type="note" xlink:label="note-380-01a" xlink:href="note-380-01"/> lis erit. </s> <s xml:id="echoid-s12303" xml:space="preserve">Cum igitur anguli B, & </s> <s xml:id="echoid-s12304" xml:space="preserve">D, ſint quoque æquales, æqualis quoque erit <lb/> <anchor type="note" xlink:label="note-380-02a" xlink:href="note-380-02"/> angulus ACD, angulo B. </s> <s xml:id="echoid-s12305" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12306" xml:space="preserve"/> </p> <div xml:id="echoid-div966" type="float" level="2" n="1"> <figure xlink:label="fig-379-02" xlink:href="fig-379-02a"> <image file="379-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/379-02"/> </figure> <note position="right" xlink:label="note-379-04" xlink:href="note-379-04a" xml:space="preserve">11. 1. Theo.</note> <note position="left" xlink:label="note-380-01" xlink:href="note-380-01a" xml:space="preserve">8. huius.</note> <note position="left" xlink:label="note-380-02" xlink:href="note-380-02a" xml:space="preserve">13. huius.</note> </div> <figure> <image file="380-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/380-01"/> </figure> <p> <s xml:id="echoid-s12307" xml:space="preserve">SINT deinde duo latera AB, AC, mi-<lb/>nora ſemicirculo BAD. </s> <s xml:id="echoid-s12308" xml:space="preserve">Dempto ergo com <lb/>muniarcu AB, erit reliquus AC, reliquo <lb/>AD, minor; </s> <s xml:id="echoid-s12309" xml:space="preserve">ac propterea angulus ACD, <lb/> <anchor type="note" xlink:label="note-380-03a" xlink:href="note-380-03"/> maior angulo D, hoc eſt, angulo B, qui an-<lb/> <anchor type="note" xlink:label="note-380-04a" xlink:href="note-380-04"/> gulo D, æqualis eſt. </s> <s xml:id="echoid-s12310" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12311" xml:space="preserve"/> </p> <div xml:id="echoid-div967" type="float" level="2" n="2"> <note position="left" xlink:label="note-380-03" xlink:href="note-380-03a" xml:space="preserve">11. huius.</note> <note position="left" xlink:label="note-380-04" xlink:href="note-380-04a" xml:space="preserve">13. huius.</note> </div> <p> <s xml:id="echoid-s12312" xml:space="preserve">SINT poſtremo latera AB, AC, ma-<lb/>iora ſemicirculo BAD. </s> <s xml:id="echoid-s12313" xml:space="preserve">Dempto igitur cõ-<lb/>muni arcu AB, erit reliquus AC, reliquo <lb/>AD, maior; </s> <s xml:id="echoid-s12314" xml:space="preserve">ac propterea angulus D, maior erit angulo ACD. </s> <s xml:id="echoid-s12315" xml:space="preserve">Cum ergo <lb/> <anchor type="note" xlink:label="note-380-05a" xlink:href="note-380-05"/> angulo D, æqualis ſit angulus B, erit quoque angulus B, maior angulo ACD, <lb/> <anchor type="note" xlink:label="note-380-06a" xlink:href="note-380-06"/> hoc eſt, angulus ACD, angulo B, minor erit. </s> <s xml:id="echoid-s12316" xml:space="preserve">Cuiuſcunque ergo trianguli, <lb/>&</s> <s xml:id="echoid-s12317" xml:space="preserve">c. </s> <s xml:id="echoid-s12318" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s12319" xml:space="preserve"/> </p> <div xml:id="echoid-div968" type="float" level="2" n="3"> <note position="left" xlink:label="note-380-05" xlink:href="note-380-05a" xml:space="preserve">11. huius.</note> <note position="left" xlink:label="note-380-06" xlink:href="note-380-06a" xml:space="preserve">13. huius.</note> </div> </div> <div xml:id="echoid-div970" type="section" level="1" n="502"> <head xml:id="echoid-head537" xml:space="preserve">THEOR. 13. PROPOS. 15.</head> <p> <s xml:id="echoid-s12320" xml:space="preserve">SI cuiuſcunque trianguli ſphærici vno latere <lb/>producto, externus angulus æqualis fuerit interno <lb/>oppoſito ſupra arcum productum, erunt duo reli-<lb/>qua latera ſimul æqualia ſemicirculo: </s> <s xml:id="echoid-s12321" xml:space="preserve">Si verò an-<lb/>gulus externus maior fuerit interno eodem, & </s> <s xml:id="echoid-s12322" xml:space="preserve">op-<lb/>poſito, erunt duo reliqua latera ſemicirculo mi-<lb/>nora: </s> <s xml:id="echoid-s12323" xml:space="preserve">Si deniq; </s> <s xml:id="echoid-s12324" xml:space="preserve">externus angulus interno oppoſi-<lb/>to dicto minor fuerit, erunt duo latera reliqua ſe-<lb/>micirculo maiora.</s> <s xml:id="echoid-s12325" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12326" xml:space="preserve">POSITO eodem triangulo ſphærico, & </s> <s xml:id="echoid-s12327" xml:space="preserve">conſtructione figuræ eadem; <lb/></s> <s xml:id="echoid-s12328" xml:space="preserve"> <anchor type="figure" xlink:label="fig-380-02a" xlink:href="fig-380-02"/> Sit primum angulus ACD, externus æqua-<lb/>lis interno oppoſito B. </s> <s xml:id="echoid-s12329" xml:space="preserve">Dico latera AB, <lb/>AC, ſemicirculo eſſe ęqualia, &</s> <s xml:id="echoid-s12330" xml:space="preserve">c. </s> <s xml:id="echoid-s12331" xml:space="preserve">Cum enim <lb/>angulus B, angulo D, æqulis ſit, erit quo-<lb/> <anchor type="note" xlink:label="note-380-07a" xlink:href="note-380-07"/> que angulus ACD, angulo D, æqualis; <lb/></s> <s xml:id="echoid-s12332" xml:space="preserve">ideoq̀ & </s> <s xml:id="echoid-s12333" xml:space="preserve">arcus AC, AD, æquales erunt. </s> <s xml:id="echoid-s12334" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-380-08a" xlink:href="note-380-08"/> Addito ergo communi arcu AB, erunt duo <lb/>arcus AB, AC, ſemicirculo BAD, æqua-<lb/>les. </s> <s xml:id="echoid-s12335" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12336" xml:space="preserve"/> </p> <div xml:id="echoid-div970" type="float" level="2" n="1"> <figure xlink:label="fig-380-02" xlink:href="fig-380-02a"> <image file="380-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/380-02"/> </figure> <note position="left" xlink:label="note-380-07" xlink:href="note-380-07a" xml:space="preserve">13. huius.</note> <note position="left" xlink:label="note-380-08" xlink:href="note-380-08a" xml:space="preserve">9. huius.</note> </div> <p> <s xml:id="echoid-s12337" xml:space="preserve">SIT deinde angulus ACD, maior angulo B, hoc eſt, angulo D, quian-<lb/> <anchor type="note" xlink:label="note-380-09a" xlink:href="note-380-09"/> gulo B, æqualis eſt; </s> <s xml:id="echoid-s12338" xml:space="preserve">eritq́ arcus AD, maior arcu AC. </s> <s xml:id="echoid-s12339" xml:space="preserve">Addito ergo commu-<lb/> <anchor type="note" xlink:label="note-380-10a" xlink:href="note-380-10"/> <pb o="369" file="381" n="381" rhead=""/> miarcu AB, erunt duo arcus AB, AC, minores ſemicirculo BAD. </s> <s xml:id="echoid-s12340" xml:space="preserve">Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s12341" xml:space="preserve"/> </p> <div xml:id="echoid-div971" type="float" level="2" n="2"> <note position="left" xlink:label="note-380-09" xlink:href="note-380-09a" xml:space="preserve">13. huius.</note> <note position="left" xlink:label="note-380-10" xlink:href="note-380-10a" xml:space="preserve">11. huius.</note> </div> <p> <s xml:id="echoid-s12342" xml:space="preserve">SIT poſtremò angulus ACD, minor angulo B, hoc eſt, angulo D, qui <lb/> <anchor type="note" xlink:label="note-381-01a" xlink:href="note-381-01"/> angulo B, æqualis eſt; </s> <s xml:id="echoid-s12343" xml:space="preserve">eritq́ue arcus AC, maior arcu AD. </s> <s xml:id="echoid-s12344" xml:space="preserve">Addito ergo com-<lb/> <anchor type="note" xlink:label="note-381-02a" xlink:href="note-381-02"/> muniarcu AB, erunt duo arcus AB, AC, maiores ſemicirculo BAD. </s> <s xml:id="echoid-s12345" xml:space="preserve">Quod <lb/>eſt propoſitum. </s> <s xml:id="echoid-s12346" xml:space="preserve">Si igitur cuiuſcunque trianguli ſphærici, &</s> <s xml:id="echoid-s12347" xml:space="preserve">c. </s> <s xml:id="echoid-s12348" xml:space="preserve">Quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s12349" xml:space="preserve"/> </p> <div xml:id="echoid-div972" type="float" level="2" n="3"> <note position="right" xlink:label="note-381-01" xlink:href="note-381-01a" xml:space="preserve">13. huius.</note> <note position="right" xlink:label="note-381-02" xlink:href="note-381-02a" xml:space="preserve">11. huius.</note> </div> </div> <div xml:id="echoid-div974" type="section" level="1" n="503"> <head xml:id="echoid-head538" xml:space="preserve">THEOR. 14. PROP. 16.</head> <p> <s xml:id="echoid-s12350" xml:space="preserve">SI cuiuſcunque trianguli ſphærici duo latera <lb/>ſimul æqualia ſint ſemicirculo, erunt duo angu-<lb/>li ſupra baſim duobus rectis æquales: </s> <s xml:id="echoid-s12351" xml:space="preserve">Si verò mi-<lb/>nora ſint ſemicirculo, erunt duobus rectis mino-<lb/>res:</s> <s xml:id="echoid-s12352" xml:space="preserve">Si denique ſemicirculo ſint maiora, erunt duo-<lb/>bus rectis maiores.</s> <s xml:id="echoid-s12353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12354" xml:space="preserve">IN triangulo ſphærico ABC, ſint primum duo latera AB, AC, ſemi-<lb/>circulo æqualia. </s> <s xml:id="echoid-s12355" xml:space="preserve">Dico duos angulos B, C, effe æquales duobus rectis, &</s> <s xml:id="echoid-s12356" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s12357" xml:space="preserve">Producto enim arcu BC, ad D, erit angulus ACD, angulo B, æqualis. </s> <s xml:id="echoid-s12358" xml:space="preserve">Cum <lb/> <anchor type="note" xlink:label="note-381-03a" xlink:href="note-381-03"/> ergo duo anguli ad C, duobus ſint rectis æquales; </s> <s xml:id="echoid-s12359" xml:space="preserve">erũt <lb/> <anchor type="note" xlink:label="note-381-04a" xlink:href="note-381-04"/> <anchor type="figure" xlink:label="fig-381-01a" xlink:href="fig-381-01"/> quoque duo anguli B, & </s> <s xml:id="echoid-s12360" xml:space="preserve">ACB, æquales duobus rectis.</s> <s xml:id="echoid-s12361" xml:space="preserve"/> </p> <div xml:id="echoid-div974" type="float" level="2" n="1"> <note position="right" xlink:label="note-381-03" xlink:href="note-381-03a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-381-04" xlink:href="note-381-04a" xml:space="preserve">5 huius.</note> <figure xlink:label="fig-381-01" xlink:href="fig-381-01a"> <image file="381-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/381-01"/> </figure> </div> <p> <s xml:id="echoid-s12362" xml:space="preserve">SINT deinde latera AB, AC, ſemicirculo mi-<lb/>nora. </s> <s xml:id="echoid-s12363" xml:space="preserve">Cum ergo duo anguli ad C, ſint duobus rectis <lb/> <anchor type="note" xlink:label="note-381-05a" xlink:href="note-381-05"/> æquales; </s> <s xml:id="echoid-s12364" xml:space="preserve">& </s> <s xml:id="echoid-s12365" xml:space="preserve">angulus B, minor ſit angulo ACD; </s> <s xml:id="echoid-s12366" xml:space="preserve">erunt <lb/> <anchor type="note" xlink:label="note-381-06a" xlink:href="note-381-06"/> anguli B, & </s> <s xml:id="echoid-s12367" xml:space="preserve">ACB, duobus rectis minores.</s> <s xml:id="echoid-s12368" xml:space="preserve"/> </p> <div xml:id="echoid-div975" type="float" level="2" n="2"> <note position="right" xlink:label="note-381-05" xlink:href="note-381-05a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-381-06" xlink:href="note-381-06a" xml:space="preserve">14. huius.</note> </div> <p> <s xml:id="echoid-s12369" xml:space="preserve">SINT tandem latera AB, AC, ſemicirculo ma-<lb/>iora. </s> <s xml:id="echoid-s12370" xml:space="preserve">Quoniam igitur duo anguli C, ſunt duobus re-<lb/> <anchor type="note" xlink:label="note-381-07a" xlink:href="note-381-07"/> ctis æquales, eſtq́ue angulus B, maior angulo ACB; <lb/></s> <s xml:id="echoid-s12371" xml:space="preserve"> <anchor type="note" xlink:label="note-381-08a" xlink:href="note-381-08"/> erunt anguli B, & </s> <s xml:id="echoid-s12372" xml:space="preserve">ACB, maiores duobus rectis. </s> <s xml:id="echoid-s12373" xml:space="preserve">Si igitur cuiuſcun que trian <lb/>guli ſphærici, &</s> <s xml:id="echoid-s12374" xml:space="preserve">c. </s> <s xml:id="echoid-s12375" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s12376" xml:space="preserve"/> </p> <div xml:id="echoid-div976" type="float" level="2" n="3"> <note position="right" xlink:label="note-381-07" xlink:href="note-381-07a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-381-08" xlink:href="note-381-08a" xml:space="preserve">14. huius.</note> </div> </div> <div xml:id="echoid-div978" type="section" level="1" n="504"> <head xml:id="echoid-head539" xml:space="preserve">THEOR. 15. PROP. 17.</head> <p> <s xml:id="echoid-s12377" xml:space="preserve">SI cuiuſcunque trianguli ſphærici duo anguli <lb/>ſupra vnum latus duobus rectis æquales fuerint, <lb/>erunt reliqua duo latera ſemicirculo æqualia: </s> <s xml:id="echoid-s12378" xml:space="preserve">Si <lb/>vero duobus rectis fuerint minores, erunt minora <lb/>ſemicirculo: </s> <s xml:id="echoid-s12379" xml:space="preserve">Si denique maiores extiterint duo-<lb/>bus rectis, erunt ſemicirculo maiora.</s> <s xml:id="echoid-s12380" xml:space="preserve"/> </p> <pb o="370" file="382" n="382" rhead=""/> <p> <s xml:id="echoid-s12381" xml:space="preserve">POSITO eodem triangulo ſphærico, & </s> <s xml:id="echoid-s12382" xml:space="preserve">conſtructione figuræ eadem; <lb/></s> <s xml:id="echoid-s12383" xml:space="preserve">Sint primum duo anguli B, C, duobus rectis æquales ſupra latus BC. </s> <s xml:id="echoid-s12384" xml:space="preserve">Dico <lb/>reliqua duo latera AB, AC, ſemicirculo æqualia eſſe, &</s> <s xml:id="echoid-s12385" xml:space="preserve">c. </s> <s xml:id="echoid-s12386" xml:space="preserve">Cum enim & </s> <s xml:id="echoid-s12387" xml:space="preserve">an-<lb/> <anchor type="figure" xlink:label="fig-382-01a" xlink:href="fig-382-01"/> guli duo ad C, æquales ſint duobus rectis; </s> <s xml:id="echoid-s12388" xml:space="preserve">dempto <lb/> <anchor type="note" xlink:label="note-382-01a" xlink:href="note-382-01"/> communi angulo ACB, remanebit angulus ACD, <lb/> <anchor type="note" xlink:label="note-382-02a" xlink:href="note-382-02"/> angulo B, æqualis. </s> <s xml:id="echoid-s12389" xml:space="preserve">Quare ſemicirculo æquales ſunt <lb/>arcus AB, AC.</s> <s xml:id="echoid-s12390" xml:space="preserve"/> </p> <div xml:id="echoid-div978" type="float" level="2" n="1"> <figure xlink:label="fig-382-01" xlink:href="fig-382-01a"> <image file="382-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/382-01"/> </figure> <note position="left" xlink:label="note-382-01" xlink:href="note-382-01a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-382-02" xlink:href="note-382-02a" xml:space="preserve">15. huius.</note> </div> <p> <s xml:id="echoid-s12391" xml:space="preserve">SINT deinde anguli B, ACB, duobus rectis mi-<lb/> <anchor type="note" xlink:label="note-382-03a" xlink:href="note-382-03"/> nores. </s> <s xml:id="echoid-s12392" xml:space="preserve">Cum ergo duo anguli ad C, ſint duobus rectis <lb/>æquales; </s> <s xml:id="echoid-s12393" xml:space="preserve">dempto communiangulo ACB, remane-<lb/>bit angulus ACD, maior angulo B. </s> <s xml:id="echoid-s12394" xml:space="preserve">Arcus ergo AB, <lb/> <anchor type="note" xlink:label="note-382-04a" xlink:href="note-382-04"/> AC, ſemicirculo ſunt minores.</s> <s xml:id="echoid-s12395" xml:space="preserve"/> </p> <div xml:id="echoid-div979" type="float" level="2" n="2"> <note position="left" xlink:label="note-382-03" xlink:href="note-382-03a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-382-04" xlink:href="note-382-04a" xml:space="preserve">15. huius.</note> </div> <p> <s xml:id="echoid-s12396" xml:space="preserve">SINT denique anguli B, ACB, duobus rectis maiores. </s> <s xml:id="echoid-s12397" xml:space="preserve">Cum ergo duo <lb/> <anchor type="note" xlink:label="note-382-05a" xlink:href="note-382-05"/> anguli ad C, ſint æquales duobus rectis; </s> <s xml:id="echoid-s12398" xml:space="preserve">ſi dematur communis angulus ACB, <lb/> <anchor type="note" xlink:label="note-382-06a" xlink:href="note-382-06"/> erit reliquus ACD, reliquo B, minor; </s> <s xml:id="echoid-s12399" xml:space="preserve">atque adeo arcus AB, AC, ſemicir-<lb/>culo maiores. </s> <s xml:id="echoid-s12400" xml:space="preserve">Quo circa ſi cuiuſcunque trianguli ſphærici, &</s> <s xml:id="echoid-s12401" xml:space="preserve">c. </s> <s xml:id="echoid-s12402" xml:space="preserve">Quod oſten-<lb/>dendum erat.</s> <s xml:id="echoid-s12403" xml:space="preserve"/> </p> <div xml:id="echoid-div980" type="float" level="2" n="3"> <note position="left" xlink:label="note-382-05" xlink:href="note-382-05a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-382-06" xlink:href="note-382-06a" xml:space="preserve">15. huius.</note> </div> </div> <div xml:id="echoid-div982" type="section" level="1" n="505"> <head xml:id="echoid-head540" xml:space="preserve">THEOR. 16. PROP. 18.</head> <p> <s xml:id="echoid-s12404" xml:space="preserve">SI duo triangula ſphærica habeant tria latera <lb/>tribus lateribus æqualia, ſingula ſingulis: </s> <s xml:id="echoid-s12405" xml:space="preserve">habebũt <lb/>& </s> <s xml:id="echoid-s12406" xml:space="preserve">tres angulos tribus angulis æquales, ſingulos <lb/>ſingulis, ſub quibus æqualia latera ſubtenduntur.</s> <s xml:id="echoid-s12407" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12408" xml:space="preserve">SINT duo triangula ſphærica ABC, DEF, habentia tria latera AB, <lb/>AC, BC, tribus lateribus DE, DF, EF, ſingula ſingulis, æqualia. </s> <s xml:id="echoid-s12409" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12410" xml:space="preserve"><lb/>angulostres A,B,C, tribus angulis D,E,F, ſingulos ſingulis, eſſe æquales, <lb/> <anchor type="figure" xlink:label="fig-382-02a" xlink:href="fig-382-02"/> ſub quibus æqualia ſubtenduntur latera. </s> <s xml:id="echoid-s12411" xml:space="preserve">Si <lb/>enim angulus A, (vt ab hoc angulo incipia-<lb/>mus.) </s> <s xml:id="echoid-s12412" xml:space="preserve">non eſt æqualis angulo D, erit vel ma-<lb/> <anchor type="note" xlink:label="note-382-07a" xlink:href="note-382-07"/> ior eo, vel minor. </s> <s xml:id="echoid-s12413" xml:space="preserve">Si maior, erit baſis BC, ma-<lb/>ior quoque baſi EF. </s> <s xml:id="echoid-s12414" xml:space="preserve">Quod eſt abſurdũ. </s> <s xml:id="echoid-s12415" xml:space="preserve">ponun <lb/>tur enim latera BC, EF, æqualia. </s> <s xml:id="echoid-s12416" xml:space="preserve">Si verò mi-<lb/>nor eſt angulus A, angulo D, erit baſis E F, <lb/> <anchor type="note" xlink:label="note-382-08a" xlink:href="note-382-08"/> maior baſi BC. </s> <s xml:id="echoid-s12417" xml:space="preserve">Quod rurſum eſt abſurdum, <lb/>cum æquales ponantur. </s> <s xml:id="echoid-s12418" xml:space="preserve">Cum ergo angulus A, <lb/>neque maior ſit, neque minor angulo D, erit vtique illi æqualis. </s> <s xml:id="echoid-s12419" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12420" xml:space="preserve">re-<lb/>liqui anguli B, C, angulis reliquis E, F, æquales erunt, nempe B, ipſi E, & </s> <s xml:id="echoid-s12421" xml:space="preserve">C, <lb/> <anchor type="note" xlink:label="note-382-09a" xlink:href="note-382-09"/> ipſi F. </s> <s xml:id="echoid-s12422" xml:space="preserve">Si duo ergo triangula ſphærica, &</s> <s xml:id="echoid-s12423" xml:space="preserve">c. </s> <s xml:id="echoid-s12424" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s12425" xml:space="preserve"/> </p> <div xml:id="echoid-div982" type="float" level="2" n="1"> <figure xlink:label="fig-382-02" xlink:href="fig-382-02a"> <image file="382-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/382-02"/> </figure> <note position="left" xlink:label="note-382-07" xlink:href="note-382-07a" xml:space="preserve">12. huius.</note> <note position="left" xlink:label="note-382-08" xlink:href="note-382-08a" xml:space="preserve">12. huius.</note> <note position="left" xlink:label="note-382-09" xlink:href="note-382-09a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div984" type="section" level="1" n="506"> <head xml:id="echoid-head541" xml:space="preserve">THEOR. 17. PROPOS. 19.</head> <p> <s xml:id="echoid-s12426" xml:space="preserve">SI duo triangula ſphærica habeant tres angu-<lb/>los tribus angulis, ſingulos ſingulis, æquales: </s> <s xml:id="echoid-s12427" xml:space="preserve">habe- <pb o="371" file="383" n="383" rhead=""/> bunt & </s> <s xml:id="echoid-s12428" xml:space="preserve">tria latera tribus lateribus æqualia, ſingu-<lb/>laſingulis, quæ æquales angulos ſubrendunt.</s> <s xml:id="echoid-s12429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12430" xml:space="preserve">HABEANT duo triangula ſphærica ABC, DEF, tres angulos A, <lb/>B, C, tribus angulis D, E, F, ſingulos ſingulis, æquales. </s> <s xml:id="echoid-s12431" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12432" xml:space="preserve">tria latera <lb/>AB, AC, BC, tribus lateribus DE, DF, EF, eſſe æqualia, ſingula ſingulis, <lb/>quæ angulos æquales ſubtendunt. </s> <s xml:id="echoid-s12433" xml:space="preserve">Sienim la-<lb/> <anchor type="figure" xlink:label="fig-383-01a" xlink:href="fig-383-01"/> tera BC, EF, (vt ab his lateribus exordiamur.) <lb/></s> <s xml:id="echoid-s12434" xml:space="preserve">non ſunt æqualia, ſit BC, ſi fieri poteſt, maius; </s> <s xml:id="echoid-s12435" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-383-01a" xlink:href="note-383-01"/> & </s> <s xml:id="echoid-s12436" xml:space="preserve">abſcindatur arcus BG, arcui EF, æqua-<lb/>lis. </s> <s xml:id="echoid-s12437" xml:space="preserve">Aut ergo arcus BA, æqualis eſt arcui ED, <lb/>aut maior, aut minor. </s> <s xml:id="echoid-s12438" xml:space="preserve">Quodcunque horũ di-<lb/>catur, ſequetur abſurdũ ex eo, quòd inæqua-<lb/>lia dicuntur eſſe latera BC, EF, nempe BC, <lb/>maius, quàm E F. </s> <s xml:id="echoid-s12439" xml:space="preserve">Sit enim primum arcus BA, <lb/> <anchor type="note" xlink:label="note-383-02a" xlink:href="note-383-02"/> arcui ED, æqualis; </s> <s xml:id="echoid-s12440" xml:space="preserve">ducaturq́ue per puncta <lb/>A, G, arcus maximi circuli AG. </s> <s xml:id="echoid-s12441" xml:space="preserve">Igitur cum latera BA, BG, æqualia ſint la-<lb/>teribus ED, EF, angulosq́ue contineãt æquales B,E, ex hypotheſi; </s> <s xml:id="echoid-s12442" xml:space="preserve">erunt an-<lb/>guli BAG, & </s> <s xml:id="echoid-s12443" xml:space="preserve">D, æquales: </s> <s xml:id="echoid-s12444" xml:space="preserve">Eſt autem angulus D, poſitus æqualis angulo BAC. <lb/></s> <s xml:id="echoid-s12445" xml:space="preserve"> <anchor type="note" xlink:label="note-383-03a" xlink:href="note-383-03"/> Angulus igitur BAG, æqualis erit quoque angulo BAC, pars toti. </s> <s xml:id="echoid-s12446" xml:space="preserve">Quod <lb/>eſt abſurdum.</s> <s xml:id="echoid-s12447" xml:space="preserve"/> </p> <div xml:id="echoid-div984" type="float" level="2" n="1"> <figure xlink:label="fig-383-01" xlink:href="fig-383-01a"> <image file="383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/383-01"/> </figure> <note position="right" xlink:label="note-383-01" xlink:href="note-383-01a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-383-02" xlink:href="note-383-02a" xml:space="preserve">20. 1. Theo.</note> <note position="right" xlink:label="note-383-03" xlink:href="note-383-03a" xml:space="preserve">7. huius.</note> </div> <note position="right" xml:space="preserve">1. huius.</note> <p> <s xml:id="echoid-s12448" xml:space="preserve">SIT deinde arcus BA, maior arcu ED, & </s> <s xml:id="echoid-s12449" xml:space="preserve">abſcindatur arcus BI, æqua-<lb/>lis ipſi ED; </s> <s xml:id="echoid-s12450" xml:space="preserve">ac per puncta G,I, arcus circuli maximi ducatur GI, conueniens <lb/> <anchor type="note" xlink:label="note-383-05a" xlink:href="note-383-05"/> cum arcu CA, protracto in H. </s> <s xml:id="echoid-s12451" xml:space="preserve">Quoniam igitur latera BI, BG, æqualia ſunt <lb/> <anchor type="note" xlink:label="note-383-06a" xlink:href="note-383-06"/> lateribus ED, EF, angulosq́ue continent æquales B, E; </s> <s xml:id="echoid-s12452" xml:space="preserve">erunt anguli BIG, <lb/>BGI, angulis D, F, hoc eſt, angulis BAC, BCA, æquales; </s> <s xml:id="echoid-s12453" xml:space="preserve">quod his duobus <lb/> <anchor type="note" xlink:label="note-383-07a" xlink:href="note-383-07"/> æquales ſint poſiti D, & </s> <s xml:id="echoid-s12454" xml:space="preserve">F; </s> <s xml:id="echoid-s12455" xml:space="preserve">ſunt autem anguli BIG, BAC, angulis HIA, <lb/>HAK, ad verticem æquales. </s> <s xml:id="echoid-s12456" xml:space="preserve">Aequales ergo ſunt & </s> <s xml:id="echoid-s12457" xml:space="preserve">anguli HAK, HIA. </s> <s xml:id="echoid-s12458" xml:space="preserve">Igi-<lb/>tur cum & </s> <s xml:id="echoid-s12459" xml:space="preserve">angulus BGH, externus æqualis ſit interno BCH, & </s> <s xml:id="echoid-s12460" xml:space="preserve">externus <lb/>HAK, interno HIK, vt oſtendimus: </s> <s xml:id="echoid-s12461" xml:space="preserve">erunt tam arcus AH, HI, quam arcus <lb/> <anchor type="note" xlink:label="note-383-08a" xlink:href="note-383-08"/> CH, HG, ſemicirculo æquales; </s> <s xml:id="echoid-s12462" xml:space="preserve">atque adeo arcus AH, HI, arcubus CH, <lb/>HG, æquales erunt, pars toti. </s> <s xml:id="echoid-s12463" xml:space="preserve">Quod eſt abſurdum.</s> <s xml:id="echoid-s12464" xml:space="preserve"/> </p> <div xml:id="echoid-div985" type="float" level="2" n="2"> <note position="right" xlink:label="note-383-05" xlink:href="note-383-05a" xml:space="preserve">20. 1. Theo.</note> <note position="right" xlink:label="note-383-06" xlink:href="note-383-06a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-383-07" xlink:href="note-383-07a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-383-08" xlink:href="note-383-08a" xml:space="preserve">15. huius.</note> </div> <p> <s xml:id="echoid-s12465" xml:space="preserve">SIT tandem arcus BA, minor arcu ED, producaturq́ue vltra A, & </s> <s xml:id="echoid-s12466" xml:space="preserve">ex eo <lb/>abſcindatur arcus BK, æqualis arcui ED; </s> <s xml:id="echoid-s12467" xml:space="preserve">atque per puncta G, K, arcus cir-<lb/> <anchor type="note" xlink:label="note-383-09a" xlink:href="note-383-09"/> culi maximi ducatur GK, ſecans arcum AC, in L. </s> <s xml:id="echoid-s12468" xml:space="preserve">Quoniam ergo latera BK, <lb/> <anchor type="note" xlink:label="note-383-10a" xlink:href="note-383-10"/> BG, lateribus ED, EF, æqualia ſunt, anguloſque continent æquales B, E, <lb/>erunt & </s> <s xml:id="echoid-s12469" xml:space="preserve">anguli BKG, BGK, angulis D, F, hoc eſt, angulis BAC, BCA, <lb/> <anchor type="note" xlink:label="note-383-11a" xlink:href="note-383-11"/> (quòd his duobus æquales ſint poſiti anguli D, F.) </s> <s xml:id="echoid-s12470" xml:space="preserve">æquales. </s> <s xml:id="echoid-s12471" xml:space="preserve">Itaque cum & </s> <s xml:id="echoid-s12472" xml:space="preserve"><lb/>angulus BAL, externus æqualis ſit interno BKL, & </s> <s xml:id="echoid-s12473" xml:space="preserve">externus BGL, inter-<lb/>no BCL, vt oſtendimus, erunt tam arcus AL, LK, quàm arcus CL, LG, <lb/> <anchor type="note" xlink:label="note-383-12a" xlink:href="note-383-12"/> ſemicirculo æquales; </s> <s xml:id="echoid-s12474" xml:space="preserve">ac proinde duo arcus AC, GK, integro circulo æqua-<lb/>les erunt. </s> <s xml:id="echoid-s12475" xml:space="preserve">Quod eſt abſurdum, cum vterque arcus AC, GK, ſemicirculo ſit <lb/> <anchor type="note" xlink:label="note-383-13a" xlink:href="note-383-13"/> minor. </s> <s xml:id="echoid-s12476" xml:space="preserve">Non ergo inæqualia ſunt latera BC, EF, ſed æqualia. </s> <s xml:id="echoid-s12477" xml:space="preserve">Eodemq́ue mo-<lb/>do oſtendemus, latera AC, DF, nec non AB, DE, æqualia eſſe. </s> <s xml:id="echoid-s12478" xml:space="preserve">Tria ergo <lb/>latera trianguli ABC, tribus lateribus trianguli DEF, æqualia ſunt. </s> <s xml:id="echoid-s12479" xml:space="preserve">Quare <lb/>ſi duo triangula ſphærica, &</s> <s xml:id="echoid-s12480" xml:space="preserve">c. </s> <s xml:id="echoid-s12481" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s12482" xml:space="preserve"/> </p> <div xml:id="echoid-div986" type="float" level="2" n="3"> <note position="right" xlink:label="note-383-09" xlink:href="note-383-09a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-383-10" xlink:href="note-383-10a" xml:space="preserve">20. 1. Theo.</note> <note position="right" xlink:label="note-383-11" xlink:href="note-383-11a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-383-12" xlink:href="note-383-12a" xml:space="preserve">15, huius.</note> <note position="right" xlink:label="note-383-13" xlink:href="note-383-13a" xml:space="preserve">2. huius.</note> </div> <pb o="372" file="384" n="384" rhead=""/> </div> <div xml:id="echoid-div988" type="section" level="1" n="507"> <head xml:id="echoid-head542" xml:space="preserve">THEOR. 18. PROPOS. 20.</head> <p> <s xml:id="echoid-s12483" xml:space="preserve">SI duo triangula ſphærica duos angulos duo-<lb/>bus angulis æquales habuerint, vtrumque vtrique, <lb/>vnumque latus vni lateri æquale, quod æqualibus <lb/>adiacet angulis: </s> <s xml:id="echoid-s12484" xml:space="preserve">Et reliqua latera reliquis lateribus <lb/>æqualia, vtrumque vtrique, & </s> <s xml:id="echoid-s12485" xml:space="preserve">reliquum angulum <lb/>reliquo angulo æqualem habebunt.</s> <s xml:id="echoid-s12486" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12487" xml:space="preserve">DVO triangula ſphærica ABC, DEF, habeant duos angulos B, C, duo-<lb/>bus angulis E, F, æquales vtrumque vtrique, & </s> <s xml:id="echoid-s12488" xml:space="preserve">latus BC, lateri EF, æquale, <lb/>quod æqualibus angulis adiacet. </s> <s xml:id="echoid-s12489" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12490" xml:space="preserve">reliqua latera AB, AC, reliquis la-<lb/> <anchor type="figure" xlink:label="fig-384-01a" xlink:href="fig-384-01"/> teribus DE, DF, æqualia eſſe, vtrumq; </s> <s xml:id="echoid-s12491" xml:space="preserve">vtri-<lb/>que, & </s> <s xml:id="echoid-s12492" xml:space="preserve">reliquum angulum A, reliquo angulo <lb/>D. </s> <s xml:id="echoid-s12493" xml:space="preserve">Si enim latera AB, DE, (vt ab his exor-<lb/>diamur.) </s> <s xml:id="echoid-s12494" xml:space="preserve">non ſunt æqualia, ſit AB, maius, & </s> <s xml:id="echoid-s12495" xml:space="preserve"><lb/>abſcindatur arcus BG, arcui DE, æqualis, <lb/> <anchor type="note" xlink:label="note-384-01a" xlink:href="note-384-01"/> & </s> <s xml:id="echoid-s12496" xml:space="preserve">per puncta C, G, arcus circuli maximi du-<lb/> <anchor type="note" xlink:label="note-384-02a" xlink:href="note-384-02"/> catur C G. </s> <s xml:id="echoid-s12497" xml:space="preserve">Quoniam igitur latera GB, B C, <lb/>æqualia ſunt lateribus DE, EF, angulosq́uc <lb/>comprehendunt æquales B, E, ex hypotheſi; <lb/></s> <s xml:id="echoid-s12498" xml:space="preserve"> <anchor type="note" xlink:label="note-384-03a" xlink:href="note-384-03"/> erunt & </s> <s xml:id="echoid-s12499" xml:space="preserve">anguli BCG, & </s> <s xml:id="echoid-s12500" xml:space="preserve">F, æquales: </s> <s xml:id="echoid-s12501" xml:space="preserve">Sed F, <lb/>æqualis ponitur ipſi BCA. </s> <s xml:id="echoid-s12502" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12503" xml:space="preserve">angulus BCG, eidem BCA, æqualis <lb/>erit, pars toti. </s> <s xml:id="echoid-s12504" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s12505" xml:space="preserve">Non ergo inæqualia ſunt latera AB, DE, <lb/>fed æqualia. </s> <s xml:id="echoid-s12506" xml:space="preserve">Quare cum latera AB, BC, lateribus DE, EF, æqualia ſint, an-<lb/>gulosq́ue comprehendantæquales B, E; </s> <s xml:id="echoid-s12507" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12508" xml:space="preserve">latera AC, DF, æqualia, & </s> <s xml:id="echoid-s12509" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-384-04a" xlink:href="note-384-04"/> anguli A, D, æquales. </s> <s xml:id="echoid-s12510" xml:space="preserve">Quapropter ſi duo triangula ſphærica duos angulos, <lb/>&</s> <s xml:id="echoid-s12511" xml:space="preserve">c. </s> <s xml:id="echoid-s12512" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s12513" xml:space="preserve"/> </p> <div xml:id="echoid-div988" type="float" level="2" n="1"> <figure xlink:label="fig-384-01" xlink:href="fig-384-01a"> <image file="384-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/384-01"/> </figure> <note position="left" xlink:label="note-384-01" xlink:href="note-384-01a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-384-02" xlink:href="note-384-02a" xml:space="preserve">20. 1. Theo.</note> <note position="left" xlink:label="note-384-03" xlink:href="note-384-03a" xml:space="preserve">7. huius.</note> <note position="left" xlink:label="note-384-04" xlink:href="note-384-04a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div990" type="section" level="1" n="508"> <head xml:id="echoid-head543" xml:space="preserve">THEOR. 19. PROPOS. 21.</head> <p> <s xml:id="echoid-s12514" xml:space="preserve">SI fuerint duo triangula ſphærica rectangula, <lb/>habuerintque duos alios angulos æquales, & </s> <s xml:id="echoid-s12515" xml:space="preserve">non <lb/>rectos, nec non duo latera æqualia, quæ ſub rectis <lb/>angulis ſubtenduntur: </s> <s xml:id="echoid-s12516" xml:space="preserve">Erunt & </s> <s xml:id="echoid-s12517" xml:space="preserve">duo reliqua latera <lb/>duobus lateribus æqualia, vtrumque vtrique, & </s> <s xml:id="echoid-s12518" xml:space="preserve">re-<lb/>liquus angulus reliquo angulo æqualis erit.</s> <s xml:id="echoid-s12519" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12520" xml:space="preserve">SINT in duobus triangulis ſphæricis ABC, DEF, anguli B, E, recti, <lb/>& </s> <s xml:id="echoid-s12521" xml:space="preserve">duo anguli C, F, æquales, & </s> <s xml:id="echoid-s12522" xml:space="preserve">non recti, nec non latera AC, DF, rectos an- <pb o="373" file="385" n="385" rhead=""/> gulos ſubtendentia, æqualia. </s> <s xml:id="echoid-s12523" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12524" xml:space="preserve">reliqua latera AB, BC, reliquis lateri-<lb/>bus DE, EF, æqualia eſſe, vtrumque vtrique; </s> <s xml:id="echoid-s12525" xml:space="preserve">Item & </s> <s xml:id="echoid-s12526" xml:space="preserve">reliquos angulos A, <lb/>D, eſſe æquales. </s> <s xml:id="echoid-s12527" xml:space="preserve">Productis enim arcubus AC, BC, abſcindatur arcus CH, ar-<lb/> <anchor type="note" xlink:label="note-385-01a" xlink:href="note-385-01"/> cui FD, hoc eſt, arcui CA, & </s> <s xml:id="echoid-s12528" xml:space="preserve">arcus CG, arcui FE, æqualis; </s> <s xml:id="echoid-s12529" xml:space="preserve">& </s> <s xml:id="echoid-s12530" xml:space="preserve">per puncta G, <lb/>H, deſcribatur arcus GH, maximi circuli. </s> <s xml:id="echoid-s12531" xml:space="preserve">Et quo-<lb/> <anchor type="note" xlink:label="note-385-02a" xlink:href="note-385-02"/> <anchor type="figure" xlink:label="fig-385-01a" xlink:href="fig-385-01"/> niã latera CH, CG, æqualia ſunt lateribus FD, <lb/>FE, angulosq́ue continent æquales GCH, & </s> <s xml:id="echoid-s12532" xml:space="preserve">F; <lb/></s> <s xml:id="echoid-s12533" xml:space="preserve">(Eſt enim ex hypotheſi angulus F, angulo ACB, <lb/>æqualis, & </s> <s xml:id="echoid-s12534" xml:space="preserve">ACB, ipſi GCH, ad verticem æqua-<lb/> <anchor type="note" xlink:label="note-385-03a" xlink:href="note-385-03"/> lis,) erunt & </s> <s xml:id="echoid-s12535" xml:space="preserve">baſes GH, ED, æquales, & </s> <s xml:id="echoid-s12536" xml:space="preserve">anguli G, <lb/> <anchor type="note" xlink:label="note-385-04a" xlink:href="note-385-04"/> H, angulis E, D, æquales; </s> <s xml:id="echoid-s12537" xml:space="preserve">ac propterea, exiſten-<lb/>te angulo E, recto, erit & </s> <s xml:id="echoid-s12538" xml:space="preserve">angulus G, rectus. </s> <s xml:id="echoid-s12539" xml:space="preserve">Duca-<lb/>tur iam per C, & </s> <s xml:id="echoid-s12540" xml:space="preserve">polum arcus BG, in vtramque <lb/> <anchor type="note" xlink:label="note-385-05a" xlink:href="note-385-05"/> partem arcus circuli maximi ICK, ſitq́ue I, po-<lb/>lus arcus BG. </s> <s xml:id="echoid-s12541" xml:space="preserve">Et quia circuli arcuum BA, HG, <lb/>tranſeunt quoque per polos eiuſdem arcus BG, <lb/> <anchor type="note" xlink:label="note-385-06a" xlink:href="note-385-06"/> ob angulos rectos B, G; </s> <s xml:id="echoid-s12542" xml:space="preserve">conuenient arcus BA, <lb/>GH, protracticum arcu CI, in polo I. </s> <s xml:id="echoid-s12543" xml:space="preserve">Conue-<lb/>niat quoque arcus GH, ex altera parte cum <lb/>eodem arcu ICK, in K, puncto, quod alter polus erit arcus BG, cum vter-<lb/> <anchor type="note" xlink:label="note-385-07a" xlink:href="note-385-07"/> que arcus ICK, IGK, per alterum polum arcus BG, tranſeat. </s> <s xml:id="echoid-s12544" xml:space="preserve">Erunt igitur <lb/> <anchor type="note" xlink:label="note-385-08a" xlink:href="note-385-08"/> tres arcus IB, IC, IG, æquales; </s> <s xml:id="echoid-s12545" xml:space="preserve">propterea quòd rectæ ſubtenſæ illis inter ſe <lb/> <anchor type="note" xlink:label="note-385-09a" xlink:href="note-385-09"/> æquales ſunt, ex definitione poli: </s> <s xml:id="echoid-s12546" xml:space="preserve">Similiterq́ue æquales erunt arcus KC, KG. <lb/></s> <s xml:id="echoid-s12547" xml:space="preserve">Quoniam verò anguli ICG, IGC, æquales ſunt angulis KCG, KGC, <lb/>cum omnes ſint recti; </s> <s xml:id="echoid-s12548" xml:space="preserve">quòd I, polus ſit arcus BG; </s> <s xml:id="echoid-s12549" xml:space="preserve">illisq́ue adiacet latus <lb/> <anchor type="note" xlink:label="note-385-10a" xlink:href="note-385-10"/> commune CG; </s> <s xml:id="echoid-s12550" xml:space="preserve">erunt latera IC, IG, lateribus KC, KG, æqualia, vtrun-<lb/> <anchor type="note" xlink:label="note-385-11a" xlink:href="note-385-11"/> que vtrique; </s> <s xml:id="echoid-s12551" xml:space="preserve">ac propterea cum IG, arcus arcui IB, æqualis ſit oſtenſus, erit <lb/>& </s> <s xml:id="echoid-s12552" xml:space="preserve">arcus KG, eidem arcui IB, æqualis. </s> <s xml:id="echoid-s12553" xml:space="preserve">Et quoniam latera IC, CA, æqualia <lb/>ſunt lateribus KC, CH, (factus enim eſt arcus CH, arcui AC, æqualis.) </s> <s xml:id="echoid-s12554" xml:space="preserve">an-<lb/>gulosq́ue ad verticẽ continent æquales; </s> <s xml:id="echoid-s12555" xml:space="preserve">erunt baſes IA, KH, & </s> <s xml:id="echoid-s12556" xml:space="preserve">anguli IAC, <lb/> <anchor type="note" xlink:label="note-385-12a" xlink:href="note-385-12"/> KHC, æquales. </s> <s xml:id="echoid-s12557" xml:space="preserve">Ablatis ergo arcubus æqualibus IA, KH, ex arcubus æqua-<lb/> <anchor type="note" xlink:label="note-385-13a" xlink:href="note-385-13"/> libus IB, KG, & </s> <s xml:id="echoid-s12558" xml:space="preserve">angulis æqualibus IAC, KHC, ex binis ad A, & </s> <s xml:id="echoid-s12559" xml:space="preserve">H, quo-<lb/>rum bini duobus rectis æquales ſunt; </s> <s xml:id="echoid-s12560" xml:space="preserve">remanebunt & </s> <s xml:id="echoid-s12561" xml:space="preserve">arcus AB, HG, & </s> <s xml:id="echoid-s12562" xml:space="preserve">angu-<lb/> <anchor type="note" xlink:label="note-385-14a" xlink:href="note-385-14"/> li BAC, GHC, æquales: </s> <s xml:id="echoid-s12563" xml:space="preserve">oſten ſus eſt autem arcus HG, arcui DE, & </s> <s xml:id="echoid-s12564" xml:space="preserve">angulus <lb/>GHC, angulo D, ęqualis. </s> <s xml:id="echoid-s12565" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12566" xml:space="preserve">arcus AB, arcui DE, & </s> <s xml:id="echoid-s12567" xml:space="preserve">angulus BAC, an-<lb/>gulo D, æqualis erit. </s> <s xml:id="echoid-s12568" xml:space="preserve">Quare cum latera AB, AC, æqualia ſint lateribus DE, <lb/>DF, angulosq́ue complectantur æquales; </s> <s xml:id="echoid-s12569" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12570" xml:space="preserve">arcus BC, EF, æquales. <lb/></s> <s xml:id="echoid-s12571" xml:space="preserve"> <anchor type="note" xlink:label="note-385-15a" xlink:href="note-385-15"/> Sunt ergo latera AB, BC, lateribus DE, EF, æqualia, & </s> <s xml:id="echoid-s12572" xml:space="preserve">angulus BAC, an-<lb/>gulo D. </s> <s xml:id="echoid-s12573" xml:space="preserve">Quamobrem, ſi fuerint duo triangula ſphærica rectangula, &</s> <s xml:id="echoid-s12574" xml:space="preserve">c. </s> <s xml:id="echoid-s12575" xml:space="preserve">Quod <lb/>demonſtrandum erat.</s> <s xml:id="echoid-s12576" xml:space="preserve"/> </p> <div xml:id="echoid-div990" type="float" level="2" n="1"> <note position="right" xlink:label="note-385-01" xlink:href="note-385-01a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-385-02" xlink:href="note-385-02a" xml:space="preserve">20. 1. Theo.</note> <figure xlink:label="fig-385-01" xlink:href="fig-385-01a"> <image file="385-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/385-01"/> </figure> <note position="right" xlink:label="note-385-03" xlink:href="note-385-03a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-385-04" xlink:href="note-385-04a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-385-05" xlink:href="note-385-05a" xml:space="preserve">20. 1. Theo.</note> <note position="right" xlink:label="note-385-06" xlink:href="note-385-06a" xml:space="preserve">13. 1. Theo.</note> <note position="right" xlink:label="note-385-07" xlink:href="note-385-07a" xml:space="preserve">Coroll. 10.</note> <note position="right" xlink:label="note-385-08" xlink:href="note-385-08a" xml:space="preserve">1. Theod.</note> <note position="right" xlink:label="note-385-09" xlink:href="note-385-09a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-385-10" xlink:href="note-385-10a" xml:space="preserve">15. 1. Theo.</note> <note position="right" xlink:label="note-385-11" xlink:href="note-385-11a" xml:space="preserve">20. huius.</note> <note position="right" xlink:label="note-385-12" xlink:href="note-385-12a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-385-13" xlink:href="note-385-13a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-385-14" xlink:href="note-385-14a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-385-15" xlink:href="note-385-15a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div992" type="section" level="1" n="509"> <head xml:id="echoid-head544" xml:space="preserve">SCHOLIVM</head> <p style="it"> <s xml:id="echoid-s12577" xml:space="preserve">_DEBENT_ autem latera æqualia ſub rectis angulis ſubtendi. </s> <s xml:id="echoid-s12578" xml:space="preserve">Alioquin, ſi alios <lb/>angulos ſubtenderent, nihil certi colligi poßet. </s> <s xml:id="echoid-s12579" xml:space="preserve">Sit enim triangulum ſphæricum quod-<lb/>cunque ABC, habens duo latera _AB, AC,_ inæqualia inter ſe, ſed ſimul ſemicircu-<lb/>lo æqualia: </s> <s xml:id="echoid-s12580" xml:space="preserve">producto verò latere _CB,_ ad partes _B,_ ducatur per _A,_ & </s> <s xml:id="echoid-s12581" xml:space="preserve">polum arcus <lb/> <anchor type="note" xlink:label="note-385-16a" xlink:href="note-385-16"/> _CD,_ arcus _AD,_ circuli maximi ſecans _CD,_ in _D;_ </s> <s xml:id="echoid-s12582" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s12583" xml:space="preserve">angulus _D,_ rectus. </s> <s xml:id="echoid-s12584" xml:space="preserve">Quo-<lb/> <anchor type="note" xlink:label="note-385-17a" xlink:href="note-385-17"/> niam igitur arcus _AB, AC,_ ſemicirculo ſunt æquales, erit angulus _ABD,_ angulo <lb/> <anchor type="note" xlink:label="note-385-18a" xlink:href="note-385-18"/> <pb o="374" file="386" n="386" rhead=""/> _C,_ æqualis. </s> <s xml:id="echoid-s12585" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s12586" xml:space="preserve">duo triangula _ADB, ADC,_ angulum rectum _D,_ habent commu-<lb/>nem, & </s> <s xml:id="echoid-s12587" xml:space="preserve">duos angulos _ABD,_ & </s> <s xml:id="echoid-s12588" xml:space="preserve">_C,_ æquales, & </s> <s xml:id="echoid-s12589" xml:space="preserve">non rectos: </s> <s xml:id="echoid-s12590" xml:space="preserve">(alias latera _AB, AC,_ <lb/> <anchor type="note" xlink:label="note-386-01a" xlink:href="note-386-01"/> æqualia eſſent, propter angulos _B, C,_ rectos, & </s> <s xml:id="echoid-s12591" xml:space="preserve">æquales:) </s> <s xml:id="echoid-s12592" xml:space="preserve">nec non latus _AD,_ æqua-<lb/> <anchor type="figure" xlink:label="fig-386-01a" xlink:href="fig-386-01"/> les angulos non rectos ſubtendens, commune: </s> <s xml:id="echoid-s12593" xml:space="preserve">Et tamen <lb/>nec reliqua latera _AB, BD,_ reliquis lateribus _AC,_ <lb/>_CD,_ æqualia ſunt, vtrumque vtrique, nec reliquus an <lb/>gulus _BAD,_ reliquo angulo _CAD,_ vt perſpicuum eſt. <lb/></s> <s xml:id="echoid-s12594" xml:space="preserve">Hoc autem indeprouenit, quòd latera æqualia non ſub-<lb/>tendunt rectos angulos, ſed latus commune _AD,_ angu-<lb/>los æquales non rectos ſubtendit.</s> <s xml:id="echoid-s12595" xml:space="preserve"/> </p> <div xml:id="echoid-div992" type="float" level="2" n="1"> <note position="right" xlink:label="note-385-16" xlink:href="note-385-16a" xml:space="preserve">_20. 1. Theo._</note> <note position="right" xlink:label="note-385-17" xlink:href="note-385-17a" xml:space="preserve">15. 1. Th</note> <note position="right" xlink:label="note-385-18" xlink:href="note-385-18a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-386-01" xlink:href="note-386-01a" xml:space="preserve">9. huius.</note> <figure xlink:label="fig-386-01" xlink:href="fig-386-01a"> <image file="386-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/386-01"/> </figure> </div> <note position="left" xml:space="preserve">Error Nico <lb/>lai Coper-<lb/>nici.</note> <p> <s xml:id="echoid-s12596" xml:space="preserve">_QVAMOBREM decipitur Nicolaus Copernicus_ <lb/>_lib. </s> <s xml:id="echoid-s12597" xml:space="preserve">I. </s> <s xml:id="echoid-s12598" xml:space="preserve">Reuolutionum propoſ. </s> <s xml:id="echoid-s12599" xml:space="preserve">6. </s> <s xml:id="echoid-s12600" xml:space="preserve">triãgulorũ ſphæricorum,_ <lb/>_vbi dicit._ </s> <s xml:id="echoid-s12601" xml:space="preserve">[Si bina triangula rectum angulum, ac in-<lb/>fuper alium ęqualem habuerint, alterum alteri, vnumq; </s> <s xml:id="echoid-s12602" xml:space="preserve">latus vni lateri æqua-<lb/>le, quod alterutri ęqualiũ angulorum (etiã non recto, vt in demonſtratione di <lb/>cit) opponitur; </s> <s xml:id="echoid-s12603" xml:space="preserve">reliqua quoque latera reliquis lateribus, alterũ alteri, acan-<lb/>gulum angulo, reliquũ reliquo æqualem habebunt.</s> <s xml:id="echoid-s12604" xml:space="preserve">] _Oppoſitum enim apparuit_ <lb/>_in triangulis rectangulis_ ADB, ADC, _in quibus latus commune_ AD, _opponitur_ <lb/>_angulis æqualibus_ ABD, ACD, _non rectis._</s> <s xml:id="echoid-s12605" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12606" xml:space="preserve">_Vnde verum non ſemper eſt, quod idem Copernicus docet ibidem propoſ. </s> <s xml:id="echoid-s12607" xml:space="preserve">4. </s> <s xml:id="echoid-s12608" xml:space="preserve">vbi ait._ <lb/></s> <s xml:id="echoid-s12609" xml:space="preserve"> <anchor type="note" xlink:label="note-386-03a" xlink:href="note-386-03"/> [In quocunq; </s> <s xml:id="echoid-s12610" xml:space="preserve">triãgulo rectum angulum habente, alius inſuper angulus fuerit <lb/>datus, cum quolibet latere, reliquus etiam angulus cum reliquis lateribus da-<lb/>bitur.</s> <s xml:id="echoid-s12611" xml:space="preserve">] _Quamuis enim angulus rectus_ D, _& </s> <s xml:id="echoid-s12612" xml:space="preserve">angulus_ ABD, _noti ſint, cum latere_ AD, <lb/>_quod angulo noto_ ABD, _non recto opponitur, non tamen proptereain cognitionem <lb/>reliqui anguli, & </s> <s xml:id="echoid-s12613" xml:space="preserve">reliquorum laterum veniemus, cum reliqua latera poſsint eſſe vel <lb/>AB, BD, _vel_ AC, CD, & </s> <s xml:id="echoid-s12614" xml:space="preserve">_reliquus angulus vel_ BAD, _vel_ CAD, _vt perſpicuum_ <lb/>_eſt. </s> <s xml:id="echoid-s12615" xml:space="preserve">Oportebit ergo aliquid aliud præterea constare, antequam reliquus angulus cum_ <lb/>_reliquis lateribus colligatur, vtin ſcholio propoſ. </s> <s xml:id="echoid-s12616" xml:space="preserve">45. </s> <s xml:id="echoid-s12617" xml:space="preserve">docebimus._</s> <s xml:id="echoid-s12618" xml:space="preserve"/> </p> <div xml:id="echoid-div993" type="float" level="2" n="2"> <note position="left" xlink:label="note-386-03" xlink:href="note-386-03a" xml:space="preserve">Alius error <lb/>Nicolai co <lb/>pernici.</note> </div> </div> <div xml:id="echoid-div995" type="section" level="1" n="510"> <head xml:id="echoid-head545" xml:space="preserve">THEOR. 20. PROPOS. 22.</head> <p> <s xml:id="echoid-s12619" xml:space="preserve">SI fuerint duo triangula ſphærica, quæ duos <lb/>angulos habeant duobus angulis æquales, vtrum-<lb/>que vtrique, vnumq; </s> <s xml:id="echoid-s12620" xml:space="preserve">latus vni lateri æquale, quod <lb/>vni æqualium angulorum ſubtenditur, duo verò <lb/>latera ſubtendẽtia reliquos angulos æquales ęqua-<lb/>lia non ſint ſemicirculo, ſed vel maiora, vel mino-<lb/>ra: </s> <s xml:id="echoid-s12621" xml:space="preserve">Erunt & </s> <s xml:id="echoid-s12622" xml:space="preserve">duo reliqua latera duobus reliquis la-<lb/>teribus æqualia, vtrum que vtrique, & </s> <s xml:id="echoid-s12623" xml:space="preserve">reliquus an-<lb/>gulus reliquo angulo æqualis erit.</s> <s xml:id="echoid-s12624" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12625" xml:space="preserve">HABEANT duo triangula ſphærica ABC, DEF, duos angulos B, <pb o="375" file="387" n="387" rhead=""/> C, duobus angulis E, F, æquales, vtrumque vtrique, & </s> <s xml:id="echoid-s12626" xml:space="preserve">latera AC, DF, ſub-<lb/>tendentia angulos æquales B, E, inter ſe æqualia, reliqua verò latera AB, <lb/>DE, ſubtendentia alios æquales angulos C,F, non æqualia ſint ſemicirculo, <lb/>ſed vel maiora, vel minora. </s> <s xml:id="echoid-s12627" xml:space="preserve">Dico reliqua latera <lb/> <anchor type="figure" xlink:label="fig-387-01a" xlink:href="fig-387-01"/> CB, BA, reliquis lateribus FE, ED, eſſe æqua-<lb/>lia, vtrumque vtrique, & </s> <s xml:id="echoid-s12628" xml:space="preserve">reliquos quoque an-<lb/>gulos A, D, eſſe æquales. </s> <s xml:id="echoid-s12629" xml:space="preserve">Si enim CB, & </s> <s xml:id="echoid-s12630" xml:space="preserve">FE, <lb/>non ſunt æqualia, ſit CB, maius, & </s> <s xml:id="echoid-s12631" xml:space="preserve">abſcindatur <lb/>CG, arcus arcui FE, æqualis, & </s> <s xml:id="echoid-s12632" xml:space="preserve">per A, G, ar-<lb/> <anchor type="note" xlink:label="note-387-01a" xlink:href="note-387-01"/> cus circuli maximi ducatur AG. </s> <s xml:id="echoid-s12633" xml:space="preserve">Quoniam igi-<lb/> <anchor type="note" xlink:label="note-387-02a" xlink:href="note-387-02"/> tur latera AC, CG, lateribus DF, FE, æqua-<lb/>lia ſunt, angulosq́ue continent æquales C, F; <lb/></s> <s xml:id="echoid-s12634" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12635" xml:space="preserve">arcus AG, DE, & </s> <s xml:id="echoid-s12636" xml:space="preserve">anguli AGC, & </s> <s xml:id="echoid-s12637" xml:space="preserve">E, æquales: </s> <s xml:id="echoid-s12638" xml:space="preserve">Poſitus eſt autem an-<lb/> <anchor type="note" xlink:label="note-387-03a" xlink:href="note-387-03"/> gulus E, angulo B, æqualis. </s> <s xml:id="echoid-s12639" xml:space="preserve">Aequalis igitur eſt etiam angulus AGC, angulo <lb/>B; </s> <s xml:id="echoid-s12640" xml:space="preserve">ac propterea arcus AB, AG, ſemicirculo æquales erunt. </s> <s xml:id="echoid-s12641" xml:space="preserve">Cum ergo arcus <lb/> <anchor type="note" xlink:label="note-387-04a" xlink:href="note-387-04"/> AG, arcui DE, oſtenſus ſit æqualis, erunt quoque arcus AB, DE, ſemicir-<lb/>culo æquales: </s> <s xml:id="echoid-s12642" xml:space="preserve">Ponuntur autem & </s> <s xml:id="echoid-s12643" xml:space="preserve">non æquales ſemicirculo. </s> <s xml:id="echoid-s12644" xml:space="preserve">Quod eſt abſur-<lb/>dum. </s> <s xml:id="echoid-s12645" xml:space="preserve">Non ergo inæquales ſunt arcus CB, FE, ſed æquales. </s> <s xml:id="echoid-s12646" xml:space="preserve">Quare cum late-<lb/>ra AC, CB, ſint æqualia lateribus DF, FE, angulosq́ue æquales contineant <lb/>C, F; </s> <s xml:id="echoid-s12647" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12648" xml:space="preserve">arcus AB, DE, & </s> <s xml:id="echoid-s12649" xml:space="preserve">anguli BAC, & </s> <s xml:id="echoid-s12650" xml:space="preserve">D, æquales. </s> <s xml:id="echoid-s12651" xml:space="preserve">Siigitur ſue-<lb/> <anchor type="note" xlink:label="note-387-05a" xlink:href="note-387-05"/> rint duo triangula ſphæica, &</s> <s xml:id="echoid-s12652" xml:space="preserve">c. </s> <s xml:id="echoid-s12653" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s12654" xml:space="preserve"/> </p> <div xml:id="echoid-div995" type="float" level="2" n="1"> <figure xlink:label="fig-387-01" xlink:href="fig-387-01a"> <image file="387-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/387-01"/> </figure> <note position="right" xlink:label="note-387-01" xlink:href="note-387-01a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-387-02" xlink:href="note-387-02a" xml:space="preserve">10. 1. Theo.</note> <note position="right" xlink:label="note-387-03" xlink:href="note-387-03a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-387-04" xlink:href="note-387-04a" xml:space="preserve">15. huius.</note> <note position="right" xlink:label="note-387-05" xlink:href="note-387-05a" xml:space="preserve">7. huius.</note> </div> </div> <div xml:id="echoid-div997" type="section" level="1" n="511"> <head xml:id="echoid-head546" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s12655" xml:space="preserve">_DIXIMVS,_ duo latera ſubtendentia reliquos angulos æquales, non debere <lb/>eſſe æqualia ſemicirculo. </s> <s xml:id="echoid-s12656" xml:space="preserve">Nam alias propoſitio vera non eſſet. </s> <s xml:id="echoid-s12657" xml:space="preserve">Sit enim triangulum <lb/>ſphæricum _ABC,_ quodcunq; </s> <s xml:id="echoid-s12658" xml:space="preserve">habens duo latera _AB, AC,_ inæqualia inter ſe, ſed <lb/> <anchor type="figure" xlink:label="fig-387-02a" xlink:href="fig-387-02"/> ſimul ſemicirculo æqualia: </s> <s xml:id="echoid-s12659" xml:space="preserve">Producto autem latere _BC,_ <lb/>vſque ad _D,_ ita tamen, vt _BD,_ ſemicirculo ſit minor, du-<lb/>caiur per _A, D,_ arcus circuli maximi _AD._ </s> <s xml:id="echoid-s12660" xml:space="preserve">Quoniam igi-<lb/> <anchor type="note" xlink:label="note-387-06a" xlink:href="note-387-06"/> tur arcus _AB, AC,_ ſemicirculo æquales ſunt, erit angu-<lb/>lus _ACD,_ angulo _B,_ æqualis. </s> <s xml:id="echoid-s12661" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s12662" xml:space="preserve">duotriangula _ABD,_ <lb/> <anchor type="note" xlink:label="note-387-07a" xlink:href="note-387-07"/> _ACD;_ </s> <s xml:id="echoid-s12663" xml:space="preserve">duos angulos _B, D,_ duobus angulis _C, D,_ æqua-<lb/>leshabent, vtrumque vtrique, & </s> <s xml:id="echoid-s12664" xml:space="preserve">latus _AD,_ commune, <lb/>quod æqualibus angulis _B, C,_ ſubtenditur; </s> <s xml:id="echoid-s12665" xml:space="preserve">& </s> <s xml:id="echoid-s12666" xml:space="preserve">tamen ne-<lb/>que reliqualatera _AB, BD,_ reliquis lateribus _AC, CD,_ <lb/>æqualia ſunt, vtrumque vtrique, neque reliquus angulus _BAD,_ reliquo angulo <lb/>_CAD,_ vt perſpicuum eſt. </s> <s xml:id="echoid-s12667" xml:space="preserve">Hoc autem ideò contingit, quod latera _AE, AC,_ ſemicir-<lb/>culo ſunt æqualia.</s> <s xml:id="echoid-s12668" xml:space="preserve"/> </p> <div xml:id="echoid-div997" type="float" level="2" n="1"> <figure xlink:label="fig-387-02" xlink:href="fig-387-02a"> <image file="387-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/387-02"/> </figure> <note position="right" xlink:label="note-387-06" xlink:href="note-387-06a" xml:space="preserve">_20. 1. Theo._</note> <note position="right" xlink:label="note-387-07" xlink:href="note-387-07a" xml:space="preserve">_14. huius._</note> </div> <p style="it"> <s xml:id="echoid-s12669" xml:space="preserve">_NICOLAVS_ ergo Copernicus lib. </s> <s xml:id="echoid-s12670" xml:space="preserve">1. </s> <s xml:id="echoid-s12671" xml:space="preserve">Reuolutionum propoſ. </s> <s xml:id="echoid-s12672" xml:space="preserve">12. </s> <s xml:id="echoid-s12673" xml:space="preserve">triangulorum <lb/> <anchor type="note" xlink:label="note-387-08a" xlink:href="note-387-08"/> ſphæricorum hallucinaiur, cum docet, omne triangulum ſphæricum, cuius duo anguli <lb/>vtcunque dati fuerint, cum aliquo latere, datorum eſſicv angulorum, & </s> <s xml:id="echoid-s12674" xml:space="preserve">laterum. <lb/></s> <s xml:id="echoid-s12675" xml:space="preserve">Nam in triangulo _ACD,_ licet duo anguli _D,_ & </s> <s xml:id="echoid-s12676" xml:space="preserve">_ACD,_ noti ſint cum latere _AD,_ <lb/>non tamen ex hoc perueniemus in notitiam reliquerum laterum, & </s> <s xml:id="echoid-s12677" xml:space="preserve">reliquianguli: </s> <s xml:id="echoid-s12678" xml:space="preserve"><lb/>cum reliqua latera eſſe poſsint vel _AC, CD,_ vel _AB, BD,_ &</s> <s xml:id="echoid-s12679" xml:space="preserve">c. </s> <s xml:id="echoid-s12680" xml:space="preserve">Oportebit ergo <lb/>præterea aliquid aliud conſtare, antequam reliquus angulus, cumreliquis lateribus <lb/>cognoſcatur, vt in ſcholio propoſ. </s> <s xml:id="echoid-s12681" xml:space="preserve">45. </s> <s xml:id="echoid-s12682" xml:space="preserve">dicemus.</s> <s xml:id="echoid-s12683" xml:space="preserve"/> </p> <div xml:id="echoid-div998" type="float" level="2" n="2"> <note position="right" xlink:label="note-387-08" xlink:href="note-387-08a" xml:space="preserve">Error Ni-<lb/>colai Co-<lb/>pernici.</note> </div> <pb o="376" file="388" n="388" rhead=""/> </div> <div xml:id="echoid-div1000" type="section" level="1" n="512"> <head xml:id="echoid-head547" xml:space="preserve">THEOR. 21. PROPOS. 23.</head> <p> <s xml:id="echoid-s12684" xml:space="preserve">Si fuerint duo triangula ſphærica, quæ duos an-<lb/>gulos duobus angulis habeantęquales, vtrumque <lb/>vtrique, duoque latera duobus lateribus circa re-<lb/>liquum angulum æqualia, vtrumque vtrique, & </s> <s xml:id="echoid-s12685" xml:space="preserve"><lb/>in reliquo angulo dicto non ſit polus reliqui late-<lb/>ris: </s> <s xml:id="echoid-s12686" xml:space="preserve">Erit & </s> <s xml:id="echoid-s12687" xml:space="preserve">reliquum latus reliquo lateri, & </s> <s xml:id="echoid-s12688" xml:space="preserve">reliquus <lb/>angulus reliquo angulo æqualis.</s> <s xml:id="echoid-s12689" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12690" xml:space="preserve">IN duobus triangulis ſphæricis ABC, DEF, ſint anguli B, C, angulis <lb/>E, F, æquales, vterque vtrique, & </s> <s xml:id="echoid-s12691" xml:space="preserve">latera AB, AC, circa reliquum angulum <lb/>A, æqualia lateribus DE, DF, vtrumque vtrique, non ſint autem A,D, po-<lb/> <anchor type="figure" xlink:label="fig-388-01a" xlink:href="fig-388-01"/> li arcuum BC, EF. </s> <s xml:id="echoid-s12692" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12693" xml:space="preserve">reliqua latera BC, <lb/>EF, æqualia eſſe, & </s> <s xml:id="echoid-s12694" xml:space="preserve">reliquos angulos A, D. </s> <s xml:id="echoid-s12695" xml:space="preserve">Si <lb/>enim arcus BC, EF, non ſunt æquales, ſit BC, <lb/> <anchor type="note" xlink:label="note-388-01a" xlink:href="note-388-01"/> maior, abſcindaturq́ue arcus CG, æqualis ipſi <lb/> <anchor type="note" xlink:label="note-388-02a" xlink:href="note-388-02"/> FE, & </s> <s xml:id="echoid-s12696" xml:space="preserve">per puncta A, G, arcus maximi circu-<lb/>li deſcribatur AG. </s> <s xml:id="echoid-s12697" xml:space="preserve">Quoniã igitur latera AC, <lb/>CG, æqualia sũt lateribus DF, FE, angulosq́; <lb/></s> <s xml:id="echoid-s12698" xml:space="preserve">æquales continent C, F; </s> <s xml:id="echoid-s12699" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12700" xml:space="preserve">arcus AG, <lb/> <anchor type="note" xlink:label="note-388-03a" xlink:href="note-388-03"/> DE, & </s> <s xml:id="echoid-s12701" xml:space="preserve">anguli AGC, & </s> <s xml:id="echoid-s12702" xml:space="preserve">E, & </s> <s xml:id="echoid-s12703" xml:space="preserve">quales: </s> <s xml:id="echoid-s12704" xml:space="preserve">Ponitur <lb/>autem arcus DE, arcui AB, & </s> <s xml:id="echoid-s12705" xml:space="preserve">angulus E, an-<lb/>gulo B, æqualis. </s> <s xml:id="echoid-s12706" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12707" xml:space="preserve">arcus AG, arcui AB, & </s> <s xml:id="echoid-s12708" xml:space="preserve">angulus AGC, angulo B, <lb/> <anchor type="note" xlink:label="note-388-04a" xlink:href="note-388-04"/> æqualis erit; </s> <s xml:id="echoid-s12709" xml:space="preserve">atque adeò, cum AGC, AGB, ſint æquales duobus rectis, erũt <lb/>& </s> <s xml:id="echoid-s12710" xml:space="preserve">B, AGB, duobus rectis æquales: </s> <s xml:id="echoid-s12711" xml:space="preserve">Sunt autem B, & </s> <s xml:id="echoid-s12712" xml:space="preserve">AGB, inter ſe æquales, <lb/> <anchor type="note" xlink:label="note-388-05a" xlink:href="note-388-05"/> ob æqualitatem arcuum AB, AG. </s> <s xml:id="echoid-s12713" xml:space="preserve">Vterque igitut rectus erit; </s> <s xml:id="echoid-s12714" xml:space="preserve">ac propterea <lb/>vterque arcus AB, AG, per polum arcus BC, tran ſibit Eſt ergo A, polus ar-<lb/> <anchor type="note" xlink:label="note-388-06a" xlink:href="note-388-06"/> cus BC. </s> <s xml:id="echoid-s12715" xml:space="preserve">Quod eſt abſurdum. </s> <s xml:id="echoid-s12716" xml:space="preserve">Ponitur enim non eſſe. </s> <s xml:id="echoid-s12717" xml:space="preserve">Non igitur inæquales <lb/>ſunt arcus BC, EF, ſed æquales; </s> <s xml:id="echoid-s12718" xml:space="preserve">atque idcirco & </s> <s xml:id="echoid-s12719" xml:space="preserve">anguli BAC, & </s> <s xml:id="echoid-s12720" xml:space="preserve">D, æqua-<lb/> <anchor type="note" xlink:label="note-388-07a" xlink:href="note-388-07"/> les erunt.</s> <s xml:id="echoid-s12721" xml:space="preserve"/> </p> <div xml:id="echoid-div1000" type="float" level="2" n="1"> <figure xlink:label="fig-388-01" xlink:href="fig-388-01a"> <image file="388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/388-01"/> </figure> <note position="left" xlink:label="note-388-01" xlink:href="note-388-01a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-388-02" xlink:href="note-388-02a" xml:space="preserve">20. 1. Theo.</note> <note position="left" xlink:label="note-388-03" xlink:href="note-388-03a" xml:space="preserve">7. huius.</note> <note position="left" xlink:label="note-388-04" xlink:href="note-388-04a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-388-05" xlink:href="note-388-05a" xml:space="preserve">8. huius.</note> <note position="left" xlink:label="note-388-06" xlink:href="note-388-06a" xml:space="preserve">13. 1. Theo.</note> <note position="left" xlink:label="note-388-07" xlink:href="note-388-07a" xml:space="preserve">18. huius.</note> </div> </div> <div xml:id="echoid-div1002" type="section" level="1" n="513"> <head xml:id="echoid-head548" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s12722" xml:space="preserve">_EST_ autem neceſſaria conditio illa, quòd in reliquo angulo polus non ſit reliqui <lb/> <anchor type="figure" xlink:label="fig-388-02a" xlink:href="fig-388-02"/> lateris Falſa enim eſſet propoſitio, ſi in illo angulo polus fo-<lb/>ret reliqui lateris. </s> <s xml:id="echoid-s12723" xml:space="preserve">Sit enim triangulum ſphæricum _ABC,_ <lb/>ſitq́; </s> <s xml:id="echoid-s12724" xml:space="preserve">in _A,_ polus arcus _BC;_ </s> <s xml:id="echoid-s12725" xml:space="preserve">& </s> <s xml:id="echoid-s12726" xml:space="preserve">ex _A,_ arcus circuli ma-<lb/>ximi deſcendat quicunque _AD,_ ſecans _BC,_ in _D._ </s> <s xml:id="echoid-s12727" xml:space="preserve">Erunt <lb/> <anchor type="note" xlink:label="note-388-08a" xlink:href="note-388-08"/> igitur anguli ad _B, C, D,_ omnes recti, atque omnes tres <lb/> <anchor type="note" xlink:label="note-388-09a" xlink:href="note-388-09"/> arcus _AB, AC, AD,_ quadrantes. </s> <s xml:id="echoid-s12728" xml:space="preserve">Itaque duo triangu-<lb/> <anchor type="note" xlink:label="note-388-10a" xlink:href="note-388-10"/> la _AB C, ADC,_ duos angulos _B, C,_ du <lb/>& </s> <s xml:id="echoid-s12729" xml:space="preserve">_C,_ æquales habent, vtrumque vtrique, & </s> <s xml:id="echoid-s12730" xml:space="preserve">duo la-<lb/>tera _AB, AC,_ duobus lateribus _AD, AC,_ circa angu- <pb o="377" file="389" n="389" rhead=""/> los _BAC, DAC,_ æqualia, vtrumque vtrique, & </s> <s xml:id="echoid-s12731" xml:space="preserve">tamen neque veliqua latera _BC,_ <lb/>_DC,_ æqualia inter ſe ſunt, neque reliqui anguli _BAC, DAC,_ vt manifeſturn eſt. <lb/></s> <s xml:id="echoid-s12732" xml:space="preserve">Hoc autem ideo accidit, quòd _A,_ polus ſit arcuum _BC, DC._</s> <s xml:id="echoid-s12733" xml:space="preserve"/> </p> <div xml:id="echoid-div1002" type="float" level="2" n="1"> <figure xlink:label="fig-388-02" xlink:href="fig-388-02a"> <image file="388-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/388-02"/> </figure> <note position="left" xlink:label="note-388-08" xlink:href="note-388-08a" xml:space="preserve">_15. 1. Theo._</note> <note position="left" xlink:label="note-388-09" xlink:href="note-388-09a" xml:space="preserve">_Coroll. 16._</note> <note position="left" xlink:label="note-388-10" xlink:href="note-388-10a" xml:space="preserve">_1. Theod._</note> </div> <p style="it"> <s xml:id="echoid-s12734" xml:space="preserve">_HINC_ perſpicuum quoque eſt, copernicũ hallucinari lib. </s> <s xml:id="echoid-s12735" xml:space="preserve">1. </s> <s xml:id="echoid-s12736" xml:space="preserve">Reuolutionum pro-<lb/> <anchor type="note" xlink:label="note-389-01a" xlink:href="note-389-01"/> poſ. </s> <s xml:id="echoid-s12737" xml:space="preserve">12. </s> <s xml:id="echoid-s12738" xml:space="preserve">cum aſſerit, omnetriangulum ſphæricum, cuius duo anguli vtcunque dati <lb/>fuerint, cum aliquo latere, datorum effici angulorum, & </s> <s xml:id="echoid-s12739" xml:space="preserve">laterum. </s> <s xml:id="echoid-s12740" xml:space="preserve">Nam in trian-<lb/>gulo _ABC,_ etiamſi dentur duo anguli _B, C,_ cum duobus lateribus _AB, AC,_ (& </s> <s xml:id="echoid-s12741" xml:space="preserve">non <lb/>cum vnotantum, vtipſe vult) non tamen ſtatim reliquum latus, & </s> <s xml:id="echoid-s12742" xml:space="preserve">reliquus angu-<lb/>lus cognoſcetur; </s> <s xml:id="echoid-s12743" xml:space="preserve">cum reliquum latus eſſe poſsit vel _BC,_ vel _DC,_ & </s> <s xml:id="echoid-s12744" xml:space="preserve">reliquus angu-<lb/>lus vel _BAC,_ vel _DAC,_ &</s> <s xml:id="echoid-s12745" xml:space="preserve">c. </s> <s xml:id="echoid-s12746" xml:space="preserve">Aliquid ergo aliud præterea cõſtet, neceſſe eſt, vt relio <lb/>quus angulus, cum reliquis lateribus cognoſcatur, vt in ſcholio propoſ. </s> <s xml:id="echoid-s12747" xml:space="preserve">45. </s> <s xml:id="echoid-s12748" xml:space="preserve">oſtẽdemus.</s> <s xml:id="echoid-s12749" xml:space="preserve"/> </p> <div xml:id="echoid-div1003" type="float" level="2" n="2"> <note position="right" xlink:label="note-389-01" xlink:href="note-389-01a" xml:space="preserve">Error Ni-<lb/>colai Co-<lb/>pernici.</note> </div> </div> <div xml:id="echoid-div1005" type="section" level="1" n="514"> <head xml:id="echoid-head549" xml:space="preserve">THEOR. 22. PROPOS. 24.</head> <p> <s xml:id="echoid-s12750" xml:space="preserve">SI fuerint duo triangula ſphærica, quæ vnum <lb/>angulum vni angulo æqualem habeant, & </s> <s xml:id="echoid-s12751" xml:space="preserve">duo la-<lb/>tera duobus lateribus circa alium angulum æqua-<lb/>lia vtrumque vtrique, atq; </s> <s xml:id="echoid-s12752" xml:space="preserve">vtrum que reliquorum <lb/>angulorum vel maiorem recto, vel minorem: </s> <s xml:id="echoid-s12753" xml:space="preserve">Erit <lb/>& </s> <s xml:id="echoid-s12754" xml:space="preserve">reliquum latus reliquo lateri æquale, & </s> <s xml:id="echoid-s12755" xml:space="preserve">reliqui <lb/>anguli reliquis angulis æquales, vterque vtrique.</s> <s xml:id="echoid-s12756" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12757" xml:space="preserve">IN duobus triangulis ſphæricis ABC, DEF, ſint anguli B, E, æquales, <lb/>& </s> <s xml:id="echoid-s12758" xml:space="preserve">duo latera BC, CA, ęqualia duobus lateribus EF, FD, vtrumque vtrique, <lb/> <anchor type="figure" xlink:label="fig-389-01a" xlink:href="fig-389-01"/> circa angulos C, F, & </s> <s xml:id="echoid-s12759" xml:space="preserve">vterq; </s> <s xml:id="echoid-s12760" xml:space="preserve">angulorũ reli-<lb/>quorum A, D, vel minor ſit, vel maior recto. <lb/></s> <s xml:id="echoid-s12761" xml:space="preserve">Dico reliqua latera AB, DE, æqualia quo-<lb/>que eſſe, & </s> <s xml:id="echoid-s12762" xml:space="preserve">reliquos duos angulos A, C, re-<lb/>liquis duobus angulis D, F, vtrũq; </s> <s xml:id="echoid-s12763" xml:space="preserve">vtrique. </s> <s xml:id="echoid-s12764" xml:space="preserve"><lb/>Si enim latera AB, DE, æqualia non ſunt, <lb/>ſit AB, maius, & </s> <s xml:id="echoid-s12765" xml:space="preserve">abſcindatur arcus BG, <lb/> <anchor type="note" xlink:label="note-389-02a" xlink:href="note-389-02"/> æqualis arcui DE, & </s> <s xml:id="echoid-s12766" xml:space="preserve">per puncta C, G, ar-<lb/>cus circuli maximi ducatur CH. </s> <s xml:id="echoid-s12767" xml:space="preserve">Quia igi-<lb/> <anchor type="note" xlink:label="note-389-03a" xlink:href="note-389-03"/> tur latera BG, BC, æqualia ſunt lateribus <lb/>ED, EF, angulosque comprehendunt æquales B, E; </s> <s xml:id="echoid-s12768" xml:space="preserve">erunt & </s> <s xml:id="echoid-s12769" xml:space="preserve">arcus GC, DF, <lb/> <anchor type="note" xlink:label="note-389-04a" xlink:href="note-389-04"/> & </s> <s xml:id="echoid-s12770" xml:space="preserve">anguli G, D, æquales: </s> <s xml:id="echoid-s12771" xml:space="preserve">Ponitur autem arcus DF, arcui AC, æqualis. </s> <s xml:id="echoid-s12772" xml:space="preserve">Ae-<lb/>qualis igitur erit quoque arcus GC, eidem arcui AC; </s> <s xml:id="echoid-s12773" xml:space="preserve">atque adeo anguli A, <lb/> <anchor type="note" xlink:label="note-389-05a" xlink:href="note-389-05"/> & </s> <s xml:id="echoid-s12774" xml:space="preserve">CGA, æquales. </s> <s xml:id="echoid-s12775" xml:space="preserve">Et quoniam anguli duo ad G, ſunt æquales duobus re-<lb/> <anchor type="note" xlink:label="note-389-06a" xlink:href="note-389-06"/> ctis, erunt queque duo anguli BGC, & </s> <s xml:id="echoid-s12776" xml:space="preserve">A, duobus rectis æquales; </s> <s xml:id="echoid-s12777" xml:space="preserve">ac proin-<lb/>de, cum angulus BGC, oſtenſus ſit æqualis angulo D, erunt & </s> <s xml:id="echoid-s12778" xml:space="preserve">duo anguli <lb/>D, & </s> <s xml:id="echoid-s12779" xml:space="preserve">A, duobus rectis æquales. </s> <s xml:id="echoid-s12780" xml:space="preserve">Quod fieri non poteſt. </s> <s xml:id="echoid-s12781" xml:space="preserve">Cum enim vterque <lb/>minor recto ponatur, vel maior, erunt ambo ſimul vel duobus rectis minores, <lb/>vel maiores. </s> <s xml:id="echoid-s12782" xml:space="preserve">Non ergo inæqualia ſunt latera AB, DE, ſed æqualia. </s> <s xml:id="echoid-s12783" xml:space="preserve">Quare <pb o="378" file="390" n="390" rhead=""/> & </s> <s xml:id="echoid-s12784" xml:space="preserve">duo Anguli A, C, duobus ãngulis D, F, æquales erunt, vterque vtrique. </s> <s xml:id="echoid-s12785" xml:space="preserve">Si <lb/> <anchor type="note" xlink:label="note-390-01a" xlink:href="note-390-01"/> fuerint igitur duo triangula, &</s> <s xml:id="echoid-s12786" xml:space="preserve">c. </s> <s xml:id="echoid-s12787" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s12788" xml:space="preserve"/> </p> <div xml:id="echoid-div1005" type="float" level="2" n="1"> <figure xlink:label="fig-389-01" xlink:href="fig-389-01a"> <image file="389-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/389-01"/> </figure> <note position="right" xlink:label="note-389-02" xlink:href="note-389-02a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-389-03" xlink:href="note-389-03a" xml:space="preserve">20.1 Theod.</note> <note position="right" xlink:label="note-389-04" xlink:href="note-389-04a" xml:space="preserve">7. huius.</note> <note position="right" xlink:label="note-389-05" xlink:href="note-389-05a" xml:space="preserve">8. huius.</note> <note position="right" xlink:label="note-389-06" xlink:href="note-389-06a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-390-01" xlink:href="note-390-01a" xml:space="preserve">18. huius.</note> </div> </div> <div xml:id="echoid-div1007" type="section" level="1" n="515"> <head xml:id="echoid-head550" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s12789" xml:space="preserve">_DIXIMVS,_ vtrumque reliquorum angulorum debere eſſe vel maiorem, vel <lb/>minorem recto. </s> <s xml:id="echoid-s12790" xml:space="preserve">Nam alias falſa eſſet propoſitio. </s> <s xml:id="echoid-s12791" xml:space="preserve">Sit enim triangulum ſphæricum <lb/> <anchor type="note" xlink:label="note-390-02a" xlink:href="note-390-02"/> quodcunque _ABC,_ habens duo latera _AB, AC,_ æqualia: </s> <s xml:id="echoid-s12792" xml:space="preserve">Producto autem latere <lb/> <anchor type="figure" xlink:label="fig-390-01a" xlink:href="fig-390-01"/> _CB,_ ad _D,_ ita vt _CD,_ ſit arcus ſemicirculo minor, duca-<lb/>tur per puncta _A, D,_ arcus circuli maximi _AD. </s> <s xml:id="echoid-s12793" xml:space="preserve">Itaque <lb/>triangula _ADB, ADC,_ angulum angulo æqualem habẽt, <lb/>nempe _D,_ communem, & </s> <s xml:id="echoid-s12794" xml:space="preserve">duo latera _AD, AB,_ æqualia <lb/>duobus lateribus _AD, AC,_ vtrumque vtrique; </s> <s xml:id="echoid-s12795" xml:space="preserve">& </s> <s xml:id="echoid-s12796" xml:space="preserve">tamen <lb/>reliqua latera _DB, DC,_ æqualia non ſunt, nec reliqui <lb/>anguli _DAB, DAC,_ immoneque anguli _ABD, ACD,_ <lb/>niſi vterq; </s> <s xml:id="echoid-s12797" xml:space="preserve">rectus ſit, vt demonſtrabimus. </s> <s xml:id="echoid-s12798" xml:space="preserve">Hoc autemideò <lb/>euenit, quòd non vterque angulus _ABD, ACD,_ maior <lb/>eſt vel minor recto, ſed vel vterq; </s> <s xml:id="echoid-s12799" xml:space="preserve">rectus, vel vnus maior <lb/>recto, & </s> <s xml:id="echoid-s12800" xml:space="preserve">alter minor: </s> <s xml:id="echoid-s12801" xml:space="preserve">quodita oſtendemus. </s> <s xml:id="echoid-s12802" xml:space="preserve">Sit primum angulus _ABD,_ rectus. <lb/></s> <s xml:id="echoid-s12803" xml:space="preserve">Dico & </s> <s xml:id="echoid-s12804" xml:space="preserve">_C,_ rectum eſſe. </s> <s xml:id="echoid-s12805" xml:space="preserve">Recto enim exiſtente angulo _ABD,_ erit & </s> <s xml:id="echoid-s12806" xml:space="preserve">_ABC,_ rectus; </s> <s xml:id="echoid-s12807" xml:space="preserve"><lb/>quòd ambo anguli ad _B,_ æquales ſint duobus rectis: </s> <s xml:id="echoid-s12808" xml:space="preserve">ſed hic æqualis eſt angulo _C,_ ob <lb/> <anchor type="note" xlink:label="note-390-03a" xlink:href="note-390-03"/> <anchor type="note" xlink:label="note-390-04a" xlink:href="note-390-04"/> æqualitatem laterum _AB, AC._ </s> <s xml:id="echoid-s12809" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12810" xml:space="preserve">_C,_ rectus erit.</s> <s xml:id="echoid-s12811" xml:space="preserve"/> </p> <div xml:id="echoid-div1007" type="float" level="2" n="1"> <note position="left" xlink:label="note-390-02" xlink:href="note-390-02a" xml:space="preserve">20.1 Theod.</note> <figure xlink:label="fig-390-01" xlink:href="fig-390-01a"> <image file="390-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/390-01"/> </figure> <note position="left" xlink:label="note-390-03" xlink:href="note-390-03a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-390-04" xlink:href="note-390-04a" xml:space="preserve">8. huius.</note> </div> <p style="it"> <s xml:id="echoid-s12812" xml:space="preserve">_SIT_ deinde angulus _ABD,_ maior recto. </s> <s xml:id="echoid-s12813" xml:space="preserve">Dico _C,_ minorem eſſe recto. </s> <s xml:id="echoid-s12814" xml:space="preserve">Cum enim <lb/>_ABD,_ ſit recto maior, erit _ABC,_ minor recto, cum ambo duobus rectis ſint æquales. <lb/></s> <s xml:id="echoid-s12815" xml:space="preserve"> <anchor type="note" xlink:label="note-390-05a" xlink:href="note-390-05"/> Igitur & </s> <s xml:id="echoid-s12816" xml:space="preserve">angulus _C,_ qui æqualis eſt angulo _ABC,_ recto minor erit.</s> <s xml:id="echoid-s12817" xml:space="preserve"/> </p> <div xml:id="echoid-div1008" type="float" level="2" n="2"> <note position="left" xlink:label="note-390-05" xlink:href="note-390-05a" xml:space="preserve">5. huius.</note> </div> <note position="left" xml:space="preserve">8. huius.</note> <p style="it"> <s xml:id="echoid-s12818" xml:space="preserve">_SIT_ tandem angulus _ABD,_ minor recto. </s> <s xml:id="echoid-s12819" xml:space="preserve">Dico _C,_ eſſe recto maiorem. </s> <s xml:id="echoid-s12820" xml:space="preserve">cum enim <lb/>_ABD,_ ſit minor recto, erit _ABC,_ hoc eſt, ſibi æqualis _C,_ maior recto.</s> <s xml:id="echoid-s12821" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Error Ni-<lb/>colai Co-<lb/>pernici.</note> <p style="it"> <s xml:id="echoid-s12822" xml:space="preserve">_HINC_ manifeſtum eſt, propoſitionem _8._ </s> <s xml:id="echoid-s12823" xml:space="preserve">Nicolai Copernici de ſphæricis triangu-<lb/>lis, lib. </s> <s xml:id="echoid-s12824" xml:space="preserve">_1._ </s> <s xml:id="echoid-s12825" xml:space="preserve">Reuolutionum falſam eſſe, quo ad eam partem, in qua dicit. </s> <s xml:id="echoid-s12826" xml:space="preserve">_Si bina trian <lb/>gula duo latera duobus lateribus æqualia habuerint, alterum alteri, & </s> <s xml:id="echoid-s12827" xml:space="preserve">angu-<lb/>lum angulo æqualem, quiad baſim fuerit; </s> <s xml:id="echoid-s12828" xml:space="preserve">baſim quoque baſi, ac reliquos an-<lb/>gulos reliquis angulis habebunt æquales. </s> <s xml:id="echoid-s12829" xml:space="preserve">Hoc enim verum non eſt, niſi ponatur <lb/>vter que reliquorum angulorum ad baſim vel maior recto, vel minor. </s> <s xml:id="echoid-s12830" xml:space="preserve">In triangulis <lb/>enim propoſitis _ADB, ADC,_ ſunt duo latera _AD, AB,_ duobus lateribus _AD, AC,_ <lb/>æqualia, angulusq́; </s> <s xml:id="echoid-s12831" xml:space="preserve">_D,_ communis eſt ſuper baſes _DB, DC;_ </s> <s xml:id="echoid-s12832" xml:space="preserve">& </s> <s xml:id="echoid-s12833" xml:space="preserve">tamen baſes non ſunt <lb/>æquales, ob cauſam dictam.</s> <s xml:id="echoid-s12834" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s12835" xml:space="preserve">_VNDE_ errat idem Nicolaus in eodem libro propoſ. </s> <s xml:id="echoid-s12836" xml:space="preserve">II. </s> <s xml:id="echoid-s12837" xml:space="preserve">vbi ait. </s> <s xml:id="echoid-s12838" xml:space="preserve">_Omne trian-<lb/> <anchor type="note" xlink:label="note-390-08a" xlink:href="note-390-08"/> gulum, cuius duo latera fuerint data cum aliquo angulo, datorum eſſicitur <lb/>angulorum, & </s> <s xml:id="echoid-s12839" xml:space="preserve">laterum. </s> <s xml:id="echoid-s12840" xml:space="preserve">Nam etiamſi latera _AD, AB,_ nota ſintcum angulo _D,_ <lb/>non tamen inde in notitiam alterius lateris, & </s> <s xml:id="echoid-s12841" xml:space="preserve">aliorum angulorum perueniemus, <lb/>cum reliquum latus poſsit eſſe vel _DB,_ vel _DC,_ &</s> <s xml:id="echoid-s12842" xml:space="preserve">c. </s> <s xml:id="echoid-s12843" xml:space="preserve">Neceſſe eſt ergo aliud quip-<lb/>piam præterea conſtare, antequam reliquum latus, cum reliquis angulis notum effi-<lb/>ciatur, vt in Scholio propoſ. </s> <s xml:id="echoid-s12844" xml:space="preserve">45. </s> <s xml:id="echoid-s12845" xml:space="preserve">perſpicuum faciemus.</s> <s xml:id="echoid-s12846" xml:space="preserve"/> </p> <div xml:id="echoid-div1009" type="float" level="2" n="3"> <note position="left" xlink:label="note-390-08" xlink:href="note-390-08a" xml:space="preserve">Alius error <lb/>Nicolai Co <lb/>pernici.</note> </div> </div> <div xml:id="echoid-div1011" type="section" level="1" n="516"> <head xml:id="echoid-head551" xml:space="preserve">THEOR. 23. PROPOS. 25.</head> <p> <s xml:id="echoid-s12847" xml:space="preserve">IN omni triangulo ſphærico Iſoſcele, ſi duo <pb o="379" file="391" n="391" rhead=""/> latera æqualia ſint quadrantes, erunt duo anguli <lb/>æquales ſuper baſim recti: </s> <s xml:id="echoid-s12848" xml:space="preserve">ſi verò vtrumque qua-<lb/>drante minus ſit, acuti: </s> <s xml:id="echoid-s12849" xml:space="preserve">ſi denique maius quadran <lb/>te, obtuſi. </s> <s xml:id="echoid-s12850" xml:space="preserve">Et ſi duo anguli æquales ad baſim ſint <lb/>recti, erunt duo latera æqualia, quadrantes: </s> <s xml:id="echoid-s12851" xml:space="preserve">ſi ve-<lb/>rò acuti, vtrumque quadrante minus erit: </s> <s xml:id="echoid-s12852" xml:space="preserve">ſi deni-<lb/>que obtuſi, vtrumque quadrante maius.</s> <s xml:id="echoid-s12853" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12854" xml:space="preserve">IN triangulo ſphærico Iſoſcelc ABC, ſint primum duo arcus æquales <lb/>AB, AC, quadrantes. </s> <s xml:id="echoid-s12855" xml:space="preserve">Dico æquales angulos B, C, ad baſim eſſe rectos, Cum <lb/>enim vterque atcus AB, AC, quadrans ſit, erunt <lb/> <anchor type="figure" xlink:label="fig-391-01a" xlink:href="fig-391-01"/> ambo ſimul ſemicirculo æquales. </s> <s xml:id="echoid-s12856" xml:space="preserve">Quare producto <lb/>arcu BC, ad D, angulus ACD, æqualis erit an-<lb/> <anchor type="note" xlink:label="note-391-01a" xlink:href="note-391-01"/> gulo B: </s> <s xml:id="echoid-s12857" xml:space="preserve">ſed angulus B, angulo ACB, æqualis eſt. <lb/></s> <s xml:id="echoid-s12858" xml:space="preserve"> <anchor type="note" xlink:label="note-391-02a" xlink:href="note-391-02"/> Igitur & </s> <s xml:id="echoid-s12859" xml:space="preserve">angulus, ACD, angulo ACB, æqualis <lb/>erit; </s> <s xml:id="echoid-s12860" xml:space="preserve">atque adeò, cum duo anguli ad C, duobus re-<lb/> <anchor type="note" xlink:label="note-391-03a" xlink:href="note-391-03"/> ctis ęquales ſint, erit vterque angulus ad C, rectus. <lb/></s> <s xml:id="echoid-s12861" xml:space="preserve">Qnare & </s> <s xml:id="echoid-s12862" xml:space="preserve">angulus B, quirecto ACB, æqualis eſt, <lb/> <anchor type="note" xlink:label="note-391-04a" xlink:href="note-391-04"/> rectus erit. </s> <s xml:id="echoid-s12863" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12864" xml:space="preserve"/> </p> <div xml:id="echoid-div1011" type="float" level="2" n="1"> <figure xlink:label="fig-391-01" xlink:href="fig-391-01a"> <image file="391-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/391-01"/> </figure> <note position="right" xlink:label="note-391-01" xlink:href="note-391-01a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-391-02" xlink:href="note-391-02a" xml:space="preserve">8. huius.</note> <note position="right" xlink:label="note-391-03" xlink:href="note-391-03a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-391-04" xlink:href="note-391-04a" xml:space="preserve">8. huius.</note> </div> <p> <s xml:id="echoid-s12865" xml:space="preserve">SIT deinde vterque arcuum AB, AC, æqua-<lb/>lium quadrante minor. </s> <s xml:id="echoid-s12866" xml:space="preserve">Dico angulos B, C, æqua-<lb/>les eſſe acutos. </s> <s xml:id="echoid-s12867" xml:space="preserve">Cum enim vterque arcus AB, AC, <lb/>quadrante minor ſit, erunt ambo ſimul ſemicircu-<lb/>lo minores. </s> <s xml:id="echoid-s12868" xml:space="preserve">Quare angulus ACD, maior erit <lb/> <anchor type="note" xlink:label="note-391-05a" xlink:href="note-391-05"/> angulo B, hoceſt, angulo ACB; </s> <s xml:id="echoid-s12869" xml:space="preserve">cum anguli B, <lb/> <anchor type="note" xlink:label="note-391-06a" xlink:href="note-391-06"/> & </s> <s xml:id="echoid-s12870" xml:space="preserve">ACB, æquales ſint. </s> <s xml:id="echoid-s12871" xml:space="preserve">Cum ergo duo anguli ad C, æquales ſint duobus <lb/> <anchor type="note" xlink:label="note-391-07a" xlink:href="note-391-07"/> rectis, erit angulus ACB, recto minor; </s> <s xml:id="echoid-s12872" xml:space="preserve">atque adeo angulus B, qui ei æqua-<lb/> <anchor type="note" xlink:label="note-391-08a" xlink:href="note-391-08"/> lis eſt, recto quoq; </s> <s xml:id="echoid-s12873" xml:space="preserve">minor erit. </s> <s xml:id="echoid-s12874" xml:space="preserve">Sunt ergo duo anguli B, & </s> <s xml:id="echoid-s12875" xml:space="preserve">ACB, acuti. </s> <s xml:id="echoid-s12876" xml:space="preserve">Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s12877" xml:space="preserve"/> </p> <div xml:id="echoid-div1012" type="float" level="2" n="2"> <note position="right" xlink:label="note-391-05" xlink:href="note-391-05a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-391-06" xlink:href="note-391-06a" xml:space="preserve">8. huius.</note> <note position="right" xlink:label="note-391-07" xlink:href="note-391-07a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-391-08" xlink:href="note-391-08a" xml:space="preserve">8. huius.</note> </div> <p> <s xml:id="echoid-s12878" xml:space="preserve">SIT poſtremo vterque arcuum AB, AC, quadrante maior. </s> <s xml:id="echoid-s12879" xml:space="preserve">Dico angu-<lb/>los æquales, B, C, eſſe obtuſos. </s> <s xml:id="echoid-s12880" xml:space="preserve">Cum enim vterque arcus AB, AC, maior ſit <lb/>quadrante, eruntambo maiores ſemicirculo. </s> <s xml:id="echoid-s12881" xml:space="preserve">Quare angulus ACD, minor <lb/> <anchor type="note" xlink:label="note-391-09a" xlink:href="note-391-09"/> erit angulo B, hoc eſt, angulo ACB, qui angulo B, æqualis eſt. </s> <s xml:id="echoid-s12882" xml:space="preserve">Cum ergo <lb/> <anchor type="note" xlink:label="note-391-10a" xlink:href="note-391-10"/> duo anguliad C, duobus rectis ſint æquales, erit angulus ACB, recto ma-<lb/> <anchor type="note" xlink:label="note-391-11a" xlink:href="note-391-11"/> ior, hoc eſt, obtuſus; </s> <s xml:id="echoid-s12883" xml:space="preserve">atque idcirco & </s> <s xml:id="echoid-s12884" xml:space="preserve">angulus B, qui ei æqualis eſt, obtuſus <lb/> <anchor type="note" xlink:label="note-391-12a" xlink:href="note-391-12"/> erit. </s> <s xml:id="echoid-s12885" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12886" xml:space="preserve"/> </p> <div xml:id="echoid-div1013" type="float" level="2" n="3"> <note position="right" xlink:label="note-391-09" xlink:href="note-391-09a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-391-10" xlink:href="note-391-10a" xml:space="preserve">8. huius.</note> <note position="right" xlink:label="note-391-11" xlink:href="note-391-11a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-391-12" xlink:href="note-391-12a" xml:space="preserve">8. huius.</note> </div> <p> <s xml:id="echoid-s12887" xml:space="preserve">SED iam vterque angulorum æqualium B, C, ſit rectus. </s> <s xml:id="echoid-s12888" xml:space="preserve">Dico vtrumque <lb/>arcum AB, AC, quadrantem eſſe. </s> <s xml:id="echoid-s12889" xml:space="preserve">Cum enim ACB, rectus ſit, & </s> <s xml:id="echoid-s12890" xml:space="preserve">duo angu-<lb/>li ad C, æquales duobus rectis, erit quoque ACD, rectus, ac proinde recto <lb/> <anchor type="note" xlink:label="note-391-13a" xlink:href="note-391-13"/> B, æqualis. </s> <s xml:id="echoid-s12891" xml:space="preserve">Suntergo duo arcus AB, AC, ſimul ſemicirculo æquales, ac pro-<lb/> <anchor type="note" xlink:label="note-391-14a" xlink:href="note-391-14"/> pterea cum ipſi æquales ponantur, vterq; </s> <s xml:id="echoid-s12892" xml:space="preserve">quadrans erit. </s> <s xml:id="echoid-s12893" xml:space="preserve">Quod eſt propoſitũ.</s> <s xml:id="echoid-s12894" xml:space="preserve"/> </p> <div xml:id="echoid-div1014" type="float" level="2" n="4"> <note position="right" xlink:label="note-391-13" xlink:href="note-391-13a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-391-14" xlink:href="note-391-14a" xml:space="preserve">15 huius.</note> </div> <p> <s xml:id="echoid-s12895" xml:space="preserve">DEINDE vterque angulorum B, C, ſit acutus. </s> <s xml:id="echoid-s12896" xml:space="preserve">Dico vtrumque arcum <lb/>AB, AC, quadrante minorem eſſe. </s> <s xml:id="echoid-s12897" xml:space="preserve">Cum enim duo anguliad C, æquales duo-<lb/> <anchor type="note" xlink:label="note-391-15a" xlink:href="note-391-15"/> <pb o="380" file="392" n="392" rhead=""/> bus rectis ſint, & </s> <s xml:id="echoid-s12898" xml:space="preserve">ACB, ponatur recto minor; </s> <s xml:id="echoid-s12899" xml:space="preserve">erit ACD, recto maior; </s> <s xml:id="echoid-s12900" xml:space="preserve">ae <lb/> <anchor type="figure" xlink:label="fig-392-01a" xlink:href="fig-392-01"/> propterea maior, quam B, qui recto etiam minor <lb/>ponitur. </s> <s xml:id="echoid-s12901" xml:space="preserve">Sunt ergo arcus AB, AC, ſimul ſemi-<lb/> <anchor type="note" xlink:label="note-392-01a" xlink:href="note-392-01"/> circulo minores; </s> <s xml:id="echoid-s12902" xml:space="preserve">atque idcirco, cum ipſi ſint ę qua-<lb/>les, vterque quadrante minor erit. </s> <s xml:id="echoid-s12903" xml:space="preserve">Quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s12904" xml:space="preserve"/> </p> <div xml:id="echoid-div1015" type="float" level="2" n="5"> <note position="right" xlink:label="note-391-15" xlink:href="note-391-15a" xml:space="preserve">5. huius.</note> <figure xlink:label="fig-392-01" xlink:href="fig-392-01a"> <image file="392-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/392-01"/> </figure> <note position="left" xlink:label="note-392-01" xlink:href="note-392-01a" xml:space="preserve">15. huius.</note> </div> <p> <s xml:id="echoid-s12905" xml:space="preserve">POSTREMO ſit vterque angulorum B, C, <lb/>obtuſus. </s> <s xml:id="echoid-s12906" xml:space="preserve">Dico utrumque arcum AB, AC, maio-<lb/>rem eſſe quadrante. </s> <s xml:id="echoid-s12907" xml:space="preserve">Cum enim duo anguli ad C, <lb/>ſint æquales duobus rectis, & </s> <s xml:id="echoid-s12908" xml:space="preserve">ACB, ponatur maior <lb/> <anchor type="note" xlink:label="note-392-02a" xlink:href="note-392-02"/> recto, erit ACD, recto minor, atque idcirco mi-<lb/>nor angulo B, qui recto quoque maior ponitur. <lb/></s> <s xml:id="echoid-s12909" xml:space="preserve"> <anchor type="note" xlink:label="note-392-03a" xlink:href="note-392-03"/> Arcus ergo AB, AC, ſimul maiores ſunt ſemicir-<lb/>culo; </s> <s xml:id="echoid-s12910" xml:space="preserve">atque adeò, cum ipſi æquales ſint, erit uterq; <lb/></s> <s xml:id="echoid-s12911" xml:space="preserve">quadrante maior. </s> <s xml:id="echoid-s12912" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s12913" xml:space="preserve">In omni <lb/>ergo triangulo ſphærico Iſoſcele, &</s> <s xml:id="echoid-s12914" xml:space="preserve">c. </s> <s xml:id="echoid-s12915" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s12916" xml:space="preserve"/> </p> <div xml:id="echoid-div1016" type="float" level="2" n="6"> <note position="left" xlink:label="note-392-02" xlink:href="note-392-02a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-392-03" xlink:href="note-392-03a" xml:space="preserve">25. huius.</note> </div> </div> <div xml:id="echoid-div1018" type="section" level="1" n="517"> <head xml:id="echoid-head552" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s12917" xml:space="preserve">EX his ſequitur, omne triangulum ſphæricum æquilaterum, ſeu æquiangulum, ſi ſin-<lb/>gula latera ſint quadrantes, habere ſingulos angulos rectos: </s> <s xml:id="echoid-s12918" xml:space="preserve">ſi verò quadrante minora, acu-<lb/>tos. </s> <s xml:id="echoid-s12919" xml:space="preserve">Si denique quadrante maiora, obtuſos. </s> <s xml:id="echoid-s12920" xml:space="preserve">Et omne triangulum ſphæricum æquiangu-<lb/>lum, ſeu æquilaterum, ſi ſinguli anguli ſint recti, habere ſingula latera quadrantes: </s> <s xml:id="echoid-s12921" xml:space="preserve">Si verò <lb/>acuti, quadrante minora: </s> <s xml:id="echoid-s12922" xml:space="preserve">ſi denique obtuſi, quadrante maiora.</s> <s xml:id="echoid-s12923" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1019" type="section" level="1" n="518"> <head xml:id="echoid-head553" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s12924" xml:space="preserve">_CAETERVM,_ quando duo latera trianguli ſpliarici ſunt quadrätes, vtrum-<lb/>que angulum ad baſim eſſe rectum: </s> <s xml:id="echoid-s12925" xml:space="preserve">Et ſi vterque angulus ad baſim rectus eſt, vtrum-<lb/>que latus eſſe quadrantem, demonſtrari etiam poterit bac ratione.</s> <s xml:id="echoid-s12926" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s12927" xml:space="preserve">_SINT_ in triangulo _ABC,_ quadrantes _AB, AC._ </s> <s xml:id="echoid-s12928" xml:space="preserve">Dico <lb/> <anchor type="figure" xlink:label="fig-392-02a" xlink:href="fig-392-02"/> angulos _B, C,_ eſſe rectos. </s> <s xml:id="echoid-s12929" xml:space="preserve">Productis enim arcubus _AB, AC,_ <lb/>donec coëantin _D,_ vt ſint ABD, ACD, ſemicirculi; </s> <s xml:id="echoid-s12930" xml:space="preserve">erunt <lb/> <anchor type="note" xlink:label="note-392-04a" xlink:href="note-392-04"/> quoque arcus _DB, DC,_ quadrantes; </s> <s xml:id="echoid-s12931" xml:space="preserve">atque adeo vterque <lb/>arcus _ABD, ACD,_ bifariam diuidetur ab arcu _B C,_ in <lb/> <anchor type="note" xlink:label="note-392-05a" xlink:href="note-392-05"/> punctis _B,_ & </s> <s xml:id="echoid-s12932" xml:space="preserve">_C._ </s> <s xml:id="echoid-s12933" xml:space="preserve">Igitur arcus _BC,_ per polos arcuum _AB,_ <lb/>_AC,_ tranſibit; </s> <s xml:id="echoid-s12934" xml:space="preserve">atqueidcirco rectos angulos ad _B,_ & </s> <s xml:id="echoid-s12935" xml:space="preserve">_C,_ <lb/> <anchor type="note" xlink:label="note-392-06a" xlink:href="note-392-06"/> efficiet.</s> <s xml:id="echoid-s12936" xml:space="preserve"/> </p> <div xml:id="echoid-div1019" type="float" level="2" n="1"> <figure xlink:label="fig-392-02" xlink:href="fig-392-02a"> <image file="392-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/392-02"/> </figure> <note position="left" xlink:label="note-392-04" xlink:href="note-392-04a" xml:space="preserve">11. 1. Theod.</note> <note position="left" xlink:label="note-392-05" xlink:href="note-392-05a" xml:space="preserve">Schol. 9. <lb/>2. Theod.</note> <note position="right" xlink:label="note-392-06" xlink:href="note-392-06a" xml:space="preserve">15. 1 Theod.</note> </div> <p style="it"> <s xml:id="echoid-s12937" xml:space="preserve">_VERVM_ iam anguli _B, C,_ recti ſint. </s> <s xml:id="echoid-s12938" xml:space="preserve">Dico latera _AB,_ <lb/>_AC,_ quadrantes eſſe. </s> <s xml:id="echoid-s12939" xml:space="preserve">cum enim anguli _B, C,_ ſint recti, <lb/> <anchor type="note" xlink:label="note-392-07a" xlink:href="note-392-07"/> tranſibit arcus _BC,_ per polos arcuum _ABD, ACD,_ qui <lb/> <anchor type="note" xlink:label="note-392-08a" xlink:href="note-392-08"/> quidem ſemicirculi ſunt; </s> <s xml:id="echoid-s12940" xml:space="preserve">atque adeò vtrumque bifariam ſe-<lb/> <anchor type="note" xlink:label="note-392-09a" xlink:href="note-392-09"/> cabit in _B, C._ </s> <s xml:id="echoid-s12941" xml:space="preserve">Sunt ergo arcus _AB, AC, DB, DC,_ qua-<lb/>drantes. </s> <s xml:id="echoid-s12942" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s12943" xml:space="preserve"/> </p> <div xml:id="echoid-div1020" type="float" level="2" n="2"> <note position="left" xlink:label="note-392-07" xlink:href="note-392-07a" xml:space="preserve">13. 1. Theod.</note> <note position="left" xlink:label="note-392-08" xlink:href="note-392-08a" xml:space="preserve">11. 1. Theod.</note> <note position="left" xlink:label="note-392-09" xlink:href="note-392-09a" xml:space="preserve">9. 2. Theod.</note> </div> </div> <div xml:id="echoid-div1022" type="section" level="1" n="519"> <head xml:id="echoid-head554" xml:space="preserve">THEOR. 24. PROPOS. 26.</head> <p> <s xml:id="echoid-s12944" xml:space="preserve">IN omni triangulo Iſoſcele ſphærico, cuius <pb o="381" file="393" n="393" rhead=""/> duo latera æqualia ſint quadrantes, ſi angulus ſub <lb/>ipſis comprehenſus fuerit rectus, erit baſis qua-<lb/>drans: </s> <s xml:id="echoid-s12945" xml:space="preserve">Si verò acutus, quadrante minor: </s> <s xml:id="echoid-s12946" xml:space="preserve">Si deni-<lb/>que obtuſus, quadrante maior. </s> <s xml:id="echoid-s12947" xml:space="preserve">Et ſi baſis fuerit <lb/>quadrans, eritangulus ſub lateribus comprehen-<lb/>ſus, rectus: </s> <s xml:id="echoid-s12948" xml:space="preserve">Si verò minor quadrante, acutus: </s> <s xml:id="echoid-s12949" xml:space="preserve">Si de-<lb/>nique maior quadrãte, obtuſus. </s> <s xml:id="echoid-s12950" xml:space="preserve">Semper autem po-<lb/>lus baſis erit in angulo ſub lateribus cõprehenſo.</s> <s xml:id="echoid-s12951" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12952" xml:space="preserve">IN triangulo ſphærico Iſoſcele ABC, ſint latera AB, AC, quadrantes, <lb/>& </s> <s xml:id="echoid-s12953" xml:space="preserve">primum angulus A, ſit rectus, vt in prima figura. </s> <s xml:id="echoid-s12954" xml:space="preserve">Dico baſim BC, quadran <lb/>tem eſſe. </s> <s xml:id="echoid-s12955" xml:space="preserve">Cum enim AB, <lb/> <anchor type="figure" xlink:label="fig-393-01a" xlink:href="fig-393-01"/> AC, ſint quadrãtes, erunt <lb/>anguli B, C, recti. </s> <s xml:id="echoid-s12956" xml:space="preserve">Quare <lb/> <anchor type="note" xlink:label="note-393-01a" xlink:href="note-393-01"/> drantes. </s> <s xml:id="echoid-s12957" xml:space="preserve">Quadrans ergo <lb/>eſt BC. </s> <s xml:id="echoid-s12958" xml:space="preserve">Quod eſt propo-<lb/>ſitum.</s> <s xml:id="echoid-s12959" xml:space="preserve"/> </p> <div xml:id="echoid-div1022" type="float" level="2" n="1"> <figure xlink:label="fig-393-01" xlink:href="fig-393-01a"> <image file="393-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/393-01"/> </figure> <note position="right" xlink:label="note-393-01" xlink:href="note-393-01a" xml:space="preserve">25. huius <lb/>omnes arcus erunt qua-<lb/>Corollar. <lb/>25. huius.</note> </div> <p> <s xml:id="echoid-s12960" xml:space="preserve">SIT deinde angulus <lb/>A, acutus, vt in ſecunda <lb/>figura. </s> <s xml:id="echoid-s12961" xml:space="preserve">Dico baſim BC, <lb/>minorem eſſe quadrante. <lb/></s> <s xml:id="echoid-s12962" xml:space="preserve">Ductoenim per A, & </s> <s xml:id="echoid-s12963" xml:space="preserve">po-<lb/> <anchor type="note" xlink:label="note-393-02a" xlink:href="note-393-02"/> lum arcus AB, arcu cir-<lb/>culi maximi AD, erit an-<lb/>gulus BAD, rectus, atque adeo maior acuto angulo BAC. </s> <s xml:id="echoid-s12964" xml:space="preserve">Occurret ergo <lb/> <anchor type="note" xlink:label="note-393-03a" xlink:href="note-393-03"/> AD, arcus arcui BC, producto, nempe in puncto D. </s> <s xml:id="echoid-s12965" xml:space="preserve">Quoniam igitur in trian <lb/>gulo ABC, vterque angulus B, C, rectus eſt, erunt in triangulo ABD, duo <lb/> <anchor type="note" xlink:label="note-393-04a" xlink:href="note-393-04"/> anguli recti B, & </s> <s xml:id="echoid-s12966" xml:space="preserve">DAB, ideoque æquales; </s> <s xml:id="echoid-s12967" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s12968" xml:space="preserve">arcus DA, DB, <lb/> <anchor type="note" xlink:label="note-393-05a" xlink:href="note-393-05"/> æquales erunt. </s> <s xml:id="echoid-s12969" xml:space="preserve">Quare Iſoſceles eſt DAB, habens ad baſim AB, duos angu-<lb/> <anchor type="note" xlink:label="note-393-06a" xlink:href="note-393-06"/> los rectos; </s> <s xml:id="echoid-s12970" xml:space="preserve">ac proinde vterque arcus AD, BD, quadrans eſt. </s> <s xml:id="echoid-s12971" xml:space="preserve">Igitur BC, qua-<lb/>drante erit minor. </s> <s xml:id="echoid-s12972" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12973" xml:space="preserve"/> </p> <div xml:id="echoid-div1023" type="float" level="2" n="2"> <note position="right" xlink:label="note-393-02" xlink:href="note-393-02a" xml:space="preserve">20. 1. Theod.</note> <note position="right" xlink:label="note-393-03" xlink:href="note-393-03a" xml:space="preserve">15. 1 Theod.</note> <note position="right" xlink:label="note-393-04" xlink:href="note-393-04a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-393-05" xlink:href="note-393-05a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-393-06" xlink:href="note-393-06a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s12974" xml:space="preserve">TERTIO ſit angulus A, obtuſus, vt in tertia figura. </s> <s xml:id="echoid-s12975" xml:space="preserve">Dico baſim BC, <lb/>eſſe quadrante maiorem. </s> <s xml:id="echoid-s12976" xml:space="preserve">Ducto enim per A, & </s> <s xml:id="echoid-s12977" xml:space="preserve">polum arcus AB, arcu circuli <lb/> <anchor type="note" xlink:label="note-393-07a" xlink:href="note-393-07"/> maximi AD, erit angulus DAB, rectus; </s> <s xml:id="echoid-s12978" xml:space="preserve">atque adeo minor obtuſo angulo <lb/> <anchor type="note" xlink:label="note-393-08a" xlink:href="note-393-08"/> BAC. </s> <s xml:id="echoid-s12979" xml:space="preserve">Occurret ergo arcus AD, arcui BC, intra triangulum, nempe in pun <lb/>cto D. </s> <s xml:id="echoid-s12980" xml:space="preserve">Quoniam ergo in triangulo ABC, rectus eſt vterque angulus B, C, <lb/> <anchor type="note" xlink:label="note-393-09a" xlink:href="note-393-09"/> erunt in triangulo DAB, duo anguli ad baſim AB, recti, & </s> <s xml:id="echoid-s12981" xml:space="preserve">propterea æqua-<lb/>les; </s> <s xml:id="echoid-s12982" xml:space="preserve">atque idcirco & </s> <s xml:id="echoid-s12983" xml:space="preserve">arcus AD, BD, æquales. </s> <s xml:id="echoid-s12984" xml:space="preserve">Quare Iſoſceles eſt DAB, ha-<lb/> <anchor type="note" xlink:label="note-393-10a" xlink:href="note-393-10"/> bens ad baſim AB, duos angulos rectos. </s> <s xml:id="echoid-s12985" xml:space="preserve">Vterque igitur arcus AD, BD, qua-<lb/> <anchor type="note" xlink:label="note-393-11a" xlink:href="note-393-11"/> drans eſt, ideoque BC, quadrante maior. </s> <s xml:id="echoid-s12986" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s12987" xml:space="preserve"/> </p> <div xml:id="echoid-div1024" type="float" level="2" n="3"> <note position="right" xlink:label="note-393-07" xlink:href="note-393-07a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-393-08" xlink:href="note-393-08a" xml:space="preserve">15. 1 Theod.</note> <note position="right" xlink:label="note-393-09" xlink:href="note-393-09a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-393-10" xlink:href="note-393-10a" xml:space="preserve">9. huius.</note> <note position="right" xlink:label="note-393-11" xlink:href="note-393-11a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s12988" xml:space="preserve">SED iam baſis BC, quadrans ſit, vt in eadem prima figura. </s> <s xml:id="echoid-s12989" xml:space="preserve">Dico angu-<lb/>lum A, rectum eſſe. </s> <s xml:id="echoid-s12990" xml:space="preserve">Quoniam enim duo arcus CA, CB, quadrantes ſunt, <lb/> <anchor type="note" xlink:label="note-393-12a" xlink:href="note-393-12"/> <pb o="382" file="394" n="394" rhead=""/> erit vterque angulus A, B, rectus. </s> <s xml:id="echoid-s12991" xml:space="preserve">Rectus igitur eſt angulus. </s> <s xml:id="echoid-s12992" xml:space="preserve">A.</s> <s xml:id="echoid-s12993" xml:space="preserve"/> </p> <div xml:id="echoid-div1025" type="float" level="2" n="4"> <note position="right" xlink:label="note-393-12" xlink:href="note-393-12a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s12994" xml:space="preserve">SIT deinde baſis BC, quadrante minor. </s> <s xml:id="echoid-s12995" xml:space="preserve">Dico angulum BAC, eſſe acu-<lb/>tum. </s> <s xml:id="echoid-s12996" xml:space="preserve">Producto enim arcu BC, ad D, vt ſit BD, quadrans, ducatur per pun-<lb/> <anchor type="figure" xlink:label="fig-394-01a" xlink:href="fig-394-01"/> cta A, D, arcus AD, cir-<lb/> <anchor type="note" xlink:label="note-394-01a" xlink:href="note-394-01"/> culi maximi. </s> <s xml:id="echoid-s12997" xml:space="preserve">Quoniã igi-<lb/>tur duo arcus BA, BD, <lb/>quadrãtes ſunt, erit vter-<lb/> <anchor type="note" xlink:label="note-394-02a" xlink:href="note-394-02"/> que angulus D, & </s> <s xml:id="echoid-s12998" xml:space="preserve">DAB, <lb/>rectus. </s> <s xml:id="echoid-s12999" xml:space="preserve">Acutus igitur eſt <lb/>angulus BAC.</s> <s xml:id="echoid-s13000" xml:space="preserve"/> </p> <div xml:id="echoid-div1026" type="float" level="2" n="5"> <figure xlink:label="fig-394-01" xlink:href="fig-394-01a"> <image file="394-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/394-01"/> </figure> <note position="left" xlink:label="note-394-01" xlink:href="note-394-01a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-394-02" xlink:href="note-394-02a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13001" xml:space="preserve">SIT tãdem baſis BC, <lb/>maior quadrante. </s> <s xml:id="echoid-s13002" xml:space="preserve">Dico <lb/>angulum BAC, obtuſum <lb/>eſſe. </s> <s xml:id="echoid-s13003" xml:space="preserve">Abſcindatur BD, ar-<lb/> <anchor type="note" xlink:label="note-394-03a" xlink:href="note-394-03"/> cus æqualis quadrãti AB; <lb/></s> <s xml:id="echoid-s13004" xml:space="preserve">& </s> <s xml:id="echoid-s13005" xml:space="preserve">per puncta A, D, arcus <lb/>circuli maximi deſcriba-<lb/> <anchor type="note" xlink:label="note-394-04a" xlink:href="note-394-04"/> tur AD. </s> <s xml:id="echoid-s13006" xml:space="preserve">Et quia duo arcus BA, BD, quadrantes ſunt, erit vterque angu-<lb/>lus BDA, DAB, rectus. </s> <s xml:id="echoid-s13007" xml:space="preserve">Obtuſus igitur eſt BAC, angulus.</s> <s xml:id="echoid-s13008" xml:space="preserve"/> </p> <div xml:id="echoid-div1027" type="float" level="2" n="6"> <note position="left" xlink:label="note-394-03" xlink:href="note-394-03a" xml:space="preserve">1. huius.</note> <note position="left" xlink:label="note-394-04" xlink:href="note-394-04a" xml:space="preserve">20. 1 Theod.</note> </div> <note position="left" xml:space="preserve">25. huius.</note> <p> <s xml:id="echoid-s13009" xml:space="preserve">DICO præterea, in omnibus his punctum A, polum eſſe baſis BC. </s> <s xml:id="echoid-s13010" xml:space="preserve">Cum <lb/>enim latera AB, AC, ponantur quadrantes, erit vterque angulus ad baſim <lb/> <anchor type="note" xlink:label="note-394-06a" xlink:href="note-394-06"/> BC, rectus; </s> <s xml:id="echoid-s13011" xml:space="preserve">ac propterea vterque arcus AB, AC, per polum arcus BC, tran-<lb/> <anchor type="note" xlink:label="note-394-07a" xlink:href="note-394-07"/> ſibit. </s> <s xml:id="echoid-s13012" xml:space="preserve">Siue igitur BC, quadrans ſit, ſiue minor, ſiue maior quadrante; </s> <s xml:id="echoid-s13013" xml:space="preserve">Et ſi-<lb/>ue angulus A, ſit rectus, ſiue acutus, ſiue obtuſus, ſemper punctum A, vbi <lb/>coëunt arcus AB, AC, polus erit baſis BC. </s> <s xml:id="echoid-s13014" xml:space="preserve">In omni igitur triangulo Ifo-<lb/>ſcele ſphærico, cuius duo latera, &</s> <s xml:id="echoid-s13015" xml:space="preserve">c. </s> <s xml:id="echoid-s13016" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s13017" xml:space="preserve"/> </p> <div xml:id="echoid-div1028" type="float" level="2" n="7"> <note position="left" xlink:label="note-394-06" xlink:href="note-394-06a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-394-07" xlink:href="note-394-07a" xml:space="preserve">23. 1 Theod.</note> </div> </div> <div xml:id="echoid-div1030" type="section" level="1" n="520"> <head xml:id="echoid-head555" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s13018" xml:space="preserve">_IMMO_ in omni triangulo ſphærico babente duos angulos rectos, demonſtrabi-<lb/>mus eodem modo, in concurſu duorum laterum, quæ rectos ſubtendunt angulos, re-<lb/>liqui lateris, quod rectis angulis adiacet, polum eſſe, etiam ſinondum ſciatur, duo <lb/>illa latera eſſe quadrantes. </s> <s xml:id="echoid-s13019" xml:space="preserve">Sint enim intrangulo ſphærico _ABC,_ duo anguli recti <lb/>_B, C._ </s> <s xml:id="echoid-s13020" xml:space="preserve">Dico _A,_ polum eſſe arcus _BC;_ </s> <s xml:id="echoid-s13021" xml:space="preserve">Nam vterque arcus _AB, AC,_ per polum arcus <lb/> <anchor type="note" xlink:label="note-394-08a" xlink:href="note-394-08"/> BC, tranſibits ac propterea A, polus erit arcus BC.</s> <s xml:id="echoid-s13022" xml:space="preserve"/> </p> <div xml:id="echoid-div1030" type="float" level="2" n="1"> <note position="left" xlink:label="note-394-08" xlink:href="note-394-08a" xml:space="preserve">13. 1. Theod.</note> </div> <p> <s xml:id="echoid-s13023" xml:space="preserve">_VERVM_ eſt tamen, duos arcus AB, AC, eſſe ſemper quadrantes, propter an-<lb/> <anchor type="note" xlink:label="note-394-09a" xlink:href="note-394-09"/> gulos rectos _B, C._</s> <s xml:id="echoid-s13024" xml:space="preserve"/> </p> <div xml:id="echoid-div1031" type="float" level="2" n="2"> <note position="left" xlink:label="note-394-09" xlink:href="note-394-09a" xml:space="preserve">25. huius.</note> </div> </div> <div xml:id="echoid-div1033" type="section" level="1" n="521"> <head xml:id="echoid-head556" xml:space="preserve">THEOR. 25. PROPOS. 27.</head> <p> <s xml:id="echoid-s13025" xml:space="preserve">IN omni triangulo ſphærico, cuius omnes ar-<lb/>cus ſint quadrante maiores, vel vnus quadrans, <lb/>& </s> <s xml:id="echoid-s13026" xml:space="preserve">reliqui duo quadrante maiores, omnes tres an-<lb/>guli ſunt obtuſi.</s> <s xml:id="echoid-s13027" xml:space="preserve"/> </p> <pb o="383" file="395" n="395" rhead=""/> <p> <s xml:id="echoid-s13028" xml:space="preserve">IN triangulo ſphærico ABC, ſint primum ſingula latera quadrante ma-<lb/>iora. </s> <s xml:id="echoid-s13029" xml:space="preserve">Dico tres angulos A, B, C, eſſe obtuſos. </s> <s xml:id="echoid-s13030" xml:space="preserve">Aut enim triangulum æquila-<lb/>terum eſt, aut Iſoſceles, aut Scalenum.</s> <s xml:id="echoid-s13031" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13032" xml:space="preserve">SI æquilaterum, perſpicuum eſt, tres angulos eſſe obtuſos.</s> <s xml:id="echoid-s13033" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Corollar. <lb/>25. huius.</note> <p> <s xml:id="echoid-s13034" xml:space="preserve">SI vero eſt Iſoſceles, habens duo latera AB, <lb/> <anchor type="figure" xlink:label="fig-395-01a" xlink:href="fig-395-01"/> <anchor type="note" xlink:label="note-395-02a" xlink:href="note-395-02"/> AC, æqualia, erunt duo anguli B, C, ad baſim ob-<lb/> <anchor type="note" xlink:label="note-395-03a" xlink:href="note-395-03"/> tuſi. </s> <s xml:id="echoid-s13035" xml:space="preserve">Sint quadrantes BD, BE, & </s> <s xml:id="echoid-s13036" xml:space="preserve">per puncta <lb/>D, E, arcus circuli maximi ducatur ED, conue-<lb/>nienscum arcu CA, protracto in F. </s> <s xml:id="echoid-s13037" xml:space="preserve">Quoniam igi-<lb/>tur BD, BE, quadrantes ſunt, & </s> <s xml:id="echoid-s13038" xml:space="preserve">angulus B, oſten <lb/> <anchor type="note" xlink:label="note-395-04a" xlink:href="note-395-04"/> ſus eſt obtuſus, erit DE, arcus quadrante maior, <lb/> <anchor type="note" xlink:label="note-395-05a" xlink:href="note-395-05"/> & </s> <s xml:id="echoid-s13039" xml:space="preserve">anguli BDE, BED, recti: </s> <s xml:id="echoid-s13040" xml:space="preserve">Ponitur autem & </s> <s xml:id="echoid-s13041" xml:space="preserve"><lb/>arcus AC, quadrante maior. </s> <s xml:id="echoid-s13042" xml:space="preserve">Igitur arcus DE, <lb/>AC, ſimul ſemicirculo maiores ſunt; </s> <s xml:id="echoid-s13043" xml:space="preserve">ac propte-<lb/>rea arcus FD, FA, ſimul minores ſemicirculo, cum <lb/>arcus FE, FC, integro circulo ſimul ſint mino-<lb/>res; </s> <s xml:id="echoid-s13044" xml:space="preserve">cum vterque arcus minor ſit ſemicirculo. </s> <s xml:id="echoid-s13045" xml:space="preserve">An-<lb/> <anchor type="note" xlink:label="note-395-06a" xlink:href="note-395-06"/> gulus igitur FDB, maior eſt angulo FAD: </s> <s xml:id="echoid-s13046" xml:space="preserve">Eſt autem angulus FDB, rectus; <lb/></s> <s xml:id="echoid-s13047" xml:space="preserve"> <anchor type="note" xlink:label="note-395-07a" xlink:href="note-395-07"/> quòd anguli FDB, BDE, duobus rectis æquales ſint, & </s> <s xml:id="echoid-s13048" xml:space="preserve">BDE, rectus oſten-<lb/> <anchor type="note" xlink:label="note-395-08a" xlink:href="note-395-08"/> ſus. </s> <s xml:id="echoid-s13049" xml:space="preserve">Ergo FAD, acutus eſt; </s> <s xml:id="echoid-s13050" xml:space="preserve">ac proinde, cum FAD, DAC, æquales ſint duo-<lb/> <anchor type="note" xlink:label="note-395-09a" xlink:href="note-395-09"/> bus rectis, angulus BAC, obtuſus erit: </s> <s xml:id="echoid-s13051" xml:space="preserve">oſtenſi ſunt autem & </s> <s xml:id="echoid-s13052" xml:space="preserve">anguli B, C, <lb/>obtuſi. </s> <s xml:id="echoid-s13053" xml:space="preserve">Omnes ergo tres anguli A, B, C, obtuſi ſunt.</s> <s xml:id="echoid-s13054" xml:space="preserve"/> </p> <div xml:id="echoid-div1033" type="float" level="2" n="1"> <figure xlink:label="fig-395-01" xlink:href="fig-395-01a"> <image file="395-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/395-01"/> </figure> <note position="right" xlink:label="note-395-02" xlink:href="note-395-02a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-395-03" xlink:href="note-395-03a" xml:space="preserve">20. 1. Theod.</note> <note position="left" xlink:label="note-395-04" xlink:href="note-395-04a" xml:space="preserve">26. huius</note> <note position="right" xlink:label="note-395-05" xlink:href="note-395-05a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-395-06" xlink:href="note-395-06a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-395-07" xlink:href="note-395-07a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-395-08" xlink:href="note-395-08a" xml:space="preserve">5 huius.</note> <note position="right" xlink:label="note-395-09" xlink:href="note-395-09a" xml:space="preserve">5. huius.</note> </div> <p> <s xml:id="echoid-s13055" xml:space="preserve">SI denique triangulum ABC, eſt Scalenum, ſit latus AC, latere AB, <lb/>maius, & </s> <s xml:id="echoid-s13056" xml:space="preserve">abſcindatur arcus AD, arcui AB, æqualis; </s> <s xml:id="echoid-s13057" xml:space="preserve">eritq́ue adhuc arcus AD, <lb/>quadrante maior, quòd & </s> <s xml:id="echoid-s13058" xml:space="preserve">arcus AB, cui æqualis eſt, maior ponatur quadran <lb/> <anchor type="note" xlink:label="note-395-10a" xlink:href="note-395-10"/> te. </s> <s xml:id="echoid-s13059" xml:space="preserve">Si igitur per puncta B, D, ducatur arcus BD, circuli maximi, erit vterq; <lb/></s> <s xml:id="echoid-s13060" xml:space="preserve"> <anchor type="note" xlink:label="note-395-11a" xlink:href="note-395-11"/> angulus ADB, ABD, obtuſus. </s> <s xml:id="echoid-s13061" xml:space="preserve">Multo ergo ma-<lb/> <anchor type="note" xlink:label="note-395-12a" xlink:href="note-395-12"/> <anchor type="figure" xlink:label="fig-395-02a" xlink:href="fig-395-02"/> gis obtuſus erit angulus ABC. </s> <s xml:id="echoid-s13062" xml:space="preserve">Sint quadrantes <lb/>BE, BF, & </s> <s xml:id="echoid-s13063" xml:space="preserve">per puncta E, F, ducatur arcus EF, <lb/> <anchor type="note" xlink:label="note-395-13a" xlink:href="note-395-13"/> circuli maximi, coiens cum arcu CA, producto in <lb/>G. </s> <s xml:id="echoid-s13064" xml:space="preserve">Quoniã igitur BE, BF, quadrantes ſunt, erunt <lb/>anguli ad E, & </s> <s xml:id="echoid-s13065" xml:space="preserve">F, recti; </s> <s xml:id="echoid-s13066" xml:space="preserve">& </s> <s xml:id="echoid-s13067" xml:space="preserve">cum angulus EBF, oſten <lb/> <anchor type="note" xlink:label="note-395-14a" xlink:href="note-395-14"/> ſus ſit obtuſus, erit arcus EF, quadrante maior: <lb/></s> <s xml:id="echoid-s13068" xml:space="preserve"> <anchor type="note" xlink:label="note-395-15a" xlink:href="note-395-15"/> Ponitur autem & </s> <s xml:id="echoid-s13069" xml:space="preserve">arcus AC, quadrante maior. <lb/></s> <s xml:id="echoid-s13070" xml:space="preserve">Igitur arcus EF, AC, ſimul ſemicirculo ſunt ma-<lb/>iores; </s> <s xml:id="echoid-s13071" xml:space="preserve">& </s> <s xml:id="echoid-s13072" xml:space="preserve">idcirco multo magis FG, CG, maiores <lb/>erũt ſemicirculo. </s> <s xml:id="echoid-s13073" xml:space="preserve">Angulus ergo BFG, quem oſten <lb/>dimus eſſe rectum, min or eſt angulo BCG; </s> <s xml:id="echoid-s13074" xml:space="preserve">ac pro-<lb/> <anchor type="note" xlink:label="note-395-16a" xlink:href="note-395-16"/> pterea angulus C, obtuſus erit. </s> <s xml:id="echoid-s13075" xml:space="preserve">Et quoniam arcus <lb/>FG, CG, ſimul integro ſunt circulo minores; <lb/></s> <s xml:id="echoid-s13076" xml:space="preserve">quòd vterque ſemicirculo min or ſit; </s> <s xml:id="echoid-s13077" xml:space="preserve">& </s> <s xml:id="echoid-s13078" xml:space="preserve">EF, AC, <lb/> <anchor type="note" xlink:label="note-395-17a" xlink:href="note-395-17"/> ſimul ſemicirculo maiores; </s> <s xml:id="echoid-s13079" xml:space="preserve">eruntarcus GE, GA, ſimul ſemicirculo mino-<lb/>res; </s> <s xml:id="echoid-s13080" xml:space="preserve">ac proinde angulus GEB, maior erit angulo GAB. </s> <s xml:id="echoid-s13081" xml:space="preserve">Cum ergo angu-<lb/> <anchor type="note" xlink:label="note-395-18a" xlink:href="note-395-18"/> lus GEB, rectus ſit, quòd duo anguli ad E, duobus ſint rectis æquales, & </s> <s xml:id="echoid-s13082" xml:space="preserve">an-<lb/> <anchor type="note" xlink:label="note-395-19a" xlink:href="note-395-19"/> gulus BEF, oſtenſus ſit rectus; </s> <s xml:id="echoid-s13083" xml:space="preserve">erit angulus GAB, acutus. </s> <s xml:id="echoid-s13084" xml:space="preserve">Quapropter cum <lb/>GAB, BAC, ęquales ſint duobus rectis, erit BAC, obtuſus. </s> <s xml:id="echoid-s13085" xml:space="preserve">Suntautem duo <lb/> <anchor type="note" xlink:label="note-395-20a" xlink:href="note-395-20"/> etiam anguli ABC, & </s> <s xml:id="echoid-s13086" xml:space="preserve">C, oſtenſi obtuſi. </s> <s xml:id="echoid-s13087" xml:space="preserve">Tres ergo anguli A, B, C, trianguli <lb/>ABC, obtuſi ſunt. </s> <s xml:id="echoid-s13088" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s13089" xml:space="preserve"/> </p> <div xml:id="echoid-div1034" type="float" level="2" n="2"> <note position="left" xlink:label="note-395-10" xlink:href="note-395-10a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-395-11" xlink:href="note-395-11a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-395-12" xlink:href="note-395-12a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-395-02" xlink:href="fig-395-02a"> <image file="395-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/395-02"/> </figure> <note position="right" xlink:label="note-395-13" xlink:href="note-395-13a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-395-14" xlink:href="note-395-14a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-395-15" xlink:href="note-395-15a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-395-16" xlink:href="note-395-16a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-395-17" xlink:href="note-395-17a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-395-18" xlink:href="note-395-18a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-395-19" xlink:href="note-395-19a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-395-20" xlink:href="note-395-20a" xml:space="preserve">5. huius.</note> </div> <pb o="384" file="396" n="396" rhead=""/> <p> <s xml:id="echoid-s13090" xml:space="preserve">SINT iam in eodem triangulo ABC, duo arcus AB, AC, quadrante <lb/>quidem maiores, at BC, quadrans. </s> <s xml:id="echoid-s13091" xml:space="preserve">Autigitur arcus AB, AC, æquales ſunt, <lb/>aut inæquales. </s> <s xml:id="echoid-s13092" xml:space="preserve">Si æquales, erunt duo anguli B, C, obtuſi. </s> <s xml:id="echoid-s13093" xml:space="preserve">Sit quadrans BD, <lb/> <anchor type="figure" xlink:label="fig-396-01a" xlink:href="fig-396-01"/> <anchor type="note" xlink:label="note-396-01a" xlink:href="note-396-01"/> & </s> <s xml:id="echoid-s13094" xml:space="preserve">per puncta C, D, arcus CD, maximi circuli du-<lb/> <anchor type="note" xlink:label="note-396-02a" xlink:href="note-396-02"/> catur conueniens cum arcu CA, protracto in E. <lb/></s> <s xml:id="echoid-s13095" xml:space="preserve">Quia igitur arcus BC, BD, quadrantes ſunt, erũt <lb/>anguli D, & </s> <s xml:id="echoid-s13096" xml:space="preserve">BCD, recti; </s> <s xml:id="echoid-s13097" xml:space="preserve">& </s> <s xml:id="echoid-s13098" xml:space="preserve">arcus CD, propter <lb/> <anchor type="note" xlink:label="note-396-03a" xlink:href="note-396-03"/> angulum B, quem obtuſum eſſe oſtendimus, qua-<lb/>drante maior: </s> <s xml:id="echoid-s13099" xml:space="preserve">Ponitur autem & </s> <s xml:id="echoid-s13100" xml:space="preserve">arcus AC, qua-<lb/> <anchor type="note" xlink:label="note-396-04a" xlink:href="note-396-04"/> drante maior. </s> <s xml:id="echoid-s13101" xml:space="preserve">Igitur arcus CD, CA, ſimul ma-<lb/>iores ſunt ſemicirculo; </s> <s xml:id="echoid-s13102" xml:space="preserve">ac propterea, cum arcus <lb/>CDE, CAE, circulum conficiant, (quòd vter-<lb/>que ſemicirculus ſit.) </s> <s xml:id="echoid-s13103" xml:space="preserve">erunt arcus ED, EA, ſemi-<lb/> <anchor type="note" xlink:label="note-396-05a" xlink:href="note-396-05"/> circulo minores. </s> <s xml:id="echoid-s13104" xml:space="preserve">Quare angulus EDB, qui rectus <lb/>eſt, (quòd duo anguli ad D, æquales ſint duobus <lb/> <anchor type="note" xlink:label="note-396-06a" xlink:href="note-396-06"/> rectis, & </s> <s xml:id="echoid-s13105" xml:space="preserve">angulus BDC, oſtenſus ſit rectus.) </s> <s xml:id="echoid-s13106" xml:space="preserve">maior <lb/>erit angulo EAD; </s> <s xml:id="echoid-s13107" xml:space="preserve">atque adeo EAD, acutus erit. <lb/></s> <s xml:id="echoid-s13108" xml:space="preserve"> <anchor type="note" xlink:label="note-396-07a" xlink:href="note-396-07"/> Cum ergo anguli EAD, DAC, duobus rectis ſint <lb/> <anchor type="note" xlink:label="note-396-08a" xlink:href="note-396-08"/> æquales, erit BAC, obtuſus. </s> <s xml:id="echoid-s13109" xml:space="preserve">Sunt etiam anguli B, C, demonſtrati obtuſi. <lb/></s> <s xml:id="echoid-s13110" xml:space="preserve">Tres igitur anguli A, B, C, trianguli ABC, obtuſi ſunt.</s> <s xml:id="echoid-s13111" xml:space="preserve"/> </p> <div xml:id="echoid-div1035" type="float" level="2" n="3"> <figure xlink:label="fig-396-01" xlink:href="fig-396-01a"> <image file="396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/396-01"/> </figure> <note position="left" xlink:label="note-396-01" xlink:href="note-396-01a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-396-02" xlink:href="note-396-02a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-396-03" xlink:href="note-396-03a" xml:space="preserve">25. huius</note> <note position="left" xlink:label="note-396-04" xlink:href="note-396-04a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-396-05" xlink:href="note-396-05a" xml:space="preserve">11. 1. Theod</note> <note position="left" xlink:label="note-396-06" xlink:href="note-396-06a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-396-07" xlink:href="note-396-07a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-396-08" xlink:href="note-396-08a" xml:space="preserve">5. huius.</note> </div> <p> <s xml:id="echoid-s13112" xml:space="preserve">SI verò AB, AC, latera, quæ quadrante maiora ſunt, non ſunt æqualia, <lb/>ſit maius AC; </s> <s xml:id="echoid-s13113" xml:space="preserve">& </s> <s xml:id="echoid-s13114" xml:space="preserve">abſcindatur arcus AD, æqualis arcui AB; </s> <s xml:id="echoid-s13115" xml:space="preserve">& </s> <s xml:id="echoid-s13116" xml:space="preserve">per puncta B, <lb/> <anchor type="note" xlink:label="note-396-09a" xlink:href="note-396-09"/> D, tranſeat arcus BD, circuli maximi: </s> <s xml:id="echoid-s13117" xml:space="preserve">eritq́ue adhuc arcus AD, maior qua-<lb/> <anchor type="figure" xlink:label="fig-396-02a" xlink:href="fig-396-02"/> <anchor type="note" xlink:label="note-396-10a" xlink:href="note-396-10"/> drante, cum ei æqualis AB, maior etiam ponatur-<lb/>Anguli igitur ADB, ABD, obtuſi ſunt. </s> <s xml:id="echoid-s13118" xml:space="preserve">Multo <lb/> <anchor type="note" xlink:label="note-396-11a" xlink:href="note-396-11"/> ergo magis obtuſus erit angulus ABC. </s> <s xml:id="echoid-s13119" xml:space="preserve">Sit qua-<lb/>drans BE, & </s> <s xml:id="echoid-s13120" xml:space="preserve">per puncta C, E, tranſeat arcus CE, <lb/> <anchor type="note" xlink:label="note-396-12a" xlink:href="note-396-12"/> circuli maximi occurrens arcui CA, producto in <lb/>F. </s> <s xml:id="echoid-s13121" xml:space="preserve">Quoniam igitur quadrantes ſunt BE, BC, & </s> <s xml:id="echoid-s13122" xml:space="preserve"><lb/>angulus EBC, oſtẽſus eſt obtuſus, erit arcus EC, <lb/> <anchor type="note" xlink:label="note-396-13a" xlink:href="note-396-13"/> maior quadrãte, ſed anguli E, & </s> <s xml:id="echoid-s13123" xml:space="preserve">BCE, recti erunt. <lb/></s> <s xml:id="echoid-s13124" xml:space="preserve"> <anchor type="note" xlink:label="note-396-14a" xlink:href="note-396-14"/> Angulus ergo ACB, obtuſus erit. </s> <s xml:id="echoid-s13125" xml:space="preserve">Et quoniam <lb/>arcus CE, oſtenſus eſt quadrante maior, & </s> <s xml:id="echoid-s13126" xml:space="preserve">arcus <lb/>AC, maior etiam ponitur, quam quadrans; </s> <s xml:id="echoid-s13127" xml:space="preserve">erunt <lb/>arcus CE, CA, ſimul ſemicirculo maiores. </s> <s xml:id="echoid-s13128" xml:space="preserve">Cũ ergo <lb/>arcus CEF, CAF, integro circulo æquales ſint, <lb/> <anchor type="note" xlink:label="note-396-15a" xlink:href="note-396-15"/> quòd vterque ſit ſemicirculus, erũt arcus FE, FA, <lb/>ſimul ſemicirculo minores. </s> <s xml:id="echoid-s13129" xml:space="preserve">Quamobrem angulus <lb/>FEB, quirectus eſt, (ſunt enim duo anguli ad E, duobus rectis æquales, & </s> <s xml:id="echoid-s13130" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-396-16a" xlink:href="note-396-16"/> angulus BEC, oſtenſus eſt rectus.) </s> <s xml:id="echoid-s13131" xml:space="preserve">maior erit angulo FAE. </s> <s xml:id="echoid-s13132" xml:space="preserve">Acutus ergo eſt <lb/> <anchor type="note" xlink:label="note-396-17a" xlink:href="note-396-17"/> angulus FAE; </s> <s xml:id="echoid-s13133" xml:space="preserve">ac propterea, cum duo anguli ad A, ſint æquales duobus rectis, <lb/> <anchor type="note" xlink:label="note-396-18a" xlink:href="note-396-18"/> angulus BAc, obtuſus erit. </s> <s xml:id="echoid-s13134" xml:space="preserve">Sunt autem etiam oſtenſi obtuſi anguli ABC, <lb/>ACB. </s> <s xml:id="echoid-s13135" xml:space="preserve">Tres igitur anguli in triangulo ABC, obtuſi ſunt. </s> <s xml:id="echoid-s13136" xml:space="preserve">In omni ergo trian <lb/>gulo ſphærico, cuius omnes arcus, &</s> <s xml:id="echoid-s13137" xml:space="preserve">c. </s> <s xml:id="echoid-s13138" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s13139" xml:space="preserve"/> </p> <div xml:id="echoid-div1036" type="float" level="2" n="4"> <note position="left" xlink:label="note-396-09" xlink:href="note-396-09a" xml:space="preserve">1. huius.</note> <figure xlink:label="fig-396-02" xlink:href="fig-396-02a"> <image file="396-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/396-02"/> </figure> <note position="left" xlink:label="note-396-10" xlink:href="note-396-10a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-396-11" xlink:href="note-396-11a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-396-12" xlink:href="note-396-12a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-396-13" xlink:href="note-396-13a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-396-14" xlink:href="note-396-14a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-396-15" xlink:href="note-396-15a" xml:space="preserve">21. 1. Theod.</note> <note position="left" xlink:label="note-396-16" xlink:href="note-396-16a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-396-17" xlink:href="note-396-17a" xml:space="preserve">14. huius.</note> <note position="left" xlink:label="note-396-18" xlink:href="note-396-18a" xml:space="preserve">5. huius.</note> </div> </div> <div xml:id="echoid-div1038" type="section" level="1" n="522"> <head xml:id="echoid-head557" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s13140" xml:space="preserve">_HAEC_ propoſitio non conuertitur. </s> <s xml:id="echoid-s13141" xml:space="preserve">Non enim omne triangulum ſphæricum, cu-<lb/>ius omnes anguli ſunt obtuſi, neceſſario habet omnes arcus quadrante maiores, vel <pb o="385" file="397" n="397" rhead=""/> duos quidem maiores quadrante, & </s> <s xml:id="echoid-s13142" xml:space="preserve">vnum quadranti æqualem: </s> <s xml:id="echoid-s13143" xml:space="preserve">Sed poſſunt eſſe duo <lb/>quidem quadrante maiores, reliquus verò quadrante minor. </s> <s xml:id="echoid-s13144" xml:space="preserve">Sint enim duo ſemicir-<lb/>culi in ſuperficie ſphæra continentes angulos _A, C,_ obtuſos. </s> <s xml:id="echoid-s13145" xml:space="preserve">Si igitur accipiantur <lb/>duo arcus æquales _AB, AD,_ quorum vterque <lb/> <anchor type="figure" xlink:label="fig-397-01a" xlink:href="fig-397-01"/> maior ſit ſesqusaltero quadrante, ita vt ambo <lb/>ſimul tres quadrantes ſuperent; </s> <s xml:id="echoid-s13146" xml:space="preserve">deſeribatur <lb/>autem per puncta _B, D,_ arcus circuli maximi <lb/> <anchor type="note" xlink:label="note-397-01a" xlink:href="note-397-01"/> _BD;_ </s> <s xml:id="echoid-s13147" xml:space="preserve">erit hic arcus _BD,_ quadrante minor. </s> <s xml:id="echoid-s13148" xml:space="preserve">Cum <lb/>enim tres arcus _AB, AD, BD,_ integro circu-<lb/> <anchor type="note" xlink:label="note-397-02a" xlink:href="note-397-02"/> lo minores ſint, ponatur autem duo arcus _AB,_ <lb/>_AD,_ tribus quadrantibus maiores; </s> <s xml:id="echoid-s13149" xml:space="preserve">erit neceſ-<lb/>ſario tertius arcus _BD,_ minor quadrãte: </s> <s xml:id="echoid-s13150" xml:space="preserve">Alias, <lb/>ſi quadrans eſſet, aut maior quadrante, ſupe-<lb/>rarent tres arcus trianguli _ABC,_ integrum <lb/>circulum. </s> <s xml:id="echoid-s13151" xml:space="preserve">Quoniam igitur duo anguli _B,_ & </s> <s xml:id="echoid-s13152" xml:space="preserve">_D,_ in triangulo _ABD,_ obtuſi ſunt, <lb/> <anchor type="note" xlink:label="note-397-03a" xlink:href="note-397-03"/> necnon & </s> <s xml:id="echoid-s13153" xml:space="preserve">tertius angulus _A,_ obtuſus quoque, ex bypotheſi; </s> <s xml:id="echoid-s13154" xml:space="preserve">erunt omnes tres anguli <lb/>_A, B, D,_ obtuſi; </s> <s xml:id="echoid-s13155" xml:space="preserve">& </s> <s xml:id="echoid-s13156" xml:space="preserve">tamen neque omnes arcus ſunt quadrante maiores; </s> <s xml:id="echoid-s13157" xml:space="preserve">neque duo <lb/>tantum, & </s> <s xml:id="echoid-s13158" xml:space="preserve">tertius quadrans; </s> <s xml:id="echoid-s13159" xml:space="preserve">ſed duo quidem _AB, AD,_ quadrante maiores ſunt, <lb/>at tertius arcus _BD,_ quadrante minor, vt oſtendimus.</s> <s xml:id="echoid-s13160" xml:space="preserve"/> </p> <div xml:id="echoid-div1038" type="float" level="2" n="1"> <figure xlink:label="fig-397-01" xlink:href="fig-397-01a"> <image file="397-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/397-01"/> </figure> <note position="right" xlink:label="note-397-01" xlink:href="note-397-01a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-397-02" xlink:href="note-397-02a" xml:space="preserve">4. huius.</note> <note position="right" xlink:label="note-397-03" xlink:href="note-397-03a" xml:space="preserve">25. huius.</note> </div> </div> <div xml:id="echoid-div1040" type="section" level="1" n="523"> <head xml:id="echoid-head558" xml:space="preserve">THEOR. 26. PROPOS. 28.</head> <p> <s xml:id="echoid-s13161" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <lb/>omnes arcus ſint quadrante minores, reliqui duo <lb/>anguli acuti ſunt. </s> <s xml:id="echoid-s13162" xml:space="preserve">Et ſi reliqui duo anguli ſint acu-<lb/>ti, erunt ſinguli arcus quadrante minores.</s> <s xml:id="echoid-s13163" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13164" xml:space="preserve">IN triangulo ſphærico ABC, ſit angulus B, rectus, & </s> <s xml:id="echoid-s13165" xml:space="preserve">ſinguli arcus qua-<lb/>drante minores. </s> <s xml:id="echoid-s13166" xml:space="preserve">Dico reliquos angulos A, C, eſſe acutos. </s> <s xml:id="echoid-s13167" xml:space="preserve">Producantur enim <lb/>arcus BA, BC, vt ſint quadrantes BD, BE; </s> <s xml:id="echoid-s13168" xml:space="preserve">& </s> <s xml:id="echoid-s13169" xml:space="preserve">per puncta C, D, arcus maxi-<lb/>mi circuli ducatur CD, necnon per puncta A, E, ar-<lb/> <anchor type="note" xlink:label="note-397-04a" xlink:href="note-397-04"/> <anchor type="figure" xlink:label="fig-397-02a" xlink:href="fig-397-02"/> cus circuli maximi AE. </s> <s xml:id="echoid-s13170" xml:space="preserve">Et quoniam quadrans BD, <lb/>ob angulum rectum B, per polos arcus BC, tranſit, <lb/> <anchor type="note" xlink:label="note-397-05a" xlink:href="note-397-05"/> abeſtq; </s> <s xml:id="echoid-s13171" xml:space="preserve">polus circuli maximi quadrate circuli maxi-<lb/>mi ab eo, erit D, polus arcus BC. </s> <s xml:id="echoid-s13172" xml:space="preserve">Igitur erit angu-<lb/> <anchor type="note" xlink:label="note-397-06a" xlink:href="note-397-06"/> lus BCD, rectus; </s> <s xml:id="echoid-s13173" xml:space="preserve">ac propterea angulus ACB, acu-<lb/> <anchor type="note" xlink:label="note-397-07a" xlink:href="note-397-07"/> tus. </s> <s xml:id="echoid-s13174" xml:space="preserve">Eodem modo, quia quadrans BE, ob angulum <lb/>rectum B, per polos arcus AB, tranſit, abeſtq́; </s> <s xml:id="echoid-s13175" xml:space="preserve">polus <lb/> <anchor type="note" xlink:label="note-397-08a" xlink:href="note-397-08"/> circuli maximi quadrante maximi circuli ab eo, erit <lb/> <anchor type="note" xlink:label="note-397-09a" xlink:href="note-397-09"/> E, polus arcus AB, Igitur angulus EAB, rectus erit; <lb/></s> <s xml:id="echoid-s13176" xml:space="preserve"> <anchor type="note" xlink:label="note-397-10a" xlink:href="note-397-10"/> ac proinde BAC, acutus.</s> <s xml:id="echoid-s13177" xml:space="preserve"/> </p> <div xml:id="echoid-div1040" type="float" level="2" n="1"> <note position="right" xlink:label="note-397-04" xlink:href="note-397-04a" xml:space="preserve">20. 1 Theod.</note> <figure xlink:label="fig-397-02" xlink:href="fig-397-02a"> <image file="397-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/397-02"/> </figure> <note position="right" xlink:label="note-397-05" xlink:href="note-397-05a" xml:space="preserve">13. 1. Theod. <lb/>Coroll. 16.</note> <note position="right" xlink:label="note-397-06" xlink:href="note-397-06a" xml:space="preserve">1. Theod.</note> <note position="right" xlink:label="note-397-07" xlink:href="note-397-07a" xml:space="preserve">15. 1 Theod.</note> <note position="right" xlink:label="note-397-08" xlink:href="note-397-08a" xml:space="preserve">13. 1 Theod.</note> <note position="right" xlink:label="note-397-09" xlink:href="note-397-09a" xml:space="preserve">Coroll. 16.</note> <note position="right" xlink:label="note-397-10" xlink:href="note-397-10a" xml:space="preserve">1. Theod.</note> </div> <note position="right" xml:space="preserve">15. 1 Theod.</note> <p> <s xml:id="echoid-s13178" xml:space="preserve">SED iam in eodem triangulo ABC, angulus B, rectus ſit, & </s> <s xml:id="echoid-s13179" xml:space="preserve">reliqui A, <lb/>C, acuti. </s> <s xml:id="echoid-s13180" xml:space="preserve">Dico ſingulos arcus eſſe quadrante minores. </s> <s xml:id="echoid-s13181" xml:space="preserve">Fiant enim recti angu-<lb/>li BCD, BAE. </s> <s xml:id="echoid-s13182" xml:space="preserve">Quia igitur vterque angulus B, BCD, rectus eſt, erit vter-<lb/> <anchor type="note" xlink:label="note-397-12a" xlink:href="note-397-12"/> que arcus BD, CD, quadrans. </s> <s xml:id="echoid-s13183" xml:space="preserve">Arcus igitur BA, quadrante minor eſt. </s> <s xml:id="echoid-s13184" xml:space="preserve">Eo-<lb/>dem modo arcus BC, minor erit quadrante; </s> <s xml:id="echoid-s13185" xml:space="preserve">propterea quòd & </s> <s xml:id="echoid-s13186" xml:space="preserve">arcus BE, AE, <lb/> <anchor type="note" xlink:label="note-397-13a" xlink:href="note-397-13"/> <pb o="386" file="398" n="398" rhead=""/> quadrantes ſunt, ob angulos rectos B, BAE. </s> <s xml:id="echoid-s13187" xml:space="preserve">Sed & </s> <s xml:id="echoid-s13188" xml:space="preserve">arcum AC, minorem eſ-<lb/>ſe quadrante, ita oſtendemus. </s> <s xml:id="echoid-s13189" xml:space="preserve">Quoniam arcus BE, ducitur per E, polum ar-<lb/> <anchor type="figure" xlink:label="fig-398-01a" xlink:href="fig-398-01"/> cus BD; </s> <s xml:id="echoid-s13190" xml:space="preserve">(oſtendemus enim E, eſſe polum arcus AB, <lb/>vt ſupra, cum BE, quadrans ſit, rectusq́ue ad arcum <lb/>AB.) </s> <s xml:id="echoid-s13191" xml:space="preserve">erit punctum C, intra peripheriam circuli ar-<lb/>cus BD, in ſuperficie ſphæræ, & </s> <s xml:id="echoid-s13192" xml:space="preserve">præter eiuſdem po-<lb/>lum. </s> <s xml:id="echoid-s13193" xml:space="preserve">Quare arcus CA, minor erit arcu CD: </s> <s xml:id="echoid-s13194" xml:space="preserve">At CD, <lb/> <anchor type="note" xlink:label="note-398-01a" xlink:href="note-398-01"/> oſtenſus eſt eſſe quadrans. </s> <s xml:id="echoid-s13195" xml:space="preserve">Igitur AC, quadrante mi-<lb/> <anchor type="note" xlink:label="note-398-02a" xlink:href="note-398-02"/> nor erit. </s> <s xml:id="echoid-s13196" xml:space="preserve">Omnes ergo arcus trianguli ABC, qua-<lb/>drante ſunt minores. </s> <s xml:id="echoid-s13197" xml:space="preserve">Quocirca in omni triangulo <lb/>ſpherico rectangulo, &</s> <s xml:id="echoid-s13198" xml:space="preserve">c. </s> <s xml:id="echoid-s13199" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s13200" xml:space="preserve"/> </p> <div xml:id="echoid-div1041" type="float" level="2" n="2"> <note position="right" xlink:label="note-397-12" xlink:href="note-397-12a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-397-13" xlink:href="note-397-13a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-398-01" xlink:href="fig-398-01a"> <image file="398-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/398-01"/> </figure> <note position="left" xlink:label="note-398-01" xlink:href="note-398-01a" xml:space="preserve">Schol. 21.</note> <note position="left" xlink:label="note-398-02" xlink:href="note-398-02a" xml:space="preserve">2. Theod.</note> </div> </div> <div xml:id="echoid-div1043" type="section" level="1" n="524"> <head xml:id="echoid-head559" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s13201" xml:space="preserve">_PRIMA_ pars buius propoſitionis vera quoque eſt, ſi ſolum vterque arcus circa <lb/>angulum rectum ponatur quadrante miner, etiamſi ignoretur, reliquum arcum, <lb/>qui rectum angulum ſubtendit, minorem eſſe quadrante. </s> <s xml:id="echoid-s13202" xml:space="preserve">Id quod liquido conſtat ex <lb/>demonſtratione prioris partis. </s> <s xml:id="echoid-s13203" xml:space="preserve">Oſtenſum eſt enim, angulos _BAC, BCA,_ eſſe acu-<lb/>tos, ex eo ſolum, quòd vterque arcus _BA, BC,_ quadrante minor ponatur, nulla <lb/>facta mentione arcus _AC._ </s> <s xml:id="echoid-s13204" xml:space="preserve">Erit tamen ſemper arcus rectum angulum ſubtendens <lb/>quadrante minor, ſi duo arcus rectum angulum continentes quadrante minores ſint, <lb/>vt ex demonſtratione manife ſtum eſt. </s> <s xml:id="echoid-s13205" xml:space="preserve">Nam cum ex eo, quòd arcus _BA, BC,_ mino-<lb/>res ſint quadrante, anguli A, C, acuti ſint, vt in priore parte demonſtratum eſt, ſit, <lb/>vt & </s> <s xml:id="echoid-s13206" xml:space="preserve">arcus AC, minor ſit quadrante, vtin parte poſteriori eſt oſtenſum. </s> <s xml:id="echoid-s13207" xml:space="preserve">Itaque <lb/>proponi poterit etiam buiuſmodi Theorema.</s> <s xml:id="echoid-s13208" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13209" xml:space="preserve">IN omni ttiangulo ſphærico rectangulo, cuius duo arcus rectum <lb/>angulum comprehendentes quadrante ſint minores, erit & </s> <s xml:id="echoid-s13210" xml:space="preserve">arcus <lb/>angulum rectum ſubtendens quadrante minor.</s> <s xml:id="echoid-s13211" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1044" type="section" level="1" n="525"> <head xml:id="echoid-head560" xml:space="preserve">THEOR. 27. PROPOS. 29.</head> <p> <s xml:id="echoid-s13212" xml:space="preserve">IN omni triangulo ſphærico, cuius omnes an-<lb/>guli ſint acuti, arcus ſinguli quadrante ſunt mi-<lb/>nores.</s> <s xml:id="echoid-s13213" xml:space="preserve"/> </p> <figure> <image file="398-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/398-02"/> </figure> <p> <s xml:id="echoid-s13214" xml:space="preserve">IN triangulo ſphærico ABC, ſint omnes an-<lb/>guli acuti. </s> <s xml:id="echoid-s13215" xml:space="preserve">Dico ſingulos arcus quadrante mino-<lb/>res eſſe. </s> <s xml:id="echoid-s13216" xml:space="preserve">Sint enim primum omnes anguli acuti <lb/>æquales. </s> <s xml:id="echoid-s13217" xml:space="preserve">Quo poſito, erunt ſinguli arcus qua-<lb/> <anchor type="note" xlink:label="note-398-03a" xlink:href="note-398-03"/> drante minores, vt ſupra demonſtratum eſt.</s> <s xml:id="echoid-s13218" xml:space="preserve"/> </p> <div xml:id="echoid-div1044" type="float" level="2" n="1"> <note position="left" xlink:label="note-398-03" xlink:href="note-398-03a" xml:space="preserve">Corollar. <lb/>25. huius.</note> </div> <p> <s xml:id="echoid-s13219" xml:space="preserve">DEINDE ſint duo tantum anguli acuti æ-<lb/>quales B, C; </s> <s xml:id="echoid-s13220" xml:space="preserve">& </s> <s xml:id="echoid-s13221" xml:space="preserve">A, minor vtroque illorum. </s> <s xml:id="echoid-s13222" xml:space="preserve">Eric <lb/> <anchor type="note" xlink:label="note-398-04a" xlink:href="note-398-04"/> igitur vterque arcus AB, AC, minor quadrante.</s> <s xml:id="echoid-s13223" xml:space="preserve"> <pb o="387" file="399" n="399" rhead=""/> Et quia angulus B, maior ponitur angulo A, <lb/> <anchor type="figure" xlink:label="fig-399-01a" xlink:href="fig-399-01"/> erit arcus AC, maior arcu BC. </s> <s xml:id="echoid-s13224" xml:space="preserve">Cum igi-<lb/> <anchor type="note" xlink:label="note-399-01a" xlink:href="note-399-01"/> tur arcus AC, oſtenſus ſit quadrante mi-<lb/>nor, erit multo magis arcus BC, minor qua-<lb/>drante.</s> <s xml:id="echoid-s13225" xml:space="preserve"/> </p> <div xml:id="echoid-div1045" type="float" level="2" n="2"> <note position="left" xlink:label="note-398-04" xlink:href="note-398-04a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-399-01" xlink:href="fig-399-01a"> <image file="399-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/399-01"/> </figure> <note position="right" xlink:label="note-399-01" xlink:href="note-399-01a" xml:space="preserve">11. huius.</note> </div> <p> <s xml:id="echoid-s13226" xml:space="preserve">TERTIO ſint duo tantum anguli acu-<lb/>ti iterum æquales B, C; </s> <s xml:id="echoid-s13227" xml:space="preserve">& </s> <s xml:id="echoid-s13228" xml:space="preserve">A, acutus vtroque <lb/>illorum maior. </s> <s xml:id="echoid-s13229" xml:space="preserve">Erit igitur rurſus vterque ar-<lb/> <anchor type="note" xlink:label="note-399-02a" xlink:href="note-399-02"/> cus AB, AC, quadrante minor. </s> <s xml:id="echoid-s13230" xml:space="preserve">Dico & </s> <s xml:id="echoid-s13231" xml:space="preserve"><lb/>BC, quadrante eſſe minorem. </s> <s xml:id="echoid-s13232" xml:space="preserve">Fiatenim angulus rectus BAD, ſirq́ue arcus <lb/>AD, vtrique arcuum AB, AC, æqualis; </s> <s xml:id="echoid-s13233" xml:space="preserve">& </s> <s xml:id="echoid-s13234" xml:space="preserve">per puncta B, D, deſcribatur ar-<lb/> <anchor type="note" xlink:label="note-399-03a" xlink:href="note-399-03"/> cus circuli maximi BD. </s> <s xml:id="echoid-s13235" xml:space="preserve">Quoniam igitur vter-<lb/> <anchor type="figure" xlink:label="fig-399-02a" xlink:href="fig-399-02"/> que arcus AB, AC, oſtenſus eſt quadrante minor <lb/>erit & </s> <s xml:id="echoid-s13236" xml:space="preserve">AD, min or quadrante. </s> <s xml:id="echoid-s13237" xml:space="preserve">Vterque ergo an-<lb/> <anchor type="note" xlink:label="note-399-04a" xlink:href="note-399-04"/> gulus ABD, & </s> <s xml:id="echoid-s13238" xml:space="preserve">D, acutus eſt. </s> <s xml:id="echoid-s13239" xml:space="preserve">Quare cum in trian <lb/>gulo ABD, angulus BAD, rectus ſit, & </s> <s xml:id="echoid-s13240" xml:space="preserve">reliqui <lb/> <anchor type="note" xlink:label="note-399-05a" xlink:href="note-399-05"/> acuti erunt omnes arcus quadrante minores. </s> <s xml:id="echoid-s13241" xml:space="preserve">Ar-<lb/>cusigitur BD, quadrante minor eſt: </s> <s xml:id="echoid-s13242" xml:space="preserve">At quia la-<lb/>tera AB, AC, lateribus AB, AD, æqualia ſunt, <lb/>eſtq́ue angulus BAD, angulo BAC, maior; </s> <s xml:id="echoid-s13243" xml:space="preserve">erit <lb/>& </s> <s xml:id="echoid-s13244" xml:space="preserve">baſis BD, baſe BC, maior: </s> <s xml:id="echoid-s13245" xml:space="preserve">Oſtenſus eſt autem <lb/> <anchor type="note" xlink:label="note-399-06a" xlink:href="note-399-06"/> arcus BD, quadrante minor. </s> <s xml:id="echoid-s13246" xml:space="preserve">Multo ergo minor quadrante erit arcus BC. <lb/></s> <s xml:id="echoid-s13247" xml:space="preserve">Omnes ergo tres arcus trianguli ABC, quadrante ſunt minores.</s> <s xml:id="echoid-s13248" xml:space="preserve"/> </p> <div xml:id="echoid-div1046" type="float" level="2" n="3"> <note position="right" xlink:label="note-399-02" xlink:href="note-399-02a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-399-03" xlink:href="note-399-03a" xml:space="preserve">20. 1 Theod.</note> <figure xlink:label="fig-399-02" xlink:href="fig-399-02a"> <image file="399-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/399-02"/> </figure> <note position="right" xlink:label="note-399-04" xlink:href="note-399-04a" xml:space="preserve">25. huius</note> <note position="right" xlink:label="note-399-05" xlink:href="note-399-05a" xml:space="preserve">28. huius.</note> <note position="right" xlink:label="note-399-06" xlink:href="note-399-06a" xml:space="preserve">12. huius.</note> </div> <p> <s xml:id="echoid-s13249" xml:space="preserve">POSTREMO ſint omnes anguli acuti A, B, C, inæquales; </s> <s xml:id="echoid-s13250" xml:space="preserve">& </s> <s xml:id="echoid-s13251" xml:space="preserve">ſit A, om-<lb/>niũ maximus. </s> <s xml:id="echoid-s13252" xml:space="preserve">Eric igitur propterea arcus BC, maior vtrouis arcuũ AB, AC. <lb/></s> <s xml:id="echoid-s13253" xml:space="preserve"> <anchor type="note" xlink:label="note-399-07a" xlink:href="note-399-07"/> Sit quoque angulus C, maior angulo B; </s> <s xml:id="echoid-s13254" xml:space="preserve">eritq́ue propterea arcus AB, maior <lb/> <anchor type="note" xlink:label="note-399-08a" xlink:href="note-399-08"/> arcu AC, Quoniam igitur arcus BC, maior eſt arcu AB, & </s> <s xml:id="echoid-s13255" xml:space="preserve">AB, maior, quam <lb/>AC; </s> <s xml:id="echoid-s13256" xml:space="preserve">abſcindatur arcus BD, æqualis arcui AB, & </s> <s xml:id="echoid-s13257" xml:space="preserve">per puncta A, D, ducatur <lb/>arcus AD, circuli maximi: </s> <s xml:id="echoid-s13258" xml:space="preserve">eruntq́ue anguli BAD, BDA, æquales: </s> <s xml:id="echoid-s13259" xml:space="preserve">Eſt au-<lb/> <anchor type="note" xlink:label="note-399-09a" xlink:href="note-399-09"/> tem angulus BAD, acutus, cum pars ſit anguli acu-<lb/> <anchor type="figure" xlink:label="fig-399-03a" xlink:href="fig-399-03"/> ti BAC. </s> <s xml:id="echoid-s13260" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s13261" xml:space="preserve">angulus BDA, acutus erit. </s> <s xml:id="echoid-s13262" xml:space="preserve">Vter-<lb/>que igitur arcus AB, BD, quadrante eſt minor. </s> <s xml:id="echoid-s13263" xml:space="preserve">Mul <lb/> <anchor type="note" xlink:label="note-399-10a" xlink:href="note-399-10"/> to igitur magis arcus AC, qui minor eſt arcu AB, <lb/>minor erit quadrante. </s> <s xml:id="echoid-s13264" xml:space="preserve">Dico & </s> <s xml:id="echoid-s13265" xml:space="preserve">arcum BC, quadran-<lb/>te minorem eſſe. </s> <s xml:id="echoid-s13266" xml:space="preserve">Fiat enim angulus BAE, rectus, & </s> <s xml:id="echoid-s13267" xml:space="preserve"><lb/>arcus AE, arcui AC, æqualis, ac per puncta B, E, <lb/>deſcribatur arcus BE, maximi circuli. </s> <s xml:id="echoid-s13268" xml:space="preserve">Et quia arcus <lb/> <anchor type="note" xlink:label="note-399-11a" xlink:href="note-399-11"/> AC, oſtenſus eſt minor quadrante, erit & </s> <s xml:id="echoid-s13269" xml:space="preserve">AE, mi-<lb/>nor quadrante. </s> <s xml:id="echoid-s13270" xml:space="preserve">In triangulo ergo ABE, angulus <lb/>BAE, rectus eſt, & </s> <s xml:id="echoid-s13271" xml:space="preserve">vterque arcuum ipſum comprehendentium quadrante <lb/>minor. </s> <s xml:id="echoid-s13272" xml:space="preserve">Igitur reliqui anguli ABE, AEB, acuti ſunt. </s> <s xml:id="echoid-s13273" xml:space="preserve">Quoniam igitur in <lb/> <anchor type="note" xlink:label="note-399-12a" xlink:href="note-399-12"/> eodem triangulo ABE, angulus BAE, rectus eſt, & </s> <s xml:id="echoid-s13274" xml:space="preserve">reliqui duo acuti, erunt <lb/>omnes arcus quadrante minores. </s> <s xml:id="echoid-s13275" xml:space="preserve">Arcus ergo BE, minor eſt quadrante. </s> <s xml:id="echoid-s13276" xml:space="preserve">Quo-<lb/> <anchor type="note" xlink:label="note-399-13a" xlink:href="note-399-13"/> niam vero duo latera AB, AE, duobus lateribus AB, AC, æqualia ſunt, <lb/>eſtque angulus BAE, maior angulo BAC; </s> <s xml:id="echoid-s13277" xml:space="preserve">erit & </s> <s xml:id="echoid-s13278" xml:space="preserve">baſis BE, baſe CE, maior: <lb/></s> <s xml:id="echoid-s13279" xml:space="preserve"> <anchor type="note" xlink:label="note-399-14a" xlink:href="note-399-14"/> Oſtenſus eſt autem arcus BE, minor quadrante. </s> <s xml:id="echoid-s13280" xml:space="preserve">Multo igitur minor quadran <lb/>te erit arcus BC. </s> <s xml:id="echoid-s13281" xml:space="preserve">Tres ergo arcus trianguli ABC, quadrante ſunt minores. <lb/></s> <s xml:id="echoid-s13282" xml:space="preserve">Quamobrẽ, In omni triangulo ſphærico, cuius, &</s> <s xml:id="echoid-s13283" xml:space="preserve">c. </s> <s xml:id="echoid-s13284" xml:space="preserve">Quod demonſtrandũ erat.</s> <s xml:id="echoid-s13285" xml:space="preserve"/> </p> <div xml:id="echoid-div1047" type="float" level="2" n="4"> <note position="right" xlink:label="note-399-07" xlink:href="note-399-07a" xml:space="preserve">11. huius.</note> <note position="right" xlink:label="note-399-08" xlink:href="note-399-08a" xml:space="preserve">11. huius.</note> <note position="right" xlink:label="note-399-09" xlink:href="note-399-09a" xml:space="preserve">8. huius.</note> <figure xlink:label="fig-399-03" xlink:href="fig-399-03a"> <image file="399-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/399-03"/> </figure> <note position="right" xlink:label="note-399-10" xlink:href="note-399-10a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-399-11" xlink:href="note-399-11a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-399-12" xlink:href="note-399-12a" xml:space="preserve">Schol. 28. <lb/>huius.</note> <note position="right" xlink:label="note-399-13" xlink:href="note-399-13a" xml:space="preserve">28. huius.</note> <note position="right" xlink:label="note-399-14" xlink:href="note-399-14a" xml:space="preserve">12. huius.</note> </div> <pb o="388" file="400" n="400" rhead=""/> </div> <div xml:id="echoid-div1049" type="section" level="1" n="526"> <head xml:id="echoid-head561" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s13286" xml:space="preserve">_PORRO_ neque hæc propoſitio conuerti poteſt. </s> <s xml:id="echoid-s13287" xml:space="preserve">Non enim omne triangulum ſphæ-<lb/>ric@m, cuius ſinguli arcus quadrante ſunt minores, neceſſario habet omnes angulos <lb/>acutos. </s> <s xml:id="echoid-s13288" xml:space="preserve">Nam vnus angulus poteſt eſſe rectus, & </s> <s xml:id="echoid-s13289" xml:space="preserve">reliqui duo acuti, vt ex propoſ. <lb/></s> <s xml:id="echoid-s13290" xml:space="preserve">præcedenti conſtat. </s> <s xml:id="echoid-s13291" xml:space="preserve">Immo & </s> <s xml:id="echoid-s13292" xml:space="preserve">vnus poteſt eſſe obtuſus, & </s> <s xml:id="echoid-s13293" xml:space="preserve">reliqui acuti. </s> <s xml:id="echoid-s13294" xml:space="preserve">Sint enim <lb/> <anchor type="figure" xlink:label="fig-400-01a" xlink:href="fig-400-01"/> duo ſemicirculi _ABC, ADC,_ continentes angulos _A,_ <lb/>C, obtuſos, accipianturq́; </s> <s xml:id="echoid-s13295" xml:space="preserve">duo arcus æquales _AB, AD,_ <lb/>quorum vterque ſesquialterum quadrantem ſuperet, & </s> <s xml:id="echoid-s13296" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-400-01a" xlink:href="note-400-01"/> per puncta _B, D,_ arcus circuli maximi deſcribatur _BD,_ <lb/>qui minor erit quadrante, vt in ſcbolio propoſ. </s> <s xml:id="echoid-s13297" xml:space="preserve">27. </s> <s xml:id="echoid-s13298" xml:space="preserve">oſten-<lb/>dimus. </s> <s xml:id="echoid-s13299" xml:space="preserve">Erunt igitur in triangulo _BCD,_ tres arcus _BC,_ <lb/>_CD, BD,_ ſinguli quadrante minores, & </s> <s xml:id="echoid-s13300" xml:space="preserve">tamen non om-<lb/> <anchor type="note" xlink:label="note-400-02a" xlink:href="note-400-02"/> nes anguli in triangulo _BCD,_ acuti ſunt, ſed _C,_ qui-<lb/>dem obtuſus, ex bypotbeſi, at verò _B, D,_ acuti, propterea quòd duo latera _CB,_ <lb/>CD, æqualia ſunt, & </s> <s xml:id="echoid-s13301" xml:space="preserve">quadrante minora.</s> <s xml:id="echoid-s13302" xml:space="preserve"/> </p> <div xml:id="echoid-div1049" type="float" level="2" n="1"> <figure xlink:label="fig-400-01" xlink:href="fig-400-01a"> <image file="400-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/400-01"/> </figure> <note position="left" xlink:label="note-400-01" xlink:href="note-400-01a" xml:space="preserve">20. i Theod.</note> <note position="left" xlink:label="note-400-02" xlink:href="note-400-02a" xml:space="preserve">25. huius.</note> </div> </div> <div xml:id="echoid-div1051" type="section" level="1" n="527"> <head xml:id="echoid-head562" xml:space="preserve">THEOR. 28. PROPOS. 30.</head> <p> <s xml:id="echoid-s13303" xml:space="preserve">IN quolibet triangulo ſphærico, cuius vnus <lb/>quidem arcus quadrante maior ſit, reliquorum <lb/>verò vterque quadrante minor, nullus anguloium <lb/>rectus erit.</s> <s xml:id="echoid-s13304" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13305" xml:space="preserve">IN triangulo ſphærico ABC, ſit quidem arcus AC, quadrante maior, at <lb/>tam AB, quam BC, minor quadrante. </s> <s xml:id="echoid-s13306" xml:space="preserve">Dico nullum angulorum eſſe rectum. <lb/></s> <s xml:id="echoid-s13307" xml:space="preserve">Sit enim ſi fieri poteſt, angulus B, qui arcui AC, quadrante maiori opponi-<lb/>tur, rcctus. </s> <s xml:id="echoid-s13308" xml:space="preserve">Abſciſlo igitur AD, quadrante, & </s> <s xml:id="echoid-s13309" xml:space="preserve">producto arcu AB, ad E, vt <lb/> <anchor type="figure" xlink:label="fig-400-02a" xlink:href="fig-400-02"/> AE, ſit etiam quadrans, & </s> <s xml:id="echoid-s13310" xml:space="preserve">per puncta D, E, arcu D E, <lb/>circuli maximi deſcripto, qui arcum BC, ſecet in <lb/> <anchor type="note" xlink:label="note-400-03a" xlink:href="note-400-03"/> F; </s> <s xml:id="echoid-s13311" xml:space="preserve">erit vterque angulus D, E, rectus: </s> <s xml:id="echoid-s13312" xml:space="preserve">Ponitur autem <lb/> <anchor type="note" xlink:label="note-400-04a" xlink:href="note-400-04"/> & </s> <s xml:id="echoid-s13313" xml:space="preserve">angulus ABC, rectus, hoc eſt, EBC; </s> <s xml:id="echoid-s13314" xml:space="preserve">ſunt enim <lb/>duo anguli ad B, duobus rectis æquales. </s> <s xml:id="echoid-s13315" xml:space="preserve">Vterque igi-<lb/> <anchor type="note" xlink:label="note-400-05a" xlink:href="note-400-05"/> tur arcus EF, BF, quadrans erit, atque adeo arcus <lb/> <anchor type="note" xlink:label="note-400-06a" xlink:href="note-400-06"/> BC, maior quadrante, quod eſf abſurdum, cum po-<lb/>natur quadrante minor. </s> <s xml:id="echoid-s13316" xml:space="preserve">Non ergo angulus B, rectus <lb/>eſſe poteſt.</s> <s xml:id="echoid-s13317" xml:space="preserve"/> </p> <div xml:id="echoid-div1051" type="float" level="2" n="1"> <figure xlink:label="fig-400-02" xlink:href="fig-400-02a"> <image file="400-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/400-02"/> </figure> <note position="left" xlink:label="note-400-03" xlink:href="note-400-03a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-400-04" xlink:href="note-400-04a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-400-05" xlink:href="note-400-05a" xml:space="preserve">5. huius.</note> <note position="left" xlink:label="note-400-06" xlink:href="note-400-06a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13318" xml:space="preserve">QVOD ſi angulus C, rectus eſſe dicatur, erit, ſi <lb/>eadem fiat conſtructio, eodem modo vterque arcus <lb/>DF, CF, quadrans: </s> <s xml:id="echoid-s13319" xml:space="preserve">(Nam & </s> <s xml:id="echoid-s13320" xml:space="preserve">angulus CDF, rectus eſt, cum vterque D, E, <lb/> <anchor type="note" xlink:label="note-400-07a" xlink:href="note-400-07"/> rectus ſit, ob quadrantes AD, AE.) </s> <s xml:id="echoid-s13321" xml:space="preserve">atque adeo arcus BC, quadrante maior. <lb/></s> <s xml:id="echoid-s13322" xml:space="preserve"> <anchor type="note" xlink:label="note-400-08a" xlink:href="note-400-08"/> quod eſt contra hypotheſim.</s> <s xml:id="echoid-s13323" xml:space="preserve"/> </p> <div xml:id="echoid-div1052" type="float" level="2" n="2"> <note position="left" xlink:label="note-400-07" xlink:href="note-400-07a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-400-08" xlink:href="note-400-08a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13324" xml:space="preserve">SI denique angulus A, rectus concedatur, ſi ex arcu CA, abſcindatur <pb o="389" file="401" n="401" rhead=""/> quadrans CG, & </s> <s xml:id="echoid-s13325" xml:space="preserve">arcus CB, producatur vſque ad H, vt & </s> <s xml:id="echoid-s13326" xml:space="preserve">CH, quadrans ſit, <lb/>deſcribaturq́ue per puncta G, H, arcus circuli maximi GH, ſecans arcum AB, <lb/> <anchor type="note" xlink:label="note-401-01a" xlink:href="note-401-01"/> in I; </s> <s xml:id="echoid-s13327" xml:space="preserve">erit vterque angulus G, H, rectus. </s> <s xml:id="echoid-s13328" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s13329" xml:space="preserve">angulus A, ponatur re-<lb/> <anchor type="note" xlink:label="note-401-02a" xlink:href="note-401-02"/> ctus, erunt in triangulo AGI, duo anguli A, G, recti. </s> <s xml:id="echoid-s13330" xml:space="preserve">Quare vterque arcus <lb/>AI, GI, quadrans eſt; </s> <s xml:id="echoid-s13331" xml:space="preserve">atq́ue adeo arcus AB, quadrante maior. </s> <s xml:id="echoid-s13332" xml:space="preserve">quod eſt con-<lb/> <anchor type="note" xlink:label="note-401-03a" xlink:href="note-401-03"/> tra hypotheſim.</s> <s xml:id="echoid-s13333" xml:space="preserve"/> </p> <div xml:id="echoid-div1053" type="float" level="2" n="3"> <note position="right" xlink:label="note-401-01" xlink:href="note-401-01a" xml:space="preserve">20.1 Theod.</note> <note position="right" xlink:label="note-401-02" xlink:href="note-401-02a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-401-03" xlink:href="note-401-03a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13334" xml:space="preserve">POSSVMVS tamen aliter demonſtrare, angulum A, non poſſe eſſe <lb/>rectum, licet non abſcindatur quadrans CG, &</s> <s xml:id="echoid-s13335" xml:space="preserve">c. </s> <s xml:id="echoid-s13336" xml:space="preserve">Si enim angulus A, rectus <lb/> <anchor type="note" xlink:label="note-401-04a" xlink:href="note-401-04"/> concedatur, erit arcus DE, quadrans. </s> <s xml:id="echoid-s13337" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s13338" xml:space="preserve">EA, quadrans ſit, erit <lb/> <anchor type="note" xlink:label="note-401-05a" xlink:href="note-401-05"/> vterque angulus A, D, rectus; </s> <s xml:id="echoid-s13339" xml:space="preserve">& </s> <s xml:id="echoid-s13340" xml:space="preserve">E, polus arcus AC, propterea quòd vter-<lb/>que arcus DE, AE, per polum arcus AD, tranſit, ob angulos rectos A, <lb/> <anchor type="note" xlink:label="note-401-06a" xlink:href="note-401-06"/> D. </s> <s xml:id="echoid-s13341" xml:space="preserve">Eodem modo D, polus erit arcus AB. </s> <s xml:id="echoid-s13342" xml:space="preserve">Quoniam igitur punctum F, eſt <lb/>intra peripheriam circuli AB, & </s> <s xml:id="echoid-s13343" xml:space="preserve">præter eius polum, duciturq́ue arcus FE, <lb/>per polum circuli AB, nempe per D, erit arcus FB, maior arcu FE. </s> <s xml:id="echoid-s13344" xml:space="preserve">Ea-<lb/>dem ratione arcus FC, maior erit arcu FD, cum FD, ducatur per E, polum <lb/>circuli AC. </s> <s xml:id="echoid-s13345" xml:space="preserve">Totus igiturarcus BC, quadrante DE, maior erit. </s> <s xml:id="echoid-s13346" xml:space="preserve">quod eſt <lb/> <anchor type="note" xlink:label="note-401-07a" xlink:href="note-401-07"/> abſurdum, cum minor quadrante ponatur. </s> <s xml:id="echoid-s13347" xml:space="preserve">Nullus ergo angulorum A, B, <lb/>C, rectus eſt. </s> <s xml:id="echoid-s13348" xml:space="preserve">Quamobrem, In quolibet triangulo ſphærico, &</s> <s xml:id="echoid-s13349" xml:space="preserve">c. </s> <s xml:id="echoid-s13350" xml:space="preserve">Quod de-<lb/>monſtrandum erat.</s> <s xml:id="echoid-s13351" xml:space="preserve"/> </p> <div xml:id="echoid-div1054" type="float" level="2" n="4"> <note position="right" xlink:label="note-401-04" xlink:href="note-401-04a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-401-05" xlink:href="note-401-05a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-401-06" xlink:href="note-401-06a" xml:space="preserve">13. 1. Theod.</note> <note position="right" xlink:label="note-401-07" xlink:href="note-401-07a" xml:space="preserve">Schol. 21. <lb/>2. Theod.</note> </div> </div> <div xml:id="echoid-div1056" type="section" level="1" n="528"> <head xml:id="echoid-head563" xml:space="preserve">THEOR. 29. PROPOS. 31.</head> <p> <s xml:id="echoid-s13352" xml:space="preserve">CVIVSCVNQVE trianguli ſphærici tres <lb/>anguli duobus quidem rectis ſunt maiores, ſex ve-<lb/>rò rectis minores.</s> <s xml:id="echoid-s13353" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13354" xml:space="preserve">SIT triangulum ſphæricum ABC. </s> <s xml:id="echoid-s13355" xml:space="preserve">Dico tres angulos A, B, C, maiores <lb/>quidem eſſe duobus rectis, minores verò ſex rectis. </s> <s xml:id="echoid-s13356" xml:space="preserve">Si enim omnes tres angu-<lb/>lirecti ſint, vel obtuſi; </s> <s xml:id="echoid-s13357" xml:space="preserve">vel duo tantũ recti, vel obtuſi; </s> <s xml:id="echoid-s13358" xml:space="preserve">vel vnus tantum rectus, <lb/>& </s> <s xml:id="echoid-s13359" xml:space="preserve">reliquorum alter obtuſus, perſpicuum eſt, omnes tres duobus eſſe rectis <lb/>maiores. </s> <s xml:id="echoid-s13360" xml:space="preserve">In quolibet autem triangulo hæc erit demonſtratio. </s> <s xml:id="echoid-s13361" xml:space="preserve">Producto late-<lb/>re BC, ad D, erit angulus ACD, vel æqualis, vel mi-<lb/> <anchor type="figure" xlink:label="fig-401-01a" xlink:href="fig-401-01"/> nor, vel maior angulo B. </s> <s xml:id="echoid-s13362" xml:space="preserve">Sit primum æqualis. </s> <s xml:id="echoid-s13363" xml:space="preserve">Erunt <lb/>igitur arcus AB, AC, ſimul ſemicirculo æquales; </s> <s xml:id="echoid-s13364" xml:space="preserve">atq; <lb/></s> <s xml:id="echoid-s13365" xml:space="preserve"> <anchor type="note" xlink:label="note-401-08a" xlink:href="note-401-08"/> adeò duo anguli ABC, ACB, duobus rectis æquales. <lb/></s> <s xml:id="echoid-s13366" xml:space="preserve"> <anchor type="note" xlink:label="note-401-09a" xlink:href="note-401-09"/> Tres ergo anguli A, B, C, duobus rectis maiores erũt. <lb/></s> <s xml:id="echoid-s13367" xml:space="preserve">Sit deinde angulus ACD, minor angulo B. </s> <s xml:id="echoid-s13368" xml:space="preserve">Erunt <lb/>igitur arcus AB, AC, ſimul maiores ſemicirculo; </s> <s xml:id="echoid-s13369" xml:space="preserve">ac <lb/> <anchor type="note" xlink:label="note-401-10a" xlink:href="note-401-10"/> propterea duo anguli ABC, ACB, duobus rectis ma-<lb/>iores. </s> <s xml:id="echoid-s13370" xml:space="preserve">Multo ergo magis tres anguli A, B, C, duobus <lb/>rectis maiores erũt. </s> <s xml:id="echoid-s13371" xml:space="preserve">Sit denique angulus ACD, maior <lb/>angulo B, & </s> <s xml:id="echoid-s13372" xml:space="preserve">fiat angulus DCE, angulo B, æqualis, occurratq́ue arcus CE, <lb/> <anchor type="note" xlink:label="note-401-11a" xlink:href="note-401-11"/> arcui BA, producto in E: </s> <s xml:id="echoid-s13373" xml:space="preserve">& </s> <s xml:id="echoid-s13374" xml:space="preserve">tandem arcus CA, protrahatur ad F. </s> <s xml:id="echoid-s13375" xml:space="preserve">Erunt igi-<lb/>tur arcus EB, EC, ſimul æquales ſemicirculo; </s> <s xml:id="echoid-s13376" xml:space="preserve">ac propterea arcus EA, EC, <lb/> <anchor type="note" xlink:label="note-401-12a" xlink:href="note-401-12"/> ſimul ſemicirculo minores. </s> <s xml:id="echoid-s13377" xml:space="preserve">Angulus igitur EAF, hoc eſt, angulus BAC, <pb o="390" file="402" n="402" rhead=""/> qui illi ad verticem æqualis eſt, maior erit angulo ACE: </s> <s xml:id="echoid-s13378" xml:space="preserve">Sed angulus ACE, <lb/> <anchor type="note" xlink:label="note-402-01a" xlink:href="note-402-01"/> & </s> <s xml:id="echoid-s13379" xml:space="preserve">anguli ACB, & </s> <s xml:id="echoid-s13380" xml:space="preserve">B, duobus rectis ſuntæquales. </s> <s xml:id="echoid-s13381" xml:space="preserve">Igitur anguli BAC, ACB, <lb/> <anchor type="note" xlink:label="note-402-02a" xlink:href="note-402-02"/> & </s> <s xml:id="echoid-s13382" xml:space="preserve">B, maiores erunt duobus rectis. </s> <s xml:id="echoid-s13383" xml:space="preserve">Semper ergo tres anguli ſimul duobus re-<lb/>ctis ſunt maiores.</s> <s xml:id="echoid-s13384" xml:space="preserve"/> </p> <div xml:id="echoid-div1056" type="float" level="2" n="1"> <figure xlink:label="fig-401-01" xlink:href="fig-401-01a"> <image file="401-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/401-01"/> </figure> <note position="right" xlink:label="note-401-08" xlink:href="note-401-08a" xml:space="preserve">15. huius.</note> <note position="right" xlink:label="note-401-09" xlink:href="note-401-09a" xml:space="preserve">16. huius.</note> <note position="right" xlink:label="note-401-10" xlink:href="note-401-10a" xml:space="preserve">15. huius.</note> <note position="right" xlink:label="note-401-11" xlink:href="note-401-11a" xml:space="preserve">10. huius.</note> <note position="right" xlink:label="note-401-12" xlink:href="note-401-12a" xml:space="preserve">15. huius.</note> <note position="left" xlink:label="note-402-01" xlink:href="note-402-01a" xml:space="preserve">@4. huius.</note> <note position="left" xlink:label="note-402-02" xlink:href="note-402-02a" xml:space="preserve">16. huius.</note> </div> <p> <s xml:id="echoid-s13385" xml:space="preserve">QVIA verò omnis angulus ſphæricus, etiam obtuſus, minor eſt duobus <lb/>rectis; </s> <s xml:id="echoid-s13386" xml:space="preserve">perſpicuum eſt, tres angulos cuiusuis trianguli ſphærici ſimul minores <lb/>eſſe ſex rectis. </s> <s xml:id="echoid-s13387" xml:space="preserve">Cuiuſcunque ergo trianguli ſphærici tres anguli, &</s> <s xml:id="echoid-s13388" xml:space="preserve">c. </s> <s xml:id="echoid-s13389" xml:space="preserve">Quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s13390" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1058" type="section" level="1" n="529"> <head xml:id="echoid-head564" xml:space="preserve">THEOR. 30. PROPOS. 32.</head> <p> <s xml:id="echoid-s13391" xml:space="preserve">IN omni triangulo ſphærico, cuius vnus an-<lb/>gulus rectus ſit, & </s> <s xml:id="echoid-s13392" xml:space="preserve">alter reliquorum acutus, ſi qui-<lb/>dem arcus illis angulis adiacẽs fuerit quadians, erit <lb/>& </s> <s xml:id="echoid-s13393" xml:space="preserve">arcus rectum ſubtendens angulum quadrans; </s> <s xml:id="echoid-s13394" xml:space="preserve">ſi <lb/>verò minor fuerit quadrante, quadrante quoque <lb/>minor erit: </s> <s xml:id="echoid-s13395" xml:space="preserve">ſi deniq; </s> <s xml:id="echoid-s13396" xml:space="preserve">quadrante fuerit maior, qua-<lb/>drante quoq; </s> <s xml:id="echoid-s13397" xml:space="preserve">maior erit: </s> <s xml:id="echoid-s13398" xml:space="preserve">Sem per autem arcus acu-<lb/>tum angulum ſubtendens minor erit quadrante.</s> <s xml:id="echoid-s13399" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13400" xml:space="preserve">IN triangulo ABC, angulus C, rectus ſit, & </s> <s xml:id="echoid-s13401" xml:space="preserve">B, acutus, ſitque primum ar-<lb/>cus BC, quadrans, Dico & </s> <s xml:id="echoid-s13402" xml:space="preserve">AB, quadrantem eſſe. </s> <s xml:id="echoid-s13403" xml:space="preserve">Fiat enim angulus CBD, <lb/> <anchor type="figure" xlink:label="fig-402-01a" xlink:href="fig-402-01"/> rectus, coëatq́ue arcus BD, cum arcu CA, <lb/>producto in D. </s> <s xml:id="echoid-s13404" xml:space="preserve">Erit igitur vterque arcus BD, <lb/> <anchor type="note" xlink:label="note-402-03a" xlink:href="note-402-03"/> CD, quadrans: </s> <s xml:id="echoid-s13405" xml:space="preserve">Ponitur autem & </s> <s xml:id="echoid-s13406" xml:space="preserve">BC, qua-<lb/>drans. </s> <s xml:id="echoid-s13407" xml:space="preserve">Ergo B, polus eſt arcus CD; </s> <s xml:id="echoid-s13408" xml:space="preserve">atque <lb/> <anchor type="note" xlink:label="note-402-04a" xlink:href="note-402-04"/> adeo rectus erit angulus ad A. </s> <s xml:id="echoid-s13409" xml:space="preserve">Quare vterque <lb/> <anchor type="note" xlink:label="note-402-05a" xlink:href="note-402-05"/> arcus BC, BA, quadrans erit. </s> <s xml:id="echoid-s13410" xml:space="preserve">Quadrans igi-<lb/> <anchor type="note" xlink:label="note-402-06a" xlink:href="note-402-06"/> tur eſt arcus AB, angulo recto C, oppoſitus.</s> <s xml:id="echoid-s13411" xml:space="preserve"/> </p> <div xml:id="echoid-div1058" type="float" level="2" n="1"> <figure xlink:label="fig-402-01" xlink:href="fig-402-01a"> <image file="402-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/402-01"/> </figure> <note position="left" xlink:label="note-402-03" xlink:href="note-402-03a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-402-04" xlink:href="note-402-04a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-402-05" xlink:href="note-402-05a" xml:space="preserve">15. 1 Theod.</note> <note position="left" xlink:label="note-402-06" xlink:href="note-402-06a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13412" xml:space="preserve">SIT deinde arcus BC, quadrante minor. <lb/></s> <s xml:id="echoid-s13413" xml:space="preserve">Dico & </s> <s xml:id="echoid-s13414" xml:space="preserve">arcum AB, quadrante eſſe minorem. </s> <s xml:id="echoid-s13415" xml:space="preserve"><lb/>Fiat enim rurſus angulus CBD, rectus, oc-<lb/>curratq́ue arcus BD, arcui CA, producto in <lb/>D; </s> <s xml:id="echoid-s13416" xml:space="preserve">eritque vt prius, vterque arcus BD, CD, <lb/> <anchor type="note" xlink:label="note-402-07a" xlink:href="note-402-07"/> quadrans. </s> <s xml:id="echoid-s13417" xml:space="preserve">Producto autem BC, ad E, vt ſit <lb/>BE, quadrans, ducatur per puncta D, E, arcus circuli maximi DE, quem BA, <lb/> <anchor type="note" xlink:label="note-402-08a" xlink:href="note-402-08"/> productus ſecet in F. </s> <s xml:id="echoid-s13418" xml:space="preserve">Quoniam igitur arcus BE, BD, quadrantes ſunt, erit <lb/>vterque angulus BDE, BED, rectus, & </s> <s xml:id="echoid-s13419" xml:space="preserve">B, polus arcus DE. </s> <s xml:id="echoid-s13420" xml:space="preserve">Rectus ergo <lb/> <anchor type="note" xlink:label="note-402-09a" xlink:href="note-402-09"/> erit angulus ad F; </s> <s xml:id="echoid-s13421" xml:space="preserve">atque adeò vterque arcus BE, BF, quadrans erit. </s> <s xml:id="echoid-s13422" xml:space="preserve">Igitur <lb/> <anchor type="note" xlink:label="note-402-10a" xlink:href="note-402-10"/> arcus BA, quadrante erit minor.</s> <s xml:id="echoid-s13423" xml:space="preserve"/> </p> <div xml:id="echoid-div1059" type="float" level="2" n="2"> <note position="left" xlink:label="note-402-07" xlink:href="note-402-07a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-402-08" xlink:href="note-402-08a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-402-09" xlink:href="note-402-09a" xml:space="preserve">25. & 26. <lb/>huius.</note> <note position="left" xlink:label="note-402-10" xlink:href="note-402-10a" xml:space="preserve">15. 1 Theod.</note> </div> <note position="left" xml:space="preserve">25. huius.</note> <p> <s xml:id="echoid-s13424" xml:space="preserve">SIT denique arcus BC, quadrante maior. </s> <s xml:id="echoid-s13425" xml:space="preserve">Dico & </s> <s xml:id="echoid-s13426" xml:space="preserve">arcum AB, maiorem <pb o="391" file="403" n="403" rhead=""/> quadrante eſſe. </s> <s xml:id="echoid-s13427" xml:space="preserve">Fiat enim rurſus angulus CBD, rectus, conuenlatq́ue arcus <lb/>BD, cum CA, protracto in D; </s> <s xml:id="echoid-s13428" xml:space="preserve">eritq́ue, vt prius, vterque arcus BD, CD, <lb/> <anchor type="note" xlink:label="note-403-01a" xlink:href="note-403-01"/> quadrans. </s> <s xml:id="echoid-s13429" xml:space="preserve">Abſciſſo autem quadrante BG, ducatur per puncta D, G, arcus cir <lb/> <anchor type="note" xlink:label="note-403-02a" xlink:href="note-403-02"/> culi maximi DG, ſecans arcum AB, in H. </s> <s xml:id="echoid-s13430" xml:space="preserve">Quoniam igitur arcus BD, BG, <lb/> <anchor type="note" xlink:label="note-403-03a" xlink:href="note-403-03"/> quadrantes ſunt, erit vterque angulus BDG, BGD, rectus, & </s> <s xml:id="echoid-s13431" xml:space="preserve">B, polus ar-<lb/>cus DG. </s> <s xml:id="echoid-s13432" xml:space="preserve">Rectus ergo erit angulus ad H; </s> <s xml:id="echoid-s13433" xml:space="preserve">ac proinde vterque arcus BG, BH, <lb/> <anchor type="note" xlink:label="note-403-04a" xlink:href="note-403-04"/> erit quadrans. </s> <s xml:id="echoid-s13434" xml:space="preserve">Quare AB, quadrante maior erit.</s> <s xml:id="echoid-s13435" xml:space="preserve"/> </p> <div xml:id="echoid-div1060" type="float" level="2" n="3"> <note position="right" xlink:label="note-403-01" xlink:href="note-403-01a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-403-02" xlink:href="note-403-02a" xml:space="preserve">20.1 Theod.</note> <note position="right" xlink:label="note-403-03" xlink:href="note-403-03a" xml:space="preserve">25. & 26. <lb/>huius.</note> <note position="right" xlink:label="note-403-04" xlink:href="note-403-04a" xml:space="preserve">25. 1 Theod.</note> </div> <note position="right" xml:space="preserve">25. huius.</note> <p> <s xml:id="echoid-s13436" xml:space="preserve">ET quoniam arcus CD, ſemper oſtenſus eſt eſſe quadrans, erit arcus AC, <lb/>quadrante minor. </s> <s xml:id="echoid-s13437" xml:space="preserve">Quapropter in omni triangulo ſphærico, &</s> <s xml:id="echoid-s13438" xml:space="preserve">c. </s> <s xml:id="echoid-s13439" xml:space="preserve">Quod oſten-<lb/>dendum erat.</s> <s xml:id="echoid-s13440" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1062" type="section" level="1" n="530"> <head xml:id="echoid-head565" xml:space="preserve">THEOR. 31. PROPOS. 33.</head> <p> <s xml:id="echoid-s13441" xml:space="preserve">IN omni triangulo ſphærico, cuius vnus an-<lb/>gulus rectus, & </s> <s xml:id="echoid-s13442" xml:space="preserve">alter reliquorum acutus, ſi quidem <lb/>arcus illis angulis adiacens ſuerit quadrans, erit re-<lb/>liquus angulus rectus: </s> <s xml:id="echoid-s13443" xml:space="preserve">ſi verò minor quadrante, <lb/>acutus: </s> <s xml:id="echoid-s13444" xml:space="preserve">ſi denique quadrante maior, obtuſus.</s> <s xml:id="echoid-s13445" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13446" xml:space="preserve">SIT in triangulo ABC, ſphærico angulus C, rectus, & </s> <s xml:id="echoid-s13447" xml:space="preserve">B, acutus, ſitq́ue <lb/>primum arcus BC, quadrans. </s> <s xml:id="echoid-s13448" xml:space="preserve">Dico reliquum angulum A, rectum eſſe. </s> <s xml:id="echoid-s13449" xml:space="preserve">Erit <lb/> <anchor type="note" xlink:label="note-403-06a" xlink:href="note-403-06"/> enim, & </s> <s xml:id="echoid-s13450" xml:space="preserve">AB, quadrans. </s> <s xml:id="echoid-s13451" xml:space="preserve">Igitur vterque angulus C, <lb/> <anchor type="note" xlink:label="note-403-07a" xlink:href="note-403-07"/> <anchor type="figure" xlink:label="fig-403-01a" xlink:href="fig-403-01"/> A, rectus.</s> <s xml:id="echoid-s13452" xml:space="preserve"/> </p> <div xml:id="echoid-div1062" type="float" level="2" n="1"> <note position="right" xlink:label="note-403-06" xlink:href="note-403-06a" xml:space="preserve">32. huius.</note> <note position="right" xlink:label="note-403-07" xlink:href="note-403-07a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-403-01" xlink:href="fig-403-01a"> <image file="403-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/403-01"/> </figure> </div> <p> <s xml:id="echoid-s13453" xml:space="preserve">SIT deinde arcus BC, quadrante minor. </s> <s xml:id="echoid-s13454" xml:space="preserve">Dico <lb/>angulum A, eſſe acutum. </s> <s xml:id="echoid-s13455" xml:space="preserve">Erit enim & </s> <s xml:id="echoid-s13456" xml:space="preserve">arcus AB, <lb/> <anchor type="note" xlink:label="note-403-08a" xlink:href="note-403-08"/> quadrante minor; </s> <s xml:id="echoid-s13457" xml:space="preserve">atque adeo arcus AB, BC, ſimul <lb/>ſemicirculo erunt minores. </s> <s xml:id="echoid-s13458" xml:space="preserve">Quare anguli A, C, duo-<lb/> <anchor type="note" xlink:label="note-403-09a" xlink:href="note-403-09"/> bus rectis ſunt minores; </s> <s xml:id="echoid-s13459" xml:space="preserve">ac proinde, cum C, ſit re-<lb/>ctus, erit A, acutus.</s> <s xml:id="echoid-s13460" xml:space="preserve"/> </p> <div xml:id="echoid-div1063" type="float" level="2" n="2"> <note position="right" xlink:label="note-403-08" xlink:href="note-403-08a" xml:space="preserve">32. huius.</note> <note position="right" xlink:label="note-403-09" xlink:href="note-403-09a" xml:space="preserve">16. huius.</note> </div> <p> <s xml:id="echoid-s13461" xml:space="preserve">SIT tandem arcus BC, maior quadrante. </s> <s xml:id="echoid-s13462" xml:space="preserve">Dico <lb/>angulum A, obtuſum eſſe. </s> <s xml:id="echoid-s13463" xml:space="preserve">Erit enim & </s> <s xml:id="echoid-s13464" xml:space="preserve">AB, quadran <lb/> <anchor type="note" xlink:label="note-403-10a" xlink:href="note-403-10"/> te maior; </s> <s xml:id="echoid-s13465" xml:space="preserve">ac proptcrea arcus AB, BC, ſimul ſemi-<lb/>circulo maiores erunt. </s> <s xml:id="echoid-s13466" xml:space="preserve">Igitur anguli A, C, duobus rectis ſunt maiores; </s> <s xml:id="echoid-s13467" xml:space="preserve">atque <lb/> <anchor type="note" xlink:label="note-403-11a" xlink:href="note-403-11"/> adeo, cum C, ſit rectus, erit A, obtuſus. </s> <s xml:id="echoid-s13468" xml:space="preserve">Quocirca in omni triangulo ſphæ-<lb/>rico, cuius vnus angulus, &</s> <s xml:id="echoid-s13469" xml:space="preserve">c. </s> <s xml:id="echoid-s13470" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s13471" xml:space="preserve"/> </p> <div xml:id="echoid-div1064" type="float" level="2" n="3"> <note position="right" xlink:label="note-403-10" xlink:href="note-403-10a" xml:space="preserve">32. huius.</note> <note position="right" xlink:label="note-403-11" xlink:href="note-403-11a" xml:space="preserve">16. huius.</note> </div> </div> <div xml:id="echoid-div1066" type="section" level="1" n="531"> <head xml:id="echoid-head566" xml:space="preserve">THEOR. 32. PROPOS. 34.</head> <p> <s xml:id="echoid-s13472" xml:space="preserve">IN omni triangulo ſphærico, cuius vnus angu-<lb/>lus rectus, ſi vteruis reliquorum angulorum ſit re-<lb/>ctus, erit arcus eum ſubtendens, quadrans: </s> <s xml:id="echoid-s13473" xml:space="preserve">ſi verò <pb o="392" file="404" n="404" rhead=""/> acutus, quadrante minor: </s> <s xml:id="echoid-s13474" xml:space="preserve">ſi denique obtuſus, ma-<lb/>ior quadrante. </s> <s xml:id="echoid-s13475" xml:space="preserve">Et ſi vteruis arcuum rectum angu-<lb/>lum continentium fuerit quadrans, erit angulus, <lb/>quem ſubtendit, rectus: </s> <s xml:id="echoid-s13476" xml:space="preserve">ſi verò minor quadrante, <lb/>acutus: </s> <s xml:id="echoid-s13477" xml:space="preserve">ſi denique quadrante maior, obtuſus.</s> <s xml:id="echoid-s13478" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13479" xml:space="preserve">SIT in triangulo ſphærico ABC, angulus C, rectus, ſitq́ue primum al-<lb/>ter reliquorum, nempe B, etiam rectus. </s> <s xml:id="echoid-s13480" xml:space="preserve">Dico arcum AC, qui eum ſubtendit, <lb/>quadrantem eſſe. </s> <s xml:id="echoid-s13481" xml:space="preserve">Cum enim vterque angulus B, C, rectus ſit, erit & </s> <s xml:id="echoid-s13482" xml:space="preserve">vterque <lb/>arcus AB, AC, quadrans.</s> <s xml:id="echoid-s13483" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">25. huius.</note> <p> <s xml:id="echoid-s13484" xml:space="preserve">SIT deinde B, angulus acutus. </s> <s xml:id="echoid-s13485" xml:space="preserve">Dieo arcum AC, eſſe quadrante mino-<lb/> <anchor type="figure" xlink:label="fig-404-01a" xlink:href="fig-404-01"/> rem. </s> <s xml:id="echoid-s13486" xml:space="preserve">Fiat enim angulus CBD, rectus, coëat-<lb/>que arcus BD, cum arcu CA, producto in <lb/>D. </s> <s xml:id="echoid-s13487" xml:space="preserve">Quoniam igitur vterq; </s> <s xml:id="echoid-s13488" xml:space="preserve">angulus C, CBD, <lb/>rectus eſt, erit & </s> <s xml:id="echoid-s13489" xml:space="preserve">vterque arcus BD, CD, <lb/> <anchor type="note" xlink:label="note-404-02a" xlink:href="note-404-02"/> quadrans; </s> <s xml:id="echoid-s13490" xml:space="preserve">atque adeo arcus AC, minor erit <lb/>quadrante.</s> <s xml:id="echoid-s13491" xml:space="preserve"/> </p> <div xml:id="echoid-div1066" type="float" level="2" n="1"> <figure xlink:label="fig-404-01" xlink:href="fig-404-01a"> <image file="404-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/404-01"/> </figure> <note position="left" xlink:label="note-404-02" xlink:href="note-404-02a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13492" xml:space="preserve">SIT poſtremo angulus B, obtuſus. </s> <s xml:id="echoid-s13493" xml:space="preserve">Dico <lb/>arcum AC, quadrante maiorem eſſe. </s> <s xml:id="echoid-s13494" xml:space="preserve">Fiat <lb/>enim angulus CBE, rectus, ſecetq́ue arcus <lb/>BE, arcum AC, in E. </s> <s xml:id="echoid-s13495" xml:space="preserve">Quoniam igitur vter-<lb/>que angulus C, CBE, rectus eſt, erit & </s> <s xml:id="echoid-s13496" xml:space="preserve">v-<lb/>terque arcus BE, CE, quadrans; </s> <s xml:id="echoid-s13497" xml:space="preserve">atque adeo <lb/> <anchor type="note" xlink:label="note-404-03a" xlink:href="note-404-03"/> arcus AC, quadrante maior erit.</s> <s xml:id="echoid-s13498" xml:space="preserve"/> </p> <div xml:id="echoid-div1067" type="float" level="2" n="2"> <note position="left" xlink:label="note-404-03" xlink:href="note-404-03a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13499" xml:space="preserve">RVRSVM ſit angulus C, rectus, & </s> <s xml:id="echoid-s13500" xml:space="preserve">ſit primum arcus AC, quadrans. <lb/></s> <s xml:id="echoid-s13501" xml:space="preserve">Dico angulum ei ſubtenſum B, eſſe rectum. </s> <s xml:id="echoid-s13502" xml:space="preserve">Erit enim A, polus arcus BC, <lb/> <anchor type="note" xlink:label="note-404-04a" xlink:href="note-404-04"/> (cum arcus CA, per polum arcus BC, tranſeat, ob angulum rectum C;) </s> <s xml:id="echoid-s13503" xml:space="preserve">atque <lb/>adeo angulus ABC, rectus.</s> <s xml:id="echoid-s13504" xml:space="preserve"/> </p> <div xml:id="echoid-div1068" type="float" level="2" n="3"> <note position="left" xlink:label="note-404-04" xlink:href="note-404-04a" xml:space="preserve">Coroll 16. <lb/>1. Theod. <lb/>13. 1. Theod. <lb/>15. 1. Theod.</note> </div> <p> <s xml:id="echoid-s13505" xml:space="preserve">DEINDE ſit arcus AC, quadrante minor. </s> <s xml:id="echoid-s13506" xml:space="preserve">Dico angulum ei ſubten-<lb/>ſum B, eſſe acutum. </s> <s xml:id="echoid-s13507" xml:space="preserve">Producto enim arcu CA, ad D, vt ſit CD, quadrans, <lb/> <anchor type="note" xlink:label="note-404-05a" xlink:href="note-404-05"/> erit eodem modo D, polus arcus CB; </s> <s xml:id="echoid-s13508" xml:space="preserve">cum arcus CA, per polum arcus BC, <lb/>tranſeat. </s> <s xml:id="echoid-s13509" xml:space="preserve">Ducto ergo per puncta D, B, arcu DB, circuli maximi, erit angu-<lb/>lus DBC, rectus; </s> <s xml:id="echoid-s13510" xml:space="preserve">ac proinde ABC, acutus.</s> <s xml:id="echoid-s13511" xml:space="preserve"/> </p> <div xml:id="echoid-div1069" type="float" level="2" n="4"> <note position="left" xlink:label="note-404-05" xlink:href="note-404-05a" xml:space="preserve">Coroll. 16. <lb/>1. Theod. <lb/>13.1 Theod. <lb/>20.1 Theod. <lb/>15.1 Theod.</note> </div> <p> <s xml:id="echoid-s13512" xml:space="preserve">POSTREMO ſit arcus AC, maior quadrante. </s> <s xml:id="echoid-s13513" xml:space="preserve">Dico angulum B, ei <lb/> <anchor type="note" xlink:label="note-404-06a" xlink:href="note-404-06"/> ſubtenſum obtuſum eſſe. </s> <s xml:id="echoid-s13514" xml:space="preserve">Abſciſſo enim quadrante CE, erit rurſum E, polus <lb/>arcus BC; </s> <s xml:id="echoid-s13515" xml:space="preserve">propterea quod arcus CA, per plum arcus BC, tranſit. </s> <s xml:id="echoid-s13516" xml:space="preserve">Ducto <lb/>ergo per puncta E, B, arcu EB, circuli maximi, erit angulus EBC, rectus; <lb/></s> <s xml:id="echoid-s13517" xml:space="preserve">atque adeò ABC, obtuſus. </s> <s xml:id="echoid-s13518" xml:space="preserve">Quapropter, in omni triangulo ſphærico, &</s> <s xml:id="echoid-s13519" xml:space="preserve">c. </s> <s xml:id="echoid-s13520" xml:space="preserve"><lb/>Quod erat demonſtrandum.</s> <s xml:id="echoid-s13521" xml:space="preserve"/> </p> <div xml:id="echoid-div1070" type="float" level="2" n="5"> <note position="left" xlink:label="note-404-06" xlink:href="note-404-06a" xml:space="preserve">Coroll. 16. <lb/>1. Theod. <lb/>13. 1. Theod. <lb/>20. 1 Theod. <lb/>25. 1. Theod.</note> </div> </div> <div xml:id="echoid-div1072" type="section" level="1" n="532"> <head xml:id="echoid-head567" xml:space="preserve">THEOR. 33. PROPOS. 35.</head> <p> <s xml:id="echoid-s13522" xml:space="preserve">IN omni triangulo ſphærico rectangulo, ſi v-<lb/>terque arcuum comprehendentium angulum re- <pb o="393" file="405" n="405" rhead=""/> ctum, vel vnus tantum, fuerit quadrans, erit & </s> <s xml:id="echoid-s13523" xml:space="preserve">ar-<lb/>cus rectum angulum ſubtendens, quadrans: </s> <s xml:id="echoid-s13524" xml:space="preserve">Si ve-<lb/>ro vterque dictorum arcuum minor fuerit qua-<lb/>drante, aut maior, erit arcus rectum angulum ſub-<lb/>rendens quadrante minor: </s> <s xml:id="echoid-s13525" xml:space="preserve">ſi denique alter illo-<lb/>rum maior fuerit quadrante, & </s> <s xml:id="echoid-s13526" xml:space="preserve">alter minor, erit ar-<lb/>cus rectum angulũ ſubtendens maior quadrante.</s> <s xml:id="echoid-s13527" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13528" xml:space="preserve">IN triangulo ſphærico rectangulo ABC, ſit angulus B, rectus, & </s> <s xml:id="echoid-s13529" xml:space="preserve">primum <lb/>vterque arcus AB, BC, vel alter illorum tantum quadrans. </s> <s xml:id="echoid-s13530" xml:space="preserve">Dico & </s> <s xml:id="echoid-s13531" xml:space="preserve">arcum <lb/>AC, qui rectum angulum ſubtendit, quadrantem <lb/> <anchor type="figure" xlink:label="fig-405-01a" xlink:href="fig-405-01"/> eſſe. </s> <s xml:id="echoid-s13532" xml:space="preserve">Si enim vterque arcus AB, BC, quadrans eſt, <lb/>cum angulus B, ponatur rectus, erit quoq; </s> <s xml:id="echoid-s13533" xml:space="preserve">arcus AC, <lb/> <anchor type="note" xlink:label="note-405-01a" xlink:href="note-405-01"/> quadrans. </s> <s xml:id="echoid-s13534" xml:space="preserve">Si verò alter tantum arcuum AB, BC, eſt <lb/>quadrans, ſit AB, quadrans. </s> <s xml:id="echoid-s13535" xml:space="preserve">Quoniã igitur arcus AB, <lb/>quadrans eſt, tranſitq́; </s> <s xml:id="echoid-s13536" xml:space="preserve">per polos arcus BC, propter <lb/> <anchor type="note" xlink:label="note-405-02a" xlink:href="note-405-02"/> angulum rectum B, erit A, polus arcus BC; </s> <s xml:id="echoid-s13537" xml:space="preserve">ac propte-<lb/> <anchor type="note" xlink:label="note-405-03a" xlink:href="note-405-03"/> rea angulus C, rectus erit. </s> <s xml:id="echoid-s13538" xml:space="preserve">Cum ergo vterque angu-<lb/>lus B, C, rectus ſit, erit vterque arcus AB, AC, qua-<lb/> <anchor type="note" xlink:label="note-405-04a" xlink:href="note-405-04"/> drans. </s> <s xml:id="echoid-s13539" xml:space="preserve">Eodem modo ſi BC, ponatur quadrans, oſten <lb/>demus AC, eſſe quadrantem. </s> <s xml:id="echoid-s13540" xml:space="preserve">Erit enim ſimiliter C, polus arcus AB; </s> <s xml:id="echoid-s13541" xml:space="preserve">ac pro-<lb/>inde angulus A, rectus. </s> <s xml:id="echoid-s13542" xml:space="preserve">Cum crgo vterque angulus B, A, rectus ſit, erit vter-<lb/> <anchor type="note" xlink:label="note-405-05a" xlink:href="note-405-05"/> que arcus BC, AC, quadrans.</s> <s xml:id="echoid-s13543" xml:space="preserve"/> </p> <div xml:id="echoid-div1072" type="float" level="2" n="1"> <figure xlink:label="fig-405-01" xlink:href="fig-405-01a"> <image file="405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/405-01"/> </figure> <note position="right" xlink:label="note-405-01" xlink:href="note-405-01a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-405-02" xlink:href="note-405-02a" xml:space="preserve">13.1 Theod.</note> <note position="right" xlink:label="note-405-03" xlink:href="note-405-03a" xml:space="preserve">Coroll. 16. <lb/>1. Theod. <lb/>15. 1 Theod.</note> <note position="right" xlink:label="note-405-04" xlink:href="note-405-04a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-405-05" xlink:href="note-405-05a" xml:space="preserve">25. huius.</note> </div> <figure> <image file="405-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/405-02"/> </figure> <p> <s xml:id="echoid-s13544" xml:space="preserve">SIT deinde vterque arcus AB, BC, quadrante <lb/>minor, vel maior. </s> <s xml:id="echoid-s13545" xml:space="preserve">Dico arcum AC, eſſe quadrante <lb/>minorem. </s> <s xml:id="echoid-s13546" xml:space="preserve">Si enim vterque eſt quadrante minor, pro-<lb/>ducto arcu CB, ad partes B, & </s> <s xml:id="echoid-s13547" xml:space="preserve">BA, ad partes A, vt <lb/>ſint CD, BE, quadrantes, ducatur per puncta D, <lb/>E, arcus circuli maximi DE, ſecans arcum CA, pro-<lb/> <anchor type="note" xlink:label="note-405-06a" xlink:href="note-405-06"/> ductum in F. </s> <s xml:id="echoid-s13548" xml:space="preserve">Quoniam igitur in triagulo BED, an-<lb/>gulus B, rectus eſt, & </s> <s xml:id="echoid-s13549" xml:space="preserve">arcus BE, quadrans, erit angulus D, quem ſubtendit, re-<lb/>ctus. </s> <s xml:id="echoid-s13550" xml:space="preserve">Rurſus quia in triangulo CDF, angu-<lb/> <anchor type="figure" xlink:label="fig-405-03a" xlink:href="fig-405-03"/> <anchor type="note" xlink:label="note-405-07a" xlink:href="note-405-07"/> lus D, rectus eſt, & </s> <s xml:id="echoid-s13551" xml:space="preserve">arcus DC, quadrans, erit <lb/>ſimiliter angulus F, rectus; </s> <s xml:id="echoid-s13552" xml:space="preserve">atque idcirco vter-<lb/>que arcus DC, FC, quadrans erit. </s> <s xml:id="echoid-s13553" xml:space="preserve">Quare <lb/> <anchor type="note" xlink:label="note-405-08a" xlink:href="note-405-08"/> arcus AC, quadrante minor erit. </s> <s xml:id="echoid-s13554" xml:space="preserve">Si verò <lb/> <anchor type="note" xlink:label="note-405-09a" xlink:href="note-405-09"/> vterque arcus AB, BC, quadrante maior <lb/>eſt, abſciſsis quadrantibus BD, CE, ducatur <lb/>per puncta D, E, arcus circuli maximi ED, <lb/> <anchor type="note" xlink:label="note-405-10a" xlink:href="note-405-10"/> ſecans arcum CA, productum in F. </s> <s xml:id="echoid-s13555" xml:space="preserve">Quoniã <lb/>igitur in triangulo DBE, angulus B, rectus <lb/>eſt, & </s> <s xml:id="echoid-s13556" xml:space="preserve">arcus BD, quadrans erit angulus E, <lb/>quem BD, ſubtendit, rectus. </s> <s xml:id="echoid-s13557" xml:space="preserve">Rurſus quia in <lb/> <anchor type="note" xlink:label="note-405-11a" xlink:href="note-405-11"/> triangulo CEF, angulus E, eſt rectus, & </s> <s xml:id="echoid-s13558" xml:space="preserve">arcus EC, quadrans, erit eodem mo-<lb/> <anchor type="note" xlink:label="note-405-12a" xlink:href="note-405-12"/> <pb o="394" file="406" n="406" rhead=""/> do angulus F, rectus. </s> <s xml:id="echoid-s13559" xml:space="preserve">Vterque ergo arcus CE, CF, quadrans erit; </s> <s xml:id="echoid-s13560" xml:space="preserve">ac propte-<lb/> <anchor type="note" xlink:label="note-406-01a" xlink:href="note-406-01"/> rea arcus AC, quadrante minor.</s> <s xml:id="echoid-s13561" xml:space="preserve"/> </p> <div xml:id="echoid-div1073" type="float" level="2" n="2"> <note position="right" xlink:label="note-405-06" xlink:href="note-405-06a" xml:space="preserve">20.1 Theod.</note> <figure xlink:label="fig-405-03" xlink:href="fig-405-03a"> <image file="405-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/405-03"/> </figure> <note position="right" xlink:label="note-405-07" xlink:href="note-405-07a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-405-08" xlink:href="note-405-08a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-405-09" xlink:href="note-405-09a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-405-10" xlink:href="note-405-10a" xml:space="preserve">20.1 Theod.</note> <note position="right" xlink:label="note-405-11" xlink:href="note-405-11a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-405-12" xlink:href="note-405-12a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-406-01" xlink:href="note-406-01a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s13562" xml:space="preserve">SIT poſtremo arcus AB, quadrante quidem maior, arcus vero BC, mi-<lb/>nor quadrante. </s> <s xml:id="echoid-s13563" xml:space="preserve">Dico arcum AC, maiorem eſſe quadrante. </s> <s xml:id="echoid-s13564" xml:space="preserve">Auferatur enim <lb/>quadrans BD; </s> <s xml:id="echoid-s13565" xml:space="preserve">& </s> <s xml:id="echoid-s13566" xml:space="preserve">arcus CB, producatur ad E, vt CE, ſit quadrans; </s> <s xml:id="echoid-s13567" xml:space="preserve">ac per pun <lb/> <anchor type="figure" xlink:label="fig-406-01a" xlink:href="fig-406-01"/> cta D, E, arcus circuli maximi ducatur ED, ſecans <lb/> <anchor type="note" xlink:label="note-406-02a" xlink:href="note-406-02"/> arcum AC, in F. </s> <s xml:id="echoid-s13568" xml:space="preserve">Quia igitur in triangulo BED, <lb/>angulus B, rectus eſt, & </s> <s xml:id="echoid-s13569" xml:space="preserve">arcus BD, quadrans, erit an-<lb/>gulus E, rectus. </s> <s xml:id="echoid-s13570" xml:space="preserve">Rurſus, quia in triangulo CEF, <lb/> <anchor type="note" xlink:label="note-406-03a" xlink:href="note-406-03"/> angulus E, rectus eſt, & </s> <s xml:id="echoid-s13571" xml:space="preserve">arcus EC, quadrans, erit <lb/>eadem ratione angulus F, rectus. </s> <s xml:id="echoid-s13572" xml:space="preserve">Vterque igitur ar-<lb/> <anchor type="note" xlink:label="note-406-04a" xlink:href="note-406-04"/> cus CE, CF, quadrans erit; </s> <s xml:id="echoid-s13573" xml:space="preserve">ac proinde arcus AC, <lb/> <anchor type="note" xlink:label="note-406-05a" xlink:href="note-406-05"/> quadrante maior. </s> <s xml:id="echoid-s13574" xml:space="preserve">In omni ergo triangulo ſphærico <lb/>rectangulo, &</s> <s xml:id="echoid-s13575" xml:space="preserve">c. </s> <s xml:id="echoid-s13576" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s13577" xml:space="preserve"/> </p> <div xml:id="echoid-div1074" type="float" level="2" n="3"> <figure xlink:label="fig-406-01" xlink:href="fig-406-01a"> <image file="406-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/406-01"/> </figure> <note position="left" xlink:label="note-406-02" xlink:href="note-406-02a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-406-03" xlink:href="note-406-03a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-406-04" xlink:href="note-406-04a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-406-05" xlink:href="note-406-05a" xml:space="preserve">25. huius.</note> </div> </div> <div xml:id="echoid-div1076" type="section" level="1" n="533"> <head xml:id="echoid-head568" xml:space="preserve">THEOR. 34. PROPOS. 36.</head> <p> <s xml:id="echoid-s13578" xml:space="preserve">IN omni triangulo ſphærico rectangulo, ſi ar-<lb/>cus rectum angulum ſubtendens fuerit quadrans, <lb/>erit vel vterq; </s> <s xml:id="echoid-s13579" xml:space="preserve">arcuum angulum rectum compre-<lb/>hendentium quadrans, vel alter illorum ſaltem: </s> <s xml:id="echoid-s13580" xml:space="preserve">ſi <lb/>verò fuerit minor quadrante, vterque reliquorum <lb/>vel minor, vel maior quadrante erit: </s> <s xml:id="echoid-s13581" xml:space="preserve">ſi deniq; </s> <s xml:id="echoid-s13582" xml:space="preserve">qua-<lb/>drante maior fuerit, erit reliquorum alter maior <lb/>quadrante, & </s> <s xml:id="echoid-s13583" xml:space="preserve">alter minor.</s> <s xml:id="echoid-s13584" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13585" xml:space="preserve">SIT in triangulo ſphærico ABC, angulus B, rectus, & </s> <s xml:id="echoid-s13586" xml:space="preserve">eum ſubtendens <lb/>arcus AC, ſit primum quadrans. </s> <s xml:id="echoid-s13587" xml:space="preserve">Dico vel vtrumque arcuum AB, BC, eſſe <lb/>quoque quadrantem, vel ſaltem alterum illorum. </s> <s xml:id="echoid-s13588" xml:space="preserve">Si enim neuter illorum eſt <lb/>quadrans, erit vel vterque illorum maior, vel minor <lb/> <anchor type="figure" xlink:label="fig-406-02a" xlink:href="fig-406-02"/> quadrante, atque adeo arcus AC, quadrante minor; <lb/></s> <s xml:id="echoid-s13589" xml:space="preserve"> <anchor type="note" xlink:label="note-406-06a" xlink:href="note-406-06"/> vel alter illorum quadrante quidem maior, alter ve-<lb/>rò minor, ac proinde arcus AC, quadrante maior; <lb/></s> <s xml:id="echoid-s13590" xml:space="preserve"> <anchor type="note" xlink:label="note-406-07a" xlink:href="note-406-07"/> quorum vtrumq; </s> <s xml:id="echoid-s13591" xml:space="preserve">abſurdum eſt, cum arcus AC, po-<lb/>natur quadrans. </s> <s xml:id="echoid-s13592" xml:space="preserve">Erit ergo vel vterque arcus AB, <lb/>BC, quadrans, vel ſaltem alter illorum.</s> <s xml:id="echoid-s13593" xml:space="preserve"/> </p> <div xml:id="echoid-div1076" type="float" level="2" n="1"> <figure xlink:label="fig-406-02" xlink:href="fig-406-02a"> <image file="406-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/406-02"/> </figure> <note position="left" xlink:label="note-406-06" xlink:href="note-406-06a" xml:space="preserve">35. huius.</note> <note position="left" xlink:label="note-406-07" xlink:href="note-406-07a" xml:space="preserve">35. huius.</note> </div> <p> <s xml:id="echoid-s13594" xml:space="preserve">DEINDE ſit arcus AC, quadrante minor. </s> <s xml:id="echoid-s13595" xml:space="preserve">Di-<lb/>co vtrumque arcum AB, BC, eſſe vel quadrante mi-<lb/>norem, vel maiorem. </s> <s xml:id="echoid-s13596" xml:space="preserve">Si enim vterque non eſt minor, <lb/>vel maior quadrante, erit vel vterque quadrans, ideoq́ue & </s> <s xml:id="echoid-s13597" xml:space="preserve">arcus AC, qua-<lb/> <anchor type="note" xlink:label="note-406-08a" xlink:href="note-406-08"/> drans; </s> <s xml:id="echoid-s13598" xml:space="preserve">vel vnus illorum quadrans, & </s> <s xml:id="echoid-s13599" xml:space="preserve">alter non, atque idcirco & </s> <s xml:id="echoid-s13600" xml:space="preserve">arcus AC, <lb/> <anchor type="note" xlink:label="note-406-09a" xlink:href="note-406-09"/> quadrans; </s> <s xml:id="echoid-s13601" xml:space="preserve">vel vnus quidem quadrante minor, alter verò maior, atque adeò &</s> <s xml:id="echoid-s13602" xml:space="preserve"> <pb o="395" file="407" n="407" rhead=""/> @rcus AC, quadrante maior: </s> <s xml:id="echoid-s13603" xml:space="preserve">quæ omnia abſurdo ſunt, eum arcus AC, po-<lb/> <anchor type="note" xlink:label="note-407-01a" xlink:href="note-407-01"/> natur quadrante minor. </s> <s xml:id="echoid-s13604" xml:space="preserve">Erit ergo vel vterque arcus AB, BC, minor quadran <lb/>te, vel maior.</s> <s xml:id="echoid-s13605" xml:space="preserve"/> </p> <div xml:id="echoid-div1077" type="float" level="2" n="2"> <note position="left" xlink:label="note-406-08" xlink:href="note-406-08a" xml:space="preserve">35. huius.</note> <note position="left" xlink:label="note-406-09" xlink:href="note-406-09a" xml:space="preserve">35. huius.</note> <note position="right" xlink:label="note-407-01" xlink:href="note-407-01a" xml:space="preserve">35. huius</note> </div> <p> <s xml:id="echoid-s13606" xml:space="preserve">TERTIO ſit arcus AC, maior quadrante. </s> <s xml:id="echoid-s13607" xml:space="preserve">Dico alterum reliquorum <lb/>AB, BC, quadrante quidem eſſe maiorem, alterum verò minorem. </s> <s xml:id="echoid-s13608" xml:space="preserve">Si enim <lb/>non eſt alter maior, & </s> <s xml:id="echoid-s13609" xml:space="preserve">alter minor quadrante, erit vel vterque quadrans, <lb/>vel alter ſaltem quadrans, & </s> <s xml:id="echoid-s13610" xml:space="preserve">alter non, ac proinde & </s> <s xml:id="echoid-s13611" xml:space="preserve">arcus AC, quadrans; <lb/></s> <s xml:id="echoid-s13612" xml:space="preserve"> <anchor type="note" xlink:label="note-407-02a" xlink:href="note-407-02"/> vel vterque minor quadrante, aut maior, atque adeo arcus AC, quadrante <lb/> <anchor type="note" xlink:label="note-407-03a" xlink:href="note-407-03"/> minor: </s> <s xml:id="echoid-s13613" xml:space="preserve">quæ omnia ſunt abſurda, cum arcus AC, maior ponatur, quam qua-<lb/>drans. </s> <s xml:id="echoid-s13614" xml:space="preserve">Erit ergo alter arcuum AB, BC, quadrante quidem maior, alter ve-<lb/>ro minor. </s> <s xml:id="echoid-s13615" xml:space="preserve">Quocirca in omni triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s13616" xml:space="preserve">c. </s> <s xml:id="echoid-s13617" xml:space="preserve">Quod de-<lb/>monſtrandum erat.</s> <s xml:id="echoid-s13618" xml:space="preserve"/> </p> <div xml:id="echoid-div1078" type="float" level="2" n="3"> <note position="right" xlink:label="note-407-02" xlink:href="note-407-02a" xml:space="preserve">35. huius.</note> <note position="right" xlink:label="note-407-03" xlink:href="note-407-03a" xml:space="preserve">35. huius.</note> </div> </div> <div xml:id="echoid-div1080" type="section" level="1" n="534"> <head xml:id="echoid-head569" xml:space="preserve">THEOR. 35. PROPOS. 37.</head> <p> <s xml:id="echoid-s13619" xml:space="preserve">IN omni triangulo ſphærico, ſi vterque reli-<lb/>quorum angulorum, vel alter ſaltem fuerit rectus, <lb/>erit arcus rectum angulum ſubtendens, quadrans: <lb/></s> <s xml:id="echoid-s13620" xml:space="preserve">ſi verò vterque reliquorum angulorum minor fue-<lb/>rit recto, vel maior, erit arcus ſubtendens angulum <lb/>rectum quadrante minor: </s> <s xml:id="echoid-s13621" xml:space="preserve">ſi deniq; </s> <s xml:id="echoid-s13622" xml:space="preserve">alter reliquo-<lb/>rum fuerit maior recto, & </s> <s xml:id="echoid-s13623" xml:space="preserve">alter minor, erit arcus <lb/>angulum rectum ſubtendens, quadrante maior.</s> <s xml:id="echoid-s13624" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13625" xml:space="preserve">IN triangulo ſphærico ABC, cuius angulus B, rectus, ſit primum vter-<lb/>que angulorum A, C, vel alter ſaltem, nempe C, rectus. </s> <s xml:id="echoid-s13626" xml:space="preserve">Dico arcum AC, qui <lb/>rectum angulum B, ſubtendit, eſſe quadrantem. </s> <s xml:id="echoid-s13627" xml:space="preserve">Si enim vterque angulus A, <lb/>C, rectus eſt, vel C, tantum, erit triangulum ABC, <lb/> <anchor type="figure" xlink:label="fig-407-01a" xlink:href="fig-407-01"/> rectangulũ habens angulum C, rectum: </s> <s xml:id="echoid-s13628" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s13629" xml:space="preserve"><lb/>angulus B, rectus. </s> <s xml:id="echoid-s13630" xml:space="preserve">Igitur arcus AC, quadrans erit.</s> <s xml:id="echoid-s13631" xml:space="preserve"/> </p> <div xml:id="echoid-div1080" type="float" level="2" n="1"> <figure xlink:label="fig-407-01" xlink:href="fig-407-01a"> <image file="407-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/407-01"/> </figure> </div> <note position="right" xml:space="preserve">34. huius.</note> <p> <s xml:id="echoid-s13632" xml:space="preserve">DEINDE ſit vterque angulus A, C, vel minor <lb/>recto, vel maior. </s> <s xml:id="echoid-s13633" xml:space="preserve">Dico arcum AC, quadrante eſſe <lb/>minorem. </s> <s xml:id="echoid-s13634" xml:space="preserve">Si namque vterque angulus A, C, eſt mi-<lb/> <anchor type="note" xlink:label="note-407-05a" xlink:href="note-407-05"/> nor recto, erit tam arcus BC, quam AB, minor qua-<lb/>drante; </s> <s xml:id="echoid-s13635" xml:space="preserve">ſi verò vterque angulus A, C, maior eſt re-<lb/> <anchor type="note" xlink:label="note-407-06a" xlink:href="note-407-06"/> cto, erit tam arcus BC, quam AB, quadrante maior. <lb/></s> <s xml:id="echoid-s13636" xml:space="preserve">Quare cum in triangulo ABc, angulus B, rectus <lb/>ſit, & </s> <s xml:id="echoid-s13637" xml:space="preserve">vterque arcus AB, BC, vel minor, vel maior quadrante, erit ſemper <lb/>arcus AC, quadrante minor.</s> <s xml:id="echoid-s13638" xml:space="preserve"/> </p> <div xml:id="echoid-div1081" type="float" level="2" n="2"> <note position="right" xlink:label="note-407-05" xlink:href="note-407-05a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-407-06" xlink:href="note-407-06a" xml:space="preserve">34. huius.</note> </div> <note position="right" xml:space="preserve">35. huius.</note> <p> <s xml:id="echoid-s13639" xml:space="preserve">TERTIO ſit angulus A, maior recto, & </s> <s xml:id="echoid-s13640" xml:space="preserve">C, minor. </s> <s xml:id="echoid-s13641" xml:space="preserve">Dico arcum AC, eſ-<lb/>ſe quadrante maiorem. </s> <s xml:id="echoid-s13642" xml:space="preserve">Cum enim angulus A, obtuſus ſit, erit arcus BC, <lb/> <anchor type="note" xlink:label="note-407-08a" xlink:href="note-407-08"/> maior quadrante: </s> <s xml:id="echoid-s13643" xml:space="preserve">Et cum angulus C, acutus ſit, erit arcus AB, quadrante mi-<lb/> <anchor type="note" xlink:label="note-407-09a" xlink:href="note-407-09"/> <pb o="396" file="408" n="408" rhead=""/> nor. </s> <s xml:id="echoid-s13644" xml:space="preserve">Igitur cum arcus BC, quadrante quidem maior ſit, & </s> <s xml:id="echoid-s13645" xml:space="preserve">AB, minor, erit <lb/>arcus AC, quadrante maior. </s> <s xml:id="echoid-s13646" xml:space="preserve">Quamobrem in omni triangulo ſphærico re-<lb/> <anchor type="note" xlink:label="note-408-01a" xlink:href="note-408-01"/> ctangulo, &</s> <s xml:id="echoid-s13647" xml:space="preserve">c. </s> <s xml:id="echoid-s13648" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s13649" xml:space="preserve"/> </p> <div xml:id="echoid-div1082" type="float" level="2" n="3"> <note position="right" xlink:label="note-407-08" xlink:href="note-407-08a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-407-09" xlink:href="note-407-09a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-408-01" xlink:href="note-408-01a" xml:space="preserve">35. huiu@.</note> </div> </div> <div xml:id="echoid-div1084" type="section" level="1" n="535"> <head xml:id="echoid-head570" xml:space="preserve">THEOR. 36. PROPOS. 38.</head> <p> <s xml:id="echoid-s13650" xml:space="preserve">IN omni triangulo ſphærico rectangulo, ſi ar-<lb/>cus rectum angulum ſubtendens fuerit quadrans, <lb/>erit ſaltem alter reliquorum angulorũ rectus quo-<lb/>que: </s> <s xml:id="echoid-s13651" xml:space="preserve">ſi verò minor quadrante, erit vterq; </s> <s xml:id="echoid-s13652" xml:space="preserve">reliquo-<lb/>rum angulorum vel maior recto, vel minor: </s> <s xml:id="echoid-s13653" xml:space="preserve">ſi de-<lb/>nique quadrante maior, erit alter reliquorum an-<lb/>gulorum maior recto, & </s> <s xml:id="echoid-s13654" xml:space="preserve">alter minor.</s> <s xml:id="echoid-s13655" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13656" xml:space="preserve">IN triangulo ſphærico ABC, cuius angulus B, rectus, ſit primum arcus <lb/>AC, ſubtendens angulum rectum B, quadrans. </s> <s xml:id="echoid-s13657" xml:space="preserve">Dico ſaltem alterum angu-<lb/>lorum A, C, rectum quoque eſſe. </s> <s xml:id="echoid-s13658" xml:space="preserve">Cum enim angulus B, ſit rectus, & </s> <s xml:id="echoid-s13659" xml:space="preserve">arcus <lb/>AC, quadrans, erit ſaltem alter arcuum AB, BC, quadrans; </s> <s xml:id="echoid-s13660" xml:space="preserve">atque adeò & </s> <s xml:id="echoid-s13661" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-408-02a" xlink:href="note-408-02"/> angulus A, vel C, quem ille arcus ſubtendit, rectus erit.</s> <s xml:id="echoid-s13662" xml:space="preserve"/> </p> <div xml:id="echoid-div1084" type="float" level="2" n="1"> <note position="left" xlink:label="note-408-02" xlink:href="note-408-02a" xml:space="preserve">36. huius.</note> </div> <note position="left" xml:space="preserve">34. huius.</note> <figure> <image file="408-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/408-01"/> </figure> <p> <s xml:id="echoid-s13663" xml:space="preserve">SIT deinde areus AC, quadrante minor. </s> <s xml:id="echoid-s13664" xml:space="preserve">Dico <lb/>vtrumque angulorum A, C, eſſe maiorem, vel mino-<lb/>rem recto. </s> <s xml:id="echoid-s13665" xml:space="preserve">Erit enim vterque arcus AB, BC, vel ma-<lb/> <anchor type="note" xlink:label="note-408-04a" xlink:href="note-408-04"/> ior quadrante, vel minor. </s> <s xml:id="echoid-s13666" xml:space="preserve">Quare vterque angulus <lb/>A, C, maior erit recto, vel minor.</s> <s xml:id="echoid-s13667" xml:space="preserve"/> </p> <div xml:id="echoid-div1085" type="float" level="2" n="2"> <note position="left" xlink:label="note-408-04" xlink:href="note-408-04a" xml:space="preserve">36. huius.</note> </div> <note position="left" xml:space="preserve">34. huius.</note> <p> <s xml:id="echoid-s13668" xml:space="preserve">POSTREMO ſit arcus AC, maior quadran-<lb/>te. </s> <s xml:id="echoid-s13669" xml:space="preserve">Dico alterum angulorum A, C, eſſe recto maio-<lb/>rem, & </s> <s xml:id="echoid-s13670" xml:space="preserve">alterum minorem. </s> <s xml:id="echoid-s13671" xml:space="preserve">Erit enim alter arcuum <lb/>AB, BC, quadrante maior, & </s> <s xml:id="echoid-s13672" xml:space="preserve">alter minor. </s> <s xml:id="echoid-s13673" xml:space="preserve">Igitur <lb/> <anchor type="note" xlink:label="note-408-06a" xlink:href="note-408-06"/> alter angulorum A, C, recto erit maior, & </s> <s xml:id="echoid-s13674" xml:space="preserve">alter mi-<lb/> <anchor type="note" xlink:label="note-408-07a" xlink:href="note-408-07"/> nor. </s> <s xml:id="echoid-s13675" xml:space="preserve">Quapropter In omni triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s13676" xml:space="preserve">c. </s> <s xml:id="echoid-s13677" xml:space="preserve">Quod oſten-<lb/>dendum erat.</s> <s xml:id="echoid-s13678" xml:space="preserve"/> </p> <div xml:id="echoid-div1086" type="float" level="2" n="3"> <note position="left" xlink:label="note-408-06" xlink:href="note-408-06a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-408-07" xlink:href="note-408-07a" xml:space="preserve">34. huius.</note> </div> </div> <div xml:id="echoid-div1088" type="section" level="1" n="536"> <head xml:id="echoid-head571" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s13679" xml:space="preserve">EX his omnibus colligitur, In omni triangulo ſphærico, cuius vnus arcuum eſt qua-<lb/>drans, vnusq́ue angulorum rectus, reliquorum quoque arcuum vnum ſal@em eſſe quadran <lb/>tem, & </s> <s xml:id="echoid-s13680" xml:space="preserve">reliquotum angulorum vnum ſaltem rectum. </s> <s xml:id="echoid-s13681" xml:space="preserve">Nam ſi vnus angulorum rectus eſt, <lb/>& </s> <s xml:id="echoid-s13682" xml:space="preserve">alter arcuum ipſum com prehendentium quadrans, erit & </s> <s xml:id="echoid-s13683" xml:space="preserve">arcus rectum angulum ſubten <lb/> <anchor type="note" xlink:label="note-408-08a" xlink:href="note-408-08"/> dens quadrans; </s> <s xml:id="echoid-s13684" xml:space="preserve">& </s> <s xml:id="echoid-s13685" xml:space="preserve">angulus quem prior quadrans ſubtendit rectus: </s> <s xml:id="echoid-s13686" xml:space="preserve">Si verò arcus angulum <lb/> <anchor type="note" xlink:label="note-408-09a" xlink:href="note-408-09"/> rectum ſubtendens quadrans eſt, erit & </s> <s xml:id="echoid-s13687" xml:space="preserve">vel vterque arcuum rectum angulum comprehen-<lb/> <anchor type="note" xlink:label="note-408-10a" xlink:href="note-408-10"/> dentium, vel alter ſaltem quadrans; </s> <s xml:id="echoid-s13688" xml:space="preserve">& </s> <s xml:id="echoid-s13689" xml:space="preserve">vel vterque reliquorum angulorum, vel alter ſaltem <lb/> <anchor type="note" xlink:label="note-408-11a" xlink:href="note-408-11"/> rectus. </s> <s xml:id="echoid-s13690" xml:space="preserve">Itaque fieri non poteſt, vt detur triangulum ſphæricum rectangulum, cuius vnus <lb/> <anchor type="note" xlink:label="note-408-12a" xlink:href="note-408-12"/> duntaxat arcus ſit quadrans, ſed vel nullus erit quadrans, vel omnes tres, vel duo quadran-<lb/>ces erunt.</s> <s xml:id="echoid-s13691" xml:space="preserve"/> </p> <div xml:id="echoid-div1088" type="float" level="2" n="1"> <note position="left" xlink:label="note-408-08" xlink:href="note-408-08a" xml:space="preserve">35. huius.</note> <note position="left" xlink:label="note-408-09" xlink:href="note-408-09a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-408-10" xlink:href="note-408-10a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-408-11" xlink:href="note-408-11a" xml:space="preserve">38. huius.</note> <note position="left" xlink:label="note-408-12" xlink:href="note-408-12a" xml:space="preserve">Nota.</note> </div> <pb o="397" file="409" n="409" rhead=""/> </div> <div xml:id="echoid-div1090" type="section" level="1" n="537"> <head xml:id="echoid-head572" xml:space="preserve">THEOR. 37. PROPOS. 39.</head> <p> <s xml:id="echoid-s13692" xml:space="preserve">ANGVLI ſphærici eandem habẽt rationem, <lb/>quam eorum arcus.</s> <s xml:id="echoid-s13693" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13694" xml:space="preserve">SINT duo anguli ſphærici BAC, EDF, quorum arcus BC, EF. </s> <s xml:id="echoid-s13695" xml:space="preserve">Dico <lb/>ita eſſe angulum A, ad angulum D, vt eſt arcus BC, ad arcum EF. </s> <s xml:id="echoid-s13696" xml:space="preserve">Erunt <lb/> <anchor type="note" xlink:label="note-409-01a" xlink:href="note-409-01"/> enim A, D, poli arcuum BC, EF; </s> <s xml:id="echoid-s13697" xml:space="preserve">& </s> <s xml:id="echoid-s13698" xml:space="preserve">arcus AB, AC, DE, DF, quadrantes. <lb/></s> <s xml:id="echoid-s13699" xml:space="preserve">Productis igitur arcu-<lb/> <anchor type="figure" xlink:label="fig-409-01a" xlink:href="fig-409-01"/> bus BC, EF, ſuman-<lb/>tur quotcunque arcus <lb/>BG, GH, arcui BC, <lb/>& </s> <s xml:id="echoid-s13700" xml:space="preserve">quotcũq; </s> <s xml:id="echoid-s13701" xml:space="preserve">arcus FI, <lb/>IK, KL, arcui EF, <lb/>æquales; </s> <s xml:id="echoid-s13702" xml:space="preserve">ac per puncta <lb/>G, H, I, K, L, & </s> <s xml:id="echoid-s13703" xml:space="preserve">po-<lb/>los A, D, arcus circu-<lb/> <anchor type="note" xlink:label="note-409-02a" xlink:href="note-409-02"/> lorum maximorũ du-<lb/>cantur AG, AH, DI, <lb/>DK, DL, qui omnes <lb/>quadrãtes erunt, nem-<lb/>pe quadrantibus AB, <lb/> <anchor type="note" xlink:label="note-409-03a" xlink:href="note-409-03"/> AC, DE, DF, æquales, propterea quòd & </s> <s xml:id="echoid-s13704" xml:space="preserve">rectæ ſubtenſæ AG, AH, DI, <lb/>DK, DL, rectis ſubtenſis AB, AC, DE, DF, æquales ſunt, ex defin. </s> <s xml:id="echoid-s13705" xml:space="preserve">poli. <lb/></s> <s xml:id="echoid-s13706" xml:space="preserve">Erunt ergo omnes anguli ad A, inter ſe æquales; </s> <s xml:id="echoid-s13707" xml:space="preserve">atque adeò quam multiplex <lb/> <anchor type="note" xlink:label="note-409-04a" xlink:href="note-409-04"/> eſt arcus CH, arcus BC, tam multiplex erit aggregatum omnium angulorũ <lb/>ad A, anguli BAC: </s> <s xml:id="echoid-s13708" xml:space="preserve">Eademque ratione tam multiplex erit aggregatum om-<lb/>nium angulorum ad D, anguli EDF, quam multiplex eſt arcus EL, arcus <lb/>EF. </s> <s xml:id="echoid-s13709" xml:space="preserve">Quoniam verò ſi arcus CH, arcui EL, æqualis fuerit, etiam angulus <lb/>HAC, angulo EDL, æqualis eſt; </s> <s xml:id="echoid-s13710" xml:space="preserve">ſi autem arcus CH, maior ſuerit arcu EL, <lb/> <anchor type="note" xlink:label="note-409-05a" xlink:href="note-409-05"/> etiam angulus HAC, angulo EDL, maior eſt; </s> <s xml:id="echoid-s13711" xml:space="preserve">& </s> <s xml:id="echoid-s13712" xml:space="preserve">ſi minor, minor; </s> <s xml:id="echoid-s13713" xml:space="preserve">deficient <lb/>propterea vnà arcus CH, & </s> <s xml:id="echoid-s13714" xml:space="preserve">angulus HAC, æquè multiplicia primæ magni-<lb/> <anchor type="note" xlink:label="note-409-06a" xlink:href="note-409-06"/> tudinis BC, & </s> <s xml:id="echoid-s13715" xml:space="preserve">tertiæ BAC, ab arcu EL, & </s> <s xml:id="echoid-s13716" xml:space="preserve">angulo EDL, æque multiplici-<lb/>bus ſecundę magnitudinis EF, & </s> <s xml:id="echoid-s13717" xml:space="preserve">quartæ EDF; </s> <s xml:id="echoid-s13718" xml:space="preserve">vel vnà æqualia erunt, vel <lb/>vnà excedent. </s> <s xml:id="echoid-s13719" xml:space="preserve">Quare quę proportio eſt arcus BC, primæ magnitudinis ad <lb/> <anchor type="note" xlink:label="note-409-07a" xlink:href="note-409-07"/> arcum EF, ſecundam magnitudinem, ea erit anguli BAC, tertiæ magnitu-<lb/>dinis ad angulum EDF, quartam magnitudinem. </s> <s xml:id="echoid-s13720" xml:space="preserve">Itaque anguli ſphærici <lb/>eandem habent rationem, quam eorum arcus. </s> <s xml:id="echoid-s13721" xml:space="preserve">Quod erat demouſtrandum.</s> <s xml:id="echoid-s13722" xml:space="preserve"/> </p> <div xml:id="echoid-div1090" type="float" level="2" n="1"> <note position="right" xlink:label="note-409-01" xlink:href="note-409-01a" xml:space="preserve">Defin. 6. <lb/>huius.</note> <figure xlink:label="fig-409-01" xlink:href="fig-409-01a"> <image file="409-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/409-01"/> </figure> <note position="right" xlink:label="note-409-02" xlink:href="note-409-02a" xml:space="preserve">20.1 Theod.</note> <note position="right" xlink:label="note-409-03" xlink:href="note-409-03a" xml:space="preserve">28. tertij.</note> <note position="right" xlink:label="note-409-04" xlink:href="note-409-04a" xml:space="preserve">18. huius.</note> <note position="right" xlink:label="note-409-05" xlink:href="note-409-05a" xml:space="preserve">18. huius.</note> <note position="right" xlink:label="note-409-06" xlink:href="note-409-06a" xml:space="preserve">12. huius.</note> <note position="right" xlink:label="note-409-07" xlink:href="note-409-07a" xml:space="preserve">Defin. 6. <lb/>quinti.</note> </div> </div> <div xml:id="echoid-div1092" type="section" level="1" n="538"> <head xml:id="echoid-head573" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s13723" xml:space="preserve">EX hoc ſequitur, @ta eſſe angulum ſphæricum quemcumque ad quatuor angulos rectos <lb/>ſphæricos, vt eſt arcus illius anguli ad totam circunferentiam circuli maximi; </s> <s xml:id="echoid-s13724" xml:space="preserve">& </s> <s xml:id="echoid-s13725" xml:space="preserve">contra. <lb/></s> <s xml:id="echoid-s13726" xml:space="preserve">Cum enim ſit angulus ſphæricus quicunque ad angulum rectum ſphæricum, vt arcus il-<lb/> <anchor type="note" xlink:label="note-409-08a" xlink:href="note-409-08"/> lius anguli ad quadrantem, nimirum ad arcum anguli recti, erit quoque idem angulus ad <lb/>quadruplum anguli recti nempe ad quatuor rectos, vt idem arcus illius anguli ad quadru-<lb/> <anchor type="note" xlink:label="note-409-09a" xlink:href="note-409-09"/> plum quadrantis, hoc eſt, ad totam circun ferentiam; </s> <s xml:id="echoid-s13727" xml:space="preserve">& </s> <s xml:id="echoid-s13728" xml:space="preserve">contra.</s> <s xml:id="echoid-s13729" xml:space="preserve"/> </p> <div xml:id="echoid-div1092" type="float" level="2" n="1"> <note position="right" xlink:label="note-409-08" xlink:href="note-409-08a" xml:space="preserve">39. huius.</note> <note position="right" xlink:label="note-409-09" xlink:href="note-409-09a" xml:space="preserve">Schol. 4. <lb/>quinti.</note> </div> <pb o="398" file="410" n="410" rhead=""/> </div> <div xml:id="echoid-div1094" type="section" level="1" n="539"> <head xml:id="echoid-head574" xml:space="preserve">THEOR. 38. PROPOS. 40.</head> <p> <s xml:id="echoid-s13730" xml:space="preserve">SI duo circuli maximi in ſphæra ſe mutuo ſe-<lb/>cent, & </s> <s xml:id="echoid-s13731" xml:space="preserve">in eorum peripherijs duo puncta ſignen-<lb/>tur, quorum vtrumque vel in eodem ſemicirculo <lb/>ſumatur; </s> <s xml:id="echoid-s13732" xml:space="preserve">vel in vno ſemicirculo vnum, & </s> <s xml:id="echoid-s13733" xml:space="preserve">alterum <lb/>in altero eiuſdem circuli; </s> <s xml:id="echoid-s13734" xml:space="preserve">vel vnum in ſemicircu-<lb/>lo vno vnius circuli, & </s> <s xml:id="echoid-s13735" xml:space="preserve">alterum in ſemicirculo <lb/>vtrolibet alterius circuli; </s> <s xml:id="echoid-s13736" xml:space="preserve">atque per vtrumque ho-<lb/>rum punctorum arcus maximi circuli ducatur fa-<lb/>ciẽs cum peripheria alterius circuli, ad quamcum-<lb/>que partem, angulum rectum: </s> <s xml:id="echoid-s13737" xml:space="preserve">habebit ſinus arcus <lb/>intercepti inter vnum illorum punctorum, & </s> <s xml:id="echoid-s13738" xml:space="preserve">al-<lb/>terutram ſectionem circulorum, ad ſinum arcus, <lb/>qui per illud punctũ ductus rectum cum periphe-<lb/>ria alterius circuli angulum facit, eandem propor-<lb/>tionem, quam habet ſinus arcus inter punctum al <lb/>terum, & </s> <s xml:id="echoid-s13739" xml:space="preserve">alterutram circulortum ſectionem inter-<lb/>iecti, ad ſinum arcus, qui per illud punctum de-<lb/>ſcriptus cum alterius circuli peripheria rectũ con-<lb/>ſtituit angulum.</s> <s xml:id="echoid-s13740" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13741" xml:space="preserve">IN ſphęra duo circuli maximi ABCD, AECF, ſe mutuo ſecentin A, & </s> <s xml:id="echoid-s13742" xml:space="preserve"><lb/>C, & </s> <s xml:id="echoid-s13743" xml:space="preserve">primum ad angulos non rectos; </s> <s xml:id="echoid-s13744" xml:space="preserve">ſignenturq́ue primum in eodem ſemicir-<lb/>culo ABC, duo puncta vtcũque B, G; </s> <s xml:id="echoid-s13745" xml:space="preserve">per quę, & </s> <s xml:id="echoid-s13746" xml:space="preserve">polum circuli AECF, qui ſit <lb/>H, circuli maximi ducantur IBHK, LGHM; </s> <s xml:id="echoid-s13747" xml:space="preserve">eruntq́ue anguli ad I, L, K, M, <lb/> <anchor type="note" xlink:label="note-410-01a" xlink:href="note-410-01"/> recti. </s> <s xml:id="echoid-s13748" xml:space="preserve">Dico eãdem habere proportionẽ ſinum arcus AB, vel CB, ad ſinũ arcus <lb/> <anchor type="note" xlink:label="note-410-02a" xlink:href="note-410-02"/> BI, vel BK, quam habet ſinus arcus AG, vel CG, ad ſinũ arcus GL, vel GM. <lb/></s> <s xml:id="echoid-s13749" xml:space="preserve">Sit enim cõmunis ſectio circulorum recta AC, ad quam ex B, G, perpẽdiculares <lb/> <anchor type="note" xlink:label="note-410-03a" xlink:href="note-410-03"/> agátur BN, GO, in plano circuli ABCD; </s> <s xml:id="echoid-s13750" xml:space="preserve">eritq; </s> <s xml:id="echoid-s13751" xml:space="preserve">BN, ſinus rectus tam arcus <lb/> <anchor type="note" xlink:label="note-410-04a" xlink:href="note-410-04"/> AB, quam arcus CB, ex definitione ſinus recti; </s> <s xml:id="echoid-s13752" xml:space="preserve">& </s> <s xml:id="echoid-s13753" xml:space="preserve">eodem modo GO, ſinus <lb/>vtriuſque arcus AG, CG. </s> <s xml:id="echoid-s13754" xml:space="preserve">Demittantur ab eiſdem punctis B, G, ad planum <lb/> <anchor type="note" xlink:label="note-410-05a" xlink:href="note-410-05"/> <pb o="399" file="411" n="411" rhead=""/> eirculi AECF, perpendiculares BP, GQ. </s> <s xml:id="echoid-s13755" xml:space="preserve">Et quoniam rectæ BP, GQ, ca-<lb/>dunt in communes ſectiones circulorum IBK, LGM, cum circulo AECF, <lb/> <anchor type="note" xlink:label="note-411-01a" xlink:href="note-411-01"/> quem bifariam ſecãtin punctis I, K; </s> <s xml:id="echoid-s13756" xml:space="preserve">L, M, hoc eſt, cadũtin diametros circulorũ <lb/>maximorum IBK, LGM; <lb/></s> <s xml:id="echoid-s13757" xml:space="preserve"> <anchor type="figure" xlink:label="fig-411-01a" xlink:href="fig-411-01"/> (quòd horum circulorum <lb/> <anchor type="note" xlink:label="note-411-02a" xlink:href="note-411-02"/> plana recta ſint ad planum <lb/>circuli AECF,) ac proin-<lb/>de rectos angulos faciunt <lb/>cum diametris circulorum <lb/>IBK, LGM, ex defin. </s> <s xml:id="echoid-s13758" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s13759" xml:space="preserve">lib. </s> <s xml:id="echoid-s13760" xml:space="preserve">11. </s> <s xml:id="echoid-s13761" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s13762" xml:space="preserve">erit quoque <lb/>tam BP, ſinus rectus arcuũ <lb/>BI, BK, quam GQ, ſinus <lb/>rectus arcuum GL, GM, <lb/>ex definitione ſinus recti. </s> <s xml:id="echoid-s13763" xml:space="preserve"><lb/>Ducantur in plano circuli <lb/>AECF, rectæ NP, OQ; </s> <s xml:id="echoid-s13764" xml:space="preserve"><lb/>eruntq; </s> <s xml:id="echoid-s13765" xml:space="preserve">per defin. </s> <s xml:id="echoid-s13766" xml:space="preserve">3. </s> <s xml:id="echoid-s13767" xml:space="preserve">lib. </s> <s xml:id="echoid-s13768" xml:space="preserve">11. </s> <s xml:id="echoid-s13769" xml:space="preserve"><lb/>Eucl. </s> <s xml:id="echoid-s13770" xml:space="preserve">anguli P, Q, recti, in <lb/>triangulis NBP, OGQ. </s> <s xml:id="echoid-s13771" xml:space="preserve"><lb/>Quia verò tam rectæ BN, <lb/> <anchor type="note" xlink:label="note-411-03a" xlink:href="note-411-03"/> GO, parallelę ſunt, propter <lb/>angulos rectos ANB, AOG, <lb/>quam rectæ BP, GQ, cum hæ perpendiculares ſint ad planũ circuli AECF; <lb/></s> <s xml:id="echoid-s13772" xml:space="preserve"> <anchor type="note" xlink:label="note-411-04a" xlink:href="note-411-04"/> erunt quoque anguli B, G, æquales in eisdem triangulis NBP, OGQ. <lb/></s> <s xml:id="echoid-s13773" xml:space="preserve"> <anchor type="note" xlink:label="note-411-05a" xlink:href="note-411-05"/> AEquiangula igitur ſunt triangula NBP, OGQ; </s> <s xml:id="echoid-s13774" xml:space="preserve">atque adeò erit, vt NB, <lb/> <anchor type="note" xlink:label="note-411-06a" xlink:href="note-411-06"/> ſinus arcus AB, vel CB, ad BP, ſinum arcus BI, vel BK, ita OG, ſinus ar-<lb/> <anchor type="note" xlink:label="note-411-07a" xlink:href="note-411-07"/> cus AG, vel CG, ad GQ, ſinum arcus GL, vel GM, quomodocunque ar-<lb/>cus ſumantur, cum cuilibet ſinui duo arcus ſemicirculũ conficientes reſpon-<lb/>deant. </s> <s xml:id="echoid-s13775" xml:space="preserve">Hcc eſt, erit, vt ſinus arcus AB, ad ſinum arcus BI, ita ſinus arcus AG, <lb/>ad ſinum arcus GL. </s> <s xml:id="echoid-s13776" xml:space="preserve">Item vt ſinus arcus AB, ad ſinum arcus BK, ita ſinus ar-<lb/>cus AG, ad ſinum arcus GM. </s> <s xml:id="echoid-s13777" xml:space="preserve">Item vt ſinus arcus CB, ad ſinum arcus BI, ita <lb/>ſinus arcus CG, ad ſinum arcus GL. </s> <s xml:id="echoid-s13778" xml:space="preserve">Item vt ſinus arcus CB, ad ſinum arcus <lb/>BK, ita ſinus arcus CG, ad ſinum arcus GM. </s> <s xml:id="echoid-s13779" xml:space="preserve">Item vt ſinus arcus AB, ad <lb/>ſinum arcus BI, ita ſinus arcus CG, ad ſinum arcus GM, &</s> <s xml:id="echoid-s13780" xml:space="preserve">c.</s> <s xml:id="echoid-s13781" xml:space="preserve"/> </p> <div xml:id="echoid-div1094" type="float" level="2" n="1"> <note position="left" xlink:label="note-410-01" xlink:href="note-410-01a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-410-02" xlink:href="note-410-02a" xml:space="preserve">15. 1. Theod.</note> <note position="left" xlink:label="note-410-03" xlink:href="note-410-03a" xml:space="preserve">3. vndee.</note> <note position="left" xlink:label="note-410-04" xlink:href="note-410-04a" xml:space="preserve">12. primi.</note> <note position="left" xlink:label="note-410-05" xlink:href="note-410-05a" xml:space="preserve">31. vndee.</note> <note position="right" xlink:label="note-411-01" xlink:href="note-411-01a" xml:space="preserve">38. vndec. <lb/>11. 1. Theod.</note> <figure xlink:label="fig-411-01" xlink:href="fig-411-01a"> <image file="411-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/411-01"/> </figure> <note position="right" xlink:label="note-411-02" xlink:href="note-411-02a" xml:space="preserve">15. 1. Theod.</note> <note position="right" xlink:label="note-411-03" xlink:href="note-411-03a" xml:space="preserve">28. primi.</note> <note position="right" xlink:label="note-411-04" xlink:href="note-411-04a" xml:space="preserve">6. vndee.</note> <note position="right" xlink:label="note-411-05" xlink:href="note-411-05a" xml:space="preserve">10. vndee.</note> <note position="right" xlink:label="note-411-06" xlink:href="note-411-06a" xml:space="preserve">32. primi.</note> <note position="right" xlink:label="note-411-07" xlink:href="note-411-07a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s13782" xml:space="preserve">DEINDE ſumatur vnum punctum, puta B, in ſemicirculo ABC, & </s> <s xml:id="echoid-s13783" xml:space="preserve">al-<lb/>terum, nempe D, in altero ſemicirculo CDA, eiuſdem circuli, ducanturq́ue <lb/>per puncta B, D, & </s> <s xml:id="echoid-s13784" xml:space="preserve">polum circuli AECF, qui ſit H, duo arcus circulorum <lb/> <anchor type="note" xlink:label="note-411-08a" xlink:href="note-411-08"/> maximorum IBK, DFS; </s> <s xml:id="echoid-s13785" xml:space="preserve">eruntq́ue anguli recti F, I, S, K. </s> <s xml:id="echoid-s13786" xml:space="preserve">Dico rurſus, vt eſt ſi-<lb/> <anchor type="note" xlink:label="note-411-09a" xlink:href="note-411-09"/> nus arcus AB, vel CB, ad ſinum arcus BI, vel BK, ita eſſe ſinum arcus AD, <lb/>vel CD, ad ſinum arcus DF, vel arcus, qui cum arcu FD, ſemicirculum per-<lb/>ficit ã puncto D, vſque ad punctum S, ſemicirculi AEC. </s> <s xml:id="echoid-s13787" xml:space="preserve">Nam arcus ab F, per <lb/>D, vſque ad S, ſemicirculus eſt, cum circuli AECF, DFS, ſe mutuo bifa-<lb/> <anchor type="note" xlink:label="note-411-10a" xlink:href="note-411-10"/> riam ſecentin F, S. </s> <s xml:id="echoid-s13788" xml:space="preserve">Sumatur enim arcui AD, arcus AG, æqualis, & </s> <s xml:id="echoid-s13789" xml:space="preserve">per G, <lb/> <anchor type="note" xlink:label="note-411-11a" xlink:href="note-411-11"/> & </s> <s xml:id="echoid-s13790" xml:space="preserve">polum circuli AECF, nempe per H, arcus maximi circuli ducatur LGM; <lb/></s> <s xml:id="echoid-s13791" xml:space="preserve"> <anchor type="note" xlink:label="note-411-12a" xlink:href="note-411-12"/> eruntq́ue anguli L, M, recti. </s> <s xml:id="echoid-s13792" xml:space="preserve">Quoniam igitur duo anguli A, L, trianguli AGL, <lb/> <anchor type="note" xlink:label="note-411-13a" xlink:href="note-411-13"/> duobus angulis A, F, trianguli ADF, æquales ſunt, (ſunt enim duo anguli <lb/>A, ad verticem æquales, & </s> <s xml:id="echoid-s13793" xml:space="preserve">anguli L, F, recti.) </s> <s xml:id="echoid-s13794" xml:space="preserve">ſuntq́ue latera AG, AD, rectos <lb/> <anchor type="note" xlink:label="note-411-14a" xlink:href="note-411-14"/> <pb o="400" file="412" n="412" rhead=""/> ſubtendentia angulos, per conſtructionem, æqualia; </s> <s xml:id="echoid-s13795" xml:space="preserve">erunt quoque arcus GL, <lb/> <anchor type="note" xlink:label="note-412-01a" xlink:href="note-412-01"/> DF, æquales, ac propterea & </s> <s xml:id="echoid-s13796" xml:space="preserve">eorum ſinus æquales erunt, necnon & </s> <s xml:id="echoid-s13797" xml:space="preserve">ſinus ar-<lb/>cuum æqualium AG, AD, erunt æquales. </s> <s xml:id="echoid-s13798" xml:space="preserve">Eadem ergo eſt proportio ſinus <lb/>arcus AG, ad ſinum arcus GL, quæ ſinus arcus AD, ad ſinum arcus DF: </s> <s xml:id="echoid-s13799" xml:space="preserve">Vt <lb/>autem ſinus arcus AG, ad ſinum arcus GL, ita demonſtratum eſt, eſſe ſinum <lb/>arcus AB, vel CB, ad ſinum arcus BI, vel BK, propterea quòd puncta B, G, <lb/>in eodem ſemicirculo ſumpta ſunt. </s> <s xml:id="echoid-s13800" xml:space="preserve">Igitur erit quoque, vt ſinus arcus AB, vel <lb/>CB, ad ſinum arcus BI, vel BK, ita ſinus arcus AD, ad ſinum arcus DF, &</s> <s xml:id="echoid-s13801" xml:space="preserve">c.</s> <s xml:id="echoid-s13802" xml:space="preserve"/> </p> <div xml:id="echoid-div1095" type="float" level="2" n="2"> <note position="right" xlink:label="note-411-08" xlink:href="note-411-08a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-411-09" xlink:href="note-411-09a" xml:space="preserve">15. 1. Theod.</note> <note position="right" xlink:label="note-411-10" xlink:href="note-411-10a" xml:space="preserve">11. 1. Theod.</note> <note position="right" xlink:label="note-411-11" xlink:href="note-411-11a" xml:space="preserve">1. huius.</note> <note position="right" xlink:label="note-411-12" xlink:href="note-411-12a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-411-13" xlink:href="note-411-13a" xml:space="preserve">15. 1. Theod.</note> <note position="right" xlink:label="note-411-14" xlink:href="note-411-14a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-412-01" xlink:href="note-412-01a" xml:space="preserve">21. huius.</note> </div> <p> <s xml:id="echoid-s13803" xml:space="preserve">POSTREMO ſumatur in ſemicirculo ABC, punctum B, & </s> <s xml:id="echoid-s13804" xml:space="preserve">in alterius <lb/>circuli ſemicirculo vtrouis <lb/>nempe in AEC, aliud pun <lb/>ctum L: </s> <s xml:id="echoid-s13805" xml:space="preserve">Et per B, & </s> <s xml:id="echoid-s13806" xml:space="preserve">polum <lb/> <anchor type="figure" xlink:label="fig-412-01a" xlink:href="fig-412-01"/> circuli AEC, arcus maxi-<lb/> <anchor type="note" xlink:label="note-412-02a" xlink:href="note-412-02"/> mi circuli ducatur IBK: <lb/></s> <s xml:id="echoid-s13807" xml:space="preserve">Item per L, & </s> <s xml:id="echoid-s13808" xml:space="preserve">per polum <lb/>circuli ABC, arcus LGM, <lb/>maximi circuli; </s> <s xml:id="echoid-s13809" xml:space="preserve">eruntq́ue <lb/>anguli I, G, recti. </s> <s xml:id="echoid-s13810" xml:space="preserve">Dico rur-<lb/> <anchor type="note" xlink:label="note-412-03a" xlink:href="note-412-03"/> ſus, vt eſt ſinus arcus AB, <lb/>ad ſinum arcus BI, ita eſſe <lb/>ſinum arcus AL, ad ſinum <lb/>arcus LG, &</s> <s xml:id="echoid-s13811" xml:space="preserve">c. </s> <s xml:id="echoid-s13812" xml:space="preserve">Per po-<lb/>los enim vtriusque circuli <lb/> <anchor type="note" xlink:label="note-412-04a" xlink:href="note-412-04"/> ABCD, AECF, arcus cir <lb/>culi maximi ducatur RE; <lb/></s> <s xml:id="echoid-s13813" xml:space="preserve">eruntq́ue anguli R, E, re-<lb/> <anchor type="note" xlink:label="note-412-05a" xlink:href="note-412-05"/> cti, diuidenturq́ue ſemicir-<lb/>culi ABC, AEC, bifa-<lb/> <anchor type="note" xlink:label="note-412-06a" xlink:href="note-412-06"/> riam in punctis R, E; </s> <s xml:id="echoid-s13814" xml:space="preserve">atque <lb/>adeo ſinus quadrantum AR, AE, æquales erunt; </s> <s xml:id="echoid-s13815" xml:space="preserve">Eademq́ue proportio erit <lb/> <anchor type="note" xlink:label="note-412-07a" xlink:href="note-412-07"/> ſinus arcus AR, ad ſinum arcus RE, quæ ſinus arcus AE, ad ſinum arcus ER. <lb/></s> <s xml:id="echoid-s13816" xml:space="preserve">Quoniam vero eſt, vt ſinus arcus AR, ad ſinũ arcus RE, ita ſinus arcus AB, <lb/>ad ſinum arcus BI, vt demonſtratum eſt; </s> <s xml:id="echoid-s13817" xml:space="preserve">(ſumpta ſunt enim duo puncta R, <lb/>B, in eodem ſemicirculo) erit quoque, vt ſinus arcus AE, ad ſinum arcus ER, <lb/>ita ſinus arcus AB, ad ſinum arcus BI: </s> <s xml:id="echoid-s13818" xml:space="preserve">Sed eadem ratione eſt, vt ſinus arcus <lb/>AE, ad ſinum arcus ER, ita ſinus arcus AL, ad ſinum arcus LG. </s> <s xml:id="echoid-s13819" xml:space="preserve">Igitur erit <lb/>quoque, vt ſinus arcus AB, ad ſinum arcus BI, ita ſinus arcus AL, ad ſinum <lb/>arcus LG, &</s> <s xml:id="echoid-s13820" xml:space="preserve">c. </s> <s xml:id="echoid-s13821" xml:space="preserve">Quòd ſi loco puncti L, ſumatur in altero ſemicirculo AFC, <lb/>eiuſdem circuli AECF, aliud punctum, nempe F, & </s> <s xml:id="echoid-s13822" xml:space="preserve">arcus FD, faciat angu-<lb/>lum D, rectum, erit adhuc, vt ſinus arcus AB, ad linum arcus BI, ita ſinus ar-<lb/>cus AF, ad ſinum arcus FD, &</s> <s xml:id="echoid-s13823" xml:space="preserve">c. </s> <s xml:id="echoid-s13824" xml:space="preserve">Vt enim proxime oſtendimus, vt ſinus ar-<lb/>cus AB, ad ſinum arcus BI, ita eſt arcus ſinus AL, ad ſinum arcus LG: </s> <s xml:id="echoid-s13825" xml:space="preserve">Vt <lb/>autem ſinus arcus AL, ad ſinum arcus LG ita demonſtratum eſt, eſſe ſinum <lb/>arcus AF, ad ſinum arcus FD, quòd puncta L, F, ſumantur in duobus ſemi-<lb/>circulis eiuſdem circuli. </s> <s xml:id="echoid-s13826" xml:space="preserve">Igitur erit quoque, vt ſinus arcus AB, ad ſinum ar-<lb/>cus BI, ita ſinus arcus AF, ad ſinum arcus FD: </s> <s xml:id="echoid-s13827" xml:space="preserve">Atque ita in vniuerſum vera <lb/>eſt propoſitio, quomodocunq; </s> <s xml:id="echoid-s13828" xml:space="preserve">duo puncta ſumãtur, quando circuli ABCD, <lb/>AECF, ſe mutuo ſecantad angulos non rectos.</s> <s xml:id="echoid-s13829" xml:space="preserve"/> </p> <div xml:id="echoid-div1096" type="float" level="2" n="3"> <figure xlink:label="fig-412-01" xlink:href="fig-412-01a"> <image file="412-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/412-01"/> </figure> <note position="left" xlink:label="note-412-02" xlink:href="note-412-02a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-412-03" xlink:href="note-412-03a" xml:space="preserve">25. 1 Theod.</note> <note position="left" xlink:label="note-412-04" xlink:href="note-412-04a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-412-05" xlink:href="note-412-05a" xml:space="preserve">15. 1. Theod.</note> <note position="left" xlink:label="note-412-06" xlink:href="note-412-06a" xml:space="preserve">9. 2. Theod.</note> <note position="left" xlink:label="note-412-07" xlink:href="note-412-07a" xml:space="preserve">7. quinti.</note> </div> <pb o="401" file="413" n="413" rhead=""/> <p> <s xml:id="echoid-s13830" xml:space="preserve">SED iam circuli ABCD, AECF, ſecent ſe mutuo ad angulos rectos in <lb/>punctis A, C; </s> <s xml:id="echoid-s13831" xml:space="preserve">ſitq́ue eorum communis ſectio recta AC. </s> <s xml:id="echoid-s13832" xml:space="preserve">Diuiſo autem v. </s> <s xml:id="echoid-s13833" xml:space="preserve">g. <lb/></s> <s xml:id="echoid-s13834" xml:space="preserve">ſemicirculo ABC, bifariam in H, vt ſint quadrantes AH, CH, ſumantur <lb/>duo puncta vtcunque B, G. </s> <s xml:id="echoid-s13835" xml:space="preserve">Dico ita eſſe rurſus <lb/>ſinum arcus AB, ad ſinum arcus, qui per B, ductus <lb/>rectos angulos facit cum circulo AECF, vt eſt <lb/> <anchor type="figure" xlink:label="fig-413-01a" xlink:href="fig-413-01"/> ſinus arcus AG, ad ſinum arcus, qui per G, ductus <lb/>cum circulo AECF, rectos facit angulos. </s> <s xml:id="echoid-s13836" xml:space="preserve">Quo-<lb/>niam enim circulus ABC, cum rectus ad circulum <lb/>AEC, ponatur, tranſit per polos circuli AEC, <lb/> <anchor type="note" xlink:label="note-413-01a" xlink:href="note-413-01"/> erit H, polus circuli AEC. </s> <s xml:id="echoid-s13837" xml:space="preserve">Quare arcus perpen-<lb/>diculares ad circulum AEC, per puncta B, G, du-<lb/>cti neceſſario per H, tranſibunt; </s> <s xml:id="echoid-s13838" xml:space="preserve">atque adeò arcus <lb/> <anchor type="note" xlink:label="note-413-02a" xlink:href="note-413-02"/> illi erunt BA, GA: </s> <s xml:id="echoid-s13839" xml:space="preserve">Perſpicuum autem eſt, vt eſt <lb/>ſinus arcus AB, ad ſinum arcus BA, ita eſſe ſinum <lb/>arcus AG, ad ſinum arcus GA, cum vtrobique <lb/>ſit proportro æqualitatis, ſeu identitatis: </s> <s xml:id="echoid-s13840" xml:space="preserve">Eſt e-<lb/>nim idem ſinus arcus AB, & </s> <s xml:id="echoid-s13841" xml:space="preserve">arcus BA, necnon <lb/>idem ſinus arcus AG, & </s> <s xml:id="echoid-s13842" xml:space="preserve">arcus GA.</s> <s xml:id="echoid-s13843" xml:space="preserve"/> </p> <div xml:id="echoid-div1097" type="float" level="2" n="4"> <figure xlink:label="fig-413-01" xlink:href="fig-413-01a"> <image file="413-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/413-01"/> </figure> <note position="right" xlink:label="note-413-01" xlink:href="note-413-01a" xml:space="preserve">13. 1. Theod. <lb/>Coroll. 16. <lb/>1. Theod.</note> <note position="right" xlink:label="note-413-02" xlink:href="note-413-02a" xml:space="preserve">13. 1 Theod.</note> </div> <p> <s xml:id="echoid-s13844" xml:space="preserve">QVOD ſi alterum punctorum ſit H, polus circuli AEC, erit quicun-<lb/>que arcus ex H, ductus, qualis eſt HE, perpendicularis, ad AEC, atque adeò <lb/> <anchor type="note" xlink:label="note-413-03a" xlink:href="note-413-03"/> quadrans. </s> <s xml:id="echoid-s13845" xml:space="preserve">Rurſus igitur manifeſtum eſt, ita eſſe ſinum arcus AB, ad ſinum ar-<lb/> <anchor type="note" xlink:label="note-413-04a" xlink:href="note-413-04"/> cus BA, vt eſt ſinus arcus AH, ad ſinum arcus HE, vel HA; </s> <s xml:id="echoid-s13846" xml:space="preserve">cum vtrobique <lb/>quoque ſit æ qualitatis proportio, &</s> <s xml:id="echoid-s13847" xml:space="preserve">c. </s> <s xml:id="echoid-s13848" xml:space="preserve">Si duo ergo circuli maximi in ſphæra <lb/>ſe mutuo ſecent, &</s> <s xml:id="echoid-s13849" xml:space="preserve">c. </s> <s xml:id="echoid-s13850" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s13851" xml:space="preserve"/> </p> <div xml:id="echoid-div1098" type="float" level="2" n="5"> <note position="right" xlink:label="note-413-03" xlink:href="note-413-03a" xml:space="preserve">15. 1 Theod.</note> <note position="right" xlink:label="note-413-04" xlink:href="note-413-04a" xml:space="preserve">Coroll. 16. <lb/>1. Theod.</note> </div> </div> <div xml:id="echoid-div1100" type="section" level="1" n="540"> <head xml:id="echoid-head575" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s13852" xml:space="preserve">PERSPICVVM eſt ex demonſtratis: </s> <s xml:id="echoid-s13853" xml:space="preserve">Si duo circuli ſe mutuo ſecent, & </s> <s xml:id="echoid-s13854" xml:space="preserve">in vno <lb/>eorum ex duobus punctis vtcunq; </s> <s xml:id="echoid-s13855" xml:space="preserve">aſſumptis ducantur ad alterius circuli planum duæ <lb/>lineæ rectæ perpendiculares; </s> <s xml:id="echoid-s13856" xml:space="preserve">ita eſſe ſinum rectum arcus intercepti inter vnum illo-<lb/>rum punctorum, & </s> <s xml:id="echoid-s13857" xml:space="preserve">alterutram circulorum ſectionem, ad perpendicularem ex illo <lb/>puncto in planum alterius circuli demiſſam, vt eſt ſinus rectus arcus inter alterum <lb/>punctum, & </s> <s xml:id="echoid-s13858" xml:space="preserve">alterutram ſectionem circulorum interiecti, ad perpendicularem ab hoc <lb/>altero puncto in planum alterius circuli demiſſam. </s> <s xml:id="echoid-s13859" xml:space="preserve">Nam in priori figura buius pro-<lb/>poſ. </s> <s xml:id="echoid-s13860" xml:space="preserve">oſtenſum eſt, ita eſſe ſinum arcus <emph style="sc">Ab</emph>, vel <emph style="sc">Cb</emph>, ad BP, ſinum rectum arcus BI, <lb/>vt eſt ſinus arcus AG, vel CG, ad GQ, ſinum rectum arcus GL. </s> <s xml:id="echoid-s13861" xml:space="preserve">Cum ergo ſinus BP, <lb/>GQ, ſint perpendiculares ex punctis B, G, in planum circuli <emph style="sc">AECf</emph>, demiſſæ, pa-<lb/>tet propoſitum. </s> <s xml:id="echoid-s13862" xml:space="preserve">Quòd ſi vnum punctorum acceptum ſit B, ex vna parte ſectionis A, <lb/>& </s> <s xml:id="echoid-s13863" xml:space="preserve">alterum punctum acceptum ſit D, ex altera parte ſectionis <emph style="sc">A</emph>, in eodem circulo; <lb/></s> <s xml:id="echoid-s13864" xml:space="preserve">erit nibilominus ita ſinus arcus AB, ad perpendicularem BP, ex B, demiſſam in pla-<lb/>num alterius circuli AECF, vt ſinus arcus AD, ad perpendicularem, quæ ex D, in <lb/>planum alterius circuli AECF, demitteretur: </s> <s xml:id="echoid-s13865" xml:space="preserve">propterea quod oſtenſum eſt, ita eſſe <lb/>ſinum arcus AB, ad ſinum arcus BI, vt @ſt ſinus arcus AD, ad ſinum arcus DF; </s> <s xml:id="echoid-s13866" xml:space="preserve">qui <lb/>quidem ſinus arcuum BI, DF, ſunt perpendiculares ex punctis B, D, in planum <lb/>circuli AECF, cadentes, vt ex demonſtratis in hac propoſ. </s> <s xml:id="echoid-s13867" xml:space="preserve">liquido conſtat. </s> <s xml:id="echoid-s13868" xml:space="preserve">Idem <lb/>perſpicitur in figura poſteriori; </s> <s xml:id="echoid-s13869" xml:space="preserve">cum ibi etiam ſit, vt ſinus arcus AB, ad perpendicu- <pb o="402" file="414" n="414" rhead=""/> larem ex B, in planum circuli <emph style="sc">AECf</emph>, demiſſam, ita ſinus arcus AG, ad perpendi-<lb/>cularem ex G, in planum circuli AECF, demiſſam: </s> <s xml:id="echoid-s13870" xml:space="preserve">propterea quòd ſinus arcuum <lb/>AB, AG, ſuntipſæmet perpendiculares ex B, G, in planum circuli AECF, demiſſæ <lb/>cadentes in rectam AC, communem circulorum ſectionem, vt patet.</s> <s xml:id="echoid-s13871" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">38. vndee.</note> <p style="it"> <s xml:id="echoid-s13872" xml:space="preserve">HINC facile demonſtrari poterunt ſequentia theoremata, quorum nonnulla pl@ <lb/>rimum ad ſphæricorum triangulerum calculum conducunt. </s> <s xml:id="echoid-s13873" xml:space="preserve">Primum autem ac ſecun-<lb/>dum ſunt duo Theoremata Ptolemæi Cyclica in primo lib. </s> <s xml:id="echoid-s13874" xml:space="preserve">Almageſti, ſed multo bre-<lb/>uius, ac facilius demonſtrata ex ijs, quæ in hoc ſcholio oſtenſa ſunt. </s> <s xml:id="echoid-s13875" xml:space="preserve">Vnde omittend@ <lb/>@on videbantur, licet eorum vſus in hiſce triangulis non appareat.</s> <s xml:id="echoid-s13876" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1101" type="section" level="1" n="541"> <head xml:id="echoid-head576" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s13877" xml:space="preserve">SI in ſphæræ ſuperficie ab vno puncto duo arcus maximorum <lb/>circulorum educantur, quorum vterque ſemicitculo ſit minor, & </s> <s xml:id="echoid-s13878" xml:space="preserve">ab <lb/>eorum terminis in ipſos reflectantur alij duo arcus maximorum cir-<lb/>culorum ſe inter duos illos priores arcus interſecantes: </s> <s xml:id="echoid-s13879" xml:space="preserve">proportio, <lb/>quam ſinus ſegmẽti vnius eductorum arcuum inter terminum eius, <lb/>& </s> <s xml:id="echoid-s13880" xml:space="preserve">arcum reflexum habet ad ſinum alterius ſegmenti eiuſdem arcus <lb/>educti, componitur ex proportione, quam ſinus ſegmenti arcus re-<lb/>flexi inter eundem terminum, & </s> <s xml:id="echoid-s13881" xml:space="preserve">alterum arcum reflexum habet ad <lb/>ſinum alterius ſegmenti eiuſdem arcus reflexi, & </s> <s xml:id="echoid-s13882" xml:space="preserve">ex proportione, <lb/>quam ſinus ſegmenti alterius eductorum arcuum inter eius termi-<lb/>num, & </s> <s xml:id="echoid-s13883" xml:space="preserve">arcum reflexum habet ad ſinũ totius eiuſdem arcus educti.</s> <s xml:id="echoid-s13884" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s13885" xml:space="preserve">E X puncto A, in ſuperficie ſpharæ educantur duo arcus AB, AC, ſemicirculis <lb/>minores, & </s> <s xml:id="echoid-s13886" xml:space="preserve">à terminis B, C, reflectantur adipſos duo arcus BD, CE, ſe interſe-<lb/>cantes in F.</s> <s xml:id="echoid-s13887" xml:space="preserve"> Dico proportionem ſinus arcus BE, ad ſinum arcus EA, componi ex pro-<lb/>portione ſinus arcus <emph style="sc">Bf</emph>, ad ſinum arcus <emph style="sc">F</emph>D, & </s> <s xml:id="echoid-s13888" xml:space="preserve">ex proportione ſinus arcus CD, ad <lb/>ſinum arcus CA. </s> <s xml:id="echoid-s13889" xml:space="preserve">Ductis enim ex punctis B, A, D, ad planum <lb/> <anchor type="figure" xlink:label="fig-414-01a" xlink:href="fig-414-01"/> circuli CE, tribus perpendicularibus BG, AH, DI; </s> <s xml:id="echoid-s13890" xml:space="preserve">quoniam <lb/>duo circuli AB, CE, ſe mutuo ſecant in E, & </s> <s xml:id="echoid-s13891" xml:space="preserve">ex punctis B, A, <lb/>in planum circuli <emph style="sc">Ce</emph>, demiſſæ ſunt perpendiculares <emph style="sc">B</emph>G, AH; <lb/></s> <s xml:id="echoid-s13892" xml:space="preserve">erit vt ſinus arcus EB, ad ſinum arcus EA, ita recta BG, ad <lb/> <anchor type="note" xlink:label="note-414-02a" xlink:href="note-414-02"/> rectam AH: </s> <s xml:id="echoid-s13893" xml:space="preserve">Item quoniam duo circuli BD, <emph style="sc">C</emph>E, ſe mutuo <lb/>ſecant in <emph style="sc">F</emph>, & </s> <s xml:id="echoid-s13894" xml:space="preserve">ex punctis B,D, in planum circuli CE, deductæ <lb/>ſunt perpendiculares BG, DI; </s> <s xml:id="echoid-s13895" xml:space="preserve">erit eadem ratione, vt ſinus ar-<lb/>cus <emph style="sc">FB</emph>, ad ſinum arcus FD, ita recta BG, ad rectam DI: </s> <s xml:id="echoid-s13896" xml:space="preserve">De-<lb/>nique quia duo circuli AC, <emph style="sc">CE</emph>, ſe interſecant in C, & </s> <s xml:id="echoid-s13897" xml:space="preserve">ex <lb/>punctis D, <emph style="sc">A,</emph> in planum circuli <emph style="sc">CE</emph>, demiſſæ ſunt perpendi-<lb/>culares rectæ lineæ DI, AH; </s> <s xml:id="echoid-s13898" xml:space="preserve">erit ſimiliter, vt ſinus arcus <lb/>CD, ad ſinum arcus CA, ita recta DI, ad rectam AH. </s> <s xml:id="echoid-s13899" xml:space="preserve">Pro-<lb/>p@rtio autem recta BG, ad rectam AH, (poſita media linea DI.) </s> <s xml:id="echoid-s13900" xml:space="preserve">componitur ex pro-<lb/>portione rectæ BG, ad rectam DI, & </s> <s xml:id="echoid-s13901" xml:space="preserve">ex proportione rectæ DI, ad rectam AH. </s> <s xml:id="echoid-s13902" xml:space="preserve">Igi-<lb/>tur & </s> <s xml:id="echoid-s13903" xml:space="preserve">proportio ſinus arcus <emph style="sc">BE</emph>, ad ſinum arcus EA, (quæ eadem eſt, quæ propor-<lb/>tio BG, ad AH.) </s> <s xml:id="echoid-s13904" xml:space="preserve">componetur ex proportione ſinus arcus BF, ad ſinum arcus FD, <lb/>(quæ eadem eſt, quæ proportio BG, ad DI.) </s> <s xml:id="echoid-s13905" xml:space="preserve">& </s> <s xml:id="echoid-s13906" xml:space="preserve">ex proportione ſinus arcus CD, ad <lb/>@inum arcus AC, (quæ eadem eſt, quæ DI, ad AH.) </s> <s xml:id="echoid-s13907" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s13908" xml:space="preserve"/> </p> <div xml:id="echoid-div1101" type="float" level="2" n="1"> <figure xlink:label="fig-414-01" xlink:href="fig-414-01a"> <image file="414-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/414-01"/> </figure> <note position="left" xlink:label="note-414-02" xlink:href="note-414-02a" xml:space="preserve">Schol. 40. <lb/>huius. & <lb/>permutan. <lb/>do.</note> </div> <pb o="403" file="415" n="415" rhead=""/> </div> <div xml:id="echoid-div1103" type="section" level="1" n="542"> <head xml:id="echoid-head577" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s13909" xml:space="preserve">IISDEM poſitis, proportio ſinus vnius arcuum eductorum ad <lb/>ſinum ſegmenti eiuſdem arcus inter punctum eductionis, & </s> <s xml:id="echoid-s13910" xml:space="preserve">arcum <lb/>reflexum, componitur ex proportione ſinus arcus reflexi à termino <lb/>dictiarcus ad ſinum ſegmenti eiuſdem arcus reflexi inter alterum ar-<lb/>cum eductum, & </s> <s xml:id="echoid-s13911" xml:space="preserve">alterum arcum reflexum, & </s> <s xml:id="echoid-s13912" xml:space="preserve">ex proportione ſi-<lb/>nus ſegmenti alterius arcus reflexi inter terminum alterius arcus <lb/>educti, & </s> <s xml:id="echoid-s13913" xml:space="preserve">priorem arcum reflexum ad ſinum totius poſterioris ar-<lb/>cus reflexi.</s> <s xml:id="echoid-s13914" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s13915" xml:space="preserve">HOC eſt, proportio ſinus arcus AB, ad ſinum arcus AE, componitur ex pr@-<lb/>portione ſinus arcus BD, ad ſinum arcus DF, & </s> <s xml:id="echoid-s13916" xml:space="preserve">ex proportione ſinus arcus CF, ad <lb/>ſinum arcus <emph style="sc">Ce</emph>. </s> <s xml:id="echoid-s13917" xml:space="preserve">Ductis enim ex punctis B, E, <emph style="sc">F</emph>, ad planum circuli AC, tribus per-<lb/>pendicularibus BG, EH, FI; </s> <s xml:id="echoid-s13918" xml:space="preserve">quoniam duo circuli <lb/> <anchor type="figure" xlink:label="fig-415-01a" xlink:href="fig-415-01"/> AB, AC, ſe mutuo ſecant in <emph style="sc">A</emph>, & </s> <s xml:id="echoid-s13919" xml:space="preserve">ex punctis <emph style="sc">B</emph>, <lb/>E, in planum circuli AC, demiſſæ ſunt perpendicu-<lb/>lares <emph style="sc">BG</emph>, EH; </s> <s xml:id="echoid-s13920" xml:space="preserve">erit, vt ſinus arcus <emph style="sc">AB</emph>, ad ſinum <lb/> <anchor type="note" xlink:label="note-415-01a" xlink:href="note-415-01"/> arcus AE, ita recta <emph style="sc">BG</emph>, ad rectam EH: </s> <s xml:id="echoid-s13921" xml:space="preserve">Item quiæ <lb/>duo circuli <emph style="sc">B</emph>D, AC, ſe interſecant in D, & </s> <s xml:id="echoid-s13922" xml:space="preserve">ex <lb/>punctis <emph style="sc">B, F,</emph> in planũ circuli AC, deductæ ſunt per-<lb/>pendiculares <emph style="sc">BG</emph>, FI; </s> <s xml:id="echoid-s13923" xml:space="preserve">erit pari ratione, vt ſinus <lb/>arcus <emph style="sc">Db</emph>, ad ſinum arcus <emph style="sc">Df</emph>, ita recta <emph style="sc">B</emph>G, ad re-<lb/>ctam <emph style="sc">FI</emph>: </s> <s xml:id="echoid-s13924" xml:space="preserve">Denique quoniam duo circuli AC, CE, ſe in C, interſecant, & </s> <s xml:id="echoid-s13925" xml:space="preserve">ex pun-<lb/>ctis F, E, in planum circuli AC, demiſſæ ſunt perpendiculares <emph style="sc">F</emph>I, EH; </s> <s xml:id="echoid-s13926" xml:space="preserve">erit eadem <lb/>argumentatione, vt ſinus arcus CF, ad ſinum arcus CE, ita recta FI, ad rectam <lb/>EH. </s> <s xml:id="echoid-s13927" xml:space="preserve">Componitur autem proportio rectæ <emph style="sc">BG</emph>, ad rectam EH, (poſita media lineæ <lb/><emph style="sc">F</emph>I.) </s> <s xml:id="echoid-s13928" xml:space="preserve">ex proportione rectæ <emph style="sc">BG</emph>, ad rectam FI, & </s> <s xml:id="echoid-s13929" xml:space="preserve">exproportione rectæ FI, ad rectam <lb/>EH. </s> <s xml:id="echoid-s13930" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s13931" xml:space="preserve">proportio ſinus arcus <emph style="sc">Ab</emph>, ad ſinum arcus AE, (quæ eadem eſt, quæ <lb/><emph style="sc">BG</emph>, ad EH.) </s> <s xml:id="echoid-s13932" xml:space="preserve">componetur ex proportione ſinus arcus BD, ad ſinum arcus DF, (quæ <lb/>@adem eſt, quæ BG, ad FI,) & </s> <s xml:id="echoid-s13933" xml:space="preserve">ex proportione ſinus arcus CF, ad ſinum arcus CE, <lb/>(quæ cadem eſt, quæ FI, ad EH.) </s> <s xml:id="echoid-s13934" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s13935" xml:space="preserve"/> </p> <div xml:id="echoid-div1103" type="float" level="2" n="1"> <figure xlink:label="fig-415-01" xlink:href="fig-415-01a"> <image file="415-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/415-01"/> </figure> <note position="right" xlink:label="note-415-01" xlink:href="note-415-01a" xml:space="preserve">Schol. 40. <lb/>huius. & <lb/>permutan-<lb/>do.</note> </div> </div> <div xml:id="echoid-div1105" type="section" level="1" n="543"> <head xml:id="echoid-head578" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s13936" xml:space="preserve">IISDEM poſitis, proportio ſinus vnius arcuum eductorum ad <lb/>ſinum ſegmenti eiuſdem arcus inter eius terminum, & </s> <s xml:id="echoid-s13937" xml:space="preserve">arcum refle-<lb/>xum, componitur ex proportione ſinus ſegmenti alterius arcus edu-<lb/>cti inter punctum eductionis, & </s> <s xml:id="echoid-s13938" xml:space="preserve">arcum reflexum ad ſinum reliqui <lb/>ſegmenti, & </s> <s xml:id="echoid-s13939" xml:space="preserve">ex proportione ſinus ſegmenti arcus reflexi à termino <lb/>poſterioris arcus educti inter terminum, & </s> <s xml:id="echoid-s13940" xml:space="preserve">alterum arcum reflexum <lb/>ad ſinum reliqui ſegmenti eiuſdem arcus reflexi.</s> <s xml:id="echoid-s13941" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s13942" xml:space="preserve">HOC eſt, (repetita figura primi theorematis) proportio ſinus arcus AC, ad ſi-<lb/>num arcus CD, componitur exproportione ſinus arcus AE, ad ſinum arcus <emph style="sc">Eb</emph>, &</s> <s xml:id="echoid-s13943" xml:space="preserve"> <pb o="404" file="416" n="416" rhead=""/> ex proportione ſinus arcus <emph style="sc">BF</emph>, ad ſinum arcus FD. </s> <s xml:id="echoid-s13944" xml:space="preserve">Quoniam enim duo circuli AC, <lb/><emph style="sc">Ce</emph>, ſe interſecant in C, & </s> <s xml:id="echoid-s13945" xml:space="preserve">ex punctis A, D, demißæ ſunt perpendiculares AH, DI, <lb/>ad planum circuli CE; </s> <s xml:id="echoid-s13946" xml:space="preserve">erit, vt ſinus arcus CA, ad ſinum arcus CD, ita recta AH, <lb/> <anchor type="note" xlink:label="note-416-01a" xlink:href="note-416-01"/> ad rectam DI: </s> <s xml:id="echoid-s13947" xml:space="preserve">Item quoniam duo circuli <emph style="sc">Ab</emph>, CE, ſe mutuo <lb/>ſecant in E, & </s> <s xml:id="echoid-s13948" xml:space="preserve">expunctis <emph style="sc">A, B</emph>, in planum circuli CE, dedu-<lb/> <anchor type="figure" xlink:label="fig-416-01a" xlink:href="fig-416-01"/> ctæ ſunt perpendiculares AH, BG; </s> <s xml:id="echoid-s13949" xml:space="preserve">erit ſimili modo, vt ſinus <lb/>arcus EA, ad ſinum arcus <emph style="sc">Eb</emph>, ita recta AH, ad rectam <emph style="sc">BG</emph>. <lb/></s> <s xml:id="echoid-s13950" xml:space="preserve">Denique quia duo circuli BD, CE, ſe mutuo ſecant in F, & </s> <s xml:id="echoid-s13951" xml:space="preserve"><lb/>ex punctis <emph style="sc">B</emph>, D, ad planum circuli CE, ductæ ſunt perpendi-<lb/>culares <emph style="sc">BG</emph>, DI; </s> <s xml:id="echoid-s13952" xml:space="preserve">erit eadem ratione, vt ſinus arcus <emph style="sc">FB</emph>, ad ſi-<lb/>num arcus FD, ita recta <emph style="sc">BG</emph>, ad rectam DI. </s> <s xml:id="echoid-s13953" xml:space="preserve">Componitur au-<lb/>tem proportio rectæ AH, ad rectam DI, (poſita media lineæ <lb/><emph style="sc">BG</emph>) ex proportione rectæ AH, ad rectam <emph style="sc">BG</emph>, & </s> <s xml:id="echoid-s13954" xml:space="preserve">ex propor <lb/>tione rectæ <emph style="sc">BG</emph>, ad rectam DI. </s> <s xml:id="echoid-s13955" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s13956" xml:space="preserve">proportio ſinus ar-<lb/>cus AC, ad ſinum arcus CD, (quæ eadem eſt, quæ AH, ad <lb/>DI.) </s> <s xml:id="echoid-s13957" xml:space="preserve">componetur ex proportione ſinus arcus AE, ad ſinum ar-<lb/>cus <emph style="sc">EB</emph>, (quæ eadem eſt, quæ AH, ad <emph style="sc">BG</emph>.) </s> <s xml:id="echoid-s13958" xml:space="preserve">& </s> <s xml:id="echoid-s13959" xml:space="preserve">ex proportione ſinus arcus <emph style="sc">BF</emph>, ad <lb/>ſinum arcus FD, (quæ eadem eſt, quæ <emph style="sc">BG</emph>, ad DI.) </s> <s xml:id="echoid-s13960" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s13961" xml:space="preserve"/> </p> <div xml:id="echoid-div1105" type="float" level="2" n="1"> <note position="left" xlink:label="note-416-01" xlink:href="note-416-01a" xml:space="preserve">Schol. 40. <lb/>huius. & <lb/>permutan-<lb/>do.</note> <figure xlink:label="fig-416-01" xlink:href="fig-416-01a"> <image file="416-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/416-01"/> </figure> </div> </div> <div xml:id="echoid-div1107" type="section" level="1" n="544"> <head xml:id="echoid-head579" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s13962" xml:space="preserve">IISDEM poſitis, proportio ſinus ſegmenti vnius arcuum re-<lb/>flexorum inter terminum arcus educti, & </s> <s xml:id="echoid-s13963" xml:space="preserve">alterum arcum reflexum <lb/>ad ſinum reliqui ſegmenti, componitur ex proportione ſinus ſegmen <lb/>ti vnius arcuum eductorum inter eundem terminum, & </s> <s xml:id="echoid-s13964" xml:space="preserve">alterum ar-<lb/>cum reflexum ad ſinum reliqui ſegmenti, & </s> <s xml:id="echoid-s13965" xml:space="preserve">ex proportione ſinus al-<lb/>terius arcus educti ad ſinum ſegmenti illius inter terminum, & </s> <s xml:id="echoid-s13966" xml:space="preserve">ar-<lb/>cum reflexum.</s> <s xml:id="echoid-s13967" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s13968" xml:space="preserve">HOC eſt, (repetita eadem figura primi theorematis) proportio ſinus arcus BF, <lb/>ad ſinum arcus FD, componitur ex proportione ſinus arcus <emph style="sc">B</emph>E, ad ſinum arcus EA, <lb/>& </s> <s xml:id="echoid-s13969" xml:space="preserve">ex proportione ſinus arcus AC, ad ſinum arcus CD. </s> <s xml:id="echoid-s13970" xml:space="preserve">Cum enim duo circuli BD, <lb/>CE, ſe mutuo ſecent in F, & </s> <s xml:id="echoid-s13971" xml:space="preserve">ex BD, ad planum circuli CE, perpendiculares BG, <lb/>DI, ſint demiſſæ; </s> <s xml:id="echoid-s13972" xml:space="preserve">erit, vt ſinus arcus FB, ad ſinum arcus FD, ita recta BG, ad re-<lb/> <anchor type="note" xlink:label="note-416-02a" xlink:href="note-416-02"/> ctam DI: </s> <s xml:id="echoid-s13973" xml:space="preserve">Quia item duo circuli BA, CE, ſe mutuo ſecantin E, & </s> <s xml:id="echoid-s13974" xml:space="preserve">ex punctis B, A, <lb/>ad circulum CE, perpendiculares ductæ ſunt BG, AH; </s> <s xml:id="echoid-s13975" xml:space="preserve">erit quoque, vt ſinus arcus <lb/>EB, ad ſinum arcus <emph style="sc">E</emph>A, ita recta BG, ad rectam AH: </s> <s xml:id="echoid-s13976" xml:space="preserve">Denique quia duo circuli <lb/>AC, CE, ſeſe in C, ſecant, & </s> <s xml:id="echoid-s13977" xml:space="preserve">ex punctis A, D, ad planum circuli CE, demiſſæ <lb/>ſunt perpendiculares AH, DI; </s> <s xml:id="echoid-s13978" xml:space="preserve">erit ſimiliter, vt ſinus arcus CA, ad ſinum arcus <lb/>CD, ita recta AH, ad rectam DI. </s> <s xml:id="echoid-s13979" xml:space="preserve">Proportio autem rectæ BG, ad rectam DI, <lb/>(poſita media linea AH) componitur exproportione rectæ BG, ad rectam AH, & </s> <s xml:id="echoid-s13980" xml:space="preserve"><lb/>ex proportione rectæ AH, ad rectam DI. </s> <s xml:id="echoid-s13981" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s13982" xml:space="preserve">proportio ſinus arcus BF, ad <lb/>ſinum arcus FD, (quæ eadem eſt, quæ BG, ad DI.) </s> <s xml:id="echoid-s13983" xml:space="preserve">componetur ex proportione ſi-<lb/>nus arcus BE, ad ſinum arcus EA, (quæ eadem eſt, quæ BG, ad AH.) </s> <s xml:id="echoid-s13984" xml:space="preserve">& </s> <s xml:id="echoid-s13985" xml:space="preserve">ex pro-<lb/>portione ſinus arcus AC, ad ſinum arcus CD, (quæ eadem eſt, quæ AH, ad DI.) <lb/></s> <s xml:id="echoid-s13986" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s13987" xml:space="preserve"/> </p> <div xml:id="echoid-div1107" type="float" level="2" n="1"> <note position="left" xlink:label="note-416-02" xlink:href="note-416-02a" xml:space="preserve">Schol. 40. <lb/>huius. & <lb/>permutan. <lb/>do.</note> </div> <pb o="405" file="417" n="417" rhead=""/> </div> <div xml:id="echoid-div1109" type="section" level="1" n="545"> <head xml:id="echoid-head580" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s13988" xml:space="preserve">IISDEM poſitis, proportio ſinus vnius arcuum reflexorum <lb/>ad ſinum ſegmenti eiuſdem inter terminum arcus educti, & </s> <s xml:id="echoid-s13989" xml:space="preserve">alterum <lb/>arcum reflexum, componitur ex proportione ſinus ſegmenti alte-<lb/>rius arcus educti inter punctum eductionis, & </s> <s xml:id="echoid-s13990" xml:space="preserve">arcum reflexum ad <lb/>ſinum totius arcus educti; </s> <s xml:id="echoid-s13991" xml:space="preserve">& </s> <s xml:id="echoid-s13992" xml:space="preserve">ex proportione ſinus alterius arcus re-<lb/>flexi ad ſinum ſegmenti eiuſdem inter priorem arcum eductum & </s> <s xml:id="echoid-s13993" xml:space="preserve"><lb/>priorem arcum reflexum.</s> <s xml:id="echoid-s13994" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s13995" xml:space="preserve">HOC eſt, (repetita figura ſecundi theorematis) proportio ſinus arcus CE, ad <lb/>ſinum arcus CF, componitur ex proportione ſinus arcus AE, ad ſinum arcus AB, <lb/>& </s> <s xml:id="echoid-s13996" xml:space="preserve">exproportione ſinus arcus BD, ad ſinum arcus DF. </s> <s xml:id="echoid-s13997" xml:space="preserve">Nam cum duo circuli AC, <lb/>CE, ſe in C, mutuò ſecent, & </s> <s xml:id="echoid-s13998" xml:space="preserve">ex punctis E, F, ad <lb/> <anchor type="figure" xlink:label="fig-417-01a" xlink:href="fig-417-01"/> planum circuli AC, ductæ ſint perpendiculares EH, <lb/>FI, erit vt ſinus arcus CE, ad ſinum arcus CF, <lb/> <anchor type="note" xlink:label="note-417-01a" xlink:href="note-417-01"/> ita recta EH, ad rectam FI: </s> <s xml:id="echoid-s13999" xml:space="preserve">Item cum duo circuli <lb/>AB, AC, ſe interſecent in A, & </s> <s xml:id="echoid-s14000" xml:space="preserve">ex punctis E, B, <lb/>ad planum circuli AC, cadant perpendiculares <lb/>EH, BG; </s> <s xml:id="echoid-s14001" xml:space="preserve">erit etiam, vt ſinus arcus AE, ad ſinum <lb/>arcus <emph style="sc">Ab</emph>, ita recta EH, ad rectam <emph style="sc">BG</emph>: </s> <s xml:id="echoid-s14002" xml:space="preserve">Quia de-<lb/>nique duo circuli AC, <emph style="sc">B</emph>D, ſe mutuo ſecant in D, <lb/>& </s> <s xml:id="echoid-s14003" xml:space="preserve">expunctis <emph style="sc">B</emph>, F, ad planum circuli AC, demiſſæ ſunt perpendiculares <emph style="sc">BG</emph>, FI; <lb/></s> <s xml:id="echoid-s14004" xml:space="preserve">erit pari ratione, vt ſinus arcus <emph style="sc">De</emph>, ad ſinum arcus DF, ita recta <emph style="sc">BG</emph>, ad rectam FI. </s> <s xml:id="echoid-s14005" xml:space="preserve"><lb/>Componitur autem proportio rectæ EH, ad rectam FI, (poſita media linea <emph style="sc">BG</emph>,) <lb/>ex proportione rectæ EH, ad rectam <emph style="sc">BG</emph>, & </s> <s xml:id="echoid-s14006" xml:space="preserve">ex proportione rectæ <emph style="sc">BG</emph>, ad rectam <lb/>FI. </s> <s xml:id="echoid-s14007" xml:space="preserve">Igitur proportio quoque ſinus arcus CE, ad ſinum arcus CF, (quæ eadem eſt, <lb/>quæ EH, ad FI.) </s> <s xml:id="echoid-s14008" xml:space="preserve">componetur ex proportione ſinus arcus AE, ad ſinum arcus <emph style="sc">Ab</emph>, <lb/>(quæ eadem eſt, quæ EH, ad <emph style="sc">BG</emph>.) </s> <s xml:id="echoid-s14009" xml:space="preserve">& </s> <s xml:id="echoid-s14010" xml:space="preserve">ex proportione ſinus arcus <emph style="sc">B</emph>D, ad ſinum arcus <lb/>DF, (quæ eadem eſt, quæ <emph style="sc">BG</emph>, ad FI.) </s> <s xml:id="echoid-s14011" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s14012" xml:space="preserve"/> </p> <div xml:id="echoid-div1109" type="float" level="2" n="1"> <figure xlink:label="fig-417-01" xlink:href="fig-417-01a"> <image file="417-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/417-01"/> </figure> <note position="right" xlink:label="note-417-01" xlink:href="note-417-01a" xml:space="preserve">Schol. 40. <lb/>huius. & <lb/>permutan-<lb/>do.</note> </div> </div> <div xml:id="echoid-div1111" type="section" level="1" n="546"> <head xml:id="echoid-head581" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s14013" xml:space="preserve">SI in ſphæræ ſuperficie duo maximi circuli ſe mutuò non ad an-<lb/>gulos rectos ſecent, & </s> <s xml:id="echoid-s14014" xml:space="preserve">à duobus punctis in vno aſſumptis ad alterum <lb/>circulum ducantur duo arcus perpendiculares: </s> <s xml:id="echoid-s14015" xml:space="preserve">Erit, vt ſinus arcus <lb/>inter punctum interſectionis, & </s> <s xml:id="echoid-s14016" xml:space="preserve">alterutrum angulorum rectorum <lb/>intercepti ad tangentem illius arcus perpendicularis, ita ſinus arcus <lb/>inter punctum interſectionis, & </s> <s xml:id="echoid-s14017" xml:space="preserve">alterum angulum rectum interie-<lb/>cti ad tangentem alterius huius arcus perpendicularis.</s> <s xml:id="echoid-s14018" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14019" xml:space="preserve">DVO circuli maximi <emph style="sc">Ab</emph>, AC, ſe mutuo ſecent in A, non ad angulos rectos, & </s> <s xml:id="echoid-s14020" xml:space="preserve"><lb/>ex punctis C, E, in circulo AC, aſſumptis ad circulum <emph style="sc">Ab</emph>, ducantur arcus perpen-<lb/>diculares <emph style="sc">GB</emph>, ED. </s> <s xml:id="echoid-s14021" xml:space="preserve">Dico ita eſſe ſinum arcus <emph style="sc">AB</emph>, ad tangentem arcus <emph style="sc">CB</emph>, vt eſt <lb/>ſinus arcus AD, ad tangentem arcus ED. </s> <s xml:id="echoid-s14022" xml:space="preserve">Productis enim arcubus <emph style="sc">BC</emph>, DE, donec <lb/>coëant in F, erunt <emph style="sc">BF</emph>, DF, quadrantes. </s> <s xml:id="echoid-s14023" xml:space="preserve">Quoniam vero à puncto <emph style="sc">B</emph>, duo arcus ma-<lb/> <anchor type="note" xlink:label="note-417-02a" xlink:href="note-417-02"/> <pb o="406" file="418" n="418" rhead=""/> æimorum circulorum <emph style="sc">BA, BF</emph>, educuntur, ab eorumq́; </s> <s xml:id="echoid-s14024" xml:space="preserve">terminis A, F, ad ipſos du@ <lb/>arcus AC, FD, reflectuntur ſe interſecantes in E;</s> <s xml:id="echoid-s14025" xml:space="preserve">componetur proportio ſinus arcus <lb/> <anchor type="note" xlink:label="note-418-01a" xlink:href="note-418-01"/> <emph style="sc">AB</emph>, ad ſinum arcus AD, ex proportione ſinus arcus <emph style="sc">BC</emph>, ad ſinum arcus CF, & </s> <s xml:id="echoid-s14026" xml:space="preserve">ex <lb/>proportione ſinus arcus EF, ad ſinum arcus DE. <lb/></s> <s xml:id="echoid-s14027" xml:space="preserve">Eſt autem, (cum CF, ſit complementũ arcus <emph style="sc">BC</emph>.) </s> <s xml:id="echoid-s14028" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-418-02a" xlink:href="note-418-02"/> vt ſinus arcus CF, ad ſinum arcus <emph style="sc">BC</emph>, ita ſinus <lb/> <anchor type="figure" xlink:label="fig-418-01a" xlink:href="fig-418-01"/> totus ad tangentem arcus <emph style="sc">BC</emph>; </s> <s xml:id="echoid-s14029" xml:space="preserve">conuertendoq́; </s> <s xml:id="echoid-s14030" xml:space="preserve">vt <lb/>ſinus arcus <emph style="sc">BC</emph>, ad ſinum arcus CF, ita tangens <lb/>arcus <emph style="sc">BC</emph>, ad ſinum totum: </s> <s xml:id="echoid-s14031" xml:space="preserve">Item (cum EF, ſit <lb/>complementum arcus DE.) </s> <s xml:id="echoid-s14032" xml:space="preserve">vt ſinus arcus EF, ad <lb/>ſinum arcus DE, ita ſinus totus ad tangentem ar-<lb/>cus DE. </s> <s xml:id="echoid-s14033" xml:space="preserve">Igitur proportio ſinus arcus <emph style="sc">Ab</emph>, ad ſi-<lb/>num arcus AD, componetur quoque ex proportio-<lb/>ne tangentis arcus BC, ad ſinum totum, & </s> <s xml:id="echoid-s14034" xml:space="preserve">ex <lb/>proportione ſinus totius ad tangentem arcus DE. <lb/></s> <s xml:id="echoid-s14035" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s14036" xml:space="preserve">proportio tangentis arcus <emph style="sc">BC</emph>, ad <lb/>tangentem arcus DE, componatur exproportio-<lb/>ne tangentis arcus <emph style="sc">BC</emph>, ad ſinum totum, & </s> <s xml:id="echoid-s14037" xml:space="preserve">ex proportione ſinus totius ad tangen-<lb/>tem arcus DE; </s> <s xml:id="echoid-s14038" xml:space="preserve">quòd ſinus totus inter dictas tangentes ſit poſitus: </s> <s xml:id="echoid-s14039" xml:space="preserve">erit, vt ſinus ar-<lb/>cus <emph style="sc">AB</emph>, ad ſinum arcus AD, ita tangens arcus <emph style="sc">BC</emph>, ad tangentem arcus DE; </s> <s xml:id="echoid-s14040" xml:space="preserve">& </s> <s xml:id="echoid-s14041" xml:space="preserve"><lb/>permutando, vt ſinus arcus <emph style="sc">AB</emph>, ad tangentem arcus <emph style="sc">CB</emph>, ita ſinus arcus AD, ad <lb/>tangentem arcus ED. </s> <s xml:id="echoid-s14042" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14043" xml:space="preserve"/> </p> <div xml:id="echoid-div1111" type="float" level="2" n="1"> <note position="right" xlink:label="note-417-02" xlink:href="note-417-02a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-418-01" xlink:href="note-418-01a" xml:space="preserve">Theorema <lb/>3. huius <lb/>ſcholij.</note> <note position="left" xlink:label="note-418-02" xlink:href="note-418-02a" xml:space="preserve">18. Sinu@.</note> <figure xlink:label="fig-418-01" xlink:href="fig-418-01a"> <image file="418-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/418-01"/> </figure> </div> </div> <div xml:id="echoid-div1113" type="section" level="1" n="547"> <head xml:id="echoid-head582" xml:space="preserve">VII.</head> <p> <s xml:id="echoid-s14044" xml:space="preserve">SI in ſphæræ ſuperficie duo quadrantes maximorum circulorum <lb/>ſe interſecent ad angulos non rectos, & </s> <s xml:id="echoid-s14045" xml:space="preserve">per extrema puncta arcus ma-<lb/>ximi circuli ducatur, necnon ab aliquo puncto vnius quadrantis ad <lb/>alterum arcus perpendicularis demittatur: </s> <s xml:id="echoid-s14046" xml:space="preserve">Erit, vt ſinus totus ad tan-<lb/>gentem huius arcus perpendicularis, ita tangens complementi arcus <lb/>per extremitates quadrantum ducti ad ſinum arcus quadrantis, ad <lb/>quem perpendicularis arcus demiſſus eſt, inter punctum ſectionis, & </s> <s xml:id="echoid-s14047" xml:space="preserve"><lb/>arcum perpendicularem interiecti.</s> <s xml:id="echoid-s14048" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14049" xml:space="preserve">DVO quadrantes maximorum circulorum <lb/> <anchor type="figure" xlink:label="fig-418-02a" xlink:href="fig-418-02"/> AD, AE, ſecent ſeſe in A, ad angulos non rectos, <lb/>& </s> <s xml:id="echoid-s14050" xml:space="preserve">per D, E, arcus circuli maximi deſeribatur <lb/>DE: </s> <s xml:id="echoid-s14051" xml:space="preserve">eruntq́ anguli D, E, recti. </s> <s xml:id="echoid-s14052" xml:space="preserve">Item ex C, pun <lb/> <anchor type="note" xlink:label="note-418-03a" xlink:href="note-418-03"/> cto quocunque demittatur ad AD, arcus perpen-<lb/>dicularis <emph style="sc">CB</emph>: </s> <s xml:id="echoid-s14053" xml:space="preserve">Productis autem arcubus DE, <emph style="sc">BC</emph>, <lb/>donec in F, coëant, erunt DF, <emph style="sc">BF</emph>, quadrantes. <lb/></s> <s xml:id="echoid-s14054" xml:space="preserve"> <anchor type="note" xlink:label="note-418-04a" xlink:href="note-418-04"/> Dico ita eſſe ſinum totuma ad tangẽtem arcus <emph style="sc">BC</emph>, <lb/>vt eſt tangens arcus EF, qui complementum eſt <lb/>arcus DE, ad ſinum arcus <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s14055" xml:space="preserve">Quoniam enim à <lb/>puncto D, duo arcus circulorũ maximorum educti <lb/>ſunt <emph style="sc">Da</emph>, DF, & </s> <s xml:id="echoid-s14056" xml:space="preserve">ab eorum terminis <emph style="sc">A</emph>, F, duo alij <lb/> <anchor type="note" xlink:label="note-418-05a" xlink:href="note-418-05"/> reflectuntur <emph style="sc">AE, FB</emph>, ſecantes ſeſe in C; </s> <s xml:id="echoid-s14057" xml:space="preserve">erit proportio ſinus arcus CF, ad ſinum <pb o="407" file="419" n="419" rhead=""/> arcus <emph style="sc">BC</emph>, compoſita ex proportione ſinus arcus <emph style="sc">EF</emph>, ad ſinum arcus DE, & </s> <s xml:id="echoid-s14058" xml:space="preserve">ex <lb/>proportione ſinus totius quadrantis AD, ad ſinum arcus <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s14059" xml:space="preserve">Eſt autem, (cum CF, <lb/>ſit complementum arcus <emph style="sc">BC</emph>.) </s> <s xml:id="echoid-s14060" xml:space="preserve">vt ſinus arcus CF, ad ſinum arcus <emph style="sc">CB</emph>, ita ſinus to-<lb/> <anchor type="note" xlink:label="note-419-01a" xlink:href="note-419-01"/> tus ad tangentem arcus <emph style="sc">BC</emph>: </s> <s xml:id="echoid-s14061" xml:space="preserve">Item, (cum DE, ſit cowplementum arcus EF.) </s> <s xml:id="echoid-s14062" xml:space="preserve">vt ſi-<lb/>nus arcus DE, ad ſinum arcus EF, ita ſinus totus ad tangentem arcus <emph style="sc">EF</emph>; </s> <s xml:id="echoid-s14063" xml:space="preserve">& </s> <s xml:id="echoid-s14064" xml:space="preserve">con-<lb/>uertendo, vt ſinus arcus <emph style="sc">EF</emph>, ad ſinum arcus <emph style="sc">De</emph>, ita tangens arcus <emph style="sc">EF</emph>, ad ſinum <lb/>totum. </s> <s xml:id="echoid-s14065" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14066" xml:space="preserve">proportio ſinus totius ad tangentem arcus <emph style="sc">BC</emph>, compoſita erit ex <lb/>proportione tangentis arcus <emph style="sc">EF</emph>, ad ſinũ totum, & </s> <s xml:id="echoid-s14067" xml:space="preserve">ex proportione ſinus totius qua-<lb/>drantis <emph style="sc">A</emph>D, ad ſinum arcus <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s14068" xml:space="preserve">Cum ergo proportio tangentis arcus <emph style="sc">EF</emph>, ad ſi-<lb/>num arcus <emph style="sc">AB</emph>, componatur quoque ex proportione tangentis arcus <emph style="sc">EF</emph>, ad ſinum <lb/>totum, & </s> <s xml:id="echoid-s14069" xml:space="preserve">ex proportione ſinus totius ad ſinum arcus <emph style="sc">AB</emph>; </s> <s xml:id="echoid-s14070" xml:space="preserve">quòd ſinus totus ſit me-<lb/>dius inter illam tangentem, & </s> <s xml:id="echoid-s14071" xml:space="preserve">hunc ſinum: </s> <s xml:id="echoid-s14072" xml:space="preserve">erit, vt ſinus totus ad tangentem arcus <lb/><emph style="sc">BC</emph>, ita tangens arcus <emph style="sc">EF</emph>, ad ſinum arcus <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s14073" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14074" xml:space="preserve"/> </p> <div xml:id="echoid-div1113" type="float" level="2" n="1"> <figure xlink:label="fig-418-02" xlink:href="fig-418-02a"> <image file="418-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/418-02"/> </figure> <note position="left" xlink:label="note-418-03" xlink:href="note-418-03a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-418-04" xlink:href="note-418-04a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-418-05" xlink:href="note-418-05a" xml:space="preserve">Theorema <lb/>4. huius <lb/>@cholij.</note> <note position="right" xlink:label="note-419-01" xlink:href="note-419-01a" xml:space="preserve">18. Sinu@.</note> </div> </div> <div xml:id="echoid-div1115" type="section" level="1" n="548"> <head xml:id="echoid-head583" xml:space="preserve">VIII.</head> <p> <s xml:id="echoid-s14075" xml:space="preserve">SI in ſphæræ ſuperficie duo maximi circuli ad angulos non rectos <lb/>ſe mutuo ſecent, & </s> <s xml:id="echoid-s14076" xml:space="preserve">à duobus punctis in vno aſſumptis ad alterum cir-<lb/>culum duo arcus perpendiculares ducantur: </s> <s xml:id="echoid-s14077" xml:space="preserve">Erit, vt ſinus arcus inter <lb/>punctum ſectionis, & </s> <s xml:id="echoid-s14078" xml:space="preserve">alterutrum punctorum ſumptorum ad ſecan-<lb/>tem complementi arcus per reliquum punctum aſſumptum ducti, ita <lb/>ſinus arcus inter punctum ſectionis, & </s> <s xml:id="echoid-s14079" xml:space="preserve">reliquum hoc punctum ſum-<lb/>ptum ad ſecantem complementi arcus per alterum illud punctum aſ-<lb/>ſumptum ducti.</s> <s xml:id="echoid-s14080" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14081" xml:space="preserve">IN proxima figura ſecent ſeſe duo maximi circuli AD, AE, in A, ad angulosno@ <lb/>rectos, & </s> <s xml:id="echoid-s14082" xml:space="preserve">ex punctis C, E, ad AD, arcus perpendiculares ducantur CB, ED, pro-<lb/>ducanturq́;</s> <s xml:id="echoid-s14083" xml:space="preserve">, donec coeant in F. </s> <s xml:id="echoid-s14084" xml:space="preserve">Erunt BF, DF, quadrantes, ac propterea CF, <emph style="sc">EF</emph>, <lb/> <anchor type="note" xlink:label="note-419-02a" xlink:href="note-419-02"/> complementa arcuum <emph style="sc">BC</emph>, <emph style="sc">De</emph>. </s> <s xml:id="echoid-s14085" xml:space="preserve">Dico ita eſſe ſinũ <lb/>arcus <emph style="sc">AE</emph>, ad ſecantem arcus CF, vt eſt ſinus ar-<lb/>cus AC, ad ſecantem arcus <emph style="sc">EF</emph>. </s> <s xml:id="echoid-s14086" xml:space="preserve">Quoniam enim à <lb/> <anchor type="figure" xlink:label="fig-419-01a" xlink:href="fig-419-01"/> puncto D, duo arcus educuntur <emph style="sc">Da</emph>, DF, à quo-<lb/>rum terminis <emph style="sc">A</emph>, F, duo alij ad ipſos reflectuntur <lb/><emph style="sc">AE, FB</emph>, ſe interſecantes in C; </s> <s xml:id="echoid-s14087" xml:space="preserve">erit proportio ſi-<lb/>nus arcus <emph style="sc">AE</emph>, ad ſinum arcus <emph style="sc">AC</emph>, compoſita ex <lb/> <anchor type="note" xlink:label="note-419-03a" xlink:href="note-419-03"/> proportione ſinus arcus <emph style="sc">De</emph>, ad ſinum totum qua-<lb/>drantis DF, & </s> <s xml:id="echoid-s14088" xml:space="preserve">ex proportione ſinus totius qua-<lb/> <anchor type="note" xlink:label="note-419-04a" xlink:href="note-419-04"/> drantis <emph style="sc">BF</emph>, ad ſinum arcus <emph style="sc">BC</emph>. </s> <s xml:id="echoid-s14089" xml:space="preserve">Eſt autem, vt <lb/>ſinus arcus <emph style="sc">De</emph>, ad ſinum totum quadrantis DF, <lb/>ita ſinus totus ad ſecantem arcus <emph style="sc">EF</emph>; </s> <s xml:id="echoid-s14090" xml:space="preserve">propterea <lb/>quòd ſinus totus medio loco proportionalis eſt in-<lb/> <anchor type="note" xlink:label="note-419-05a" xlink:href="note-419-05"/> ter ſinum rectum arcus <emph style="sc">De</emph>, & </s> <s xml:id="echoid-s14091" xml:space="preserve">ſecantem arcus <emph style="sc">EF</emph>, qui complementum eſt arcus <emph style="sc">De</emph>: <lb/></s> <s xml:id="echoid-s14092" xml:space="preserve"><emph style="sc">E</emph>ademq́; </s> <s xml:id="echoid-s14093" xml:space="preserve">ratione ita eſt ſecans arcus CF, ad ſinum totum, vt ſinus totus quadrantis <lb/><emph style="sc">BF</emph>, ad ſinum arcus <emph style="sc">BC</emph>; </s> <s xml:id="echoid-s14094" xml:space="preserve">quòd ſinus totus medio quoque loco ſit proportionalis inter <lb/>ſecantem arcus CF, qui complementum eſt arcus <emph style="sc">BC</emph>, & </s> <s xml:id="echoid-s14095" xml:space="preserve">ſinum rectum arcus <emph style="sc">BC</emph>. </s> <s xml:id="echoid-s14096" xml:space="preserve"><lb/>Igitur proportio ſinus arcus <emph style="sc">AE</emph>, ad ſinum arcus <emph style="sc">AC</emph>, componetur quoque ex pro-<lb/>portione ſecantis arcus CF, ad ſinum totum, & </s> <s xml:id="echoid-s14097" xml:space="preserve">exproportione ſinus totius ad ſecan- <pb o="408" file="420" n="420" rhead=""/> tem arcus <emph style="sc">EF</emph>. </s> <s xml:id="echoid-s14098" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s14099" xml:space="preserve">proportio ſecantis arcus CF, ad ſecantem arcus <emph style="sc">E</emph>F, <lb/>componatur ex proportione ſecantis arcus CF, ad ſinum totum, & </s> <s xml:id="echoid-s14100" xml:space="preserve">ex proportione <lb/>ſinus totius ad ſecantem arcus <emph style="sc">EF</emph>; </s> <s xml:id="echoid-s14101" xml:space="preserve">quòd ſinus totus ſit medius inter has ſecãtes: </s> <s xml:id="echoid-s14102" xml:space="preserve">erit, <lb/>vt ſinus arcus <emph style="sc">AE</emph>, ad ſinum arcus <emph style="sc">AC</emph>, ita ſecans arcus CF, ad ſecantem arcus <emph style="sc">EF</emph>; <lb/></s> <s xml:id="echoid-s14103" xml:space="preserve">& </s> <s xml:id="echoid-s14104" xml:space="preserve">permutando, vt ſinus arcus <emph style="sc">AE</emph>, ad ſecantem arcus CF, ita ſinus arcus AC, ad <lb/>ſecantem arcus <emph style="sc">EF</emph>. </s> <s xml:id="echoid-s14105" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14106" xml:space="preserve"/> </p> <div xml:id="echoid-div1115" type="float" level="2" n="1"> <note position="right" xlink:label="note-419-02" xlink:href="note-419-02a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-419-01" xlink:href="fig-419-01a"> <image file="419-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/419-01"/> </figure> <note position="right" xlink:label="note-419-03" xlink:href="note-419-03a" xml:space="preserve">Theorema <lb/>5. huius <lb/>ſcholij.</note> <note position="right" xlink:label="note-419-04" xlink:href="note-419-04a" xml:space="preserve">18. Sinuũ.</note> <note position="right" xlink:label="note-419-05" xlink:href="note-419-05a" xml:space="preserve">18. Sinuú.</note> </div> <p style="it"> <s xml:id="echoid-s14107" xml:space="preserve">ALITER. </s> <s xml:id="echoid-s14108" xml:space="preserve">Quoniam eſt, vt ſinus arcus <emph style="sc">AE</emph>, ad ſinum arcus <emph style="sc">E</emph>D, ita ſinus ar-<lb/> <anchor type="note" xlink:label="note-420-01a" xlink:href="note-420-01"/> cus <emph style="sc">AC</emph>, ad ſinum arcus <emph style="sc">CB</emph>; </s> <s xml:id="echoid-s14109" xml:space="preserve">hoc eſt, permutando, vt ſinus arcus <emph style="sc">AE</emph>, ad ſinum ar-<lb/>cus <emph style="sc">AC</emph>, ita ſinus arcus <emph style="sc">E</emph>D, ad ſinum arcus <emph style="sc">CB</emph>: </s> <s xml:id="echoid-s14110" xml:space="preserve"><emph style="sc">E</emph>ſt autemita ſecans arcus <emph style="sc">CF</emph>, <lb/> <anchor type="note" xlink:label="note-420-02a" xlink:href="note-420-02"/> ad ſecantem arcus <emph style="sc">E</emph>F, vt ſinus <emph style="sc">E</emph>D, qui complementum eſt poſterioris arcus <emph style="sc">EF</emph>, ad <lb/>ſinum arcus <emph style="sc">CB</emph>, qui complementum eſt arcus prioris <emph style="sc">CF</emph>; </s> <s xml:id="echoid-s14111" xml:space="preserve">erit quoque, vt ſinus ar-<lb/>cus <emph style="sc">AE</emph>, ad ſinum arcus <emph style="sc">AC</emph>, ita ſecans arcus CF, ad ſecantem arcus <emph style="sc">EF</emph>. </s> <s xml:id="echoid-s14112" xml:space="preserve"><emph style="sc">E</emph>t per-<lb/>mutando, vt ſinus arcus <emph style="sc">AE</emph>, ad ſecantem arcus CF, ita ſinus arcus <emph style="sc">AC</emph>, ad ſecan-<lb/>tem arcus <emph style="sc">EF</emph>. </s> <s xml:id="echoid-s14113" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14114" xml:space="preserve"/> </p> <div xml:id="echoid-div1116" type="float" level="2" n="2"> <note position="left" xlink:label="note-420-01" xlink:href="note-420-01a" xml:space="preserve">40. huius.</note> <note position="left" xlink:label="note-420-02" xlink:href="note-420-02a" xml:space="preserve">22. Sinuũ.</note> </div> <p style="it"> <s xml:id="echoid-s14115" xml:space="preserve">EADEM hæc demonſtratio locum etiam habet, licet duo puncta aſſumpta ſint ad <lb/>diuerſas partes puncti ſectionis. </s> <s xml:id="echoid-s14116" xml:space="preserve">Secent enim rurſum ſeſe duo circuli maximi <emph style="sc">EF, BA</emph>, <lb/>in D; </s> <s xml:id="echoid-s14117" xml:space="preserve">& </s> <s xml:id="echoid-s14118" xml:space="preserve">à punctis <emph style="sc">F, E</emph>, arcus <emph style="sc">EF</emph>, ducantur ad <emph style="sc">BA</emph>, arcus perpendiculares <emph style="sc">FA, EB</emph>. <lb/></s> <s xml:id="echoid-s14119" xml:space="preserve">Dicoita eſſe ſinum arcus <emph style="sc">E</emph>D, ad ſecantem complementi <lb/>arcus <emph style="sc">FA</emph>, vt est ſinus arcus DF, ad ſecantem comple-<lb/> <anchor type="figure" xlink:label="fig-420-01a" xlink:href="fig-420-01"/> menti arcus <emph style="sc">EB</emph>. </s> <s xml:id="echoid-s14120" xml:space="preserve">Nam quoniam eſt, vt ſinus arcus ED, <lb/> <anchor type="note" xlink:label="note-420-03a" xlink:href="note-420-03"/> ad ſinum arcus <emph style="sc">EB</emph>, ita ſinus arcus DF, ad ſinum arcus <lb/><emph style="sc">FA</emph>; </s> <s xml:id="echoid-s14121" xml:space="preserve">& </s> <s xml:id="echoid-s14122" xml:space="preserve">permutando, vt ſinus arcus <emph style="sc">E</emph>D, ad ſinum arcus <lb/>DF, ita ſinus arcus <emph style="sc">EB</emph>, ad ſinum arcus <emph style="sc">FA</emph>; </s> <s xml:id="echoid-s14123" xml:space="preserve">Vt autem <lb/>ſinus arcus <emph style="sc">EB</emph>, ad ſinum arcus <emph style="sc">FA</emph>, ita eſt ſecans com-<lb/> <anchor type="note" xlink:label="note-420-04a" xlink:href="note-420-04"/> plementi arcus <emph style="sc">FA</emph>, ad ſecantem complementi arcus <emph style="sc">EB</emph>: <lb/></s> <s xml:id="echoid-s14124" xml:space="preserve">erit quoque, vt ſinus arcus <emph style="sc">E</emph>D, ad ſinum arcus DF, ita <lb/>ſecans complementi arcus <emph style="sc">FA</emph>, ad ſecantem complementi <lb/>arcus <emph style="sc">EB</emph>; </s> <s xml:id="echoid-s14125" xml:space="preserve">& </s> <s xml:id="echoid-s14126" xml:space="preserve">permutando, vt ſinus arcus <emph style="sc">E</emph>D, ad ſecantem complementi arcus <emph style="sc">FA</emph>, <lb/>ita ſinus arcus DF, ad ſecantem complementi arcus <emph style="sc">EB</emph>. </s> <s xml:id="echoid-s14127" xml:space="preserve">Quod eſt propoſi@um.</s> <s xml:id="echoid-s14128" xml:space="preserve"/> </p> <div xml:id="echoid-div1117" type="float" level="2" n="3"> <figure xlink:label="fig-420-01" xlink:href="fig-420-01a"> <image file="420-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/420-01"/> </figure> <note position="left" xlink:label="note-420-03" xlink:href="note-420-03a" xml:space="preserve">40. huius.</note> <note position="left" xlink:label="note-420-04" xlink:href="note-420-04a" xml:space="preserve">22. Sinuũ.</note> </div> <p style="it"> <s xml:id="echoid-s14129" xml:space="preserve">REPETIVIMVS autem hic figuram quartam propoſ. </s> <s xml:id="echoid-s14130" xml:space="preserve">35. </s> <s xml:id="echoid-s14131" xml:space="preserve">licet arcuum BC, <lb/>CF, nulla fiat mentio, ne nouam figuram cogeremur extruere.</s> <s xml:id="echoid-s14132" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1119" type="section" level="1" n="549"> <head xml:id="echoid-head584" xml:space="preserve">THEOR. 39. PROPOS. 41.</head> <p> <s xml:id="echoid-s14133" xml:space="preserve">IN omni triangulo ſphærico, ſinus cuiuſlibet <lb/>arcus ad ſinum anguli, quem ſubtendit, eandem <lb/>habet proportionem, quam ſinus vtriuſque reli-<lb/>quorum arcuum ad ſinũ anguli, quem ſubtendit.</s> <s xml:id="echoid-s14134" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14135" xml:space="preserve">SIT triangulum ſphæricum quodcunque ABC. </s> <s xml:id="echoid-s14136" xml:space="preserve">Dico ita eſſe ſinum ar-<lb/>cus AB, ad ſinum anguli C, quem ſubtendit, vt eſt ſinus arcus AC, ad ſinum <lb/>auguli B, quem ſubtendit, & </s> <s xml:id="echoid-s14137" xml:space="preserve">vt ſinus arcus BC, ad ſinum anguli A, quem <lb/>ſubtendit. </s> <s xml:id="echoid-s14138" xml:space="preserve">Sint enim primum omnes tres anguli recti; </s> <s xml:id="echoid-s14139" xml:space="preserve">eruntq́ue propterea <lb/> <anchor type="note" xlink:label="note-420-05a" xlink:href="note-420-05"/> omnes arcus quadrantes. </s> <s xml:id="echoid-s14140" xml:space="preserve">Manifeſtum igitur eſt, vt eſt ſinus totus quadrantis <pb o="409" file="421" n="421" rhead=""/> AB, ad ſinum totum anguli recti C, ita eſſe totum quadrantis A C, ad ſinum <lb/>totum anguli recti B, & </s> <s xml:id="echoid-s14141" xml:space="preserve">ſinum totum quadrantis B C, <lb/> <anchor type="figure" xlink:label="fig-421-01a" xlink:href="fig-421-01"/> ad ſinum totum anguli recti A.</s> <s xml:id="echoid-s14142" xml:space="preserve"/> </p> <div xml:id="echoid-div1119" type="float" level="2" n="1"> <note position="left" xlink:label="note-420-05" xlink:href="note-420-05a" xml:space="preserve">Coroll. 25. <lb/>huius.</note> <figure xlink:label="fig-421-01" xlink:href="fig-421-01a"> <image file="421-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/421-01"/> </figure> </div> <p> <s xml:id="echoid-s14143" xml:space="preserve">DEINDE ſint duo tantum anguli A, B, recti, <lb/>eruntq́; </s> <s xml:id="echoid-s14144" xml:space="preserve">idcirco arcus AC, BC, quadrantes, & </s> <s xml:id="echoid-s14145" xml:space="preserve">C, po-<lb/> <anchor type="note" xlink:label="note-421-01a" xlink:href="note-421-01"/> lus arcus AB. </s> <s xml:id="echoid-s14146" xml:space="preserve">Itaque rurſus perſpicuum eſt, vt eſt ſi-<lb/>nus arcus AB, ad ſinum anguli C, hoc eſt, ad ſinum <lb/>arcus AB, (Eſt enim A B, arcus anguli C, cum C, ſit <lb/>polus arcus AB, vt oſtenſum eſt) ita eſſe ſinum totum <lb/>quadrantis AC, ad ſinum totum anguli recti B, & </s> <s xml:id="echoid-s14147" xml:space="preserve">ſi-<lb/>num totum quadrantis BC, ad ſinum totum anguli recti A; </s> <s xml:id="echoid-s14148" xml:space="preserve">cum ſemper ſit <lb/>æqualitatis proportio.</s> <s xml:id="echoid-s14149" xml:space="preserve"/> </p> <div xml:id="echoid-div1120" type="float" level="2" n="2"> <note position="right" xlink:label="note-421-01" xlink:href="note-421-01a" xml:space="preserve">25. huius. <lb/>Schol. 26. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s14150" xml:space="preserve">TERTIO ſit angulus duntaxat C, rectus, & </s> <s xml:id="echoid-s14151" xml:space="preserve">reliquorum angulorum <lb/>A, B, vterque recto minor, vel maior; </s> <s xml:id="echoid-s14152" xml:space="preserve">vel alter recto maior, & </s> <s xml:id="echoid-s14153" xml:space="preserve">alter minor. </s> <s xml:id="echoid-s14154" xml:space="preserve">Si <lb/>igitur vterque recto minor eſt, erunt omnes arcus quadrante minores. </s> <s xml:id="echoid-s14155" xml:space="preserve">Produ-<lb/> <anchor type="note" xlink:label="note-421-02a" xlink:href="note-421-02"/> cantur omnes, & </s> <s xml:id="echoid-s14156" xml:space="preserve">fiant quadrantes BD, AE, BF, AG, <lb/>& </s> <s xml:id="echoid-s14157" xml:space="preserve">per puncta D, F, arcus maximi circuli DF, & </s> <s xml:id="echoid-s14158" xml:space="preserve">per <lb/> <anchor type="note" xlink:label="note-421-03a" xlink:href="note-421-03"/> <anchor type="figure" xlink:label="fig-421-02a" xlink:href="fig-421-02"/> puncta E, G, arcus maximi circuli E G, ducatur; </s> <s xml:id="echoid-s14159" xml:space="preserve">e-<lb/>runtq́ue anguli D, F, E, G, recti, & </s> <s xml:id="echoid-s14160" xml:space="preserve">B, polus arcus <lb/> <anchor type="note" xlink:label="note-421-04a" xlink:href="note-421-04"/> DF, & </s> <s xml:id="echoid-s14161" xml:space="preserve">A, polus arcus EG; </s> <s xml:id="echoid-s14162" xml:space="preserve">ac proinde arcus DF, <lb/>EG, arcus erunt angulorum B, A. </s> <s xml:id="echoid-s14163" xml:space="preserve">Tam verò qua-<lb/>drans BD, quam AE, arcus eſt anguli recti C, vt ex <lb/>defin. </s> <s xml:id="echoid-s14164" xml:space="preserve">6. </s> <s xml:id="echoid-s14165" xml:space="preserve">perſpicuum eſt. </s> <s xml:id="echoid-s14166" xml:space="preserve">Quoniam igitur duo circu-<lb/>li maximi BD, BF, ſe mutuo ſecant in ſphæra in pun <lb/>cto B, & </s> <s xml:id="echoid-s14167" xml:space="preserve">in arcu BD, ſumpta ſunt duo puncta A, D, <lb/>à quibus ad arcum BF, ducti ſunt arcus perpendiculares AC, DF; </s> <s xml:id="echoid-s14168" xml:space="preserve">erit vt ſi-<lb/>nus arcus AB, ad ſinum arcus AC, ita ſinus arcus BD, ad ſinum arcus DF: <lb/></s> <s xml:id="echoid-s14169" xml:space="preserve"> <anchor type="note" xlink:label="note-421-05a" xlink:href="note-421-05"/> & </s> <s xml:id="echoid-s14170" xml:space="preserve">permutando, vt ſinus arcus AB, trianguli ABC, ad ſinum quadrantis BD, <lb/>hoc eſt, ad ſinum totum anguli recti C, in eodem triangulo ABC, ita ſinus <lb/>arcus AC, trianguli eiuſdem ABC, ad ſinum arcus DF, hoc eſt, ad ſinum an-<lb/>guli B, eiuſdem trianguli ABC. </s> <s xml:id="echoid-s14171" xml:space="preserve">Eodem modo erit, vt ſinus arcus AB, in <lb/>triangulo ABC, ad ſinum quadrantis AE, hoc eſt, ad ſinum totum anguli <lb/>recti C, eiuſdem trianguli ABC, ita ſinus arcus BC, eiuſdem trianguli ABC, <lb/>ad ſinum arcus EG, hoc eſt, ad ſinum anguli A, in eodem triangulo ABC. <lb/></s> <s xml:id="echoid-s14172" xml:space="preserve">Patet ergo propoſitum.</s> <s xml:id="echoid-s14173" xml:space="preserve"/> </p> <div xml:id="echoid-div1121" type="float" level="2" n="3"> <note position="right" xlink:label="note-421-02" xlink:href="note-421-02a" xml:space="preserve">28. huius.</note> <note position="right" xlink:label="note-421-03" xlink:href="note-421-03a" xml:space="preserve">20. i Theod.</note> <figure xlink:label="fig-421-02" xlink:href="fig-421-02a"> <image file="421-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/421-02"/> </figure> <note position="right" xlink:label="note-421-04" xlink:href="note-421-04a" xml:space="preserve">25. huius. <lb/>Schol. 26. <lb/>huius.</note> <note position="right" xlink:label="note-421-05" xlink:href="note-421-05a" xml:space="preserve">40. huius.</note> </div> <p> <s xml:id="echoid-s14174" xml:space="preserve">SI verò vterque angulorum A, B, eſt re-<lb/> <anchor type="figure" xlink:label="fig-421-03a" xlink:href="fig-421-03"/> cto maior, erit arcus AB, quadrante minor: <lb/></s> <s xml:id="echoid-s14175" xml:space="preserve"> <anchor type="note" xlink:label="note-421-06a" xlink:href="note-421-06"/> & </s> <s xml:id="echoid-s14176" xml:space="preserve">tam arcus AC, quam BC, quadrante ma-<lb/> <anchor type="note" xlink:label="note-421-07a" xlink:href="note-421-07"/> ior. </s> <s xml:id="echoid-s14177" xml:space="preserve">Producto igitur arcu AB, in vtramque <lb/>partem, vt ſint quadrantes AE, BD, abſciſ-<lb/>ſisq́ue quadrantibus AG, BF, ducatur per <lb/>puncta D, F, arcus maximi circuli DF, & </s> <s xml:id="echoid-s14178" xml:space="preserve">per <lb/> <anchor type="note" xlink:label="note-421-08a" xlink:href="note-421-08"/> E, G, maximi circuli arcus EG; </s> <s xml:id="echoid-s14179" xml:space="preserve">eritq́ue rur-<lb/> <anchor type="note" xlink:label="note-421-09a" xlink:href="note-421-09"/> ſum B, polus arcus DF, & </s> <s xml:id="echoid-s14180" xml:space="preserve">A, polus arcus EG. <lb/></s> <s xml:id="echoid-s14181" xml:space="preserve">Igitur DF, EG, arcus erunt angulorum B, <lb/>A; </s> <s xml:id="echoid-s14182" xml:space="preserve">necnon tam quadrans BD, quam AE, ar-<lb/>cus anguli recti C, ex defin. </s> <s xml:id="echoid-s14183" xml:space="preserve">6. </s> <s xml:id="echoid-s14184" xml:space="preserve">Item propter <lb/>quadrantes BD, BF, vterque angulus D, F; </s> <s xml:id="echoid-s14185" xml:space="preserve">& </s> <s xml:id="echoid-s14186" xml:space="preserve">propter quadrantes AE, AG, <lb/> <anchor type="note" xlink:label="note-421-10a" xlink:href="note-421-10"/> <pb o="410" file="422" n="422" rhead=""/> vterque angulus E, G, rectus erit. </s> <s xml:id="echoid-s14187" xml:space="preserve">Quia igitur duo maximi circuli BD, BC, <lb/>ſe mutuo in ſphæra ſecant in B, ſumptaq́ue ſunt in BD, duo puncta A, D, à <lb/>quibus ad BC, ducti ſunt duo arcus AC, DF, perpendiculares, erit vt ſinus <lb/>arcus AB, ad ſinum arcus AC, ita ſinus arcus BD, ad ſin um arcus DF: </s> <s xml:id="echoid-s14188" xml:space="preserve">& </s> <s xml:id="echoid-s14189" xml:space="preserve">per-<lb/> <anchor type="note" xlink:label="note-422-01a" xlink:href="note-422-01"/> mutando, vt ſinus arcus AB, trianguli ABC, ad ſinum quadrantis BD, hoc <lb/>eſt, ad ſinum totum anguli recti C, in eodem triangulo ABC, ita ſinus arcus <lb/>AC, eiuſdem trianguli ABC, ad ſinum arcus DF, hoc eſt, ad ſinum anguli <lb/>B, in eodem triangulo ABC. </s> <s xml:id="echoid-s14190" xml:space="preserve">Eademq; </s> <s xml:id="echoid-s14191" xml:space="preserve">ratione erit, vt ſinus arcus AB, trian <lb/>guli ABC, ad ſinum quadrantis AE, hoc eſt, ad ſinum totum anguli recti C, <lb/>in eodem triangulo ABC, ita finus arcus BC, eiuſdem trianguli ABC, ad <lb/>finum arcus EG, hoc eſt, ad ſinum anguli A, in eodem triangulo ABC. </s> <s xml:id="echoid-s14192" xml:space="preserve">Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s14193" xml:space="preserve"/> </p> <div xml:id="echoid-div1122" type="float" level="2" n="4"> <figure xlink:label="fig-421-03" xlink:href="fig-421-03a"> <image file="421-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/421-03"/> </figure> <note position="right" xlink:label="note-421-06" xlink:href="note-421-06a" xml:space="preserve">37. huius.</note> <note position="right" xlink:label="note-421-07" xlink:href="note-421-07a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-421-08" xlink:href="note-421-08a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-421-09" xlink:href="note-421-09a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-421-10" xlink:href="note-421-10a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-422-01" xlink:href="note-422-01a" xml:space="preserve">40. huius.</note> </div> <p> <s xml:id="echoid-s14194" xml:space="preserve">SI denique alter angulorum A, B, recto maior eſt, & </s> <s xml:id="echoid-s14195" xml:space="preserve">alter minor; </s> <s xml:id="echoid-s14196" xml:space="preserve">ſit B, ma-<lb/>ior, & </s> <s xml:id="echoid-s14197" xml:space="preserve">A, minor. </s> <s xml:id="echoid-s14198" xml:space="preserve">Erit igitur arcus AB, quadrante maior: </s> <s xml:id="echoid-s14199" xml:space="preserve">Item arcus AC, <lb/> <anchor type="note" xlink:label="note-422-02a" xlink:href="note-422-02"/> <anchor type="note" xlink:label="note-422-03a" xlink:href="note-422-03"/> quadrante etiam maior, at verò BC, minor <lb/>quadrante. </s> <s xml:id="echoid-s14200" xml:space="preserve">Abſcindantur ergo quadrantes <lb/> <anchor type="figure" xlink:label="fig-422-01a" xlink:href="fig-422-01"/> BD, AE, & </s> <s xml:id="echoid-s14201" xml:space="preserve">AG, productoq́ue arcu BC, fiat <lb/> <anchor type="note" xlink:label="note-422-04a" xlink:href="note-422-04"/> quadrans BF; </s> <s xml:id="echoid-s14202" xml:space="preserve">& </s> <s xml:id="echoid-s14203" xml:space="preserve">per puncta D, F, ducatur ar-<lb/>cus DF, circuli maximi, necnon per E, G, <lb/>arcus circuli maximi EG; </s> <s xml:id="echoid-s14204" xml:space="preserve">eritq́ue rurſus B, <lb/>polus arcus DF, & </s> <s xml:id="echoid-s14205" xml:space="preserve">A, polus arcus EG. </s> <s xml:id="echoid-s14206" xml:space="preserve">Igi-<lb/> <anchor type="note" xlink:label="note-422-05a" xlink:href="note-422-05"/> tur DF, EG, arcus erunt angulorum B, A; <lb/></s> <s xml:id="echoid-s14207" xml:space="preserve">necnon tam quadrans BD, quam AE, arcus <lb/>anguli C, recti, ex defin. </s> <s xml:id="echoid-s14208" xml:space="preserve">6. </s> <s xml:id="echoid-s14209" xml:space="preserve">Item propter qua-<lb/>drantes AE, AG, vterque angulus E, G, re-<lb/> <anchor type="note" xlink:label="note-422-06a" xlink:href="note-422-06"/> ctus erit. </s> <s xml:id="echoid-s14210" xml:space="preserve">Quoniam igitur duo circuli maxi-<lb/>mi BA, BF, in ſphæra ſe mutuo ſecant in B, <lb/>ſumptaq́ue ſunt in BA, duo puncta A, D, à <lb/>quibus ad BF, ducti ſunt duo arcus perpendiculares AC, DF; </s> <s xml:id="echoid-s14211" xml:space="preserve">erit, vt ſinus <lb/>arcus AB, ad ſinum arcus AC, ita ſinus arcus BD, ad ſinum arcus DF: </s> <s xml:id="echoid-s14212" xml:space="preserve">& </s> <s xml:id="echoid-s14213" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-422-07a" xlink:href="note-422-07"/> permutando, vt ſinus arcus AB, trianguli ABC, ad ſinum quadrantis BD, <lb/>hoc eſt, ad ſinum totum anguli recti C, in eodem triangulo ABC, ita ſinus <lb/>arcus AC, trianguli eiuſdem ABC, ad ſinum arcus DF, hoc eſt, ad ſinum an-<lb/>guli B, in triangulo eodem ABC. </s> <s xml:id="echoid-s14214" xml:space="preserve">Eodemq́ue modo erit, vt ſinus arcus AB, <lb/>trianguli ABC, ad ſinum quadrantis AE, hoc eſt, ad ſinum totum recti an-<lb/>guli C, in eodẽ triangulo ABC, ita ſinus arcus BC, eiuſdem trianguli ABC, <lb/>ad ſinum arcus EG, hoc eſt, ad ſinum anguli A, eiuſdem trianguli ABC, <lb/>Quod eſt propoſitum.</s> <s xml:id="echoid-s14215" xml:space="preserve"/> </p> <div xml:id="echoid-div1123" type="float" level="2" n="5"> <note position="left" xlink:label="note-422-02" xlink:href="note-422-02a" xml:space="preserve">37. huius.</note> <note position="left" xlink:label="note-422-03" xlink:href="note-422-03a" xml:space="preserve">34. huius.</note> <figure xlink:label="fig-422-01" xlink:href="fig-422-01a"> <image file="422-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/422-01"/> </figure> <note position="left" xlink:label="note-422-04" xlink:href="note-422-04a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-422-05" xlink:href="note-422-05a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-422-06" xlink:href="note-422-06a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-422-07" xlink:href="note-422-07a" xml:space="preserve">40. huius.</note> </div> <p> <s xml:id="echoid-s14216" xml:space="preserve">QVARTO ac vltimo nullus angulorum A, B, C, rectus ſit. </s> <s xml:id="echoid-s14217" xml:space="preserve">Per pun-<lb/> <anchor type="figure" xlink:label="fig-422-02a" xlink:href="fig-422-02"/> ctum A, & </s> <s xml:id="echoid-s14218" xml:space="preserve">polum circuli BC, ducatur arcus circu-<lb/> <anchor type="note" xlink:label="note-422-08a" xlink:href="note-422-08"/> li maximi AD, cadatq́ue primum in latus BC, in-<lb/> <anchor type="note" xlink:label="note-422-09a" xlink:href="note-422-09"/> tra triangulum; </s> <s xml:id="echoid-s14219" xml:space="preserve">eruntq́; </s> <s xml:id="echoid-s14220" xml:space="preserve">anguli ad D, recti. </s> <s xml:id="echoid-s14221" xml:space="preserve">Quoniam <lb/>igitur in triangulo ABD, angulus D, rectus eſt; </s> <s xml:id="echoid-s14222" xml:space="preserve">erit, <lb/>vt iam demonſtratum eſt, vt ſinus arcus AB, ad ſi-<lb/>num anguli ADB, ita ſinus arcus AD, ad ſinum an-<lb/>guli B: </s> <s xml:id="echoid-s14223" xml:space="preserve">& </s> <s xml:id="echoid-s14224" xml:space="preserve">permutando, vt ſinus arcus AB, ad ſinum <lb/>arcus AD, ita ſinus anguli ADB, ad ſinum anguli <lb/>B. </s> <s xml:id="echoid-s14225" xml:space="preserve">Sed eodem modo, cum in triangulo ADC, an- <pb o="411" file="423" n="423" rhead=""/> gulus D, rectus ſit; </s> <s xml:id="echoid-s14226" xml:space="preserve">eſt, vt ſinus arcus AD, ad ſinum anguli ACD, ita ſinus <lb/>arcus AC, ad ſinum anguli ADC: </s> <s xml:id="echoid-s14227" xml:space="preserve">& </s> <s xml:id="echoid-s14228" xml:space="preserve">permutando, vt ſinus arcus AD, ad ſi-<lb/>num arcus AC, ita ſinus anguli ACD, ad ſinum anguli ADC, hoc eſt, ad <lb/>ſinum anguli ADB, cum anguli ad D, ſint recti. </s> <s xml:id="echoid-s14229" xml:space="preserve">Ex æqualitate ergo, & </s> <s xml:id="echoid-s14230" xml:space="preserve">per-<lb/>turbata proportione, erit, vt ſinus arcus AB, ad ſinum arcus AC, ita ſinus an-<lb/>guli ACD, ad ſinum anguli B; </s> <s xml:id="echoid-s14231" xml:space="preserve">vt in appoſita formula apparet. </s> <s xml:id="echoid-s14232" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14233" xml:space="preserve">per-<lb/>mutando erit, vt ſinus arcus AB, in triangulo ABC, ad ſinum anguli ACB, <lb/>in eodem triangulo ABC, ita ſinus arcus AC, eiuſdem <lb/>trianguli ABC, ad ſinum anguli B, in eodem triangulo ABC.</s> <s xml:id="echoid-s14234" xml:space="preserve"/> </p> <div xml:id="echoid-div1124" type="float" level="2" n="6"> <figure xlink:label="fig-422-02" xlink:href="fig-422-02a"> <image file="422-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/422-02"/> </figure> <note position="left" xlink:label="note-422-08" xlink:href="note-422-08a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-422-09" xlink:href="note-422-09a" xml:space="preserve">25. 1. Theod.</note> </div> <note position="right" xml:space="preserve"> <lb/>arcus # anguli <lb/>AB. # ACD. <lb/>AD. # ADB. <lb/>AC. # B. <lb/></note> <p> <s xml:id="echoid-s14235" xml:space="preserve">CADAT deinde arcus per A, & </s> <s xml:id="echoid-s14236" xml:space="preserve">polum circuli BC, <lb/>ductus in arcum BC, productum ad E, eritq́; </s> <s xml:id="echoid-s14237" xml:space="preserve">angulus E, re-<lb/> <anchor type="note" xlink:label="note-423-02a" xlink:href="note-423-02"/> ctus. </s> <s xml:id="echoid-s14238" xml:space="preserve">Quoniam igitur in triangulo ABE, angulus E, rectus <lb/>eſt; </s> <s xml:id="echoid-s14239" xml:space="preserve">erit, vt demonſtratum eſt, vt ſinus arcus AB, ad ſinum <lb/>anguli E, ita ſinus arcus AE, ad ſinum anguli B: </s> <s xml:id="echoid-s14240" xml:space="preserve">& </s> <s xml:id="echoid-s14241" xml:space="preserve">permu-<lb/>tando, vt ſinus arcus AB, ad ſinum arcus AE, ita ſinus anguli E, ad ſinum <lb/>anguli B. </s> <s xml:id="echoid-s14242" xml:space="preserve">Sed eadem ratione, cum in triangulo ACE, angulus E, rectus ſit, <lb/>eſt, vt ſinus arcus AE, ad ſinum anguli ACE, ita ſinus arcus AC, ad ſi-<lb/>num anguli E: </s> <s xml:id="echoid-s14243" xml:space="preserve">& </s> <s xml:id="echoid-s14244" xml:space="preserve">permutando, vt ſinus arcus AE, ad ſinum arcus AC, ita <lb/>ſinus anguli ACE, ad ſinum anguli E. </s> <s xml:id="echoid-s14245" xml:space="preserve">Igitur ex æqualitate, & </s> <s xml:id="echoid-s14246" xml:space="preserve">perturbata <lb/>proportione, erit vt ſinus arcus AB, ad ſinum arcus AC, <lb/>ita ſinus anguli ACE, hoc eſt, anguli ACB, (cum idem ſit <lb/> <anchor type="note" xlink:label="note-423-03a" xlink:href="note-423-03"/> ſinus vtriuſq; </s> <s xml:id="echoid-s14247" xml:space="preserve">anguli ad C, quòd eorum arcus ſemicirculum <lb/>conſtituant, vt conſtat ex coroll. </s> <s xml:id="echoid-s14248" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14249" xml:space="preserve">5. </s> <s xml:id="echoid-s14250" xml:space="preserve">huius tracta-<lb/>tus. </s> <s xml:id="echoid-s14251" xml:space="preserve">Perſpicuum autem eſt ex ijs, quæ in tractatione ſinuum <lb/>diximus, duos arcus ſemicirculum conficientes, eundem ha-<lb/>bere ſinum.) </s> <s xml:id="echoid-s14252" xml:space="preserve">ad ſinum anguli B; </s> <s xml:id="echoid-s14253" xml:space="preserve">vt in appoſita ſormula ap-<lb/>paret. </s> <s xml:id="echoid-s14254" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14255" xml:space="preserve">permutando erit, vt ſinus arcus AB, in triangulo ABC, <lb/>ad ſinum anguli ACB, eiuſdem trianguli ABC, ita ſinus arcus AC, in <lb/>eodem triangulo ABC, ad ſinum anguli B, eiuſdem trianguli ABC. </s> <s xml:id="echoid-s14256" xml:space="preserve">Quod <lb/>ſi ex B, ad arcum AC, ducatur alius arcus perpendicularis, qui vel intra trian <lb/>gulum cadet, vel in arcum productum, oſtendemus eodem modo, ita eſſe ſinũ <lb/>arcus AB, ad ſinum anguli ACB, vt eſt ſinus arcus BC, ad ſinum anguli <lb/>BAC. </s> <s xml:id="echoid-s14257" xml:space="preserve">Itaque in omni triangulo ſphærico, ſinus cuiuſlibet arcus, &</s> <s xml:id="echoid-s14258" xml:space="preserve">c. </s> <s xml:id="echoid-s14259" xml:space="preserve">Quod <lb/>erat oſtendendum.</s> <s xml:id="echoid-s14260" xml:space="preserve"/> </p> <div xml:id="echoid-div1125" type="float" level="2" n="7"> <note position="right" xlink:label="note-423-02" xlink:href="note-423-02a" xml:space="preserve">15. 1. Theod.</note> <note position="right" xlink:label="note-423-03" xlink:href="note-423-03a" xml:space="preserve"> <lb/>arcus # anguli <lb/>AB. # ACE. <lb/>AE. # E. <lb/>AC. # B. <lb/></note> </div> </div> <div xml:id="echoid-div1127" type="section" level="1" n="550"> <head xml:id="echoid-head585" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s14261" xml:space="preserve">HINC perſpicuum eſt, in omni triangulo ſphærico rectangulo, vt eſt ſinus arcus rectum <lb/>angulum ſubtendentis ad ſinum totum, nempe ad ſinum anguli recti, quem ſubtendit, ita <lb/>eſſe ſinum cuiuſ@@bet reliquorum arcuum ad ſinum anguli, quem ſubtendit. </s> <s xml:id="echoid-s14262" xml:space="preserve">Quod idcir. <lb/></s> <s xml:id="echoid-s14263" xml:space="preserve">co dixerim, quia plerique ſcriptores hoc corollarium, tanquàm propoſitionem ab hac no-<lb/>ftra propoſitione 41. </s> <s xml:id="echoid-s14264" xml:space="preserve">diuerſam, demonſtrant: </s> <s xml:id="echoid-s14265" xml:space="preserve">ſed placuit nobis propoſitionem hanc magis <lb/>vniuerſalem reddere, prout nimirum complectitur & </s> <s xml:id="echoid-s14266" xml:space="preserve">triangulum ſphæricum rectangulum, <lb/>& </s> <s xml:id="echoid-s14267" xml:space="preserve">non rectangulum.</s> <s xml:id="echoid-s14268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1128" type="section" level="1" n="551"> <head xml:id="echoid-head586" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s14269" xml:space="preserve">IN hac, & </s> <s xml:id="echoid-s14270" xml:space="preserve">ſequentibus propoſitionibus adducemus problemata, quibus ſphæri-<lb/>corum triangulorum rectangulorum calculus perficitur, quæq́; </s> <s xml:id="echoid-s14271" xml:space="preserve">ex ipſis propoſitioni-<lb/>bus eliciuntur, Quanquam autem nonnunquam in problemate aliquo plura propo- <pb o="412" file="424" n="424" rhead=""/> nantur inueſtiganda, primum tamen ſemper potiſsimum eſt, quod quæritur, infer-<lb/>turq́; </s> <s xml:id="echoid-s14272" xml:space="preserve">primò ac per ſe ex ipſo problemate. </s> <s xml:id="echoid-s14273" xml:space="preserve">Ex hac igitur propoſitione ſequentia triæ <lb/>problemata colliguntur.</s> <s xml:id="echoid-s14274" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1129" type="section" level="1" n="552"> <head xml:id="echoid-head587" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s14275" xml:space="preserve">IN triangulo ſphęrico rectangulo, dato arcu, qui recto angulo <lb/>opponitur & </s> <s xml:id="echoid-s14276" xml:space="preserve">alterutro arcuum angulum rectum ambientium; </s> <s xml:id="echoid-s14277" xml:space="preserve">in-<lb/>uenire angulum huic arcui oppoſitum.</s> <s xml:id="echoid-s14278" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14279" xml:space="preserve">IN triangulo ABC, cuius angulus C, rectus, dati ſint <lb/> <anchor type="figure" xlink:label="fig-424-01a" xlink:href="fig-424-01"/> arcus <emph style="sc">Ab</emph>, AC. </s> <s xml:id="echoid-s14280" xml:space="preserve">Dico dari quoque angulum <emph style="sc">B</emph>, arcui <lb/> <anchor type="note" xlink:label="note-424-01a" xlink:href="note-424-01"/> AC, oppoſitum. </s> <s xml:id="echoid-s14281" xml:space="preserve">Quoniam enim eſt, vt ſinus arcus <emph style="sc">Ab</emph>, <lb/>ad ſinum totum anguli recti C, ita ſinus arcus <emph style="sc">Ac</emph>, ad ſi-<lb/>num anguli B:</s> <s xml:id="echoid-s14282" xml:space="preserve"/> </p> <div xml:id="echoid-div1129" type="float" level="2" n="1"> <figure xlink:label="fig-424-01" xlink:href="fig-424-01a"> <image file="424-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/424-01"/> </figure> <note position="left" xlink:label="note-424-01" xlink:href="note-424-01a" xml:space="preserve">41. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14283" xml:space="preserve">S I fiat, vt ſinus arcus dati recto angulo <lb/>oppoſiti ad ſinum totum, ita ſinus arcus circa <lb/> <anchor type="note" xlink:label="note-424-02a" xlink:href="note-424-02"/> angulum rectum dati ad aliud, reperietur ſinus <lb/>anguli quæſiti.</s> <s xml:id="echoid-s14284" xml:space="preserve"/> </p> <div xml:id="echoid-div1130" type="float" level="2" n="2"> <note position="left" xlink:label="note-424-02" xlink:href="note-424-02a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s14285" xml:space="preserve">VERVM hic diligenter attendendum eſt, num angulus quæſitus <emph style="sc">B,</emph> ſit acutus, <lb/>an obtuſus. </s> <s xml:id="echoid-s14286" xml:space="preserve">Si enim acutus eſt, dabit arcus ſinui inuento reſpondens angulum B: </s> <s xml:id="echoid-s14287" xml:space="preserve">Si <lb/>vero eſt obtuſus, relinquet idem arcus ex ſemicirculo ſublatus angulum B. </s> <s xml:id="echoid-s14288" xml:space="preserve">Pulchre <lb/>autem arcus datus <emph style="sc">AC</emph>, circa angulum rectum C, docebit, an angulus <emph style="sc">B</emph>, acutus ſit, <lb/> <anchor type="note" xlink:label="note-424-03a" xlink:href="note-424-03"/> vel obtuſus. </s> <s xml:id="echoid-s14289" xml:space="preserve">Nam ſi AC, eſt minor quadrante, erit angulus <emph style="sc">B</emph>, acutus: </s> <s xml:id="echoid-s14290" xml:space="preserve">Si vero <lb/>quadrante maior, obtuſus. </s> <s xml:id="echoid-s14291" xml:space="preserve">Sumimus autem hic triangulum ſphæricum, in quo vnus <lb/>tantum angulus rectus eſt, & </s> <s xml:id="echoid-s14292" xml:space="preserve">proinde nullus arcus Quadrans, vt in propoſ. </s> <s xml:id="echoid-s14293" xml:space="preserve">dictum <lb/>eſt: </s> <s xml:id="echoid-s14294" xml:space="preserve">quod etiam in ſequentibus intelligatur.</s> <s xml:id="echoid-s14295" xml:space="preserve"/> </p> <div xml:id="echoid-div1131" type="float" level="2" n="3"> <note position="left" xlink:label="note-424-03" xlink:href="note-424-03a" xml:space="preserve">24. huius.</note> </div> </div> <div xml:id="echoid-div1133" type="section" level="1" n="553"> <head xml:id="echoid-head588" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s14296" xml:space="preserve">IN triangulo ſphęrico rectangulo, dato arcu, qui recto angulo <lb/>opponitur, & </s> <s xml:id="echoid-s14297" xml:space="preserve">alterutro angulorum non rectorum; </s> <s xml:id="echoid-s14298" xml:space="preserve">inuenire arcum <lb/>huic angulo oppoſitum.</s> <s xml:id="echoid-s14299" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14300" xml:space="preserve">IN eodem triangulo datus ſit arcus <emph style="sc">Ab</emph>, recto angulo C, oppoſitus, & </s> <s xml:id="echoid-s14301" xml:space="preserve">in ſuper <lb/>angulus B. </s> <s xml:id="echoid-s14302" xml:space="preserve">Dico dari quoque arcum AC, angulo B, oppoſitum. </s> <s xml:id="echoid-s14303" xml:space="preserve">Cum enim ſit, vt ſinu@ <lb/> <anchor type="note" xlink:label="note-424-04a" xlink:href="note-424-04"/> arcus <emph style="sc">Ab</emph>, ad ſinum totum anguli recti C, ita ſinus arcus <emph style="sc">A</emph>C, ad ſinum anguli <emph style="sc">B</emph> @ <lb/>erit conuertendo, vt ſinus totus ad ſinum arcus <emph style="sc">Ab</emph>, ita ſinus anguli B, ad ſinum <lb/>arcus <emph style="sc">Ac</emph>.</s> <s xml:id="echoid-s14304" xml:space="preserve"/> </p> <div xml:id="echoid-div1133" type="float" level="2" n="1"> <note position="left" xlink:label="note-424-04" xlink:href="note-424-04a" xml:space="preserve">41. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14305" xml:space="preserve">SI igitur fiat, vt ſinus totus ad ſinum arcus angulo recto oppoſiti, ita <lb/> <anchor type="note" xlink:label="note-424-05a" xlink:href="note-424-05"/> ſinus anguli dati ad aliud, inuenietur ſinus arcus quæſiti.</s> <s xml:id="echoid-s14306" xml:space="preserve"/> </p> <div xml:id="echoid-div1134" type="float" level="2" n="2"> <note position="left" xlink:label="note-424-05" xlink:href="note-424-05a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s14307" xml:space="preserve">Hic autem arcus erit quadrante minor, ſi datus angulus eſt acutus: </s> <s xml:id="echoid-s14308" xml:space="preserve">quadrante <lb/> <anchor type="note" xlink:label="note-424-06a" xlink:href="note-424-06"/> autem maior, ſi obtuſus.</s> <s xml:id="echoid-s14309" xml:space="preserve"/> </p> <div xml:id="echoid-div1135" type="float" level="2" n="3"> <note position="left" xlink:label="note-424-06" xlink:href="note-424-06a" xml:space="preserve">34. huius.</note> </div> </div> <div xml:id="echoid-div1137" type="section" level="1" n="554"> <head xml:id="echoid-head589" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s14310" xml:space="preserve">IN triangulo ſphærico rectangulo, dato alterutro arcuũ circa an-<lb/>gulum rectũ, & </s> <s xml:id="echoid-s14311" xml:space="preserve">angulo, qui ei opponitur; </s> <s xml:id="echoid-s14312" xml:space="preserve">inuenire arcũ recto angulo <lb/>oppoſitum. </s> <s xml:id="echoid-s14313" xml:space="preserve">Oportet autem conſtare, num tertius angulus ſitacutus, <lb/>an obtuſus: </s> <s xml:id="echoid-s14314" xml:space="preserve">vel an tertius arcus ſit quadrante minor, aut maior.</s> <s xml:id="echoid-s14315" xml:space="preserve"/> </p> <pb o="413" file="425" n="425" rhead=""/> <p style="it"> <s xml:id="echoid-s14316" xml:space="preserve">IN eodem triangulo datus ſit arcus AC, circa angulum C, rectum, & </s> <s xml:id="echoid-s14317" xml:space="preserve">angulus <lb/>præterea B, illi oppoſitus. </s> <s xml:id="echoid-s14318" xml:space="preserve">Dico dari quoque arcum AB, recto angulo oppoſitum, <lb/>Cum enim ſit, vt ſinus arcus AC, ad ſinum anguli <emph style="sc">B</emph>, ita ſinus arcus AB, ad ſinum <lb/> <anchor type="note" xlink:label="note-425-01a" xlink:href="note-425-01"/> totum anguli recti C; </s> <s xml:id="echoid-s14319" xml:space="preserve">erit conuertendo, vt ſinus anguli B, dati ad ſinum arcus <emph style="sc">AC</emph>, <lb/>dati, ita ſinus totus ad ſinum arcus AB, recto angulo oppoſiti, qui quæritur.</s> <s xml:id="echoid-s14320" xml:space="preserve"/> </p> <div xml:id="echoid-div1137" type="float" level="2" n="1"> <note position="right" xlink:label="note-425-01" xlink:href="note-425-01a" xml:space="preserve">41. huius</note> </div> <p style="it"> <s xml:id="echoid-s14321" xml:space="preserve">SI igitur fiat, vt ſinus anguli dati ad ſinum dati arcus, ita ſinus to-<lb/> <anchor type="note" xlink:label="note-425-02a" xlink:href="note-425-02"/> tus ad aliud, reperietur ſinus arcus quæſiti, qui recto angulo opponitur.</s> <s xml:id="echoid-s14322" xml:space="preserve"/> </p> <div xml:id="echoid-div1138" type="float" level="2" n="2"> <note position="right" xlink:label="note-425-02" xlink:href="note-425-02a" xml:space="preserve">Praxia.</note> </div> <p style="it"> <s xml:id="echoid-s14323" xml:space="preserve">OPORTET autem conſtare, num tertius angulus A, acutus ſit, an obtuſus: <lb/></s> <s xml:id="echoid-s14324" xml:space="preserve">vel an tertius arcus <emph style="sc">C</emph>B, quadrante minor ſit, aut maior. </s> <s xml:id="echoid-s14325" xml:space="preserve">Hinc enim diſcemus, quan <lb/>do arcus quæſitus <emph style="sc">Ab</emph>, eſt quadrante minor, & </s> <s xml:id="echoid-s14326" xml:space="preserve">quando maior; </s> <s xml:id="echoid-s14327" xml:space="preserve">ſi aliunde id non con-<lb/>ſtiterit. </s> <s xml:id="echoid-s14328" xml:space="preserve">Nam ſi angulus A, fuerit acutus, ſi quidem & </s> <s xml:id="echoid-s14329" xml:space="preserve">B, datus ſit acutus: </s> <s xml:id="echoid-s14330" xml:space="preserve">Vel ſi A, <lb/>fuerit obtuſus, ſi quidem & </s> <s xml:id="echoid-s14331" xml:space="preserve"><emph style="sc">B</emph>, datus ſit obtuſus; </s> <s xml:id="echoid-s14332" xml:space="preserve">erit arcus <emph style="sc">Ab</emph>, recto angulo op-<lb/> <anchor type="note" xlink:label="note-425-03a" xlink:href="note-425-03"/> poſitus quadrante minor. </s> <s xml:id="echoid-s14333" xml:space="preserve">Si vero angulus A, fuerit acutus, at <emph style="sc">B</emph>, datus obtuſus: </s> <s xml:id="echoid-s14334" xml:space="preserve">Vel <lb/>ſi A, fuerit obtuſus, at B, datus acutus; </s> <s xml:id="echoid-s14335" xml:space="preserve">erit arcus <emph style="sc">A</emph>B, maior quadrante. </s> <s xml:id="echoid-s14336" xml:space="preserve">Ita etiã, <lb/>ſi arcus <emph style="sc">Cb</emph>, fuerit quadrante minor, ſi quidem & </s> <s xml:id="echoid-s14337" xml:space="preserve"><emph style="sc">A</emph>C, datus ſit quadrante minor: <lb/></s> <s xml:id="echoid-s14338" xml:space="preserve">Vel ſi CB, fuerit quadrante maior, ſi quidem & </s> <s xml:id="echoid-s14339" xml:space="preserve"><emph style="sc">Ac</emph>, datus ſit maior quadrante; </s> <s xml:id="echoid-s14340" xml:space="preserve"><lb/>erit arcus AB, recto angulo oppoſitus quadrante minor. </s> <s xml:id="echoid-s14341" xml:space="preserve">Si vero arcus <emph style="sc">Cb</emph>, fuerit <lb/> <anchor type="note" xlink:label="note-425-04a" xlink:href="note-425-04"/> minor quadrante, at AC, datus quadrante maior: </s> <s xml:id="echoid-s14342" xml:space="preserve">Vel ſi CB, fuerit quadrante ma-<lb/>ior, at AC, datus quadrante minor; </s> <s xml:id="echoid-s14343" xml:space="preserve">erit arcus AB, maior quadrante. </s> <s xml:id="echoid-s14344" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s14345" xml:space="preserve">non <lb/>ſatis eſt, dari vnum arcum circa rectum angulum, cum angulo oppoſito, vt vult Co-<lb/>pernicus propoſ. </s> <s xml:id="echoid-s14346" xml:space="preserve">4. </s> <s xml:id="echoid-s14347" xml:space="preserve">de triangulis ſphæricis: </s> <s xml:id="echoid-s14348" xml:space="preserve">Id quod in ſcholio propoſ. </s> <s xml:id="echoid-s14349" xml:space="preserve">21. </s> <s xml:id="echoid-s14350" xml:space="preserve">ſupra ad-<lb/>monuimus; </s> <s xml:id="echoid-s14351" xml:space="preserve">ſed dari etiam debet ſpecies tertij anguli, vel tertij arcus. </s> <s xml:id="echoid-s14352" xml:space="preserve">Qua in re etiam <lb/>lapſus eſt Ioan. </s> <s xml:id="echoid-s14353" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s14354" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14355" xml:space="preserve">27. </s> <s xml:id="echoid-s14356" xml:space="preserve">lib. </s> <s xml:id="echoid-s14357" xml:space="preserve">4. </s> <s xml:id="echoid-s14358" xml:space="preserve">triangulorum.</s> <s xml:id="echoid-s14359" xml:space="preserve"/> </p> <div xml:id="echoid-div1139" type="float" level="2" n="3"> <note position="right" xlink:label="note-425-03" xlink:href="note-425-03a" xml:space="preserve">37. huius.</note> <note position="right" xlink:label="note-425-04" xlink:href="note-425-04a" xml:space="preserve">35. huius.</note> </div> </div> <div xml:id="echoid-div1141" type="section" level="1" n="555"> <head xml:id="echoid-head590" xml:space="preserve">THEOR. 40. PROPOS. 42.</head> <p> <s xml:id="echoid-s14360" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius nullus arcuum quadrans ſit, ſinus vtriuſlibet <lb/>reliquorum angulorum eandem habet propor-<lb/>tionem ad ſinum totum, quam ſinus complemen-<lb/>ti reliqui anguli ad ſinum complementi arcus ip-<lb/>ſum ſubtendentis.</s> <s xml:id="echoid-s14361" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14362" xml:space="preserve">IN triangulo ſphærico ABC, angulus B, <lb/>ſit rectus, & </s> <s xml:id="echoid-s14363" xml:space="preserve">nullus arcuum quadrans. </s> <s xml:id="echoid-s14364" xml:space="preserve">Dico ita <lb/> <anchor type="figure" xlink:label="fig-425-01a" xlink:href="fig-425-01"/> eſſe ſinum anguli C, ad ſinum totum, vt eſt ſi-<lb/>nus complementi reliqui anguli A, ad ſinum <lb/>complementi arcus BC, angulum A, ſubten-<lb/>dentis. </s> <s xml:id="echoid-s14365" xml:space="preserve">Quoniam enim nullus arcuum ponitur <lb/>quadrans, nullus reliquorum angulorum re-<lb/>ctus erit: </s> <s xml:id="echoid-s14366" xml:space="preserve">Alias triangulum ABC, duos habens <lb/> <anchor type="note" xlink:label="note-425-05a" xlink:href="note-425-05"/> angulos rectos haberet duos arcus quadrantes; <lb/></s> <s xml:id="echoid-s14367" xml:space="preserve">quod non ponitur. </s> <s xml:id="echoid-s14368" xml:space="preserve">Sit ergo primum angulus <lb/>A, acutus, & </s> <s xml:id="echoid-s14369" xml:space="preserve">arcus AB, ipſi, & </s> <s xml:id="echoid-s14370" xml:space="preserve">recto angulo B, <pb o="414" file="426" n="426" rhead=""/> adiacens, quadrante minor. </s> <s xml:id="echoid-s14371" xml:space="preserve">Quo poſito, erit & </s> <s xml:id="echoid-s14372" xml:space="preserve">angulus C, acutus: </s> <s xml:id="echoid-s14373" xml:space="preserve">atque adeo <lb/> <anchor type="note" xlink:label="note-426-01a" xlink:href="note-426-01"/> omnes arcus trianguli ABC, quadrante minores. </s> <s xml:id="echoid-s14374" xml:space="preserve">Producantur arcus AB, <lb/> <anchor type="note" xlink:label="note-426-02a" xlink:href="note-426-02"/> AC, & </s> <s xml:id="echoid-s14375" xml:space="preserve">fiant quadrantes AD, AE, ac per <lb/>puncta D, E, arcus DE, circuli maximi <lb/>ducatur DE, conueniens cum arcu BC, <lb/> <anchor type="figure" xlink:label="fig-426-01a" xlink:href="fig-426-01"/> <anchor type="note" xlink:label="note-426-03a" xlink:href="note-426-03"/> <anchor type="note" xlink:label="note-426-04a" xlink:href="note-426-04"/> producto in F; </s> <s xml:id="echoid-s14376" xml:space="preserve">Erit ergo vterq; </s> <s xml:id="echoid-s14377" xml:space="preserve">angulus D, <lb/>E, rectus, ob quadrantes AD, AE; </s> <s xml:id="echoid-s14378" xml:space="preserve">atque <lb/>adeo, cum & </s> <s xml:id="echoid-s14379" xml:space="preserve">angulus B, rectus ſit, vterque <lb/>arcus BF, DF, quadrans erit, ob angulos <lb/> <anchor type="note" xlink:label="note-426-05a" xlink:href="note-426-05"/> rectos B, D. </s> <s xml:id="echoid-s14380" xml:space="preserve">Erit quoque DE, arcus anguli <lb/>A; </s> <s xml:id="echoid-s14381" xml:space="preserve">propterea quòd A, polus eſt arcus DE, <lb/> <anchor type="note" xlink:label="note-426-06a" xlink:href="note-426-06"/> ob quadrantes AD, AE. </s> <s xml:id="echoid-s14382" xml:space="preserve">Item arcus EF, <lb/>complementum erit arcus DE, & </s> <s xml:id="echoid-s14383" xml:space="preserve">arcus <lb/>CF, complementum arcus BC, ob qua-<lb/>drantes DF, BF. </s> <s xml:id="echoid-s14384" xml:space="preserve">Quoniam vero in triangu-<lb/>lo CEF, angulus E, rectus eſt; </s> <s xml:id="echoid-s14385" xml:space="preserve">erit vt ſinus arcus CF, ad ſinum totum, ita <lb/> <anchor type="note" xlink:label="note-426-07a" xlink:href="note-426-07"/> ſinus arcus EF, ad ſinum anguli ECF: </s> <s xml:id="echoid-s14386" xml:space="preserve">& </s> <s xml:id="echoid-s14387" xml:space="preserve">conuertendo, vt ſinus anguli <lb/> <anchor type="note" xlink:label="note-426-08a" xlink:href="note-426-08"/> ECF, hoc eſt, anguli ACB, qui illi æqualis eſt ad verticem, ad ſinum ar-<lb/>cus EF, ita ſinus totus ad ſinum arcus CF: </s> <s xml:id="echoid-s14388" xml:space="preserve">& </s> <s xml:id="echoid-s14389" xml:space="preserve">permutando, vt ſinus anguli <lb/>ACB, ad ſinum totum, ita ſinus arcus EF, hoc eſt, ſinus complemen-<lb/>ti anguli A, ad ſinum arcus CF, id eſt, ad ſinum complementi arcus CB. <lb/></s> <s xml:id="echoid-s14390" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14391" xml:space="preserve"/> </p> <div xml:id="echoid-div1141" type="float" level="2" n="1"> <figure xlink:label="fig-425-01" xlink:href="fig-425-01a"> <image file="425-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/425-01"/> </figure> <note position="right" xlink:label="note-425-05" xlink:href="note-425-05a" xml:space="preserve">Schol. 25 <lb/>huius.</note> <note position="left" xlink:label="note-426-01" xlink:href="note-426-01a" xml:space="preserve">33. huius.</note> <note position="left" xlink:label="note-426-02" xlink:href="note-426-02a" xml:space="preserve">28. huius.</note> <figure xlink:label="fig-426-01" xlink:href="fig-426-01a"> <image file="426-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/426-01"/> </figure> <note position="left" xlink:label="note-426-03" xlink:href="note-426-03a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-426-04" xlink:href="note-426-04a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-426-05" xlink:href="note-426-05a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-426-06" xlink:href="note-426-06a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-426-07" xlink:href="note-426-07a" xml:space="preserve">Coroll. 41. <lb/>huius.</note> <note position="left" xlink:label="note-426-08" xlink:href="note-426-08a" xml:space="preserve">6. huius.</note> </div> <p> <s xml:id="echoid-s14392" xml:space="preserve">SIT deinde angulus A, obtuſus, & </s> <s xml:id="echoid-s14393" xml:space="preserve">adhuc arcus AB, quadrante minor. <lb/></s> <s xml:id="echoid-s14394" xml:space="preserve">Fiat angulus BAD, rectus, ſecetq́ue arcus AD, arcum BC, in D. </s> <s xml:id="echoid-s14395" xml:space="preserve">Producto <lb/>quoque arcu AB, fiat quadrans AE, & </s> <s xml:id="echoid-s14396" xml:space="preserve">per puncta E, D, ducatur arcus ED, <lb/> <anchor type="note" xlink:label="note-426-09a" xlink:href="note-426-09"/> circuli maximi ſecans arcum AC, in F. </s> <s xml:id="echoid-s14397" xml:space="preserve">Et quia <lb/> <anchor type="figure" xlink:label="fig-426-02a" xlink:href="fig-426-02"/> duo anguli DAB, DBA, recti ſunt, erunt ar-<lb/> <anchor type="note" xlink:label="note-426-10a" xlink:href="note-426-10"/> cus AD, BD, quadrantes; </s> <s xml:id="echoid-s14398" xml:space="preserve">atque adeo cum <lb/>AE, quadrans quoque ſit, & </s> <s xml:id="echoid-s14399" xml:space="preserve">angulus DAE, <lb/> <anchor type="note" xlink:label="note-426-11a" xlink:href="note-426-11"/> rectus, erit & </s> <s xml:id="echoid-s14400" xml:space="preserve">DE, quadrans, & </s> <s xml:id="echoid-s14401" xml:space="preserve">A, polus arcus <lb/>DE. </s> <s xml:id="echoid-s14402" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14403" xml:space="preserve">arcus AF, quadrans erit, cum ar-<lb/> <anchor type="note" xlink:label="note-426-12a" xlink:href="note-426-12"/> cus EF, quadrante ſem per ab ſit à ſuo polo. </s> <s xml:id="echoid-s14404" xml:space="preserve">An <lb/> <anchor type="note" xlink:label="note-426-13a" xlink:href="note-426-13"/> gulus item vterque ad F, cum arcus AF, tran-<lb/> <anchor type="note" xlink:label="note-426-14a" xlink:href="note-426-14"/> ſeat per A, polum arcus EF, rectus erit. </s> <s xml:id="echoid-s14405" xml:space="preserve">Præ-<lb/>terea EF, erit arcus anguli BAC, ob quadran-<lb/>tes AE, AF. </s> <s xml:id="echoid-s14406" xml:space="preserve">Erit quoque arcus DF, comple-<lb/>tum arcus EF, ſeu anguli BAC; </s> <s xml:id="echoid-s14407" xml:space="preserve">& </s> <s xml:id="echoid-s14408" xml:space="preserve">arcus CD, <lb/>complementum arcus BC, ob quadrantes DE, <lb/>BD. </s> <s xml:id="echoid-s14409" xml:space="preserve">Quoniam igitur in triangulo CDF, an-<lb/> <anchor type="note" xlink:label="note-426-15a" xlink:href="note-426-15"/> gulus F, rectus eſt, erit vt ſinus arcus CD, ad ſinum totum, ita ſinus ar-<lb/>cus DF, ad ſinum anguli C: </s> <s xml:id="echoid-s14410" xml:space="preserve">& </s> <s xml:id="echoid-s14411" xml:space="preserve">conuertendo, vt ſinus anguli C, ad ſi-<lb/>num arcus DF. </s> <s xml:id="echoid-s14412" xml:space="preserve">ita ſinus totus ad ſinum arcus CD: </s> <s xml:id="echoid-s14413" xml:space="preserve">& </s> <s xml:id="echoid-s14414" xml:space="preserve">permutando, vt ſinus <lb/>anguli C, ad ſinum totum, ita ſinus arcus DE, hoc eſt, ſinus complemen ti <lb/>anguli BAC, ad ſinum arcus CD, id eſt, ad ſinum complementi arcus BC. <lb/></s> <s xml:id="echoid-s14415" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14416" xml:space="preserve"/> </p> <div xml:id="echoid-div1142" type="float" level="2" n="2"> <note position="left" xlink:label="note-426-09" xlink:href="note-426-09a" xml:space="preserve">20. 1 Theod.</note> <figure xlink:label="fig-426-02" xlink:href="fig-426-02a"> <image file="426-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/426-02"/> </figure> <note position="left" xlink:label="note-426-10" xlink:href="note-426-10a" xml:space="preserve">Schol. 25. <lb/>huius.</note> <note position="left" xlink:label="note-426-11" xlink:href="note-426-11a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-426-12" xlink:href="note-426-12a" xml:space="preserve">Coroll. 16.</note> <note position="left" xlink:label="note-426-13" xlink:href="note-426-13a" xml:space="preserve">2. Theod.</note> <note position="left" xlink:label="note-426-14" xlink:href="note-426-14a" xml:space="preserve">25. 1. Theod.</note> <note position="left" xlink:label="note-426-15" xlink:href="note-426-15a" xml:space="preserve">Corollar. <lb/>41. huius.</note> </div> <p> <s xml:id="echoid-s14417" xml:space="preserve">SIT tertio angulus A, acutus, & </s> <s xml:id="echoid-s14418" xml:space="preserve">arcus AB, quadrante maior. </s> <s xml:id="echoid-s14419" xml:space="preserve">Quo poſi-<lb/>to, erit reliquus angulus C, obtuſus; </s> <s xml:id="echoid-s14420" xml:space="preserve">ac proinde arcus AC, rectum angulum <lb/> <anchor type="note" xlink:label="note-426-16a" xlink:href="note-426-16"/> B, ſubtendens quadrante quoque maior. </s> <s xml:id="echoid-s14421" xml:space="preserve">Abſcindantur quadrantes AD, AE, <lb/> <anchor type="note" xlink:label="note-426-17a" xlink:href="note-426-17"/> <pb o="415" file="427" n="427" rhead=""/> & </s> <s xml:id="echoid-s14422" xml:space="preserve">per puncta D, E, ducatur arcus DE, circuli maximi coiens cum arcu BC, <lb/> <anchor type="note" xlink:label="note-427-01a" xlink:href="note-427-01"/> protracto in F; </s> <s xml:id="echoid-s14423" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s14424" xml:space="preserve">vterq; </s> <s xml:id="echoid-s14425" xml:space="preserve">angulus D, E, re-<lb/> <anchor type="note" xlink:label="note-427-02a" xlink:href="note-427-02"/> ctus, ob quadrantes AD, AE; </s> <s xml:id="echoid-s14426" xml:space="preserve">atque idcirco, <lb/> <anchor type="figure" xlink:label="fig-427-01a" xlink:href="fig-427-01"/> cum & </s> <s xml:id="echoid-s14427" xml:space="preserve">angulus B, ſit rectus, quadrantes erunt <lb/> <anchor type="note" xlink:label="note-427-03a" xlink:href="note-427-03"/> BF, DF. </s> <s xml:id="echoid-s14428" xml:space="preserve">Erit quoque DE, arcus anguli A, <lb/>quòd A, polus ſit arcus DE. </s> <s xml:id="echoid-s14429" xml:space="preserve">Quare EF, com-<lb/> <anchor type="note" xlink:label="note-427-04a" xlink:href="note-427-04"/> plementum eſt anguli A; </s> <s xml:id="echoid-s14430" xml:space="preserve">& </s> <s xml:id="echoid-s14431" xml:space="preserve">CF, complemen-<lb/>tum arcus BC, ob quadrantes DF, BF. <lb/></s> <s xml:id="echoid-s14432" xml:space="preserve">Quoniam igitur in triangulo CEF, angulus <lb/>E, rectus eſt, erit vt ſinus arcus CF, ad ſinum <lb/> <anchor type="note" xlink:label="note-427-05a" xlink:href="note-427-05"/> totum, ita ſinus arcus EF, ad ſinum anguli <lb/>ECF: </s> <s xml:id="echoid-s14433" xml:space="preserve">& </s> <s xml:id="echoid-s14434" xml:space="preserve">conuertendo vt ſinus anguli ECF, <lb/>hoc eſt, anguli ACB. </s> <s xml:id="echoid-s14435" xml:space="preserve">(Habent enim duo an-<lb/>guli ad C, eundem ſinum, cum eorum arcus <lb/>ſemicirculum conficiant, ex coroll. </s> <s xml:id="echoid-s14436" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s14437" xml:space="preserve">5.) </s> <s xml:id="echoid-s14438" xml:space="preserve">ad ſinum complementi anguli A, ita ſinus totus ad ſinum arcus CF, hoc <lb/>eſt, ad ſinum complementi arcus BC: </s> <s xml:id="echoid-s14439" xml:space="preserve">& </s> <s xml:id="echoid-s14440" xml:space="preserve">per mutãdo, vt ſinus anguli ACB, ad <lb/>ſinum totum, ita ſinus arcus EF, ſiue complementi anguli A, ad ſinum arcus <lb/>CF, ſeu complementi arcus BC. </s> <s xml:id="echoid-s14441" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14442" xml:space="preserve"/> </p> <div xml:id="echoid-div1143" type="float" level="2" n="3"> <note position="left" xlink:label="note-426-16" xlink:href="note-426-16a" xml:space="preserve">33. huius.</note> <note position="left" xlink:label="note-426-17" xlink:href="note-426-17a" xml:space="preserve">37. huius.</note> <note position="right" xlink:label="note-427-01" xlink:href="note-427-01a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-427-02" xlink:href="note-427-02a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-427-01" xlink:href="fig-427-01a"> <image file="427-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/427-01"/> </figure> <note position="right" xlink:label="note-427-03" xlink:href="note-427-03a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-427-04" xlink:href="note-427-04a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-427-05" xlink:href="note-427-05a" xml:space="preserve">Coroll. 41. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s14443" xml:space="preserve">QVARTO ac vltimo ſit angulus A, obtuſus, & </s> <s xml:id="echoid-s14444" xml:space="preserve">adhuc arcus AB, qua-<lb/>drante maior. </s> <s xml:id="echoid-s14445" xml:space="preserve">Fiat angulus rectus BAD, ſecetq́; </s> <s xml:id="echoid-s14446" xml:space="preserve">arcus AD, arcum BC, in <lb/>D. </s> <s xml:id="echoid-s14447" xml:space="preserve">Abſcindatur quoque ex AB, quadrans AE, & </s> <s xml:id="echoid-s14448" xml:space="preserve">per puncta E, D, ducatur <lb/> <anchor type="note" xlink:label="note-427-06a" xlink:href="note-427-06"/> arcus ED, circuli maximi ſecans arcum AC, productum in F. </s> <s xml:id="echoid-s14449" xml:space="preserve">Et quia an-<lb/>gulus B, rectus ponitur, & </s> <s xml:id="echoid-s14450" xml:space="preserve">angulus BAD, rectus quoque ex conſtructio-<lb/>ne eſt, erunt arcus AD, BD, quadrantes. </s> <s xml:id="echoid-s14451" xml:space="preserve">Rurſus quia arcus AD, AE, <lb/> <anchor type="note" xlink:label="note-427-07a" xlink:href="note-427-07"/> quadrantes ſunt, continentque angulum DAE, rectum, erit arcus DE, qua-<lb/> <anchor type="note" xlink:label="note-427-08a" xlink:href="note-427-08"/> drans, & </s> <s xml:id="echoid-s14452" xml:space="preserve">A, polus arcus DE; </s> <s xml:id="echoid-s14453" xml:space="preserve">ac proinde cum <lb/> <anchor type="figure" xlink:label="fig-427-02a" xlink:href="fig-427-02"/> arcus AF, tranſeat per A, polum arcus EF, <lb/>erit angulus F, rectus. </s> <s xml:id="echoid-s14454" xml:space="preserve">Item EF, erit arcus an-<lb/> <anchor type="note" xlink:label="note-427-09a" xlink:href="note-427-09"/> guli BAC. </s> <s xml:id="echoid-s14455" xml:space="preserve">Præterea arcus DF, complemen-<lb/>tum erit arcus EF, ſeu anguli BAC; </s> <s xml:id="echoid-s14456" xml:space="preserve">& </s> <s xml:id="echoid-s14457" xml:space="preserve">arcus <lb/>CD, complementum arcus BC, ob quadran-<lb/>tes DE, BD. </s> <s xml:id="echoid-s14458" xml:space="preserve">Quoniam igitur in triangulo <lb/>CDF, angulus F, rectus eſt, erit vt ſinus arcus <lb/>CD, ad ſinum totũ ita ſinus arcus DF, ad ſinum <lb/> <anchor type="note" xlink:label="note-427-10a" xlink:href="note-427-10"/> anguli DCF: </s> <s xml:id="echoid-s14459" xml:space="preserve">Et conuertendo, vt ſinus anguli <lb/>DCF, hoc eſt, anguli ACB, (Habent enim ar-<lb/>cus angulorum DCF, ACB, eundem ſinum, <lb/>cum ſemicirculum cõſtituant) ad ſinum arcus <lb/> <anchor type="note" xlink:label="note-427-11a" xlink:href="note-427-11"/> DF, hoc eſt, ad ſinum complementi anguli BAC, ita ſinus totus ad ſinum ar-<lb/>cus CD, hoc eſt, ad ſinum complementi arcus BC: </s> <s xml:id="echoid-s14460" xml:space="preserve">Et permutando, vt ſinus <lb/>anguli ACB, ad ſinum totum, ita ſinus arcus DF, ſiue complementi anguli <lb/>BAC, ad ſinum arcus CD, ſeu complementi arcus BC. </s> <s xml:id="echoid-s14461" xml:space="preserve">Quod eſt propoſi-<lb/>lum. </s> <s xml:id="echoid-s14462" xml:space="preserve">Igitur in omni triangulo ſphærico rectangulo. </s> <s xml:id="echoid-s14463" xml:space="preserve">&</s> <s xml:id="echoid-s14464" xml:space="preserve">c. </s> <s xml:id="echoid-s14465" xml:space="preserve">Quod demonſtran-<lb/>dum erat.</s> <s xml:id="echoid-s14466" xml:space="preserve"/> </p> <div xml:id="echoid-div1144" type="float" level="2" n="4"> <note position="right" xlink:label="note-427-06" xlink:href="note-427-06a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-427-07" xlink:href="note-427-07a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-427-08" xlink:href="note-427-08a" xml:space="preserve">26. huius.</note> <figure xlink:label="fig-427-02" xlink:href="fig-427-02a"> <image file="427-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/427-02"/> </figure> <note position="right" xlink:label="note-427-09" xlink:href="note-427-09a" xml:space="preserve">15. 1. Theo.</note> <note position="right" xlink:label="note-427-10" xlink:href="note-427-10a" xml:space="preserve">Coroll. 41. <lb/>huius.</note> <note position="right" xlink:label="note-427-11" xlink:href="note-427-11a" xml:space="preserve">Coroll. 5. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1146" type="section" level="1" n="556"> <head xml:id="echoid-head591" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s14467" xml:space="preserve">COLLIGEMVS ex hac propoſitione duo hæc problemata.</s> <s xml:id="echoid-s14468" xml:space="preserve"/> </p> <pb o="416" file="428" n="428" rhead=""/> </div> <div xml:id="echoid-div1147" type="section" level="1" n="557"> <head xml:id="echoid-head592" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s14469" xml:space="preserve">IN triangulo ſphętico rectangulo, datis duobus angulis non <lb/>rectis; </s> <s xml:id="echoid-s14470" xml:space="preserve">inuenire arcum vtrilibet eorum oppoſitum, vna cum arcu, <lb/>qui recto angulo opponitur.</s> <s xml:id="echoid-s14471" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14472" xml:space="preserve">IN triangulo <emph style="sc">Ab</emph>C, cuius angulus C, rectus, dati ſint anguli A, <emph style="sc">B</emph>. </s> <s xml:id="echoid-s14473" xml:space="preserve">Dice <lb/>vtrumuis arcuum <emph style="sc">AC</emph>, BC, quoque dari, cum arcu <lb/> <anchor type="figure" xlink:label="fig-428-01a" xlink:href="fig-428-01"/> AB. </s> <s xml:id="echoid-s14474" xml:space="preserve">Quoniam enim eſt, vt ſinus anguli A, ad ſinum <lb/>totum, ita ſinus complementi anguli B, ad ſinum com-<lb/> <anchor type="note" xlink:label="note-428-01a" xlink:href="note-428-01"/> plementi arcus <emph style="sc">A</emph>C. </s> <s xml:id="echoid-s14475" xml:space="preserve">Item, vt ſinus anguli <emph style="sc">B</emph>, ad ſinum <lb/>totum, ita ſinus complementi anguli <emph style="sc">A</emph>, ad ſinum com-<lb/>plementi arcus BC;</s> <s xml:id="echoid-s14476" xml:space="preserve"/> </p> <div xml:id="echoid-div1147" type="float" level="2" n="1"> <figure xlink:label="fig-428-01" xlink:href="fig-428-01a"> <image file="428-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/428-01"/> </figure> <note position="left" xlink:label="note-428-01" xlink:href="note-428-01a" xml:space="preserve">41. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14477" xml:space="preserve">SI fiat, vt ſinus anguli dati, qui quæſito la <lb/> <anchor type="note" xlink:label="note-428-02a" xlink:href="note-428-02"/> teri adiacet, ad ſinum totum, ita ſinus com-<lb/>plementi reliqui anguli dati ad aliud, produ-<lb/>cetur ſinus complementi arcus huic posteriori angulo oppoſiti, qui quæ-<lb/>ritur. </s> <s xml:id="echoid-s14478" xml:space="preserve">Inuento autem vtroque arcu circa angulum rectum, reperietur <lb/>quoque ex vtrolibet illorum, & </s> <s xml:id="echoid-s14479" xml:space="preserve">ex angulo, qui ei opponuntur dato, ar-<lb/>cus recto angulo oppoſitus, vt in problemate 3. </s> <s xml:id="echoid-s14480" xml:space="preserve">propoſitionis 41. </s> <s xml:id="echoid-s14481" xml:space="preserve">oſten-<lb/>dimus.</s> <s xml:id="echoid-s14482" xml:space="preserve"/> </p> <div xml:id="echoid-div1148" type="float" level="2" n="2"> <note position="left" xlink:label="note-428-02" xlink:href="note-428-02a" xml:space="preserve">Praxit.</note> </div> <p style="it"> <s xml:id="echoid-s14483" xml:space="preserve">VTRVM autem arcus AC, <emph style="sc">BC</emph>, ſint minores quadrante, aut maiores, ita <lb/>diſcemus. </s> <s xml:id="echoid-s14484" xml:space="preserve">Si angulus <emph style="sc">B</emph>, eſt acutus, erit arcus AC, ei oppoſitus quadrante minor: </s> <s xml:id="echoid-s14485" xml:space="preserve">Si <lb/> <anchor type="note" xlink:label="note-428-03a" xlink:href="note-428-03"/> vero obtuſus, quadrante maior. </s> <s xml:id="echoid-s14486" xml:space="preserve">Eadem ratione ſi angulus A, fuerit acutus, erit ar-<lb/>cus ei oppoſitus <emph style="sc">B</emph>C, quadrante minor: </s> <s xml:id="echoid-s14487" xml:space="preserve">ſi vero obtuſus, quadrante maior.</s> <s xml:id="echoid-s14488" xml:space="preserve"/> </p> <div xml:id="echoid-div1149" type="float" level="2" n="3"> <note position="left" xlink:label="note-428-03" xlink:href="note-428-03a" xml:space="preserve">14. huius.</note> </div> </div> <div xml:id="echoid-div1151" type="section" level="1" n="558"> <head xml:id="echoid-head593" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s14489" xml:space="preserve">IN triangulo ſphærico rectangulo, dato alterutro angulorum <lb/>non rectorum, cum alterutro arcuum circa angulum rectum; </s> <s xml:id="echoid-s14490" xml:space="preserve">inue-<lb/>nire alium angulum non rectum, & </s> <s xml:id="echoid-s14491" xml:space="preserve">reliquos duos arcus.</s> <s xml:id="echoid-s14492" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14493" xml:space="preserve">IN eodem triangulo datus ſit primum arcus AC, cum angulo A, ſibi adiacente. <lb/></s> <s xml:id="echoid-s14494" xml:space="preserve">Dico dari quoque angulum B, cum arcubus <emph style="sc">BC</emph>, <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s14495" xml:space="preserve">Cum enim ſit, vt ſinus an-<lb/>guli <emph style="sc">A</emph>, ad ſinum totum, ita ſinus complementi anguli <emph style="sc">B</emph>, ad ſinum complementi ar-<lb/> <anchor type="note" xlink:label="note-428-04a" xlink:href="note-428-04"/> cus <emph style="sc">A</emph>C; </s> <s xml:id="echoid-s14496" xml:space="preserve">erit conuertendo, vt ſinus totus ad ſinum anguli A, dati, ita ſinus comple-<lb/>menti dati arcus AC, ad ſinum complementi anguli <emph style="sc">B</emph>, qui quæritur.</s> <s xml:id="echoid-s14497" xml:space="preserve"/> </p> <div xml:id="echoid-div1151" type="float" level="2" n="1"> <note position="left" xlink:label="note-428-04" xlink:href="note-428-04a" xml:space="preserve">42. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14498" xml:space="preserve">QVANDO ergo datur arcus cum angulo ſibi adiacẽte, ſi fiat, vt ſi-<lb/> <anchor type="note" xlink:label="note-428-05a" xlink:href="note-428-05"/> nus totus ad ſinum anguli dati, ita ſinus complementi arcus dati ad aliud, <lb/>reperietur ſinus complementi alterius anguli, qui quæritur. </s> <s xml:id="echoid-s14499" xml:space="preserve">Hinc ex duo-<lb/>bus angulis non rectis iam cognitis, cognoſcentur reliqui duo arcus, vt <lb/>in proximè antecedenti problemate demonſtratum eſt: </s> <s xml:id="echoid-s14500" xml:space="preserve">Tertius autem da-<lb/>tus eſt ex hypotheſi.</s> <s xml:id="echoid-s14501" xml:space="preserve"/> </p> <div xml:id="echoid-div1152" type="float" level="2" n="2"> <note position="left" xlink:label="note-428-05" xlink:href="note-428-05a" xml:space="preserve">Praxis, quã <lb/>do datur <lb/>areus cum <lb/>angulo a-<lb/>diacente.</note> </div> <p style="it"> <s xml:id="echoid-s14502" xml:space="preserve">NVM vero angulus <emph style="sc">B</emph>, quæſitus ſit acutus, obtuſusue, docebit datus arcus AC. <lb/></s> <s xml:id="echoid-s14503" xml:space="preserve">Si enim fuerit quadrante minor, erit angulus <emph style="sc">B</emph>, acutus: </s> <s xml:id="echoid-s14504" xml:space="preserve">ſi vero maior quadrante, <lb/> <anchor type="note" xlink:label="note-428-06a" xlink:href="note-428-06"/> @btuſus.</s> <s xml:id="echoid-s14505" xml:space="preserve"/> </p> <div xml:id="echoid-div1153" type="float" level="2" n="3"> <note position="left" xlink:label="note-428-06" xlink:href="note-428-06a" xml:space="preserve">44. huius.</note> </div> <pb o="417" file="429" n="429" rhead=""/> <p style="it"> <s xml:id="echoid-s14506" xml:space="preserve">DATVS deinde ſit arcus AC, cum angulo B, ſibi oppoſito, conſtetq́; </s> <s xml:id="echoid-s14507" xml:space="preserve">de reliqu@ <lb/>angulo A, num acutus ſit, an obtuſus: </s> <s xml:id="echoid-s14508" xml:space="preserve">vel de altero arcu B <emph style="sc">C</emph>, circa rectum angulum, <lb/>qualis ſit. </s> <s xml:id="echoid-s14509" xml:space="preserve">Dico rur ſum dari & </s> <s xml:id="echoid-s14510" xml:space="preserve">reliquum angulũ A, & </s> <s xml:id="echoid-s14511" xml:space="preserve">reliquos arcus <emph style="sc">Bc</emph>, AB. </s> <s xml:id="echoid-s14512" xml:space="preserve">Nam <lb/>cum ſit, vt ſinus anguli A, ad ſinum totum, ita ſinus complementi anguli B, ad ſinum <lb/> <anchor type="note" xlink:label="note-429-01a" xlink:href="note-429-01"/> complementi arcus AC; </s> <s xml:id="echoid-s14513" xml:space="preserve">erit conuertendo, vt ſinus complementi arcus AC, dati ad <lb/>ſinum complementi anguli B, dati, ita ſinus totus ad ſinum anguli A, quæſiti.</s> <s xml:id="echoid-s14514" xml:space="preserve"/> </p> <div xml:id="echoid-div1154" type="float" level="2" n="4"> <note position="right" xlink:label="note-429-01" xlink:href="note-429-01a" xml:space="preserve">42. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14515" xml:space="preserve">IGITVR cum datur arcus cum angulo ſibi oppoſito, ſi fiat, vt ſi. <lb/></s> <s xml:id="echoid-s14516" xml:space="preserve"> <anchor type="note" xlink:label="note-429-02a" xlink:href="note-429-02"/> nus complementi arcus dati ad ſinum complementi anguli dati, ita ſinus <lb/>totus ad aliud, procreabitur ſinus reliqui anguli, qui quæritur. </s> <s xml:id="echoid-s14517" xml:space="preserve">Ex duo-<lb/>bus ergo angulis non rectis iam cognitis, cognoſcentur reliqui duo arcus, <lb/>vt in præcedenti problemate monſtrauimus. </s> <s xml:id="echoid-s14518" xml:space="preserve">Tertius autem per hypothe-<lb/>ſim datus est.</s> <s xml:id="echoid-s14519" xml:space="preserve"/> </p> <div xml:id="echoid-div1155" type="float" level="2" n="5"> <note position="right" xlink:label="note-429-02" xlink:href="note-429-02a" xml:space="preserve">Praxis, quã <lb/>do datur <lb/>arcus cum <lb/>angulo op-<lb/>poſito.</note> </div> <p style="it"> <s xml:id="echoid-s14520" xml:space="preserve">OPORTET autem conſtare, num reliquus angulus A, ſit acutus, an obtuſus, <lb/>vt ſciatur, qualis angulus ſinui inuento reſpondens ſit accipiendus, acutuſne, an obtu-<lb/>ſus. </s> <s xml:id="echoid-s14521" xml:space="preserve">Quòd ſi conſtaret de arcu BC, qualis ſit, illico cognoſceretur quoque ſpecies an-<lb/>guli A. </s> <s xml:id="echoid-s14522" xml:space="preserve">Nam ſi arcus <emph style="sc">B</emph> C, fuerit quadrante minor, erit angulus A, acutus: </s> <s xml:id="echoid-s14523" xml:space="preserve">ſi autem <lb/>quadrante maior, obtuſus. </s> <s xml:id="echoid-s14524" xml:space="preserve">Pari ratione, ſi ſciretur, qualis ſit arcus AB, angulo re-<lb/> <anchor type="note" xlink:label="note-429-03a" xlink:href="note-429-03"/> cto oppoſitus, continuò ſpeciem anguli A, cognoſceremus. </s> <s xml:id="echoid-s14525" xml:space="preserve">Nam ſi arcus AB, fuerit <lb/>minor quadrante, & </s> <s xml:id="echoid-s14526" xml:space="preserve">datus quidem angulus B, acutus, erit quoque angulus A, acu-<lb/> <anchor type="note" xlink:label="note-429-04a" xlink:href="note-429-04"/> tus; </s> <s xml:id="echoid-s14527" xml:space="preserve">Si vero datus angulus B, ſit obtuſus, erit quoque obtuſus angulus A. </s> <s xml:id="echoid-s14528" xml:space="preserve">At ſi arcus <lb/>AB, fuerit maior quadrante, & </s> <s xml:id="echoid-s14529" xml:space="preserve">datus quidem angulus B, acutus, erit angulus A, <lb/>obtuſus: </s> <s xml:id="echoid-s14530" xml:space="preserve">Si vero datus angulus B, ſit obtuſus, erit angulus A, acutus. </s> <s xml:id="echoid-s14531" xml:space="preserve">Itaque non <lb/>eſt ſatis, dari angulum non rectum, cum arcu oppoſito, vt vult Copernicus propoſ. </s> <s xml:id="echoid-s14532" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s14533" xml:space="preserve">de triangulis ſphæricis: </s> <s xml:id="echoid-s14534" xml:space="preserve">Id quod ſupra quoque monuimus in ſcholio propoſ. </s> <s xml:id="echoid-s14535" xml:space="preserve">21. </s> <s xml:id="echoid-s14536" xml:space="preserve">ſed <lb/>debet etiam dari ſpecies tertij anguli, vel ſpecies arcus alterius circa rectum angu-<lb/>lum; </s> <s xml:id="echoid-s14537" xml:space="preserve">vel certe ſpecies arcus recto angulo oppoſiti. </s> <s xml:id="echoid-s14538" xml:space="preserve">Qua in re lapſus eſt Nicolaus Co-<lb/> <anchor type="note" xlink:label="note-429-05a" xlink:href="note-429-05"/> pernicus, qui voluit in propoſ 4. </s> <s xml:id="echoid-s14539" xml:space="preserve">de triangulis ſphæricis, ſatis eſſe, vt detur arcus cir-<lb/>ca rectum angulum, cum alterutro angulorum non rectorum. </s> <s xml:id="echoid-s14540" xml:space="preserve">Falſum enim hoc eſt <lb/>de angulo dato arcui oppoſito, niſi aliud præterea conſtet, vt hic diximus, & </s> <s xml:id="echoid-s14541" xml:space="preserve">in ſcho-<lb/>lio propoſ. </s> <s xml:id="echoid-s14542" xml:space="preserve">21. </s> <s xml:id="echoid-s14543" xml:space="preserve">monuimus.</s> <s xml:id="echoid-s14544" xml:space="preserve"/> </p> <div xml:id="echoid-div1156" type="float" level="2" n="6"> <note position="right" xlink:label="note-429-03" xlink:href="note-429-03a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-429-04" xlink:href="note-429-04a" xml:space="preserve">38. huius.</note> <note position="right" xlink:label="note-429-05" xlink:href="note-429-05a" xml:space="preserve">Error Co-<lb/>pernici.</note> </div> </div> <div xml:id="echoid-div1158" type="section" level="1" n="559"> <head xml:id="echoid-head594" xml:space="preserve">THEOR. 41. PROPOS. 43.</head> <p> <s xml:id="echoid-s14545" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <lb/>nullus arcuum quadrans ſit, ſinus complementi <lb/>arcus rectum angulum ſubtendẽtis ad ſinum com <lb/>plementi vtriuſve reliquorum arcuum eandem ha <lb/>bet proportionem, quam ſinus complementi re-<lb/>liqui arcus ad ſinum totum.</s> <s xml:id="echoid-s14546" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14547" xml:space="preserve">IN triangulo ſphærico rectangulo ABC, angulus B, ſit rectus, & </s> <s xml:id="echoid-s14548" xml:space="preserve">nullus <pb o="418" file="430" n="430" rhead=""/> arcuum quadrans. </s> <s xml:id="echoid-s14549" xml:space="preserve">Dico ita eſſe ſinum complementi arcus AC, ad ſinum com <lb/>plementi arcus v. </s> <s xml:id="echoid-s14550" xml:space="preserve">g. </s> <s xml:id="echoid-s14551" xml:space="preserve">AB, vt eſt, ſinus complementi reliqui arcus BC, ad ſi-<lb/>num totum. </s> <s xml:id="echoid-s14552" xml:space="preserve">Quoniam enim nullus arcuum <lb/> <anchor type="figure" xlink:label="fig-430-01a" xlink:href="fig-430-01"/> ponitur quadrans, nullus reliquorum angu-<lb/>lorum erit rectus. </s> <s xml:id="echoid-s14553" xml:space="preserve">Alias triangulum ABC, <lb/>duos angulos habens rectos haberet duos ar-<lb/> <anchor type="note" xlink:label="note-430-01a" xlink:href="note-430-01"/> cus quadrantes. </s> <s xml:id="echoid-s14554" xml:space="preserve">quod non ponitur. </s> <s xml:id="echoid-s14555" xml:space="preserve">Sit ergo <lb/>primum angulus A, acutus, & </s> <s xml:id="echoid-s14556" xml:space="preserve">arcus AB, ipſi <lb/>& </s> <s xml:id="echoid-s14557" xml:space="preserve">recto angulo B, adiacens quadrante minor. <lb/></s> <s xml:id="echoid-s14558" xml:space="preserve">Quo poſito, erit & </s> <s xml:id="echoid-s14559" xml:space="preserve">angulus C, acutus; </s> <s xml:id="echoid-s14560" xml:space="preserve">atque <lb/> <anchor type="note" xlink:label="note-430-02a" xlink:href="note-430-02"/> adeo omnes arcus trianguli ABC, quadran-<lb/> <anchor type="note" xlink:label="note-430-03a" xlink:href="note-430-03"/> te minores. </s> <s xml:id="echoid-s14561" xml:space="preserve">Producantur arcus AB, AC, & </s> <s xml:id="echoid-s14562" xml:space="preserve"><lb/>fiant quadrantes AD, AE; </s> <s xml:id="echoid-s14563" xml:space="preserve">ac per puncta D, <lb/>E, arcus DE, circuli maximi ducatur DE, <lb/> <anchor type="note" xlink:label="note-430-04a" xlink:href="note-430-04"/> conueniens cum arcu BC, producto in F. </s> <s xml:id="echoid-s14564" xml:space="preserve">Erit <lb/>ergo vterque angulus D, E, rectus, ob quadrantes AD, AE; </s> <s xml:id="echoid-s14565" xml:space="preserve">atque adeo, cum <lb/> <anchor type="note" xlink:label="note-430-05a" xlink:href="note-430-05"/> & </s> <s xml:id="echoid-s14566" xml:space="preserve">angulus B, ponatur rectus, erit vterq; </s> <s xml:id="echoid-s14567" xml:space="preserve">arcus BF, DF, quadrans, ob rectos <lb/>angulos B, D. </s> <s xml:id="echoid-s14568" xml:space="preserve">Præterea BD, erit arcus anguli F; </s> <s xml:id="echoid-s14569" xml:space="preserve">propterea quòd F, polus eſt <lb/> <anchor type="note" xlink:label="note-430-06a" xlink:href="note-430-06"/> arcus BD, ob quadrantes BF, DF. </s> <s xml:id="echoid-s14570" xml:space="preserve">Item CF, complementum erit arcus BC; <lb/></s> <s xml:id="echoid-s14571" xml:space="preserve">& </s> <s xml:id="echoid-s14572" xml:space="preserve">BD, CE, complementa arcuum AB, AC, ob quadrantes BF, AD, AE. </s> <s xml:id="echoid-s14573" xml:space="preserve"><lb/>Manifeſtum autem eſt in triangulo CEF, ita eſſe ſinum arcus CE, hoc eſt, ſi-<lb/> <anchor type="note" xlink:label="note-430-07a" xlink:href="note-430-07"/> num complementi arcus AC, ad ſinum anguli F, hoc eſt, ad ſinum arcus BD, <lb/>ſeu complementi arcus AB, vt eſt ſinus arcus CF, hoc eſt, ſinus complemen <lb/>ti arcus BC, ad ſinum anguli recti E, id eſt, ad ſinum totum. </s> <s xml:id="echoid-s14574" xml:space="preserve">Quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s14575" xml:space="preserve"/> </p> <div xml:id="echoid-div1158" type="float" level="2" n="1"> <figure xlink:label="fig-430-01" xlink:href="fig-430-01a"> <image file="430-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/430-01"/> </figure> <note position="left" xlink:label="note-430-01" xlink:href="note-430-01a" xml:space="preserve">Schol. 25. <lb/>huius.</note> <note position="left" xlink:label="note-430-02" xlink:href="note-430-02a" xml:space="preserve">33. huius.</note> <note position="left" xlink:label="note-430-03" xlink:href="note-430-03a" xml:space="preserve">28. huius.</note> <note position="left" xlink:label="note-430-04" xlink:href="note-430-04a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-430-05" xlink:href="note-430-05a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-430-06" xlink:href="note-430-06a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-430-07" xlink:href="note-430-07a" xml:space="preserve">41. huius.</note> </div> <p> <s xml:id="echoid-s14576" xml:space="preserve">SIT deinde angulus A, obtuſus, & </s> <s xml:id="echoid-s14577" xml:space="preserve">adhuc arcus AB, quadrante minor. <lb/></s> <s xml:id="echoid-s14578" xml:space="preserve">Fiat angulus BAD, rectus, ſecetq́; </s> <s xml:id="echoid-s14579" xml:space="preserve">arcus AD, arcum BC, in D. </s> <s xml:id="echoid-s14580" xml:space="preserve">Producto <lb/>quoque arcu AB, fiat quadrans AE, & </s> <s xml:id="echoid-s14581" xml:space="preserve">per puncta E, D, ducatur arcus ED, <lb/> <anchor type="note" xlink:label="note-430-08a" xlink:href="note-430-08"/> circuli maximi ſecans arcum AC, in F. </s> <s xml:id="echoid-s14582" xml:space="preserve">Et quia duo anguli DAB, DBA, <lb/>recti ſunt, erunt arcus AD, BD, quadrantes; </s> <s xml:id="echoid-s14583" xml:space="preserve">atque adeo cum AE, quoque <lb/> <anchor type="note" xlink:label="note-430-09a" xlink:href="note-430-09"/> ſit quadrans, & </s> <s xml:id="echoid-s14584" xml:space="preserve">angulus DAE, rectus, erit & </s> <s xml:id="echoid-s14585" xml:space="preserve"><lb/>arcus DE, quadrans; </s> <s xml:id="echoid-s14586" xml:space="preserve">ac proinde BE, ob qua-<lb/> <anchor type="figure" xlink:label="fig-430-02a" xlink:href="fig-430-02"/> <anchor type="note" xlink:label="note-430-10a" xlink:href="note-430-10"/> drantes BD, ED, erit arcus anguli BDE, <lb/>hoc eſt, anguli CDF, qui illi ad verticem eſt <lb/> <anchor type="note" xlink:label="note-430-11a" xlink:href="note-430-11"/> æqualis. </s> <s xml:id="echoid-s14587" xml:space="preserve">Quoniam vero A, polus eſt arcus <lb/> <anchor type="note" xlink:label="note-430-12a" xlink:href="note-430-12"/> ED, erit & </s> <s xml:id="echoid-s14588" xml:space="preserve">arcus AF, quadrans, cum arcus <lb/> <anchor type="note" xlink:label="note-430-13a" xlink:href="note-430-13"/> EF, quadrante ſemper abſit à ſuo polo; </s> <s xml:id="echoid-s14589" xml:space="preserve">nec-<lb/> <anchor type="note" xlink:label="note-430-14a" xlink:href="note-430-14"/> non & </s> <s xml:id="echoid-s14590" xml:space="preserve">angulus AFE, & </s> <s xml:id="echoid-s14591" xml:space="preserve">angulus CFD, re-<lb/> <anchor type="note" xlink:label="note-430-15a" xlink:href="note-430-15"/> ctus. </s> <s xml:id="echoid-s14592" xml:space="preserve">Præterea erit arcus CE, complemen-<lb/>tum arcus AC; </s> <s xml:id="echoid-s14593" xml:space="preserve">& </s> <s xml:id="echoid-s14594" xml:space="preserve">arcus BE, complementum <lb/>arcus AB; </s> <s xml:id="echoid-s14595" xml:space="preserve">& </s> <s xml:id="echoid-s14596" xml:space="preserve">arcus CD, complementum ar-<lb/>cus BC, ob quadrantes AF, AE, BD. </s> <s xml:id="echoid-s14597" xml:space="preserve">Per-<lb/>ſpicuum autem eſt in triãgulo CDF, ita eſſe <lb/>ſinum arcus CF, hoc eſt, ſinum complementi <lb/> <anchor type="note" xlink:label="note-430-16a" xlink:href="note-430-16"/> arcus AC, ad ſinum anguli CDF, hoc eſt, ad ſinum arcus BE, ſiue comple-<lb/>menti arcus AB, vt eſt ſinus arcus CD, nempe ſinus complementi arcus BC, <lb/>ad ſinum anguli recti F, hoc eſt, ad ſinum totum. </s> <s xml:id="echoid-s14598" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14599" xml:space="preserve"/> </p> <div xml:id="echoid-div1159" type="float" level="2" n="2"> <note position="left" xlink:label="note-430-08" xlink:href="note-430-08a" xml:space="preserve">20. 1 Theod.</note> <note position="left" xlink:label="note-430-09" xlink:href="note-430-09a" xml:space="preserve">Schol. 25. <lb/>huius.</note> <figure xlink:label="fig-430-02" xlink:href="fig-430-02a"> <image file="430-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/430-02"/> </figure> <note position="left" xlink:label="note-430-10" xlink:href="note-430-10a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-430-11" xlink:href="note-430-11a" xml:space="preserve">6. huius.</note> <note position="left" xlink:label="note-430-12" xlink:href="note-430-12a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-430-13" xlink:href="note-430-13a" xml:space="preserve">Coroll. 16.</note> <note position="left" xlink:label="note-430-14" xlink:href="note-430-14a" xml:space="preserve">1. Theod.</note> <note position="left" xlink:label="note-430-15" xlink:href="note-430-15a" xml:space="preserve">25. 1. Theod.</note> <note position="left" xlink:label="note-430-16" xlink:href="note-430-16a" xml:space="preserve">41. huius.</note> </div> <p> <s xml:id="echoid-s14600" xml:space="preserve">TERTIO ſit angulus A, acutus, & </s> <s xml:id="echoid-s14601" xml:space="preserve">arcus AB, quadrante maior. </s> <s xml:id="echoid-s14602" xml:space="preserve">Quo <pb o="419" file="431" n="431" rhead=""/> poſito, erit reliquus angulus C, obtuſus;</s> <s xml:id="echoid-s14603" xml:space="preserve">; ac proinde arcus AC, rectum angu-<lb/> <anchor type="note" xlink:label="note-431-01a" xlink:href="note-431-01"/> lum B, ſubtendens quadrante quoque maior. <lb/></s> <s xml:id="echoid-s14604" xml:space="preserve"> <anchor type="note" xlink:label="note-431-02a" xlink:href="note-431-02"/> Abſcindantur quadrantes AD, AE, & </s> <s xml:id="echoid-s14605" xml:space="preserve">per <lb/> <anchor type="figure" xlink:label="fig-431-01a" xlink:href="fig-431-01"/> puncta D, E, ducatur arcus DE, circuli ma-<lb/> <anchor type="note" xlink:label="note-431-03a" xlink:href="note-431-03"/> ximi conueniens cum arcu BC, producto in <lb/>F; </s> <s xml:id="echoid-s14606" xml:space="preserve">Eritá; </s> <s xml:id="echoid-s14607" xml:space="preserve">vterque angulus D, E, rectus, ob <lb/> <anchor type="note" xlink:label="note-431-04a" xlink:href="note-431-04"/> quadrantes AD, AE; </s> <s xml:id="echoid-s14608" xml:space="preserve">atque adeo, cum & </s> <s xml:id="echoid-s14609" xml:space="preserve">an-<lb/>gulus B, rectus ſit, quadrantes erunt arcus <lb/> <anchor type="note" xlink:label="note-431-05a" xlink:href="note-431-05"/> BF, DF; </s> <s xml:id="echoid-s14610" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s14611" xml:space="preserve">BD, arcus erit anguli <lb/>F. </s> <s xml:id="echoid-s14612" xml:space="preserve">Item arcus CF, complementum erit arcus <lb/> <anchor type="note" xlink:label="note-431-06a" xlink:href="note-431-06"/> BC, & </s> <s xml:id="echoid-s14613" xml:space="preserve">arcus DB, EC, complementa arcuum <lb/>AB, AC, ob quadrantes BF, AD, AE. </s> <s xml:id="echoid-s14614" xml:space="preserve">Per-<lb/>ſpicuum eſt autem in triangulo CEF, ita eſ-<lb/>ſe ſinum arcus EC, id eſt, ſinum complemen-<lb/>ti arcus AC, ad ſinum anguli F, hoc eſt, ad ſi-<lb/>num arcus DB, hoc eſt, ad ſinum complementi arcus AB, vt eſt ſinus arcus <lb/>CF, nempe ſinus complementi arcus BC, ad ſinum anguli recti E, hoc eſt, ad <lb/>ſinum totum. </s> <s xml:id="echoid-s14615" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s14616" xml:space="preserve"/> </p> <div xml:id="echoid-div1160" type="float" level="2" n="3"> <note position="right" xlink:label="note-431-01" xlink:href="note-431-01a" xml:space="preserve">33. huius.</note> <note position="right" xlink:label="note-431-02" xlink:href="note-431-02a" xml:space="preserve">37. huius.</note> <figure xlink:label="fig-431-01" xlink:href="fig-431-01a"> <image file="431-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/431-01"/> </figure> <note position="right" xlink:label="note-431-03" xlink:href="note-431-03a" xml:space="preserve">20. 1 Theod.</note> <note position="right" xlink:label="note-431-04" xlink:href="note-431-04a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-431-05" xlink:href="note-431-05a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-431-06" xlink:href="note-431-06a" xml:space="preserve">41. huius.</note> </div> <p> <s xml:id="echoid-s14617" xml:space="preserve">POSTREMO ſit angulus A, obtuſus, & </s> <s xml:id="echoid-s14618" xml:space="preserve">adhuc arcus AB, quadrante <lb/>maior. </s> <s xml:id="echoid-s14619" xml:space="preserve">Fiat angulus rectus BAD, ſecetq́; </s> <s xml:id="echoid-s14620" xml:space="preserve">arcus AD, arcum BC, in D. </s> <s xml:id="echoid-s14621" xml:space="preserve">Ab-<lb/>ſcindatur quoque ex AB, quadrans AE, & </s> <s xml:id="echoid-s14622" xml:space="preserve">per puncta E, D, deſcribatur ar-<lb/> <anchor type="note" xlink:label="note-431-07a" xlink:href="note-431-07"/> cus ED, circuli maximi ſecans arcum AC, <lb/>productum in F. </s> <s xml:id="echoid-s14623" xml:space="preserve">Et quia angulus B, ponitur <lb/> <anchor type="figure" xlink:label="fig-431-02a" xlink:href="fig-431-02"/> rectus, & </s> <s xml:id="echoid-s14624" xml:space="preserve">angulus BAD, rectus factus eſt, e-<lb/> <anchor type="note" xlink:label="note-431-08a" xlink:href="note-431-08"/> runt arcus AD, BD, quadrantes. </s> <s xml:id="echoid-s14625" xml:space="preserve">Rurſus <lb/>quia arcus AD, AE, quadrantes ſunt, con-<lb/>tinentq́; </s> <s xml:id="echoid-s14626" xml:space="preserve">angulum rectum DAE, erit & </s> <s xml:id="echoid-s14627" xml:space="preserve">arcus <lb/>DE, quadrans, & </s> <s xml:id="echoid-s14628" xml:space="preserve">A, polus arcus ED; </s> <s xml:id="echoid-s14629" xml:space="preserve">atque <lb/> <anchor type="note" xlink:label="note-431-09a" xlink:href="note-431-09"/> adeo angulus F, rectus erit Præterea quia <lb/> <anchor type="note" xlink:label="note-431-10a" xlink:href="note-431-10"/> DB, DE, quadrantes ſunt oſtenſi, erit EB, <lb/>arcus anguli BDE, hoc eſt, anguli CDF, qui <lb/> <anchor type="note" xlink:label="note-431-11a" xlink:href="note-431-11"/> illi ad verticem eſt æqualis. </s> <s xml:id="echoid-s14630" xml:space="preserve">Item cum A, po-<lb/>lus ſit arcus EF, erit arcus AF, quadrans, <lb/> <anchor type="note" xlink:label="note-431-12a" xlink:href="note-431-12"/> quòd arcus EF, quadrãte ſemper abſit à ſuo <lb/>polo. </s> <s xml:id="echoid-s14631" xml:space="preserve">Arcus item CF, complementum erit <lb/>arcus AC; </s> <s xml:id="echoid-s14632" xml:space="preserve">& </s> <s xml:id="echoid-s14633" xml:space="preserve">arcus EB, complementum arcus AB; </s> <s xml:id="echoid-s14634" xml:space="preserve">& </s> <s xml:id="echoid-s14635" xml:space="preserve">arcus CD, complemen <lb/>tum arcus BC, ob quadrantes AF, AE, BD. </s> <s xml:id="echoid-s14636" xml:space="preserve">Maniſeſtum eſt autem in trian <lb/>gulo CDF, ita eſſe ſinum arcus CF, id eſt, ſinum complementi arcus AC, ad <lb/> <anchor type="note" xlink:label="note-431-13a" xlink:href="note-431-13"/> ſinum anguli CDF, hoc eſt, ad ſinum arcus BE, ſiue complementi arcus AB, <lb/>vt eſt ſinus arcus CD, nempe ſinus complementi arcus BC, ad ſinum anguli <lb/>recti F, hoc eſt, ad ſinum totum. </s> <s xml:id="echoid-s14637" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s14638" xml:space="preserve">In omni ergo trian-<lb/>gulo ſphærico rectangulo, &</s> <s xml:id="echoid-s14639" xml:space="preserve">c. </s> <s xml:id="echoid-s14640" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s14641" xml:space="preserve"/> </p> <div xml:id="echoid-div1161" type="float" level="2" n="4"> <note position="right" xlink:label="note-431-07" xlink:href="note-431-07a" xml:space="preserve">20. 1 Theod.</note> <figure xlink:label="fig-431-02" xlink:href="fig-431-02a"> <image file="431-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/431-02"/> </figure> <note position="right" xlink:label="note-431-08" xlink:href="note-431-08a" xml:space="preserve">25 huius.</note> <note position="right" xlink:label="note-431-09" xlink:href="note-431-09a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-431-10" xlink:href="note-431-10a" xml:space="preserve">15. 1. Theo.</note> <note position="right" xlink:label="note-431-11" xlink:href="note-431-11a" xml:space="preserve">6. huius.</note> <note position="right" xlink:label="note-431-12" xlink:href="note-431-12a" xml:space="preserve">Coroll. 16. <lb/>1. Theod.</note> <note position="right" xlink:label="note-431-13" xlink:href="note-431-13a" xml:space="preserve">41. huius</note> </div> </div> <div xml:id="echoid-div1163" type="section" level="1" n="560"> <head xml:id="echoid-head595" xml:space="preserve">SCHOLIVM. I.</head> <p style="it"> <s xml:id="echoid-s14642" xml:space="preserve">SEQVENS problema ex hac propoſ. </s> <s xml:id="echoid-s14643" xml:space="preserve">colligemus hunc in modum.</s> <s xml:id="echoid-s14644" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14645" xml:space="preserve">IN triangulo ſphærico rectangulo, datis duobus arcubus qui-<lb/>buſlibet, inuenire tertium arcũ, & </s> <s xml:id="echoid-s14646" xml:space="preserve">reliquos duos angulos non rectos.</s> <s xml:id="echoid-s14647" xml:space="preserve"/> </p> <pb o="420" file="432" n="432" rhead=""/> <p style="it"> <s xml:id="echoid-s14648" xml:space="preserve">IN triangulo ABC, cuius angulus C, rectus, dat@ <lb/> <anchor type="figure" xlink:label="fig-432-01a" xlink:href="fig-432-01"/> ſint primum duo arcus AC, CB, circa angulum rectum <lb/>C. </s> <s xml:id="echoid-s14649" xml:space="preserve">Dico dari quoque tertium arcum AB, cum duobus an-<lb/>gulis A, <emph style="sc">B</emph>. </s> <s xml:id="echoid-s14650" xml:space="preserve">Quoniam enim eſt, vt ſinus complementi ar-<lb/> <anchor type="note" xlink:label="note-432-01a" xlink:href="note-432-01"/> cus <emph style="sc">Ab</emph>, ad ſinum complementi arcus AC, ita ſinus com-<lb/>plementi arcus CB, ad ſinum totum; </s> <s xml:id="echoid-s14651" xml:space="preserve">erit conuertendo, vt <lb/>ſinus totus ad ſinum complementi arcus CB, ita ſinus com <lb/>plementi arcus AC, ad ſinum complementi arcus AB.</s> <s xml:id="echoid-s14652" xml:space="preserve"/> </p> <div xml:id="echoid-div1163" type="float" level="2" n="1"> <figure xlink:label="fig-432-01" xlink:href="fig-432-01a"> <image file="432-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/432-01"/> </figure> <note position="left" xlink:label="note-432-01" xlink:href="note-432-01a" xml:space="preserve">43. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14653" xml:space="preserve">QV AMOBREM, datis duobus arcubus rectum angulum ambien <lb/> <anchor type="note" xlink:label="note-432-02a" xlink:href="note-432-02"/> tibus, ſi fiat, vt ſinus totus ad ſinum complementi vtriuſlibet arcuum <lb/>datorum, ita ſinus complementi alterius arcus dati ad aliud, producetur <lb/>ſinus complementi arcus recto angulo oppoſiti, qui quæritur. </s> <s xml:id="echoid-s14654" xml:space="preserve">Ex dato au-<lb/>tem arcu, quirecto angulo opponitur, cum vtrouis arcu cir ca rectum an-<lb/>gulum, inuenietur angulus ci oppoſitus, vt in probtemate 1. </s> <s xml:id="echoid-s14655" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14656" xml:space="preserve">41. <lb/></s> <s xml:id="echoid-s14657" xml:space="preserve">tradidimus.</s> <s xml:id="echoid-s14658" xml:space="preserve"/> </p> <div xml:id="echoid-div1164" type="float" level="2" n="2"> <note position="left" xlink:label="note-432-02" xlink:href="note-432-02a" xml:space="preserve">Praxis, quã <lb/>do dantut <lb/>duo arcus <lb/>circa angu <lb/>lũ rectum.</note> </div> <p style="it"> <s xml:id="echoid-s14659" xml:space="preserve">VTRVM vero quæſitus arcus AB, quadrante minor ſit, aut maior, docebunt <lb/> <anchor type="note" xlink:label="note-432-03a" xlink:href="note-432-03"/> duo arcus dati, Si enim vter que fuerit minor, aut maior quadrante, erit arcus AB, <lb/>quadrante minor: </s> <s xml:id="echoid-s14660" xml:space="preserve">Si vero vnus ſit quadrante minor, & </s> <s xml:id="echoid-s14661" xml:space="preserve">alter maior, erit arcus AB, <lb/>quadrante maior.</s> <s xml:id="echoid-s14662" xml:space="preserve"/> </p> <div xml:id="echoid-div1165" type="float" level="2" n="3"> <note position="left" xlink:label="note-432-03" xlink:href="note-432-03a" xml:space="preserve">35. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14663" xml:space="preserve">DATVS deinde ſit arcus AB, recto angulo oppoſitus, cum alterutro arcuum <lb/>circa angulum rectum, vt cum AC. </s> <s xml:id="echoid-s14664" xml:space="preserve">Dico rurſum dari reliquum arcum CB, cum <lb/>duobus angulis A, B. </s> <s xml:id="echoid-s14665" xml:space="preserve">Nam cum ſit, vt ſinus complementi arcus <emph style="sc">Ab</emph>, ad ſinum com-<lb/> <anchor type="note" xlink:label="note-432-04a" xlink:href="note-432-04"/> plementi arcus AC, ita ſinus complementi arcus <emph style="sc">Cb</emph>, ad ſinum totum; </s> <s xml:id="echoid-s14666" xml:space="preserve">erit conuer-<lb/>tendo, vt ſinus complementi arcus AC, ad ſinum complementi arcus <emph style="sc">Ab</emph>, ita ſinus <lb/>totus ad ſinum complementi arcus <emph style="sc">Cb</emph>.</s> <s xml:id="echoid-s14667" xml:space="preserve"/> </p> <div xml:id="echoid-div1166" type="float" level="2" n="4"> <note position="left" xlink:label="note-432-04" xlink:href="note-432-04a" xml:space="preserve">43. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14668" xml:space="preserve">QV APROPTER, dato arcu, qui recto angulo opponitur, cum <lb/> <anchor type="note" xlink:label="note-432-05a" xlink:href="note-432-05"/> alterutro arcuum circa angulum rectum, ſi ſiat, vt ſinus complementi ar-<lb/>cus dati circa angulum rectum ad ſinum complementi arcus angulo recto <lb/>oppoſiti, ita ſinus totus ad aliud, inuenietur ſinus complementi alterius <lb/>arcus circa angulum rectum, qui quæritur. </s> <s xml:id="echoid-s14669" xml:space="preserve">Ex quouis autem arcu dato <lb/>cir ca rectum angulum, cum arcu, quirecto angulo opponitur, reperietur <lb/>angulus illi arcui oppoſitus, vt in problemate 1. </s> <s xml:id="echoid-s14670" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14671" xml:space="preserve">41. </s> <s xml:id="echoid-s14672" xml:space="preserve">demonſtra-<lb/>tum est.</s> <s xml:id="echoid-s14673" xml:space="preserve"/> </p> <div xml:id="echoid-div1167" type="float" level="2" n="5"> <note position="left" xlink:label="note-432-05" xlink:href="note-432-05a" xml:space="preserve">Praxis, quã <lb/>do datur <lb/>arcus recto <lb/>angulo op-<lb/>poſitus, cũ <lb/>alterutro <lb/>circa angu <lb/>lũ rectum.</note> </div> <p style="it"> <s xml:id="echoid-s14674" xml:space="preserve">AN vero tertius arcus CB, quæſitus ſit quadrante minor, aut maior, intellige-<lb/>mus ex duobus arcubus datis. </s> <s xml:id="echoid-s14675" xml:space="preserve">Si namque arcus <emph style="sc">AB</emph>, angulo recto oppoſitus fuerit <lb/>quadrante minor, ſi quidem & </s> <s xml:id="echoid-s14676" xml:space="preserve">alter datus AC, ſit quadrante minor, erit & </s> <s xml:id="echoid-s14677" xml:space="preserve">arcus <lb/>CB, quadrante minor; </s> <s xml:id="echoid-s14678" xml:space="preserve">ſi vero AC, ſit quadrante maior, erit & </s> <s xml:id="echoid-s14679" xml:space="preserve">CB, maior qua-<lb/> <anchor type="note" xlink:label="note-432-06a" xlink:href="note-432-06"/> drante. </s> <s xml:id="echoid-s14680" xml:space="preserve">Si autem <emph style="sc">AB</emph>, fuerit quadrante maior, ſi quidem & </s> <s xml:id="echoid-s14681" xml:space="preserve"><emph style="sc">A</emph>C, ſit quadrante <lb/>maior, erit CB, quadrante minor; </s> <s xml:id="echoid-s14682" xml:space="preserve">ſi vero AC, ſit minor quadrante, erit CB, <lb/>quadrante maior.</s> <s xml:id="echoid-s14683" xml:space="preserve"/> </p> <div xml:id="echoid-div1168" type="float" level="2" n="6"> <note position="left" xlink:label="note-432-06" xlink:href="note-432-06a" xml:space="preserve">36. huius.</note> </div> </div> <div xml:id="echoid-div1170" type="section" level="1" n="561"> <head xml:id="echoid-head596" xml:space="preserve">SCHOLIVM. II.</head> <p style="it"> <s xml:id="echoid-s14684" xml:space="preserve">QVAMVIS & </s> <s xml:id="echoid-s14685" xml:space="preserve">hanc propoſ. </s> <s xml:id="echoid-s14686" xml:space="preserve">43. </s> <s xml:id="echoid-s14687" xml:space="preserve">& </s> <s xml:id="echoid-s14688" xml:space="preserve">antecedentem 42. </s> <s xml:id="echoid-s14689" xml:space="preserve">quadrimembrem feco-<lb/> <anchor type="note" xlink:label="note-432-07a" xlink:href="note-432-07"/> <pb o="421" file="433" n="433" rhead=""/> rimus, vt vtraque in omnibus caſibus demonſtraretur: </s> <s xml:id="echoid-s14690" xml:space="preserve">ſatis tamen fuißet, ſi vtraq; <lb/></s> <s xml:id="echoid-s14691" xml:space="preserve"> <anchor type="note" xlink:label="note-433-01a" xlink:href="note-433-01"/> in primo caſu, exiſtentibus nimirum omnibus arcubus quadrante minoribus, demon <lb/>ſtratione fuiſſet confirmata. </s> <s xml:id="echoid-s14692" xml:space="preserve">Eo enim caſu demonſtrato, facile demonſtrationem om-<lb/>nibus alijs caſibus accommodabimus. </s> <s xml:id="echoid-s14693" xml:space="preserve">Sit nam-<lb/>que triangulum ſphæricum quodcunq; </s> <s xml:id="echoid-s14694" xml:space="preserve">rectan <lb/> <anchor type="figure" xlink:label="fig-433-01a" xlink:href="fig-433-01"/> gulum ACD, habens angulum C, rectum. </s> <s xml:id="echoid-s14695" xml:space="preserve">Aut <lb/>igitur duo arcus AC, CD, circa angulum re-<lb/>ctum quadrãte ſunt minores, ac proinde & </s> <s xml:id="echoid-s14696" xml:space="preserve">ter <lb/>tius arcus AD, quadrante quoque minor; </s> <s xml:id="echoid-s14697" xml:space="preserve">aut <lb/>vnus quadrante maior, & </s> <s xml:id="echoid-s14698" xml:space="preserve">alter minor; </s> <s xml:id="echoid-s14699" xml:space="preserve">aut <lb/> <anchor type="note" xlink:label="note-433-02a" xlink:href="note-433-02"/> denique ambo quadrante maiores: </s> <s xml:id="echoid-s14700" xml:space="preserve">Nam de eo <lb/>ſolo ſphærico triangulo rectangulo agimus, in <lb/>quo nullus arcus eſt quadrãs. </s> <s xml:id="echoid-s14701" xml:space="preserve">Sint primum duo <lb/>arcus AC, CD, circa angulum rectum quadrante minores: </s> <s xml:id="echoid-s14702" xml:space="preserve">quo poſito, erit vterque <lb/> <anchor type="note" xlink:label="note-433-03a" xlink:href="note-433-03"/> angulus D, A, acutus, proptereaque triangulo ACD, demonſtratio vtriuſque pro-<lb/>poſitionis conueniet, quo ad primum caſum.</s> <s xml:id="echoid-s14703" xml:space="preserve"/> </p> <div xml:id="echoid-div1170" type="float" level="2" n="1"> <note position="left" xlink:label="note-432-07" xlink:href="note-432-07a" xml:space="preserve">Quicquid <lb/>demonſtra <lb/>tur de triã-<lb/>gulo ſphæ.</note> <note position="right" xlink:label="note-433-01" xlink:href="note-433-01a" xml:space="preserve">tico rectan <lb/>gulo, cuius <lb/>omnes ar-<lb/>cus ſint qua <lb/>drante mi-<lb/>nores, locũ <lb/>etiã habet i <lb/>omni trian <lb/>gulo ſphæ-<lb/>rico rectan <lb/>gulo.</note> <figure xlink:label="fig-433-01" xlink:href="fig-433-01a"> <image file="433-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/433-01"/> </figure> <note position="right" xlink:label="note-433-02" xlink:href="note-433-02a" xml:space="preserve">35. huius.,</note> <note position="right" xlink:label="note-433-03" xlink:href="note-433-03a" xml:space="preserve">34. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14704" xml:space="preserve">SIT deinde arcus DC, quadrante maior, & </s> <s xml:id="echoid-s14705" xml:space="preserve">CA, minor. </s> <s xml:id="echoid-s14706" xml:space="preserve">Productis arcubus DC, <lb/>DA, donec coeant in B; </s> <s xml:id="echoid-s14707" xml:space="preserve">erunt <emph style="sc">DAb</emph>, DCB, ſemicirculi; </s> <s xml:id="echoid-s14708" xml:space="preserve">atque adeo CB, qua-<lb/> <anchor type="note" xlink:label="note-433-04a" xlink:href="note-433-04"/> drante minor. </s> <s xml:id="echoid-s14709" xml:space="preserve">Sunt ergo in triangulo ACB, duo arcus <emph style="sc">AC, CB</emph>, circa angulum re-<lb/>ctum C, quadrante minores. </s> <s xml:id="echoid-s14710" xml:space="preserve">Quare, vt proxime oſtendimus, ei vtriuſque propoſi-<lb/>tionis demonſtratio, quo ad primum caſum, conueniet. </s> <s xml:id="echoid-s14711" xml:space="preserve">Cum ergo ijdem ſinus tam re-<lb/>cti, quam complementorum, ſint arcuum, & </s> <s xml:id="echoid-s14712" xml:space="preserve">angulorum trianguli <emph style="sc">AC</emph>B, qui arcuum, <lb/>& </s> <s xml:id="echoid-s14713" xml:space="preserve">angulorum trianguli <emph style="sc">Ac</emph>D; </s> <s xml:id="echoid-s14714" xml:space="preserve">(Nam, vt in ſinubus diximus, arcus CD, CB, eun-<lb/>dem ſinum habent tam rectum, quam complementi, necnon & </s> <s xml:id="echoid-s14715" xml:space="preserve">arcus <emph style="sc">A</emph>D, AB. </s> <s xml:id="echoid-s14716" xml:space="preserve">Item <lb/>tam recti anguli ad C, eundem ſinum habent, nempe totum, quam anguli obliqui ad <lb/>A, cum duobus rectis ſint æquales. </s> <s xml:id="echoid-s14717" xml:space="preserve">Denique & </s> <s xml:id="echoid-s14718" xml:space="preserve">anguli D, B, eundem ſinum habent, <lb/> <anchor type="note" xlink:label="note-433-05a" xlink:href="note-433-05"/> cum ſint inter ſe æquales: </s> <s xml:id="echoid-s14719" xml:space="preserve">Arcus autem <emph style="sc">Ac</emph>, vtrique triangulo communis eſt.) </s> <s xml:id="echoid-s14720" xml:space="preserve">li-<lb/> <anchor type="note" xlink:label="note-433-06a" xlink:href="note-433-06"/> quido conſtat, quicquid de ſinubus arcuum, angulorumq́; </s> <s xml:id="echoid-s14721" xml:space="preserve">trianguli ACB, fuerit <lb/>oſtenſum, idem in ſinubus arcuum, & </s> <s xml:id="echoid-s14722" xml:space="preserve">angulorum trianguli <emph style="sc">AC</emph>D, locum habere.</s> <s xml:id="echoid-s14723" xml:space="preserve"/> </p> <div xml:id="echoid-div1171" type="float" level="2" n="2"> <note position="right" xlink:label="note-433-04" xlink:href="note-433-04a" xml:space="preserve">11. 1 Theod.</note> <note position="right" xlink:label="note-433-05" xlink:href="note-433-05a" xml:space="preserve">5. huius.</note> <note position="right" xlink:label="note-433-06" xlink:href="note-433-06a" xml:space="preserve">13. primi.</note> </div> <p style="it"> <s xml:id="echoid-s14724" xml:space="preserve">POSTREMO ſint duo arcus DC, <emph style="sc">CA</emph>, quadrante maiores: </s> <s xml:id="echoid-s14725" xml:space="preserve">quo poſito, erit <lb/>arcus CB, minor quadrante. </s> <s xml:id="echoid-s14726" xml:space="preserve">Habet igitur triangulum <emph style="sc">A</emph>CB, arcum <emph style="sc">AC</emph>, circa <lb/>angulum rectum <emph style="sc">C</emph>, quadrante maiorem, & </s> <s xml:id="echoid-s14727" xml:space="preserve"><emph style="sc">Cb</emph>, minorem. </s> <s xml:id="echoid-s14728" xml:space="preserve">Quare ei, vt proxime <lb/>eſt demonſtratum, vtraque propoſitio conueniet. </s> <s xml:id="echoid-s14729" xml:space="preserve">Cum ergo ijdem ſinus tam recti, <lb/>quam complementorum, ſint arcuum, & </s> <s xml:id="echoid-s14730" xml:space="preserve">angulorum trianguli ACB, qui arcuum, <lb/>& </s> <s xml:id="echoid-s14731" xml:space="preserve">angulorum trianguli ACD, vt paulo ante diximus, liquet eaſdem propoſitiones <lb/>triangulo quoque ACD, conuenire. </s> <s xml:id="echoid-s14732" xml:space="preserve">Perſpicuum ergo eſt, quicquid de ſinubus arcuum, <lb/>angulorumq́; </s> <s xml:id="echoid-s14733" xml:space="preserve">trianguli ſphærici rectanguli, cuius duo arcus circa angulum rectum <lb/>quadrante ſint minores, demonſtratum fuerit, locum etiam habere in quocunq; </s> <s xml:id="echoid-s14734" xml:space="preserve">alio <lb/>triangulo ſphærico rectangulo.</s> <s xml:id="echoid-s14735" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14736" xml:space="preserve">IDEM prorſus dicendum eſt de tertio caſu propoſ. </s> <s xml:id="echoid-s14737" xml:space="preserve">41. </s> <s xml:id="echoid-s14738" xml:space="preserve">Satis enim fuiſſet illum <lb/>demonſtraſſe in triangulo rectangulo, cuius omnes arcus ſunt quadrante minores, <lb/>quale est triangulum ſecundæ figuræ propoſ. </s> <s xml:id="echoid-s14739" xml:space="preserve">41 dictæ; </s> <s xml:id="echoid-s14740" xml:space="preserve">cum eius trianguli demon-<lb/>ſtratio omnibus alijs conueniat, vt ex demonſtratis in hoc ſcholio eſt manifeſtum.</s> <s xml:id="echoid-s14741" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14742" xml:space="preserve">EX his, quæ proximis tribus propoſitionibus demonſtrauimus, abſolutus iam per <lb/>ſinus eſt calculus triangulorum ſphæricorum rectangulorum: </s> <s xml:id="echoid-s14743" xml:space="preserve">quareiam non rectan <lb/>gulorum calculus ſequi deberet. </s> <s xml:id="echoid-s14744" xml:space="preserve">Sed quia per lineas tangentes, ac ſecantes breuius <lb/>plerunque triangulorum rectangulorum calculus, quam per ſinus, expeditur, adiun <pb o="422" file="434" n="434" rhead=""/> gemus ſequentes propoſitiones ad triangula quoque ſphærica rectangula ſpectantes, <lb/>antequam triangulorum ſphæricorum non rectangulorum calculum exponamus. <lb/></s> <s xml:id="echoid-s14745" xml:space="preserve">Vt autem clariores fiant demonſtrationes, & </s> <s xml:id="echoid-s14746" xml:space="preserve">minus confuſæ, proponemus ſemper <lb/>triangulum ſphæricum rectangulum, cuius duo arcus circa angulum rectum, ac pro-<lb/>inde omnes tres, minores ſint quadrante. </s> <s xml:id="echoid-s14747" xml:space="preserve">Nam eædem demonſtrationes alijs omnibus <lb/>conuenient, vt in hoc ſcbolio demonſtrauimus: </s> <s xml:id="echoid-s14748" xml:space="preserve">quippe cum & </s> <s xml:id="echoid-s14749" xml:space="preserve">tam duo arcus ſemicir <lb/>culum conſicientes, quàm duo anguli duobus rectis æquales, eandem habeant tangen <lb/>tem, ac ſecantem, quemadmodum & </s> <s xml:id="echoid-s14750" xml:space="preserve">eundem ſinum, vt in tractatione tangentium, <lb/>& </s> <s xml:id="echoid-s14751" xml:space="preserve">ſecantium monuimus.</s> <s xml:id="echoid-s14752" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1173" type="section" level="1" n="562"> <head xml:id="echoid-head597" xml:space="preserve">THEOR. 42. PROPOS. 44.</head> <p> <s xml:id="echoid-s14753" xml:space="preserve">IN omni triangulo ſphætico rectangulo, cu-<lb/>ius omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s14754" xml:space="preserve">ſinus to-<lb/>tus ad ſinum vtriuſuis arcuum circarectum angu-<lb/>lum eandem habet proportionem, quam tangens <lb/>anguli non recti dicto arcui adiacentis ad tangen-<lb/>tem reliqui arcus circa angulum rectum huic an-<lb/>gulo oppoſiti.</s> <s xml:id="echoid-s14755" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14756" xml:space="preserve">IN triangulo ſph ærico ABC, cuius omnes arcus quadrante minores, ſit <lb/>angulus C, rectus. </s> <s xml:id="echoid-s14757" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum arcus BC, vt eſt tan-<lb/>gens anguli B, ad tangentem arcus AC. </s> <s xml:id="echoid-s14758" xml:space="preserve">Productis <lb/> <anchor type="figure" xlink:label="fig-434-01a" xlink:href="fig-434-01"/> enim arcubus BC, BA, donec fiant quadrantes BF, <lb/>BD, ac per puncta F, D, arcu FD, circuli maximi <lb/>deſcripto; </s> <s xml:id="echoid-s14759" xml:space="preserve">erit vterque angulus F, D, rectus, ob qua-<lb/> <anchor type="note" xlink:label="note-434-01a" xlink:href="note-434-01"/> drantes BF, BD: </s> <s xml:id="echoid-s14760" xml:space="preserve">& </s> <s xml:id="echoid-s14761" xml:space="preserve">DF, arcus erit anguli B; </s> <s xml:id="echoid-s14762" xml:space="preserve">cum <lb/>B, polus ſit arcus DF. </s> <s xml:id="echoid-s14763" xml:space="preserve">Quia igitur duo circuli ma-<lb/> <anchor type="note" xlink:label="note-434-02a" xlink:href="note-434-02"/> ximi in ſphæra BF, BD, ſecant ſeſe in B, ductiq́ue <lb/>ſunt ex A, D, ad BF, arcus perpendiculares AC, <lb/>DF; </s> <s xml:id="echoid-s14764" xml:space="preserve">erit, vt ſinus quadrantis BF, hoc eſt, ſinus to-<lb/> <anchor type="note" xlink:label="note-434-03a" xlink:href="note-434-03"/> tus, ad tangentem arcus FD, hoc eſt, ad tangentem <lb/>anguli B, ita ſinus arcus BC, ad tangentem arcus AC: </s> <s xml:id="echoid-s14765" xml:space="preserve">Et permutando, vt <lb/>ſinus totus ad ſinum arcus BC, ita tangens anguli B, ad tangentem arcus AC. <lb/></s> <s xml:id="echoid-s14766" xml:space="preserve">Non aliter demonſtrabimus, ita eſſe ſinum totum ad ſinum arcus AC, vt eſt <lb/>tangens anguli A, ad tangentem arcus BC: </s> <s xml:id="echoid-s14767" xml:space="preserve">vt patet, ſi arcus AC, AB, pro-<lb/>ducantur, donec fiant quadrantes AG, AE, perque G, E, arcus maximi cir-<lb/>culi deſcribatur GE. </s> <s xml:id="echoid-s14768" xml:space="preserve">Erit enim rurſus, vt ſinus quadrantis AG, id eſt, ſinus <lb/> <anchor type="note" xlink:label="note-434-04a" xlink:href="note-434-04"/> totus, ad tangentem arcus EG, ſeu anguli A, ita ſinus arcus AC, ad tangen <lb/>tem arcus BC: </s> <s xml:id="echoid-s14769" xml:space="preserve">Et permutãdo, vt ſinus totus ad ſinum arcus AC, ita tangens <lb/>anguli A, ad tangentem arcus BC. </s> <s xml:id="echoid-s14770" xml:space="preserve">In omni ergo triangulo ſphærico rectan-<lb/>gulo, &</s> <s xml:id="echoid-s14771" xml:space="preserve">c. </s> <s xml:id="echoid-s14772" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s14773" xml:space="preserve"/> </p> <div xml:id="echoid-div1173" type="float" level="2" n="1"> <figure xlink:label="fig-434-01" xlink:href="fig-434-01a"> <image file="434-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/434-01"/> </figure> <note position="left" xlink:label="note-434-01" xlink:href="note-434-01a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-434-02" xlink:href="note-434-02a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-434-03" xlink:href="note-434-03a" xml:space="preserve">Theor. 6. <lb/>ſcholij. 40. <lb/>huius.</note> <note position="left" xlink:label="note-434-04" xlink:href="note-434-04a" xml:space="preserve">Theor. 6. <lb/>ſcholij 40. <lb/>huius.</note> </div> <pb o="423" file="435" n="435" rhead=""/> </div> <div xml:id="echoid-div1175" type="section" level="1" n="563"> <head xml:id="echoid-head598" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s14774" xml:space="preserve">HINC colligemus duo ſequentia problemata.</s> <s xml:id="echoid-s14775" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1176" type="section" level="1" n="564"> <head xml:id="echoid-head599" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s14776" xml:space="preserve">IN triangulo ſphærico rectangulo, dato alterutro arcuum circa <lb/>angulum rectum, cum alterutro angulorum non rectorum, reperire <lb/>alium arcum circa rectum angulum, & </s> <s xml:id="echoid-s14777" xml:space="preserve">reliquum angulum non re-<lb/>ctum, cum arcu, qui recto angulo opponitur: </s> <s xml:id="echoid-s14778" xml:space="preserve">dum modo, quando <lb/>angulus datus opponitur arcui dato, conſtet, an reliquus arcus circa <lb/>rectum angulum ſit quadrante minor, maiorve; </s> <s xml:id="echoid-s14779" xml:space="preserve">vel an reliquus an-<lb/>gulus non rectus ſit acutus, obtuſusve.</s> <s xml:id="echoid-s14780" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14781" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus <emph style="sc">C</emph>, rectus, da-<lb/> <anchor type="figure" xlink:label="fig-435-01a" xlink:href="fig-435-01"/> tus ſit primum arcus AC, cum angulo <emph style="sc">A</emph>, ſibi adia-<lb/>cente. </s> <s xml:id="echoid-s14782" xml:space="preserve">Dico dari quoque arcum BC, vnà cum angu-<lb/>lo B, & </s> <s xml:id="echoid-s14783" xml:space="preserve">arcu <emph style="sc">Ab</emph>. </s> <s xml:id="echoid-s14784" xml:space="preserve">Quoniam enim eſt, vt ſinus totus <lb/> <anchor type="note" xlink:label="note-435-01a" xlink:href="note-435-01"/> ad ſinum arcus AC, ita tangens anguli A, ad tangen-<lb/>tem arcus <emph style="sc">BC</emph>:</s> <s xml:id="echoid-s14785" xml:space="preserve"/> </p> <div xml:id="echoid-div1176" type="float" level="2" n="1"> <figure xlink:label="fig-435-01" xlink:href="fig-435-01a"> <image file="435-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/435-01"/> </figure> <note position="right" xlink:label="note-435-01" xlink:href="note-435-01a" xml:space="preserve">44. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14786" xml:space="preserve">SI (quando datur arcus cum angulo adia-<lb/> <anchor type="note" xlink:label="note-435-02a" xlink:href="note-435-02"/> cente) fiat, vt ſinus totus ad ſinum dati arcus, <lb/>ita tangens anguli dati ad aliud, producetur <lb/>tangens arcus quæſiti. </s> <s xml:id="echoid-s14787" xml:space="preserve">Ex eodem vero arcu dato, & </s> <s xml:id="echoid-s14788" xml:space="preserve">argulo dato, inue-<lb/>nietur alter angulus non rectus, et arcus recto angulo oppoſitus, vt in pro-<lb/>blemate 2. </s> <s xml:id="echoid-s14789" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14790" xml:space="preserve">42. </s> <s xml:id="echoid-s14791" xml:space="preserve">demonſtrauimus.</s> <s xml:id="echoid-s14792" xml:space="preserve"/> </p> <div xml:id="echoid-div1177" type="float" level="2" n="2"> <note position="right" xlink:label="note-435-02" xlink:href="note-435-02a" xml:space="preserve">Praxis, cũ <lb/>datur arcus <lb/>cum angu-<lb/>lo adiacẽte.</note> </div> <p style="it"> <s xml:id="echoid-s14793" xml:space="preserve">AN vero arcus quæſitus <emph style="sc">BC</emph>, ſit quadrante minor, maiorue, indicabit angulus <lb/>datus <emph style="sc">A</emph>. </s> <s xml:id="echoid-s14794" xml:space="preserve">Nam ſi fuerit acutus, erit arcus <emph style="sc">BC</emph>, quadrante minor; </s> <s xml:id="echoid-s14795" xml:space="preserve">ſi vero obtuſus, <lb/> <anchor type="note" xlink:label="note-435-03a" xlink:href="note-435-03"/> quadrante maior.</s> <s xml:id="echoid-s14796" xml:space="preserve"/> </p> <div xml:id="echoid-div1178" type="float" level="2" n="3"> <note position="right" xlink:label="note-435-03" xlink:href="note-435-03a" xml:space="preserve">34. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14797" xml:space="preserve">SIT deinde datus arcus <emph style="sc">AC</emph>, cum angulo <emph style="sc">B</emph>, ſibi oppoſito, conſtetq́; </s> <s xml:id="echoid-s14798" xml:space="preserve">præterea de <lb/>altero arcu <emph style="sc">BC</emph>, num quadrante minor ſit, an maior; </s> <s xml:id="echoid-s14799" xml:space="preserve">vel an alter angulus A, acu-<lb/>tus ſit, an obtuſus. </s> <s xml:id="echoid-s14800" xml:space="preserve">Dico rurſum dari arcum <emph style="sc">BC</emph>, vnà cum angulo <emph style="sc">B</emph>, & </s> <s xml:id="echoid-s14801" xml:space="preserve">arcu <emph style="sc">AB</emph>. <lb/></s> <s xml:id="echoid-s14802" xml:space="preserve">Cum enim ſit, vt tangens anguli <emph style="sc">B</emph>, ad tangentem arcus <emph style="sc">AC</emph>, ita ſinus totus ad ſi-<lb/> <anchor type="note" xlink:label="note-435-04a" xlink:href="note-435-04"/> num arcus <emph style="sc">BC</emph>:</s> <s xml:id="echoid-s14803" xml:space="preserve"/> </p> <div xml:id="echoid-div1179" type="float" level="2" n="4"> <note position="right" xlink:label="note-435-04" xlink:href="note-435-04a" xml:space="preserve">44. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14804" xml:space="preserve">SI (quando datur arcus cum angulo oppoſito) fiat, vt tangens angu-<lb/> <anchor type="note" xlink:label="note-435-05a" xlink:href="note-435-05"/> li dati ad tangentem dati arcus, ita ſinus totus ad aliud, reperietur ſinus <lb/>arcus quæſiti. </s> <s xml:id="echoid-s14805" xml:space="preserve">Ex dato vero arcu, & </s> <s xml:id="echoid-s14806" xml:space="preserve">angulo dato dabitur & </s> <s xml:id="echoid-s14807" xml:space="preserve">alter angu-<lb/>lus non rectus, & </s> <s xml:id="echoid-s14808" xml:space="preserve">arcus recto angulo oppoſitus, vt in problemate 2. </s> <s xml:id="echoid-s14809" xml:space="preserve">pro-<lb/>poſ. </s> <s xml:id="echoid-s14810" xml:space="preserve">42. </s> <s xml:id="echoid-s14811" xml:space="preserve">diximus.</s> <s xml:id="echoid-s14812" xml:space="preserve"/> </p> <div xml:id="echoid-div1180" type="float" level="2" n="5"> <note position="right" xlink:label="note-435-05" xlink:href="note-435-05a" xml:space="preserve">Praxis. cũ <lb/>datur arcus <lb/>cũ angulo <lb/>oppoſito.</note> </div> <p style="it"> <s xml:id="echoid-s14813" xml:space="preserve">OPORTET autem conſtare, an arcus <emph style="sc">BC</emph>, ſit quadrante minor, an maior, vt <lb/>ſciamus, qualis arcus inuento ſinui reſpondens accipiẽdus ſit, an videlicet minor qua-<lb/>dr ante, an vero maior. </s> <s xml:id="echoid-s14814" xml:space="preserve">Quòd ſi conſtaret de angulo <emph style="sc">A</emph>, qualis ſit, ſtatim cognoſce-<lb/>remus, qualis ſit arcus <emph style="sc">BC</emph>. </s> <s xml:id="echoid-s14815" xml:space="preserve">Exiſtente enim angulo A, acuto, erit arcus BC, qua-<lb/> <anchor type="note" xlink:label="note-435-06a" xlink:href="note-435-06"/> drante minor: </s> <s xml:id="echoid-s14816" xml:space="preserve">exiſtente vero obtuſo, quadrante maior. </s> <s xml:id="echoid-s14817" xml:space="preserve">Sic etiam, ſi ſciretur, qualis <pb o="424" file="436" n="436" rhead=""/> ſit arcus <emph style="sc">Ab</emph>, recto angulo oppoſitus, ſpeciem quoque arcus BC, cognoſceremus. </s> <s xml:id="echoid-s14818" xml:space="preserve">Nam <lb/>ſi AB, ſit quadrante minor, erit vterque AC, BC, vet <lb/> <anchor type="figure" xlink:label="fig-436-01a" xlink:href="fig-436-01"/> minor quadrante, vel maior: </s> <s xml:id="echoid-s14819" xml:space="preserve">qualis ergo eſt datus arcus <lb/> <anchor type="note" xlink:label="note-436-01a" xlink:href="note-436-01"/> AC, talis quoque erit arcus BC. </s> <s xml:id="echoid-s14820" xml:space="preserve">Si vero AB, fuerit <lb/>maior quadrante, & </s> <s xml:id="echoid-s14821" xml:space="preserve">datus arcus AC, minor quidem <lb/>quadrante, erit BC, quadrante maior; </s> <s xml:id="echoid-s14822" xml:space="preserve">ſi vero datus ar-<lb/>cus AC, ſit quadrante maior, erit <emph style="sc">BC</emph>, quadrante mi-<lb/>nor. </s> <s xml:id="echoid-s14823" xml:space="preserve">Itaque non ſatis eſt, dari arcum, cum angulo oppo-<lb/>ſito, vt vult Copernicus propoſ 4. </s> <s xml:id="echoid-s14824" xml:space="preserve">de triangulis ſphæri-<lb/>cis. </s> <s xml:id="echoid-s14825" xml:space="preserve">Id quod ſupra in ſcholio propoſ. </s> <s xml:id="echoid-s14826" xml:space="preserve">21. </s> <s xml:id="echoid-s14827" xml:space="preserve">monuimus.</s> <s xml:id="echoid-s14828" xml:space="preserve"/> </p> <div xml:id="echoid-div1181" type="float" level="2" n="6"> <note position="right" xlink:label="note-435-06" xlink:href="note-435-06a" xml:space="preserve">34. huius.</note> <figure xlink:label="fig-436-01" xlink:href="fig-436-01a"> <image file="436-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/436-01"/> </figure> <note position="left" xlink:label="note-436-01" xlink:href="note-436-01a" xml:space="preserve">36. huius.</note> </div> </div> <div xml:id="echoid-div1183" type="section" level="1" n="565"> <head xml:id="echoid-head600" xml:space="preserve">II.</head> <p style="it"> <s xml:id="echoid-s14829" xml:space="preserve">IN triangulo ſphærico rectangulo, datis duobus arcubus circa <lb/>rectum angulum, vtrumlibetangulorum non rectorum, vnà cum ar-<lb/>cu reliquo, qui angulo recto opponitur, explorare.</s> <s xml:id="echoid-s14830" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14831" xml:space="preserve">IN eodem triangulo dati ſint duo arcus AC, <emph style="sc">BC</emph>. </s> <s xml:id="echoid-s14832" xml:space="preserve">Dico dari quoque vtrum vis <lb/>angulorum A, <emph style="sc">B</emph>, & </s> <s xml:id="echoid-s14833" xml:space="preserve">arcum <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s14834" xml:space="preserve">Cum enim ſit, vt ſinus totus ad ſinum arcus AC, <lb/> <anchor type="note" xlink:label="note-436-02a" xlink:href="note-436-02"/> ita tangens anguli A, ad tangentem arcus <emph style="sc">BC</emph>: </s> <s xml:id="echoid-s14835" xml:space="preserve">Et conuertendo, vt ſinus arcus AC, <lb/>ad ſinum totum, ita tangens arcus <emph style="sc">BC</emph>, ad tangentem anguli A; </s> <s xml:id="echoid-s14836" xml:space="preserve">Eademq́; </s> <s xml:id="echoid-s14837" xml:space="preserve">ratione, <lb/>vt ſinus arcus <emph style="sc">BC</emph>, ad ſinum totum, ita tangens arcus <emph style="sc">AC</emph>, ad tangentem anguli <emph style="sc">B</emph>.</s> <s xml:id="echoid-s14838" xml:space="preserve"/> </p> <div xml:id="echoid-div1183" type="float" level="2" n="1"> <note position="left" xlink:label="note-436-02" xlink:href="note-436-02a" xml:space="preserve">44. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14839" xml:space="preserve">SI fiat, vt ſinus vtriuſuis ar cuum circa angulum rectum ad ſinurn <lb/> <anchor type="note" xlink:label="note-436-03a" xlink:href="note-436-03"/> totum, ita tangens alterius arcus ad aliud, inuenietur tangens anguli huic <lb/>poſteriori arcui oppoſiti. </s> <s xml:id="echoid-s14840" xml:space="preserve">Ex datis quoque duobus ar cubus circa angulum <lb/>rectum cognoſcetur & </s> <s xml:id="echoid-s14841" xml:space="preserve">tertius arcus recto angulo oppoſitus, vt in proble-<lb/>mate propoſ. </s> <s xml:id="echoid-s14842" xml:space="preserve">43. </s> <s xml:id="echoid-s14843" xml:space="preserve">traditum eſt. </s> <s xml:id="echoid-s14844" xml:space="preserve">Vel certe ex dato vno arcu, & </s> <s xml:id="echoid-s14845" xml:space="preserve">alterutro <lb/>angulor um inuento, vt in problemate 2. </s> <s xml:id="echoid-s14846" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14847" xml:space="preserve">42. </s> <s xml:id="echoid-s14848" xml:space="preserve">oſtenſum eſt.</s> <s xml:id="echoid-s14849" xml:space="preserve"/> </p> <div xml:id="echoid-div1184" type="float" level="2" n="2"> <note position="left" xlink:label="note-436-03" xlink:href="note-436-03a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s14850" xml:space="preserve">NVM autem angulus quæſitus ſit acutus, obtuſuſve, docebit arcus ei oppoſitus. <lb/></s> <s xml:id="echoid-s14851" xml:space="preserve">Hic enim ſi minor quadrante ſuerit, erit angulus ei oppoſitus, acutus, ſi vero ma-<lb/> <anchor type="note" xlink:label="note-436-04a" xlink:href="note-436-04"/> ior, obtuſus.</s> <s xml:id="echoid-s14852" xml:space="preserve"/> </p> <div xml:id="echoid-div1185" type="float" level="2" n="3"> <note position="left" xlink:label="note-436-04" xlink:href="note-436-04a" xml:space="preserve">54. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14853" xml:space="preserve">QVONIAM verò in ſcholio 2. </s> <s xml:id="echoid-s14854" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14855" xml:space="preserve">præcedentis diximus, per lineas tangen-<lb/>tes, ac ſecantes breuius nonnulla expediri, quam per ſinus, intelligendum id eſt de ijs, <lb/>quæ primo loco in problematibus quæruntur, non autem, quæ ſecundo loco inueſti-<lb/>gantur. </s> <s xml:id="echoid-s14856" xml:space="preserve">Quod vt planius fiat, exponemus, quo paõto vtrumque problema hic pro-<lb/>poſitum abſoluendum ſit per ſinus. </s> <s xml:id="echoid-s14857" xml:space="preserve">Itaque, vt ex arcu circa angulum rectum dato, <lb/>cum alterutro angulorum acutorum, inueniatur alter arcus circa angulum rectums <lb/>qui primo loco in primo problemate inueſtigandus proponitur: </s> <s xml:id="echoid-s14858" xml:space="preserve">ita progrediendum <lb/>erit. </s> <s xml:id="echoid-s14859" xml:space="preserve">Si arcus circa rectum angulum detur cum angulo oppoſito, inquirendus pri-<lb/>mum erit arcus recto angulo oppoſitus, ex problemate 3. </s> <s xml:id="echoid-s14860" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14861" xml:space="preserve">41. </s> <s xml:id="echoid-s14862" xml:space="preserve">Deinde ex hoc <lb/>arcu inuento, & </s> <s xml:id="echoid-s14863" xml:space="preserve">dato arcu, eliciendus erit, per problema propoſ. </s> <s xml:id="echoid-s14864" xml:space="preserve">43. </s> <s xml:id="echoid-s14865" xml:space="preserve">alter arcus cir <lb/>ca angulum rectum, qui quæritur. </s> <s xml:id="echoid-s14866" xml:space="preserve">Si vero detur arcus circa angulum rectum cum <lb/>angulo adiacente, quærendus eſt primum per problema 2. </s> <s xml:id="echoid-s14867" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14868" xml:space="preserve">42. </s> <s xml:id="echoid-s14869" xml:space="preserve">alter angulus <lb/>acutus. </s> <s xml:id="echoid-s14870" xml:space="preserve">Deinde per problema 1. </s> <s xml:id="echoid-s14871" xml:space="preserve">eiuſdem propoſ. </s> <s xml:id="echoid-s14872" xml:space="preserve">42. </s> <s xml:id="echoid-s14873" xml:space="preserve">ex hoc angulo inuento, & </s> <s xml:id="echoid-s14874" xml:space="preserve">angulo <lb/>dato, arcus dato angulo oppoſitus eliciendus. </s> <s xml:id="echoid-s14875" xml:space="preserve">At, vt ex duobus arcubus circa angu-<lb/>lum rectum datis, vteruis angulorum acutorum eruatur; </s> <s xml:id="echoid-s14876" xml:space="preserve">qui primo loco in ſecun-<lb/>do problemate inquiritur: </s> <s xml:id="echoid-s14877" xml:space="preserve">reperiendus erit primum arcus recto angulo oppoſitus per <pb o="425" file="437" n="437" rhead=""/> problema propoſ. </s> <s xml:id="echoid-s14878" xml:space="preserve">43. </s> <s xml:id="echoid-s14879" xml:space="preserve">ex datis duobus arcubus. </s> <s xml:id="echoid-s14880" xml:space="preserve">Deinde per problema 1. </s> <s xml:id="echoid-s14881" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14882" xml:space="preserve">41. <lb/></s> <s xml:id="echoid-s14883" xml:space="preserve">ex hoc arcu inuento, & </s> <s xml:id="echoid-s14884" xml:space="preserve">alterutro circa angulum rectum dato, inueniendus angulus <lb/>huic dato arcui oppoſitus. </s> <s xml:id="echoid-s14885" xml:space="preserve">Vides igitur, id, quod primo loco in vtroque problemate <lb/>quæritur, duplici opere inueſtigari per ſinus, quod ſimplici per tangentes inuenimus. </s> <s xml:id="echoid-s14886" xml:space="preserve"><lb/>Eadem ratio eſt in ſequentibus problematibus, quod ſemel hic monuiſſe ſatis ſit.</s> <s xml:id="echoid-s14887" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1187" type="section" level="1" n="566"> <head xml:id="echoid-head601" xml:space="preserve">THEOR. 43. PROPOS. 45.</head> <p> <s xml:id="echoid-s14888" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s14889" xml:space="preserve">ſinus to-<lb/>tus ad ſinum complementi vtriuſuis angulorum <lb/>acutorum eandem proportionem habet, quam tan <lb/>gens arcus recto angulo oppoſiti ad tangentem ar-<lb/>cus dicto acuto angulo adiacentis.</s> <s xml:id="echoid-s14890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14891" xml:space="preserve">IN triangulo ſphærico ABC, cuius omnes arcus quadrante minores, ſit <lb/>angulus B, rectus. </s> <s xml:id="echoid-s14892" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum complementi anguli A, <lb/>vt eſt tangens arcus AC, ad tangentem arcus AB. </s> <s xml:id="echoid-s14893" xml:space="preserve">Productis enim arcubus <lb/>AB, AC, dictum angulum comprehendenti-<lb/>bus, donec quadrantes ſiant AD, AE; </s> <s xml:id="echoid-s14894" xml:space="preserve">de-<lb/> <anchor type="figure" xlink:label="fig-437-01a" xlink:href="fig-437-01"/> ſcriptoq́; </s> <s xml:id="echoid-s14895" xml:space="preserve">per D, E, arcu circuli maximi DE, <lb/>productoq́ue, donec cum arcu BC, produ-<lb/>cto coëat in F: </s> <s xml:id="echoid-s14896" xml:space="preserve">erit vterque angulus D, E, <lb/> <anchor type="note" xlink:label="note-437-01a" xlink:href="note-437-01"/> rectus, ob quadrantes AD, AE; </s> <s xml:id="echoid-s14897" xml:space="preserve">& </s> <s xml:id="echoid-s14898" xml:space="preserve">DE, ar-<lb/>cus erit anguli A, cum A, ſit polus arcus DE. <lb/></s> <s xml:id="echoid-s14899" xml:space="preserve"> <anchor type="note" xlink:label="note-437-02a" xlink:href="note-437-02"/> Item arcus DF, BF, quadrantes erunt, ob re-<lb/>ctos angulos B, D; </s> <s xml:id="echoid-s14900" xml:space="preserve">ac proinde arcus EF, com <lb/>plementum anguli A. </s> <s xml:id="echoid-s14901" xml:space="preserve">Quoniam igitur duo <lb/>circuli maximi in ſphæra BF, DF, ſe in terſe-<lb/>cant in F; </s> <s xml:id="echoid-s14902" xml:space="preserve">ductiq́; </s> <s xml:id="echoid-s14903" xml:space="preserve">ſunt ex punctis B, C, arcus <lb/>BF, ad arcum DF, arcus perpẽdiculares BD, <lb/>CE; </s> <s xml:id="echoid-s14904" xml:space="preserve">erit vt ſinus totus quadrantis DF, ad tangentem arcus BD, ita ſinus ar-<lb/> <anchor type="note" xlink:label="note-437-03a" xlink:href="note-437-03"/> cus EF, hoc eſt, ſinus complementi anguli A, ad tangentem arcus CE: </s> <s xml:id="echoid-s14905" xml:space="preserve">Et <lb/>permutando, vt ſinus totus ad ſinum complementi anguli A, ita tangens ar-<lb/>cus BD, ad tangentem arcus CE. </s> <s xml:id="echoid-s14906" xml:space="preserve">Eſt autem (cum AC, AB, ſint complemen <lb/>ta arcuum CE, BD.) </s> <s xml:id="echoid-s14907" xml:space="preserve">vt tangens arcus BD, ad tangentem arcus CE, ita tan <lb/> <anchor type="note" xlink:label="note-437-04a" xlink:href="note-437-04"/> gens arcus AC, ad tangentem arcus AB. </s> <s xml:id="echoid-s14908" xml:space="preserve">Igitur erit quoque, vt ſinus totus <lb/>ad ſinum complementi anguli A, ita tangens arcus AC, recto angulo oppoſi-<lb/>ti ad tangentem arcus AB, acuto angulo A, adiacentis. </s> <s xml:id="echoid-s14909" xml:space="preserve">Eodem modo oſten-<lb/>demus, ita eſſe ſinum totum ad ſinum complementi anguli C, vt eſt tangens <lb/>arcus AC, recto angulo oppoſiti ad tangentem arcus BC, angulo acuto C, <lb/>adiacentis, ſi nimirum arcus CA, CB, angulum C, continentes producantur, <lb/>&</s> <s xml:id="echoid-s14910" xml:space="preserve">c. </s> <s xml:id="echoid-s14911" xml:space="preserve">In omni ergo triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s14912" xml:space="preserve">c. </s> <s xml:id="echoid-s14913" xml:space="preserve">Quod oſtendendũ erat.</s> <s xml:id="echoid-s14914" xml:space="preserve"/> </p> <div xml:id="echoid-div1187" type="float" level="2" n="1"> <figure xlink:label="fig-437-01" xlink:href="fig-437-01a"> <image file="437-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/437-01"/> </figure> <note position="right" xlink:label="note-437-01" xlink:href="note-437-01a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-437-02" xlink:href="note-437-02a" xml:space="preserve">26. huius.</note> <note position="right" xlink:label="note-437-03" xlink:href="note-437-03a" xml:space="preserve">Theor. 6. <lb/>ſcholij 40. <lb/>huius.</note> <note position="right" xlink:label="note-437-04" xlink:href="note-437-04a" xml:space="preserve">21. Sinuũ.</note> </div> <pb o="426" file="438" n="438" rhead=""/> </div> <div xml:id="echoid-div1189" type="section" level="1" n="567"> <head xml:id="echoid-head602" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s14915" xml:space="preserve">EX hoc theoremate abſoluemus ſequentia tria problemata.</s> <s xml:id="echoid-s14916" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1190" type="section" level="1" n="568"> <head xml:id="echoid-head603" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s14917" xml:space="preserve">IN triangulo ſphærico rectangulo, dato alterutro arcuum circa <lb/>angulum rectum, cum angulo non recto adiacente, inuenire arcum <lb/>recto angulo oppoſitum, & </s> <s xml:id="echoid-s14918" xml:space="preserve">reliquum arcum circa angulum rectum, <lb/>cum reliquo angulo non recto.</s> <s xml:id="echoid-s14919" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14920" xml:space="preserve">IN triangulo <emph style="sc">ABc</emph>, cuius angulus C, rectus, datus ſit arcus AC, & </s> <s xml:id="echoid-s14921" xml:space="preserve">angulus <lb/>A. </s> <s xml:id="echoid-s14922" xml:space="preserve">Dico dari quoq; </s> <s xml:id="echoid-s14923" xml:space="preserve">arcum AB, cum arcu <emph style="sc">BC</emph>, & </s> <s xml:id="echoid-s14924" xml:space="preserve">angulo B. </s> <s xml:id="echoid-s14925" xml:space="preserve">Cum enim ſit, vt ſinus <lb/>totus ad ſinũ complementi anguli A, ita tangens arcus <emph style="sc">Ab</emph>, ad tangentem arcus AC; <lb/></s> <s xml:id="echoid-s14926" xml:space="preserve"> <anchor type="note" xlink:label="note-438-01a" xlink:href="note-438-01"/> Et conuertendo, vt ſinus complementi anguli <emph style="sc">A</emph>, ad ſinum totum, ita tangens arcus <lb/>AC, ad tangentem arcus AB:</s> <s xml:id="echoid-s14927" xml:space="preserve"/> </p> <div xml:id="echoid-div1190" type="float" level="2" n="1"> <note position="left" xlink:label="note-438-01" xlink:href="note-438-01a" xml:space="preserve">45. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14928" xml:space="preserve">SI fiat, vt ſinus complementi anguli dati <lb/> <anchor type="note" xlink:label="note-438-02a" xlink:href="note-438-02"/> ad ſinum totum, ita tangens arcus dati ad aliud, <lb/> <anchor type="figure" xlink:label="fig-438-01a" xlink:href="fig-438-01"/> reperietur tangens arcus recto angulo oppoſiti, <lb/>qui quæritur. </s> <s xml:id="echoid-s14929" xml:space="preserve">Ex arcu vero AB, & </s> <s xml:id="echoid-s14930" xml:space="preserve">angulo A, <lb/>inuenietur arcus BC, per problema 2. </s> <s xml:id="echoid-s14931" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s14932" xml:space="preserve">41. </s> <s xml:id="echoid-s14933" xml:space="preserve">Et ex arcubus AB, AC, angulus B, arcui <lb/>AC, oppoſitus, per problema 1. </s> <s xml:id="echoid-s14934" xml:space="preserve">eiuſdem pro-<lb/>poſ. </s> <s xml:id="echoid-s14935" xml:space="preserve">41.</s> <s xml:id="echoid-s14936" xml:space="preserve"/> </p> <div xml:id="echoid-div1191" type="float" level="2" n="2"> <note position="left" xlink:label="note-438-02" xlink:href="note-438-02a" xml:space="preserve">Praxis.</note> <figure xlink:label="fig-438-01" xlink:href="fig-438-01a"> <image file="438-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/438-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s14937" xml:space="preserve">ITA autem ſciemus, an arcus quæſitus AB, ſit quadrante maior, an minor. </s> <s xml:id="echoid-s14938" xml:space="preserve">Si datus <lb/>angulus A, fuerit acutus, erit arcus BC, quadrante minor. </s> <s xml:id="echoid-s14939" xml:space="preserve">Si ergo datus arcus <emph style="sc">Ac</emph>, <lb/> <anchor type="note" xlink:label="note-438-03a" xlink:href="note-438-03"/> ſit quoq; </s> <s xml:id="echoid-s14940" xml:space="preserve">minor, erit & </s> <s xml:id="echoid-s14941" xml:space="preserve">arcus AB, minor quadrante, Si vero <emph style="sc">A</emph>C, ſit quadrante ma-<lb/> <anchor type="note" xlink:label="note-438-04a" xlink:href="note-438-04"/> ior, erit & </s> <s xml:id="echoid-s14942" xml:space="preserve"><emph style="sc">Ab</emph>, maior. </s> <s xml:id="echoid-s14943" xml:space="preserve">At ſi datus angulus <emph style="sc">A</emph>, ſuerit, obtuſus, erit arcus BC, qua-<lb/> <anchor type="note" xlink:label="note-438-05a" xlink:href="note-438-05"/> drante maior: </s> <s xml:id="echoid-s14944" xml:space="preserve">Si ergo datus arcus <emph style="sc">Ac</emph>, ſit quoque maior, erit arcus AB, minor qua-<lb/> <anchor type="note" xlink:label="note-438-06a" xlink:href="note-438-06"/> drante; </s> <s xml:id="echoid-s14945" xml:space="preserve">Si vero AC, ſit minor quadrante, erit AB, maior.</s> <s xml:id="echoid-s14946" xml:space="preserve"/> </p> <div xml:id="echoid-div1192" type="float" level="2" n="3"> <note position="left" xlink:label="note-438-03" xlink:href="note-438-03a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-438-04" xlink:href="note-438-04a" xml:space="preserve">35. huius.</note> <note position="left" xlink:label="note-438-05" xlink:href="note-438-05a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-438-06" xlink:href="note-438-06a" xml:space="preserve">35. huius.</note> </div> </div> <div xml:id="echoid-div1194" type="section" level="1" n="569"> <head xml:id="echoid-head604" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s14947" xml:space="preserve">IN triangulo ſphęrico rectangulo, dato alterutro arcuum circa <lb/>angulum rectum, cum arcu, qui recto angulo opponitur, inueſtigare <lb/>angulum à dictis arcubus comprehenſum, hoc eſt, arcui, qui circa <lb/>angulum rectum datus eſt, adiacentem, cum reliquo arcu, & </s> <s xml:id="echoid-s14948" xml:space="preserve">angulo.</s> <s xml:id="echoid-s14949" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14950" xml:space="preserve">IN eodem triangulo dati ſint arcus AC, AB. </s> <s xml:id="echoid-s14951" xml:space="preserve">Dico dari etiam angulum A, cum <lb/> <anchor type="note" xlink:label="note-438-07a" xlink:href="note-438-07"/> arcu BC, & </s> <s xml:id="echoid-s14952" xml:space="preserve">angulo B. </s> <s xml:id="echoid-s14953" xml:space="preserve">Quoniam enim eſt, vt ſinus totus ad ſinum complementi angu <lb/>li A, ita tangens arcus AB, ad tangentem arcus AC: </s> <s xml:id="echoid-s14954" xml:space="preserve">Hoc eſt, vt tangens arcus AB, <lb/>ad tangentem arcus AC, ita ſinus totus ad ſinum complementi anguli A:</s> <s xml:id="echoid-s14955" xml:space="preserve"/> </p> <div xml:id="echoid-div1194" type="float" level="2" n="1"> <note position="left" xlink:label="note-438-07" xlink:href="note-438-07a" xml:space="preserve">45. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14956" xml:space="preserve">SI fiat, vt tangens arcus recto angulo oppoſiti ad tangentem dati <lb/> <anchor type="note" xlink:label="note-438-08a" xlink:href="note-438-08"/> arcus circa rectum angulum, ita ſinus totus ad aliud, procreabitur ſinus <lb/>complementi anguli quæſiti. </s> <s xml:id="echoid-s14957" xml:space="preserve">Hinc reliqua inuenientur, vt in præcedenti <lb/>problemate.</s> <s xml:id="echoid-s14958" xml:space="preserve"/> </p> <div xml:id="echoid-div1195" type="float" level="2" n="2"> <note position="left" xlink:label="note-438-08" xlink:href="note-438-08a" xml:space="preserve">Praxis.</note> </div> <pb o="427" file="439" n="439" rhead=""/> <p style="it"> <s xml:id="echoid-s14959" xml:space="preserve">VTRVM vero angulus A, quæſitus ſit acutus, obtuſusue, ita diſcemus. </s> <s xml:id="echoid-s14960" xml:space="preserve">Si arcus <lb/>AB, recto angulo oppoſitus fuerit quadrante minor, erit vterq; </s> <s xml:id="echoid-s14961" xml:space="preserve">arcus AC, <emph style="sc">B</emph>C, vel <lb/> <anchor type="note" xlink:label="note-439-01a" xlink:href="note-439-01"/> minor quadrante, vel maior. </s> <s xml:id="echoid-s14962" xml:space="preserve">Si ergo datus arcus AC, ſit minor, erit quoque <emph style="sc">Bc</emph>, mi-<lb/>nor, ac proinde angulus A, acutus; </s> <s xml:id="echoid-s14963" xml:space="preserve">ſi vero AC, ſit quadrante maior, erit & </s> <s xml:id="echoid-s14964" xml:space="preserve">BC, <lb/> <anchor type="note" xlink:label="note-439-02a" xlink:href="note-439-02"/> maior, ac propterea angulus A, obtuſus. </s> <s xml:id="echoid-s14965" xml:space="preserve">At ſi arcus AB, fuerit quadrante maior, erit <lb/> <anchor type="note" xlink:label="note-439-03a" xlink:href="note-439-03"/> alter reliquorum arcuum maior, & </s> <s xml:id="echoid-s14966" xml:space="preserve">alter minor: </s> <s xml:id="echoid-s14967" xml:space="preserve">Si igitur datus arcus AC, ſit ma-<lb/> <anchor type="note" xlink:label="note-439-04a" xlink:href="note-439-04"/> ior, erit BC, minor, proptereaq́; </s> <s xml:id="echoid-s14968" xml:space="preserve">angulus A, acutus; </s> <s xml:id="echoid-s14969" xml:space="preserve">Si vero AC, ſit quadrante mi-<lb/>nor, erit BC, maior, & </s> <s xml:id="echoid-s14970" xml:space="preserve">angulus A, obtuſus:</s> <s xml:id="echoid-s14971" xml:space="preserve"/> </p> <div xml:id="echoid-div1196" type="float" level="2" n="3"> <note position="right" xlink:label="note-439-01" xlink:href="note-439-01a" xml:space="preserve">36. huius.</note> <note position="right" xlink:label="note-439-02" xlink:href="note-439-02a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-439-03" xlink:href="note-439-03a" xml:space="preserve">36. huius.</note> <note position="right" xlink:label="note-439-04" xlink:href="note-439-04a" xml:space="preserve">34. huius.</note> </div> </div> <div xml:id="echoid-div1198" type="section" level="1" n="570"> <head xml:id="echoid-head605" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s14972" xml:space="preserve">IN triangulo ſphærico rectangulo, dato arcu, qui recto angulo <lb/>opponitur, cum alterurro angulorum non rectorum, inuenire arcum <lb/>huic angulo adiacentem, cum reliquo arcu, & </s> <s xml:id="echoid-s14973" xml:space="preserve">angulo.</s> <s xml:id="echoid-s14974" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s14975" xml:space="preserve">IN eodem triangulo datus ſit arcus AB, cum angulo A. </s> <s xml:id="echoid-s14976" xml:space="preserve">Dico dari quoq; </s> <s xml:id="echoid-s14977" xml:space="preserve">arcum <lb/>AC, &</s> <s xml:id="echoid-s14978" xml:space="preserve">c. </s> <s xml:id="echoid-s14979" xml:space="preserve">Nam cum ſit, vt ſinus totus ad ſinum complementi anguli A, ita tangens <lb/> <anchor type="note" xlink:label="note-439-05a" xlink:href="note-439-05"/> arcus AB, ad tangentem arcus AC:</s> <s xml:id="echoid-s14980" xml:space="preserve"/> </p> <div xml:id="echoid-div1198" type="float" level="2" n="1"> <note position="right" xlink:label="note-439-05" xlink:href="note-439-05a" xml:space="preserve">45. huius.</note> </div> <p style="it"> <s xml:id="echoid-s14981" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum complementi anguli dati, ita tangens <lb/> <anchor type="note" xlink:label="note-439-06a" xlink:href="note-439-06"/> arcus recto angulo oppoſiti ad aliud, producetur tangens arcus quæſiti. <lb/></s> <s xml:id="echoid-s14982" xml:space="preserve">Reliqua inuer. </s> <s xml:id="echoid-s14983" xml:space="preserve">ientur, vt in primo problemate huius propoſ.</s> <s xml:id="echoid-s14984" xml:space="preserve"/> </p> <div xml:id="echoid-div1199" type="float" level="2" n="2"> <note position="right" xlink:label="note-439-06" xlink:href="note-439-06a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s14985" xml:space="preserve">NVM autem quæſitus arcus AC, ſit minor quadrante, maiorue, hinc cognoſce-<lb/>mus. </s> <s xml:id="echoid-s14986" xml:space="preserve">Si arcus AB, angulo recto oppoſitus fuerit minor quadrante, erit vterq; </s> <s xml:id="echoid-s14987" xml:space="preserve">angu-<lb/>lus A, B, vel acutus, vel obtuſus. </s> <s xml:id="echoid-s14988" xml:space="preserve">Quare ſi datus angulus A, ſit acutus, erit quoque <lb/> <anchor type="note" xlink:label="note-439-07a" xlink:href="note-439-07"/> B, acutus, atque adeo arcus AC, quadrante minor; </s> <s xml:id="echoid-s14989" xml:space="preserve">Si vero A, ſit obtuſus, erit & </s> <s xml:id="echoid-s14990" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-439-08a" xlink:href="note-439-08"/> B, obtuſus, ideoq́; </s> <s xml:id="echoid-s14991" xml:space="preserve">arcus <emph style="sc">AC</emph>, quadrante maior. </s> <s xml:id="echoid-s14992" xml:space="preserve">At ſi arcus AB, ſuuerit maior qua-<lb/>drante, erit alter reliquorum angulorum acutus, & </s> <s xml:id="echoid-s14993" xml:space="preserve">alter obtuſus. </s> <s xml:id="echoid-s14994" xml:space="preserve">Siergo A, datus <lb/> <anchor type="note" xlink:label="note-439-09a" xlink:href="note-439-09"/> ſit acutus, erit B, obtuſus, & </s> <s xml:id="echoid-s14995" xml:space="preserve">idcirco arcus AC, quadrante maior; </s> <s xml:id="echoid-s14996" xml:space="preserve">Si vero A, ſit <lb/> <anchor type="note" xlink:label="note-439-10a" xlink:href="note-439-10"/> obtuſus, erit B, acutus, & </s> <s xml:id="echoid-s14997" xml:space="preserve">arcus <emph style="sc">A</emph>C, quadrante minor.</s> <s xml:id="echoid-s14998" xml:space="preserve"/> </p> <div xml:id="echoid-div1200" type="float" level="2" n="3"> <note position="right" xlink:label="note-439-07" xlink:href="note-439-07a" xml:space="preserve">38 huius.</note> <note position="right" xlink:label="note-439-08" xlink:href="note-439-08a" xml:space="preserve">34 huius.</note> <note position="right" xlink:label="note-439-09" xlink:href="note-439-09a" xml:space="preserve">38. huius.</note> <note position="right" xlink:label="note-439-10" xlink:href="note-439-10a" xml:space="preserve">34. huius.</note> </div> </div> <div xml:id="echoid-div1202" type="section" level="1" n="571"> <head xml:id="echoid-head606" xml:space="preserve">THEOR. 44. PROPOS. 46.</head> <p> <s xml:id="echoid-s14999" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s15000" xml:space="preserve">ſinus to-<lb/>tus ad ſinum complementi vtriuſuis angulorum <lb/>acutorum eandem proportionem habet, quam <lb/>tangens complementi arcus circa angulum rectũ <lb/>dicto angulo adiacentis ad tangentem comple-<lb/>menti arcus recto angulo oppoſiti.</s> <s xml:id="echoid-s15001" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15002" xml:space="preserve">IN triangulo ABC, cuius omnes arcus quadrante minores, ſit angulus B, <lb/>rectus. </s> <s xml:id="echoid-s15003" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinũ complemẽti anguli A, vt eſt tangens <pb o="428" file="440" n="440" rhead=""/> complementi arcus AB, ad tangentem complementi arcus AC. </s> <s xml:id="echoid-s15004" xml:space="preserve">Facta namque <lb/>conſtructione, vt in pręcedẽti propoſ. </s> <s xml:id="echoid-s15005" xml:space="preserve">quoniam duo circuli maximi in ſphæra <lb/>BF, DF, ſe mutuo ſecãt in F, productiq́; </s> <s xml:id="echoid-s15006" xml:space="preserve">ſunt <lb/> <anchor type="figure" xlink:label="fig-440-01a" xlink:href="fig-440-01"/> ex pũctis B, C, arcus BF, ad arcum DF, arcus <lb/>perpendiculares BD, CE; </s> <s xml:id="echoid-s15007" xml:space="preserve">erit, vt ſinus totus <lb/>quadrantis DF, ad rangentem arcus BD, ita <lb/>ſinus arcus EF, ad tangentem arcus CE: </s> <s xml:id="echoid-s15008" xml:space="preserve">Et <lb/> <anchor type="note" xlink:label="note-440-01a" xlink:href="note-440-01"/> permutando, vt ſinus totus ad ſinũ arcus EF, <lb/>hoc eſt, ad ſinum complementi anguli A, ita <lb/>tangens arcus BD, hoc eſt, ita tangens com-<lb/>plementi arcus AB, ad tangentem arcus CE, <lb/>hoc eſt, ad tangentẽ complementi arcus AC. <lb/></s> <s xml:id="echoid-s15009" xml:space="preserve">Non aliter demonſtrabimus, ita eſſe ſinum to <lb/>tum ad ſinum complementi anguli C, vt eſt <lb/>tangens complementi arcus BC, ad tangentem complementi arcus AC, ſi ni-<lb/>mirum arcus CB, CA, angulum C, continentes producantur, &</s> <s xml:id="echoid-s15010" xml:space="preserve">c. </s> <s xml:id="echoid-s15011" xml:space="preserve">In omni <lb/>igitur triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s15012" xml:space="preserve">c. </s> <s xml:id="echoid-s15013" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s15014" xml:space="preserve"/> </p> <div xml:id="echoid-div1202" type="float" level="2" n="1"> <figure xlink:label="fig-440-01" xlink:href="fig-440-01a"> <image file="440-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/440-01"/> </figure> <note position="left" xlink:label="note-440-01" xlink:href="note-440-01a" xml:space="preserve">Theor. 6. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1204" type="section" level="1" n="572"> <head xml:id="echoid-head607" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s15015" xml:space="preserve">INFEREMVS hinc problema ſequens, quod quamuis in problemate prima <lb/>antecedentis propoſ. </s> <s xml:id="echoid-s15016" xml:space="preserve">demonſtratum quoque ſit, facilius tamen hic abſoluitur, cùm in <lb/>aurea regula primum locum ſortiatur ſinus totus.</s> <s xml:id="echoid-s15017" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15018" xml:space="preserve">IN triangulo ſphærico rectangulo, dato alterutto arcuum cir-<lb/>ca angulum rectum, cum angulo non recto adiacente, inuenire ar-<lb/>cum recto angulo oppoſitum, vnà cum reliquo arcu circa angulum <lb/>rectum, & </s> <s xml:id="echoid-s15019" xml:space="preserve">reliquo angulo non recto.</s> <s xml:id="echoid-s15020" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s15021" xml:space="preserve">IN triangulo <emph style="sc">A</emph>BC, cuius angulus C, rectus, datus <lb/> <anchor type="figure" xlink:label="fig-440-02a" xlink:href="fig-440-02"/> ſit arcus AC, cum angulo A, ſibi adiacente. </s> <s xml:id="echoid-s15022" xml:space="preserve">Dico dari <lb/>quoque arcum <emph style="sc">AB</emph>, vnà cum arcu <emph style="sc">BC</emph>, & </s> <s xml:id="echoid-s15023" xml:space="preserve">angulo B.</s> <s xml:id="echoid-s15024" xml:space="preserve"> <lb/>Nam cum ſit, vt ſinus totus ad ſinum complementi angu-<lb/>li A, ita tangens complementi arcus <emph style="sc">AC</emph>, ad tangentem <lb/> <anchor type="note" xlink:label="note-440-02a" xlink:href="note-440-02"/> complementi arcus <emph style="sc">AB</emph>:</s> <s xml:id="echoid-s15025" xml:space="preserve"/> </p> <div xml:id="echoid-div1204" type="float" level="2" n="1"> <figure xlink:label="fig-440-02" xlink:href="fig-440-02a"> <image file="440-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/440-02"/> </figure> <note position="left" xlink:label="note-440-02" xlink:href="note-440-02a" xml:space="preserve">46. huius.</note> </div> <p style="it"> <s xml:id="echoid-s15026" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum complemen-<lb/> <anchor type="note" xlink:label="note-440-03a" xlink:href="note-440-03"/> ti anguli dati, ita tangens complementi arcus da-<lb/>ti ad aliud, producetur tangens complementi ar-<lb/>cus recto angulo oppoſiti, qui quæritur. </s> <s xml:id="echoid-s15027" xml:space="preserve">Reliqua inuenientur, vt in pro-<lb/>blemate 1. </s> <s xml:id="echoid-s15028" xml:space="preserve">propoſitionis antecedentis dictum eſt.</s> <s xml:id="echoid-s15029" xml:space="preserve"/> </p> <div xml:id="echoid-div1205" type="float" level="2" n="2"> <note position="left" xlink:label="note-440-03" xlink:href="note-440-03a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s15030" xml:space="preserve"><emph style="sc">A</emph>RCVM autem <emph style="sc">AB</emph>, quæſitum eße quadrante minorem, maioremve, cognoſce-<lb/>mus, vt in dicto problemate 1. </s> <s xml:id="echoid-s15031" xml:space="preserve">ſuperioris propoſ. </s> <s xml:id="echoid-s15032" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s15033" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1207" type="section" level="1" n="573"> <head xml:id="echoid-head608" xml:space="preserve">THEOR. 45. PROPOS. 47.</head> <p> <s xml:id="echoid-s15034" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <lb/>omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s15035" xml:space="preserve">ſinus totus <pb o="429" file="441" n="441" rhead=""/> ad ſinum complementi arcus recto angulo oppo-<lb/>ſiti eandem proportionem habet, quam tangens <lb/>vtriusvis angulorum non rectorum ad tangentem <lb/>complementi reliqui anguli.</s> <s xml:id="echoid-s15036" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15037" xml:space="preserve">IN triangulo ABC, cuius omnes arcus quadrante minores, ſit angulus <lb/>B, rectus. </s> <s xml:id="echoid-s15038" xml:space="preserve">Dico, ita eſſe ſinum totum ad ſinum complementi arcus AC, vt <lb/>eſt tangens anguli C, ad tangentem complementi anguli A. </s> <s xml:id="echoid-s15039" xml:space="preserve">Facta conſtru-<lb/>ctione, vt in propoſ. </s> <s xml:id="echoid-s15040" xml:space="preserve">45. </s> <s xml:id="echoid-s15041" xml:space="preserve">productoq́; </s> <s xml:id="echoid-s15042" xml:space="preserve">arcu CE, ad G, vt CG, ſit quadrans, <lb/>deſcribatur ex polo C, ad interual lum quadran <lb/> <anchor type="figure" xlink:label="fig-441-01a" xlink:href="fig-441-01"/> tis CG, arcus circuli maximi GH, ſecans ar-<lb/>cus CF, EF, productos in I, H: </s> <s xml:id="echoid-s15043" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s15044" xml:space="preserve">CI, qua-<lb/>drans quoque; </s> <s xml:id="echoid-s15045" xml:space="preserve">cum circulus GH, à polo C, ab-<lb/> <anchor type="note" xlink:label="note-441-01a" xlink:href="note-441-01"/> ſit quadrante. </s> <s xml:id="echoid-s15046" xml:space="preserve">Arcus item GH, EH, quadran-<lb/>tes erunt, propter rectos angulos G, E. </s> <s xml:id="echoid-s15047" xml:space="preserve">Eſt enim <lb/>angulus E, rectus, vt propoſ. </s> <s xml:id="echoid-s15048" xml:space="preserve">45. </s> <s xml:id="echoid-s15049" xml:space="preserve">oſtenſum eſt; <lb/></s> <s xml:id="echoid-s15050" xml:space="preserve">at G, rectus eſt, propterea quòd circulus CG, <lb/>ad circulum GH, rectus eſt. </s> <s xml:id="echoid-s15051" xml:space="preserve">Rurſus IG, ar-<lb/> <anchor type="note" xlink:label="note-441-02a" xlink:href="note-441-02"/> cus eſt anguli C; </s> <s xml:id="echoid-s15052" xml:space="preserve">& </s> <s xml:id="echoid-s15053" xml:space="preserve">CE, complementum arcus AC, recto angulo oppoſiti; <lb/></s> <s xml:id="echoid-s15054" xml:space="preserve">& </s> <s xml:id="echoid-s15055" xml:space="preserve">FE, complementum arcus DE, id eſt, anguli A. </s> <s xml:id="echoid-s15056" xml:space="preserve">Quoniam igitur duo cir-<lb/>culi maximi CG, CI, in ſphæra ſe interſecant in C, ductiq́; </s> <s xml:id="echoid-s15057" xml:space="preserve">ſunt ex arcus CI, <lb/>punctis F, I, ad arcum CG, arcus perpendiculares FE, IG; </s> <s xml:id="echoid-s15058" xml:space="preserve">erit, vt ſinus to-<lb/> <anchor type="note" xlink:label="note-441-03a" xlink:href="note-441-03"/> tus quadrantis CG, ad tangentem arcus IG, hoc eſt, anguli C, ita ſinus ar-<lb/>cus CE, hoc eſt, complementiarcus AC, ad tangentem arcus FE, hoc eſt, <lb/>complementi anguli A: </s> <s xml:id="echoid-s15059" xml:space="preserve">Et permutando erit, vt ſinus totus ad ſinum comple-<lb/>mentiarcus AC, recto angulo oppoſiti, ita tangens anguli C, ad tangentem <lb/>complementi anguli A. </s> <s xml:id="echoid-s15060" xml:space="preserve">Similimodo, aliter conſtructa figura, demonſtrabi-<lb/>mus, ita eſſe ſinum totum ad ſinum complementi arcus AC vt eſt tangens <lb/>anguli A, ad tangentem complementi anguli C. </s> <s xml:id="echoid-s15061" xml:space="preserve">In omni igitur triangulo <lb/>ſphærico rectangulo, &</s> <s xml:id="echoid-s15062" xml:space="preserve">c. </s> <s xml:id="echoid-s15063" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s15064" xml:space="preserve"/> </p> <div xml:id="echoid-div1207" type="float" level="2" n="1"> <figure xlink:label="fig-441-01" xlink:href="fig-441-01a"> <image file="441-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/441-01"/> </figure> <note position="right" xlink:label="note-441-01" xlink:href="note-441-01a" xml:space="preserve">Coroll. 16. <lb/>1. Theod. <lb/>25. huius.</note> <note position="right" xlink:label="note-441-02" xlink:href="note-441-02a" xml:space="preserve">15.1. Theod.</note> <note position="right" xlink:label="note-441-03" xlink:href="note-441-03a" xml:space="preserve">Theor. 6. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1209" type="section" level="1" n="574"> <head xml:id="echoid-head609" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s15065" xml:space="preserve">EX hoc theoremate ſequens problema colligitur.</s> <s xml:id="echoid-s15066" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15067" xml:space="preserve">IN triangulo ſphærico rectangulo, dato arcu, qui recto angulo <lb/>opponitur, cum alterutro angulorum non rectorum, inuenire alte-<lb/>rum angulum non rectum, & </s> <s xml:id="echoid-s15068" xml:space="preserve">duos arcus circa angulum rectum.</s> <s xml:id="echoid-s15069" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15070" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus C, rectus, datus <lb/> <anchor type="figure" xlink:label="fig-441-02a" xlink:href="fig-441-02"/> ſit arcus <emph style="sc">AB</emph>, cum angulo <emph style="sc">B</emph>. </s> <s xml:id="echoid-s15071" xml:space="preserve">Dico dari quoque reliquum <lb/>angulum <emph style="sc">A</emph>, & </s> <s xml:id="echoid-s15072" xml:space="preserve">duos arcus <emph style="sc">AC, CB</emph>. </s> <s xml:id="echoid-s15073" xml:space="preserve">Cum enim ſit, vt <lb/> <anchor type="note" xlink:label="note-441-04a" xlink:href="note-441-04"/> ſinus totus ad ſinum complementi arcus <emph style="sc">AB</emph>, ita tangens <lb/>anguli <emph style="sc">B</emph>, ad tangentem complementi anguli <emph style="sc">A</emph>:</s> <s xml:id="echoid-s15074" xml:space="preserve"/> </p> <div xml:id="echoid-div1209" type="float" level="2" n="1"> <figure xlink:label="fig-441-02" xlink:href="fig-441-02a"> <image file="441-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/441-02"/> </figure> <note position="right" xlink:label="note-441-04" xlink:href="note-441-04a" xml:space="preserve">47. huius.</note> </div> <p style="it"> <s xml:id="echoid-s15075" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum complementi <lb/> <anchor type="note" xlink:label="note-441-05a" xlink:href="note-441-05"/> arcus recto angulo oppoſiti, & </s> <s xml:id="echoid-s15076" xml:space="preserve">dati, ita tangens <lb/>anguli dati ad aliud, reperietur tangens comple- <pb o="430" file="442" n="442" rhead=""/> menti anguli quæſiti. </s> <s xml:id="echoid-s15077" xml:space="preserve">Hincex arcu AB, & </s> <s xml:id="echoid-s15078" xml:space="preserve">vtroque angulo B, A, vter-<lb/> <anchor type="figure" xlink:label="fig-442-01a" xlink:href="fig-442-01"/> que arcus AC, CB, inuenietur, vt in 2. </s> <s xml:id="echoid-s15079" xml:space="preserve">proble-<lb/>mate propoſ. </s> <s xml:id="echoid-s15080" xml:space="preserve">41. </s> <s xml:id="echoid-s15081" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s15082" xml:space="preserve"/> </p> <div xml:id="echoid-div1210" type="float" level="2" n="2"> <note position="right" xlink:label="note-441-05" xlink:href="note-441-05a" xml:space="preserve">Praxis.</note> <figure xlink:label="fig-442-01" xlink:href="fig-442-01a"> <image file="442-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/442-01"/> </figure> </div> <p> <s xml:id="echoid-s15083" xml:space="preserve">AN vero angulus quæſitus <emph style="sc">A</emph>, acutus ſit, obtuſusve, <lb/>diſcemus exarcu dato <emph style="sc">AB</emph>, & </s> <s xml:id="echoid-s15084" xml:space="preserve">dato angulo <emph style="sc">B</emph>. </s> <s xml:id="echoid-s15085" xml:space="preserve">Nam ſi <emph style="sc">AB</emph>, <lb/>eſt quadrante minor, & </s> <s xml:id="echoid-s15086" xml:space="preserve">angulus <emph style="sc">B</emph>, acutus quidem, erit <lb/> <anchor type="note" xlink:label="note-442-01a" xlink:href="note-442-01"/> & </s> <s xml:id="echoid-s15087" xml:space="preserve"><emph style="sc">A</emph>, acutus; </s> <s xml:id="echoid-s15088" xml:space="preserve">ſi autem <emph style="sc">B</emph>, eſt obtuſus, erit & </s> <s xml:id="echoid-s15089" xml:space="preserve"><emph style="sc">A</emph>, obtuſus. <lb/></s> <s xml:id="echoid-s15090" xml:space="preserve">At ſi <emph style="sc">AB</emph>, eſt maior quadrante, & </s> <s xml:id="echoid-s15091" xml:space="preserve"><emph style="sc">B</emph>, quidem acutus, erit <lb/><emph style="sc">A</emph>, obtuſus; </s> <s xml:id="echoid-s15092" xml:space="preserve">ſi vero <emph style="sc">B</emph>, eſt obtuſus, erit <emph style="sc">A</emph>, acutus.</s> <s xml:id="echoid-s15093" xml:space="preserve"/> </p> <div xml:id="echoid-div1211" type="float" level="2" n="3"> <note position="left" xlink:label="note-442-01" xlink:href="note-442-01a" xml:space="preserve">38. huius</note> </div> </div> <div xml:id="echoid-div1213" type="section" level="1" n="575"> <head xml:id="echoid-head610" xml:space="preserve">THEOR. 46. PROPOS. 48.</head> <p> <s xml:id="echoid-s15094" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <lb/>omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s15095" xml:space="preserve">Sinus totus <lb/>ad ſinum vtriusvis arcuum circa angulum rectum <lb/>eandem habet proportionem, quam tangens com <lb/>plementi alterius arcus circa angulum rectum ad <lb/>tangentem complementi anguli oppoſiti.</s> <s xml:id="echoid-s15096" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15097" xml:space="preserve">IN triangulo ſphærico ABC, cuius omnes arcus minores quadrante, ſit <lb/>rectus angulus B. </s> <s xml:id="echoid-s15098" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum arcus AB, vt eſt tangens <lb/> <anchor type="figure" xlink:label="fig-442-02a" xlink:href="fig-442-02"/> complementi arcus BC, ad tangentem com-<lb/>plementi anguli A. </s> <s xml:id="echoid-s15099" xml:space="preserve">Facta enim conſtructio-<lb/>ne, vt in propoſ. </s> <s xml:id="echoid-s15100" xml:space="preserve">45. </s> <s xml:id="echoid-s15101" xml:space="preserve">erit angulus D, rectus, <lb/>& </s> <s xml:id="echoid-s15102" xml:space="preserve"><emph style="sc">Cf</emph>, complementum arcus BC; </s> <s xml:id="echoid-s15103" xml:space="preserve">& </s> <s xml:id="echoid-s15104" xml:space="preserve">EF, com <lb/>plementum anguli A; </s> <s xml:id="echoid-s15105" xml:space="preserve">& </s> <s xml:id="echoid-s15106" xml:space="preserve">AD, quadrans, vt ibi <lb/>oſtenſum eſt. </s> <s xml:id="echoid-s15107" xml:space="preserve">Quoniam igitur duo circuli ma-<lb/>ximi AD, AE, in ſphæra ſe mutuo ſecãt in A, <lb/>ductiq́; </s> <s xml:id="echoid-s15108" xml:space="preserve">ſunt ex punctis C, E, ad arcum AD, <lb/>arcus perpendiculares CB, ED; </s> <s xml:id="echoid-s15109" xml:space="preserve">erit, vt ſinus <lb/>totus quadrantis AD, ad tangentem arcus <lb/> <anchor type="note" xlink:label="note-442-02a" xlink:href="note-442-02"/> DE, ita ſinus arcus AB, ad tangentem arcus <lb/>BC: </s> <s xml:id="echoid-s15110" xml:space="preserve">Et permutando, vt ſinus totus ad ſinum <lb/>arcus AB, ita tangens arcus DE, ad tangen-<lb/>tem arcus BC. </s> <s xml:id="echoid-s15111" xml:space="preserve">Eſt autem, (cum CF, EF, ſint complementa arcuum BC, DE,) <lb/>vt tangens arcus DE, ad tangentem arcus BC, ita tangens arcus CF, ad tan <lb/> <anchor type="note" xlink:label="note-442-03a" xlink:href="note-442-03"/> gentem arcus EF. </s> <s xml:id="echoid-s15112" xml:space="preserve">Igitur erit quoque, vt ſinus totus ad ſinum arcus AB, ita <lb/>tangens arcus CF, hoc eſt, complementi arcus BC, ad tangentem arcus EF, <lb/>hoceſt, complementi anguli A, arcui BC, oppoſiti. </s> <s xml:id="echoid-s15113" xml:space="preserve">Non aliter oſtendemus, <lb/>ſi aliter figura conſtruatur, ita eſſe ſinum totum ad ſinum arcus BC, vt eſt tan <lb/>gens complementi arcus AB, ad tangentem complementi anguli C. </s> <s xml:id="echoid-s15114" xml:space="preserve">In omni <lb/>triangulo ergo ſphærico rectangulo, &</s> <s xml:id="echoid-s15115" xml:space="preserve">c. </s> <s xml:id="echoid-s15116" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s15117" xml:space="preserve"/> </p> <div xml:id="echoid-div1213" type="float" level="2" n="1"> <figure xlink:label="fig-442-02" xlink:href="fig-442-02a"> <image file="442-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/442-02"/> </figure> <note position="left" xlink:label="note-442-02" xlink:href="note-442-02a" xml:space="preserve">Theor. 6. <lb/>ſcholij 40. <lb/>huius.</note> <note position="left" xlink:label="note-442-03" xlink:href="note-442-03a" xml:space="preserve">81. Sinuũ</note> </div> <pb o="431" file="443" n="443" rhead=""/> </div> <div xml:id="echoid-div1215" type="section" level="1" n="576"> <head xml:id="echoid-head611" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15118" xml:space="preserve">INFERTVR ex theoremate hoc ſequens problema: </s> <s xml:id="echoid-s15119" xml:space="preserve">quod licet demonſtratum <lb/>quoque ſit problemate 2. </s> <s xml:id="echoid-s15120" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15121" xml:space="preserve">44. </s> <s xml:id="echoid-s15122" xml:space="preserve">facilius tamen hic abſoluitur, cumin aurea re-<lb/>gula ſinus to tus primum obtineat locum.</s> <s xml:id="echoid-s15123" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15124" xml:space="preserve">IN triangulo ſphærico rectangulo, datis duobus arcubus circa <lb/>angulum rectum, vtrumlibet angulorum non rectorum, vnà cum <lb/>arcu reliquo, qui angulo recto opponitur, indagare.</s> <s xml:id="echoid-s15125" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15126" xml:space="preserve">IN triangulo ABC, cuius angulus C, rectus, <lb/> <anchor type="figure" xlink:label="fig-443-01a" xlink:href="fig-443-01"/> ſint dati duo arcus AC, CB. </s> <s xml:id="echoid-s15127" xml:space="preserve">Dico vtrumuis angulo-<lb/>rum A, B, & </s> <s xml:id="echoid-s15128" xml:space="preserve">arcum AB, quoque dari. </s> <s xml:id="echoid-s15129" xml:space="preserve">Nam cum ſit, <lb/>vt ſinus totus ad ſinum arcus AC, ita tangens comple-<lb/> <anchor type="note" xlink:label="note-443-01a" xlink:href="note-443-01"/> menti arcus CB, ad tangentem complementi anguli <lb/>A. </s> <s xml:id="echoid-s15130" xml:space="preserve">Item vt ſinus totus ad ſinum arcus CB, ita tan-<lb/>gens complementi arcus AC, ad tangentem complemen <lb/>ti anguli B:</s> <s xml:id="echoid-s15131" xml:space="preserve"/> </p> <div xml:id="echoid-div1215" type="float" level="2" n="1"> <figure xlink:label="fig-443-01" xlink:href="fig-443-01a"> <image file="443-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/443-01"/> </figure> <note position="right" xlink:label="note-443-01" xlink:href="note-443-01a" xml:space="preserve">48. huius.</note> </div> <p style="it"> <s xml:id="echoid-s15132" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum vtrius vis arcuum circa angulum re-<lb/> <anchor type="note" xlink:label="note-443-02a" xlink:href="note-443-02"/> ctum, ita tangens complementi alterius arcus circa rectum angulum ad <lb/>aliud, reperietur tangens complementi anguli huic poſterioriarcui oppo-<lb/>ſiti. </s> <s xml:id="echoid-s15133" xml:space="preserve">Ex datis quoque duobus arcubus circa angulum rectum cognoſcetur <lb/>& </s> <s xml:id="echoid-s15134" xml:space="preserve">tertius arcus angulo recto oppoſitus, vt in problemate propoſ. </s> <s xml:id="echoid-s15135" xml:space="preserve">43. </s> <s xml:id="echoid-s15136" xml:space="preserve">oſten <lb/>dimus. </s> <s xml:id="echoid-s15137" xml:space="preserve">Vel certe ex dato vtrolibet arcu, & </s> <s xml:id="echoid-s15138" xml:space="preserve">angulo, qui ei opponitur, in-<lb/>uento, vt in problemate 3. </s> <s xml:id="echoid-s15139" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15140" xml:space="preserve">41. </s> <s xml:id="echoid-s15141" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s15142" xml:space="preserve"/> </p> <div xml:id="echoid-div1216" type="float" level="2" n="2"> <note position="right" xlink:label="note-443-02" xlink:href="note-443-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s15143" xml:space="preserve">VTRVM autem angulus quæſitus ſit acutus, obtuſusve, docebit arcus ei oppo-<lb/>ſitus. </s> <s xml:id="echoid-s15144" xml:space="preserve">Hic enim ſi minor fuerit quadrante, erit angulus ei oppoſitus, acutus; </s> <s xml:id="echoid-s15145" xml:space="preserve">ſi vero <lb/> <anchor type="note" xlink:label="note-443-03a" xlink:href="note-443-03"/> maior, obtuſus.</s> <s xml:id="echoid-s15146" xml:space="preserve"/> </p> <div xml:id="echoid-div1217" type="float" level="2" n="3"> <note position="right" xlink:label="note-443-03" xlink:href="note-443-03a" xml:space="preserve">34. huius.</note> </div> </div> <div xml:id="echoid-div1219" type="section" level="1" n="577"> <head xml:id="echoid-head612" xml:space="preserve">THEOR. 47. PROPOS. 49.</head> <p> <s xml:id="echoid-s15147" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius omnes arcus ſint minores quadrante: </s> <s xml:id="echoid-s15148" xml:space="preserve">ſinus to-<lb/>tus ad tangentem vtriusvis arcuum circa angulum <lb/>rectum eandem proportionem habet, quam tan-<lb/>gens complementi anguli oppoſiti ad ſinum alte-<lb/>rius arcus circa rectum angulum.</s> <s xml:id="echoid-s15149" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15150" xml:space="preserve">IN ſphærico triangulo ADE, cuius arcus omnes quadrante minores, ſit <lb/>angulus D, rectus. </s> <s xml:id="echoid-s15151" xml:space="preserve">Dico ita eſſe ſinum totum ad tangentem arcus DE, vt eſt <pb o="432" file="444" n="444" rhead=""/> tangens complementi anguli A, ad ſinum arcus AD. </s> <s xml:id="echoid-s15152" xml:space="preserve">Repetita enim conſtru-<lb/> <anchor type="figure" xlink:label="fig-444-01a" xlink:href="fig-444-01"/> ctione figuræ propoſ. </s> <s xml:id="echoid-s15153" xml:space="preserve">45. </s> <s xml:id="echoid-s15154" xml:space="preserve">erunt AB, AC, <lb/>quadrantes, & </s> <s xml:id="echoid-s15155" xml:space="preserve">CF, complementum arcus <lb/>BC, id eſt, anguli A, vt ibi oſtenſum eſt. <lb/></s> <s xml:id="echoid-s15156" xml:space="preserve">Igitur quoniam quadrantes ſunt AB, AC, <lb/>& </s> <s xml:id="echoid-s15157" xml:space="preserve">arcus ED, ad AB, perpendicularis; </s> <s xml:id="echoid-s15158" xml:space="preserve">erit, <lb/>vt ſinus totus ad tangentem arcus ED, ita <lb/>tangens complementi arcus CB, hoc eſt, <lb/> <anchor type="note" xlink:label="note-444-01a" xlink:href="note-444-01"/> anguli A, ad ſinum arcus AD. </s> <s xml:id="echoid-s15159" xml:space="preserve">Eodem mo <lb/>do oſtendetur, ita eſſe ſinum totum ad tan <lb/>gentem arcus AD, vt eſt tangens comple-<lb/>menti anguli E, ad ſinum arcus DE: </s> <s xml:id="echoid-s15160" xml:space="preserve">ſi ni-<lb/>mirum aliter figura conſtruatur. </s> <s xml:id="echoid-s15161" xml:space="preserve">In omni <lb/>ergo trianguloſphærico rectangulo, &</s> <s xml:id="echoid-s15162" xml:space="preserve">c. </s> <s xml:id="echoid-s15163" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s15164" xml:space="preserve"/> </p> <div xml:id="echoid-div1219" type="float" level="2" n="1"> <figure xlink:label="fig-444-01" xlink:href="fig-444-01a"> <image file="444-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/444-01"/> </figure> <note position="left" xlink:label="note-444-01" xlink:href="note-444-01a" xml:space="preserve">Theor. 7. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1221" type="section" level="1" n="578"> <head xml:id="echoid-head613" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15165" xml:space="preserve">HINC tale problema colligitur, quod per problema 1. </s> <s xml:id="echoid-s15166" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15167" xml:space="preserve">44. </s> <s xml:id="echoid-s15168" xml:space="preserve">alio modo ab-<lb/>ſelui quoque potest.</s> <s xml:id="echoid-s15169" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15170" xml:space="preserve">IN triangulo ſphærico rectangulo, dato alterutro arcuum circa <lb/>angulum rectum, cum angulo oppoſito, reliquum arcum circa re-<lb/>ctum angulum, & </s> <s xml:id="echoid-s15171" xml:space="preserve">arcum recto angulo oppoſitum, cum reliquo an-<lb/>gulo non recto inquirere: </s> <s xml:id="echoid-s15172" xml:space="preserve">ſi modo conſtet, num arcus quæſitus ſit <lb/>maior quadrante, minorve: </s> <s xml:id="echoid-s15173" xml:space="preserve">Vel an reliquus angulus non rectus ſit <lb/>acutus, obtuſusve.</s> <s xml:id="echoid-s15174" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15175" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus <emph style="sc">C</emph>, rectus, datus ſit arcus <emph style="sc">AC</emph>, cum angulo <lb/> <anchor type="figure" xlink:label="fig-444-02a" xlink:href="fig-444-02"/> oppoſito <emph style="sc">B</emph>. </s> <s xml:id="echoid-s15176" xml:space="preserve">Dico dari quoque arcum <emph style="sc">BC</emph>, &</s> <s xml:id="echoid-s15177" xml:space="preserve">c. </s> <s xml:id="echoid-s15178" xml:space="preserve">Cumenim <lb/>ſit, vt ſinus totus ad tangentem arcus <emph style="sc">AC</emph>, ita tangens <lb/> <anchor type="note" xlink:label="note-444-02a" xlink:href="note-444-02"/> complementi anguli <emph style="sc">B</emph>, ad ſinum arcus <emph style="sc">BC</emph>:</s> <s xml:id="echoid-s15179" xml:space="preserve"/> </p> <div xml:id="echoid-div1221" type="float" level="2" n="1"> <figure xlink:label="fig-444-02" xlink:href="fig-444-02a"> <image file="444-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/444-02"/> </figure> <note position="left" xlink:label="note-444-02" xlink:href="note-444-02a" xml:space="preserve">49.huius.</note> </div> <p style="it"> <s xml:id="echoid-s15180" xml:space="preserve">SI fiat, vt ſinus totus ad tangentem datiar-<lb/> <anchor type="note" xlink:label="note-444-03a" xlink:href="note-444-03"/> cus, ita tangens complemẽti anguli dati ad aliud, <lb/>producetur ſinus arcus quæſiti. </s> <s xml:id="echoid-s15181" xml:space="preserve">Ex duobus por-<lb/>ro arcubus circa rectum angulum cognitis in co-<lb/>gnitionem reliqui arcus, & </s> <s xml:id="echoid-s15182" xml:space="preserve">reliqui angulinon re-<lb/>cti perueniemus, vt in problemate propoſ. </s> <s xml:id="echoid-s15183" xml:space="preserve">43. </s> <s xml:id="echoid-s15184" xml:space="preserve">demonſtrauimus; </s> <s xml:id="echoid-s15185" xml:space="preserve">vel cer-<lb/>te ex alterutro arcuum circa angulum rectum, & </s> <s xml:id="echoid-s15186" xml:space="preserve">dato angulo, vt in pro-<lb/>blemate 2. </s> <s xml:id="echoid-s15187" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15188" xml:space="preserve">42. </s> <s xml:id="echoid-s15189" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s15190" xml:space="preserve"/> </p> <div xml:id="echoid-div1222" type="float" level="2" n="2"> <note position="left" xlink:label="note-444-03" xlink:href="note-444-03a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s15191" xml:space="preserve">OPORTET autem hic conſtare, num arcus quæſitus <emph style="sc">BC</emph>, ſit quadrante maior, <lb/>minor ve; </s> <s xml:id="echoid-s15192" xml:space="preserve">vel an angulus <emph style="sc">A</emph>, reliquus ſit acutus, obtuſus ve: </s> <s xml:id="echoid-s15193" xml:space="preserve">quemadmodũ in poſteriore <lb/>parte problematis 1. </s> <s xml:id="echoid-s15194" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15195" xml:space="preserve">44. </s> <s xml:id="echoid-s15196" xml:space="preserve">traditum eſt: </s> <s xml:id="echoid-s15197" xml:space="preserve">Vbi etiam errorẽ Copernici deteximus.</s> <s xml:id="echoid-s15198" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1224" type="section" level="1" n="579"> <head xml:id="echoid-head614" xml:space="preserve">THEOR. 48. PROPOS. 50.</head> <p> <s xml:id="echoid-s15199" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <pb o="433" file="445" n="445" rhead=""/> omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s15200" xml:space="preserve">ſinus totus <lb/>ad tangentem complementi vtriusvis angulorum <lb/>non rectorum habet proportionem eãdem, quam <lb/>tangens complemẽti reliqui anguli ad ſinum com <lb/>plementi arcus recto angulo oppoſiti.</s> <s xml:id="echoid-s15201" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15202" xml:space="preserve">IN triangulo ABC, cuius omnes arcus quadrante minores, ſit angulus <lb/>B, rectus. </s> <s xml:id="echoid-s15203" xml:space="preserve">Dico ita eſſe ſinum totum ad tangentem complementi anguli A, vt <lb/>eſt tangens complementi anguli C, ad ſinum complementi arcus AC. </s> <s xml:id="echoid-s15204" xml:space="preserve">Repe-<lb/>tita namq; </s> <s xml:id="echoid-s15205" xml:space="preserve">figura propoſ. </s> <s xml:id="echoid-s15206" xml:space="preserve">47. </s> <s xml:id="echoid-s15207" xml:space="preserve">cum CG, CI, qua-<lb/> <anchor type="figure" xlink:label="fig-445-01a" xlink:href="fig-445-01"/> drãtes ſint ſe interſecãtes in C, & </s> <s xml:id="echoid-s15208" xml:space="preserve">arcus IG, FE, <lb/>ad CG, perpendiculares, vt ex conſtructione ibi-<lb/>dem facta perſpicuum eſt; </s> <s xml:id="echoid-s15209" xml:space="preserve">erit, vt ſinus totus ad <lb/> <anchor type="note" xlink:label="note-445-01a" xlink:href="note-445-01"/> tangentem arcus EF, qui complementũ eſt arcus <lb/>DE, hoc eſt, anguli A, ita tangens complementi <lb/>arcus IG, id eſt, anguli C, ad ſinum arcus CE, <lb/>hoc eſt, complementi arcus AC, recto angulo <lb/>oppoſiti. </s> <s xml:id="echoid-s15210" xml:space="preserve">Simili ratione oſtendemus, ſi aliter figuræ conſtructio inſtituatur, <lb/>ita eſſe ſinum totum ad tangentem complementi anguli C, vt eſt tangens com <lb/>plementi anguli A, ad ſinum complementi arcus AC. </s> <s xml:id="echoid-s15211" xml:space="preserve">Quam ob rem in omni <lb/>triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s15212" xml:space="preserve">c. </s> <s xml:id="echoid-s15213" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s15214" xml:space="preserve"/> </p> <div xml:id="echoid-div1224" type="float" level="2" n="1"> <figure xlink:label="fig-445-01" xlink:href="fig-445-01a"> <image file="445-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/445-01"/> </figure> <note position="right" xlink:label="note-445-01" xlink:href="note-445-01a" xml:space="preserve">Theor. 7. <lb/>ſcholij 40, <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1226" type="section" level="1" n="580"> <head xml:id="echoid-head615" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15215" xml:space="preserve">INFEREMVS ex hac propoſ. </s> <s xml:id="echoid-s15216" xml:space="preserve">theorema ſequens.</s> <s xml:id="echoid-s15217" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15218" xml:space="preserve">IN triangulo ſphærico rectangulo, datis duobus angulis non re-<lb/>ctis, inquirere arcum angulo recto oppoſitum, & </s> <s xml:id="echoid-s15219" xml:space="preserve">reliquos duos ar-<lb/>cus circa angulum rectum.</s> <s xml:id="echoid-s15220" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15221" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus <emph style="sc">C</emph>, rectus, dati <lb/> <anchor type="figure" xlink:label="fig-445-02a" xlink:href="fig-445-02"/> ſint duo anguli non recti <emph style="sc">A, B</emph>. </s> <s xml:id="echoid-s15222" xml:space="preserve">Dico dari quoque arcum <lb/><emph style="sc">AB</emph>, vnà cum arcubus <emph style="sc">AC, BC</emph>. </s> <s xml:id="echoid-s15223" xml:space="preserve">Quoniam enim eſt, vt <lb/>ſinus totus ad tangentem complementi anguli <emph style="sc">A</emph>, ita tan-<lb/> <anchor type="note" xlink:label="note-445-02a" xlink:href="note-445-02"/> gens complementi anguli <emph style="sc">B</emph>, ad ſinum complementi ar-<lb/>cus <emph style="sc">AB</emph>:</s> <s xml:id="echoid-s15224" xml:space="preserve"/> </p> <div xml:id="echoid-div1226" type="float" level="2" n="1"> <figure xlink:label="fig-445-02" xlink:href="fig-445-02a"> <image file="445-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/445-02"/> </figure> <note position="right" xlink:label="note-445-02" xlink:href="note-445-02a" xml:space="preserve">50. huius.</note> </div> <note position="right" xml:space="preserve">Praxis.</note> <p style="it"> <s xml:id="echoid-s15225" xml:space="preserve">SI fiat, vt ſinus totus ad tangentem comple-<lb/>menti vtriusvis angulorum datorum, ita tangens <lb/>complementi alterius dati anguli ad aliud, procre abitur ſinus complemen <lb/>ti arcus recto angulo oppoſiti. </s> <s xml:id="echoid-s15226" xml:space="preserve">Iam ex arcu, qui recto angulo opponitur, <lb/>& </s> <s xml:id="echoid-s15227" xml:space="preserve">vtrolibet angulorum non rectorum, inuenietur arcus ei oppoſitus, vt <lb/>in 2. </s> <s xml:id="echoid-s15228" xml:space="preserve">problemate propoſ. </s> <s xml:id="echoid-s15229" xml:space="preserve">41. </s> <s xml:id="echoid-s15230" xml:space="preserve">monſtr auimus.</s> <s xml:id="echoid-s15231" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15232" xml:space="preserve">PORRO an arcus quæſitus quadrante ſit maior, aut minor, ita diſcemus. </s> <s xml:id="echoid-s15233" xml:space="preserve">Si <lb/>vterq; </s> <s xml:id="echoid-s15234" xml:space="preserve">angulorum <emph style="sc">A, B</emph>, fuerit obtuſus, vel acutus, erit arcus <emph style="sc">AB</emph>, quadrante minor, <lb/> <anchor type="note" xlink:label="note-445-04a" xlink:href="note-445-04"/> ſi vero alter eorum acutus fuerit, et alter obtuſus, erit idem arcus quadrante maior.</s> <s xml:id="echoid-s15235" xml:space="preserve"/> </p> <div xml:id="echoid-div1227" type="float" level="2" n="2"> <note position="right" xlink:label="note-445-04" xlink:href="note-445-04a" xml:space="preserve">37. huius.</note> </div> <pb o="434" file="446" n="446" rhead=""/> </div> <div xml:id="echoid-div1229" type="section" level="1" n="581"> <head xml:id="echoid-head616" xml:space="preserve">THEOR. 49. PROPOS. 51.</head> <p> <s xml:id="echoid-s15236" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <lb/>arcus omnes ſint minores quadrante: </s> <s xml:id="echoid-s15237" xml:space="preserve">ſinus totus <lb/>ad tangentem complementi arcus recto angulo <lb/>oppoſiti proportionem habet eandem, quam tan <lb/>gens vtriusvis arcuum circa angulum rectum ad <lb/>ſinum complementi anguli non recti adiacentis.</s> <s xml:id="echoid-s15238" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15239" xml:space="preserve">IN triangulo ſphærico ABC, cuius omnes arcus quadrante minores, re-<lb/>ctus ſit angulus B. </s> <s xml:id="echoid-s15240" xml:space="preserve">Dico ita eſſe ſinum totum ad tangentem complementi ar-<lb/> <anchor type="figure" xlink:label="fig-446-01a" xlink:href="fig-446-01"/> cus AC, vt eſt tangens arcus AB, ad ſinum <lb/>complementi anguli A. </s> <s xml:id="echoid-s15241" xml:space="preserve">Repetita namq; </s> <s xml:id="echoid-s15242" xml:space="preserve">con-<lb/>ſtructione figuræ propoſ. </s> <s xml:id="echoid-s15243" xml:space="preserve">45. </s> <s xml:id="echoid-s15244" xml:space="preserve">erunt AD, AE, <lb/>quadrantes, & </s> <s xml:id="echoid-s15245" xml:space="preserve">anguli D, E, recti, necnon & </s> <s xml:id="echoid-s15246" xml:space="preserve"><lb/>BF, DF, quadrantes, vt ibi eſt oſtenſum. <lb/></s> <s xml:id="echoid-s15247" xml:space="preserve">Quia igitur in ſphæra arcus DB, per extremi-<lb/>tates quadrantum BF, DF, ſeſe in F, ſecan-<lb/>tium ducitur, & </s> <s xml:id="echoid-s15248" xml:space="preserve">CE, ad DF, perpendicularis <lb/>eſt; </s> <s xml:id="echoid-s15249" xml:space="preserve">erit vt ſinus totus ad tangentẽ arcus CE, <lb/> <anchor type="note" xlink:label="note-446-01a" xlink:href="note-446-01"/> qui complementum eſt arcus AC, recto angu-<lb/>lo oppoſiti, ita tangens complementi arcus <lb/>DB, hoc eſt, tangens arcus AB, ad ſinum ar-<lb/>cus EF, qui complementum eſt arcus DE, <lb/>ſeu anguli A. </s> <s xml:id="echoid-s15250" xml:space="preserve">Non aliter demonſtrabitur, ita eſſe ſinum totum ad tangen <lb/>tem complementi arcus AC, vt eſt tangens arcus BC, ad ſinum complementi <lb/>anguli C, ſi aliter inſtituatur conſtructio figuræ. </s> <s xml:id="echoid-s15251" xml:space="preserve">Quocirca in omni triangulo <lb/>ſphærico rectangulo, &</s> <s xml:id="echoid-s15252" xml:space="preserve">c. </s> <s xml:id="echoid-s15253" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s15254" xml:space="preserve"/> </p> <div xml:id="echoid-div1229" type="float" level="2" n="1"> <figure xlink:label="fig-446-01" xlink:href="fig-446-01a"> <image file="446-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/446-01"/> </figure> <note position="left" xlink:label="note-446-01" xlink:href="note-446-01a" xml:space="preserve">Theor. 7. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1231" type="section" level="1" n="582"> <head xml:id="echoid-head617" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15255" xml:space="preserve">ORITVR ex hoc theoremate problema huiuſmodi, quod problemate 2. </s> <s xml:id="echoid-s15256" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s15257" xml:space="preserve">45. </s> <s xml:id="echoid-s15258" xml:space="preserve">declaratum quoque fuit.</s> <s xml:id="echoid-s15259" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15260" xml:space="preserve">IN triangulo ſphærico rectangulo, dato arcu, quirecto angulo <lb/>opponitur, cum alterutro arcuum circa eundem rectum angulum, <lb/>reperire angulum non rectum huic arcui adiacentem, hoc eſt, à da-<lb/>tis arcubus comprehenſum, cum reliquo arcu, & </s> <s xml:id="echoid-s15261" xml:space="preserve">angulo non recto.</s> <s xml:id="echoid-s15262" xml:space="preserve"/> </p> <figure> <image file="446-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/446-02"/> </figure> <p> <s xml:id="echoid-s15263" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus <emph style="sc">C</emph>, rectus, datus <lb/>ſit arcus <emph style="sc">AB</emph>, cum arcu <emph style="sc">AC</emph>. </s> <s xml:id="echoid-s15264" xml:space="preserve">Dico dari quoque angulum <lb/><emph style="sc">A</emph>, cum arcu <emph style="sc">BC</emph>, & </s> <s xml:id="echoid-s15265" xml:space="preserve">angulo <emph style="sc">B</emph>. </s> <s xml:id="echoid-s15266" xml:space="preserve">Quoniam enim eſt, vt <lb/>ſinus totus ad tangentem complementi arcus <emph style="sc">AB</emph>, ita tan-<lb/> <anchor type="note" xlink:label="note-446-02a" xlink:href="note-446-02"/> gens arcus <emph style="sc">AC</emph>, ad ſinum complementi anguli <emph style="sc">A</emph>:</s> <s xml:id="echoid-s15267" xml:space="preserve"/> </p> <div xml:id="echoid-div1231" type="float" level="2" n="1"> <note position="left" xlink:label="note-446-02" xlink:href="note-446-02a" xml:space="preserve">51.huius.</note> </div> <p style="it"> <s xml:id="echoid-s15268" xml:space="preserve">SI fiat, vt ſinus totus ad tangentem comple-<lb/> <anchor type="note" xlink:label="note-446-03a" xlink:href="note-446-03"/> menti arcus recto angulo oppoſiti, ita tangens da- <pb o="435" file="447" n="447" rhead=""/> ticrcus circa rectum angulum ad aliud, inuenietur ſinus complementi an-<lb/>g@l adiacentis, qui quæritur. </s> <s xml:id="echoid-s15269" xml:space="preserve">Hinc reliqua inuenientur, vt in problema-<lb/>te 1. </s> <s xml:id="echoid-s15270" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15271" xml:space="preserve">45. </s> <s xml:id="echoid-s15272" xml:space="preserve">traditum est.</s> <s xml:id="echoid-s15273" xml:space="preserve"/> </p> <div xml:id="echoid-div1232" type="float" level="2" n="2"> <note position="left" xlink:label="note-446-03" xlink:href="note-446-03a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s15274" xml:space="preserve">NVM vero quæſitus angulus acutus ſit, nec ne, addiſcemus, vt in problemate 2. <lb/></s> <s xml:id="echoid-s15275" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15276" xml:space="preserve">45. </s> <s xml:id="echoid-s15277" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s15278" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1234" type="section" level="1" n="583"> <head xml:id="echoid-head618" xml:space="preserve">THEOR. 50. PROPOS 52.</head> <p> <s xml:id="echoid-s15279" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s15280" xml:space="preserve">ſinus to-<lb/>tus ad ſinum vtriusvis angulorum non rectorum <lb/>habet proportionem eandem, quam ſecans alte-<lb/>rius anguli non recti ad ſecantem arcus huic angu <lb/>lo oppoſiti.</s> <s xml:id="echoid-s15281" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15282" xml:space="preserve">IN triangulo ſphærico ABC, cuius omnes arcus quadrante minores, ſit <lb/>angulus B, rectus. </s> <s xml:id="echoid-s15283" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum anguli A, vt eſt ſecans <lb/>anguli C, ad ſecantem arcus AB. </s> <s xml:id="echoid-s15284" xml:space="preserve">Facta conſtructione, vt in propoſ. </s> <s xml:id="echoid-s15285" xml:space="preserve">47. </s> <s xml:id="echoid-s15286" xml:space="preserve">erunt <lb/>GH, HE, AE, AD, DF, quadrantes, & </s> <s xml:id="echoid-s15287" xml:space="preserve">GI, arcus anguli C, & </s> <s xml:id="echoid-s15288" xml:space="preserve">DE, ar-<lb/>cus anguli A, vt partim in propoſ. </s> <s xml:id="echoid-s15289" xml:space="preserve">45. </s> <s xml:id="echoid-s15290" xml:space="preserve">partim vero in 47. </s> <s xml:id="echoid-s15291" xml:space="preserve">oſtenſum eſt. </s> <s xml:id="echoid-s15292" xml:space="preserve">Item <lb/>angulus I, rectus erit, propterea quòd arcus CI, <lb/> <anchor type="figure" xlink:label="fig-447-01a" xlink:href="fig-447-01"/> tranſiens per C, polum arcus GH, rectus eſt <lb/> <anchor type="note" xlink:label="note-447-01a" xlink:href="note-447-01"/> ad GH. </s> <s xml:id="echoid-s15293" xml:space="preserve">Itaque quonlam duo circuli maximi <lb/>BI, DH, in ſphæra ſe mutuo ſecant in F, & </s> <s xml:id="echoid-s15294" xml:space="preserve">ex <lb/>punctis D, H, arcus DH, ad arcum BI, ducti <lb/>ſunt arcus perpendiculares DB, HI; </s> <s xml:id="echoid-s15295" xml:space="preserve">erit, vt ſi-<lb/> <anchor type="note" xlink:label="note-447-02a" xlink:href="note-447-02"/> nustotus quadrantis DF, ad ſecantem arcus <lb/>GI, qui complementum eſt arcus HI, ita ſinus <lb/>arcus FH, ad ſecantem arcus AB, qui complementum eſt arcus DB: </s> <s xml:id="echoid-s15296" xml:space="preserve">Et per-<lb/>mutando, vt ſinus totus ad ſinum arcus FH, vel arcus DE, (ſunt enim arcus <lb/>FH, DE, æquales, quod & </s> <s xml:id="echoid-s15297" xml:space="preserve">toti quadrantes EH, DF, æquales ſint) hoc eſt, <lb/>anguli A, ita ſecans arcus GI, id eſt, anguli C, ad ſecantem arcus AB, angu-<lb/>lo C, oppoſiti. </s> <s xml:id="echoid-s15298" xml:space="preserve">Pari ratione, ſi aliter conſtruatur figura, demonſtrabimus, ita <lb/>eſſe ſinum totum ad ſinum anguli C, vt eſt ſecans anguli A, ad ſecantem arcus <lb/>BC. </s> <s xml:id="echoid-s15299" xml:space="preserve">In omni ergo triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s15300" xml:space="preserve">c. </s> <s xml:id="echoid-s15301" xml:space="preserve">Quod erat oſtendendũ.</s> <s xml:id="echoid-s15302" xml:space="preserve"/> </p> <div xml:id="echoid-div1234" type="float" level="2" n="1"> <figure xlink:label="fig-447-01" xlink:href="fig-447-01a"> <image file="447-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/447-01"/> </figure> <note position="right" xlink:label="note-447-01" xlink:href="note-447-01a" xml:space="preserve">15. 1. Theod.</note> <note position="right" xlink:label="note-447-02" xlink:href="note-447-02a" xml:space="preserve">Theor. 8. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1236" type="section" level="1" n="584"> <head xml:id="echoid-head619" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15303" xml:space="preserve">ELICITVR hinc ſequens problema, quod aliter etiam in probl. </s> <s xml:id="echoid-s15304" xml:space="preserve">1. </s> <s xml:id="echoid-s15305" xml:space="preserve">ſcholij <lb/>propoſ. </s> <s xml:id="echoid-s15306" xml:space="preserve">42. </s> <s xml:id="echoid-s15307" xml:space="preserve">ſolutum fuit.</s> <s xml:id="echoid-s15308" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15309" xml:space="preserve">IN triangulo ſphærico rectangulo, datis duobus angulis non re-<lb/>ctis, elicere arcum vtrilibet eorum oppoſitum, vnà cum arcu, qui <lb/>recto angulo opponitur.</s> <s xml:id="echoid-s15310" xml:space="preserve"/> </p> <pb o="436" file="448" n="448" rhead=""/> <p> <s xml:id="echoid-s15311" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus <emph style="sc">C</emph>, rectus, dati ſint duo anguli <emph style="sc">A, B</emph>. </s> <s xml:id="echoid-s15312" xml:space="preserve">Dic@ <lb/> <anchor type="figure" xlink:label="fig-448-01a" xlink:href="fig-448-01"/> dari quoque vtrumuis arcuum <emph style="sc">BC, AC</emph>, vnà cum arcu <lb/> <anchor type="note" xlink:label="note-448-01a" xlink:href="note-448-01"/> <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s15313" xml:space="preserve">Nam cum ſit, vt ſinus totus ad ſinum anguli <emph style="sc">A</emph>, ita <lb/>ſecans anguli <emph style="sc">B</emph>, ad ſecantem arcus <emph style="sc">AC</emph>: </s> <s xml:id="echoid-s15314" xml:space="preserve">Item, vt ſinus <lb/>totus ad ſinum angult <emph style="sc">B</emph>, ita ſecans anguli <emph style="sc">A</emph>, ad ſecan-<lb/>tem arcus <emph style="sc">BC</emph>;</s> <s xml:id="echoid-s15315" xml:space="preserve"/> </p> <div xml:id="echoid-div1236" type="float" level="2" n="1"> <figure xlink:label="fig-448-01" xlink:href="fig-448-01a"> <image file="448-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/448-01"/> </figure> <note position="left" xlink:label="note-448-01" xlink:href="note-448-01a" xml:space="preserve">52. huius.</note> </div> <p style="it"> <s xml:id="echoid-s15316" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum anguli non re <lb/> <anchor type="note" xlink:label="note-448-02a" xlink:href="note-448-02"/> cti quæſito lateri adiacentis, ita ſecans alterius <lb/>anguli non recti ad aliud, reperietur ſecans arcus <lb/>huic poſteriori angulo oppoſiti, qui quæritur. </s> <s xml:id="echoid-s15317" xml:space="preserve">In-<lb/>uento autem vtroque arcu circa angulum rectum, reperietur ex ipſis ter-<lb/>tius arcus recto angulo oppoſitus, vt in problemate propoſ. </s> <s xml:id="echoid-s15318" xml:space="preserve">43. </s> <s xml:id="echoid-s15319" xml:space="preserve">oſtendi-<lb/>mus: </s> <s xml:id="echoid-s15320" xml:space="preserve">Vel certe ex inuento alterutro arcu, & </s> <s xml:id="echoid-s15321" xml:space="preserve">angulo dato, quiei opponi-<lb/>tur, vt in problemate 3. </s> <s xml:id="echoid-s15322" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15323" xml:space="preserve">41. </s> <s xml:id="echoid-s15324" xml:space="preserve">diximus.</s> <s xml:id="echoid-s15325" xml:space="preserve"/> </p> <div xml:id="echoid-div1237" type="float" level="2" n="2"> <note position="left" xlink:label="note-448-02" xlink:href="note-448-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s15326" xml:space="preserve">NVM vero duo arcus quæſiti circa angulum rectum minores quadrante ſint, ma-<lb/>isresve, ſciemus, vt in problemate 1. </s> <s xml:id="echoid-s15327" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15328" xml:space="preserve">42. </s> <s xml:id="echoid-s15329" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s15330" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1239" type="section" level="1" n="585"> <head xml:id="echoid-head620" xml:space="preserve">THEOR. 51. PROPOS. 53.</head> <p> <s xml:id="echoid-s15331" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius <lb/>omnes arcus quadrante minores ſint: </s> <s xml:id="echoid-s15332" xml:space="preserve">ſinus totus <lb/>ad ſinum complementi vtriusvis arcuum circa an <lb/>gulum rectum habet eandem proportionẽ, quam <lb/>ſecans arcus recto angulo oppoſiti ad ſecantem re-<lb/>liqui arcus.</s> <s xml:id="echoid-s15333" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15334" xml:space="preserve">IN triangulo ſphærico ABC, cuius omnes arcus quadrante minores, ſit <lb/>angulus B, rectus. </s> <s xml:id="echoid-s15335" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum complementi arcus <lb/> <anchor type="figure" xlink:label="fig-448-02a" xlink:href="fig-448-02"/> BC, vt eſt ſecans arcus AC, ad ſecantem ar-<lb/>cus AB. </s> <s xml:id="echoid-s15336" xml:space="preserve">Facta namque conſtructione figuræ, <lb/>vt in propoſ. </s> <s xml:id="echoid-s15337" xml:space="preserve">45. </s> <s xml:id="echoid-s15338" xml:space="preserve">erit BF, quadrans; </s> <s xml:id="echoid-s15339" xml:space="preserve">CE, BD, <lb/>ad DF, perpendiculares, vt ibi eſt demon-<lb/>ſtratum. </s> <s xml:id="echoid-s15340" xml:space="preserve">Quia ergo in ſphæra duo circuli ma-<lb/>ximi BF, DF, ſe mutuo ſecant in F, ductiq́; <lb/></s> <s xml:id="echoid-s15341" xml:space="preserve">ſunt ex punctis B, C, ad DF, perpendiculares <lb/>arcus BD, CE; </s> <s xml:id="echoid-s15342" xml:space="preserve">erit, vt ſinus totus quadran-<lb/>tis BF, ad ſecantem complementi arcus CE, <lb/> <anchor type="note" xlink:label="note-448-03a" xlink:href="note-448-03"/> hoc eſt, ad ſecantem arcus AC, ita ſinus ar-<lb/>cus FC, qui complementum eſt arcus BC, ad <lb/>ſecantem complementi arcus BD, hoc eſt, ad <lb/>ſecantem arcus AB: </s> <s xml:id="echoid-s15343" xml:space="preserve">Et permutando, vt ſi-<lb/>custotus ad ſinum complementi arcus BC, ita ſecans arcus AC, ad ſecantem <pb o="437" file="449" n="449" rhead=""/> arcus AB. </s> <s xml:id="echoid-s15344" xml:space="preserve">Simili modo oſtendemus, ita eſſe ſinum totum ad ſinum comple-<lb/>menti arcus AB, vt eſt ſecans arcus AC, ad ſecantem arcus BC, ſi nimitum <lb/>figura paulo aliter conſtruatur. </s> <s xml:id="echoid-s15345" xml:space="preserve">In omni ergo triangulo ſphærico rectangu-<lb/>lo, &</s> <s xml:id="echoid-s15346" xml:space="preserve">c. </s> <s xml:id="echoid-s15347" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s15348" xml:space="preserve"/> </p> <div xml:id="echoid-div1239" type="float" level="2" n="1"> <figure xlink:label="fig-448-02" xlink:href="fig-448-02a"> <image file="448-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/448-02"/> </figure> <note position="left" xlink:label="note-448-03" xlink:href="note-448-03a" xml:space="preserve">Theot. 8. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1241" type="section" level="1" n="586"> <head xml:id="echoid-head621" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15349" xml:space="preserve">SEQVENS problema ex hoc theoremate colligitur.</s> <s xml:id="echoid-s15350" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15351" xml:space="preserve">IN triangulo ſphærico rectangulo, dato arcu, quirecto angulo <lb/>opponitur, cum alterutro arcuum circa rectum angulum, inueſtiga-<lb/>re tertium arcum, cum duobus angulis non rectis.</s> <s xml:id="echoid-s15352" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15353" xml:space="preserve">IN triangulo <emph style="sc">ABC</emph>, cuius angulus <emph style="sc">C</emph>, rectus, datus ſit arcus <emph style="sc">AB</emph>, vnà cum ar-<lb/>cu <emph style="sc">AC</emph>. </s> <s xml:id="echoid-s15354" xml:space="preserve">Dico dari quoque arcum <emph style="sc">BC</emph>, cum angulis <emph style="sc">A, B</emph>. <lb/></s> <s xml:id="echoid-s15355" xml:space="preserve"> <anchor type="figure" xlink:label="fig-449-01a" xlink:href="fig-449-01"/> Camenimſit, vt ſinus totus ad ſinum complementi arcus <lb/> <anchor type="note" xlink:label="note-449-01a" xlink:href="note-449-01"/> <emph style="sc">AC</emph>, ita ſecans arcus <emph style="sc">AB</emph>, ad ſecantem arcus <emph style="sc">BC</emph>:</s> <s xml:id="echoid-s15356" xml:space="preserve"/> </p> <div xml:id="echoid-div1241" type="float" level="2" n="1"> <figure xlink:label="fig-449-01" xlink:href="fig-449-01a"> <image file="449-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/449-01"/> </figure> <note position="right" xlink:label="note-449-01" xlink:href="note-449-01a" xml:space="preserve">53. huius.</note> </div> <p style="it"> <s xml:id="echoid-s15357" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum complementi <lb/> <anchor type="note" xlink:label="note-449-02a" xlink:href="note-449-02"/> dati arcus circa angulum rectum, ita ſecans arcus <lb/>angulo recto oppoſiti ad aliud, producetur ſecans <lb/>tertij arcus, qui inquiritur. </s> <s xml:id="echoid-s15358" xml:space="preserve">Hinc ex duobus ar-<lb/>cubus circa rectum angulum cognitis, vterlibet <lb/>angulorum non rectorum cognoſcetur, vt in 5. </s> <s xml:id="echoid-s15359" xml:space="preserve">problemate ſcholij propoſ. <lb/></s> <s xml:id="echoid-s15360" xml:space="preserve">44. </s> <s xml:id="echoid-s15361" xml:space="preserve">vel in problemate ſcholij propoſ. </s> <s xml:id="echoid-s15362" xml:space="preserve">48. </s> <s xml:id="echoid-s15363" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s15364" xml:space="preserve"/> </p> <div xml:id="echoid-div1242" type="float" level="2" n="2"> <note position="right" xlink:label="note-449-02" xlink:href="note-449-02a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s15365" xml:space="preserve">VTRVM vero quæſitus arcus <emph style="sc">BC</emph>, ſit quadrante maior, minorve, diſcemus eæ <lb/>datis duobus arcubus, vt ad finem problematis ſcholij 1. </s> <s xml:id="echoid-s15366" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15367" xml:space="preserve">43. </s> <s xml:id="echoid-s15368" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s15369" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1244" type="section" level="1" n="587"> <head xml:id="echoid-head622" xml:space="preserve">THEOR. 52. PROPOS. 54.</head> <p> <s xml:id="echoid-s15370" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius omnes arcus quadrante ſint minores: </s> <s xml:id="echoid-s15371" xml:space="preserve">ſinus to-<lb/>tus ad ſinum vtriuſlibet angulorum non rectorum <lb/>proportionem habet eandem, quam ſecans com-<lb/>plementi arcus illi angulo oppoſiti ad ſecantem <lb/>complementi arcus recto angulo oppoſiti.</s> <s xml:id="echoid-s15372" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15373" xml:space="preserve">IN triangulo ABC, cuius arcus omnes ſint minores quadrante, ſit angu-<lb/>lus B, rectus. </s> <s xml:id="echoid-s15374" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum anguli A, vt eſt ſecans com-<lb/>plementi arcus BC, ad ſecantem complementi arcus AC. </s> <s xml:id="echoid-s15375" xml:space="preserve">Repetita enim <lb/>conſtructione figuræ propoſ. </s> <s xml:id="echoid-s15376" xml:space="preserve">47. </s> <s xml:id="echoid-s15377" xml:space="preserve">erit angulus I, rectus, vt in propoſ. </s> <s xml:id="echoid-s15378" xml:space="preserve">52. <lb/></s> <s xml:id="echoid-s15379" xml:space="preserve">monſtratum eſt; </s> <s xml:id="echoid-s15380" xml:space="preserve">necnon & </s> <s xml:id="echoid-s15381" xml:space="preserve">angulus G. </s> <s xml:id="echoid-s15382" xml:space="preserve">Item GH, EH, DF, BF, AE, qua-<lb/>drantes, vt ex demonſtratis in propoſ. </s> <s xml:id="echoid-s15383" xml:space="preserve">45. </s> <s xml:id="echoid-s15384" xml:space="preserve">& </s> <s xml:id="echoid-s15385" xml:space="preserve">47. </s> <s xml:id="echoid-s15386" xml:space="preserve">conſtat. </s> <s xml:id="echoid-s15387" xml:space="preserve">Quia igitur in ſphæra <pb o="438" file="450" n="450" rhead=""/> duo circuli maximi EH, GH, ſe mutuo ſecant in H, & </s> <s xml:id="echoid-s15388" xml:space="preserve">ex punctis E, F, arcus <lb/>EH, ad arcum GH, ducti ſunt arcus perpendiculares EG, FI; </s> <s xml:id="echoid-s15389" xml:space="preserve">erit, vt ſinus <lb/> <anchor type="figure" xlink:label="fig-450-01a" xlink:href="fig-450-01"/> totus quadrantis EH, ad ſecantẽ complementi <lb/> <anchor type="note" xlink:label="note-450-01a" xlink:href="note-450-01"/> arcus FI, hoc eſt, ad ſecãtem arcus CF, qui com <lb/>plementum etiam eſt arcus BC, ita ſinus arcus <lb/>FH, hoc eſt, arcus DE, (eſt enim arcus FH, ar-<lb/>cui DE, æqualis, ob quadrantes EH, DF, æqua <lb/>les) qui arcus eſt anguli A, ad ſecantem comple-<lb/>menti arcus EG, id eſt, ad ſecantem arcus EC, <lb/>qui complementum quoque eſt arcus AC: </s> <s xml:id="echoid-s15390" xml:space="preserve">Et <lb/>permutãdo, vt ſinus totus ad ſinum arcus DE, <lb/>hoc eſt, anguli A, ita ſecans complementi arcus BC, ad ſecantem comple-<lb/>menti arcus AC. </s> <s xml:id="echoid-s15391" xml:space="preserve">Non ſecus oſtendemus, ſi aliter figura conſtruatur, ita <lb/>eſſe ſinum totum ad ſinum anguli C, vt eſt ſecans complementi arcus AB, ad <lb/>ſecantem complementi arcus AC. </s> <s xml:id="echoid-s15392" xml:space="preserve">In omni igitur triangulo ſphærico rectan-<lb/>gulo, &</s> <s xml:id="echoid-s15393" xml:space="preserve">c. </s> <s xml:id="echoid-s15394" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s15395" xml:space="preserve"/> </p> <div xml:id="echoid-div1244" type="float" level="2" n="1"> <figure xlink:label="fig-450-01" xlink:href="fig-450-01a"> <image file="450-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/450-01"/> </figure> <note position="left" xlink:label="note-450-01" xlink:href="note-450-01a" xml:space="preserve">Theor. 8. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1246" type="section" level="1" n="588"> <head xml:id="echoid-head623" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15396" xml:space="preserve">SEQVITVR ex hoc theoremate ſequens problema, quod aliter etiam abſol-<lb/>nimus in problemate. </s> <s xml:id="echoid-s15397" xml:space="preserve">3. </s> <s xml:id="echoid-s15398" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15399" xml:space="preserve">41.</s> <s xml:id="echoid-s15400" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15401" xml:space="preserve">IN triangulo ſphęrico rectangulo, dato vtrolibet angulorum non <lb/>rectorum, cum arcu oppoſito, inueſtigare arcum recto angulo op-<lb/>poſitum, vnà cum tertio arcu, & </s> <s xml:id="echoid-s15402" xml:space="preserve">reliquo angulo non recto: </s> <s xml:id="echoid-s15403" xml:space="preserve">dummo-<lb/>do conſtet, num arcus angulo recto oppoſitus ſit maior quadrante, <lb/>minorve: </s> <s xml:id="echoid-s15404" xml:space="preserve">aut an alter angulus non rectus ſit acutus, obtuſusve.</s> <s xml:id="echoid-s15405" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15406" xml:space="preserve">IN triangulo ABC, rectum habente angulum C, datus ſit angulus <emph style="sc">B</emph>, cum ar-<lb/> <anchor type="figure" xlink:label="fig-450-02a" xlink:href="fig-450-02"/> cu <emph style="sc">AC</emph>. </s> <s xml:id="echoid-s15407" xml:space="preserve">Dico dari quoque arcum AB, vnà cum arcu <lb/>BC, & </s> <s xml:id="echoid-s15408" xml:space="preserve">angulo A. </s> <s xml:id="echoid-s15409" xml:space="preserve">Cum namque ſit, vt ſinus totus ad <lb/> <anchor type="note" xlink:label="note-450-02a" xlink:href="note-450-02"/> ſinum anguli B, dati, ita ſecans complementi arcus da-<lb/>ti <emph style="sc">AC</emph>, ad ſecantem complementiarcus <emph style="sc">AB</emph>:</s> <s xml:id="echoid-s15410" xml:space="preserve"/> </p> <div xml:id="echoid-div1246" type="float" level="2" n="1"> <figure xlink:label="fig-450-02" xlink:href="fig-450-02a"> <image file="450-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/450-02"/> </figure> <note position="left" xlink:label="note-450-02" xlink:href="note-450-02a" xml:space="preserve">54.huius.</note> </div> <p style="it"> <s xml:id="echoid-s15411" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum dati anguli, <lb/> <anchor type="note" xlink:label="note-450-03a" xlink:href="note-450-03"/> ita ſecans complementi dati arcus ad aliud, pro-<lb/>ducetur ſecans complementi arcus recto angulo <lb/>oppoſiti, qui inquiritur. </s> <s xml:id="echoid-s15412" xml:space="preserve">Ex arcubus vero AB, <lb/>AC, cognitis notus fiet tertius arcus BC, ex problemate propoſ. </s> <s xml:id="echoid-s15413" xml:space="preserve">43. <lb/></s> <s xml:id="echoid-s15414" xml:space="preserve">Item ex arcubus AB, BC, notis cognitus fiet angulus A, ex problema-<lb/>te 1. </s> <s xml:id="echoid-s15415" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15416" xml:space="preserve">41.</s> <s xml:id="echoid-s15417" xml:space="preserve"/> </p> <div xml:id="echoid-div1247" type="float" level="2" n="2"> <note position="left" xlink:label="note-450-03" xlink:href="note-450-03a" xml:space="preserve">Praxis.</note> </div> <p> <s xml:id="echoid-s15418" xml:space="preserve">OPORTET autem hic conſtare, num arcus quæſitus <emph style="sc">AB</emph>, ſit quadrante ma-<lb/>ior, an minor: </s> <s xml:id="echoid-s15419" xml:space="preserve">Vel an reliquus angulus non rectus <emph style="sc">A</emph>, ſit acutus, obtuſus ve, alioquin <lb/>neſciremus, qualis arcus pro <emph style="sc">AB</emph>, aſſumendus ſit, cum poßit eße maior quadrante, vel <lb/>minor, vt perſpicuum eſt. </s> <s xml:id="echoid-s15420" xml:space="preserve">Id quod ad problema 3. </s> <s xml:id="echoid-s15421" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15422" xml:space="preserve">41. </s> <s xml:id="echoid-s15423" xml:space="preserve">monuimus: </s> <s xml:id="echoid-s15424" xml:space="preserve">Vbi etiam <lb/>copernici, atque Ioan Regiom. </s> <s xml:id="echoid-s15425" xml:space="preserve">errorem aperuimus.</s> <s xml:id="echoid-s15426" xml:space="preserve"/> </p> <pb o="439" file="451" n="451" rhead=""/> </div> <div xml:id="echoid-div1249" type="section" level="1" n="589"> <head xml:id="echoid-head624" xml:space="preserve">THEOR. 53. PROPOS. 55.</head> <p> <s xml:id="echoid-s15427" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius omnes arcus minores quadrante ſint: </s> <s xml:id="echoid-s15428" xml:space="preserve">ſinus to-<lb/>tus ad ſinum arcus recto angulo oppoſiti eandem <lb/>proportionem habet, quam ſecans complementi <lb/>vtriuſlibet arcuum circa angulum rectum ad ſecã-<lb/>tem complementi anguli huic arcui oppoſiti.</s> <s xml:id="echoid-s15429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15430" xml:space="preserve">IN ſphærico triangulo ABC, cuius omnes arcus ſint minores quadrante, <lb/>angulus B, rectus ſit. </s> <s xml:id="echoid-s15431" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum arcus AC, vt eſt ſe-<lb/>cans complementi arcus BC, ad ſecantem <lb/>complementi anguli A, arcui BC, oppoſiti. <lb/></s> <s xml:id="echoid-s15432" xml:space="preserve">Repetita enim conſtructione figuræ propoſ. </s> <s xml:id="echoid-s15433" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-451-01a" xlink:href="fig-451-01"/> 45. </s> <s xml:id="echoid-s15434" xml:space="preserve">erit AE, quadrans; </s> <s xml:id="echoid-s15435" xml:space="preserve">DE, arcus anguli A, <lb/>& </s> <s xml:id="echoid-s15436" xml:space="preserve">EF, eius complementum; </s> <s xml:id="echoid-s15437" xml:space="preserve">atque CF, com-<lb/>plementum arcus BC, vt ibi demonſtratum <lb/>eſt. </s> <s xml:id="echoid-s15438" xml:space="preserve">Quia ergo in ſphæra duo maximi circuli <lb/>AE, AD, ſe interſecant in A, & </s> <s xml:id="echoid-s15439" xml:space="preserve">ex punctis <lb/> <anchor type="note" xlink:label="note-451-01a" xlink:href="note-451-01"/> C, E, arcus AE, ad arcum AD, ducti ſunt per-<lb/>pendiculares arcus CB, ED; </s> <s xml:id="echoid-s15440" xml:space="preserve">erit vt ſinus to <lb/>tus quadrantis AE, ad ſecantem complemen <lb/>ti arcus CB, hoc eſt, ad ſecantem arcus CF, <lb/>ita ſinus arcus AC, ad ſecantem complemen-<lb/>ti arcus DE, ſiue anguli A, id eſt, ad ſecan-<lb/>tem arcus EF: </s> <s xml:id="echoid-s15441" xml:space="preserve">Et permutando, vt ſinus totus ad ſinum arcus AC, ita ſecans <lb/>complementi arcus BC, ad ſecantem complementi anguli A. </s> <s xml:id="echoid-s15442" xml:space="preserve">Pari ratione, ſi <lb/>aliter figura extruatur, erit, vt ſinus totus ad ſinum arcus AC, ita ſecans com <lb/>plementi arcus AB, ad ſecantem complementi anguli C. </s> <s xml:id="echoid-s15443" xml:space="preserve">Quare in omni trian <lb/>gulo ſphærico rectangulo, &</s> <s xml:id="echoid-s15444" xml:space="preserve">c. </s> <s xml:id="echoid-s15445" xml:space="preserve">Quod erat oſtendendum.</s> <s xml:id="echoid-s15446" xml:space="preserve"/> </p> <div xml:id="echoid-div1249" type="float" level="2" n="1"> <figure xlink:label="fig-451-01" xlink:href="fig-451-01a"> <image file="451-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/451-01"/> </figure> <note position="right" xlink:label="note-451-01" xlink:href="note-451-01a" xml:space="preserve">Theor. 8. <lb/>ſcholij. 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1251" type="section" level="1" n="590"> <head xml:id="echoid-head625" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s15447" xml:space="preserve">EX his ſequens problema diſſoluemus, quod alio quoque modo in problemate 1. <lb/></s> <s xml:id="echoid-s15448" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15449" xml:space="preserve">41. </s> <s xml:id="echoid-s15450" xml:space="preserve">abſolutum fuit.</s> <s xml:id="echoid-s15451" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15452" xml:space="preserve">IN triangulo ſphærico rectangulo, dato arcu, qui recto angulo <lb/>opponitur, cum alterutro arcuum circa rectum angulum, inuenire <lb/>angulum huic arcui oppoſitum, cum reliquo arcu, & </s> <s xml:id="echoid-s15453" xml:space="preserve">angulo.</s> <s xml:id="echoid-s15454" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15455" xml:space="preserve">IN triangulo ABC, cuius rectus angulus C, datus ſit tam arcus <emph style="sc">AB</emph>, quam AC. <lb/></s> <s xml:id="echoid-s15456" xml:space="preserve">Dico dari quo que angulum <emph style="sc">B</emph>, vnà cum arcu <emph style="sc">B</emph> C, & </s> <s xml:id="echoid-s15457" xml:space="preserve">angulo <emph style="sc">A</emph>. </s> <s xml:id="echoid-s15458" xml:space="preserve">Quia enim eſt, vt ſi-<lb/>nus totus ad ſinum arcus <emph style="sc">Ab</emph>, ita ſecans complementi arcus <emph style="sc">A</emph>C, ad ſecantem com-<lb/> <anchor type="note" xlink:label="note-451-02a" xlink:href="note-451-02"/> plementi anguli <emph style="sc">B</emph>:</s> <s xml:id="echoid-s15459" xml:space="preserve"/> </p> <div xml:id="echoid-div1251" type="float" level="2" n="1"> <note position="right" xlink:label="note-451-02" xlink:href="note-451-02a" xml:space="preserve">55. huius.</note> </div> <pb o="440" file="452" n="452" rhead=""/> <p style="it"> <s xml:id="echoid-s15460" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum arcus angulo recto oppoſiti, ita ſecans <lb/> <anchor type="note" xlink:label="note-452-01a" xlink:href="note-452-01"/> complementi arcus circa rectum angulum dati ad aliud, producetur ſe-<lb/>cans complementi anguli quæſiti, qui dicto arcui opponitur. </s> <s xml:id="echoid-s15461" xml:space="preserve">Iam ex da-<lb/>tis duobus arcubus tertium inueniemus, vt in problemate propoſ. </s> <s xml:id="echoid-s15462" xml:space="preserve">43. </s> <s xml:id="echoid-s15463" xml:space="preserve">vel <lb/>in problemate propoſ. </s> <s xml:id="echoid-s15464" xml:space="preserve">53. </s> <s xml:id="echoid-s15465" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s15466" xml:space="preserve">Item ex arcu, qui recto angulo <lb/>opponitur, & </s> <s xml:id="echoid-s15467" xml:space="preserve">hoc arcu inuento, reperiemus reliquum angulum huic inuen <lb/>to arcui oppoſitum, vt dictum eſt in hoc problemate, vel certe, vt in pro-<lb/>blemate 1. </s> <s xml:id="echoid-s15468" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15469" xml:space="preserve">41. </s> <s xml:id="echoid-s15470" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s15471" xml:space="preserve"/> </p> <div xml:id="echoid-div1252" type="float" level="2" n="2"> <note position="left" xlink:label="note-452-01" xlink:href="note-452-01a" xml:space="preserve">@raxis.</note> </div> <p> <s xml:id="echoid-s15472" xml:space="preserve">AN vero quæſitus angulus <emph style="sc">B</emph>, acutus ſit, an obtuſus, docebit arcus <emph style="sc">AC</emph>, circa an-<lb/>gulum rectum datus, vt in problemate 1. </s> <s xml:id="echoid-s15473" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15474" xml:space="preserve">41. </s> <s xml:id="echoid-s15475" xml:space="preserve">præcepimus.</s> <s xml:id="echoid-s15476" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1254" type="section" level="1" n="591"> <head xml:id="echoid-head626" xml:space="preserve">THEOR. 54. PROPOS. 56.</head> <p> <s xml:id="echoid-s15477" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cu-<lb/>ius arcus ſint omnes quadrante minores: </s> <s xml:id="echoid-s15478" xml:space="preserve">ſinus to-<lb/>tus ad ſinum complementi vtriuſlibet arcuum cir <lb/>ca rectum angulum eandem proportionem ha-<lb/>bet, quam ſecans anguli huic arcui oppoſiti ad ſe-<lb/>cantem complementi reliqui anguli non recti.</s> <s xml:id="echoid-s15479" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15480" xml:space="preserve">IN triangulo ſphærico ABC, angulum B, rectum habente, ſint omnes ar-<lb/>cus quadrante minores. </s> <s xml:id="echoid-s15481" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinũ complementi ar-<lb/>cus BC, vt eſt ſecans anguli non recti A, ad ſecantem complementi anguli <lb/>C. </s> <s xml:id="echoid-s15482" xml:space="preserve">Repetita enim conſtructione figuræ propoſ. </s> <s xml:id="echoid-s15483" xml:space="preserve">47. </s> <s xml:id="echoid-s15484" xml:space="preserve">erunt anguli G, E, re-<lb/>cti, & </s> <s xml:id="echoid-s15485" xml:space="preserve">arcus BF, DF, CI, EH, GH, quadran-<lb/>tes, & </s> <s xml:id="echoid-s15486" xml:space="preserve">DE, arcus anguli A, & </s> <s xml:id="echoid-s15487" xml:space="preserve">GI, arcus anguli <lb/> <anchor type="figure" xlink:label="fig-452-01a" xlink:href="fig-452-01"/> C, vt ex demõſtratis in propoſ. </s> <s xml:id="echoid-s15488" xml:space="preserve">45. </s> <s xml:id="echoid-s15489" xml:space="preserve">& </s> <s xml:id="echoid-s15490" xml:space="preserve">47. </s> <s xml:id="echoid-s15491" xml:space="preserve">liquet. <lb/></s> <s xml:id="echoid-s15492" xml:space="preserve">Igitur quia duo maximi in ſphæra circuli CG, <lb/> <anchor type="note" xlink:label="note-452-02a" xlink:href="note-452-02"/> CI, ſe in C, interſecant, ductiq́; </s> <s xml:id="echoid-s15493" xml:space="preserve">ſunt ex pun-<lb/>ctis F, I, arcus CI, ad arcum CG, arcus perpen-<lb/>diculares FE, IG; </s> <s xml:id="echoid-s15494" xml:space="preserve">erit, vt ſinus totus quadran <lb/>tis CI, ad ſecantem complementi arcus FE, hoc <lb/>eſt, ad ſecantem arcus DE, anguli A, ita ſinus <lb/>arcus CF, qui complementum eſt arcus BC, ad ſecantem complementi arcus <lb/>GI, anguli C: </s> <s xml:id="echoid-s15495" xml:space="preserve">Et permutando, vt ſinus totus ad ſinum complementi arcus <lb/>BC, ita ſecans anguli A, ad ſecantem complementi anguli C. </s> <s xml:id="echoid-s15496" xml:space="preserve">Non ſecus o-<lb/>ſtendemus, ſi aliter conſtruatur figura, ita eſſe ſinum totum ad ſinum comple <lb/>menti arcus AB, vt eſt ſecans anguli C, ad ſecantem complementi anguli A. <lb/></s> <s xml:id="echoid-s15497" xml:space="preserve">Quapropter in omni triangulo ſphærico rectangulo, &</s> <s xml:id="echoid-s15498" xml:space="preserve">c. </s> <s xml:id="echoid-s15499" xml:space="preserve">Quod demonſtran-<lb/>dum erat.</s> <s xml:id="echoid-s15500" xml:space="preserve"/> </p> <div xml:id="echoid-div1254" type="float" level="2" n="1"> <figure xlink:label="fig-452-01" xlink:href="fig-452-01a"> <image file="452-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/452-01"/> </figure> <note position="left" xlink:label="note-452-02" xlink:href="note-452-02a" xml:space="preserve">Theor 8. <lb/>ſcholij 40. <lb/>huius.</note> </div> </div> <div xml:id="echoid-div1256" type="section" level="1" n="592"> <head xml:id="echoid-head627" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15501" xml:space="preserve">INFERTVR hinc problema huiuſmodi.</s> <s xml:id="echoid-s15502" xml:space="preserve"/> </p> <pb o="441" file="453" n="453" rhead=""/> <p> <s xml:id="echoid-s15503" xml:space="preserve">IN triangulo ſphærico rectangulo, dato vtrouis arcuum circa an-<lb/>gulum rectum, cum angulo non recto oppoſito, inquirere teliquum <lb/>angulum non rectum, & </s> <s xml:id="echoid-s15504" xml:space="preserve">iuſuper reliquos duos arcus: </s> <s xml:id="echoid-s15505" xml:space="preserve">modo conſter, <lb/>an quæſitus angulus ſit acutus, obtuſusve: </s> <s xml:id="echoid-s15506" xml:space="preserve">Vel certe, an alter arcus <lb/>circa angulum rectum ſit minor quadrante, an maior.</s> <s xml:id="echoid-s15507" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15508" xml:space="preserve">IN triangulo ABC, angulum C, rectum habente, datus ſit arcus <emph style="sc">AC</emph>, cum an-<lb/>gulo <emph style="sc">B</emph>. </s> <s xml:id="echoid-s15509" xml:space="preserve">Dico dari quoque engulum <emph style="sc">A</emph>, cum arcubus <emph style="sc">BC</emph>, <lb/> <anchor type="figure" xlink:label="fig-453-01a" xlink:href="fig-453-01"/> <emph style="sc">AB</emph>. </s> <s xml:id="echoid-s15510" xml:space="preserve">Cum enim ſit, vt ſinus totus ad ſinum complementi <lb/>arcus <emph style="sc">AC</emph>, dati, ita ſecans anguli dati <emph style="sc">B</emph>, ad ſecantem <lb/> <anchor type="note" xlink:label="note-453-01a" xlink:href="note-453-01"/> complementi anguli <emph style="sc">A</emph>:</s> <s xml:id="echoid-s15511" xml:space="preserve"/> </p> <div xml:id="echoid-div1256" type="float" level="2" n="1"> <figure xlink:label="fig-453-01" xlink:href="fig-453-01a"> <image file="453-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/453-01"/> </figure> <note position="right" xlink:label="note-453-01" xlink:href="note-453-01a" xml:space="preserve">56. huius.</note> </div> <p style="it"> <s xml:id="echoid-s15512" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum complementi <lb/> <anchor type="note" xlink:label="note-453-02a" xlink:href="note-453-02"/> arcus dati, ita ſecans dati anguli ad aliud, repe-<lb/>rietur ſecans complementi anguli alterius non re-<lb/>cti. </s> <s xml:id="echoid-s15513" xml:space="preserve">Hinc ex duobus angulis non rectis notis inue-<lb/>ſtigabitur arcus recto oppoſitus angulo, vt in problemate propoſ. </s> <s xml:id="echoid-s15514" xml:space="preserve">50. </s> <s xml:id="echoid-s15515" xml:space="preserve">mon <lb/>ſtrauimus, ac proinde & </s> <s xml:id="echoid-s15516" xml:space="preserve">reliquus arcus, ex arcu, qui recto angulo oppo-<lb/>nitur, & </s> <s xml:id="echoid-s15517" xml:space="preserve">ex noto angulo, qui reliquo arcui opponitur, vt in problemate 2. <lb/></s> <s xml:id="echoid-s15518" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15519" xml:space="preserve">41. </s> <s xml:id="echoid-s15520" xml:space="preserve">diximus.</s> <s xml:id="echoid-s15521" xml:space="preserve"/> </p> <div xml:id="echoid-div1257" type="float" level="2" n="2"> <note position="right" xlink:label="note-453-02" xlink:href="note-453-02a" xml:space="preserve">Praxis.</note> </div> <p style="it"> <s xml:id="echoid-s15522" xml:space="preserve">OTORTET autem conſtare, an reliquus angulus non rectus, qui quæritur ſit <lb/>acutus, obtuſusve; </s> <s xml:id="echoid-s15523" xml:space="preserve">vel, an reliquus arcus circa angulum rectum ſit minor, aut maior <lb/>quadrante, vt ad calcem problematis 2. </s> <s xml:id="echoid-s15524" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15525" xml:space="preserve">42. </s> <s xml:id="echoid-s15526" xml:space="preserve">monuimus, vbi errorem etiam <lb/>Nicolai Copernici deteximus.</s> <s xml:id="echoid-s15527" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s15528" xml:space="preserve">QVONIAM vero abſolutus iam eſt triangulorum ſphæricorum rectangulorum <lb/>calculus, libet hoc loco omnia problemata hactenus explicata in tabulam quandam <lb/>referre, vt facilius quilibet id, quod maxime ſcire deſiderat, poſsit inuenire. </s> <s xml:id="echoid-s15529" xml:space="preserve">Itaque <lb/>cum in omni triangulo ſphærico rectangulo id, quod primo loco quæritur, ſit vel ar-<lb/>cus recto angulo oppoſitus, vel vterlibet arcuum circa rectum angulum, vel denique <lb/>alteruter angulorum non rectorum, (quamuis eo, quod potiſsimum quæritur, inuen-<lb/>to, cætera quoque reperiantur, vt ad praxes ſingulorum problematum monuimus) <lb/>trimembrem tabulam, pro numero quæſitorum, conſecimus, appoſuimuſque problema-<lb/>ta propoſitionum, in quibus inuentiones quæſitorum demonſtratæ ſunt.</s> <s xml:id="echoid-s15530" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1259" type="section" level="1" n="593"> <head xml:id="echoid-head628" xml:space="preserve">Sequuntur problemata ſuperiorum propoſi-<lb/>tionum in trimembrem tabel-<lb/>lam digeſta.</head> <pb o="442" file="454" n="454" rhead=""/> </div> <div xml:id="echoid-div1260" type="section" level="1" n="594"> <head xml:id="echoid-head629" xml:space="preserve">Inuentio arcus recto angulo oppoſiti.</head> <p> <s xml:id="echoid-s15531" xml:space="preserve">Quando datur {1. </s> <s xml:id="echoid-s15532" xml:space="preserve">Arcus circa angulum rectum: </s> <s xml:id="echoid-s15533" xml:space="preserve">Et an- \ġulus non rectus ei oppoſitus. </s> <s xml:id="echoid-s15534" xml:space="preserve">\\ 2. </s> <s xml:id="echoid-s15535" xml:space="preserve">Vterque arcus circa angulum re- \ċtum. </s> <s xml:id="echoid-s15536" xml:space="preserve">\\ 3. </s> <s xml:id="echoid-s15537" xml:space="preserve">Arcus circa angulum rectum: </s> <s xml:id="echoid-s15538" xml:space="preserve">Et an- \ġulus non rectus ei adiacens. </s> <s xml:id="echoid-s15539" xml:space="preserve">\\ 4. </s> <s xml:id="echoid-s15540" xml:space="preserve">Vterque angulus non rectus. </s> <s xml:id="echoid-s15541" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15542" xml:space="preserve">3. </s> <s xml:id="echoid-s15543" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15544" xml:space="preserve">41. </s> <s xml:id="echoid-s15545" xml:space="preserve">\\ (& </s> <s xml:id="echoid-s15546" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15547" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15548" xml:space="preserve">54. </s> <s xml:id="echoid-s15549" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15550" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15551" xml:space="preserve">43. </s> <s xml:id="echoid-s15552" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15553" xml:space="preserve">1. </s> <s xml:id="echoid-s15554" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15555" xml:space="preserve">45. </s> <s xml:id="echoid-s15556" xml:space="preserve">\\ (& </s> <s xml:id="echoid-s15557" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15558" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15559" xml:space="preserve">46. </s> <s xml:id="echoid-s15560" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15561" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15562" xml:space="preserve">50.</s> <s xml:id="echoid-s15563" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1261" type="section" level="1" n="595"> <head xml:id="echoid-head630" xml:space="preserve">Inuentio arcus vtriuſlibet circa angu-<lb/>lum rectum.</head> <p> <s xml:id="echoid-s15564" xml:space="preserve">Quando datur {1. </s> <s xml:id="echoid-s15565" xml:space="preserve">Arcus recto angulo oppoſitus: </s> <s xml:id="echoid-s15566" xml:space="preserve">Et an- \ġulus non rectus quęſito arcui op- \ṗoſitus. </s> <s xml:id="echoid-s15567" xml:space="preserve">\\ 2. </s> <s xml:id="echoid-s15568" xml:space="preserve">Vterq; </s> <s xml:id="echoid-s15569" xml:space="preserve">angulus non rectus- \\ 3. </s> <s xml:id="echoid-s15570" xml:space="preserve">Arcus recto angulo oppoſitus: </s> <s xml:id="echoid-s15571" xml:space="preserve">Et al- \ṫer arcus circa rectum angulum. </s> <s xml:id="echoid-s15572" xml:space="preserve">\\ 4. </s> <s xml:id="echoid-s15573" xml:space="preserve">Arcus alter circa angulum rectum: </s> <s xml:id="echoid-s15574" xml:space="preserve">\\ Et vteruis angulorum non recto- \ṙum. </s> <s xml:id="echoid-s15575" xml:space="preserve">\\ 5. </s> <s xml:id="echoid-s15576" xml:space="preserve">Arcus recto angulo oppoſitus: </s> <s xml:id="echoid-s15577" xml:space="preserve">Et \ȧngulus non rectus quæſito arcui \ȧdiacens. </s> <s xml:id="echoid-s15578" xml:space="preserve">\\ 6. </s> <s xml:id="echoid-s15579" xml:space="preserve">Arcus alter circa angulum rectum: </s> <s xml:id="echoid-s15580" xml:space="preserve">\\ Et alter angulus non rectus ei op- \ṗoſitus. </s> <s xml:id="echoid-s15581" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15582" xml:space="preserve">2. </s> <s xml:id="echoid-s15583" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15584" xml:space="preserve">41. </s> <s xml:id="echoid-s15585" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15586" xml:space="preserve">1. </s> <s xml:id="echoid-s15587" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15588" xml:space="preserve">42. </s> <s xml:id="echoid-s15589" xml:space="preserve">\\ (& </s> <s xml:id="echoid-s15590" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15591" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15592" xml:space="preserve">52. </s> <s xml:id="echoid-s15593" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15594" xml:space="preserve">{pro}poſ. </s> <s xml:id="echoid-s15595" xml:space="preserve">43. </s> <s xml:id="echoid-s15596" xml:space="preserve">& </s> <s xml:id="echoid-s15597" xml:space="preserve">53. </s> <s xml:id="echoid-s15598" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15599" xml:space="preserve">1. </s> <s xml:id="echoid-s15600" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15601" xml:space="preserve">44. </s> <s xml:id="echoid-s15602" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15603" xml:space="preserve">3. </s> <s xml:id="echoid-s15604" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15605" xml:space="preserve">45. </s> <s xml:id="echoid-s15606" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15607" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15608" xml:space="preserve">49. </s> <s xml:id="echoid-s15609" xml:space="preserve">\\ (& </s> <s xml:id="echoid-s15610" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15611" xml:space="preserve">1. </s> <s xml:id="echoid-s15612" xml:space="preserve">{pro}poſ. </s> <s xml:id="echoid-s15613" xml:space="preserve">44.</s> <s xml:id="echoid-s15614" xml:space="preserve"/> </p> <pb o="443" file="455" n="455" rhead=""/> </div> <div xml:id="echoid-div1262" type="section" level="1" n="596"> <head xml:id="echoid-head631" xml:space="preserve">Inuentio anguli non recti vtriusvis.</head> <p> <s xml:id="echoid-s15615" xml:space="preserve">Quando datur {1. </s> <s xml:id="echoid-s15616" xml:space="preserve">Arcus recto angulo oppoſitus: </s> <s xml:id="echoid-s15617" xml:space="preserve">Et \ȧrcus circa angulum rectum quę- \\ ſito angulo oppoſitus. </s> <s xml:id="echoid-s15618" xml:space="preserve">\\ 2. </s> <s xml:id="echoid-s15619" xml:space="preserve">Arcus circa angulum rectum: </s> <s xml:id="echoid-s15620" xml:space="preserve">Et \ȧlter angulus non rectus. </s> <s xml:id="echoid-s15621" xml:space="preserve">\\ 3. </s> <s xml:id="echoid-s15622" xml:space="preserve">Vterq; </s> <s xml:id="echoid-s15623" xml:space="preserve">arcus circa rectum angulum. </s> <s xml:id="echoid-s15624" xml:space="preserve">\\ 4. </s> <s xml:id="echoid-s15625" xml:space="preserve">Arcus recto angulo oppoſitus: </s> <s xml:id="echoid-s15626" xml:space="preserve">Et \ȧrcus circa rectum angulum quę- \\ ſito angulo adiacens. </s> <s xml:id="echoid-s15627" xml:space="preserve">\\ 5. </s> <s xml:id="echoid-s15628" xml:space="preserve">Arcus recto angulo oppoſitus: </s> <s xml:id="echoid-s15629" xml:space="preserve">Et \ȧlter angulus non rectus. </s> <s xml:id="echoid-s15630" xml:space="preserve">\\ 6. </s> <s xml:id="echoid-s15631" xml:space="preserve">Arcus circa angulum rectum quæ- \\ ſito angulo adiacens: </s> <s xml:id="echoid-s15632" xml:space="preserve">Et alter an- \ġulus non rectus huic arcui op- \ṗoſitus: </s> <s xml:id="echoid-s15633" xml:space="preserve">Probl 1. </s> <s xml:id="echoid-s15634" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15635" xml:space="preserve">41. </s> <s xml:id="echoid-s15636" xml:space="preserve">& </s> <s xml:id="echoid-s15637" xml:space="preserve">\\ (Probl. </s> <s xml:id="echoid-s15638" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15639" xml:space="preserve">55. </s> <s xml:id="echoid-s15640" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15641" xml:space="preserve">2. </s> <s xml:id="echoid-s15642" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15643" xml:space="preserve">42. </s> <s xml:id="echoid-s15644" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15645" xml:space="preserve">2. </s> <s xml:id="echoid-s15646" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15647" xml:space="preserve">44. </s> <s xml:id="echoid-s15648" xml:space="preserve">\\ (& </s> <s xml:id="echoid-s15649" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15650" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15651" xml:space="preserve">48. </s> <s xml:id="echoid-s15652" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15653" xml:space="preserve">2. </s> <s xml:id="echoid-s15654" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15655" xml:space="preserve">45. </s> <s xml:id="echoid-s15656" xml:space="preserve">\\ (& </s> <s xml:id="echoid-s15657" xml:space="preserve">Probl. </s> <s xml:id="echoid-s15658" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15659" xml:space="preserve">51. </s> <s xml:id="echoid-s15660" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15661" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15662" xml:space="preserve">47. </s> <s xml:id="echoid-s15663" xml:space="preserve">\\ Probl. </s> <s xml:id="echoid-s15664" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15665" xml:space="preserve">56. </s> <s xml:id="echoid-s15666" xml:space="preserve">& </s> <s xml:id="echoid-s15667" xml:space="preserve">\\ (Probl. </s> <s xml:id="echoid-s15668" xml:space="preserve">2. </s> <s xml:id="echoid-s15669" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15670" xml:space="preserve">42.</s> <s xml:id="echoid-s15671" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15672" xml:space="preserve">SED quia hactenus de eo ſolum triangulo rectangulo egimus, cuius nullus ar-<lb/> <anchor type="note" xlink:label="note-455-01a" xlink:href="note-455-01"/> cuum quadrans eſt, doceamus breuiter, (rem quidem cuilibet perfacilem ex demon-<lb/>ſtratis) quo pacto nos gerere debeamus in eo, quod duos ſaltem arcus habet quadran <lb/>tes, & </s> <s xml:id="echoid-s15673" xml:space="preserve">duos angulos rectos. </s> <s xml:id="echoid-s15674" xml:space="preserve">Nullum enim triangulum eſſe poteſt rectangulum, cuius <lb/>vnus duntaxat arcus ſit quadrans, ſed vel nullus erit quadrans, vel omnes tres qua-<lb/> <anchor type="note" xlink:label="note-455-02a" xlink:href="note-455-02"/> drantes erunt, vel duo, &</s> <s xml:id="echoid-s15675" xml:space="preserve">c. </s> <s xml:id="echoid-s15676" xml:space="preserve">Sit ergo triangulum ſphæricum <emph style="sc">ABC</emph>, in quo angulus <lb/><emph style="sc">B</emph>, ponatur rectus, & </s> <s xml:id="echoid-s15677" xml:space="preserve">arcus <emph style="sc">AB</emph>, circa angulum rectum quadrans. </s> <s xml:id="echoid-s15678" xml:space="preserve">Hoc poſito, erit <lb/>& </s> <s xml:id="echoid-s15679" xml:space="preserve">arcus <emph style="sc">AC</emph>, recto angulo oppoſitus, quadrans. </s> <s xml:id="echoid-s15680" xml:space="preserve">Quare cum duo arcus <emph style="sc">AB, AC</emph>, <lb/> <anchor type="note" xlink:label="note-455-03a" xlink:href="note-455-03"/> quadrantes ſint, erunt duo anguli <emph style="sc">B</emph>, C, recti; </s> <s xml:id="echoid-s15681" xml:space="preserve">ac propte-<lb/> <anchor type="note" xlink:label="note-455-04a" xlink:href="note-455-04"/> rea <emph style="sc">A</emph>, polus erit arcus <emph style="sc">BC</emph>; </s> <s xml:id="echoid-s15682" xml:space="preserve">& </s> <s xml:id="echoid-s15683" xml:space="preserve"><emph style="sc">BC</emph>, arcus anguli <emph style="sc">A</emph>, ex <lb/> <anchor type="note" xlink:label="note-455-05a" xlink:href="note-455-05"/> <anchor type="figure" xlink:label="fig-455-01a" xlink:href="fig-455-01"/> definitione 6. </s> <s xml:id="echoid-s15684" xml:space="preserve">Igitur ſi datus ſit tertius angulus A, datus <lb/>etiam erit tertius arcus <emph style="sc">BC</emph>: </s> <s xml:id="echoid-s15685" xml:space="preserve">Et contra, ſi datus ſit ter-<lb/>tius arcus <emph style="sc">BC</emph>, datus quoque erit tertius angulus <emph style="sc">A</emph>. </s> <s xml:id="echoid-s15686" xml:space="preserve">Eo-<lb/>dem modo, ſi alter arcus BC, circa angulum rectum qua-<lb/>drans ponatur, ostendemus & </s> <s xml:id="echoid-s15687" xml:space="preserve">arcum <emph style="sc">A</emph>C, recto angulo <lb/>oppoſitum quadrantem eſſe, & </s> <s xml:id="echoid-s15688" xml:space="preserve">angulum <emph style="sc">A</emph>, rectum. </s> <s xml:id="echoid-s15689" xml:space="preserve">Si ergo <lb/>detur tertius angulus <emph style="sc">C</emph>, dabitur quoque tertius arcus <lb/><emph style="sc">AB</emph>, & </s> <s xml:id="echoid-s15690" xml:space="preserve">contra, vt prius. </s> <s xml:id="echoid-s15691" xml:space="preserve">Quòd ſi quantitas tertij angu-<lb/>li, aut arcus datanon fuerit, nihilcerti colligi poterit, licet duo alij anguli recti ſint, <pb o="444" file="456" n="456" rhead=""/> duoque arcus illis oppoſiti, quadrantes, vt manifeſtum eſt.</s> <s xml:id="echoid-s15692" xml:space="preserve"/> </p> <div xml:id="echoid-div1262" type="float" level="2" n="1"> <note position="right" xlink:label="note-455-01" xlink:href="note-455-01a" xml:space="preserve">Quid agen <lb/>dũ in trian <lb/>gulo rectã-<lb/>gulo, ĩ quo <lb/>quadrãtes <lb/>ſunt.</note> <note position="right" xlink:label="note-455-02" xlink:href="note-455-02a" xml:space="preserve">Coroll. 38. <lb/>huius.</note> <note position="right" xlink:label="note-455-03" xlink:href="note-455-03a" xml:space="preserve">35. huius.</note> <note position="right" xlink:label="note-455-04" xlink:href="note-455-04a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-455-05" xlink:href="note-455-05a" xml:space="preserve">Coroll. 26. <lb/>huius.</note> <figure xlink:label="fig-455-01" xlink:href="fig-455-01a"> <image file="455-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/455-01"/> </figure> </div> <p> <s xml:id="echoid-s15693" xml:space="preserve">PONATVR iam arcus AC, recto angulo <emph style="sc">B</emph>, oppoſitus quadrans. </s> <s xml:id="echoid-s15694" xml:space="preserve">Quo poſito, <lb/> <anchor type="note" xlink:label="note-456-01a" xlink:href="note-456-01"/> erit & </s> <s xml:id="echoid-s15695" xml:space="preserve">alter ſaltem arcuum <emph style="sc">AB, B</emph>C, circa angulum rectum quadrans. </s> <s xml:id="echoid-s15696" xml:space="preserve">Quamobrem <lb/>reliqua conſequentur, vt proxime demonſtratum eſt.</s> <s xml:id="echoid-s15697" xml:space="preserve"/> </p> <div xml:id="echoid-div1263" type="float" level="2" n="2"> <note position="left" xlink:label="note-456-01" xlink:href="note-456-01a" xml:space="preserve">36. huius.</note> </div> <p> <s xml:id="echoid-s15698" xml:space="preserve">QVOD ſi quando duo arcus <emph style="sc">AB, BC</emph>, circa angulum rectum quadrantes ponan <lb/> <anchor type="note" xlink:label="note-456-02a" xlink:href="note-456-02"/> tur, erit quoque tertius arcus <emph style="sc">AC</emph>, recto angulo oppoſitus, quadrans. </s> <s xml:id="echoid-s15699" xml:space="preserve">Quocirca cum <lb/>omnes arcus quadrantes ſint, erunt omnes anguli recti.</s> <s xml:id="echoid-s15700" xml:space="preserve"/> </p> <div xml:id="echoid-div1264" type="float" level="2" n="3"> <note position="left" xlink:label="note-456-02" xlink:href="note-456-02a" xml:space="preserve">26. vel 35. <lb/>huius.</note> </div> <note position="left" xml:space="preserve">Coroll. 25. <lb/>huius.</note> <p style="it"> <s xml:id="echoid-s15701" xml:space="preserve">EX his facile quiuis intelliget, quid agere debeat, quando aliquis arcus in trian <lb/>gulo rectangulo quadrans ponitur: </s> <s xml:id="echoid-s15702" xml:space="preserve">præſertim ſi propoſ. </s> <s xml:id="echoid-s15703" xml:space="preserve">25. </s> <s xml:id="echoid-s15704" xml:space="preserve">26. </s> <s xml:id="echoid-s15705" xml:space="preserve">27. </s> <s xml:id="echoid-s15706" xml:space="preserve">28. </s> <s xml:id="echoid-s15707" xml:space="preserve">29. </s> <s xml:id="echoid-s15708" xml:space="preserve">30. </s> <s xml:id="echoid-s15709" xml:space="preserve">31. <lb/></s> <s xml:id="echoid-s15710" xml:space="preserve">32. </s> <s xml:id="echoid-s15711" xml:space="preserve">33. </s> <s xml:id="echoid-s15712" xml:space="preserve">34. </s> <s xml:id="echoid-s15713" xml:space="preserve">35. </s> <s xml:id="echoid-s15714" xml:space="preserve">36. </s> <s xml:id="echoid-s15715" xml:space="preserve">37. </s> <s xml:id="echoid-s15716" xml:space="preserve">38. </s> <s xml:id="echoid-s15717" xml:space="preserve">attente conſiderentur.</s> <s xml:id="echoid-s15718" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s15719" xml:space="preserve">SEQVITVR iam, vt calculum triangulorum non rectangulorum tandem ex-<lb/>ponamus: </s> <s xml:id="echoid-s15720" xml:space="preserve">Verũ prius aliquot theoremata ad hanc rem perutilia demonſtranda ſunt.</s> <s xml:id="echoid-s15721" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1266" type="section" level="1" n="597"> <head xml:id="echoid-head632" xml:space="preserve">THEOR. 55. PROPOS. 57.</head> <p> <s xml:id="echoid-s15722" xml:space="preserve">SI in triangulo ſphærico ſupra vnum arcum <lb/>duo anguli acuti, aut obtuſi conſiſtant; </s> <s xml:id="echoid-s15723" xml:space="preserve">Perpendi-<lb/>cularis arcus à tertio angulo in eum arcum demiſ-<lb/>ſus intra triangulum cadit. </s> <s xml:id="echoid-s15724" xml:space="preserve">Si vero duorum angu-<lb/>lorum ſupra vnum arcum conſiſtentium vnus ſit <lb/>acutus, & </s> <s xml:id="echoid-s15725" xml:space="preserve">obtuſus alter; </s> <s xml:id="echoid-s15726" xml:space="preserve">Perpendicularis arcus à <lb/>tertio angulo in eum arcum demiſſus extra trian-<lb/>gulum cadit.</s> <s xml:id="echoid-s15727" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15728" xml:space="preserve">IN triangulo ſphærico ABC, ſint duo anguli B, C, ſupra arcum BC, acu-<lb/>ti, vel obtuſi. </s> <s xml:id="echoid-s15729" xml:space="preserve">Dico arcum perpendicularem ex A, ad arcum BC, demiſſum <lb/>cadere intra triangulum, cuiuſmodi eſt arcus AD. </s> <s xml:id="echoid-s15730" xml:space="preserve">Si enim dicatur cadere ex-<lb/>tra, cadat, ſi fieri poteſt, arcus AE, ad BC, arcum <lb/>productum perpendicularis extra triangulum, & </s> <s xml:id="echoid-s15731" xml:space="preserve">po <lb/> <anchor type="figure" xlink:label="fig-456-01a" xlink:href="fig-456-01"/> nantur primum duo anguli B, C, acuti, ac proinde <lb/>angulus ACE, obtuſus. </s> <s xml:id="echoid-s15732" xml:space="preserve">Quoniam igitur in trian-<lb/>gulo ACE, angulum E, habente rectum, angulus <lb/>ACE, obtuſus eſt, erit arcus AE, quadrante ma-<lb/> <anchor type="note" xlink:label="note-456-04a" xlink:href="note-456-04"/> lor. </s> <s xml:id="echoid-s15733" xml:space="preserve">Rurſus quia in triangulo ABE, habente angu-<lb/>lum rectum E, angulus B, acutus eſt, erit arcus AE, <lb/> <anchor type="note" xlink:label="note-456-05a" xlink:href="note-456-05"/> quadrante minor: </s> <s xml:id="echoid-s15734" xml:space="preserve">Sed & </s> <s xml:id="echoid-s15735" xml:space="preserve">quadrante maior oſten-<lb/>ſus eſt; </s> <s xml:id="echoid-s15736" xml:space="preserve">quod eſt abſurdum.</s> <s xml:id="echoid-s15737" xml:space="preserve"/> </p> <div xml:id="echoid-div1266" type="float" level="2" n="1"> <figure xlink:label="fig-456-01" xlink:href="fig-456-01a"> <image file="456-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/456-01"/> </figure> <note position="left" xlink:label="note-456-04" xlink:href="note-456-04a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-456-05" xlink:href="note-456-05a" xml:space="preserve">34. huius.</note> </div> <p> <s xml:id="echoid-s15738" xml:space="preserve">PONANTVR deinde anguli B, C, obtuſi, atque adeo angulus ACE, <lb/>acutus. </s> <s xml:id="echoid-s15739" xml:space="preserve">Quia ergo in triangulo ACE, habente rectum angulum E, angulus <lb/>ACE, acutus eſt, erit arcus AE, minor quadrante. </s> <s xml:id="echoid-s15740" xml:space="preserve">Rurſus quoniam in trian <lb/> <anchor type="note" xlink:label="note-456-06a" xlink:href="note-456-06"/> gulo ABE, rectum habente E, angulum, angulus B, obtuſus eſt, erit arcus <lb/>AE, quadrante maior: </s> <s xml:id="echoid-s15741" xml:space="preserve">Sed & </s> <s xml:id="echoid-s15742" xml:space="preserve">quadrante minor oſtenſus eſt; </s> <s xml:id="echoid-s15743" xml:space="preserve">quod eſt abſur-<lb/> <anchor type="note" xlink:label="note-456-07a" xlink:href="note-456-07"/> dum. </s> <s xml:id="echoid-s15744" xml:space="preserve">Non cadit ergo arcus perpendicularis extra triangulum: </s> <s xml:id="echoid-s15745" xml:space="preserve">ſed neque cum <pb o="445" file="457" n="457" rhead=""/> altero arcuum AB, AC, coincidet, quòd neuter angulorum B, C, ponatur <lb/>rectus. </s> <s xml:id="echoid-s15746" xml:space="preserve">Cadit ergo intra triangulum.</s> <s xml:id="echoid-s15747" xml:space="preserve"/> </p> <div xml:id="echoid-div1267" type="float" level="2" n="2"> <note position="left" xlink:label="note-456-06" xlink:href="note-456-06a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-456-07" xlink:href="note-456-07a" xml:space="preserve">34. huius.</note> </div> <p> <s xml:id="echoid-s15748" xml:space="preserve">IAM vero ponatur in eodem triangulo ABC, angulus B, acutus, & </s> <s xml:id="echoid-s15749" xml:space="preserve">C, <lb/>obtuſus. </s> <s xml:id="echoid-s15750" xml:space="preserve">Dico perpendicularem arcum ex A, ad arcum BC, demiſſum extra <lb/>triangulum cadere, cuiuſmodi eſt arcus AE. </s> <s xml:id="echoid-s15751" xml:space="preserve">Nam ſi intra dicatur cadere, ca-<lb/>dat, ſi fi fieri poteſt, arcus AD, ad BC, perpendicularis intra triangulum. </s> <s xml:id="echoid-s15752" xml:space="preserve">Ita-<lb/>que quia in triangulo ACD, angulum rectum habente D, angulus C, obtu-<lb/>ſus eſt, erit arcus AD, maior quadrante. </s> <s xml:id="echoid-s15753" xml:space="preserve">Rurſus cum in triangulo ABD, re-<lb/> <anchor type="note" xlink:label="note-457-01a" xlink:href="note-457-01"/> ctum habente angulum D, angulus B, acutus eſt, erit arcus AD, quadrante <lb/> <anchor type="note" xlink:label="note-457-02a" xlink:href="note-457-02"/> minor: </s> <s xml:id="echoid-s15754" xml:space="preserve">Sed & </s> <s xml:id="echoid-s15755" xml:space="preserve">quadrante oſtenſus eſt maior; </s> <s xml:id="echoid-s15756" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s15757" xml:space="preserve">Arcus ergo <lb/>perpendicularis non cadit intra triangulum; </s> <s xml:id="echoid-s15758" xml:space="preserve">ſed neque cum altero arcuum <lb/>AB, AC, coincidit, cum neuter angulorum B, C, rectus ponatur. </s> <s xml:id="echoid-s15759" xml:space="preserve">Cadit ergo <lb/>extra triangulum. </s> <s xml:id="echoid-s15760" xml:space="preserve">Quapropter, ſi in triangulo ſphærico ſupra vnum arcum <lb/>duo anguli acuti, &</s> <s xml:id="echoid-s15761" xml:space="preserve">c. </s> <s xml:id="echoid-s15762" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s15763" xml:space="preserve"/> </p> <div xml:id="echoid-div1268" type="float" level="2" n="3"> <note position="right" xlink:label="note-457-01" xlink:href="note-457-01a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-457-02" xlink:href="note-457-02a" xml:space="preserve">34. huius.</note> </div> </div> <div xml:id="echoid-div1270" type="section" level="1" n="598"> <head xml:id="echoid-head633" xml:space="preserve">THEOR. 56. PROPOS. 58.</head> <p> <s xml:id="echoid-s15764" xml:space="preserve">IN omni triangulo ſphærico, cuius duo arcus <lb/>ſint inæquales; </s> <s xml:id="echoid-s15765" xml:space="preserve">quadratum ſinus totius ad rectan-<lb/>gulum ſub ſinubus rectis duorum arcuum inęqua <lb/>lium contentum, eandem proportionem habet, <lb/>quam ſinus verſus anguli à dictis arcubus compre-<lb/>henſi ad differentiam duorum ſinuum verſorum, <lb/>quorum vnus differentiæ eorundem arcuum de-<lb/>betur, alter vero tertio arcui, qui prædicto angulo <lb/>oppoſitus eſt, reſpondet.</s> <s xml:id="echoid-s15766" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15767" xml:space="preserve">IN triangulo ſphærico ABC, ſint duo arcus AB, AC, inęquales, ille <lb/>minor, & </s> <s xml:id="echoid-s15768" xml:space="preserve">hic maior. </s> <s xml:id="echoid-s15769" xml:space="preserve">Dico ita eſſe quadratum ſinus totius ad rectangulum ſub <lb/>ſinubus rectis arcuum AB, AC, contentum, vt eſt ſinus verſus anguli A, ad <lb/>differentiam inter ſinum verſum arcus, quo arcus AB, AC, inter ſe differunt, <lb/>& </s> <s xml:id="echoid-s15770" xml:space="preserve">ſinum verſum arcus BC. </s> <s xml:id="echoid-s15771" xml:space="preserve">Cu-<lb/> <anchor type="figure" xlink:label="fig-457-01a" xlink:href="fig-457-01"/> ius rei demonſtrationem, vt cla-<lb/>rior fiat, & </s> <s xml:id="echoid-s15772" xml:space="preserve">generalior, in quin-<lb/>decim caſus diuidemus.</s> <s xml:id="echoid-s15773" xml:space="preserve"/> </p> <div xml:id="echoid-div1270" type="float" level="2" n="1"> <figure xlink:label="fig-457-01" xlink:href="fig-457-01a"> <image file="457-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/457-01"/> </figure> </div> <p> <s xml:id="echoid-s15774" xml:space="preserve">1. </s> <s xml:id="echoid-s15775" xml:space="preserve">SINT omnes tres arcus <lb/>quadrante minores. </s> <s xml:id="echoid-s15776" xml:space="preserve">Complea-<lb/> <anchor type="note" xlink:label="note-457-03a" xlink:href="note-457-03"/> tur minoris arcus AB, circulus <lb/>ABDGH, productisq́; </s> <s xml:id="echoid-s15777" xml:space="preserve">arcu-<lb/>bus AC, BC, fiant quadrantes <lb/>AL, BM; </s> <s xml:id="echoid-s15778" xml:space="preserve">& </s> <s xml:id="echoid-s15779" xml:space="preserve">polis A, B, inter-<lb/>uallis autem quadrantum AL, <lb/>BM, circuli maximi deſcribantur DLEF, GMEH: </s> <s xml:id="echoid-s15780" xml:space="preserve">Et eiſdem polis, inter- <pb o="446" file="458" n="458" rhead=""/> uallis autem AC, BC, circuli non maximi delineentur KCN, OCP, qui il-<lb/>lis maximis paralleli erunt: </s> <s xml:id="echoid-s15781" xml:space="preserve">& </s> <s xml:id="echoid-s15782" xml:space="preserve">tam hi, quam illi ad circulum ABDGH, re-<lb/> <anchor type="note" xlink:label="note-458-01a" xlink:href="note-458-01"/> cti erunt, cum ille per horũ polos trãſiens ad ipſos ſit rectus. </s> <s xml:id="echoid-s15783" xml:space="preserve">Poſt hæc, vt con-<lb/> <anchor type="note" xlink:label="note-458-02a" xlink:href="note-458-02"/> fuſio vitetur, in circulo ABDGH, ſeorſum deſcripto ſint communes ſectio-<lb/>nes ipſius, & </s> <s xml:id="echoid-s15784" xml:space="preserve">circulorum ex polis A, B, deſcriptorum, nempe DF, GH, com-<lb/>munes ſectiones ipſius, & </s> <s xml:id="echoid-s15785" xml:space="preserve">maximorum circulorum DLEF, GMEH, quæ <lb/>ipſorum diametri erunt ſeſe in centro ſphæræ X, interſecantes: </s> <s xml:id="echoid-s15786" xml:space="preserve">At KN, OP, <lb/>communes ſectiones eiuſdem, & </s> <s xml:id="echoid-s15787" xml:space="preserve">circulorum KCN, OCP, ſe interſecantes <lb/>in S; </s> <s xml:id="echoid-s15788" xml:space="preserve">quæ ipſis DF, GH, parallelæ erunt; </s> <s xml:id="echoid-s15789" xml:space="preserve">& </s> <s xml:id="echoid-s15790" xml:space="preserve">diametri circulorum KCN, <lb/> <anchor type="note" xlink:label="note-458-03a" xlink:href="note-458-03"/> OCP; </s> <s xml:id="echoid-s15791" xml:space="preserve">quòd maximus circulus ABDGH, per eorum polos tranſiens eos <lb/>bifariam ſecet, nimirum per eorum diametros. </s> <s xml:id="echoid-s15792" xml:space="preserve">Ducantur quoque ſemidiame-<lb/> <anchor type="note" xlink:label="note-458-04a" xlink:href="note-458-04"/> tri AX, ſecans KN, in Y; </s> <s xml:id="echoid-s15793" xml:space="preserve">& </s> <s xml:id="echoid-s15794" xml:space="preserve">BX, ſecans KN, OP, in I, R. </s> <s xml:id="echoid-s15795" xml:space="preserve">Eruntque ſemi-<lb/>diametri AX, BX, perpendiculares ad circulos per DF, KN, GH, OP, <lb/> <anchor type="note" xlink:label="note-458-05a" xlink:href="note-458-05"/> ductos; </s> <s xml:id="echoid-s15796" xml:space="preserve">cum ab eorum polis A, <lb/> <anchor type="note" xlink:label="note-458-06a" xlink:href="note-458-06"/> B, ducantur per X, ſphæræcen-<lb/> <anchor type="figure" xlink:label="fig-458-01a" xlink:href="fig-458-01"/> trum: </s> <s xml:id="echoid-s15797" xml:space="preserve">ac proinde anguli ad Y, & </s> <s xml:id="echoid-s15798" xml:space="preserve"><lb/>R, recti erunt, ex defin. </s> <s xml:id="echoid-s15799" xml:space="preserve">3. </s> <s xml:id="echoid-s15800" xml:space="preserve">lib. </s> <s xml:id="echoid-s15801" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s15802" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s15803" xml:space="preserve">Ducantur denique ad BX, <lb/>OP, perpendiculares AV, KQ, <lb/>KT. </s> <s xml:id="echoid-s15804" xml:space="preserve">Erit igitur, per ea, quæ in <lb/>tractatione ſinuum ſcripſimus, <lb/>AV, ſinus rectus arcus AB; </s> <s xml:id="echoid-s15805" xml:space="preserve">& </s> <s xml:id="echoid-s15806" xml:space="preserve"><lb/>KY, ſinus rectus arcus AK, hoc <lb/>eſt, arcus AC, cum arcus AK, <lb/>AC, ex defin. </s> <s xml:id="echoid-s15807" xml:space="preserve">poli, æquales ſint, vt in primo circulo apparet. </s> <s xml:id="echoid-s15808" xml:space="preserve">BR, erit ſinus <lb/>verſus arcus BO, id eſt, arcus BC, cum arcus BO, BC, æquales ſint, ex defin. </s> <s xml:id="echoid-s15809" xml:space="preserve"><lb/>poli. </s> <s xml:id="echoid-s15810" xml:space="preserve">BQ, ſinus verſus erit arcus BK, qui differentia eſt arcuum inæqualium <lb/>AB, AC, propterea quod, ex defin. </s> <s xml:id="echoid-s15811" xml:space="preserve">poli, arcus AK, arcum AB, arcu BQ, ſupe-<lb/>rans, æqualis eſt arcui AC: </s> <s xml:id="echoid-s15812" xml:space="preserve">ac proinde QR, vel KT, differentia erit inter BR, <lb/>ſinum verſum tertij arcus BC, & </s> <s xml:id="echoid-s15813" xml:space="preserve">BQ, ſinum verſum differentiæ arcuum inæ-<lb/>qualium AB, AC, hoc eſt, ſinum verſum arcus BK. </s> <s xml:id="echoid-s15814" xml:space="preserve">Poſtremo erit KS, ſinus <lb/>verſus arcus KC, in circulo non maximo KCN, cum recta ex C, in commu-<lb/>nes ſectiones circulorum KCN, OCP, cum circulo ABDGH, hoc eſt, in <lb/>punctum S, cadens, (quæ quidem ad circulũ ABDGH, recta eſt, vtpote com-<lb/>munis ſectio circulorum KCN, OCP, ad eundem circulum ABDGH, re-<lb/> <anchor type="note" xlink:label="note-458-07a" xlink:href="note-458-07"/> ctorum) ſinus rectus ſit eiuſdem arcus KC. </s> <s xml:id="echoid-s15815" xml:space="preserve">Sumatur quoque DZ, ſinus ver-<lb/>ſus arcus DL, hoc eſt, anguli A, qui quidem arcus arcui KC, ſimilis eſt. </s> <s xml:id="echoid-s15816" xml:space="preserve">De-<lb/> <anchor type="note" xlink:label="note-458-08a" xlink:href="note-458-08"/> monſtrandum igitur eſt, ita eſſe quadratum ſinus totius, hoc eſt, rectangulum <lb/>ſub DX, XA, contentum, ad rectangulum ſub ſinubus rectis AV, KY, ar-<lb/>cuum AB, AC, contentum, vt eſt ſinus verſus DZ, anguli A, ad KT, diſ-<lb/>fer entiam inter BR, ſinum verſum arcus BC, & </s> <s xml:id="echoid-s15817" xml:space="preserve">BQ, ſinum verſum arcus BK, <lb/>differentiæ arcuum inæqualium AB, AC. </s> <s xml:id="echoid-s15818" xml:space="preserve">quod ita fiet.</s> <s xml:id="echoid-s15819" xml:space="preserve"/> </p> <div xml:id="echoid-div1271" type="float" level="2" n="2"> <note position="right" xlink:label="note-457-03" xlink:href="note-457-03a" xml:space="preserve">1. caſus.</note> <note position="left" xlink:label="note-458-01" xlink:href="note-458-01a" xml:space="preserve">2. 2. Theod.</note> <note position="left" xlink:label="note-458-02" xlink:href="note-458-02a" xml:space="preserve">15. 1. Theo.</note> <note position="left" xlink:label="note-458-03" xlink:href="note-458-03a" xml:space="preserve">16. vndec.</note> <note position="left" xlink:label="note-458-04" xlink:href="note-458-04a" xml:space="preserve">15.1 Theod.</note> <note position="left" xlink:label="note-458-05" xlink:href="note-458-05a" xml:space="preserve">Schol. 10.</note> <note position="left" xlink:label="note-458-06" xlink:href="note-458-06a" xml:space="preserve">1. Theod.</note> <figure xlink:label="fig-458-01" xlink:href="fig-458-01a"> <image file="458-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/458-01"/> </figure> <note position="left" xlink:label="note-458-07" xlink:href="note-458-07a" xml:space="preserve">19. vndec.</note> <note position="left" xlink:label="note-458-08" xlink:href="note-458-08a" xml:space="preserve">10.2. Theo.</note> </div> <p> <s xml:id="echoid-s15820" xml:space="preserve">QVONIAM angulus XIY, angulo RIS, æqualis eſt, & </s> <s xml:id="echoid-s15821" xml:space="preserve">angulus re-<lb/> <anchor type="note" xlink:label="note-458-09a" xlink:href="note-458-09"/> ctus Y, angulo recto R; </s> <s xml:id="echoid-s15822" xml:space="preserve">erit reliquus angulus IXY, trianguli IXY, reliquo <lb/> <anchor type="note" xlink:label="note-458-10a" xlink:href="note-458-10"/> angulo ISR, trianguli ISR, æqualis, hoc eſt, angulo ad verticem KST. <lb/></s> <s xml:id="echoid-s15823" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s15824" xml:space="preserve">angulus rectus V, recto angulo T, æqualis ſit, erit & </s> <s xml:id="echoid-s15825" xml:space="preserve">reliquus <lb/>angulus XAV, trianguli XAV, reliquo angulo SKT, trianguli SKT, <lb/>æqualis Quam ob rem erit, vt XA, ad AV, ita SK, ad KT. </s> <s xml:id="echoid-s15826" xml:space="preserve">Rurſus quia DZ, <lb/> <anchor type="note" xlink:label="note-458-11a" xlink:href="note-458-11"/> <pb o="447" file="459" n="459" rhead=""/> KS, ſinus verſi ſunt arcuum ſimilium DL, KC; </s> <s xml:id="echoid-s15827" xml:space="preserve">erit, vt DX, ad KY, ſinus <lb/>totus ad ſinum totum, ita DZ, ad KS, per lem ma propoſ. </s> <s xml:id="echoid-s15828" xml:space="preserve">1. </s> <s xml:id="echoid-s15829" xml:space="preserve">noſtræ Gnomo-<lb/>nices. </s> <s xml:id="echoid-s15830" xml:space="preserve">Quia vero proportio rectanguli ſub DX, XA, ad rectangulum ſub KY, <lb/>AV, componitur ex proportione DX, ad KY, hoc eſt, DZ, ad KS, & </s> <s xml:id="echoid-s15831" xml:space="preserve">ex <lb/> <anchor type="note" xlink:label="note-459-01a" xlink:href="note-459-01"/> proportione XA, ad AV: </s> <s xml:id="echoid-s15832" xml:space="preserve">Et proportio DZ, ad KT, componitur ex eiſdẽ <lb/>proportionibus, nempe (poſita media recta KS) ex proportione DZ, ad KS, <lb/>& </s> <s xml:id="echoid-s15833" xml:space="preserve">ex proportione KS, ad KT, hoc eſt, ex proportione XA, ad AV; </s> <s xml:id="echoid-s15834" xml:space="preserve">erit, vt <lb/>rectangulum ſub DX, XA, id eſt, quadratum ſinus totius, ad rectangulum <lb/>ſub KY, AV, ſinubus rectis arcuum inæqualium AC, AB, ita DZ, ſinus ver <lb/>ſus anguli A, ad KT, differentiam inter BR, ſinum verſum arcus BC, angu-<lb/>lo A, oppoſiti, & </s> <s xml:id="echoid-s15835" xml:space="preserve">BQ, ſinum verſum differentiæ arcuum inæqualium AC, <lb/>AB. </s> <s xml:id="echoid-s15836" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s15837" xml:space="preserve"/> </p> <div xml:id="echoid-div1272" type="float" level="2" n="3"> <note position="left" xlink:label="note-458-09" xlink:href="note-458-09a" xml:space="preserve">15. primi.</note> <note position="left" xlink:label="note-458-10" xlink:href="note-458-10a" xml:space="preserve">32. primi.</note> <note position="left" xlink:label="note-458-11" xlink:href="note-458-11a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-459-01" xlink:href="note-459-01a" xml:space="preserve">23. ſexti.</note> </div> <p> <s xml:id="echoid-s15838" xml:space="preserve">2. </s> <s xml:id="echoid-s15839" xml:space="preserve">SINT duo arcus inæquales AB, AC, quadrante quidem minores, at <lb/> <anchor type="note" xlink:label="note-459-02a" xlink:href="note-459-02"/> BC, quadrans. </s> <s xml:id="echoid-s15840" xml:space="preserve">Compleatur minoris arcus AB, circulus ABDGH, & </s> <s xml:id="echoid-s15841" xml:space="preserve">pro-<lb/>ducto arcu AC, vt fiat quadrans AL, deſcribantur ex polis A, B, ad inter-<lb/>ualla quadrantum AL, BC, cir-<lb/>culi maximi DELF, GECH: <lb/></s> <s xml:id="echoid-s15842" xml:space="preserve"> <anchor type="figure" xlink:label="fig-459-01a" xlink:href="fig-459-01"/> Item ex polo A, ad interuallum <lb/>AC, circulus non maximus <lb/>KCN, qui ipſi DELF, paral-<lb/> <anchor type="note" xlink:label="note-459-03a" xlink:href="note-459-03"/> lelus erit, ſecabitq́ue circulus <lb/>ABDGH, circulos DELF, <lb/> <anchor type="note" xlink:label="note-459-04a" xlink:href="note-459-04"/> GECH, KCN, ad angulos re-<lb/>ctos, & </s> <s xml:id="echoid-s15843" xml:space="preserve">bifariam: </s> <s xml:id="echoid-s15844" xml:space="preserve">ac proinde ho-<lb/>rum cum illo communes ſectio-<lb/>nes DF, GH, KN, diametri eo-<lb/>rum erunt, & </s> <s xml:id="echoid-s15845" xml:space="preserve">DF, GH, ſe in X, centro ſphæræ interſeca bunt, parallelæq́ue <lb/>erunt DF, KN. </s> <s xml:id="echoid-s15846" xml:space="preserve">Reliqua fiant, vt in præcedenti caſu, niſi quòd hic punctum <lb/> <anchor type="note" xlink:label="note-459-05a" xlink:href="note-459-05"/> R, idem eſt, quod X, propterea quòd circulus OCP, à circulo GEH, atq; <lb/></s> <s xml:id="echoid-s15847" xml:space="preserve">adeo recta ORP, à recta GH, non differt. </s> <s xml:id="echoid-s15848" xml:space="preserve">Erit, vt prius, AV, ſinus rectus ar-<lb/>cus AB; </s> <s xml:id="echoid-s15849" xml:space="preserve">& </s> <s xml:id="echoid-s15850" xml:space="preserve">KY, ſinus rectus arcus AK, hoc eſt, arcus AC, ipſi AK, ex de-<lb/>fin. </s> <s xml:id="echoid-s15851" xml:space="preserve">poli, æqualis. </s> <s xml:id="echoid-s15852" xml:space="preserve">Item BR, ſinus verſus erit arcus BG, id eſt, arcus BC, <lb/>ipſi BG, æqualis. </s> <s xml:id="echoid-s15853" xml:space="preserve">At BQ, ſinus erit verſus arcus BK, differentiæ arcuum <lb/>AB, AC; </s> <s xml:id="echoid-s15854" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s15855" xml:space="preserve">QR, vel KT, differentia erit inter ſinus verſos BR, BQ, <lb/>arcuum BC, BK. </s> <s xml:id="echoid-s15856" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s15857" xml:space="preserve">KS, erit ſinus verſus arcus KC. </s> <s xml:id="echoid-s15858" xml:space="preserve">Sumpto ergo DZ, <lb/>ſinu verſo arcus DL, hoc eſt, anguli A, demonſtrandum eſt, ita eſſe quadra-<lb/>tum ſinus totias, id eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV, <lb/>KY, ſinubus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad <lb/>KT, differentiam ſinuum verſorum BR, BQ, arcuum BC, BK. </s> <s xml:id="echoid-s15859" xml:space="preserve">quod quidem <lb/>demonſtrabitur, vt in præcedenti caſu, niſi quod triangulum XAV, oſtende-<lb/>tur hic æquiangulum eſſe triangulo SKT, ex eo quòd angulus IXY, angu-<lb/>lo YSX, æqualis eſt, propterea quòd triangula IXY, YSX, ſimilia ſunt inter <lb/> <anchor type="note" xlink:label="note-459-06a" xlink:href="note-459-06"/> ſe. </s> <s xml:id="echoid-s15860" xml:space="preserve">Hinc enim fit, rectangula triangula XAV, SKT, inter ſe omnino æquian-<lb/>gula eſſe.</s> <s xml:id="echoid-s15861" xml:space="preserve"/> </p> <div xml:id="echoid-div1273" type="float" level="2" n="4"> <note position="right" xlink:label="note-459-02" xlink:href="note-459-02a" xml:space="preserve">2. caſus.</note> <figure xlink:label="fig-459-01" xlink:href="fig-459-01a"> <image file="459-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/459-01"/> </figure> <note position="right" xlink:label="note-459-03" xlink:href="note-459-03a" xml:space="preserve">2.2. Theo.</note> <note position="right" xlink:label="note-459-04" xlink:href="note-459-04a" xml:space="preserve">15.1. Theod.</note> <note position="right" xlink:label="note-459-05" xlink:href="note-459-05a" xml:space="preserve">16. vndec.</note> <note position="right" xlink:label="note-459-06" xlink:href="note-459-06a" xml:space="preserve">8. ſexti.</note> </div> <p> <s xml:id="echoid-s15862" xml:space="preserve">3. </s> <s xml:id="echoid-s15863" xml:space="preserve">SINT rurſus AB, AC, quadrante minores, at BC, maior. </s> <s xml:id="echoid-s15864" xml:space="preserve">Com-<lb/> <anchor type="note" xlink:label="note-459-07a" xlink:href="note-459-07"/> pleatur minoris arcus AB, circulus, & </s> <s xml:id="echoid-s15865" xml:space="preserve">ex BC, abſcindatur quadrans BM, <lb/>producaturq́; </s> <s xml:id="echoid-s15866" xml:space="preserve">AC, vt fiat quadrans AL. </s> <s xml:id="echoid-s15867" xml:space="preserve">Reliqua conſtruantur, vt in prmo <lb/>caſu. </s> <s xml:id="echoid-s15868" xml:space="preserve">Erunt hic ſinus, vt ibi. </s> <s xml:id="echoid-s15869" xml:space="preserve">Demonſtrandum ergo eſt, ita eſſe quadratũ ſinus <pb o="448" file="460" n="460" rhead=""/> totius, nempe rectangulum ſub <emph style="sc">Dx</emph>, XA, ad rectangulum ſub AV, KY, ſinu-<lb/>bus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad KT, diſſe-<lb/>rentiam ſinuum verſorum BR, <lb/>BQ, arcuum BC, BK. </s> <s xml:id="echoid-s15870" xml:space="preserve">quod <lb/> <anchor type="figure" xlink:label="fig-460-01a" xlink:href="fig-460-01"/> quidem oſtendetur, vt in primo <lb/>caſu, niſi quòd triangulũ XAV, <lb/>oſtendemus hic triangulo SkT, <lb/>æquiangulum eſſe, ex eo, quòd <lb/>angulus XIY, angulo SKT, <lb/>externus interno, æqualis eſt. <lb/></s> <s xml:id="echoid-s15871" xml:space="preserve"> <anchor type="note" xlink:label="note-460-01a" xlink:href="note-460-01"/> Hinc enim efficitur, in triangu-<lb/>lis rectangulis XIY, SkT, re-<lb/>liquos angulos IXY, kST, æ-<lb/>quales eſſe; </s> <s xml:id="echoid-s15872" xml:space="preserve">atq; </s> <s xml:id="echoid-s15873" xml:space="preserve">idcirco rectangula triangula XAV, SkT, eſſe æquiangula.</s> <s xml:id="echoid-s15874" xml:space="preserve"/> </p> <div xml:id="echoid-div1274" type="float" level="2" n="5"> <note position="right" xlink:label="note-459-07" xlink:href="note-459-07a" xml:space="preserve">3. caſus.</note> <figure xlink:label="fig-460-01" xlink:href="fig-460-01a"> <image file="460-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/460-01"/> </figure> <note position="left" xlink:label="note-460-01" xlink:href="note-460-01a" xml:space="preserve">29. primi.</note> </div> <p> <s xml:id="echoid-s15875" xml:space="preserve">4. </s> <s xml:id="echoid-s15876" xml:space="preserve">SIT arcus AC, quadrans, atque adeo AB, quadrante minor; </s> <s xml:id="echoid-s15877" xml:space="preserve">ſtatua-<lb/> <anchor type="note" xlink:label="note-460-02a" xlink:href="note-460-02"/> tur quoque BC, minor quadrante. </s> <s xml:id="echoid-s15878" xml:space="preserve">Completo circulo ABDGH, minoris <lb/>arcus AB; </s> <s xml:id="echoid-s15879" xml:space="preserve">productoq́; </s> <s xml:id="echoid-s15880" xml:space="preserve">arcu BC, vt fiat quadrans BM, deſcribantur ex polis <lb/>A, B, ad interualla quadrantum <lb/> <anchor type="figure" xlink:label="fig-460-02a" xlink:href="fig-460-02"/> AC, BM, circuli maximi DCEF, <lb/>GMEH: </s> <s xml:id="echoid-s15881" xml:space="preserve">Item ex polo B, ad in-<lb/>teruallum arcus BC, circulus <lb/>non maximus OCP. </s> <s xml:id="echoid-s15882" xml:space="preserve">Reliqua <lb/>conſtruantur, vt in primo caſu, <lb/>niſi quòd hic duo circuli paral-<lb/>leli DEF, kCN, inter ſe non <lb/>differunt, propter quadrantem <lb/>AC. </s> <s xml:id="echoid-s15883" xml:space="preserve">Ex quo fit, rectas DF, kN, <lb/>inter ſe quoq; </s> <s xml:id="echoid-s15884" xml:space="preserve">non differre. </s> <s xml:id="echoid-s15885" xml:space="preserve">quod etiam de ſinubus verſis kS, DZ, dicendum eſt. <lb/></s> <s xml:id="echoid-s15886" xml:space="preserve">Alij ſinus ſunt, vt prius. </s> <s xml:id="echoid-s15887" xml:space="preserve">Iam verò, ita eſſe quadratum ſinus totius, ſiue rectan <lb/>gulum ſub DX, XA, ad rectangulum ſub AV, kV, ſinubus rectis arcuum AB, <lb/>AC, vt eſt DZ, ſinus verſus anguli A, ſiue arcus KC, ad KT, differentiam ſi-<lb/>nuum verſorum BR, BQ, arcuum BC, BK, oſtendemus, vt in primo caſu; </s> <s xml:id="echoid-s15888" xml:space="preserve">ex-<lb/>cepto, quod hic triangulum XAV, triangulo SkT, æquiangulum eſſe de-<lb/>monſtrabimus, ex eo, quòd angulus AXV, angulo YSR, æqualis eſt, (pro-<lb/>pterea quod triangula IXR, RSY, ſimilia ſunt) atque adeo angulo <emph style="sc">K</emph>ST. </s> <s xml:id="echoid-s15889" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-460-03a" xlink:href="note-460-03"/> Hinc enim fit, rectangula triangula XAV, SkT, eſſe ęquiangula.</s> <s xml:id="echoid-s15890" xml:space="preserve"/> </p> <div xml:id="echoid-div1275" type="float" level="2" n="6"> <note position="left" xlink:label="note-460-02" xlink:href="note-460-02a" xml:space="preserve">4. caſus.</note> <figure xlink:label="fig-460-02" xlink:href="fig-460-02a"> <image file="460-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/460-02"/> </figure> <note position="left" xlink:label="note-460-03" xlink:href="note-460-03a" xml:space="preserve">8. ſexti.</note> </div> <p> <s xml:id="echoid-s15891" xml:space="preserve">5. </s> <s xml:id="echoid-s15892" xml:space="preserve">SIT rurſus AC, quadrans, proptereaq́; </s> <s xml:id="echoid-s15893" xml:space="preserve">AB, quadrante minor, ſed <lb/> <anchor type="note" xlink:label="note-460-04a" xlink:href="note-460-04"/> BC, ponatur quoque quadrans. <lb/></s> <s xml:id="echoid-s15894" xml:space="preserve"> <anchor type="figure" xlink:label="fig-460-03a" xlink:href="fig-460-03"/> Completo circulo ABDGH, mi-<lb/>noris arcus AB, deſcribantur ex <lb/>polis A, B, ad interualla quadran-<lb/>tũ AC, BC, circuli maximi DCF, <lb/>GCH, & </s> <s xml:id="echoid-s15895" xml:space="preserve">reliqua fiant, vt prius, <lb/>niſi quod hic circuli kN, OP, non <lb/>maximi à maximis DF, GH, non <lb/>differunt, &</s> <s xml:id="echoid-s15896" xml:space="preserve">c. </s> <s xml:id="echoid-s15897" xml:space="preserve">Demonſtrandum <lb/>igitur eſt, ita eſſe quadratum ſinus <lb/>totius, hoc eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV, kY, ſi- <pb o="449" file="461" n="461" rhead=""/> nubus rectis arcuum AB, AC, vt eſt DZ, ſinus verſus anguli A, ſeu arcus <lb/>kC, ad kT, differentiam ſinuum verſorum BY, BQ, quorum ille arcui BC, <lb/>hic autem arcui Bk, debetur. </s> <s xml:id="echoid-s15898" xml:space="preserve">quod quidem oſtendemus, vt in primo caſu. </s> <s xml:id="echoid-s15899" xml:space="preserve">So-<lb/>lum triangulum XAV, ita demonſtrabitur triangulo SkT, æquiangulum. <lb/></s> <s xml:id="echoid-s15900" xml:space="preserve">Quoniam anguli DSA, BSH, recti ſunt, cum AS, BS, axes ſint circulo-<lb/>rum DF, GH; </s> <s xml:id="echoid-s15901" xml:space="preserve">erunt, dempto communi ASB, reliqui DSB, ASH, æqua-<lb/>les: </s> <s xml:id="echoid-s15902" xml:space="preserve">ſed ille angulo alterno SkT, & </s> <s xml:id="echoid-s15903" xml:space="preserve">hic alterno angulo XAV, æqualis eſt. </s> <s xml:id="echoid-s15904" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-461-01a" xlink:href="note-461-01"/> Igitur & </s> <s xml:id="echoid-s15905" xml:space="preserve">anguli SkT, XAV, æquales erunt: </s> <s xml:id="echoid-s15906" xml:space="preserve">ac proinde triangula rectangu. <lb/></s> <s xml:id="echoid-s15907" xml:space="preserve">la XAV, SkT, æquiangula erunt.</s> <s xml:id="echoid-s15908" xml:space="preserve"/> </p> <div xml:id="echoid-div1276" type="float" level="2" n="7"> <note position="left" xlink:label="note-460-04" xlink:href="note-460-04a" xml:space="preserve">5. caſus.</note> <figure xlink:label="fig-460-03" xlink:href="fig-460-03a"> <image file="460-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/460-03"/> </figure> <note position="right" xlink:label="note-461-01" xlink:href="note-461-01a" xml:space="preserve">29. primi.</note> </div> <p> <s xml:id="echoid-s15909" xml:space="preserve">6. </s> <s xml:id="echoid-s15910" xml:space="preserve">SIT adhuc AC, quadrans, ideoq́; </s> <s xml:id="echoid-s15911" xml:space="preserve">AB, minor quadrante, ſed BC, <lb/> <anchor type="note" xlink:label="note-461-02a" xlink:href="note-461-02"/> quadrante ſtatuatur maior. </s> <s xml:id="echoid-s15912" xml:space="preserve">Completo circulo ABDGH, arcus minoris AB; <lb/></s> <s xml:id="echoid-s15913" xml:space="preserve">& </s> <s xml:id="echoid-s15914" xml:space="preserve">ex BC, abſciſſo quadrante BM, deſcribantur ex polis A, B, ad interualla <lb/>quadrantum AC, BM, maximi <lb/>circuli DEF, GEH: </s> <s xml:id="echoid-s15915" xml:space="preserve">Item ex <lb/> <anchor type="figure" xlink:label="fig-461-01a" xlink:href="fig-461-01"/> polo B, ad interuallum BC, cir-<lb/>culus non maximus OCP, qui <lb/>ipſi GEH, parallelus crit. </s> <s xml:id="echoid-s15916" xml:space="preserve">Reli-<lb/> <anchor type="note" xlink:label="note-461-03a" xlink:href="note-461-03"/> qua fiant, vt prius, niſi quòd hic <lb/>inter ſe non difterũt circuli DF, <lb/>kN, &</s> <s xml:id="echoid-s15917" xml:space="preserve">c. </s> <s xml:id="echoid-s15918" xml:space="preserve">Iam demon ſtrabimus, <lb/>vt in primo caſu, ita eſſe quadra-<lb/>tum ſinus totius, id eſt, rectan-<lb/>gulum ſub DX, XA, ad rectan-<lb/>gulum ſub AV, kY, ſinubus rectis arcuum AB, AC, vt eſt DZ, ſinus ver-<lb/>ſus anguli A, ſeu arcus DC, ad kT, differentiam ſinuum verſorum BR, BQ, <lb/>arcuum BC, Bk. </s> <s xml:id="echoid-s15919" xml:space="preserve">Verum triangulum XAV, triangulo SkT, æquiangulum <lb/>eſſe, ita monſtrabimus. </s> <s xml:id="echoid-s15920" xml:space="preserve">Cum anguli recti ſint AXk, BXG, reliqui æquales <lb/>erunt AXV, kXG: </s> <s xml:id="echoid-s15921" xml:space="preserve">ſed hic æqualis eſt oppoſito, & </s> <s xml:id="echoid-s15922" xml:space="preserve">interno TSk. </s> <s xml:id="echoid-s15923" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s15924" xml:space="preserve">an-<lb/> <anchor type="note" xlink:label="note-461-04a" xlink:href="note-461-04"/> gulus AXB, angulo TSk, æqualis erit: </s> <s xml:id="echoid-s15925" xml:space="preserve">atque adeo rectangula triangula <lb/>XAV. </s> <s xml:id="echoid-s15926" xml:space="preserve">SKT, æquiangula erunt.</s> <s xml:id="echoid-s15927" xml:space="preserve"/> </p> <div xml:id="echoid-div1277" type="float" level="2" n="8"> <note position="right" xlink:label="note-461-02" xlink:href="note-461-02a" xml:space="preserve">6. caſus.</note> <figure xlink:label="fig-461-01" xlink:href="fig-461-01a"> <image file="461-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/461-01"/> </figure> <note position="right" xlink:label="note-461-03" xlink:href="note-461-03a" xml:space="preserve">2. 2. Theod.</note> <note position="right" xlink:label="note-461-04" xlink:href="note-461-04a" xml:space="preserve">29. primi.</note> </div> <p> <s xml:id="echoid-s15928" xml:space="preserve">7. </s> <s xml:id="echoid-s15929" xml:space="preserve">SIT arcus AC, quadrante maior, & </s> <s xml:id="echoid-s15930" xml:space="preserve">AB, BC, quadrante minores. <lb/></s> <s xml:id="echoid-s15931" xml:space="preserve"> <anchor type="note" xlink:label="note-461-05a" xlink:href="note-461-05"/> Completo circulo ABDGH, & </s> <s xml:id="echoid-s15932" xml:space="preserve">abſciſſo quadrante AL, ex AC, producto-<lb/>q́ue arcu BC, vt fiat quadrans <lb/> <anchor type="figure" xlink:label="fig-461-02a" xlink:href="fig-461-02"/> BM, fiant reliqua omnia, vt in <lb/>primo caſu. </s> <s xml:id="echoid-s15933" xml:space="preserve">Demonſtrabimus <lb/>enim, vt ibi, ita eſſe quadratum <lb/>ſinus totius, ſiue rectangulũ ſub <lb/>DX, XA, ad rectangulum ſub <lb/>AV, kY, ſinubus rectis arcuum <lb/>AB, AC, vt eſt DZ, ſinus ver-<lb/>ſus anguli A, ſiue arcus DL, ad <lb/>kT, differentiam ſinuum verſo-<lb/>rum BR, BQ, arcuum BC, BK: <lb/></s> <s xml:id="echoid-s15934" xml:space="preserve">ſed triangulum AXV, ita probabitur æquiangulum eſſe triangulo SKT. </s> <s xml:id="echoid-s15935" xml:space="preserve">An-<lb/>gulus KST, angulo ISR, æqualis eſt. </s> <s xml:id="echoid-s15936" xml:space="preserve">Igitur in triangulis rectangulis SkT, <lb/> <anchor type="note" xlink:label="note-461-06a" xlink:href="note-461-06"/> ISR, reliqui anguli SkT, SIR, æquales erunt; </s> <s xml:id="echoid-s15937" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s15938" xml:space="preserve">in rectangulis <lb/>triangulis SKT, XIY, reliqui anguli KST, IXY, æquales erunt. </s> <s xml:id="echoid-s15939" xml:space="preserve">Cum <lb/>ergo angulus IXY, angulo AXV, æqualis ſit, erit quoq; </s> <s xml:id="echoid-s15940" xml:space="preserve">kST, eidem AXV, <lb/> <anchor type="note" xlink:label="note-461-07a" xlink:href="note-461-07"/> <pb o="450" file="462" n="462" rhead=""/> æqualis; </s> <s xml:id="echoid-s15941" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s15942" xml:space="preserve">in triangulis rectangulis SkT, XAV, reliqui anguli <lb/><emph style="sc">Sk</emph>T, XAV, æquales erunt.</s> <s xml:id="echoid-s15943" xml:space="preserve"/> </p> <div xml:id="echoid-div1278" type="float" level="2" n="9"> <note position="right" xlink:label="note-461-05" xlink:href="note-461-05a" xml:space="preserve">7. caſus.</note> <figure xlink:label="fig-461-02" xlink:href="fig-461-02a"> <image file="461-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/461-02"/> </figure> <note position="right" xlink:label="note-461-06" xlink:href="note-461-06a" xml:space="preserve">15. primi.</note> <note position="right" xlink:label="note-461-07" xlink:href="note-461-07a" xml:space="preserve">15. primi.</note> </div> <p> <s xml:id="echoid-s15944" xml:space="preserve">8. </s> <s xml:id="echoid-s15945" xml:space="preserve">SIT adhuc AC, quadrante maior, & </s> <s xml:id="echoid-s15946" xml:space="preserve">AB, minor quadrante, ſed BC, <lb/> <anchor type="note" xlink:label="note-462-01a" xlink:href="note-462-01"/> quadrans. </s> <s xml:id="echoid-s15947" xml:space="preserve">Completo circulo ABDGH, & </s> <s xml:id="echoid-s15948" xml:space="preserve">abſciſſo quadrante AL, ex AC, <lb/>deſcribantur ex polis A, B, ad in-<lb/>terualla quadrantum AL, BC, <lb/> <anchor type="figure" xlink:label="fig-462-01a" xlink:href="fig-462-01"/> maximi circuli DEF, <emph style="sc">GEh</emph>: <lb/></s> <s xml:id="echoid-s15949" xml:space="preserve">Item ex polo A, ad interuallum <lb/>AC, circulus nõ maximus <emph style="sc">K</emph>CN, <lb/>& </s> <s xml:id="echoid-s15950" xml:space="preserve">alia fiãt, vt in primo caſu. </s> <s xml:id="echoid-s15951" xml:space="preserve">De-<lb/>monſtrabitur, vt ibi, ita eſſe qua-<lb/>dratum ſinus totius, nimirum re <lb/>ctãgulum ſub DX, XA, ad re-<lb/>ctangulum ſub AV, <emph style="sc">K</emph>Y, ſinubus <lb/>rectis arcuum AB, AC, vt eſt <lb/>DZ, ſinus verſus arcus DL, ſiue anguli A, ad <emph style="sc">K</emph>T, differentiam ſinuum verſo-<lb/>rum BR, BQ, arcuum BC, BK: </s> <s xml:id="echoid-s15952" xml:space="preserve">ſi tamen triangula XAV, <emph style="sc">Sk</emph>T, oſtenda-<lb/>mus æquiangula eſſe, vtin ſeptimo caſu.</s> <s xml:id="echoid-s15953" xml:space="preserve"/> </p> <div xml:id="echoid-div1279" type="float" level="2" n="10"> <note position="left" xlink:label="note-462-01" xlink:href="note-462-01a" xml:space="preserve">3. caſus.</note> <figure xlink:label="fig-462-01" xlink:href="fig-462-01a"> <image file="462-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/462-01"/> </figure> </div> <p> <s xml:id="echoid-s15954" xml:space="preserve">9. </s> <s xml:id="echoid-s15955" xml:space="preserve">SIT rurſus AC, maior quadrante, & </s> <s xml:id="echoid-s15956" xml:space="preserve">AB, quadrante minor, ſed BC, <lb/> <anchor type="note" xlink:label="note-462-02a" xlink:href="note-462-02"/> maior etiam quadrante. </s> <s xml:id="echoid-s15957" xml:space="preserve">Completo circulo ABDGH, & </s> <s xml:id="echoid-s15958" xml:space="preserve">abſciſsis quadran-<lb/>tibus AL, BM, ex AC, BC, reliqua conſtruantur, vt in primo caſu. </s> <s xml:id="echoid-s15959" xml:space="preserve">Nam, <lb/>vtibi, ita hic demonſtrabitur, ita <lb/>eſſe quadratum ſinus totius, re-<lb/> <anchor type="figure" xlink:label="fig-462-02a" xlink:href="fig-462-02"/> ctangulum videlicet ſub DX, <lb/>XA, ad rectangulum ſub AV, <lb/><emph style="sc">K</emph>Y, ſinubus rectis arcuum AB, <lb/>AC, vt eſt DZ, ſinus verſus ar-<lb/>cus DL, ſiue anguli A, ad <emph style="sc">K</emph>T, <lb/>differentiam ſinuum verſorum <lb/>BR, BQ arcuum BC, <emph style="sc">Bk</emph>. </s> <s xml:id="echoid-s15960" xml:space="preserve">Sed <lb/>triangula XAV, <emph style="sc">Sk</emph>T, eſſe æ-<lb/>quiangula, ita confirmabitur. <lb/></s> <s xml:id="echoid-s15961" xml:space="preserve">Angulus <emph style="sc">K</emph>ST, angulo ISR, æqualis eſt. </s> <s xml:id="echoid-s15962" xml:space="preserve">Igitur in rectangulis triangulis SKT, <lb/> <anchor type="note" xlink:label="note-462-03a" xlink:href="note-462-03"/> SIR, & </s> <s xml:id="echoid-s15963" xml:space="preserve">reliqui anguli <emph style="sc">Sk</emph>T, SIR, æquales erunt; </s> <s xml:id="echoid-s15964" xml:space="preserve">ac proinde in triangulis <lb/>rectangulis <emph style="sc">Sk</emph>T, IXY, reliqui quoque anguli <emph style="sc">K</emph>ST, IXY, hoc eſt, AXV, <lb/>(cum hic ipſi IXY, ſit æqualis) inter ſe æquales erunt. </s> <s xml:id="echoid-s15965" xml:space="preserve">Quare & </s> <s xml:id="echoid-s15966" xml:space="preserve">reliqui angu-<lb/> <anchor type="note" xlink:label="note-462-04a" xlink:href="note-462-04"/> li <emph style="sc">Sk</emph>T, XAV, in triangulis rectangulis <emph style="sc">Sk</emph>T, XAV, erunt æquales.</s> <s xml:id="echoid-s15967" xml:space="preserve"/> </p> <div xml:id="echoid-div1280" type="float" level="2" n="11"> <note position="left" xlink:label="note-462-02" xlink:href="note-462-02a" xml:space="preserve">5. caſus.</note> <figure xlink:label="fig-462-02" xlink:href="fig-462-02a"> <image file="462-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/462-02"/> </figure> <note position="left" xlink:label="note-462-03" xlink:href="note-462-03a" xml:space="preserve">15. primi.</note> <note position="left" xlink:label="note-462-04" xlink:href="note-462-04a" xml:space="preserve">15. primi.</note> </div> <p> <s xml:id="echoid-s15968" xml:space="preserve">10. </s> <s xml:id="echoid-s15969" xml:space="preserve">SIT arcus AC, maior <lb/> <anchor type="figure" xlink:label="fig-462-03a" xlink:href="fig-462-03"/> <anchor type="note" xlink:label="note-462-05a" xlink:href="note-462-05"/> quadrante, & </s> <s xml:id="echoid-s15970" xml:space="preserve">AB, quadrans, at <lb/>BC, quadrante minor. </s> <s xml:id="echoid-s15971" xml:space="preserve">Comple-<lb/>to circulo ABGF, abſciſſoq́ue <lb/>quadrante AL, ex AC, & </s> <s xml:id="echoid-s15972" xml:space="preserve">pro-<lb/>ducto BC, vt fiat quadrans BM, <lb/>deſcribãtur ex polis A, B, ad in-<lb/>terualla quadrantum AL, BM, <lb/>circuli maximi BLEF, AEMG, <lb/>incedetq́; </s> <s xml:id="echoid-s15973" xml:space="preserve">ille per punctum B, & </s> <s xml:id="echoid-s15974" xml:space="preserve">hic per punctum A, ob quadrantem AB; </s> <s xml:id="echoid-s15975" xml:space="preserve">pro-<lb/> <anchor type="note" xlink:label="note-462-06a" xlink:href="note-462-06"/> pterea quòd maximus circulus à polo abeſt quadrante maximi circuli. </s> <s xml:id="echoid-s15976" xml:space="preserve">Item <lb/> <anchor type="note" xlink:label="note-462-07a" xlink:href="note-462-07"/> <pb o="451" file="463" n="463" rhead=""/> ex eiſdem polis A, B, ad interualla AC, BC, delineentur circuli non maxi-<lb/>mi KCN, OCP, qui prioribus erunt paralleli. </s> <s xml:id="echoid-s15977" xml:space="preserve">Deſcriptis deinde in alio cir <lb/> <anchor type="note" xlink:label="note-463-01a" xlink:href="note-463-01"/> culo communibus ſectionibus horum circulorum cum circulo ABGF, quæ <lb/>inter ſe parallelę erunt, ſeſeq́; </s> <s xml:id="echoid-s15978" xml:space="preserve">ad angulos rectos ſecabunt; </s> <s xml:id="echoid-s15979" xml:space="preserve">(Nam AX, ex A, <lb/> <anchor type="note" xlink:label="note-463-02a" xlink:href="note-463-02"/> polo circuli BF, in ſphæræ centrum X, cadens ad ipſum circulum recta eſt; <lb/></s> <s xml:id="echoid-s15980" xml:space="preserve"> <anchor type="note" xlink:label="note-463-03a" xlink:href="note-463-03"/> ac propterea, ex defin. </s> <s xml:id="echoid-s15981" xml:space="preserve">3. </s> <s xml:id="echoid-s15982" xml:space="preserve">lib. </s> <s xml:id="echoid-s15983" xml:space="preserve">11. </s> <s xml:id="echoid-s15984" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s15985" xml:space="preserve">anguli ad X, recti erunt. </s> <s xml:id="echoid-s15986" xml:space="preserve">Ex quo fit, etiam <lb/> <anchor type="note" xlink:label="note-463-04a" xlink:href="note-463-04"/> angulos ad R, S, Y, rectos eſſe, ob parallelas lineas BF, <emph style="sc">K</emph>N, & </s> <s xml:id="echoid-s15987" xml:space="preserve">AG, OP.) </s> <s xml:id="echoid-s15988" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-463-05a" xlink:href="note-463-05"/> AV, ſinus rectus quadrantis AB; </s> <s xml:id="echoid-s15989" xml:space="preserve"><emph style="sc">K</emph>Y, ſinus rectus arcus AC, ſiue arcus AK, <lb/>illi, ex defin. </s> <s xml:id="echoid-s15990" xml:space="preserve">poli, æqualis; </s> <s xml:id="echoid-s15991" xml:space="preserve">BR, ſinus verſus arcus BC, ſeu arcus BO, illi, ex <lb/>poli defin. </s> <s xml:id="echoid-s15992" xml:space="preserve">æqualis; </s> <s xml:id="echoid-s15993" xml:space="preserve">BQ, (ducta <emph style="sc">K</emph>Q, ad BF, perpendiculari) ſinus verſus ar-<lb/>cus <emph style="sc">Bk</emph>, quo arcus AB, AC, inter ſe differunt; </s> <s xml:id="echoid-s15994" xml:space="preserve">Denique <emph style="sc">K</emph>S, ſinus verſus ar-<lb/>cus <emph style="sc">K</emph>C. </s> <s xml:id="echoid-s15995" xml:space="preserve">Itaque ſi ſumatur DZ, ſinus verſus anguli A, ſiue arcus BL, qui ar-<lb/>cui <emph style="sc">K</emph>C, ſimilis eſt, demonſtrabimus, vt in primo caſu, ita eſſe quadratum ſi-<lb/>nus totius, nempe rectangulum ſub DX, XA, ad rectangulum ſub AV, <emph style="sc">K</emph>Y, <lb/>ſinubus rectis arcuum AB, AC, vt eſt DZ, ſinus verſus anguli A, ſiue arcus <lb/>BL, ad <emph style="sc">K</emph>T, ſiue ad QR, differentiam ſinuum verſorum BR, BQ, arcuum <lb/>BC, <emph style="sc">Bk</emph>, niſi quòd hic non inueniuntur triangula æquiangula, ſed AV, ab <lb/>XA, non differt, quemadmodum nec <emph style="sc">K</emph>S, à <emph style="sc">K</emph>T.</s> <s xml:id="echoid-s15996" xml:space="preserve"/> </p> <div xml:id="echoid-div1281" type="float" level="2" n="12"> <figure xlink:label="fig-462-03" xlink:href="fig-462-03a"> <image file="462-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/462-03"/> </figure> <note position="left" xlink:label="note-462-05" xlink:href="note-462-05a" xml:space="preserve">10. caſus.</note> <note position="left" xlink:label="note-462-06" xlink:href="note-462-06a" xml:space="preserve">Coroll. 16.</note> <note position="left" xlink:label="note-462-07" xlink:href="note-462-07a" xml:space="preserve">1. Theod.</note> <note position="right" xlink:label="note-463-01" xlink:href="note-463-01a" xml:space="preserve">2. 2. Theod.</note> <note position="right" xlink:label="note-463-02" xlink:href="note-463-02a" xml:space="preserve">16. vndec.</note> <note position="right" xlink:label="note-463-03" xlink:href="note-463-03a" xml:space="preserve">Schol. 10.</note> <note position="right" xlink:label="note-463-04" xlink:href="note-463-04a" xml:space="preserve">1. Theod.</note> <note position="right" xlink:label="note-463-05" xlink:href="note-463-05a" xml:space="preserve">29. primi.</note> </div> <p> <s xml:id="echoid-s15997" xml:space="preserve">11. </s> <s xml:id="echoid-s15998" xml:space="preserve">SIT iterum AC, quadrante maior, attam AB, quam BC, quadrans. <lb/></s> <s xml:id="echoid-s15999" xml:space="preserve"> <anchor type="note" xlink:label="note-463-06a" xlink:href="note-463-06"/> Completo circulo ABGF, & </s> <s xml:id="echoid-s16000" xml:space="preserve">reſecto quadrante AE, ex AC, deſcribantur <lb/>ex polo A, ad interualla AE, <lb/>AC, circuli BEF, KCN, & </s> <s xml:id="echoid-s16001" xml:space="preserve">ex <lb/> <anchor type="figure" xlink:label="fig-463-01a" xlink:href="fig-463-01"/> polo B, ad interuallum BC, cir-<lb/>culus AEG, aliaque fiant, vt in <lb/>præcedenti caſu. </s> <s xml:id="echoid-s16002" xml:space="preserve">Oſtendemus <lb/>ergo, vt in primo caſu, ita eſſe <lb/>quadratum ſinus totius, hoc eſt, <lb/>rectangulum ſub DX, XA, ad re-<lb/>ctangulum ſub AV, KY, ſinu-<lb/>bus rectis arcuum AB, AC, vt <lb/>eſt DZ, ſinus verſus anguli A, ſi-<lb/>ue arcus BE, ad KT, ſeu QR, differentiam ſinuum verſorum BR, BQ, ar-<lb/>cuum BC, BK, niſi quod hic nulla adſint æquiangula triangula, quemadmo-<lb/>dum nec in præcedenti caſu, atque AV, ab XA, & </s> <s xml:id="echoid-s16003" xml:space="preserve">DZ, à BR, & </s> <s xml:id="echoid-s16004" xml:space="preserve">KS, à KT, <lb/>non differt.</s> <s xml:id="echoid-s16005" xml:space="preserve"/> </p> <div xml:id="echoid-div1282" type="float" level="2" n="13"> <note position="right" xlink:label="note-463-06" xlink:href="note-463-06a" xml:space="preserve">11. caſus.</note> <figure xlink:label="fig-463-01" xlink:href="fig-463-01a"> <image file="463-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/463-01"/> </figure> </div> <p> <s xml:id="echoid-s16006" xml:space="preserve">12. </s> <s xml:id="echoid-s16007" xml:space="preserve">SIT arcus AC, quadrante maior, & </s> <s xml:id="echoid-s16008" xml:space="preserve">AB, quadrans, ſed BC, maior <lb/> <anchor type="note" xlink:label="note-463-07a" xlink:href="note-463-07"/> etiam quadrante. </s> <s xml:id="echoid-s16009" xml:space="preserve">Completo cir-<lb/>culo ABGF, & </s> <s xml:id="echoid-s16010" xml:space="preserve">ablatis quadran <lb/> <anchor type="figure" xlink:label="fig-463-02a" xlink:href="fig-463-02"/> tibus AL, BM, ex arcubus AC, <lb/>BC, deſcribantur circuli ex po-<lb/>lis A, B, ad interualla quadran-<lb/>tum AL, BM, & </s> <s xml:id="echoid-s16011" xml:space="preserve">arcuum AC, <lb/>BC, cæteraq́ue fiant, vt in præ-<lb/>cedentibus. </s> <s xml:id="echoid-s16012" xml:space="preserve">Erit ergo rurſus, vt <lb/>in primo caſu demonſtratum eſt, <lb/>ita quadratum ſinus totius, id <lb/>eſt, rectangulum ſub DX, XA, <lb/>ad rectangulum ſub AV, KY, ſinubus rectis arcuum AB, AC, vt eſt DZ, <pb o="452" file="464" n="464" rhead=""/> ſinus verſus anguli A, ſiue arcus BL, ad KT, differentiam ſinuum verſorum <lb/>BR, BQ, arcuum BC, BK; </s> <s xml:id="echoid-s16013" xml:space="preserve">quamuis nulla hic appareant triangula æquian-<lb/>gula, ſed XA, AV, inter ſe non differant, quemadmodum neque KS, KT.</s> <s xml:id="echoid-s16014" xml:space="preserve"/> </p> <div xml:id="echoid-div1283" type="float" level="2" n="14"> <note position="right" xlink:label="note-463-07" xlink:href="note-463-07a" xml:space="preserve">12. caſus.</note> <figure xlink:label="fig-463-02" xlink:href="fig-463-02a"> <image file="463-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/463-02"/> </figure> </div> <p> <s xml:id="echoid-s16015" xml:space="preserve">13. </s> <s xml:id="echoid-s16016" xml:space="preserve">SINT arcus AC, AB, quadrante maiores, at BC, minor quadran-<lb/> <anchor type="note" xlink:label="note-464-01a" xlink:href="note-464-01"/> te. </s> <s xml:id="echoid-s16017" xml:space="preserve">Completo circulo ABGF, & </s> <s xml:id="echoid-s16018" xml:space="preserve">reſecto quadrante AL, ex AC, producto <lb/>item arcu BC, vt fiat quadrans BM, reliqua fiant, vt in ſuperioribus. </s> <s xml:id="echoid-s16019" xml:space="preserve">Demon <lb/>ſtrabimus iam, vt in primo caſu, <lb/>ita eſſe quadratum ſinus totius, <lb/> <anchor type="figure" xlink:label="fig-464-01a" xlink:href="fig-464-01"/> nimirum rectangulum ſub DX, <lb/>XA, ad rectangulum ſub AV, <lb/>KY, ſinubus rectis arcuum AB, <lb/>AC, vt eſt DZ, ſinus verſus an-<lb/>guli A, ſeuarcus DL, ad KT, <lb/>differentiam ſinuum verſorum <lb/>BR, BQ, arcuum BC, Bk; </s> <s xml:id="echoid-s16020" xml:space="preserve">ni-<lb/>ſi quòd triagulum XAV, trian <lb/>gulo <emph style="sc">Sk</emph>T, demonſtrandum eſt <lb/>eſſe æquiangulum hac ratione. </s> <s xml:id="echoid-s16021" xml:space="preserve">Quoniam angulus <emph style="sc">K</emph>ST, angulo oppoſito, & </s> <s xml:id="echoid-s16022" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-464-02a" xlink:href="note-464-02"/> interno YIX, æqualis eſt, erit in triangulis rectangulis <emph style="sc">Sk</emph>T, IXY, & </s> <s xml:id="echoid-s16023" xml:space="preserve">re-<lb/>liquus angulus <emph style="sc">Sk</emph>T, reliquo angulo <emph style="sc">IXy</emph>, hoc eſt, angulo oppoſito, & </s> <s xml:id="echoid-s16024" xml:space="preserve">inter-<lb/>no VAX, (cum parallelæ ſint AV, GH.) </s> <s xml:id="echoid-s16025" xml:space="preserve">æqualis. </s> <s xml:id="echoid-s16026" xml:space="preserve">Igitur in triangulis rectan <lb/>gulis <emph style="sc">Sk</emph>T, XAV, anguli quoque reliqui <emph style="sc">K</emph>ST, AXV, æquales erunt, ac <lb/>proinde æquiangula erunt triangula <emph style="sc">Sk</emph>T, XAV.</s> <s xml:id="echoid-s16027" xml:space="preserve"/> </p> <div xml:id="echoid-div1284" type="float" level="2" n="15"> <note position="left" xlink:label="note-464-01" xlink:href="note-464-01a" xml:space="preserve">13. caſus.</note> <figure xlink:label="fig-464-01" xlink:href="fig-464-01a"> <image file="464-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/464-01"/> </figure> <note position="left" xlink:label="note-464-02" xlink:href="note-464-02a" xml:space="preserve">29 primi.</note> </div> <p> <s xml:id="echoid-s16028" xml:space="preserve">14. </s> <s xml:id="echoid-s16029" xml:space="preserve">SINT rurſum AC, AB, maiores quadrante, at BC, quadrans. </s> <s xml:id="echoid-s16030" xml:space="preserve">Com-<lb/> <anchor type="note" xlink:label="note-464-03a" xlink:href="note-464-03"/> pleto circulo ABGF, & </s> <s xml:id="echoid-s16031" xml:space="preserve">abſciſſo quadrante AL, ex AC, necnon deſcriptis <lb/>circulis DEF, <emph style="sc">K</emph>CN, ex polo A, ad interualla AL, AC, deſcribatur quoq; <lb/></s> <s xml:id="echoid-s16032" xml:space="preserve">ex polo B, ad interuallum quadrantis BC, circulus maximus GEH, atq; </s> <s xml:id="echoid-s16033" xml:space="preserve">alia <lb/>frant, vt ſupra. </s> <s xml:id="echoid-s16034" xml:space="preserve">Demonſtrandum <lb/>ergo eſt, ita eſſe quadratum ſinus <lb/> <anchor type="figure" xlink:label="fig-464-02a" xlink:href="fig-464-02"/> totius, id eſt, rectágulum ſub DX, <lb/>XA, ad rectangulum ſub AV, <emph style="sc">KY</emph>, <lb/>ſinubus arcuum AB, AC, vt eſt <lb/>DZ, ſinus verſus anguli A, arcuſ-<lb/>ve DL, ad <emph style="sc">K</emph>T, differentiam in-<lb/>ter ſinus verſos BR, BQ, ar-<lb/>cuum BC, <emph style="sc">Bk</emph>. </s> <s xml:id="echoid-s16035" xml:space="preserve">Quod quidem o-<lb/>ſtendemus, vtin primo caſu. </s> <s xml:id="echoid-s16036" xml:space="preserve">So-<lb/>lum triangula <emph style="sc">Sk</emph>T, XAV, pro-<lb/>babũtur æquiangula eſſe, hoc mo <lb/>do. </s> <s xml:id="echoid-s16037" xml:space="preserve">Angulus S, communis eſt vtrique triangulo rectangulo <emph style="sc">Sk</emph>T, <emph style="sc">SRy</emph>. </s> <s xml:id="echoid-s16038" xml:space="preserve">Igi-<lb/>tur angulus reliquus <emph style="sc">Sk</emph>T, reliquo angulo <emph style="sc">SRy</emph>, hoceſt, angulo oppoſito, & </s> <s xml:id="echoid-s16039" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-464-04a" xlink:href="note-464-04"/> interno VAX, (cum parallelę ſint AV, GH.) </s> <s xml:id="echoid-s16040" xml:space="preserve">æqualis erit. </s> <s xml:id="echoid-s16041" xml:space="preserve">Quare rectangu <lb/>la triangula <emph style="sc">Sk</emph>T, XAV, æquiangula erunt.</s> <s xml:id="echoid-s16042" xml:space="preserve"/> </p> <div xml:id="echoid-div1285" type="float" level="2" n="16"> <note position="left" xlink:label="note-464-03" xlink:href="note-464-03a" xml:space="preserve">14. caſus.</note> <figure xlink:label="fig-464-02" xlink:href="fig-464-02a"> <image file="464-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/464-02"/> </figure> <note position="left" xlink:label="note-464-04" xlink:href="note-464-04a" xml:space="preserve">29. primi.</note> </div> <p> <s xml:id="echoid-s16043" xml:space="preserve">15. </s> <s xml:id="echoid-s16044" xml:space="preserve">SINT poſtremo omnes tres arcus trianguli ABC, quadrante ma-<lb/> <anchor type="note" xlink:label="note-464-05a" xlink:href="note-464-05"/> iores. </s> <s xml:id="echoid-s16045" xml:space="preserve">Completo circulo ABGF, & </s> <s xml:id="echoid-s16046" xml:space="preserve">reſectis quadrantibus AL, BM, ex ar-<lb/>cubus AC, BC, fiant omnia alia, vt prius. </s> <s xml:id="echoid-s16047" xml:space="preserve">Oſtendemus non ſecus, ac in pri-<lb/>mo caſu, ita eſſe quadratum ſinus totius, nempe rectangulum ſub DX, <emph style="sc">Xa</emph>, <lb/>ad rectangulum ſub AV, <emph style="sc">KY</emph>, ſinubus rectis arcuum AB, AC, vt eſt DZ, ſi- <pb o="453" file="465" n="465" rhead=""/> nus verſus anguli A, ſeu arcus DL, ad <emph style="sc">K</emph>T, differentiam inter ſinus verſos <lb/>BR, BQ, arcuum BC, <emph style="sc">Bk</emph>; </s> <s xml:id="echoid-s16048" xml:space="preserve">ſi modo triangula <emph style="sc">Sk</emph>T, XAV, æquiangula eſſe <lb/>concludamus, hac argumétatio-<lb/> <anchor type="figure" xlink:label="fig-465-01a" xlink:href="fig-465-01"/> ne. </s> <s xml:id="echoid-s16049" xml:space="preserve">Angulus <emph style="sc">Y</emph>IX, æqualis eſt <lb/>interno & </s> <s xml:id="echoid-s16050" xml:space="preserve">oppoſito <emph style="sc">K</emph>ST. </s> <s xml:id="echoid-s16051" xml:space="preserve">Igitur <lb/> <anchor type="note" xlink:label="note-465-01a" xlink:href="note-465-01"/> in triangulis rectangulis <emph style="sc">Sk</emph>T, <lb/><emph style="sc">IXy</emph>, reliquus angulus <emph style="sc">Sk</emph>T, re-<lb/>liquo angulo <emph style="sc">IXy</emph>, hoc eſt, an-<lb/>gulo interno, & </s> <s xml:id="echoid-s16052" xml:space="preserve">oppoſito XAV, <lb/>(cum parallelæ ſint AV, GH.) <lb/></s> <s xml:id="echoid-s16053" xml:space="preserve">æqualis erit; </s> <s xml:id="echoid-s16054" xml:space="preserve">ac proinde triãgula <lb/>rectangula <emph style="sc">Sk</emph>T, XAV, æquian-<lb/>gula erunt. </s> <s xml:id="echoid-s16055" xml:space="preserve">Quapropter In omni triangulo ſphærico, cuius duo arcus ſint <lb/>inæquales, &</s> <s xml:id="echoid-s16056" xml:space="preserve">c. </s> <s xml:id="echoid-s16057" xml:space="preserve">Quod demonſtrandum erat.</s> <s xml:id="echoid-s16058" xml:space="preserve"/> </p> <div xml:id="echoid-div1286" type="float" level="2" n="17"> <note position="left" xlink:label="note-464-05" xlink:href="note-464-05a" xml:space="preserve">25. caſus.</note> <figure xlink:label="fig-465-01" xlink:href="fig-465-01a"> <image file="465-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/465-01"/> </figure> <note position="right" xlink:label="note-465-01" xlink:href="note-465-01a" xml:space="preserve">29. primi.</note> </div> </div> <div xml:id="echoid-div1288" type="section" level="1" n="599"> <head xml:id="echoid-head634" xml:space="preserve">SCHOLIVM. I.</head> <p style="it"> <s xml:id="echoid-s16059" xml:space="preserve">FX omnibus quindecim caſibus buius demonſtrationis liquet, arcum BC, angus <lb/>lo A, ſub arcubus inæqualibus comprehenſo oppoſitum ſemper maiorem eſſe arcu BK, <lb/>hoc eſt, d'fferentia arcuum inæqualium. </s> <s xml:id="echoid-s16060" xml:space="preserve">In omnibus enim figuris arcus BC, per de-<lb/>fin. </s> <s xml:id="echoid-s16061" xml:space="preserve">poli, arcui BO, (vel arcui BG, quando BC, quadrans eſt, vt in caſu 2. </s> <s xml:id="echoid-s16062" xml:space="preserve">5. </s> <s xml:id="echoid-s16063" xml:space="preserve">8. </s> <s xml:id="echoid-s16064" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s16065" xml:space="preserve">& </s> <s xml:id="echoid-s16066" xml:space="preserve">14.) </s> <s xml:id="echoid-s16067" xml:space="preserve">æqualis eſt. </s> <s xml:id="echoid-s16068" xml:space="preserve">Conſtat autem arcum BO, (vel arcum BH, in dictis quinque <lb/>caſibus) maiorem eſſe arcu BK: </s> <s xml:id="echoid-s16069" xml:space="preserve">quod tamen ita eſſe, facile ſequens quoque theore-<lb/>ma demonſtrabit.</s> <s xml:id="echoid-s16070" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16071" xml:space="preserve">IN omni triangulo ſphærico, cuius duo arcus ſint inæquales; </s> <s xml:id="echoid-s16072" xml:space="preserve">ar-<lb/>cus reliquus maior eſt arcu, quo inæquales arcus inter ſe differunt.</s> <s xml:id="echoid-s16073" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16074" xml:space="preserve">IN triangulo enim <emph style="sc">ABC</emph>, ſit arcus AB, maior arcu <lb/>AC, & </s> <s xml:id="echoid-s16075" xml:space="preserve">ex polo A, ad interuallum AC, arcus circuli de-<lb/> <anchor type="figure" xlink:label="fig-465-02a" xlink:href="fig-465-02"/> ſcribatur CD. </s> <s xml:id="echoid-s16076" xml:space="preserve">Erit ergo arcus AD, arcui AC, per <lb/>deſin. </s> <s xml:id="echoid-s16077" xml:space="preserve">poli, æqualis, atque adeo arcus BD, differentia <lb/>arcuum inæqualium AB, AC. </s> <s xml:id="echoid-s16078" xml:space="preserve">Dico arcum BC, arcu <lb/>BD, maiorem eſſe. </s> <s xml:id="echoid-s16079" xml:space="preserve">Quoniam enim duo arcus <emph style="sc">Ac</emph>, CB, <lb/>ſimul maiores ſunt arcu AB; </s> <s xml:id="echoid-s16080" xml:space="preserve">ablatis æqualibus arcubus <lb/> <anchor type="note" xlink:label="note-465-02a" xlink:href="note-465-02"/> AC, AD, reliquus quoq; </s> <s xml:id="echoid-s16081" xml:space="preserve">CB, reliquo BD, maior erit. <lb/></s> <s xml:id="echoid-s16082" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s16083" xml:space="preserve"/> </p> <div xml:id="echoid-div1288" type="float" level="2" n="1"> <figure xlink:label="fig-465-02" xlink:href="fig-465-02a"> <image file="465-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/465-02"/> </figure> <note position="right" xlink:label="note-465-02" xlink:href="note-465-02a" xml:space="preserve">3. huius.</note> </div> <p style="it"> <s xml:id="echoid-s16084" xml:space="preserve">ITAQVE in omni ſphærico triangulo, cuius duo arcus inæquales ſint, ſinus ver <lb/> <anchor type="note" xlink:label="note-465-03a" xlink:href="note-465-03"/> ſus reliqui arcus ſemper maior eſt ſinu verſo differentiæ arcuum inæqualium. </s> <s xml:id="echoid-s16085" xml:space="preserve">Cum <lb/>enim arcus ille reliquus oſtenſus ſit maior, quam ea differentia, maior autem arcus <lb/>habeat ſemper maiorem ſinum verſum, vt ex tractatione ſinuum conſtat, perſpicuum <lb/>fit, reliqui arcus ſinum verſum maiorem eſſe ſinu verſo differentiæ arcuum inæ-<lb/>qualium.</s> <s xml:id="echoid-s16086" xml:space="preserve"/> </p> <div xml:id="echoid-div1289" type="float" level="2" n="2"> <note position="right" xlink:label="note-465-03" xlink:href="note-465-03a" xml:space="preserve">In triangu <lb/>lo ſphęrico <lb/>duorum ar <lb/>cuum inæ-<lb/>qualium, <lb/>ſinus uer-<lb/>ſuster@j ar <lb/>cus ma@or <lb/>eſt ſinu ver <lb/>ſo differen <lb/>tię arcuum <lb/>inæqualiũ.</note> </div> <p style="it"> <s xml:id="echoid-s16087" xml:space="preserve">HOC idcirco dixerim, vt rationem videas, quare in praxipropoſ. </s> <s xml:id="echoid-s16088" xml:space="preserve">64. </s> <s xml:id="echoid-s16089" xml:space="preserve">differentia <lb/>inter ſinus verſos, quorum vnus reliquo tertio arcui, alter vero differentiæ inæqua-<lb/>lium arcuum debetur, adijcienda præcipiatur ſinui verſo differentiæ arcuum inæqua <lb/>lium, vt componatur ſinus verſus reliqui tertij arcus, nunquam autem detrahenda à <lb/>ſinu verſo dictæ differentiæ, vt ſinus verſus reliqui arcus relinquatur.</s> <s xml:id="echoid-s16090" xml:space="preserve"/> </p> <pb o="454" file="466" n="466" rhead=""/> </div> <div xml:id="echoid-div1291" type="section" level="1" n="600"> <head xml:id="echoid-head635" xml:space="preserve">SCHOLIVM. II.</head> <p style="it"> <s xml:id="echoid-s16091" xml:space="preserve">CATERVM ex hac propoſ. </s> <s xml:id="echoid-s16092" xml:space="preserve">58. </s> <s xml:id="echoid-s16093" xml:space="preserve">colligemus ſequens theorema ad calculum trian <lb/>gulorum ſphæricorum non rectangulorum perutile, videlicet.</s> <s xml:id="echoid-s16094" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16095" xml:space="preserve">IN omni triangulo ſphærico, cuius duo arcus ſint inæquales: </s> <s xml:id="echoid-s16096" xml:space="preserve">ſi-<lb/>nus totus ad quantitatem, quæ ſinui toti, & </s> <s xml:id="echoid-s16097" xml:space="preserve">duobus ſinubus arcuum <lb/>inæqualium quarto loco proportionalis eſt, eandem proportionem <lb/>habet, quam ſinus verſus anguli ſub dictis arcubus comprehenſi ad <lb/>differentiam inter ſinum verſum reliqui tertij arcus, & </s> <s xml:id="echoid-s16098" xml:space="preserve">ſinum ver-<lb/>ſum arcus, quo duo inæquales arcus inter ſe differunt.</s> <s xml:id="echoid-s16099" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16100" xml:space="preserve">IN triangulo ſphærico ABC, proxime antecedenti <lb/> <anchor type="figure" xlink:label="fig-466-01a" xlink:href="fig-466-01"/> ſit arcus AB, maior arcu AC, & </s> <s xml:id="echoid-s16101" xml:space="preserve">ex polo A, ad interual <lb/>lum AC, deſcribatur arcus circuli CD, vt arcus AC, <lb/>AD, per defin. </s> <s xml:id="echoid-s16102" xml:space="preserve">poli, ſint æquales, atque adeo arcus BD, <lb/>exceſſus ſit, ſeu differentia arcuum AB, AC. </s> <s xml:id="echoid-s16103" xml:space="preserve">Fiat iam, <lb/>vt ſinus totus ad ſinum arcus AB, ita ſinus arcus AC, ad <lb/>aliud, quod quantitas quarta proportionalis vocetur, <lb/>vt hic vides: <lb/></s> <s xml:id="echoid-s16104" xml:space="preserve"> <anchor type="note" xlink:label="note-466-01a" xlink:href="note-466-01"/> Dico ita eſſe ſinum totum ad quantitatem quartam ſinui toti, & </s> <s xml:id="echoid-s16105" xml:space="preserve">duobus ſinubus ar-<lb/>cuum inæqualium proportionalem, vt eſt ſinus verſus anguli A, ad differentiam inter <lb/>ſinum verſum arcus BC, & </s> <s xml:id="echoid-s16106" xml:space="preserve">ſinum verſum arcus BD, quo inter ſe arcus <emph style="sc">AB</emph>, <emph style="sc">Ac</emph>, <lb/>differunt. </s> <s xml:id="echoid-s16107" xml:space="preserve">Quoiniam enim proportio ſinus totius ad quantitatem illam quartam pro-<lb/>portionalem componitur (poſito ſinu arcus AB, medio) ex proportione ſinus totius ad <lb/>ſinum arcus AB, & </s> <s xml:id="echoid-s16108" xml:space="preserve">ex proportione ſinus arcus AB, ad quantitatem quartam pro-<lb/>portionalem: </s> <s xml:id="echoid-s16109" xml:space="preserve">Et proportio quadrati ſinus totius ad rectangulum ſub ſinubus rectis ar-<lb/>cuum AB, AC, componitur ex eiſdem proportionibus, nempe ex proportione ſinus to <lb/>tius ad ſinum arcus AB, & </s> <s xml:id="echoid-s16110" xml:space="preserve">ex proportione ſinus totius ad ſinum arcus AC, quæ ea-<lb/> <anchor type="note" xlink:label="note-466-02a" xlink:href="note-466-02"/> dem eſt, quæ proportio ſinus arcus AB, ad quantitatem quartam proportionalem: <lb/></s> <s xml:id="echoid-s16111" xml:space="preserve">(Nam cum ſit, vt ſinus totus ad ſinum arcus <emph style="sc">A</emph>B, ita ſinus arcus AC, ad quantita-<lb/>tem quartam proportionalem; </s> <s xml:id="echoid-s16112" xml:space="preserve">erit permutando, vt ſinus totus ad ſinum arcus AC, <lb/>ita ſinus arcus AB, ad quantitatem quartam proportionalem.) </s> <s xml:id="echoid-s16113" xml:space="preserve">erit, vt ſinus totus <lb/>ad quantitatem quartam proportionalem, ita quadratum ſinus totius ad rectangu-<lb/>lum ſub ſinubus arcuum AB, AC, contentum. </s> <s xml:id="echoid-s16114" xml:space="preserve">Cum ergo ſit, vt quadratum ſinus to-<lb/>tius ad rectangulum ſub ſinubus arcuum AB, AC, ita ſinus verſus anguli A, ad dif-<lb/> <anchor type="note" xlink:label="note-466-03a" xlink:href="note-466-03"/> ferentiam ſinuum verſorum arcuum BC, BD; </s> <s xml:id="echoid-s16115" xml:space="preserve">erit quoque, vt ſinus totus ad quan-<lb/>titatem quartam proportionalem, ita ſinus verſus anguli A, ad differentiam inter ſi-<lb/>nus verſos arcuum BC, BD. </s> <s xml:id="echoid-s16116" xml:space="preserve">quod eſt propoſitum,</s> </p> <div xml:id="echoid-div1291" type="float" level="2" n="1"> <figure xlink:label="fig-466-01" xlink:href="fig-466-01a"> <image file="466-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/466-01"/> </figure> <note position="right" xlink:label="note-466-01" xlink:href="note-466-01a" xml:space="preserve"> <lb/>Sinus \ṫotus. # ſinus arcus \\ AB. # ſinus arcus \\ AC. # quantitas quarta \ṗroportionalis. <lb/></note> <note position="left" xlink:label="note-466-02" xlink:href="note-466-02a" xml:space="preserve">23. ſexti.</note> <note position="left" xlink:label="note-466-03" xlink:href="note-466-03a" xml:space="preserve">38. huius.</note> </div> </div> <div xml:id="echoid-div1293" type="section" level="1" n="601"> <head xml:id="echoid-head636" xml:space="preserve">THEOR. 57. PROPOS. 59.</head> <p> <s xml:id="echoid-s16117" xml:space="preserve">SI duo triangula ſphærica duos angulos duo-<lb/>bus angulis æquales habeant, vtrumque vtrique:</s> <s xml:id="echoid-s16118" xml:space="preserve"> <pb o="455" file="467" n="467" rhead=""/> crũt ſinus arcuum circa reliquum angulum vnius <lb/>ſinubus arcuum circa reliquum angulum alterius <lb/>proportionales, homologiq́; </s> <s xml:id="echoid-s16119" xml:space="preserve">erunt ſinus arcuum <lb/>æquales angulos ſubtendentium. </s> <s xml:id="echoid-s16120" xml:space="preserve">Et ſi vnus angu-<lb/>lus vnius vniangulo alterius ſit æqualis, ſinusq́; </s> <s xml:id="echoid-s16121" xml:space="preserve">ar-<lb/>cuum circa alium angulum vnius ſinubus arcuum <lb/>circa alium angulum alterius proportionales, ita <lb/>vt ſinus arcuum æquales angulos ſubtendentium <lb/>ſint homologi: </s> <s xml:id="echoid-s16122" xml:space="preserve">erunt & </s> <s xml:id="echoid-s16123" xml:space="preserve">anguli arcubus reliquo-<lb/>rum ſinuum homologorum oppoſiti inter ſe æ-<lb/>quales, vel æquales duobus rectis.</s> <s xml:id="echoid-s16124" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16125" xml:space="preserve">SINT in duobus triangulis ſphæricis ABC, DEF, duo anguli inter ſe <lb/>æquales B, E, necnon duo C, F. </s> <s xml:id="echoid-s16126" xml:space="preserve">Dico ita eſſe ſinum arcus AB, ad ſinum arcus <lb/>AC, vt eſt ſinus arcus DE, ad ſinum arcus DF. </s> <s xml:id="echoid-s16127" xml:space="preserve">Quia enim eſt, vt ſinus arcus <lb/>AB, ad ſinum anguli C, ita ſinus arcus AC, ad ſinum <lb/> <anchor type="note" xlink:label="note-467-01a" xlink:href="note-467-01"/> anguli B; </s> <s xml:id="echoid-s16128" xml:space="preserve">erit permutando, vt ſinus arcus AB, ad ſi-<lb/> <anchor type="figure" xlink:label="fig-467-01a" xlink:href="fig-467-01"/> num arcus AC, ita ſinus anguli C, ad ſinum anguli B, <lb/>hoc eſt, ita ſinus anguli F, ad ſinum anguli E, cum hi <lb/>anguli illis ponantur æquales. </s> <s xml:id="echoid-s16129" xml:space="preserve">Item quia eſt, vt ſi-<lb/>nus arcus DE, ad ſinum anguli F, ita ſinus arcus DF, <lb/> <anchor type="note" xlink:label="note-467-02a" xlink:href="note-467-02"/> ad ſinum anguli E; </s> <s xml:id="echoid-s16130" xml:space="preserve">erit permutando, vt ſinus arcus <lb/>DE, ad ſinum arcus DF, ita ſinus anguli F, ad ſinum <lb/>anguli E. </s> <s xml:id="echoid-s16131" xml:space="preserve">Oſtenſum autem eſt, ita etiam eſſe ſinum <lb/>arcus AB, ad ſinum arcus AC, vt eſt ſinus anguli F, <lb/>ad ſinum anguli E. </s> <s xml:id="echoid-s16132" xml:space="preserve">Igitur erit, vt ſinus arcus AB, ad <lb/>ſinum arcus AC, ita ſinus arcus DE, ad ſinum arcus DF. </s> <s xml:id="echoid-s16133" xml:space="preserve">Quod eſt propoſitũ.</s> <s xml:id="echoid-s16134" xml:space="preserve"/> </p> <div xml:id="echoid-div1293" type="float" level="2" n="1"> <note position="right" xlink:label="note-467-01" xlink:href="note-467-01a" xml:space="preserve">41. huius.</note> <figure xlink:label="fig-467-01" xlink:href="fig-467-01a"> <image file="467-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/467-01"/> </figure> <note position="right" xlink:label="note-467-02" xlink:href="note-467-02a" xml:space="preserve">41. huius.</note> </div> <p> <s xml:id="echoid-s16135" xml:space="preserve">SED ſint iam anguli B, E, æquales, & </s> <s xml:id="echoid-s16136" xml:space="preserve">ita ſit ſinus arcus AB, ad ſinus ar-<lb/>cus AC, vt eſt ſinus arcus DE, ad ſinum arcus DF. </s> <s xml:id="echoid-s16137" xml:space="preserve">Dico angulos quoq; </s> <s xml:id="echoid-s16138" xml:space="preserve">C, <lb/>F, æquales eſſe, vel certe duobus rectis æquales. </s> <s xml:id="echoid-s16139" xml:space="preserve">Oſtendemus enim, vt prius, <lb/>ita eſſe ſinum arcus AB, ad ſinum arcus AC, vt eſt ſinus anguli C, ad ſinum <lb/> <anchor type="note" xlink:label="note-467-03a" xlink:href="note-467-03"/> anguli B. </s> <s xml:id="echoid-s16140" xml:space="preserve">Item ita eſſe ſinum arcus DE, ad ſinum arcus DF, vt eſt ſinus angu-<lb/>li F, ad ſinum anguli E. </s> <s xml:id="echoid-s16141" xml:space="preserve">Quare cum ponatur, vt ſinus arcus AB, ad ſinum ar-<lb/>cus AC, ita ſinus arcus DE, ad ſinum arcus DF; </s> <s xml:id="echoid-s16142" xml:space="preserve">erit, vt ſinus anguli C, ad ſi-<lb/>num anguli B, ita ſinus anguli F, ad ſinum anguli E: </s> <s xml:id="echoid-s16143" xml:space="preserve">Et conuertendo, vt ſinus <lb/>anguli B, ad ſinum anguli C, ita ſinus anguli E, ad ſinum anguli F. </s> <s xml:id="echoid-s16144" xml:space="preserve">Cum ergo <lb/>ſinus æqualium angulorum B, E, æquales ſint, erunt & </s> <s xml:id="echoid-s16145" xml:space="preserve">ſinus angulorum C, <lb/> <anchor type="note" xlink:label="note-467-04a" xlink:href="note-467-04"/> F, æquales; </s> <s xml:id="echoid-s16146" xml:space="preserve">ac proinde vel anguli C, F, æquales erunt, vel duobus rectis æqua-<lb/>les. </s> <s xml:id="echoid-s16147" xml:space="preserve">Quod eſt propoſitum. </s> <s xml:id="echoid-s16148" xml:space="preserve">Itaque ſi duo triangula ſphærica duos angulos, <lb/>&</s> <s xml:id="echoid-s16149" xml:space="preserve">c. </s> <s xml:id="echoid-s16150" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s16151" xml:space="preserve"/> </p> <div xml:id="echoid-div1294" type="float" level="2" n="2"> <note position="right" xlink:label="note-467-03" xlink:href="note-467-03a" xml:space="preserve">41. huius. <lb/>Et permu-<lb/>tando.</note> <note position="right" xlink:label="note-467-04" xlink:href="note-467-04a" xml:space="preserve">14. quinti.</note> </div> <pb o="456" file="468" n="468" rhead=""/> </div> <div xml:id="echoid-div1296" type="section" level="1" n="602"> <head xml:id="echoid-head637" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s16152" xml:space="preserve">QVOD ſi anguli B, E, vel C, F, non forent æquales, ſed ſolum æquales duobus <lb/>rectis, adhuc theorematis veritas vetineretur: </s> <s xml:id="echoid-s16153" xml:space="preserve">propterea quod anguli B, E, ſemper <lb/>æquales ſinus rectos habent, ſiue ipſi inter ſe æquales ſint, ſiue æquales duobus re-<lb/>ctis. </s> <s xml:id="echoid-s16154" xml:space="preserve">quod etiam de angulis C, F, dicendum eſt. </s> <s xml:id="echoid-s16155" xml:space="preserve">Id quod perſpicue conſtare poteſt ex <lb/>ijs, quæ in tractatione ſinuum tradidimus.</s> <s xml:id="echoid-s16156" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1297" type="section" level="1" n="603"> <head xml:id="echoid-head638" xml:space="preserve">THEOR. 58. PROPOS. 60.</head> <p> <s xml:id="echoid-s16157" xml:space="preserve">SI ab angulo ſphærici trianguli ad baſim arcus <lb/>maximi circuli demittatur diuidens angulum bi-<lb/>fariam: </s> <s xml:id="echoid-s16158" xml:space="preserve">habebunt ſinus ſegmentorum baſis ean-<lb/>dem proportionem, quam ſinus reliquorum duo-<lb/>rum arcuum. </s> <s xml:id="echoid-s16159" xml:space="preserve">Et ſi ſinus ſegmentorum baſis ean-<lb/>dem proportionem habeant, quam ſinus reliquo-<lb/>rum duorum arcuum: </s> <s xml:id="echoid-s16160" xml:space="preserve">diuidet arcus demiſſus an-<lb/>gulum bifariam.</s> <s xml:id="echoid-s16161" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16162" xml:space="preserve">IN triangulo ſphærico ABC, ſecet arcus AD, angulum A, bifariam. </s> <s xml:id="echoid-s16163" xml:space="preserve">Di-<lb/>co ita eſſe ſinum arcus AB, ad ſinum arcus AC, vt eſt, ſinus arcus BD, ad ſi-<lb/>num arcus DC. </s> <s xml:id="echoid-s16164" xml:space="preserve">Quia enim triangula ABD, ACD, angulos ad A, habent <lb/> <anchor type="note" xlink:label="note-468-01a" xlink:href="note-468-01"/> æquales, & </s> <s xml:id="echoid-s16165" xml:space="preserve">angulos ad D, æquales duobus rectis; </s> <s xml:id="echoid-s16166" xml:space="preserve">erit, vt ſinus arcus AB, ad <lb/>ſinum arcus BD, ita ſinus arcus AC, ad ſinum arcus CD: <lb/></s> <s xml:id="echoid-s16167" xml:space="preserve">Et permutando, vt ſinus arcus AB, ad ſinum arcus AC, <lb/> <anchor type="figure" xlink:label="fig-468-01a" xlink:href="fig-468-01"/> ita ſinus arcus BD, ad ſinum arcus DC. </s> <s xml:id="echoid-s16168" xml:space="preserve">quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s16169" xml:space="preserve"/> </p> <div xml:id="echoid-div1297" type="float" level="2" n="1"> <note position="left" xlink:label="note-468-01" xlink:href="note-468-01a" xml:space="preserve">59. huius. <lb/>& eius ſcho <lb/>hum.</note> <figure xlink:label="fig-468-01" xlink:href="fig-468-01a"> <image file="468-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/468-01"/> </figure> </div> <p> <s xml:id="echoid-s16170" xml:space="preserve">SED iam ſit, vt ſinus arcus AB, ad ſinum arcus AC, <lb/>ita ſinus arcus BD, ad ſinum arcus DC. </s> <s xml:id="echoid-s16171" xml:space="preserve">Dico angulum <lb/>A, ſectum eſſe bifariam. </s> <s xml:id="echoid-s16172" xml:space="preserve">Erit enim permutando quoque, <lb/>vt ſinus arcus AB, ad ſinum arcus BD, ita ſinus arcus AC, <lb/>ad ſinũ arcus CD. </s> <s xml:id="echoid-s16173" xml:space="preserve">Habent igitur triangula ABD, ACD, <lb/>angulos ad D, ęquales duobus rectis, & </s> <s xml:id="echoid-s16174" xml:space="preserve">ſinus arcuum circa <lb/>angulos B, C, proportionales, homologiq́; </s> <s xml:id="echoid-s16175" xml:space="preserve">ſunt finus arcuũ angulis ad D, op-<lb/>poſitorũ. </s> <s xml:id="echoid-s16176" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s16177" xml:space="preserve">anguli ad A, vel æquales erunt inter ſe, vel duobus rectis æ-<lb/> <anchor type="note" xlink:label="note-468-02a" xlink:href="note-468-02"/> quales: </s> <s xml:id="echoid-s16178" xml:space="preserve">Non poſſunt autẽ duobus rectis eſſe æquales, quod angulus A, ſit duo-<lb/>bus rectis minor. </s> <s xml:id="echoid-s16179" xml:space="preserve">Igitur æquales inter ſe erunt. </s> <s xml:id="echoid-s16180" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s16181" xml:space="preserve">Si igitur <lb/>ab angulo ſphærici trianguli ad baſim, &</s> <s xml:id="echoid-s16182" xml:space="preserve">c. </s> <s xml:id="echoid-s16183" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s16184" xml:space="preserve"/> </p> <div xml:id="echoid-div1298" type="float" level="2" n="2"> <note position="left" xlink:label="note-468-02" xlink:href="note-468-02a" xml:space="preserve">59. huius. <lb/>& eius ſeho <lb/>lium.</note> </div> </div> <div xml:id="echoid-div1300" type="section" level="1" n="604"> <head xml:id="echoid-head639" xml:space="preserve">THEOR. 59. PROPOS. 61.</head> <p> <s xml:id="echoid-s16185" xml:space="preserve">SI ab angulo ſphærici triáguli ad baſim, etiam <pb o="457" file="469" n="469" rhead=""/> productam, arcus perpendicularis deducatur: </s> <s xml:id="echoid-s16186" xml:space="preserve">ha-<lb/>bebunt ſinus angulorum, quos arcus perpendicu-<lb/>laris cum duobus arcubus dictum angulum com-<lb/>prehendentibus facit, eandem proportionẽ, quam <lb/>ſinus complemẽtorum reliquorum duorum trian <lb/>guli angulorum.</s> <s xml:id="echoid-s16187" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16188" xml:space="preserve">IN triangulo ABC, deducatur ex angulo A, ad baſin BC, arcus perpen <lb/>dicularis AD, cadens ſiue intra triangulum, <lb/>ſiue extra. </s> <s xml:id="echoid-s16189" xml:space="preserve">Dico ita eſſe ſinum anguli BAD, <lb/> <anchor type="figure" xlink:label="fig-469-01a" xlink:href="fig-469-01"/> ad ſinum anguli DAC, vt eſt ſinus comple-<lb/>menti anguli B, ad ſinum complementi angu <lb/>li C. </s> <s xml:id="echoid-s16190" xml:space="preserve">Nam in triangulo ABD, cuius angulus <lb/>D, rectus, erit, vt ſinus anguli BAD, ad ſi-<lb/> <anchor type="note" xlink:label="note-469-01a" xlink:href="note-469-01"/> num totum, ita ſinus complementi anguli B, <lb/>ad ſinum complementi arcus AD. </s> <s xml:id="echoid-s16191" xml:space="preserve">Item in <lb/>triangulo CAD, habente angulum D, rectum, <lb/>erit, vt ſinus anguli DAC, ad ſinum totum, <lb/>ita ſinus complementi anguli C, (habent au-<lb/>tem duo anguli ad C, in ſecundo triangulo <lb/>eundem ſinum) ad ſinum complementi arcus <lb/>AD: </s> <s xml:id="echoid-s16192" xml:space="preserve">Et conuertendo, vt ſinus totus ad ſinum anguli DAC, ita ſinus com. <lb/></s> <s xml:id="echoid-s16193" xml:space="preserve"> <anchor type="note" xlink:label="note-469-02a" xlink:href="note-469-02"/> plementi arcus AD, ad ſinum com-<lb/>plementi anguli C. </s> <s xml:id="echoid-s16194" xml:space="preserve">Ex æqualitate <lb/>ergo (vt in appoſita formula vides) <lb/>erit, vt ſinus anguli BAD, ad ſinum <lb/>anguli DAC, ita ſinus complemen-<lb/>tianguli B, ad ſinum complementi anguli C. </s> <s xml:id="echoid-s16195" xml:space="preserve">Si igitur ab angulo ſphærici <lb/>trianguli ad baſin, &</s> <s xml:id="echoid-s16196" xml:space="preserve">c. </s> <s xml:id="echoid-s16197" xml:space="preserve">Quod oſtendendum erat.</s> <s xml:id="echoid-s16198" xml:space="preserve"/> </p> <div xml:id="echoid-div1300" type="float" level="2" n="1"> <figure xlink:label="fig-469-01" xlink:href="fig-469-01a"> <image file="469-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/469-01"/> </figure> <note position="right" xlink:label="note-469-01" xlink:href="note-469-01a" xml:space="preserve">42. huius.</note> <note position="right" xlink:label="note-469-02" xlink:href="note-469-02a" xml:space="preserve"> <lb/>Sin. ang. BAD. # Sin. compl. ang. B. <lb/>Sinus totus. # Sin. cõpl. arcus AD. <lb/>Sin. ang. DAC. # Sin. compl. ang C. <lb/></note> </div> </div> <div xml:id="echoid-div1302" type="section" level="1" n="605"> <head xml:id="echoid-head640" xml:space="preserve">PROBL. 3. PROP. 62.</head> <p> <s xml:id="echoid-s16199" xml:space="preserve">DATIS omnibus angulis trianguli ſphærici <lb/>non rectanguli, omnes tres arcus efficere notos.</s> <s xml:id="echoid-s16200" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16201" xml:space="preserve">IN triangulo ſphærico non rectangulo ABC, <lb/> <anchor type="figure" xlink:label="fig-469-02a" xlink:href="fig-469-02"/> <anchor type="note" xlink:label="note-469-03a" xlink:href="note-469-03"/> dati ſint omnes anguli A, B, C: </s> <s xml:id="echoid-s16202" xml:space="preserve">ſintq́; </s> <s xml:id="echoid-s16203" xml:space="preserve">primum omnes <lb/>tres anguli inæquales. </s> <s xml:id="echoid-s16204" xml:space="preserve">Oportet ex his tres cius arcus <lb/>perſcrutari. </s> <s xml:id="echoid-s16205" xml:space="preserve">Quoniam nullus angulus ponitur rectus, <lb/>erunt ſaltem duo vel acuti, vel obtuſi: </s> <s xml:id="echoid-s16206" xml:space="preserve">ſint B, C, vel <lb/>ambo acuti, vel obtuſi, quicquid ſit de reliquo A, ȧ <lb/>quo ad arcum BC, arcus perpendicularis ducatur, qui <lb/>neceſſario cadet intra triangulum. </s> <s xml:id="echoid-s16207" xml:space="preserve">Et quia eſt, vt ſi-<lb/> <anchor type="note" xlink:label="note-469-04a" xlink:href="note-469-04"/> nus anguli BAD, ad ſinum anguli DAC, ita ſinus complementi anguli B, ad <pb o="458" file="470" n="470" rhead=""/> ſinum complementi anguli C: </s> <s xml:id="echoid-s16208" xml:space="preserve">proportio autem hæc poſterior data eſt in ſinu-<lb/>bus complementorum angulorum B, C, datorum; </s> <s xml:id="echoid-s16209" xml:space="preserve">erit quoque proportio ſi-<lb/>nus anguli BAD, ad ſinum anguli DAC, data, nempe in ſinubus complemen <lb/>torum angulorum B,C: </s> <s xml:id="echoid-s16210" xml:space="preserve">Sed & </s> <s xml:id="echoid-s16211" xml:space="preserve">aggregatum eorun-<lb/> <anchor type="figure" xlink:label="fig-470-01a" xlink:href="fig-470-01"/> dem duorum angulorum BAD, DAC, datum eſt, & </s> <s xml:id="echoid-s16212" xml:space="preserve"><lb/>minus ſemicirculo, nempe totus angulus BAC, qui <lb/>duobus rectis minor eſt. </s> <s xml:id="echoid-s16213" xml:space="preserve">Sigillatim igitur vterque an-<lb/>gulorum BAD, DAC, cognitus erit. </s> <s xml:id="echoid-s16214" xml:space="preserve">Quoniam ergo <lb/> <anchor type="note" xlink:label="note-470-01a" xlink:href="note-470-01"/> in triangulo ABD, cuius angulus D, rectus, dati ſunt <lb/>duo anguli non recti B, & </s> <s xml:id="echoid-s16215" xml:space="preserve">BAD; </s> <s xml:id="echoid-s16216" xml:space="preserve">dabitur quoque ar-<lb/> <anchor type="note" xlink:label="note-470-02a" xlink:href="note-470-02"/> cus AB, recto angulo oppoſitus. </s> <s xml:id="echoid-s16217" xml:space="preserve">Hinc, quia in eo-<lb/>dem triangulo ABD, angulum habente rectum D, co <lb/>gnitus eſt arcus AB, recto angulo oppoſitus, & </s> <s xml:id="echoid-s16218" xml:space="preserve">inſu-<lb/> <anchor type="note" xlink:label="note-470-03a" xlink:href="note-470-03"/> per angulus non rectus BAD: <lb/></s> <s xml:id="echoid-s16219" xml:space="preserve">VEL certe, quoniam dati ſunt duo anguli non <lb/> <anchor type="note" xlink:label="note-470-04a" xlink:href="note-470-04"/> recti B, & </s> <s xml:id="echoid-s16220" xml:space="preserve">BAD; <lb/></s> <s xml:id="echoid-s16221" xml:space="preserve">notus quoque fiet, ex ſcholijs in margine citatis, arcus BD, circa angulum re-<lb/>ctum angulo BAD, oppoſitus. </s> <s xml:id="echoid-s16222" xml:space="preserve">Eadem ratione, quia in triangulo ACD, cu-<lb/>ius angulus D, rectus, dati ſunt duo anguli non recti C, & </s> <s xml:id="echoid-s16223" xml:space="preserve">CAD; </s> <s xml:id="echoid-s16224" xml:space="preserve">dabitur quo-<lb/> <anchor type="note" xlink:label="note-470-05a" xlink:href="note-470-05"/> que arcus AC, angulo recto oppoſitus. </s> <s xml:id="echoid-s16225" xml:space="preserve">Hinc, quoniam in eodem triangulo <lb/>ACD, habente rectum angulum D, cognitus iam eſt arcus AC, recto angulo <lb/>oppoſitus, cum angulo non recto CAD: <lb/></s> <s xml:id="echoid-s16226" xml:space="preserve"> <anchor type="note" xlink:label="note-470-06a" xlink:href="note-470-06"/> AVT certe, quia datiſunt duo anguli non re-<lb/> <anchor type="note" xlink:label="note-470-07a" xlink:href="note-470-07"/> cti C, & </s> <s xml:id="echoid-s16227" xml:space="preserve">CAD; <lb/></s> <s xml:id="echoid-s16228" xml:space="preserve">cognoſcetur etiam, ex eiſdem ſcholijs in margine adductis, arcus CD, circa <lb/>angulum rectum angulo CAD, oppoſitus. </s> <s xml:id="echoid-s16229" xml:space="preserve">Atque ita iam duo arcus AB, AC, <lb/>cogniti ſunt: </s> <s xml:id="echoid-s16230" xml:space="preserve">Aggregatum vero duorum arcuum BD, CD, inuentorum ter-<lb/>tium arcum BC, notum etiam efficiet.</s> <s xml:id="echoid-s16231" xml:space="preserve"/> </p> <div xml:id="echoid-div1302" type="float" level="2" n="1"> <figure xlink:label="fig-469-02" xlink:href="fig-469-02a"> <image file="469-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/469-02"/> </figure> <note position="right" xlink:label="note-469-03" xlink:href="note-469-03a" xml:space="preserve">Quãdo on@ <lb/>nes tres an <lb/>guli ſunt <lb/>inæquales.</note> <note position="right" xlink:label="note-469-04" xlink:href="note-469-04a" xml:space="preserve">57. huius. <lb/>61. huius.</note> <figure xlink:label="fig-470-01" xlink:href="fig-470-01a"> <image file="470-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/470-01"/> </figure> <note position="left" xlink:label="note-470-01" xlink:href="note-470-01a" xml:space="preserve">@. triang. <lb/>rectil.</note> <note position="left" xlink:label="note-470-02" xlink:href="note-470-02a" xml:space="preserve">Schol. 50. <lb/>huius.</note> <note position="left" xlink:label="note-470-03" xlink:href="note-470-03a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="left" xlink:label="note-470-04" xlink:href="note-470-04a" xml:space="preserve">Schol. 42. <lb/>vel 52. huiꝰ.</note> <note position="left" xlink:label="note-470-05" xlink:href="note-470-05a" xml:space="preserve">Schol. 50. <lb/>huius.</note> <note position="left" xlink:label="note-470-06" xlink:href="note-470-06a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="left" xlink:label="note-470-07" xlink:href="note-470-07a" xml:space="preserve">Schol. 42. <lb/>vel 52. huiꝰ.</note> </div> <p> <s xml:id="echoid-s16232" xml:space="preserve">QVOD ſi quando alter angulorum ad A, nempe BAD, inuentus fuerit <lb/>rectus, cum & </s> <s xml:id="echoid-s16233" xml:space="preserve">D, rectus ſit, erit vterque arcus AB, BD, quadrans: </s> <s xml:id="echoid-s16234" xml:space="preserve">atque ita <lb/> <anchor type="note" xlink:label="note-470-08a" xlink:href="note-470-08"/> ſine vlla moleſtia inuenti erunt dicti arcus. </s> <s xml:id="echoid-s16235" xml:space="preserve">Pari ratione, ſi angulus CAD, <lb/>deprehenſus fuerit rectus, non autem BAD, (fieri enim non poteſt, vt vter-<lb/>que angulus ad A, rectus ſit, cum angulus BAD, duobus rectis ſit minor.) <lb/></s> <s xml:id="echoid-s16236" xml:space="preserve">erunt arcus AC, CD, quadrantes; </s> <s xml:id="echoid-s16237" xml:space="preserve">atque adeo noti, ſine alio labore.</s> <s xml:id="echoid-s16238" xml:space="preserve"/> </p> <div xml:id="echoid-div1303" type="float" level="2" n="2"> <note position="left" xlink:label="note-470-08" xlink:href="note-470-08a" xml:space="preserve">25. huius.</note> </div> <p style="it"> <s xml:id="echoid-s16239" xml:space="preserve">PRAXIS huius problematis, cum ex propoſ. </s> <s xml:id="echoid-s16240" xml:space="preserve">6. </s> <s xml:id="echoid-s16241" xml:space="preserve">triang. </s> <s xml:id="echoid-s16242" xml:space="preserve">rectil. </s> <s xml:id="echoid-s16243" xml:space="preserve">& </s> <s xml:id="echoid-s16244" xml:space="preserve">ex <lb/> <anchor type="note" xlink:label="note-470-09a" xlink:href="note-470-09"/> ſcholijs in margine ſcriptis petẽda ſit, nõ eſt, quòd hic pluribus explicetur. <lb/></s> <s xml:id="echoid-s16245" xml:space="preserve">Nam ſi statuãtur duo ſinus complementorum angulorum B, C, acutorum, <lb/>vel obtuſorũ, pro terminis proportionis ſinus anguli BAD, ad ſinũ angu-<lb/>li CAD, inueniemus vtrumq; </s> <s xml:id="echoid-s16246" xml:space="preserve">angulũ BAD, CAD, per primã, vel ſecun <lb/>dam praxim propoſ. </s> <s xml:id="echoid-s16247" xml:space="preserve">6. </s> <s xml:id="echoid-s16248" xml:space="preserve">triangulorum rectilineorum, quòd bæ expeditio-<lb/>res ſinò, quam tertia. </s> <s xml:id="echoid-s16249" xml:space="preserve">Nam licet propoſitio illa 6. </s> <s xml:id="echoid-s16250" xml:space="preserve">de arcubus, & </s> <s xml:id="echoid-s16251" xml:space="preserve">angulis <lb/> <anchor type="note" xlink:label="note-470-10a" xlink:href="note-470-10"/> rectilineis antum propoſita ſit, intelligẽda tamen etiã eſt de angulis ſphæ <lb/>ricis, cumillorum ſinus à ſinubus arcuum eorũdem angulorum non diſcre-<lb/>pent. </s> <s xml:id="echoid-s16252" xml:space="preserve">Innento antem vtroque angulo BAD, CAD, adhibenda erit pra-<lb/>xis problematis ſcholy propoſ. </s> <s xml:id="echoid-s16253" xml:space="preserve">50. </s> <s xml:id="echoid-s16254" xml:space="preserve">huius, vt tam arcus AB, recto angu- <pb o="459" file="471" n="471" rhead=""/> lo D, intriangulo ABD, oppoſitus, quam arcus AC, angulo recto D, in <lb/>triangulo ACD, oppoſitus inueniatur. </s> <s xml:id="echoid-s16255" xml:space="preserve">Poſtremo adducenda est praxis <lb/>problematis 2. </s> <s xml:id="echoid-s16256" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16257" xml:space="preserve">41. </s> <s xml:id="echoid-s16258" xml:space="preserve">vel problematis 1. </s> <s xml:id="echoid-s16259" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16260" xml:space="preserve">42. <lb/></s> <s xml:id="echoid-s16261" xml:space="preserve">vel certe praxis ſcholij propoſ. </s> <s xml:id="echoid-s16262" xml:space="preserve">52. </s> <s xml:id="echoid-s16263" xml:space="preserve">ad eruendum tam arcum BD, angulo <lb/>non recto BAD, in triangulo ABD, oppoſitum, quam arcum CD, an-<lb/>gulo non recto CAD, oppoſitum in triangulo ACD.</s> <s xml:id="echoid-s16264" xml:space="preserve"/> </p> <div xml:id="echoid-div1304" type="float" level="2" n="3"> <note position="left" xlink:label="note-470-09" xlink:href="note-470-09a" xml:space="preserve">Praxis, quã <lb/>do omnes <lb/>tres dati an <lb/>guli inæ-<lb/>quales sũt.</note> <note position="left" xlink:label="note-470-10" xlink:href="note-470-10a" xml:space="preserve">Pro@oſitio <lb/>6. triag. re-<lb/>ctil. intelli-<lb/>genda eti á <lb/>eſt de angu <lb/>lis ſphæri-<lb/>cis.</note> </div> <p style="it"> <s xml:id="echoid-s16265" xml:space="preserve">QVOD ſi in hoc problemate enodando ſolis ſinubus vti libeat, inue-<lb/> <anchor type="note" xlink:label="note-471-01a" xlink:href="note-471-01"/> niendus erit vterque angulus BAD, CAD, per praxim tertiam propoſ. <lb/></s> <s xml:id="echoid-s16266" xml:space="preserve">6. </s> <s xml:id="echoid-s16267" xml:space="preserve">triang. </s> <s xml:id="echoid-s16268" xml:space="preserve">rectil. </s> <s xml:id="echoid-s16269" xml:space="preserve">non autem per primam, vel ſecundam. </s> <s xml:id="echoid-s16270" xml:space="preserve">Deinde cx praxi <lb/>problematis 1. </s> <s xml:id="echoid-s16271" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16272" xml:space="preserve">42, huius, eliciendus tam arcus BD, angu-<lb/>lo non recto BAD, oppoſitus in triangulo ABD, quam arcus CD, an-<lb/>gulo nonrecto CAD, in triangulo ACD, oppoſitus. </s> <s xml:id="echoid-s16273" xml:space="preserve">Ad extremum, per <lb/>praxim problematis 3. </s> <s xml:id="echoid-s16274" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16275" xml:space="preserve">41. </s> <s xml:id="echoid-s16276" xml:space="preserve">inuestigandus tam arcus AB, <lb/>quam arcus AC, recto angulo D, quilibet in ſuo triangulo oppoſitus: </s> <s xml:id="echoid-s16277" xml:space="preserve">quia <lb/>præter inuentum arcum BD, & </s> <s xml:id="echoid-s16278" xml:space="preserve">oppoſitum angulum BAD; </s> <s xml:id="echoid-s16279" xml:space="preserve">necnon præ-<lb/>ter arcum inuentum CD, & </s> <s xml:id="echoid-s16280" xml:space="preserve">angulum CAD, oppoſitum, conſtat etiam <lb/> <anchor type="note" xlink:label="note-471-02a" xlink:href="note-471-02"/> ſpecies tam anguli B, quam anguli C, cum vterque datus ſit.</s> <s xml:id="echoid-s16281" xml:space="preserve"/> </p> <div xml:id="echoid-div1305" type="float" level="2" n="4"> <note position="right" xlink:label="note-471-01" xlink:href="note-471-01a" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quãdo om-<lb/>nes tres an <lb/>guli dati <lb/>ſunt inæ-<lb/>quales.</note> <note position="right" xlink:label="note-471-02" xlink:href="note-471-02a" xml:space="preserve">Quãdo oẽs <lb/>tres anguli <lb/>dati@, vel <lb/>duo ſalté, <lb/>sũc ęquales.</note> </div> <p> <s xml:id="echoid-s16282" xml:space="preserve">LONGE facilius fit hoc problema, quando omnes tres anguli dati, vel <lb/>duo ſaltem, ſunt æquales. </s> <s xml:id="echoid-s16283" xml:space="preserve">Nam ſi ſint duo v. </s> <s xml:id="echoid-s16284" xml:space="preserve">g. </s> <s xml:id="echoid-s16285" xml:space="preserve">anguli B, C, æquales, quic-<lb/>quid ſit de reliquo A; </s> <s xml:id="echoid-s16286" xml:space="preserve">erunt & </s> <s xml:id="echoid-s16287" xml:space="preserve">arcus AB, AC, æquales. </s> <s xml:id="echoid-s16288" xml:space="preserve">Et quoniam trian-<lb/> <anchor type="note" xlink:label="note-471-03a" xlink:href="note-471-03"/> gulum ABC, ponitur non rectangulum, erit vterque an-<lb/>gulorum æqualium B, C, vel acutus, vel obtuſus. </s> <s xml:id="echoid-s16289" xml:space="preserve">Quare <lb/> <anchor type="figure" xlink:label="fig-471-01a" xlink:href="fig-471-01"/> arcus perpendicularis AD, ex tertio angulo A, ad arcum <lb/>BC, demiſſus intra triangulum cadet. </s> <s xml:id="echoid-s16290" xml:space="preserve">Quia ergo triangu-<lb/> <anchor type="note" xlink:label="note-471-04a" xlink:href="note-471-04"/> la ABD, ACD, angulos ad D, rectos habent, & </s> <s xml:id="echoid-s16291" xml:space="preserve">angulos <lb/>B, C, non rectos, æquales; </s> <s xml:id="echoid-s16292" xml:space="preserve">necnon & </s> <s xml:id="echoid-s16293" xml:space="preserve">arcus AB, AC, rectis <lb/>angulis oppoſitos, æquales, vt oſtendimus; </s> <s xml:id="echoid-s16294" xml:space="preserve">erunt & </s> <s xml:id="echoid-s16295" xml:space="preserve">arcus <lb/>BD, CD, & </s> <s xml:id="echoid-s16296" xml:space="preserve">anguli BAD, CAD, æquales; </s> <s xml:id="echoid-s16297" xml:space="preserve">ac proinde <lb/> <anchor type="note" xlink:label="note-471-05a" xlink:href="note-471-05"/> vterque angulus BAD, CAD, cum dimidium ſit dati an-<lb/>guli BAC, notus erit. </s> <s xml:id="echoid-s16298" xml:space="preserve">Poſt hæc, quoniam in triangulo <lb/>ABD, rectum habente angulum D, datus eſt vterque an-<lb/>gulus non rectus B, & </s> <s xml:id="echoid-s16299" xml:space="preserve">BAD; </s> <s xml:id="echoid-s16300" xml:space="preserve">dabitur quoque arcus AB, recto angulo oppo-<lb/> <anchor type="note" xlink:label="note-471-06a" xlink:href="note-471-06"/> ſitus; </s> <s xml:id="echoid-s16301" xml:space="preserve">proptereaque & </s> <s xml:id="echoid-s16302" xml:space="preserve">illi æqualis AC, notus erit. </s> <s xml:id="echoid-s16303" xml:space="preserve">Atque ita iam duo arcus <lb/> <anchor type="note" xlink:label="note-471-07a" xlink:href="note-471-07"/> AB, AC, noti facti ſunt. </s> <s xml:id="echoid-s16304" xml:space="preserve">Rurſus quia in eodem triangulo ABD, dati ſunt <lb/>duo anguli non recti B, & </s> <s xml:id="echoid-s16305" xml:space="preserve">BAD: <lb/></s> <s xml:id="echoid-s16306" xml:space="preserve">VEL, quoniam datus eſt arcus AB, angulo recto op-<lb/> <anchor type="note" xlink:label="note-471-08a" xlink:href="note-471-08"/> poſitus, & </s> <s xml:id="echoid-s16307" xml:space="preserve">angulus non rectus B: <lb/></s> <s xml:id="echoid-s16308" xml:space="preserve">VEL denique, quia datus eſt arcus AB, recto angulo <lb/> <anchor type="note" xlink:label="note-471-09a" xlink:href="note-471-09"/> oppoſitus, cum angulo non recto BAD; <lb/></s> <s xml:id="echoid-s16309" xml:space="preserve">cognitus etiam erit arcus BD, circa angulum rectum: </s> <s xml:id="echoid-s16310" xml:space="preserve">qui duplicatus totum <lb/>tertium arcum BC, notum exhibebit. </s> <s xml:id="echoid-s16311" xml:space="preserve">Omnes ergo tres arcus, qui quæruntur, <lb/>noti effecti ſunt.</s> <s xml:id="echoid-s16312" xml:space="preserve"/> </p> <div xml:id="echoid-div1306" type="float" level="2" n="5"> <note position="right" xlink:label="note-471-03" xlink:href="note-471-03a" xml:space="preserve">9. huius.</note> <figure xlink:label="fig-471-01" xlink:href="fig-471-01a"> <image file="471-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/471-01"/> </figure> <note position="right" xlink:label="note-471-04" xlink:href="note-471-04a" xml:space="preserve">57. huius.</note> <note position="right" xlink:label="note-471-05" xlink:href="note-471-05a" xml:space="preserve">21. huius.</note> <note position="right" xlink:label="note-471-06" xlink:href="note-471-06a" xml:space="preserve">Schol. 50. <lb/>huius.</note> <note position="right" xlink:label="note-471-07" xlink:href="note-471-07a" xml:space="preserve">Schol. 42. <lb/>vel 52. huiꝰ.</note> <note position="right" xlink:label="note-471-08" xlink:href="note-471-08a" xml:space="preserve">Schol. 45. <lb/>huius.</note> <note position="right" xlink:label="note-471-09" xlink:href="note-471-09a" xml:space="preserve">Schol. 41. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s16313" xml:space="preserve">NON est obſcura praxis huius rei. </s> <s xml:id="echoid-s16314" xml:space="preserve">Pendet enim ex ſcholijs in mar- <pb o="460" file="472" n="472" rhead=""/> gine citatis. </s> <s xml:id="echoid-s16315" xml:space="preserve">AT ſi ſolis ſinubus quis vti velit, inquirendus erit per pro-<lb/> <anchor type="note" xlink:label="note-472-01a" xlink:href="note-472-01"/> blema 1. </s> <s xml:id="echoid-s16316" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16317" xml:space="preserve">42. </s> <s xml:id="echoid-s16318" xml:space="preserve">arcus BD, in triangulo ABD, in quo da-<lb/>tus est angulus B, & </s> <s xml:id="echoid-s16319" xml:space="preserve">angulus BAD, nempe dimidium anguli dati BAC: <lb/></s> <s xml:id="echoid-s16320" xml:space="preserve">qui arcus BD, duplicatus dabit totum ar cum BC. </s> <s xml:id="echoid-s16321" xml:space="preserve">Deinde per problema <lb/>3. </s> <s xml:id="echoid-s16322" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16323" xml:space="preserve">41. </s> <s xml:id="echoid-s16324" xml:space="preserve">in eodem triangulo, in quo repertus eſt arcus BD, <lb/>& </s> <s xml:id="echoid-s16325" xml:space="preserve">angulus oppoſitus BAD, constat{quam} ſpecies alterius anguli non recti B, <lb/>dati, eliciendus erit arcus AB, angulo recto oppoſitus: </s> <s xml:id="echoid-s16326" xml:space="preserve">quo inuento, in-<lb/>uentus quoque erit ei æqualis AC.</s> <s xml:id="echoid-s16327" xml:space="preserve"/> </p> <div xml:id="echoid-div1307" type="float" level="2" n="6"> <note position="left" xlink:label="note-472-01" xlink:href="note-472-01a" xml:space="preserve">Praxis pet <lb/>folo@ ſinus, <lb/>quãdo om-<lb/>nes @res an <lb/>guli dati, <lb/>vel duo ſal <lb/>tem, ſunt <lb/>æquales.</note> </div> <p> <s xml:id="echoid-s16328" xml:space="preserve">DATIS igitur omnibus angulis trianguli ſphærici non rectanguli, om-<lb/>nes tres arcus effecimus notos. </s> <s xml:id="echoid-s16329" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s16330" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1309" type="section" level="1" n="606"> <head xml:id="echoid-head641" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s16331" xml:space="preserve">DIFFERT ergo, vt vides, ſphæricum triangulum non rectangulum à rectili-<lb/>neo non rectangulo; </s> <s xml:id="echoid-s16332" xml:space="preserve">quòd in ſphærico exſolis angulis datis inueniuntur omnes arcus, <lb/>vt in hoc problemate oſtenſum eſt; </s> <s xml:id="echoid-s16333" xml:space="preserve">in rectilineo vero ex datis ſolis angulis latera co-<lb/>gnoſci nequeunt, niſi vnum ſaltem latus etiam detur. </s> <s xml:id="echoid-s16334" xml:space="preserve">Cuius rei cauſa hæc eſt, quòd <lb/>duo triangula rect@linea ſimilia, quamuis latera vnius lateribus alterius valde ſint <lb/>inæqualia, ſingula ſingulis, angulos tamen habeant angulis aquales, ſingulos ſingu-<lb/>lis; </s> <s xml:id="echoid-s16335" xml:space="preserve">ita vt dari poſsint duo triangula rectilinea inter ſe quidem æquiangula, non ta-<lb/>men æquilatera: </s> <s xml:id="echoid-s16336" xml:space="preserve">At vero duo triangula ſphærica inter ſe æquiangula eſſe non poſ-<lb/> <anchor type="note" xlink:label="note-472-02a" xlink:href="note-472-02"/> ſunt, quin etiam æquilatera exiſtant. </s> <s xml:id="echoid-s16337" xml:space="preserve">Ex quo fit, in ſphærico triangulo ex datis an-<lb/>gulis dari etiam arcus, cum angulis determinati reſpondeant arcus; </s> <s xml:id="echoid-s16338" xml:space="preserve">in rectilineo ve <lb/>r<unsure/>o ex datis angulis latera dari non poſſe, cum angulis determinata latera non reſpon <lb/>deant, ſed poſsint eiſdem maiora, vel minora later a ſubtendi.</s> <s xml:id="echoid-s16339" xml:space="preserve"/> </p> <div xml:id="echoid-div1309" type="float" level="2" n="1"> <note position="left" xlink:label="note-472-02" xlink:href="note-472-02a" xml:space="preserve">39. huius.</note> </div> </div> <div xml:id="echoid-div1311" type="section" level="1" n="607"> <head xml:id="echoid-head642" xml:space="preserve">PROBL. 4. PROPOS. 63.</head> <p> <s xml:id="echoid-s16340" xml:space="preserve">DATIS omnibus arcubus trianguli ſphæri-<lb/>ci non rectanguli, omnes tres eius angulos inue-<lb/>ſtigare.</s> <s xml:id="echoid-s16341" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16342" xml:space="preserve">IN triangulo ſphærico non rectangulo ABC, dati ſint omnes tres arcus. <lb/></s> <s xml:id="echoid-s16343" xml:space="preserve">Oportet ex ipſis omnes tres angulos reperire. </s> <s xml:id="echoid-s16344" xml:space="preserve">Sit primo loco quærendus an-<lb/>gulus A: </s> <s xml:id="echoid-s16345" xml:space="preserve">Neque enim ſemper omnibus angulis indigemus; </s> <s xml:id="echoid-s16346" xml:space="preserve">ſed ſæpenumero <lb/>vnus, aut alter ex datis arcubus inquirendus eſt. </s> <s xml:id="echoid-s16347" xml:space="preserve">Aut igitur duo arcus AB, <lb/>AC, angulum A, qui quæritur, complectentes, ſunt inęquales, aut æquales: </s> <s xml:id="echoid-s16348" xml:space="preserve"><lb/>Si inæquales, aut ambo ſunt quadrante minores; </s> <s xml:id="echoid-s16349" xml:space="preserve">aut maiores; </s> <s xml:id="echoid-s16350" xml:space="preserve">aut vnus maior, <lb/>& </s> <s xml:id="echoid-s16351" xml:space="preserve">alter minor; </s> <s xml:id="echoid-s16352" xml:space="preserve">aut vnus quadrans, & </s> <s xml:id="echoid-s16353" xml:space="preserve">alter quadrante minor; </s> <s xml:id="echoid-s16354" xml:space="preserve">aut deniq; </s> <s xml:id="echoid-s16355" xml:space="preserve">vnus <lb/>quadrans, & </s> <s xml:id="echoid-s16356" xml:space="preserve">alter maior quadrante. </s> <s xml:id="echoid-s16357" xml:space="preserve">Neque enim ambo eſſe poſſunt quadran <lb/>tes: </s> <s xml:id="echoid-s16358" xml:space="preserve">quia duo anguli ipſis oppoſiti eſſent recti. </s> <s xml:id="echoid-s16359" xml:space="preserve">quod eſſet abſurdum, cum trian <lb/> <anchor type="note" xlink:label="note-472-03a" xlink:href="note-472-03"/> gulum ponatur non rectangulum. </s> <s xml:id="echoid-s16360" xml:space="preserve">Sint primum duo arcus AB, AC, inæqua-<lb/> <anchor type="note" xlink:label="note-472-04a" xlink:href="note-472-04"/> les, & </s> <s xml:id="echoid-s16361" xml:space="preserve">quadrante minores, quicquid ſit de arcu BC. </s> <s xml:id="echoid-s16362" xml:space="preserve">Productis arcubus AB, <lb/>AC, vt fiant quadrantes AD, AE, deſcribatur per D, E, arcus circuli maxi-<lb/>mi DE, occurrens arcui BC, in vtramuis partem producto in F: </s> <s xml:id="echoid-s16363" xml:space="preserve">Hortarer <pb o="461" file="473" n="473" rhead=""/> autem, vt produceretur verſus maiorem arcum, qui hic ſit AC. </s> <s xml:id="echoid-s16364" xml:space="preserve">Erunt au-<lb/> <anchor type="note" xlink:label="note-473-01a" xlink:href="note-473-01"/> tem anguli D, E, recti, ob quadrantes AD, AE. </s> <s xml:id="echoid-s16365" xml:space="preserve">Quoniam igitur duo maximi <lb/>circuli BF, DF, ſe interſecant in F, & </s> <s xml:id="echoid-s16366" xml:space="preserve">à pun-<lb/> <anchor type="figure" xlink:label="fig-473-01a" xlink:href="fig-473-01"/> ctis B, C, arcus BF, ad arcum DF, demiſsi <lb/>ſunt perpendiculares arcus BD, CE; </s> <s xml:id="echoid-s16367" xml:space="preserve">erit, vt <lb/>ſinus arcus BF, ad ſinum arcus BD, ita ſinus <lb/> <anchor type="note" xlink:label="note-473-02a" xlink:href="note-473-02"/> arcus CF, ad ſinum arcus CE: </s> <s xml:id="echoid-s16368" xml:space="preserve">Et permutan <lb/>do, vt ſinus arcus BF, ad ſinum arcus CF, ita <lb/>ſinus arcus BD, ad ſinum arcus CE. </s> <s xml:id="echoid-s16369" xml:space="preserve">Eſt au-<lb/>tem proportio ſinus arcus BD, ad ſinum ar-<lb/>cus CE, data, quòd arcus BD, CE, dati ſint, <lb/>vtpotè complementa datorum arcuum AB, <lb/>AC. </s> <s xml:id="echoid-s16370" xml:space="preserve">Igitur proportio ſinus arcus BF, ad ſi-<lb/>num arcus CF, data quoque erit, nempe in <lb/>ſinubus complementorum, arcuum datorum <lb/>AB, AC: </s> <s xml:id="echoid-s16371" xml:space="preserve">Sed & </s> <s xml:id="echoid-s16372" xml:space="preserve">eorundem arcuum BF, CF, <lb/>quorum ſinguli ſemicirculo minores ſunt, differentia data eſt, nempe arcus <lb/> <anchor type="note" xlink:label="note-473-03a" xlink:href="note-473-03"/> BC. </s> <s xml:id="echoid-s16373" xml:space="preserve">Vterque ergo arcus BF, CF, notus reddetur. </s> <s xml:id="echoid-s16374" xml:space="preserve">Itaque quoniam in trian <lb/> <anchor type="note" xlink:label="note-473-04a" xlink:href="note-473-04"/> gulo BFD, habente angulum D, rectum, datus eſt arcus BF, recto angulo op-<lb/>poſitus cum arcu BD, complemento videlicet arcus AB, dati; </s> <s xml:id="echoid-s16375" xml:space="preserve">cognitus erit <lb/> <anchor type="note" xlink:label="note-473-05a" xlink:href="note-473-05"/> & </s> <s xml:id="echoid-s16376" xml:space="preserve">tertius arcus DF. </s> <s xml:id="echoid-s16377" xml:space="preserve">Eadem ratione, cum in triangulo CFE, angulum ha-<lb/>bente rectum E, datus ſit arcus CF, angulo recto oppoſitus, cum arcu CE, <lb/>complemento nimirum arcus dati AC; </s> <s xml:id="echoid-s16378" xml:space="preserve">cognoſcetur, etiã tertius arcus EF: </s> <s xml:id="echoid-s16379" xml:space="preserve">qui <lb/> <anchor type="note" xlink:label="note-473-06a" xlink:href="note-473-06"/> ſubtractus ex inuento arcu DF, notum reddet arcum reliquum DE, anguli A; <lb/></s> <s xml:id="echoid-s16380" xml:space="preserve">ac proinde angulus A, cognitus erit. </s> <s xml:id="echoid-s16381" xml:space="preserve">Rurſus in triangulo priore BFD, cuius <lb/> <anchor type="note" xlink:label="note-473-07a" xlink:href="note-473-07"/> angulus D, rectus, cum datus ſit arcus BF, recto angulo oppoſitus, cum arcu <lb/>BD, complemento videlicet arcus dati AB: <lb/></s> <s xml:id="echoid-s16382" xml:space="preserve">VEL, cum duo arcus BD, DF, circa angulum re-<lb/> <anchor type="note" xlink:label="note-473-08a" xlink:href="note-473-08"/> ctum dati ſint: <lb/></s> <s xml:id="echoid-s16383" xml:space="preserve">AVT denique, cum datus ſit arcus BF, recto angu-<lb/> <anchor type="note" xlink:label="note-473-09a" xlink:href="note-473-09"/> lo oppoſitus, & </s> <s xml:id="echoid-s16384" xml:space="preserve">arcus DF; <lb/></s> <s xml:id="echoid-s16385" xml:space="preserve">inuenietur quoque, ex ſcholijs in margine citatis, angulus DBF: </s> <s xml:id="echoid-s16386" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s16387" xml:space="preserve"><lb/>reliquus duorum rectorum ABC, notus erit. </s> <s xml:id="echoid-s16388" xml:space="preserve">Eadem ratione, cum in poſte-<lb/> <anchor type="note" xlink:label="note-473-10a" xlink:href="note-473-10"/> riore triangulo CFE, angulum E, habente rectum, datus ſit arcus CF, angu-<lb/>lo recto oppoſitus, cum arcu CE, complemento nimirum arcus dati AC: <lb/></s> <s xml:id="echoid-s16389" xml:space="preserve">VEL, cum duo arcus CE, EF, circa rectum angu-<lb/> <anchor type="note" xlink:label="note-473-11a" xlink:href="note-473-11"/> lum dati ſint: <lb/></s> <s xml:id="echoid-s16390" xml:space="preserve">AVT denique, cum datus ſit arcus CF, recto angu-<lb/> <anchor type="note" xlink:label="note-473-12a" xlink:href="note-473-12"/> lo oppoſitus, & </s> <s xml:id="echoid-s16391" xml:space="preserve">inſuper arcus EF; <lb/></s> <s xml:id="echoid-s16392" xml:space="preserve">cognoſcetur etiam, ex ſcholijs in margine poſitis, angu-<lb/> <anchor type="figure" xlink:label="fig-473-02a" xlink:href="fig-473-02"/> lus ECF: </s> <s xml:id="echoid-s16393" xml:space="preserve">ideoq́ue & </s> <s xml:id="echoid-s16394" xml:space="preserve">angulus ACB, qui ei ad verticem <lb/> <anchor type="note" xlink:label="note-473-13a" xlink:href="note-473-13"/> æqualis eſt, notus erit. </s> <s xml:id="echoid-s16395" xml:space="preserve">Tres ergo anguli trianguli ABC, <lb/>omnes noti facti ſunt.</s> <s xml:id="echoid-s16396" xml:space="preserve"/> </p> <div xml:id="echoid-div1311" type="float" level="2" n="1"> <note position="left" xlink:label="note-472-03" xlink:href="note-472-03a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-472-04" xlink:href="note-472-04a" xml:space="preserve">Quuando <lb/>duo arcus <lb/>angulũ pri <lb/>mo loco in <lb/>ueniẽdum</note> <note position="right" xlink:label="note-473-01" xlink:href="note-473-01a" xml:space="preserve">comprehe<unsure/> <lb/>dẽtes ſunt <lb/>inæquales.</note> <figure xlink:label="fig-473-01" xlink:href="fig-473-01a"> <image file="473-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/473-01"/> </figure> <note position="right" xlink:label="note-473-02" xlink:href="note-473-02a" xml:space="preserve">40. huius.</note> <note position="right" xlink:label="note-473-03" xlink:href="note-473-03a" xml:space="preserve">2. huius.</note> <note position="right" xlink:label="note-473-04" xlink:href="note-473-04a" xml:space="preserve">7. triãg. re-<lb/>ctil.</note> <note position="right" xlink:label="note-473-05" xlink:href="note-473-05a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ.</note> <note position="right" xlink:label="note-473-06" xlink:href="note-473-06a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ.</note> <note position="right" xlink:label="note-473-07" xlink:href="note-473-07a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="right" xlink:label="note-473-08" xlink:href="note-473-08a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="right" xlink:label="note-473-09" xlink:href="note-473-09a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="right" xlink:label="note-473-10" xlink:href="note-473-10a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="right" xlink:label="note-473-11" xlink:href="note-473-11a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="right" xlink:label="note-473-12" xlink:href="note-473-12a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <figure xlink:label="fig-473-02" xlink:href="fig-473-02a"> <image file="473-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/473-02"/> </figure> <note position="right" xlink:label="note-473-13" xlink:href="note-473-13a" xml:space="preserve">6. huius.</note> </div> <p> <s xml:id="echoid-s16397" xml:space="preserve">SINT deinde duo arcus inæquales AB, AC, maio-<lb/>res quadrante. </s> <s xml:id="echoid-s16398" xml:space="preserve">Prodocantur, donec coeant in D. </s> <s xml:id="echoid-s16399" xml:space="preserve">Erunt <lb/>iu triangulo DBC, duo arcus DB, DC, quadrante mi-<lb/>nores, atq; </s> <s xml:id="echoid-s16400" xml:space="preserve">adeo noti, cum reliqui ſint ex arcubus ABD, <lb/>ACD, qui ſemicirculi ſunt. </s> <s xml:id="echoid-s16401" xml:space="preserve">Igitur, vt proxime demon-<lb/> <anchor type="note" xlink:label="note-473-14a" xlink:href="note-473-14"/> <pb o="462" file="474" n="474" rhead=""/> ſtrauimus, omnes eius tres anguli noti fient, ac proinde & </s> <s xml:id="echoid-s16402" xml:space="preserve">reliqui duorum re-<lb/>ctorum ABC, ACB, necnon & </s> <s xml:id="echoid-s16403" xml:space="preserve">angulus A, cum angulo D, ſit æqualis.</s> <s xml:id="echoid-s16404" xml:space="preserve"/> </p> <div xml:id="echoid-div1312" type="float" level="2" n="2"> <note position="right" xlink:label="note-473-14" xlink:href="note-473-14a" xml:space="preserve">11. 1 Theod.</note> </div> <note position="right" xml:space="preserve">13. huius.</note> <p> <s xml:id="echoid-s16405" xml:space="preserve">SIT tertio arcus quidem AB, quandrante minor, at AC, maior. </s> <s xml:id="echoid-s16406" xml:space="preserve">Produ-<lb/>cto arcu AB, vt fiat quadrans AD, & </s> <s xml:id="echoid-s16407" xml:space="preserve">reſecto quadrante AE, ex AC, vt in <lb/>prima harum figurarum, ducatur per D, E, arcus circuli maximi DE, ſecans <lb/>arcum BC, in F. </s> <s xml:id="echoid-s16408" xml:space="preserve">Eruntq́; </s> <s xml:id="echoid-s16409" xml:space="preserve">anguli D, E, recti, ob quadrantes AD, AE. </s> <s xml:id="echoid-s16410" xml:space="preserve">Quia <lb/> <anchor type="note" xlink:label="note-474-02a" xlink:href="note-474-02"/> ergo duo maximi circuli BC, DE, ſecant ſe-<lb/> <anchor type="figure" xlink:label="fig-474-01a" xlink:href="fig-474-01"/> ſe in F, & </s> <s xml:id="echoid-s16411" xml:space="preserve">à punctis B, C, arcus BC, ad arcum <lb/>DE, ducti ſunt arcus perpendiculares BD, <lb/> <anchor type="note" xlink:label="note-474-03a" xlink:href="note-474-03"/> CE; </s> <s xml:id="echoid-s16412" xml:space="preserve">erit, vt ſinus arcus BF, ad ſinum arcus <lb/>BD, ita ſinus arcus CF, ad ſinum arcus CE: <lb/></s> <s xml:id="echoid-s16413" xml:space="preserve">Et permutando, vt ſinus arcus BF, ad ſinum <lb/>arcus CF, ita ſinus arcus BD, ad ſinum ar-<lb/>cus CE. </s> <s xml:id="echoid-s16414" xml:space="preserve">Eſt autem proportio ſinus arcus BD, <lb/>ad ſinũ arcus CE, cognita, quod arcus BD, <lb/>CE, dati ſint, cũ ſint complemẽta datorũ ar-<lb/>cuũ AB, AC. </s> <s xml:id="echoid-s16415" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s16416" xml:space="preserve">proportio ſinus arcus <lb/>BF, ad ſinũ arcus CF, cognita erit, vtpote in <lb/>ſinubus complementorum arcuum AB, AC, <lb/>datorum: </s> <s xml:id="echoid-s16417" xml:space="preserve">Sed & </s> <s xml:id="echoid-s16418" xml:space="preserve">eorundem arcuum BF, CF, <lb/>aggregatum datum eſt, (nimirũ totus arcus BC.) </s> <s xml:id="echoid-s16419" xml:space="preserve">& </s> <s xml:id="echoid-s16420" xml:space="preserve">minus ſemicirculo; </s> <s xml:id="echoid-s16421" xml:space="preserve">quod <lb/>latus quodlibet trianguli ſphærici ſemicirculo ſit minus. </s> <s xml:id="echoid-s16422" xml:space="preserve">Igitur vterque arcus <lb/> <anchor type="note" xlink:label="note-474-04a" xlink:href="note-474-04"/> BF, CF, cognitus erit. </s> <s xml:id="echoid-s16423" xml:space="preserve">Quoniam ergo in triangulo BFD, cuius angulus D, <lb/> <anchor type="note" xlink:label="note-474-05a" xlink:href="note-474-05"/> rectus, datus eſt arcus BF, angulo recto oppoſitus, cum arcu BD, qui com-<lb/> <anchor type="note" xlink:label="note-474-06a" xlink:href="note-474-06"/> plementum eſt arcus AB, dati; </s> <s xml:id="echoid-s16424" xml:space="preserve">notus erit quoque tertius arcus DF. </s> <s xml:id="echoid-s16425" xml:space="preserve">Simili <lb/>modo, quia in triangulo CFE, rectum habente angulum E, datus eſt arcus <lb/>CF, angulo recto oppoſitus, & </s> <s xml:id="echoid-s16426" xml:space="preserve">arcus CE, complementum ſcilicet arcus AC; <lb/></s> <s xml:id="echoid-s16427" xml:space="preserve">reperietur quoque tertius arcus EF: </s> <s xml:id="echoid-s16428" xml:space="preserve">qui additus arcui DF, inuento, notum <lb/> <anchor type="note" xlink:label="note-474-07a" xlink:href="note-474-07"/> efficiet totum arcum DE, anguli A; </s> <s xml:id="echoid-s16429" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s16430" xml:space="preserve">angulus A, notus erit. </s> <s xml:id="echoid-s16431" xml:space="preserve">Rur-<lb/>ſus in triangulo priori BFD, cuius angulus D, rectus, quoniam datus eſt ar-<lb/>cus BF, recto angulo oppoſitus, & </s> <s xml:id="echoid-s16432" xml:space="preserve">arcus BD, complementum nimirum da-<lb/> <anchor type="note" xlink:label="note-474-08a" xlink:href="note-474-08"/> ti arcus AB: <lb/></s> <s xml:id="echoid-s16433" xml:space="preserve"> <anchor type="note" xlink:label="note-474-09a" xlink:href="note-474-09"/> AVT quia duo arcus BD, DF, circa rectum angu-<lb/>lum dati ſunt: <lb/></s> <s xml:id="echoid-s16434" xml:space="preserve">VEL certe, quia datus eſt arcus BF, recto angulo <lb/> <anchor type="note" xlink:label="note-474-10a" xlink:href="note-474-10"/> oppoſitus, cum arcu DF; <lb/></s> <s xml:id="echoid-s16435" xml:space="preserve">notus efficietur quoque angulus DBF, ex ſcholijs in margine adductis; </s> <s xml:id="echoid-s16436" xml:space="preserve">atque <lb/>adeo & </s> <s xml:id="echoid-s16437" xml:space="preserve">reliquus duorum rectorum ABC, notus erit. </s> <s xml:id="echoid-s16438" xml:space="preserve">Pari ratione, cum in po <lb/>ſteriori triangulo CFE, cuius angulus E, rectus, datus ſit arcus CF, recto an-<lb/> <anchor type="note" xlink:label="note-474-11a" xlink:href="note-474-11"/> gulo oppoſitus, cum arcu CE, complemento videlicet arcus AC, dati; <lb/></s> <s xml:id="echoid-s16439" xml:space="preserve">VEL cum duo arcus CE, EF, circa angulum re-<lb/> <anchor type="note" xlink:label="note-474-12a" xlink:href="note-474-12"/> ctum dati ſint: <lb/></s> <s xml:id="echoid-s16440" xml:space="preserve">VEL certe, cum datus ſit arcus CF, recto angulo <lb/> <anchor type="note" xlink:label="note-474-13a" xlink:href="note-474-13"/> oppoſitus, cum arcu EF; <lb/></s> <s xml:id="echoid-s16441" xml:space="preserve">dabitur etiam angulus C, per ſcholia in margine deſcripta. </s> <s xml:id="echoid-s16442" xml:space="preserve">Atque ita omnes <lb/>tres anguli ABC, noti facti ſunt.</s> <s xml:id="echoid-s16443" xml:space="preserve"/> </p> <div xml:id="echoid-div1313" type="float" level="2" n="3"> <note position="left" xlink:label="note-474-02" xlink:href="note-474-02a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-474-01" xlink:href="fig-474-01a"> <image file="474-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/474-01"/> </figure> <note position="left" xlink:label="note-474-03" xlink:href="note-474-03a" xml:space="preserve">40. huius.</note> <note position="left" xlink:label="note-474-04" xlink:href="note-474-04a" xml:space="preserve">2. huius.</note> <note position="left" xlink:label="note-474-05" xlink:href="note-474-05a" xml:space="preserve">6. triang. <lb/>rectil.</note> <note position="left" xlink:label="note-474-06" xlink:href="note-474-06a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ.</note> <note position="left" xlink:label="note-474-07" xlink:href="note-474-07a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ.</note> <note position="left" xlink:label="note-474-08" xlink:href="note-474-08a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="left" xlink:label="note-474-09" xlink:href="note-474-09a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ</note> <note position="left" xlink:label="note-474-10" xlink:href="note-474-10a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ</note> <note position="left" xlink:label="note-474-11" xlink:href="note-474-11a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="left" xlink:label="note-474-12" xlink:href="note-474-12a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ</note> <note position="left" xlink:label="note-474-13" xlink:href="note-474-13a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> </div> <p> <s xml:id="echoid-s16444" xml:space="preserve">SIT quarto arcus AB, quadrans, & </s> <s xml:id="echoid-s16445" xml:space="preserve">AC, minor, vt in poſteriore proxi-<lb/>marum figurarum. </s> <s xml:id="echoid-s16446" xml:space="preserve">Producto arcu AC, vt fiat quadrans AD, ducatur per B, <pb o="463" file="475" n="475" rhead=""/> D, arcus circuli maximi BD. </s> <s xml:id="echoid-s16447" xml:space="preserve">Etuntq́; </s> <s xml:id="echoid-s16448" xml:space="preserve">anguli ABD, & </s> <s xml:id="echoid-s16449" xml:space="preserve">D, recti, ob quadran-<lb/> <anchor type="note" xlink:label="note-475-01a" xlink:href="note-475-01"/> tes AB, AD. </s> <s xml:id="echoid-s16450" xml:space="preserve">Et quoniam in triangulo BCD, cuius angulus D, rectus, datus <lb/>eſt arcus BC, angulo recto oppoſitus, & </s> <s xml:id="echoid-s16451" xml:space="preserve">inſuper arcus CD, quippequicom-<lb/>plementum ſit dati arcus AC; </s> <s xml:id="echoid-s16452" xml:space="preserve">dabitur quoque arcus teriius BD, anguli A, <lb/> <anchor type="note" xlink:label="note-475-02a" xlink:href="note-475-02"/> ideoq́ue angulus A, notus erit. </s> <s xml:id="echoid-s16453" xml:space="preserve">Deinde quia in eodem triangulo BCD, haben <lb/>te angulum rectum D, datus eſt arcus BC, recto angulo oppoſitus, cum arcu <lb/> <anchor type="note" xlink:label="note-475-03a" xlink:href="note-475-03"/> CD, complemento ſcilicet arcus dati AC: <lb/></s> <s xml:id="echoid-s16454" xml:space="preserve">VEL, quia duo arcus BD, CD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-475-04a" xlink:href="note-475-04"/> ctum dati ſunt: <lb/></s> <s xml:id="echoid-s16455" xml:space="preserve">VEL certe, quoniam datus eſt arcus BC, recto an-<lb/> <anchor type="note" xlink:label="note-475-05a" xlink:href="note-475-05"/> gulo oppo ſitus, cum arcu BD; <lb/></s> <s xml:id="echoid-s16456" xml:space="preserve">inuenietur etiam ex ſcholijs notatis in margine, angulus BCD: </s> <s xml:id="echoid-s16457" xml:space="preserve">ac proinde <lb/>& </s> <s xml:id="echoid-s16458" xml:space="preserve">duorum rectorum reliquus ACB, notus erit. </s> <s xml:id="echoid-s16459" xml:space="preserve">Poſtremo, cum in eodem pro-<lb/>ximo triangulo BCD, angulum rectum habente D, datus ſit arcus BC, an-<lb/> <anchor type="note" xlink:label="note-475-06a" xlink:href="note-475-06"/> gulo recto oppoſitus, & </s> <s xml:id="echoid-s16460" xml:space="preserve">præterea arcus CD, complementnm videlicet dati ar-<lb/>cus AC: <lb/></s> <s xml:id="echoid-s16461" xml:space="preserve">AVT cum dati ſint duo arcus BD, CD, circa an-<lb/> <anchor type="note" xlink:label="note-475-07a" xlink:href="note-475-07"/> gulum rectum: <lb/></s> <s xml:id="echoid-s16462" xml:space="preserve">VEL cum datus ſit arcus BC, recto angulo oppo-<lb/> <anchor type="note" xlink:label="note-475-08a" xlink:href="note-475-08"/> ſitus, cum arcu BD; <lb/></s> <s xml:id="echoid-s16463" xml:space="preserve">AVT cum datus ſit angulus BCD, cum arcu CD, <lb/> <anchor type="note" xlink:label="note-475-09a" xlink:href="note-475-09"/> vel BD; </s> <s xml:id="echoid-s16464" xml:space="preserve">Nam quando datur arcus BD, conſtat de al-<lb/>tero arcu CD, circa rectum angulum@, cum datus ſit, <lb/>an ſit maior quadrante, vel minor: <lb/></s> <s xml:id="echoid-s16465" xml:space="preserve">VEL denique, quia datus eſt arcus BC, recto an-<lb/> <anchor type="note" xlink:label="note-475-10a" xlink:href="note-475-10"/> gulo oppoſitus, cum angulo BCD; <lb/></s> <s xml:id="echoid-s16466" xml:space="preserve">notus fiet quoque ex ſcholijs in margine citatis, angulus CBD; </s> <s xml:id="echoid-s16467" xml:space="preserve">atque adeo <lb/>& </s> <s xml:id="echoid-s16468" xml:space="preserve">eius complementum, angulus ſcilicet ABC, cognoſcetur, Omnes ergo tres <lb/>anguli trianguli ABC, cogniti ſunt.</s> <s xml:id="echoid-s16469" xml:space="preserve"/> </p> <div xml:id="echoid-div1314" type="float" level="2" n="4"> <note position="right" xlink:label="note-475-01" xlink:href="note-475-01a" xml:space="preserve">25. huius</note> <note position="right" xlink:label="note-475-02" xlink:href="note-475-02a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ.</note> <note position="right" xlink:label="note-475-03" xlink:href="note-475-03a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ</note> <note position="right" xlink:label="note-475-04" xlink:href="note-475-04a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="right" xlink:label="note-475-05" xlink:href="note-475-05a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="right" xlink:label="note-475-06" xlink:href="note-475-06a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="right" xlink:label="note-475-07" xlink:href="note-475-07a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="right" xlink:label="note-475-08" xlink:href="note-475-08a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="right" xlink:label="note-475-09" xlink:href="note-475-09a" xml:space="preserve">Schol. 42. <lb/>huius.</note> <note position="right" xlink:label="note-475-10" xlink:href="note-475-10a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s16470" xml:space="preserve">SIT quinto, & </s> <s xml:id="echoid-s16471" xml:space="preserve">vltimo arcus AC, quadrans, & </s> <s xml:id="echoid-s16472" xml:space="preserve">AB, maior, vt in eadem <lb/>poſteriore proximarum figurarum. </s> <s xml:id="echoid-s16473" xml:space="preserve">Abſciſſo quadrante AE, ex AB, ducatur <lb/>per C, E, arcus circuli maximi CE. </s> <s xml:id="echoid-s16474" xml:space="preserve">Eruntq́; </s> <s xml:id="echoid-s16475" xml:space="preserve">anguli ACE, & </s> <s xml:id="echoid-s16476" xml:space="preserve">E, recti, ob quadran <lb/> <anchor type="note" xlink:label="note-475-11a" xlink:href="note-475-11"/> tes AC, AE. </s> <s xml:id="echoid-s16477" xml:space="preserve">Quia ergo in triangulo BCE, angulum rectum habente E, da-<lb/>tus eſt arcus BC, recto angulo oppoſitus, & </s> <s xml:id="echoid-s16478" xml:space="preserve">præterea arcus BE, nempe com-<lb/>plementum dati arcus AB; </s> <s xml:id="echoid-s16479" xml:space="preserve">dabitur quoque tertius arcus CE, anguli A; </s> <s xml:id="echoid-s16480" xml:space="preserve">pro-<lb/> <anchor type="note" xlink:label="note-475-12a" xlink:href="note-475-12"/> indeq́; </s> <s xml:id="echoid-s16481" xml:space="preserve">& </s> <s xml:id="echoid-s16482" xml:space="preserve">angulus A, cognitus fiet. </s> <s xml:id="echoid-s16483" xml:space="preserve">Rurſus, cum in eodem triangulo BCE, <lb/>cuius angulus E, rectus, datus ſit arcus BC, recto angulo oppoſitus, cum ar-<lb/> <anchor type="note" xlink:label="note-475-13a" xlink:href="note-475-13"/> cu BE, complemento nimirum dati arcus AB: <lb/></s> <s xml:id="echoid-s16484" xml:space="preserve">AVT cum dati ſint duo arcus BE, CE, circa an-<lb/> <anchor type="note" xlink:label="note-475-14a" xlink:href="note-475-14"/> gulum rectum. <lb/></s> <s xml:id="echoid-s16485" xml:space="preserve">VEL denique, cum datus ſit arcus BC, angulo <lb/> <anchor type="note" xlink:label="note-475-15a" xlink:href="note-475-15"/> recto oppoſitus, cum arcu CE; <lb/></s> <s xml:id="echoid-s16486" xml:space="preserve">dabitur etiam angulus CBE, ex ſcholijs in margine adductis. </s> <s xml:id="echoid-s16487" xml:space="preserve">Denique quia <lb/>in triangulo eodem BCE, angulum rectum habente E, datus eſt arcus BC, <lb/> <anchor type="note" xlink:label="note-475-16a" xlink:href="note-475-16"/> angulo recto oppoſitus, & </s> <s xml:id="echoid-s16488" xml:space="preserve">arcus etiam BE, cum ſit complementum arcus <lb/>AB, dati: <lb/></s> <s xml:id="echoid-s16489" xml:space="preserve">VEL, quiæ duo arcus BE, CE, circa angulum re-<lb/> <anchor type="note" xlink:label="note-475-17a" xlink:href="note-475-17"/> ctum dati ſunt:</s> <s xml:id="echoid-s16490" xml:space="preserve"> <pb o="464" file="476" n="476" rhead=""/> AVT, quoniam notus eſt arcus BC, recto angulo <lb/> <anchor type="note" xlink:label="note-476-01a" xlink:href="note-476-01"/> oppoſitus, & </s> <s xml:id="echoid-s16491" xml:space="preserve">arcus CE: <lb/></s> <s xml:id="echoid-s16492" xml:space="preserve">VEL, quia datus eſt angulus CBE, & </s> <s xml:id="echoid-s16493" xml:space="preserve">arcus BE, <lb/> <anchor type="note" xlink:label="note-476-02a" xlink:href="note-476-02"/> vel CE; </s> <s xml:id="echoid-s16494" xml:space="preserve">Nam quando datur arcus CE, conſtat etiam, <lb/>an alter arcus BE, circa angulum rectum datus, ma-<lb/>ior ſit quadrante, vel minor: <lb/></s> <s xml:id="echoid-s16495" xml:space="preserve">AVT denique, quia notus eſt arcus BC, angulo <lb/> <anchor type="note" xlink:label="note-476-03a" xlink:href="note-476-03"/> recto oppoſitus, cum angulo CBE; <lb/></s> <s xml:id="echoid-s16496" xml:space="preserve">notus quoq; </s> <s xml:id="echoid-s16497" xml:space="preserve">ſiet, ex ſcholijs in margine citatis angulus BCE; </s> <s xml:id="echoid-s16498" xml:space="preserve">atque idcirco, <lb/>addito recto angulo ACE, totus angulus ACB, datus erit. </s> <s xml:id="echoid-s16499" xml:space="preserve">Rurſus ergo om <lb/>nes tres anguli trianguli ABC, inuenti ſunt.</s> <s xml:id="echoid-s16500" xml:space="preserve"/> </p> <div xml:id="echoid-div1315" type="float" level="2" n="5"> <note position="right" xlink:label="note-475-11" xlink:href="note-475-11a" xml:space="preserve">25. huius.</note> <note position="right" xlink:label="note-475-12" xlink:href="note-475-12a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ</note> <note position="right" xlink:label="note-475-13" xlink:href="note-475-13a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="right" xlink:label="note-475-14" xlink:href="note-475-14a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="right" xlink:label="note-475-15" xlink:href="note-475-15a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="right" xlink:label="note-475-16" xlink:href="note-475-16a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="right" xlink:label="note-475-17" xlink:href="note-475-17a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="left" xlink:label="note-476-01" xlink:href="note-476-01a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="left" xlink:label="note-476-02" xlink:href="note-476-02a" xml:space="preserve">Schol. 42. <lb/>huius.</note> <note position="left" xlink:label="note-476-03" xlink:href="note-476-03a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s16501" xml:space="preserve">PRAXIS huius problematis petenda eſt ex ſcholijs in margine ci-<lb/> <anchor type="note" xlink:label="note-476-04a" xlink:href="note-476-04"/> tatis. </s> <s xml:id="echoid-s16502" xml:space="preserve">Solum, vt cognoſcantur arcus BF, CF, in primo caſu, & </s> <s xml:id="echoid-s16503" xml:space="preserve">tertio, <lb/>ſtatuendi erunt ſinus complementorum arcuum datorum AB, AC, pro <lb/>terminis proportionis ſinus arcus BF, ad ſinum arcus CF, & </s> <s xml:id="echoid-s16504" xml:space="preserve">in primo <lb/>quidem caſu adhibenda vel prima praxis propoſ. </s> <s xml:id="echoid-s16505" xml:space="preserve">7. </s> <s xml:id="echoid-s16506" xml:space="preserve">triangulorum recti-<lb/>lineorum, vel aliqua ex alijs eiuſdem propoſ. </s> <s xml:id="echoid-s16507" xml:space="preserve">prout res exiget; </s> <s xml:id="echoid-s16508" xml:space="preserve">in tertio <lb/>vero caſu adducenda erit prima, vel ſecunda praxis propoſ. </s> <s xml:id="echoid-s16509" xml:space="preserve">6. </s> <s xml:id="echoid-s16510" xml:space="preserve">triangulo-<lb/>rum rectilineorum, &</s> <s xml:id="echoid-s16511" xml:space="preserve">c.</s> <s xml:id="echoid-s16512" xml:space="preserve"/> </p> <div xml:id="echoid-div1316" type="float" level="2" n="6"> <note position="left" xlink:label="note-476-04" xlink:href="note-476-04a" xml:space="preserve">Praxis, quã <lb/>do duo at-<lb/>cus quæſi-<lb/>tum angu-<lb/>lum conti-<lb/>nentes ſunt <lb/>inæquales.</note> </div> <p style="it"> <s xml:id="echoid-s16513" xml:space="preserve">QVOD ſi ſolis vti libeat ſinubus, inueſtigandi erunt arcus BF, CF, <lb/> <anchor type="note" xlink:label="note-476-05a" xlink:href="note-476-05"/> in primo caſu, per ſecundam praxim propoſ. </s> <s xml:id="echoid-s16514" xml:space="preserve">7. </s> <s xml:id="echoid-s16515" xml:space="preserve">triang. </s> <s xml:id="echoid-s16516" xml:space="preserve">rectil. </s> <s xml:id="echoid-s16517" xml:space="preserve">In tertio ve-<lb/>ro per praxim tertiam propoſ. </s> <s xml:id="echoid-s16518" xml:space="preserve">6. </s> <s xml:id="echoid-s16519" xml:space="preserve">Deinde in triangulo BFD, per praxim <lb/>ſcholij 1. </s> <s xml:id="echoid-s16520" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16521" xml:space="preserve">43. </s> <s xml:id="echoid-s16522" xml:space="preserve">eruendus arcus DF: </s> <s xml:id="echoid-s16523" xml:space="preserve">Et eodem modo in triangulo <lb/>CFE, arcus EF; </s> <s xml:id="echoid-s16524" xml:space="preserve">vt reliquus arcus DE, in primo caſu, vel totus arcus <lb/>DE, in tertio caſu habeatur, qui quidem eſt arcus anguli A. </s> <s xml:id="echoid-s16525" xml:space="preserve">Poſt hæc per <lb/>praxim problematis 1. </s> <s xml:id="echoid-s16526" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16527" xml:space="preserve">41. </s> <s xml:id="echoid-s16528" xml:space="preserve">inueniendus in triangulo BFD, <lb/>angulus DBF: </s> <s xml:id="echoid-s16529" xml:space="preserve">ex quo reliquus duorum rectorum ABC, notus fiet. </s> <s xml:id="echoid-s16530" xml:space="preserve">At-<lb/>que eodem pacto in triangulo CFE, eliciendus angulus ECF, ex quo in <lb/>primo caſu angulus quoque ACB, ad verticem cognitus erit.</s> <s xml:id="echoid-s16531" xml:space="preserve"/> </p> <div xml:id="echoid-div1317" type="float" level="2" n="7"> <note position="left" xlink:label="note-476-05" xlink:href="note-476-05a" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quando @ar <lb/>cus duo an <lb/>gulũ quæ-<lb/>ſitum am-<lb/>biẽtes ſunt <lb/>inæquales.</note> </div> <p style="it"> <s xml:id="echoid-s16532" xml:space="preserve">AT vero in quarto caſu ex praxi ſcholij 1. </s> <s xml:id="echoid-s16533" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16534" xml:space="preserve">43. </s> <s xml:id="echoid-s16535" xml:space="preserve">inueniendus <lb/>eſt arcus BD, anguli A, in triangulo BCD: </s> <s xml:id="echoid-s16536" xml:space="preserve">Et eodem modo in quinto ca <lb/>ſu arcus CE, anguli eiuſdem A, in triangulo BCE. </s> <s xml:id="echoid-s16537" xml:space="preserve">Deinde in quarto ca-<lb/>ſu, per praxim problematis 1. </s> <s xml:id="echoid-s16538" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16539" xml:space="preserve">41. </s> <s xml:id="echoid-s16540" xml:space="preserve">in triangulo BCD, inda-<lb/>gandus angulus BCD; </s> <s xml:id="echoid-s16541" xml:space="preserve">ex quo reliquus duorum rectorum ACB, notus <lb/>fiet: </s> <s xml:id="echoid-s16542" xml:space="preserve">Atque eadem ratione in quinto caſu, angulus EBC, in triangulo <lb/>BCE, inueniendus. </s> <s xml:id="echoid-s16543" xml:space="preserve">Ad extremum in quarto caſu, per praxim problema-<lb/>tis 2. </s> <s xml:id="echoid-s16544" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16545" xml:space="preserve">42. </s> <s xml:id="echoid-s16546" xml:space="preserve">in triangulo BCD, exquirendus angulus CBD; </s> <s xml:id="echoid-s16547" xml:space="preserve">ex quo <lb/>& </s> <s xml:id="echoid-s16548" xml:space="preserve">ABC, reliquus recti ABD, notus erit: </s> <s xml:id="echoid-s16549" xml:space="preserve">Et ſimiliter in quinto caſu, <lb/>eliciendus angulus BCE; </s> <s xml:id="echoid-s16550" xml:space="preserve">qui additus recto angulo ACE, totum angu-<lb/>lum ACB, notum exhibebit.</s> <s xml:id="echoid-s16551" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16552" xml:space="preserve">ALITER, & </s> <s xml:id="echoid-s16553" xml:space="preserve">multo breuius. </s> <s xml:id="echoid-s16554" xml:space="preserve">Sint rurſum dati tres arcus trianguli ABC, <lb/> <anchor type="note" xlink:label="note-476-06a" xlink:href="note-476-06"/> <pb o="465" file="477" n="477" rhead=""/> arcusq́; </s> <s xml:id="echoid-s16555" xml:space="preserve">AB, AC, angulum A, inquirendum continentes, inæquales. </s> <s xml:id="echoid-s16556" xml:space="preserve">Quo-<lb/> <anchor type="note" xlink:label="note-477-01a" xlink:href="note-477-01"/> niam igitur eſt, vt ſinus totus ad quantitatem quartam proportionalem ſinui <lb/>toti, & </s> <s xml:id="echoid-s16557" xml:space="preserve">duobus ſinubus arcuum AB, AC, inæqua-<lb/> <anchor type="figure" xlink:label="fig-477-01a" xlink:href="fig-477-01"/> lium, ita ſinus verſus anguli A, ad differentiam in-<lb/> <anchor type="note" xlink:label="note-477-02a" xlink:href="note-477-02"/> ter ſinum verſum arcus BC, angulo A, oppoſiti, & </s> <s xml:id="echoid-s16558" xml:space="preserve"><lb/>ſinum verſum arcus, quo ſe mutuo excedunt arcus <lb/>inæquales AB, AC: </s> <s xml:id="echoid-s16559" xml:space="preserve">Et conuertendo, vt dicta quan <lb/>titas quarta proportionalis ad ſinum totum, ita dif <lb/>ferentia inter ſinũ verſum arcus BC, & </s> <s xml:id="echoid-s16560" xml:space="preserve">ſinum ver-<lb/>ſum arcus, quo inter ſe arcus inæquales AB, AC, <lb/>differunt, ad ſinum verſum anguli A, quæſiti:</s> <s xml:id="echoid-s16561" xml:space="preserve"/> </p> <div xml:id="echoid-div1318" type="float" level="2" n="8"> <note position="left" xlink:label="note-476-06" xlink:href="note-476-06a" xml:space="preserve">Alia demõ <lb/>ftratio bre.</note> <note position="right" xlink:label="note-477-01" xlink:href="note-477-01a" xml:space="preserve">uior, & per <lb/>ſolos ſinus.</note> <figure xlink:label="fig-477-01" xlink:href="fig-477-01a"> <image file="477-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/477-01"/> </figure> <note position="right" xlink:label="note-477-02" xlink:href="note-477-02a" xml:space="preserve">Schol. 2. <lb/>58. huius.</note> </div> <note position="right" xml:space="preserve">Praxis, bre-<lb/>uior, & per <lb/>ſolos ſinus, <lb/>quãdo duo <lb/>arcus dati <lb/>quæſitũ an <lb/>gulum cõ-<lb/>prehenden <lb/>tes ſunt in-<lb/>æquales.</note> <p style="it"> <s xml:id="echoid-s16562" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum vtriuslibet arcuum inæqualium quæ-<lb/>ſitum angulum comprehendentium, it a ſinus alterius arcus circa eundem <lb/>angulum ad aliud, inuenietur numerus quartus proportionalis ſinui toti, <lb/>& </s> <s xml:id="echoid-s16563" xml:space="preserve">duobus ſinubus dictorum duorum arcuum. </s> <s xml:id="echoid-s16564" xml:space="preserve">Si ergo rurſum fiat, vt nu-<lb/>merus quartus proportionalis proxime inuentus ad ſinum totum, ita dif-<lb/>ferentia inter ſinum verſum arcus quæſito angulo oppoſiti, & </s> <s xml:id="echoid-s16565" xml:space="preserve">ſinum ver-<lb/>ſum arcus, quo duo arcus quæſitum angulum ambientes inter ſe differunt, <lb/>ad aliud, producetur ſinus verſus anguli, qui quæritur: </s> <s xml:id="echoid-s16566" xml:space="preserve">Ex quo arcum an-<lb/>guli quæſiti, atque adeo ipſum angulum, elicies, vt in explicatione, atque <lb/>vſu tabulæ Sinuum docuimus.</s> <s xml:id="echoid-s16567" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16568" xml:space="preserve">CAETERVM differentia inter ſinum verſum arcus quæſito angu <lb/> <anchor type="note" xlink:label="note-477-04a" xlink:href="note-477-04"/> lo oppoſiti, & </s> <s xml:id="echoid-s16569" xml:space="preserve">ſinum verſum differentiæ arcuum eundem angulum conti-<lb/>nentium, ita facile reperietur. </s> <s xml:id="echoid-s16570" xml:space="preserve">Quando arcus angulo quæſito oppoſitus <lb/>quadrante minor eſt, tetrahendus erit ſinus eius complementi ex ſinu com <lb/>plementi differĕtiæ arcuum quæſitum angulum ambientium. </s> <s xml:id="echoid-s16571" xml:space="preserve">Reliqua enim <lb/>erit differentia, quæinquiritur. </s> <s xml:id="echoid-s16572" xml:space="preserve">Id quod liquido conſtat ex figuris caſuum <lb/>1. </s> <s xml:id="echoid-s16573" xml:space="preserve">4. </s> <s xml:id="echoid-s16574" xml:space="preserve">7. </s> <s xml:id="echoid-s16575" xml:space="preserve">10. </s> <s xml:id="echoid-s16576" xml:space="preserve">& </s> <s xml:id="echoid-s16577" xml:space="preserve">13. </s> <s xml:id="echoid-s16578" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16579" xml:space="preserve">58. </s> <s xml:id="echoid-s16580" xml:space="preserve">Quando vero arcus quæſito angulo oppo-<lb/>ſitus quadrans eſt; </s> <s xml:id="echoid-s16581" xml:space="preserve">dabit ſinus complementi differentiæ arcuum angulum <lb/>quæſitum comprehendentium differentiam inter dictos ſinus verſos quæſi-<lb/>tam: </s> <s xml:id="echoid-s16582" xml:space="preserve">vt manifeſtum ex figuris caſuum 2. </s> <s xml:id="echoid-s16583" xml:space="preserve">5. </s> <s xml:id="echoid-s16584" xml:space="preserve">8. </s> <s xml:id="echoid-s16585" xml:space="preserve">11. </s> <s xml:id="echoid-s16586" xml:space="preserve">& </s> <s xml:id="echoid-s16587" xml:space="preserve">14. </s> <s xml:id="echoid-s16588" xml:space="preserve">eiuſdem pro-<lb/>poſ. </s> <s xml:id="echoid-s16589" xml:space="preserve">58. </s> <s xml:id="echoid-s16590" xml:space="preserve">Quendo denique arcus quæſito angulo oppoſitus quadrante maior <lb/>eſt; </s> <s xml:id="echoid-s16591" xml:space="preserve">adijciendus erit ſinus eius complementi ad ſinum complementi diffe-<lb/>rentiæ arcuum quæſitum angulum continentium. </s> <s xml:id="echoid-s16592" xml:space="preserve">Compoſitus namque nu-<lb/>merus erit differentia quæſita: </s> <s xml:id="echoid-s16593" xml:space="preserve">Vt facile apparere poteſt ex figuris caſuum <lb/>3. </s> <s xml:id="echoid-s16594" xml:space="preserve">6. </s> <s xml:id="echoid-s16595" xml:space="preserve">9. </s> <s xml:id="echoid-s16596" xml:space="preserve">12. </s> <s xml:id="echoid-s16597" xml:space="preserve">& </s> <s xml:id="echoid-s16598" xml:space="preserve">15. </s> <s xml:id="echoid-s16599" xml:space="preserve">eiuſdem propoſ, 58.</s> <s xml:id="echoid-s16600" xml:space="preserve"/> </p> <div xml:id="echoid-div1319" type="float" level="2" n="9"> <note position="right" xlink:label="note-477-04" xlink:href="note-477-04a" xml:space="preserve">Inuétio dif <lb/>ferentię in-<lb/>ter ſinum <lb/>verſum ar-<lb/>cus angulo <lb/>quæſito op <lb/>poſiti, & ſi-<lb/>nũ verſum <lb/>differentiæ <lb/>arcuũ eun-<lb/>dem angu-<lb/>lum ambi<unsure/>ẽ <lb/>tium.</note> </div> <p> <s xml:id="echoid-s16601" xml:space="preserve">EODEM modo inueſtigabimus angulos B, C, ſi arcus illos continentes <lb/>fuerint inæquales.</s> <s xml:id="echoid-s16602" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16603" xml:space="preserve">PORRO, inuento vno angulo, nullo fere negotio reliqui duo inueniren <lb/>tur, ſi conſtaret, qualis quiſque eorum ſit, acutuſne, an obtuſus. </s> <s xml:id="echoid-s16604" xml:space="preserve">Nam inuento <lb/>v. </s> <s xml:id="echoid-s16605" xml:space="preserve">g. </s> <s xml:id="echoid-s16606" xml:space="preserve">angulo A, ſi eſſet inueniendus angulus B, ſumeremus pro eius ſinu nume <lb/>rum, qui quartus proportionalis eſt ſinui arcus BC, inuento angulo A, op- <pb o="466" file="478" n="478" rhead=""/> poſiti; </s> <s xml:id="echoid-s16607" xml:space="preserve">ſinui anguli inuenti A; </s> <s xml:id="echoid-s16608" xml:space="preserve">& </s> <s xml:id="echoid-s16609" xml:space="preserve">ſinui arcus AC, quæſito angulo B, oppoſiti: <lb/></s> <s xml:id="echoid-s16610" xml:space="preserve">Siautem quærendus eſſet angulus C, acciperemus pro eius ſinu numerum, qui <lb/>quartus proportionalis eſt ſinui arcus BC, inuento angulo A, oppoſiti; </s> <s xml:id="echoid-s16611" xml:space="preserve">ſinui <lb/>anguli inuenti A; </s> <s xml:id="echoid-s16612" xml:space="preserve">& </s> <s xml:id="echoid-s16613" xml:space="preserve">ſinui arcus AB, angulo quæſito C, oppoſiti: </s> <s xml:id="echoid-s16614" xml:space="preserve">propterea <lb/>quòd eſt, vt ſinus arcus BC, ad ſinum anguli A, ita tam ſinus arcus AC, ad <lb/> <anchor type="note" xlink:label="note-478-01a" xlink:href="note-478-01"/> ſinum anguli B, quam ſinus arcus AB, ad ſinum anguli C. </s> <s xml:id="echoid-s16615" xml:space="preserve">Quocirca ſi con-<lb/>ſtaret, qualis ſit tam angulus B, quam angulus C, illico ex ſinu illo quarto <lb/>proportionali angulum quæſitum in tabula ſinuum reperi@emus.</s> <s xml:id="echoid-s16616" xml:space="preserve"/> </p> <div xml:id="echoid-div1320" type="float" level="2" n="10"> <note position="left" xlink:label="note-478-01" xlink:href="note-478-01a" xml:space="preserve">41. huius. <lb/>Quádo oés <lb/>tres arcus, <lb/>vel duo tã-<lb/>tũ angulũ <lb/>primo loco <lb/>inueſtigan <lb/>dum conti <lb/>nentes ſunt <lb/>æquales.</note> </div> <p> <s xml:id="echoid-s16617" xml:space="preserve">IAM vero ſi omnes tres arcus dati, vel duo tantum AB, AC, angulum <lb/>A, complectentes, æquales fint, quicquid ſit de reliquo arcu BC, longe faci-<lb/>lius angulum A, & </s> <s xml:id="echoid-s16618" xml:space="preserve">reliquos duos B, C, inquiremus. </s> <s xml:id="echoid-s16619" xml:space="preserve">Quoniam duo arcus AB, <lb/>AC, æquales ſunt, erunt & </s> <s xml:id="echoid-s16620" xml:space="preserve">duo anguli B, C, æquales inter ſe: </s> <s xml:id="echoid-s16621" xml:space="preserve">propterea quod <lb/>Iſoſcelium triangulorum ſphæricorum, qui ad baſin ſunt, anguli inter ſe ſunt <lb/> <anchor type="note" xlink:label="note-478-02a" xlink:href="note-478-02"/> æquales. </s> <s xml:id="echoid-s16622" xml:space="preserve">Cum ergo triangulum ABC, ponatur non rectangulum, neuter an-<lb/>gulorum B, C, rectus erit, ac proinde neuter arcuum AB, AC, quadrans: </s> <s xml:id="echoid-s16623" xml:space="preserve">quia <lb/> <anchor type="note" xlink:label="note-478-03a" xlink:href="note-478-03"/> alias duo anguli B, C, eſſent recti. </s> <s xml:id="echoid-s16624" xml:space="preserve">Erit igitur vterque angulus B, C, vel acu-<lb/>tus, vel obtuſus. </s> <s xml:id="echoid-s16625" xml:space="preserve">Demiſſus ergo ex A, ad arcum BC, ar-<lb/> <anchor type="figure" xlink:label="fig-478-01a" xlink:href="fig-478-01"/> cus perpendicularis AD, intra triangulum cadet. </s> <s xml:id="echoid-s16626" xml:space="preserve">Duo <lb/> <anchor type="note" xlink:label="note-478-04a" xlink:href="note-478-04"/> ergo triangula ABD, ACD, angulos ad D, rectos ha-<lb/>bent, & </s> <s xml:id="echoid-s16627" xml:space="preserve">angulos B, C, non rectos æquales, necnon & </s> <s xml:id="echoid-s16628" xml:space="preserve">ar-<lb/>cus AB, AC, rectis oppoſitos angulis æquales. </s> <s xml:id="echoid-s16629" xml:space="preserve">Quare & </s> <s xml:id="echoid-s16630" xml:space="preserve"><lb/>arcus BD, CD, & </s> <s xml:id="echoid-s16631" xml:space="preserve">anguliad A, inter ſe æquales erunt. <lb/></s> <s xml:id="echoid-s16632" xml:space="preserve"> <anchor type="note" xlink:label="note-478-05a" xlink:href="note-478-05"/> Itaque quoniam in triangulo ABD, cuius angulus <lb/>D, rectus eſt, arcus AB, recto angulo oppoſitus datus eſt, <lb/>& </s> <s xml:id="echoid-s16633" xml:space="preserve">præterea arcus BD, quippe qui dimidium ſit dati arcus <lb/>BC; </s> <s xml:id="echoid-s16634" xml:space="preserve">dabitur quoque angulus BAD, arcui BD, circa an-<lb/> <anchor type="note" xlink:label="note-478-06a" xlink:href="note-478-06"/> gulum rectum dato oppoſitus: </s> <s xml:id="echoid-s16635" xml:space="preserve">qui duplicatus totum an-<lb/>gulum quæſitum BAC, notum efficiet; </s> <s xml:id="echoid-s16636" xml:space="preserve">cum anguli ad A, oſtenſi ſint æqua-<lb/>les. </s> <s xml:id="echoid-s16637" xml:space="preserve">Rurſus, quia in eodem triangulo ABD, angulum rectum habente D, ar-<lb/>cus AB, angulo recto oppoſitus datus eſt, cum arcu BD, nempe cum dimi-<lb/> <anchor type="note" xlink:label="note-478-07a" xlink:href="note-478-07"/> dio dati arcus BC: <lb/></s> <s xml:id="echoid-s16638" xml:space="preserve">VEL, quia datus eſt angulus non rectus BAD, cum <lb/> <anchor type="note" xlink:label="note-478-08a" xlink:href="note-478-08"/> arcu oppoſito BD, circa angulum rectum; </s> <s xml:id="echoid-s16639" xml:space="preserve">conſtatq́; <lb/></s> <s xml:id="echoid-s16640" xml:space="preserve">præterea ſpecies reliqui anguli nõ recti B. </s> <s xml:id="echoid-s16641" xml:space="preserve">Nam ſi AB, <lb/>quadrãte minor ſit, erit angulus B, acutus, quemadmo-<lb/> <anchor type="note" xlink:label="note-478-09a" xlink:href="note-478-09"/> dum, & </s> <s xml:id="echoid-s16642" xml:space="preserve">BAD, acutus eſt: </s> <s xml:id="echoid-s16643" xml:space="preserve">Si vero AB, ſit maior quadran <lb/>te, erit idem angulus B, obtuſus, cum BAD acutus ſit: <lb/></s> <s xml:id="echoid-s16644" xml:space="preserve"> <anchor type="note" xlink:label="note-478-10a" xlink:href="note-478-10"/> VEL certe, quoniam datus eſt arcus AB, recto an-<lb/>gulo oppoſitus, cum angulo non recto BAD; <lb/></s> <s xml:id="echoid-s16645" xml:space="preserve">datus quoque erit angulus B; </s> <s xml:id="echoid-s16646" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s16647" xml:space="preserve">reliquus angulus C, ipſi B, æqua-<lb/>lis notus erit. </s> <s xml:id="echoid-s16648" xml:space="preserve">Atq; </s> <s xml:id="echoid-s16649" xml:space="preserve">ita omnes tres anguli in triangulo ABC, inuenti ſunt.</s> <s xml:id="echoid-s16650" xml:space="preserve"/> </p> <div xml:id="echoid-div1321" type="float" level="2" n="11"> <note position="left" xlink:label="note-478-02" xlink:href="note-478-02a" xml:space="preserve">8. huius.</note> <note position="left" xlink:label="note-478-03" xlink:href="note-478-03a" xml:space="preserve">25. huius.</note> <figure xlink:label="fig-478-01" xlink:href="fig-478-01a"> <image file="478-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/478-01"/> </figure> <note position="left" xlink:label="note-478-04" xlink:href="note-478-04a" xml:space="preserve">57. huius.</note> <note position="left" xlink:label="note-478-05" xlink:href="note-478-05a" xml:space="preserve">21. huius.</note> <note position="left" xlink:label="note-478-06" xlink:href="note-478-06a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="left" xlink:label="note-478-07" xlink:href="note-478-07a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="left" xlink:label="note-478-08" xlink:href="note-478-08a" xml:space="preserve">Schol. 42. <lb/>vel 56. huiꝰ.</note> <note position="left" xlink:label="note-478-09" xlink:href="note-478-09a" xml:space="preserve">38. huius.</note> <note position="left" xlink:label="note-478-10" xlink:href="note-478-10a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s16651" xml:space="preserve">QVANDO ergo duo arcus ſunt æquales, adhibenda erit praxis ſcho <lb/> <anchor type="note" xlink:label="note-478-11a" xlink:href="note-478-11"/> lij propoſ. </s> <s xml:id="echoid-s16652" xml:space="preserve">55. </s> <s xml:id="echoid-s16653" xml:space="preserve">vel problematis 1. </s> <s xml:id="echoid-s16654" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16655" xml:space="preserve">41. </s> <s xml:id="echoid-s16656" xml:space="preserve">vt ex altero arcuum æqua-<lb/>lium, & </s> <s xml:id="echoid-s16657" xml:space="preserve">ex dimidio tertij arcus eliciatur angulus, qui duplicatus angu-<lb/>lum tertio arcui oppoſitum exhibeat. </s> <s xml:id="echoid-s16658" xml:space="preserve">Deinde adhibenda praxis ſcholij <lb/>propoſ. </s> <s xml:id="echoid-s16659" xml:space="preserve">51. </s> <s xml:id="echoid-s16660" xml:space="preserve">vel 45. </s> <s xml:id="echoid-s16661" xml:space="preserve">vt ex eiſdem arcubus inueniatur alter angulorum æ-<lb/>qualium ſupratertium arcum. </s> <s xml:id="echoid-s16662" xml:space="preserve">Vel aduocanda praxis problematis 2. </s> <s xml:id="echoid-s16663" xml:space="preserve">ſcho <pb o="467" file="479" n="479" rhead=""/> lij propoſ. </s> <s xml:id="echoid-s16664" xml:space="preserve">42. </s> <s xml:id="echoid-s16665" xml:space="preserve">vel propoſ. </s> <s xml:id="echoid-s16666" xml:space="preserve">56. </s> <s xml:id="echoid-s16667" xml:space="preserve">aut certe praxis ſcholij propoſ. </s> <s xml:id="echoid-s16668" xml:space="preserve">47.</s> <s xml:id="echoid-s16669" xml:space="preserve"/> </p> <div xml:id="echoid-div1322" type="float" level="2" n="12"> <note position="left" xlink:label="note-478-11" xlink:href="note-478-11a" xml:space="preserve">Praxis, quã <lb/>do duo at-<lb/>cus quæſi-<lb/>tum angu-<lb/>lum amb@ẽ <lb/>tes ſunt æ-<lb/>quales.</note> </div> <note position="right" xml:space="preserve">Praxis pet <lb/>ſolos ſinus, <lb/>quãdo duo <lb/>arcus da@ũ <lb/>quæſitũ an <lb/>gulũ conti-<lb/>nentes ſunt <lb/>æquales.</note> <p style="it"> <s xml:id="echoid-s16670" xml:space="preserve">PER ſolos ſinus ita rem peragemus. </s> <s xml:id="echoid-s16671" xml:space="preserve">Ex praxi problematis 1. </s> <s xml:id="echoid-s16672" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s16673" xml:space="preserve">41. </s> <s xml:id="echoid-s16674" xml:space="preserve">inueniemus angulum BAD; </s> <s xml:id="echoid-s16675" xml:space="preserve">qui duplicatus totum BAC, dabit. </s> <s xml:id="echoid-s16676" xml:space="preserve">De-<lb/>inde per praxim problematis 2. </s> <s xml:id="echoid-s16677" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16678" xml:space="preserve">42. </s> <s xml:id="echoid-s16679" xml:space="preserve">reperiemus angulum <lb/>B, qui ipſi C, æqualis est.</s> <s xml:id="echoid-s16680" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1324" type="section" level="1" n="608"> <head xml:id="echoid-head643" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s16681" xml:space="preserve">IOANNES Regiom. </s> <s xml:id="echoid-s16682" xml:space="preserve">& </s> <s xml:id="echoid-s16683" xml:space="preserve">Nicolaus Copernicus alio etiam modo, datis omnibus <lb/>arcubus trianguli ſphærici, omnes tres angulos inquirunt, inueſtigantes nimirum <lb/>angulum quendam rectilineum in centro ſphæræ, cuius arcus angulum ſphæricum <lb/>quæſitum exhibet notum. </s> <s xml:id="echoid-s16684" xml:space="preserve">Sed eam rationem, quamuis acutam, & </s> <s xml:id="echoid-s16685" xml:space="preserve">ſubtilem, quoniam <lb/>obſcurior eſt, & </s> <s xml:id="echoid-s16686" xml:space="preserve">longior, dedita opera hic omiſimus: </s> <s xml:id="echoid-s16687" xml:space="preserve">præſertim, cum eam quilibev <lb/>apud Regiom. </s> <s xml:id="echoid-s16688" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16689" xml:space="preserve">34. </s> <s xml:id="echoid-s16690" xml:space="preserve">lib. </s> <s xml:id="echoid-s16691" xml:space="preserve">4. </s> <s xml:id="echoid-s16692" xml:space="preserve">triangulorum, & </s> <s xml:id="echoid-s16693" xml:space="preserve">apud Copernicum lib. </s> <s xml:id="echoid-s16694" xml:space="preserve">1. </s> <s xml:id="echoid-s16695" xml:space="preserve">Reuo-<lb/>lutionum propoſ 13. </s> <s xml:id="echoid-s16696" xml:space="preserve">de triangulis ſphæricis, legere poſsit.</s> <s xml:id="echoid-s16697" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16698" xml:space="preserve">MALVIMVS in ſecund@ demonſtratione huius problematis vſurpare theoremæ <lb/>ſcholij 2. </s> <s xml:id="echoid-s16699" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16700" xml:space="preserve">58. </s> <s xml:id="echoid-s16701" xml:space="preserve">quam cum Ioan. </s> <s xml:id="echoid-s16702" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s16703" xml:space="preserve">theorema eiuſdem propoſ. </s> <s xml:id="echoid-s16704" xml:space="preserve">58. </s> <s xml:id="echoid-s16705" xml:space="preserve">vt labo-<lb/>ris difficultatem effugeremus. </s> <s xml:id="echoid-s16706" xml:space="preserve">Nam cum ſit, vt rectangulum ſub ſinubus arcuum in-<lb/> <anchor type="note" xlink:label="note-479-02a" xlink:href="note-479-02"/> æqualium angulum quæſitum ambientium ad quadratum ſinus totius, ita differen-<lb/>tiainter ſinum verſum arcus eidem angulo oppoſiti, & </s> <s xml:id="echoid-s16707" xml:space="preserve">ſinum verſum differentiæ ar-<lb/>cuum illorum inæqualium, ad ſinum verſum anguli quæſiti: </s> <s xml:id="echoid-s16708" xml:space="preserve">ſi vellemus hoc theore-<lb/>mate propoſ. </s> <s xml:id="echoid-s16709" xml:space="preserve">58. </s> <s xml:id="echoid-s16710" xml:space="preserve">vti, obtineret rectangulum illud primum aureæ regulæ locum. </s> <s xml:id="echoid-s16711" xml:space="preserve">Qua-<lb/>re laborioſa redderetur diuiſio, vt patet. </s> <s xml:id="echoid-s16712" xml:space="preserve">Facilior autem fit diuiſio ſecundum theore-<lb/>ma ſcholij 2. </s> <s xml:id="echoid-s16713" xml:space="preserve">eiuſdem propoſ. </s> <s xml:id="echoid-s16714" xml:space="preserve">58. </s> <s xml:id="echoid-s16715" xml:space="preserve">cum primum locum aureæ regulæ quantitas quartæ <lb/>proportionalis occupet, quæ multo minor eſt illo rectangulo, facileq́; </s> <s xml:id="echoid-s16716" xml:space="preserve">inuenitur per <lb/>abiectionem ſolam tot figurarum ad dexteram ex eo rectangulo, quot cifræ in ſinu to <lb/>to cominentur; </s> <s xml:id="echoid-s16717" xml:space="preserve">propterea quod dictum rectangulum per ſinum totum ſit diuidendum, <lb/>vt illa quantitas quarta proportionalis producatur.</s> <s xml:id="echoid-s16718" xml:space="preserve"/> </p> <div xml:id="echoid-div1324" type="float" level="2" n="1"> <note position="right" xlink:label="note-479-02" xlink:href="note-479-02a" xml:space="preserve">58. huius. <lb/>& permu-<lb/>tando.</note> </div> </div> <div xml:id="echoid-div1326" type="section" level="1" n="609"> <head xml:id="echoid-head644" xml:space="preserve">PROBL. 5. PROPOS. 64.</head> <p> <s xml:id="echoid-s16719" xml:space="preserve">DATIS duobus arcubus trianguli ſphærici <lb/>non rectanguli, cum angulo ab ipſis comprehen-<lb/>ſo; </s> <s xml:id="echoid-s16720" xml:space="preserve">reliquum arcum, cum reliquis angulis reperire.</s> <s xml:id="echoid-s16721" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16722" xml:space="preserve">IN ſphærico triangulo ABC, non rectãgulo dati ſint duo arcus AB, BC, <lb/> <anchor type="note" xlink:label="note-479-03a" xlink:href="note-479-03"/> cum angulo B. </s> <s xml:id="echoid-s16723" xml:space="preserve">Oportet ex his & </s> <s xml:id="echoid-s16724" xml:space="preserve">reliquum arcum AC, & </s> <s xml:id="echoid-s16725" xml:space="preserve">reliquos angulos <lb/>BAC, & </s> <s xml:id="echoid-s16726" xml:space="preserve">ACB, exquirere. </s> <s xml:id="echoid-s16727" xml:space="preserve">Sint primum dati arcus inæquales, & </s> <s xml:id="echoid-s16728" xml:space="preserve">ex termino <lb/>vnius eorum, nempe ex termino A, arcus AB, ad alterum arcum BC, demit-<lb/>tatur arcus per pendicularis AD: </s> <s xml:id="echoid-s16729" xml:space="preserve">qu@an intra triangulum, an vero extra ca-<lb/>dat, calculus, & </s> <s xml:id="echoid-s16730" xml:space="preserve">operatio docebit. </s> <s xml:id="echoid-s16731" xml:space="preserve">Quoniam enim in triangulo ABD, cu-<lb/>ius angulus D, rectus, datus eſt arcus AB, recto angulo oppoſitus, cum an-<lb/>gulo B; </s> <s xml:id="echoid-s16732" xml:space="preserve">dabitur quoque arcus perpendicularis AD, dato angulo B, oppoſi-<lb/> <anchor type="note" xlink:label="note-479-04a" xlink:href="note-479-04"/> tus. </s> <s xml:id="echoid-s16733" xml:space="preserve">Rurſus, quia in eodem triangulo datus eſt arcus AB, recto angulo oppo-<lb/> <anchor type="note" xlink:label="note-479-05a" xlink:href="note-479-05"/> ſitus, cum augulo B:</s> <s xml:id="echoid-s16734" xml:space="preserve"> <pb o="468" file="480" n="480" rhead=""/> VEL, quia cognitus eſt arcus AD, & </s> <s xml:id="echoid-s16735" xml:space="preserve">præterea an-<lb/> <anchor type="note" xlink:label="note-480-01a" xlink:href="note-480-01"/> gulus B, datus: </s> <s xml:id="echoid-s16736" xml:space="preserve">conſtatq́; </s> <s xml:id="echoid-s16737" xml:space="preserve">ſpecies alterius arcus BD, <lb/>circa angulum rectum. </s> <s xml:id="echoid-s16738" xml:space="preserve">Nam quando arcus AB, eſt mi-<lb/>nor quadrante, ſi quidem & </s> <s xml:id="echoid-s16739" xml:space="preserve">inuentus AD, ſit minor, <lb/>erit & </s> <s xml:id="echoid-s16740" xml:space="preserve">BD, minor; </s> <s xml:id="echoid-s16741" xml:space="preserve">Si vero AD, ſit quadrante maior, erit <lb/> <anchor type="note" xlink:label="note-480-02a" xlink:href="note-480-02"/> & </s> <s xml:id="echoid-s16742" xml:space="preserve">BD, maior: </s> <s xml:id="echoid-s16743" xml:space="preserve">At ſi AB, eſt maior quadrante, ſi quidem <lb/>& </s> <s xml:id="echoid-s16744" xml:space="preserve">inuentus AD, ſit maior, erit BD, minor; </s> <s xml:id="echoid-s16745" xml:space="preserve">ſi autem <lb/>AD, ſit minor, erit BD, maior:</s> <s xml:id="echoid-s16746" xml:space="preserve"/> </p> <div xml:id="echoid-div1326" type="float" level="2" n="1"> <note position="right" xlink:label="note-479-03" xlink:href="note-479-03a" xml:space="preserve">Quãdo duo <lb/>arcus dati <lb/>inæquales <lb/>ſunr, & neu <lb/>t@r quadrãs</note> <note position="right" xlink:label="note-479-04" xlink:href="note-479-04a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="right" xlink:label="note-479-05" xlink:href="note-479-05a" xml:space="preserve">Schol. 45. <lb/>huius.</note> <note position="left" xlink:label="note-480-01" xlink:href="note-480-01a" xml:space="preserve">Schol. 49. <lb/>vel 44. huiꝰ.</note> <note position="left" xlink:label="note-480-02" xlink:href="note-480-02a" xml:space="preserve">36. huius.</note> </div> <p> <s xml:id="echoid-s16747" xml:space="preserve">VEL denique, quia datus eſt arcus AB, recto angu <lb/> <anchor type="note" xlink:label="note-480-03a" xlink:href="note-480-03"/> lo oppoſitus, & </s> <s xml:id="echoid-s16748" xml:space="preserve">arcus AD, circa rectum angulum; <lb/></s> <s xml:id="echoid-s16749" xml:space="preserve">inuenietur quoq;</s> <s xml:id="echoid-s16750" xml:space="preserve">, ex ſcholijs in margine ad-<lb/> <anchor type="figure" xlink:label="fig-480-01a" xlink:href="fig-480-01"/> ductis, arcus BD. </s> <s xml:id="echoid-s16751" xml:space="preserve">Si igitur arcus hic BD, in-<lb/>uentus fuerit minor dato arcu BC, argumen <lb/>to eſt, arcum perpendicularem AD, intra <lb/>triangulum cecidiſſe; </s> <s xml:id="echoid-s16752" xml:space="preserve">extra vero, ſi maior. </s> <s xml:id="echoid-s16753" xml:space="preserve">Et <lb/>quoniam ad vtramque partem arcus AB, du-<lb/>ci poteſt arcus perpendicularis ad BC, nos, <lb/>quãdo is extra triangulum cadit, eum in hac, <lb/>& </s> <s xml:id="echoid-s16754" xml:space="preserve">ſequentibus propoſitionibus eligimus, qui <lb/>angulum ABC, ſubtendit. </s> <s xml:id="echoid-s16755" xml:space="preserve">Iam ablato arcu <lb/>inuento BD, ſi minor eſt, quam datus arcus <lb/>BC, ex arcu BC; </s> <s xml:id="echoid-s16756" xml:space="preserve">vel ſi maior eſt, ſublato ar-<lb/>cu dato BC, ex inuento arcu BD, notus fiet <lb/>reliquus arcus CD. </s> <s xml:id="echoid-s16757" xml:space="preserve">Quare cum in triangulo ADC, angulum habente rectum <lb/>D, arcus duo AD, CD, circa rectum angulum cogniti ſint; </s> <s xml:id="echoid-s16758" xml:space="preserve">dabitur quoque ar-<lb/> <anchor type="note" xlink:label="note-480-04a" xlink:href="note-480-04"/> cus AC, recto angulo oppoſitus, qui in triangulo ABC, quærebatur. <lb/></s> <s xml:id="echoid-s16759" xml:space="preserve">POST hæc, quoniam in triangulo ABD, rectum <lb/> <anchor type="note" xlink:label="note-480-05a" xlink:href="note-480-05"/> habente angulum D, datus eſt arcus AB, recto angulo <lb/>oppoſitus, cum angulo B: <lb/></s> <s xml:id="echoid-s16760" xml:space="preserve">VEL, quia notus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-480-06a" xlink:href="note-480-06"/> ctum, cum angulo B, non recto; </s> <s xml:id="echoid-s16761" xml:space="preserve">conſtatq́; </s> <s xml:id="echoid-s16762" xml:space="preserve">præterea de <lb/>reliquo arcu BD, circa rectum angulum inuento, an <lb/>maior ſit quadrante, minorue. <lb/></s> <s xml:id="echoid-s16763" xml:space="preserve">AVT quia datus eſt arcus AB, recto angulo oppo-<lb/> <anchor type="note" xlink:label="note-480-07a" xlink:href="note-480-07"/> ſitus, & </s> <s xml:id="echoid-s16764" xml:space="preserve">inſuper arcus AD, circa rectum angulum: <lb/></s> <s xml:id="echoid-s16765" xml:space="preserve">VEL deniq;</s> <s xml:id="echoid-s16766" xml:space="preserve">, quia datus eſt arcus AB, oppoſitus an <lb/> <anchor type="note" xlink:label="note-480-08a" xlink:href="note-480-08"/> gulo recto, & </s> <s xml:id="echoid-s16767" xml:space="preserve">præterea arcus BD, circa rectum angulũ; <lb/></s> <s xml:id="echoid-s16768" xml:space="preserve">cognitus quoque erit, per ſcholia in margine notata, angulus BAD, Sic quo-<lb/>que, quia in triangulo ACD, rectum habente angulum D, datus eſt arcus <lb/> <anchor type="note" xlink:label="note-480-09a" xlink:href="note-480-09"/> AC, recto angulo oppoſitus, cum arcu AD, circa rectum angulum: <lb/></s> <s xml:id="echoid-s16769" xml:space="preserve">VEL certe, quia datur arcus AC, recto angulo op-<lb/> <anchor type="note" xlink:label="note-480-10a" xlink:href="note-480-10"/> poſitus, & </s> <s xml:id="echoid-s16770" xml:space="preserve">præterea arcus CD, circa angulum rectum; <lb/></s> <s xml:id="echoid-s16771" xml:space="preserve">notus efficietur etiam, per ſcholia in margine appoſita, angulus CAD. </s> <s xml:id="echoid-s16772" xml:space="preserve">Addi-<lb/>tus autem angulus CAD, proxime inuentus angulo BAD, nuper etiam in-<lb/>uento, quando arcus AD, intra triangulum cadit; </s> <s xml:id="echoid-s16773" xml:space="preserve">vel quando cadit extra, <lb/>ablatus angulus CAD, ex angulo BAD, notum efficiet angulum BAC, qui <lb/>in triangulo ABC, quærebatur.</s> <s xml:id="echoid-s16774" xml:space="preserve"/> </p> <div xml:id="echoid-div1327" type="float" level="2" n="2"> <note position="left" xlink:label="note-480-03" xlink:href="note-480-03a" xml:space="preserve">Schol. 53. <lb/>vel 43. huiꝰ.</note> <figure xlink:label="fig-480-01" xlink:href="fig-480-01a"> <image file="480-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/480-01"/> </figure> <note position="left" xlink:label="note-480-04" xlink:href="note-480-04a" xml:space="preserve">Schol. 1. <lb/>43. huius.</note> <note position="left" xlink:label="note-480-05" xlink:href="note-480-05a" xml:space="preserve">Schol. 47. <lb/>huius.</note> <note position="left" xlink:label="note-480-06" xlink:href="note-480-06a" xml:space="preserve">Schol. 56. <lb/>vel 42. huiꝰ.</note> <note position="left" xlink:label="note-480-07" xlink:href="note-480-07a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="left" xlink:label="note-480-08" xlink:href="note-480-08a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> <note position="left" xlink:label="note-480-09" xlink:href="note-480-09a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="left" xlink:label="note-480-10" xlink:href="note-480-10a" xml:space="preserve">Schol. 55. <lb/>vel 41. huiꝰ.</note> </div> <p> <s xml:id="echoid-s16775" xml:space="preserve">AD extremum, cum in triangulo ACD, rectum habente angulum D, da-<lb/> <anchor type="note" xlink:label="note-480-11a" xlink:href="note-480-11"/> <pb o="469" file="481" n="481" rhead=""/> tus ſit arcus AC, angulo recto oppoſitus, & </s> <s xml:id="echoid-s16776" xml:space="preserve">angulus CAD, iam inuentus: <lb/></s> <s xml:id="echoid-s16777" xml:space="preserve">VEL cum ſit cognitus arcus CD, circa angulum. </s> <s xml:id="echoid-s16778" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-481-01a" xlink:href="note-481-01"/> rectum, ac præterea angulus CAD; </s> <s xml:id="echoid-s16779" xml:space="preserve">conſtetq́; </s> <s xml:id="echoid-s16780" xml:space="preserve">de re-<lb/>liquo arcu AD, circa rectum angulum noto iam facto, <lb/>an minor quadrante ſit, an maior: <lb/></s> <s xml:id="echoid-s16781" xml:space="preserve">AVT, cum datus ſit arcus AC, angulo recto oppo-<lb/> <anchor type="note" xlink:label="note-481-02a" xlink:href="note-481-02"/> ſitus, cum arcu CD, circa angulum rectum: <lb/></s> <s xml:id="echoid-s16782" xml:space="preserve">VEL denique, quoniam notus eſt arcus AC, recto <lb/> <anchor type="note" xlink:label="note-481-03a" xlink:href="note-481-03"/> angulo oppoſitus, vnà cum arcu AD, circa angu-<lb/>lum rectum; <lb/></s> <s xml:id="echoid-s16783" xml:space="preserve">cognitus quoque fiet, per ſcholia in margine adducta, angulus ACD; </s> <s xml:id="echoid-s16784" xml:space="preserve">qui qui-<lb/>dem in priori triangulo, vbi arcus AD, intra triangulum cadit, quærebatur: </s> <s xml:id="echoid-s16785" xml:space="preserve"><lb/>in poſterioriautem, vbi arcus AD, extra triangulum cadit, idem angulus <lb/>ACD, ex duobus rectis ablatus, notum relinquit quæſitum angulum ACB. </s> <s xml:id="echoid-s16786" xml:space="preserve"><lb/>Atque ita inuentus eſt & </s> <s xml:id="echoid-s16787" xml:space="preserve">arcus reliquus AC, & </s> <s xml:id="echoid-s16788" xml:space="preserve">reliqui anguli BAC, ACB.</s> <s xml:id="echoid-s16789" xml:space="preserve"/> </p> <div xml:id="echoid-div1328" type="float" level="2" n="3"> <note position="left" xlink:label="note-480-11" xlink:href="note-480-11a" xml:space="preserve">Schol. 47. <lb/>huius.</note> <note position="right" xlink:label="note-481-01" xlink:href="note-481-01a" xml:space="preserve">Schol. 56. <lb/>vel 42. huiꝰ.</note> <note position="right" xlink:label="note-481-02" xlink:href="note-481-02a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="right" xlink:label="note-481-03" xlink:href="note-481-03a" xml:space="preserve">Schol. 55. <lb/>vel 41. hui ꝰ.</note> </div> <p> <s xml:id="echoid-s16790" xml:space="preserve">NVLLA porro ratione alteruter arcuum AD, BD, eſſe poteſt quadrans: <lb/></s> <s xml:id="echoid-s16791" xml:space="preserve">quia alias & </s> <s xml:id="echoid-s16792" xml:space="preserve">arcus AB, recto angulo oppoſitus quadrans foret: </s> <s xml:id="echoid-s16793" xml:space="preserve">quod eſt con <lb/> <anchor type="note" xlink:label="note-481-04a" xlink:href="note-481-04"/> tra hypotheſim.</s> <s xml:id="echoid-s16794" xml:space="preserve"/> </p> <div xml:id="echoid-div1329" type="float" level="2" n="4"> <note position="right" xlink:label="note-481-04" xlink:href="note-481-04a" xml:space="preserve">35. huius.</note> </div> <p> <s xml:id="echoid-s16795" xml:space="preserve">QVOD ſi quando arcus CD, deprehenſus fuerit quadrans; </s> <s xml:id="echoid-s16796" xml:space="preserve">erit & </s> <s xml:id="echoid-s16797" xml:space="preserve">arcus <lb/>AC, quæſitus, & </s> <s xml:id="echoid-s16798" xml:space="preserve">recto angulo oppoſitus, quadrans; </s> <s xml:id="echoid-s16799" xml:space="preserve">& </s> <s xml:id="echoid-s16800" xml:space="preserve">angulus CAD, rectus. <lb/></s> <s xml:id="echoid-s16801" xml:space="preserve"> <anchor type="note" xlink:label="note-481-05a" xlink:href="note-481-05"/> Atque ita ſine vllo labore inuentus erit & </s> <s xml:id="echoid-s16802" xml:space="preserve">arcus AC, qui quæritur, & </s> <s xml:id="echoid-s16803" xml:space="preserve">angulus <lb/> <anchor type="note" xlink:label="note-481-06a" xlink:href="note-481-06"/> CAD. </s> <s xml:id="echoid-s16804" xml:space="preserve">Reliqua reperientur, vt prius.</s> <s xml:id="echoid-s16805" xml:space="preserve"/> </p> <div xml:id="echoid-div1330" type="float" level="2" n="5"> <note position="right" xlink:label="note-481-05" xlink:href="note-481-05a" xml:space="preserve">35. huius.</note> <note position="right" xlink:label="note-481-06" xlink:href="note-481-06a" xml:space="preserve">34. huius.</note> </div> <p style="it"> <s xml:id="echoid-s16806" xml:space="preserve">PRAXIS ad enodandum hoc problema petenda eſt ex ſcholijs in <lb/>margine citatis.</s> <s xml:id="echoid-s16807" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16808" xml:space="preserve">VERVM per ſolos ſinus ita progrediendum erit. </s> <s xml:id="echoid-s16809" xml:space="preserve">Ex praxi proble-<lb/> <anchor type="note" xlink:label="note-481-07a" xlink:href="note-481-07"/> matis 2. </s> <s xml:id="echoid-s16810" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16811" xml:space="preserve">41. </s> <s xml:id="echoid-s16812" xml:space="preserve">inquirendus erit arcus AD.</s> <s xml:id="echoid-s16813" xml:space="preserve"/> </p> <div xml:id="echoid-div1331" type="float" level="2" n="6"> <note position="right" xlink:label="note-481-07" xlink:href="note-481-07a" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quãdo duo <lb/>dati arcus <lb/>ſunt inæ-<lb/>quales, & <lb/>neuter qua <lb/>drans.</note> </div> <p style="it"> <s xml:id="echoid-s16814" xml:space="preserve">DEINDE ex praxi ſcholij 1. </s> <s xml:id="echoid-s16815" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16816" xml:space="preserve">43. </s> <s xml:id="echoid-s16817" xml:space="preserve">arcus BD; </s> <s xml:id="echoid-s16818" xml:space="preserve">ex quo arcus CD, <lb/>notus efficietur, auferendo inuentum arcum BD, ex dato arcu BC, vel <lb/>datum arcum BC, ex ipſo inuento arcu BD, prout minor inuentus fuerit, <lb/>quam datus arcus BC, aut maior.</s> <s xml:id="echoid-s16819" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16820" xml:space="preserve">AD hæc, in triãgulo BAD, explorãdus erit angulus BAD, per praxim <lb/>problematis 1. </s> <s xml:id="echoid-s16821" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16822" xml:space="preserve">41. </s> <s xml:id="echoid-s16823" xml:space="preserve">vel per praxim problematis 2. </s> <s xml:id="echoid-s16824" xml:space="preserve">ſcholij propoſ. <lb/></s> <s xml:id="echoid-s16825" xml:space="preserve">42. </s> <s xml:id="echoid-s16826" xml:space="preserve">Similiter in triangulo ACD, eliciendus angulus CAD, ex praxi <lb/>problematis 1. </s> <s xml:id="echoid-s16827" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16828" xml:space="preserve">41. </s> <s xml:id="echoid-s16829" xml:space="preserve">Ex duobus autem angulis BAD, CAD, <lb/>inuentis notus euadet angulus BAC, trianguli propoſiti; </s> <s xml:id="echoid-s16830" xml:space="preserve">addendo ſcili-<lb/>cet vnum alteri, vt in prioritriangulo, vel auferendo angulum CAD, ex <lb/>angulo BAD, vt in triangulo poſteriori.</s> <s xml:id="echoid-s16831" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s16832" xml:space="preserve">PER praxim denique problematis 1. </s> <s xml:id="echoid-s16833" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16834" xml:space="preserve">41. </s> <s xml:id="echoid-s16835" xml:space="preserve">vel problema-<lb/>tis 2. </s> <s xml:id="echoid-s16836" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16837" xml:space="preserve">42. </s> <s xml:id="echoid-s16838" xml:space="preserve">in triangulo ACD, eodem indagandus angulus <lb/>ACD. </s> <s xml:id="echoid-s16839" xml:space="preserve">Hic enim in priori triangulo propoſito eſt quæſitus, in poſteriori <lb/>veroreliquus duorum rectorum eſt is, qui quæritur.</s> <s xml:id="echoid-s16840" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16841" xml:space="preserve">ALITER, & </s> <s xml:id="echoid-s16842" xml:space="preserve">quidem magis expeditè. </s> <s xml:id="echoid-s16843" xml:space="preserve">Sint rurſus in triangulo ABC, dati <lb/> <anchor type="note" xlink:label="note-481-08a" xlink:href="note-481-08"/> duo arcus inæquales AB, AC, cum angulo A. </s> <s xml:id="echoid-s16844" xml:space="preserve">Quoniam igitur eſt, vt ſinus <pb o="470" file="482" n="482" rhead=""/> totus ad quantitatcm quartam proportionalem ſinui toti, & </s> <s xml:id="echoid-s16845" xml:space="preserve">duobus ſinubus <lb/> <anchor type="note" xlink:label="note-482-01a" xlink:href="note-482-01"/> arcuum inæqualium AB, AC, ita ſinus verſus an-<lb/>guli A, ad difterétiam inter ſinum verſum arcus BC, <lb/> <anchor type="figure" xlink:label="fig-482-01a" xlink:href="fig-482-01"/> angulo A, oppoſiti, & </s> <s xml:id="echoid-s16846" xml:space="preserve">ſinum verſum differentiæ ar-<lb/>cuum AB, AC:</s> <s xml:id="echoid-s16847" xml:space="preserve"/> </p> <div xml:id="echoid-div1332" type="float" level="2" n="7"> <note position="right" xlink:label="note-481-08" xlink:href="note-481-08a" xml:space="preserve">Alia demõ <lb/>ſtratio bre-<lb/>uior.</note> <note position="left" xlink:label="note-482-01" xlink:href="note-482-01a" xml:space="preserve">chol. 58. <lb/>huius.</note> <figure xlink:label="fig-482-01" xlink:href="fig-482-01a"> <image file="482-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/482-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s16848" xml:space="preserve">SI fiat, vt ſinus totus ad ſinum vtriuslibet <lb/> <anchor type="note" xlink:label="note-482-02a" xlink:href="note-482-02"/> arcuum inæqualium datorum, ita ſinus alterius <lb/>arcus dati ad aliud, producetur numcrus quartus <lb/>proportionalis ſinui toti, & </s> <s xml:id="echoid-s16849" xml:space="preserve">duobus ſinubus dicto-<lb/>rum duorum arcuum. </s> <s xml:id="echoid-s16850" xml:space="preserve">Si ergo rurſus fiat, vt ſinus totus ad numerum quar <lb/>tum proportionalem proxime inuentum, ita ſinus verſus anguli A, dati <lb/>ad aliud, reperietur differentia inter ſinum verſum tertij arcus, qui quæ-<lb/>ritur, & </s> <s xml:id="echoid-s16851" xml:space="preserve">ſinum verſum differentiæ ar cuum datorum inæqualium. </s> <s xml:id="echoid-s16852" xml:space="preserve">Et quia <lb/>ſupra monſtrauimus, ſinum verſum tertij arcus maiorem ſemper eſſe ſinu <lb/> <anchor type="note" xlink:label="note-482-03a" xlink:href="note-482-03"/> verſo differentiæ duorum arcuum inæqualium; </s> <s xml:id="echoid-s16853" xml:space="preserve">ſi differentia nuper inuen-<lb/>ta adijciatur ad ſinum verſum differentiæ datorum arcuum inæqualium, <lb/>componetur ſinus verſus tertij arcus dato angulo oppoſiti, qui quæritur, <lb/>ex quo arcum ipſum eliciemus, vt in explicatione, atque vſu tabulæ ſinuum <lb/>dictum eſt. </s> <s xml:id="echoid-s16854" xml:space="preserve">Angulum porro C, inueniemus ex cognitis arcubus AC, CB; <lb/></s> <s xml:id="echoid-s16855" xml:space="preserve">& </s> <s xml:id="echoid-s16856" xml:space="preserve">angulum B, ex notis arcubus AB, BC, vt in praxi ſecundæ demon-<lb/>strationis præcedentis propoſ. </s> <s xml:id="echoid-s16857" xml:space="preserve">præcepimus, ſi duo arcus angulum quemli-<lb/>bet quæſitum continentes fuerint inæquales. </s> <s xml:id="echoid-s16858" xml:space="preserve">Nam ſi aliquando æquales <lb/>ſint, adhibenda erit praxis poſtremæ demonstrationis eiuſdem propoſitio-<lb/>nis antecedentis. </s> <s xml:id="echoid-s16859" xml:space="preserve">Quòd ſi ſciremus, an anguli B, C, ſint acuti, vel obtuſi, <lb/> <anchor type="note" xlink:label="note-482-04a" xlink:href="note-482-04"/> facili negotio, inuento arcu BC, ipſos inueniremus, vt ad finem ſecundæ <lb/>demonstrationis antecedentis propoſ. </s> <s xml:id="echoid-s16860" xml:space="preserve">monuimus.</s> <s xml:id="echoid-s16861" xml:space="preserve"/> </p> <div xml:id="echoid-div1333" type="float" level="2" n="8"> <note position="left" xlink:label="note-482-02" xlink:href="note-482-02a" xml:space="preserve">Praxis bre-<lb/>uior, per ſo <lb/>los ſinus, <lb/>quádo dati <lb/>duo arcus <lb/>sũt inæqua <lb/>les, & neu-<lb/>ter qua-<lb/>drans.</note> <note position="left" xlink:label="note-482-03" xlink:href="note-482-03a" xml:space="preserve">Schol. 1. <lb/>58. huius.</note> <note position="left" xlink:label="note-482-04" xlink:href="note-482-04a" xml:space="preserve">Quando <lb/>alter arcuũ <lb/>datorũ inę-<lb/>qualiũ eſt <lb/>quadrans.</note> </div> <p style="it"> <s xml:id="echoid-s16862" xml:space="preserve">QVOD ſi alter inæqualium arcuum datorũ ſit quadrans, nempe AB, <lb/>ducemus ab eius extremo A, ad alterum arcum BC, arcum perpendicu-<lb/>larem AD. </s> <s xml:id="echoid-s16863" xml:space="preserve">Eritq́ alter ſaltem arcuum AD, BD, quadrans quoque. </s> <s xml:id="echoid-s16864" xml:space="preserve">Non <lb/> <anchor type="note" xlink:label="note-482-05a" xlink:href="note-482-05"/> poteſt autem AD, eſſe quadrans; </s> <s xml:id="echoid-s16865" xml:space="preserve">quia alias, cum & </s> <s xml:id="echoid-s16866" xml:space="preserve">AB, quadrans po-<lb/>natur, eſſent anguli B, D, recti, atq; </s> <s xml:id="echoid-s16867" xml:space="preserve">adeo triangulum ABC, eſſet rectan-<lb/> <anchor type="note" xlink:label="note-482-06a" xlink:href="note-482-06"/> gulum. </s> <s xml:id="echoid-s16868" xml:space="preserve">quod non ponitur. </s> <s xml:id="echoid-s16869" xml:space="preserve">Erit ergo BD, quadrans, ideoq́ angulus oppo-<lb/> <anchor type="note" xlink:label="note-482-07a" xlink:href="note-482-07"/> ſitus BAD, rectus. </s> <s xml:id="echoid-s16870" xml:space="preserve">Polus quoq; </s> <s xml:id="echoid-s16871" xml:space="preserve">arcus AD, erit B, ob quadrantes AB, <lb/> <anchor type="note" xlink:label="note-482-08a" xlink:href="note-482-08"/> BD: </s> <s xml:id="echoid-s16872" xml:space="preserve">proptereaq́, arcus AD, ex angulo ipſo B, dato cognitus erit. </s> <s xml:id="echoid-s16873" xml:space="preserve">Atq; <lb/></s> <s xml:id="echoid-s16874" xml:space="preserve">ita duo arcus AD, BD, cum angulo BAD, facti erunt noti ſine vllo ne-<lb/>gotio multiplicationis. </s> <s xml:id="echoid-s16875" xml:space="preserve">Reliqua inuenientur, vt prius.</s> <s xml:id="echoid-s16876" xml:space="preserve"/> </p> <div xml:id="echoid-div1334" type="float" level="2" n="9"> <note position="left" xlink:label="note-482-05" xlink:href="note-482-05a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-482-06" xlink:href="note-482-06a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-482-07" xlink:href="note-482-07a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-482-08" xlink:href="note-482-08a" xml:space="preserve">26. huius.</note> </div> <p> <s xml:id="echoid-s16877" xml:space="preserve">SED ſint iam dati arcus AB, AC, datum angulum A, comprehendentes, <lb/> <anchor type="note" xlink:label="note-482-09a" xlink:href="note-482-09"/> æquales. </s> <s xml:id="echoid-s16878" xml:space="preserve">Erunt igitur duo anguli B, C, æquales, nempe vel acuti, vel obtuſi, <lb/>& </s> <s xml:id="echoid-s16879" xml:space="preserve">neuter arcuũ AB, AC, quadrans; </s> <s xml:id="echoid-s16880" xml:space="preserve">arcusq́; </s> <s xml:id="echoid-s16881" xml:space="preserve">perpendicularis AD, ex A, in BC, <lb/>demiſſus intra triangulum cadet, necnon & </s> <s xml:id="echoid-s16882" xml:space="preserve">arcus BD, CD, & </s> <s xml:id="echoid-s16883" xml:space="preserve">anguli ad A, <pb o="471" file="483" n="483" rhead=""/> æquales erunt, vt in vltima figura præcedentis propoſ. </s> <s xml:id="echoid-s16884" xml:space="preserve">oſtendimus: </s> <s xml:id="echoid-s16885" xml:space="preserve">ac proin-<lb/>de vterque angulus ad A, datus erit, cum dimidium ſit an-<lb/> <anchor type="figure" xlink:label="fig-483-01a" xlink:href="fig-483-01"/> guli BAC, dati. </s> <s xml:id="echoid-s16886" xml:space="preserve">Quoniam ergo in triangulo ABD, an-<lb/>gulum habente rectum D, datus eſt arcus AB, recto angu-<lb/>lo oppoſitus, cum angulo BAD, nimirum cum dimidio <lb/> <anchor type="note" xlink:label="note-483-01a" xlink:href="note-483-01"/> datianguli BAC; </s> <s xml:id="echoid-s16887" xml:space="preserve">cognitus erit arcus BD, dato angulo <lb/>BAD, oppoſitus: </s> <s xml:id="echoid-s16888" xml:space="preserve">qui duplicatus totum arcum BC, quæ-<lb/>ſitum reddet notum. </s> <s xml:id="echoid-s16889" xml:space="preserve">Rurſus quia in eodem triangulo ABD, <lb/> <anchor type="note" xlink:label="note-483-02a" xlink:href="note-483-02"/> rectum habente angulum D, datus eſt arcus AB, angulo <lb/>recto opppoſitus, cum arcu BD, circa angulum rectum: <lb/></s> <s xml:id="echoid-s16890" xml:space="preserve">VEL, quia datus eſt arcus AB, recto angulo op-<lb/> <anchor type="note" xlink:label="note-483-03a" xlink:href="note-483-03"/> poſitus, & </s> <s xml:id="echoid-s16891" xml:space="preserve">præterea angulus non rectus BAD: <lb/></s> <s xml:id="echoid-s16892" xml:space="preserve">VEL denique, quia datus eſt arcus BD, circa re-<lb/> <anchor type="note" xlink:label="note-483-04a" xlink:href="note-483-04"/> ctum angulum, vnà cum angulo non recto BAD, qui <lb/>dato arcui BD, opponitur, conſtatq́; </s> <s xml:id="echoid-s16893" xml:space="preserve">pręterea ſpecies <lb/>reliqui anguli non recti B. </s> <s xml:id="echoid-s16894" xml:space="preserve">Nam ſi AB, fuerit quadran <lb/>te minor, erit angulus B, acutus, ſicut & </s> <s xml:id="echoid-s16895" xml:space="preserve">BAD, acutus <lb/>eſt: </s> <s xml:id="echoid-s16896" xml:space="preserve">Si vero AB, maior quadrante extiterit, erit angu-<lb/>lus B, obtuſus, quandoquidem BAD, acutus eſt; <lb/></s> <s xml:id="echoid-s16897" xml:space="preserve">notus erit quoque, ex ſcholijs in margine adductis, angulus B; </s> <s xml:id="echoid-s16898" xml:space="preserve">ideoq́; </s> <s xml:id="echoid-s16899" xml:space="preserve">& </s> <s xml:id="echoid-s16900" xml:space="preserve">an-<lb/>gulus C, illi æqualis.</s> <s xml:id="echoid-s16901" xml:space="preserve"/> </p> <div xml:id="echoid-div1335" type="float" level="2" n="10"> <note position="left" xlink:label="note-482-09" xlink:href="note-482-09a" xml:space="preserve">Quãdoduo <lb/>arcus dati <lb/>funt æqua. <lb/>les.</note> <figure xlink:label="fig-483-01" xlink:href="fig-483-01a"> <image file="483-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/483-01"/> </figure> <note position="right" xlink:label="note-483-01" xlink:href="note-483-01a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="right" xlink:label="note-483-02" xlink:href="note-483-02a" xml:space="preserve">Schol. 51. <lb/>vel 45. huiꝰ.</note> <note position="right" xlink:label="note-483-03" xlink:href="note-483-03a" xml:space="preserve">Schol. 47. <lb/>huius.</note> <note position="right" xlink:label="note-483-04" xlink:href="note-483-04a" xml:space="preserve">Schol. 56. <lb/>vel 42. huiꝰ.</note> </div> <note position="right" xml:space="preserve">38. huius.</note> <p style="it"> <s xml:id="echoid-s16902" xml:space="preserve">PRAXIS petatur ex ſcholijs in margine adductis.</s> <s xml:id="echoid-s16903" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Praxis.</note> <p style="it"> <s xml:id="echoid-s16904" xml:space="preserve">SOLIS ſinubus ita vtemur. </s> <s xml:id="echoid-s16905" xml:space="preserve">Per praxim problematis 2. </s> <s xml:id="echoid-s16906" xml:space="preserve">ſcholij pro <lb/> <anchor type="note" xlink:label="note-483-07a" xlink:href="note-483-07"/> poſ. </s> <s xml:id="echoid-s16907" xml:space="preserve">41. </s> <s xml:id="echoid-s16908" xml:space="preserve">exquiremus arum BD; </s> <s xml:id="echoid-s16909" xml:space="preserve">qui duplicatus totum BC, qui quæritur, <lb/>dabit. </s> <s xml:id="echoid-s16910" xml:space="preserve">Deinde ex praxi problemat is 2. </s> <s xml:id="echoid-s16911" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s16912" xml:space="preserve">42. </s> <s xml:id="echoid-s16913" xml:space="preserve">quæremus an-<lb/>gulum B; </s> <s xml:id="echoid-s16914" xml:space="preserve">cui æqualis eſt alter angulus C.</s> <s xml:id="echoid-s16915" xml:space="preserve"/> </p> <div xml:id="echoid-div1336" type="float" level="2" n="11"> <note position="right" xlink:label="note-483-07" xlink:href="note-483-07a" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quádo dati <lb/>duo arcus <lb/>ſunt æqua-<lb/>les.</note> </div> <p> <s xml:id="echoid-s16916" xml:space="preserve">DATIS igitur duobus arcubus trianguli ſphærici non rectanguli, cum <lb/>angulo ab ipſis comprehenſo; </s> <s xml:id="echoid-s16917" xml:space="preserve">reliquum arcum, cum reliquis angulis reperi-<lb/>mus. </s> <s xml:id="echoid-s16918" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s16919" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1338" type="section" level="1" n="610"> <head xml:id="echoid-head645" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s16920" xml:space="preserve">HIC quoque potius vti voluimus theoremate ſcholij 2. </s> <s xml:id="echoid-s16921" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16922" xml:space="preserve">58 in demonſtra-<lb/>tione ſecunda huius problemat is, quam theoremate eiuſdem propoſ. </s> <s xml:id="echoid-s16923" xml:space="preserve">58. </s> <s xml:id="echoid-s16924" xml:space="preserve">vt praxis mi-<lb/>nus fieret laborioſa. </s> <s xml:id="echoid-s16925" xml:space="preserve">Nam cum ſit, vt quadratum ſinus totius ad rectangulum ſub ſi-<lb/> <anchor type="note" xlink:label="note-483-08a" xlink:href="note-483-08"/> nubus datorum arcuum inæqualium contentum, ita ſinus verſus anguli dati à dictis <lb/>arcubus comprehenſi ad differentiam inter ſinum verſum arcus dato angulo oppoſiti, <lb/>& </s> <s xml:id="echoid-s16926" xml:space="preserve">ſinum verſum differentiæ duorum arcuum datorum inæqualium: </s> <s xml:id="echoid-s16927" xml:space="preserve">ſi vellemus vti <lb/>hoc theoremate propoſ. </s> <s xml:id="echoid-s16928" xml:space="preserve">58. </s> <s xml:id="echoid-s16929" xml:space="preserve">moleſta redderetur multiplicatio in aurea regula, cum <lb/>ſinus verſus dati anguli multiplicandus eſſet per dictum rectangulum. </s> <s xml:id="echoid-s16930" xml:space="preserve">At in noſtra <lb/>praxi multo breuior fit muliiplicatio, vt patet, quamuis bis regulam auream adhi-<lb/>beamus.</s> <s xml:id="echoid-s16931" xml:space="preserve"/> </p> <div xml:id="echoid-div1338" type="float" level="2" n="1"> <note position="right" xlink:label="note-483-08" xlink:href="note-483-08a" xml:space="preserve">58. huius.</note> </div> </div> <div xml:id="echoid-div1340" type="section" level="1" n="611"> <head xml:id="echoid-head646" xml:space="preserve">PROBL. 6. PROP. 65.</head> <p> <s xml:id="echoid-s16932" xml:space="preserve">DATIS duobus angulis triáguli ſphærici non <pb o="472" file="484" n="484" rhead=""/> rectanguli, vnà cum arcu ipſis adiacente; </s> <s xml:id="echoid-s16933" xml:space="preserve">reliquos <lb/>arcus, cum reliquo angulo ſcrutari.</s> <s xml:id="echoid-s16934" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16935" xml:space="preserve">IN triangulo ſphærico ABC, non rectangulo dati ſint duo anguli B, & </s> <s xml:id="echoid-s16936" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-484-01a" xlink:href="note-484-01"/> BAC, cum arcu adiacente AB. </s> <s xml:id="echoid-s16937" xml:space="preserve">Oportet ex his reliquos arcus AC, BC, cum <lb/>reliquo angulo C, ſcrutari. </s> <s xml:id="echoid-s16938" xml:space="preserve">Sit prim um datus arcus AB, non quadrans, ſed <lb/>vel maior, vel minor quadrante, & </s> <s xml:id="echoid-s16939" xml:space="preserve">dati anguli B, BAC, inæquales, à quorum <lb/>vno, nempe à BAC, ad arcum oppoſitum BC, arcus perpendicularis demit-<lb/>tatur AD: </s> <s xml:id="echoid-s16940" xml:space="preserve">qui an intra triangulum, an ve-<lb/>ro extra cadat, calculus, atque operatio in-<lb/> <anchor type="figure" xlink:label="fig-484-01a" xlink:href="fig-484-01"/> dicabit. </s> <s xml:id="echoid-s16941" xml:space="preserve">Nam cum in triangulo ABD, rectũ <lb/>habente angulum D, datus ſit arcus AB, an-<lb/>gulo recto oppoſitus, & </s> <s xml:id="echoid-s16942" xml:space="preserve">angulus B; </s> <s xml:id="echoid-s16943" xml:space="preserve">dabitur <lb/>etiam angulus BAD: </s> <s xml:id="echoid-s16944" xml:space="preserve">qui ſi minor repertus <lb/> <anchor type="note" xlink:label="note-484-02a" xlink:href="note-484-02"/> fuerit dato angulo BAC, cadet arcus AD, <lb/>intra triangulum; </s> <s xml:id="echoid-s16945" xml:space="preserve">extra vero, ſi maior. </s> <s xml:id="echoid-s16946" xml:space="preserve">Iam <lb/>ablato angulo BAD, inuento, ſi minor eſt <lb/>dato angulo BAC, ex angulo BAC; </s> <s xml:id="echoid-s16947" xml:space="preserve">vel ſi <lb/>maior eſt, ſubducto angulo dato BAC, ex <lb/>inuento angulo BAD, notus euadet reliquus <lb/>angulus CAD.</s> <s xml:id="echoid-s16948" xml:space="preserve"/> </p> <div xml:id="echoid-div1340" type="float" level="2" n="1"> <note position="left" xlink:label="note-484-01" xlink:href="note-484-01a" xml:space="preserve">Quãdo duo <lb/>anguli dati <lb/>sũt inæqua <lb/>les, & arcus <lb/>adiacẽs da-<lb/>tus maior, <lb/>aut minor <lb/>quadrante.</note> <figure xlink:label="fig-484-01" xlink:href="fig-484-01a"> <image file="484-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/484-01"/> </figure> <note position="left" xlink:label="note-484-02" xlink:href="note-484-02a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s16949" xml:space="preserve">NVNQVAM vero inuentus angulus BAD, eſſe poteſt rectus: </s> <s xml:id="echoid-s16950" xml:space="preserve">quia <lb/>duo arcus AB, BD, eſſent quadrantes, ob angulos rectos BAD, ADB; </s> <s xml:id="echoid-s16951" xml:space="preserve">cum <lb/> <anchor type="note" xlink:label="note-484-03a" xlink:href="note-484-03"/> tamen AB, ponatur eſſe nõ quadrans: </s> <s xml:id="echoid-s16952" xml:space="preserve">ſed CAD, poterit aliquando eſſe rectus.</s> <s xml:id="echoid-s16953" xml:space="preserve"/> </p> <div xml:id="echoid-div1341" type="float" level="2" n="2"> <note position="left" xlink:label="note-484-03" xlink:href="note-484-03a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s16954" xml:space="preserve">RVRSVS, quia in eodem triangulo ABD, rectum habente angulum <lb/> <anchor type="note" xlink:label="note-484-04a" xlink:href="note-484-04"/> D, datus eſt arcus AB, angulo recto oppoſitus, & </s> <s xml:id="echoid-s16955" xml:space="preserve">angulus non rectus B: <lb/></s> <s xml:id="echoid-s16956" xml:space="preserve">VEL, quia notus eſt vterque angulus non rectus <lb/> <anchor type="note" xlink:label="note-484-05a" xlink:href="note-484-05"/> B, & </s> <s xml:id="echoid-s16957" xml:space="preserve">BAD: <lb/></s> <s xml:id="echoid-s16958" xml:space="preserve">AVT denique, quoniam datus eſt arcus AB, recto <lb/> <anchor type="note" xlink:label="note-484-06a" xlink:href="note-484-06"/> angulo oppoſitus, vna cum angulo non recto BAD; <lb/></s> <s xml:id="echoid-s16959" xml:space="preserve">cognoſcetur quoque, per ſcholia adducta in margine, arcus AD. </s> <s xml:id="echoid-s16960" xml:space="preserve">Eodemq́ue <lb/> <anchor type="note" xlink:label="note-484-07a" xlink:href="note-484-07"/> pacto, quia in eodem triangulo BAD, cuius angulus D, rectus, datus eſt ar-<lb/>cus AB, recto angulo oppoſitus, vna cum angulo BAD: <lb/></s> <s xml:id="echoid-s16961" xml:space="preserve">VEL, quia cognitus eſt vterque angulus non re-<lb/> <anchor type="note" xlink:label="note-484-08a" xlink:href="note-484-08"/> ctus B, & </s> <s xml:id="echoid-s16962" xml:space="preserve">BAD: <lb/></s> <s xml:id="echoid-s16963" xml:space="preserve">VEL, quoniam notus eſt arcus AB, angulo recto <lb/> <anchor type="note" xlink:label="note-484-09a" xlink:href="note-484-09"/> oppoſitus, vna cum angulo non recto B: <lb/></s> <s xml:id="echoid-s16964" xml:space="preserve">AVT, quia datus eſt arcus AB, angulo recto oppo <lb/> <anchor type="note" xlink:label="note-484-10a" xlink:href="note-484-10"/> ſitus, & </s> <s xml:id="echoid-s16965" xml:space="preserve">præterea arcus AD, circa rectum angulum: <lb/></s> <s xml:id="echoid-s16966" xml:space="preserve">VEL, quoniam notus eſt arcus AD, circa angu-<lb/> <anchor type="note" xlink:label="note-484-11a" xlink:href="note-484-11"/> lum rectum, vna cum angulo non recto B, ei oppoſito; <lb/></s> <s xml:id="echoid-s16967" xml:space="preserve">conſtatq́; </s> <s xml:id="echoid-s16968" xml:space="preserve">præterea, an alter arcus BD, circa rectum <lb/>angulum ſit maior, minorue quadrante. </s> <s xml:id="echoid-s16969" xml:space="preserve">Nam ſi inuen <lb/>tus angulus BAD, eſt acutus, erit arcus BD, quadran <lb/>te minor; </s> <s xml:id="echoid-s16970" xml:space="preserve">maior autem, ſi obtuſus: </s> <s xml:id="echoid-s16971" xml:space="preserve"><lb/>VEL denique, quoniam notus eſt arcus AD, circa <lb/> <anchor type="note" xlink:label="note-484-12a" xlink:href="note-484-12"/> rectum angum, & </s> <s xml:id="echoid-s16972" xml:space="preserve">præterea angulus non rectus BAD, <lb/>ci adiacens;</s> <s xml:id="echoid-s16973" xml:space="preserve"> <pb o="473" file="485" n="485" rhead=""/> cognoſcetur quoque, ex ſcholijs in margine citatis, arcus BD. </s> <s xml:id="echoid-s16974" xml:space="preserve">Præterea, quia <lb/>in triangulo ACD, habente rectum angulum D, cognitus eſt arcus AD, circa <lb/>angulum rectum, vnà cum angulo non recto CAD, ei adiacente; </s> <s xml:id="echoid-s16975" xml:space="preserve">inuenietur <lb/>quoque arcus AC, recto angulo oppoſitus. </s> <s xml:id="echoid-s16976" xml:space="preserve">Atque ita iam vnus reliquorum <lb/> <anchor type="note" xlink:label="note-485-01a" xlink:href="note-485-01"/> arcuum repertus eſt AC.</s> <s xml:id="echoid-s16977" xml:space="preserve"/> </p> <div xml:id="echoid-div1342" type="float" level="2" n="3"> <note position="left" xlink:label="note-484-04" xlink:href="note-484-04a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="left" xlink:label="note-484-05" xlink:href="note-484-05a" xml:space="preserve">Schol. 52. <lb/>vel 42. huiꝰ.</note> <note position="left" xlink:label="note-484-06" xlink:href="note-484-06a" xml:space="preserve">School. 45. <lb/>huius.</note> <note position="left" xlink:label="note-484-07" xlink:href="note-484-07a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="left" xlink:label="note-484-08" xlink:href="note-484-08a" xml:space="preserve">Schol. 52. <lb/>vel 42. huiꝰ.</note> <note position="left" xlink:label="note-484-09" xlink:href="note-484-09a" xml:space="preserve">Schol. 45. <lb/>huius.</note> <note position="left" xlink:label="note-484-10" xlink:href="note-484-10a" xml:space="preserve">Schol. 53. <lb/>ve 43. huiꝰ.</note> <note position="left" xlink:label="note-484-11" xlink:href="note-484-11a" xml:space="preserve">Schol. 49. <lb/>vel 44. huiꝰ.</note> <note position="left" xlink:label="note-484-12" xlink:href="note-484-12a" xml:space="preserve">Schol. 44. <lb/>huius.</note> <note position="right" xlink:label="note-485-01" xlink:href="note-485-01a" xml:space="preserve">Schol. 46. <lb/>vel 45. huiꝰ.</note> </div> <p> <s xml:id="echoid-s16978" xml:space="preserve">POST hæc, quoniam in eodem triangulo ACD, cuius angulus D, rectus, <lb/> <anchor type="note" xlink:label="note-485-02a" xlink:href="note-485-02"/> datus eſt arcus AC, recto angulo oppoſitus, vna cum angulo non recto CAD: <lb/></s> <s xml:id="echoid-s16979" xml:space="preserve">VEL, quia datus eſt arcus AC, recto angulo op-<lb/> <anchor type="note" xlink:label="note-485-03a" xlink:href="note-485-03"/> poſitus, & </s> <s xml:id="echoid-s16980" xml:space="preserve">præterea arcus AD, circa eundem angu-<lb/>lum rectum: <lb/></s> <s xml:id="echoid-s16981" xml:space="preserve">VEL denique, quia datus eſt arcus AD, circa angu <lb/> <anchor type="note" xlink:label="note-485-04a" xlink:href="note-485-04"/> lum rectum, vnà cum angulo non recto CAD, ei adia-<lb/>cente: <lb/></s> <s xml:id="echoid-s16982" xml:space="preserve">notus quoque fiet, ex ſcholijs in margine deſcriptis, arcus CD: </s> <s xml:id="echoid-s16983" xml:space="preserve">qui adiectus <lb/>arcui inuento BD, quando perpendicularis arcus AD, intra triangulum ca-<lb/>dit; </s> <s xml:id="echoid-s16984" xml:space="preserve">vel, quando extra cadit, ſublatus ex arcu inuento BD, notum exhibebit <lb/>arcum BC, qui eſt alter reliquorum arcuum, qui quæruntur.</s> <s xml:id="echoid-s16985" xml:space="preserve"/> </p> <div xml:id="echoid-div1343" type="float" level="2" n="4"> <note position="right" xlink:label="note-485-02" xlink:href="note-485-02a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="right" xlink:label="note-485-03" xlink:href="note-485-03a" xml:space="preserve">Schol. 43. <lb/>vel 53. huiꝰ.</note> <note position="right" xlink:label="note-485-04" xlink:href="note-485-04a" xml:space="preserve">Schol. 44. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s16986" xml:space="preserve">AD extremum in eodem triangulo ACD, quoniam datus eſt arcus AC, re-<lb/> <anchor type="note" xlink:label="note-485-05a" xlink:href="note-485-05"/> cto angulo oppoſitus, cum arcu AD, circa rectum angulum: <lb/></s> <s xml:id="echoid-s16987" xml:space="preserve">VEL, quia datus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-485-06a" xlink:href="note-485-06"/> ctum, & </s> <s xml:id="echoid-s16988" xml:space="preserve">angulus non rectus CAD, ei adiacens: <lb/></s> <s xml:id="echoid-s16989" xml:space="preserve">VEL, quia datus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-485-07a" xlink:href="note-485-07"/> ctum, & </s> <s xml:id="echoid-s16990" xml:space="preserve">angulus non rectus ACD, ei oppoſitus; </s> <s xml:id="echoid-s16991" xml:space="preserve">con-<lb/>ſtatq; </s> <s xml:id="echoid-s16992" xml:space="preserve">præterea, an reliquus arcus CD, circa rectum an-<lb/>gulum inuentus ſit maior quadrante, aut minor: <lb/></s> <s xml:id="echoid-s16993" xml:space="preserve">AVT, quia datus eſt vterque arcus AD, CD, cir-<lb/> <anchor type="note" xlink:label="note-485-08a" xlink:href="note-485-08"/> ca angulum rectum: <lb/></s> <s xml:id="echoid-s16994" xml:space="preserve">AVT, quia datus eſt arcus AC, angulo recto op-<lb/> <anchor type="note" xlink:label="note-485-09a" xlink:href="note-485-09"/> poſitus, & </s> <s xml:id="echoid-s16995" xml:space="preserve">inſuper arcus CD, circa rectum angulum: <lb/></s> <s xml:id="echoid-s16996" xml:space="preserve">AVT denique, quoniam datus eſt arcus AC, recto <lb/> <anchor type="note" xlink:label="note-485-10a" xlink:href="note-485-10"/> angulo oppoſitus, cum angulo non recto CAD; <lb/></s> <s xml:id="echoid-s16997" xml:space="preserve">notus quoque fiet, ex ſcholijs in margine nominatis, angulus ACD, qui in <lb/>priori triangulo eſt is, qui quæritur; </s> <s xml:id="echoid-s16998" xml:space="preserve">in poſteriori vero ſubductus ex duobus <lb/>rectis reliquum facit quæſitum angulum ACB. </s> <s xml:id="echoid-s16999" xml:space="preserve">Atque ita iam omnia, quæ <lb/>propoſita ſunt, inuenimus.</s> <s xml:id="echoid-s17000" xml:space="preserve"/> </p> <div xml:id="echoid-div1344" type="float" level="2" n="5"> <note position="right" xlink:label="note-485-05" xlink:href="note-485-05a" xml:space="preserve">Schol. 41. <lb/>vel 55. huiꝰ.</note> <note position="right" xlink:label="note-485-06" xlink:href="note-485-06a" xml:space="preserve">Schol. 42. <lb/>huius.</note> <note position="right" xlink:label="note-485-07" xlink:href="note-485-07a" xml:space="preserve">Schol. 56. <lb/>vel 42. huiꝰ.</note> <note position="right" xlink:label="note-485-08" xlink:href="note-485-08a" xml:space="preserve">Schol. 48. <lb/>vel 44. huiꝰ.</note> <note position="right" xlink:label="note-485-09" xlink:href="note-485-09a" xml:space="preserve">Schol. 45. <lb/>vel 51. huiꝰ.</note> <note position="right" xlink:label="note-485-10" xlink:href="note-485-10a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s17001" xml:space="preserve">DE praxi nibil noui præcipimus, ſed recurrendum erit ad praxes ſcho <lb/>liorum, quæ in margine citata ſunt.</s> <s xml:id="echoid-s17002" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17003" xml:space="preserve">PER ſolos autem ſinus ita propoſitum exequemur. </s> <s xml:id="echoid-s17004" xml:space="preserve">Per praxim pro-<lb/> <anchor type="note" xlink:label="note-485-11a" xlink:href="note-485-11"/> blematis 2. </s> <s xml:id="echoid-s17005" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17006" xml:space="preserve">41. </s> <s xml:id="echoid-s17007" xml:space="preserve">in triangulo ABD, rectangulo inueſtiga-<lb/>bimus arcum AD: </s> <s xml:id="echoid-s17008" xml:space="preserve">Et per praxim problematis ſcholij 1. </s> <s xml:id="echoid-s17009" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s17010" xml:space="preserve">43. </s> <s xml:id="echoid-s17011" xml:space="preserve">ar-<lb/>cum BD. </s> <s xml:id="echoid-s17012" xml:space="preserve">Deinde per praxim problematis 1. </s> <s xml:id="echoid-s17013" xml:space="preserve">ſcholij 1. </s> <s xml:id="echoid-s17014" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s17015" xml:space="preserve">41. </s> <s xml:id="echoid-s17016" xml:space="preserve">angu-<lb/>lum BAD: </s> <s xml:id="echoid-s17017" xml:space="preserve">quem, ſi minor est dato angulo BAC, auferemus ex angulo <lb/>BAC, dato; </s> <s xml:id="echoid-s17018" xml:space="preserve">vel, ſimaior eſt, ab eo datum angulum BAC, detrahemus, <lb/>vt notus fiat angulus CAD.</s> <s xml:id="echoid-s17019" xml:space="preserve"/> </p> <div xml:id="echoid-div1345" type="float" level="2" n="6"> <note position="right" xlink:label="note-485-11" xlink:href="note-485-11a" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quãdo dati <lb/>duo anguli <lb/>inæquales <lb/>ſunt, & arcꝰ <lb/>datus illis <lb/>adiacẽs nõ <lb/>eſt quadrãs.</note> </div> <p style="it"> <s xml:id="echoid-s17020" xml:space="preserve">HINC per praxim problematis 2. </s> <s xml:id="echoid-s17021" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17022" xml:space="preserve">42. </s> <s xml:id="echoid-s17023" xml:space="preserve">eliciemus in <pb o="474" file="486" n="486" rhead=""/> triangul o rectangulo ACD, angulum ACD: </s> <s xml:id="echoid-s17024" xml:space="preserve">qui erit quæſitus ACB, <lb/>in triangulo ABC, ſi inuentus angulus BAD, fuerit minor angulo dato <lb/>BAC: </s> <s xml:id="echoid-s17025" xml:space="preserve">Si autem maior, idem angulus ACD, ex duobus rectis demptus <lb/>reliquum faciet angulum quæſitum ACB.</s> <s xml:id="echoid-s17026" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17027" xml:space="preserve">IAM vero per praxim problematis 3. </s> <s xml:id="echoid-s17028" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17029" xml:space="preserve">41. </s> <s xml:id="echoid-s17030" xml:space="preserve">inueniemus <lb/>in triangulo eodẽ ACD, arcum AC, recto angulo oppoſitum. </s> <s xml:id="echoid-s17031" xml:space="preserve">Datur enim <lb/>arcus AD, circa angulum rectum, & </s> <s xml:id="echoid-s17032" xml:space="preserve">angulus nõrectus ACD, constat{quam} <lb/>præterea, qualis ſit alter angulus non rectus CAD, iamdudũ inuentus: </s> <s xml:id="echoid-s17033" xml:space="preserve">qui <lb/>quidem arcus AC, eſt vnus reliquorũ, qui in triangulo ABC, quæruntur.</s> <s xml:id="echoid-s17034" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17035" xml:space="preserve">PER praxim tandem problematis 2. </s> <s xml:id="echoid-s17036" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17037" xml:space="preserve">41. </s> <s xml:id="echoid-s17038" xml:space="preserve">reperietur <lb/>arcus CD: </s> <s xml:id="echoid-s17039" xml:space="preserve">Vel per praxim problematis 1. </s> <s xml:id="echoid-s17040" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17041" xml:space="preserve">42. </s> <s xml:id="echoid-s17042" xml:space="preserve">Vel certe <lb/>per praxim problematis ſcholij 1. </s> <s xml:id="echoid-s17043" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s17044" xml:space="preserve">43. </s> <s xml:id="echoid-s17045" xml:space="preserve">eundem arcum CD, cogno-<lb/>ſcemus: </s> <s xml:id="echoid-s17046" xml:space="preserve">qui additus inuento arcui BD, quando angulus BAD, inuentus <lb/>minor fuerit dato angulo BAC; </s> <s xml:id="echoid-s17047" xml:space="preserve">vel, quando maior ſuerit, ab eodem ar-<lb/>cu BD, ſubtractus, notum efficiet arcum BC, qui eſt alter eorum in trian <lb/>gulo ABC, qui inueſtigari debent.</s> <s xml:id="echoid-s17048" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17049" xml:space="preserve">QVOD ſi quando angulus inuentus CAD, fuerit rectus, cum & </s> <s xml:id="echoid-s17050" xml:space="preserve"><lb/>ADC, rectus ſit, erunt AC, CD, quadrantes; </s> <s xml:id="echoid-s17051" xml:space="preserve">& </s> <s xml:id="echoid-s17052" xml:space="preserve">AD, arcus anguli C; <lb/></s> <s xml:id="echoid-s17053" xml:space="preserve"> <anchor type="note" xlink:label="note-486-01a" xlink:href="note-486-01"/> ac proinde angulus C, ex arcu inuento AD, cognitus erit. </s> <s xml:id="echoid-s17054" xml:space="preserve">Reliquus autem <lb/> <anchor type="note" xlink:label="note-486-02a" xlink:href="note-486-02"/> arcus BC, cognoſcetur ex quadrante CD, & </s> <s xml:id="echoid-s17055" xml:space="preserve">arcu BD, inuento, vt prius.</s> <s xml:id="echoid-s17056" xml:space="preserve"/> </p> <div xml:id="echoid-div1346" type="float" level="2" n="7"> <note position="left" xlink:label="note-486-01" xlink:href="note-486-01a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-486-02" xlink:href="note-486-02a" xml:space="preserve">Quãdoduo <lb/>anguli ſunt <lb/>inæquales, <lb/>& arcus ad-<lb/>iacẽs datus <lb/>quadrans.</note> </div> <p> <s xml:id="echoid-s17057" xml:space="preserve">IAM vero ſi datus arcus AB, ſit quadrans, exiſtentibus adhuc angulis B, <lb/>& </s> <s xml:id="echoid-s17058" xml:space="preserve">BAC, inæqualibus, erit angulus BAD, rectus, & </s> <s xml:id="echoid-s17059" xml:space="preserve">arcus etiam BD, quadrãs. <lb/></s> <s xml:id="echoid-s17060" xml:space="preserve">Nam cum in triangulo rectangulo ABD, arcus AB, angulo recto oppoſitus <lb/>ponatur quadrans, erit ſaltem alter reliquorum arcuum quadrans. </s> <s xml:id="echoid-s17061" xml:space="preserve">Non po-<lb/> <anchor type="note" xlink:label="note-486-03a" xlink:href="note-486-03"/> teſt autem AD, eſſe quadrans: </s> <s xml:id="echoid-s17062" xml:space="preserve">quia duo anguli B, D, eſſent recti, ob quadran <lb/> <anchor type="note" xlink:label="note-486-04a" xlink:href="note-486-04"/> tes AB, AD, cum tamen triangulum ABC, ponatur non rectangulum. </s> <s xml:id="echoid-s17063" xml:space="preserve">Igi-<lb/>tur BD, quadrans erit; </s> <s xml:id="echoid-s17064" xml:space="preserve">ac propterea oppoſitus angulus BAD, rectus. </s> <s xml:id="echoid-s17065" xml:space="preserve">Polus <lb/> <anchor type="note" xlink:label="note-486-05a" xlink:href="note-486-05"/> quoque arcus AD, erit B, ob quadrantes BA, BD; </s> <s xml:id="echoid-s17066" xml:space="preserve">ac proinde AD, arcus <lb/> <anchor type="note" xlink:label="note-486-06a" xlink:href="note-486-06"/> erit dati anguli B, ideoq́ datus. </s> <s xml:id="echoid-s17067" xml:space="preserve">Inuentis autem arcubus AD, BD, & </s> <s xml:id="echoid-s17068" xml:space="preserve">angu-<lb/>lo recto BAD, ſine vllo labore, cum in eis inueſtigandis nullo problemate ex <lb/>præcedentibus egeamus, reliqua inueniemus, vt prius.</s> <s xml:id="echoid-s17069" xml:space="preserve"/> </p> <div xml:id="echoid-div1347" type="float" level="2" n="8"> <note position="left" xlink:label="note-486-03" xlink:href="note-486-03a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-486-04" xlink:href="note-486-04a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-486-05" xlink:href="note-486-05a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-486-06" xlink:href="note-486-06a" xml:space="preserve">26. huius.</note> </div> <note position="left" xml:space="preserve">Quãdo duo <lb/>angul@dati <lb/>aũt ęquales.</note> <p> <s xml:id="echoid-s17070" xml:space="preserve">SINT deinde in triangulo ABC, dati duo anguli B, & </s> <s xml:id="echoid-s17071" xml:space="preserve">C, æquales, cum <lb/>arcu BC, illis adiacente, ſiue quadrans is ſit, fiue qua-<lb/>drante maior, aut minor. </s> <s xml:id="echoid-s17072" xml:space="preserve">Erunt arcus AB, AC, æqua-<lb/> <anchor type="figure" xlink:label="fig-486-01a" xlink:href="fig-486-01"/> <anchor type="note" xlink:label="note-486-08a" xlink:href="note-486-08"/> les; </s> <s xml:id="echoid-s17073" xml:space="preserve">ideoq́; </s> <s xml:id="echoid-s17074" xml:space="preserve">arcus perpendicularis AD, ad datum arcum <lb/>BC, ex oppoſito angulo A, demiſſus intra triangulum ca-<lb/>det, ſecabitque & </s> <s xml:id="echoid-s17075" xml:space="preserve">arcum datum BC, & </s> <s xml:id="echoid-s17076" xml:space="preserve">angulum BAC, <lb/>oppoſitum bifariam, vt in poſteriore caſu propoſ. </s> <s xml:id="echoid-s17077" xml:space="preserve">62. <lb/></s> <s xml:id="echoid-s17078" xml:space="preserve">monſtrauimus. </s> <s xml:id="echoid-s17079" xml:space="preserve">Quoniam ergo in triangulo ABD, re-<lb/>ctum habente angulum D, datus eſt arcus BD, circa re-<lb/>ctum angulum, quippe qui dimidium ſit datiarcus BC, & </s> <s xml:id="echoid-s17080" xml:space="preserve"><lb/>inſuper angulus B, ei adiacens; </s> <s xml:id="echoid-s17081" xml:space="preserve">dabitur & </s> <s xml:id="echoid-s17082" xml:space="preserve">arcus AB, re-<lb/> <anchor type="note" xlink:label="note-486-09a" xlink:href="note-486-09"/> cto angulo oppoſitus, ideoq́; </s> <s xml:id="echoid-s17083" xml:space="preserve">& </s> <s xml:id="echoid-s17084" xml:space="preserve">AC, illi æqualis datus erit. </s> <s xml:id="echoid-s17085" xml:space="preserve">Atque ita duo ar- <pb o="475" file="487" n="487" rhead=""/> cus reliqui iam noti facti ſunt. </s> <s xml:id="echoid-s17086" xml:space="preserve">Rurſus quia in eodem triangulo datus eſt, per <lb/> <anchor type="note" xlink:label="note-487-01a" xlink:href="note-487-01"/> inuentionem, arcus AB, recto angulo oppoſitus, cum arcu BD, circa re-<lb/>ctum angulum: <lb/></s> <s xml:id="echoid-s17087" xml:space="preserve">VEL, quia datus eſt arcus BD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-487-02a" xlink:href="note-487-02"/> ctum, vnà cum angulo non recto B, ei adiacente: <lb/></s> <s xml:id="echoid-s17088" xml:space="preserve">VEL certe, quia datus eſt arcus AB, angulo recto <lb/> <anchor type="note" xlink:label="note-487-03a" xlink:href="note-487-03"/> oppoſitus, cum angulo non recto B; <lb/></s> <s xml:id="echoid-s17089" xml:space="preserve">reperietur, per ſcholia in margine adducta, angulus quoque BAD: </s> <s xml:id="echoid-s17090" xml:space="preserve">qui du-<lb/>plicatus totum angulum BAC, quæſitum efficiet cognitum.</s> <s xml:id="echoid-s17091" xml:space="preserve"/> </p> <div xml:id="echoid-div1348" type="float" level="2" n="9"> <figure xlink:label="fig-486-01" xlink:href="fig-486-01a"> <image file="486-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/486-01"/> </figure> <note position="left" xlink:label="note-486-08" xlink:href="note-486-08a" xml:space="preserve">9. huius.</note> <note position="left" xlink:label="note-486-09" xlink:href="note-486-09a" xml:space="preserve">Schol. 45. <lb/>vel 46. huiꝰ.</note> <note position="right" xlink:label="note-487-01" xlink:href="note-487-01a" xml:space="preserve">Schol. 41. <lb/>vel 55. huiꝰ.</note> <note position="right" xlink:label="note-487-02" xlink:href="note-487-02a" xml:space="preserve">Schol. 42. <lb/>huius.</note> <note position="right" xlink:label="note-487-03" xlink:href="note-487-03a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s17092" xml:space="preserve">SED per ſolos ſinus ita praxis ſe habet. </s> <s xml:id="echoid-s17093" xml:space="preserve">Per praxim problematis 2. <lb/></s> <s xml:id="echoid-s17094" xml:space="preserve"> <anchor type="note" xlink:label="note-487-04a" xlink:href="note-487-04"/> ſcholij propoſ. </s> <s xml:id="echoid-s17095" xml:space="preserve">42. </s> <s xml:id="echoid-s17096" xml:space="preserve">ex arcu BD, circa angulum recturn dato, & </s> <s xml:id="echoid-s17097" xml:space="preserve">ex angu-<lb/>lo B, ei adiacente dato, inueniemus angulum BAD, qui duplicatus totum <lb/>angulum quæſitum BAC, dabit. </s> <s xml:id="echoid-s17098" xml:space="preserve">Deinde per praxim problematis 3. </s> <s xml:id="echoid-s17099" xml:space="preserve">ſcho <lb/>lij propoſ. </s> <s xml:id="echoid-s17100" xml:space="preserve">41, ex arcu BD, circa rectum angulum, & </s> <s xml:id="echoid-s17101" xml:space="preserve">angulo BAD, op-<lb/>poſito iam inuento, eruemus ar cum AB, recto angulo oppoſitum, ideo{quam} <lb/>& </s> <s xml:id="echoid-s17102" xml:space="preserve">arcum AC, illi æqualem. </s> <s xml:id="echoid-s17103" xml:space="preserve">Nam præter data conſiat etiam ſpecies re-<lb/>liqui anguli B, dati non recti.</s> <s xml:id="echoid-s17104" xml:space="preserve"/> </p> <div xml:id="echoid-div1349" type="float" level="2" n="10"> <note position="right" xlink:label="note-487-04" xlink:href="note-487-04a" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quãdo duo <lb/>anguli dati <lb/>ſunt æqua-<lb/>les.</note> </div> <p> <s xml:id="echoid-s17105" xml:space="preserve">DATIS igitur duobus angulis trianguli ſphærici non rectanguli, vnà <lb/>cum arcu ipſis adiacente; </s> <s xml:id="echoid-s17106" xml:space="preserve">reliquos arcus, cum reliquo angulo ſcrutati ſumus. <lb/></s> <s xml:id="echoid-s17107" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s17108" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1351" type="section" level="1" n="612"> <head xml:id="echoid-head647" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s17109" xml:space="preserve">IN triangulis rectilineis non rectãgulis problema huic ſimile propoſitum non fuit: <lb/></s> <s xml:id="echoid-s17110" xml:space="preserve">propterea quod, datis duobus angulis, datur & </s> <s xml:id="echoid-s17111" xml:space="preserve">tertius: </s> <s xml:id="echoid-s17112" xml:space="preserve">qui nimirum relinquitur, <lb/> <anchor type="note" xlink:label="note-487-05a" xlink:href="note-487-05"/> ſi duo illi ex duobus rectis tollantur. </s> <s xml:id="echoid-s17113" xml:space="preserve">Quare cum vnum etiam latus detur, duo reli-<lb/>qua latera per propoſ. </s> <s xml:id="echoid-s17114" xml:space="preserve">10. </s> <s xml:id="echoid-s17115" xml:space="preserve">triang. </s> <s xml:id="echoid-s17116" xml:space="preserve">rectil. </s> <s xml:id="echoid-s17117" xml:space="preserve">efficientur nota.</s> <s xml:id="echoid-s17118" xml:space="preserve"/> </p> <div xml:id="echoid-div1351" type="float" level="2" n="1"> <note position="right" xlink:label="note-487-05" xlink:href="note-487-05a" xml:space="preserve">32. primi.</note> </div> </div> <div xml:id="echoid-div1353" type="section" level="1" n="613"> <head xml:id="echoid-head648" xml:space="preserve">PROBL. 7. PROPOS. 66.</head> <p> <s xml:id="echoid-s17119" xml:space="preserve">DATIS duobus angulis triãguli ſphærici non <lb/>rectanguli, cum arcu, qui alteri illorum opponi-<lb/>tur; </s> <s xml:id="echoid-s17120" xml:space="preserve">reliquos arcus, cum reliquo angulo indagare. <lb/></s> <s xml:id="echoid-s17121" xml:space="preserve">Oporter autem conſtare, num arcus alteri angulo <lb/>dato oppoſitus maior ſit quadrante, an minor, aut <lb/>certe quadrans.</s> <s xml:id="echoid-s17122" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Quãdoduo <lb/>anguli dati <lb/>inæquales <lb/>sũt, & arcns <lb/>datꝰ qui v-<lb/>ni eorũ op-<lb/>ponitur, nõ <lb/>quadrans.</note> <p> <s xml:id="echoid-s17123" xml:space="preserve">IN triangulo ABC, non rectangulo dati ſint duo anguli B, C, cum ar-<lb/>cu AB, qui angulo C, opponitur, conſtetq́ue, an arcus AC, maior quadrante <lb/>ſit, minorve, an quadrans. </s> <s xml:id="echoid-s17124" xml:space="preserve">Oportet ex his & </s> <s xml:id="echoid-s17125" xml:space="preserve">reliquos arcus AC, BC, & </s> <s xml:id="echoid-s17126" xml:space="preserve">reliquũ <lb/>angulum BAC, inuenire. </s> <s xml:id="echoid-s17127" xml:space="preserve">Sint primum dati duo anguli B, C, inæquales, &</s> <s xml:id="echoid-s17128" xml:space="preserve"> <pb o="476" file="488" n="488" rhead=""/> datus arcus AB, non quadrans. </s> <s xml:id="echoid-s17129" xml:space="preserve">Ducatur ex angulo A, ad arcum BC, datis <lb/>angulis adiacentẽ arcus perpendicularis AD: <lb/></s> <s xml:id="echoid-s17130" xml:space="preserve"> <anchor type="note" xlink:label="note-488-01a" xlink:href="note-488-01"/> qui intra triangulum cadet, ſi vterque angu-<lb/> <anchor type="figure" xlink:label="fig-488-01a" xlink:href="fig-488-01"/> lorum B, C, datorum fuerit acutus, vel obtu-<lb/>ſus; </s> <s xml:id="echoid-s17131" xml:space="preserve">extra vero, ſi vnus fuerit acutus, & </s> <s xml:id="echoid-s17132" xml:space="preserve">obtu <lb/>ſus alter. </s> <s xml:id="echoid-s17133" xml:space="preserve">Quia ergo in triangulo ABD, re-<lb/>ctum habẽte angulum D, datus eſt arcus AB, <lb/>angulo recto oppoſitus, cum angulo non re-<lb/>cto B; </s> <s xml:id="echoid-s17134" xml:space="preserve">notus fiet arcus AD, circa angulum <lb/> <anchor type="note" xlink:label="note-488-02a" xlink:href="note-488-02"/> rectum dato angulo B, oppoſitus. </s> <s xml:id="echoid-s17135" xml:space="preserve">Hinc in <lb/>eodem triangulo ABD, quoniam datus eſt <lb/> <anchor type="note" xlink:label="note-488-03a" xlink:href="note-488-03"/> arcus AB, recto angulo oppoſitus, cum arcu <lb/>AD, circa angulum rectum: <lb/></s> <s xml:id="echoid-s17136" xml:space="preserve">VEL, quia datus eſt arcus AB, angulo recto op-<lb/> <anchor type="note" xlink:label="note-488-04a" xlink:href="note-488-04"/> poſitus, & </s> <s xml:id="echoid-s17137" xml:space="preserve">præterea angulus B, non rectus: <lb/></s> <s xml:id="echoid-s17138" xml:space="preserve">VEL denique, quoniam datus eſt arcus AD, circa <lb/> <anchor type="note" xlink:label="note-488-05a" xlink:href="note-488-05"/> rectum angulum, cum angulo B, non recto ei oppoſito; <lb/></s> <s xml:id="echoid-s17139" xml:space="preserve">conſtatq́; </s> <s xml:id="echoid-s17140" xml:space="preserve">præterea ſpecies arcus BD. </s> <s xml:id="echoid-s17141" xml:space="preserve">Nam ſi AB, da-<lb/>tus fuerit minor quadrante; </s> <s xml:id="echoid-s17142" xml:space="preserve">ſi quidem & </s> <s xml:id="echoid-s17143" xml:space="preserve">AD, inuentus <lb/>ſit minor, erit quoque BD, minor; </s> <s xml:id="echoid-s17144" xml:space="preserve">ſi autem maior, ma-<lb/> <anchor type="note" xlink:label="note-488-06a" xlink:href="note-488-06"/> ior. </s> <s xml:id="echoid-s17145" xml:space="preserve">Si vero AB, datus fuerit quadrante maior; </s> <s xml:id="echoid-s17146" xml:space="preserve">ſi qui-<lb/>dem & </s> <s xml:id="echoid-s17147" xml:space="preserve">AD, inuentus ſit maior, erit BD, minor; </s> <s xml:id="echoid-s17148" xml:space="preserve">ſi ve-<lb/>ro AD, ſit minor, erit BD, maior; <lb/></s> <s xml:id="echoid-s17149" xml:space="preserve">reperietur quoque, ex ſcholijs in margine citatis, alter arcus BD, circa an-<lb/>gulum rectum. </s> <s xml:id="echoid-s17150" xml:space="preserve">Hinc rurſus in eodem triangulo ABD, quoniam datus eſt ar-<lb/> <anchor type="note" xlink:label="note-488-07a" xlink:href="note-488-07"/> cus AB, recto angulo oppoſitus, & </s> <s xml:id="echoid-s17151" xml:space="preserve">præterea arcus BD, circa angulũ rectum: <lb/></s> <s xml:id="echoid-s17152" xml:space="preserve">AVT, quia datus eſt vterque arcus AD, BD, cir-<lb/> <anchor type="note" xlink:label="note-488-08a" xlink:href="note-488-08"/> ca angulum rectum: <lb/></s> <s xml:id="echoid-s17153" xml:space="preserve">VEL, quia datus eſt arcus AB, recto angulo oppo <lb/> <anchor type="note" xlink:label="note-488-09a" xlink:href="note-488-09"/> ſitus, cum arcu AD, circa angulum rectum: <lb/></s> <s xml:id="echoid-s17154" xml:space="preserve">VEL, quia datus eſt arcus AD, circa rectum angu-<lb/> <anchor type="note" xlink:label="note-488-10a" xlink:href="note-488-10"/> lum, & </s> <s xml:id="echoid-s17155" xml:space="preserve">inſuper angulus non rectus B, ei oppoſitus; </s> <s xml:id="echoid-s17156" xml:space="preserve">con <lb/>ſtatq́; </s> <s xml:id="echoid-s17157" xml:space="preserve">præterea ſpecies anguli BAD. </s> <s xml:id="echoid-s17158" xml:space="preserve">Nam ſi BD, arcus <lb/>inuentus ſit quadrante maior, erit angulus BAD, ob-<lb/> <anchor type="note" xlink:label="note-488-11a" xlink:href="note-488-11"/> tuſus; </s> <s xml:id="echoid-s17159" xml:space="preserve">ſi vero minor, acutus. <lb/></s> <s xml:id="echoid-s17160" xml:space="preserve">VEL denique, quia datus eſt arcus AD, angulo <lb/> <anchor type="note" xlink:label="note-488-12a" xlink:href="note-488-12"/> recto oppoſitus, cum angulo non recto B; <lb/></s> <s xml:id="echoid-s17161" xml:space="preserve">notus fiet quoq; </s> <s xml:id="echoid-s17162" xml:space="preserve">angulus non rectus BAD, ex ſcholijs in margine appoſitis.</s> <s xml:id="echoid-s17163" xml:space="preserve"/> </p> <div xml:id="echoid-div1353" type="float" level="2" n="1"> <note position="left" xlink:label="note-488-01" xlink:href="note-488-01a" xml:space="preserve">57. huius.</note> <figure xlink:label="fig-488-01" xlink:href="fig-488-01a"> <image file="488-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/488-01"/> </figure> <note position="left" xlink:label="note-488-02" xlink:href="note-488-02a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="left" xlink:label="note-488-03" xlink:href="note-488-03a" xml:space="preserve">Schol. 43. <lb/>vel 53. huiꝰ.</note> <note position="left" xlink:label="note-488-04" xlink:href="note-488-04a" xml:space="preserve">Schol. 45. <lb/>huius.</note> <note position="left" xlink:label="note-488-05" xlink:href="note-488-05a" xml:space="preserve">Schol. 49. <lb/>vel 44. huiꝰ.</note> <note position="left" xlink:label="note-488-06" xlink:href="note-488-06a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-488-07" xlink:href="note-488-07a" xml:space="preserve">Schol. 41. <lb/>vel 55. huiꝰ.</note> <note position="left" xlink:label="note-488-08" xlink:href="note-488-08a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="left" xlink:label="note-488-09" xlink:href="note-488-09a" xml:space="preserve">Schol. 45. <lb/>vel 51. huiꝰ.</note> <note position="left" xlink:label="note-488-10" xlink:href="note-488-10a" xml:space="preserve">Schol. 56. <lb/>vel 42. huiꝰ.</note> <note position="left" xlink:label="note-488-11" xlink:href="note-488-11a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-488-12" xlink:href="note-488-12a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17164" xml:space="preserve">DEINDE in triangulo ACD, rectum habente angulum D, quoniam <lb/>datus eſt arcus AD, circa rectum angulum, cum angulo C, oppoſito; </s> <s xml:id="echoid-s17165" xml:space="preserve">(Nam <lb/>quando perpendicularis arcus AD, extra triangulum cadit, dabitur angulus <lb/>ACD, ſi datus angulus ACB, ex duobus rectis ſubducatur.) </s> <s xml:id="echoid-s17166" xml:space="preserve">poniturq; </s> <s xml:id="echoid-s17167" xml:space="preserve">præ-<lb/>terea conſtare ſpecies arcus AC, qui in propoſito triangulo ABC, alteri da-<lb/>to angulo B, opponitur, in hoc vero triangulo ACD, recto angulo D, op-<lb/>poſitus eſt; </s> <s xml:id="echoid-s17168" xml:space="preserve">notus quoque euadet arcus AC; </s> <s xml:id="echoid-s17169" xml:space="preserve">qui vnus eſt reliquorum arcuum, <lb/> <anchor type="note" xlink:label="note-488-13a" xlink:href="note-488-13"/> qui inueſtigandi proponuntur in triangulo ABC. </s> <s xml:id="echoid-s17170" xml:space="preserve">Hinc quia in eodem trian <lb/>gulo ACD, datus eſt arcus AC, recto angulo oppoſitus, & </s> <s xml:id="echoid-s17171" xml:space="preserve">arcus AD, circa <lb/> <anchor type="note" xlink:label="note-488-14a" xlink:href="note-488-14"/> rectum angulum:</s> <s xml:id="echoid-s17172" xml:space="preserve"> <pb o="477" file="489" n="489" rhead=""/> VEL, quia datus eſt arcus AC, angulo recto op-<lb/> <anchor type="note" xlink:label="note-489-01a" xlink:href="note-489-01"/> poſitus, cum angulo non recto C: <lb/></s> <s xml:id="echoid-s17173" xml:space="preserve">VEL denique, quoniam datus eſt arcus AD, circa <lb/> <anchor type="note" xlink:label="note-489-02a" xlink:href="note-489-02"/> angulum rectum, cum angulo C, ei oppoſito, conſtatq́; <lb/></s> <s xml:id="echoid-s17174" xml:space="preserve">præterea ſpecies arcus CD. </s> <s xml:id="echoid-s17175" xml:space="preserve">Nam ſi arcus AC, recto an-<lb/>gulo oppoſitus, inuentus fuerit minor quadrante, erit <lb/>vterque arcus AD, CD, vel minor etiam, vel maior; </s> <s xml:id="echoid-s17176" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-489-03a" xlink:href="note-489-03"/> atque ita ex cognito arcu AD, ſciemus, an CD, minor <lb/>ſit, vel maior quadrante: </s> <s xml:id="echoid-s17177" xml:space="preserve">Si vero inuentus arcus AC, <lb/>fuerit quadrante maior, & </s> <s xml:id="echoid-s17178" xml:space="preserve">AD, minor, erit CD, maior; <lb/></s> <s xml:id="echoid-s17179" xml:space="preserve">at ſi AD, maior fuerit, erit CD, minor; </s> <s xml:id="echoid-s17180" xml:space="preserve"><lb/>cognoſcetur etiam, per ſcholia in margine poſita, arcus CD: </s> <s xml:id="echoid-s17181" xml:space="preserve">qui additus ar-<lb/>cui iamdudum inuento BD, ſi perpendicularis arcus AD, intra triangulum <lb/>cadit; </s> <s xml:id="echoid-s17182" xml:space="preserve">vel, ſi extra, ablatus ex arcu BD, inuento; </s> <s xml:id="echoid-s17183" xml:space="preserve">notum efficiet arcum BC, <lb/>quæſitum. </s> <s xml:id="echoid-s17184" xml:space="preserve">Atque ita iam reliqui duo arcus AC, BC, inuenti erunt.</s> <s xml:id="echoid-s17185" xml:space="preserve"/> </p> <div xml:id="echoid-div1354" type="float" level="2" n="2"> <note position="left" xlink:label="note-488-13" xlink:href="note-488-13a" xml:space="preserve">Schol. 41. <lb/>vel 54. huiꝰ.</note> <note position="left" xlink:label="note-488-14" xlink:href="note-488-14a" xml:space="preserve">Schol. 43. <lb/>vel 53. huiꝰ.</note> <note position="right" xlink:label="note-489-01" xlink:href="note-489-01a" xml:space="preserve">Schol. 45. <lb/>huius.</note> <note position="right" xlink:label="note-489-02" xlink:href="note-489-02a" xml:space="preserve">Schol. 49. <lb/>vel 44. huiꝰ.</note> <note position="right" xlink:label="note-489-03" xlink:href="note-489-03a" xml:space="preserve">56. huius.</note> </div> <p> <s xml:id="echoid-s17186" xml:space="preserve">POSTREMO, quia in eodem proximo triangulo ACD, datus eſt ar-<lb/> <anchor type="note" xlink:label="note-489-04a" xlink:href="note-489-04"/> cus AC, angulo recto oppoſitus, cum arcu CD, circa rectum angulum: <lb/></s> <s xml:id="echoid-s17187" xml:space="preserve">VEL, quoniam datus eſt arcus CD, circa rectum <lb/> <anchor type="note" xlink:label="note-489-05a" xlink:href="note-489-05"/> angulum, & </s> <s xml:id="echoid-s17188" xml:space="preserve">præterea angulus non rectus C: <lb/></s> <s xml:id="echoid-s17189" xml:space="preserve">VEL, quia datus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-489-06a" xlink:href="note-489-06"/> ctum, vnà cum angulo non recto C, oppoſito; </s> <s xml:id="echoid-s17190" xml:space="preserve">conſtatq́; <lb/></s> <s xml:id="echoid-s17191" xml:space="preserve">præterea ſpecies alterius anguli CAD. </s> <s xml:id="echoid-s17192" xml:space="preserve">Nam ſi arcus <lb/>inuentus CD, minor eſt quadrante, erit angulus CAD, <lb/> <anchor type="note" xlink:label="note-489-07a" xlink:href="note-489-07"/> acutus; </s> <s xml:id="echoid-s17193" xml:space="preserve">obtuſus vero, ſi CD, quadrante maior eſt: <lb/></s> <s xml:id="echoid-s17194" xml:space="preserve">VEL, quia datus eſt vterq; </s> <s xml:id="echoid-s17195" xml:space="preserve">arcus AD, CD, circ@ <lb/> <anchor type="note" xlink:label="note-489-08a" xlink:href="note-489-08"/> angulum rectum. <lb/></s> <s xml:id="echoid-s17196" xml:space="preserve">VEL, quoniam datus eſt arcus AC, recto angulo <lb/> <anchor type="note" xlink:label="note-489-09a" xlink:href="note-489-09"/> oppoſitus, & </s> <s xml:id="echoid-s17197" xml:space="preserve">arcus AD, circa rectum angulum; <lb/></s> <s xml:id="echoid-s17198" xml:space="preserve">VEL denique, quia datus eſt arcus AC, angulo re-<lb/> <anchor type="note" xlink:label="note-489-10a" xlink:href="note-489-10"/> cto oppoſitus, vnà cum angulo C, non recto; <lb/></s> <s xml:id="echoid-s17199" xml:space="preserve">fiet quoque notus angulus CAD, ex ſcholijs in margine adductis. </s> <s xml:id="echoid-s17200" xml:space="preserve">Hic autem <lb/>angulus CAD, additus angulo BAD, iam antea inuento, ſi arcus perpendi-<lb/>cularis AD, intra triangulum cadit; </s> <s xml:id="echoid-s17201" xml:space="preserve">vel ſi extra, ablatus ex inuento angulo <lb/>BAD, cognitum exhibebit angulum BAC, quæſitum.</s> <s xml:id="echoid-s17202" xml:space="preserve"/> </p> <div xml:id="echoid-div1355" type="float" level="2" n="3"> <note position="right" xlink:label="note-489-04" xlink:href="note-489-04a" xml:space="preserve">Schol. 47. <lb/>vel 55. huiꝰ.</note> <note position="right" xlink:label="note-489-05" xlink:href="note-489-05a" xml:space="preserve">Schol. 42. <lb/>huius.</note> <note position="right" xlink:label="note-489-06" xlink:href="note-489-06a" xml:space="preserve">Schol. 42. <lb/>vel 56. huiꝰ.</note> <note position="right" xlink:label="note-489-07" xlink:href="note-489-07a" xml:space="preserve">34. huius.</note> <note position="right" xlink:label="note-489-08" xlink:href="note-489-08a" xml:space="preserve">Schol. 44. <lb/>vel 48. huiꝰ.</note> <note position="right" xlink:label="note-489-09" xlink:href="note-489-09a" xml:space="preserve">Schol. 45. <lb/>vel 51. huiꝰ.</note> <note position="right" xlink:label="note-489-10" xlink:href="note-489-10a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17203" xml:space="preserve">CAETERVM nullo modo alteruter arcuum AD, BD, quadrans eſſe <lb/>poteſt in hoc caſu: </s> <s xml:id="echoid-s17204" xml:space="preserve">quia ſi alter illorum eſſet quadrans, eſſet quoq; </s> <s xml:id="echoid-s17205" xml:space="preserve">arcus AB, <lb/> <anchor type="note" xlink:label="note-489-11a" xlink:href="note-489-11"/> angulo recto oppoſitus, quadrans. </s> <s xml:id="echoid-s17206" xml:space="preserve">quod eſt contra hypotheſim.</s> <s xml:id="echoid-s17207" xml:space="preserve"/> </p> <div xml:id="echoid-div1356" type="float" level="2" n="4"> <note position="right" xlink:label="note-489-11" xlink:href="note-489-11a" xml:space="preserve">35. huius.</note> </div> <p style="it"> <s xml:id="echoid-s17208" xml:space="preserve">PRAXIS huius problematis pendet ex ſcholijs in margine not atis.</s> <s xml:id="echoid-s17209" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17210" xml:space="preserve">SOLIS autem ſinubus it a rem perficiemus. </s> <s xml:id="echoid-s17211" xml:space="preserve">Per praxim problema-<lb/> <anchor type="note" xlink:label="note-489-12a" xlink:href="note-489-12"/> tis 2. </s> <s xml:id="echoid-s17212" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17213" xml:space="preserve">41. </s> <s xml:id="echoid-s17214" xml:space="preserve">inueniemus arcum AD: </s> <s xml:id="echoid-s17215" xml:space="preserve">Et per praxim proble-<lb/>matis ſcholij 1. </s> <s xml:id="echoid-s17216" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s17217" xml:space="preserve">43. </s> <s xml:id="echoid-s17218" xml:space="preserve">arcum BD: </s> <s xml:id="echoid-s17219" xml:space="preserve">Et per praxim problematis 1. <lb/></s> <s xml:id="echoid-s17220" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17221" xml:space="preserve">41. </s> <s xml:id="echoid-s17222" xml:space="preserve">angulum BAD.</s> <s xml:id="echoid-s17223" xml:space="preserve"/> </p> <div xml:id="echoid-div1357" type="float" level="2" n="5"> <note position="right" xlink:label="note-489-12" xlink:href="note-489-12a" xml:space="preserve">Praxis pet <lb/>ſolos ſinus, <lb/>quãdo dati <lb/>anguli inę-<lb/>quales sũt, <lb/>& datus at <lb/>cus, qui vni <lb/>eorum op-<lb/>ponitur, nõ <lb/>quadrans.</note> </div> <p style="it"> <s xml:id="echoid-s17224" xml:space="preserve">DEINDE per praxim problematis 3. </s> <s xml:id="echoid-s17225" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17226" xml:space="preserve">41. </s> <s xml:id="echoid-s17227" xml:space="preserve">cognoſce-<lb/>mus arcum AC, cum conſtet ex bypotheſi eius ſpecies. </s> <s xml:id="echoid-s17228" xml:space="preserve">Hinc per praxim <lb/>problematis ſcholij 1. </s> <s xml:id="echoid-s17229" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s17230" xml:space="preserve">43. </s> <s xml:id="echoid-s17231" xml:space="preserve">arcus CD, notus fiet; </s> <s xml:id="echoid-s17232" xml:space="preserve">ex quo, ſi adda-<lb/>tur arcui inuento BD, vel abe@dem ſubtrahatur, prout perpendicularis <pb o="478" file="490" n="490" rhead=""/> arcus AD, intra, vel extra triangulum ceciderit, cognitus fiet arcus BC.</s> <s xml:id="echoid-s17233" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17234" xml:space="preserve">AD extremum, per praxim problematis 1. </s> <s xml:id="echoid-s17235" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17236" xml:space="preserve">41. </s> <s xml:id="echoid-s17237" xml:space="preserve">erue-<lb/>mus angulum CAD; </s> <s xml:id="echoid-s17238" xml:space="preserve">qui additus angulo inuento BAD, vel ab eo ſubtra-<lb/>ctus, prout arcus perpendicularis AD, intra triangulum ceciderit, vel <lb/>extra, notum faciet angulum BAC.</s> <s xml:id="echoid-s17239" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17240" xml:space="preserve">QVOD ſi quando arcus AC, alteriangulo B, dato oppoſitus, ſit qua-<lb/>drans, quod euenire poteſt, non exiſtente quadrante AB; </s> <s xml:id="echoid-s17241" xml:space="preserve">erit alter ſaltem <lb/>reliquorum quoque arcuum AD, CD, in triangulo ACD, quadrans. </s> <s xml:id="echoid-s17242" xml:space="preserve">Cum <lb/> <anchor type="note" xlink:label="note-490-01a" xlink:href="note-490-01"/> ergo AD, eſſe non poſsit quadrans, erit CD, quadrans; </s> <s xml:id="echoid-s17243" xml:space="preserve">ac proinde angulus <lb/> <anchor type="note" xlink:label="note-490-02a" xlink:href="note-490-02"/> ei oppoſitus CAD, rectus. </s> <s xml:id="echoid-s17244" xml:space="preserve">Itaque tunc inuentus erit & </s> <s xml:id="echoid-s17245" xml:space="preserve">arcus CD, & </s> <s xml:id="echoid-s17246" xml:space="preserve">angu-<lb/> <anchor type="note" xlink:label="note-490-03a" xlink:href="note-490-03"/> lus CAD, ſine vllo alio labore: </s> <s xml:id="echoid-s17247" xml:space="preserve">ex quibus & </s> <s xml:id="echoid-s17248" xml:space="preserve">arcus BC, & </s> <s xml:id="echoid-s17249" xml:space="preserve">angulus BAC, de-<lb/>prehendentur, vt dictum eſt.</s> <s xml:id="echoid-s17250" xml:space="preserve"/> </p> <div xml:id="echoid-div1358" type="float" level="2" n="6"> <note position="left" xlink:label="note-490-01" xlink:href="note-490-01a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-490-02" xlink:href="note-490-02a" xml:space="preserve">46. huius.</note> <note position="left" xlink:label="note-490-03" xlink:href="note-490-03a" xml:space="preserve">Quãdo duo <lb/>anguli dati <lb/>inæquales <lb/>sũt, & datus <lb/>arcꝰ vni eo-<lb/>rũ oppoſitꝰ, <lb/>quadrans.</note> </div> <p> <s xml:id="echoid-s17251" xml:space="preserve">SIT iam arcus datus AB, quadrans, & </s> <s xml:id="echoid-s17252" xml:space="preserve">adhuc duo anguli dati B, C, inæ-<lb/>quales. </s> <s xml:id="echoid-s17253" xml:space="preserve">Erit arcus BD, quadrans etiam, & </s> <s xml:id="echoid-s17254" xml:space="preserve">angulus BAD, rectus. </s> <s xml:id="echoid-s17255" xml:space="preserve">Cum enim <lb/>in triangulo ABD, arcus AB, angulo recto oppoſitus quadrans ponatur; </s> <s xml:id="echoid-s17256" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-490-04a" xlink:href="note-490-04"/> ſaltem & </s> <s xml:id="echoid-s17257" xml:space="preserve">alter reliquorum arcuum AD, BD, quadrans. </s> <s xml:id="echoid-s17258" xml:space="preserve">Non poteſt autẽ AD, <lb/> <anchor type="note" xlink:label="note-490-05a" xlink:href="note-490-05"/> eſſe quadrans: </s> <s xml:id="echoid-s17259" xml:space="preserve">quia duo anguli B, D, ob quadrantes AB, AD, recti eſſent, <lb/>ideoq́; </s> <s xml:id="echoid-s17260" xml:space="preserve">triangulum ABC, rectangulum, quod non ponitur. </s> <s xml:id="echoid-s17261" xml:space="preserve">Erit ergo BD, <lb/>quadrans, ac proinde angulus oppoſitus BAD, rectus. </s> <s xml:id="echoid-s17262" xml:space="preserve">Erit quoque B, po-<lb/> <anchor type="note" xlink:label="note-490-06a" xlink:href="note-490-06"/> lus arcus AD, ob quadrantes AB, BD; </s> <s xml:id="echoid-s17263" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s17264" xml:space="preserve">datus angulus B, arcum <lb/> <anchor type="note" xlink:label="note-490-07a" xlink:href="note-490-07"/> BD, notum efficiet. </s> <s xml:id="echoid-s17265" xml:space="preserve">Inuẽtis autem arcubus AD, BD, cum angulo recto BAD, <lb/>ſine vlla multiplicationis moleſtia, in uenientur reliqua, vt prius. </s> <s xml:id="echoid-s17266" xml:space="preserve">In hoc ta-<lb/>men caſu arcus AC, nullo pacto quadrans erit, ne duo quadrantes ſint AB, <lb/> <anchor type="note" xlink:label="note-490-08a" xlink:href="note-490-08"/> AC, in triangulo ABC, ac proinde duo anguli B, C, recti. </s> <s xml:id="echoid-s17267" xml:space="preserve">Quod eſtet contra <lb/>hypotheſim, cum triangulum ponatur non rectangulum.</s> <s xml:id="echoid-s17268" xml:space="preserve"/> </p> <div xml:id="echoid-div1359" type="float" level="2" n="7"> <note position="left" xlink:label="note-490-04" xlink:href="note-490-04a" xml:space="preserve">36. huius.</note> <note position="left" xlink:label="note-490-05" xlink:href="note-490-05a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-490-06" xlink:href="note-490-06a" xml:space="preserve">34. huius.</note> <note position="left" xlink:label="note-490-07" xlink:href="note-490-07a" xml:space="preserve">26. huius.</note> <note position="left" xlink:label="note-490-08" xlink:href="note-490-08a" xml:space="preserve">25. huius.</note> </div> <p> <s xml:id="echoid-s17269" xml:space="preserve">VERVM ſint iam in triangulo ABC, dati duo anguli B, C, æquales. <lb/></s> <s xml:id="echoid-s17270" xml:space="preserve"> <anchor type="note" xlink:label="note-490-09a" xlink:href="note-490-09"/> Erunt duo arcus AB, AC, a quales; </s> <s xml:id="echoid-s17271" xml:space="preserve">atq; </s> <s xml:id="echoid-s17272" xml:space="preserve">adeo neuter <lb/>eorum quadrans, ne duo anguli B, C, recti exiſtant. <lb/></s> <s xml:id="echoid-s17273" xml:space="preserve"> <anchor type="figure" xlink:label="fig-490-01a" xlink:href="fig-490-01"/> Demiſſus igitur arcus perpendicularis AD, ex tertio <lb/>angulo A, intra triangulum cadet, diuidetq; </s> <s xml:id="echoid-s17274" xml:space="preserve">tam ar-<lb/>cum BC, quam angulum BAC, bifariam, vt ſupra in <lb/>ſecundo caſu propoſ. </s> <s xml:id="echoid-s17275" xml:space="preserve">62. </s> <s xml:id="echoid-s17276" xml:space="preserve">oſtendimus. </s> <s xml:id="echoid-s17277" xml:space="preserve">Igitur quia in <lb/>triangulo ABD, rectum habente angulum D, datus <lb/>eſt arcus AB, angulo recto oppoſitus, cum angulo <lb/> <anchor type="note" xlink:label="note-490-10a" xlink:href="note-490-10"/> B; </s> <s xml:id="echoid-s17278" xml:space="preserve">cognitus erit & </s> <s xml:id="echoid-s17279" xml:space="preserve">arcus BD: </s> <s xml:id="echoid-s17280" xml:space="preserve">qui duplicatus totum <lb/>arcum BC, notum efficiet: </s> <s xml:id="echoid-s17281" xml:space="preserve">Sed & </s> <s xml:id="echoid-s17282" xml:space="preserve">AC, notus eſt, <lb/>cum dato arcui AB, æqualis ſit. </s> <s xml:id="echoid-s17283" xml:space="preserve">Deinde quoniam in eodem triangulo ABD, <lb/> <anchor type="note" xlink:label="note-490-11a" xlink:href="note-490-11"/> datus eſt arcus AB, angulo recto oppoſitus, cum arcu BD, circa angulum re-<lb/>ctum proximè inuento; <lb/></s> <s xml:id="echoid-s17284" xml:space="preserve">VEL, quia datus eſt arcus BD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-490-12a" xlink:href="note-490-12"/> ctum, cum angulo non recto adiacente B: <lb/></s> <s xml:id="echoid-s17285" xml:space="preserve">VEL denique, quoniam datus eſt arcus AB, recto <lb/> <anchor type="note" xlink:label="note-490-13a" xlink:href="note-490-13"/> angulo oppoſitus, cum angulo non recto B; <lb/></s> <s xml:id="echoid-s17286" xml:space="preserve">dabitur quoque, per ſcholia in margine adducta, angulus BAD: </s> <s xml:id="echoid-s17287" xml:space="preserve">qui duplica-<lb/>tus totum BAC, quæſitum præbebit.</s> <s xml:id="echoid-s17288" xml:space="preserve"/> </p> <div xml:id="echoid-div1360" type="float" level="2" n="8"> <note position="left" xlink:label="note-490-09" xlink:href="note-490-09a" xml:space="preserve">Quãdo duc <lb/>dati anguli <lb/>squales sũt.</note> <figure xlink:label="fig-490-01" xlink:href="fig-490-01a"> <image file="490-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/490-01"/> </figure> <note position="left" xlink:label="note-490-10" xlink:href="note-490-10a" xml:space="preserve">Schol. 45. <lb/>huius.</note> <note position="left" xlink:label="note-490-11" xlink:href="note-490-11a" xml:space="preserve">Schol. 41. <lb/>vel 55. huiꝰ.</note> <note position="left" xlink:label="note-490-12" xlink:href="note-490-12a" xml:space="preserve">Schol. 42. <lb/>huius.</note> <note position="left" xlink:label="note-490-13" xlink:href="note-490-13a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p style="it"> <s xml:id="echoid-s17289" xml:space="preserve">ITA autem ſolis ſinubus in hoc caſu vtemur. </s> <s xml:id="echoid-s17290" xml:space="preserve">Per praxim proble- <pb o="479" file="491" n="491" rhead=""/> matis 2. </s> <s xml:id="echoid-s17291" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17292" xml:space="preserve">41. </s> <s xml:id="echoid-s17293" xml:space="preserve">reperiemus arcum AD: </s> <s xml:id="echoid-s17294" xml:space="preserve">Ethinc per pra-<lb/> <anchor type="note" xlink:label="note-491-01a" xlink:href="note-491-01"/> xim problematis ſcholij 1. </s> <s xml:id="echoid-s17295" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s17296" xml:space="preserve">43. </s> <s xml:id="echoid-s17297" xml:space="preserve">arcum BD; </s> <s xml:id="echoid-s17298" xml:space="preserve">qui duplicatus totum <lb/>arcum BC, dabit notum. </s> <s xml:id="echoid-s17299" xml:space="preserve">Deinde per praxim problematis 1. </s> <s xml:id="echoid-s17300" xml:space="preserve">ſcholij pro-<lb/>poſ. </s> <s xml:id="echoid-s17301" xml:space="preserve">41. </s> <s xml:id="echoid-s17302" xml:space="preserve">Vel per praxim problematis 2. </s> <s xml:id="echoid-s17303" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17304" xml:space="preserve">42. </s> <s xml:id="echoid-s17305" xml:space="preserve">inueniemus an-<lb/>gulum BAD, ac proinde eius duplum BAC, qui quæritur. </s> <s xml:id="echoid-s17306" xml:space="preserve">Tertius au-<lb/>tem arcus AC, dato arcui AB, æqualis eſt, atque adeo cognitus.</s> <s xml:id="echoid-s17307" xml:space="preserve"/> </p> <div xml:id="echoid-div1361" type="float" level="2" n="9"> <note position="right" xlink:label="note-491-01" xlink:href="note-491-01a" xml:space="preserve">Praxis pet <lb/>ſolos ſinus, <lb/>quádo dati <lb/>duo anguli <lb/>ęquales sũt.</note> </div> <p> <s xml:id="echoid-s17308" xml:space="preserve">DATIS igitur duobus angulis trianguli ſphæricinon rectanguli, cum <lb/>vno arcu, qui alteri illorum opponitur, &</s> <s xml:id="echoid-s17309" xml:space="preserve">c. </s> <s xml:id="echoid-s17310" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s17311" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1363" type="section" level="1" n="614"> <head xml:id="echoid-head649" xml:space="preserve">SCHOLIVM.</head> <p style="it"> <s xml:id="echoid-s17312" xml:space="preserve">HVIC etiam problemati nullam propoſitionem reſpondentem attulimus in triam <lb/>gulis rectilineis, propter cauſam in ſcholio antecedentis propoſ. </s> <s xml:id="echoid-s17313" xml:space="preserve">allatam.</s> <s xml:id="echoid-s17314" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17315" xml:space="preserve">OPORTET autem in primo caſu huiuſce problematis dari etiam neceſſario ſpe <lb/>ciem arcus AC, alteri angulo dato B, oppoſiti. </s> <s xml:id="echoid-s17316" xml:space="preserve">Alioquin in triangulo ACD, exda-<lb/>to arcu AD, & </s> <s xml:id="echoid-s17317" xml:space="preserve">angulo C, oppoſito, (cum nihil certi adhuc exploratum habeamus de <lb/>arcu CD, vel angulo CAD, qualesnã ſint.) </s> <s xml:id="echoid-s17318" xml:space="preserve">non inueniretur arcus <emph style="sc">Ac</emph>, recto angu-<lb/>lo oppoſitus, cum is poſsit eſſe vel maior quadrante, vel minor, & </s> <s xml:id="echoid-s17319" xml:space="preserve">nondum ex datis, <lb/>vel demonſtratis conſtet, qualis futurus ſit. </s> <s xml:id="echoid-s17320" xml:space="preserve">Caterum non ſatis eſſe, ſi dentur anguli <lb/>duo, cum arcu vnieorum oppoſito, ad eliciendos reliquos arcus, & </s> <s xml:id="echoid-s17321" xml:space="preserve">reliquum angu-<lb/> <anchor type="note" xlink:label="note-491-02a" xlink:href="note-491-02"/> lum, iampridem admonuimus in ſcholio propoſ 22. </s> <s xml:id="echoid-s17322" xml:space="preserve">& </s> <s xml:id="echoid-s17323" xml:space="preserve">23. </s> <s xml:id="echoid-s17324" xml:space="preserve">Vbi etiam Copernicum hal-<lb/>lucinatum ea in re eſſe lib. </s> <s xml:id="echoid-s17325" xml:space="preserve">1. </s> <s xml:id="echoid-s17326" xml:space="preserve">Reuolutionum propoſ. </s> <s xml:id="echoid-s17327" xml:space="preserve">12. </s> <s xml:id="echoid-s17328" xml:space="preserve">triang. </s> <s xml:id="echoid-s17329" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s17330" xml:space="preserve">indicauimus. <lb/></s> <s xml:id="echoid-s17331" xml:space="preserve"> <anchor type="note" xlink:label="note-491-03a" xlink:href="note-491-03"/> Quod tamen hic breuiter ita rurſum demon§trabimus. <lb/></s> <s xml:id="echoid-s17332" xml:space="preserve"> <anchor type="figure" xlink:label="fig-491-01a" xlink:href="fig-491-01"/> Sint duo arcus inæquales <emph style="sc">Ab</emph>, AC, angulum <emph style="sc">BAc</emph>, con <lb/>tinentes, & </s> <s xml:id="echoid-s17333" xml:space="preserve">ſemicirculo ſimul æquales; </s> <s xml:id="echoid-s17334" xml:space="preserve">atque adeo vnus <lb/>quadrante maior, & </s> <s xml:id="echoid-s17335" xml:space="preserve">alter minor. </s> <s xml:id="echoid-s17336" xml:space="preserve">Ducto autem per B, <lb/>C, arcu circuli maximi BC, ducatur ad eum productum <lb/>ex A, alius arcus <emph style="sc">A</emph>D, neque per polos arcus <emph style="sc">Ac</emph>, neque <lb/>per polos arcus BC; </s> <s xml:id="echoid-s17337" xml:space="preserve">ita vt anguli D, & </s> <s xml:id="echoid-s17338" xml:space="preserve"><emph style="sc">C</emph>AD, ſint non <lb/>recti. </s> <s xml:id="echoid-s17339" xml:space="preserve">Sed neque angulus ACD, rectus eſt. </s> <s xml:id="echoid-s17340" xml:space="preserve">Nam ſi foret <lb/>rectus, eſſet angulus <emph style="sc">Ab</emph>C, cui ille æquælis eſt, rectus quo <lb/> <anchor type="note" xlink:label="note-491-04a" xlink:href="note-491-04"/> que; </s> <s xml:id="echoid-s17341" xml:space="preserve">atque ita duo arcus <emph style="sc">Ab</emph>, AC, propter rectos angu-<lb/> <anchor type="note" xlink:label="note-491-05a" xlink:href="note-491-05"/> los B, C, æquales eſſent, & </s> <s xml:id="echoid-s17342" xml:space="preserve">quadrantes. </s> <s xml:id="echoid-s17343" xml:space="preserve">Quod eſt contra hypotheſim. </s> <s xml:id="echoid-s17344" xml:space="preserve">Triangulum <lb/>ergo ACD, non rectangulum eſt; </s> <s xml:id="echoid-s17345" xml:space="preserve">in quo licet duo anguli <emph style="sc">A</emph>CD, & </s> <s xml:id="echoid-s17346" xml:space="preserve">D, dentur, cum <lb/>arcu AD, qui angulo ACD, opponitur; </s> <s xml:id="echoid-s17347" xml:space="preserve">non tameninde colligemus arcum <emph style="sc">Ac</emph>, alte-<lb/>ridato angulo D, oppoſitum, cum eidem opponatur in triangulo ABD, etiam arcus <lb/><emph style="sc">Ab</emph>, ipſi <emph style="sc">AC</emph>, inæqualis; </s> <s xml:id="echoid-s17348" xml:space="preserve">propterea quod eadem hypotheſis manet in triangulo ABD, <lb/>nempe anguli dati B, D, (cum angulus B, angulo <emph style="sc">Ac</emph>D, æqualis ſit, vt oſtendimus) <lb/>& </s> <s xml:id="echoid-s17349" xml:space="preserve">arcus datus <emph style="sc">A</emph>D, angulo <emph style="sc">B</emph>, oppoſitus. </s> <s xml:id="echoid-s17350" xml:space="preserve">Neceſſe eſt ergo, vt detur ſpecies arcus an-<lb/>gulo D, oppoſiti, vt ſciamus, num maior quadranteis ſit, an minor, hoc eſt, num ar-<lb/>cus <emph style="sc">AB,</emph> an AC, ſumendus ſit, cum vnus eorum maior quadrante ſit, & </s> <s xml:id="echoid-s17351" xml:space="preserve">alter mi-<lb/>nor, &</s> <s xml:id="echoid-s17352" xml:space="preserve">c.</s> <s xml:id="echoid-s17353" xml:space="preserve"/> </p> <div xml:id="echoid-div1363" type="float" level="2" n="1"> <note position="right" xlink:label="note-491-02" xlink:href="note-491-02a" xml:space="preserve">Etror Co-<lb/>pernici.</note> <note position="right" xlink:label="note-491-03" xlink:href="note-491-03a" xml:space="preserve">Nõ ſatis eſ-<lb/>ſe, dari duo@ <lb/>angulos, cũ <lb/>arcu vni co <lb/>rũ oppoſi-<lb/>to, ad reli-<lb/>qua ĩuenié <lb/>da in trian <lb/>gulo nõ re-<lb/>ctangulo.</note> <figure xlink:label="fig-491-01" xlink:href="fig-491-01a"> <image file="491-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/491-01"/> </figure> <note position="right" xlink:label="note-491-04" xlink:href="note-491-04a" xml:space="preserve">14. huius.</note> <note position="right" xlink:label="note-491-05" xlink:href="note-491-05a" xml:space="preserve">25. huius.</note> </div> <p style="it"> <s xml:id="echoid-s17354" xml:space="preserve"><emph style="sc">Ha</emph>C inre lapſus etiam eſt Ioan. </s> <s xml:id="echoid-s17355" xml:space="preserve">Regiom. </s> <s xml:id="echoid-s17356" xml:space="preserve">lib. </s> <s xml:id="echoid-s17357" xml:space="preserve">4. </s> <s xml:id="echoid-s17358" xml:space="preserve">triangulorum propoſ. </s> <s xml:id="echoid-s17359" xml:space="preserve">32. </s> <s xml:id="echoid-s17360" xml:space="preserve">cum <lb/> <anchor type="note" xlink:label="note-491-06a" xlink:href="note-491-06"/> vult ex duobus angulis datis, cum vno latere oppoſito, reliqua inuenire. </s> <s xml:id="echoid-s17361" xml:space="preserve">quod tamen <lb/>non ſatis eſſe, hic demonſtrauimus.</s> <s xml:id="echoid-s17362" xml:space="preserve"/> </p> <div xml:id="echoid-div1364" type="float" level="2" n="2"> <note position="right" xlink:label="note-491-06" xlink:href="note-491-06a" xml:space="preserve">Error Re-<lb/>giom.</note> </div> <pb o="480" file="492" n="492" rhead=""/> </div> <div xml:id="echoid-div1366" type="section" level="1" n="615"> <head xml:id="echoid-head650" xml:space="preserve">PROBL. 8. PROP. 67.</head> <p> <s xml:id="echoid-s17363" xml:space="preserve">DATIS duobus arcubus trianguli ſphærici <lb/>non rectanguli, cum angulo, qui alteri eorum op-<lb/>ponitur; </s> <s xml:id="echoid-s17364" xml:space="preserve">reliquos angulos, cum reliquo arcu inue-<lb/>nire. </s> <s xml:id="echoid-s17365" xml:space="preserve">Oportet autem conſtare, num angulus alte-<lb/>ri arcui dato oppoſitus acutus ſit, an obtuſus.</s> <s xml:id="echoid-s17366" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17367" xml:space="preserve">IN triangulo ſphærico non rectangulo ABC, dati ſint duo arcus AB, AC, <lb/>cum angulo B, qui arcui AC, opponitur, <lb/> <anchor type="note" xlink:label="note-492-01a" xlink:href="note-492-01"/> <anchor type="figure" xlink:label="fig-492-01a" xlink:href="fig-492-01"/> conſtetq́;</s> <s xml:id="echoid-s17368" xml:space="preserve">, an angulus C, acutus ſit, an obtu-<lb/>ſus. </s> <s xml:id="echoid-s17369" xml:space="preserve">Oportet ex his & </s> <s xml:id="echoid-s17370" xml:space="preserve">reliquos angulos C, <lb/>BAC, & </s> <s xml:id="echoid-s17371" xml:space="preserve">reliquum arcum BC, ſcrutari. </s> <s xml:id="echoid-s17372" xml:space="preserve">Sint <lb/>primum dati duo arcus AB, AC, inæquales, & </s> <s xml:id="echoid-s17373" xml:space="preserve"><lb/>neuter eorum quadrans. </s> <s xml:id="echoid-s17374" xml:space="preserve">Ducantur ab angulo <lb/>A, tertio arcui oppoſito ad ipſum arcum ter-<lb/>tium BC, arcus perpendicularis AD: </s> <s xml:id="echoid-s17375" xml:space="preserve">qui in-<lb/>tra triangulum cadet, ſi vterque angulus B, <lb/> <anchor type="note" xlink:label="note-492-02a" xlink:href="note-492-02"/> C, acutus eſt, vel obtuſus; </s> <s xml:id="echoid-s17376" xml:space="preserve">extra vero, ſi vnus <lb/>acutus, & </s> <s xml:id="echoid-s17377" xml:space="preserve">alter obtuſus fuerit: </s> <s xml:id="echoid-s17378" xml:space="preserve">conſtat autem <lb/>ex datis, an vterque angulus acutus ſit, obtu-<lb/>ſusve, an vnus acutus, & </s> <s xml:id="echoid-s17379" xml:space="preserve">obtuſus alter; </s> <s xml:id="echoid-s17380" xml:space="preserve">cum <lb/>datus ſit angulus B, cum ſpecie anguli C. </s> <s xml:id="echoid-s17381" xml:space="preserve">Ita-<lb/>que quoniam in triangulo ABD, rectum habente angulum D, datus eſt arcus <lb/> <anchor type="note" xlink:label="note-492-03a" xlink:href="note-492-03"/> AB, angulo recto oppoſitus, cum angulo B; </s> <s xml:id="echoid-s17382" xml:space="preserve">datus etiam erit arcus AD, circa <lb/>rectum angulum dato angulo B, oppoſitus. </s> <s xml:id="echoid-s17383" xml:space="preserve">Hinc in eodem triangulo ABD, <lb/>quia datus eſt arcus AB, recto angulo oppoſitus, & </s> <s xml:id="echoid-s17384" xml:space="preserve">inſuper arcus AD, circa <lb/> <anchor type="note" xlink:label="note-492-04a" xlink:href="note-492-04"/> eundem angulum rectum:</s> <s xml:id="echoid-s17385" xml:space="preserve"/> </p> <div xml:id="echoid-div1366" type="float" level="2" n="1"> <note position="left" xlink:label="note-492-01" xlink:href="note-492-01a" xml:space="preserve">Quando <lb/>aeuter da-<lb/>torũ arcuũ <lb/>inæqualiũ <lb/>oſt quadrás.</note> <figure xlink:label="fig-492-01" xlink:href="fig-492-01a"> <image file="492-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/492-01"/> </figure> <note position="left" xlink:label="note-492-02" xlink:href="note-492-02a" xml:space="preserve">57. huius.</note> <note position="left" xlink:label="note-492-03" xlink:href="note-492-03a" xml:space="preserve">Schol. 41. <lb/>huius.</note> <note position="left" xlink:label="note-492-04" xlink:href="note-492-04a" xml:space="preserve">Schol. 43. <lb/>vel 53. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17386" xml:space="preserve">VEL, quia datus eſt arcus AB, recto angulo op-<lb/> <anchor type="note" xlink:label="note-492-05a" xlink:href="note-492-05"/> poſitus, & </s> <s xml:id="echoid-s17387" xml:space="preserve">præterea angulus non rectus B:</s> <s xml:id="echoid-s17388" xml:space="preserve"/> </p> <div xml:id="echoid-div1367" type="float" level="2" n="2"> <note position="left" xlink:label="note-492-05" xlink:href="note-492-05a" xml:space="preserve">Schol. 45. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17389" xml:space="preserve">VEL denique, quia datus eſt arcus AD, circa an-<lb/> <anchor type="note" xlink:label="note-492-06a" xlink:href="note-492-06"/> gulum rectum, cum angulo B, oppoſito; </s> <s xml:id="echoid-s17390" xml:space="preserve">conſtatq́; </s> <s xml:id="echoid-s17391" xml:space="preserve">ſpe-<lb/>cies præterea arcus BD. </s> <s xml:id="echoid-s17392" xml:space="preserve">Nam ſi AB, datus fuerit mi-<lb/>nor quadrante; </s> <s xml:id="echoid-s17393" xml:space="preserve">ſi quidem & </s> <s xml:id="echoid-s17394" xml:space="preserve">AD, inuentus minor ſit, <lb/>erit quoque BD, minor; </s> <s xml:id="echoid-s17395" xml:space="preserve">ſi autem maior, maior. </s> <s xml:id="echoid-s17396" xml:space="preserve">At ſi <lb/> <anchor type="note" xlink:label="note-492-07a" xlink:href="note-492-07"/> AB, datus fuerit maior quadrante; </s> <s xml:id="echoid-s17397" xml:space="preserve">ſi quidem & </s> <s xml:id="echoid-s17398" xml:space="preserve">AD, <lb/>inuentus maior ſit, erit BD, minor; </s> <s xml:id="echoid-s17399" xml:space="preserve">ſi autem AD, mi-<lb/>nor ſit, erit BD, maior;</s> <s xml:id="echoid-s17400" xml:space="preserve"/> </p> <div xml:id="echoid-div1368" type="float" level="2" n="3"> <note position="left" xlink:label="note-492-06" xlink:href="note-492-06a" xml:space="preserve">Schol 49. <lb/>vel 44. huiꝰ.</note> <note position="left" xlink:label="note-492-07" xlink:href="note-492-07a" xml:space="preserve">36. huius.</note> </div> <p> <s xml:id="echoid-s17401" xml:space="preserve">cognitus etiam erit, ex adductis ſcholijs in margine, alter arcus BD, circa an-<lb/>gulum rectum. </s> <s xml:id="echoid-s17402" xml:space="preserve">Hinc rurſus in eodem triangulo ABD, quia datus eſt arcus <lb/> <anchor type="note" xlink:label="note-492-08a" xlink:href="note-492-08"/> AB, angulo recto oppoſitus, cum arcu BD, circa eundem rectum angulum:</s> <s xml:id="echoid-s17403" xml:space="preserve"/> </p> <div xml:id="echoid-div1369" type="float" level="2" n="4"> <note position="left" xlink:label="note-492-08" xlink:href="note-492-08a" xml:space="preserve">Schol. 41. <lb/>vel 55. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17404" xml:space="preserve">VEL, quia datus eſt vterque arcus AD, BD, cir-<lb/> <anchor type="note" xlink:label="note-492-09a" xlink:href="note-492-09"/> ca angulum rectum:</s> <s xml:id="echoid-s17405" xml:space="preserve"/> </p> <div xml:id="echoid-div1370" type="float" level="2" n="5"> <note position="left" xlink:label="note-492-09" xlink:href="note-492-09a" xml:space="preserve">Schol 44. <lb/>vel 48. huiꝰ.</note> </div> <pb o="481" file="493" n="493" rhead=""/> <p> <s xml:id="echoid-s17406" xml:space="preserve">VEL, quoniam datus eſt arcus AB, recto angulo <lb/> <anchor type="note" xlink:label="note-493-01a" xlink:href="note-493-01"/> oppoſitus, cum arcu AD, circa eundem angulũ rectum:</s> <s xml:id="echoid-s17407" xml:space="preserve"/> </p> <div xml:id="echoid-div1371" type="float" level="2" n="6"> <note position="right" xlink:label="note-493-01" xlink:href="note-493-01a" xml:space="preserve">Schol. 45. <lb/>vel 51. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17408" xml:space="preserve">VEL, quia datus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-493-02a" xlink:href="note-493-02"/> ctum, cum angulo oppoſito B, conſtatq́; </s> <s xml:id="echoid-s17409" xml:space="preserve">præterea ſpe-<lb/>cies anguli BAD. </s> <s xml:id="echoid-s17410" xml:space="preserve">Nam ſi BD, arcus inuentus ſit mi-<lb/>nor quadrante, erit angulus BAD, acutus; </s> <s xml:id="echoid-s17411" xml:space="preserve">obtuſus <lb/> <anchor type="note" xlink:label="note-493-03a" xlink:href="note-493-03"/> vero, ſi maior:</s> <s xml:id="echoid-s17412" xml:space="preserve"/> </p> <div xml:id="echoid-div1372" type="float" level="2" n="7"> <note position="right" xlink:label="note-493-02" xlink:href="note-493-02a" xml:space="preserve">Schol. 56. <lb/>vel 42. huiꝰ.</note> <note position="right" xlink:label="note-493-03" xlink:href="note-493-03a" xml:space="preserve">34. huius.</note> </div> <p> <s xml:id="echoid-s17413" xml:space="preserve">VEL denique, quoniam datus eſt arcus AB, an-<lb/> <anchor type="note" xlink:label="note-493-04a" xlink:href="note-493-04"/> gulo recto oppoſitus, & </s> <s xml:id="echoid-s17414" xml:space="preserve">inſuper angulus non rectus B;</s> <s xml:id="echoid-s17415" xml:space="preserve"/> </p> <div xml:id="echoid-div1373" type="float" level="2" n="8"> <note position="right" xlink:label="note-493-04" xlink:href="note-493-04a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17416" xml:space="preserve">efficietur quoq; </s> <s xml:id="echoid-s17417" xml:space="preserve">notus, ex ſcholijs in margine poſitis, angulus nõ rectus BAD.</s> <s xml:id="echoid-s17418" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17419" xml:space="preserve">DEINDE, quia in triangulo ACD, rectum habente angulum D, da-<lb/>tus eſt arcus AC, recto angulo oppoſitus, & </s> <s xml:id="echoid-s17420" xml:space="preserve">inuentus arcus AD, circa angu-<lb/>lum rectum; </s> <s xml:id="echoid-s17421" xml:space="preserve">cognoſcetur quoque angulus CAD, à dictis arcubus comprehen <lb/> <anchor type="note" xlink:label="note-493-05a" xlink:href="note-493-05"/> ſus: </s> <s xml:id="echoid-s17422" xml:space="preserve">qui additus inuento angulo BAD, vel ab eo ſubtractus, prout arcus per-<lb/>pendicularis AD, intra triangulum cadit, aut extra, (quod quidem cogno-<lb/>ſcemus, vt ad initium diximus, ex dato angulo B, & </s> <s xml:id="echoid-s17423" xml:space="preserve">ſpecie data anguli C.) </s> <s xml:id="echoid-s17424" xml:space="preserve">da-<lb/>bit quæſitum angulum BAC.</s> <s xml:id="echoid-s17425" xml:space="preserve"/> </p> <div xml:id="echoid-div1374" type="float" level="2" n="9"> <note position="right" xlink:label="note-493-05" xlink:href="note-493-05a" xml:space="preserve">Schol. 51. <lb/>vel45. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17426" xml:space="preserve">RVRSVS, quoniam in eodem triangulo ACD, datus eſt arcus AC, an-<lb/> <anchor type="note" xlink:label="note-493-06a" xlink:href="note-493-06"/> gulo recto oppoſitus, & </s> <s xml:id="echoid-s17427" xml:space="preserve">inuentus arcus AD, circa rectum angulum:</s> <s xml:id="echoid-s17428" xml:space="preserve"/> </p> <div xml:id="echoid-div1375" type="float" level="2" n="10"> <note position="right" xlink:label="note-493-06" xlink:href="note-493-06a" xml:space="preserve">Schol. 55. <lb/>vel41. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17429" xml:space="preserve">VEL, quia datus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-493-07a" xlink:href="note-493-07"/> ctum, & </s> <s xml:id="echoid-s17430" xml:space="preserve">inſuper angulus non rectus CAD:</s> <s xml:id="echoid-s17431" xml:space="preserve"/> </p> <div xml:id="echoid-div1376" type="float" level="2" n="11"> <note position="right" xlink:label="note-493-07" xlink:href="note-493-07a" xml:space="preserve">Schol. 42. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17432" xml:space="preserve">AVT denique, quoniam datus eſt arcus AC, an-<lb/> <anchor type="note" xlink:label="note-493-08a" xlink:href="note-493-08"/> gulo recto oppoſitus, & </s> <s xml:id="echoid-s17433" xml:space="preserve">præterea angulus non re-<lb/>ctus CAD;</s> <s xml:id="echoid-s17434" xml:space="preserve"/> </p> <div xml:id="echoid-div1377" type="float" level="2" n="12"> <note position="right" xlink:label="note-493-08" xlink:href="note-493-08a" xml:space="preserve">Schol. 47. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17435" xml:space="preserve">cognitus quoque erit angulus ACD. </s> <s xml:id="echoid-s17436" xml:space="preserve">Si igitur arcus perpendicularis AD, ca-<lb/>dit intra triangulum, inuentus angulus erit ACB, qui quæritur; </s> <s xml:id="echoid-s17437" xml:space="preserve">ſi vero ca-<lb/>dit extra, angulus inuentus ACD, demptus ex duobus rectis, notum relin-<lb/>quet quæſitum angulum ACB. </s> <s xml:id="echoid-s17438" xml:space="preserve">Qui quidem angulus ACB, ita quoque re-<lb/>perietur, licet arcus AD, non adeſſet. </s> <s xml:id="echoid-s17439" xml:space="preserve">Quoniam eſt, vt ſinus arcus AC, ad <lb/> <anchor type="note" xlink:label="note-493-09a" xlink:href="note-493-09"/> ſinum anguli B, ita ſinus arcus AB, ad ſinum arcus ACB: </s> <s xml:id="echoid-s17440" xml:space="preserve">ſi fiat, vt ſinus da-<lb/>ti arcus dato angulo oppoſiti ad ſinum dati anguli, ita ſinus alterius arcus da-<lb/>ti ad aliud, producetur ſinus anguli huic arcui oppoſiti; </s> <s xml:id="echoid-s17441" xml:space="preserve">ac proinde angulus <lb/>ipſe ACB, cognitus erit, cum conſtet eius ſpecies. </s> <s xml:id="echoid-s17442" xml:space="preserve">Atq; </s> <s xml:id="echoid-s17443" xml:space="preserve">ita inuenti iam ſunt <lb/>reliqui duo anguli BAC, ACB.</s> <s xml:id="echoid-s17444" xml:space="preserve"/> </p> <div xml:id="echoid-div1378" type="float" level="2" n="13"> <note position="right" xlink:label="note-493-09" xlink:href="note-493-09a" xml:space="preserve">41. huius.</note> </div> <p> <s xml:id="echoid-s17445" xml:space="preserve">QVONIAM denique in eodem triangulo ACD, datus eſt arcus AC, <lb/> <anchor type="note" xlink:label="note-493-10a" xlink:href="note-493-10"/> angulo recto oppoſitus, cum angulo CAD, proxime inuento:</s> <s xml:id="echoid-s17446" xml:space="preserve"/> </p> <div xml:id="echoid-div1379" type="float" level="2" n="14"> <note position="right" xlink:label="note-493-10" xlink:href="note-493-10a" xml:space="preserve">Schol. 41. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17447" xml:space="preserve">VEL, quia datus eſt vterque angulus non rectus <lb/> <anchor type="note" xlink:label="note-493-11a" xlink:href="note-493-11"/> ACD, CAD:</s> <s xml:id="echoid-s17448" xml:space="preserve"/> </p> <div xml:id="echoid-div1380" type="float" level="2" n="15"> <note position="right" xlink:label="note-493-11" xlink:href="note-493-11a" xml:space="preserve">Schol. 42. <lb/>vel 52. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17449" xml:space="preserve">VEL, quia datus eſt arcus AC, recto angulo oppo-<lb/> <anchor type="note" xlink:label="note-493-12a" xlink:href="note-493-12"/> ſitus, cum arcu AD, circa angulum rectum:</s> <s xml:id="echoid-s17450" xml:space="preserve"/> </p> <div xml:id="echoid-div1381" type="float" level="2" n="16"> <note position="right" xlink:label="note-493-12" xlink:href="note-493-12a" xml:space="preserve">Schol. 43. <lb/>vel 53. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17451" xml:space="preserve">VEL, quia datus eſt arcus AD, circa angulum re-<lb/> <anchor type="note" xlink:label="note-493-13a" xlink:href="note-493-13"/> ctum, cum angulo non recto CAD:</s> <s xml:id="echoid-s17452" xml:space="preserve"/> </p> <div xml:id="echoid-div1382" type="float" level="2" n="17"> <note position="right" xlink:label="note-493-13" xlink:href="note-493-13a" xml:space="preserve">Schol. 44. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17453" xml:space="preserve">VEL, quoniam datus eſt arcus AD, circa rectum <lb/> <anchor type="note" xlink:label="note-493-14a" xlink:href="note-493-14"/> angulum, cum angulo oppoſito ACD; </s> <s xml:id="echoid-s17454" xml:space="preserve">conſtatq́; </s> <s xml:id="echoid-s17455" xml:space="preserve">præ-<lb/>terea ſpecies alterius arcus CD, circa rectum angulum. <lb/></s> <s xml:id="echoid-s17456" xml:space="preserve">Exiſtente enim angulo inuento CAD, acuto, erit ar-<lb/>cus CD, quadrante minor; </s> <s xml:id="echoid-s17457" xml:space="preserve">maior autem, ſi obtuſus.</s> <s xml:id="echoid-s17458" xml:space="preserve"/> </p> <div xml:id="echoid-div1383" type="float" level="2" n="18"> <note position="right" xlink:label="note-493-14" xlink:href="note-493-14a" xml:space="preserve">Schol. 49. <lb/>vel 44. huiꝰ.</note> </div> <note position="right" xml:space="preserve">34. huius.</note> <p> <s xml:id="echoid-s17459" xml:space="preserve">VEL denique, quia datus eſt arcus AC, recto an-<lb/> <anchor type="note" xlink:label="note-493-16a" xlink:href="note-493-16"/> <pb o="482" file="494" n="494" rhead=""/> gulo oppoſitus, cum angulo non recto ACD;</s> <s xml:id="echoid-s17460" xml:space="preserve"/> </p> <div xml:id="echoid-div1384" type="float" level="2" n="19"> <note position="right" xlink:label="note-493-16" xlink:href="note-493-16a" xml:space="preserve">Schol. 45. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17461" xml:space="preserve">reperietur quoque, per ſcholia in margine adducta, arcus CD, circa rectum <lb/>angulum: </s> <s xml:id="echoid-s17462" xml:space="preserve">qui vel additus arcui BD, iam dudum inuento, vel ab eo ſubductus, <lb/>(prout nimirum arcus perpendicularis AD, intra triangulum ceciderit, vel <lb/>extra) dabit arcum BC, in propoſito triangulo ABC, quæſitum.</s> <s xml:id="echoid-s17463" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17464" xml:space="preserve">PORRO nulla ratione alteruter arcuum AD, BD, in hoc caſu, qua-<lb/>drans eſſe poteſt: </s> <s xml:id="echoid-s17465" xml:space="preserve">quia alioquin & </s> <s xml:id="echoid-s17466" xml:space="preserve">arcus AB, recto angulo D, oppoſitus eſſet <lb/> <anchor type="note" xlink:label="note-494-01a" xlink:href="note-494-01"/> quadrans, quod eſt contra hypotheſim. </s> <s xml:id="echoid-s17467" xml:space="preserve">Eadem ratione neque CD, quadrans <lb/>erit, ne & </s> <s xml:id="echoid-s17468" xml:space="preserve">arcus AC, angulo recto oppoſitus quadrans ſit, quod eſſet etiam <lb/>contra hypotheſim.</s> <s xml:id="echoid-s17469" xml:space="preserve"/> </p> <div xml:id="echoid-div1385" type="float" level="2" n="20"> <note position="left" xlink:label="note-494-01" xlink:href="note-494-01a" xml:space="preserve">35. huius.</note> </div> <p style="it"> <s xml:id="echoid-s17470" xml:space="preserve">PRAXIS huius problematis petatur ex ſcholijs in margine poſitis.</s> <s xml:id="echoid-s17471" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Praxis per <lb/>ſolos ſinus, <lb/>quãdoneu-<lb/>ter datorũ <lb/>arcuum in <lb/>ęqualiũ eſt <lb/>quadrans.</note> <p style="it"> <s xml:id="echoid-s17472" xml:space="preserve">SED per ſolos ſinus ita problema abſoluetur. </s> <s xml:id="echoid-s17473" xml:space="preserve">Per praxim problema-<lb/>tis 2. </s> <s xml:id="echoid-s17474" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17475" xml:space="preserve">41. </s> <s xml:id="echoid-s17476" xml:space="preserve">inuenietur arcus AD, in triangulo ABD: </s> <s xml:id="echoid-s17477" xml:space="preserve">Et hinc <lb/>per praxim problematis ſcholij propoſ. </s> <s xml:id="echoid-s17478" xml:space="preserve">43. </s> <s xml:id="echoid-s17479" xml:space="preserve">arcus BD. </s> <s xml:id="echoid-s17480" xml:space="preserve">Deinde per praxim <lb/>problematis 2. </s> <s xml:id="echoid-s17481" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17482" xml:space="preserve">42. </s> <s xml:id="echoid-s17483" xml:space="preserve">reperietur angulus BAD.</s> <s xml:id="echoid-s17484" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17485" xml:space="preserve">POST hæc in triangulo ACD, per praxim problematis 1. </s> <s xml:id="echoid-s17486" xml:space="preserve">ſcholij pro-<lb/>poſ. </s> <s xml:id="echoid-s17487" xml:space="preserve">41. </s> <s xml:id="echoid-s17488" xml:space="preserve">cognitus erit angulus ACD, qui e§t vnus quæſitorum, ſi conſtet, <lb/>angulum C, eiuſdem eſſe ſpeciei cum angulo dato B; </s> <s xml:id="echoid-s17489" xml:space="preserve">ſi vero diuerſæ, veli-<lb/>quus duorum rectorum erit angulus ACB, quæſitus, quia ibi arcus per-<lb/> <anchor type="note" xlink:label="note-494-03a" xlink:href="note-494-03"/> pendicularis intra triangulum cadit, hic vero extra. </s> <s xml:id="echoid-s17490" xml:space="preserve">Rurſus per praxim <lb/>problematis ſcholij propoſ. </s> <s xml:id="echoid-s17491" xml:space="preserve">43, notus efficietur arcus CD, qui in priori <lb/>triangulo additus inuento arcui BD, in poſteriori vero ex eodem ſublatus <lb/>exhibebit reliquum arcum BC, in propoſito triangulo notum. </s> <s xml:id="echoid-s17492" xml:space="preserve">Ad extre-<lb/>mum, per praxim problematis 1. </s> <s xml:id="echoid-s17493" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17494" xml:space="preserve">41. </s> <s xml:id="echoid-s17495" xml:space="preserve">reperietur angu-<lb/>lus CAD, qui additus, vel ſubductus ex inuento angulo BAD, tertium <lb/>angulum BAC, qui quæritur, notum efficiet.</s> <s xml:id="echoid-s17496" xml:space="preserve"/> </p> <div xml:id="echoid-div1386" type="float" level="2" n="21"> <note position="left" xlink:label="note-494-03" xlink:href="note-494-03a" xml:space="preserve">57. huius.</note> </div> <note position="left" xml:space="preserve">Quando al <lb/>ter duorũ <lb/>arcuũ inæ-<lb/>qualiũ da-<lb/>torum eſt <lb/>guadrans.</note> <p> <s xml:id="echoid-s17497" xml:space="preserve">QVOD ſi alter arcuum datorum inæqualium AB, AC, ſit quadrans; </s> <s xml:id="echoid-s17498" xml:space="preserve">ſi <lb/>quidem AB, quadrans fuerit, erit quoque BD, quadrans, & </s> <s xml:id="echoid-s17499" xml:space="preserve">angulus BAD, <lb/>rectus, necnon B, polus arcus AD; </s> <s xml:id="echoid-s17500" xml:space="preserve">atque adeo angulus datus B, eundem ar-<lb/>cum AD, notum exhibebit, vt in præcedenti propoſ. </s> <s xml:id="echoid-s17501" xml:space="preserve">oſtendimus, quando ar-<lb/>cus AB, ponebatur eſſe quadrans. </s> <s xml:id="echoid-s17502" xml:space="preserve">Inuentis igitur arcubus AD, BD, & </s> <s xml:id="echoid-s17503" xml:space="preserve">angu <lb/>lo BAD, ſine vllo negotio, reliqua inueniemus, vt prius. </s> <s xml:id="echoid-s17504" xml:space="preserve">Si vero arcus AC, <lb/>ſit quadrans, erit eadem ratione CD, quadrans, & </s> <s xml:id="echoid-s17505" xml:space="preserve">angulus CAD, rectus, nec <lb/>non C, polus arcus AD; </s> <s xml:id="echoid-s17506" xml:space="preserve">atque adeo inuentus arcus AD, angulum ACD, no-<lb/> <anchor type="figure" xlink:label="fig-494-01a" xlink:href="fig-494-01"/> tum faciet: </s> <s xml:id="echoid-s17507" xml:space="preserve">qui vnus erit ex quæſitis, quando arcus AD, <lb/>cadit intra triangulum; </s> <s xml:id="echoid-s17508" xml:space="preserve">ſi vero extra, reliquus duorum re <lb/>ctorum dabit angulum quæſitum ACB. </s> <s xml:id="echoid-s17509" xml:space="preserve">Atq; </s> <s xml:id="echoid-s17510" xml:space="preserve">ita inuen-<lb/>tus tunc erit, ſine multiplicatione vlla, & </s> <s xml:id="echoid-s17511" xml:space="preserve">arcus CD, & </s> <s xml:id="echoid-s17512" xml:space="preserve"><lb/>angulus CAD, necnon angulus ACD: </s> <s xml:id="echoid-s17513" xml:space="preserve">ex quibus repe-<lb/>rientur reliqua, vt prius.</s> <s xml:id="echoid-s17514" xml:space="preserve"/> </p> <div xml:id="echoid-div1387" type="float" level="2" n="22"> <figure xlink:label="fig-494-01" xlink:href="fig-494-01a"> <image file="494-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/494-01"/> </figure> </div> <p> <s xml:id="echoid-s17515" xml:space="preserve">SINT iam dati duo arcus AB, AC, æquales. </s> <s xml:id="echoid-s17516" xml:space="preserve">Erunt <lb/> <anchor type="note" xlink:label="note-494-05a" xlink:href="note-494-05"/> duo anguli B, C, æquales; </s> <s xml:id="echoid-s17517" xml:space="preserve">& </s> <s xml:id="echoid-s17518" xml:space="preserve">arcus perpendicularis AD, ex <lb/>A, in BC, demiſlus intra triangulum cadet; </s> <s xml:id="echoid-s17519" xml:space="preserve">necnon & </s> <s xml:id="echoid-s17520" xml:space="preserve">ar-<lb/>cus BD, CD, & </s> <s xml:id="echoid-s17521" xml:space="preserve">anguli ad A, ęquales erunt, vt in vltimo caſu propoſ. </s> <s xml:id="echoid-s17522" xml:space="preserve">63. </s> <s xml:id="echoid-s17523" xml:space="preserve">oſten <pb o="483" file="495" n="495" rhead=""/> dimus. </s> <s xml:id="echoid-s17524" xml:space="preserve">Cum ergo angulus B, datus ſit, erit quoque C, illi æqualis, datus. </s> <s xml:id="echoid-s17525" xml:space="preserve">Dein-<lb/>de quia in triangulo ABD, habente rectum angulum D, datus eſt arcus AB, <lb/>angulo recto oppoſitus, cum angulo B; </s> <s xml:id="echoid-s17526" xml:space="preserve">dabitur quoque angulus BAD: </s> <s xml:id="echoid-s17527" xml:space="preserve">qui <lb/> <anchor type="note" xlink:label="note-495-01a" xlink:href="note-495-01"/> duplicatus totum angulum BAC, quæſitum offeret notum. </s> <s xml:id="echoid-s17528" xml:space="preserve">Hinc, quoniam in <lb/>eodem triangulo ABD, datus eſt arcus AB, recto angulo oppoſitus, cum an-<lb/> <anchor type="note" xlink:label="note-495-02a" xlink:href="note-495-02"/> gulo BAD, inuento:</s> <s xml:id="echoid-s17529" xml:space="preserve"/> </p> <div xml:id="echoid-div1388" type="float" level="2" n="23"> <note position="left" xlink:label="note-494-05" xlink:href="note-494-05a" xml:space="preserve">Quãdo da <lb/>ti duo arcꝰ <lb/>ęquales sũt.</note> <note position="right" xlink:label="note-495-01" xlink:href="note-495-01a" xml:space="preserve">Schol. 47. <lb/>huius.</note> <note position="right" xlink:label="note-495-02" xlink:href="note-495-02a" xml:space="preserve">Schol. 41. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17530" xml:space="preserve">VEL, quia vterq; </s> <s xml:id="echoid-s17531" xml:space="preserve">angulus non rectus B, & </s> <s xml:id="echoid-s17532" xml:space="preserve">BAD, <lb/> <anchor type="note" xlink:label="note-495-03a" xlink:href="note-495-03"/> datus eſt:</s> <s xml:id="echoid-s17533" xml:space="preserve"/> </p> <div xml:id="echoid-div1389" type="float" level="2" n="24"> <note position="right" xlink:label="note-495-03" xlink:href="note-495-03a" xml:space="preserve">Schol. 42. <lb/>vel52. huiꝰ.</note> </div> <p> <s xml:id="echoid-s17534" xml:space="preserve">VEL denique, quia datus eſt arcus AB, angulo re-<lb/> <anchor type="note" xlink:label="note-495-04a" xlink:href="note-495-04"/> cto oppoſitus, cum angulo B, non recto;</s> <s xml:id="echoid-s17535" xml:space="preserve"/> </p> <div xml:id="echoid-div1390" type="float" level="2" n="25"> <note position="right" xlink:label="note-495-04" xlink:href="note-495-04a" xml:space="preserve">Schol. 45. <lb/>huius.</note> </div> <p> <s xml:id="echoid-s17536" xml:space="preserve">cognoſcetur quoque, per ſcholia in margine allata, arcus BD, circa angulum <lb/>rectum; </s> <s xml:id="echoid-s17537" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s17538" xml:space="preserve">eius duplus BC, qui in quirendus proponitur.</s> <s xml:id="echoid-s17539" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17540" xml:space="preserve">PRAXIS facile colligi poteſt ex ſcholijs in margine appoſitis.</s> <s xml:id="echoid-s17541" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s17542" xml:space="preserve">SI vero ſolos ſinus adhibere malueris; </s> <s xml:id="echoid-s17543" xml:space="preserve">inueniendus primum erit ar-<lb/> <anchor type="note" xlink:label="note-495-05a" xlink:href="note-495-05"/> cus AD, per praxim problematis 2. </s> <s xml:id="echoid-s17544" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17545" xml:space="preserve">41. </s> <s xml:id="echoid-s17546" xml:space="preserve">Atque hinc per <lb/>praxim problematis ſcholij propoſ. </s> <s xml:id="echoid-s17547" xml:space="preserve">43. </s> <s xml:id="echoid-s17548" xml:space="preserve">arcus BD: </s> <s xml:id="echoid-s17549" xml:space="preserve">qui duplicatus totum <lb/>quæſitum BC, dabit. </s> <s xml:id="echoid-s17550" xml:space="preserve">Deinde per praxim problematis 1. </s> <s xml:id="echoid-s17551" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17552" xml:space="preserve">41. <lb/></s> <s xml:id="echoid-s17553" xml:space="preserve">vel per praxim problematis 2. </s> <s xml:id="echoid-s17554" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s17555" xml:space="preserve">42. </s> <s xml:id="echoid-s17556" xml:space="preserve">reperiendus angulus <lb/>BAD; </s> <s xml:id="echoid-s17557" xml:space="preserve">ex quo eius duplus BAC, quem quærimus, notus erit: </s> <s xml:id="echoid-s17558" xml:space="preserve">tertius au-<lb/>tem angulus C, iam datus eſt, cum æqualis ſit dato angulo B.</s> <s xml:id="echoid-s17559" xml:space="preserve"/> </p> <div xml:id="echoid-div1391" type="float" level="2" n="26"> <note position="right" xlink:label="note-495-05" xlink:href="note-495-05a" xml:space="preserve">Praxis, per <lb/>ſolos ſinus, <lb/>quãdo duo <lb/>arcus dati <lb/>ęquales sũt.</note> </div> <p> <s xml:id="echoid-s17560" xml:space="preserve">DATIS igitur duobus arcubus trianguli ſphærici non rectanguli, cuns <lb/>vno angulo, qui alteri eorum opponitur, &</s> <s xml:id="echoid-s17561" xml:space="preserve">c. </s> <s xml:id="echoid-s17562" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s17563" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1393" type="section" level="1" n="616"> <head xml:id="echoid-head651" xml:space="preserve">SCHOLIVM. I.</head> <p style="it"> <s xml:id="echoid-s17564" xml:space="preserve">NECESSE eſt autem conſtare in hoc problemate, num angulus <emph style="sc">C</emph>, alteri da-<lb/>to arcui oppoſitus ſit acutus, obtuſus ve, vt ſciatur, num perpendicularis arcus AD, <lb/>intra triangulum cadat, nec ne. </s> <s xml:id="echoid-s17565" xml:space="preserve">Hoc enim ignorato, neſciremus, an angulus CAD, <lb/>addendus ſit angulo <emph style="sc">Ba</emph>D, an ab eo ſubtrahendus, vt inueniatur angulus <emph style="sc">Ba</emph>C, quæ-<lb/>ſitus: </s> <s xml:id="echoid-s17566" xml:space="preserve">Item an arcus CD, arcui BD, ſit adij ciendus, an ſubducendus ex eo, vt ar-<lb/>cus quæſitus <emph style="sc">B</emph>C, reperiatur. </s> <s xml:id="echoid-s17567" xml:space="preserve">Vel denique, num angulus inuentus ACD, ſit is, qui <lb/>quæritur, an vero reliquus duorum rectorũ, vt manife§tum eſt. </s> <s xml:id="echoid-s17568" xml:space="preserve">Non eſſe porro ſatis, <lb/>ſi duo arcus dentur, cum angulo vni eorum oppoſito, ad inquirendos reliquos angu-<lb/>los, cum reliquo arcu, iam dudum ſupra docuimus in ſcholio propoſ. </s> <s xml:id="echoid-s17569" xml:space="preserve">24. </s> <s xml:id="echoid-s17570" xml:space="preserve">Qua in re <lb/> <anchor type="note" xlink:label="note-495-06a" xlink:href="note-495-06"/> Nicolaum Copernicum erraſſe lib. </s> <s xml:id="echoid-s17571" xml:space="preserve">1. </s> <s xml:id="echoid-s17572" xml:space="preserve">Reuolutionum, pro-<lb/>poſ. </s> <s xml:id="echoid-s17573" xml:space="preserve">11. </s> <s xml:id="echoid-s17574" xml:space="preserve">triang. </s> <s xml:id="echoid-s17575" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s17576" xml:space="preserve">ibidem monuimus. </s> <s xml:id="echoid-s17577" xml:space="preserve">Quod tamen bre <lb/> <anchor type="figure" xlink:label="fig-495-01a" xlink:href="fig-495-01"/> <anchor type="note" xlink:label="note-495-07a" xlink:href="note-495-07"/> uiter ita hic rurſus oſtendemus. </s> <s xml:id="echoid-s17578" xml:space="preserve">Sint duo arcus æquales <lb/>AD, AC, angulum DAC, ambientes, & </s> <s xml:id="echoid-s17579" xml:space="preserve">vterque qua-<lb/>drante minor, aut maior. </s> <s xml:id="echoid-s17580" xml:space="preserve">Ducto autem per C, D, arcu <lb/>circuli maximi CD, ducatur ad eum productum alius ar <lb/>cus <emph style="sc">AB</emph>, ex A, neque per polos arcus CD, neque per po-<lb/>los arcus AD, ita vt anguli B, & </s> <s xml:id="echoid-s17581" xml:space="preserve">DAB, ſint non recti. <lb/></s> <s xml:id="echoid-s17582" xml:space="preserve">Sed neque angulus ADB, rectus eſt. </s> <s xml:id="echoid-s17583" xml:space="preserve">Si namque vterque <lb/>arcus AD, AC, minor eſt quadrante, erunt duo anguli C, & </s> <s xml:id="echoid-s17584" xml:space="preserve">ADC, acuti; </s> <s xml:id="echoid-s17585" xml:space="preserve">ſi vero <lb/> <anchor type="note" xlink:label="note-495-08a" xlink:href="note-495-08"/> vterq; </s> <s xml:id="echoid-s17586" xml:space="preserve">arcus AD, <emph style="sc">A</emph>C, quadrante maior eſt, erunt duo anguli C, & </s> <s xml:id="echoid-s17587" xml:space="preserve">ADC, obtuſi.</s> <s xml:id="echoid-s17588" xml:space="preserve"> <pb o="484" file="496" n="496" rhead=""/> Ex quo fit, angulum <emph style="sc">ADb</emph>, eſſe vel obtuſum, quando nimirum <emph style="sc">A</emph>DC, acutus eſts <lb/> <anchor type="note" xlink:label="note-496-01a" xlink:href="note-496-01"/> vel acutum, quando videlicet ADC, eſt obtuſus: </s> <s xml:id="echoid-s17589" xml:space="preserve">cum duo anguli ad D, duobus re-<lb/>ctis ſint æquales. </s> <s xml:id="echoid-s17590" xml:space="preserve">Triangulum ergo <emph style="sc">Ab</emph>D, non rectangulum e§t: </s> <s xml:id="echoid-s17591" xml:space="preserve">in quo licet duo ar-<lb/> <anchor type="figure" xlink:label="fig-496-01a" xlink:href="fig-496-01"/> cus <emph style="sc">A</emph>B, AD, dentur, cum angulo B, qui arcui AD, oppo <lb/>nitur; </s> <s xml:id="echoid-s17592" xml:space="preserve">non tamen inde colligere poterimus reliquum an-<lb/>gulum alteri arcui dato AB, oppoſitum eſſe <emph style="sc">A</emph><emph style="sc">Db</emph>; </s> <s xml:id="echoid-s17593" xml:space="preserve">cum <lb/>eidem arcui AB, opponatur quoque in triangulo <emph style="sc">Ab</emph>C, <lb/>angulus C, ipſi ADC, inæqualis: </s> <s xml:id="echoid-s17594" xml:space="preserve">propterea quòd in trian <lb/>gulo <emph style="sc">Abc</emph>, eadem hypotheſis manet, nempe arcus duo da <lb/>ti AB, <emph style="sc">AC</emph>, (ponitur enim arcus AC, arcui <emph style="sc">A</emph>D, æqua-<lb/>lis) & </s> <s xml:id="echoid-s17595" xml:space="preserve">angulus datus <emph style="sc">B</emph>, arcui <emph style="sc">AC</emph>, oppoſitus. </s> <s xml:id="echoid-s17596" xml:space="preserve">Oportet <lb/>ergo conſtare, num angulus alteri arcui <emph style="sc">Ab</emph>, oppoſitus <lb/>ſit acutus, aut obtuſus, hoc eſt, an ſumendus ſit angulus <emph style="sc">A</emph><emph style="sc">Db</emph>, an vero C, cum vnus <lb/>obtuſus ſit, & </s> <s xml:id="echoid-s17597" xml:space="preserve">alter acutus, &</s> <s xml:id="echoid-s17598" xml:space="preserve">c.</s> <s xml:id="echoid-s17599" xml:space="preserve"/> </p> <div xml:id="echoid-div1393" type="float" level="2" n="1"> <note position="right" xlink:label="note-495-06" xlink:href="note-495-06a" xml:space="preserve">Error Co-<lb/>pernici.</note> <figure xlink:label="fig-495-01" xlink:href="fig-495-01a"> <image file="495-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/495-01"/> </figure> <note position="right" xlink:label="note-495-07" xlink:href="note-495-07a" xml:space="preserve">Nõ ſatis eſ-<lb/>ſe, dari duos <lb/>arcus, cum <lb/>angulo vni <lb/>eorũ oppo <lb/>ſito, in triã <lb/>gulo nõ re-<lb/>ctãgulo, vt <lb/>reliqua in-<lb/>ueniantur.</note> <note position="right" xlink:label="note-495-08" xlink:href="note-495-08a" xml:space="preserve">25. huius.</note> <note position="left" xlink:label="note-496-01" xlink:href="note-496-01a" xml:space="preserve">5. huius.</note> <figure xlink:label="fig-496-01" xlink:href="fig-496-01a"> <image file="496-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/496-01"/> </figure> </div> </div> <div xml:id="echoid-div1395" type="section" level="1" n="617"> <head xml:id="echoid-head652" xml:space="preserve">SCHOLIVM. II.</head> <p style="it"> <s xml:id="echoid-s17600" xml:space="preserve"><emph style="sc">Hactenvs</emph> demonſtrauimus ea, quæ ad triangulorum calculum requirun-<lb/>tur, pluribus ſane propoſitionibus, & </s> <s xml:id="echoid-s17601" xml:space="preserve">fortaſſe longioribus, quam in calculo, qui fa-<lb/>cilis & </s> <s xml:id="echoid-s17602" xml:space="preserve">breuis eſſe debet, quis deſideret. </s> <s xml:id="echoid-s17603" xml:space="preserve">Quare operæ pretium me facturum arbitror, <lb/>ſi Epilogi loco praxes omnium problematum, quæ in triangulis rectilineis, & </s> <s xml:id="echoid-s17604" xml:space="preserve">ſphæ-<lb/>ricis demon ſtratæ ſunt, ſeorſum hic, in vnum quaſi locum congeſtas, deſcribam; </s> <s xml:id="echoid-s17605" xml:space="preserve">vt in <lb/>promptu eas ſemper, & </s> <s xml:id="echoid-s17606" xml:space="preserve">quaſi ad manus habeamus, quando vſurpandæ ſunt, ne ſru-<lb/>ſtrain eis è tanta propoſitionum multitudine ſeligendis tempus teramus. </s> <s xml:id="echoid-s17607" xml:space="preserve">In margine <lb/>porrò propoſitiones, ac problemata, in quibus earum demonſtrationes continentur, <lb/>adducemus, vt facile à quovis, cum res exiget, poſsint reperiri. </s> <s xml:id="echoid-s17608" xml:space="preserve">Itaque quod ad cal <lb/>culum triangulorum attinet, ſatis erit, ſi pauca hæc, quæ ſequuntur, attente, cum <lb/>opus fuerit, perlegantur. </s> <s xml:id="echoid-s17609" xml:space="preserve">In eis enim ſumma omnium, quæ de triangulis demonſtra-<lb/>uimus, comprehenditur. </s> <s xml:id="echoid-s17610" xml:space="preserve">Quamuis autem in triangulis ſphæricis non rectangulis ple-<lb/>runque arcus, & </s> <s xml:id="echoid-s17611" xml:space="preserve">angules triangulorum, in quæ triangula non rectangula reſol-<lb/>uimus, pluribus vijs inueſtigauerimus, in praxibus tamen ſequẽtibus, vt omnem con-<lb/>fuſionem vitaremus, vnam tantum in quouis arcu, ſiue angulo inquirendo, quam vi-<lb/>delicet iudicauimus eſſe commodiorem, delegimus.</s> <s xml:id="echoid-s17612" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17613" xml:space="preserve">SEQVVNTVR PRAXES PRO-<lb/>blematum omnium triangulorum ex de-<lb/>monſtrationibus ſuperioribus excerptæ, <lb/>in quibus totus fructus noſtrorum trian-<lb/>gulorum tam rectilineorum, quam ſphæ-<lb/>ricorum conſiſtit.</s> <s xml:id="echoid-s17614" xml:space="preserve"/> </p> <pb o="485" file="497" n="497"/> </div> <div xml:id="echoid-div1396" type="section" level="1" n="618"> <head xml:id="echoid-head653" xml:space="preserve">TRIANGVLORVM RECTI-<lb/>LINEORVM RECTANGVLORVM <lb/><emph style="sc">PROBLEMATA, E</emph>T <emph style="sc">Praxes.</emph></head> <p> <s xml:id="echoid-s17615" xml:space="preserve">1. </s> <s xml:id="echoid-s17616" xml:space="preserve">DATIS angulis omnibus cuiuſcunq; </s> <s xml:id="echoid-s17617" xml:space="preserve">trian <lb/> <anchor type="note" xlink:label="note-497-01a" xlink:href="note-497-01"/> guli; </s> <s xml:id="echoid-s17618" xml:space="preserve">inuentire omnium laterũ proportiones.</s> <s xml:id="echoid-s17619" xml:space="preserve"/> </p> <div xml:id="echoid-div1396" type="float" level="2" n="1"> <note position="right" xlink:label="note-497-01" xlink:href="note-497-01a" xml:space="preserve">Quærũtur <lb/>proportio-<lb/>nes laterũ.</note> </div> <p> <s xml:id="echoid-s17620" xml:space="preserve">ADSCRIBANTVR fingulis lateribus ſinus recti angulorum oppo-<lb/> <anchor type="note" xlink:label="note-497-02a" xlink:href="note-497-02"/> ſitorum. </s> <s xml:id="echoid-s17621" xml:space="preserve">Latera enim eas inter ſe proportiones habent, quæ inter dictos ſi-<lb/>nus angulorum lateribus oppoſitis adſcriptos reperiuntur. </s> <s xml:id="echoid-s17622" xml:space="preserve">Quod ſi duo tan-<lb/>tum anguli dati ſint, in ueniendus primum erit tertius angulus, per ſubt ractio <lb/>nem duorum datorum ex duobus rectis, ac tum demum eodem modo propor-<lb/>tiones laterum indagandæ.</s> <s xml:id="echoid-s17623" xml:space="preserve"/> </p> <div xml:id="echoid-div1397" type="float" level="2" n="2"> <note position="right" xlink:label="note-497-02" xlink:href="note-497-02a" xml:space="preserve">Schol. pro-<lb/>poſ. 1. triãg. <lb/>rectil.</note> </div> </div> <div xml:id="echoid-div1399" type="section" level="1" n="619"> <head xml:id="echoid-head654" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17624" xml:space="preserve">DVPLICETVR ſinus rectus cuiusvis anguli acuti, habebiturq́; </s> <s xml:id="echoid-s17625" xml:space="preserve">la-<lb/> <anchor type="note" xlink:label="note-497-03a" xlink:href="note-497-03"/> tus illi angulo oppoſitum in partibus ſinus totius, quem refert ſemidiameter <lb/>circuli triangulo circumſcripti. </s> <s xml:id="echoid-s17626" xml:space="preserve">Pro latere vero, quod recto angulo opponi-<lb/>tur, ſi forte triangulum eſt rectangulum, ſumatur ſinus totus duplicatus. </s> <s xml:id="echoid-s17627" xml:space="preserve">Pro <lb/>latere denique, quod angulo obtuſo opponitur, ſi forte obtuſangulum eſt <lb/>triangulum, accipiatur duplũ ſinus recti, qui ſemiſsi aggregati ex duplis duo-<lb/>rum angulorum acutorum debetur.</s> <s xml:id="echoid-s17628" xml:space="preserve"/> </p> <div xml:id="echoid-div1399" type="float" level="2" n="1"> <note position="right" xlink:label="note-497-03" xlink:href="note-497-03a" xml:space="preserve">Schol. pro-<lb/>poſ 1. triãg. <lb/>rectil.</note> </div> <note position="right" xml:space="preserve">Quætitur <lb/>latus, circa <lb/>angulum re <lb/>ctum vtrili <lb/>bet angulo <lb/>rum acuto <lb/>rum oppo-<lb/>ſitum.</note> </div> <div xml:id="echoid-div1401" type="section" level="1" n="620"> <head xml:id="echoid-head655" xml:space="preserve">2. DATO latere in triangulo rectágulo, quod <lb/>recto angulo opponitur, cum vno angulo-<lb/>rum acutorum, ac proinde & cum altero acu <lb/>to: (cum ambo ſint vni recto æquales) inue-<lb/>nire latus circa angulum rectum vtrilibet a-<lb/>cutorum angulorum oppoſitum.</head> <p> <s xml:id="echoid-s17629" xml:space="preserve">FIAT, vt ſinus totus ad datum latus recto angulo oppoſitum, ita ſinus <lb/> <anchor type="note" xlink:label="note-497-05a" xlink:href="note-497-05"/> vtriuſvis anguli acuti dati ad a liud, produceturq́; </s> <s xml:id="echoid-s17630" xml:space="preserve">latus illi dato acuto angu-<lb/>lo oppoſitum in partibus menſuræ, ſecundum quam datum eſt latus angulo <lb/>recto opppoſitum.</s> <s xml:id="echoid-s17631" xml:space="preserve"/> </p> <div xml:id="echoid-div1401" type="float" level="2" n="1"> <note position="right" xlink:label="note-497-05" xlink:href="note-497-05a" xml:space="preserve">Propoſ. 2. <lb/>triang. re-<lb/>ctil.</note> </div> <note position="right" xml:space="preserve">Quætitur <lb/>latus recto <lb/>angulo op-<lb/>poſitum, & <lb/>alterutrum <lb/>duorũ cir. <lb/>ca eundem <lb/>angulũ re-<lb/>ctum.</note> </div> <div xml:id="echoid-div1403" type="section" level="1" n="621"> <head xml:id="echoid-head656" xml:space="preserve">3. DATO vno latere trianguli rectanguli cir-<lb/>ca rectum angulum, cum vno acutorum an <lb/>gulorum, atque adeo & cum altero acuto: <pb o="486" file="498" n="498" rhead=""/> (quòd ambo vni recto ſint æquales) inueni-<lb/>realia duo latera.</head> <p> <s xml:id="echoid-s17632" xml:space="preserve">FIAT, vt ſinus totus ad latus datum circa angulum rectum, ita tangens <lb/> <anchor type="note" xlink:label="note-498-01a" xlink:href="note-498-01"/> acuti anguli dato lateri adiacentis ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17633" xml:space="preserve">alterum latus circa <lb/>angulum rectum: </s> <s xml:id="echoid-s17634" xml:space="preserve">Fiat item, vt ſinus totus ad latus idem circa angulum re-<lb/>ctum datum, ita ſecans eiuſdem anguli acuti dato lateri adiacentis ad aliud, <lb/>produceturq́; </s> <s xml:id="echoid-s17635" xml:space="preserve">latus recto angulo oppoſitum, in partibus menſuræ, ſecun-<lb/>dum quam latus circa angulum rectum eſt datum.</s> <s xml:id="echoid-s17636" xml:space="preserve"/> </p> <div xml:id="echoid-div1403" type="float" level="2" n="1"> <note position="left" xlink:label="note-498-01" xlink:href="note-498-01a" xml:space="preserve">Propoſ. 1. <lb/>triang. re-<lb/>ctil.</note> </div> </div> <div xml:id="echoid-div1405" type="section" level="1" n="622"> <head xml:id="echoid-head657" xml:space="preserve">Aliter per ſolos ſinus.</head> <p style="it"> <s xml:id="echoid-s17637" xml:space="preserve"><emph style="sc">FIa</emph>T, vt ſinus anguli acuti dato lateri oppoſiti ad latus datum circa angulum <lb/> <anchor type="note" xlink:label="note-498-02a" xlink:href="note-498-02"/> rectum, ita ſinus alterius anguli acuti ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17638" xml:space="preserve">latus huic alteri acuto <lb/>angulo opp oſitum circa angulum rectum: </s> <s xml:id="echoid-s17639" xml:space="preserve">Fiat item, vt ſinus anguli acuti dato lateri <lb/>oppoſiti ad datum latus circa rectum angulum, ita ſinus totus ad aliud, produceturq́; <lb/></s> <s xml:id="echoid-s17640" xml:space="preserve">latus angulo recto oppoſitum, in partibus menſuræ, ſecundum quam latus circa an-<lb/>gulum rectum datum eſt.</s> <s xml:id="echoid-s17641" xml:space="preserve"/> </p> <div xml:id="echoid-div1405" type="float" level="2" n="1"> <note position="left" xlink:label="note-498-02" xlink:href="note-498-02a" xml:space="preserve">Propoſ. 1. <lb/>triang. re-<lb/>ctil.</note> </div> <note position="left" xml:space="preserve">Quærũtur <lb/>duo anguli <lb/>acuti, & v-<lb/>num latus <lb/>circa angu <lb/>lum rectũ.</note> </div> <div xml:id="echoid-div1407" type="section" level="1" n="623"> <head xml:id="echoid-head658" xml:space="preserve">4. DATO latere in triangulo rectãgulo, quod <lb/>angulo recto opponitur, cum alterutro reli-<lb/>quorum duorum laterum circa angulum re <lb/>ctum, reperire duos angulos acutos, & alte-<lb/>rum latus circa angulum rectum.</head> <p> <s xml:id="echoid-s17642" xml:space="preserve">FIAT, vt datum latus recto angulo oppoſitum ad ſinum totum, ita da-<lb/> <anchor type="note" xlink:label="note-498-04a" xlink:href="note-498-04"/> tum latus circa angulum rectum ad aliud, procreabiturq́; </s> <s xml:id="echoid-s17643" xml:space="preserve">ſinus anguli acuti <lb/>huic poſteriori lateri dato oppoſiti: </s> <s xml:id="echoid-s17644" xml:space="preserve">Ex hoc autem angulo inuẽto, alter quo-<lb/>que acutus notus fiet, cum ambo vni recto ſint æquales. </s> <s xml:id="echoid-s17645" xml:space="preserve">Fiat rurſus, vt ſinus <lb/>totus ad datum latus angulo recto oppoſitum, ita ſinus acuti anguli inuenti <lb/>quæſito tertio lateri oppoſitiad aliud, inuenieturq́; </s> <s xml:id="echoid-s17646" xml:space="preserve">alterum hoc latus circa <lb/>angulũ rectum, in partibus menſuræ, ſecundũ quam duo alia latera data ſunt.</s> <s xml:id="echoid-s17647" xml:space="preserve"/> </p> <div xml:id="echoid-div1407" type="float" level="2" n="1"> <note position="left" xlink:label="note-498-04" xlink:href="note-498-04a" xml:space="preserve">Propoſ. 3. <lb/>triang. re-<lb/>ctil.</note> </div> <note position="left" xml:space="preserve">Quætũtur <lb/>duo acuti <lb/>anguli, &@la <lb/>tus recto an <lb/>gulo oppo-<lb/>ſitum.</note> </div> <div xml:id="echoid-div1409" type="section" level="1" n="624"> <head xml:id="echoid-head659" xml:space="preserve">5. DATIS duobus lateribus circa angulum <lb/>rectum, inuenire duos angulos acutos, & la-<lb/>tus recto angulo oppoſitum.</head> <p> <s xml:id="echoid-s17648" xml:space="preserve">FIAT, vt alterutrum laterum datorum ad ſinum totum, ita alterum la-<lb/> <anchor type="note" xlink:label="note-498-06a" xlink:href="note-498-06"/> tus datum ad aliud, prodibitq́; </s> <s xml:id="echoid-s17649" xml:space="preserve">tangens anguli acuti huic poſteriori lateri op <lb/>poſiti. </s> <s xml:id="echoid-s17650" xml:space="preserve">Ex hoc autem angulo inuento notus euadet alter acutus angulus, cum <lb/>ambo acuti vni recto ſint æquales. </s> <s xml:id="echoid-s17651" xml:space="preserve">Fiat rurſum, vt ſinus totus ad vtrumuis la-<lb/>terum circa angulum rectum datum, ita ſecans anguli acuti accepto huic la-<lb/>teri a diacentis ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17652" xml:space="preserve">latus angulo recto oppoſitum, in par-<lb/>tibus, in quibus data ſunt duo latera circa rectum angulum.</s> <s xml:id="echoid-s17653" xml:space="preserve"/> </p> <div xml:id="echoid-div1409" type="float" level="2" n="1"> <note position="left" xlink:label="note-498-06" xlink:href="note-498-06a" xml:space="preserve">Propoſ. 3. <lb/>triang. re-<lb/>ctil.</note> </div> <pb o="487" file="499" n="499" rhead=""/> </div> <div xml:id="echoid-div1411" type="section" level="1" n="625"> <head xml:id="echoid-head660" xml:space="preserve">Aliter per ſolos ſinus.</head> <p style="it"> <s xml:id="echoid-s17654" xml:space="preserve">ADDANTVR ſimul quadrata duorum laterum circa angulum rectum date-<lb/> <anchor type="note" xlink:label="note-499-01a" xlink:href="note-499-01"/> rum. </s> <s xml:id="echoid-s17655" xml:space="preserve">Nam huius aggregatir adix erit latus angulo recto oppoſitum. </s> <s xml:id="echoid-s17656" xml:space="preserve">Fiat rurſus, vt <lb/>latus recto angulo oppoſitum, quod iam inuentum eſt, ad ſinum totum, ita alteru-<lb/>trum datorum laterum circa angulum rectum ad aliud, prouenietq́; </s> <s xml:id="echoid-s17657" xml:space="preserve">ſinus acuti angu-<lb/>li aſſumpto lateri circa angulum rectum oppoſiti. </s> <s xml:id="echoid-s17658" xml:space="preserve">Ex hoc autem angulo inuento fiet <lb/>quoque alter cognitus, cum vni recto ambo acuti ſint æquales.</s> <s xml:id="echoid-s17659" xml:space="preserve"/> </p> <div xml:id="echoid-div1411" type="float" level="2" n="1"> <note position="right" xlink:label="note-499-01" xlink:href="note-499-01a" xml:space="preserve">Propof. 3. <lb/>triang. re-<lb/>ctil.</note> </div> </div> <div xml:id="echoid-div1413" type="section" level="1" n="626"> <head xml:id="echoid-head661" xml:space="preserve">TRIANGVLORVM RECTILI-<lb/>NEORVM NON RECTANGVLORVM <lb/>PROBLEMATA, ET PRAXES.</head> <note position="right" xml:space="preserve">Quærũtur <lb/>duo arcus, <lb/>vel anguli, <lb/>ex eorũ ag-<lb/>gregato.</note> </div> <div xml:id="echoid-div1414" type="section" level="1" n="627"> <head xml:id="echoid-head662" xml:space="preserve">6. DATO aggregato duorum arcuum, vel an-<lb/>gulorum, quod minus ſit, quam grad. 180 vna <lb/>cum proportione, quam eorum ſinus habẽt, <lb/>vtrumqueillorum exhibere notum.</head> <p> <s xml:id="echoid-s17660" xml:space="preserve">FIAT, vt ſemiſsis aggregati terminorum proportionis datæ, quam ſinus <lb/> <anchor type="note" xlink:label="note-499-03a" xlink:href="note-499-03"/> arcuum, vel angulorum habent, ad tangentem ſemiſsis aggregati arcuum, vel <lb/>angulorum dati, (quærendo tangentem per partem proportionalem reſpon-<lb/>dentem 30. </s> <s xml:id="echoid-s17661" xml:space="preserve">ſecundis, ſi forte aggregatum arcuum, vel angulorum bifariam <lb/>diuidi nequeat ſine ſecundis.) </s> <s xml:id="echoid-s17662" xml:space="preserve">ita differentia inter ſemiſſem aggregati ter-<lb/>minorum datæ proportionis, & </s> <s xml:id="echoid-s17663" xml:space="preserve">alterutrum terminorum, ad aliud. </s> <s xml:id="echoid-s17664" xml:space="preserve">Inue-<lb/>nietur enim tangens arcus, vel anguli, quo vterque arcus, vel angulus quæſi-<lb/>tus à ſemiſſe aggregati eorundem arcuum, vel angulorum dati differt: </s> <s xml:id="echoid-s17665" xml:space="preserve">atque <lb/>adeo arcus, vel angulus tangentis huius inuentæ additus ad ſemiſſem dati ag-<lb/>gregati arcuum, vel angulorum dabit maiorem arcum, vel angulum quęſitum; <lb/></s> <s xml:id="echoid-s17666" xml:space="preserve">ablatus vero ab eadem ſemiſſe relinquet arcum, vel angulum minorem.</s> <s xml:id="echoid-s17667" xml:space="preserve"/> </p> <div xml:id="echoid-div1414" type="float" level="2" n="1"> <note position="right" xlink:label="note-499-03" xlink:href="note-499-03a" xml:space="preserve">Propoſ. 6. <lb/>triang. re-<lb/>ctil.</note> </div> </div> <div xml:id="echoid-div1416" type="section" level="1" n="628"> <head xml:id="echoid-head663" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17668" xml:space="preserve">FIAT, vt ſemiſsis differentiæ inter duos terminos proportionis datæ ad <lb/> <anchor type="note" xlink:label="note-499-04a" xlink:href="note-499-04"/> tangentem ſemiſsis differentiæ inter datum aggregatum arcuum, vel angulo-<lb/>rum, & </s> <s xml:id="echoid-s17669" xml:space="preserve">ſemicirculum, ita aggregatum ex ſemiſſe differentiæ inter duos termi-<lb/>nos datæ proportionis, & </s> <s xml:id="echoid-s17670" xml:space="preserve">conſequente termino eiuſdem proportionis, ad <lb/>aliud. </s> <s xml:id="echoid-s17671" xml:space="preserve">Producetur enim tangens arcus, ſeu anguli, à quo ſi detrahatur ſemiſ-<lb/>ſis differentiæ inter datum aggregatum arcuum, vel angulorum, & </s> <s xml:id="echoid-s17672" xml:space="preserve">ſemicircu-<lb/>lum, reliquus fiet arcus, ſiue angulus minor quæſitus: </s> <s xml:id="echoid-s17673" xml:space="preserve">hic autem ex dato ag-<lb/>gregato ſubductus relinquet arcum, vel angulum quæſitum maiorem.</s> <s xml:id="echoid-s17674" xml:space="preserve"/> </p> <div xml:id="echoid-div1416" type="float" level="2" n="1"> <note position="right" xlink:label="note-499-04" xlink:href="note-499-04a" xml:space="preserve">Propoſ. 6. <lb/>triang. re. <lb/>ctil.</note> </div> <pb o="488" file="500" n="500" rhead=""/> </div> <div xml:id="echoid-div1418" type="section" level="1" n="629"> <head xml:id="echoid-head664" xml:space="preserve">Aliter per ſolos ſinus.</head> <p style="it"> <s xml:id="echoid-s17675" xml:space="preserve">FIAT, vt ſemiſsis aggregati terminorum proportionis datæ ad ſinum ſemiſsis aga <lb/> <anchor type="note" xlink:label="note-500-01a" xlink:href="note-500-01"/> gregati arcuum, ſeu angulorum, ita differentia inter ſemiſſem aggregati terminorum <lb/>datæ proportionis, & </s> <s xml:id="echoid-s17676" xml:space="preserve">alterutrum terminorum, ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17677" xml:space="preserve">quartus qui-<lb/>dam numerus; </s> <s xml:id="echoid-s17678" xml:space="preserve">cuius quadratum ſi adijciatur quadrato ſinus complementi ſemißis ag-<lb/>gregati arcuum, ſeu angulorum: </s> <s xml:id="echoid-s17679" xml:space="preserve">Et rur ſum fiat, vt radix quadrata aggregati duo-<lb/>rum dictorum quadratorum ad ſinum totum, ita quartus ille numerus inuentus ad <lb/>aliud, producetur ſinus arcus, ſiue anguli, quo vterque arcus, angulus ve ab eorun <lb/>dem aggregati dati ſemiſſe differt. </s> <s xml:id="echoid-s17680" xml:space="preserve">Additus ergo hic arcus, ſeu angulus ad ſemiſſem <lb/>aggregati datipræbebit maiorem arcum, vel angulum; </s> <s xml:id="echoid-s17681" xml:space="preserve">ablatus vero ex eadem ſemiſ-<lb/>ſe minorem arcum, ſeu angulum relinquet.</s> <s xml:id="echoid-s17682" xml:space="preserve"/> </p> <div xml:id="echoid-div1418" type="float" level="2" n="1"> <note position="left" xlink:label="note-500-01" xlink:href="note-500-01a" xml:space="preserve">Propoſ. 6. <lb/>triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17683" xml:space="preserve">QVOD ſi quando proportio ſinuum data ſit æqualitatis, dabit ſemiſsis <lb/>dati aggregati arcuum, ſeu angulorum, vtrumque arcum, ſiue angulum.</s> <s xml:id="echoid-s17684" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quærũtur <lb/>duo arcus, <lb/>ſeu anguli, <lb/>ex eorũ ag-<lb/>gregato.</note> </div> <div xml:id="echoid-div1420" type="section" level="1" n="630"> <head xml:id="echoid-head665" xml:space="preserve">7. DATO duorum arcuum, quorum vterq; <lb/>ſemicirculo minor ſit, vel duorum angulo-<lb/>rum aggregato, quod maius ſit, quam grad. <lb/>180. vnà cum proportione, quam eorũ ſinus <lb/>habent, vtrumque illorum reddere notum.</head> <p> <s xml:id="echoid-s17685" xml:space="preserve">DETRACTO dato aggregato ex grad. </s> <s xml:id="echoid-s17686" xml:space="preserve">360. </s> <s xml:id="echoid-s17687" xml:space="preserve">inueniatur per proble-<lb/> <anchor type="note" xlink:label="note-500-03a" xlink:href="note-500-03"/> ma 6. </s> <s xml:id="echoid-s17688" xml:space="preserve">vterque arcus, ſiue angulus reſidui aggregati, quod minus eſt ſemper, <lb/>quam grad. </s> <s xml:id="echoid-s17689" xml:space="preserve">180. </s> <s xml:id="echoid-s17690" xml:space="preserve">remanetq́; </s> <s xml:id="echoid-s17691" xml:space="preserve">eadem proportio ſinuum. </s> <s xml:id="echoid-s17692" xml:space="preserve">Nam ſi ambo inuenti <lb/>ſeorſum ex ſemicirculo ſubtrahantur, reliqui erunt arcus, vel anguli quęſiti.</s> <s xml:id="echoid-s17693" xml:space="preserve"/> </p> <div xml:id="echoid-div1420" type="float" level="2" n="1"> <note position="left" xlink:label="note-500-03" xlink:href="note-500-03a" xml:space="preserve">Propoſ. 6. <lb/>triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17694" xml:space="preserve">QVANDO proportio data eſt æqualitatis, dabit quoque ſemiſsis dati <lb/>aggregati vtrumque arcum, ſiue angulum quæſitum.</s> <s xml:id="echoid-s17695" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17696" xml:space="preserve">QVOD ſi forte datum aggregatum contineat præciſe grad. </s> <s xml:id="echoid-s17697" xml:space="preserve">180. </s> <s xml:id="echoid-s17698" xml:space="preserve">proble-<lb/>maſolui non poteſt.</s> <s xml:id="echoid-s17699" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quærũtur <lb/>duo arcus, <lb/>ſiue angu-<lb/>li, ex eorum <lb/>differẽtia.</note> </div> <div xml:id="echoid-div1422" type="section" level="1" n="631"> <head xml:id="echoid-head666" xml:space="preserve">8. DATA differentia duorum arcuum, quo-<lb/>rum vterque ſemicirculo ſit minor, vel duo-<lb/>rum angulorum, vna cum proportione, <lb/>quam eorum ſinus habent, vtrum que illo-<lb/>rum notum efficere.</head> <p> <s xml:id="echoid-s17700" xml:space="preserve">SI proportio ſinus maioris arcus, vel anguli, ad ſinum minoris eſt maio-<lb/> <anchor type="note" xlink:label="note-500-05a" xlink:href="note-500-05"/> ris inæqualitatis; </s> <s xml:id="echoid-s17701" xml:space="preserve">fiat, vt ſemiſsis differentiæ ter minorum proportionis datæ <lb/>ad tangentem ſemiſsis datæ differentiæ arcuum, vel angulorum, ita aggrega-<lb/>tum ex ſemiſſe differentiæ terminorum proportionis, & </s> <s xml:id="echoid-s17702" xml:space="preserve">conſequente termino <lb/>proportionis eiuſdem, ad aliud, produceturq́; </s> <s xml:id="echoid-s17703" xml:space="preserve">tangens arcus, ſiue anguli, qui <lb/>ſemiſsi differentiæ arcuum, vel angulorum date additus componet maiorem <lb/>arcum, ſeu angulum; </s> <s xml:id="echoid-s17704" xml:space="preserve">ſi vero ab eodem ſemiſsis differentiæ arcuum, vel angu- <pb o="489" file="501" n="501" rhead=""/> lorum ſubducatur, reliquus erit arcus, vel angulus minor.</s> <s xml:id="echoid-s17705" xml:space="preserve"/> </p> <div xml:id="echoid-div1422" type="float" level="2" n="1"> <note position="left" xlink:label="note-500-05" xlink:href="note-500-05a" xml:space="preserve">Propoſ. 7. <lb/>triang. re. <lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17706" xml:space="preserve">SI vero proportio ſinus maioris arcus, vel anguli, ad ſinum minoris eſt <lb/>minoris inæqualitatis; </s> <s xml:id="echoid-s17707" xml:space="preserve">inuertantur eius termini, vt fiat proportio maioris in-<lb/>æqualitatis; </s> <s xml:id="echoid-s17708" xml:space="preserve">atque ex hac, & </s> <s xml:id="echoid-s17709" xml:space="preserve">data differentia arcuum, ſeu angulorum inue-<lb/>niantur duo arcus, vel anguli, vt dictum eſt. </s> <s xml:id="echoid-s17710" xml:space="preserve">Nam maior eotum ex ſemicircu-<lb/>lo ſublatus dabit minorem arcum, ſeu angulum quæſitum; </s> <s xml:id="echoid-s17711" xml:space="preserve">minor vero ſubdu-<lb/>ctus ex ſemicirculo offeret maiorem.</s> <s xml:id="echoid-s17712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1424" type="section" level="1" n="632"> <head xml:id="echoid-head667" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17713" xml:space="preserve">QVANDO ſinus maioris arcus, vel anguli ad ſinũ minoris habet pro-<lb/> <anchor type="note" xlink:label="note-501-01a" xlink:href="note-501-01"/> portionem maioris inæqualitatis; </s> <s xml:id="echoid-s17714" xml:space="preserve">inquirantur ex data illa proportione maio-<lb/>ris inæqualitatis, & </s> <s xml:id="echoid-s17715" xml:space="preserve">ex arcu, ſeu angulo, qui poſt detractionem datæ differen-<lb/>tiæ ex ſemicirculo relinquitur, tanquam ex aggregato duorum arcuum, ſiue <lb/>angulorum, duo arcus, ſiue anguli huius aggregati, vt in problemate 6. </s> <s xml:id="echoid-s17716" xml:space="preserve">præ-<lb/>cepimus. </s> <s xml:id="echoid-s17717" xml:space="preserve">Nam maior arcus, ſeu angulus inuentus, ſi ex ſemicirculo aufera-<lb/>tur, dabit maiorem arcum ſiue angulum quæſitum: </s> <s xml:id="echoid-s17718" xml:space="preserve">Minor autem inuentus <lb/>erit minor quæſitus.</s> <s xml:id="echoid-s17719" xml:space="preserve"/> </p> <div xml:id="echoid-div1424" type="float" level="2" n="1"> <note position="right" xlink:label="note-501-01" xlink:href="note-501-01a" xml:space="preserve">Propoſ. 7. <lb/>triang re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17720" xml:space="preserve">QVANDO autem proportio data eſt minoris inæqualitatis; </s> <s xml:id="echoid-s17721" xml:space="preserve">inuertan-<lb/>tur eius termini, vt fiat proportio maioris inæqualitatis: </s> <s xml:id="echoid-s17722" xml:space="preserve">atque ex hac, & </s> <s xml:id="echoid-s17723" xml:space="preserve">da-<lb/>ta differentia arcuum, ſeu angulorum, inueſtigentur duo arcus, ſiue anguli, <lb/>vt iam dictum eſt. </s> <s xml:id="echoid-s17724" xml:space="preserve">Maior enim eorum ex ſemicirculo ſubtractus dabit arcum, <lb/>ſeu angulum quæſitum minorem; </s> <s xml:id="echoid-s17725" xml:space="preserve">Minor vero maiorem.</s> <s xml:id="echoid-s17726" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1426" type="section" level="1" n="633"> <head xml:id="echoid-head668" xml:space="preserve">Aliter per ſolos ſinus.</head> <p style="it"> <s xml:id="echoid-s17727" xml:space="preserve">SI data proportio ſinus maioris arcus, ſiue anguli ad ſinum minoris eſt maioris in-<lb/> <anchor type="note" xlink:label="note-501-02a" xlink:href="note-501-02"/> æqualitatis; </s> <s xml:id="echoid-s17728" xml:space="preserve">fiat, vt ſemiſsis differentiæ terminorum proportionis datæ ad ſinum ſe-<lb/>miſsis datæ differentiæ arcuum, vel angulorum, ita aggregatum ex ſemiſſe differen <lb/>tiæ terminorum proportionis, & </s> <s xml:id="echoid-s17729" xml:space="preserve">conſequẽte termino eiuſdem proportionis, ad aliud, <lb/>inuenieturq́; </s> <s xml:id="echoid-s17730" xml:space="preserve">quartus quidam numerus; </s> <s xml:id="echoid-s17731" xml:space="preserve">cuius quadratum ſi adijciatur quadrato ſi-<lb/>nus complementi ſemiſsis differentiæ arcuum, ſeu angulorum datæ: </s> <s xml:id="echoid-s17732" xml:space="preserve">Et rurſum fiat, <lb/>vt radix quadrata aggregati dictorum duorum quadratorum ad ſinum totum, ita <lb/>numerus ille quartus inuentus ad aliud, reperietur ſinus arcus, ſiue anguli, cui ſi <lb/>addatur ſemißis datæ differentiæ arcuum, ſeu angulorum, notus fiet maior arcus, ſi-<lb/>ue angulus: </s> <s xml:id="echoid-s17733" xml:space="preserve">ab eodem vero ſi eadem ſemiſsis detrahatur, reliquus erit minor.</s> <s xml:id="echoid-s17734" xml:space="preserve"/> </p> <div xml:id="echoid-div1426" type="float" level="2" n="1"> <note position="right" xlink:label="note-501-02" xlink:href="note-501-02a" xml:space="preserve">Propoſ. 7. <lb/>trtrang. re-<lb/>ctil.</note> </div> <p style="it"> <s xml:id="echoid-s17735" xml:space="preserve">QVOD ſi data proportio ſit minoris inæqualitatis, agẽdũ erit, vt ſupra diximus.</s> <s xml:id="echoid-s17736" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17737" xml:space="preserve">IAM vero ſi forte proportio data ſit æqualitatis, detrahatur differentia <lb/>data arcuum, ſiue angulorum ex ſemicirculo. </s> <s xml:id="echoid-s17738" xml:space="preserve">Nam reſidui ſemiſsis erit minor <lb/>arcus, ſeu angulus quæſitus: </s> <s xml:id="echoid-s17739" xml:space="preserve">eadem vero ſemiſsis ad datam differentiam adie-<lb/>cta dabit maiorem.</s> <s xml:id="echoid-s17740" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Quærũtur <lb/>caſus lineæ <lb/>perpẽdicu-<lb/>laris.</note> </div> <div xml:id="echoid-div1428" type="section" level="1" n="634"> <head xml:id="echoid-head669" xml:space="preserve">9. SI ab vno angulo trianguli cuiuſvis dato-<lb/>rum laterum ad latus oppoſitum perpendicu <lb/>laris demittatur, quãta ſit recta inter perpen-<lb/>dicularem, & vtrum vis reliquorum angulo-<lb/>rum, inuenire.</head> <pb o="490" file="502" n="502" rhead=""/> <p> <s xml:id="echoid-s17741" xml:space="preserve">DIFFERENTIA inter quadrata duorum laterum ambientium an-<lb/> <anchor type="note" xlink:label="note-502-01a" xlink:href="note-502-01"/> gulum, à quo perpendicularis demiſſa eſt, diuidatur per latus tertium, produ <lb/>ceturq́; </s> <s xml:id="echoid-s17742" xml:space="preserve">numerus; </s> <s xml:id="echoid-s17743" xml:space="preserve">qui ſi minor fuerit tertio latere, indicabit, perpendicularem <lb/>intra triangulum cecidiſſe; </s> <s xml:id="echoid-s17744" xml:space="preserve">idemq́; </s> <s xml:id="echoid-s17745" xml:space="preserve">ex tertio eodem latere ſubductus relinquet <lb/>numerum, cuius ſemiſsis dabit minus ſegmentum baſis, hoc autem ex toto ter <lb/>tio latere ſubtractum dabit ſegmentum maius. </s> <s xml:id="echoid-s17746" xml:space="preserve">Idem vero numerus ex diui-<lb/>ſione productus, ſi fuerit maior tertio latere, argumento erit, perpẽdicularem <lb/>extra triangulum cecidiſſe. </s> <s xml:id="echoid-s17747" xml:space="preserve">Quare ſi ex eo tertium latus auſeratur, reliquus <lb/>erit numerus, cuius ſemiſsis dabit rectam extra triangulum inter perpendicu-<lb/>larem, & </s> <s xml:id="echoid-s17748" xml:space="preserve">angulum obtuſum; </s> <s xml:id="echoid-s17749" xml:space="preserve">eadem vero ſemiſsis tertio lateri appoſita dabit <lb/>alteram rectam inter perpendicularem, & </s> <s xml:id="echoid-s17750" xml:space="preserve">angulum acutum.</s> <s xml:id="echoid-s17751" xml:space="preserve"/> </p> <div xml:id="echoid-div1428" type="float" level="2" n="1"> <note position="left" xlink:label="note-502-01" xlink:href="note-502-01a" xml:space="preserve">Propoſ. 9. <lb/>triang. re-<lb/>ctil.</note> </div> </div> <div xml:id="echoid-div1430" type="section" level="1" n="635"> <head xml:id="echoid-head670" xml:space="preserve">Aliter, & facilius.</head> <p> <s xml:id="echoid-s17752" xml:space="preserve">FIAT, vt tertium latus, in quod demiſſa eſt perpendicularis, ad ſummam <lb/> <anchor type="note" xlink:label="note-502-02a" xlink:href="note-502-02"/> aliorum duorum laterum, ita differentia eorundem ad aliud, prouenietq́; </s> <s xml:id="echoid-s17753" xml:space="preserve">nu-<lb/>merus, ex quo rectam inter perpendicularem, & </s> <s xml:id="echoid-s17754" xml:space="preserve">angulum vtrumq; </s> <s xml:id="echoid-s17755" xml:space="preserve">inuenie-<lb/>mus, vt nuper diximus.</s> <s xml:id="echoid-s17756" xml:space="preserve"/> </p> <div xml:id="echoid-div1430" type="float" level="2" n="1"> <note position="left" xlink:label="note-502-02" xlink:href="note-502-02a" xml:space="preserve">Propoſ. 9. <lb/>triang. re-<lb/>ctil.</note> </div> </div> <div xml:id="echoid-div1432" type="section" level="1" n="636"> <head xml:id="echoid-head671" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17757" xml:space="preserve">CADENTE perpendiculari intra triangulum; </s> <s xml:id="echoid-s17758" xml:space="preserve">diuidatur ſemiſsis diffe-<lb/> <anchor type="note" xlink:label="note-502-03a" xlink:href="note-502-03"/> rentiæ inter quadratum vttiuſvis laterum ambientium angulum, à quo per-<lb/>pendicularis eſt demiſſa, & </s> <s xml:id="echoid-s17759" xml:space="preserve">ſummam quadratorum ex alijs duobus lateribus de-<lb/>ſcriptorum, per latus, in quod perpendicularis cadit, produceturq́; </s> <s xml:id="echoid-s17760" xml:space="preserve">ſegmen-<lb/>tum baſis prope angulum, quem continẽt duo latera, quorum ſumma quadra-<lb/>torum fuit accepta; </s> <s xml:id="echoid-s17761" xml:space="preserve">hoc autem ſegmentum ex tota baſi detractum relinquet <lb/>alterum ſegmentum.</s> <s xml:id="echoid-s17762" xml:space="preserve"/> </p> <div xml:id="echoid-div1432" type="float" level="2" n="1"> <note position="left" xlink:label="note-502-03" xlink:href="note-502-03a" xml:space="preserve">Schol. pro-<lb/>poſ 9. triãg-<lb/>rectil.</note> </div> <p> <s xml:id="echoid-s17763" xml:space="preserve">CADENTE vero perpendiculari extra triangulum, diuidatur ſemiſsis <lb/>differentiæ inter quadratum lateris angulo obtuſo oppoſiti, & </s> <s xml:id="echoid-s17764" xml:space="preserve">ſummam qua <lb/>dratorum ex alijs duobus lateribus deſcriptorum, per latus, in quod produ-<lb/>ctum perpendicularis cadit, procreabiturq; </s> <s xml:id="echoid-s17765" xml:space="preserve">linea extra triangulum inter per-<lb/>pendicularem, & </s> <s xml:id="echoid-s17766" xml:space="preserve">angulum obtuſum; </s> <s xml:id="echoid-s17767" xml:space="preserve">hæc vero toti baſi adiecta conficiet alte-<lb/>ram rectam inter perpendicularem, & </s> <s xml:id="echoid-s17768" xml:space="preserve">acutum angulum baſis.</s> <s xml:id="echoid-s17769" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17770" xml:space="preserve">QVOD ſi duo latera circa perpendicularem ſint æqualia, ſecabit perpen <lb/> <anchor type="note" xlink:label="note-502-04a" xlink:href="note-502-04"/> dicularis baſim bifariam. </s> <s xml:id="echoid-s17771" xml:space="preserve">Quare dimidiũ baſis dabit vtramq; </s> <s xml:id="echoid-s17772" xml:space="preserve">rectã quęſitam.</s> <s xml:id="echoid-s17773" xml:space="preserve"/> </p> <div xml:id="echoid-div1433" type="float" level="2" n="2"> <note position="left" xlink:label="note-502-04" xlink:href="note-502-04a" xml:space="preserve">Coroll. pro-<lb/>poſ. S. triãg. <lb/>rectil.</note> </div> <note position="left" xml:space="preserve">Quærũtur <lb/>duo latera.</note> </div> <div xml:id="echoid-div1435" type="section" level="1" n="637"> <head xml:id="echoid-head672" xml:space="preserve">10. DATIS omnibus angulis trianguli non <lb/>rectanguli, cum vno latere, inuenire alia duo <lb/>latera.</head> <p> <s xml:id="echoid-s17774" xml:space="preserve">FIAT, vt ſinus anguli dato lateri oppoſiti ad ſinum vtriusvis reliquo-<lb/> <anchor type="note" xlink:label="note-502-06a" xlink:href="note-502-06"/> rum angulorum, ita latus datum ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17775" xml:space="preserve">latus poſteriori huic <lb/>angulo oppoſitum. </s> <s xml:id="echoid-s17776" xml:space="preserve">Fiat rurſus, vt ſinus anguli dato lateri oppoſiti ad ſinum <lb/>tertij anguli, ita latus datum ad aliud, produceturq́; </s> <s xml:id="echoid-s17777" xml:space="preserve">tertium latus huic ter-<lb/>tio angulo oppoſitum.</s> <s xml:id="echoid-s17778" xml:space="preserve"/> </p> <div xml:id="echoid-div1435" type="float" level="2" n="1"> <note position="left" xlink:label="note-502-06" xlink:href="note-502-06a" xml:space="preserve">Propoſ. 10. <lb/>triang. re <lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17779" xml:space="preserve">SI triangulum ſit Iſoſceles, vnius tantum lateris inuentione opus eſt, ſi <lb/>vnum datum ſit, cum angulis. </s> <s xml:id="echoid-s17780" xml:space="preserve">Idem dicendum eſt de Scaleno, ſi duo eius late-<lb/>ra cum angulis data ſint. </s> <s xml:id="echoid-s17781" xml:space="preserve">In Aequilatero vero, ſi vnum latus detur, data e-<lb/>runt & </s> <s xml:id="echoid-s17782" xml:space="preserve">reliqua illi æqualia.</s> <s xml:id="echoid-s17783" xml:space="preserve"/> </p> <pb o="491" file="503" n="503" rhead=""/> <note position="right" xml:space="preserve">Quærũtur <lb/>anguli.</note> </div> <div xml:id="echoid-div1437" type="section" level="1" n="638"> <head xml:id="echoid-head673" xml:space="preserve">11. DATIS omnibus lateribus trianguli non <lb/>rectanguli, reperire omnes eius angulos.</head> <p> <s xml:id="echoid-s17784" xml:space="preserve">DVCTA ad maximum latus perpẽdiculari ex angulo oppoſito, (vt per-<lb/> <anchor type="note" xlink:label="note-503-02a" xlink:href="note-503-02"/> pendicularis ſemper intra triangulum cadat) inueniãtur, per antecedens pro-<lb/>blema, rectæ inter perpendicularem, & </s> <s xml:id="echoid-s17785" xml:space="preserve">duos angulos maximi lateris poſitæ. <lb/></s> <s xml:id="echoid-s17786" xml:space="preserve">Deinde fiat, vt minimum latus ad ſinum totum, ita minus ſegmentum baſis ad <lb/>aliud, gigneturq́; </s> <s xml:id="echoid-s17787" xml:space="preserve">ſinus, cuius arcus complementum dabit angulum baſis mini-<lb/>mo lateri adiacentem. </s> <s xml:id="echoid-s17788" xml:space="preserve">Rurſus fiat, vt medium latus ad ſinum totum, ita ma-<lb/>ius ſegmentum baſis ad aliud, procreabiturq́; </s> <s xml:id="echoid-s17789" xml:space="preserve">ſinus, cuius arcus complemen-<lb/>tum dabit angulum baſis medio lateri adiacentem. </s> <s xml:id="echoid-s17790" xml:space="preserve">Tertius vero angulus ma-<lb/>ximo lateri oppoſitus conflabitur ex duobus arcubus duorum ſinuum inuen-<lb/>torum: </s> <s xml:id="echoid-s17791" xml:space="preserve">Vel certe relinquetur poſt detractionem duorum angulorum inuen-<lb/>torum ex duobus rectis.</s> <s xml:id="echoid-s17792" xml:space="preserve"/> </p> <div xml:id="echoid-div1437" type="float" level="2" n="1"> <note position="right" xlink:label="note-503-02" xlink:href="note-503-02a" xml:space="preserve">Propoſ. 11. <lb/>triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17793" xml:space="preserve">SI triangulum ſit Iſoſceles, ducenda erit perpendicularis ad baſim, quam <lb/>bifariam ſecabit. </s> <s xml:id="echoid-s17794" xml:space="preserve">Nam ſi tunc fiat, vt vnum æqualium laterum ad ſinum to-<lb/>tum, ita dimidium baſis ad aliud, reperietur ſinus, cuius arcus complementum <lb/>dabit vnum æqualium angulorum ſupra baſim, ac proinde & </s> <s xml:id="echoid-s17795" xml:space="preserve">alterum. </s> <s xml:id="echoid-s17796" xml:space="preserve">Ter-<lb/>tius ex his duobus elicietur.</s> <s xml:id="echoid-s17797" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17798" xml:space="preserve">IN æquilatero dabuntur anguli, etiam ſi latera non dentur, cum quilibet <lb/>ſit tertia pars duorum rectorum, vel duæ tertiæ vnius recti.</s> <s xml:id="echoid-s17799" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Quæritur <lb/>latus, cum <lb/>eius angu-<lb/>lis duobus.</note> </div> <div xml:id="echoid-div1439" type="section" level="1" n="639"> <head xml:id="echoid-head674" xml:space="preserve">12. DATIS duobus lateribus trianguli non <lb/>rectanguli, cum angulo ab ipſis comprehen <lb/>ſo, inuenire tertiũ latus, & reliquos angulos.</head> <p> <s xml:id="echoid-s17800" xml:space="preserve">SVBDVCTO angulo dato ex duobus rectis, vt aggregatum aliorum <lb/> <anchor type="note" xlink:label="note-503-04a" xlink:href="note-503-04"/> duorum habeatur, inueniatur, per 6. </s> <s xml:id="echoid-s17801" xml:space="preserve">problema triang. </s> <s xml:id="echoid-s17802" xml:space="preserve">rectil. </s> <s xml:id="echoid-s17803" xml:space="preserve">ex hoc aggrega-<lb/>to, & </s> <s xml:id="echoid-s17804" xml:space="preserve">proportione laterum da torum eis oppoſitorum, (quæ eadem eſt, quæ <lb/>inter ſinus eorum reperitur) vterque eorum. </s> <s xml:id="echoid-s17805" xml:space="preserve">Deinde fiat, vt ſinus vtriusvis <lb/>horum angulorum inuẽtorum ad ſinum anguli in principio dati, ita latus in-<lb/>uento angulo, qui in aurea regula acceptus fuerit, oppoſitum ad aliud, inue-<lb/>nieturq́; </s> <s xml:id="echoid-s17806" xml:space="preserve">tertium latus.</s> <s xml:id="echoid-s17807" xml:space="preserve"/> </p> <div xml:id="echoid-div1439" type="float" level="2" n="1"> <note position="right" xlink:label="note-503-04" xlink:href="note-503-04a" xml:space="preserve">Propoſ. 12. <lb/>triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17808" xml:space="preserve">QVOD ſi data duo latera ſint æqualia, ablato angulo dato ex duobus <lb/>rectis, dabit ſemiſsis reſidui vtrumque angulorum æqualium: </s> <s xml:id="echoid-s17809" xml:space="preserve">Et ſi fiat, vt ſi-<lb/>nus vnius illorum ad ſinum anguli dati, ita vnum laterum æqualium ad aliud, <lb/>prodibit tertium latus.</s> <s xml:id="echoid-s17810" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Quærũtur <lb/>duo angu-<lb/>li, cum vno <lb/>latere.</note> </div> <div xml:id="echoid-div1441" type="section" level="1" n="640"> <head xml:id="echoid-head675" xml:space="preserve">13. DATIS duobus lateribus trianguli non <lb/>rectanguli, cum angulo, qui vni eorum oppo <lb/>nitur, inueſtigare reliquos angulos, & ter-<lb/>tium latus: ſi modo, quando datus angulus <lb/>eſt acutus, conſtet, num angulus alteri da- <pb o="492" file="504" n="504" rhead=""/> to lateri oppoſitus ſit acutus etiam, an vero <lb/>obtuſus.</head> <p> <s xml:id="echoid-s17811" xml:space="preserve">FIAT, vt latus datum angulo dato oppoſitum ad alterum latus da-<lb/> <anchor type="note" xlink:label="note-504-01a" xlink:href="note-504-01"/> tum, ita ſinus anguli dati ad aliud, reperieturq́ue ſinus anguli alteri dato la-<lb/>teri oppoſiti, qui ſi acutus fuerit, (ſemper autem acutus erit, ſi datus eſt ob-<lb/>tuſus) ex ipſo ſinu inuento notus fiet; </s> <s xml:id="echoid-s17812" xml:space="preserve">ſi vero obtuſus, ſinus inuentus dabit <lb/>angulum, qui ex duobus rectis ſubductus quæſitum angulum alteri dato la-<lb/>teri oppoſitum relinquet: </s> <s xml:id="echoid-s17813" xml:space="preserve">Summa autem ex dato angulo, & </s> <s xml:id="echoid-s17814" xml:space="preserve">inuento angulo <lb/>conflata, ſi ex duobus rectis ſubtrahatur, indicabit tertium angulum à datis <lb/>lateribus comprehenſum. </s> <s xml:id="echoid-s17815" xml:space="preserve">Fiat deinde, vt ſinus anguli dati ad ſinum huius ter <lb/>tij anguli inuenti, ita latus datum dato angulo oppoſitum ad aliud, gigne-<lb/>turq́; </s> <s xml:id="echoid-s17816" xml:space="preserve">tertium latus quæſitum.</s> <s xml:id="echoid-s17817" xml:space="preserve"/> </p> <div xml:id="echoid-div1441" type="float" level="2" n="1"> <note position="left" xlink:label="note-504-01" xlink:href="note-504-01a" xml:space="preserve">Propoſ. 13. <lb/>triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s17818" xml:space="preserve">SI data latera ſint æqualia, datus etiam erit angulus alteri dato lateri op-<lb/>poſitus, cum dato angulo vt æqualis. </s> <s xml:id="echoid-s17819" xml:space="preserve">Hinc tertius angulus, & </s> <s xml:id="echoid-s17820" xml:space="preserve">tertium latus <lb/>reperietur, vt prius.</s> <s xml:id="echoid-s17821" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1443" type="section" level="1" n="641"> <head xml:id="echoid-head676" xml:space="preserve">TRIANGVLORVM SPHAERI-<lb/>CORVM RECTANGVLORVM <lb/>PROBLEMATA, ACPRAXES.</head> <note position="left" xml:space="preserve">Quæritur <lb/>angulus nõ <lb/>rectus.</note> </div> <div xml:id="echoid-div1444" type="section" level="1" n="642"> <head xml:id="echoid-head677" xml:space="preserve">1. DATO arcu in triangulo rectangulo, qui <lb/>recto angulo opponitur, cum alterutro ar-<lb/>cuum circa rectum angulum, inuenire angu <lb/>lum huic arcui oppoſitum.</head> <p> <s xml:id="echoid-s17822" xml:space="preserve">FIAT, vt ſinus arcus dati recto angulo oppoſiti ad ſinum totum, ita ſi-<lb/> <anchor type="note" xlink:label="note-504-03a" xlink:href="note-504-03"/> nus arcus dati circa angulum rectum ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17823" xml:space="preserve">ſin us anguli huic <lb/>arcui oppoſiti, qui quæritur. </s> <s xml:id="echoid-s17824" xml:space="preserve">Hic autem angulus erit acutus, ſi datus ar-<lb/>cus ei oppoſitus circa rectum angulum fuerit quadrante minor; </s> <s xml:id="echoid-s17825" xml:space="preserve">obtuſus au-<lb/>tem, ſi maior.</s> <s xml:id="echoid-s17826" xml:space="preserve"/> </p> <div xml:id="echoid-div1444" type="float" level="2" n="1"> <note position="left" xlink:label="note-504-03" xlink:href="note-504-03a" xml:space="preserve">Probl. 1. pro <lb/>poſ. 41. tri-<lb/>ang. ſphær.</note> </div> </div> <div xml:id="echoid-div1446" type="section" level="1" n="643"> <head xml:id="echoid-head678" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17827" xml:space="preserve">FIAT, vt ſinus totus ad ſinum arcus angulo recto oppoſiti, ita ſecans <lb/> <anchor type="note" xlink:label="note-504-04a" xlink:href="note-504-04"/> complementi arcus circa rectum angulum dati ad aliud, produceturq́; </s> <s xml:id="echoid-s17828" xml:space="preserve">ſecans <lb/>complementi anguli quæſiti, qui huic arcui opponitur.</s> <s xml:id="echoid-s17829" xml:space="preserve"/> </p> <div xml:id="echoid-div1446" type="float" level="2" n="1"> <note position="left" xlink:label="note-504-04" xlink:href="note-504-04a" xml:space="preserve">Probl. pro-<lb/>poſ. 55. tri-<lb/>ang. ſphær.</note> </div> </div> <div xml:id="echoid-div1448" type="section" level="1" n="644"> <head xml:id="echoid-head679" xml:space="preserve">2. DATO arcu in triangulo rectangulo, qui <pb o="493" file="505" n="505" rhead=""/> recto angulo opponitur, cum alterutro an-<lb/> gulorum nõ rectorum, inuenire arcum huic angulo oppoſitum.</head> <note position="right" xml:space="preserve">Quæritue <lb/>arcus circa <lb/>rectum an-<lb/>gulum.</note> <p> <s xml:id="echoid-s17830" xml:space="preserve">FIAT, vt ſinus totus ad ſinum arcus angulo recto oppoſiti, ita ſinus an-<lb/> <anchor type="note" xlink:label="note-505-02a" xlink:href="note-505-02"/> guli dati ad aliud, reperieturq́; </s> <s xml:id="echoid-s17831" xml:space="preserve">ſinus arcus huic angulo oppoſiti, qui quæri-<lb/>tur. </s> <s xml:id="echoid-s17832" xml:space="preserve">Hic autem arcus quadrante minor erit, ſi datus angulus ei oppoſitus fue <lb/>rit acutus; </s> <s xml:id="echoid-s17833" xml:space="preserve">maior vero, ſi obtuſus.</s> <s xml:id="echoid-s17834" xml:space="preserve"/> </p> <div xml:id="echoid-div1448" type="float" level="2" n="1"> <note position="right" xlink:label="note-505-02" xlink:href="note-505-02a" xml:space="preserve">Probl. 2. pro <lb/>poſ. 41. tri-<lb/>ang. ſphær.</note> </div> <note position="right" xml:space="preserve">Quæritur <lb/>arcus recto <lb/>angulo op-<lb/>poſitus.</note> </div> <div xml:id="echoid-div1450" type="section" level="1" n="645"> <head xml:id="echoid-head680" xml:space="preserve">3. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum angu-<lb/>lo ei oppoſito, reperire arcum recto angulo <lb/>oppoſitum: ſi modo conſtet, num quadran-<lb/>te minor ſit, an maior; vel an alter angulus <lb/>dato arcui adiacens ſit acutus, obtuſusve; vel <lb/>denique, an alter arcus circa rectum angu-<lb/>lum ſit minor quadrante, aut maior.</head> <p> <s xml:id="echoid-s17835" xml:space="preserve">FIAT, vt ſinus anguli dati ad ſinum dati arcus, ita ſinus totus ad aliud, <lb/> <anchor type="note" xlink:label="note-505-04a" xlink:href="note-505-04"/> produceturq́; </s> <s xml:id="echoid-s17836" xml:space="preserve">ſinus arcus recto angulo oppoſiti: </s> <s xml:id="echoid-s17837" xml:space="preserve">qui ex inuento ſinu cogno-<lb/>ſci non poterit, niſi conſtet, num ſit quadrante minor, vel maior; </s> <s xml:id="echoid-s17838" xml:space="preserve">aut an al-<lb/>ter angulus non rectus ſit acutus, obtuſusve; </s> <s xml:id="echoid-s17839" xml:space="preserve">aut an alter arcus circa angu-<lb/>lum rectum ſit minor, aut maior quadrante. </s> <s xml:id="echoid-s17840" xml:space="preserve">Nam ſi alter angulus eſt acutus, <lb/>ſi quidem & </s> <s xml:id="echoid-s17841" xml:space="preserve">angulus datus acutus ſit; </s> <s xml:id="echoid-s17842" xml:space="preserve">aut ſi tam ille, quam hic eſt obtuſus, <lb/>erit quæſitus arcus recto angulo oppoſitus, quadrante minor: </s> <s xml:id="echoid-s17843" xml:space="preserve">ſi vero alter <lb/>ille angulus eſt acutus, & </s> <s xml:id="echoid-s17844" xml:space="preserve">datus obtuſus; </s> <s xml:id="echoid-s17845" xml:space="preserve">aut ille obtuſus, & </s> <s xml:id="echoid-s17846" xml:space="preserve">hic acutus, erit <lb/>idem arcus quæſitus, & </s> <s xml:id="echoid-s17847" xml:space="preserve">angulo recto oppoſitus, quadrante maior. </s> <s xml:id="echoid-s17848" xml:space="preserve">Sic etiam, <lb/>ſi alter arcus circa angulum rectum, & </s> <s xml:id="echoid-s17849" xml:space="preserve">datus arcus, ſunt eiuſdem ſpeciei, nem <lb/>pe ambo minores, aut maiores quadrante, erit arcus quæſitus recto angulo op <lb/>poſitus quadrante minor; </s> <s xml:id="echoid-s17850" xml:space="preserve">ſi vero diuerſarum ſpecierum, nimirum vnus qua-<lb/>drante minor, & </s> <s xml:id="echoid-s17851" xml:space="preserve">altet maior, erit idem arcus quæſitus quadrante maior.</s> <s xml:id="echoid-s17852" xml:space="preserve"/> </p> <div xml:id="echoid-div1450" type="float" level="2" n="1"> <note position="right" xlink:label="note-505-04" xlink:href="note-505-04a" xml:space="preserve">Probl. 3. pro <lb/>poſ 41. tri-<lb/>ang. ſphær.</note> </div> </div> <div xml:id="echoid-div1452" type="section" level="1" n="646"> <head xml:id="echoid-head681" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17853" xml:space="preserve">FIAT, vt ſinus totus ad ſinum dati anguli, ita ſecans complementi arcus <lb/> <anchor type="note" xlink:label="note-505-05a" xlink:href="note-505-05"/> dati ad aliud, produceturq́; </s> <s xml:id="echoid-s17854" xml:space="preserve">ſecans complementi arcus recto angulo oppoſiti.</s> <s xml:id="echoid-s17855" xml:space="preserve"/> </p> <div xml:id="echoid-div1452" type="float" level="2" n="1"> <note position="right" xlink:label="note-505-05" xlink:href="note-505-05a" xml:space="preserve">Probl. pro-<lb/>poſ. 54. tri-<lb/>ang. ſphær.</note> </div> <note position="right" xml:space="preserve">Quæritur <lb/>vterque ar-<lb/>cuscirca an <lb/>gulũ rectũ. <lb/>Deinde ar-<lb/>cus recto an <lb/>gulo oppo-<lb/>ſitus.</note> </div> <div xml:id="echoid-div1454" type="section" level="1" n="647"> <head xml:id="echoid-head682" xml:space="preserve">4. DATIS duobus angulis non rectis in trian <lb/>gulo rectãgulo, inuenire arcum vtrilibet eo-<lb/>rum oppoſitum, vna cum arcu rectum angu <lb/>lum ſubrendente.</head> <pb o="494" file="506" n="506" rhead=""/> <p> <s xml:id="echoid-s17856" xml:space="preserve">FIAT, vt ſinus anguli dati quæſito arcui adiacentis ad ſinum totum, ita <lb/> <anchor type="note" xlink:label="note-506-01a" xlink:href="note-506-01"/> ſinus complementi alterius anguli dati ad aliud, produceturq́; </s> <s xml:id="echoid-s17857" xml:space="preserve">ſinus comple-<lb/>menti arcus huic poſteriori angulo oppoſiti. </s> <s xml:id="echoid-s17858" xml:space="preserve">Erit autem vterlibet arcus inuen <lb/>tus quadrante minor, ſi datus angulus ei oppoſitus fuerit acutus; </s> <s xml:id="echoid-s17859" xml:space="preserve">maior vero, <lb/>ſi obtuſus.</s> <s xml:id="echoid-s17860" xml:space="preserve"/> </p> <div xml:id="echoid-div1454" type="float" level="2" n="1"> <note position="left" xlink:label="note-506-01" xlink:href="note-506-01a" xml:space="preserve">Probl. 1. pro <lb/>poſ. 42. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17861" xml:space="preserve">IAM inuento vtroque arcu circa angulum rectum, inuenietur, per pro-<lb/>blema 3. </s> <s xml:id="echoid-s17862" xml:space="preserve">ex vtrolibet illorum, & </s> <s xml:id="echoid-s17863" xml:space="preserve">angulo ei oppoſito dato, arcus quoque recto <lb/>angulo oppoſitus.</s> <s xml:id="echoid-s17864" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1456" type="section" level="1" n="648"> <head xml:id="echoid-head683" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17865" xml:space="preserve">FIAT, vt ſinus totus ad ſinum anguli non recti quæſito arcui adiacen-<lb/> <anchor type="note" xlink:label="note-506-02a" xlink:href="note-506-02"/> tis, ita ſecans alterius anguli non recti ad aliud, reperieturq́; </s> <s xml:id="echoid-s17866" xml:space="preserve">ſecans arcus huic <lb/>poſteriori angulo oppoſiti, qui quæritur.</s> <s xml:id="echoid-s17867" xml:space="preserve"/> </p> <div xml:id="echoid-div1456" type="float" level="2" n="1"> <note position="left" xlink:label="note-506-02" xlink:href="note-506-02a" xml:space="preserve">Probl. pro-<lb/>poſ 52. tri <lb/>ang. ſphær.</note> </div> <note position="left" xml:space="preserve">Quæritur <lb/>angulus nõ <lb/>rectus, Dein <lb/>de alij duo <lb/>arcus.</note> </div> <div xml:id="echoid-div1458" type="section" level="1" n="649"> <head xml:id="echoid-head684" xml:space="preserve">5. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum angu-<lb/>lo ei adiacente, inueſtigare alium angulum <lb/>eidẽ arcui oppoſitũ, & reliquos duos arcus.</head> <p> <s xml:id="echoid-s17868" xml:space="preserve">FIAT, vt ſinus totus ad ſinum anguli dati, ita ſinus complementi arcus <lb/> <anchor type="note" xlink:label="note-506-04a" xlink:href="note-506-04"/> dati ad aliud, procreabiturq́; </s> <s xml:id="echoid-s17869" xml:space="preserve">ſinus complementi alterius anguli, quem quæri-<lb/>mus. </s> <s xml:id="echoid-s17870" xml:space="preserve">Hic autem angulus erit acutus, ſi datus arcus ſuerit quadrante minor; <lb/></s> <s xml:id="echoid-s17871" xml:space="preserve">obtuſus vero, ſi maior.</s> <s xml:id="echoid-s17872" xml:space="preserve"/> </p> <div xml:id="echoid-div1458" type="float" level="2" n="1"> <note position="left" xlink:label="note-506-04" xlink:href="note-506-04a" xml:space="preserve">Probl 2 pro <lb/>poſ. 42. tri-<lb/>ang ſphær.</note> </div> <p> <s xml:id="echoid-s17873" xml:space="preserve">EX vtroque autem angulo non recto, quorum vnus datus eſt, & </s> <s xml:id="echoid-s17874" xml:space="preserve">alter in-<lb/>uentus, reperientur reliqui duo arcus, vt in præcedenti problemate dictũ eſt.</s> <s xml:id="echoid-s17875" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quæritur <lb/>angulus nõ <lb/>rectus. Dein <lb/>de alij duo <lb/>arcus.</note> </div> <div xml:id="echoid-div1460" type="section" level="1" n="650"> <head xml:id="echoid-head685" xml:space="preserve">6. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum angu-<lb/>lo ei oppoſito, in ueſtigare alium angulum <lb/>non rectum eidem arcui adiacentem, & reli-<lb/>quos duos arcus: ſi modo conſtet, num alius <lb/>ille angulus non rectus quæſitus ſit acutus, <lb/>obtuſusve; vel an alteruter arcuum quæſito-<lb/>rum quadrante minor ſit, vel maior.</head> <p> <s xml:id="echoid-s17876" xml:space="preserve">FIAT, vt ſinus complementi arcus dati ad ſinum complementi anguli da <lb/> <anchor type="note" xlink:label="note-506-06a" xlink:href="note-506-06"/> ti, ita ſinus totus ad aliud, reperieturq́; </s> <s xml:id="echoid-s17877" xml:space="preserve">ſinus alterius anguli non recti quæſi-<lb/>ti: </s> <s xml:id="echoid-s17878" xml:space="preserve">qui ex inuento ſinu non elicietur, niſi prius conſtet, an acutus ſit, an ob-<lb/>tuſus: </s> <s xml:id="echoid-s17879" xml:space="preserve">Aut, an alteruter reliquorum duorum arcuum non datorum ſit qua-<lb/>drante minor, aut maior. </s> <s xml:id="echoid-s17880" xml:space="preserve">Nam ſi alter arcus circa angulum rectum non da-<lb/>tus, & </s> <s xml:id="echoid-s17881" xml:space="preserve">quæſito angulo oppoſitus, fuerit minor quadrante, erit quæſitus angu <lb/>lus acutus; </s> <s xml:id="echoid-s17882" xml:space="preserve">ſi vero maior, obtuſus. </s> <s xml:id="echoid-s17883" xml:space="preserve">Pari ratione, ſi arcus recto angulo oppo- <pb o="495" file="507" n="507" rhead=""/> ſitus, & </s> <s xml:id="echoid-s17884" xml:space="preserve">non datus, fuerit quadrante minor; </s> <s xml:id="echoid-s17885" xml:space="preserve">ſi quidem angulus datus ſit acu-<lb/>tus, erit quæſitus quoque angulus acutus; </s> <s xml:id="echoid-s17886" xml:space="preserve">ſi vero obtuſus, obtuſus: </s> <s xml:id="echoid-s17887" xml:space="preserve">At ſi ar-<lb/>cus angulo recto oppoſitus fuerit maior quadrante; </s> <s xml:id="echoid-s17888" xml:space="preserve">ſi quidem datus angulus <lb/>ſit acutus, erit quæſitus angulus obtuſus; </s> <s xml:id="echoid-s17889" xml:space="preserve">ſi vero obtuſus, acutus.</s> <s xml:id="echoid-s17890" xml:space="preserve"/> </p> <div xml:id="echoid-div1460" type="float" level="2" n="1"> <note position="left" xlink:label="note-506-06" xlink:href="note-506-06a" xml:space="preserve">Probl 2. pro <lb/>poſ. 42. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17891" xml:space="preserve">EX vtroque porro angulo non recto, quorum vnus datus eſt, & </s> <s xml:id="echoid-s17892" xml:space="preserve">alter in-<lb/>uentus, inuenientur reliqui duo arcus, vt in problemate 4. </s> <s xml:id="echoid-s17893" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s17894" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1462" type="section" level="1" n="651"> <head xml:id="echoid-head686" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17895" xml:space="preserve">FIAT, vt ſinus totus ad ſinum complementi arcus dati, ita ſecans dati <lb/> <anchor type="note" xlink:label="note-507-01a" xlink:href="note-507-01"/> anguliad aliud, reperieturq́; </s> <s xml:id="echoid-s17896" xml:space="preserve">ſecans complementi alterius anguli non recti, <lb/>qui quæritur. </s> <s xml:id="echoid-s17897" xml:space="preserve">Reliqua in uenientur, vt ſupra dictum eſt.</s> <s xml:id="echoid-s17898" xml:space="preserve"/> </p> <div xml:id="echoid-div1462" type="float" level="2" n="1"> <note position="right" xlink:label="note-507-01" xlink:href="note-507-01a" xml:space="preserve">Probl. pro-<lb/>poſ. 56. tri-<lb/>ang. ſphær.</note> </div> <note position="right" xml:space="preserve">Quæritur <lb/>arcus recto <lb/>angulo op-<lb/>poſitus. De-<lb/>inde duo <lb/>anguli non <lb/>recti.</note> </div> <div xml:id="echoid-div1464" type="section" level="1" n="652"> <head xml:id="echoid-head687" xml:space="preserve">7. DATIS duobus arcubus in triangulo re-<lb/>ctangulo circa angulum rectum, reperire ter <lb/>tium arcum angulo recto oppoſitũ, & duos <lb/>angulos non rectos.</head> <p> <s xml:id="echoid-s17899" xml:space="preserve">FIAT, vt ſinus totus ad ſinũ complementi vtriuſlibet arcuum datorum, <lb/> <anchor type="note" xlink:label="note-507-03a" xlink:href="note-507-03"/> ita ſinus complementi alterius arcus dati ad aliud, produceturq́; </s> <s xml:id="echoid-s17900" xml:space="preserve">ſinus com-<lb/>plementi arcus recto angulo oppoſiti. </s> <s xml:id="echoid-s17901" xml:space="preserve">Hic autem arcus quadrante erit mi-<lb/>nor, ſi vterque arcus circa rectum angulum datus fuerit minor, aut maior qua <lb/>drante; </s> <s xml:id="echoid-s17902" xml:space="preserve">quadrante vero maior, ſi vnus datorum arcuum fuerit quadrante mi-<lb/>nor, & </s> <s xml:id="echoid-s17903" xml:space="preserve">alter maior.</s> <s xml:id="echoid-s17904" xml:space="preserve"/> </p> <div xml:id="echoid-div1464" type="float" level="2" n="1"> <note position="right" xlink:label="note-507-03" xlink:href="note-507-03a" xml:space="preserve">Probl. pro-<lb/>poſ. 43. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17905" xml:space="preserve">EX arcu autem rectum angulum ſubtẽdente inuento, & </s> <s xml:id="echoid-s17906" xml:space="preserve">alterutro arcuum <lb/>circa angulum rectum datorum, inuenietur angulus ei oppoſitus, vt in pro-<lb/>blemate 1. </s> <s xml:id="echoid-s17907" xml:space="preserve">diximus.</s> <s xml:id="echoid-s17908" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Quæritur <lb/>arcus circa <lb/>angulũ re-<lb/>ctum. Dein <lb/>de duo an-<lb/>guli non re <lb/>cti.</note> </div> <div xml:id="echoid-div1466" type="section" level="1" n="653"> <head xml:id="echoid-head688" xml:space="preserve">8. DATO arcu in triangulo rectangulo, qui <lb/>recto angulo opponitur, cum alterutro ar <lb/>cuum circa angulũ rectum, inquirere alium <lb/>arcum circa rectum angulum, & duos angu-<lb/>los non rectos.</head> <p> <s xml:id="echoid-s17909" xml:space="preserve">FIAT, vt ſinus complementi arcus dati circa angulum rectum ad ſinum <lb/> <anchor type="note" xlink:label="note-507-05a" xlink:href="note-507-05"/> complementi arcus recto angulo oppoſiti, ita ſinus totus ad aliud, gigneturq́; <lb/></s> <s xml:id="echoid-s17910" xml:space="preserve">ſinus complementi alterius arcus circa rectum angulum, qui quæritur. </s> <s xml:id="echoid-s17911" xml:space="preserve">Hic au <lb/>tem arcus erit quadrante minor, ſi vterque arcus datus minor quadrante fue-<lb/>rit, aut maior; </s> <s xml:id="echoid-s17912" xml:space="preserve">maior vero, ſi alter datorum arcuum fuerit quadrante minor, <lb/>& </s> <s xml:id="echoid-s17913" xml:space="preserve">alter maior.</s> <s xml:id="echoid-s17914" xml:space="preserve"/> </p> <div xml:id="echoid-div1466" type="float" level="2" n="1"> <note position="right" xlink:label="note-507-05" xlink:href="note-507-05a" xml:space="preserve">Probl. pro-<lb/>poſ. 43. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17915" xml:space="preserve">INVENTO autem arcu rectum angulum ſubtendente, reperientur an-<lb/>guli, vt in præcedenti problemate dictum eſt.</s> <s xml:id="echoid-s17916" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1468" type="section" level="1" n="654"> <head xml:id="echoid-head689" xml:space="preserve">Aliter.</head> <note position="right" xml:space="preserve">Probl. pro-<lb/>poſ. 53. tri-<lb/>ang. ſphær.</note> <p> <s xml:id="echoid-s17917" xml:space="preserve">FIAT, vt ſinus totus ad ſinum complementi dati arcus circa angulum re <pb o="496" file="508" n="508" rhead=""/> ctum, ita ſecans arcus angulo recto oppoſiti ad aliud, produceturq́; </s> <s xml:id="echoid-s17918" xml:space="preserve">ſecans ter <lb/>tij arcus, qui quæritur, &</s> <s xml:id="echoid-s17919" xml:space="preserve">c.</s> <s xml:id="echoid-s17920" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quæritur <lb/>arcus eirca <lb/>angulũ re-<lb/>ctum. Dein <lb/>de alter an <lb/>gulus non <lb/>rectus, & at <lb/>cus recto <lb/>angulo op-<lb/>poſitus.</note> </div> <div xml:id="echoid-div1469" type="section" level="1" n="655"> <head xml:id="echoid-head690" xml:space="preserve">9. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum angu-<lb/>lo non recto ei adiacente, ſcrutari alterum <lb/>arcum circa angulum rectum, & alium angu <lb/>lum non rectum, cum arcu rectum angulum <lb/>ſubtendente.</head> <p> <s xml:id="echoid-s17921" xml:space="preserve">FIAT, vt ſinus totus ad ſinum dati arcus, ita tangẽs dati anguli ad aliud, <lb/> <anchor type="note" xlink:label="note-508-02a" xlink:href="note-508-02"/> produceturq́; </s> <s xml:id="echoid-s17922" xml:space="preserve">tangens arcus quæſiti. </s> <s xml:id="echoid-s17923" xml:space="preserve">Qui arcus min or quadrante erit, ſi datus <lb/>angulus ei oppoſitus fuerit acutus; </s> <s xml:id="echoid-s17924" xml:space="preserve">maior autem, ſi obtuſus.</s> <s xml:id="echoid-s17925" xml:space="preserve"/> </p> <div xml:id="echoid-div1469" type="float" level="2" n="1"> <note position="left" xlink:label="note-508-02" xlink:href="note-508-02a" xml:space="preserve">Probl. 1. pro <lb/>poſ. 44. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17926" xml:space="preserve">EX eodem porrò arcu circa angulum rectum dato, & </s> <s xml:id="echoid-s17927" xml:space="preserve">angulo adiacente, <lb/>reperietur & </s> <s xml:id="echoid-s17928" xml:space="preserve">alter angulus non rectus, & </s> <s xml:id="echoid-s17929" xml:space="preserve">arcus recto angulo oppoſitus, vt ſu-<lb/>pra in 5. </s> <s xml:id="echoid-s17930" xml:space="preserve">problemate docuimus.</s> <s xml:id="echoid-s17931" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quæritur <lb/>arcus circa <lb/>angulũ re-<lb/>ctum. Dein <lb/>de alter an <lb/>gulus non <lb/>rectus, & ar <lb/>cus recto <lb/>angulo op-<lb/>poſitus.</note> </div> <div xml:id="echoid-div1471" type="section" level="1" n="656"> <head xml:id="echoid-head691" xml:space="preserve">10. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum angu-<lb/>lo ei oppoſito, indagare alterum arcum circa <lb/>rectum angulum, & alium angulum non re-<lb/>ctum, cum arcu rectum angulum ſubtenden <lb/>te: ſi modo conſtet, an reliquus arcus circa <lb/>angulum rectum quęſitus quadrante minor <lb/>ſit, aut maior; vel an alter angulus non rectus <lb/>ſit acutus, obtuſusve; vel denique num arcus <lb/>angulo recto oppoſitus ſit minor quadran-<lb/>te, aut maior.</head> <p> <s xml:id="echoid-s17932" xml:space="preserve">FIAT, vt tangens anguli dati ad tangentem dati arcus, ita ſinus totus ad <lb/> <anchor type="note" xlink:label="note-508-04a" xlink:href="note-508-04"/> aliud, reperieturq́; </s> <s xml:id="echoid-s17933" xml:space="preserve">ſinus arcus quæſiti: </s> <s xml:id="echoid-s17934" xml:space="preserve">qui ex inuento ſinu non cognoſcetur, <lb/>niſi cõſtet, num quadrante minor ſit, aut maior; </s> <s xml:id="echoid-s17935" xml:space="preserve">vel an alter angulus non re-<lb/>ctus ſit acutus, obtuſusve; </s> <s xml:id="echoid-s17936" xml:space="preserve">vel denique, an arcus recto angulo oppoſitus ſit <lb/>minor quadrante, aut maior. </s> <s xml:id="echoid-s17937" xml:space="preserve">Nam ſi alter angulus fuerit acutus, erit quæſi-<lb/>tus arcus ei oppoſitus, quadrante minor; </s> <s xml:id="echoid-s17938" xml:space="preserve">ſi vero obtuſus, maior. </s> <s xml:id="echoid-s17939" xml:space="preserve">Sic etiam, ſi <lb/>arcus recto angulo oppoſitus fuerit minor quadrante, ſi quidem & </s> <s xml:id="echoid-s17940" xml:space="preserve">datus ar-<lb/>cus ſit quadrante minor, erit quæſitus arcus minor quoque quadrante; </s> <s xml:id="echoid-s17941" xml:space="preserve">ſi vero <lb/>quadrãte maior, maior quoque: </s> <s xml:id="echoid-s17942" xml:space="preserve">At ſi arcus recto angulo oppoſitus fuerit qua- <pb o="497" file="509" n="509" rhead=""/> drante maior, ſi quidem datus arcus maior quoque ſit, erit quęſitus arcus mi-<lb/>nor quadrante; </s> <s xml:id="echoid-s17943" xml:space="preserve">ſi vero quadrante minor, maior.</s> <s xml:id="echoid-s17944" xml:space="preserve"/> </p> <div xml:id="echoid-div1471" type="float" level="2" n="1"> <note position="left" xlink:label="note-508-04" xlink:href="note-508-04a" xml:space="preserve">Probl. 1. pro <lb/>poſ. 44. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17945" xml:space="preserve">IAM vero ex eodem arcu circa angulum rectum dato, & </s> <s xml:id="echoid-s17946" xml:space="preserve">angulo oppoſi-<lb/>to, reperietur & </s> <s xml:id="echoid-s17947" xml:space="preserve">alter angulus non rectus, & </s> <s xml:id="echoid-s17948" xml:space="preserve">arcus recto angulo oppoſitus, vt <lb/>in problemate 6. </s> <s xml:id="echoid-s17949" xml:space="preserve">traditum eſt. </s> <s xml:id="echoid-s17950" xml:space="preserve">Vel certe, ex duobus arcubus circa angulum <lb/>rectum, quorum vnus datus eſt, & </s> <s xml:id="echoid-s17951" xml:space="preserve">alter inuentus, inuenietur arcus recto angu <lb/>lo oppoſitus, cum duobus angulis non rectis, vt in problemate 7. </s> <s xml:id="echoid-s17952" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s17953" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1473" type="section" level="1" n="657"> <head xml:id="echoid-head692" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17954" xml:space="preserve">FIAT, vt ſinus totus ad tangentem dati arcus, ita tangens complemen-<lb/> <anchor type="note" xlink:label="note-509-01a" xlink:href="note-509-01"/> ti anguli dati ad aliud, reperieturq́; </s> <s xml:id="echoid-s17955" xml:space="preserve">ſinus arcus quæſiti. </s> <s xml:id="echoid-s17956" xml:space="preserve">Reliqua inuenientur, <lb/>vt proxime præcepimus.</s> <s xml:id="echoid-s17957" xml:space="preserve"/> </p> <div xml:id="echoid-div1473" type="float" level="2" n="1"> <note position="right" xlink:label="note-509-01" xlink:href="note-509-01a" xml:space="preserve">Probl. pro-<lb/>poſ. 49. tri-<lb/>ang. ſphær.</note> </div> <note position="right" xml:space="preserve">Quæritur <lb/>vterque an <lb/>gulus non <lb/>rectus. Dein <lb/>de arcus re <lb/>cto angulo <lb/>oppoſitus.</note> </div> <div xml:id="echoid-div1475" type="section" level="1" n="658"> <head xml:id="echoid-head693" xml:space="preserve">11. DATIS duobus arcubus in triangulo re-<lb/>ctangulo circa angulum rectum, inuenire <lb/>vtrumlibet angulorum non rectorum, & ar-<lb/>cum præterea recto angulo oppoſitum.</head> <p> <s xml:id="echoid-s17958" xml:space="preserve">FIAT, vt ſinus vtriusvis arcuum datorum ad ſinum totum, ita tangens <lb/> <anchor type="note" xlink:label="note-509-03a" xlink:href="note-509-03"/> alterius arcus dati ad aliud, procreabiturq́; </s> <s xml:id="echoid-s17959" xml:space="preserve">tangens anguli huic poſteriori ar-<lb/>cui oppoſiti. </s> <s xml:id="echoid-s17960" xml:space="preserve">Qui angulus acutus erit, ſi datus arcus oppoſitus fuerit quadran <lb/>te minor; </s> <s xml:id="echoid-s17961" xml:space="preserve">obtuſus autem, ſi maior.</s> <s xml:id="echoid-s17962" xml:space="preserve"/> </p> <div xml:id="echoid-div1475" type="float" level="2" n="1"> <note position="right" xlink:label="note-509-03" xlink:href="note-509-03a" xml:space="preserve">Probl. 2. pro <lb/>poſ. 44. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s17963" xml:space="preserve">EX eiſdem duobus arcubus datis inuenietur, per 7. </s> <s xml:id="echoid-s17964" xml:space="preserve">problema, arcus ter-<lb/>tius recto angulo oppoſitus: </s> <s xml:id="echoid-s17965" xml:space="preserve">Vel certe, per problema 3. </s> <s xml:id="echoid-s17966" xml:space="preserve">ex alterutro arcuum <lb/>datorum, & </s> <s xml:id="echoid-s17967" xml:space="preserve">angulo oppoſito inuento.</s> <s xml:id="echoid-s17968" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1477" type="section" level="1" n="659"> <head xml:id="echoid-head694" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17969" xml:space="preserve">FIAT, vt ſinus totus ad ſinum vtriusvis arcuum datorum, ita tangens <lb/> <anchor type="note" xlink:label="note-509-04a" xlink:href="note-509-04"/> complementi alterius arcus dati ad aliud, prodibitq́; </s> <s xml:id="echoid-s17970" xml:space="preserve">tangens complemẽti an-<lb/>guli poſteriori huic arcui oppoſiti. </s> <s xml:id="echoid-s17971" xml:space="preserve">Reliqua inueniẽtur, vt proxime dictum eſt.</s> <s xml:id="echoid-s17972" xml:space="preserve"/> </p> <div xml:id="echoid-div1477" type="float" level="2" n="1"> <note position="right" xlink:label="note-509-04" xlink:href="note-509-04a" xml:space="preserve">Probl. pro-<lb/>poſ. 48. tri-<lb/>ang. ſphær.</note> </div> <note position="right" xml:space="preserve">Quęritur ar <lb/>cus recto an <lb/>gulo oppoſi <lb/>tus. Deinde <lb/>alter arcus <lb/>circa rectũ <lb/>angulum, <lb/>cum altero <lb/>angulo nõ <lb/>recto.</note> </div> <div xml:id="echoid-div1479" type="section" level="1" n="660"> <head xml:id="echoid-head695" xml:space="preserve">12. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum angu-<lb/>lo non recto ei adiacente, inuenire arcum re <lb/>cto angulo oppoſitum, & reliquum arcum <lb/>circa angulum rectum, cum altero angulo <lb/>non recto.</head> <p> <s xml:id="echoid-s17973" xml:space="preserve">FIAT, vt ſinus complementi anguli dati ad ſinum totum, ita tangens <lb/> <anchor type="note" xlink:label="note-509-06a" xlink:href="note-509-06"/> dati arcus ad aliud, reperieturq́; </s> <s xml:id="echoid-s17974" xml:space="preserve">tangens arcus angulo recto oppoſiti. </s> <s xml:id="echoid-s17975" xml:space="preserve">Hic <lb/>autem arcus quadrante erit minor, ſi datus angulus fuerit acutus, & </s> <s xml:id="echoid-s17976" xml:space="preserve">datus <lb/>arcus ei adiacens quadrante minor; </s> <s xml:id="echoid-s17977" xml:space="preserve">aut ſi angulus datus obtuſus fuerit, & </s> <s xml:id="echoid-s17978" xml:space="preserve">ar-<lb/>cus datus quadrante maior: </s> <s xml:id="echoid-s17979" xml:space="preserve">Maior autem quadrante erit idem arcus quæſitus, <pb o="498" file="510" n="510" rhead=""/> ſi datus angulus ſuerit acutus, & </s> <s xml:id="echoid-s17980" xml:space="preserve">arcus datus quadrante maior; </s> <s xml:id="echoid-s17981" xml:space="preserve">aut ſi datus <lb/>angulus fuerit obtuſus, & </s> <s xml:id="echoid-s17982" xml:space="preserve">arcus datus minor quadrante.</s> <s xml:id="echoid-s17983" xml:space="preserve"/> </p> <div xml:id="echoid-div1479" type="float" level="2" n="1"> <note position="right" xlink:label="note-509-06" xlink:href="note-509-06a" xml:space="preserve">Probl. 1. pro <lb/>poſ. 45. tri-<lb/>ang ſphær.</note> </div> <p> <s xml:id="echoid-s17984" xml:space="preserve">IAM vero, per 2. </s> <s xml:id="echoid-s17985" xml:space="preserve">problema, ex arcu rectum angulum ſubtendente inuen <lb/>to, & </s> <s xml:id="echoid-s17986" xml:space="preserve">angulo dato, reperietur alter arcus circa angulum rectum dato angulo <lb/>oppoſitus. </s> <s xml:id="echoid-s17987" xml:space="preserve">Ex eodem vero arcu rectum angulum ſubtendente, & </s> <s xml:id="echoid-s17988" xml:space="preserve">arcu in prin <lb/>cipio dato, inuenietur, per 1. </s> <s xml:id="echoid-s17989" xml:space="preserve">problema, alter angulus non rectus dato ar-<lb/>cui oppoſitus.</s> <s xml:id="echoid-s17990" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1481" type="section" level="1" n="661"> <head xml:id="echoid-head696" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s17991" xml:space="preserve">FIAT, vt ſinus totus ad ſinum complementi anguli dati, ita tangens com <lb/> <anchor type="note" xlink:label="note-510-01a" xlink:href="note-510-01"/> plementi arcus dati ad aliud, inuenieturq́; </s> <s xml:id="echoid-s17992" xml:space="preserve">tangens complementi arcus recto <lb/>angulo oppoſiti. </s> <s xml:id="echoid-s17993" xml:space="preserve">Reliqua reperientur, vt prius.</s> <s xml:id="echoid-s17994" xml:space="preserve"/> </p> <div xml:id="echoid-div1481" type="float" level="2" n="1"> <note position="left" xlink:label="note-510-01" xlink:href="note-510-01a" xml:space="preserve">Probl. pro-<lb/>poſ. 46. tri-<lb/>ang. ſphær.</note> </div> <note position="left" xml:space="preserve">Quæritur <lb/>angulius nõ <lb/>rectus. De-<lb/>inde alter <lb/>arcus circa <lb/>rectum an-<lb/>gulũ, & al. <lb/>ter angulus <lb/>non rectus.</note> </div> <div xml:id="echoid-div1483" type="section" level="1" n="662"> <head xml:id="echoid-head697" xml:space="preserve">13. DATO alterutro arcuum in triangulo re-<lb/>ctangulo circa angulum rectum, cum arcu re <lb/>ctum angulum ſubtendente, reperire angu-<lb/>lum à dictis arcubus comprehenſum, ſiue da <lb/>to arcui circa rectum angulum adiacentem, <lb/>& inſuper reliquum arcum, & angulum.</head> <p> <s xml:id="echoid-s17995" xml:space="preserve">FIAT, vt tangens arcus recto angulo oppoſiti ad tangentem dati arcus <lb/> <anchor type="note" xlink:label="note-510-03a" xlink:href="note-510-03"/> circa angulum rectum, ita ſinus totus ad aliud, produceturq́; </s> <s xml:id="echoid-s17996" xml:space="preserve">ſinus comple-<lb/>menti anguli à dictis arcubus comprehenſi, qui quæritur. </s> <s xml:id="echoid-s17997" xml:space="preserve">Hic autem acutus <lb/>erit, ſi datus arcus recto angulo oppoſitus fuerit quadrante minor, & </s> <s xml:id="echoid-s17998" xml:space="preserve">arcus <lb/>circa rectum angulum datus minor quoque; </s> <s xml:id="echoid-s17999" xml:space="preserve">aut ſi tam ille, quam hic quadran <lb/>te maior fuerit: </s> <s xml:id="echoid-s18000" xml:space="preserve">Idem vero angulus quæſitus erit obtuſus, ſi datus arcus an-<lb/>gulo recto oppoſitus fuerit minor quadrante, & </s> <s xml:id="echoid-s18001" xml:space="preserve">datus arcus circa rectum an-<lb/>gulum quadrante maior; </s> <s xml:id="echoid-s18002" xml:space="preserve">aut ſi ille fuerit quadrante maior, & </s> <s xml:id="echoid-s18003" xml:space="preserve">hic minor.</s> <s xml:id="echoid-s18004" xml:space="preserve"/> </p> <div xml:id="echoid-div1483" type="float" level="2" n="1"> <note position="left" xlink:label="note-510-03" xlink:href="note-510-03a" xml:space="preserve">Probl. 2. pro <lb/>poſ. 45. tri-<lb/>ang. ſphær.</note> </div> <p> <s xml:id="echoid-s18005" xml:space="preserve">RELIQVA inueſtigabũtur, vt in præcedenti problemate traditum eſt.</s> <s xml:id="echoid-s18006" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1485" type="section" level="1" n="663"> <head xml:id="echoid-head698" xml:space="preserve">Aliter.</head> <p> <s xml:id="echoid-s18007" xml:space="preserve">FIAT, vt ſinus totus ad tangentem complementi arcus angulo recto op <lb/> <anchor type="note" xlink:label="note-510-04a" xlink:href="note-510-04"/> poſiti, ita tangens dati arcus circa rectum angulum ad aliud, inuenieturq́; </s> <s xml:id="echoid-s18008" xml:space="preserve">ſi-<lb/>nus complementi anguli adia centis, qui deſideratur.</s> <s xml:id="echoid-s18009" xml:space="preserve"/> </p> <div xml:id="echoid-div1485" type="float" level="2" n="1"> <note position="left" xlink:label="note-510-04" xlink:href="note-510-04a" xml:space="preserve">Probl. pro-<lb/>poſ. 51. tri-<lb/>ang. ſphær.</note> </div> <note position="left" xml:space="preserve">Quæritur <lb/>arcus circa <lb/>angulũ re-<lb/>ctum. Dein <lb/>de alter ar-<lb/>cus circa an <lb/>gulum re-<lb/>ctum, cum <lb/>reliquo an-<lb/>gulo non <lb/>recto.</note> </div> <div xml:id="echoid-div1487" type="section" level="1" n="664"> <head xml:id="echoid-head699" xml:space="preserve">14. DATO arcu rectum angulum ſubtenden <lb/>te in triangulo rectangulo, cum alterutro an-<lb/>gulorum non rectorum, reperire arcum cir-<lb/>ca angulum rectum huic angulo adiacẽtem, <lb/>ac præterea alterum arcum circa angulum re <lb/>ctum, cum altero angulo non recto.</head> <pb o="499" file="511" n="511" rhead=""/> <p> <s xml:id="echoid-s18010" xml:space="preserve">FIAT, vt ſinus totus ad ſinum complementianguli dati, ita tangens ar-<lb/> <anchor type="note" xlink:label="note-511-01a" xlink:href="note-511-01"/> cus recto angulo oppoſiti ad aliud, procreabiturq́; </s> <s xml:id="echoid-s18011" xml:space="preserve">tangens arcus quæſiti. </s> <s xml:id="echoid-s18012" xml:space="preserve">Qui <lb/>quadrãte minor erit, ſi arcus datus recto angulo oppoſitus fuerit minor qua-<lb/>drante, & </s> <s xml:id="echoid-s18013" xml:space="preserve">datus angulus acutus; </s> <s xml:id="echoid-s18014" xml:space="preserve">aut ſi arcus datus quadrante fuerit maior, & </s> <s xml:id="echoid-s18015" xml:space="preserve"><lb/>angulus datus obtuſus: </s> <s xml:id="echoid-s18016" xml:space="preserve">Idem vero arcus quæſitus erit quadrante maior, ſi da-<lb/>tus arcus angulo recto oppoſitus fuerit minor quadrãte, & </s> <s xml:id="echoid-s18017" xml:space="preserve">datus angulus ob-<lb/>tuſus; </s> <s xml:id="echoid-s18018" xml:space="preserve">aut ſi arcus datus fuerit quadrante maior, & </s> <s xml:id="echoid-s18019" xml:space="preserve">datus angulus acutus.</s> <s xml:id="echoid-s18020" xml:space="preserve"/> </p> <div xml:id="echoid-div1487" type="float" level="2" n="1"> <note position="right" xlink:label="note-511-01" xlink:href="note-511-01a" xml:space="preserve">Probl. 3. <lb/>propoſ. 45. <lb/>triãg. ſphęr.</note> </div> <p> <s xml:id="echoid-s18021" xml:space="preserve">CAETERA explorabuntur, vt in problemate 12. </s> <s xml:id="echoid-s18022" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s18023" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18024" xml:space="preserve">15. </s> <s xml:id="echoid-s18025" xml:space="preserve">DATO arcu in triangulo rectangulo, qui <lb/> <anchor type="note" xlink:label="note-511-02a" xlink:href="note-511-02"/> recto angulo opponitur, cum alterutro angu <lb/>lorum non rectorum, inquirere alterum an-<lb/>gulum non rectum, & </s> <s xml:id="echoid-s18026" xml:space="preserve">duos arcus circa re-<lb/>ctum angulum.</s> <s xml:id="echoid-s18027" xml:space="preserve"/> </p> <div xml:id="echoid-div1488" type="float" level="2" n="2"> <note position="right" xlink:label="note-511-02" xlink:href="note-511-02a" xml:space="preserve">Quæritur <lb/>angulus nõ <lb/>rectus. De-<lb/>inde duo <lb/>reliqui ar-<lb/>cus.</note> </div> <p> <s xml:id="echoid-s18028" xml:space="preserve">FIAT, vt ſinus totus ad ſinum complementi dati arcus recto angulo op <lb/> <anchor type="note" xlink:label="note-511-03a" xlink:href="note-511-03"/> poſiti, ita tangens anguli dati ad aliud, reperieturq́; </s> <s xml:id="echoid-s18029" xml:space="preserve">tangens complementian-<lb/>guli quæſiti. </s> <s xml:id="echoid-s18030" xml:space="preserve">Hic vero erit acutus, ſi arcus recto angulo oppoſitus fuerit qua-<lb/>drante minor, & </s> <s xml:id="echoid-s18031" xml:space="preserve">datus angulus acutus; </s> <s xml:id="echoid-s18032" xml:space="preserve">aut ſi datus arcus fuerit maior quadran <lb/>te, & </s> <s xml:id="echoid-s18033" xml:space="preserve">datus angulus obtuſus: </s> <s xml:id="echoid-s18034" xml:space="preserve">At angulus idem quæſitus erit obtuſus, ſi arcus <lb/>angulo recto oppoſitus quadrante minor fuerit, & </s> <s xml:id="echoid-s18035" xml:space="preserve">angulus datus obtuſus; </s> <s xml:id="echoid-s18036" xml:space="preserve">aut <lb/>ſi arcus ille fuerit quadrante maior, & </s> <s xml:id="echoid-s18037" xml:space="preserve">datus angulus acutus.</s> <s xml:id="echoid-s18038" xml:space="preserve"/> </p> <div xml:id="echoid-div1489" type="float" level="2" n="3"> <note position="right" xlink:label="note-511-03" xlink:href="note-511-03a" xml:space="preserve">Probl. pro-<lb/>poſ 47. tri-<lb/>ang ſphær.</note> </div> <p> <s xml:id="echoid-s18039" xml:space="preserve">HINC ex dato arcu angulum rectum ſubtendente, & </s> <s xml:id="echoid-s18040" xml:space="preserve">vtroque angulo <lb/>non recto, quorum vnus datus eſt, & </s> <s xml:id="echoid-s18041" xml:space="preserve">alterinuentus, reperietur, per 2. </s> <s xml:id="echoid-s18042" xml:space="preserve">pro-<lb/>blema, vterque arcus circa rectum angulum.</s> <s xml:id="echoid-s18043" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18044" xml:space="preserve">16. </s> <s xml:id="echoid-s18045" xml:space="preserve">DATIS duobus angulis non rectis in trian <lb/> <anchor type="note" xlink:label="note-511-04a" xlink:href="note-511-04"/> gulo rectangulo, inuenire arcum recto angu-<lb/>lo oppoſitum, & </s> <s xml:id="echoid-s18046" xml:space="preserve">reliquos duos arcus circa <lb/>angulum rectum.</s> <s xml:id="echoid-s18047" xml:space="preserve"/> </p> <div xml:id="echoid-div1490" type="float" level="2" n="4"> <note position="right" xlink:label="note-511-04" xlink:href="note-511-04a" xml:space="preserve">Quæritur <lb/>arcus angu <lb/>lo recto op <lb/>poſitus. De <lb/>inde duo <lb/>arcus circa <lb/>angulum <lb/>rectum.</note> </div> <p> <s xml:id="echoid-s18048" xml:space="preserve">FIAT, vt ſinus totus ad tangentem complementi vtriusvis angulorum <lb/> <anchor type="note" xlink:label="note-511-05a" xlink:href="note-511-05"/> datorum, ita tangens complementi alterius dati anguli ad aliud, procreabi-<lb/>turq́; </s> <s xml:id="echoid-s18049" xml:space="preserve">ſinus complementi arcus angulo recto oppoſiti, quem deſideramus. </s> <s xml:id="echoid-s18050" xml:space="preserve">Hic <lb/>arcus erit quadrante minor, ſi vterque angulorum datorum acutus fuerit, ob-<lb/>tuſusve; </s> <s xml:id="echoid-s18051" xml:space="preserve">quadrante vero maior, ſi alter acutus fuerit, & </s> <s xml:id="echoid-s18052" xml:space="preserve">alter obtuſus.</s> <s xml:id="echoid-s18053" xml:space="preserve"/> </p> <div xml:id="echoid-div1491" type="float" level="2" n="5"> <note position="right" xlink:label="note-511-05" xlink:href="note-511-05a" xml:space="preserve">Probl. pro-<lb/>poſ. 50. tri-<lb/>ang ſphær.</note> </div> <p> <s xml:id="echoid-s18054" xml:space="preserve">PORRO ex arcu rectum angulum ſubtendente inuento, & </s> <s xml:id="echoid-s18055" xml:space="preserve">vtrouis an-<lb/>gulorum datorũ, reperietur arcus ei oppoſitus, vt in 2. </s> <s xml:id="echoid-s18056" xml:space="preserve">problem. </s> <s xml:id="echoid-s18057" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s18058" xml:space="preserve"/> </p> <pb o="500" file="512" n="512"/> </div> <div xml:id="echoid-div1493" type="section" level="1" n="665"> <head xml:id="echoid-head700" xml:space="preserve">TRIANGVLORVM SPHAERI-<lb/>CORVM NON RECTANGVLORVM</head> <head xml:id="echoid-head701" xml:space="preserve">PROBLEMATA, ETPRAXES.</head> <p> <s xml:id="echoid-s18059" xml:space="preserve">17. </s> <s xml:id="echoid-s18060" xml:space="preserve">DATIS omnibus angulis trianguli non <lb/> <anchor type="note" xlink:label="note-512-01a" xlink:href="note-512-01"/> rectanguli, inuenire omnes eius arcus.</s> <s xml:id="echoid-s18061" xml:space="preserve"/> </p> <div xml:id="echoid-div1493" type="float" level="2" n="1"> <note position="left" xlink:label="note-512-01" xlink:href="note-512-01a" xml:space="preserve">Quætũtur <lb/>omucs ar-<lb/>cus.</note> </div> <p> <s xml:id="echoid-s18062" xml:space="preserve">SINT primum omnes anguli dati in triangulo ABC, inæquales, quo-<lb/> <anchor type="note" xlink:label="note-512-02a" xlink:href="note-512-02"/> rum duo B, C, acuti, vel obtuſi, & </s> <s xml:id="echoid-s18063" xml:space="preserve">ex tertio angulo A, ad BC, ducatur arcus <lb/> <anchor type="figure" xlink:label="fig-512-01a" xlink:href="fig-512-01"/> perdendicularis AD, qui intra triangulum cadet. </s> <s xml:id="echoid-s18064" xml:space="preserve">Sta <lb/>tuantur finus complementorum angulorum B, C, pro <lb/>terminis proportionis ſinus anguli BAD, ad ſinum <lb/> <anchor type="note" xlink:label="note-512-03a" xlink:href="note-512-03"/> anguli CAD. </s> <s xml:id="echoid-s18065" xml:space="preserve">Atque ex hac proportione, & </s> <s xml:id="echoid-s18066" xml:space="preserve">ag-<lb/>gregato angulorum BAD, CAD, hoc eſt, ex dato <lb/>angulo BAC, inquiratur, per problema 6. </s> <s xml:id="echoid-s18067" xml:space="preserve">triang. </s> <s xml:id="echoid-s18068" xml:space="preserve">re-<lb/>ctil. </s> <s xml:id="echoid-s18069" xml:space="preserve">vterque angulus BAD, CAD. </s> <s xml:id="echoid-s18070" xml:space="preserve">Deinde, per pro-<lb/>blema 16. </s> <s xml:id="echoid-s18071" xml:space="preserve">triang. </s> <s xml:id="echoid-s18072" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18073" xml:space="preserve">tam ex duobus angulis B, <lb/>BAD, non rectis inueſtigetur arcus AB, angulo re-<lb/>cto D, oppoſitus in triangulo ABD, quam ex duo-<lb/>bus angulis non rectis C, CAD, arcus AC, recto angulo D, in triãgulo ACD, <lb/>oppoſitus. </s> <s xml:id="echoid-s18074" xml:space="preserve">Poſtremo, per problema 2. </s> <s xml:id="echoid-s18075" xml:space="preserve">tam ex arcu AB, rectum angulum D, <lb/>ſubtendente, & </s> <s xml:id="echoid-s18076" xml:space="preserve">angulo BAD, inuentis reperiatur arcus BD, quam ex arcu <lb/>AC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18077" xml:space="preserve">angulo CAD, inuentis arcus CD. <lb/></s> <s xml:id="echoid-s18078" xml:space="preserve">Summa enim arcuum BD, CD, totum arcum BC, efficiet notum. </s> <s xml:id="echoid-s18079" xml:space="preserve">Atque ita <lb/>omnes tres arcus AB, AC, BC, noti facti erunt.</s> <s xml:id="echoid-s18080" xml:space="preserve"/> </p> <div xml:id="echoid-div1494" type="float" level="2" n="2"> <note position="left" xlink:label="note-512-02" xlink:href="note-512-02a" xml:space="preserve">Quãdo om <lb/>nes anguli <lb/>dati sũt in-<lb/>æquales.</note> <figure xlink:label="fig-512-01" xlink:href="fig-512-01a"> <image file="512-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/512-01"/> </figure> <note position="left" xlink:label="note-512-03" xlink:href="note-512-03a" xml:space="preserve">Propoſ. 62. <lb/>triãg. ſphęr.</note> </div> <p style="it"> <s xml:id="echoid-s18081" xml:space="preserve">_PER_ ſolos ſinus ita problema abſoluemus. </s> <s xml:id="echoid-s18082" xml:space="preserve">_V_ter que angulus _<emph style="sc">B</emph>AD, <emph style="sc">Ca</emph>D,_ in-<lb/> <anchor type="note" xlink:label="note-512-04a" xlink:href="note-512-04"/> ueniatur per _3._ </s> <s xml:id="echoid-s18083" xml:space="preserve">praxim problematis 6. </s> <s xml:id="echoid-s18084" xml:space="preserve">triang. </s> <s xml:id="echoid-s18085" xml:space="preserve">rectil. </s> <s xml:id="echoid-s18086" xml:space="preserve">_D_einde, per 1. </s> <s xml:id="echoid-s18087" xml:space="preserve">praxim proble-<lb/>matis _4._ </s> <s xml:id="echoid-s18088" xml:space="preserve">triang. </s> <s xml:id="echoid-s18089" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18090" xml:space="preserve">tam ex duobus angulis _<emph style="sc">B, B</emph>AD,_ inueſtigetur arcus _<emph style="sc">B</emph>D,_ <lb/>quam ex duobus angulis _C, <emph style="sc">Ca</emph>D,_ arcus _CD._ </s> <s xml:id="echoid-s18091" xml:space="preserve">Summa enim arcuum _<emph style="sc">B</emph>D, CD,_ to-<lb/>tum arcum _<emph style="sc">B</emph>C,_ notum efficiet. </s> <s xml:id="echoid-s18092" xml:space="preserve">_P_oſtremo, per problema _3._ </s> <s xml:id="echoid-s18093" xml:space="preserve">triang. </s> <s xml:id="echoid-s18094" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18095" xml:space="preserve">reperia-<lb/>tur tam arcus _<emph style="sc">Ab</emph>,_ recto angulo _D,_ oppoſitus, ex arcu _<emph style="sc">B</emph>D,_ & </s> <s xml:id="echoid-s18096" xml:space="preserve">angulo eioppoſito <lb/>_<emph style="sc">B</emph>AD,_ inuentis, quam arcus _<emph style="sc">A</emph>C,_ recto angulo _D,_ oppoſitus, ex arcu _CD,_ & </s> <s xml:id="echoid-s18097" xml:space="preserve">an-<lb/>gulo _CAD,_ ei oppoſito inuentis: </s> <s xml:id="echoid-s18098" xml:space="preserve">quia preter data conſtat etiam ſpecies tam alterius <lb/>anguli _B,_ quam anguli alterius _C,_ cum vterque datus ſit.</s> <s xml:id="echoid-s18099" xml:space="preserve"/> </p> <div xml:id="echoid-div1495" type="float" level="2" n="3"> <note position="left" xlink:label="note-512-04" xlink:href="note-512-04a" xml:space="preserve">Per ſolos ſi <lb/>nus, quãdo <lb/>omnes dati <lb/>anguli inę-<lb/>quales sũt.</note> </div> <p> <s xml:id="echoid-s18100" xml:space="preserve">QVOD ſi quando alter angulorum ad A, inuentus fuerit rectus, nempe <lb/>BAD; </s> <s xml:id="echoid-s18101" xml:space="preserve">inuenti erunt duo arcus AB, BD, cum vterque ſit quadrans, ob rectos <lb/>angulos D, DAB. </s> <s xml:id="echoid-s18102" xml:space="preserve">Eadem ratione, ſi deprehenſus fuerit angulus CAD, re-<lb/>ctus, non autem BAD, (fieri enim non poteſt, vt angulus vterque ad A, re-<lb/>ctus ſit, cum totus BAC, minor ſit duobus rectis.) </s> <s xml:id="echoid-s18103" xml:space="preserve">inuenti erunt duo arcus <lb/>AC, CD, vtpote quadrantes, ob angulos rectos D, DAC.</s> <s xml:id="echoid-s18104" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18105" xml:space="preserve">SINT deinde duo ſaltem anguli dati B, C, æquales, quicquid ſit de ter-<lb/> <anchor type="note" xlink:label="note-512-05a" xlink:href="note-512-05"/> tio A, à quo arcus perpendicularis AD, ad BC, ducatur. </s> <s xml:id="echoid-s18106" xml:space="preserve">Erunt tam duo ar-<lb/>cus AB, AC, quam duo BD, CD, & </s> <s xml:id="echoid-s18107" xml:space="preserve">duo anguliad A, æquales; </s> <s xml:id="echoid-s18108" xml:space="preserve">ac proinde <lb/>vterque angulus ad A, cognitus, tanquam dimidium dati anguli BAC. </s> <s xml:id="echoid-s18109" xml:space="preserve">Inue- <pb o="501" file="513" n="513" rhead=""/> niatur ergo, per 16. </s> <s xml:id="echoid-s18110" xml:space="preserve">problema, triang. </s> <s xml:id="echoid-s18111" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18112" xml:space="preserve">arcus AB, recto angulo D, op-<lb/>poſitus, ex duobus angulis B, BAD; </s> <s xml:id="echoid-s18113" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s18114" xml:space="preserve">proinde & </s> <s xml:id="echoid-s18115" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-513-01a" xlink:href="fig-513-01"/> AC, illi æqualis, cognitus. </s> <s xml:id="echoid-s18116" xml:space="preserve">Deinde, per problema 14. <lb/></s> <s xml:id="echoid-s18117" xml:space="preserve">triang. </s> <s xml:id="echoid-s18118" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18119" xml:space="preserve">ex inuento arcu AB, rectum angulum ſub-<lb/>tendente, & </s> <s xml:id="echoid-s18120" xml:space="preserve">dato angulo B, reperiatur arcus BD; </s> <s xml:id="echoid-s18121" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s18122" xml:space="preserve"><lb/>propterea & </s> <s xml:id="echoid-s18123" xml:space="preserve">CD, illi æqualis, cognitus; </s> <s xml:id="echoid-s18124" xml:space="preserve">ideoq́; </s> <s xml:id="echoid-s18125" xml:space="preserve">& </s> <s xml:id="echoid-s18126" xml:space="preserve">totus <lb/>BC, notus. </s> <s xml:id="echoid-s18127" xml:space="preserve">Inuentiq́; </s> <s xml:id="echoid-s18128" xml:space="preserve">iam erunt omnes tres arcus AB, <lb/>AC, BC.</s> <s xml:id="echoid-s18129" xml:space="preserve"/> </p> <div xml:id="echoid-div1496" type="float" level="2" n="4"> <note position="left" xlink:label="note-512-05" xlink:href="note-512-05a" xml:space="preserve">Quãdo da <lb/>ti duo an-<lb/>guli ſunt <lb/>æquales.</note> <figure xlink:label="fig-513-01" xlink:href="fig-513-01a"> <image file="513-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/513-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s18130" xml:space="preserve">_PER_ ſolos autem ſinus ita rem exequemur. </s> <s xml:id="echoid-s18131" xml:space="preserve">_P_er 1. </s> <s xml:id="echoid-s18132" xml:space="preserve">praxim <lb/> <anchor type="note" xlink:label="note-513-01a" xlink:href="note-513-01"/> problematis _4._ </s> <s xml:id="echoid-s18133" xml:space="preserve">triang. </s> <s xml:id="echoid-s18134" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18135" xml:space="preserve">inquiratur arcus _<emph style="sc">B</emph>D,_ ex duobus <lb/>angulis _<emph style="sc">B, BA</emph>D;_ </s> <s xml:id="echoid-s18136" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s18137" xml:space="preserve">idcirco & </s> <s xml:id="echoid-s18138" xml:space="preserve">_CD,_ illi æqualis, cognitus, proptereaq́; </s> <s xml:id="echoid-s18139" xml:space="preserve">& </s> <s xml:id="echoid-s18140" xml:space="preserve">totus <lb/>_<emph style="sc">BC</emph>,_ notus. </s> <s xml:id="echoid-s18141" xml:space="preserve">_D_einde, per problema _3._ </s> <s xml:id="echoid-s18142" xml:space="preserve">triang. </s> <s xml:id="echoid-s18143" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18144" xml:space="preserve">ex arcuinuento _<emph style="sc">B</emph>D,_ & </s> <s xml:id="echoid-s18145" xml:space="preserve">an-<lb/>gulo ei oppoſito _<emph style="sc">Bad</emph>,_ reperiatur arcus _<emph style="sc">AB</emph>,_ recto angulo oppoſitus: </s> <s xml:id="echoid-s18146" xml:space="preserve">quia præter da-<lb/>ta conſtat etiam ſpecies alterius anguli _<emph style="sc">B</emph>,_ cum datus ſit: </s> <s xml:id="echoid-s18147" xml:space="preserve">eritq́; </s> <s xml:id="echoid-s18148" xml:space="preserve">propterea & </s> <s xml:id="echoid-s18149" xml:space="preserve">arcus <lb/>_<emph style="sc">A</emph>C,_ ipſi _<emph style="sc">Ab</emph>,_ æqualis, cognitus.</s> <s xml:id="echoid-s18150" xml:space="preserve"/> </p> <div xml:id="echoid-div1497" type="float" level="2" n="5"> <note position="right" xlink:label="note-513-01" xlink:href="note-513-01a" xml:space="preserve">Per ſolos G <lb/>nus, quãdo <lb/>duo dati an <lb/>guli ſunt <lb/>æquales.</note> </div> <p> <s xml:id="echoid-s18151" xml:space="preserve">18. </s> <s xml:id="echoid-s18152" xml:space="preserve">DATIS omnibus arcubus trianguli non <lb/> <anchor type="note" xlink:label="note-513-02a" xlink:href="note-513-02"/> rectanguli, inueſtigare omnes eius angulos.</s> <s xml:id="echoid-s18153" xml:space="preserve"/> </p> <div xml:id="echoid-div1498" type="float" level="2" n="6"> <note position="right" xlink:label="note-513-02" xlink:href="note-513-02a" xml:space="preserve">Quætũtur <lb/>omnes an-<lb/>guli.</note> </div> <p> <s xml:id="echoid-s18154" xml:space="preserve">SINT omnes arcus in triangulo ABC, dati, ſitq́; </s> <s xml:id="echoid-s18155" xml:space="preserve">primo loco inquiren <lb/> <anchor type="note" xlink:label="note-513-03a" xlink:href="note-513-03"/> dus angulus A, & </s> <s xml:id="echoid-s18156" xml:space="preserve">duo arcus AB, AC, eum continentes ſint inæquales, quadran <lb/>teq́; </s> <s xml:id="echoid-s18157" xml:space="preserve">minores, quicquid ſit de arcu BC. </s> <s xml:id="echoid-s18158" xml:space="preserve">Productis arcubus AB, AC, vt fiant <lb/>quadrantes AD, AE, deſcribatur per D, E, arcus circuli maximi DE, occur-<lb/>rens arcui BC, producto verſus maiorem arcum, qui ſit AC, in puncto F. </s> <s xml:id="echoid-s18159" xml:space="preserve">Sta-<lb/> <anchor type="note" xlink:label="note-513-04a" xlink:href="note-513-04"/> tuantur ſinus complementorum arcuum datorum AB, AC, pro terminis pro-<lb/>portionis ſinus arcus BF, ad ſinum arcus CF. </s> <s xml:id="echoid-s18160" xml:space="preserve">Atque ex hac proportione, & </s> <s xml:id="echoid-s18161" xml:space="preserve"><lb/>arcu dato BC, qui differentia eſt arcuum BF, CF, inueſtigetur, per proble-<lb/>ma 8. </s> <s xml:id="echoid-s18162" xml:space="preserve">triang. </s> <s xml:id="echoid-s18163" xml:space="preserve">rectil. </s> <s xml:id="echoid-s18164" xml:space="preserve">vterque arcus BF, CF. <lb/></s> <s xml:id="echoid-s18165" xml:space="preserve"> <anchor type="figure" xlink:label="fig-513-02a" xlink:href="fig-513-02"/> Deinde, per problema 8. </s> <s xml:id="echoid-s18166" xml:space="preserve">triang. </s> <s xml:id="echoid-s18167" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18168" xml:space="preserve">inue-<lb/>ſtigetur tam arcus DF, ex arcu inuento BF, <lb/>rectum angulum D, ſubtẽdente, & </s> <s xml:id="echoid-s18169" xml:space="preserve">arcu BD, <lb/>qui complementum eſt dati arcus AB; </s> <s xml:id="echoid-s18170" xml:space="preserve">quam <lb/>arcus EF, ex arcu inuento CF, rectum angu <lb/>lum E, ſubtendente, & </s> <s xml:id="echoid-s18171" xml:space="preserve">arcu CE, qui comple <lb/>mentum eſt dati arcus AC. </s> <s xml:id="echoid-s18172" xml:space="preserve">Subducto enim <lb/>arcu EF, inuento, ex inuento arcu DF, no-<lb/>tus remanebit arcus DE, anguli, A; </s> <s xml:id="echoid-s18173" xml:space="preserve">ac proin <lb/>de angulus A, notus erit. </s> <s xml:id="echoid-s18174" xml:space="preserve">Poſt hæc, per pro-<lb/>blema 11. </s> <s xml:id="echoid-s18175" xml:space="preserve">triang. </s> <s xml:id="echoid-s18176" xml:space="preserve">ſphæ. </s> <s xml:id="echoid-s18177" xml:space="preserve">ex arcubus notis BD, <lb/>DF, circa rectum angulum D, inueniatur an-<lb/>gulus DBF, ac proinde & </s> <s xml:id="echoid-s18178" xml:space="preserve">reliquus duorum <lb/>rectorum ABC. </s> <s xml:id="echoid-s18179" xml:space="preserve">Eadem denique ratione, ex arcubus CE, EF, notis circa an-<lb/>gulum rectum E, eruatur angulus ECF, atque adeo & </s> <s xml:id="echoid-s18180" xml:space="preserve">angulus ACB, ei ad <lb/>verticem æqualis. </s> <s xml:id="echoid-s18181" xml:space="preserve">Atque ita iam omnes tres anguli A, B, C, inuenti erunt.</s> <s xml:id="echoid-s18182" xml:space="preserve"/> </p> <div xml:id="echoid-div1499" type="float" level="2" n="7"> <note position="right" xlink:label="note-513-03" xlink:href="note-513-03a" xml:space="preserve">Quãdo duo <lb/>dati arcus <lb/>ſunt inæ-<lb/>quales, & <lb/>quadrante <lb/>minores.</note> <note position="right" xlink:label="note-513-04" xlink:href="note-513-04a" xml:space="preserve">Prop. 63. <lb/>triãg. ſphęr.</note> <figure xlink:label="fig-513-02" xlink:href="fig-513-02a"> <image file="513-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/513-02"/> </figure> </div> <p style="it"> <s xml:id="echoid-s18183" xml:space="preserve">_PER_ ſolos ſinus ita progrediemur. </s> <s xml:id="echoid-s18184" xml:space="preserve">Vterque arcus _<emph style="sc">B</emph>F, CF,_ reperiatur per _3._ <lb/></s> <s xml:id="echoid-s18185" xml:space="preserve"> <anchor type="note" xlink:label="note-513-05a" xlink:href="note-513-05"/> praxim problematis _8._ </s> <s xml:id="echoid-s18186" xml:space="preserve">triang. </s> <s xml:id="echoid-s18187" xml:space="preserve">rectil. </s> <s xml:id="echoid-s18188" xml:space="preserve">_D_einde, per _1._ </s> <s xml:id="echoid-s18189" xml:space="preserve">praxim problematis _8._ </s> <s xml:id="echoid-s18190" xml:space="preserve">triang. <lb/></s> <s xml:id="echoid-s18191" xml:space="preserve">ſphar. </s> <s xml:id="echoid-s18192" xml:space="preserve">tam arcus _DF,_ ex arcu inuento _<emph style="sc">B</emph>F,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18193" xml:space="preserve">ar-<lb/>cu _<emph style="sc">B</emph>D,_ complemento dati arcus _<emph style="sc">Ab</emph>,_ inueniatur, quam arcus _EF,_ ex inuento ar- <pb o="502" file="514" n="514" rhead=""/> cu _<emph style="sc">CF</emph>,_ rectum angulum _E,_ ſubtendente, & </s> <s xml:id="echoid-s18194" xml:space="preserve">arcu _CE,_ complemento dati arcus _<emph style="sc">Ac</emph>._ <lb/></s> <s xml:id="echoid-s18195" xml:space="preserve"> <anchor type="note" xlink:label="note-514-01a" xlink:href="note-514-01"/> Subducto enim arcu _EF,_ ex arcu _DF,_ notus relinquetur _DE,_ arcus anguli A; </s> <s xml:id="echoid-s18196" xml:space="preserve">atq; <lb/></s> <s xml:id="echoid-s18197" xml:space="preserve">adeo angulus _A,_ notus erit. </s> <s xml:id="echoid-s18198" xml:space="preserve">Poſt hæc, per problema _1._ </s> <s xml:id="echoid-s18199" xml:space="preserve">triang. </s> <s xml:id="echoid-s18200" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18201" xml:space="preserve">ex arcu inuen-<lb/>to _<emph style="sc">BF</emph>,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18202" xml:space="preserve">inuento arcu _DF,_ inquiratur angu-<lb/>lus _<emph style="sc">Db</emph>F,_ arcui _DF,_ oppoſitus: </s> <s xml:id="echoid-s18203" xml:space="preserve">_E_x quo notus quoque fiet reliquus angulus duorum <lb/>rectorum, nempe _<emph style="sc">AB</emph>C._ </s> <s xml:id="echoid-s18204" xml:space="preserve">Ad extremum eadem ratione, ex arcuinuento _<emph style="sc">C</emph>F,_ rectum <lb/>angulum _E,_ ſubtendente, & </s> <s xml:id="echoid-s18205" xml:space="preserve">inuento arcu _EF,_ inueſtigetur angulus _ECF,_ arcui <lb/>_EF,_ oppoſitus: </s> <s xml:id="echoid-s18206" xml:space="preserve">Ex quo notus etiam fiet angulus ei ad verticem æqualis _ACB._</s> <s xml:id="echoid-s18207" xml:space="preserve"/> </p> <div xml:id="echoid-div1500" type="float" level="2" n="8"> <note position="right" xlink:label="note-513-05" xlink:href="note-513-05a" xml:space="preserve">Per ſolos fi-<lb/>nus, quãdo <lb/>dati duo ar <lb/>cus sũt in-<lb/>æquales, &</note> <note position="left" xlink:label="note-514-01" xlink:href="note-514-01a" xml:space="preserve">quadrante <lb/>minores.</note> </div> <p> <s xml:id="echoid-s18208" xml:space="preserve">SINT deinde duo arcus inæquales AB, AC, qua-<lb/> <anchor type="note" xlink:label="note-514-02a" xlink:href="note-514-02"/> <anchor type="figure" xlink:label="fig-514-01a" xlink:href="fig-514-01"/> drante maiores; </s> <s xml:id="echoid-s18209" xml:space="preserve">qui producantur, donec conueniant <lb/>in D: </s> <s xml:id="echoid-s18210" xml:space="preserve">Eruntq́; </s> <s xml:id="echoid-s18211" xml:space="preserve">in triangulo DBC, duo arcus DB, DC, <lb/>inæquales, & </s> <s xml:id="echoid-s18212" xml:space="preserve">quadrante minores. </s> <s xml:id="echoid-s18213" xml:space="preserve">Quare, vt proxime <lb/>diximus, omnes eius tres anguli reperientur; </s> <s xml:id="echoid-s18214" xml:space="preserve">ac pro-<lb/>inde & </s> <s xml:id="echoid-s18215" xml:space="preserve">reliqui duorum rectorum ABC, ACB, noti <lb/>erunt, nec non & </s> <s xml:id="echoid-s18216" xml:space="preserve">A, ipſi D, æqualis.</s> <s xml:id="echoid-s18217" xml:space="preserve"/> </p> <div xml:id="echoid-div1501" type="float" level="2" n="9"> <note position="left" xlink:label="note-514-02" xlink:href="note-514-02a" xml:space="preserve">Quãdo duo <lb/>dati arcus <lb/>sũt inęqua-<lb/>les, & qua-<lb/>drante ma-<lb/>iores.</note> <figure xlink:label="fig-514-01" xlink:href="fig-514-01a"> <image file="514-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/514-01"/> </figure> </div> <p> <s xml:id="echoid-s18218" xml:space="preserve">SIT tertio arcus AB, quadrante minor, & </s> <s xml:id="echoid-s18219" xml:space="preserve">AC, <lb/> <anchor type="note" xlink:label="note-514-03a" xlink:href="note-514-03"/> maior quadrante. </s> <s xml:id="echoid-s18220" xml:space="preserve">Producto AB, vt fiat quadrans <lb/>AD, & </s> <s xml:id="echoid-s18221" xml:space="preserve">abſciſſo ex AC, quadrante AE, ducatur per D, E, arcus circuli maxi-<lb/>mi DE, ſecans BC, in F, vt in priore harum duarum figurarum. </s> <s xml:id="echoid-s18222" xml:space="preserve">Statuantur <lb/> <anchor type="figure" xlink:label="fig-514-02a" xlink:href="fig-514-02"/> ſinus complemẽtorum arcuum datorum AB, <lb/>AC, pro terminis proportionis ſinus arcus <lb/>BF, ad ſinum arcus CF. </s> <s xml:id="echoid-s18223" xml:space="preserve">Atque ex hac pro-<lb/>portione, & </s> <s xml:id="echoid-s18224" xml:space="preserve">aggregato arcuum BF, CF, hoc <lb/>eſt, ex dato arcu BC, indagetur, per 6. </s> <s xml:id="echoid-s18225" xml:space="preserve">pro-<lb/>blema triang. </s> <s xml:id="echoid-s18226" xml:space="preserve">rectil. </s> <s xml:id="echoid-s18227" xml:space="preserve">vterque arcus BF, CF. <lb/></s> <s xml:id="echoid-s18228" xml:space="preserve">Deinde, per problema 8. </s> <s xml:id="echoid-s18229" xml:space="preserve">triang. </s> <s xml:id="echoid-s18230" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18231" xml:space="preserve">inue-<lb/>niatur tam arcus DF, ex arcu BF, inuento <lb/>rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18232" xml:space="preserve">arcu BD, <lb/>complemẽto dati arcus AB; </s> <s xml:id="echoid-s18233" xml:space="preserve">quam arcus EF, <lb/>ex inuento arcu CF, rectũ angulum E, ſubten <lb/>dente, & </s> <s xml:id="echoid-s18234" xml:space="preserve">arcu CE, complemento arcus AC, <lb/>dati. </s> <s xml:id="echoid-s18235" xml:space="preserve">Summa enim inuentorum arcuum DF, <lb/>EF, dabit totum arcum DE, anguli A; </s> <s xml:id="echoid-s18236" xml:space="preserve">ac proinde angulus A, cognitus erit. </s> <s xml:id="echoid-s18237" xml:space="preserve"><lb/>Poſt hæc, per problema 11. </s> <s xml:id="echoid-s18238" xml:space="preserve">triang. </s> <s xml:id="echoid-s18239" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18240" xml:space="preserve">perueſtigetur ex arcubus DB, DF, <lb/>notis circa angulum rectum D, angulus DBF, ac proinde & </s> <s xml:id="echoid-s18241" xml:space="preserve">duorum recto-<lb/>rum reliquus ABC. </s> <s xml:id="echoid-s18242" xml:space="preserve">Ac tandem eodem modo ex arcubus CE, EF, circa an-<lb/>gulum rectum E, notis eliciatur angulus C: </s> <s xml:id="echoid-s18243" xml:space="preserve">Inuentiq́; </s> <s xml:id="echoid-s18244" xml:space="preserve">erunt omnes tres an-<lb/>guli A, B, C.</s> <s xml:id="echoid-s18245" xml:space="preserve"/> </p> <div xml:id="echoid-div1502" type="float" level="2" n="10"> <note position="left" xlink:label="note-514-03" xlink:href="note-514-03a" xml:space="preserve">Quãdo duo <lb/>arcus dati <lb/>inæquales <lb/>sũt, & vnus <lb/>quadrante <lb/>maior, & al <lb/>ter minor.</note> <figure xlink:label="fig-514-02" xlink:href="fig-514-02a"> <image file="514-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/514-02"/> </figure> </div> <p style="it"> <s xml:id="echoid-s18246" xml:space="preserve">_PER_ ſolos ſinus ita agendum erit. </s> <s xml:id="echoid-s18247" xml:space="preserve">_V_terque arcus _BF, CF,_ per _3._ </s> <s xml:id="echoid-s18248" xml:space="preserve">praxim pro-<lb/> <anchor type="note" xlink:label="note-514-04a" xlink:href="note-514-04"/> blematis 6. </s> <s xml:id="echoid-s18249" xml:space="preserve">triang. </s> <s xml:id="echoid-s18250" xml:space="preserve">rectil, inueniatur. </s> <s xml:id="echoid-s18251" xml:space="preserve">_D_einde per _1._ </s> <s xml:id="echoid-s18252" xml:space="preserve">praxim problematis _8._ </s> <s xml:id="echoid-s18253" xml:space="preserve">triang. <lb/></s> <s xml:id="echoid-s18254" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18255" xml:space="preserve">tam arcus _DF,_ ex arcus _<emph style="sc">B</emph>F,_ inuento, rectumq́; </s> <s xml:id="echoid-s18256" xml:space="preserve">angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18257" xml:space="preserve"><lb/>arcu _<emph style="sc">B</emph>D,_ complemento arcus dati _<emph style="sc">Ab</emph>;_ </s> <s xml:id="echoid-s18258" xml:space="preserve">quam arcus _EF,_ ex inuento arcu _CF,_ qui <lb/>recto angulo _E,_ opponitur, & </s> <s xml:id="echoid-s18259" xml:space="preserve">arcu _CE,_ complemento dati arcus _AC,_ eruatur. </s> <s xml:id="echoid-s18260" xml:space="preserve"><lb/>_N_am ſumma inuentorum arcuum _DF, EF,_ totum arcum _DE,_ anguli _A,_ dabit. </s> <s xml:id="echoid-s18261" xml:space="preserve">Poſt <lb/>hæc, per problema _1._ </s> <s xml:id="echoid-s18262" xml:space="preserve">triang. </s> <s xml:id="echoid-s18263" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18264" xml:space="preserve">reperiatur ex arcu _<emph style="sc">Bf</emph>,_ rectum angulum _D,_ ſub-<lb/>tendente, & </s> <s xml:id="echoid-s18265" xml:space="preserve">arcu _DF,_ notis, angulus _<emph style="sc">Db</emph>F,_ ac proinde & </s> <s xml:id="echoid-s18266" xml:space="preserve">duorum rectorum re-<lb/>liquus _<emph style="sc">AB</emph>C._ </s> <s xml:id="echoid-s18267" xml:space="preserve">Et tandẽ eodem modo ex notis arcubus _CF, EF,_ angulus _C,_ inueniatur.</s> <s xml:id="echoid-s18268" xml:space="preserve"/> </p> <div xml:id="echoid-div1503" type="float" level="2" n="11"> <note position="left" xlink:label="note-514-04" xlink:href="note-514-04a" xml:space="preserve">Per ſolos ſi-<lb/>nus, quãdo <lb/>dati duo ar <lb/>cus inæqua <lb/>les ſunt, & <lb/>vnus qua-<lb/>drante ma <lb/>ior, & alter <lb/>minor.</note> </div> <p> <s xml:id="echoid-s18269" xml:space="preserve">SIT quarto maior arcus AB, quadrans, & </s> <s xml:id="echoid-s18270" xml:space="preserve">AC, minor quadrante, vtin <pb o="503" file="515" n="515" rhead=""/> poſteriore proximarum duarum ſigurarum. </s> <s xml:id="echoid-s18271" xml:space="preserve">Producto arcu AC, vt fiat qua-<lb/> <anchor type="note" xlink:label="note-515-01a" xlink:href="note-515-01"/> drans AD, ducatur per B, D, arcus circuli maximi BD. </s> <s xml:id="echoid-s18272" xml:space="preserve">Deinde, per proble-<lb/>ma 8. </s> <s xml:id="echoid-s18273" xml:space="preserve">triang. </s> <s xml:id="echoid-s18274" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18275" xml:space="preserve">ex arcu dato BC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18276" xml:space="preserve">ar-<lb/>cu CD, complemento dati arcus AC, inueniatur arcus BD, anguli A; </s> <s xml:id="echoid-s18277" xml:space="preserve">ex quo <lb/>angulus ipſe A, notus erit. </s> <s xml:id="echoid-s18278" xml:space="preserve">Poſt hæc, per problema 11. </s> <s xml:id="echoid-s18279" xml:space="preserve">triang. </s> <s xml:id="echoid-s18280" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18281" xml:space="preserve">inueſti-<lb/>getur ex duobus arcubus notis BD, CD, circa rectum angulum D, angulus <lb/>BCD; </s> <s xml:id="echoid-s18282" xml:space="preserve">ex quo notus quoque erit duorum rectorum reliquus ACB. </s> <s xml:id="echoid-s18283" xml:space="preserve">Denique, <lb/>per idem problema 11. </s> <s xml:id="echoid-s18284" xml:space="preserve">ex eiſdem arcubus BD, CD, reperiatur angulus CBD; <lb/></s> <s xml:id="echoid-s18285" xml:space="preserve">qui ex recto ABD, detractus notum relinquet angulum ABC.</s> <s xml:id="echoid-s18286" xml:space="preserve"/> </p> <div xml:id="echoid-div1504" type="float" level="2" n="12"> <note position="right" xlink:label="note-515-01" xlink:href="note-515-01a" xml:space="preserve">Quãdo duo <lb/>arcus dati <lb/>inæquales <lb/>sũt, & ma-<lb/>ior arcus <lb/>quadrans, <lb/>minor au-<lb/>tem qua-<lb/>dr ante mi-<lb/>nor.</note> </div> <p style="it"> <s xml:id="echoid-s18287" xml:space="preserve">_PER_ ſolos ſinus ſic procedemus _P_er _1_ praxim problematis _8._ </s> <s xml:id="echoid-s18288" xml:space="preserve">triang ſphær ex da <lb/> <anchor type="note" xlink:label="note-515-02a" xlink:href="note-515-02"/> toarcu _<emph style="sc">B</emph>C,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18289" xml:space="preserve">arcu _CD,_ complemento dati arcus <lb/>_AC,_ reperiatur _<emph style="sc">B</emph>D,_ arcus anguli _A:_ </s> <s xml:id="echoid-s18290" xml:space="preserve">ex quo angulus ipſe _A,_ cognitus erit. </s> <s xml:id="echoid-s18291" xml:space="preserve">_D_einde <lb/>ex arcubus notis _<emph style="sc">B</emph>C, <emph style="sc">B</emph>D,_ per problema _1._ </s> <s xml:id="echoid-s18292" xml:space="preserve">triang ſphær. </s> <s xml:id="echoid-s18293" xml:space="preserve">eruitur angulus _<emph style="sc">B</emph>CD;_ <lb/></s> <s xml:id="echoid-s18294" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s18295" xml:space="preserve">duorum rectorum reliquus _<emph style="sc">ACb</emph>._ </s> <s xml:id="echoid-s18296" xml:space="preserve">_E_adem tandem ratione, ex notis <lb/>arcubus _<emph style="sc">B</emph>C, <emph style="sc">C</emph>D,_ inquiratur angulus _<emph style="sc">Cb</emph>D,_ qui ex recto _<emph style="sc">Ab</emph>D,_ demptus notum <lb/>relinquet angulum _ABC._</s> <s xml:id="echoid-s18297" xml:space="preserve"/> </p> <div xml:id="echoid-div1505" type="float" level="2" n="13"> <note position="right" xlink:label="note-515-02" xlink:href="note-515-02a" xml:space="preserve">Per ſolcs ſi-<lb/>nus, quan-<lb/>do maior <lb/>arcus qua-<lb/>draus eſt.</note> </div> <p> <s xml:id="echoid-s18298" xml:space="preserve">SIT quinto, & </s> <s xml:id="echoid-s18299" xml:space="preserve">vltimo maior arcus AB, quadrante maior, & </s> <s xml:id="echoid-s18300" xml:space="preserve">minor AC, <lb/> <anchor type="note" xlink:label="note-515-03a" xlink:href="note-515-03"/> quadrans, vt in eadem poſteriore proximarum duarum ſigurarum. </s> <s xml:id="echoid-s18301" xml:space="preserve">Abſciſſo <lb/>quadrante AE, ex AB, ducatur per C, E, arcus circuli maximi CE, Deinde, <lb/>per problema 8. </s> <s xml:id="echoid-s18302" xml:space="preserve">triang. </s> <s xml:id="echoid-s18303" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18304" xml:space="preserve">ex dato arcu BC, rectum angulum E, ſubten-<lb/>dente, & </s> <s xml:id="echoid-s18305" xml:space="preserve">arcu BE, complemento arcus dati AB, inueniatur arcus CE, angu-<lb/>li A; </s> <s xml:id="echoid-s18306" xml:space="preserve">ex quo angulus ipſe A, cognoſcetur. </s> <s xml:id="echoid-s18307" xml:space="preserve">Poſt hæc, per problema 11. </s> <s xml:id="echoid-s18308" xml:space="preserve">triang. <lb/></s> <s xml:id="echoid-s18309" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18310" xml:space="preserve">ex notis duobus arcubus BE, EC, circa rectum angulum E, eliciatur <lb/>angulus BCE; </s> <s xml:id="echoid-s18311" xml:space="preserve">cui ſi addatur rectus ACE, notus fiet totus angulus ACB. </s> <s xml:id="echoid-s18312" xml:space="preserve"><lb/>Eadem tãdem ratione ex eiſdem arcubus BE, EC, inueniatur angulus EBC.</s> <s xml:id="echoid-s18313" xml:space="preserve"/> </p> <div xml:id="echoid-div1506" type="float" level="2" n="14"> <note position="right" xlink:label="note-515-03" xlink:href="note-515-03a" xml:space="preserve">Quãdo ma <lb/>ior arcus da <lb/>tus quadrã <lb/>te maior <lb/>eſt, & mi-<lb/>nor qua-<lb/>drans.</note> </div> <p style="it"> <s xml:id="echoid-s18314" xml:space="preserve">_PER_ ſolos ſinus ita propoſitum exequemur. </s> <s xml:id="echoid-s18315" xml:space="preserve">_P_er _1._ </s> <s xml:id="echoid-s18316" xml:space="preserve">praxim problematis 8 triang. <lb/></s> <s xml:id="echoid-s18317" xml:space="preserve"> <anchor type="note" xlink:label="note-515-04a" xlink:href="note-515-04"/> ſphæ. </s> <s xml:id="echoid-s18318" xml:space="preserve">ex dato arcu _<emph style="sc">B</emph>C,_ rectum angulũ _E,_ ſubtendente, & </s> <s xml:id="echoid-s18319" xml:space="preserve">arcu _<emph style="sc">B</emph>E,_ complemento da-<lb/>ti arcus _<emph style="sc">Ab</emph>,_ inquiratur arcus _CE,_ anguli _A:_ </s> <s xml:id="echoid-s18320" xml:space="preserve">fietque ita notus angulus _A. </s> <s xml:id="echoid-s18321" xml:space="preserve">D_einde <lb/>per problema _1._ </s> <s xml:id="echoid-s18322" xml:space="preserve">triang ſphær. </s> <s xml:id="echoid-s18323" xml:space="preserve">ex notis arcubus _<emph style="sc">BC, B</emph>E,_ reperiatur angulus _<emph style="sc">B</emph>CE;_ <lb/></s> <s xml:id="echoid-s18324" xml:space="preserve">cui ſi addatur rectus _ACE,_ totus _<emph style="sc">ACb</emph>,_ cognitus erit. </s> <s xml:id="echoid-s18325" xml:space="preserve">_P_ari ratione tandem ex ar-<lb/>cubus notis _<emph style="sc">B</emph>C, CE,_ indagetur angulus _<emph style="sc">Cb</emph>E._</s> <s xml:id="echoid-s18326" xml:space="preserve"/> </p> <div xml:id="echoid-div1507" type="float" level="2" n="15"> <note position="right" xlink:label="note-515-04" xlink:href="note-515-04a" xml:space="preserve">Per ſolos ſi-<lb/>nus, quãdo <lb/>maior ar--<lb/>cus datus <lb/>quadiante <lb/>maior eſt, <lb/>& minor <lb/>quadrans.</note> </div> <p> <s xml:id="echoid-s18327" xml:space="preserve">ALITER, & </s> <s xml:id="echoid-s18328" xml:space="preserve">facilius, per ſolos ſinus, quan-<lb/>do duo arcus quæſitum angulũ comprehen-<lb/>dentes ſunt inæquales quo modocunque.</s> <s xml:id="echoid-s18329" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s18330" xml:space="preserve">_FIAT,_ vt ſinus totus ad ſinum vtriuslibet arcuum inæqualiũ quæſitum angulum <lb/> <anchor type="note" xlink:label="note-515-05a" xlink:href="note-515-05"/> comprehendentium, ita ſinus alterius arcus circa eundem angulum ad aliud, inue-<lb/>nieturq́; </s> <s xml:id="echoid-s18331" xml:space="preserve">numerus quidam quartus. </s> <s xml:id="echoid-s18332" xml:space="preserve">_D_einde rurſus fiat, vt numerus ille quartus <lb/>inuentus ad ſinum totum, ita differentia inter ſinum verſum arcus quæſito angulo <lb/>oppoſiti, & </s> <s xml:id="echoid-s18333" xml:space="preserve">ſinum verſum arcus, quo duo arcus quæſitum angulum ambientes inter <lb/>ſe differunt, ad aliud, produceturq́; </s> <s xml:id="echoid-s18334" xml:space="preserve">ſinus verſus anguli, qui quærnur: </s> <s xml:id="echoid-s18335" xml:space="preserve">ex quo an-<lb/>gulus ipſe elicietur.</s> <s xml:id="echoid-s18336" xml:space="preserve"/> </p> <div xml:id="echoid-div1508" type="float" level="2" n="16"> <note position="right" xlink:label="note-515-05" xlink:href="note-515-05a" xml:space="preserve">Praxis faci <lb/>lior, & gene <lb/>ralis, per ſo <lb/>los ſinus, <lb/>quãdo duo <lb/>arcus angu <lb/>lum quæſi-<lb/>tum conti-<lb/>nentes ſunt <lb/>inæquales.</note> </div> <p style="it"> <s xml:id="echoid-s18337" xml:space="preserve">_EODEM_ modo alij duo anguli inueſtigabuntur, ſi arcus illos continentes fue-<lb/>rint inæquales.</s> <s xml:id="echoid-s18338" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18339" xml:space="preserve">SINT iam duo arcus AB, AC, quæſitum angulum A, comprehenden-<lb/> <anchor type="note" xlink:label="note-515-06a" xlink:href="note-515-06"/> tes, æquales. </s> <s xml:id="echoid-s18340" xml:space="preserve">Secabit arcus perpendicularis AD, & </s> <s xml:id="echoid-s18341" xml:space="preserve">angulum A, & </s> <s xml:id="echoid-s18342" xml:space="preserve">baſim BC, <pb o="504" file="516" n="516" rhead=""/> bifariam. </s> <s xml:id="echoid-s18343" xml:space="preserve">Inueniatur ergo, per problema 1. </s> <s xml:id="echoid-s18344" xml:space="preserve">triang. </s> <s xml:id="echoid-s18345" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18346" xml:space="preserve">ex dato arcu AB, <lb/> <anchor type="figure" xlink:label="fig-516-01a" xlink:href="fig-516-01"/> rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18347" xml:space="preserve">arcu BD, dimidio <lb/>dati arcus BC, angulus BAD, qui duplicatus totum <lb/>angulum BAC, dabit. </s> <s xml:id="echoid-s18348" xml:space="preserve">Deinde, per problema 13. </s> <s xml:id="echoid-s18349" xml:space="preserve">triang. <lb/></s> <s xml:id="echoid-s18350" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18351" xml:space="preserve">ex eiſdem notis arcubus AB, BD, reperiatur an-<lb/>gulus B, cui æqualis eſt angulus C, (ob æquales arcus <lb/>AB, AC,) ac proinde cognitus quoque.</s> <s xml:id="echoid-s18352" xml:space="preserve"/> </p> <div xml:id="echoid-div1509" type="float" level="2" n="17"> <note position="right" xlink:label="note-515-06" xlink:href="note-515-06a" xml:space="preserve">Quãdoduo <lb/>arcus dati <lb/>sũt ęquales.</note> <figure xlink:label="fig-516-01" xlink:href="fig-516-01a"> <image file="516-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/516-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s18353" xml:space="preserve">_PER_ ſolos ſinus ita agemus. </s> <s xml:id="echoid-s18354" xml:space="preserve">_P_er _1._ </s> <s xml:id="echoid-s18355" xml:space="preserve">praxim problematis <lb/> <anchor type="note" xlink:label="note-516-01a" xlink:href="note-516-01"/> _1._ </s> <s xml:id="echoid-s18356" xml:space="preserve">triang. </s> <s xml:id="echoid-s18357" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18358" xml:space="preserve">inueniatur ex dato arcu _<emph style="sc">Ab</emph>,_ rectum angu <lb/>lum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18359" xml:space="preserve">arcu _<emph style="sc">B</emph>D,_ dimidio arcus dati _<emph style="sc">B</emph>C,_ an-<lb/>gulus _<emph style="sc">B</emph>AD,_ qui duplicatus totum _<emph style="sc">BA</emph>C,_ notum efficiet. <lb/></s> <s xml:id="echoid-s18360" xml:space="preserve">Deinde per _1._ </s> <s xml:id="echoid-s18361" xml:space="preserve">praxim problematis 6. </s> <s xml:id="echoid-s18362" xml:space="preserve">triang. </s> <s xml:id="echoid-s18363" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18364" xml:space="preserve">ex arcu _<emph style="sc">B</emph>D,_ dimidio arcus da-<lb/>ti _<emph style="sc">B</emph>C,_ & </s> <s xml:id="echoid-s18365" xml:space="preserve">angulo oppoſito _<emph style="sc">BA</emph>D,_ inuento, (cum ſpecies alterius anguli _B,_ conſtet. </s> <s xml:id="echoid-s18366" xml:space="preserve"><lb/>_N_am ſi datus arcus _<emph style="sc">AB</emph>,_ recto angulo _D,_ oppoſitus eſt quadrante minor, angulus _<emph style="sc">B</emph>,_ <lb/>acutus erit, quemadmodum & </s> <s xml:id="echoid-s18367" xml:space="preserve">_<emph style="sc">BA</emph>D,_ acutus eſt: </s> <s xml:id="echoid-s18368" xml:space="preserve">Si vero _<emph style="sc">Ab</emph>,_ quadrante maior eſt, <lb/>erit angulus _<emph style="sc">B</emph>,_ obtuſus, cum _<emph style="sc">BA</emph>D,_ acutus ſit.) </s> <s xml:id="echoid-s18369" xml:space="preserve">reperiatur angulus _<emph style="sc">B</emph>,_ cui æqualis <lb/>eſt angulus _C,_ ob æquales arcus _<emph style="sc">Ab, ac</emph>._</s> <s xml:id="echoid-s18370" xml:space="preserve"/> </p> <div xml:id="echoid-div1510" type="float" level="2" n="18"> <note position="left" xlink:label="note-516-01" xlink:href="note-516-01a" xml:space="preserve">Per ſolos ſi <lb/>nus, quãdo <lb/>dati duo ar <lb/>cus æquales <lb/>ſunt.</note> </div> <p> <s xml:id="echoid-s18371" xml:space="preserve">NEQVE vero duo duo æquales arcus eſſe poſſunt quadrãtes. </s> <s xml:id="echoid-s18372" xml:space="preserve">Nam alias <lb/>duo anguli ſupra baſim eſſent recti; </s> <s xml:id="echoid-s18373" xml:space="preserve">atque adeo triangulum eſſet rectangulum. <lb/></s> <s xml:id="echoid-s18374" xml:space="preserve">quod eſt contra hypotheſin.</s> <s xml:id="echoid-s18375" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quæritur <lb/>arcus, cum <lb/>duobus an <lb/>gulis adia-<lb/>centibus.</note> <p> <s xml:id="echoid-s18376" xml:space="preserve">19. </s> <s xml:id="echoid-s18377" xml:space="preserve">DATIS duobus arcubus trianguli non re-<lb/>ctanguli cum angulo ab ipſis comprehen ſo, <lb/>inueſtigare reliquum arcum, cum reliquis <lb/>duobus angulis.</s> <s xml:id="echoid-s18378" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18379" xml:space="preserve">SINT in triangulo ABC, duo arcus AB, BC, dati, cum angulo B: </s> <s xml:id="echoid-s18380" xml:space="preserve">ſintq́; <lb/></s> <s xml:id="echoid-s18381" xml:space="preserve"> <anchor type="note" xlink:label="note-516-03a" xlink:href="note-516-03"/> primum inæquales, & </s> <s xml:id="echoid-s18382" xml:space="preserve">neuter eorum quadrans. </s> <s xml:id="echoid-s18383" xml:space="preserve">Ex A, termino vnius eorum ad <lb/>alterum demittatur arcus perpendicularis AD, qui an intra triangulum, an <lb/>vero extra cadat, ex operatione ipſa diſcemus. </s> <s xml:id="echoid-s18384" xml:space="preserve">Nam inueniatur, per proble-<lb/> <anchor type="figure" xlink:label="fig-516-02a" xlink:href="fig-516-02"/> ma 2. </s> <s xml:id="echoid-s18385" xml:space="preserve">triang. </s> <s xml:id="echoid-s18386" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18387" xml:space="preserve">ex arcu dato AB, rectum <lb/>angulum D, ſubtendente, & </s> <s xml:id="echoid-s18388" xml:space="preserve">dato angulo B, <lb/>arcus AD, angulo B, oppoſitus. </s> <s xml:id="echoid-s18389" xml:space="preserve">Rurſus ex da-<lb/> <anchor type="note" xlink:label="note-516-04a" xlink:href="note-516-04"/> to arcu AB, rectum angulum D, ſubtenden-<lb/>te, & </s> <s xml:id="echoid-s18390" xml:space="preserve">inuento arcu AD, reperiatur, per pro-<lb/>blema 8. </s> <s xml:id="echoid-s18391" xml:space="preserve">triang. </s> <s xml:id="echoid-s18392" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18393" xml:space="preserve">tertius arcus BD. </s> <s xml:id="echoid-s18394" xml:space="preserve">Si <lb/>igitur arcus hic BD, inuentus fuerit minor <lb/>dato arcu BC, cadet arcus AD, intra trian-<lb/>gulum, extra vero, ſi maior. </s> <s xml:id="echoid-s18395" xml:space="preserve">Sublato autem <lb/>inuento arcu BD, ex dato arcu BC, (ſi ille <lb/>hoc minor eſt) vel dempto arcu BC, dato ex <lb/>inuento arcu BD, (ſi hic illo maior eſt) no-<lb/>tus relinquetur arcus CD. </s> <s xml:id="echoid-s18396" xml:space="preserve">Ex arcubus deni-<lb/>que AD, CD, circa angulum rectum D, inueniatur, per problema 7. </s> <s xml:id="echoid-s18397" xml:space="preserve">triang. <lb/></s> <s xml:id="echoid-s18398" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18399" xml:space="preserve">tertius arcus AC, qui quæritur.</s> <s xml:id="echoid-s18400" xml:space="preserve"/> </p> <div xml:id="echoid-div1511" type="float" level="2" n="19"> <note position="left" xlink:label="note-516-03" xlink:href="note-516-03a" xml:space="preserve">Quãdoduo <lb/>arcus dati <lb/>ſunt inæ-<lb/>quales, & <lb/>neuter eo-<lb/>rum qua-<lb/>drans.</note> <figure xlink:label="fig-516-02" xlink:href="fig-516-02a"> <image file="516-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/516-02"/> </figure> <note position="left" xlink:label="note-516-04" xlink:href="note-516-04a" xml:space="preserve">Proproſ. 64. <lb/>@riãg. ſphęr.</note> </div> <p> <s xml:id="echoid-s18401" xml:space="preserve">DEINDE, per problema 13. </s> <s xml:id="echoid-s18402" xml:space="preserve">triang. </s> <s xml:id="echoid-s18403" xml:space="preserve">ſphær, ex dato arcu AB, rectum <pb o="505" file="517" n="517" rhead=""/> angulum D, ſubtendente, & </s> <s xml:id="echoid-s18404" xml:space="preserve">arcu inuento AD, reperiatur angulus BAD, à <lb/>dictis arcubus comprehenſus. </s> <s xml:id="echoid-s18405" xml:space="preserve">Eademq́; </s> <s xml:id="echoid-s18406" xml:space="preserve">ratione, ex inuento arcu AC, rectum <lb/>angulum D, ſubtendente, & </s> <s xml:id="echoid-s18407" xml:space="preserve">inuento arcu AD, inueniatur angulus CAD, à <lb/>dictis arcubus comprehenſus. </s> <s xml:id="echoid-s18408" xml:space="preserve">Nam angulus CAD, additus angulo BAD, <lb/>(quando arcus AD, intra triangulum cadit) vel angulus CAD, ex angulo <lb/>BAD, ſublatus, (quando arcus AD, cadit extra triangulum) conſiciet, aut <lb/>relinquet angulum quæſitum BAC.</s> <s xml:id="echoid-s18409" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18410" xml:space="preserve">AD extremum, per problema 13. </s> <s xml:id="echoid-s18411" xml:space="preserve">triang. </s> <s xml:id="echoid-s18412" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18413" xml:space="preserve">ex inuento arcu AC, re-<lb/>ctum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18414" xml:space="preserve">arcu inuento CD, eliciatur angulus C: </s> <s xml:id="echoid-s18415" xml:space="preserve">qui, <lb/>vbiarcus AD, intra triangulum cadit, quæritur; </s> <s xml:id="echoid-s18416" xml:space="preserve">at, quando arcus AD, ca-<lb/>dit extra triangulum, ſubductus ex duobus rectis relinquit angulum ACB, <lb/>quæſitum.</s> <s xml:id="echoid-s18417" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s18418" xml:space="preserve">_PER_ ſolos ſinus ſic agemus. </s> <s xml:id="echoid-s18419" xml:space="preserve">_P_er problema 2. </s> <s xml:id="echoid-s18420" xml:space="preserve">triang. </s> <s xml:id="echoid-s18421" xml:space="preserve">ſphær reperiatur ex dato ar-<lb/> <anchor type="note" xlink:label="note-517-01a" xlink:href="note-517-01"/> cu _AB,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18422" xml:space="preserve">dato angulo _<emph style="sc">B</emph>,_ oppoſitus arcus _AD: </s> <s xml:id="echoid-s18423" xml:space="preserve">E_t <lb/>hinc per _1._ </s> <s xml:id="echoid-s18424" xml:space="preserve">praxim problematis _8._ </s> <s xml:id="echoid-s18425" xml:space="preserve">triang. </s> <s xml:id="echoid-s18426" xml:space="preserve">ſphær arcus _<emph style="sc">B</emph>D. </s> <s xml:id="echoid-s18427" xml:space="preserve">H_ic enim ablatus ex dato <lb/>arcu _<emph style="sc">BC</emph>,_ (ſi ille hoc minor eſt) vel ex inuento arcu _<emph style="sc">B</emph>D,_ ablatus adtus arcus _<emph style="sc">BC</emph>,_ <lb/>(ſi hic illo minor eſt) notum relinquet arcum _CD. </s> <s xml:id="echoid-s18428" xml:space="preserve">D_einde per _1._ </s> <s xml:id="echoid-s18429" xml:space="preserve">praxim problematis _1._ <lb/></s> <s xml:id="echoid-s18430" xml:space="preserve">triang. </s> <s xml:id="echoid-s18431" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18432" xml:space="preserve">tam ex dato arcu _<emph style="sc">AB</emph>,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18433" xml:space="preserve">arcu in-<lb/>uento _<emph style="sc">B</emph>D,_ eruatur angulus oppoſitus _<emph style="sc">BA</emph>D;_ </s> <s xml:id="echoid-s18434" xml:space="preserve">quam ex inuento arcu _<emph style="sc">AC</emph>,_ rectum <lb/>angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18435" xml:space="preserve">inuento arcu _CD,_ oppoſitus angulus _<emph style="sc">Ca</emph>D. </s> <s xml:id="echoid-s18436" xml:space="preserve">N_am ex duo-<lb/>bus angulis _<emph style="sc">BA</emph>D, <emph style="sc">Ca</emph>D,_ inuentis quæſitus angulus _<emph style="sc">Bac</emph>,_ cognoſietur, ſi vnus al-<lb/>vi addatur, quando arcus _<emph style="sc">A</emph>D,_ intra triangulum cadit, vel, quando cadit extra, ſi <lb/>ex _<emph style="sc">BA</emph>D,_ detrahatur _<emph style="sc">CA</emph>D. </s> <s xml:id="echoid-s18437" xml:space="preserve">P_oſtremo, per _1._ </s> <s xml:id="echoid-s18438" xml:space="preserve">praxim problematis _1._ </s> <s xml:id="echoid-s18439" xml:space="preserve">triang. </s> <s xml:id="echoid-s18440" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18441" xml:space="preserve"><lb/>inquiratur ex inuento arcu _<emph style="sc">AC</emph>,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18442" xml:space="preserve">arcu inuente <lb/>_<emph style="sc">A</emph>D,_ oppoſitus angulus _C. </s> <s xml:id="echoid-s18443" xml:space="preserve">H_ic enim in priori triangulo eſt quæſitus, in poſteriori ve <lb/>ro reliquus duorum rectorum _<emph style="sc">AC</emph>B,_ eſt is, qui quæritur.</s> <s xml:id="echoid-s18444" xml:space="preserve"/> </p> <div xml:id="echoid-div1512" type="float" level="2" n="20"> <note position="right" xlink:label="note-517-01" xlink:href="note-517-01a" xml:space="preserve">Per ſolos ſi <lb/>nus, quan-<lb/>do dati duo <lb/>arcus inæ-<lb/>quales sũt, <lb/>& neuter eo <lb/>rum qua-<lb/>drans.</note> </div> <p> <s xml:id="echoid-s18445" xml:space="preserve">QVOD ſi forte arcus CD, deprehendatur quadrans, (nunquam autem <lb/>BD, erit quadrans, poſito AB, non quadrãte) erit tunc & </s> <s xml:id="echoid-s18446" xml:space="preserve">arcus quæſitus AC, <lb/>quadrans, & </s> <s xml:id="echoid-s18447" xml:space="preserve">angulus CAD, rectus. </s> <s xml:id="echoid-s18448" xml:space="preserve">Atque ita ſine moleſtia inuentus erit ar-<lb/>cus AC, qui quæritur, & </s> <s xml:id="echoid-s18449" xml:space="preserve">angulus CAD: </s> <s xml:id="echoid-s18450" xml:space="preserve">ex quibus quæſitos angulos BAC, <lb/>ACB, inueniemus, vt prius.</s> <s xml:id="echoid-s18451" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18452" xml:space="preserve">SIT iam alter datorum arcuum inæqualium quadrans, nempe AB, à cu-<lb/> <anchor type="note" xlink:label="note-517-02a" xlink:href="note-517-02"/> ius extremo A, ad alterum arcus perpendicularis AD, demittatur. </s> <s xml:id="echoid-s18453" xml:space="preserve">Erit tunc <lb/>arcus quoque BD, quadrans, & </s> <s xml:id="echoid-s18454" xml:space="preserve">angulus BAD, rectus: </s> <s xml:id="echoid-s18455" xml:space="preserve">nec non B, polus ar-<lb/>cus AD; </s> <s xml:id="echoid-s18456" xml:space="preserve">ac proinde arcus AD, ex dato angulo B, notus ſiet. </s> <s xml:id="echoid-s18457" xml:space="preserve">Atque ita in <lb/>hoc caſu duo arcus BD, AD, cum angulo BAD, noti facti erunt, ſine alio la-<lb/>bore: </s> <s xml:id="echoid-s18458" xml:space="preserve">ex quibus reliqua inueſtigabuntur, vt prius.</s> <s xml:id="echoid-s18459" xml:space="preserve"/> </p> <div xml:id="echoid-div1513" type="float" level="2" n="21"> <note position="right" xlink:label="note-517-02" xlink:href="note-517-02a" xml:space="preserve">Quando al <lb/>ter datotũ <lb/>inæqualiũ <lb/>arcuum eſt <lb/>quadrans.</note> </div> <p> <s xml:id="echoid-s18460" xml:space="preserve">ALITER, & </s> <s xml:id="echoid-s18461" xml:space="preserve">facilius, per ſolos ſinus, quando <lb/> <anchor type="note" xlink:label="note-517-03a" xlink:href="note-517-03"/> dati duo arcus inæquales ſunt quomodo-<lb/>cunque.</s> <s xml:id="echoid-s18462" xml:space="preserve"/> </p> <div xml:id="echoid-div1514" type="float" level="2" n="22"> <note position="right" xlink:label="note-517-03" xlink:href="note-517-03a" xml:space="preserve">Praxis faci-<lb/>lior, & gene <lb/>ralis, per ſo <lb/>los ſinus, <lb/>quando da <lb/>ti duo ar-<lb/>cus ſunt in <lb/>æquales.</note> </div> <p style="it"> <s xml:id="echoid-s18463" xml:space="preserve">_FIAT,_ vt ſinus totus ad ſinum vtriuslibet datorum arcuum inæqualium, ita <lb/>ſinus alterius arcus dati ad aliud, produceturq́; </s> <s xml:id="echoid-s18464" xml:space="preserve">quidam quartus numerus. </s> <s xml:id="echoid-s18465" xml:space="preserve">_D_einde <lb/>vurſus ſiat, vt ſinus totus ad inuentum illum quartum numerum, ita ſinus verſus <lb/>anguli dati ad aliud, reperieturq́; </s> <s xml:id="echoid-s18466" xml:space="preserve">differentia inter ſinum verſum tertij arcus, qui <lb/>quæritur, & </s> <s xml:id="echoid-s18467" xml:space="preserve">ſinum verſum differentiæ datorum arcuum inæqualium: </s> <s xml:id="echoid-s18468" xml:space="preserve">quæ differen- <pb o="506" file="518" n="518" rhead=""/> tiainuenta, ſi adijciatur ad ſinum verſum differentiæ datorum arcuum, componet ſi-<lb/>num verſum tertij arcus quæſiti. </s> <s xml:id="echoid-s18469" xml:space="preserve">_C_ognitis iam tribus arcubus propoſiti trianguli, re-<lb/>perientur alij duo anguli ex præcedenti problemate, præſertim ex praxi illa facilio-<lb/>ri, ſi arcus duo quemlibet illorum continẽtes fuerint inæquales. </s> <s xml:id="echoid-s18470" xml:space="preserve">Quòd ſi quando æ qua-<lb/>les ſint, adhibenda erit poſtrema praxis eiuſdem problematis præcedentis.</s> <s xml:id="echoid-s18471" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18472" xml:space="preserve">SED iam duo arcus dati AB, AC, datum angulum A, com prehendentes <lb/> <anchor type="note" xlink:label="note-518-01a" xlink:href="note-518-01"/> <anchor type="figure" xlink:label="fig-518-01a" xlink:href="fig-518-01"/> ſint æquales, ac proinde neuter quadrans. </s> <s xml:id="echoid-s18473" xml:space="preserve">Secabit arcus <lb/>perpendicularis AD, bifariam & </s> <s xml:id="echoid-s18474" xml:space="preserve">datum angulum A, & </s> <s xml:id="echoid-s18475" xml:space="preserve"><lb/>baſim BC. </s> <s xml:id="echoid-s18476" xml:space="preserve">Inueniatur ergo, per problema 2. </s> <s xml:id="echoid-s18477" xml:space="preserve">triãg. </s> <s xml:id="echoid-s18478" xml:space="preserve">ſpher. <lb/></s> <s xml:id="echoid-s18479" xml:space="preserve">ex dato arcu AB, angulum rectum D, ſubtendente, & </s> <s xml:id="echoid-s18480" xml:space="preserve">ex <lb/>angulo BAD, dimidio dati anguli BAC, arcus oppoſi-<lb/>tus BD: </s> <s xml:id="echoid-s18481" xml:space="preserve">qui duplicatus totum arcum BC, quæſitum da-<lb/>bit. </s> <s xml:id="echoid-s18482" xml:space="preserve">Deinde, per problema 13. </s> <s xml:id="echoid-s18483" xml:space="preserve">triang. </s> <s xml:id="echoid-s18484" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18485" xml:space="preserve">ex dato ar-<lb/>cu AB, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18486" xml:space="preserve">inuento arcu <lb/>BD, inquiratur angulus B, à dictis arcubus comprehen-<lb/>ſus; </s> <s xml:id="echoid-s18487" xml:space="preserve">cui ęqualis eſt angulus C, ob ęquales arcus AB, AC.</s> <s xml:id="echoid-s18488" xml:space="preserve"/> </p> <div xml:id="echoid-div1515" type="float" level="2" n="23"> <note position="left" xlink:label="note-518-01" xlink:href="note-518-01a" xml:space="preserve">Quando <lb/>duo arcus <lb/>dati sũt æ-<lb/>quales.</note> <figure xlink:label="fig-518-01" xlink:href="fig-518-01a"> <image file="518-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/518-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s18489" xml:space="preserve">_PER_ ſolos ſinus ita res peragetur. </s> <s xml:id="echoid-s18490" xml:space="preserve">_E_x dato arcu _AB,_ rectum angulum _D,_ ſub-<lb/> <anchor type="note" xlink:label="note-518-02a" xlink:href="note-518-02"/> tendente, & </s> <s xml:id="echoid-s18491" xml:space="preserve">angulo _<emph style="sc">Ba</emph>D,_ dimidio dati anguli _<emph style="sc">B</emph>AC,_ reperiatur, per problema 2, <lb/>triang. </s> <s xml:id="echoid-s18492" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18493" xml:space="preserve">arcus oppoſitus _BD:_ </s> <s xml:id="echoid-s18494" xml:space="preserve">qui duplicatus quæſitum totum _BC,_ offeret. </s> <s xml:id="echoid-s18495" xml:space="preserve">_D_e-<lb/>inde, per _1._ </s> <s xml:id="echoid-s18496" xml:space="preserve">praxim problematis 6. </s> <s xml:id="echoid-s18497" xml:space="preserve">triang. </s> <s xml:id="echoid-s18498" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18499" xml:space="preserve">ex inuento arcu _BD,_ & </s> <s xml:id="echoid-s18500" xml:space="preserve">angulo <lb/>_<emph style="sc">Ba</emph>D,_ dimidio anguli _<emph style="sc">Ba</emph>C,_ dati (cum præterea conſtet ſpecies alterius anguli _B._ <lb/></s> <s xml:id="echoid-s18501" xml:space="preserve">_N_am ſi arcus _<emph style="sc">AB</emph>,_ fuerit minor quadrante, erit angulus _B,_ acutus, ſicut & </s> <s xml:id="echoid-s18502" xml:space="preserve">_<emph style="sc">BA</emph>D,_ <lb/>acutus est: </s> <s xml:id="echoid-s18503" xml:space="preserve">Si vere _<emph style="sc">Ab</emph>,_ ſit quadrante maior, erit _B,_ obtuſus, cum _<emph style="sc">B</emph>AD,,_ acutus <lb/>ſit.) </s> <s xml:id="echoid-s18504" xml:space="preserve">eliciatur angulus _<emph style="sc">B</emph>;_ </s> <s xml:id="echoid-s18505" xml:space="preserve">cui angulus _C,_ æqualis eſt.</s> <s xml:id="echoid-s18506" xml:space="preserve"/> </p> <div xml:id="echoid-div1516" type="float" level="2" n="24"> <note position="left" xlink:label="note-518-02" xlink:href="note-518-02a" xml:space="preserve">Per ſolos ſi-<lb/>nus, quan-<lb/>do duo ar-<lb/>cus datisũt <lb/>æquales.</note> </div> <p> <s xml:id="echoid-s18507" xml:space="preserve">20. </s> <s xml:id="echoid-s18508" xml:space="preserve">DATIS duobus angulis trianguli non re-<lb/> <anchor type="note" xlink:label="note-518-03a" xlink:href="note-518-03"/> ctanguli, cum arcu ipſis adiacente, indagare <lb/>reliquos arcus, cum angulo reliquo.</s> <s xml:id="echoid-s18509" xml:space="preserve"/> </p> <div xml:id="echoid-div1517" type="float" level="2" n="25"> <note position="left" xlink:label="note-518-03" xlink:href="note-518-03a" xml:space="preserve">Quætũtur <lb/>duo arcus, <lb/>cũ angulo <lb/>ab ipſis cõ-<lb/>prehenſo.</note> </div> <p> <s xml:id="echoid-s18510" xml:space="preserve">SINT in triangulo ABC, dati duo anguli B, BAC, cum arcu AB, adia-<lb/> <anchor type="note" xlink:label="note-518-04a" xlink:href="note-518-04"/> cente: </s> <s xml:id="echoid-s18511" xml:space="preserve">ſintq́; </s> <s xml:id="echoid-s18512" xml:space="preserve">primum dati anguli inæquales, & </s> <s xml:id="echoid-s18513" xml:space="preserve">arcus AB, non quadrans. </s> <s xml:id="echoid-s18514" xml:space="preserve">Ex <lb/> <anchor type="figure" xlink:label="fig-518-02a" xlink:href="fig-518-02"/> altero datorum angulorum, nempe ex A, ad <lb/>arcum oppoſitum BC, demittatur arcus per-<lb/>pendicularis, quian intra triangulum cadat, <lb/>an extra, operatio ipſa docebit. </s> <s xml:id="echoid-s18515" xml:space="preserve">Inueniatur <lb/>enim per problema 15. </s> <s xml:id="echoid-s18516" xml:space="preserve">triang. </s> <s xml:id="echoid-s18517" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18518" xml:space="preserve">ex dato <lb/> <anchor type="note" xlink:label="note-518-05a" xlink:href="note-518-05"/> arcu AB, angulum rectum D, ſubtendente, & </s> <s xml:id="echoid-s18519" xml:space="preserve"><lb/>dato angulo B, alter angulus nõ rectus BAD: <lb/></s> <s xml:id="echoid-s18520" xml:space="preserve">qui ſi minor fuerit angulo dato BAC, cadet <lb/>arcus AD, intra triangulum; </s> <s xml:id="echoid-s18521" xml:space="preserve">extra vero, ſi <lb/>maior. </s> <s xml:id="echoid-s18522" xml:space="preserve">In priori caſu ſubductus angulus BAD, <lb/>inuentus ex dato angulo BAC; </s> <s xml:id="echoid-s18523" xml:space="preserve">in poſterio-<lb/>ri vero datus angulus BAC, ex inuẽto BAD, <lb/>detractus, notum relinquet angulum CAD. </s> <s xml:id="echoid-s18524" xml:space="preserve"><lb/>Rurſus, per problema 2. </s> <s xml:id="echoid-s18525" xml:space="preserve">triang. </s> <s xml:id="echoid-s18526" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18527" xml:space="preserve">ex da-<lb/>to arcu AB, angulum rectum D, ſubtendente, & </s> <s xml:id="echoid-s18528" xml:space="preserve">angulo B, dato, reperiatur ar-<lb/>cus AD, oppoſitus, Item, per problema 12. </s> <s xml:id="echoid-s18529" xml:space="preserve">triang. </s> <s xml:id="echoid-s18530" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18531" xml:space="preserve">ex inuento arcu AD, <pb o="507" file="519" n="519" rhead=""/> & </s> <s xml:id="echoid-s18532" xml:space="preserve">angulo adiacente CAD, inuento, eruatur arcus AC, recto angulo D, op-<lb/>poſitus; </s> <s xml:id="echoid-s18533" xml:space="preserve">qui quidem eſt vnus ex quæſitis.</s> <s xml:id="echoid-s18534" xml:space="preserve"/> </p> <div xml:id="echoid-div1518" type="float" level="2" n="26"> <note position="left" xlink:label="note-518-04" xlink:href="note-518-04a" xml:space="preserve">Quãdo da-<lb/>ti anguli sũt <lb/>inæquales, <lb/>& arcus ad-<lb/>iacens non <lb/>quadrans.</note> <figure xlink:label="fig-518-02" xlink:href="fig-518-02a"> <image file="518-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/518-02"/> </figure> <note position="left" xlink:label="note-518-05" xlink:href="note-518-05a" xml:space="preserve">Propoſ. 65. <lb/>triãg. ſphęt.</note> </div> <p> <s xml:id="echoid-s18535" xml:space="preserve">DEINDE, per problema 8. </s> <s xml:id="echoid-s18536" xml:space="preserve">triang. </s> <s xml:id="echoid-s18537" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18538" xml:space="preserve">tam ex dato arcu AB, rectum <lb/>angulum D, ſubtendente, & </s> <s xml:id="echoid-s18539" xml:space="preserve">inuento arcu AD, indagetur arcus BD; </s> <s xml:id="echoid-s18540" xml:space="preserve">quam <lb/>ex inuento arcu AC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18541" xml:space="preserve">arcu inuento AD, <lb/>arcus CD: </s> <s xml:id="echoid-s18542" xml:space="preserve">qui adiectus ad inuentum arcum BD, cadente arcu AD, intra <lb/>triangulum, vel ſubductus ex eodem arcu BD, cadente arcu AD, extra trian <lb/>gulum, notum dabit alterum arcum BC, quæſitum.</s> <s xml:id="echoid-s18543" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18544" xml:space="preserve">AD extremum, per problema 15. </s> <s xml:id="echoid-s18545" xml:space="preserve">triang. </s> <s xml:id="echoid-s18546" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18547" xml:space="preserve">inueſtigetur ex inuento <lb/>arcu AC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18548" xml:space="preserve">angulo inuento CAD, angulus <lb/>ACD: </s> <s xml:id="echoid-s18549" xml:space="preserve">qui in priori triangulo eſt is, qui quæritur; </s> <s xml:id="echoid-s18550" xml:space="preserve">in poſteriori autem ſubdu-<lb/>ctus ex duobus rectis reliquum facit ACB, quæſitum.</s> <s xml:id="echoid-s18551" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s18552" xml:space="preserve">_PER_ ſolos ſinus ſic negotium abſoluetur. </s> <s xml:id="echoid-s18553" xml:space="preserve">_P_er problema _2._ </s> <s xml:id="echoid-s18554" xml:space="preserve">triang. </s> <s xml:id="echoid-s18555" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18556" xml:space="preserve">inue-<lb/> <anchor type="note" xlink:label="note-519-01a" xlink:href="note-519-01"/> niatur ex dato arcu _AB,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18557" xml:space="preserve">angulo dato _B,_ arcus <lb/>oppoſitus _AD:_ </s> <s xml:id="echoid-s18558" xml:space="preserve">_E_tper _1._ </s> <s xml:id="echoid-s18559" xml:space="preserve">praxim problematis _8._ </s> <s xml:id="echoid-s18560" xml:space="preserve">triang. </s> <s xml:id="echoid-s18561" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18562" xml:space="preserve">reperiatur ex dato ar-<lb/>cu _AB,_ rectum angulum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18563" xml:space="preserve">inuento arcu _AD,_ tertius arcus _BD._ <lb/></s> <s xml:id="echoid-s18564" xml:space="preserve">_I_tem per _1._ </s> <s xml:id="echoid-s18565" xml:space="preserve">praxim problematis _1._ </s> <s xml:id="echoid-s18566" xml:space="preserve">triang. </s> <s xml:id="echoid-s18567" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18568" xml:space="preserve">inquiratur ex dato arcu _<emph style="sc">Ab</emph>,_ an-<lb/>gulum rectum _D,_ ſubtendente, & </s> <s xml:id="echoid-s18569" xml:space="preserve">inuento arcu _BD,_ angulus oppoſitus _<emph style="sc">B</emph>AD:_ </s> <s xml:id="echoid-s18570" xml:space="preserve">qui <lb/>ablatus ex dato _BAC,_ (ſi ille hoc minor eſt) vel ex eo datus _BAC,_ detractus, (ſi hic <lb/>illo minor eſt) notũ relinquet angulum _<emph style="sc">Ca</emph>D. </s> <s xml:id="echoid-s18571" xml:space="preserve">R_urſus per problema _5._ </s> <s xml:id="echoid-s18572" xml:space="preserve">triang. </s> <s xml:id="echoid-s18573" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18574" xml:space="preserve"><lb/>ex inuento arcu _AD,_ & </s> <s xml:id="echoid-s18575" xml:space="preserve">angulo adiacẽte _CAD,_ eruatur angulus _<emph style="sc">A</emph>CD;_ </s> <s xml:id="echoid-s18576" xml:space="preserve">qui in priori <lb/>triangulo eſt quæſitus, in poſteriori vero reliquus duorũ rectorum _<emph style="sc">Ac</emph>B,_ quæſitus eſt.</s> <s xml:id="echoid-s18577" xml:space="preserve"/> </p> <div xml:id="echoid-div1519" type="float" level="2" n="27"> <note position="right" xlink:label="note-519-01" xlink:href="note-519-01a" xml:space="preserve">Per ſolos ſi <lb/>nus, quãdo <lb/>dati anguli <lb/>sũt inęqua-<lb/>les, & arcus <lb/>adiacẽs nõ <lb/>quadrans.</note> </div> <p style="it"> <s xml:id="echoid-s18578" xml:space="preserve">_POST_ hæc, per _1._ </s> <s xml:id="echoid-s18579" xml:space="preserve">praxim problematis _4._ </s> <s xml:id="echoid-s18580" xml:space="preserve">triang. </s> <s xml:id="echoid-s18581" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18582" xml:space="preserve">ex vtroque angulo <lb/>_CAD, ACD,_ inuento reperiatur arcus _CD:_ </s> <s xml:id="echoid-s18583" xml:space="preserve">qui in priori triangulo additus iam-<lb/>dudum inuento arcui _<emph style="sc">B</emph>D,_ vel in poſteriori ab eo ablatus, notum faciet arcum _BC,_ <lb/>quæſitum.</s> <s xml:id="echoid-s18584" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s18585" xml:space="preserve">_DENIQVE,_ per problema _7._ </s> <s xml:id="echoid-s18586" xml:space="preserve">triang. </s> <s xml:id="echoid-s18587" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18588" xml:space="preserve">inueniatur exinuentis arcubus <lb/>_<emph style="sc">A</emph>D, CD,_ circa angulum rectum _D,_ arcus tertius _<emph style="sc">A</emph>C,_ recto angulo _D,_ oppoſitus, <lb/>qui quæritur. </s> <s xml:id="echoid-s18589" xml:space="preserve">_<emph style="sc">A</emph>_tqueita inuenti erunt duo reliqui arcus _<emph style="sc">B</emph>C, <emph style="sc">Ac</emph>,_ cum reliquo an-<lb/>gulo _ACB._</s> <s xml:id="echoid-s18590" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18591" xml:space="preserve">QVOD ſi quando angulus inuentus CAD, fuerit rectus, (BAD, nun-<lb/>quam poteſt eſſe rectus, poſito AB, non quadrante) erunt AC, CD, qua-<lb/>drantes; </s> <s xml:id="echoid-s18592" xml:space="preserve">& </s> <s xml:id="echoid-s18593" xml:space="preserve">AD, arcus anguli C; </s> <s xml:id="echoid-s18594" xml:space="preserve">ac proinde angulus C, notus fiet ex inuento <lb/>arcu AD. </s> <s xml:id="echoid-s18595" xml:space="preserve">Reliquus autem arcus BC, cognoſcetur ex inuento arcu BD, & </s> <s xml:id="echoid-s18596" xml:space="preserve"><lb/>quadrante CD, veluti prius.</s> <s xml:id="echoid-s18597" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18598" xml:space="preserve">IAM vero ſi datus arcus AB, ſit quadrans, exiſten tibus adhuc angulis B, <lb/> <anchor type="note" xlink:label="note-519-02a" xlink:href="note-519-02"/> BAC, datis inæqualibus, erit angulus BAD, rectus, & </s> <s xml:id="echoid-s18599" xml:space="preserve">arcus quoque BD, <lb/>quadrans. </s> <s xml:id="echoid-s18600" xml:space="preserve">Item B, erit polus arcus AD; </s> <s xml:id="echoid-s18601" xml:space="preserve">proptereaq́; </s> <s xml:id="echoid-s18602" xml:space="preserve">arcus ipſe AD, ex dato <lb/>angulo B, cognitus erit. </s> <s xml:id="echoid-s18603" xml:space="preserve">Inuentis autem tunc tanta facilitate arcubus AD, <lb/>BD, & </s> <s xml:id="echoid-s18604" xml:space="preserve">angulo recto BAD, reperiemus cætera, vt prius.</s> <s xml:id="echoid-s18605" xml:space="preserve"/> </p> <div xml:id="echoid-div1520" type="float" level="2" n="28"> <note position="right" xlink:label="note-519-02" xlink:href="note-519-02a" xml:space="preserve">Quãdo da-<lb/>tus arcus eſt <lb/>quadrans.</note> </div> <p> <s xml:id="echoid-s18606" xml:space="preserve">SINT deinde in triangulo ABC, dati duo anguli B, <lb/> <anchor type="note" xlink:label="note-519-03a" xlink:href="note-519-03"/> <anchor type="figure" xlink:label="fig-519-01a" xlink:href="fig-519-01"/> C, æquales, cum arcu BC, adiacente, ſiue quadrans is ſit, <lb/>ſiue non. </s> <s xml:id="echoid-s18607" xml:space="preserve">Erunt arcus AB, AC, æquales, & </s> <s xml:id="echoid-s18608" xml:space="preserve">arcus per-<lb/>pendicularis AD, ex tertio angulo A, ad BC, demiſſus <lb/>ſecabit & </s> <s xml:id="echoid-s18609" xml:space="preserve">arcum BC, & </s> <s xml:id="echoid-s18610" xml:space="preserve">angulum A, bifariam. </s> <s xml:id="echoid-s18611" xml:space="preserve">Inuenia-<lb/>tur ergo, per problema 12. </s> <s xml:id="echoid-s18612" xml:space="preserve">triang. </s> <s xml:id="echoid-s18613" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18614" xml:space="preserve">ex arcu BD, di-<lb/>midio dati arcus BC, & </s> <s xml:id="echoid-s18615" xml:space="preserve">dato angulo B, adiacente, arcus <lb/>AB, recto angulo D, oppoſitus; </s> <s xml:id="echoid-s18616" xml:space="preserve">cui cum æqualis ſit AC, <pb o="508" file="520" n="520" rhead=""/> inuenti erunt reliqui duo arcus. </s> <s xml:id="echoid-s18617" xml:space="preserve">Rurſus, per problema 5. </s> <s xml:id="echoid-s18618" xml:space="preserve">triang. </s> <s xml:id="echoid-s18619" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18620" xml:space="preserve">ex <lb/>eodem arcu BD, dimidio dati arcus BC, & </s> <s xml:id="echoid-s18621" xml:space="preserve">dato angulo B, adiacente reperia-<lb/>tur alter angulus non rectus BAD. </s> <s xml:id="echoid-s18622" xml:space="preserve">Hic namque duplicatus totum quæſi-<lb/>tum angulum BAC, dabit.</s> <s xml:id="echoid-s18623" xml:space="preserve"/> </p> <div xml:id="echoid-div1521" type="float" level="2" n="29"> <note position="right" xlink:label="note-519-03" xlink:href="note-519-03a" xml:space="preserve">Quãdo da-<lb/>ti duo an -<lb/>guli sũt æ-<lb/>quales.</note> <figure xlink:label="fig-519-01" xlink:href="fig-519-01a"> <image file="519-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/519-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s18624" xml:space="preserve">_PER_ ſolos ſinus ita operabimur. </s> <s xml:id="echoid-s18625" xml:space="preserve">_P_er problema _5._ </s> <s xml:id="echoid-s18626" xml:space="preserve">triang. </s> <s xml:id="echoid-s18627" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18628" xml:space="preserve">inueniatur ex <lb/> <anchor type="note" xlink:label="note-520-01a" xlink:href="note-520-01"/> arcu _BD,_ dimidio dati arcus _BD,_ & </s> <s xml:id="echoid-s18629" xml:space="preserve">dato angulo _B,_ adiacente angulus _BAD:_ </s> <s xml:id="echoid-s18630" xml:space="preserve">qui <lb/>duplicatus dabit totum _<emph style="sc">BAc</emph>,_ quæſitum. </s> <s xml:id="echoid-s18631" xml:space="preserve">_D_einde per _1._ </s> <s xml:id="echoid-s18632" xml:space="preserve">praxim problematis _3._ </s> <s xml:id="echoid-s18633" xml:space="preserve">triang. <lb/></s> <s xml:id="echoid-s18634" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18635" xml:space="preserve">ex arcu _BD,_ dimidio dati arcus _BC,_ & </s> <s xml:id="echoid-s18636" xml:space="preserve">inuento angulo _BAD,_ oppoſito re-<lb/>periatur (cum præterea conſtet ſpecies alterius anguli _B,_ qui datus eſt) arcus _AB,_ <lb/>recto angulo _D,_ oppoſitus: </s> <s xml:id="echoid-s18637" xml:space="preserve">cui æqualis eſt _AC._</s> <s xml:id="echoid-s18638" xml:space="preserve"/> </p> <div xml:id="echoid-div1522" type="float" level="2" n="30"> <note position="left" xlink:label="note-520-01" xlink:href="note-520-01a" xml:space="preserve">Per ſolos ſi-<lb/>nus, quãdo <lb/>duo anguli <lb/>dati sũt æ-<lb/>quales.</note> </div> <p> <s xml:id="echoid-s18639" xml:space="preserve">21. </s> <s xml:id="echoid-s18640" xml:space="preserve">DATIS duobus angulis trianguli non re-<lb/> <anchor type="note" xlink:label="note-520-02a" xlink:href="note-520-02"/> ctanguli, cum arcu qui alteri illorum oppo-<lb/>nitur, reliquos arcus, cum reliquo angulo in-<lb/>ueſtigare: </s> <s xml:id="echoid-s18641" xml:space="preserve">ſi modo conſtet, num arcus alteri <lb/>angulo dato oppoſitus quadrante maior ſit, <lb/>aut minor, aut certe quadrans.</s> <s xml:id="echoid-s18642" xml:space="preserve"/> </p> <div xml:id="echoid-div1523" type="float" level="2" n="31"> <note position="left" xlink:label="note-520-02" xlink:href="note-520-02a" xml:space="preserve">Quærũtur <lb/>duo arcus, <lb/>cũ vno an-<lb/>gulo.</note> </div> <p> <s xml:id="echoid-s18643" xml:space="preserve">SINT in triangulo ABC, dati duo anguli B, C, primum inæquales, cum <lb/> <anchor type="note" xlink:label="note-520-03a" xlink:href="note-520-03"/> arcu AB, qui angulo C, opponitur, non quadrante, conſtetq́; </s> <s xml:id="echoid-s18644" xml:space="preserve">præterea ſpe-<lb/> <anchor type="figure" xlink:label="fig-520-01a" xlink:href="fig-520-01"/> cies arcus AC, alteri angulo dato B, oppoſiti. </s> <s xml:id="echoid-s18645" xml:space="preserve">Du-<lb/>catur ex tertio angulo A, ad arcum BC, arcus per-<lb/>pendicularis AD; </s> <s xml:id="echoid-s18646" xml:space="preserve">qui intra triangulum cadet, ſi v-<lb/>terque angulus datus B, & </s> <s xml:id="echoid-s18647" xml:space="preserve">C, eſt acutus, vel obtu-<lb/>ſus; </s> <s xml:id="echoid-s18648" xml:space="preserve">extra vero, ſi vnus acutus, & </s> <s xml:id="echoid-s18649" xml:space="preserve">alter obtuſus eſt. <lb/></s> <s xml:id="echoid-s18650" xml:space="preserve">Inueſtigetur, per problema 2. </s> <s xml:id="echoid-s18651" xml:space="preserve">triang. </s> <s xml:id="echoid-s18652" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18653" xml:space="preserve">ex dato <lb/> <anchor type="note" xlink:label="note-520-04a" xlink:href="note-520-04"/> arcu AB, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18654" xml:space="preserve">dato <lb/>angulo B, arcus oppofitus AD. </s> <s xml:id="echoid-s18655" xml:space="preserve">Item, per proble-<lb/>ma 14. </s> <s xml:id="echoid-s18656" xml:space="preserve">triang. </s> <s xml:id="echoid-s18657" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18658" xml:space="preserve">ex eodem dato arcu AB, an-<lb/>gulum rectum D, ſubtendente, & </s> <s xml:id="echoid-s18659" xml:space="preserve">dato angulo B, eli-<lb/>ciatur arcus BD. </s> <s xml:id="echoid-s18660" xml:space="preserve">Rurſus reperiatur, per problema <lb/>15. </s> <s xml:id="echoid-s18661" xml:space="preserve">triang. </s> <s xml:id="echoid-s18662" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18663" xml:space="preserve">ex eodem dato arcu AB, rectum <lb/>angulum D, fubtendente, & </s> <s xml:id="echoid-s18664" xml:space="preserve">dato angulo B, angu-<lb/>lus BAD. </s> <s xml:id="echoid-s18665" xml:space="preserve">Ad hæc, per problema 3. </s> <s xml:id="echoid-s18666" xml:space="preserve">triang. </s> <s xml:id="echoid-s18667" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18668" xml:space="preserve">inueniatur quoque ex in-<lb/>uento arcu AD, & </s> <s xml:id="echoid-s18669" xml:space="preserve">dato angulo C, oppoſito (Nam, cadente arcu AD, extra <lb/>triangulum, angulus ACD, arcui AD, oppoſitus relinquitur norus poſt ſub-<lb/>tractionem dati anguli ACB, ex duobus rectis) arcus AC, recto angulo D, <lb/>oppoſitus, cum eius ſpecies conſtare ponatur. </s> <s xml:id="echoid-s18670" xml:space="preserve">Atque ita inuentus erit arcus <lb/>AC, vnus ex quæſitis.</s> <s xml:id="echoid-s18671" xml:space="preserve"/> </p> <div xml:id="echoid-div1524" type="float" level="2" n="32"> <note position="left" xlink:label="note-520-03" xlink:href="note-520-03a" xml:space="preserve">Quãdo da-<lb/>ti duo an-<lb/>guli inęqua <lb/>les ſunt, & <lb/>arcus datus <lb/>non qua-<lb/>drans.</note> <figure xlink:label="fig-520-01" xlink:href="fig-520-01a"> <image file="520-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/520-01"/> </figure> <note position="left" xlink:label="note-520-04" xlink:href="note-520-04a" xml:space="preserve">Propoſ. 66. <lb/>triãg. ſphęr.</note> </div> <p> <s xml:id="echoid-s18672" xml:space="preserve">DEINDE, per problema 14. </s> <s xml:id="echoid-s18673" xml:space="preserve">triang. </s> <s xml:id="echoid-s18674" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18675" xml:space="preserve">reperiatur ex inuento arcu <lb/>AC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18676" xml:space="preserve">dato angulo C, arcus CD: </s> <s xml:id="echoid-s18677" xml:space="preserve">quiad-<lb/>ditus arcui BD, ſupra inuento, vel ex eo detractus, (prout nimirũ arcus AD, <lb/>mtra triangulum cadit, aut extra) notum faciet arcum BC, qui eſt alter ex <lb/>quæſitis.</s> <s xml:id="echoid-s18678" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18679" xml:space="preserve">AD extremum, per problema 15. </s> <s xml:id="echoid-s18680" xml:space="preserve">triang. </s> <s xml:id="echoid-s18681" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18682" xml:space="preserve">ex inuento eodem arcu <pb o="509" file="521" n="521" rhead=""/> AC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18683" xml:space="preserve">dato angulo C, inueniatur angulus <lb/>CAD: </s> <s xml:id="echoid-s18684" xml:space="preserve">qui additus angulo BAD, ſi arcus intra triangulum cadit, vel ſi ex-<lb/>tra, ex eodem ſubductus, cognitum efficiet angulum BAC, quæſitum.</s> <s xml:id="echoid-s18685" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18686" xml:space="preserve">SOLIS _ſinubus vtemur ſic. </s> <s xml:id="echoid-s18687" xml:space="preserve">Per problema 2. </s> <s xml:id="echoid-s18688" xml:space="preserve">triang. </s> <s xml:id="echoid-s18689" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18690" xml:space="preserve">ex dato arcu_ <emph style="sc">A</emph>B, <lb/> <anchor type="note" xlink:label="note-521-01a" xlink:href="note-521-01"/> _rectum angulum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18691" xml:space="preserve">dato angulo_ B, _inquiratur arcus oppoſitus_ <emph style="sc">AD</emph>. <lb/></s> <s xml:id="echoid-s18692" xml:space="preserve">_Et hinc, per 1. </s> <s xml:id="echoid-s18693" xml:space="preserve">praxim problematis_ 8. </s> <s xml:id="echoid-s18694" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18695" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18696" xml:space="preserve">ex dato arcu_ <emph style="sc">A</emph>B, _angulum <lb/>_rectum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18697" xml:space="preserve">inuento arcu_ <emph style="sc">A</emph>D, _reperiatur tertius arcus_ <emph style="sc">B</emph>D. </s> <s xml:id="echoid-s18698" xml:space="preserve">_Et_ <lb/>_rurſus, per_ 1. </s> <s xml:id="echoid-s18699" xml:space="preserve">_praxim problematis_ 1. </s> <s xml:id="echoid-s18700" xml:space="preserve">_traing ſphær. </s> <s xml:id="echoid-s18701" xml:space="preserve">ex dato arcu_ <emph style="sc">AB</emph>, _rectum an-_ <lb/>_gulum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18702" xml:space="preserve">inuento arcu_ <emph style="sc">B</emph>D, _eruatur angulus oppoſitus_ <emph style="sc">BA</emph>D. </s> <s xml:id="echoid-s18703" xml:space="preserve">_Po§t_ <lb/>_hæc, per_ 1 _praxim problematis_ 3. </s> <s xml:id="echoid-s18704" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18705" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18706" xml:space="preserve">eliciatur ex inuento arcu_ <emph style="sc">A</emph>D, & </s> <s xml:id="echoid-s18707" xml:space="preserve"><lb/>_oppoſito angulo dato_ C, _arcus_ <emph style="sc">AC</emph>, _recto angulo_ D, _oppoſitus, cum eius ſpecies con-_ <lb/>_ſtet ex hypotheſi: </s> <s xml:id="echoid-s18708" xml:space="preserve">qui arcus_ <emph style="sc">A</emph>C, _ex quæſitis vnus eſt._</s> <s xml:id="echoid-s18709" xml:space="preserve"/> </p> <div xml:id="echoid-div1525" type="float" level="2" n="33"> <note position="right" xlink:label="note-521-01" xlink:href="note-521-01a" xml:space="preserve">Per ſolos ſi <lb/>nus, quãdo <lb/>dati duoan <lb/>guli sũt inę <lb/>quales, & <lb/>arcus datus <lb/>non qua-<lb/>drans.</note> </div> <p> <s xml:id="echoid-s18710" xml:space="preserve"><emph style="sc">Deinde</emph>, _per_ 1. </s> <s xml:id="echoid-s18711" xml:space="preserve">_praxim problematis_ 8 _triang. </s> <s xml:id="echoid-s18712" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18713" xml:space="preserve">ex inuento arcu_ AC, <lb/>_rectum angulum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18714" xml:space="preserve">arcu_ AD, _reperiatur tertius arcus_ CD: </s> <s xml:id="echoid-s18715" xml:space="preserve">_ex quo,_ <lb/>_ſi in priori triangulo arcui_ <emph style="sc">BD</emph>, _inuento addatur, velin poſteriori ex eodem ſub-_ <lb/>_trahatur, cognitus fiet alter arcus quæſitus_ BC.</s> <s xml:id="echoid-s18716" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18717" xml:space="preserve">PBR 1. </s> <s xml:id="echoid-s18718" xml:space="preserve">_praxim denique problematis_ 1. </s> <s xml:id="echoid-s18719" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18720" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18721" xml:space="preserve">ex arcu_ <emph style="sc">A</emph>C, _angulum_ <lb/>_rectum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18722" xml:space="preserve">arcu_ <emph style="sc">C</emph>D, _inuento, inquiratur angulus oppoſitus_ <emph style="sc">CA</emph>D. <lb/></s> <s xml:id="echoid-s18723" xml:space="preserve">_Nam hic in priori triangulo additus inuento angulo_ <emph style="sc">BA</emph>D, _vel in poſteriori ab eo-<lb/>_dem demptus, notum faciet angulum_ <emph style="sc">BA</emph>C, _quæſitum._</s> <s xml:id="echoid-s18724" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18725" xml:space="preserve">QVOD ſi quando arcus AC, alteri angulo B, dato oppoſitus ſit quadrãs, <lb/>quod euenire poteſt, non exiſtẽte quadrante AB, (quo in caſu nunquam qua-<lb/>drans eſſe poterit AD, vel BD,) erit quoque CD, quadrans, & </s> <s xml:id="echoid-s18726" xml:space="preserve">angulus CAD, <lb/>rectus. </s> <s xml:id="echoid-s18727" xml:space="preserve">Quare non laborandum tunc erit in inquiſitione arcuum AC, CD, <lb/>& </s> <s xml:id="echoid-s18728" xml:space="preserve">anguli CAD: </s> <s xml:id="echoid-s18729" xml:space="preserve">ſed ex ijs inueniendus erit arcus BC, & </s> <s xml:id="echoid-s18730" xml:space="preserve">angulus BAC, vt <lb/>diximus.</s> <s xml:id="echoid-s18731" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18732" xml:space="preserve">VERVM ſit iam datus arcus AB, quadrans, & </s> <s xml:id="echoid-s18733" xml:space="preserve">adhuc dati duo anguli B, <lb/> <anchor type="note" xlink:label="note-521-02a" xlink:href="note-521-02"/> C, inæquales. </s> <s xml:id="echoid-s18734" xml:space="preserve">Quo poſito, erit & </s> <s xml:id="echoid-s18735" xml:space="preserve">BD, quadrans, & </s> <s xml:id="echoid-s18736" xml:space="preserve">angulus BAD, rectus; </s> <s xml:id="echoid-s18737" xml:space="preserve">nec <lb/>non B, polus arcus AD; </s> <s xml:id="echoid-s18738" xml:space="preserve">ac proinde arcus AD, ex dato angulo B, cum eius <lb/>arcus ſit, notus fiet. </s> <s xml:id="echoid-s18739" xml:space="preserve">Cognitis autem tanta facilitate arcubus BD, AD, cum <lb/>angulo recto BAD, inuenientur reliqua, vt prius.</s> <s xml:id="echoid-s18740" xml:space="preserve"/> </p> <div xml:id="echoid-div1526" type="float" level="2" n="34"> <note position="right" xlink:label="note-521-02" xlink:href="note-521-02a" xml:space="preserve">Quãdo da-<lb/>tus arcus <lb/>quadrãs eſt, <lb/>& dati duo <lb/>anguli inę-<lb/>quales.</note> </div> <p> <s xml:id="echoid-s18741" xml:space="preserve">SINT tandem dati duo anguli B, C, æquales. </s> <s xml:id="echoid-s18742" xml:space="preserve">Diui-<lb/> <anchor type="note" xlink:label="note-521-03a" xlink:href="note-521-03"/> <anchor type="figure" xlink:label="fig-521-01a" xlink:href="fig-521-01"/> det arcus AD, & </s> <s xml:id="echoid-s18743" xml:space="preserve">baſim BC, & </s> <s xml:id="echoid-s18744" xml:space="preserve">angulum A, bifariam; </s> <s xml:id="echoid-s18745" xml:space="preserve">& </s> <s xml:id="echoid-s18746" xml:space="preserve"><lb/>arcus AB, AC, æquales erunt. </s> <s xml:id="echoid-s18747" xml:space="preserve">Inquiratur, per pro-<lb/>blema 14. </s> <s xml:id="echoid-s18748" xml:space="preserve">triang. </s> <s xml:id="echoid-s18749" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18750" xml:space="preserve">ex dato arcu AB, angulum re-<lb/>ctum D, ſubtendente, & </s> <s xml:id="echoid-s18751" xml:space="preserve">dato angulo B, arcus BD: </s> <s xml:id="echoid-s18752" xml:space="preserve">qui <lb/>duplicatus totum quæſitum BC, offeret. </s> <s xml:id="echoid-s18753" xml:space="preserve">Alter autem <lb/>quæſitus AC, datus erit, cum dato AB, ſit æqualis. </s> <s xml:id="echoid-s18754" xml:space="preserve">Rur-<lb/>ſus, per problema 15. </s> <s xml:id="echoid-s18755" xml:space="preserve">triang. </s> <s xml:id="echoid-s18756" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18757" xml:space="preserve">ex eodem arcu dato <lb/>AB, & </s> <s xml:id="echoid-s18758" xml:space="preserve">angulo B, eliciatur angulus BAD; </s> <s xml:id="echoid-s18759" xml:space="preserve">quo duplica-<lb/>to, habebitur totus BAC, qui quæritur.</s> <s xml:id="echoid-s18760" xml:space="preserve"/> </p> <div xml:id="echoid-div1527" type="float" level="2" n="35"> <note position="left" xlink:label="note-521-03" xlink:href="note-521-03a" xml:space="preserve">Quãdoduo <lb/>anguli dati <lb/>ſunt æqua-<lb/>les.</note> <figure xlink:label="fig-521-01" xlink:href="fig-521-01a"> <image file="521-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/521-01"/> </figure> </div> <p> <s xml:id="echoid-s18761" xml:space="preserve">PER _ſolos ſinus ita abſoluemus problema. </s> <s xml:id="echoid-s18762" xml:space="preserve">Per problema_ 2. </s> <s xml:id="echoid-s18763" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18764" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18765" xml:space="preserve">inve-_ <lb/> <anchor type="note" xlink:label="note-521-04a" xlink:href="note-521-04"/> _ſtigetur arcus_ <emph style="sc">A</emph>D, _ex dato arcu_ <emph style="sc">A</emph>B, _rectum angulum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18766" xml:space="preserve">dato an-_ <lb/>_gulo_ B, _arcui_ <emph style="sc">A</emph>D, _oppoſito. </s> <s xml:id="echoid-s18767" xml:space="preserve">Atque hinc, per_ 1. </s> <s xml:id="echoid-s18768" xml:space="preserve">_praxim problematis_ 8. </s> <s xml:id="echoid-s18769" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18770" xml:space="preserve">ſphær._ <lb/></s> <s xml:id="echoid-s18771" xml:space="preserve">_reperiatur ex dato arcu_ <emph style="sc">A</emph>B, _angulum rectum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18772" xml:space="preserve">inuento arcu_ AD, <lb/>_tertius arcus_ <emph style="sc">B</emph>D: </s> <s xml:id="echoid-s18773" xml:space="preserve">_qui duplicatus totum quæſitum_ <emph style="sc">BC</emph>, _dabit. </s> <s xml:id="echoid-s18774" xml:space="preserve">Deinde, per_ 1. </s> <s xml:id="echoid-s18775" xml:space="preserve">_pra-_ <lb/>_xim problematis_ 1. </s> <s xml:id="echoid-s18776" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18777" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18778" xml:space="preserve">ex dato arcu_ <emph style="sc">AB</emph>, _rectum angulum_ D, _ſubtendente,_ <pb o="510" file="522" n="522" rhead=""/> _& </s> <s xml:id="echoid-s18779" xml:space="preserve">arcu_ <emph style="sc">BD</emph>, _inuento indagetur angulus oppoſitus_ BA:</s> <s xml:id="echoid-s18780" xml:space="preserve">D _qui duplicatus offeres_ <lb/>_totum_ <emph style="sc">BA</emph>C, _quæſitum._</s> <s xml:id="echoid-s18781" xml:space="preserve"/> </p> <div xml:id="echoid-div1528" type="float" level="2" n="36"> <note position="right" xlink:label="note-521-04" xlink:href="note-521-04a" xml:space="preserve">Per ſolos ſi <lb/>nus, quãdo <lb/>duo anguli <lb/>dati sũt æ-<lb/>quales.</note> </div> <p> <s xml:id="echoid-s18782" xml:space="preserve">IN hoc porro caſu non poteſt datus arcus AB, eſſe quadrans.</s> <s xml:id="echoid-s18783" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18784" xml:space="preserve">22. </s> <s xml:id="echoid-s18785" xml:space="preserve">DATIS duobus arcubus trianguli non <lb/> <anchor type="note" xlink:label="note-522-01a" xlink:href="note-522-01"/> rectanguli, cum angulo, qui alteri eorum <lb/>opponitur, reliquos angulos, cum reliquo <lb/>arcu ſcrutari: </s> <s xml:id="echoid-s18786" xml:space="preserve">ſi modo conſter, num angu-<lb/>lus alteri arcui dato oppoſitus acutus ſit, aut <lb/>obtuſus.</s> <s xml:id="echoid-s18787" xml:space="preserve"/> </p> <div xml:id="echoid-div1529" type="float" level="2" n="37"> <note position="left" xlink:label="note-522-01" xlink:href="note-522-01a" xml:space="preserve">Quætũtur <lb/>luo angu-<lb/>li cum vno <lb/>ercu.</note> </div> <p> <s xml:id="echoid-s18788" xml:space="preserve">SINT in triangulo ABC, dati duo arcus AB, AC, cum angulo B, qui <lb/> <anchor type="note" xlink:label="note-522-02a" xlink:href="note-522-02"/> arcui AC, opponitur: </s> <s xml:id="echoid-s18789" xml:space="preserve">ſint autem primum illi arcus inæquales, & </s> <s xml:id="echoid-s18790" xml:space="preserve">neuter qua-<lb/>drans, conſtetq́; </s> <s xml:id="echoid-s18791" xml:space="preserve">præterea ſpecies anguli C, alteri arcui dato AB, oppoſiti. <lb/></s> <s xml:id="echoid-s18792" xml:space="preserve">Ducatur ex angulo A, à datis arcubus comprehenſo ad arcum BC, arcus per-<lb/> <anchor type="figure" xlink:label="fig-522-01a" xlink:href="fig-522-01"/> pendicularis, qui intra triangulum cadet, ſi <lb/>vterque angulus B, C, ſit acutus, vel obtu-<lb/>ſus; </s> <s xml:id="echoid-s18793" xml:space="preserve">extra vero, ſi vnus acutus ſit, & </s> <s xml:id="echoid-s18794" xml:space="preserve">alter ob-<lb/> <anchor type="note" xlink:label="note-522-03a" xlink:href="note-522-03"/> tuſus. </s> <s xml:id="echoid-s18795" xml:space="preserve">Conſtat autem, an vterque angulus <lb/>acutus ſit, obtuſusve, an non; </s> <s xml:id="echoid-s18796" xml:space="preserve">quia angulus <lb/>B, datus eſt, cum ſpecie anguli C. </s> <s xml:id="echoid-s18797" xml:space="preserve">inquiratur <lb/>ergo, per problema 2. </s> <s xml:id="echoid-s18798" xml:space="preserve">triang. </s> <s xml:id="echoid-s18799" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18800" xml:space="preserve">ex dato <lb/>arcu AB, angulum rectum D, ſubtendente, & </s> <s xml:id="echoid-s18801" xml:space="preserve"><lb/>angulo dato B, arcus oppoſitus AD. </s> <s xml:id="echoid-s18802" xml:space="preserve">Et hinc, <lb/>per problema 8. </s> <s xml:id="echoid-s18803" xml:space="preserve">triang. </s> <s xml:id="echoid-s18804" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18805" xml:space="preserve">ex eodem ar-<lb/>cu dato AB, & </s> <s xml:id="echoid-s18806" xml:space="preserve">arcu inuento AD, eliciatur <lb/>tertius arcus BD. </s> <s xml:id="echoid-s18807" xml:space="preserve">Hinc rurſus, per proble-<lb/>ma 1. </s> <s xml:id="echoid-s18808" xml:space="preserve">triang. </s> <s xml:id="echoid-s18809" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18810" xml:space="preserve">ex dato arcu AB, rectum <lb/>angulum D, ſubtendente, & </s> <s xml:id="echoid-s18811" xml:space="preserve">arcu BD, inuen-<lb/>to reperiatur angulus BAD, arcui BD, oppoſitus: </s> <s xml:id="echoid-s18812" xml:space="preserve">Et per problema 13. </s> <s xml:id="echoid-s18813" xml:space="preserve">triãg. <lb/></s> <s xml:id="echoid-s18814" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18815" xml:space="preserve">ex dato arcu AC, rectum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18816" xml:space="preserve">arcu AD, in-<lb/>uento eruatur angulus CAD, à dictis arcubus comprehenſus. </s> <s xml:id="echoid-s18817" xml:space="preserve">Nam hic an-<lb/>gulus adiectus ad inuentum angulum BAD, vel ab eodem ſubtractus, (prout <lb/>arcus AD, cadit intra, vel extra triangulum) dabit quæſitum angulum BAC. </s> <s xml:id="echoid-s18818" xml:space="preserve"><lb/>Inueniatur præterea, per problema 15. </s> <s xml:id="echoid-s18819" xml:space="preserve">triang. </s> <s xml:id="echoid-s18820" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18821" xml:space="preserve">ex dato arcu AC, re-<lb/>ctum angulum D, ſubtendente, & </s> <s xml:id="echoid-s18822" xml:space="preserve">inuento angulo CAD, angulus ACD: </s> <s xml:id="echoid-s18823" xml:space="preserve">qui <lb/>erit is, quem quærimus, ſi arcus AD, intra triangulum cadit, ſi vero extra, <lb/>ablatus ex duobus rectis dabit angulum ACB, quæſitum: </s> <s xml:id="echoid-s18824" xml:space="preserve">ſicq́; </s> <s xml:id="echoid-s18825" xml:space="preserve">duo reliqui <lb/>anguli BAC, ACB, erunt cogniti.</s> <s xml:id="echoid-s18826" xml:space="preserve"/> </p> <div xml:id="echoid-div1530" type="float" level="2" n="38"> <note position="left" xlink:label="note-522-02" xlink:href="note-522-02a" xml:space="preserve">Quãdo da-<lb/>ti duo ar-<lb/>cus sũt inę <lb/>quales, & <lb/>neuter eo-<lb/>rum qua-<lb/>drans.</note> <figure xlink:label="fig-522-01" xlink:href="fig-522-01a"> <image file="522-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/522-01"/> </figure> <note position="left" xlink:label="note-522-03" xlink:href="note-522-03a" xml:space="preserve">Propoſ. 67. <lb/>triãg. ſphęr.</note> </div> <p> <s xml:id="echoid-s18827" xml:space="preserve">DEINDE, per problema 8. </s> <s xml:id="echoid-s18828" xml:space="preserve">triang. </s> <s xml:id="echoid-s18829" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18830" xml:space="preserve">ex dato arcu AC, angulum <lb/>rectum D, ſubtendente, & </s> <s xml:id="echoid-s18831" xml:space="preserve">inuento arcu AD, inquira tur arcus CD. </s> <s xml:id="echoid-s18832" xml:space="preserve">Hic enim <lb/>additus arcui inuento BD, vel ab eodem ſubductus (prout arcus AD, intra <lb/>triangulum cadit, vel extra) notum offeret quæſitum arcum BC.</s> <s xml:id="echoid-s18833" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18834" xml:space="preserve">IN hoc porro caſu, nullus arcuum AD, BD, CD; </s> <s xml:id="echoid-s18835" xml:space="preserve">quadrans eſſe poteſt.</s> <s xml:id="echoid-s18836" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18837" xml:space="preserve">PER _ſolos ſinus ita erit agendum. </s> <s xml:id="echoid-s18838" xml:space="preserve">Per problema_ 2. </s> <s xml:id="echoid-s18839" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18840" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18841" xml:space="preserve">ex dato arcu_ <pb o="511" file="523" n="523" rhead=""/> <emph style="sc">Ab</emph>, _angulum rectum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18842" xml:space="preserve">angulo dato_ <emph style="sc">B</emph>, _inueniatur oppoſitus ar-_ <lb/> <anchor type="note" xlink:label="note-523-01a" xlink:href="note-523-01"/> _cus_ <emph style="sc">A</emph>D: </s> <s xml:id="echoid-s18843" xml:space="preserve">_Atque hinc, per_ 1. </s> <s xml:id="echoid-s18844" xml:space="preserve">_praxim problematis_ 8. </s> <s xml:id="echoid-s18845" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18846" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18847" xml:space="preserve">ex dato arcu_ <emph style="sc">AB</emph>, <lb/>_dato rectum ſubtendente angulum_ D, _& </s> <s xml:id="echoid-s18848" xml:space="preserve">inuento arcu_ <emph style="sc">A</emph>D, _eruatur tertius arcus_ <lb/><emph style="sc">B</emph>D: </s> <s xml:id="echoid-s18849" xml:space="preserve">_Atque hinc rurſus, per_ 1. </s> <s xml:id="echoid-s18850" xml:space="preserve">_praxim problematis_ 1. </s> <s xml:id="echoid-s18851" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18852" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18853" xml:space="preserve">ex arcu_ <emph style="sc">Ab</emph>, <lb/>_angulum rectum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18854" xml:space="preserve">arcu inuento_ <emph style="sc">BD</emph>, _reperiatur angulus oppoſitus_ <lb/>BAD: </s> <s xml:id="echoid-s18855" xml:space="preserve">_Nec non, per_ 1. </s> <s xml:id="echoid-s18856" xml:space="preserve">_praxim problematis_ 8. </s> <s xml:id="echoid-s18857" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18858" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18859" xml:space="preserve">ex dato arcu_ AC, <lb/>_rectum angulum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18860" xml:space="preserve">arcu inuento_ AD, _eruatnr tertius arcus_ CD; <lb/></s> <s xml:id="echoid-s18861" xml:space="preserve">_qui vel additus arcui inuento_ <emph style="sc">BD</emph>, _vel ex eo ſubtractus, (prout arcus_ <emph style="sc">AD</emph>, _cadit_ <lb/>_intra, vel extra triangulum) dabit quæſitum arcum_ <emph style="sc">B</emph>C.</s> <s xml:id="echoid-s18862" xml:space="preserve"/> </p> <div xml:id="echoid-div1531" type="float" level="2" n="39"> <note position="right" xlink:label="note-523-01" xlink:href="note-523-01a" xml:space="preserve">Per ſolos ſi <lb/>nus, quan-<lb/>do duo ar-<lb/>cus dati sũt <lb/>inæquales, <lb/>& neuter <lb/>eorum qua <lb/>drans.</note> </div> <p> <s xml:id="echoid-s18863" xml:space="preserve"><emph style="sc">Deinde</emph> _inueſtigetur per_ 1. </s> <s xml:id="echoid-s18864" xml:space="preserve">_praxim problematis_ 1. </s> <s xml:id="echoid-s18865" xml:space="preserve">_triang ſphær. </s> <s xml:id="echoid-s18866" xml:space="preserve">ex dato ar-_ <lb/>_cu_ AC, _rectum angulum_ D, _ſubtendente, & </s> <s xml:id="echoid-s18867" xml:space="preserve">inuento arcu_ CD, _angulus oppoſitus_ <lb/>CAD: </s> <s xml:id="echoid-s18868" xml:space="preserve">_qui angulo_ <emph style="sc">BAD</emph>, _adiunctus, vel ab eo demptus, (prout arcus_ AD, _intra_ <lb/>_triangulum, aut extra cadit) exhibebit quæſitum angulum_ <emph style="sc">BA</emph>C. </s> <s xml:id="echoid-s18869" xml:space="preserve">_Denique per_ <lb/>_problema_ 5. </s> <s xml:id="echoid-s18870" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18871" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18872" xml:space="preserve">ex arcu inuento_ AD, _& </s> <s xml:id="echoid-s18873" xml:space="preserve">inuento angulo adiacente_ CAD, <lb/>_reperiatur angulus alter_ ACD: </s> <s xml:id="echoid-s18874" xml:space="preserve">_qui erit ex quæſitis alter, ſi arcus_ <emph style="sc">A</emph>D, _intra trians-_ <lb/>_gulum cadit, ſi vero cadit extra, detrahendus erit ex duobus rectis, vt reliquus_ <lb/>_fiat alter angulus quæſitus_ <emph style="sc">ACb</emph>.</s> <s xml:id="echoid-s18875" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18876" xml:space="preserve">QVOD ſi alter datorum arcuum ſit quadrans; </s> <s xml:id="echoid-s18877" xml:space="preserve">ſi quidem AB, quadrans <lb/> <anchor type="note" xlink:label="note-523-02a" xlink:href="note-523-02"/> fuerit, erit quoque BD, quadrans, & </s> <s xml:id="echoid-s18878" xml:space="preserve">angulus BAD, rectus, nec non B, po-<lb/>lus arcus AD, ac proinde arcus AD, cognoſcetur ex dato angulo B. </s> <s xml:id="echoid-s18879" xml:space="preserve">Atque <lb/>ita cognitis arcubus AD, BD, & </s> <s xml:id="echoid-s18880" xml:space="preserve">angulo recto BAD, reliqua inueniemus, vt <lb/>prius. </s> <s xml:id="echoid-s18881" xml:space="preserve">Pariratione, ſi AC, fuerit quadrans, erit quoque CD, quadrans, & </s> <s xml:id="echoid-s18882" xml:space="preserve"><lb/>angulus CAD, rectus, nec non C, polus arcus AD; </s> <s xml:id="echoid-s18883" xml:space="preserve">atque adeo inuentus ar-<lb/>cus AD, notum faciet angulum ſuum ACD; </s> <s xml:id="echoid-s18884" xml:space="preserve">qui vnus erit ex quæſitis, ſi ar-<lb/>cus AD, intra triangulum cadit; </s> <s xml:id="echoid-s18885" xml:space="preserve">ſi vero cadit extra, idem ex duobus rectis <lb/>detractus relinquet quæſitum angulum ACB. </s> <s xml:id="echoid-s18886" xml:space="preserve">Inuentis autem tanta facilita-<lb/>te angulis CAD, ACD, & </s> <s xml:id="echoid-s18887" xml:space="preserve">arcu CD, reperientur cætera, vt prius.</s> <s xml:id="echoid-s18888" xml:space="preserve"/> </p> <div xml:id="echoid-div1532" type="float" level="2" n="40"> <note position="right" xlink:label="note-523-02" xlink:href="note-523-02a" xml:space="preserve">Quãdo al-<lb/>ter datorũ <lb/>arcuum eſt <lb/>quadrans.</note> </div> <p> <s xml:id="echoid-s18889" xml:space="preserve">SINT iam dati duo arcus AB, AC, æquales. </s> <s xml:id="echoid-s18890" xml:space="preserve">Secabit arcus AD, & </s> <s xml:id="echoid-s18891" xml:space="preserve">ba-<lb/> <anchor type="note" xlink:label="note-523-03a" xlink:href="note-523-03"/> ſim BC, & </s> <s xml:id="echoid-s18892" xml:space="preserve">angulum A, bifariam, anguliq́; </s> <s xml:id="echoid-s18893" xml:space="preserve">B, C, &</s> <s xml:id="echoid-s18894" xml:space="preserve">qua-<lb/> <anchor type="figure" xlink:label="fig-523-01a" xlink:href="fig-523-01"/> les erunt; </s> <s xml:id="echoid-s18895" xml:space="preserve">atque ita inquirendus erit tantum angulus <lb/>BAC, cum arcu BC. </s> <s xml:id="echoid-s18896" xml:space="preserve">Inquiratur ergo, per problema <lb/>15. </s> <s xml:id="echoid-s18897" xml:space="preserve">triang. </s> <s xml:id="echoid-s18898" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18899" xml:space="preserve">ex dato arcu AB, rectum angulum D, <lb/>ſubtendente, & </s> <s xml:id="echoid-s18900" xml:space="preserve">dato angulo B, angulus BAD; </s> <s xml:id="echoid-s18901" xml:space="preserve">qui du-<lb/>plicatus offeret totum quæſitum BAC. </s> <s xml:id="echoid-s18902" xml:space="preserve">Rurſus, per pro-<lb/>blema 2. </s> <s xml:id="echoid-s18903" xml:space="preserve">triang. </s> <s xml:id="echoid-s18904" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18905" xml:space="preserve">ex arcu AB, angulum rectum D, <lb/>ſubtendente, & </s> <s xml:id="echoid-s18906" xml:space="preserve">inuento angulo BAD, reperiatur arcus <lb/>oppoſitus BD: </s> <s xml:id="echoid-s18907" xml:space="preserve">qui duplicatus totum quæſitum BC, <lb/>dabit.</s> <s xml:id="echoid-s18908" xml:space="preserve"/> </p> <div xml:id="echoid-div1533" type="float" level="2" n="41"> <note position="right" xlink:label="note-523-03" xlink:href="note-523-03a" xml:space="preserve">Quãdoduo <lb/>arcus dati <lb/>sũt ęquales.</note> <figure xlink:label="fig-523-01" xlink:href="fig-523-01a"> <image file="523-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/523-01"/> </figure> </div> <p> <s xml:id="echoid-s18909" xml:space="preserve">PER _ſolos ſinus ſic. </s> <s xml:id="echoid-s18910" xml:space="preserve">Per problema_ 2. </s> <s xml:id="echoid-s18911" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18912" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18913" xml:space="preserve">inueniatur ex dato arcu_ <emph style="sc">Ab</emph>, <lb/> <anchor type="note" xlink:label="note-523-04a" xlink:href="note-523-04"/> _angulum rectum_ D, _ſubtendente & </s> <s xml:id="echoid-s18914" xml:space="preserve">dato angulo_ <emph style="sc">B</emph>, _arcus oppoſitus_ AD: </s> <s xml:id="echoid-s18915" xml:space="preserve">_Atque hinc <lb/>_per_ 1. </s> <s xml:id="echoid-s18916" xml:space="preserve">_praxim problematis_ 8. </s> <s xml:id="echoid-s18917" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18918" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18919" xml:space="preserve">ex dato arcu_ <emph style="sc">AB</emph>, _rectum angulum_ D, <lb/>_ſubtendente, & </s> <s xml:id="echoid-s18920" xml:space="preserve">inuento arcu_ AD, _reperiatur tertius arcus_ <emph style="sc">B</emph>D; </s> <s xml:id="echoid-s18921" xml:space="preserve">_qui duplicatus_ <lb/>_totum quæſitum_ <emph style="sc">B</emph>C, _exhibebit. </s> <s xml:id="echoid-s18922" xml:space="preserve">Per problema tandcm_ 5. </s> <s xml:id="echoid-s18923" xml:space="preserve">_triang. </s> <s xml:id="echoid-s18924" xml:space="preserve">ſphær. </s> <s xml:id="echoid-s18925" xml:space="preserve">inueſtige-_ <lb/>_tur ex inuento arcu_ <emph style="sc">B</emph>D, _& </s> <s xml:id="echoid-s18926" xml:space="preserve">dato angulo_ <emph style="sc">B</emph>, _adiacente angulus_ <emph style="sc">B</emph>AD, _arcui_ <emph style="sc">B</emph>D, <lb/>_oppoſitus. </s> <s xml:id="echoid-s18927" xml:space="preserve">Hicenim duplicatus dabit totum_ <emph style="sc">B</emph><emph style="sc">Ac</emph>, _quem deſideramus._</s> <s xml:id="echoid-s18928" xml:space="preserve"/> </p> <div xml:id="echoid-div1534" type="float" level="2" n="42"> <note position="right" xlink:label="note-523-04" xlink:href="note-523-04a" xml:space="preserve">Per ſolos fi <lb/>nus, quan-<lb/>do dati duo <lb/>arcus ſunt <lb/>æquales.</note> </div> <p> <s xml:id="echoid-s18929" xml:space="preserve">CAETERVM, vt facilius problema illud, quod maxime optamus, prę-<lb/>ſertim in ſphæricis triangulis, inuenire poſsimus, confecimus hic indicem om- <pb o="512" file="524" n="524" rhead=""/> nium problematum ad calculum neceſſariorum: </s> <s xml:id="echoid-s18930" xml:space="preserve">quibus quidem numeros prę-<lb/>fiximus, qui indicent, quem ordinem quodlibet inter problemata, quorum <lb/>praxes proxime expoſuimus, obtineat; </s> <s xml:id="echoid-s18931" xml:space="preserve">quemadmodum & </s> <s xml:id="echoid-s18932" xml:space="preserve">ſupra problematibus <lb/>ipſis in margine adſcripſimus propoſitiones, & </s> <s xml:id="echoid-s18933" xml:space="preserve">problemata, in quibus praxes <lb/>demonſtrantur in noſtris triangulis rectilineis, & </s> <s xml:id="echoid-s18934" xml:space="preserve">ſphæricis. </s> <s xml:id="echoid-s18935" xml:space="preserve">Quanquam autem <lb/>in indice triangulorum ſphæricorum rectangulorum proponantur tantum <lb/>ſingula in ſingulis problematibus inuenienda: </s> <s xml:id="echoid-s18936" xml:space="preserve">ijs tamen inuentis, pleraque <lb/>etiam alia in eiſdem reperiuntur, vt ex ſuperioribus liquet.</s> <s xml:id="echoid-s18937" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1536" type="section" level="1" n="666"> <head xml:id="echoid-head702" xml:space="preserve">INDEX PROBLEMATVM, ET <lb/>PRAXIVM TRIANGVLORVM.</head> <head xml:id="echoid-head703" xml:space="preserve">IN TRIANGVLIS RECTILINEIS <lb/>RECTANGVLIS</head> <head xml:id="echoid-head704" xml:space="preserve">Inuenitur</head> <p> <s xml:id="echoid-s18938" xml:space="preserve">2. </s> <s xml:id="echoid-s18939" xml:space="preserve">Latus circa angulum rectum vtrilibet angulorum acutorum op-<lb/>poſitum; </s> <s xml:id="echoid-s18940" xml:space="preserve">ex latere rectum angulum ſubtendente, & </s> <s xml:id="echoid-s18941" xml:space="preserve">alterutro <lb/>acutorum angulorum.</s> <s xml:id="echoid-s18942" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18943" xml:space="preserve">3. </s> <s xml:id="echoid-s18944" xml:space="preserve">Latus angulo recto oppoſitum, & </s> <s xml:id="echoid-s18945" xml:space="preserve">alterutrum duorum circa eun-<lb/>dem rectum angulum; </s> <s xml:id="echoid-s18946" xml:space="preserve">ex altero latere circa angulum rectum, <lb/>& </s> <s xml:id="echoid-s18947" xml:space="preserve">vno acutorum angulorum.</s> <s xml:id="echoid-s18948" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18949" xml:space="preserve">4. </s> <s xml:id="echoid-s18950" xml:space="preserve">Vterque angulus acutus, & </s> <s xml:id="echoid-s18951" xml:space="preserve">alterutrum duorum laterum circa an-<lb/>gulum rectum; </s> <s xml:id="echoid-s18952" xml:space="preserve">ex latere angulum rectum ſubtendente, & </s> <s xml:id="echoid-s18953" xml:space="preserve">al-<lb/>tero latere circa eundem rectum angulum.</s> <s xml:id="echoid-s18954" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18955" xml:space="preserve">5. </s> <s xml:id="echoid-s18956" xml:space="preserve">Vterque angulus acutus, & </s> <s xml:id="echoid-s18957" xml:space="preserve">latus recto angulo oppoſitum; </s> <s xml:id="echoid-s18958" xml:space="preserve">ex duo-<lb/>bus lateribus circa eundem angulum rectum.</s> <s xml:id="echoid-s18959" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1537" type="section" level="1" n="667"> <head xml:id="echoid-head705" xml:space="preserve">IN TRIANGVLIS RECTILINEIS NON <lb/>RECTANGVLIS</head> <head xml:id="echoid-head706" xml:space="preserve">Inueniuntur</head> <p> <s xml:id="echoid-s18960" xml:space="preserve">10. </s> <s xml:id="echoid-s18961" xml:space="preserve">Duo latera; </s> <s xml:id="echoid-s18962" xml:space="preserve">ex omnibus angulis, & </s> <s xml:id="echoid-s18963" xml:space="preserve">reliquo latere.</s> <s xml:id="echoid-s18964" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18965" xml:space="preserve">11. </s> <s xml:id="echoid-s18966" xml:space="preserve">Omnes anguli; </s> <s xml:id="echoid-s18967" xml:space="preserve">ex omnibus lateribus.</s> <s xml:id="echoid-s18968" xml:space="preserve"/> </p> <pb o="513" file="525" n="525" rhead=""/> <p> <s xml:id="echoid-s18969" xml:space="preserve">12. </s> <s xml:id="echoid-s18970" xml:space="preserve">Vnum latus, & </s> <s xml:id="echoid-s18971" xml:space="preserve">duo anguli illi adiacentes; </s> <s xml:id="echoid-s18972" xml:space="preserve">ex reliquis duobus la-<lb/>teribus, & </s> <s xml:id="echoid-s18973" xml:space="preserve">reliquo angulo ab ipſis comprehenſo.</s> <s xml:id="echoid-s18974" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18975" xml:space="preserve">13. </s> <s xml:id="echoid-s18976" xml:space="preserve">Duo anguli, & </s> <s xml:id="echoid-s18977" xml:space="preserve">vnum latus vni eorum oppoſitum; </s> <s xml:id="echoid-s18978" xml:space="preserve">ex reliquis duo-<lb/>bus lateribus, & </s> <s xml:id="echoid-s18979" xml:space="preserve">reliquo angulo, qui vni eorum opponitur.</s> <s xml:id="echoid-s18980" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1538" type="section" level="1" n="668"> <head xml:id="echoid-head707" xml:space="preserve">IN TRIANGVLIS SPHÆRICIS <lb/>RECTANGVLIS</head> <head xml:id="echoid-head708" xml:space="preserve">Inuenitur arcus angulo recto opponſitus</head> <p> <s xml:id="echoid-s18981" xml:space="preserve">3. </s> <s xml:id="echoid-s18982" xml:space="preserve">Ex arcu circa rectum angulum, & </s> <s xml:id="echoid-s18983" xml:space="preserve">angulo ei oppoſito.</s> <s xml:id="echoid-s18984" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18985" xml:space="preserve">12. </s> <s xml:id="echoid-s18986" xml:space="preserve">Ex arcu circa angulum rectum, & </s> <s xml:id="echoid-s18987" xml:space="preserve">angulo ei adiacente.</s> <s xml:id="echoid-s18988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18989" xml:space="preserve">7. </s> <s xml:id="echoid-s18990" xml:space="preserve">Ex vtroque arcu circa angulum rectum.</s> <s xml:id="echoid-s18991" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18992" xml:space="preserve">16. </s> <s xml:id="echoid-s18993" xml:space="preserve">Ex vtroque angulo non recto.</s> <s xml:id="echoid-s18994" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1539" type="section" level="1" n="669"> <head xml:id="echoid-head709" xml:space="preserve">Inuenitur arcus circa angulum rectum</head> <p> <s xml:id="echoid-s18995" xml:space="preserve">2. </s> <s xml:id="echoid-s18996" xml:space="preserve">Ex arcu rectum angulum ſubtendente, & </s> <s xml:id="echoid-s18997" xml:space="preserve">angulo, qui quæſito ar-<lb/>cui opponitur.</s> <s xml:id="echoid-s18998" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18999" xml:space="preserve">14. </s> <s xml:id="echoid-s19000" xml:space="preserve">Ex arcu rectum angulum ſubtendente, & </s> <s xml:id="echoid-s19001" xml:space="preserve">angulo, qui quæſito ar-<lb/>cui adiacet.</s> <s xml:id="echoid-s19002" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19003" xml:space="preserve">8. </s> <s xml:id="echoid-s19004" xml:space="preserve">Ex arcu angulum rectum ſubtendente, & </s> <s xml:id="echoid-s19005" xml:space="preserve">altero arcu circa angu-<lb/>lum rectum.</s> <s xml:id="echoid-s19006" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19007" xml:space="preserve">10. </s> <s xml:id="echoid-s19008" xml:space="preserve">Ex altero arcu circa rectum angulum, & </s> <s xml:id="echoid-s19009" xml:space="preserve">angulo ei oppoſito.</s> <s xml:id="echoid-s19010" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19011" xml:space="preserve">9. </s> <s xml:id="echoid-s19012" xml:space="preserve">Ex altero arcu circa angulum rectum, & </s> <s xml:id="echoid-s19013" xml:space="preserve">angulo ei adiacente.</s> <s xml:id="echoid-s19014" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19015" xml:space="preserve">4. </s> <s xml:id="echoid-s19016" xml:space="preserve">Ex vtroque angulo non recto.</s> <s xml:id="echoid-s19017" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1540" type="section" level="1" n="670"> <head xml:id="echoid-head710" xml:space="preserve">Inuenitur angulus non rectus</head> <p> <s xml:id="echoid-s19018" xml:space="preserve">1. </s> <s xml:id="echoid-s19019" xml:space="preserve">Ex arcu rectum angulum ſubtendente, & </s> <s xml:id="echoid-s19020" xml:space="preserve">arcu circa angulum re <lb/>ctum, qui quæſito angulo opponitur.</s> <s xml:id="echoid-s19021" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19022" xml:space="preserve">13. </s> <s xml:id="echoid-s19023" xml:space="preserve">Ex arcu angulum rectum ſubtendente, & </s> <s xml:id="echoid-s19024" xml:space="preserve">arcu circa angulum re-<lb/>ctum, qui quæſito angulo adiacet.</s> <s xml:id="echoid-s19025" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19026" xml:space="preserve">15. </s> <s xml:id="echoid-s19027" xml:space="preserve">Ex arcu rectum angulũ ſubtendente, & </s> <s xml:id="echoid-s19028" xml:space="preserve">altero angulo non recto.</s> <s xml:id="echoid-s19029" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19030" xml:space="preserve">11. </s> <s xml:id="echoid-s19031" xml:space="preserve">Ex vtroque arcu circa angulum rectum.</s> <s xml:id="echoid-s19032" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19033" xml:space="preserve">5. </s> <s xml:id="echoid-s19034" xml:space="preserve">Ex arcu circa rectum angulum, qui angulo quæſito opponitur, &</s> <s xml:id="echoid-s19035" xml:space="preserve"> <pb o="514" file="526" n="526" rhead=""/> altero angulo non recto illi arcui adiacente.</s> <s xml:id="echoid-s19036" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19037" xml:space="preserve">6. </s> <s xml:id="echoid-s19038" xml:space="preserve">Ex arcu circa angulum rectum, qui angulo quæſito adiacet, & </s> <s xml:id="echoid-s19039" xml:space="preserve">al-<lb/>tero angulo non recto illi arcui oppoſito.</s> <s xml:id="echoid-s19040" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1541" type="section" level="1" n="671"> <head xml:id="echoid-head711" xml:space="preserve">IN TRIANGVLIS SPHÆRICIS <lb/>NON RECTANGVLIS</head> <head xml:id="echoid-head712" xml:space="preserve">Inueniuntur</head> <p> <s xml:id="echoid-s19041" xml:space="preserve">17. </s> <s xml:id="echoid-s19042" xml:space="preserve">Omnes tres arcus; </s> <s xml:id="echoid-s19043" xml:space="preserve">ex omnibus tribus angulis.</s> <s xml:id="echoid-s19044" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19045" xml:space="preserve">18. </s> <s xml:id="echoid-s19046" xml:space="preserve">Omnes tres anguli; </s> <s xml:id="echoid-s19047" xml:space="preserve">ex omnibus tribus arcubus.</s> <s xml:id="echoid-s19048" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19049" xml:space="preserve">19. </s> <s xml:id="echoid-s19050" xml:space="preserve">Vnus arcus, & </s> <s xml:id="echoid-s19051" xml:space="preserve">duo anguli illi adiacentes; </s> <s xml:id="echoid-s19052" xml:space="preserve">ex alijs duobus arcubus, <lb/>& </s> <s xml:id="echoid-s19053" xml:space="preserve">reliquo angulo ab ipſis comprehenſo.</s> <s xml:id="echoid-s19054" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19055" xml:space="preserve">20. </s> <s xml:id="echoid-s19056" xml:space="preserve">Duo arcus, & </s> <s xml:id="echoid-s19057" xml:space="preserve">angulus ab ipſis comprehenſus; </s> <s xml:id="echoid-s19058" xml:space="preserve">ex reliquo arcu, & </s> <s xml:id="echoid-s19059" xml:space="preserve"><lb/>alijs duobus angulis huic arcui adiacentibus.</s> <s xml:id="echoid-s19060" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19061" xml:space="preserve">21. </s> <s xml:id="echoid-s19062" xml:space="preserve">Duo arcus, & </s> <s xml:id="echoid-s19063" xml:space="preserve">vnus angulus vni eorum oppoſitus; </s> <s xml:id="echoid-s19064" xml:space="preserve">ex reliquo arcu, <lb/>& </s> <s xml:id="echoid-s19065" xml:space="preserve">alijs duobus angulis, quorum vni hic arcus opponitur: </s> <s xml:id="echoid-s19066" xml:space="preserve">ſi mo-<lb/>do conſtet ſpecies arcus alteri angulo dato oppoſiti.</s> <s xml:id="echoid-s19067" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19068" xml:space="preserve">22. </s> <s xml:id="echoid-s19069" xml:space="preserve">Duo anguli, & </s> <s xml:id="echoid-s19070" xml:space="preserve">vnus arcus vni eorum oppoſitus; </s> <s xml:id="echoid-s19071" xml:space="preserve">ex reliquo angu-<lb/>lo, & </s> <s xml:id="echoid-s19072" xml:space="preserve">alijs duobus arcubus, quorum vni hic angulus opponi-<lb/>tur: </s> <s xml:id="echoid-s19073" xml:space="preserve">ſi modo conſtet ſpecies anguli alteri arcui dato oppoſiti.</s> <s xml:id="echoid-s19074" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s19075" xml:space="preserve">ATQVE hic finis ſit noſtrorum triangulorum, in quibus omnia ea <lb/>videor eſſe complexus, quæ ad calculum ipſorum requiruntur. </s> <s xml:id="echoid-s19076" xml:space="preserve">His ergo, <lb/>benigne Lector, interea fruere feliciter, dum tres integros libros triangu-<lb/>lorum ſphæricorum Menelai, cum duobus Franciſci Maurolyci, in quibus <lb/>multo plura, quam hic à nobis explanata ſunt, & </s> <s xml:id="echoid-s19077" xml:space="preserve">quidem ſcitu iucundiſ-<lb/>ſima, continentur, clarioribus demonſtrationibus illuſtratos in lucem, Deo <lb/>noſtris cœptis bene fauente, prodire ſinamus.</s> <s xml:id="echoid-s19078" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1542" type="section" level="1" n="672"> <head xml:id="echoid-head713" xml:space="preserve">FINIS TRIANGVLORVM <lb/>SPHÆRICORVM.</head> <pb file="527" n="527" rhead=""/> <p> <s xml:id="echoid-s19079" xml:space="preserve">ABCDEFGHIKLMNOPQRST <lb/>VXYZ.</s> <s xml:id="echoid-s19080" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19081" xml:space="preserve">Aa Bb Cc Dd Ee Ff Gg Hh Ii Kk Ll Mm <lb/>Nn Oo Pp Qq Rr Sſ Tt Vu Xx Yy Zz.</s> <s xml:id="echoid-s19082" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19083" xml:space="preserve">Aaa Bbb Ccc Ddd Eee Fff Ggg Hhh Iii Kkk <lb/>Lll Mmm Nnn Ooo Ppp Qqq Rrr.</s> <s xml:id="echoid-s19084" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s19085" xml:space="preserve">Omnes ſunt Duerni. </s> <s xml:id="echoid-s19086" xml:space="preserve">Solum Gg Sſ Nn Terni ſunt.</s> <s xml:id="echoid-s19087" xml:space="preserve"/> </p> <figure> <image file="527-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/527-01"/> </figure> </div> <div xml:id="echoid-div1543" type="section" level="1" n="673"> <head xml:id="echoid-head714" xml:space="preserve">ROMAE, <lb/>Ex Typographia Dominici Baſæ. <lb/>MDLXXXVI.</head> <pb file="528" n="528"/> <pb file="529" n="529"/> <pb file="530" n="530"/> <pb file="531" n="531"/> <pb file="532" n="532"/> </div></text> </echo>