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Removing DESpecs directory which deserted to git
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> <metadata> <dcterms:identifier>ECHO:FBVYV7EH.xml</dcterms:identifier> <dcterms:creator identifier="GND:118880632">Clavius, Christoph</dcterms:creator> <dcterms:title xml:lang="la">Geometria practica</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1606</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> <parameters>despecs = 1.1.2</parameters> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"> <pb file="001" n="1"/> <figure> <image file="001-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/001-01"/> </figure> <pb file="002" n="2"/> <figure> <image file="002-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/002-01"/> </figure> <handwritten/> <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"><emph style="red">CHRISTOPHORI <lb/>CLAVII BAMBER-</emph> <lb/>GENSISE SOCIETATE <lb/><emph style="sc">Iesv.</emph> <lb/><emph style="red">GEOMETRIA</emph> <lb/>PRACTICA.</head> <figure> <image file="005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/005-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head2" style="it" xml:space="preserve"><emph style="red">Cum gratia & Priuilegio Sac. Cæſ. Maieſtat.</emph> <lb/>Superiorum Permiſu. <lb/><emph style="red"><emph style="sc">Mogvntia</emph>,</emph> <lb/>Ex Typographeo l<emph style="sc">OANNIS</emph> <emph style="sc">Albini.</emph> <lb/><emph style="red">ANNO M. DC. VI.</emph></head> <pb file="006" n="6"/> <handwritten/> <pb file="007" n="7"/> <figure> <image file="007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/007-01"/> </figure> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head3" xml:space="preserve"><emph style="sc">Perillvstri</emph> <emph style="sc">Ac</emph> <emph style="sc">Generoso</emph> D. <lb/>GEORGIO FVGGERO <lb/>SENIORI</head> <head xml:id="echoid-head4" style="it" xml:space="preserve">BARONI IN KIRCHBERG, <lb/>ET VVEISSENHORN</head> <head xml:id="echoid-head5" xml:space="preserve">Chriſtophorus Clauius è Societate <lb/>IESV S.P.D.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">VT <emph style="sc">Veniet</emph> <emph style="sc">In</emph> <emph style="sc">Manvs</emph> <emph style="sc">Tvas</emph> <lb/>liber hic meus, Generoſe Domine, ve-<lb/>niet & </s> <s xml:id="echoid-s2" xml:space="preserve">in mentẽ, vt opinor, Theocritę-<lb/>um illud, βάρ{δι}{στα}ς μακάρων ὧ{ρα}ι φίλ{αι}: </s> <s xml:id="echoid-s3" xml:space="preserve">cum <lb/>lentiſſimo gradu, quo mira celeritate <lb/>aduolare debuerat, vix aliquando <lb/>peruenerit. </s> <s xml:id="echoid-s4" xml:space="preserve">Serius omnino merito tuo, & </s> <s xml:id="echoid-s5" xml:space="preserve">vo-<lb/>to meo partus hic noſter properat ad Patronum <lb/>ſuum: </s> <s xml:id="echoid-s6" xml:space="preserve">verum ſi editum illum iam grauibus annis re-<lb/>putaueris, facile apudte conſtitues, à tarda ſene-<lb/>ctute non. </s> <s xml:id="echoid-s7" xml:space="preserve">niſi fœtum tardum expectari potuiſſe. <lb/></s> <s xml:id="echoid-s8" xml:space="preserve">Nunquam alias tam frequentibus, moleſtiſque mor-<lb/>bis conflictatus, nunquã tanto arſi deſiderio, vt quod <lb/>iam diu conceperam, aliquando in lucem exponerẽ, <pb file="008" n="8"/> experientia, & </s> <s xml:id="echoid-s9" xml:space="preserve">Pindarica voce doctus, ἀ{προ}{σί}κτων ἐρώτων <lb/>ὀζυ{τέ}{ρα}ς {εἶ}ν{αι} μ{αν}ί{ας}. </s> <s xml:id="echoid-s10" xml:space="preserve">Obuerſabaris enim noctes, dieſque <lb/>oculis meis, vir ampliſſimè: </s> <s xml:id="echoid-s11" xml:space="preserve">& </s> <s xml:id="echoid-s12" xml:space="preserve">iure tuo, vt iam affe-<lb/>cta perfecta tibi traderem, flagitare videbaris. </s> <s xml:id="echoid-s13" xml:space="preserve">Non <lb/>exiguam vim pecuniæ gentilitia liberalitate ad men-<lb/>ſas depoſueras, paratam in vſus, expenſaſq; </s> <s xml:id="echoid-s14" xml:space="preserve">cudendi li-<lb/>bri: </s> <s xml:id="echoid-s15" xml:space="preserve">res monebat, vt diligentia mea minus prompta <lb/>non eſſet munificentia tua. </s> <s xml:id="echoid-s16" xml:space="preserve">Accedebat quod onus <lb/>Ætna grauius impoſuerat obſeruantia tua ſingularis <lb/>in me: </s> <s xml:id="echoid-s17" xml:space="preserve">quo aliqua ex parte qua ratione leuari poſſem <lb/>alia, non videbam. </s> <s xml:id="echoid-s18" xml:space="preserve">Vir enim in Mathematicis non <lb/>vulgari cum laude exercitatiſſimus, tanto in honore <lb/>habuiſti ſemper labores, & </s> <s xml:id="echoid-s19" xml:space="preserve">libros meos, vt ijs, quite <lb/>probe, meque norunt penè miraculo poſſis eſſe. </s> <s xml:id="echoid-s20" xml:space="preserve">Sie-<lb/>nim Clauij libros in manibus haberes eo conſilio: </s> <s xml:id="echoid-s21" xml:space="preserve">vt <lb/>amici tui ſcripta, quod potes, redderes meliora: </s> <s xml:id="echoid-s22" xml:space="preserve">doctri <lb/>nam tuam omnes agnoſcerent, laudarẽt beneuolen-<lb/>tiã; </s> <s xml:id="echoid-s23" xml:space="preserve">Sed à te tanto viro in Mathematicis partus tenues, <lb/>nec operæ multæ aſſidua verſari manu, hoc illud eſt, <lb/>quod jure mirari quis poſſit: </s> <s xml:id="echoid-s24" xml:space="preserve">hoc non obſcurum indi-<lb/>cium, à te mea, quaſi egregium, eximiumque aliquid <lb/>ſint, exiſtimari. </s> <s xml:id="echoid-s25" xml:space="preserve">A qua præſtantia quantum abſim, <lb/>quia video: </s> <s xml:id="echoid-s26" xml:space="preserve">ideo intelligo maximum eſſe ſtudium in <lb/>me tuum: </s> <s xml:id="echoid-s27" xml:space="preserve">à quo ſi præter meritum honeſtor, pro me-<lb/>rito certe ita obſtringor, vt quæ maxima eſt viribus <lb/>meis grati animi teſtificatio, minima ſit pro benefi-<lb/>cio tuo. </s> <s xml:id="echoid-s28" xml:space="preserve">Hanc tamen qualemcunque gratæ mentis in <lb/>te ſignificationẽ extare quam primumvolebam, cum <lb/>præſertim ad has rationes acceſſio fieret aliarum, quæ <lb/>apud optimum quemq; </s> <s xml:id="echoid-s29" xml:space="preserve">plurimum ſemper valere cõ- <pb file="009" n="9"/> ſueuerunt. </s> <s xml:id="echoid-s30" xml:space="preserve">Ego enim has viuendi rationes, & </s> <s xml:id="echoid-s31" xml:space="preserve">inſtitu-<lb/>ta ſecutus, ad cuius alterius patrocinium meos libros <lb/>adlegarem, niſi ad illum, quem vniuerſus noſtræ So-<lb/>cietatis ordo acerrimũ ſui propugnatorem, & </s> <s xml:id="echoid-s32" xml:space="preserve">aman-<lb/>tiſſimum patronum ſemper expertus eſt? </s> <s xml:id="echoid-s33" xml:space="preserve">Familiaris <lb/>igitur tibi cum ſit tutela noſtrum, mea hæc ab inui-<lb/>dorum, ignarorumq; </s> <s xml:id="echoid-s34" xml:space="preserve">morſibus vindicabis, non au-<lb/>ctoritate modo vrgens aduerſarios, ſed etiam doctri-<lb/>na. </s> <s xml:id="echoid-s35" xml:space="preserve">Qua cogitatione ſum permotus, vt tibi præcipuè <lb/>Geometriam Practicam deſtinarem. </s> <s xml:id="echoid-s36" xml:space="preserve">Iis etenim, qui <lb/>noſtris in ſtudiis non admodum deſudarunt, ſi quid <lb/>offeras tale, magna cura explicanda prius eſt, & </s> <s xml:id="echoid-s37" xml:space="preserve">com-<lb/>mendanda per epiſtolam libri doctrina, ne ſe contem-<lb/>ptos humilitate muneris arbitrentur. </s> <s xml:id="echoid-s38" xml:space="preserve">Tu, vt hoc dun-<lb/>taxat vnum legis, Geometria Practica; </s> <s xml:id="echoid-s39" xml:space="preserve">quantum vtili-<lb/>tatis, voluptatiſque lateat in toto libro, optimè perui-<lb/>des. </s> <s xml:id="echoid-s40" xml:space="preserve">His igitur aſſiduis cogitationibus, quaſi tot faci-<lb/>bus inflammatus, ætatem, valetudinemq; </s> <s xml:id="echoid-s41" xml:space="preserve">aduerſam, <lb/>Deo propitio, tandem vici; </s> <s xml:id="echoid-s42" xml:space="preserve">& </s> <s xml:id="echoid-s43" xml:space="preserve">per hæc locorum in-<lb/>terualla opus ad te tuũ venire iubeo. </s> <s xml:id="echoid-s44" xml:space="preserve">Tuum, inquam: <lb/></s> <s xml:id="echoid-s45" xml:space="preserve">& </s> <s xml:id="echoid-s46" xml:space="preserve">quia virtus, liberalitaſque tua illud aſſeruere tibi: </s> <s xml:id="echoid-s47" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s48" xml:space="preserve">quia ab eo ſcriptum eſt, qui tuus eſt. </s> <s xml:id="echoid-s49" xml:space="preserve">Quod ſi gra-<lb/>tum fuiſſe tibi intelligam, laborum meorum fructum <lb/>cumulatum feram, cum amantiſſimum mei re non <lb/>ingrata donauerim. </s> <s xml:id="echoid-s50" xml:space="preserve">Vale.</s> <s xml:id="echoid-s51" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head6" xml:space="preserve"><emph style="sc">Romæ pridie</emph> <emph style="sc">Idvs</emph> <emph style="sc">Septemb.</emph> <lb/>cIↄ. cI. <emph style="sc">CIIII.</emph></head> <pb file="010" n="10"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head7" xml:space="preserve"><emph style="sc">Clavdivs</emph> <emph style="sc">Aqvaviva</emph> <emph style="sc">Societatis</emph> <lb/><emph style="sc">Iesv</emph> Præpoſitus Generalis.</head> <p> <s xml:id="echoid-s52" xml:space="preserve"><emph style="sc">CVm</emph> opus hoc Geometriæ Practicæ, à P.</s> <s xml:id="echoid-s53" xml:space="preserve">Chriſto-<lb/>phoro Clauio Societatis noſtræ Theologo com-<lb/>poſitum, & </s> <s xml:id="echoid-s54" xml:space="preserve">in octo libros diſtributum, tres eiuſdem <lb/>Societatis Theologi, quibus id commiſimus, recog-<lb/>nouerint, ac in lucem edi poſſe probauerint; </s> <s xml:id="echoid-s55" xml:space="preserve">faculta-<lb/>tem concedimus, vt typis mandetur, ſi ita Reueren-<lb/>diſſimo D. </s> <s xml:id="echoid-s56" xml:space="preserve">Viceſgerenti, ac Reuerendiſs. </s> <s xml:id="echoid-s57" xml:space="preserve">P.</s> <s xml:id="echoid-s58" xml:space="preserve">M. </s> <s xml:id="echoid-s59" xml:space="preserve">ſacri <lb/>Palatij videbitur. </s> <s xml:id="echoid-s60" xml:space="preserve">In quorum fidem has litteras manu <lb/>noſtra ſubſcriptas, & </s> <s xml:id="echoid-s61" xml:space="preserve">ſigillo noſtro munitas dedimus. <lb/></s> <s xml:id="echoid-s62" xml:space="preserve">Romæ 23. </s> <s xml:id="echoid-s63" xml:space="preserve">Maij 1604.</s> <s xml:id="echoid-s64" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s65" xml:space="preserve">Claudius Aquauiua Societatis <emph style="sc">Iesv</emph> Præp.</s> <s xml:id="echoid-s66" xml:space="preserve">Gen.</s> <s xml:id="echoid-s67" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s68" xml:space="preserve">Imprimatur ſi placet R.</s> <s xml:id="echoid-s69" xml:space="preserve">P.</s> <s xml:id="echoid-s70" xml:space="preserve">M.</s> <s xml:id="echoid-s71" xml:space="preserve">S. </s> <s xml:id="echoid-s72" xml:space="preserve">Palatii.</s> <s xml:id="echoid-s73" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s74" xml:space="preserve">B. </s> <s xml:id="echoid-s75" xml:space="preserve">Gypſius Viceſgerens.</s> <s xml:id="echoid-s76" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s77" xml:space="preserve">EGO Theodoſius Rubeus Priuernas S. </s> <s xml:id="echoid-s78" xml:space="preserve">Theol. </s> <s xml:id="echoid-s79" xml:space="preserve">& </s> <s xml:id="echoid-s80" xml:space="preserve">I.</s> <s xml:id="echoid-s81" xml:space="preserve">V.</s> <s xml:id="echoid-s82" xml:space="preserve">D. </s> <s xml:id="echoid-s83" xml:space="preserve">ex com-<lb/>miſſione Reuerendiſs. </s> <s xml:id="echoid-s84" xml:space="preserve">P.</s> <s xml:id="echoid-s85" xml:space="preserve">M. </s> <s xml:id="echoid-s86" xml:space="preserve">Ioannis Mariæ Braſichellenſ. </s> <s xml:id="echoid-s87" xml:space="preserve">Sac. </s> <s xml:id="echoid-s88" xml:space="preserve">Palat. <lb/></s> <s xml:id="echoid-s89" xml:space="preserve">Apoſt. </s> <s xml:id="echoid-s90" xml:space="preserve">Mag. </s> <s xml:id="echoid-s91" xml:space="preserve">ſedula, qua potui diligentia, euolui Geometriæ Practicæ li-<lb/>bros octo, grauiſſimi, & </s> <s xml:id="echoid-s92" xml:space="preserve">doctiſſimi viri Chriſtophori Clauii Bamber-<lb/>genſis Societatis IESV, in qua talis inuentionum vbertas, & </s> <s xml:id="echoid-s93" xml:space="preserve">ſubtiliſſi-<lb/>ma inuentorum demonſtratio elucet, vttanto viro, tantoque ingenio <lb/>digna cenſeatur. </s> <s xml:id="echoid-s94" xml:space="preserve">Ex quo facilè deprehen di poteſt, in eanihil Catholicæ <lb/>Religioni, boniſque moribus diſſonum inueniri, eamque in lucem non <lb/>edere, eſſet patrisfamilias creditum talentum in terra defodere. </s> <s xml:id="echoid-s95" xml:space="preserve">Et ita <lb/>ego cenſeo, ſaluo ſemper ſaniori iudicio. </s> <s xml:id="echoid-s96" xml:space="preserve">Dat. </s> <s xml:id="echoid-s97" xml:space="preserve">ex meo ſtudio hac die 16. </s> <s xml:id="echoid-s98" xml:space="preserve"><lb/>Iulii 1604.</s> <s xml:id="echoid-s99" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s100" xml:space="preserve">Ego Theodoſius Rubeus, qui ſupra, manu propria.</s> <s xml:id="echoid-s101" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s102" xml:space="preserve">Imprimatur.</s> <s xml:id="echoid-s103" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s104" xml:space="preserve">Fr. </s> <s xml:id="echoid-s105" xml:space="preserve">Paulus de Francis de Neap. </s> <s xml:id="echoid-s106" xml:space="preserve">ſocius Reuerendiſs. </s> <s xml:id="echoid-s107" xml:space="preserve">P.</s> <s xml:id="echoid-s108" xml:space="preserve">Magiſtri ſacri <lb/>Palatii Apoſt.</s> <s xml:id="echoid-s109" xml:space="preserve"/> </p> <pb file="011" n="11"/> <figure> <image file="011-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/011-01"/> </figure> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head8" xml:space="preserve">INDEX <lb/>CAPITVM, PROBLE-<lb/>MATVM, AC PROPOSITIONVM <lb/>HORVM VIII. LIBRORVM.</head> <head xml:id="echoid-head9" xml:space="preserve">PRIMI LIBRI CAPITA.</head> <note style="it" position="right" xml:space="preserve"> <lb/>I. <emph style="sc">Instrvmenti</emph> partium conſtructio, atque vſus multiplex. # 4. vſ{q́ue} ad 14 <lb/>II. Conſtructio Qu@adrantis, in quo Minuta quoque ac Secunda deprehendantur, <lb/>e<unsure/>tiamſi gradus in ea ſecti non ſint. Et quo pacto eadens Min. & Sec. obtineri poſſint in <lb/>Quadrante in 90. gradus diſtributo. Ac deni qua ratione ex data recta in pauciſſimas <lb/>part{es} æqual{es} diuiſa abſcindi poſſint part{es} m@lleſimæ, & c. # 14. vſ ad 44 <lb/>III. Problemata varia triangulorum rectilineorum. # 44. vſque ad 50 <lb/></note> </div> <div xml:id="echoid-div8" type="section" level="1" n="8"> <head xml:id="echoid-head10" xml:space="preserve">SECVNDI LIBRI PROBLEMATA.</head> <note style="it" position="right" xml:space="preserve"> <lb/>I. <emph style="sc">Distantiam</emph> in plano, ſiue acceſſibilis ea ſit, ſiue inacceſſibilis, per du{as} ſta-<lb/>tion{es} in eodem plano factas, per quadrantem metiri, quando in ei{us} extremo erecta est <lb/>alitudo aliqua perpendicularis, etiamſi infimum @i{us} extre@um non cernatur. Atque <lb/>hinc altitudinem quoque ipſam elicere. # 51 <lb/><emph style="sc">Lemma.</emph> Datis duabus rectis ad inuicem inclinatis, punctum, in quo con-<lb/>ueniant, inuenire. # 55 <lb/>II. Altitudinem inacceſſibilem, quando diſtantia à loco menſoris ad baſem altitu-<lb/>dinis ignota eſt, per du{as} ſtation{es} in plano fact{as}, per quadrantem dimetiri. Atque hinc <lb/>diſtantiam quoque ipſam eruere, {et}iamſi extrem{us} ei{us} termin{us} non cernatur. # 57 <lb/>III. Ex vertice montis, aut turris, in cui{us} ſummitate duæ ſtation{es} fieri poſſint, <lb/>èquib{us} ſignum aliquod in Horizonte æppareat, altitudinem ipſi{us} montis turriſue di-<lb/>m{et}iri. Atque hinc ipſam quoque diſtantiam à turris baſi, vel perpendiculo mo<unsure/>ntis ad <lb/>ſignum illud inueſtigare. # 59 <lb/>IV. Ex vertice montis, vel turris, per du{as} ſtation{es} in aliqua haſtaerecta, velin <lb/>duab{us} feneſtris turris, quarum vna ſupra aliam exiſtat, fact{as}, è quib{us} ſignum ali-<lb/>quod in Horizonte videri poſſit, altitudinem ipſi{us} montis, aut turris per quadrantem. <lb/>m{et}@ri. At{q́ue} hinc diſtantiam quoque a perpendiculo montis, velturris, vſque ad ſignum <lb/>viſum cognoſcere. # 62 <lb/>V. Ex vertice montis, aut turris, altitudinem ipſi{us}, ſi in plano, cui inſiſtit, ſpatium. <lb/>aliquodè directo menſoris notum ſit, per quadr antem deprehendere. # 64 <lb/>VI. Diſtantiam ab oculo, vel pede menſoris ad quoduis punctum in aliqua altitu-<lb/>dine notatum, per du{as} ſtation{es} in plano fact{as}, per quadrantem metiri. # 65 <lb/>VII. Interuallum inter duo puncta in quolibet plano eleuato, ſiue illud ad Horizon-<lb/>tem rectum ſit, ſiue inclinatum, per quadrantem metiri. # 67 <pb file="012" n="12" rhead="INDEX."/> VIII. Longitudinem lineæ rectæ, quando menſor in vno ei{us} extremo, vel in ali-<lb/>qua altitudine nota<unsure/>, quæ perpendicularis ſit in eo extremo ad planum in quo linea iacet, <lb/>exiſtens alterum extremum videre potest, per Quadrantem comprehendere. # 68 <lb/>IX. Longitudinem, ad cuius extrema accedere non liceat, dummodo ea appareant, <lb/>& ipſa longitudo producta ad ped{es} menſoris pertingat, ex altitudine aliqua nota per qua-<lb/>drantem dimetiri. # 68 <lb/>X. Longitudinem tranſuerſam in Horizonte, cui{us} vtrumque extremum inſpici <lb/>potest, notam efficere per quadr antem: # 69 <lb/>XI. Longitudinem in Horizonte inter turrim aliquam, & aliud quodpiam ſi-<lb/>gnum, exturri per du{as} ſtation{es} in faſtigio fact{as}, velin duab{us} feneſtris, quarum vna <lb/>ſit ſub altera ad perpendiculum, quando ſpatium inter ill{as} feneſtr{as} notum eſt, etiamſi <lb/>toti{us} turris altitudo ignota ſit, per quadrantem dimetiri. Atque hinc obiter altitudi-<lb/>nem turris patefacere. # 70 <lb/>XII. Longitudinem rectæ è directo menſoris poſitæ, cui{us} extremum vtrum{q́ue}, vel <lb/>alterum non appareat, niſi ad dexteram, vel ſiniſtram recedat menſor, per quadrantem. <lb/>comprehendere. # 71 <lb/>XIII. Diſtantiam alicui{us} ſigni in Horizonte poſiti à ſummitate turris, vel muri <lb/>alicui{us}, licet ad ipſum ſignum acceſſ{us} non pateat, per quadrantem colligere. # 72 <lb/>XIIII. Altitudinem inacceſſibilem, cui{us} baſis non videatur, & ad quam per <lb/>nullum ſpatium ſecundum lineam rectam accedere poſſim{us}, aut recedere, vt duæ ſta-<lb/>tion{es} fieri poſſint, ſed ſolum ad dexteram, ſiniſlramue ad locum, è quo ei{us} baſis appareat, <lb/>per quadrantem explorare. # 72 <lb/>XV. Altitudinem inacceſſibilem, quando neque diſtantia à loco menſoris ad ei{us} <lb/>baſem nota eſt, neque è directo ipſi{us} duæ ſtation{es} in plano fieri poſſunt, neque denique <lb/>baſis appareat, per quadrantem notam reddere. Atque hinc obiter ipſam quoque diſtan-<lb/>tiam elicere. # 73 <lb/>XVI. Altitudinem maiorem ex minori cognita per du{as} ſtation{es} in ſummitate, vel <lb/>in duab{us} feneſtris fact{as}, {et}iamſi ſolum maioris altitudinis vertex cernatur, per qua-<lb/>drantem adinuenire. Atque hinc diſtantiam quoque inter altitudin{es} colligere. # 74 <lb/>XVII. Altitudinem maiorem ex minori incognita, dummodo baſis maioris cerni <lb/>poſſit, per quadrantem perſcrutari. # 75 <lb/>XVIII. Altitudinem minorem ex maiori cognita, licet baſis minoris non cerni <lb/>poſſit, ope quadrantis perueſtigare. Atque hinc diſtantiam quoque inter du{as} altitu di-<lb/>n{es} eruere. # 75 <lb/>XIX. Altitudinem minorem ex maiori incognita, dummodo baſis minoris vide-<lb/>ri poſſit per quadrantem explorare. Atque hinc diſtantiam quoque inter du{as} altitudi-<lb/>n{es} coniicere. # 76 <lb/>XX. Portionem altitudinis maioris ex minore altitudine, & minoris portionem <lb/>ex maiore, per quadrantem cognoſcere. # 76 <lb/>XXI. Altitudinem, cui{us} baſis impoſita ſit alteri altitudini, & vtr aque illi{us} ex-<lb/>tremitas cerni poſſit, {et}iamſi infimum punctum alieri{us}, cui imponitur, lateat, & eiuſ-<lb/>dem puncti infimi diſtantia à loco menſoris cognita nonſit, per quadrantem ex valle, aut <lb/>ex plano Horizontis explorare. # 77 <lb/>XXII. Diſtantiam accliuem montis à loco menſoris vſque ad baſem altitudinis <lb/>monti impoſitæ, {et}iam non viſam, vna cum ipſa altitudine, quando menſor in aſcenſu <lb/>montis conſiſtit, prope verum beneficio quadrantis efficere cognitam. # 79 <lb/>XXIII. Profunditatem putei, vel ædificii cuiuſcunque ad perpendiculum erecti, <pb file="013" n="13" rhead="INDEX."/> ſi modo angul{us} fundi, vel ſignum aliquod in fundo poſitum conſpiciatur, per quadran-<lb/>tem reperire. # 80 <lb/>XXIV. Profunditatem vallis, eiuſdemque deſcenſum obliquum, ſi non ſit valdè <lb/>inæqualis, eiuſque termin{us}, vel aliquod in ea ſignum conſpici poſſit, per quadrantem. <lb/>ſcrutari. # 82 <lb/></note> </div> <div xml:id="echoid-div9" type="section" level="1" n="9"> <head xml:id="echoid-head11" xml:space="preserve">TERTII LIBRI PROBLEMATA.</head> <note style="it" position="right" xml:space="preserve"> <lb/>Quadrati Geometrici conſtructio. # 84 <lb/>I. Altitudinem Solis, vel ſtellæ cuiuſuis per quadratum Geometricum obſer-<lb/>uare. # 87 <lb/>Tabula Gnomonica. # 91 <lb/>II. Diſtantiam interte, & ſignum quodcunque in plano Horizontis poſitum, per <lb/>quadratum perueſtigare. # 96 <lb/>Eandem beneficio baculi, vel arundinis cognoſcere. # 100 <lb/>III. Diſtantiam in plano per du{as} ſtation{es} in eodem plano fact{as}, per quadr atum <lb/>m{et}iri, quando in ei{us} extremo erecta eſt altitudo aliqua perpendicularis, {et}iam ſi infi-<lb/>mum ei{us} extremum non cernatur. # 100 <lb/>IIII. Diſtantiam eandem per du{as} ſtation{es} in aliqua altitudine erecta fact{as}, ope <lb/>quadrati perſcrutari. # 103 <lb/>V. Altitudinem cuiuslibet rei erectæ per ei{us} diſtantiam ab oculo menſoris, benefi-<lb/>cio quadrati coniicere. # 106 <lb/>VI. Altitudinem eandem, {et}iamſi ei{us} diſtantia ab oculo menſoris neque data <lb/>ſit, neque inuenta, per du{as} ſtation{es} in plano fact{as}, auxilio quadrati patefa-<lb/>cere. # 107 <lb/>VII. Altitudinem eandem, quando diſtantia ab oculo menſoris neque data est, <lb/>neque inuenta, neque è directo altitudinis duæ ſtation{es} fieri poſſunt, per du{as} ſt ation{es} in <lb/>haſt a aliqua erecta fact{as}, per quadratum indagare. # 111 <lb/><emph style="sc">Scholivm.</emph> Eandem altitudinem, eiuſque diſtantiam ab oculo menſo-<lb/>ris, vna cum hypotenuſa ab oculo ad faſtigium altitudinis extenſa, ope qua-<lb/>drati ſtabilis per vnicam ſtationem venari, etiamſi ſolum faſtigium rei erectæ <lb/>cernatur adeò vt ſcholium hoc omnia præſtet, quæ in problem. 3. 4. 5. 6. & 7. per <lb/>plures ſtationes inueſtigauimus. # 112 <lb/>VIII. Altitudinem turris, aut montis, ex ei{us} ſummitate per quadratum dim{et}i-<lb/>ri quando in plano ſummitatis Horizonti æquidiſtante duæ ſtation{es} fieri poſſunt, & <lb/>ſignum aliquodin Horizonte cernitur. # 114 <lb/>IX. Altitudinem turris, vel montis, ex ei{us} ſummitate per du{as} ſtation{es} in ha-<lb/>ſta aliquaerecta fact{as} per quadratum inueſtigare, quando ſignum aliquodin Horizon-<lb/>te poſitum videri potest. # 116 <lb/><emph style="sc">Scholivm.</emph> Eandem altitudinem ex eius vertice per vnicam ſtationem, <lb/>vna cum diſtantia à turre vel perpendiculo montis ad ſignum in Horizonte po-<lb/>ſitum, per quadratum ſtabile metiri. # 117 <lb/>X. Ex ſummitate turris, vel aliqua ei{us} feneſtra, diſtantiam à baſe turris ad ſi-<lb/>gnum propoſitum in Horizonte per quadratum cognoſcere. # 119 <lb/>XI Ex altitudinis alicui{us} faſtigio, {et}iamſi altitudo ſit menſoris ſtatura, diſtan- <pb file="014" n="14" rhead="INDEX."/> tiam inter duo ſigna in plano, cui altitudo inſiſtit, ſi ea diſtantia è directo menſoris <lb/>iaceat, & vtrumque ei{us} extremum cerni poſſit, per quadratum compreben-<lb/>dere. # 121 <lb/>XII. Longitudinem in Horizonte extenſam, per quadratum metiri, quando men-<lb/>ſor in vno ei{us} extremo exiſtens alterum extremum videre non potest, propter tumo-<lb/>rem aliquem interiectum, neque altitudo in promptu est, ſed ſolum ad dextram, vel ſi-<lb/>niſtram per lineam perpendicularem recedere potest adlocum, è quo alterum extremum <lb/>appare{at}<unsure/>. # 121 <lb/>XIII. Longitudinem in Horizonte è directo menſoris iacentem, per quadratum. <lb/>cognoſcere, ad cui{us} extrema neque accedere liceat, neque è loco menſoris eam d@meti-<lb/>ri, neque vlla adſit altitudo, dummodo ad dextram, vel ſiniſtram per lineam perpen-<lb/>dicularem ad locum aliquem ire poſſit menſor, ex quo vtrumque extremum appa-<lb/>reat. # 122 <lb/>XIV. Altitudinem montis, vel turris, ex ei{us}faſtigio, quando è directo menſoris in-<lb/>teruallum aliquod inter duo ſigna, vel {et}iam inter ſignum quodpi@m, ac turrim cogni-<lb/>tum est, per quadratum coniicere. # 122 <lb/>XV. Diſtantiam ab oculo, vel pede menſoris, (vbicunque exiſtat,) ad quoduis <lb/>punctum in aliqua altitudine notatum, per quadratum exquirere. # 123 <lb/>XVI. Interuallum inter duo ſigna, vel punctain quolib{et} plano, ſiue recto ad Hori-<lb/>Zontem, ſiue inclinato, per quadratum metiri. # 126 <lb/>XVII. Interuallum tranſuerſum in Horizonte, cui{us} vtrumque extremum videri <lb/>potest, per quadratum metiri. # 127 <lb/>XVIII. Diſtantiam alicui{us} ſigni in Horizonte poſiti à ſummitate turris, vel muri<unsure/> <lb/>alicui{us}, licet ad ipſum ſignum acceſſ{us} non pateat, per quadr atum eruere, vbicunque <lb/>menſor exiſtat. # 128 <lb/>XIX. Altitudinem inacceſſibilem, cui{us} baſis non videatur, & ad quam per nul-<lb/>lum ſpatium ſecundum rectam lineam accedere poßit menſor aut recedere, vt duæ ſta-<lb/>tion{es} fieri poßint, ſed ſolum ad dextr am, ſiniſtramue ad locum, è quo ei{us} baſis cernatur, <lb/>per quadratum explorare. # 128 <lb/>XX. Altitudinem maiorem ex minori cognita, etiamſi ſolum maioris altitudinis <lb/>vertex cernatur, per quadratum efficere notam. # 129 <lb/>XXI. Altitudinem maiorem ex minori incognita, ſi tamen baſis maioris cerni poſ-<lb/>ſit, per quadratum venari. # 130 <lb/>XXII. Altitudinem minorem ex maiori cognita, licet baſis minoris cerni non poſ-<lb/>ſit, per quadratum ſc<unsure/>rutari. # 130 <lb/>XXIII. Altitudinem minorem ex maiori incognita, dummodo baſis minoris ap-<lb/>pareat, per quadratum elicere. # 130 <lb/>XXIV. Portionem altitudinis maioris ex minore altitudine, & minoris portionem <lb/>ex maiore, per quadratum percipere. # 131 <lb/>XXV. Altitudinem, cui{us} baſis impoſita ſit monti, vel alteri cuipiam altitudini, & <lb/>vtraque ill{us} extremit{as} cerni poßit, etiamſi infimum punctum alteri{us}, cuiimponi-<lb/>tur, lateat, & eiuſdem puncti infimi diſtantia à loco menſoris cognita non ſit, per quadra-<lb/>tum ex valle, aut ex plano Horizontis explorare. # 131 <lb/>XXVI. Diſtantiam accliuem montis à loco menſoris vſ ad baſem altitudinis mon-<lb/>ti impoſitæ, etiam non viſam, vnà cum ipſa altitudine, quando menſor in aſcenſu mon-<lb/>tis conſiſtit, propè verum, beneficio quadrati efficere cognitam. # 132 <pb file="015" n="15" rhead="INDEX."/> XXVII. Profunditatem putei, vel ædific<unsure/>ii cuiuſuis ad perpendiculum erecti, ß <lb/>modo angul{us} fundi, vel ſignum aliquod in fundo poſitum conſpiciatur, per quadratum. <lb/>efficere notam. # 134 <lb/>XXVIII. Profunditatem vallis, eiuſdemque deſcenſum obliquum, ſi non ſit valdè <lb/>inæqualis, & ei{us} termin{us}, vel aliquod in ea ſignum conſpici poſſit, per quadratum. <lb/>cognoſcere. # 135 <lb/>XXIX. Diſtantiam inter ped{es} menſoris, & ſignum aliquod in plano Hori-<lb/>Zontis, beneficio baculi m{et}iri, quando extrem{us} termin{us} diſtantiæ videri po-<lb/>test, # 137 <lb/>XXX. Altitudinem turris, aut alteri{us} rei, per baculum indagare. # 137 <lb/>XXXI. Diſtantiam in plano Horizontis inter menſorem, & ſignum quoduis, be-<lb/>neficio Normæ adinuenire. # 138 <lb/>XXXII. Altitudinem turris, aut alteri{us} rei, per Normam inueſtigare. # 139 <lb/>XXXIII. Diſtantiam in plano Horizontis, quæ non ſit valdè magna, alio modo <lb/>facill@mo dimetiri # 139 <lb/>XXXIV. Altitudinem cuiuſque rei erectæ ex ei{us} vmbra, quam, Sole lucente, <lb/>proiicit, ſi nota fuerit, per quadratum deprehendere. # 140 <lb/>XXXV. Longitudinem vmbræ ab altitudine, Sole lucente, proiectæ, quando alti-<lb/>tudo est cognita ope quadrati apertam, & manifeſtam facere. # 141 <lb/>XXXVI. Diſtantiam in Horizonte inter menſorem, & ſignum aliquod viſum. <lb/>beneficio ſimpliciſſimi cuiuſdam inſtrumenti comperire. # 141 <lb/>XXXVII. Diſtantiam inter duo montium, autturrium cacumina, ope prædicti in-<lb/>ſtrumenti coniicere. # 142 <lb/>XXXVIII. Longitudinem aſcenſ{us} alicui{us} montis, ſi ei{us} cacumen ab oculo in <lb/>radice conſtituto videatur, eiuſdem inſtrumenti beneficio cognoſcere. # 143 <lb/>XXXIX. Altitudinem, ad cui{us} baſem pateat acceſſ{us}, beneficio ſpeculi plani, vna <lb/>cum d@ſtantia ſpeculi a cacumine altitudinis deprehendere. # 144 <lb/>XL. Altitudinem inacceßibilem beneficio ſpeculi plani, vnà cum ſpeculi diſtantia <lb/>tam a baſe, etiam non viſa, quam a cacumine ali@tudinis, cognoſcere. # 145 <lb/>XLI. Altitudinem monti impoſitam, ſi modo altitudinis baſis poßit conſpici; vel por-<lb/>tionem ſuperiorem alicui{us} turris, beneficio ſpeculi plani efficere nota<unsure/>m. # 147 <lb/>XLII. Situm cuiuſlibet campi, aut atru, vel templi, vel {et}iam vrbis, aut regionis cu-<lb/>iuſuis in plano deſcribere, ſi è duob{us} locis intra ipſum ſitum aſſumptis baculi ex omnib{us} <lb/>campi angulis erecti, vel certè ipſi anguli in ædificio, aut vrbe, vel loc@regionis videri poſ-<lb/>ſint: ſimulque latitudin{es} laterum campi, vel ædificii, nec non diſtanti{as} inter angu-<lb/>los, & vtrumuis locorum aſſumptorum, in data menſura cognoſcere. Quod ſi ta<unsure/>-<lb/>lia duo loca intra ſitum eligi nequeant, idem efficere, dummodo ſitum poſſim{us} cir-<lb/>cumire. # 147 <lb/>XLIII. Longitudinem trabis ad Horizontem inclinatæ, cui{us} portio ſuperior tan-<lb/>tum c<unsure/>onſpiciatur, vna cum angulo inclinationis, diſtantia baſis à menſore, & altitudine <lb/>faſtigii ſupra Horizontem, per quadratum metiri. # 151 <lb/>XLIV. Viſis duarum turrium ſummitatib{us}, {et}iamſi baſ{es} propter ædificia interie-<lb/>cta occultentur, d@ſtantiam tam inter earum baſ{es}, quam inter earundem faſtigia, vnà <lb/>cum ipſarum altitudinib{us}, ac diſt antiis à menſor@ coniicere. # 151 <lb/><emph style="sc">Scholivm.</emph> Vnica regula ad omnes rectas dimetiendas quando earum <lb/>extrema videntur. # 152 <pb file="016" n="16" rhead="INDEX."/> XLV. Spatium terræ inæquale pro ducendis aquis librare: aut {et}iam, ſilubet, Ho-<lb/>rizonti æquidiſtans eff<unsure/>icere. # 153 <lb/></note> </div> <div xml:id="echoid-div10" type="section" level="1" n="10"> <head xml:id="echoid-head12" xml:space="preserve">QVARTI LIBRI CAPITA.</head> <note position="right" xml:space="preserve"> <lb/>I. De area Rectangulorum. # 157 <lb/>II. De area Triangulorum. # 158 <lb/>III. De area Quadrilaterorum non rectangulorum. # 169 <lb/>IIII. De area multilaterarum figurarum irregularium. # 171 <lb/>V. De area multilaterarum figurarum regularium. # 175 <lb/>VI. De dimenſio<unsure/>ne circuli ex Archimede. # 181 <lb/><emph style="sc">Propositio</emph> I. Area cuiuslibet circuli æqualis eſt triangulo, cuius vnum <lb/>quidemla@us circa angulum rectum ſemidiametrum circuli, alterum verò pe-<lb/>ripheriæ eiuſdem circuli æquale eſt. # 182 <lb/><emph style="sc">Iosephi<unsure/></emph> Scaligeri error hoc in loco. # 184 <lb/><emph style="sc">Propositio</emph> II. Cuiuslibet circuli peripheria tripla eſt diametri, & ad-<lb/>huc ſuperat parte, quæ quidem minor eſt decem ſeptuageſimis, hoc eſt, ſeptima <lb/>parte diametri, maior verò decem ſeptuageſimis primis. # 185 <lb/><emph style="sc">Corollarivm.</emph> Diameter per 3 {1/7}. multiplicata gignit numerum maio-<lb/>rem circumferentia: multiplicata verò per 3 {10/71}. facit numerum circumferen-<lb/>tia minorem. E contrario circumferentia diuiſa per 3 {1/7}. procreat numerum mi-<lb/>norem diametro: diuiſa verò per 3 {10/71}. producit numerum diametro maio-<lb/>rem. # 191 <lb/><emph style="sc">Propositio</emph> III. Circulus quilibet ad quadratum diametri proportio-<lb/>nem habet, quam 11. ad 14. proximè. # 191 <lb/>VII. De area circuli, in@entioneque circumferentiæ ex diam{et}ro & diam{et}ri ex <lb/>circumferentia. # 192 <lb/><emph style="sc">Propositio</emph> I. Circulorum diametri inter ſe ſunt, vt circumferentiæ. # 194 <lb/><emph style="sc">Propositio</emph> II. Proportio quadrati ex diametro cuiuslibet circuli deſcri-<lb/>pti ad circuli aream maior eſt, quam 14. ad 11. minor autem, quam 284. <lb/>ad 223. # 195 <lb/><emph style="sc">Propositio</emph> III. Proportio quadrati à circumferentia circuli cuiuſuis <lb/>deſcripti ad circuli aream maior eſt, quam 892. ad 71. minor autem, quam 88. <lb/>ad 7. # 196 <lb/>VIII. De area ſegmentorum circuli. # 199 <lb/>I. Data circuliarea, circumferentiam, ac diametrum cognoſcere. # 201 <lb/>II. Dato arcu cuiuſuis circuli, diametrũ circuli in numeris inueſtigare. # 201 <lb/>III. Datis diametris duorum circulorum, vel circumferentiis: Aut duobus <lb/>lateribus homologis duarum figurarum ſ<unsure/>imilium, ſimiliter que poſitarum; quam <lb/>proportionem circuli, vel figuræ inter ſe habeant, cognoſcere. # 201 <lb/>IIII. Datis pluribus circulis, quorum diametri, vel circumferentiæ cognitæ <lb/>ſint: Item pluribus figuris ſimilibus, ſimiliter que poſitis, quarum latera homo-<lb/>loga ſint nota; inuenire diametrum, vel circumferentiam, cuius circulus omni-<lb/>bus circulis propoſitis æqualis ſit: Item latus reperire, cuius figura ſimilis, ſimi-<lb/>literque poſita, æqualis ſit omnibus propoſitis figuris. # 202 <pb file="017" n="17" rhead="INDEX."/> V. Aream propoſitæ Ellipſis indagare. # 202 <lb/>VI. Aream propoſitæ Parabolæ inueſtigare. # 203 <lb/></note> </div> <div xml:id="echoid-div11" type="section" level="1" n="11"> <head xml:id="echoid-head13" xml:space="preserve">QVINTI LIBRI CAPITA.</head> <note position="right" xml:space="preserve"> <lb/>I. De area Parallelepipedorum, Priſmatum, & Cylindrorum. # 204 <lb/>II. De area Pyramidum, & Conorum. # 206 <lb/>III. De Area fruſti Pyramidis, & Coni. # 207 <lb/>SCHOLIVM. De area variorum ſolidorum. # 209 <lb/>IV. De Area quinque corporum Regularium. # 210 <lb/>V. De Area ſphæræ, inuentio<unsure/>neque ſuperficiei conuexæ eiuſdem ſphæræ. # 218 <lb/><emph style="sc">Propositio</emph> I. Quam proportionem habent duæ quælibet partes ali-<lb/>quotæ magnitudinis cuiuſcunque, eandem habent duæ ſimiles partes alterius <lb/>cuiuſuis magnitudinis. # 218 <lb/><emph style="sc">Propositio</emph> II. Rectangulum ſub diametro, & circumferentia maxi-<lb/>mi circuli in ſphæra comprehenſum, quadruplum eſt circuli maximi, & ſuper-<lb/>ficiei conuexæ eiuſdem ſphæræ æquale. # 219 <lb/><emph style="sc">Propositio</emph> III. Eadem eſt proportio quadrati circumferentiæ circu-<lb/>limaximi in ſphæra ad ſuperficiem ſphæræ, quæ circumferentiæ maximi circuli <lb/>ad diametrum. Item eadem eſt proportio quadrati diametri maximi circuli in <lb/>ſphæra ad ſuperficiem ſphæræ, quæ diametri ad circumferentiam eiuſdem circu-<lb/>li maximi. # 220 <lb/><emph style="sc">Propositio</emph> IV. Quadratum circumferentiæ circuli maximi in ſphæra <lb/>ad ſuperficiem ſphæræ conuexam, maiorem proportionem habet, quam 223. ad <lb/>71. minorem vero, quam 22. ad 7. # 221 <lb/><emph style="sc">Propositio</emph> V. Quadratum diametri circuli in ſphæra maximi ad ſu-<lb/>perficiem ſphæræ conuexam, maiorem proportionem habet, quam 7. ad 22. <lb/>minorem vero, quam 71. ad 223. # 221 <lb/><emph style="sc">Propositio</emph> VI. Proportio cubi ex circumferentia maximi in ſphæra <lb/>circuli deſcripti ad ſphæram, maior eſt, quam 298374. ad 5041. minor autem, <lb/>quam 2904. ad 49. # 221 <lb/><emph style="sc">Propositio</emph> VII. Cubus diametri ſphæræ ad ſphæram, maiorem pro-<lb/>portionem habet,<unsure/> quam 21. ad 11. minorem verò, quam 426. ad 223. # 222 <lb/>VI. De Area ſegmentorum ſphæræ. # 229 <lb/>VII. De Area ſphæroidis, eiuſdemque portionum. # 232 <lb/>VIII. De Area Conoidis Parabolici. # 232 <lb/>IX. De Area Conoidis Hyperbolici. # 233 <lb/>X. De Area Doli<unsure/>orum. # 233 <lb/>XI. De Area corporum omnino regularium. # 234 <lb/>XII. De ſuperficie conuexa coni, & cylindri recti. # 235 <lb/></note> <pb file="018" n="18" rhead="INDEX."/> </div> <div xml:id="echoid-div12" type="section" level="1" n="12"> <head xml:id="echoid-head14" xml:space="preserve">SEXTI LIBRI PROPOSITIONES.</head> <note style="it" position="right" xml:space="preserve"> <lb/>I. Si magnitudo in quotuis part{es} ſec{et}ur vtcunque, & alia quæpiam magnitudo in <lb/>totidem part{es} or dine illis proportional{es}; habebunt quotlib{et} part{es} prioris magnitudi-<lb/>nis ſimul ad reliqu{as} omn{es} part{es} ſimul, eandem proportionen<unsure/>s, quam totidem part{es} po-<lb/>ſterioris magnitudinis ſimul, ad reliqu{as} omn{es} part{es} ſimul. Et ſi quælib{et} pars prio-<lb/>ris magnitudinis ſec{et}ur in du{as} part{es} vto<unsure/>unque, ſecetur autem & pars poſterioris ma-<lb/>gnitudinis illi parti reſpondens in ali{as} du{as} part{es} duab{us} illis proportional{es}; erunt quo-<lb/>que ibidem totæ magnitudin{es} ſectæ proportionaliter. # 237 <lb/>II. Dato rectilineo, ſuper datam rectam inter ali{as} du{as} interceptam, conſtituere <lb/>quadrilaterum æquale, cui{us} lat{us} oppoſitum inter du{as} eaſdem rect{as} interceptum, datæ <lb/>rectæ ſit parallelum. Et datis duob{us} rectilineis inæqualib{us} quibuſcunque, ex ma-<lb/>iore per lineam vni lateri parallelam detrahere rectilineum minori æquale, quando id <lb/>fieri poteſt. quod ex ipſa problematis ſolutione cognoſcetur. # 239 <lb/>III. Diui<unsure/>ſo rectilineo quolib{et} in triangula ex vno aliquo puncto; rect{as} line{as} ipſis <lb/>triangulis ordine proportional{es} inuenire. # 246 <lb/>IV. Datum rectilineum per rectam à quouis angulo, vel puncto in aliquo latere du-<lb/>ctam in proportionem datam diuidere: ita vt antecedens proportionis in quam malueris <lb/>partem verg{at}. # 248 <lb/>SCHOLIVM. Datum rectilineum ex dato angulo, vel puncto in latere, <lb/>in quotuis partes æquales ſecare. # 252 <lb/>V. Datum rectilineum per rectam lineam datæ rectæ parallelam, in datam propor-<lb/>tionem diuidere: ita vt antecedens proportionis in quam elegeris partem verg{at}. # 253 <lb/>SCHOLIVM. Datum rectilineum in quotuis partes æquales per lineas <lb/>cuilibet rectæ parallelas diſtribuere. # 260 <lb/>VI. Si duo triangula æquæ<unsure/>lia habeant vnum lat{us} commune, & in diuerſ{as} part{es} <lb/>vergant: Recta oppoſitos angulos connectens a latere illo communi bifariam ſecatur. # 260 <lb/>VII. Si in triangulo baſi parallela ducatur, & extrema parallelarum rectis iun-<lb/>gantur ſeſeinterſecantib{us}: Habebit vtriuſuis harum rectarum ſegmentum ab angu-<lb/>lo incipiens ad reliquum in latere terminatum, eandem proportionem, quam lat{us} ab il-<lb/>la recta diuiſum ad partem ei{us} ſuperiorem. Recta autem ex tertio angulo per interſe-<lb/>ctionem dictarum rectarum extenſa ſecabit vtramque parallelam bifariam. # 261 <lb/>VIII. Si in triangulo à duob{us} angulis duærectæ ducantur ad media puncta oppoſi-<lb/>torum laterum: Recta ex angulo reliquo per interſectionem earum deducta ſecat quo-<lb/>que reliquum lat{us} bifariam. Cui{us}lib{et} autem illarum trium linearum ſegmentum <lb/>prope angulum ad reliquum ſegmentum duplam hab{et} proportionem. Triangulum de-<lb/>nique per rect{as} ab interſectione ad angulos duct{as} in tria triangula æqualia diuiditur. <lb/># 261 <lb/>IX. Si in triangulo ducatur recta vtcunque duo latera ſecans: Erit totum trian-<lb/>gulum ad abſciſſum triangulum, vt rectangulum ſub duob{us} laterib{us} ſectis toti{us} trian-<unsure/> <lb/>guli comprehenſum, ad rectangulum ſub duob{us} laterib{us} trianguli abſciſſi, quæ prio-<lb/>rum ſegmenta ſunt, comprehenſum. # 262 <pb file="019" n="19" rhead="INDEX."/> X. Datum triangulum ex dato puncto in ei{us} latere in quotlib{et} part{es} æqual{es} di-<lb/>uidere. # 262 <lb/>XI. Datum triangulum per line{as} vni lateri parallel{as} in quotlib{et} æqual{es} part{es} <lb/>diuidere. # 263 <lb/>XII. Datum triangulum per rectam ex puncto extra triangulum dato ductam in <lb/>du{as} part{es} æqual{es} diuidere. # 264 <lb/>XIII. Datum par allelogrammum in quotcunque part{es} æqual{es} per line{as} duob{us} <lb/>laterib{us} oppoſitis æquidiſtant{es} diuidere. # 265 <lb/>XIV. Datum parallelogrammum per rectam ex puncto ſiue extra, ſiue intra ipſum, <lb/>ſiue in aliquo latere dato ductam bifariam diuidere. # 265 <lb/>XV. Inter du{as} rect{as}, du{as} medi{as} proportional{es}, prope verum, inuenire: ex He-<lb/>rone & Apollonio Pergæo: ex Philone Byſantio, ac Philopono: ex Diocle: ac poſtre-<lb/>mo ex Nicomede per lineam conc<unsure/>hoideos. # 266. vſque ad 272. <lb/>XVI. Datam figuram planam, vel circulum augere, vel minuere in data propor-<lb/>tione. # 272 <lb/>XVII. Datam figuram ſolidam qualemcunque ex ijs, de quib{us} Eucl. in libris Ste-<lb/>reometriæ agi<unsure/>t, augere, vel minuere in proportione data. # 273 <lb/>XVIII. Inter duos numeros datos tum vnum, tum duos medios proportional{es} <lb/>reperire. # 274 <lb/>LEMMA. Si ſint quatuor lineæ continuè proportionales: parallelepipe-<lb/>dum ſub quadrato alterutrius extremarum, & altera extrema comprehenſum, <lb/>æquale eſt cubo mediæ proportionalis, quæ priori extremæ propinquior eſt. <lb/># 275 <lb/>XIX. Radicem cui{us} lib{et} generis extro<unsure/>bere. # 276 <lb/>EXTRACTIO radicis quadratæ. # 279 <lb/>EXTRACTIO radicis cubicæ. # 280 <lb/>EXTRACTIO radicis ſurdeſolidæ. # 281 <lb/>REGVLA propria extra ctionis radicis cubicæ. # 283 <lb/>XX. In numeris non quadratis, non cubis, non zenſizenſis, non ſurdeſolidis, & c. <lb/>radicem veræ propinquam inuenire. # 284 <lb/>XXI. Radicem cuiuſque generis ex data minutia extrahere. # 286 <lb/>XXII. Radicem quadratam, & cubicam in numeris non quadratis, & non cubi-<lb/>cis per line{as} Geometricè inuenire. # 289 <lb/></note> </div> <div xml:id="echoid-div13" type="section" level="1" n="13"> <head xml:id="echoid-head15" xml:space="preserve">SEP TIMI LIBRI <lb/>Propoſitiones.</head> <note style="it" position="right" xml:space="preserve"> <lb/>I. Area cui{us} lib{et} trianguli æqualis est rectangulo comprehenſo ſub perpendicula-<lb/>ria<unsure/> vertice ad baſem protracta, & dimidiaparte baſis. Item rectangulo comprehenſo <lb/>ſub ſemiſſe perpendicularis, & tota baſe. Ve<unsure/>ldenique ſemiſſirectanguli<unsure/> ſub tota perpen-<lb/>diculari, & tota baſe comprehenſi. # 292 <pb file="020" n="20" rhead="INDEX."/> II. Area cul{us}lib{et} figuræ regularis æqualis est rectangulo contento ſub perpendicu-<lb/>lari à centro figuræ ad vnum lat{us} ducta, & ſub dimidiato ambitu eiuſdem figuræ. # 293 <lb/>III. Area cui{us}libet figuræ regularis æqualis eſt triangulo rectangulo, cui{us} vnum <lb/>lat{us} circa angulum rectum æquale eſt perpendiculari à centro figuræ ad vnum lat{us} <lb/>ductæ, alterum verò æquale ambitui eiuſdem figuræ. # 294 <lb/>IV. Area cui{us}libet circuli æqualis eſt rectangulo comprehenſo ſub ſemidiametro, <lb/>& dimidiata circumferentia circuli. # 294 <lb/>V. In omni triangulo rectangulo, ſi ab vno acutorum angulorum vtcumque adlat{us} <lb/>oppoſitum linearecta ducatur, erit maior proportio hui{us} lateris adei{us} ſegmentum, quod <lb/>prope angulum rectum exiſtit, quam anguli acuti prædicti ad ei{us} partem dicto ſegmen-<lb/>to lateris oppoſitam. # 295 <lb/>VI. Iſoperimetrarum figurarum regularium maior eſt illa, quæ plur{es} continet an-<lb/>gulos, plurave latera. # 296 <lb/>VII. Propoſito triangulo, cui{us} duo latera ſint inæqualia, ſupra reliquum lat{us} tri-<lb/>angulum priori Iſoperimetrum, ac duo habens latera æqualia, deſcribere. # 297 <lb/>VIII. Duorum triangulorum Iſoperimetrorum eandem habentium baſem, quorum <lb/>vni{us} duo latera ſint æqualia, alteri{us} verò inæqualia; mai{us} erit illud, cui{us} duo latera <lb/>æqualia ſunt. # 297 <lb/>IX. In ſimilib{us} triangulis rectangulis quadratum à laterib{us}, quæ angulis rectis ſub-<lb/>tenduntur, tanquam ab vna linea, deſcriptum, æquale eſt quadratis duob{us} ſimul, quæ <lb/>à reliquis homologis laterib{us}, tanquam ex duab{us} lineis, ita vt quæli<unsure/>bet duo latera ho-<lb/>mologa conficiant vnam lineam rectam, deſcribuntur. # 298 <lb/>X. Datis duob{us} triangulis Iſoſcelib{us}, quorum baſ{es} inæqual{es} exiſtant, duoque late-<lb/>ra vni{us} æqualia ſint duob{us} laterib{us} alteri{us}; ſuper eiſdem baſib{us} duo alia triangula <lb/>Iſoſcelia inter ſe quidem ſimilia, priorib{us} verò ſimul ſumptis Iſoperimetra ſimul ſum-<lb/>pta, conſtituere. # 299 <lb/>XI. Duo triangula Iſoſcelia ſimilia ſuper inæqualib{us} baſib{us} conſtit{us} conſtituta, vtraque ſi-<lb/>mul maiora ſunt duob{us} triangulis Iſoſcelib{us}, vtriſque ſimul, quæ habeant eaſdem ba-<lb/>ſ{es} cum priorib{us}, ſintque diſſimilia quidem inter ſe, at Iſoperimetra priorib{us} duob{us}, <lb/>necnon quatuor latera inter ſe habeant æqualia. # 300 <lb/>XII. Iſoperimetrarum figurarum latera numero æqualia habentium maxima & æ-<lb/>quilatera eſt, & æquiangula. # 303 <lb/>XIII. Circul{us} omnib{us} figuris rectilineis regularib{us} ſibi Iſoperimetris maior eſt. <lb/># 306 <lb/><emph style="sc">Corollarivm.</emph> Circulus ablolute omnium figurarum rectilinearum ſi-<lb/>bi Iſoperimetrarum maximus eſt. # 306 <lb/>XIV. Area cui{us}libet pyramidis æqualis est ſolido rectangulo contento ſubperpen-<lb/>diculari à vertice ad baſem protracta & tertia parte baſis. # 307 <lb/>XV. Area cui{us}li<unsure/>b{et} corporis planis ſuperficieb{us} contenti, & circa ſphæram aliquã <lb/>circumſcriptibilis, hoc eſt, à cui{us} puncto aliquo medio omn{es} perpendicular{es} ad ei{us} ba-<lb/>ſ{es} produc@æ ſunt æqual{es}, æqualis est ſolido rectangulo contento ſub vna perpendiculariũ, <lb/>& tertia parte ambit{us} corporis. # 307 <lb/>XVI. Area cui{us}lib{et} ſphæræ æqualis eſt ſolido rectangulo comprehenſo ſub ſemi-<lb/>diametro ſphæræ, & tertia parte ambit{us} ſphæræ. # 308 <lb/>XVII. Sphæra omnib{us} corporib{us} ſibi Iſoperimetris, quæ planis ſuperficieb{us} conti-<lb/>neantur, circaque ali{as} ſphær{as} circumſcriptibilia ſint, hoc eſt, quorum omn{es} perpendi- <pb file="021" n="21" rhead="INDEX."/> cular{es} ad baſ{es} productæ ab aliquo puncto medio ſint æqual{es}, maior eſt. # 310 <lb/>XVIII. Sphæra omnib{us} corporib{us} ſibi Iſoperimetris, & circa ali{as} ſphær{as} cir-<lb/>cumſcriptibilib{us}, quæ ſuperficieb{us} conicis contineantur, ita vt latera omnia conica ſint <lb/>æqualia, maior eſt. # 311 <lb/>XIX. Sphæra quolibet cono, & cylindro ſibi Iſoperimetro maior eſt. # 313 <lb/>XX. Dato ſemicirculo, vel quadranti, veloctauæ parti circuli, aut decimæ ſextæ, <lb/>&c. rectangulum conſtituere Iſoperimetrum, & æquale, ſi linea recta peripheriæ detur <lb/>æqualis. # 313 <lb/>XXI. Dato triangulo cuicunque parallelogrammum æquale, atque Iſoperime-<lb/>trum conſtituere. # 314 <lb/>XXII. Dato rectilineo parallelogrammum rectangulum æquale, & Iſoperime-<lb/>trum conſtituere. Oportet autem lat{us} quadratirectilineo æqualis mai{us} non eſſe ſemiſ@ <lb/>ſe dimidiati ambit{us} dati rectilinei. # 316 <lb/><emph style="sc">Appendix</emph> De circulo per lineas quadrando. # 317 <lb/><emph style="sc">Qvadratvra</emph> Arabum, quam Ioſephus Scaliger in ſuis cyclometricis <lb/>approbat, Alberti Dureri, & quæ Campano perperam aſcribitur, falſa eſt. # 318 <lb/><emph style="sc">Qvadratvra</emph> Hipocratis Chijper lunulas, acuta quidẽ, ſed falſa quo-<lb/>que eſt. # 318 <lb/>I. QVADRATRICEM lineam deſcribere. # 320 <lb/>COROLLARIVM. Si ex centro Quadratricis recta ducatur ſecans <lb/>quadrantem, & quadratricem: ita ſe habebit arcus quadrantis ad eius arcum <lb/>abſciſſum, vtſemidiameter ad perpendicularem ex puncto quadratricis demiſ-<lb/>ſam. # 323 <lb/>II. Siquadrantis, & quadratricisidem centrum ſit: erunt arcus quadran-<lb/>tis, ſemidiameter, & baſis quadratricis continuè proportionales. # 324 <lb/>COROLLARIVM I. Rectam reperire arcui quadrãtis, ac proinde & <lb/>ſemicircumferentiæ, immo & toti circumferentiæ æqualem. # 325 <lb/>COROLLARIVM II. Si baſis Quadratricis ſtatuatur ſemidiameter <lb/>alicuius circuli: erit eius latus quartæ parti circumferentiæ illius circuli æquale, <lb/>&c. # 326 <lb/>COROLLARIVM III. Siduæ lineæ eandem proportionem habeant, <lb/>quam latus Quadratricis, eiuſque baſis, minor autem fiat ſemidiameter alicu-<lb/>ius circuli: erit maior quartæ parti circumferentiæ illius circuli æqualis, &c. <lb/># 326 <lb/>III. Dato circulo quadratum æquale conſtituere. # 327 <lb/>FACILIS inuentio rectæ lineæ, quæ quartæ parti circumferentiæ dati cir-<lb/>culi ſit æqualis. # 327 <lb/>FACILIS inuentio quadrati dato circulo æqualis. # 328 <lb/>IV. Dato quadrato circulum æqualem deſcribere. # 329 <lb/>COROLLARIVM. Circulum cuicunque figuræ rectilineæ æqualem: <lb/>Et contra, cuicunque circulo figuram rectilineam qualemcunq; æqualem con-<lb/>ſtituere. # 329 <lb/>V. Datæ rectæ lineæ circumferentiam circulireperire æqualem. # 329 <lb/></note> <pb file="022" n="22" rhead="INDEX."/> <note position="right" xml:space="preserve"> <lb/>## OCTAVI LIBRI <lb/>## Propoſitiones. <lb/>I. Figura regularis circulo circumſcripta maiorem ambitum habet, quam <lb/>circul{us}. # 330 <lb/>LEMMA I. Si fuerint quatuor quantitates, & minor ſit exceſ-<lb/>ſus inter primam & ſecundam, quam inter tertiam & quartam, ſit-<lb/>que prima non minor, quam tertia, maior verò, quam ſecunda, itẽ <lb/>tertia maior, quam quarta: Erit minor proportio primæ quantita-<lb/>tis ad ſecundam, quam tertiæ ad quartam. # 331 <lb/>LEMMA II. Si circuli arcum duæ rectæ tangant, in vno pun-<lb/>cto coeuntes, & in eodem arcu aptentur quotlibet rectæ æquales di-<lb/>uidentes ipſum in partes totidem æquales: Erunt duæ illæ tangen-<lb/>tes omnibus hiſce chordis ſimul maiores. # 332 <lb/>LEMMA III. Si circuli arcum tres rectæ tangant, in duobus <lb/>punctis coeuntes, ita vt contactus punctum medium diuidat arcum <lb/>bifariam, in eodem autem arcu accommodentur quotlibet rectæ <lb/>numero pares, & inter ſe æquales; Erunt tres illæ tangentes omni-<lb/>bus his ſimul ſumptis maiores. # 332 <lb/>CARDANI demonſtratio figuræ regularis circulo circum-<lb/>ſcriptæ ambitum maiorem eſſe, quam circuliam bitum. # 333 <lb/>II. Circulorum diametri inter ſe ſunt, vt circumferentiæ Ex Pappo. # 334 <lb/>III. Arc{us} cuiuſuis circuli ad arcum ſimilem alteri{us} circuli eandem ha-<lb/>bet proportionem, quam chorda adchordam. Et contra, arc{us} eandem habentes <lb/>proportionem, quam chordæ, ſimiles ſunt. # 335 <lb/>IV. Dato quadrilatero æquale parallelogrammum in dato angulo, facili{us}, <lb/>quam per propoſ. 45. lib. 1. Eucl. conſtituere. # 336 <lb/>V. Dato Rectangulo ſupra datam rectam æquale rectangulum, facili{us}, <lb/>quam per propoſ. 45. lib. 1. Euclid. conſtituere. # 339 <lb/>VI. Dato rectilineo æquale rectangulum, facili{us}, quam per propoſ. 45. lib. <lb/>1. Euclid. conſtituere. # 339 <lb/>VII. Si ex duob{us} punctis ad vnum punctum cuiuſuis lineæ rectæ quæ com-<lb/>munis ſectio ſit plani per duo illa puncta ducti cum alio quopiam plano, duæ re-<lb/>ctæ ducantur facientes cum illa duos angulos æquales: Erunt duæ hæ rectæ bre-<lb/>uiores quibuſcunque alijs duab{us} rectis, quæ exijſdem duob{us} punctis ad aliud <lb/>punctum ciuſdem lineæ rectæ ducuntur. # 340 <pb file="023" n="23" rhead="INDEX."/> SCHOLIVM. Angulus incidentię apud Perſpectiuos angulo re-<lb/>flexionis æ qualis eſt. # 340 <lb/>VIII. Si quis numerum mente conceperit, quot ei vnitates post tres ope-<lb/>rationes imperat{as} reliquæ ſint, conijcere. # 341 <lb/>IX. Datum numerum quadratum in quotuis quadratos numeros par-<lb/>tiri. # 342 <lb/>X. Propoſitis duab{us} minutijs inæqualib{us}: minutia cuius numerator <lb/>ex illorum numeratorib{us}, denominator autem ex denominatorib{us} conſla-<lb/>tur, maior quidem est minore, minor vero maiore. # 343 <lb/>XI. Si duo numeri inter ſe primi non ſint ambo quadrati, aut cubi: Ne-<lb/>que eorum æquè multiplices vlli, quadrati erunt, aut cubi. Et ſi eorum æque <lb/>multiplices aliqui ſint ambo quadrati, aut cubi: etiam ipſi erunt quadrati, aut <lb/>cubi. # 343 <lb/>XII. In omni quadrilatera figura rectilinea, tria latera, vt libet, aſſum-<lb/>pta, maiora ſunt reliquo latere. # 344 <lb/>XIII. Datis trib{us} punctis, per quæ circul{us} deſcribend{us} ſit, inuenire alia <lb/>puncta, per quæ idem circul{us} tranſire debeat. # 344 <lb/>XIV. Dato exceſſu diametri Quadrati ſupra lat{us}: Item dato exceſ-<lb/>ſu diametri Rhombi ſupra lat{us}, vel lateris ſupra diametrum, vnà cum v. <lb/>no Rhombi angulo: Dato præterea exceſſu diametri Rectanguli ſupra vtrum-<lb/>libet laterum inæqualium, vnà cum angulo, quem diameter cum eo latere fa-<lb/>cit, vel vnà cum proportione eorundem inæqualium laterum: Dato denique <lb/>exceſſu diametri Rhomboidis ſupra vtrumuis laterum inæqualium, vel vtri <lb/>uſuis inæqualium laterum ſupra diametrum, vnà cum angulo Rhomboidis, <lb/>& inſuper cum angulo, quem diameter cum eo latere facit, vel inſuper cum <lb/>proportione duorum laterum inæqualium: Quadratum ipſum, Rhombum, <lb/>Rectangulum, & Rhomboides conſtituere. # 345 <lb/>XV. In rectangulo parallelogrammo ſumptis exceſſib{us}, quib{us} dia-<lb/>meter duo latera ſuperat: Rectangulum ſub differentia exceſſuum, & mi-<lb/>nore exceſſu bis ſumptum, vnà cum quadrato minoris exceſſ{us} bis ſumpto, <lb/>æquale est quadrato rectæ, qua min{us} lat{us} minorem exceſſum ſuperat. <lb/># 348. & 349 <lb/>XVI. Datis exceſſib{us}, quib{us} diameter Rectanguli vtrumque lat{us} ſu-<lb/>perat: vtrumque lat{us}, & diametrum inuenire. # 349 <lb/>XVII. Dato exceſſu diametri Rectanguli ſupra mai{us} lat{us}, & exceſſu <lb/>maioris lateris ſupra min{us}: vtrumque lat{us}, ac diametrum inuenire. # 351 <pb file="024" n="24" rhead="INDEX."/> XVIII. Secta linea recta vtcunque, adiungere ei verſ{us} vtramuis partem <lb/>lineam rectam, ita vt quadratum toti{us} rectæ compoſitæ æquale ſit quadrato re-<lb/>ctæ adiunctæ, vnà cum quadrato rectæ, quæ ex adiuncta, & proximo ſegmen-<lb/>to prioris lineæ conflatur. # 351 <lb/>XIX. Datis duab{us} rectis inæqualib{us}, quarum maior diametrum qua-<lb/>drati ex minore deſcriptinon ſuperat: Maiorem ita ſecare in du{as} partes in-<lb/>æquales, vt earum quadrata ſimul ſumpta quadrato minoris lineæ ſint æqualia. <lb/># 352 <lb/>XX. Data chorda alicui{us} arc{us}, vnà cum perpendiculari, quæ ex medio <lb/>puncto chordæ ad arcum vſque educitur: Quot grad{us}, vel palmos tam arc{us}, <lb/>quam ſemidiameter circuli complectitur, inuenire. # 353 <lb/>XXI. In omni triangulo quadratum maximi lateris min{us} eſt, quam du-<lb/>plum ſummæ quadratorum ex reliquis duob{us} laterib{us} deſcriptorum. # 353 <lb/>XXII. Datis trib{us} rectis vtcunque in plano non parallelis, niſi quando <lb/>extremæ à media æqualiter diſtant, rectam lineam ducere, & quidem per datum <lb/>punctum in media, ſi omnes tres in vno puncto conueniant, ita vt ei{us} ſegmen-<lb/>ta inter mediam, & extrem{as} ſint inter ſe æqualia, vel datam habeant propor-<lb/>tionem. # 354 <lb/>XXIII. Cui{us}libet lineæ, quamuis minimæ, exhibere multiplicem quam-<lb/>cunque, etiamſi circino non accipiatur. # 355 <lb/>XXIV. Ex qualibet lineola quamuis minima, auferre partem, vel partes <lb/>imperat{as}. # 355 <lb/>XXV. Angulum datum rectilineum in tres æquales partes partiri. # 356 <lb/>XXVI. Si per idem punctum diametri in rectangulo duæ lineæ ducantur <lb/>laterib{us} parallelæ: Erit rectangulum ſub ſegmentis diametri comprehenſum <lb/>æquale duob{us} rectangulis ſub ſegmentis duorum laterum comprehenſis. # 357 <lb/>COROLLARIVM. In quadrato rectangulum ſub ſegmentis <lb/>diametri comprehenſum, æquale eſt duobus complementis. # 357 <lb/>XXVII. Dato centro Ellipſis in linea axis in infinitum producta vnà <lb/>cum duob{us} punctis ad eaſdem partes axis, vel centri, per quæ tranſire dicatur <lb/>Ellipſis: Vtrumque axis vtriuſque extremum inuenire. # 357 <lb/>XXVIII. Si in circuli diametro producta punctum ſumatur, ab eoque re-<lb/>cta circulum tangens ducatur, à puncto autem contact{us} chorda ducatur ad dia-<lb/>metrum perpendicularis: Recta ex eodem contact{us} puncto ad vtrumlibet ex-<lb/>tremum diametri ducta diuidet angulum à tangente, & prædicta perpendicu-<lb/>lari comprehenſum bifariam. Item ſi ab eodem puncto in diametro producta <lb/>aſſumpto recta ducatur circulum ſecans, & ab alterutro ſectionis puncto ad in-<lb/>terſectionem diametri cum prædicta chorda perpendiculari recta iungatur: Re-<lb/>cta ex eoaem ſectionis puncto ad vtrumlibet diametri extremum ducta ſecabit <pb file="025" n="25" rhead="INDEX."/> quoque angulum à linea ſecante, & illa alia, quæ per interſectionem diametri <lb/>cum prædicta chorda perpendiculari ducitur, bifariam. # 258 <lb/>XXIX. Deſcriptionem pentagoni æquilateri, & æquianguli ſupra datam <lb/>rectam ab Alberto Durero traditam, & quam omnes ferè architecti, atque arti-<lb/>fices approbant, falſam eſſe, demonſtrare. # 360 <lb/>SCHOLIVM. Deſcriptionem eiuſdem pentagoniab aliis nonnul-<lb/>lis traditam, falſam quoque eſſe, demonſtrare. # 362 <lb/>XXX. Inuentionem lateris heptagoni in dato circulo non rectè à quibuſ-<lb/>dam tradi, demonſtrare. # 362 <lb/>XXXI. Octogonum æquilaterum, & æquiangulum circulo inſcriptum, me-<lb/>dio loco proportionale eſt inter quadratum eidem circulo circumſcriptum, & <lb/>quadratum inſcriptum. # 364 <lb/>XXXII. Si ex diametro quadrati detrahatur ipſi{us} lat{us}: Reliqua linea <lb/>erit lat{us} alteri{us} quadrati, cui{us} diameter eſt linea, quæ relinquitur ſi lat{us} in-<lb/>uentum bis ex diametro prioris quadrati auferatur: vel ſi idem lat{us} inuentum <lb/>ex prioris quadrati latere tollatur. # 365 <lb/>XXXIII. Octogonum æquilaterum, & æquiangulum ad datam altitudi-<lb/>nem, latitudinemue conſtituere. # 365 <lb/>XXXIV. Ambitumterræ ex edito aliquo monte metiri. # 366 <lb/>XXXV. Priſmati cuicunque cylindrum æqualem, & Pyramidi conum æ-<lb/>qualem: Ac viciſſim cylindro Priſma æquale, & cono æqualem Pyramidem <lb/>conſtituere. # 367 <lb/>XXXVI. Dato cylindro, aut Priſmati æqualem conum, vel Pyramidem <lb/>ſub eadem altitudine: Et viciſſim dato cono, vel pyramidi æqualem cylindrum, <lb/>aut Priſma eiuſdem altitudinis conſtituere. # 368 <lb/>COROLLARIVM I. Tam Cylindrum, quam Priſma, mu-<lb/>tare in Pyramidem, aut conum; Et Pyramidem tam in cylindrum, quam <lb/>in priſma æquale conuertere. # 368 <lb/>COROLLARIVM II. Cylindrum, Priſma, Conum, ac Pyrami-<lb/>dem commutare in parallelepipedum rectangulum æquale, cuius baſis <lb/>ſit quadrata. # 369 <lb/>XXXVII. Datum cylindrum, vel Priſma: Similiter datum conum, <lb/>vel pyramidem cuiuſcunque altitudinis, in æqualem ſub data qualibet <lb/>alia altitudine, & ſupra baſem quotcunque angulorum reuocare. # 369 <lb/>XXXVIII. Dato parallelepipedo rectangulo cubum æqualem deſcri-<lb/>bere. # 369 <pb file="026" n="26" rhead="INDEX."/> COROLLARIVM. Cylindro, priſmati, cono, ac pyramidi cu-<lb/>bum æqualem exhibere. # 369 <lb/>XXXIX. Dato cubo æquale parallelepipedum rectangulum ſub data alti-<lb/>tudine, vel ſupra datam baſem conſtr@ere. # 370 <lb/>COROLLARIVM. Cylindrum, priſma, conum, ac pyramidem <lb/>in parallelepipedum rectangulum æquale datæ altitudinis, vel baſis <lb/>commutare. # 370 <lb/>XL. Sphæræ datæ cubum æqualem; Et dato cubo æqualem ſphæram con-<lb/>ſtituere. # 370 <lb/>COROLLARIVM I. Sphæræ datæ ſolidum rectangulum ſupra <lb/>baſem quotlibet angulorum æquale, & pyramidem cuiuſcunque baſis <lb/>æqualem, vel etiam conum æqualem conſtruere. Et viciſsim cuilibet <lb/>priſmati ſphæram æqualem exhibere. # 371 <lb/>COROLLARIVM II. Sphæram cuilibet corpori regulari con-<lb/>ſtituere æqualem. # 371 <lb/>XLI. Duob{us}, aut plurib{us} cubis vnum cubum æqualem efficere. <lb/># 372 <lb/>SCHOLIVM. Quotlibet figuris ſolidis non cubis cubum æqua-<lb/>lem conſtruere. # 372 <lb/>XLII. Dato cubo corp{us} regulare, quod ex quinque elegeris, æqualem con-<lb/>ſtruere. # 372 <lb/>XLIII. Ex maiori cubo detrahere minorem, reſiduoque cubum æquale <lb/>exhibere. # 373 <lb/>SCHOLIVM. Ex quauis figura ſolida maiori minorem quamcun-<lb/>que auferre, reſiduoque cubum æqualem conſtituere. # 373 <lb/>XLIV. Datis duab{us}, aut plurib{us} ſphæris ſphæram vnam æqualem con-<lb/>ſtituere. # 373 <lb/>XLV. Ex maiori ſphæra minorem ſphæram detrahere, reſiduoque ſphæram <lb/>æqualem exhibere. # 373 <lb/>XLVI. Datum cubum, aut par allelepipedum, ſecundum datam proportio-<lb/>nem ſecare. # 373 <lb/>SCHOLIVM. Priſma, vel cylindrum, ſecundum datam propor-<lb/>tionem diuidere. # 373 <lb/>XLVI. Figuram Ellipſi ſimilem, quam ouatam dicunt, circino deſ@ri-<lb/>bere: Eiuſque aream, ſi beneficio trianguli æquilateri deſcripta est, explo-<lb/>rare. # 374 <pb file="027" n="27" rhead="INDEX."/> SCHOLIVM. Tabula quadratorum, & cuborum vſque ad radi-<lb/>cem 1000. # 378 <lb/>Differentiæ quadratorum, & cuborum. # 387 <lb/>Dato cubo, eiuſque radice, qui numeri impares illum componant, <lb/>inquirere. # 390 <lb/>Vſus tabulæ quadratorum, & cuborum in extrahendis radicibus <lb/>quadratis, atque cubis. # 391 <lb/></note> </div> <div xml:id="echoid-div14" type="section" level="1" n="14"> <head xml:id="echoid-head16" xml:space="preserve">FINIS.</head> <figure> <image file="027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/027-01"/> </figure> <pb file="028" n="28"/> <pb o="1." file="029" n="29"/> <figure> <image file="029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/029-01"/> </figure> </div> <div xml:id="echoid-div15" type="section" level="1" n="15"> <head xml:id="echoid-head17" xml:space="preserve">PRÆFATIO.</head> <p> <s xml:id="echoid-s110" xml:space="preserve"><emph style="sc">QVandoqvidem</emph> Mathematica-<lb/>rum diſciplinarum ſtadium ſcribendo ingreſſi, <lb/>nonnullam eius partem, fauente Deo, percurri-<lb/>mus: </s> <s xml:id="echoid-s111" xml:space="preserve">Geometriæ practicæ tractatio omittenda <lb/>non fuit, vt niſi metam tangere licuerit, ab illa cer-<lb/>tè quam minimè diſtemus. </s> <s xml:id="echoid-s112" xml:space="preserve">Ætatem ſenectute <lb/>grauem allicit operis iucunditas, laborem leuat <lb/>varietas, amicorum preces tantum non cogunt, quò mea ferebar <lb/>ſponte, ſumma cumfeſtinatione properare. </s> <s xml:id="echoid-s113" xml:space="preserve">Et verò, cum perpetua <lb/>multorum annorum experientia compererim, admodum paucos eſ-<lb/>ſe, qui non in Mathematicis exerceantur eo conſilio, vt quæ didicerint, <lb/>ad aliquem vſum trahant: </s> <s xml:id="echoid-s114" xml:space="preserve">in hoc quicquid eſt laboris veniebam ala-<lb/>cer, vt qui fructus è Mathematicis percipi poſſint ad humanæ vitæ <lb/>commoda, non inani venditatione, ſed re ipſa conſtaret. </s> <s xml:id="echoid-s115" xml:space="preserve">Etenim dum <lb/>certa ratio traditur, qua camporum longitudines, altitudines mon-<lb/>tium, vallium depreſſiones, locorum omnium inæqualitates inter ſe, <lb/>& </s> <s xml:id="echoid-s116" xml:space="preserve">interualla deprehendere metiendo debeamus: </s> <s xml:id="echoid-s117" xml:space="preserve">cuilibet liquet, vt <lb/>arbitror, quantum commodi, vtilitatiſque ſubſtructioni ædificiorum, <lb/>cultui agrorum, armorum tractationi, contemplationi ſiderum, aliiſ-<lb/>que artibus, & </s> <s xml:id="echoid-s118" xml:space="preserve">diſciplinis ex horum cognitione manare poſſit. </s> <s xml:id="echoid-s119" xml:space="preserve">Hæc e-<lb/>nim vna Mathematicarum rerum ſcientiæ pars, ſicut ab artificibus ob <lb/>ſui neceſſitatem auidè ſemper eſt arrepta: </s> <s xml:id="echoid-s120" xml:space="preserve">ita ob inſignes vtilitates, <lb/>quas in retota militari ſuppeditat, in maximorum Principum, Regum-<lb/>que aulis omni tempeſtate verſata eſt. </s> <s xml:id="echoid-s121" xml:space="preserve">Quamobrem & </s> <s xml:id="echoid-s122" xml:space="preserve">multos, & </s> <s xml:id="echoid-s123" xml:space="preserve">eru-<lb/>ditos viros habuit, qui partes illius omnes accurata, & </s> <s xml:id="echoid-s124" xml:space="preserve">diligenti ſcri-<lb/>ptione perſecuti ſunt: </s> <s xml:id="echoid-s125" xml:space="preserve">Inter quos, vt Leonhardus Piſanus, Frater Lucas <lb/>Pacciolus, Nicolaus Tartalea, Orontius, Cardanus, aliique præcipuas <lb/>obtinuerunt: </s> <s xml:id="echoid-s126" xml:space="preserve">ita eximia in cæteris laude floruerunt. </s> <s xml:id="echoid-s127" xml:space="preserve">Primas tamẽ adiu-<lb/>dicarim 10. </s> <s xml:id="echoid-s128" xml:space="preserve">Antonio Magino præſtanti Mathematico; </s> <s xml:id="echoid-s129" xml:space="preserve">qui tam etſi tantũ <lb/>lin earum dimenſiones docuit, ea tamen copia, doctrina, perſpicacita-<lb/>te cuncta tradidit, vt locum non modo iis, qui ante ſcripſerunt, ſed <lb/>ſpem poſteris æqualis gloriæ, ne dum maioris, ademiſſe videatur. </s> <s xml:id="echoid-s130" xml:space="preserve">Ve- <pb o="2" file="030" n="30"/> rum quoniam & </s> <s xml:id="echoid-s131" xml:space="preserve">hic de vnica tantum parte fuit ſollicitus: </s> <s xml:id="echoid-s132" xml:space="preserve">& </s> <s xml:id="echoid-s133" xml:space="preserve">alii, <lb/>quamuis aggreſſi omnia, multa tamen inter ſcriben dum præterierunt: <lb/></s> <s xml:id="echoid-s134" xml:space="preserve">decreui, ſi qua poſſem, perficere: </s> <s xml:id="echoid-s135" xml:space="preserve">vt, quicquid vtiliter in Geometria <lb/>practica ab aliis traditum, à me etiam inuentum eſt, vnius operis gyro <lb/>clauderetur. </s> <s xml:id="echoid-s136" xml:space="preserve">Quod opus, cum ſpecies tres quantitatis continuæ ſint, <lb/>in tria membra, parteſ{q́ue} præcipuas ſecuimus: </s> <s xml:id="echoid-s137" xml:space="preserve">In prima rectas lineas, <lb/>in altera ſuperficies, corpora metientes in poſtrema: </s> <s xml:id="echoid-s138" xml:space="preserve">cui annectuntur <lb/>alia, quæ non tam ad quantitatis dimenſionem, quam ad alias Geo-<lb/>met@iæ praxes, ac demonſtrationes pertinent, à noſtro inſtituto non <lb/>aliena.</s> <s xml:id="echoid-s139" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s140" xml:space="preserve">VNIVERSAM autem tractationem in octo libros partiti ſum{us}.</s> <s xml:id="echoid-s141" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s142" xml:space="preserve">PRIMVS propoſitiones tres omnino neceſſari{as}, & </s> <s xml:id="echoid-s143" xml:space="preserve">perquam vtiles ad <lb/>omnium magnitudinum dimenſionem accuratè perficiendam continet.</s> <s xml:id="echoid-s144" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s145" xml:space="preserve">IN Secundo dimenſio linearum rectarum per Quadrantem Aſtronomi-<lb/>cum tam pendulum, quàm ſtabilem abſoluitur.</s> <s xml:id="echoid-s146" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s147" xml:space="preserve">TERTIVS de earundem rectarum linearum dimenſione per Quadra-<lb/>tum Geometricum tum pendulum, tum ſtabile, etiam per vnicam ſtationem, <lb/>agit. </s> <s xml:id="echoid-s148" xml:space="preserve">Vbietiam, qua ratione ſine huiuſmodi inſtrumento earundem recta-<lb/>rum linearum Dimenſiones nonnullæ fieri poſſint, traditur.</s> <s xml:id="echoid-s149" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s150" xml:space="preserve">QVARTVS ſuperficierum are{as} inquirit:</s> <s xml:id="echoid-s151" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s152" xml:space="preserve">QVINTVS ſolid{as} magnitudines metitur.</s> <s xml:id="echoid-s153" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s154" xml:space="preserve">ATQVE hiſce quinque libris omnes tres partes Geometriæ practicæ à <lb/>nobis propoſitæ explicantur.</s> <s xml:id="echoid-s155" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s156" xml:space="preserve">IN Sexto deinde libro de Geodæſia, ideſt, de ſuperficierum rectiline<unsure/>arum <lb/>cuiuſque generis Diuiſione tam per rect{as} ex certo aliquo puncto duct{as}, <lb/>quam per line{as} parallel{as}, diſſeritur. </s> <s xml:id="echoid-s157" xml:space="preserve">Vbinonnulla etiam alia problemata ad <lb/>idem argumentum ſpectantia ſoluuntur. </s> <s xml:id="echoid-s158" xml:space="preserve">Item qua ratione figuræ tam pla-<lb/>næ, quam ſolidæ, vnà<unsure/> cum circulo ac ſphæra in data proportione augendæ ſint, <lb/>minuendæúe. </s> <s xml:id="echoid-s159" xml:space="preserve">In cui{us} rei gratiam modi aliquot proponuntur inuenienda-<lb/>rum duarum mediarum proportionalium inter du{as} rect{as} dat{as}. </s> <s xml:id="echoid-s160" xml:space="preserve">Denique <lb/>ars facilis, & </s> <s xml:id="echoid-s161" xml:space="preserve">expedita pro extrabendis radicib{us} cuiuſque generis præſcri-<lb/>bitur.</s> <s xml:id="echoid-s162" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s163" xml:space="preserve">SEPTIMVS de figuris Iſopemetris diſputat.</s> <s xml:id="echoid-s164" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s165" xml:space="preserve">IN Octauo deniq{ue} varia problemata, ac theoremata Geometrica per-<lb/>tractantur.</s> <s xml:id="echoid-s166" xml:space="preserve"/> </p> <pb file="031" n="31"/> <pb file="032" n="32"/> <figure> <image file="032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/032-01"/> </figure> <pb o="3" file="033" n="33"/> <figure> <image file="033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/033-01"/> </figure> </div> <div xml:id="echoid-div16" type="section" level="1" n="16"> <head xml:id="echoid-head18" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ. <lb/>LIBER PRIMVS.</head> <head xml:id="echoid-head19" xml:space="preserve">Tria capita ad dimenſionem linearum ſum-<lb/>me neceſſaria complectens.</head> <p style="it"> <s xml:id="echoid-s167" xml:space="preserve">VT magnitudinum dimenſio omnib{us} ſuis numeris ab-<lb/>ſoluta, perfectaque reddatur, tria primo hoc libro dili-<lb/>genter explicanda pri{us} erunt. </s> <s xml:id="echoid-s168" xml:space="preserve">Primum conſtruenda <lb/>est norma quædam variarum partium, quam non in-<lb/>congruè Inſtrumentum partium vocare poſſum{us}; <lb/></s> <s xml:id="echoid-s169" xml:space="preserve">quòdineo variæ partes & </s> <s xml:id="echoid-s170" xml:space="preserve">ad line{as} rect{as}, & </s> <s xml:id="echoid-s171" xml:space="preserve">ad circu-<lb/>los diuidendas, tum etiam ad ali{as} operationes ſiue <lb/>Geometric{as}, ſiue Aſtronomic{as} ritè perficiend{as} con-<lb/>tineantur. </s> <s xml:id="echoid-s172" xml:space="preserve">Hui{us} enim vſ{us} credi vix potest, quàm latè pateat tum in di-<lb/>metiendis magnitudinib{us} ſine numerorum multiplicatione, tum verò ma-<lb/>ximè in horologiis Solarib{us} ea ratione, quam per line{as} Tangentes in noua <lb/>horologiorum deſcriptione tradidimus, deſcribendis, & </s> <s xml:id="echoid-s173" xml:space="preserve">in aliis reb{us} tam <lb/>Geometricis, quam Aſtronomicis, vt ex iis, quæ capite primo hui{us} libri, & </s> <s xml:id="echoid-s174" xml:space="preserve"><lb/>alibitradituriſum{us}, perſpicuum fiet. </s> <s xml:id="echoid-s175" xml:space="preserve">Secundo loco conficiendus eſt quadrans, <lb/>in quo præter grad{us}, Minuta quoque ac Secunda (quamuis in eo deſignata <lb/>non ſint) cognoſci, ac diſcerni queant; </s> <s xml:id="echoid-s176" xml:space="preserve">docendumque, qua ratione idem præ-<lb/>ſtaripoſſit in quolibet quadrante in 90. </s> <s xml:id="echoid-s177" xml:space="preserve">grad{us} exquiſitè diſtributo: </s> <s xml:id="echoid-s178" xml:space="preserve">parteſ-<lb/>que centeſimæ, atque milleſimæ in recta quauis linea in pauciſſim{as} partes æ-<lb/>quales diuiſa dignoſcendæ ſint. </s> <s xml:id="echoid-s179" xml:space="preserve">Tertio atque poſtremo loco proponenda erunt, <lb/>acſoluenda varia problemata triangulorum recti lineorum, vt & </s> <s xml:id="echoid-s180" xml:space="preserve">latera eo-<lb/>rum, atque anguli ex quibuſdam datis, & </s> <s xml:id="echoid-s181" xml:space="preserve">cognitis facilè poſſunt cognoſci. </s> <s xml:id="echoid-s182" xml:space="preserve"><lb/>Quamuis enim eadem hæc problemata ad finem Lemmatis 53. </s> <s xml:id="echoid-s183" xml:space="preserve">lib. </s> <s xml:id="echoid-s184" xml:space="preserve">1. </s> <s xml:id="echoid-s185" xml:space="preserve">noſtri <lb/>Aſtrolabii expoſita ſint, tamen ne ſtudioſ{us} Lector adillud Lemmaſæpi{us}, & </s> <s xml:id="echoid-s186" xml:space="preserve"><lb/>non ſine moleſtia recurrere cogatur, libet ea hic repetere totidem penè ver bis, <lb/>quot in prædicto Lemmate præſcripta ſunt. </s> <s xml:id="echoid-s187" xml:space="preserve">Sedecce tria hæc, quæ <lb/>præmittenda eſſe dixim{us}, trib{us} capitib{us} explicata <lb/>ſequuntur.</s> <s xml:id="echoid-s188" xml:space="preserve"/> </p> <pb o="4" file="034" n="34" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div17" type="section" level="1" n="17"> <head xml:id="echoid-head20" xml:space="preserve">INSTRVMENTI PARTIVM <lb/>Conſtructio, atque vſus. <lb/>CAPVT I.</head> <p> <s xml:id="echoid-s189" xml:space="preserve"><emph style="sc">FIant</emph> ex orichalco, vel alia materia ſolida duæ regulæ ABD, <lb/> <anchor type="note" xlink:label="note-034-01a" xlink:href="note-034-01"/> AEC, æquales omnino, quæ in A, ita coniungantur clauo <lb/>aliquo tereti, vt circa A, vniformiter poſsint moueri, quem-<lb/>admodum in Norma vulgari, quæ, prout opus eſt, conſtrin-<lb/>gi poteſt, & </s> <s xml:id="echoid-s190" xml:space="preserve">dilatari, fieriſolet. </s> <s xml:id="echoid-s191" xml:space="preserve">Deinde ex A, in planis dicta-<lb/>rum regularum duæ rectæ ducantur AF, AG, eæquein 100. </s> <s xml:id="echoid-s192" xml:space="preserve">particulasæ-<lb/>quales diſtribuantur, velin 1000. </s> <s xml:id="echoid-s193" xml:space="preserve">ſi longiores ſint. </s> <s xml:id="echoid-s194" xml:space="preserve">Ita enim ex qualibet recta <lb/>quotuis partes centeſimæ, aut milleſimæ abſcindi poterunt. </s> <s xml:id="echoid-s195" xml:space="preserve">Immo ſi ſumatur <lb/>linea KL, continens 11. </s> <s xml:id="echoid-s196" xml:space="preserve">particulas ex illis 100. </s> <s xml:id="echoid-s197" xml:space="preserve">vel 1000. </s> <s xml:id="echoid-s198" xml:space="preserve">diuidaturq; </s> <s xml:id="echoid-s199" xml:space="preserve">in 10. <lb/></s> <s xml:id="echoid-s200" xml:space="preserve">partesæquales, ſi quidem ſecta ſit vtraque regulain 100. </s> <s xml:id="echoid-s201" xml:space="preserve">partes æquales, po-<lb/>terunt beneficio rectæ KL, continentis 11. </s> <s xml:id="echoid-s202" xml:space="preserve">particulas eiuſmodi, & </s> <s xml:id="echoid-s203" xml:space="preserve">in 10. </s> <s xml:id="echoid-s204" xml:space="preserve">par-<lb/>tes æquales diuiſæ, ex data recta qualibet accipi quotuis milleſimæ partes, <lb/>perinde ac ſi partes ſingulæ centeſimæ in vtraque regula ſectæ eſſent in de-<lb/>nas particulas æquales: </s> <s xml:id="echoid-s205" xml:space="preserve">ſi vero vtraqueregula in 1000. </s> <s xml:id="echoid-s206" xml:space="preserve">particulas diſtributa <lb/>ſit, & </s> <s xml:id="echoid-s207" xml:space="preserve">linea KL, talium 11. </s> <s xml:id="echoid-s208" xml:space="preserve">partium in 10. </s> <s xml:id="echoid-s209" xml:space="preserve">particulas diſſecta, poterunt ex qua-<lb/>uis linea recta propoſita partes, quot quis voluerit, milleſimarum decimæ <lb/>auferri, non ſecus ac ſi ſingulæ partes milleſimæ in regula diſtributæ eſſent in <lb/>10. </s> <s xml:id="echoid-s210" xml:space="preserve">particulas æquales, vt in vſu inſtrumenti dicemus.</s> <s xml:id="echoid-s211" xml:space="preserve"/> </p> <div xml:id="echoid-div17" type="float" level="2" n="1"> <note position="left" xlink:label="note-034-01" xlink:href="note-034-01a" xml:space="preserve">Inſtrumentũ <lb/>partium quo <lb/>pacto cõſtrua-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s212" xml:space="preserve"><emph style="sc">Rvrsvs</emph> ſiregula contineat 100. </s> <s xml:id="echoid-s213" xml:space="preserve">partes, & </s> <s xml:id="echoid-s214" xml:space="preserve">recta quæpiam MN, con-<lb/>ſtans ex 101. </s> <s xml:id="echoid-s215" xml:space="preserve">eiuſmodi particulis diſtribuatur in 100. </s> <s xml:id="echoid-s216" xml:space="preserve">partes, poterimus ex <lb/>quauis data recta accipere partes decimas milleſimarum. </s> <s xml:id="echoid-s217" xml:space="preserve">At ſi inregula no-<lb/>tatæ ſint 1000. </s> <s xml:id="echoid-s218" xml:space="preserve">partes, & </s> <s xml:id="echoid-s219" xml:space="preserve">linea quæpiam continens eiuſmo dipartes 101. </s> <s xml:id="echoid-s220" xml:space="preserve">ſece-<lb/>tur in 100. </s> <s xml:id="echoid-s221" xml:space="preserve">partes, deprehendipoterunt in qualibet recta quotcunque par-<lb/>tes centeſimæ milleſimarum, ac ſi partes ſingulæ milleſimæ in regula comple-<lb/>cterentur partes 100. </s> <s xml:id="echoid-s222" xml:space="preserve">Si denique linea earum partium 1001. </s> <s xml:id="echoid-s223" xml:space="preserve">diuidatur in <lb/>1000. </s> <s xml:id="echoid-s224" xml:space="preserve">partes, capiemus in quauis recta partes milleſimas milleſimarum, per-<lb/>inde ac ſi partes milleſimę ſingulæ in regula partes 1000. </s> <s xml:id="echoid-s225" xml:space="preserve">comprehende-<lb/>rent, vt ex vſu inſtrumenticonſtabit. </s> <s xml:id="echoid-s226" xml:space="preserve">Atque hæc eſt conſtructio inſtrumenti <lb/>in vna facie pro partibus linearum rectarum inquirendis.</s> <s xml:id="echoid-s227" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s228" xml:space="preserve"><emph style="sc">In</emph> altera vero inſtrumenti facie deſignantur chordæ omnium arcuum <lb/>quadrantis hoc modo. </s> <s xml:id="echoid-s229" xml:space="preserve">Ductis ex centro A, rectis AF, AG, vt in priori fa-<lb/>cie, ſumendus eſt quadrans circuli chordam habens æqualem rectæ, AF, & </s> <s xml:id="echoid-s230" xml:space="preserve"><lb/>in rectas AF, AG, transferenda chorda gradus 1. </s> <s xml:id="echoid-s231" xml:space="preserve">illius quadrantis, deinde <lb/>chorda grad. </s> <s xml:id="echoid-s232" xml:space="preserve">2. </s> <s xml:id="echoid-s233" xml:space="preserve">3. </s> <s xml:id="echoid-s234" xml:space="preserve">4. </s> <s xml:id="echoid-s235" xml:space="preserve">5. </s> <s xml:id="echoid-s236" xml:space="preserve">& </s> <s xml:id="echoid-s237" xml:space="preserve">ſic deinceps vſque ad chordam 89. </s> <s xml:id="echoid-s238" xml:space="preserve">graduum: </s> <s xml:id="echoid-s239" xml:space="preserve">ita <lb/>enim ex quolibet quadrante abſcindere licebit arcum quotcunque gra-<lb/>duum, vt Num. </s> <s xml:id="echoid-s240" xml:space="preserve">16. </s> <s xml:id="echoid-s241" xml:space="preserve">dicetur: </s> <s xml:id="echoid-s242" xml:space="preserve">quamuis nos beneficio particularum æ qualium <lb/>in priori facie poſitarũ capiemus ex quadrante propoſito non ſolum gradus <lb/>integros, ſed etiam minuta, quod Num. </s> <s xml:id="echoid-s243" xml:space="preserve">14. </s> <s xml:id="echoid-s244" xml:space="preserve">docebimus. </s> <s xml:id="echoid-s245" xml:space="preserve">Atq; </s> <s xml:id="echoid-s246" xml:space="preserve">ita abſoluta eſt <lb/>conſtru ctio inſtrumenti in altera facie. </s> <s xml:id="echoid-s247" xml:space="preserve">Huius inſtrumenti vſus ampliſsimus <lb/>eſt, vt diximus, & </s> <s xml:id="echoid-s248" xml:space="preserve">non obſcurè exiis, quæ ſequuntur, intelligipoteſt.</s> <s xml:id="echoid-s249" xml:space="preserve"/> </p> <pb o="5" file="035" n="35" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s250" xml:space="preserve">1. </s> <s xml:id="echoid-s251" xml:space="preserve"><emph style="sc">Qvando</emph> enim linea recta propoſita rectæ AF, vel AG, in priorifa-<lb/> <anchor type="note" xlink:label="note-035-01a" xlink:href="note-035-01"/> cie inſtrumenti æqualis eſt, nullo negotio ex ea abſcindẽtur quotcunq; </s> <s xml:id="echoid-s252" xml:space="preserve">par-<lb/>tes centeſimæ, aut milleſi-<lb/> <anchor type="figure" xlink:label="fig-035-01a" xlink:href="fig-035-01"/> mæ, prout inſtrumentum in <lb/>100. </s> <s xml:id="echoid-s253" xml:space="preserve">aut 1000. </s> <s xml:id="echoid-s254" xml:space="preserve">partes fuerit <lb/>diuiſum; </s> <s xml:id="echoid-s255" xml:space="preserve">ſi nimirum partes, <lb/>quæ deſiderantur, exinſtru-<lb/>mento in datam rectam <lb/>transferantur.</s> <s xml:id="echoid-s256" xml:space="preserve"/> </p> <div xml:id="echoid-div18" type="float" level="2" n="2"> <note position="right" xlink:label="note-035-01" xlink:href="note-035-01a" xml:space="preserve">Centeſimæ, <lb/>vel milleſimæ <lb/>part{es} in recta <lb/>linea quo mo-<lb/>do accipian-<lb/>tur.</note> <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/xxxxxxxx/figures/035-01"/> </figure> </div> <p> <s xml:id="echoid-s257" xml:space="preserve">2. </s> <s xml:id="echoid-s258" xml:space="preserve"><emph style="sc">Qvand</emph> doero pro-<lb/>poſita linea non eſt æqualis <lb/>rectæ AF, vel AG, in inſtru-<lb/>mento, dilatandum inſtru-<lb/>mentum eſt, vel conſtrin-<lb/>gendum, donec interual-<lb/>lum inter F, G, datę lineę ſit <lb/>æquale. </s> <s xml:id="echoid-s259" xml:space="preserve">Nam circinus inter <lb/>partes rectarum AF, AG, <lb/>quę deſiderantur, extenſus <lb/>dabit in recta propoſita <lb/>partes quęſitas. </s> <s xml:id="echoid-s260" xml:space="preserve">Vt ſi data <lb/>recta æqualis ſitipſi FG, & </s> <s xml:id="echoid-s261" xml:space="preserve"><lb/>deſiderẽtur 50 partes cen-<lb/>teſimæ, continebit interual-<lb/>lum inter partes 50. </s> <s xml:id="echoid-s262" xml:space="preserve">& </s> <s xml:id="echoid-s263" xml:space="preserve">50. <lb/></s> <s xml:id="echoid-s264" xml:space="preserve">in lineis AF, AG, partes 50. </s> <s xml:id="echoid-s265" xml:space="preserve"><lb/>ex 100. </s> <s xml:id="echoid-s266" xml:space="preserve">in quas recta FG, <lb/>cogitatur eſſe diuiſa. </s> <s xml:id="echoid-s267" xml:space="preserve">quod <lb/>ſic demonſtratur. </s> <s xml:id="echoid-s268" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Rectæ <anchor type="note" xlink:label="note-035-02a" xlink:href="note-035-02"/> FG, & </s> <s xml:id="echoid-s269" xml:space="preserve">50. </s> <s xml:id="echoid-s270" xml:space="preserve">50. </s> <s xml:id="echoid-s271" xml:space="preserve">(ſi concipia-<lb/>tur ducta recta à parte 50. </s> <s xml:id="echoid-s272" xml:space="preserve">ad <lb/>partem 50.) </s> <s xml:id="echoid-s273" xml:space="preserve">parallelæ ſunt; </s> <s xml:id="echoid-s274" xml:space="preserve">propterea quod latera AF, AG, proportionali-<lb/>ter ſecta ſuntin 50. </s> <s xml:id="echoid-s275" xml:space="preserve">& </s> <s xml:id="echoid-s276" xml:space="preserve">50. </s> <s xml:id="echoid-s277" xml:space="preserve">Sunt enim tam AF, AG, quam A 50, A 50, æquales. <lb/></s> <s xml:id="echoid-s278" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/> Igitur erit vt A 50. </s> <s xml:id="echoid-s279" xml:space="preserve">ad rectam 50. </s> <s xml:id="echoid-s280" xml:space="preserve">50. </s> <s xml:id="echoid-s281" xml:space="preserve">ita AF, ad FG. </s> <s xml:id="echoid-s282" xml:space="preserve">Et permutando vt A 50.</s> <s xml:id="echoid-s283" xml:space="preserve"> <anchor type="note" xlink:label="note-035-03a" xlink:href="note-035-03"/> ad AF, ita recta 50. </s> <s xml:id="echoid-s284" xml:space="preserve">50. </s> <s xml:id="echoid-s285" xml:space="preserve">ad FG. </s> <s xml:id="echoid-s286" xml:space="preserve">Cum ergo A 50. </s> <s xml:id="echoid-s287" xml:space="preserve">contineat partes 50. </s> <s xml:id="echoid-s288" xml:space="preserve">ex 100. <lb/></s> <s xml:id="echoid-s289" xml:space="preserve">totius AF, continebit quoque recta 50. </s> <s xml:id="echoid-s290" xml:space="preserve">50. </s> <s xml:id="echoid-s291" xml:space="preserve">partes 50. </s> <s xml:id="echoid-s292" xml:space="preserve">ex 100. </s> <s xml:id="echoid-s293" xml:space="preserve">in quas diuiſa <lb/>eſſe concipitur FG. </s> <s xml:id="echoid-s294" xml:space="preserve">Eademque ratio eſt de cæteris. </s> <s xml:id="echoid-s295" xml:space="preserve">Nam verbi gratia inter-<lb/>uallum quo que interpuncta 80. </s> <s xml:id="echoid-s296" xml:space="preserve">& </s> <s xml:id="echoid-s297" xml:space="preserve">80. </s> <s xml:id="echoid-s298" xml:space="preserve">partes 80. </s> <s xml:id="echoid-s299" xml:space="preserve">comple ctetur ex 100. </s> <s xml:id="echoid-s300" xml:space="preserve">totius <lb/>FG, &</s> <s xml:id="echoid-s301" xml:space="preserve">c. </s> <s xml:id="echoid-s302" xml:space="preserve">quæ demonſtratio locum etiam habet, ſi in AF, contineantur 1000. </s> <s xml:id="echoid-s303" xml:space="preserve"><lb/> <anchor type="handwritten" xlink:label="hd-035-02a" xlink:href="hd-035-02"/> partes, vt conſtat.</s> <s xml:id="echoid-s304" xml:space="preserve"/> </p> <div xml:id="echoid-div19" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-035-02" xlink:href="note-035-02a" xml:space="preserve">2. ſext.</note> <note symbol="b" position="right" xlink:label="note-035-03" xlink:href="note-035-03a" xml:space="preserve">4. ſexti.</note> <handwritten xlink:label="hd-035-02" xlink:href="hd-035-02a"/> </div> <p> <s xml:id="echoid-s305" xml:space="preserve"><emph style="sc">Non</emph> aliter, propoſitis duabus rectis, quarum altera in quotlibet partes <lb/> <anchor type="note" xlink:label="note-035-04a" xlink:href="note-035-04"/> æquales cogitetur eſſe diuiſa, cognoſcemus, quotnam ex illis partibus alte-<lb/>ra recta contineat; </s> <s xml:id="echoid-s306" xml:space="preserve">hac ſcilicet ratione. </s> <s xml:id="echoid-s307" xml:space="preserve">Aperto inſtrumento, ſtatuatur inter-<lb/>uallum rectæ diuiſæ inter partes, in quas diuiſa intelligitur. </s> <s xml:id="echoid-s308" xml:space="preserve">Nam ſi altera per <lb/>circinum tran sferatur inter duas alias partes eaſdem, vel inter duo puncta ab <lb/>eiſdem duabus partibus æqualiter diſtantia, continebit illa recta tot partes, <pb o="6" file="036" n="36" rhead="GEOMETR. PRACT."/> quot inregula A F, includuntur inter centrum A, & </s> <s xml:id="echoid-s309" xml:space="preserve">circinipedem: </s> <s xml:id="echoid-s310" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>propte- <anchor type="note" xlink:label="note-036-01a" xlink:href="note-036-01"/> rea quod eandem proportionem habet ſegmentum regulæ A F, vſque ad in-<lb/>teruallum rectæ diuiſæ, ad ſegmentum ejuſdem regulæ vſque adinteruallum <lb/>alterius lineę, quam interuallum rectę diuiſæ ad interuallum alterius lineę ha-<lb/>bet, &</s> <s xml:id="echoid-s311" xml:space="preserve">c. </s> <s xml:id="echoid-s312" xml:space="preserve">Quod ſi linea hæc altera eſſet nimis longa, auferendũ ex ea primum <lb/> <anchor type="handwritten" xlink:label="hd-036-01a" xlink:href="hd-036-01"/> eſſet interuallum inter 100. </s> <s xml:id="echoid-s313" xml:space="preserve">& </s> <s xml:id="echoid-s314" xml:space="preserve">100. </s> <s xml:id="echoid-s315" xml:space="preserve">quoties fieri poteſt. </s> <s xml:id="echoid-s316" xml:space="preserve">Deinde reliquum ſe-<lb/>gmentum transferendum in inſtrumentum, vt dictum eſt. </s> <s xml:id="echoid-s317" xml:space="preserve">Verbi gratia ſi alte-<lb/>rarectarum diuiſa ſit in 50. </s> <s xml:id="echoid-s318" xml:space="preserve">partes æquales, ſumemus ei æquale interuallum <lb/>inter 50. </s> <s xml:id="echoid-s319" xml:space="preserve">& </s> <s xml:id="echoid-s320" xml:space="preserve">50. </s> <s xml:id="echoid-s321" xml:space="preserve">Si ergo altera habuerit interuallum æquale rectę F G, inter <lb/>100. </s> <s xml:id="echoid-s322" xml:space="preserve">& </s> <s xml:id="echoid-s323" xml:space="preserve">100. </s> <s xml:id="echoid-s324" xml:space="preserve">continebit ea 100. </s> <s xml:id="echoid-s325" xml:space="preserve">partes ęquales. </s> <s xml:id="echoid-s326" xml:space="preserve">Et ſi in ea ſupereſſet ſegmen-<lb/>tum ęquale interuallo 30.</s> <s xml:id="echoid-s327" xml:space="preserve">30. </s> <s xml:id="echoid-s328" xml:space="preserve">contineret eadem recta partes 130. </s> <s xml:id="echoid-s329" xml:space="preserve">Quod ſi in-<lb/>teruallum inter 100. </s> <s xml:id="echoid-s330" xml:space="preserve">& </s> <s xml:id="echoid-s331" xml:space="preserve">100. </s> <s xml:id="echoid-s332" xml:space="preserve">terin data recta contineretur. </s> <s xml:id="echoid-s333" xml:space="preserve">& </s> <s xml:id="echoid-s334" xml:space="preserve">inſuper ſegmẽ-<lb/> <anchor type="note" xlink:label="note-036-02a" xlink:href="note-036-02"/> tum ęquale interuallo 40.</s> <s xml:id="echoid-s335" xml:space="preserve">40. </s> <s xml:id="echoid-s336" xml:space="preserve">complecteretur ea recta particulas 340. </s> <s xml:id="echoid-s337" xml:space="preserve">&</s> <s xml:id="echoid-s338" xml:space="preserve">c.</s> <s xml:id="echoid-s339" xml:space="preserve"/> </p> <div xml:id="echoid-div20" type="float" level="2" n="4"> <note position="right" xlink:label="note-035-04" xlink:href="note-035-04a" xml:space="preserve">Diuiſa recta <lb/>in quotuis par <lb/>tes æqual{es}, <lb/>quot eiuſmo-<lb/>di part{es} in <lb/>quauis al ia <lb/>contineantur.</note> <note symbol="c" position="left" xlink:label="note-036-01" xlink:href="note-036-01a" xml:space="preserve">4. ſexti.</note> <handwritten xlink:label="hd-036-01" xlink:href="hd-036-01a"/> <note position="left" xlink:label="note-036-02" xlink:href="note-036-02a" xml:space="preserve">Tangent{es} <lb/>quo modo ac-<lb/>cipiantur re-<lb/>ſpectu ſin{us} <lb/>toti{us} 100.</note> </div> <p> <s xml:id="echoid-s340" xml:space="preserve">3. </s> <s xml:id="echoid-s341" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi in Tangentibus nouę deſcriptionis horologiorum (vt hu-<lb/>ius inſtrumenti vtilitatem quo que in deſcribendis horologijs aperiamus) ſi-<lb/>nus totus ſtatuatur 100. </s> <s xml:id="echoid-s342" xml:space="preserve">quantuſcunque ille ſit, eique interuallum F G, po-<lb/>natur ęquale, capie@@s commodiſsime quamcumque Tangentem tabulę <lb/> <anchor type="handwritten" xlink:label="hd-036-01a" xlink:href="hd-036-01"/> in noua deſcriptione p@ſitę, ſi ea, abiecta prima tantum figura ad dexteram, <lb/>minor fuerit quam 100. </s> <s xml:id="echoid-s343" xml:space="preserve">Vt ſi quęratur Tangens Grad. </s> <s xml:id="echoid-s344" xml:space="preserve">39. </s> <s xml:id="echoid-s345" xml:space="preserve">min. </s> <s xml:id="echoid-s346" xml:space="preserve">57. </s> <s xml:id="echoid-s347" xml:space="preserve">quoniam <lb/>ea in tabula eſt 838. </s> <s xml:id="echoid-s348" xml:space="preserve">ſi abijciatur prima figura 8. </s> <s xml:id="echoid-s349" xml:space="preserve">ad dexteram, erit Tangens 83. <lb/></s> <s xml:id="echoid-s350" xml:space="preserve">reſpectu ſinus totius 100. </s> <s xml:id="echoid-s351" xml:space="preserve">vel potius 84. </s> <s xml:id="echoid-s352" xml:space="preserve">propterea quod figura 8. </s> <s xml:id="echoid-s353" xml:space="preserve">abiecta <lb/>maior eſt, quam 5. </s> <s xml:id="echoid-s354" xml:space="preserve">ac proinde pro ea vnitas adijcienda eſt, cum conſtituat <lb/>{8/1<unsure/>0} hoc eſt, pluſquam @. </s> <s xml:id="echoid-s355" xml:space="preserve">Itaque ſi accipiantur in inſtrumento partes 84. </s> <s xml:id="echoid-s356" xml:space="preserve">pau-<lb/>lo minus, vt dictum eſt, habebitur Tangens quęſita: </s> <s xml:id="echoid-s357" xml:space="preserve">ſi vero Tangens in ta-<lb/>bula, abjecta prima figura ad dexteram, maior fuerit quam 100. </s> <s xml:id="echoid-s358" xml:space="preserve">accipienda <lb/>eſt Tangens per denas, at que vnitates expreſſa, relictis centenis, & </s> <s xml:id="echoid-s359" xml:space="preserve">illi Tan-<lb/>genti poſtea ſinus totus adij ciendus eſt toties, quoties vnitas in centenis re-<lb/>peritur. </s> <s xml:id="echoid-s360" xml:space="preserve">Vt ſi quis velit Tangentem Grad. </s> <s xml:id="echoid-s361" xml:space="preserve">68. </s> <s xml:id="echoid-s362" xml:space="preserve">min. </s> <s xml:id="echoid-s363" xml:space="preserve">50. </s> <s xml:id="echoid-s364" xml:space="preserve">quoniam ea in tabula <lb/>eſt 2583. </s> <s xml:id="echoid-s365" xml:space="preserve">& </s> <s xml:id="echoid-s366" xml:space="preserve">abiecta prima figura, 258. </s> <s xml:id="echoid-s367" xml:space="preserve">fumenda eſt Tangens 58. </s> <s xml:id="echoid-s368" xml:space="preserve">eiq; </s> <s xml:id="echoid-s369" xml:space="preserve">ſinus to-<lb/>tus F G, bis adij ciendus, & </s> <s xml:id="echoid-s370" xml:space="preserve">ſic de reliquis.</s> <s xml:id="echoid-s371" xml:space="preserve"/> </p> <div xml:id="echoid-div21" type="float" level="2" n="5"> <handwritten xlink:label="hd-036-01" xlink:href="hd-036-01a"/> </div> <p> <s xml:id="echoid-s372" xml:space="preserve">4. </s> <s xml:id="echoid-s373" xml:space="preserve"><emph style="sc">Qvod</emph> ſi rectam K L, partium 11. </s> <s xml:id="echoid-s374" xml:space="preserve">in 10. </s> <s xml:id="echoid-s375" xml:space="preserve">ęquales partes diuiſam adhi-<lb/> <anchor type="note" xlink:label="note-036-03a" xlink:href="note-036-03"/> bere velimus accipere poterimus ex data recta partes milleſimas. </s> <s xml:id="echoid-s376" xml:space="preserve">Quoniam <lb/>enim ita ſe habet linea K L, ad vnam eius partem, vt portio A 10. </s> <s xml:id="echoid-s377" xml:space="preserve">rectę A F, <lb/>decem partium ad vnam, cum vtrobique proportio ſit decupla; </s> <s xml:id="echoid-s378" xml:space="preserve">erit permu-<lb/>tando quo que K L, ad A 10. </s> <s xml:id="echoid-s379" xml:space="preserve">vt vna particulaipſius K L, ad vnam particulam <lb/>ipſius A 10. </s> <s xml:id="echoid-s380" xml:space="preserve">Cum ergo K L, contineat ipſam A 10. </s> <s xml:id="echoid-s381" xml:space="preserve">ſemel, & </s> <s xml:id="echoid-s382" xml:space="preserve">inſuper partem <lb/>ipſius decimam, (ſumpta eſt enim K L, partiũ 11. </s> <s xml:id="echoid-s383" xml:space="preserve">qualium 10. </s> <s xml:id="echoid-s384" xml:space="preserve">eſt A 10.) </s> <s xml:id="echoid-s385" xml:space="preserve">con-<lb/>tinebit quo que vna particula ipſius K L, vnam particulam ipſius A 10. </s> <s xml:id="echoid-s386" xml:space="preserve">ſemel, <lb/>& </s> <s xml:id="echoid-s387" xml:space="preserve">decimam inſuper eius partem: </s> <s xml:id="echoid-s388" xml:space="preserve">Atque adeo duę illius includent duas hu-<lb/>ius cum {2/10}. </s> <s xml:id="echoid-s389" xml:space="preserve">& </s> <s xml:id="echoid-s390" xml:space="preserve">tres continebunt tres cum {3/10}. </s> <s xml:id="echoid-s391" xml:space="preserve">& </s> <s xml:id="echoid-s392" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s393" xml:space="preserve">Quare ſi verbi <lb/>gratia deſiderentur {87/1000}. </s> <s xml:id="echoid-s394" xml:space="preserve">accipiendę erunt octo partes ex 100. </s> <s xml:id="echoid-s395" xml:space="preserve">totius A F, & </s> <s xml:id="echoid-s396" xml:space="preserve"><lb/>inſuper {7/10}. </s> <s xml:id="echoid-s397" xml:space="preserve">ſequentis partis nonę, cum decima pars vnius centeſimę ſit {1/1000}. <lb/></s> <s xml:id="echoid-s398" xml:space="preserve"> <anchor type="handwritten" xlink:label="hd-036-01a" xlink:href="hd-036-01"/> quodita fiet. </s> <s xml:id="echoid-s399" xml:space="preserve">Circino aliquo ſumantur 7. </s> <s xml:id="echoid-s400" xml:space="preserve">partes ex KL, eęque in A F, transfe-<lb/>rantur ex quavis<unsure/> parte. </s> <s xml:id="echoid-s401" xml:space="preserve">Nam pes circinimobilis auferet {7/10}. </s> <s xml:id="echoid-s402" xml:space="preserve">ex octaua parte <lb/>poſt pedem circini immobilem, quę particula in nonam partem eſt transfe-<lb/>renda. </s> <s xml:id="echoid-s403" xml:space="preserve">Ita enim, cum octo partes complectantur {80/10}. </s> <s xml:id="echoid-s404" xml:space="preserve">vnius centeſimę, (quod <pb o="7" file="037" n="37" rhead="LIBER PRIMVS."/> quælibet pars contineat {10/10}@ vnius centeſimæ.) </s> <s xml:id="echoid-s405" xml:space="preserve">hoc eſt, {80/1000}. </s> <s xml:id="echoid-s406" xml:space="preserve">& </s> <s xml:id="echoid-s407" xml:space="preserve">{7/10}. </s> <s xml:id="echoid-s408" xml:space="preserve">vnius cen-<lb/>teſimæ contineant {7/1000}. </s> <s xml:id="echoid-s409" xml:space="preserve">comprehendet tota linea abſciſſa {87/1000}. </s> <s xml:id="echoid-s410" xml:space="preserve">Item ſi quis <lb/>cupiat {17/1000}. </s> <s xml:id="echoid-s411" xml:space="preserve">accipienda erit vna pá<unsure/>rs ipſius A F, & </s> <s xml:id="echoid-s412" xml:space="preserve">{7/10}. </s> <s xml:id="echoid-s413" xml:space="preserve">ſequentis partis ſe-<lb/>cundæ, vt paulo ante dictum eſt. </s> <s xml:id="echoid-s414" xml:space="preserve">Vel ſic agemus. </s> <s xml:id="echoid-s415" xml:space="preserve">Sumptis 7. </s> <s xml:id="echoid-s416" xml:space="preserve">partibus ex <lb/>K L, transferemus easin A F. </s> <s xml:id="echoid-s417" xml:space="preserve">vbicunque libuerit. </s> <s xml:id="echoid-s418" xml:space="preserve">Nam abſciſſæ partes erunt <lb/>7 {7/17}. </s> <s xml:id="echoid-s419" xml:space="preserve">Vna ergo pars cum {7/10}. </s> <s xml:id="echoid-s420" xml:space="preserve">continebit {17/1000}. </s> <s xml:id="echoid-s421" xml:space="preserve">Denique ſi optentur {457/1000}. </s> <s xml:id="echoid-s422" xml:space="preserve">ſu-<lb/>mendæ erunt exrecta A F, partes 45. </s> <s xml:id="echoid-s423" xml:space="preserve">cum hæ æquiualeant {450/1000}. </s> <s xml:id="echoid-s424" xml:space="preserve">Deinde {7/10}. <lb/></s> <s xml:id="echoid-s425" xml:space="preserve">ex ſequenti parte quadrageſima ſexta, beneficio ſeptem particularum rectæ <lb/>K L, &</s> <s xml:id="echoid-s426" xml:space="preserve">c.</s> <s xml:id="echoid-s427" xml:space="preserve"/> </p> <div xml:id="echoid-div22" type="float" level="2" n="6"> <note position="left" xlink:label="note-036-03" xlink:href="note-036-03a" xml:space="preserve">Milleſimæ <lb/>part{es} quo mo <lb/>do capiantur, <lb/>etiamſi in in-<lb/>ſtrumento cõ-<lb/>tineantur tã-<lb/>tum part{es} <lb/>100.</note> <handwritten xlink:label="hd-036-01" xlink:href="hd-036-01a"/> </div> <p> <s xml:id="echoid-s428" xml:space="preserve">5. </s> <s xml:id="echoid-s429" xml:space="preserve"><emph style="sc">Si</emph> ergo ſinus totus ponatur 1000. </s> <s xml:id="echoid-s430" xml:space="preserve">habebuntur Tangentes, vt in ta-<lb/>bula nouæ deſcriptionis horologiorum poſitæ ſunt, nulla figura abiecta. </s> <s xml:id="echoid-s431" xml:space="preserve">Sed <lb/> <anchor type="note" xlink:label="note-037-01a" xlink:href="note-037-01"/> quando Tangens maior eſt quam 1000. </s> <s xml:id="echoid-s432" xml:space="preserve">relictis millenis, accipendæ ſuntre-<lb/>liquæ partes milleſimæ pro Tangente, eiq; </s> <s xml:id="echoid-s433" xml:space="preserve">toties ſinus totus addendus, quo-<lb/> <anchor type="handwritten" xlink:label="hd-037-01a" xlink:href="hd-037-01"/> ties vnitas in millenis relictis reperitur. </s> <s xml:id="echoid-s434" xml:space="preserve">Vt ſi quis velit Tangentem Grad. </s> <s xml:id="echoid-s435" xml:space="preserve">40. <lb/></s> <s xml:id="echoid-s436" xml:space="preserve">min. </s> <s xml:id="echoid-s437" xml:space="preserve">30. </s> <s xml:id="echoid-s438" xml:space="preserve">quæ in tabula eſt 854. </s> <s xml:id="echoid-s439" xml:space="preserve">accipiendæ ſunt in regula A F, partes 85. </s> <s xml:id="echoid-s440" xml:space="preserve">& </s> <s xml:id="echoid-s441" xml:space="preserve"><lb/>{4/10}. </s> <s xml:id="echoid-s442" xml:space="preserve">vnius. </s> <s xml:id="echoid-s443" xml:space="preserve">Ita enim Tangens continebit partes 854. </s> <s xml:id="echoid-s444" xml:space="preserve">ex 1000. </s> <s xml:id="echoid-s445" xml:space="preserve">At ſi quæratur <lb/>Tangens Grad. </s> <s xml:id="echoid-s446" xml:space="preserve">80. </s> <s xml:id="echoid-s447" xml:space="preserve">min. </s> <s xml:id="echoid-s448" xml:space="preserve">0. </s> <s xml:id="echoid-s449" xml:space="preserve">quæ in tabula eſt 5671. </s> <s xml:id="echoid-s450" xml:space="preserve">relictis millenis accipien-<lb/>dæ ſunt partes 67. </s> <s xml:id="echoid-s451" xml:space="preserve">& </s> <s xml:id="echoid-s452" xml:space="preserve">{1/10}. </s> <s xml:id="echoid-s453" xml:space="preserve">vnius partis, & </s> <s xml:id="echoid-s454" xml:space="preserve">Tangenti 671. </s> <s xml:id="echoid-s455" xml:space="preserve">addendus ſinus to-<lb/>tus quinquies.</s> <s xml:id="echoid-s456" xml:space="preserve"/> </p> <div xml:id="echoid-div23" type="float" level="2" n="7"> <note position="right" xlink:label="note-037-01" xlink:href="note-037-01a" xml:space="preserve">Tangentes <lb/>quo modo in-<lb/>inueniantur, <lb/>poſito ſinu toto <lb/>100.</note> <handwritten xlink:label="hd-037-01" xlink:href="hd-037-01a"/> </div> <p> <s xml:id="echoid-s457" xml:space="preserve">6. </s> <s xml:id="echoid-s458" xml:space="preserve"><emph style="sc">Pari</emph> ratione ſi adhibeatur recta M N, partium 101. </s> <s xml:id="echoid-s459" xml:space="preserve">diuiſa in 100. </s> <s xml:id="echoid-s460" xml:space="preserve">de-<lb/> <anchor type="note" xlink:label="note-037-02a" xlink:href="note-037-02"/> promemus ex recta A F, partes decimas milleſimarum; </s> <s xml:id="echoid-s461" xml:space="preserve">cum quælibet parti-<lb/>cula rectæ M N, contineat vnam particulam ipſius A F, ſemel, & </s> <s xml:id="echoid-s462" xml:space="preserve">inſuper <lb/>{1/100}. </s> <s xml:id="echoid-s463" xml:space="preserve">Ita vt quælibet particula rectæ A F, diuiſa eſſe cogitetur in 100. </s> <s xml:id="echoid-s464" xml:space="preserve">particu-<lb/>las; </s> <s xml:id="echoid-s465" xml:space="preserve">ac proinde tota A F, ſit 10000. </s> <s xml:id="echoid-s466" xml:space="preserve">particularum, quod eodẽ modo demon-<lb/>ſtrabitur. </s> <s xml:id="echoid-s467" xml:space="preserve">Eſt enim eadem proportio M N, ad vnam particulam ſuam cente-<lb/>ſimam, quæ rectæ A F, ad vnam ſuam centeſimam, &</s> <s xml:id="echoid-s468" xml:space="preserve">c.</s> <s xml:id="echoid-s469" xml:space="preserve"/> </p> <div xml:id="echoid-div24" type="float" level="2" n="8"> <note position="right" xlink:label="note-037-02" xlink:href="note-037-02a" xml:space="preserve">Decimæ par-<lb/>tes milleſima-<lb/>rum quo mo-<lb/>do ſumantur, <lb/>etiam ſi inſtru <lb/>mentumdiui-<lb/>ſum ſit in 100. <lb/>part{es} dunta-<lb/>x{at}.</note> </div> <p> <s xml:id="echoid-s470" xml:space="preserve">7. </s> <s xml:id="echoid-s471" xml:space="preserve"><emph style="sc">A@qve</emph> hac ratione haberi poterunt Tangentes, poſito ſinu toto <lb/>10000. </s> <s xml:id="echoid-s472" xml:space="preserve">abiectis nimirum tribus figuris primis ex Tangentibus tabulæ in no-<lb/>ſtro Theodoſio deſcriptæ. </s> <s xml:id="echoid-s473" xml:space="preserve">Vtſi velimus Tangentem Grad. </s> <s xml:id="echoid-s474" xml:space="preserve">78. </s> <s xml:id="echoid-s475" xml:space="preserve">Min. </s> <s xml:id="echoid-s476" xml:space="preserve">30. </s> <s xml:id="echoid-s477" xml:space="preserve">quæ <lb/>in tabula (abiectis tribus figuris) eſt 49151. </s> <s xml:id="echoid-s478" xml:space="preserve">relictis denis millenarum, acci-<lb/> <anchor type="note" xlink:label="note-037-03a" xlink:href="note-037-03"/> piemus partes 9151. </s> <s xml:id="echoid-s479" xml:space="preserve">nimirum partes 91. </s> <s xml:id="echoid-s480" xml:space="preserve">ex 100. </s> <s xml:id="echoid-s481" xml:space="preserve">regulæ A F, & </s> <s xml:id="echoid-s482" xml:space="preserve">{51/100}. </s> <s xml:id="echoid-s483" xml:space="preserve">vnius. <lb/></s> <s xml:id="echoid-s484" xml:space="preserve">quod fiet, ſi partes 51. </s> <s xml:id="echoid-s485" xml:space="preserve">rectæ M N, transferantur in A F. </s> <s xml:id="echoid-s486" xml:space="preserve">Circinus enim vltra <lb/>partes 51. </s> <s xml:id="echoid-s487" xml:space="preserve">abſcindet {51/100}. </s> <s xml:id="echoid-s488" xml:space="preserve">Nam quia ſingulæ particulæ rectæ A F, concipiun-<lb/>tur ſectæ in 100. </s> <s xml:id="echoid-s489" xml:space="preserve">particulas, continebuntur in 91. </s> <s xml:id="echoid-s490" xml:space="preserve">partibus particulæ 9100. </s> <s xml:id="echoid-s491" xml:space="preserve"><lb/>quibus ſi addantur 51. </s> <s xml:id="echoid-s492" xml:space="preserve">habebitur Tangens 9151. </s> <s xml:id="echoid-s493" xml:space="preserve">Huic tandem apponendus <lb/>eſt ſinus totus quater, propter quatuor denas millenarum relictas, Facile au-<lb/>tem ad 91. </s> <s xml:id="echoid-s494" xml:space="preserve">partes adijcies particulam continentem {51/160}. </s> <s xml:id="echoid-s495" xml:space="preserve">vnius centeſimæ, ſi <lb/>eam, (quæ nimirum vltra partes 51. </s> <s xml:id="echoid-s496" xml:space="preserve">regulæ A F, exiſtit) cum vna parte regu-<lb/>læ A F, transferas, vt pes circini inter partes 91. </s> <s xml:id="echoid-s497" xml:space="preserve">& </s> <s xml:id="echoid-s498" xml:space="preserve">92. </s> <s xml:id="echoid-s499" xml:space="preserve">cadat, hoc eſt, ſi eam <lb/>cum vna parte transferas ex parte 90. </s> <s xml:id="echoid-s500" xml:space="preserve">vel cum duabus partibus, ex parte 89. </s> <s xml:id="echoid-s501" xml:space="preserve"><lb/>&</s> <s xml:id="echoid-s502" xml:space="preserve">c.</s> <s xml:id="echoid-s503" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="9"> <note position="right" xlink:label="note-037-03" xlink:href="note-037-03a" xml:space="preserve">Tangent{es} p@-<lb/>ſito ſinu toto <lb/>10000 quo pa-<lb/>cto ſumãtur.</note> </div> <p> <s xml:id="echoid-s504" xml:space="preserve">8. </s> <s xml:id="echoid-s505" xml:space="preserve"><emph style="sc">Inventa</emph> porro Tangente in vtraqueregula A F, A G, dabit inter-<lb/> <anchor type="note" xlink:label="note-037-04a" xlink:href="note-037-04"/> uallum inter Tangentem regulæ A F, & </s> <s xml:id="echoid-s506" xml:space="preserve">Tangentem regulæ A G, eandem <lb/>Tangentem reſpectu ſinus totius F G.</s> <s xml:id="echoid-s507" xml:space="preserve"/> </p> <div xml:id="echoid-div26" type="float" level="2" n="10"> <note position="right" xlink:label="note-037-04" xlink:href="note-037-04a" xml:space="preserve">Qua ratione <lb/>ex inuenta <lb/>parte milleſi-<lb/>ma, vel deci-<lb/>{es} mill@-</note> </div> <p> <s xml:id="echoid-s508" xml:space="preserve">9. </s> <s xml:id="echoid-s509" xml:space="preserve"><emph style="sc">Qvando</emph> autem Tangens tam exigua eſt, vt eius interuallum prope <lb/>punctum A, accipinequeat, vtemur hoc artificio. </s> <s xml:id="echoid-s510" xml:space="preserve">Sit verbi gratia ſumenda <pb o="8" file="038" n="38" rhead="GEOMETR. PRACT."/> Tangens 7. </s> <s xml:id="echoid-s511" xml:space="preserve">partium reſpectu ſinus totius FG. </s> <s xml:id="echoid-s512" xml:space="preserve">Sumptis duabus Tangentibus <lb/> <anchor type="note" xlink:label="note-038-01a" xlink:href="note-038-01"/> majoribus, quarum maior minorem ſeptem vnitatibus ſuperet, nimirum 30. <lb/></s> <s xml:id="echoid-s513" xml:space="preserve">& </s> <s xml:id="echoid-s514" xml:space="preserve">37. </s> <s xml:id="echoid-s515" xml:space="preserve">vel 80. </s> <s xml:id="echoid-s516" xml:space="preserve">& </s> <s xml:id="echoid-s517" xml:space="preserve">87. </s> <s xml:id="echoid-s518" xml:space="preserve">&</s> <s xml:id="echoid-s519" xml:space="preserve">c. </s> <s xml:id="echoid-s520" xml:space="preserve">dabit earum differentia (ſi nimirũ vtraquein aliquam <lb/>rectam lineam transferatur) Tangentem 7. </s> <s xml:id="echoid-s521" xml:space="preserve">quæ quæritur: </s> <s xml:id="echoid-s522" xml:space="preserve">Atque ita ſemper <lb/>ſumendæ erunt duæ Tangẽtes maiores prope medium inſtrumenti quarum <lb/> <anchor type="handwritten" xlink:label="hd-038-01a" xlink:href="hd-038-01"/> differentia æqualis ſit Tangenti exiguæ propoſitæ.</s> <s xml:id="echoid-s523" xml:space="preserve"/> </p> <div xml:id="echoid-div27" type="float" level="2" n="11"> <note position="left" xlink:label="note-038-01" xlink:href="note-038-01a" xml:space="preserve">ſima in AF, <lb/>eadem reperi-<lb/>atur reſpectu <lb/>dati ſin{us} to-<lb/>ti{us}.</note> <handwritten xlink:label="hd-038-01" xlink:href="hd-038-01a"/> </div> <p> <s xml:id="echoid-s524" xml:space="preserve">10. </s> <s xml:id="echoid-s525" xml:space="preserve"><emph style="sc">Si</emph> vtraque regula AF, AG, contineat 1000. </s> <s xml:id="echoid-s526" xml:space="preserve">particulas, & </s> <s xml:id="echoid-s527" xml:space="preserve">ſinus to-<lb/>tupropoſitus conſtituatur 1000. </s> <s xml:id="echoid-s528" xml:space="preserve">eique interuallũ F G, æquale ſumatur, com-<lb/>modiſsime accipientur omnes Tangentes, vtin Tabula nouæ deſcriptionis <lb/>horologiorum poſitæ ſunt. </s> <s xml:id="echoid-s529" xml:space="preserve">Nam verbi gratia Tangens 2430. </s> <s xml:id="echoid-s530" xml:space="preserve">Grad. </s> <s xml:id="echoid-s531" xml:space="preserve">67. </s> <s xml:id="echoid-s532" xml:space="preserve">min. <lb/></s> <s xml:id="echoid-s533" xml:space="preserve">38. </s> <s xml:id="echoid-s534" xml:space="preserve">habebitur, ſi relictis millenis, ſumaturinteruallũ inter partes 430. </s> <s xml:id="echoid-s535" xml:space="preserve">vtriuſq; </s> <s xml:id="echoid-s536" xml:space="preserve"><lb/>regulæ A F, A G, eique ſinus totus F G, bis adijciatur.</s> <s xml:id="echoid-s537" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s538" xml:space="preserve">11. </s> <s xml:id="echoid-s539" xml:space="preserve"><emph style="sc">Et</emph> ſi adhibeas lineam K L, partium 11. </s> <s xml:id="echoid-s540" xml:space="preserve">diuiſam in 10. </s> <s xml:id="echoid-s541" xml:space="preserve">accipere poteris <lb/>Tangentes reſpectu ſinus totius 10000. </s> <s xml:id="echoid-s542" xml:space="preserve">Item ſi rectam partium 101. </s> <s xml:id="echoid-s543" xml:space="preserve">in 100. <lb/></s> <s xml:id="echoid-s544" xml:space="preserve">particulas diſtributam adhibeas, habebis Tangentes reſpectu ſinus totius <lb/>100000. </s> <s xml:id="echoid-s545" xml:space="preserve">Si denique rectam partium 1001. </s> <s xml:id="echoid-s546" xml:space="preserve">partiaris in 1000. </s> <s xml:id="echoid-s547" xml:space="preserve">particulas, obti-<lb/>nebis Tangentes, poſito ſinu toto 1000000. </s> <s xml:id="echoid-s548" xml:space="preserve">vt ex dictis patet.</s> <s xml:id="echoid-s549" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s550" xml:space="preserve">12. </s> <s xml:id="echoid-s551" xml:space="preserve"><emph style="sc">Qvod</emph> dictum eſt de Tangentibus, intelligendum eſt etiam de ſinu-<lb/> <anchor type="handwritten" xlink:label="hd-038-01a" xlink:href="hd-038-01"/> bus, & </s> <s xml:id="echoid-s552" xml:space="preserve">ſecantibus. </s> <s xml:id="echoid-s553" xml:space="preserve">Nam ſi interuallum F G, æquale ſit ſinuialicuitoti, ſiueis <lb/>partium ſit 100. </s> <s xml:id="echoid-s554" xml:space="preserve">ſiue 1000. </s> <s xml:id="echoid-s555" xml:space="preserve">ſiue plurium, dabunt interualla inter ſinus, vel ſe-<lb/>cantes in vtraqueregula A F, A G, acceptas, per ea, quæ Num. </s> <s xml:id="echoid-s556" xml:space="preserve">4. </s> <s xml:id="echoid-s557" xml:space="preserve">docuimus, <lb/>ſinus & </s> <s xml:id="echoid-s558" xml:space="preserve">ſecantes reſpectu ſinus totius F G, accepti; </s> <s xml:id="echoid-s559" xml:space="preserve">propterea quod per Lem-<lb/>ma 5. </s> <s xml:id="echoid-s560" xml:space="preserve">lib. </s> <s xml:id="echoid-s561" xml:space="preserve">1. </s> <s xml:id="echoid-s562" xml:space="preserve">noſtri Aſtrolabij, eandem proportionem habet ſinus totus A F, ad <lb/>ſinum totum F G, quam ſinus verbi gratia A 50. </s> <s xml:id="echoid-s563" xml:space="preserve">ad ſinum arcus circuli, cuius <lb/>ſemidiameter F G, quiarcus arcuiſinus A 50. </s> <s xml:id="echoid-s564" xml:space="preserve">ſimilis eſt. </s> <s xml:id="echoid-s565" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum ergo ſit, vt A F, <anchor type="note" xlink:label="note-038-02a" xlink:href="note-038-02"/> ad F G, ita A 50. </s> <s xml:id="echoid-s566" xml:space="preserve">ad rectam 50. </s> <s xml:id="echoid-s567" xml:space="preserve">50. </s> <s xml:id="echoid-s568" xml:space="preserve">erit recta 50. </s> <s xml:id="echoid-s569" xml:space="preserve">50. </s> <s xml:id="echoid-s570" xml:space="preserve">ſinus arcus, quiarcuiſinus <lb/>A 50. </s> <s xml:id="echoid-s571" xml:space="preserve">ſimilis eſt. </s> <s xml:id="echoid-s572" xml:space="preserve">Eademqueratio eſt de ſecantibus.</s> <s xml:id="echoid-s573" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="12"> <handwritten xlink:label="hd-038-01" xlink:href="hd-038-01a"/> <note symbol="a" position="left" xlink:label="note-038-02" xlink:href="note-038-02a" xml:space="preserve">4. ſexti.</note> </div> <note position="left" xml:space="preserve">Quo pacto co-<lb/>gnoſcatur, <lb/>quot decimæ <lb/>in particula <lb/>cuiuſuis cen-<lb/>teſimæ partis <lb/>contineantur.</note> <p> <s xml:id="echoid-s574" xml:space="preserve">13. </s> <s xml:id="echoid-s575" xml:space="preserve"><emph style="sc">Vicissim</emph> cognoſcemus, quot particulas ex 1000. </s> <s xml:id="echoid-s576" xml:space="preserve">quælibet parti-<lb/>cula vnius partis rectæ A F, complectatur: </s> <s xml:id="echoid-s577" xml:space="preserve">hoc ſcilicet modo. </s> <s xml:id="echoid-s578" xml:space="preserve">Circino ſuma-<lb/>tur data particula, vna cum vna parte centeſima, vel duabus, vel tribus, qua-<lb/>tuorue; </s> <s xml:id="echoid-s579" xml:space="preserve">circinuſque decies repetatur in recta A F, diligenterque notetur ſe-<lb/>gmentum rectæ A F, quod circinus percucurrit. </s> <s xml:id="echoid-s580" xml:space="preserve">Nam ſi ex partibus centeſimis <lb/>in eo ſegmento contentis abijciantur to ties 10. </s> <s xml:id="echoid-s581" xml:space="preserve">quot partes vna cum particu-<lb/>la data ſumptę fuerũt, reliquus numerusindicabit partes decimas vnius cen-<lb/>teſimæ, hoc eſt, milleſimas in data particula comprehenſas. </s> <s xml:id="echoid-s582" xml:space="preserve">Et ſi cumreliqua <lb/>particula eius ſegmenti (ſi qua forte ſuperſit) ſimiliter agemus, reperiemus <lb/>partes decimas vnius milleſimæ, hoc eſt, partes {1/10000}. </s> <s xml:id="echoid-s583" xml:space="preserve">Et ſi iterum operatio-<lb/>nemrepetemus, inueniemus partes {1/10000}. </s> <s xml:id="echoid-s584" xml:space="preserve">quod quidem ad finem libelli de <lb/>fabrica, & </s> <s xml:id="echoid-s585" xml:space="preserve">vſu inſtrumenti horologiorum demoſtrauimus, eademq; </s> <s xml:id="echoid-s586" xml:space="preserve">demon-<lb/>ſtrationem breuiter capite in ſequenti Num. </s> <s xml:id="echoid-s587" xml:space="preserve">14. </s> <s xml:id="echoid-s588" xml:space="preserve">repetemus. </s> <s xml:id="echoid-s589" xml:space="preserve">Exempli cauſa. </s> <s xml:id="echoid-s590" xml:space="preserve">Si <lb/>particula data cum tribus centeſimis deciesrepetita percurrat partes 37. </s> <s xml:id="echoid-s591" xml:space="preserve">abie-<lb/>ctis 30. </s> <s xml:id="echoid-s592" xml:space="preserve">continebunturin data particula {7/10}. </s> <s xml:id="echoid-s593" xml:space="preserve">vnius centeſimæ. </s> <s xml:id="echoid-s594" xml:space="preserve">Quare ſi ea par-<lb/>ticula data fuerit V. </s> <s xml:id="echoid-s595" xml:space="preserve">g. </s> <s xml:id="echoid-s596" xml:space="preserve">poſt vigeſimam partem centeſimam, continebit illud <lb/>ſegmentum rectæ A F, {207/1000}. </s> <s xml:id="echoid-s597" xml:space="preserve">Nam {7/10} vniuscenteſimæ faciunt {7/1000}. </s> <s xml:id="echoid-s598" xml:space="preserve">& </s> <s xml:id="echoid-s599" xml:space="preserve">20. </s> <s xml:id="echoid-s600" xml:space="preserve">cen-<lb/>teſimæ, ſi ſingulæ in decem particulas congitentur eſſe ſectæ, efficiunt {200/1000}. <lb/></s> <s xml:id="echoid-s601" xml:space="preserve">quippe cum omnes centum partes æqui valeant 1000. </s> <s xml:id="echoid-s602" xml:space="preserve">particulis. </s> <s xml:id="echoid-s603" xml:space="preserve">Quod ſi <pb o="9" file="039" n="39" rhead="LIBER PRIMVS."/> idem fiat cum reliqua particula (ſi qua forte ſuperfuerit poſt 37. </s> <s xml:id="echoid-s604" xml:space="preserve">partes per-<lb/>curſas) vna cum tribus centeſimis, & </s> <s xml:id="echoid-s605" xml:space="preserve">inciderimus verbi gratia in particulam <lb/>34. </s> <s xml:id="echoid-s606" xml:space="preserve">abiectis 30. </s> <s xml:id="echoid-s607" xml:space="preserve">continebit ea particula {4/10}. </s> <s xml:id="echoid-s608" xml:space="preserve">vnius milleſimæ, hoc eſt, {4/10000}. </s> <s xml:id="echoid-s609" xml:space="preserve">Et <lb/>quia {207/1000} æquiualent {2070/10000}. </s> <s xml:id="echoid-s610" xml:space="preserve">ſi addantur {4/10000}. </s> <s xml:id="echoid-s611" xml:space="preserve">vltimo loco deprehenſæ, ha-<lb/>bebimus {2074/10000}. </s> <s xml:id="echoid-s612" xml:space="preserve">Si denique cumreliqua particula (ſi qua forte remanſerit) vna <lb/>cum tribus centeſimis idem fiat, percurſæque verbi gratia ſint 39. </s> <s xml:id="echoid-s613" xml:space="preserve">partes, ab-<lb/>iectis 30. </s> <s xml:id="echoid-s614" xml:space="preserve">ſupererunt {9/10}. </s> <s xml:id="echoid-s615" xml:space="preserve">vnius partis {1/10000}. </s> <s xml:id="echoid-s616" xml:space="preserve">hoc eſt {9/100000}. </s> <s xml:id="echoid-s617" xml:space="preserve">Cum ergo {2074/10000} ef-<lb/>ficiant {20740/100000} ſi addantur {9/100000}. </s> <s xml:id="echoid-s618" xml:space="preserve">habebimus {20749/100000}. </s> <s xml:id="echoid-s619" xml:space="preserve">Atque adeo ſi recta A F, <lb/>ſtatuatur ſinus totus partium 100000. </s> <s xml:id="echoid-s620" xml:space="preserve">erit ſegmentum 20. </s> <s xml:id="echoid-s621" xml:space="preserve">partium cum par-<lb/>ticula data, ſinus partium 20749. </s> <s xml:id="echoid-s622" xml:space="preserve">Si particula data commode per cir cinum <lb/>poſsit comprehendi, & </s> <s xml:id="echoid-s623" xml:space="preserve">decies repetatur, dabunt centeſimæ partes percurſæ <lb/>partes decimas vnius centeſimæ. </s> <s xml:id="echoid-s624" xml:space="preserve">&</s> <s xml:id="echoid-s625" xml:space="preserve">c. </s> <s xml:id="echoid-s626" xml:space="preserve">Si quo que nonnunquam nulla ſuper-<lb/>ſit particula, ita vt verbi gratia inuentæ ſint præcisè {207/2000} multiplicandus erit <lb/>tam numerator, quam denominator per 100. </s> <s xml:id="echoid-s627" xml:space="preserve">vt habeatur ſinus 20700. </s> <s xml:id="echoid-s628" xml:space="preserve">reſpe-<lb/>ctu ſinus totius 100000. </s> <s xml:id="echoid-s629" xml:space="preserve">quemadmodũ & </s> <s xml:id="echoid-s630" xml:space="preserve">40. </s> <s xml:id="echoid-s631" xml:space="preserve">centeſimæ conſtituunt ſinum <lb/>40000. </s> <s xml:id="echoid-s632" xml:space="preserve">reſpectu ſinus totius 100000. </s> <s xml:id="echoid-s633" xml:space="preserve">Sinamque vterque numerus minutiæ <lb/>{40/100} ducatur in 1000. </s> <s xml:id="echoid-s634" xml:space="preserve">fiet minutia {40000/100000}.</s> <s xml:id="echoid-s635" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s636" xml:space="preserve">14. </s> <s xml:id="echoid-s637" xml:space="preserve"><emph style="sc">Hoc</emph> eodem inſtrumento, & </s> <s xml:id="echoid-s638" xml:space="preserve">in eadem facie partium æqualium, ex <lb/> <anchor type="note" xlink:label="note-039-01a" xlink:href="note-039-01"/> data qualibet circumferentia auferemus arcum quotuis graduum & </s> <s xml:id="echoid-s639" xml:space="preserve">minuto-<lb/>rum, hac arte. </s> <s xml:id="echoid-s640" xml:space="preserve">Sit ex quadrante, cuius ſemidiameter interuallo F G, æqualis <lb/>fit, abſcindẽdus verbigratia arcus grad. </s> <s xml:id="echoid-s641" xml:space="preserve">53. </s> <s xml:id="echoid-s642" xml:space="preserve">hoc eſt, chorda huius arcusinue-<lb/>nienda. </s> <s xml:id="echoid-s643" xml:space="preserve">Sumatur ex tabula ſinuum ſinus ſemiſsis propoſiti arcus, graduum <lb/>videlicet 16. </s> <s xml:id="echoid-s644" xml:space="preserve">min. </s> <s xml:id="echoid-s645" xml:space="preserve">45. </s> <s xml:id="echoid-s646" xml:space="preserve">quiſinus, abiectis quatuor figuris ad dexteram, eſt 450. <lb/></s> <s xml:id="echoid-s647" xml:space="preserve">reſpectu ſinus totius 1000. </s> <s xml:id="echoid-s648" xml:space="preserve">Siergo ſinus hicreſpectu ſinus totius A F, accipia-<lb/>turinrecta A F, vſque ad 45. </s> <s xml:id="echoid-s649" xml:space="preserve">perea, quæſupra Num. </s> <s xml:id="echoid-s650" xml:space="preserve">4. </s> <s xml:id="echoid-s651" xml:space="preserve">& </s> <s xml:id="echoid-s652" xml:space="preserve">12. </s> <s xml:id="echoid-s653" xml:space="preserve">tradita ſunt, da-<lb/>bit interuallum inter 45. </s> <s xml:id="echoid-s654" xml:space="preserve">& </s> <s xml:id="echoid-s655" xml:space="preserve">45. </s> <s xml:id="echoid-s656" xml:space="preserve">ſinum quoque eundemreſpectu ſinus totius <lb/>F G, quodinteruallum duplicatum dabit chordamdupli arcus grad. </s> <s xml:id="echoid-s657" xml:space="preserve">26. </s> <s xml:id="echoid-s658" xml:space="preserve">min. </s> <s xml:id="echoid-s659" xml:space="preserve"><lb/>45. </s> <s xml:id="echoid-s660" xml:space="preserve">id eſt, chordam arcus grad. </s> <s xml:id="echoid-s661" xml:space="preserve">53. </s> <s xml:id="echoid-s662" xml:space="preserve">min. </s> <s xml:id="echoid-s663" xml:space="preserve">30. </s> <s xml:id="echoid-s664" xml:space="preserve">qui quæritur. </s> <s xml:id="echoid-s665" xml:space="preserve">Si ſinus totus ſtatua-<lb/>tur 10000. </s> <s xml:id="echoid-s666" xml:space="preserve">erit ſinus grad. </s> <s xml:id="echoid-s667" xml:space="preserve">26. </s> <s xml:id="echoid-s668" xml:space="preserve">min. </s> <s xml:id="echoid-s669" xml:space="preserve">45. </s> <s xml:id="echoid-s670" xml:space="preserve">in tabula ſinuũ 4501. </s> <s xml:id="echoid-s671" xml:space="preserve">abiectis nimirum <lb/>tribus figuris ad dexteram: </s> <s xml:id="echoid-s672" xml:space="preserve">quiin A F, capietur, vt Num. </s> <s xml:id="echoid-s673" xml:space="preserve">6. </s> <s xml:id="echoid-s674" xml:space="preserve">& </s> <s xml:id="echoid-s675" xml:space="preserve">12. </s> <s xml:id="echoid-s676" xml:space="preserve">dictum eſt.</s> <s xml:id="echoid-s677" xml:space="preserve"/> </p> <div xml:id="echoid-div29" type="float" level="2" n="13"> <note position="right" xlink:label="note-039-01" xlink:href="note-039-01a" xml:space="preserve">Ex circulo <lb/>qua ratione <lb/>abſcindatur <lb/>arc{us} datorũ <lb/>grad. ac min.</note> </div> <p> <s xml:id="echoid-s678" xml:space="preserve"><emph style="sc">Qvod</emph> ſi vtraque regula A F, A G, contineat 1000. </s> <s xml:id="echoid-s679" xml:space="preserve">partes, ſtatui poterit ſi-<lb/>nustotus 100000. </s> <s xml:id="echoid-s680" xml:space="preserve">& </s> <s xml:id="echoid-s681" xml:space="preserve">etiam plurium partium, ſi nimirum adhibeatur recta <lb/>M N, partium 101. </s> <s xml:id="echoid-s682" xml:space="preserve">diuiſa in 100. </s> <s xml:id="echoid-s683" xml:space="preserve">particulas, vel alia recta partium 1001. </s> <s xml:id="echoid-s684" xml:space="preserve">in 1000. <lb/></s> <s xml:id="echoid-s685" xml:space="preserve">particulas ſecta.</s> <s xml:id="echoid-s686" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s687" xml:space="preserve">15. </s> <s xml:id="echoid-s688" xml:space="preserve"><emph style="sc">EContrario</emph> facile etiam cognoſcemus, quot gradus, & </s> <s xml:id="echoid-s689" xml:space="preserve">minu-<lb/> <anchor type="note" xlink:label="note-039-02a" xlink:href="note-039-02"/> tain propoſito arcu cuiuſuis quadrantis contineantur. </s> <s xml:id="echoid-s690" xml:space="preserve">Sit enim in quadran-<lb/>te, cuius ſemidiameter F G, cuiæqualis ſitrecta R S, datus aliquis arcus, cuius <lb/>chordæſemiſsis ſir R T. </s> <s xml:id="echoid-s691" xml:space="preserve">Huic R T, æqualis inueniatur recta 40. </s> <s xml:id="echoid-s692" xml:space="preserve">40. </s> <s xml:id="echoid-s693" xml:space="preserve">inter rectas <lb/>A F, A G, ita vt puncta 40. </s> <s xml:id="echoid-s694" xml:space="preserve">40. </s> <s xml:id="echoid-s695" xml:space="preserve">vel abſcindant æquales partes, vtin dato exem-<lb/>plo, vel æqualiter diſtent à duabus partibus æqualibus. </s> <s xml:id="echoid-s696" xml:space="preserve">Deinde per ea, quæ <lb/>Num. </s> <s xml:id="echoid-s697" xml:space="preserve">13. </s> <s xml:id="echoid-s698" xml:space="preserve">ſcripſimus, inquiratur, quot partes ex 1000, vel 10000. </s> <s xml:id="echoid-s699" xml:space="preserve">vel 100000. <lb/></s> <s xml:id="echoid-s700" xml:space="preserve">in ſegmento ab A, vſque ad punctum inuẽtum 40. </s> <s xml:id="echoid-s701" xml:space="preserve">comprehendantur. </s> <s xml:id="echoid-s702" xml:space="preserve">In da-<lb/>to exemplo reperiuntur partes 400. </s> <s xml:id="echoid-s703" xml:space="preserve">vel 4000. </s> <s xml:id="echoid-s704" xml:space="preserve">vel 40000. </s> <s xml:id="echoid-s705" xml:space="preserve">prout ſinus totus <lb/>conſtituitur 1000. </s> <s xml:id="echoid-s706" xml:space="preserve">vel 10000. </s> <s xml:id="echoid-s707" xml:space="preserve">vel 100000. </s> <s xml:id="echoid-s708" xml:space="preserve">atque tantus eſt ſinus 40. </s> <s xml:id="echoid-s709" xml:space="preserve">40. </s> <s xml:id="echoid-s710" xml:space="preserve">re-<lb/>ſpectu ſinus totius F G, cuireſpondent Grad. </s> <s xml:id="echoid-s711" xml:space="preserve">23. </s> <s xml:id="echoid-s712" xml:space="preserve">min. </s> <s xml:id="echoid-s713" xml:space="preserve">35. </s> <s xml:id="echoid-s714" xml:space="preserve">Duplus ergo arcus <lb/>gr. </s> <s xml:id="echoid-s715" xml:space="preserve">47. </s> <s xml:id="echoid-s716" xml:space="preserve">min. </s> <s xml:id="echoid-s717" xml:space="preserve">10. </s> <s xml:id="echoid-s718" xml:space="preserve">quichordæipſius R T, duplæ debetur, eritis, qui quæritur.</s> <s xml:id="echoid-s719" xml:space="preserve"/> </p> <div xml:id="echoid-div30" type="float" level="2" n="14"> <note position="right" xlink:label="note-039-02" xlink:href="note-039-02a" xml:space="preserve">Quot grad{us}, <lb/>ac minuta in <lb/>dato arcu cõ-<lb/>tineãtur, quo <lb/>pacto cogno-<lb/>ſcatur.</note> </div> <pb o="10" file="040" n="40" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s720" xml:space="preserve">16. </s> <s xml:id="echoid-s721" xml:space="preserve"><emph style="sc">In</emph> altera facie inſtrumenti, in quam chordę arcuum quadrantis <lb/> <anchor type="note" xlink:label="note-040-01a" xlink:href="note-040-01"/> ſunt translatę, facilius arcum quotcunque graduum accipiemus, hoc mo-<lb/>do. </s> <s xml:id="echoid-s722" xml:space="preserve">Chordæ quadrantis propoſiti ſumatur ęquale interuallum F G: </s> <s xml:id="echoid-s723" xml:space="preserve">Vel <lb/>etiam ſemidiametro Quadrantis, chordę nimirum grad. </s> <s xml:id="echoid-s724" xml:space="preserve">60. </s> <s xml:id="echoid-s725" xml:space="preserve">capiatur inter-<lb/>uallum 60. </s> <s xml:id="echoid-s726" xml:space="preserve">60. </s> <s xml:id="echoid-s727" xml:space="preserve">ęquale. </s> <s xml:id="echoid-s728" xml:space="preserve">Siigitur verbigratia deſideret quis arcum Grad. </s> <s xml:id="echoid-s729" xml:space="preserve">56. <lb/></s> <s xml:id="echoid-s730" xml:space="preserve">ſumendum erit per circinum interuallum inter puncta 56. </s> <s xml:id="echoid-s731" xml:space="preserve">& </s> <s xml:id="echoid-s732" xml:space="preserve">56. </s> <s xml:id="echoid-s733" xml:space="preserve">Huic enim <lb/>ęqualis eſt chorda grad. </s> <s xml:id="echoid-s734" xml:space="preserve">56, Si præter gradus accipienda ſint etiam minuta, <lb/>oportebit per ęſtimationem in ſequentiparticula accipere talem partem ipſi-<lb/>us, qualem minuta propoſita partem vnius gradus conſtituunt, Vt ſi cupiat <lb/>quis min. </s> <s xml:id="echoid-s735" xml:space="preserve">30. </s> <s xml:id="echoid-s736" xml:space="preserve">ſumenda eſt ſemiſsis, ſi 20. </s> <s xml:id="echoid-s737" xml:space="preserve">tertia pars, &</s> <s xml:id="echoid-s738" xml:space="preserve">c. </s> <s xml:id="echoid-s739" xml:space="preserve">Interuallum enim <lb/>inter partem regulę A F, & </s> <s xml:id="echoid-s740" xml:space="preserve">partemregulę A G, acceptum dabit chordam <lb/> <anchor type="note" xlink:label="note-040-02a" xlink:href="note-040-02"/> quęſiti arcus.</s> <s xml:id="echoid-s741" xml:space="preserve"/> </p> <div xml:id="echoid-div31" type="float" level="2" n="15"> <note position="left" xlink:label="note-040-01" xlink:href="note-040-01a" xml:space="preserve">Quo pacto a-<lb/>liter ex circu-<lb/>lo abſcindan-<lb/>tur arc{us} da-<lb/>torum gr. & <lb/>min.</note> <note position="left" xlink:label="note-040-02" xlink:href="note-040-02a" xml:space="preserve">Quo pacto a-<lb/>liter cogno-<lb/>ſcatur, quot <lb/>grad. & min. <lb/>in dato arcu <lb/>comprehen-<lb/>dantur.</note> </div> <p> <s xml:id="echoid-s742" xml:space="preserve"><emph style="sc">Vicissim</emph> ſi cognoſcere velimus, quot gradus, ac minuta in dato arcu <lb/>exiſtant, inueſtiganda erit eius chorda inter rectas A F, A G, ita vt puncta eius <lb/>cadant vel in duas partes eaſdem, vel ęqualiter à duabus eiſdẽ diſtent. </s> <s xml:id="echoid-s743" xml:space="preserve">Nam <lb/>tot gradus continebuntur in dato arcu, quotgradus cõtinentur in recta A F, <lb/>à centro A, vſque ad punctum, è quo chorda datiarcus in rectam A G, trãs-<lb/>lata eſt: </s> <s xml:id="echoid-s744" xml:space="preserve">ita vt ſi dati arcus chorda extiterit inter grad. </s> <s xml:id="echoid-s745" xml:space="preserve">70. </s> <s xml:id="echoid-s746" xml:space="preserve">& </s> <s xml:id="echoid-s747" xml:space="preserve">70. </s> <s xml:id="echoid-s748" xml:space="preserve">propoſitus ar-<lb/>cus complectatur gr. </s> <s xml:id="echoid-s749" xml:space="preserve">70. </s> <s xml:id="echoid-s750" xml:space="preserve">&</s> <s xml:id="echoid-s751" xml:space="preserve">c.</s> <s xml:id="echoid-s752" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quaratione <lb/>ex datarecta <lb/>pars imperata <lb/>abſcindatur.</note> <handwritten/> <p> <s xml:id="echoid-s753" xml:space="preserve">17. </s> <s xml:id="echoid-s754" xml:space="preserve"><emph style="sc">Iam</emph> vero nemo neſcit, ſi ex linea aliqua abſcindenda ſit {1/2}. </s> <s xml:id="echoid-s755" xml:space="preserve">vel {1/3}. </s> <s xml:id="echoid-s756" xml:space="preserve">vel <lb/>{1/17}. </s> <s xml:id="echoid-s757" xml:space="preserve">vel denique quęcunq; </s> <s xml:id="echoid-s758" xml:space="preserve">pars, cuius denominator maior non ſit quam 100. <lb/></s> <s xml:id="echoid-s759" xml:space="preserve">quo pactoid fieri debeat. </s> <s xml:id="echoid-s760" xml:space="preserve">Sinamque interuallum F G, in priori facie inſtru-<lb/>menti ęquale fuerit datę rectę, dabit interuallum inter 50. </s> <s xml:id="echoid-s761" xml:space="preserve">& </s> <s xml:id="echoid-s762" xml:space="preserve">50. </s> <s xml:id="echoid-s763" xml:space="preserve">partem {1/2}. </s> <s xml:id="echoid-s764" xml:space="preserve"><lb/>Interuallum autem inter 25. </s> <s xml:id="echoid-s765" xml:space="preserve">& </s> <s xml:id="echoid-s766" xml:space="preserve">25. </s> <s xml:id="echoid-s767" xml:space="preserve">partem. </s> <s xml:id="echoid-s768" xml:space="preserve">{1/4}. </s> <s xml:id="echoid-s769" xml:space="preserve">Interuallum vero inter 20. </s> <s xml:id="echoid-s770" xml:space="preserve">& </s> <s xml:id="echoid-s771" xml:space="preserve"><lb/>20. </s> <s xml:id="echoid-s772" xml:space="preserve">partem {1/5}. </s> <s xml:id="echoid-s773" xml:space="preserve">Item ſi interuallum inter 90. </s> <s xml:id="echoid-s774" xml:space="preserve">& </s> <s xml:id="echoid-s775" xml:space="preserve">90. </s> <s xml:id="echoid-s776" xml:space="preserve">vel inter 60. </s> <s xml:id="echoid-s777" xml:space="preserve">& </s> <s xml:id="echoid-s778" xml:space="preserve">60. </s> <s xml:id="echoid-s779" xml:space="preserve">vel in-<lb/>ter 30. </s> <s xml:id="echoid-s780" xml:space="preserve">& </s> <s xml:id="echoid-s781" xml:space="preserve">30. </s> <s xml:id="echoid-s782" xml:space="preserve">fiat datę rectę ęquale, dabitinteruallum inter 30. </s> <s xml:id="echoid-s783" xml:space="preserve">& </s> <s xml:id="echoid-s784" xml:space="preserve">30. </s> <s xml:id="echoid-s785" xml:space="preserve">vel in-<lb/>ter 20. </s> <s xml:id="echoid-s786" xml:space="preserve">& </s> <s xml:id="echoid-s787" xml:space="preserve">20. </s> <s xml:id="echoid-s788" xml:space="preserve">vel inter 10. </s> <s xml:id="echoid-s789" xml:space="preserve">& </s> <s xml:id="echoid-s790" xml:space="preserve">0. </s> <s xml:id="echoid-s791" xml:space="preserve">partem {1/3}. </s> <s xml:id="echoid-s792" xml:space="preserve">Rurſus ſi interuallum inter 17. </s> <s xml:id="echoid-s793" xml:space="preserve">& </s> <s xml:id="echoid-s794" xml:space="preserve">17. </s> <s xml:id="echoid-s795" xml:space="preserve"><lb/>vel inter 34. </s> <s xml:id="echoid-s796" xml:space="preserve">& </s> <s xml:id="echoid-s797" xml:space="preserve">34. </s> <s xml:id="echoid-s798" xml:space="preserve">datæ rectę ſumatur ęquale, dabit interuallum inter 1. </s> <s xml:id="echoid-s799" xml:space="preserve">& </s> <s xml:id="echoid-s800" xml:space="preserve">1. </s> <s xml:id="echoid-s801" xml:space="preserve"><lb/>vel inter 2. </s> <s xml:id="echoid-s802" xml:space="preserve">& </s> <s xml:id="echoid-s803" xml:space="preserve">2. </s> <s xml:id="echoid-s804" xml:space="preserve">partem {1/1@}. </s> <s xml:id="echoid-s805" xml:space="preserve">Atinteruallum inter 5. </s> <s xml:id="echoid-s806" xml:space="preserve">& </s> <s xml:id="echoid-s807" xml:space="preserve">5. </s> <s xml:id="echoid-s808" xml:space="preserve">velinter 10. </s> <s xml:id="echoid-s809" xml:space="preserve">& </s> <s xml:id="echoid-s810" xml:space="preserve">10. </s> <s xml:id="echoid-s811" xml:space="preserve">da-<lb/>bit {5/17}. </s> <s xml:id="echoid-s812" xml:space="preserve">&</s> <s xml:id="echoid-s813" xml:space="preserve">c. </s> <s xml:id="echoid-s814" xml:space="preserve">Ex hiſceporro exemplis adductis facileintelliges, quo modo te <lb/>in aliis partibus imperatis gerere debeas.</s> <s xml:id="echoid-s815" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Quaratione <lb/>ex dato circu-<lb/>lo lat{us} poly-<lb/>goni propoſiti <lb/>inueniatur.</note> <p> <s xml:id="echoid-s816" xml:space="preserve">18. </s> <s xml:id="echoid-s817" xml:space="preserve"><emph style="sc">Non</emph> aliter in poſteriori facie inſtrumenti Iatera polygonorũ in quo-<lb/>uis circulo reperiemus. </s> <s xml:id="echoid-s818" xml:space="preserve">Nam gradus 120. </s> <s xml:id="echoid-s819" xml:space="preserve">(quifacile accipientur, ſi quadran-<lb/>ti graduum 90. </s> <s xml:id="echoid-s820" xml:space="preserve">adij ciantur gradus 30.) </s> <s xml:id="echoid-s821" xml:space="preserve">dabuntlatus trianguli ęquilateri. </s> <s xml:id="echoid-s822" xml:space="preserve">Gra-<lb/>dus 90. </s> <s xml:id="echoid-s823" xml:space="preserve">latus quadrati. </s> <s xml:id="echoid-s824" xml:space="preserve">Gradus 72. </s> <s xml:id="echoid-s825" xml:space="preserve">latus pentagoni. </s> <s xml:id="echoid-s826" xml:space="preserve">Gradus 51 {3/7}. </s> <s xml:id="echoid-s827" xml:space="preserve">hoc eſt, <lb/>gradus 51. </s> <s xml:id="echoid-s828" xml:space="preserve">min. </s> <s xml:id="echoid-s829" xml:space="preserve">26. </s> <s xml:id="echoid-s830" xml:space="preserve">paulo minus, latus heptagoni. </s> <s xml:id="echoid-s831" xml:space="preserve">&</s> <s xml:id="echoid-s832" xml:space="preserve">c. </s> <s xml:id="echoid-s833" xml:space="preserve">qui quidem gradus <lb/>habebuntur, ſi gradus 360. </s> <s xml:id="echoid-s834" xml:space="preserve">totius circuli per numerum laterum polygoni <lb/>propoſiti diuidantur.</s> <s xml:id="echoid-s835" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s836" xml:space="preserve">19. </s> <s xml:id="echoid-s837" xml:space="preserve"><emph style="sc">Qvando</emph> in dato quadrante cognoſcere lubet, (quod non raro v-<lb/> <anchor type="note" xlink:label="note-040-05a" xlink:href="note-040-05"/> ſu venit) in quodnam punctum ſemidiametriperpendicularis ex quouis gra-<lb/>du ab altera ſemidiametro numerato demiſſa cadat, ita agendum erit. </s> <s xml:id="echoid-s838" xml:space="preserve">Sit <lb/>verbi gratia ſemidiameter alicuius quadrantis F G, quærendumq; </s> <s xml:id="echoid-s839" xml:space="preserve">ſit punctũ, <lb/>in quod cadat perpendicularis ex grad. </s> <s xml:id="echoid-s840" xml:space="preserve">26. </s> <s xml:id="echoid-s841" xml:space="preserve">min. </s> <s xml:id="echoid-s842" xml:space="preserve">45. </s> <s xml:id="echoid-s843" xml:space="preserve">demiſſa. </s> <s xml:id="echoid-s844" xml:space="preserve">Sinus grad. </s> <s xml:id="echoid-s845" xml:space="preserve">26. <lb/></s> <s xml:id="echoid-s846" xml:space="preserve">min. </s> <s xml:id="echoid-s847" xml:space="preserve">45. </s> <s xml:id="echoid-s848" xml:space="preserve">eſt 45. </s> <s xml:id="echoid-s849" xml:space="preserve">poſito ſinutoto 100. </s> <s xml:id="echoid-s850" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s851" xml:space="preserve">recta interpartes 45. </s> <s xml:id="echoid-s852" xml:space="preserve">& </s> <s xml:id="echoid-s853" xml:space="preserve">45. </s> <s xml:id="echoid-s854" xml:space="preserve">erit <lb/>ſinus grad. </s> <s xml:id="echoid-s855" xml:space="preserve">26. </s> <s xml:id="echoid-s856" xml:space="preserve">min. </s> <s xml:id="echoid-s857" xml:space="preserve">45. </s> <s xml:id="echoid-s858" xml:space="preserve">reſpectu ſinus totius F G, vt Num. </s> <s xml:id="echoid-s859" xml:space="preserve">12. </s> <s xml:id="echoid-s860" xml:space="preserve">oſtenſum eſt. </s> <s xml:id="echoid-s861" xml:space="preserve"><lb/>Quocirca recta 45. </s> <s xml:id="echoid-s862" xml:space="preserve">45. </s> <s xml:id="echoid-s863" xml:space="preserve">in ſemidiametrum dati quadrantis ex centro translata <lb/>indicabit punctum quæſitum.</s> <s xml:id="echoid-s864" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="16"> <note position="left" xlink:label="note-040-05" xlink:href="note-040-05a" xml:space="preserve">Quopacto co-<lb/>gnoſcatur, in <lb/>quodpunctũ <lb/>ſemidiametri <lb/>cadat perpen-<lb/>dicularis ex <lb/>quolibet gra-<lb/>du quadran-<lb/>tisdemiſſa.</note> </div> <pb o="11" file="041" n="41" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s865" xml:space="preserve">20. </s> <s xml:id="echoid-s866" xml:space="preserve"><emph style="sc">Non</emph> aliter reperiemus in diametro Aſtrolabij (quod notatu di-<lb/> <anchor type="note" xlink:label="note-041-01a" xlink:href="note-041-01"/> gnum eſt) punctum cuiuſcunque declinationis. </s> <s xml:id="echoid-s867" xml:space="preserve">Poſito enim ſinu toto ſemi-<lb/>diametro Aequatoris, ſi declinatio eſt Borealis, tranferenda eſt in diametrum <lb/>ex centro Tangens ſemiſsis complementi declinationis: </s> <s xml:id="echoid-s868" xml:space="preserve">ſi vero auſtralis eſt, <lb/>Tangens ſemiſsis arcus ex quadrante, & </s> <s xml:id="echoid-s869" xml:space="preserve">declinatione compoſiti. </s> <s xml:id="echoid-s870" xml:space="preserve">Vt Tan-<lb/>gens grad. </s> <s xml:id="echoid-s871" xml:space="preserve">33. </s> <s xml:id="echoid-s872" xml:space="preserve">min. </s> <s xml:id="echoid-s873" xml:space="preserve">15. </s> <s xml:id="echoid-s874" xml:space="preserve">qui ſemiſſem complementi declinationis ♋. </s> <s xml:id="echoid-s875" xml:space="preserve">conſtitu-<lb/>unt, dabit punctumextremum ſemidiametri paralleli ♋, quod videlicet ab <lb/>Aequatore in Boream grad. </s> <s xml:id="echoid-s876" xml:space="preserve">23. </s> <s xml:id="echoid-s877" xml:space="preserve">min. </s> <s xml:id="echoid-s878" xml:space="preserve">30. </s> <s xml:id="echoid-s879" xml:space="preserve">declinat. </s> <s xml:id="echoid-s880" xml:space="preserve">Tangens vero grad. </s> <s xml:id="echoid-s881" xml:space="preserve">56. <lb/></s> <s xml:id="echoid-s882" xml:space="preserve">min. </s> <s xml:id="echoid-s883" xml:space="preserve">45. </s> <s xml:id="echoid-s884" xml:space="preserve">qui ſemiſſem conſtituunt arcus ex quadrante, & </s> <s xml:id="echoid-s885" xml:space="preserve">declinatione ♑, cõ-<lb/>flati, dabit extremum punctum ſemidiametri paralleli ♑, quod ab Aequatore <lb/>in Auſtrum grad. </s> <s xml:id="echoid-s886" xml:space="preserve">23. </s> <s xml:id="echoid-s887" xml:space="preserve">min. </s> <s xml:id="echoid-s888" xml:space="preserve">30. </s> <s xml:id="echoid-s889" xml:space="preserve">recedit. </s> <s xml:id="echoid-s890" xml:space="preserve">Ratio huiuſcerei eſt, quod recta in A-<lb/>ſtrolabio ab extremitate diametri rectum Horizontem referẽtis vſque ad in-<lb/>terſectionem paralleli borealis cum altera diametro Meridianum repræſen-<lb/>tãte ducta conſtituit cũ altera diametro angulũ ſemiſsis complementi decli-<lb/>nationis borealis; </s> <s xml:id="echoid-s891" xml:space="preserve">ad interſectionem vero paralleli auſtralis cum eadem po-<lb/>ſteriore diametro educta effi cit angulum ſemiſsis arcus ex quadrante, & </s> <s xml:id="echoid-s892" xml:space="preserve">de-<lb/>clinatione auſtrali conflati: </s> <s xml:id="echoid-s893" xml:space="preserve">atque vtriuslibet anguli Tangens ſemidiameter <lb/>eſt paralleli, vt ex Aſtrolabio liquet.</s> <s xml:id="echoid-s894" xml:space="preserve"/> </p> <div xml:id="echoid-div33" type="float" level="2" n="17"> <note position="right" xlink:label="note-041-01" xlink:href="note-041-01a" xml:space="preserve">Quo pacto in <lb/>diametro A-<lb/>ſtrolabij pun-<lb/>ctum cuiuſuis <lb/>declinationis <lb/>reperiatur.</note> </div> <p> <s xml:id="echoid-s895" xml:space="preserve"><emph style="sc">Eodemqve</emph> modo, ſi conſtiterit, quem angulum in extremitate ſemi-<lb/>diametri Aequatoris in Aſtrolabio recta ad quodcunque punctum diametri, <lb/>quæ ad illam ſemidiametrum perpendicularis eſt, ducta conſtituat, reperie-<lb/>mus punctum illud per Tangentem illius anguli, ſicut in parallelis ♋, & </s> <s xml:id="echoid-s896" xml:space="preserve">♑. <lb/></s> <s xml:id="echoid-s897" xml:space="preserve">factum eſt.</s> <s xml:id="echoid-s898" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s899" xml:space="preserve">21. </s> <s xml:id="echoid-s900" xml:space="preserve"><emph style="sc">Si</emph> etiam quæcunque linea ex centro inſtrumenti huius egrediens <lb/> <anchor type="note" xlink:label="note-041-02a" xlink:href="note-041-02"/> ſecetur quomo do cunque, vt verbi gratia extrema, & </s> <s xml:id="echoid-s901" xml:space="preserve">media ratione, vel <lb/>(quod operæ pretium eſſet, vt expeditius horologia deſcribantur) ſicutæ-<lb/>quinoctialis linea in horologio horizontali diuiſa eſt: </s> <s xml:id="echoid-s902" xml:space="preserve">ſecabitur quæuis alia ſi-<lb/>militer, ſi nimirum ei æquale interuallum F G, ſumatur, vt ex dictis liquido <lb/>conſtat. </s> <s xml:id="echoid-s903" xml:space="preserve">Satis tamen eſt, ſi horæ ex vna parte lineę meridianæ, nimirum vel <lb/>horæ antemeridianæ, vel pomeridianæ duntaxat in inſtrumentum transfe-<lb/>rantur.</s> <s xml:id="echoid-s904" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="18"> <note position="right" xlink:label="note-041-02" xlink:href="note-041-02a" xml:space="preserve">Præceptum <lb/>generale ad <lb/>diuidendam <lb/>lineam datã, <lb/>vt alia quæ-<lb/>cunque diui-<lb/>ſaest.</note> </div> <p> <s xml:id="echoid-s905" xml:space="preserve">22. </s> <s xml:id="echoid-s906" xml:space="preserve"><emph style="sc">Præterea</emph> aperto inſtrumento quomodocunque, cognoſcemus <lb/> <anchor type="note" xlink:label="note-041-03a" xlink:href="note-041-03"/> quantitatem anguli F A G, in centro A, conſtituti, hoc modo. </s> <s xml:id="echoid-s907" xml:space="preserve">Circino ſuma-<lb/>tur interualluminter gradus 60. </s> <s xml:id="echoid-s908" xml:space="preserve">& </s> <s xml:id="echoid-s909" xml:space="preserve">60. </s> <s xml:id="echoid-s910" xml:space="preserve">in poſteriore inſtrumenti facie, tranſ-<lb/>feraturque ex centro in alterutram lineam chordarum. </s> <s xml:id="echoid-s911" xml:space="preserve">Nam quot gradus in <lb/>co interuallo includuntur, tot gradus continebit angulus F A G. </s> <s xml:id="echoid-s912" xml:space="preserve">Ratio eſt, <lb/>quòd arcus ex centro A, per gradus 60. </s> <s xml:id="echoid-s913" xml:space="preserve">& </s> <s xml:id="echoid-s914" xml:space="preserve">60. </s> <s xml:id="echoid-s915" xml:space="preserve">deſcriptus portio eſt quadran-<lb/>tis, cuius chorda eſt tota linea A F: </s> <s xml:id="echoid-s916" xml:space="preserve">propterea quod corda 60. </s> <s xml:id="echoid-s917" xml:space="preserve">graduum ſe-<lb/>midiameter eſt quadrantis dicti, cuius chorda eſt A F, vt ex inſtrumenti con-<lb/>ſtructione manifeſtum eſt. </s> <s xml:id="echoid-s918" xml:space="preserve">Igitur interuallum inter gr. </s> <s xml:id="echoid-s919" xml:space="preserve">60. </s> <s xml:id="echoid-s920" xml:space="preserve">& </s> <s xml:id="echoid-s921" xml:space="preserve">60. </s> <s xml:id="echoid-s922" xml:space="preserve">eſt chorda <lb/>anguli F A G, propoſiti, &</s> <s xml:id="echoid-s923" xml:space="preserve">c.</s> <s xml:id="echoid-s924" xml:space="preserve"/> </p> <div xml:id="echoid-div35" type="float" level="2" n="19"> <note position="right" xlink:label="note-041-03" xlink:href="note-041-03a" xml:space="preserve">Qua ratione <lb/>quantitas an-<lb/>guli quem la-<lb/>tera inſtru-<lb/>menti conti-<lb/>nent cogno-<lb/>ſcatur.</note> </div> <p> <s xml:id="echoid-s925" xml:space="preserve">23. </s> <s xml:id="echoid-s926" xml:space="preserve"><emph style="sc">Sed</emph> neque hoc omittendum eſt, (quando quidem de Tangentibus, <lb/> <anchor type="note" xlink:label="note-041-04a" xlink:href="note-041-04"/> ſinubus, & </s> <s xml:id="echoid-s927" xml:space="preserve">ſecantibus in hoc inſtrumento reſpectu datiſinus totius accipi-<lb/>endis verba fecimus) ſinum totum interdum eſſe tam exiguum, vt ex F, in G, <lb/>transferri nequeat, etiamſi inſtrumẽtum prorſus claudatur. </s> <s xml:id="echoid-s928" xml:space="preserve">Vt ergo reſpectu <lb/>illius ſinus totius Tangens, ſinus, ac ſecantes accipere poſsimus ex inſtrumẽ <pb o="12" file="042" n="42" rhead="GEOMETR. PRACT."/> to, ſumendus eſtille ſinus totusin aliqua recta bis, ter, aut quater, &</s> <s xml:id="echoid-s929" xml:space="preserve">c. </s> <s xml:id="echoid-s930" xml:space="preserve">atque <lb/> <anchor type="note" xlink:label="note-042-01a" xlink:href="note-042-01"/> ita ex F, in G, transferendus. </s> <s xml:id="echoid-s931" xml:space="preserve">Nam ſi Tangentium quæſitarum, vel ſinuũ, aut <lb/>ſecantium reſpectu ſinus totius 100. </s> <s xml:id="echoid-s932" xml:space="preserve">capiantur ſemiſſes, vel tertiæ partes, aut <lb/>quartæ, &</s> <s xml:id="echoid-s933" xml:space="preserve">c. </s> <s xml:id="echoid-s934" xml:space="preserve">prout videlicet ſinus totus bis, ter, quateruè, &</s> <s xml:id="echoid-s935" xml:space="preserve">c. </s> <s xml:id="echoid-s936" xml:space="preserve">acceptus fuit, <lb/>habebũtur Tangẽtes quæſitæ, vel ſinus, aut ſecãtes. </s> <s xml:id="echoid-s937" xml:space="preserve">Vt ſi ſinus totus duplice-<lb/>tur, & </s> <s xml:id="echoid-s938" xml:space="preserve">poſito ſinu toto 100. </s> <s xml:id="echoid-s939" xml:space="preserve">Tangens verbigratia ſit 378. </s> <s xml:id="echoid-s940" xml:space="preserve">ſumenda eſt Tangẽs <lb/>189. </s> <s xml:id="echoid-s941" xml:space="preserve">&</s> <s xml:id="echoid-s942" xml:space="preserve">c. </s> <s xml:id="echoid-s943" xml:space="preserve">Sed commodiſsime res hæc peragetur, ſi ſinus totus, quiperpuſil-<lb/>lus eſt, decupletur. </s> <s xml:id="echoid-s944" xml:space="preserve">Ita enim poſito ſinu toto 100. </s> <s xml:id="echoid-s945" xml:space="preserve">ſi ex Tangente verbi gra-<lb/>tia propoſita (relicta prima figura ad dexteram) abijciatur vna figura ad de-<lb/>xteram, quæ eſt ſecunda in tota Tangente, habebitur decima eius pars. </s> <s xml:id="echoid-s946" xml:space="preserve">Ha-<lb/>benda tamen ſemper eſt ratio figuræ abiectæ, vt ſcilicet pro ea ſumatur 1. </s> <s xml:id="echoid-s947" xml:space="preserve">ſi <lb/>maior eſt quam 5. </s> <s xml:id="echoid-s948" xml:space="preserve">&</s> <s xml:id="echoid-s949" xml:space="preserve">c. </s> <s xml:id="echoid-s950" xml:space="preserve">Hacratione Tangente 2414. </s> <s xml:id="echoid-s951" xml:space="preserve">grad 67. </s> <s xml:id="echoid-s952" xml:space="preserve">min 30. </s> <s xml:id="echoid-s953" xml:space="preserve">propo-<lb/>ſita (relictis {4/1000}.) </s> <s xml:id="echoid-s954" xml:space="preserve">transferenda erit Tangens 24 paulo amplius, nimirum <lb/>pars decima Tangentis 241. </s> <s xml:id="echoid-s955" xml:space="preserve">reſpectu ſinus totius 100.</s> <s xml:id="echoid-s956" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="20"> <note position="right" xlink:label="note-041-04" xlink:href="note-041-04a" xml:space="preserve">Quandoſinus <lb/>tot{us} tam par-<lb/>u@@ eſt, vt in <lb/>inſtrumen-<lb/>tum tranſ-</note> <note position="left" xlink:label="note-042-01" xlink:href="note-042-01a" xml:space="preserve">ferrinequeat, <lb/>quid agen-<lb/>dum.</note> </div> <p> <s xml:id="echoid-s957" xml:space="preserve">24. </s> <s xml:id="echoid-s958" xml:space="preserve"><emph style="sc">Sic</emph> etiam, quando Tangens aliqua ſinum totum ſuperat, ne coga-<lb/>mur primum ſinum totum aliquoties transferre, deinde vero reliquas partes, <lb/> <anchor type="note" xlink:label="note-042-02a" xlink:href="note-042-02"/> vel contra; </s> <s xml:id="echoid-s959" xml:space="preserve">ſed vt ſtatim punctum, quod quæritur, per vnam translationem <lb/>poſsimus inuenire, diuidenda eſt tota Tangens tabulæ ad finem nouæ deſcri-<lb/>ptionis Horologiorum poſitæ per 2. </s> <s xml:id="echoid-s960" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s961" xml:space="preserve">vel per talem denique numerum, <lb/>vt producatur in Quotiente Tangens trium figurarum. </s> <s xml:id="echoid-s962" xml:space="preserve">Tunc enim abiecta <lb/>prima figura ad dexteram, reliqua Tangens transferenda eſt reſpectu ſinus <lb/>totius 100. </s> <s xml:id="echoid-s963" xml:space="preserve">multiplicatiper eundem numerum, per quem Tangẽs diuiſa fuit. <lb/></s> <s xml:id="echoid-s964" xml:space="preserve">Vt Tangens hor. </s> <s xml:id="echoid-s965" xml:space="preserve">4. </s> <s xml:id="echoid-s966" xml:space="preserve">& </s> <s xml:id="echoid-s967" xml:space="preserve">8. </s> <s xml:id="echoid-s968" xml:space="preserve">reſpectu ſinus totius 1000. </s> <s xml:id="echoid-s969" xml:space="preserve">eſt 1732. </s> <s xml:id="echoid-s970" xml:space="preserve">quæ diuiſa per <lb/>2. </s> <s xml:id="echoid-s971" xml:space="preserve">facit 866. </s> <s xml:id="echoid-s972" xml:space="preserve">Ergo transferenda eſt Tangens 86. </s> <s xml:id="echoid-s973" xml:space="preserve">{1/2}. </s> <s xml:id="echoid-s974" xml:space="preserve">paulo amplius reſpectu ſi-<lb/>nus totius 100. </s> <s xml:id="echoid-s975" xml:space="preserve">duplicati. </s> <s xml:id="echoid-s976" xml:space="preserve">Item Tangens hor. </s> <s xml:id="echoid-s977" xml:space="preserve">5. </s> <s xml:id="echoid-s978" xml:space="preserve">& </s> <s xml:id="echoid-s979" xml:space="preserve">7. </s> <s xml:id="echoid-s980" xml:space="preserve">eſt 3732. </s> <s xml:id="echoid-s981" xml:space="preserve">quæ diuiſa per <lb/>quatuor ſacit 933. </s> <s xml:id="echoid-s982" xml:space="preserve">Ergo Tangens 93 {3/10}. </s> <s xml:id="echoid-s983" xml:space="preserve">transferenda eſt reſpectu ſinus totius <lb/>100. </s> <s xml:id="echoid-s984" xml:space="preserve">quadruplicati. </s> <s xml:id="echoid-s985" xml:space="preserve">Atqueita de cæteris.</s> <s xml:id="echoid-s986" xml:space="preserve"/> </p> <div xml:id="echoid-div37" type="float" level="2" n="21"> <note position="left" xlink:label="note-042-02" xlink:href="note-042-02a" xml:space="preserve">Quando Tã-<lb/>gens ſuper{at} <lb/>ſinum totum, <lb/>quid agendũ, <lb/>vt per vnicã <lb/>tranſlationem <lb/>punctũ quæſi-<lb/>tum inuenia-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s987" xml:space="preserve">25. </s> <s xml:id="echoid-s988" xml:space="preserve"><emph style="sc">In</emph> hoc eodem denique inſtrumento facile duabus rectis tertiam pro-<lb/>portionalem, & </s> <s xml:id="echoid-s989" xml:space="preserve">tribus quartam adiungemus. </s> <s xml:id="echoid-s990" xml:space="preserve">Nam ſi, duabus propoſitis, pri-<lb/> <anchor type="note" xlink:label="note-042-03a" xlink:href="note-042-03"/> mæinrecta A F, regulæ A B, æqualis capiatur: </s> <s xml:id="echoid-s991" xml:space="preserve">& </s> <s xml:id="echoid-s992" xml:space="preserve">ſecunda à fine huius, aperto <lb/>inſtrumento, per circinum transferatur in regulam A C, ad numerum ſimilem <lb/>illi, qui in extremo primæ in regula AB, appoſitus eſt (ita vt pedes circini ſta-<lb/>tuantur vel in ſimilibus partibus vtriuſque regulæ, vel in duobus punctis æ-<lb/>qualiter diſtantibus à ſimilibus partibus) eidemque ſecundæ in regula A B, æ-<lb/> <anchor type="handwritten" xlink:label="hd-042-1a" xlink:href="hd-042-1"/> qualis ſumatur, dabitinterualluminter finem huius ſecundæ, & </s> <s xml:id="echoid-s993" xml:space="preserve">numerumin <lb/>regula A C, ſimilemilli, qui prope finem ſecundæ ſcriptus eſt, tertiam propor-<lb/>tionalem; </s> <s xml:id="echoid-s994" xml:space="preserve">vt ex demonſtratis Num. </s> <s xml:id="echoid-s995" xml:space="preserve">2. </s> <s xml:id="echoid-s996" xml:space="preserve">conſtare poteſt.</s> <s xml:id="echoid-s997" xml:space="preserve"/> </p> <div xml:id="echoid-div38" type="float" level="2" n="22"> <note position="left" xlink:label="note-042-03" xlink:href="note-042-03a" xml:space="preserve">Quo pacto <lb/>tertia, & <lb/>quarta pro-<lb/>portionalis <lb/>reperiatur.</note> <handwritten xlink:label="hd-042-1" xlink:href="hd-042-1a"/> </div> <p> <s xml:id="echoid-s998" xml:space="preserve"><emph style="sc">Eadem</emph> ratione, ſi, tribus rectis propoſitis, prima & </s> <s xml:id="echoid-s999" xml:space="preserve">ſecunda in inſtru-<lb/>mentum transferantur, vt dictum eſt, tertiæ autem in regula A B, æqualis quo-<lb/> <anchor type="handwritten" xlink:label="hd-042-1a" xlink:href="hd-042-1"/> que capiatur, dabit interuallum inter finem tertiæ, & </s> <s xml:id="echoid-s1000" xml:space="preserve">numerum regulæ A C, ſi-<lb/>milem illi, qui ad extremum tertiæ in regula A B, notatus eſt, quartam propor-<lb/>tionalem.</s> <s xml:id="echoid-s1001" xml:space="preserve"/> </p> <div xml:id="echoid-div39" type="float" level="2" n="23"> <handwritten xlink:label="hd-042-1" xlink:href="hd-042-1a"/> </div> <p> <s xml:id="echoid-s1002" xml:space="preserve"><emph style="sc">Qvod</emph> ſi lineæ propoſitæ tam magnæ ſint, velaliqua illarum, vt in inſtru-<lb/>mentum tranſportari nequeant, ſumendæ erunt omnium ſemiſſes, vel tertiæ <lb/>partes, vel quartæ &</s> <s xml:id="echoid-s1003" xml:space="preserve">c. </s> <s xml:id="echoid-s1004" xml:space="preserve">at que cum illis procedendum, vt dictum eſt. </s> <s xml:id="echoid-s1005" xml:space="preserve">Inuenta <lb/>enim duplicata, vel trip licata, vel quadruplicata, &</s> <s xml:id="echoid-s1006" xml:space="preserve">c. </s> <s xml:id="echoid-s1007" xml:space="preserve">offeret tertiã aut quar-<lb/>tam proportionalem quæſitam.</s> <s xml:id="echoid-s1008" xml:space="preserve"/> </p> <pb o="13" file="043" n="43" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s1009" xml:space="preserve">26. </s> <s xml:id="echoid-s1010" xml:space="preserve"><emph style="sc">Loco</emph> prędicti inſtrumenti conſtrui poteſt in lamina aliqua, vel <lb/> <anchor type="note" xlink:label="note-043-01a" xlink:href="note-043-01"/> plano quolibet, figura eundem vſum habens, facillima hac ratione. </s> <s xml:id="echoid-s1011" xml:space="preserve">Fiatan-<lb/>gulus B A C, cuiuſcunque magnitudinis; </s> <s xml:id="echoid-s1012" xml:space="preserve">quo autem maior fuerit, eo maio-<lb/>res ſinus toti in figura aſſumi poterunt: </s> <s xml:id="echoid-s1013" xml:space="preserve">ita vt non malè feceris ſi rectum <lb/>conſtituas. </s> <s xml:id="echoid-s1014" xml:space="preserve">Ita namque quadrantem quoque recto angulo oppoſitum ob-<lb/>tinebis: </s> <s xml:id="echoid-s1015" xml:space="preserve">Recta autem A B, in 100. </s> <s xml:id="echoid-s1016" xml:space="preserve">particulas æquales ſecta, (poſſet etiam ſe-<lb/>cari in 1000. </s> <s xml:id="echoid-s1017" xml:space="preserve">ſi commodè fieri poſſet, vt de ſuperiore inſtrumento diximus) <lb/>deſcribantur ex centro A, per ſingulas partes 100. </s> <s xml:id="echoid-s1018" xml:space="preserve">arcus circulorum, qui <lb/>rectam quo que A C, in 100. </s> <s xml:id="echoid-s1019" xml:space="preserve">particulas ęquales diſtinguent: </s> <s xml:id="echoid-s1020" xml:space="preserve">parataque erit <lb/>figura.</s> <s xml:id="echoid-s1021" xml:space="preserve"/> </p> <div xml:id="echoid-div40" type="float" level="2" n="24"> <note position="right" xlink:label="note-043-01" xlink:href="note-043-01a" xml:space="preserve">Conſtructio <lb/>alteri{us} inſtris <lb/>menti pro eo-<lb/>dem vſu.</note> </div> <p> <s xml:id="echoid-s1022" xml:space="preserve"><emph style="sc">Nam</emph> ſi in infimo arcu B C, ſumatur interuallũ B D, dato ſinui toti æqua-<lb/>le ducaturque recta occulta A D, (hæc in ęnea tabella ducenda erit atramen-<lb/>mento non admodum nigro, vel alio colore, vt poſtea deleri poſſit) fun-<lb/>gentur rectę A B, A D, officio regularum A F. </s> <s xml:id="echoid-s1023" xml:space="preserve">A G, ſuperioris inſtrumenti ad <lb/>propoſitam magnitudinem B D, aperti, & </s> <s xml:id="echoid-s1024" xml:space="preserve">dilatati. </s> <s xml:id="echoid-s1025" xml:space="preserve">Quamobrem inuenien-<lb/>tur in hac figura omnes Tangentes reſpectu ſinus totius B D, vtſupra. </s> <s xml:id="echoid-s1026" xml:space="preserve">Vt <lb/>Tangens verbi gratia partium 40. </s> <s xml:id="echoid-s1027" xml:space="preserve">erit mteruallum E F, <anchor type="note" xlink:href="" symbol="a"/> cum ducta recta <anchor type="note" xlink:label="note-043-02a" xlink:href="note-043-02"/> E F, parallela ſit rectę ductę B D, propterea quodlatera A B, A D, ſecta ſunt <lb/>in E, F, proportionaliter. </s> <s xml:id="echoid-s1028" xml:space="preserve">Alij vſus ſupra explicati facile quo que ad hanc <lb/>figuram aptabuntur: </s> <s xml:id="echoid-s1029" xml:space="preserve">pręſertim ſi in alteram faciem laminæ transferantur <lb/>chordæ omnium arcuum quadrantis alicuius, vt ex dato circulo quotcun-<lb/>que gradus poſsint abſcindi, &</s> <s xml:id="echoid-s1030" xml:space="preserve">c. </s> <s xml:id="echoid-s1031" xml:space="preserve">Habet figura hæc id commodi, quod pe-<lb/>riculum non eſt, ne clauus in centro atteratur, ſicut in ſuperiore inſtrumento. <lb/></s> <s xml:id="echoid-s1032" xml:space="preserve">Deinde in eadem hac figura poſſunt accipi particulę etiam minimę, prope <lb/>centrum, & </s> <s xml:id="echoid-s1033" xml:space="preserve">in extremo quadrante ſinus totus quamuis perpuſillus, quod in <lb/>ſuperiore inſtrumento non licebat.</s> <s xml:id="echoid-s1034" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="25"> <note symbol="a" position="right" xlink:label="note-043-02" xlink:href="note-043-02a" xml:space="preserve">2. ſexti.</note> </div> <p> <s xml:id="echoid-s1035" xml:space="preserve"><emph style="sc">Qvamvis</emph> autem ad magnitudinum dimenſiones non omnes huius in-<lb/>ſtrumenti partium vſus neceſſarij ſint, ſed ſolum ille, quem Num. </s> <s xml:id="echoid-s1036" xml:space="preserve">1. </s> <s xml:id="echoid-s1037" xml:space="preserve">& </s> <s xml:id="echoid-s1038" xml:space="preserve">2. </s> <s xml:id="echoid-s1039" xml:space="preserve">ex-<lb/>plicauimus, potiſsimum requiratur; </s> <s xml:id="echoid-s1040" xml:space="preserve">placuit tamen tam varios eius vſus in <lb/>vnum hunclocum congerere, tum vtinſtrumenti pręſtantia magis eluceat, <lb/>tum vtſtudioſus lector habeat, vbi alios vſus, quos deſiderat, inquirere de-<lb/>beat. </s> <s xml:id="echoid-s1041" xml:space="preserve">Non ſum etiam neſcius, quam plurimos alios pręclari huius inſtru-<lb/>menti vſus poſſe excogitari, quos proprio Marte, atque induſtria qui-<lb/>uis facile, quando idres poſtulauerit, cogitando inueniet: <lb/></s> <s xml:id="echoid-s1042" xml:space="preserve">nos præcipuos ſolum indicare voluimus <lb/>hoc loco.</s> <s xml:id="echoid-s1043" xml:space="preserve"/> </p> <pb o="14" file="044" n="44" rhead="GEOMETR. PRACT."/> <figure> <image file="044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/044-01"/> </figure> <p> <s xml:id="echoid-s1044" xml:space="preserve">Conſtructio Quadrantis, in quo minuta quoque, <lb/>acſecunda deprehendantur, etiamſi gradus in eaſecti non ſint. </s> <s xml:id="echoid-s1045" xml:space="preserve">Et <lb/>quo pacto eadem minuta, & </s> <s xml:id="echoid-s1046" xml:space="preserve">ſec. </s> <s xml:id="echoid-s1047" xml:space="preserve">obtineri poſſint in quadrante in <lb/>90. </s> <s xml:id="echoid-s1048" xml:space="preserve">gradus diſtributo. </s> <s xml:id="echoid-s1049" xml:space="preserve">Ac denique qua ratione ex data recta in <lb/>pauciſſimas partes æquales diuiſa abſcindi poſſint partes milleſi-<lb/>mæ, &</s> <s xml:id="echoid-s1050" xml:space="preserve">c.</s> <s xml:id="echoid-s1051" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div43" type="section" level="1" n="18"> <head xml:id="echoid-head21" xml:space="preserve">CAPVT II.</head> <p> <s xml:id="echoid-s1052" xml:space="preserve">HOc eſt ſecundum, quod præmittendum eſſe diximus, qua vi-<lb/>delicet via cognoſcere poſsimus, quotminuta, & </s> <s xml:id="echoid-s1053" xml:space="preserve">ſecunda in <lb/>propoſita particula cuiuſuis gradus contineantur: </s> <s xml:id="echoid-s1054" xml:space="preserve">Et quot <lb/>partes milleſimas quęlibet particula datęrectę comprehendat, <lb/>licet in pauciſsimas ea partes ſit diſtributa. </s> <s xml:id="echoid-s1055" xml:space="preserve">Quod vt aſſequa-<lb/>mur, conſtruendus eſt quadrans, quem anno 1586. </s> <s xml:id="echoid-s1056" xml:space="preserve">in Fabrica, & </s> <s xml:id="echoid-s1057" xml:space="preserve">vſu inſtru- <pb o="15" file="045" n="45" rhead="LIBER PRIMV"/> menti Horologiorũ confecimus, hoc modo. </s> <s xml:id="echoid-s1058" xml:space="preserve">Deſcriptis ex A, centro qua-<lb/>drantis B C, intra eundem quadrantem aliis 36. </s> <s xml:id="echoid-s1059" xml:space="preserve">quadrantibus æqualiter, ſi pla-<lb/>cet, inter ſe diſtantibus, vt venuſtior appareat figura: </s> <s xml:id="echoid-s1060" xml:space="preserve">ita vt in vniuerſum ſint <lb/> <anchor type="note" xlink:label="note-045-01a" xlink:href="note-045-01"/> 40. </s> <s xml:id="echoid-s1061" xml:space="preserve">quadrantes, quorum extimus in 90. </s> <s xml:id="echoid-s1062" xml:space="preserve">gradus more ſolito ſecetur: </s> <s xml:id="echoid-s1063" xml:space="preserve">proxi-<lb/>mus deinde in 128. </s> <s xml:id="echoid-s1064" xml:space="preserve">partes æquales, primum videlicet in duas, & </s> <s xml:id="echoid-s1065" xml:space="preserve">vtra que pars <lb/>rurſus in duas, & </s> <s xml:id="echoid-s1066" xml:space="preserve">quęlibet harum quatuor partium iterum in duas, & </s> <s xml:id="echoid-s1067" xml:space="preserve">ita <lb/>deinceps, donec 7. </s> <s xml:id="echoid-s1068" xml:space="preserve">diuiſiones abſolutæ ſint, atque adeò totus quadrans in <lb/>128. </s> <s xml:id="echoid-s1069" xml:space="preserve">partes ęquales diſtributus. </s> <s xml:id="echoid-s1070" xml:space="preserve">Poſt hæc producantur alij quadrantes vltra <lb/>ſemidiametrum A B: </s> <s xml:id="echoid-s1071" xml:space="preserve">ille quidem, qui tertius eſt ab extremo B C, vſque ad <lb/>gradum 91. </s> <s xml:id="echoid-s1072" xml:space="preserve">extremi quadrantis CB, producti, hoc eſt, vſque ad lineam ex A, <lb/>ad gradum 91. </s> <s xml:id="echoid-s1073" xml:space="preserve">ductam @ ſequens deinde vſque ad gradum 92. </s> <s xml:id="echoid-s1074" xml:space="preserve">& </s> <s xml:id="echoid-s1075" xml:space="preserve">inſequens <lb/>ad gradum 93. </s> <s xml:id="echoid-s1076" xml:space="preserve">atque ita deinceps vſque ad alios gradus, ita vt quadrageſi-<lb/>mus quadrans vſque ad gradum 128. </s> <s xml:id="echoid-s1077" xml:space="preserve">producatur. </s> <s xml:id="echoid-s1078" xml:space="preserve">Hiarcus ita producti diui-<lb/>dantur ſinguliin 128. </s> <s xml:id="echoid-s1079" xml:space="preserve">partes ęquales, ſicuti quadrans extimo quadranti pro-<lb/>ximus: </s> <s xml:id="echoid-s1080" xml:space="preserve">qua diuiſione peracta, partes ſupra ſemidiametrum A B, reſecentur, <lb/>tanquam ſuperuacaneæ.</s> <s xml:id="echoid-s1081" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="1"> <note position="right" xlink:label="note-045-01" xlink:href="note-045-01a" xml:space="preserve">Quadrantis<unsure/> <lb/>conſtructio <lb/>ad m. & ſec<unsure/>. <lb/>cognoſcenda.</note> </div> <p> <s xml:id="echoid-s1082" xml:space="preserve">2. </s> <s xml:id="echoid-s1083" xml:space="preserve"><emph style="sc">Qvod</emph> ſi quadrantes vltra ſemidiametrum A B, produci commodè <lb/>non poſsint, ob ſpacij anguſtias, inſtituenda erit diuiſio hoc modo. </s> <s xml:id="echoid-s1084" xml:space="preserve">In qua-<lb/>drante extremo B C, ſumatur ſemiſſis numeri graduum, quem quilibet arcus <lb/>productus continere deberet, & </s> <s xml:id="echoid-s1085" xml:space="preserve">ex A, ad illam ſemiſſem linea occulta duca-<lb/>tur. </s> <s xml:id="echoid-s1086" xml:space="preserve">Hęc enim ſecabit quadrantem propoſitum in puncto, vbi arcus produ-<lb/>ctus prima diuiſione bifariam ſecaretur. </s> <s xml:id="echoid-s1087" xml:space="preserve">Quare ſi arcus inter hoc punctum <lb/>& </s> <s xml:id="echoid-s1088" xml:space="preserve">ſemidiametrum A C, comprehendens 04. </s> <s xml:id="echoid-s1089" xml:space="preserve">partes ex illis 128. </s> <s xml:id="echoid-s1090" xml:space="preserve">totius arcus <lb/>producendi, ſecetur bifariam continuè ſex diuiſionibus, parteſque illius in <lb/>a@@um interidem punctum, & </s> <s xml:id="echoid-s1091" xml:space="preserve">ſemidiametrum A B, transferantur, quę tranſ-<lb/>ſ@rri poſſunt, habebuntur in dato quadrante omnes partes, quæ ex illis 128. <lb/></s> <s xml:id="echoid-s1092" xml:space="preserve">in quas totus arcus pro ductus diuideretur, in quadrantem cadunt. </s> <s xml:id="echoid-s1093" xml:space="preserve">Vt ſi di-<lb/>uidendus ſit quadrans M N, vſque ad grad. </s> <s xml:id="echoid-s1094" xml:space="preserve">104. </s> <s xml:id="echoid-s1095" xml:space="preserve">producendus, ducemus ad <lb/>grad. </s> <s xml:id="echoid-s1096" xml:space="preserve">52. </s> <s xml:id="echoid-s1097" xml:space="preserve">nimirum ad ſemiſſem grad. </s> <s xml:id="echoid-s1098" xml:space="preserve">104. </s> <s xml:id="echoid-s1099" xml:space="preserve">rectam, quę ſecet Quadrantem M N, <lb/>in O. </s> <s xml:id="echoid-s1100" xml:space="preserve">Nam ſi arcus O N, continens partes 64. </s> <s xml:id="echoid-s1101" xml:space="preserve">ex illis 128. </s> <s xml:id="echoid-s1102" xml:space="preserve">totius arcus produ-<lb/>cti, ſecetur continuè bifariam ſex diuiſio nibus, parteſque eius in arcum O M, <lb/>transferantur, habebuntur omnes partes in quadrantem M N, cadentes, non <lb/>ſecus, ac ſi totus arcus productus in 128. </s> <s xml:id="echoid-s1103" xml:space="preserve">partes diſtributus eſſet. </s> <s xml:id="echoid-s1104" xml:space="preserve">Sic etiam, <lb/>ſi quadrans ad gradum 125. </s> <s xml:id="echoid-s1105" xml:space="preserve">producendus, diuidendus ſit, ducenda erit linea <lb/>occulta ad gradum 62 {1/2} nimirum ad ſemiſſem graduum 125. </s> <s xml:id="echoid-s1106" xml:space="preserve">Item ſi qua-<lb/>drans D E, 120. </s> <s xml:id="echoid-s1107" xml:space="preserve">producendus, diuidendus ſit, ducenda erit linea occulta ad <lb/>gradum 60. </s> <s xml:id="echoid-s1108" xml:space="preserve">&</s> <s xml:id="echoid-s1109" xml:space="preserve">c.</s> <s xml:id="echoid-s1110" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1111" xml:space="preserve">3. </s> <s xml:id="echoid-s1112" xml:space="preserve"><emph style="sc">Hisce</emph> quadrantibus ita diuiſis duplices numeri aſſcribendi ſunt, pro-<lb/> <anchor type="note" xlink:label="note-045-02a" xlink:href="note-045-02"/> pe ſemidiametrum quidem A C, numeri quadrantum, vt 1. </s> <s xml:id="echoid-s1113" xml:space="preserve">prope extre-<lb/>mum; </s> <s xml:id="echoid-s1114" xml:space="preserve">2. </s> <s xml:id="echoid-s1115" xml:space="preserve">iuxta ſequentem; </s> <s xml:id="echoid-s1116" xml:space="preserve">& </s> <s xml:id="echoid-s1117" xml:space="preserve">3. </s> <s xml:id="echoid-s1118" xml:space="preserve">iuxta tertium, &</s> <s xml:id="echoid-s1119" xml:space="preserve">c. </s> <s xml:id="echoid-s1120" xml:space="preserve">Ita vides quadranti, qui <lb/>vſque ad gradum 96. </s> <s xml:id="echoid-s1121" xml:space="preserve">productus eſt, appoſitum eſſe numerum 8. </s> <s xml:id="echoid-s1122" xml:space="preserve">cum is octa-<lb/>uus ſit. </s> <s xml:id="echoid-s1123" xml:space="preserve">Primus enim eſt quadrans B C; </s> <s xml:id="echoid-s1124" xml:space="preserve">ſecundus, qui ſequitur, 90. </s> <s xml:id="echoid-s1125" xml:space="preserve">graduũ <lb/>Tertius graduum 91. </s> <s xml:id="echoid-s1126" xml:space="preserve">quartus graduum 92. </s> <s xml:id="echoid-s1127" xml:space="preserve">quintus graduum 93. </s> <s xml:id="echoid-s1128" xml:space="preserve">Sextus <lb/>graduum 94. </s> <s xml:id="echoid-s1129" xml:space="preserve">Septimus graduum 95. </s> <s xml:id="echoid-s1130" xml:space="preserve">& </s> <s xml:id="echoid-s1131" xml:space="preserve">Octauus graduum 96. </s> <s xml:id="echoid-s1132" xml:space="preserve">Sic etiam qua-<lb/>dranti vſque ad grad. </s> <s xml:id="echoid-s1133" xml:space="preserve">100. </s> <s xml:id="echoid-s1134" xml:space="preserve">preducto cernis aſſcriptum eſſe numerum 12. </s> <s xml:id="echoid-s1135" xml:space="preserve">&</s> <s xml:id="echoid-s1136" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1137" xml:space="preserve">At verò iuxta ſemidiametrum A B, numeriillorum graduum ſcribendi ſunt, ad <lb/>quos vſque quilibet quadrans extenditur, vt in figura vides. </s> <s xml:id="echoid-s1138" xml:space="preserve">Ita enim caden- <pb file="046" n="46"/> <anchor type="handwritten" xlink:label="hd-046-1a" xlink:href="hd-046-1"/> <anchor type="figure" xlink:label="fig-046-01a" xlink:href="fig-046-01"/> <anchor type="handwritten" xlink:label="hd-046-2a" xlink:href="hd-046-2"/> <pb o="17" file="047" n="47" rhead="LIBER PRIMVS."/> filo perpendiculi in partem aliquam integram alicuius quadrantis, illico iuxta <lb/>ſemidiametrum A B, apparebit, ad quem gradum vſq; </s> <s xml:id="echoid-s1139" xml:space="preserve">quadransille productus <lb/>fuit. </s> <s xml:id="echoid-s1140" xml:space="preserve">Qui quidem graduum numerus in regula trium tertium occupatlocum, vt <lb/>minuta, atque ſecunda inquirantur, vt paulo poſt Num. </s> <s xml:id="echoid-s1141" xml:space="preserve">7. </s> <s xml:id="echoid-s1142" xml:space="preserve">dicemus.</s> <s xml:id="echoid-s1143" xml:space="preserve"/> </p> <div xml:id="echoid-div44" type="float" level="2" n="2"> <note position="right" xlink:label="note-045-02" xlink:href="note-045-02a" xml:space="preserve">Quinumeri <lb/>quæ<unsure/>drantib{us} <lb/>aſſcribend@ <lb/>ſint.</note> <handwritten xlink:label="hd-046-1" xlink:href="hd-046-1a"/> <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/xxxxxxxx/figures/046-01"/> </figure> <handwritten xlink:label="hd-046-2" xlink:href="hd-046-2a"/> </div> <p> <s xml:id="echoid-s1144" xml:space="preserve">4. </s> <s xml:id="echoid-s1145" xml:space="preserve"><emph style="sc">Ivxta</emph> ſemidiametrum A B, affigenda ſunt duo pinnacidia ad angulos <lb/> <anchor type="note" xlink:label="note-047-01a" xlink:href="note-047-01"/> rectos, ita vt foramina, per quę radius ſolis, vel viſualis tranſire debet, ad per-<lb/>pendiculum rectę AB, exiſtant; </s> <s xml:id="echoid-s1146" xml:space="preserve">alio quin non paruus error in dimenſione linea-<lb/>rum committeretur.</s> <s xml:id="echoid-s1147" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="3"> <note position="right" xlink:label="note-047-01" xlink:href="note-047-01a" xml:space="preserve">Pinnacidia <lb/>quo pacto af-<lb/>figenda.</note> </div> <p> <s xml:id="echoid-s1148" xml:space="preserve">5. </s> <s xml:id="echoid-s1149" xml:space="preserve"><emph style="sc">Qvando</emph> porro per radium viſualem altitudo ſt ellę inueſtiganda eſt, <lb/> <anchor type="note" xlink:label="note-047-02a" xlink:href="note-047-02"/> vel punctum aliquod lineæ dimetiendæ inſpiciendum, conſtrui debent duo pin-<lb/>nacidia hoc modo. </s> <s xml:id="echoid-s1150" xml:space="preserve">In tabella ænea quadrata IK, fiat foramen rotundum medio-<lb/>cris magnitudinis, in cuius medio relinquatur foramen L, quod ſuſtineatur à <lb/>diametro quadam tenui; </s> <s xml:id="echoid-s1151" xml:space="preserve">Et circa I, circumuertatur alia tabella ænea quadrata <lb/>ſubtilis, priori ęqualis, in cuius medio ſit etiam perexiguum foramen M, reſp on-<lb/>dens foramini L, quando hęc tabella priori ſup erimponitur. </s> <s xml:id="echoid-s1152" xml:space="preserve">Huiuſmodi duo <lb/>pinnacidia ſi fiant, dici vix poteſt, quàm expeditè quamcunque ſtellam, aut a-<lb/>liam quamlibet rem contueri liceat. </s> <s xml:id="echoid-s1153" xml:space="preserve">Nam pinnacidium, quod ab oculo propius <lb/>abeſt, claudendum eſt tabella illa quadrata circumducta circa punctum I, aliud <lb/>autem aperiendum. </s> <s xml:id="echoid-s1154" xml:space="preserve">Sic enim fiet, vt radius viſualis per foramen M, prope ocu-<lb/>lumimmiſſus, illico conſpiciat per maius foramen L, in pinnacidio remotiore <lb/>ſtellam, vel aliam rem propoſitam: </s> <s xml:id="echoid-s1155" xml:space="preserve">quia foramen illud maius apertum facilè <lb/>remipſam intueri, & </s> <s xml:id="echoid-s1156" xml:space="preserve">ſine vllo negotio foramen exiguum L, in eodem pinnaci-<lb/>dio remotiore in ipſam rem viſum dirigere nosſinit.</s> <s xml:id="echoid-s1157" xml:space="preserve"/> </p> <div xml:id="echoid-div46" type="float" level="2" n="4"> <note position="right" xlink:label="note-047-02" xlink:href="note-047-02a" xml:space="preserve">Pinnacidia <lb/>pro radio vi-<lb/>ſuali quo pa-<lb/>cto conſtruẽ-<lb/>da.</note> </div> <p> <s xml:id="echoid-s1158" xml:space="preserve">6. </s> <s xml:id="echoid-s1159" xml:space="preserve"><emph style="sc">Postremo</emph> ex centro A, egrediatur filum ſub tiliſsimum cum appenſo <lb/> <anchor type="note" xlink:label="note-047-03a" xlink:href="note-047-03"/> perpendiculo. </s> <s xml:id="echoid-s1160" xml:space="preserve">Aut certè loco fili conſtruatur regula ænea admodum tenuis <lb/>cum linea fiducię, in cuius extremitate promineat laminula, ex qua ſuſpendatur <lb/>perpendiculum hac conditione, vt regula liberè pendente, filum aliquod cum <lb/>perpendiculo demiſſum, ad vnguem lineę fiducię reſpondeat. </s> <s xml:id="echoid-s1161" xml:space="preserve">Atq; </s> <s xml:id="echoid-s1162" xml:space="preserve">in hoc ſum̃a <lb/>diligentia adhibenda eſt, alioquin gradus non rectè à linea fiducię indicarentur.</s> <s xml:id="echoid-s1163" xml:space="preserve"/> </p> <div xml:id="echoid-div47" type="float" level="2" n="5"> <note position="right" xlink:label="note-047-03" xlink:href="note-047-03a" xml:space="preserve">Conſtructio <lb/>regulæ, loco <lb/>fili.</note> </div> <p> <s xml:id="echoid-s1164" xml:space="preserve"><emph style="sc">Atqve</emph> hoc modo Quadrans in ſuo vſu erit pendulus, ſiueres in ſublimi <lb/> <anchor type="note" xlink:label="note-047-04a" xlink:href="note-047-04"/> exiſtens ex B, per A, ſiueres in plano poſita ex A, per B, inſpiciatur.</s> <s xml:id="echoid-s1165" xml:space="preserve"/> </p> <div xml:id="echoid-div48" type="float" level="2" n="6"> <note position="right" xlink:label="note-047-04" xlink:href="note-047-04a" xml:space="preserve">Quadrans <lb/>pendul{us}.</note> </div> <p> <s xml:id="echoid-s1166" xml:space="preserve"><emph style="sc">Qvod</emph> ſi circa centrum A, regula affigatur cum linea fiducię AB, & </s> <s xml:id="echoid-s1167" xml:space="preserve">duobus <lb/>pinnacidijs c, b quorum foramina lineę fiduclę reſpondeant, ipſa queregula ita <lb/> <anchor type="note" xlink:label="note-047-05a" xlink:href="note-047-05"/> firmetur, vt cir ca centrum circumducta ad quemcunque gradum immota per-<lb/>maneat, erit Quadrans in ſuo vſu ſtabilis eundem ſemper ſitum habens, ſiueres <lb/>in ſublimi exiſtens inſpiciatur ex A, per b, poſito nimirum latere A C, Horizonti <lb/>parallelo, in plano Horizontali, ſiue rem in plano poſitam quis intueatur ex A, <lb/>per b, latere A C, ad Horizontem exiſtente perpendiculari, & </s> <s xml:id="echoid-s1168" xml:space="preserve">latere A B, eidem <lb/>Horizonti parallelo, ſuperioremq; </s> <s xml:id="echoid-s1169" xml:space="preserve">locũ occupante. </s> <s xml:id="echoid-s1170" xml:space="preserve">Verum hęc planius intel-<lb/>ligentur, cum de vtro que vſu lib. </s> <s xml:id="echoid-s1171" xml:space="preserve">2. </s> <s xml:id="echoid-s1172" xml:space="preserve">agemus.</s> <s xml:id="echoid-s1173" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="7"> <note position="right" xlink:label="note-047-05" xlink:href="note-047-05a" xml:space="preserve">Quadrans <lb/>ſtabilis.</note> </div> <p> <s xml:id="echoid-s1174" xml:space="preserve">7. </s> <s xml:id="echoid-s1175" xml:space="preserve"><emph style="sc">Vsvs</emph> quadrantis hoc modo conſtructi in minutis, ac ſecundis exquiren-<lb/> <anchor type="note" xlink:label="note-047-06a" xlink:href="note-047-06"/> dis, pręclarus eſt. </s> <s xml:id="echoid-s1176" xml:space="preserve">Nam cadente filo perpendiculi, aut linea fiducię AB, in par-<lb/>tem aliquamintegram alicuius Quadrantis (quod ferè ſemper accidet propter <lb/>diuerſitatem partium in tanta quadrantum multitudine) ſi fiat vt 128. </s> <s xml:id="echoid-s1177" xml:space="preserve">nimirum <lb/>vt numerus partium, in quas quilibet arcus productus diuiditur, ad partes à filo <lb/>abſciſſas, ita numerus graduum in toto arcu pro ducto comprehenſorum, in <lb/>cuius partem aliquam integram filum incidit, ad aliud, reperietur numerus gra- <pb o="18" file="048" n="48" rhead="GEOMETR. PRACT."/> duumin arcu abſciſſo contentorum. </s> <s xml:id="echoid-s1178" xml:space="preserve">Et ſi quid in diuiſione fueritreſidui, illud <lb/>per 60. </s> <s xml:id="echoid-s1179" xml:space="preserve">multiplicatum, atq; </s> <s xml:id="echoid-s1180" xml:space="preserve">in eundem diuiſorem, hoc eſt, in 128. </s> <s xml:id="echoid-s1181" xml:space="preserve">diuiſum, dabit <lb/>minuta graduum. </s> <s xml:id="echoid-s1182" xml:space="preserve">Et ſi adhuc quippiam remanſerit in hac diuiſione, illud eodem <lb/>modo per 60. </s> <s xml:id="echoid-s1183" xml:space="preserve">multiplicatum, & </s> <s xml:id="echoid-s1184" xml:space="preserve">in eundem diuiſorem 128, diuiſum, exhibebit <lb/>ſecunda. </s> <s xml:id="echoid-s1185" xml:space="preserve">Atq; </s> <s xml:id="echoid-s1186" xml:space="preserve">hoc modo progrediendo, reperientur Tertia, Quarta, &</s> <s xml:id="echoid-s1187" xml:space="preserve">c. </s> <s xml:id="echoid-s1188" xml:space="preserve">do-<lb/>nec nihil in diuiſione ſuperſit. </s> <s xml:id="echoid-s1189" xml:space="preserve">Tunc enim vlterius progrediendum non eſt; </s> <s xml:id="echoid-s1190" xml:space="preserve">Sed <lb/>fatis eſt, ad ſecunda vſq; </s> <s xml:id="echoid-s1191" xml:space="preserve">progredi. </s> <s xml:id="echoid-s1192" xml:space="preserve">Exempligratia. </s> <s xml:id="echoid-s1193" xml:space="preserve">Ponatur ex Quadrante P Q, <lb/>vſq; </s> <s xml:id="echoid-s1194" xml:space="preserve">ad gradum 100. </s> <s xml:id="echoid-s1195" xml:space="preserve">producto, qualis eſt duo decimus, filum perpendiculi ab-<lb/> <anchor type="handwritten" xlink:label="hd-048-1a" xlink:href="hd-048-1"/> ſcidiſſe partes 20. </s> <s xml:id="echoid-s1196" xml:space="preserve">ex illis 128. </s> <s xml:id="echoid-s1197" xml:space="preserve">in quas totus arcus productus diſtributus eſt. </s> <s xml:id="echoid-s1198" xml:space="preserve">Fiat <lb/>ergo, vt 128. </s> <s xml:id="echoid-s1199" xml:space="preserve">ad 20. </s> <s xml:id="echoid-s1200" xml:space="preserve">ita 100. </s> <s xml:id="echoid-s1201" xml:space="preserve">ad aliud; </s> <s xml:id="echoid-s1202" xml:space="preserve">inuenienturq; </s> <s xml:id="echoid-s1203" xml:space="preserve">gradus 15. </s> <s xml:id="echoid-s1204" xml:space="preserve">ſupereruntq; </s> <s xml:id="echoid-s1205" xml:space="preserve">in di-<lb/> <anchor type="handwritten" xlink:label="hd-048-1a" xlink:href="hd-048-1"/> uiſione 80. </s> <s xml:id="echoid-s1206" xml:space="preserve">quę ducta in 60. </s> <s xml:id="echoid-s1207" xml:space="preserve">faciunt 4800. </s> <s xml:id="echoid-s1208" xml:space="preserve">quę diuiſa per 128. </s> <s xml:id="echoid-s1209" xml:space="preserve">dant minuta 37, & </s> <s xml:id="echoid-s1210" xml:space="preserve"><lb/>ſuperſunt adhuc 64. </s> <s xml:id="echoid-s1211" xml:space="preserve">quæ ſi ducãtur in 60. </s> <s xml:id="echoid-s1212" xml:space="preserve">& </s> <s xml:id="echoid-s1213" xml:space="preserve">productus numerus 3840. </s> <s xml:id="echoid-s1214" xml:space="preserve">diuida-<lb/>tur per 128. </s> <s xml:id="echoid-s1215" xml:space="preserve">prodibunt Sec. </s> <s xml:id="echoid-s1216" xml:space="preserve">30. </s> <s xml:id="echoid-s1217" xml:space="preserve">nihilq; </s> <s xml:id="echoid-s1218" xml:space="preserve">in diuiſione ſupereſt. </s> <s xml:id="echoid-s1219" xml:space="preserve">Arcus ergo Q 20, <lb/>vel arcus Quadrantis BC, inter C, & </s> <s xml:id="echoid-s1220" xml:space="preserve">filum perpendiculi includit gr. </s> <s xml:id="echoid-s1221" xml:space="preserve">15. </s> <s xml:id="echoid-s1222" xml:space="preserve">Min. </s> <s xml:id="echoid-s1223" xml:space="preserve">37. <lb/></s> <s xml:id="echoid-s1224" xml:space="preserve">Sec. </s> <s xml:id="echoid-s1225" xml:space="preserve">30. </s> <s xml:id="echoid-s1226" xml:space="preserve">Rurſus ponamus ex octauo quadrante RS, vſq; </s> <s xml:id="echoid-s1227" xml:space="preserve">ad gradum 96. </s> <s xml:id="echoid-s1228" xml:space="preserve">produ-<lb/>cto filum perpẽdiculi ab ſcidiſſe partes 96. </s> <s xml:id="echoid-s1229" xml:space="preserve">ex illis 128. </s> <s xml:id="echoid-s1230" xml:space="preserve">quæ in toto arcu producto <lb/>continentur. </s> <s xml:id="echoid-s1231" xml:space="preserve">Fiat ergo, vt 128. </s> <s xml:id="echoid-s1232" xml:space="preserve">ad 96. </s> <s xml:id="echoid-s1233" xml:space="preserve">ita 96. </s> <s xml:id="echoid-s1234" xml:space="preserve">ad aliud: </s> <s xml:id="echoid-s1235" xml:space="preserve">reperientur que gradus 72. </s> <s xml:id="echoid-s1236" xml:space="preserve"><lb/>pręcisè arcui abſciſſo conuenire. </s> <s xml:id="echoid-s1237" xml:space="preserve">Item ceciderit filum in partem 64. </s> <s xml:id="echoid-s1238" xml:space="preserve">Quadrantis <lb/>ſextidecimi MN, vſque ad gradum 104. </s> <s xml:id="echoid-s1239" xml:space="preserve">producti. </s> <s xml:id="echoid-s1240" xml:space="preserve">Si ergo fiat, vt 128. </s> <s xml:id="echoid-s1241" xml:space="preserve">ad 64. </s> <s xml:id="echoid-s1242" xml:space="preserve">ita <lb/>104. </s> <s xml:id="echoid-s1243" xml:space="preserve">ad aliud, producentur quo que grad. </s> <s xml:id="echoid-s1244" xml:space="preserve">52. </s> <s xml:id="echoid-s1245" xml:space="preserve">præcisè. </s> <s xml:id="echoid-s1246" xml:space="preserve">atque ita de cæteris; </s> <s xml:id="echoid-s1247" xml:space="preserve">dum-<lb/>modo ſis memor, vt ſi quid in diuiſi onibus ſuperfuerit, reſidua diuiſionum mul-<lb/>tiplicentur per 60. </s> <s xml:id="echoid-s1248" xml:space="preserve">& </s> <s xml:id="echoid-s1249" xml:space="preserve">producti numeri per 128. </s> <s xml:id="echoid-s1250" xml:space="preserve">diuidantur, vt dictum eſt.</s> <s xml:id="echoid-s1251" xml:space="preserve"/> </p> <div xml:id="echoid-div50" type="float" level="2" n="8"> <note position="right" xlink:label="note-047-06" xlink:href="note-047-06a" xml:space="preserve">Vſ{us} quadrã-<lb/>tis proximè <lb/>conſtruct in <lb/>minutis ex-<lb/>quirendis. <lb/><lb/></note> <handwritten xlink:label="hd-048-1" xlink:href="hd-048-1a"/> <handwritten xlink:label="hd-048-1" xlink:href="hd-048-1a"/> </div> <p> <s xml:id="echoid-s1252" xml:space="preserve"><emph style="sc">Demonstratio</emph> huius operationis perſpicua eſt. </s> <s xml:id="echoid-s1253" xml:space="preserve">Quoniam enim eſt, (in <lb/>vltimo exemplo) vt arcus N M, vſque ad grad. </s> <s xml:id="echoid-s1254" xml:space="preserve">104. </s> <s xml:id="echoid-s1255" xml:space="preserve">productus, quatenus in <lb/> <anchor type="handwritten" xlink:label="hd-048-1a" xlink:href="hd-048-1"/> 128. </s> <s xml:id="echoid-s1256" xml:space="preserve">partes ſectus eſt, ad arcum N O, earundem partium 64. </s> <s xml:id="echoid-s1257" xml:space="preserve">vt idem arcus N M, <lb/>totus productus, quatenus grad. </s> <s xml:id="echoid-s1258" xml:space="preserve">104. </s> <s xml:id="echoid-s1259" xml:space="preserve">complectitur, ad eundem arcum N O, re-<lb/>ſpectu eorundem graduum; </s> <s xml:id="echoid-s1260" xml:space="preserve">efficitur, vt ſi fiat quemadmodum partes 128, totius <lb/>arcus N M, vſque ad grad. </s> <s xml:id="echoid-s1261" xml:space="preserve">104. </s> <s xml:id="echoid-s1262" xml:space="preserve">producti ad partes 64. </s> <s xml:id="echoid-s1263" xml:space="preserve">in arcu N O, contentas ita <lb/>idem arcus N M, productus graduum 104. </s> <s xml:id="echoid-s1264" xml:space="preserve">ad aliud, reperiantur gradus in eo dem <lb/>arcu N O, contenti, &</s> <s xml:id="echoid-s1265" xml:space="preserve">c.</s> <s xml:id="echoid-s1266" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="9"> <handwritten xlink:label="hd-048-1" xlink:href="hd-048-1a"/> </div> <p> <s xml:id="echoid-s1267" xml:space="preserve">8. </s> <s xml:id="echoid-s1268" xml:space="preserve"><emph style="sc">In</emph> gratiam autem ſtudio ſorum placet hic tabellam inſerere, in qua ex re-<lb/> <anchor type="note" xlink:label="note-048-01a" xlink:href="note-048-01"/> ſiduo primæ operationis regulæ aureæ, qua gradus eliciuntur, mox apparet, <lb/>quot minuta, & </s> <s xml:id="echoid-s1269" xml:space="preserve">ſecunda illi reſi duo reſpondeant: </s> <s xml:id="echoid-s1270" xml:space="preserve">Ita vt opus ſit ſemel tantum <lb/>regulam auream adhibere. </s> <s xml:id="echoid-s1271" xml:space="preserve">Conſtruitur autem tabella, ſi ſingula reſidua, quæ <lb/>plura, quam 127, eſſe nequeunt, per 60. </s> <s xml:id="echoid-s1272" xml:space="preserve">multiplicentur, productique numeri per <lb/>128. </s> <s xml:id="echoid-s1273" xml:space="preserve">diuidantur. </s> <s xml:id="echoid-s1274" xml:space="preserve">Atque vt ſtructura, & </s> <s xml:id="echoid-s1275" xml:space="preserve">vſus huiuſce tabellæ facilius intelliga-<lb/>tur, apponemus vnum exemplum. </s> <s xml:id="echoid-s1276" xml:space="preserve">Cadat verbi gratia filum perpendiculi in <lb/> <anchor type="handwritten" xlink:label="hd-048-1a" xlink:href="hd-048-1"/> partem 29. </s> <s xml:id="echoid-s1277" xml:space="preserve">Quadrantis 32. </s> <s xml:id="echoid-s1278" xml:space="preserve">ad gradum vſque 120. </s> <s xml:id="echoid-s1279" xml:space="preserve">producti. </s> <s xml:id="echoid-s1280" xml:space="preserve">Fiat igitur vt 128. <lb/></s> <s xml:id="echoid-s1281" xml:space="preserve">ad 29. </s> <s xml:id="echoid-s1282" xml:space="preserve">ita 120. </s> <s xml:id="echoid-s1283" xml:space="preserve">ad aliud; </s> <s xml:id="echoid-s1284" xml:space="preserve">producenturque grad. </s> <s xml:id="echoid-s1285" xml:space="preserve">27. </s> <s xml:id="echoid-s1286" xml:space="preserve">Quia verò in diuiſione ſu-<lb/> <anchor type="handwritten" xlink:label="hd-048-1a" xlink:href="hd-048-1"/> perſunt 24. </s> <s xml:id="echoid-s1287" xml:space="preserve">ſub quo numero in tabella ponuntur duo hi numeri 11. </s> <s xml:id="echoid-s1288" xml:space="preserve">15. </s> <s xml:id="echoid-s1289" xml:space="preserve">Prior <lb/> <anchor type="handwritten" xlink:label="hd-048-1a" xlink:href="hd-048-1"/> ergo dat minuta, & </s> <s xml:id="echoid-s1290" xml:space="preserve">poſterior ſecunda; </s> <s xml:id="echoid-s1291" xml:space="preserve">Ita vt arcus à filo abſciſſus complecta-<lb/>tur grad. </s> <s xml:id="echoid-s1292" xml:space="preserve">27: </s> <s xml:id="echoid-s1293" xml:space="preserve">Min. </s> <s xml:id="echoid-s1294" xml:space="preserve">11. </s> <s xml:id="echoid-s1295" xml:space="preserve">ſec. </s> <s xml:id="echoid-s1296" xml:space="preserve">15. </s> <s xml:id="echoid-s1297" xml:space="preserve">Atque hæc minuta, & </s> <s xml:id="echoid-s1298" xml:space="preserve">Secunda producuntur, ſi reſi-<lb/>duum diuiſionis, nimirum 24. </s> <s xml:id="echoid-s1299" xml:space="preserve">ducatur in 60. </s> <s xml:id="echoid-s1300" xml:space="preserve">& </s> <s xml:id="echoid-s1301" xml:space="preserve">productus numerus per 128. </s> <s xml:id="echoid-s1302" xml:space="preserve">di-<lb/>uidatur, &</s> <s xml:id="echoid-s1303" xml:space="preserve">c. </s> <s xml:id="echoid-s1304" xml:space="preserve">Eadem ratio eſt dereliquis tabellę numeris. </s> <s xml:id="echoid-s1305" xml:space="preserve">Nam ſemper ſuperior <lb/>numerus eſt ille, qui in diuiſione remanſit: </s> <s xml:id="echoid-s1306" xml:space="preserve">Inferiorum autem numerorum prior <lb/>ad minuta, & </s> <s xml:id="echoid-s1307" xml:space="preserve">poſterior ad ſecunda ſpectat.</s> <s xml:id="echoid-s1308" xml:space="preserve"/> </p> <div xml:id="echoid-div52" type="float" level="2" n="10"> <note position="left" xlink:label="note-048-01" xlink:href="note-048-01a" xml:space="preserve">Conſtructio <lb/>& vſ{us} tabel-<lb/>la pro minutis <lb/>& ſecundis.</note> <handwritten xlink:label="hd-048-1" xlink:href="hd-048-1a"/> <handwritten xlink:label="hd-048-1" xlink:href="hd-048-1a"/> <handwritten xlink:label="hd-048-1" xlink:href="hd-048-1a"/> </div> </div> <div xml:id="echoid-div54" type="section" level="1" n="19"> <head xml:id="echoid-head22" xml:space="preserve">SEQVITVR TABELLA.</head> <pb o="19" file="049" n="49" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s1309" xml:space="preserve">TABELLA INDICANS, QVOT MI-<lb/>nuta, ac Secunda reſiduo primæ operationis regu-<lb/>læ aureæ, qua gradus in ſupra nominatæ tabulæ <lb/>conſtructione eruuntur, reſpondeant.</s> <s xml:id="echoid-s1310" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>## 1 ## 2 ## 3 ## 4 ## 5 ## 6 ## 7 ## 8 ## 9 ## 10 ## 11 ## 12 <lb/> <lb/>M # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>0. # 28 # 0. # 56 # 1. # 24 # 1. # 52 # 2. # 21 # 2. # 49 # 3. # 17 # 3. # 45 # 4. # 13 # 4. # 41 # 5. # 9 # 5. # 37 <lb/>## 13 ## 14 ## 15 ## 16 ## 17 ## 18 ## 19 ## 20 ## 21 ## 22 ## 23 ## 24 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/><lb/>6. # 6 # 6. # 34 # 7. # 2 # 7. # 30 # 7. # 58 # 8. # 26 # 8. # 54 # 9. # 22 # 9. # 51 # 10. # 19 # 10. # 47 # 11. # 15 <lb/>## 25 ## 26 ## 27 ## 28 ## 29 ## 30 ## 31 ## 32 ## 33 ## 34 ## 35 ## 36 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>11. # 43 # 12. # 11 # 12. # 39 # 13. # 7 # 13. # 36 # 14. # 4 # 14. # 32 # 15. # 0 # 15. # 28 # 15. # 56 # 16. # 24 # 16. # 52 <lb/>## 37 ## 38 ## 39 ## 40 ## 41 ## 42 ## 43 ## 44 ## 45 ## 46 ## 47 ## 48 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>17. # 21 # 17. # 49 # 18. # 17 # 18. # 45 # 9. # 13 # 19. # 41 # 20. # 9 # 20. # 37 # 21. # 6 # 21. # 34 # 22. # 2 # 22. # 30 <lb/>## 49 ## 50 ## 51 ## 52 ## 53 ## 54 ## 55 ## 56 ## 57 ## 58 ## 59 ## 60 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>22. # 58 # 23. # 26 # 23. # 54 # 24 # 22 # 24. # 51 # 25. # 19 # 25. # 47 # 26. # 15 # 26. # 43 # 27. # 11 # 27. # 39 # 28. # 7 <lb/>## 61 ## 62 ## 63 ## 64 ## 65 ## 66 ## 67 ## 68 ## 69 ## 70 ## 71 ## 72 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>28. # 36 # 29. # 4 # 29. # 32 # 30. # 0 # 30. # 28 # 30. # 56 # 31. # 24 # 31 # 52 # 32. # 21 # 32. # 49 # 33. # 17 # 33. # 45 <lb/>## 73 ## 74 ## 75 ## 76 ## 77 ## 78 ## 79 ## 80 ## 81 ## 82 ## 83 ## 84 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>34. # 13 # 34. # 41 # 35. # 9 # 35. # 37 # 36. # 6 # 36. # 34 # 37. # 2 # 37. # 30 # 37. # 58 # 38. # 26 # 38. # 54 # 39. # 22 <lb/>## 85 ## 86 ## 87 ## 88 ## 89 ## 90 ## 91 ## 92 ## 93 ## 94 ## 95 ## 96 <lb/>M. # S # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>39. # 51 # 40. # 19 # 40. # 47 # 41. # 15 # 41. # 43 # 42. # 11 # 42. # 39 # 43. # 7 # 43. # 36 # 44. # 4 # 44. # 32 # 45. # 0 <lb/>## 97 ## 98 ## 99 ## 100 ## 101 ## 102 ## 103 ## 104 ## 105 ## 106 ## 107 ## 108 <lb/>M. # S. # M. # S. # M. # S. # M. # S # M. # S # M. # S. # M. # S. # M. # S. # M. # S. # M. # S # M. # S. # M. # S. <lb/>45. # 28 # 45 # 56 # 46. # 24 # 46. # 52 # 47. # 21 # 47. # 49 # 48. # 17 # 48. # 45 # 49. # 13 # 49 # 41 # 50 # 9 # 50. # 37 <lb/>## 109 ## 110 ## 111 ## 112 ## 113 ## 114 ## 115 ## 116 ## 117 ## 118 ## 119 ## 120 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>51. # 6 # 51. # 34 # 52. # 2 # 52. # 30 # 52. # 58 # 53. # 26 # 53. # 54 # 54. # 22 # 54. # 5 # 55. # 19 # 55. # 47 # 56. # 15 <lb/>## 121 ## 122 ## 123 ## 124 ## 125 ## 126 ## 127 ## 128 <lb/>M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. # M. # S. <lb/>56. # 43 # 57 # 1 # 5@. # 39 # 58. # 7 # 8. # 36 # 59. # 4 # 59. # 32 # 60. # 0 <lb/></note> <pb o="20" file="050" n="50" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s1311" xml:space="preserve">9. </s> <s xml:id="echoid-s1312" xml:space="preserve"><emph style="sc">Porro</emph> vt ſtudioſos omni labore ſupputandi leuaremus, compoſita <lb/> <anchor type="note" xlink:label="note-050-01a" xlink:href="note-050-01"/> à nobis eſt ſequens tabula, in qua confeſtim apparet, quot gradus, minuta, ac <lb/>ſecunda cuilibet parti cuiuſuis Quadrantis reſpondeant. </s> <s xml:id="echoid-s1313" xml:space="preserve">Nam ſi in latere tabu-<lb/>læſiniſtro ſumatur numerus illius quadrantis, in cuius partem aliquam integram <lb/>filum perpendiculi cecidit, numerus, inquam, iuxta ſemidiametrum A C, illi <lb/>quadranti appoſitus, in vertice verò eiuſdem tabulæ acc@piatur numerus par-<lb/>tium à filo abſciſſarum, reperientur in angulo communigrad. </s> <s xml:id="echoid-s1314" xml:space="preserve">Min. </s> <s xml:id="echoid-s1315" xml:space="preserve">& </s> <s xml:id="echoid-s1316" xml:space="preserve">Sec. </s> <s xml:id="echoid-s1317" xml:space="preserve">ar-<lb/>cus abſciſsi. </s> <s xml:id="echoid-s1318" xml:space="preserve">Exemplum. </s> <s xml:id="echoid-s1319" xml:space="preserve">Ceciderit filum in partem 30. </s> <s xml:id="echoid-s1320" xml:space="preserve">Quadrantis 16. </s> <s xml:id="echoid-s1321" xml:space="preserve">qui vſ-<lb/>que ad grad. </s> <s xml:id="echoid-s1322" xml:space="preserve">104. </s> <s xml:id="echoid-s1323" xml:space="preserve">productus fuit. </s> <s xml:id="echoid-s1324" xml:space="preserve">Si ergo in vertice tabulæ ſumatur numerus <lb/>30. </s> <s xml:id="echoid-s1325" xml:space="preserve">partium, & </s> <s xml:id="echoid-s1326" xml:space="preserve">in ſiniſtro latere numerus quadrantis 16. </s> <s xml:id="echoid-s1327" xml:space="preserve">deprehendentur in com-<lb/>muni angulo grad. </s> <s xml:id="echoid-s1328" xml:space="preserve">24. </s> <s xml:id="echoid-s1329" xml:space="preserve">Min. </s> <s xml:id="echoid-s1330" xml:space="preserve">3. </s> <s xml:id="echoid-s1331" xml:space="preserve">Sec. </s> <s xml:id="echoid-s1332" xml:space="preserve">30. </s> <s xml:id="echoid-s1333" xml:space="preserve">Item cadente filo in partem 111. </s> <s xml:id="echoid-s1334" xml:space="preserve">Qua-<lb/>drantis 15. </s> <s xml:id="echoid-s1335" xml:space="preserve">qui vſque ad grad. </s> <s xml:id="echoid-s1336" xml:space="preserve">103. </s> <s xml:id="echoid-s1337" xml:space="preserve">fuit productus; </s> <s xml:id="echoid-s1338" xml:space="preserve">ſi in vertice tabulę accipia-<lb/>tur numerus 111. </s> <s xml:id="echoid-s1339" xml:space="preserve">partium, & </s> <s xml:id="echoid-s1340" xml:space="preserve">in latere ſiniſtro numerus Quadrantis 15. </s> <s xml:id="echoid-s1341" xml:space="preserve">reperien-<lb/>tur in angulo communi gradus 89. </s> <s xml:id="echoid-s1342" xml:space="preserve">min. </s> <s xml:id="echoid-s1343" xml:space="preserve">19. </s> <s xml:id="echoid-s1344" xml:space="preserve">ſec. </s> <s xml:id="echoid-s1345" xml:space="preserve">13. </s> <s xml:id="echoid-s1346" xml:space="preserve">Atq; </s> <s xml:id="echoid-s1347" xml:space="preserve">ita de cæteris. </s> <s xml:id="echoid-s1348" xml:space="preserve">Conſtru-<lb/>ctio tabulę ex dictis obſcura non eſt. </s> <s xml:id="echoid-s1349" xml:space="preserve">Nam ſi fiat, vt 128, ad 1. </s> <s xml:id="echoid-s1350" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s1351" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s1352" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s1353" xml:space="preserve">& </s> <s xml:id="echoid-s1354" xml:space="preserve"><lb/> <anchor type="handwritten" xlink:label="hd-050-1a" xlink:href="hd-050-1"/> ita deinceps, vſque ad 128. </s> <s xml:id="echoid-s1355" xml:space="preserve">ita numerus graduum cuiuslibet arcus to tius pro-<lb/>ducti ad aliud, reperientur gradus. </s> <s xml:id="echoid-s1356" xml:space="preserve">Minuta & </s> <s xml:id="echoid-s1357" xml:space="preserve">Sec. </s> <s xml:id="echoid-s1358" xml:space="preserve">pro partibus cuiuſque Qua-<lb/>drantis. </s> <s xml:id="echoid-s1359" xml:space="preserve">Continentur autem in tabula tantummodo 40. </s> <s xml:id="echoid-s1360" xml:space="preserve">Quadrantes, quod hi <lb/>ſatis eſſe videantur: </s> <s xml:id="echoid-s1361" xml:space="preserve">Si quis tamen plures deſcribere velit facilè tabulam exten-<lb/>dere poterit ſecundum do ctrinam traditam hoc loco ad quotuis Quadrantes. <lb/></s> <s xml:id="echoid-s1362" xml:space="preserve">In eadem tabula quando in tertia operatione regulæ aureæ, qua ſecunda inqui-<lb/> <anchor type="handwritten" xlink:label="hd-050-1a" xlink:href="hd-050-1"/> runtur, numerus reliquus fuit maior quam 64. </s> <s xml:id="echoid-s1363" xml:space="preserve">maior nimirum dimidio Diuiſo-<lb/>ris 128. </s> <s xml:id="echoid-s1364" xml:space="preserve">aſſumpſimus vnum ſecundum integrum.</s> <s xml:id="echoid-s1365" xml:space="preserve"/> </p> <div xml:id="echoid-div54" type="float" level="2" n="1"> <note position="left" xlink:label="note-050-01" xlink:href="note-050-01a" xml:space="preserve">Conſtructio <lb/>& vſ{us} tabulæ <lb/>ſequentis.</note> <handwritten xlink:label="hd-050-1" xlink:href="hd-050-1a"/> <handwritten xlink:label="hd-050-1" xlink:href="hd-050-1a"/> </div> <p> <s xml:id="echoid-s1366" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi quis tabulam extendere velit ad plures Quadrantes, facere <lb/> <anchor type="note" xlink:label="note-050-02a" xlink:href="note-050-02"/> id poterit ſine vlla operatione regulæ aureæ, hoc modo. </s> <s xml:id="echoid-s1367" xml:space="preserve">Gradibus, Minutis <lb/>ac ſecundis quadrageſimi Quadrantis, quivſque ad grad. </s> <s xml:id="echoid-s1368" xml:space="preserve">128. </s> <s xml:id="echoid-s1369" xml:space="preserve">productus fuit, <lb/>adiiciantur differentiæ inter gradus, minuta, ac ſecunda quadrageſimi Qua-<lb/>drantis, & </s> <s xml:id="echoid-s1370" xml:space="preserve">gradus, Minuta, ac ſecunda aliorum quadrantum infra Quadrage-<lb/>ſimum. </s> <s xml:id="echoid-s1371" xml:space="preserve">Ita namque conficientur gradus, minuta, ac ſecunda qua drantum ſu-<lb/>pra quadrageſimum. </s> <s xml:id="echoid-s1372" xml:space="preserve">Nam gradus, minuta, ac ſecunda trium quorumlibet <lb/>quadrantum, quorum vnus ſit quadrageſimus, alij verò duo æqualiter ab eo <lb/>diſtent, obſeruant proportionem Arithmeticam continuam, vt hic apparet, <lb/> <anchor type="handwritten" xlink:label="hd-050-1a" xlink:href="hd-050-1"/> <anchor type="note" xlink:label="note-050-03a" xlink:href="note-050-03"/> Numeri enim Quadrantum 39. </s> <s xml:id="echoid-s1373" xml:space="preserve">40. </s> <s xml:id="echoid-s1374" xml:space="preserve">41. </s> <s xml:id="echoid-s1375" xml:space="preserve">in prima columna ſuperant ſe conti-<lb/>nuè ſecundis 28. </s> <s xml:id="echoid-s1376" xml:space="preserve">In ſecunda verò columna ſecundis 56. </s> <s xml:id="echoid-s1377" xml:space="preserve">& </s> <s xml:id="echoid-s1378" xml:space="preserve">in tertia Minuto 1. <lb/></s> <s xml:id="echoid-s1379" xml:space="preserve">Secundis 24. </s> <s xml:id="echoid-s1380" xml:space="preserve">&</s> <s xml:id="echoid-s1381" xml:space="preserve">c. </s> <s xml:id="echoid-s1382" xml:space="preserve">Ita quoq; </s> <s xml:id="echoid-s1383" xml:space="preserve">Numeri Quadrantum 38. </s> <s xml:id="echoid-s1384" xml:space="preserve">40. </s> <s xml:id="echoid-s1385" xml:space="preserve">42. </s> <s xml:id="echoid-s1386" xml:space="preserve">in prima columna <pb o="21" file="051" n="51" rhead="LIBER PRIMVS."/> ſuperant ſe continue ſecundis 56. </s> <s xml:id="echoid-s1387" xml:space="preserve">In ſecunda vero Minuto 1. </s> <s xml:id="echoid-s1388" xml:space="preserve">ſecundis 53. </s> <s xml:id="echoid-s1389" xml:space="preserve">& </s> <s xml:id="echoid-s1390" xml:space="preserve">in <lb/>tertia Minutis 2. </s> <s xml:id="echoid-s1391" xml:space="preserve">ſecundis 49. </s> <s xml:id="echoid-s1392" xml:space="preserve">& </s> <s xml:id="echoid-s1393" xml:space="preserve">c. </s> <s xml:id="echoid-s1394" xml:space="preserve">Quareſi differentiæ inter gradus, minuta, ac <lb/>ſecunda Quadrantis 39. </s> <s xml:id="echoid-s1395" xml:space="preserve">& </s> <s xml:id="echoid-s1396" xml:space="preserve">Quadrantis 40. </s> <s xml:id="echoid-s1397" xml:space="preserve">adijciantur ordine ad gradus, minu-<lb/>ta, ac ſecunda Quandrantis 40. </s> <s xml:id="echoid-s1398" xml:space="preserve">componentur gradus, Minuta ac ſecunda Qua-<lb/>drantis 41. </s> <s xml:id="echoid-s1399" xml:space="preserve">Differentiæ autem inter gradus, Minuta ac Secun. </s> <s xml:id="echoid-s1400" xml:space="preserve">Quadrantis 38. </s> <s xml:id="echoid-s1401" xml:space="preserve">& </s> <s xml:id="echoid-s1402" xml:space="preserve"><lb/>Quadrantis 40. </s> <s xml:id="echoid-s1403" xml:space="preserve">additæ ordinatim gradibus, Minutis, ac ſecundis Quadrãtis 40. <lb/></s> <s xml:id="echoid-s1404" xml:space="preserve">conficient gradus minuta, ac Secunda Quadrantis 42. </s> <s xml:id="echoid-s1405" xml:space="preserve">Sic quo que differentiæ <lb/>inter gradus, Minuta, ac Secunda Quadrantis 30. </s> <s xml:id="echoid-s1406" xml:space="preserve">& </s> <s xml:id="echoid-s1407" xml:space="preserve">Quadrantis 40. </s> <s xml:id="echoid-s1408" xml:space="preserve">appoſitæ <lb/>gradibus, Minutis, & </s> <s xml:id="echoid-s1409" xml:space="preserve">ſecundis Quadrantis 40. </s> <s xml:id="echoid-s1410" xml:space="preserve">component gradus, minuta, & </s> <s xml:id="echoid-s1411" xml:space="preserve"><lb/>ſecunda Quadrantis 50. </s> <s xml:id="echoid-s1412" xml:space="preserve">& </s> <s xml:id="echoid-s1413" xml:space="preserve">c.</s> <s xml:id="echoid-s1414" xml:space="preserve"/> </p> <div xml:id="echoid-div55" type="float" level="2" n="2"> <note position="left" xlink:label="note-050-02" xlink:href="note-050-02a" xml:space="preserve">Quo pacto ta-<lb/>bula 40 Qua <lb/>drantum ex-<lb/>tendatur ad <lb/>plur{es} Qua-<lb/>drant{es} ſine <lb/>ope aureæ re-<lb/>gulæ.</note> <handwritten xlink:label="hd-050-1" xlink:href="hd-050-1a"/> <note position="right" xlink:label="note-050-03" xlink:href="note-050-03a" xml:space="preserve"> <lb/>Par \\ tes ### 1 ### 2 ### 3 ### 4 ### 5 ### 6 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>38 # 0 # 59 # 4 # 1 # 58 # 7 # 2 # 57 # 11 # 3 # 56 # 15 # 4 # 55 # 19 # 5 # 54 # 22 <lb/>39 # 0 # 59 # 32 # 1 # 59 # 4 # 2 # 58 # 36 # 3 # 58 # 7 # 4 # 57 # 39 # 5 # 57 # 11 <lb/>40 # 1 # 0 # 0 # 2 # 0 # 0 # 3 # 0 # 0 # 4 # 0 # 0 # 5 # 0 # 0 # 6 # 0 # 0 <lb/>41 # 1 # 0 # 28 # 2 # 0 # 56 # 3 # 1 # 24 # 4 # 1 # 53 # 5 # 2 # 21 # 6 # 2 # 49 <lb/>42 # 1 # 0 # 56 # 2 # 1 # 53 # 3 # 2 # 49 # 4 # 3 # 45 # 5 # 4 # 41 # 6 # 5 # 38 <lb/></note> </div> <p> <s xml:id="echoid-s1415" xml:space="preserve">SEQVITVR TABVLA QVADRANTIS PAVLO AN-<lb/>te conſtructi, vbi ſinguli arcus producti diſtribuuntur in 128. </s> <s xml:id="echoid-s1416" xml:space="preserve">partes <lb/>æquales: </s> <s xml:id="echoid-s1417" xml:space="preserve">in qua ſtatim apparet, quot Gradus, Minuta, ac Secunda <lb/>ſingulis particulis cuiuſuis quadrantis reſpondeant: </s> <s xml:id="echoid-s1418" xml:space="preserve">cuius <lb/>quidem vſum ſupra expoſuimus.</s> <s xml:id="echoid-s1419" xml:space="preserve"/> </p> <figure> <image file="051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/051-01"/> </figure> <pb o="22" file="052" n="52" rhead="GEOMETR. PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ es. ### 1 ### 2 ### 3 ### 4 ### 5 ### 6 ### 7 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 1 # 0 # 0 # 2 # 0 # 0 # 3 # 0 # 0 # 4 # 0 # 0 # 5 # 0 # 0 # 6 # 0 # 0 # 7 # 0 # 0 <lb/>2 # 0 # 42 # 11 # 1 # 24 # 22 # 2 # 6 # 34 # 2 # 48 # 45 # 3 # 30 # 56 # 4 # 13 # 7 # 4 # 55 # 19 <lb/>3 # 0 # 42 # 39 # 1 # 25 # 19 # 2 # 7 # 58 # 2 # 50 # 37 # 3 # 33 # 17 # 4 # 15 # 56 # 4 # 58 # 36 <lb/>4 # 0 # 43 # 7 # 1 # 26 # 15 # 2 # 9 # 22 # 2 # 52 # 30 # 3 # 35 # 37 # 4 # 18 # 45 # 5 # 1 # 52 <lb/>5 # 0 # 43 # 36 # 1 # 27 # 11 # 2 # 10 # 47 # 2 # 54 # 22 # 3 # 37 # 58 # 4 # 21 # 34 # 5 # 5 # 9 <lb/>6 # 0 # 44 # 4 # 1 # 28 # 7 # 2 # 12 # 11 # 2 # 56 # 15 # 3 # 40 # 19 # 4 # 24 # 22 # 5 # 8 # 26 <lb/>7 # 0 # 44 # 32 # 1 # 29 # 4 # 2 # 13 # 36 # 2 # 58 # 7 # 3 # 42 # 39 # 4 # 27 # 11 # 5 # 11 # 43 <lb/>8 # 0 # 45 # 0 # 1 # 30 # 0 # 2 # 15 # 0 # 3 # 0 # 0 # 3 # 45 # 0 # 4 # 30 # 0 # 5 # 15 # 0 <lb/>9 # 0 # 45 # 28 # 1 # 30 # 56 # 2 # 16 # 24 # 3 # 1 # 53 # 3 # 47 # 21 # 4 # 32 # 49 # 5 # 18 # 17 <lb/>10 # 0 # 45 # 56 # 1 # 31 # 52 # 2 # 17 # 49 # 3 # 3 # 45 # 3 # 49 # 41 # 4 # 35 # 37 # 5 # 21 # 34 <lb/>11 # 0 # 46 # 24 # 1 # 32 # 49 # 2 # 19 # 13 # 3 # 5 # 37 # 3 # 52 # 2 # 4 # 38 # 26 # 5 # 24 # 51 <lb/>12 # 0 # 46 # 52 # 1 # 33 # 45 # 2 # 20 # 37 # 3 # 7 # 30 # 3 # 54 # 22 # 4 # 41 # 15 # 5 # 28 # 7 <lb/>13 # 0 # 47 # 21 # 1 # 34 # 41 # 2 # 22 # 2 # 3 # 9 # 22 # 3 # 56 # 43 # 4 # 44 # 4 # 5 # 31 # 24 <lb/>14 # 0 # 47 # 49 # 1 # 35 # 37 # 2 # 23 # 26 # 3 # 11 # 15 # 3 # 59 # 4 # 4 # 46 # 52 # 5 # 34 # 41 <lb/>15 # 0 # 48 # 17 # 1 # 36 # 34 # 2 # 24 # 51 # 3 # 13 # 7 # 4 # 1 # 24 # 4 # 49 # 41 # 5 # 37 # 58 <lb/>16 # 0 # 48 # 45 # 1 # 37 # 30 # 2 # 26 # 15 # 3 # 15 # 0 # 4 # 3 # 45 # 4 # 52 # 30 # 5 # 41 # 15 <lb/>17 # 0 # 49 # 13 # 1 # 38 # 26 # 2 # 27 # 39 # 3 # 16 # 52 # 4 # 6 # 6 # 4 # 55 # 19 # 5 # 44 # 32 <lb/>18 # 0 # 49 # 41 # 1 # 39 # 22 # 2 # 29 # 4 # 3 # 18 # 45 # 4 # 8 # 26 # 4 # 58 # 7 # 5 # 47 # 49 <lb/>19 # 0 # 50 # 9 # 1 # 40 # 19 # 2 # 30 # 28 # 3 # 20 # 37 # 4 # 10 # 47 # 5 # 0 # 56 # 5 # 51 # 6 <lb/>20 # 0 # 50 # 37 # 1 # 41 # 15 # 2 # 31 # 52 # 3 # 22 # 30 # 4 # 13 # 7 # 5 # 3 # 45 # 5 # 54 # 22 <lb/>21 # 0 # 51 # 6 # 1 # 42 # 11 # 2 # 33 # 17 # 3 # 24 # 22 # 4 # 15 # 28 # 5 # 6 # 34 # 5 # 57 # 39 <lb/>22 # 0 # 51 # 34 # 1 # 43 # 7 # 2 # 34 # 41 # 3 # 26 # 15 # 4 # 17 # 49 # 5 # 9 # 22 # 6 # 0 # 56 <lb/>23 # 0 # 52 # 2 # 1 # 44 # 4 # 2 # 36 # 6 # 3 # 28 # 7 # 4 # 20 # 9 # 5 # 11 # 11 # 6 # 4 # 13 <lb/>24 # 0 # 52 # 30 # 1 # 45 # 0 # 2 # 37 # 30 # 3 # 30 # 0 # 4 # 22 # 30 # 5 # 15 # 0 # 6 # 7 # 30 <lb/>25 # 0 # 52 # 58 # 1 # 45 # 56 # 2 # 38 # 54 # 3 # 31 # 52 # 4 # 24 # 51 # 5 # 17 # 49 # 6 # 10 # 47 <lb/>26 # 0 # 53 # 26 # 1 # 46 # 52 # 2 # 40 # 19 # 3 # 33 # 45 # 4 # 27 # 11 # 5 # 20 # 37 # 6 # 14 # 4 <lb/>27 # 0 # 53 # 54 # 1 # 47 # 49 # 2 # 41 # 43 # 3 # 35 # 37 # 4 # 29 # 32 # 5 # 23 # 26 # 6 # 17 # 21 <lb/>28 # 0 # 54 # 22 # 1 # 48 # 45 # 2 # 43 # 7 # 3 # 37 # 30 # 4 # 31 # 52 # 5 # 26 # 15 # 6 # 20 # 37 <lb/>29 # 0 # 54 # 51 # 1 # 49 # 41 # 2 # 44 # 32 # 3 # 39 # 22 # 4 # 34 # 13 # 5 # 29 # 4 # 6 # 23 # 54 <lb/>30 # 0 # 55 # 19 # 1 # 50 # 37 # 2 # 45 # 56 # 3 # 41 # 15 # 4 # 36 # 34 # 5 # 31 # 52 # 6 # 27 # 11 <lb/>31 # 0 # 55 # 47 # 1 # 51 # 34 # 2 # 47 # 21 # 3 # 43 # 7 # 4 # 38 # 54 # 5 # 34 # 41 # 6 # 30 # 28 <lb/>32 # 0 # 56 # 15 # 1 # 52 # 30 # 2 # 48 # 45 # 3 # 45 # 0 # 4 # 41 # 15 # 5 # 37 # 30 # 6 # 33 # 45 <lb/>33 # 0 # 56 # 43 # 1 # 53 # 26 # 2 # 50 # 9 # 3 # 46 # 52 # 4 # 43 # 36 # 5 # 40 # 19 # 6 # 37 # 2 <lb/>34 # 0 # 57 # 11 # 1 # 54 # 22 # 2 # 51 # 34 # 3 # 48 # 45 # 4 # 45 # 56 # 5 # 43 # 7 # 6 # 40 # 19 <lb/>35 # 0 # 57 # 39 # 1 # 55 # 19 # 2 # 52 # 58 # 3 # 50 # 37 # 4 # 48 # 17 # 5 # 45 # 56 # 6 # 43 # 36 <lb/>36 # 0 # 58 # 7 # 1 # 56 # 15 # 2 # 54 # 22 # 3 # 52 # 30 # 4 # 50 # 37 # 5 # 48 # 45 # 6 # 46 # 52 <lb/>37 # 0 # 58 # 36 # 1 # 57 # 11 # 2 # 55 # 47 # 3 # 54 # 22 # 4 # 52 # 58 # 5 # 51 # 34 # 6 # 50 # 9 <lb/>38 # 0 # 59 # 4 # 1 # 58 # 7 # 2 # 57 # 11 # 3 # 56 # 15 # 4 # 55 # 19 # 5 # 54 # 22 # 6 # 53 # 26 <lb/>39 # 0 # 59 # 32 # 1 # 59 # 4 # 2 # 58 # 36 # 3 # 58 # 7 # 4 # 57 # 39 # 5 # 57 # 11 # 6 # 56 # 43 <lb/>40 # 1 # 0 # 0 # 2 # 0 # 0 # 3 # 0 # 0 # 4 # 0 # 0 # 5 # 0 # 0 # 6 # 0 # 0 # 7 # 0 # 0 <lb/></note> <pb o="23" file="053" n="53" rhead="LIBER PRIMVS."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 8 ### 9 ### 10 ### 11 ### 12 ### 13 ### 14 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 8 # 0 # 0 # 9 # 0 # 0 # 10 # 0 # 0 # 11 # 0 # 0 # 12 # 0 # 0 # 13 # 0 # 0 # 14 # 0 # 0 <lb/>2 # 5 # 37 # 30 # 6 # 19 # 41 # 7 # 1 # 52 # 7 # 44 # 4 # 8 # 26 # 15 # 9 # 8 # 26 # 9 # 50 # 37 <lb/>3 # 5 # 41 # 15 # 6 # 23 # 54 # 7 # 6 # 34 # 7 # 49 # 13 # 8 # 31 # 52 # 9 # 14 # 32 # 9 # 57 # 11 <lb/>4 # 5 # 45 # 0 # 6 # 28 # 7 # 7 # 11 # 15 # 7 # 54 # 22 # 8 # 37 # 30 # 9 # 20 # 37 # 10 # 3 # 45 <lb/>5 # 5 # 48 # 45 # 6 # 32 # 21 # 7 # 15 # 56 # 7 # 59 # 32 # 8 # 43 # 7 # 9 # 26 # 43 # 10 # 10 # 19 <lb/>6 # 5 # 52 # 30 # 6 # 36 # 34 # 7 # 20 # 37 # 8 # 4 # 41 # 8 # 48 # 45 # 9 # 32 # 49 # 10 # 16 # 52 <lb/>7 # 5 # 56 # 15 # 6 # 40 # 47 # 7 # 25 # 19 # 8 # 9 # 51 # 8 # 54 # 22 # 9 # 38 # 54 # 10 # 23 # 26 <lb/>8 # 6 # 0 # 0 # 6 # 45 # 0 # 7 # 30 # 0 # 8 # 15 # 0 # 9 # 0 # 0 # 9 # 45 # 0 # 10 # 30 # 0 <lb/>9 # 6 # 3 # 45 # 6 # 49 # 13 # 7 # 34 # 41 # 8 # 20 # 9 # 9 # 5 # 37 # 9 # 51 # 6 # 10 # 36 # 34 <lb/>10 # 6 # 7 # 30 # 6 # 53 # 26 # 7 # 39 # 22 # 8 # 25 # 19 # 9 # 11 # 15 # 9 # 57 # 11 # 10 # 43 # 7 <lb/>11 # 6 # 11 # 15 # 6 # 57 # 39 # 7 # 44 # 4 # 8 # 30 # 28 # 9 # 16 # 52 # 10 # 3 # 17 # 10 # 49 # 41 <lb/>12 # 6 # 15 # 0 # 7 # 1 # 52 # 7 # 48 # 45 # 8 # 35 # 37 # 9 # 22 # 30 # 10 # 9 # 22 # 10 # 56 # 15 <lb/>13 # 6 # 18 # 45 # 7 # 6 # 6 # 7 # 53 # 26 # 8 # 40 # 47 # 9 # 28 # 7 # 10 # 15 # 28 # 11 # 2 # 49 <lb/>14 # 6 # 22 # 30 # 7 # 10 # 19 # 7 # 58 # 7 # 8 # 45 # 57 # 9 # 33 # 45 # 10 # 21 # 34 # 11 # 9 # 22 <lb/>15 # 6 # 26 # 15 # 7 # 14 # 32 # 8 # 2 # 49 # 8 # 51 # 6 # 9 # 39 # 22 # 10 # 27 # 39 # 11 # 15 # 56 <lb/>16 # 6 # 30 # 0 # 7 # 18 # 45 # 8 # 7 # 30 # 8 # 56 # 15 # 9 # 45 # 0 # 10 # 33 # 45 # 11 # 22 # 30 <lb/>17 # 6 # 33 # 45 # 7 # 22 # 58 # 8 # 12 # 11 # 9 # 1 # 24 # 9 # 50 # 37 # 10 # 39 # 51 # 11 # 29 # 4 <lb/>18 # 6 # 37 # 30 # 7 # 27 # 11 # 8 # 16 # 52 # 9 # 6 # 34 # 9 # 56 # 15 # 10 # 45 # 56 # 11 # 35 # 37 <lb/>19 # 6 # 41 # 15 # 7 # 31 # 24 # 8 # 21 # 34 # 9 # 11 # 43 # 10 # 1 # 52 # 10 # 52 # 2 # 11 # 42 # 11 <lb/>20 # 6 # 45 # 0 # 7 # 35 # 37 # 8 # 26 # 15 # 9 # 16 # 52 # 10 # 7 # 30 # 10 # 58 # 7 # 11 # 48 # 45 <lb/>21 # 6 # 48 # 45 # 7 # 39 # 51 # 8 # 30 # 56 # 9 # 22 # 2 # 10 # 13 # 7 # 11 # 4 # 13 # 11 # 55 # 19 <lb/>22 # 6 # 52 # 30 # 7 # 44 # 4 # 8 # 35 # 37 # 9 # 27 # 11 # 10 # 18 # 45 # 11 # 10 # 19 # 12 # 1 # 52 <lb/>23 # 6 # 56 # 15 # 7 # 48 # 17 # 8 # 40 # 19 # 9 # 32 # 21 # 10 # 24 # 22 # 11 # 16 # 24 # 12 # 8 # 26 <lb/>24 # 7 # 0 # 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PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ es. ### 15 ### 16 ### 17 ### 18 ### 19 ### 20 ### 21 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 15 # 0 # 0 # 16 # 0 # 0 # 17 # 0 # 0 # 18 # 0 # 0 # 19 # 0 # 0 # 20 # 0 # 0 # 21 # 0 # 0 <lb/>2 # 10 # 32 # 49 # 11 # 15 # 0 # 11 # 57 # 11 # 12 # 39 # 22 # 13 # 21 # 34 # 14 # 3 # 45 # 14 # 45 # 56 <lb/>3 # 10 # 39 # 51 # 11 # 22 # 30 # 12 # 5 # 9 # 12 # 47 # 49 # 13 # 30 # 28 # 14 # 13 # 7 # 14 # 55 # 47 <lb/>4 # 10 # 46 # 52 # 11 # 30 # 0 # 12 # 13 # 7 # 12 # 56 # 15 # 13 # 39 # 22 # 14 # 22 # 30 # 15 # 5 # 37 <lb/>5 # 10 # 53 # 54 # 11 # 37 # 30 # 12 # 21 # 6 # 13 # 4 # 41 # 13 # 48 # 17 # 14 # 31 # 52 # 15 # 15 # 28 <lb/>6 # 11 # 0 # 56 # 11 # 45 # 0 # 12 # 29 # 4 # 13 # 13 # 7 # 13 # 57 # 11 # 14 # 41 # 15 # 15 # 25 # 19 <lb/>7 # 11 # 7 # 58 # 11 # 52 # 30 # 12 # 37 # 2 # 13 # 21 # 34 # 14 # 6 # 6 # 14 # 50 # 37 # 15 # 35 # 9 <lb/>8 # 11 # 15 # 0 # 12 # 0 # 0 # 12 # 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PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 29 ### 30 ### 31 ### 32 ### 33 ### 34 ### 35 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 29 # 0 # 0 # 30 # 0 # 0 # 31 # 0 # 0 # 32 # 0 # 0 # 33 # 0 # 0 # 34 # 0 # 0 # 35 # 0 # 0 <lb/>2 # 20 # 23 # 26 # 21 # 5 # 37 # 21 # 47 # 49 # 22 # 30 # 0 # 23 # 12 # 11 # 23 # 54 # 22 # 24 # 36 # 34 <lb/>3 # 20 # 37 # 2 # 21 # 19 # 41 # 22 # 2 # 21 # 22 # 45 # 0 # 23 # 27 # 39 # 24 # 10 # 19 # 24 # 52 # 58 <lb/>4 # 20 # 50 # 37 # 21 # 33 # 45 # 22 # 16 # 52 # 23 # 0 # 0 # 23 # 43 # 7 # 24 # 26 # 15 # 25 # 9 # 22 <lb/>5 # 21 # 4 # 13 # 21 # 47 # 49 # 22 # 31 # 24 # 23 # 15 # 0 # 23 # 58 # 36 # 24 # 42 # 11 # 25 # 25 # 47 <lb/>6 # 21 # 17 # 49 # 22 # 1 # 52 # 22 # 45 # 56 # 23 # 30 # 0 # 24 # 14 # 4 # 24 # 58 # 7 # 25 # 42 # 11 <lb/>7 # 21 # 31 # 24 # 22 # 15 # 56 # 23 # 0 # 28 # 23 # 45 # 0 # 24 # 29 # 32 # 25 # 14 # 4 # 25 # 58 # 36 <lb/>8 # 21 # 45 # 0 # 22 # 30 # 0 # 23 # 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16 # 33 # 37 # 58 <lb/>36 # 28 # 5 # 37 # 29 # 3 # 45 # 30 # 1 # 52 # 31 # 0 # 0 # 31 # 58 # 7 # 32 # 56 # 15 # 33 # 54 # 22 <lb/>37 # 28 # 19 # 13 # 29 # 17 # 49 # 30 # 16 # 24 # 31 # 15 # 0 # 32 # 13 # 36 # 33 # 12 # 11 # 34 # 10 # 47 <lb/>38 # 28 # 32 # 49 # 29 # 31 # 52 # 30 # 30 # 56 # 31 # 30 # 0 # 32 # 29 # 4 # 33 # 28 # 7 # 34 # 27 # 11 <lb/>39 # 28 # 46 # 24 # 29 # 45 # 56 # 30 # 45 # 28 # 31 # 45 # 0 # 32 # 44 # 32 # 33 # 44 # 4 # 34 # 43 # 36 <lb/>40 # 29 # 0 # 0 # 30 # 0 # 0 # 31 # 0 # 0 # 32 # 0 # 0 # 33 # 0 # 0 # 34 # 0 # 0 # 35 # 0 # 0 <lb/></note> <pb o="27" file="057" n="57" rhead="LIBER PRIMVS."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 36 ### 37 ### 38 ### 39 ### 40 ### 41 ### 42 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 36 # 0 # 0 # 37 # 0 # 0 # 38 # 0 # 0 # 39 # 0 # 0 # 40 # 0 # 0 # 41 # 0 # 0 # 42 # 0 # 0 <lb/>2 # 25 # 18 # 45 # 26 # 0 # 56 # 26 # 43 # 7 # 27 # 25 # 19 # 28 # 7 # 30 # 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33 # 11 # 15 # 34 # 6 # 34 # 35 # 1 # 52 # 35 # 57 # 11 # 36 # 52 # 30 # 37 # 47 # 49 # 38 # 43 # 7 <lb/>31 # 33 # 28 # 7 # 34 # 23 # 54 # 35 # 19 # 41 # 36 # 15 # 28 # 37 # 11 # 15 # 38 # 7 # 2 # 39 # 2 # 49 <lb/>32 # 33 # 45 # 0 # 34 # 41 # 15 # 35 # 37 # 30 # 36 # 33 # 45 # 37 # 30 # 0 # 38 # 26 # 15 # 39 # 22 # 30 <lb/>33 # 34 # 1 # 52 # 34 # 58 # 36 # 35 # 55 # 19 # 36 # 52 # 2 # 37 # 48 # 45 # 38 # 45 # 28 # 39 # 42 # 11 <lb/>34 # 34 # 18 # 45 # 35 # 15 # 56 # 36 # 13 # 7 # 37 # 10 # 19 # 38 # 7 # 30 # 39 # 4 # 41 # 40 # 1 # 52 <lb/>35 # 34 # 35 # 37 # 35 # 33 # 17 # 36 # 30 # 56 # 37 # 28 # 36 # 38 # 26 # 15 # 39 # 23 # 54 # 40 # 21 # 34 <lb/>36 # 34 # 52 # 30 # 35 # 50 # 37 # 36 # 48 # 45 # 37 # 46 # 52 # 38 # 45 # 0 # 39 # 43 # 7 # 40 # 41 # 15 <lb/>37 # 35 # 9 # 22 # 35 # 7 # 38 # 37 # 6 # 34 # 38 # 5 # 9 # 39 # 3 # 45 # 40 # 2 # 21 # 41 # 0 # 56 <lb/>38 # 35 # 26 # 15 # 36 # 25 # 19 # 37 # 24 # 22 # 38 # 23 # 26 # 39 # 22 # 30 # 40 # 21 # 34 # 41 # 20 # 37 <lb/>39 # 35 # 43 # 7 # 36 # 42 # 39 # 37 # 32 # 11 # 38 # 41 # 43 # 39 # 41 # 15 # 40 # 40 # 47 # 41 # 40 # 19 <lb/>40 # 36 # 0 # 0 # 37 # 0 # 0 # 38 # 0 # 0 # 39 # 0 # 0 # 40 # 0 # 0 # 41 # 0 # 0 # 42 # 0 # 0 <lb/></note> <pb o="28" file="058" n="58" rhead="GEOMETR. PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 43 ### 44 ### 45 ### 46 ### 47 ### 48 ### 49 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 43 # 0 # 0 # 44 # 0 # 0 # 45 # 0 # 0 # 46 # 0 # 0 # 47 # 0 # 0 # 48 # 0 # 0 # 49 # 0 # 0 <lb/>2 # 30 # 14 # 4 # 30 # 56 # 15 # 31 # 38 # 26 # 33 # 20 # 37 # 33 # 2 # 49 # 33 # 45 # 0 # 34 # 27 # 11 <lb/>3 # 30 # 34 # 13 # 31 # 16 # 52 # 31 # 59 # 32 # 32 # 42 # 11 # 33 # 24 # 51 # 34 # 7 # 30 # 34 # 50 # 9 <lb/>4 # 30 # 54 # 22 # 31 # 37 # 30 # 32 # 20 # 37 # 33 # 3 # 45 # 33 # 46 # 52 # 34 # 30 # 0 # 35 # 13 # 7 <lb/>5 # 31 # 14 # 32 # 31 # 58 # 7 # 32 # 41 # 43 # 33 # 25 # 19 # 34 # 8 # 54 # 34 # 52 # 30 # 35 # 36 # 6 <lb/>6 # 31 # 34 # 41 # 32 # 18 # 45 # 33 # 2 # 49 # 33 # 46 # 52 # 54 # 30 # 56 # 35 # 15 # 0 # 35 # 59 # 4 <lb/>7 # 31 # 54 # 51 # 32 # 39 # 22 # 33 # 23 # 54 # 34 # 8 # 26 # 34 # 52 # 58 # 33 # 37 # 30 # 36 # 22 # 2 <lb/>8 # 32 # 15 # 0 # 33 # 0 # 0 # 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7 # 41 # 51 # 34 # 42 # 45 # 0 # 43 # 38 # 26 <lb/>27 # 38 # 37 # 58 # 39 # 31 # 52 # 40 # 25 # 47 # 41 # 19 # 41 # 42 # 13 # 36 # 43 # 7 # 30 # 44 # 1 # 24 <lb/>28 # 38 # 58 # 7 # 39 # 52 # 30 # 40 # 46 # 52 # 41 # 41 # 15 # 42 # 35 # 37 # 43 # 30 # 0 # 44 # 24 # 22 <lb/>29 # 39 # 18 # 17 # 40 # 13 # 7 # 41 # 7 # 58 # 42 # 2 # 49 # 42 # 57 # 39 # 43 # 52 # 30 # 44 # 47 # 21 <lb/>30 # 39 # 38 # 26 # 40 # 33 # 45 # 41 # 29 # 4 # 42 # 24 # 22 # 43 # 19 # 41 # 44 # 15 # 0 # 45 # 10 # 19 <lb/>31 # 39 # 58 # 36 # 40 # 54 # 22 # 41 # 50 # 9 # 42 # 45 # 56 # 43 # 41 # 43 # 44 # 37 # 30 # 45 # 33 # 17 <lb/>32 # 40 # 18 # 45 # 41 # 15 # 0 # 42 # 11 # 15 # 43 # 7 # 30 # 44 # 3 # 45 # 45 # 0 # 0 # 45 # 56 # 15 <lb/>33 # 40 # 38 # 54 # 41 # 35 # 37 # 42 # 32 # 21 # 43 # 29 # 4 # 44 # 25 # 47 # 45 # 22 # 30 # 46 # 19 # 13 <lb/>34 # 40 # 59 # 4 # 41 # 56 # 15 # 42 # 53 # 26 # 43 # 50 # 37 # 44 # 47 # 49 # 45 # 45 # 0 # 46 # 42 # 11 <lb/>35 # 41 # 19 # 13 # 42 # 16 # 52 # 43 # 14 # 32 # 44 # 12 # 11 # 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PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes ### 85 ### 86 ### 87 ### 88 ### 89 ### 90 ### 91 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 85 # 0 # 0 # 86 # 0 # 0 # 87 # 0 # 0 # 88 # 0 # 0 # 89 # 0 # 0 # 90 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 59 # 45 # 50 # 60 # 28 # 7 # 61 # 10 # 19 # 61 # 52 # 30 # 62 # 34 # 41 # 63 # 16 # 52 # 63 # 59 # 4 <lb/>3 # 60 # 25 # 47 # 61 # 8 # 26 # 61 # 51 # 6 # 62 # 33 # 45 # 63 # 16 # 24 # 63 # 59 # 4 # 64 # 41 # 43 <lb/>4 # 61 # 5 # 37 # 61 # 48 # 45 # 62 # 31 # 52 # 63 # 15 # 0 # 63 # 58 # 7 # 64 # 41 # 15 # 65 # 24 # 22 <lb/>5 # 61 # 45 # 28 # 62 # 29 # 4 # 63 # 12 # 39 # 63 # 56 # 15 # 64 # 30 # 51 # 65 # 23 # 26 # 66 # 7 # 2 <lb/>6 # 62 # 25 # 19 # 63 # 9 # 22 # 63 # 53 # 26 # 64 # 37 # 30 # 65 # 21 # 34 # 66 # 5 # 37 # 66 # 49 # 41 <lb/>7 # 63 # 5 # 9 # 63 # 49 # 41 # 64 # 34 # 13 # 65 # 18 # 45 # 66 # 3 # 17 # 66 # 47 # 49 # 67 # 32 # 21 <lb/>8 # 63 # 45 # 0 # 64 # 30 # 0 # 65 # 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30 # 79 # 15 # 56 # 80 # 9 # 22 # 81 # 2 # 49 <lb/>27 # 76 # 22 # 2 # 77 # 15 # 56 # 78 # 9 # 51 # 79 # 3 # 45 # 79 # 57 # 39 # 80 # 51 # 34 # 81 # 45 # 28 <lb/>28 # 77 # 1 # 52 # 77 # 56 # 15 # 78 # 50 # 37 # 79 # 45 # 0 # 80 # 39 # 22 # 81 # 33 # 45 # 82 # 28 # 7 <lb/>29 # 77 # 41 # 43 # 78 # 36 # 3 # 79 # 31 # 24 # 80 # 26 # 15 # 81 # 21 # 6 # 82 # 15 # 56 # 83 # 10 # 47 <lb/>30 # 78 # 21 # 34 # 79 # 16 # 52 # 80 # 12 # 11 # 81 # 7 # 30 # 82 # 2 # 49 # 82 # 58 # 7 # 83 # 53 # 27 <lb/>31 # 79 # 1 # 24 # 79 # 57 # 11 # 80 # 52 # 58 # 81 # 48 # 45 # 82 # 44 # 32 # 83 # 40 # 19 # 84 # 36 # 6 <lb/>32 # 79 # 41 # 15 # 80 # 37 # 30 # 81 # 33 # 45 # 82 # 30 # 0 # 83 # 26 # 15 # 84 # 22 # 30 # 85 # 18 # 45 <lb/>33 # 80 # 21 # 6 # 81 # 17 # 49 # 82 # 14 # 32 # 83 # 11 # 15 # 84 # 7 # 58 # 85 # 4 # 41 # 86 # 1 # 24 <lb/>34 # 81 # 0 # 56 # 81 # 58 # 7 # 82 # 55 # 19 # 83 # 52 # 30 # 84 # 49 # 41 # 85 # 46 # 52 # 86 # 44 # 4 <lb/>35 # 81 # 40 # 47 # 82 # 38 # 26 # 83 # 36 # 6 # 84 # 33 # 45 # 85 # 31 # 24 # 86 # 29 # 4 # 87 # 26 # 43 <lb/>36 # 82 # 20 # 37 # 83 # 18 # 45 # 84 # 16 # 52 # 85 # 15 # 0 # 86 # 13 # 7 # 86 # 11 # 15 # 88 # 9 # 22 <lb/>37 # 83 # 0 # 28 # 83 # 59 # 4 # 84 # 57 # 39 # 85 # 56 # 15 # 86 # 54 # 51 # 87 # 53 # 26 # 88 # 52 # 2 <lb/>38 # 83 # 40 # 19 # 84 # 39 # 22 # 85 # 38 # 26 # 86 # 37 # 30 # 87 # 36 # 34 # 88 # 35 # 37 # 89 # 34 # 41 <lb/>39 # 84 # 20 # 9 # 85 # 19 # 41 # 86 # 19 # 13 # 87 # 18 # 45 # 88 # 18 # 17 # 89 # 17 # 49 # 0 # 0 # 0 <lb/>40 # 85 # 0 # 0 # 86 # 0 # 0 # 87 # 0 # 0 # 88 # 0 # 0 # 89 # 0 # 0 # 90 # 0 # 0 # 0 # 0 # 0 <lb/></note> <pb o="35" file="065" n="65" rhead="LIBER PRIMVS."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 92 ### 93 ### 94 ### 95 ### 96 ### 97 ### 98 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 64 # 41 # 15 # 65 # 23 # 26 # 66 # 5 # 37 # 66 # 47 # 49 # 67 # 30 # 0 # 68 # 12 # 11 # 68 # 54 # 22 <lb/>3 # 65 # 24 # 22 # 66 # 7 # 2 # 66 # 49 # 41 # 67 # 32 # 21 # 68 # 15 # 0 # 68 # 57 # 39 # 69 # 40 # 19 <lb/>4 # 66 # 7 # 30 # 66 # 50 # 37 # 67 # 33 # 45 # 68 # 16 # 52 # 69 # 0 # 0 # 69 # 43 # 7 # 70 # 26 # 15 <lb/>5 # 66 # 50 # 37 # 67 # 34 # 13 # 68 # 17 # 49 # 69 # 1 # 24 # 69 # 45 # 0 # 70 # 28 # 36 # 71 # 12 # 11 <lb/>6 # 67 # 33 # 45 # 68 # 17 # 49 # 69 # 1 # 52 # 69 # 45 # 56 # 70 # 30 # 0 # 71 # 14 # 4 # 71 # 58 # 7 <lb/>7 # 68 # 16 # 52 # 69 # 1 # 24 # 69 # 45 # 56 # 70 # 30 # 28 # 71 # 15 # 0 # 71 # 59 # 32 # 72 # 44 # 4 <lb/>8 # 69 # 0 # 0 # 69 # 45 # 0 # 70 # 30 # 0 # 71 # 15 # 0 # 72 # 0 # 0 # 72 # 45 # 0 # 73 # 30 # 0 <lb/>9 # 69 # 43 # 7 # 70 # 28 # 36 # 71 # 14 # 4 # 71 # 59 # 32 # 72 # 45 # 0 # 73 # 30 # 28 # 74 # 15 # 56 <lb/>10 # 70 # 26 # 15 # 71 # 12 # 11 # 71 # 58 # 7 # 72 # 44 # 4 # 73 # 30 # 0 # 74 # 15 # 56 # 75 # 1 # 52 <lb/>11 # 71 # 9 # 22 # 71 # 55 # 47 # 72 # 42 # 11 # 73 # 28 # 36 # 74 # 15 # 0 # 75 # 1 # 24 # 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15 <lb/>21 # 78 # 20 # 37 # 79 # 11 # 43 # 80 # 2 # 49 # 80 # 53 # 54 # 81 # 45 # 0 # 82 # 36 # 6 # 83 # 27 # 11 <lb/>22 # 79 # 3 # 45 # 79 # 55 # 19 # 80 # 46 # 50 # 81 # 38 # 26 # 82 # 30 # 0 # 83 # 21 # 34 # 84 # 13 # 7 <lb/>23 # 79 # 46 # 52 # 80 # 38 # 54 # 81 # 30 # 56 # 82 # 22 # 58 # 83 # 15 # 0 # 84 # 7 # 2 # 84 # 59 # 4 <lb/>24 # 80 # 30 # 0 # 81 # 22 # 30 # 82 # 15 # 0 # 83 # 7 # 30 # 84 # 0 # 0 # 84 # 52 # 30 # 85 # 45 # 0 <lb/>25 # 81 # 13 # 7 # 82 # 6 # 6 # 82 # 59 # 4 # 83 # 52 # 2 # 84 # 45 # 0 # 85 # 37 # 58 # 86 # 30 # 56 <lb/>26 # 81 # 56 # 15 # 82 # 49 # 41 # 83 # 43 # 7 # 84 # 36 # 34 # 85 # 30 # 0 # 86 # 23 # 26 # 87 # 16 # 52 <lb/>27 # 82 # 39 # 22 # 83 # 33 # 17 # 84 # 27 # 11 # 85 # 21 # 6 # 86 # 15 # 0 # 87 # 8 # 54 # 88 # 2 # 49 <lb/>28 # 83 # 22 # 30 # 84 # 16 # 52 # 85 # 11 # 15 # 86 # 5 # 37 # 87 # 0 # 0 # 87 # 54 # 22 # 88 # 48 # 45 <lb/>29 # 84 # 5 # 37 # 85 # 0 # 28 # 85 # 55 # 19 # 86 # 50 # 9 # 87 # 45 # 0 # 88 # 39 # 51 # 89 # 34 # 4@ <lb/>30 # 84 # 48 # 45 # 85 # 44 # 4 # 86 # 39 # 22 # 87 # 34 # 41 # 88 # 30 # 0 # 89 # 25 # 19 # 0 # 0 # 0 <lb/>31 # 85 # 31 # 52 # 86 # 27 # 39 # 87 # 23 # 26 # 88 # 19 # 13 # 89 # 15 # 0 # 0 # 0 # 0 <lb/>32 # 86 # 15 # 0 # 87 # 11 # 15 # 88 # 7 # 30 # 89 # 3 # 45 # 90 # 0 # 0 # 0 # 0 # 8 <lb/>33 # 86 # 58 # 7 # 87 # 54 # 51 # 88 # 51 # 34 # 89 # 48 # 17 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>34 # 87 # 41 # 15 # 88 # 38 # 26 # 89 # 35 # 37 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>35 # 88 # 24 # 22 # 89 # 22 # 2 # 0 # 0 # 0 <lb/>36 # 89 # 7 # 30 # 0 # 0 # 0 <lb/>37 # 89 # 50 # 37 # 0 # 0 # 0 <lb/>38 # 0 # 0 # 0 # 0 # 0 # 0 <lb/></note> <pb o="36" file="066" n="66" rhead="GEOMETR. PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes ### 99 ### 100 ### 101 ### 102 ### 103 ### 104 ### 105 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 69 # 36 # 34 # 70 # 18 # 45 # 71 # 0 # 56 # 71 # 43 # 7 # 72 # 25 # 19 # 73 # 7 # 30 # 73 # 49 # 41 <lb/>3 # 70 # 22 # 58 # 71 # 5 # 37 # 71 # 48 # 17 # 72 # 30 # 56 # 73 # 13 # 36 # 73 # 56 # 15 # 74 # 37 # 58 <lb/>4 # 71 # 9 # 22 # 71 # 52 # 30 # 72 # 35 # 37 # 73 # 18 # 45 # 74 # 1 # 52 # 74 # 45 # 0 # 75 # 28 # 7 <lb/>5 # 71 # 55 # 47 # 72 # 39 # 22 # 73 # 22 # 58 # 74 # 6 # 34 # 74 # 50 # 9 # 75 # 33 # 45 # 76 # 17 # 21 <lb/>6 # 72 # 42 # 11 # 73 # 26 # 15 # 74 # 10 # 19 # 74 # 54 # 22 # 75 # 38 # 26 # 76 # 22 # 30 # 77 # 6 # 34 <lb/>7 # 73 # 28 # 36 # 74 # 13 # 7 # 74 # 57 # 39 # 75 # 42 # 11 # 76 # 26 # 43 # 77 # 11 # 15 # 77 # 55 # 47 <lb/>8 # 74 # 15 # 0 # 75 # 0 # 0 # 75 # 45 # 0 # 76 # 30 # 0 # 77 # 15 # 0 # 78 # 0 # 0 # 78 # 45 # 0 <lb/>9 # 75 # 1 # 24 # 75 # 46 # 52 # 76 # 32 # 21 # 77 # 17 # 49 # 78 # 3 # 17 # 78 # 48 # 45 # 79 # 34 # 13 <lb/>10 # 75 # 47 # 49 # 76 # 33 # 45 # 77 # 19 # 41 # 78 # 5 # 37 # 78 # 51 # 34 # 79 # 37 # 30 # 80 # 23 # 26 <lb/>11 # 76 # 34 # 13 # 77 # 20 # 37 # 78 # 9 # 2 # 78 # 53 # 26 # 79 # 39 # 51 # 80 # 26 # 15 # 81 # 12 # 39 <lb/>12 # 77 # 20 # 37 # 78 # 7 # 30 # 78 # 54 # 22 # 79 # 41 # 15 # 80 # 28 # 7 # 81 # 15 # 0 # 82 # 1 # 52 <lb/>13 # 78 # 7 # 2 # 78 # 54 # 22 # 79 # 41 # 43 # 80 # 29 # 4 # 81 # 16 # 24 # 82 # 3 # 45 # 82 # 51 # 6 <lb/>14 # 78 # 53 # 26 # 79 # 41 # 15 # 80 # 29 # 4 # 81 # 16 # 52 # 82 # 4 # 41 # 82 # 52 # 30 # 83 # 40 # 19 <lb/>15 # 79 # 39 # 51 # 80 # 28 # 7 # 81 # 16 # 24 # 82 # 4 # 41 # 82 # 52 # 58 # 83 # 41 # 15 # 84 # 20 # 32 <lb/>16 # 80 # 26 # 15 # 81 # 15 # 0 # 82 # 3 # 45 # 82 # 52 # 30 # 83 # 41 # 15 # 84 # 30 # 0 # 85 # 18 # 45 <lb/>17 # 81 # 12 # 39 # 82 # 1 # 52 # 82 # 51 # 6 # 83 # 40 # 19 # 84 # 29 # 32 # 85 # 18 # 45 # 86 # 7 # 58 <lb/>18 # 81 # 59 # 4 # 82 # 48 # 45 # 83 # 37 # 30 # 84 # 28 # 7 # 85 # 17 # 49 # 86 # 7 # 30 # 86 # 57 # 11 <lb/>19 # 82 # 45 # 28 # 83 # 35 # 37 # 84 # 25 # 47 # 85 # 15 # 56 # 86 # 6 # 6 # 86 # 56 # 15 # 87 # 46 # 24 <lb/>20 # 83 # 31 # 52 # 84 # 22 # 30 # 85 # 13 # 7 # 86 # 3 # 45 # 86 # 54 # 22 # 87 # 45 # 0 # 88 # 35 # 37 <lb/>21 # 84 # 18 # 17 # 85 # 9 # 22 # 86 # 0 # 82 # 86 # 51 # 34 # 87 # 42 # 39 # 88 # 33 # 45 # 80 # 24 # 51 <lb/>22 # 85 # 4 # 41 # 85 # 56 # 15 # 86 # 47 # 49 # 87 # 39 # 22 # 88 # 30 # 56 # 89 # 22 # 30 # 0 # 0 # 0 <lb/>23 # 85 # 51 # 6 # 86 # 43 # 7 # 87 # 35 # 9 # 88 # 27 # 11 # 89 # 19 # 13 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>24 # 86 # 37 # 30 # 87 # 30 # 0 # 88 # 22 # 30 # 89 # 15 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>25 # 87 # 23 # 54 # 88 # 16 # 52 # 89 # 9 # 51 # 0 # 0 # 0 <lb/>26 # 88 # 10 # 19 # 89 # 3 # 45 # 89 # 57 # 11 # 0 # 0 # 0 <lb/>27 # 88 # 56 # 43 # 89 # 50 # 37 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>28 # 89 # 43 # 7 # 0 # 0 # 0 <lb/>29 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>30 # 0 # 0 # 0 <lb/></note> <pb o="37" file="067" n="67" rhead="LIBER PRIMVS."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 106 ### 107 ### 108 ### 109 ### 110 ### 111 ### 112 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 74 # 31 # 52 # 75 # 14 # 4 # 75 # 56 # 15 # 75 # 38 # 26 # 77 # 20 # 37 # 78 # 2 # 49 # 78 # 45 # 0 <lb/>3 # 75 # 21 # 34 # 76 # 4 # 13 # 76 # 46 # 52 # 77 # 29 # 32 # 78 # 12 # 11 # 78 # 54 # 51 # 79 # 52 # 30 <lb/>4 # 76 # 11 # 15 # 76 # 54 # 22 # 77 # 37 # 30 # 78 # 20 # 37 # 79 # 3 # 45 # 79 # 46 # 52 # 80 # 30 # 0 <lb/>5 # 77 # 0 # 56 # 77 # 44 # 32 # 78 # 28 # 7 # 79 # 11 # 43 # 79 # 55 # 19 # 80 # 38 # 54 # 81 # 22 # 30 <lb/>6 # 77 # 50 # 37 # 78 # 34 # 41 # 79 # 18 # 45 # 80 # 2 # 49 # 80 # 46 # 52 # 81 # 30 # 56 # 82 # 15 # 0 <lb/>7 # 78 # 40 # 19 # 79 # 24 # 51 # 80 # 9 # 22 # 80 # 53 # 54 # 81 # 38 # 26 # 82 # 22 # 58 # 83 # 7 # 30 <lb/>8 # 79 # 30 # 0 # 80 # 15 # 0 # 81 # 0 # 0 # 81 # 45 # 0 # 82 # 30 # 0 # 83 # 15 # 0 # 84 # 0 # 0 <lb/>9 # 80 # 10 # 41 # 81 # 5 # 9 # 81 # 50 # 37 # 82 # 36 # 6 # 83 # 21 # 34 # 84 # 7 # 2 # 84 # 52 # 30 <lb/>10 # 81 # 9 # 22 # 81 # 55 # 19 # 82 # 41 # 15 # 83 # 27 # 11 # 84 # 13 # 7 # 84 # 59 # 4 # 85 # 45 # 0 <lb/>11 # 81 # 59 # 4 # 82 # 45 # 28 # 83 # 31 # 52 # 84 # 18 # 17 # 85 # 4 # 41 # 85 # 51 # 6 # 86 # 37 # 30 <lb/>12 # 82 # 48 # 45 # 83 # 35 # 37 # 84 # 22 # 30 # 85 # 9 # 22 # 85 # 56 # 15 # 86 # 43 # 7 # 87 # 30 # 0 <lb/>13 # 83 # 38 # 26 # 84 # 25 # 47 # 85 # 13 # 7 # 86 # 0 # 28 # 86 # 47 # 49 # 87 # 35 # 9 # 88 # 22 # 30 <lb/>14 # 84 # 28 # 7 # 85 # 15 # 56 # 86 # 3 # 45 # 86 # 51 # 34 # 87 # 39 # 22 # 88 # 27 # 11 # 89 # 15 # 0 <lb/>15 # 85 # 17 # 49 # 86 # 6 # 6 # 86 # 54 # 22 # 87 # 42 # 39 # 88 # 30 # 56 # 89 # 19 # 13 # 0 # 0 # 0 <lb/>16 # 86 # 7 # 30 # 86 # 56 # 15 # 87 # 45 # 0 # 88 # 33 # 45 # 89 # 22 # 30 # 0 # 0 # 0 <lb/>17 # 86 # 57 # 11 # 87 # 46 # 24 # 88 # 35 # 37 # 89 # 24 # 51 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>18 # 87 # 46 # 52 # 88 # 36 # 34 # 89 # 26 # 15 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>19 # 88 # 36 # 34 # 89 # 26 # 43 # 0 # 0 # 0 <lb/>20 # 89 # 26 # 15 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>21 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/></note> <pb o="38" file="068" n="68" rhead="GEOMETR. PRACT."/> <note position="right" xml:space="preserve"> <lb/>Par \\ tes. ### 113 ### 114 ### 115 ### 116 ### 117 ### 118 ### 119 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 79 # 27 # 11 # 80 # 9 # 22 # 80 # 51 # 34 # 81 # 33 # 45 # 82 # 15 # 56 # 82 # 58 # 7 # 83 # 40 # 19 <lb/>3 # 80 # 20 # 9 # 81 # 2 # 49 # 81 # 45 # 28 # 82 # 28 # 7 # 83 # 10 # 47 # 83 # 53 # 26 # 84 # 36 # 6 <lb/>4 # 81 # 13 # 7 # 81 # 56 # 15 # 82 # 39 # 22 # 83 # 22 # 30 # 84 # 5 # 37 # 84 # 48 # 45 # 85 # 31 # 52 <lb/>5 # 83 # 6 # 6 # 82 # 49 # 41 # 83 # 33 # 17 # 84 # 16 # 52 # 85 # 0 # 28 # 85 # 44 # 4 # 86 # 27 # 39 <lb/>6 # 82 # 59 # 4 # 83 # 43 # 7 # 84 # 27 # 11 # 85 # 11 # 15 # 85 # 55 # 19 # 86 # 39 # 22 # 87 # 23 # 26 <lb/>7 # 83 # 52 # 2 # 84 # 36 # 34 # 85 # 21 # 6 # 86 # 5 # 37 # 86 # 50 # 9 # 87 # 34 # 41 # 88 # 19 # 13 <lb/>8 # 84 # 45 # 0 # 85 # 30 # 0 # 86 # 15 # 0 # 87 # 0 # 0 # 87 # 45 # 0 # 88 # 30 # 0 # 89 # 15 # 0 <lb/>9 # 85 # 37 # 58 # 86 # 23 # 26 # 87 # 8 # 54 # 87 # 54 # 22 # 88 # 39 # 51 # 89 # 25 # 19 # 0 # 0 # 0 <lb/>10 # 86 # 30 # 56 # 87 # 16 # 52 # 88 # 2 # 49 # 88 # 48 # 45 # 89 # 34 # 41 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>11 # 87 # 23 # 54 # 88 # 10 # 19 # 88 # 56 # 43 # 89 # 43 # 7 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>12 # 88 # 16 # 52 # 89 # 3 # 45 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>13 # 89 # 9 # 51 # 89 # 57 # 11 # 0 # 0 # 0 <lb/>14 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>15 # 0 # 0 # 0 <lb/>Par \\ tes. ### 120 ### 121 ### 122 ### 123 ### 124 ### 125 ### 126 <lb/># G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S # G # M # S <lb/>1 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 84 # 22 # 30 # 85 # 4 # 41 # 85 # 46 # 52 # 86 # 29 # 4 # 87 # 11 # 15 # 87 # 53 # 27 # 88 # 35 # 37 <lb/>3 # 85 # 18 # 45 # 86 # 1 # 24 # 86 # 44 # 4 # 87 # 26 # 43 # 88 # 9 # 22 # 88 # 52 # 2 # 89 # 34 # 41 <lb/>4 # 86 # 15 # 0 # 86 # 58 # 7 # 87 # 41 # 15 # 88 # 24 # 22 # 89 # 7 # 30 # 89 # 50 # 37 # 0 # 0 # 0 <lb/>5 # 87 # 11 # 15 # 87 # 54 # 51 # 88 # 38 # 26 # 89 # 22 # 2 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 8 <lb/>6 # 88 # 7 # 30 # 88 # 51 # 34 # 89 # 35 # 37 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>7 # 89 # 3 # 45 # 89 # 48 # 17 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>8 # 90 # 0 # 0 # 0 # 0 # 0 <lb/>9 # 0 # 0 # 0 <lb/>Par \\ tes. ### 127 ### 128 <lb/># G # M # S # G # M # S <lb/>1 # 0 # 0 # 0 # 0 # 0 # 0 <lb/>2 # 89 # 17 # 49 # 90 # 0 # 0 <lb/>3 # 0 # 0 # 0 # 0 # 0 # 0 <lb/></note> <pb o="39" file="069" n="69" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s1420" xml:space="preserve">10. </s> <s xml:id="echoid-s1421" xml:space="preserve"><emph style="sc">Qvando</emph> intra Quadrantem non deſcripti ſunt plures quadrantes, <lb/> <anchor type="note" xlink:label="note-069-01a" xlink:href="note-069-01"/> ſed vnicus tantum adeſt in 90. </s> <s xml:id="echoid-s1422" xml:space="preserve">gradus exquiſitè diſtributus, cognoſcemus ſo-<lb/>lo circinibeneſicio, quot Minuta, ac ſecunda in quauis gradus particula conti-<lb/>neantur: </s> <s xml:id="echoid-s1423" xml:space="preserve">id quod etiam in libello de vſu, & </s> <s xml:id="echoid-s1424" xml:space="preserve">fabrica inſtrumenti horologiorũ, <lb/>& </s> <s xml:id="echoid-s1425" xml:space="preserve">in Aſtrolabio fecimus, hoc ſcilicet modo. </s> <s xml:id="echoid-s1426" xml:space="preserve">Sit data verbigratia particulatu, <lb/>in gradu 20. </s> <s xml:id="echoid-s1427" xml:space="preserve">ſuperioris Quadrantis BC. </s> <s xml:id="echoid-s1428" xml:space="preserve">Sumatur ea diligentiſsimè beneſicio <lb/>circini, & </s> <s xml:id="echoid-s1429" xml:space="preserve">à principio Quadrantis, id eſt, à puncto C, incipiendo, eadem aper-<lb/>tura circini accipiantur 60. </s> <s xml:id="echoid-s1430" xml:space="preserve">æquales particulæ vſque ad punctum X, ita vt arcus <lb/>CX, ſexagecuplus ſit arcus t u: </s> <s xml:id="echoid-s1431" xml:space="preserve">Quot. </s> <s xml:id="echoid-s1432" xml:space="preserve">n. </s> <s xml:id="echoid-s1433" xml:space="preserve">gradus integri in hoc arcu ſexagecu-<lb/>plo comprehenduntur, tot minuta complectetur particula data tu. </s> <s xml:id="echoid-s1434" xml:space="preserve">Et ſi vl-<lb/>tra gradus integros in arcu CX, ſuperſit aliqua particula, accipiatur ea ſexagies <lb/>quoque, initio facto à puncto C. </s> <s xml:id="echoid-s1435" xml:space="preserve">Nam quot gradus integri in hoc arcu ſexa-<lb/>gecuplo continentur, tot ſecunda vltra Minuta inuenta continebuntur in da-<lb/> <anchor type="handwritten" xlink:label="hd-069-1a" xlink:href="hd-069-1"/> ta particula t u. </s> <s xml:id="echoid-s1436" xml:space="preserve">Quod ſi adhuc aliquid ſuperſit, reperientur eodem modo Ter-<lb/>tia, &</s> <s xml:id="echoid-s1437" xml:space="preserve">c. </s> <s xml:id="echoid-s1438" xml:space="preserve">Ita que cumin arcu CX, quiſexagecuplus eſt particulæ t u, continean-<lb/>tur 40. </s> <s xml:id="echoid-s1439" xml:space="preserve">gradus integri, comprehendet particula t u, quadraginta Minuta, & </s> <s xml:id="echoid-s1440" xml:space="preserve">in-<lb/>ſuper tot ſecunda, quot gradus continenturin arcu, qui ſexagecuplus ſit par-<lb/>ticulæ vltra 40. </s> <s xml:id="echoid-s1441" xml:space="preserve">gradus in arcu CX, contentæ, &</s> <s xml:id="echoid-s1442" xml:space="preserve">c.</s> <s xml:id="echoid-s1443" xml:space="preserve"/> </p> <div xml:id="echoid-div56" type="float" level="2" n="3"> <note position="right" xlink:label="note-069-01" xlink:href="note-069-01a" xml:space="preserve">Quo pact@ <lb/>per circinum <lb/>deprehendan-<lb/>tur Minuta, <lb/>ac ſecunda in <lb/>quauis propo-<lb/>ſita grad{us} <lb/>particula. <lb/><lb/></note> <handwritten xlink:label="hd-069-1" xlink:href="hd-069-1a"/> </div> <p> <s xml:id="echoid-s1444" xml:space="preserve"><emph style="sc">Hoc</emph> autem ita demonſtro. </s> <s xml:id="echoid-s1445" xml:space="preserve">Quã proportionem habet arcus 60. </s> <s xml:id="echoid-s1446" xml:space="preserve">graduum <lb/> <anchor type="handwritten" xlink:label="hd-069-1a" xlink:href="hd-069-1"/> ad 1. </s> <s xml:id="echoid-s1447" xml:space="preserve">gradum, eam habetarcus CX. </s> <s xml:id="echoid-s1448" xml:space="preserve">ad particulam tu, cum vtrobique propor-<lb/>tio ſit ſexagecupla. </s> <s xml:id="echoid-s1449" xml:space="preserve">Igitur permutando erit, vt arcus 60. </s> <s xml:id="echoid-s1450" xml:space="preserve">graduum ad arcum <lb/>CX, ita 1. </s> <s xml:id="echoid-s1451" xml:space="preserve">gradus ad particulam t u: </s> <s xml:id="echoid-s1452" xml:space="preserve">Ac proinde quot partes ſexageſimæ arcus <lb/>60. </s> <s xml:id="echoid-s1453" xml:space="preserve">graduum, hoc eſt, quot gradus in arcu C X, continentur, tot ſexageſimæ <lb/>partes vnius gradus, id eſt, tot minuta in particula tu, exiſtent. </s> <s xml:id="echoid-s1454" xml:space="preserve">Item quam <lb/>proportionem haber arcus 60. </s> <s xml:id="echoid-s1455" xml:space="preserve">Minutorum adi. </s> <s xml:id="echoid-s1456" xml:space="preserve">Minutum eam habet arcus ſe-<lb/>xagecuplus particulæ, quæ vltra gradus integros vſque ad X, ſupereſt ad hanc <lb/>ipſam particulam. </s> <s xml:id="echoid-s1457" xml:space="preserve">Permutando igitur erit, vt arcus 60. </s> <s xml:id="echoid-s1458" xml:space="preserve">Minutorum ad arcum <lb/>ſexagecuplum dictæ particulæ reliquæ, ita 1. </s> <s xml:id="echoid-s1459" xml:space="preserve">Minutum ad dictam particulam re-<lb/>liquam. </s> <s xml:id="echoid-s1460" xml:space="preserve">Quare quot partes ſexageſimæ arcus 60. </s> <s xml:id="echoid-s1461" xml:space="preserve">Minutorum, hoc eſt, quot <lb/>minuta in arcu dictæ particulæ ſexagecuplo (ſumendo nunc gradus Quadran-<lb/>tis BC, pro Minutis) continentur, tot partes ſexageſimæ vnius Minuti, id eſt, tot <lb/>Secunda, in reliqua illa particula includentur, & </s> <s xml:id="echoid-s1462" xml:space="preserve">ſic deinceps, ſi opus ſit, de <lb/>Tertijs, Quartis, &</s> <s xml:id="echoid-s1463" xml:space="preserve">c. </s> <s xml:id="echoid-s1464" xml:space="preserve">intelligatur. </s> <s xml:id="echoid-s1465" xml:space="preserve">Sed ſatis eſt, meo iudicio, ſi minuta diligen-<lb/>ter inquirantur. </s> <s xml:id="echoid-s1466" xml:space="preserve">Etſi quidem particula remanens maior fuerit ſemiſſe gradus, <lb/> <anchor type="handwritten" xlink:label="hd-069-1a" xlink:href="hd-069-1"/> pro illa Minutum aſſumatur, Minutiſque inuentis adij ciatur, quod tunc in illa <lb/>particula contineantur plura Secunda quam 30. </s> <s xml:id="echoid-s1467" xml:space="preserve">Si verò eadem particula ſe-<lb/>miſſe gradus fuerit minor, nihil Minutis inuẽtis adijciatur, quod tuncin illa par-<lb/>ticula pauciora Secũ da includantur, ꝗ̃ 30. </s> <s xml:id="echoid-s1468" xml:space="preserve">Si deniq; </s> <s xml:id="echoid-s1469" xml:space="preserve">dicta particula ſemiſsi grad. <lb/></s> <s xml:id="echoid-s1470" xml:space="preserve">fuerit æqualis, liberũ ſit addere Minutis inuẽtis vnum Minutũ, vel non addere.</s> <s xml:id="echoid-s1471" xml:space="preserve"/> </p> <div xml:id="echoid-div57" type="float" level="2" n="4"> <handwritten xlink:label="hd-069-1" xlink:href="hd-069-1a"/> <handwritten xlink:label="hd-069-1" xlink:href="hd-069-1a"/> </div> <p> <s xml:id="echoid-s1472" xml:space="preserve">11. </s> <s xml:id="echoid-s1473" xml:space="preserve"><emph style="sc">Qvia</emph> verò fa cilè error committi poteſt, ſi circino particulam dictam <lb/> <anchor type="handwritten" xlink:label="hd-069-1a" xlink:href="hd-069-1"/> gradus, vel minuti ſexagies ordine ſumere velimus, rectius feceris, quando <lb/>particula data ſemiſſe gradus minor eſt, ſi eam vna cum præcedente graduacce-<lb/>ptam primo loco quintuples, deinde hunc arcum quintuplũ duples, tertio hũc <lb/> <anchor type="handwritten" xlink:label="hd-069-1a" xlink:href="hd-069-1"/> arcum duplum triples, ac tandem quarto hunc arcum triplum iterum duples. <lb/></s> <s xml:id="echoid-s1474" xml:space="preserve">Vltimus enim hic arcus erit datæ particulæ vna cum gradu accepto ſexagecu-<lb/>plus. </s> <s xml:id="echoid-s1475" xml:space="preserve">Quareſi ex eo demãtur 60. </s> <s xml:id="echoid-s1476" xml:space="preserve">gradus, in dicabuntreliqui gradus numerũ mi-<lb/>nutorumin data particula contentorum. </s> <s xml:id="echoid-s1477" xml:space="preserve">Qnod ſi per æſtimationẽ cognoueris, <lb/> <anchor type="handwritten" xlink:label="hd-069-1a" xlink:href="hd-069-1"/> <pb o="40" file="070" n="70" rhead="GEOMETR. PRACT."/> <anchor type="handwritten" xlink:label="hd-070-1a" xlink:href="hd-070-1"/> particulam ſemiſſe gradus minorem non ſuperare min. </s> <s xml:id="echoid-s1478" xml:space="preserve">24. </s> <s xml:id="echoid-s1479" xml:space="preserve">(Nam ſi ſuperaret <lb/>24. </s> <s xml:id="echoid-s1480" xml:space="preserve">minuta, non poſſemus ratione iam explicanda in ouirere minuta; </s> <s xml:id="echoid-s1481" xml:space="preserve">propterea <lb/> <anchor type="handwritten" xlink:label="hd-070-1a" xlink:href="hd-070-1"/> quod circumducendo circinum, to tum Quadrantem excederemus, vt patebit.) <lb/></s> <s xml:id="echoid-s1482" xml:space="preserve">commodius minuta datæ particulæ cognoſcemus hoc modo. </s> <s xml:id="echoid-s1483" xml:space="preserve">Datam particu-<lb/>lam cum gradu præcedenti primo loco quadruplicabimus: </s> <s xml:id="echoid-s1484" xml:space="preserve">deinde huncarcum <lb/>quadruplum duplabimus, vt hab eamus octuplum arcus ex particula, & </s> <s xml:id="echoid-s1485" xml:space="preserve">vno <lb/>gradu compoſiti; </s> <s xml:id="echoid-s1486" xml:space="preserve">tertio arcum hunc octuplum iterum duplabimus, vt fiat ar-<lb/>c<emph style="sub">9</emph> ſedecupl<emph style="sub">9</emph> arcus ex data particula, & </s> <s xml:id="echoid-s1487" xml:space="preserve">vno gradu cõpoſiti; </s> <s xml:id="echoid-s1488" xml:space="preserve">quarto hunc arcũ <lb/>rurſus duplabimus; </s> <s xml:id="echoid-s1489" xml:space="preserve">& </s> <s xml:id="echoid-s1490" xml:space="preserve">quinto tandem hunc duplum iterum duplabimus; </s> <s xml:id="echoid-s1491" xml:space="preserve">vt <lb/>habeatur arcus continens arcum ex particula pro poſita, & </s> <s xml:id="echoid-s1492" xml:space="preserve">vno gradu ſexagies <lb/>& </s> <s xml:id="echoid-s1493" xml:space="preserve">quater, cuius extremum punctũ<unsure/>m diligenter notetur. </s> <s xml:id="echoid-s1494" xml:space="preserve">Nam ſi ex toto arcu <lb/>ab eo puncto incipiendo, auferatur arcus quadruplus particulæ datæ vnâ cum <lb/>vno gradu, ſupererit arcus ſexagecuplus eiuſdem particulæ vnâ cum vno gra-<lb/>du; </s> <s xml:id="echoid-s1495" xml:space="preserve">ex quo deniqueſi demantur 60. </s> <s xml:id="echoid-s1496" xml:space="preserve">gradus, reliqui gradus numerum minu-<lb/>torum indicabunt. </s> <s xml:id="echoid-s1497" xml:space="preserve">Quòd ſi data particula ſemiſſe gradus fuerit maior, explo-<lb/>tabimus eodem modo minuta inreliqua particula minore comprehenſa. </s> <s xml:id="echoid-s1498" xml:space="preserve">Hæc <lb/>namque minuta ex 60. </s> <s xml:id="echoid-s1499" xml:space="preserve">detracta relinquent minutain particula illa maiore com-<lb/> <anchor type="handwritten" xlink:label="hd-070-1a" xlink:href="hd-070-1"/> prehenſa. </s> <s xml:id="echoid-s1500" xml:space="preserve">Benè autem vides, quando data particula minor parum à ſemiſſe <lb/>gradus differt, tutius eſſe quintuplare illam vnâ cum vno gradu; </s> <s xml:id="echoid-s1501" xml:space="preserve">deinde hunc <lb/>arcum duplare, tertio hunc arcum triplare, & </s> <s xml:id="echoid-s1502" xml:space="preserve">poſtremò hunc iterum duplare. <lb/></s> <s xml:id="echoid-s1503" xml:space="preserve">Hac enim ratione ex vltimo arcu duplato auferenditantum ſunt 60. </s> <s xml:id="echoid-s1504" xml:space="preserve">gradus, & </s> <s xml:id="echoid-s1505" xml:space="preserve"><lb/>nunquam totus quadrans exhauritur, vt patet.</s> <s xml:id="echoid-s1506" xml:space="preserve"/> </p> <div xml:id="echoid-div58" type="float" level="2" n="5"> <handwritten xlink:label="hd-069-1" xlink:href="hd-069-1a"/> <handwritten xlink:label="hd-069-1" xlink:href="hd-069-1a"/> <handwritten xlink:label="hd-069-1" xlink:href="hd-069-1a"/> <handwritten xlink:label="hd-070-1" xlink:href="hd-070-1a"/> <handwritten xlink:label="hd-070-1" xlink:href="hd-070-1a"/> <handwritten xlink:label="hd-070-1" xlink:href="hd-070-1a"/> </div> <p> <s xml:id="echoid-s1507" xml:space="preserve">12. </s> <s xml:id="echoid-s1508" xml:space="preserve"><emph style="sc">Neqve</emph> verò ſemper opus eſt, vt particula illa gradus, vel particula mi-<lb/>nor vnâ cum vno gradu ſexagies repetatur, ſed ſatis eſt, vt ea aliquoties repeti-<lb/> <anchor type="handwritten" xlink:label="hd-070-1a" xlink:href="hd-070-1"/> ta incidat præciſè in aliquem gradum; </s> <s xml:id="echoid-s1509" xml:space="preserve">quod non raro accidere ſolet. </s> <s xml:id="echoid-s1510" xml:space="preserve">Nam <lb/>tunc conſtituetur fractio, cuius numerator eſt numerus graduum percurſorũ: <lb/></s> <s xml:id="echoid-s1511" xml:space="preserve">Denominator autem numerus tot vnitatum, quoties particula circino repetita <lb/>fuerit. </s> <s xml:id="echoid-s1512" xml:space="preserve">Verbi gratia, ſi aliqua grad<emph style="sub">9</emph> particula vicies repetita incidat in 6. </s> <s xml:id="echoid-s1513" xml:space="preserve">gradũ, cõ <lb/>plectetur particula illa {6/20}. </s> <s xml:id="echoid-s1514" xml:space="preserve">vnius gradus. </s> <s xml:id="echoid-s1515" xml:space="preserve">Quare ſi numerator 6. </s> <s xml:id="echoid-s1516" xml:space="preserve">per 60. </s> <s xml:id="echoid-s1517" xml:space="preserve">multi-<lb/>plicetur, & </s> <s xml:id="echoid-s1518" xml:space="preserve">pro ductus numerus 360. </s> <s xml:id="echoid-s1519" xml:space="preserve">per denominatorem 20. </s> <s xml:id="echoid-s1520" xml:space="preserve">diuidatur, in dica-<lb/>bit Quotiens 18. </s> <s xml:id="echoid-s1521" xml:space="preserve">particulam illam continere 18. </s> <s xml:id="echoid-s1522" xml:space="preserve">minuta. </s> <s xml:id="echoid-s1523" xml:space="preserve">Sic ſi alia particula ſe-<lb/>pties repetita incidat in tertium gradum, comprehendet ea {3/7}. </s> <s xml:id="echoid-s1524" xml:space="preserve">vnius gradus. </s> <s xml:id="echoid-s1525" xml:space="preserve">Si <lb/>igitur numerator 3. </s> <s xml:id="echoid-s1526" xml:space="preserve">per 60. </s> <s xml:id="echoid-s1527" xml:space="preserve">multiplicetur, & </s> <s xml:id="echoid-s1528" xml:space="preserve">numerus productus 180. </s> <s xml:id="echoid-s1529" xml:space="preserve">per de-<lb/>nominatorem 7. </s> <s xml:id="echoid-s1530" xml:space="preserve">diuidatur, reperientur 25. </s> <s xml:id="echoid-s1531" xml:space="preserve">min. </s> <s xml:id="echoid-s1532" xml:space="preserve">Et quia in diuiſione ſuperſunt <lb/>5. </s> <s xml:id="echoid-s1533" xml:space="preserve">ſi ea rurſum multiplicentur per 60. </s> <s xml:id="echoid-s1534" xml:space="preserve">productuſque numerus 300. </s> <s xml:id="echoid-s1535" xml:space="preserve">per eundem <lb/>denominatorem 7. </s> <s xml:id="echoid-s1536" xml:space="preserve">diuidatur, dabit quotiens adhuc 42 {6/7}. </s> <s xml:id="echoid-s1537" xml:space="preserve">ſecunda.</s> <s xml:id="echoid-s1538" xml:space="preserve"/> </p> <div xml:id="echoid-div59" type="float" level="2" n="6"> <handwritten xlink:label="hd-070-1" xlink:href="hd-070-1a"/> </div> <p> <s xml:id="echoid-s1539" xml:space="preserve"><emph style="sc">Demonstratio</emph> huius praxis hæc eſt. </s> <s xml:id="echoid-s1540" xml:space="preserve">Quoniam in priori exemplo, ita <lb/>ſe habent 20. </s> <s xml:id="echoid-s1541" xml:space="preserve">gradus ad 1. </s> <s xml:id="echoid-s1542" xml:space="preserve">gradum, vt particula vicies repetita ad vnam parti-<lb/>culam; </s> <s xml:id="echoid-s1543" xml:space="preserve">erit permutando arcus 20. </s> <s xml:id="echoid-s1544" xml:space="preserve">graduum ad arcum continentem particulam <lb/>vicies, hoc eſt, ad arcum 6. </s> <s xml:id="echoid-s1545" xml:space="preserve">graduum, vt 1. </s> <s xml:id="echoid-s1546" xml:space="preserve">gradus ad vnam particulam: </s> <s xml:id="echoid-s1547" xml:space="preserve">& </s> <s xml:id="echoid-s1548" xml:space="preserve">con-<lb/>uertendo 6. </s> <s xml:id="echoid-s1549" xml:space="preserve">grad. </s> <s xml:id="echoid-s1550" xml:space="preserve">ad 20. </s> <s xml:id="echoid-s1551" xml:space="preserve">grad. </s> <s xml:id="echoid-s1552" xml:space="preserve">vt 1 particula ad 1. </s> <s xml:id="echoid-s1553" xml:space="preserve">grad. </s> <s xml:id="echoid-s1554" xml:space="preserve">Cum ergo 6. </s> <s xml:id="echoid-s1555" xml:space="preserve">grad. </s> <s xml:id="echoid-s1556" xml:space="preserve">ſint <lb/>{6/20}. </s> <s xml:id="echoid-s1557" xml:space="preserve">graduum 20. </s> <s xml:id="echoid-s1558" xml:space="preserve">continebit quo que vna particula {6/20}. </s> <s xml:id="echoid-s1559" xml:space="preserve">vnius gradus, quod eſt <lb/>propoſitum. </s> <s xml:id="echoid-s1560" xml:space="preserve">In poſteriori verò exemplo, quia ita ſe habent 7. </s> <s xml:id="echoid-s1561" xml:space="preserve">grad. </s> <s xml:id="echoid-s1562" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s1563" xml:space="preserve">gra-<lb/>dum, vt particula ſepties repetita, hoc eſt, vt 3. </s> <s xml:id="echoid-s1564" xml:space="preserve">gradus ad 1. </s> <s xml:id="echoid-s1565" xml:space="preserve">particuiam; </s> <s xml:id="echoid-s1566" xml:space="preserve">erit per-<lb/>mutando arcus 7. </s> <s xml:id="echoid-s1567" xml:space="preserve">graduum ad 3 grad. </s> <s xml:id="echoid-s1568" xml:space="preserve">vt 1. </s> <s xml:id="echoid-s1569" xml:space="preserve">grad. </s> <s xml:id="echoid-s1570" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s1571" xml:space="preserve">particulam: </s> <s xml:id="echoid-s1572" xml:space="preserve">& </s> <s xml:id="echoid-s1573" xml:space="preserve">conuerten-<lb/>do 3. </s> <s xml:id="echoid-s1574" xml:space="preserve">gradus ad 7. </s> <s xml:id="echoid-s1575" xml:space="preserve">grad. </s> <s xml:id="echoid-s1576" xml:space="preserve">vt vna particula ad 1. </s> <s xml:id="echoid-s1577" xml:space="preserve">grad. </s> <s xml:id="echoid-s1578" xml:space="preserve">Cum ergo 3. </s> <s xml:id="echoid-s1579" xml:space="preserve">gradus ſint {3/7}. <lb/></s> <s xml:id="echoid-s1580" xml:space="preserve">ſeptem graduum, complectetur quoque 1. </s> <s xml:id="echoid-s1581" xml:space="preserve">particula {5/7}. </s> <s xml:id="echoid-s1582" xml:space="preserve">vnius gradus; </s> <s xml:id="echoid-s1583" xml:space="preserve">quod eſt <lb/>propoſitum. </s> <s xml:id="echoid-s1584" xml:space="preserve">eadem que in cęteris ratio eſt.</s> <s xml:id="echoid-s1585" xml:space="preserve"/> </p> <pb o="41" file="071" n="71" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s1586" xml:space="preserve"><emph style="sc">Qvando</emph> particula minor cum vno gradu repetita incidit in gradum ali-<lb/>quem præcisè, auferendi erunt ex gradibus percurſis tot gradus, quoties parti-<lb/>cula illa cum vno gradurepetita fuit. </s> <s xml:id="echoid-s1587" xml:space="preserve">Reliquus enim numerus erit Numerator <lb/>fractionis: </s> <s xml:id="echoid-s1588" xml:space="preserve">Denominator autem erit, qui prius. </s> <s xml:id="echoid-s1589" xml:space="preserve">Vtſi particula illa minor cum <lb/>1. </s> <s xml:id="echoid-s1590" xml:space="preserve">gradu repetita ſepties inciditin 10. </s> <s xml:id="echoid-s1591" xml:space="preserve">gradum; </s> <s xml:id="echoid-s1592" xml:space="preserve">demendi erunt 7. </s> <s xml:id="echoid-s1593" xml:space="preserve">gradus repetiti. <lb/></s> <s xml:id="echoid-s1594" xml:space="preserve">Atq; </s> <s xml:id="echoid-s1595" xml:space="preserve">ita habebuntur iterum {3/7}. </s> <s xml:id="echoid-s1596" xml:space="preserve">vnius gradus in data particula.</s> <s xml:id="echoid-s1597" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1598" xml:space="preserve">13. </s> <s xml:id="echoid-s1599" xml:space="preserve"><emph style="sc">Si</emph> viciſsim ex quouis gradu auferre velimus particulam quotlibet mi-<lb/> <anchor type="note" xlink:label="note-071-01a" xlink:href="note-071-01"/> nutorum, ita agendum erit. </s> <s xml:id="echoid-s1600" xml:space="preserve">In quadrante ſuperiore B C, accipiatur circino arcus <lb/>tot graduum, quot minuta deſiderantur; </s> <s xml:id="echoid-s1601" xml:space="preserve">atq; </s> <s xml:id="echoid-s1602" xml:space="preserve">(vt confuſionem vitemus) in ar-<lb/>cuminteruallo ſemidiametri quadrantis B C, deſcriptum transferatur. </s> <s xml:id="echoid-s1603" xml:space="preserve">Si enim <lb/>hic arcus in 60. </s> <s xml:id="echoid-s1604" xml:space="preserve">partes æquales ſecetur, (primum, videlicet in duas: </s> <s xml:id="echoid-s1605" xml:space="preserve">deinde vna <lb/>harũ iterum in duas: </s> <s xml:id="echoid-s1606" xml:space="preserve">Tertio vna harũ in tres; </s> <s xml:id="echoid-s1607" xml:space="preserve">ac poſtremo vna harũ in quinq;) <lb/></s> <s xml:id="echoid-s1608" xml:space="preserve">cõtinebit vna particula ſexageſima numerum minutorum propoſitũ. </s> <s xml:id="echoid-s1609" xml:space="preserve">Verbigra-<lb/>tia, ſi particula quæratur continens 50. </s> <s xml:id="echoid-s1610" xml:space="preserve">min. </s> <s xml:id="echoid-s1611" xml:space="preserve">accipiemus in arcu F G, adinterual-<lb/>lum ſemidiametri A C, deſcripto arcum F G, arcui C Z, graduum 50. </s> <s xml:id="echoid-s1612" xml:space="preserve">æqualem, <lb/>eumque in 60. </s> <s xml:id="echoid-s1613" xml:space="preserve">partes æquales ſecabimus, primum in duas in puncto K: </s> <s xml:id="echoid-s1614" xml:space="preserve">Deinde <lb/>arcum F K, iterum in duas in puncto L; </s> <s xml:id="echoid-s1615" xml:space="preserve">tertio arcum F L, in tres in punctis T, V. </s> <s xml:id="echoid-s1616" xml:space="preserve"><lb/>ac tandem arcum F V, in quinque. </s> <s xml:id="echoid-s1617" xml:space="preserve">Vna namque harum 5. </s> <s xml:id="echoid-s1618" xml:space="preserve">particularum compre-<lb/>hendet 50. </s> <s xml:id="echoid-s1619" xml:space="preserve">min. </s> <s xml:id="echoid-s1620" xml:space="preserve">ac proinde ſi transferatur circino in quemlibet gradum Quadrã-<lb/>tis B C, abſciſſa erunt 50. </s> <s xml:id="echoid-s1621" xml:space="preserve">Min. </s> <s xml:id="echoid-s1622" xml:space="preserve">ex eo gradu. </s> <s xml:id="echoid-s1623" xml:space="preserve">Hoc ita demonſtro. </s> <s xml:id="echoid-s1624" xml:space="preserve">Quoniam eſt, <lb/>vt arcus 60. </s> <s xml:id="echoid-s1625" xml:space="preserve">graduum ad 1. </s> <s xml:id="echoid-s1626" xml:space="preserve">gradum, ita arcus C Z, vel F G, graduum 50. </s> <s xml:id="echoid-s1627" xml:space="preserve">ad ſexa-<lb/>geſimam partem eiuſdem arcus F G; </s> <s xml:id="echoid-s1628" xml:space="preserve">erit permutando, vt arcus 60. </s> <s xml:id="echoid-s1629" xml:space="preserve">graduum ad <lb/>arcum F G, graduum 50; </s> <s xml:id="echoid-s1630" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s1631" xml:space="preserve">gradus ad ſexageſimam particulam arcus F G: </s> <s xml:id="echoid-s1632" xml:space="preserve">& </s> <s xml:id="echoid-s1633" xml:space="preserve"><lb/>conuertendo, vt arcus F G, graduum 50. </s> <s xml:id="echoid-s1634" xml:space="preserve">ad arcum graduum 60. </s> <s xml:id="echoid-s1635" xml:space="preserve">ita particula ſe-<lb/>xageſima arcus F G, ad 1. </s> <s xml:id="echoid-s1636" xml:space="preserve">gradum. </s> <s xml:id="echoid-s1637" xml:space="preserve">Cum ergo arcus F G, contineat {50/60}, arcus 60. </s> <s xml:id="echoid-s1638" xml:space="preserve"><lb/>graduum, continebit quoq; </s> <s xml:id="echoid-s1639" xml:space="preserve">particula ſexageſima arcus F G, {50/60}. </s> <s xml:id="echoid-s1640" xml:space="preserve">vnius gradus, <lb/>hoc eſt, 50. </s> <s xml:id="echoid-s1641" xml:space="preserve">Min. </s> <s xml:id="echoid-s1642" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s1643" xml:space="preserve"/> </p> <div xml:id="echoid-div60" type="float" level="2" n="7"> <note position="right" xlink:label="note-071-01" xlink:href="note-071-01a" xml:space="preserve">Quo pacto ex <lb/>quouis gradu <lb/>particula quot <lb/>libet Minuto-<lb/>rum abſcin-<lb/>datur.</note> </div> <p> <s xml:id="echoid-s1644" xml:space="preserve"><emph style="sc">Sed</emph> hæcres incommodiſsima eſt in paruis Quadrantibus, præſertim ſi pau-<lb/>ca Minuta, vtpote 1. </s> <s xml:id="echoid-s1645" xml:space="preserve">2. </s> <s xml:id="echoid-s1646" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s1647" xml:space="preserve">abſcindenda ſint. </s> <s xml:id="echoid-s1648" xml:space="preserve">Quis enimin Quadrante exiguo <lb/>arcum 1. </s> <s xml:id="echoid-s1649" xml:space="preserve">gradus, vel 2. </s> <s xml:id="echoid-s1650" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s1651" xml:space="preserve">in 60. </s> <s xml:id="echoid-s1652" xml:space="preserve">particulas diſtribuat? </s> <s xml:id="echoid-s1653" xml:space="preserve">Quamobrem commo-<lb/>diusid, quod proponitur, efficiemus hacratione. </s> <s xml:id="echoid-s1654" xml:space="preserve">Ex eo dem quadranteſupe-<lb/> <anchor type="handwritten" xlink:label="hd-071-1a" xlink:href="hd-071-1"/> riore B C, arcus grad. </s> <s xml:id="echoid-s1655" xml:space="preserve">61. </s> <s xml:id="echoid-s1656" xml:space="preserve">transferatur in arcum X Y, ad interuallum ſemidiame-<lb/>tri A C. </s> <s xml:id="echoid-s1657" xml:space="preserve">deſcriptum, ab X, vſque ad Y. </s> <s xml:id="echoid-s1658" xml:space="preserve">Atque hic arcus X Y, in 6<emph style="sub">0</emph>. </s> <s xml:id="echoid-s1659" xml:space="preserve">partes æquales <lb/>ſecetur; </s> <s xml:id="echoid-s1660" xml:space="preserve">Primum ſcilicet in 2. </s> <s xml:id="echoid-s1661" xml:space="preserve">deinde vtraq; </s> <s xml:id="echoid-s1662" xml:space="preserve">ſemiſsis in 3. </s> <s xml:id="echoid-s1663" xml:space="preserve">Deinde prima pars in <lb/> <anchor type="handwritten" xlink:label="hd-071-1a" xlink:href="hd-071-1"/> 10. </s> <s xml:id="echoid-s1664" xml:space="preserve">particulas æquales diuidatur, ita vt quælibet harum particularũ ſit {1/60}. </s> <s xml:id="echoid-s1665" xml:space="preserve">arcus <lb/>XY. </s> <s xml:id="echoid-s1666" xml:space="preserve">Et quoniam vna harum particularum eſt ad arcum XY, vt 1. </s> <s xml:id="echoid-s1667" xml:space="preserve">gradus Qua-<lb/>drantis B C, ad arcum 60. </s> <s xml:id="echoid-s1668" xml:space="preserve">graduum; </s> <s xml:id="echoid-s1669" xml:space="preserve">cum vtrobique proportio ſit ſubſexagecu-<lb/>pla; </s> <s xml:id="echoid-s1670" xml:space="preserve">erit permutando vna illarum particularum ad 1.</s> <s xml:id="echoid-s1671" xml:space="preserve"><unsure/> grad. </s> <s xml:id="echoid-s1672" xml:space="preserve">vt arcus XY, ad arcum <lb/>60. </s> <s xml:id="echoid-s1673" xml:space="preserve">graduum. </s> <s xml:id="echoid-s1674" xml:space="preserve">Quocirca quemadmodum arcus X Y, arcum grad. </s> <s xml:id="echoid-s1675" xml:space="preserve">60. </s> <s xml:id="echoid-s1676" xml:space="preserve">continet <lb/>ſemel, & </s> <s xml:id="echoid-s1677" xml:space="preserve">inſuper vnam eius partem ſexageſimam, id eſt, 1. </s> <s xml:id="echoid-s1678" xml:space="preserve">ad grad. </s> <s xml:id="echoid-s1679" xml:space="preserve">ex conſtru-<lb/>ctione; </s> <s xml:id="echoid-s1680" xml:space="preserve">Ita quoq; </s> <s xml:id="echoid-s1681" xml:space="preserve">vna illarum particularum comprehendet 1. </s> <s xml:id="echoid-s1682" xml:space="preserve">gradum ſemel, & </s> <s xml:id="echoid-s1683" xml:space="preserve"><lb/>inſuper vnam partem ſexageſimam vnius gradus, hoc eſt, 1. </s> <s xml:id="echoid-s1684" xml:space="preserve">Minutum. </s> <s xml:id="echoid-s1685" xml:space="preserve">Ex quo <lb/>fit, vt duæ particulæ complectantur 2. </s> <s xml:id="echoid-s1686" xml:space="preserve">grad. </s> <s xml:id="echoid-s1687" xml:space="preserve">& </s> <s xml:id="echoid-s1688" xml:space="preserve">inſuper 2. </s> <s xml:id="echoid-s1689" xml:space="preserve">Minuta. </s> <s xml:id="echoid-s1690" xml:space="preserve">Atvero 3. </s> <s xml:id="echoid-s1691" xml:space="preserve">par-<lb/>ticulæ contineant 3. </s> <s xml:id="echoid-s1692" xml:space="preserve">grad. </s> <s xml:id="echoid-s1693" xml:space="preserve">& </s> <s xml:id="echoid-s1694" xml:space="preserve">3. </s> <s xml:id="echoid-s1695" xml:space="preserve">Min. </s> <s xml:id="echoid-s1696" xml:space="preserve">& </s> <s xml:id="echoid-s1697" xml:space="preserve">ſic deinceps.</s> <s xml:id="echoid-s1698" xml:space="preserve"/> </p> <div xml:id="echoid-div61" type="float" level="2" n="8"> <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a"/> <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a"/> </div> <p> <s xml:id="echoid-s1699" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi in Quadrantem BC, transferatur vna particula ſexageſima arcus <lb/> <anchor type="handwritten" xlink:label="hd-071-1a" xlink:href="hd-071-1"/> XY, à puncto C, vel à quouis gradu, habebitur 1. </s> <s xml:id="echoid-s1700" xml:space="preserve">Min. </s> <s xml:id="echoid-s1701" xml:space="preserve">in 2. </s> <s xml:id="echoid-s1702" xml:space="preserve">gradu, vel quouis a-<lb/>lio. </s> <s xml:id="echoid-s1703" xml:space="preserve">Et ſi duæ particulæ transferantur, habebuntur in tertio gradu duo Minuta:</s> <s xml:id="echoid-s1704" xml:space="preserve"> <pb o="42" file="072" n="72" rhead="GEOMETR. PRACT."/> tria autem Min. </s> <s xml:id="echoid-s1705" xml:space="preserve">in 4. </s> <s xml:id="echoid-s1706" xml:space="preserve">gradu, ſi tres particulæ transferantur, & </s> <s xml:id="echoid-s1707" xml:space="preserve">ſic decæteris.</s> <s xml:id="echoid-s1708" xml:space="preserve"/> </p> <div xml:id="echoid-div62" type="float" level="2" n="9"> <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a"/> </div> <p> <s xml:id="echoid-s1709" xml:space="preserve"><emph style="sc">Pari</emph> ratione, ſi quis deſideret quotlibet gradus, ac Minuta, inquirenda <lb/>prius erit particula Minutorum, quæ deſiderantur, eaque ad gradus propoſitos <lb/>adijcienda. </s> <s xml:id="echoid-s1710" xml:space="preserve">Quod ſi particula minutorum inuentorum tam exigua fuerit, vt <lb/>circino vix accipi poſsit, accipienda ea erit vnà cum 1. </s> <s xml:id="echoid-s1711" xml:space="preserve">gradu: </s> <s xml:id="echoid-s1712" xml:space="preserve">& </s> <s xml:id="echoid-s1713" xml:space="preserve">hic arcus ex 1. <lb/></s> <s xml:id="echoid-s1714" xml:space="preserve">gradu & </s> <s xml:id="echoid-s1715" xml:space="preserve">particula conflatus adijciendus ad numerum graduum propoſitum <lb/>minus vno. </s> <s xml:id="echoid-s1716" xml:space="preserve">Vt ſi velit quis grad. </s> <s xml:id="echoid-s1717" xml:space="preserve">89. </s> <s xml:id="echoid-s1718" xml:space="preserve">Min. </s> <s xml:id="echoid-s1719" xml:space="preserve">59. </s> <s xml:id="echoid-s1720" xml:space="preserve">Inuenienda prius erunt 59. </s> <s xml:id="echoid-s1721" xml:space="preserve">Mi-<lb/>nuta. </s> <s xml:id="echoid-s1722" xml:space="preserve">quod fiet, ſi 59. </s> <s xml:id="echoid-s1723" xml:space="preserve">particulæ arcus X Y, in Quadrantem B C, transferantur. </s> <s xml:id="echoid-s1724" xml:space="preserve"><lb/>Nam particula in 60. </s> <s xml:id="echoid-s1725" xml:space="preserve">gradu complectetur 59. </s> <s xml:id="echoid-s1726" xml:space="preserve">Min. </s> <s xml:id="echoid-s1727" xml:space="preserve">vt dictum eſt. </s> <s xml:id="echoid-s1728" xml:space="preserve">Siigitur arcus <lb/>ex illa particula, & </s> <s xml:id="echoid-s1729" xml:space="preserve">1. </s> <s xml:id="echoid-s1730" xml:space="preserve">grad. </s> <s xml:id="echoid-s1731" xml:space="preserve">conflatus adijciatur ad arcum 88. </s> <s xml:id="echoid-s1732" xml:space="preserve">grad. </s> <s xml:id="echoid-s1733" xml:space="preserve">conficietur <lb/>arcus grad. </s> <s xml:id="echoid-s1734" xml:space="preserve">89. </s> <s xml:id="echoid-s1735" xml:space="preserve">Min. </s> <s xml:id="echoid-s1736" xml:space="preserve">59. </s> <s xml:id="echoid-s1737" xml:space="preserve">Eademque ratio eſt de cæteris. </s> <s xml:id="echoid-s1738" xml:space="preserve">Accipientur autem in ar-<lb/>cu XY, particulæ 59. </s> <s xml:id="echoid-s1739" xml:space="preserve">ſi vnus pes circini in puncto 50. </s> <s xml:id="echoid-s1740" xml:space="preserve">ſtatuatur, & </s> <s xml:id="echoid-s1741" xml:space="preserve">alter in nona <lb/>particula primæ partis ſextæ totius arcus XY, verſus X. </s> <s xml:id="echoid-s1742" xml:space="preserve">Ita accipientur quoque <lb/>particulæ 49. </s> <s xml:id="echoid-s1743" xml:space="preserve">48. </s> <s xml:id="echoid-s1744" xml:space="preserve">39. </s> <s xml:id="echoid-s1745" xml:space="preserve">34. </s> <s xml:id="echoid-s1746" xml:space="preserve">&</s> <s xml:id="echoid-s1747" xml:space="preserve">c. </s> <s xml:id="echoid-s1748" xml:space="preserve">vt perſpicuum eſt.</s> <s xml:id="echoid-s1749" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1750" xml:space="preserve"><emph style="sc">Iam</emph> vero ſi Minutanonin Quadrante BC, ſed in maiori, minoriue accipien-<lb/>da ſint, inquirenda ea erunt in Quadrante B C, beneficio arcus X Y, vt docui-<lb/>mus; </s> <s xml:id="echoid-s1751" xml:space="preserve">Deinde arcuiinter C, & </s> <s xml:id="echoid-s1752" xml:space="preserve">finem particulæ inuentæ auferendus ex Quadrã-<lb/>te propoſito arcus ſimilis. </s> <s xml:id="echoid-s1753" xml:space="preserve">quod fiet, ſi ille Quadrans ex centro A, deſcribatur, <lb/>rectaque ex A, per finem particulæ in B C, inuentæ educatur, &</s> <s xml:id="echoid-s1754" xml:space="preserve">c.</s> <s xml:id="echoid-s1755" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1756" xml:space="preserve">14. </s> <s xml:id="echoid-s1757" xml:space="preserve"><emph style="sc">Qvæ</emph> Num. </s> <s xml:id="echoid-s1758" xml:space="preserve">13. </s> <s xml:id="echoid-s1759" xml:space="preserve">præcedenti diximus, perbelle etiam quadrant in lineas <lb/> <anchor type="note" xlink:label="note-072-01a" xlink:href="note-072-01"/> rectas. </s> <s xml:id="echoid-s1760" xml:space="preserve">Nam eadem ratione cognoſcemus, ſi linea recta in quotuis partes æqua-<lb/>les ſecetur, quantam fractionem quælibet particula vnius partis contineat: </s> <s xml:id="echoid-s1761" xml:space="preserve">Et <lb/>viciſsim quo pacto ex vna parte abſcindenda ſit quæcun que fractio propoſita. <lb/></s> <s xml:id="echoid-s1762" xml:space="preserve">Quæ res incredibile eſt, quantam vtilitatem cum alijs rebus Geometricis, tum <lb/>ver ò maxime Dimenſionibus, quæ per ſcalam altimetram fieri ſolent, afferat, vt <lb/>lib. </s> <s xml:id="echoid-s1763" xml:space="preserve">3. </s> <s xml:id="echoid-s1764" xml:space="preserve">cum de Quadrato Geometrico, vbiſcalæ altimetræ vſus apparebit, perſpi-<lb/>cuum erit. </s> <s xml:id="echoid-s1765" xml:space="preserve">Sit enim recta linea A B, vt ad pedem Quadrantis ſuperioris vides, <lb/>ſecta in 10. </s> <s xml:id="echoid-s1766" xml:space="preserve">partes æquales. </s> <s xml:id="echoid-s1767" xml:space="preserve">(In totenim partes libet tam vmbiam rectam, quam <lb/> <anchor type="handwritten" xlink:label="hd-072-1a" xlink:href="hd-072-1"/> verſam ſcalæ altimetræ diſtribuere: </s> <s xml:id="echoid-s1768" xml:space="preserve">quamuis ab alijs vtraque in 12. </s> <s xml:id="echoid-s1769" xml:space="preserve">diuidatur: <lb/></s> <s xml:id="echoid-s1770" xml:space="preserve">quod per illam diuiſionem facilius Dimenſiones perficiantur, vt ſuo loco pate-<lb/>bit. </s> <s xml:id="echoid-s1771" xml:space="preserve">Magis tamen probarem, ſi vtrumque vmbræ latus in 100. </s> <s xml:id="echoid-s1772" xml:space="preserve">partes ſecare-<lb/>tur, ſi id magnitudo inſtrumenti commode permittit) propoſitumque ſit, quot <lb/>partes decimas contineat particula D C, partis quintæ. </s> <s xml:id="echoid-s1773" xml:space="preserve">Beneficio circini ſum-<lb/>pta paiticula D C, decupletur ab A, vſque ad E. </s> <s xml:id="echoid-s1774" xml:space="preserve">Et quoniamin A E, continen-<lb/>tur ſex partes totius lineæ A B, continebit propterea particula D C. </s> <s xml:id="echoid-s1775" xml:space="preserve">{6/10}. </s> <s xml:id="echoid-s1776" xml:space="preserve">vnius <lb/>partis decimæ, hoceſt, {6/100}. </s> <s xml:id="echoid-s1777" xml:space="preserve">totius lineæ. </s> <s xml:id="echoid-s1778" xml:space="preserve">Ita vt ſirecta A B, diuiſa cogitetur in <lb/>100. </s> <s xml:id="echoid-s1779" xml:space="preserve">partes, tribuendo ſingulis decimis partibus denas particulas, ſegmentum <lb/>A C, comprehendat {46/100}. </s> <s xml:id="echoid-s1780" xml:space="preserve">Quia vero vltra {6/10}. </s> <s xml:id="echoid-s1781" xml:space="preserve">ſupereſt adhuc particula F E, <lb/>vnius decimæ, ſi ea rurſum decupletur ab A, vſque ad G, reperientur in A G, <lb/>octo partes totius lineæ A B. </s> <s xml:id="echoid-s1782" xml:space="preserve">Continet ergo particula F E, {8/10}. </s> <s xml:id="echoid-s1783" xml:space="preserve">vnius decimæ, <lb/>hoc eſt, propoſita particula D C, vltra {6/10}. </s> <s xml:id="echoid-s1784" xml:space="preserve">vnius partis rectæ A B, continet in <lb/>ſuper {8/10}, vnius decimæ, (vnius inquam decimæ ex illis {6/10}. </s> <s xml:id="echoid-s1785" xml:space="preserve">quas in particula <lb/>D C, diximus comprehendi) nimirum {8/100}. </s> <s xml:id="echoid-s1786" xml:space="preserve">vnius partis. </s> <s xml:id="echoid-s1787" xml:space="preserve">ſi ſingulæ partes deci-<lb/>mærectæ A B, diuiſæ eſſent in 100. </s> <s xml:id="echoid-s1788" xml:space="preserve">particulas; </s> <s xml:id="echoid-s1789" xml:space="preserve">atque adeo, ſi recta A B, ſecta <lb/>intelligatur in 1000. </s> <s xml:id="echoid-s1790" xml:space="preserve">partes, tribuendo ſingulis decimis partibus centenas par-<lb/>ticulas ſegmentum A C, complectetur {468/1000}. </s> <s xml:id="echoid-s1791" xml:space="preserve">quippe cum in A D, continean- <pb o="43" file="073" n="73" rhead="LIBER PRIMVS."/> @ur {400/1000}. </s> <s xml:id="echoid-s1792" xml:space="preserve">Et in D C, {68/100}. </s> <s xml:id="echoid-s1793" xml:space="preserve">vnius partis rectæ AB, ſiue {63/1000}. </s> <s xml:id="echoid-s1794" xml:space="preserve">totius lineæ AB; </s> <s xml:id="echoid-s1795" xml:space="preserve">cum <lb/>quælibet centeſima particula vnius partis decimæ ſit {1/1000}. </s> <s xml:id="echoid-s1796" xml:space="preserve">Atque in hunc mo-<lb/>dum progredilicebit ad decimas vnius decimæ ex illis {6/10}. </s> <s xml:id="echoid-s1797" xml:space="preserve">quæin particula DC,<unsure/> <lb/>continentur. </s> <s xml:id="echoid-s1798" xml:space="preserve">nimirum ad fractionem à 10000. </s> <s xml:id="echoid-s1799" xml:space="preserve">denominatam, &</s> <s xml:id="echoid-s1800" xml:space="preserve">c. </s> <s xml:id="echoid-s1801" xml:space="preserve">ſed ſatis mi-<lb/>hi videtur ad partes milleſimas peruenire.</s> <s xml:id="echoid-s1802" xml:space="preserve"/> </p> <div xml:id="echoid-div63" type="float" level="2" n="10"> <note position="left" xlink:label="note-072-01" xlink:href="note-072-01a" xml:space="preserve">Quo pactore-<lb/>peri<unsure/>atur fra-<lb/>ctio cuiuſque <lb/>particulæ in <lb/>parte qualibet <lb/>lineæ rectæ in <lb/>part{es} æqual{es} <lb/>diuiſæ.</note> <handwritten xlink:label="hd-072-1" xlink:href="hd-072-1a"/> </div> <p> <s xml:id="echoid-s1803" xml:space="preserve"><emph style="sc">Demonstratio</emph> hic eadem eſt, quæin gradibus, ac Minutis. </s> <s xml:id="echoid-s1804" xml:space="preserve">Eandem <lb/>enim proportionem habet recta 10. </s> <s xml:id="echoid-s1805" xml:space="preserve">partium ad 1. </s> <s xml:id="echoid-s1806" xml:space="preserve">partem, quamrecta A E, ad <lb/>particulam D C, cum vtrobique proportio ſit decupla, ideoque permutando <lb/>erit vt recta 10. </s> <s xml:id="echoid-s1807" xml:space="preserve">partium ad rectam A E, ita 1. </s> <s xml:id="echoid-s1808" xml:space="preserve">pars ad particulam D C. </s> <s xml:id="echoid-s1809" xml:space="preserve">Quamo-<lb/>brem ſicut in A E, continentur {6/10}. </s> <s xml:id="echoid-s1810" xml:space="preserve">rectæ A B, decem partium, & </s> <s xml:id="echoid-s1811" xml:space="preserve">inſuper par-<lb/>ticula F E, reſpectu vnius partis lineæ A B, decem partium, ita quoque in par-<lb/>ticula data D C, continebuntur {6/10}. </s> <s xml:id="echoid-s1812" xml:space="preserve">vnius partis, & </s> <s xml:id="echoid-s1813" xml:space="preserve">inſuper talis particula re-<lb/>ſpectu vnius decimæ qualis eſt F E, reſpectu vnius partis lineæ A B, decem par-<lb/>tium, &</s> <s xml:id="echoid-s1814" xml:space="preserve">c.</s> <s xml:id="echoid-s1815" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1816" xml:space="preserve"><emph style="sc">Itaqve</emph>, vt vides, duabus operationibus ad milleſimas partes perueni-<lb/> <anchor type="note" xlink:label="note-073-01a" xlink:href="note-073-01"/> tur, ac ſi latus totum A B, in 1000. </s> <s xml:id="echoid-s1817" xml:space="preserve">partes ſectum eſſet, quod conſideratione <lb/>dignum eſt. </s> <s xml:id="echoid-s1818" xml:space="preserve">Et duo quidem numeratores decimarum eo c<unsure/>rdine poſiti, quo <lb/>inuenti ſunt, dant Numeratorem centeſimarum: </s> <s xml:id="echoid-s1819" xml:space="preserve">cui ſi præponatur ad ſiniſtram <lb/>numerus integrarum partium ante datam particulam exiſtentium, conflabitur <lb/>Numerator milleſimarum. </s> <s xml:id="echoid-s1820" xml:space="preserve">Vt in ſuperiori exemplo, quia ante datam parti-<lb/>culam D C, reperiuntur 4. </s> <s xml:id="echoid-s1821" xml:space="preserve">partes, inuentæque ſunt {6/10}. </s> <s xml:id="echoid-s1822" xml:space="preserve">& </s> <s xml:id="echoid-s1823" xml:space="preserve">{8/10}. </s> <s xml:id="echoid-s1824" xml:space="preserve">erit tota fra ctio <lb/>{468/1000}. </s> <s xml:id="echoid-s1825" xml:space="preserve">Sic ſi inuentæ eſſent pro aliqua particula in octaua parte, {7/10}. </s> <s xml:id="echoid-s1826" xml:space="preserve">{0/10}. </s> <s xml:id="echoid-s1827" xml:space="preserve">conſti-<lb/>tueretur fractio {870/1000}. </s> <s xml:id="echoid-s1828" xml:space="preserve">Item ſi in tertia parte inuentæ eſſent {0/10}. </s> <s xml:id="echoid-s1829" xml:space="preserve">{7/10}. </s> <s xml:id="echoid-s1830" xml:space="preserve">fieret fractio <lb/> <anchor type="handwritten" xlink:label="hd-073-1a" xlink:href="hd-073-1"/> {307/1000}. </s> <s xml:id="echoid-s1831" xml:space="preserve">& </s> <s xml:id="echoid-s1832" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s1833" xml:space="preserve">Quod ſirecta A B, id eſt, latus vtriuſque vmbræ diui-<lb/>deretur in 100. </s> <s xml:id="echoid-s1834" xml:space="preserve">partes, reperientur partes milleſimæ vnica operatione; </s> <s xml:id="echoid-s1835" xml:space="preserve">ſi nimi-<lb/>rum particula data in vna centeſima decuplaretur: </s> <s xml:id="echoid-s1836" xml:space="preserve">quia tunc intelligerentur ſin-<lb/>gulæ centeſimę in denas particulas ſub diuiſæ. </s> <s xml:id="echoid-s1837" xml:space="preserve">Sed quia particula data in cen-<lb/>teſima aliqua parte perparua eſt, vt vix circino capi poſsit, accipiemus eam cũ <lb/> <anchor type="handwritten" xlink:label="hd-073-1a" xlink:href="hd-073-1"/> vna centeſima, vel duabus, & </s> <s xml:id="echoid-s1838" xml:space="preserve">ex decuplo abijciemus 10. </s> <s xml:id="echoid-s1839" xml:space="preserve">vel 20. </s> <s xml:id="echoid-s1840" xml:space="preserve">centeſimas, vt <lb/>reliquę centeſimæ exhibeant decimas vnius centeſimę. </s> <s xml:id="echoid-s1841" xml:space="preserve">Eadem ratione, quan-<lb/>do data particula in linea 10. </s> <s xml:id="echoid-s1842" xml:space="preserve">partium eſt perpuſilla accipienda erit reliqua parti-<lb/>cula maior decies. </s> <s xml:id="echoid-s1843" xml:space="preserve">Nam reliqua pars rectę AB, à B, vſque ad finem illius par-<lb/>ticulę decuplę dabit numerum decimarum in propoſita minore particula, &</s> <s xml:id="echoid-s1844" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1845" xml:space="preserve">Hac ratione, ſi particulam CH, decuples, in cides in punctum K Ergo ſegmẽ-<lb/>tum B K, dabit {6/10}. </s> <s xml:id="echoid-s1846" xml:space="preserve">& </s> <s xml:id="echoid-s1847" xml:space="preserve">inſuper particulam DK, vt ſupra; </s> <s xml:id="echoid-s1848" xml:space="preserve">cum DK, particulę FE, <lb/>ſit ęqualis. </s> <s xml:id="echoid-s1849" xml:space="preserve">Ratio huius reieſt, quod ambę particulę DC, CH, decuplatę con-<lb/>ficere debeanttotam A B, vt conſtat.</s> <s xml:id="echoid-s1850" xml:space="preserve"/> </p> <div xml:id="echoid-div64" type="float" level="2" n="11"> <note position="right" xlink:label="note-073-01" xlink:href="note-073-01a" xml:space="preserve">Denominatio <lb/>facilis fractio-<lb/>num milleſi. <lb/>marum.</note> <handwritten xlink:label="hd-073-1" xlink:href="hd-073-1a"/> <handwritten xlink:label="hd-073-1" xlink:href="hd-073-1a"/> </div> <p> <s xml:id="echoid-s1851" xml:space="preserve"><emph style="sc">Ad</emph> Maiorem quo que commo ditatem pro inueſtigandis partibus decimis, <lb/> <anchor type="handwritten" xlink:label="hd-073-1a" xlink:href="hd-073-1"/> hoc eſt, pro decuplanda particula propoſita, conſtrui poterit circinus dupli-<lb/>cis aperturę, in quo ſcilicet crura producta ſe mutuo interſecent, at que vna a-<lb/>pertura alterius ſit ſemper decupla, inſtar circini, quo linea duas in partes ęqua-<lb/>les diuidi ſolet. </s> <s xml:id="echoid-s1852" xml:space="preserve">Ita enim fiet, vt accepta per minorem aperturam particula ab-<lb/> <anchor type="handwritten" xlink:label="hd-073-1a" xlink:href="hd-073-1"/> ſciſſa, particula maior exhibeat eam particulam decies ſumptam, vt non opus <lb/>ſit toties circinum circumducere: </s> <s xml:id="echoid-s1853" xml:space="preserve">qua quidem in re facile error committi <lb/>poteſt, quiillo circino, ſi recte fabricatus ſit, facilius vitatur. </s> <s xml:id="echoid-s1854" xml:space="preserve">Sed ſine hoc <lb/>circino, idem fieri poteſt per inſtrumentum partium, quod capite pręceden-<lb/>ti conſtruximus. </s> <s xml:id="echoid-s1855" xml:space="preserve">Nam ſi particula data circino capiatur, & </s> <s xml:id="echoid-s1856" xml:space="preserve">ſumma diligentia ei <lb/> <anchor type="handwritten" xlink:label="hd-073-1a" xlink:href="hd-073-1"/> <pb o="44" file="074" n="74" rhead="GEOMETR. PRACT."/> ſumatur in inſtrumento interuallum inter 10. </s> <s xml:id="echoid-s1857" xml:space="preserve">& </s> <s xml:id="echoid-s1858" xml:space="preserve">10. </s> <s xml:id="echoid-s1859" xml:space="preserve">æquale, erit interuallum in-<lb/>ter 100. </s> <s xml:id="echoid-s1860" xml:space="preserve">& </s> <s xml:id="echoid-s1861" xml:space="preserve">100. </s> <s xml:id="echoid-s1862" xml:space="preserve">datæ particulæ decuplum.</s> <s xml:id="echoid-s1863" xml:space="preserve"/> </p> <div xml:id="echoid-div65" type="float" level="2" n="12"> <handwritten xlink:label="hd-073-1" xlink:href="hd-073-1a"/> <handwritten xlink:label="hd-073-1" xlink:href="hd-073-1a"/> <handwritten xlink:label="hd-073-1" xlink:href="hd-073-1a"/> </div> <p> <s xml:id="echoid-s1864" xml:space="preserve">15. </s> <s xml:id="echoid-s1865" xml:space="preserve"><emph style="sc">Iam</emph> vero ſi viciſsim ex qualibet parte rectæ A B, auferendæ ſint quot-<lb/> <anchor type="note" xlink:label="note-074-01a" xlink:href="note-074-01"/> cunque decimæ partes vnius, diuidendum erit ſegmentum continens numerum <lb/>partium decimarum in 10. </s> <s xml:id="echoid-s1866" xml:space="preserve">pattes æquales. </s> <s xml:id="echoid-s1867" xml:space="preserve">Nam vna pars decima huius ſegmen-<lb/>ti continebit decimas partes quæſitas. </s> <s xml:id="echoid-s1868" xml:space="preserve">Vt ſi cupiat quis {7/10}, diuidendum erit <lb/>ſegmentum rectæ A B, includens 7. </s> <s xml:id="echoid-s1869" xml:space="preserve">partes, in 10. </s> <s xml:id="echoid-s1870" xml:space="preserve">particulas. </s> <s xml:id="echoid-s1871" xml:space="preserve">Quælibet namq; <lb/></s> <s xml:id="echoid-s1872" xml:space="preserve">harum comprehendet {7/10}. </s> <s xml:id="echoid-s1873" xml:space="preserve">vnius partis rectæ A B. </s> <s xml:id="echoid-s1874" xml:space="preserve">Eademq; </s> <s xml:id="echoid-s1875" xml:space="preserve">ratio eſt de cæteris. </s> <s xml:id="echoid-s1876" xml:space="preserve"><lb/>Sed commodius hunc vſum nobis præſtabit inſtrumentum partium ſupra con-<lb/> <anchor type="handwritten" xlink:label="hd-074-1a" xlink:href="hd-074-1"/> ſtructum. </s> <s xml:id="echoid-s1877" xml:space="preserve">Eius enim beneficio ex quauis recta abſcindemus non ſolum quot-<lb/>cunque decimas, ſed etiam centeſimas, decimas nonas, nonageſimas octauas, <lb/>& </s> <s xml:id="echoid-s1878" xml:space="preserve">ſic deinceps, vſque ad dimidiatam partem. </s> <s xml:id="echoid-s1879" xml:space="preserve">Nam ſi velimus {3/100}. </s> <s xml:id="echoid-s1880" xml:space="preserve">alicuius <lb/>lineæ, capiemus ei lineæ interuallum inter 100. </s> <s xml:id="echoid-s1881" xml:space="preserve">& </s> <s xml:id="echoid-s1882" xml:space="preserve">100. </s> <s xml:id="echoid-s1883" xml:space="preserve">æquale. </s> <s xml:id="echoid-s1884" xml:space="preserve">Circinus nam-<lb/>que extenſus inter 3. </s> <s xml:id="echoid-s1885" xml:space="preserve">& </s> <s xml:id="echoid-s1886" xml:space="preserve">3. </s> <s xml:id="echoid-s1887" xml:space="preserve">dabit {3/100}. </s> <s xml:id="echoid-s1888" xml:space="preserve">quæſitas. </s> <s xml:id="echoid-s1889" xml:space="preserve">Ita quo que circini pedes inter <lb/>50. </s> <s xml:id="echoid-s1890" xml:space="preserve">& </s> <s xml:id="echoid-s1891" xml:space="preserve">50. </s> <s xml:id="echoid-s1892" xml:space="preserve">dabunt {50/100}. </s> <s xml:id="echoid-s1893" xml:space="preserve">id eſt, {1/2}. </s> <s xml:id="echoid-s1894" xml:space="preserve">Rurſus pedes circini inter 20, & </s> <s xml:id="echoid-s1895" xml:space="preserve">20. </s> <s xml:id="echoid-s1896" xml:space="preserve">dabunt <lb/>{20/100}. </s> <s xml:id="echoid-s1897" xml:space="preserve">hoc eſt, {1/5}. </s> <s xml:id="echoid-s1898" xml:space="preserve">& </s> <s xml:id="echoid-s1899" xml:space="preserve">ſic de cæteris, vt fuſius in vſu prædicti inſtrumenti partium <lb/>cap. </s> <s xml:id="echoid-s1900" xml:space="preserve">1. </s> <s xml:id="echoid-s1901" xml:space="preserve">expoſuimus.</s> <s xml:id="echoid-s1902" xml:space="preserve"/> </p> <div xml:id="echoid-div66" type="float" level="2" n="13"> <note position="left" xlink:label="note-074-01" xlink:href="note-074-01a" xml:space="preserve">Quo modo ex <lb/>data linea au-<lb/>ferendæ ſint <lb/>quotcunque <lb/>part{es} decim æ <lb/>velaliæ part{es}.</note> <handwritten xlink:label="hd-074-1" xlink:href="hd-074-1a"/> </div> </div> <div xml:id="echoid-div68" type="section" level="1" n="20"> <head xml:id="echoid-head23" xml:space="preserve">PROBLEMATA VARIA TRIANGV-<lb/>lorum rectilineorum. <lb/><emph style="sc">Capvt</emph> III.</head> <p> <s xml:id="echoid-s1903" xml:space="preserve">TERTIO loco præmiſſuros nos polliciti ſumus varia problema-<lb/>ta triangulorum rectilineorum, vt & </s> <s xml:id="echoid-s1904" xml:space="preserve">latera eorum atque anguli <lb/>ex quibuſdam datis, & </s> <s xml:id="echoid-s1905" xml:space="preserve">cognitis promptè, atque expeditè, cum id <lb/>res poſtulauerit, poſsint erui: </s> <s xml:id="echoid-s1906" xml:space="preserve">quod hoc tertio capite exequemur. <lb/></s> <s xml:id="echoid-s1907" xml:space="preserve">Prioriautem loco de triangulis rectangulis: </s> <s xml:id="echoid-s1908" xml:space="preserve">poſteriori vero de ob-<lb/>liquangulis agemus. </s> <s xml:id="echoid-s1909" xml:space="preserve">In margine porrò adſcripſimus propoſitiones noſtri <lb/>tractatus triangulorum rectilineorum, in Theodoſio noſtro editi, in quibus <lb/>problemata hæc demonſtrantur, vt ſtudioſi intelligant, vnde eorum demon-<lb/>ſtrationes, quando libuerit, petere debeant.</s> <s xml:id="echoid-s1910" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div69" type="section" level="1" n="21"> <head xml:id="echoid-head24" xml:space="preserve">TRIANGVLORVM RECTILINEORVM RECTAN-<lb/>gulorum problemata. <lb/>I. PROPORTIONES LATERVM</head> <p> <s xml:id="echoid-s1911" xml:space="preserve">Ex datis omnibus angulis cuiuſuis trianguli patefacere.</s> <s xml:id="echoid-s1912" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">1. Triang. re-<lb/>ctil.</note> <p style="it"> <s xml:id="echoid-s1913" xml:space="preserve">Singulis laterib{us} adſcribantur ſin{us} angulorum oppoſitorum. </s> <s xml:id="echoid-s1914" xml:space="preserve">Latera enim eaſ-<lb/>dem proportion{es} habent, quæ inter ſin{us} angulorum laterib{us} oppoſitorum reperiuntur.</s> <s xml:id="echoid-s1915" xml:space="preserve"/> </p> <pb o="45" file="075" n="75" rhead="LIBER PRIMVS."/> </div> <div xml:id="echoid-div70" type="section" level="1" n="22"> <head xml:id="echoid-head25" xml:space="preserve">II. LATVS.</head> <p> <s xml:id="echoid-s1916" xml:space="preserve">Ex baſe, & </s> <s xml:id="echoid-s1917" xml:space="preserve">alterutro angulorum acutorum, ac proinde & </s> <s xml:id="echoid-s1918" xml:space="preserve">altero, notum <lb/>efficere.</s> <s xml:id="echoid-s1919" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad baſem # Ita ſin{us} anguli lateriquæ- \\ ſito oppoſiti # ad lat{us} quæſitum in par- \\ tib{us} baſis. <lb/></note> <note position="right" xml:space="preserve">2. Triang. <lb/>rectil.</note> </div> <div xml:id="echoid-div71" type="section" level="1" n="23"> <head xml:id="echoid-head26" xml:space="preserve">III. LATVS.</head> <p> <s xml:id="echoid-s1920" xml:space="preserve">Ex baſe, & </s> <s xml:id="echoid-s1921" xml:space="preserve">altero latere cognoſcere.</s> <s xml:id="echoid-s1922" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt baſis # ad ſinum totum # ita datum lat{us}. # ad ſinum anguli dato late- \\ ri oppoſiti. <lb/></note> <note position="right" xml:space="preserve">3. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s1923" xml:space="preserve">Deinde, ſumpto complemento anguli inuenti pro reliquo angulo.</s> <s xml:id="echoid-s1924" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad baſem # Ita ſin{us} anguli, qui la- \\ teri quæſito opponi- \\ tur, # Adlat{us} quæſitum in par- \\ tib{us} baſis, & alteri{us} \\ lateris. <lb/></note> </div> <div xml:id="echoid-div72" type="section" level="1" n="24"> <head xml:id="echoid-head27" xml:space="preserve">IIII. LATVS.</head> <p> <s xml:id="echoid-s1925" xml:space="preserve">Ex altero latere, & </s> <s xml:id="echoid-s1926" xml:space="preserve">alterutro angulo acuto, ac proinde & </s> <s xml:id="echoid-s1927" xml:space="preserve">altero, eruere.</s> <s xml:id="echoid-s1928" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">2. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad lat{us} \\ datum. # Ita tangens anguli quæſito \\ lateri oppoſiti # Ad lat{us} quæſitum in par- \\ tib{us} dati lateris. <lb/> #### Vel ## Vt ſin{us} anguli dato \\ lateri oppoſiti. # ad lat{us} \\ datum. # Ita ſin{us} alteri{us} \\ anguli # ad lat{us} quæſitum in par. \\ tib{us} dati lateris.</note> <note position="right" xml:space="preserve">2. Triang. <lb/>rectil.</note> </div> <div xml:id="echoid-div73" type="section" level="1" n="25"> <head xml:id="echoid-head28" xml:space="preserve">V. BASEM.</head> <p> <s xml:id="echoid-s1929" xml:space="preserve">Ex vno latere, & </s> <s xml:id="echoid-s1930" xml:space="preserve">vno angulo acuto, ac proinde & </s> <s xml:id="echoid-s1931" xml:space="preserve">altero, inueſtigare.</s> <s xml:id="echoid-s1932" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">2. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad lat{us} \\ datum: # Ita ſecans angulo dato late- \\ ri adiacentis. # Ad baſem in partib{us} la- \\ ris dati. <lb/> #### Vel ## Vt ſin{us} anguli dato la- \\ teri oppoſiti # ad ſinum to- \\ tum # Ita lat{us} da- \\ tum # ad baſem in partib{us} \\ lateris dati.</note> <note position="right" xml:space="preserve">2. Triang. <lb/>rectil.</note> </div> <div xml:id="echoid-div74" type="section" level="1" n="26"> <head xml:id="echoid-head29" xml:space="preserve">VI. BASEM.</head> <p> <s xml:id="echoid-s1933" xml:space="preserve">Ex vtro que latere perſcrutari, vna cum angulis acutis.</s> <s xml:id="echoid-s1934" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">3. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vilat{us} alterutrum \\ datum # ad ſinum \\ totum: # Ita alterum \\ lat{us} datum # ad tangentem anguli huic \\ alteri lateri oppoſiti. <lb/> </note> <p> <s xml:id="echoid-s1935" xml:space="preserve">Deinde, ſumpto complemento anguli inuenti pro reliquo angulo.</s> <s xml:id="echoid-s1936" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad lat{us} alteru- \\ trum datum: # ita ſecans anguli acce- \\ pto lateri adiacentis # ad baſem in partib{us} \\ lateris dati. <lb/></note> <pb o="46" file="076" n="76" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div75" type="section" level="1" n="27"> <head xml:id="echoid-head30" xml:space="preserve">VII. ANGVLVM.</head> <p> <s xml:id="echoid-s1937" xml:space="preserve">Ex baſe, & </s> <s xml:id="echoid-s1938" xml:space="preserve">vno latere inquirere.</s> <s xml:id="echoid-s1939" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt baſis # ad ſinum to- \\ tum # ita lat{us} da- \\ tum # ad ſinum anguli dato lateri oppo- \\ ſiti <lb/></note> <note position="left" xml:space="preserve">3. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s1940" xml:space="preserve">Complementum anguli inuenti dabit alterum angulum.</s> <s xml:id="echoid-s1941" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div76" type="section" level="1" n="28"> <head xml:id="echoid-head31" xml:space="preserve">VIII. ANGVLVM.</head> <p> <s xml:id="echoid-s1942" xml:space="preserve">Ex vtro que latere reddere cognitum.</s> <s xml:id="echoid-s1943" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} alte- \\ rutrum # ad ſinum \\ totum: # ita alterum la- \\ t{us} datum # ad tangentem anguli huic al- \\ teri laterioppoſiti. <lb/></note> <note position="left" xml:space="preserve">3. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s1944" xml:space="preserve">Complementum anguli inuenti dabit alterum angulum.</s> <s xml:id="echoid-s1945" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div77" type="section" level="1" n="29"> <head xml:id="echoid-head32" xml:space="preserve">TRIANGVLORVM RECTILINEO-<lb/>rum obliquangulorum Problemata. <lb/>IX. SEGMENTA LATERIS A <lb/>Perpendiculari facta.</head> <p> <s xml:id="echoid-s1946" xml:space="preserve">Ex datis tribus lateribus cognoſcere.</s> <s xml:id="echoid-s1947" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us}, in quod \\ cadit perpen- \\ dicularis # ad ſummam alio- \\ rum duorum la- \\ terum # ita differentia \\ eorundẽ duo- \\ rum laterum # ad quartum quen- \\ dam numerum. <lb/></note> <note position="left" xml:space="preserve">9. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s1948" xml:space="preserve">Et ſi quidem quartus numerus inuentus minor fuerit latere, in quod cadit <lb/>perpendicularis, auferendus is erit exillo latere. </s> <s xml:id="echoid-s1949" xml:space="preserve">Semiſsis enim reliqui numeri <lb/>dabit minus ſegmentum, quod ex toto latere ſubductum relinquet ſegmentum <lb/>maius.</s> <s xml:id="echoid-s1950" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1951" xml:space="preserve">Si verò quartus numerus inuentus maior fuerit latere, in quod cadit perpen-<lb/>dicularis, auferendum erit hoc latus ex illo numero. </s> <s xml:id="echoid-s1952" xml:space="preserve">Semiſsis enim reliqui nu-<lb/>meri dabit ſegmentum minus, exterius videlicet inter perpendicularem, & </s> <s xml:id="echoid-s1953" xml:space="preserve">an-<lb/>gulum obtuſum: </s> <s xml:id="echoid-s1954" xml:space="preserve">quod additum eidem lateri conflabit aliud ſegmentum maius <lb/>inter perpendicularem, & </s> <s xml:id="echoid-s1955" xml:space="preserve">angulum acutum.</s> <s xml:id="echoid-s1956" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div78" type="section" level="1" n="30"> <head xml:id="echoid-head33" xml:space="preserve">X. LATERA DVO.</head> <p> <s xml:id="echoid-s1957" xml:space="preserve">Ex tertio latere, & </s> <s xml:id="echoid-s1958" xml:space="preserve">duobus quibuſuis angulis, ac proinde omnibus tribus, <lb/>cumtertius ſit aliorum complementum ad ſemicirculum, hoc eſt, ad grad. </s> <s xml:id="echoid-s1959" xml:space="preserve">180. <lb/></s> <s xml:id="echoid-s1960" xml:space="preserve">inuenire.</s> <s xml:id="echoid-s1961" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>1. Vt ſin{us} an- \\ guli dato la- \\ teri oppoſiti # ad lat{us} da- \\ tuns<unsure/>: # ita ſin{us} alteru- \\ tri{us} reliquorũ \\ angulorum # ad lat{us} huic langulo \\ oppoſitum. <lb/></note> <note position="left" xml:space="preserve">10. Triang. <lb/>rectil.</note> </div> <div xml:id="echoid-div79" type="section" level="1" n="31"> <head xml:id="echoid-head34" xml:space="preserve">Rurſus</head> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli dato \\ laterioppoſiti # ad lat{us} da- \\ tum: # ita ſin{us} tertii \\ anguli # ad lat{us} huic tertio an- \\ gulo oppoſitum. <lb/></note> <note position="left" xml:space="preserve">1. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s1962" xml:space="preserve">2. </s> <s xml:id="echoid-s1963" xml:space="preserve"><emph style="sc">In</emph> Iſoſcele vnius tantum lateris inuentione opus eſt, cum vnum datum <pb o="47" file="077" n="77" rhead="LIBER PRIMVS."/> ſit cum angulis. </s> <s xml:id="echoid-s1964" xml:space="preserve">In æquilatero vero triangulo, ſi vnumlatus datum ſit, erunt & </s> <s xml:id="echoid-s1965" xml:space="preserve"><lb/>reliquailli æqualia, data.</s> <s xml:id="echoid-s1966" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div80" type="section" level="1" n="32"> <head xml:id="echoid-head35" xml:space="preserve">XI. LATVS.</head> <p> <s xml:id="echoid-s1967" xml:space="preserve">Ex duobus reliquis lateribus, & </s> <s xml:id="echoid-s1968" xml:space="preserve">duobus quibuſuis angulis, ac proinde o-<lb/>mnibus tribus, cum tertius ſit aliorum complementum ad ſemicirculum, id eſt, <lb/>ad grad. </s> <s xml:id="echoid-s1969" xml:space="preserve">180. </s> <s xml:id="echoid-s1970" xml:space="preserve">addiſcere.</s> <s xml:id="echoid-s1971" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli alteru- \\ tri lateri dato oppo- \\ ſiti # ad lat{us} oppo- \\ ſitum da@um: # ita ſin{us} anguli \\ quæſito lateri \\ oppoſiti # ad lat{us} quæſi- \\ tum. <lb/></note> <note position="right" xml:space="preserve">10. Triang. <lb/>rectil.</note> </div> <div xml:id="echoid-div81" type="section" level="1" n="33"> <head xml:id="echoid-head36" xml:space="preserve">XII. LATVS.</head> <p> <s xml:id="echoid-s1972" xml:space="preserve">Ex duobus lateribus, & </s> <s xml:id="echoid-s1973" xml:space="preserve">angulo ab ipſis comprehenſo colligere.</s> <s xml:id="echoid-s1974" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>1. Vt ſin{us} \\ tot{us} # ad ſecantem complementi ar- \\ c{us}, qui ſemiſſi aggregati da- \\ torum laterum ad ſin{us} re- \\ uocatorum, vt ſinui, debe- \\ tur: # Ita differentia in- \\ ter eam ſemiſ- \\ ſem, & alteru- \\ trum<unsure/> datorum \\ laterum ad ſin{us} \\ reuocatorum. # ad quar- \\ tũ quen \\ dam nu- \\ merum. <lb/></note> <note position="right" xml:space="preserve">Problema <lb/>17. triang. <lb/>ſphæric.</note> <p> <s xml:id="echoid-s1975" xml:space="preserve">Latera data ad ſinus reuo cabuntur, ſi vtrumq; </s> <s xml:id="echoid-s1976" xml:space="preserve">multiplicetur per 10. </s> <s xml:id="echoid-s1977" xml:space="preserve">vel 100. <lb/></s> <s xml:id="echoid-s1978" xml:space="preserve">vel 1000. </s> <s xml:id="echoid-s1979" xml:space="preserve">&</s> <s xml:id="echoid-s1980" xml:space="preserve">c. </s> <s xml:id="echoid-s1981" xml:space="preserve">ita vt maioris lateris numerus habeat tot figuras, quot continen-<lb/>tur in maioribus ſinubus tabulæ ſinuum, nimirum 5. </s> <s xml:id="echoid-s1982" xml:space="preserve">ſi ſinus totus ſtatuatur <lb/>100000. </s> <s xml:id="echoid-s1983" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s1984" xml:space="preserve">ſi ſinus totus ponatur 10000000.</s> <s xml:id="echoid-s1985" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div82" type="section" level="1" n="34"> <head xml:id="echoid-head37" xml:space="preserve">Deinde.</head> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad tangentem ſemiſſis arc{us}, \\ qui d{et}racto dato angulo ex \\ ſemicirculo relinquitur: # ita quart{us} \\ numer{us} \\ inuent{us} # ad tangentem diffe- \\ rentiæ inter ſem ßẽ \\ eiuſdem arc{us}, & \\ alterutrum angulo- \\ rum non datorum. <lb/></note> </div> <div xml:id="echoid-div83" type="section" level="1" n="35"> <head xml:id="echoid-head38" xml:space="preserve">Hæc autem tangens hoc etiam modo inuenietur, qui priori <lb/>præferendus videtur.</head> <note style="it" position="right" xml:space="preserve"> <lb/>2. Vt ſemiſ- \\ ſis aggre- \\ gati duo- \\ rum late- \\ rum da- \\ torum. # ad tangentem ſe- \\ miſſis arc{us}, qui \\ detracto dato an \\ gulo ex ſemici<unsure/>r- \\ culo relinquitur. # ita differentia in- \\ ter ſemiſſem ag- \\ gregati duorum \\ datorum laterũ, \\ & vtrumlib{et} \\ laterum # ad tangentem diffe- \\ rentiæ inter ſemiſ@ \\ ſem arc{us} prædi- \\ cti, & alterutrum \\ argulorum non \\ datorum. <lb/></note> <note position="right" xml:space="preserve">Alio modo <lb/>qui priori <lb/>præferendus <lb/>videtur.</note> <p> <s xml:id="echoid-s1986" xml:space="preserve">Arcus huius tangentis inuentæ additus ad ſemiſſem eiuſdem arcus (eſt au-<lb/>tem hic arcus ſumma duorum angulorum non datorum, nimirum dati anguli <lb/>complementum ad ſemicir culum, hoc eſt, ad grad. </s> <s xml:id="echoid-s1987" xml:space="preserve">180.) </s> <s xml:id="echoid-s1988" xml:space="preserve">dabit ma@orem angu-<lb/>lum non datum, qui videlicet maiorilateri dato opponitur: </s> <s xml:id="echoid-s1989" xml:space="preserve">ex eadem verò ſe-<lb/>miſſe detractus reliquum faciet minorem angulum non datum, quinimirum <lb/>lateri minori dato opponitur.</s> <s xml:id="echoid-s1990" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div84" type="section" level="1" n="36"> <head xml:id="echoid-head39" xml:space="preserve">Poſt hæc.</head> <pb o="48" file="078" n="78" rhead="GEOMETR. PRACT."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} vtriuſl@bet \\ anguli inuenti # ad lat{us} oppo- \\ ſitum: # ita angul{us} \\ dat{us} # ad lat{us} oppoſitum quod \\ quæritur. <lb/></note> <note position="left" xml:space="preserve">1. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s1991" xml:space="preserve">Itaque antequam tertium latus inueniatur, diſquirendi prius ſunt reliqui <lb/>duo anguli: </s> <s xml:id="echoid-s1992" xml:space="preserve">qui commodius videntur poſteriori via Num. </s> <s xml:id="echoid-s1993" xml:space="preserve">2. </s> <s xml:id="echoid-s1994" xml:space="preserve">exp oſita indagari, <lb/>quam priori illa ratione Num. </s> <s xml:id="echoid-s1995" xml:space="preserve">1 explicata.</s> <s xml:id="echoid-s1996" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1997" xml:space="preserve">3. </s> <s xml:id="echoid-s1998" xml:space="preserve">Siduo latera ſint æqualia; </s> <s xml:id="echoid-s1999" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erunt reliqui duo anguli æquales. </s> <s xml:id="echoid-s2000" xml:space="preserve">Semiſsis <anchor type="note" xlink:label="note-078-03a" xlink:href="note-078-03"/> ergo arcus, qui detracto angulo dato ex ſemicirculo relinquitur, dabit vtrum-<lb/>que, &</s> <s xml:id="echoid-s2001" xml:space="preserve">c.</s> <s xml:id="echoid-s2002" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-078-03" xlink:href="note-078-03a" xml:space="preserve">@5. primi.</note> </div> </div> <div xml:id="echoid-div86" type="section" level="1" n="37"> <head xml:id="echoid-head40" xml:space="preserve">XIII. LATVS.</head> <p> <s xml:id="echoid-s2003" xml:space="preserve">Ex duobus lareribus, & </s> <s xml:id="echoid-s2004" xml:space="preserve">angulo vni eorum oppoſito (ſi modo conſtet ſpe-<lb/>cies anguli alteri lateri dato oppoſiti, quando datus angulus acutus eſt) ex-<lb/>quirere.</s> <s xml:id="echoid-s2005" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>1. Vt lat{us} datum \\ dato angulo oppo- \\ ſitum # ad ſinum angu- \\ li dati: # ita alterum lat{us} \\ datum, # ad ſinum anguli \\ huic alteri late- \\ rioppoſiti. <lb/></note> <note position="left" xml:space="preserve">13. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2006" xml:space="preserve">Hic ſinus inuentus dabit angulum alteri dato lateri oppoſitum, ſi acutus <lb/>fuerit: </s> <s xml:id="echoid-s2007" xml:space="preserve">(erit autem ſemper acutus, quando datus angulus eſt obtuſus) ſi verò <lb/>fuerit obtuſus, arcus ſinus inuenti ex ſemicirculo demptus, reliquum faciet eum <lb/>angulum: </s> <s xml:id="echoid-s2008" xml:space="preserve">Propterea quando datus angulus eſt acutus, oportet dari huius al-<lb/>terius ſpeciem, vt ſciamus, num acutus ſit, vel obtuſus. </s> <s xml:id="echoid-s2009" xml:space="preserve">Summa autem horum <lb/>angulorum ex ſemicirculo ſubtracta relinquet tertium angulum quæſito lateri <lb/>oppoſitum. </s> <s xml:id="echoid-s2010" xml:space="preserve">Ergo.</s> <s xml:id="echoid-s2011" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} dati \\ anguli # ad datum lat{us} \\ ei oppoſitum; # ita ſin{us} tertii anguli inuenti \\ quæſito lateri oppoſiti. # ad lat{us} quæ- \\ ſitum. <lb/></note> <note position="left" xml:space="preserve">1. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2012" xml:space="preserve">2. </s> <s xml:id="echoid-s2013" xml:space="preserve">Si duo latera data ſint æqualia: </s> <s xml:id="echoid-s2014" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit angulus alteri dato lateri oppoſitus, <anchor type="note" xlink:label="note-078-08a" xlink:href="note-078-08"/> dato angulo æqualis.</s> <s xml:id="echoid-s2015" xml:space="preserve"/> </p> <div xml:id="echoid-div86" type="float" level="2" n="1"> <note position="left" xlink:label="note-078-08" xlink:href="note-078-08a" xml:space="preserve">@5. rimi.</note> </div> </div> <div xml:id="echoid-div88" type="section" level="1" n="38"> <head xml:id="echoid-head41" xml:space="preserve">XIIII. ANGVLOS DVOS.</head> <p> <s xml:id="echoid-s2016" xml:space="preserve">Ex duobus lateribus, & </s> <s xml:id="echoid-s2017" xml:space="preserve">angulo ab ipſis comprehenſo, reperire.</s> <s xml:id="echoid-s2018" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2019" xml:space="preserve">Inuenientur ex iis, quæ data ſunt, duo anguli, vt in priori parte problem atis 12. </s> <s xml:id="echoid-s2020" xml:space="preserve">di-<lb/>ctum eſt: </s> <s xml:id="echoid-s2021" xml:space="preserve">ſi nimirum inquiratur tangens differentiæ inter ſemiſſem arc{us}, qui detracto <lb/>angulo dato ex ſemicirculo relinquitur, & </s> <s xml:id="echoid-s2022" xml:space="preserve">alterutrum angulorum, qui quæruntur, &</s> <s xml:id="echoid-s2023" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s2024" xml:space="preserve">quæ quidem tangens duob{us} modis inuenta est in priori parte problematis 12. </s> <s xml:id="echoid-s2025" xml:space="preserve">in quo la-<lb/>t{us} proponitur inueſtigandum ex duob{us} laterib{us}, & </s> <s xml:id="echoid-s2026" xml:space="preserve">angulo ab ipſis comprehenſo. </s> <s xml:id="echoid-s2027" xml:space="preserve"><lb/>Quod vt fier{et}, inuenti pri{us} fuerunt alii duo anguli, qui in hoc problemate 14. </s> <s xml:id="echoid-s2028" xml:space="preserve">quæ-<lb/>runtur.</s> <s xml:id="echoid-s2029" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div89" type="section" level="1" n="39"> <head xml:id="echoid-head42" xml:space="preserve">XV. ANGVLOS DVOS.</head> <p> <s xml:id="echoid-s2030" xml:space="preserve">Ex duobus lateribus, & </s> <s xml:id="echoid-s2031" xml:space="preserve">angulo vni eorum oppoſito (ſi modo conſtet ſpe-<lb/>cies anguli alteri lateri dato oppoſiti, quando datus angulus acutus eſt) ex-<lb/>piſcari.</s> <s xml:id="echoid-s2032" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2033" xml:space="preserve">Hic {et}iam adhibenda eſt prior operatio problematis 13. </s> <s xml:id="echoid-s2034" xml:space="preserve">in quo lat{us} opponitur inqui-<lb/>rendum ex iiſdem datis: </s> <s xml:id="echoid-s2035" xml:space="preserve">quod vt fier{et}, inuenti pri{us} fuerereliqui<unsure/> duo anguli, qui in hoc <lb/>problemate 15. </s> <s xml:id="echoid-s2036" xml:space="preserve">indagandi proponuntur.</s> <s xml:id="echoid-s2037" xml:space="preserve"/> </p> <pb o="49" file="079" n="79" rhead="LIBER PRIMVS."/> </div> <div xml:id="echoid-div90" type="section" level="1" n="40"> <head xml:id="echoid-head43" xml:space="preserve">XVI. ANGVLOS OMNES TRES. <lb/>Ex tribus omnibus lateribus perueſtigare.</head> <p style="it"> <s xml:id="echoid-s2038" xml:space="preserve">1. </s> <s xml:id="echoid-s2039" xml:space="preserve">Ducta ad maximum lat{us} perpendiculari ex angulo oppoſito <anchor type="note" xlink:href="" symbol="a"/> (vt ni@irum per- <anchor type="note" xlink:label="note-079-01a" xlink:href="note-079-01"/> pendicularis ſemper intratriangulum cadat) inueniantur per problema 9. </s> <s xml:id="echoid-s2040" xml:space="preserve">ſegmenta duo <lb/>@aximi lateris facta à perpendiculari. </s> <s xml:id="echoid-s2041" xml:space="preserve">Deinde.</s> <s xml:id="echoid-s2042" xml:space="preserve"/> </p> <div xml:id="echoid-div90" type="float" level="2" n="1"> <note position="right" xlink:label="note-079-01" xlink:href="note-079-01a" xml:space="preserve">11. triang. <lb/>rectil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt minimum \\ lat{us} # ad ſinum \\ totum: # ita min{us} ſegmen- \\ tum maximi late- \\ ris # ad ſinum complementi \\ anguli medi@ lateri \\ oppoſiti. <lb/></note> </div> <div xml:id="echoid-div92" type="section" level="1" n="41"> <head xml:id="echoid-head44" xml:space="preserve">Rurſus.</head> <note position="right" xml:space="preserve">1. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt me- \\ dium \\ lat{us} # ad ſinum \\ totum: # ita mai{us} ſegmen- \\ tum maximi la- \\ teris # ad ſinum complementi angu- \\ li medio lateri oppoſiti. <lb/></note> <note position="right" xml:space="preserve">1. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2043" xml:space="preserve">Inuentis duobus angulis ad maximum latus, qui medio lateri, & </s> <s xml:id="echoid-s2044" xml:space="preserve">minimo <lb/>opponuntur; </s> <s xml:id="echoid-s2045" xml:space="preserve">ſi eorum ſumma ex ſemicirculo dematur, reliquus fiet tertius an-<lb/>gulus lateri maximo oppoſitus.</s> <s xml:id="echoid-s2046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2047" xml:space="preserve">2. </s> <s xml:id="echoid-s2048" xml:space="preserve">In Iſoſcele, ducta perpendiculari ad baſem <anchor type="note" xlink:href="" symbol="a"/>, quam bifariam ſecabit,</s> </p> <note position="right" xml:space="preserve">Schol. 26. <lb/>lib. 1. Eucl.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt alterum \\ laterum æ- \\ qualium # ad ſinum \\ totum: # ita ſemiſſis \\ baſis # ad ſinum complementi vni{us} \\ angulorum æqualium ad ba- \\ ſem. <lb/></note> <p> <s xml:id="echoid-s2049" xml:space="preserve">Summa duorum angulorum æqualium inuentorum ex ſemicirculo detra-<lb/>cta, reliquum faciet tertium angulum.</s> <s xml:id="echoid-s2050" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2051" xml:space="preserve">3. </s> <s xml:id="echoid-s2052" xml:space="preserve"><emph style="sc">In</emph> æquilatero triangulo dabuntur anguli, etiamſi latera non dentur, e<unsure/>um <lb/>quilibet gradus 60. </s> <s xml:id="echoid-s2053" xml:space="preserve">tertiam videlicet partem duorum rectorum, vel duas tertias <lb/>partes vnius recti, complectatur.</s> <s xml:id="echoid-s2054" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div93" type="section" level="1" n="42"> <head xml:id="echoid-head45" xml:space="preserve">XVII. PERPENDICVLAREM IN LATVS <lb/>quodcunque ex angulo oppoſito cadentem. <lb/>Ex tribus omnibus lateribus efficere notam.</head> <p style="it"> <s xml:id="echoid-s2055" xml:space="preserve">Per problema 9. </s> <s xml:id="echoid-s2056" xml:space="preserve">inquirantur ſegmenta lateris facta à perpendiculari. </s> <s xml:id="echoid-s2057" xml:space="preserve">Deinde diffe-<lb/>rentia inter vtrumuis ſegmentum, & </s> <s xml:id="echoid-s2058" xml:space="preserve">lat{us} adiacens ducatur in ſummam eiuſdem ſeg-<lb/>menti, & </s> <s xml:id="echoid-s2059" xml:space="preserve">lateris adiacentis. </s> <s xml:id="echoid-s2060" xml:space="preserve">Radix namque quadrata numeriproducti perpendicula-<lb/>rem quæſitam indicabit.</s> <s xml:id="echoid-s2061" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2062" xml:space="preserve">In triangulo enim A B C, ſit A B, 10. </s> <s xml:id="echoid-s2063" xml:space="preserve">A C, 17. </s> <s xml:id="echoid-s2064" xml:space="preserve">& </s> <s xml:id="echoid-s2065" xml:space="preserve">B C, 21. </s> <s xml:id="echoid-s2066" xml:space="preserve">inueſtiganda{q́ue} ſit perpen-<lb/>dicularis A D. </s> <s xml:id="echoid-s2067" xml:space="preserve">Per problema 9. </s> <s xml:id="echoid-s2068" xml:space="preserve">reperi{et}ur ſegmen-<lb/> <anchor type="figure" xlink:label="fig-079-01a" xlink:href="fig-079-01"/> tum B D, 6. </s> <s xml:id="echoid-s2069" xml:space="preserve">& </s> <s xml:id="echoid-s2070" xml:space="preserve">C D, 15. </s> <s xml:id="echoid-s2071" xml:space="preserve">Differentia inter B D, & </s> <s xml:id="echoid-s2072" xml:space="preserve"><lb/>A B, eſt 4. </s> <s xml:id="echoid-s2073" xml:space="preserve">quæducta in 16. </s> <s xml:id="echoid-s2074" xml:space="preserve">ſummam rectarum B D <lb/>& </s> <s xml:id="echoid-s2075" xml:space="preserve">A B, faci{et} 64. </s> <s xml:id="echoid-s2076" xml:space="preserve">cui{us} radix quadrata 8. </s> <s xml:id="echoid-s2077" xml:space="preserve">d{at} <lb/>perpendicularem A D. </s> <s xml:id="echoid-s2078" xml:space="preserve">Quod quia in noſtro tra-<lb/>ctatis triangulorum rectilineorum demonſtratum <lb/>non est, demonſtro hoc propoſito Theoremate.</s> <s xml:id="echoid-s2079" xml:space="preserve"/> </p> <div xml:id="echoid-div93" type="float" level="2" n="1"> <figure xlink:label="fig-079-01" xlink:href="fig-079-01a"> <image file="079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/079-01"/> </figure> </div> <pb o="50" file="080" n="80" rhead="GEOMETR. PRACT. LIB. I."/> <p> <s xml:id="echoid-s2080" xml:space="preserve"><emph style="sc">In</emph> triangulo rectangulo rectangulum ſub dif-<lb/> <anchor type="note" xlink:label="note-080-01a" xlink:href="note-080-01"/> <anchor type="figure" xlink:label="fig-080-01a" xlink:href="fig-080-01"/> ferentia baſis, & </s> <s xml:id="echoid-s2081" xml:space="preserve">alterutrius lateris circa rectum <lb/>angulum, & </s> <s xml:id="echoid-s2082" xml:space="preserve">ſub ſumma baſis, & </s> <s xml:id="echoid-s2083" xml:space="preserve">eiuſdem lateris, <lb/>æquale eſt quadrato alterius lateris circa angulum <lb/>rectum.</s> <s xml:id="echoid-s2084" xml:space="preserve"/> </p> <div xml:id="echoid-div94" type="float" level="2" n="2"> <note position="left" xlink:label="note-080-01" xlink:href="note-080-01a" xml:space="preserve">Theorema.</note> <figure xlink:label="fig-080-01" xlink:href="fig-080-01a"> <image file="080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/080-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s2085" xml:space="preserve">Nam in triangulo rectangulo A B D, cui{us} angul{us} <lb/>D, rect{us}, ſiex B, per D, ſemicircul{us} deſcribatur E F D, <lb/>erit A E, differentia inter baſem A B, & </s> <s xml:id="echoid-s2086" xml:space="preserve">lat{us} B D: </s> <s xml:id="echoid-s2087" xml:space="preserve">At A F, ſumma erit baſis A B, <lb/>& </s> <s xml:id="echoid-s2088" xml:space="preserve">eiuſdem lateris B D, cum B D, B E, B F, rectæ ſint æqual{es}. </s> <s xml:id="echoid-s2089" xml:space="preserve">Dico igitur rectang ulum <lb/>ſub A E, A F, æquale eſſe quadrato lateris A D. </s> <s xml:id="echoid-s2090" xml:space="preserve">Recta enim A D, cum perpendicu-<lb/>laris ſit ad ſemidiam{et}rum B D, <anchor type="note" xlink:href="" symbol="a"/> ſemicirculum tang{et} in D. </s> <s xml:id="echoid-s2091" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> lgitur rectan- <anchor type="note" xlink:label="note-080-02a" xlink:href="note-080-02"/> gulum ſub A E, A F, quadrato tangentis A D, æquale erit, <lb/>quod erat demonſtrandum.</s> <s xml:id="echoid-s2092" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-080-02" xlink:href="note-080-02a" xml:space="preserve">Coroll. 16. <lb/>ter.</note> </div> <note symbol="b" position="left" xml:space="preserve">36. ter.</note> </div> <div xml:id="echoid-div97" type="section" level="1" n="43"> <head xml:id="echoid-head46" xml:space="preserve">FINIS LIBRI PRIMI.</head> <figure> <image file="080-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/080-02"/> </figure> <pb o="51" file="081" n="81"/> <figure> <image file="081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/081-01"/> </figure> </div> <div xml:id="echoid-div98" type="section" level="1" n="44"> <head xml:id="echoid-head47" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ <lb/>LIBER SECVNDVS.</head> <p> <s xml:id="echoid-s2093" xml:space="preserve">Linearum rectarum per Quadrantem Aſtrono-<lb/>micum Dimenſionem explicans.</s> <s xml:id="echoid-s2094" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2095" xml:space="preserve">NOMINE linearum rectarum intelligim{us} diſtanti{as} loc@-<lb/>rum, ſeu interualla, longitudineſue: </s> <s xml:id="echoid-s2096" xml:space="preserve">altitudines turrium, <lb/>ædificio<unsure/>rum, arborum, & </s> <s xml:id="echoid-s2097" xml:space="preserve">montium: </s> <s xml:id="echoid-s2098" xml:space="preserve">ac poſtremo profun-<lb/>ditates puteorum, vallium, atque foſſarum. </s> <s xml:id="echoid-s2099" xml:space="preserve">Ad harum Di-<lb/>menſiones varii varia adhibent inſtrumenta; </s> <s xml:id="echoid-s2100" xml:space="preserve">quibuſdam <lb/>enim placet ſcala altimetra in dorſo Aſtrolabii, ſeu plani-<lb/>ſphærii deſcripta. </s> <s xml:id="echoid-s2101" xml:space="preserve">Aliis radi{us} Aſtronomic{us} Gemmæ Friſii, vel radi{us} dict{us} <lb/>Latin{us}, qu@d à Domino Latino Vrſino nobili Romano excogitat{us} ſit; </s> <s xml:id="echoid-s2102" xml:space="preserve">vel <lb/>bacul{us} Iacob: </s> <s xml:id="echoid-s2103" xml:space="preserve">Aliis annul{us} Aſtronomic{us}, vel Holometrum; </s> <s xml:id="echoid-s2104" xml:space="preserve">Aliis deniq{ue} <lb/>alia inſtrumenta arrident. </s> <s xml:id="echoid-s2105" xml:space="preserve">Mihi vero præ cæteris probatur Quadrans Aſtro-<lb/>nomic{us} in 90. </s> <s xml:id="echoid-s2106" xml:space="preserve">grad{us} diſtribut{us}: </s> <s xml:id="echoid-s2107" xml:space="preserve">& </s> <s xml:id="echoid-s2108" xml:space="preserve">Quadratum Geometricum tum ſtabile, <lb/>tum pendulum, cui{us} duo latera in cert{as} quaſdam partes æquales ſint diuiſa. <lb/></s> <s xml:id="echoid-s2109" xml:space="preserve">Hoc autem 2. </s> <s xml:id="echoid-s2110" xml:space="preserve">lib. </s> <s xml:id="echoid-s2111" xml:space="preserve">qua ratione per Quadrantem Aſtronomicum Dimenſio <lb/>linearum rectarum perficiatur, docebim{us}. </s> <s xml:id="echoid-s2112" xml:space="preserve">Quæ ratio vt intelligatur, in <lb/>promptu, & </s> <s xml:id="echoid-s2113" xml:space="preserve">ad manum eſſe debent tabulæ ſinuum, Tangentium, atque ſecan-<lb/>tium in noſtro Theodoſio, & </s> <s xml:id="echoid-s2114" xml:space="preserve">in aliorum auctorum libris deſcripræ, vnde peten-<lb/>dæ erunt. </s> <s xml:id="echoid-s2115" xml:space="preserve">Superuacaneum enim eſſe duxim{us}, eaſdem hic repetere, ne op{us} in <lb/>maiorem formam excreſcat.</s> <s xml:id="echoid-s2116" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2117" xml:space="preserve">DISTANTIAM in plano, ſiue acceſſibilis ea ſit, ſiue inacceſſibilis <lb/>per duas ſtationes in eodem plano factas per Quadrantem metiri, <lb/>quando in eius extremo erecta eſt altitudo aliqua perpendicularis, <lb/>etiamſi infimum eius extremum non cernatur. </s> <s xml:id="echoid-s2118" xml:space="preserve">Atque hinc altitudi-<lb/>nem quoque ipſam elicere.</s> <s xml:id="echoid-s2119" xml:space="preserve"/> </p> <pb o="52" file="082" n="82" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div99" type="section" level="1" n="45"> <head xml:id="echoid-head48" xml:space="preserve">PROBLEMA I.</head> <p> <s xml:id="echoid-s2120" xml:space="preserve"><emph style="sc">Sit</emph> diſtantia ſiue longitu-<lb/> <anchor type="figure" xlink:label="fig-082-01a" xlink:href="fig-082-01"/> do inueſtiganda A B, in <lb/>plano C B, erectaque ſit in <lb/>extremo B, altitudo quæ-<lb/>piam perpendicularis B G, <lb/>licet extremum B, non ap-<lb/>pareat. </s> <s xml:id="echoid-s2121" xml:space="preserve">Statura menſoris <lb/>ſit D A, ab oculo ad pedes <lb/>vſque. </s> <s xml:id="echoid-s2122" xml:space="preserve">Neautem hæc ſta-<lb/>tura mutetur, ſed eadem <lb/>ſemper maneat, recte fe-<lb/>ceris, ſi pro ea ſtatura ba-<lb/>culum eidem æqualem ac-<lb/>cipias, ad cuius extremum <lb/>oculum applices. </s> <s xml:id="echoid-s2123" xml:space="preserve">Ducta autem cogitatione per D, ipſi C B, parallela E F, fiat <lb/>prima ſtatio in D, Secunda verò in E, puncto remotiore: </s> <s xml:id="echoid-s2124" xml:space="preserve">ſitquerecta D E, quæ <lb/> <anchor type="note" xlink:label="note-082-01a" xlink:href="note-082-01"/> differentia ſtationum dicitur, nota ſecundum aliquam menſuram vulgarem. <lb/></s> <s xml:id="echoid-s2125" xml:space="preserve">Deinde dirigatur latus quadrantis H K, in quo ſunt pinnacidia, verſus faſti-<lb/>gium G, ita vt oculus in D, poſitus per vtriuſque pinnacidij foramina, fafti-<lb/>gium G videat, libere pendente perpendiculo H I: </s> <s xml:id="echoid-s2126" xml:space="preserve">diligenterque per ea, <lb/>quæ cap. </s> <s xml:id="echoid-s2127" xml:space="preserve">2. </s> <s xml:id="echoid-s2128" xml:space="preserve">libr. </s> <s xml:id="echoid-s2129" xml:space="preserve">1. </s> <s xml:id="echoid-s2130" xml:space="preserve">Num. </s> <s xml:id="echoid-s2131" xml:space="preserve">7. </s> <s xml:id="echoid-s2132" xml:space="preserve">& </s> <s xml:id="echoid-s2133" xml:space="preserve">10. </s> <s xml:id="echoid-s2134" xml:space="preserve">tradita ſunt, notetur in gradibus, ac minu-<lb/>tis angulus G D F, quem arcus I L, in Quadrante manifeſtabit, complemen-<lb/>tum videlicet arcus I K, Cum enim filum perpendiculi H I, ſit ad D F, rectum, <lb/>erit angulus G D F, complementum anguli D H I, ęqualis nimirum angulo <lb/>I H L, qui eiuſdem anguli D H I, complementum etiam eſt. </s> <s xml:id="echoid-s2135" xml:space="preserve">Atque hunc an-<lb/>gulum G D F, angulum obſeruationis dicemus. </s> <s xml:id="echoid-s2136" xml:space="preserve">Eodem modo obſeruetur in <lb/> <anchor type="note" xlink:label="note-082-02a" xlink:href="note-082-02"/> ſecunda ſtatione angulus G E F, per radium viſualem ab oculo, & </s> <s xml:id="echoid-s2137" xml:space="preserve">per pinna-<lb/>cidia Quadrantis ad faſtigium G, directum. </s> <s xml:id="echoid-s2138" xml:space="preserve">Sumptis autem E M, D N, æquali-<lb/>bus, erigantur perpendiculares M N, N O, (in figura conincidit M H, cum fi-<lb/>lo perpendiculi; </s> <s xml:id="echoid-s2139" xml:space="preserve">quod nihil refert.) </s> <s xml:id="echoid-s2140" xml:space="preserve">Si igitur E M, D N, ſtatuantur ſinus to-<lb/>ti, erunt M H, N O, Tangentes angulorum obſeruationum E, & </s> <s xml:id="echoid-s2141" xml:space="preserve">D. </s> <s xml:id="echoid-s2142" xml:space="preserve">Ducta <lb/>quoque D Q, ipſi E G, parallela ſecante N O, in P, <anchor type="note" xlink:href="" symbol="a"/> erit angulus N D P, an- <anchor type="note" xlink:label="note-082-03a" xlink:href="note-082-03"/> gulo E, æqualis. </s> <s xml:id="echoid-s2143" xml:space="preserve">Cum ergo duo anguli N. </s> <s xml:id="echoid-s2144" xml:space="preserve">D, trianguli N D P, duobus angu-<lb/>lis M, E, trianguli M E H, ſint æquales, (eſt enim & </s> <s xml:id="echoid-s2145" xml:space="preserve">rectus N, recto M, æqua-<lb/>lis) lateraque D N, E M, quibus adiacent, æqualia; </s> <s xml:id="echoid-s2146" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erunt latera N P, M H;</s> <s xml:id="echoid-s2147" xml:space="preserve"> <anchor type="note" xlink:label="note-082-04a" xlink:href="note-082-04"/> æqualia: </s> <s xml:id="echoid-s2148" xml:space="preserve">ac proinde O P, differentia erit inter Tangentes angulorum obſer-<lb/>uationum. </s> <s xml:id="echoid-s2149" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quia verò eſt, vt O P, ad P N, ita G Q, ad Q F: </s> <s xml:id="echoid-s2150" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et vt G Q, <anchor type="note" xlink:label="note-082-05a" xlink:href="note-082-05"/> ad Q F, ita E D, ad D F; </s> <s xml:id="echoid-s2151" xml:space="preserve">erit quo que vt O P, differentia Tangentium angulis <lb/>obſeruationum reſpondentium ad P N, ſiue ad H M, Tangentem remotioris <lb/> <anchor type="note" xlink:label="note-082-06a" xlink:href="note-082-06"/> ſtationis, ita E D, differentia ſtationum ad D F, diſtantiam quæſitam. </s> <s xml:id="echoid-s2152" xml:space="preserve">Quo-<lb/>cir ca ſi fiat,</s> </p> <div xml:id="echoid-div99" type="float" level="2" n="1"> <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/xxxxxxxx/figures/082-01"/> </figure> <note position="left" xlink:label="note-082-01" xlink:href="note-082-01a" xml:space="preserve">Differentia <lb/>ſtationum.</note> <note position="left" xlink:label="note-082-02" xlink:href="note-082-02a" xml:space="preserve">Angul{us} ob-<lb/>ſeruationis.</note> <note symbol="a" position="left" xlink:label="note-082-03" xlink:href="note-082-03a" xml:space="preserve">29. primi.</note> <note symbol="b" position="left" xlink:label="note-082-04" xlink:href="note-082-04a" xml:space="preserve">26. primi.</note> <note symbol="c" position="left" xlink:label="note-082-05" xlink:href="note-082-05a" xml:space="preserve">Schol. 4. <lb/>lib. 6.</note> <note symbol="d" position="left" xlink:label="note-082-06" xlink:href="note-082-06a" xml:space="preserve">2. ſext.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt O P, differentia \\ inter Tangent{es} an- \\ gulorum obſeruatio- \\ num. # ad P N, vel \\ H M, Tan- \\ gentem mi- \\ norem: # Ita E D, differentia \\ ſtationum nota in \\ menſura aliqua vul- \\ gari # ad aliud; \\ hoc eſt ad \\ D F, <lb/></note> <note position="left" xml:space="preserve">Diſtantiæ in-<lb/>uentio per <lb/>tangentes.</note> <pb o="53" file="083" n="83" rhead="LIBER SECVNDVS."/> <p> <s xml:id="echoid-s2153" xml:space="preserve">prodibit diſtantia D F, quæſita, ſiue A B, in eadem menſura differentiæ ſtatio-<lb/>num: </s> <s xml:id="echoid-s2154" xml:space="preserve">cui ſi adij ciatur differentia ſtationum E D, cognita etiam fiet diſtantia E F; <lb/></s> <s xml:id="echoid-s2155" xml:space="preserve">vel C B, à remotiori ſtatione.</s> <s xml:id="echoid-s2156" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div101" type="section" level="1" n="46"> <head xml:id="echoid-head49" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s2157" xml:space="preserve">2. </s> <s xml:id="echoid-s2158" xml:space="preserve"><emph style="sc">Posito</emph> ſinu toto G F, erit D F, Tangens anguli D G F, complementi <lb/>anguli obſeruationis G D F. </s> <s xml:id="echoid-s2159" xml:space="preserve">quem angulum D G F, in dicat arcus Quadrantis IK, <lb/> <anchor type="note" xlink:label="note-083-01a" xlink:href="note-083-01"/> à perpendiculo verſus oculum, <anchor type="note" xlink:href="" symbol="a"/> cum angulus D H I, æqualis ſit angulo D G F, externus interno. </s> <s xml:id="echoid-s2160" xml:space="preserve">Eodem modo E F, Tangens erit anguli E G F, complementi <lb/>alterius anguli obſeruationis GEF. </s> <s xml:id="echoid-s2161" xml:space="preserve">At ED, differentia inter eas Tangentes exi-<lb/>ſtet. </s> <s xml:id="echoid-s2162" xml:space="preserve">Si igitur fiat,</s> </p> <div xml:id="echoid-div101" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-083-01" xlink:href="note-083-01a" xml:space="preserve">29. primi.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>vt E D, differentia \\ inter Tangent{es} \\ angulorũ, qui cõ- \\ plemẽta ſunt angu- \\ lorũ obſeruationũ, # ad D F, Tangentem com- \\ plementi anguli obſerua- \\ tionis G D F, in propin- \\ quiore ſtatione, hoc eſt, ad \\ Tangentem minorem: # Ita E D, diffe- \\ rentia ſtatio- \\ num nota in \\ aliqua men- \\ ſura vulgari. # ad aliud \\ hoc eſt, \\ ad DF, <lb/></note> <note position="right" xml:space="preserve">Diſtantiæ in-<lb/>uentio alia <lb/>per tangen-<lb/>tes.</note> <p> <s xml:id="echoid-s2163" xml:space="preserve">procreabitur diſtantia minor quæſita DF, vel A B, in eadem menſura differentiæ <lb/>ſtationum: </s> <s xml:id="echoid-s2164" xml:space="preserve">cui ſi addatur differentia ſtationum E D, nota quo que fiet diſtantia <lb/>maior E F.</s> <s xml:id="echoid-s2165" xml:space="preserve"/> </p> <note symbol="b" position="right" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>#### 3. Rurſus <anchor type="note" xlink:href="" symbol="b"/> ſi fiat, Vt D N, ſi- \\ n{us} tot{us} # ad N O, Tangentem anguli \\ G D F, in propinquiore ſtatione: # Ita D F, diſtantia \\ inuenta minor # ad aliud, hoc \\ eſt, ad F G; </note> <p> <s xml:id="echoid-s2166" xml:space="preserve">@ Inuenietur altitudo F G, in menſura diſtantiæ inuentæ D F; </s> <s xml:id="echoid-s2167" xml:space="preserve">minoris, cui ſi ad-<lb/> <anchor type="note" xlink:label="note-083-06a" xlink:href="note-083-06"/> datur menſ@ris ſtatura F B, cognita erit tota altitudo B G. </s> <s xml:id="echoid-s2168" xml:space="preserve">Item <anchor type="note" xlink:href="" symbol="c"/> ſi fiat,</s> </p> <div xml:id="echoid-div102" type="float" level="2" n="2"> <note position="right" xlink:label="note-083-06" xlink:href="note-083-06a" xml:space="preserve">Altitudinis in <lb/>uentio per tã-<lb/>gent{es}.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt E M, ſi- \\ n{us} tot{us} # ad M H, tangentem anguli G E F, \\ in remotiore ſtatione: # ita E F, diſtantia \\ inuenta maior # ad aliud, hoc \\ est, ad FG, <lb/></note> <note symbol="c" position="right" xml:space="preserve">4. ſexti.</note> <p> <s xml:id="echoid-s2169" xml:space="preserve">reperietur eadem altitudo F G, in menſura diſtantiæ inuentæ E F, maioris, cui ſi <lb/> <anchor type="note" xlink:label="note-083-09a" xlink:href="note-083-09"/> addatur ſtatura menſoris FB, nota fiet tota altitudo B G.</s> <s xml:id="echoid-s2170" xml:space="preserve"/> </p> <div xml:id="echoid-div103" type="float" level="2" n="3"> <note position="right" xlink:label="note-083-09" xlink:href="note-083-09a" xml:space="preserve">Altitudinis <lb/>inuentio alia <lb/>per tangent{es}.</note> </div> </div> <div xml:id="echoid-div105" type="section" level="1" n="47"> <head xml:id="echoid-head50" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s2171" xml:space="preserve">4. </s> <s xml:id="echoid-s2172" xml:space="preserve">Si per ſolos ſinus idem expedire lubeat, erit operatio aliquando longior. <lb/></s> <s xml:id="echoid-s2173" xml:space="preserve">Primum enim inuenienda eſt vtra que hypotenuſa E G, D G, in aliqua menſura <lb/>nota, hoc modo. </s> <s xml:id="echoid-s2174" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam angulus G D F, æqualis eſt duobus angulis E, <anchor type="note" xlink:label="note-083-10a" xlink:href="note-083-10"/> EGD: </s> <s xml:id="echoid-s2175" xml:space="preserve">ſi angulus E, in remotiore ſtatione obſeruatus dematur ex angulo GDF, <lb/>in propinquiore ſtatione deprehenſo, reliquus fiet angulus EGD, differentia ni-<lb/>miruminter duosangulos obſeruationum. </s> <s xml:id="echoid-s2176" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quod ſi fiat,</s> </p> <div xml:id="echoid-div105" type="float" level="2" n="1"> <note symbol="d" position="right" xlink:label="note-083-10" xlink:href="note-083-10a" xml:space="preserve">32. primi.</note> </div> <note symbol="c" position="right" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli E G D, \\ differentiæ inter an- \\ gulos duos obſeruatio- \\ num # ad E D dif- \\ ferentiam \\ ſtationum \\ notam: # Ita ſin{us} anguli D E G, \\ vel. Ita ſin{us} anguli \\ E D G, cõplementian- \\ guli G D F, ad duos rectos, # ad D G, \\ vel \\ ad E G, <lb/></note> <p> <s xml:id="echoid-s2177" xml:space="preserve">pro ducetur tam D G, quam E G, nota in partibus differentiæ ſtationum. </s> <s xml:id="echoid-s2178" xml:space="preserve">Igi-<lb/> <anchor type="note" xlink:label="note-083-13a" xlink:href="note-083-13"/> tur <anchor type="note" xlink:href="" symbol="f"/> ſi fiat,</s> </p> <div xml:id="echoid-div106" type="float" level="2" n="2"> <note position="right" xlink:label="note-083-13" xlink:href="note-083-13a" xml:space="preserve">Inuentio Hy-<lb/>potenuſarum.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} anguli \\ recti F, # ad hypoten@- \\ ſam D G, \\ proxime inuentam # Ita ſin{us} anguli D G F, complementi anguli ob- \\ ſeruationis in propinquiore ſtatione # ad D F, <lb/></note> <note symbol="f" position="right" xml:space="preserve">10. triang re-<lb/>ctil.</note> <pb o="54" file="084" n="84" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s2179" xml:space="preserve">nota fiet diſtantia D F, cui ſi a dij ciatur differentia ſtationum E D, cognita etiam <lb/> <anchor type="note" xlink:label="note-084-01a" xlink:href="note-084-01"/> erit longior diſtantia EF. </s> <s xml:id="echoid-s2180" xml:space="preserve">quaminuenies quo que, <anchor type="note" xlink:href="" symbol="a"/> ſi fiat,</s> </p> <div xml:id="echoid-div107" type="float" level="2" n="3"> <note position="left" xlink:label="note-084-01" xlink:href="note-084-01a" xml:space="preserve">Diſtantiæ in-<lb/>@entio per ſo-<lb/>los ſin{us}.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} anguli \\ recti F, # ad hypotenuſam \\ E G, nuper inuen- \\ tam # Ita ſin{us} anguli E G F, complemen- \\ ti anguli obſeruationis in rem@- \\ tiore ſtatione # ad E F. <lb/></note> <note symbol="a" position="left" xml:space="preserve">10. triang. re-<lb/>ctil.</note> <p> <s xml:id="echoid-s2181" xml:space="preserve">Altitudo autem F G, per ſolos ſinus inuenietur, <anchor type="note" xlink:href="" symbol="b"/> ſi fiat,</s> </p> <note symbol="b" position="left" xml:space="preserve">10. triang. <lb/>rectil. <lb/>Altitudinis in <lb/>uentio per ſo-<lb/>los ſin{us}.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} anguli \\ recti F, # ad hypotenuſam EG, \\ vel ad hypotenuſam \\ D G: # Ita ſin{us} anguli ε, obſeruati \\ minoris, vel ita ſin{us} anguli \\ G D F, obſeruati maioris # ad F G, \\ ad F G. <lb/></note> <p> <s xml:id="echoid-s2182" xml:space="preserve">5. </s> <s xml:id="echoid-s2183" xml:space="preserve"><emph style="sc">Per</emph> Quadrantem ſtabilem eo dem modo dimenſio fit: </s> <s xml:id="echoid-s2184" xml:space="preserve">ſolum anguli ob-<lb/>ſeruationum in duabus ſtationibus hac ra-<lb/> <anchor type="figure" xlink:label="fig-084-01a" xlink:href="fig-084-01"/> tione inueſtigantur. </s> <s xml:id="echoid-s2185" xml:space="preserve">Collocato Quadran-<lb/>te ſupra baſem aliquam planam Horizonti <lb/>æquidiſtãtem, ita vtrectus ſit ad Horizon-<lb/>tem; </s> <s xml:id="echoid-s2186" xml:space="preserve">quod beneficio alicuius perpendiculi <lb/>efficies. </s> <s xml:id="echoid-s2187" xml:space="preserve">Collocato, inquam, hoc modo <lb/>Quadrante eleua dioptram, donec per fo-<lb/>ramina pinnacidiorum faſtigiũ G, videas. <lb/></s> <s xml:id="echoid-s2188" xml:space="preserve">Ita enim in propinquiore ſtatione D, angulus obſeruationis erit GDF; </s> <s xml:id="echoid-s2189" xml:space="preserve">In remo-<lb/>tiore vero GEF. </s> <s xml:id="echoid-s2190" xml:space="preserve">Vtrumque autem metietur arcus Quadrantis inter rectam EF, <lb/>& </s> <s xml:id="echoid-s2191" xml:space="preserve">dioptræ lineam fiduciæ. </s> <s xml:id="echoid-s2192" xml:space="preserve">Reliqua omnia fient, ſicut in Quadrante pendulo, <lb/>vt figura demonſtrat. </s> <s xml:id="echoid-s2193" xml:space="preserve">Solum ad altitudinem F G, inuentam adij cienda erit alti-<lb/>tudo baſis; </s> <s xml:id="echoid-s2194" xml:space="preserve">cuiimpoſitus eſt Quadrans, non autem ſtatura menſoris, niſi altitu-<lb/>dini baſis ſit æqualis.</s> <s xml:id="echoid-s2195" xml:space="preserve"/> </p> <div xml:id="echoid-div108" type="float" level="2" n="4"> <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/xxxxxxxx/figures/084-01"/> </figure> </div> <p> <s xml:id="echoid-s2196" xml:space="preserve">6. </s> <s xml:id="echoid-s2197" xml:space="preserve"><emph style="sc">Iam</emph> vero ſine numeris, id eſt, ſine multiplicatione ac diuiſione numero-<lb/> <anchor type="note" xlink:label="note-084-06a" xlink:href="note-084-06"/> rum omnia hæc explorari poterunt, (quod ijs, qui parum in numeris, & </s> <s xml:id="echoid-s2198" xml:space="preserve">vſu ſi-<lb/>nuum, Tangentium, ac ſecantium exercitati ſunt, pergratum fore non dubi-<lb/>to.) </s> <s xml:id="echoid-s2199" xml:space="preserve">hoc modo. </s> <s xml:id="echoid-s2200" xml:space="preserve">In charta aliqua, vel plano conſtruatur figura illi omnino ſi-<lb/>milis, quam ab oculo vſque ad altitudinem ſupra concepimus eſſe conſtructã; <lb/></s> <s xml:id="echoid-s2201" xml:space="preserve">quod ita fiet. </s> <s xml:id="echoid-s2202" xml:space="preserve">Ex inſtrumento partium cap. </s> <s xml:id="echoid-s2203" xml:space="preserve">1. </s> <s xml:id="echoid-s2204" xml:space="preserve">ſuperioris libri fabricato, vel (ſi <lb/>inſtrumentum non adſit) ex aliqua recta in particulas plurimas æquales diuiſa <lb/>capiantur circino tot particulæ æquales, quot palmi, vel pedes inter duas ſtati-<lb/>ones comprehenduntur; </s> <s xml:id="echoid-s2205" xml:space="preserve">transferaturque interuallum illud circini in rectam <lb/>quamcunque ex E, in D: </s> <s xml:id="echoid-s2206" xml:space="preserve">atque in D, & </s> <s xml:id="echoid-s2207" xml:space="preserve">E, omni adhibita diligentia, anguli ob-<lb/>ſeruatioi um GDF, GEF, primæ, & </s> <s xml:id="echoid-s2208" xml:space="preserve">ſecundæ ſtationis fiant: </s> <s xml:id="echoid-s2209" xml:space="preserve">Punctumque G, in <lb/>quo rectæ D G, E G, conueniunt, diligenter notetur, (Ne in hoc puncto erretur, <lb/>propter obliquam ſectionem, docebimus in ſequenti lemmate, quo pacto ex-<lb/>quiſitiſsime deprehen di poſsit. </s> <s xml:id="echoid-s2210" xml:space="preserve">Niſi enim hoc fiat, <lb/> <anchor type="figure" xlink:label="fig-084-02a" xlink:href="fig-084-02"/> mẽſuræ nõ inueniẽtur accurate) ex quo perp ẽdicu-<lb/>laris demittatur G F Figura ita conſtructa, ſi rectæ <lb/>D F, EF, F G, D G, E G, per circinum tranſporten-<lb/>tur in latus 100. </s> <s xml:id="echoid-s2211" xml:space="preserve">partium prædicti inſtrumenti, vel in <lb/>dictam rectam in plurimas partes æquales diuiſam, <lb/>illico apparebit, quot parriculæ inter pedes circini <pb o="55" file="085" n="85" rhead="LIBER SECVNDVS."/> includantur. </s> <s xml:id="echoid-s2212" xml:space="preserve">Atque tot palmos, aut pedes quælibet illarum rectarum comple-<lb/>ctetur, quot particulæ in earum interuallis deprehenſæ fuerint.</s> <s xml:id="echoid-s2213" xml:space="preserve"/> </p> <div xml:id="echoid-div109" type="float" level="2" n="5"> <note position="left" xlink:label="note-084-06" xlink:href="note-084-06a" xml:space="preserve">Problema hoc <lb/>1. quo pacto ſi-<lb/>ne numeris ab <lb/>ſolu@@ur.</note> <figure xlink:label="fig-084-02" xlink:href="fig-084-02a"> <image file="084-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/084-02"/> </figure> </div> </div> <div xml:id="echoid-div111" type="section" level="1" n="48"> <head xml:id="echoid-head51" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s2214" xml:space="preserve">7. </s> <s xml:id="echoid-s2215" xml:space="preserve"><emph style="sc">Idem</emph> aſſequentur via quadam generali, quæ in omnes dimenſiones <lb/>quadrat, videlicet. </s> <s xml:id="echoid-s2216" xml:space="preserve">Fiat angulus quicunque B A C. </s> <s xml:id="echoid-s2217" xml:space="preserve">Deinde ſumpta verbigra-<lb/>tia, in exemplum regula trium Num. </s> <s xml:id="echoid-s2218" xml:space="preserve">1. </s> <s xml:id="echoid-s2219" xml:space="preserve">huius propoſitionis, accipiatur primæ <lb/>quantitati O P, (hoc eſt, differentiæ Tangentium angulorum obſeruatorum) <lb/> <anchor type="figure" xlink:label="fig-085-01a" xlink:href="fig-085-01"/> æqualis; </s> <s xml:id="echoid-s2220" xml:space="preserve">vel ſi nimis paru<unsure/>a eſt, multiplex A D. <lb/></s> <s xml:id="echoid-s2221" xml:space="preserve"> <anchor type="note" xlink:label="note-085-01a" xlink:href="note-085-01"/> (Nos duplam accepimus) Item ſecũ dæ P N, (hoc <lb/>eſt, Tangenti minoris anguli) æqualis, vel ęque <lb/>multiplex cum A D, nimirum D B. </s> <s xml:id="echoid-s2222" xml:space="preserve">Poſt hæc ex <lb/>inſtrumento partium capiantur tot particulę A E, <lb/>quot palmi, aut pedes in ED, differentia ſtationum <lb/>continentur. </s> <s xml:id="echoid-s2223" xml:space="preserve">Ducta autemrecta D E, agatur ei pa-<lb/>rallela BC. </s> <s xml:id="echoid-s2224" xml:space="preserve">Nam quot partes inſtrumenti partium includetinteruallum EC, tot <lb/>palmos, aut pedes diſtantia DF, complectetur; </s> <s xml:id="echoid-s2225" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cum quatuor quantitates AD, <anchor type="note" xlink:label="note-085-02a" xlink:href="note-085-02"/> DB, AE, EC, proportionales ſint.</s> <s xml:id="echoid-s2226" xml:space="preserve"/> </p> <div xml:id="echoid-div111" type="float" level="2" n="1"> <figure xlink:label="fig-085-01" xlink:href="fig-085-01a"> <image file="085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/085-01"/> </figure> <note position="right" xlink:label="note-085-01" xlink:href="note-085-01a" xml:space="preserve">Problema hoc <lb/>1. qua ratione <lb/>aliter ſine nu-<lb/>meris abſolua <lb/>tur</note> <note symbol="a" position="right" xlink:label="note-085-02" xlink:href="note-085-02a" xml:space="preserve">4. Sexti.</note> </div> <p> <s xml:id="echoid-s2227" xml:space="preserve"><emph style="sc">Eodem</emph> modo procedes in alijs exemplis, hoc obſeruato, vt quando ſinus <lb/>alicuius anguli in regula trium reperitur, accipias ex tabula ſinuum ſinum, abie-<lb/>ctis quinque figuris, vtſinus totus ſit 100. </s> <s xml:id="echoid-s2228" xml:space="preserve">Verbi gratia. </s> <s xml:id="echoid-s2229" xml:space="preserve">In vltimo exemplo <lb/>Num. </s> <s xml:id="echoid-s2230" xml:space="preserve">4. </s> <s xml:id="echoid-s2231" xml:space="preserve">recta AD, ſumenda eſſet æqualis 100. </s> <s xml:id="echoid-s2232" xml:space="preserve">particulis inſtrumenti partium, ni-<lb/>mirum ſinui tori. </s> <s xml:id="echoid-s2233" xml:space="preserve">At D B, æqualis hypotenuſę E G, vel D G, in figura Num. </s> <s xml:id="echoid-s2234" xml:space="preserve">6. <lb/></s> <s xml:id="echoid-s2235" xml:space="preserve">Et A E, ſi angulus E, eſt grad. </s> <s xml:id="echoid-s2236" xml:space="preserve">30. </s> <s xml:id="echoid-s2237" xml:space="preserve">Min. </s> <s xml:id="echoid-s2238" xml:space="preserve">15. </s> <s xml:id="echoid-s2239" xml:space="preserve">æqualis 50 {4/10}. </s> <s xml:id="echoid-s2240" xml:space="preserve">ferme particulis: </s> <s xml:id="echoid-s2241" xml:space="preserve">quia <lb/>tantus eſt ſinus grad. </s> <s xml:id="echoid-s2242" xml:space="preserve">30. </s> <s xml:id="echoid-s2243" xml:space="preserve">Min. </s> <s xml:id="echoid-s2244" xml:space="preserve">15. </s> <s xml:id="echoid-s2245" xml:space="preserve">Vel ſi angulus G D F, eſt grad. </s> <s xml:id="echoid-s2246" xml:space="preserve">53. </s> <s xml:id="echoid-s2247" xml:space="preserve">Min. </s> <s xml:id="echoid-s2248" xml:space="preserve">20. </s> <s xml:id="echoid-s2249" xml:space="preserve">ac-<lb/>cipienda eſſet AE, æqualis particulis 80 {3/10}. </s> <s xml:id="echoid-s2250" xml:space="preserve">fere. </s> <s xml:id="echoid-s2251" xml:space="preserve">Ita enim interuallum E C, <lb/>dabit tot palmos, aut pedes rectę F G, quot particulę in eo comprehenduntur, <lb/>Et ſic de cęteris.</s> <s xml:id="echoid-s2252" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2253" xml:space="preserve"><emph style="sc">Qvando</emph> autem tota regula 100. </s> <s xml:id="echoid-s2254" xml:space="preserve">partium eſt nimis longa, ſumi poteſt pro <lb/>ſinu toto quoduis interuallum inter 100, & </s> <s xml:id="echoid-s2255" xml:space="preserve">100. </s> <s xml:id="echoid-s2256" xml:space="preserve">dummodo reſpectu huius ſi-<lb/>nus totius accipiantur poſtea ſinus, vt cap. </s> <s xml:id="echoid-s2257" xml:space="preserve">1. </s> <s xml:id="echoid-s2258" xml:space="preserve">lib. </s> <s xml:id="echoid-s2259" xml:space="preserve">1. </s> <s xml:id="echoid-s2260" xml:space="preserve">Num. </s> <s xml:id="echoid-s2261" xml:space="preserve">12. </s> <s xml:id="echoid-s2262" xml:space="preserve">declarauimus.</s> <s xml:id="echoid-s2263" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2264" xml:space="preserve"><emph style="sc">Poteris</emph> autem nonnunquam ordinemimmutare, ponendo nimirum ſe-<lb/>cundamquantitatem DB, in recta AC; </s> <s xml:id="echoid-s2265" xml:space="preserve">& </s> <s xml:id="echoid-s2266" xml:space="preserve">tertiam AE, in recta DB, prout videli-<lb/>cetid expedire cognoueris ad parallelas DE, BC, ducendas.</s> <s xml:id="echoid-s2267" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div113" type="section" level="1" n="49"> <head xml:id="echoid-head52" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s2268" xml:space="preserve">DATIS duabus rectis ad inuicem inclinatis, punctum, in quo con-<lb/>ueniant, inuenire.</s> <s xml:id="echoid-s2269" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2270" xml:space="preserve">QVOD hic proponitur, demonſtratum à <lb/> <anchor type="figure" xlink:label="fig-085-02a" xlink:href="fig-085-02"/> nobis fuit lẽmate 13. </s> <s xml:id="echoid-s2271" xml:space="preserve">lib. </s> <s xml:id="echoid-s2272" xml:space="preserve">1. </s> <s xml:id="echoid-s2273" xml:space="preserve">noſtri Aſtrolabij plu-<lb/>rib{us} viis. </s> <s xml:id="echoid-s2274" xml:space="preserve">Sed quia ei{us} inſignis eſt vtilit{as} in <lb/>puncto concurſ{us} duarum rectarum exquiren-<lb/>do demonſtrabim{us} illud ipſum hoc loco paulo <lb/>aliter. </s> <s xml:id="echoid-s2275" xml:space="preserve">Sint ergo duærectæ A B, C D, oblique <lb/>ſe in concurſu B, ſecant{es}. </s> <s xml:id="echoid-s2276" xml:space="preserve">Ex quotli-<lb/>lib{et} punctis E, F, G, vtcunque in alte- <pb o="56" file="086" n="86" rhead="GEOMETR. PRACT."/> ra earum aſſumptis deſcribantur verſ{us} alteram ad quodcunque idem interuallum ar-<lb/>c{us} H I, K L, M N, ex quib{us} arc{us} quadrante minor{es} abſcindantur in 1, L N, punctis, <lb/>per quæex centris rectæ egrediantur @ ſecant{es} C D, in O, P, Q. </s> <s xml:id="echoid-s2277" xml:space="preserve">Sumptis deinde in E I, pro-<lb/>ducta ipſi E O, tot partib{us} æqualib{us} vſque ad R, quot ſatis eſſe videbuntur, vt recta ex <lb/>R, verſ{us} concurſum ducta non valde oblique ipſ{as} rect{as} ſec{et}, accipiantur in F L, G N, <lb/> <anchor type="handwritten" xlink:label="hd-086-1a" xlink:href="hd-086-1"/> productis totidem part{es} ipſis F P, P Q, æqual{es} vſque ad S, T. </s> <s xml:id="echoid-s2278" xml:space="preserve">Dico tam rectam R S, <lb/>quam RT, & </s> <s xml:id="echoid-s2279" xml:space="preserve">quam ST, in punctum B, concurſ{us} cadere, ita vt puncta R, S, T, B, in vna <lb/>recta linea iaceant. </s> <s xml:id="echoid-s2280" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim anguli E, F, G, æqual{es} ſunt, <anchor type="note" xlink:href="" symbol="b"/> eruntrectæ E R, F S, <anchor type="note" xlink:label="note-086-01a" xlink:href="note-086-01"/> GT, parallelæ. </s> <s xml:id="echoid-s2281" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Cum ergo ER, FS, GT, eaſdem proportion{es} habeant, qu{as} EO, FE<unsure/>P, G Q, <anchor type="note" xlink:label="note-086-02a" xlink:href="note-086-02"/> hoc eſte <anchor type="note" xlink:href="" symbol="d"/> EB, FB, GB, habent, <anchor type="note" xlink:href="" symbol="e"/> cadentrectæ RS, RT, ST, in punctum B, quod eſt propo- <anchor type="note" xlink:label="note-086-03a" xlink:href="note-086-03"/> ſitum.</s> <s xml:id="echoid-s2282" xml:space="preserve"/> </p> <div xml:id="echoid-div113" type="float" level="2" n="1"> <figure xlink:label="fig-085-02" xlink:href="fig-085-02a"> <image file="085-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/085-02"/> </figure> <handwritten xlink:label="hd-086-1" xlink:href="hd-086-1a"/> <note symbol="a" position="left" xlink:label="note-086-01" xlink:href="note-086-01a" xml:space="preserve">27. Tertij.</note> <note symbol="b" position="left" xlink:label="note-086-02" xlink:href="note-086-02a" xml:space="preserve">28. Primi</note> <note symbol="c" position="left" xlink:label="note-086-03" xlink:href="note-086-03a" xml:space="preserve">15. Quinti</note> </div> <note symbol="d" position="left" xml:space="preserve">4. Sexti & <lb/>permutãdo.</note> <p style="it"> <s xml:id="echoid-s2283" xml:space="preserve">HOC ergo lemma ſi adhibeatur, ſatis exquiſitè in ſuperiore figura Num. </s> <s xml:id="echoid-s2284" xml:space="preserve">6. </s> <s xml:id="echoid-s2285" xml:space="preserve">punctum <lb/>concurſ{us} G, deprehendetur, proindeque menſura rectarum, qu{as} ſine numeris inuenire <lb/> <anchor type="note" xlink:label="note-086-05a" xlink:href="note-086-05"/> docuim{us}, non multum à vero aberunt.</s> <s xml:id="echoid-s2286" xml:space="preserve"/> </p> <div xml:id="echoid-div114" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-086-05" xlink:href="note-086-05a" xml:space="preserve">ſchol. 4. ſex <lb/>ti.</note> </div> <p style="it"> <s xml:id="echoid-s2287" xml:space="preserve">8. </s> <s xml:id="echoid-s2288" xml:space="preserve">VERVM commode obliquam illam ſectionem in concurſu G, vitabim{us}, ſi figu-<lb/> <anchor type="note" xlink:label="note-086-06a" xlink:href="note-086-06"/> ram hoc alio modo conſtruem{us}, Fiat in figura Num. </s> <s xml:id="echoid-s2289" xml:space="preserve">6, angul{us} rect{us} EFG, & </s> <s xml:id="echoid-s2290" xml:space="preserve">in quoli-<lb/>b{et} puncto G, vbi concurſum eſſe volum{us}, conſtituantur anguli F G D, F G E, aqual{es} <lb/>complementis angulorum obſeruationum. </s> <s xml:id="echoid-s2291" xml:space="preserve">Ita enim DE, reſpondebit differentiæ ſtatio-<lb/>num, & </s> <s xml:id="echoid-s2292" xml:space="preserve">c. </s> <s xml:id="echoid-s2293" xml:space="preserve">Quocirca ſi cogitetur DE, ſectain tot part{es} æqual{es}, quod palmi, vel ped{es}<unsure/> in <lb/>differentia ſtationum fuerunt aſſumpti, cognoſcem{us} per ea, quæ ad finem Num. </s> <s xml:id="echoid-s2294" xml:space="preserve">1. </s> <s xml:id="echoid-s2295" xml:space="preserve">cap. </s> <s xml:id="echoid-s2296" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s2297" xml:space="preserve">lib. </s> <s xml:id="echoid-s2298" xml:space="preserve">1. </s> <s xml:id="echoid-s2299" xml:space="preserve">docuim{us}, quot ex ijs partib{us} in diſtantijs DF, EF, & </s> <s xml:id="echoid-s2300" xml:space="preserve">in altitudine F G, at que hy-<lb/>potenuſis GD, GE, comprehendantur. </s> <s xml:id="echoid-s2301" xml:space="preserve">Atque hoc modo punctum concurſ{us} G, dubium <lb/>aut incertum eſſe non poteſt, cum illud ante omnia elegerim{us}<unsure/>.</s> <s xml:id="echoid-s2302" xml:space="preserve"/> </p> <div xml:id="echoid-div115" type="float" level="2" n="3"> <note position="left" xlink:label="note-086-06" xlink:href="note-086-06a" xml:space="preserve">Quo pacto <lb/>infig. Num. <lb/>6. obliqua ſe-<lb/>ctio in pun-<lb/>cto concur-<lb/>ſus G, vite-<lb/>tur.</note> </div> </div> <div xml:id="echoid-div117" type="section" level="1" n="50"> <head xml:id="echoid-head53" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s2303" xml:space="preserve">Vt etiam pro tyronibus ſemel explicemus, quid per noſtrum loquendimo-<lb/> <anchor type="handwritten" xlink:label="hd-086-1a" xlink:href="hd-086-1"/> dum intelligamus, cum dicimus verbi gratia, Num. </s> <s xml:id="echoid-s2304" xml:space="preserve">1. </s> <s xml:id="echoid-s2305" xml:space="preserve">huius problematis, Fiat.</s> <s xml:id="echoid-s2306" xml:space="preserve"/> </p> <div xml:id="echoid-div117" type="float" level="2" n="1"> <handwritten xlink:label="hd-086-1" xlink:href="hd-086-1a"/> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt O P, differentia inter \\ tangent{es} angulorum \\ obſeruationum # ad P N, vel H M, \\ Tangentem mino- \\ rem. # It a E D, diffe- \\ rentia ſtatio- \\ num # ad D F: <lb/></note> <p> <s xml:id="echoid-s2307" xml:space="preserve">Sciendum eſt, nos hoc modo redigere opus ad termino s regulæ trium. </s> <s xml:id="echoid-s2308" xml:space="preserve">Qua <lb/>propter ſi iuxta tenorem regulæ numerus in tertio loco poſitus ducatur in eum, <lb/>qui ſecundum locum occupat, hoceſt, differentia ſtationum in propoſito e-<lb/>xemplo multip licetur per Tangentem minorem, productuſque numerus per <lb/>eum, qui in primo loco collocatur, id eſt, per differentiam Tangentium, diui-<lb/>datur: </s> <s xml:id="echoid-s2309" xml:space="preserve">(niſi quando primus numerus eſt ſinus totus Tunc enim diuiſio non fit, <lb/> <anchor type="note" xlink:label="note-086-08a" xlink:href="note-086-08"/> ſed ex producto quinque figuræ abijciuntur, vel ſeptem, prout ſinus totus ſta-<lb/>tuitur 100.</s> <s xml:id="echoid-s2310" xml:space="preserve">000. </s> <s xml:id="echoid-s2311" xml:space="preserve">vel 10,000.</s> <s xml:id="echoid-s2312" xml:space="preserve">000.) </s> <s xml:id="echoid-s2313" xml:space="preserve">pro creabitur in quotiente quartus numerus, <lb/>qui quæritur, nimirum diſtantia D F. </s> <s xml:id="echoid-s2314" xml:space="preserve">Eademque eſt ratio de cæteris.</s> <s xml:id="echoid-s2315" xml:space="preserve"/> </p> <div xml:id="echoid-div118" type="float" level="2" n="2"> <note position="left" xlink:label="note-086-08" xlink:href="note-086-08a" xml:space="preserve">Altitudinis <lb/>inuentio per <lb/>vnicam ſtatio <lb/>nem, quando <lb/>diſtantia nota <lb/>eſt.</note> </div> </div> <div xml:id="echoid-div120" type="section" level="1" n="51"> <head xml:id="echoid-head54" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s2316" xml:space="preserve"><emph style="sc">Itaqve</emph> quando diſtantia à loco menſoris vſque ad altitudinem ignotam <lb/>cognita eſt, inuenietur altitudo per vnicam ſtationem. </s> <s xml:id="echoid-s2317" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Sifiat.</s> <s xml:id="echoid-s2318" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} # ad Tangentem anguli \\ obſeruationis: # Ita diſtantia \\ nota # ad altitudinem. <lb/></note> <note symbol="f" position="left" xml:space="preserve">4. Triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2319" xml:space="preserve">Hoc enim demonſtratum eſt Num. </s> <s xml:id="echoid-s2320" xml:space="preserve">3. </s> <s xml:id="echoid-s2321" xml:space="preserve">huius problematis 1. </s> <s xml:id="echoid-s2322" xml:space="preserve">tam per angulum ob- <pb o="57" file="087" n="87" rhead="LIBER SECVNDVS."/> ſeruationis GDF, & </s> <s xml:id="echoid-s2323" xml:space="preserve">diſtantiam DF, quam per angulum obſeruationis GEF, & </s> <s xml:id="echoid-s2324" xml:space="preserve"><lb/>diſtantiam EF. </s> <s xml:id="echoid-s2325" xml:space="preserve">Vtroque enim modo inuenta eſt altitudo GF.</s> <s xml:id="echoid-s2326" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div121" type="section" level="1" n="52"> <head xml:id="echoid-head55" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s2327" xml:space="preserve"><emph style="sc">Perspicvvm</emph> etiam eſt, ſi G, ſit cacumen alicuius montis, nos per hoc <lb/> <anchor type="note" xlink:label="note-087-01a" xlink:href="note-087-01"/> problema 1. </s> <s xml:id="echoid-s2328" xml:space="preserve">eius altitudinem poſſe metiri per duas ſtationes D, E, in plano fa-<lb/>ctas: </s> <s xml:id="echoid-s2329" xml:space="preserve">ſi nimirum prius inueſtigetur recta D F, vel E F, ab oculo menſoris vſque <lb/>ad perpendicularem GF, quæ à cacumine G, in planum Horizontis cadit, etiãſi <lb/>eius extremum F, non videamus.</s> <s xml:id="echoid-s2330" xml:space="preserve"/> </p> <div xml:id="echoid-div121" type="float" level="2" n="1"> <note position="right" xlink:label="note-087-01" xlink:href="note-087-01a" xml:space="preserve">Altitudo mõ-<lb/>tis quo pacto <lb/>inueſtigetur.</note> </div> <p> <s xml:id="echoid-s2331" xml:space="preserve">ALTITVDINEM inacceſſibilem, quando diſtantia à loco mẽ-<lb/>ſoris ad baſem altitudinis ignota eſt, per duas ſtationes in plano factas, <lb/>per quadrantem dimetiri. </s> <s xml:id="echoid-s2332" xml:space="preserve">Atque hinc diſtantiam quoque ipſam erue-<lb/>re, etiam ſi extremus eius terminus non cernatur.</s> <s xml:id="echoid-s2333" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div123" type="section" level="1" n="53"> <head xml:id="echoid-head56" xml:space="preserve">PROBLEMA II.</head> <p> <s xml:id="echoid-s2334" xml:space="preserve">1. </s> <s xml:id="echoid-s2335" xml:space="preserve"><emph style="sc">Sit</emph> inquirenda altitudo AB, ſiue ea turris ſit, ſiue mons, ſiue aliquid ali-<lb/>ud, licetnon cernatur eius perpendiculi infimus terminus B, vt in omni monte <lb/>contingit: </s> <s xml:id="echoid-s2336" xml:space="preserve">planum autem, cui perpendicularis eſt altitudo, ſit CB. </s> <s xml:id="echoid-s2337" xml:space="preserve">Statura mẽ-<lb/>ſoris D E. </s> <s xml:id="echoid-s2338" xml:space="preserve">Ducta autem cogitatione per E, ipſi CB, parallela GF, fiat prima ſta-<lb/>tio in D, propinquior, ſecunda vero in G, remotior, vt differentia ſtationum ſit <lb/>GE. </s> <s xml:id="echoid-s2339" xml:space="preserve">Deinde per radios viſuales EA, GA, ad verticem A, directos diligenter ob-<lb/>ſeruentur anguli AEF, AGF, ſiue per quadrantem pendulum, vt Num. </s> <s xml:id="echoid-s2340" xml:space="preserve">1. </s> <s xml:id="echoid-s2341" xml:space="preserve">pro-<lb/>blematis præcedentis do cuimus, ſiue per ſtabilem, vt Num. </s> <s xml:id="echoid-s2342" xml:space="preserve">5. </s> <s xml:id="echoid-s2343" xml:space="preserve">eiuſdem proble-<lb/>matis præcepimus. </s> <s xml:id="echoid-s2344" xml:space="preserve">Eodem enim ſemper modo dicti anguli obſeruantur, quan-<lb/>do è loco inferiori altitu dinis faſtigium inſpicitur. </s> <s xml:id="echoid-s2345" xml:space="preserve">Cogitetur quo que ducta HI, <lb/> <anchor type="figure" xlink:label="fig-087-01a" xlink:href="fig-087-01"/> ipſi G F, parallela, <anchor type="note" xlink:href="" symbol="a"/> vt demiſſæ perpendiculares H L, I K, in <anchor type="note" xlink:label="note-087-02a" xlink:href="note-087-02"/> parallelogrammo LI, ſint æquales, pro ſinubus totis: </s> <s xml:id="echoid-s2346" xml:space="preserve">quo-<lb/>rum tangentes ſunt EK, GL, angulis, I, H, qui complemen-<lb/>ta ſunt angulorum obſeruationum E, G, debitæ. </s> <s xml:id="echoid-s2347" xml:space="preserve">Et quo-<lb/>niam angulus G A F, maior eſt angulo E A F, <anchor type="note" xlink:href="" symbol="b"/> eſt que priori <anchor type="note" xlink:label="note-087-03a" xlink:href="note-087-03"/> angulus G H L, & </s> <s xml:id="echoid-s2348" xml:space="preserve">poſteriori angulus E I K, æqualis: </s> <s xml:id="echoid-s2349" xml:space="preserve">erit <lb/>quo que GHL, maior quam EIK, ideo que tãgens G L, ma-<lb/>ior Tangente EK, quòd ſinus toti H L, I K, æquales ſint. </s> <s xml:id="echoid-s2350" xml:space="preserve">Ab-<lb/>ſcindatur LM, ipſi EK, æqualis, vt GM, ſit differẽtia Tangẽ-<lb/>tium G L, E K, <anchor type="note" xlink:href="" symbol="c"/> Et quia eſt vt G L, ad L H, ita G F, ad F A, erit permutando, vt <anchor type="note" xlink:label="note-087-04a" xlink:href="note-087-04"/> GL, ad GF, ita LH, vel IK, ad FA; </s> <s xml:id="echoid-s2351" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Vtautem IK, ad F A, ita quoque eſt EK, ad <anchor type="note" xlink:label="note-087-05a" xlink:href="note-087-05"/> EF. </s> <s xml:id="echoid-s2352" xml:space="preserve">Igitur erit, vttota GL, ad totam GF, ita EK, vel LM, ex GL, ablata, ad EF, <lb/>ex GF, ablatam: </s> <s xml:id="echoid-s2353" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ac proinde erit etiam vt GM, ex GL, reliqua ad G E, ex G F, <anchor type="note" xlink:label="note-087-06a" xlink:href="note-087-06"/> reliquam, ita tota G L, ad totam G F, hoc eſt, <anchor type="note" xlink:href="" symbol="f"/> ita L H, ſinus totus, a d F A.</s> <s xml:id="echoid-s2354" xml:space="preserve"> <anchor type="note" xlink:label="note-087-07a" xlink:href="note-087-07"/> Quamobrem ſi fiat.</s> <s xml:id="echoid-s2355" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="1"> <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/xxxxxxxx/figures/087-01"/> </figure> <note symbol="a" position="right" xlink:label="note-087-02" xlink:href="note-087-02a" xml:space="preserve">34. prim.</note> <note symbol="b" position="right" xlink:label="note-087-03" xlink:href="note-087-03a" xml:space="preserve">29. primi.</note> <note symbol="c" position="right" xlink:label="note-087-04" xlink:href="note-087-04a" xml:space="preserve">4. ſexti.</note> <note symbol="d" position="right" xlink:label="note-087-05" xlink:href="note-087-05a" xml:space="preserve">4. ſexti & <lb/>permutando</note> <note symbol="e" position="right" xlink:label="note-087-06" xlink:href="note-087-06a" xml:space="preserve">19. quinti</note> <note symbol="f" position="right" xlink:label="note-087-07" xlink:href="note-087-07a" xml:space="preserve">4 ſexti & <lb/>permutando.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt G M, differentia Tangentium \\ G L, E K, complementorum an- \\ gulorum obſeruationum # ad G E, diffe- \\ rentiam ſt a- \\ tionum. # ita L H, \\ ſin{us} i<unsure/>o- \\ t{us}. # ad FA, <lb/></note> <pb o="58" file="088" n="88" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s2356" xml:space="preserve">inuenta erit altitudo F A, in partibus differentiæ ſtationum, cuiſi adij ciatur F B, <lb/> <anchor type="note" xlink:label="note-088-01a" xlink:href="note-088-01"/> ſtatura menſoris, tota altitudo AB, nota euadet.</s> <s xml:id="echoid-s2357" xml:space="preserve"/> </p> <div xml:id="echoid-div124" type="float" level="2" n="2"> <note position="left" xlink:label="note-088-01" xlink:href="note-088-01a" xml:space="preserve">Altitudinis <lb/>inuentio per <lb/>Tangent{es}.</note> </div> </div> <div xml:id="echoid-div126" type="section" level="1" n="54"> <head xml:id="echoid-head57" xml:space="preserve">2. ITEM ſi fiat.</head> <note style="it" position="right" xml:space="preserve"> <lb/>Vt G M, differentia Tangen- \\ tium complementorum an- \\ gulorum obſeruationum. # ad G E, diffe- \\ rentiam ſta- \\ tionum: # ita G L, Tangens \\ complementi angu- \\ li obſeruationis in \\ remotiori ſtatione # ad \\ G F, <lb/></note> <p> <s xml:id="echoid-s2358" xml:space="preserve">effi cietur nota G F, diſtantia maior, quando quidem paulo ante demonſtratum <lb/> <anchor type="note" xlink:label="note-088-03a" xlink:href="note-088-03"/> eſt, eſſe GM, ad GE, vt GL, ad GF. </s> <s xml:id="echoid-s2359" xml:space="preserve">Quod ſi ex GF, inuenta detrahatur GE, dif-<lb/>ferentia ſtationum, nota relinquetur EF, diſtantia minor.</s> <s xml:id="echoid-s2360" xml:space="preserve"/> </p> <div xml:id="echoid-div126" type="float" level="2" n="1"> <note position="left" xlink:label="note-088-03" xlink:href="note-088-03a" xml:space="preserve">Diſtantiæ in-<lb/>uentio.</note> </div> </div> <div xml:id="echoid-div128" type="section" level="1" n="55"> <head xml:id="echoid-head58" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2361" xml:space="preserve">3. </s> <s xml:id="echoid-s2362" xml:space="preserve"><emph style="sc">Si</emph> per ſolos ſinus dimenſio inſtituatur, inueſtiganda primum erit alteru-<lb/> <anchor type="note" xlink:label="note-088-04a" xlink:href="note-088-04"/> tra hypotenuſarum GA, EA, vel vtraque. </s> <s xml:id="echoid-s2363" xml:space="preserve">hoc ſcilicet modo. </s> <s xml:id="echoid-s2364" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam an- gulus A E F, duobus G, G A E, æqualis eſt, ſi angulus G, in remotiore ſtatione <lb/>tollatur ex angulo AEF, in ſtatione propinquiore, reliquus fiet angulus GAE, <lb/>differentia inter duos angulos G, AEF obſeruationum.</s> <s xml:id="echoid-s2365" xml:space="preserve"/> </p> <div xml:id="echoid-div128" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-088-04" xlink:href="note-088-04a" xml:space="preserve">32. primi.</note> </div> </div> <div xml:id="echoid-div130" type="section" level="1" n="56"> <head xml:id="echoid-head59" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Ergo ſi fiat,</head> <note symbol="b" position="left" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli G A E, \\ differentiæ inter duos \\ angulos obſeruationum # ad G E, dif- \\ ferentiam \\ ſtationum: # ita ſin{us} anguli A G E, \\ vel \\ it a ſin{us}(<unsure/>anguli G E A, \\ complementi)<unsure/>anguli \\ A E F, ad duosrectos. # ad AE, \\ ad G A, <lb/></note> <note position="left" xml:space="preserve">Hypotenuſa-<lb/>rum inuentio.</note> <p> <s xml:id="echoid-s2366" xml:space="preserve">nota fiet tam A E, quam A G, in partibus differentiæ ſtationum.</s> <s xml:id="echoid-s2367" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div131" type="section" level="1" n="57"> <head xml:id="echoid-head60" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Si igitur fiat.</head> <note symbol="c" position="left" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} anguli \\ recti F, # ad rectam \\ E A: \\ vel \\ ad rectam \\ G A, proxi- \\ mè inuentam # ita ſin{us} anguli AEF, in pro- \\ pinquiore ſtatione, \\ vel \\ ita ſin{us} anguli A G F, in re- \\ motiore ſtatione # ad A F, \\ ad A F, <lb/></note> <note position="left" xml:space="preserve">Altitudinis <lb/>inuentio per <lb/>ſolos ſin{us}.</note> <p> <s xml:id="echoid-s2368" xml:space="preserve">gignetur altitudo A F, & </s> <s xml:id="echoid-s2369" xml:space="preserve">ſi adiungetur F B, ſtatura menſoris, tota altitudo AB, <lb/>efficietur nota.</s> <s xml:id="echoid-s2370" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2371" xml:space="preserve"><emph style="sc">Distantia</emph> autem vtraque E F, G F, per ſolos etiam ſinus inuenietur, <anchor type="note" xlink:href="" symbol="d"/> ſi <anchor type="note" xlink:label="note-088-11a" xlink:href="note-088-11"/> fiat.</s> <s xml:id="echoid-s2372" xml:space="preserve"/> </p> <div xml:id="echoid-div131" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-088-11" xlink:href="note-088-11a" xml:space="preserve">10. triang. <lb/>rectil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} an- \\ guli recti \\ F. # ad hypotenu- \\ ſam E A. \\ vel \\ ad hypotenu- \\ ſam G A: # ita ſin{us} anguli E A F, complementi \\ anguli in propinquiore ſtatione, \\ vel \\ ita ſin{us} anguli G A F, complementi \\ anguli in remotiore ſtatione. # ad E F. <lb/></note> <note position="left" xml:space="preserve">Diſtantiæ in-<lb/>uẽtis per ſo-<lb/>los ſinus.</note> <figure> <image file="088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/088-01"/> </figure> <p> <s xml:id="echoid-s2373" xml:space="preserve">4. </s> <s xml:id="echoid-s2374" xml:space="preserve"><emph style="sc">Sine</emph> numerorum multiplicatione ac diui-<lb/> <anchor type="note" xlink:label="note-088-14a" xlink:href="note-088-14"/> ſione eadem inueſtigabuntur, ſi illi figuræ, quam ab <lb/>oculo menſoris ad altitu dinem vſque concepimus <lb/>eſſe conſtruendam, ſimilem omni diligentia conſti-<lb/>tuamus, vt Num. </s> <s xml:id="echoid-s2375" xml:space="preserve">6. </s> <s xml:id="echoid-s2376" xml:space="preserve">& </s> <s xml:id="echoid-s2377" xml:space="preserve">8. </s> <s xml:id="echoid-s2378" xml:space="preserve">præcedentis problematis 1.</s> <s xml:id="echoid-s2379" xml:space="preserve"> <pb o="59" file="089" n="89" rhead="LIBER SSECVNDVS."/> diximus, vt manifeſtum eſt, ſi in ea figura Num. </s> <s xml:id="echoid-s2380" xml:space="preserve">6. </s> <s xml:id="echoid-s2381" xml:space="preserve">altitudo intelligatur F G, & </s> <s xml:id="echoid-s2382" xml:space="preserve"><lb/>diſtantiæ FD, FE, & </s> <s xml:id="echoid-s2383" xml:space="preserve">c. </s> <s xml:id="echoid-s2384" xml:space="preserve">Idemque efficies per ea, quæ Num. </s> <s xml:id="echoid-s2385" xml:space="preserve">7. </s> <s xml:id="echoid-s2386" xml:space="preserve">eiuſdem proble-<lb/>matis 1. </s> <s xml:id="echoid-s2387" xml:space="preserve">tradita ſunt.</s> <s xml:id="echoid-s2388" xml:space="preserve"/> </p> <div xml:id="echoid-div132" type="float" level="2" n="2"> <note position="left" xlink:label="note-088-14" xlink:href="note-088-14a" xml:space="preserve">Problemaque <lb/>pacti ſi ſine <lb/>@ imeris ſol@ <lb/>u<unsure/>atur.</note> </div> </div> <div xml:id="echoid-div134" type="section" level="1" n="58"> <head xml:id="echoid-head61" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s2389" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi altitudo fuerit nota, inuenietur diſtantia per vnicam ſtationem, <lb/> <anchor type="note" xlink:label="note-089-01a" xlink:href="note-089-01"/> <anchor type="note" xlink:href="" symbol="a"/>ſi fiat.</s> <s xml:id="echoid-s2390" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="1"> <note position="right" xlink:label="note-089-01" xlink:href="note-089-01a" xml:space="preserve">Diſtantiæ in-<lb/>uentio per v-<lb/>uicam ſtatio-<lb/>nem, quand@ <lb/>altitudo notæ <lb/>est.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} A F, # ad Tangentem F E, complementi \\ anguli in propinquiore ſtatione: \\ vel \\ ad Tangentem F G, complementi \\ anguli in remotiore ſtatione: # ita altitu- \\ do nota \\ A F, # ad diſtan- \\ tiam F E, \\ vel \\ ad diſtan- \\ tiam F G. <lb/></note> <note symbol="a" position="right" xml:space="preserve">4. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2391" xml:space="preserve"><emph style="sc">Item</emph> ſi diſtantia nota fuerit, reperietur altitudo per vnicam quoq; </s> <s xml:id="echoid-s2392" xml:space="preserve">ſtatio-<lb/>nem, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} G F, # ad A F, Tangentem anguli G, \\ obſeruati in remotiori ſtatione: # ita G F, diſtan- \\ tia nota # ad AF, al- \\ titudinem <lb/>#### Vel <lb/>Vt ſin{us} to- \\ t{us} E F, # ad AF, Tangentem anguli E, \\ obſeruati in ſtatione propin- \\ quiore: # ita E F, diſtan- \\ tia nota, # ad A F, alti- \\ tudinem. <lb/></note> </div> <div xml:id="echoid-div136" type="section" level="1" n="59"> <head xml:id="echoid-head62" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s2393" xml:space="preserve"><emph style="sc">Manifestvm</emph> etiam eſt, ſi punctum A, ſit cacumen alicuius montis eo-<lb/>dem pacto inueſtigari poſſe lineam perpendicularem A F, quæ ex cacumine in <lb/>Horizontis planum demittitur: </s> <s xml:id="echoid-s2394" xml:space="preserve">ſi nimirum duæ ſtationes fiant in E, G.</s> <s xml:id="echoid-s2395" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2396" xml:space="preserve">EX VERTICE montis, aut turris, in cuius ſummitate duæ ſtationes <lb/>fieri poſſint, è quibus ſignum aliquod in Horizonte appareat, altitu-<lb/>din em ipſius montis, turriſue dimetiri. </s> <s xml:id="echoid-s2397" xml:space="preserve">Atque hinc ipſam quoque di-<lb/>ſtantiam à turris baſi, vel perpendiculo montis ad ſignum illud inue-<lb/>ſtigare.</s> <s xml:id="echoid-s2398" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div137" type="section" level="1" n="60"> <head xml:id="echoid-head63" xml:space="preserve">PROBLEMA III.</head> <p> <s xml:id="echoid-s2399" xml:space="preserve">1. </s> <s xml:id="echoid-s2400" xml:space="preserve"><emph style="sc">Sit</emph> mons, aut turris G H N M, cuius altitudo perpendiculum E F, vel <lb/>etiam latus turris G M, vel HN. </s> <s xml:id="echoid-s2401" xml:space="preserve">Eligantur duæ ſtationes in G, H; </s> <s xml:id="echoid-s2402" xml:space="preserve">& </s> <s xml:id="echoid-s2403" xml:space="preserve">ab oculo <lb/>menſoris tamin I, quam in K, poſito cerni poſsit ſignum L, in Horizonte, vel <lb/>plano, cui turris, aut mons inſiſtit. </s> <s xml:id="echoid-s2404" xml:space="preserve">Dirigatur latus Quadrantis penduli, in quo <lb/>pinnacidia, verſus L, diligenterq; </s> <s xml:id="echoid-s2405" xml:space="preserve">angulus K, notetur, quem arcus Quadrantis <lb/>inter latus pinnacidiorum, & </s> <s xml:id="echoid-s2406" xml:space="preserve">filum perpendiculi libere pendentis determinat. <lb/></s> <s xml:id="echoid-s2407" xml:space="preserve">Eodemque pacto angulus I, obſeruetur. </s> <s xml:id="echoid-s2408" xml:space="preserve">Per Quadrantem ſtabilem vterque <lb/>obſeruabitur, ſi vnum eius latus ad Horizontem ſitrectũ, (quod tum demũ fiet, <pb o="60" file="090" n="90" rhead="GEOMETR. PRACT."/> quando filum perpendiculi ex centro pendentis illi lateri adhærebit) & </s> <s xml:id="echoid-s2409" xml:space="preserve">dio-<lb/> <anchor type="figure" xlink:label="fig-090-01a" xlink:href="fig-090-01"/> ptra attollatur, donec per pinnacidio-<lb/>rum foramina ſignum L, cerni poſsit. <lb/></s> <s xml:id="echoid-s2410" xml:space="preserve">Nam arcus inter dictum latus, & </s> <s xml:id="echoid-s2411" xml:space="preserve">lineam <lb/>fiduciæ in dioptra angulum obſeruatio-<lb/>nis meti@tur. </s> <s xml:id="echoid-s2412" xml:space="preserve">Intelligatur quoque filum <lb/>perpendiculi in vtro que ſitu Quadrantis <lb/>penduli, vel latus quandrantis ſtabilis ad <lb/>Horizontem rectum productum vſque <lb/>ad baſem montis, aut turris ad M, N. </s> <s xml:id="echoid-s2413" xml:space="preserve">Sũ-<lb/>ptis deinde rectis æqualibus K O, I P, pro <lb/>ſinubus totis, ducantur P Q, O R, ad IN, <lb/> <anchor type="note" xlink:label="note-090-01a" xlink:href="note-090-01"/> KM, perpendiculares, <anchor type="note" xlink:href="" symbol="a"/> quæ parallelæ e- runt plano Horizontis, hoc eſt, rectę NL; <lb/></s> <s xml:id="echoid-s2414" xml:space="preserve">& </s> <s xml:id="echoid-s2415" xml:space="preserve">Tangentes angulorũ obſeruationum <lb/>I, K <anchor type="note" xlink:href="" symbol="b"/> Ipſæ autem HN, GM, altitudini EF, <anchor type="note" xlink:label="note-090-02a" xlink:href="note-090-02"/> æquales erunt. </s> <s xml:id="echoid-s2416" xml:space="preserve">Et quoniam in triangulis <lb/>I N L, K M L, anguli recti N, M, ſunt æ-<lb/>quales, & </s> <s xml:id="echoid-s2417" xml:space="preserve">ILN, minor quam KLM, pars <lb/>toto; </s> <s xml:id="echoid-s2418" xml:space="preserve">eritreliquus N I L, reliquo MKL, maior, ideo que & </s> <s xml:id="echoid-s2419" xml:space="preserve">Tangens P Q, Tan-<lb/> <anchor type="note" xlink:label="note-090-03a" xlink:href="note-090-03"/> gente O R, maior. </s> <s xml:id="echoid-s2420" xml:space="preserve">Abſciſſa ergo P T, ipſi O R, æquali; </s> <s xml:id="echoid-s2421" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> quia eſt vt IP, ſinus totus ad P Q, Tangentem maioris anguli obſeruationis, ita IN, ad NL: </s> <s xml:id="echoid-s2422" xml:space="preserve">erit per-<lb/>mutando, vt IP, ad IN, ita P Q, ad NL. </s> <s xml:id="echoid-s2423" xml:space="preserve">Eademque ratione erit, vt K O, ad KM, <lb/>hoc eſt, vt IP, ad IN, <anchor type="note" xlink:href="" symbol="d"/> (cum K O, KM, ipſis IP, IN, ſint æquales) ita OR, hoc eſt, <anchor type="note" xlink:label="note-090-04a" xlink:href="note-090-04"/> ita PT, ad ML; </s> <s xml:id="echoid-s2424" xml:space="preserve">at que ideo erit, vt PQ, ad NL, ita PT, ad ML. </s> <s xml:id="echoid-s2425" xml:space="preserve">Quia ergo eſt, vt <lb/>tota P Q, ad totam N L, ita P T, ablata ad M L, ablatam, <anchor type="note" xlink:href="" symbol="e"/> erit quo que reliqua <anchor type="note" xlink:label="note-090-05a" xlink:href="note-090-05"/> T Q, differentia Tangentium, adreliquam NM, differentiam ſtationum, <anchor type="note" xlink:href="" symbol="f"/> (quod <anchor type="note" xlink:label="note-090-06a" xlink:href="note-090-06"/> NM, HG, æquales ſint.) </s> <s xml:id="echoid-s2426" xml:space="preserve">vttota P Q, ad totam NL, <anchor type="note" xlink:href="" symbol="g"/> hoc eſt, vt IP, ad IN. </s> <s xml:id="echoid-s2427" xml:space="preserve">Quã- <anchor type="note" xlink:label="note-090-07a" xlink:href="note-090-07"/> obrem ſi fiat.</s> <s xml:id="echoid-s2428" xml:space="preserve"/> </p> <div xml:id="echoid-div137" type="float" level="2" n="1"> <figure xlink:label="fig-090-01" xlink:href="fig-090-01a"> <image file="090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/090-01"/> </figure> <note symbol="a" position="left" xlink:label="note-090-01" xlink:href="note-090-01a" xml:space="preserve">28. primi.</note> <note symbol="b" position="left" xlink:label="note-090-02" xlink:href="note-090-02a" xml:space="preserve">34. primi.</note> <note symbol="c" position="left" xlink:label="note-090-03" xlink:href="note-090-03a" xml:space="preserve">4. ſexti.</note> <note symbol="d" position="left" xlink:label="note-090-04" xlink:href="note-090-04a" xml:space="preserve">34. primi.</note> <note symbol="e" position="left" xlink:label="note-090-05" xlink:href="note-090-05a" xml:space="preserve">19. quinti.</note> <note symbol="f" position="left" xlink:label="note-090-06" xlink:href="note-090-06a" xml:space="preserve">34. primi.</note> <note symbol="g" position="left" xlink:label="note-090-07" xlink:href="note-090-07a" xml:space="preserve">4. ſexti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt T Q, differentia \\ inter Tangentes an- \\ gulorum obſeruationum # ad N M, vel H G, diffe- \\ rentiam ſtationum: # ita ſin{us} to- \\ t{us} I P, # ad IN, <lb/></note> <p> <s xml:id="echoid-s2429" xml:space="preserve">reperietur recta I N, ex qua ſi dematur I H, ſtatura menſoris, nota relinquetur <lb/> <anchor type="note" xlink:label="note-090-09a" xlink:href="note-090-09"/> HN, vel EF, altitudo quæſita. </s> <s xml:id="echoid-s2430" xml:space="preserve">Et ſi rurſus fiat,</s> </p> <div xml:id="echoid-div138" type="float" level="2" n="2"> <note position="left" xlink:label="note-090-09" xlink:href="note-090-09a" xml:space="preserve">Altitudinis <lb/>inuentio.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt T Q, differen- \\ tia Tangentium # ad N M, vel G H, differen- \\ tiam ſtationum: # ita P Q, Tan- \\ gens maior # ad NL, di- ſtantiam. <lb/></note> <p> <s xml:id="echoid-s2431" xml:space="preserve">nota fiet diſtantia NL; </s> <s xml:id="echoid-s2432" xml:space="preserve">à qua ſi ſubtrahatur NF, vel HE, quam metiri licebit, co-<lb/> <anchor type="note" xlink:label="note-090-11a" xlink:href="note-090-11"/> gnita relinquetur FL, diſtantia à perpendiculo montis. </s> <s xml:id="echoid-s2433" xml:space="preserve">Vel ſi dematur NM, dif-<lb/>ferentia ſtationum, nota relinquetur diſtantia ML, à turri vſque ad L. </s> <s xml:id="echoid-s2434" xml:space="preserve">Item <anchor type="note" xlink:href="" symbol="h"/> ſi fiat.</s> <s xml:id="echoid-s2435" xml:space="preserve"/> </p> <div xml:id="echoid-div139" type="float" level="2" n="3"> <note position="left" xlink:label="note-090-11" xlink:href="note-090-11a" xml:space="preserve">Diſtantiæ in-<lb/>uentio.</note> </div> <note symbol="h" position="left" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt IP, ſin{us} \\ tot{us} # ad P Q. Tangen- \\ tem maiorem: # ita IN, paulo ante \\ inuenta # ad NL, diſtantiam, <lb/></note> <p> <s xml:id="echoid-s2436" xml:space="preserve">inuenietur rurſus diſtantia NL, & </s> <s xml:id="echoid-s2437" xml:space="preserve">c.</s> <s xml:id="echoid-s2438" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2439" xml:space="preserve">2. </s> <s xml:id="echoid-s2440" xml:space="preserve"><emph style="sc">Vervm</emph> ita eſſe T Q. </s> <s xml:id="echoid-s2441" xml:space="preserve">differentiam Tangentium ad NM, vel H G, diffe- <pb o="61" file="091" n="91" rhead="LIBER SECVNDVS."/> rentiam ſtationum, vt eſt P Q, Tangens maior ad diſtantiam N L, vt paulo an-<lb/>te oſtenſum eſt, facilius demonſtrabimus, ſi ducatur recta I T, quæ producta <lb/>ſecet N L, in S. </s> <s xml:id="echoid-s2442" xml:space="preserve">Nam in triangulis IP T, K O R, duo latera IP, P T, duobus la-<lb/>teribus KO, OR, æqualia ſunt, angulo ſque continent æquales, vtp ote rectos. <lb/></s> <s xml:id="echoid-s2443" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> Igitur anguli T, R, ęquales ſunt; </s> <s xml:id="echoid-s2444" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ideo que KL, IS, parallelæ. </s> <s xml:id="echoid-s2445" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Et quia ducta <anchor type="note" xlink:label="note-091-01a" xlink:href="note-091-01"/> I K, æqualis eſt & </s> <s xml:id="echoid-s2446" xml:space="preserve">ipſi SL, & </s> <s xml:id="echoid-s2447" xml:space="preserve">differentiæ ſtationum HG, vel NM, <anchor type="note" xlink:href="" symbol="d"/> liquido con- <anchor type="note" xlink:label="note-091-02a" xlink:href="note-091-02"/> ſtat, ita eſſe T Q, differentiam Tangentium ad S L, quæ differentię ſtationum <lb/> <anchor type="note" xlink:label="note-091-03a" xlink:href="note-091-03"/> I K, vel H G, æqualis eſt, vt eſt P Q, Tangens maior ad diſtantiam NL,</s> </p> <div xml:id="echoid-div140" type="float" level="2" n="4"> <note symbol="a" position="right" xlink:label="note-091-01" xlink:href="note-091-01a" xml:space="preserve">4. primi.</note> <note symbol="b" position="right" xlink:label="note-091-02" xlink:href="note-091-02a" xml:space="preserve">28. primi.</note> <note symbol="c" position="right" xlink:label="note-091-03" xlink:href="note-091-03a" xml:space="preserve">34. primi.</note> </div> <note symbol="d" position="right" xml:space="preserve">ſchol. 4. <lb/>ſext.</note> </div> <div xml:id="echoid-div142" type="section" level="1" n="61"> <head xml:id="echoid-head64" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2448" xml:space="preserve">3. </s> <s xml:id="echoid-s2449" xml:space="preserve"><emph style="sc">Per</emph> ſolos ſinus ita Problema efficiemus. </s> <s xml:id="echoid-s2450" xml:space="preserve">Primum inquiremus hypote-<lb/> <anchor type="note" xlink:label="note-091-05a" xlink:href="note-091-05"/> nuſas I L, K L, hoc modo. </s> <s xml:id="echoid-s2451" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Quoniam angulus L K V, duobus angulis L I K, I L K, æqualis eſt; </s> <s xml:id="echoid-s2452" xml:space="preserve">ſi L I K, angulus complementi maioris anguli obſeruationis <lb/>L I N, auferatur ex L K V, angulo complementi minoris anguli obſeruationis <lb/>L K M, reliquus fiet angulus I K L. </s> <s xml:id="echoid-s2453" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Si ergo fiat.</s> <s xml:id="echoid-s2454" xml:space="preserve"/> </p> <div xml:id="echoid-div142" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-091-05" xlink:href="note-091-05a" xml:space="preserve">32. primi.</note> </div> <note symbol="f" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli I K L, dif- \\ ferentiæ inter duos an- \\ gulos complementorum \\ angulorum obſeruatio- \\ num # ad I K, \\ differẽ- \\ tiam \\ ſtatio- \\ num: # Ita ſin{us} anguli I K L, conflati ex \\ recto, & angulo obſeruationis \\ minore, vel # ad I L, <lb/># # Ita ſin{us} anguli LIK, complemen- \\ ti maioris anguli obſeruationis. # ad K L, <lb/></note> <note position="right" xml:space="preserve">Hypotenu-<lb/>ſarum inuẽ-<lb/>tio.</note> <p> <s xml:id="echoid-s2455" xml:space="preserve">euadet nota tam I L, quam K L, in menſura differentiæ ſtationum. </s> <s xml:id="echoid-s2456" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Igitur ſi fiat,</s> </p> <note symbol="g" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} \\ angulire- \\ cti N, # ad Hypotenuſam \\ proxime inuen- \\ tam I L, # Ita ſin{us} anguli I L N, complementi \\ maioris anguli obſeruationis # ad I N, <lb/>#### Vel <lb/>Vt ſin{us} tot{us} an- \\ guli recti M, # ad hypotenuſam K L, \\ nuper inuentam: # Ita ſin{us} anguli K L M, complemen- \\ ti minoris anguli obſeruationis # ad \\ K M, <lb/></note> <note position="right" xml:space="preserve">Altitudinis <lb/>inuentio per <lb/>ſolos ſinus.</note> <p> <s xml:id="echoid-s2457" xml:space="preserve">cognita fiet altitudò<unsure/> I N, vel KM, ex qua ſi dematur ſtatura menſoris, altitudo <lb/>quæſita relinquetur H N, vel G M, hoc eſt, E F.</s> <s xml:id="echoid-s2458" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2459" xml:space="preserve"><emph style="sc">Distantia</emph> autem vtraque N L, M L, reperietur, <anchor type="note" xlink:href="" symbol="h"/> ſi fiat.</s> <s xml:id="echoid-s2460" xml:space="preserve"/> </p> <note symbol="h" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note position="right" xml:space="preserve">Diſtantiæ in-<lb/>uentio per <lb/>ſolos ſinus.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} \\ anguli recti \\ N, vel M, # ad Hypotenuſam \\ inuentam I L, \\ vel K L, # Ita ſin{us} anguli maioris ob- \\ ſeruati NIL, vel \\ ad N L, diſtan- \\ tiam <lb/> # # Ita ſin{us} anguli minoris ob- \\ ſeruati M K L, # ad M L, diſtan- \\ tiam </note> <p> <s xml:id="echoid-s2461" xml:space="preserve"><emph style="sc">Sine</emph> numeris eædem rectæ IN, N L, IL, K L, &</s> <s xml:id="echoid-s2462" xml:space="preserve">c. </s> <s xml:id="echoid-s2463" xml:space="preserve">reperientur, vt in ſupe-<lb/> <anchor type="note" xlink:label="note-091-15a" xlink:href="note-091-15"/> rioribus: </s> <s xml:id="echoid-s2464" xml:space="preserve">ſi nimirum (vtemur figura huius problematis, ne nouam conſtruere <lb/>cogamur) ſumptarecta IK, tot partium, quot palmi, pedeſuein differentia ſta-<lb/>tionum exiſtunt, fiant anguli VKL, VIL, complementorum angulorum MKL, <lb/>NIL, obſeruationum, & </s> <s xml:id="echoid-s2465" xml:space="preserve">concurſus L, notetur, ex quo ducatur LN, ipſi IK, pa-<lb/>rallela, & </s> <s xml:id="echoid-s2466" xml:space="preserve">ad hanc perpendicularis in I, excitetur IN, &</s> <s xml:id="echoid-s2467" xml:space="preserve">c. </s> <s xml:id="echoid-s2468" xml:space="preserve">Vel ſi angulus rectus <lb/>conſtituatur INL, & </s> <s xml:id="echoid-s2469" xml:space="preserve">in aſſumpto puncto L, vbi concurſum eſſe volumus, fiat <pb o="62" file="092" n="92" rhead="GEOMETR. PRACT."/> tam angulus N L I complementi maioris anguli obſeruationis, quam angulus <lb/>NLK, complementi minoris anguli obſeruationis, &</s> <s xml:id="echoid-s2470" xml:space="preserve">c. </s> <s xml:id="echoid-s2471" xml:space="preserve">Reliqua autem fiant, vt <lb/>in problemate 1. </s> <s xml:id="echoid-s2472" xml:space="preserve">Num. </s> <s xml:id="echoid-s2473" xml:space="preserve">6. </s> <s xml:id="echoid-s2474" xml:space="preserve">& </s> <s xml:id="echoid-s2475" xml:space="preserve">8. </s> <s xml:id="echoid-s2476" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s2477" xml:space="preserve">Idemque efficies per ea, quæ Num. </s> <s xml:id="echoid-s2478" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s2479" xml:space="preserve">in eodem problemate 1. </s> <s xml:id="echoid-s2480" xml:space="preserve">ſcripſimus.</s> <s xml:id="echoid-s2481" xml:space="preserve"/> </p> <div xml:id="echoid-div143" type="float" level="2" n="2"> <note position="right" xlink:label="note-091-15" xlink:href="note-091-15a" xml:space="preserve">Quo pacto <lb/>problema <lb/>conficiatur ſi-<lb/>ne numeris.</note> </div> <p> <s xml:id="echoid-s2482" xml:space="preserve">EX vertice montis, vel turris per duas ſtationes in aliqua haſta erecta, <lb/>vel in duabus feneſtris turris, quarum vna ſupra aliam exiſtat, factas, <lb/>è quibus ſignum aliquod in Horizonte videri poſſit, altitudinem <lb/>ipſius montis, aut turris metiri. </s> <s xml:id="echoid-s2483" xml:space="preserve">Atque hinc diſtantiam quoque à <lb/>perpendiculo montis, vel turris vſque ad ſignum viſum cogno-<lb/>ſcere.</s> <s xml:id="echoid-s2484" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div145" type="section" level="1" n="62"> <head xml:id="echoid-head65" xml:space="preserve">PROBLEMA IV.</head> <p> <s xml:id="echoid-s2485" xml:space="preserve">@. </s> <s xml:id="echoid-s2486" xml:space="preserve"><emph style="sc">Non</emph> poſſunt aliquando commodè duæ ſtationes in ſummitate montis, <lb/>vel turris fieri. </s> <s xml:id="echoid-s2487" xml:space="preserve">Quare tunc ita agendum erit. </s> <s xml:id="echoid-s2488" xml:space="preserve">Sit altitu-<lb/> <anchor type="figure" xlink:label="fig-092-01a" xlink:href="fig-092-01"/> do A B, ſupra planum BC, menſuranda ex ſummitate <lb/>A. </s> <s xml:id="echoid-s2489" xml:space="preserve">Erigatur haſta aliqua quotlibet palmorum, aut pe-<lb/>dum. </s> <s xml:id="echoid-s2490" xml:space="preserve">Et primum inſpiciatur ſignum C, in plano per an-<lb/>gulum B D C; </s> <s xml:id="echoid-s2491" xml:space="preserve">Deinde idem inſpiciatur ex loco ſupe-<lb/>riore haſtæ per angulum BEC. </s> <s xml:id="echoid-s2492" xml:space="preserve">Vterque autem angulus <lb/>ob ſeruabitur vel per quadrantem pendulum, vt in prio-<lb/>ri angulo vides, vel per ſtabilem cum dio ptra, vt in po-<lb/>ſteriori. </s> <s xml:id="echoid-s2493" xml:space="preserve">Sumptis quo que æqualibus DF, EG, pro ſinu-<lb/>bus totis, ducantur ad EB, perpendiculares F H, G I, <lb/>pro Tangentibus angulorum obſeruationum. </s> <s xml:id="echoid-s2494" xml:space="preserve">Ducta <lb/>autem DL ipſi E C, parallela ſecante F H, in K: </s> <s xml:id="echoid-s2495" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quo- <anchor type="note" xlink:label="note-092-01a" xlink:href="note-092-01"/> niam anguli BDL, & </s> <s xml:id="echoid-s2496" xml:space="preserve">E, æquales ſunt, & </s> <s xml:id="echoid-s2497" xml:space="preserve">F, G, recti, nec <lb/>non & </s> <s xml:id="echoid-s2498" xml:space="preserve">latera adia centia DF, EG, æqualia: </s> <s xml:id="echoid-s2499" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erunt late- <anchor type="note" xlink:label="note-092-02a" xlink:href="note-092-02"/> ra F K, GI, æqualia: </s> <s xml:id="echoid-s2500" xml:space="preserve">atque adeo K H, differentia Tangentium. </s> <s xml:id="echoid-s2501" xml:space="preserve">Quia vero ex <lb/>ſchol. </s> <s xml:id="echoid-s2502" xml:space="preserve">propoſitionis 4. </s> <s xml:id="echoid-s2503" xml:space="preserve">lib. </s> <s xml:id="echoid-s2504" xml:space="preserve">6. </s> <s xml:id="echoid-s2505" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s2506" xml:space="preserve">eſt, vt KH, ad FK, ita LC, ad BL: </s> <s xml:id="echoid-s2507" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Vt autem <anchor type="note" xlink:label="note-092-03a" xlink:href="note-092-03"/> LC, ad BL, ita eſt, ED, ad BD; </s> <s xml:id="echoid-s2508" xml:space="preserve">erit quo que vt KH, ad FK, ita ED, ad DB. </s> <s xml:id="echoid-s2509" xml:space="preserve">Siigi-<lb/>tur fiat,</s> </p> <div xml:id="echoid-div145" 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/xxxxxxxx/figures/092-01"/> </figure> <note symbol="a" position="left" xlink:label="note-092-01" xlink:href="note-092-01a" xml:space="preserve">29. primi.</note> <note symbol="b" position="left" xlink:label="note-092-02" xlink:href="note-092-02a" xml:space="preserve">26. primi.</note> <note symbol="c" position="left" xlink:label="note-092-03" xlink:href="note-092-03a" xml:space="preserve">2. ſexti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt KH, differentia \\ inter Tangent{es} \\ angulorum ob- \\ ſeruationum # ad FK, vel ad \\ G I, Tan- \\ gentem mi- \\ norem: # Ita E D, ſpatium inter angulos obſer- \\ uationum, quod notum eſſe potest \\ per aliquam menſuram, # ad D B, <lb/></note> <p> <s xml:id="echoid-s2510" xml:space="preserve">inuenietur D B, ex qua ſi dematur portio haſtæ A D, inter altitudinis faſti-<lb/> <anchor type="note" xlink:label="note-092-05a" xlink:href="note-092-05"/> gium A, & </s> <s xml:id="echoid-s2511" xml:space="preserve">inferiorem angulum obſeruationis D, nota relinquetur altitudo pro-<lb/>poſita A B.</s> <s xml:id="echoid-s2512" xml:space="preserve"/> </p> <div xml:id="echoid-div146" type="float" level="2" n="2"> <note position="left" xlink:label="note-092-05" xlink:href="note-092-05a" xml:space="preserve">Altitudinis <lb/>inuentio.</note> </div> <p> <s xml:id="echoid-s2513" xml:space="preserve"><emph style="sc">Eodemqve</emph> modo in turri eadem altitudo deprehendetur, ſi pro haſta <lb/>A E, duæ feneſtræ eligantur, è quibus ſignum C, ſub iiſdem angulis videatur, vt <lb/>patet. </s> <s xml:id="echoid-s2514" xml:space="preserve">Sed tunc altitudini D B, inuentæ (ſi feneſtræ ſint D E,) adiicienda eſt por-<lb/>tio turris inter punctum D, feneſtræ inferioris, & </s> <s xml:id="echoid-s2515" xml:space="preserve">turris faſtigium E, vt tota alti-<lb/>tudo turris habeatur E B. </s> <s xml:id="echoid-s2516" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et ſi fiat,</s> </p> <note symbol="d" position="left" xml:space="preserve">4. ſexti.</note> <pb o="63" file="093" n="93" rhead="LIBER SECVNDVS."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt D F, ſin{us} \\ tot{us} # ad F H, Tangentem \\ maiorem: # Ita D B, altitudo inuen- \\ ta # ad B C, <lb/></note> <note position="right" xml:space="preserve">Diſtantiæ in-<lb/>uentio.</note> <p> <s xml:id="echoid-s2517" xml:space="preserve">inuenta erit diſtantia B C.</s> <s xml:id="echoid-s2518" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div148" type="section" level="1" n="63"> <head xml:id="echoid-head66" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2519" xml:space="preserve">2. </s> <s xml:id="echoid-s2520" xml:space="preserve"><emph style="sc">Posito</emph> ſinu toto C B, erit B E, Tangens anguli B C E, complementi <lb/>minoris anguli obſeruationis E, & </s> <s xml:id="echoid-s2521" xml:space="preserve">BD, Tangens anguli B C D, complementi <lb/>maioris anguli obſeruationis D; </s> <s xml:id="echoid-s2522" xml:space="preserve">ideo que D E, differentia illarum Tangentium. <lb/></s> <s xml:id="echoid-s2523" xml:space="preserve">Quamobrem ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt D E, differentia Tangen- \\ tium complementorum an- \\ gulorũ obſer uationum # ad D B, tangen- \\ tem minorem: # Ita D E, differentia ſta- \\ tionum oculorum # ad D B, <lb/></note> <p> <s xml:id="echoid-s2524" xml:space="preserve">prodibit altitudo D B, cognita ab oculo in prima ſtatione vſque ad baſem alti-<lb/> <anchor type="note" xlink:label="note-093-04a" xlink:href="note-093-04"/> tudinis, &</s> <s xml:id="echoid-s2525" xml:space="preserve">c.</s> <s xml:id="echoid-s2526" xml:space="preserve"/> </p> <div xml:id="echoid-div148" type="float" level="2" n="1"> <note position="right" xlink:label="note-093-04" xlink:href="note-093-04a" xml:space="preserve">Altitudinis <lb/>inuentio aliæ.</note> </div> </div> <div xml:id="echoid-div150" type="section" level="1" n="64"> <head xml:id="echoid-head67" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2527" xml:space="preserve">3. </s> <s xml:id="echoid-s2528" xml:space="preserve"><emph style="sc">Vt</emph> per ſolos ſinus inſtituatur operatio, inueſtiganda prius eſt vtraque <lb/>hypotenuſa C D, C E, vel alterutra earum, hoc modo. </s> <s xml:id="echoid-s2529" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam angulus <anchor type="note" xlink:label="note-093-05a" xlink:href="note-093-05"/> BDC, duobus E, & </s> <s xml:id="echoid-s2530" xml:space="preserve">DCE, æqualis eſt, ſi minor angulus obſeruationis E, ex ma-<lb/>iori angulo obſeruationis BDC, ſubtrahatur, notus relinquetur angulus DCE. <lb/></s> <s xml:id="echoid-s2531" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/>Itaque ſi fiat,</s> </p> <div xml:id="echoid-div150" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-093-05" xlink:href="note-093-05a" xml:space="preserve">29. primi.</note> </div> <note symbol="b" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli D C E differen- \\ tiæ angulorum obſeruatio- \\ num # ad D E, differen- \\ tiam ſtationum \\ oculorum: # Ita ſin{us} minoris \\ anguli obſeruatio- \\ nis E, # ad C D-<lb/># # Vel <lb/># # Ita ſin{us} anguli \\ E D C, complemen- \\ ti maioris anguli \\ obſeruationis ad \\ duos rectos, # ad C E, <lb/></note> <note position="right" xml:space="preserve">Hypotenu-<lb/>ſarum inuẽ-<lb/>tio.</note> <p> <s xml:id="echoid-s2532" xml:space="preserve">reperietur tam hypotenuſa C D, quam C E, in partibus differentiæ ſtationum <lb/>D E. </s> <s xml:id="echoid-s2533" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quapropter ſi iam fiat,</s> </p> <note symbol="c" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} anguli \\ vecti B, # ad hypotenu- \\ ſam C D, # Ita ſin{us} anguli B C D, complementi ma- \\ ioris anguli obſeruationis B D C, # ad D B, <lb/># Vel # Vel <lb/># ad hypotenuſam \\ C E, proximè \\ inuentam: # Ita ſin{us} anguli B C E, comple- \\ menti minoris anguli obſerua- \\ tionis E; # ad E B, <lb/></note> <p> <s xml:id="echoid-s2534" xml:space="preserve">cognita erit vtraque altitudo D B, E B, &</s> <s xml:id="echoid-s2535" xml:space="preserve">c. </s> <s xml:id="echoid-s2536" xml:space="preserve">Siautem fiat,</s> </p> <note position="right" xml:space="preserve">Altitudinis <lb/>inuentio per<unsure/> <lb/>ſolos ſin{us}.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} an- \\ guli recti B, # ad Hypotenuſam \\ C D, # Ita ſin{us} anguli D, maio- \\ ris obſeruati, # ad B C, <lb/># Vel # Vel <lb/># ad hypotenuſam C E, \\ nuper inuentam # Ita ſin{us} anguli E, mino- \\ ris obſeruati. # ad B C, <lb/></note> <p> <s xml:id="echoid-s2537" xml:space="preserve">c<unsure/>ognoſcetur quo que diſtantia B C, per ſolos ſinus,</s> </p> <note position="right" xml:space="preserve">4. A@s-</note> <pb o="64" file="094" n="94" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s2538" xml:space="preserve">4. </s> <s xml:id="echoid-s2539" xml:space="preserve"><emph style="sc">Absqve</emph> numeris problema efficiemus, vtin pręcedentibus, ſi nimi-<lb/> <anchor type="note" xlink:label="note-094-01a" xlink:href="note-094-01"/> rum in recta EB, ſumatur portio ED, tot partium æqualium, quot palmi pedeſ-<lb/>uein E D, differentia ſtationum oculorum exiſtunt, & </s> <s xml:id="echoid-s2540" xml:space="preserve">tam angulus E, minor <lb/>obſeruationis, quam BDC, maior conſtituatur, concurſuſque C, notetur, à quo <lb/>ad EB, perpendi cularis ducatur CB, &</s> <s xml:id="echoid-s2541" xml:space="preserve">c. </s> <s xml:id="echoid-s2542" xml:space="preserve">vel ſi angulus rectus efficiatur B, & </s> <s xml:id="echoid-s2543" xml:space="preserve">in <lb/>quouis puncto C, vbi optamus eſſe concurſum, cõſtituatur tam angulus BCD, <lb/> <anchor type="note" xlink:label="note-094-02a" xlink:href="note-094-02"/> complemento maioris anguli obſeruationis, quam angulus BCE, complemen-<lb/>to minoris anguli obſeruationis æqualis, &</s> <s xml:id="echoid-s2544" xml:space="preserve">c. </s> <s xml:id="echoid-s2545" xml:space="preserve">Reliqua autem abſoluantur, vt in <lb/>Problemate 1. </s> <s xml:id="echoid-s2546" xml:space="preserve">Num. </s> <s xml:id="echoid-s2547" xml:space="preserve">6. </s> <s xml:id="echoid-s2548" xml:space="preserve">& </s> <s xml:id="echoid-s2549" xml:space="preserve">8. </s> <s xml:id="echoid-s2550" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s2551" xml:space="preserve">Idem que efficies per ea, quæ ibid. </s> <s xml:id="echoid-s2552" xml:space="preserve">Num. <lb/></s> <s xml:id="echoid-s2553" xml:space="preserve">7. </s> <s xml:id="echoid-s2554" xml:space="preserve">explicata ſunt.</s> <s xml:id="echoid-s2555" xml:space="preserve"/> </p> <div xml:id="echoid-div151" type="float" level="2" n="2"> <note position="left" xlink:label="note-094-01" xlink:href="note-094-01a" xml:space="preserve">Diſtantiæ in-<lb/>uentio per ſo-<lb/>los ſin{us}.</note> <note position="left" xlink:label="note-094-02" xlink:href="note-094-02a" xml:space="preserve">Problematis <lb/>ſoluti@ ſine <lb/>numeris.</note> </div> </div> <div xml:id="echoid-div153" type="section" level="1" n="65"> <head xml:id="echoid-head68" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s2556" xml:space="preserve"><emph style="sc">Igitvr</emph> ſi diſtantia ſigni ex turre viſi vſque ad turrim nota fuerit, nimirum <lb/>recta CB, rep erietur altitudo turris per vnicam ſtationem in faſtigio A factam: <lb/></s> <s xml:id="echoid-s2557" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> ſi videlicet fiat,</s> </p> <note symbol="a" position="left" xml:space="preserve">4. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} C B, # ad B D, Tangentem anguli BCD, qui comple- \\ mentum eſt anguli obſeruationis ‘D, # Ita diſt antia \\ nota C B, # ad B D, <lb/></note> <p> <s xml:id="echoid-s2558" xml:space="preserve">Et ſi ex inuenta B D, auferatur menſoris ſtatura A D, nota relin quetur altitudo <lb/>turris B A.</s> <s xml:id="echoid-s2559" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Altitudinis <lb/>inuentio per <lb/>vnicam ſta-<lb/>tionem quan-<lb/>do diſtantia <lb/>@ota est.</note> </div> <div xml:id="echoid-div154" type="section" level="1" n="66"> <head xml:id="echoid-head69" xml:space="preserve">VEL PER SOLOS SINVS.</head> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli ob- \\ ſeruationis D, # ad diſtantiam \\ C B, notam # Ita ſin{us} anguli B C D, complemen- \\ ti anguli obſeruationis D. # ad B D <lb/></note> <p> <s xml:id="echoid-s2560" xml:space="preserve"><emph style="sc">Qvod</emph> ſi oculus D, ſtatuatur in aliqua feneſtra turris, adiicienda erit portio <lb/>turris inter oculum, & </s> <s xml:id="echoid-s2561" xml:space="preserve">faſtigium ad altitudinem DB, inuentam. </s> <s xml:id="echoid-s2562" xml:space="preserve">Ita namq; </s> <s xml:id="echoid-s2563" xml:space="preserve">con-<lb/>ficietur tota altitudo turris E B, ſi faſtigium ſit E, vt perſpicuum eſt.</s> <s xml:id="echoid-s2564" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2565" xml:space="preserve">EX VERTICE MONTIS, AVT TVRRIS ALTITVDI-<lb/>nem ipſius, ſi in plano, cui inſiſtit, ſpatium aliquod è directo menſoris <lb/>notum ſit, deprehendere.</s> <s xml:id="echoid-s2566" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div155" type="section" level="1" n="67"> <head xml:id="echoid-head70" xml:space="preserve">PROBLEMA V.</head> <p> <s xml:id="echoid-s2567" xml:space="preserve">1. </s> <s xml:id="echoid-s2568" xml:space="preserve"><emph style="sc">Qvando</emph> ſpatium aliquod D E, è directo <lb/> <anchor type="figure" xlink:label="fig-094-01a" xlink:href="fig-094-01"/> menſoris à monte vel turri remotum fuerit notum, <lb/>metiemur ipſam altitudinem F G, è faſtigio G, hac <lb/>r<unsure/>atione. </s> <s xml:id="echoid-s2569" xml:space="preserve">Inſpiciantur termini D, E, per angulos <lb/>FGD, FGE, ſiue per Quadrantem, pendulum, ſiue <lb/>ſiue per ſtabilem. </s> <s xml:id="echoid-s2570" xml:space="preserve">Et quoniam, poſito ſinu toto <lb/>GF, Tangentes angulorum obſeruationum ſunt <lb/>D F, E F, ipſarumque differentia per ſpatium propoſitum: </s> <s xml:id="echoid-s2571" xml:space="preserve">Si fiat,</s> </p> <div xml:id="echoid-div155" type="float" level="2" n="1"> <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/xxxxxxxx/figures/094-01"/> </figure> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt D E, differentia Tangentium angu- \\ gulis obſeruationum debitarum # ad ſinum to- \\ tum G F: # Ita D E, ſpatium \\ notum # ad G F, <lb/></note> <p> <s xml:id="echoid-s2572" xml:space="preserve">manifeſta erit altitudo G F, quæſita in partibus ſpatij noti D E,</s> </p> <pb o="65" file="095" n="95" rhead="LIBER SECVNDVS."/> </div> <div xml:id="echoid-div157" type="section" level="1" n="68"> <head xml:id="echoid-head71" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s2573" xml:space="preserve">2. </s> <s xml:id="echoid-s2574" xml:space="preserve">Per ſolos ſinus eandem altitudinem G F, adipiſcemur, ſi prius hypote-<lb/>nuſam G D, venabimur, hoc modo. </s> <s xml:id="echoid-s2575" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Fiat,</s> </p> <note symbol="a" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli D G E, diffe- \\ rentiæ angulorum obſer- \\ uationum # ad D E, ſpa- \\ tium cogni- \\ tum # Ita ſin{us} anguli E, com- \\ plementi maioris an- \\ guli obſeruationis # ad hypotenu- \\ ſam G D, <lb/></note> <p> <s xml:id="echoid-s2576" xml:space="preserve">Numerus enim productus dabit hypotenuſam GD, in partibus ſpatii DE, no-<lb/>tam. </s> <s xml:id="echoid-s2577" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Si ergo rurſus fiat,</s> </p> <note symbol="b" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} an- \\ guli recti F, # ad hypotenuſam \\ G D, proximè \\ inuentam: # Ita ſin{us} anguli D, comple- \\ menti minoris anguli ob- \\ ſeruationis # ad G F, altitu- \\ dinem, <lb/></note> <p> <s xml:id="echoid-s2578" xml:space="preserve">producetur altitudo GF, in partibus hypotenuſæ GD, ſiue ſpatij DE, nota.</s> <s xml:id="echoid-s2579" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Solutio pro-<lb/>blematis ſine <lb/>numeris.</note> <p> <s xml:id="echoid-s2580" xml:space="preserve">3. </s> <s xml:id="echoid-s2581" xml:space="preserve"><emph style="sc">Sine</emph> numeris agendum erit, vt in problemate 1. </s> <s xml:id="echoid-s2582" xml:space="preserve">declaratum eſt.</s> <s xml:id="echoid-s2583" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2584" xml:space="preserve">DISTANTIAM ab oculo, vel pede menſoris ad quoduis punctum <lb/>in aliqua altitudine notatum, per duas ſtationes in plano factas me-<lb/>tiri.</s> <s xml:id="echoid-s2585" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div158" type="section" level="1" n="69"> <head xml:id="echoid-head72" xml:space="preserve">PROBLEMA VI.</head> <p> <s xml:id="echoid-s2586" xml:space="preserve">1 <emph style="sc">Sit</emph> inqui@enda diſtantia puncti A, in muro G H, ſiue perpendiculari ad <lb/>Horizontem, ſiue inclinato, vel etiam in tecto quo piam; </s> <s xml:id="echoid-s2587" xml:space="preserve">ab oculo B, vel pe-<lb/>de C, poſita ſtatura menſoris B C. </s> <s xml:id="echoid-s2588" xml:space="preserve">Concipiatur ducta B D, ipſi plano C E, pa-<lb/>ſallela, ſitque primo punctum A, altius, <lb/> <anchor type="figure" xlink:label="fig-095-01a" xlink:href="fig-095-01"/> quam oculus B. </s> <s xml:id="echoid-s2589" xml:space="preserve">Inſpecto puncto A, no-<lb/>tetur angulus ABD, quem latus pinnaci-<lb/>diorum in quadrante pendulo, vel linea <lb/>fiduciæ in dioptra Quadrantis ſtabilis <lb/>cum recta BD, facit. </s> <s xml:id="echoid-s2590" xml:space="preserve">Deinde accede ver-<lb/>ſrus punctum A, quotlibet palmis aut pe-<lb/>dibus vſque ad D, vt BD, differentia ſta-<lb/>tionum nota ſit. </s> <s xml:id="echoid-s2591" xml:space="preserve">Rurſumque ex D, pun-<lb/>ctum A, inſpiciatur, notato diligenter angulo ADF: </s> <s xml:id="echoid-s2592" xml:space="preserve">exiſtentque rectæ AB, AD, <lb/>B D, B C, D E, in eodem plano, in eo videlicet, quod per ſtaturas menſoris B C, <lb/>D E, & </s> <s xml:id="echoid-s2593" xml:space="preserve">per punctum A, ducitur, <anchor type="note" xlink:href="" symbol="c"/> Et quia angulus A D F, duobus B, & </s> <s xml:id="echoid-s2594" xml:space="preserve">B A D, <anchor type="note" xlink:label="note-095-06a" xlink:href="note-095-06"/> eſt æqualis: </s> <s xml:id="echoid-s2595" xml:space="preserve">ſi angulus B, in remotiore ſtatione auferatur ex angulo D, in pro-<lb/>pinquiore, notus fiet reliquus angulus A. </s> <s xml:id="echoid-s2596" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Siigitur fiat,</s> </p> <div xml:id="echoid-div158" type="float" level="2" n="1"> <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/xxxxxxxx/figures/095-01"/> </figure> <note symbol="c" position="right" xlink:label="note-095-06" xlink:href="note-095-06a" xml:space="preserve">32. primi.</note> </div> <note symbol="d" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli \\ A, quo angu- \\ l{us} minor B, à \\ maiori differt, # ad B D, diffe- \\ rentiam ſta- \\ tionum: # Ita ſin{us} anguli A D B, qui comple- \\ mentum eſt maioris anguli D, ad \\ duos rectos, # ad A B, di- \\ ſtantiã quæ- \\ ſitam, <lb/></note> <p> <s xml:id="echoid-s2597" xml:space="preserve">reperta erit quæſita diſtantia A B, in partibus differentiæ ſtationum.</s> <s xml:id="echoid-s2598" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2599" xml:space="preserve"><emph style="sc">Qvod</emph> ſi oculus exiſtat in D, & </s> <s xml:id="echoid-s2600" xml:space="preserve">recedatur à puncto D, vſque ad B, inue-<lb/>nietur eadem arte diſtantia D A, ſi loco anguli A D B, ſumatur angulus B, in <lb/> <anchor type="note" xlink:label="note-095-09a" xlink:href="note-095-09"/> remotiore ſtatione, vt liquet. </s> <s xml:id="echoid-s2601" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Eſt enim vt ſinus anguli A, quo angulus mi- <pb o="68" file="096" n="96" rhead="GEOMETR. PRACT."/> nor B, à maiori ADF, differt, ad BD, differentiam ſtationum: </s> <s xml:id="echoid-s2602" xml:space="preserve">ita ſinus anguli B, <lb/>in remotiori ſtatione. </s> <s xml:id="echoid-s2603" xml:space="preserve">ad D A.</s> <s xml:id="echoid-s2604" xml:space="preserve"/> </p> <div xml:id="echoid-div159" type="float" level="2" n="2"> <note symbol="e" position="right" xlink:label="note-095-09" xlink:href="note-095-09a" xml:space="preserve">10. triang, <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s2605" xml:space="preserve">2. </s> <s xml:id="echoid-s2606" xml:space="preserve"><emph style="sc">Distantia</emph> verò à puncto A, ad pedem menſoris C, hoc eſt, recta AC, <lb/>cognoſcetur per Problema 12. </s> <s xml:id="echoid-s2607" xml:space="preserve">triang. </s> <s xml:id="echoid-s2608" xml:space="preserve">rectil. </s> <s xml:id="echoid-s2609" xml:space="preserve">cap. </s> <s xml:id="echoid-s2610" xml:space="preserve">3. </s> <s xml:id="echoid-s2611" xml:space="preserve">lib. </s> <s xml:id="echoid-s2612" xml:space="preserve">1. </s> <s xml:id="echoid-s2613" xml:space="preserve">cum in triangulo obli-<lb/>quangulo ABC, duo latera AB, BC, nota ſint, nimirum diſtantia inuenta, & </s> <s xml:id="echoid-s2614" xml:space="preserve">ſta-<lb/>tura menſoris, comprehendantque angulum ABC, notum, vtpote conflatum <lb/>ex recto CBD, & </s> <s xml:id="echoid-s2615" xml:space="preserve">angulo obſeruationis ABD.</s> <s xml:id="echoid-s2616" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2617" xml:space="preserve">3. </s> <s xml:id="echoid-s2618" xml:space="preserve"><emph style="sc">Sit</emph> deinde punctum A, vt in muro HI, infra oculum B. </s> <s xml:id="echoid-s2619" xml:space="preserve">Inſpecto pun-<lb/>cto A, obſeruetur angulus CBA. </s> <s xml:id="echoid-s2620" xml:space="preserve">quem latus pinnacidiorum cum perpendiculi <lb/>filo, vel dio ptræ linea fiduciæ cum BC, facit: </s> <s xml:id="echoid-s2621" xml:space="preserve">Deinde accede verſus A. </s> <s xml:id="echoid-s2622" xml:space="preserve">vſque ad <lb/>D, & </s> <s xml:id="echoid-s2623" xml:space="preserve">iterum conſidera angulum EDA: </s> <s xml:id="echoid-s2624" xml:space="preserve">exiſtentque rectæ AB, AD BD, BC, DE, <lb/>in vno eodemque plano, in eo videlicet, quod per ſta-<lb/> <anchor type="figure" xlink:label="fig-096-01a" xlink:href="fig-096-01"/> turas menſoris BC, DE, & </s> <s xml:id="echoid-s2625" xml:space="preserve">per punctum A, ducitur. <lb/></s> <s xml:id="echoid-s2626" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> Et quoniam angulus FDA, complementi anguli ob- <anchor type="note" xlink:label="note-096-01a" xlink:href="note-096-01"/> ſeruationis in propinquiore ſtatione æqualis eſt duo-<lb/>bus DBA, DAB; </s> <s xml:id="echoid-s2627" xml:space="preserve">ſi DBA, angulus complementi an-<lb/>guli remotioris ſtationis dematur ex angulo A D F, <lb/>complementi anguli ſtationis propinquioris, reliquus <lb/>fiet notus BAD. </s> <s xml:id="echoid-s2628" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Si ergo fiat,</s> </p> <div xml:id="echoid-div160" type="float" level="2" n="3"> <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/xxxxxxxx/figures/096-01"/> </figure> <note symbol="a" position="left" xlink:label="note-096-01" xlink:href="note-096-01a" xml:space="preserve">32. primi.</note> </div> <note symbol="b" position="left" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus anguli BAD, dif- \\ ferentiæ inter angulos com \\ plementorum angulorum \\ obſeruationum # ad B D, diffe- \\ rentiam ſta- \\ tionum: # Ita ſinus anguli ADB, con- \\ flati ex recto B D E, & \\ ex angulo obſeruationis \\ A D E, in propinquiore \\ ſtatione # ad AB di- \\ ſtantiam \\ quæſitã. <lb/></note> <p> <s xml:id="echoid-s2629" xml:space="preserve">cognita erit diſtantia A B, quam quærimus, in partibus differentiæ ſtatio-<lb/>num B D.</s> <s xml:id="echoid-s2630" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2631" xml:space="preserve"><emph style="sc">Qvod</emph> ſi oculus ponatur in D, & </s> <s xml:id="echoid-s2632" xml:space="preserve">recedatur à puncto D, vſque ad B, repe-<lb/>rietur eodem modo diſtantia DA, ſi pro angulo BDA, aſſumes angulum DBA, <lb/>complementianguli ABC, obſeruationis in remotiore ſtatione, vt manifeſtum <lb/>eſt. </s> <s xml:id="echoid-s2633" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Nam eſt, vt ſinus anguli BAD, differentiæ inter angulos complemento- <anchor type="note" xlink:label="note-096-04a" xlink:href="note-096-04"/> rum angulorum obſeruationum, ad BD, differentiam ſtationum: </s> <s xml:id="echoid-s2634" xml:space="preserve">ita ſinus angu-<lb/>li DBA, complementi anguli ABC, in remotiore ſtatione, ad DA.</s> <s xml:id="echoid-s2635" xml:space="preserve"/> </p> <div xml:id="echoid-div161" type="float" level="2" n="4"> <note symbol="c" position="left" xlink:label="note-096-04" xlink:href="note-096-04a" xml:space="preserve">10. Triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s2636" xml:space="preserve">4. </s> <s xml:id="echoid-s2637" xml:space="preserve"><emph style="sc">Vt</emph> autem diſtantia CA, à pede ad punctum A, inueniatur, ita progredie-<lb/>mur. </s> <s xml:id="echoid-s2638" xml:space="preserve">Quoniam in triangulo rectangulo ABG, (ſi ex puncto A, concipiatur <lb/>ducta ad BC, ſtaturam menſoris perpendicularis AG,) baſis AB, nota eſt per in-<lb/>uentionem, & </s> <s xml:id="echoid-s2639" xml:space="preserve">angulus BAG, notus, quippe cum ſit complementum anguli <lb/>obſeruationis ABG; </s> <s xml:id="echoid-s2640" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Si fiat,</s> </p> <note symbol="d" position="left" xml:space="preserve">2. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus \\ totus # ad baſem A B, proximè \\ inuentam: # Ita ſinus anguli B A G, complemen- \\ ti anguli obſeruationis, # ad B G, <lb/></note> <p> <s xml:id="echoid-s2641" xml:space="preserve">cognoſcetur BG, in partibus baſis AB, hoc eſt, in partibus differentiæ ſtationum <lb/>BD, in quibus AB, inuenta fuit. </s> <s xml:id="echoid-s2642" xml:space="preserve">Ablata autem BG, ex menſoris ſtatura BC, no-<lb/>ta fiet reliqua CG. </s> <s xml:id="echoid-s2643" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Item ſi fiat,</s> </p> <note symbol="e" position="left" xml:space="preserve">2. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus to- \\ tus # ad baſem A B, nu- \\ per inuentam: # Ita ſinus anguli obſeruatio- \\ nis A B G, # ad A G, <lb/></note> <p> <s xml:id="echoid-s2644" xml:space="preserve">nota etiam fiet A G, in partibus eiuſdem baſis A B, vel differentiæ ſtationum <pb o="67" file="097" n="97" rhead="LIBER SECVNDVS."/> BD. </s> <s xml:id="echoid-s2645" xml:space="preserve">Quia ergo in triangulo rectangulo ACG, duo latera AG, GC, per inuen-<lb/>tionem nota fa cta ſunt; </s> <s xml:id="echoid-s2646" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cognoſcetur quoq; </s> <s xml:id="echoid-s2647" xml:space="preserve">baſis AC, quod eſt propoſitum.</s> <s xml:id="echoid-s2648" xml:space="preserve"/> </p> <note symbol="a" position="right" xml:space="preserve">3. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2649" xml:space="preserve">5. </s> <s xml:id="echoid-s2650" xml:space="preserve"><emph style="sc">Manifestvm</emph> autem eſt, eodem pacto vtramque diſtantiam AB, AC, <lb/>reperiri, etiamſi punctum A, in plano ſit, in quo menſor conſiſtit, nimirum in <lb/>Horizonte, qui ponatur tranſire per rectam AG, ita vt ſtatura menſoris ſit B G, <lb/>Immo tunc per vnicam ſtationem vtraque diſtantia AB, AG, rep erietur. </s> <s xml:id="echoid-s2651" xml:space="preserve">Nam <lb/>poſito ſinu toto B G, ſtatura menſoris; </s> <s xml:id="echoid-s2652" xml:space="preserve">A B, ſecans eſt anguli obſeruationis <lb/>ABG, & </s> <s xml:id="echoid-s2653" xml:space="preserve">AG, Tangens. </s> <s xml:id="echoid-s2654" xml:space="preserve">Quo circa ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt BG, ſinus \\ totus # ad B A, ſecantem anguli obſer- \\ @ationis A B G, # Ita BG, ſtatura \\ menſoris # ad B A, <lb/>#### Vel <lb/># ad GA, Tangentem anguli ob- \\ ſeruationis A B G: # # ad G A, <lb/></note> <p> <s xml:id="echoid-s2655" xml:space="preserve">vtraque diſtantia & </s> <s xml:id="echoid-s2656" xml:space="preserve">BA, ab oculo B, & </s> <s xml:id="echoid-s2657" xml:space="preserve">G A, à G, pede menſoris cognita fiet.</s> <s xml:id="echoid-s2658" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2659" xml:space="preserve">6. </s> <s xml:id="echoid-s2660" xml:space="preserve"><emph style="sc">Iam</emph> verò ſine numeris operabimur, vt in præcedentibus, vt manifeſtum <lb/> <anchor type="note" xlink:label="note-097-03a" xlink:href="note-097-03"/> eſt, ſi rectè figura conſtruatur, quemadmodum Num. </s> <s xml:id="echoid-s2661" xml:space="preserve">6. </s> <s xml:id="echoid-s2662" xml:space="preserve">& </s> <s xml:id="echoid-s2663" xml:space="preserve">8. </s> <s xml:id="echoid-s2664" xml:space="preserve">problematis 1. </s> <s xml:id="echoid-s2665" xml:space="preserve">di-<lb/>ctum eſt.</s> <s xml:id="echoid-s2666" xml:space="preserve"/> </p> <div xml:id="echoid-div162" type="float" level="2" n="5"> <note position="right" xlink:label="note-097-03" xlink:href="note-097-03a" xml:space="preserve">Quo pacto ꝓ-<lb/>blema ſine nu-<lb/>meris ſit ſol-<lb/>uendum.</note> </div> <p> <s xml:id="echoid-s2667" xml:space="preserve">INTER VALLVM inter duo puncta in quolibet plano eleuato, ſiuc <lb/>illud ad Horizontem rectum ſit, ſiue inclinatum, metiri.</s> <s xml:id="echoid-s2668" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div164" type="section" level="1" n="70"> <head xml:id="echoid-head73" xml:space="preserve">PROBLEMA VII.</head> <p> <s xml:id="echoid-s2669" xml:space="preserve">1. </s> <s xml:id="echoid-s2670" xml:space="preserve"><emph style="sc">In</emph> plano quolibet eleuato AB. </s> <s xml:id="echoid-s2671" xml:space="preserve">prop oſitum ſit interuallum CD, quod <lb/>ex plano EB, inueſtigandum ſit. </s> <s xml:id="echoid-s2672" xml:space="preserve">Poſito oculo in G, vt ſtatura menſoris ſit, GE, <lb/>inueſtiget ur per præcedens problema 6. </s> <s xml:id="echoid-s2673" xml:space="preserve">vtraque diſtantia G C, G D, in parti-<lb/>bus ſtaturæ menſoris G E, ſiue differentiæ <lb/> <anchor type="figure" xlink:label="fig-097-01a" xlink:href="fig-097-01"/> duarum ſtationum, è quibus ipſæ diſtan-<lb/>tiæ inueſtigantur. </s> <s xml:id="echoid-s2674" xml:space="preserve">Deinde applicato Qua-<lb/>drante ſtabili ad oculum G, ita vt eius pla-<lb/>num per puncta CD, tranſeat, & </s> <s xml:id="echoid-s2675" xml:space="preserve">vna eius <lb/>ſemidiameter ad punctum D, vergat, <lb/>(quod fiet, ſi poſita linea fiduciæ dioptræ <lb/>ſupra illam ſemidiametrum, punctum D, <lb/>per foramina pinnacidiorum conſpicia-<lb/>tur) vertatur dioptra, donec per eam punctum C, appareat, arcuſque in-<lb/>ter dictam ſemidiametrum, & </s> <s xml:id="echoid-s2676" xml:space="preserve">lineam fiduciæ interceptus notetur. </s> <s xml:id="echoid-s2677" xml:space="preserve">hic enim <lb/>angulum G, metietur. </s> <s xml:id="echoid-s2678" xml:space="preserve">Quod ſi altera ſemidiameter Quadrantis vltra re-<lb/>ctam G C, exiſtat, erit angulus acutus C G D: </s> <s xml:id="echoid-s2679" xml:space="preserve">Si verò altera illa ſemidia-<lb/>meter citra rectam G C, extiterit, dictus angulus erit obtuſus, qui cogno-<lb/>ſcetur, ſi ad rectum adiiciatur reliquus angulus inter alteram illam ſemidia-<lb/>metrum, & </s> <s xml:id="echoid-s2680" xml:space="preserve">rectam G C; </s> <s xml:id="echoid-s2681" xml:space="preserve">quem quidem reliquum inueſtigabimus per <lb/>Quadrantem, vt de acuto C G D, diximus, ſi videlicet in recta C D, <lb/>mente notemus punctum, in quod altera illa ſemidiameter incurre-<lb/>ret producta. </s> <s xml:id="echoid-s2682" xml:space="preserve">Si namque tunc ſemidiameter illa rectæ G C, congruat, <pb o="68" file="098" n="98" rhead="GEOMETR. PRACT."/> & </s> <s xml:id="echoid-s2683" xml:space="preserve">dioptra ad illud punctum mente notatum dirigatur, indicabunt gradus in-<lb/>ter illam ſemidiametrum, & </s> <s xml:id="echoid-s2684" xml:space="preserve">dioptram prædictum angulum reliquum. </s> <s xml:id="echoid-s2685" xml:space="preserve">Si deni-<lb/>que altera illa ſemidiameter præcisè in C, tendat, angulus C G D, rectus erit. <lb/></s> <s xml:id="echoid-s2686" xml:space="preserve">Quia ergo in triangulo G C D, obliquangulo latera nota G C, G D, continent <lb/>angulum notum G; </s> <s xml:id="echoid-s2687" xml:space="preserve">cognoſcetur latus C D, per problema 12. </s> <s xml:id="echoid-s2688" xml:space="preserve">triang. </s> <s xml:id="echoid-s2689" xml:space="preserve">rectil. </s> <s xml:id="echoid-s2690" xml:space="preserve"><lb/>cap. </s> <s xml:id="echoid-s2691" xml:space="preserve">3. </s> <s xml:id="echoid-s2692" xml:space="preserve">lib. </s> <s xml:id="echoid-s2693" xml:space="preserve">1.</s> <s xml:id="echoid-s2694" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="1"> <figure xlink:label="fig-097-01" xlink:href="fig-097-01a"> <image file="097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/097-01"/> </figure> </div> <p> <s xml:id="echoid-s2695" xml:space="preserve">2. </s> <s xml:id="echoid-s2696" xml:space="preserve"><emph style="sc">Absqve</emph> numeris facile problema ſoluetur, ſi fiat angulus G, æqualis <lb/>ei, qui per Quadrantem obſeruatus fuit, & </s> <s xml:id="echoid-s2697" xml:space="preserve">in rectis G C, GD, ex inſtrumento <lb/> <anchor type="note" xlink:label="note-098-01a" xlink:href="note-098-01"/> partium tot particulæ ſumantur, quot palmi, aut pedes in diſtantiis G C, GD, <lb/>inuenti ſunt, & </s> <s xml:id="echoid-s2698" xml:space="preserve">c.</s> <s xml:id="echoid-s2699" xml:space="preserve"/> </p> <div xml:id="echoid-div165" type="float" level="2" n="2"> <note position="left" xlink:label="note-098-01" xlink:href="note-098-01a" xml:space="preserve">Problematis <lb/>ſolutio ſine <lb/>numeris.</note> </div> <p> <s xml:id="echoid-s2700" xml:space="preserve">LONGITVDINEM lineæ rectę, quando menſor in vno eius extre-<lb/>mo, vel in aliqua altitudine nota, quę perpendicularis ſit in eo extre-<lb/>mo ad planum, in quo linea iacet, exiſtens alterum extremum videre <lb/>poteſt, per Quadrantem comprehendere.</s> <s xml:id="echoid-s2701" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div167" type="section" level="1" n="71"> <head xml:id="echoid-head74" xml:space="preserve">PROBLEMA VIII.</head> <p> <s xml:id="echoid-s2702" xml:space="preserve">1. </s> <s xml:id="echoid-s2703" xml:space="preserve"><emph style="sc">Sit</emph> exquirenda longitudo A B, hoc eſt, diſtan-<lb/> <anchor type="figure" xlink:label="fig-098-01a" xlink:href="fig-098-01"/> tia inter A, & </s> <s xml:id="echoid-s2704" xml:space="preserve">B, etiamſi puncta intermedia ſiue propter <lb/>tumores interiectos, ſiue propter valles, cerninequeant, <lb/>dummodo in extremo A, exiſtens menſor, vel in aliqua <lb/>altitudine cognita ad planum, in quo linea A B, perpen-<lb/>diculari, ita vt A C, ſit vel ſtatura menſoris, vel haſta ali-<lb/>qua erecta, vel turris. </s> <s xml:id="echoid-s2705" xml:space="preserve">Inſpecto extremo B, obſeruetur angulus C. </s> <s xml:id="echoid-s2706" xml:space="preserve">Et quia po-<lb/>ſito ſinu toto AC, diſtantia AB, eſt Tangens anguli obſeruati C: </s> <s xml:id="echoid-s2707" xml:space="preserve">ſi fiat,</s> </p> <div xml:id="echoid-div167" type="float" level="2" n="1"> <figure xlink:label="fig-098-01" xlink:href="fig-098-01a"> <image file="098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/098-01"/> </figure> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus totus \\ AC, # ad AB, tangentem anguli \\ obſeruati C. # Ita AC, ſtatura menſoris, \\ vel altitudo nota, # Ad AB, longi- \\ tudinem. <lb/></note> <p> <s xml:id="echoid-s2708" xml:space="preserve">procreabitur longitudo A B, in partibus altitudinis notæ A C. </s> <s xml:id="echoid-s2709" xml:space="preserve">Quæ per ſolos <lb/> <anchor type="note" xlink:label="note-098-03a" xlink:href="note-098-03"/> ſinus etiam producetur, <anchor type="note" xlink:href="" symbol="a"/> ſi fiat,</s> </p> <div xml:id="echoid-div168" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-098-03" xlink:href="note-098-03a" xml:space="preserve">4. Triang. <lb/>rectil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus anguli B, com- \\ plenti anguli C, ob- \\ ſeruati # Ad AC, ſtaturam menſo- \\ ris, vel altitudinem no- \\ tam # Ita ſinus angu- \\ li C, obſerua- \\ uationis # ad AB, longi- \\ tudinem <lb/></note> <p> <s xml:id="echoid-s2710" xml:space="preserve">2. </s> <s xml:id="echoid-s2711" xml:space="preserve"><emph style="sc">Sine</emph> numeris eadem longitudo AB, cognoſcetur, vt in præcedentibus <lb/> <anchor type="note" xlink:label="note-098-05a" xlink:href="note-098-05"/> dictum eſt: </s> <s xml:id="echoid-s2712" xml:space="preserve">ſi videlicet ex inſtrumento partium accipiatur A C, tot particula-<lb/>rum, quot palmi, pedeſue in altitudine AC, exiſtunt, conſtituatur que angulus <lb/>obſeruationis C, ac tandem ad AC, perpendicularis excitetur AB, & </s> <s xml:id="echoid-s2713" xml:space="preserve">c.</s> <s xml:id="echoid-s2714" xml:space="preserve"/> </p> <div xml:id="echoid-div169" type="float" level="2" n="3"> <note position="left" xlink:label="note-098-05" xlink:href="note-098-05a" xml:space="preserve">Solutio pro-<lb/>blematis ſine <lb/>numeris.</note> </div> <p> <s xml:id="echoid-s2715" xml:space="preserve">LONGITVDINEM, ad cuius extrema accedere non liceat, dum-<lb/>modo ea appareant, & </s> <s xml:id="echoid-s2716" xml:space="preserve">ipſa longitudo producta ad pedes menſoris <lb/>pertingat, ex altitudine aliqua nota dim@tiri:</s> <s xml:id="echoid-s2717" xml:space="preserve"/> </p> <pb o="69" file="099" n="99" rhead="LIBER SECVNDVS."/> </div> <div xml:id="echoid-div171" type="section" level="1" n="72"> <head xml:id="echoid-head75" xml:space="preserve">PROBLEMA IX.</head> <p> <s xml:id="echoid-s2718" xml:space="preserve">2. </s> <s xml:id="echoid-s2719" xml:space="preserve"><emph style="sc">Sit</emph> longitudo AB, è directo menſoris in C, exiſtentis, ita vt recta B A, <lb/>pro ducta tranſeat per C. </s> <s xml:id="echoid-s2720" xml:space="preserve">Sit quo que CD, vel ſta-<lb/> <anchor type="figure" xlink:label="fig-099-01a" xlink:href="fig-099-01"/> tura menſoris, vel altitudo quæpiam nota. </s> <s xml:id="echoid-s2721" xml:space="preserve">Siigi-<lb/>tur per præcedens problema 8. </s> <s xml:id="echoid-s2722" xml:space="preserve">inquiratur vtra que <lb/>longitudo CB, CA, & </s> <s xml:id="echoid-s2723" xml:space="preserve">CA, ex CB, detrahatur, reli-<lb/>qua fiet AB, ac proinde cognita.</s> <s xml:id="echoid-s2724" xml:space="preserve"/> </p> <div xml:id="echoid-div171" type="float" level="2" n="1"> <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/xxxxxxxx/figures/099-01"/> </figure> </div> </div> <div xml:id="echoid-div173" type="section" level="1" n="73"> <head xml:id="echoid-head76" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s2725" xml:space="preserve">2. </s> <s xml:id="echoid-s2726" xml:space="preserve"><emph style="sc">Posito</emph> ſinu toto CD, ſi termini A, B, per angulos CDA, CDB, ſpectẽ-<lb/>tur, erit C A, Tangens minoris anguli, & </s> <s xml:id="echoid-s2727" xml:space="preserve">C B, maioris, at A B, differentia earum <lb/>Tangentium. </s> <s xml:id="echoid-s2728" xml:space="preserve">Quare ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus to- \\ tus CD, # Ad AB, differentiam Tangentium \\ angulorum obſeruationum # Ita C D, alti- \\ tudo nota # Ad AB, lon- \\ gitudinem. <lb/></note> <p> <s xml:id="echoid-s2729" xml:space="preserve">efficietur longitudo AB, nota in partibus altitudinis notæ C D.</s> <s xml:id="echoid-s2730" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div174" type="section" level="1" n="74"> <head xml:id="echoid-head77" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s2731" xml:space="preserve">3. </s> <s xml:id="echoid-s2732" xml:space="preserve"><emph style="sc">Per</emph> ſolos ſinus eandem longitudinem A B, cognoſcemus: </s> <s xml:id="echoid-s2733" xml:space="preserve">ſed prius in-<lb/>uenienda eſt AD, <anchor type="note" xlink:href="" symbol="a"/> hac ratione fiat,</s> </p> <note symbol="a" position="right" xml:space="preserve">5. triang. re-<lb/>ctil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus anguli CAD, comple- \\ menti minoris anguli obſerua- \\ tionis # ad C D, altitudi- \\ nem notam: # Ita ſinus totus \\ anguli recti C, # ad A D, <lb/></note> <p> <s xml:id="echoid-s2734" xml:space="preserve">Productus enim numerus dabit A D, notam in partibus altitudinis notæ CD, <anchor type="note" xlink:href="" symbol="b"/> <anchor type="note" xlink:label="note-099-04a" xlink:href="note-099-04"/> Siergo rurſus fiat,</s> </p> <div xml:id="echoid-div174" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-099-04" xlink:href="note-099-04a" xml:space="preserve">10. triang re-<lb/>ctil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſinus anguli CBD, com- \\ plementi maioris anguli \\ obſeruationis # Ad A D, pro- \\ xime inuen- \\ tam # Ita ſinus anguli A D B, \\ differentia inter duos an- \\ gulos obſeruatos # ad AB, \\ longi- \\ tudinẽ, <lb/></note> <p> <s xml:id="echoid-s2735" xml:space="preserve">prodibit nota longitudo AB, in partibus rectę AD, hoc eſt, altitudinis notæ CD, <lb/>in quibus recta AD, fuit inuenta.</s> <s xml:id="echoid-s2736" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2737" xml:space="preserve">4. </s> <s xml:id="echoid-s2738" xml:space="preserve"><emph style="sc">Sine</emph> numeris rem perficies, quemadmodum in præcedentibus, vt li-<lb/>quet.</s> <s xml:id="echoid-s2739" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2740" xml:space="preserve">LONGITVDINEM tranſuerſam in Horizonte, cuius vtrum{q́ue} ex-<lb/>tremum inſpici poteſt, notam efficere.</s> <s xml:id="echoid-s2741" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div176" type="section" level="1" n="75"> <head xml:id="echoid-head78" xml:space="preserve">PROBLEMA X.</head> <p> <s xml:id="echoid-s2742" xml:space="preserve">1. </s> <s xml:id="echoid-s2743" xml:space="preserve"><emph style="sc">Sit</emph> planum Horizontis AB, in quo jaceat longitudo CD, in tranſuerſum, <lb/>pes autem menſoris ſit in E, ita vt recta DC, per pedes menſoris in E, non tranſe-<lb/>at. </s> <s xml:id="echoid-s2744" xml:space="preserve">Quando namque longitudo D C, è directo menſoris ſita eſt, inueſtigabitur <lb/>ea per problema 9. </s> <s xml:id="echoid-s2745" xml:space="preserve">præcedens. </s> <s xml:id="echoid-s2746" xml:space="preserve">Vt ergo tranuserſa longitudo CD, nota effi-<lb/>ciatur, inueſtiganda primum erit vtriuſque puncti extremi C, D, diſtantia à pede <lb/>menſoris E, & </s> <s xml:id="echoid-s2747" xml:space="preserve">qu@dem per vnicam ſtationem in E, factã, vt problemate 6. </s> <s xml:id="echoid-s2748" xml:space="preserve">Num.</s> <s xml:id="echoid-s2749" xml:space="preserve"> <pb o="70" file="100" n="100" rhead="GEOMETR. PRACT."/> 5. </s> <s xml:id="echoid-s2750" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s2751" xml:space="preserve">Deinde per Quadrantem cum dioptra angulus C E D, exquiren-<lb/>dus in plano Horizontis. </s> <s xml:id="echoid-s2752" xml:space="preserve">quod fiet, ſi Quadrantis <lb/> <anchor type="figure" xlink:label="fig-100-01a" xlink:href="fig-100-01"/> planum erectum tranſeat ſemel per puncta E, C, & </s> <s xml:id="echoid-s2753" xml:space="preserve"><lb/>iterum per puncta E, D, vt deſignari poſsint partes <lb/>rectarum E C, E D. </s> <s xml:id="echoid-s2754" xml:space="preserve">Quadrantis enim vno latere in-<lb/>cumbente rectæ E C, dioptra vero rectæ E D, ſi an-<lb/>gulus eſt acutus, indicabit arcus inter illud latus, ac <lb/>dioptram, angulum C E D. </s> <s xml:id="echoid-s2755" xml:space="preserve">Quod ſi alterum Qua-<lb/>drantis latus rectæ ED, congruet erit angulus CED rectus: </s> <s xml:id="echoid-s2756" xml:space="preserve">Si vero recta ED. </s> <s xml:id="echoid-s2757" xml:space="preserve">vl-<lb/>tra alterum latus Quadrantis extiterit, dictus angulus obtuſus erit, qui cogni-<lb/>tus erit, ſi recto angulo ad datur reliquus inter alterum latus, & </s> <s xml:id="echoid-s2758" xml:space="preserve">rectam ED: </s> <s xml:id="echoid-s2759" xml:space="preserve">Qui <lb/>quidem reliquus angulus per Quadrantem explorabitur vt de acuto diximus in <lb/>problemate 7. </s> <s xml:id="echoid-s2760" xml:space="preserve">Atq; </s> <s xml:id="echoid-s2761" xml:space="preserve">ita habebimus triangulum ECD, cuius duo latera nota ſunt <lb/>EC, E D, angulumq; </s> <s xml:id="echoid-s2762" xml:space="preserve">notum comprehendunt C E D. </s> <s xml:id="echoid-s2763" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur, & </s> <s xml:id="echoid-s2764" xml:space="preserve">tertiumlatus <anchor type="note" xlink:label="note-100-01a" xlink:href="note-100-01"/> C D, cognitum erit.</s> <s xml:id="echoid-s2765" xml:space="preserve"/> </p> <div xml:id="echoid-div176" type="float" level="2" n="1"> <figure xlink:label="fig-100-01" xlink:href="fig-100-01a"> <image file="100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/100-01"/> </figure> <note symbol="a" position="left" xlink:label="note-100-01" xlink:href="note-100-01a" xml:space="preserve">12. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s2766" xml:space="preserve">2. </s> <s xml:id="echoid-s2767" xml:space="preserve"><emph style="sc">Qvo</emph> pacto autem ſine ope numerorum problema perſiciendum ſit, tra-<lb/>ditum eſt in problemate 7. </s> <s xml:id="echoid-s2768" xml:space="preserve">Num. </s> <s xml:id="echoid-s2769" xml:space="preserve">2.</s> <s xml:id="echoid-s2770" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2771" xml:space="preserve">LONGITVDINEM in Horizonte inter turrim aliquam, & </s> <s xml:id="echoid-s2772" xml:space="preserve">aliud <lb/>quodpiam ſignum, ex turri per duas ſtationes in faſtigio factas: </s> <s xml:id="echoid-s2773" xml:space="preserve">vel in <lb/>duabus feneſtris, quarum vna ſit ſub altera ad perpendiculum, quan-<lb/>do ſpatium inter illas feneſtras notum eſt, etiamſi totius turris altitu-<lb/>do ignota ſit, demetiri. </s> <s xml:id="echoid-s2774" xml:space="preserve">Atque hinc obiter altitudinem turris patefa-<lb/>cere.</s> <s xml:id="echoid-s2775" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div178" type="section" level="1" n="76"> <head xml:id="echoid-head79" xml:space="preserve">PROBLEMA XI.</head> <p> <s xml:id="echoid-s2776" xml:space="preserve">1. </s> <s xml:id="echoid-s2777" xml:space="preserve"><emph style="sc">Qvamvis</emph> problema hoc ſolutum iam ſit in <lb/> <anchor type="figure" xlink:label="fig-100-02a" xlink:href="fig-100-02"/> problemate 3. </s> <s xml:id="echoid-s2778" xml:space="preserve">& </s> <s xml:id="echoid-s2779" xml:space="preserve">4. </s> <s xml:id="echoid-s2780" xml:space="preserve">occaſione altitu dinis inquirendæ, <lb/>libet tamen idem hic per ſe, & </s> <s xml:id="echoid-s2781" xml:space="preserve">paulo aliter expedire. <lb/></s> <s xml:id="echoid-s2782" xml:space="preserve">Sit ergo turris A B CD, & </s> <s xml:id="echoid-s2783" xml:space="preserve">longitudo propoſita C E: </s> <s xml:id="echoid-s2784" xml:space="preserve"><lb/>ſtatura autem menſoris B G, vel AF. </s> <s xml:id="echoid-s2785" xml:space="preserve">Inſpecto extre-<lb/>mo E, in prima ſtatione per angulum C G E, & </s> <s xml:id="echoid-s2786" xml:space="preserve">in ſe-<lb/>cũda per angulum DFE, ducatur FH, ipſi G E, paral-<lb/>lela. </s> <s xml:id="echoid-s2787" xml:space="preserve">Et quia in triangulis F D H, G C E, anguli D, C, <lb/> <anchor type="note" xlink:label="note-100-02a" xlink:href="note-100-02"/> recti ſunt, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s2788" xml:space="preserve">H, E, æquales <anchor type="note" xlink:href="" symbol="c"/> latuſque FD, lateri GC.</s> <s xml:id="echoid-s2789" xml:space="preserve"> <anchor type="note" xlink:label="note-100-03a" xlink:href="note-100-03"/> æquale: </s> <s xml:id="echoid-s2790" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erunt quo que & </s> <s xml:id="echoid-s2791" xml:space="preserve">anguli G, DFH, æquales, <anchor type="note" xlink:label="note-100-04a" xlink:href="note-100-04"/> & </s> <s xml:id="echoid-s2792" xml:space="preserve">latera D H, C E. </s> <s xml:id="echoid-s2793" xml:space="preserve">Poſito autem ſinu toto F D, erit <lb/>DE, Tangens maioris anguli obſeruationis D F E, & </s> <s xml:id="echoid-s2794" xml:space="preserve"><lb/>D H, Tangens anguli D F H, hoc eſt, anguli æqualis <lb/>G, in prima ſtatione obſeruati: </s> <s xml:id="echoid-s2795" xml:space="preserve">ac proinde H E, differentia erit earum Tangen-<lb/>tium, <anchor type="note" xlink:href="" symbol="e"/> quæ quidem æqualis eſt differentiæ ſtationum F G, vel AB, vel DC. </s> <s xml:id="echoid-s2796" xml:space="preserve">Si <anchor type="note" xlink:label="note-100-05a" xlink:href="note-100-05"/> igitur fiat,</s> </p> <div xml:id="echoid-div178" type="float" level="2" n="1"> <figure xlink:label="fig-100-02" xlink:href="fig-100-02a"> <image file="100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/100-02"/> </figure> <note symbol="b" position="left" xlink:label="note-100-02" xlink:href="note-100-02a" xml:space="preserve">29. primi.</note> <note symbol="c" position="left" xlink:label="note-100-03" xlink:href="note-100-03a" xml:space="preserve">34. primi.</note> <note symbol="d" position="left" xlink:label="note-100-04" xlink:href="note-100-04a" xml:space="preserve">26. primi.</note> <note symbol="e" position="left" xlink:label="note-100-05" xlink:href="note-100-05a" xml:space="preserve">34. primi.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt H E, differentia Tan- \\ gentium angulorumob- \\ ſeruationum # Ad CE, Tangentem \\ min@ris anguli ob- \\ ſeruationis CGE. # Ita HE, vel FG, \\ differentia ſtatio- \\ num # ad G E, \\ longitu- \\ dinem, \\ gigne-<lb/></note> <pb o="71" file="101" n="101" rhead="LIBER SECVNDVS."/> <p> <s xml:id="echoid-s2797" xml:space="preserve">gignetur longitudo optata C E, in partibus differentiæ ſtationum H E, vel F G, <lb/>& </s> <s xml:id="echoid-s2798" xml:space="preserve">c. </s> <s xml:id="echoid-s2799" xml:space="preserve">Quod ſi rurſus fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt H E, differentia Tangentium \\ angulorum obſeruationum # Ad FD, ſi- \\ num totum # Ita H E, vel F G, differen- \\ tia ſtationum # ad F D, <lb/></note> <p> <s xml:id="echoid-s2800" xml:space="preserve">inuenietur recta F D, à qua ſi tollatur ſtatura menſoris A F, reliqua fiet altitudo <lb/>turris A D.</s> <s xml:id="echoid-s2801" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2802" xml:space="preserve">2. </s> <s xml:id="echoid-s2803" xml:space="preserve">Sediam ex duabus feneſtris F, G, inſpiciatur extremum E, per angulos <lb/>CFE, CGE, ducaturq; </s> <s xml:id="echoid-s2804" xml:space="preserve">FH, ipſi GE, parallela. </s> <s xml:id="echoid-s2805" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniã igitur angulus C F H, <anchor type="note" xlink:label="note-101-02a" xlink:href="note-101-02"/> angulo CGE, æqualis eſt: </s> <s xml:id="echoid-s2806" xml:space="preserve">ſi ponatur ſinus totus C E, erit C G, Tangens anguli <lb/>CEG, complementi minoris anguli obſeruationis G, & </s> <s xml:id="echoid-s2807" xml:space="preserve">CF, Tangens anguli <lb/>CEF, complementi maioris anguli obſeruationis C F E: </s> <s xml:id="echoid-s2808" xml:space="preserve">at F G, differentia ſta-<lb/>tionum, hoc eſt, interuallum inter feneſtras, differentia erit dictarum Tangen-<lb/>tium. </s> <s xml:id="echoid-s2809" xml:space="preserve"># Quamobrem ſi fiat,</s> </p> <div xml:id="echoid-div179" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-101-02" xlink:href="note-101-02a" xml:space="preserve">29. primi.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>VFG, differentia \\ Tangentium cõ- \\ plemẽtorum an- \\ gulorum obſer- \\ uationum, # Ad CE, \\ ſinum to- \\ tum # Ita F G, diffe- \\ rentia ſtatio- \\ num interfe- \\ neſtr{as}, # ad C E, \\ longitudi- \\ nem pro- \\ poſitam, <lb/></note> <figure> <image file="101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/101-01"/> </figure> <p> <s xml:id="echoid-s2810" xml:space="preserve">cognita erit longitudo CE, deſiderata. </s> <s xml:id="echoid-s2811" xml:space="preserve">Et ſirurſum fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt F G, differentia \\ Tangentium com- \\ plementorũ angu- \\ lorũ obſeruationũ # ad G C, Tangen- \\ tem complemen- \\ ti minoris angu- \\ li obſeruati # Ita F G, ad GC, \\ differentia ſta- \\ tionum inter \\ feneſtr{as} <lb/></note> <p> <s xml:id="echoid-s2812" xml:space="preserve">inuenta erit altitudo G C, à ſup eriori feneſtra ad baſem, in partibus differentiæ <lb/>ſtationum, cui ſi addatur portio GB, à ſuperiori feneſtra ad faſtigium vſque, to-<lb/>ta turris altitudo BC, non ignorabitur.</s> <s xml:id="echoid-s2813" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2814" xml:space="preserve">3. </s> <s xml:id="echoid-s2815" xml:space="preserve"><emph style="sc">Sine</emph> auxilio numerorum procedendum eſt, vtin ſuperioribus.</s> <s xml:id="echoid-s2816" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2817" xml:space="preserve">LONGITVDINEM rectæè directo menſoris poſitæ, cuius extre-<lb/>mum vtrumque, vel alterum non appareat, niſi ad dextram, vel fini-<lb/>ſtram accedat menſor, per quadrantem comprehendere.</s> <s xml:id="echoid-s2818" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div181" type="section" level="1" n="77"> <head xml:id="echoid-head80" xml:space="preserve">PROBLEMA XII.</head> <p> <s xml:id="echoid-s2819" xml:space="preserve">1. </s> <s xml:id="echoid-s2820" xml:space="preserve"><emph style="sc">Sit</emph> longitudo A B, & </s> <s xml:id="echoid-s2821" xml:space="preserve">menſor in extremo A, conſtitutus videre non poſ-<lb/>fit alterum extremum, niſi ad dextram ſiniſtramue <lb/> <anchor type="figure" xlink:label="fig-101-02a" xlink:href="fig-101-02"/> recedat vſq; </s> <s xml:id="echoid-s2822" xml:space="preserve">adD, punctũ, è quo vtrumq; </s> <s xml:id="echoid-s2823" xml:space="preserve">extremũ <lb/>cerni poſsit. </s> <s xml:id="echoid-s2824" xml:space="preserve">Eritigitur longitudo AB, menſori in <lb/>D, exiſtenti poſita in tranſuerſum. </s> <s xml:id="echoid-s2825" xml:space="preserve">Quare ea per <lb/>problema 10. </s> <s xml:id="echoid-s2826" xml:space="preserve">inueſtigabitur.</s> <s xml:id="echoid-s2827" xml:space="preserve"/> </p> <div xml:id="echoid-div181" type="float" level="2" n="1"> <figure xlink:label="fig-101-02" xlink:href="fig-101-02a"> <image file="101-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/101-02"/> </figure> </div> <p> <s xml:id="echoid-s2828" xml:space="preserve">2. </s> <s xml:id="echoid-s2829" xml:space="preserve">Eodẽ modo, ſi mẽſor exiſtat in C, è directo lõgi-<lb/>tudinis, ſed vel neutrũ extremũ, vel alterũ dũtaxat <pb o="72" file="102" n="102" rhead="GEOMETR. PRACT."/> intueri poſsit, longitu dinem A B, venabimur. </s> <s xml:id="echoid-s2830" xml:space="preserve">Sinamque ex C, ad ſiniſtram, <lb/>vel dextram procedemus, donec in D, vtrumq; </s> <s xml:id="echoid-s2831" xml:space="preserve">extremũ videamus, inuenietur <lb/>tranſuerſa longitu do AB, per problema 10. </s> <s xml:id="echoid-s2832" xml:space="preserve">vt prius. </s> <s xml:id="echoid-s2833" xml:space="preserve">Neque vero refert, ſiue per <lb/>angulum rectum BCD, recedatur in latus, ſiue per angulum acutum BAD, & </s> <s xml:id="echoid-s2834" xml:space="preserve">c.</s> <s xml:id="echoid-s2835" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2836" xml:space="preserve">3. </s> <s xml:id="echoid-s2837" xml:space="preserve"><emph style="sc">Operatio</emph> ſine numeris inſtituenda eſt, vt in ſuperioribus.</s> <s xml:id="echoid-s2838" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2839" xml:space="preserve">4. </s> <s xml:id="echoid-s2840" xml:space="preserve"><emph style="sc">Qvando</emph> menſor in A, exiſtens videre poteſt extremum B, inueſtigabi-<lb/>tur longitudo AB, per problema 8.</s> <s xml:id="echoid-s2841" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2842" xml:space="preserve"><emph style="sc">Si</emph> autem è directo longitudinis exiſtat in C, & </s> <s xml:id="echoid-s2843" xml:space="preserve">vtrumque etremum cernat, <lb/>explorabitur per problema 9. </s> <s xml:id="echoid-s2844" xml:space="preserve">eadem longitudo A B.</s> <s xml:id="echoid-s2845" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2846" xml:space="preserve">DISTANTIAM alicuius ſigni in Horizonte poſiti, à ſummitate tur-<lb/>ris, vel muri alicuius, licet ad ipſum ſignum acceſſus non pateat, per <lb/>quadrantem colligere.</s> <s xml:id="echoid-s2847" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div183" type="section" level="1" n="78"> <head xml:id="echoid-head81" xml:space="preserve">PROBLEMA XIII.</head> <p> <s xml:id="echoid-s2848" xml:space="preserve">1. </s> <s xml:id="echoid-s2849" xml:space="preserve"><emph style="sc">In</emph> Horizontis plano punctum A, diſtet à ſummi-<lb/> <anchor type="figure" xlink:label="fig-102-01a" xlink:href="fig-102-01"/> tate D, alicuius altitudinis CD, per rectam AD, quam me-<lb/>tiri iubemur, Vbicunq; </s> <s xml:id="echoid-s2850" xml:space="preserve">oculus menſoris exiſtat, nimirum <lb/>in B, vt ſit ſtatura menſoris BG, inueſtigentur per proble-<lb/>ma 6. </s> <s xml:id="echoid-s2851" xml:space="preserve">diſtantiæ punctorum A, D, ab oculo menſoris B.</s> <s xml:id="echoid-s2852" xml:space="preserve"/> </p> <div xml:id="echoid-div183" type="float" level="2" n="1"> <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/xxxxxxxx/figures/102-01"/> </figure> </div> <p> <s xml:id="echoid-s2853" xml:space="preserve">Deinde angulus exploretur A B D, quem nobis præ-<lb/>bebit Quadrãs cum dioptra, ſi ad oculum ita applicetur, <lb/>vt eius planum per tria puncta B, A, D, tranſeat, poſito centro in B; </s> <s xml:id="echoid-s2854" xml:space="preserve">atque vnum <lb/>eius latus rectæ B A, incumbat, dioptra vero ad punctum D, dirigatur, & </s> <s xml:id="echoid-s2855" xml:space="preserve">c. </s> <s xml:id="echoid-s2856" xml:space="preserve">Itaq; <lb/></s> <s xml:id="echoid-s2857" xml:space="preserve">cum in triangulo B A D, duo latera nota B A, B D, angulum notum contineant <lb/>B; </s> <s xml:id="echoid-s2858" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cognoſcetur quo que latus AD.</s> <s xml:id="echoid-s2859" xml:space="preserve"/> </p> <note symbol="a" position="left" xml:space="preserve">10. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2860" xml:space="preserve">2. </s> <s xml:id="echoid-s2861" xml:space="preserve"><emph style="sc">Qva</emph> ratione eadem diſtantia AD, exquirenda ſit abſque numeris, do-<lb/>cuimus Num. </s> <s xml:id="echoid-s2862" xml:space="preserve">5. </s> <s xml:id="echoid-s2863" xml:space="preserve">problematis 7.</s> <s xml:id="echoid-s2864" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2865" xml:space="preserve">ALTITVDINEM inacceſſibilem, cuius baſis non videatur, & </s> <s xml:id="echoid-s2866" xml:space="preserve">ad <lb/>quam per nullum ſpatium ſecundum lineam rectam accedere poſſi-<lb/>mus, aut recedere, vt duæ ſtationes fieri poſſint, ſed ſolũ ad dextram, <lb/>ſiniſtramue ad locum, è quo eius baſis appareat, per Quadrantem ex-<lb/>plorare.</s> <s xml:id="echoid-s2867" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div185" type="section" level="1" n="79"> <head xml:id="echoid-head82" xml:space="preserve">PROBLEMA XIV.</head> <figure> <image file="102-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/102-02"/> </figure> <p> <s xml:id="echoid-s2868" xml:space="preserve">1. </s> <s xml:id="echoid-s2869" xml:space="preserve">Sit altitudo A B, ad quam ex C, loco menſoris <lb/>non liceat accedere, aut ab earecedere ſecundum li-<lb/>neam rectam, ſed ſolum in latus, verbigratia vſque ad <lb/>D, vnde baſem videre poſsimus. </s> <s xml:id="echoid-s2870" xml:space="preserve">Inquiratur per pro-<lb/>blema 10. </s> <s xml:id="echoid-s2871" xml:space="preserve">longitudo tranſuerſa A C, ex loco D: </s> <s xml:id="echoid-s2872" xml:space="preserve">inſpi-<lb/>ciatur que vertex B, ex C, per angulum ACB. </s> <s xml:id="echoid-s2873" xml:space="preserve">Et quia, <lb/>poſito ſinu toto A C, altitudo A B, Tangens eſt an-<lb/>guli obſeruationis ACB, <anchor type="note" xlink:href="" symbol="b"/> ſi fiat.</s> <s xml:id="echoid-s2874" xml:space="preserve"/> </p> <note symbol="b" position="left" xml:space="preserve">4. Triang. <lb/>rectil.</note> <pb o="73" file="103" n="103" rhead="LIBER SECVNDVS."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} t@- \\ t{us} AC, # ad AB, Tangentem an- \\ guli obſeruati ACB: # Ita longitudo A C, per \\ problema 10. inuenta # Ad AB, alt@- \\ tudinem, <lb/></note> <p> <s xml:id="echoid-s2875" xml:space="preserve">prodibit nota altitudo AB, in partibus, in quibus AC, inuenta fuit.</s> <s xml:id="echoid-s2876" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2877" xml:space="preserve">2. </s> <s xml:id="echoid-s2878" xml:space="preserve"><emph style="sc">Non</emph> erit autem difficile problema hoc ſine numerorum auxilio exequi, <lb/>ſi ſuperiora præcepta conſulantur.</s> <s xml:id="echoid-s2879" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2880" xml:space="preserve">ALTITVDINEM inacceſſibilem, quando neque diſtantia à loco <lb/>menſoris ad eius baſem nota eſt, neque è directo ipſius duæ ſtationes <lb/>in plano fieri poſſunt, neque denique baſis appareat, per Quadran-<lb/>tem notam reddere. </s> <s xml:id="echoid-s2881" xml:space="preserve">Atque hinc obiter ipſam quoque diſtantiam e-<lb/>licere.</s> <s xml:id="echoid-s2882" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div186" type="section" level="1" n="80"> <head xml:id="echoid-head83" xml:space="preserve">PROBLEMA XV.</head> <p> <s xml:id="echoid-s2883" xml:space="preserve">1. </s> <s xml:id="echoid-s2884" xml:space="preserve"><emph style="sc">Sit</emph> altitudo A B, locus menſoris C, diſtantia CB, incognita: </s> <s xml:id="echoid-s2885" xml:space="preserve">baſis B, non <lb/>appareat, & </s> <s xml:id="echoid-s2886" xml:space="preserve">à loco C, non liceat accedere ad A B, nequerecedere, vt duæ ſta-<lb/>tiones fiantin plano. </s> <s xml:id="echoid-s2887" xml:space="preserve">Erigatur haſta C E, ſi non præſtò ſit turris aliqua C O, & </s> <s xml:id="echoid-s2888" xml:space="preserve"><lb/>ſtatura menſoris ſit CD. </s> <s xml:id="echoid-s2889" xml:space="preserve">Sumpta deinde portione haſtæ D E, notarum partium, <lb/>concipiantur ductæ DM, EN, ipſi CB, pa-<lb/> <anchor type="figure" xlink:label="fig-103-01a" xlink:href="fig-103-01"/> rallelæ, obſeruenturq; </s> <s xml:id="echoid-s2890" xml:space="preserve">per Quadrantem <lb/>anguli ADM, AEN. </s> <s xml:id="echoid-s2891" xml:space="preserve">Sumptis quoq; </s> <s xml:id="echoid-s2892" xml:space="preserve">rectis <lb/>æqualibus EF, DG, pro ſinubus totis, eri-<lb/>gantur perpendiculares FH, GI, pro Tã-<lb/>gentibus angulorum obſeruationũ. </s> <s xml:id="echoid-s2893" xml:space="preserve">Sum <lb/>pta item A L, æqualiipſi D E, ducaturre-<lb/>cta D L, <anchor type="note" xlink:href="" symbol="a"/> quæ parallela erit ipſi E A, ſeca- <anchor type="note" xlink:label="note-103-02a" xlink:href="note-103-02"/> bitq; </s> <s xml:id="echoid-s2894" xml:space="preserve">GI, in K. </s> <s xml:id="echoid-s2895" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam vero angulus <anchor type="note" xlink:label="note-103-03a" xlink:href="note-103-03"/> DKG, angulo DLB, & </s> <s xml:id="echoid-s2896" xml:space="preserve">hic angulo EAB, <lb/>& </s> <s xml:id="echoid-s2897" xml:space="preserve">hic angulo EHF, æqualis eſt; </s> <s xml:id="echoid-s2898" xml:space="preserve">erit angu-<lb/>lus DKG, angulo EHF, æqualis: </s> <s xml:id="echoid-s2899" xml:space="preserve">Eſt au-<lb/>tem & </s> <s xml:id="echoid-s2900" xml:space="preserve">rectus G, recto F, æqualis, & </s> <s xml:id="echoid-s2901" xml:space="preserve">latus <lb/>DG, lateri EF, æquale; </s> <s xml:id="echoid-s2902" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt quoq; </s> <s xml:id="echoid-s2903" xml:space="preserve">late- <anchor type="note" xlink:label="note-103-04a" xlink:href="note-103-04"/> ra GK, FH, æqualia, & </s> <s xml:id="echoid-s2904" xml:space="preserve">anguli D, E, æqua-<lb/>les: </s> <s xml:id="echoid-s2905" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s2906" xml:space="preserve">IK, differentia erit inter Tan-<lb/>gentes GI, GK, angulorum GDI, GDK, <lb/>ſiue F E H. </s> <s xml:id="echoid-s2907" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et quia eſt, vt I K, ad K G. </s> <s xml:id="echoid-s2908" xml:space="preserve">ita A L, ad L M; </s> <s xml:id="echoid-s2909" xml:space="preserve">erit per contrariam compoſitionem à nobis in ſcholio propoſ. </s> <s xml:id="echoid-s2910" xml:space="preserve">18. <lb/></s> <s xml:id="echoid-s2911" xml:space="preserve"> <anchor type="note" xlink:label="note-103-05a" xlink:href="note-103-05"/> lib. </s> <s xml:id="echoid-s2912" xml:space="preserve">5. </s> <s xml:id="echoid-s2913" xml:space="preserve">demonſtratam, ita quo que IK, ad IG, vt AL, hoc eſt, vt ED, differentia ſta-<lb/>tionum, ad altitudinem AM. </s> <s xml:id="echoid-s2914" xml:space="preserve"># Igitur ſi fiat,</s> </p> <div xml:id="echoid-div186" type="float" level="2" n="1"> <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/xxxxxxxx/figures/103-01"/> </figure> <note symbol="a" position="right" xlink:label="note-103-02" xlink:href="note-103-02a" xml:space="preserve">33. primi.</note> <note symbol="b" position="right" xlink:label="note-103-03" xlink:href="note-103-03a" xml:space="preserve">29. primi.</note> <note symbol="c" position="right" xlink:label="note-103-04" xlink:href="note-103-04a" xml:space="preserve">26. primi.</note> <note symbol="d" position="right" xlink:label="note-103-05" xlink:href="note-103-05a" xml:space="preserve">ſchol. 4. ſe-<lb/>xti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt I K, differen- \\ tia Tangentium # ad I G, Tangen- \\ te maiorem: # Ita AL, vel ED, differen- \\ tia ſtationum # Ad A M, alti- \\ tudinem, <lb/></note> <p> <s xml:id="echoid-s2915" xml:space="preserve">gignetur altitudo AM, cui ſi apponaturſtatura menſoris MB, tota altitudo AB, <lb/>nota efficietur. </s> <s xml:id="echoid-s2916" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Iam ſi fiat,</s> </p> <note symbol="e" position="right" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt G I, ſi- \\ n{us} tot{us} # ad G D, Tangentem complementi ma- \\ ioris anguli obſeruationis: # Ita altitudo in- \\ uenta A M, # ad M D, di- \\ ſtantiam, <lb/></note> <p> <s xml:id="echoid-s2917" xml:space="preserve">inuenta erit diſtantia D M, vel C B.</s> <s xml:id="echoid-s2918" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2919" xml:space="preserve">2. </s> <s xml:id="echoid-s2920" xml:space="preserve"><emph style="sc">Per</emph> ſolos ſinus idem problema conficiemus, ſi prius inueſtigetur hy-<lb/> <anchor type="note" xlink:label="note-103-09a" xlink:href="note-103-09"/> potenuſa AD, hoc modo. </s> <s xml:id="echoid-s2921" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Fiat,</s> </p> <div xml:id="echoid-div187" type="float" level="2" n="2"> <note symbol="f" position="right" xlink:label="note-103-09" xlink:href="note-103-09a" xml:space="preserve">10. Triang. <lb/>rectil.</note> </div> <pb o="74" file="104" n="104" rhead="GEOMETR. PRACT."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli A D L, \\ differentiæ inter an- \\ gulos obſeruationum \\ ADM, LDM, # ad A L, hoc \\ eſt, ad D E, \\ differentiam \\ ſtationum: # Ita ſin{us} anguli ALD, qui \\ relinquitur, ſi DLM, comple- \\ mentum minoris anguli \\ L D M, obſeruati ex duob{us} \\ rectis ſubtrahatur, # ad AD. <lb/></note> <p> <s xml:id="echoid-s2922" xml:space="preserve">Nam productus numerus offeret hypotenuſam AD. </s> <s xml:id="echoid-s2923" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Si ergo rurſus fiat,</s> </p> <note symbol="a" position="left" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} an- \\ guli recti M, # ad hypotenuſam \\ AD, inuentam: # Ita ſin{us} maioris anguli \\ obſer uationis ADM, # ad AM, <lb/></note> <note symbol="b" position="left" xml:space="preserve">10. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s2924" xml:space="preserve">procreabitur altitudo A M, & </s> <s xml:id="echoid-s2925" xml:space="preserve">c. </s> <s xml:id="echoid-s2926" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Siautem fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} anguli \\ recti M, # ad hypotenuſam AD, \\ proxime inuentam, # Ita ſin{us} anguli D A M, \\ complementi maioris an- \\ guli obſeruationis, # ad DM, <lb/></note> <p> <s xml:id="echoid-s2927" xml:space="preserve">effi cietur diſtantia D M, cognita.</s> <s xml:id="echoid-s2928" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2929" xml:space="preserve">3. </s> <s xml:id="echoid-s2930" xml:space="preserve"><emph style="sc">Si</emph> ſine numeris problema ſoluen dum eſt, recurrendum erit ad ſuperiora, <lb/>præſertim ad Num. </s> <s xml:id="echoid-s2931" xml:space="preserve">6. </s> <s xml:id="echoid-s2932" xml:space="preserve">8. </s> <s xml:id="echoid-s2933" xml:space="preserve">& </s> <s xml:id="echoid-s2934" xml:space="preserve">7. </s> <s xml:id="echoid-s2935" xml:space="preserve">problematis 1.</s> <s xml:id="echoid-s2936" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2937" xml:space="preserve">ALTITVDINEM maiorem ex minori cognita, per duas ſtationes <lb/>in ſummitate, vel in duabus feneſtris factas, etiamſi ſolum maioris alti-<lb/>tudinis vertex cernatur, per Quadrantem adinuenire. </s> <s xml:id="echoid-s2938" xml:space="preserve">Atque hinc <lb/>diſtantiam quoque inter altitudines colligere.</s> <s xml:id="echoid-s2939" xml:space="preserve"/> </p> <figure> <image file="104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/104-01"/> </figure> </div> <div xml:id="echoid-div189" type="section" level="1" n="81"> <head xml:id="echoid-head84" xml:space="preserve">PROBLEMA XVI.</head> <p> <s xml:id="echoid-s2940" xml:space="preserve">1. </s> <s xml:id="echoid-s2941" xml:space="preserve"><emph style="sc">Maior</emph> altitudo ſit AN, minor tur-<lb/>ris C O, cognita, ex qua ſolum cacumen <lb/>A, non autem baſis N, appareat. </s> <s xml:id="echoid-s2942" xml:space="preserve">Fiantin <lb/>ſummitate duæ ſtationes in C, D, menſo-<lb/>ris que ſtatura ſit C G, vel D E, & </s> <s xml:id="echoid-s2943" xml:space="preserve">ad A N, <lb/>intelligatur ducta perpendicularis G E F. <lb/></s> <s xml:id="echoid-s2944" xml:space="preserve">Atque inſpecto cacumine A, obſeruentur <lb/>anguli A E F, AGF. </s> <s xml:id="echoid-s2945" xml:space="preserve">Reliqua fiant, vt in 2. </s> <s xml:id="echoid-s2946" xml:space="preserve"><lb/>problemate. </s> <s xml:id="echoid-s2947" xml:space="preserve">Si ergo fiat, ſicut in eo pro-<lb/>blemate demonſtrauimus,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt GM, differentia Tangentium GL, EK, angu- \\ lorum H, I, qui complementa ſunt angulorũ ob- \\ ſeruationum, poſitis ſinub{us} totis HL, IK, # ad GE, diffe- \\ rentiam ſta- \\ tionum: # Ita LH, ſi- \\ n{us} tot{us} # ad FA. <lb/></note> <p> <s xml:id="echoid-s2948" xml:space="preserve">Inuenietur altitudo A F, cuiſi adijciatur F N, conflata ex altitudine turris C O, <lb/>& </s> <s xml:id="echoid-s2949" xml:space="preserve">ſtatura menſoris, tota maior altitudo AN, nota euadet. </s> <s xml:id="echoid-s2950" xml:space="preserve">Item ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt GM, differentiæ Tan- \\ gentium earundem. # ad GE, differenti- \\ am ſtationum: # Ita GL, Tangens complementi \\ minoris anguli obſeruationis # ad GF, <lb/></note> <p> <s xml:id="echoid-s2951" xml:space="preserve">pro creabitur diſtantia GF, à qua ſi dematur latitudo turris C O, reliqua erit di-<lb/>ſtantia inter duas turres.</s> <s xml:id="echoid-s2952" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2953" xml:space="preserve"><emph style="sc">Hæc</emph> omnia in 2. </s> <s xml:id="echoid-s2954" xml:space="preserve">problemate demonſtrauimus: </s> <s xml:id="echoid-s2955" xml:space="preserve">& </s> <s xml:id="echoid-s2956" xml:space="preserve">ob hanc cauſam eiſdem <lb/>prorſus literis hic vſi ſumus, quas ibivſurpauimus, vt demonſtratio ex illo loco <lb/>in hunc transferri poſsit.</s> <s xml:id="echoid-s2957" xml:space="preserve"/> </p> <pb o="75" file="105" n="105" rhead="LIBER SECVNDVS."/> <p> <s xml:id="echoid-s2958" xml:space="preserve">2. </s> <s xml:id="echoid-s2959" xml:space="preserve"><emph style="sc">Qvando</emph> in ſummitate turris minoris fieri duæ ſtationes nequeunt, eli-<lb/>gantur duæ feneſtræ, in quibus duæ ſtationes fiant. </s> <s xml:id="echoid-s2960" xml:space="preserve">Vt in figura præcedentis pro-<lb/>blematis 15. </s> <s xml:id="echoid-s2961" xml:space="preserve">in minoriturri CE, deligantur duæ feneſtræ D, E, & </s> <s xml:id="echoid-s2962" xml:space="preserve">reliqua fiant, vt <lb/>in haſta C E. </s> <s xml:id="echoid-s2963" xml:space="preserve">Solum pro ſtatura menſoris ad altitu dinem inuentam AM, adijci-<lb/>enda eſt portio turris inter inferiorem feneſtram D, & </s> <s xml:id="echoid-s2964" xml:space="preserve">baſem C, vttota maior al-<lb/>titudo AB, nota effi ciatur.</s> <s xml:id="echoid-s2965" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2966" xml:space="preserve">3. </s> <s xml:id="echoid-s2967" xml:space="preserve"><emph style="sc">Sine</emph> numeris nihil noui p̃cipimus, ſed ad ſuperiora lectorem amandamus.</s> <s xml:id="echoid-s2968" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2969" xml:space="preserve">4. </s> <s xml:id="echoid-s2970" xml:space="preserve"><emph style="sc">Qvo</emph> ẽt pacto ꝓblema hoc ꝑ ſolos ſinus poſsit effici, docuim’ ꝓbl. </s> <s xml:id="echoid-s2971" xml:space="preserve">2. </s> <s xml:id="echoid-s2972" xml:space="preserve">& </s> <s xml:id="echoid-s2973" xml:space="preserve">15.</s> <s xml:id="echoid-s2974" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2975" xml:space="preserve">ALTITVDINEM maiorem ex minori incognita, dummodo baſis <lb/>maioris cerni poſſit per Quadrantem perſcrutari.</s> <s xml:id="echoid-s2976" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div190" type="section" level="1" n="82"> <head xml:id="echoid-head85" xml:space="preserve">PROBLEMA XVII.</head> <figure> <image file="105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/105-01"/> </figure> <p> <s xml:id="echoid-s2977" xml:space="preserve">1. </s> <s xml:id="echoid-s2978" xml:space="preserve"><emph style="sc">Sit</emph> maior altitudo A B, & </s> <s xml:id="echoid-s2979" xml:space="preserve">minor C D, incognita, <lb/>poſsitq; </s> <s xml:id="echoid-s2980" xml:space="preserve">baſis maioris B, videri ex minori altitudine. </s> <s xml:id="echoid-s2981" xml:space="preserve">Pri-<lb/>mum per duas ſtationes in ſummitate turris minoris, vel <lb/>in duabus feneſtris, inquiratur tam altitudo turris mino -<lb/>ris C D, quam diſtantia D B, vt problemate 11. </s> <s xml:id="echoid-s2982" xml:space="preserve">vel etiam <lb/>3. </s> <s xml:id="echoid-s2983" xml:space="preserve">& </s> <s xml:id="echoid-s2984" xml:space="preserve">4. </s> <s xml:id="echoid-s2985" xml:space="preserve">traditum eſt. </s> <s xml:id="echoid-s2986" xml:space="preserve">Namtunc ex minori altitudine nota <lb/>C D, maior A B, explorabitur perea, quæ in antecedent@ <lb/>problemate 16. </s> <s xml:id="echoid-s2987" xml:space="preserve">ſcripſimus.</s> <s xml:id="echoid-s2988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2989" xml:space="preserve">2. </s> <s xml:id="echoid-s2990" xml:space="preserve"><emph style="sc">At</emph> ex altitudine CD, & </s> <s xml:id="echoid-s2991" xml:space="preserve">diſtantia D B, cognitis di-<lb/>ſcemus altitudinem maiorem A B, per ſolos ſinus, hoc <lb/>modo. </s> <s xml:id="echoid-s2992" xml:space="preserve">Ex aliqua feneſtra C, minoris altitudinisin ſpici-<lb/>antur extrema A, B, maioris altitudinis per ãgulos ACE, <lb/>BCD, (ducta prius CE, ipſi DB, parallela, vel ad vtramq; <lb/></s> <s xml:id="echoid-s2993" xml:space="preserve">altitu dinem perpendiculari.) </s> <s xml:id="echoid-s2994" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Deinde fiat,</s> </p> <note symbol="a" position="right" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli C B D, complemen- \\ ti illi{us}, quo baſis inſpicitur # ad inuentam altitu- \\ altitudinem CD: # Ita ſin{us} tot{us} \\ anguli recti D, # ad B C, <lb/>#### Vel <lb/>Vt ſin{us} anguli B C D, quo \\ baſis inſpicitur, # ad inuentam diſtan- \\ ſtantiam B D: # Ita ſin{us} tot{us} an- \\ gulirecti D, # ad BC, <lb/></note> <p> <s xml:id="echoid-s2995" xml:space="preserve">Vtro que enim modo cognita erit hypotenuſa BC. </s> <s xml:id="echoid-s2996" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Siergo rurſus fiat,</s> </p> <note symbol="b" position="right" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli A, com- \\ plementi illi{us} quo ca- \\ cumen inſpicitur, # ad inuentam \\ hypotenuſam \\ BC: # Ita ſin{us} anguli ACB, conflati ex com- \\ plemento anguli, quo baſem intue- \\ mur, & ex angulo, quo faſtigium \\ cernitur, # ad \\ AB, <lb/></note> <p> <s xml:id="echoid-s2997" xml:space="preserve">manifeſtabitur altitudo maior AB.</s> <s xml:id="echoid-s2998" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2999" xml:space="preserve">3. </s> <s xml:id="echoid-s3000" xml:space="preserve"><emph style="sc">De</emph> ſolutione problematis ſine numeris nihil noui hic præcipimus, ſed ea <lb/>ex ſuperioribus petenda eſt.</s> <s xml:id="echoid-s3001" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3002" xml:space="preserve">ALTITVDINEM minorem ex maiori cognita, licet baſis minoris <lb/>non cerni poſſit, ope Quadrantis perueſtigare. </s> <s xml:id="echoid-s3003" xml:space="preserve">Atque hinc diſtantiam <lb/>quoque inter altitudines duas eruere.</s> <s xml:id="echoid-s3004" xml:space="preserve"/> </p> <pb o="76" file="106" n="106" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div191" type="section" level="1" n="83"> <head xml:id="echoid-head86" xml:space="preserve">PROBLEMA XVIII.</head> <p> <s xml:id="echoid-s3005" xml:space="preserve">1. </s> <s xml:id="echoid-s3006" xml:space="preserve"><emph style="sc">Minor</emph> altitudo A B, ex maiore C D, co-<lb/> <anchor type="figure" xlink:label="fig-106-01a" xlink:href="fig-106-01"/> gnita proponatur addiſcenda, etiamſi baſis B, non <lb/>cernatur. </s> <s xml:id="echoid-s3007" xml:space="preserve">Concipiatur ducta recta AE, ipſi BD, pa-<lb/>rallela, vt E D, minorialtitudini AB, ſit æqualis. </s> <s xml:id="echoid-s3008" xml:space="preserve">Si <lb/>igitur ex duabus ſtationibus in ſummitate maioris <lb/>altitu dinis C D, factis, per problema 3. </s> <s xml:id="echoid-s3009" xml:space="preserve">vel ex dua-<lb/>bus feneſtris, per problema 4. </s> <s xml:id="echoid-s3010" xml:space="preserve">inueſtigetur tam alti-<lb/>tudo C E, quam diſtantia A E, inſpecto@cacumine <lb/>A, ac ſi eſſet ſignum aliquod in Horizonte A E, vi-<lb/>ſum, & </s> <s xml:id="echoid-s3011" xml:space="preserve">CE, ex tota altitu dine C D, auferatur, reli-<lb/>qua ED, hoc eſt, minor altitudo fiet nota. </s> <s xml:id="echoid-s3012" xml:space="preserve">Diſtan-<lb/>tia autem AE, inuenta quæſitæ BD, eſt æqualis: </s> <s xml:id="echoid-s3013" xml:space="preserve">ac <lb/>proinde DB, cognita erit.</s> <s xml:id="echoid-s3014" xml:space="preserve"/> </p> <div xml:id="echoid-div191" 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/xxxxxxxx/figures/106-01"/> </figure> </div> <p> <s xml:id="echoid-s3015" xml:space="preserve">ALTITVDINEM minorem ex maiori incognita, dummodo baſis <lb/>minoris videri poſſit, per Quadrantem explorare. </s> <s xml:id="echoid-s3016" xml:space="preserve">Atque hinc diſtan-<lb/>tiam quoque inter duas altitudines coniicere.</s> <s xml:id="echoid-s3017" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div193" type="section" level="1" n="84"> <head xml:id="echoid-head87" xml:space="preserve">PROBLEMA XIX.</head> <p> <s xml:id="echoid-s3018" xml:space="preserve">1. </s> <s xml:id="echoid-s3019" xml:space="preserve"><emph style="sc">Repetatvr</emph> figura præcedentis problematis. </s> <s xml:id="echoid-s3020" xml:space="preserve">Et quia baſis B, minoris <lb/>altitudinis ex maiore apparet; </s> <s xml:id="echoid-s3021" xml:space="preserve">ſi punctum B, ex duabus ſtationibus in ſummitate <lb/>maioris altitudinis C D, factis inſpiciatur, reperietur per problema 3. </s> <s xml:id="echoid-s3022" xml:space="preserve">tã altitudo <lb/>maior CD. </s> <s xml:id="echoid-s3023" xml:space="preserve">quam diſtantia BD. </s> <s xml:id="echoid-s3024" xml:space="preserve">Quod etiam efficies per problema 4. </s> <s xml:id="echoid-s3025" xml:space="preserve">ſi punctum <lb/>B, ex duabus feneſtris maioris altitudinis C D, inſpiciatur. </s> <s xml:id="echoid-s3026" xml:space="preserve">Cognita ergo altitu-<lb/>dine maiori CD, inuenietur minor altitudo AB, vtin præcedẽti problemate tra-<lb/>ditũ eſt. </s> <s xml:id="echoid-s3027" xml:space="preserve">Cũ igitur & </s> <s xml:id="echoid-s3028" xml:space="preserve">diſtãtia BD, ſit explorata, patet ſolutio ꝓblematis ꝓpoſiti.</s> <s xml:id="echoid-s3029" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3030" xml:space="preserve">PORTIONEM altitudinis maioris ex minore altitudine, & </s> <s xml:id="echoid-s3031" xml:space="preserve">m@noris <lb/>portionem ex maiori cognoſcere per Quadrantem.</s> <s xml:id="echoid-s3032" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div194" type="section" level="1" n="85"> <head xml:id="echoid-head88" xml:space="preserve">PROBLEMA XX.</head> <p> <s xml:id="echoid-s3033" xml:space="preserve">1. </s> <s xml:id="echoid-s3034" xml:space="preserve"><emph style="sc">Sit</emph> portio A C, maioris altitudinis A B, exquirenda <lb/> <anchor type="figure" xlink:label="fig-106-02a" xlink:href="fig-106-02"/> ex minore altitudine DE: </s> <s xml:id="echoid-s3035" xml:space="preserve">Item portio FG, minoris altitudi-<lb/>nis FB, ex altitudine maiore DE. </s> <s xml:id="echoid-s3036" xml:space="preserve">SiDE, altitudo minor eſt <lb/>portione C B, inueſtigetur tam altitudo maior A B, quam <lb/>CB, ex minore altitudine DE, per problema 16. </s> <s xml:id="echoid-s3037" xml:space="preserve">vel 17. </s> <s xml:id="echoid-s3038" xml:space="preserve">pro-<lb/>ut videlicet D E, cognita fuerit, aut incognita. </s> <s xml:id="echoid-s3039" xml:space="preserve">Nam <lb/>C B, ablata ex A B, notam relinquet portionem A C, quæ-<lb/>ſitam.</s> <s xml:id="echoid-s3040" xml:space="preserve"/> </p> <div xml:id="echoid-div194" type="float" level="2" n="1"> <figure xlink:label="fig-106-02" xlink:href="fig-106-02a"> <image file="106-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/106-02"/> </figure> </div> <p> <s xml:id="echoid-s3041" xml:space="preserve">2. </s> <s xml:id="echoid-s3042" xml:space="preserve"><emph style="sc">Si</emph> vero D E, maior eſt portione F B, explorandaq; <lb/></s> <s xml:id="echoid-s3043" xml:space="preserve">ſit portio AF; </s> <s xml:id="echoid-s3044" xml:space="preserve">in quirẽda quidem erit maior altitudo A B, ex <lb/>minore D E, per problema 16. </s> <s xml:id="echoid-s3045" xml:space="preserve">vel 17. </s> <s xml:id="echoid-s3046" xml:space="preserve">At vero altitudo mi- <pb o="77" file="107" n="107" rhead="LIBER SECVNDVS."/> nor FB, ex maiore DE, per problema 18. </s> <s xml:id="echoid-s3047" xml:space="preserve">vel 19. </s> <s xml:id="echoid-s3048" xml:space="preserve">exploranda erit. </s> <s xml:id="echoid-s3049" xml:space="preserve">Nam rurſus FB, <lb/>detracta ex AB, notam relinquet portionem AF.</s> <s xml:id="echoid-s3050" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3051" xml:space="preserve">3. </s> <s xml:id="echoid-s3052" xml:space="preserve"><emph style="sc">Non</emph> ſecus per problema 18. </s> <s xml:id="echoid-s3053" xml:space="preserve">vel 19. </s> <s xml:id="echoid-s3054" xml:space="preserve">indaganda erit vtraque altitudo mi-<lb/>nor F B, GB, ex maiore D E, vtillarum differentia F G, quæ quæritur, colliga-<lb/>tur nota.</s> <s xml:id="echoid-s3055" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3056" xml:space="preserve">ALTITVDINEM, cuius baſis impoſita ſit alteri altitudini, & </s> <s xml:id="echoid-s3057" xml:space="preserve">vtra-<lb/>que illius extremitas cerni poſſit, etiamſi infimum punctum alterius, <lb/>cui imponitur, lateat, & </s> <s xml:id="echoid-s3058" xml:space="preserve">eiuſdem puncti infimi diſtantia à loco men-<lb/>ſoris cognita non ſit, per Quadrantem ex valle, aut ex plano Horizon-<lb/>tis explorare.</s> <s xml:id="echoid-s3059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div196" type="section" level="1" n="86"> <head xml:id="echoid-head89" xml:space="preserve">PROBLEMA XXI.</head> <p> <s xml:id="echoid-s3060" xml:space="preserve">1. </s> <s xml:id="echoid-s3061" xml:space="preserve"><emph style="sc">Hvivscemodi</emph> altitudo eſt tur-<lb/> <anchor type="figure" xlink:label="fig-107-01a" xlink:href="fig-107-01"/> ris ſupra montem poſita, & </s> <s xml:id="echoid-s3062" xml:space="preserve">portio ali-<lb/>cuius ædificij inter duas feneſtras, vel duo <lb/>ſigna, quorum alterum altero ſuperius eſt. <lb/></s> <s xml:id="echoid-s3063" xml:space="preserve">Sit igitur ſupra montem altitudo turris <lb/>A B, propoſita. </s> <s xml:id="echoid-s3064" xml:space="preserve">Ex aliquoloco E, in pla-<lb/>nitie, aut valle, vnde vtrumque extremum <lb/>A, B, videatur, obſeruentur per Quadran-<lb/>tem anguli A D C, B D C, quos radij D A, <lb/>DB, cũ recta D C, quæ ex D, ad AB, pro-<lb/>ductam intra montẽ eſt perpendicularis: </s> <s xml:id="echoid-s3065" xml:space="preserve"><lb/>ita vt ſtatura menſoris ſit D E. </s> <s xml:id="echoid-s3066" xml:space="preserve">Deinde <lb/>per problema 6. </s> <s xml:id="echoid-s3067" xml:space="preserve">inueſtigetur longitudo <lb/>vtriuſque radij D A, D B. </s> <s xml:id="echoid-s3068" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Nam ſi fiat in <anchor type="note" xlink:label="note-107-01a" xlink:href="note-107-01"/> triangulo A B D,</s> </p> <div xml:id="echoid-div196" 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/xxxxxxxx/figures/107-01"/> </figure> <note symbol="a" position="right" xlink:label="note-107-01" xlink:href="note-107-01a" xml:space="preserve">10. Triang. <lb/>rectil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli B A D, \\ qui complèmentum eſt \\ maioris anguli ADC, \\ obſeruati, # adradium DB, \\ proximè inuẽ- \\ tum: # Ita ſin{us} anguli A D B, \\ qui differentia est an- \\ gulorum obſeruationũ \\ ADC, BDC, # ad A B, alti- \\ tudinem, <lb/></note> <p> <s xml:id="echoid-s3069" xml:space="preserve">inuenta erit altitudo AB, quæſita, in partibus, in quibus cognitus eſt radius <lb/>DB. </s> <s xml:id="echoid-s3070" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Sic etiam, ſi fiat,</s> </p> <note symbol="b" position="right" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli A B D, vel (quod \\ idem eſt) anguli D B C, qui \\ complementum est minoris \\ anguli obſeruati B D C, # ad radium \\ D A, proxi- \\ mè inuen- \\ tum # Ita ſin{us} anguli A B D, \\ differentiæ angulorum \\ obſeruatorum A D C, \\ B D C, # ad A B, \\ altitu- \\ dinem <lb/></note> <p> <s xml:id="echoid-s3071" xml:space="preserve">pro dibit rurſus altitudo quæſita A B, in partibus radij inuenti D A.</s> <s xml:id="echoid-s3072" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div198" type="section" level="1" n="87"> <head xml:id="echoid-head90" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3073" xml:space="preserve">2, <emph style="sc">Per</emph> problema 2. </s> <s xml:id="echoid-s3074" xml:space="preserve">vel 15. </s> <s xml:id="echoid-s3075" xml:space="preserve">inueſtigetur tam altitudo inacceſsibilis A C, <pb o="78" file="108" n="108" rhead="GEOMETR. PRACT."/> quam BC, ſecluſa menſoris ſtatura CF. </s> <s xml:id="echoid-s3076" xml:space="preserve">Altitudo namq; </s> <s xml:id="echoid-s3077" xml:space="preserve">BC, exaltitudine A C, <lb/>detracta notam relinquet altitudinem AB, quæ quæritur.</s> <s xml:id="echoid-s3078" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div199" type="section" level="1" n="88"> <head xml:id="echoid-head91" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3079" xml:space="preserve">3. </s> <s xml:id="echoid-s3080" xml:space="preserve"><emph style="sc">Inventa</emph> diſtantia DC, per ea, quæ in problemate 1. </s> <s xml:id="echoid-s3081" xml:space="preserve">& </s> <s xml:id="echoid-s3082" xml:space="preserve">2. </s> <s xml:id="echoid-s3083" xml:space="preserve">tradidimus, <lb/> <anchor type="note" xlink:href="" symbol="a"/> ſi fiat,</s> </p> <note symbol="a" position="left" xml:space="preserve">4. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} DC, # ad diſtantiam cogni- \\ tam DC, # Ita A C, Tangens maioris anguli \\ obſeruati ADC, # ad AC, <lb/></note> <p> <s xml:id="echoid-s3084" xml:space="preserve">reperietur|altitudo maior A C, in partibus diſtantiæ inuentæ D C. </s> <s xml:id="echoid-s3085" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et ſirur- <anchor type="note" xlink:label="note-108-03a" xlink:href="note-108-03"/> ſus fiat,</s> </p> <div xml:id="echoid-div199" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-108-03" xlink:href="note-108-03a" xml:space="preserve">4. Triang. <lb/>rectil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} DC, # ad diſtantiam in- \\ uentam DC: # Ita BC, Tangens minoris anguli ob- \\ ſeruati BDC, # ad B C, <lb/></note> <p> <s xml:id="echoid-s3086" xml:space="preserve">cognita fiet altitudo minor BC, in partibus eiuſdem diſtantiæ inuentæ DC, quæ <lb/>dempta ex maiore altitudine A C, notam relinquet altitudinem turris B A, <lb/>quæſitam.</s> <s xml:id="echoid-s3087" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3088" xml:space="preserve"><emph style="sc">Atqve</emph> hæc ratio commodiſsima eſt, quando in turri aliqua, vel ædificio, <lb/>cuius diſtantia à menſore cognita ſit, metiendum eſt interuallum perpendicu-<lb/>lare inter duas feneſtras.</s> <s xml:id="echoid-s3089" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div201" type="section" level="1" n="89"> <head xml:id="echoid-head92" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3090" xml:space="preserve">4. </s> <s xml:id="echoid-s3091" xml:space="preserve"><emph style="sc">Inventa</emph> rurſum diſtantia DC, per ea, quæin problemate 1. </s> <s xml:id="echoid-s3092" xml:space="preserve">& </s> <s xml:id="echoid-s3093" xml:space="preserve">2. </s> <s xml:id="echoid-s3094" xml:space="preserve">ſcripſi-<lb/>mus; </s> <s xml:id="echoid-s3095" xml:space="preserve">detrahatur Tangens B C, (poſito ſinu toto D C,) minoris anguli obſer-<lb/>uati A D C, vt nota remaneat AB, differen@a dictarum Tangentium. </s> <s xml:id="echoid-s3096" xml:space="preserve">Nam ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} DC, # ad diſtantiam in- \\ uentam DC, # ita A B, differentia Tangen- \\ tium AC, BC, # ad A B, altitu- \\ dinem, <lb/></note> <p> <s xml:id="echoid-s3097" xml:space="preserve">procreabitur altitudo quæſita AB, in partibus diſtantiæ inuentæ D C.</s> <s xml:id="echoid-s3098" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div202" type="section" level="1" n="90"> <head xml:id="echoid-head93" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3099" xml:space="preserve">5. </s> <s xml:id="echoid-s3100" xml:space="preserve"><emph style="sc">Inventa</emph> per problema 2. </s> <s xml:id="echoid-s3101" xml:space="preserve">vel 15. </s> <s xml:id="echoid-s3102" xml:space="preserve">altitudine montis B C, obſeruen-<lb/>tur anguli B D C, A D C, per radios D B, D A. </s> <s xml:id="echoid-s3103" xml:space="preserve">Poſito namque ſinu toto <lb/>D C, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt B C, Tangens \\ minoris anguli \\ obſeruati B D C, # ad B C, altitu- \\ dinem inuen- \\ tam: # Ita A B, differentia Tangen- \\ tium AC, CB, angulorum \\ obſeruatorum ADC, B DC. # ad A B <lb/></note> <p> <s xml:id="echoid-s3104" xml:space="preserve">prodibitrurſus altitudo nota AB, quam quærimus.</s> <s xml:id="echoid-s3105" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div203" type="section" level="1" n="91"> <head xml:id="echoid-head94" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s3106" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi AB, portio ſuperior totius alicuius altitudinis AC, deſideretur, in-<lb/>ueſtiganda erit per Problema 2. </s> <s xml:id="echoid-s3107" xml:space="preserve">vel 15. </s> <s xml:id="echoid-s3108" xml:space="preserve">tam tota altitudo AC, quam eius inferior <lb/>portio BC. </s> <s xml:id="echoid-s3109" xml:space="preserve">Earum enim differentia notam dabit ſuperiorem portionem AB.</s> <s xml:id="echoid-s3110" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3111" xml:space="preserve"><emph style="sc">Si</emph> autem media aliqua portio IB, cognoſcenda eſt, coniicienda rurſus erit <lb/>vtraque altitudo IC, BC, vt earum differentia IB, nota red datur.</s> <s xml:id="echoid-s3112" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3113" xml:space="preserve"><emph style="sc">Si</emph> deniqueinferior pars B C, proponitur inquirenda, fiet id per Problema <lb/>2. </s> <s xml:id="echoid-s3114" xml:space="preserve">vel 15.</s> <s xml:id="echoid-s3115" xml:space="preserve"/> </p> <pb o="79" file="109" n="109" rhead="LIBER SECVNDVS."/> <p> <s xml:id="echoid-s3116" xml:space="preserve">DISTANTIAM accliuem montis à loco menſoris vſque ad baſem <lb/>altitudinis monti impoſitæ, etiam non viſam, vna cumipſa altitudine, <lb/>quando menſor in aſcenſu montis conſiſtit, prope verum efficere <lb/>cognitam, beneficio Quadrantis.</s> <s xml:id="echoid-s3117" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div204" type="section" level="1" n="92"> <head xml:id="echoid-head95" xml:space="preserve">PROBLEMA XXII.</head> <figure> <image file="109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/109-01"/> </figure> <p> <s xml:id="echoid-s3118" xml:space="preserve">1. </s> <s xml:id="echoid-s3119" xml:space="preserve"><emph style="sc">Sit</emph> turris AB, monti impoſita, <lb/>in cuius aſcenſu ſeu latere menſor con-<lb/>ſiſtat in C, ex quo loco baſem turris <lb/> <anchor type="note" xlink:label="note-109-01a" xlink:href="note-109-01"/> videre non poſsit. </s> <s xml:id="echoid-s3120" xml:space="preserve">Erigatur haſta aliqua <lb/>C G, ad Horizontem, non autem ad la-<lb/>tus montis perpendicularis, ſitque men-<lb/>ſoris ſtatura CE. </s> <s xml:id="echoid-s3121" xml:space="preserve">Cogitetur ducta E K, <lb/>ad altitudinem perpendicularis: </s> <s xml:id="echoid-s3122" xml:space="preserve">Et in-<lb/>ſpecto cacumine A, per angulum A EK, <lb/>fiat alia ſtatio ſuperior in F, ductaque <lb/>FI, ad altitu dinem perpendiculari, inſpi-<lb/>ciatur idem cacumen A, per angulum <lb/> <anchor type="note" xlink:label="note-109-02a" xlink:href="note-109-02"/> A F I, <anchor type="note" xlink:href="" symbol="a"/> Et quia angulus A F G, duobus angulis FEA, FAE, eſt æqualis: </s> <s xml:id="echoid-s3123" xml:space="preserve">ſi dema-<lb/>tur A EF, complementum maioris anguli <lb/>A EK, in prima ſtatione obſeruati, reli-<lb/> <anchor type="note" xlink:label="note-109-03a" xlink:href="note-109-03"/> quus fiet angulus EAF, <anchor type="note" xlink:href="" symbol="b"/> Igitur ſi fiat,</s> </p> <div xml:id="echoid-div204" type="float" level="2" n="1"> <note position="right" xlink:label="note-109-01" xlink:href="note-109-01a" xml:space="preserve">In figura duc <lb/>rectam F A.</note> <note symbol="a" position="right" xlink:label="note-109-02" xlink:href="note-109-02a" xml:space="preserve">32. primi.</note> <note symbol="b" position="right" xlink:label="note-109-03" xlink:href="note-109-03a" xml:space="preserve">10. triang. <lb/>rectil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt ſin{us} anguli \\ EAF, differen- \\ tiæ complemen- \\ torũ angulorũ \\ obſeruationum # ad EF, diffe- \\ rentiam ſta- \\ tionum: # Ita ſin{us} anguli AFE, confta- \\ ti ex recto E F I, & minore \\ angulo AFI, in ſeounda ſta- \\ tione obſeruati, # ad A E \\ hypote- \\ nuſam. <lb/></note> <p> <s xml:id="echoid-s3124" xml:space="preserve">efficietur nota hypotenuſa AE. </s> <s xml:id="echoid-s3125" xml:space="preserve">Poſt hæc erigatur alius baculus DH, ad Hori-<lb/>zontem rectus, ſumptaque menſoris ſtatura DH, ipſi CE, æquali, & </s> <s xml:id="echoid-s3126" xml:space="preserve">ducta recta <lb/>HL, ad altitudinem perpendiculari, inſpiciatur punctum E, per angulum EHL, <lb/>& </s> <s xml:id="echoid-s3127" xml:space="preserve">perradium HEM, <anchor type="note" xlink:href="" symbol="c"/> quiipſi BCD, lateri montis parallelus erit. </s> <s xml:id="echoid-s3128" xml:space="preserve">Concipienda <anchor type="note" xlink:label="note-109-05a" xlink:href="note-109-05"/> enim ſunt tria puncta B, C, D, in vna recta iacere, ac ſi DC, producta ad baſem <lb/>turris perueniret. </s> <s xml:id="echoid-s3129" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quia verò angulus EHL, angulo MEK, æqualis eſt:</s> <s xml:id="echoid-s3130" xml:space="preserve">ſi hic ex <anchor type="note" xlink:label="note-109-06a" xlink:href="note-109-06"/> maiori angulo obſeruato AEN, inprima ſtatione dematur, reliquus fiet angulus <lb/>AEM. </s> <s xml:id="echoid-s3131" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s3132" xml:space="preserve">angulus EAN, cognitus, quippe cum ſit complementum <lb/>maioris anguli obſeruati AEN. </s> <s xml:id="echoid-s3133" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s3134" xml:space="preserve">AME, reliquus duorum rectorum cog-<lb/>nitus erit: </s> <s xml:id="echoid-s3135" xml:space="preserve">qui quidem etiam relinquitur, ſi complementum anguli M HL, in <lb/>ſtatione D, obſeruati ex duobusrectis detrahatur. </s> <s xml:id="echoid-s3136" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igitur ſi fiat,</s> </p> <div xml:id="echoid-div205" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-109-05" xlink:href="note-109-05a" xml:space="preserve">33: primi.</note> <note symbol="d" position="right" xlink:label="note-109-06" xlink:href="note-109-06a" xml:space="preserve">29. primi.</note> </div> <note symbol="e" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vtſin{us} anguli A M E, qui \\ velinquitur, ſicomplemen- \\ tum poſtremi anguli obſer- \\ uati MHL, ex duob{us} re- \\ ctis dematur # ad hypotenu- \\ ſam A E, \\ nuper inuẽ- \\ tam: # Ita ſin{us} anguli, \\ E A N, comple- \\ mẽti maioris an- \\ guli obſeruati AEN # ad M E, <lb/></note> <pb o="80" file="110" n="110" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s3137" xml:space="preserve">inuenta erit recta E M, hoc eſt, diſtantia quæſita CB. </s> <s xml:id="echoid-s3138" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et rurſum ſi fiat.</s> <s xml:id="echoid-s3139" xml:space="preserve"/> </p> <note symbol="a" position="left" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vtſin{us} eiuſ- \\ dem anguli \\ A M E. # adeanďem hy- \\ petenuſam \\ A E, nuper \\ inuentam: # Ita ſin{us} anguli AEM, quirelin- \\ quitur, ſi angulus poſtremo loco \\ obſeruat{us} MHL, vel MEK, ex \\ ængulo maiori obſeruato A E K, \\ detrahatur. # ad AM <lb/></note> <p> <s xml:id="echoid-s3140" xml:space="preserve">producetur AM, cuiſi addatur menſoris ſtatura MB, tota altitudo AB, cogni-<lb/>ta erit.</s> <s xml:id="echoid-s3141" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3142" xml:space="preserve"><emph style="sc">Danda</emph> autem erit opera diligenter, vt tria puncta B, C, D, invna recta ia-<lb/>ceant, hoc eſt, vtrecta ab vltima ſtatione D, per primam C, ducta per baſem B, <lb/>tranſeat, quod plus minusiudicio ſenſuum aſſequemur. </s> <s xml:id="echoid-s3143" xml:space="preserve">Nam per ea, quæ dicta <lb/>ſunt hoc loco, ſolum reperitur diſtantia à loco C, vſque ad punctum altitudi-<lb/>nis, in quod recta DC, protracta incidit, & </s> <s xml:id="echoid-s3144" xml:space="preserve">altitudo ab A, vſque ad idem pun-<lb/>ctum, quod non multum à puncto B, diſtabit, ſi diligentia adhibeatur in ſtatio-<lb/>nibus C, D, captandis. </s> <s xml:id="echoid-s3145" xml:space="preserve">Propter hanc cauſam in propoſitione diximus (prope <lb/>verum efficere cognitam) quia neque diſtantia C B, neque altitudo AB, præ-<lb/>cisè cognoſcitur, niſi quando tria puncta B, C, D, in recta linea iacent.</s> <s xml:id="echoid-s3146" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3147" xml:space="preserve">2. </s> <s xml:id="echoid-s3148" xml:space="preserve"><emph style="sc">Vt</emph> ſine numerorum auxilio problema effi cias, recurrendum erit ad ſu-<lb/>periora.</s> <s xml:id="echoid-s3149" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3150" xml:space="preserve">PROFVNDITATEM putei, vel ædificii cuiuſcunque ad perpen-<lb/>diculum erecti, ſi modo angulus fundi, vel ſignum aliquod in fundo <lb/>poſitum conſpiciatur, per Quadrantem reperire.</s> <s xml:id="echoid-s3151" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div207" type="section" level="1" n="93"> <head xml:id="echoid-head96" xml:space="preserve">PROBLEMA XXIII.</head> <figure> <image file="110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/110-01"/> </figure> <p> <s xml:id="echoid-s3152" xml:space="preserve">1. </s> <s xml:id="echoid-s3153" xml:space="preserve"><emph style="sc">Hoc</emph> nihil eſt aliud, niſi turrim ex eius vertice, quan-<lb/>do in Horizonte ſignum aliquod apparet, per duas ſta-<lb/>tiones in haſta aliqua factas metiri, vt problemate 4. </s> <s xml:id="echoid-s3154" xml:space="preserve">fa-<lb/>ctum eſt. </s> <s xml:id="echoid-s3155" xml:space="preserve">Operationem ergo eius problematis hic repe-<lb/>temus. </s> <s xml:id="echoid-s3156" xml:space="preserve">Sit puteus, ſeu ædificium erectum ABCM, cuius <lb/>angulus C, in fundo, vel ſignum C, in fundo poſitum <lb/>conſpicipoſsit. </s> <s xml:id="echoid-s3157" xml:space="preserve">Erecta haſta A E, in orificio putei, vel <lb/>ſummitate ædificij, fiant duæ ſtationes oculi menſoris in <lb/>D, E, inſpiciatur que punctum C, perradios DC, EC, fa-<lb/>cientes angulos BDC, BEC: </s> <s xml:id="echoid-s3158" xml:space="preserve">Sumptis deinde D F, E G, <lb/>æqualibus pro ſinubustotis, ducantur ad EB, perpendicu-<lb/>lares FH, GI, pro Tangentibus angulorũ B D C, B E C, <lb/>obſeruatorum. </s> <s xml:id="echoid-s3159" xml:space="preserve">Ducta quoque D L, ipſi E C, parallela <lb/>ſecãte FH, in K, vt KH, differentia ſit Tangentium, quem-<lb/>admodum problemate 4. </s> <s xml:id="echoid-s3160" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s3161" xml:space="preserve">Si igitur, vt ibi, fiat hic. <lb/></s> <s xml:id="echoid-s3162" xml:space="preserve"> <anchor type="note" xlink:label="note-110-03a" xlink:href="note-110-03"/> <pb o="81" file="111" n="111" rhead="LIBER SECVNDVS."/> prodibitrecta DB, nota, vtproblemate 4. </s> <s xml:id="echoid-s3163" xml:space="preserve">oſtendimus: </s> <s xml:id="echoid-s3164" xml:space="preserve">ex qua ſi ſubtrahatur <lb/>ſegmentum haſtæ AD. </s> <s xml:id="echoid-s3165" xml:space="preserve">inter orificium, & </s> <s xml:id="echoid-s3166" xml:space="preserve">oculum in prima ſtatione, reliqua fiet <lb/>profunditas, ſiue altitudo putei, velædificij AB.</s> <s xml:id="echoid-s3167" xml:space="preserve"/> </p> <div xml:id="echoid-div207" type="float" level="2" n="1"> <note style="it" position="right" xlink:label="note-110-03" xlink:href="note-110-03a" xml:space="preserve"> <lb/>Vt K H, differentia in- \\ ter Tangentes an- \\ gulorum obſeruato- \\ rum # ad F K, velad \\ G I, Tangentẽ \\ minorem: # ita D E, differentia ſtatio- \\ num, hoc eſt, ſpatium \\ inter oculos, velangulos \\ obſeruationum. # ad D B, <lb/></note> </div> </div> <div xml:id="echoid-div209" type="section" level="1" n="94"> <head xml:id="echoid-head97" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3168" xml:space="preserve">2. </s> <s xml:id="echoid-s3169" xml:space="preserve"><emph style="sc">Posito</emph> ſinu toto BC, fiat.</s> <s xml:id="echoid-s3170" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt DE, differentia Tangentium BD, \\ BE, quæ complementis angulorum \\ obſeruationum debentur, # ad DB, Tan- \\ gentem mi- \\ norem # ita DE, diffe- \\ rentia ſtatio- \\ nũ oculorum # ad DE, <lb/></note> <p> <s xml:id="echoid-s3171" xml:space="preserve">Nam numerus procreatus notam exhibebit eandem rectam DB, &</s> <s xml:id="echoid-s3172" xml:space="preserve">c.</s> <s xml:id="echoid-s3173" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div210" type="section" level="1" n="95"> <head xml:id="echoid-head98" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s3174" xml:space="preserve">3. </s> <s xml:id="echoid-s3175" xml:space="preserve"><emph style="sc">Si</emph> per ſolos ſinus operarilubeat, ita agendum erit. </s> <s xml:id="echoid-s3176" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam angulus <anchor type="note" xlink:label="note-111-02a" xlink:href="note-111-02"/> maior obſeruatus BDC, duobus angulis E, DCE, æqualis eſt; </s> <s xml:id="echoid-s3177" xml:space="preserve">ſi angulus E, <lb/>minor obſeruatus tollatur ex maiore BDC, notus remanebit angulus DCE, dif-<lb/>ferentia angulorum obſeruatorum. <lb/></s> <s xml:id="echoid-s3178" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/>Igitur ſi fiat, <anchor type="note" xlink:label="note-111-03a" xlink:href="note-111-03"/> <anchor type="note" xlink:label="note-111-04a" xlink:href="note-111-04"/> proſiliet nota hypotenuſa D C. </s> <s xml:id="echoid-s3179" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quapropter ſi rurſum fiat, <anchor type="note" xlink:label="note-111-05a" xlink:href="note-111-05"/> <anchor type="note" xlink:label="note-111-06a" xlink:href="note-111-06"/> euadet iterum cognita recta DB, &</s> <s xml:id="echoid-s3180" xml:space="preserve">c.</s> <s xml:id="echoid-s3181" xml:space="preserve"/> </p> <div xml:id="echoid-div210" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-111-02" xlink:href="note-111-02a" xml:space="preserve">32. primi.</note> <note symbol="b" position="right" xlink:label="note-111-03" xlink:href="note-111-03a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-111-04" xlink:href="note-111-04a" xml:space="preserve"> <lb/>Vt ſin{us} anguli D C E, \\ differentiæ angulo- \\ rum obſeruatorum. # ad D E, differen- \\ rentiam ſtationũ \\ oculorum: # ita ſin{us} anguli \\ E, minoris ob- \\ ſeruati # ad D C, <lb/></note> <note symbol="e" position="right" xlink:label="note-111-05" xlink:href="note-111-05a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-111-06" xlink:href="note-111-06a" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} \\ anguli re- \\ cti B, # ad hypotenuſam \\ DC, proximè in- \\ uentam: # ita ſin{us} anguli BCD, com- \\ plementi maioris anguli ob- \\ ſeruati # ad D B, <lb/></note> </div> <p> <s xml:id="echoid-s3182" xml:space="preserve">4. </s> <s xml:id="echoid-s3183" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi latitudo oriſicij AM, vel fundi BC, cognita fuerit, (Non erit <lb/>autem diffi cile eam aliqua menſura nota metiri) facilius in cognitionem altitu-<lb/>dinis, profunditatiſue AB, veniemus, pervnicam videlicet ſtationem in D, fa-<lb/>ctam, hoc modo. </s> <s xml:id="echoid-s3184" xml:space="preserve">Fiat, <lb/> <anchor type="note" xlink:label="note-111-07a" xlink:href="note-111-07"/> Numerus enim procreatus offeret DB, notam, vt ſupra, &</s> <s xml:id="echoid-s3185" xml:space="preserve">c.</s> <s xml:id="echoid-s3186" xml:space="preserve"/> </p> <div xml:id="echoid-div211" type="float" level="2" n="2"> <note style="it" position="right" xlink:label="note-111-07" xlink:href="note-111-07a" xml:space="preserve"> <lb/>Vt ſin{us} \\ tot{us} \\ CB, # ad BD, Tangentem anguli BCD, \\ complementi anguli obſeruatio- \\ nis: # ita latitudo co- \\ gnita CB, # ad DB. <lb/></note> </div> <p> <s xml:id="echoid-s3187" xml:space="preserve"><emph style="sc">Vel</emph> per ſolos ſinus, <anchor type="note" xlink:href="" symbol="d"/> ſi fiat, <anchor type="note" xlink:label="note-111-08a" xlink:href="note-111-08"/> <anchor type="note" xlink:label="note-111-09a" xlink:href="note-111-09"/> inuenietur rurſus DB, &</s> <s xml:id="echoid-s3188" xml:space="preserve">c.</s> <s xml:id="echoid-s3189" xml:space="preserve"/> </p> <div xml:id="echoid-div212" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-111-08" xlink:href="note-111-08a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-111-09" xlink:href="note-111-09a" xml:space="preserve"> <lb/>Vt ſin{us} anguli \\ B D C, obſer- \\ uationis # ad latitudinem co- \\ gnitam BC: # ita ſin{us} anguli BCD, \\ complemeti anguli ob- \\ ſeruationis. # ad DB, <lb/></note> </div> <pb o="82" file="112" n="112" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s3190" xml:space="preserve">5. </s> <s xml:id="echoid-s3191" xml:space="preserve"><emph style="sc">Vt</emph> idem aſſequaris ſine auxilio numerorum, conſule ea, quę problema-<lb/>te 4. </s> <s xml:id="echoid-s3192" xml:space="preserve">num. </s> <s xml:id="echoid-s3193" xml:space="preserve">4. </s> <s xml:id="echoid-s3194" xml:space="preserve">ſcripſimus.</s> <s xml:id="echoid-s3195" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3196" xml:space="preserve">PROFVNDITATEM vallis, eiuſdemque deſcenſum obliquum, ſi <lb/>non ſit valdè inæqualis, eiuſque terminus, vel aliquod in ea ſignum <lb/>conſpici poſſit, per Quadrantem ſcrutari.</s> <s xml:id="echoid-s3197" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div214" type="section" level="1" n="96"> <head xml:id="echoid-head99" xml:space="preserve">PROBLEMA XXIV.</head> <p> <s xml:id="echoid-s3198" xml:space="preserve">1. </s> <s xml:id="echoid-s3199" xml:space="preserve"><emph style="sc">Hoc</emph> etiam nihil aliud eſt, niſi altitudinem quampiam ex eius ſum-<lb/>mo faſtigio per duas ſtationes dimetiri, vt problemate 4. </s> <s xml:id="echoid-s3200" xml:space="preserve">oſtenſum eſt. </s> <s xml:id="echoid-s3201" xml:space="preserve">Sit enim <lb/>vallis inter duosmontes AB, FG, poſita, & </s> <s xml:id="echoid-s3202" xml:space="preserve">terminus ipſius C, ex monte AB, <lb/>poſsit conſpici. </s> <s xml:id="echoid-s3203" xml:space="preserve">Ere cta haſta aliqua AE, fiantin D, & </s> <s xml:id="echoid-s3204" xml:space="preserve">E, duæ ſtationes, obſer-<lb/>uenturque anguli D, & </s> <s xml:id="echoid-s3205" xml:space="preserve">E, inſpecto termino C, per radios DC, EC. </s> <s xml:id="echoid-s3206" xml:space="preserve">Intelliga-<lb/>tur autem recta EA, vſque ad baſem montis extenſa in B: </s> <s xml:id="echoid-s3207" xml:space="preserve">& </s> <s xml:id="echoid-s3208" xml:space="preserve">recta excurrens <lb/>AF, ipſi BG, parallela, vel ad EB, perpendicularis: </s> <s xml:id="echoid-s3209" xml:space="preserve">& </s> <s xml:id="echoid-s3210" xml:space="preserve">denique CH ipſi AB, pa-<lb/>rallela, <anchor type="note" xlink:href="" symbol="a"/> quæ altitudini AB, ęqualis erit, ita vt AB, vel HC, ſit profunditas val- <anchor type="note" xlink:label="note-112-01a" xlink:href="note-112-01"/> lis ab A, vſque ad baſem montis, & </s> <s xml:id="echoid-s3211" xml:space="preserve">IC, eius deſcenſus obliquus. </s> <s xml:id="echoid-s3212" xml:space="preserve">Liquidò au-<lb/> <anchor type="figure" xlink:label="fig-112-01a" xlink:href="fig-112-01"/> tem eonſtat, profunditatem AB, vel CH, exquiri poſſe, vt problemate 4. </s> <s xml:id="echoid-s3213" xml:space="preserve">alti-<lb/>tudo turris AB, ex duabus ſtationibus in haſta AE, factis, vel in duabus fe-<lb/>neſtris, vel certè, vtin præcedenti problemate profunditas putei AB, inuen-<lb/>tafuit.</s> <s xml:id="echoid-s3214" xml:space="preserve"/> </p> <div xml:id="echoid-div214" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-112-01" xlink:href="note-112-01a" xml:space="preserve">34. primi.</note> <figure xlink:label="fig-112-01" xlink:href="fig-112-01a"> <image file="112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/112-01"/> </figure> </div> <p> <s xml:id="echoid-s3215" xml:space="preserve"><emph style="sc">Descensvs</emph> autem obliquus IC, ita notus euadet. </s> <s xml:id="echoid-s3216" xml:space="preserve">Quoniam, vt in ante-<lb/>cedente problemate monſtratum eſt, angulus DCE, differentia eſt angulorum <lb/>BDC, BEC, obſeruatorum: </s> <s xml:id="echoid-s3217" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ſi fiat, <anchor type="note" xlink:label="note-112-02a" xlink:href="note-112-02"/> <anchor type="note" xlink:label="note-112-03a" xlink:href="note-112-03"/> cognita fiet recta DC, ex qua ſi detrahatur DI, (quam facilè ab oculo vſque ad <pb o="83" file="113" n="113" rhead="LIBER SECVNDVS."/> rectam AI, menſurare poteris, cum ſit exigua) notus remanebit deſcenſus ob-<lb/>liquus IC.</s> <s xml:id="echoid-s3218" xml:space="preserve"/> </p> <div xml:id="echoid-div215" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-112-02" xlink:href="note-112-02a" xml:space="preserve">10. Triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-112-03" xlink:href="note-112-03a" xml:space="preserve"> <lb/>Vt ſin{us} anguli D C E, \\ differentiæ inter angu- \\ losobſeruatos # ad D E, differen- \\ tiam ſtationum \\ oculorum: # ita ſin{us} anguli \\ E, minoris ob- \\ ſeruati # ad DC, <lb/></note> </div> <p> <s xml:id="echoid-s3219" xml:space="preserve">2. </s> <s xml:id="echoid-s3220" xml:space="preserve"><emph style="sc">Qvod</emph> ſi terminus C, in fundo cerni nequeat, inſpiciendum erit ex D, <lb/>& </s> <s xml:id="echoid-s3221" xml:space="preserve">E, aliquod aliud ſignum K, in valle, & </s> <s xml:id="echoid-s3222" xml:space="preserve">obſeruandi anguli BDK, BEK, per ra-<lb/>dios DK, EK. </s> <s xml:id="echoid-s3223" xml:space="preserve">Ex his enim rurſus profunditas AB, vel KL, deprehendetur, vt in <lb/>problemate 4. </s> <s xml:id="echoid-s3224" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s3225" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3226" xml:space="preserve"><emph style="sc">Qvin</emph> etiam, ſi in plano vallis commodè duæ ſtationes fieri poſsint: </s> <s xml:id="echoid-s3227" xml:space="preserve">cog-<lb/>noſci ex illis poterit altitudo montis AB, vel IM, perea, quę in problem. </s> <s xml:id="echoid-s3228" xml:space="preserve">2. </s> <s xml:id="echoid-s3229" xml:space="preserve">ſcri-<lb/>pſimus: </s> <s xml:id="echoid-s3230" xml:space="preserve">deſcenſus vero obliquus IC, hoceſt, interualluminter I, & </s> <s xml:id="echoid-s3231" xml:space="preserve">C, ex iis, <lb/>quæ in problemate 7. </s> <s xml:id="echoid-s3232" xml:space="preserve">tradita ſunt.</s> <s xml:id="echoid-s3233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3234" xml:space="preserve">3. </s> <s xml:id="echoid-s3235" xml:space="preserve"><emph style="sc">Eandem</emph> denique profunditatem CH, perſcrutari licebit ex altiore <lb/>monte N G, dummodo infimus terminus C, minoris montis ex cacumine N, <lb/>appareat, vel aliquod aliud ſignum in valle; </s> <s xml:id="echoid-s3236" xml:space="preserve">non aliter, quam in problemate 18. <lb/></s> <s xml:id="echoid-s3237" xml:space="preserve">vel 19. </s> <s xml:id="echoid-s3238" xml:space="preserve">altitudinem minorem ex maiori incognita indagare docuimus. </s> <s xml:id="echoid-s3239" xml:space="preserve">Nam <lb/>hic maior altitudo eſt NG, & </s> <s xml:id="echoid-s3240" xml:space="preserve">minor CH, cuius terminum C, ex N, cerni poſſe <lb/>ſtatuimus.</s> <s xml:id="echoid-s3241" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3242" xml:space="preserve">4. </s> <s xml:id="echoid-s3243" xml:space="preserve"><emph style="sc">Absqve</emph> numerorum multiplicatione, ac diuiſioneres peragetur, vt in <lb/>antecedentibus dictum eſt.</s> <s xml:id="echoid-s3244" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div217" type="section" level="1" n="97"> <head xml:id="echoid-head100" xml:space="preserve">FINIS LIBRI SECVNDI.</head> <figure> <image file="113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/113-01"/> </figure> <pb o="84" file="114" n="114"/> <figure> <image file="114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/114-01"/> </figure> </div> <div xml:id="echoid-div218" type="section" level="1" n="98"> <head xml:id="echoid-head101" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ <lb/>LIBER TERTIVS.</head> <p> <s xml:id="echoid-s3245" xml:space="preserve">Earundem linearum rectarum dimenſionem per <lb/>Quadratum Geometricum exequens.</s> <s xml:id="echoid-s3246" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3247" xml:space="preserve">QVONIAM dimenſio rectarum linearum per Qua-<lb/>drantem Aſtronomicum ſuperiori lib. </s> <s xml:id="echoid-s3248" xml:space="preserve">expoſita requirit <lb/>tabul{as} Sinuum, Tangentium, & </s> <s xml:id="echoid-s3249" xml:space="preserve">ſecantium, non ſem-<lb/>per autem eiuſmodi tabul{as} ad manum habere poſſu-<lb/>m{us}, immo neque omnes in illis verſati ſunt, atque exer-<lb/>citati: </s> <s xml:id="echoid-s3250" xml:space="preserve">propoſitum nobis tertio hoc libro eſt, line{as} rect{as}, <lb/>longitudines videlicet, latitudines, altitudines, & </s> <s xml:id="echoid-s3251" xml:space="preserve">pro-<lb/>funditates dimeriri per Quadratum Geometricum, vbi prædictis tabulis non <lb/>indigem{us}, ſed omnia per vmbram rectam, & </s> <s xml:id="echoid-s3252" xml:space="preserve">verſam; </s> <s xml:id="echoid-s3253" xml:space="preserve">vt vocant, expediuntur. <lb/></s> <s xml:id="echoid-s3254" xml:space="preserve">Qua tamen in re non nihil ab aliis ſcriptorib{us} diſſidebim{us}, quippe cum aliter <lb/>tam vmbram rectam, quam verſam in partes diuiſuri ſim{us}, quam ab illis fieri <lb/>ſolet: </s> <s xml:id="echoid-s3255" xml:space="preserve">vt nimirum per noſtram partitionem expediti{us} dimenſiones perfician-<lb/>tur; </s> <s xml:id="echoid-s3256" xml:space="preserve">quod prudens Lector facilè iudicabit, ſi noſtrã hanc diuiſionis rationem cum <lb/>aliorum partitione contulerit. </s> <s xml:id="echoid-s3257" xml:space="preserve">Sed principio Quadr atum Geometricum con-<lb/>ſtruendum est, explicandumque quo pacto tam in Quadrato ſtabili, quam in <lb/>pendulo vtraque vmbra, recta videlicet, ac verſa conſiderari debeat. </s> <s xml:id="echoid-s3258" xml:space="preserve">Neque <lb/>enim ſemper eundem ſitum prædictæ vmbræ in inſtrumento ſeruant, ſed pro va-<lb/>rietate vſuum non raro eum permutare ſolent, vt exiis, quæſequuntur, liquido <lb/>conſtabit.</s> <s xml:id="echoid-s3259" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div219" type="section" level="1" n="99"> <head xml:id="echoid-head102" xml:space="preserve">QVADRATI GEOMETRICI CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3260" xml:space="preserve">1. </s> <s xml:id="echoid-s3261" xml:space="preserve"><emph style="sc">Ex</emph> quauis materia ſolida & </s> <s xml:id="echoid-s3262" xml:space="preserve">dura conficiatur quadratum A B C D, ſiue <lb/> <anchor type="note" xlink:label="note-114-01a" xlink:href="note-114-01"/> ſolidum totum, ſiue excauatum: </s> <s xml:id="echoid-s3263" xml:space="preserve">vel potius ex quatuor regulis æqualibus AB, <lb/>BC, CD, DA, ita compactum, vt omnes in vno eo demqueplano exiſtant. </s> <s xml:id="echoid-s3264" xml:space="preserve">De- <pb o="85" file="115" n="115" rhead="LIBER TERTIVS."/> indein duab. </s> <s xml:id="echoid-s3265" xml:space="preserve">regulis BC, CD, ducantur tres parallelæ extremitatibus quadrati, <lb/> <anchor type="note" xlink:label="note-115-01a" xlink:href="note-115-01"/> pro partibus & </s> <s xml:id="echoid-s3266" xml:space="preserve">numeris vtriuſque vmbræ deſignandis, vt in figura apparet. </s> <s xml:id="echoid-s3267" xml:space="preserve">La-<lb/>tus BC, vmbræ rectæ, & </s> <s xml:id="echoid-s3268" xml:space="preserve">CD, vmbræ verſæ deſtinatur à Geometris: </s> <s xml:id="echoid-s3269" xml:space="preserve">Vtrumque <lb/> <anchor type="figure" xlink:label="fig-115-01a" xlink:href="fig-115-01"/> autem in 12. </s> <s xml:id="echoid-s3270" xml:space="preserve">partes æquales partiri ſolent omnes, qui de vſu Quadrati Geome-<lb/> <anchor type="note" xlink:label="note-115-02a" xlink:href="note-115-02"/> triciſcrip ſerunt: </s> <s xml:id="echoid-s3271" xml:space="preserve">Et ſi capacitas inſtrumenti permittit, ſingulas partes in 60. </s> <s xml:id="echoid-s3272" xml:space="preserve">ſub-<lb/>diuidunt, vt tota vmbra partes 720. </s> <s xml:id="echoid-s3273" xml:space="preserve">complectatur, vel in 100. </s> <s xml:id="echoid-s3274" xml:space="preserve">vt partes 1200. </s> <s xml:id="echoid-s3275" xml:space="preserve">in <lb/>vtraque vmbra exiſtant. </s> <s xml:id="echoid-s3276" xml:space="preserve">Ego vtramque vmbram in 10. </s> <s xml:id="echoid-s3277" xml:space="preserve">partes duntaxat æqua-<lb/>les diuido, niſi inſtrumentum ſit tantæ magnitudinis, vt commode vtra que re-<lb/>cta BC, CD, in 100, aut 1000. </s> <s xml:id="echoid-s3278" xml:space="preserve">partes ſecari poſsit, ſub diuiſis videlicet ſingulis <lb/>decimis partibus in 10. </s> <s xml:id="echoid-s3279" xml:space="preserve">vel 100. </s> <s xml:id="echoid-s3280" xml:space="preserve">particulas. </s> <s xml:id="echoid-s3281" xml:space="preserve">Figura porro ex vtraque vmbra cõ-<lb/>ſtans dici ſolet à Geometris Scala altimetra.</s> <s xml:id="echoid-s3282" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="1"> <note position="left" xlink:label="note-114-01" xlink:href="note-114-01a" xml:space="preserve">Compoſitio <lb/>Quadrati <lb/>Geometrici.</note> <note position="right" xlink:label="note-115-01" xlink:href="note-115-01a" xml:space="preserve">Vmbra recta, <lb/>& verſa in <lb/>quadrato quæ, <lb/>& in quot par <lb/>tes à Geome-<lb/>tris vtraque <lb/>ſecetur.</note> <figure xlink:label="fig-115-01" xlink:href="fig-115-01a"> <image file="115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/115-01"/> </figure> <note position="right" xlink:label="note-115-02" xlink:href="note-115-02a" xml:space="preserve">In quot part{es} <lb/>vtraque vm-<lb/>brain noſtro <lb/>quadrato di-<lb/>uidatur. <lb/>Scala altime-<lb/>tra quid.</note> </div> <p> <s xml:id="echoid-s3283" xml:space="preserve">2. </s> <s xml:id="echoid-s3284" xml:space="preserve"><emph style="sc">Antepono</emph> autem diuiſioni conſuetæ in 12. </s> <s xml:id="echoid-s3285" xml:space="preserve">vel 720. </s> <s xml:id="echoid-s3286" xml:space="preserve">vel 1200. </s> <s xml:id="echoid-s3287" xml:space="preserve">partes <lb/> <anchor type="note" xlink:label="note-115-03a" xlink:href="note-115-03"/> diuiſionem noſtram in partes 10. </s> <s xml:id="echoid-s3288" xml:space="preserve">vel 100. </s> <s xml:id="echoid-s3289" xml:space="preserve">vel 1000. </s> <s xml:id="echoid-s3290" xml:space="preserve">æquales, propterea quòd, ſi <lb/>inſtrumentum propter paruitatem ſectum ſit tantummodo in 10. </s> <s xml:id="echoid-s3291" xml:space="preserve">partes, facili <lb/>admodum negotio cognoſcere poſſumus, quot centeſimæ, vel etiam milleſi-<lb/>mæ partes in qualibet particula cuiuſcunque partis decimæ rectarum B C, CD, <lb/>aſsignata comprehendantur: </s> <s xml:id="echoid-s3292" xml:space="preserve">non ſecus ac ſi vtra que recta in 100. </s> <s xml:id="echoid-s3293" xml:space="preserve">vel 1000. <lb/></s> <s xml:id="echoid-s3294" xml:space="preserve">partes ſecta foret, vt Num. </s> <s xml:id="echoid-s3295" xml:space="preserve">14. </s> <s xml:id="echoid-s3296" xml:space="preserve">cap. </s> <s xml:id="echoid-s3297" xml:space="preserve">2. </s> <s xml:id="echoid-s3298" xml:space="preserve">lib. </s> <s xml:id="echoid-s3299" xml:space="preserve">1. </s> <s xml:id="echoid-s3300" xml:space="preserve">copioſè expoſuimus. </s> <s xml:id="echoid-s3301" xml:space="preserve">Huc accedit, <lb/>quod in dimetiendis lineis per Quadratum Geometricum fieri ſemper debeat <lb/>multiplicatio, aut diuiſio per omnes partes lateris BC, vel CD: </s> <s xml:id="echoid-s3302" xml:space="preserve">Manifeſtum au-<lb/>tem eſt, faciliorem eſſe multiplicationẽ, diuiſionemue per 10. </s> <s xml:id="echoid-s3303" xml:space="preserve">aut 100. </s> <s xml:id="echoid-s3304" xml:space="preserve">vel 1000. </s> <s xml:id="echoid-s3305" xml:space="preserve"><lb/>quàm per 12. </s> <s xml:id="echoid-s3306" xml:space="preserve">aut 720. </s> <s xml:id="echoid-s3307" xml:space="preserve">vel 1200. </s> <s xml:id="echoid-s3308" xml:space="preserve">cum illa fiat per ſolam appoſitionem, vel <lb/>detractionem 0. </s> <s xml:id="echoid-s3309" xml:space="preserve">vel 00. </s> <s xml:id="echoid-s3310" xml:space="preserve">vel 000. </s> <s xml:id="echoid-s3311" xml:space="preserve">vtin noſtra Arithmetica practica declaraui-<lb/>mus.</s> <s xml:id="echoid-s3312" xml:space="preserve"/> </p> <div xml:id="echoid-div220" type="float" level="2" n="2"> <note position="right" xlink:label="note-115-03" xlink:href="note-115-03a" xml:space="preserve">Quare noſtr<emph style="sub">a</emph> <lb/>diuiſio vmbr<emph style="sub">æ</emph> <lb/>præferatur a-<lb/>liorum diui-<lb/>ſioni.</note> </div> <p> <s xml:id="echoid-s3313" xml:space="preserve"><emph style="sc">Invenio</emph> quidem latus quadrati à do ctiſsimo<unsure/> 10. </s> <s xml:id="echoid-s3314" xml:space="preserve">Antonio Magino diui-<lb/>ſum quo que eſſe, & </s> <s xml:id="echoid-s3315" xml:space="preserve">quidem optimo conſilio, in 100. </s> <s xml:id="echoid-s3316" xml:space="preserve">partes æquales; </s> <s xml:id="echoid-s3317" xml:space="preserve">quamuis <lb/>ab eo regula non tradatur, qua cognoſcendũ ſit, quot partes mill eſimæ in qua-<lb/>uis particula vnius centeſimæ comprehendantur, quod tamen omnino neceſ-<lb/>ſarium eſt, vt dimenſiones, ac ſtellarum altitudines exquiſitè obſeruentur: </s> <s xml:id="echoid-s3318" xml:space="preserve">præ-<lb/>ſertim ſi propter inſtrumenti paruitatem latus in 10. </s> <s xml:id="echoid-s3319" xml:space="preserve">partes duntaxat commode <lb/>poſsit diuidi. </s> <s xml:id="echoid-s3320" xml:space="preserve">Id quod per noſtram do ctrinam, vt diximus, ſine magno laborè <lb/>effici poteſt.</s> <s xml:id="echoid-s3321" xml:space="preserve"/> </p> <pb o="86" file="116" n="116" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s3322" xml:space="preserve">3. </s> <s xml:id="echoid-s3323" xml:space="preserve"><emph style="sc">Post</emph> hæc è centro A, procedat filum cum perpendiculo, aut certere-<lb/> <anchor type="note" xlink:label="note-116-01a" xlink:href="note-116-01"/> gula tenuis, cum linea fiduciæ, quæ pendens libere moueatur, vt lib. </s> <s xml:id="echoid-s3324" xml:space="preserve">1. </s> <s xml:id="echoid-s3325" xml:space="preserve">capit. </s> <s xml:id="echoid-s3326" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s3327" xml:space="preserve">Num 6. </s> <s xml:id="echoid-s3328" xml:space="preserve">tradidimus: </s> <s xml:id="echoid-s3329" xml:space="preserve">& </s> <s xml:id="echoid-s3330" xml:space="preserve">in latere A B, duo pinnacidia affigantur, de quibus lib. </s> <s xml:id="echoid-s3331" xml:space="preserve">1. </s> <s xml:id="echoid-s3332" xml:space="preserve"><lb/>cap. </s> <s xml:id="echoid-s3333" xml:space="preserve">2. </s> <s xml:id="echoid-s3334" xml:space="preserve">Num. </s> <s xml:id="echoid-s3335" xml:space="preserve">5. </s> <s xml:id="echoid-s3336" xml:space="preserve">dictum eſt, ſi quadratum in ſuo vſu debeat eſſe pendulum. </s> <s xml:id="echoid-s3337" xml:space="preserve">Nam <lb/>ſi illud ſtabile eſſe velis, affigenda eſt circa centrum A, dioptra, hoc eſt, regula <lb/>cum linea fiduciæ, ac duobus pinnacidijs, eodem artificio conſtructis, vt libere <lb/>poſsit circumduci, & </s> <s xml:id="echoid-s3338" xml:space="preserve">in omni ſitu firmari, vt loco ſupra citato in Quadrante ſta-<lb/>bili faciendum eſſe præcepimus.</s> <s xml:id="echoid-s3339" xml:space="preserve"/> </p> <div xml:id="echoid-div221" type="float" level="2" n="3"> <note position="left" xlink:label="note-116-01" xlink:href="note-116-01a" xml:space="preserve">Quadratum <lb/>pendulum, ac <lb/>ſtabile.</note> </div> <p> <s xml:id="echoid-s3340" xml:space="preserve"><emph style="sc">Postremo</emph> iuxta latus A D, in plano quadrati (Nam ſi hoc fieret extra, <lb/>prope craſsitiem inſtrumenti, non poſſet quadratum in plano Horizontis lo-<lb/>carierectum ſupra latus A D. </s> <s xml:id="echoid-s3341" xml:space="preserve">quod tamen vt fiat, non raro vſus quadratipoſtu-<lb/>lat:) </s> <s xml:id="echoid-s3342" xml:space="preserve">apponàtur duæ tabellæ perforatæ cumfilo, & </s> <s xml:id="echoid-s3343" xml:space="preserve">perpendiculo, vt eius bene-<lb/>ficio dignoſcere poſsis, numinſtrumentum rectum ſit ad Horizontem, necne. <lb/></s> <s xml:id="echoid-s3344" xml:space="preserve">quod omnino neceſſarium eſt.</s> <s xml:id="echoid-s3345" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3346" xml:space="preserve">4. </s> <s xml:id="echoid-s3347" xml:space="preserve"><emph style="sc">Sed</emph> deceamus, quem ſitum vtraq; </s> <s xml:id="echoid-s3348" xml:space="preserve">vmbra in vſu quadrati habeat. </s> <s xml:id="echoid-s3349" xml:space="preserve">Res <lb/> <anchor type="note" xlink:label="note-116-02a" xlink:href="note-116-02"/> enim hæc non parui momenti eſt, vt in dimenſionibus nulla confuſio inter vm-<lb/>bras oriatur. </s> <s xml:id="echoid-s3350" xml:space="preserve">In quadrato ergo pendulo vmbra verſa opponitur ſemper lateri <lb/>pinna cidiorum: </s> <s xml:id="echoid-s3351" xml:space="preserve">recta autem cum eodem latere concurrit in puncto à centro A, <lb/>remotiori, vt in figura latus vmbræ rectæ eſt B C, verſæ autem C D. </s> <s xml:id="echoid-s3352" xml:space="preserve">At in qua-<lb/>drato ſtabili, ſi metienda ſit altitudo, & </s> <s xml:id="echoid-s3353" xml:space="preserve">centrum A, inferiorem obtineatlo cum, <lb/>in eoque oculus ponatur, latus vmbræ rectæ ſupremam occupabit ſedem; </s> <s xml:id="echoid-s3354" xml:space="preserve">la-<lb/>tus vero vmbræ verſæ vergere debebit verſusip ſam altitudinem: </s> <s xml:id="echoid-s3355" xml:space="preserve">ita vt tunc ba-<lb/>ſis inſtrumenti ſit A D. </s> <s xml:id="echoid-s3356" xml:space="preserve">quo pacto iterum vmbra recta eſt B C, & </s> <s xml:id="echoid-s3357" xml:space="preserve">verſa C D. </s> <s xml:id="echoid-s3358" xml:space="preserve">Si <lb/>autem centrum A, ſuperiorem locum poſsideat, & </s> <s xml:id="echoid-s3359" xml:space="preserve">oculus in extremitate dio-<lb/>ptræ exiſtat, (quod etiam fieri poteſt) latus vmbræ rectæ erit baſis C D, & </s> <s xml:id="echoid-s3360" xml:space="preserve">latus <lb/>vmbræ verſæ B C, prope oculum, & </s> <s xml:id="echoid-s3361" xml:space="preserve">ab altitudine metienda remotius. </s> <s xml:id="echoid-s3362" xml:space="preserve">Siau-<lb/>tem longitudo metienda proponatur, centrumque A, in ſuperiori loco ſitum <lb/>ſit, & </s> <s xml:id="echoid-s3363" xml:space="preserve">in eo oculus collocetur, erit quoque baſis C D, latus vmbræ rectæ: </s> <s xml:id="echoid-s3364" xml:space="preserve">latus <lb/>autem B C, vmbræ verſæ deputabitur, quod quidem verſus longitudinem me-<lb/>tiendam vergere debebit, & </s> <s xml:id="echoid-s3365" xml:space="preserve">longius ab oculo abeſſe. </s> <s xml:id="echoid-s3366" xml:space="preserve">At vero ſi centrum A, <lb/>ponaturinloco inferiori, ita vt baſis ſit A D, oculus autem in extremitate dio-<lb/>ptræ conſiſtat, (quod etiam fieri poteſt, præſertim ſi quadratum in ſublimi fue-<lb/>rit poſitum, vt in monte, vel turri aliqua) latus vmbræ rectæ erit B C, baſi A D, <lb/>oppoſitum, vmbrævero verſæ latus erit C D, quod ſcilicet longius à longitudi-<lb/>ne metienda recedit. </s> <s xml:id="echoid-s3367" xml:space="preserve">In vtro que porro quadrato centrum A, per diametrum <lb/>opponitur puncto, in quo vmbra recta cum verſa concurrit: </s> <s xml:id="echoid-s3368" xml:space="preserve">& </s> <s xml:id="echoid-s3369" xml:space="preserve">in ſtabili vm-<lb/>brarecta perpetuo vel ſupremum locum occupat, quando videlicet centrum <lb/>A, infimam ſedem tenet; </s> <s xml:id="echoid-s3370" xml:space="preserve">(vel infimum locum, quando ſcilicet centrum A, in <lb/>ſuperioriloco exiſtit, vt ex dictis liquet. </s> <s xml:id="echoid-s3371" xml:space="preserve">Hæcnon negligenter conſideranda <lb/>ſunt, ne in vario inſtrumentivſu vmbram rectam pro verſa accipias, aut contra; <lb/></s> <s xml:id="echoid-s3372" xml:space="preserve">quando quidem pro diuerſo ſitu quadrati ſtabilis tamlatus BC, quam CD, mo-<lb/>do vmbrærectæ, modo verſæ munus obire poteſt, vt diximus.</s> <s xml:id="echoid-s3373" xml:space="preserve"/> </p> <div xml:id="echoid-div222" type="float" level="2" n="4"> <note position="left" xlink:label="note-116-02" xlink:href="note-116-02a" xml:space="preserve">Vmbrarecta, <lb/>& verſa, quo <lb/>pacto in vtro-<lb/>que quadrato <lb/>cognoſcenda <lb/>ſit.</note> </div> <p> <s xml:id="echoid-s3374" xml:space="preserve">5. </s> <s xml:id="echoid-s3375" xml:space="preserve"><emph style="sc">Qvamvis</emph> autem vel ſola vmbrarecta, vel verſa ſatis ſit ad altitudines, <lb/>longitudines, profunditateſque perueſtigandas, vt ex ſequentibus fiet mani-<lb/> <anchor type="handwritten" xlink:label="hd-116-1a" xlink:href="hd-116-1"/> feſtum: </s> <s xml:id="echoid-s3376" xml:space="preserve">vtraque tamen aſſumitur à Geometris, eo quod interdum vmbrarecta <lb/>excedit latus B C, nimirum quando filum perpendiculi, aut linea fiduciæ ſecat <pb o="87" file="117" n="117" rhead="LIBER TERTIVS."/> latus CD: </s> <s xml:id="echoid-s3377" xml:space="preserve">Tunc enim neceſſario latus BC, produci debet, vt ſecaripoſsit. </s> <s xml:id="echoid-s3378" xml:space="preserve">Item <lb/>ſæpe numero vmbra verſa ſuperat latus C D, quando videlicet filum perpendi-<lb/>culi, aut linea fiduciæ interſecat latus B C: </s> <s xml:id="echoid-s3379" xml:space="preserve">Tunc enim neceſſario latus D C, <lb/>productum verſus C, ſecabitur, vt perſpicuum eſt. </s> <s xml:id="echoid-s3380" xml:space="preserve">Ne ergo cogamur vel la-<lb/>tus B C, vel C D, producere, aſſumenda eſt vmbra quidem verſa, quando recta <lb/>latus BC, excedit: </s> <s xml:id="echoid-s3381" xml:space="preserve">recta autem, quando verſa ſuo latere C D, maior eſt.</s> <s xml:id="echoid-s3382" xml:space="preserve"/> </p> <div xml:id="echoid-div223" type="float" level="2" n="5"> <handwritten xlink:label="hd-116-1" xlink:href="hd-116-1a"/> </div> <p> <s xml:id="echoid-s3383" xml:space="preserve">6. </s> <s xml:id="echoid-s3384" xml:space="preserve"><emph style="sc">Est</emph> autem perpetuo latus qua-<lb/> <anchor type="figure" xlink:label="fig-117-01a" xlink:href="fig-117-01"/> <anchor type="note" xlink:label="note-117-01a" xlink:href="note-117-01"/> drati, quod Gnomonem appellant, me-<lb/>dio loco proportionale inter vmbrã re-<lb/>ctam ac verſam. </s> <s xml:id="echoid-s3385" xml:space="preserve">Secet namque in qua-<lb/>drato pendulo filum perpendiculi, vel in <lb/>ſtabili linea fiduciæ, latus vmbræ BC, in E, <lb/>& </s> <s xml:id="echoid-s3386" xml:space="preserve">latus vmbræ DC, productũ in F. </s> <s xml:id="echoid-s3387" xml:space="preserve">Erunt <lb/>igitur triangula ABE, ADF, æquiangula, <lb/> <anchor type="note" xlink:label="note-117-02a" xlink:href="note-117-02"/> cum anguli B, D. </s> <s xml:id="echoid-s3388" xml:space="preserve">recti ſint,<anchor type="note" xlink:href="" symbol="a"/> & </s> <s xml:id="echoid-s3389" xml:space="preserve">tam alterni <anchor type="note" xlink:label="note-117-03a" xlink:href="note-117-03"/> BAE, DFA, quam BEA, DAF, æquales.</s> <s xml:id="echoid-s3390" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quamobrem erit vt B E, vmbra abſciſſa <lb/>ad gnomonem B A, ita gnomon A D, ad vmbram abſciſſam D F: </s> <s xml:id="echoid-s3391" xml:space="preserve">hoc eſt gno-<lb/>mon B A, vel A D, medio loco eſt proportionalis inter duas vmbras B E, D F, <lb/>quarum vna recta eſt, & </s> <s xml:id="echoid-s3392" xml:space="preserve">altera verſa. <lb/></s> <s xml:id="echoid-s3393" xml:space="preserve"> <anchor type="note" xlink:label="note-117-04a" xlink:href="note-117-04"/> </s> </p> <div xml:id="echoid-div224" type="float" level="2" n="6"> <figure xlink:label="fig-117-01" xlink:href="fig-117-01a"> <image file="117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/117-01"/> </figure> <note position="right" xlink:label="note-117-01" xlink:href="note-117-01a" xml:space="preserve">Gnomon me-<lb/>dio loco pro-<lb/>portionalis eſt <lb/>inter vmbrã <lb/>rectam, & <lb/>verſam. <lb/> <lb/></note> <note symbol="a" position="right" xlink:label="note-117-02" xlink:href="note-117-02a" xml:space="preserve">29. primi.</note> <note symbol="b" position="right" xlink:label="note-117-03" xlink:href="note-117-03a" xml:space="preserve">4. ſexti.</note> <note position="right" xlink:label="note-117-04" xlink:href="note-117-04a" xml:space="preserve">Redactio vm-<lb/>bræ rectæ ad <lb/>verſam, & <lb/>contra.</note> </div> <p> <s xml:id="echoid-s3394" xml:space="preserve">7. </s> <s xml:id="echoid-s3395" xml:space="preserve"><emph style="sc">Hinc</emph> facilis eſt reductio vnius vmbræ ad aliam, quod non raro vſu ve-<lb/>nit. </s> <s xml:id="echoid-s3396" xml:space="preserve">Nam ſi gnomon complectens partes 1000. </s> <s xml:id="echoid-s3397" xml:space="preserve">(in tot namq; </s> <s xml:id="echoid-s3398" xml:space="preserve">partes latus qua-<lb/>drati diuiſum concipere lubet) in ſe mu@tiplicetur, & </s> <s xml:id="echoid-s3399" xml:space="preserve">productus numerus qua-<lb/>dratus 1000000. </s> <s xml:id="echoid-s3400" xml:space="preserve">lateris AB, per alterutram vmbram diuidatur, indicabit Quo-<lb/>tiens partes alterius vmbræ: </s> <s xml:id="echoid-s3401" xml:space="preserve">hoc eſt, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt alterutra vmbra # ad gnomonem # itagnomon # ad alteram vmbram: <lb/></note> <p> <s xml:id="echoid-s3402" xml:space="preserve">hoc eſt, ſi quadratus numerus lateris quadrati, vel gnomonis, videlicet <lb/>1000000. </s> <s xml:id="echoid-s3403" xml:space="preserve">per alterutram vmbram diuidatur. </s> <s xml:id="echoid-s3404" xml:space="preserve">Verbi gratia ſi ponatur B E, vm-<lb/>bra recta partium 700. </s> <s xml:id="echoid-s3405" xml:space="preserve">diuidatur que numerus quadratus 1000000. </s> <s xml:id="echoid-s3406" xml:space="preserve">lateris A B, <lb/>per 700. </s> <s xml:id="echoid-s3407" xml:space="preserve">producetur vmbra verſa DF, partium 1428 {2/3}. </s> <s xml:id="echoid-s3408" xml:space="preserve">Sic etiam, ſi BE, ſtatua-<lb/>tur vmbra verſa partium 700. </s> <s xml:id="echoid-s3409" xml:space="preserve">reperietur vmbra recta DF, partium 1428 {2/3}. </s> <s xml:id="echoid-s3410" xml:space="preserve">Quod <lb/>ſi vna vmbra ſit 400. </s> <s xml:id="echoid-s3411" xml:space="preserve">erit altera 2500. </s> <s xml:id="echoid-s3412" xml:space="preserve">& </s> <s xml:id="echoid-s3413" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s3414" xml:space="preserve">Sediam ad vſum vtri-<lb/>uſque quadrati accedamus.</s> <s xml:id="echoid-s3415" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3416" xml:space="preserve">ALTITVDINEM Solis, vel ſtellę cuiuſuis per quadratum Geome-<lb/> <anchor type="handwritten" xlink:label="hd-117-2a" xlink:href="hd-117-2"/> tricum obſeruare.</s> <s xml:id="echoid-s3417" xml:space="preserve"/> </p> <div xml:id="echoid-div225" type="float" level="2" n="7"> <handwritten xlink:label="hd-117-2" xlink:href="hd-117-2a"/> </div> </div> <div xml:id="echoid-div227" type="section" level="1" n="100"> <head xml:id="echoid-head103" xml:space="preserve">PROBLEMA I.</head> <p> <s xml:id="echoid-s3418" xml:space="preserve">1. </s> <s xml:id="echoid-s3419" xml:space="preserve"><emph style="sc">Præparetvr</emph> baſis plana Horizontiæ quidiſtans, vt ſupra illam Qua-<lb/> <anchor type="note" xlink:label="note-117-06a" xlink:href="note-117-06"/> dratum ſtabile erectum, ſit ad Horizontem perpendiculare. </s> <s xml:id="echoid-s3420" xml:space="preserve">Eleuetur deinde <lb/>pendulum qua dratum, centro A, ad Solem, vel ſtellam verſo, ita vt eius pla-<lb/>num per centrum Solis, aut ſtellæ tranſeat, donec radius Solis per duo fo-<lb/>ramina pinnacidiorum tranſire deprehendatur: </s> <s xml:id="echoid-s3421" xml:space="preserve">vel radius viſualis per eadem <pb o="88" file="118" n="118" rhead="GEOMETR. PRACT."/> foramina pinnacidiorum ſtellam videat. </s> <s xml:id="echoid-s3422" xml:space="preserve">Idemque fiat cum quadrato ſtabili, <lb/>collocando nimirum eius latus AD, vel CD, ſupra baſem præparatam, ip ſumq; <lb/></s> <s xml:id="echoid-s3423" xml:space="preserve">circumducendo, ita vt eius planum per centrum Solis autſtellæ incedat; </s> <s xml:id="echoid-s3424" xml:space="preserve">ac de-<lb/>nique eleuando dioptram, donecradius Solis per foramina pinnacidiorũ tran-<lb/> <anchor type="note" xlink:label="note-118-01a" xlink:href="note-118-01"/> ſeat, vel radius viſualis per eadem foramina ſtellam conſpiciat. </s> <s xml:id="echoid-s3425" xml:space="preserve">Quo peracto, <lb/>conſideretur ſumma diligentia angulus, quem filum perpendiculi, vel linea fi-<lb/>duciæ in dioptra cum proximo latere quadrati conſtituit, inueſtigando magna <lb/>cum cura, & </s> <s xml:id="echoid-s3426" xml:space="preserve">diligentia perea, quæ lib. </s> <s xml:id="echoid-s3427" xml:space="preserve">1. </s> <s xml:id="echoid-s3428" xml:space="preserve">cap. </s> <s xml:id="echoid-s3429" xml:space="preserve">2. </s> <s xml:id="echoid-s3430" xml:space="preserve">Num. </s> <s xml:id="echoid-s3431" xml:space="preserve">14. </s> <s xml:id="echoid-s3432" xml:space="preserve">tradidimus, quot par-<lb/>tes milleſimæ ex vmbra ſiue recta, ſine verſa abſciſſæ ſint à filo perpendiculi, vel <lb/>linea fiduciæ. </s> <s xml:id="echoid-s3433" xml:space="preserve">Ille enim angulus, ſi quidem vmbra recta interſecetur, (quæin <lb/>quadrato ſtabili vel ſupremum locum, vel infimum occupat: </s> <s xml:id="echoid-s3434" xml:space="preserve">Inpendulo vero <lb/>cumlatere pinnacidiorum coniungitur, vt ſupra diximus) dabit, vt Num. </s> <s xml:id="echoid-s3435" xml:space="preserve">2. </s> <s xml:id="echoid-s3436" xml:space="preserve">de-<lb/>monſtrabimus, complementum altitudinis Solis autſtellæ: </s> <s xml:id="echoid-s3437" xml:space="preserve">Si vero vmbram <lb/> <anchor type="figure" xlink:label="fig-118-01a" xlink:href="fig-118-01"/> verſam filum, aut dioptra interſecet, ipſemet angulus altitu-<lb/>dinem exhibebit. </s> <s xml:id="echoid-s3438" xml:space="preserve">Quantitatem porro huius anguli ita cogno-<lb/> <anchor type="handwritten" xlink:label="hd-118-2a" xlink:href="hd-118-2"/> ſcemus. </s> <s xml:id="echoid-s3439" xml:space="preserve">Sit quadratum ſiue pendulum ſiue ſtabile (eadem e-<lb/>nim eſt in vtroque ratio) A B C D, abſciſſaque ſit vmbra recta <lb/>B E, quæ in partibus milleſimis lateris B C, reperietur, vt cap. </s> <s xml:id="echoid-s3440" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s3441" xml:space="preserve">Num. </s> <s xml:id="echoid-s3442" xml:space="preserve">14. </s> <s xml:id="echoid-s3443" xml:space="preserve">lib. </s> <s xml:id="echoid-s3444" xml:space="preserve">1. </s> <s xml:id="echoid-s3445" xml:space="preserve">docuimus, etiamſi latusipſum ſit ſolumin 10. </s> <s xml:id="echoid-s3446" xml:space="preserve"><lb/>vel 100. </s> <s xml:id="echoid-s3447" xml:space="preserve">partes diuiſum: </s> <s xml:id="echoid-s3448" xml:space="preserve">quæ quidem vmbra B E, ſtatuatur <lb/>verbi gratia, eſſe partium 850. </s> <s xml:id="echoid-s3449" xml:space="preserve">Quia ergo duo latera AB, BE, triangulirectanguli <lb/> <anchor type="note" xlink:label="note-118-02a" xlink:href="note-118-02"/> ABE, nota ſunt, cum AB, ſit 1000. </s> <s xml:id="echoid-s3450" xml:space="preserve">& </s> <s xml:id="echoid-s3451" xml:space="preserve">BE, 850. </s> <s xml:id="echoid-s3452" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ſi fiat,</s> </p> <div xml:id="echoid-div227" type="float" level="2" n="1"> <note position="right" xlink:label="note-117-06" xlink:href="note-117-06a" xml:space="preserve">Altitudo So-<lb/>lis, velſtellæ, <lb/>quo pacto per <lb/>quadratum <lb/>cognoſcatur.</note> <note position="left" xlink:label="note-118-01" xlink:href="note-118-01a" xml:space="preserve">Quando an-<lb/>gul{us}, quem <lb/>filism cum <lb/>pr@ximo qua-<lb/>drati latere <lb/>facit, offerat <lb/>altitudinem <lb/>ſolis; & quan-<lb/>do complemẽ-<lb/>tum altitudi-<lb/>nis.</note> <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/xxxxxxxx/figures/118-01"/> </figure> <handwritten xlink:label="hd-118-2" xlink:href="hd-118-2a"/> <note symbol="a" position="left" xlink:label="note-118-02" xlink:href="note-118-02a" xml:space="preserve">8. trìang. re-<lb/>ctil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} quadra- \\ ti A B, 1000. # ad ſinum to- \\ tum 100000. # Ita lat{us}, vel vmbra \\ BE, 850. # ad Tangentem \\ anguli BAE, \\ 85000 <lb/></note> <p> <s xml:id="echoid-s3453" xml:space="preserve">(quod quidem factum erit, ſi ad latus B E, hoc eſt, ad 850. </s> <s xml:id="echoid-s3454" xml:space="preserve">duæ cifræ apponan-<lb/>tur) reperietur Tangens anguli B A E, 85000. </s> <s xml:id="echoid-s3455" xml:space="preserve">quæ in tabula Tangentium non <lb/>reperitur; </s> <s xml:id="echoid-s3456" xml:space="preserve">ſed proxime minor eſt 84956. </s> <s xml:id="echoid-s3457" xml:space="preserve">cuireſpondent gradus 40. </s> <s xml:id="echoid-s3458" xml:space="preserve">min. </s> <s xml:id="echoid-s3459" xml:space="preserve">21. </s> <s xml:id="echoid-s3460" xml:space="preserve">Et <lb/>quia differentia inter Tangentem inuentam 85000. </s> <s xml:id="echoid-s3461" xml:space="preserve">& </s> <s xml:id="echoid-s3462" xml:space="preserve">84956. </s> <s xml:id="echoid-s3463" xml:space="preserve">in tabula ſum-<lb/> <anchor type="handwritten" xlink:label="hd-118-2a" xlink:href="hd-118-2"/> ptam, eſt 44. </s> <s xml:id="echoid-s3464" xml:space="preserve">Differentia autẽ inter duas Tangentes proximas 84956. </s> <s xml:id="echoid-s3465" xml:space="preserve">& </s> <s xml:id="echoid-s3466" xml:space="preserve">85006. <lb/></s> <s xml:id="echoid-s3467" xml:space="preserve">eſt 50. </s> <s xml:id="echoid-s3468" xml:space="preserve">cui debetur 1. </s> <s xml:id="echoid-s3469" xml:space="preserve">minutum, ſiue Sec. </s> <s xml:id="echoid-s3470" xml:space="preserve">60. </s> <s xml:id="echoid-s3471" xml:space="preserve">propterea quòd poſteriori Tãgen-<lb/>ti 85006. </s> <s xml:id="echoid-s3472" xml:space="preserve">reſpondet vnum minutum amplius, quàm Tangenti priori 84956. </s> <s xml:id="echoid-s3473" xml:space="preserve">re-<lb/>periemus perregulam trium, quot ſecunda differentiæ 44. </s> <s xml:id="echoid-s3474" xml:space="preserve">congruant; </s> <s xml:id="echoid-s3475" xml:space="preserve">ſi dica-<lb/>mus. </s> <s xml:id="echoid-s3476" xml:space="preserve">Si differentia 50. </s> <s xml:id="echoid-s3477" xml:space="preserve">poſcit, ſec. </s> <s xml:id="echoid-s3478" xml:space="preserve">60. </s> <s xml:id="echoid-s3479" xml:space="preserve">quot ſecunda poſcit differentia 44? </s> <s xml:id="echoid-s3480" xml:space="preserve">inue-<lb/>niemuſque ſec. </s> <s xml:id="echoid-s3481" xml:space="preserve">52 {4/5}. </s> <s xml:id="echoid-s3482" xml:space="preserve">hoc eſt, ſec. </s> <s xml:id="echoid-s3483" xml:space="preserve">53. </s> <s xml:id="echoid-s3484" xml:space="preserve">fere. </s> <s xml:id="echoid-s3485" xml:space="preserve">Angulus ergo BAE, continent grad. </s> <s xml:id="echoid-s3486" xml:space="preserve"><lb/>40. </s> <s xml:id="echoid-s3487" xml:space="preserve">Min. </s> <s xml:id="echoid-s3488" xml:space="preserve">21. </s> <s xml:id="echoid-s3489" xml:space="preserve">ſec. </s> <s xml:id="echoid-s3490" xml:space="preserve">53. </s> <s xml:id="echoid-s3491" xml:space="preserve">paulo minus. </s> <s xml:id="echoid-s3492" xml:space="preserve">Complementum igitur, nimirum grad. </s> <s xml:id="echoid-s3493" xml:space="preserve">49. </s> <s xml:id="echoid-s3494" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s3495" xml:space="preserve">38. </s> <s xml:id="echoid-s3496" xml:space="preserve">ſec. </s> <s xml:id="echoid-s3497" xml:space="preserve">7. </s> <s xml:id="echoid-s3498" xml:space="preserve">fere, oſtendet altitudinem Solis, vel ſtellæ. </s> <s xml:id="echoid-s3499" xml:space="preserve">Et ſi vmbra D F, eſt <lb/>verſa partium quo que 850. </s> <s xml:id="echoid-s3500" xml:space="preserve">eritip ſemet angulus DAF, inuentus grad. </s> <s xml:id="echoid-s3501" xml:space="preserve">40. </s> <s xml:id="echoid-s3502" xml:space="preserve">Min. </s> <s xml:id="echoid-s3503" xml:space="preserve"><lb/>21. </s> <s xml:id="echoid-s3504" xml:space="preserve">ſec. </s> <s xml:id="echoid-s3505" xml:space="preserve">53. </s> <s xml:id="echoid-s3506" xml:space="preserve">altitudo Solis, vel ſtellæ. </s> <s xml:id="echoid-s3507" xml:space="preserve">Atque hoc modo ſi diligenter partes vm-<lb/>brę explorabimus reſpectu lateris BC, vel CD, 1000. </s> <s xml:id="echoid-s3508" xml:space="preserve">quemadmodum lib. </s> <s xml:id="echoid-s3509" xml:space="preserve">1. </s> <s xml:id="echoid-s3510" xml:space="preserve">cap. </s> <s xml:id="echoid-s3511" xml:space="preserve"><lb/>2. </s> <s xml:id="echoid-s3512" xml:space="preserve">Numer. </s> <s xml:id="echoid-s3513" xml:space="preserve">14. </s> <s xml:id="echoid-s3514" xml:space="preserve">traditum eſt, inuenietur ſemper altitudo Solis, autſtellæ in grad. </s> <s xml:id="echoid-s3515" xml:space="preserve"><lb/>Min. </s> <s xml:id="echoid-s3516" xml:space="preserve">& </s> <s xml:id="echoid-s3517" xml:space="preserve">ſec. </s> <s xml:id="echoid-s3518" xml:space="preserve">Sed quia moleſtũ eſt, ac laborioſum, per differentias Tangentium <lb/>Secunda inquirere, ſatis erit Tangentem anguli inuentam in tabula quærere, & </s> <s xml:id="echoid-s3519" xml:space="preserve"><lb/>ſi quidem reperta fuerit, accipere gradus, & </s> <s xml:id="echoid-s3520" xml:space="preserve">minuta reſpondentia; </s> <s xml:id="echoid-s3521" xml:space="preserve">ſi vero non <lb/>fuerit inuenta, ſumere Tangentem, vel minorem, vel maiorem quæ nimirum mi-<lb/>nus ab inuenta differat, &</s> <s xml:id="echoid-s3522" xml:space="preserve">c. </s> <s xml:id="echoid-s3523" xml:space="preserve">Hac ratione ſumenda erit in noſtro exemplo Tan-<lb/>gens 86006. </s> <s xml:id="echoid-s3524" xml:space="preserve">cui reſp ondent grad. </s> <s xml:id="echoid-s3525" xml:space="preserve">40. </s> <s xml:id="echoid-s3526" xml:space="preserve">Min. </s> <s xml:id="echoid-s3527" xml:space="preserve">22. </s> <s xml:id="echoid-s3528" xml:space="preserve">complementum autem erit grad. </s> <s xml:id="echoid-s3529" xml:space="preserve"><lb/> <anchor type="handwritten" xlink:label="hd-118-2a" xlink:href="hd-118-2"/> <pb o="89" file="119" n="119" rhead="LIBER TERTIVS."/> 49. </s> <s xml:id="echoid-s3530" xml:space="preserve">Min 38. </s> <s xml:id="echoid-s3531" xml:space="preserve">veluti prius. </s> <s xml:id="echoid-s3532" xml:space="preserve">ſolum deſunt ſec. </s> <s xml:id="echoid-s3533" xml:space="preserve">7. </s> <s xml:id="echoid-s3534" xml:space="preserve">quæ nullius ſunt momenti. </s> <s xml:id="echoid-s3535" xml:space="preserve">Vt au-<lb/> <anchor type="handwritten" xlink:label="hd-119-1a" xlink:href="hd-119-1"/> tem ſtudioſos moleſtia hac ſupputandi liberem, conſtruxi ſequentem tabulam <lb/>pro ſingulis partibus milleſimis vtriuſque vmbræ: </s> <s xml:id="echoid-s3536" xml:space="preserve">In qua ſi centenæ in vertice, <lb/>& </s> <s xml:id="echoid-s3537" xml:space="preserve">reliquæ vnitates in latere ſumantur, illico in angulo communi reperientur <lb/>gradus, ac Min. </s> <s xml:id="echoid-s3538" xml:space="preserve">pro angulo quæſito: </s> <s xml:id="echoid-s3539" xml:space="preserve">qui videlicet altitudinem Solis aut ſtellæ <lb/> <anchor type="note" xlink:label="note-119-01a" xlink:href="note-119-01"/> indicabit, ſi partes milleſimæ ad vmbram verſam ſpectent: </s> <s xml:id="echoid-s3540" xml:space="preserve">ſi autem partes ad <lb/>vmbram rectam pertinent Complementum eius anguli altitu dinem Solis ſtel-<lb/>læue oſtendet, vt paulo infra demonſtrabimus. </s> <s xml:id="echoid-s3541" xml:space="preserve">Componetur autẽ tabula hęc, <lb/>ſi ſingulis partibus milleſimis vmbræ apponantur ad dextram quinque cifræ, & </s> <s xml:id="echoid-s3542" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-119-02a" xlink:href="note-119-02"/> ex toto illo numero abijciantur tres cifræ: </s> <s xml:id="echoid-s3543" xml:space="preserve">quod idem eſt, ac ſi apponantur tã-<lb/>tum duæ cifræ, vt Tangentes angulorum habeantur, Ita factum eſſe vides ſupra: <lb/></s> <s xml:id="echoid-s3544" xml:space="preserve">Nam ad partes vmbræ 850. </s> <s xml:id="echoid-s3545" xml:space="preserve">adiectæ ſunt duæ cifræ, vt Tangens fieret 85000. </s> <s xml:id="echoid-s3546" xml:space="preserve"><lb/>Hac enim ratione fit multip licatio vmbræ abſciſſæ per ſinum totum 100000. </s> <s xml:id="echoid-s3547" xml:space="preserve">& </s> <s xml:id="echoid-s3548" xml:space="preserve"><lb/> <anchor type="handwritten" xlink:label="hd-119-1a" xlink:href="hd-119-1"/> diuiſio per 1000. </s> <s xml:id="echoid-s3549" xml:space="preserve">vtin Arithmetica noſtra practica ſcripſimus. </s> <s xml:id="echoid-s3550" xml:space="preserve">Quod ſi quatuor <lb/>cifræ partibus milleſimis apponantur, habentur eadem ratione Tangẽtes, poſi-<lb/>to ſinu toto 10000000.</s> <s xml:id="echoid-s3551" xml:space="preserve"/> </p> <div xml:id="echoid-div228" type="float" level="2" n="2"> <handwritten xlink:label="hd-118-2" xlink:href="hd-118-2a"/> <handwritten xlink:label="hd-118-2" xlink:href="hd-118-2a"/> <handwritten xlink:label="hd-119-1" xlink:href="hd-119-1a"/> <note position="right" xlink:label="note-119-01" xlink:href="note-119-01a" xml:space="preserve">Vſus tabule <lb/>gnomonicæ <lb/>ſequentis. <lb/> <lb/></note> <note position="right" xlink:label="note-119-02" xlink:href="note-119-02a" xml:space="preserve">Compoſiti@ <lb/>tabulæ gno-<lb/>monicæfacil-<lb/>lima.</note> <handwritten xlink:label="hd-119-1" xlink:href="hd-119-1a"/> </div> <p> <s xml:id="echoid-s3552" xml:space="preserve">2. </s> <s xml:id="echoid-s3553" xml:space="preserve"><emph style="sc">Vervm</emph> demonſtremus prius, angulum, quem filum perpendiculi, vel <lb/>linea fiduciæ ſecans vmbramrectam cum pro ximo latere facit, in vtro que Qua-<lb/>drato eſſe complementum altitudinis Solis, ſeu ſtellæ; </s> <s xml:id="echoid-s3554" xml:space="preserve">illum vero, quem filum <lb/> <anchor type="handwritten" xlink:label="hd-119-1a" xlink:href="hd-119-1"/> aut fiduciæ linea vmbram verſam ſecans efficit, exhibere ipſam altitudinem. </s> <s xml:id="echoid-s3555" xml:space="preserve">Sit <lb/>ergo primum quadratum pendulum ABCD, & </s> <s xml:id="echoid-s3556" xml:space="preserve">Quadrans circuli Verticalis per <lb/>ſtellam ducti AEF, ita vt AE, Horizonti æquidiſter: </s> <s xml:id="echoid-s3557" xml:space="preserve">ſintq; </s> <s xml:id="echoid-s3558" xml:space="preserve">tres altitudines ſtel-<lb/>læ, EG, grad. </s> <s xml:id="echoid-s3559" xml:space="preserve">45. </s> <s xml:id="echoid-s3560" xml:space="preserve">EH, maior, & </s> <s xml:id="echoid-s3561" xml:space="preserve">EK, minor. </s> <s xml:id="echoid-s3562" xml:space="preserve">Quando ergo latus pinnacidiorum <lb/>AB, cum radio GA, coincidit, erit angulus altitudinis ſemirectus GAE, <anchor type="note" xlink:href="" symbol="a"/> quiæ- <anchor type="note" xlink:label="note-119-03a" xlink:href="note-119-03"/> qualis eſt ſuo complemento FAG, propter æquales arcus EG, FG, <anchor type="note" xlink:href="" symbol="b"/> hoc eſt, an- <anchor type="note" xlink:label="note-119-04a" xlink:href="note-119-04"/> <anchor type="figure" xlink:label="fig-119-01a" xlink:href="fig-119-01"/> <anchor type="note" xlink:label="note-119-05a" xlink:href="note-119-05"/> <anchor type="note" xlink:label="note-119-06a" xlink:href="note-119-06"/> gulo quem filum cumlatere AB, facit. </s> <s xml:id="echoid-s3563" xml:space="preserve">Cum ergo diameter quadrati<anchor type="note" xlink:href="" symbol="c"/> ſecet an- gulos eius rectos bifariam, in ſemirectos nimirum, tranſibit filum per angulum <lb/>C: </s> <s xml:id="echoid-s3564" xml:space="preserve">Ac proinde tam CAB, quam CAD, æqualis tunc erit angulo altitudinis E-<lb/>AG. </s> <s xml:id="echoid-s3565" xml:space="preserve">Quando autem pinnacidiorum latus cum radio HA, coincidit, erit angu-<lb/>lus altitudinis EAH, ſemirecto maior, & </s> <s xml:id="echoid-s3566" xml:space="preserve">angulus complementi FAH, ſemirecto <lb/>minor,<anchor type="note" xlink:href="" symbol="d"/> cui æqualis eſt angulus IAB: </s> <s xml:id="echoid-s3567" xml:space="preserve">Acpropterea filum vmbram rectam in- <anchor type="note" xlink:label="note-119-07a" xlink:href="note-119-07"/> <pb o="90" file="120" n="120" rhead="GEOMETR. PRACT."/> terſecabit, facietque angulum complementi altitudinis IAB, ſemirecto minorẽ. <lb/></s> <s xml:id="echoid-s3568" xml:space="preserve">Quando denique latus AB, idem efficitur cum radio KA, erit angulus altitudinis <lb/>EAK, ſemirecto minor, & </s> <s xml:id="echoid-s3569" xml:space="preserve">angulus complementi FAK, ſemirecto maior,<anchor type="note" xlink:href="" symbol="a"/> cui æ- <anchor type="note" xlink:label="note-120-01a" xlink:href="note-120-01"/> qualis eſt angulus IAB: </s> <s xml:id="echoid-s3570" xml:space="preserve">acproinde filum vmbram verſam abrumpet, conſti-<lb/>tuetque angulum altitudinis I A D, ſemirecto minorem. </s> <s xml:id="echoid-s3571" xml:space="preserve">Idem prorſits cernitur <lb/>in quadrato ſtabili ABCD, in quo vmbrærectæ latus eſt infimum CD, cum cen-<lb/>trum A, ſupremum occupet locum, vt ſuprâ in conſtructione Quadrati Num. <lb/></s> <s xml:id="echoid-s3572" xml:space="preserve">4. </s> <s xml:id="echoid-s3573" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s3574" xml:space="preserve">Vbi rurſus perſpicitur, quando altitudo EG, eſt grad. </s> <s xml:id="echoid-s3575" xml:space="preserve">45. </s> <s xml:id="echoid-s3576" xml:space="preserve">radium <lb/>G A, cadere in angulum oppoſitum C, ac proinde tam angulum C A B, quam <lb/>CAD, æqualem eſſe angulo altitudinis, nimirum ſemirectum. </s> <s xml:id="echoid-s3577" xml:space="preserve">Quando autem <lb/>altitudo E H, maior eſt, quam grad. </s> <s xml:id="echoid-s3578" xml:space="preserve">45. </s> <s xml:id="echoid-s3579" xml:space="preserve">angulum I A D, quem linea fiduciæ AI, <lb/> <anchor type="note" xlink:label="note-120-02a" xlink:href="note-120-02"/> vmbram rectam auferens DI, cum latere AD, facit, <anchor type="note" xlink:href="" symbol="b"/> æqualem eſſe angulo FAH, qui complementum eſt anguli altitudinis EAH. </s> <s xml:id="echoid-s3580" xml:space="preserve">Quando denique altitudo EK, <lb/>minor eſt, quam grad. </s> <s xml:id="echoid-s3581" xml:space="preserve">45. </s> <s xml:id="echoid-s3582" xml:space="preserve">angulum IAB, quemlinea fiduciæ AI, vmbram verſam <lb/>BI, aſcindens cumlatere AB, conſtituit,<anchor type="note" xlink:href="" symbol="c"/> æqualem eſſe angulo ipſi altitudinis E- <anchor type="note" xlink:label="note-120-03a" xlink:href="note-120-03"/> AK: </s> <s xml:id="echoid-s3583" xml:space="preserve">quæ omnia demonſtranda erant. </s> <s xml:id="echoid-s3584" xml:space="preserve">Sed ecce tibitabulam, de qua dixi. </s> <s xml:id="echoid-s3585" xml:space="preserve">cõ-<lb/>tinentem gradus, ac minuta angulorum, quos filum perpendiculi, vel linea fi-<lb/>duciæ in omnibus partibus milleſimis vtriuſq; </s> <s xml:id="echoid-s3586" xml:space="preserve">vmbræ cum proximo latere qua-<lb/>drati efficit. </s> <s xml:id="echoid-s3587" xml:space="preserve">In qua vides, angulum ſub parte 800. </s> <s xml:id="echoid-s3588" xml:space="preserve">in vertice ſumpta, & </s> <s xml:id="echoid-s3589" xml:space="preserve">ère-<lb/>gione partis 50. </s> <s xml:id="echoid-s3590" xml:space="preserve">continere gradus 40. </s> <s xml:id="echoid-s3591" xml:space="preserve">Min. </s> <s xml:id="echoid-s3592" xml:space="preserve">22. </s> <s xml:id="echoid-s3593" xml:space="preserve">fere, vt ſupra diximus: </s> <s xml:id="echoid-s3594" xml:space="preserve">ac tantus <lb/>erit angulus altitudinis, ſi partes 850. </s> <s xml:id="echoid-s3595" xml:space="preserve">ſpectent ad vmbram verſam: </s> <s xml:id="echoid-s3596" xml:space="preserve">eius verò <lb/>complementum grad 49. </s> <s xml:id="echoid-s3597" xml:space="preserve">Min. </s> <s xml:id="echoid-s3598" xml:space="preserve">38. </s> <s xml:id="echoid-s3599" xml:space="preserve">fere altitudinem exhibebit, ſi dictæ partes ex <lb/> <anchor type="note" xlink:label="note-120-04a" xlink:href="note-120-04"/> vmbra recta abſciſſæ fuerint. </s> <s xml:id="echoid-s3600" xml:space="preserve">Tabula porro hæc dici poteſt Gnomoni-<lb/>ca, quodin quadrato ſtabilidicti anguli in tabula com-<lb/>prehenſi efficiantur à gnomone AD, vel AB, <lb/>cum linea fiduciæ, vt patet.</s> <s xml:id="echoid-s3601" xml:space="preserve"/> </p> <div xml:id="echoid-div229" type="float" level="2" n="3"> <handwritten xlink:label="hd-119-1" xlink:href="hd-119-1a"/> <note symbol="a" position="right" xlink:label="note-119-03" xlink:href="note-119-03a" xml:space="preserve">27. tertij.</note> <note symbol="b" position="right" xlink:label="note-119-04" xlink:href="note-119-04a" xml:space="preserve">15. primi.</note> <figure xlink:label="fig-119-01" xlink:href="fig-119-01a"> <image file="119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/119-01"/> </figure> <note position="right" xlink:label="note-119-05" xlink:href="note-119-05a" xml:space="preserve">Filum per-<lb/>pendiculi ſe-<lb/>cansvmbra@ <lb/>rectam facit <lb/>angulum cõ-<lb/>plementi alti-<lb/>tudinis; ſecans <lb/>vero vmbram <lb/>verſam angu-<lb/>lum conſtituit <lb/>ipſi{us} altitu-<lb/>dinis.</note> <note symbol="c" position="right" xlink:label="note-119-06" xlink:href="note-119-06a" xml:space="preserve">ſchol 34. <lb/>primi.</note> <note symbol="d" position="right" xlink:label="note-119-07" xlink:href="note-119-07a" xml:space="preserve">15. primi.</note> <note symbol="a" position="left" xlink:label="note-120-01" xlink:href="note-120-01a" xml:space="preserve">15. primi.</note> <note symbol="b" position="left" xlink:label="note-120-02" xlink:href="note-120-02a" xml:space="preserve">15. primi.</note> <note symbol="c" position="left" xlink:label="note-120-03" xlink:href="note-120-03a" xml:space="preserve">15. primi.</note> <note position="left" xlink:label="note-120-04" xlink:href="note-120-04a" xml:space="preserve">Tabula Gno-<lb/>monica cur <lb/>ſic dicatur.</note> </div> </div> <div xml:id="echoid-div231" type="section" level="1" n="101"> <head xml:id="echoid-head104" xml:space="preserve">SEQVITVR TABVLA <lb/>Gnomonica.</head> <pb o="91" file="121" n="121" rhead="LIBER TERTIVS."/> </div> <div xml:id="echoid-div232" type="section" level="1" n="102"> <head xml:id="echoid-head105" xml:space="preserve">Tabula Gnomonica.</head> <note position="right" xml:space="preserve"> <lb/># ## 0 ## 100 ## 200 ## 300 ## 400 ## 500 ## 600 ## 700 <lb/># G # M # G # M # G # M # G # M # G # M # G # M # G # M # G # M <lb/>0 # 0 # 0 # 5 # 43 # 11 # 19 # 16 # 42 # 21 # 48 # 26 # 34 # 30 # 58 # 35 # 0 <lb/>1 # 0 # 3 # 5 # 46 # 11 # 22 # 16 # 45 # 21 # 51 # 26 # 37 # 31 # 0 # 35 # 2 <lb/>2 # 0 # 7 # 5 # 49 # 11 # 25 # 16 # 48 # 21 # 54 # 26 # 39 # 31 # 3 # 35 # 4 <lb/>3 # 0 # 10 # 5 # 53 # 11 # 29 # 16 # 51 # 21 # 57 # 26 # 42 # 31 # 5 # 35 # 6 <lb/>4 # 0 # 14 # 5 # 56 # 11 # 32 # 16 # 55 # 22 # 0 # 26 # 45 # 31 # 8 # 35 # 9 <lb/>5 # 0 # 17 # 6 # 0 # 11 # 35 # 16 # 58 # 22 # 3 # 26 # 48 # 31 # 10 # 35 # 11 <lb/>6 # 0 # 21 # 6 # 3 # 11 # 38 # 17 # 1 # 22 # 6 # 26 # 50 # 31 # 13 # 35 # 13 <lb/>7 # 0 # 24 # 6 # 6 # 11 # 42 # 17 # 4 # 22 # 9 # 26 # 53 # 31 # 15 # 35 # 16 <lb/>8 # 0 # 28 # 6 # 10 # 11 # 45 # 17 # 7 # 22 # 12 # 26 # 56 # 31 # 18 # 35 # 18 <lb/>9 # 0 # 31 # 6 # 13 # 11 # 48 # 17 # 10 # 22 # 15 # 26 # 59 # 31 # 20 # 35 # 20 <lb/>10 # 0 # 34 # 6 # 17 # 11 # 52 # 17 # 13 # 22 # 18 # 27 # 1 # 31 # 23 # 35 # 22 <lb/>11 # 0 # 38 # 6 # 20 # 11 # 55 # 17 # 17 # 22 # 21 # 27 # 4 # 31 # 26 # 53 # 25 <lb/>12 # 0 # 41 # 6 # 23 # 11 # 58 # 17 # 20 # 22 # 24 # 27 # 7 # 31 # 29 # 35 # 27 <lb/>13 # 0 # 45 # 6 # 27 # 12 # 1 # 17 # 23 # 22 # 26 # 27 # 9 # 31 # 31 # 35 # 29 <lb/>14 # 0 # 48 # 6 # 30 # 12 # 5 # 17 # 26 # 22 # 29 # 27 # 12 # 31 # 34 # 35 # 32 <lb/>15 # 0 # 52 # 6 # 34 # 12 # 8 # 17 # 29 # 22 # 32 # 27 # 15 # 31 # 36 # 35 # 34 <lb/>16 # 0 # 55 # 6 # 37 # 12 # 11 # 17 # 32 # 22 # 35 # 27 # 18 # 31 # 38 # 35 # 36 <lb/>17 # 0 # 58 # 6 # 40 # 12 # 15 # 17 # 35 # 22 # 38 # 27 # 20 # 31 # 40 # 35 # 38 <lb/>18 # 1 # 2 # 6 # 44 # 12 # 18 # 17 # 38 # 22 # 41 # 27 # 23 # 31 # 43 # 35 # 41 <lb/>19 # 1 # 5 # 6 # 47 # 12 # 21 # 17 # 42 # 22 # 44 # 27 # 26 # 31 # 45 # 35 # 43 <lb/>20 # 1 # 9 # 6 # 51 # 12 # 24 # 17 # 45 # 22 # 47 # 27 # 28 # 31 # 48 # 35 # 45 <lb/>21 # 1 # 12 # 6 # 54 # 12 # 28 # 17 # 48 # 22 # 50 # 27 # 31 # 31 # 50 # 35 # 48 <lb/>22 # 1 # 16 # 6 # 57 # 12 # 31 # 17 # 51 # 22 # 53 # 27 # 34 # 31 # 53 # 35 # 50 <lb/>23 # 1 # 19 # 7 # 1 # 12 # 34 # 17 # 54 # 22 # 56 # 27 # 37 # 31 # 55 # 35 # 52 <lb/>24 # 1 # 22 # 7 # 4 # 12 # 38 # 17 # 57 # 22 # 59 # 27 # 39 # 31 # 58 # 35 # 54 <lb/>25 # 1 # 26 # 7 # 8 # 12 # 41 # 18 # 0 # 23 # 2 # 27 # 42 # 32 # 0 # 35 # 57 <lb/>26 # 1 # 29 # 7 # 11 # 12 # 44 # 18 # 3 # 23 # 4 # 27 # 45 # 32 # 3 # 35 # 59 <lb/>27 # 1 # 33 # 7 # 14 # 12 # 47 # 18 # 6 # 23 # 7 # 27 # 47 # 32 # 5 # 36 # 1 <lb/>28 # 1 # 36 # 7 # 18 # 12 # 51 # 18 # 10 # 23 # 10 # 27 # 50 # 32 # 8 # 36 # 3 <lb/>29 # 1 # 40 # 7 # 21 # 12 # 54 # 18 # 13 # 23 # 13 # 27 # 53 # 32 # 10 # 36 # 6 <lb/>30 # 1 # 43 # 7 # 24 # 12 # 57 # 18 # 16 # 23 # 16 # 27 # 55 # 32 # 13 # 36 # 8 <lb/>31 # 1 # 47 # 7 # 28 # 13 # 0 # 18 # 19 # 23 # 19 # 27 # 58 # 32 # 15 # 36 # 10 <lb/>32 # 1 # 50 # 7 # 31 # 13 # 4 # 18 # 22 # 23 # 22 # 28 # 1 # 32 # 18 # 36 # 12 <lb/>33 # 1 # 53 # 7 # 35 # 13 # 7 # 18 # 25 # 23 # 25 # 28 # 3 # 32 # 20 # 36 # 14 <lb/></note> <pb o="92" file="122" n="122" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div233" type="section" level="1" n="103"> <head xml:id="echoid-head106" xml:space="preserve">Tabula Gnomonica.</head> <note position="right" xml:space="preserve"> <lb/># ## 0 ## 100 ## 200 ## 300 ## 400 ## 500 ## 600 ## 700 <lb/># G # M # G # M # G # M # G # M # G # M # G # M # G # M # G # M <lb/>34 # 1 # 57 # 7 # 38 # 13 # 10 # 18 # 28 # 23 # 28 # 28 # 6 # 32 # 22 # 36 # 17 <lb/>35 # 2 # 0 # 7 # 41 # 13 # 13 # 18 # 31 # 23 # 31 # 28 # 9 # 32 # 25 # 36 # 19 <lb/>36 # 2 # 4 # 7 # 45 # 13 # 17 # 18 # 34 # 23 # 33 # 28 # 11 # 32 # 27 # 36 # 21 <lb/>37 # 2 # 7 # 7 # 48 # 13 # 20 # 18 # 37 # 23 # 36 # 28 # 14 # 32 # 30 # 36 # 23 <lb/>38 # 2 # 11 # 7 # 51 # 13 # 23 # 18 # 41 # 23 # 39 # 28 # 17 # 32 # 32 # 36 # 26 <lb/>39 # 2 # 14 # 7 # 55 # 13 # 27 # 18 # 44 # 23 # 42 # 28 # 19 # 32 # 35 # 36 # 28 <lb/>40 # 2 # 17 # 7 # 58 # 13 # 30 # 18 # 47 # 23 # 45 # 28 # 22 # 32 # 37 # 36 # 30 <lb/>41 # 2 # 21 # 8 # 2 # 13 # 33 # 18 # 50 # 23 # 48 # 28 # 25 # 32 # 40 # 36 # 32 <lb/>42 # 2 # 24 # 8 # 5 # 13 # 36 # 18 # 53 # 23 # 51 # 28 # 27 # 32 # 42 # 36 # 35 <lb/>43 # 2 # 28 # 8 # 8 # 13 # 40 # 18 # 56 # 23 # 54 # 28 # 30 # 32 # 44 # 36 # 37 <lb/>44 # 2 # 31 # 8 # 12 # 13 # 43 # 18 # 59 # 23 # 56 # 28 # 33 # 32 # 47 # 36 # 39 <lb/>45 # 2 # 35 # 8 # 15 # 13 # 46 # 19 # 2 # 23 # 59 # 28 # 35 # 32 # 49 # 36 # 41 <lb/>46 # 2 # 38 # 8 # 18 # 13 # 49 # 19 # 5 # 24 # 2 # 28 # 38 # 32 # 52 # 36 # 43 <lb/>47 # 2 # 41 # 8 # 22 # 13 # 52 # 19 # 8 # 24 # 5 # 28 # 41 # 32 # 54 # 36 # 46 <lb/>48 # 2 # 45 # 8 # 25 # 13 # 56 # 19 # 11 # 24 # 8 # 28 # 43 # 32 # 57 # 36 # 48 <lb/>49 # 2 # 48 # 8 # 28 # 13 # 59 # 19 # 14 # 24 # 11 # 28 # 46 # 32 # 59 # 36 # 50 <lb/>50 # 2 # 52 # 8 # 32 # 14 # 2 # 19 # 17 # 24 # 14 # 28 # 49 # 33 # 1 # 36 # 52 <lb/>51 # 2 # 55 # 8 # 35 # 14 # 5 # 19 # 20 # 24 # 17 # 28 # 51 # 33 # 4 # 36 # 54 <lb/>52 # 2 # 59 # 8 # 39 # 14 # 9 # 19 # 24 # 24 # 19 # 28 # 54 # 33 # 6 # 36 # 57 <lb/>53 # 3 # 2 # 8 # 42 # 14 # 12 # 19 # 27 # 24 # 22 # 28 # 57 # 33 # 9 # 36 # 59 <lb/>54 # 3 # 5 # 8 # 45 # 14 # 15 # 19 # 30 # 24 # 25 # 28 # 59 # 33 # 11 # 37 # 1 <lb/>55 # 3 # 9 # 8 # 49 # 14 # 18 # 19 # 33 # 24 # 28 # 29 # 2 # 33 # 13 # 37 # 3 <lb/>56 # 3 # 12 # 8 # 52 # 14 # 22 # 19 # 36 # 24 # 31 # 29 # 4 # 33 # 16 # 37 # 5 <lb/>57 # 3 # 16 # 8 # 55 # 14 # 25 # 19 # 39 # 24 # 34 # 29 # 7 # 33 # 18 # 37 # 8 <lb/>58 # 3 # 19 # 8 # 59 # 14 # 28 # 19 # 42 # 24 # 36 # 29 # 10 # 33 # 21 # 37 # 10 <lb/>59 # 3 # 23 # 9 # 2 # 14 # 31 # 19 # 45 # 24 # 39 # 29 # 12 # 33 # 23 # 37 # 12 <lb/>60 # 3 # 26 # 9 # 5 # 14 # 34 # 19 # 48 # 24 # 42 # 29 # 15 # 33 # 25 # 37 # 14 <lb/>61 # 3 # 29 # 9 # 9 # 14 # 38 # 19 # 51 # 24 # 45 # 29 # 18 # 33 # 28 # 37 # 16 <lb/>62 # 3 # 33 # 9 # 12 # 14 # 41 # 19 # 54 # 24 # 48 # 29 # 20 # 33 # 30 # 37 # 18 <lb/>63 # 3 # 36 # 9 # 15 # 14 # 44 # 19 # 57 # 24 # 51 # 29 # 23 # 33 # 33 # 37 # 21 <lb/>64 # 3 # 40 # 9 # 19 # 14 # 47 # 20 # 0 # 24 # 53 # 29 # 25 # 33 # 35 # 37 # 23 <lb/>65 # 3 # 43 # 9 # 22 # 14 # 51 # 20 # 3 # 24 # 56 # 29 # 28 # 33 # 37 # 37 # 25 <lb/>66 # 3 # 47 # 9 # 26 # 14 # 54 # 20 # 6 # 24 # 59 # 29 # 31 # 33 # 40 # 37 # 27 <lb/>67 # 3 # 50 # 9 # 29 # 14 # 57 # 20 # 9 # 25 # 2 # 29 # 33 # 33 # 42 # 37 # 29 <lb/></note> <pb o="93" file="123" n="123" rhead="LIBER TERTIVS."/> </div> <div xml:id="echoid-div234" type="section" level="1" n="104"> <head xml:id="echoid-head107" xml:space="preserve">Tabula Gnomonica.</head> <note position="right" xml:space="preserve"> <lb/># ## 0 ## 100 ## 200 ## 300 ## 400 ## 500 ## 600 ## 700 <lb/># G # M # G # M # G # M # G # M # G # M # G # M # G # M # G # M <lb/>68 # 3 # 53 # 9 # 32 # 15 # 0 # 20 # 12 # 25 # 5 # 29 # 36 # 33 # 45 # 37 # 31 <lb/>69 # 3 # 57 # 9 # 36 # 15 # 3 # 20 # 15 # 25 # 8 # 29 # 38 # 33 # 47 # 37 # 34 <lb/>70 # 4 # 0 # 9 # 39 # 15 # 7 # 20 # 18 # 25 # 10 # 29 # 41 # 33 # 49 # 37 # 36 <lb/>71 # 4 # 4 # 9 # 42 # 15 # 10 # 20 # 21 # 25 # 13 # 29 # 44 # 33 # 52 # 37 # 38 <lb/>72 # 4 # 7 # 9 # 46 # 15 # 13 # 20 # 24 # 25 # 16 # 29 # 46 # 33 # 54 # 37 # 40 <lb/>73 # 4 # 11 # 9 # 49 # 15 # 16 # 20 # 27 # 25 # 19 # 29 # 49 # 33 # 56 # 37 # 42 <lb/>74 # 4 # 14 # 9 # 52 # 15 # 19 # 20 # 30 # 25 # 22 # 29 # 51 # 33 # 59 # 37 # 44 <lb/>75 # 4 # 17 # 9 # 56 # 15 # 23 # 20 # 33 # 25 # 24 # 29 # 54 # 34 # 1 # 37 # 47 <lb/>76 # 4 # 21 # 9 # 59 # 15 # 26 # 20 # 36 # 25 # 27 # 29 # 57 # 34 # 4 # 37 # 49 <lb/>77 # 4 # 24 # 10 # 2 # 15 # 29 # 20 # 39 # 25 # 30 # 29 # 59 # 34 # 6 # 37 # 51 <lb/>78 # 4 # 28 # 10 # 6 # 15 # 32 # 20 # 42 # 25 # 33 # 30 # 2 # 34 # 8 # 37 # 53 <lb/>79 # 4 # 31 # 10 # 9 # 15 # 35 # 20 # 45 # 25 # 36 # 30 # 4 # 34 # 11 # 37 # 55 <lb/>80 # 4 # 34 # 10 # 12 # 15 # 39 # 20 # 48 # 25 # 38 # 30 # 7 # 34 # 13 # 37 # 57 <lb/>81 # 4 # 38 # 10 # 16 # 15 # 42 # 20 # 51 # 25 # 41 # 30 # 9 # 34 # 15 # 37 # 59 <lb/>82 # 4 # 41 # 10 # 19 # 15 # 45 # 20 # 54 # 25 # 44 # 30 # 12 # 34 # 18 # 38 # 2 <lb/>83 # 4 # 45 # 10 # 22 # 15 # 48 # 20 # 57 # 25 # 47 # 30 # 15 # 34 # 20 # 38 # 4 <lb/>84 # 4 # 48 # 10 # 26 # 15 # 51 # 21 # 0 # 25 # 50 # 30 # 17 # 34 # 22 # 38 # 6 <lb/>85 # 4 # 52 # 10 # 29 # 15 # 54 # 21 # 3 # 25 # 52 # 30 # 20 # 34 # 25 # 38 # 8 <lb/>86 # 4 # 55 # 10 # 32 # 15 # 58 # 21@ # 25 # 55 # 30 # 22 # 34 # 27 # 38 # 10 <lb/>87 # 4 # 58 # 10 # 36 # 16 # 1 # 21 # 9 # 25 # 58 # 30 # 25 # 34 # 29 # 38 # 12 <lb/>88 # 5 # 2 # 10 # 39 # 16 # 4 # 21 # 12 # 26 # 1 # 30 # 27 # 34 # 32 # 38 # 14 <lb/>89 # 5 # 5 # 10 # 42 # 16 # 7 # 21 # 15 # 26 # 4 # 30 # 30 # 34 # 34 # 38 # 16 <lb/>90 # 5 # 9 # 10 # 45 # 16 # 10 # 21 # 18 # 26 # 6 # 30 # 32 # 34 # 36 # 38 # 19 <lb/>91 # 5 # 12 # 10 # 49 # 16 # 14 # 21 # 21 # 26 # 9 # 30 # 35 # 34 # 39 # 38 # 21 <lb/>92 # 5 # 15 # 10 # 52 # 16 # 17 # 21 # 24 # 26 # 12 # 30 # 38 # 34 # 41 # 38 # 23 <lb/>93 # 5 # 19 # 10 # 55 # 16 # 20 # 21 # 27 # 26 # 15 # 30 # 40 # 34 # 43 # 38 # 25 <lb/>94 # 5 # 22 # 10 # 59 # 16 # 23 # 21 # 30 # 26 # 17 # 30 # 43 # 34 # 46 # 38 # 27 <lb/>95 # 5 # 26 # 11 # 2 # 16 # 26 # 21 # 33 # 26 # 20 # 30 # 45 # 34 # 48 # 38 # 29 <lb/>96 # 5 # 29 # 11 # 5 # 16 # 29 # 21 # 36 # 26 # 23 # 30 # 48 # 34 # 50 # 38 # 31 <lb/>97 # 5 # 32 # 11 # 9 # 16 # 32 # 21 # 39 # 26 # 26 # 30 # 50 # 34 # 53 # 38 # 33 <lb/>98 # 5 # 36 # 11 # 12 # 16 # 36 # 21 # 42 # 26 # 28 # 30 # 53 # 34 # 55 # 38 # 35 <lb/>99 # 5 # 39 # 11 # 15 # 16 # 39 # 21 # 45 # 26 # 31 # 30 # 55 # 34 # 57 # 38 # 37 <lb/>100 # 5 # 43 # 11 # 19 # 16 # 42 # 21 # 48 # 26 # 34 # 30 # 58 # 34 # 0 # 38 # 40 <lb/></note> <pb o="96" file="124" n="124" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div235" type="section" level="1" n="105"> <head xml:id="echoid-head108" xml:space="preserve">Tabula Gnomonica.</head> <note position="right" xml:space="preserve"> <lb/># ## 800 ## 900 ## 800 ## 900 ## 800 ## 900 <lb/># G # M # G # M # # G # M # G # M # # G # M # G # M <lb/>0 # 38 # 40 # 41 # 59 # 34 # 39 # 50 # 43 # 3 # 68 # 40 # 57 # 44 # 4 <lb/>1 # 38 # 42 # 42 # 1 # 35 # 39 # 52 # 43 # 5 # 69 # 40 # 59 # 44 # 6 <lb/>2 # 38 # 44 # 42 # 3 # 36 # 39 # 54 # 43 # 6 # 70 # 41 # 1 # 44 # 8 <lb/>3 # 38 # 46 # 42 # 5 # 37 # 39 # 56 # 43 # 8 # 71 # 41 # 3 # 44 # 9 <lb/>4 # 38 # 48 # 42 # 7 # 38 # 39 # 58 # 43 # 10 # 72 # 41 # 5 # 44 # 11 <lb/>5 # 38 # 50 # 42 # 9 # 39 # 40 # 0 # 43 # 12 # 73 # 41 # 7 # 44 # 13 <lb/>6 # 38 # 52 # 42 # 11 # 40 # 40 # 2 # 43 # 14 # 74 # 41 # 9 # 44 # 15 <lb/>7 # 38 # 54 # 42 # 12 # 41 # 40 # 4 # 43 # 16 # 75 # 41 # 11 # 44 # 16 <lb/>8 # 38 # 56 # 42 # 14 # 42 # 40 # 6 # 43 # 17 # 76 # 41 # 13 # 44 # 18 <lb/>9 # 38 # 58 # 42 # 16 # 43 # 40 # 8 # 43 # 19 # 77 # 41 # 15 # 44 # 20 <lb/>10 # 39 # 0 # 42 # 18 # 44 # 40 # 10 # 43 # 21 # 78 # 41 # 17 # 44 # 22 <lb/>11 # 39 # 3 # 42 # 20 # 45 # 40 # 12 # 43 # 23 # 79 # 41 # 19 # 44 # 24 <lb/>12 # 39 # 5 # 42 # 22 # 46 # 40 # 14 # 43 # 25 # 80 # 41 # 21 # 44 # 25 <lb/>13 # 39 # 7 # 42 # 24 # 47 # 40 # 16 # 43 # 26 # 81 # 41 # 23 # 44 # 27 <lb/>14 # 39 # 9 # 42 # 26 # 48 # 40 # 18 # 43 # 28 # 82 # 41 # 25 # 44 # 29 <lb/>15 # 39 # 11 # 42 # 28 # 49 # 40 # 20 # 43 # 30 # 83 # 41 # 27 # 44 # 31 <lb/>16 # 39 # 13 # 42 # 29 # 50 # 40 # 22 # 43 # 32 # 84 # 41 # 29 # 44 # 32 <lb/>17 # 39 # 15 # 42 # 31 # 51 # 40 # 24 # 43 # 34 # 85 # 41 # 31 # 44 # 34 <lb/>18 # 39 # 17 # 42 # 33 # 52 # 40 # 26 # 43 # 35 # 86 # 41 # 32 # 44 # 36 <lb/>19 # 39 # 19 # 42 # 35 # 53 # 40 # 28 # 43 # 37 # 87 # 41 # 34 # 44 # 37 <lb/>20 # 39 # 21 # 42 # 37 # 54 # 40 # 30 # 43 # 39 # 88 # 41 # 36 # 44 # 39 <lb/>21 # 39 # 23 # 42 # 39 # 55 # 40 # 32 # 43 # 41 # 89 # 41 # 38 # 44 # 41 <lb/>22 # 39 # 25 # 42 # 41 # 56 # 40 # 34 # 43 # 43 # 90 # 41 # 40 # 44 # 43 <lb/>23 # 39 # 27 # 42 # 42 # 57 # 40 # 36 # 43 # 44 # 91 # 41 # 42 # 44 # 44 <lb/>24 # 39 # 29 # 42 # 44 # 58 # 40 # 38 # 43 # 46 # 92 # 41 # 44 # 44 # 46 <lb/>25 # 39 # 31 # 42 # 46 # 59 # 40 # 40 # 43 # 48 # 93 # 41 # 46 # 44 # 48 <lb/>26 # 39 # 33 # 42 # 48 # 60 # 40 # 42 # 43 # 50 # 94 # 41 # 48 # 44 # 50 <lb/>27 # 39 # 35 # 42 # 50 # 61 # 40 # 44 # 43 # 52 # 95 # 41 # 50 # 44 # 51 <lb/>28 # 39 # 37 # 42 # 52 # 62 # 40 # 46 # 43 # 53 # 96 # 41 # 52 # 44 # 53 <lb/>29 # 39 # 40 # 42 # 54 # 63 # 40 # 48 # 43 # 55 # 97 # 41 # 54 # 44 # 55 <lb/>30 # 39 # 42 # 42 # 55 # 64 # 40 # 50 # 43 # 57 # 98 # 41 # 55 # 44 # 57 <lb/>31 # 39 # 44 # 42 # 57 # 65 # 40 # 52 # 43 # 59 # 99 # 41 # 57 # 44 # 58 <lb/>32 # 39 # 46 # 42 # 59 # 66 # 40 # 54 # 44 # 1 # 100 # 41 # 59 # 45 # 0 <lb/>33 # 39 # 48 # 43 # 1 # 67 # 40 # 56 # 44 # 2 <lb/></note> <pb o="95" file="125" n="125" rhead="LIBER TERTIVS."/> <p> <s xml:id="echoid-s3602" xml:space="preserve">3. </s> <s xml:id="echoid-s3603" xml:space="preserve"><emph style="sc">Similem</emph> tabulam gnomonicam compoſuit quoque Georgius Purba-<lb/>chius in ſuo quadrato Geometrico, poſito latere partium 1200. </s> <s xml:id="echoid-s3604" xml:space="preserve">eamque ad ſe-<lb/>cunda extendit: </s> <s xml:id="echoid-s3605" xml:space="preserve">quod etiam fecit 10. </s> <s xml:id="echoid-s3606" xml:space="preserve">Antonius Maginus, conſtituto quadrati <lb/>latere partium 1000. </s> <s xml:id="echoid-s3607" xml:space="preserve">quod quidem ſerius animaduerti. </s> <s xml:id="echoid-s3608" xml:space="preserve">Incidi enim caſu quo-<lb/>dam in eam, cum hanc penè meam abſolueram. </s> <s xml:id="echoid-s3609" xml:space="preserve">alio quin in hoc ſupputandi la-<lb/>bore ſuperſediſſem, tabulamque Magini huc tranſtuliſſem. </s> <s xml:id="echoid-s3610" xml:space="preserve">Itaque ſi quis in alti-<lb/>tudinibus aſtrorũ deſideret etiam ſecunda, petere ea debebit ex Maginitabula: <lb/></s> <s xml:id="echoid-s3611" xml:space="preserve"> <anchor type="handwritten" xlink:label="hd-125-1a" xlink:href="hd-125-1"/> quæ ſanè fideliter, & </s> <s xml:id="echoid-s3612" xml:space="preserve">accuratè ab eo ſupputata eſt, vel certè per calculum eadem <lb/>elicere, vt ſupra Num. </s> <s xml:id="echoid-s3613" xml:space="preserve">1. </s> <s xml:id="echoid-s3614" xml:space="preserve">docuimus in hoc probl. </s> <s xml:id="echoid-s3615" xml:space="preserve">Sed meo iudicio contenti eſ-<lb/> <anchor type="note" xlink:label="note-125-01a" xlink:href="note-125-01"/> ſe poſſumus hac noſtra, quæ ad Minuta ſolum vltra gradus progreditur. </s> <s xml:id="echoid-s3616" xml:space="preserve">In <lb/>qua, ſi eam cumilla Maginiconferre quis volet, deprehendet, in noſtra Minu-<lb/>tis graduum additum eſſe ſemper vnum minutum, quando in ea reperiuntur <lb/>plura ſecunda, quam 30. </s> <s xml:id="echoid-s3617" xml:space="preserve">Sed vt verum fatear, neque noſtra, nequeilla Magini <lb/> <anchor type="note" xlink:label="note-125-02a" xlink:href="note-125-02"/> omnino neceſſaria eſt, cumipſæmet partes milleſimę lateris quadrati, ſi appo-<lb/> <anchor type="handwritten" xlink:label="hd-125-1a" xlink:href="hd-125-1"/> nantur duæ cifræ, ſint tangentes altitudinum reſpectu ſinus totius 100.</s> <s xml:id="echoid-s3618" xml:space="preserve">000. </s> <s xml:id="echoid-s3619" xml:space="preserve">& </s> <s xml:id="echoid-s3620" xml:space="preserve"><lb/>eædem, ſi addantur quatuor cifræ, Tangẽtes eaſdẽ, poſito ſinutoto 10.</s> <s xml:id="echoid-s3621" xml:space="preserve">000.</s> <s xml:id="echoid-s3622" xml:space="preserve">000 <lb/> <anchor type="note" xlink:label="note-125-03a" xlink:href="note-125-03"/> exhibeant: </s> <s xml:id="echoid-s3623" xml:space="preserve">ac proinde ex tabula Tangentium altitudines excerpi poſsint cui-<lb/>cunque partimilleſimæ congruentes, ſi prius ei adiungantur duæ, aut quatuor <lb/>cifræ, vt ſupra ad finem Num. </s> <s xml:id="echoid-s3624" xml:space="preserve">1. </s> <s xml:id="echoid-s3625" xml:space="preserve">oſtendimus. </s> <s xml:id="echoid-s3626" xml:space="preserve">Quia tamen moleſtia non caret, <lb/>per Tangentes hac ratione formatas ex tabula Tangentium angulos altitudi-<lb/>num eruere, quod rarò admodum Tangentes illę in tabula præcisè reperian-<lb/>tur, ac propterea pro illis accipiendæ ſint vel proximè minores, vel maiores, il-<lb/> <anchor type="note" xlink:label="note-125-04a" xlink:href="note-125-04"/> læ videlicet, quæ paucioribus vnitatibus ab inuentis differunt: </s> <s xml:id="echoid-s3627" xml:space="preserve">non abs re fue-<lb/>rit, vel tabulam hanc noſtram Gnomonicam, vel illam Magini, ſi ſecunda etiam <lb/>deſiderentur, adſciſcere.</s> <s xml:id="echoid-s3628" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="1"> <handwritten xlink:label="hd-125-1" xlink:href="hd-125-1a"/> <note position="right" xlink:label="note-125-01" xlink:href="note-125-01a" xml:space="preserve">Tabulã gno-<lb/>monicam non <lb/>eſſe abſolutè <lb/>neceſſariam.</note> <note position="right" xlink:label="note-125-02" xlink:href="note-125-02a" xml:space="preserve">Quo pacto <lb/>part{es} milleſi-<lb/>mæ lateris</note> <handwritten xlink:label="hd-125-1" xlink:href="hd-125-1a"/> <note position="right" xlink:label="note-125-03" xlink:href="note-125-03a" xml:space="preserve">Quadrati of-<lb/>ferant tangẽ-<lb/>t{es} poſito ſinu <lb/>toto 10000. <lb/>vel <lb/>100′00000.</note> <note position="right" xlink:label="note-125-04" xlink:href="note-125-04a" xml:space="preserve">Non magn{us} <lb/>error in alti-<lb/>tudinib{us} cõ-<lb/>mittitur, etiãſi <lb/>per integr{as} <lb/>milleſim{as} ta-<lb/>bula progre-<lb/>diatur, & quo <lb/>pacto error <lb/>hic corrigen-<lb/>d{us} ſit.</note> </div> <p> <s xml:id="echoid-s3629" xml:space="preserve">4. </s> <s xml:id="echoid-s3630" xml:space="preserve"><emph style="sc">Qva<gap/>is</emph> autem tabula Gnonomica perſolas partes milleſimas inte-<lb/>gras progrediatur; </s> <s xml:id="echoid-s3631" xml:space="preserve">tamen ſi quando ex do ctrina cap. </s> <s xml:id="echoid-s3632" xml:space="preserve">2. </s> <s xml:id="echoid-s3633" xml:space="preserve">Num. </s> <s xml:id="echoid-s3634" xml:space="preserve">14. </s> <s xml:id="echoid-s3635" xml:space="preserve">lib. </s> <s xml:id="echoid-s3636" xml:space="preserve">1. </s> <s xml:id="echoid-s3637" xml:space="preserve">tradita <lb/>vltra milleſimas integras ſuperſit adhuc aliqua particula, non magnus error in <lb/>altitudinibus aſtrorum obſeruandis committi poteſt. </s> <s xml:id="echoid-s3638" xml:space="preserve">Cum enim anguli in ta-<lb/>bula creſcant ordine per tria duntaxat minuta, vel duo, vel vnum, non fiet er-<lb/>ror niſi vnius autalterius minuri, etiamſi fractio illa milleſimæ partis negligatur. <lb/></s> <s xml:id="echoid-s3639" xml:space="preserve">Quod ſi errorem hunc, licet minimum, vitare cupis, conſidera fra ctionem mil-<lb/>leſimæ, an ſit tertia pars, an ſemiſsis, an verò duæ tertię partes. </s> <s xml:id="echoid-s3640" xml:space="preserve">quo diudicio ſen-<lb/>ſus facilè cognoſces ex particula illa, quæ vltra partes decimas circino percur-<lb/>ſas ſupereſt. </s> <s xml:id="echoid-s3641" xml:space="preserve">Nam vbi anguli in tabula per tria Minuta augentur, addendum <lb/>erit vnum minutum, vel duo, prout particula illa reliqua fuerit {1/3}. </s> <s xml:id="echoid-s3642" xml:space="preserve">vel {2/3}. </s> <s xml:id="echoid-s3643" xml:space="preserve">Idem <lb/>faciendum erit ſi dicta particula fuerit maior, quam {1/3}. </s> <s xml:id="echoid-s3644" xml:space="preserve">minortamen, quam ſe-<lb/>miſsis. </s> <s xml:id="echoid-s3645" xml:space="preserve">Tunc enim addendum erit etiam vnum Minutum. </s> <s xml:id="echoid-s3646" xml:space="preserve">Item quando parti-<lb/>cula illa maior fuerit, quam ſemiſsis, addipoſſunt duo minuta. </s> <s xml:id="echoid-s3647" xml:space="preserve">At quando an-<lb/>guli tabulæ augentur per duo Minuta, addendum erit vnum minutum pro ſe-<lb/>miſſe vnius partis milleſimæ.</s> <s xml:id="echoid-s3648" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3649" xml:space="preserve">DISTANTIAM interte, & </s> <s xml:id="echoid-s3650" xml:space="preserve">ſignum quodcunque in plano Horizon-<lb/>tis poſitum, per Quadratum Geometricum inueſtigare.</s> <s xml:id="echoid-s3651" xml:space="preserve"/> </p> <pb o="96" file="126" n="126" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div237" type="section" level="1" n="106"> <head xml:id="echoid-head109" xml:space="preserve">PROBLEMA II.</head> <p> <s xml:id="echoid-s3652" xml:space="preserve">1. </s> <s xml:id="echoid-s3653" xml:space="preserve"><emph style="sc">Sit</emph> diſtantia metienda FG. </s> <s xml:id="echoid-s3654" xml:space="preserve">Hocper quadratum pendulum ſic fiet. </s> <s xml:id="echoid-s3655" xml:space="preserve">Eri-<lb/>gatur ex F, altitudo quæpiam nota F A, Horizonti ad angulos rectos, ſiue ea ſit <lb/>ſtatura menſoris ab oculo ad planum, ſiue maior quædam altitudo. </s> <s xml:id="echoid-s3656" xml:space="preserve">Inſpiciatur <lb/>extremũ G, per radium AG, ab oculo A, per foramina pinna @ diorum inceden-<lb/>tem, filo perpendiculari liberè pendente, & </s> <s xml:id="echoid-s3657" xml:space="preserve">inſtrumentum radente, punctum-<lb/> <anchor type="figure" xlink:label="fig-126-01a" xlink:href="fig-126-01"/> que E, notetur, vbi filum latus quadrati interſecat, quod fiet vel in latere B C, <lb/>vmbræ rectæ, vel in angulo C, vel in latere DC, vmbræ verſæ. </s> <s xml:id="echoid-s3658" xml:space="preserve">Cadente namque <lb/>filo in vmbram rectam, vel in punctum C, vt in 1. </s> <s xml:id="echoid-s3659" xml:space="preserve">& </s> <s xml:id="echoid-s3660" xml:space="preserve">2. </s> <s xml:id="echoid-s3661" xml:space="preserve">figura, fiet triangulum <lb/>ABE, triangulo AFG, æquiangulum, cum anguli B, F, ſint recti, & </s> <s xml:id="echoid-s3662" xml:space="preserve">angulus A, <lb/>communis. </s> <s xml:id="echoid-s3663" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Si igitur fiat,</s> </p> <div xml:id="echoid-div237" 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/xxxxxxxx/figures/126-01"/> </figure> </div> <note symbol="a" position="left" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} A B, par- \\ tium 1000. # ad vmbramrectam \\ abſciſſam B E, # Ita altitudo no- \\ ta A F, # ad F G, diſtan- \\ tiam, <lb/></note> <p> <s xml:id="echoid-s3664" xml:space="preserve">euadet nota diſtantia quæſita F G, in partibus altitudinis AF. </s> <s xml:id="echoid-s3665" xml:space="preserve">Cadente autem <lb/>puncto E, in vmbram verſam, vt in 3. </s> <s xml:id="echoid-s3666" xml:space="preserve">figura, abſcindetur rurſus triangulum ADE, <lb/>triangulo AF G, æquiangulum, cum anguli D, F, rectiſint, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s3667" xml:space="preserve">alterni AED, <anchor type="note" xlink:label="note-126-03a" xlink:href="note-126-03"/> FAG, æquales. </s> <s xml:id="echoid-s3668" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Ergo ſi fiat.</s> <s xml:id="echoid-s3669" xml:space="preserve"/> </p> <div xml:id="echoid-div238" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-126-03" xlink:href="note-126-03a" xml:space="preserve">29. primi.</note> </div> <note symbol="c" position="left" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra verſa \\ abſciſſa D E, # ad lat{us} D A, partium \\ 1000. # Ita altitudo \\ nota AF. # ad FG, diſtantiam ′ <lb/></note> <note position="left" xml:space="preserve">Quando al-<lb/>titudo maior <lb/>eſt, quam di-<lb/>ſtantia, & <lb/>quando æqua-<lb/>lis, & quando <lb/>minor.</note> <p> <s xml:id="echoid-s3670" xml:space="preserve">nota quoque effi cietur diſtantia quæſita F G, in partibus altitudinis A F. </s> <s xml:id="echoid-s3671" xml:space="preserve">Ex <lb/>quo illud intelligere poteris, quando punctum E, interſectionis fili cum latere <lb/>quadrati cadit in vmbram rectam, altitudinem AF, maiorem eſſe diſtantia F G: <lb/></s> <s xml:id="echoid-s3672" xml:space="preserve">quando in punctum C, æqualem; </s> <s xml:id="echoid-s3673" xml:space="preserve">quando denique in vmbram verſam, mi-<lb/>norem.</s> <s xml:id="echoid-s3674" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3675" xml:space="preserve">2. </s> <s xml:id="echoid-s3676" xml:space="preserve"><emph style="sc">Qvadrato</emph> ſtabili ita eandem diſtantiam explorabimus. </s> <s xml:id="echoid-s3677" xml:space="preserve">Erigatur in <lb/>extremo diſtantiæ E, altitudo quædam perpendicularis notarum partium, & </s> <s xml:id="echoid-s3678" xml:space="preserve"><lb/>quadratum ita accommodetur, vt centrum A, ſuperiorem locum occupet, & </s> <s xml:id="echoid-s3679" xml:space="preserve"><lb/>vmbra recta DC, Horizonti æquidiſtet. </s> <s xml:id="echoid-s3680" xml:space="preserve">Directa igitur dioptra ad ſignum pro-<lb/>poſitum, notetur diligenter vmbra abſciſſa: </s> <s xml:id="echoid-s3681" xml:space="preserve">quæ ſi recta fuerit, nimirum D I, <lb/>erit altitudo AE, maior, quam diſtantia EG, <anchor type="note" xlink:href="" symbol="d"/> triangulumque A D I, triangulo <anchor type="note" xlink:label="note-126-07a" xlink:href="note-126-07"/> AEG, ſimile erit. </s> <s xml:id="echoid-s3682" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Si gitur fiat.</s> <s xml:id="echoid-s3683" xml:space="preserve"/> </p> <div xml:id="echoid-div239" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-126-07" xlink:href="note-126-07a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> </div> <note symbol="e" position="left" xml:space="preserve">4. ſexti.</note> <note position="right" xml:space="preserve"> <lb/>Vt lat{us} quadrati \\ AD, 1000. # ad part{es} vmbræ \\ rectæ D I: # Ita altitudo A E, \\ cognita # ad E G, diſtan- \\ tiam <lb/></note> <p> <s xml:id="echoid-s3684" xml:space="preserve">reperietur diſtantia quæſita EG, in partibus altitudinis aſſumptæ AE.</s> <s xml:id="echoid-s3685" xml:space="preserve"/> </p> <pb o="97" file="127" n="127" rhead="LIBER TERTIVS."/> <p> <s xml:id="echoid-s3686" xml:space="preserve"><emph style="sc">Si</emph> autem dioptra per C, tranſierit, erit <lb/>altitudo A E, diſtantiæ quęſitæ E H, æqua-<lb/> <anchor type="figure" xlink:label="fig-127-01a" xlink:href="fig-127-01"/> lis: </s> <s xml:id="echoid-s3687" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cum ſit vt AD, ad DC, æqualem, ita <anchor type="note" xlink:label="note-127-01a" xlink:href="note-127-01"/> A E, ad EH.</s> <s xml:id="echoid-s3688" xml:space="preserve"/> </p> <div xml:id="echoid-div240" type="float" level="2" n="4"> <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/xxxxxxxx/figures/127-01"/> </figure> <note symbol="a" position="right" xlink:label="note-127-01" xlink:href="note-127-01a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s3689" xml:space="preserve"><emph style="sc">Si</emph> denique vmbra abſciſſa fuerit ver-<lb/>ſa, vt BK, erit altitudo AE, minor, quam <lb/>diſtantia E F, quod plerunq; </s> <s xml:id="echoid-s3690" xml:space="preserve">in diſtantiis <lb/>w<unsure/>etiendis accidere ſolet; </s> <s xml:id="echoid-s3691" xml:space="preserve">eritq; </s> <s xml:id="echoid-s3692" xml:space="preserve">triangu-<lb/>lum ABK, triangulo AEF, æquiangulum, <lb/>cum anguli B, E, recti ſint, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s3693" xml:space="preserve">alterni <anchor type="note" xlink:label="note-127-02a" xlink:href="note-127-02"/> BAK, AFE, æquales. </s> <s xml:id="echoid-s3694" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quare ſi fiat,</s> </p> <div xml:id="echoid-div241" type="float" level="2" n="5"> <note symbol="b" position="right" xlink:label="note-127-02" xlink:href="note-127-02a" xml:space="preserve">29. primi.</note> </div> <note symbol="c" position="right" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt part{es} vmbræ \\ verſæ BK, # ad lat{us} quadrati \\ AB, 1000. # Ita altitudo no- \\ ta AE, # ad E F, diſtan- \\ tiam, <lb/></note> <p> <s xml:id="echoid-s3695" xml:space="preserve">producetur quæſita diſtantia EF, in partibus altitudinis erectæ AE.</s> <s xml:id="echoid-s3696" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3697" xml:space="preserve">3. </s> <s xml:id="echoid-s3698" xml:space="preserve"><emph style="sc">Qvando</emph> diſtantia non valdè magna eſt, vel extremum eius punctum <lb/>facilè videri poteſt ſatis erit, ſi qua dratum ſupra planum Horizontis con-<lb/>ſtituatur, ita vt vmbrę verſę latus B C, ad punctum illud recta vergat. </s> <s xml:id="echoid-s3699" xml:space="preserve">Vt ſi di-<lb/>ſtantia horizontalis D M, metienda ſit, eique imponatur Quadratum erectum, <lb/>mouenda eſt dioptra, donec linea fiduciæ in extremum M, dirigatur. </s> <s xml:id="echoid-s3700" xml:space="preserve">Quara-<lb/>tione ſemper vmbra verſa BC, abſcindetur. </s> <s xml:id="echoid-s3701" xml:space="preserve">Nam ſi linea fiducię per C, tranſi-<lb/>ret, aut vmbramrectam C D, interſecaret, eſſet diſtantia vel æqualis lateri C D, <lb/>vel minor: </s> <s xml:id="echoid-s3702" xml:space="preserve">ac proinde dimenſione nonindigeret. </s> <s xml:id="echoid-s3703" xml:space="preserve">Quoniam igitur rurſus trian-<lb/>gulum NBA, triangulo A D M, ęquiangulum eſt, propter rectos angulos B, <lb/>D, <anchor type="note" xlink:href="" symbol="d"/> & </s> <s xml:id="echoid-s3704" xml:space="preserve">alternos æquales BAN, AMD; </s> <s xml:id="echoid-s3705" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Si fiat,</s> </p> <note symbol="d" position="right" xml:space="preserve">29. primi.</note> <note symbol="e" position="right" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt part{es} vmbræ \\ verſæ BN, # ad lat{us} quadrati \\ AB, 1000. # Ita lat{us} quadrati \\ AD, 1000. # ad DM, diſtan- \\ tiam, <lb/></note> <p> <s xml:id="echoid-s3706" xml:space="preserve">cognita erit diſtantia DM, in partibus milleſimis Iateris quadrati AD.</s> <s xml:id="echoid-s3707" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3708" xml:space="preserve">4. </s> <s xml:id="echoid-s3709" xml:space="preserve"><emph style="sc">Solent</emph> nonnulli Scriptores non inquirere diſtantiam propoſitam in <lb/>partibus altitudinis aſſumptę AE, vel in partibus milleſimis lateris quadrati AD; <lb/></s> <s xml:id="echoid-s3710" xml:space="preserve">ſed ſolum inueſtigant, quoties altitudo electa AE, vellatus quadrati AD, in di-<lb/>ſtantia propoſita contineatur: </s> <s xml:id="echoid-s3711" xml:space="preserve">quod idem eſt, ac ſi altitudo, autlatus vmbræ <lb/>ſtatuatur 1. </s> <s xml:id="echoid-s3712" xml:space="preserve">Atq; </s> <s xml:id="echoid-s3713" xml:space="preserve">ita diuidunt vel partes vmbrę rectę abſciſſas per totum latus <lb/>partium 1000. </s> <s xml:id="echoid-s3714" xml:space="preserve">vel totum latus vmbræ verſæ per partes vmbrę verſę abſciſlas. </s> <s xml:id="echoid-s3715" xml:space="preserve"><lb/>Nam Quotiens numerus indicat, quoties altitudo A C, vel latus Quadrati in <lb/>propoſita diſtantia comprehendatur: </s> <s xml:id="echoid-s3716" xml:space="preserve">cum ſit,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt totum lat{us} A D, \\ partium 1000. # ad part{es} vmbræ \\ rectæ D I, # ita altitudo A E, vel la- \\ t{us} A D, vt 1. # ad diſtantiam \\ E G, vel D I. <lb/></note> <note style="it" position="right" xml:space="preserve"> <lb/>#### Item. <lb/>Vt part{es} vmbræ ver- \\ ſæ B N, # ad totum lat{us} AB, \\ 1000. # Ita altitudo A E, vel \\ lat{us} AD, vt 1. # ad diſtantiam \\ EF, vel DM, <lb/></note> <p> <s xml:id="echoid-s3717" xml:space="preserve">Hinc enim fit, vt cum ſecundum pręceptum regulę trium tertius numerus in ſe-<lb/>cundum ſit ducendus, productuſq; </s> <s xml:id="echoid-s3718" xml:space="preserve">numerus per primum diuidendus, ſatis ſit, <lb/>ſi ſecundus per primum diuidatur: </s> <s xml:id="echoid-s3719" xml:space="preserve">quando quidem vnitas in tertio loco poſita, <lb/>ſi multiplicet ſecundum numerum, eundem ſecundum numerum procreat, &</s> <s xml:id="echoid-s3720" xml:space="preserve">c.</s> <s xml:id="echoid-s3721" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3722" xml:space="preserve"><emph style="sc">Hac</emph> ratione, ſi duæ paites milleſimę abſcindantur ex vmbra verſa, conti- <pb o="98" file="128" n="128" rhead="GEOMETR. PRACT."/> nebitur altitudo A E, vel latus AD, in diſtantia ſecundum hunc numerum 500. <lb/></s> <s xml:id="echoid-s3723" xml:space="preserve">quòd 1000. </s> <s xml:id="echoid-s3724" xml:space="preserve">diuiſa per 2. </s> <s xml:id="echoid-s3725" xml:space="preserve">dent Quotientem 500. </s> <s xml:id="echoid-s3726" xml:space="preserve">At vmbra verſa trium mille-<lb/>ſimarum dabunt Quotientem 333 {1/3}. </s> <s xml:id="echoid-s3727" xml:space="preserve">quià priori differt hoc numero, 166 {2/3}. </s> <s xml:id="echoid-s3728" xml:space="preserve">Ex <lb/>quo intelligi licet, quando vmbra verſa abſciſſa valde parua eſt, magnum poſ-<lb/>ſe errorem committi in diſtantia inueſtiganda. </s> <s xml:id="echoid-s3729" xml:space="preserve">Cum enim partes milleſimæ ſint <lb/>perexiguæ, facilè decipi poſſumus, vt nimirum putemus, ab ſciſſas eſſe tres mil-<lb/>leſimas, cum fortaſsis ſolum duæ abſciſſæ ſint: </s> <s xml:id="echoid-s3730" xml:space="preserve">ac proinde err or committi po-<lb/>terit 166 {2/3}. </s> <s xml:id="echoid-s3731" xml:space="preserve">altitudinem AE, vellaterum A D, qui error contemnendus non eſt. </s> <s xml:id="echoid-s3732" xml:space="preserve"><lb/>At quando partes vmbræ verſæ plures milleſimas continent, non tantus error <lb/>committitur, etiamſi vnam milleſimam pro altera accipiamus. </s> <s xml:id="echoid-s3733" xml:space="preserve">Nam ſi verbi <lb/>gratia putemus, ab ſciſſas eſſe partes {30/1000}. </s> <s xml:id="echoid-s3734" xml:space="preserve">ex vmbra verſa, cum verè abſciſſæ <lb/>ſint {31/1000}. </s> <s xml:id="echoid-s3735" xml:space="preserve">error fieri poterit ſolum in 1. </s> <s xml:id="echoid-s3736" xml:space="preserve">altitudine A E, vellatere A D, & </s> <s xml:id="echoid-s3737" xml:space="preserve">{7/93}. </s> <s xml:id="echoid-s3738" xml:space="preserve"><lb/>cum {30/1000}. </s> <s xml:id="echoid-s3739" xml:space="preserve">dent Quotientem 33 {1/3}. </s> <s xml:id="echoid-s3740" xml:space="preserve">at {31/1000}. </s> <s xml:id="echoid-s3741" xml:space="preserve">Quotientem offerant 32 {8/31}. </s> <s xml:id="echoid-s3742" xml:space="preserve"><lb/>Itaque magis probo, vt Quadratum in altiori loco ſtatuatur, quam in plano <lb/>Horizontis, quia ibi plures partes vmbræ verſę abſcinduntur, quam hic, vt con-<lb/>ſtat in diſtantiis æqualibus E F, D M. </s> <s xml:id="echoid-s3743" xml:space="preserve">Vides ergo magnam eſſe adhibendam <lb/>diligentiam, vt accuratè partes milleſimæ rep eriantur per ea, quæ lib. </s> <s xml:id="echoid-s3744" xml:space="preserve">1. </s> <s xml:id="echoid-s3745" xml:space="preserve">cap. </s> <s xml:id="echoid-s3746" xml:space="preserve">2. </s> <s xml:id="echoid-s3747" xml:space="preserve"><lb/>Num. </s> <s xml:id="echoid-s3748" xml:space="preserve">14. </s> <s xml:id="echoid-s3749" xml:space="preserve">ſcripſimus.</s> <s xml:id="echoid-s3750" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3751" xml:space="preserve"><emph style="sc">Iam</emph> verò, ſi quadratum conſtructum ſit ad certam aliquam menſuram, hoc <lb/>eſt, vt latus contineat vel 1. </s> <s xml:id="echoid-s3752" xml:space="preserve">paſſum, vel 1. </s> <s xml:id="echoid-s3753" xml:space="preserve">cubitũ, vel 2. </s> <s xml:id="echoid-s3754" xml:space="preserve">pedes, aut palmos, aut 3. <lb/></s> <s xml:id="echoid-s3755" xml:space="preserve">aut 4. </s> <s xml:id="echoid-s3756" xml:space="preserve">&</s> <s xml:id="echoid-s3757" xml:space="preserve">c. </s> <s xml:id="echoid-s3758" xml:space="preserve">res erit expeditiſsima, ſi ſemel tantum latus particularum 1000. </s> <s xml:id="echoid-s3759" xml:space="preserve">du-<lb/>catur in menſuras, quibus latus æquiualet. </s> <s xml:id="echoid-s3760" xml:space="preserve">Ita enim in longitudinibus ex-<lb/>quirendis, ſi quadratum ſupra planum, in quo eſt longitudo, ſtatuatur, (vt fit <lb/>in omnibus hypotenuſis, ſiue diſtantiis à loco menſoris vſque ad aliquod pun-<lb/>ctum menſore ſublimus, depreſsiuſue, quemadmo dum patuitin proxima figu-<lb/>ra, quando longitudo D M, inquiſita eſt, patebitque in ſcholio problematis 7. </s> <s xml:id="echoid-s3761" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s3762" xml:space="preserve">in ſcholio problem. </s> <s xml:id="echoid-s3763" xml:space="preserve">9. </s> <s xml:id="echoid-s3764" xml:space="preserve">Item in problem. </s> <s xml:id="echoid-s3765" xml:space="preserve">15. </s> <s xml:id="echoid-s3766" xml:space="preserve">Num. </s> <s xml:id="echoid-s3767" xml:space="preserve">5. </s> <s xml:id="echoid-s3768" xml:space="preserve">& </s> <s xml:id="echoid-s3769" xml:space="preserve">in problem. </s> <s xml:id="echoid-s3770" xml:space="preserve">26. </s> <s xml:id="echoid-s3771" xml:space="preserve"><lb/>Num. </s> <s xml:id="echoid-s3772" xml:space="preserve">3. </s> <s xml:id="echoid-s3773" xml:space="preserve">& </s> <s xml:id="echoid-s3774" xml:space="preserve">in problem. </s> <s xml:id="echoid-s3775" xml:space="preserve">27. </s> <s xml:id="echoid-s3776" xml:space="preserve">Num. </s> <s xml:id="echoid-s3777" xml:space="preserve">3. </s> <s xml:id="echoid-s3778" xml:space="preserve">nec non in problem. </s> <s xml:id="echoid-s3779" xml:space="preserve">37. </s> <s xml:id="echoid-s3780" xml:space="preserve">& </s> <s xml:id="echoid-s3781" xml:space="preserve">38. </s> <s xml:id="echoid-s3782" xml:space="preserve">Num. </s> <s xml:id="echoid-s3783" xml:space="preserve">3. </s> <s xml:id="echoid-s3784" xml:space="preserve">& </s> <s xml:id="echoid-s3785" xml:space="preserve"><lb/>denique in problem. </s> <s xml:id="echoid-s3786" xml:space="preserve">43. </s> <s xml:id="echoid-s3787" xml:space="preserve">& </s> <s xml:id="echoid-s3788" xml:space="preserve">44. </s> <s xml:id="echoid-s3789" xml:space="preserve">& </s> <s xml:id="echoid-s3790" xml:space="preserve">in aliis locis) diuidendus ſolum eſt numerus <lb/>productus per partes vmbræ verſæ (quæ ſemper abſcinditur) notatas. </s> <s xml:id="echoid-s3791" xml:space="preserve">Quo-<lb/>tiens enim dabit longitudinem quæſitám in data menſura. </s> <s xml:id="echoid-s3792" xml:space="preserve">Verbigratia, ſi in vno <lb/>latere comprehendantur 3. </s> <s xml:id="echoid-s3793" xml:space="preserve">pedes, multiplicabimus latus 1000. </s> <s xml:id="echoid-s3794" xml:space="preserve">per 3. </s> <s xml:id="echoid-s3795" xml:space="preserve">efficie-<lb/>muſque 3000. </s> <s xml:id="echoid-s3796" xml:space="preserve">Ita que ſi in inueſtiganda longitudine quapiam abſciſſæ ſint par-<lb/>tes 200. </s> <s xml:id="echoid-s3797" xml:space="preserve">ex vmbra verſa, partiemur 3000. </s> <s xml:id="echoid-s3798" xml:space="preserve">per 200. </s> <s xml:id="echoid-s3799" xml:space="preserve">Nam Quotiens 15. </s> <s xml:id="echoid-s3800" xml:space="preserve">dabit 15. </s> <s xml:id="echoid-s3801" xml:space="preserve"><lb/>pedes pro quæſita longitudine. </s> <s xml:id="echoid-s3802" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Nam cum in proxima figura ſit, vt vmbra <anchor type="note" xlink:label="note-128-01a" xlink:href="note-128-01"/> verſa NB, 200. </s> <s xml:id="echoid-s3803" xml:space="preserve">ad latus BA, 1000. </s> <s xml:id="echoid-s3804" xml:space="preserve">Italatus AD, 3. </s> <s xml:id="echoid-s3805" xml:space="preserve">pedum ad DM, multiplican-<lb/>dum eſt latus 1000. </s> <s xml:id="echoid-s3806" xml:space="preserve">per 3. </s> <s xml:id="echoid-s3807" xml:space="preserve">pedes, & </s> <s xml:id="echoid-s3808" xml:space="preserve">productus numerus 3000. </s> <s xml:id="echoid-s3809" xml:space="preserve">per 200. </s> <s xml:id="echoid-s3810" xml:space="preserve">diuiden-<lb/>dus. </s> <s xml:id="echoid-s3811" xml:space="preserve">Eademque decæteris ratio eſt. </s> <s xml:id="echoid-s3812" xml:space="preserve">Sic ſi latus contineret {1/2} cubiti, diuidendu@ <lb/>eſſet numerus 500. </s> <s xml:id="echoid-s3813" xml:space="preserve">ex {1/2}. </s> <s xml:id="echoid-s3814" xml:space="preserve">in 1000. </s> <s xml:id="echoid-s3815" xml:space="preserve">procreatus per vmbram verſam. </s> <s xml:id="echoid-s3816" xml:space="preserve">Pari ratio-<lb/>ne ſi latus complecteretur 10. </s> <s xml:id="echoid-s3817" xml:space="preserve">palmos, diuidendus eſſet numerus 1000. </s> <s xml:id="echoid-s3818" xml:space="preserve">genitu@ <lb/>ex 10. </s> <s xml:id="echoid-s3819" xml:space="preserve">in 1000. </s> <s xml:id="echoid-s3820" xml:space="preserve">per vmbram verſam: </s> <s xml:id="echoid-s3821" xml:space="preserve">vt in Quotiente prodeant palmi in longi-<lb/>@udine contenti, &</s> <s xml:id="echoid-s3822" xml:space="preserve">c. </s> <s xml:id="echoid-s3823" xml:space="preserve">quod non indiligenter notandum eſt.</s> <s xml:id="echoid-s3824" xml:space="preserve"/> </p> <div xml:id="echoid-div242" type="float" level="2" n="6"> <note symbol="a" position="left" xlink:label="note-128-01" xlink:href="note-128-01a" xml:space="preserve">4. ſexti.</note> </div> </div> <div xml:id="echoid-div244" type="section" level="1" n="107"> <head xml:id="echoid-head110" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s3825" xml:space="preserve">5. </s> <s xml:id="echoid-s3826" xml:space="preserve"><emph style="sc">Qvando</emph> diſtantia metienda magna eſt, & </s> <s xml:id="echoid-s3827" xml:space="preserve">adeſt bona planities campi, <lb/>vti poterimus Quadrato ſtabili commodiſsimè hoc modo. </s> <s xml:id="echoid-s3828" xml:space="preserve">Sit diſtantia obla- <pb o="99" file="129" n="129" rhead="LIBER TERTIVS."/> ta<unsure/> A E. </s> <s xml:id="echoid-s3829" xml:space="preserve">Erigatur quadratum ad Horizontem ad rectos angulos, circumduca-<lb/>turque, donec per eius planum oculus in A, conſtitutus ſignum E, perſpiciat, <lb/> <anchor type="figure" xlink:label="fig-129-01a" xlink:href="fig-129-01"/> atque linea notetur AG. </s> <s xml:id="echoid-s3830" xml:space="preserve">Demiſſo deinde Qua-<lb/>dratovſquead Horizontis planum, ita vt latus <lb/>A B, recta tendat ad ſignum E, hoc eſt, à recta <lb/>notata A G, non recedat, protendatur linea re-<lb/>cta per latus A D, in qua accipiantur quotcun-<lb/>que partes lateri A D, æquales: </s> <s xml:id="echoid-s3831" xml:space="preserve">vel quotcun-<lb/>quealiæ menſuræ æquales, vt paſſus, vel cubiti, <lb/>vſque ad punctum a, in quo rurſus erigatur qua-<lb/>dratum, (In hoc ſecundo quadrato aſſcripſi-<lb/>mus Iiteras minuſculas, ne literæ vnius quadra-<lb/>ti cum literis alterius confundantur, quod in <lb/>ſequentibus etiam obſeruabimus.) </s> <s xml:id="echoid-s3832" xml:space="preserve">vertatur-<lb/>que, donec per eius planum idem ſignum E, ap-<lb/>pareat per rectam a H. </s> <s xml:id="echoid-s3833" xml:space="preserve">Poſt hæc quadratum <lb/>Horizonti incumbat, latuſque a d, rectæ A a <lb/>congruat, & </s> <s xml:id="echoid-s3834" xml:space="preserve">dioptra circumducatur, donec <lb/>eius linea fiduciæ rectæ a H, præcisè ſit ſuppoſita, notenturque partes milleſi-<lb/>mæin portione vmbræ verſæ b F, quam diligentiſsimè. </s> <s xml:id="echoid-s3835" xml:space="preserve">Abſcindetur autem ſem-<lb/>per vmbra verſa, propterea quod diſtantia AE, proponitur magna, ſaltem ma-<lb/>ior, quam A a: </s> <s xml:id="echoid-s3836" xml:space="preserve">alio quin menſurari poſſet ſine quadrato, quemadmodum & </s> <s xml:id="echoid-s3837" xml:space="preserve"><lb/>A a. </s> <s xml:id="echoid-s3838" xml:space="preserve">Quibus peractis, erunt triangula a b F, a A E, æquiangula, cum angulos <lb/> <anchor type="note" xlink:label="note-129-01a" xlink:href="note-129-01"/> habeant rectos <anchor type="note" xlink:href="" symbol="a"/> b, A, & </s> <s xml:id="echoid-s3839" xml:space="preserve">alternos æquales b a F, A E a. </s> <s xml:id="echoid-s3840" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Si ergo fiat,</s> </p> <div xml:id="echoid-div244" type="float" level="2" n="1"> <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/xxxxxxxx/figures/129-01"/> </figure> <note symbol="a" position="right" xlink:label="note-129-01" xlink:href="note-129-01a" xml:space="preserve">29. primi.</note> </div> <note symbol="b" position="right" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra b F, # ad lat{us} b a, 1000, # Ita a A, nota # ad A E, diſtantiam, <lb/></note> <p> <s xml:id="echoid-s3841" xml:space="preserve">inuenietur diſtantia A E, in partibus rectæ A a. </s> <s xml:id="echoid-s3842" xml:space="preserve">Vel ſi diuidatur latus b a, 1000. <lb/></s> <s xml:id="echoid-s3843" xml:space="preserve">per partes milleſimas vmbræ B F, procreabitur numerus, ſecundum quem re-<lb/>cta A a, in diſtantia eadem A E, continetur: </s> <s xml:id="echoid-s3844" xml:space="preserve">poſita nimirum recta A a, vt 1. </s> <s xml:id="echoid-s3845" xml:space="preserve">vt <lb/>Num. </s> <s xml:id="echoid-s3846" xml:space="preserve">4. </s> <s xml:id="echoid-s3847" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s3848" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3849" xml:space="preserve">6. </s> <s xml:id="echoid-s3850" xml:space="preserve"><emph style="sc">Qvod</emph> ſi ad manum non habeamus Quadratum, obtinebimus eandem <lb/>diſtantiam beneficio baculi, vel arundinis hoc artificio. </s> <s xml:id="echoid-s3851" xml:space="preserve">Figatur baculus, vel <lb/>arundo in G, ad rectos angulos plano Horizontis, quod per filum aliquod cum <lb/>perpendiculo facile fiet. </s> <s xml:id="echoid-s3852" xml:space="preserve">Deinderecede per quotlibet paſlus vſque ad A, ita vt <lb/>viſus per baculum incedens feratur in ſignũ E. </s> <s xml:id="echoid-s3853" xml:space="preserve">Poſt hæc ducatur linea A a, ad <lb/>AE, perpẽdicularis, in qua numera quotlibet etiam paſſus vſq; </s> <s xml:id="echoid-s3854" xml:space="preserve">ad a. </s> <s xml:id="echoid-s3855" xml:space="preserve">Ducta tan-<lb/>dem GI, ad A E, quoque perpendiculari, & </s> <s xml:id="echoid-s3856" xml:space="preserve">ipſi A a, æquali, figatur rurſum ba-<lb/>culus ad angulos rectos in tali puncto rectæ GI, nimirumin H, vt oculus iterum <lb/>ex a, per baculum in cedens feratur in ſignum E: </s> <s xml:id="echoid-s3857" xml:space="preserve">inquiranturque exquiſitiſsi-<lb/>me paſſus vna cum fragmentis vnius paſſus in H I, contenti. </s> <s xml:id="echoid-s3858" xml:space="preserve">His enim pera-<lb/>ctis, quo niam rurſus triangula H I a, a A E, æquiangula ſunt, ob angulos re-<lb/>ctos I, A, & </s> <s xml:id="echoid-s3859" xml:space="preserve">alternos æquales I a H, A E a, <anchor type="note" xlink:href="" symbol="c"/> ſi fiat,</s> </p> <note symbol="c" position="right" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt H I, nota # ad I a, notam # Ita a A, nota # ad A E, diſtantiam ′ <lb/></note> <p> <s xml:id="echoid-s3860" xml:space="preserve">effi cietur nota diſtantia A E, in partibus rectarum G A, A a, a I. </s> <s xml:id="echoid-s3861" xml:space="preserve">Vel ſi diuidatur <lb/>a I, nota per IH, notam, reperiemus, quoties Aa, in diſtantia AE, contineatur, ſi <lb/>nimirum recta A a, ponatur vt 1.</s> <s xml:id="echoid-s3862" xml:space="preserve"/> </p> <pb o="100" file="130" n="130" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s3863" xml:space="preserve">7. </s> <s xml:id="echoid-s3864" xml:space="preserve"><emph style="sc">Iam</emph> verò abſq; </s> <s xml:id="echoid-s3865" xml:space="preserve">numer orum auxilio problema abſoluemus, ſi attentè ea <lb/>conſiderentur, quælib. </s> <s xml:id="echoid-s3866" xml:space="preserve">2. </s> <s xml:id="echoid-s3867" xml:space="preserve">problem. </s> <s xml:id="echoid-s3868" xml:space="preserve">1. </s> <s xml:id="echoid-s3869" xml:space="preserve">Num. </s> <s xml:id="echoid-s3870" xml:space="preserve">7. </s> <s xml:id="echoid-s3871" xml:space="preserve">ſcripſimus. </s> <s xml:id="echoid-s3872" xml:space="preserve">Nam ſi fiat angulus <lb/>quicunque B A C, & </s> <s xml:id="echoid-s3873" xml:space="preserve">pro primo exemplo Num. </s> <s xml:id="echoid-s3874" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s3875" xml:space="preserve"> <anchor type="figure" xlink:label="fig-130-01a" xlink:href="fig-130-01"/> huius problem. </s> <s xml:id="echoid-s3876" xml:space="preserve">ſumatur A D, æqualis lateri quadrati <lb/>AB, vel ſi eſſet nimis magnum, æqualis ſemiſsi, vel tertiæ <lb/>parti, aut quartæ, & </s> <s xml:id="echoid-s3877" xml:space="preserve">c. </s> <s xml:id="echoid-s3878" xml:space="preserve">eiuſdem lateris. </s> <s xml:id="echoid-s3879" xml:space="preserve">Deinde D B, æ-<lb/>qualis vmbræ abſciſſę B E, (quæ ſumma cura per circi-<lb/>numin quadrato accipienda eſt) vel eius ſemiſsi, vel ter-<lb/>tiæ parti, aut quartæ & </s> <s xml:id="echoid-s3880" xml:space="preserve">c. </s> <s xml:id="echoid-s3881" xml:space="preserve">vt nimirum A D, D B, ſintæquè <lb/>ſub-multiplices lateris AB, & </s> <s xml:id="echoid-s3882" xml:space="preserve">vmbræ BE, ſi eisæquales non ſunt. </s> <s xml:id="echoid-s3883" xml:space="preserve">Poſt hęc exin-<lb/>ſtrumento partium ſumatur AE, tot particularum, quotpalmi, aut pedes in al-<lb/>titudine AF, continentur. </s> <s xml:id="echoid-s3884" xml:space="preserve">Si enim iunctæ rectę DE, parallela agatur BC, conti-<lb/>nebit diſtantia F G, tot palmos, aut pedes, quotparticulę inſtrumenti partium <lb/>in interuallo EC, comprehenduntur. </s> <s xml:id="echoid-s3885" xml:space="preserve">Vt ſi altitudo AF, ſit pedum 5. </s> <s xml:id="echoid-s3886" xml:space="preserve">erit diſtan-<lb/>tia F G, pedum 3. </s> <s xml:id="echoid-s3887" xml:space="preserve">& </s> <s xml:id="echoid-s3888" xml:space="preserve">ſic de cęteris. </s> <s xml:id="echoid-s3889" xml:space="preserve">Atque hoc modo procedendum erit in aliis <lb/>exemplis omnibus, conſiderando videlicet attentè primas tres magnitudines <lb/>Regulætrium, & </s> <s xml:id="echoid-s3890" xml:space="preserve">c. </s> <s xml:id="echoid-s3891" xml:space="preserve">quod ſemel monuiſſe ſatis eſt.</s> <s xml:id="echoid-s3892" xml:space="preserve"/> </p> <div xml:id="echoid-div245" type="float" level="2" n="2"> <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/xxxxxxxx/figures/130-01"/> </figure> </div> <p> <s xml:id="echoid-s3893" xml:space="preserve">DISTANTIAM in plano per duas ſtationes in eodem plano factas <lb/>per Quadratum Geometricum metiri, quando in eius extremo erecta <lb/>eſt altitudo aliqua perpendicularis, etiamſi infimum eius extremum <lb/>non cernatur.</s> <s xml:id="echoid-s3894" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div247" type="section" level="1" n="108"> <head xml:id="echoid-head111" xml:space="preserve">PROBLEMA III.</head> <p> <s xml:id="echoid-s3895" xml:space="preserve">1. </s> <s xml:id="echoid-s3896" xml:space="preserve"><emph style="sc">Distantia</emph> inueſtiganda ſit A F. </s> <s xml:id="echoid-s3897" xml:space="preserve">Intelligatur planum Horizontis <lb/>KL; </s> <s xml:id="echoid-s3898" xml:space="preserve">erecta altitudo G L: </s> <s xml:id="echoid-s3899" xml:space="preserve">ſtatura menſoris AK, vel alia quædam menſura cog-<lb/>nita. </s> <s xml:id="echoid-s3900" xml:space="preserve">Supra aliquam baſem Horizonti æquidiſtantem collocetur quadratum <lb/>ſtabile, ita vt eius latus AD, cogitatione productum occurrat altitudini erectæ <lb/>in F, ad angulos rectos. </s> <s xml:id="echoid-s3901" xml:space="preserve">Inſpiciatur ſummitas G, per dioptram, ſecetque eius li-<lb/> <anchor type="figure" xlink:label="fig-130-02a" xlink:href="fig-130-02"/> nea fiduciæ vmbram verſam <lb/>CD, in E, quod ſemper con-<lb/>tinget, quando diſtantia A F, <lb/>maior eſt altitudine F G. <lb/></s> <s xml:id="echoid-s3902" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> quod tunc angulus A, mi- <anchor type="note" xlink:label="note-130-01a" xlink:href="note-130-01"/> nor fiat angulo G, ac proin-<lb/>de ſemirecto minor, quem <lb/>cum AD, in A, faceret radius <lb/>per G, emiſſus. </s> <s xml:id="echoid-s3903" xml:space="preserve">Deindeacce-<lb/>de magis, vel recede per lon-<lb/>gitudinem notam Aa, & </s> <s xml:id="echoid-s3904" xml:space="preserve">col-<lb/>locato quadrato ſupra ean-<lb/>dem baſem, ſecetiterum dio-<lb/>ptra ad punctum G, directa <lb/>vmbram verſam c d, in H: </s> <s xml:id="echoid-s3905" xml:space="preserve">eritq; </s> <s xml:id="echoid-s3906" xml:space="preserve">vmbra verſa d H, in propinquioriſtatione ma-<lb/>ior, quam vmbra verſa D E, inſtatione remotiore, <anchor type="note" xlink:href="" symbol="b"/> quod angulus a, maior ſit <anchor type="note" xlink:label="note-130-02a" xlink:href="note-130-02"/> angulo A. </s> <s xml:id="echoid-s3907" xml:space="preserve">Auferatur dI, ipſi D E, æqualis, vt HI, differentia ſit vmbrarum ver- <pb o="101" file="131" n="131" rhead="LIBER TERTIVS."/> ſarum. </s> <s xml:id="echoid-s3908" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quia eſt, vt AD, ad DE, ita AF, ad FG: </s> <s xml:id="echoid-s3909" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> eritrectangulo ſub A D, FG, <anchor type="note" xlink:label="note-131-01a" xlink:href="note-131-01"/> æquale rectangulum ſub D E, AF. </s> <s xml:id="echoid-s3910" xml:space="preserve">Eadem ratione erit rectangulo ſub a d, F G. <lb/></s> <s xml:id="echoid-s3911" xml:space="preserve"> <anchor type="note" xlink:label="note-131-02a" xlink:href="note-131-02"/> rectangulum ſub dH, aF, æquale: </s> <s xml:id="echoid-s3912" xml:space="preserve">propterea quod etiam eſt <anchor type="note" xlink:href="" symbol="c"/> vta d, ad dH, ita <anchor type="note" xlink:label="note-131-03a" xlink:href="note-131-03"/> aF, ad F G. </s> <s xml:id="echoid-s3913" xml:space="preserve">Cum ergo rectangulum ſub AD, FG, æquale ſit rectangulo ſub a d, <lb/>F G, quod rectæ A D, a d, æquales ſint: </s> <s xml:id="echoid-s3914" xml:space="preserve">erit quo que rectangulum ſub dH, aF, <lb/>rectangulo ſub DE, AF, hoc eſt, ſub dI, AF, æquale; </s> <s xml:id="echoid-s3915" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> ideo que erit, vt dH, prima <anchor type="note" xlink:label="note-131-04a" xlink:href="note-131-04"/> ad d I, ſecundam, ita AF, tertia ad AF, quartam: </s> <s xml:id="echoid-s3916" xml:space="preserve">Et permutando, vt tota dH, ad <lb/>totam AF, ita ablata dI, ad ablatam aF. </s> <s xml:id="echoid-s3917" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igitur erit quoque reliqua H I, ad reli- <anchor type="note" xlink:label="note-131-05a" xlink:href="note-131-05"/> quam Aa, vt tota d H, ad totam A F. </s> <s xml:id="echoid-s3918" xml:space="preserve">Quocirca ſit fiat,</s> </p> <div xml:id="echoid-div247" type="float" level="2" n="1"> <figure xlink:label="fig-130-02" xlink:href="fig-130-02a"> <image file="130-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/130-02"/> </figure> <note symbol="a" position="left" xlink:label="note-130-01" xlink:href="note-130-01a" xml:space="preserve">18. primi.</note> <note symbol="b" position="left" xlink:label="note-130-02" xlink:href="note-130-02a" xml:space="preserve">16. ſexti.</note> <note symbol="a" position="right" xlink:label="note-131-01" xlink:href="note-131-01a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="right" xlink:label="note-131-02" xlink:href="note-131-02a" xml:space="preserve">16. ſexti.</note> <note symbol="c" position="right" xlink:label="note-131-03" xlink:href="note-131-03a" xml:space="preserve">4. ſexti.</note> <note symbol="d" position="right" xlink:label="note-131-04" xlink:href="note-131-04a" xml:space="preserve">16. ſexti.</note> <note symbol="e" position="right" xlink:label="note-131-05" xlink:href="note-131-05a" xml:space="preserve">19. quinti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt H I, differentia \\ vmbrarum verſa- \\ rum # ad Aa, differentiam \\ ſtationum: # Ita dH, vmbra verſa \\ propinquioris ſtatio- \\ nis, ſiue maior, # ad A F, \\ diſtantiã <lb/></note> <p> <s xml:id="echoid-s3919" xml:space="preserve">producetur A F, diſtantia nota in partibus differentiæ ſtationum Aa, notæ.</s> <s xml:id="echoid-s3920" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3921" xml:space="preserve">2. </s> <s xml:id="echoid-s3922" xml:space="preserve"><emph style="sc">Eadem</emph> prorſus ratio eſt in quadrato pendulo. </s> <s xml:id="echoid-s3923" xml:space="preserve">Nam filum perpendiculi <lb/>abſcindit quo que triangula ADE, ad H, triangulis AFG, aFG, æquiangula: </s> <s xml:id="echoid-s3924" xml:space="preserve">pro-<lb/> <anchor type="note" xlink:label="note-131-07a" xlink:href="note-131-07"/> pterea, quod tam anguli D, F, rectiſunt, <anchor type="note" xlink:href="" symbol="f"/> & </s> <s xml:id="echoid-s3925" xml:space="preserve">angulus F, hoc eſt, alternus B A E, angulo AGF, externus interno æqualis; </s> <s xml:id="echoid-s3926" xml:space="preserve">quam anguli d, F, recti, & </s> <s xml:id="echoid-s3927" xml:space="preserve">angulus H, id <lb/>eſt, alternus ba H, angulo a G F, externus interno æqualis. </s> <s xml:id="echoid-s3928" xml:space="preserve">Reliqua demonſtra-<lb/>buntur, vt prius. </s> <s xml:id="echoid-s3929" xml:space="preserve">ſunt enim vmbræ verſæ in pendulo quadrato vmbris in ſtabili <lb/>æquales. </s> <s xml:id="echoid-s3930" xml:space="preserve">Nam cum duo anguli D, E, in triangulo ADE, quadrati penduli æqua-<lb/>les ſint duobus angulis D, E, in triangulo ADE, quadrati ſtabilis, quod ea trian-<lb/> <anchor type="note" xlink:label="note-131-08a" xlink:href="note-131-08"/> gula ſint, vt oſtendimus, æquiangula: </s> <s xml:id="echoid-s3931" xml:space="preserve">ſint autem & </s> <s xml:id="echoid-s3932" xml:space="preserve">latera AD, AD æqualia; </s> <s xml:id="echoid-s3933" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> e- runt & </s> <s xml:id="echoid-s3934" xml:space="preserve">rectæ D E, D E, hoc eſt, vmbræ verſæ, æquales. </s> <s xml:id="echoid-s3935" xml:space="preserve">Eademque ratione verſæ <lb/>vmbræ d H, d H, æquales erunt, &</s> <s xml:id="echoid-s3936" xml:space="preserve">c.</s> <s xml:id="echoid-s3937" xml:space="preserve"/> </p> <div xml:id="echoid-div248" type="float" level="2" n="2"> <note symbol="f" position="right" xlink:label="note-131-07" xlink:href="note-131-07a" xml:space="preserve">29. primi.</note> <note symbol="g" position="right" xlink:label="note-131-08" xlink:href="note-131-08a" xml:space="preserve">26. primi.</note> </div> <p> <s xml:id="echoid-s3938" xml:space="preserve">3. </s> <s xml:id="echoid-s3939" xml:space="preserve"><emph style="sc">Si</emph> in vtraq; </s> <s xml:id="echoid-s3940" xml:space="preserve">ſtatione vmbra recta abſcindatur à linea fiduciæ, vel à filo per-<lb/>pendiculi, vtin E, & </s> <s xml:id="echoid-s3941" xml:space="preserve">H, quod quidem ſemper continget, quando diſtantia A F, <lb/>minor eſt altitudine FG, <anchor type="note" xlink:href="" symbol="h"/> quod tunc angulus A, maior fiat angulo G, ac proin- <anchor type="note" xlink:label="note-131-09a" xlink:href="note-131-09"/> de ſemirecto maior, quem cum AD, conſtitueret radius per C, emiſſus. </s> <s xml:id="echoid-s3942" xml:space="preserve">Eritque <lb/>vmbra recta BE, in remotiore ſtatione ma-<lb/> <anchor type="figure" xlink:label="fig-131-01a" xlink:href="fig-131-01"/> ior, quã vmbra recta b H, in ſtatione pro-<lb/>pinquiore, <anchor type="note" xlink:href="" symbol="i"/> quòd angulus FaG, maior ſit <anchor type="note" xlink:label="note-131-10a" xlink:href="note-131-10"/> angulo F A G; </s> <s xml:id="echoid-s3943" xml:space="preserve">ac proinde baH, minor an-<lb/>gulo BAE. </s> <s xml:id="echoid-s3944" xml:space="preserve">Auferatur BI, ipſi bH, æqualis, <lb/>vt I E, differentia ſit vmbrarum rectarum. <lb/></s> <s xml:id="echoid-s3945" xml:space="preserve">Et quia triangula ABE, AFG, æquiangula <lb/>ſunt, propter angulos rectos B, F, <anchor type="note" xlink:href="" symbol="k"/> & </s> <s xml:id="echoid-s3946" xml:space="preserve">al- <anchor type="note" xlink:label="note-131-11a" xlink:href="note-131-11"/> ternos æquales B A E, A G F: </s> <s xml:id="echoid-s3947" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> erit vt A B, <anchor type="note" xlink:label="note-131-12a" xlink:href="note-131-12"/> ad B E, ita F G, ad A F, & </s> <s xml:id="echoid-s3948" xml:space="preserve">permutando, vt <lb/>AB, ad F G, ita BE, ad A F. </s> <s xml:id="echoid-s3949" xml:space="preserve">Eademratione, <lb/>quia triãgula a b H, a F G, æquiãgula ſunt, <lb/> <anchor type="note" xlink:label="note-131-13a" xlink:href="note-131-13"/> propter rectos angulos b, F, <anchor type="note" xlink:href="" symbol="m"/> & </s> <s xml:id="echoid-s3950" xml:space="preserve">alternos æquales b a H, a G F, <anchor type="note" xlink:href="" symbol="n"/> erit vt ab, ad b H, <anchor type="note" xlink:label="note-131-14a" xlink:href="note-131-14"/> ita F G, ad a F: </s> <s xml:id="echoid-s3951" xml:space="preserve">Et permutando vt a b, ad <lb/>FG, ita b H, ad A F. </s> <s xml:id="echoid-s3952" xml:space="preserve">Cum ergo ſit, vt AB, ad <lb/>F G, ita a b, ad FG, propter rectas æquales <lb/>AB, a b. </s> <s xml:id="echoid-s3953" xml:space="preserve"><anchor type="note" xlink:href="" symbol="o"/> erit quo que vt BE, tota ad A F, <anchor type="note" xlink:label="note-131-15a" xlink:href="note-131-15"/> <pb o="102" file="132" n="132" rhead="GEOMETR. PRACT."/> totam ita BI, ablata ipſi b H, æqualis, ad ablatum a F. </s> <s xml:id="echoid-s3954" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur erit & </s> <s xml:id="echoid-s3955" xml:space="preserve">reliqua I E, <anchor type="note" xlink:label="note-132-01a" xlink:href="note-132-01"/> ad reliquam A a, vt tota B E, ad totam AF. </s> <s xml:id="echoid-s3956" xml:space="preserve">Quapropter ſi fiat.</s> <s xml:id="echoid-s3957" xml:space="preserve"/> </p> <div xml:id="echoid-div249" type="float" level="2" n="3"> <note symbol="h" position="right" xlink:label="note-131-09" xlink:href="note-131-09a" xml:space="preserve">18. primi.</note> <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/xxxxxxxx/figures/131-01"/> </figure> <note symbol="i" position="right" xlink:label="note-131-10" xlink:href="note-131-10a" xml:space="preserve">16. primi.</note> <note symbol="k" position="right" xlink:label="note-131-11" xlink:href="note-131-11a" xml:space="preserve">29. primi.</note> <note symbol="l" position="right" xlink:label="note-131-12" xlink:href="note-131-12a" xml:space="preserve">4. ſexti.</note> <note symbol="m" position="right" xlink:label="note-131-13" xlink:href="note-131-13a" xml:space="preserve">29. primi.</note> <note symbol="n" position="right" xlink:label="note-131-14" xlink:href="note-131-14a" xml:space="preserve">4. ſexti.</note> <note symbol="o" position="right" xlink:label="note-131-15" xlink:href="note-131-15a" xml:space="preserve">11. quinti.</note> <note symbol="a" position="left" xlink:label="note-132-01" xlink:href="note-132-01a" xml:space="preserve">19. quinti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt I E, differen- \\ tia vmbrarum \\ rectarum # ad A a, differen- \\ tiam ſtationum: # Ita B E, vmbra recta re- \\ motioris ſtationis, ſiue \\ maior # ad AF, di- \\ ſtantiam, <lb/></note> <p> <s xml:id="echoid-s3958" xml:space="preserve">procreabitur AF, diſtantia nota in partibus differentiæ ſtationum A a, notæ.</s> <s xml:id="echoid-s3959" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3960" xml:space="preserve">4. </s> <s xml:id="echoid-s3961" xml:space="preserve"><emph style="sc">Eadem</emph> omnino in quadrato pendulo eſt ratio. </s> <s xml:id="echoid-s3962" xml:space="preserve">Nam filum perpendi-<lb/>culi ab ſcindit quo que triangula ABE, ab H, triangulis A F G, aFG, æquiangula; <lb/></s> <s xml:id="echoid-s3963" xml:space="preserve"> <anchor type="note" xlink:label="note-132-03a" xlink:href="note-132-03"/> quod tam anguli B, F, recti ſint, <anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s3964" xml:space="preserve">angulus BAE angulo AGF, externus inter- no æqualis, quam anguli b, F, recti, & </s> <s xml:id="echoid-s3965" xml:space="preserve">angulus b a H, angulo a GF, æqualis, ex-<lb/>ternus interno. </s> <s xml:id="echoid-s3966" xml:space="preserve">Reliqua demonſtrabuntur, vt in ſtabili quadrato. </s> <s xml:id="echoid-s3967" xml:space="preserve">Sunt enim <lb/>vmbræ rectæ in quadrato pendulo vmbris rectis in ſtabili æquales. </s> <s xml:id="echoid-s3968" xml:space="preserve">Nam cum <lb/>duo anguli B, E, in triangulo A B E, quadrati penduli, æquales ſint duobus an-<lb/>gulis B, E, in triangulo A B E, quadrati ſtabilis; </s> <s xml:id="echoid-s3969" xml:space="preserve">quod hæc triangula ſint, vt o-<lb/>ſtenſum eſt, æquiangula, vt pote æquiangula triangulo A G F; </s> <s xml:id="echoid-s3970" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt & </s> <s xml:id="echoid-s3971" xml:space="preserve">rectæ <anchor type="note" xlink:label="note-132-04a" xlink:href="note-132-04"/> B E, B E, hoc eſt, vmbræ rectæ æquales. <lb/></s> <s xml:id="echoid-s3972" xml:space="preserve">Eademque ratione vmbrærectæ b H, b H, æ-<lb/> <anchor type="figure" xlink:label="fig-132-01a" xlink:href="fig-132-01"/> quales erunt &</s> <s xml:id="echoid-s3973" xml:space="preserve">c.</s> <s xml:id="echoid-s3974" xml:space="preserve"/> </p> <div xml:id="echoid-div250" type="float" level="2" n="4"> <note symbol="b" position="left" xlink:label="note-132-03" xlink:href="note-132-03a" xml:space="preserve">29. primi.</note> <note symbol="c" position="left" xlink:label="note-132-04" xlink:href="note-132-04a" xml:space="preserve">26. primi.</note> <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/xxxxxxxx/figures/132-01"/> </figure> </div> <p> <s xml:id="echoid-s3975" xml:space="preserve">5. </s> <s xml:id="echoid-s3976" xml:space="preserve"><emph style="sc">Si</emph> denique in reniotiore ſtatione ſece-<lb/> <anchor type="note" xlink:label="note-132-05a" xlink:href="note-132-05"/> turvmbra verſa C D, in E, & </s> <s xml:id="echoid-s3977" xml:space="preserve">recta b c, in H, in <lb/>ſtatione propinquiore, reducenda erit alteru-<lb/>tra earum ad alteram, vt habeantur ſimiles vm-<lb/>bræ, per ea, quæ in quadrati conſtructione Nu. <lb/></s> <s xml:id="echoid-s3978" xml:space="preserve">7. </s> <s xml:id="echoid-s3979" xml:space="preserve">ad initium huius libritradidimus, diuidendo <lb/>nimirum quadratum lateris A B, per vmbram, <lb/>quæreduci debet, &</s> <s xml:id="echoid-s3980" xml:space="preserve">c. </s> <s xml:id="echoid-s3981" xml:space="preserve">Nam ſi fiat vt I N, differentia vmbrarum ſiue rectarum, <lb/>ſiue verſarum, ad A a, differentiam ſtationum: </s> <s xml:id="echoid-s3982" xml:space="preserve">Ita B N, maior vmbra recta vel <lb/>d N, vmbra verſa maior ad aliud, gignetur diſtantia A F, vt demonſtratum eſt <lb/>Numero 3. </s> <s xml:id="echoid-s3983" xml:space="preserve">& </s> <s xml:id="echoid-s3984" xml:space="preserve">1.</s> <s xml:id="echoid-s3985" xml:space="preserve"/> </p> <div xml:id="echoid-div251" type="float" level="2" n="5"> <note position="left" xlink:label="note-132-05" xlink:href="note-132-05a" xml:space="preserve">Ad dexterum <lb/>angulum ſu-<lb/>perioremprio-<lb/>ris quadrati <lb/>pone C. ad de-<lb/>xterum infe-<lb/>riorem poſte-<lb/>rioris d.</note> </div> <p> <s xml:id="echoid-s3986" xml:space="preserve">6. </s> <s xml:id="echoid-s3987" xml:space="preserve"><emph style="sc">Qvod</emph> ſi quando in vna ſtatione linea fiduciæ tranſierit per C, aſſumi <lb/>poterit vel latus vmbrærectæ BC, vel verſæ C D, prout in altera ſtatione abſciſſa <lb/>erit vmbra recta, vel verſa; </s> <s xml:id="echoid-s3988" xml:space="preserve">vt nimirum vmbræ ſint ſimiles.</s> <s xml:id="echoid-s3989" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div253" type="section" level="1" n="109"> <head xml:id="echoid-head112" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s3990" xml:space="preserve"><emph style="sc">Colligitvr</emph> ex demonſtratis, eundem eſſe operandi modum in vtroq; <lb/></s> <s xml:id="echoid-s3991" xml:space="preserve"> <anchor type="note" xlink:label="note-132-06a" xlink:href="note-132-06"/> quadrato: </s> <s xml:id="echoid-s3992" xml:space="preserve">quando quidem eædem vmbræ in quadrato pendulo, quæ in ſtabili, <lb/>abſcinduntur, vt oſtendimus: </s> <s xml:id="echoid-s3993" xml:space="preserve">Ita que præcepta, quæ in vno præſcribuntur, in <lb/>altero quo que obſeruanda ſunt.</s> <s xml:id="echoid-s3994" xml:space="preserve"/> </p> <div xml:id="echoid-div253" type="float" level="2" n="1"> <note position="left" xlink:label="note-132-06" xlink:href="note-132-06a" xml:space="preserve">Eundem eſſe <lb/>modum ope-<lb/>randi in vtro-<lb/>quequadrato.</note> </div> </div> <div xml:id="echoid-div255" type="section" level="1" n="110"> <head xml:id="echoid-head113" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s3995" xml:space="preserve"><emph style="sc">Patet</emph> etiam ex dictis, operationem non variari, ſiue per vmbras verſas, <lb/> <anchor type="note" xlink:label="note-132-07a" xlink:href="note-132-07"/> ſiue per rectas inſtituatur: </s> <s xml:id="echoid-s3996" xml:space="preserve">quando quidem ſemper eſt, vt differentia vmbrarum <lb/>ad differentiam ſtationum, ita vmbra maior, ad diſtantiam, quæ inueſtiganda <lb/>proponitur, vt demonſtratum eſt.</s> <s xml:id="echoid-s3997" xml:space="preserve"/> </p> <div xml:id="echoid-div255" type="float" level="2" n="1"> <note position="left" xlink:label="note-132-07" xlink:href="note-132-07a" xml:space="preserve">Eundem eſſe <lb/>operandi mo-<lb/>dum per vm-<lb/>br{as}<unsure/> verſ{as}, & <lb/>per rect{as}.</note> </div> <pb o="103" file="133" n="133" rhead="LIBER TERTIVS."/> <p> <s xml:id="echoid-s3998" xml:space="preserve"><emph style="sc">Cætervm</emph> in ſcholio problematis 7. </s> <s xml:id="echoid-s3999" xml:space="preserve">præſcribemus rationẽ, qua eandem <lb/>diſtantiam A F, metiri licebit per vnicam ſtationem.</s> <s xml:id="echoid-s4000" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4001" xml:space="preserve">DISTANTIAM eandem per duas ſtationes in aliqua altitudine ere-<lb/>cta factas, ope quadrati perſcrutari.</s> <s xml:id="echoid-s4002" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div257" type="section" level="1" n="111"> <head xml:id="echoid-head114" xml:space="preserve">PROBLEMA IV.</head> <p> <s xml:id="echoid-s4003" xml:space="preserve">1. </s> <s xml:id="echoid-s4004" xml:space="preserve"><emph style="sc">Qvando</emph> in plano commode fieri nequeunt duæ ſtationes, erigatu@ <lb/>haſta aliqua K b, in qua fiant duæ ſtationes oculi menſoris in A, a: </s> <s xml:id="echoid-s4005" xml:space="preserve">Vel in dua-<lb/>bus feneſtris alicuius turris, quarum vna ſuperſtet alteri ad perpendiculum: </s> <s xml:id="echoid-s4006" xml:space="preserve">ab-<lb/>ſcindatur que primum latus vmbræ verſæ in vtraque ſtatione in E, H, Erit que in-<lb/>ferioris ſtationis vmbra verſa D E, maior, quam vmbra verſa d H, ſtationis ſu-<lb/>perioris, quod angulus A, maior ſit angulus A, maior ſit angulo a; </s> <s xml:id="echoid-s4007" xml:space="preserve">quip pe cũ <lb/>A G F, minor ſit, quama G M; </s> <s xml:id="echoid-s4008" xml:space="preserve">& </s> <s xml:id="echoid-s4009" xml:space="preserve">angu-<lb/>li F, M, recti. </s> <s xml:id="echoid-s4010" xml:space="preserve">Sumatur D I, ipſi d H, <lb/> <anchor type="figure" xlink:label="fig-133-01a" xlink:href="fig-133-01"/> <anchor type="note" xlink:label="note-133-01a" xlink:href="note-133-01"/> æqualis; </s> <s xml:id="echoid-s4011" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quoniam eſt, vt A D, ad D E, ita A F, ad F G: </s> <s xml:id="echoid-s4012" xml:space="preserve">erit permutando, <lb/>vt A D, ad A F, ita D E, ad F G. </s> <s xml:id="echoid-s4013" xml:space="preserve">Eadem-<lb/>que ratione erit, vt ad@ ad a M, ita d H, ad <lb/>M G. </s> <s xml:id="echoid-s4014" xml:space="preserve">Cum ergo eadem ſit proportio <lb/>A D, ad A F, quę a d, ad a M, propter ę-<lb/>qualitatem linearum AD, a d, & </s> <s xml:id="echoid-s4015" xml:space="preserve">AF, a M; <lb/></s> <s xml:id="echoid-s4016" xml:space="preserve"> <anchor type="note" xlink:label="note-133-02a" xlink:href="note-133-02"/> <anchor type="note" xlink:href="" symbol="b"/> Erit vt D E, tota ad totam FG, ita dH, <anchor type="note" xlink:label="note-133-03a" xlink:href="note-133-03"/> hoceſt DI, ablata ad ablatam MG. </s> <s xml:id="echoid-s4017" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igi- turerit quoque reliqua I E, ad reliquam F M, hoc eſt, ad reliquam A a, vt tota <lb/>D E, ad totam F G, hoc eſt, vt A D, ad A F; </s> <s xml:id="echoid-s4018" xml:space="preserve">cum oſtenſum ſit eſſe A D, ad A F, vt <lb/>DE, ad FG. </s> <s xml:id="echoid-s4019" xml:space="preserve">Quocirca ſi fiat,</s> </p> <div xml:id="echoid-div257" type="float" level="2" n="1"> <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/xxxxxxxx/figures/133-01"/> </figure> <note symbol="a" position="right" xlink:label="note-133-01" xlink:href="note-133-01a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="right" xlink:label="note-133-02" xlink:href="note-133-02a" xml:space="preserve">11. quinti.</note> <note symbol="c" position="right" xlink:label="note-133-03" xlink:href="note-133-03a" xml:space="preserve">19. quinti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt IE, differentia vm- \\ brarum verſarum # ad A a, differen- \\ tiam ſtationum, # Ita AD, lat{us} \\ quadrati # ad A F, @diſtan- \\ tiam, <lb/></note> <p> <s xml:id="echoid-s4020" xml:space="preserve">manifeſta colligetur diſtantia AF, in partibus diffe-<lb/> <anchor type="figure" xlink:label="fig-133-02a" xlink:href="fig-133-02"/> rentię ſtationum Aa.</s> <s xml:id="echoid-s4021" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="2" n="2"> <figure xlink:label="fig-133-02" xlink:href="fig-133-02a"> <image file="133-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/133-02"/> </figure> </div> <p> <s xml:id="echoid-s4022" xml:space="preserve">2. </s> <s xml:id="echoid-s4023" xml:space="preserve"><emph style="sc">Si</emph> in vtraque ſtatione latus vmbrę rectę in-<lb/>terſecetur in E, H, reducenda eſt vtraque vmbra re-<lb/>cta B@ E, b H, ad verſam, per ea, quę tradita ſunt ad <lb/>initium huius libri in conſtructi@ne quadrati <lb/>Numero 7. </s> <s xml:id="echoid-s4024" xml:space="preserve">diuidendo videlicet quadratum nu-<lb/>merum lateris quadrati per vtramque vmbram <lb/>rectam abſciſſam ſigillatim. </s> <s xml:id="echoid-s4025" xml:space="preserve">Nam ſi produ-<lb/>cantur latera D C, d c, vmbrę verſę vſque ad <lb/>radios A G, a G, ad puncta I, K, erit iterum, <lb/>vt demonſtrauimus Numero 1. </s> <s xml:id="echoid-s4026" xml:space="preserve">vt I N, differentia <lb/>vmbrarum verſarum ad A a, differentiam ſtationũ <pb o="104" file="134" n="134" rhead="GEOMETR. PRACT."/> ita AD, latus quadrati ad A F, diſtantiam. </s> <s xml:id="echoid-s4027" xml:space="preserve">Ergo vt Num. </s> <s xml:id="echoid-s4028" xml:space="preserve">1. </s> <s xml:id="echoid-s4029" xml:space="preserve">oſtenſum eſt, inue-<lb/>nietur diſtantia AF, in partibus differentiæ ſtationum Aa.</s> <s xml:id="echoid-s4030" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div260" type="section" level="1" n="112"> <head xml:id="echoid-head115" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s4031" xml:space="preserve"><emph style="sc">Sine</emph> reductione vmbrarum rectarum ad verſas hoc alio modo eandem di-<lb/>ſtantiam AF, eliciemus. </s> <s xml:id="echoid-s4032" xml:space="preserve">Ex OH, differentia vmbrarum rectarum in latus a b, fiat <lb/>P: </s> <s xml:id="echoid-s4033" xml:space="preserve">& </s> <s xml:id="echoid-s4034" xml:space="preserve">ex b H, vmbra recta maiore in minorem B E, fiat Q. </s> <s xml:id="echoid-s4035" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quia eſt, vt I D, ad <anchor type="note" xlink:label="note-134-01a" xlink:href="note-134-01"/> DA, ita AB, ad BE; </s> <s xml:id="echoid-s4036" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit rectangulum ſub ID, B E, æquale quadrato ex A D, vel <anchor type="note" xlink:label="note-134-02a" xlink:href="note-134-02"/> AB. </s> <s xml:id="echoid-s4037" xml:space="preserve">Eodemque modo rectangulum ſub K d, b H, quadrato ex a d, vel a b, hoc <lb/>eſt, eidem quadrato ex AD, vel AB, æquale erit: </s> <s xml:id="echoid-s4038" xml:space="preserve">ac proinde rectangula ſub ID, <lb/> <anchor type="note" xlink:label="note-134-03a" xlink:href="note-134-03"/> B E: </s> <s xml:id="echoid-s4039" xml:space="preserve">& </s> <s xml:id="echoid-s4040" xml:space="preserve">ſub K d, b H, æqualia inter ſe erunt. </s> <s xml:id="echoid-s4041" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur erit, vt I D, ad K d, ita b H, ad BE, & </s> <s xml:id="echoid-s4042" xml:space="preserve">permutando vt I D, ad b H, ita K d, ad B E, hoc eſt, vt I D, tota ad totam <lb/> <anchor type="note" xlink:label="note-134-04a" xlink:href="note-134-04"/> b H, ita K d, hoc eſt, ita D N, ablata ad B E, hoc eſt, ad b O, ablatam: </s> <s xml:id="echoid-s4043" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Ideoque erit & </s> <s xml:id="echoid-s4044" xml:space="preserve">reliqua I N, ad reliquam O H, vt tota I D, ad totam b H; </s> <s xml:id="echoid-s4045" xml:space="preserve">& </s> <s xml:id="echoid-s4046" xml:space="preserve">permutando <lb/>I N, ad ID, vt O H, ad b H. </s> <s xml:id="echoid-s4047" xml:space="preserve">Quia vero proportio I N, ad D A, (poſita media ID,) <lb/>componitur ex proportionibus I N, ad ID, & </s> <s xml:id="echoid-s4048" xml:space="preserve">I D, ad D A: </s> <s xml:id="echoid-s4049" xml:space="preserve">Eſt autem, vt pro-<lb/>xime monſtratum eſt, vt I N, ad I D, ita O H, ad b H; </s> <s xml:id="echoid-s4050" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> & </s> <s xml:id="echoid-s4051" xml:space="preserve">vt I D, ad D A, ita A B, <anchor type="note" xlink:label="note-134-05a" xlink:href="note-134-05"/> ad B E: </s> <s xml:id="echoid-s4052" xml:space="preserve">componetur quoque proportio IN, ad DA, ex proportionibus O H, ad <lb/>b H, & </s> <s xml:id="echoid-s4053" xml:space="preserve">A B, ad B E. </s> <s xml:id="echoid-s4054" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Sed proportio etiam producti P, ad productum Q@ com- <anchor type="note" xlink:label="note-134-06a" xlink:href="note-134-06"/> ponitur ex eiſdem proportionibus, nimirum ex lateribus. </s> <s xml:id="echoid-s4055" xml:space="preserve">Igitur eadem eſt pro-<lb/>portio P, ad Q, q̃ IN, ad D A. </s> <s xml:id="echoid-s4056" xml:space="preserve">Cum ergo, vtin 1. </s> <s xml:id="echoid-s4057" xml:space="preserve">modo huius Num. </s> <s xml:id="echoid-s4058" xml:space="preserve">2. </s> <s xml:id="echoid-s4059" xml:space="preserve">oſtendi-<lb/>mus, ſit I N, ad A a, vt AD, ad A F, hoc eſt, permutando vt I N, ad D A, ita A a, ad <lb/>A F: </s> <s xml:id="echoid-s4060" xml:space="preserve">Erit quoque P, ad Q, vt A a, ad A F. </s> <s xml:id="echoid-s4061" xml:space="preserve">Quapropter ſi fiat,</s> </p> <div xml:id="echoid-div260" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-134-01" xlink:href="note-134-01a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="left" xlink:label="note-134-02" xlink:href="note-134-02a" xml:space="preserve">17. ſexti.</note> <note symbol="c" position="left" xlink:label="note-134-03" xlink:href="note-134-03a" xml:space="preserve">16. ſexti.</note> <note symbol="d" position="left" xlink:label="note-134-04" xlink:href="note-134-04a" xml:space="preserve">19. quinti.</note> <note symbol="e" position="left" xlink:label="note-134-05" xlink:href="note-134-05a" xml:space="preserve">4. ſexti.</note> <note symbol="f" position="left" xlink:label="note-134-06" xlink:href="note-134-06a" xml:space="preserve">23. ſexti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt P, numer{us}, qui fit ex O H, \\ differentia vmbrarum recta- \\ rum in a b, lat{us} quadrati, # ad Q, numerum, qui \\ fit ex vmbra recta \\ b H, maiore in mino- \\ rem B E, # Ita A a, dif- \\ ferentiaſta- \\ tionum # ad A F, \\ diſtan- \\ tiam, <lb/></note> <p> <s xml:id="echoid-s4062" xml:space="preserve">producetur eadem diſtantia quæſita A F, in partibus differentiæ ſtationum A a.</s> <s xml:id="echoid-s4063" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div262" type="section" level="1" n="113"> <head xml:id="echoid-head116" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s4064" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> <emph style="sc">Qvoniam</emph> eſt, vt b H, ad a b, ita a M, ad M G: </s> <s xml:id="echoid-s4065" xml:space="preserve">ſi fiat,</s> </p> <note symbol="g" position="left" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt b H, vmbra \\ recta, # ad d b, lat{us} qua- \\ drati 1000. # Ita a M, qua- \\ ten{us} 1. # ad M G, <lb/></note> <p> <s xml:id="echoid-s4066" xml:space="preserve">hoc eſt, (quia 1. </s> <s xml:id="echoid-s4067" xml:space="preserve">multiplicans latus quadrati 1000. </s> <s xml:id="echoid-s4068" xml:space="preserve">producit idem latus 1000.) <lb/></s> <s xml:id="echoid-s4069" xml:space="preserve">ſi quadrati latus 1000. </s> <s xml:id="echoid-s4070" xml:space="preserve">ab, diuidatur per vmbram rectam b H, exibit Quotiens <lb/>M G, indicans, quoties a M, quatenus 1. </s> <s xml:id="echoid-s4071" xml:space="preserve">in M G, comprehendatur. </s> <s xml:id="echoid-s4072" xml:space="preserve">Eodem pa-<lb/>cto, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt B E, vmbrarecta # ad A B, lat{us} qua- \\ drati 1000. # Ita A F, quaten{us} 1. # ad F G, <lb/></note> <p> <s xml:id="echoid-s4073" xml:space="preserve">hoc eſt, ſi quadrati latus 1000. </s> <s xml:id="echoid-s4074" xml:space="preserve">A B, diuidatur per vmbram rectam B E, fiet Quo-<lb/>tiens F G, ſignificans, quoties A F, quatenus. </s> <s xml:id="echoid-s4075" xml:space="preserve">1. </s> <s xml:id="echoid-s4076" xml:space="preserve">contineatur in F G. </s> <s xml:id="echoid-s4077" xml:space="preserve">Si igitur ex <lb/>poſteriore hoc Quotiente F G, priorille Quotiens M G, detrahatur, reliqua fiet <lb/>F M, vel A a, differentia ſtationum cognita in partibus, quarum A F, eſt 1. </s> <s xml:id="echoid-s4078" xml:space="preserve">Quo-<lb/>circa ſi fiat,</s> </p> <pb o="105" file="135" n="135" rhead="LIBER TERTIVS."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt A a, differentia Quotientum \\ qui fiunt, ſilat{us} quadrati \\ per vtramque vmbram rectam \\ diuidatur, # ad A a, differentiam \\ ſtationum notam in \\ menſura aliqua: # ita A F, \\ vt 1. # ad AF, \\ diſtan- \\ tiam,@ <lb/></note> <figure> <image file="135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/135-01"/> </figure> <p> <s xml:id="echoid-s4079" xml:space="preserve">hoc eſt, ſi differentia ſtationum diuidatur per dif-<lb/>ferentiam Quotientum, efficietur nota diſtantia <lb/>AF, in partibus differentiæ ſtationum A a, &</s> <s xml:id="echoid-s4080" xml:space="preserve">c.</s> <s xml:id="echoid-s4081" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4082" xml:space="preserve">3. </s> <s xml:id="echoid-s4083" xml:space="preserve"><emph style="sc">Si</emph> denique in ſtatione inferiori latus vm-<lb/>brærectæ ſecetur in E, & </s> <s xml:id="echoid-s4084" xml:space="preserve">in ſuperiori ſtatione latus <lb/>vmbræ verſæ in H, reducenda quo que erit vmbra <lb/>recta ad verſam, vt diximus, & </s> <s xml:id="echoid-s4085" xml:space="preserve">producendum la-<lb/>tus D C, vmbræ verſæ vſque ad punctum N, ſu-<lb/>mendaque D I, ipſi d H, æqualis. </s> <s xml:id="echoid-s4086" xml:space="preserve">Nam vt Num. </s> <s xml:id="echoid-s4087" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s4088" xml:space="preserve">oſtendimus, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt N I, differentia vm- \\ brarum verſarum # ad Aa, differentiam \\ ſtationum: # Ita A D, lat{us} \\ quadrati # ad A F, di- \\ ſtantiam, <lb/></note> <p> <s xml:id="echoid-s4089" xml:space="preserve">cognita rurſus erit diſtantia A F, in partibus differentiæ ſtationum A a.</s> <s xml:id="echoid-s4090" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div263" type="section" level="1" n="114"> <head xml:id="echoid-head117" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s4091" xml:space="preserve"><emph style="sc">Sine</emph> reductione vmbræ rectæ ad verſam ita quoque agemus. </s> <s xml:id="echoid-s4092" xml:space="preserve">Numerus <lb/>qui fit ex recta vmbra B E, in vmbram verſam d H, auferatur ex 1000000. </s> <s xml:id="echoid-s4093" xml:space="preserve">qua-<lb/>drato lateris 1000. </s> <s xml:id="echoid-s4094" xml:space="preserve">reſiduumque ſit, O. </s> <s xml:id="echoid-s4095" xml:space="preserve">Item ex vmbra recta B E, in latus qua-<lb/>drati 1000. </s> <s xml:id="echoid-s4096" xml:space="preserve">fiat P. </s> <s xml:id="echoid-s4097" xml:space="preserve">Et quoniam, vt initio huius libro in conſtructione quadrati <lb/>Numer. </s> <s xml:id="echoid-s4098" xml:space="preserve">6. </s> <s xml:id="echoid-s4099" xml:space="preserve">oſtendimus, latus quadrati medio loco proportionale eſt inter vm-<lb/>bras B E, D N: </s> <s xml:id="echoid-s4100" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Erit rectangulum ſub B E, D N, quadrato lateris æquale. </s> <s xml:id="echoid-s4101" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cũ <anchor type="note" xlink:label="note-135-03a" xlink:href="note-135-03"/> <anchor type="note" xlink:label="note-135-04a" xlink:href="note-135-04"/> ergo rectangulum ſub B E, D N, æquale ſitrectangulis ſub D E, D I, & </s> <s xml:id="echoid-s4102" xml:space="preserve">ſub B E, <lb/>N I: </s> <s xml:id="echoid-s4103" xml:space="preserve">ſi rectangulum ſub B E, D I, auferatur ex rectangulo ſub B E, DN, id eſt, <lb/>ex 1000000. </s> <s xml:id="echoid-s4104" xml:space="preserve">quadrato lateris A D, reliquum fiet rectangulum ſub <lb/>B E, N I; </s> <s xml:id="echoid-s4105" xml:space="preserve">ac proinde cum D I, ipſi d H, ſi æqualis, fiet rectangulum ſub B E, D I, <lb/>ex vmbra B E, in vmbram d H; </s> <s xml:id="echoid-s4106" xml:space="preserve">atque idcirco reliquum rectangulum ſub B E, <lb/>N I, (quod videlicet relin quitur, ſi rectangulum ſub B E, D I, ex quadrato late-<lb/>ris detrahatur, vt dictum eſt,) numero O, æquale erit. </s> <s xml:id="echoid-s4107" xml:space="preserve">Eſt autem ex conſtru-<lb/>ctione rectangulum quo que ſub B E, vmbrarecta, & </s> <s xml:id="echoid-s4108" xml:space="preserve">latere A D, numero P, æ-<lb/>quale. </s> <s xml:id="echoid-s4109" xml:space="preserve">Igitur cum numerus B E, multiplicans N I, A D, producat O, P, <anchor type="note" xlink:href="" symbol="c"/> <anchor type="note" xlink:label="note-135-05a" xlink:href="note-135-05"/> erit O, ad P, vt N I, ad AD. </s> <s xml:id="echoid-s4110" xml:space="preserve">Sed vt Numero 1. </s> <s xml:id="echoid-s4111" xml:space="preserve">huius problematis demonſtra-<lb/>uimus, vt NI, differentia vmbrarum verſarum ad A a, differentiam ſtationum, ita <lb/>eſt AD, latus quadrati ad AF, & </s> <s xml:id="echoid-s4112" xml:space="preserve">permutando vt NI, ad AD, ita A a, ad AF. </s> <s xml:id="echoid-s4113" xml:space="preserve">Igi-<lb/>tur erit quoque O, ad P, vt A a, ad A F. </s> <s xml:id="echoid-s4114" xml:space="preserve">Quamobrem ſi fiat,</s> </p> <div xml:id="echoid-div263" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-135-03" xlink:href="note-135-03a" xml:space="preserve">16. ſexti.</note> <note symbol="b" position="right" xlink:label="note-135-04" xlink:href="note-135-04a" xml:space="preserve">1. ſecundum.</note> <note symbol="c" position="right" xlink:label="note-135-05" xlink:href="note-135-05a" xml:space="preserve">17. ſept.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt O, numer{us}, quirelinquitur \\ ſinumer{us} genit{us} ex vmbra \\ recta in verſam ex quadrato \\ lateris de@rahatur, # ad numerum P, \\ qui ex vmbra re- \\ cta B E, in lat{us} AD, \\ producitur: # Ita A a, diffe- \\ @entiaſtatio- \\ tionum # Ad AF, \\ diſtan- \\ tiam <lb/></note> <pb o="106" file="136" n="136" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s4115" xml:space="preserve">procreabitur diſtantia quæſita AF, in partibus differentiæ ſtationum Aa,</s> </p> <p> <s xml:id="echoid-s4116" xml:space="preserve">4. </s> <s xml:id="echoid-s4117" xml:space="preserve"><emph style="sc">Hic</emph> etiam ſi forte in vna ſtatione linea fiduciæ, aut filum perpendiculi, <lb/>tranſierit per punctum, C, aſſumendum erit vellatus vmbræ rectæ, vel verſæ, vt <lb/>Num. </s> <s xml:id="echoid-s4118" xml:space="preserve">6. </s> <s xml:id="echoid-s4119" xml:space="preserve">præcedentis problematis dictum eſt.</s> <s xml:id="echoid-s4120" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4121" xml:space="preserve">ALTITVDINEM cuiuslibet rei erectæ per eius diſtantiam ab ocu-<lb/>lo menſoris, beneficio quadrati coniicere.</s> <s xml:id="echoid-s4122" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div265" type="section" level="1" n="115"> <head xml:id="echoid-head118" xml:space="preserve">PROBLEMA V.</head> <figure> <image file="136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/136-01"/> </figure> <p> <s xml:id="echoid-s4123" xml:space="preserve">1. </s> <s xml:id="echoid-s4124" xml:space="preserve"><emph style="sc">Sit</emph> altitudo conijcienda GL, vel ML, vel IL, ad Horizontem perpen-<lb/>dicularis. </s> <s xml:id="echoid-s4125" xml:space="preserve">Statura menſoris, vel aliqua alia menſura nota ſit F L, vel A K, pare-<lb/>turque planum Horizonti parallelum A F, in quo quadratum erigatur, cuius la-<lb/>tus A D, intelligatur productum occurrens altitudini in F. </s> <s xml:id="echoid-s4126" xml:space="preserve">Inſpecto cacumine <lb/>I, ſecet primum dioptra vel filum perpendiculi latus C D, vmbræ verſæ in E. <lb/></s> <s xml:id="echoid-s4127" xml:space="preserve">quod accidet, quando diſtantia A F, maior eſt, altitudine F I, vt problemate @. </s> <s xml:id="echoid-s4128" xml:space="preserve"><lb/>Num. </s> <s xml:id="echoid-s4129" xml:space="preserve">1. </s> <s xml:id="echoid-s4130" xml:space="preserve">oſtendimus. </s> <s xml:id="echoid-s4131" xml:space="preserve">Quoniam igitur triangulum ADE, triangulo AFI, æqui-<lb/>angulum eſt, vtibidem monſtratum eſt;</s> <s xml:id="echoid-s4132" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit vt AD, ad DE, ita AF, ad FI. </s> <s xml:id="echoid-s4133" xml:space="preserve">Quã- <anchor type="note" xlink:label="note-136-01a" xlink:href="note-136-01"/> obrem ſi fiat,</s> </p> <div xml:id="echoid-div265" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-136-01" xlink:href="note-136-01a" xml:space="preserve">4. ſexti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} qua- \\ drati 1000. # Ad vmbram \\ verſam DE: # Ita diſtantia A F, vel data, vel \\ inuenta per probl. 3. aut 4. # AD F I <lb/></note> <p> <s xml:id="echoid-s4134" xml:space="preserve">patefacta erit altitudo F I, in partibus diſtantiæ A F, quæ ſi data non eſt, exqui-<lb/>renda erit vel per problema 3. </s> <s xml:id="echoid-s4135" xml:space="preserve">vel 4. </s> <s xml:id="echoid-s4136" xml:space="preserve">Et ſi ad F I, adijcietur menſura, vel ſtatura <lb/>menſoris F L, tota altitudo propoſita IL, nota fiet,</s> </p> <p> <s xml:id="echoid-s4137" xml:space="preserve">2. </s> <s xml:id="echoid-s4138" xml:space="preserve"><emph style="sc">Deinde</emph> linea fiduciæ tranſeat per punctum C. </s> <s xml:id="echoid-s4139" xml:space="preserve">quod fiet, quando di-<lb/> <anchor type="note" xlink:label="note-136-03a" xlink:href="note-136-03"/> ſtantia AF, altitudini FM, æqualis eſt;</s> <s xml:id="echoid-s4140" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> quod tunc angulus CAD, ſemirectus ſit, ac proinde & </s> <s xml:id="echoid-s4141" xml:space="preserve">reliquus M, in triangulo F A M. </s> <s xml:id="echoid-s4142" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Ideoque latera AF, FM, æqua- <anchor type="note" xlink:label="note-136-04a" xlink:href="note-136-04"/> lia ſint. </s> <s xml:id="echoid-s4143" xml:space="preserve">Quanta ergo eſt diſtantia AF, quæ vel data eſt, velinuenienda per pro-<lb/>blema 3. </s> <s xml:id="echoid-s4144" xml:space="preserve">vel 4. </s> <s xml:id="echoid-s4145" xml:space="preserve">tanta erit altitudo F M, cui addita F L, nota patefaciet totam al-<lb/>titudinem M L.</s> <s xml:id="echoid-s4146" xml:space="preserve"/> </p> <div xml:id="echoid-div266" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-136-03" xlink:href="note-136-03a" xml:space="preserve">ſchol. 34. <lb/>primi.</note> <note symbol="c" position="left" xlink:label="note-136-04" xlink:href="note-136-04a" xml:space="preserve">6. primi.</note> </div> <p> <s xml:id="echoid-s4147" xml:space="preserve">3. </s> <s xml:id="echoid-s4148" xml:space="preserve"><emph style="sc">Postremo</emph> interſeceturvmbræ rectæ latus in H. </s> <s xml:id="echoid-s4149" xml:space="preserve">quod eueniet, quan- <pb o="107" file="137" n="137" rhead="LIBER TERTIVS."/> do diſtantia A F, minor eſt altitudine F G: </s> <s xml:id="echoid-s4150" xml:space="preserve">eritque triangulum A B H, triangu-<lb/>lo AFG, æquiangulum, vt problemate 3. </s> <s xml:id="echoid-s4151" xml:space="preserve">Num. </s> <s xml:id="echoid-s4152" xml:space="preserve">3. </s> <s xml:id="echoid-s4153" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s4154" xml:space="preserve">Igitur erit, <lb/>vt BH, ad AB, ita AF, ad F G: </s> <s xml:id="echoid-s4155" xml:space="preserve">ac proinde, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra \\ recta B H, # ad AB, lat{us} qua- \\ drati 1000. # Ita diſtantia AF, veldata, vel \\ inuenta per probl. 3. aut 4. # ad F G, <lb/></note> <p> <s xml:id="echoid-s4156" xml:space="preserve">nota euadet FG, cuiſi addetur menſura vel ſtatura menſoris F L, efficietur nota <lb/>tota altitudo G L, propoſita.</s> <s xml:id="echoid-s4157" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4158" xml:space="preserve">ALTITVDINEM eandem, etiamſi eius diſtantia ab oculo menſo-<lb/>ris neque data ſit, neq; </s> <s xml:id="echoid-s4159" xml:space="preserve">inuenta, per duas ſtationes in plano factas pa-<lb/>tefacere auxilio quadrati.</s> <s xml:id="echoid-s4160" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div268" type="section" level="1" n="116"> <head xml:id="echoid-head119" xml:space="preserve">PROBLEMA VI.</head> <p> <s xml:id="echoid-s4161" xml:space="preserve">1. </s> <s xml:id="echoid-s4162" xml:space="preserve"><emph style="sc">Proposita</emph> ſit altitudo metienda FG, ſecluſa menſoris ſtatura FL. </s> <s xml:id="echoid-s4163" xml:space="preserve">Fi-<lb/> <anchor type="figure" xlink:label="fig-137-01a" xlink:href="fig-137-01"/> ant duæ ſtationes, vt in problemate 3. </s> <s xml:id="echoid-s4164" xml:space="preserve">cuius figura 1. </s> <s xml:id="echoid-s4165" xml:space="preserve">hicrepetatur, ſeceturque <lb/>primum vmbra verſa in vtraque ſtatione in punctis E, H. </s> <s xml:id="echoid-s4166" xml:space="preserve">Et quia propter ſimili-<lb/>tudinem triangulorum ADE, AFG, <anchor type="note" xlink:href="" symbol="a"/> eſt vt DE, ad A D, ita F G, ad A F: </s> <s xml:id="echoid-s4167" xml:space="preserve">ſi fiat,</s> </p> <div xml:id="echoid-div268" type="float" level="2" n="1"> <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/xxxxxxxx/figures/137-01"/> </figure> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt D E, vmbra ver- \\ ſa abſciſſa # ad AD, lat{us} qua- \\ drati 1000. # Ita FG, quaten{us} 1. # ad A F. <lb/></note> <p> <s xml:id="echoid-s4168" xml:space="preserve">hoc eſt, (cum 1. </s> <s xml:id="echoid-s4169" xml:space="preserve">multiplicans latus quadrati 1000. </s> <s xml:id="echoid-s4170" xml:space="preserve">producat idem latus 1000.) <lb/></s> <s xml:id="echoid-s4171" xml:space="preserve">ſi quadrati latus 1000. </s> <s xml:id="echoid-s4172" xml:space="preserve">diuidatur per vmbram verſam D E, prodibit Quotiens <lb/>AF, indicans, quoties FG, in AF, contineatur. </s> <s xml:id="echoid-s4173" xml:space="preserve">Eodem pacto ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra verſa d H, # ad a d, lat{us} quadra, \\ ti 1000. # Ita F G, quaten{us} \\ 1. # ad a F, <lb/></note> <p> <s xml:id="echoid-s4174" xml:space="preserve">hoc eſt, ſi quadrati latus 1000. </s> <s xml:id="echoid-s4175" xml:space="preserve">diuidatur per vmbram verſam d H, gignetur <lb/>Quotiens a F, monſtrans, quoties FG, in a F, contineatur. </s> <s xml:id="echoid-s4176" xml:space="preserve">Siigitur poſterior hic <lb/>Quotiens a F, ex priori Quotiente a F, detrahatur, relinquetur differentia A a, <lb/>cognita in partibus, quarum F G, eſt 1. </s> <s xml:id="echoid-s4177" xml:space="preserve">Siergo fiat,</s> </p> <pb o="108" file="138" n="138" rhead="GEOMETR. PRACT."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt A a, differentia Quotientum. diuiſo \\ latere Quadrati per vtramque vm- \\ bram verſam, # ad A a, dif- \\ ferentiam \\ ſtationum. # Ita F G, vt 1. # ad FG, <lb/></note> <p> <s xml:id="echoid-s4178" xml:space="preserve">producetur altitudo F G, inpartibus differentiæ ſtationum A a, nota, & </s> <s xml:id="echoid-s4179" xml:space="preserve">adiecta <lb/>menſoris ſtatura F L, tota altitudo GL, nota fiet.</s> <s xml:id="echoid-s4180" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div270" type="section" level="1" n="117"> <head xml:id="echoid-head120" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s4181" xml:space="preserve">EX differentia H I, vmbrarum verſarum in latus quadrati gignatur nume-<lb/>rus N: </s> <s xml:id="echoid-s4182" xml:space="preserve">& </s> <s xml:id="echoid-s4183" xml:space="preserve">ex vmbra verſa D E, in verſam d H, fiat O. </s> <s xml:id="echoid-s4184" xml:space="preserve">Productis autem lateribus <lb/>BC, bc, vſque ad R, P, in radijs, abſcindatur B Q, ipſi b P, æqualis. </s> <s xml:id="echoid-s4185" xml:space="preserve">Et quia, vt pro-<lb/>blemate 3. </s> <s xml:id="echoid-s4186" xml:space="preserve">Num. </s> <s xml:id="echoid-s4187" xml:space="preserve">3. </s> <s xml:id="echoid-s4188" xml:space="preserve">demonſtratum eſt, ita eſt OR, differentia vmbrarumrectarum <lb/> <anchor type="note" xlink:label="note-138-02a" xlink:href="note-138-02"/> ad A a, differentiam ſtationum, vt vmbra recta maior BR, ad AF, diſtantiam: </s> <s xml:id="echoid-s4189" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Vt autem B R, ad B A, ita eſt A F, ad F G: </s> <s xml:id="echoid-s4190" xml:space="preserve">& </s> <s xml:id="echoid-s4191" xml:space="preserve">permutando, vt B R, ad A F, ita B A, ad <lb/>F G; </s> <s xml:id="echoid-s4192" xml:space="preserve">erit quoque QR, ad A a, vt B A, ad F G, & </s> <s xml:id="echoid-s4193" xml:space="preserve">permutando Q R, ad B A, vt A a, <lb/> <anchor type="note" xlink:label="note-138-03a" xlink:href="note-138-03"/> ad F G. </s> <s xml:id="echoid-s4194" xml:space="preserve">Dico iam, vt eſt QR, ad AB, ita eſſe numerum N, ad numerum O.</s> <s xml:id="echoid-s4195" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cum <anchor type="note" xlink:label="note-138-04a" xlink:href="note-138-04"/> enim ſit, vt DE, ad DA, ita AB, ad BR;</s> <s xml:id="echoid-s4196" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit rectangulum ſub DE, BR, rectangulo ſub D A, A B, hoc eſt quadrato lateris æquale. </s> <s xml:id="echoid-s4197" xml:space="preserve">Eademque ratione erit rectangu-<lb/>lum ſub d H, b P, eidem quadrato laterisæquale; </s> <s xml:id="echoid-s4198" xml:space="preserve">ideo que rectangulum ſub DE, <lb/>BR, rectangulo ſub d H, b P, æquale erit. </s> <s xml:id="echoid-s4199" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Igitur erit, vt D E, ad D H, ita b P, ad <anchor type="note" xlink:label="note-138-05a" xlink:href="note-138-05"/> B R: </s> <s xml:id="echoid-s4200" xml:space="preserve">& </s> <s xml:id="echoid-s4201" xml:space="preserve">conuertendo, vt d H, ad D E, ita B R, ad b P, vel ad B Q: </s> <s xml:id="echoid-s4202" xml:space="preserve">& </s> <s xml:id="echoid-s4203" xml:space="preserve">permutando <lb/> <anchor type="note" xlink:label="note-138-06a" xlink:href="note-138-06"/> vttota d H, ad totam BR, ita DE, vel d I, ablata ad b P, vel ad BQ@ ablatam. </s> <s xml:id="echoid-s4204" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igi- tur & </s> <s xml:id="echoid-s4205" xml:space="preserve">reliqua HI, ad reliquam QR, erit, vt tota dH, adtotam BR, vel vtablata dI, <lb/>ad ablatam BQ;</s> <s xml:id="echoid-s4206" xml:space="preserve"><unsure/> & </s> <s xml:id="echoid-s4207" xml:space="preserve">permutando, vt H I, ad d I, ita QR, ad B Q, vel ad b P. </s> <s xml:id="echoid-s4208" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Pro- <anchor type="note" xlink:label="note-138-07a" xlink:href="note-138-07"/> portio autem numeri N, ad numerum O, componitur ex proportionibus HI, ad <lb/>d I, vel ad DE, & </s> <s xml:id="echoid-s4209" xml:space="preserve">lateris a d, ad d H: </s> <s xml:id="echoid-s4210" xml:space="preserve">propterea quod N, factus eſt ex HI, differen-<lb/>tia vmbrarum verſarum in latus quadratia d; </s> <s xml:id="echoid-s4211" xml:space="preserve">Atvero O, ex vmbra verſa D E, in <lb/>verſam d H, ex conſtructione. </s> <s xml:id="echoid-s4212" xml:space="preserve">Cum ergo ſit, vt paulo ante oſtendimus, quem-<lb/> <anchor type="note" xlink:label="note-138-08a" xlink:href="note-138-08"/> admodum HI, ad d I, ita QR. </s> <s xml:id="echoid-s4213" xml:space="preserve">ad b P,<anchor type="note" xlink:href="" symbol="g"/> Itẽ vtlatus a d, ad DH, ita b P, ad b a, com- ponetur quoque proportio N, ad O, ex proportionibus QR, ad b P, & </s> <s xml:id="echoid-s4214" xml:space="preserve">b P, ad <lb/>b a. </s> <s xml:id="echoid-s4215" xml:space="preserve">Sed ex his eiſdem componitur proportio QR, ad ba. </s> <s xml:id="echoid-s4216" xml:space="preserve">Igitur eadem eſt pro-<lb/>portio N, ad O, quæ QR, ad b a, vel BA. </s> <s xml:id="echoid-s4217" xml:space="preserve">quod oſtendere volebamus eſt autem <lb/>vt QR, ad BA, ita A a, differentiaſtationum ad altitudinem F G, vt ſupra oſtendi-<lb/>mus prope initium huius demonſtrationis. </s> <s xml:id="echoid-s4218" xml:space="preserve">Igitur ſi fiat,</s> </p> <div xml:id="echoid-div270" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-138-02" xlink:href="note-138-02a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="left" xlink:label="note-138-03" xlink:href="note-138-03a" xml:space="preserve">4. ſexti.</note> <note symbol="c" position="left" xlink:label="note-138-04" xlink:href="note-138-04a" xml:space="preserve">16. ſexti.</note> <note symbol="d" position="left" xlink:label="note-138-05" xlink:href="note-138-05a" xml:space="preserve">16. ſexti.</note> <note symbol="e" position="left" xlink:label="note-138-06" xlink:href="note-138-06a" xml:space="preserve">19. quinti.</note> <note symbol="f" position="left" xlink:label="note-138-07" xlink:href="note-138-07a" xml:space="preserve">23. ſexti.</note> <note symbol="g" position="left" xlink:label="note-138-08" xlink:href="note-138-08a" xml:space="preserve">4. ſexti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt N, numer{us}, qui fit ex HI, \\ differentia vmbrarum verſa- \\ rum in lat{us} quadrati 1000. # ad numerum O, fa- \\ ctum ex vmbra ver- \\ ſa D E, in verſam dH. # Ita A a, dif- \\ ferentiaſta- \\ tionum # ad F G, \\ altitudi- \\ dinem, <lb/></note> <p> <s xml:id="echoid-s4219" xml:space="preserve">deprehenſa erit altitudo FG, in partibus differentiæ ſtationum A a, &</s> <s xml:id="echoid-s4220" xml:space="preserve">c.</s> <s xml:id="echoid-s4221" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div272" type="section" level="1" n="118"> <head xml:id="echoid-head121" xml:space="preserve">ALITER</head> <p> <s xml:id="echoid-s4222" xml:space="preserve"><emph style="sc">Redvcatvr</emph> vtraque vmbra verſa ad rectam, vt adinitium huius libri in <lb/>quadrati conſtructione Num. </s> <s xml:id="echoid-s4223" xml:space="preserve">7. </s> <s xml:id="echoid-s4224" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s4225" xml:space="preserve">Nam per has vmbras rectas altitudi-<lb/>nem FG, nanciſcemur, vti Num. </s> <s xml:id="echoid-s4226" xml:space="preserve">2. </s> <s xml:id="echoid-s4227" xml:space="preserve">in ſequenti tradetur.</s> <s xml:id="echoid-s4228" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4229" xml:space="preserve">2. </s> <s xml:id="echoid-s4230" xml:space="preserve"><emph style="sc">Secetvr</emph> deinde vtraque vmbra recta in E, H, vt in 2. </s> <s xml:id="echoid-s4231" xml:space="preserve">figura problema-<lb/>tis 3. </s> <s xml:id="echoid-s4232" xml:space="preserve">quæ hic repetatur. </s> <s xml:id="echoid-s4233" xml:space="preserve">Et quoniam ob ſimilem triangulorum A B E, A F G, <pb o="109" file="139" n="139" rhead="LIBER TERTIVS."/> <anchor type="note" xlink:href="" symbol="a"/> eſt, vt BE, ad AB, ita AF, ad GF; </s> <s xml:id="echoid-s4234" xml:space="preserve">erit permutando, vt BE, ad AF, ita AB, ad a <anchor type="note" xlink:label="note-139-01a" xlink:href="note-139-01"/> FG. </s> <s xml:id="echoid-s4235" xml:space="preserve">Eademque ratione erit vt b H, ad a F, ita a b, ad FG; </s> <s xml:id="echoid-s4236" xml:space="preserve">ac proinde erit, vt tota <lb/>BE, ad totam AF, ita ablata b H, vel B I, ad ablatam a F, cum vtraque propor-<lb/>tio ſit, quæ AB, vel a b, ad FG.</s> <s xml:id="echoid-s4237" xml:space="preserve"/> </p> <div xml:id="echoid-div272" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-139-01" xlink:href="note-139-01a" xml:space="preserve">4. ſexti.</note> </div> <figure> <image file="139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/139-01"/> </figure> <p> <s xml:id="echoid-s4238" xml:space="preserve">Igitur erit quo que reliqua IE, ad reliquam A a, vt tota BE, ad totam AF, hoc <lb/>eſt, vt AB, ad FG, cum duæ hę proportiones eædem inter ſe ſint, vt paulò ant@ <lb/>demonſtrauimus. </s> <s xml:id="echoid-s4239" xml:space="preserve">Quo circa ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt IE, differentia vmbra- \\ rum rectarum # Ad Aa, differentiam ſta- \\ tionum: # Ita A B, quadrati<unsure/> \\ lat{us} 1000. # ad F G, <lb/></note> <p> <s xml:id="echoid-s4240" xml:space="preserve">proſiliet nota altitudo FG, & </s> <s xml:id="echoid-s4241" xml:space="preserve">adiuncta menſoris ſtatura FL, tota altitudo G L, <lb/>quęſita nota efficietur.</s> <s xml:id="echoid-s4242" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div274" type="section" level="1" n="119"> <head xml:id="echoid-head122" xml:space="preserve">ALITER.</head> <p style="it"> <s xml:id="echoid-s4243" xml:space="preserve">DIVIDE quadrati lat{us} 1000. </s> <s xml:id="echoid-s4244" xml:space="preserve">per vtramque vmbram rectam, & </s> <s xml:id="echoid-s4245" xml:space="preserve">minorem Quo-<lb/>tientem duc in differ entiam ſtationum, numerumque productum partire per differen-<lb/>tiam Quotientum. </s> <s xml:id="echoid-s4246" xml:space="preserve">Si namque hic vltim{us} Quotiens ducatur in Quotientem maiorem <lb/>ex priorib{us} duob{us} Quotientib{us}, produc{et}ur altitudo quæſi@a in partib{us} differentiæ <lb/>ſtationum. </s> <s xml:id="echoid-s4247" xml:space="preserve">Quod ita perſpicuum fi{et}.</s> <s xml:id="echoid-s4248" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4249" xml:space="preserve"><emph style="sc">Sit</emph> vmbra recta minor b H. </s> <s xml:id="echoid-s4250" xml:space="preserve">50. </s> <s xml:id="echoid-s4251" xml:space="preserve">verbi gratia, & </s> <s xml:id="echoid-s4252" xml:space="preserve">maior B E, 125. </s> <s xml:id="echoid-s4253" xml:space="preserve">at differentia <lb/>ſtationum Aa, ſit 16. </s> <s xml:id="echoid-s4254" xml:space="preserve">paſſuum. </s> <s xml:id="echoid-s4255" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et quia eſt, vt b H, ad a b, ita a F, quatenus 1. </s> <s xml:id="echoid-s4256" xml:space="preserve">ad <anchor type="note" xlink:label="note-139-03a" xlink:href="note-139-03"/> F G; </s> <s xml:id="echoid-s4257" xml:space="preserve">ſi a b, 1000. </s> <s xml:id="echoid-s4258" xml:space="preserve">diuidatur per b H, 50. </s> <s xml:id="echoid-s4259" xml:space="preserve">indicabit Quotiens 20. </s> <s xml:id="echoid-s4260" xml:space="preserve">rectam a F, in <lb/>F G, contineri vicies.</s> <s xml:id="echoid-s4261" xml:space="preserve"/> </p> <div xml:id="echoid-div274" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-139-03" xlink:href="note-139-03a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s4262" xml:space="preserve"><emph style="sc">Sic</emph> etiam,<anchor type="note" xlink:href="" symbol="c"/> quia eſt, vt BE, ad A B, ita AF, quatenus 1, ad FG; </s> <s xml:id="echoid-s4263" xml:space="preserve">ſi A B, 1000.</s> <s xml:id="echoid-s4264" xml:space="preserve"> <anchor type="note" xlink:label="note-139-04a" xlink:href="note-139-04"/> diuidatur per BE, 125. </s> <s xml:id="echoid-s4265" xml:space="preserve">monſtrabit Quotiens 8. </s> <s xml:id="echoid-s4266" xml:space="preserve">rectam AF, in FG, contineri o-<lb/>cties. </s> <s xml:id="echoid-s4267" xml:space="preserve">Quia itaque A a, 16. </s> <s xml:id="echoid-s4268" xml:space="preserve">paſſuum vna cum a F, continetur in FG, octies, fit vt <lb/>Aa, 16. </s> <s xml:id="echoid-s4269" xml:space="preserve">paſſuum octies ſumpta, vna cum a F, octies quoque ſumpta, faciat 128. <lb/></s> <s xml:id="echoid-s4270" xml:space="preserve">paſſus, & </s> <s xml:id="echoid-s4271" xml:space="preserve">a F, octies ſumptam, nimirum numerum ipſi F G, æqualem. </s> <s xml:id="echoid-s4272" xml:space="preserve">Eſt autem <pb o="110" file="140" n="140" rhead="GEOMETR. PRACT."/> vicies ſumpta eidem FG, æqualis. </s> <s xml:id="echoid-s4273" xml:space="preserve">Igitur 128. </s> <s xml:id="echoid-s4274" xml:space="preserve">paſſus, vna cum a F, octies ſumpta <lb/>æquiualent ipſi a F, vicies ſumptæ. </s> <s xml:id="echoid-s4275" xml:space="preserve">Si ergo vtrobique auferatur a F, octies ſum-<lb/>pta, reliqui erunt 128. </s> <s xml:id="echoid-s4276" xml:space="preserve">paſſus æquales ipſi a F, duodecies ſumptæ. </s> <s xml:id="echoid-s4277" xml:space="preserve">Quare ſi 128. <lb/></s> <s xml:id="echoid-s4278" xml:space="preserve">diuiduntur per 12. </s> <s xml:id="echoid-s4279" xml:space="preserve">prodibunt 10 {2/3}. </s> <s xml:id="echoid-s4280" xml:space="preserve">paſſus pro recta, a F, cum hic Quotiens in <lb/>12. </s> <s xml:id="echoid-s4281" xml:space="preserve">ductus producat 128. </s> <s xml:id="echoid-s4282" xml:space="preserve">Cumigitur a F, contineatur vicies in F G: </s> <s xml:id="echoid-s4283" xml:space="preserve">ſi a F, 10 {2/3}. </s> <s xml:id="echoid-s4284" xml:space="preserve"><lb/>paſſuum ducatur in 20. </s> <s xml:id="echoid-s4285" xml:space="preserve">pro creabitur altitudo F G, 213 {1/3}. </s> <s xml:id="echoid-s4286" xml:space="preserve">paſſuum. </s> <s xml:id="echoid-s4287" xml:space="preserve">Vides ergo <lb/>minorem Quotientem 8. </s> <s xml:id="echoid-s4288" xml:space="preserve">in differentiam ſtationum paſſuum 16. </s> <s xml:id="echoid-s4289" xml:space="preserve">ductam eſſe, vt <lb/>gignerentur 128. </s> <s xml:id="echoid-s4290" xml:space="preserve">& </s> <s xml:id="echoid-s4291" xml:space="preserve">hunc numerum diuiſum eſſe per 12. </s> <s xml:id="echoid-s4292" xml:space="preserve">differentiam Quotien-<lb/>tum, vt pro dirent paſſus 10 {2/3}. </s> <s xml:id="echoid-s4293" xml:space="preserve">pro recta a F: </s> <s xml:id="echoid-s4294" xml:space="preserve">ac tandem hunc Quotientem 10 {2/3}. </s> <s xml:id="echoid-s4295" xml:space="preserve"><lb/>ductum eſſe in 20. </s> <s xml:id="echoid-s4296" xml:space="preserve">Quotientem maiorem, vt produceretur altitudo F G, paſ-<lb/>ſuum 213 {1/3}. </s> <s xml:id="echoid-s4297" xml:space="preserve">vt in regula pręcepimus. </s> <s xml:id="echoid-s4298" xml:space="preserve">Quam altitudinem etiam reperies per præ-<lb/>cedentem viam, ſi nimirum fiat, vt 75. </s> <s xml:id="echoid-s4299" xml:space="preserve">differentia vmbrarum rectarum, ad diffe-<lb/>rentiam ſtationum, nimirum ad 16. </s> <s xml:id="echoid-s4300" xml:space="preserve">paſſus. </s> <s xml:id="echoid-s4301" xml:space="preserve">ita quadrati latus 1000. </s> <s xml:id="echoid-s4302" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s4303" xml:space="preserve"><lb/>Nam 16. </s> <s xml:id="echoid-s4304" xml:space="preserve">ducta in 1000. </s> <s xml:id="echoid-s4305" xml:space="preserve">& </s> <s xml:id="echoid-s4306" xml:space="preserve">productus numerus 16000. </s> <s xml:id="echoid-s4307" xml:space="preserve">diuiſus per 75. </s> <s xml:id="echoid-s4308" xml:space="preserve">facit Quo-<lb/>tientem 213 {1/3}. </s> <s xml:id="echoid-s4309" xml:space="preserve">veluti prius.</s> <s xml:id="echoid-s4310" xml:space="preserve"/> </p> <div xml:id="echoid-div275" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-139-04" xlink:href="note-139-04a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s4311" xml:space="preserve">3. </s> <s xml:id="echoid-s4312" xml:space="preserve"><emph style="sc">Deniqve</emph> in remotiore ſtatione inter-<lb/> <anchor type="figure" xlink:label="fig-140-01a" xlink:href="fig-140-01"/> ſecetur vmbra verſa in E, & </s> <s xml:id="echoid-s4313" xml:space="preserve">in propinquiore <lb/>vmbra recta in H, vt in 3. </s> <s xml:id="echoid-s4314" xml:space="preserve">figura problem. </s> <s xml:id="echoid-s4315" xml:space="preserve">3. </s> <s xml:id="echoid-s4316" xml:space="preserve">Re-<lb/>ducta vmbra verſa adrectam, vt in Quadrati <lb/>conſtructione Num. </s> <s xml:id="echoid-s4317" xml:space="preserve">7. </s> <s xml:id="echoid-s4318" xml:space="preserve">principio huius lib. </s> <s xml:id="echoid-s4319" xml:space="preserve">ſcri-<lb/>pſimus: </s> <s xml:id="echoid-s4320" xml:space="preserve">ſi fiat, vt IN, differentia vmbrarum re-<lb/>ctarum ad A a, differentiam ſtationum, ita A B, <lb/>latus Quadrati 1000. </s> <s xml:id="echoid-s4321" xml:space="preserve">ad aliud, procreabitur al-<lb/>titudo F G, quemadmodum Num. </s> <s xml:id="echoid-s4322" xml:space="preserve">2. </s> <s xml:id="echoid-s4323" xml:space="preserve">oſten-<lb/>dimus.</s> <s xml:id="echoid-s4324" xml:space="preserve"/> </p> <div xml:id="echoid-div276" type="float" level="2" n="3"> <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/xxxxxxxx/figures/140-01"/> </figure> </div> <p> <s xml:id="echoid-s4325" xml:space="preserve">4. </s> <s xml:id="echoid-s4326" xml:space="preserve"><emph style="sc">Si</emph> fortè in vna ſtatione tranſiret linea fiduciæ, aut filum perpendiculi per <lb/>punctum C, aſſumendum eſſet in primo caſu latus C D, vmbrę verſę in duo-<lb/>bus autem aliis caſibus latus BC, vmbrę rectæ.</s> <s xml:id="echoid-s4327" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div278" type="section" level="1" n="120"> <head xml:id="echoid-head123" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s4328" xml:space="preserve">Sine reductione vmbrę verſę ad rectam ita agendum erit. </s> <s xml:id="echoid-s4329" xml:space="preserve">Numerus, qui fit <lb/>ex vmbra verſa D E, in vmbram rectam b H, auferatur ex 1000000. </s> <s xml:id="echoid-s4330" xml:space="preserve">quadrato <lb/>lateris 1000@ reſi duumque ſit P. </s> <s xml:id="echoid-s4331" xml:space="preserve">Item ex vmbra verſa DE, in latus 1000. </s> <s xml:id="echoid-s4332" xml:space="preserve">fiat Q. <lb/></s> <s xml:id="echoid-s4333" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> Et quoniam ob triangulorum ſimilitudinem eſt, vt BN, ad BA, ita AF, ad F G:</s> <s xml:id="echoid-s4334" xml:space="preserve"> <anchor type="note" xlink:label="note-140-01a" xlink:href="note-140-01"/> Etpermutando, vt BN, ad AF, ita BA, ad FG. </s> <s xml:id="echoid-s4335" xml:space="preserve">Sed vt BN, vmbra recta maior ad <lb/>A F, diſtantiam, ita eſt I N, differentia vmbrarum rectarum ad A a, differentiam <lb/>ſtationum, vt Num. </s> <s xml:id="echoid-s4336" xml:space="preserve">5. </s> <s xml:id="echoid-s4337" xml:space="preserve">in problem. </s> <s xml:id="echoid-s4338" xml:space="preserve">3. </s> <s xml:id="echoid-s4339" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s4340" xml:space="preserve">Igitur erit quoque IN, ad Aa, <lb/>ſicuti AB, ad FG: </s> <s xml:id="echoid-s4341" xml:space="preserve">Et permutando IN, ad AB, vt Aa, ad FG. </s> <s xml:id="echoid-s4342" xml:space="preserve">Dico iam ita eſſe <lb/>P, ad Q, vt IN, ad AB. </s> <s xml:id="echoid-s4343" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam enim eſt, vt DE, ad AD, ita AB, vel AD, ad <anchor type="note" xlink:label="note-140-02a" xlink:href="note-140-02"/> BN:</s> <s xml:id="echoid-s4344" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit rectangulum ſub DE, BN, ęquale quadrato numero 1000000. </s> <s xml:id="echoid-s4345" xml:space="preserve">lateris <anchor type="note" xlink:label="note-140-03a" xlink:href="note-140-03"/> AB, 1000. </s> <s xml:id="echoid-s4346" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Eſt autemrectangulum ſub DE, BN, æquale rectangulis ſub DE, <anchor type="note" xlink:label="note-140-04a" xlink:href="note-140-04"/> BI, & </s> <s xml:id="echoid-s4347" xml:space="preserve">ſub DE, IN. </s> <s xml:id="echoid-s4348" xml:space="preserve">Igitur ſi productum ex vmbra verſa DE, in B I, hoc eſt, in <lb/>vmbram rectamb H, dematur ex rectangulo ſub DE, BN, hoc eſt, ex quadrato <lb/>1000000. </s> <s xml:id="echoid-s4349" xml:space="preserve">remanebit rectangulum ſub DE, IN. </s> <s xml:id="echoid-s4350" xml:space="preserve">Ac propterea P, fiet ex DE, in <lb/>IN. </s> <s xml:id="echoid-s4351" xml:space="preserve">Fit autem Q, ex eadem vmbra verſa DE, in latus AB, 1000. </s> <s xml:id="echoid-s4352" xml:space="preserve">Igitur cum DE, <lb/>multiplicans IN, & </s> <s xml:id="echoid-s4353" xml:space="preserve">AB; </s> <s xml:id="echoid-s4354" xml:space="preserve">producat P, Q, <anchor type="note" xlink:href="" symbol="e"/> erit P, ad Q, ſicut IN, ad AB; </s> <s xml:id="echoid-s4355" xml:space="preserve">hoc <anchor type="note" xlink:label="note-140-05a" xlink:href="note-140-05"/> eſt, ſicut Aa, ad FG. </s> <s xml:id="echoid-s4356" xml:space="preserve">Quocirca ſi fiat.</s> <s xml:id="echoid-s4357" xml:space="preserve"/> </p> <div xml:id="echoid-div278" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-140-01" xlink:href="note-140-01a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="left" xlink:label="note-140-02" xlink:href="note-140-02a" xml:space="preserve">4. ſexti.</note> <note symbol="c" position="left" xlink:label="note-140-03" xlink:href="note-140-03a" xml:space="preserve">17. ſexti.</note> <note symbol="d" position="left" xlink:label="note-140-04" xlink:href="note-140-04a" xml:space="preserve">1. ſecundi.</note> <note symbol="e" position="left" xlink:label="note-140-05" xlink:href="note-140-05a" xml:space="preserve">17. ſexti.@</note> </div> <pb o="111" file="141" n="141" rhead="LIBER TERTIVS."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt P, numer{us}, qui relinquitur, ſi \\ product{us} ex vmbra verſa D E, \\ in vmbrã rectam b H, ex qua- \\ drato 1,000,000. d{et}rahatur. # ad Q, numerum, \\ qui fit ex vmbra \\ verſa D E, in la- \\ t{us} A B, 1000. # Ita Aa, diffe- \\ rentia ſtatio- \\ num # ad F G, al- \\ titudinem, <lb/></note> <p> <s xml:id="echoid-s4358" xml:space="preserve">producetur altitudo F G, nota in partibus differentię ſtationum A a: </s> <s xml:id="echoid-s4359" xml:space="preserve">cui ſi ad-<lb/>iicietur ſtatura menſoris F L, tota altitudo GL, nota efficietur.</s> <s xml:id="echoid-s4360" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4361" xml:space="preserve"><emph style="sc">In</emph> ſcholio porro ſequentis problematis idem hoc problema per vnicam <lb/>ſtationem abſoluemus.</s> <s xml:id="echoid-s4362" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4363" xml:space="preserve">ALTITVDINEM eandem, quando diſtantia ab oculo menſoris ne-<lb/>que data eſt, neque inuenta, neque è directo altitudinis duæ ſtationes <lb/>fieri poſſunt, per duas ſtationes in aliqua haſta erecta factas, indagare <lb/>per Quadratum.</s> <s xml:id="echoid-s4364" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div280" type="section" level="1" n="121"> <head xml:id="echoid-head124" xml:space="preserve">PROBLEMA VII.</head> <p> <s xml:id="echoid-s4365" xml:space="preserve">1. </s> <s xml:id="echoid-s4366" xml:space="preserve"><emph style="sc">Cvm</emph> in plano duæ ſtationes fieri commodè nequeunt erigatur haſta ali-<lb/>qua K b, ad Horizontem recta, niſi fortè ad ſit aliquod ædificium erectum, ibi-<lb/> <anchor type="figure" xlink:label="fig-141-01a" xlink:href="fig-141-01"/> que fiant duæ ſtationes in A, & </s> <s xml:id="echoid-s4367" xml:space="preserve">a, vt in <lb/>problemate 4. </s> <s xml:id="echoid-s4368" xml:space="preserve">Cadat autem primum <lb/>dioptra, vel filum perpendiculi in vtra-<lb/>que ſtatione in vmbram verſam, vt in 1. <lb/></s> <s xml:id="echoid-s4369" xml:space="preserve">figura problematis 4. </s> <s xml:id="echoid-s4370" xml:space="preserve">quę hic repetita eſt. </s> <s xml:id="echoid-s4371" xml:space="preserve"><lb/> <anchor type="note" xlink:href="" symbol="a"/> Et quoniã propter triangulorũ ſimili- <anchor type="note" xlink:label="note-141-02a" xlink:href="note-141-02"/> tudinẽ eſt, vt A D, ad D E, ita AF, ad F G; <lb/></s> <s xml:id="echoid-s4372" xml:space="preserve">erit permutando vt AD, ad AF, ita DE, ad <lb/>FG. </s> <s xml:id="echoid-s4373" xml:space="preserve">Eademque ratione erit vt, a d, ad <lb/>a M, ita d H, ad M G. </s> <s xml:id="echoid-s4374" xml:space="preserve">Cumergo eadem <lb/>ſit proportio A D, ad A F, quæ a d, ad a M, <lb/>propter ęqualitatem rectarum AD, ad, & </s> <s xml:id="echoid-s4375" xml:space="preserve">AF, a M; </s> <s xml:id="echoid-s4376" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit vt tota D E, ad totam <anchor type="note" xlink:label="note-141-03a" xlink:href="note-141-03"/> F G, ita d H, hoc eſt, ita D I, ablata ad ablatam MG. </s> <s xml:id="echoid-s4377" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur erit quo que reliqua <anchor type="note" xlink:label="note-141-04a" xlink:href="note-141-04"/> I E, ad reliquam F M, hoc eſt, ad D d, vt tota D E, ad totam F G. </s> <s xml:id="echoid-s4378" xml:space="preserve">Quocirca ſi <lb/>fiat,</s> </p> <div xml:id="echoid-div280" type="float" level="2" n="1"> <figure xlink:label="fig-141-01" xlink:href="fig-141-01a"> <image file="141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/141-01"/> </figure> <note symbol="a" position="right" xlink:label="note-141-02" xlink:href="note-141-02a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="right" xlink:label="note-141-03" xlink:href="note-141-03a" xml:space="preserve">11. quinti.</note> <note symbol="c" position="right" xlink:label="note-141-04" xlink:href="note-141-04a" xml:space="preserve">19. quinti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt IE, differentia vm- \\ brarum verſarum # ad Dd, differentiam \\ ſtationum: # Ita D E, mæior vm- \\ bra verſa # ad F G, altitu- \\ dinem <lb/></note> <p> <s xml:id="echoid-s4379" xml:space="preserve">euadet cognita altitudo F G, in partibus differentiæ ſtationum D d. </s> <s xml:id="echoid-s4380" xml:space="preserve">Appoſita <lb/>autem ſtatura menſoris FL, tota altitudo G L, quæſita cognita erit.</s> <s xml:id="echoid-s4381" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4382" xml:space="preserve">2. </s> <s xml:id="echoid-s4383" xml:space="preserve"><emph style="sc">Abscindat</emph> deinde dioptra in vtraque ſtatione vmbram rectam, vt <lb/>in 2. </s> <s xml:id="echoid-s4384" xml:space="preserve">figura problematis 4. </s> <s xml:id="echoid-s4385" xml:space="preserve">quæ poſita eſt in pagina 103. </s> <s xml:id="echoid-s4386" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et quia propter trian- <anchor type="note" xlink:label="note-141-06a" xlink:href="note-141-06"/> gulorum ſimilitudinem eſt vt b H, ad ab, ita a M, ad M G: </s> <s xml:id="echoid-s4387" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit rectangu- <anchor type="note" xlink:label="note-141-07a" xlink:href="note-141-07"/> lum ſub b H, M G, æquale rectangulo ſub a b, a M. </s> <s xml:id="echoid-s4388" xml:space="preserve">Eadem ratione erit rectan-<lb/>gulum ſub B E, F G, æquale rectangulo ſub A B, A F, <anchor type="note" xlink:href="" symbol="f"/> quod eadem quo que <anchor type="note" xlink:label="note-141-08a" xlink:href="note-141-08"/> fit proportio B E, ad A B, quæ A F, ad F G. </s> <s xml:id="echoid-s4389" xml:space="preserve">Cum ergo rectangulum ſub a b, <lb/>a M, rectangulo ſub A B, AF, æquale ſit, ob æqualitatem rectarum a b, A B, & </s> <s xml:id="echoid-s4390" xml:space="preserve"><lb/>a M, AF, erit etiam rectangulum ſub b H, M G; </s> <s xml:id="echoid-s4391" xml:space="preserve">rectangulo ſub BE, FG, æquale, <pb o="112" file="142" n="142" rhead="GEOMETR. PRACT."/> <anchor type="note" xlink:href="" symbol="a"/> Quare erit vt tota BH, ad totam FG, ita BE, vel b O, ablata ad ablatam MG;</s> <s xml:id="echoid-s4392" xml:space="preserve"> <anchor type="note" xlink:label="note-142-01a" xlink:href="note-142-01"/> <anchor type="note" xlink:href="" symbol="b"/> Ac propterea erit quo que reliqua O H, ad reliquam F M, ſiue ad A a, vt tota <anchor type="note" xlink:label="note-142-02a" xlink:href="note-142-02"/> b H, ad totam FG. </s> <s xml:id="echoid-s4393" xml:space="preserve">Quamobrem ſi fiat,</s> </p> <div xml:id="echoid-div281" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-141-06" xlink:href="note-141-06a" xml:space="preserve">4. ſexti.</note> <note symbol="e" position="right" xlink:label="note-141-07" xlink:href="note-141-07a" xml:space="preserve">16. ſexti.</note> <note symbol="f" position="right" xlink:label="note-141-08" xlink:href="note-141-08a" xml:space="preserve">4. ſexti.</note> <note symbol="a" position="left" xlink:label="note-142-01" xlink:href="note-142-01a" xml:space="preserve">17. ſexti.</note> <note symbol="b" position="left" xlink:label="note-142-02" xlink:href="note-142-02a" xml:space="preserve">19. ſexti.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt O H, differentia vmbra- \\ rum rectarum # ad A a, differentiam \\ ſtationum: # Ita b H, maior \\ vmbrarecta # ad F G, alti- \\ tudinem, <lb/></note> <p> <s xml:id="echoid-s4394" xml:space="preserve">altitudo F G, prodibit cognita, quę cum ſtatura menſoris F L, totam altitudi-<lb/>nem LG, efficiet notam. </s> <s xml:id="echoid-s4395" xml:space="preserve">Vbi vides eundem prorſus eſſe operandi modum in <lb/>vmbris rectis, & </s> <s xml:id="echoid-s4396" xml:space="preserve">in verſis.</s> <s xml:id="echoid-s4397" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4398" xml:space="preserve">3. </s> <s xml:id="echoid-s4399" xml:space="preserve"><emph style="sc">Postremo</emph> in ſtatione inferiori reſecetur vmbra recta BE, & </s> <s xml:id="echoid-s4400" xml:space="preserve">verſa d H, <lb/>in ſuperiori ſtatione, vt in 3. </s> <s xml:id="echoid-s4401" xml:space="preserve">figura problematis 4. </s> <s xml:id="echoid-s4402" xml:space="preserve">quæ poſita eſt in pagina 105. <lb/></s> <s xml:id="echoid-s4403" xml:space="preserve">Reducta ergo vmbra recta ad verſam, vt ad initium huius libri in conſtructione <lb/>Quadrati Num. </s> <s xml:id="echoid-s4404" xml:space="preserve">7. </s> <s xml:id="echoid-s4405" xml:space="preserve">declarauimus: </s> <s xml:id="echoid-s4406" xml:space="preserve">ſi fiat, vt IN, differentia vmbrarum verſarum <lb/>ad A a, differentiã ſtationum, ita DN, vmbra verſa maior ad aliud, pro creabitur <lb/>F G, altitudo in partibus differentiæ ſtationum A a, quemadmodum Num. </s> <s xml:id="echoid-s4407" xml:space="preserve">1. </s> <s xml:id="echoid-s4408" xml:space="preserve"><lb/>demonſtrauimus. </s> <s xml:id="echoid-s4409" xml:space="preserve">Quod ſi malueris reducere vmbram verſam ad rectam, ean-<lb/>dem altitudinem reperies per duas vmbras rectas, ceu Num. </s> <s xml:id="echoid-s4410" xml:space="preserve">2. </s> <s xml:id="echoid-s4411" xml:space="preserve">monſtrauimus.</s> <s xml:id="echoid-s4412" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4413" xml:space="preserve">4. </s> <s xml:id="echoid-s4414" xml:space="preserve"><emph style="sc">Qvando</emph> fortaſſe in altera ſtationum linea fiduciæ tranſiret per pun-<lb/>ctum C, in arbitrio tuo erit, vel totum latus vmbræ verſæ, vel rectæ aſſumere.</s> <s xml:id="echoid-s4415" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div283" type="section" level="1" n="122"> <head xml:id="echoid-head125" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s4416" xml:space="preserve">1. </s> <s xml:id="echoid-s4417" xml:space="preserve"><emph style="sc">Eandem</emph> altitudinem, eiuſque diſtantiam ab oculo menſoris, vna cum <lb/>hypotenuſa ab oculo ad faſtigium altitudinis extenſa, venari licebit ope qua-<lb/>drati ſtabilis per vnicam ſtationem, etiamſi ſolum faſtigium rei erectæ cerna-<lb/>tur: </s> <s xml:id="echoid-s4418" xml:space="preserve">adeò vt ſcholium hoc omnia præſtet, quæ in problemate 3. </s> <s xml:id="echoid-s4419" xml:space="preserve">4. </s> <s xml:id="echoid-s4420" xml:space="preserve">5. </s> <s xml:id="echoid-s4421" xml:space="preserve">6. </s> <s xml:id="echoid-s4422" xml:space="preserve">& </s> <s xml:id="echoid-s4423" xml:space="preserve">7. </s> <s xml:id="echoid-s4424" xml:space="preserve">per <lb/>plures ſtationes inueſtigauimus, ac proinde diligenter perdiſcendum.</s> <s xml:id="echoid-s4425" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4426" xml:space="preserve"><emph style="sc">Sit</emph> enim altitudo metienda IF; </s> <s xml:id="echoid-s4427" xml:space="preserve">ſtatura menſoris I G, vel H A. </s> <s xml:id="echoid-s4428" xml:space="preserve">Paretur pla-<lb/> <anchor type="note" xlink:label="note-142-04a" xlink:href="note-142-04"/> num Horizonti æquidiſtans per A, tranſi-<lb/> <anchor type="figure" xlink:label="fig-142-01a" xlink:href="fig-142-01"/> ens, in quo Quadratum firmari poſsit, aut <lb/>collo cari. </s> <s xml:id="echoid-s4429" xml:space="preserve">Primum ergo inueſtiganda eſt <lb/>hypotenuſa A F, hoc modo. </s> <s xml:id="echoid-s4430" xml:space="preserve">Inclinetur <lb/>quadratum, ita vt centrum dioptræ oc-<lb/>cupet ſuperiorem locum lateris a A, cuius <lb/>inferius punctum ſtatuatur in A, oculo <lb/>menſoris. </s> <s xml:id="echoid-s4431" xml:space="preserve">Et totum quadratum tamdiu <lb/>eleuetur, aut deprimatur, doneclatus <lb/>A e, recta in cacumen F, vergat. </s> <s xml:id="echoid-s4432" xml:space="preserve">Manen-<lb/>te ſic Quadrato, menſor oculum transferat ad a, & </s> <s xml:id="echoid-s4433" xml:space="preserve">per dioptram inſpiciat ca-<lb/>cumen F, notenturque partes vmbræ verſæ abſciſſæ b c. </s> <s xml:id="echoid-s4434" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Deinde fiat,</s> </p> <div xml:id="echoid-div283" type="float" level="2" n="1"> <note position="left" xlink:label="note-142-04" xlink:href="note-142-04a" xml:space="preserve">Quo pacto <lb/>præcedentia <lb/>5. problemata, <lb/>@pe quadrati <lb/>ſtabilis per v-<lb/>nicam ſtatio-<lb/>nem abſol-<lb/>uantur.</note> <figure xlink:label="fig-142-01" xlink:href="fig-142-01a"> <image file="142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/142-01"/> </figure> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt c b, vmbra \\ verſa # ad lat{us} a b, \\ 1000. # Ita lat{us} a A, notum in data men- \\ ſura, quod nunc ponam{us} conti- \\ nere 3. ped{es}, # ad A F, <lb/></note> <p> <s xml:id="echoid-s4435" xml:space="preserve">Nam inuentus numerus dabit hypotenuſam A F, cognitam in partibus lateris <lb/>a A. </s> <s xml:id="echoid-s4436" xml:space="preserve">Verbi gratia, ſi c b, contineat 50. </s> <s xml:id="echoid-s4437" xml:space="preserve">partes vmbræ verſę, & </s> <s xml:id="echoid-s4438" xml:space="preserve">a A, tribus pedi-<lb/>bus ſit æqualis: </s> <s xml:id="echoid-s4439" xml:space="preserve">ſi fiat vt 50. </s> <s xml:id="echoid-s4440" xml:space="preserve">ad 1000. </s> <s xml:id="echoid-s4441" xml:space="preserve">ita 3. </s> <s xml:id="echoid-s4442" xml:space="preserve">ad aliud, reperietur A F, complecti <pb o="113" file="143" n="143" rhead="LIBER TERTIVS."/> 60. </s> <s xml:id="echoid-s4443" xml:space="preserve">pedes. </s> <s xml:id="echoid-s4444" xml:space="preserve">Si verò a A, ponatur 1000. </s> <s xml:id="echoid-s4445" xml:space="preserve">comperietur eadem AF, particularum <lb/>200000. </s> <s xml:id="echoid-s4446" xml:space="preserve">qualium 1000. </s> <s xml:id="echoid-s4447" xml:space="preserve">in a A, comprehenduntur.</s> <s xml:id="echoid-s4448" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4449" xml:space="preserve">2. </s> <s xml:id="echoid-s4450" xml:space="preserve"><emph style="sc">Post</emph> hæc deprimatur Quadratum, ita vt latus AD, Horizonti æquidi-<lb/>ſtet, centrum que dioptrę ſit in A. </s> <s xml:id="echoid-s4451" xml:space="preserve">quo poſito, videbitur cacumen F, per inuen-<lb/>tam hypotenuſam AF, quæ primum ſecet vmbram verſam C D, in E, vt in pri-<lb/>mo quadrato, inquiratur que portio dioptrę AE, in partibus laterum A D, D E, <lb/> <anchor type="note" xlink:label="note-143-01a" xlink:href="note-143-01"/> <anchor type="note" xlink:href="" symbol="a"/> quod fiet vel ex vtro que latere A D, D E, cognito: </s> <s xml:id="echoid-s4452" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Vel ex cognito per pro- blema 1. </s> <s xml:id="echoid-s4453" xml:space="preserve">ex vmbra D E, angulo D A E, ex latere DE, noto: </s> <s xml:id="echoid-s4454" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Vel denique ſi ex <anchor type="note" xlink:label="note-143-02a" xlink:href="note-143-02"/> ſumma quadratorum ex AD, DE, deſcriptorum radix quadrata extrahatur. </s> <s xml:id="echoid-s4455" xml:space="preserve">In-<lb/>uenta portione A E, in partibus milleſimis lateris A D, ductaque EM, ipſi A G, <lb/> <anchor type="note" xlink:label="note-143-03a" xlink:href="note-143-03"/> parallela, <anchor type="note" xlink:href="" symbol="d"/> ſi fiat,</s> </p> <div xml:id="echoid-div284" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-143-01" xlink:href="note-143-01a" xml:space="preserve">6. Triang. <lb/>rectil.</note> <note symbol="b" position="right" xlink:label="note-143-02" xlink:href="note-143-02a" xml:space="preserve">5. Triang. <lb/>rectil.</note> <note symbol="c" position="right" xlink:label="note-143-03" xlink:href="note-143-03a" xml:space="preserve">47. primi.</note> </div> <note symbol="d" position="right" xml:space="preserve">2. ſexti & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt AE, inuenta in partib{us} \\ milleſimis AD, # ad AF, in iiſdem par- \\ tib{us} inuentam # Ita G M, in iiſdem par- \\ tib{us} cognita, cum æ- \\ qualis ſit ipſi D E, # ad GF, <lb/></note> <p> <s xml:id="echoid-s4456" xml:space="preserve">nota effi cietur GF, in partibus milleſimis lateris AD. </s> <s xml:id="echoid-s4457" xml:space="preserve">Quod ſi rurſus fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} AD, \\ 1000. # ad 3. ped{es}, quos \\ ponim{us} in AD, \\ contineri: # Ita G F, in milleſimis parti- \\ b{us} lateris AD, inuenta, # ad GF, <lb/></note> <p> <s xml:id="echoid-s4458" xml:space="preserve">reperietur eadem GF, in menſura pedum. </s> <s xml:id="echoid-s4459" xml:space="preserve">Et ſi adiiciatur, ſtatura menſoris GI, <lb/>nota in eadem menſura, cognita fiet tota altitudo I F, in menſura pedum.</s> <s xml:id="echoid-s4460" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4461" xml:space="preserve">Vbi vides, nos eadem opera inueniſſe quo que diſtantiam AF, ab oculo A, ad <lb/>cacumen vſque F.</s> <s xml:id="echoid-s4462" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4463" xml:space="preserve">3. </s> <s xml:id="echoid-s4464" xml:space="preserve"><emph style="sc">Secet</emph> deinde hypotenuſa a F, inuenta (inuenietur autem vt Num. </s> <s xml:id="echoid-s4465" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s4466" xml:space="preserve">dictum eſt) vtrumque latus vmbræ in C, vt in 2. </s> <s xml:id="echoid-s4467" xml:space="preserve">quadrato. </s> <s xml:id="echoid-s4468" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Si igitur fiat,</s> </p> <note symbol="e" position="right" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ a c, inuenta in par- \\ tib{us} a d, lateris, 1000. vt Num. 2. \\ dixim{us}. # ad a F, hypotenuſam \\ in iiſdem partib{us} \\ inuentam: # Ita GK, \\ 1000. # ad GF, <lb/></note> <p> <s xml:id="echoid-s4469" xml:space="preserve">prodibit GF, nota in partibus milleſimis lateris d c: </s> <s xml:id="echoid-s4470" xml:space="preserve">Et ſi rurſus fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} d c, \\ 1000. # ad 3. ped{es}, quib{us} æquale \\ ponim{us} lat{us} d c, # Ita GK, 1000. cum æqua- \\ le ſit ipſi d c, # ad GF, <lb/></note> <p> <s xml:id="echoid-s4471" xml:space="preserve">inuenta erit eadem GF, in menſura pedum. </s> <s xml:id="echoid-s4472" xml:space="preserve">Et ſi addatur ſtatura menſoris G I, <lb/>nota in eadem menſura, cognoſcetur tota altitudo IF, in menſura pedum.</s> <s xml:id="echoid-s4473" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4474" xml:space="preserve">4. </s> <s xml:id="echoid-s4475" xml:space="preserve"><emph style="sc">Tertio</emph> ſecet hypotenuſa inuenta AF, (quę inuenietur vt Num. </s> <s xml:id="echoid-s4476" xml:space="preserve">7. </s> <s xml:id="echoid-s4477" xml:space="preserve">do-<lb/>cuimus) vmbram rectam in E, vt in tertio quadrato. </s> <s xml:id="echoid-s4478" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Siergo fiat,</s> </p> <note symbol="f" position="right" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ A E, inuenta, \\ vt Num. 2. docuim{us}. # ad AF, hypotenuſam \\ inuentam # Ita G K, \\ 1000. # ad G F, <lb/></note> <p> <s xml:id="echoid-s4479" xml:space="preserve">exibit GF, nota in partibus milleſimis lateris DC. </s> <s xml:id="echoid-s4480" xml:space="preserve">Et ſi iterum fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} D C, \\ 1000. # ad 3. ped{es} in D C, \\ contentos # Ita GK, 1000. cum ſit æqua- \\ lis ipſi D C, # ad G F, <lb/></note> <p> <s xml:id="echoid-s4481" xml:space="preserve">cognita erit eadem GF, in menſura pedum, & </s> <s xml:id="echoid-s4482" xml:space="preserve">c.</s> <s xml:id="echoid-s4483" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4484" xml:space="preserve"><emph style="sc">Atqve</emph> hac eadem ratione omnes aliæ lineæ inuentę in partibus milleſimis <lb/>lateris Quadratireuo cabuntur ad quamcunque aliam menſuram, vt ad pedes, <lb/>vel cubitos, &</s> <s xml:id="echoid-s4485" xml:space="preserve">c.</s> <s xml:id="echoid-s4486" xml:space="preserve"/> </p> <pb o="114" file="144" n="144" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s4487" xml:space="preserve">5. </s> <s xml:id="echoid-s4488" xml:space="preserve"><emph style="sc">Distantia</emph> autem A G, vel HI, cognita euadet, <anchor type="note" xlink:href="" symbol="a"/> ſi fiat,</s> </p> <note symbol="a" position="left" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ A E, in pri- \\ mo Quadrato, vel a c, in 2. \\ Quadrato, # ad hypotenuſam \\ AF, vela F, # Ita lat{us} A D, \\ in 1. Quadrato \\ vel a d, in 2. \\ Quadrato # ad A G, in \\ 1. quadrato \\ vel ad a G, \\ in ſecundo. <lb/></note> </div> <div xml:id="echoid-div286" type="section" level="1" n="123"> <head xml:id="echoid-head126" xml:space="preserve"><emph style="sc">Vel</emph></head> <p> <s xml:id="echoid-s4489" xml:space="preserve">Ducta EL, lateri AB, parallela in tertio quadrato,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt A E, portio dio- \\ ptræ, # ad hypotenuſam \\ A F, # Ita vmbrarecta B E, ipſi \\ AL, æqualis, # ad AG. <lb/></note> <p> <s xml:id="echoid-s4490" xml:space="preserve"><emph style="sc">Vides</emph> igitur, in hac ratione dimetiendę altitudinis, ac diſtantię, etiam inac-<lb/>ceſsibilis, opus non eſſe duabus ſtationibus ſiue in plano, ſiue in haſta aliqua <lb/>erecta faciendis; </s> <s xml:id="echoid-s4491" xml:space="preserve">Quod non parum moleſtiæ plerunque afferre ſolet: </s> <s xml:id="echoid-s4492" xml:space="preserve">ſed ſo-<lb/>lum inueſtigandam eſſe prius hypotenuſam per quadratum ſtabile inclinatum: <lb/></s> <s xml:id="echoid-s4493" xml:space="preserve">Deinde portionem dioptræ inter eius centrum, & </s> <s xml:id="echoid-s4494" xml:space="preserve">latus Quadrati, quod inter-<lb/>ſecat: </s> <s xml:id="echoid-s4495" xml:space="preserve">quæinuentio diffi cilis non eſt, vt patuit, cum menſor nunquam locum <lb/>mutare cogatur.</s> <s xml:id="echoid-s4496" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4497" xml:space="preserve">ALTITVDINEM turris aut montis, ex eius ſummitate per Quadra-<lb/>tum dimetiri, quando in plano ſummitatis Horizonti æquidiſtan-<lb/>te duæ ſtationes fieri poſſunt, & </s> <s xml:id="echoid-s4498" xml:space="preserve">ſignum aliquod in Horizonte cer-<lb/>nitur.</s> <s xml:id="echoid-s4499" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div287" type="section" level="1" n="124"> <head xml:id="echoid-head127" xml:space="preserve">PROBLEMA VIII.</head> <p> <s xml:id="echoid-s4500" xml:space="preserve">1. </s> <s xml:id="echoid-s4501" xml:space="preserve"><emph style="sc">Sit</emph> altitudo D F M d, in cuius ſummitate, <lb/> <anchor type="figure" xlink:label="fig-144-01a" xlink:href="fig-144-01"/> nimirum in plano D d, duę ſtationes fieri poſsint, <lb/>ex quibus ſignum aliquod in Horizonte, videli-<lb/>cet G, videri poſsit. </s> <s xml:id="echoid-s4502" xml:space="preserve">(Nos ne nouam cogeremur <lb/>deſcribere figuram, repetiuimus primam proble-<lb/>matis 4. </s> <s xml:id="echoid-s4503" xml:space="preserve">inuerſo tamen ordine poſitam, ita vt lite-<lb/>ræ rectum ſitum non habeant; </s> <s xml:id="echoid-s4504" xml:space="preserve">Et quamuis ſignum <lb/>G, non cernatur ex A, propter planum Dd, eadem <lb/>tamen erit ratio, ſi aliud ſignum longius diſtans <lb/>eligatur, quod ex A, inſpici poſsit.) </s> <s xml:id="echoid-s4505" xml:space="preserve">Collocetur <lb/>inſtrumentum in vtra que ſtatione, vt latus vmbrę <lb/>rectæ D C, vergat deorſum. </s> <s xml:id="echoid-s4506" xml:space="preserve">Et primum vtraque <lb/>vmbra recta ſecetur in E, H, (In hac enim inuer-<lb/>ſione latus D C, vmbræ rectę, & </s> <s xml:id="echoid-s4507" xml:space="preserve">BC, verſę deputa-<lb/>tur, vt in conſtructione Quadrati Num. </s> <s xml:id="echoid-s4508" xml:space="preserve">4. </s> <s xml:id="echoid-s4509" xml:space="preserve">initio huius libri declaratum eſt,) at-<lb/>que vmbræ d H, in propinquiore ſtatione, quæ ſemper minor eſt, æqualis ab-<lb/>ſcindatur DI. </s> <s xml:id="echoid-s4510" xml:space="preserve">Itaque ſi fiat,</s> </p> <div xml:id="echoid-div287" type="float" level="2" n="1"> <figure xlink:label="fig-144-01" xlink:href="fig-144-01a"> <image file="144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/144-01"/> </figure> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt I E, differentia vm- \\ brarum rectarum # ad Aa, differentiam \\ ſtationum # Ita A D, lat{us} qua- \\ drati 1000. # ad A F, <lb/></note> <p> <s xml:id="echoid-s4511" xml:space="preserve">cognita erit recta A F, ex qua ſi dematur Iatus quadrati A D, notum in partibus <pb o="115" file="145" n="145" rhead="LIBER TERTIVS."/> differentiæ ſtationum, nota relinquetur altitudo D F, vel d M, quæſita. </s> <s xml:id="echoid-s4512" xml:space="preserve">Inueniri <lb/>autem hac ratione rectam AF, demonſtratum eſt in problemate 4. </s> <s xml:id="echoid-s4513" xml:space="preserve">Num. </s> <s xml:id="echoid-s4514" xml:space="preserve">1.</s> <s xml:id="echoid-s4515" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4516" xml:space="preserve">2. </s> <s xml:id="echoid-s4517" xml:space="preserve"><emph style="sc">Qvod</emph> ſi in vtraque ſtatione latus vmbræ verſæ ſecetur in F, H, vt in 2. <lb/></s> <s xml:id="echoid-s4518" xml:space="preserve">figura problematis 4. </s> <s xml:id="echoid-s4519" xml:space="preserve">hic repetita, inuerſa tamen, habebimus tres vias inueſti-<lb/>gandi altitudinem A F, ex qua ſi tollatur latus quadrati A D, manifeſta relin-<lb/>quetur quęſita altitudo DF, vel d M. </s> <s xml:id="echoid-s4520" xml:space="preserve">Nam reducta vtraque vmbra verſa ad re-<lb/>ctam, ſi fiat,</s> </p> <figure> <image file="145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/145-01"/> </figure> <note style="it" position="right" xml:space="preserve"> <lb/>Vt I N, differentia vmbra- \\ rum rectarum, # ad A a, differentiam \\ ſtationum: # Ita A D, lat{us} qua- \\ drati 1000. # ad AF. <lb/># Vel ſine reductione. <lb/>Vt P, numer{us}, qui fit ex \\ O H, differentia vmbra- \\ rum verſarum in a b, la- \\ t{us} quadrati, # ad numerum, qui fit ex vm- \\ bra verſa b H, maiore in \\ minorem B E, # Ita A a, diffe- \\ rentia ſtatio- \\ num # ad A F, <lb/>#### Vel <lb/>Vt A a, differentia Quotientum, qui \\ fiunt, ſi lat{us} quadrati per vtram \\ vmbram verſam diuidatur, # ad A a, differentiam ſta- \\ tionum notam in menſu- \\ ra aliqua # Ita A F, \\ vt 1. # ad A F, <lb/></note> <p> <s xml:id="echoid-s4521" xml:space="preserve">procreabitur ſemper altitudo AF, ab oculo inſpectoris A, numerata: </s> <s xml:id="echoid-s4522" xml:space="preserve">quemad-<lb/>modumin problemate 4. </s> <s xml:id="echoid-s4523" xml:space="preserve">Num. </s> <s xml:id="echoid-s4524" xml:space="preserve">2. </s> <s xml:id="echoid-s4525" xml:space="preserve">demonſtratum eſt.</s> <s xml:id="echoid-s4526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4527" xml:space="preserve">3. </s> <s xml:id="echoid-s4528" xml:space="preserve"><emph style="sc">Si</emph> denique in vna ſtatione ſecetur latus vmbræ verſę in E, & </s> <s xml:id="echoid-s4529" xml:space="preserve">in altera la-<lb/>tus vmbrę rectæ in H, vt in 3. </s> <s xml:id="echoid-s4530" xml:space="preserve">figura problematis 4. </s> <s xml:id="echoid-s4531" xml:space="preserve">hic inuerſo ordine repetita: <lb/></s> <s xml:id="echoid-s4532" xml:space="preserve">ſi vtin eodem problemate 4. </s> <s xml:id="echoid-s4533" xml:space="preserve">Num. </s> <s xml:id="echoid-s4534" xml:space="preserve">3. </s> <s xml:id="echoid-s4535" xml:space="preserve">demonſtrauimus, reducatur vmbra ver-<lb/>ſa ad rectam, & </s> <s xml:id="echoid-s4536" xml:space="preserve">fiat,</s> </p> <figure> <image file="145-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/145-02"/> </figure> <pb o="116" file="146" n="146" rhead="GEOMETR. PRACT."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt NI, differentia vm- \\ brarumrectarum # ad Aa, differentiam \\ ſtationum: # Ita AD, latus qua- \\ drati # ad A F, <lb/>#### Vel ſine reductione. <lb/>Vt Onumerus, qui relinquitur, \\ ſi numerus genitus ex vmbra \\ verſa in rectam ex quadrato \\ lateris dematur, # ad numerum P, qui \\ ex vmbra verſa BE, \\ in latus AD, produ- \\ citur: # Ita Aa, \\ differẽ- \\ tia ſta- \\ tionum # ad AF, <lb/></note> <p> <s xml:id="echoid-s4537" xml:space="preserve">producetur AF, altitudo in partibus differentiæ ſtationum Aa, & </s> <s xml:id="echoid-s4538" xml:space="preserve">c.</s> <s xml:id="echoid-s4539" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4540" xml:space="preserve">4. </s> <s xml:id="echoid-s4541" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi diſtantia FG, à turri ad ſignum G, in Horizonte viſum nota <lb/>fuerit, facilius per vnicam ſtationem in A, factam altitudinem coniiciemus. </s> <s xml:id="echoid-s4542" xml:space="preserve">Nam <lb/>ſi vmbra recta DC, ſecetur in E, vt in 1. </s> <s xml:id="echoid-s4543" xml:space="preserve">figura: </s> <s xml:id="echoid-s4544" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Fiat autem.</s> <s xml:id="echoid-s4545" xml:space="preserve"/> </p> <note symbol="a" position="left" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra recta D E, # ad latus A D: # Ita diſtantia cognita FG, # ad A F, <lb/></note> <note symbol="b" position="left" xml:space="preserve">4. ſexti.</note> <p> <s xml:id="echoid-s4546" xml:space="preserve">Vel quando vmbra verſa BC, ſecetur in E, vt in 2. </s> <s xml:id="echoid-s4547" xml:space="preserve">figura; </s> <s xml:id="echoid-s4548" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ſi fiat.</s> <s xml:id="echoid-s4549" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} A B, # ad vmbram ver- \\ ſam B E, # Ita diſtantia cognita \\ F G, # ad A F. <lb/></note> <p> <s xml:id="echoid-s4550" xml:space="preserve">effi cietnr nota recta AF, in partibus diſtantiæ F G, à qua ſi dematur latus quadra-<lb/>ti AD, notum in eiſdem partibus diſtantiæ F G, nota relin quetur altitudo DF.</s> <s xml:id="echoid-s4551" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4552" xml:space="preserve"><emph style="sc">Qvod</emph> ſi dio ptra per C, tranſeat, <anchor type="note" xlink:href="" symbol="c"/> erit diſtantia F G, altitudini AF, æqualis.</s> <s xml:id="echoid-s4553" xml:space="preserve"> <anchor type="note" xlink:label="note-146-06a" xlink:href="note-146-06"/> Dempto ergo latere quadrati AD, altitudo turris DF, cognita fiet.</s> <s xml:id="echoid-s4554" xml:space="preserve"/> </p> <div xml:id="echoid-div288" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-146-06" xlink:href="note-146-06a" xml:space="preserve">6. primi.</note> </div> <p> <s xml:id="echoid-s4555" xml:space="preserve"><emph style="sc">Hoc</emph> idem problema in ſcholio ſequentis problematis abſoluemus per vni-<lb/>cam ſtationem.</s> <s xml:id="echoid-s4556" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4557" xml:space="preserve">ALTITVDINEM turris vel montis ex eius ſummitate per duas ſta-<lb/>tiones in haſta aliqua erecta factas, inueſtigare per quadratum, quan-<lb/>do ſignum aliquod in Horizonte videri poteſt.</s> <s xml:id="echoid-s4558" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div290" type="section" level="1" n="125"> <head xml:id="echoid-head128" xml:space="preserve">PROBLEMA IX.</head> <figure> <image file="146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/146-01"/> </figure> <p> <s xml:id="echoid-s4559" xml:space="preserve">1. </s> <s xml:id="echoid-s4560" xml:space="preserve"><emph style="sc">Qvando</emph> in ſummitate non eſt pla-<lb/>nities tanta, vt in ea duæ ſtationes poſsint <lb/>fieri, erigatur haſta aliqua in qua duæ ſta-<lb/>tiones fiant. </s> <s xml:id="echoid-s4561" xml:space="preserve">Vt ſi altitudo inueſtiganda <lb/>ſit d F, erigatur haſta d A, & </s> <s xml:id="echoid-s4562" xml:space="preserve">centro dio-<lb/>ptræ applicato ad haſtam primum in a, <lb/>deinde in A, inſpiciatur ſignum aliquod <lb/>G, in Horizonte per radios a G, A G, ſe-<lb/>ceturque primum vtraque vmbra recta in <lb/>H E, vt in prima figura problematis 3. <lb/></s> <s xml:id="echoid-s4563" xml:space="preserve">quam hic inuerſo ordine repetiuimus. </s> <s xml:id="echoid-s4564" xml:space="preserve"><lb/>Nam vt in præcedenti problemate dixi-<lb/>mus, in hac inuerſione D C, fit latus vm-<lb/>bræ rectæ, & </s> <s xml:id="echoid-s4565" xml:space="preserve">B C, verſæ. </s> <s xml:id="echoid-s4566" xml:space="preserve">Si ergo, vt in <lb/>problemate 1. </s> <s xml:id="echoid-s4567" xml:space="preserve">demonſtrauimus Num. </s> <s xml:id="echoid-s4568" xml:space="preserve">1. </s> <s xml:id="echoid-s4569" xml:space="preserve"><lb/>fiat,</s> </p> <pb o="117" file="147" n="147" rhead="LIBER TERTIVS."/> <note style="it" position="right" xml:space="preserve"> <lb/>Vt H I, differentia vm- \\ brarum rectarum # ad A a, differentiam \\ ſtationum; # Ita d H, vmbra \\ recta maior # ad AF, <lb/></note> <p> <s xml:id="echoid-s4570" xml:space="preserve">nota fiet tota recta AF, ex quaſi dematur portio haſtæ A d, quod duo quadra-<lb/>ta occupant, cognita fiet reliqua altitudo d F.</s> <s xml:id="echoid-s4571" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4572" xml:space="preserve">2. </s> <s xml:id="echoid-s4573" xml:space="preserve"><emph style="sc">Si</emph> vero in vtraque ſtatione vmbra verſa à radijs interſecetur, vt in 2. </s> <s xml:id="echoid-s4574" xml:space="preserve">figu-<lb/>ra problematis 3. </s> <s xml:id="echoid-s4575" xml:space="preserve">hic repetita ordine inuerſo, & </s> <s xml:id="echoid-s4576" xml:space="preserve">fiat,</s> </p> <figure> <image file="147-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/147-01"/> </figure> <note style="it" position="right" xml:space="preserve"> <lb/>vt I E, differentia vm- \\ brarum verſarum, # ad Aa, differentiam \\ ſtationum: # Ita B E, vmbra ver- \\ ſa maior # ad AF, <lb/></note> <p> <s xml:id="echoid-s4577" xml:space="preserve">pro dibit recta A F, ex qua ſi detrahatur portio haſtæ <lb/> <anchor type="figure" xlink:label="fig-147-02a" xlink:href="fig-147-02"/> A d, à principio primi quadrati ad finem ſecundi, re-<lb/> <anchor type="note" xlink:label="note-147-03a" xlink:href="note-147-03"/> liqua d F, altitudo fiet quo que nota: </s> <s xml:id="echoid-s4578" xml:space="preserve">veluti in pro-<lb/>blemate 3. </s> <s xml:id="echoid-s4579" xml:space="preserve">Num. </s> <s xml:id="echoid-s4580" xml:space="preserve">3. </s> <s xml:id="echoid-s4581" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s4582" xml:space="preserve"/> </p> <div xml:id="echoid-div290" type="float" level="2" n="1"> <figure xlink:label="fig-147-02" xlink:href="fig-147-02a"> <image file="147-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/147-02"/> </figure> <note position="right" xlink:label="note-147-03" xlink:href="note-147-03a" xml:space="preserve">Ad ſiniſtrum <lb/>angulum in-<lb/>feriorem, in-<lb/>ferioris qua-<lb/>drati, poned</note> </div> <p> <s xml:id="echoid-s4583" xml:space="preserve">3. </s> <s xml:id="echoid-s4584" xml:space="preserve">At ſiin vna ſtatione ſecetur latus vmbræ rectæ, <lb/>& </s> <s xml:id="echoid-s4585" xml:space="preserve">in altera latus vmbræ verſæ, vtin 3. </s> <s xml:id="echoid-s4586" xml:space="preserve">figura proble-<lb/>matis 3. </s> <s xml:id="echoid-s4587" xml:space="preserve">quæ inuerſa huc translata eſt, reducenda erit <lb/>vel recta vmbra ad verſam, vel verſa ad rectam. </s> <s xml:id="echoid-s4588" xml:space="preserve">Nam <lb/>vt in problemate 3. </s> <s xml:id="echoid-s4589" xml:space="preserve">Num. </s> <s xml:id="echoid-s4590" xml:space="preserve">5. </s> <s xml:id="echoid-s4591" xml:space="preserve">oſtendimus, ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt I N, differentia vm- \\ brarum ſiue verſarum, \\ ſiue rectarum, # ad Aa, differen- \\ tiam ſtationum: # Ita B N, maior vmbra \\ verſa, veld N, maior \\ vmbrarecta, # ad AF, <lb/></note> <p> <s xml:id="echoid-s4592" xml:space="preserve">effi cietur nota recta A F, ex qua ſi dematur portio haſtæ A d, nota relinquetur <lb/>altitudo d F.</s> <s xml:id="echoid-s4593" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4594" xml:space="preserve">4. </s> <s xml:id="echoid-s4595" xml:space="preserve"><emph style="sc">Qvod</emph> ſi turris eſſet A F, & </s> <s xml:id="echoid-s4596" xml:space="preserve">ex duabus feneſtris d, D, obſeruatio fieret, <lb/>deprehenderetur eo dem modo altitudo turris A F, vt patet.</s> <s xml:id="echoid-s4597" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div292" type="section" level="1" n="126"> <head xml:id="echoid-head129" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s4598" xml:space="preserve">1. </s> <s xml:id="echoid-s4599" xml:space="preserve"><emph style="sc">Qvod</emph> præcedentia duo problemata per duas ſtationes ſiue in plano <lb/>ſummitatis turris, vel montis, ſiue in haſta aliqua erecta factas docuerunt, poſſu- <pb o="118" file="148" n="148" rhead="GEOMETR. PRACT."/> mus per vnicam ſtationem quo que efficere, & </s> <s xml:id="echoid-s4600" xml:space="preserve">ſimul diſtantiam à perpendiculo <lb/>montis, vel à turre vſque ad ſignum in Horizonte propoſitum inuenire.</s> <s xml:id="echoid-s4601" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4602" xml:space="preserve"><emph style="sc">Sit</emph> enim montis alicuius, aut turris <lb/> <anchor type="note" xlink:label="note-148-01a" xlink:href="note-148-01"/> <anchor type="figure" xlink:label="fig-148-01a" xlink:href="fig-148-01"/> altitudo D E, & </s> <s xml:id="echoid-s4603" xml:space="preserve">ſignum in Horizonte vi-<lb/>ſum F. </s> <s xml:id="echoid-s4604" xml:space="preserve">Erigatur ex D, haſta aliqua D A, & </s> <s xml:id="echoid-s4605" xml:space="preserve"><lb/>ſumpta D A, æquali lateri quadrati ſtabi-<lb/>lis, accommodetur quadratum a b c d, in <lb/>A, ita vt centrum dio ptræ a, ſit ſuperius, & </s> <s xml:id="echoid-s4606" xml:space="preserve"><lb/>latus inferius recta ad ſignum F, vergat. <lb/></s> <s xml:id="echoid-s4607" xml:space="preserve">Poſito deinde oculo in a, dirigatur dio-<lb/>ptra verſus F, notentur que partes in vm-<lb/>bra verſa b e. </s> <s xml:id="echoid-s4608" xml:space="preserve">Nam ſi fiat,</s> </p> <div xml:id="echoid-div292" type="float" level="2" n="1"> <note position="left" xlink:label="note-148-01" xlink:href="note-148-01a" xml:space="preserve">Altitudinem <lb/>montis, vel <lb/>turris ex eius <lb/>vertice per v-<lb/>nicam ſtatio-<lb/>nem, vna cũ <lb/>d ſtãtia à tur-<lb/>re, vel perpen-<lb/>diculo mo@tis <lb/>ad ſignum in <lb/>Horizonte <lb/>propoſitum <lb/>metiri.</note> <figure xlink:label="fig-148-01" xlink:href="fig-148-01a"> <image file="148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/148-01"/> </figure> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt be, vmbra verſa # ad latus b a, 1000. # Ita latus a d, 1000. # ad A F, <lb/></note> <p> <s xml:id="echoid-s4609" xml:space="preserve">reperietur hypotenuſa A F, in partibus lateris a d. </s> <s xml:id="echoid-s4610" xml:space="preserve">Accommo detur rurſus qua-<lb/>dratum A B C D, vt centrum dioptræ A, ſit ſuperius, & </s> <s xml:id="echoid-s4611" xml:space="preserve">latus C D, Horizontiæ-<lb/>quidiſtet, hoc eſt, latus AD, haſtæ ere ctæ congruat. </s> <s xml:id="echoid-s4612" xml:space="preserve">Quo poſito, videbitur ſignũ <lb/>F, per inuentam hypotenuſam A F, quæ primum ſecet latus vmbræ verſæ B C, <lb/>in G; </s> <s xml:id="echoid-s4613" xml:space="preserve">in quiratur que portio dio ptræ A G, vt in ſchol. </s> <s xml:id="echoid-s4614" xml:space="preserve">problem. </s> <s xml:id="echoid-s4615" xml:space="preserve">7. </s> <s xml:id="echoid-s4616" xml:space="preserve">Num. </s> <s xml:id="echoid-s4617" xml:space="preserve">2. </s> <s xml:id="echoid-s4618" xml:space="preserve">docui-<lb/>mus. </s> <s xml:id="echoid-s4619" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Nam ſi fiat,</s> </p> <note symbol="a" position="left" xml:space="preserve">4. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ \\ A G, inuenta # adinuentam hy- \\ potenuſam A F, # Ita A L, vmbræ verſæ B G, æ- \\ qualis (ducta G L, paralle- \\ la ipſi E F,) # ad A E, <lb/></note> <p> <s xml:id="echoid-s4620" xml:space="preserve">comperta erit AE, in partibus hypotenuſæ AF. </s> <s xml:id="echoid-s4621" xml:space="preserve">Et ſi dematur latus quadrati A D, <lb/>in ijſdem partibus notum, reliqua fiet D E, altitudo montis, aut turris. </s> <s xml:id="echoid-s4622" xml:space="preserve">Quòd ſi <lb/>rurſus fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ \\ AG, inuenta # ad inuentam hy- \\ potenuſam AF, # Ita E M, (producta BC, vſ- \\ que ad M,) lateri CD, æqua- \\ lis, # ad EF, <lb/></note> <p> <s xml:id="echoid-s4623" xml:space="preserve">exibit nota diſtantia EF, in ijſdem partibus hypotenuſæ A F.</s> <s xml:id="echoid-s4624" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4625" xml:space="preserve"><emph style="sc">Vbi</emph> vides eadem opera inueniri diſtantiam ab oculo A, vſque ad ſignum F, <lb/>in Horizonte.</s> <s xml:id="echoid-s4626" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4627" xml:space="preserve"><emph style="sc">Non</emph> erit autem difficile ipſam DE, vel EF, ſi inuenta fuerit in partibus mil-<lb/>leſimis lateris AD, vel CD, in alia menſura, vt in pedibus, efficere notam; </s> <s xml:id="echoid-s4628" xml:space="preserve">ſi fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt AD, la- \\ tus 1000. # ad AD, notam in pedibus, ver- \\ bigratia in 3. vel in ſe miſſe \\ vnius cubiti: # Ita D E, vel E F, in- \\ uenta in milleſimis \\ partibus, # ad aliud <lb/></note> <p> <s xml:id="echoid-s4629" xml:space="preserve">Atque hac eadem ratione omnes aliæ lineæ inuentæ in milleſimis partibus late-<lb/>ris quadrati, reducentur ad aliam menſuram vel pedum, vel cubitorum, & </s> <s xml:id="echoid-s4630" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s4631" xml:space="preserve">Quod etiam in ſcholio problem. </s> <s xml:id="echoid-s4632" xml:space="preserve">7. </s> <s xml:id="echoid-s4633" xml:space="preserve">monuimus.</s> <s xml:id="echoid-s4634" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4635" xml:space="preserve">2. </s> <s xml:id="echoid-s4636" xml:space="preserve"><emph style="sc">Secet</emph> deinde hypotenuſa AI, (quæ reperietur vt AF, inuenta eſt) qua- <pb o="119" file="149" n="149" rhead="LIBER TERTIVS."/> dratum in C: </s> <s xml:id="echoid-s4637" xml:space="preserve">inueſtigetur que portio dioptræ A C, vt in ſchol. </s> <s xml:id="echoid-s4638" xml:space="preserve">problem. </s> <s xml:id="echoid-s4639" xml:space="preserve">7. </s> <s xml:id="echoid-s4640" xml:space="preserve">Num. <lb/></s> <s xml:id="echoid-s4641" xml:space="preserve">2. </s> <s xml:id="echoid-s4642" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s4643" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Deinde fiat,</s> </p> <note symbol="a" position="right" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt p@rtio dioptræ AC, \\ inuenta # ad inuentam hypotenu- \\ ſam A I, # Ita A D, latus \\ 1000. # ad AE, <lb/></note> <p> <s xml:id="echoid-s4644" xml:space="preserve">Nam numerus productus dabit A E, in partibus hypotenuſæ AI. </s> <s xml:id="echoid-s4645" xml:space="preserve">Atque totidem <lb/>partes complectetur diſtantia E I; </s> <s xml:id="echoid-s4646" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> quod A E, EI, æquales ſint, ob angulos ſe- <anchor type="note" xlink:label="note-149-03a" xlink:href="note-149-03"/> mirectos EAI, EIA, æquales.</s> <s xml:id="echoid-s4647" xml:space="preserve"/> </p> <div xml:id="echoid-div293" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-149-03" xlink:href="note-149-03a" xml:space="preserve">6. primi.</note> </div> <p> <s xml:id="echoid-s4648" xml:space="preserve">3. </s> <s xml:id="echoid-s4649" xml:space="preserve"><emph style="sc">Postremo</emph> hypotenuſa A K, inuenta eo modo, quo A F, cognita eſt, <lb/>ſecet latus CD, vmbræ rectæ in H, reperiatur que portio dioptræ AH, vt in ſchol. <lb/></s> <s xml:id="echoid-s4650" xml:space="preserve">problem. </s> <s xml:id="echoid-s4651" xml:space="preserve">7. </s> <s xml:id="echoid-s4652" xml:space="preserve">Num. </s> <s xml:id="echoid-s4653" xml:space="preserve">2. </s> <s xml:id="echoid-s4654" xml:space="preserve">do cuimus. </s> <s xml:id="echoid-s4655" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Nam ſi fiat.</s> <s xml:id="echoid-s4656" xml:space="preserve"/> </p> <note symbol="c" position="right" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ \\ A H, inuenta # ad inuentam hypotenu- \\ ſam A K: # Ita A D, lat{us} qua- \\ drati # ad A E, <lb/></note> <p> <s xml:id="echoid-s4657" xml:space="preserve">cognoſcetur A E, in partibus hypotenuſæ A K, & </s> <s xml:id="echoid-s4658" xml:space="preserve">c. </s> <s xml:id="echoid-s4659" xml:space="preserve">Et ſi rurſus fiat, ducta prius <lb/>H N, ipſi A E, parallela,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt portio dioptræ \\ A H, inuenta # Ad inuentam hy- \\ potenuſam A K, # Ita E N, vmbræ rectæ \\ D H, æqualis # ad E K, <lb/></note> <p> <s xml:id="echoid-s4660" xml:space="preserve">nota quo que red detur diſtantia E K, in ijſdem partibus hypotenuſæ A K, & </s> <s xml:id="echoid-s4661" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s4662" xml:space="preserve">Quod ſi ex vertice D, appareret radix montis, inueniretur eo dem modo diſtan-<lb/>tia à radice vſque ad E, quæ ablata ex diſtantia F E, inuenta notam quo que re-<lb/>lin quet diſtantiam ab F, vſque ad radicem montis, quæ nonnunquam poſſet <lb/>deſiderari.</s> <s xml:id="echoid-s4663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4664" xml:space="preserve">EX ſummitate turris, vel aliqua eius feneſtra, diſtantiam à baſe turris ad <lb/>ſignum propoſitum in Horizonte per quadratum cognoſcere.</s> <s xml:id="echoid-s4665" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div295" type="section" level="1" n="127"> <head xml:id="echoid-head130" xml:space="preserve">PROBLEMA X.</head> <p> <s xml:id="echoid-s4666" xml:space="preserve">1. </s> <s xml:id="echoid-s4667" xml:space="preserve"><emph style="sc">Sit</emph> turris aliqua A F, & </s> <s xml:id="echoid-s4668" xml:space="preserve">diſtantia metienda F G. </s> <s xml:id="echoid-s4669" xml:space="preserve">Si igitur altitudo turris <lb/>cognita eſt, aut eius portio inter feneſtram A, & </s> <s xml:id="echoid-s4670" xml:space="preserve">baſem F; </s> <s xml:id="echoid-s4671" xml:space="preserve">inſpiciatur ſignum G, <lb/>per pinnacidia quadrati penduli. </s> <s xml:id="echoid-s4672" xml:space="preserve">Et ſi quidem filum perpendiculi interſecet <lb/>latus vmbræ rectæ in E, vel per punctum C, tranſeat, Fiat autem</s> </p> <figure> <image file="149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/149-01"/> </figure> <note style="it" position="right" xml:space="preserve"> <lb/>Vt latus AB, par- \\ @ium 1000. # ad vmbram re- \\ ctam BE, # Ita A F, altitudo turris \\ nota # ad FG, <lb/></note> <pb o="120" file="150" n="150" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s4673" xml:space="preserve">cognita erit diſtantia F G, quæſita in partibus turris notæ.</s> <s xml:id="echoid-s4674" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4675" xml:space="preserve"><emph style="sc">Si</emph> autem vmbra verſa ſecetur in E, & </s> <s xml:id="echoid-s4676" xml:space="preserve">fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra verſa \\ D E, # ad latus D A, par- \\ tium 1000. # Ita A F, altitudo turris co- \\ gnita # ad F G, <lb/></note> <p> <s xml:id="echoid-s4677" xml:space="preserve">efficietur quo que nota eadem diſtantia F G, in partibus turris cognitæ.</s> <s xml:id="echoid-s4678" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4679" xml:space="preserve">2. </s> <s xml:id="echoid-s4680" xml:space="preserve"><emph style="sc">Per</emph> quadratum ſtabile eandem quo que diſtantiam adipiſceris. </s> <s xml:id="echoid-s4681" xml:space="preserve">Site-<lb/>nim rurſus turris quæpiam D E, & </s> <s xml:id="echoid-s4682" xml:space="preserve">diſtantia metienda E G, vel E H, vel E F. </s> <s xml:id="echoid-s4683" xml:space="preserve">Si <lb/>igitur dio ptra ſecet latus vmbræ rectæ in I. </s> <s xml:id="echoid-s4684" xml:space="preserve">Et fiat,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt latus quadrati \\ AD, 1000. # ad vmbram \\ rectam D I: # Ita altitudo A E, ex turre & la- \\ tere quadrati A D, conflata. # ad E G, <lb/></note> <figure> <image file="150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/150-01"/> </figure> <p> <s xml:id="echoid-s4685" xml:space="preserve">prodibit diſtantia quæſita E G.</s> <s xml:id="echoid-s4686" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4687" xml:space="preserve"><emph style="sc">Si</emph> autem dioptra per C. </s> <s xml:id="echoid-s4688" xml:space="preserve">tranſeat, erit <lb/>diſtantia E H, altitudini A E, notæ æqua-<lb/>lis.</s> <s xml:id="echoid-s4689" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4690" xml:space="preserve"><emph style="sc">Si</emph> denique vmbra verſa interſecetur <lb/>in K; </s> <s xml:id="echoid-s4691" xml:space="preserve">fiat autem,</s> </p> <note style="it" position="right" xml:space="preserve"> <lb/>Vt vmbra verſa \\ B K, # ad latus quadrati \\ 1000. # Ita altitudo nota A E, # ad E F, <lb/></note> <p> <s xml:id="echoid-s4692" xml:space="preserve">inuenietur diſtantia EF, in partibus altitudinis A E,</s> </p> <p> <s xml:id="echoid-s4693" xml:space="preserve"><emph style="sc">Hæc</emph> enim omnia in 2. </s> <s xml:id="echoid-s4694" xml:space="preserve">problem. </s> <s xml:id="echoid-s4695" xml:space="preserve">Num. </s> <s xml:id="echoid-s4696" xml:space="preserve">1. </s> <s xml:id="echoid-s4697" xml:space="preserve">& </s> <s xml:id="echoid-s4698" xml:space="preserve">2. </s> <s xml:id="echoid-s4699" xml:space="preserve">demonſtrata ſunt.</s> <s xml:id="echoid-s4700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4701" xml:space="preserve">3. </s> <s xml:id="echoid-s4702" xml:space="preserve"><emph style="sc">Qvod</emph> ſi altitudo turris cognita non ſit, inueſtiganda ea primum erit per <lb/>problema 8. </s> <s xml:id="echoid-s4703" xml:space="preserve">vel 9. </s> <s xml:id="echoid-s4704" xml:space="preserve">vel potius per ſcholium problem. </s> <s xml:id="echoid-s4705" xml:space="preserve">9. </s> <s xml:id="echoid-s4706" xml:space="preserve">aut certe, ſi commode <lb/>fieri poſsit, per chordam aliquam cum appenſo perpendiculo demiſſam explo-<lb/>randa. </s> <s xml:id="echoid-s4707" xml:space="preserve">Deinde procedendum erit, vt Num 1. </s> <s xml:id="echoid-s4708" xml:space="preserve">& </s> <s xml:id="echoid-s4709" xml:space="preserve">2. </s> <s xml:id="echoid-s4710" xml:space="preserve">dictum eſt.</s> <s xml:id="echoid-s4711" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4712" xml:space="preserve"><emph style="sc">Vervm</emph> ſine cognitione turris idem aſſequemur ea ratione, quain problem. <lb/></s> <s xml:id="echoid-s4713" xml:space="preserve">6. </s> <s xml:id="echoid-s4714" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s4715" xml:space="preserve">altitudinem propoſitam indagauimus, etiamſi diſtantia vſque ad alti-<lb/>tudinem ſit ignota. </s> <s xml:id="echoid-s4716" xml:space="preserve">Nam ſi in 6. </s> <s xml:id="echoid-s4717" xml:space="preserve">problemate turris ſit d f, diſtantia autem me-<lb/>tien da FG, effi ciemus illud, ſi in haſta aliqua erecta d A, fiant duæ ſtationes o-<lb/>culi, in a A. </s> <s xml:id="echoid-s4718" xml:space="preserve">Solum quod ibi dictum eſt de vmbra verſa, hic de recta intelligatur, <lb/>& </s> <s xml:id="echoid-s4719" xml:space="preserve">contra.</s> <s xml:id="echoid-s4720" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4721" xml:space="preserve"><emph style="sc">Pari</emph> ratione, ſi in 7. </s> <s xml:id="echoid-s4722" xml:space="preserve">problemate turris ſit d F, & </s> <s xml:id="echoid-s4723" xml:space="preserve">diſtantia metienda F G, <lb/>inueniemus eam, ſi in plano ſummitatis turris fiant duæ ſtationes oculi in A, a. <lb/></s> <s xml:id="echoid-s4724" xml:space="preserve">Solum quod ibi dicitur de vmbra verſa, hic etiam de recta intelligatur. </s> <s xml:id="echoid-s4725" xml:space="preserve">Nam ſi <lb/>figuræ problem. </s> <s xml:id="echoid-s4726" xml:space="preserve">6. </s> <s xml:id="echoid-s4727" xml:space="preserve">& </s> <s xml:id="echoid-s4728" xml:space="preserve">7. </s> <s xml:id="echoid-s4729" xml:space="preserve">hoc eſt, figuræ problem. </s> <s xml:id="echoid-s4730" xml:space="preserve">3. </s> <s xml:id="echoid-s4731" xml:space="preserve">& </s> <s xml:id="echoid-s4732" xml:space="preserve">4. </s> <s xml:id="echoid-s4733" xml:space="preserve">inuertantur, vt altitu-<lb/>dines fiant A F, A F, latus quadrati D C, ad vmbram rectam, & </s> <s xml:id="echoid-s4734" xml:space="preserve">B C, ad verſam <lb/>pertinebit, vt initio huius libri in conſtru ctione quadrati Num. </s> <s xml:id="echoid-s4735" xml:space="preserve">4. </s> <s xml:id="echoid-s4736" xml:space="preserve">explicauimus.</s> <s xml:id="echoid-s4737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4738" xml:space="preserve"><emph style="sc">Sed</emph> hanc diſtantiam longè facilius in ſcholio præcedentis problematis in-<lb/>uenimus per vnicam ſtationem, vt patuit in diſtantia EF, & </s> <s xml:id="echoid-s4739" xml:space="preserve">c.</s> <s xml:id="echoid-s4740" xml:space="preserve"/> </p> <pb o="121" file="151" n="151" rhead="LIBER TERTIVS."/> <p> <s xml:id="echoid-s4741" xml:space="preserve">EX ALTITVDINIS alicuius faſtigio, etiamſi altitudo ſit menſoris <lb/>ſtatura, diſtantiam inter duo ſigna in plano, cui altitudo inſiſtit, ſiea <lb/>diſtantia è directo menſoris iaceat, & </s> <s xml:id="echoid-s4742" xml:space="preserve">vtrumque eius extremum cerni <lb/>poſſit, per quadratum comprehendere.</s> <s xml:id="echoid-s4743" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div296" type="section" level="1" n="128"> <head xml:id="echoid-head131" xml:space="preserve">PROBLEMA XI.</head> <p> <s xml:id="echoid-s4744" xml:space="preserve">1. </s> <s xml:id="echoid-s4745" xml:space="preserve"><emph style="sc">Sit</emph> diſtantia metienda AB, è directo altitu-<lb/> <anchor type="figure" xlink:label="fig-151-01a" xlink:href="fig-151-01"/> dinis C D, in qua oculus menſoris exiſtat in D, fa-<lb/>ſtigio. </s> <s xml:id="echoid-s4746" xml:space="preserve">Per problema antecedens inueſtigetur ex <lb/>vertice D, tam diſtantia C B, quam C A. </s> <s xml:id="echoid-s4747" xml:space="preserve">Minore-<lb/>nim hæc ex illa maiore detracta notam relinquet <lb/>diſtantiam A B, inter ſigna A, & </s> <s xml:id="echoid-s4748" xml:space="preserve">B, in partibus al-<lb/>titudinis C D, in quibus videlicet diſtantiæ etiam <lb/>C B, C A, inuentæſunt.</s> <s xml:id="echoid-s4749" xml:space="preserve"/> </p> <div xml:id="echoid-div296" type="float" level="2" n="1"> <figure xlink:label="fig-151-01" xlink:href="fig-151-01a"> <image file="151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/151-01"/> </figure> </div> <p> <s xml:id="echoid-s4750" xml:space="preserve">2. </s> <s xml:id="echoid-s4751" xml:space="preserve"><emph style="sc">Si</emph> altitudo C D, ſit ſtatura menſoris, reperietur eodem modo diſtantia, <lb/>A B, ſi oculus menſoris in D, vtrum que extremum A, B, cernere poſsit, vt liquet.</s> <s xml:id="echoid-s4752" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4753" xml:space="preserve">LONGITVDINEM in Horizonte extenſam metiri per Quadratũ, <lb/>quando menſor in vno eius extremo exiſtens alterum extremum vi-<lb/>dere non poteſt, propter tumorem aliquem interiectum, neque alti-<lb/>tudo in promptu eſt, ſed ſolum ad dextram, vel ſiniſtram per lineam <lb/>perpendicularem recedere poteſt ad locum, è quo alterum extremũ <lb/>appareat.</s> <s xml:id="echoid-s4754" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div298" type="section" level="1" n="129"> <head xml:id="echoid-head132" xml:space="preserve">PROBLEMA XII.</head> <p> <s xml:id="echoid-s4755" xml:space="preserve">1. </s> <s xml:id="echoid-s4756" xml:space="preserve"><emph style="sc">Sit</emph> longitudo metienda A E, cuius extre-<lb/> <anchor type="figure" xlink:label="fig-151-02a" xlink:href="fig-151-02"/> mum E, ex A, menſorvidere non poſsit, neq; </s> <s xml:id="echoid-s4757" xml:space="preserve">ad-<lb/>ſit altitudo, ſed tamen ſi ad dextram, vel ſiniſtrã <lb/>recedat per lineam perpendicularem A a, vſque <lb/>ad a, illud videre poſsit. </s> <s xml:id="echoid-s4758" xml:space="preserve">Quadratum ſtabile ita <lb/>erigatur, vt eius planum longitudini A E, con-<lb/>gruat. </s> <s xml:id="echoid-s4759" xml:space="preserve">Debet namque conſtare, quænam recta <lb/>ad extrema A, E, pertineat, hoc eſt, rectam con-<lb/>ſtituat, cum data longitudine. </s> <s xml:id="echoid-s4760" xml:space="preserve">Deinde colloca-<lb/>to quadrato in Horizontis plano, ita vt latus <lb/>A B, a longitudine non recedat, extendaturrecta <lb/>per latus A D, vſq; </s> <s xml:id="echoid-s4761" xml:space="preserve">ad a, vnde extremum, E, ap-<lb/>pareat, ſitque ſpatium A a, per aliquam menſu-<lb/>ram notum. </s> <s xml:id="echoid-s4762" xml:space="preserve">Erectum autem in a, quadratum <lb/>circumducatur, donec per eius planum extre-<lb/>mum E, cernatur. </s> <s xml:id="echoid-s4763" xml:space="preserve">Poſt hæcidem quadratum in <lb/>Horizonte collocetur, latuſque a d, perpendiculari A a, congruat: </s> <s xml:id="echoid-s4764" xml:space="preserve">Et dioptra <pb o="122" file="152" n="152" rhead="GEOMETR. PRACT."/> circumuoluatur, donec eius linea fiduciæ rectæ a H, per quam extremum E, in-<lb/>ſpectum fuit, reſpondeat, notetur que vmbra verſa b F, abſciſſa. </s> <s xml:id="echoid-s4765" xml:space="preserve">Eruntque tri-<lb/>angula a b F, a A E, æquiangula, propter rectos angulos b, A, & </s> <s xml:id="echoid-s4766" xml:space="preserve">alternos b a F, <lb/> <anchor type="note" xlink:label="note-152-01a" xlink:href="note-152-01"/> A E a, æquales. </s> <s xml:id="echoid-s4767" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quamobrem ſi fiat, <anchor type="note" xlink:label="note-152-02a" xlink:href="note-152-02"/> cognita erit longitudo A E, in partibus ſpatij A a.</s> <s xml:id="echoid-s4768" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="2" n="1"> <figure xlink:label="fig-151-02" xlink:href="fig-151-02a"> <image file="151-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/151-02"/> </figure> <note symbol="a" position="left" xlink:label="note-152-01" xlink:href="note-152-01a" xml:space="preserve">4. ſexts.</note> <note style="it" position="right" xlink:label="note-152-02" xlink:href="note-152-02a" xml:space="preserve"> <lb/>Vt vmbra verſa b F, # ad quadrati lat{us} a b, \\ 1000. # ita ſpatium A a, \\ notum # ad A e, <lb/></note> </div> <p> <s xml:id="echoid-s4769" xml:space="preserve"><emph style="sc">Si</emph> forte dioptra latus d c, vmbræ rectæ interſecet, (quod raro continget, cũ <lb/> <anchor type="note" xlink:label="note-152-03a" xlink:href="note-152-03"/> plerunque AE, maior, ſit, quam A a,) <anchor type="note" xlink:href="" symbol="b"/> erit tunc.</s> <s xml:id="echoid-s4770" xml:space="preserve"> <anchor type="note" xlink:label="note-152-04a" xlink:href="note-152-04"/> vt perſpicuum eſt, ſi ducatur ex a, recta ſecans latus d c, &</s> <s xml:id="echoid-s4771" xml:space="preserve">c.</s> <s xml:id="echoid-s4772" xml:space="preserve"/> </p> <div xml:id="echoid-div299" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-152-03" xlink:href="note-152-03a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-152-04" xlink:href="note-152-04a" xml:space="preserve"> <lb/>Vt lat{us} a d, 1000. # ad vmbram rectam \\ abſciſſam: # Ita ſpatium A a, # ad longitu- \\ dinem. <lb/></note> </div> <note symbol="c" position="left" xml:space="preserve">6. primi.</note> <p> <s xml:id="echoid-s4773" xml:space="preserve"><emph style="sc">Si</emph> denique dioptra fortaſsis per c, tranſiret, <anchor type="note" xlink:href="" symbol="c"/> eſſet ſpatium A a, longitudini quæſitæ æquale; </s> <s xml:id="echoid-s4774" xml:space="preserve">propterea quod tunc fieret angulus ſemirectus d a c, ideoque <lb/>& </s> <s xml:id="echoid-s4775" xml:space="preserve">recta a c, ſi duceretur, faceret cum AE, angulum ſemirectum, atque adeo an-<lb/>gulo d a c, æqualem.</s> <s xml:id="echoid-s4776" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4777" xml:space="preserve">2. </s> <s xml:id="echoid-s4778" xml:space="preserve"><emph style="sc">Eadem</emph> diſtantia longitudone A E, cognoſcetur, ſi tamin G, quam in H, <lb/>baculus, ſeu arundo adangulos rectos figatur, ita vt ex A, a, radij per arundinem <lb/>incedentes ad E, ferantur, ſpatiumque A a, cognitum ſit: </s> <s xml:id="echoid-s4779" xml:space="preserve">vt in 2. </s> <s xml:id="echoid-s4780" xml:space="preserve">probl. </s> <s xml:id="echoid-s4781" xml:space="preserve">Num. <lb/></s> <s xml:id="echoid-s4782" xml:space="preserve">6. </s> <s xml:id="echoid-s4783" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s4784" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4785" xml:space="preserve">LONGITVDINEM in Horizonte è directo menſoris iacentem co-<lb/>gnoſcere, ad cuius extrema neque accedere liceat, neque è loco men-<lb/>ſoris eam dimetiri, neque vlla adſit altitudo, dummodo ad dextram <lb/>vel ſiniſtram per lineam perpendicularem ad locum aliquem ire poſ-<lb/>ſit menſor, ex quo vtrumque extremum appareat.</s> <s xml:id="echoid-s4786" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div301" type="section" level="1" n="130"> <head xml:id="echoid-head133" xml:space="preserve">PROBLEMA XIII.</head> <p> <s xml:id="echoid-s4787" xml:space="preserve">1. </s> <s xml:id="echoid-s4788" xml:space="preserve"><emph style="sc">Longitvdo</emph> metienda ſit E D, è directo <lb/> <anchor type="figure" xlink:label="fig-152-01a" xlink:href="fig-152-01"/> menſoris in F, exiſtentis, ita vt neque ad eam acce-<lb/>dere liceat, neque eam è loco F, metiri, neque vlla <lb/>adſit altitudo: </s> <s xml:id="echoid-s4789" xml:space="preserve">Sed ſolum per lineam perpendicu-<lb/>larem F G, ad locum G, vnde vtrumque extremum <lb/>D, E, videatur, poſsit accedere. </s> <s xml:id="echoid-s4790" xml:space="preserve">Per problema præ-<lb/>cedens in quiratur ex G, tam longitudo F E, quam <lb/>F D. </s> <s xml:id="echoid-s4791" xml:space="preserve">Hæc enim ex illa detracta notam relinquet propoſitam longitudinem D E,</s> </p> <div xml:id="echoid-div301" type="float" level="2" n="1"> <figure xlink:label="fig-152-01" xlink:href="fig-152-01a"> <image file="152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/152-01"/> </figure> </div> <p> <s xml:id="echoid-s4792" xml:space="preserve">ALTITVDINEM montis, vel turris ex eius faſtigio, quando è di-<lb/>recto menſoris interuallum aliquod inter duo ſigna, vel etiam inter <lb/>fignum quodpiam ac turrim cognitum eſt, per quadratum coniicere.</s> <s xml:id="echoid-s4793" xml:space="preserve"/> </p> <pb o="123" file="153" n="153" rhead="LIBER TERTIVS."/> </div> <div xml:id="echoid-div303" type="section" level="1" n="131"> <head xml:id="echoid-head134" xml:space="preserve">PROBLEMA XIV.</head> <p> <s xml:id="echoid-s4794" xml:space="preserve">1. </s> <s xml:id="echoid-s4795" xml:space="preserve"><emph style="sc">Sit</emph> mons, aut turris D E, ſitque pri-<lb/> <anchor type="figure" xlink:label="fig-153-01a" xlink:href="fig-153-01"/> mum è directo menſoris in faſtigio D, exi-<lb/>ſtentis interuallum F G, notum. </s> <s xml:id="echoid-s4796" xml:space="preserve">Accom-<lb/>modetur quadratum ſtabile in ſummitate <lb/>D, ita vt latus A D, perpendiculare ſit ad <lb/>Horizontem, & </s> <s xml:id="echoid-s4797" xml:space="preserve">C D, Horizonti paralle-<lb/>lum. </s> <s xml:id="echoid-s4798" xml:space="preserve">Inſpecto igitur per dioptram vtroq; <lb/></s> <s xml:id="echoid-s4799" xml:space="preserve"> <anchor type="note" xlink:label="note-153-01a" xlink:href="note-153-01"/> termino F, G, ſecetur vmbra recta in H, I. </s> <s xml:id="echoid-s4800" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam igitur eſt, vt I H, ad H D, ita G F, <lb/> <anchor type="note" xlink:label="note-153-02a" xlink:href="note-153-02"/> ad F E: </s> <s xml:id="echoid-s4801" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Item ob triangulorum ſimilitu- dinem, vt H D, ad D A, ita FE, ad EA; </s> <s xml:id="echoid-s4802" xml:space="preserve">erit ex <lb/>æquo, vt I H, ad D A, ita G F, ad E A. </s> <s xml:id="echoid-s4803" xml:space="preserve">Igitur ſi fiat, <lb/> <anchor type="note" xlink:label="note-153-03a" xlink:href="note-153-03"/> exibit nota recta E A. </s> <s xml:id="echoid-s4804" xml:space="preserve">Et ſi dematur Iatus quadrati D A, (quod fieri debet notum <lb/>in partibus interualli G F,) reliqua fiet nota altitudo D E, in partibus interualli <lb/>G F.</s> <s xml:id="echoid-s4805" xml:space="preserve"/> </p> <div xml:id="echoid-div303" type="float" level="2" n="1"> <figure xlink:label="fig-153-01" xlink:href="fig-153-01a"> <image file="153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/153-01"/> </figure> <note symbol="a" position="right" xlink:label="note-153-01" xlink:href="note-153-01a" xml:space="preserve">ſchol. 4. ſe-<lb/>xti.</note> <note symbol="b" position="right" xlink:label="note-153-02" xlink:href="note-153-02a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-153-03" xlink:href="note-153-03a" xml:space="preserve"> <lb/>Vt I H, d@fferentia vm- \\ brarum rectarum # ad D A, lat{us} qua- \\ drati # Ita interuallum G F, \\ cognitum # ad E A <lb/></note> </div> <p> <s xml:id="echoid-s4806" xml:space="preserve">2. </s> <s xml:id="echoid-s4807" xml:space="preserve"><emph style="sc">Sit</emph> tota deinde diſtantia E G, nota. </s> <s xml:id="echoid-s4808" xml:space="preserve">Inſpiciendum ergo ſolum eſt extre-<lb/>mum G. </s> <s xml:id="echoid-s4809" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Nam ſi fiat, <anchor type="note" xlink:label="note-153-04a" xlink:href="note-153-04"/> <anchor type="note" xlink:label="note-153-05a" xlink:href="note-153-05"/> efficietur rurſus nota recta E A, &</s> <s xml:id="echoid-s4810" xml:space="preserve">c.</s> <s xml:id="echoid-s4811" xml:space="preserve"/> </p> <div xml:id="echoid-div304" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-153-04" xlink:href="note-153-04a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-153-05" xlink:href="note-153-05a" xml:space="preserve"> <lb/>Vt I D, vmbrarecta # ad D A, lat{us} quadrati # Ita diſtantia no- \\ ta G E, # ad E A, <lb/></note> </div> <p> <s xml:id="echoid-s4812" xml:space="preserve">3. </s> <s xml:id="echoid-s4813" xml:space="preserve"><emph style="sc">Qvando</emph> vmbra verſa B C, interſecatur, vt ſi ſpatium notum ſit L N, re-<lb/>ducenda eſt vtra que vmbra verſa ad rectas D K, D M. </s> <s xml:id="echoid-s4814" xml:space="preserve">Nam rurſus erit, vt K M, <lb/>differentia vmbrarum rectarum ad latus D A, ita ſpatium notum L N, ad E A.</s> <s xml:id="echoid-s4815" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4816" xml:space="preserve"><emph style="sc">Sic</emph> etiam quando vnus radius vmbram rectam, & </s> <s xml:id="echoid-s4817" xml:space="preserve">alter verſam interſecat, <lb/>vt in ſpatio L G, contingit, reuo canda erit vmbra verſa ad rectam DK, vtiterum <lb/>ſit K I, differentia vmbrarum rectarum ad D A, latus, vt ſpatium notum L G, ad <lb/>E A.</s> <s xml:id="echoid-s4818" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4819" xml:space="preserve">4. </s> <s xml:id="echoid-s4820" xml:space="preserve"><emph style="sc">Si</emph> denique radius per C, tranſiret, ſumendum eſſet totumlatus CD, pro <lb/>vmbra recta, at vmbra verſa, ſi qua eſſet, ad rectam reducenda.</s> <s xml:id="echoid-s4821" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4822" xml:space="preserve">DISTANTIAM ab oculo, vel pede menſoris (vbicunque exiſtat) <lb/>ad quoduis punctum in aliqua altitudine notatum per quadratum ex-<lb/>quirere.</s> <s xml:id="echoid-s4823" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div306" type="section" level="1" n="132"> <head xml:id="echoid-head135" xml:space="preserve">PROBLEMA XV.</head> <p> <s xml:id="echoid-s4824" xml:space="preserve">1. </s> <s xml:id="echoid-s4825" xml:space="preserve"><emph style="sc">Exploranda</emph> ſit diſtantia puncti F, in muro aliquo ſiue perpendicu-<lb/>lariad Horizontem, ſiue in clinato, vel etiam in tecto quo piam ab oculo A, vel à <lb/>pede E, poſita ſtatura menſoris A E. </s> <s xml:id="echoid-s4826" xml:space="preserve">Et ſit primum altius punctum F, quam ocu-<lb/>lus A. </s> <s xml:id="echoid-s4827" xml:space="preserve">Concipiatur ex F, demiſſa perpendicularis F G, & </s> <s xml:id="echoid-s4828" xml:space="preserve">ad hanc ducta ab ocu-<lb/>lo A, alia perpendicularis A H. </s> <s xml:id="echoid-s4829" xml:space="preserve">Collocato ergo ita quadrato, vt latus A D, Ho-<lb/>rizonti æquidiſtet, inſpiciatur punctum F, radiuſque A F, vel dioptra auferat <pb o="124" file="154" n="154" rhead="GEOMETR. PRACT."/> vmbram verſam D I. </s> <s xml:id="echoid-s4830" xml:space="preserve">Per problema 3. </s> <s xml:id="echoid-s4831" xml:space="preserve">vel 4. </s> <s xml:id="echoid-s4832" xml:space="preserve">vel potius per ſcholium problem. <lb/></s> <s xml:id="echoid-s4833" xml:space="preserve">7. </s> <s xml:id="echoid-s4834" xml:space="preserve">inueſtigetur diſtantia A H, etiamſi pun-<lb/> <anchor type="figure" xlink:label="fig-154-01a" xlink:href="fig-154-01"/> ctum H, non appareat: </s> <s xml:id="echoid-s4835" xml:space="preserve">diligenter que in-<lb/>quiratur, quot partes milleſimæ lateris A <lb/>D, in ſegmento dioptræ A I, comprehen-<lb/>dantur. </s> <s xml:id="echoid-s4836" xml:space="preserve">quod multis modis, vt in ſchol. <lb/></s> <s xml:id="echoid-s4837" xml:space="preserve">probl. </s> <s xml:id="echoid-s4838" xml:space="preserve">7. </s> <s xml:id="echoid-s4839" xml:space="preserve">Num. </s> <s xml:id="echoid-s4840" xml:space="preserve">2. </s> <s xml:id="echoid-s4841" xml:space="preserve">docuimus, exequemur <lb/>hoc modo. </s> <s xml:id="echoid-s4842" xml:space="preserve">Primum quoniam duo latera <lb/>A D, D I, in rectangulo triangulo ADI, da-<lb/>ta ſunt, <anchor type="note" xlink:href="" symbol="a"/> ignorarinon poterit baſis in par- <anchor type="note" xlink:label="note-154-01a" xlink:href="note-154-01"/> tibus laterum. </s> <s xml:id="echoid-s4843" xml:space="preserve">Deinde quia per probl. </s> <s xml:id="echoid-s4844" xml:space="preserve">1. </s> <s xml:id="echoid-s4845" xml:space="preserve">ex vmbra D I, notus fit angulus D A I, <lb/> <anchor type="note" xlink:href="" symbol="b"/> cognoſcetur rurſus baſis A I. </s> <s xml:id="echoid-s4846" xml:space="preserve">Tertio <anchor type="note" xlink:href="" symbol="c"/> quia quadrata AD, DI, quadrato AI, æ- <anchor type="note" xlink:label="note-154-02a" xlink:href="note-154-02"/> qualia ſunt; </s> <s xml:id="echoid-s4847" xml:space="preserve">ſi ex aggregato eorũ radix quadrata eruatur, exhibebit earadix ba-<lb/>ſem A I, notam. </s> <s xml:id="echoid-s4848" xml:space="preserve">His peractis, <anchor type="note" xlink:href="" symbol="d"/> ſi fiat, <anchor type="note" xlink:label="note-154-03a" xlink:href="note-154-03"/> <anchor type="note" xlink:label="note-154-04a" xlink:href="note-154-04"/> <anchor type="note" xlink:label="note-154-05a" xlink:href="note-154-05"/> cognita erit diſtantia A F, quæſita in partibus inuentæ diſtantiæ A H.</s> <s xml:id="echoid-s4849" xml:space="preserve"/> </p> <div xml:id="echoid-div306" type="float" level="2" n="1"> <figure xlink:label="fig-154-01" xlink:href="fig-154-01a"> <image file="154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/154-01"/> </figure> <note symbol="a" position="left" xlink:label="note-154-01" xlink:href="note-154-01a" xml:space="preserve">6. triang. re-<lb/>ctil.</note> <note symbol="b" position="left" xlink:label="note-154-02" xlink:href="note-154-02a" xml:space="preserve">5. triang. re-<lb/>ctil.</note> <note symbol="c" position="left" xlink:label="note-154-03" xlink:href="note-154-03a" xml:space="preserve">47. primi.</note> <note symbol="d" position="left" xlink:label="note-154-04" xlink:href="note-154-04a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-154-05" xlink:href="note-154-05a" xml:space="preserve"> <lb/>Vt lat{us} \\ A D, 1000 # ad portionem dioptræ \\ A I, nuper inuentam: # ita diſtantia A H, \\ nuper etiam inuenta # ad A F, <lb/></note> </div> <p> <s xml:id="echoid-s4850" xml:space="preserve"><emph style="sc">Distantia</emph> autem E F, à pede ad datũ punctum F, ita reperiemus. </s> <s xml:id="echoid-s4851" xml:space="preserve">Quo-<lb/>niam in triangulo AEF, duo latera AF, AE, cognita ſunt, cum illud proxime ſit <lb/>inuentum, & </s> <s xml:id="echoid-s4852" xml:space="preserve">hoc ſtaturæ menſoris æquale ſit; </s> <s xml:id="echoid-s4853" xml:space="preserve">comprehenduntq; </s> <s xml:id="echoid-s4854" xml:space="preserve">angulum no-<lb/>tum EAF, vt pote conflatum ex recto EAH, & </s> <s xml:id="echoid-s4855" xml:space="preserve">DAI, qui per problema 1. </s> <s xml:id="echoid-s4856" xml:space="preserve">inuen-<lb/>tus eſt ex vmbra verſa D I: </s> <s xml:id="echoid-s4857" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> notum efficietur latus quo que E F, quod quæri- <anchor type="note" xlink:label="note-154-06a" xlink:href="note-154-06"/> tur.</s> <s xml:id="echoid-s4858" xml:space="preserve"/> </p> <div xml:id="echoid-div307" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-154-06" xlink:href="note-154-06a" xml:space="preserve">12. trian. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s4859" xml:space="preserve">2. </s> <s xml:id="echoid-s4860" xml:space="preserve"><emph style="sc">Qvod</emph> ſi vmbra recta ſecetur in L, vt in altero quadrato, vbiiterum mẽ-<lb/>ſoris ſtatura eſt A K: </s> <s xml:id="echoid-s4861" xml:space="preserve">Inuenta portione dio ptræ A L, in partibus milleſimis late-<lb/>ris quadrati ex angulo BAL, &</s> <s xml:id="echoid-s4862" xml:space="preserve">c. </s> <s xml:id="echoid-s4863" xml:space="preserve">necnon diſtantia AH, ex problem. </s> <s xml:id="echoid-s4864" xml:space="preserve">3. </s> <s xml:id="echoid-s4865" xml:space="preserve">vel 4. </s> <s xml:id="echoid-s4866" xml:space="preserve">vel <lb/>ex ſcholio probl. </s> <s xml:id="echoid-s4867" xml:space="preserve">7. </s> <s xml:id="echoid-s4868" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Sifiat, <anchor type="note" xlink:label="note-154-07a" xlink:href="note-154-07"/> <anchor type="note" xlink:label="note-154-08a" xlink:href="note-154-08"/> pro dibit rurſus nota diſtantia A F, in partibus diſtantiæ inuentæ A H.</s> <s xml:id="echoid-s4869" xml:space="preserve"/> </p> <div xml:id="echoid-div308" type="float" level="2" n="3"> <note symbol="f" position="left" xlink:label="note-154-07" xlink:href="note-154-07a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-154-08" xlink:href="note-154-08a" xml:space="preserve"> <lb/>Vt B L, vm- \\ brarecta # ad L A, portionem \\ dioptræ inuentam: # Ita diſtantia A H, \\ inuenta # ad A F, <lb/></note> </div> <p> <s xml:id="echoid-s4870" xml:space="preserve"><emph style="sc">Non</emph> aliter procedes ſi dioptra per C, tranſeat: </s> <s xml:id="echoid-s4871" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Cum tunc etiam ſit, vt <anchor type="note" xlink:label="note-154-09a" xlink:href="note-154-09"/> A D, latus ad partem dioptræ A C, inuentam, vt prius, ita diſtantia inuenta A H, <lb/>ad diſtantiam quæſitam A F.</s> <s xml:id="echoid-s4872" xml:space="preserve"/> </p> <div xml:id="echoid-div309" type="float" level="2" n="4"> <note symbol="g" position="left" xlink:label="note-154-09" xlink:href="note-154-09a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s4873" xml:space="preserve"><emph style="sc">Distantia</emph> autem K F, à pede K, vſque ad F, <anchor type="note" xlink:href="" symbol="h"/> inuenietur, vt prius, ex <anchor type="note" xlink:label="note-154-10a" xlink:href="note-154-10"/> duobus lateribus notis A F, A K, & </s> <s xml:id="echoid-s4874" xml:space="preserve">angulo ab ipſis comprehenſo F A K; </s> <s xml:id="echoid-s4875" xml:space="preserve">qui ni-<lb/>mirum conflatur exrecto A, & </s> <s xml:id="echoid-s4876" xml:space="preserve">D A L, complemento anguli B A L, quem per <lb/>problema 1. </s> <s xml:id="echoid-s4877" xml:space="preserve">cognitum efficit vmbra recta B L.</s> <s xml:id="echoid-s4878" xml:space="preserve"/> </p> <div xml:id="echoid-div310" type="float" level="2" n="5"> <note symbol="h" position="left" xlink:label="note-154-10" xlink:href="note-154-10a" xml:space="preserve">12. trian. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s4879" xml:space="preserve">3. </s> <s xml:id="echoid-s4880" xml:space="preserve"><emph style="sc">Sed</emph> ſit iam punctum F, oculo A, depreſsius, & </s> <s xml:id="echoid-s4881" xml:space="preserve">ſtatura menſoris ſit A E. <lb/></s> <s xml:id="echoid-s4882" xml:space="preserve">Concipiatur ex F, duci F K, Horizonti parallela, vel ad A E, perpendicularis. </s> <s xml:id="echoid-s4883" xml:space="preserve"><lb/>Item per F, recta G H, ipſi AE, parallela. </s> <s xml:id="echoid-s4884" xml:space="preserve">Accommodato autem quadrato, vt la-<lb/>tus AD, rectum ſit ad Horizontem, & </s> <s xml:id="echoid-s4885" xml:space="preserve">D C, Horizonti æquidiſtans, cogitetur <lb/>DC, latus productum vſque ad H, punctum perpendicularis G H. </s> <s xml:id="echoid-s4886" xml:space="preserve">Primum ita-<lb/>que reperiatur altitudo A K, per problema 8. </s> <s xml:id="echoid-s4887" xml:space="preserve">vel 9. </s> <s xml:id="echoid-s4888" xml:space="preserve">duabus ſtationibus factis in <lb/>recta D H, vel in haſta D A, protracta, vel certe perſcholium problem. </s> <s xml:id="echoid-s4889" xml:space="preserve">9. </s> <s xml:id="echoid-s4890" xml:space="preserve"><lb/>Quamuis enim ſtatura menſoris AE, cognita ſit, ignoratur tamẽ, quanta ſit eius <pb o="125" file="155" n="155" rhead="LIBER TERTIVS."/> pars AK, quam parallela FK, per imaginationem ducta abſcindit: </s> <s xml:id="echoid-s4891" xml:space="preserve">ita vt omni-<lb/>no neceſſarium ſit altitudinem A K, inquirere. </s> <s xml:id="echoid-s4892" xml:space="preserve">quod per 8. </s> <s xml:id="echoid-s4893" xml:space="preserve">problema facilè <lb/>exequemur, ſi in vtra que ſtatione vmbra verſa ſecetur in I, N, (quod plerunque <lb/>hic continget (& </s> <s xml:id="echoid-s4894" xml:space="preserve">vtraque abſciſſa B I, b N, ad rectas D M, d O, reuocetur: <lb/></s> <s xml:id="echoid-s4895" xml:space="preserve">Nam ſi fiat, <lb/> <anchor type="figure" xlink:label="fig-155-01a" xlink:href="fig-155-01"/> <anchor type="note" xlink:label="note-155-01a" xlink:href="note-155-01"/> inuenta erit altitudo A K, oculo poſito in eius ſummitate A; </s> <s xml:id="echoid-s4896" xml:space="preserve">vt in dicto pro-<lb/>blem. </s> <s xml:id="echoid-s4897" xml:space="preserve">8. </s> <s xml:id="echoid-s4898" xml:space="preserve">Num. </s> <s xml:id="echoid-s4899" xml:space="preserve">2. </s> <s xml:id="echoid-s4900" xml:space="preserve">diximus, &</s> <s xml:id="echoid-s4901" xml:space="preserve">c. </s> <s xml:id="echoid-s4902" xml:space="preserve">Quætamen altitudo A K, facilius per ſcholium <lb/>problem. </s> <s xml:id="echoid-s4903" xml:space="preserve">9. </s> <s xml:id="echoid-s4904" xml:space="preserve">reperiri poteſt.</s> <s xml:id="echoid-s4905" xml:space="preserve"/> </p> <div xml:id="echoid-div311" type="float" level="2" n="6"> <figure xlink:label="fig-155-01" xlink:href="fig-155-01a"> <image file="155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/155-01"/> </figure> <note style="it" position="right" xlink:label="note-155-01" xlink:href="note-155-01a" xml:space="preserve"> <lb/>Vt L M, differentia vm- \\ brarum rectarum # ad D d, differentiam \\ ſtationum: # ita A D, \\ lat{us} # ad A K, <lb/></note> </div> <p> <s xml:id="echoid-s4906" xml:space="preserve"><emph style="sc">Itaqve</emph> quia vmbra B I, per 1. </s> <s xml:id="echoid-s4907" xml:space="preserve">problema patefacit angulum B A I, hoc eſt, <lb/>alternum A F K, ſibi æqualem, nec non & </s> <s xml:id="echoid-s4908" xml:space="preserve">eius complementum F A K; </s> <s xml:id="echoid-s4909" xml:space="preserve">erunt in <lb/>triangulo rectangulo AKF, duo anguliacuti cogniti, vna cumlatere AK, proxi-<lb/> <anchor type="note" xlink:label="note-155-02a" xlink:href="note-155-02"/> mè inuento; </s> <s xml:id="echoid-s4910" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Ita que ſi fiat, <anchor type="note" xlink:label="note-155-03a" xlink:href="note-155-03"/> cognita fiet A F, in partibus lateris inuenti A K. </s> <s xml:id="echoid-s4911" xml:space="preserve">Vel inuenta parte dioptræ A I, <lb/>in partibus milleſimis lateris quadrati, vt ſupra dictum eſt prope initium huius <lb/> <anchor type="note" xlink:label="note-155-04a" xlink:href="note-155-04"/> problematis, <anchor type="note" xlink:href="" symbol="b"/> ſi fiat, <anchor type="note" xlink:label="note-155-05a" xlink:href="note-155-05"/> nota rurſus efficietur diſtantia A F, in partibus rectæ AK, inuentæ.</s> <s xml:id="echoid-s4912" xml:space="preserve"/> </p> <div xml:id="echoid-div312" type="float" level="2" n="7"> <note symbol="a" position="right" xlink:label="note-155-02" xlink:href="note-155-02a" xml:space="preserve">5. Triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-155-03" xlink:href="note-155-03a" xml:space="preserve"> <lb/>Vt ſin{us} to- \\ t{us} # ad lat{us} A K, in- \\ uentum: # ita A F, ſecans anguli \\ FAK, # ad A F, <lb/></note> <note symbol="b" position="right" xlink:label="note-155-04" xlink:href="note-155-04a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-155-05" xlink:href="note-155-05a" xml:space="preserve"> <lb/>Vt B I, vmbra \\ verſa # ad I A, partem dioptræ \\ inuentæ: # ita K A, altitudo \\ inuenta # ad A F, <lb/></note> </div> <p> <s xml:id="echoid-s4913" xml:space="preserve"><emph style="sc">Porro</emph> diſtantiam E F, à pede menſoris ad pun ctum F, <anchor type="note" xlink:href="" symbol="c"/> inueniemus, vt ſupra: </s> <s xml:id="echoid-s4914" xml:space="preserve">propterea quo din triangulo AEF, duo latera AE, AF, nota ſunt, cum il-<lb/> <anchor type="note" xlink:label="note-155-06a" xlink:href="note-155-06"/> lud ſit ſtatura menſoris, hoc autem ſit proximè inuentum, angulumque conti-<lb/>nent notum FAK, vt paulò ante diximus</s> </p> <div xml:id="echoid-div313" type="float" level="2" n="8"> <note symbol="c" position="right" xlink:label="note-155-06" xlink:href="note-155-06a" xml:space="preserve">12. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s4915" xml:space="preserve">4. </s> <s xml:id="echoid-s4916" xml:space="preserve"><emph style="sc">Non</emph> aliter vtra que diſtantia cognoſcetur, ſi punctum F, in Horizonte <lb/>ſit poſitum, qui Horizon per FK, intelligatur tranſire, ita vt ſtatura menſoris, vel <lb/>aliqua alia altitudo nota, ſit AK. </s> <s xml:id="echoid-s4917" xml:space="preserve">Nam cognita portione dioptrę A I, vt ſupra <lb/> <anchor type="note" xlink:label="note-155-07a" xlink:href="note-155-07"/> traditum eſt; </s> <s xml:id="echoid-s4918" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Si fiat, <anchor type="note" xlink:label="note-155-08a" xlink:href="note-155-08"/> <pb o="126" file="156" n="156" rhead="GEOMETR. PRACT."/> nota euadet diſtantia A F, in partibus ſtaturæ menſoris, vel altitudinis no-<lb/>tæ A K.</s> <s xml:id="echoid-s4919" xml:space="preserve"/> </p> <div xml:id="echoid-div314" type="float" level="2" n="9"> <note symbol="d" position="right" xlink:label="note-155-07" xlink:href="note-155-07a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-155-08" xlink:href="note-155-08a" xml:space="preserve"> <lb/>Vt B I, vmbra \\ abſciſſa # ad I A, portionem dio- \\ ptra inuentans: # ita A K, altitudo, \\ vel ſtatura men- \\ſoris # ad A F, <lb/></note> </div> <p> <s xml:id="echoid-s4920" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/>Rurſus ſi fiat.</s> <s xml:id="echoid-s4921" xml:space="preserve"> <anchor type="note" xlink:label="note-156-01a" xlink:href="note-156-01"/> <anchor type="note" xlink:label="note-156-02a" xlink:href="note-156-02"/> cognita etiam fiet diſtantia K F, à pede menſoris vſque ad punctum F, in par-<lb/>tibus eiuſdem ſtaturæ menſoris, vel altitudinis notæ A K. </s> <s xml:id="echoid-s4922" xml:space="preserve">Vbi vides, vtramque <lb/>diſtantiam cognoſci per vnicam ſtationem, quando punctum datum eſt in Ho-<lb/>rizonte.</s> <s xml:id="echoid-s4923" xml:space="preserve"/> </p> <div xml:id="echoid-div315" type="float" level="2" n="10"> <note symbol="a" position="left" xlink:label="note-156-01" xlink:href="note-156-01a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-156-02" xlink:href="note-156-02a" xml:space="preserve"> <lb/>Vt B I, vmbra \\ abſciſſa # ad A B, lat{us} qua- \\ drati 1000. # ita A K, altitudo, vel ſta- \\ tura menſoris # ad K F, <lb/></note> </div> <p> <s xml:id="echoid-s4924" xml:space="preserve">5. </s> <s xml:id="echoid-s4925" xml:space="preserve"><emph style="sc">Sed</emph> vbicunque punctum F, exiſtat ſiue in Horizonte, ſiue in ſublimi, vel <lb/>infra oculum, inueniemus nihil ominus vtramque diſtantiam per vnicam ſtatio-<lb/>nem, hoc modo. </s> <s xml:id="echoid-s4926" xml:space="preserve">Quando punctum F, eſt in ſublimi, vt in prima figura, accom-<lb/>detur quadratum ita, vt dioptræ centrum a, ſit ſuperius, & </s> <s xml:id="echoid-s4927" xml:space="preserve">latus d c, verſus <lb/>punctum F, dirigatur. </s> <s xml:id="echoid-s4928" xml:space="preserve">Inſpecto enim per dioptram puncto F, notetur vmbra <lb/>verſa eb, <anchor type="note" xlink:href="" symbol="b"/> Et fiat, <anchor type="note" xlink:label="note-156-03a" xlink:href="note-156-03"/> <anchor type="note" xlink:label="note-156-04a" xlink:href="note-156-04"/> Nam numerus productus notam faciet diſtantiam A F, in partibus lateris qua-<lb/>drati. </s> <s xml:id="echoid-s4929" xml:space="preserve">Non aliter diſtantiam K F, à pede menſoris elicies, ſi eodem pacto qua-<lb/>dratum ad punctum K, applicabis, vt conſtat.</s> <s xml:id="echoid-s4930" xml:space="preserve"/> </p> <div xml:id="echoid-div316" type="float" level="2" n="11"> <note symbol="b" position="left" xlink:label="note-156-03" xlink:href="note-156-03a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-156-04" xlink:href="note-156-04a" xml:space="preserve"> <lb/>Vt vmbra ver- \\ ſa e b, # ad lat{us} \\ b a; # ita lat{us} \\ a d, # ad A F, <lb/></note> </div> <p> <s xml:id="echoid-s4931" xml:space="preserve"><emph style="sc">Qvando</emph> autem punctum F, depreſsius eſt oculo, vt in 2. </s> <s xml:id="echoid-s4932" xml:space="preserve">figura, erigen-<lb/>dus eſt baculus a d, minor, quam ſtatura menſoris, & </s> <s xml:id="echoid-s4933" xml:space="preserve">in a, applicandum qua-<lb/>dratum, vt rurſus centrum dioptræ A, ſit ſuperius, & </s> <s xml:id="echoid-s4934" xml:space="preserve">latus a c, ad punctum F, <lb/>vergat. </s> <s xml:id="echoid-s4935" xml:space="preserve">Nam viſo puncto F, ex A, per dioptram, notataque vmbra verſa e B, <lb/>ſi fiat, <lb/> <anchor type="note" xlink:label="note-156-05a" xlink:href="note-156-05"/> exibit diſtantia a F, nota in partibus lateris A a. </s> <s xml:id="echoid-s4936" xml:space="preserve">Diſtantia porrò E F, à pede <lb/>menſoris coniicietur, ſi quadratum in E, applicetur, vt in prima figura dictum eſt. <lb/></s> <s xml:id="echoid-s4937" xml:space="preserve">Quando denique punctum F, eſt in Horizonte, inuenietur vtraque diſtantia, vt <lb/>Num. </s> <s xml:id="echoid-s4938" xml:space="preserve">4. </s> <s xml:id="echoid-s4939" xml:space="preserve">tradidimus.</s> <s xml:id="echoid-s4940" xml:space="preserve"/> </p> <div xml:id="echoid-div317" type="float" level="2" n="12"> <note style="it" position="right" xlink:label="note-156-05" xlink:href="note-156-05a" xml:space="preserve"> <lb/>Vt vmbra verſa \\ e B, # ad lat{us} \\ B A: # Ita lat{us} \\ A a, # ad a F, <lb/></note> </div> <p> <s xml:id="echoid-s4941" xml:space="preserve">INTERVALLVM inter duo ſigna, vel puncta in quolibet planoſiue <lb/>recto ad Horizontem, ſiue inclinato, per quadratum metiri.</s> <s xml:id="echoid-s4942" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div319" type="section" level="1" n="133"> <head xml:id="echoid-head136" xml:space="preserve">PROBLEMA XVI.</head> <p> <s xml:id="echoid-s4943" xml:space="preserve">1. </s> <s xml:id="echoid-s4944" xml:space="preserve"><emph style="sc">In</emph> quolibet plano eleuato AB, inquirendum ſit interuallum CD, ex pla-<lb/>no EF. </s> <s xml:id="echoid-s4945" xml:space="preserve">Poſito oculo in G, vt ſtatura menſoris ſit G E, inueſtigetur per præce-<lb/>dens problema, vtra que diſtantia G C, G D, in partibus ſtaturæ menſoris G E. <lb/></s> <s xml:id="echoid-s4946" xml:space="preserve">Deinde applicato quadrato ſtabili ad oculum G, ita vt eius planum per puncta <lb/>C, D, tranſeat, atque vnum eius latus ad punctum D, recta vergat, (quod fiet, <lb/>ſi poſita linea fiduciæ dioptræ ſupra illud latus, punctum D, per foramina pin- <pb o="127" file="157" n="157" rhead="LIBER TERTIVS."/> nacidiorum conſpiciatur) vertatur dioptra, donec per eam punctum quo que <lb/>C, appareat, & </s> <s xml:id="echoid-s4947" xml:space="preserve">vmbra abſciſſa notetur, per quam ex problem. </s> <s xml:id="echoid-s4948" xml:space="preserve">1. </s> <s xml:id="echoid-s4949" xml:space="preserve">anguli C G D, <lb/>magnitudo addiſcatur. </s> <s xml:id="echoid-s4950" xml:space="preserve">Quod ſi alterum latus quadrati vltra rectam G C, exi-<lb/>ſtat, erit angulus CGD, acutus: </s> <s xml:id="echoid-s4951" xml:space="preserve">ſi verò citra rectam GC, extiterit, dictus angu-<lb/>lus erit obtuſus: </s> <s xml:id="echoid-s4952" xml:space="preserve">quem cognoſcemus, ſi ad rectum adiiciemus reliquum angu-<lb/>lum, inter alterum illud latus, & </s> <s xml:id="echoid-s4953" xml:space="preserve">rectam GC; </s> <s xml:id="echoid-s4954" xml:space="preserve">quem quidem inueſtigabimus, vt <lb/> <anchor type="figure" xlink:label="fig-157-01a" xlink:href="fig-157-01"/> de acuto C G D, diximus, ſi nimirum in recta CD, mente notemus punctum, in <lb/>quod alterum illud latus productum incideret. </s> <s xml:id="echoid-s4955" xml:space="preserve">Nam ſi tunc latus illud rectæ <lb/>G C, congruat, & </s> <s xml:id="echoid-s4956" xml:space="preserve">dioptra ad punctum notatum vergat, indicabit vmbra inter <lb/>latus illud, & </s> <s xml:id="echoid-s4957" xml:space="preserve">lineam fiduciæ angulum prædictum reliquum, vt in problemate 1. <lb/></s> <s xml:id="echoid-s4958" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s4959" xml:space="preserve">Si denique alterum illud latus præcisè rectæ GC, congruat, angulus <lb/>CGD, rectus erit. </s> <s xml:id="echoid-s4960" xml:space="preserve">His peractis, quia in triangulo CGD, latera GC, GD, nota <lb/>continent angulum G, etiam cognitum; </s> <s xml:id="echoid-s4961" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cognoſcetur quoque tertium latus <anchor type="note" xlink:label="note-157-01a" xlink:href="note-157-01"/> CD, quod quæritur.</s> <s xml:id="echoid-s4962" xml:space="preserve"/> </p> <div xml:id="echoid-div319" type="float" level="2" n="1"> <figure xlink:label="fig-157-01" xlink:href="fig-157-01a"> <image file="157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/157-01"/> </figure> <note symbol="a" position="right" xlink:label="note-157-01" xlink:href="note-157-01a" xml:space="preserve">12. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s4963" xml:space="preserve">2. </s> <s xml:id="echoid-s4964" xml:space="preserve"><emph style="sc">Vervm</emph> inuentis diſtantiis GC, G D, in aliqua menſura, vna cum an-<lb/>gulo G, fa cilius, licet non tam certò, interuallum CD, co<unsure/>gnoſcemus hoc pacto. <lb/></s> <s xml:id="echoid-s4965" xml:space="preserve">Deſcripto angulo G, ſeorſum ſumatur in GC, portio quotlibet partium late-<lb/>ris quadrati, verbi gratia 500. </s> <s xml:id="echoid-s4966" xml:space="preserve">id eſt, ſemiſsis lateris. </s> <s xml:id="echoid-s4967" xml:space="preserve">Et fiat, vt diſtantia inuen-<lb/>ta G C, ad diſtantiam inuentam G D, ita partes acceptæ 500. </s> <s xml:id="echoid-s4968" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s4969" xml:space="preserve">Nam <lb/>Quotiens dabit numerum quartum proportionalem earundem partium la-<lb/>teris quadrati; </s> <s xml:id="echoid-s4970" xml:space="preserve">quibus ſi capiatur in G D, portio æqualis, <anchor type="note" xlink:href="" symbol="b"/> erit interuallum <anchor type="note" xlink:label="note-157-02a" xlink:href="note-157-02"/> inter extrema puncta dictarum portionum, æquidiſtans interuallo CD; </s> <s xml:id="echoid-s4971" xml:space="preserve">vt con-<lb/>ſtat, ſi illæ portiones in rectas GC, GD, in figura transferrentur; </s> <s xml:id="echoid-s4972" xml:space="preserve">propterea ꝙ in <lb/>illis extremis punctis rectæ GC, GD, ſectæ eſſent proportionaliter. </s> <s xml:id="echoid-s4973" xml:space="preserve">Quare ſi il-<lb/>lud interuallum circino menſuretur in eiſdem partibus lateris quadrati, con-<lb/>tinebit interuallum C D, totidem particulas rectæ G C, in 500. </s> <s xml:id="echoid-s4974" xml:space="preserve">partes æqua-<lb/>les diuiſæ, quot in illo interuallo comprehen duntur ex particulis lateris qua-<lb/>drati: </s> <s xml:id="echoid-s4975" xml:space="preserve">cum ſit vt portio particularum 500. </s> <s xml:id="echoid-s4976" xml:space="preserve">ad interuallum illud, ita G C, di-<lb/>uiſa in partes æquales 500. </s> <s xml:id="echoid-s4977" xml:space="preserve">ad CD. </s> <s xml:id="echoid-s4978" xml:space="preserve">Quæ particulæ reducentur ad menſuram, in <lb/>qua inuentæ ſunt GC, GD: </s> <s xml:id="echoid-s4979" xml:space="preserve">ſi fiat, vt GC, quatenus 500. </s> <s xml:id="echoid-s4980" xml:space="preserve">ad menſuras in G C, <lb/>inuentas, ita C D, inuenta in particulis rectæ GC, in 500. </s> <s xml:id="echoid-s4981" xml:space="preserve">partes diuiſæ, ad <lb/>aliud.</s> <s xml:id="echoid-s4982" xml:space="preserve"/> </p> <div xml:id="echoid-div320" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-157-02" xlink:href="note-157-02a" xml:space="preserve">2. ſexti.</note> </div> <p> <s xml:id="echoid-s4983" xml:space="preserve"><emph style="sc">Qvia</emph> verò vix per circinum accurate reperiri poteſt interuallum illud inter <lb/>extremitates proportionalium, magis ex quiſitè, licet laborioſius, interuallum <lb/>propoſitum cognoſcetur per 12. </s> <s xml:id="echoid-s4984" xml:space="preserve">triang. </s> <s xml:id="echoid-s4985" xml:space="preserve">rectilineorum.</s> <s xml:id="echoid-s4986" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4987" xml:space="preserve">INTERVALLVM tranſuerſum in Horizonte, cuius vtrumque ex-<lb/>tremum videri poteſt, per quadratum metiri.</s> <s xml:id="echoid-s4988" xml:space="preserve"/> </p> <pb o="128" file="158" n="158" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div322" type="section" level="1" n="134"> <head xml:id="echoid-head137" xml:space="preserve">PROBLEMA XVII.</head> <p> <s xml:id="echoid-s4989" xml:space="preserve">1. </s> <s xml:id="echoid-s4990" xml:space="preserve"><emph style="sc">In</emph> Plano Horizontis AB, iaceat interuallum C D, in tranſuerſum, pesau-<lb/>tem menſoris in E, ita vt longitudo C D, in vtramque partem producta per E, <lb/>non tranſeat. </s> <s xml:id="echoid-s4991" xml:space="preserve">Nam quando recta C D, è directo menſoris iacet, inueſtigabitur <lb/>ea, per problema 11. </s> <s xml:id="echoid-s4992" xml:space="preserve">Itaque vt tranſuerſum interuallum C D, cognoſcatur, in-<lb/>quirenda erit primum vtriuſque extremi puncti C, <lb/> <anchor type="figure" xlink:label="fig-158-01a" xlink:href="fig-158-01"/> D, diſtantia à pede menſoris E, vt Num. </s> <s xml:id="echoid-s4993" xml:space="preserve">4. </s> <s xml:id="echoid-s4994" xml:space="preserve">proble-<lb/>matis 15. </s> <s xml:id="echoid-s4995" xml:space="preserve">traditum eſt, per vnicam ſtationem. </s> <s xml:id="echoid-s4996" xml:space="preserve">Dein-<lb/>de angulus C E D, explorandus, quod fiet, ſi vnum <lb/>latus quadrati rectæ E C, congruat, & </s> <s xml:id="echoid-s4997" xml:space="preserve">dioptra rectæ <lb/>E D. </s> <s xml:id="echoid-s4998" xml:space="preserve">Nam vmbra abſciſſa inter latus illud, ac dio-<lb/>ptram oſtendet quantitatem anguli CED, vt in pro-<lb/>blemate 1. </s> <s xml:id="echoid-s4999" xml:space="preserve">dictũ eſt: </s> <s xml:id="echoid-s5000" xml:space="preserve">qui quidem acutus erit, ſi alterũ latus vltra rectam E D, exi-<lb/>ſtet: </s> <s xml:id="echoid-s5001" xml:space="preserve">rectus verò ſi præcisè rectę E D, congruet: </s> <s xml:id="echoid-s5002" xml:space="preserve">obtuſus denique, ſi citra re-<lb/>ctam E D, cadet; </s> <s xml:id="echoid-s5003" xml:space="preserve">quem cognoſcemus, ſi recto angulo adiiciemus reliquum <lb/>acutum, qui deprehendetur, vt in pręcedenti problemate docuimus. </s> <s xml:id="echoid-s5004" xml:space="preserve">Quoniam <lb/>ergo triangulum habemus C E D, cuius duo latera E C, E D, cognita ſunt, vna <lb/>cum angulo comprehenſo E: </s> <s xml:id="echoid-s5005" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cognitum quo que erit tertium latus C D, in <anchor type="note" xlink:label="note-158-01a" xlink:href="note-158-01"/> partibus rectarum E C, ED.</s> <s xml:id="echoid-s5006" xml:space="preserve"/> </p> <div xml:id="echoid-div322" type="float" level="2" n="1"> <figure xlink:label="fig-158-01" xlink:href="fig-158-01a"> <image file="158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/158-01"/> </figure> <note symbol="a" position="left" xlink:label="note-158-01" xlink:href="note-158-01a" xml:space="preserve">12. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s5007" xml:space="preserve"><emph style="sc">Eadem</emph> recta C D, cognita erit, ſi in rectis E C, E D, ſeorſum deſcriptis cum <lb/>angulo E, inuento ſumantur partes ipſis EC, ED, proportionales, &</s> <s xml:id="echoid-s5008" xml:space="preserve">c. </s> <s xml:id="echoid-s5009" xml:space="preserve">vt Num. <lb/></s> <s xml:id="echoid-s5010" xml:space="preserve">2. </s> <s xml:id="echoid-s5011" xml:space="preserve">pręcedentis problematis factum eſt.</s> <s xml:id="echoid-s5012" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5013" xml:space="preserve">DISTANTIAM alicuius ſigni in Horizonte poſiti à ſummitate turris, <lb/>vel muri alicuius, licet ad ipſum ſignum acceſſus non pateat, per qua-<lb/>dratum eruere, vbicunque menſor exiſtat.</s> <s xml:id="echoid-s5014" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div324" type="section" level="1" n="135"> <head xml:id="echoid-head138" xml:space="preserve">PROBLEMA XVIII.</head> <p> <s xml:id="echoid-s5015" xml:space="preserve">1. </s> <s xml:id="echoid-s5016" xml:space="preserve"><emph style="sc">In</emph> Horizontis plano punctum A, diſtet à ſummitate D, alicuius altitudi-<lb/>nis per rectam A D, quam ſic venabimur. </s> <s xml:id="echoid-s5017" xml:space="preserve">Vbicunque oculus menſoris exiſtat, <lb/>nimirum in B, indagentur per problema 15. </s> <s xml:id="echoid-s5018" xml:space="preserve">diſtantię <lb/> <anchor type="figure" xlink:label="fig-158-02a" xlink:href="fig-158-02"/> punctorum A, D, ab oculo menſoris B. </s> <s xml:id="echoid-s5019" xml:space="preserve">Deinde angu-<lb/>lus exploretur A B D, vt in problemate 16. </s> <s xml:id="echoid-s5020" xml:space="preserve">do cuimus. <lb/></s> <s xml:id="echoid-s5021" xml:space="preserve">Nam ſic habebimus triangulum A B D, cuius duo la-<lb/>tera nota ſunt BA, BD, vna cum angulo B. </s> <s xml:id="echoid-s5022" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Igitur ter- <anchor type="note" xlink:label="note-158-02a" xlink:href="note-158-02"/> tium quo que latus AD, cognitum erit.</s> <s xml:id="echoid-s5023" xml:space="preserve"/> </p> <div xml:id="echoid-div324" type="float" level="2" n="1"> <figure xlink:label="fig-158-02" xlink:href="fig-158-02a"> <image file="158-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/158-02"/> </figure> <note symbol="b" position="left" xlink:label="note-158-02" xlink:href="note-158-02a" xml:space="preserve">12. triang. <lb/>rectil,</note> </div> <p> <s xml:id="echoid-s5024" xml:space="preserve"><emph style="sc">Qvod</emph> etiam inuenietur, vt Num. </s> <s xml:id="echoid-s5025" xml:space="preserve">2. </s> <s xml:id="echoid-s5026" xml:space="preserve">problem. </s> <s xml:id="echoid-s5027" xml:space="preserve">16. </s> <s xml:id="echoid-s5028" xml:space="preserve">docuimus, ſi in rectis B A, <lb/>B D, cum angulo B, ſeorſum ductis ſumentur partes ipſis B A, B D, proportio-<lb/>nales, &</s> <s xml:id="echoid-s5029" xml:space="preserve">c.</s> <s xml:id="echoid-s5030" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5031" xml:space="preserve">ALTITVDINEM inacceſſibile<unsure/>m, cuius baſis non videatur; </s> <s xml:id="echoid-s5032" xml:space="preserve">& </s> <s xml:id="echoid-s5033" xml:space="preserve">ad <lb/>quam per nullum ſpatium ſecundum rectam lineam accedere poſſit <lb/>menſor, autrecedere, vt duæſtationes fieri poſſint, ſed ſolum ad dex- <pb o="129" file="159" n="159" rhead="LIBER TERTIVS."/> tram, ſiniſtramue ad locum, è quo eius baſis cernatur, per quadratum <lb/>explorare.</s> <s xml:id="echoid-s5034" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div326" type="section" level="1" n="136"> <head xml:id="echoid-head139" xml:space="preserve">PROBLEMA XIX.</head> <p> <s xml:id="echoid-s5035" xml:space="preserve">1. </s> <s xml:id="echoid-s5036" xml:space="preserve"><emph style="sc">Altitvdo</emph> metienda ſit AB, inacceſsibilis, ad <lb/> <anchor type="figure" xlink:label="fig-159-01a" xlink:href="fig-159-01"/> quam ex C, loco menſoris non liceat accedere, aut <lb/>ab ea recedere ſecundum lineam rectam, ſed ſolum <lb/>in tranſuerſum vſque ad D, vnde baſem A, videre <lb/>poſsimus. </s> <s xml:id="echoid-s5037" xml:space="preserve">Per problema 17. </s> <s xml:id="echoid-s5038" xml:space="preserve">inueſtigetur ex D, in-<lb/>teruallum tranſuerſum A C. </s> <s xml:id="echoid-s5039" xml:space="preserve">Nam per 5. </s> <s xml:id="echoid-s5040" xml:space="preserve">problema, <lb/>quando iam diſtantia C A, manifeſta eſt, altitudo AB, <lb/>reddetur nota, quam quærimus, in partibus diſtantiæ inuentæ C A.</s> <s xml:id="echoid-s5041" xml:space="preserve"/> </p> <div xml:id="echoid-div326" type="float" level="2" n="1"> <figure xlink:label="fig-159-01" xlink:href="fig-159-01a"> <image file="159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/159-01"/> </figure> </div> <p> <s xml:id="echoid-s5042" xml:space="preserve"><emph style="sc">Cætervm</emph> ex ſcholio problem. </s> <s xml:id="echoid-s5043" xml:space="preserve">7. </s> <s xml:id="echoid-s5044" xml:space="preserve">facilius per vnicam ſtationem in C, fa-<lb/>ctam inueſtigabitur & </s> <s xml:id="echoid-s5045" xml:space="preserve">altitudo AB, & </s> <s xml:id="echoid-s5046" xml:space="preserve">diſtantia A C, vna cum hypotenuſa BC.</s> <s xml:id="echoid-s5047" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5048" xml:space="preserve">ALTIT VDINEM maiorem ex minori cognita, etiamſi ſolum ma-<lb/>ioris altitudinis vertex cernatur, per quadratum efficere notam.</s> <s xml:id="echoid-s5049" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div328" type="section" level="1" n="137"> <head xml:id="echoid-head140" xml:space="preserve">PROBLEMA XX.</head> <p> <s xml:id="echoid-s5050" xml:space="preserve">1. </s> <s xml:id="echoid-s5051" xml:space="preserve"><emph style="sc">Maior</emph> altitudo A N, metienda proponatur ex minori aliqua turri C O, <lb/>cognita, ex qua ſolum cacumen A, non autem baſis N, appareat. </s> <s xml:id="echoid-s5052" xml:space="preserve">Fiant in ſum-<lb/>mitate duæ ſtationes in C, & </s> <s xml:id="echoid-s5053" xml:space="preserve">D, ſi planum ſummitatis id permittat, ſtaturaque <lb/>menſoris ſit C G, vel D E, & </s> <s xml:id="echoid-s5054" xml:space="preserve">ad A N, intel-<lb/> <anchor type="figure" xlink:label="fig-159-02a" xlink:href="fig-159-02"/> ligatur ducta perpendicularis GEF. </s> <s xml:id="echoid-s5055" xml:space="preserve">Per <lb/>6. </s> <s xml:id="echoid-s5056" xml:space="preserve">problema reperietur altitudo A F, ad <lb/>quam ſi adiicietur altitudo minor D O, <lb/>vna cum menſoris ſtatura C G, hoc eſt, <lb/>recta F N, cognita fiet tota altitudo ma-<lb/>ior A N.</s> <s xml:id="echoid-s5057" xml:space="preserve"/> </p> <div xml:id="echoid-div328" type="float" level="2" n="1"> <figure xlink:label="fig-159-02" xlink:href="fig-159-02a"> <image file="159-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/159-02"/> </figure> </div> <p> <s xml:id="echoid-s5058" xml:space="preserve">2. </s> <s xml:id="echoid-s5059" xml:space="preserve"><emph style="sc">Qvod</emph> ſi in ſummitate tanta pla-<lb/>nities non ſit, vt duæ ſtationes fieri poſ-<lb/>ſint, erigenda eſt haſta aliqua ad Hori-<lb/>zontem recta, niſi forte adſit ædificium <lb/>quo dpiam erectum, ibiq; </s> <s xml:id="echoid-s5060" xml:space="preserve">faciendæ duæ <lb/>ſtationes. </s> <s xml:id="echoid-s5061" xml:space="preserve">Nam tunc per problema 7. </s> <s xml:id="echoid-s5062" xml:space="preserve">in-<lb/>uenietur altitudo inter cacumen maioris <lb/>altitudinis, & </s> <s xml:id="echoid-s5063" xml:space="preserve">rectam, quæ ab inferiori <lb/>ſtatione Horizonti ducitur parallela. <lb/></s> <s xml:id="echoid-s5064" xml:space="preserve">Cuiſi apponitur altitudo minor ab inferiore ſtatione vſque ad eius baſem, con-<lb/>flabitur tota altitudo maior. </s> <s xml:id="echoid-s5065" xml:space="preserve">Vt in figuris problematis 4. </s> <s xml:id="echoid-s5066" xml:space="preserve">ſi maior altitudo ſit <lb/>G L, & </s> <s xml:id="echoid-s5067" xml:space="preserve">minor A K, fiantq; </s> <s xml:id="echoid-s5068" xml:space="preserve">duæ ſtationes in A, a, inferior vna, & </s> <s xml:id="echoid-s5069" xml:space="preserve">altera ſu-<lb/>perior, reperietur per 7. </s> <s xml:id="echoid-s5070" xml:space="preserve">problema altitudo G F, cui ſi addetur minor altitu-<lb/>do A K, ab inferiori ſtationevſque ad baſem, componetur tota altitudo ma-<lb/>or G L.</s> <s xml:id="echoid-s5071" xml:space="preserve"/> </p> <pb o="130" file="160" n="160" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s5072" xml:space="preserve"><emph style="sc">Porro</emph> facilius ex ſcholio problem. </s> <s xml:id="echoid-s5073" xml:space="preserve">7. </s> <s xml:id="echoid-s5074" xml:space="preserve">altitu dinem A F, eruemus, ſi in G, <lb/>bis quadratum accommodetur, vt in eo ſcholio diximus. </s> <s xml:id="echoid-s5075" xml:space="preserve">Vt ſi in figura eius <lb/>ſcholij maior altitudo foret I F, & </s> <s xml:id="echoid-s5076" xml:space="preserve">minor cognita A H, inueniretur G F, vt ibi <lb/>oſtenſum eſt, &</s> <s xml:id="echoid-s5077" xml:space="preserve">c.</s> <s xml:id="echoid-s5078" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5079" xml:space="preserve">ALTITVDINEM maiorem ex minori incognita, ſi tamen baſis ma-<lb/>ioris cerni poſſit, per quadratum venari.</s> <s xml:id="echoid-s5080" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div330" type="section" level="1" n="138"> <head xml:id="echoid-head141" xml:space="preserve">PROBLEMA XXI.</head> <figure> <image file="160-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/160-01"/> </figure> <p> <s xml:id="echoid-s5081" xml:space="preserve">1. </s> <s xml:id="echoid-s5082" xml:space="preserve"><emph style="sc">In</emph> figura problematis 17. </s> <s xml:id="echoid-s5083" xml:space="preserve">lib. </s> <s xml:id="echoid-s5084" xml:space="preserve">2. </s> <s xml:id="echoid-s5085" xml:space="preserve">addiſcatur altitu-<lb/>do A E, vel per problema 6. </s> <s xml:id="echoid-s5086" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s5087" xml:space="preserve">aut potius per ſcho-<lb/>lium problem. </s> <s xml:id="echoid-s5088" xml:space="preserve">7. </s> <s xml:id="echoid-s5089" xml:space="preserve">ſi C, ſit ſummitas minoris altitudinis <lb/>C D. </s> <s xml:id="echoid-s5090" xml:space="preserve">Deinde quia baſis B, maioris altitudinis ponitur <lb/>poſſe videri ex C, inquiratur etiam altitudo minor C D, <lb/>per problema 8. </s> <s xml:id="echoid-s5091" xml:space="preserve">aut 9, vel potius per ſcholiũ probl. </s> <s xml:id="echoid-s5092" xml:space="preserve">9. </s> <s xml:id="echoid-s5093" xml:space="preserve">ſi <lb/>C, fuerit ſummitas minoris altitudinis C D. </s> <s xml:id="echoid-s5094" xml:space="preserve">Hæc enim <lb/>adiecta ad inuentam altitudinem A E, conficiet totam <lb/>maiorem altitudinem A B, notam, quæ deſidera-<lb/>tur.</s> <s xml:id="echoid-s5095" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5096" xml:space="preserve">ALTITVDINEM minorem ex maiori cognita, licet baſis minoris <lb/>cerninon poſſit@, per quadratum ſcrutari.</s> <s xml:id="echoid-s5097" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div331" type="section" level="1" n="139"> <head xml:id="echoid-head142" xml:space="preserve">PROBLEMA XXII.</head> <p> <s xml:id="echoid-s5098" xml:space="preserve">1. </s> <s xml:id="echoid-s5099" xml:space="preserve"><emph style="sc">Minor</emph> altitudo AB, ex maiore C D, cog-<lb/> <anchor type="figure" xlink:label="fig-160-02a" xlink:href="fig-160-02"/> nita proponatur dimetienda. </s> <s xml:id="echoid-s5100" xml:space="preserve">Intelligatur ducta <lb/>recta A E, Horizonti B D, æquidiſtans, vt E D, fiat <lb/>minori altitudini AB, æqualis. </s> <s xml:id="echoid-s5101" xml:space="preserve">Si igitur ex ſummi-<lb/>tate C, per problema 8. </s> <s xml:id="echoid-s5102" xml:space="preserve">vel 9. </s> <s xml:id="echoid-s5103" xml:space="preserve">aut potius per ſcho-<lb/>lium problem. </s> <s xml:id="echoid-s5104" xml:space="preserve">9. </s> <s xml:id="echoid-s5105" xml:space="preserve">exploretur altitudo C E, inſpe-<lb/>cto nimirum cacumine A, ac ſi eſſet ſignum ali-<lb/>quod in Horizonte A E, ex C, viſum: </s> <s xml:id="echoid-s5106" xml:space="preserve">atque hæc <lb/>altitudo inuenta C E, ex maiore altitudine C D, quę <lb/>cognita ponitur, detrahatur, reliqua fiet minor al-<lb/>titudo AB, quam in quirimus.</s> <s xml:id="echoid-s5107" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="2" n="1"> <figure xlink:label="fig-160-02" xlink:href="fig-160-02a"> <image file="160-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/160-02"/> </figure> </div> <p> <s xml:id="echoid-s5108" xml:space="preserve">ALTITVDINEM minorem ex maiori incognita, dummodo baſis <lb/>minoris appareat, per quadratum elicere.</s> <s xml:id="echoid-s5109" xml:space="preserve"/> </p> <pb o="131" file="161" n="161" rhead="LIBER TERTIVS."/> </div> <div xml:id="echoid-div333" type="section" level="1" n="140"> <head xml:id="echoid-head143" xml:space="preserve">PROBLEMA XXIII.</head> <p> <s xml:id="echoid-s5110" xml:space="preserve">1. </s> <s xml:id="echoid-s5111" xml:space="preserve"><emph style="sc">Figvra</emph> præcedentis problematis repetatur. </s> <s xml:id="echoid-s5112" xml:space="preserve">Et quia baſis B, minoris<unsure/> <lb/>altitudinis, tanquam ſignum quodpiam in Horizonte poſitum, videri poteſt, <lb/>ex hypotheſi, nota efficietur per problema 8. </s> <s xml:id="echoid-s5113" xml:space="preserve">aut 9. </s> <s xml:id="echoid-s5114" xml:space="preserve">vel potius per ſcholium <lb/>problem. </s> <s xml:id="echoid-s5115" xml:space="preserve">9. </s> <s xml:id="echoid-s5116" xml:space="preserve">maior altitudo C D. </s> <s xml:id="echoid-s5117" xml:space="preserve">Quare, vtin præcedenti problemate dictum <lb/>eſt, minor altitudo AB, ex maiore CD, proximè inuenta cognoſcetur; </s> <s xml:id="echoid-s5118" xml:space="preserve">quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s5119" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5120" xml:space="preserve">PORTIONEM altitudinis maioris ex minore altitudine, & </s> <s xml:id="echoid-s5121" xml:space="preserve">minoris <lb/>portionem ex maiore, per quadratum percipere.</s> <s xml:id="echoid-s5122" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div334" type="section" level="1" n="141"> <head xml:id="echoid-head144" xml:space="preserve">PROBLEMA XXIV.</head> <p> <s xml:id="echoid-s5123" xml:space="preserve">1. </s> <s xml:id="echoid-s5124" xml:space="preserve"><emph style="sc">Sit</emph> portio A G, maioris altitudinis A B, exquirenda ex minore altitudi-<lb/> <anchor type="figure" xlink:label="fig-161-01a" xlink:href="fig-161-01"/> ne D E; </s> <s xml:id="echoid-s5125" xml:space="preserve">Item portio GF, altitudinis minoris FB, ex ma-<lb/>iori altitudine D E. </s> <s xml:id="echoid-s5126" xml:space="preserve">Si altitudo D E, minor eſt reliqua <lb/>portione C B, maioris altitudinis, inueſtiganda erit per <lb/>problema 20. </s> <s xml:id="echoid-s5127" xml:space="preserve">vel 21. </s> <s xml:id="echoid-s5128" xml:space="preserve">vtraque altitudo maior A B, C B, <lb/>ex minori D E, prout videlicet D E, cognita fuerit, aut <lb/>incognita. </s> <s xml:id="echoid-s5129" xml:space="preserve">Nam C B, ablata ex A B, notam reliquet <lb/>porpoſitam portionem A C. </s> <s xml:id="echoid-s5130" xml:space="preserve">Si verò D E, maior eſt re-<lb/>liqua portione F B, ſi nimirum maioris altitudinis por-<lb/>tio A F, metienda proponatur: </s> <s xml:id="echoid-s5131" xml:space="preserve">inueſtiganda quidem <lb/>erit maior altitudo A B, ex minore D E, perproblema <lb/>20. </s> <s xml:id="echoid-s5132" xml:space="preserve">vel 21. </s> <s xml:id="echoid-s5133" xml:space="preserve">At vero minor altitudo F B, ex maiore D E, <lb/>per problema 22. </s> <s xml:id="echoid-s5134" xml:space="preserve">vel 23. </s> <s xml:id="echoid-s5135" xml:space="preserve">exploranda erit, prout videli-<lb/>cet D E, nota fuerit, autignota. </s> <s xml:id="echoid-s5136" xml:space="preserve">Nam rurſus FB, detracta <lb/>ex AB, notam relinquet portionem propoſitam A F.</s> <s xml:id="echoid-s5137" xml:space="preserve"/> </p> <div xml:id="echoid-div334" type="float" level="2" n="1"> <figure xlink:label="fig-161-01" xlink:href="fig-161-01a"> <image file="161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/161-01"/> </figure> </div> <p> <s xml:id="echoid-s5138" xml:space="preserve">2. </s> <s xml:id="echoid-s5139" xml:space="preserve"><emph style="sc">Non</emph> ſecus per problema 22. </s> <s xml:id="echoid-s5140" xml:space="preserve">vel 23. </s> <s xml:id="echoid-s5141" xml:space="preserve">indaganda erit vtraque altitudo mi-<lb/>nor FB, GB, ex maiore DE, ſi fortè illarum differentia F G, inuenienda ſit.</s> <s xml:id="echoid-s5142" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5143" xml:space="preserve">ALTITVDINEM, cuius baſis impoſita ſit monti, vel alteri cui-<lb/>piam altitudini, & </s> <s xml:id="echoid-s5144" xml:space="preserve">vtraque illius extremitas cerni poſſit, etiamſi infi-<lb/>mum punctum alterius, cui imponitur, lateat, & </s> <s xml:id="echoid-s5145" xml:space="preserve">eiuſdem puncti in-<lb/>fimi diſtantia à loco menſoris cognita non ſit, per quadratum ex val-<lb/>le, aut ex plano Horizontis explorare.</s> <s xml:id="echoid-s5146" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div336" type="section" level="1" n="142"> <head xml:id="echoid-head145" xml:space="preserve">PROBLEMA XXV.</head> <p> <s xml:id="echoid-s5147" xml:space="preserve">1. </s> <s xml:id="echoid-s5148" xml:space="preserve"><emph style="sc">Hvivscemodi</emph> altitudo eſt turris ſupra montem poſita, & </s> <s xml:id="echoid-s5149" xml:space="preserve">portio ali-<lb/>cuius ædificij inter duas feneſtras, vel duo ſigna, quorum alterum altero ſupe-<lb/>rius eſt. </s> <s xml:id="echoid-s5150" xml:space="preserve">Sit igitur ſupra montem B F, altitudo turris A B, propoſita. </s> <s xml:id="echoid-s5151" xml:space="preserve">Ex ali-<lb/>quo loco E, in planitie, aut valle, vnde vtrumque turris extremum conſpicia- <pb o="132" file="162" n="162" rhead="GEOMETR. PRACT."/> tur, inueſtigetur perproblema 6. </s> <s xml:id="echoid-s5152" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s5153" xml:space="preserve">tam altitudo AF, quam B F, proutſcili-<lb/>cet duæ ſtationes fiunt aut in plano, aut in aliqua haſta erecta. </s> <s xml:id="echoid-s5154" xml:space="preserve">Vtraque tamen <lb/> <anchor type="figure" xlink:label="fig-162-01a" xlink:href="fig-162-01"/> altitudo A F, BF, facilius inuenietur per ſcholium problem. </s> <s xml:id="echoid-s5155" xml:space="preserve">7. </s> <s xml:id="echoid-s5156" xml:space="preserve">ſi in E, bis qua-<lb/>dratum accommodetur, vtin eo ſcholio factum eſt. </s> <s xml:id="echoid-s5157" xml:space="preserve">Vtraque altitudine inuen-<lb/>ta, altitudo montis BF, dempta ex tota altitudine AF, notam relinquet opta-<lb/>tam altitudinem A B.</s> <s xml:id="echoid-s5158" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="1"> <figure xlink:label="fig-162-01" xlink:href="fig-162-01a"> <image file="162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/162-01"/> </figure> </div> </div> <div xml:id="echoid-div338" type="section" level="1" n="143"> <head xml:id="echoid-head146" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s5159" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi AB, portio ſuperior totius alicuius altitudinis AF, deſideretur, <lb/> <anchor type="note" xlink:label="note-162-01a" xlink:href="note-162-01"/> indaganda erit per problem. </s> <s xml:id="echoid-s5160" xml:space="preserve">6. </s> <s xml:id="echoid-s5161" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s5162" xml:space="preserve">aut potius per ſcholium probl. </s> <s xml:id="echoid-s5163" xml:space="preserve">7. </s> <s xml:id="echoid-s5164" xml:space="preserve">tam tota <lb/>altitudo AF, quàm eius portio BF. </s> <s xml:id="echoid-s5165" xml:space="preserve">Earum enim differentia notam dabitſupe-<lb/>riorem portionem A B, deſideratam.</s> <s xml:id="echoid-s5166" xml:space="preserve"/> </p> <div xml:id="echoid-div338" type="float" level="2" n="1"> <note position="left" xlink:label="note-162-01" xlink:href="note-162-01a" xml:space="preserve">Suprema me-<lb/>dia atque in-<lb/>fima pars alti-<lb/>tudinis quo <lb/>pacto metien-<lb/>da ſit.</note> </div> <p> <s xml:id="echoid-s5167" xml:space="preserve"><emph style="sc">Si</emph> autem media aliqua portio IB, cognoſcenda ſit, coniicienda rurſum erit <lb/>per problema 6. </s> <s xml:id="echoid-s5168" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s5169" xml:space="preserve">aut potius per ſcholium problem. </s> <s xml:id="echoid-s5170" xml:space="preserve">7. </s> <s xml:id="echoid-s5171" xml:space="preserve">vtraque altitudo <lb/>IF, BF, vt earum differentia IB, nota reddatur.</s> <s xml:id="echoid-s5172" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5173" xml:space="preserve"><emph style="sc">Si</emph> denique infima pars B F, proponatur inquirenda, exploranda ea erit per <lb/>problema quoque 6. </s> <s xml:id="echoid-s5174" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s5175" xml:space="preserve">aut potius per ſcholium problem. </s> <s xml:id="echoid-s5176" xml:space="preserve">7.</s> <s xml:id="echoid-s5177" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5178" xml:space="preserve">DISTANTIAM accliuem montis à loco menſoris vſque ad baſem <lb/>altitudinis monti impoſitæ, etiam non viſam, vna cum ipſa altitudine <lb/>quando menſor in aſcenſu montis conſiſtit, propè verum, beneficio <lb/>quadrati efficere cognitam.</s> <s xml:id="echoid-s5179" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div340" type="section" level="1" n="144"> <head xml:id="echoid-head147" xml:space="preserve">PROBLEMA XXVI.</head> <p> <s xml:id="echoid-s5180" xml:space="preserve">1. </s> <s xml:id="echoid-s5181" xml:space="preserve"><emph style="sc">Sit</emph> turris AB, monti impoſita, in cuius aſcenſu, ſeu latere menſor conſi-<lb/>ſtat in C, ex quo loco baſem turris videre nõ poſsit. </s> <s xml:id="echoid-s5182" xml:space="preserve">Erigatur haſta aliqua C G, <lb/>ad Horizontem, non autem ad latus montis, perpendicularis, ſitque men-<lb/>ſoris ſtatura C E. </s> <s xml:id="echoid-s5183" xml:space="preserve">Cogitetur ducta EK, ad altitudinem perpendicularis, aut <pb o="133" file="163" n="163" rhead="LIBER TERTIVS."/> Horizonti æquidiſtans. </s> <s xml:id="echoid-s5184" xml:space="preserve">Applicato autem quadrato ſtabili ad haſtam in puncto <lb/>E, (ſi pendulum adhibeatur, cõſtituendus etiam erit oculus in E.) </s> <s xml:id="echoid-s5185" xml:space="preserve">dirigatur dio-<lb/>ptra verſus cacumen A; </s> <s xml:id="echoid-s5186" xml:space="preserve">& </s> <s xml:id="echoid-s5187" xml:space="preserve">per vmbram <lb/> <anchor type="figure" xlink:label="fig-163-01a" xlink:href="fig-163-01"/> abſciſſaminueniatur angulus AEK, vt in <lb/>problemate 1. </s> <s xml:id="echoid-s5188" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s5189" xml:space="preserve">Deinde fiat <lb/>alia ſtatio ſuperior in F, in haſta, ductaq; <lb/></s> <s xml:id="echoid-s5190" xml:space="preserve">per cogitationẽ recta F I, ad altitudinem <lb/> <anchor type="note" xlink:label="note-163-01a" xlink:href="note-163-01"/> perpendiculari, dirigatur rurſus dioptra <lb/>verſus A, atque eo dem modo angulus e-<lb/>ruatur AFI, <anchor type="note" xlink:href="" symbol="a"/> Et quia angulus AFG, duo- bus angulis FEA, FAE, æqualis eſt, ſi au-<lb/> <anchor type="note" xlink:label="note-163-02a" xlink:href="note-163-02"/> feratur AEF, complementum prioris an-<lb/>guli AEK, in ſtatione inferiori E, inuenti, <lb/>ex A F G, complemento poſterioris an-<lb/>guli in ſuperiori ſtatione F, deprehenſi, <lb/>reliquus fiet angulus E A F, cognitus. <lb/></s> <s xml:id="echoid-s5191" xml:space="preserve">Quoniam igitur in trangulo A E F, duo <lb/>anguli A, E, cogniti ſunt, vna cum latere <lb/>E F, hoc eſt, cum differentia ſtationum, <anchor type="note" xlink:href="" symbol="b"/> nota fientreliqua latera AE, AF.</s> <s xml:id="echoid-s5192" xml:space="preserve"/> </p> <div xml:id="echoid-div340" type="float" level="2" n="1"> <figure xlink:label="fig-163-01" xlink:href="fig-163-01a"> <image file="163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/163-01"/> </figure> <note position="right" xlink:label="note-163-01" xlink:href="note-163-01a" xml:space="preserve">In figura duc <lb/>lineam ex F. <lb/>ad A.</note> <note symbol="a" position="right" xlink:label="note-163-02" xlink:href="note-163-02a" xml:space="preserve">32. primi.</note> </div> <note symbol="b" position="right" xml:space="preserve">10. triang. <lb/>rectil.</note> <p> <s xml:id="echoid-s5193" xml:space="preserve">2. </s> <s xml:id="echoid-s5194" xml:space="preserve"><emph style="sc">Post</emph> hæc erigatur alius baculus D H, ad Horizontem rectus, ſumptaq; <lb/></s> <s xml:id="echoid-s5195" xml:space="preserve">menſoris ſtatura ipſi C E, æquali, & </s> <s xml:id="echoid-s5196" xml:space="preserve">ducta recta HL, ad altitu dinem perpendicu-<lb/>lari, applicetur ad H, quadratum, ac dioptra in punctum E, dirigatur, rurſumque <lb/>per problema 1. </s> <s xml:id="echoid-s5197" xml:space="preserve">ex vmbra abſciſſa angulus eliciatur EHL, acradius viſualis HE, <lb/>altitudini occurrere concipiatur in M, <anchor type="note" xlink:href="" symbol="c"/> qui ipſi B C D, lateri montis parallelus erit. </s> <s xml:id="echoid-s5198" xml:space="preserve">Concipienda etenim ſunt tria puncta B, C, D, in vna recta linea iacere, ac ſi <lb/> <anchor type="note" xlink:label="note-163-04a" xlink:href="note-163-04"/> D C, producta ad baſem turris pertineret. </s> <s xml:id="echoid-s5199" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quia vero angulus E H L, angulo <anchor type="note" xlink:label="note-163-05a" xlink:href="note-163-05"/> M E K, internus externo æqualis eſt, ſi E H L, cognitus hoc eſt M E K, <lb/>ex angulo A E N, in priori ſtatione E, obſeruato, detrahatur, notus relin-<lb/>quetur angulus AEM: </s> <s xml:id="echoid-s5200" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s5201" xml:space="preserve">angulus EAM, cum complementum ſit an-<lb/>guli AEN, in priori ſtatione E, obſeruati, cognitus. </s> <s xml:id="echoid-s5202" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s5203" xml:space="preserve">AME, reliquus duo-<lb/>rum rectorum cognitus erit: </s> <s xml:id="echoid-s5204" xml:space="preserve">quiquidem etiam relinquitur, ſi N M H, comple-<lb/>mentum anguli M H L, in ſtatione poſtrema H, inuenti ex duobus rectis detra-<lb/>hatur. </s> <s xml:id="echoid-s5205" xml:space="preserve">Quapropter cumin triangulo AEM, omnes anguli noti ſint, vna cumla-<lb/>tere A E, quod paulo ante inuenimus, <anchor type="note" xlink:href="" symbol="e"/> cognoſcentur quoque reliqua duo la- <anchor type="note" xlink:label="note-163-06a" xlink:href="note-163-06"/> tera ME, AM: </s> <s xml:id="echoid-s5206" xml:space="preserve">ac propterea diſtantia E M, vel C B, inuenta erit. </s> <s xml:id="echoid-s5207" xml:space="preserve">Et ſi rectæ A M, <lb/>inuentæ addatur menſoris ſtatura MB, tota turris altitudo A B, cognita erit.</s> <s xml:id="echoid-s5208" xml:space="preserve"/> </p> <div xml:id="echoid-div341" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-163-04" xlink:href="note-163-04a" xml:space="preserve">33. primi.</note> <note symbol="d" position="right" xlink:label="note-163-05" xlink:href="note-163-05a" xml:space="preserve">29. primi.</note> <note symbol="e" position="right" xlink:label="note-163-06" xlink:href="note-163-06a" xml:space="preserve">10. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s5209" xml:space="preserve"><emph style="sc">Cvrandvm</emph> autem eſt magnopere, vt tria puncta B, C, D, in vna linea re-<lb/>ctaiaceant, id quod in problemate 22. </s> <s xml:id="echoid-s5210" xml:space="preserve">lib 2. </s> <s xml:id="echoid-s5211" xml:space="preserve">faciendum eſſe monuimus.</s> <s xml:id="echoid-s5212" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5213" xml:space="preserve">3. </s> <s xml:id="echoid-s5214" xml:space="preserve"><emph style="sc">Qvod</emph> ſi baſis altitudinis ex latere montis videri poſsit, nullo ferme la-<lb/>bore problema per vnicam ſtationem efficiemus. </s> <s xml:id="echoid-s5215" xml:space="preserve">Nam ſi in D, ſtatuatur qua-<lb/>dratum, ita vt centrum dioptræ ſit ſuperius, & </s> <s xml:id="echoid-s5216" xml:space="preserve">latus infimum recta in B, feratur, <lb/> <anchor type="note" xlink:label="note-163-07a" xlink:href="note-163-07"/> dirigenda erit dioptra verſus idem punctum B. </s> <s xml:id="echoid-s5217" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Nam ſi fiat, <anchor type="note" xlink:label="note-163-08a" xlink:href="note-163-08"/> inuenietur diſtantia DB, in partibus lateris quadrati, vt liquido cõſtat ex ijs, quæ <lb/>in problem. </s> <s xml:id="echoid-s5218" xml:space="preserve">15. </s> <s xml:id="echoid-s5219" xml:space="preserve">Num. </s> <s xml:id="echoid-s5220" xml:space="preserve">5. </s> <s xml:id="echoid-s5221" xml:space="preserve">ſcripſimus, patetque in 1. </s> <s xml:id="echoid-s5222" xml:space="preserve">figura eiuſdem problematis; </s> <s xml:id="echoid-s5223" xml:space="preserve">ſi <pb o="134" file="164" n="164" rhead="GEOMETR. PRACT."/> i<unsure/>n ea cogitetur aſcenſus montis AF, in ſecundo quadrato, & </s> <s xml:id="echoid-s5224" xml:space="preserve">F, baſis altitudinis <lb/>monti impoſitæ.</s> <s xml:id="echoid-s5225" xml:space="preserve"/> </p> <div xml:id="echoid-div342" type="float" level="2" n="3"> <note symbol="f" position="right" xlink:label="note-163-07" xlink:href="note-163-07a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-163-08" xlink:href="note-163-08a" xml:space="preserve"> <lb/>Vt vmbra verſa \\ abſciſſa # Ad lat{us} quadra- \\ ti: # Ita lat{us} qua- \\ drati # a<unsure/>daliud, <lb/></note> </div> <p> <s xml:id="echoid-s5226" xml:space="preserve"><emph style="sc">Deinde</emph> ſi idem quadratum in D, ita ſtatuatur, vt rurſus centrum dioptræ <lb/>ſit in ſublimi, & </s> <s xml:id="echoid-s5227" xml:space="preserve">latus infimum ad faſtigium altitu dinis A, recta tendat, idemque <lb/>punctum A, per dioptraminſpiciatur, reperietur eodem pacto diſtantia à D, vſ-<lb/>que ad A; </s> <s xml:id="echoid-s5228" xml:space="preserve">ſi fiat, <lb/> <anchor type="note" xlink:label="note-164-01a" xlink:href="note-164-01"/> </s> </p> <div xml:id="echoid-div343" type="float" level="2" n="4"> <note style="it" position="right" xlink:label="note-164-01" xlink:href="note-164-01a" xml:space="preserve"> <lb/>Vt vmbra verſa abſciſſa # ad lat{us} quadrati: # Ita lat{us} quadrati # ad alin<unsure/>d. <lb/></note> </div> <p> <s xml:id="echoid-s5229" xml:space="preserve"><emph style="sc">Postremo</emph>, accommodato quadrato, ita vt vnum latus rectæ D B, con-<lb/>gruat, & </s> <s xml:id="echoid-s5230" xml:space="preserve">dioptra in A, dirigatur, inuenietur angulus, quem rectæ DB, DA, effi-<lb/>ciunt, vt in problem 16. </s> <s xml:id="echoid-s5231" xml:space="preserve">Num. </s> <s xml:id="echoid-s5232" xml:space="preserve">1. </s> <s xml:id="echoid-s5233" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s5234" xml:space="preserve">Si ergo hic angulus ſeorſum deſcri-<lb/>batur, & </s> <s xml:id="echoid-s5235" xml:space="preserve">in rectis D B, D A, capiantur duæ portiones proportionales, vt in eo-<lb/>dem problem. </s> <s xml:id="echoid-s5236" xml:space="preserve">16. </s> <s xml:id="echoid-s5237" xml:space="preserve">Num. </s> <s xml:id="echoid-s5238" xml:space="preserve">2. </s> <s xml:id="echoid-s5239" xml:space="preserve">tradidimus, reperietur altitudo A B, per interuallum <lb/>inter duas illas portiones, vt ibi interuallum C D, indagauimus.</s> <s xml:id="echoid-s5240" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5241" xml:space="preserve">PROFVNDITATEM putei, vel ædificii cuiuſuis ad perpendicu-<lb/>lum erecti, ſi modo angulus fundi, vel ſignum aliquod in fundo poſi-<lb/>tum conſpiciatur, per quadratum efficere cognitam.</s> <s xml:id="echoid-s5242" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div345" type="section" level="1" n="145"> <head xml:id="echoid-head148" xml:space="preserve">PROBLEMA XXVII.</head> <p> <s xml:id="echoid-s5243" xml:space="preserve">1. </s> <s xml:id="echoid-s5244" xml:space="preserve"><emph style="sc">Hoc</emph> nihil eſt aliud, niſi turrim ex eius vertice, quando in Horizonte ſi-<lb/>gnum aliquod apparet, per duas ſtationes in haſta aliqua <lb/> <anchor type="figure" xlink:label="fig-164-01a" xlink:href="fig-164-01"/> factas metiri, vt in problem. </s> <s xml:id="echoid-s5245" xml:space="preserve">9. </s> <s xml:id="echoid-s5246" xml:space="preserve">factum eſt. </s> <s xml:id="echoid-s5247" xml:space="preserve">Quare eius pro-<lb/>blematis praxim hic breuiter repetemus. </s> <s xml:id="echoid-s5248" xml:space="preserve">Sit puteus, ſeu <lb/>ædificium erectum ABCM, cuius angulus C, in fundo, vel <lb/>ſignum quodpiam C, in fundo poſitum conſpici poſsit. <lb/></s> <s xml:id="echoid-s5249" xml:space="preserve">Erecta haſta A E, in orificio putei, vel ſummitate ædificij, <lb/>fiant duæſtationes oculi menſoris in D, E, & </s> <s xml:id="echoid-s5250" xml:space="preserve">applicato la-<lb/>tere quadrati ad haſtam bis, vt modo centrum dioptræ in <lb/>D, & </s> <s xml:id="echoid-s5251" xml:space="preserve">modo in E, ſtatuatur, dirigatur dioptra verſus C. </s> <s xml:id="echoid-s5252" xml:space="preserve">Si <lb/>igitur in vtraque ſtatione dioptra vmbram rectam interſe-<lb/>cet, quod plerumque in puteorum dimenſi onefieri ſolet: </s> <s xml:id="echoid-s5253" xml:space="preserve"><lb/>Fiat autem, <lb/> <anchor type="note" xlink:label="note-164-02a" xlink:href="note-164-02"/> exibit recta E B, nota in partibus differentiæ ſtationum D E. </s> <s xml:id="echoid-s5254" xml:space="preserve">Si ergo auferatur <lb/>recta E A, compoſita ex differentia ſtationum D E, & </s> <s xml:id="echoid-s5255" xml:space="preserve">portione haſtæ D A, quæ <lb/>plerunque ſtaturæ menſoris eſſe ſolet æqualis, vel certe facile meſurari poteſt, <lb/>nota relinquetur altitudo AB, quæſita.</s> <s xml:id="echoid-s5256" xml:space="preserve"/> </p> <div xml:id="echoid-div345" type="float" level="2" n="1"> <figure xlink:label="fig-164-01" xlink:href="fig-164-01a"> <image file="164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/164-01"/> </figure> <note style="it" position="right" xlink:label="note-164-02" xlink:href="note-164-02a" xml:space="preserve"> <lb/>Vt differentia vmbra- \\ rum rectarum # Ad D E, differen- \\ tia ſtationum # Ita vmbra re- \\ cta maior # ad E B, <lb/></note> </div> <p> <s xml:id="echoid-s5257" xml:space="preserve">2. </s> <s xml:id="echoid-s5258" xml:space="preserve"><emph style="sc">Si</emph> vero in vtra que ſtatione vmbra verſa interſecetur, & </s> <s xml:id="echoid-s5259" xml:space="preserve">fiat, <pb o="135" file="165" n="165" rhead="LIBER TERTIVS."/> <anchor type="note" xlink:label="note-165-01a" xlink:href="note-165-01"/> fietrurſus nota recta E B, &</s> <s xml:id="echoid-s5260" xml:space="preserve">c.</s> <s xml:id="echoid-s5261" xml:space="preserve"/> </p> <div xml:id="echoid-div346" type="float" level="2" n="2"> <note style="it" position="right" xlink:label="note-165-01" xlink:href="note-165-01a" xml:space="preserve"> <lb/>Vt differentia vmbrarum \\ verſarum # ad D E, differen- \\ tiam ſtationum # Ita vmbra verſa \\ maior # ad EB, <lb/></note> </div> <p> <s xml:id="echoid-s5262" xml:space="preserve">3. </s> <s xml:id="echoid-s5263" xml:space="preserve"><emph style="sc">Si</emph> denique in vna ſtatione latus vmbræ rectæ ſecetur, & </s> <s xml:id="echoid-s5264" xml:space="preserve">in altera latus <lb/>vmbræ verſæ, reducenda erit vel vmbra recta ad verſam, vel verſa ad rectam. <lb/></s> <s xml:id="echoid-s5265" xml:space="preserve">Nam ſi rurſus fiat. </s> <s xml:id="echoid-s5266" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-165-02a" xlink:href="note-165-02"/> iterum producetur recta E B, in partibus differentiæ ſtationum D E, &</s> <s xml:id="echoid-s5267" xml:space="preserve">c.</s> <s xml:id="echoid-s5268" xml:space="preserve"/> </p> <div xml:id="echoid-div347" type="float" level="2" n="3"> <note style="it" position="right" xlink:label="note-165-02" xlink:href="note-165-02a" xml:space="preserve"> <lb/>Vt differentia vmbrarum \\ ſiue verſarum, ſiue rectarum, # ad D E, differen- \\ tiam ſtationum: # Ita vmbr a verſa \\ velrecta maior # ad E B. <lb/></note> </div> <p> <s xml:id="echoid-s5269" xml:space="preserve"><emph style="sc">Per</emph> vnicam quoque ſtationem aſſe quemur altitudinem putei, quemad-<lb/>modum in ſcholio probl. </s> <s xml:id="echoid-s5270" xml:space="preserve">9. </s> <s xml:id="echoid-s5271" xml:space="preserve">ex montis vertice eius altitudinem menſi ſumus: </s> <s xml:id="echoid-s5272" xml:space="preserve">ſi <lb/>videlicet in A, quadratum ita ſtatuatur, vt centruma, dioptræ <lb/> <anchor type="figure" xlink:label="fig-165-01a" xlink:href="fig-165-01"/> ſuperius ſit, & </s> <s xml:id="echoid-s5273" xml:space="preserve">latus infimum A C, ex A, ad punctum C, ver-<lb/>gat, ad inueniendam diſtantiam, vel hypotenuſam A C, &</s> <s xml:id="echoid-s5274" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s5275" xml:space="preserve">vtin eo ſcholio factum eſt, atque hæc figura appoſita decla-<lb/>rat. </s> <s xml:id="echoid-s5276" xml:space="preserve">Secundo enim quadratumita locandum eſt, vt A, cen-<lb/>trum dioptræſit in A, & </s> <s xml:id="echoid-s5277" xml:space="preserve">latus A D, lateri pueri A B, adhæreat; </s> <s xml:id="echoid-s5278" xml:space="preserve"><lb/>Adeo vt per dioptram puncto C, inſpecto, radius viſualis ab <lb/>hypotenuſa A C, non differat. </s> <s xml:id="echoid-s5279" xml:space="preserve">Itaque ſi fiat. </s> <s xml:id="echoid-s5280" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-165-03a" xlink:href="note-165-03"/> inuenietur hypotenuſa A C. </s> <s xml:id="echoid-s5281" xml:space="preserve">Inuenta deinde portione dioptræ A E, in ſecundo <lb/>quadrato, vt in ſcholio problem. </s> <s xml:id="echoid-s5282" xml:space="preserve">7. </s> <s xml:id="echoid-s5283" xml:space="preserve">docuimus: </s> <s xml:id="echoid-s5284" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Si rurſus fiat, <anchor type="note" xlink:label="note-165-04a" xlink:href="note-165-04"/> <anchor type="note" xlink:label="note-165-05a" xlink:href="note-165-05"/> prodibit altitudo, ſiue profunditas A B.</s> <s xml:id="echoid-s5285" xml:space="preserve"/> </p> <div xml:id="echoid-div348" type="float" level="2" n="4"> <figure xlink:label="fig-165-01" xlink:href="fig-165-01a"> <image file="165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/165-01"/> </figure> <note style="it" position="right" xlink:label="note-165-03" xlink:href="note-165-03a" xml:space="preserve"> <lb/>Vt eb, vmbra verſa # ad lat{us} ba, # ita lat{us}a A, # ad aliud, <lb/></note> <note symbol="a" position="right" xlink:label="note-165-04" xlink:href="note-165-04a" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> <note style="it" position="right" xlink:label="note-165-05" xlink:href="note-165-05a" xml:space="preserve"> <lb/>Vt portio dioptræ A E, \\ iuuenta # ad hypotenuſam inuen- \\ tam A C: # {it}a lat{us} \\ A D, # ad aliud, <lb/></note> </div> <p> <s xml:id="echoid-s5286" xml:space="preserve">4. </s> <s xml:id="echoid-s5287" xml:space="preserve"><emph style="sc">Qvod</emph> ſi latitudo orificij A M, vel fundi B C, cognita fuerit, quæ facile <lb/>per aliquam menſuram cognoſci poterit facilius per vnam duntaxat ſtationem <lb/>in D, factam, & </s> <s xml:id="echoid-s5288" xml:space="preserve">per vnicam applicationem quadrati, profunditatem A B, conij-<lb/>ciemus. </s> <s xml:id="echoid-s5289" xml:space="preserve">Nam ſi fiat, <lb/> <anchor type="note" xlink:label="note-165-06a" xlink:href="note-165-06"/> pro ducetur AB, profunditas nota in partibus latitudinis.</s> <s xml:id="echoid-s5290" xml:space="preserve"/> </p> <div xml:id="echoid-div349" type="float" level="2" n="5"> <note style="it" position="right" xlink:label="note-165-06" xlink:href="note-165-06a" xml:space="preserve"> <lb/>Vt vmbra recta, ſi ea abſciſſa \\ fuerit, # ad lat{us} quadrati: # Ita latitudo cogni- \\ ta B C, # ad AB, <lb/>#### Vel <lb/>Vt lat{us} qua- \\ drati # ad vmbram verſam, ſiea \\ fuerit abſciſſa: # Ita latitudo cogni- \\ ta B C, # ad AB, <lb/></note> </div> <p> <s xml:id="echoid-s5291" xml:space="preserve">Et ſi forte dioptra per punctum C, in quadrato tranſeat, erit latitudo BC, re-<lb/>ctæ A B, æqualis.</s> <s xml:id="echoid-s5292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5293" xml:space="preserve"><emph style="sc">Hæ</emph> porro praxes demonſtratæ ſunt omnes in prædicto problemate 9. </s> <s xml:id="echoid-s5294" xml:space="preserve">hac <lb/>vltima Num. </s> <s xml:id="echoid-s5295" xml:space="preserve">4. </s> <s xml:id="echoid-s5296" xml:space="preserve">excepta, quamin problemate 8. </s> <s xml:id="echoid-s5297" xml:space="preserve">Num. </s> <s xml:id="echoid-s5298" xml:space="preserve">4. </s> <s xml:id="echoid-s5299" xml:space="preserve">demonſtrauimus.</s> <s xml:id="echoid-s5300" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5301" xml:space="preserve">PROFVNDITATEM vallis, eiuſdemque deſcenſum obliquum, <lb/>ſi non ſit valde inæqualis, & </s> <s xml:id="echoid-s5302" xml:space="preserve">eius terminus, vel aliquod in ea ſignum <lb/>conſpici poſſit, per quadratum cognoſcere.</s> <s xml:id="echoid-s5303" xml:space="preserve"/> </p> <pb o="163" file="166" n="166" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div351" type="section" level="1" n="146"> <head xml:id="echoid-head149" xml:space="preserve">PROBLEMA XXVIII.</head> <p> <s xml:id="echoid-s5304" xml:space="preserve">1. </s> <s xml:id="echoid-s5305" xml:space="preserve">Hoc etiam aliud nihil eſt, niſi altitu dinem quampiam ex eius ſummo fa-<lb/>ſtigio per duas ſtationes in haſta aliqua erecta dimetiri, vt in problem. </s> <s xml:id="echoid-s5306" xml:space="preserve">9. </s> <s xml:id="echoid-s5307" xml:space="preserve">factum <lb/>eſt, & </s> <s xml:id="echoid-s5308" xml:space="preserve">in præcedenti problemate repetitum. </s> <s xml:id="echoid-s5309" xml:space="preserve">Sit enim vallis inter duos montes <lb/>AB, NG, poſita, & </s> <s xml:id="echoid-s5310" xml:space="preserve">terminus ipſius C, ex monte A B, conſpici poſsit, Erecta ha-<lb/> <anchor type="figure" xlink:label="fig-166-01a" xlink:href="fig-166-01"/> ſta A E, in qua duæ ſtationes oculimenſoris fieri poſsint in D, E, ſi reliqua con-<lb/>ſtruantur, vt in problemate 24. </s> <s xml:id="echoid-s5311" xml:space="preserve">lib. </s> <s xml:id="echoid-s5312" xml:space="preserve">2. </s> <s xml:id="echoid-s5313" xml:space="preserve">cuius hic figuram iterauimus; </s> <s xml:id="echoid-s5314" xml:space="preserve">inueſtigabi-<lb/>tur recta EB, vtin 1. </s> <s xml:id="echoid-s5315" xml:space="preserve">figura præcedentis probl. </s> <s xml:id="echoid-s5316" xml:space="preserve">recta EB, fuit inuenta. </s> <s xml:id="echoid-s5317" xml:space="preserve">Etſi dema-<lb/>tur portio haſtæ A E, altitudo montis A B, vel profunditas vallis H C, reliqua <lb/>fiet nota.</s> <s xml:id="echoid-s5318" xml:space="preserve"/> </p> <div xml:id="echoid-div351" type="float" level="2" n="1"> <figure xlink:label="fig-166-01" xlink:href="fig-166-01a"> <image file="166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/166-01"/> </figure> </div> <p> <s xml:id="echoid-s5319" xml:space="preserve">2. </s> <s xml:id="echoid-s5320" xml:space="preserve"><emph style="sc">Descensvs</emph> autem obliquus IC, ita colligetur. </s> <s xml:id="echoid-s5321" xml:space="preserve">Per vmbram à dioptra <lb/>ad C, directa abſciſſam eliciatur angulus BDC, ex 1. </s> <s xml:id="echoid-s5322" xml:space="preserve">problemate, hoc eſt, angu-<lb/>lus CIM, <anchor type="note" xlink:href="" symbol="a"/> qui ei æqualis eſt, externus interno; </s> <s xml:id="echoid-s5323" xml:space="preserve">eritque proinde reliquus angu- <anchor type="note" xlink:label="note-166-01a" xlink:href="note-166-01"/> lus I C M, notus, vtpote illius complementum. </s> <s xml:id="echoid-s5324" xml:space="preserve">Quocirca cum in triangulo <lb/>C I M, anguli acuti cogniti ſint, vna cum latere I M, nuper inuento; </s> <s xml:id="echoid-s5325" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> baſis I C, <anchor type="note" xlink:label="note-166-02a" xlink:href="note-166-02"/> ignorari non poterit.</s> <s xml:id="echoid-s5326" xml:space="preserve"/> </p> <div xml:id="echoid-div352" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-166-01" xlink:href="note-166-01a" xml:space="preserve">29. primi.</note> <note symbol="b" position="left" xlink:label="note-166-02" xlink:href="note-166-02a" xml:space="preserve">5. triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s5327" xml:space="preserve"><emph style="sc">Sed</emph> hæc omnia facilius per vnicam ſtationem inueſtigabuntur, ſi in D, pun-<lb/>cto haſtæ quadratum bis applicetur, vt in præcedenti problemate Num. </s> <s xml:id="echoid-s5328" xml:space="preserve">2. </s> <s xml:id="echoid-s5329" xml:space="preserve">& </s> <s xml:id="echoid-s5330" xml:space="preserve">in <lb/>ſcholio problem. </s> <s xml:id="echoid-s5331" xml:space="preserve">9. </s> <s xml:id="echoid-s5332" xml:space="preserve">factum eſt in puncto A, &</s> <s xml:id="echoid-s5333" xml:space="preserve">c.</s> <s xml:id="echoid-s5334" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5335" xml:space="preserve">3. </s> <s xml:id="echoid-s5336" xml:space="preserve"><emph style="sc">Qvod</emph> ſi terminus C, non cernatur, eligatur aliquod ſignum K, in valle, <lb/>quod ex ſtationibus D, & </s> <s xml:id="echoid-s5337" xml:space="preserve">E, appareat. </s> <s xml:id="echoid-s5338" xml:space="preserve">Ita enim per problema 9. </s> <s xml:id="echoid-s5339" xml:space="preserve">vel potius per <lb/>eius ſcholium, eadem altitudo I M, vel H C, inuenietur.</s> <s xml:id="echoid-s5340" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5341" xml:space="preserve"><emph style="sc">Immo</emph> ſi in plano vallis duæ ſtationes commode fieri poſsint, percipietur <lb/>ex illis eadem altitudo I M, per problema 6. </s> <s xml:id="echoid-s5342" xml:space="preserve">vel per ſcholium problem. </s> <s xml:id="echoid-s5343" xml:space="preserve">7. </s> <s xml:id="echoid-s5344" xml:space="preserve">De-<lb/>ſcenſus verò obliquus IC, tanquam interuallũ inter duo ſigna I, & </s> <s xml:id="echoid-s5345" xml:space="preserve">C, per pro-<lb/>blema 16. </s> <s xml:id="echoid-s5346" xml:space="preserve">indagandus erit.</s> <s xml:id="echoid-s5347" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5348" xml:space="preserve">4. </s> <s xml:id="echoid-s5349" xml:space="preserve"><emph style="sc">Eandem</emph> denique profunditatem C H, venarilicebit ex altiore monte <lb/>N G, ſi modo terminus C, minoris montis, vel aliquod ſignum in valle appare-<lb/>at ex cacumine N: </s> <s xml:id="echoid-s5350" xml:space="preserve">non aliter, quam in problemate 22. </s> <s xml:id="echoid-s5351" xml:space="preserve">vel 23. </s> <s xml:id="echoid-s5352" xml:space="preserve">minorem alti- <pb o="137" file="167" n="167" rhead="LIBER TERTIVS."/> tudinem ex maiore deprehendimus. </s> <s xml:id="echoid-s5353" xml:space="preserve">Nam hic minor altitudo inquirenda eſt <lb/>CH, & </s> <s xml:id="echoid-s5354" xml:space="preserve">maior NG, ex cuius vertice N, terminum C, videri poſſe ſtatuimus.</s> <s xml:id="echoid-s5355" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5356" xml:space="preserve">DISTANTIAM inter pedes menſoris, & </s> <s xml:id="echoid-s5357" xml:space="preserve">ſignum aliquod in plano <lb/>Horizontis beneficio baculi metiri, quando extremus terminus di-<lb/>ſtantiæ videri poteſt.</s> <s xml:id="echoid-s5358" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div354" type="section" level="1" n="147"> <head xml:id="echoid-head150" xml:space="preserve">PROBLEMA XXIX.</head> <p> <s xml:id="echoid-s5359" xml:space="preserve">1. </s> <s xml:id="echoid-s5360" xml:space="preserve"><emph style="sc">Absolvtis</emph> dimenſionibus, quæ per quadrantem, & </s> <s xml:id="echoid-s5361" xml:space="preserve">quadratum ſieri <lb/>ſolent, libet nonnullas alias rationes dimetiendi a diungere, vt illis, quando ne-<lb/>que quadrans, neque quadratum adeſt, vti poſsimus. </s> <s xml:id="echoid-s5362" xml:space="preserve">Ex pluribus autem me-<lb/>dis illis ſolum ſeligemus, quo faciliorem vſum habent.</s> <s xml:id="echoid-s5363" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5364" xml:space="preserve"><emph style="sc">Sit</emph> ergo diſtantia metienda D B. </s> <s xml:id="echoid-s5365" xml:space="preserve">In <lb/> <anchor type="figure" xlink:label="fig-167-01a" xlink:href="fig-167-01"/> D, erigatur baculus DE, minor altitudine <lb/>AC, ab oculo menſoris ad pedes, rectus <lb/>ad Horizontem. </s> <s xml:id="echoid-s5366" xml:space="preserve">quod fiet, ſi filum cum <lb/>perpendiculo baculo adhærebit, vella-<lb/>pillus ex E, demiſſus in punctum D, ca-<lb/>det. </s> <s xml:id="echoid-s5367" xml:space="preserve">Deinde retro cedat menſor vſque ad <lb/>A, donec radius viſualis ex C, prodiens, <lb/>& </s> <s xml:id="echoid-s5368" xml:space="preserve">per extremum E, baculitranſiens oc-<lb/>currat puncto B; </s> <s xml:id="echoid-s5369" xml:space="preserve">intelligaturque duci recta E F, ipſi A B, parallela. </s> <s xml:id="echoid-s5370" xml:space="preserve">Quoniam <lb/>igitur triangula C F E, E D B, æquiangula ſunt; </s> <s xml:id="echoid-s5371" xml:space="preserve">quodanguli F, D, ſintrecti, <anchor type="note" xlink:href="" symbol="a"/> &</s> <s xml:id="echoid-s5372" xml:space="preserve"> <anchor type="note" xlink:label="note-167-01a" xlink:href="note-167-01"/> ECF, BED, æquales, internus, & </s> <s xml:id="echoid-s5373" xml:space="preserve">externus, &</s> <s xml:id="echoid-s5374" xml:space="preserve">c. </s> <s xml:id="echoid-s5375" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Siigitur fiat.</s> <s xml:id="echoid-s5376" xml:space="preserve"> <anchor type="note" xlink:label="note-167-02a" xlink:href="note-167-02"/> <anchor type="note" xlink:label="note-167-03a" xlink:href="note-167-03"/> nota prodibit diſtantia quæſita D B, in partibus baculi D E, vel ſtaturæ menſoris<unsure/> <lb/>AC. </s> <s xml:id="echoid-s5377" xml:space="preserve">Debent enim baculus, & </s> <s xml:id="echoid-s5378" xml:space="preserve">ſtatura menſoris per vnam eandemque menſu-<lb/>ram eſſe cognita.</s> <s xml:id="echoid-s5379" xml:space="preserve"/> </p> <div xml:id="echoid-div354" type="float" level="2" n="1"> <figure xlink:label="fig-167-01" xlink:href="fig-167-01a"> <image file="167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/167-01"/> </figure> <note symbol="a" position="right" xlink:label="note-167-01" xlink:href="note-167-01a" xml:space="preserve">29. primi.</note> <note symbol="b" position="right" xlink:label="note-167-02" xlink:href="note-167-02a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-167-03" xlink:href="note-167-03a" xml:space="preserve"> <lb/>Vt CF, differentia inter ba- \\ culum D E, & menſoris ſta- \\ tur am AC. # ad FE, ſpatium inter \\ menſerem & baculum: # Ita E D, lon- \\ gitudo bacu- \\ linoti. # ad D B, <lb/></note> </div> <p> <s xml:id="echoid-s5380" xml:space="preserve">ALTITVDINEM turris, aut alterius rei per baculum indagare.</s> <s xml:id="echoid-s5381" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div356" type="section" level="1" n="148"> <head xml:id="echoid-head151" xml:space="preserve">PROBLEMA XXX.</head> <p> <s xml:id="echoid-s5382" xml:space="preserve">1. </s> <s xml:id="echoid-s5383" xml:space="preserve"><emph style="sc">Sit</emph> in figura præcedentis problematis metienda altitudo A C. </s> <s xml:id="echoid-s5384" xml:space="preserve">Figatur <lb/>in terra baculus G H, rectus ad Horizontem, & </s> <s xml:id="echoid-s5385" xml:space="preserve">aliquãtulum maior ſtatura mẽ-<lb/>ſoris ab oculo ad pedes quæ ſit IK. </s> <s xml:id="echoid-s5386" xml:space="preserve">Deinderetro cedat menſor vſque ad I, ita vt <lb/>eius oculus in K, conſtitutus faſtigium C, inſpiciat: </s> <s xml:id="echoid-s5387" xml:space="preserve">intelligatur que ducta recta <lb/> <anchor type="note" xlink:label="note-167-04a" xlink:href="note-167-04"/> KL, Horizonti AB, parallela, ſecans baculumin M. </s> <s xml:id="echoid-s5388" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quoniam igitur triangu- la KMH, KLC, ſimilia ſunt, propter parallelas M H, L C: </s> <s xml:id="echoid-s5389" xml:space="preserve">ſi fiat, <pb o="138" file="168" n="168" rhead="GEOMETR. PRACT."/> <anchor type="figure" xlink:label="fig-168-01a" xlink:href="fig-168-01"/> <anchor type="note" xlink:label="note-168-01a" xlink:href="note-168-01"/> producetur recta L C, cuiſi adijcietur ſtatura menſoris A L, tota altitudo pro-<lb/>poſita A C, cognita erit in partibus ſtaturæ menſoris, vel diſtantiæ K L.</s> <s xml:id="echoid-s5390" xml:space="preserve"/> </p> <div xml:id="echoid-div356" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-167-04" xlink:href="note-167-04a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> <figure xlink:label="fig-168-01" xlink:href="fig-168-01a"> <image file="168-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/168-01"/> </figure> <note style="it" position="right" xlink:label="note-168-01" xlink:href="note-168-01a" xml:space="preserve"> <lb/>Vt K M, ſpatium inter men- \\ ſorem, & baculum, # ad M H, differentiam \\ inter baculum & ſta- \\ tur am menſoris: # Ita diſtantia K L, \\ quæ nota fiat per \\ aliquam menſu- \\ ram, # ad L C, <lb/></note> </div> <p> <s xml:id="echoid-s5391" xml:space="preserve">2. </s> <s xml:id="echoid-s5392" xml:space="preserve"><emph style="sc">Qvod</emph> ſi acceſſus non pateat ad altitudinem A C, vt diſtantiam IA, me-<lb/>tiri poſsimus, figatur idem baculus, vel alius illi æqualis N O, accedendo vide-<lb/>licet propius ad altitudinem. </s> <s xml:id="echoid-s5393" xml:space="preserve">Deinde menſor retrocedat ad P, vt eius oculus in <lb/>Q@exiſtens per ſummitatem baculi O, iterum ſaſtigium C, videat: </s> <s xml:id="echoid-s5394" xml:space="preserve">ſeceturque <lb/> <anchor type="note" xlink:label="note-168-02a" xlink:href="note-168-02"/> baculus NO, à recta KL in R. </s> <s xml:id="echoid-s5395" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quia eſt, vt KM, ad MH, ita KL, ad L C: </s> <s xml:id="echoid-s5396" xml:space="preserve">Item <anchor type="note" xlink:label="note-168-03a" xlink:href="note-168-03"/> vt QR, ad RO, ita QL, ad LC: </s> <s xml:id="echoid-s5397" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Eſt autem maior proportio KL, ad L C, quam QL, ad eandem LC; </s> <s xml:id="echoid-s5398" xml:space="preserve">erit quo que maior proportio K M, ad M H, quam Q R, ad <lb/>RO, ipſi MH, æqualem. </s> <s xml:id="echoid-s5399" xml:space="preserve">(Cum enim & </s> <s xml:id="echoid-s5400" xml:space="preserve">totæ G H, N O, & </s> <s xml:id="echoid-s5401" xml:space="preserve">ablatæ G M, N R, æ-<lb/> <anchor type="note" xlink:label="note-168-04a" xlink:href="note-168-04"/> quales ſint, erunt quo que reliquæ M H, R O, æquales) <anchor type="note" xlink:href="" symbol="c"/> Igitur K M, maior erit, quàm QR. </s> <s xml:id="echoid-s5402" xml:space="preserve">Abſciſſa ergo K T, æquali ipſi QR, <anchor type="note" xlink:href="" symbol="d"/> quoniam eſt, vt K M, ad M H, <anchor type="note" xlink:label="note-168-05a" xlink:href="note-168-05"/> ita KL, ad LC; </s> <s xml:id="echoid-s5403" xml:space="preserve">erit permutando, vt KM, tota ad totam K L, ita M H, ad L C. </s> <s xml:id="echoid-s5404" xml:space="preserve">Ea-<lb/>demratione erit QR, hoc eſt, KT, ablata ex K M, ad QL, ablatã ex KL, vt RO, <lb/>ad L C, hoc eſt, vt M H, ad L C. </s> <s xml:id="echoid-s5405" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s5406" xml:space="preserve">reliqua T M, ad reliquam K Q erit vt <lb/>tota KM, ad totam KL: </s> <s xml:id="echoid-s5407" xml:space="preserve">Vt autem K M, ad K L, ita paulo ante oſtendimus eſſe <lb/>MH, ad LC. </s> <s xml:id="echoid-s5408" xml:space="preserve">Quapropter ſi fiat. <lb/></s> <s xml:id="echoid-s5409" xml:space="preserve"> <anchor type="note" xlink:label="note-168-06a" xlink:href="note-168-06"/> efficietur nota LC, cuiſi addetur ſtatura menſoris AL, nota quoq; </s> <s xml:id="echoid-s5410" xml:space="preserve">euadet pro-<lb/>poſita altitudo AC.</s> <s xml:id="echoid-s5411" xml:space="preserve"/> </p> <div xml:id="echoid-div357" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-168-02" xlink:href="note-168-02a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="left" xlink:label="note-168-03" xlink:href="note-168-03a" xml:space="preserve">8. quinti.</note> <note symbol="c" position="left" xlink:label="note-168-04" xlink:href="note-168-04a" xml:space="preserve">10. quinti.</note> <note symbol="d" position="left" xlink:label="note-168-05" xlink:href="note-168-05a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-168-06" xlink:href="note-168-06a" xml:space="preserve"> <lb/>Vt T M, differentia inter ſpatia \\ K M, & Q R, vel K T, à \\ menſore vſque ad baculos, # ad K Q, diffe- \\ rentiam ſta- \\ tionum. # Ita MH, differentia \\ inter baculum & \\ menſoris ſtatur am, # ad L C, <lb/></note> </div> <p> <s xml:id="echoid-s5412" xml:space="preserve">3. </s> <s xml:id="echoid-s5413" xml:space="preserve"><emph style="sc">Iam</emph> ſi prima ſtatio fiatin Q. </s> <s xml:id="echoid-s5414" xml:space="preserve">& </s> <s xml:id="echoid-s5415" xml:space="preserve">ſecunda in K, recedendo magis ab altitu-<lb/>dine, non variabitur praxis & </s> <s xml:id="echoid-s5416" xml:space="preserve">demonſtratio, vt liquet.</s> <s xml:id="echoid-s5417" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5418" xml:space="preserve">DISTANTIAM in plano Horizontis inter menſorem, & </s> <s xml:id="echoid-s5419" xml:space="preserve">ſignum <lb/>quoduis beneficio normæ adinuenire.</s> <s xml:id="echoid-s5420" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div359" type="section" level="1" n="149"> <head xml:id="echoid-head152" xml:space="preserve">PROBLEMA XXXI.</head> <p> <s xml:id="echoid-s5421" xml:space="preserve">1. </s> <s xml:id="echoid-s5422" xml:space="preserve"><emph style="sc">Proposita</emph> ſit diſtantia AB. </s> <s xml:id="echoid-s5423" xml:space="preserve">In A, loco menſoris figatur ad angulos re-<lb/>ctosbaculus A C, paulò minor ſtatura menſoris, & </s> <s xml:id="echoid-s5424" xml:space="preserve">per menſurã, per quam mẽ-<lb/>foris ſtatura cognita eſt, notus. </s> <s xml:id="echoid-s5425" xml:space="preserve">Angulus deinde rectus normæ C, ſummitati ba- <pb o="139" file="169" n="169" rhead="LIBER TERTIVS."/> culi applicetur, & </s> <s xml:id="echoid-s5426" xml:space="preserve">eius latus CE, circa C, paulatim at-<lb/> <anchor type="figure" xlink:label="fig-169-01a" xlink:href="fig-169-01"/> tollatur, deprimaturue, donec radius viſualis per la-<lb/>tus interius normæ C E, incedens ſeratur in punctum <lb/>extremum B, diligenter que notetur punctum F, in <lb/>quod radius viſualis per alterum latus interius C D, <lb/>tranſiens incidit. </s> <s xml:id="echoid-s5427" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quoniam A C, media eſt pro- <anchor type="note" xlink:label="note-169-01a" xlink:href="note-169-01"/> portio nalis inter A F, quæ in partibus baculi nota effi-<lb/>ciatur, & </s> <s xml:id="echoid-s5428" xml:space="preserve">diſtantiam AB: </s> <s xml:id="echoid-s5429" xml:space="preserve">ſi fiat, <lb/> <anchor type="note" xlink:label="note-169-02a" xlink:href="note-169-02"/> Hoc eſt, ſi quadratus numerus baculi per A F, diuidatur, prodibitin Quotiente <lb/>diſtantia A B, nota in partibus baculi.</s> <s xml:id="echoid-s5430" xml:space="preserve"/> </p> <div xml:id="echoid-div359" type="float" level="2" n="1"> <figure xlink:label="fig-169-01" xlink:href="fig-169-01a"> <image file="169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/169-01"/> </figure> <note symbol="a" position="right" xlink:label="note-169-01" xlink:href="note-169-01a" xml:space="preserve">coroll. @. <lb/>ſexti.</note> <note style="it" position="right" xlink:label="note-169-02" xlink:href="note-169-02a" xml:space="preserve"> <lb/>Vt A F, inter baculum, & pun- \\ ctum F, cognita # ad baculum \\ A C: # Ita bacul{us}<unsure/> A C, # ad A B, <lb/></note> </div> <p> <s xml:id="echoid-s5431" xml:space="preserve">2. </s> <s xml:id="echoid-s5432" xml:space="preserve"><emph style="sc">Non</emph> absre ſoret, ſi duo clauiculi in vtroque latere normæ interioriaffi-<lb/>gerentur, vt radius viſualis per illos incedens rectius in B, & </s> <s xml:id="echoid-s5433" xml:space="preserve">F, feratur.</s> <s xml:id="echoid-s5434" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5435" xml:space="preserve">ALTITVDINEM turris, aut alterius rei per normam inueſtigare.</s> <s xml:id="echoid-s5436" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div361" type="section" level="1" n="150"> <head xml:id="echoid-head153" xml:space="preserve">PROBLEMA XXXII.</head> <p> <s xml:id="echoid-s5437" xml:space="preserve">1. </s> <s xml:id="echoid-s5438" xml:space="preserve">IN ſigura præcedentis problematis ſit metienda altitudo G H. </s> <s xml:id="echoid-s5439" xml:space="preserve">Figatur <lb/>rurſus in A, loco menſoris baculus AC, cuius ſummitati C, angulus rectus nor-<lb/>mæ applicetur, eiuſq; </s> <s xml:id="echoid-s5440" xml:space="preserve">latus CE, paulatim eleuetur, deprimaturue, donec per la-<lb/>tus interius CE, vertex altitudinis H, conſpiciatur punctumq; </s> <s xml:id="echoid-s5441" xml:space="preserve">F, in quo alterum <lb/>latus CD, incurrit, notetur. </s> <s xml:id="echoid-s5442" xml:space="preserve">Et quoniam ducta recta CI, Horizonti parallela, tri-<lb/>angula ACF, HCI, æquiãgula ſunt; </s> <s xml:id="echoid-s5443" xml:space="preserve">ꝙ anguli A, I, recti ſint, & </s> <s xml:id="echoid-s5444" xml:space="preserve">ACF, HCI, reliqui <lb/>ex rectis ACI, HCD, (ſi nimirum communis auſeratur DCI,) æquales: </s> <s xml:id="echoid-s5445" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ſi fiat.</s> <s xml:id="echoid-s5446" xml:space="preserve"> <anchor type="note" xlink:label="note-169-03a" xlink:href="note-169-03"/> <anchor type="note" xlink:label="note-169-04a" xlink:href="note-169-04"/> procreabitur IH, nota, ad quam ſi adij cietur longitudo baculi AC, vel GI, pro-<lb/>poſita altitudo GH, cognita erit.</s> <s xml:id="echoid-s5447" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-169-03" xlink:href="note-169-03a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-169-04" xlink:href="note-169-04a" xml:space="preserve"> <lb/>Vt bacul{us} \\ A C, # ad A F, inter bacu- \\ lum, & punctum \\ F, cognitam # Ita diſtantia CI, vel A G, quæ \\ nota ſiat per aliquam menſu- \\ ram. # ad IH, <lb/></note> </div> <p> <s xml:id="echoid-s5448" xml:space="preserve">DISTANTIAM in plano Horizontis, quæ non ſit valde magna, alio <lb/>modo facillimo dimetiri.</s> <s xml:id="echoid-s5449" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div363" type="section" level="1" n="151"> <head xml:id="echoid-head154" xml:space="preserve">PROBLEMA XXXIII.</head> <p> <s xml:id="echoid-s5450" xml:space="preserve">1. </s> <s xml:id="echoid-s5451" xml:space="preserve"><emph style="sc">Qvando</emph> planum aliquod exiguam habet longitudinem, ſed tamen <lb/>menſurari non poteſt, ob aliquod impedimentum, cuiuſmodi ſunt latitu dines <lb/> <anchor type="figure" xlink:label="fig-169-02a" xlink:href="fig-169-02"/> fluminum, & </s> <s xml:id="echoid-s5452" xml:space="preserve">ſtagnorum quorum latitudines, quas <lb/>nunc pro longitu dinibus accipimus, menſurarine-<lb/>que<unsure/>unt, ob interiectã aquam, vtemur inter alia hoc <lb/>etiam artificio. </s> <s xml:id="echoid-s5453" xml:space="preserve">Properipam fluminis, aut ſtagni, vel <lb/>alterius cuiuſcunque reimetiendæ, figatur baculus <lb/>A B, ad Horizontem rectus cui in puncto B, accõ- <pb o="140" file="170" n="170" rhead="GEOMETR. PRACT."/> modetur virgula C D, ita vt circa B, deprimi aut eleuari poſsit, donec radius vi-<lb/>ſualis per CD, tranſiens occurrat extremo E, diſtantiæ A E, metiendæ. </s> <s xml:id="echoid-s5454" xml:space="preserve">Deinde <lb/>virgula C D, manente immobili, ne angulus D B A, mutetur, vertatur baculus <lb/>AC, ad rectos ſemper angulos Horizonti inſiſtens, donec radius viſualis per e-<lb/>andem virgulam CD, incedens occurrat plano alicui prope flumen, aut ſtagnũ, <lb/>in quo nimirum menſor exiſtit, & </s> <s xml:id="echoid-s5455" xml:space="preserve">quod menſuraripoſsit, in puncto F. </s> <s xml:id="echoid-s5456" xml:space="preserve">Quoniam <lb/>ergo duo anguli A, B, trianguli ABE, duobus angulis A, B, trianguli ABF, æqua-<lb/>les ſunt, & </s> <s xml:id="echoid-s5457" xml:space="preserve">latus AB, commune: </s> <s xml:id="echoid-s5458" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erunt latera AE, AF, æqualia: </s> <s xml:id="echoid-s5459" xml:space="preserve">atque idcirco, <anchor type="note" xlink:label="note-170-01a" xlink:href="note-170-01"/> ſi AF, in plano per aliquam menſuram efficiatur nota; </s> <s xml:id="echoid-s5460" xml:space="preserve">diſtantia quo que AE, co-<lb/>gnita erit.</s> <s xml:id="echoid-s5461" xml:space="preserve"/> </p> <div xml:id="echoid-div363" type="float" level="2" n="1"> <figure xlink:label="fig-169-02" xlink:href="fig-169-02a"> <image file="169-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/169-02"/> </figure> <note symbol="a" position="left" xlink:label="note-170-01" xlink:href="note-170-01a" xml:space="preserve">26. primi.</note> </div> <p> <s xml:id="echoid-s5462" xml:space="preserve">2. </s> <s xml:id="echoid-s5463" xml:space="preserve"><emph style="sc">Alii</emph> hoc ipſum efficiunt ſine baculo, hac ratione. </s> <s xml:id="echoid-s5464" xml:space="preserve">Erigunt ſe ad angu-<lb/>losrectos cum Horizonte: </s> <s xml:id="echoid-s5465" xml:space="preserve">Deinde deprimunt pileum, qui ſit aliquantulũ la-<lb/>tus, donecradius viſualis per extremitatem pilei excurrens in cidat in terminum <lb/>E, diſtantiæ AE, metiendæ: </s> <s xml:id="echoid-s5466" xml:space="preserve">Pileo vero immobili manente, vertunt ſe verſus pla-<lb/>num aliquod menſurabile, notantque in eo punctum F, in quo radius viſualis <lb/>ipſi plano occurrit. </s> <s xml:id="echoid-s5467" xml:space="preserve">Nam rurſus longitudo AF, quæ per menſuram aliquam co-<lb/>gnoſcenda eſt, diſtantiæ A E, propoſitæ æqualis eſt: </s> <s xml:id="echoid-s5468" xml:space="preserve">propterea quod ſtatura <lb/>menſoris fungitur tunc munere baculi AB, & </s> <s xml:id="echoid-s5469" xml:space="preserve">pileus depreſſus vices gerit virgu-<lb/>læ C D, vt conſtat.</s> <s xml:id="echoid-s5470" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5471" xml:space="preserve">ALTITVDINEM cuiuſque rei erectæ ex eius vmbra, quam Sole lu-<lb/>cente proiicit, ſi nota fuerit, per quadratum deprehendere.</s> <s xml:id="echoid-s5472" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div365" type="section" level="1" n="152"> <head xml:id="echoid-head155" xml:space="preserve">PROBLEMA XXXIV.</head> <p> <s xml:id="echoid-s5473" xml:space="preserve">1. </s> <s xml:id="echoid-s5474" xml:space="preserve">Hoc problema à quinto non differt, niſi quòd hic vtimur radio Solis <lb/>pro radio viſuali, & </s> <s xml:id="echoid-s5475" xml:space="preserve">longitudine vmbræ, quam altitudo, Sole lucente, proiicit, <lb/>pro diſtantia à menſore ad altitu dinem vſque. </s> <s xml:id="echoid-s5476" xml:space="preserve">Sit ergo altitudo FG, vel FI, vel <lb/>FM, eiuſque vmbra proiecta FA. </s> <s xml:id="echoid-s5477" xml:space="preserve">Dirigatur qua dratum verſus Solem (licet hoc <lb/>non fiat è directo altitudinis) & </s> <s xml:id="echoid-s5478" xml:space="preserve">tranſeunte radio Solis per foramina pinnaci-<lb/>diorum, notetur quanta vmbra ſiue recta, ſiue verſa à dioptra, vel filo perpen-<lb/>diculi in quadrato abſcindatur. </s> <s xml:id="echoid-s5479" xml:space="preserve">Nam ita ſeſe ha-<lb/> <anchor type="figure" xlink:label="fig-170-01a" xlink:href="fig-170-01"/> bebit vmbra recta H B, in quadrato ad latus A B, <lb/>vt vmbra proiecta AF, ad altitudinem F G. </s> <s xml:id="echoid-s5480" xml:space="preserve">Item <lb/>ea eſt proportio lateris A D, ad vmbram verſam <lb/>DE, in quadrato, quæ vmbræ proiectæ AF, ad alti-<lb/>tudinem FI. </s> <s xml:id="echoid-s5481" xml:space="preserve">Denique dio ptra, vel filo perpendi-<lb/>culi per punctum C, incedente, tanta erit vmbra <lb/>proiecta A F, quanta eſt altitudo F M. </s> <s xml:id="echoid-s5482" xml:space="preserve">quæ omnia <lb/>in problemate 5. </s> <s xml:id="echoid-s5483" xml:space="preserve">demonſtrata ſunt. </s> <s xml:id="echoid-s5484" xml:space="preserve">Igitur ſi fiat.</s> <s xml:id="echoid-s5485" xml:space="preserve"> <pb o="141" file="171" n="171" rhead="LIBER TERTIVS."/> <anchor type="note" xlink:label="note-171-01a" xlink:href="note-171-01"/> altitudo propoſita F G, vel F I, proueniet nota &</s> <s xml:id="echoid-s5486" xml:space="preserve">c.</s> <s xml:id="echoid-s5487" xml:space="preserve"/> </p> <div xml:id="echoid-div365" type="float" level="2" n="1"> <figure xlink:label="fig-170-01" xlink:href="fig-170-01a"> <image file="170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/170-01"/> </figure> <note style="it" position="right" xlink:label="note-171-01" xlink:href="note-171-01a" xml:space="preserve"> <lb/>Vt vmbrarecta H B, in \\ quadrato abſciſſa # ad lat{us} A B, # Ita longitudo vmbræ pro- \\ iectæ A F, # ad F G, <lb/>#### Vel <lb/>Vt lat{us} A D, # ad vmbram verſam D E, ab- \\ ſciſſam in quadrato # Ita longitudo vmbræ \\ proiectæ A F, # ad F I, <lb/></note> </div> <p> <s xml:id="echoid-s5488" xml:space="preserve">LONGITVDINEM vmbræ ab altitudine, Sole lucente, quando <lb/>altitudo eſt cognita, ope quadrati apertam, & </s> <s xml:id="echoid-s5489" xml:space="preserve">manifeſtam facere.</s> <s xml:id="echoid-s5490" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div367" type="section" level="1" n="153"> <head xml:id="echoid-head156" xml:space="preserve">PROBLEMA XXXV.</head> <figure> <image file="171-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/171-01"/> </figure> <p> <s xml:id="echoid-s5491" xml:space="preserve">1. </s> <s xml:id="echoid-s5492" xml:space="preserve">Hoc etiam problema à ſecundo diuerſum non eſt: </s> <s xml:id="echoid-s5493" xml:space="preserve">ſolum hic pro diſtan-<lb/>tia accipimus longitudinem vmbræ, quam altitudo nota, Sole lucente, proii-<lb/>cit. </s> <s xml:id="echoid-s5494" xml:space="preserve">Sit ergo altitudo A F, eiuſque vmbra proiecta F G. </s> <s xml:id="echoid-s5495" xml:space="preserve">Dirigatur quadratum <lb/>verſus Solem, vbicunque menſor extra vmbram conſtiterit, & </s> <s xml:id="echoid-s5496" xml:space="preserve">tranſeunte ra-<lb/>dio Solari per pinna cidiorum foramina, obſeruetur vmbra ſiue recta, ſiue ver-<lb/>ſa, quam filum perpendiculi, vel dioptra abſcindit. </s> <s xml:id="echoid-s5497" xml:space="preserve">Ita namque ſe habebit latus <lb/>quadrati A B, ad vmbram eius rectam B E, abſciſſam, vt altitudo A F, ad vmbram <lb/>proiectam F G. </s> <s xml:id="echoid-s5498" xml:space="preserve">Item ſic erit vmbra verſa D E, in quadrato ad latus A D, vt al-<lb/>titudo A F, ad vmbram proiectam F G. </s> <s xml:id="echoid-s5499" xml:space="preserve">Dioptra denique, vel filo perpendicu-<lb/>li per punctum C, tranſeunte, altitudo A F, longitu dini vmbræ proiectæ F G, <lb/>æqualis eſt. </s> <s xml:id="echoid-s5500" xml:space="preserve">quæ omnia in problemate 2. </s> <s xml:id="echoid-s5501" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s5502" xml:space="preserve">Quamobrem ſi fiat, <lb/> <anchor type="note" xlink:label="note-171-02a" xlink:href="note-171-02"/> exibit nota longitudo vmbræ proiectæ F G, &</s> <s xml:id="echoid-s5503" xml:space="preserve">c.</s> <s xml:id="echoid-s5504" xml:space="preserve"/> </p> <div xml:id="echoid-div367" type="float" level="2" n="1"> <note style="it" position="right" xlink:label="note-171-02" xlink:href="note-171-02a" xml:space="preserve"> <lb/>Vt lat{us} A B, # ad vmbram rectam abſciſ- \\ ſam B E, # Ita altitudo nota \\ A F, # ad F G: <lb/>#### Vel <lb/>Vt vmbra verſa D E, abſciſſa \\ in quadrato # ad lat{us} A D, # Ita altitudo nota \\ A F, # ad F G, <lb/></note> </div> <p> <s xml:id="echoid-s5505" xml:space="preserve">DISTANTIAM in Horizonte inter menſorem, & </s> <s xml:id="echoid-s5506" xml:space="preserve">ſignum aliquod <lb/>viſum, beneficio ſimpliciſſimi cuiuſdam inſtrumenti comperire.</s> <s xml:id="echoid-s5507" xml:space="preserve"/> </p> <pb o="142" file="172" n="172" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div369" type="section" level="1" n="154"> <head xml:id="echoid-head157" xml:space="preserve">PROBLEMA XXXVI.</head> <p> <s xml:id="echoid-s5508" xml:space="preserve">1. </s> <s xml:id="echoid-s5509" xml:space="preserve"><emph style="sc">Constrvatvr</emph> inſtrumentum hoc modo. </s> <s xml:id="echoid-s5510" xml:space="preserve">Accipiatur baculus rectus <lb/>A B, paulò minor, quàm menſoris ſtatura, diuidaturque in 5. </s> <s xml:id="echoid-s5511" xml:space="preserve">partes, vel etiam <lb/>plures, paucioreſue æquales, & </s> <s xml:id="echoid-s5512" xml:space="preserve">in prima eius parte C, velin alia quacunque, in-<lb/>figatur alius baculus C D, ad rectos angulos, in <lb/> <anchor type="figure" xlink:label="fig-172-01a" xlink:href="fig-172-01"/> quo ſumantur ordine quotcun que palmi, aut pe-<lb/>des. </s> <s xml:id="echoid-s5513" xml:space="preserve">In extremitate quo que baculi A B, infigatur <lb/>alius baculus B E, ad angulos rectos: </s> <s xml:id="echoid-s5514" xml:space="preserve">conſtru-<lb/>ctumque erit inſtrumentum, quo varias magnitu-<lb/>dines licebit metiri, vt patebit: </s> <s xml:id="echoid-s5515" xml:space="preserve">Sit enim primum <lb/>metienda diſtantia B F, cuius terminum F, menſor <lb/>in altero termino B, exiſtens videre poſsit ex A, etiamſi in medio ſint valles, & </s> <s xml:id="echoid-s5516" xml:space="preserve"><lb/>alia impedimenta. </s> <s xml:id="echoid-s5517" xml:space="preserve">Conſtituatur inſtrumentum in tali ſitu, vt A B, ſit ad Hori-<lb/>zontem perpendicularis, & </s> <s xml:id="echoid-s5518" xml:space="preserve">B E, eidem æquidiſtans. </s> <s xml:id="echoid-s5519" xml:space="preserve">Inſpecto extremo F, ex A, <lb/>notetur ſumma cura ac diligentia punctum D, in baculo C D, per quod radius <lb/>viſualis tranſit. </s> <s xml:id="echoid-s5520" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Nam cum triangula A C D, A B F, ſint ſimilia; </s> <s xml:id="echoid-s5521" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ſi fiat, <anchor type="note" xlink:label="note-172-01a" xlink:href="note-172-01"/> <anchor type="note" xlink:label="note-172-02a" xlink:href="note-172-02"/> <anchor type="note" xlink:label="note-172-03a" xlink:href="note-172-03"/> inuenta erit diſtantia B F, in partibus C D. </s> <s xml:id="echoid-s5522" xml:space="preserve">Itaque ſi A C, eſt quinta pars, verbi <lb/>gratia, baculi A B, erit quoque C D, quinta pars diſtantiæ B F. </s> <s xml:id="echoid-s5523" xml:space="preserve">Et quia in noſtro <lb/>exemplo C D, continet 2. </s> <s xml:id="echoid-s5524" xml:space="preserve">pedes, ſi eos multiplicemus per 5. </s> <s xml:id="echoid-s5525" xml:space="preserve">producentur 10. <lb/></s> <s xml:id="echoid-s5526" xml:space="preserve">pedes pro diſtantia B F.</s> <s xml:id="echoid-s5527" xml:space="preserve"/> </p> <div xml:id="echoid-div369" type="float" level="2" n="1"> <figure xlink:label="fig-172-01" xlink:href="fig-172-01a"> <image file="172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/172-01"/> </figure> <note symbol="a" position="left" xlink:label="note-172-01" xlink:href="note-172-01a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> <note style="it" position="right" xlink:label="note-172-02" xlink:href="note-172-02a" xml:space="preserve"> <lb/>Vt A C, # Ad A B, # Ita C D, # ad B F, <lb/></note> <note symbol="b" position="left" xlink:label="note-172-03" xlink:href="note-172-03a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s5528" xml:space="preserve">2. </s> <s xml:id="echoid-s5529" xml:space="preserve"><emph style="sc">Cvrandvm</emph> autem erit diligenter, vt quando radius viſualis non tran-<lb/>ſit per aliquem pedem integrum baculi C D, inueſtigetur, quot decimæ, vel <lb/>centeſimæ vnius pedis in particula abſciſſa contineantur, per ea, quæ lib. </s> <s xml:id="echoid-s5530" xml:space="preserve">1. </s> <s xml:id="echoid-s5531" xml:space="preserve">cap. <lb/></s> <s xml:id="echoid-s5532" xml:space="preserve">2. </s> <s xml:id="echoid-s5533" xml:space="preserve">Num. </s> <s xml:id="echoid-s5534" xml:space="preserve">14. </s> <s xml:id="echoid-s5535" xml:space="preserve">ſcripſimus, quod fiet, ſi in regula C D, decem pedes comprehen-<lb/>dantur. </s> <s xml:id="echoid-s5536" xml:space="preserve">Idemque de quacunque alia men<unsure/>ſura intelligendum eſt, ſi ea in C D, <lb/>loco pedum ſignietur.</s> <s xml:id="echoid-s5537" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5538" xml:space="preserve">3. </s> <s xml:id="echoid-s5539" xml:space="preserve"><emph style="sc">Vervm</emph> hoc eadem facilitate præſtitimus ope quadrati, tum penduli, <lb/>tum ſtabilis.</s> <s xml:id="echoid-s5540" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5541" xml:space="preserve">DISTANTIAM inter duo montium aut turrium cacumina, ope <lb/>prædicti inſtrumenti coniicere.</s> <s xml:id="echoid-s5542" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div371" type="section" level="1" n="155"> <head xml:id="echoid-head158" xml:space="preserve">PROBLEMA XXXVII.</head> <p> <s xml:id="echoid-s5543" xml:space="preserve">1. </s> <s xml:id="echoid-s5544" xml:space="preserve"><emph style="sc">Sint</emph> duo cacumina montium F, G. </s> <s xml:id="echoid-s5545" xml:space="preserve">Po-<lb/> <anchor type="figure" xlink:label="fig-172-02a" xlink:href="fig-172-02"/> ſito puncto B, prædicti inſtrumenti in cacumi-<lb/>ne F, minoris montis, deprimatur baculus A B, <lb/>donec baculus B E, recta in cacumen G, ten-<lb/>dat, quod per duos clauiculos in B, E, infixos, <lb/>facile fiet, vt Num. </s> <s xml:id="echoid-s5546" xml:space="preserve">2. </s> <s xml:id="echoid-s5547" xml:space="preserve">problem. </s> <s xml:id="echoid-s5548" xml:space="preserve">31. </s> <s xml:id="echoid-s5549" xml:space="preserve">diximus. </s> <s xml:id="echoid-s5550" xml:space="preserve">Ma-<lb/>nente in hoc ſitu inſtrumento, inſpiciatur ca-<lb/>cumen G, ex A, obſeruetur que punctum D, interſectionis radij A G, cum bacu-<lb/> <anchor type="note" xlink:label="note-172-04a" xlink:href="note-172-04"/> lo C D. </s> <s xml:id="echoid-s5551" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Nam cumæquiangula ſint triangula A C D, A B G, ſi hat, <pb o="143" file="173" n="173" rhead="LIBER TERTIVS."/> <anchor type="note" xlink:label="note-173-01a" xlink:href="note-173-01"/> cognoſcetur diſtantia B G, in partibus C D. </s> <s xml:id="echoid-s5552" xml:space="preserve">Vt ſi A C, eſt quinta pars ipſius A B, <lb/>& </s> <s xml:id="echoid-s5553" xml:space="preserve">C D, contineat 1. </s> <s xml:id="echoid-s5554" xml:space="preserve">pedem, & </s> <s xml:id="echoid-s5555" xml:space="preserve">inſuper {2/3}. </s> <s xml:id="echoid-s5556" xml:space="preserve">ſi 1 {2/3}. </s> <s xml:id="echoid-s5557" xml:space="preserve">quinquies ſumatur, efficietur di-<lb/>ſtantia B G, pedum 8 {1/3}.</s> <s xml:id="echoid-s5558" xml:space="preserve"/> </p> <div xml:id="echoid-div371" type="float" level="2" n="1"> <figure xlink:label="fig-172-02" xlink:href="fig-172-02a"> <image file="172-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/172-02"/> </figure> <note symbol="c" position="left" xlink:label="note-172-04" xlink:href="note-172-04a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> <note style="it" position="right" xlink:label="note-173-01" xlink:href="note-173-01a" xml:space="preserve"> <lb/>Vt A C, nota in parti- \\ b{us} A B, # ad A B, notarum \\ partium: # Ita C D, nota in data \\ menſura # ad B G, <lb/></note> </div> <p> <s xml:id="echoid-s5559" xml:space="preserve">2. </s> <s xml:id="echoid-s5560" xml:space="preserve"><emph style="sc">Non</emph> aliter diſtantia b F, ex maioris montis cacumine G, mueſtigabitur, <lb/>vt figura indicat.</s> <s xml:id="echoid-s5561" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5562" xml:space="preserve"><emph style="sc">Eodem</emph> modo procedes, ſi puncta F, G, ſint faſtigia duarum turrium, aut ſi <lb/>vnum ſit faſtigium turris, & </s> <s xml:id="echoid-s5563" xml:space="preserve">alterum, cacumen montis, vt conſtat.</s> <s xml:id="echoid-s5564" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5565" xml:space="preserve">3. </s> <s xml:id="echoid-s5566" xml:space="preserve"><emph style="sc">Idem</emph> hoc per quadratum fiet hoc modo. </s> <s xml:id="echoid-s5567" xml:space="preserve">Sit quadratum a b f g, ſtatua-<lb/>turque angulus b, in cacumine G, & </s> <s xml:id="echoid-s5568" xml:space="preserve">latus b f, deprimatur, ita vtrecta in cacu-<lb/>men F, tendat, ſi productum intelligatur, obſeruetur que vmbra verſa abſciſ-<lb/>ſa g h, à radio viſuali a B. </s> <s xml:id="echoid-s5569" xml:space="preserve">Nam quia triangula a g h, B b a, æquiangula ſunt, <lb/> <anchor type="note" xlink:href="" symbol="a"/> ſi fiat.</s> <s xml:id="echoid-s5570" xml:space="preserve"> <anchor type="note" xlink:label="note-173-02a" xlink:href="note-173-02"/> <anchor type="note" xlink:label="note-173-03a" xlink:href="note-173-03"/> hoc eſt, ſi quadratus numerus lateris, nimirum 1000000. </s> <s xml:id="echoid-s5571" xml:space="preserve">diuidatur per vmbram <lb/>verſam, gignetur in Quotiente diſtantia B b.</s> <s xml:id="echoid-s5572" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-173-02" xlink:href="note-173-02a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-173-03" xlink:href="note-173-03a" xml:space="preserve"> <lb/>Vt g h, vmbra \\ verſa # ad g a, lat{us} \\ quadrati # ita lat{us} quadrati \\ a b, # ad B b, <lb/></note> </div> <p> <s xml:id="echoid-s5573" xml:space="preserve"><emph style="sc">Idemqve</emph> fieret, ſi angulus quadrati in B, collocaretur, infimum que latus <lb/>recta à B, in cacumen G, tenderet, &</s> <s xml:id="echoid-s5574" xml:space="preserve">c.</s> <s xml:id="echoid-s5575" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5576" xml:space="preserve">LONGITVDINEM aſcenſus alicuius montis, ſi eius cacumen ab <lb/>oculo in radice conſtituto videatur, eiuſdem inſtrumenti beneficio <lb/>cognoſcere.</s> <s xml:id="echoid-s5577" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div374" type="section" level="1" n="156"> <head xml:id="echoid-head159" xml:space="preserve">PROBLEMA XXXVIII.</head> <p> <s xml:id="echoid-s5578" xml:space="preserve">1. </s> <s xml:id="echoid-s5579" xml:space="preserve"><emph style="sc">Accommodetvr</emph> prædictum inſtrumen-<lb/> <anchor type="figure" xlink:label="fig-173-01a" xlink:href="fig-173-01"/> tum, vt punctum B, in radice montis ſtatuatur, <lb/>& </s> <s xml:id="echoid-s5580" xml:space="preserve">baculus B E, beneficio duorum clauiculorum <lb/>infixorum, vt Num. </s> <s xml:id="echoid-s5581" xml:space="preserve">2. </s> <s xml:id="echoid-s5582" xml:space="preserve">problem. </s> <s xml:id="echoid-s5583" xml:space="preserve">31. </s> <s xml:id="echoid-s5584" xml:space="preserve">dictum eſt, re-<lb/>cta in cacumen F, vergat; </s> <s xml:id="echoid-s5585" xml:space="preserve">notetur que interſe-<lb/>ctio D, radij viſualis cum baculo C D, inſpecto <lb/>cacumine F, ex A. </s> <s xml:id="echoid-s5586" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Nam iterum triangula A C D, <anchor type="note" xlink:label="note-173-04a" xlink:href="note-173-04"/> A B F. </s> <s xml:id="echoid-s5587" xml:space="preserve">Similia erunt. </s> <s xml:id="echoid-s5588" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quare ſi fiat, <anchor type="note" xlink:label="note-173-05a" xlink:href="note-173-05"/> <anchor type="note" xlink:label="note-173-06a" xlink:href="note-173-06"/> cognoſcetur aſcenſus obliquus montis B, F, in partibus C D.</s> <s xml:id="echoid-s5589" xml:space="preserve"/> </p> <div xml:id="echoid-div374" type="float" level="2" n="1"> <figure xlink:label="fig-173-01" xlink:href="fig-173-01a"> <image file="173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/173-01"/> </figure> <note symbol="b" position="right" xlink:label="note-173-04" xlink:href="note-173-04a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> <note symbol="c" position="right" xlink:label="note-173-05" xlink:href="note-173-05a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-173-06" xlink:href="note-173-06a" xml:space="preserve"> <lb/>Vt A C, nota in parti- \\ b{us} A B, # ad A B, notarum \\ partium: # Ita C D, nota in data \\ menſura # ad B F, <lb/></note> </div> <p> <s xml:id="echoid-s5590" xml:space="preserve">2. </s> <s xml:id="echoid-s5591" xml:space="preserve"><emph style="sc">Pari</emph> ratione, ſi F, ſit faſtigium alicuius turris, inueſtigabis diſtantiam à <lb/>puncto B, in terra, vel alibi poſito, vſque ad F, hoc eſt, hypotenuſam B F, vt per-<lb/>ſpicuum eſt. </s> <s xml:id="echoid-s5592" xml:space="preserve">Atque hoc eodem inſtrumento complures aliæ dimenſiones ab-<lb/>ſolui poterunt, quod prudens Lector facilè perſpiciet.</s> <s xml:id="echoid-s5593" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5594" xml:space="preserve">3. </s> <s xml:id="echoid-s5595" xml:space="preserve"><emph style="sc">Idem</emph> aſſequemur quadrato A B G H, ſi eius angulus B, in radice ſta-<lb/>@uatur, & </s> <s xml:id="echoid-s5596" xml:space="preserve">latus B G, eleuetur ita, vtrecta tendat in cacumen F. </s> <s xml:id="echoid-s5597" xml:space="preserve">Obſeruara enim <pb o="144" file="174" n="174" rhead="GEOMETR. PRACT."/> vmbra verſa H I, quam dioptra in cacumen F, directa abſcindit, fiunt trlangula <lb/>A H I, F B A, æquiangula. </s> <s xml:id="echoid-s5598" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quamobrem ſi fiat, <anchor type="note" xlink:label="note-174-01a" xlink:href="note-174-01"/> <anchor type="note" xlink:label="note-174-02a" xlink:href="note-174-02"/> id eſt, ſi quadratus numerus 1000000. </s> <s xml:id="echoid-s5599" xml:space="preserve">lateris diuidatur per vmbram verſam, da-<lb/>bit Quotiens numerus longitudinem aſcenſus obliqui B F.</s> <s xml:id="echoid-s5600" xml:space="preserve"/> </p> <div xml:id="echoid-div375" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-174-01" xlink:href="note-174-01a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-174-02" xlink:href="note-174-02a" xml:space="preserve"> <lb/>Vt H I, vmbra ver- \\ ſa # ad A H, lat{us} qua- \\ drati # Ita lat{us} quadrati \\ A B, # ad B F, <lb/></note> </div> <p> <s xml:id="echoid-s5601" xml:space="preserve">ALTITVDINEM, ad cuius baſem pateat acceſſus, beneficio ſpe-<lb/>culi plani, vna cum diſtantia ſpeculi à cacumine altitudinis depre-<lb/>hendere.</s> <s xml:id="echoid-s5602" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div377" type="section" level="1" n="157"> <head xml:id="echoid-head160" xml:space="preserve">PROBLEMA XXXIX.</head> <p> <s xml:id="echoid-s5603" xml:space="preserve">1. </s> <s xml:id="echoid-s5604" xml:space="preserve"><emph style="sc">Sit</emph> altitudo A B, à cuius baſe B, recedatur per quotuis paſſus, aut pedes, <lb/>vſque ad C, punctum, in quo ſpeculi plani centrum collocetur, & </s> <s xml:id="echoid-s5605" xml:space="preserve">ſecundum <lb/> <anchor type="figure" xlink:label="fig-174-01a" xlink:href="fig-174-01"/> rectam B C, retrocedatur, donec menſoris oculus <lb/>in E, conſtitutus cacumen A, intueri poſsit per ra-<lb/>dium reflexum E C A, ita vt D E, ſit ſtatura menſo-<lb/>ris ab oculo vſque ad planum. </s> <s xml:id="echoid-s5606" xml:space="preserve">Et quoniam an-<lb/>gulus incidentiæ D C E, æqualis eſt angulo refle-<lb/>xionis A C B, vt Perſpectiui docent, & </s> <s xml:id="echoid-s5607" xml:space="preserve">anguli D, <lb/>B, recti ſunt; </s> <s xml:id="echoid-s5608" xml:space="preserve">erunt triangula D C E, B C A, æquian-<lb/>gula; </s> <s xml:id="echoid-s5609" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ideoque erit, vt C D, ad D E, ita C B, ad B A. </s> <s xml:id="echoid-s5610" xml:space="preserve">Quocirca ſi fiat, <anchor type="note" xlink:label="note-174-03a" xlink:href="note-174-03"/> <anchor type="note" xlink:label="note-174-04a" xlink:href="note-174-04"/> producetur altitudo B A, quam quærimus, nota in partibus ſtaturæ menſo-<lb/>ris D E.</s> <s xml:id="echoid-s5611" xml:space="preserve"/> </p> <div xml:id="echoid-div377" type="float" level="2" n="1"> <figure xlink:label="fig-174-01" xlink:href="fig-174-01a"> <image file="174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/174-01"/> </figure> <note symbol="b" position="left" xlink:label="note-174-03" xlink:href="note-174-03a" xml:space="preserve">4. ſexti.</note> <note style="it" position="right" xlink:label="note-174-04" xlink:href="note-174-04a" xml:space="preserve"> <lb/>Vt C D, diſtantia men- \\ ſoris à ſpeculo C, # ad D E, ſtaturam \\ menſoris: # Ita C B, diſtantia ſpecu- \\ li ab altitudine. # ad B A, <lb/></note> </div> </div> <div xml:id="echoid-div379" type="section" level="1" n="158"> <head xml:id="echoid-head161" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s5612" xml:space="preserve">2. </s> <s xml:id="echoid-s5613" xml:space="preserve"><emph style="sc">Mensvretvr</emph> per quadrantem angulus D C E, vel B C A, (Hoc <lb/>fiet, ſi angulus rectus conſtruatur F G H, & </s> <s xml:id="echoid-s5614" xml:space="preserve">recta F G, tot particulas æquales con-<lb/>tineat, quot paſſus, vel pedes in C D, diſtantia continentur: </s> <s xml:id="echoid-s5615" xml:space="preserve">Item recta G H, <lb/>tot particulas eaſdem, quot paſſus aut pedes ſtatura menſoris D E, complecti-<lb/>tur. </s> <s xml:id="echoid-s5616" xml:space="preserve">Iuncta namque recta F H, <anchor type="note" xlink:href="" symbol="c"/> erit angulus F, angulo C, æqualis. </s> <s xml:id="echoid-s5617" xml:space="preserve">quem an- <anchor type="note" xlink:label="note-174-05a" xlink:href="note-174-05"/> gulum F, nullo negotio per quadrantem aliquem in gradus diuiſum cognoſce-<lb/>mus.) </s> <s xml:id="echoid-s5618" xml:space="preserve">Nam ſi poſito ſinu toto C B, fiat. <lb/></s> <s xml:id="echoid-s5619" xml:space="preserve"> <anchor type="note" xlink:label="note-174-06a" xlink:href="note-174-06"/> prodibit altitudo B A, nota in partibus diſtantiæ C B. </s> <s xml:id="echoid-s5620" xml:space="preserve">Et ſi rurſum fiat, <lb/> <anchor type="note" xlink:label="note-174-07a" xlink:href="note-174-07"/> cognita etiam erit C A, diſtantia à ſpeculo C, vſque ad cacumen A, in partibus <lb/>diſtantiæ C B.</s> <s xml:id="echoid-s5621" xml:space="preserve"/> </p> <div xml:id="echoid-div379" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-174-05" xlink:href="note-174-05a" xml:space="preserve">4. primi.</note> <note style="it" position="right" xlink:label="note-174-06" xlink:href="note-174-06a" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} \\ C B, # ad B A, tangentem anguli reflexio- \\ nis B C A, vel incidentiæ D C E, \\ quem proximè inuenim{us} # Ita C B, diſtan- \\ tia cognita # ad B A, <lb/></note> <note style="it" position="right" xlink:label="note-174-07" xlink:href="note-174-07a" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} C B, # ad C A, ſecantem eiuſdem \\ anguli B C A, # Ita C B, diſtan- \\ tia cognita # ad C A, <lb/></note> </div> <pb o="145" file="175" n="175" rhead="LIBER TERTIVS."/> </div> <div xml:id="echoid-div381" type="section" level="1" n="159"> <head xml:id="echoid-head162" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s5622" xml:space="preserve">3. </s> <s xml:id="echoid-s5623" xml:space="preserve"><emph style="sc">Per</emph> ſolos ſinus ita progrediemur, ſi libet, <anchor type="note" xlink:href="" symbol="a"/> Fiat, <anchor type="note" xlink:label="note-175-01a" xlink:href="note-175-01"/> <anchor type="note" xlink:label="note-175-02a" xlink:href="note-175-02"/> Nam productus numerus dabit altitudinem B A, notam in partibus diſtantiæ <lb/> <anchor type="note" xlink:label="note-175-03a" xlink:href="note-175-03"/> C B. </s> <s xml:id="echoid-s5624" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Rurſus fiat, <anchor type="note" xlink:label="note-175-04a" xlink:href="note-175-04"/> Numerus enim proueniens dabit hypotenuſam C A, in partibus diſtantiæ C B, <lb/>cognitam.</s> <s xml:id="echoid-s5625" xml:space="preserve"/> </p> <div xml:id="echoid-div381" type="float" level="2" n="1"> <note style="it" position="right" xlink:label="note-175-01" xlink:href="note-175-01a" xml:space="preserve"> <lb/>Vt ſin{us} anguli A, complemen- \\ ti anguli C, reflexionis, velinci- \\ cidentiæ, # ad C B, diſtantiam \\ cognitam # Ita ſin{us} anguli C, \\ incidentiæ, velre- \\ flexionis, # ad B A. <lb/></note> <note symbol="a" position="right" xlink:label="note-175-02" xlink:href="note-175-02a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note symbol="b" position="right" xlink:label="note-175-03" xlink:href="note-175-03a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-175-04" xlink:href="note-175-04a" xml:space="preserve"> <lb/>Vt ſin{us} anguli A, complemen- \\ ti anguli eiuſdem C, # ad C B, diſt antiam \\ cognitam: # Ita ſin{us} tot{us} recti \\ anguli B, # ad C A. <lb/></note> </div> <p> <s xml:id="echoid-s5626" xml:space="preserve">ALTITVDINEM in acceſſibilem beneficio ſpeculi plani, vnà cum <lb/>ſpeculi diſtantia tam à baſe, etiam non viſa, quàm à cacumine altitudi-<lb/>nis cognoſcere.</s> <s xml:id="echoid-s5627" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div383" type="section" level="1" n="160"> <head xml:id="echoid-head163" xml:space="preserve">PROBLEMA XL.</head> <p> <s xml:id="echoid-s5628" xml:space="preserve">1. </s> <s xml:id="echoid-s5629" xml:space="preserve"><emph style="sc">Sit</emph> rurſus in præcedenti figura altitudo A B, ſupra planum B D, erecta. <lb/></s> <s xml:id="echoid-s5630" xml:space="preserve">Collocato ſpeculo plano in C, recedatur ab altitudine ad D, donec cacumen <lb/>A, per radium refl exum E C A, cerni poſsit, obſeruetur que per quadrantem an-<lb/>gulus E C D, ideo que & </s> <s xml:id="echoid-s5631" xml:space="preserve">angulus A C B, vt in problemate præcedenti Num. </s> <s xml:id="echoid-s5632" xml:space="preserve">2. </s> <s xml:id="echoid-s5633" xml:space="preserve">di-<lb/>ctum eſt. </s> <s xml:id="echoid-s5634" xml:space="preserve">Deinde collocato ſpeculo in K, puncto, quotuis paſsibus à C, verſus <lb/>altitudinem diſtante, recedatur iterum ad I, donec cacumen A, inſpiciatur rur-<lb/>ſn<unsure/>m per radium reflexum L K A; </s> <s xml:id="echoid-s5635" xml:space="preserve">inquiratur que magnitudo anguli I K L, ideo-<lb/>que & </s> <s xml:id="echoid-s5636" xml:space="preserve">anguli A K B, Et quoniam poſito ſinu toto A B, rectæ B K, B C, tangen-<lb/>tes ſunt angulorum B A K, B A C, qui complementa ſunt angulorum K, C, refle-<lb/>xionum per quadrantem cognitorum, cognita erit K C, earum tangentium <lb/>differentia. </s> <s xml:id="echoid-s5637" xml:space="preserve">Si ergo fiat. </s> <s xml:id="echoid-s5638" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-175-05a" xlink:href="note-175-05"/> <anchor type="figure" xlink:label="fig-175-01a" xlink:href="fig-175-01"/> pro creabitur numerus, qui altitudinem A B, notam exhibebit in partibus diffe-<lb/>rentiæ poſitionum ſpeculi K C. <lb/></s> <s xml:id="echoid-s5639" xml:space="preserve"> <anchor type="note" xlink:label="note-175-06a" xlink:href="note-175-06"/> <pb o="146" file="176" n="176" rhead="GEOMETR. PRACT."/> reperietur CB, maior diſtantia ſpeculi C, ab altitudine. </s> <s xml:id="echoid-s5640" xml:space="preserve">Ex qua ſi auferatur K C, <lb/>differentia poſitionum ſpeculi, nota remanebit KB, minor diſtantia ſpeculiK, <lb/>ab eadem altitudine. </s> <s xml:id="echoid-s5641" xml:space="preserve">Quæ etiam inuenietur, ſi fiat, vt KC, differentia prædicto-<lb/>rum angulorum, qui complementa ſunt angulorum incidentiæ in ſpeculo, ad <lb/>KB, tangentem minorem: </s> <s xml:id="echoid-s5642" xml:space="preserve">Ita KC, differentia poſitionum ſpeculi ad aliud, vt <lb/>perſpicuum eſt.</s> <s xml:id="echoid-s5643" xml:space="preserve"/> </p> <div xml:id="echoid-div383" type="float" level="2" n="1"> <note style="it" position="right" xlink:label="note-175-05" xlink:href="note-175-05a" xml:space="preserve"> <lb/>Vt K C, differentia tangentium \\ quæ complemẽtis angulorum in- \\ cidentiæ debentur, # ad A B, ſi- \\ num to- \\ tum # Ita K C, differentia po- \\ ſitionum ſpeculi # ad A B, <lb/></note> <figure xlink:label="fig-175-01" xlink:href="fig-175-01a"> <image file="175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/175-01"/> </figure> <note style="it" position="right" xlink:label="note-175-06" xlink:href="note-175-06a" xml:space="preserve"> <lb/># <emph style="sc">Rvrsvs</emph> ſi fiat, # # <lb/>Vt K C, differentia complemen- \\ torum angulorum incidenti<unsure/>æ in \\ ſpeculo, # ad C B, tangento<unsure/>m \\ maiorem: # Ita K C, differen- \\ tia poſitionum \\ ſpeculi # ad C B, <lb/></note> </div> <p> <s xml:id="echoid-s5644" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/><emph style="sc">Deinde</emph> quia angulus AKB, in propinquiore ſpeculi poſitione duobus <anchor type="note" xlink:label="note-176-01a" xlink:href="note-176-01"/> angulis ACK, CAK, æqualis eſt, ſi angulus ACK, remotioris poſitionis detra-<lb/>hatur ex angulo AKB, poſitionis propinquioris: </s> <s xml:id="echoid-s5645" xml:space="preserve">remanebit angulus CAK, no-<lb/>tus. </s> <s xml:id="echoid-s5646" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Si igitur fiat, <anchor type="note" xlink:label="note-176-02a" xlink:href="note-176-02"/> <anchor type="note" xlink:label="note-176-03a" xlink:href="note-176-03"/> gignetur hypotenuſa CA, remotioris poſitionis ſpeculi, in partibus differentiæ <lb/>poſitionum ſpeculi KC. </s> <s xml:id="echoid-s5647" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Et ſi rurſus fiat, <anchor type="note" xlink:label="note-176-04a" xlink:href="note-176-04"/> <anchor type="note" xlink:label="note-176-05a" xlink:href="note-176-05"/> procreabitur quoquehypotenuſa KA, propinquioris poſitionis ſpeculi, in eiſ-<lb/>dem partibus differentiæ poſitionum ſpeculi KC.</s> <s xml:id="echoid-s5648" xml:space="preserve"/> </p> <div xml:id="echoid-div384" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-176-01" xlink:href="note-176-01a" xml:space="preserve">22. primi.</note> <note symbol="b" position="left" xlink:label="note-176-02" xlink:href="note-176-02a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-176-03" xlink:href="note-176-03a" xml:space="preserve"> <lb/>Vt ſin{us} anguli C A K, \\ differentiæ angulorum \\ incidentiæ # ad K C, differen- \\ tiam poſitionum \\ ſpeculi: # Ita ſin{us} anguli AKC, com- \\ plementi anguli AKB, ad \\ duos rectos in propinquio- \\ re poſitione ſpeculi # ad C A, <lb/></note> <note symbol="c" position="left" xlink:label="note-176-04" xlink:href="note-176-04a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-176-05" xlink:href="note-176-05a" xml:space="preserve"> <lb/>Vt ſin{us} anguli C A K, \\ differentiæ angulorum \\ incidentiæ # ad K C, differentiam \\ poſitionum ſpeculi: # Ita ſin{us} anguli re- \\ flexionis ACK, in \\ remotiori poſitione \\ ſpeculi # ad K A, <lb/></note> </div> </div> <div xml:id="echoid-div386" type="section" level="1" n="161"> <head xml:id="echoid-head164" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s5649" xml:space="preserve">2. </s> <s xml:id="echoid-s5650" xml:space="preserve"><emph style="sc">Per</emph> ſolos ſinus idem aſſequemur hocmodo. </s> <s xml:id="echoid-s5651" xml:space="preserve">Inuenta hypotenuſa CA, <lb/>vt proximè diximus, per ſinus. </s> <s xml:id="echoid-s5652" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> fiat, <anchor type="note" xlink:label="note-176-06a" xlink:href="note-176-06"/> <anchor type="note" xlink:label="note-176-07a" xlink:href="note-176-07"/> Prodibit enim in Quotiente altitudo AB, nota in partibus hypotenuſæ inuen-<lb/>tæ CA. </s> <s xml:id="echoid-s5653" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quod ſirurſus fiat, <anchor type="note" xlink:label="note-176-08a" xlink:href="note-176-08"/> <anchor type="note" xlink:label="note-176-09a" xlink:href="note-176-09"/> producetur CB, maior ſpeculi diſtantia ab altitudine. </s> <s xml:id="echoid-s5654" xml:space="preserve">Ex qua ſi ſubtrahatur KC, <lb/>differentia poſitionum ſpeculi, cognita etiam relinquetur diſtantia minor KB. <lb/></s> <s xml:id="echoid-s5655" xml:space="preserve">Quæetiam, ſi inueſtigetur hypotenuſa KB, vt ſupra traditum eſt, reperietur: </s> <s xml:id="echoid-s5656" xml:space="preserve">ſi <lb/>fiat, vt ſinus totus angulirecti B, ad hypotenuſam inuentam KB, ita ſinus anguli <lb/>BAK, complementi anguli in propinquiore poſitione ſpeculi, ad aliud, vt ma-<lb/>nifeſtum eſt.</s> <s xml:id="echoid-s5657" xml:space="preserve"/> </p> <div xml:id="echoid-div386" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-176-06" xlink:href="note-176-06a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-176-07" xlink:href="note-176-07a" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} angu- \\ lirecti B, # ad hypotenuſam \\ inuentam C A, # Ita ſin{us} anguli A C B, \\ remotioris poſitionis \\ ſpeculi # ad A B, <lb/></note> <note symbol="e" position="left" xlink:label="note-176-08" xlink:href="note-176-08a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note style="it" position="right" xlink:label="note-176-09" xlink:href="note-176-09a" xml:space="preserve"> <lb/>Vt ſin{us} tot{us} an- \\ gulirecti B, # ad hypotenuſam in- \\ uentam C A, # Ita ſin{us} anguli B A C, comple- \\ menti anguli in remotiore poſi- \\ tione ſpeculi # ad C B, <lb/></note> </div> <pb o="147" file="177" n="177" rhead="LIBER TERTIVS."/> <p> <s xml:id="echoid-s5658" xml:space="preserve">ALTITVDINEM monti impoſitam, ſi modo altitudinis baſis poſſit <lb/>conſpici, vel portionem ſuperiorem alicuius turris, beneficio ſpeculi <lb/>plani efficere notam.</s> <s xml:id="echoid-s5659" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div388" type="section" level="1" n="162"> <head xml:id="echoid-head165" xml:space="preserve">PROBLEMA XLI.</head> <p> <s xml:id="echoid-s5660" xml:space="preserve">1. </s> <s xml:id="echoid-s5661" xml:space="preserve"><emph style="sc">Qvando</emph> ad turrim patet acceſſus, vt eius à menſore diſtantia cogno-<lb/>ſcipoſsit; </s> <s xml:id="echoid-s5662" xml:space="preserve">ſi per probl. </s> <s xml:id="echoid-s5663" xml:space="preserve">39. </s> <s xml:id="echoid-s5664" xml:space="preserve">inueſtigetur tam altitudo à ſummitate portionis pro-<lb/>poſitæ, vſque ad baſem turris, quam altitudo ab infima parte eiuſdem portio-<lb/>nis, vſque ad eandem turris baſem: </s> <s xml:id="echoid-s5665" xml:space="preserve">& </s> <s xml:id="echoid-s5666" xml:space="preserve">minor hæc altitudo ab illa maiore de-<lb/>matur, reliqua fiet portio, quæ inquiritur.</s> <s xml:id="echoid-s5667" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5668" xml:space="preserve">2. </s> <s xml:id="echoid-s5669" xml:space="preserve"><emph style="sc">At</emph> verò, quando altitudo monti eſt impoſita, & </s> <s xml:id="echoid-s5670" xml:space="preserve">baſis altitudinis appa-<lb/>ret, aut ad turrim nonpatet acceſſ<unsure/>us: </s> <s xml:id="echoid-s5671" xml:space="preserve">exquirenda erit per præcedens problema <lb/>vtraquealtitudo prædicta. </s> <s xml:id="echoid-s5672" xml:space="preserve">Namrurſus minor detracta exmaiore, reliquam fa-<lb/>ciet altitudinem, vel portionem, quæ deſideratur.</s> <s xml:id="echoid-s5673" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5674" xml:space="preserve">SITVM cuiuslibet campi, aut atrii, vel templi, vel etiam vrbis, aut re-<lb/>g@onis cuiuſuis in plano deſcribere, ſi è duobus locis intra ipſum ſi-<lb/>tum aſſumptis baculi ex omnibus campi angulis erecti, vel certè <lb/>ipſi anguli in ædificio, aut vrbe, vel loca regionis videri poſſint: </s> <s xml:id="echoid-s5675" xml:space="preserve">ſi-<lb/>mulque longitudines laterum campi, vel ædificii, nec non diſtan-<lb/>tias inter angulos, & </s> <s xml:id="echoid-s5676" xml:space="preserve">vtrumuis locorum aſſumptorum in data men-<lb/>ſura cognoſcere. </s> <s xml:id="echoid-s5677" xml:space="preserve">Quod ſi talia duo loca intra ſitum eliginequeant, <lb/>idem efficere, dummodo ſitum poſſimus circumire.</s> <s xml:id="echoid-s5678" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div389" type="section" level="1" n="163"> <head xml:id="echoid-head166" xml:space="preserve">PROBLEMA XLII.</head> <p> <s xml:id="echoid-s5679" xml:space="preserve">1. </s> <s xml:id="echoid-s5680" xml:space="preserve"><emph style="sc">Etsi</emph> problema hocvel Geographicum eſt, vel Architectonicum; </s> <s xml:id="echoid-s5681" xml:space="preserve">ta-<lb/> <anchor type="note" xlink:label="note-177-01a" xlink:href="note-177-01"/> men quia ſine dimenſione linearum abſolui non poteſt, lubet illud hocloco <lb/>paucis explicare. </s> <s xml:id="echoid-s5682" xml:space="preserve">Sit ergo campus quinque lateribus AB, BC, CD, DE, EA, <lb/>cinctus. </s> <s xml:id="echoid-s5683" xml:space="preserve">Figantur in quinque angulis A, B, C, D, E, quinque baculi ad angu-<lb/>losrectos cum Horizonte, paretur que circa medium areæ planum aliquantu-<lb/>lum altum Horizonti æquidiſtans, in quo duo puncta F, G, quantumlibetin-<lb/>ter ſe diſtantia, verbigratia 100. </s> <s xml:id="echoid-s5684" xml:space="preserve">pedibus, è quibus omnes quinque baculi cerni <lb/>poſsint. </s> <s xml:id="echoid-s5685" xml:space="preserve">Per F, G, ducatur recta F G, ad vtraſque partes; </s> <s xml:id="echoid-s5686" xml:space="preserve">continebitque ſe-<lb/>gmentum F G, 100. </s> <s xml:id="echoid-s5687" xml:space="preserve">pedes ex hypotheſi. </s> <s xml:id="echoid-s5688" xml:space="preserve">Affixa deinde dioptra volubili cum <lb/>pinna cidiis in vtroq; </s> <s xml:id="echoid-s5689" xml:space="preserve">puncto F, & </s> <s xml:id="echoid-s5690" xml:space="preserve">G, deſcriptiſque circulis duobus ex F, & </s> <s xml:id="echoid-s5691" xml:space="preserve">G, vt <lb/>per eorum circumferentias angulorum magnitudines, qui in F, G, conſtituẽtur, <lb/>cognoſcipo ſsint, inſpiciantur ex F, & </s> <s xml:id="echoid-s5692" xml:space="preserve">G, perforamina pinnacidiorum (circum-<lb/>ducta dioptra) baculi ex angulis A, B, C, D, E, erecti, & </s> <s xml:id="echoid-s5693" xml:space="preserve">anguli, quos linea fiducię <lb/>cũ recta HI, facit, aut quos rectæ à linea fiduciæ deſignatę inter ſefaciunt, tranſ-<lb/>ferantur ordine ad puncta K, L, quomodocunq; </s> <s xml:id="echoid-s5694" xml:space="preserve">inter ſe diſtantia in recta, KL, <pb o="148" file="178" n="178" rhead="GEOMETR. PRACT."/> quæ ſeorſum in charta aliqua ſit deſcripta, productis lineis, quæ illos angulos in <lb/>K, & </s> <s xml:id="echoid-s5695" xml:space="preserve">L, efficiunt vt in 2. </s> <s xml:id="echoid-s5696" xml:space="preserve">figura apparet. </s> <s xml:id="echoid-s5697" xml:space="preserve">Si namque puncta, vbi dictæ lineæ ex <lb/> <anchor type="figure" xlink:label="fig-178-01a" xlink:href="fig-178-01"/> K, & </s> <s xml:id="echoid-s5698" xml:space="preserve">L, prodeuntes concurrunt, lineis rectis coniungantur, deſcripta erit figu-<lb/>ra O P Q R S, ſimilis omnino figuræ campi A B C D E, Quod ſic demonſtro. <lb/></s> <s xml:id="echoid-s5699" xml:space="preserve">Triangula AGF, OLK, ſimilia ſunt, quòd anguli AGF, AFG, angulis OLK, <lb/>OKL, æquales ſint, ex conſtructione. </s> <s xml:id="echoid-s5700" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur erit AG, ad GF, vt OL, ad LK.</s> <s xml:id="echoid-s5701" xml:space="preserve"> <anchor type="note" xlink:label="note-178-01a" xlink:href="note-178-01"/> Eademque ratione, ob ſimilitudinem triangulorum FGE, KLS, erit GF, ad GE, <lb/>vt LK, ad LS: </s> <s xml:id="echoid-s5702" xml:space="preserve">ac proinde exæquo erit AG, ad GE, vt OI, ad LS. </s> <s xml:id="echoid-s5703" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s5704" xml:space="preserve"><lb/>anguli AGE, OLS, circa quos latera illa ſunt proportionalia, æquales ſint, <lb/>quippe cum angulo AGE, factus ſit æ-<lb/> <anchor type="figure" xlink:label="fig-178-02a" xlink:href="fig-178-02"/> qualis ex conſtructione angulus O L S: <lb/></s> <s xml:id="echoid-s5705" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/> ſimilia erunt triangula AGE, OLS, hoc <anchor type="note" xlink:label="note-178-02a" xlink:href="note-178-02"/> eſt, æquiangula. </s> <s xml:id="echoid-s5706" xml:space="preserve">Pari ratione ob ſimili-<lb/>tudinem triangulorum FGD, KLR, <anchor type="note" xlink:href="" symbol="c"/> erit <anchor type="note" xlink:label="note-178-03a" xlink:href="note-178-03"/> GD, ad GF, vt LR, ad LK. </s> <s xml:id="echoid-s5707" xml:space="preserve">Item ob ſimi-<lb/>litudinem triangulorum FGE, KLS, erit <lb/>FG, ad GE, vt KL, ad LS. </s> <s xml:id="echoid-s5708" xml:space="preserve">Igitur erit ex <lb/>æquo GD, ad GE, vt LR, ad LS: </s> <s xml:id="echoid-s5709" xml:space="preserve">Ac <lb/>proinde cum anguli DGE, RLS, ex con-<lb/>ſtructione ſint æquales; </s> <s xml:id="echoid-s5710" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> æquiangula quoque erunt triangula D G E, R L S.</s> <s xml:id="echoid-s5711" xml:space="preserve"> <anchor type="note" xlink:label="note-178-04a" xlink:href="note-178-04"/> Non aliter oſtendemus, triangula CGD, BGC, AGB, triangulis, QLR, PLQ, <lb/>OLP, eſſe æquiangula. </s> <s xml:id="echoid-s5712" xml:space="preserve">Immo eiſdem argumentis concludemus, quamuis non <lb/>ſitneceſſarium, triangula, quæ in F, ſupra latera campi conſtructa ſunt, æqui-<lb/>angula eſſe triangulis in K, ſupra latera figuræ O P Q R S, conſtitutis. </s> <s xml:id="echoid-s5713" xml:space="preserve">Ex his <lb/>ſequitur, figuram A B C D E, figuræ O P Q R S, æquiangulam eſſe: </s> <s xml:id="echoid-s5714" xml:space="preserve">quippe cum <lb/>earum anguli coagmentati ſint ex angulis æqualibus, nimirum angulus AED, <lb/>ex angulis AEG, GED, ipſum componentibus, qui angulis OSL, LSR, an-<lb/>gulum OSR, componentibus æquales ſunt; </s> <s xml:id="echoid-s5715" xml:space="preserve">& </s> <s xml:id="echoid-s5716" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s5717" xml:space="preserve">Sequitur et-<lb/>iam latera earundem figurarum circa æquales angulos eſſe proportionalia. <lb/></s> <s xml:id="echoid-s5718" xml:space="preserve">Nam propter triangulorum ſimilitudinem, <anchor type="note" xlink:href="" symbol="e"/> eſt AE, ad EG, vt OS, ad SL:</s> <s xml:id="echoid-s5719" xml:space="preserve"> <anchor type="note" xlink:label="note-178-05a" xlink:href="note-178-05"/> <pb o="149" file="179" n="179" rhead="LIBER TERTIVS."/> & </s> <s xml:id="echoid-s5720" xml:space="preserve">EG, ad ED, vt SL, ad SR. </s> <s xml:id="echoid-s5721" xml:space="preserve">Ideoque exæquo AE, ad ED, vt OS, ad SR: </s> <s xml:id="echoid-s5722" xml:space="preserve">Atque <lb/>ita de alijs. </s> <s xml:id="echoid-s5723" xml:space="preserve">Similis ergo ſunt figuræ ABCDE, OPQRS.</s> <s xml:id="echoid-s5724" xml:space="preserve"/> </p> <div xml:id="echoid-div389" type="float" level="2" n="1"> <note position="right" xlink:label="note-177-01" xlink:href="note-177-01a" xml:space="preserve">Sit{us} camp@ <lb/>cuiuſuis, quo<unsure/> <lb/>pacto ex duo-<lb/>b{us} locis in-<lb/>tra ipſum aſ-<lb/>ſumptis deli-<lb/>neetur.</note> <figure xlink:label="fig-178-01" xlink:href="fig-178-01a"> <image file="178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/178-01"/> </figure> <note symbol="a" position="left" xlink:label="note-178-01" xlink:href="note-178-01a" xml:space="preserve">4. ſexti.</note> <figure xlink:label="fig-178-02" xlink:href="fig-178-02a"> <image file="178-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/178-02"/> </figure> <note symbol="b" position="left" xlink:label="note-178-02" xlink:href="note-178-02a" xml:space="preserve">6. ſexti.</note> <note symbol="c" position="left" xlink:label="note-178-03" xlink:href="note-178-03a" xml:space="preserve">4. ſexti.</note> <note symbol="d" position="left" xlink:label="note-178-04" xlink:href="note-178-04a" xml:space="preserve">4. ſexti.</note> <note symbol="c" position="left" xlink:label="note-178-05" xlink:href="note-178-05a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s5725" xml:space="preserve">2. </s> <s xml:id="echoid-s5726" xml:space="preserve"><emph style="sc">Iam</emph> vero vt longitudines laterum AB, BC, CD, DE, EA, & </s> <s xml:id="echoid-s5727" xml:space="preserve">rectarum ex <lb/>F, vel G, ad angulos in prima figura ductarum inueniamus, diuidendum erit in fi-<lb/>gura inuenta, id eſt, in ſecunda, interuallum KL, in quotcunq; </s> <s xml:id="echoid-s5728" xml:space="preserve">partes æquales. <lb/></s> <s xml:id="echoid-s5729" xml:space="preserve">Deinde inquirendum, quotnam exillis partibus in ſingulis lateribus, & </s> <s xml:id="echoid-s5730" xml:space="preserve">rectis <lb/>eiuſdem figuræ ſecundæ ex K, vel L, prodeuntibus contineantur. </s> <s xml:id="echoid-s5731" xml:space="preserve">quod vel per <lb/>cir cinum fieri poteſt, repetendo ſæpius vnam particulam in dictis lateribus re-<lb/>ctis: </s> <s xml:id="echoid-s5732" xml:space="preserve">vel (quod magis probo) hoc modo. </s> <s xml:id="echoid-s5733" xml:space="preserve">Repetatur tota KL, in quolibetlate-<lb/>re, velrecta, quoties fieri poteſt, & </s> <s xml:id="echoid-s5734" xml:space="preserve">in reliquo ſegmento v@a etiam particula in-<lb/>terualli KL, circino iteretur, quoties fieri poteſt. </s> <s xml:id="echoid-s5735" xml:space="preserve">Nam quoties repetita fuerit <lb/>KL: </s> <s xml:id="echoid-s5736" xml:space="preserve">toties numerus particularum ipſius KL, in latere continebitur, cum tot in-<lb/>ſuper particulis, quot per circinum in reliquo r<unsure/>egmento fuerint deprehenſæ. </s> <s xml:id="echoid-s5737" xml:space="preserve"><lb/>Aut certe per ea, quæ lib 1. </s> <s xml:id="echoid-s5738" xml:space="preserve">cap. </s> <s xml:id="echoid-s5739" xml:space="preserve">1. </s> <s xml:id="echoid-s5740" xml:space="preserve">ad finem N@m. </s> <s xml:id="echoid-s5741" xml:space="preserve">2. </s> <s xml:id="echoid-s5742" xml:space="preserve">ſcripſimus, inueſtigetur in in-<lb/>ſtrumento partium, quot particulæ inter@alli KL, in dictis lateribus, & </s> <s xml:id="echoid-s5743" xml:space="preserve">rectis cõ-<lb/>prehendantur. </s> <s xml:id="echoid-s5744" xml:space="preserve">Deinde fiat, vtnumerus particularum interualli K L, aſſumptus <lb/>in 2. </s> <s xml:id="echoid-s5745" xml:space="preserve">figura, ad numerum pedum inter puncta F, & </s> <s xml:id="echoid-s5746" xml:space="preserve">G, in prima figura aſſumptum, <lb/>ita numerus particularum in quolibet latere, vel recta in ſecunda figura inuen-<lb/>tarum, ad aliud. </s> <s xml:id="echoid-s5747" xml:space="preserve">Quotiens enim numerus indicabit, quot pedes in aſſumpto <lb/>latere, vel recta contineantur. </s> <s xml:id="echoid-s5748" xml:space="preserve">Ratio eſt, quia cum eandem proportionem ha-<lb/>beat KL, in 2. </s> <s xml:id="echoid-s5749" xml:space="preserve">figura ad quodlibet latus, vel rectam eiuſdem figuræ, quam habet <lb/>FG, in prima figura ad reſpondens latus, vel rectam, propter ſimilitudinẽ figu-<lb/>rarũ, erit ꝑmutando KL, ad interuallũ FG, vt latus adlatus, &</s> <s xml:id="echoid-s5750" xml:space="preserve">c. </s> <s xml:id="echoid-s5751" xml:space="preserve">Verbi gratia, In <lb/>2. </s> <s xml:id="echoid-s5752" xml:space="preserve">figura interuallum KL, ſectum eſt in 5. </s> <s xml:id="echoid-s5753" xml:space="preserve">particulas, qualium 17. </s> <s xml:id="echoid-s5754" xml:space="preserve">in latere OP, in-<lb/>uentæ ſunt: </s> <s xml:id="echoid-s5755" xml:space="preserve">Et quia ſpatium FG, in 1. </s> <s xml:id="echoid-s5756" xml:space="preserve">figura poſitum eſt 100. </s> <s xml:id="echoid-s5757" xml:space="preserve">pedum: </s> <s xml:id="echoid-s5758" xml:space="preserve">ſi fiat, <lb/> <anchor type="note" xlink:label="note-179-01a" xlink:href="note-179-01"/> hoc eſt, ſi, vt regula aurea præcipit, 100. </s> <s xml:id="echoid-s5759" xml:space="preserve">ducantur in 17. </s> <s xml:id="echoid-s5760" xml:space="preserve">ſecundus numerus in <lb/>tertium, & </s> <s xml:id="echoid-s5761" xml:space="preserve">productus numerus 1700. </s> <s xml:id="echoid-s5762" xml:space="preserve">diuidatur per 5. </s> <s xml:id="echoid-s5763" xml:space="preserve">id eſt, per primum nume-<lb/>rum, reperientur in Quotiente 340. </s> <s xml:id="echoid-s5764" xml:space="preserve">pedes pro latere AB, & </s> <s xml:id="echoid-s5765" xml:space="preserve">ſic de cæteris.</s> <s xml:id="echoid-s5766" xml:space="preserve"/> </p> <div xml:id="echoid-div390" type="float" level="2" n="2"> <note style="it" position="right" xlink:label="note-179-01" xlink:href="note-179-01a" xml:space="preserve"> <lb/>Vt KL, quinque parti- \\ cularum # ad FG. 100. pedum: # Ita OP, 17. particu- \\ larum # ad AB, <lb/></note> </div> <p> <s xml:id="echoid-s5767" xml:space="preserve">3. </s> <s xml:id="echoid-s5768" xml:space="preserve"><emph style="sc">Evndem</emph> ſitum campi propoſiti A B C D E, delineabimus etiam ex vno <lb/> <anchor type="note" xlink:label="note-179-02a" xlink:href="note-179-02"/> tantumloco F, intra ipſum aſſumpto, hacratione. </s> <s xml:id="echoid-s5769" xml:space="preserve">Dioptra ad ſingulos bacu-<lb/>los ex angulis erectos, dirigatur, notatis angulis, quos lineæ, p<unsure/>er dioptram de-<lb/>ſignatæ inter ſe faciunt; </s> <s xml:id="echoid-s5770" xml:space="preserve">diſtantiæ que ab F, ad ſingulos angulos inquirantur in <lb/>aliqua menſura, vel per catenulã aliquã ferream, quæ nec intendi poſsit, nec re-<lb/>mitti, vel per chordam ex F, ad ſingulos angulos extenſam vel certe, ſi diſtantiæ <lb/>illæ magnæ ſint, per problema 2. </s> <s xml:id="echoid-s5771" xml:space="preserve">vel 36. </s> <s xml:id="echoid-s5772" xml:space="preserve">beneficio quadrati, alteriuſue inſtrumẽ-<lb/>ti. </s> <s xml:id="echoid-s5773" xml:space="preserve">Nam ſi in charta aliqua ad quo dlibet punctum K, ijdem anguli conſtituan-<lb/>tur, & </s> <s xml:id="echoid-s5774" xml:space="preserve">in rectis illos angulos effi cientibus accipiantur tot particulæ inter ſe æ-<lb/>quales cuiuſuis magnitudinis, quot menſuræ inuentæ ſunt in rectis reſponden-<lb/>tibus, quæ ex F, exeunt: </s> <s xml:id="echoid-s5775" xml:space="preserve">extrema autem puncta vltimarum particularum rectis <lb/>lineis coniungantur, deſcripta erit figura OPQRS, ſimilis omnino campo AB-<lb/>CDE: </s> <s xml:id="echoid-s5776" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> propterea quod triangula ad punctum F, collecta ſimilia ſunt triangulis <anchor type="note" xlink:label="note-179-03a" xlink:href="note-179-03"/> ad punctum K, collectis, ob æqualitatem angulorum in F, & </s> <s xml:id="echoid-s5777" xml:space="preserve">K, conſtitutorum, <lb/>& </s> <s xml:id="echoid-s5778" xml:space="preserve">latera circa illos angulos proportionalia, ex conſtructione.</s> <s xml:id="echoid-s5779" xml:space="preserve"/> </p> <div xml:id="echoid-div391" type="float" level="2" n="3"> <note position="right" xlink:label="note-179-02" xlink:href="note-179-02a" xml:space="preserve">Sit{us} campi <lb/>cuiſuis qua <lb/>ratione ex v-<lb/>no loco intra <lb/>ipſum aſſum-<lb/>pto deſcriba-<lb/>tur.</note> <note symbol="a" position="right" xlink:label="note-179-03" xlink:href="note-179-03a" xml:space="preserve">6. ſexti.</note> </div> <p> <s xml:id="echoid-s5780" xml:space="preserve">4. </s> <s xml:id="echoid-s5781" xml:space="preserve"><emph style="sc">Latervm</emph> autem longitudines in campo ABCDE, cognoſcentur, ſi <pb o="150" file="180" n="180" rhead="GEOMETR. PRACT."/> per circinum inquiratur, quot particulæ in lateribus figuræ OPQRS, continean-<lb/>tur ex illis, quæ in quauis recta ex K, emiſſa ſumptæ fuere. </s> <s xml:id="echoid-s5782" xml:space="preserve">Totidem namq; </s> <s xml:id="echoid-s5783" xml:space="preserve">men-<lb/>ſuræ in lateribus campi comprehendentur, vt perſpicuum eſt ex figurarum ſimi-<lb/>litudine.</s> <s xml:id="echoid-s5784" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Vrbis cuiuſ-<lb/>uis, ac regio-<lb/>nis ſit{us} quo <lb/>pacto deſcri-<lb/>batur.</note> <p> <s xml:id="echoid-s5785" xml:space="preserve">5. </s> <s xml:id="echoid-s5786" xml:space="preserve"><emph style="sc">Eadem</emph> prorſus ratio tenenda eſt in ſitu alicuius atrij vel templi, vel vr-<lb/>bis, autregionis explorando. </s> <s xml:id="echoid-s5787" xml:space="preserve">Solum in vrbe deſcribenda pro punctis F, & </s> <s xml:id="echoid-s5788" xml:space="preserve">G, <lb/>eligendæ ſunt duæ turres altæ, è quibus omnes vrbis anguli conſpici poſsint, & </s> <s xml:id="echoid-s5789" xml:space="preserve"><lb/>quarum diſtantia vnius ab altera vel cognita ſit, vel per præcedentia problema-<lb/>ta inueſtigata. </s> <s xml:id="echoid-s5790" xml:space="preserve">In regione autem delineanda pro ijſdem punctis F, & </s> <s xml:id="echoid-s5791" xml:space="preserve">G, duo op-<lb/>pida deligenda ſunt, & </s> <s xml:id="echoid-s5792" xml:space="preserve">in quolibet ex altiſsima turre circumia centia oppida in-<lb/>ſpicienda, vt anguli habeantur, quos rectæ ab oculo menſoris ad ſingula oppi-<lb/>da eductæ conſtituunt. </s> <s xml:id="echoid-s5793" xml:space="preserve">Hiautem facilius obſeruabuntur, ſi loco dioptrę, quia <lb/>nimis alta eſt, ſtatuatur planumerectum in punctis F, & </s> <s xml:id="echoid-s5794" xml:space="preserve">G, ita vt cir cumductum <lb/>tranſeat per oppida circumiacentia, ſi intelligatur eſſe productum. </s> <s xml:id="echoid-s5795" xml:space="preserve">Ita namque <lb/>planum ipſum rectas deſignabit, quæ angulos prædictos conſtituant. </s> <s xml:id="echoid-s5796" xml:space="preserve">Atq; </s> <s xml:id="echoid-s5797" xml:space="preserve">hoc <lb/>etiam in ſitu vrbium perueſtigando faciendum erit. </s> <s xml:id="echoid-s5798" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s5799" xml:space="preserve">ſi anguli vrbis cuiuſ-<lb/>piam, aut oppida alicuius regionis ſint A, B, C, D, E, & </s> <s xml:id="echoid-s5800" xml:space="preserve">duæ turres in F, & </s> <s xml:id="echoid-s5801" xml:space="preserve">G, o-<lb/>mnia perficienda erunt, vt ſupra de campi ſitu dictum eſt. </s> <s xml:id="echoid-s5802" xml:space="preserve">Nam ſitum vrbis, vel <lb/>regionis exhibebit figura OPQRS, diſtantiæ que locorum A, B, C, D, E, vnius ab <lb/>altero cognoſcentur, vt de lateribus campi dictum eſt.</s> <s xml:id="echoid-s5803" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5804" xml:space="preserve">6. </s> <s xml:id="echoid-s5805" xml:space="preserve"><emph style="sc">Qvod</emph> ſi intra ſitum propoſitum duoloca, vel vnus ſaltem non exiſtat, <lb/>vnde omnes anguli conſpici poſsint, vt in omnibus ædificijs contingit, oporte-<lb/>bit ſitum circumire, & </s> <s xml:id="echoid-s5806" xml:space="preserve">inueſtigare angulos per rectas, quæ in campo à baculo ad <lb/>baculum ducendæ ſunt per catenulam ferream, vel chordam, aut certe erigen-<lb/> <anchor type="figure" xlink:label="fig-180-01a" xlink:href="fig-180-01"/> dum planum, quod per binos baculos tranſire conſpiciatur. <lb/></s> <s xml:id="echoid-s5807" xml:space="preserve">Ipſum enim planum dictas rectas exhibebit. </s> <s xml:id="echoid-s5808" xml:space="preserve">In ædificijs au-<lb/>tem, ac templisipſi muriexteriores dictos angulos conſtitu-<lb/>unt, quorum amplitudo inueſtigari ſolet ab artificibus inſtru-<lb/>mento quodam ex duabus regulis compacto, quarum vna <lb/>ſub alteram ingreſſa moueatur, (quod Italis Squadrazoppa <lb/>dicitur) vt hæc figura indicat. </s> <s xml:id="echoid-s5809" xml:space="preserve">Aperto namque inſtrumento, <lb/>ſi crura duobus muris angulum effi cientibus congruent, da-<lb/>buntinteriora crurium latera in cõcurſu angulum quæſitum. </s> <s xml:id="echoid-s5810" xml:space="preserve"><lb/>Et ſi idem inſtrumentum interioribus angulis ædificiorum applicetur, dabunt <lb/>eadem latera interiora crurium angulos, qui à muris effi ciuntur. </s> <s xml:id="echoid-s5811" xml:space="preserve">Inuentis angu-<lb/>lis, ac notatis, menſuranda erunt interualla inter baculos in campo erectos, vel <lb/>inter angulos ædificiorum, per catenulam ferream, vel chordã, aut certe in cam-<lb/>pis, ſi ea interualla longa ſint, per problema 2. </s> <s xml:id="echoid-s5812" xml:space="preserve">vel 36. </s> <s xml:id="echoid-s5813" xml:space="preserve">exploranda in aliqua men-<lb/>ſura nota, vtin cubitis. </s> <s xml:id="echoid-s5814" xml:space="preserve">Nam ſi in charta ducatur linea OP, tot particulas æqua-<lb/>les continens, quot cubiti, verbi gratia, in interuallo AB, deprehenſi ſunt, & </s> <s xml:id="echoid-s5815" xml:space="preserve">an-<lb/>gulus POS, angulo BAE, inuento fiat æqualis: </s> <s xml:id="echoid-s5816" xml:space="preserve">& </s> <s xml:id="echoid-s5817" xml:space="preserve">in recta O S, accipiantur tot <lb/>particulæ prioribus æquales vſq; </s> <s xml:id="echoid-s5818" xml:space="preserve">ad S, quot cubiti in interuallo AE, repertiſunt: </s> <s xml:id="echoid-s5819" xml:space="preserve"><lb/>acrurſum angulus OSR, angulo AED, æqualis fiat, & </s> <s xml:id="echoid-s5820" xml:space="preserve">ſic deinceps, repræſenta-<lb/>bitur ſitus campi per figuram O P Q R S. </s> <s xml:id="echoid-s5821" xml:space="preserve">Idemque de ſitu ædificiorum <lb/>tam exteriori, quam interiori intelligendum eſt, ſi diligenter anguli, ac diſtantię <lb/>obſeruentur.</s> <s xml:id="echoid-s5822" xml:space="preserve"/> </p> <div xml:id="echoid-div392" type="float" level="2" n="4"> <figure xlink:label="fig-180-01" xlink:href="fig-180-01a"> <image file="180-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/180-01"/> </figure> </div> <pb o="151" file="181" n="181" rhead="LIBER TERTIVS."/> <p> <s xml:id="echoid-s5823" xml:space="preserve">LONGITVDINEM trabis ad Horizontem in clinatæ, cuius portio <lb/>ſuperior tantum conſpiciatur, vna cum angulo inclinationis, diſtan-<lb/>tia baſis à menſore, & </s> <s xml:id="echoid-s5824" xml:space="preserve">altitudine faſtigii ſupra Horizontem, per Qua-<lb/>dratum metiri.</s> <s xml:id="echoid-s5825" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div394" type="section" level="1" n="164"> <head xml:id="echoid-head167" xml:space="preserve">PROBLEMA XLIII.</head> <p> <s xml:id="echoid-s5826" xml:space="preserve">1. </s> <s xml:id="echoid-s5827" xml:space="preserve">Trabs inclinata ſupra murum CD, ſit AB, & </s> <s xml:id="echoid-s5828" xml:space="preserve">menſor in E, conſtitutus è di-<lb/>recto ipſius trabis, ita, vt ipſe & </s> <s xml:id="echoid-s5829" xml:space="preserve">trabs in eodem ſint plano, metiri debeat lon-<lb/>gitudinẽ A B, angulum in clinationis ABE, diſtantiam B E, & </s> <s xml:id="echoid-s5830" xml:space="preserve">altitudinem A F, <lb/>etiamſi tantum portionem ſupremam A G, videat, baſem ocultante muro C D <lb/>Si quadratum ad E, bis applicetur, ſemel videlicet in clinatum, vt vnum latus re-<lb/>cta tendat ad punctum A, & </s> <s xml:id="echoid-s5831" xml:space="preserve">iterum demiſſum ad Horizontem, ita vt centrum <lb/>dioptræ ſit in E, & </s> <s xml:id="echoid-s5832" xml:space="preserve">latus E L, Horizontiincumbat. </s> <s xml:id="echoid-s5833" xml:space="preserve">Nam inſpecto puncto A, ex <lb/>a, <anchor type="note" xlink:href="" symbol="a"/> ſi ſ<unsure/>iat, vt vmbra verſa e b, ad latus b a, ita latus a <anchor type="figure" xlink:label="fig-181-01a" xlink:href="fig-181-01"/> <anchor type="note" xlink:label="note-181-01a" xlink:href="note-181-01"/> E, ad aliud, gignetur diſtãtia, ſiue hypotenuſa E A. <lb/></s> <s xml:id="echoid-s5834" xml:space="preserve">Deinde explorata portione dioptræ E I, vt in ſcho-<lb/>lio Probl. </s> <s xml:id="echoid-s5835" xml:space="preserve">7. </s> <s xml:id="echoid-s5836" xml:space="preserve">Num. </s> <s xml:id="echoid-s5837" xml:space="preserve">2. </s> <s xml:id="echoid-s5838" xml:space="preserve">docui; </s> <s xml:id="echoid-s5839" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Si fiat, vt E I, portio <anchor type="note" xlink:label="note-181-02a" xlink:href="note-181-02"/> dio ptræ inuenta ad repertam hypotenuſam EA, ita <lb/>fI, lateri quadrati æqualis ad aliud, producetur al-<lb/>titudo AF, quæſita. </s> <s xml:id="echoid-s5840" xml:space="preserve">quod eſt quartum.</s> <s xml:id="echoid-s5841" xml:space="preserve"/> </p> <div xml:id="echoid-div394" type="float" level="2" n="1"> <figure xlink:label="fig-181-01" xlink:href="fig-181-01a"> <image file="181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/181-01"/> </figure> <note symbol="a" position="right" xlink:label="note-181-01" xlink:href="note-181-01a" xml:space="preserve">4. ſexti.</note> <note symbol="b" position="right" xlink:label="note-181-02" xlink:href="note-181-02a" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> </div> <p> <s xml:id="echoid-s5842" xml:space="preserve">2. </s> <s xml:id="echoid-s5843" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <emph style="sc">Et</emph> ſi rurſus fiat, vt EI, portio dioptræ ad hypotenuſam EA, ita Ef, ipſi cI, <anchor type="note" xlink:label="note-181-03a" xlink:href="note-181-03"/> æqualis ad aliud, nota fiet diſtantia EF. </s> <s xml:id="echoid-s5844" xml:space="preserve">Non aliter, ſi aliud punctum G, inſpi-<lb/>ciatur, cognita fiet hypotenuſa E G, & </s> <s xml:id="echoid-s5845" xml:space="preserve">altitudo G H, vna cum diſtantia E H, ſi <lb/>nimirum quadratum primo ita applicetur ad E, vt vnum latus tendat recta ad <lb/>G, &</s> <s xml:id="echoid-s5846" xml:space="preserve">c. </s> <s xml:id="echoid-s5847" xml:space="preserve">quemadmo dum in ſcholio probl. </s> <s xml:id="echoid-s5848" xml:space="preserve">7. </s> <s xml:id="echoid-s5849" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s5850" xml:space="preserve"/> </p> <div xml:id="echoid-div395" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-181-03" xlink:href="note-181-03a" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> </div> <p> <s xml:id="echoid-s5851" xml:space="preserve">3. </s> <s xml:id="echoid-s5852" xml:space="preserve"><emph style="sc">Iam</emph> ſi E F, inuenta ex cognita E H, ſubducatur, nota relinquetur F H, id <lb/>eſt, GI, quæ in ſublimi duci cogitetur parallela Horizonti Eodem modo ſi G H, <lb/>cognita, vel illi æqualis FI, ex inuenta AF, dematur, reliqua AI, nota fiet. </s> <s xml:id="echoid-s5853" xml:space="preserve">Cum <lb/>igitur in triangulo rectangulo AGI, in ſublimi conſtituto duo latera AI, IG, co-<lb/> <anchor type="note" xlink:label="note-181-04a" xlink:href="note-181-04"/> gnita ſint, <anchor type="note" xlink:href="" symbol="d"/> cognoſcetur & </s> <s xml:id="echoid-s5854" xml:space="preserve">baſis A G, & </s> <s xml:id="echoid-s5855" xml:space="preserve">angulus AGI, <anchor type="note" xlink:href="" symbol="e"/> quiæ qualis eſt angu- lo in clinationis ABE, qui quæritur. </s> <s xml:id="echoid-s5856" xml:space="preserve">quod eſt ſecundum.</s> <s xml:id="echoid-s5857" xml:space="preserve"/> </p> <div xml:id="echoid-div396" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-181-04" xlink:href="note-181-04a" xml:space="preserve">6. triang. re-<lb/>ctil.</note> </div> <note symbol="e" position="right" xml:space="preserve">29. primi.</note> <p> <s xml:id="echoid-s5858" xml:space="preserve">4. </s> <s xml:id="echoid-s5859" xml:space="preserve"><emph style="sc">Itaqve</emph> cum in triangulo rectangulo ABF, latus A F, notũ ſit factum, <lb/> <anchor type="note" xlink:label="note-181-06a" xlink:href="note-181-06"/> vna cum angulo B, ac proinde & </s> <s xml:id="echoid-s5860" xml:space="preserve">cum eius complemento BAF, <anchor type="note" xlink:href="" symbol="f"/> cognitum fiet alterum quo que latus B F, cui ſi adij cietur diſtantia E F, inuenta, tota diſtantia <lb/>quæſita E B, manifeſta erit. </s> <s xml:id="echoid-s5861" xml:space="preserve">quod eſt tertium.</s> <s xml:id="echoid-s5862" xml:space="preserve"/> </p> <div xml:id="echoid-div397" type="float" level="2" n="4"> <note symbol="f" position="right" xlink:label="note-181-06" xlink:href="note-181-06a" xml:space="preserve">4. triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s5863" xml:space="preserve">5. </s> <s xml:id="echoid-s5864" xml:space="preserve"><emph style="sc">Deniqve</emph> in eodem triangulo ABF, rectangulo, ex latere A F, & </s> <s xml:id="echoid-s5865" xml:space="preserve">angu-<lb/>lo B, ac proinde & </s> <s xml:id="echoid-s5866" xml:space="preserve">eius complemento BAF, cognitis, <anchor type="note" xlink:href="" symbol="g"/> cognoſcetur etiam baſis <anchor type="note" xlink:label="note-181-07a" xlink:href="note-181-07"/> AB, longitudo videlicet trabis. </s> <s xml:id="echoid-s5867" xml:space="preserve">quod eſt primum. </s> <s xml:id="echoid-s5868" xml:space="preserve">Hæc autem nota etiam effi-<lb/>cietur, <anchor type="note" xlink:href="" symbol="h"/> ſi fiat, vt AI, differentia altitudinum cognitarum AF, GH, ad AF, maio- <anchor type="note" xlink:label="note-181-08a" xlink:href="note-181-08"/> rem altitudinem, ita AG, paulo ante nota effecta, ad A B.</s> <s xml:id="echoid-s5869" xml:space="preserve"/> </p> <div xml:id="echoid-div398" type="float" level="2" n="5"> <note symbol="g" position="right" xlink:label="note-181-07" xlink:href="note-181-07a" xml:space="preserve">4. triang re-<lb/>ctil.</note> <note symbol="h" position="right" xlink:label="note-181-08" xlink:href="note-181-08a" xml:space="preserve">2. ſexti. & <lb/>componendo.</note> </div> <p> <s xml:id="echoid-s5870" xml:space="preserve">VISIS duarum turrium ſummitatibus, etiamſi baſes propter ædificia <lb/>interiecta occultentur, diſtantiam tam inter earum baſes, quam inter <lb/>earundem faſtigia, vna cum ipſarum altitudinibus, ac diſtãtiis à men-<lb/>ſore coniicere.</s> <s xml:id="echoid-s5871" xml:space="preserve"/> </p> <pb o="152" file="182" n="182" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div400" type="section" level="1" n="165"> <head xml:id="echoid-head168" xml:space="preserve">PROBLEMA XLIV.</head> <p> <s xml:id="echoid-s5872" xml:space="preserve">1. </s> <s xml:id="echoid-s5873" xml:space="preserve"><emph style="sc">Sint</emph> duæ turres A F, GH, quarum ſola faſtigia A, G, cernantur exloco <lb/>Horizontis B. </s> <s xml:id="echoid-s5874" xml:space="preserve">Oportet inueſtigare & </s> <s xml:id="echoid-s5875" xml:space="preserve">diſtantiam F H, & </s> <s xml:id="echoid-s5876" xml:space="preserve">interuallum A G, & </s> <s xml:id="echoid-s5877" xml:space="preserve">v-<lb/>triuſq; </s> <s xml:id="echoid-s5878" xml:space="preserve">turris altitudinem. </s> <s xml:id="echoid-s5879" xml:space="preserve">Sit primum minor turris A F, inter maiorem, & </s> <s xml:id="echoid-s5880" xml:space="preserve">men-<lb/>ſorem, ita vt menſor in eodem cum turribus ſit plano, & </s> <s xml:id="echoid-s5881" xml:space="preserve">minor non occultet fa-<lb/>ſtigium G, maioris. </s> <s xml:id="echoid-s5882" xml:space="preserve">Per ſchol. </s> <s xml:id="echoid-s5883" xml:space="preserve">probl. </s> <s xml:id="echoid-s5884" xml:space="preserve">7. </s> <s xml:id="echoid-s5885" xml:space="preserve">inuenietur & </s> <s xml:id="echoid-s5886" xml:space="preserve">vtraque diſtantia B F, BH, <lb/>& </s> <s xml:id="echoid-s5887" xml:space="preserve">vtra que altitudo A F, G H: </s> <s xml:id="echoid-s5888" xml:space="preserve">ſi nimirum in B, Quadratum ita locetur, vt vnum <lb/>eius latus cum hypotenuſis BA, BG, coincidat, &</s> <s xml:id="echoid-s5889" xml:space="preserve">c. </s> <s xml:id="echoid-s5890" xml:space="preserve">quod eſt quartum, ac terti-<lb/>um. </s> <s xml:id="echoid-s5891" xml:space="preserve">Et quia tria puncta B, F, H, ponuntur in eadem recta, erit diſtantiarum diffe-<lb/>rentia F H, cognita, hoc eſt, diſtantia inter turrium baſes, quod eſt primum. </s> <s xml:id="echoid-s5892" xml:space="preserve">Rur-<lb/>ſus differentia altitudinum GC, nota erit, <anchor type="note" xlink:href="" symbol="a"/> ac propterea in triangulo rectangu- <anchor type="note" xlink:label="note-182-01a" xlink:href="note-182-01"/> lo ACG, ex duobus lateribus. </s> <s xml:id="echoid-s5893" xml:space="preserve">A C, C G, cognitis, baſis quo que A G, efficietur <lb/>nota. </s> <s xml:id="echoid-s5894" xml:space="preserve">quod eſt ſecundum.</s> <s xml:id="echoid-s5895" xml:space="preserve"/> </p> <div xml:id="echoid-div400" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-182-01" xlink:href="note-182-01a" xml:space="preserve">6. triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s5896" xml:space="preserve">2. </s> <s xml:id="echoid-s5897" xml:space="preserve"><emph style="sc">Deinde</emph> conſiſtat menſor in D, ita vtipſe, ac baſes F, H, non iaceant in <lb/>vna linea recta. </s> <s xml:id="echoid-s5898" xml:space="preserve">Per ſcholium problem. </s> <s xml:id="echoid-s5899" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s5900" xml:space="preserve">iterum tam altitu dines AF, GH, quam di-<lb/>ſtantiæ D F, D H, congitæ fient, ſi videlicet <lb/> <anchor type="figure" xlink:label="fig-182-01a" xlink:href="fig-182-01"/> quadrati vnum latus hypotenuſis D A, <lb/>D G, congruet, &</s> <s xml:id="echoid-s5901" xml:space="preserve">c. </s> <s xml:id="echoid-s5902" xml:space="preserve">quod eſt tertium, ac <lb/>quartum. </s> <s xml:id="echoid-s5903" xml:space="preserve">Inueſtigatis autem hypotenuſis <lb/>DA, DG, vt in eodem ſcholio traditũ eſt, <lb/>cognoſcetur per problema 16. </s> <s xml:id="echoid-s5904" xml:space="preserve">præſertim <lb/>per ea, quæ Num. </s> <s xml:id="echoid-s5905" xml:space="preserve">2. </s> <s xml:id="echoid-s5906" xml:space="preserve">eiuſdem problematis <lb/>ſcripſimus, interuallum A G, ſi nimirum in <lb/>hypotenuſis accipentur portiones D I, <lb/>D E, ipſis hypotenuſis proportionales, vt in illo Num. </s> <s xml:id="echoid-s5907" xml:space="preserve">2. </s> <s xml:id="echoid-s5908" xml:space="preserve">diximus, &</s> <s xml:id="echoid-s5909" xml:space="preserve">c. </s> <s xml:id="echoid-s5910" xml:space="preserve">quod eſt <lb/>ſecundum. </s> <s xml:id="echoid-s5911" xml:space="preserve">Et quoniam altitudines AF, GH, notæ factæ ſunt, erit etiam earum <lb/>differentia G C, nota. </s> <s xml:id="echoid-s5912" xml:space="preserve">Quam obrem ex baſe A G, & </s> <s xml:id="echoid-s5913" xml:space="preserve">latere G C, in triangulo re-<lb/>ctangulo ACG, cognitis, latus quoq; </s> <s xml:id="echoid-s5914" xml:space="preserve">AC, hoc eſt, diſtantia F H, inter baſes no-<lb/>ta erit: </s> <s xml:id="echoid-s5915" xml:space="preserve">quod eſt primum.</s> <s xml:id="echoid-s5916" xml:space="preserve"/> </p> <div xml:id="echoid-div401" type="float" level="2" n="2"> <figure xlink:label="fig-182-01" xlink:href="fig-182-01a"> <image file="182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/182-01"/> </figure> </div> <p> <s xml:id="echoid-s5917" xml:space="preserve">SI turres eſſent AF, CH, æquales, eſſet diſtantia A C, inter faſtigia diſtantiæ <lb/>FH, inter baſes æqualis.</s> <s xml:id="echoid-s5918" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div403" type="section" level="1" n="166"> <head xml:id="echoid-head169" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s5919" xml:space="preserve">1. </s> <s xml:id="echoid-s5920" xml:space="preserve">Ex omnibus, quæ demonſtrata ſuntin hoc 3. </s> <s xml:id="echoid-s5921" xml:space="preserve">libro, colligi poteſt regula <lb/> <anchor type="note" xlink:label="note-182-02a" xlink:href="note-182-02"/> generalis ad dimetiendas omnes longitudines, ſiue eæ ſint diſtantię in Horizon-<lb/>te, ſiue altitudines, profunditateſue, ſiue hypotenuſę, id eſt, diſtãtię ab oculo ad <lb/>quo dlibet punctum ſiue interualla inter duo puncta, vbicunq; </s> <s xml:id="echoid-s5922" xml:space="preserve">exiſtant: </s> <s xml:id="echoid-s5923" xml:space="preserve">dum-<lb/>modo vtrumq; </s> <s xml:id="echoid-s5924" xml:space="preserve">extremum longitudinis dimetiendæ videri poſsit à menſore, v-<lb/>bicun que etiam ipſe exiſtat. </s> <s xml:id="echoid-s5925" xml:space="preserve">Nam ſi per problema 15. </s> <s xml:id="echoid-s5926" xml:space="preserve">præſertim per ea, quæ Nu. <lb/></s> <s xml:id="echoid-s5927" xml:space="preserve">5. </s> <s xml:id="echoid-s5928" xml:space="preserve">eius problematis ſcripſimus, diſtantiæ à menſore vſque ad duo extrema lon-<lb/>gitu dinis explorentur, inueſtigato prius angulo, quem duæ illæ diſtantiæ, ſiue <lb/>hypotenuſæ effi ciunt, vt in ſcholio probl. </s> <s xml:id="echoid-s5929" xml:space="preserve">7. </s> <s xml:id="echoid-s5930" xml:space="preserve">Num. </s> <s xml:id="echoid-s5931" xml:space="preserve">2. </s> <s xml:id="echoid-s5932" xml:space="preserve">docuimus; </s> <s xml:id="echoid-s5933" xml:space="preserve">&</s> <s xml:id="echoid-s5934" xml:space="preserve">c. </s> <s xml:id="echoid-s5935" xml:space="preserve">factum e-<lb/>rit, quod proponitur. </s> <s xml:id="echoid-s5936" xml:space="preserve">Itaque ſi diligenter ea, quæ in problem. </s> <s xml:id="echoid-s5937" xml:space="preserve">15. </s> <s xml:id="echoid-s5938" xml:space="preserve">ac 16. </s> <s xml:id="echoid-s5939" xml:space="preserve">ſcripſi- <pb o="153" file="183" n="183" rhead="LIBER TERTIVS."/> mus, percepta fuerint, eadem ſemper ratio metien darum rectarum tenenda erit, <lb/>ſi vtra que extremitas rectæ propoſitæ cerni poteſt, vt diximus.</s> <s xml:id="echoid-s5940" xml:space="preserve"/> </p> <div xml:id="echoid-div403" type="float" level="2" n="1"> <note position="left" xlink:label="note-182-02" xlink:href="note-182-02a" xml:space="preserve">Vnica regula <lb/>adomnes re-<lb/>ct{as} dimetien-<lb/>d{as}, quando <lb/>earum extre-<lb/>ma videntur.</note> </div> <p> <s xml:id="echoid-s5941" xml:space="preserve">2. </s> <s xml:id="echoid-s5942" xml:space="preserve"><emph style="sc">Sed</emph> neque hoc omittendum puto, quando inuentæ ſunt duæ diſtantiæ <lb/>à menſoris loco vſque ad duas extremitates rectæ metiendæ, vna cum angulo <lb/>ab ipſis comprehenſo, certius (quam quam laborioſius) interuallum inter duo <lb/>illa extrema, hoc eſt, rectam propoſitam inueniri poſſe ex duabus illis diſtantijs, <lb/>& </s> <s xml:id="echoid-s5943" xml:space="preserve">angulo comprehenſo, per 12. </s> <s xml:id="echoid-s5944" xml:space="preserve">triang. </s> <s xml:id="echoid-s5945" xml:space="preserve">rectil. </s> <s xml:id="echoid-s5946" xml:space="preserve">quam per duas dictas portiones <lb/>illis diſtantijs proportionales: </s> <s xml:id="echoid-s5947" xml:space="preserve">propterea quod interuallum inter extrema illa-<lb/>rum portionum vix accurate per circinum reperiri poſsit. </s> <s xml:id="echoid-s5948" xml:space="preserve">Id quod etiam in pro-<lb/>blem. </s> <s xml:id="echoid-s5949" xml:space="preserve">16. </s> <s xml:id="echoid-s5950" xml:space="preserve">ad finem Num. </s> <s xml:id="echoid-s5951" xml:space="preserve">2. </s> <s xml:id="echoid-s5952" xml:space="preserve">monuimus.</s> <s xml:id="echoid-s5953" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5954" xml:space="preserve">SPATIVM terræ inæquale pro ducendis aquis librare: </s> <s xml:id="echoid-s5955" xml:space="preserve">aut etiam ſi lu-<lb/>bet, Horizonti æquidiſtans efficere.</s> <s xml:id="echoid-s5956" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div405" type="section" level="1" n="167"> <head xml:id="echoid-head170" xml:space="preserve">PROBLEMA XLV.</head> <p> <s xml:id="echoid-s5957" xml:space="preserve">1. </s> <s xml:id="echoid-s5958" xml:space="preserve"><emph style="sc">Qvando</emph> oblatum ſpatium non eſt valde magnum, excogitauit Ioan-<lb/> <anchor type="note" xlink:label="note-183-01a" xlink:href="note-183-01"/> nes Ferrerius Hiſpanus nobilis Architectus, & </s> <s xml:id="echoid-s5959" xml:space="preserve">Mathematicus, inſtrumentum <lb/> <anchor type="figure" xlink:label="fig-183-01a" xlink:href="fig-183-01"/> percommodum pro librationibus, hoc modo. </s> <s xml:id="echoid-s5960" xml:space="preserve">Compingantur duæ regulæ AB, <lb/>AC, ex ligno aliquo ſolido, ac duro, æqualium crurum, quæ longitudinem ha-<lb/>beant ſatis longam, ita vt diſtantiæ inter extrema B, C, contineat 10. </s> <s xml:id="echoid-s5961" xml:space="preserve">palmos prę-<lb/>ciſe, aut etiam plures. </s> <s xml:id="echoid-s5962" xml:space="preserve">Deinde ducta recta AG, ad BC, perpendiculari, deſcriba-<lb/>tur ex A, ſemicirculus quantuſcunque I D K, cuius ſemidiameter A D, in totæ-<lb/>quales partes ſecetur, quot palmi in diſtantia B C, comprehenduntur. </s> <s xml:id="echoid-s5963" xml:space="preserve">Deſcri-<lb/>pto quoq; </s> <s xml:id="echoid-s5964" xml:space="preserve">circa A D, ſemicirculo occulto A E D, transferantur ex D, in eius pe-<lb/>rip heriam omnia interualla inter D, & </s> <s xml:id="echoid-s5965" xml:space="preserve">puncta rectæ A D: </s> <s xml:id="echoid-s5966" xml:space="preserve">ac tandem ex A, per <lb/>ſingula puncta ſemicirculi AED, rectæ occultæ emittantur, notenturq; </s> <s xml:id="echoid-s5967" xml:space="preserve">interſe-<lb/>ctiones earum cum peripheria D I, at que in alteram peripheriam D K, tranſpor- <pb o="154" file="184" n="184" rhead="GEOMETR. PRACT."/> tentur. </s> <s xml:id="echoid-s5968" xml:space="preserve">Si nam que ex A; </s> <s xml:id="echoid-s5969" xml:space="preserve">filum cum perpendiculo egrediatur, & </s> <s xml:id="echoid-s5970" xml:space="preserve">omnes partes <lb/>excindantur, relictis ſolum cruribus inſtrumenti AB, AC, vna cum perip heria ſe-<lb/>micirculi IDK, cõſtru ctum erit inſtrumentum ad liberationes per opportunum.</s> <s xml:id="echoid-s5971" xml:space="preserve"/> </p> <div xml:id="echoid-div405" type="float" level="2" n="1"> <note position="right" xlink:label="note-183-01" xlink:href="note-183-01a" xml:space="preserve">Inſtrumenti <lb/>conſtructio <lb/>pro librationi-<lb/>b{us}</note> <figure xlink:label="fig-183-01" xlink:href="fig-183-01a"> <image file="183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/183-01"/> </figure> </div> <p> <s xml:id="echoid-s5972" xml:space="preserve">2. </s> <s xml:id="echoid-s5973" xml:space="preserve"><emph style="sc">Nam</emph> in campo aliquo, vel horto, poſitis punctis B, C, in terra, ſi filum <lb/> <anchor type="note" xlink:label="note-184-01a" xlink:href="note-184-01"/> perpendiculi tranſit per D, erunt puncta B, C, in terra eiuſdem altitu dinis, ita vt ſi <lb/>ſpatium in teriectum B C, complanetur, ſpatium illud horti, vel campi ſit libratũ, <lb/>hoc eſt, Horizonti parallelum.</s> <s xml:id="echoid-s5974" xml:space="preserve"/> </p> <div xml:id="echoid-div406" type="float" level="2" n="2"> <note position="left" xlink:label="note-184-01" xlink:href="note-184-01a" xml:space="preserve">Spatium in æ-<lb/>quale quo pa-<lb/>cto libretur.</note> </div> <p> <s xml:id="echoid-s5975" xml:space="preserve"><emph style="sc">Sivero</emph> filum perpendiculi AH, abſcindet ex quadrante DI, aliquot partes, <lb/>nimirum 3. </s> <s xml:id="echoid-s5976" xml:space="preserve">erit punctum C, tribus palmis altius puncto B, atque ita fo diendum <lb/>ibi erit ad altitudinem trium palmorum, vt complanatum ſpatium inter B & </s> <s xml:id="echoid-s5977" xml:space="preserve">in-<lb/>fimum punctum eff oſſum Horizonti ſit parallelum. </s> <s xml:id="echoid-s5978" xml:space="preserve">Quod ſi filum perpendi-<lb/>culi abſcin deret ex alio quadrante DK, quotcunq; </s> <s xml:id="echoid-s5979" xml:space="preserve">partes nimirum 5. </s> <s xml:id="echoid-s5980" xml:space="preserve">eſlet pun-<lb/>ctum C, depreſsius quinque palmis puncto B. </s> <s xml:id="echoid-s5981" xml:space="preserve">Quare tunc ſuperimp onenda, <lb/>foret puncto C, terra ad altitudinem 5. </s> <s xml:id="echoid-s5982" xml:space="preserve">palmorum, vt ſpatium inter B, & </s> <s xml:id="echoid-s5983" xml:space="preserve">ſupre-<lb/>mum punctum terræ ſuperimp oſitæ complanatum Horizonti æquidiſter. </s> <s xml:id="echoid-s5984" xml:space="preserve">Com-<lb/>planato ſpatio inter B, & </s> <s xml:id="echoid-s5985" xml:space="preserve">aliud punctum prope C, ſiue effo ſſum, ſiue eleuatum, <lb/>iteranda erit eadem operatio, poſito crure A B, in puncto inuento, &</s> <s xml:id="echoid-s5986" xml:space="preserve">c. </s> <s xml:id="echoid-s5987" xml:space="preserve">Atque <lb/>ita deinceps procedendum eſt vſque ad vltimum ſignum in horto, vel campo <lb/>propoſitum. </s> <s xml:id="echoid-s5988" xml:space="preserve">Hoc ita demonſtratur. </s> <s xml:id="echoid-s5989" xml:space="preserve">Concipiatur ducta recta BC, & </s> <s xml:id="echoid-s5990" xml:space="preserve">recta CF, <lb/>filo perpendiculi AH, duci parallela, quæ ad Horizontem erit perpendicularis, <lb/>ac proinde ducta BF, ad CF, perpendicularis Horizonti æquidiſtabit. </s> <s xml:id="echoid-s5991" xml:space="preserve">Et quo-<lb/>niam in triangulis A G H, B F C, recti anguli E, F, æquales ſunt, <anchor type="note" xlink:href="" symbol="a"/> nec non & </s> <s xml:id="echoid-s5992" xml:space="preserve">al- <anchor type="note" xlink:label="note-184-02a" xlink:href="note-184-02"/> terni C, H, <anchor type="note" xlink:href="" symbol="b"/> æquiangula erunt triangula; </s> <s xml:id="echoid-s5993" xml:space="preserve">Eſt autem A G H, triangulo A D E, <anchor type="note" xlink:label="note-184-03a" xlink:href="note-184-03"/> æquiangulum, quod & </s> <s xml:id="echoid-s5994" xml:space="preserve">recti G, E, æquales ſint, & </s> <s xml:id="echoid-s5995" xml:space="preserve">A, communis. </s> <s xml:id="echoid-s5996" xml:space="preserve">Igitur tri-<lb/>angula quoque A D E, B C F, æquiangula erunt. </s> <s xml:id="echoid-s5997" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Ideo que erit vt A D, 10.</s> <s xml:id="echoid-s5998" xml:space="preserve"> <anchor type="note" xlink:label="note-184-04a" xlink:href="note-184-04"/> partium ad D E, 3. </s> <s xml:id="echoid-s5999" xml:space="preserve">partium, ita B C, 10. </s> <s xml:id="echoid-s6000" xml:space="preserve">palmorum, ad C E, ac proinde C F, 3. <lb/></s> <s xml:id="echoid-s6001" xml:space="preserve">palmos continebit, tot nimirum, quot partes filum perpen diculi abſcindit ex <lb/>ſemicirculo I D K. </s> <s xml:id="echoid-s6002" xml:space="preserve">Quod ſi quadratum C F, nimirum in dato exemplo 9. </s> <s xml:id="echoid-s6003" xml:space="preserve">pal-<lb/>mi (cum latus C F, ſit 3. </s> <s xml:id="echoid-s6004" xml:space="preserve">palmorum) dematur ex 100. </s> <s xml:id="echoid-s6005" xml:space="preserve">id eſt, ex quadrato BC. </s> <s xml:id="echoid-s6006" xml:space="preserve">10. </s> <s xml:id="echoid-s6007" xml:space="preserve"><lb/>palmorum, reliquum fiet quadratum 91. </s> <s xml:id="echoid-s6008" xml:space="preserve">lineæ horizontalis BF, cuius quadrata <lb/>radix 9 {10/19}. </s> <s xml:id="echoid-s6009" xml:space="preserve">dabit horizontalem diſtantiam B F, à puncto B, vſque ad perpendi-<lb/>cularem CF.</s> <s xml:id="echoid-s6010" xml:space="preserve"/> </p> <div xml:id="echoid-div407" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-184-02" xlink:href="note-184-02a" xml:space="preserve">29. primi.</note> <note symbol="b" position="left" xlink:label="note-184-03" xlink:href="note-184-03a" xml:space="preserve">32. primi.</note> <note symbol="c" position="left" xlink:label="note-184-04" xlink:href="note-184-04a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s6011" xml:space="preserve"><emph style="sc">Hæc</emph> eadem diſtantia horizontalis B F, cogno ſcetur quo que ſine nume-<lb/>rorum ſupputatione, hoc modo. </s> <s xml:id="echoid-s6012" xml:space="preserve">Ex A, deſcribatur alius ſemicirculus, & </s> <s xml:id="echoid-s6013" xml:space="preserve">in ſe-<lb/>micir culum AED, transferantur omnia interualla inter A, & </s> <s xml:id="echoid-s6014" xml:space="preserve">punctarectæ AD, ac <lb/>tandem ex A, rectis occultis emiſsis per puncta in ſemicirculo notata, obſeruẽ-<lb/>tur earum interſectiones cum ſemicir culo ex A, deſcripto, transferantur que in <lb/>alterum quadrantem verſus K. </s> <s xml:id="echoid-s6015" xml:space="preserve">Nam quot partes filum perpen diculi AH, ex vl-<lb/>timo hoc ſemicirculo ex A, deſcripto abſcindet, tot palmos continebit hori-<lb/>zontalis longitudo BF: </s> <s xml:id="echoid-s6016" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> propterea quod eadem eſt pro portio DA, ad AE, quę <anchor type="note" xlink:label="note-184-05a" xlink:href="note-184-05"/> CB, ad BF, quippe cum triangula D A E, CBF, oſtenſa ſint ſimilia. </s> <s xml:id="echoid-s6017" xml:space="preserve">Cum ergo ex <lb/>conſtructione, recta AE, complectatur tot partes rectę A D, quot ex A, in ſemi-<lb/>circulum AED, vſque ad filum perpendiculiſunt translatæ, (vt in noſtro exem-<lb/>plo partes propemodum 9 {1/2}.) </s> <s xml:id="echoid-s6018" xml:space="preserve">comprehendentur totidem palmirectæ CB, in re-<lb/>cta BF. </s> <s xml:id="echoid-s6019" xml:space="preserve">Eſt autem conſideratione dignum, partes poſterioris ſemicirculi contra-<lb/>rio ordine ſimiles eſſe partibus prioris ſemicir culi IDK. </s> <s xml:id="echoid-s6020" xml:space="preserve">Nam ſi verbi gratia rectæ <lb/>D E, quæ æqualis eſt tribus partibus rectæ D A, initium ſumentibus à D, accipia- <pb o="155" file="185" n="185" rhead="LIBER TERTIVS."/> tur æqualis AL, trium quo que partium rectæ AD, initium ſumentium à puncto <lb/>A, ducaturq; </s> <s xml:id="echoid-s6021" xml:space="preserve">recta AL, fiet angulus IAL, æqualis angulo DAE. </s> <s xml:id="echoid-s6022" xml:space="preserve">Quia enim arcus <lb/>DE, AL, æquales ſunt, erunt quoq; </s> <s xml:id="echoid-s6023" xml:space="preserve">reliqui AE, DL, æquales. </s> <s xml:id="echoid-s6024" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur anguli A- <anchor type="note" xlink:label="note-185-01a" xlink:href="note-185-01"/> DE, DAL, æquales erunt: </s> <s xml:id="echoid-s6025" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s6026" xml:space="preserve">& </s> <s xml:id="echoid-s6027" xml:space="preserve">reliqui D A E, I A L, æquales inter ſe erunt: <lb/></s> <s xml:id="echoid-s6028" xml:space="preserve">propterea quod tam duo A D E, D A E, propter rectum E, in ſemicirculo, quam <lb/>duo DAL, IAL, vni recto ſunt æquales. </s> <s xml:id="echoid-s6029" xml:space="preserve">Ex quo fit, rectas AI, AL, ex poſteriori <lb/>ſemicirculo ex A, deſcripto auferre arcum ſimilem arcui in priori ſemicirculo <lb/>IDK, inter rectas AD, AE, inter cepto: </s> <s xml:id="echoid-s6030" xml:space="preserve">Eademque ratio eſt de alijs.</s> <s xml:id="echoid-s6031" xml:space="preserve"/> </p> <div xml:id="echoid-div408" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-184-05" xlink:href="note-184-05a" xml:space="preserve">4. ſexti.</note> <note symbol="a" position="right" xlink:label="note-185-01" xlink:href="note-185-01a" xml:space="preserve">27. tertij.</note> </div> <p> <s xml:id="echoid-s6032" xml:space="preserve"><emph style="sc">Qvando</emph> inſtrumentum ſæpius repetitum fuit, quæritur autem, quanto al-<lb/>tius, vel depreſsius ſit primum punctum, quam vltimum, ſcietur hoc per alti-<lb/>tudines, depreſsioneſue intermedias. </s> <s xml:id="echoid-s6033" xml:space="preserve">Vt ſi primus locus fuerit altior quam ſe-<lb/>cundus, quinque palmis; </s> <s xml:id="echoid-s6034" xml:space="preserve">& </s> <s xml:id="echoid-s6035" xml:space="preserve">hic altior quam tertius, duobus palmis; </s> <s xml:id="echoid-s6036" xml:space="preserve">hic au-<lb/>tem depreſsior, quam quartus locus, tribus palmis; </s> <s xml:id="echoid-s6037" xml:space="preserve">& </s> <s xml:id="echoid-s6038" xml:space="preserve">hic denique altior quam <lb/>vltimus locus, vno palmo, colligemus primum locum altiorem eſſe vltimo lo-<lb/>co quin que palmis. </s> <s xml:id="echoid-s6039" xml:space="preserve">Nam primus locus erit altior tertio ſeptem palmis, cum pri-<lb/>mus ſecundum quin que palmis ſuperet, & </s> <s xml:id="echoid-s6040" xml:space="preserve">ſecundus tertium, duobus. </s> <s xml:id="echoid-s6041" xml:space="preserve">Et quia <lb/>tertius ſuperarur à quarto, tribus palmis, ſuperabit primus quartum quatuor <lb/>palmis. </s> <s xml:id="echoid-s6042" xml:space="preserve">Cum ergo hic altior ſit, quam vltimus, vno palmo, erit primus altior, <lb/>quam vltimus, quin que palmis, & </s> <s xml:id="echoid-s6043" xml:space="preserve">ſic de cæteris.</s> <s xml:id="echoid-s6044" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6045" xml:space="preserve">3. </s> <s xml:id="echoid-s6046" xml:space="preserve"><emph style="sc">Cætervm</emph> quando duo loca multum interſe diſtant, explorabimus per <lb/> <anchor type="note" xlink:label="note-185-02a" xlink:href="note-185-02"/> quadratum ſtabile, quanto alter altero ſit altior, vel humilior, hac ratione. </s> <s xml:id="echoid-s6047" xml:space="preserve">Sit <lb/>primus locus A, ſecundus B, in prima figura. </s> <s xml:id="echoid-s6048" xml:space="preserve">Erecto baculo A D, ſtaturæ men-<lb/>soris æquali, concipiatur ex B, ad perpendiculum loci A, altioris perpendicula-<lb/>ris B C, (per quadratum autem facilè cognoſces, vter locorum ſit altior. </s> <s xml:id="echoid-s6049" xml:space="preserve">Nam <lb/>quando per latus ſupremum Horizonti æquidiſtans, ex A, locus B, cernitur, ea-<lb/>dem eſt altitudo vtriuſque loci Sivero deprimendum eſt illud, vt B, videri poſ-<lb/>ſit, erit locus A, altior: </s> <s xml:id="echoid-s6050" xml:space="preserve">Si denique idem latus attollendum eſt, erit locus B, alti-<lb/>or.) </s> <s xml:id="echoid-s6051" xml:space="preserve">Nam ſi per do ctrinam ſcholij probl. </s> <s xml:id="echoid-s6052" xml:space="preserve">9. </s> <s xml:id="echoid-s6053" xml:space="preserve">inueſtigetur altitudo D C, & </s> <s xml:id="echoid-s6054" xml:space="preserve">aufe-<lb/>ratur menſoris ſtatura A D, reliqua erit altitudo A C: </s> <s xml:id="echoid-s6055" xml:space="preserve">Ac tanto erit altior locus <lb/>A, loco B. </s> <s xml:id="echoid-s6056" xml:space="preserve">Et ſi è contrario in loco humiliori B, erigatur baculus BF, ſtaturæ mẽ-<lb/>ſoris æqualis, & </s> <s xml:id="echoid-s6057" xml:space="preserve">per do cttinam ſcholij probl. </s> <s xml:id="echoid-s6058" xml:space="preserve">7. </s> <s xml:id="echoid-s6059" xml:space="preserve">inquiratur altitudo A E, vſque <lb/>ad perpendicularem FE, per cogitationem ad AC, ductam, & </s> <s xml:id="echoid-s6060" xml:space="preserve">apponatur ſtatu-<lb/>ra menſoris B F, nota euadet tota altitudo A C; </s> <s xml:id="echoid-s6061" xml:space="preserve">Ac tantò depreſsior erit locus <lb/>B, quam locus A. </s> <s xml:id="echoid-s6062" xml:space="preserve">Atque ita ex loco A, ad locum B, conduci poteſt aqua, non <lb/>autem contra.</s> <s xml:id="echoid-s6063" xml:space="preserve"/> </p> <div xml:id="echoid-div409" type="float" level="2" n="5"> <note position="right" xlink:label="note-185-02" xlink:href="note-185-02a" xml:space="preserve">Libration{es} <lb/>pro conducen <lb/>dis aquis, quo <lb/>modo fiant.</note> </div> <p> <s xml:id="echoid-s6064" xml:space="preserve">4. </s> <s xml:id="echoid-s6065" xml:space="preserve"><emph style="sc">Qvod</emph> ſi inter primum locum A, & </s> <s xml:id="echoid-s6066" xml:space="preserve">ſecundum B, interp oſitus ſit mons <lb/>C, ita vt ex A, locus B, videri nequeat, vt in 2. </s> <s xml:id="echoid-s6067" xml:space="preserve">figura, ita procedemus. </s> <s xml:id="echoid-s6068" xml:space="preserve">Ex A, per <lb/>ſcholium problem. </s> <s xml:id="echoid-s6069" xml:space="preserve">7. </s> <s xml:id="echoid-s6070" xml:space="preserve">indagabimus altitudinem CD. </s> <s xml:id="echoid-s6071" xml:space="preserve">Deinde ex C, per ſcholi-<lb/>um problem. </s> <s xml:id="echoid-s6072" xml:space="preserve">9. </s> <s xml:id="echoid-s6073" xml:space="preserve">explorabimus altitudinem C E, ductis nimirum ad CE, perpen-<lb/>dicularibus AD, BE. </s> <s xml:id="echoid-s6074" xml:space="preserve">Nam hac ratione concludes locum A, altiorem eſſe loco <lb/>B, quantitate D E. </s> <s xml:id="echoid-s6075" xml:space="preserve">Quamobrem ſi perfodiatur mons C, vel aquæductus circa <lb/>ipſum extruatur, conduci poterit a qua ex A, ad B, & </s> <s xml:id="echoid-s6076" xml:space="preserve">non contra.</s> <s xml:id="echoid-s6077" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6078" xml:space="preserve">5. </s> <s xml:id="echoid-s6079" xml:space="preserve"><emph style="sc">Postremo</emph> ſi in monte aliquo, vel in eius latere ſit aqua vel in puteo <lb/>aliquo profundo, vel in foſſa aliqua profunda C D; </s> <s xml:id="echoid-s6080" xml:space="preserve">& </s> <s xml:id="echoid-s6081" xml:space="preserve">ſcire deſideres, an ex D, <lb/>ad B, conduci poſsit aqua per aquæductum, ita erit agendũ. </s> <s xml:id="echoid-s6082" xml:space="preserve">Primũ ſi CD, puteus <lb/>eſt, inueſtiga eius profunditatem CD, per probl. </s> <s xml:id="echoid-s6083" xml:space="preserve">27. </s> <s xml:id="echoid-s6084" xml:space="preserve">ſi vero foſſa, aut vorago ali- <pb o="156" file="186" n="186" rhead="GEOMETR. PRACT."/> qua, perproblema 28. </s> <s xml:id="echoid-s6085" xml:space="preserve">Deinde concipiatur ex B, ad perpendiculum C D, duci <lb/>perpendicularis BE; </s> <s xml:id="echoid-s6086" xml:space="preserve">atq; </s> <s xml:id="echoid-s6087" xml:space="preserve">per ſcholium probl. </s> <s xml:id="echoid-s6088" xml:space="preserve">7. </s> <s xml:id="echoid-s6089" xml:space="preserve">ex B, inueſtigetur altitudo CE. <lb/></s> <s xml:id="echoid-s6090" xml:space="preserve"> <anchor type="figure" xlink:label="fig-186-01a" xlink:href="fig-186-01"/> Sinamq; </s> <s xml:id="echoid-s6091" xml:space="preserve">hęc deprehenſa fuerit ęqualis profunditati inuentę CD, habebit fun-<lb/>dus aquę D, eandem altitudinem cumloco B, ac proinde aqua ad B, ex D, de-<lb/>fluere non poterit per a quęductum: </s> <s xml:id="echoid-s6092" xml:space="preserve">Sivero altitudo C E, reperta fuerit maior <lb/>quam profunditas CD, conduci poterit a qua ex D, in locum B: </s> <s xml:id="echoid-s6093" xml:space="preserve">Si denique alti-<lb/>tudo CE, inuenta fuerit minor profunditate CD, erit locus B, altior quam a qua <lb/>iuxta D, at que idcirco defluere non poterit ad B.</s> <s xml:id="echoid-s6094" xml:space="preserve"/> </p> <div xml:id="echoid-div410" type="float" level="2" n="6"> <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/xxxxxxxx/figures/186-01"/> </figure> </div> <p> <s xml:id="echoid-s6095" xml:space="preserve">Ex his non obſcurè intelliges, vbiaquęductus vtiliter ſint extruendi, & </s> <s xml:id="echoid-s6096" xml:space="preserve">vbi <lb/>non.</s> <s xml:id="echoid-s6097" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div412" type="section" level="1" n="168"> <head xml:id="echoid-head171" xml:space="preserve">FINIS LIBRI TERTII.</head> <figure> <image file="186-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/186-02"/> </figure> <pb o="157" file="187" n="187"/> <figure> <image file="187-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/187-01"/> </figure> </div> <div xml:id="echoid-div413" type="section" level="1" n="169"> <head xml:id="echoid-head172" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ <lb/>LIBER QVARTVS.</head> <figure> <image file="187-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/187-02"/> </figure> </div> <div xml:id="echoid-div414" type="section" level="1" n="170"> <head xml:id="echoid-head173" xml:space="preserve">AREAS</head> <p> <s xml:id="echoid-s6098" xml:space="preserve">Superficierum planarum inueſtigans.</s> <s xml:id="echoid-s6099" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6100" xml:space="preserve">QVEMADMODVM linea recta rect{as} line{as} meti-<lb/> <anchor type="note" xlink:label="note-187-01a" xlink:href="note-187-01"/> tur, ita Geometræ ſuperficies plan{as} per ſuperficiem qua-<lb/>dratam, & </s> <s xml:id="echoid-s6101" xml:space="preserve">corpora, ſiue ſolida, per corp{us} cubicum me-<lb/>tiri ſolent. </s> <s xml:id="echoid-s6102" xml:space="preserve">Nam ſicut linea recta dicitur 100. </s> <s xml:id="echoid-s6103" xml:space="preserve">palmorum <lb/>in qua linea vni{us} palmi centies continetur, ita ſuperfi-<lb/>cies plana dicitur 100. </s> <s xml:id="echoid-s6104" xml:space="preserve">palmorum, quæ centies quadra-<lb/>tum continet, cui{us} lat{us} palmo æquale est: </s> <s xml:id="echoid-s6105" xml:space="preserve">& </s> <s xml:id="echoid-s6106" xml:space="preserve">ſolidum <lb/>100. </s> <s xml:id="echoid-s6107" xml:space="preserve">palmorum illud dicitur, quod complectitur 100. </s> <s xml:id="echoid-s6108" xml:space="preserve">cubos, quorum quilibet la-<lb/>t{us} habet vni{us} palmi: </s> <s xml:id="echoid-s6109" xml:space="preserve">quod de aliis etiam menſuris, vt de pede, cubito, paſſu, <lb/>milliario, &</s> <s xml:id="echoid-s6110" xml:space="preserve">c. </s> <s xml:id="echoid-s6111" xml:space="preserve">intelligendum est. </s> <s xml:id="echoid-s6112" xml:space="preserve">Quia vero quælibet ſuperficies tot quadrata <lb/>cuiuſque menſuræ comprehendere dicitur, quot in parallelogrammo rectangulo, <lb/>quod illi æquale est, continentur, explicandum primo loco erit, quaratione area <lb/>cuiuſlibet rectanguli cognoſcatur. </s> <s xml:id="echoid-s6113" xml:space="preserve">Deinde de area triangulorum, quadrilatero-<lb/>rum non rectangulorum, cæterarumque figurarum plurium laterum @gem{us}: <lb/></s> <s xml:id="echoid-s6114" xml:space="preserve">ac denique circulum, eiuſque partes metiemur.</s> <s xml:id="echoid-s6115" xml:space="preserve"/> </p> <div xml:id="echoid-div414" type="float" level="2" n="1"> <note position="right" xlink:label="note-187-01" xlink:href="note-187-01a" xml:space="preserve">Pen{es} quid <lb/>menſuræ li-<lb/>nearum re-<lb/>ctarum, pla-<lb/>narum ſuper-<lb/>ficierum & <lb/>ſolidorum ſu-<lb/>mantur.</note> </div> </div> <div xml:id="echoid-div416" type="section" level="1" n="171"> <head xml:id="echoid-head174" xml:space="preserve">DE AREA RECTANGVLORVM <lb/><emph style="sc">Capvt</emph> I.</head> <note position="right" xml:space="preserve">Area qua-<lb/>drati, & alte-<lb/>ra parte lon-<lb/>gioris quo pa-<lb/>cto cognoſca-<lb/>tur.</note> <p> <s xml:id="echoid-s6116" xml:space="preserve"><emph style="sc">QVoniam</emph> Euclides defin. </s> <s xml:id="echoid-s6117" xml:space="preserve">1. </s> <s xml:id="echoid-s6118" xml:space="preserve">lib. </s> <s xml:id="echoid-s6119" xml:space="preserve">2. </s> <s xml:id="echoid-s6120" xml:space="preserve">docet, omne parallelogram-<lb/>mum rectangulum contineri ſub rectis duabus lineis, quæ rectum <lb/>comprehendunt angulum; </s> <s xml:id="echoid-s6121" xml:space="preserve">manifeſtum eſt, aream cuiuſque re-<lb/>ctanguli produci ex multip licatione duorum laterum circa rectum <pb o="158" file="188" n="188" rhead="GEOMETR. PRACT."/> angulum, vnius in alterum; </s> <s xml:id="echoid-s6122" xml:space="preserve">adeo vt in quadrato ſatis ſit ducere vnum latus in <lb/>ſe, vt eius area cognoſcatur: </s> <s xml:id="echoid-s6123" xml:space="preserve">quippe cum duo latera circa vnum angulum re-<lb/> <anchor type="figure" xlink:label="fig-188-01a" xlink:href="fig-188-01"/> ctum æqualia ſint. </s> <s xml:id="echoid-s6124" xml:space="preserve">Vtin quadrato A B C D, <lb/>cuius ſingula latera quinos palmos con-<lb/>tinent, ſi latus A B, quinque palmorum <lb/>in ſe ducatur, producentur 25. </s> <s xml:id="echoid-s6125" xml:space="preserve">quadrata, <lb/>quorum quodlibet habet latus vnius pal-<lb/>mi; </s> <s xml:id="echoid-s6126" xml:space="preserve">atque tot palmos quadratos conti-<lb/>nere dicetur area quadrati A B C D. </s> <s xml:id="echoid-s6127" xml:space="preserve">At <lb/>area rectanguli altera parte longioris <lb/>EFGH, cuius vnum latus circa rectum an-<lb/>gulum continet 5. </s> <s xml:id="echoid-s6128" xml:space="preserve">pedes, & </s> <s xml:id="echoid-s6129" xml:space="preserve">alterum 3. </s> <s xml:id="echoid-s6130" xml:space="preserve">dicetur continere 15. </s> <s xml:id="echoid-s6131" xml:space="preserve">palmos quadra-<lb/>tos, propterea quod ex mutua laterum 5. </s> <s xml:id="echoid-s6132" xml:space="preserve">& </s> <s xml:id="echoid-s6133" xml:space="preserve">3. </s> <s xml:id="echoid-s6134" xml:space="preserve">multiplicatione numerus 15. <lb/></s> <s xml:id="echoid-s6135" xml:space="preserve">procreatur.</s> <s xml:id="echoid-s6136" xml:space="preserve"/> </p> <div xml:id="echoid-div416" type="float" level="2" n="1"> <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/xxxxxxxx/figures/188-01"/> </figure> </div> <note position="left" xml:space="preserve">Vt camp{us} <lb/>rectangul{us} <lb/>menſuretur, <lb/>quid agen-<lb/>dum.</note> <p> <s xml:id="echoid-s6137" xml:space="preserve">2. </s> <s xml:id="echoid-s6138" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi campum aliquem rectangulum, vel parallelo grammum re-<lb/>ctangulum metiri iubeamur, menſuranda erunt per aliquam menſuram notam, <lb/>vt per palmum, vel pedem, &</s> <s xml:id="echoid-s6139" xml:space="preserve">c. </s> <s xml:id="echoid-s6140" xml:space="preserve">duo latera circa angulum rectum. </s> <s xml:id="echoid-s6141" xml:space="preserve">Nam vno <lb/>in alterum ducto, area propoſiti campi, vel parallelogrammi rectanguli produ-<lb/>cetur, vt dictum eſt.</s> <s xml:id="echoid-s6142" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div418" type="section" level="1" n="172"> <head xml:id="echoid-head175" xml:space="preserve">DE AREA TRIANGVLORVM <lb/><emph style="sc">Capvt</emph> II.</head> <p> <s xml:id="echoid-s6143" xml:space="preserve">1. </s> <s xml:id="echoid-s6144" xml:space="preserve"><emph style="sc">Qvando</emph> trianguli omnia tria latera cognita ſunt, duabus viis eius area <lb/> <anchor type="handwritten" xlink:label="hd-188-2a" xlink:href="hd-188-2"/> cognoſci poteſt. </s> <s xml:id="echoid-s6145" xml:space="preserve">Prima, quæ accuratiſsima eſt, ita ſe habet.</s> <s xml:id="echoid-s6146" xml:space="preserve"><unsure/> Colligantur omnia <lb/>latera in vnam ſummam: </s> <s xml:id="echoid-s6147" xml:space="preserve">Ex hui{us} ſummæ ſemiſſe ſubtr ahantur ſingula latera, vt ha-<lb/>beantur tr{es} differentiæ inter illam ſemiſſem, & </s> <s xml:id="echoid-s6148" xml:space="preserve">latera ſingula: </s> <s xml:id="echoid-s6149" xml:space="preserve">Poſtremo tr{es} hæ diffe-<lb/>rentiæ, & </s> <s xml:id="echoid-s6150" xml:space="preserve">dicta ſemiſſis inter ſe mutuo multiplicentur. </s> <s xml:id="echoid-s6151" xml:space="preserve">Producti enim numeriradix <lb/>quadrata erit area trianguli quæſita.</s> <s xml:id="echoid-s6152" xml:space="preserve"/> </p> <div xml:id="echoid-div418" type="float" level="2" n="1"> <handwritten xlink:label="hd-188-2" xlink:href="hd-188-2a"/> </div> <handwritten/> <p> <s xml:id="echoid-s6153" xml:space="preserve">Verbigratia, ſi latera ſint 10. </s> <s xml:id="echoid-s6154" xml:space="preserve">17. </s> <s xml:id="echoid-s6155" xml:space="preserve">21. </s> <s xml:id="echoid-s6156" xml:space="preserve">erit ſumma ex illis collecta 48. </s> <s xml:id="echoid-s6157" xml:space="preserve">& </s> <s xml:id="echoid-s6158" xml:space="preserve">ſemiſsis <lb/>24. </s> <s xml:id="echoid-s6159" xml:space="preserve">Differentiæ autem inter hanc ſemiſſem, & </s> <s xml:id="echoid-s6160" xml:space="preserve">latera erunt 147. </s> <s xml:id="echoid-s6161" xml:space="preserve">3. </s> <s xml:id="echoid-s6162" xml:space="preserve">Hæ inter ſe <lb/>multip licatæ) ducendo primum 14. </s> <s xml:id="echoid-s6163" xml:space="preserve">in 7. </s> <s xml:id="echoid-s6164" xml:space="preserve">deinde productum in 3.) </s> <s xml:id="echoid-s6165" xml:space="preserve">faciunt <lb/>294. </s> <s xml:id="echoid-s6166" xml:space="preserve">quæ ducta in 24. </s> <s xml:id="echoid-s6167" xml:space="preserve">ſemiſſem prædictam, producunt 7056. </s> <s xml:id="echoid-s6168" xml:space="preserve">cuius numeriradix <lb/>quadrata 84. </s> <s xml:id="echoid-s6169" xml:space="preserve">erit area dicti trianguli, cuius latera ſunt 10. </s> <s xml:id="echoid-s6170" xml:space="preserve">17. </s> <s xml:id="echoid-s6171" xml:space="preserve">21. </s> <s xml:id="echoid-s6172" xml:space="preserve">Rurſus ſi in alio <lb/>quopiam triangulo latera ſint 13. </s> <s xml:id="echoid-s6173" xml:space="preserve">14. </s> <s xml:id="echoid-s6174" xml:space="preserve">15. </s> <s xml:id="echoid-s6175" xml:space="preserve">inueniemus eandem a@eam. </s> <s xml:id="echoid-s6176" xml:space="preserve">Nam ſ@m̃a <lb/>laterum eſt 42. </s> <s xml:id="echoid-s6177" xml:space="preserve">ſemiſsis 21. </s> <s xml:id="echoid-s6178" xml:space="preserve">Differẽtię inter hanc ſemiſſem, & </s> <s xml:id="echoid-s6179" xml:space="preserve">tria latera ſunt 8. </s> <s xml:id="echoid-s6180" xml:space="preserve">7. </s> <s xml:id="echoid-s6181" xml:space="preserve">6. <lb/></s> <s xml:id="echoid-s6182" xml:space="preserve">quæ inter ſe multip licatæ faciunt 336. </s> <s xml:id="echoid-s6183" xml:space="preserve">quæ ducta in 21. </s> <s xml:id="echoid-s6184" xml:space="preserve">ſemiſſem prædictam effi-<lb/>ciunt 7056. </s> <s xml:id="echoid-s6185" xml:space="preserve">cuius numeri quadrata radix 84. </s> <s xml:id="echoid-s6186" xml:space="preserve">dabit aream trianguli, quod lateri-<lb/>bus 13. </s> <s xml:id="echoid-s6187" xml:space="preserve">14. </s> <s xml:id="echoid-s6188" xml:space="preserve">15. </s> <s xml:id="echoid-s6189" xml:space="preserve">continetur. </s> <s xml:id="echoid-s6190" xml:space="preserve">Denique ſi detur triangulum A B C, in quo latus A B, <lb/>7. </s> <s xml:id="echoid-s6191" xml:space="preserve">B C, 10. </s> <s xml:id="echoid-s6192" xml:space="preserve">& </s> <s xml:id="echoid-s6193" xml:space="preserve">A C, 11. </s> <s xml:id="echoid-s6194" xml:space="preserve">ſumma omnium eſt 28. </s> <s xml:id="echoid-s6195" xml:space="preserve">& </s> <s xml:id="echoid-s6196" xml:space="preserve">ſemiſsis 14. </s> <s xml:id="echoid-s6197" xml:space="preserve">quæ latera ſuperat hiſ-<lb/>ce numeris 7. </s> <s xml:id="echoid-s6198" xml:space="preserve">4. </s> <s xml:id="echoid-s6199" xml:space="preserve">3. </s> <s xml:id="echoid-s6200" xml:space="preserve">qui inter ſemultip licati faciunt 84. </s> <s xml:id="echoid-s6201" xml:space="preserve">quę ducta in 14. </s> <s xml:id="echoid-s6202" xml:space="preserve">ſemiſſem <lb/>ſummæ gignunt numerum 1176. </s> <s xml:id="echoid-s6203" xml:space="preserve">cuius radix quadrata 34 {20/69}. </s> <s xml:id="echoid-s6204" xml:space="preserve">ferè dabit aream <lb/>trianguli A B C. </s> <s xml:id="echoid-s6205" xml:space="preserve">Ex quo colliges, non omnis trianguli aream eſſe numerum ra-<lb/>tionalem: </s> <s xml:id="echoid-s6206" xml:space="preserve">propterea quod numerus vltimo loco productus non eſt ſemper <lb/>quadratus, vt in poſtremo hoc exemplo @ontigit.</s> <s xml:id="echoid-s6207" xml:space="preserve"/> </p> <pb o="159" file="189" n="189" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s6208" xml:space="preserve"><emph style="sc">Hanc</emph> praxim, ſiueregulam, quæ exquiſitiſsima eſt, vt dixi, ita in triangu-<lb/> <anchor type="handwritten" xlink:label="hd-189-1a" xlink:href="hd-189-1"/> lo A B C, demonſtrabimus. </s> <s xml:id="echoid-s6209" xml:space="preserve">Diuiſis angulis A B C, A C B, bifariam per rectas <lb/>BD, CD, coeuntes in D, ducantur ex D, ad ſingula latera perpendiculares D E, <lb/>DF, DG, iungatur que recta AD. </s> <s xml:id="echoid-s6210" xml:space="preserve">Quoniamigitur duo anguli E, D B E, in trian-<lb/>gulo DEB, æquales ſunt duobus angulis G, D B G, in triangulo DGB, & </s> <s xml:id="echoid-s6211" xml:space="preserve">latus <lb/>DB; </s> <s xml:id="echoid-s6212" xml:space="preserve">commune; </s> <s xml:id="echoid-s6213" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erunt tam latera DE, DG, quam BE, BG, æqualia. </s> <s xml:id="echoid-s6214" xml:space="preserve">Eodemq;</s> <s xml:id="echoid-s6215" xml:space="preserve"> <anchor type="note" xlink:label="note-189-01a" xlink:href="note-189-01"/> modo tamlatera DF, DG, æqualia eruntin triangulis DFC, DGC: </s> <s xml:id="echoid-s6216" xml:space="preserve">acproinde <lb/>DE, DF, (cum vtraque ipſi D G, ſit oſtenſa æqualis) inter ſe æquales erunt: </s> <s xml:id="echoid-s6217" xml:space="preserve">ideo-<lb/>que omnes tres perpendiculares DE, DF, DG, æquales inter ſe erunt.</s> <s xml:id="echoid-s6218" xml:space="preserve"/> </p> <div xml:id="echoid-div419" type="float" level="2" n="2"> <handwritten xlink:label="hd-189-1" xlink:href="hd-189-1a"/> <note symbol="a" position="right" xlink:label="note-189-01" xlink:href="note-189-01a" xml:space="preserve">26. primi.</note> </div> <p> <s xml:id="echoid-s6219" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> <emph style="sc">Deinde</emph> quia quadrato ex AD, æqualia ſunt tam quadrata ex A E, E D, <anchor type="note" xlink:label="note-189-02a" xlink:href="note-189-02"/> quam quadrata ex A F, F D; </s> <s xml:id="echoid-s6220" xml:space="preserve">æqualia erunt quadrata ex A E, E D, quadratis ex <lb/>AF, FD, Ac proinde ablatis æqualibus quadratis rectarum ED, FD, æqualium, <lb/>reliqua quadrata rectarum A E, A F, æqualia erunt: </s> <s xml:id="echoid-s6221" xml:space="preserve">proptereaque & </s> <s xml:id="echoid-s6222" xml:space="preserve">rectæ <lb/> <anchor type="figure" xlink:label="fig-189-01a" xlink:href="fig-189-01"/> ipſæ A E, A F, æquales erunt. </s> <s xml:id="echoid-s6223" xml:space="preserve">Igitur cum latera A E, <lb/>@A D, trianguli A D E, lateribus A F, A D, trianguli <lb/>A D F, æqualia ſint, & </s> <s xml:id="echoid-s6224" xml:space="preserve">baſis E D, baſi F D; </s> <s xml:id="echoid-s6225" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit an- <anchor type="note" xlink:label="note-189-03a" xlink:href="note-189-03"/> gulus D A E, angulo D A F, æqualis.</s> <s xml:id="echoid-s6226" xml:space="preserve"/> </p> <div xml:id="echoid-div420" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-189-02" xlink:href="note-189-02a" xml:space="preserve">47. primi.</note> <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/xxxxxxxx/figures/189-01"/> </figure> <note symbol="c" position="right" xlink:label="note-189-03" xlink:href="note-189-03a" xml:space="preserve">8. primi.</note> </div> <p> <s xml:id="echoid-s6227" xml:space="preserve"><emph style="sc">Qvia</emph> verò A E, ipſi A F, & </s> <s xml:id="echoid-s6228" xml:space="preserve">E B, ipſi B G, ęqua-<lb/>lis eſt oſtenſa, erit tota A B, duabus A F, B G, ęqua-<lb/>lis: </s> <s xml:id="echoid-s6229" xml:space="preserve">additiſque æqualibus C G, C F, duę A B, C G, <lb/>duabus A C, B G, æquales erunt. </s> <s xml:id="echoid-s6230" xml:space="preserve">Tam ergo duę A B, <lb/>C G, quam duæ A C, B G, ſemiſſem trium laterum <lb/>A B, B C, A C, conſtituent. </s> <s xml:id="echoid-s6231" xml:space="preserve">Quocirca C G, vel C F, <lb/>diifferentia erit inter ſemiſſem laterum, & </s> <s xml:id="echoid-s6232" xml:space="preserve">latus A B. </s> <s xml:id="echoid-s6233" xml:space="preserve">Item B G, vel BE, differen-<lb/>tia inter eandem ſemiſſem, & </s> <s xml:id="echoid-s6234" xml:space="preserve">latus A C. </s> <s xml:id="echoid-s6235" xml:space="preserve">Denique cum A B, C G, ſemiſſem late-<lb/>rum efficiant, ſitque B G, ipſi B E, æqualis, vt oſtendimus, conſtituent quo que <lb/>B C, A E, ſemiſſem eorundem laterum: </s> <s xml:id="echoid-s6236" xml:space="preserve">ideo que A E, differentia erit inter late-<lb/>rum ſemiſſem, & </s> <s xml:id="echoid-s6237" xml:space="preserve">latus B C. </s> <s xml:id="echoid-s6238" xml:space="preserve">Tres ergo rectę A E, E B, C G, & </s> <s xml:id="echoid-s6239" xml:space="preserve">ſemiſſem late-<lb/>rum conſtituunt, & </s> <s xml:id="echoid-s6240" xml:space="preserve">tres differentias inter ſemiſſem laterum, & </s> <s xml:id="echoid-s6241" xml:space="preserve">tria latera trian-<lb/>guli.</s> <s xml:id="echoid-s6242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6243" xml:space="preserve"><emph style="sc">Prodvctis</emph> iam A B, A C, ſit B H, ipſi C G, & </s> <s xml:id="echoid-s6244" xml:space="preserve">C I, ipſi B G, æqualis; </s> <s xml:id="echoid-s6245" xml:space="preserve">ita vt <lb/>tam A H, ſemiſsi laterum, rectis videlicet A B, C G, quam A I, eidem ſemiſsi late-<lb/>rum, rectis nimirum A C, B G, ſit ęqu<gap/>, conſtet que ex tribus differentiis an-<lb/>te dictis. </s> <s xml:id="echoid-s6246" xml:space="preserve">Ducta quo que H K, ad A H, perpendiculari, quę cum A D, producta <lb/>conueniat in K; </s> <s xml:id="echoid-s6247" xml:space="preserve">connectantur rectę K I, K B, K C. </s> <s xml:id="echoid-s6248" xml:space="preserve">Et quia duo latera A H, A K, <lb/>trianguli AHK, duobus lateribus AI, AK, trianguli AIK, ęqualia ſunt, anguloſ-<lb/>que ad A, continent ęquales, vt ſupra oſtendimus, <anchor type="note" xlink:href="" symbol="d"/> æquales quo que erunt &</s> <s xml:id="echoid-s6249" xml:space="preserve"> <anchor type="note" xlink:label="note-189-04a" xlink:href="note-189-04"/> baſes HK, IK, & </s> <s xml:id="echoid-s6250" xml:space="preserve">anguli H, I. </s> <s xml:id="echoid-s6251" xml:space="preserve">Cum ergo H, per conſtructionem ſit rectus, rectus <lb/>etiam erit I.</s> <s xml:id="echoid-s6252" xml:space="preserve"/> </p> <div xml:id="echoid-div421" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-189-04" xlink:href="note-189-04a" xml:space="preserve">4. primi.</note> </div> <p> <s xml:id="echoid-s6253" xml:space="preserve"><emph style="sc">Abscindatvr</emph> pręterea BL, ipſi C G, vel B H, æqualis, vt proinde reli-<lb/>qua C L@ reliquę B G, vel ipſi C I, æqualis ſit, iungaturq; </s> <s xml:id="echoid-s6254" xml:space="preserve">recta KL. </s> <s xml:id="echoid-s6255" xml:space="preserve">Producta au-<lb/>tem B H, ſumatur H M, ipſi C I, æqualis, connectatur querecta L M. </s> <s xml:id="echoid-s6256" xml:space="preserve">Et quia duo <lb/>latera KH, HM, trianguli HMK, duobus later bus KI, IC, trianguli CIK, æqua-<lb/>lia ſunt, angulo ſque H, I, continent ęquales, vt pote rectos: </s> <s xml:id="echoid-s6257" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erunt quo que ba- <anchor type="note" xlink:label="note-189-05a" xlink:href="note-189-05"/> ſes K M, K C, ęquales: </s> <s xml:id="echoid-s6258" xml:space="preserve">at que adeò cum duo latera BM, BK, trianguli BMK, duo-<lb/>bus lateribus B C, B K, trianguli B C K, ęqualia ſint, (eſt nam que B M, ipſi B C, <lb/>æqualis, quod partes B H, H M, partibus B L, L C, ſint æquales) ſit que baſis <pb o="160" file="190" n="190" rhead="GGOMETR. PRACT."/> K M, baſi K C, oſtenſa æqualis; </s> <s xml:id="echoid-s6259" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erunt quo que anguli K B M, KBC, æquales.</s> <s xml:id="echoid-s6260" xml:space="preserve"> <anchor type="note" xlink:label="note-190-01a" xlink:href="note-190-01"/> Itaque quoniam duo latera B H, B K, trianguli B H K, duo bus lateribus B L, B K, <lb/>trianguli B L K, æqualia ſunt, æqualeſq; </s> <s xml:id="echoid-s6261" xml:space="preserve">continent angulos ad B, vt oſtendimus, <lb/> <anchor type="note" xlink:href="" symbol="b"/> erunt & </s> <s xml:id="echoid-s6262" xml:space="preserve">baſes HK, KL, & </s> <s xml:id="echoid-s6263" xml:space="preserve">anguli H, L, æquales. </s> <s xml:id="echoid-s6264" xml:space="preserve">Cum ergo H, ex conſtructio- <anchor type="note" xlink:label="note-190-02a" xlink:href="note-190-02"/> ne rectus ſit, erit quo que L, rectus. </s> <s xml:id="echoid-s6265" xml:space="preserve">Quare cum latera KH, KB, trianguli KBH, <lb/>lateribus KL, KB, trianguli KBL, æqualia ſint, & </s> <s xml:id="echoid-s6266" xml:space="preserve">baſis BH, baſi BL, <anchor type="note" xlink:href="" symbol="c"/> erunt etiam <anchor type="note" xlink:label="note-190-03a" xlink:href="note-190-03"/> anguli BKH, BKL, æquales.</s> <s xml:id="echoid-s6267" xml:space="preserve"/> </p> <div xml:id="echoid-div422" type="float" level="2" n="5"> <note symbol="e" position="right" xlink:label="note-189-05" xlink:href="note-189-05a" xml:space="preserve">8. primi.</note> <note symbol="a" position="left" xlink:label="note-190-01" xlink:href="note-190-01a" xml:space="preserve">8. primi.</note> <note symbol="b" position="left" xlink:label="note-190-02" xlink:href="note-190-02a" xml:space="preserve">4. primi.</note> <note symbol="c" position="left" xlink:label="note-190-03" xlink:href="note-190-03a" xml:space="preserve">8. primi.</note> </div> <p> <s xml:id="echoid-s6268" xml:space="preserve"><emph style="sc">Qvoniam</emph> autem ex iis, quæ ad prop. </s> <s xml:id="echoid-s6269" xml:space="preserve">32. </s> <s xml:id="echoid-s6270" xml:space="preserve">lib. </s> <s xml:id="echoid-s6271" xml:space="preserve">1. </s> <s xml:id="echoid-s6272" xml:space="preserve">Euclidis demonſtrauimus, <lb/>quatuor anguli quadrilateri BHKL, quatuor rectis ſunt æquales: </s> <s xml:id="echoid-s6273" xml:space="preserve">erunt <lb/>demptis duobus rectis H, L, duo HBL, HKL, duobus rectis æquales; </s> <s xml:id="echoid-s6274" xml:space="preserve">ideo que <lb/>duobus angulis HBL, EBL, æquales, <anchor type="note" xlink:href="" symbol="d"/> quod hi quo que duobus ſint rectis æ- <anchor type="note" xlink:label="note-190-04a" xlink:href="note-190-04"/> quales, ablato que communi HBL, reliquus HKL, reliquo EBL, æqualis erit: <lb/></s> <s xml:id="echoid-s6275" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s6276" xml:space="preserve">HKB, ipſi E B D, dimidius dimidio, æqualis erit. </s> <s xml:id="echoid-s6277" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s6278" xml:space="preserve"><lb/>rectus H, recto E, ſit ęqualis, <anchor type="note" xlink:href="" symbol="e"/> erit quo que reliquus H B K, in triangulo H B K, <anchor type="note" xlink:label="note-190-05a" xlink:href="note-190-05"/> reliquo EDB, in triangulo EDB, æqualis; </s> <s xml:id="echoid-s6279" xml:space="preserve">ac proinde triangula BHK, DEB, æ-<lb/>quiangula erunt. </s> <s xml:id="echoid-s6280" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Quapropter erit vt DE, ad EB, ita BH, ad HK; </s> <s xml:id="echoid-s6281" xml:space="preserve">at que idcir- <anchor type="note" xlink:label="note-190-06a" xlink:href="note-190-06"/> co ſi lineæ hę ad numeros contrahantur, <anchor type="note" xlink:href="" symbol="g"/> erit numerus, qui fit ex D E, in H K, <anchor type="note" xlink:label="note-190-07a" xlink:href="note-190-07"/> æqualis numero, qui fit ex EB, in BH. </s> <s xml:id="echoid-s6282" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Eandem ergo proportionem habebit <anchor type="note" xlink:label="note-190-08a" xlink:href="note-190-08"/> quadratus ex DE, ad productum ex DE, in HK, & </s> <s xml:id="echoid-s6283" xml:space="preserve">ad productum ex EB, in BH. <lb/></s> <s xml:id="echoid-s6284" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="i"/> Sedita eſt quadratus ex D E, ad productum ex DE, in HK, vt DE, ad HK:</s> <s xml:id="echoid-s6285" xml:space="preserve"> <anchor type="note" xlink:label="note-190-09a" xlink:href="note-190-09"/> proptera quod DE, multiplicans DE, & </s> <s xml:id="echoid-s6286" xml:space="preserve">HK, fecit & </s> <s xml:id="echoid-s6287" xml:space="preserve">quadratum ex DE, & </s> <s xml:id="echoid-s6288" xml:space="preserve">pro-<lb/>ductum ex D E, in H K. </s> <s xml:id="echoid-s6289" xml:space="preserve">Eritigitur quadratus quo que ex D E, ad productum ex <lb/>EB, in BH, vt DE, ad HK. </s> <s xml:id="echoid-s6290" xml:space="preserve">Vtautem DE, ad HK, ita eſt AE, ad AH. </s> <s xml:id="echoid-s6291" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Nam cum <anchor type="note" xlink:label="note-190-10a" xlink:href="note-190-10"/> parallelæ ſint DE, HK, æquiangula erunt triangula AED, AHK, ex coroll. </s> <s xml:id="echoid-s6292" xml:space="preserve">prop. <lb/></s> <s xml:id="echoid-s6293" xml:space="preserve">4. </s> <s xml:id="echoid-s6294" xml:space="preserve">lib. </s> <s xml:id="echoid-s6295" xml:space="preserve">6. </s> <s xml:id="echoid-s6296" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s6297" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Igitur erit vt AE, ad ED, ita AH, ad HK, & </s> <s xml:id="echoid-s6298" xml:space="preserve">permutando, vt <anchor type="note" xlink:label="note-190-11a" xlink:href="note-190-11"/> AE, ad AH, ita ED, ad HK. </s> <s xml:id="echoid-s6299" xml:space="preserve">Igitur erit quadratus quo que ex D E, ad produ-<lb/>ctum ex EB, in BH, vt AE, ad AH. </s> <s xml:id="echoid-s6300" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> Qui ergo fit ex quadrato ipſius DE, in <anchor type="note" xlink:label="note-190-12a" xlink:href="note-190-12"/> AH, æqualis erit ei, qui fit AE, in productum ex EB, in BH. </s> <s xml:id="echoid-s6301" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s6302" xml:space="preserve">numerus, <lb/>qui ex producto ex quadrato ipſius D E, in A H, multiplicato in A H, gignitur, <lb/>æqualis erit numero, qui ex producto ex AE, in productum ex EB, in BH, mul-<lb/>tiplicato in eundem A H, procreatur. </s> <s xml:id="echoid-s6303" xml:space="preserve">(Nam quia ęquales numeri, nimirum <lb/>productus ex quadrato ipſius DE, in AH, & </s> <s xml:id="echoid-s6304" xml:space="preserve">productus ex AE, in productum ex <lb/>EB, in BH, eundem numerum AH, multiplicant <anchor type="note" xlink:href="" symbol="n"/> habebunt producti, nimirum <anchor type="note" xlink:label="note-190-13a" xlink:href="note-190-13"/> numerus, qui ex producto ex quadrato ipſius DE, in AH, multiplicato in AH, <lb/>gignitur, & </s> <s xml:id="echoid-s6305" xml:space="preserve">numerus, qui ex producto ex AE, in productum ex EB, in BH, mul-<lb/>tiplicato in eundem A H, procreatur, eandem proportionem, quam multipli-<lb/>cantes. </s> <s xml:id="echoid-s6306" xml:space="preserve">Cum ergo hiæquales ſint, erunt & </s> <s xml:id="echoid-s6307" xml:space="preserve">illi producti æquales) hoc eſt, nu-<lb/>merus productus ex AH, in AH, id eſt, quadratus ipſius AH, ductus in quadra-<lb/>tum ipſius DE, (Per ſcholium enim propoſ. </s> <s xml:id="echoid-s6308" xml:space="preserve">19. </s> <s xml:id="echoid-s6309" xml:space="preserve">lib. </s> <s xml:id="echoid-s6310" xml:space="preserve">8. </s> <s xml:id="echoid-s6311" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s6312" xml:space="preserve">quomodocun-<lb/>que tres numeri inter ſe multiplicentur, idem ſemper numerus procreatur) <lb/>æqualis erit numero, qui ex producto ex A E, in productum ex E B, in B H, <lb/>nempe ex producto trium differentiarum A E, E B, B H, inter ſe multipli-<lb/>catarum, ducto in A H, id eſt, in ſemiſſem laterum gignitur. </s> <s xml:id="echoid-s6313" xml:space="preserve">At ex quadra-<lb/>to ipſius D E, in quadratum ipſius A H, producitur quadratus numerus areæ <lb/>trianguli A B C, vt mox oſtendemus. </s> <s xml:id="echoid-s6314" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s6315" xml:space="preserve">ex producto trium exceſ-<lb/>ſuum A E, EB, B H, inter ſe multiplicatorum, ducto in A H, ſemiſſem late-<lb/>rum A B, B C, A C, producitur idem quadratus numerus areæ @@ianguli A B C:</s> <s xml:id="echoid-s6316" xml:space="preserve"> <pb o="161" file="191" n="191" rhead="LIBER QVARTVS."/> ac proinde radix quadratihuius numeri erit dicti trianguli area: </s> <s xml:id="echoid-s6317" xml:space="preserve">quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s6318" xml:space="preserve"/> </p> <div xml:id="echoid-div423" type="float" level="2" n="6"> <note symbol="d" position="left" xlink:label="note-190-04" xlink:href="note-190-04a" xml:space="preserve">13. primi.</note> <note symbol="e" position="left" xlink:label="note-190-05" xlink:href="note-190-05a" xml:space="preserve">32. primi.</note> <note symbol="f" position="left" xlink:label="note-190-06" xlink:href="note-190-06a" xml:space="preserve">4. ſexti.</note> <note symbol="g" position="left" xlink:label="note-190-07" xlink:href="note-190-07a" xml:space="preserve">19. ſeptim.</note> <note symbol="h" position="left" xlink:label="note-190-08" xlink:href="note-190-08a" xml:space="preserve">7. quinti.</note> <note symbol="i" position="left" xlink:label="note-190-09" xlink:href="note-190-09a" xml:space="preserve">17. ſeptim.</note> <note symbol="k" position="left" xlink:label="note-190-10" xlink:href="note-190-10a" xml:space="preserve">28. primi.</note> <note symbol="l" position="left" xlink:label="note-190-11" xlink:href="note-190-11a" xml:space="preserve">4. ſexti.</note> <note symbol="m" position="left" xlink:label="note-190-12" xlink:href="note-190-12a" xml:space="preserve">19. ſept.</note> <note symbol="n" position="left" xlink:label="note-190-13" xlink:href="note-190-13a" xml:space="preserve">18. ſept.</note> </div> <p> <s xml:id="echoid-s6319" xml:space="preserve"><emph style="sc">Qvod</emph> autem ex quadrato ipſius D E, in quadratum ipſius A H, produca-<lb/>tur quadratusnumerus areætrianguli ABC, in hunc modum demonſtro. </s> <s xml:id="echoid-s6320" xml:space="preserve">Quo-<lb/>niam vt Num. </s> <s xml:id="echoid-s6321" xml:space="preserve">2. </s> <s xml:id="echoid-s6322" xml:space="preserve">oſtendemus, ex D E, in ſemiſſem lateris A B, producitur area <lb/>trianguli ADB; </s> <s xml:id="echoid-s6323" xml:space="preserve">Et ex eadem D E, hoc eſt, ex DG, in ſemiſſemlateris BC, effi-<lb/>citur area trianguli B D C; </s> <s xml:id="echoid-s6324" xml:space="preserve">Item ex eadem D E, id eſt, ex D F, in ſemiſſem late-<lb/>ris A C, gignitur area trianguli A D C: </s> <s xml:id="echoid-s6325" xml:space="preserve">Quod autem fit ex D E, in ſemiſſes late-<lb/>rum AB, BC, AC, <anchor type="note" xlink:href="" symbol="a"/> æquale eſt ei, quod fit ex DE, in AH, ex illis ſemiſsibu<unsure/>s con- <anchor type="note" xlink:label="note-191-01a" xlink:href="note-191-01"/> ſlatam. </s> <s xml:id="echoid-s6326" xml:space="preserve">fiet propterea area trianguli A B C, ex DE, in AH, ac propterea (con-<lb/>tractis hiſcelineis ad numeros) quadratus numerus areæ eiuſdem trianguli pro-<lb/>creabitur ex quadrato ipſius DE, in quadratumipſius A H. </s> <s xml:id="echoid-s6327" xml:space="preserve">Quando enim duo <lb/>numeri ſe mutuo multiplicantes fecerint aliquem, producent eorum quadrati <lb/>ſe mutuo multiplicantes quadratum illius producti, quod ita perſpicuum fiet. <lb/></s> <s xml:id="echoid-s6328" xml:space="preserve">Duo numeri A, & </s> <s xml:id="echoid-s6329" xml:space="preserve">B, ſemultiplicantes faciant D; </s> <s xml:id="echoid-s6330" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s6331" xml:space="preserve">ambo ſeipſos multiplicantes faciant C, & </s> <s xml:id="echoid-s6332" xml:space="preserve">E: </s> <s xml:id="echoid-s6333" xml:space="preserve"><lb/>Denique hi quadrati C, & </s> <s xml:id="echoid-s6334" xml:space="preserve">E, ſemultiplicantes <lb/> <anchor type="figure" xlink:label="fig-191-01a" xlink:href="fig-191-01"/> faciant F. </s> <s xml:id="echoid-s6335" xml:space="preserve">Dico F, eſſe quadratum ipſius D. <lb/></s> <s xml:id="echoid-s6336" xml:space="preserve">Cum enim A, multiplicans ſeipſum, & </s> <s xml:id="echoid-s6337" xml:space="preserve">B, faciat <lb/>C, & </s> <s xml:id="echoid-s6338" xml:space="preserve">D: </s> <s xml:id="echoid-s6339" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit vt A, ad B, ita C, ad D: </s> <s xml:id="echoid-s6340" xml:space="preserve">Eadem- <anchor type="note" xlink:label="note-191-02a" xlink:href="note-191-02"/> queratione, cum B, multiplicans A, & </s> <s xml:id="echoid-s6341" xml:space="preserve">ſeipſum, <lb/>faciat D, & </s> <s xml:id="echoid-s6342" xml:space="preserve">E, erit vt A, ad B, ita D, ad E: </s> <s xml:id="echoid-s6343" xml:space="preserve">ideoque C, D, E, continuè propor-<lb/>tionales erunt. </s> <s xml:id="echoid-s6344" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quare qui fit ex C, in E, numerus videlicet F, æqualis erit ei, <anchor type="note" xlink:label="note-191-03a" xlink:href="note-191-03"/> qui fit ex D, in ſe: </s> <s xml:id="echoid-s6345" xml:space="preserve">ac proinde F, quadratus erit ipſius D. </s> <s xml:id="echoid-s6346" xml:space="preserve">Quæ cumita ſint, cum <lb/>ex DE, in AH, producatur area trianguli A B C, vt oſtendimus, fiet ex quadrato <lb/>ipſius DE, in quadratum ipſius AH, quadratus numerus areæ eiuſdem triangu-<lb/>li ABC. </s> <s xml:id="echoid-s6347" xml:space="preserve">Quod erat demonſtrandum.</s> <s xml:id="echoid-s6348" xml:space="preserve"/> </p> <div xml:id="echoid-div424" type="float" level="2" n="7"> <note symbol="a" position="right" xlink:label="note-191-01" xlink:href="note-191-01a" xml:space="preserve">1. ſecun<unsure/>di.</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/xxxxxxxx/figures/191-01"/> </figure> <note symbol="b" position="right" xlink:label="note-191-02" xlink:href="note-191-02a" xml:space="preserve">17. ſept.</note> <note symbol="c" position="right" xlink:label="note-191-03" xlink:href="note-191-03a" xml:space="preserve">20. ſept.</note> </div> <p> <s xml:id="echoid-s6349" xml:space="preserve">2. </s> <s xml:id="echoid-s6350" xml:space="preserve"><emph style="sc">Altera</emph> via, qua ex datis lateribus area trianguli colligitur, <lb/> <anchor type="note" xlink:label="note-191-04a" xlink:href="note-191-04"/> hæc eſt.</s> <s xml:id="echoid-s6351" xml:space="preserve"/> </p> <div xml:id="echoid-div425" type="float" level="2" n="8"> <note position="right" xlink:label="note-191-04" xlink:href="note-191-04a" xml:space="preserve">Area trian-<lb/>guli quo pacto <lb/>aliter ex datis <lb/>laterib{us} colli-<lb/>gatur.</note> </div> <p style="it"> <s xml:id="echoid-s6352" xml:space="preserve">Ex quouis angulo ad lat{us} oppoſitum, etiam protractum, ſiop{us} eſt, perpendicularis <lb/>ducatur. </s> <s xml:id="echoid-s6353" xml:space="preserve">Hæc enim (ſi ei{us} quantit{as} cognita fuerit) multiplicata in ſemiſſem baſis, ſeu <lb/>dicti lateris, vel ei{us} ſemiſſis in totam baſem producet aream trianguli, Velſimauis, tota <lb/>perpendicularis ducta in totam baſem, numerum procreabit, cui{us} ſemiſſis aream trian-<lb/>guli offeret:</s> <s xml:id="echoid-s6354" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6355" xml:space="preserve"><emph style="sc">Nam</emph> vtlib. </s> <s xml:id="echoid-s6356" xml:space="preserve">7. </s> <s xml:id="echoid-s6357" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s6358" xml:space="preserve">1. </s> <s xml:id="echoid-s6359" xml:space="preserve">demonſtrauimus, eſt area trianguli æqualis rectan-<lb/>gulo comprehenſo ſub perpendiculari, & </s> <s xml:id="echoid-s6360" xml:space="preserve">ſemiſſe baſis, vel ſub ſemiſſe perpen-<lb/>dicularis, actotabaſe; </s> <s xml:id="echoid-s6361" xml:space="preserve">Item ſemiſsi rectanguli ſub perpendiculari, ac tota baſe <lb/>comprehenſi. </s> <s xml:id="echoid-s6362" xml:space="preserve">Cum ergo per cap. </s> <s xml:id="echoid-s6363" xml:space="preserve">1. </s> <s xml:id="echoid-s6364" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s6365" xml:space="preserve">area rectanguli illius producatur <lb/>ex multiplicatione vnius lateris circa angulum rectum in alterum: </s> <s xml:id="echoid-s6366" xml:space="preserve">hoc eſt, ex <lb/>perpendiculari (<anchor type="note" xlink:href="" symbol="d"/> quæ vnilateriæqualis eſt) in ſemiſſem baſis trianguli, vel ex <anchor type="note" xlink:label="note-191-05a" xlink:href="note-191-05"/> ſemiſſe perpendicularis (<anchor type="note" xlink:href="" symbol="e"/> quæ ſemiſsi lateris eſt æqualis) in totam baſem: </s> <s xml:id="echoid-s6367" xml:space="preserve">vel <anchor type="note" xlink:label="note-191-06a" xlink:href="note-191-06"/> deniquerectangulum trianguli duplum ex perpendiculariin totam baſem trian-<lb/>guli: </s> <s xml:id="echoid-s6368" xml:space="preserve">conſtat propoſitum.</s> <s xml:id="echoid-s6369" xml:space="preserve"/> </p> <div xml:id="echoid-div426" type="float" level="2" n="9"> <note symbol="d" position="right" xlink:label="note-191-05" xlink:href="note-191-05a" xml:space="preserve">34. primi.</note> <note symbol="e" position="right" xlink:label="note-191-06" xlink:href="note-191-06a" xml:space="preserve">34. primi.</note> </div> <p> <s xml:id="echoid-s6370" xml:space="preserve"><emph style="sc">Magnitvdo</emph> autem dictę perpendicularis, ſicuti & </s> <s xml:id="echoid-s6371" xml:space="preserve">baſis, in metiendis <lb/>campis inueſtiganda eſt per catenulam ferream, quodhęc nequeintendatur, <lb/>neque remittatur, aut certè, ſi omnialatera nota ſint, Geometrice hoc modo. <lb/></s> <s xml:id="echoid-s6372" xml:space="preserve">Sit tr<unsure/>iangulum ABC, cuius latus AB, ſit 10. </s> <s xml:id="echoid-s6373" xml:space="preserve">& </s> <s xml:id="echoid-s6374" xml:space="preserve">B C, 21. </s> <s xml:id="echoid-s6375" xml:space="preserve">& </s> <s xml:id="echoid-s6376" xml:space="preserve">A C, 17. </s> <s xml:id="echoid-s6377" xml:space="preserve">Primuminue- <pb o="162" file="192" n="192" rhead="GEOMETR. PRACT."/> ſtiganda ſunt ſegmenta BD, CD, inter perpendicularem, perea, quæ lib. </s> <s xml:id="echoid-s6378" xml:space="preserve">1. </s> <s xml:id="echoid-s6379" xml:space="preserve">cap. <lb/></s> <s xml:id="echoid-s6380" xml:space="preserve"> <anchor type="note" xlink:label="note-192-01a" xlink:href="note-192-01"/> 3. </s> <s xml:id="echoid-s6381" xml:space="preserve">Nũ. </s> <s xml:id="echoid-s6382" xml:space="preserve">9. </s> <s xml:id="echoid-s6383" xml:space="preserve">ſcrip ſimus, hac ratione. </s> <s xml:id="echoid-s6384" xml:space="preserve">Fiat vt latus BC, in quod cadit perpendicularis <lb/>AD, (Semper autem eſſet demittenda perpendi-<lb/> <anchor type="figure" xlink:label="fig-192-01a" xlink:href="fig-192-01"/> cularis in maximumlatus, vtintra triangulum ca-<lb/>deret) ad ſummam aliorum duorum laterum A B, <lb/>AC, nimirum vt 21. </s> <s xml:id="echoid-s6385" xml:space="preserve">ad 27. </s> <s xml:id="echoid-s6386" xml:space="preserve">ita differentia eorundem <lb/> <anchor type="handwritten" xlink:label="hd-192-2a" xlink:href="hd-192-2"/> laterum, videlicet 7. </s> <s xml:id="echoid-s6387" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s6388" xml:space="preserve">Producetur enim <lb/>numerus 9. </s> <s xml:id="echoid-s6389" xml:space="preserve">(qui quoniam minor eſt latere BC, ar-<lb/>gumento eſt, perpendicularem intra triangulum <lb/>cadere. </s> <s xml:id="echoid-s6390" xml:space="preserve">Si enim maior eſſet, caderet extra, vt prop oſ. </s> <s xml:id="echoid-s6391" xml:space="preserve">9. </s> <s xml:id="echoid-s6392" xml:space="preserve">noſtrorum triangulo-<lb/>rumrectil. </s> <s xml:id="echoid-s6393" xml:space="preserve">oſtendimus) qui ablatus ex latere BC, id eſt, ex 21. </s> <s xml:id="echoid-s6394" xml:space="preserve">relinquit 12. </s> <s xml:id="echoid-s6395" xml:space="preserve">cuius <lb/>ſemiſsis 6. </s> <s xml:id="echoid-s6396" xml:space="preserve">dabit minus ſegmentum B D, prope minus latus AB, quę eadem ſe-<lb/>miſsis ex latere BC, ſubtracta reliquum faciet maius ſegmentum C D, nimirum <lb/>15. </s> <s xml:id="echoid-s6397" xml:space="preserve">iuxta maius latus AC.</s> <s xml:id="echoid-s6398" xml:space="preserve"/> </p> <div xml:id="echoid-div427" type="float" level="2" n="10"> <note position="left" xlink:label="note-192-01" xlink:href="note-192-01a" xml:space="preserve">Segmenta ba-<lb/>ſis, quæ à per-<lb/>pendiculari <lb/>abſcinduntur, <lb/>quo pacto cog-<lb/>noſcatur.</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/xxxxxxxx/figures/192-01"/> </figure> <handwritten xlink:label="hd-192-2" xlink:href="hd-192-2a"/> </div> <handwritten/> <p> <s xml:id="echoid-s6399" xml:space="preserve"><emph style="sc">Sit</emph> rurſus triangulum ABD, cuius latus AB, 12. </s> <s xml:id="echoid-s6400" xml:space="preserve">AD, 11. </s> <s xml:id="echoid-s6401" xml:space="preserve">& </s> <s xml:id="echoid-s6402" xml:space="preserve">B D, 20. </s> <s xml:id="echoid-s6403" xml:space="preserve">demit-<lb/>tenda autem ſit perpendicularis ex D, ad AB. </s> <s xml:id="echoid-s6404" xml:space="preserve">Fiat, vtlatus AB, 12. </s> <s xml:id="echoid-s6405" xml:space="preserve">ad ſummam <lb/>aliorum duorum 31. </s> <s xml:id="echoid-s6406" xml:space="preserve">ita differentia eorundem, ni-<lb/> <anchor type="figure" xlink:label="fig-192-02a" xlink:href="fig-192-02"/> mirum 9. </s> <s xml:id="echoid-s6407" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s6408" xml:space="preserve">Producetur enim numerus <lb/>23 {1/4}. </s> <s xml:id="echoid-s6409" xml:space="preserve">(qui quo niam maior eſt latere AB, argumen-<lb/>to eſt, perpendicularem cadere extra triangulum, <lb/>angulumque A, eſſe obtuſum,) ex quo latus A B, <lb/>12. </s> <s xml:id="echoid-s6410" xml:space="preserve">ſubtractum, relinquit, 11 {1/4}. </s> <s xml:id="echoid-s6411" xml:space="preserve">Huius autem nu-<lb/>meri ſemiſsis 5 {5/8}. </s> <s xml:id="echoid-s6412" xml:space="preserve">dabit ſegmentum exterius A C, <lb/>inter perpendicularem, & </s> <s xml:id="echoid-s6413" xml:space="preserve">angulum obtuſum: </s> <s xml:id="echoid-s6414" xml:space="preserve">eademque ſemiſsis addita lateri <lb/>AB, efficiet aliud ſegmentum BC, inter perpendicularem, & </s> <s xml:id="echoid-s6415" xml:space="preserve">angulum acutum <lb/>17 {5/8}. </s> <s xml:id="echoid-s6416" xml:space="preserve">Atque hæc ratio expeditiſsima eſt.</s> <s xml:id="echoid-s6417" xml:space="preserve"/> </p> <div xml:id="echoid-div428" type="float" level="2" n="11"> <figure xlink:label="fig-192-02" xlink:href="fig-192-02a"> <image file="192-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/192-02"/> </figure> </div> <handwritten/> <p> <s xml:id="echoid-s6418" xml:space="preserve">A<emph style="sc">LITER.</emph> Sit ducenda perpendicularis ad maximum latus B C, in priori <lb/>triangulo ABC, quæ neceſſario cadet intra triangulum; </s> <s xml:id="echoid-s6419" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> propterea, quod an- <anchor type="note" xlink:label="note-192-02a" xlink:href="note-192-02"/> gulus A, maior eſt vtrolibet BC; </s> <s xml:id="echoid-s6420" xml:space="preserve">ac proindevterque horum acutus eſt. </s> <s xml:id="echoid-s6421" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quo- <anchor type="note" xlink:label="note-192-03a" xlink:href="note-192-03"/> niam autem quadratum rectę A B, minus eſt quadratis rectarum A C, BC, re-<lb/> <anchor type="handwritten" xlink:label="hd-192-3a" xlink:href="hd-192-3"/> ctangulo bis comprehenſo ſub BC, & </s> <s xml:id="echoid-s6422" xml:space="preserve">ſegmento CD, inter C, & </s> <s xml:id="echoid-s6423" xml:space="preserve">perpendicula-<lb/>rem: </s> <s xml:id="echoid-s6424" xml:space="preserve">ſi quadratum rectę AB, 100. </s> <s xml:id="echoid-s6425" xml:space="preserve">detrahatur ex ſumma quadratorum rectarum <lb/>A C, B C, id eſt ex 730. </s> <s xml:id="echoid-s6426" xml:space="preserve">reliquum fiet rectangulum 630. </s> <s xml:id="echoid-s6427" xml:space="preserve">bis comprehenſum <lb/>ſub B C, & </s> <s xml:id="echoid-s6428" xml:space="preserve">ſegmento CD, inter C, & </s> <s xml:id="echoid-s6429" xml:space="preserve">perpendicularem. </s> <s xml:id="echoid-s6430" xml:space="preserve">Quare eius ſemiſ-<lb/>ſis 315. </s> <s xml:id="echoid-s6431" xml:space="preserve">æqualis erit rectangulo illi ſemel ſumpto: </s> <s xml:id="echoid-s6432" xml:space="preserve">ac proinde hoc rectangu-<lb/>lo 315. </s> <s xml:id="echoid-s6433" xml:space="preserve">diuiſo per latus B C, 21. </s> <s xml:id="echoid-s6434" xml:space="preserve">dabit Quotiens 15. </s> <s xml:id="echoid-s6435" xml:space="preserve">ſegmentum C D, prope <lb/>angulum acutum C, cui opponitur latus A B, cuius quadratum ex ſumma <lb/>reliquorum quadratorum detractum fuit. </s> <s xml:id="echoid-s6436" xml:space="preserve">Ablato autem ſegmento C D, 15. <lb/></s> <s xml:id="echoid-s6437" xml:space="preserve">ex latere B C, 21. </s> <s xml:id="echoid-s6438" xml:space="preserve">remanebit alterum ſegmentum B D, 6. </s> <s xml:id="echoid-s6439" xml:space="preserve">Eodem pacto, <anchor type="note" xlink:href="" symbol="c"/> quia <anchor type="note" xlink:label="note-192-04a" xlink:href="note-192-04"/> quadratum rectę A C, minus eſt quadratis rectarum A B, B C, rectangulo <lb/> <anchor type="handwritten" xlink:label="hd-192-3a" xlink:href="hd-192-3"/> bis comprehenſo ſub CB, & </s> <s xml:id="echoid-s6440" xml:space="preserve">ſegmento B D, inter B, & </s> <s xml:id="echoid-s6441" xml:space="preserve">perpendicularem: </s> <s xml:id="echoid-s6442" xml:space="preserve">ſi <lb/>quadratum rectę A C, 289. </s> <s xml:id="echoid-s6443" xml:space="preserve">ſubducatur ex ſumma 541. </s> <s xml:id="echoid-s6444" xml:space="preserve">quadratorum recta-<lb/>rum A B, B C, reliquum fiet rectangulum 252. </s> <s xml:id="echoid-s6445" xml:space="preserve">comprehenſum bis ſub C B, <lb/>& </s> <s xml:id="echoid-s6446" xml:space="preserve">ſegmento B D, inter B, & </s> <s xml:id="echoid-s6447" xml:space="preserve">perpendicularem. </s> <s xml:id="echoid-s6448" xml:space="preserve">Quare eius ſemiſsis 126. </s> <s xml:id="echoid-s6449" xml:space="preserve">æ-<lb/>qualis erit illi rectangulo ſemel ſumpto: </s> <s xml:id="echoid-s6450" xml:space="preserve">ac proinde hoc rectangulo 126. </s> <s xml:id="echoid-s6451" xml:space="preserve">di-<lb/>uiſo per latus CB, 21. </s> <s xml:id="echoid-s6452" xml:space="preserve">dabit Quotiens 6. </s> <s xml:id="echoid-s6453" xml:space="preserve">ſegmentum BD, iuxta acutum angu-<lb/>lum B, cui latus A C, quadrati ex duobus aliis quadratis detracti opponitur.</s> <s xml:id="echoid-s6454" xml:space="preserve"> <pb o="163" file="193" n="193" rhead="LIBER QVARTVS."/> Quo ſegmento 6. </s> <s xml:id="echoid-s6455" xml:space="preserve">dempto ex latere B C, 21. </s> <s xml:id="echoid-s6456" xml:space="preserve">remanebit alterum ſegmentum <lb/>C D, 15.</s> <s xml:id="echoid-s6457" xml:space="preserve"/> </p> <div xml:id="echoid-div429" type="float" level="2" n="12"> <note symbol="a" position="left" xlink:label="note-192-02" xlink:href="note-192-02a" xml:space="preserve">18. primi.</note> <note symbol="b" position="left" xlink:label="note-192-03" xlink:href="note-192-03a" xml:space="preserve">13. ſecundi.</note> <handwritten xlink:label="hd-192-3" xlink:href="hd-192-3a"/> <note symbol="c" position="left" xlink:label="note-192-04" xlink:href="note-192-04a" xml:space="preserve">13. ſecundi.</note> <handwritten xlink:label="hd-192-3" xlink:href="hd-192-3a"/> </div> <p> <s xml:id="echoid-s6458" xml:space="preserve"><emph style="sc">Deinde</emph> in poſteriori triangulo ABD, ducenda ſit perpendicularis ad latus <lb/> <anchor type="note" xlink:label="note-193-01a" xlink:href="note-193-01"/> A B, non maximum. </s> <s xml:id="echoid-s6459" xml:space="preserve">Et quia latus DB, latere AD, maius eſt; </s> <s xml:id="echoid-s6460" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit angulus A, <anchor type="note" xlink:label="note-193-02a" xlink:href="note-193-02"/> maior angulo B. </s> <s xml:id="echoid-s6461" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cum ambo ergo ſimul ſint duobus rectis minores, erit ſaltem <anchor type="note" xlink:label="note-193-03a" xlink:href="note-193-03"/> minor B, acutus: </s> <s xml:id="echoid-s6462" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ac proinde quadratum rectæ AD, minus erit quadratis recta- <anchor type="handwritten" xlink:label="hd-193-1a" xlink:href="hd-193-1"/> rum A B, B D, rectangulo bis comprehenſo ſub latere AB, & </s> <s xml:id="echoid-s6463" xml:space="preserve">ſegmento inter B, <lb/>& </s> <s xml:id="echoid-s6464" xml:space="preserve">perpendicularem. </s> <s xml:id="echoid-s6465" xml:space="preserve">Siergo quadratum rectę AD, 121. </s> <s xml:id="echoid-s6466" xml:space="preserve">ſubtrahatur ex ſumma <lb/>quadratorum rectarum AB, B D, id eſt, ex 544. </s> <s xml:id="echoid-s6467" xml:space="preserve">reliquum fiet rectangulum bis <lb/>comprehenſum ſub A B, & </s> <s xml:id="echoid-s6468" xml:space="preserve">ſegmento, inter B, & </s> <s xml:id="echoid-s6469" xml:space="preserve">perpendicularem, nimirum <lb/>423. </s> <s xml:id="echoid-s6470" xml:space="preserve">ideoque eius ſemiſsis 211 {1/2}. </s> <s xml:id="echoid-s6471" xml:space="preserve">æqualis erit illi rectangulo ſemel ſumpto. </s> <s xml:id="echoid-s6472" xml:space="preserve">Qua-<lb/>re ſi rectangulum hoc 211 {1/2}. </s> <s xml:id="echoid-s6473" xml:space="preserve">diuidatur per latus AB, 12. </s> <s xml:id="echoid-s6474" xml:space="preserve">dabit Quotiens 17 {5/8}. </s> <s xml:id="echoid-s6475" xml:space="preserve">ſe-<lb/>gmentum inter B, & </s> <s xml:id="echoid-s6476" xml:space="preserve">perpendicularem. </s> <s xml:id="echoid-s6477" xml:space="preserve">quod quia maius eſt latere A B, argu-<lb/>mento eſt, perpendicularem DC, cadere extra triangulum: </s> <s xml:id="echoid-s6478" xml:space="preserve">ac proinde angu-<lb/>lum A, obtuſum eſſe. </s> <s xml:id="echoid-s6479" xml:space="preserve">Quod ſi ex hoc ſegmento 17 {5/8}. </s> <s xml:id="echoid-s6480" xml:space="preserve">dematur latus AB, 12. </s> <s xml:id="echoid-s6481" xml:space="preserve">re-<lb/>manebit exterius ſegmentum 5 {5/8}.</s> <s xml:id="echoid-s6482" xml:space="preserve"/> </p> <div xml:id="echoid-div430" type="float" level="2" n="13"> <note position="right" xlink:label="note-193-01" xlink:href="note-193-01a" xml:space="preserve">18. primi.</note> <note position="right" xlink:label="note-193-02" xlink:href="note-193-02a" xml:space="preserve">17. primi.</note> <note position="right" xlink:label="note-193-03" xlink:href="note-193-03a" xml:space="preserve">13. primi.</note> <handwritten xlink:label="hd-193-1" xlink:href="hd-193-1a"/> </div> <handwritten/> <p> <s xml:id="echoid-s6483" xml:space="preserve"><emph style="sc">Qvando</emph> conſtat, angulum A, eſſe obtuſum, ideoque perpendicularem <lb/>DC, extra triangulum cadere, reperiemus eadem ſegmenta BC, CA, hoc etiam <lb/> <anchor type="note" xlink:label="note-193-04a" xlink:href="note-193-04"/> modo. </s> <s xml:id="echoid-s6484" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam quadratum lateris BD, ſuperat quadrata laterum A B, AD, <anchor type="handwritten" xlink:label="hd-193-1a" xlink:href="hd-193-1"/> rectangulo bis comprehenſo ſub latere A B, & </s> <s xml:id="echoid-s6485" xml:space="preserve">ſegmento exteriore AC; </s> <s xml:id="echoid-s6486" xml:space="preserve">ſi ſum-<lb/>mam quadratorum rectarum AB, AD, 265. </s> <s xml:id="echoid-s6487" xml:space="preserve">detrahatur ex quadrato lateris BD, <lb/>400. </s> <s xml:id="echoid-s6488" xml:space="preserve">reliquum erit rectangulum 135. </s> <s xml:id="echoid-s6489" xml:space="preserve">bis comprehenſum ſub AB, AC: </s> <s xml:id="echoid-s6490" xml:space="preserve">& </s> <s xml:id="echoid-s6491" xml:space="preserve">eius ſe-<lb/>miſsis 67 {1/2}. </s> <s xml:id="echoid-s6492" xml:space="preserve">illi rectangulo ſemel ſumpto æqualis erit; </s> <s xml:id="echoid-s6493" xml:space="preserve">ac proinde hocrectan-<lb/>gulo 67 {1/2}. </s> <s xml:id="echoid-s6494" xml:space="preserve">diuiſo per latus A B, 12. </s> <s xml:id="echoid-s6495" xml:space="preserve">indicabit Quotiens 5 {5/8}. </s> <s xml:id="echoid-s6496" xml:space="preserve">ſegmentum exterius <lb/> <anchor type="note" xlink:label="note-193-05a" xlink:href="note-193-05"/> CD; </s> <s xml:id="echoid-s6497" xml:space="preserve">cui ſi addatur latus A B, 12. </s> <s xml:id="echoid-s6498" xml:space="preserve">conflabitur ſegmentum BC, 17 {5/8}. </s> <s xml:id="echoid-s6499" xml:space="preserve">Sed prior <lb/>ratio, quæ exlibr. </s> <s xml:id="echoid-s6500" xml:space="preserve">2. </s> <s xml:id="echoid-s6501" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s6502" xml:space="preserve">non pendet, expeditior eſt, ac proinde tenenda: <lb/></s> <s xml:id="echoid-s6503" xml:space="preserve">quamuis auctores alij poſteriorem hanc viam plerunque ſequantur.</s> <s xml:id="echoid-s6504" xml:space="preserve"/> </p> <div xml:id="echoid-div431" type="float" level="2" n="14"> <note symbol="d" position="right" xlink:label="note-193-04" xlink:href="note-193-04a" xml:space="preserve">12. ſecundi.</note> <handwritten xlink:label="hd-193-1" xlink:href="hd-193-1a"/> <note position="right" xlink:label="note-193-05" xlink:href="note-193-05a" xml:space="preserve">Quæ ratio te-<lb/>nenda in ſe-<lb/>gmentis ex-<lb/>quirendis. <lb/>Perpendicu-<lb/>laris in trian-<lb/>gulo quo pa-<lb/>cto reperia-<lb/>tur.</note> </div> <p> <s xml:id="echoid-s6505" xml:space="preserve"><emph style="sc">Inventis</emph> ſegmentis à perpendiculari factis, ita magnitudinem perpendi-<lb/>cularis cognoſcemus.</s> <s xml:id="echoid-s6506" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s6507" xml:space="preserve">DIFFERENTIA inter vtrumuis ſegmentum, & </s> <s xml:id="echoid-s6508" xml:space="preserve">lat{us} adiacens ducatur in <lb/>ſummam ex eodem ſegmento & </s> <s xml:id="echoid-s6509" xml:space="preserve">later@ conflatam. </s> <s xml:id="echoid-s6510" xml:space="preserve">Radix enim quadrata nume-<lb/>ri producti perpendicularem exhibebit notam, vt lib. </s> <s xml:id="echoid-s6511" xml:space="preserve">1. </s> <s xml:id="echoid-s6512" xml:space="preserve">cap. </s> <s xml:id="echoid-s6513" xml:space="preserve">3. </s> <s xml:id="echoid-s6514" xml:space="preserve">Num. </s> <s xml:id="echoid-s6515" xml:space="preserve">17. </s> <s xml:id="echoid-s6516" xml:space="preserve">demonſtra-<lb/>uim{us}.</s> <s xml:id="echoid-s6517" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6518" xml:space="preserve"><emph style="sc">Verbi</emph> gratia. </s> <s xml:id="echoid-s6519" xml:space="preserve">In priori triangulo ABC, ſi differentia 4. </s> <s xml:id="echoid-s6520" xml:space="preserve">inter ſegmentum B D, <lb/> <anchor type="handwritten" xlink:label="hd-193-1a" xlink:href="hd-193-1"/> & </s> <s xml:id="echoid-s6521" xml:space="preserve">latus AB, hoc eſt, inter 6. </s> <s xml:id="echoid-s6522" xml:space="preserve">& </s> <s xml:id="echoid-s6523" xml:space="preserve">10. </s> <s xml:id="echoid-s6524" xml:space="preserve">multip licetur per 16. </s> <s xml:id="echoid-s6525" xml:space="preserve">nempe per ſummam eiuſ-<lb/>dem ſegmenti BD, & </s> <s xml:id="echoid-s6526" xml:space="preserve">lateris AB; </s> <s xml:id="echoid-s6527" xml:space="preserve">gignetur numerus 64. </s> <s xml:id="echoid-s6528" xml:space="preserve">cuius radix quadrata <lb/>8. </s> <s xml:id="echoid-s6529" xml:space="preserve">dabit perpendicularem AD. </s> <s xml:id="echoid-s6530" xml:space="preserve">Pari ratione ſi diſſerentia 2. </s> <s xml:id="echoid-s6531" xml:space="preserve">inter ſegmentum <lb/>CD, & </s> <s xml:id="echoid-s6532" xml:space="preserve">latus AC, hoc eſt, 15. </s> <s xml:id="echoid-s6533" xml:space="preserve">& </s> <s xml:id="echoid-s6534" xml:space="preserve">17. </s> <s xml:id="echoid-s6535" xml:space="preserve">ducaturin 32. </s> <s xml:id="echoid-s6536" xml:space="preserve">id eſt, in ſummam eiuſdem ſeg-<lb/>menti CD, & </s> <s xml:id="echoid-s6537" xml:space="preserve">lateris AC: </s> <s xml:id="echoid-s6538" xml:space="preserve">procreabitur numerus 64. </s> <s xml:id="echoid-s6539" xml:space="preserve">cuius radix quadrata 8. <lb/></s> <s xml:id="echoid-s6540" xml:space="preserve">præbebit perpendicularem AD, vt prius.</s> <s xml:id="echoid-s6541" xml:space="preserve"/> </p> <div xml:id="echoid-div432" type="float" level="2" n="15"> <handwritten xlink:label="hd-193-1" xlink:href="hd-193-1a"/> </div> <p> <s xml:id="echoid-s6542" xml:space="preserve"><emph style="sc">In</emph> poſteriori autem triangulo ABD, ſi differentia 5 {3/8}. </s> <s xml:id="echoid-s6543" xml:space="preserve">inter <lb/>ſegmentum AC, & </s> <s xml:id="echoid-s6544" xml:space="preserve">latus AD, nimirũ inter 5 {5/8}. </s> <s xml:id="echoid-s6545" xml:space="preserve">& </s> <s xml:id="echoid-s6546" xml:space="preserve">11. </s> <s xml:id="echoid-s6547" xml:space="preserve">ducatur <lb/> <anchor type="figure" xlink:label="fig-193-01a" xlink:href="fig-193-01"/> in 16 {5/8}. </s> <s xml:id="echoid-s6548" xml:space="preserve">hoc eſt, in ſum̃am eiuſdem ſegmenti AC, & </s> <s xml:id="echoid-s6549" xml:space="preserve">latus AD: <lb/></s> <s xml:id="echoid-s6550" xml:space="preserve">producetur numerus {5719/64}. </s> <s xml:id="echoid-s6551" xml:space="preserve">ſiue 89 {23/64}. </s> <s xml:id="echoid-s6552" xml:space="preserve">cuius radix quadrata <lb/>in numeris exhiberi non poteſt, ſed paulo maior eſt, quãap-<lb/>poſita fractio cuius numerator eſt 75 {94/151}. </s> <s xml:id="echoid-s6553" xml:space="preserve">denominator aũt 8.</s> <s xml:id="echoid-s6554" xml:space="preserve"> <pb o="164" file="194" n="194" rhead="GEOMETR. PRACT."/> quæad huncnumerum 9 {547/1208}. </s> <s xml:id="echoid-s6555" xml:space="preserve">reducetur, ſi numerator per denominatorem <lb/>diuidatur: </s> <s xml:id="echoid-s6556" xml:space="preserve">atque tanta eſt fermè perpendicularis DC, nimirum 9 {547/1208}. </s> <s xml:id="echoid-s6557" xml:space="preserve">Sic et-<lb/>iam ſi differentia 2 {3/8}. </s> <s xml:id="echoid-s6558" xml:space="preserve">inter ſegmentum B C, 17 {5/8}. </s> <s xml:id="echoid-s6559" xml:space="preserve">& </s> <s xml:id="echoid-s6560" xml:space="preserve">latus B D, 20. </s> <s xml:id="echoid-s6561" xml:space="preserve">multiplice-<lb/>tur per 37 {5/8}. </s> <s xml:id="echoid-s6562" xml:space="preserve">ſummam eiuſdem ſegmenti B C, & </s> <s xml:id="echoid-s6563" xml:space="preserve">latus B D: </s> <s xml:id="echoid-s6564" xml:space="preserve">gignetur idem <lb/>numerus, qui prius, {5719/64}. </s> <s xml:id="echoid-s6565" xml:space="preserve">hoc eſt, 89 {23/64}. </s> <s xml:id="echoid-s6566" xml:space="preserve">cuius radix quadrata paulo ma-<lb/>ior eſt, quam 9 {547/1208}. </s> <s xml:id="echoid-s6567" xml:space="preserve">velutiprius. </s> <s xml:id="echoid-s6568" xml:space="preserve">Atque hęcratio ſatis expedita eſt.</s> <s xml:id="echoid-s6569" xml:space="preserve"/> </p> <div xml:id="echoid-div433" type="float" level="2" n="16"> <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/xxxxxxxx/figures/193-01"/> </figure> </div> <p> <s xml:id="echoid-s6570" xml:space="preserve">A<emph style="sc">LTER.</emph> <anchor type="note" xlink:href="" symbol="a"/> Quoniam quadratum rectę A B, in priori triangulo A B C, <anchor type="note" xlink:label="note-194-01a" xlink:href="note-194-01"/> æquale eſt duobus quadratis rectarum A D, B D: </s> <s xml:id="echoid-s6571" xml:space="preserve">ſi quadratum 36. </s> <s xml:id="echoid-s6572" xml:space="preserve">ſegmen-<lb/>ti B D, tollatur ex 100. </s> <s xml:id="echoid-s6573" xml:space="preserve">quadrato lateris adiacentis A B, relinquetur quadra-<lb/>tum 64. </s> <s xml:id="echoid-s6574" xml:space="preserve">perpendicularis A D. </s> <s xml:id="echoid-s6575" xml:space="preserve">Radix ergo eius quadrata 8. </s> <s xml:id="echoid-s6576" xml:space="preserve">erit magnitudo per-<lb/>pendicula@is A D, vt ſupra. </s> <s xml:id="echoid-s6577" xml:space="preserve">Similiter ſi quadratum 225. </s> <s xml:id="echoid-s6578" xml:space="preserve">ſegmenti C D, dema-<lb/>tur ex 289. </s> <s xml:id="echoid-s6579" xml:space="preserve">quadrato lateris adiacentis A C, reliquum fiet qudratum 64. </s> <s xml:id="echoid-s6580" xml:space="preserve">per-<lb/>pendicularis A D, cuius radix 8. </s> <s xml:id="echoid-s6581" xml:space="preserve">magnitudo erit perpendicularis A D, vt <lb/>prius.</s> <s xml:id="echoid-s6582" xml:space="preserve"/> </p> <div xml:id="echoid-div434" type="float" level="2" n="17"> <note symbol="a" position="left" xlink:label="note-194-01" xlink:href="note-194-01a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s6583" xml:space="preserve"><emph style="sc">In</emph> triangulo verò poſteriori A B D, ſi quadratum 31 {45<unsure/>/64}. </s> <s xml:id="echoid-s6584" xml:space="preserve">ſegmenti AC, 5 {5/8}. <lb/></s> <s xml:id="echoid-s6585" xml:space="preserve">ſubtrahatur ex 121. </s> <s xml:id="echoid-s6586" xml:space="preserve">quadrato lateris adiacentis A D, 11. </s> <s xml:id="echoid-s6587" xml:space="preserve">remanebit quadratum <lb/>89 {23/64}. </s> <s xml:id="echoid-s6588" xml:space="preserve">perpendicularis DC, cuius radix eſt paulò maior quam 9 {547/1208}. </s> <s xml:id="echoid-s6589" xml:space="preserve">vt ſupra. </s> <s xml:id="echoid-s6590" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-194-02a" xlink:href="note-194-02"/> Sic etiam ſi quadratum 310 {41/64}. </s> <s xml:id="echoid-s6591" xml:space="preserve">ſegmenti B C, 17 {5/8}. </s> <s xml:id="echoid-s6592" xml:space="preserve">ſubtrahatur à quadrato 400. <lb/></s> <s xml:id="echoid-s6593" xml:space="preserve">lateris adiacentis B D, 20. </s> <s xml:id="echoid-s6594" xml:space="preserve">remanebitrurſus quadratum 89 {23/64}. </s> <s xml:id="echoid-s6595" xml:space="preserve">perpendicularis <lb/>DC, cuiusradix paulo maior eſt quam 9 {547/1208}. </s> <s xml:id="echoid-s6596" xml:space="preserve">vt ſupra. </s> <s xml:id="echoid-s6597" xml:space="preserve">Verum priorratio ma-<lb/>gis expedita videtur, quanquam alij poſteri@rem hanc viam tradant.</s> <s xml:id="echoid-s6598" xml:space="preserve"/> </p> <div xml:id="echoid-div435" type="float" level="2" n="18"> <note position="left" xlink:label="note-194-02" xlink:href="note-194-02a" xml:space="preserve">Quæ via in-<lb/>ueſtigandæ <lb/>perpendicu-<lb/>laris ſit expe-<lb/>ditior.</note> </div> <p> <s xml:id="echoid-s6599" xml:space="preserve"><emph style="sc">In</emph> triangulo porro æquilatero perpendicularis hoc alio modo inuenie-<lb/>tur. </s> <s xml:id="echoid-s6600" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam quadratum lateris ſeſquitertium eſt quadrati perpendicula- <anchor type="note" xlink:label="note-194-03a" xlink:href="note-194-03"/> ris: </s> <s xml:id="echoid-s6601" xml:space="preserve">ſi fiat vt 4. </s> <s xml:id="echoid-s6602" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s6603" xml:space="preserve">ita quadratum lateris ad aliud, proueniet quadratum per-<lb/>pendicularis. </s> <s xml:id="echoid-s6604" xml:space="preserve">Radix ergo huius quadrati notam exhibebit perpendicularem. <lb/></s> <s xml:id="echoid-s6605" xml:space="preserve">Vtſi latus eſt 10. </s> <s xml:id="echoid-s6606" xml:space="preserve">& </s> <s xml:id="echoid-s6607" xml:space="preserve">fiat vt 4. </s> <s xml:id="echoid-s6608" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s6609" xml:space="preserve">ita 100. </s> <s xml:id="echoid-s6610" xml:space="preserve">quadratum lateris ad aliud gignetur <lb/>quadratum perpendicularis 75. </s> <s xml:id="echoid-s6611" xml:space="preserve">cuius radix 8 {11/17}. </s> <s xml:id="echoid-s6612" xml:space="preserve">proximè erit perpendicula-<lb/>ris quæſita.</s> <s xml:id="echoid-s6613" xml:space="preserve"/> </p> <div xml:id="echoid-div436" type="float" level="2" n="19"> <note position="left" xlink:label="note-194-03" xlink:href="note-194-03a" xml:space="preserve">12. quarti <lb/>decimi.</note> </div> <p> <s xml:id="echoid-s6614" xml:space="preserve"><emph style="sc">Deniqve</emph> inuentis ſegmentis baſis à perpendiculari factis, perpendicu-<lb/>laris ipſa per ſinusita fiet cognita in priori triangulo. </s> <s xml:id="echoid-s6615" xml:space="preserve">Fiat vt 10. </s> <s xml:id="echoid-s6616" xml:space="preserve">latus A B, re-<lb/>cto angulo oppoſitum ad ſinum totum rectianguli D; </s> <s xml:id="echoid-s6617" xml:space="preserve">ita 6. </s> <s xml:id="echoid-s6618" xml:space="preserve">ſegmentum B D, <lb/>ad aliud; </s> <s xml:id="echoid-s6619" xml:space="preserve">produceturque ſinus anguli oppoſiti B A D, 60000. </s> <s xml:id="echoid-s6620" xml:space="preserve">Ergo angulus <lb/>B A D, erit grad. </s> <s xml:id="echoid-s6621" xml:space="preserve">36. </s> <s xml:id="echoid-s6622" xml:space="preserve">Min. </s> <s xml:id="echoid-s6623" xml:space="preserve">52. </s> <s xml:id="echoid-s6624" xml:space="preserve">ac proinde angulus B, eius complementum erit <lb/>grad. </s> <s xml:id="echoid-s6625" xml:space="preserve">53. </s> <s xml:id="echoid-s6626" xml:space="preserve">Min. </s> <s xml:id="echoid-s6627" xml:space="preserve">8. </s> <s xml:id="echoid-s6628" xml:space="preserve">Siergo rurſus fiat, vt ſinustotus ad 10. </s> <s xml:id="echoid-s6629" xml:space="preserve">latus A B: </s> <s xml:id="echoid-s6630" xml:space="preserve">ita 80003. <lb/></s> <s xml:id="echoid-s6631" xml:space="preserve">ſinus anguli B A D, ad aliud, procreabitur latus A D, hoc eſt, perpendicula-<lb/>ris 8 {30/100000}. </s> <s xml:id="echoid-s6632" xml:space="preserve">fermè, quæ paulo maior eſt, quam 8. </s> <s xml:id="echoid-s6633" xml:space="preserve">ſupra inuenta. </s> <s xml:id="echoid-s6634" xml:space="preserve">quod di-<lb/>ſcrimen oritur exeo, quodſinus nonſunt omnino tales, qualesin tabula deſcri-<lb/>buntur: </s> <s xml:id="echoid-s6635" xml:space="preserve">quodcamen in menſuratione camporumnon inducit ſenſibilem ad-<lb/>modumerrorem.</s> <s xml:id="echoid-s6636" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6637" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi in priori triangulo A B C, perpendicularis A D, 8. </s> <s xml:id="echoid-s6638" xml:space="preserve">ducatur in <lb/> <anchor type="note" xlink:label="note-194-04a" xlink:href="note-194-04"/> 10 {1/2}. </s> <s xml:id="echoid-s6639" xml:space="preserve">ſemiſem baſis BC, vel 4. </s> <s xml:id="echoid-s6640" xml:space="preserve">ſemiſsis perpendicularis AD, in 21. </s> <s xml:id="echoid-s6641" xml:space="preserve">totam baſem, <lb/>procreabitur area trianguli ABC, 84. </s> <s xml:id="echoid-s6642" xml:space="preserve">Quę etiam producetur, ſi tota perpendi-<lb/>culari@ 8. </s> <s xml:id="echoid-s6643" xml:space="preserve">in totam baſem 21. </s> <s xml:id="echoid-s6644" xml:space="preserve">multiplicetur, & </s> <s xml:id="echoid-s6645" xml:space="preserve">producti numeri 168. </s> <s xml:id="echoid-s6646" xml:space="preserve">ſemiſsis ac-<lb/>cipi@tur 84.</s> <s xml:id="echoid-s6647" xml:space="preserve"/> </p> <div xml:id="echoid-div437" type="float" level="2" n="20"> <note position="left" xlink:label="note-194-04" xlink:href="note-194-04a" xml:space="preserve">Area prioris <lb/>trianguli <lb/>ABC.</note> </div> <p> <s xml:id="echoid-s6648" xml:space="preserve"><emph style="sc">Item</emph> in poſteriori triangulo ABD, ſi perpendicularis DC, 9 {547/1208}. </s> <s xml:id="echoid-s6649" xml:space="preserve">ducatur <lb/> <anchor type="note" xlink:label="note-194-05a" xlink:href="note-194-05"/> i@ 6. </s> <s xml:id="echoid-s6650" xml:space="preserve">ſemiſſem baſis AB, vel ſemiſsis perpendicularis, nimirum {11419/2416}. </s> <s xml:id="echoid-s6651" xml:space="preserve">vel <pb o="165" file="195" n="195" rhead="LIBER QVARTVS."/> 4 {1755/2416}. </s> <s xml:id="echoid-s6652" xml:space="preserve">in totam baſem 12. </s> <s xml:id="echoid-s6653" xml:space="preserve">conficietur area trianguli ABD, 56 {866/1208}. </s> <s xml:id="echoid-s6654" xml:space="preserve">vel 56 {433/604}. <lb/></s> <s xml:id="echoid-s6655" xml:space="preserve">Quæ etiam producetur, ſi tota perpendicularis in totam baſem ducatur, & </s> <s xml:id="echoid-s6656" xml:space="preserve"><lb/>producti capiatur ſemiſsis.</s> <s xml:id="echoid-s6657" xml:space="preserve"/> </p> <div xml:id="echoid-div438" type="float" level="2" n="21"> <note position="left" xlink:label="note-194-05" xlink:href="note-194-05a" xml:space="preserve">Area poſte-<lb/>rioris trian-<lb/>guli ABD.</note> </div> <p> <s xml:id="echoid-s6658" xml:space="preserve"><emph style="sc">Vt</emph> autem fractiones, quoad eius fieri poteſt, vitentur, curabis, vt quando <lb/> <anchor type="note" xlink:label="note-195-01a" xlink:href="note-195-01"/> perpendicularis eſt numerus par, & </s> <s xml:id="echoid-s6659" xml:space="preserve">baſis numerus impar, accipias ſemiſſem per-<lb/>pendicularis, eamquein totam baſem ducas: </s> <s xml:id="echoid-s6660" xml:space="preserve">quando vero perpendicularis eſt <lb/>numerus impar, & </s> <s xml:id="echoid-s6661" xml:space="preserve">baſis numerus par, ſumas ſemiſſem baſis, eamque ducas in <lb/>totam perpendicularem. </s> <s xml:id="echoid-s6662" xml:space="preserve">Quod ſi tam perpendicularis, quam baſis fuerit nu-<lb/>merus par, velimpar, nihil intereſt, vtrius ſemiſſem capias.</s> <s xml:id="echoid-s6663" xml:space="preserve"/> </p> <div xml:id="echoid-div439" type="float" level="2" n="22"> <note position="right" xlink:label="note-195-01" xlink:href="note-195-01a" xml:space="preserve">Vt fractiones <lb/>vitentur quid <lb/>agendum.</note> </div> <p> <s xml:id="echoid-s6664" xml:space="preserve"><emph style="sc">Qvando</emph> etiam perpendicularis eſt radix ſurda, quæ videlicet numero ex-<lb/> <anchor type="note" xlink:label="note-195-02a" xlink:href="note-195-02"/> primi nequeat, qualis fuit DC, in poſteriori triangulo ABD, rectè feceris, ſi eius <lb/>quadratum (non extracta radiceilla ſurda) multiplices per quadratum ſemiſsis <lb/>baſis. </s> <s xml:id="echoid-s6665" xml:space="preserve">Numerus enim productus erit quadratus numerus areæ trianguli: </s> <s xml:id="echoid-s6666" xml:space="preserve">adeo <lb/>vt radix eius ſit ipſa trianguli area. </s> <s xml:id="echoid-s6667" xml:space="preserve">Hac enimratione minus à vero aberrabimus. <lb/></s> <s xml:id="echoid-s6668" xml:space="preserve">Vtin dicto poſteriori triangulo ABD, ſi quadratum perpendicularis DC, {5719/64}. </s> <s xml:id="echoid-s6669" xml:space="preserve"><lb/>ducamus in 36. </s> <s xml:id="echoid-s6670" xml:space="preserve">quadratum ſemiſsis baſis, producemus {@@@884/64}. </s> <s xml:id="echoid-s6671" xml:space="preserve">quadratuma-<lb/>reæ, cuius radix eſt 56 {2605/3628}. </s> <s xml:id="echoid-s6672" xml:space="preserve">area videlicet trianguli ABD, paulo maior, quam <lb/>priusinuenta. </s> <s xml:id="echoid-s6673" xml:space="preserve">Pariratione, ſi in aliquo triangulo quadratum perpendicularis <lb/>foret 72. </s> <s xml:id="echoid-s6674" xml:space="preserve">& </s> <s xml:id="echoid-s6675" xml:space="preserve">ſemiſsis baſis 6. </s> <s xml:id="echoid-s6676" xml:space="preserve">ſi radicem numeri 72. </s> <s xml:id="echoid-s6677" xml:space="preserve">nimirum 8 {8/17}. </s> <s xml:id="echoid-s6678" xml:space="preserve">hoc eſt, ipſam <lb/>perpendicularem, ducamus in 6. </s> <s xml:id="echoid-s6679" xml:space="preserve">producemus aream 50 {14/17}. </s> <s xml:id="echoid-s6680" xml:space="preserve">At ſi ipſũmet qua-<lb/>dratum 72. </s> <s xml:id="echoid-s6681" xml:space="preserve">multiplicemus per 36. </s> <s xml:id="echoid-s6682" xml:space="preserve">quadratum videlicet ſemiſsis baſis, procrea-<lb/>bimus 2592. </s> <s xml:id="echoid-s6683" xml:space="preserve">quadratum areæ, cuius radix paulo maior eſt, quam 50 {92/101}. </s> <s xml:id="echoid-s6684" xml:space="preserve">quinu-<lb/>merus aliquanto maior eſt, quam area prius inuenta 50 {14/17}. </s> <s xml:id="echoid-s6685" xml:space="preserve">Ratio huius noſtræ <lb/>regulæ eſt, quòd, vt paulò ante ad finem Num. </s> <s xml:id="echoid-s6686" xml:space="preserve">1. </s> <s xml:id="echoid-s6687" xml:space="preserve">demonſtrauimus, duo nume-<lb/>riſeſe multiplicantes producantradicem numeri ex eorum quadratis producti.</s> <s xml:id="echoid-s6688" xml:space="preserve"/> </p> <div xml:id="echoid-div440" type="float" level="2" n="23"> <note position="right" xlink:label="note-195-02" xlink:href="note-195-02a" xml:space="preserve">Quid agen-<lb/>dum, quando <lb/>perpendicula-<lb/>ris eſt nume-<lb/>r{us} ſurd{us}.</note> </div> <p> <s xml:id="echoid-s6689" xml:space="preserve">3. </s> <s xml:id="echoid-s6690" xml:space="preserve"><emph style="sc">Expositis</emph> duabusregulis generalibus, per quas trianguli cuiuslibet <lb/>area ex cognitis lateribus inueſtigatur, proponemus nunc particularia quædam <lb/> <anchor type="note" xlink:label="note-195-03a" xlink:href="note-195-03"/> præcepta pro particularibus triangulis nonnullis, quæ nõnuquam magno vſui <lb/>erunt, cumper ea ſæpenumero expeditius in aliquibus triangulis areæ reperi-<lb/>antur, quam perillas generales regulas. </s> <s xml:id="echoid-s6691" xml:space="preserve">Area ergo triangulirectanguli produ-<lb/>cetur,<unsure/> ſi duo latera circa rectum angulum inter ſe multiplicentur, & </s> <s xml:id="echoid-s6692" xml:space="preserve">numeri pro-<lb/>ductiſemiſsis capiatur. </s> <s xml:id="echoid-s6693" xml:space="preserve">Nam ex multiplicatione illa gignitur parallelogrãmum <lb/>rectangulum ſub duobus lateribus circa angulum rectum comprehenſum, vt c. <lb/></s> <s xml:id="echoid-s6694" xml:space="preserve">1. </s> <s xml:id="echoid-s6695" xml:space="preserve">dictum eſt, <anchor type="note" xlink:href="" symbol="a"/> cuius rectanguli triangulum ſemiſsis eſt, Quod perinde eſt, ac ſi <anchor type="note" xlink:label="note-195-04a" xlink:href="note-195-04"/> ſemiſsis vtriuſuis lateris in totum alterum, tamquam in baſem, multiplicetur. </s> <s xml:id="echoid-s6696" xml:space="preserve">Vt <lb/>in præcedentitriangulo ABC, diuiſo in duo triangula rectangula ADB, ADC; </s> <s xml:id="echoid-s6697" xml:space="preserve">ſi <lb/>AD, 8.</s> <s xml:id="echoid-s6698" xml:space="preserve"><unsure/> ducaturin BD, 6. </s> <s xml:id="echoid-s6699" xml:space="preserve">producetur numerus 48. </s> <s xml:id="echoid-s6700" xml:space="preserve">cuius ſemiſsis 24. </s> <s xml:id="echoid-s6701" xml:space="preserve">erit area tri-<lb/>anguli ADB. </s> <s xml:id="echoid-s6702" xml:space="preserve">Sic ſi AD, 8, ducatur in DC, 15. </s> <s xml:id="echoid-s6703" xml:space="preserve">fiet numerus 120. </s> <s xml:id="echoid-s6704" xml:space="preserve">cuus ſemiſsis 60. <lb/></s> <s xml:id="echoid-s6705" xml:space="preserve">eritarea trianguli ADC: </s> <s xml:id="echoid-s6706" xml:space="preserve">vbi vides, duo triangula 24. </s> <s xml:id="echoid-s6707" xml:space="preserve">& </s> <s xml:id="echoid-s6708" xml:space="preserve">60. </s> <s xml:id="echoid-s6709" xml:space="preserve">componere totum <lb/>triangulum ABC, 84. </s> <s xml:id="echoid-s6710" xml:space="preserve">vtſuprainuenimus.</s> <s xml:id="echoid-s6711" xml:space="preserve"/> </p> <div xml:id="echoid-div441" type="float" level="2" n="24"> <note position="right" xlink:label="note-195-03" xlink:href="note-195-03a" xml:space="preserve">Area triangu <lb/>li rectanguli:</note> <note symbol="a" position="right" xlink:label="note-195-04" xlink:href="note-195-04a" xml:space="preserve">41. primi.</note> </div> <figure> <image file="195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/195-01"/> </figure> <p> <s xml:id="echoid-s6712" xml:space="preserve">4. </s> <s xml:id="echoid-s6713" xml:space="preserve"><emph style="sc">Area</emph> trianguli Iſoſcelis, vel etiam æquilate-<lb/> <anchor type="note" xlink:label="note-195-05a" xlink:href="note-195-05"/> ri, procreabitur, ſi quadratum ſemiſsis baſis ex qua-<lb/>drato lateris auferatur, & </s> <s xml:id="echoid-s6714" xml:space="preserve">reliquus numerusinidem <lb/>quadratum ſemiſsis baſis ducatur, ac denique huius <lb/>ꝓducti radix quadrata eruatur. </s> <s xml:id="echoid-s6715" xml:space="preserve">Vtin Iſoſcele ABC, <lb/>cuius æqualia latera AB, AC, ſint 32. </s> <s xml:id="echoid-s6716" xml:space="preserve">32. </s> <s xml:id="echoid-s6717" xml:space="preserve">& </s> <s xml:id="echoid-s6718" xml:space="preserve">baſis BC, <lb/>24. </s> <s xml:id="echoid-s6719" xml:space="preserve">ſi qua dratum 144. </s> <s xml:id="echoid-s6720" xml:space="preserve">ſemiſsis baſis dematur ex 1024. </s> <s xml:id="echoid-s6721" xml:space="preserve">quadrato lateris AC, vel <pb o="166" file="196" n="196" rhead="GEOMETR. PRACT."/> AB, & </s> <s xml:id="echoid-s6722" xml:space="preserve">reliquus numerus 880. </s> <s xml:id="echoid-s6723" xml:space="preserve">ducatur in 144. </s> <s xml:id="echoid-s6724" xml:space="preserve">quadratũ ſemiſsis baſis, erit pro-<lb/>du @ti 126720. </s> <s xml:id="echoid-s6725" xml:space="preserve">radix quadrata 355 {695/711}. </s> <s xml:id="echoid-s6726" xml:space="preserve">(quæ paulo minor eſt vera radice) area <lb/>trianguli ABC. </s> <s xml:id="echoid-s6727" xml:space="preserve">Nam ſi quadratum ſemiſsis baſis DC, auferatur ex quadrato la-<lb/> <anchor type="note" xlink:label="note-196-01a" xlink:href="note-196-01"/> teris AC,<anchor type="note" xlink:href="" symbol="a"/> reliquum fit quadratum perpendicularis AD, quod ex ſcholio pro- poſ. </s> <s xml:id="echoid-s6728" xml:space="preserve">26. </s> <s xml:id="echoid-s6729" xml:space="preserve">lib. </s> <s xml:id="echoid-s6730" xml:space="preserve">1. </s> <s xml:id="echoid-s6731" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s6732" xml:space="preserve">perpendicularis AD, baſem BC, ſecet bifariam in D. </s> <s xml:id="echoid-s6733" xml:space="preserve">Quare <lb/>vt circa finem Num 2. </s> <s xml:id="echoid-s6734" xml:space="preserve">oſtendimus, quadratum perpendicularis AD, ductum in <lb/>quadratum D C, ſemiſsis baſis producet quadratum areæ trianguli A B C. </s> <s xml:id="echoid-s6735" xml:space="preserve">Ea-<lb/>demq; </s> <s xml:id="echoid-s6736" xml:space="preserve">ratio eſt in triãgulo æquilatero, cum hoc habeat etiã duo latera æqualia. <lb/></s> <s xml:id="echoid-s6737" xml:space="preserve"> <anchor type="note" xlink:label="note-196-02a" xlink:href="note-196-02"/> </s> </p> <div xml:id="echoid-div442" type="float" level="2" n="25"> <note position="right" xlink:label="note-195-05" xlink:href="note-195-05a" xml:space="preserve">Areatrian-<lb/>guli Iſoſcelis.</note> <note symbol="a" position="left" xlink:label="note-196-01" xlink:href="note-196-01a" xml:space="preserve">47. primi.</note> <note position="left" xlink:label="note-196-02" xlink:href="note-196-02a" xml:space="preserve">Area trian-<lb/>guliæ quilate-<lb/>ri.</note> </div> <p> <s xml:id="echoid-s6738" xml:space="preserve">5. </s> <s xml:id="echoid-s6739" xml:space="preserve"><emph style="sc">Pro</emph> area tamen trianguliæ quilateri hæc etiam regula ab auctoribus tra-<lb/>ditur, quamuis à nemine (quod ſciam) demonſtrata ſit. </s> <s xml:id="echoid-s6740" xml:space="preserve">Quadratum lateris du-<lb/>catur in 13. </s> <s xml:id="echoid-s6741" xml:space="preserve">productuſque numerus per 30. </s> <s xml:id="echoid-s6742" xml:space="preserve">diuidatur. </s> <s xml:id="echoid-s6743" xml:space="preserve">Quotiens enim erit area <lb/>trianguli æquilateri. </s> <s xml:id="echoid-s6744" xml:space="preserve">Vt ſi vnum latus æquilateri trianguli ſit 10. </s> <s xml:id="echoid-s6745" xml:space="preserve">ducatur qua-<lb/>dratum lateris 10. </s> <s xml:id="echoid-s6746" xml:space="preserve">nimirum 100, in 13. </s> <s xml:id="echoid-s6747" xml:space="preserve">productuſque numerus 1300. </s> <s xml:id="echoid-s6748" xml:space="preserve">per 30. </s> <s xml:id="echoid-s6749" xml:space="preserve">diui-<lb/>datur. </s> <s xml:id="echoid-s6750" xml:space="preserve">Quotiens enim 43 {1/3}. </s> <s xml:id="echoid-s6751" xml:space="preserve">erit area trianguli. </s> <s xml:id="echoid-s6752" xml:space="preserve">Hanc regulam ita demonſtro. <lb/></s> <s xml:id="echoid-s6753" xml:space="preserve">Area trianguli æquilateri, cuius ſingula latera ſunt 1. </s> <s xml:id="echoid-s6754" xml:space="preserve">eſt radix quadrata huius <lb/>numeri {3/16}. </s> <s xml:id="echoid-s6755" xml:space="preserve">(Nam perregulam præcedentem Nume. </s> <s xml:id="echoid-s6756" xml:space="preserve">4. </s> <s xml:id="echoid-s6757" xml:space="preserve">explicatam, ſi {1/4}. </s> <s xml:id="echoid-s6758" xml:space="preserve">qua-<lb/>dratum ſemiſsis lateris dematur ex 1. </s> <s xml:id="echoid-s6759" xml:space="preserve">quadrato lateris, & </s> <s xml:id="echoid-s6760" xml:space="preserve">reliquus numerus {3/4}. </s> <s xml:id="echoid-s6761" xml:space="preserve"><lb/>ducatur in idem quadratum {1/4}. </s> <s xml:id="echoid-s6762" xml:space="preserve">ſemiſsis lateris, producetur quadratum areæ <lb/>trianguli {3/16}.) </s> <s xml:id="echoid-s6763" xml:space="preserve">nimirum {13/30}. </s> <s xml:id="echoid-s6764" xml:space="preserve">proximè. </s> <s xml:id="echoid-s6765" xml:space="preserve">Cum ergo quadratum lateris 1. </s> <s xml:id="echoid-s6766" xml:space="preserve">ad qua-<lb/>dratum lateris 10. </s> <s xml:id="echoid-s6767" xml:space="preserve">hoc eſt, 1. </s> <s xml:id="echoid-s6768" xml:space="preserve">ad 100. </s> <s xml:id="echoid-s6769" xml:space="preserve">eandem proportionem habeat, quam area <lb/>{13/30}. </s> <s xml:id="echoid-s6770" xml:space="preserve">trianguli, cuius vnumlatus eſt 1. </s> <s xml:id="echoid-s6771" xml:space="preserve">ad aream trianguli, cuius vnum latus eſt 10. </s> <s xml:id="echoid-s6772" xml:space="preserve"><lb/> <anchor type="note" xlink:href="" symbol="b"/> quod vtraque proportio proportionis lateris 1. </s> <s xml:id="echoid-s6773" xml:space="preserve">ad latus 10. </s> <s xml:id="echoid-s6774" xml:space="preserve">ſit duplicata: </s> <s xml:id="echoid-s6775" xml:space="preserve">ſi ſi- <anchor type="note" xlink:label="note-196-03a" xlink:href="note-196-03"/> at vt 1. </s> <s xml:id="echoid-s6776" xml:space="preserve">(quadratum lateris 1.) </s> <s xml:id="echoid-s6777" xml:space="preserve">ad 100. </s> <s xml:id="echoid-s6778" xml:space="preserve">(quadratum lateris 10.) </s> <s xml:id="echoid-s6779" xml:space="preserve">ita area {13/30}. </s> <s xml:id="echoid-s6780" xml:space="preserve">ad a-<lb/>liud, pro ducetur area trianguli, cuius vnum latus eſt 10. </s> <s xml:id="echoid-s6781" xml:space="preserve">Hocautem fit, ducen-<lb/>do ſecundum numerum 100. </s> <s xml:id="echoid-s6782" xml:space="preserve">in tertium {13/30}. </s> <s xml:id="echoid-s6783" xml:space="preserve">hoc eſt, (vt conſtat ex regula mul-<lb/>tiplicationis fra ctorum, ducendo 100. </s> <s xml:id="echoid-s6784" xml:space="preserve">in numeratorem 13. </s> <s xml:id="echoid-s6785" xml:space="preserve">& </s> <s xml:id="echoid-s6786" xml:space="preserve">productum per <lb/>denominatorem 30. </s> <s xml:id="echoid-s6787" xml:space="preserve">diuidendo. </s> <s xml:id="echoid-s6788" xml:space="preserve">Neque vero opus eſt pro ductum hunc nume-<lb/>rum 43 {1/3}. </s> <s xml:id="echoid-s6789" xml:space="preserve">per primum 1. </s> <s xml:id="echoid-s6790" xml:space="preserve">partiri, cum vnitas diuidens, aut multiplicans quem-<lb/>cunq; </s> <s xml:id="echoid-s6791" xml:space="preserve">numerũ producat numerum eundem. </s> <s xml:id="echoid-s6792" xml:space="preserve">Sic etiam ſi latus vnum trianguli <lb/>æquilateri ſit 6. </s> <s xml:id="echoid-s6793" xml:space="preserve">ducemus eius quadratum 36. </s> <s xml:id="echoid-s6794" xml:space="preserve">in {13/30}. </s> <s xml:id="echoid-s6795" xml:space="preserve">hoc eſt in 13. </s> <s xml:id="echoid-s6796" xml:space="preserve">numeratorem, <lb/>productumq; </s> <s xml:id="echoid-s6797" xml:space="preserve">468. </s> <s xml:id="echoid-s6798" xml:space="preserve">per 30. </s> <s xml:id="echoid-s6799" xml:space="preserve">partiemur. </s> <s xml:id="echoid-s6800" xml:space="preserve">Quotiens namq; </s> <s xml:id="echoid-s6801" xml:space="preserve">15 {3/5}. </s> <s xml:id="echoid-s6802" xml:space="preserve">erit trianguli pro-<lb/>poſiti area.</s> <s xml:id="echoid-s6803" xml:space="preserve"/> </p> <div xml:id="echoid-div443" type="float" level="2" n="26"> <note symbol="b" position="left" xlink:label="note-196-03" xlink:href="note-196-03a" xml:space="preserve">20. & 19. <lb/>ſexti.</note> </div> <p> <s xml:id="echoid-s6804" xml:space="preserve"><emph style="sc">Qvod</emph> autem {13/30}. </s> <s xml:id="echoid-s6805" xml:space="preserve">ſitradix quadrata numeri {3/16}, patet ex regula qua cuiuſ-<lb/> <anchor type="note" xlink:label="note-196-04a" xlink:href="note-196-04"/> uis fracti numeri radix extrahitur: </s> <s xml:id="echoid-s6806" xml:space="preserve">quæ talis eſt. </s> <s xml:id="echoid-s6807" xml:space="preserve">Numerator in denominatorem <lb/>ducatur, & </s> <s xml:id="echoid-s6808" xml:space="preserve">productiradix propinqua inueniatur. </s> <s xml:id="echoid-s6809" xml:space="preserve">Sienim per hanc radicem <lb/>diuidemus numeratorem: </s> <s xml:id="echoid-s6810" xml:space="preserve">velipſam radicem per denominatorem partiemur; <lb/></s> <s xml:id="echoid-s6811" xml:space="preserve">exibit radix fractionis propoſitæ: </s> <s xml:id="echoid-s6812" xml:space="preserve">priori quidem modo maior quam vera, po-<lb/>ſterioriautem minor, ſi radix illa propinqua producti ex numeratore in deno-<lb/>minatorem fuerit minor, quam vera: </s> <s xml:id="echoid-s6813" xml:space="preserve">quia in priori illo modo fit diuiſio per nu-<lb/>merum vero minorem, in poſteriori autẽ numerus vero minor diuiditur. </s> <s xml:id="echoid-s6814" xml:space="preserve">Quod <lb/>ſi radixilla propinqua foret maior, quam vera, produceretur priori modo radix <lb/>fractionis minor, quam vera, poſterioriautem maior, vtliquet. </s> <s xml:id="echoid-s6815" xml:space="preserve">Verbi gratia. </s> <s xml:id="echoid-s6816" xml:space="preserve">In-<lb/>uenienda ſitradix quadrata fra ctionis {3/16}. </s> <s xml:id="echoid-s6817" xml:space="preserve">quam diximus eſſe quadratum areæ <lb/>trianguli æquilateri, cuius vnum latus eſt. </s> <s xml:id="echoid-s6818" xml:space="preserve">1. </s> <s xml:id="echoid-s6819" xml:space="preserve">Ex 3. </s> <s xml:id="echoid-s6820" xml:space="preserve">in 16. </s> <s xml:id="echoid-s6821" xml:space="preserve">fit numerus 48. </s> <s xml:id="echoid-s6822" xml:space="preserve">cu-<lb/>ius radix propinqua 6 {12/13}. </s> <s xml:id="echoid-s6823" xml:space="preserve">minor quam vera, per quam ſi diuidatur numerator 3. </s> <s xml:id="echoid-s6824" xml:space="preserve"><lb/>prodibit radix {13/30}. </s> <s xml:id="echoid-s6825" xml:space="preserve">fractionis {3/16}. </s> <s xml:id="echoid-s6826" xml:space="preserve">maior, quam vera: </s> <s xml:id="echoid-s6827" xml:space="preserve">at ſi radicem eandem pro-<lb/>pinquam 6 {12/13}. </s> <s xml:id="echoid-s6828" xml:space="preserve">partiamur per denominatorem 16. </s> <s xml:id="echoid-s6829" xml:space="preserve">reperietur radix {45/104}. </s> <s xml:id="echoid-s6830" xml:space="preserve">eiuſdem <pb o="167" file="197" n="197" rhead="LIBER QVARTVS."/> fractionis {3/16}. </s> <s xml:id="echoid-s6831" xml:space="preserve">minor, quam vera. </s> <s xml:id="echoid-s6832" xml:space="preserve">Ratio huius extractionis hæc eſt. </s> <s xml:id="echoid-s6833" xml:space="preserve">Quando <lb/>numerator 3. </s> <s xml:id="echoid-s6834" xml:space="preserve">denominatorem 16. </s> <s xml:id="echoid-s6835" xml:space="preserve">multiplicat, <anchor type="note" xlink:href="" symbol="a"/> erit producti 48. </s> <s xml:id="echoid-s6836" xml:space="preserve">radix quadra- <anchor type="note" xlink:label="note-197-01a" xlink:href="note-197-01"/> ta medio loco proportionalis inter 3. </s> <s xml:id="echoid-s6837" xml:space="preserve">& </s> <s xml:id="echoid-s6838" xml:space="preserve">16. </s> <s xml:id="echoid-s6839" xml:space="preserve">quod radix hæc in ſe ducta produ-<lb/>cat numerum 48. </s> <s xml:id="echoid-s6840" xml:space="preserve">æqualemei, qui ab extremis 3. </s> <s xml:id="echoid-s6841" xml:space="preserve">& </s> <s xml:id="echoid-s6842" xml:space="preserve">16. </s> <s xml:id="echoid-s6843" xml:space="preserve">inter ſe multiplicatis gi-<lb/>gnitur: </s> <s xml:id="echoid-s6844" xml:space="preserve">ac proinde proportio 3. </s> <s xml:id="echoid-s6845" xml:space="preserve">ad 16. </s> <s xml:id="echoid-s6846" xml:space="preserve">erit duplicata tam proportionis 3. </s> <s xml:id="echoid-s6847" xml:space="preserve">ad il-<lb/>lam radicem, quam illius radicis ad 16. </s> <s xml:id="echoid-s6848" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cum ergo proportio 3. </s> <s xml:id="echoid-s6849" xml:space="preserve">ad 16. </s> <s xml:id="echoid-s6850" xml:space="preserve">ſit quo- <anchor type="note" xlink:label="note-197-02a" xlink:href="note-197-02"/> que duplicata proportionis, quam radix numeri 3. </s> <s xml:id="echoid-s6851" xml:space="preserve">ad radicem numeri 16. </s> <s xml:id="echoid-s6852" xml:space="preserve">habet: <lb/></s> <s xml:id="echoid-s6853" xml:space="preserve">erit vt 3. </s> <s xml:id="echoid-s6854" xml:space="preserve">ad radicem producti 48. </s> <s xml:id="echoid-s6855" xml:space="preserve">Vel vtradix huius producti ad 16. </s> <s xml:id="echoid-s6856" xml:space="preserve">ita radix <lb/>numeri 3. </s> <s xml:id="echoid-s6857" xml:space="preserve">ad radicem numeri 16. </s> <s xml:id="echoid-s6858" xml:space="preserve">Quapropter cum fractio, cuius numerator eſt <lb/>radix numeri 3. </s> <s xml:id="echoid-s6859" xml:space="preserve">denominator autem radix numeri 16. </s> <s xml:id="echoid-s6860" xml:space="preserve">ſit radix fractionis {3/16}. </s> <s xml:id="echoid-s6861" xml:space="preserve">erit <lb/>quo quetam fractio, cuius numerator 3. </s> <s xml:id="echoid-s6862" xml:space="preserve">& </s> <s xml:id="echoid-s6863" xml:space="preserve">denominator radix producti 48. </s> <s xml:id="echoid-s6864" xml:space="preserve">quã <lb/>fractio cuius numeratorradix producti 48. </s> <s xml:id="echoid-s6865" xml:space="preserve">denominator autem 16. </s> <s xml:id="echoid-s6866" xml:space="preserve">hoc eſt, tam <lb/>Quotiens, qui fit ex diuiſione 3. </s> <s xml:id="echoid-s6867" xml:space="preserve">per radicem producti 48. </s> <s xml:id="echoid-s6868" xml:space="preserve">quam quotiens, qui <lb/>fit ex diuiſione radicis producti 48. </s> <s xml:id="echoid-s6869" xml:space="preserve">per 16. </s> <s xml:id="echoid-s6870" xml:space="preserve">radix propinqua fractionis {3/16}. </s> <s xml:id="echoid-s6871" xml:space="preserve">Ea-<lb/>demque de cæteris ratio eſt.</s> <s xml:id="echoid-s6872" xml:space="preserve"/> </p> <div xml:id="echoid-div444" type="float" level="2" n="27"> <note position="left" xlink:label="note-196-04" xlink:href="note-196-04a" xml:space="preserve">Radix qua-<lb/>drata numeri <lb/>fracti quo pa-<lb/>cto eruatur.</note> <note symbol="a" position="right" xlink:label="note-197-01" xlink:href="note-197-01a" xml:space="preserve">20. ſept.</note> <note symbol="b" position="right" xlink:label="note-197-02" xlink:href="note-197-02a" xml:space="preserve">11. octaus<unsure/>.</note> </div> <p> <s xml:id="echoid-s6873" xml:space="preserve"><emph style="sc">Alii</emph> hanc tra dunt regulam ad aream trianguli æ quilateri inueniendam. </s> <s xml:id="echoid-s6874" xml:space="preserve">Ex <lb/>quadrato lateris ſumatur tam pars decima, quàm tertia. </s> <s xml:id="echoid-s6875" xml:space="preserve">Harum enim parti-<lb/>um ſumma erit area trianguli. </s> <s xml:id="echoid-s6876" xml:space="preserve">Quod ſic oſtendo. </s> <s xml:id="echoid-s6877" xml:space="preserve">Hæfractiones {1/10}. </s> <s xml:id="echoid-s6878" xml:space="preserve">& </s> <s xml:id="echoid-s6879" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s6880" xml:space="preserve">in <lb/>vnam collectæ ſummam effi ciunt {13/30}. </s> <s xml:id="echoid-s6881" xml:space="preserve">ac proin deidem eſt ex quadrato lateris <lb/>auferre {1/10}. </s> <s xml:id="echoid-s6882" xml:space="preserve">& </s> <s xml:id="echoid-s6883" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s6884" xml:space="preserve">atque {13/30}. </s> <s xml:id="echoid-s6885" xml:space="preserve">Sed quando auferentur {13/30}. </s> <s xml:id="echoid-s6886" xml:space="preserve">multiplicatur qua-<lb/>dratum lateris per {13/30}. </s> <s xml:id="echoid-s6887" xml:space="preserve">vt in 6. </s> <s xml:id="echoid-s6888" xml:space="preserve">quæſtiuncula fractorum docui. </s> <s xml:id="echoid-s6889" xml:space="preserve">Igitur cum, vt <lb/>Num. </s> <s xml:id="echoid-s6890" xml:space="preserve">5. </s> <s xml:id="echoid-s6891" xml:space="preserve">explicatum eſt, ex multiplicatione quadratilateris in {13/30}. </s> <s xml:id="echoid-s6892" xml:space="preserve">producatur <lb/>area trianguli æquilateri, liquido conſtat, partem decimam, & </s> <s xml:id="echoid-s6893" xml:space="preserve">partem tertiam <lb/>qua drati lateris conficere eandem aream. </s> <s xml:id="echoid-s6894" xml:space="preserve">Itaque ſi latus ſit 30. </s> <s xml:id="echoid-s6895" xml:space="preserve">erit eius quadra-<lb/>tum 900. </s> <s xml:id="echoid-s6896" xml:space="preserve">cuius {1/10}. </s> <s xml:id="echoid-s6897" xml:space="preserve">eſt 90. </s> <s xml:id="echoid-s6898" xml:space="preserve">& </s> <s xml:id="echoid-s6899" xml:space="preserve">{1/3}. </s> <s xml:id="echoid-s6900" xml:space="preserve">eſt 300. </s> <s xml:id="echoid-s6901" xml:space="preserve">quæ partes ſimul conficiuntnumerum <lb/>390. </s> <s xml:id="echoid-s6902" xml:space="preserve">pro area illius trianguli æquilateri.</s> <s xml:id="echoid-s6903" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6904" xml:space="preserve">6. </s> <s xml:id="echoid-s6905" xml:space="preserve"><emph style="sc">Hactenvs</emph> expoſuimus regulas, quæ nosin cognitionem areæ cuiuſ-<lb/> <anchor type="note" xlink:label="note-197-03a" xlink:href="note-197-03"/> cunquetrianguli ducunt, ſi ſingula latera cognita ſint. </s> <s xml:id="echoid-s6906" xml:space="preserve">Nunc triangulorum <lb/>areas per doctrinam ſinuum, Tangentium, ſecantium que inueſtig abimus, licet <lb/>non omnia latera ſint cognita, ſed vnum duntaxat, vel duo, vna cum duobus <lb/>angulis, vel vno. </s> <s xml:id="echoid-s6907" xml:space="preserve">In triangulis ergo rectangulis ita procedemus.</s> <s xml:id="echoid-s6908" xml:space="preserve"/> </p> <div xml:id="echoid-div445" type="float" level="2" n="28"> <note position="right" xlink:label="note-197-03" xlink:href="note-197-03a" xml:space="preserve">Area trian-<lb/>guli rectan-<lb/>guli ex later@ <lb/>quod recto <lb/>angulo oppo-<lb/>nitur, & vno <lb/>angulo acuto <lb/>cognito, quo <lb/>pacto inueſti-<lb/>getur.</note> </div> <p> <s xml:id="echoid-s6909" xml:space="preserve"><emph style="sc">Qvando</emph> in triangulo rectangulo latus recto angulo oppoſitum cogni-<lb/>tum eſt, cum vno angulo acuto, cognoſcemus aream hoc modo. </s> <s xml:id="echoid-s6910" xml:space="preserve">Detracto <lb/>angulo acuto dato exrecto, id eſt, ex grad. </s> <s xml:id="echoid-s6911" xml:space="preserve">90. </s> <s xml:id="echoid-s6912" xml:space="preserve">relin quetur alter acutus angu-<lb/>lus etiam notus. </s> <s xml:id="echoid-s6913" xml:space="preserve">Vtintriangulo DCB, habente an-<lb/>gulum rectum C, & </s> <s xml:id="echoid-s6914" xml:space="preserve">latus BD, notum, vna cum an-<lb/> <anchor type="figure" xlink:label="fig-197-01a" xlink:href="fig-197-01"/> gulo acuto B. </s> <s xml:id="echoid-s6915" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Et quoniam duo B, & </s> <s xml:id="echoid-s6916" xml:space="preserve">B D C, vni <anchor type="note" xlink:label="note-197-04a" xlink:href="note-197-04"/> recto, id eſt, gradibus 90. </s> <s xml:id="echoid-s6917" xml:space="preserve">ſunt æquales, ſi angulus <lb/>B, ex grad. </s> <s xml:id="echoid-s6918" xml:space="preserve">90. </s> <s xml:id="echoid-s6919" xml:space="preserve">detrahatur, reliquus fiet angulus B-<lb/>D C. </s> <s xml:id="echoid-s6920" xml:space="preserve">Si ergo fiat, vt ſinus totus angulirecti C, ad <lb/>oppoſitum latus BD, in qualibet menſura cogni-<lb/>tum; </s> <s xml:id="echoid-s6921" xml:space="preserve">itatam ſinus anguli B, quàm anguli BDC, ad aliud; </s> <s xml:id="echoid-s6922" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> nota fient latera D C, <anchor type="note" xlink:label="note-197-05a" xlink:href="note-197-05"/> & </s> <s xml:id="echoid-s6923" xml:space="preserve">CB, in partibus lateris BD: </s> <s xml:id="echoid-s6924" xml:space="preserve">atque ita omnia tria latera cognita erunt. </s> <s xml:id="echoid-s6925" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s6926" xml:space="preserve"><lb/>area cognoſcetur vel ex Num. </s> <s xml:id="echoid-s6927" xml:space="preserve">1. </s> <s xml:id="echoid-s6928" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s6929" xml:space="preserve">vel ex Num. </s> <s xml:id="echoid-s6930" xml:space="preserve">3.</s> <s xml:id="echoid-s6931" xml:space="preserve"/> </p> <div xml:id="echoid-div446" type="float" level="2" n="29"> <figure xlink:label="fig-197-01" xlink:href="fig-197-01a"> <image file="197-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/197-01"/> </figure> <note symbol="c" position="right" xlink:label="note-197-04" xlink:href="note-197-04a" xml:space="preserve">32. primi.</note> <note symbol="d" position="right" xlink:label="note-197-05" xlink:href="note-197-05a" xml:space="preserve">2. triang re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s6932" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi campus menſurandus triangularis eſt habẽs vnum angulum re-<lb/>ctum, ſatis erit, ſi ſumma diligentia latus recto angulo oppoſitum menſuretur, <lb/>& </s> <s xml:id="echoid-s6933" xml:space="preserve">inſuper vnus angulus acutus, beneficio alicuius quadrantis in gradus diuiſi, <pb o="168" file="198" n="198" rhead="GEOMETR. PRACT."/> qualis eſt à nobis conſtructus cap. </s> <s xml:id="echoid-s6934" xml:space="preserve">2. </s> <s xml:id="echoid-s6935" xml:space="preserve">lib. </s> <s xml:id="echoid-s6936" xml:space="preserve">1. </s> <s xml:id="echoid-s6937" xml:space="preserve">Nam ex his cognitis, vt proxime di-<lb/>ximus, tota area trianguli cognoſcetur, etiamſi ad alia duo latera accedere non <lb/>poſsimus.</s> <s xml:id="echoid-s6938" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6939" xml:space="preserve"><emph style="sc">Qvando</emph> in eodem triangulo rectangulo B D C, alterutrum latus datur <lb/>circa angulum rectum vna cumlatere, quod recto angulo opponitur, nimirum <lb/>ſi DC, & </s> <s xml:id="echoid-s6940" xml:space="preserve">DB, cognita ſint; </s> <s xml:id="echoid-s6941" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cognoſcetur quoque alterum latus B C, ſi fiat, vt <anchor type="note" xlink:label="note-198-01a" xlink:href="note-198-01"/> latus DB, angulo recto oppoſitum ad ſinum to tum angulirecti C, ita Iatus datũ <lb/>DC, ad aliud. </s> <s xml:id="echoid-s6942" xml:space="preserve">Productus enim numerus erit ſinus anguli B, quo cognito ex ta-<lb/> <anchor type="note" xlink:label="note-198-02a" xlink:href="note-198-02"/> bula ſinuum, cognitũ etiam erit eius complementum BDC. </s> <s xml:id="echoid-s6943" xml:space="preserve">Siergo rurſus fi-<lb/>at, vt ſinus totus angulirecti C, ad latus oppoſitum datum DB, ita ſinus anguli <lb/>BDC, inuentiad aliud, exibit latus oppoſitum B C. </s> <s xml:id="echoid-s6944" xml:space="preserve">Ex duobus igitur lateribus <lb/>DC, CB, cognitis, area triangulinota fiet ex ijs, quæ Num. </s> <s xml:id="echoid-s6945" xml:space="preserve">3. </s> <s xml:id="echoid-s6946" xml:space="preserve">paulo ante ſcripſi-<lb/>mus.</s> <s xml:id="echoid-s6947" xml:space="preserve"/> </p> <div xml:id="echoid-div447" type="float" level="2" n="30"> <note symbol="a" position="left" xlink:label="note-198-01" xlink:href="note-198-01a" xml:space="preserve">3. triang. re-<lb/>ctil.</note> <note position="left" xlink:label="note-198-02" xlink:href="note-198-02a" xml:space="preserve">Area trian-<lb/>guli rectan-<lb/>guli ex vno <lb/>latere circa <lb/>angulum re-<lb/>ctum, & late-<lb/>re quodrecto <lb/>angulo oppo-<lb/>nitur.</note> </div> <p> <s xml:id="echoid-s6948" xml:space="preserve"><emph style="sc">Atqve</emph> ita in campo, ſi detur portio triangularis, habens angulum rectum; <lb/></s> <s xml:id="echoid-s6949" xml:space="preserve">ſatis eſt, ſi diligenter menſuretur vnum latus circa angulum rectum, vna cum la-<lb/>tere, quodrecto angulo opponitur, etiamſi ad tertium latus non pateat acceſ-<lb/>ſus. </s> <s xml:id="echoid-s6950" xml:space="preserve">Exillis enim duobus area cognoſcetur, vt dictum eſt.</s> <s xml:id="echoid-s6951" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6952" xml:space="preserve"><emph style="sc">Qvod</emph> ſi in eodem triangulo rectangulo B D C, notum fuerit vnum latus <lb/>circa rectum angulum, videlicet DC, vna cum alterutro angulo acuto, vt pote <lb/>cum C, <anchor type="note" xlink:href="" symbol="b"/> notum effi cietur alterum latus B C: </s> <s xml:id="echoid-s6953" xml:space="preserve">ſi fiat, vt ſinus totus ad datum la- <anchor type="note" xlink:label="note-198-03a" xlink:href="note-198-03"/> tus DC: </s> <s xml:id="echoid-s6954" xml:space="preserve">Ita Tangens anguli BDC, quæſito lateri CB, oppoſiti, (cognoſcetur <lb/>autem alter hic angulus BDC, ſi angulus C, ex gradibus 90. </s> <s xml:id="echoid-s6955" xml:space="preserve">dematur) ad aliud. <lb/></s> <s xml:id="echoid-s6956" xml:space="preserve"> <anchor type="note" xlink:label="note-198-04a" xlink:href="note-198-04"/> Nam inuentus numerus dabit latus CB, quæſitum. </s> <s xml:id="echoid-s6957" xml:space="preserve">Vel ſi fiat, vt ſinus anguli C, <lb/>dato lateri DC, oppoſiti ad latus datum D C: </s> <s xml:id="echoid-s6958" xml:space="preserve">Ita ſinus alterius anguli B D C, ad <lb/>aliud. </s> <s xml:id="echoid-s6959" xml:space="preserve">Nam rurſus producetur latus quæſitum B C. </s> <s xml:id="echoid-s6960" xml:space="preserve">Ex duobus ergo lateribus <lb/>D C, C B, aream cognoſcemus, vt Num. </s> <s xml:id="echoid-s6961" xml:space="preserve">3. </s> <s xml:id="echoid-s6962" xml:space="preserve">traditur.</s> <s xml:id="echoid-s6963" xml:space="preserve"/> </p> <div xml:id="echoid-div448" type="float" level="2" n="31"> <note symbol="b" position="left" xlink:label="note-198-03" xlink:href="note-198-03a" xml:space="preserve">4. triang. <lb/>rectil.</note> <note position="left" xlink:label="note-198-04" xlink:href="note-198-04a" xml:space="preserve">Areatrian-<lb/>gulirectangu-<lb/>liex vno late-<lb/>re circa angu-<lb/>lum rectum <lb/>& vno angu-<lb/>lo acuto.</note> </div> <p> <s xml:id="echoid-s6964" xml:space="preserve">IN Campo ergo aliquo, ſi proponatur portio triangularis angulum habẽs <lb/>rectum, ſatis erit vnum latus circa rectum angulum, & </s> <s xml:id="echoid-s6965" xml:space="preserve">vnum angulum acutum <lb/>metiri, vt eius trianguli area reperiatur, etiamſi ad alia duo latera acceſſus dene-<lb/>getur. </s> <s xml:id="echoid-s6966" xml:space="preserve">Atque hæc de rectangulis triangulis: </s> <s xml:id="echoid-s6967" xml:space="preserve">veniamusiam ab obliquangula.</s> <s xml:id="echoid-s6968" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6969" xml:space="preserve">7. </s> <s xml:id="echoid-s6970" xml:space="preserve"><emph style="sc">Si</emph> ergo in triangulo non rectangulo A B D, notum ſit vnum latus, cum <lb/>duobus angulis quibuſcunque, perueniemus in cognitionem areæ hoc modo. <lb/></s> <s xml:id="echoid-s6971" xml:space="preserve">Ex duobus angulis cognitus erit quoque tertius, cum ſit complementum alio-<lb/> <anchor type="note" xlink:label="note-198-05a" xlink:href="note-198-05"/> rum duorum ad gr. </s> <s xml:id="echoid-s6972" xml:space="preserve">180. </s> <s xml:id="echoid-s6973" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur alia duo latera cognolcentur: </s> <s xml:id="echoid-s6974" xml:space="preserve">ac proinde ex tri- bus lateribus cognitis area fiet nota exijs, quæ Num. </s> <s xml:id="echoid-s6975" xml:space="preserve">1. </s> <s xml:id="echoid-s6976" xml:space="preserve">& </s> <s xml:id="echoid-s6977" xml:space="preserve">2. </s> <s xml:id="echoid-s6978" xml:space="preserve">tradita ſunt. <lb/></s> <s xml:id="echoid-s6979" xml:space="preserve"> <anchor type="note" xlink:label="note-198-06a" xlink:href="note-198-06"/> </s> </p> <div xml:id="echoid-div449" type="float" level="2" n="32"> <note symbol="c" position="left" xlink:label="note-198-05" xlink:href="note-198-05a" xml:space="preserve">10. triang. <lb/>rectil.</note> <note position="left" xlink:label="note-198-06" xlink:href="note-198-06a" xml:space="preserve">Area trian-<lb/>guli obliquã-<lb/>guli ex vno <lb/>latere ac duo-<lb/>bus angulis.</note> </div> <p> <s xml:id="echoid-s6980" xml:space="preserve"><emph style="sc">Vt</emph> ergo campus triangularis nullum habens angulũ rectum cognitus fiat, <lb/>ſatis erit, ſi vnum latus cum duobus angulis accuratè menſuretur. </s> <s xml:id="echoid-s6981" xml:space="preserve">Exijs enim <lb/>duo reliqua latera nota efficientur, &</s> <s xml:id="echoid-s6982" xml:space="preserve">c. </s> <s xml:id="echoid-s6983" xml:space="preserve">vt dictum eſt.</s> <s xml:id="echoid-s6984" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s6985" xml:space="preserve"><emph style="sc">Rvrsvs</emph> ſi in eodem triangulo ABD, nonrectangulo nota ſint duo late-<lb/>ra, vna cum angulo abipſis comprehenſo; </s> <s xml:id="echoid-s6986" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> inuenietur tertium latus: </s> <s xml:id="echoid-s6987" xml:space="preserve">ac pro- <anchor type="note" xlink:label="note-198-07a" xlink:href="note-198-07"/> inde, vt prius, ex omnibus tribus lateribus area trianguli efficietur cognita.</s> <s xml:id="echoid-s6988" xml:space="preserve"/> </p> <div xml:id="echoid-div450" type="float" level="2" n="33"> <note symbol="d" position="left" xlink:label="note-198-07" xlink:href="note-198-07a" xml:space="preserve">12. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s6989" xml:space="preserve"><emph style="sc">Itaqve</emph> ſatis erit, ſi in campo quouis triangulari duo latera, vna eum an-<lb/> <anchor type="note" xlink:label="note-198-08a" xlink:href="note-198-08"/> gulo abipſis comprehenſo menſurentur: </s> <s xml:id="echoid-s6990" xml:space="preserve">vt areaipſius nota reddatur.</s> <s xml:id="echoid-s6991" xml:space="preserve"/> </p> <div xml:id="echoid-div451" type="float" level="2" n="34"> <note position="left" xlink:label="note-198-08" xlink:href="note-198-08a" xml:space="preserve">Area trian-<lb/>guliobliquan-<lb/>guli ex duo-<lb/>b{us} laterib{us} <lb/>& angulo ab <lb/>ipſis compre-<lb/>henſo.</note> </div> <p> <s xml:id="echoid-s6992" xml:space="preserve">8. </s> <s xml:id="echoid-s6993" xml:space="preserve"><emph style="sc">Neqve</emph> vero hoc omittendum videtur: </s> <s xml:id="echoid-s6994" xml:space="preserve">ſi videlicet vnum latus trian-<lb/>guli, vel cuiuſuis figuræ rectilineæ in partes quotlibet æquales ſecetur, reliqua <lb/>latera in eiſdem partibus fieri poſſe cognita, beneficio inſtrumenti partium, vt ad <lb/>finem Num. </s> <s xml:id="echoid-s6995" xml:space="preserve">1. </s> <s xml:id="echoid-s6996" xml:space="preserve">cap. </s> <s xml:id="echoid-s6997" xml:space="preserve">1. </s> <s xml:id="echoid-s6998" xml:space="preserve">lib. </s> <s xml:id="echoid-s6999" xml:space="preserve">1. </s> <s xml:id="echoid-s7000" xml:space="preserve">declarauimus. </s> <s xml:id="echoid-s7001" xml:space="preserve">Ve@um vt magis exquiſite reperiantur, <pb o="169" file="199" n="199" rhead="LIBER QVARTVS."/> inquirendum erit fragmentum vltimæ particulæ (ſi quod ſuperfuerit) in parti-<lb/> <anchor type="note" xlink:label="note-199-01a" xlink:href="note-199-01"/> bus milleſimis, per ea, quæ Num. </s> <s xml:id="echoid-s7002" xml:space="preserve">14. </s> <s xml:id="echoid-s7003" xml:space="preserve">cap. </s> <s xml:id="echoid-s7004" xml:space="preserve">2. </s> <s xml:id="echoid-s7005" xml:space="preserve">lib. </s> <s xml:id="echoid-s7006" xml:space="preserve">1. </s> <s xml:id="echoid-s7007" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s7008" xml:space="preserve">Ita enim in dimen-<lb/>ſionibus figurarum minus à vero aberrabimus.</s> <s xml:id="echoid-s7009" xml:space="preserve"/> </p> <div xml:id="echoid-div452" type="float" level="2" n="35"> <note position="right" xlink:label="note-199-01" xlink:href="note-199-01a" xml:space="preserve">Diuiſovno la-<lb/>tere figuræ in <lb/>quotuis part{es} <lb/>æquales, quo <lb/>pacto alia la-<lb/>ter ain eiſdem <lb/>partib{us} fiant <lb/>nota.</note> </div> <p> <s xml:id="echoid-s7010" xml:space="preserve">9. </s> <s xml:id="echoid-s7011" xml:space="preserve"><emph style="sc">Neminem</emph> autẽ moueat, aut perturbet, quod rectas dixerimus metien-<lb/>das eſſe nonnunquam mechanice per catenulam aliquam ſerreã, aut per inſtru-<lb/>mentum partium. </s> <s xml:id="echoid-s7012" xml:space="preserve">Nam in hoc dimetiendi negotio, præſertimin campis, & </s> <s xml:id="echoid-s7013" xml:space="preserve">agris <lb/>admittenda omnino eſt huiuſmo dimechanica linearum dimenſio, tum quia a-<lb/>pud omues agrimenſores hic mos eſt: </s> <s xml:id="echoid-s7014" xml:space="preserve">tum quia non ſemper via Geometrica id <lb/>præſtare poteſt; </s> <s xml:id="echoid-s7015" xml:space="preserve">tum vero maximè, quia in dimenſi onibus agrorum, ſiue figu-<lb/> <anchor type="note" xlink:label="note-199-02a" xlink:href="note-199-02"/> rarum ſatis eſt rem prope verum attingere, dum modo notabilis error non cõ-<lb/>mitatur. </s> <s xml:id="echoid-s7016" xml:space="preserve">Quod ſi hæc dimenſio quarundem linearum alicuinõ probetur, is pro-<lb/>fecto è medio tollat, neceſſe eſt, omnem agrorum, figurarumue dimenſionem. <lb/></s> <s xml:id="echoid-s7017" xml:space="preserve">Vnde enim conſtat, agrum propoſitum, vel figuram habere latera cognita, niſi <lb/>hæcipſa per menſuram aliquam materialem ſint explorata? </s> <s xml:id="echoid-s7018" xml:space="preserve">Siigitur laterum di-<lb/>menſio mechanica, tanquam à vero parum aberrans, ab omnibus vſurpatur, <lb/>cur eamin lineisintra figuras metiendis reij ciendam cenſeamus, nõ video. </s> <s xml:id="echoid-s7019" xml:space="preserve">Non <lb/>nego tamen, viam Geometricam, quando fieri poteſt, adhiben dam eſſe. </s> <s xml:id="echoid-s7020" xml:space="preserve">In fi-<lb/>guris quoque, vbilatera non ſunt nimis magna, vtendũ cenſeo doctrina, quam <lb/>in inſtrumento partium lib. </s> <s xml:id="echoid-s7021" xml:space="preserve">1. </s> <s xml:id="echoid-s7022" xml:space="preserve">cap. </s> <s xml:id="echoid-s7023" xml:space="preserve">1. </s> <s xml:id="echoid-s7024" xml:space="preserve">ad finem Num. </s> <s xml:id="echoid-s7025" xml:space="preserve">2. </s> <s xml:id="echoid-s7026" xml:space="preserve">tradidimus, non neglectis <lb/>etiamijs, quæ in eodem lib. </s> <s xml:id="echoid-s7027" xml:space="preserve">1. </s> <s xml:id="echoid-s7028" xml:space="preserve">cap. </s> <s xml:id="echoid-s7029" xml:space="preserve">2. </s> <s xml:id="echoid-s7030" xml:space="preserve">Nume 14. </s> <s xml:id="echoid-s7031" xml:space="preserve">de quauis particula lineæ cogno-<lb/>ſcenda, in partibus ſaltem milleſimis, ſcripſimus, quod hac ratione vix à vero <lb/>quis aberrare poſsit.</s> <s xml:id="echoid-s7032" xml:space="preserve"/> </p> <div xml:id="echoid-div453" type="float" level="2" n="36"> <note position="right" xlink:label="note-199-02" xlink:href="note-199-02a" xml:space="preserve">In negotio di-<lb/>menſionum <lb/>admittendam <lb/>eſſe in nonnul-<lb/>lis lineis & <lb/>angulis me-<lb/>chanicam <lb/>menſuratio-<lb/>nem.</note> </div> <p> <s xml:id="echoid-s7033" xml:space="preserve"><emph style="sc">Idem</emph> de mechanica angulorum dimenſione per quadrantem intelligen-<lb/>dum eſt: </s> <s xml:id="echoid-s7034" xml:space="preserve">præſertim ſi præter gradus inueſtigentur quo que minuta, vt lib. </s> <s xml:id="echoid-s7035" xml:space="preserve">1. </s> <s xml:id="echoid-s7036" xml:space="preserve">cap. <lb/></s> <s xml:id="echoid-s7037" xml:space="preserve">2. </s> <s xml:id="echoid-s7038" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s7039" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div455" type="section" level="1" n="173"> <head xml:id="echoid-head176" xml:space="preserve">DE AREA QVADRILATERORVM <lb/>non rectangulorum.</head> <head xml:id="echoid-head177" xml:space="preserve"><emph style="sc">Capvt</emph> III.</head> <p> <s xml:id="echoid-s7040" xml:space="preserve">1. </s> <s xml:id="echoid-s7041" xml:space="preserve">TRIA ſunt genera quadril aterarum figurarum, quæ vel nullum angu-<lb/> <anchor type="note" xlink:label="note-199-03a" xlink:href="note-199-03"/> lum rectum habent, vel certe non omnes rectos: </s> <s xml:id="echoid-s7042" xml:space="preserve">Rhombus, Rhom-<lb/>boides, & </s> <s xml:id="echoid-s7043" xml:space="preserve">Trapezium. </s> <s xml:id="echoid-s7044" xml:space="preserve">Primæ duæ figuræ nullum habent angulum re-<lb/>ctum: </s> <s xml:id="echoid-s7045" xml:space="preserve">poſterior autem poteſt habere vel vnum rectum, vel duos, veletiam <lb/>nullum: </s> <s xml:id="echoid-s7046" xml:space="preserve">Item duo latera oppoſita parallela, vel non parallela. </s> <s xml:id="echoid-s7047" xml:space="preserve">Rhom-<lb/>bi & </s> <s xml:id="echoid-s7048" xml:space="preserve">Rhomboidis, quorum latera nota ſint, area pro-<lb/> <anchor type="figure" xlink:label="fig-199-01a" xlink:href="fig-199-01"/> ducitur ex ductu perpendicularis in latus, in quod per-<lb/>pendicularis cadit. </s> <s xml:id="echoid-s7049" xml:space="preserve">Ita vt magnitudo perpendicularis accu-<lb/>rate ſit prius exploranda vel per inſtrumentum partium initio <lb/>huius operis conſtructi, vt paulo ante cap. </s> <s xml:id="echoid-s7050" xml:space="preserve">2. </s> <s xml:id="echoid-s7051" xml:space="preserve">Num. </s> <s xml:id="echoid-s7052" xml:space="preserve">8. </s> <s xml:id="echoid-s7053" xml:space="preserve">monui-<lb/>mus, vel alio modo, vt mox dicam. </s> <s xml:id="echoid-s7054" xml:space="preserve">Verbi gratia, in Rhombo <lb/>& </s> <s xml:id="echoid-s7055" xml:space="preserve">Rhomboide A B C D, producetur area ex multiplicatione <lb/>perpendicularis AE, in latus B C, <anchor type="note" xlink:href="" symbol="a"/> Nam rectangulum A F, <anchor type="note" xlink:label="note-199-04a" xlink:href="note-199-04"/> ſub A E, & </s> <s xml:id="echoid-s7056" xml:space="preserve">A D, comprehenſum æquale eſt parallelogram-<lb/>mo B D, quòd hæc duo parallelogramma ſint inter paralle- <pb o="170" file="200" n="200" rhead="GEOMETR. PRACT."/> las AD, BC, & </s> <s xml:id="echoid-s7057" xml:space="preserve">ſuper eandẽ baſem AD. </s> <s xml:id="echoid-s7058" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s7059" xml:space="preserve">fruſtra alij p̃cipiunt, vt diameter du <lb/>catur AC, & </s> <s xml:id="echoid-s7060" xml:space="preserve">beneficio perpendicularis AE, area trianguli ABC, inquiratur, quod <lb/>hæc duplicata aream exhibeat totius parallelogrammi, <anchor type="note" xlink:href="" symbol="a"/> quippe cum triangu- <anchor type="note" xlink:label="note-200-01a" xlink:href="note-200-01"/> lum ABC, ſemiſsis ſit parallelogrammi. </s> <s xml:id="echoid-s7061" xml:space="preserve">Fruſtra, inquam hoc præcipiunt, cum <lb/>expeditius area inueniatur ſi perpendicularis in totum latus BC, ducatur, quam <lb/>ſi in ſemiſſem multiplicetur, ac productus deinde numerus dupletur.</s> <s xml:id="echoid-s7062" xml:space="preserve"/> </p> <div xml:id="echoid-div455" type="float" level="2" n="1"> <note position="right" xlink:label="note-199-03" xlink:href="note-199-03a" xml:space="preserve">Rhombi & <lb/>Rhomboidis <lb/>area</note> <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/xxxxxxxx/figures/199-01"/> </figure> <note symbol="a" position="right" xlink:label="note-199-04" xlink:href="note-199-04a" xml:space="preserve">35. primi.</note> <note symbol="a" position="left" xlink:label="note-200-01" xlink:href="note-200-01a" xml:space="preserve">34. primi.</note> </div> <p> <s xml:id="echoid-s7063" xml:space="preserve"><emph style="sc">Si</emph> per quadrantem cap. </s> <s xml:id="echoid-s7064" xml:space="preserve">2. </s> <s xml:id="echoid-s7065" xml:space="preserve">lib. </s> <s xml:id="echoid-s7066" xml:space="preserve">1. </s> <s xml:id="echoid-s7067" xml:space="preserve">conſtructum inueſtigetur quantitas anguli <lb/> <anchor type="note" xlink:label="note-200-02a" xlink:href="note-200-02"/> B, reperietur perpendicularis AE, perſinus, hacratione. </s> <s xml:id="echoid-s7068" xml:space="preserve">Fiat vt ſinus totus an-<lb/>gulirecti E, ad latus oppoſitum AB, ita ſinus anguli B, ad aliud. </s> <s xml:id="echoid-s7069" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Productus e- <anchor type="note" xlink:label="note-200-03a" xlink:href="note-200-03"/> nim numerus erit perpendicularis AE, cognita in partibus lateris dati AB.</s> <s xml:id="echoid-s7070" xml:space="preserve"/> </p> <div xml:id="echoid-div456" type="float" level="2" n="2"> <note position="left" xlink:label="note-200-02" xlink:href="note-200-02a" xml:space="preserve">Perpendicu-<lb/>laris inuentio.</note> <note symbol="b" position="left" xlink:label="note-200-03" xlink:href="note-200-03a" xml:space="preserve">2. triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s7071" xml:space="preserve">2. </s> <s xml:id="echoid-s7072" xml:space="preserve"><emph style="sc">Trapezii</emph>, in quo duo latera oppoſita ſint parallela AB, BC, & </s> <s xml:id="echoid-s7073" xml:space="preserve">omnia <lb/>latera nota, area producitur ex perpendiculari A E, inter duo latera parallela <lb/>multiplicata in ſemiſſem ſummæ ex lateribus parallelis conflatæ. </s> <s xml:id="echoid-s7074" xml:space="preserve">Nam ducta <lb/> <anchor type="note" xlink:label="note-200-04a" xlink:href="note-200-04"/> diametro AC, area trianguli ABC, producitur ex perpendiculari A E, in ſemiſ-<lb/>ſem baſis BC, vt cap. </s> <s xml:id="echoid-s7075" xml:space="preserve">2. </s> <s xml:id="echoid-s7076" xml:space="preserve">Num. </s> <s xml:id="echoid-s7077" xml:space="preserve">2. </s> <s xml:id="echoid-s7078" xml:space="preserve">dictum eſt: </s> <s xml:id="echoid-s7079" xml:space="preserve">Item area trian-<lb/> <anchor type="figure" xlink:label="fig-200-01a" xlink:href="fig-200-01"/> guli ACD, ex eadem perpendiculari A E, in ſemiſſem baſis <lb/>AD: </s> <s xml:id="echoid-s7080" xml:space="preserve">Acproinde hæ duæ areæ ſimul aream totius Trapezij A-<lb/>BCD, conficient. </s> <s xml:id="echoid-s7081" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Cum igitur idem fiat ex AE, in ſummam ex <anchor type="note" xlink:label="note-200-05a" xlink:href="note-200-05"/> ſemiſſe rectæ B C, & </s> <s xml:id="echoid-s7082" xml:space="preserve">ex ſemiſſe rectæ A D, conflatam, id eſt, in <lb/>ſemiſſem rectarum BC, AD, ſimul: </s> <s xml:id="echoid-s7083" xml:space="preserve">quod ex A E, in ſemiſſem <lb/>lateris B C, & </s> <s xml:id="echoid-s7084" xml:space="preserve">ex A E, in ſemiſſem lateris A D; </s> <s xml:id="echoid-s7085" xml:space="preserve">liquidò conſtat, <lb/>aream Trapezij gigni ex perpendiculari AE, in ſemiſſem ſum-<lb/>mæ laterum AD, BC. </s> <s xml:id="echoid-s7086" xml:space="preserve">Atque hæc ratio locum etiam habetin <lb/>Trapezio habente vnum angulum rectum, vel duos rectos.</s> <s xml:id="echoid-s7087" xml:space="preserve"/> </p> <div xml:id="echoid-div457" type="float" level="2" n="3"> <note position="left" xlink:label="note-200-04" xlink:href="note-200-04a" xml:space="preserve">Areatrapezii <lb/>habentis duo <lb/>lateraparalle-<lb/>la.</note> <figure xlink:label="fig-200-01" xlink:href="fig-200-01a"> <image file="200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/200-01"/> </figure> <note symbol="c" position="left" xlink:label="note-200-05" xlink:href="note-200-05a" xml:space="preserve">1. ſecundi.</note> </div> <p> <s xml:id="echoid-s7088" xml:space="preserve"><emph style="sc">Perpendicvlaris</emph> vero AE, inuenietur, vt in Rhombo, & </s> <s xml:id="echoid-s7089" xml:space="preserve">Rhomboi-<lb/>de diximus, duobus modis, ſi per quadrantem angulus B, inueſtigetur, &</s> <s xml:id="echoid-s7090" xml:space="preserve">c.</s> <s xml:id="echoid-s7091" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7092" xml:space="preserve"><emph style="sc">In</emph> Trapezio autem FGHI, in quo nulla ſunt latera parallela, omnia tamen <lb/> <anchor type="note" xlink:label="note-200-06a" xlink:href="note-200-06"/> latera ſunt nota, menſuranda primum eſt diameter. </s> <s xml:id="echoid-s7093" xml:space="preserve">IG, per inſtrumentum par-<lb/>tium. </s> <s xml:id="echoid-s7094" xml:space="preserve">Deinde vtriuſque trianguli FGI, GHI, area inuenienda, vt cap. </s> <s xml:id="echoid-s7095" xml:space="preserve">2. </s> <s xml:id="echoid-s7096" xml:space="preserve">Nume. <lb/></s> <s xml:id="echoid-s7097" xml:space="preserve">1. </s> <s xml:id="echoid-s7098" xml:space="preserve">& </s> <s xml:id="echoid-s7099" xml:space="preserve">2. </s> <s xml:id="echoid-s7100" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s7101" xml:space="preserve">Ambæ enim areæ ſimul conficient aream totius Trapezij.</s> <s xml:id="echoid-s7102" xml:space="preserve"/> </p> <div xml:id="echoid-div458" type="float" level="2" n="4"> <note position="left" xlink:label="note-200-06" xlink:href="note-200-06a" xml:space="preserve">Areatrapezii <lb/>nulla haben-<lb/>tis latera pa-<lb/>rallela.</note> </div> <p> <s xml:id="echoid-s7103" xml:space="preserve"><emph style="sc">Qvod</emph> ſi malueri angulum F, vel H, per quadrantem inuenire, cognoſce-<lb/>mus diametri GI, magnitudinem, per doctrinam ſinuum, ac Tangentium, <lb/> <anchor type="note" xlink:href="" symbol="d"/> vt lib. </s> <s xml:id="echoid-s7104" xml:space="preserve">1. </s> <s xml:id="echoid-s7105" xml:space="preserve">capit. </s> <s xml:id="echoid-s7106" xml:space="preserve">3. </s> <s xml:id="echoid-s7107" xml:space="preserve">docuimus, ex duobus lateribus F G, F I, & </s> <s xml:id="echoid-s7108" xml:space="preserve">angulo F, ab <anchor type="note" xlink:label="note-200-07a" xlink:href="note-200-07"/> ipſis comprehenſo, vel ex duobus lateribus HG, HI, & </s> <s xml:id="echoid-s7109" xml:space="preserve">angulo H, quem con-<lb/>tinent.</s> <s xml:id="echoid-s7110" xml:space="preserve"/> </p> <div xml:id="echoid-div459" type="float" level="2" n="5"> <note symbol="d" position="left" xlink:label="note-200-07" xlink:href="note-200-07a" xml:space="preserve">12. trian. re-<lb/>ctil. Num. 2.</note> </div> <p> <s xml:id="echoid-s7111" xml:space="preserve">3. </s> <s xml:id="echoid-s7112" xml:space="preserve"><emph style="sc">Non</emph> aliter aream conſequemur cuiuſcun que quadrilateri irregularis, et-<lb/> <anchor type="note" xlink:label="note-200-08a" xlink:href="note-200-08"/> iamſi non habeat omnes angulos introrſum, ſicut Trapezium. </s> <s xml:id="echoid-s7113" xml:space="preserve">Vt ſi in Trape-<lb/>zio FGHI, ducantur ex G, & </s> <s xml:id="echoid-s7114" xml:space="preserve">I, duæ rectæ GK, IK, conſtituetur quadrilaterum <lb/>GHIK, irregulare, cum ſolum habeat tres angulos GHI, HIK, HGK. </s> <s xml:id="echoid-s7115" xml:space="preserve">Nam ad <lb/>K, non fit angulus GKI, introrſum verſus H, cum illud ſpatium ſit duo-<lb/>bus rectis maius, ſed verſus F, extrorſum. </s> <s xml:id="echoid-s7116" xml:space="preserve">Huius ergo figuræ qua-<lb/>drilateræirregularis aream colligemus, ducta diametro <lb/>K H, ex duabus areis triangulorum IKH, <lb/>GKH, vt de Trapezio FGHI, <lb/>dictum eſt.</s> <s xml:id="echoid-s7117" xml:space="preserve"/> </p> <div xml:id="echoid-div460" type="float" level="2" n="6"> <note position="left" xlink:label="note-200-08" xlink:href="note-200-08a" xml:space="preserve">Area figuræ <lb/>quadrilateræ <lb/>irregularis.</note> </div> <pb o="171" file="201" n="201" rhead="LIBER QVARTVS."/> </div> <div xml:id="echoid-div462" type="section" level="1" n="174"> <head xml:id="echoid-head178" xml:space="preserve">DE AREA MVLTIL ATERARVM <lb/>figurarum irregularium.</head> <head xml:id="echoid-head179" xml:space="preserve"><emph style="sc">Capvt</emph> IV.</head> <p> <s xml:id="echoid-s7118" xml:space="preserve">1. </s> <s xml:id="echoid-s7119" xml:space="preserve">FIGVRAS multilateras irregulares, quæ videlicet plura latera habent <lb/> <anchor type="note" xlink:label="note-201-01a" xlink:href="note-201-01"/> inæqualia, quam quatuor, etiamſi valdè irregulares ſint, metiemur, vt <lb/>trapezia irregularia, reſoluendo nimirum illasin triangula, & </s> <s xml:id="echoid-s7120" xml:space="preserve">ſingulorũ <lb/>triangulorum areas inueſtigando. </s> <s xml:id="echoid-s7121" xml:space="preserve">Nam omnes hæ areæ in vnam ſummam col-<lb/>lectæ æquales ſunt areæ totius figuræ propoſitæ. </s> <s xml:id="echoid-s7122" xml:space="preserve">Vt ſi figura ſeptem laterum A-<lb/>BCDEFG, reſoluatur in quin que triangula ABG, GBD, DBC, DEF, FDG, ita <lb/> <anchor type="note" xlink:label="note-201-02a" xlink:href="note-201-02"/> vteorum latera ſe mutuo non interſecent, in quirendæ ſunt areæ ſingulorũ hoc <lb/> <anchor type="figure" xlink:label="fig-201-01a" xlink:href="fig-201-01"/> modo. </s> <s xml:id="echoid-s7123" xml:space="preserve">Quando omnia latera triangulorum nota effici poſſunt per aliquam <lb/>menſuram, ſiue figura agrum aliquem repræſentet, ſiue in charta ſolum ſit de-<lb/>ſcripta, demittantur ex angulis ad latera oppoſita perpendiculares AH, DI, CK, <lb/>DM, FL, ſingulæ in ſingulis triangulis. </s> <s xml:id="echoid-s7124" xml:space="preserve">Deinde in triangulo ABG, <anchor type="note" xlink:href="" symbol="a"/> inquirantur <anchor type="note" xlink:label="note-201-03a" xlink:href="note-201-03"/> ex tribus lateribus notis ſegmenta B H, H G; </s> <s xml:id="echoid-s7125" xml:space="preserve">& </s> <s xml:id="echoid-s7126" xml:space="preserve">ex his perpendicularis A H, vt <lb/>cap. </s> <s xml:id="echoid-s7127" xml:space="preserve">2. </s> <s xml:id="echoid-s7128" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s7129" xml:space="preserve">Num. </s> <s xml:id="echoid-s7130" xml:space="preserve">2. </s> <s xml:id="echoid-s7131" xml:space="preserve">declarauimus. </s> <s xml:id="echoid-s7132" xml:space="preserve">Nam AH, in ſemiſſem baſis B G, ducta <lb/>producet aream trianguli A B G. </s> <s xml:id="echoid-s7133" xml:space="preserve">Eadem que ratione aliorum triangulorum areæ <lb/>perueſtigentur: </s> <s xml:id="echoid-s7134" xml:space="preserve">atque omnes areæ in vnam redigantur ſummam, vt area toti-<lb/>us figuræ habeatur. </s> <s xml:id="echoid-s7135" xml:space="preserve">Quod ſi malueris, poteris omnium triangulorum areas in-<lb/>dagare ex tribus lateribus cognitis, per ea, quæ capit. </s> <s xml:id="echoid-s7136" xml:space="preserve">2. </s> <s xml:id="echoid-s7137" xml:space="preserve">Numer. </s> <s xml:id="echoid-s7138" xml:space="preserve">1. </s> <s xml:id="echoid-s7139" xml:space="preserve">ſcripſimus, <lb/>etiamſi neque perpendiculares ductæ ſint, neque ſegmenta B H, G H, in-<lb/>uenta.</s> <s xml:id="echoid-s7140" xml:space="preserve"/> </p> <div xml:id="echoid-div462" type="float" level="2" n="1"> <note position="right" xlink:label="note-201-01" xlink:href="note-201-01a" xml:space="preserve">Area multi <lb/>lateræ figuræ.</note> <note position="right" xlink:label="note-201-02" xlink:href="note-201-02a" xml:space="preserve">Quando figu-<lb/>ra in triangu-<lb/>la reſolui po-<lb/>teſt, quo m@-<lb/>do ei{us} area <lb/>colligatur.</note> <figure xlink:label="fig-201-01" xlink:href="fig-201-01a"> <image file="201-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/201-01"/> </figure> <note symbol="a" position="right" xlink:label="note-201-03" xlink:href="note-201-03a" xml:space="preserve">9. triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s7141" xml:space="preserve">2. </s> <s xml:id="echoid-s7142" xml:space="preserve"><emph style="sc">Qvando</emph> latera triangulorum interiora menſurarinequeunt, immo ne-<lb/> <anchor type="note" xlink:label="note-201-04a" xlink:href="note-201-04"/> que duci, vt non raro accidit in campis, aut agris, qui vel propter arbores, vel <lb/>paludes interiectas, rectis itineribus pertranſirinon poſſunt; </s> <s xml:id="echoid-s7143" xml:space="preserve">alia ratione ſco-<lb/>pum attingemus, hac videlicet. </s> <s xml:id="echoid-s7144" xml:space="preserve">Cognitis lateribus figuram ambientibus <lb/>per aliquam menſuram, inueſtigentur quoque anguli ab ipſis comprehenſi <lb/>beneficio quadrantis alicuius in gradus diuiſi. </s> <s xml:id="echoid-s7145" xml:space="preserve">In propoſita figura angulus <lb/>C D E, indagandus non eſt, quod ſit extra figuram. </s> <s xml:id="echoid-s7146" xml:space="preserve">Reſoluta deinde figura <lb/>mente. </s> <s xml:id="echoid-s7147" xml:space="preserve">aut cogitatione in triangula, ac ſi latera interiora ducta eſſent, vt p@us:</s> <s xml:id="echoid-s7148" xml:space="preserve"> <pb o="172" file="202" n="202" rhead="GEOMETR. PRACT."/> <anchor type="note" xlink:href="" symbol="a"/> explorentur in triangulo ABG, duo anguli B, G, ex duobus lateribus AB, A G, <anchor type="note" xlink:label="note-202-01a" xlink:href="note-202-01"/> angulo que ab ipſis comprehenſo; </s> <s xml:id="echoid-s7149" xml:space="preserve">atque inſuper latus B G. </s> <s xml:id="echoid-s7150" xml:space="preserve">Hinc enim in tri-<lb/>angulo rectangulo ABH, vel AGH, <anchor type="note" xlink:href="" symbol="b"/> demiſl<unsure/>a perpendicularis ex A, cognoſce- <anchor type="note" xlink:label="note-202-02a" xlink:href="note-202-02"/> tur ex baſe AB, & </s> <s xml:id="echoid-s7151" xml:space="preserve">angulo B, vel ex baſe A G, & </s> <s xml:id="echoid-s7152" xml:space="preserve">angulo G: </s> <s xml:id="echoid-s7153" xml:space="preserve">ac proinde area trian-<lb/>guli reperietur, vt antea, exijs, quæ c. </s> <s xml:id="echoid-s7154" xml:space="preserve">2. </s> <s xml:id="echoid-s7155" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s7156" xml:space="preserve">Nume. </s> <s xml:id="echoid-s7157" xml:space="preserve">1. </s> <s xml:id="echoid-s7158" xml:space="preserve">& </s> <s xml:id="echoid-s7159" xml:space="preserve">2. </s> <s xml:id="echoid-s7160" xml:space="preserve">tradita ſunt. <lb/></s> <s xml:id="echoid-s7161" xml:space="preserve">Non aliter area trianguli B C D, nota fiet ex duobus lateribus CB, CD, notis an-<lb/> <anchor type="note" xlink:label="note-202-03a" xlink:href="note-202-03"/> gulum C, notum ambientibus, <anchor type="note" xlink:href="" symbol="c"/> ſi nimirum inueſtigentur prius anguli B, D, vna cum latere B D, <anchor type="note" xlink:href="" symbol="d"/> & </s> <s xml:id="echoid-s7162" xml:space="preserve">ex his demiſla perpendicularis C K. </s> <s xml:id="echoid-s7163" xml:space="preserve">Poſt hæc in triangulo <anchor type="note" xlink:label="note-202-04a" xlink:href="note-202-04"/> BD G, ſi anguli AB G, CB D, iam cogniti detrahantur ex toto angulo ABC, no-<lb/>to, remanebit angulus D B G, notus. </s> <s xml:id="echoid-s7164" xml:space="preserve">Cum ergo latera B D, B G, ipſum inclu-<lb/> <anchor type="note" xlink:label="note-202-05a" xlink:href="note-202-05"/> dentia facta ſint etiam nota; </s> <s xml:id="echoid-s7165" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> cognoſcentur eodem modo & </s> <s xml:id="echoid-s7166" xml:space="preserve">anguli D, G, &</s> <s xml:id="echoid-s7167" xml:space="preserve"> latus D G; </s> <s xml:id="echoid-s7168" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> atque in ſuper perpendicularis ex B, in D G, demiſſa, &</s> <s xml:id="echoid-s7169" xml:space="preserve">c. </s> <s xml:id="echoid-s7170" xml:space="preserve">Similiter <anchor type="note" xlink:label="note-202-06a" xlink:href="note-202-06"/> in triangulo GDF, ſi ex angulo noto A G F, tollantur anguli A G B, B G D, iam <lb/>noti effecti, relinquetur angulus D G F, notus. </s> <s xml:id="echoid-s7171" xml:space="preserve">Cumigitur & </s> <s xml:id="echoid-s7172" xml:space="preserve">latus G D, notum <lb/> <anchor type="note" xlink:label="note-202-07a" xlink:href="note-202-07"/> factum ſit: </s> <s xml:id="echoid-s7173" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> cognoſcemus & </s> <s xml:id="echoid-s7174" xml:space="preserve">duos angulos GDF, GFD, & </s> <s xml:id="echoid-s7175" xml:space="preserve">inſuper latus D F, <anchor type="note" xlink:href="" symbol="h"/> vna cum perpendiculari ex G, in D F, demiſſa, &</s> <s xml:id="echoid-s7176" xml:space="preserve">c. </s> <s xml:id="echoid-s7177" xml:space="preserve">In triangulo denique DEF, <lb/> <anchor type="note" xlink:label="note-202-08a" xlink:href="note-202-08"/> cum omnia latera ſint nota; </s> <s xml:id="echoid-s7178" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> efficientur quoq; </s> <s xml:id="echoid-s7179" xml:space="preserve">noti omnes tres anguli: </s> <s xml:id="echoid-s7180" xml:space="preserve">ac pro- inde demiſſa perpendicularis D M, ex D, demiſſa, vel ex quocunque alio angu-<lb/> <anchor type="note" xlink:label="note-202-09a" xlink:href="note-202-09"/> lo, <anchor type="note" xlink:href="" symbol="k"/> nota fiet, &</s> <s xml:id="echoid-s7181" xml:space="preserve">c. </s> <s xml:id="echoid-s7182" xml:space="preserve">Ex his facile intelliges, quomodo in alijs figuris irregulari- buste gerere debeas. <lb/></s> <s xml:id="echoid-s7183" xml:space="preserve"> <anchor type="note" xlink:label="note-202-10a" xlink:href="note-202-10"/> </s> </p> <div xml:id="echoid-div463" type="float" level="2" n="2"> <note position="right" xlink:label="note-201-04" xlink:href="note-201-04a" xml:space="preserve">Quando figu-<lb/>ra in triangu-<lb/>la reſolui non <lb/>poteſt quo mo <lb/>do ei{us} area <lb/>deprehenda-<lb/>tur.</note> <note symbol="a" position="left" xlink:label="note-202-01" xlink:href="note-202-01a" xml:space="preserve">12. triang. <lb/>rectil.</note> <note symbol="b" position="left" xlink:label="note-202-02" xlink:href="note-202-02a" xml:space="preserve">2. triang. re-<lb/>ctil.</note> <note symbol="c" position="left" xlink:label="note-202-03" xlink:href="note-202-03a" xml:space="preserve">12. triang. <lb/>rectil. Nu 2.</note> <note symbol="d" position="left" xlink:label="note-202-04" xlink:href="note-202-04a" xml:space="preserve">2. triang. re-<lb/>ctil.</note> <note symbol="e" position="left" xlink:label="note-202-05" xlink:href="note-202-05a" xml:space="preserve">12 triang. <lb/>rectil. Nu 2.</note> <note symbol="f" position="left" xlink:label="note-202-06" xlink:href="note-202-06a" xml:space="preserve">2. triang. re-<lb/>ctil.</note> <note symbol="g" position="left" xlink:label="note-202-07" xlink:href="note-202-07a" xml:space="preserve">12. triang. <lb/>rectil.</note> <note symbol="h" position="left" xlink:label="note-202-08" xlink:href="note-202-08a" xml:space="preserve">2. triang. <lb/>rectil.</note> <note symbol="i" position="left" xlink:label="note-202-09" xlink:href="note-202-09a" xml:space="preserve">16. triang. <lb/>rectil.</note> <note symbol="k" position="left" xlink:label="note-202-10" xlink:href="note-202-10a" xml:space="preserve">2. triang. re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s7184" xml:space="preserve"><emph style="sc">Potes</emph> etiam, ſi vis, deſcribere in charta aliqua figuram agro ſimilem: </s> <s xml:id="echoid-s7185" xml:space="preserve">ſi ni-<lb/>mirum ſumas rectam A B, tot particularum æqualium, quot menſuræ in latere <lb/> <anchor type="note" xlink:label="note-202-11a" xlink:href="note-202-11"/> agrireſp ondente includuntur, angulum que conſtituas ABC, æqualẽ ei, quem <lb/>in ambitu agriinueniſti. </s> <s xml:id="echoid-s7186" xml:space="preserve">Dein de in B C, tot particulas accipias, quot in latere <lb/>agrireſpondente continentur, iterumque angulum B C D, æqualem illi conſti-<lb/>tuas, quem in figura deprehendiſti. </s> <s xml:id="echoid-s7187" xml:space="preserve">Denique ſi idem facias de angulo CD E, & </s> <s xml:id="echoid-s7188" xml:space="preserve"><lb/>reliquis, necnon de rectis CD, DE, & </s> <s xml:id="echoid-s7189" xml:space="preserve">alijs, deſcripta erit figura ſimilis agro: </s> <s xml:id="echoid-s7190" xml:space="preserve">quę <lb/>ſi reſoluetur in triangula, quorum latera intra figura per inſtrumentum partium <lb/>menſurentur, reperietur eius area, ſicuti prius, quãdo agerreſoluip oterat in tri-<lb/>angula.</s> <s xml:id="echoid-s7191" xml:space="preserve"/> </p> <div xml:id="echoid-div464" type="float" level="2" n="3"> <note position="left" xlink:label="note-202-11" xlink:href="note-202-11a" xml:space="preserve">Quomodo fi-<lb/>gura agro pro <lb/>poſito ſimilis <lb/>deſeribi poſſit.</note> </div> <p> <s xml:id="echoid-s7192" xml:space="preserve"><emph style="sc">Qvando</emph> campi planities non eſt impedita, magis exquiſitè figura eiſimi-<lb/>lis deſcribetur per ea, quæ lib. </s> <s xml:id="echoid-s7193" xml:space="preserve">3. </s> <s xml:id="echoid-s7194" xml:space="preserve">Problem. </s> <s xml:id="echoid-s7195" xml:space="preserve">42. </s> <s xml:id="echoid-s7196" xml:space="preserve">Num. </s> <s xml:id="echoid-s7197" xml:space="preserve">3. </s> <s xml:id="echoid-s7198" xml:space="preserve">ſcripſimus: </s> <s xml:id="echoid-s7199" xml:space="preserve">ſi videlicet <lb/>intra campum eligatur punctum quodpiam, ex quo ad omnes angulos rectæ <lb/>ducantur, notatis angulis, quos efficiunt. </s> <s xml:id="echoid-s7200" xml:space="preserve">Nam ſi illæ rectæ menſurentur, an-<lb/>gulique ad aliquod punctum in charta transferantur, & </s> <s xml:id="echoid-s7201" xml:space="preserve">in rectis angulorum ca-<lb/>piantur tot particulæ æquales, quot menſuræ in rectis angulorum circa punctũ <lb/>in campo electum conſiſtentium deprehenſæ ſunt, continebunt rectæ extrema <lb/>puncta connectentes figuram ſimilem campo, vt in loco citato demonſtraui-<lb/>mus. </s> <s xml:id="echoid-s7202" xml:space="preserve">Quod ſi duo puncta eligantur in campo, è quibusrectæ ad angulos du-<lb/>cantur, notatis punctis, vbibinæ rectæ, conueniunt, deſcribetur etiam figura cã-<lb/>po ſimilis, quamuis rectæillæ non menſurentur: </s> <s xml:id="echoid-s7203" xml:space="preserve">quemadmodum ibidem Num. <lb/></s> <s xml:id="echoid-s7204" xml:space="preserve">1. </s> <s xml:id="echoid-s7205" xml:space="preserve">declarauimus.</s> <s xml:id="echoid-s7206" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7207" xml:space="preserve"><emph style="sc">Manifestvm</emph> autem eſt, eodem pacto menſurari poſſe campum, in quo <lb/> <anchor type="note" xlink:label="note-202-12a" xlink:href="note-202-12"/> lacus, vel ſylua comprehendatur, licet in triangula reſoluinon poſsit, dummo-<lb/>do eius latera exteriora cum angulis cognoſcipoſsint. </s> <s xml:id="echoid-s7208" xml:space="preserve">Quod ſi in circuitu agri <lb/>fuerit aliqua portio curua, & </s> <s xml:id="echoid-s7209" xml:space="preserve">nonrecta, ſecanda ea erit in tot partes, donec à re-<lb/>ctis lineis parum differant, eæque pro lateribus rectis aſſumendæ.</s> <s xml:id="echoid-s7210" xml:space="preserve"/> </p> <div xml:id="echoid-div465" type="float" level="2" n="4"> <note position="left" xlink:label="note-202-12" xlink:href="note-202-12a" xml:space="preserve">Quaratione <lb/>camp{us} intra <lb/>quem lac{us} <lb/>velſylua ex-<lb/>@ſt at menſure-<lb/>tur.</note> </div> <pb o="173" file="203" n="203" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s7211" xml:space="preserve">3. </s> <s xml:id="echoid-s7212" xml:space="preserve"><emph style="sc">Agrimensores</emph> ne cogantur totum cam-<lb/> <anchor type="figure" xlink:label="fig-203-01a" xlink:href="fig-203-01"/> <anchor type="note" xlink:label="note-203-01a" xlink:href="note-203-01"/> pum ſæpius perambulare, vt perpendiculares in trian-<lb/>gulis ducant, anguloſque metiantur, hanc ineunt ra-<lb/>tionem. </s> <s xml:id="echoid-s7213" xml:space="preserve">In agro, ſeu figura conſtituunt, quam poſ-<lb/>ſunt, maximum rectangulum, atque ad eius latera ex <lb/>angulis figuræ perpendiculares concipiunt demitti, <lb/>quod faciunt, applicando vnum latus normæ ad latus <lb/>rectanguli, & </s> <s xml:id="echoid-s7214" xml:space="preserve">aliud ad angulum figuræ oppoſitum di-<lb/>rigendo. </s> <s xml:id="echoid-s7215" xml:space="preserve">Ita namque tota figura reſoluta erit in rectan-<lb/>gulum illud conſtitutum, & </s> <s xml:id="echoid-s7216" xml:space="preserve">in trapezia duorum late-<lb/>rum parallelorum, at que in triangula rectangula. </s> <s xml:id="echoid-s7217" xml:space="preserve">Dein-<lb/>de vel ipſimet metiuntur latera rectanguli, & </s> <s xml:id="echoid-s7218" xml:space="preserve">perpen-<lb/>diculares, vel vt ab aliis menſurentut, præcipiunt, quod <lb/>quidem per catenulam ferream exequuntur. </s> <s xml:id="echoid-s7219" xml:space="preserve">Poſtre-<lb/>mo triangula quidem rectangula metiuntur, vt cap. </s> <s xml:id="echoid-s7220" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s7221" xml:space="preserve">Num. </s> <s xml:id="echoid-s7222" xml:space="preserve">3. </s> <s xml:id="echoid-s7223" xml:space="preserve">tradidimus, trapezia verò duorum laterum parallelorum, vt cap. </s> <s xml:id="echoid-s7224" xml:space="preserve">3. </s> <s xml:id="echoid-s7225" xml:space="preserve"><lb/>Num. </s> <s xml:id="echoid-s7226" xml:space="preserve">2. </s> <s xml:id="echoid-s7227" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s7228" xml:space="preserve">Rectangulum denique per ea, quæ cap. </s> <s xml:id="echoid-s7229" xml:space="preserve">1. </s> <s xml:id="echoid-s7230" xml:space="preserve">ſcripſimus, men-<lb/>ſurant. </s> <s xml:id="echoid-s7231" xml:space="preserve">Horum enim areæ in vnam ſummam collectæ conficiunt aream totius <lb/>agri, ſeu figuræ. </s> <s xml:id="echoid-s7232" xml:space="preserve">In propoſito octãgulo ABCDEFGH, continetur rectangulum <lb/>IKLM, trapezia HSTG, GTQF, OLRE; </s> <s xml:id="echoid-s7233" xml:space="preserve">& </s> <s xml:id="echoid-s7234" xml:space="preserve">triangula rectangula ANK, ANI, <lb/>ISH, FQM, MOE, DPR, CDP, BCV, BKV.</s> <s xml:id="echoid-s7235" xml:space="preserve"/> </p> <div xml:id="echoid-div466" type="float" level="2" n="5"> <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/xxxxxxxx/figures/203-01"/> </figure> <note position="right" xlink:label="note-203-01" xlink:href="note-203-01a" xml:space="preserve">Ratio com-<lb/>munis men-<lb/>ſorum in area <lb/>cuiuſuis figu-<lb/>ræinueſtigan-<lb/>da.</note> </div> <p> <s xml:id="echoid-s7236" xml:space="preserve"><emph style="sc">Alii</emph> non conſtituunt rectangulum intra campum, vel figuram, ſed lineam, <lb/>quam poſſunt, longiſsimam ducunt, nimirum à puncto A, adlatus E F, quam <lb/>fundamentalem appellant. </s> <s xml:id="echoid-s7237" xml:space="preserve">Ad hanc ex angulis demittunt perpendiculares: <lb/></s> <s xml:id="echoid-s7238" xml:space="preserve">atque ita totam rurſus figuramin trapezia duorum laterum parallelorum, & </s> <s xml:id="echoid-s7239" xml:space="preserve">in <lb/>triangula rectangula diſpertiunt, &</s> <s xml:id="echoid-s7240" xml:space="preserve">c. </s> <s xml:id="echoid-s7241" xml:space="preserve">Sed prior ratio commodior videtur: </s> <s xml:id="echoid-s7242" xml:space="preserve">quip-<lb/>pein qua perpendiculares ex angulis deductæ breuiores ſint, ac propterea fa-<lb/>cilius menſurentur, ac certius.</s> <s xml:id="echoid-s7243" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7244" xml:space="preserve"><emph style="sc">Qvando</emph> intra agrum dictæ operationes fierinequeunt, ſolent etiam men-<lb/>ſores circa agrum includentem ſyluas, lacus, & </s> <s xml:id="echoid-s7245" xml:space="preserve">ædificia, vel alia impedimenta, <lb/>formare rectangulum. </s> <s xml:id="echoid-s7246" xml:space="preserve">Nam ſi ad eius latera ducantur perpendiculares ab an-<lb/>gulis exterioribus agri conſtituentur iterum trapezia rectangula duorum la-<lb/>terum parallelorum, vel parallelogramma, & </s> <s xml:id="echoid-s7247" xml:space="preserve">triangula rectangula extra agrum: <lb/></s> <s xml:id="echoid-s7248" xml:space="preserve">quorum areæ ſi ex area totius rectanguli ſub ducantur, reliqua fiet area propoſi-<lb/>ti agri.</s> <s xml:id="echoid-s7249" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7250" xml:space="preserve">4. </s> <s xml:id="echoid-s7251" xml:space="preserve"><emph style="sc">Sed</emph> neque aſpernanda mihi videtur ea ratio, quam olimmeis auditori-<lb/> <anchor type="note" xlink:label="note-203-02a" xlink:href="note-203-02"/> bus explicare ſolebam. </s> <s xml:id="echoid-s7252" xml:space="preserve">Vt nimirum toti figuræ (reducto prius agro ad ſimilem <lb/>figuram, vt paulo ante Num. </s> <s xml:id="echoid-s7253" xml:space="preserve">2. </s> <s xml:id="echoid-s7254" xml:space="preserve">præcepi) conſtituatur quadratum æquale, vel <lb/>certè latus eius quadrati inueniatur. </s> <s xml:id="echoid-s7255" xml:space="preserve">Nam ſi vnum huius quadrati latus men-<lb/>ſuretur, atque in ſeipſum ducatur, pro dibit area figurę propoſitę. </s> <s xml:id="echoid-s7256" xml:space="preserve">Menſuran-<lb/>dum porro eſt latus in particulis laterum figuræ, quę menſuris laterum agrire-<lb/>ſpondent, quod facilè fiet per inſtrumentum partium.</s> <s xml:id="echoid-s7257" xml:space="preserve"/> </p> <div xml:id="echoid-div467" type="float" level="2" n="6"> <note position="right" xlink:label="note-203-02" xlink:href="note-203-02a" xml:space="preserve">Pulchra ratio <lb/>areæ inueni-<lb/>endæ cuiuſcũ-<lb/>que figuræ.</note> </div> <p> <s xml:id="echoid-s7258" xml:space="preserve"><emph style="sc">Qvo</emph> pacto autem cuilibet figuræ quadratum ęquale ſine magno labore <lb/> <anchor type="note" xlink:label="note-203-03a" xlink:href="note-203-03"/> conſtruipoſsit, do cui in ſcholio propoſ. </s> <s xml:id="echoid-s7259" xml:space="preserve">14. </s> <s xml:id="echoid-s7260" xml:space="preserve">lib. </s> <s xml:id="echoid-s7261" xml:space="preserve">2. </s> <s xml:id="echoid-s7262" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7263" xml:space="preserve">quod hocloco, pau-<lb/>cis in melius mutatis, repetendum cenſeo. </s> <s xml:id="echoid-s7264" xml:space="preserve">Sit ergo heptagonum irregulare <lb/>ABCDEFG, quo reſoluto in 5. </s> <s xml:id="echoid-s7265" xml:space="preserve">triãgula ABG, BCG, CDG, DEG, EFG, ducan-<lb/>tur ad B G, baſem communem duorũ triangulorum ab angulis oppoſitis A, C, <pb o="174" file="204" n="204" rhead="GEOMETR. PRACT."/> perpendiculares AL, CH, quarũ poſterior in GB, protractam cadit in noſtra fi-<lb/>gura. </s> <s xml:id="echoid-s7266" xml:space="preserve">Deinde in recta quacunque OP, ſumantur OQ, QR, ipſis AL, CH, æqua-<lb/>les. </s> <s xml:id="echoid-s7267" xml:space="preserve">Item RP, ipſi BI, ſemiſsi baſis BG, æqualis; </s> <s xml:id="echoid-s7268" xml:space="preserve">at circa OP, exmedio puncto S, <lb/>ſemicirculus deſcribatur O T P: </s> <s xml:id="echoid-s7269" xml:space="preserve">ac denique ex R, termino rectę OR, quę dua-<lb/>bus perpendicularibus AL, CH, æqualis eſt, ad O P, perpendicularis excitetur <lb/>R T, ſemicirculum ſecans in T. </s> <s xml:id="echoid-s7270" xml:space="preserve">Dico quadratum rectę R T, duobus triangulis <lb/>ſimul ABG, BCG, eſſe æquale. </s> <s xml:id="echoid-s7271" xml:space="preserve">Quia enim rectangulum ſub BI, ſemiſſe baſis, & </s> <s xml:id="echoid-s7272" xml:space="preserve"><lb/>perpendiculari AL, æquale eſt triangulo ABG, ex propoſ. </s> <s xml:id="echoid-s7273" xml:space="preserve">1. </s> <s xml:id="echoid-s7274" xml:space="preserve">lib. </s> <s xml:id="echoid-s7275" xml:space="preserve">7. </s> <s xml:id="echoid-s7276" xml:space="preserve">huius; </s> <s xml:id="echoid-s7277" xml:space="preserve">& </s> <s xml:id="echoid-s7278" xml:space="preserve">re-<lb/>ctangulum ſub eadem B I, & </s> <s xml:id="echoid-s7279" xml:space="preserve">perpendiculari G H, æquale eſt triangulo B C G: <lb/></s> <s xml:id="echoid-s7280" xml:space="preserve">Quod autem ſub B I, & </s> <s xml:id="echoid-s7281" xml:space="preserve">aggregato ex AL, CH, hoc eſt, ſub R P, O R, (quod <lb/> <anchor type="note" xlink:label="note-204-01a" xlink:href="note-204-01"/> RP, ipſi BI, ſumpta ſit æqualis, & </s> <s xml:id="echoid-s7282" xml:space="preserve">OR, ipſis AL, CH, ſimul) <anchor type="note" xlink:href="" symbol="a"/> æquale eſt eis, quę ſub BI, & </s> <s xml:id="echoid-s7283" xml:space="preserve">AL, CH, comprehenduntur, rectangulis; </s> <s xml:id="echoid-s7284" xml:space="preserve">erit rectangulum ſub <lb/> <anchor type="note" xlink:label="note-204-02a" xlink:href="note-204-02"/> OR, RP, duobus triangulis ABG, BCG, æquale. </s> <s xml:id="echoid-s7285" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cum ergo quadratum ex R T, rectangulo ſub OR, RP, ſit æquale, (quod ex ſchol. </s> <s xml:id="echoid-s7286" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7287" xml:space="preserve">13. </s> <s xml:id="echoid-s7288" xml:space="preserve">lib. </s> <s xml:id="echoid-s7289" xml:space="preserve">6. </s> <s xml:id="echoid-s7290" xml:space="preserve">Eu-<lb/>clid R T, media proportionalis ſit inter O R, RP,) erit quo que quadratum ex <lb/>R T, duobus triangulis ABG, BCG, æquale, quod eſt propoſitum. </s> <s xml:id="echoid-s7291" xml:space="preserve">Immo qua-<lb/> <anchor type="figure" xlink:label="fig-204-01a" xlink:href="fig-204-01"/> dratum ex R T, rectangulo ſub O R, R P, æquale eſſe, demonſtrabitur hoc et-<lb/>iam modo ſine ope lib. </s> <s xml:id="echoid-s7292" xml:space="preserve">6. </s> <s xml:id="echoid-s7293" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7294" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Rectangulum ſub O R, R P, vna cum <anchor type="note" xlink:label="note-204-03a" xlink:href="note-204-03"/> quadrato ex S R, æquale eſt quadrato ex S P, hoc eſt, (ducta S T,) quadrato <lb/>ex S T, <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, quadratis ex S R, R T. </s> <s xml:id="echoid-s7295" xml:space="preserve">Ablato ergo communi quadrato <anchor type="note" xlink:label="note-204-04a" xlink:href="note-204-04"/> rectę S R, reliquum rectangulum ſub O R, R P, reliquo quadrato ex R T, erit <lb/>æquale.</s> <s xml:id="echoid-s7296" xml:space="preserve"/> </p> <div xml:id="echoid-div468" type="float" level="2" n="7"> <note position="right" xlink:label="note-203-03" xlink:href="note-203-03a" xml:space="preserve">Quadratum <lb/>datæ figuræ <lb/>æquale qua <lb/>ratione con-<lb/>ſtruatur.</note> <note symbol="a" position="left" xlink:label="note-204-01" xlink:href="note-204-01a" xml:space="preserve">1. ſecundi.</note> <note symbol="b" position="left" xlink:label="note-204-02" xlink:href="note-204-02a" xml:space="preserve">17. ſexti.</note> <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/xxxxxxxx/figures/204-01"/> </figure> <note symbol="c" position="left" xlink:label="note-204-03" xlink:href="note-204-03a" xml:space="preserve">5. ſecundi.</note> <note symbol="d" position="left" xlink:label="note-204-04" xlink:href="note-204-04a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s7297" xml:space="preserve"><emph style="sc">Eodem</emph> modo reperiemus quadratum duobus triangulis C D G, DEG, æ-<lb/>quale, ſi ad baſem communem D G, ducantur perpendiculares C D, E M, qua-<lb/>rum prior in noſtra figura cum latere C D, coincidit, &</s> <s xml:id="echoid-s7298" xml:space="preserve">c. </s> <s xml:id="echoid-s7299" xml:space="preserve">Atque ita deinceps <lb/>ſi plura fuerint triangula, reperiemus ſemper binis triangulis ſingula quadrata <lb/>æqualia. </s> <s xml:id="echoid-s7300" xml:space="preserve">Sed quia in noſtra figura ſupereſt vnum tantum triangulum EF G, in- <pb o="175" file="205" n="205" rhead="LIBER QVARTVS."/> ueniemus ei quadratum æquale, ſi, ducta perpendiculari F N, circa rectam ex <lb/>F N, & </s> <s xml:id="echoid-s7301" xml:space="preserve">ſemiſſe baſis E Z, conflatam ſemicirculus deſcribatur, &</s> <s xml:id="echoid-s7302" xml:space="preserve">c. </s> <s xml:id="echoid-s7303" xml:space="preserve">Sint ergo a, b, <lb/>c, latera quadratorum trapeziis, ABCG, CDEG, & </s> <s xml:id="echoid-s7304" xml:space="preserve">triangulo EFG, æqualium; <lb/></s> <s xml:id="echoid-s7305" xml:space="preserve">quibus omnibus quadratis vnum ęquale exhibebimus hac arte. </s> <s xml:id="echoid-s7306" xml:space="preserve">Fiat angulus <lb/>rectus def, & </s> <s xml:id="echoid-s7307" xml:space="preserve">lateribus a, b, æquales ſumantur rectę ed, eg, <anchor type="note" xlink:href="" symbol="a"/> eritque quadra- <anchor type="note" xlink:label="note-205-01a" xlink:href="note-205-01"/> tum ductę rectę d g, quadratis rectarum ed, eg, hoc eſt, laterum a, b, æquale. <lb/></s> <s xml:id="echoid-s7308" xml:space="preserve">Capiatur rurſus e k, lateric, & </s> <s xml:id="echoid-s7309" xml:space="preserve">recta e h, rectę d g, ęqualis; </s> <s xml:id="echoid-s7310" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> eritque rurſus qua- <anchor type="note" xlink:label="note-205-02a" xlink:href="note-205-02"/> dratum ex kh, ęquale quadratis ex k e, e h, id eſt ex c, eh, nimirum tribus ex a, <lb/>b, c. </s> <s xml:id="echoid-s7311" xml:space="preserve">Siigitur latus ex k h, menſuretur, & </s> <s xml:id="echoid-s7312" xml:space="preserve">in ſe ducatur, ginetur area figurę pro-<lb/>poſitæ A B C D E F G.</s> <s xml:id="echoid-s7313" xml:space="preserve"/> </p> <div xml:id="echoid-div469" type="float" level="2" n="8"> <note symbol="a" position="right" xlink:label="note-205-01" xlink:href="note-205-01a" xml:space="preserve">47. primi.</note> <note symbol="b" position="right" xlink:label="note-205-02" xlink:href="note-205-02a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s7314" xml:space="preserve"><emph style="sc">Eodem</emph> artificio, ſi plura ſint latera, inueniemus quadratum omnibus qua-<lb/>dratis ęquale. </s> <s xml:id="echoid-s7315" xml:space="preserve">Vt ſi foret alterum latus q, acciperemus ei æqualem rectam <lb/>e p. </s> <s xml:id="echoid-s7316" xml:space="preserve">Item rectam ef, rectę k h, æqualem. </s> <s xml:id="echoid-s7317" xml:space="preserve">Nam quadratum ex p f, quadratis ex <lb/>e p, hoc eſt, ex q, & </s> <s xml:id="echoid-s7318" xml:space="preserve">ex ef, id eſt, ex a, b, c, erit ęquale, & </s> <s xml:id="echoid-s7319" xml:space="preserve">ſic de pluribus. <lb/></s> <s xml:id="echoid-s7320" xml:space="preserve"> <anchor type="note" xlink:label="note-205-03a" xlink:href="note-205-03"/> </s> </p> <div xml:id="echoid-div470" type="float" level="2" n="9"> <note position="right" xlink:label="note-205-03" xlink:href="note-205-03a" xml:space="preserve">Facilis ratio <lb/>menſurandi <lb/>trapezii irre-<lb/>gularis.</note> </div> <p> <s xml:id="echoid-s7321" xml:space="preserve"><emph style="sc">Ex</emph> his colligitur facilis ratio metiendi trapezij irregularis, cuiuſmo di eſt in <lb/>proxima figura trapezium A B C G. </s> <s xml:id="echoid-s7322" xml:space="preserve">Nam ducta diametro B G, ſi ad eam duę <lb/>perpendiculares demiſſę A L, CH, menſurentur, earumque aggregatum in me-<lb/>dietatem diametri B G, multiplicetur, procreabitur area trapezij, vt demonſtra-<lb/>tum eſt.</s> <s xml:id="echoid-s7323" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7324" xml:space="preserve"><emph style="sc">In</emph> octauo porrò lib. </s> <s xml:id="echoid-s7325" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7326" xml:space="preserve">6. </s> <s xml:id="echoid-s7327" xml:space="preserve">docebimus quo que, qua ratione datę figu-<lb/>rę rectilineę rectangulum æquale conſtruatur. </s> <s xml:id="echoid-s7328" xml:space="preserve">quod ſi fiat hoc loco, effi cietur <lb/>illi rectangulo quadratum ęquale, per vltimam propoſ. </s> <s xml:id="echoid-s7329" xml:space="preserve">lib. </s> <s xml:id="echoid-s7330" xml:space="preserve">2. </s> <s xml:id="echoid-s7331" xml:space="preserve">Euclidis, ſine vl-<lb/>lo negotio, aut moleſtia.</s> <s xml:id="echoid-s7332" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div472" type="section" level="1" n="175"> <head xml:id="echoid-head180" xml:space="preserve">DE AREA MVLTILATERA-<lb/>rum figurarum regularium.</head> <head xml:id="echoid-head181" xml:space="preserve"><emph style="sc">Capvt</emph> V.</head> <p> <s xml:id="echoid-s7333" xml:space="preserve">1. </s> <s xml:id="echoid-s7334" xml:space="preserve"><emph style="sc">QVanqvam</emph> regulares figurę, quę ſcilicet ſunt & </s> <s xml:id="echoid-s7335" xml:space="preserve">æquilaterę & </s> <s xml:id="echoid-s7336" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s7337" xml:space="preserve">æquiangulę, menſurari poſ@int, vt irregulares pręcedentis capi-<lb/>tis, reſoluendo eas in triangula, &</s> <s xml:id="echoid-s7338" xml:space="preserve">c. </s> <s xml:id="echoid-s7339" xml:space="preserve">ſolet tamen dari propria ac pe-<lb/>culiaris regula, qua cuiuſque figurę regularis area inuenitur: </s> <s xml:id="echoid-s7340" xml:space="preserve">quę ita ſe habet.</s> <s xml:id="echoid-s7341" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7342" xml:space="preserve">SEMISSIS ambit{us} figuræ multiplicetur in perpendicularem è centro figuræ ad <lb/>vnum lat{us} cadentem. </s> <s xml:id="echoid-s7343" xml:space="preserve">Numer{us} enim product{us} area erit figuræ.</s> <s xml:id="echoid-s7344" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7345" xml:space="preserve"><emph style="sc">Nam</emph> vt lib. </s> <s xml:id="echoid-s7346" xml:space="preserve">7. </s> <s xml:id="echoid-s7347" xml:space="preserve">de Iſoperimetris propoſ. </s> <s xml:id="echoid-s7348" xml:space="preserve">2. </s> <s xml:id="echoid-s7349" xml:space="preserve">demonſtrabimus, area cuiuslibet <lb/> <anchor type="note" xlink:label="note-205-04a" xlink:href="note-205-04"/> figurę regularis ęqualis eſt rectangulo contento ſub perpendiculari à centro fi-<lb/>gurę ad vnum latus ducta, & </s> <s xml:id="echoid-s7350" xml:space="preserve">ſub dimidiato ambitu eiuſdem figurę.</s> <s xml:id="echoid-s7351" xml:space="preserve"/> </p> <div xml:id="echoid-div472" type="float" level="2" n="1"> <note position="right" xlink:label="note-205-04" xlink:href="note-205-04a" xml:space="preserve">Area figura <lb/>regularis.</note> </div> <p> <s xml:id="echoid-s7352" xml:space="preserve">2. </s> <s xml:id="echoid-s7353" xml:space="preserve"><emph style="sc">Perpendicvlaris</emph> porrò è centro figurę in vnum latus cadens, <lb/> <anchor type="note" xlink:label="note-205-05a" xlink:href="note-205-05"/> vna cum ſemidiametro circuli figuram ambientis ſic reperietur. </s> <s xml:id="echoid-s7354" xml:space="preserve">Numerus late-<lb/>rum, ſiue angulorum duplicetur, & </s> <s xml:id="echoid-s7355" xml:space="preserve">à duplo auferantur 4. </s> <s xml:id="echoid-s7356" xml:space="preserve">Nam reliquus nu-<lb/>merus indicabit, quot rectis angulis omnes anguli figurę ęquiualeant, per ea, <lb/>quę in ſcholio propoſ. </s> <s xml:id="echoid-s7357" xml:space="preserve">32. </s> <s xml:id="echoid-s7358" xml:space="preserve">lib. </s> <s xml:id="echoid-s7359" xml:space="preserve">1. </s> <s xml:id="echoid-s7360" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7361" xml:space="preserve">demonſtrata ſunt. </s> <s xml:id="echoid-s7362" xml:space="preserve">Hic idem nume-<lb/>@us reliquus, videlicet, numerus angulorum rectorum, per numerum angulo-<lb/>rum diuidatur, vt Quotiens vnius anguli figurę magnitudinem exhibeat, qui <pb o="176" file="206" n="206" rhead="GEOMETR. PRACT."/> in hexagono continet rectum cum parte tertia, hoc eſt, grad. </s> <s xml:id="echoid-s7363" xml:space="preserve">120. </s> <s xml:id="echoid-s7364" xml:space="preserve">Et quoniam <lb/>ſemidiameter ſecat angulum figuræ bifariam, vt conſtat ex demonſtratione <lb/>propoſ. </s> <s xml:id="echoid-s7365" xml:space="preserve">12. </s> <s xml:id="echoid-s7366" xml:space="preserve">lib. </s> <s xml:id="echoid-s7367" xml:space="preserve">4. </s> <s xml:id="echoid-s7368" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7369" xml:space="preserve">fit vt ſi ſemiſsis lateris, in quod perpendicularis cadit, <lb/>ponatur ſinus totus, perpendicularis ſit Tangens ſemiſsis anguli figuræ, & </s> <s xml:id="echoid-s7370" xml:space="preserve">ſemi-<lb/>diameter, Secans. </s> <s xml:id="echoid-s7371" xml:space="preserve">Si fiat ergo, <lb/> <anchor type="note" xlink:label="note-206-01a" xlink:href="note-206-01"/> prodibit tam perpendicularis, quàm ſemidiameter in partibus lateris figuræ. <lb/></s> <s xml:id="echoid-s7372" xml:space="preserve">Verbi gratia, in Hexagono regulari, cuius vnum latus ſit 12. </s> <s xml:id="echoid-s7373" xml:space="preserve">ſi fiat, vt 100000. </s> <s xml:id="echoid-s7374" xml:space="preserve">ſi-<lb/>nus totus ad 6. </s> <s xml:id="echoid-s7375" xml:space="preserve">ſemiſſem lateris: </s> <s xml:id="echoid-s7376" xml:space="preserve">ita 173205. </s> <s xml:id="echoid-s7377" xml:space="preserve">tangens ſemiſsis anguli Hexagoni <lb/>ad aliud, reperietur perpendicularis 10 {39230/100000}. </s> <s xml:id="echoid-s7378" xml:space="preserve">vel 10 {3923/10000}. </s> <s xml:id="echoid-s7379" xml:space="preserve">Item ſi fiat, vt <lb/>100000. </s> <s xml:id="echoid-s7380" xml:space="preserve">ſinus totus ad 6. </s> <s xml:id="echoid-s7381" xml:space="preserve">ſemiſſem lateris; </s> <s xml:id="echoid-s7382" xml:space="preserve">ita 200000. </s> <s xml:id="echoid-s7383" xml:space="preserve">Secans ſemiſsis anguli <lb/>liexagoni ad aliud, exibit ſemidiameter figuræ 12.</s> <s xml:id="echoid-s7384" xml:space="preserve"/> </p> <div xml:id="echoid-div473" type="float" level="2" n="2"> <note position="right" xlink:label="note-205-05" xlink:href="note-205-05a" xml:space="preserve">Perpendicu-<lb/>laris & ſemi-<lb/>diameter fi-<lb/>guræ regula-<lb/>ris quo pacto <lb/>inueniatur.</note> <note style="it" position="right" xlink:label="note-206-01" xlink:href="note-206-01a" xml:space="preserve"> <lb/>Vt lat{us} 100000, ſin{us} \\ tot{us}. # ad ſemiſſem \\ lateris: # ita Tangens ſemiſſis an- \\ guli, vel ſecans. # ad aliud, <lb/></note> </div> <p> <s xml:id="echoid-s7385" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi perpendicularis 10 {3923/10000}. </s> <s xml:id="echoid-s7386" xml:space="preserve">ducatur in 36. </s> <s xml:id="echoid-s7387" xml:space="preserve">ſemiſſem ambitus He-<lb/>xagoni ex tribus lateribus conflatam, producetur area Hexagoni 374 {1228/10000}. </s> <s xml:id="echoid-s7388" xml:space="preserve">vel <lb/>374 {307/2500}. </s> <s xml:id="echoid-s7389" xml:space="preserve">Eodemque modo procedendum eſt in aliis figuris regularibus, in <lb/>quibus angulorum ſemiſſes in tabula Tangentium, ac ſecantium accipi poſ-<lb/>ſunt.</s> <s xml:id="echoid-s7390" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7391" xml:space="preserve">3. </s> <s xml:id="echoid-s7392" xml:space="preserve"><emph style="sc">Qvoniam</emph> vero circa quamlibet figuram regularem circulus deſcribi <lb/>poteſt, vt ex lib. </s> <s xml:id="echoid-s7393" xml:space="preserve">4. </s> <s xml:id="echoid-s7394" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7395" xml:space="preserve">conſtet, proponemus hic plurimarum figurarum regu-<lb/>larium latera in partibus diametri circuli ambientis 20000000. </s> <s xml:id="echoid-s7396" xml:space="preserve">vel ſemidiame-<lb/>tri, ſiue ſinus totius 10000000. </s> <s xml:id="echoid-s7397" xml:space="preserve">ex probatis auctoribus, vt earum areæ magis ex-<lb/> <anchor type="note" xlink:label="note-206-02a" xlink:href="note-206-02"/> quiſitè inueniri poſsint per regulam Num. </s> <s xml:id="echoid-s7398" xml:space="preserve">1. </s> <s xml:id="echoid-s7399" xml:space="preserve">propoſitam. </s> <s xml:id="echoid-s7400" xml:space="preserve">Nam ſi ex ſolo vno <lb/>latere in aliqua menſura cognito aream inueſtigare velimus, quemadmodum in <lb/>Hexagono factum eſt, occurrẽt multæ difficultates acmagnę, propterea quod <lb/>ſemidiametri, perpendiculareſque ex ſinubus, Tangentibus, ac ſecantibus erui <lb/>non poſſunt, niſi quando ſemiſsis anguli figuræ comprehendit pręcisè gradus, <lb/>vel gradus cum minutis, vel gradus cumminutis & </s> <s xml:id="echoid-s7401" xml:space="preserve">ſecundis: </s> <s xml:id="echoid-s7402" xml:space="preserve">(quamuis ſi ſe-<lb/>cunda adſint, neceſſe ſit partem proportionalem adhibere) ſicut in Hexago-<lb/>no paulo ante factum eſt, cuius anguli ſemiſsis continet grad. </s> <s xml:id="echoid-s7403" xml:space="preserve">60. </s> <s xml:id="echoid-s7404" xml:space="preserve">quod in <lb/>quam plurimis figuris non contingit. </s> <s xml:id="echoid-s7405" xml:space="preserve">Nam, verbi gratia, in Heptagono omnes <lb/>7. </s> <s xml:id="echoid-s7406" xml:space="preserve">anguli ex ſcholio propoſ. </s> <s xml:id="echoid-s7407" xml:space="preserve">32. </s> <s xml:id="echoid-s7408" xml:space="preserve">lib. </s> <s xml:id="echoid-s7409" xml:space="preserve">1. </s> <s xml:id="echoid-s7410" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7411" xml:space="preserve">æquiualent 10. </s> <s xml:id="echoid-s7412" xml:space="preserve">rectis, ideo que v-<lb/>nus angulus complectitur vnum rectum cum {3/7}. </s> <s xml:id="echoid-s7413" xml:space="preserve">hoc eſt, gradus 128. </s> <s xml:id="echoid-s7414" xml:space="preserve">Min. </s> <s xml:id="echoid-s7415" xml:space="preserve">34. <lb/></s> <s xml:id="echoid-s7416" xml:space="preserve">Sec. </s> <s xml:id="echoid-s7417" xml:space="preserve">17. </s> <s xml:id="echoid-s7418" xml:space="preserve">Ter. </s> <s xml:id="echoid-s7419" xml:space="preserve">8. </s> <s xml:id="echoid-s7420" xml:space="preserve">Quar. </s> <s xml:id="echoid-s7421" xml:space="preserve">34. </s> <s xml:id="echoid-s7422" xml:space="preserve">&</s> <s xml:id="echoid-s7423" xml:space="preserve">c. </s> <s xml:id="echoid-s7424" xml:space="preserve">atque eius ſemiſsis grad. </s> <s xml:id="echoid-s7425" xml:space="preserve">64. </s> <s xml:id="echoid-s7426" xml:space="preserve">min. </s> <s xml:id="echoid-s7427" xml:space="preserve">17. </s> <s xml:id="echoid-s7428" xml:space="preserve">Secun. </s> <s xml:id="echoid-s7429" xml:space="preserve">8. </s> <s xml:id="echoid-s7430" xml:space="preserve"><lb/>Ter. </s> <s xml:id="echoid-s7431" xml:space="preserve">34. </s> <s xml:id="echoid-s7432" xml:space="preserve">Quar. </s> <s xml:id="echoid-s7433" xml:space="preserve">17. </s> <s xml:id="echoid-s7434" xml:space="preserve">&</s> <s xml:id="echoid-s7435" xml:space="preserve">c. </s> <s xml:id="echoid-s7436" xml:space="preserve">ex qua ſemiſſe nequit Tangens pro perpendiculari, ne-<lb/>que ſecans pro ſemidiametro per tabulas Tangentium, atque ſecantium excer-<lb/>pi poteſt. </s> <s xml:id="echoid-s7437" xml:space="preserve">Idemque in aliis figuris innumeris accidere comperies. </s> <s xml:id="echoid-s7438" xml:space="preserve">Quod ſi <lb/>ſcientia inuenta eſſet conſtruendi omnes figuras regulares, ſuperari aliquo mo-<lb/>do poſſet hæc diffi cultas: </s> <s xml:id="echoid-s7439" xml:space="preserve">propterea quod cognito latere vno in quantiſcun-<lb/>que partibus, cognoſci quoque poſſet tam perpendicularis, quam ſemidiame-<lb/>ter in iiſdem partibus, beneficio inſtrumenti partium, vt lib. </s> <s xml:id="echoid-s7440" xml:space="preserve">1. </s> <s xml:id="echoid-s7441" xml:space="preserve">cap. </s> <s xml:id="echoid-s7442" xml:space="preserve">1. </s> <s xml:id="echoid-s7443" xml:space="preserve">ad finem <lb/>Num. </s> <s xml:id="echoid-s7444" xml:space="preserve">1. </s> <s xml:id="echoid-s7445" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s7446" xml:space="preserve">Sed quia pauciſsimas figuras æquilateras deſcribere no-<lb/>uimus intra circulum, areas plurimarum ignorari ncceſſe eſt. </s> <s xml:id="echoid-s7447" xml:space="preserve">Hanc ob cau-<lb/>ſam nonnulli Geometræ, inter quos ſtrenuam operam nauauit Ludolphus, <lb/>à Collen, ingenti labore latera figurarum, ſiue deſcriptæ eæ ſint, ſiue non, ex-<lb/>plorarunt: </s> <s xml:id="echoid-s7448" xml:space="preserve">quamuis earum anguli præter gradus, ac minuta comprehendant</s> </p> <div xml:id="echoid-div474" type="float" level="2" n="3"> <note position="left" xlink:label="note-206-02" xlink:href="note-206-02a" xml:space="preserve">Quare ex la-<lb/>tere figuræ re-<lb/>gularis dato <lb/>non ſemper <lb/>poſſit inueniri <lb/>area, niſi figu-<lb/>raipſa deſcri-<lb/>pta ſit.</note> </div> <pb o="177" file="207" n="207" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s7449" xml:space="preserve">TABVLA CONTINENS LATERA FI-<lb/>gurarum regularium à triangulo vſque ad figuram <lb/>80. </s> <s xml:id="echoid-s7450" xml:space="preserve">laterum, poſita diametro 20000000. </s> <s xml:id="echoid-s7451" xml:space="preserve">vel ſinu <lb/>toto 10000000.</s> <s xml:id="echoid-s7452" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>Num. lat. vel angulor. # Latera figurarum \\ regularium, po- \\ ſita diametro \\ 20000000. \\ vel ſinu toto \\ 10000000. <lb/>3 # 17320508 <lb/>4 # 14142135 <lb/>5 # 11755705 <lb/>6 # 10000000 <lb/>7 # 8677674 <lb/>8 # 7653668 <lb/>9 # 6840402 <lb/>10 # 6180339 <lb/>11 # 5634651 <lb/>12 # 5176380 <lb/>13 # 4786313 <lb/>14 # 4450418 <lb/>15 # 4158233 <lb/>16 # 3901806 <lb/>17 # 3674990 <lb/>18 # 3472963 <lb/>19 # 3291891 <lb/>20 # 3128689 <lb/>21 # 2980845 <lb/>22 # 2846296 <lb/>23 # 2723332 <lb/>24 # 2610523 <lb/>25 # 2506664 <lb/>26 # 2410733 <lb/>27 # 2321858 <lb/>28 # 2239289 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Num. lat. vel angulor. # Latera figurarum \\ regularium, po- \\ ſita diametro \\ 20000000. \\ vel ſinu toto \\ 10000000. <lb/>29 # 2162380 <lb/>30 # 2090569 <lb/>31 # 2023366 <lb/>32 # 1960341 <lb/>33 # 1901120 <lb/>34 # 1845367 <lb/>35 # 1792786 <lb/>36 # 1743114 <lb/>37 # 1696118 <lb/>38 # 1651586 <lb/>39 # 1609331 <lb/>40 # 1569181 <lb/>41 # 1530985 <lb/>42 # 1494601 <lb/>43 # 1459906 <lb/>44 # 1426783 <lb/>45 # 1395129 <lb/>46 # 1364848 <lb/>47 # 1335852 <lb/>48 # 1308062 <lb/>49 # 1281404 <lb/>50 # 1255810 <lb/>51 # 1231218 <lb/>52 # 1207569 <lb/>53 # 1184812 <lb/>54 # 1162896 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Num. lat. vel angulor. # Latera figurarum \\ regularium, po- \\ ſita diametro \\ 20000000. \\ vel ſinu toto \\ 10000000. <lb/>55 # 1141776 <lb/>56 # 1121408 <lb/>57 # 1101755 <lb/>58 # 1082778 <lb/>59 # 1064443 <lb/>60 # 1046719 <lb/>61 # 1029575 <lb/>62 # 1012983 <lb/>63 # 996917 <lb/>64 # 981353 <lb/>65 # 966275 <lb/>66 # 951638 <lb/>67 # 937445 <lb/>68 # 923669 <lb/>69 # 910291 <lb/>70 # 897296 <lb/>71 # 884666 <lb/>72 # 872387 <lb/>73 # 860444 <lb/>74 # 848824 <lb/>75 # 837513 <lb/>76 # 826499 <lb/>77 # 815771 <lb/>78 # 805318 <lb/>79 # 795130 <lb/>80 # 785196 <lb/></note> <pb o="178" file="208" n="208" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s7453" xml:space="preserve"> inſuper Secunda, Tertia, Quarta, &</s> <s xml:id="echoid-s7454" xml:space="preserve">c. </s> <s xml:id="echoid-s7455" xml:space="preserve">vt earum areas conſequi poſsimus. </s> <s xml:id="echoid-s7456" xml:space="preserve">Nam <lb/>ex latere cuiuſcunque figurę regularis cognito in partibus diametri circuli cir-<lb/>cumſcripti, vel ſinus totius, veniemus per ea, quæ cap. </s> <s xml:id="echoid-s7457" xml:space="preserve">2. </s> <s xml:id="echoid-s7458" xml:space="preserve">Num. </s> <s xml:id="echoid-s7459" xml:space="preserve">2. </s> <s xml:id="echoid-s7460" xml:space="preserve">ſcripſimus, in <lb/>cognitionem lineæ perpendicularis ex centro in vnum latus deductę, ac proin-<lb/>de totam aream nanciſcemur, vt paulo ante Num. </s> <s xml:id="echoid-s7461" xml:space="preserve">1. </s> <s xml:id="echoid-s7462" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s7463" xml:space="preserve">Latera igitur in <lb/>quam plurimis figuris in tabula pręcedenti expoſita habes, quæ omnia veris la-<lb/>teribus ſunt paulò minora; </s> <s xml:id="echoid-s7464" xml:space="preserve">& </s> <s xml:id="echoid-s7465" xml:space="preserve">ſi adieceris vnitatem, fient paulò maiora veris; <lb/></s> <s xml:id="echoid-s7466" xml:space="preserve">ita vt verum latus trianguli æquilateri inter hos duos numeros 17320508. </s> <s xml:id="echoid-s7467" xml:space="preserve"><lb/>17320509. </s> <s xml:id="echoid-s7468" xml:space="preserve">conſiſtat.</s> <s xml:id="echoid-s7469" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Ex cognita ſe-<lb/>midiametro <lb/>circuli inue-<lb/>nire lat{us} fi-<lb/>guræregula-<lb/>ris in eo circu-<lb/>lo deſcriptæ.</note> <p> <s xml:id="echoid-s7470" xml:space="preserve">4. </s> <s xml:id="echoid-s7471" xml:space="preserve"><emph style="sc">Iam</emph> verò cognita ſemidiametro alicuius circuli in partibus cuiuſcunque <lb/>menſuræ, reperiemus in iiſdem partibus latus figurę regularis, cuius laterum nu-<lb/>merus maior non eſt, quam 80. </s> <s xml:id="echoid-s7472" xml:space="preserve">beneficio præcedentis tabulę: </s> <s xml:id="echoid-s7473" xml:space="preserve">ſi nimirum fiat, <lb/>vt ſinus totus 10000000. </s> <s xml:id="echoid-s7474" xml:space="preserve">ad latus figurę propoſitæ in præcedenti tabula, ita ſe-<lb/>midiamer circuli propoſiti data ad aliud. </s> <s xml:id="echoid-s7475" xml:space="preserve">Sit verbi gratia, ſemidiameter alicuius <lb/>circuli 12. </s> <s xml:id="echoid-s7476" xml:space="preserve">& </s> <s xml:id="echoid-s7477" xml:space="preserve">inueniendum ſit latus decagoni reſpectu dictæ ſemidiametri: </s> <s xml:id="echoid-s7478" xml:space="preserve">Fiat <lb/>vt 10000000. </s> <s xml:id="echoid-s7479" xml:space="preserve">ſinus totus ad 6180339. </s> <s xml:id="echoid-s7480" xml:space="preserve">latus Decagoni; </s> <s xml:id="echoid-s7481" xml:space="preserve">ita 12. </s> <s xml:id="echoid-s7482" xml:space="preserve">ſemidiameter da-<lb/>ta ad aliud; </s> <s xml:id="echoid-s7483" xml:space="preserve">exibitque latus quæſitum 8 {164068/10000000}. </s> <s xml:id="echoid-s7484" xml:space="preserve">vel in minoribus numeris <lb/>8 {41017/2500000}.</s> <s xml:id="echoid-s7485" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Fractionem <lb/>magnam ad <lb/>minorem ferè <lb/>æquiualentem <lb/>reducere.</note> <p> <s xml:id="echoid-s7486" xml:space="preserve"><emph style="sc">Et</emph> ſi moleſtum videatur operari cum fractione tam magna, reduces eam ad <lb/>minorem quaſi æquiualentem hoc modo. </s> <s xml:id="echoid-s7487" xml:space="preserve">Elige pro Numeratore quemuis nu-<lb/>merum, vt 10. </s> <s xml:id="echoid-s7488" xml:space="preserve">Et fiat vt Numerator 164068. </s> <s xml:id="echoid-s7489" xml:space="preserve">ad ſuum Denominatorẽ 10000000. <lb/></s> <s xml:id="echoid-s7490" xml:space="preserve">ita Numerator electus 10. </s> <s xml:id="echoid-s7491" xml:space="preserve">ad aliud, reperieſque Denominatorem 609 {82588/164068}. </s> <s xml:id="echoid-s7492" xml:space="preserve"><lb/>Ita vt relicta hac fractione, Denominator 609. </s> <s xml:id="echoid-s7493" xml:space="preserve">ſit minor quam verus; </s> <s xml:id="echoid-s7494" xml:space="preserve">& </s> <s xml:id="echoid-s7495" xml:space="preserve">610. </s> <s xml:id="echoid-s7496" xml:space="preserve"><lb/>maior, hoc eſt, fractio {10/609}. </s> <s xml:id="echoid-s7497" xml:space="preserve">ſit maior fractione {164068/10000000}. </s> <s xml:id="echoid-s7498" xml:space="preserve">fractio autem <lb/> <anchor type="note" xlink:label="note-208-03a" xlink:href="note-208-03"/> {10/610}. </s> <s xml:id="echoid-s7499" xml:space="preserve">minor Inter has autẽ duas fractiones {10/609}. </s> <s xml:id="echoid-s7500" xml:space="preserve">{10/610}. </s> <s xml:id="echoid-s7501" xml:space="preserve">produces mediã {20/1219}. </s> <s xml:id="echoid-s7502" xml:space="preserve">cuius <lb/>Numerator ex Numeratoribus, & </s> <s xml:id="echoid-s7503" xml:space="preserve">Denominator ex Denominatoribus confla-<lb/>tus eſt. </s> <s xml:id="echoid-s7504" xml:space="preserve">Erit que fractio inuenta ferè maiori illi æqualis.</s> <s xml:id="echoid-s7505" xml:space="preserve"/> </p> <div xml:id="echoid-div475" type="float" level="2" n="4"> <note position="left" xlink:label="note-208-03" xlink:href="note-208-03a" xml:space="preserve">Inter du{as} <lb/>fractiones in-<lb/>uenire mediã.</note> </div> <p> <s xml:id="echoid-s7506" xml:space="preserve"><emph style="sc">Fractionem</emph> porrò, cuius Numerator ex duobus Numeratoribus, & </s> <s xml:id="echoid-s7507" xml:space="preserve">De-<lb/>nominator ex Denominatoribus duarum minutiarum componitur, eſſe maio-<lb/>rem minore, & </s> <s xml:id="echoid-s7508" xml:space="preserve">minorem maiore, demonſtrabimus lib. </s> <s xml:id="echoid-s7509" xml:space="preserve">8. </s> <s xml:id="echoid-s7510" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7511" xml:space="preserve">10.</s> <s xml:id="echoid-s7512" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Ex cognito <lb/>latere figuræ <lb/>regularis, in-<lb/>uenire ſemi-<lb/>diam{et}rum <lb/>circuli cir-<lb/>cumſcripti.</note> <p> <s xml:id="echoid-s7513" xml:space="preserve"><emph style="sc">Vicissim</emph> ex dato latere cuiuslibet figuræ regularis cognoſcemus ſemi-<lb/>diametrum circuli circumſcribentis: </s> <s xml:id="echoid-s7514" xml:space="preserve">ſi fiat, vt latus propoſitæ figuræ in tabula <lb/>antecedente ad 10000000. </s> <s xml:id="echoid-s7515" xml:space="preserve">ita latus datum ad aliud. </s> <s xml:id="echoid-s7516" xml:space="preserve">Vt ſi latus Pentagoni de-<lb/>tur 12. </s> <s xml:id="echoid-s7517" xml:space="preserve">& </s> <s xml:id="echoid-s7518" xml:space="preserve">fiat, vt 11755705. </s> <s xml:id="echoid-s7519" xml:space="preserve">ad 10000000. </s> <s xml:id="echoid-s7520" xml:space="preserve">ita 12. </s> <s xml:id="echoid-s7521" xml:space="preserve">ad aliud, reperietur ſemidiame-<lb/>ter circuli circumſcripti 10 {2442950/11755705}. </s> <s xml:id="echoid-s7522" xml:space="preserve">Et ſi fiat, vt Numerator huius fractionis ad <lb/>ſuum Denominatorem: </s> <s xml:id="echoid-s7523" xml:space="preserve">Ita Numerator electus quicunque, nimirum 1. </s> <s xml:id="echoid-s7524" xml:space="preserve">ad a-<lb/>liud, inuenietur Denominator ſequens 4 {1983905/2442950}. </s> <s xml:id="echoid-s7525" xml:space="preserve">ita vt fractio {1/4}. </s> <s xml:id="echoid-s7526" xml:space="preserve">ſit maior, <lb/>quã {2442950/11755705}. </s> <s xml:id="echoid-s7527" xml:space="preserve">at {1/5}. </s> <s xml:id="echoid-s7528" xml:space="preserve">minor Ex additione numeratorũ 1. </s> <s xml:id="echoid-s7529" xml:space="preserve">1. </s> <s xml:id="echoid-s7530" xml:space="preserve">inter ſe, & </s> <s xml:id="echoid-s7531" xml:space="preserve">Denomina-<lb/>rum 4. </s> <s xml:id="echoid-s7532" xml:space="preserve">5. </s> <s xml:id="echoid-s7533" xml:space="preserve">inter ſe, efficies fractionem {2/9}. </s> <s xml:id="echoid-s7534" xml:space="preserve">mediam, quæ adhuc maior eſt, quam <lb/>{2442950/11755705}. </s> <s xml:id="echoid-s7535" xml:space="preserve">Media autem inter {1/5}. </s> <s xml:id="echoid-s7536" xml:space="preserve">& </s> <s xml:id="echoid-s7537" xml:space="preserve">{2/9}. </s> <s xml:id="echoid-s7538" xml:space="preserve">eſt {3/14}. </s> <s xml:id="echoid-s7539" xml:space="preserve">quæ parum ab illa differt: </s> <s xml:id="echoid-s7540" xml:space="preserve">ita vt <lb/>ſemidiameter quæſita dici poſsit eſſe 10 {3/14}. </s> <s xml:id="echoid-s7541" xml:space="preserve">Atque in hunc modum per mino-<lb/>res numeros operationes fieri poſſunt, quamuis non omnino ex quiſitè, quod <lb/>fractiones aſſumptę non ſint omnino veræ; </s> <s xml:id="echoid-s7542" xml:space="preserve">ſed hic error in dimenſionibus cam-<lb/>porum tolerabilis eſt.</s> <s xml:id="echoid-s7543" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7544" xml:space="preserve">5. </s> <s xml:id="echoid-s7545" xml:space="preserve"><emph style="sc">Anteqvam</emph> rectilinearum figurarum dimenſionem concludam, lubet <lb/>regulam attexere, qua ex cognita area cuiuſcunque figuræ latus habentis no- <pb o="179" file="209" n="209" rhead="LIBER QVARTVS."/> tum venire poſsimus in cognitionem alterius figurę ſimilis illi, ſimiliterque po-<lb/>ſitę latus homologum etiam notum habentis: </s> <s xml:id="echoid-s7546" xml:space="preserve">quæ ſic ſe habet.</s> <s xml:id="echoid-s7547" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s7548" xml:space="preserve">Quadrat{us} numer{us} denominatoris proportionis, quam lat{us} figuræ ignotæ ad lat{us} <lb/> <anchor type="note" xlink:label="note-209-01a" xlink:href="note-209-01"/> figuræ cognitæ hab{et}, (qui denominator habebitur, ſi lat{us} figuræ ignotæ per lat{us} figuræ <lb/>cognitæ diuidatur) ſi ducatur in aream cognitam, produc{et}ur area alteri{us} figuræ quæſi-<lb/>tæ. </s> <s xml:id="echoid-s7549" xml:space="preserve">Debent autem figuræ eſſe ſimil{es}, ſimiliter que poſitæ, & </s> <s xml:id="echoid-s7550" xml:space="preserve">earum latera homologa ſumi <lb/>vt dictum est.</s> <s xml:id="echoid-s7551" xml:space="preserve"/> </p> <div xml:id="echoid-div476" type="float" level="2" n="5"> <note position="right" xlink:label="note-209-01" xlink:href="note-209-01a" xml:space="preserve">Quaratione <lb/>ex area cu-<lb/>iusl@bet figuræ <lb/>eruatur areæ <lb/>alteriu figuræ <lb/>ſimilis.</note> </div> <p> <s xml:id="echoid-s7552" xml:space="preserve">Nam denominator proportionis lateris figurę quæſitę ad latus figuræ datę <lb/>in ſe multiplicatus gignit denominatorem proportionis duplicatę eorum late-<lb/> <anchor type="note" xlink:label="note-209-02a" xlink:href="note-209-02"/> rum, vt ad defin. </s> <s xml:id="echoid-s7553" xml:space="preserve">10. </s> <s xml:id="echoid-s7554" xml:space="preserve">lib. </s> <s xml:id="echoid-s7555" xml:space="preserve">5. </s> <s xml:id="echoid-s7556" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7557" xml:space="preserve">ſcripſimus. </s> <s xml:id="echoid-s7558" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum ergo figurę ſimiles ſimili- terque poſitę habeant etiam proportionem duplicatam laterum homologo-<lb/>rum; </s> <s xml:id="echoid-s7559" xml:space="preserve">fit vt denominator proportionis duplicatę laterum prædictorum multi-<lb/>plicans aream cognitam producat aream quęſitam, hoc eſt, numerum, qui ad <lb/>aream cognitam proportionem habeat duplicatam proportionis datorum la-<lb/>terum, denominatam ſcilicet à denominatore, qui ex denominatore proportio-<lb/>nis eorum laterum in ſe multip licato producitur. </s> <s xml:id="echoid-s7560" xml:space="preserve">Verbi gratia, Trianguli ABC, <lb/>cuius latus A B, 10. </s> <s xml:id="echoid-s7561" xml:space="preserve">AC, 17. </s> <s xml:id="echoid-s7562" xml:space="preserve">& </s> <s xml:id="echoid-s7563" xml:space="preserve">B C, 21. </s> <s xml:id="echoid-s7564" xml:space="preserve">area eſt 84. </s> <s xml:id="echoid-s7565" xml:space="preserve">Si ergo ſit aliud triangulum <lb/>huic ſimile habens latus ipſi A B, homologum 70. </s> <s xml:id="echoid-s7566" xml:space="preserve">ipſi verò A C, homologum <lb/> <anchor type="figure" xlink:label="fig-209-01a" xlink:href="fig-209-01"/> 119. </s> <s xml:id="echoid-s7567" xml:space="preserve">& </s> <s xml:id="echoid-s7568" xml:space="preserve">ipſi B C, homologum 147. </s> <s xml:id="echoid-s7569" xml:space="preserve">diuidaturque latus 70. </s> <s xml:id="echoid-s7570" xml:space="preserve">per 10. </s> <s xml:id="echoid-s7571" xml:space="preserve">vt denomi-<lb/>nator 7. </s> <s xml:id="echoid-s7572" xml:space="preserve">proportionis lateris 70. </s> <s xml:id="echoid-s7573" xml:space="preserve">ad latus 10. </s> <s xml:id="echoid-s7574" xml:space="preserve">procreetur, & </s> <s xml:id="echoid-s7575" xml:space="preserve">quadratus nume-<lb/>rus huius denominatoris; </s> <s xml:id="echoid-s7576" xml:space="preserve">nimirum 49. </s> <s xml:id="echoid-s7577" xml:space="preserve">ducatur in 84. </s> <s xml:id="echoid-s7578" xml:space="preserve">areã trianguli A B C, pro-<lb/>ducetur area 4116. </s> <s xml:id="echoid-s7579" xml:space="preserve">poſterioris trianguli. </s> <s xml:id="echoid-s7580" xml:space="preserve">Rurſus quia area trianguli æquila-<lb/>teri, cuius ſingula latera ſint, 1. </s> <s xml:id="echoid-s7581" xml:space="preserve">area eſt {13/30}. </s> <s xml:id="echoid-s7582" xml:space="preserve">fermè, vt ſupra patuit, ſi de-<lb/>tur aliud triangulumæquilaterum, cuius ſingula latera ſint 70. </s> <s xml:id="echoid-s7583" xml:space="preserve">inueniemus eius <lb/>aream hoc modo. </s> <s xml:id="echoid-s7584" xml:space="preserve">Denominator proportionis laterum eſt ipſummet latus 70. <lb/></s> <s xml:id="echoid-s7585" xml:space="preserve">quod 70. </s> <s xml:id="echoid-s7586" xml:space="preserve">diuiſa per 1. </s> <s xml:id="echoid-s7587" xml:space="preserve">faciant 70. </s> <s xml:id="echoid-s7588" xml:space="preserve">Ducemus ergo 4900. </s> <s xml:id="echoid-s7589" xml:space="preserve">quadratum lateris 70. </s> <s xml:id="echoid-s7590" xml:space="preserve"><lb/>in {13/30}. </s> <s xml:id="echoid-s7591" xml:space="preserve">aream cognitam. </s> <s xml:id="echoid-s7592" xml:space="preserve">Productus enim numerus 2123 {1/3}. </s> <s xml:id="echoid-s7593" xml:space="preserve">erit area poſt erioris <lb/>trianguli.</s> <s xml:id="echoid-s7594" xml:space="preserve"/> </p> <div xml:id="echoid-div477" type="float" level="2" n="6"> <note symbol="a" position="right" xlink:label="note-209-02" xlink:href="note-209-02a" xml:space="preserve">18. vel 20. <lb/>ſexti.</note> <figure xlink:label="fig-209-01" xlink:href="fig-209-01a"> <image file="209-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/209-01"/> </figure> </div> <p> <s xml:id="echoid-s7595" xml:space="preserve">6. </s> <s xml:id="echoid-s7596" xml:space="preserve"><emph style="sc">Hæc</emph> regula ita quoque proponi poterit. </s> <s xml:id="echoid-s7597" xml:space="preserve">Fiat vt quadrat{us} nu-<lb/>mer{us} lateris figuræ cognitæ ad quadratum numerum lateris figuræ quæſitæ, ita a <lb/> <anchor type="note" xlink:label="note-209-03a" xlink:href="note-209-03"/> rea figuræ cognitæ ad aliud. </s> <s xml:id="echoid-s7598" xml:space="preserve">Product{us} enim numer{us} erit area figuræ quæſitæ. <lb/></s> <s xml:id="echoid-s7599" xml:space="preserve">Propterea quod eadem eſt proportio quadrati lateris cognitæ figuræ ad qua-<lb/>dratum lateris figuræ quæſitę, quæ figurę notæ ad figuram quæſitam: </s> <s xml:id="echoid-s7600" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> quip- <anchor type="note" xlink:label="note-209-04a" xlink:href="note-209-04"/> pe cum vtraque proportio ſit duplicata proportionis laterum homologo-<lb/>rum. </s> <s xml:id="echoid-s7601" xml:space="preserve">Et quoniam quadratum lateris 1. </s> <s xml:id="echoid-s7602" xml:space="preserve">eſt 1. </s> <s xml:id="echoid-s7603" xml:space="preserve">fit, vt quotieſcunque latus figu-<lb/>ræ aream cognitam habentis fuerit 1. </s> <s xml:id="echoid-s7604" xml:space="preserve">ſatis ſit, quadratum numerum lateris fi-<lb/>guræ quæſitæ multiplicare in datam aream, vt quęſita area producatur: </s> <s xml:id="echoid-s7605" xml:space="preserve">Adeo <lb/>vt operæ pretium ſit areas inueſtigare plurimarum figurarum regularium, qua-<lb/>rum latera ſint 1. </s> <s xml:id="echoid-s7606" xml:space="preserve">Ex his enim ſine magno labore areæ aliarum figurarum <pb o="180" file="210" n="210" rhead="GEOMETR. PRACT."/> ſimilium elicientur, quarum latera cognita ſint. </s> <s xml:id="echoid-s7607" xml:space="preserve">Areas decem figurarum regu-<lb/> <anchor type="note" xlink:label="note-210-01a" xlink:href="note-210-01"/> larium, quarum latera ſunt 1. </s> <s xml:id="echoid-s7608" xml:space="preserve">hic ſubiecimus, vt per eas ſimilium figurarum, qua-<lb/>rum latera vnitatem ſuperant, inueſtigari poſsint ex regula Num. </s> <s xml:id="echoid-s7609" xml:space="preserve">5. </s> <s xml:id="echoid-s7610" xml:space="preserve">vel 6. </s> <s xml:id="echoid-s7611" xml:space="preserve">præ-<lb/>ſcripta; </s> <s xml:id="echoid-s7612" xml:space="preserve">quamuis in quadrato id neceſſarium non ſit, cum latus datum in ſe du-<lb/>ctum producat ſuum quadratum.</s> <s xml:id="echoid-s7613" xml:space="preserve"/> </p> <div xml:id="echoid-div478" type="float" level="2" n="7"> <note position="right" xlink:label="note-209-03" xlink:href="note-209-03a" xml:space="preserve">Regula ſupra-<lb/>dicta aliter <lb/>propoſita.</note> <note symbol="b" position="right" xlink:label="note-209-04" xlink:href="note-209-04a" xml:space="preserve">19. vel 20. <lb/>ſexti.</note> <note position="left" xlink:label="note-210-01" xlink:href="note-210-01a" xml:space="preserve">Facilit{as} præ <lb/>dictæ regulæ, <lb/>quando lat{us} <lb/>figuræ notæ eſt <lb/>vnit{as}.</note> </div> <note position="right" xml:space="preserve"> <lb/>Quarum latera ſunt 1. # Figuræ regulares. # Areæ prædicta- \\ rum figurarum ſunt \\ ferè hæ. <lb/># Trigonum # {1875000/4330127} Vel {13/30} <lb/># Tetragonum # 1 <lb/># Pentagonum # 1 {8469719/11775706} <lb/># Hexagonum # 2 {2990381/5000000} <lb/># Heptagonum # 3 {5507221/8677674} <lb/># Octogonum # 4 {1585127/1913417} <lb/># Enneagonum # 6 {1243755/6840402} <lb/># Decagonum # 7 {858089/1236068} <lb/># Vndecagonum # 9 {517050/1408663} <lb/># Duodecagonum # 11 {84614/431365} <lb/></note> <p> <s xml:id="echoid-s7614" xml:space="preserve"><emph style="sc">Possvnt</emph> autem hę fractiones ad minores ferè æquiualentes, ſi placet, re-<lb/> <anchor type="note" xlink:label="note-210-03a" xlink:href="note-210-03"/> uo cari, vt ſupra Num. </s> <s xml:id="echoid-s7615" xml:space="preserve">4. </s> <s xml:id="echoid-s7616" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s7617" xml:space="preserve"/> </p> <div xml:id="echoid-div479" type="float" level="2" n="8"> <note position="left" xlink:label="note-210-03" xlink:href="note-210-03a" xml:space="preserve">Qua ratione <lb/>beneficio late-<lb/>rum ſuperio-<lb/>ris tabulæ a-<lb/>reæ figurarum <lb/>regularium <lb/>inueniantur.</note> </div> <p> <s xml:id="echoid-s7618" xml:space="preserve">7. </s> <s xml:id="echoid-s7619" xml:space="preserve"><emph style="sc">Areæ</emph> porro hæ procreatę ſunt per regulam Num. </s> <s xml:id="echoid-s7620" xml:space="preserve">1. </s> <s xml:id="echoid-s7621" xml:space="preserve">præſcriptam, inuen-<lb/>tis prius perpendicularibus ex centris in latera cadentibus, licet in nonnullis <lb/>figuris anguli contineant ſecunda, ac tertia, pręter gradus, & </s> <s xml:id="echoid-s7622" xml:space="preserve">minuta; </s> <s xml:id="echoid-s7623" xml:space="preserve">hoc mo-<lb/>do. </s> <s xml:id="echoid-s7624" xml:space="preserve">Pro heptagono, verbi gratia, ex ſup eriori tabula ſumpta eſt ſemiſsis lateris <lb/>heptagoni 4338837. </s> <s xml:id="echoid-s7625" xml:space="preserve">(quando numerus lateris eſt impar, addenda eſt 1. </s> <s xml:id="echoid-s7626" xml:space="preserve">vt fiat <lb/>par, ac proinde ſemiſſem habeat, quandoquidem minus eſt vero latere, vt dixi-<lb/>mus. </s> <s xml:id="echoid-s7627" xml:space="preserve">Deinde, quia, vtin tractatione ſinuum dictum eſt, ſemiſsis hęc ſinus eſt an-<lb/>guli oppoſiti in centro, quęſitus eſt is angulus ex tabula ſinuum (adhibita parte <lb/>proportionali, quemadmodum ad finem Lemmatis 53. </s> <s xml:id="echoid-s7628" xml:space="preserve">noſtri Aſtrolabij docui-<lb/>mus, vtangulus reperiretur in gradibus, minutis, ac ſecundis.) </s> <s xml:id="echoid-s7629" xml:space="preserve">inuentuſque eſt <lb/>grad. </s> <s xml:id="echoid-s7630" xml:space="preserve">25. </s> <s xml:id="echoid-s7631" xml:space="preserve">min. </s> <s xml:id="echoid-s7632" xml:space="preserve">42. </s> <s xml:id="echoid-s7633" xml:space="preserve">ſec. </s> <s xml:id="echoid-s7634" xml:space="preserve">51.</s> <s xml:id="echoid-s7635" xml:space="preserve"/> </p> <pb o="181" file="211" n="211" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s7636" xml:space="preserve">POST hæc, <anchor type="note" xlink:href="" symbol="a"/> quia eſt, vt ſinus huius anguliinuenti, nimirum ſemiſsis ipſa <anchor type="note" xlink:label="note-211-01a" xlink:href="note-211-01"/> 4338837. </s> <s xml:id="echoid-s7637" xml:space="preserve">lateris ex ſuperiori tabula excerpti, ad {1/2}. </s> <s xml:id="echoid-s7638" xml:space="preserve">ſemiſſem lateris, id eſt, ita ſi-<lb/>nus complementi eiuſdem anguliinuenti, hoc eſt, ita ſinus grad. </s> <s xml:id="echoid-s7639" xml:space="preserve">64. </s> <s xml:id="echoid-s7640" xml:space="preserve">min. </s> <s xml:id="echoid-s7641" xml:space="preserve">17. </s> <s xml:id="echoid-s7642" xml:space="preserve">ſec. <lb/></s> <s xml:id="echoid-s7643" xml:space="preserve">9. </s> <s xml:id="echoid-s7644" xml:space="preserve">(adhibita quoque parte proportionali, propter ſecunda.) </s> <s xml:id="echoid-s7645" xml:space="preserve">nimirum 9011398. </s> <s xml:id="echoid-s7646" xml:space="preserve"><lb/>ad perpendicularem huic complemento inuenti anguli oppoſitam in triangulo <lb/>rectangulo: </s> <s xml:id="echoid-s7647" xml:space="preserve">Inuenta eſt hæc perpendicularis {9011398/8677678} quæ tandem ducta in {7/2}. </s> <s xml:id="echoid-s7648" xml:space="preserve"><lb/>ſemiſſem ambitus heptagoni produxit aream heptagoni 3 {11014442/17355348}. </s> <s xml:id="echoid-s7649" xml:space="preserve">velin mino-<lb/>ribus numeris 3 {5507221/3677674} At que hac eadem ratione aream cuiuſcunque figurære-<lb/>gularis latus habentis 1. </s> <s xml:id="echoid-s7650" xml:space="preserve">dummodo numerus laterum maior non ſit, quam 80. </s> <s xml:id="echoid-s7651" xml:space="preserve"><lb/>reperies: </s> <s xml:id="echoid-s7652" xml:space="preserve">Ex qua deinde aream ſimilis figuræ latus habentis maius, quam 1. </s> <s xml:id="echoid-s7653" xml:space="preserve">per <lb/>regulam Num. </s> <s xml:id="echoid-s7654" xml:space="preserve">5. </s> <s xml:id="echoid-s7655" xml:space="preserve">vel 6. </s> <s xml:id="echoid-s7656" xml:space="preserve">traditam elicies.</s> <s xml:id="echoid-s7657" xml:space="preserve"/> </p> <div xml:id="echoid-div480" type="float" level="2" n="9"> <note symbol="a" position="right" xlink:label="note-211-01" xlink:href="note-211-01a" xml:space="preserve">4. triang re-<lb/>ctil.</note> </div> <p> <s xml:id="echoid-s7658" xml:space="preserve"><emph style="sc">Eodem</emph> tamen artificio hoc Num. </s> <s xml:id="echoid-s7659" xml:space="preserve">7. </s> <s xml:id="echoid-s7660" xml:space="preserve">expoſito aream cuiuſuis figuræ regu-<lb/>laris, etiamſi latus habeat maius, quam 1. </s> <s xml:id="echoid-s7661" xml:space="preserve">(ſi placet) colligere licebit, quamuis <lb/>non reperiatur prius area figuræ ſimilis, cuius latus ſit 1. </s> <s xml:id="echoid-s7662" xml:space="preserve">ſi nimirum loco {1/2}. </s> <s xml:id="echoid-s7663" xml:space="preserve">ſemiſ-<lb/>ſis lateris 1. </s> <s xml:id="echoid-s7664" xml:space="preserve">accipiatur ſemiſsis lateris dati, quod maius ſit quam 1. </s> <s xml:id="echoid-s7665" xml:space="preserve">vt perſpicuum <lb/>eſt.</s> <s xml:id="echoid-s7666" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7667" xml:space="preserve">8. </s> <s xml:id="echoid-s7668" xml:space="preserve">Cognita area figuræ regularis, cuius numerus laterum maior non ſit, quã <lb/> <anchor type="note" xlink:label="note-211-02a" xlink:href="note-211-02"/> 12. </s> <s xml:id="echoid-s7669" xml:space="preserve">cognoſcetur eius latus hoc modo. </s> <s xml:id="echoid-s7670" xml:space="preserve">Fiat vt area figuræ ſimilis latus habentis <lb/>1. </s> <s xml:id="echoid-s7671" xml:space="preserve">ex præce dentitabula deſumpta ad aream figuræ propoſitæ: </s> <s xml:id="echoid-s7672" xml:space="preserve">Ita 1. </s> <s xml:id="echoid-s7673" xml:space="preserve">quadratum <lb/>lateris 1. </s> <s xml:id="echoid-s7674" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s7675" xml:space="preserve">Productus enim numerus erit quadratus lateris quæſiti. </s> <s xml:id="echoid-s7676" xml:space="preserve">Ra-<lb/>dix ergo eius qua drata dabit latus quæſitum. </s> <s xml:id="echoid-s7677" xml:space="preserve">Nam ita eſt area ad aream, vt qua-<lb/>dratum lateris ad quadratum lateris; </s> <s xml:id="echoid-s7678" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> quod vtraque proportio ſit proportio- <anchor type="note" xlink:label="note-211-03a" xlink:href="note-211-03"/> nis laterum duplicata.</s> <s xml:id="echoid-s7679" xml:space="preserve"/> </p> <div xml:id="echoid-div481" type="float" level="2" n="10"> <note position="right" xlink:label="note-211-02" xlink:href="note-211-02a" xml:space="preserve">Ex area co-<lb/>gnita quo pa-<lb/>cto lat{us} erua-<lb/>tur.</note> <note symbol="b" position="right" xlink:label="note-211-03" xlink:href="note-211-03a" xml:space="preserve">10. vel 20. <lb/>ſexti.</note> </div> <p> <s xml:id="echoid-s7680" xml:space="preserve"><emph style="sc">Qvod</emph> ſi area cognita ſit figuræ regularis plura latera habentis, quam 12. </s> <s xml:id="echoid-s7681" xml:space="preserve">nõ <lb/>plura tamen, quam 80. </s> <s xml:id="echoid-s7682" xml:space="preserve">inuenienda primum erit area figuræ ſimilis latus habentis <lb/>1. </s> <s xml:id="echoid-s7683" xml:space="preserve">beneficio ſuperioris tabulæ laterum, vt Num. </s> <s xml:id="echoid-s7684" xml:space="preserve">7. </s> <s xml:id="echoid-s7685" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s7686" xml:space="preserve">Deinde latus ex-<lb/>quirendum, vt hoc Num. </s> <s xml:id="echoid-s7687" xml:space="preserve">8. </s> <s xml:id="echoid-s7688" xml:space="preserve">declaratum eſt. </s> <s xml:id="echoid-s7689" xml:space="preserve">Et ſi tabula ſuperior laterum exten-<lb/>ſa eſſet ad plura latera, inueniretur eodem modo latus figuræ plurium laterum, <lb/>quam 80. </s> <s xml:id="echoid-s7690" xml:space="preserve">ex eius area. </s> <s xml:id="echoid-s7691" xml:space="preserve">Exempli gratia. </s> <s xml:id="echoid-s7692" xml:space="preserve">Sit area alicuius trianguliæ quilateri 15 {3/5}. <lb/></s> <s xml:id="echoid-s7693" xml:space="preserve">Et quia area æquilateritrianguli, cuiuslatus eſt 1. </s> <s xml:id="echoid-s7694" xml:space="preserve">inuenta eſt ſupra {13/30}. </s> <s xml:id="echoid-s7695" xml:space="preserve">ſi fiat vt <lb/>{13/30}. </s> <s xml:id="echoid-s7696" xml:space="preserve">ad aream propoſitam. </s> <s xml:id="echoid-s7697" xml:space="preserve">15 {3/5}. </s> <s xml:id="echoid-s7698" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s7699" xml:space="preserve">quadratum lateris 1. </s> <s xml:id="echoid-s7700" xml:space="preserve">ad aliud, reperietur qua-<lb/>dratum lateris quæſiti 36. </s> <s xml:id="echoid-s7701" xml:space="preserve">cuius radix quadrata 6. </s> <s xml:id="echoid-s7702" xml:space="preserve">dabit latus, quod quæritur.</s> <s xml:id="echoid-s7703" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div483" type="section" level="1" n="176"> <head xml:id="echoid-head182" xml:space="preserve">De dimenſione circuli ex Archimede.</head> <head xml:id="echoid-head183" xml:space="preserve"><emph style="sc">Capvt</emph> VI.</head> <p> <s xml:id="echoid-s7704" xml:space="preserve">1. </s> <s xml:id="echoid-s7705" xml:space="preserve">VT circulum quemlibet propoſitum, eiuſq; </s> <s xml:id="echoid-s7706" xml:space="preserve">partes metiri poſsimus, ne-<lb/>ceſſe eſt probè noſſe, quæ Archimedes de circuli dimenſione tradidit. <lb/></s> <s xml:id="echoid-s7707" xml:space="preserve">Non absre ergo erit, ſi eius libellum de circuli dimenſione acutiſsimũ <lb/>ſane, & </s> <s xml:id="echoid-s7708" xml:space="preserve">ſubtiliſsimum hic interſeram, tum quia breuiſsimus eſt, quippe quitri-<lb/>bus duntaxat propoſitionibus conſtet: </s> <s xml:id="echoid-s7709" xml:space="preserve">tum ne ſtudio ſus, vt rem tam vtilem, <lb/>atque apud omnes artifices peruulgatam intelligat, Archimedem ipſum adire <lb/>cogatur: </s> <s xml:id="echoid-s7710" xml:space="preserve">tum vero maximè, quod cum Archimedis ſcripta ob affectatam bre-<lb/>uitatem, ſint paulo obſcuriora, illis nos lucem aliquam allaturos ſperamus. </s> <s xml:id="echoid-s7711" xml:space="preserve">Nec <pb o="182" file="212" n="212" rhead="GEOMETR. PRACT."/> dubitamus etiam, quin res hæc ſtudioſo lectorigrata, ac iucunda ſit futura.</s> <s xml:id="echoid-s7712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div484" type="section" level="1" n="177"> <head xml:id="echoid-head184" xml:space="preserve">PROPOSITIO I.</head> <p> <s xml:id="echoid-s7713" xml:space="preserve">AREA cuiuslibet circuli æqualis eſt triangulo rectangulo, cuius vnum <lb/>quidem latus circa angulum rectum ſemidiametro circuli, alterũ ve-<lb/>rò peripheriæ eiuſdem circuli æquale eſt.</s> <s xml:id="echoid-s7714" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7715" xml:space="preserve"><emph style="sc">Sit</emph> circulus ABCD, cuius centrum E, ſemidiameter EA: </s> <s xml:id="echoid-s7716" xml:space="preserve">ſitque triangulum <lb/>rectangulum FGH, angulum habens rectum G, latus verò F G, ſemidiametro <lb/>circuli EA, & </s> <s xml:id="echoid-s7717" xml:space="preserve">latus GH, peripheriæ eiuſdem circuli æquale. </s> <s xml:id="echoid-s7718" xml:space="preserve">Dico circulum AB-<lb/>CD, triangulo FGH, æqualem eſſe. </s> <s xml:id="echoid-s7719" xml:space="preserve">Si enim dicatur non eſſe æqualis, ſit primũ, <lb/>ſi fieri poteſt, circulus maior quam triangulum, magnitu dinez: </s> <s xml:id="echoid-s7720" xml:space="preserve">adeo vt circulus <lb/>æqualis ſit triangulo, & </s> <s xml:id="echoid-s7721" xml:space="preserve">magnitudiniz. </s> <s xml:id="echoid-s7722" xml:space="preserve">ſimul; </s> <s xml:id="echoid-s7723" xml:space="preserve">propterea que maior, quam z. </s> <s xml:id="echoid-s7724" xml:space="preserve">Si <lb/>igitur ex circulo auferatur plus, quàm dimidium, & </s> <s xml:id="echoid-s7725" xml:space="preserve">à reſiduo plus etiam, quam <lb/>dimidium, & </s> <s xml:id="echoid-s7726" xml:space="preserve">ita deinceps: </s> <s xml:id="echoid-s7727" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> relinquetur tandem magnitudo minor, quam z.</s> <s xml:id="echoid-s7728" xml:space="preserve"> <anchor type="note" xlink:label="note-212-01a" xlink:href="note-212-01"/> </s> </p> <div xml:id="echoid-div484" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-212-01" xlink:href="note-212-01a" xml:space="preserve">1. decimi.</note> </div> <p style="it"> <s xml:id="echoid-s7729" xml:space="preserve">Hæc autem detractio continua fiet, ſi primo loco auferatur ex circulo quadr atum in-<lb/>ſcriptum A B C D. </s> <s xml:id="echoid-s7730" xml:space="preserve">Hoc enim cum dimidium ſit quadrati I K L M, circulo circumſcri-<lb/>pti, vt in ſchol. </s> <s xml:id="echoid-s7731" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7732" xml:space="preserve">9. </s> <s xml:id="echoid-s7733" xml:space="preserve">lib. </s> <s xml:id="echoid-s7734" xml:space="preserve">4. </s> <s xml:id="echoid-s7735" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7736" xml:space="preserve">oſtendim{us}: </s> <s xml:id="echoid-s7737" xml:space="preserve">circul{us} autem ipſi{us} quadrati I K L M, <lb/>pars ſit: </s> <s xml:id="echoid-s7738" xml:space="preserve">erit quadratum inſcriptum A B C D mai{us} quam dimidium circuli. </s> <s xml:id="echoid-s7739" xml:space="preserve">Deinde ſi <lb/>auferantur à<unsure/> reſiduis quatuor ſegmentis quatuor triangula Iſoſcelia AOB, BPC, CQD, <lb/>DNA, ductis rectis ad media puncta arcuum. </s> <s xml:id="echoid-s7740" xml:space="preserve">Hæc enim ſimul maiora ſunt, quam di-<lb/> <anchor type="figure" xlink:label="fig-212-01a" xlink:href="fig-212-01"/> midium quatuor ſegmentorum ſimul, cum vnum quod <lb/>mai{us} ſit, quam dimidium ſegmenti, in quo exiſtit. </s> <s xml:id="echoid-s7741" xml:space="preserve">Com-<lb/>pleto enim rectangulo A R, <anchor type="note" xlink:href="" symbol="b"/> erit ei{us} dimidium trianga- <anchor type="note" xlink:label="note-212-02a" xlink:href="note-212-02"/> lum A N D: </s> <s xml:id="echoid-s7742" xml:space="preserve">ac proinde idem triangulum mai{us} erit <lb/>quam dimidium ſegmenti A N D. </s> <s xml:id="echoid-s7743" xml:space="preserve">Eademque ratio est <lb/>de aliis. </s> <s xml:id="echoid-s7744" xml:space="preserve">Pari ratione, ſi à reſiduis octo ſegmentis aufe-<lb/>rantur octo alia triangula Iſoſcelia in illis conſtituta, &</s> <s xml:id="echoid-s7745" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s7746" xml:space="preserve">atque ita deinceps.</s> <s xml:id="echoid-s7747" xml:space="preserve"/> </p> <div xml:id="echoid-div485" type="float" level="2" n="2"> <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/xxxxxxxx/figures/212-01"/> </figure> <note symbol="b" position="left" xlink:label="note-212-02" xlink:href="note-212-02a" xml:space="preserve">41. primi.</note> </div> <figure> <image file="212-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/212-02"/> </figure> <p> <s xml:id="echoid-s7748" xml:space="preserve">Ponantur ergo iam octo ſegmenta A O, O B, B P, P C, C Q, Q D, D N, N A, <lb/>relicta eſſe minora magnitudine z. </s> <s xml:id="echoid-s7749" xml:space="preserve">& </s> <s xml:id="echoid-s7750" xml:space="preserve">quoniam circulus æqualis conceditur tri-<lb/>angulo F G H, & </s> <s xml:id="echoid-s7751" xml:space="preserve">magnitudiniz, ſimul: </s> <s xml:id="echoid-s7752" xml:space="preserve">ſi demantur inæqualia@, nimirum iſta ſe-<lb/>gmenta ex circulo, & </s> <s xml:id="echoid-s7753" xml:space="preserve">magnitudo z, ex aggregato trianguli cum z, reliqua erit <lb/>figura inſcripta, Octo gona videlicet, maior triangulo F G H, quod eſt abſu@ dũ, <lb/>quippe cum multo minor ſit. </s> <s xml:id="echoid-s7754" xml:space="preserve">Sinamque ex centro E, ad latus B O, ducatur <lb/>perpendicularis E T, & </s> <s xml:id="echoid-s7755" xml:space="preserve">in triangulo ſumatur G K, ipſi E T, & </s> <s xml:id="echoid-s7756" xml:space="preserve">recta G i, ambitui <pb o="183" file="213" n="213" rhead="LIBER QVARTVS."/> Octogoniæqualis, cadet punctumk, citra F, & </s> <s xml:id="echoid-s7757" xml:space="preserve">i, citra H, quod E T, minor ſit <lb/>ſemidiametro circuli, & </s> <s xml:id="echoid-s7758" xml:space="preserve">ambitus Octogoni minor peripheria eiuſdem circuli. </s> <s xml:id="echoid-s7759" xml:space="preserve">I-<lb/>gitur ducta recta ki, erit triangulum G k i, minus triangulo F G H, pars toto. </s> <s xml:id="echoid-s7760" xml:space="preserve">Eſt <lb/>autem triangulum k Gi, Octogono æquale: </s> <s xml:id="echoid-s7761" xml:space="preserve">quippe cum ex ſcholio propoſ. </s> <s xml:id="echoid-s7762" xml:space="preserve">41. <lb/></s> <s xml:id="echoid-s7763" xml:space="preserve">lib. </s> <s xml:id="echoid-s7764" xml:space="preserve">1, Euclid. </s> <s xml:id="echoid-s7765" xml:space="preserve">æquale ſit rectangulo ſub G k, & </s> <s xml:id="echoid-s7766" xml:space="preserve">ſemiſſe ipſius Gi, comprehenſo, <lb/>quod per propoſitionem 2. </s> <s xml:id="echoid-s7767" xml:space="preserve">lib. </s> <s xml:id="echoid-s7768" xml:space="preserve">7. </s> <s xml:id="echoid-s7769" xml:space="preserve">de Iſoperimetris Octogono æquale eſt. </s> <s xml:id="echoid-s7770" xml:space="preserve">O-<lb/>ctogonum ergo minus eſt triangulo F G H. </s> <s xml:id="echoid-s7771" xml:space="preserve">Non ergo maius eſt: </s> <s xml:id="echoid-s7772" xml:space="preserve">ac proinde cir-<lb/>culus triangulo maius eſſe nequit.</s> <s xml:id="echoid-s7773" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7774" xml:space="preserve"><emph style="sc">Sit</emph> deinde, ſi fieri poteſt, circulus ABCD, minor quam triangulum FGH, <lb/>magnitudinez. </s> <s xml:id="echoid-s7775" xml:space="preserve">Circumſcribatur circulo quadratum IKL M, cuius latera cir-<lb/>culum tangantin punctis A, B, C, D. </s> <s xml:id="echoid-s7776" xml:space="preserve">quod maius erit triangulo FGH. </s> <s xml:id="echoid-s7777" xml:space="preserve">Cum <lb/>enim eius ambitus (vt lib. </s> <s xml:id="echoid-s7778" xml:space="preserve">8. </s> <s xml:id="echoid-s7779" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7780" xml:space="preserve">1. </s> <s xml:id="echoid-s7781" xml:space="preserve">probabimus) maior ſit peripheria circuli, <lb/>hoc eſt, recta G H, & </s> <s xml:id="echoid-s7782" xml:space="preserve">perpendicularis E A, ipſi F G, æqualis, erit triangulum re-<lb/>ctangulum latus vnum habens æqualeipſi F G, & </s> <s xml:id="echoid-s7783" xml:space="preserve">alterum maius latere GH, (æ-<lb/>quale nimirum ambitui quadrati I K L M.) </s> <s xml:id="echoid-s7784" xml:space="preserve">maius triangulo FGH. </s> <s xml:id="echoid-s7785" xml:space="preserve">Cum ergo <lb/>triangulum illud, per ſcholium propoſ. </s> <s xml:id="echoid-s7786" xml:space="preserve">45. </s> <s xml:id="echoid-s7787" xml:space="preserve">lib. </s> <s xml:id="echoid-s7788" xml:space="preserve">1. </s> <s xml:id="echoid-s7789" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7790" xml:space="preserve">ſit æquale rectangulo <lb/>ſub FG, & </s> <s xml:id="echoid-s7791" xml:space="preserve">ſemiſſe ambitus quadrati IKLM, comprehenſo: </s> <s xml:id="echoid-s7792" xml:space="preserve">hoc autem rectan-<lb/>gulum per propoſ. </s> <s xml:id="echoid-s7793" xml:space="preserve">2. </s> <s xml:id="echoid-s7794" xml:space="preserve">lib. </s> <s xml:id="echoid-s7795" xml:space="preserve">7. </s> <s xml:id="echoid-s7796" xml:space="preserve">de Iſoperimetris, qua drato IKLM, æquale; </s> <s xml:id="echoid-s7797" xml:space="preserve">erit quo-<lb/>que quadratum IKLM, maius triangulo F G H. </s> <s xml:id="echoid-s7798" xml:space="preserve">Et quia triangulum F G H, po-<lb/>nitur æquale circulo, & </s> <s xml:id="echoid-s7799" xml:space="preserve">magnitudini z. </s> <s xml:id="echoid-s7800" xml:space="preserve">ſimul, ac proinde maius quã z, erit quo-<lb/>que quadratum IKLM, (quod maius eſſe oſtendimus triangulo FGH,) maius, <lb/>quam z. </s> <s xml:id="echoid-s7801" xml:space="preserve">Siigitur ex quadrato IKLM, auferatur plus, quam dimidium, & </s> <s xml:id="echoid-s7802" xml:space="preserve">à reſi-<lb/>dio plus etiam quam dimidium, at queita deinceps, <anchor type="note" xlink:href="" symbol="a"/> relin quetur tandem ma- <anchor type="note" xlink:label="note-213-01a" xlink:href="note-213-01"/> gnitudo minor, quam z.</s> <s xml:id="echoid-s7803" xml:space="preserve"/> </p> <div xml:id="echoid-div486" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-213-01" xlink:href="note-213-01a" xml:space="preserve">1. decimi.</note> </div> <p style="it"> <s xml:id="echoid-s7804" xml:space="preserve">Hæc autem detractio continua fiet, ſi primo loco auferatur circul{us} A B C D: </s> <s xml:id="echoid-s7805" xml:space="preserve">Hic <lb/> <anchor type="figure" xlink:label="fig-213-01a" xlink:href="fig-213-01"/> enim maior eſt ſemiſſe quadrati I K L M, propterea quod <lb/>quadratum inſcriptum (quod min{us} eſt circulo, pars toto) <lb/>ſemiſſis eſt quadrati circumſcripti, exſcholio propoſ. </s> <s xml:id="echoid-s7806" xml:space="preserve">9. </s> <s xml:id="echoid-s7807" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s7808" xml:space="preserve">4. </s> <s xml:id="echoid-s7809" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7810" xml:space="preserve">Quod ſi ducta recta E K, ſecante circulum in <lb/>O, ducatur per O, ad E K, perpendicularis V X,<anchor type="note" xlink:href="" symbol="b"/> quæ cir- <anchor type="note" xlink:label="note-213-02a" xlink:href="note-213-02"/> culum tanget in O: </s> <s xml:id="echoid-s7811" xml:space="preserve">idemque fiat, ductis rectis EL, EM, <lb/>EI, &</s> <s xml:id="echoid-s7812" xml:space="preserve">c. </s> <s xml:id="echoid-s7813" xml:space="preserve">deſcriptum erit Octogonum a quilaterum, & </s> <s xml:id="echoid-s7814" xml:space="preserve">æ-<lb/> <anchor type="figure" xlink:label="fig-213-02a" xlink:href="fig-213-02"/> quiangulũ VXY a b c d e V, vt conſtat ex conſtructione, demonſtratione propoſ. </s> <s xml:id="echoid-s7815" xml:space="preserve">12. </s> <s xml:id="echoid-s7816" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s7817" xml:space="preserve">4. </s> <s xml:id="echoid-s7818" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7819" xml:space="preserve">quippe cum ad E A, E O, & </s> <s xml:id="echoid-s7820" xml:space="preserve">adreliqu{as} ſemidiametros Octogoni inſcripti ductæ <lb/>ſint perpendiculares ve, V X, &</s> <s xml:id="echoid-s7821" xml:space="preserve">c. </s> <s xml:id="echoid-s7822" xml:space="preserve">Quoniã vero v A, v O, per 2. </s> <s xml:id="echoid-s7823" xml:space="preserve">coroll propoſ. </s> <s xml:id="echoid-s7824" xml:space="preserve">36. </s> <s xml:id="echoid-s7825" xml:space="preserve">lib. </s> <s xml:id="echoid-s7826" xml:space="preserve">3. </s> <s xml:id="echoid-s7827" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7828" xml:space="preserve"><lb/>æqual{es} ſunt; </s> <s xml:id="echoid-s7829" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> & </s> <s xml:id="echoid-s7830" xml:space="preserve">eſt K V, maior quam v O: </s> <s xml:id="echoid-s7831" xml:space="preserve">erit quoque K V, maior quam v A, ideoque <anchor type="note" xlink:label="note-213-03a" xlink:href="note-213-03"/> & </s> <s xml:id="echoid-s7832" xml:space="preserve">triangulum K v O, triangulo v A O, mai{us} erit; </s> <s xml:id="echoid-s7833" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> cum ſit triangulum ad triangulum, <anchor type="note" xlink:label="note-213-04a" xlink:href="note-213-04"/> vt baſis ad baſem. </s> <s xml:id="echoid-s7834" xml:space="preserve">Igitur triangulum K V O, mai{us} erit, quam dimidium trianguli <pb o="184" file="214" n="214" rhead="GEOMETR. PRACT."/> KAO; </s> <s xml:id="echoid-s7835" xml:space="preserve">ac proinde multo mai{us}, quam dimidium trianguli mixti KAO, cui{us} vnum la-<lb/>t{us} eſt arc{us} A O. </s> <s xml:id="echoid-s7836" xml:space="preserve">Eadem ratione erit K O X, mai{us}, quam dimidium trianguli mixti <lb/>KOB, cui{us} vnum lat{us} eſt arc{us} O B. </s> <s xml:id="echoid-s7837" xml:space="preserve">Auferendo ergo triangulum KVX, ex figura mi-<lb/>ſtilinea K A B, in qua vnum lat{us} eſt arc{us} A O B, ablatum erit pl{us} quam dimidium. <lb/></s> <s xml:id="echoid-s7838" xml:space="preserve">Subtractis igitur quatuor eiuſmodi triangulis K V X, L Y a, M b c, I d e, ablatum erit <lb/>plus, quam dimidium ex quatuor reſiduis extra circulum, & </s> <s xml:id="echoid-s7839" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s7840" xml:space="preserve">Ponantur <lb/>igituriam octo triãgula mixta reſidua, quorũ baſes ſunt arcus AO, OB, BP, &</s> <s xml:id="echoid-s7841" xml:space="preserve">c. </s> <s xml:id="echoid-s7842" xml:space="preserve"><lb/>minora magnitudine z. </s> <s xml:id="echoid-s7843" xml:space="preserve">Cum ergo circulus cum z, æqualis poſitus ſit triangulo <lb/>F G H, erit circulus cum illis octo reſiduis, hoc eſt, figura Octogona V X Y, <lb/>a b c d e V, minor eodem triangulo F G H. </s> <s xml:id="echoid-s7844" xml:space="preserve">quod eſt ab ſurdum, cum maius fit: </s> <s xml:id="echoid-s7845" xml:space="preserve"><lb/>quippe cum perpẽdicularis EO, æqualis ſit lateri F G, & </s> <s xml:id="echoid-s7846" xml:space="preserve">ambitus Octogini ma-<lb/>ior circumferentia circuli, hoc eſt, recta GH. </s> <s xml:id="echoid-s7847" xml:space="preserve">Hinc enim fit, triangulum rectan-<lb/>gulum, cuius latus F G, æquale perpendiculari EO, & </s> <s xml:id="echoid-s7848" xml:space="preserve">alterum latus æquale am-<lb/>bitui Octogoni, maius videlicet, quam GH, maius eſſe triangulo FGH. </s> <s xml:id="echoid-s7849" xml:space="preserve">Cum er-<lb/>go illud triangulum ſit, ex ſcholio propoſ. </s> <s xml:id="echoid-s7850" xml:space="preserve">41. </s> <s xml:id="echoid-s7851" xml:space="preserve">lib. </s> <s xml:id="echoid-s7852" xml:space="preserve">1. </s> <s xml:id="echoid-s7853" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7854" xml:space="preserve">æquale rectangulo ſub <lb/>FG, & </s> <s xml:id="echoid-s7855" xml:space="preserve">ſemiſſe ambitus Octogoni comprehenſo; </s> <s xml:id="echoid-s7856" xml:space="preserve">hoc autem rectangulum O-<lb/>ctogono æquale, ex propoſ. </s> <s xml:id="echoid-s7857" xml:space="preserve">2. </s> <s xml:id="echoid-s7858" xml:space="preserve">lib. </s> <s xml:id="echoid-s7859" xml:space="preserve">7. </s> <s xml:id="echoid-s7860" xml:space="preserve">de Iſoperimetris: </s> <s xml:id="echoid-s7861" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s7862" xml:space="preserve">Octogonum <lb/>maius triangulo FGH. </s> <s xml:id="echoid-s7863" xml:space="preserve">Nõ ergo minus eſſe poteſt, ac proinde circulus ABCD, <lb/>minor non eſt triangulo FGH: </s> <s xml:id="echoid-s7864" xml:space="preserve">Sed neque maior eſt, vt demonſtrauimus. </s> <s xml:id="echoid-s7865" xml:space="preserve">Igi-<lb/>tur æqualis eſt, quod erat demonſtrandum.</s> <s xml:id="echoid-s7866" xml:space="preserve"/> </p> <div xml:id="echoid-div487" type="float" level="2" n="4"> <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/xxxxxxxx/figures/213-01"/> </figure> <note symbol="b" position="right" xlink:label="note-213-02" xlink:href="note-213-02a" xml:space="preserve">16. tertij.</note> <figure xlink:label="fig-213-02" xlink:href="fig-213-02a"> <image file="213-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/213-02"/> </figure> <note symbol="c" position="right" xlink:label="note-213-03" xlink:href="note-213-03a" xml:space="preserve">19. primi.</note> <note symbol="d" position="right" xlink:label="note-213-04" xlink:href="note-213-04a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div489" type="section" level="1" n="178"> <head xml:id="echoid-head185" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s7867" xml:space="preserve"><emph style="sc">Iosephvs</emph> Scaliger, vel quia vim huius demonſtrationis non perpendit, <lb/>vel quia ſuæ circuli quadrandi rationi vidit eſſe contrariam, non eſt veritus Ar-<lb/>chimedem hoc loco falſitatis arguere: </s> <s xml:id="echoid-s7868" xml:space="preserve">conaturque oſtendere, non rectè ab eo <lb/>demonſtratum, circulum æqualem eſſe triangulo rectangulo, cuius vnum latus <lb/>ſemidiametro, & </s> <s xml:id="echoid-s7869" xml:space="preserve">alterum circumferentiæ circuli eſt æquale. </s> <s xml:id="echoid-s7870" xml:space="preserve">Nam, ait, ſi de-<lb/>monſtratio Archimedis bona eſt, demonſtrabitur eodem modo, circulũ æqua-<lb/>lem eſſe triangulo rectangulo, cuius vnum latus circa angulum rectum ſemidi-<lb/>ametro æquale eſt, & </s> <s xml:id="echoid-s7871" xml:space="preserve">alterum peripheria circuli maius. </s> <s xml:id="echoid-s7872" xml:space="preserve">Sit enim in triangulo <lb/>lmn, latus quidem lm, trianguli ſemidiametro circuli E A, æquale, at mn, peri-<lb/>pheria maius. </s> <s xml:id="echoid-s7873" xml:space="preserve">Concedit ergo Scaliger, circulum non eſſe maiorem triangulo <lb/>FGH, rectè eſſe ab Archimede demonſtratum, hoc eſt, triangulum F G H, cuius <lb/>latus GH, peripheriæ eſt æquale, non eſſe minus circulo, ac proinde neque tri-<lb/>angulum lmn, cuius latus m n, maius eſt peripheria, circulo minus eſſe. </s> <s xml:id="echoid-s7874" xml:space="preserve">Con-<lb/>cedititem, rectè probatum eſſe, circulũ non eſſe minorem triangulo FGH, ſi la-<lb/>tus GH, peripheriæ ſit ęquale, hoc eſt, triangulum FGH, non eſſe maius circu-<lb/>lo. </s> <s xml:id="echoid-s7875" xml:space="preserve">Sed negat, ex hoc ſequi, triangulum FGH, eſſe æquale circulo. </s> <s xml:id="echoid-s7876" xml:space="preserve">Cur? </s> <s xml:id="echoid-s7877" xml:space="preserve">quia, <lb/>inquit, eodem modo, ſi baſis m n, maior eſt peripheria, ſed minor circumſcripti <lb/>polygoni ambitu, (hoc enim contingere, ait, nihil prohibet) polygonum erit <lb/>quidem maius triangulo l m n, quod ambitus polygoni maior ſit recta m n, & </s> <s xml:id="echoid-s7878" xml:space="preserve"><lb/>ſemidiameter EA, rectæ l m, æqualis. </s> <s xml:id="echoid-s7879" xml:space="preserve">Sed reſectis portionibus, ſequeretur, idẽ <lb/>polygonum eſſe triangulo l m n, minus, quod eſt ineptum. </s> <s xml:id="echoid-s7880" xml:space="preserve">Ita ne verò mi Sca-<lb/>liger? </s> <s xml:id="echoid-s7881" xml:space="preserve">Non aduertis, te cum hypotheſi pugnare? </s> <s xml:id="echoid-s7882" xml:space="preserve">Nam poſito latere m n, ma-<lb/>iore, quam peripheria; </s> <s xml:id="echoid-s7883" xml:space="preserve">quando eo peruentum erit, polygonum eſſe minus tri-<lb/>angulo l m n, (ſi nimirumrelictæ portiones minores fuerint magnitudine z,) ſe- <pb o="185" file="215" n="215" rhead="LIBER QVARTVS."/> quitur neceſſariò, ambitum polygoni minorem eſſelatere m n. </s> <s xml:id="echoid-s7884" xml:space="preserve">Cum enim tri-<lb/>angulum rectangulum, cuius altitudo ſemidiametro polygoni, & </s> <s xml:id="echoid-s7885" xml:space="preserve">baſis ambi-<lb/>tui æqualis eſt, æquale ſit, ex ſcholio propoſ. </s> <s xml:id="echoid-s7886" xml:space="preserve">41. </s> <s xml:id="echoid-s7887" xml:space="preserve">lib. </s> <s xml:id="echoid-s7888" xml:space="preserve">1. </s> <s xml:id="echoid-s7889" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s7890" xml:space="preserve">rectangulo ſub ea-<lb/>dem ſemidiametro, & </s> <s xml:id="echoid-s7891" xml:space="preserve">ſemiſſe ambitus polygoni comprehenſa; </s> <s xml:id="echoid-s7892" xml:space="preserve">hoc autem, per <lb/>propoſ. </s> <s xml:id="echoid-s7893" xml:space="preserve">2. </s> <s xml:id="echoid-s7894" xml:space="preserve">lib. </s> <s xml:id="echoid-s7895" xml:space="preserve">7. </s> <s xml:id="echoid-s7896" xml:space="preserve">huius de Iſo perimetris, polygono æquale: </s> <s xml:id="echoid-s7897" xml:space="preserve">erit quoque trian-<lb/>gulum illud minus triangulo l m n. </s> <s xml:id="echoid-s7898" xml:space="preserve">Quare cum hæc triangula habeant æquales <lb/>altitudines; </s> <s xml:id="echoid-s7899" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit vtillud triangulum ad l m n, ita baſis illius ad baſem m n: </s> <s xml:id="echoid-s7900" xml:space="preserve">ac <anchor type="note" xlink:label="note-215-01a" xlink:href="note-215-01"/> proinde illa baſis, hoc eſt, ambitus polygoni, baſe m n, minor erit. </s> <s xml:id="echoid-s7901" xml:space="preserve">Non ergo <lb/>ponere potes baſem trianguli l m n, ſi maior eſt, quam peripheria circuli, mino-<lb/>rem ambitu polygoni: </s> <s xml:id="echoid-s7902" xml:space="preserve">In demonſtratione autem Archimedis conſtat, ambitũ <lb/>polygonimaiorem eſſe baſe trianguli F G H, ſi G H, æqualis eſt peripheriæ cir-<lb/>culi, cum maior ſit, quam perip heria: </s> <s xml:id="echoid-s7903" xml:space="preserve">ac propterea rectè concluſit Archimedes, <lb/>polygonum eſſe maius triangulo FGH, cum tamen ex hypotheſi aduerſarij o-<lb/>ſtenſum ſit eſſe minus. </s> <s xml:id="echoid-s7904" xml:space="preserve">Itaque potuiſſet Archimedes ita quo que propoſitum <lb/>colligere. </s> <s xml:id="echoid-s7905" xml:space="preserve">Polygonum minus eſt triangulo F G H, propter relictas ſectiones <lb/>minores magnitudine z. </s> <s xml:id="echoid-s7906" xml:space="preserve">Ergo eius ambitus minor eſt baſe G H, (quemadmo-<lb/>dum proximè demonſtrauimus.) </s> <s xml:id="echoid-s7907" xml:space="preserve">hoc eſt, peripheria circuli. </s> <s xml:id="echoid-s7908" xml:space="preserve">quod eſt abſurdũ, <lb/>cum ambitus polygoni maior ſit, quam peripheria. </s> <s xml:id="echoid-s7909" xml:space="preserve">Quod abſurdum, doctiſsi-<lb/>mè Scaliger, colligere non potes in tuo triangulo l m n, cum ſtatuas baſem mn, <lb/>perip heria circuli maiorem. </s> <s xml:id="echoid-s7910" xml:space="preserve">Et ſane miror te, Mathematicus cũ ſis, negare quã-<lb/>titatẽ aliquam illi eſſe æqualem, qua neque maior eſt, neque minor. </s> <s xml:id="echoid-s7911" xml:space="preserve">Si enim æ-<lb/>qualis non eſt, erit inæqualis. </s> <s xml:id="echoid-s7912" xml:space="preserve">Igitur vel maior vel minor, contra hypotheſim, <lb/>cum dicatur neque maior eſſe, neque minor. </s> <s xml:id="echoid-s7913" xml:space="preserve">An non vides, non ſolum Archi-<lb/>medem, ſed etiam Euclidem lib. </s> <s xml:id="echoid-s7914" xml:space="preserve">12. </s> <s xml:id="echoid-s7915" xml:space="preserve">hunc argumentandi modum frequentiſsimè <lb/>vſurpare?</s> <s xml:id="echoid-s7916" xml:space="preserve"/> </p> <div xml:id="echoid-div489" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-215-01" xlink:href="note-215-01a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div491" type="section" level="1" n="179"> <head xml:id="echoid-head186" xml:space="preserve">PROPOSITIO II.</head> <p> <s xml:id="echoid-s7917" xml:space="preserve">CVIVSLIBET circuli peripheria tripla eſt diametri, & </s> <s xml:id="echoid-s7918" xml:space="preserve">adhuc ſupe-<lb/>rat parte, quæ quidem minor eſt decem ſeptuageſimis, hoc eſt, ſepti-<lb/>ma parte diametri, maior verò decem ſeptuageſimis primis.</s> <s xml:id="echoid-s7919" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7920" xml:space="preserve"><emph style="sc">Hæc</emph> eſt Archimedis propoſitio 3. </s> <s xml:id="echoid-s7921" xml:space="preserve">quam nos ſecundam facimus, vt do ctri-<lb/>næ ordo ſeruetur, quando quidem ſequens propoſitio 3. </s> <s xml:id="echoid-s7922" xml:space="preserve">quamipſe 2. </s> <s xml:id="echoid-s7923" xml:space="preserve">facit, hãc <lb/>noſtram propoſitionem 2. </s> <s xml:id="echoid-s7924" xml:space="preserve">in demonſtrationem adhibet.</s> <s xml:id="echoid-s7925" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s7926" xml:space="preserve">Sit igitur circulus ABCD, cuius centrum E, diameter AB, quam ad rectos an-<lb/>gulos ſecet ſemidiameter E c, & </s> <s xml:id="echoid-s7927" xml:space="preserve">e c F, ad E c, perpendicularis ducatur, <anchor type="note" xlink:href="" symbol="b"/> quæ cir- <anchor type="note" xlink:label="note-215-02a" xlink:href="note-215-02"/> culum tangetin c. </s> <s xml:id="echoid-s7928" xml:space="preserve">Ducatur latus hexagoni AD, <anchor type="note" xlink:href="" symbol="c"/> quod ſemidiametro æquale e- <anchor type="note" xlink:label="note-215-03a" xlink:href="note-215-03"/> rit, & </s> <s xml:id="echoid-s7929" xml:space="preserve">arcus AD, grad. </s> <s xml:id="echoid-s7930" xml:space="preserve">60. </s> <s xml:id="echoid-s7931" xml:space="preserve">Ideoq; </s> <s xml:id="echoid-s7932" xml:space="preserve">D c. </s> <s xml:id="echoid-s7933" xml:space="preserve">grad. </s> <s xml:id="echoid-s7934" xml:space="preserve">30. </s> <s xml:id="echoid-s7935" xml:space="preserve">Ducta ergo recta E D e, erit angu-<lb/>lus e E c, tertia pars recti, cum rectus angulus contineat grad. </s> <s xml:id="echoid-s7936" xml:space="preserve">90. </s> <s xml:id="echoid-s7937" xml:space="preserve">Fiat quo que <lb/>angulus c E F, angulo c E e, æqualis: </s> <s xml:id="echoid-s7938" xml:space="preserve">eruntq; </s> <s xml:id="echoid-s7939" xml:space="preserve">angulie, F, inter ſe æquales, quod <lb/>vterque complementum ſit tertiæ partis angulirecti, ac proinde vter que duas <lb/>tertias partes vnius recti comprehendet. </s> <s xml:id="echoid-s7940" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Cum ergo omnes tres anguliin trian- <anchor type="note" xlink:label="note-215-04a" xlink:href="note-215-04"/> gulo e E F, contineant {6/3}. </s> <s xml:id="echoid-s7941" xml:space="preserve">vnius recti, continebit quo que e E F, {2/3}. </s> <s xml:id="echoid-s7942" xml:space="preserve">vnius recti <lb/>ipſumq; </s> <s xml:id="echoid-s7943" xml:space="preserve">triangulum æquiangulum erit, hoc eſt, per coroll. </s> <s xml:id="echoid-s7944" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s7945" xml:space="preserve">6. </s> <s xml:id="echoid-s7946" xml:space="preserve">lib. </s> <s xml:id="echoid-s7947" xml:space="preserve">1. </s> <s xml:id="echoid-s7948" xml:space="preserve">Euc. <lb/></s> <s xml:id="echoid-s7949" xml:space="preserve">æquilaterũ; </s> <s xml:id="echoid-s7950" xml:space="preserve">proptereaq; </s> <s xml:id="echoid-s7951" xml:space="preserve">perpendicularis E c, baſem e F, bifariã ſecabit, ex ſcho-<lb/>lio propoſ. </s> <s xml:id="echoid-s7952" xml:space="preserve">26. </s> <s xml:id="echoid-s7953" xml:space="preserve">lib. </s> <s xml:id="echoid-s7954" xml:space="preserve">1. </s> <s xml:id="echoid-s7955" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s7956" xml:space="preserve">atq; </s> <s xml:id="echoid-s7957" xml:space="preserve">ideo E@e, ipſius e c, dupla erit. </s> <s xml:id="echoid-s7958" xml:space="preserve">Poſita igitur c e, <pb o="186" file="216" n="216" rhead="GEOMETR. PRACT."/> 153. </s> <s xml:id="echoid-s7959" xml:space="preserve">erit E e, 306. </s> <s xml:id="echoid-s7960" xml:space="preserve">Et ſi quadratum ip ſius c e, 23409. </s> <s xml:id="echoid-s7961" xml:space="preserve">dematur ex 93636. </s> <s xml:id="echoid-s7962" xml:space="preserve">quadra-<lb/>to ipſius E e, <anchor type="note" xlink:href="" symbol="a"/> reliquum fiet quadratum ipſius E c, 70227. </s> <s xml:id="echoid-s7963" xml:space="preserve">cuius radix eſt pau- <anchor type="note" xlink:label="note-216-01a" xlink:href="note-216-01"/> lo maior, quam 265. </s> <s xml:id="echoid-s7964" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ac proinde E c, ad c e, maiorem habebit proportionem, <anchor type="note" xlink:label="note-216-02a" xlink:href="note-216-02"/> quam 265. </s> <s xml:id="echoid-s7965" xml:space="preserve">ad 153.</s> <s xml:id="echoid-s7966" xml:space="preserve"/> </p> <div xml:id="echoid-div491" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-215-02" xlink:href="note-215-02a" xml:space="preserve">16. tertij.</note> <note symbol="c" position="right" xlink:label="note-215-03" xlink:href="note-215-03a" xml:space="preserve">15. quar.</note> <note symbol="d" position="right" xlink:label="note-215-04" xlink:href="note-215-04a" xml:space="preserve">32. primi.</note> <note symbol="a" position="left" xlink:label="note-216-01" xlink:href="note-216-01a" xml:space="preserve">47. primi.</note> <note symbol="b" position="left" xlink:label="note-216-02" xlink:href="note-216-02a" xml:space="preserve">8. quinti.</note> </div> <figure> <image file="216-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/216-01"/> </figure> <p> <s xml:id="echoid-s7967" xml:space="preserve"><emph style="sc">Secto</emph> iam angulo e E c, bifariam per rectã E d; </s> <s xml:id="echoid-s7968" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erite E, ad E c, vt ed, ad <anchor type="note" xlink:label="note-216-03a" xlink:href="note-216-03"/> d c. </s> <s xml:id="echoid-s7969" xml:space="preserve">Et componendo e E, Ec ſimul ad E c, vt e c, ad d c. </s> <s xml:id="echoid-s7970" xml:space="preserve">Et permutando e E, E c, <lb/>fimul ad e c, vt E c, ad c d. </s> <s xml:id="echoid-s7971" xml:space="preserve">Quia verò e E, E c, ſimul maiores ſunt, quam 571. <lb/></s> <s xml:id="echoid-s7972" xml:space="preserve">(quippe cum E e, ſit 306. </s> <s xml:id="echoid-s7973" xml:space="preserve">& </s> <s xml:id="echoid-s7974" xml:space="preserve">E c, paulo maior, quam 265.) </s> <s xml:id="echoid-s7975" xml:space="preserve">& </s> <s xml:id="echoid-s7976" xml:space="preserve">e c, poſita eſt <lb/>153. </s> <s xml:id="echoid-s7977" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> habebunt e E, E c, ſimul ad e c, maiorem proportionem, quam 571. </s> <s xml:id="echoid-s7978" xml:space="preserve">ad <anchor type="note" xlink:label="note-216-04a" xlink:href="note-216-04"/> 153. </s> <s xml:id="echoid-s7979" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s7980" xml:space="preserve">proportio E c, ad c d, maior erit, quam 571. </s> <s xml:id="echoid-s7981" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s7982" xml:space="preserve">ac proin-<lb/>de ſi c d, ponatur 153. </s> <s xml:id="echoid-s7983" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit E c, paulo maior quam 571. </s> <s xml:id="echoid-s7984" xml:space="preserve">Igitur quadrantum ip ſi- <anchor type="note" xlink:label="note-216-05a" xlink:href="note-216-05"/> us E c, paulo maius erit, quam 326041. </s> <s xml:id="echoid-s7985" xml:space="preserve">atque idcirco, cum quadratum ipſius <lb/>c d, ſit 23409. </s> <s xml:id="echoid-s7986" xml:space="preserve">erit qua dratum ipſius E d, <anchor type="note" xlink:href="" symbol="f"/> quod quadratis rectarum E c, c d, eſt <anchor type="note" xlink:label="note-216-06a" xlink:href="note-216-06"/> æquale, paulo maius, quam 349450. </s> <s xml:id="echoid-s7987" xml:space="preserve">eiuſqueradix maior, quam 591 {1/8}. </s> <s xml:id="echoid-s7988" xml:space="preserve">quippe <lb/>cum huius radicis quadratum ſit tantum 349428 {49/64}. </s> <s xml:id="echoid-s7989" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Habebit igitur E d, ad <anchor type="note" xlink:label="note-216-07a" xlink:href="note-216-07"/> d c, maiorem proportionem quam 591 {1/8}. </s> <s xml:id="echoid-s7990" xml:space="preserve">ad 153.</s> <s xml:id="echoid-s7991" xml:space="preserve"/> </p> <div xml:id="echoid-div492" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-216-03" xlink:href="note-216-03a" xml:space="preserve">3. ſexti.</note> <note symbol="d" position="left" xlink:label="note-216-04" xlink:href="note-216-04a" xml:space="preserve">8. quinti.</note> <note symbol="e" position="left" xlink:label="note-216-05" xlink:href="note-216-05a" xml:space="preserve">10. quinti.</note> <note symbol="f" position="left" xlink:label="note-216-06" xlink:href="note-216-06a" xml:space="preserve">47. primi.</note> <note symbol="g" position="left" xlink:label="note-216-07" xlink:href="note-216-07a" xml:space="preserve">8. quinti.</note> </div> <p> <s xml:id="echoid-s7992" xml:space="preserve"><emph style="sc">Secto</emph> rurſus angulo d E c, bifariam per rectam E b; </s> <s xml:id="echoid-s7993" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> erit rurſus vt d E, E c, <anchor type="note" xlink:label="note-216-08a" xlink:href="note-216-08"/> ſimul ad d c, ita E c, ad c b. </s> <s xml:id="echoid-s7994" xml:space="preserve">Quia verò d E, E c, ſimul maiores ſunt, quam 1162 {1/8}. <lb/></s> <s xml:id="echoid-s7995" xml:space="preserve">(quip pe cum E d, maior ſit, quam 591 {1/8}. </s> <s xml:id="echoid-s7996" xml:space="preserve">& </s> <s xml:id="echoid-s7997" xml:space="preserve">E c, maior, quam 571.) </s> <s xml:id="echoid-s7998" xml:space="preserve">& </s> <s xml:id="echoid-s7999" xml:space="preserve">d c, po-<lb/>ſita eſt 153. </s> <s xml:id="echoid-s8000" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> habebunt d E, E c, ad d c, maiorem proportionem, quam 1162 {1/8}.</s> <s xml:id="echoid-s8001" xml:space="preserve"> <anchor type="note" xlink:label="note-216-09a" xlink:href="note-216-09"/> ad 153. </s> <s xml:id="echoid-s8002" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s8003" xml:space="preserve">E c, ad c b, proportionem habebit maiorem, quam 1162 {1/8}. <lb/></s> <s xml:id="echoid-s8004" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s8005" xml:space="preserve">ac proinde ſi ponatur c b, 153. </s> <s xml:id="echoid-s8006" xml:space="preserve">erit E c, maior, quam 1162 {1/8}. </s> <s xml:id="echoid-s8007" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Igitur <anchor type="note" xlink:label="note-216-10a" xlink:href="note-216-10"/> quadratum ipſius E c, maius erit quam 1350534 {3/6} {3/4}. </s> <s xml:id="echoid-s8008" xml:space="preserve">cui ſi ad datur quadratum <lb/>23409. </s> <s xml:id="echoid-s8009" xml:space="preserve">ipſius c b, erit quadratum ipſius E b, <anchor type="note" xlink:href="" symbol="l"/> quod quadratis rectarum E c, <anchor type="note" xlink:label="note-216-11a" xlink:href="note-216-11"/> c b, æquale eſt, maius, quam 1373943 {3/6} {3/4}. </s> <s xml:id="echoid-s8010" xml:space="preserve">eiuſque radix propterea, id eſt, re- <pb o="187" file="217" n="217" rhead="LIBER QVARTVS."/> cta Eb, paulo maior quam 1172 {1/8}. </s> <s xml:id="echoid-s8011" xml:space="preserve">quip pè cum huius radicis quadratum ſit tan-<lb/>tum 1373877 {1/64}. </s> <s xml:id="echoid-s8012" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Habebit ergo E b, ad b c, maiorem proportionem, quam 1172 <anchor type="note" xlink:label="note-217-01a" xlink:href="note-217-01"/> {1/8}. </s> <s xml:id="echoid-s8013" xml:space="preserve">ad 153.</s> <s xml:id="echoid-s8014" xml:space="preserve"/> </p> <div xml:id="echoid-div493" type="float" level="2" n="3"> <note symbol="h" position="left" xlink:label="note-216-08" xlink:href="note-216-08a" xml:space="preserve">3. ſexti. & <lb/>componendo <lb/>permutando-<lb/>que.</note> <note symbol="i" position="right" xlink:label="note-216-09" xlink:href="note-216-09a" xml:space="preserve">8. quinti.</note> <note symbol="k" position="left" xlink:label="note-216-10" xlink:href="note-216-10a" xml:space="preserve">10. quinti.</note> <note symbol="l" position="left" xlink:label="note-216-11" xlink:href="note-216-11a" xml:space="preserve">47. primi.</note> <note symbol="a" position="right" xlink:label="note-217-01" xlink:href="note-217-01a" xml:space="preserve">8. quinti.</note> </div> <figure> <image file="217-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/217-01"/> </figure> <p> <s xml:id="echoid-s8015" xml:space="preserve"><emph style="sc">Secto</emph> item angulo b E c, bifariam per rectam E a; </s> <s xml:id="echoid-s8016" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erunt b E, E c, ſimul ad <anchor type="note" xlink:label="note-217-02a" xlink:href="note-217-02"/> b c, vt E c, ad c a. </s> <s xml:id="echoid-s8017" xml:space="preserve">Quia verò b E, maior eſt, quam 1172 {1/8}. </s> <s xml:id="echoid-s8018" xml:space="preserve">& </s> <s xml:id="echoid-s8019" xml:space="preserve">E c, maior, quã <lb/>1162 {1/8}. </s> <s xml:id="echoid-s8020" xml:space="preserve">erunt b E, E c, ſimul maiores, quam 2334 {2/8}. </s> <s xml:id="echoid-s8021" xml:space="preserve">vel 2334 {1/4}. </s> <s xml:id="echoid-s8022" xml:space="preserve">Cum ergo <lb/>c b, poſita ſit 153. </s> <s xml:id="echoid-s8023" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>habebunt b E, E c, ſimul ad c b, maiorem proportionem, quam 2334 {1/4}. </s> <s xml:id="echoid-s8024" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s8025" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s8026" xml:space="preserve">E c, ad c a, proportionem habebit maiorem, <lb/> <anchor type="note" xlink:label="note-217-03a" xlink:href="note-217-03"/> quam 2334 {1/4}. </s> <s xml:id="echoid-s8027" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s8028" xml:space="preserve">ac proinde, ſi c a, ponatur 153. </s> <s xml:id="echoid-s8029" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erit E c, maior, quam <anchor type="note" xlink:label="note-217-04a" xlink:href="note-217-04"/> 2334 {1/4}. </s> <s xml:id="echoid-s8030" xml:space="preserve">Igitur quadratum ipſius E c, maius erit, quam 5448723 {1/16}. </s> <s xml:id="echoid-s8031" xml:space="preserve">cui ſi ad <lb/> <anchor type="note" xlink:label="note-217-05a" xlink:href="note-217-05"/> datur quadratum 23409. </s> <s xml:id="echoid-s8032" xml:space="preserve">ipſius c a, erit quadratum ipſius E a, <anchor type="note" xlink:href="" symbol="e"/> quod quadra- tis rectarum E c, ca, æquale eſt, maius quam 5472132 {1/16}. </s> <s xml:id="echoid-s8033" xml:space="preserve">eiuſqueradix maior, <lb/> <anchor type="note" xlink:label="note-217-06a" xlink:href="note-217-06"/> quam 2339 {1/4}. </s> <s xml:id="echoid-s8034" xml:space="preserve">cum huius radicis quadratum ſit tantum 5472090 {9/16}. </s> <s xml:id="echoid-s8035" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Ergo E a, ad a c, maiorem proportionem habebit, quam 2339 {1/4}. </s> <s xml:id="echoid-s8036" xml:space="preserve">ad 153. <lb/></s> <s xml:id="echoid-s8037" xml:space="preserve"> <anchor type="note" xlink:label="note-217-07a" xlink:href="note-217-07"/> </s> </p> <div xml:id="echoid-div494" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-217-02" xlink:href="note-217-02a" xml:space="preserve">3. ſexti. & <lb/>componendo, <lb/>permutando-<lb/>que.</note> <note symbol="c" position="right" xlink:label="note-217-03" xlink:href="note-217-03a" xml:space="preserve">8. quinti.</note> <note symbol="d" position="right" xlink:label="note-217-04" xlink:href="note-217-04a" xml:space="preserve">10. quinti.</note> <note symbol="e" position="right" xlink:label="note-217-05" xlink:href="note-217-05a" xml:space="preserve">47. primi.</note> <note symbol="f" position="right" xlink:label="note-217-06" xlink:href="note-217-06a" xml:space="preserve">8. quinti.</note> <note symbol="g" position="right" xlink:label="note-217-07" xlink:href="note-217-07a" xml:space="preserve">3. ſexti. & <lb/>componendo, <lb/>permutando-<lb/>que.</note> </div> <p> <s xml:id="echoid-s8038" xml:space="preserve"><emph style="sc">Secto</emph> denique angulo quoque a E c, bifariam per rectam E o, <anchor type="note" xlink:href="" symbol="g"/> erunt a E, E c, ſimul ad a c, vt E c, ad c o. </s> <s xml:id="echoid-s8039" xml:space="preserve">Quia verò a E, maior eſt, quam 2339 {1/4}. </s> <s xml:id="echoid-s8040" xml:space="preserve">a & </s> <s xml:id="echoid-s8041" xml:space="preserve"><lb/>E c, maior quam 2334 {1/4}. </s> <s xml:id="echoid-s8042" xml:space="preserve">erunta E, E c, ſimul maiores, quam 4673 {1/2}. </s> <s xml:id="echoid-s8043" xml:space="preserve">Cum er-<lb/>go c a, poſita ſit 153. </s> <s xml:id="echoid-s8044" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> habebunt a E, E c, ſimul ad c a, hoc eſt, E c, ad c o, ma- <anchor type="note" xlink:label="note-217-08a" xlink:href="note-217-08"/> iorem proportionem, quam 4673 {1/2}. </s> <s xml:id="echoid-s8045" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s8046" xml:space="preserve">ac propterea ſi ponatur c o, 153. <lb/></s> <s xml:id="echoid-s8047" xml:space="preserve"> <anchor type="note" xlink:label="note-217-09a" xlink:href="note-217-09"/> <anchor type="note" xlink:href="" symbol="i"/> erit E c, maior quam 4673 {1/2}.</s> <s xml:id="echoid-s8048" xml:space="preserve"/> </p> <div xml:id="echoid-div495" type="float" level="2" n="5"> <note symbol="h" position="right" xlink:label="note-217-08" xlink:href="note-217-08a" xml:space="preserve">8. quinti.</note> <note symbol="i" position="right" xlink:label="note-217-09" xlink:href="note-217-09a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s8049" xml:space="preserve"><emph style="sc">Qvoniam</emph> igitur angulus e E c, tertia pars eſt recti, erit eius ſemiſsis d E c, ſexta <lb/>pars recti, & </s> <s xml:id="echoid-s8050" xml:space="preserve">huius ſemiſsis b E c, {1/@}. </s> <s xml:id="echoid-s8051" xml:space="preserve">recti, & </s> <s xml:id="echoid-s8052" xml:space="preserve">huius ſemiſsis a E c, {1/24}. </s> <s xml:id="echoid-s8053" xml:space="preserve">recti, & </s> <s xml:id="echoid-s8054" xml:space="preserve">deni-<lb/> <anchor type="note" xlink:label="note-217-10a" xlink:href="note-217-10"/> que huius ſemiſsis o E c, {1/48}. </s> <s xml:id="echoid-s8055" xml:space="preserve">recti. </s> <s xml:id="echoid-s8056" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Qualium ergo partium 48. </s> <s xml:id="echoid-s8057" xml:space="preserve">eſt quadrans A c, talium 1. </s> <s xml:id="echoid-s8058" xml:space="preserve">erit arcus c o: </s> <s xml:id="echoid-s8059" xml:space="preserve">& </s> <s xml:id="echoid-s8060" xml:space="preserve">id circo erit c o, {1/192}. </s> <s xml:id="echoid-s8061" xml:space="preserve">to tius circumferentiæ. </s> <s xml:id="echoid-s8062" xml:space="preserve">Fiat angulus <lb/> <anchor type="note" xlink:label="note-217-11a" xlink:href="note-217-11"/> cEi, angulo cEo, æqualis; </s> <s xml:id="echoid-s8063" xml:space="preserve">eritq; </s> <s xml:id="echoid-s8064" xml:space="preserve">totus angulus oEi, {1/96}. </s> <s xml:id="echoid-s8065" xml:space="preserve">quatuor rectorũ: </s> <s xml:id="echoid-s8066" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> ideoq;</s> <s xml:id="echoid-s8067" xml:space="preserve"> arc<emph style="sub">9</emph> o i, {1/98}. </s> <s xml:id="echoid-s8068" xml:space="preserve">totius circũferentiæ. </s> <s xml:id="echoid-s8069" xml:space="preserve">Per do ctrinã ergo ꝓpoſ. </s> <s xml:id="echoid-s8070" xml:space="preserve">12. </s> <s xml:id="echoid-s8071" xml:space="preserve">lib. </s> <s xml:id="echoid-s8072" xml:space="preserve">4. </s> <s xml:id="echoid-s8073" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s8074" xml:space="preserve">recta o i, <pb o="188" file="218" n="218" rhead="GEOMETR. PRACT."/> latus erit polygoni circulo circumſcripti, quod lateribus æqualibus 96. </s> <s xml:id="echoid-s8075" xml:space="preserve">conti-<lb/>netur. </s> <s xml:id="echoid-s8076" xml:space="preserve">Et quia oſtenſum eſt, E c, ad c o, maiorem habere proportionem, quam <lb/>4673 {1/2}. </s> <s xml:id="echoid-s8077" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s8078" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> habebit quo que diameter AB, ipſius E c, dupla ad o i, ipſius c o, <anchor type="note" xlink:label="note-218-01a" xlink:href="note-218-01"/> duplam maiorem proportionem, quam 4673 {1/2}. </s> <s xml:id="echoid-s8079" xml:space="preserve">ad 153. </s> <s xml:id="echoid-s8080" xml:space="preserve">Si ergo o i, latus polygo-<lb/>ni ponatur 153. </s> <s xml:id="echoid-s8081" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit diameter AB, maior, quam 4673 {1/2}. </s> <s xml:id="echoid-s8082" xml:space="preserve">Multiplicentur 153. </s> <s xml:id="echoid-s8083" xml:space="preserve">per <anchor type="note" xlink:label="note-218-02a" xlink:href="note-218-02"/> 96. </s> <s xml:id="echoid-s8084" xml:space="preserve">vt totus ambitus polygoni producatur 14688. </s> <s xml:id="echoid-s8085" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> habebitque ambitus poly- <anchor type="note" xlink:label="note-218-03a" xlink:href="note-218-03"/> goni ad diametrum AB, minorem proportionem, quam 14688. </s> <s xml:id="echoid-s8086" xml:space="preserve">ad 4673 {1/2}. </s> <s xml:id="echoid-s8087" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Eſt <anchor type="note" xlink:label="note-218-04a" xlink:href="note-218-04"/> autem proportio 14688. </s> <s xml:id="echoid-s8088" xml:space="preserve">ad 4673 {1/2}. </s> <s xml:id="echoid-s8089" xml:space="preserve">minor, quam tripla ſeſquiſeptima; </s> <s xml:id="echoid-s8090" xml:space="preserve">quod <lb/>14688. </s> <s xml:id="echoid-s8091" xml:space="preserve">ad 4673. </s> <s xml:id="echoid-s8092" xml:space="preserve">{5/11}. </s> <s xml:id="echoid-s8093" xml:space="preserve">(quinumerus paulo minor eſt quam 4673 {1/2}.) </s> <s xml:id="echoid-s8094" xml:space="preserve">habeant pro-<lb/>portionem triplam ſeſquiſeptimam. </s> <s xml:id="echoid-s8095" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/>Igitur & </s> <s xml:id="echoid-s8096" xml:space="preserve">ambitus polygoni ad diametrũ <anchor type="note" xlink:label="note-218-05a" xlink:href="note-218-05"/> AB, proportionem habet minorem tripla ſeſquiſeptima: </s> <s xml:id="echoid-s8097" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> atque adeo circum- ferentia, quæ (vt lib. </s> <s xml:id="echoid-s8098" xml:space="preserve">8. </s> <s xml:id="echoid-s8099" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s8100" xml:space="preserve">1. </s> <s xml:id="echoid-s8101" xml:space="preserve">probabimus) minor eſt ambitu polygoni, mul-<lb/> <anchor type="note" xlink:label="note-218-06a" xlink:href="note-218-06"/> to minorem proportionem tripla ſeſquiſeptima ad diametrum habebit; </s> <s xml:id="echoid-s8102" xml:space="preserve">ideoq; <lb/></s> <s xml:id="echoid-s8103" xml:space="preserve">circumferentia tripla eſt diametri, & </s> <s xml:id="echoid-s8104" xml:space="preserve">ad huc ſuperat parte, quæ minor eſt {30/70}. </s> <s xml:id="echoid-s8105" xml:space="preserve">hoc <lb/>eſt, {1/7}. </s> <s xml:id="echoid-s8106" xml:space="preserve">diametri. </s> <s xml:id="echoid-s8107" xml:space="preserve">Nam ſi contineret ter, & </s> <s xml:id="echoid-s8108" xml:space="preserve">{1/7}. </s> <s xml:id="echoid-s8109" xml:space="preserve">haberet cir cumferentia ad diame-<lb/>trum proportionem triplam ſeſquiſeptimam: </s> <s xml:id="echoid-s8110" xml:space="preserve">ſi vero contineret ter, & </s> <s xml:id="echoid-s8111" xml:space="preserve">pluſquã <lb/>{1/7}. </s> <s xml:id="echoid-s8112" xml:space="preserve">haberet maiorem proportionem, quam triplam ſeſquiſep timam, cum tamen <lb/>minorem habeat, vt demonſtratum eſt.</s> <s xml:id="echoid-s8113" xml:space="preserve"/> </p> <div xml:id="echoid-div496" type="float" level="2" n="6"> <note symbol="k" position="right" xlink:label="note-217-10" xlink:href="note-217-10a" xml:space="preserve">33 ſexti.</note> <note symbol="l" position="right" xlink:label="note-217-11" xlink:href="note-217-11a" xml:space="preserve">33. ſexti.</note> <note symbol="a" position="left" xlink:label="note-218-01" xlink:href="note-218-01a" xml:space="preserve">15. quinti.</note> <note symbol="b" position="left" xlink:label="note-218-02" xlink:href="note-218-02a" xml:space="preserve">10. quinti.</note> <note symbol="c" position="left" xlink:label="note-218-03" xlink:href="note-218-03a" xml:space="preserve">8. quinti.</note> <note symbol="d" position="left" xlink:label="note-218-04" xlink:href="note-218-04a" xml:space="preserve">8. quinti.</note> <note symbol="e" position="left" xlink:label="note-218-05" xlink:href="note-218-05a" xml:space="preserve">ſchol. 13. <lb/>quinti.</note> <note symbol="f" position="left" xlink:label="note-218-06" xlink:href="note-218-06a" xml:space="preserve">8 quinti.</note> </div> <p> <s xml:id="echoid-s8114" xml:space="preserve"><emph style="sc">Iam</emph> vero in eodem circulo ſit latus hexagoni B G, ſemidiametro æquale, <lb/>per coroll. </s> <s xml:id="echoid-s8115" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s8116" xml:space="preserve">15. </s> <s xml:id="echoid-s8117" xml:space="preserve">lib. </s> <s xml:id="echoid-s8118" xml:space="preserve">4. </s> <s xml:id="echoid-s8119" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s8120" xml:space="preserve">iunganturque rectæ A G, E G. </s> <s xml:id="echoid-s8121" xml:space="preserve">Et quia trian-<lb/>gulum E B G, eſt æquilaterum conſtans ex tribus ſemidiametris; </s> <s xml:id="echoid-s8122" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit angulus <anchor type="note" xlink:label="note-218-07a" xlink:href="note-218-07"/> B E G, {2/3}. </s> <s xml:id="echoid-s8123" xml:space="preserve">vnius recti, <anchor type="note" xlink:href="" symbol="h"/> ac proinde eius ſemiſsis B A G, erit {1/3}. </s> <s xml:id="echoid-s8124" xml:space="preserve">vnius recti. </s> <s xml:id="echoid-s8125" xml:space="preserve">Et quia diameter AB, dupla eſt ſemidiametri BG, ſi BG, ponatur 780. </s> <s xml:id="echoid-s8126" xml:space="preserve">erit AB, 1560. </s> <s xml:id="echoid-s8127" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Cũ <anchor type="note" xlink:label="note-218-08a" xlink:href="note-218-08"/> ergo quadratumipſius A B, æquale ſit quadratis rectarum B G, G A; </s> <s xml:id="echoid-s8128" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> quod an- <anchor type="note" xlink:label="note-218-09a" xlink:href="note-218-09"/> gulus AGB, in ſemicirculo rectus ſit: </s> <s xml:id="echoid-s8129" xml:space="preserve">ſi quadratum 608400. </s> <s xml:id="echoid-s8130" xml:space="preserve">ipſius B G, dema-<lb/> <anchor type="note" xlink:label="note-218-10a" xlink:href="note-218-10"/> tur ex 2433600. </s> <s xml:id="echoid-s8131" xml:space="preserve">quadrato ipſius A B, reliquum fiet quadratum 1825200. </s> <s xml:id="echoid-s8132" xml:space="preserve">ipſius <lb/>AG, cuius radix paulo minor eſt, quam 1351. </s> <s xml:id="echoid-s8133" xml:space="preserve">cum huius quadratum 1825201. <lb/></s> <s xml:id="echoid-s8134" xml:space="preserve">maius ſit, quam 1825200. </s> <s xml:id="echoid-s8135" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Igitur AG, ad GB, minorem habebit proportionem, <anchor type="note" xlink:label="note-218-11a" xlink:href="note-218-11"/> quam 1351. </s> <s xml:id="echoid-s8136" xml:space="preserve">ad 780. </s> <s xml:id="echoid-s8137" xml:space="preserve">ac proinde ſi B G, ponatur 780. </s> <s xml:id="echoid-s8138" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> erit A G, minor, quam 1351.</s> <s xml:id="echoid-s8139" xml:space="preserve"/> </p> <div xml:id="echoid-div497" type="float" level="2" n="7"> <note symbol="g" position="left" xlink:label="note-218-07" xlink:href="note-218-07a" xml:space="preserve">coroll. 3 pro-<lb/>poſ. 32. lib. 1.</note> <note symbol="h" position="left" xlink:label="note-218-08" xlink:href="note-218-08a" xml:space="preserve">20. tertij.</note> <note symbol="i" position="left" xlink:label="note-218-09" xlink:href="note-218-09a" xml:space="preserve">47. primi.</note> <note symbol="k" position="left" xlink:label="note-218-10" xlink:href="note-218-10a" xml:space="preserve">31. tertij.</note> <note symbol="l" position="left" xlink:label="note-218-11" xlink:href="note-218-11a" xml:space="preserve">8. quinti.</note> </div> <note symbol="m" position="left" xml:space="preserve">10. quinti.</note> <p> <s xml:id="echoid-s8140" xml:space="preserve"><emph style="sc">Secto</emph> iam angulo BAG, bifariam per rectam AH, ſecantem BG, in M, du-<lb/>ctaque HB, erunt triangula BHM, AHB, æquiangula: </s> <s xml:id="echoid-s8141" xml:space="preserve"><anchor type="note" xlink:href="" symbol="n"/> propterea quod angu- <anchor type="note" xlink:label="note-218-13a" xlink:href="note-218-13"/> lus HBM, æqualis eſt angulo HAG, ob eandem baſem GH; </s> <s xml:id="echoid-s8142" xml:space="preserve">ideoque per con-<lb/>ſtru ctionem angulo HAB; </s> <s xml:id="echoid-s8143" xml:space="preserve">& </s> <s xml:id="echoid-s8144" xml:space="preserve">angulus rectus H, in ſemicirculo communis. </s> <s xml:id="echoid-s8145" xml:space="preserve"><anchor type="note" xlink:href="" symbol="o"/>I- <anchor type="note" xlink:label="note-218-14a" xlink:href="note-218-14"/> gitur erit AH, ad HB, vt HB, ad HM. </s> <s xml:id="echoid-s8146" xml:space="preserve">Item AB, ad BH, vt BM, ad MH: </s> <s xml:id="echoid-s8147" xml:space="preserve">& </s> <s xml:id="echoid-s8148" xml:space="preserve">per-<lb/>mutando AB, ad BM, vt BH, ad HM: </s> <s xml:id="echoid-s8149" xml:space="preserve">ideoque erunt tres hæ proportiones AH, <lb/>ad HB; </s> <s xml:id="echoid-s8150" xml:space="preserve">HB, ad HM: </s> <s xml:id="echoid-s8151" xml:space="preserve">& </s> <s xml:id="echoid-s8152" xml:space="preserve">AB, ad BM, æquales. </s> <s xml:id="echoid-s8153" xml:space="preserve">Sed vt AB, ad BM, ita eſt vtraq; <lb/></s> <s xml:id="echoid-s8154" xml:space="preserve">ſimul BA, AG, ad BG. </s> <s xml:id="echoid-s8155" xml:space="preserve"><anchor type="note" xlink:href="" symbol="p"/> Nam vt AG, ad AB, ita eſt GM, ad MB; </s> <s xml:id="echoid-s8156" xml:space="preserve">& </s> <s xml:id="echoid-s8157" xml:space="preserve">componendo vt <anchor type="note" xlink:label="note-218-15a" xlink:href="note-218-15"/> AG, AB, ſimul ad AB, ita GM, MB, ſimul id eſt, tota G B, ad M B: </s> <s xml:id="echoid-s8158" xml:space="preserve">Et permut ando vt <lb/>A G, A B, ſimulad G B, ita A B, ad M B. </s> <s xml:id="echoid-s8159" xml:space="preserve">Igitur erit quoque, vt vtraque AG, AB, ſi-<lb/>mul ad GB, ita AH, ad HB. </s> <s xml:id="echoid-s8160" xml:space="preserve">Eſt autem AG, oſtenſa minor, quam 1351. </s> <s xml:id="echoid-s8161" xml:space="preserve">& </s> <s xml:id="echoid-s8162" xml:space="preserve">AB, po-<lb/>ſita eſt 1560. </s> <s xml:id="echoid-s8163" xml:space="preserve">& </s> <s xml:id="echoid-s8164" xml:space="preserve">GB, 780. </s> <s xml:id="echoid-s8165" xml:space="preserve">Igitur vtraque AG, AB, ſimul (cum minus effi ciant, quã <lb/>2911.) </s> <s xml:id="echoid-s8166" xml:space="preserve"><anchor type="note" xlink:href="" symbol="q"/> minorem habebit proportionem ad GB, quam 2911. </s> <s xml:id="echoid-s8167" xml:space="preserve">ad 780. </s> <s xml:id="echoid-s8168" xml:space="preserve">Quare et- <anchor type="note" xlink:label="note-218-16a" xlink:href="note-218-16"/> iam proportio AH, ad HB, minor erit, quam 2911. </s> <s xml:id="echoid-s8169" xml:space="preserve">ad 780. </s> <s xml:id="echoid-s8170" xml:space="preserve">ac proinde ſi HB, po-<lb/>natur 780. </s> <s xml:id="echoid-s8171" xml:space="preserve"><anchor type="note" xlink:href="" symbol="r"/> erit AH, minor, quam 2911. </s> <s xml:id="echoid-s8172" xml:space="preserve">ideoque eius quadratum minus, quam <anchor type="note" xlink:label="note-218-17a" xlink:href="note-218-17"/> 8473921. </s> <s xml:id="echoid-s8173" xml:space="preserve">cui ſi addatur quadratum 608400. </s> <s xml:id="echoid-s8174" xml:space="preserve">ipſius B H, fiet quadratum ipſius <lb/>AB, <anchor type="note" xlink:href="" symbol="s"/> (quod quadratis rectarum AH, HB, æquale eſt) minus, quam 9082321. </s> <s xml:id="echoid-s8175" xml:space="preserve">id- <anchor type="note" xlink:label="note-218-18a" xlink:href="note-218-18"/> co que eius radix, vel recta A B, minor, quam 3013 {3/4}. </s> <s xml:id="echoid-s8176" xml:space="preserve">cum huius quadratum <pb o="189" file="219" n="219" rhead="LIBER QVARTVS."/> 9082689 {1/16}. </s> <s xml:id="echoid-s8177" xml:space="preserve">maius ſit, quam 9082321. </s> <s xml:id="echoid-s8178" xml:space="preserve">Igitur AB, ad BH, minorem proportio-<lb/>nem habebit, quam 3013 {3/4}. </s> <s xml:id="echoid-s8179" xml:space="preserve">ad 780. </s> <s xml:id="echoid-s8180" xml:space="preserve">ac proinde ſi BH, ponatur 780. </s> <s xml:id="echoid-s8181" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit AB, mi- <anchor type="note" xlink:label="note-219-01a" xlink:href="note-219-01"/> nor quam 3013 {3/4}.</s> <s xml:id="echoid-s8182" xml:space="preserve"/> </p> <div xml:id="echoid-div498" type="float" level="2" n="8"> <note symbol="n" position="left" xlink:label="note-218-13" xlink:href="note-218-13a" xml:space="preserve">21. tertij.</note> <note symbol="o" position="left" xlink:label="note-218-14" xlink:href="note-218-14a" xml:space="preserve">4. ſexti.</note> <note symbol="p" position="left" xlink:label="note-218-15" xlink:href="note-218-15a" xml:space="preserve">3. ſexti.</note> <note symbol="q" position="left" xlink:label="note-218-16" xlink:href="note-218-16a" xml:space="preserve">8. quinti.</note> <note symbol="r" position="left" xlink:label="note-218-17" xlink:href="note-218-17a" xml:space="preserve">10. quinti.</note> <note symbol="s" position="left" xlink:label="note-218-18" xlink:href="note-218-18a" xml:space="preserve">47. primi.</note> <note symbol="a" position="right" xlink:label="note-219-01" xlink:href="note-219-01a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s8183" xml:space="preserve"><emph style="sc">Secto</emph> rurſus angulo HAB, bifariam per rectam AI, ſecantem HB, in N; <lb/></s> <s xml:id="echoid-s8184" xml:space="preserve">erunt vt prius, triangula BIN, AIB, æquiangula. </s> <s xml:id="echoid-s8185" xml:space="preserve">Ergo, vt ſupra, demonſtrabi-<lb/>mus, vtramque BA, AH, ſimul ad HB, habere eandem proportionem quam AI, <lb/>ad IB. </s> <s xml:id="echoid-s8186" xml:space="preserve">Eſt autem BA, oſtenſa minor, quam 3013 {3/4}. </s> <s xml:id="echoid-s8187" xml:space="preserve">& </s> <s xml:id="echoid-s8188" xml:space="preserve">AH, minor, quam 2911. </s> <s xml:id="echoid-s8189" xml:space="preserve">& </s> <s xml:id="echoid-s8190" xml:space="preserve"><lb/>ob id earum ſumma minor, quam 5924 {3/4}. </s> <s xml:id="echoid-s8191" xml:space="preserve">ipſa autem HB, poſita eſt 780. </s> <s xml:id="echoid-s8192" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Igi- <anchor type="note" xlink:label="note-219-02a" xlink:href="note-219-02"/> tur vtraque BA, AH, ſimul ad HB, hoc eſt, AI, ad IB, minorem habebit propor-<lb/> <anchor type="note" xlink:label="note-219-03a" xlink:href="note-219-03"/> tionem, quam 5924 {3/4}. </s> <s xml:id="echoid-s8193" xml:space="preserve">ad 780. </s> <s xml:id="echoid-s8194" xml:space="preserve">Siergo IB, ponatur 780. </s> <s xml:id="echoid-s8195" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit AI, minor, quam 5924 {3/4}. </s> <s xml:id="echoid-s8196" xml:space="preserve">Et quoniam eſt, vt 5924 {3/4}. </s> <s xml:id="echoid-s8197" xml:space="preserve">ad 780. </s> <s xml:id="echoid-s8198" xml:space="preserve">ita 1823. </s> <s xml:id="echoid-s8199" xml:space="preserve">ad 240, quod idem nu-<lb/>merus fiat ex primo in quartum, qui ex ſecundo in tertium, quæ quidem pro-<lb/>portio denominatur à 7 {143/240}. </s> <s xml:id="echoid-s8200" xml:space="preserve">habebit quoque AI, ad IB, minorem proportio-<lb/>nem, quam 1823. </s> <s xml:id="echoid-s8201" xml:space="preserve">ad 240. </s> <s xml:id="echoid-s8202" xml:space="preserve">ideoque poſita I B, 240. </s> <s xml:id="echoid-s8203" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erit A I, minor, <anchor type="note" xlink:label="note-219-04a" xlink:href="note-219-04"/> quam 1823. </s> <s xml:id="echoid-s8204" xml:space="preserve">atque ob id quadratum ipſius A I, minus, quam 3323329. </s> <s xml:id="echoid-s8205" xml:space="preserve">cui <lb/>ſi addatur quadratum 57600. </s> <s xml:id="echoid-s8206" xml:space="preserve">ipſius IB, fiet quadratum ipſius AB, <anchor type="note" xlink:href="" symbol="e"/> (quod qua- <anchor type="note" xlink:label="note-219-05a" xlink:href="note-219-05"/> dratis rectarum A I, IB, æquale eſt) minus quam 3380929. </s> <s xml:id="echoid-s8207" xml:space="preserve">eiuſque radix pro-<lb/>pterea, vel recta AB, minor quam 1838. </s> <s xml:id="echoid-s8208" xml:space="preserve">{9/11}. </s> <s xml:id="echoid-s8209" xml:space="preserve">cum huius quadratum 3381252 {37/121}. <lb/></s> <s xml:id="echoid-s8210" xml:space="preserve">maius ſit, quam 3380929. </s> <s xml:id="echoid-s8211" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Igitur AB, ad BI, minorem proportionem habebit, <anchor type="note" xlink:label="note-219-06a" xlink:href="note-219-06"/> quam 1838 {9/11}. </s> <s xml:id="echoid-s8212" xml:space="preserve">ad 240. </s> <s xml:id="echoid-s8213" xml:space="preserve">ac proinde poſita BI, 240. </s> <s xml:id="echoid-s8214" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit A B, minor, quam <anchor type="note" xlink:label="note-219-07a" xlink:href="note-219-07"/> 1838 {9/11}.</s> <s xml:id="echoid-s8215" xml:space="preserve"/> </p> <div xml:id="echoid-div499" type="float" level="2" n="9"> <note symbol="b" position="right" xlink:label="note-219-02" xlink:href="note-219-02a" xml:space="preserve">8. quinti.</note> <note symbol="c" position="right" xlink:label="note-219-03" xlink:href="note-219-03a" xml:space="preserve">10. quinti.</note> <note symbol="d" position="right" xlink:label="note-219-04" xlink:href="note-219-04a" xml:space="preserve">10. quinti.</note> <note symbol="e" position="right" xlink:label="note-219-05" xlink:href="note-219-05a" xml:space="preserve">47. primi.</note> <note symbol="f" position="right" xlink:label="note-219-06" xlink:href="note-219-06a" xml:space="preserve">8. quinti.</note> <note symbol="g" position="right" xlink:label="note-219-07" xlink:href="note-219-07a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s8216" xml:space="preserve"><emph style="sc">Secto</emph> item angulo IAB, bifariam per rectam AK, oſtendemus eodem mo-<lb/>do, vtramque BA, AI, ſimul ad IB, habere eandem pro portionem, quam AK, ad <lb/>KB. </s> <s xml:id="echoid-s8217" xml:space="preserve">Sunt autem BA, AI, ambæ ſimul minores, quam 3661 {9/11}. </s> <s xml:id="echoid-s8218" xml:space="preserve">(quod B A, oſ-<lb/> <anchor type="figure" xlink:label="fig-219-01a" xlink:href="fig-219-01"/> tenſa ſit minor, quam 1838 {9/11}. </s> <s xml:id="echoid-s8219" xml:space="preserve">& </s> <s xml:id="echoid-s8220" xml:space="preserve">AI, minor, quam 1823.) </s> <s xml:id="echoid-s8221" xml:space="preserve">& </s> <s xml:id="echoid-s8222" xml:space="preserve">IB, poſita eſt 240. <lb/></s> <s xml:id="echoid-s8223" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="h"/> Vtraq; </s> <s xml:id="echoid-s8224" xml:space="preserve">ergo BA, AI, ſimulad IB, hoc eſt, AK, ad KB, minorem habebit propor- <anchor type="note" xlink:label="note-219-08a" xlink:href="note-219-08"/> <pb o="190" file="220" n="220" rhead="GEOMETR. PRACT."/> tionem, quam 3661 {9/11}. </s> <s xml:id="echoid-s8225" xml:space="preserve">ad 240. </s> <s xml:id="echoid-s8226" xml:space="preserve">Vt autem 3661 {9/11}. </s> <s xml:id="echoid-s8227" xml:space="preserve">ad 240. </s> <s xml:id="echoid-s8228" xml:space="preserve">ita eſt 1007. </s> <s xml:id="echoid-s8229" xml:space="preserve">ad <lb/>66. </s> <s xml:id="echoid-s8230" xml:space="preserve">quod idem gignatur numerus ex primo in quartum, qui ex ſecundo in ter-<lb/>tium, quę quidem proportio denominatur à 15 {17/66}. </s> <s xml:id="echoid-s8231" xml:space="preserve">Igitur AK, ad KB, minorem <lb/>quo queprop ortionem habebit, quam 1007. </s> <s xml:id="echoid-s8232" xml:space="preserve">ad 66. </s> <s xml:id="echoid-s8233" xml:space="preserve">ideoque poſita K B, 66. <lb/></s> <s xml:id="echoid-s8234" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> erit A K, minor, quam 1007. </s> <s xml:id="echoid-s8235" xml:space="preserve">ac propterea eius quadratum minus, quam <anchor type="note" xlink:label="note-220-01a" xlink:href="note-220-01"/> 1014049. </s> <s xml:id="echoid-s8236" xml:space="preserve">cuiſi ad datur quadratum 4356. </s> <s xml:id="echoid-s8237" xml:space="preserve">ipſius KB; </s> <s xml:id="echoid-s8238" xml:space="preserve">fiet qua dratum ipſius A B, <lb/> <anchor type="note" xlink:href="" symbol="b"/> (quod illis duobus æquale eſt) minus quam 1018405. </s> <s xml:id="echoid-s8239" xml:space="preserve">eiuſque radix propter- <anchor type="note" xlink:label="note-220-02a" xlink:href="note-220-02"/> ea, id eſt, recta AB, minor, quam 1009 {1/4}. </s> <s xml:id="echoid-s8240" xml:space="preserve">cum huius quadratum 1018417 {13/36}. </s> <s xml:id="echoid-s8241" xml:space="preserve">ſit <lb/>maius, quam 1018405. </s> <s xml:id="echoid-s8242" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quocirca AB, ad BK, minorem proportionem habe- <anchor type="note" xlink:label="note-220-03a" xlink:href="note-220-03"/> bit, quam 1009 {1/6}. </s> <s xml:id="echoid-s8243" xml:space="preserve">ad 66. </s> <s xml:id="echoid-s8244" xml:space="preserve">atqueidcirco poſita BK, 66. </s> <s xml:id="echoid-s8245" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erit AB, minor, quam <anchor type="note" xlink:label="note-220-04a" xlink:href="note-220-04"/> 1009 {1/6}.</s> <s xml:id="echoid-s8246" xml:space="preserve"/> </p> <div xml:id="echoid-div500" type="float" level="2" n="10"> <figure xlink:label="fig-219-01" xlink:href="fig-219-01a"> <image file="219-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/219-01"/> </figure> <note symbol="h" position="right" xlink:label="note-219-08" xlink:href="note-219-08a" xml:space="preserve">8. quinti.</note> <note symbol="a" position="left" xlink:label="note-220-01" xlink:href="note-220-01a" xml:space="preserve">10. quinti.</note> <note symbol="b" position="left" xlink:label="note-220-02" xlink:href="note-220-02a" xml:space="preserve">47. primi.</note> <note symbol="c" position="left" xlink:label="note-220-03" xlink:href="note-220-03a" xml:space="preserve">8. quinti.</note> <note symbol="d" position="left" xlink:label="note-220-04" xlink:href="note-220-04a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s8247" xml:space="preserve"><emph style="sc">Secto</emph> denique angulo KAB, bifariam, perrectam AL, demonſtrabimus <lb/>eadem ratione, vtramque BA, AK, ſimul ad CK, eſſe, vt AL, ad LB. </s> <s xml:id="echoid-s8248" xml:space="preserve">Sunt autem <lb/>ambę B A, A K, ſimul minores, quam 2016 {1/6}. </s> <s xml:id="echoid-s8249" xml:space="preserve">(quod BA, ſit oſtenſa minor, <lb/>quam 1009 {1/6}. </s> <s xml:id="echoid-s8250" xml:space="preserve">& </s> <s xml:id="echoid-s8251" xml:space="preserve">AK, minor, quam 1007.) </s> <s xml:id="echoid-s8252" xml:space="preserve">& </s> <s xml:id="echoid-s8253" xml:space="preserve">BK, poſita eſt 66. </s> <s xml:id="echoid-s8254" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/>Igitur vtra- <anchor type="note" xlink:label="note-220-05a" xlink:href="note-220-05"/> que BA, AK, ſimul ad CK, hoc eſt, AL, ad LB, habebit proportionem minorem, <lb/>quam 2016 {1/6}. </s> <s xml:id="echoid-s8255" xml:space="preserve">ad 66. </s> <s xml:id="echoid-s8256" xml:space="preserve">atque idcirco ſi LB, ponatur 66. </s> <s xml:id="echoid-s8257" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> erit AL, minor, quam <anchor type="note" xlink:label="note-220-06a" xlink:href="note-220-06"/> 2016 {1/6}. </s> <s xml:id="echoid-s8258" xml:space="preserve">ideo que quadratum eius minus quam 4064928 {1/36}. </s> <s xml:id="echoid-s8259" xml:space="preserve">cui ſi ad datur qua-<lb/>dratum 4356. </s> <s xml:id="echoid-s8260" xml:space="preserve">ipſius LB; </s> <s xml:id="echoid-s8261" xml:space="preserve">fiet quadratum ipſius A B, <anchor type="note" xlink:href="" symbol="g"/> quod duobus illis eſt æ- <anchor type="note" xlink:label="note-220-07a" xlink:href="note-220-07"/> quale) minus, quam numerus 4069284 {1/36}. </s> <s xml:id="echoid-s8262" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s8263" xml:space="preserve">eius radix, id eſt, recta AB, <lb/>minor, quam 2017 {1/4}. </s> <s xml:id="echoid-s8264" xml:space="preserve">cum huius quadratum 4069297 {9/36}. </s> <s xml:id="echoid-s8265" xml:space="preserve">ſuperet 4069284 {1/36}. <lb/></s> <s xml:id="echoid-s8266" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="h"/>Quamobrem AB, ad BL, minorem proportionem habebit, quam 2017 {1/4}. </s> <s xml:id="echoid-s8267" xml:space="preserve">ad <anchor type="note" xlink:label="note-220-08a" xlink:href="note-220-08"/> 66. </s> <s xml:id="echoid-s8268" xml:space="preserve">ideoque ſi BL, ponatur 66. </s> <s xml:id="echoid-s8269" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> erit AB, minor, quam 2017 {1/4}.</s> <s xml:id="echoid-s8270" xml:space="preserve"/> </p> <div xml:id="echoid-div501" type="float" level="2" n="11"> <note symbol="e" position="left" xlink:label="note-220-05" xlink:href="note-220-05a" xml:space="preserve">8. quinti.</note> <note symbol="f" position="left" xlink:label="note-220-06" xlink:href="note-220-06a" xml:space="preserve">10. quinti.</note> <note symbol="g" position="left" xlink:label="note-220-07" xlink:href="note-220-07a" xml:space="preserve">47. primi.</note> <note symbol="h" position="left" xlink:label="note-220-08" xlink:href="note-220-08a" xml:space="preserve">8. quinti.</note> </div> <note symbol="i" position="left" xml:space="preserve">10. quinti.</note> <p> <s xml:id="echoid-s8271" xml:space="preserve"><emph style="sc">Qvoniam</emph> igitur angulus G A H, angulo H A B, æqualis eſt; </s> <s xml:id="echoid-s8272" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/>erit arcus <anchor type="note" xlink:label="note-220-10a" xlink:href="note-220-10"/> <anchor type="figure" xlink:label="fig-220-01a" xlink:href="fig-220-01"/> GH, arcui HB, æqualis: </s> <s xml:id="echoid-s8273" xml:space="preserve">eademqueratione arcus HI, arcui IB, & </s> <s xml:id="echoid-s8274" xml:space="preserve">IK, ipſi KB, & </s> <s xml:id="echoid-s8275" xml:space="preserve"><lb/>KL, ipſi LB, æqualis erit. </s> <s xml:id="echoid-s8276" xml:space="preserve">Cum ergo GB, ſit {1/6}. </s> <s xml:id="echoid-s8277" xml:space="preserve">totius circumſerentiæ, erit HB.</s> <s xml:id="echoid-s8278" xml:space="preserve"> <pb o="191" file="221" n="221" rhead="LIBER QVARTVS."/> @ {1/2<unsure/>}. </s> <s xml:id="echoid-s8279" xml:space="preserve">& </s> <s xml:id="echoid-s8280" xml:space="preserve">IB, {1/24}. </s> <s xml:id="echoid-s8281" xml:space="preserve">& </s> <s xml:id="echoid-s8282" xml:space="preserve">KB, {1/48}. </s> <s xml:id="echoid-s8283" xml:space="preserve">& </s> <s xml:id="echoid-s8284" xml:space="preserve">LB, {1/96}. </s> <s xml:id="echoid-s8285" xml:space="preserve">ac proinde recta BL, latus erit Polygoni cir-<lb/>culo inſcripti, quod 96. </s> <s xml:id="echoid-s8286" xml:space="preserve">lateribus æqualibus continetur. </s> <s xml:id="echoid-s8287" xml:space="preserve">Quoniam vero BL, <lb/>poſita eſt 66. </s> <s xml:id="echoid-s8288" xml:space="preserve">ſi 66. </s> <s xml:id="echoid-s8289" xml:space="preserve">ducantur in 96. </s> <s xml:id="echoid-s8290" xml:space="preserve">fiet ambitus Polygoni 6336. </s> <s xml:id="echoid-s8291" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quapropter <anchor type="note" xlink:label="note-221-01a" xlink:href="note-221-01"/> ambitus Polygoniad diametrum AB, maiorem proportionem habebit, quam <lb/>6336. </s> <s xml:id="echoid-s8292" xml:space="preserve">ad 2017 {1/4}. </s> <s xml:id="echoid-s8293" xml:space="preserve">cum diameter A B, oſtenſa ſit minor, quam 2017 {1/4}. </s> <s xml:id="echoid-s8294" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Eſt autem <anchor type="note" xlink:label="note-221-02a" xlink:href="note-221-02"/> proportio 6336. </s> <s xml:id="echoid-s8295" xml:space="preserve">ad 2017 {1/4}. </s> <s xml:id="echoid-s8296" xml:space="preserve">maior, quam tripla ſuperdecupartiens ſeptuageſi-<lb/>mas primas, quod proportio 6336. </s> <s xml:id="echoid-s8297" xml:space="preserve">ad 2017 {65/223}. </s> <s xml:id="echoid-s8298" xml:space="preserve">(qui numerus maior eſt, <lb/>quam 2107 {1/4}) ſit tripla ſuperdecupartiens ſeptuageſimas primas. </s> <s xml:id="echoid-s8299" xml:space="preserve">Ergo ambi-<lb/>tus Polygoni maiorem habet proportionem ad diametrum, quam 3 {10/71}. <lb/></s> <s xml:id="echoid-s8300" xml:space="preserve">Cumigitur circumferentia circuliſit maior ambitu Polygoni, <anchor type="note" xlink:href="" symbol="c"/> habebit circum- <anchor type="note" xlink:label="note-221-03a" xlink:href="note-221-03"/> ferentia ad diametrum multo maiorem proportionem, quam 3 {10/71}. </s> <s xml:id="echoid-s8301" xml:space="preserve">Ac proin-<lb/>de circumferentia diametrum continebit ter, & </s> <s xml:id="echoid-s8302" xml:space="preserve">inſuper partem maiorem, <lb/>quam {10/75}. </s> <s xml:id="echoid-s8303" xml:space="preserve">Nam ſi contineret {10/71}. </s> <s xml:id="echoid-s8304" xml:space="preserve">haberet circumferentia ad diametrum propor-<lb/>tionem 3 {10/71}. </s> <s xml:id="echoid-s8305" xml:space="preserve">Si autem contineret maiorem partem, quam {10/71}. </s> <s xml:id="echoid-s8306" xml:space="preserve">haberet propor-<lb/>tionemminorem, quam 3 {10/71}. </s> <s xml:id="echoid-s8307" xml:space="preserve">cumtamen maiorem habere demonſtratum ſit. <lb/></s> <s xml:id="echoid-s8308" xml:space="preserve">Et quoniam {1/8}. </s> <s xml:id="echoid-s8309" xml:space="preserve">minor eſt, quam {10/71}. </s> <s xml:id="echoid-s8310" xml:space="preserve">liquet ex hac ſecunda parte propoſitionis, <lb/>circumferentiam continere diametrum ter, & </s> <s xml:id="echoid-s8311" xml:space="preserve">plus, quam {1/8}. </s> <s xml:id="echoid-s8312" xml:space="preserve">diametri. </s> <s xml:id="echoid-s8313" xml:space="preserve"><lb/>Cum ergo ex prima parte conſtet, circumferentiam continere diametrum ter, <lb/>& </s> <s xml:id="echoid-s8314" xml:space="preserve">minus, quam {1/7}. </s> <s xml:id="echoid-s8315" xml:space="preserve">diametri: </s> <s xml:id="echoid-s8316" xml:space="preserve">perſpicuum eſt, veram proportionem circumfe-<lb/>rentiæ ad diametrum conſiſtere inter triplam ſeſquiſeptimam, & </s> <s xml:id="echoid-s8317" xml:space="preserve">triplam ſeſqui-<lb/>octauam.</s> <s xml:id="echoid-s8318" xml:space="preserve"/> </p> <div xml:id="echoid-div502" type="float" level="2" n="12"> <note symbol="k" position="left" xlink:label="note-220-10" xlink:href="note-220-10a" xml:space="preserve">26. tertij.</note> <figure xlink:label="fig-220-01" xlink:href="fig-220-01a"> <image file="220-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/220-01"/> </figure> <note symbol="a" position="right" xlink:label="note-221-01" xlink:href="note-221-01a" xml:space="preserve">8. quinti.</note> <note symbol="b" position="right" xlink:label="note-221-02" xlink:href="note-221-02a" xml:space="preserve">8. quinti.</note> <note symbol="c" position="right" xlink:label="note-221-03" xlink:href="note-221-03a" xml:space="preserve">8. quinti.</note> </div> </div> <div xml:id="echoid-div504" type="section" level="1" n="180"> <head xml:id="echoid-head187" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s8319" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi diameter per 3 {1/7}. </s> <s xml:id="echoid-s8320" xml:space="preserve">multiplicetur, gignetur numerus maior, quam <lb/>circumferentia: </s> <s xml:id="echoid-s8321" xml:space="preserve">ſi verò multiplicetur per 3 {10/71}. </s> <s xml:id="echoid-s8322" xml:space="preserve">pro creabitur minor numerus, <lb/>quam circumferentia. </s> <s xml:id="echoid-s8323" xml:space="preserve">Econtrario, ſi circumferentia diuidatur per 3 {1/7}. </s> <s xml:id="echoid-s8324" xml:space="preserve">produ-<lb/>cetur numerus minor, quam diameter: </s> <s xml:id="echoid-s8325" xml:space="preserve">ſi vero diuidatur per 3 {10/71}. </s> <s xml:id="echoid-s8326" xml:space="preserve">prodibit nume-<lb/>rus maior, quam diameter.</s> <s xml:id="echoid-s8327" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div505" type="section" level="1" n="181"> <head xml:id="echoid-head188" xml:space="preserve">PROPOSITIO III.</head> <p> <s xml:id="echoid-s8328" xml:space="preserve">CIRCVLVS quilibet ad quadratum diametri proportionem habet, <lb/>quam ad 11. </s> <s xml:id="echoid-s8329" xml:space="preserve">ad 14. </s> <s xml:id="echoid-s8330" xml:space="preserve">proximé.</s> <s xml:id="echoid-s8331" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8332" xml:space="preserve"><emph style="sc">Sit</emph> circulus A B C D, & </s> <s xml:id="echoid-s8333" xml:space="preserve">quadratum diametri B D, circulo circumſcri-<lb/> <anchor type="figure" xlink:label="fig-221-01a" xlink:href="fig-221-01"/> ptum E G H I. </s> <s xml:id="echoid-s8334" xml:space="preserve">Dico circulum ad quadra-<lb/>tum eſſe, vt 11. </s> <s xml:id="echoid-s8335" xml:space="preserve">ad 14. </s> <s xml:id="echoid-s8336" xml:space="preserve">proximè. </s> <s xml:id="echoid-s8337" xml:space="preserve">Pro-<lb/>ducto enim latere E G, ſumatur E K, ip-<lb/>ſius E G, tripla, & </s> <s xml:id="echoid-s8338" xml:space="preserve">K F, ſeptima pars dia-<lb/>metri B D, vel lateris E G; </s> <s xml:id="echoid-s8339" xml:space="preserve">iunganturque <pb o="192" file="222" n="222" rhead="GEOMETR. PRACT."/> rectę B G, BF. </s> <s xml:id="echoid-s8340" xml:space="preserve">Quoniamigitur E F. </s> <s xml:id="echoid-s8341" xml:space="preserve">ad diametrum E G, proportionem habet <lb/>triplam ſeſquiſeptimam, ex conſtructione; </s> <s xml:id="echoid-s8342" xml:space="preserve">erit per pręcedentem E F, circum-<lb/>ferentiæ circuli fermè æqualis. </s> <s xml:id="echoid-s8343" xml:space="preserve">Cum ergo BE, ęqualis ſit ſemidiametro: </s> <s xml:id="echoid-s8344" xml:space="preserve">erit per 1. <lb/></s> <s xml:id="echoid-s8345" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s8346" xml:space="preserve">triangulum BEF, circulo æquale proximè: </s> <s xml:id="echoid-s8347" xml:space="preserve">Triangulum autem B E G, <lb/>quarta pars erit quadrati E H. </s> <s xml:id="echoid-s8348" xml:space="preserve">Quia verò poſito latere E G, 7. </s> <s xml:id="echoid-s8349" xml:space="preserve">recta E F, eſt 22. </s> <s xml:id="echoid-s8350" xml:space="preserve"><lb/> <anchor type="note" xlink:href="" symbol="a"/> erit triangulum BEF, hoc eſt, circulus ABCD, ad triangulum BEG, vt 22. </s> <s xml:id="echoid-s8351" xml:space="preserve">ad 7.</s> <s xml:id="echoid-s8352" xml:space="preserve"> <anchor type="note" xlink:label="note-222-01a" xlink:href="note-222-01"/> Sed poſito triangulo B E G, 7. </s> <s xml:id="echoid-s8353" xml:space="preserve">quadratum EGHI, ipſius quadruplum, eſt 28. <lb/></s> <s xml:id="echoid-s8354" xml:space="preserve">Igitur circulus ad quadratum, eſt fermè, vt 22. </s> <s xml:id="echoid-s8355" xml:space="preserve">ad 28. </s> <s xml:id="echoid-s8356" xml:space="preserve">hoc eſt, vt 11. </s> <s xml:id="echoid-s8357" xml:space="preserve">ad 14. </s> <s xml:id="echoid-s8358" xml:space="preserve">quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s8359" xml:space="preserve"/> </p> <div xml:id="echoid-div505" type="float" level="2" n="1"> <figure xlink:label="fig-221-01" xlink:href="fig-221-01a"> <image file="221-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/221-01"/> </figure> <note symbol="a" position="left" xlink:label="note-222-01" xlink:href="note-222-01a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div507" type="section" level="1" n="182"> <head xml:id="echoid-head189" xml:space="preserve">DE AREA CIRCVLI, INVENTIONE-<lb/>que circumferentiæ ex diametro, & diametri <lb/>ex circumfetentia.</head> <head xml:id="echoid-head190" xml:space="preserve"><emph style="sc">Capvt</emph> VII.</head> <p> <s xml:id="echoid-s8360" xml:space="preserve">1. </s> <s xml:id="echoid-s8361" xml:space="preserve"><emph style="sc">QVoniam</emph> triangulum rectangulum, cuius vnum latus circa angu-<lb/>lumrectum ſemidiametro circuli, & </s> <s xml:id="echoid-s8362" xml:space="preserve">alterum peripheriæ eiuſdem æ-<lb/>quale eſt, <anchor type="note" xlink:href="" symbol="b"/> areæ circuli adæquatur: </s> <s xml:id="echoid-s8363" xml:space="preserve">huius autem trianguli area ex <anchor type="note" xlink:label="note-222-02a" xlink:href="note-222-02"/> ductu perpendicularis in ſemiſlem baſis producitur, vt cap. </s> <s xml:id="echoid-s8364" xml:space="preserve">2. </s> <s xml:id="echoid-s8365" xml:space="preserve">Num. </s> <s xml:id="echoid-s8366" xml:space="preserve">2. </s> <s xml:id="echoid-s8367" xml:space="preserve">huius li-<lb/>bri ſcripſimus: </s> <s xml:id="echoid-s8368" xml:space="preserve">Fit vt area circuli producatur ex multiplicatione ſemidiam{et}ri in <lb/> <anchor type="note" xlink:label="note-222-03a" xlink:href="note-222-03"/> <anchor type="figure" xlink:label="fig-222-01a" xlink:href="fig-222-01"/> ſemiſſem peripheriæ: </s> <s xml:id="echoid-s8369" xml:space="preserve">(ſi nimirum bæſis illi{us} trianguli <lb/>ſtatuatur lat{us}, quod peripheriæ æquale eſt) Vel ex du-<lb/>ctutoti{us} peripheriæ in ſemiſſem ſemidiam{et}ri, hoc est, <lb/>in quartam partem diam{et}ri: </s> <s xml:id="echoid-s8370" xml:space="preserve">ſumendo videlicet in eo-<lb/>dem triangulo pro baſe lat{us}, quod ſemidiam{et}ro est æ. <lb/></s> <s xml:id="echoid-s8371" xml:space="preserve">quale.) </s> <s xml:id="echoid-s8372" xml:space="preserve">Vel denique ex ductu toti{us} diam{et}ri in quartam peripheriæ partem, quod ita <lb/>perſpicuum faciemus.</s> <s xml:id="echoid-s8373" xml:space="preserve"/> </p> <div xml:id="echoid-div507" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-222-02" xlink:href="note-222-02a" xml:space="preserve">1. de Dimẽſ. <lb/>circuli.</note> <note position="left" xlink:label="note-222-03" xlink:href="note-222-03a" xml:space="preserve">Area circuli <lb/>trib. viis, ex <lb/>cognita dia-<lb/>metro, & cir-<lb/>cumferentia.</note> <figure xlink:label="fig-222-01" xlink:href="fig-222-01a"> <image file="222-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/222-01"/> </figure> </div> <p> <s xml:id="echoid-s8374" xml:space="preserve"><emph style="sc">Repetatvr</emph> figura pręcedentis propoſitionis, diuidaturque EF, quę pe-<lb/>ripheriæ circuli eſt æqualis, bifariam in L, ita vt EL, ſemiperipherię ſit æqualis: <lb/></s> <s xml:id="echoid-s8375" xml:space="preserve">Item EL, bifariam ſecetur in M, vt EM, æqualis ſit quartę parti peripherię. </s> <s xml:id="echoid-s8376" xml:space="preserve">Et <lb/>tandem BE, bifariam quo que ſecetur in N, vt EN, ſemiſsis ſit ſemidiametri BE, <lb/>hoc eſt, quarta pars totius diametri. </s> <s xml:id="echoid-s8377" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Et quia triangulum BEF, æquale eſt cir- <anchor type="note" xlink:label="note-222-04a" xlink:href="note-222-04"/> culo ABCD; </s> <s xml:id="echoid-s8378" xml:space="preserve">erit quo que rectangulũ ſub ſemidiametro BE, & </s> <s xml:id="echoid-s8379" xml:space="preserve">ſemiperip heria <lb/>EL, comprehenſum (quod per propoſitionem 1. </s> <s xml:id="echoid-s8380" xml:space="preserve">lib. </s> <s xml:id="echoid-s8381" xml:space="preserve">7. </s> <s xml:id="echoid-s8382" xml:space="preserve">huius, triangulo ęquale <lb/>eſt.) </s> <s xml:id="echoid-s8383" xml:space="preserve">eidem circulo ęquale; </s> <s xml:id="echoid-s8384" xml:space="preserve">quod eſt primum.</s> <s xml:id="echoid-s8385" xml:space="preserve"/> </p> <div xml:id="echoid-div508" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-222-04" xlink:href="note-222-04a" xml:space="preserve">1. de Dimẽſ. <lb/>circuli.</note> </div> <p> <s xml:id="echoid-s8386" xml:space="preserve"><emph style="sc">Non</emph> aliter rectangulum comprehenſum ſub tota peripheria EF, & </s> <s xml:id="echoid-s8387" xml:space="preserve">EN, <lb/>quarta parte d@ametri (quod per propoſ. </s> <s xml:id="echoid-s8388" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s8389" xml:space="preserve">7. </s> <s xml:id="echoid-s8390" xml:space="preserve">huius, eidem triangulo æquale <lb/>eſt) eidem circulo erit æquale, quod eſt ſecundum.</s> <s xml:id="echoid-s8391" xml:space="preserve"/> </p> <pb o="193" file="223" n="223" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s8392" xml:space="preserve"><emph style="sc">Deniqve</emph> quia quatuor lineæ EI, EB, EL, EM, proportionales ſunt, quod <lb/> <anchor type="note" xlink:label="note-223-01a" xlink:href="note-223-01"/> tam priores duę, quam poſteriores duę habeant proportionem duplam; </s> <s xml:id="echoid-s8393" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit rectangulum ſub EI, diametro, & </s> <s xml:id="echoid-s8394" xml:space="preserve">EM, quarta parte perip heriæ comprehen-<lb/>ſum, rectangulo ſub EB, ſemidiametro, & </s> <s xml:id="echoid-s8395" xml:space="preserve">EL, ſemiperipheria comprehenſo <lb/>æquale. </s> <s xml:id="echoid-s8396" xml:space="preserve">Cum ergo hoc in prima parte oſtenſum ſit æquale circulo, erit quo que <lb/>illud eidem circulo æquale. </s> <s xml:id="echoid-s8397" xml:space="preserve">quod eſt tertium.</s> <s xml:id="echoid-s8398" xml:space="preserve"/> </p> <div xml:id="echoid-div509" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-223-01" xlink:href="note-223-01a" xml:space="preserve">16. ſexti.</note> </div> <p> <s xml:id="echoid-s8399" xml:space="preserve"><emph style="sc">Vt</emph> fra ctiones interdum vitentur, ducenda erit tota diameter in totam cir-<lb/> <anchor type="note" xlink:label="note-223-02a" xlink:href="note-223-02"/> cumferentiam. </s> <s xml:id="echoid-s8400" xml:space="preserve">Quarta enim pars numeri producti area erit circuli: </s> <s xml:id="echoid-s8401" xml:space="preserve">propterea <lb/>quod numerus productus quadruplus eſt numeri producti ex ſemidiametro in <lb/>ſemicircumferentiam, vt liquet.</s> <s xml:id="echoid-s8402" xml:space="preserve"/> </p> <div xml:id="echoid-div510" type="float" level="2" n="4"> <note position="right" xlink:label="note-223-02" xlink:href="note-223-02a" xml:space="preserve">Area ſemi-<lb/>circuli, Qua-<lb/>drantis, octa-<lb/>uæpartis, & c.</note> </div> <p> <s xml:id="echoid-s8403" xml:space="preserve"><emph style="sc">Seqvitvr</emph> ex prima parte, aream ſemicirculi produci ex ſemidiametro in <lb/>quartam partem circumferentię: </s> <s xml:id="echoid-s8404" xml:space="preserve">quia nimirum producitur ſemiſsis eius, quod <lb/>fit ex ſemidiametro in ſemiſſem peripheriæ. </s> <s xml:id="echoid-s8405" xml:space="preserve">item aream Quadrantis procreari <lb/>ex ſemidiametro in octauam partem circumferentię: </s> <s xml:id="echoid-s8406" xml:space="preserve">Et aream octauę partis ex <lb/>ſemidiametro in ſextamdecimam partem circumferentię: </s> <s xml:id="echoid-s8407" xml:space="preserve">Et aream ſextę de-<lb/>cimę partis ex ſemidiametro in {1/32}. </s> <s xml:id="echoid-s8408" xml:space="preserve">circumferentię, & </s> <s xml:id="echoid-s8409" xml:space="preserve">ſic deinceps: </s> <s xml:id="echoid-s8410" xml:space="preserve">quia ſem-<lb/>per producitur ſemiſsis pręcedentis producti; </s> <s xml:id="echoid-s8411" xml:space="preserve">quemadmodum & </s> <s xml:id="echoid-s8412" xml:space="preserve">partes cir-<lb/>culi ſemiſſes ſunt pręcedentium partium: </s> <s xml:id="echoid-s8413" xml:space="preserve">nimirum Quadrans ſemiſsis eſt ſe-<lb/>micirculi; </s> <s xml:id="echoid-s8414" xml:space="preserve">& </s> <s xml:id="echoid-s8415" xml:space="preserve">octaua pars ſemiſsis Quadrantis; </s> <s xml:id="echoid-s8416" xml:space="preserve">& </s> <s xml:id="echoid-s8417" xml:space="preserve">ſextadecima pars ſemiſsis o-<lb/>ctauę partis, & </s> <s xml:id="echoid-s8418" xml:space="preserve">c.</s> <s xml:id="echoid-s8419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8420" xml:space="preserve">1. </s> <s xml:id="echoid-s8421" xml:space="preserve"><emph style="sc">Igitvr</emph> vt area circuli reperiatur, neceſſe eſt tam eius diametrum, quam <lb/>circumferentiam eſſe cognitam. </s> <s xml:id="echoid-s8422" xml:space="preserve">Quare trademus hic regulas nonnullas, per <lb/>quas ex data diametro circumferentia tum maior, tum minor, quam vera, ex <lb/>propoſ. </s> <s xml:id="echoid-s8423" xml:space="preserve">2. </s> <s xml:id="echoid-s8424" xml:space="preserve">de Dimenſione circuli inueniatur. </s> <s xml:id="echoid-s8425" xml:space="preserve">Deinde alias regulas pręſcri-<lb/>bam, per quas area circulitum maior, tum minor, quam vera, vel ex ſola diame-<lb/>tro, vel ex ſola circumferentia cognita eruatur.</s> <s xml:id="echoid-s8426" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div512" type="section" level="1" n="183"> <head xml:id="echoid-head191" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s8427" xml:space="preserve">Ex data diametro circuli circumferentiam vera maiorem reperire.</s> <s xml:id="echoid-s8428" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8429" xml:space="preserve">FIAT vt 7 ad 22 ita data diameter, verbigratia, 28. </s> <s xml:id="echoid-s8430" xml:space="preserve">ad aliud, procreabitur que cir-<lb/>cumferentia 88. </s> <s xml:id="echoid-s8431" xml:space="preserve">maior quam vera. </s> <s xml:id="echoid-s8432" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam enim proportio circumferentiæ ad <anchor type="note" xlink:label="note-223-03a" xlink:href="note-223-03"/> diametrum minor eſt, quam tripla ſeſquiſeptima; </s> <s xml:id="echoid-s8433" xml:space="preserve">proportio autem 88. </s> <s xml:id="echoid-s8434" xml:space="preserve">ad 28. <lb/></s> <s xml:id="echoid-s8435" xml:space="preserve">eſt tripla ſeſquiſeptima, <anchor type="note" xlink:href="" symbol="c"/> eadem videlicet, quæ 22. </s> <s xml:id="echoid-s8436" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8437" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit numerus 88. </s> <s xml:id="echoid-s8438" xml:space="preserve">ma- <anchor type="note" xlink:label="note-223-04a" xlink:href="note-223-04"/> ior, quam circumferentia circuli, cuius diameter eſt 28.</s> <s xml:id="echoid-s8439" xml:space="preserve"/> </p> <div xml:id="echoid-div512" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-223-03" xlink:href="note-223-03a" xml:space="preserve">2. de Dimeſ. <lb/>circuli:</note> <note symbol="c" position="right" xlink:label="note-223-04" xlink:href="note-223-04a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div514" type="section" level="1" n="184"> <head xml:id="echoid-head192" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s8440" xml:space="preserve">EX data diametro circuli circumferentiam vera minorem elicere.</s> <s xml:id="echoid-s8441" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8442" xml:space="preserve">FIAT vt 71. </s> <s xml:id="echoid-s8443" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8444" xml:space="preserve">ita data diameter 28. </s> <s xml:id="echoid-s8445" xml:space="preserve">ad aliud, produc{et}urque circumferentia <lb/>87 {67/71}. </s> <s xml:id="echoid-s8446" xml:space="preserve">minor quam vera. </s> <s xml:id="echoid-s8447" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam enim proportio circumferentię ad diame- <anchor type="note" xlink:label="note-223-05a" xlink:href="note-223-05"/> trum maior eſt, quam tripla ſuperdecupartiens ſeptuageſimas primas; </s> <s xml:id="echoid-s8448" xml:space="preserve">propor-<lb/>tio autem 87 {67/71}. </s> <s xml:id="echoid-s8449" xml:space="preserve">ad 28. </s> <s xml:id="echoid-s8450" xml:space="preserve">eſt tripla ſuperdecupartiens ſeptuageſimas primas, ni-<lb/>mirum eadem, quæ 223. </s> <s xml:id="echoid-s8451" xml:space="preserve">ad 28. </s> <s xml:id="echoid-s8452" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit circumferentia circuli, cuius diameter 28.</s> <s xml:id="echoid-s8453" xml:space="preserve"> <anchor type="note" xlink:label="note-223-06a" xlink:href="note-223-06"/> maior, quam 87 {@/71}.</s> <s xml:id="echoid-s8454" xml:space="preserve"/> </p> <div xml:id="echoid-div514" type="float" level="2" n="1"> <note symbol="d" position="right" xlink:label="note-223-05" xlink:href="note-223-05a" xml:space="preserve">2. de Dimeſ. <lb/>circuli.</note> <note symbol="e" position="right" xlink:label="note-223-06" xlink:href="note-223-06a" xml:space="preserve">10. quinti.</note> </div> <pb o="194" file="224" n="224" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div516" type="section" level="1" n="185"> <head xml:id="echoid-head193" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s8455" xml:space="preserve">Ex data circuli circumferentia, diametrum vera maiorem indagare.</s> <s xml:id="echoid-s8456" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s8457" xml:space="preserve">Fiat vt 223. </s> <s xml:id="echoid-s8458" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8459" xml:space="preserve">ita data circumferentia, verbigratia 88. </s> <s xml:id="echoid-s8460" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s8461" xml:space="preserve">Product{us} <lb/> <anchor type="note" xlink:label="note-224-01a" xlink:href="note-224-01"/> enim numer{us} 28 {4/223}. </s> <s xml:id="echoid-s8462" xml:space="preserve">dabit diametrum vera maiorem. </s> <s xml:id="echoid-s8463" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum enim circumferentiæ ad diam{et}rum habeat maiorem proportionem, quam triplam ſuperdecupartientem ſe-<lb/>ptuageſim{as} prim{as}, hoc eſt, maiorem quam 223. </s> <s xml:id="echoid-s8464" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8465" xml:space="preserve">habebit quoque data circumfe-<lb/>rentia 88. </s> <s xml:id="echoid-s8466" xml:space="preserve">ad ſuam diam{et}rum proportionem maiorem, quam ad 28 {4/223}. </s> <s xml:id="echoid-s8467" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ac proinde diam{et}er circumferentiæ 88. </s> <s xml:id="echoid-s8468" xml:space="preserve">minor erit, quam 28 {4/223}:</s> <s xml:id="echoid-s8469" xml:space="preserve"/> </p> <div xml:id="echoid-div516" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-224-01" xlink:href="note-224-01a" xml:space="preserve">2. de Di-<lb/>menſ. circu-<lb/>li.</note> </div> <note symbol="b" position="left" xml:space="preserve">10. quinti.</note> </div> <div xml:id="echoid-div518" type="section" level="1" n="186"> <head xml:id="echoid-head194" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s8470" xml:space="preserve">Ex data circuli circumferentia diametrum inueſtigare vera minorem.</s> <s xml:id="echoid-s8471" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8472" xml:space="preserve">Fiat vt 22. </s> <s xml:id="echoid-s8473" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8474" xml:space="preserve">ita circumferentia data 88. </s> <s xml:id="echoid-s8475" xml:space="preserve">ad aliud. </s> <s xml:id="echoid-s8476" xml:space="preserve">Numer{us} enim procreat{us} <lb/>28. </s> <s xml:id="echoid-s8477" xml:space="preserve">offeret diam{et}rum vera maiorem. </s> <s xml:id="echoid-s8478" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Cum enim circumferentia ad diametrum <anchor type="note" xlink:label="note-224-03a" xlink:href="note-224-03"/> habeat minorem proportionem, quam triplam ſeſquiſeptimam, hoc eſt, quam <lb/>22. </s> <s xml:id="echoid-s8479" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8480" xml:space="preserve">habebit quo que circumferentia data 88. </s> <s xml:id="echoid-s8481" xml:space="preserve">ad ſuam diametrum propor-<lb/>tionem minorem, quam ad 28. </s> <s xml:id="echoid-s8482" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> atque idcirco diameter circumferentię 88. </s> <s xml:id="echoid-s8483" xml:space="preserve">ma- <anchor type="note" xlink:label="note-224-04a" xlink:href="note-224-04"/> ior erit, quam 28.</s> <s xml:id="echoid-s8484" xml:space="preserve"/> </p> <div xml:id="echoid-div518" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-224-03" xlink:href="note-224-03a" xml:space="preserve">2. de Dimẽſ. <lb/>circuli.</note> <note symbol="d" position="left" xlink:label="note-224-04" xlink:href="note-224-04a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s8485" xml:space="preserve">3. </s> <s xml:id="echoid-s8486" xml:space="preserve"><emph style="sc">Iam</emph> verò vt do ceamus, qua ratione area circuli vel ex ſola diametro cog-<lb/>nita, vel ex ſola circumferentia cognoſcatur, ita vt neceſſe non ſit ex diametro <lb/>circumferentiam, vel ex circumferentia diametrum inueſtigare, demonſtrandæ <lb/>prius erunt ſequentes tres propoſitiones, quarum primam deinde lib. </s> <s xml:id="echoid-s8487" xml:space="preserve">8. </s> <s xml:id="echoid-s8488" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s8489" xml:space="preserve">2. </s> <s xml:id="echoid-s8490" xml:space="preserve">ex Pappo Alexandrino aliter quo que demonſtrabimus.</s> <s xml:id="echoid-s8491" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div520" type="section" level="1" n="187"> <head xml:id="echoid-head195" xml:space="preserve">PROPOSITIO I.</head> <p> <s xml:id="echoid-s8492" xml:space="preserve">Circulorum diametri inter ſe ſunt, vt circumferentiæ.</s> <s xml:id="echoid-s8493" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8494" xml:space="preserve"><emph style="sc">Sint</emph> diametri duorum circulorum AB, DE, & </s> <s xml:id="echoid-s8495" xml:space="preserve">rectæ circumferentiis æqua-<lb/>les BC, EF, quę cum diametris angulos rectos efficiant B, E, compleantur que <lb/>triangula ABC, DEF. </s> <s xml:id="echoid-s8496" xml:space="preserve">Dico ita eſſe diametrum AB, ad diametrum DE, vt eſt cir-<lb/>cumferentia BC, ad circumferentiam EF. </s> <s xml:id="echoid-s8497" xml:space="preserve">Diametris enim AB, DE, inueniatur <lb/>tertia proportionalis G: </s> <s xml:id="echoid-s8498" xml:space="preserve">Et tribus rectis BC, EF, DE, quarta proportionalis <lb/>H. </s> <s xml:id="echoid-s8499" xml:space="preserve">Et quia continuè proportionales ſunt AB, DE, G; </s> <s xml:id="echoid-s8500" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit quadratum ex AB, <anchor type="note" xlink:label="note-224-05a" xlink:href="note-224-05"/> <anchor type="figure" xlink:label="fig-224-01a" xlink:href="fig-224-01"/> ad quadratum ex DE, vt AB, ad G. </s> <s xml:id="echoid-s8501" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Sed vt quadratum ex AB, ad quadratum <anchor type="note" xlink:label="note-224-06a" xlink:href="note-224-06"/> <pb o="195" file="225" n="225" rhead="LIBER QVARTVS."/> ex DE, ita eſt circulus diametri AB, ad circulum diametri DE. </s> <s xml:id="echoid-s8502" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et vt circulus <anchor type="note" xlink:label="note-225-01a" xlink:href="note-225-01"/> ad circulum, ita eſt triangulum ABC, ad triangulum DEF, quòd hęc triangula <lb/>circulorum ſint dupla. </s> <s xml:id="echoid-s8503" xml:space="preserve">(Nam diuiſa diametro AB, bifariam in I, ducta que recta <lb/>IC, b erit triangulũ IBC, circulo æquale: </s> <s xml:id="echoid-s8504" xml:space="preserve">ac proinde cum <anchor type="note" xlink:href="" symbol="b"/> triangula AIC, IBC, <anchor type="note" xlink:label="note-225-02a" xlink:href="note-225-02"/> æqualia ſint; </s> <s xml:id="echoid-s8505" xml:space="preserve">erit triangulum ABC, duplum trianguli IBC, hoc eſt, circuli, cu-<lb/>ius diameter AB. </s> <s xml:id="echoid-s8506" xml:space="preserve">Eademque ratione triangulum DEF, duplum erit circuli, cu-<lb/> <anchor type="note" xlink:label="note-225-03a" xlink:href="note-225-03"/> ius diameter DE.) </s> <s xml:id="echoid-s8507" xml:space="preserve">Igitur erit quo que triangulum A B C, ad triangulum D E F, <lb/>vt AB, ad G. </s> <s xml:id="echoid-s8508" xml:space="preserve">At vt triangulum ABC, ad triangulum DEF, ita eſt A B, ad H. <lb/></s> <s xml:id="echoid-s8509" xml:space="preserve">quod vtraque proportio compoſita ſit ex iiſdem proportionibus. </s> <s xml:id="echoid-s8510" xml:space="preserve">(<anchor type="note" xlink:href="" symbol="d"/> Nam <anchor type="note" xlink:label="note-225-04a" xlink:href="note-225-04"/> proportio trianguli A B C, ad triangulum DEF, compoſita eſt ex proportione <lb/>baſis AB, ad baſem D E, & </s> <s xml:id="echoid-s8511" xml:space="preserve">ex proportione altitudinis BC, ad altitudinem EF, <lb/>hoc eſt, ex proportione DE, ad H, quæ ex conſtru ctione eadem eſt, quæ BC, ad <lb/>EF. </s> <s xml:id="echoid-s8512" xml:space="preserve">Proportio autem AB, ad H, componitur quo que ex proportionibus A B, <lb/>ad DE, & </s> <s xml:id="echoid-s8513" xml:space="preserve">DE, ad H, ex defin.) </s> <s xml:id="echoid-s8514" xml:space="preserve">Igitur erit, vt AB, ad G, ita AB, ad H. </s> <s xml:id="echoid-s8515" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ideo que <anchor type="note" xlink:label="note-225-05a" xlink:href="note-225-05"/> G, & </s> <s xml:id="echoid-s8516" xml:space="preserve">H, æquales erunt: </s> <s xml:id="echoid-s8517" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> ac proinde erit DE, ad G, vt DE, ad H. </s> <s xml:id="echoid-s8518" xml:space="preserve">Eſt autem per <anchor type="note" xlink:label="note-225-06a" xlink:href="note-225-06"/> conſtru ctionem AB, diameter ad diametrum DE, vt DE, ad G, hoc eſt, vt DE, <lb/>ad H. </s> <s xml:id="echoid-s8519" xml:space="preserve">Et vt DE, ad H, ita per conſtructionem, circumferentia BC, ad circumfe-<lb/>rentiam EF. </s> <s xml:id="echoid-s8520" xml:space="preserve">Igitur erit quo que diameter AB, ad diametrum DE, vt circumfe-<lb/>rentia BC, ad circumferentiam EF. </s> <s xml:id="echoid-s8521" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s8522" xml:space="preserve"/> </p> <div xml:id="echoid-div520" type="float" level="2" n="1"> <note symbol="e" position="left" xlink:label="note-224-05" xlink:href="note-224-05a" xml:space="preserve">20. coroll. <lb/>ſexti.</note> <figure xlink:label="fig-224-01" xlink:href="fig-224-01a"> <image file="224-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/224-01"/> </figure> <note symbol="f" position="left" xlink:label="note-224-06" xlink:href="note-224-06a" xml:space="preserve">2. duodec.</note> <note symbol="a" position="right" xlink:label="note-225-01" xlink:href="note-225-01a" xml:space="preserve">15. quinti.</note> <note symbol="b" position="right" xlink:label="note-225-02" xlink:href="note-225-02a" xml:space="preserve">1. de Dimẽſ. <lb/>circuli.</note> <note symbol="c" position="right" xlink:label="note-225-03" xlink:href="note-225-03a" xml:space="preserve">38. primi.</note> <note symbol="d" position="right" xlink:label="note-225-04" xlink:href="note-225-04a" xml:space="preserve">ſchol. 23. <lb/>ſexti.</note> <note symbol="e" position="right" xlink:label="note-225-05" xlink:href="note-225-05a" xml:space="preserve">9. quinti.</note> <note symbol="f" position="right" xlink:label="note-225-06" xlink:href="note-225-06a" xml:space="preserve">7. quinti.</note> </div> </div> <div xml:id="echoid-div522" type="section" level="1" n="188"> <head xml:id="echoid-head196" xml:space="preserve">PROPOSITIO II.</head> <p> <s xml:id="echoid-s8523" xml:space="preserve">PROPORTIO quadrati ex diametro cuiuslibet circuli deſcripti ad <lb/>circuli aream maior eſt, quam 14. </s> <s xml:id="echoid-s8524" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8525" xml:space="preserve">minor autem, quam 284. <lb/></s> <s xml:id="echoid-s8526" xml:space="preserve">ad 223.</s> <s xml:id="echoid-s8527" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8528" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> <emph style="sc">Qvoniam</emph> enim quadratum diametri cuiuſuis circuli ad quadratum <anchor type="note" xlink:label="note-225-07a" xlink:href="note-225-07"/> diametri alterius circuli eſt, vt circulus ad circulum: </s> <s xml:id="echoid-s8529" xml:space="preserve">erit permutando quadra-<lb/>tum diametri ad circulum eiuſdem diametri, vt qua dratum alterius diametriad <lb/>circulum eiuſdem diametri. </s> <s xml:id="echoid-s8530" xml:space="preserve">Poſita autem diametro alicuius circuli 1. </s> <s xml:id="echoid-s8531" xml:space="preserve">propor-<lb/>tio quadrati ipſius ad circulum maior eſt, quam 14. </s> <s xml:id="echoid-s8532" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8533" xml:space="preserve">minorautem, quam <lb/>284. </s> <s xml:id="echoid-s8534" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8535" xml:space="preserve">Igitur proportio quadrati diametri cuiuſuis alterius circuli ad <lb/>ipſum circulum, maior quo que erit quam 14. </s> <s xml:id="echoid-s8536" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8537" xml:space="preserve">minor autem, quam 284. <lb/></s> <s xml:id="echoid-s8538" xml:space="preserve">ad 223.</s> <s xml:id="echoid-s8539" xml:space="preserve"/> </p> <div xml:id="echoid-div522" type="float" level="2" n="1"> <note symbol="g" position="right" xlink:label="note-225-07" xlink:href="note-225-07a" xml:space="preserve">2. duodec.</note> </div> <p> <s xml:id="echoid-s8540" xml:space="preserve"><emph style="sc">Qvod</emph> autem proportio quadrati diametri 1. </s> <s xml:id="echoid-s8541" xml:space="preserve">ad ſuum circulum maior ſit, <lb/>quam 14. </s> <s xml:id="echoid-s8542" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8543" xml:space="preserve">minor verò, quàm 284. </s> <s xml:id="echoid-s8544" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8545" xml:space="preserve">ita perſpicuum fiet. </s> <s xml:id="echoid-s8546" xml:space="preserve">Sifiat vt 7. </s> <s xml:id="echoid-s8547" xml:space="preserve">ad <lb/>22. </s> <s xml:id="echoid-s8548" xml:space="preserve">ita diameter 1. </s> <s xml:id="echoid-s8549" xml:space="preserve">ad aliud, prodibit ex regula prima Num. </s> <s xml:id="echoid-s8550" xml:space="preserve">2. </s> <s xml:id="echoid-s8551" xml:space="preserve">circumferentia 3 {1/7}. <lb/></s> <s xml:id="echoid-s8552" xml:space="preserve">vel {22/7}. </s> <s xml:id="echoid-s8553" xml:space="preserve">maior, quam vera Igitur ex eius ſemiſſe {11/7}. </s> <s xml:id="echoid-s8554" xml:space="preserve">in {1/2}. </s> <s xml:id="echoid-s8555" xml:space="preserve">ſemidiametrum procrea-<lb/>bitur, vt Num. </s> <s xml:id="echoid-s8556" xml:space="preserve">1. </s> <s xml:id="echoid-s8557" xml:space="preserve">dictum eſt, area circuli {11/14}. </s> <s xml:id="echoid-s8558" xml:space="preserve">maior tamen, quam vera: </s> <s xml:id="echoid-s8559" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Ac proin- <anchor type="note" xlink:label="note-225-08a" xlink:href="note-225-08"/> de quadratum diametri 1. </s> <s xml:id="echoid-s8560" xml:space="preserve">quod eſt 1. </s> <s xml:id="echoid-s8561" xml:space="preserve">ad veram aream circuli, quæ minor eſt, <lb/>quam {11/14}. </s> <s xml:id="echoid-s8562" xml:space="preserve">maiorem proportionem habebit, quam ad {11/14}. </s> <s xml:id="echoid-s8563" xml:space="preserve">Cum ergo ſit 1. </s> <s xml:id="echoid-s8564" xml:space="preserve">ad {11/14}. </s> <s xml:id="echoid-s8565" xml:space="preserve">vt <lb/>14. </s> <s xml:id="echoid-s8566" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8567" xml:space="preserve">(Quoniamenim ex propoſitione 2. </s> <s xml:id="echoid-s8568" xml:space="preserve">Minutiarum ad finem lib. </s> <s xml:id="echoid-s8569" xml:space="preserve">9. </s> <s xml:id="echoid-s8570" xml:space="preserve">Eucl, <lb/>eadem proportio eſt numeratoris 11. </s> <s xml:id="echoid-s8571" xml:space="preserve">ad denominatorem 14. </s> <s xml:id="echoid-s8572" xml:space="preserve">quæ minutæ {11/14}. </s> <s xml:id="echoid-s8573" xml:space="preserve">ad <lb/>ſuum integrũ 1. </s> <s xml:id="echoid-s8574" xml:space="preserve">erit conuertendo, vt 14. </s> <s xml:id="echoid-s8575" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8576" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s8577" xml:space="preserve">ad {11/14}.) </s> <s xml:id="echoid-s8578" xml:space="preserve">habebit quoq; </s> <s xml:id="echoid-s8579" xml:space="preserve">quadra-<lb/>tum diametri 1. </s> <s xml:id="echoid-s8580" xml:space="preserve">ad aream circuli veram, minorem proportionem, quam 14. </s> <s xml:id="echoid-s8581" xml:space="preserve">ad 11. <lb/></s> <s xml:id="echoid-s8582" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s8583" xml:space="preserve">Conſtat ergo prima propoſitionis pars.</s> <s xml:id="echoid-s8584" xml:space="preserve"/> </p> <div xml:id="echoid-div523" type="float" level="2" n="2"> <note symbol="h" position="right" xlink:label="note-225-08" xlink:href="note-225-08a" xml:space="preserve">8. quinti.</note> </div> <pb o="196" file="226" n="226" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s8585" xml:space="preserve"><emph style="sc">Rvrsvs</emph> ſi fiat, vt 71. </s> <s xml:id="echoid-s8586" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8587" xml:space="preserve">ita diameter 1. </s> <s xml:id="echoid-s8588" xml:space="preserve">ad aliud, reperietur perregulam <lb/>2. </s> <s xml:id="echoid-s8589" xml:space="preserve">Num. </s> <s xml:id="echoid-s8590" xml:space="preserve">2. </s> <s xml:id="echoid-s8591" xml:space="preserve">circumſerentia circuli 3 {10/71}. </s> <s xml:id="echoid-s8592" xml:space="preserve">vel {223/71}. </s> <s xml:id="echoid-s8593" xml:space="preserve">minor, quam vera. </s> <s xml:id="echoid-s8594" xml:space="preserve">Igitur, vt <lb/>Num. </s> <s xml:id="echoid-s8595" xml:space="preserve">1. </s> <s xml:id="echoid-s8596" xml:space="preserve">dictum eſt ex eius ſemiſſe {223/142}. </s> <s xml:id="echoid-s8597" xml:space="preserve">in {1/2}. </s> <s xml:id="echoid-s8598" xml:space="preserve">ſemidiametrum producetur area <lb/>circuli {223/284}. </s> <s xml:id="echoid-s8599" xml:space="preserve">maior tamen, quam vera: </s> <s xml:id="echoid-s8600" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Ac proinde quadratum diametri 1.</s> <s xml:id="echoid-s8601" xml:space="preserve"> <anchor type="note" xlink:label="note-226-01a" xlink:href="note-226-01"/> quod eſt 1. </s> <s xml:id="echoid-s8602" xml:space="preserve">ad veram circuli aream, quę maior eſt, quam {223/284}. </s> <s xml:id="echoid-s8603" xml:space="preserve">minorem habebit <lb/>proportionem, quam ad {223/284}. </s> <s xml:id="echoid-s8604" xml:space="preserve">Cum ergo ſit 1. </s> <s xml:id="echoid-s8605" xml:space="preserve">ad {223/284}. </s> <s xml:id="echoid-s8606" xml:space="preserve">vt 284. </s> <s xml:id="echoid-s8607" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8608" xml:space="preserve">(Nam quia <lb/>ex propoſ. </s> <s xml:id="echoid-s8609" xml:space="preserve">2. </s> <s xml:id="echoid-s8610" xml:space="preserve">Minutiarum, eadem eſt proportio Numeratoris 223. </s> <s xml:id="echoid-s8611" xml:space="preserve">ad denomi-<lb/>natorem 284. </s> <s xml:id="echoid-s8612" xml:space="preserve">quę minutię {223/284}. </s> <s xml:id="echoid-s8613" xml:space="preserve">ad ſuum integrum 1. </s> <s xml:id="echoid-s8614" xml:space="preserve">erit conuertendo, vt 284. <lb/></s> <s xml:id="echoid-s8615" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8616" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s8617" xml:space="preserve">ad {223/284}.) </s> <s xml:id="echoid-s8618" xml:space="preserve">habebit quoque quadratum diametri 1. </s> <s xml:id="echoid-s8619" xml:space="preserve">ad aream veram <lb/>circuli minorem proportionem, quam 284. </s> <s xml:id="echoid-s8620" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8621" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s8622" xml:space="preserve">Con-<lb/>ſtat ergo ſecunda etiam pars propoſitionis.</s> <s xml:id="echoid-s8623" xml:space="preserve"/> </p> <div xml:id="echoid-div524" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-226-01" xlink:href="note-226-01a" xml:space="preserve">8. quinti.</note> </div> </div> <div xml:id="echoid-div526" type="section" level="1" n="189"> <head xml:id="echoid-head197" xml:space="preserve">PROPOSITIO III.</head> <p> <s xml:id="echoid-s8624" xml:space="preserve">PROPORTIO quadrati à circumferentia circuli cuiuſuis deſcripti <lb/>ad circuli aream maior eſt, quam 892. </s> <s xml:id="echoid-s8625" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8626" xml:space="preserve">minor autem, quam <lb/>88. </s> <s xml:id="echoid-s8627" xml:space="preserve">ad 7.</s> <s xml:id="echoid-s8628" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8629" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/><emph style="sc">Qvoniam</emph> enim circumferentia cuiuſuis circuli ad circumferentiam al- <anchor type="note" xlink:label="note-226-02a" xlink:href="note-226-02"/> terius circuli eſt, vt diameter ad diametrum: </s> <s xml:id="echoid-s8630" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit quoque quadratum circum- ferentiæ ad quadratum circumferentię, vt quadratum diametri, ad quadratum <lb/> <anchor type="note" xlink:label="note-226-03a" xlink:href="note-226-03"/> diametri. </s> <s xml:id="echoid-s8631" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Sed vt quadratrum diametri ad quadratum diametri, ita eſt circulus <anchor type="note" xlink:label="note-226-04a" xlink:href="note-226-04"/> ad circulum. </s> <s xml:id="echoid-s8632" xml:space="preserve">Igitur erit quadratum quo que circumferentię ad quadratum cir-<lb/>cumferentię; </s> <s xml:id="echoid-s8633" xml:space="preserve">vt circulus ad circulum: </s> <s xml:id="echoid-s8634" xml:space="preserve">Et permutando quadratum circumfe-<lb/>rentię, ad ſuum circulũ, vt quadratum alterius circumferentię ad ſuum circu-<lb/>lum. </s> <s xml:id="echoid-s8635" xml:space="preserve">Poſita autem circumferentia alicuius circuli 1. </s> <s xml:id="echoid-s8636" xml:space="preserve">proportio quadrati circum-<lb/>ferentię illius circuli ad circulum maior eſt, quam 892. </s> <s xml:id="echoid-s8637" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8638" xml:space="preserve">minor verò, quàm <lb/>88. </s> <s xml:id="echoid-s8639" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8640" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s8641" xml:space="preserve">proportio quadrati circumferentię cuiuslibet alterius circuli <lb/>ad ipſum circulum maior erit, quam 892. </s> <s xml:id="echoid-s8642" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8643" xml:space="preserve">minor autem, quam 88. </s> <s xml:id="echoid-s8644" xml:space="preserve">ad 7.</s> <s xml:id="echoid-s8645" xml:space="preserve"/> </p> <div xml:id="echoid-div526" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-226-02" xlink:href="note-226-02a" xml:space="preserve">1. hui{us} <lb/>Num. 3.</note> <note symbol="c" position="left" xlink:label="note-226-03" xlink:href="note-226-03a" xml:space="preserve">22. ſexti.</note> <note symbol="d" position="left" xlink:label="note-226-04" xlink:href="note-226-04a" xml:space="preserve">2. duodec.</note> </div> <p> <s xml:id="echoid-s8646" xml:space="preserve"><emph style="sc">Qvod</emph> autem proportio quadrati ex circumferentia 1. </s> <s xml:id="echoid-s8647" xml:space="preserve">deſcripti, ad ſuum <lb/>circulum maior ſit, quam 892. </s> <s xml:id="echoid-s8648" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8649" xml:space="preserve">minor verò, quam 88. </s> <s xml:id="echoid-s8650" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8651" xml:space="preserve">ſic demonſtra-<lb/>bimus. </s> <s xml:id="echoid-s8652" xml:space="preserve">Quoniã ſi fiat, vt 223. </s> <s xml:id="echoid-s8653" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8654" xml:space="preserve">ita data circumferentia 1. </s> <s xml:id="echoid-s8655" xml:space="preserve">ad aliud, diameter <lb/>procreatur {71/223}. </s> <s xml:id="echoid-s8656" xml:space="preserve">maior, quam vera, vt ex 3. </s> <s xml:id="echoid-s8657" xml:space="preserve">regula Num. </s> <s xml:id="echoid-s8658" xml:space="preserve">2. </s> <s xml:id="echoid-s8659" xml:space="preserve">conſtat. </s> <s xml:id="echoid-s8660" xml:space="preserve">fit vt {1/2}. </s> <s xml:id="echoid-s8661" xml:space="preserve">ſe-<lb/>miſsis circumferentię ducta in {71/446}. </s> <s xml:id="echoid-s8662" xml:space="preserve">ſemiſſem diametri inuentę producat aream <lb/>circuli {71/892}. </s> <s xml:id="echoid-s8663" xml:space="preserve">vera maiorem, vt Num. </s> <s xml:id="echoid-s8664" xml:space="preserve">1. </s> <s xml:id="echoid-s8665" xml:space="preserve">dictum eſt. </s> <s xml:id="echoid-s8666" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igitur quadratum circumfe- <anchor type="note" xlink:label="note-226-05a" xlink:href="note-226-05"/> rentię 1. </s> <s xml:id="echoid-s8667" xml:space="preserve">quod eſt 1. </s> <s xml:id="echoid-s8668" xml:space="preserve">ad veram aream circuli, quę minor eſt, quam {71/892}. </s> <s xml:id="echoid-s8669" xml:space="preserve">maiorem <lb/>proportionem habebit, quam ad {71/892}. </s> <s xml:id="echoid-s8670" xml:space="preserve">Vt autem 1. </s> <s xml:id="echoid-s8671" xml:space="preserve">ad {71/892}. </s> <s xml:id="echoid-s8672" xml:space="preserve">ita eſt, ex propoſ. </s> <s xml:id="echoid-s8673" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s8674" xml:space="preserve">minutiarum, & </s> <s xml:id="echoid-s8675" xml:space="preserve">conuertendo, 892. </s> <s xml:id="echoid-s8676" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8677" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s8678" xml:space="preserve">quadratum circumferentiæ <lb/>1. </s> <s xml:id="echoid-s8679" xml:space="preserve">ad veram circuli aream maiorem proportionem habebit, quam 892. </s> <s xml:id="echoid-s8680" xml:space="preserve">ad 71 <lb/>quod eſt propoſitum. </s> <s xml:id="echoid-s8681" xml:space="preserve">Vera ergo eſt prior propoſitio nis pars.</s> <s xml:id="echoid-s8682" xml:space="preserve"/> </p> <div xml:id="echoid-div527" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-226-05" xlink:href="note-226-05a" xml:space="preserve">8. quinti.</note> </div> <p> <s xml:id="echoid-s8683" xml:space="preserve"><emph style="sc">Rvrsvs</emph>, quia ſi fiat, vt 22. </s> <s xml:id="echoid-s8684" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8685" xml:space="preserve">ita data circumferentia 1. </s> <s xml:id="echoid-s8686" xml:space="preserve">ad aliud, diame-<lb/>ter producitur {7/22}. </s> <s xml:id="echoid-s8687" xml:space="preserve">minor quam vera: </s> <s xml:id="echoid-s8688" xml:space="preserve">fit vt {1/2}. </s> <s xml:id="echoid-s8689" xml:space="preserve">ſemiſsis circumferentię ducta in <lb/>{7/44}. </s> <s xml:id="echoid-s8690" xml:space="preserve">ſemiſſem diametri inuentæ producat aream circuli {7/88}. </s> <s xml:id="echoid-s8691" xml:space="preserve">vera minorem, vt ex <lb/>iis, conſtat, quæ Num. </s> <s xml:id="echoid-s8692" xml:space="preserve">1. </s> <s xml:id="echoid-s8693" xml:space="preserve">diximus. </s> <s xml:id="echoid-s8694" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Igitur quadratum circumferentię 1. </s> <s xml:id="echoid-s8695" xml:space="preserve">quod eſt <anchor type="note" xlink:label="note-226-06a" xlink:href="note-226-06"/> 1. </s> <s xml:id="echoid-s8696" xml:space="preserve">ad aream veram circuli, quæ maior eſt, quam {7/88}. </s> <s xml:id="echoid-s8697" xml:space="preserve">minorem habebit proportio-<lb/>nem, quam ad {7/88}. </s> <s xml:id="echoid-s8698" xml:space="preserve">Vtautem 1. </s> <s xml:id="echoid-s8699" xml:space="preserve">ad {7/88}. </s> <s xml:id="echoid-s8700" xml:space="preserve">ita eſt, ex propoſitione 2. </s> <s xml:id="echoid-s8701" xml:space="preserve">Minutiarum, <pb o="197" file="227" n="227" rhead="LIBER QVARTVS."/> & </s> <s xml:id="echoid-s8702" xml:space="preserve">conuertendo, 88. </s> <s xml:id="echoid-s8703" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8704" xml:space="preserve">Ergo etiam quadratum circumferentiæ 1. </s> <s xml:id="echoid-s8705" xml:space="preserve">ad aream cir-<lb/>culi minorem proportionem habebit, quam 88. </s> <s xml:id="echoid-s8706" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8707" xml:space="preserve">quod eſt propoſitum. </s> <s xml:id="echoid-s8708" xml:space="preserve">Ve-<lb/>ra igitur etiam eſt poſterior pars propoſitio nis.</s> <s xml:id="echoid-s8709" xml:space="preserve"/> </p> <div xml:id="echoid-div528" type="float" level="2" n="3"> <note symbol="f" position="left" xlink:label="note-226-06" xlink:href="note-226-06a" xml:space="preserve">8. quinti.</note> </div> <p> <s xml:id="echoid-s8710" xml:space="preserve">4. </s> <s xml:id="echoid-s8711" xml:space="preserve"><emph style="sc">His</emph> ita demonſtratis, ſequunturiam quatuor regulæ, per quas aream cir-<lb/>culi propoſiti ſiue maiorem, ſiue minorem vera, vel ex ſola diametro, vel ex ſo-<lb/>la circumferentia cognita conijcere licebit.</s> <s xml:id="echoid-s8712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div530" type="section" level="1" n="190"> <head xml:id="echoid-head198" xml:space="preserve">I.</head> <head xml:id="echoid-head199" xml:space="preserve">EX diametro aream circuli vera maiorem inueſtigare.</head> <p> <s xml:id="echoid-s8713" xml:space="preserve">Fiat vt 14. </s> <s xml:id="echoid-s8714" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8715" xml:space="preserve">ita quadratum datæ diametri ad aliud. </s> <s xml:id="echoid-s8716" xml:space="preserve">Product{us} enim numer{us} <lb/>dabit aream circuli veramaiorem. </s> <s xml:id="echoid-s8717" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum enim maior ſit proportio quadrati dia- <anchor type="note" xlink:label="note-227-01a" xlink:href="note-227-01"/> metri ad aream circuli, quam 14. </s> <s xml:id="echoid-s8718" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8719" xml:space="preserve">Sit autem quadratum diametri datæ ad a-<lb/>ream inuentam, vt 14. </s> <s xml:id="echoid-s8720" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s8721" xml:space="preserve">erit quo que maior proportio quadrati diametri da-<lb/> <anchor type="note" xlink:label="note-227-02a" xlink:href="note-227-02"/> tæ ad veram aream circuli, quam ad aream inuentam. </s> <s xml:id="echoid-s8722" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Ac proinde vera circuli area erit minor, quaminuenta: </s> <s xml:id="echoid-s8723" xml:space="preserve">hoc eſt, area inuenta maior erit, quam vera.</s> <s xml:id="echoid-s8724" xml:space="preserve"/> </p> <div xml:id="echoid-div530" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-227-01" xlink:href="note-227-01a" xml:space="preserve">2. Num. 3.</note> <note symbol="b" position="right" xlink:label="note-227-02" xlink:href="note-227-02a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div532" type="section" level="1" n="191"> <head xml:id="echoid-head200" xml:space="preserve">II.</head> <head xml:id="echoid-head201" xml:space="preserve">EX diametro aream circuli vera minorem inueſtigare.</head> <p> <s xml:id="echoid-s8725" xml:space="preserve">Fiat vt 284. </s> <s xml:id="echoid-s8726" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8727" xml:space="preserve">ita quadratum datæ diametri ad aliud.</s> <s xml:id="echoid-s8728" xml:space="preserve">’ Numer{us} enim procrea-<lb/>t{us} aream circuli vera minorem offeret. </s> <s xml:id="echoid-s8729" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Cum enim minor ſit proportio quadrati <anchor type="note" xlink:label="note-227-03a" xlink:href="note-227-03"/> diametri datæ ad aream circuli, quam 284. </s> <s xml:id="echoid-s8730" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8731" xml:space="preserve">ſit autem quadratum diametri <lb/>datæ ad aream inuentam, vt 284. </s> <s xml:id="echoid-s8732" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s8733" xml:space="preserve">erit quo que minor proportio quadrati <lb/> <anchor type="note" xlink:label="note-227-04a" xlink:href="note-227-04"/> diametri datæ ad veram aream circuli, quam ad areã inuentam: </s> <s xml:id="echoid-s8734" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Atque idcirco vera area circuli maior erit, quaminuenta, hoc eſt, inuenta area erit minor, quam <lb/>vera.</s> <s xml:id="echoid-s8735" xml:space="preserve"/> </p> <div xml:id="echoid-div532" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-227-03" xlink:href="note-227-03a" xml:space="preserve">2. Num. 3.</note> <note symbol="d" position="right" xlink:label="note-227-04" xlink:href="note-227-04a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div534" type="section" level="1" n="192"> <head xml:id="echoid-head202" xml:space="preserve">III.</head> <head xml:id="echoid-head203" xml:space="preserve">EX circumferentia aream circuli vera maiorem colligere.</head> <p> <s xml:id="echoid-s8736" xml:space="preserve">Fiat vt 892. </s> <s xml:id="echoid-s8737" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8738" xml:space="preserve">ita quadratum datæ circumferentiæ ad aliud. </s> <s xml:id="echoid-s8739" xml:space="preserve">Procreat{us} nam-<lb/>que numer{us} aream circuli vera maiorem indicabit. </s> <s xml:id="echoid-s8740" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Cum enim maior ſit propor- <anchor type="note" xlink:label="note-227-05a" xlink:href="note-227-05"/> tio quadrati circumferentiæ ad aream circuli, quam 892. </s> <s xml:id="echoid-s8741" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8742" xml:space="preserve">Sit autem quadra-<lb/>tnm datæ circumferentię ad aream inuentam vt 892. </s> <s xml:id="echoid-s8743" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s8744" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s8745" xml:space="preserve">maior pro-<lb/>portio quadrati datæ circumferentiæ ad veram aream circuli, quam ad aream in-<lb/>uentam: </s> <s xml:id="echoid-s8746" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> ideo que vera circuli area minor erit, quam inuenta; </s> <s xml:id="echoid-s8747" xml:space="preserve">hoc eſt, inuenta <anchor type="note" xlink:label="note-227-06a" xlink:href="note-227-06"/> area maior erit, quam vera.</s> <s xml:id="echoid-s8748" xml:space="preserve"/> </p> <div xml:id="echoid-div534" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-227-05" xlink:href="note-227-05a" xml:space="preserve">3. Num. 3.</note> <note symbol="f" position="right" xlink:label="note-227-06" xlink:href="note-227-06a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div536" type="section" level="1" n="193"> <head xml:id="echoid-head204" xml:space="preserve">IV.</head> <head xml:id="echoid-head205" xml:space="preserve">EX circumferentia aream circuli vera minorem concludere.</head> <p> <s xml:id="echoid-s8749" xml:space="preserve">Fiat vt 88. </s> <s xml:id="echoid-s8750" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8751" xml:space="preserve">ita quadratum circumferentiæ datæ ad aliud. </s> <s xml:id="echoid-s8752" xml:space="preserve">Numer{us} namque, <pb o="198" file="228" n="228" rhead="GEOMETR. PRACT."/> qui gignitur, erit area circuli minor, quam vera. </s> <s xml:id="echoid-s8753" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum enim minor ſit proportio <anchor type="note" xlink:label="note-228-01a" xlink:href="note-228-01"/> quadrati circumferentiæ datæ ad aream circuli, quam 88. </s> <s xml:id="echoid-s8754" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8755" xml:space="preserve">ſit autem quadra-<lb/>tum circumferentiæ datæ ad aream inuentam, vt 88. </s> <s xml:id="echoid-s8756" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s8757" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s8758" xml:space="preserve">minor pro-<lb/>portio quadrati datæ circumferentiæ ad veram aream circuli, quam ad areamin-<lb/>uentam: </s> <s xml:id="echoid-s8759" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Ac proinde area circulivera erit maior, quam inuenta: </s> <s xml:id="echoid-s8760" xml:space="preserve">hoc eſt, area <anchor type="note" xlink:label="note-228-02a" xlink:href="note-228-02"/> inuenta minor erit, quam vera.</s> <s xml:id="echoid-s8761" xml:space="preserve"/> </p> <div xml:id="echoid-div536" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-228-01" xlink:href="note-228-01a" xml:space="preserve">3. Num. 3.</note> <note symbol="b" position="left" xlink:label="note-228-02" xlink:href="note-228-02a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s8762" xml:space="preserve">5. </s> <s xml:id="echoid-s8763" xml:space="preserve"><emph style="sc">Omnes</emph> hæviæ, quibus area circuliin quiritur, pendent ex proportione <lb/>circumſerentiæ circuliad diametrum, quam Archimedes inuenit eſſe quidẽ mi-<lb/>norem tripla ſeſquiſeptima, maiorem verò tripla ſuperdecupartiente ſeptuage-<lb/>ſimas primas: </s> <s xml:id="echoid-s8764" xml:space="preserve">ac proinde cum hæproportiones accuratæ non ſint, neceſſe eſt, <lb/>aream inuentam vel veram magnitudinem circuli ſuperare, vel ab ea deficere, <lb/>vt ex ſuperioribus regulis liquet. </s> <s xml:id="echoid-s8765" xml:space="preserve">Et quamuis differentia inter veram aream, at-<lb/>queinuentam in paruis circulis, perexigna ſit, in magnis tamen circulis negligẽ-<lb/> <anchor type="note" xlink:label="note-228-03a" xlink:href="note-228-03"/> da non videtur. </s> <s xml:id="echoid-s8766" xml:space="preserve">Quamobrem quimagis accuratam circuli aream deſiderat, aſ-<lb/>ſumat proportionem diametri ad circumferentiam verò propinquiorem, quàm <lb/>poſteriores Geometræ, p̃ſertim Ludolphus à Collen, & </s> <s xml:id="echoid-s8767" xml:space="preserve">Chriſtophorus Gruẽ-<lb/>bergerus inuenerunt, vt ſequitur.</s> <s xml:id="echoid-s8768" xml:space="preserve"/> </p> <div xml:id="echoid-div537" type="float" level="2" n="2"> <note position="left" xlink:label="note-228-03" xlink:href="note-228-03a" xml:space="preserve">Accuratior <lb/>proportio dia-<lb/>metri ad cir-<lb/>cumferentiã.</note> </div> <note position="right" xml:space="preserve"> <lb/>Diameter # Circumſerentia maior quam vera <lb/>100000000000000000000. # 314159265358979323847. <lb/>Diameter # Circumferentia minor quam vera <lb/>100000000000000000000. # 314159265358979323846. <lb/></note> <p> <s xml:id="echoid-s8769" xml:space="preserve">Ita vt proportio circumferentiæ ad diametrum <lb/>3 {14159265358979323847/100000000000000000000}. <lb/></s> <s xml:id="echoid-s8770" xml:space="preserve">maior, quam vera, denominetur ab hoc primo <lb/>numero, quæ minor eſt, quam tripla ſeſquiſe-<lb/>ptima. </s> <s xml:id="echoid-s8771" xml:space="preserve">Proportio verò minor, quam vera, de-<lb/>3 {142<unsure/>59265358979323846/100000000000000000000}. </s> <s xml:id="echoid-s8772" xml:space="preserve"><lb/>nominetur ab hoc ſecundo numero quæ ma-<lb/>ior eſt, quam tripla ſuperdecupartiens ſeptua-<lb/>geſimas primas: </s> <s xml:id="echoid-s8773" xml:space="preserve">qui denominatores habentur, <lb/>ſi vtraque circumferentia per diametrum diuidatur.</s> <s xml:id="echoid-s8774" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8775" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi fiat, vt diameter prædicta ad circumſerentiam vera maiorem, ita <lb/> <anchor type="note" xlink:label="note-228-05a" xlink:href="note-228-05"/> diameter alicuius circuli data ad aliud, reperietur circumferentia maior, quam <lb/>vera, minus tamen differens à vera, quam illa Archimedis, quæ Num. </s> <s xml:id="echoid-s8776" xml:space="preserve">2. </s> <s xml:id="echoid-s8777" xml:space="preserve">exregu-<lb/>la 1. </s> <s xml:id="echoid-s8778" xml:space="preserve">inuenta fuit. </s> <s xml:id="echoid-s8779" xml:space="preserve">Siautem fiat, vt eadem diameter prædicta ad circumferentiã <lb/>vera minorem, ita data diameter alicuius circuli ad aliud, reperietur circumferẽ-<lb/>tia minor, quam vera, magis tamen ad veram accedens, quam illa Archimedis, <lb/>quæ Nu. </s> <s xml:id="echoid-s8780" xml:space="preserve">2. </s> <s xml:id="echoid-s8781" xml:space="preserve">exreg. </s> <s xml:id="echoid-s8782" xml:space="preserve">2. </s> <s xml:id="echoid-s8783" xml:space="preserve">inuenitur. </s> <s xml:id="echoid-s8784" xml:space="preserve">Econtrario verò, ſi fiat vt prædicta circumferen-<lb/>tia minor, quam vera, ad diametrum, ita circũferentia alicuius circuli data ad ali-<lb/>ud, prodibit diameter maior quam vera: </s> <s xml:id="echoid-s8785" xml:space="preserve">Si autem fiat, vt circumferentia prædi-<lb/>cta maior, quam vera, ad diametrum, ita circumferentia alicuius circuli data ad <lb/>aliud, proueniet dameter minor, quam vera. </s> <s xml:id="echoid-s8786" xml:space="preserve">Vtraq; </s> <s xml:id="echoid-s8787" xml:space="preserve">tamen diameter inuẽta ma-<lb/>gis ad veram accedet, quã illa Archimedis, quæ Nu. </s> <s xml:id="echoid-s8788" xml:space="preserve">2. </s> <s xml:id="echoid-s8789" xml:space="preserve">per reg. </s> <s xml:id="echoid-s8790" xml:space="preserve">3. </s> <s xml:id="echoid-s8791" xml:space="preserve">& </s> <s xml:id="echoid-s8792" xml:space="preserve">4. </s> <s xml:id="echoid-s8793" xml:space="preserve">inuẽta fuit.</s> <s xml:id="echoid-s8794" xml:space="preserve"/> </p> <div xml:id="echoid-div538" type="float" level="2" n="3"> <note position="left" xlink:label="note-228-05" xlink:href="note-228-05a" xml:space="preserve">Accur atior <lb/>inuentio cir-<lb/>cumferentiæ <lb/>ex data dia-<lb/>metro: & dia-<lb/>metri ex data <lb/>circumferen-<lb/>tia.</note> </div> <pb o="199" file="229" n="229" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s8795" xml:space="preserve"><emph style="sc">Inventa</emph> circumferentia ex diametro, vel diametro ex circumferentia, re-<lb/>perietur area circuli, vt Num. </s> <s xml:id="echoid-s8796" xml:space="preserve">1. </s> <s xml:id="echoid-s8797" xml:space="preserve">tra ditum eſt: </s> <s xml:id="echoid-s8798" xml:space="preserve">ſi nimirum ſemidiameter in ſemicir-<lb/>cumferentiam ducatur: </s> <s xml:id="echoid-s8799" xml:space="preserve">vel tota circumferentia in ſemiſſem ſemidiametri: </s> <s xml:id="echoid-s8800" xml:space="preserve">vel <lb/>deniq; </s> <s xml:id="echoid-s8801" xml:space="preserve">tota diameter in quartam partem circumferentiæ. </s> <s xml:id="echoid-s8802" xml:space="preserve">quæ quidem area mi-<lb/>nus à vera diſtabit, quam illa, quæ ex proportione Archimedis inuenitur. </s> <s xml:id="echoid-s8803" xml:space="preserve">Sed <lb/>quia diffi cilius eſt per magnos numeros calculum inſtituere, quam per minores, <lb/>vſus artificum obtinuit, vt proportio Archimedis ad calculum ad hibeatur. </s> <s xml:id="echoid-s8804" xml:space="preserve">Quã-<lb/>do tamẽ deſideratur accuratior calculus, vtendum erit poſteriori hac propor-<lb/>tione Ludolphi, præſertimin maioribus circulis.</s> <s xml:id="echoid-s8805" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div540" type="section" level="1" n="194"> <head xml:id="echoid-head206" xml:space="preserve">DE AREA SEGMENTORVM CIRCVLI.</head> <head xml:id="echoid-head207" xml:space="preserve"><emph style="sc">Capvt</emph> VIII.</head> <handwritten/> <p> <s xml:id="echoid-s8806" xml:space="preserve">1. </s> <s xml:id="echoid-s8807" xml:space="preserve"><emph style="sc">Sit</emph> primum propoſitus ſector circuli ABCD, comprehenſus duabus ſe-<lb/>midiametris AB, AD, &</s> <s xml:id="echoid-s8808" xml:space="preserve">arcu BCD. </s> <s xml:id="echoid-s8809" xml:space="preserve">Huius aream ita explorabimus. </s> <s xml:id="echoid-s8810" xml:space="preserve">Si tam <lb/>ſemidiameter AB, nota ſit, nimirum palmorum 7. </s> <s xml:id="echoid-s8811" xml:space="preserve">quam arcus B C D, palmorum <lb/>videlicet 3 {2/3}. </s> <s xml:id="echoid-s8812" xml:space="preserve">ducatur ſemidiameter 7. </s> <s xml:id="echoid-s8813" xml:space="preserve">in {11/6}. </s> <s xml:id="echoid-s8814" xml:space="preserve">id eſt, in ſemiſſem arcus, Produ-<lb/>ctus enim numerus 12 {5/6}. </s> <s xml:id="echoid-s8815" xml:space="preserve">palm. </s> <s xml:id="echoid-s8816" xml:space="preserve">erit area ſectoris ABCD, vt demonſtrabimus. </s> <s xml:id="echoid-s8817" xml:space="preserve">Si <lb/>autem neque ſemidiameter AB, ne que perip heria BCD, data ſit, menſuranda erit <lb/>ſemidiameter aliqua menſura nota, & </s> <s xml:id="echoid-s8818" xml:space="preserve">ſecundum <lb/> <anchor type="figure" xlink:label="fig-229-01a" xlink:href="fig-229-01"/> eandem menſuram inuenienda circumſerẽtia cir-<lb/>culi per regulas antecedentis capit. </s> <s xml:id="echoid-s8819" xml:space="preserve">necnonrecta <lb/>BD. </s> <s xml:id="echoid-s8820" xml:space="preserve">Deinde ſiat, vt AB, nota in aſſumpta menſu-<lb/>ra ad ſinum totum 100000. </s> <s xml:id="echoid-s8821" xml:space="preserve">ita BD, nota in eadem <lb/>menſura aſſumpta ad aliud. </s> <s xml:id="echoid-s8822" xml:space="preserve">Numerus enim pro-<lb/>creatus dabit rectam B D, cognitam in partibus ſi-<lb/>nus totius. </s> <s xml:id="echoid-s8823" xml:space="preserve">Huius autem medietas ſinus erit ſemiſsis arcus B D: </s> <s xml:id="echoid-s8824" xml:space="preserve">ac proinde ex <lb/>tabula ſinuũ ſemiſsis BC, in gradib. </s> <s xml:id="echoid-s8825" xml:space="preserve">nota erit, ideoq; </s> <s xml:id="echoid-s8826" xml:space="preserve">totus arcus BD, nõ ignora-<lb/>bitur. </s> <s xml:id="echoid-s8827" xml:space="preserve">Et quiatota circuli circumferentia nota facta eſt in aſſumpta menſura: </s> <s xml:id="echoid-s8828" xml:space="preserve">ſi <lb/>fiat vt grad. </s> <s xml:id="echoid-s8829" xml:space="preserve">360. </s> <s xml:id="echoid-s8830" xml:space="preserve">ad totam circumferentia in aſſumpta menſura cognitam, ita ar-<lb/>cus BD, in gradibus cognitus ad aliud, cognoſcetur idem arcus B D, in menſura <lb/>aſſumpta. </s> <s xml:id="echoid-s8831" xml:space="preserve">Quare, vt prius, area ſectoris A B C D, reperietur. </s> <s xml:id="echoid-s8832" xml:space="preserve">Poſlent quoque <lb/>gradus in arcu BD, contenti inueſtigari beneficio quadrantis alicuius in gradus <lb/>diuiſi, adhibita doctrina cap. </s> <s xml:id="echoid-s8833" xml:space="preserve">2. </s> <s xml:id="echoid-s8834" xml:space="preserve">lib. </s> <s xml:id="echoid-s8835" xml:space="preserve">1. </s> <s xml:id="echoid-s8836" xml:space="preserve">Nume. </s> <s xml:id="echoid-s8837" xml:space="preserve">10. </s> <s xml:id="echoid-s8838" xml:space="preserve">tradita, vt minuta etiam cogno-<lb/>ſcantur, quando in arcu BD, vltra gradus aliqua particula ſupereſt.</s> <s xml:id="echoid-s8839" xml:space="preserve"/> </p> <div xml:id="echoid-div540" type="float" level="2" n="1"> <figure xlink:label="fig-229-01" xlink:href="fig-229-01a"> <image file="229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/229-01"/> </figure> </div> <p> <s xml:id="echoid-s8840" xml:space="preserve"><emph style="sc">Aream</emph> porro ſectoris produci ex ſemidiametro in ſemiſſem arcus ſectoris, <lb/> <anchor type="handwritten" xlink:label="hd-229-2a" xlink:href="hd-229-2"/> ſic demonſtro. </s> <s xml:id="echoid-s8841" xml:space="preserve">Sit quadrans B E, & </s> <s xml:id="echoid-s8842" xml:space="preserve">ſemicirculus BEF. </s> <s xml:id="echoid-s8843" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quoniam eſt, vt ar- <anchor type="note" xlink:label="note-229-01a" xlink:href="note-229-01"/> cus B D, ad quadrantem BE, ita ſector ABCD, ad ſectorem ABDE: </s> <s xml:id="echoid-s8844" xml:space="preserve">erit quo que <lb/>ex ſcholio propoſ. </s> <s xml:id="echoid-s8845" xml:space="preserve">22. </s> <s xml:id="echoid-s8846" xml:space="preserve">lib. </s> <s xml:id="echoid-s8847" xml:space="preserve">5. </s> <s xml:id="echoid-s8848" xml:space="preserve">Eucli. </s> <s xml:id="echoid-s8849" xml:space="preserve">vt arcus BD, ad quadruplum quadrantis BE, <lb/>hoc eſt, ad totam circumferentiam, ita ſector A B C D, ad quadruplum ſectoris <lb/>A B D E, hoc eſt, ad totum circulum. </s> <s xml:id="echoid-s8850" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Vt autem arcus BD, ad totam circum- <anchor type="note" xlink:label="note-229-02a" xlink:href="note-229-02"/> ferentiam, ita eſt BC, ſemiſsis arcus BD, ad BEF, ſemiſſem totius circumferentiæ. <lb/></s> <s xml:id="echoid-s8851" xml:space="preserve">Igitur erit quo que vt B C, ad B E F, ita ſector A B C D, ad totum circulum. </s> <s xml:id="echoid-s8852" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <anchor type="note" xlink:label="note-229-03a" xlink:href="note-229-03"/> Sed vt B C, ad B E F, ita eſt rectangulum ſub A B, B C, ad rectangulum ſub <lb/>AB, BEF. </s> <s xml:id="echoid-s8853" xml:space="preserve">Ergo erit quoq; </s> <s xml:id="echoid-s8854" xml:space="preserve">ſector ABCD, ad totũ circulum, vtrectangulum ſub <lb/>AB, BC, ad rectangulum ſub AB, BEF. </s> <s xml:id="echoid-s8855" xml:space="preserve">Cum ergo vt cap. </s> <s xml:id="echoid-s8856" xml:space="preserve">7. </s> <s xml:id="echoid-s8857" xml:space="preserve">Num. </s> <s xml:id="echoid-s8858" xml:space="preserve">1. </s> <s xml:id="echoid-s8859" xml:space="preserve">tradidimus, <pb o="200" file="230" n="230" rhead="GEOMETR. PRACT."/> circulus æqualis ſitrectangulo ſub AB, BEF, <anchor type="note" xlink:href="" symbol="a"/> erit quo que ſector A B C D, re- <anchor type="note" xlink:label="note-230-01a" xlink:href="note-230-01"/> ctangulo ſub AB, BC, æqualis. </s> <s xml:id="echoid-s8860" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s8861" xml:space="preserve"/> </p> <div xml:id="echoid-div541" type="float" level="2" n="2"> <handwritten xlink:label="hd-229-2" xlink:href="hd-229-2a"/> <note symbol="a" position="right" xlink:label="note-229-01" xlink:href="note-229-01a" xml:space="preserve">33. ſexti.</note> <note symbol="b" position="right" xlink:label="note-229-02" xlink:href="note-229-02a" xml:space="preserve">15. quinti.</note> <note symbol="c" position="right" xlink:label="note-229-03" xlink:href="note-229-03a" xml:space="preserve">1. ſexti.</note> <note symbol="a" position="left" xlink:label="note-230-01" xlink:href="note-230-01a" xml:space="preserve">14. quinti.</note> </div> <p> <s xml:id="echoid-s8862" xml:space="preserve"><emph style="sc">Eadem</emph> ratione procreabitur ſector A B F D A, ex ſemidiametro A B, in ſe-<lb/>miſſem arcus BFD.</s> <s xml:id="echoid-s8863" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8864" xml:space="preserve">2. </s> <s xml:id="echoid-s8865" xml:space="preserve"><emph style="sc">Sit</emph> deinde ſegmentum BCD. </s> <s xml:id="echoid-s8866" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Inuento centro A, arcus B C D, & </s> <s xml:id="echoid-s8867" xml:space="preserve">co- <anchor type="note" xlink:label="note-230-02a" xlink:href="note-230-02"/> gnitis per aliquam menſuram lateribus trianguli A B D, & </s> <s xml:id="echoid-s8868" xml:space="preserve">arcu B C D, in eadem <lb/>menſura, vt Num. </s> <s xml:id="echoid-s8869" xml:space="preserve">1. </s> <s xml:id="echoid-s8870" xml:space="preserve">diximus, inueſtigetur tam area ſectoris ABCD, quam trian-<lb/>guli ABD. </s> <s xml:id="echoid-s8871" xml:space="preserve">Hæc enim detracta ex illa relinquet aream ſegmenti propoſiti BCD.</s> <s xml:id="echoid-s8872" xml:space="preserve"/> </p> <div xml:id="echoid-div542" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-230-02" xlink:href="note-230-02a" xml:space="preserve">25. tertij.</note> </div> <p> <s xml:id="echoid-s8873" xml:space="preserve">3. </s> <s xml:id="echoid-s8874" xml:space="preserve"><emph style="sc">Sit</emph> præterea figura lenticularis duobus arcubus G H I, GKI. </s> <s xml:id="echoid-s8875" xml:space="preserve">contenta. <lb/></s> <s xml:id="echoid-s8876" xml:space="preserve">Ducta recta GI, inquiratur, vt Nu. </s> <s xml:id="echoid-s8877" xml:space="preserve">2. </s> <s xml:id="echoid-s8878" xml:space="preserve">docuimus, vtriuſque ſegmenti GHI, GKI, <lb/>area. </s> <s xml:id="echoid-s8879" xml:space="preserve">Summa enim ex duabus hiſce areis conflata, erit area propoſitæ figuræ <lb/>GHIK. </s> <s xml:id="echoid-s8880" xml:space="preserve">Quod ſi ſegmenta GHI, GKI, ſint æqualia, ſatis erit vnius areaminue-<lb/>ſtigare. </s> <s xml:id="echoid-s8881" xml:space="preserve">Hæc namque duplicata dabit propoſitæ figuræ GHIK, aream.</s> <s xml:id="echoid-s8882" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8883" xml:space="preserve">4. </s> <s xml:id="echoid-s8884" xml:space="preserve"><emph style="sc">Non</emph> aliter metiemur figuras ex varijs circulorum ſegmentis coagmen-<lb/>tatas, ſiue omnes circumferentiæ extrorſus vergant, ſiue introrſum, ſiue partim <lb/>introrſum, & </s> <s xml:id="echoid-s8885" xml:space="preserve">partium extrorſum. </s> <s xml:id="echoid-s8886" xml:space="preserve">Vtin tribus his figuris, ſi arcubus ſubten-<lb/>dantur chordæ, metiemur in prima quadrilaterum ABCD, vt cap. </s> <s xml:id="echoid-s8887" xml:space="preserve">1. </s> <s xml:id="echoid-s8888" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s8889" xml:space="preserve">do cui-<lb/>mus: </s> <s xml:id="echoid-s8890" xml:space="preserve">& </s> <s xml:id="echoid-s8891" xml:space="preserve">ſegmenta AEB, BFC, CGD, DHA, vthoccap. </s> <s xml:id="echoid-s8892" xml:space="preserve">Num. </s> <s xml:id="echoid-s8893" xml:space="preserve">2. </s> <s xml:id="echoid-s8894" xml:space="preserve">traditũ eſt. </s> <s xml:id="echoid-s8895" xml:space="preserve">Sie-<lb/>nim hæc ſegmenta quadrilatero adijciantur, quod omnia extrorſum tendant, <lb/> <anchor type="figure" xlink:label="fig-230-01a" xlink:href="fig-230-01"/> conſlabitur area figuræ A E B F C G D H, ex quatuor arcubus compoſitæ.</s> <s xml:id="echoid-s8896" xml:space="preserve"/> </p> <div xml:id="echoid-div543" type="float" level="2" n="4"> <figure xlink:label="fig-230-01" xlink:href="fig-230-01a"> <image file="230-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/230-01"/> </figure> </div> <p> <s xml:id="echoid-s8897" xml:space="preserve"><emph style="sc">In</emph> ſecunda autem metiemur pentagonum ABCDE, per ea, quæ cap. </s> <s xml:id="echoid-s8898" xml:space="preserve">4. </s> <s xml:id="echoid-s8899" xml:space="preserve">ſcri-<lb/>pta ſunt: </s> <s xml:id="echoid-s8900" xml:space="preserve">Ex quo ſi dememus quinque ſegmenta introrſum vergentia, quæ qui-<lb/>dem ex ijs, quæ Num. </s> <s xml:id="echoid-s8901" xml:space="preserve">2. </s> <s xml:id="echoid-s8902" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s8903" xml:space="preserve">ſcripſimus, cognoſcentur, reliqua fiet area ſi-<lb/>guræ A F B G C H D I E K, ex quinque arcubus conflatæ.</s> <s xml:id="echoid-s8904" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8905" xml:space="preserve"><emph style="sc">In</emph> tertia denique pentagono A B C D E, adij ciemus tria ſegmenta, A F B, <lb/>A G E, C H D, extrorſum vergentia, & </s> <s xml:id="echoid-s8906" xml:space="preserve">ex compoſito numero duo ſegmenta <lb/>B I C, D K E, introrſum vergentia tollemus, vt area relinquatur figuræ AFBIC-<lb/>HDKEG, ex quinque arcubus compoſitæ. </s> <s xml:id="echoid-s8907" xml:space="preserve">Atque hoc modo agrum quantum-<lb/>uis irregularem metirilicebit.</s> <s xml:id="echoid-s8908" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8909" xml:space="preserve">5. </s> <s xml:id="echoid-s8910" xml:space="preserve"><emph style="sc">Sit</emph> denique in prima figura huius cap. </s> <s xml:id="echoid-s8911" xml:space="preserve">ſegmentum circuli LMON, com-<lb/>prehenſum duabus rectis L M, N O, & </s> <s xml:id="echoid-s8912" xml:space="preserve">duobus arcubus LN, MO. </s> <s xml:id="echoid-s8913" xml:space="preserve">Exploretur <lb/>vt Num. </s> <s xml:id="echoid-s8914" xml:space="preserve">2. </s> <s xml:id="echoid-s8915" xml:space="preserve">declaratum eſt, area vtriuſque ſegmenti P L M, P N O. </s> <s xml:id="echoid-s8916" xml:space="preserve">Minor enim <lb/>area P N O, detracta ex maiori PLM, reliquam ſaciet aream propoſiti ſegmenti <lb/>L M O N.</s> <s xml:id="echoid-s8917" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8918" xml:space="preserve">6. </s> <s xml:id="echoid-s8919" xml:space="preserve"><emph style="sc">Vt</emph> quartus hic liber concludatur, lubet hic appẽdicis loco regulas quaſ-<lb/>dam alias à noſtro inſtituto non alienas ſubiungere.</s> <s xml:id="echoid-s8920" xml:space="preserve"/> </p> <pb o="201" file="231" n="231" rhead="LIBER QVARTVS."/> </div> <div xml:id="echoid-div545" type="section" level="1" n="195"> <head xml:id="echoid-head208" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s8921" xml:space="preserve">DATA circuli area, circumferentiam, ac diametrum cognoſcere.</s> <s xml:id="echoid-s8922" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8923" xml:space="preserve"><emph style="sc">Fiat</emph> vt 7. </s> <s xml:id="echoid-s8924" xml:space="preserve">ad 88. </s> <s xml:id="echoid-s8925" xml:space="preserve">ita data area ad aliud. </s> <s xml:id="echoid-s8926" xml:space="preserve">Productus enim numerus erit qua-<lb/>dratum circumferentiæ vero maius, vt ex 4. </s> <s xml:id="echoid-s8927" xml:space="preserve">reg. </s> <s xml:id="echoid-s8928" xml:space="preserve">Num. </s> <s xml:id="echoid-s8929" xml:space="preserve">4. </s> <s xml:id="echoid-s8930" xml:space="preserve">cap. </s> <s xml:id="echoid-s8931" xml:space="preserve">7. </s> <s xml:id="echoid-s8932" xml:space="preserve">liquet. </s> <s xml:id="echoid-s8933" xml:space="preserve">Radix <lb/>ergo quadrata numeri producti dabit circumferentiam vera maiorem. </s> <s xml:id="echoid-s8934" xml:space="preserve">Quod ſi <lb/>fiat, vt 71. </s> <s xml:id="echoid-s8935" xml:space="preserve">ad 892. </s> <s xml:id="echoid-s8936" xml:space="preserve">ita data area ad aliud, gignetur quadratum circumferentiæ ve-<lb/>to minus, vt conſtat ex 3. </s> <s xml:id="echoid-s8937" xml:space="preserve">reg. </s> <s xml:id="echoid-s8938" xml:space="preserve">Num. </s> <s xml:id="echoid-s8939" xml:space="preserve">4. </s> <s xml:id="echoid-s8940" xml:space="preserve">capit. </s> <s xml:id="echoid-s8941" xml:space="preserve">7. </s> <s xml:id="echoid-s8942" xml:space="preserve">Ac proinde eius radix quadrata <lb/>circumferentiam vera minorem indicabit.</s> <s xml:id="echoid-s8943" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8944" xml:space="preserve"><emph style="sc">Fiat</emph> rurſus, vt 223. </s> <s xml:id="echoid-s8945" xml:space="preserve">ad 284. </s> <s xml:id="echoid-s8946" xml:space="preserve">ita area propoſita ad aliud. </s> <s xml:id="echoid-s8947" xml:space="preserve">Procreatus nam-<lb/>que numerus erit quadratum diametri verò maius, vt ex 2. </s> <s xml:id="echoid-s8948" xml:space="preserve">reg. </s> <s xml:id="echoid-s8949" xml:space="preserve">Num. </s> <s xml:id="echoid-s8950" xml:space="preserve">4. </s> <s xml:id="echoid-s8951" xml:space="preserve">cap. </s> <s xml:id="echoid-s8952" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s8953" xml:space="preserve">perſpicuum eſt. </s> <s xml:id="echoid-s8954" xml:space="preserve">Radix ergo quadrata numeri producti diametrum exhibebit <lb/>vera maiorem. </s> <s xml:id="echoid-s8955" xml:space="preserve">Quod ſi fiat, vt 11. </s> <s xml:id="echoid-s8956" xml:space="preserve">ad 14. </s> <s xml:id="echoid-s8957" xml:space="preserve">ita area data ad aliud, reperietur qua-<lb/>dratum diametri verò minus, vt ex reg. </s> <s xml:id="echoid-s8958" xml:space="preserve">1. </s> <s xml:id="echoid-s8959" xml:space="preserve">Num 4. </s> <s xml:id="echoid-s8960" xml:space="preserve">cap. </s> <s xml:id="echoid-s8961" xml:space="preserve">7. </s> <s xml:id="echoid-s8962" xml:space="preserve">colligitur. </s> <s xml:id="echoid-s8963" xml:space="preserve">Ac proinde <lb/>tadix eius quadrata diametrum offeret vera minorem.</s> <s xml:id="echoid-s8964" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div546" type="section" level="1" n="196"> <head xml:id="echoid-head209" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s8965" xml:space="preserve">DATO arcu cuiuſuis circuli, diametrum circuli in numeris inueſti-<lb/>gare.</s> <s xml:id="echoid-s8966" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8967" xml:space="preserve"><emph style="sc">Sit</emph> datus arcus A B C. </s> <s xml:id="echoid-s8968" xml:space="preserve">Ducta chorda A C, ſectaque bifariam in F, ducatur <lb/> <anchor type="note" xlink:label="note-231-01a" xlink:href="note-231-01"/> per F, perpendicularis FB,<anchor type="note" xlink:href="" symbol="a"/> quæ per centrum circuli tranſibit, <anchor type="note" xlink:href="" symbol="b"/> ideo querectan- gulum ſub C F, A F, hoc eſt, quadratum ex A F, æquale eritre-<lb/> <anchor type="figure" xlink:label="fig-231-01a" xlink:href="fig-231-01"/> ctangulo ſub B F, & </s> <s xml:id="echoid-s8969" xml:space="preserve">reliqua portione diametri. </s> <s xml:id="echoid-s8970" xml:space="preserve">Si igitur A F, <lb/> <anchor type="note" xlink:label="note-231-02a" xlink:href="note-231-02"/> FB, per aliquam menſuram fiant notæ, & </s> <s xml:id="echoid-s8971" xml:space="preserve">quadratus numerus <lb/>rectæ A F, diuidatur per F B, prodibit reliqua portio diametri <lb/>F D, quæ addita perpendiculari FB, conficiet totam diametrum <lb/>BD, notam in eadem menſura, in qua A F, FB, cognitæ ſunt.</s> <s xml:id="echoid-s8972" xml:space="preserve"/> </p> <div xml:id="echoid-div546" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-231-01" xlink:href="note-231-01a" xml:space="preserve">coroll. 1. <lb/>tertij.</note> <figure xlink:label="fig-231-01" xlink:href="fig-231-01a"> <image file="231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/231-01"/> </figure> <note symbol="b" position="right" xlink:label="note-231-02" xlink:href="note-231-02a" xml:space="preserve">35. tertij.</note> </div> <p> <s xml:id="echoid-s8973" xml:space="preserve"><emph style="sc">Geometrice</emph> eadem portio FD, reperietur, ſi duabus <lb/>F B, A F, inueniatur tertia proportionalis F D: </s> <s xml:id="echoid-s8974" xml:space="preserve">propterea quod ex ſcholio pro-<lb/>poſ. </s> <s xml:id="echoid-s8975" xml:space="preserve">13. </s> <s xml:id="echoid-s8976" xml:space="preserve">lib. </s> <s xml:id="echoid-s8977" xml:space="preserve">6. </s> <s xml:id="echoid-s8978" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s8979" xml:space="preserve">AF, media proportionalis eſt inter diametri ſegmenta.</s> <s xml:id="echoid-s8980" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div548" type="section" level="1" n="197"> <head xml:id="echoid-head210" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s8981" xml:space="preserve">DATIS diametris duorum circulorum, vel circumferentiis: </s> <s xml:id="echoid-s8982" xml:space="preserve">Aut duo-<lb/>bus lateribus homologis duarum figurarum ſimilium, ſimilium, ſimiliterq; </s> <s xml:id="echoid-s8983" xml:space="preserve">po-<lb/>ſitarum: </s> <s xml:id="echoid-s8984" xml:space="preserve">quam proportionem proportionem circuli, vel figuræ inter ſe habeant, co-<lb/>gnoſcere.</s> <s xml:id="echoid-s8985" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s8986" xml:space="preserve"><emph style="sc">Qvoniam</emph> circuli, & </s> <s xml:id="echoid-s8987" xml:space="preserve">figuræ ſimiles ſimiliterque poſitæ, habent duplica-<lb/>tam proportionem diametrorum, vel circumferentiarum, & </s> <s xml:id="echoid-s8988" xml:space="preserve">laterum homolo-<lb/>gorum: </s> <s xml:id="echoid-s8989" xml:space="preserve">ſi maior diameter, vel circumferentia per minorem, & </s> <s xml:id="echoid-s8990" xml:space="preserve">maius latus ho-<lb/>mologum per minus diuidatur, prodibit denominator proportionis, quam ma-<lb/>ior diameter, circumferentiaue ad minorem, vel maius latus homologum ad mi-<lb/>nus habet. </s> <s xml:id="echoid-s8991" xml:space="preserve">Siigitur hic denominator in ſe ducatur, producetur denominator <pb o="202" file="232" n="232" rhead="GEOMETR. PRACT."/> duplicatæ proportionis, quam videlicet circulus, vel figura ad minorem habet <lb/>Vt ſi diameter vnius circuli ſit 56. </s> <s xml:id="echoid-s8992" xml:space="preserve">& </s> <s xml:id="echoid-s8993" xml:space="preserve">circumſerentia 176. </s> <s xml:id="echoid-s8994" xml:space="preserve">Alterius autem circuli <lb/>diameter 14.</s> <s xml:id="echoid-s8995" xml:space="preserve">| & </s> <s xml:id="echoid-s8996" xml:space="preserve">| circumferentia 44. </s> <s xml:id="echoid-s8997" xml:space="preserve">Diuiſis 56. </s> <s xml:id="echoid-s8998" xml:space="preserve">per 14. </s> <s xml:id="echoid-s8999" xml:space="preserve">vel 176. </s> <s xml:id="echoid-s9000" xml:space="preserve">per 44. </s> <s xml:id="echoid-s9001" xml:space="preserve">fit Quo-<lb/>tiens 4. </s> <s xml:id="echoid-s9002" xml:space="preserve">qui ductus in ſe producit 16. </s> <s xml:id="echoid-s9003" xml:space="preserve">denominatorem proportionis maioris cir-<lb/>culi ad minorem. </s> <s xml:id="echoid-s9004" xml:space="preserve">Eandemque proportionem habebitfigura ad minorem ſimi-<lb/>lem, ſimiliter que poſitam, ſi latera homologa ſint 56. </s> <s xml:id="echoid-s9005" xml:space="preserve">& </s> <s xml:id="echoid-s9006" xml:space="preserve">14. </s> <s xml:id="echoid-s9007" xml:space="preserve">vel 176. </s> <s xml:id="echoid-s9008" xml:space="preserve">& </s> <s xml:id="echoid-s9009" xml:space="preserve">44.</s> <s xml:id="echoid-s9010" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div549" type="section" level="1" n="198"> <head xml:id="echoid-head211" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s9011" xml:space="preserve">DATIS pluribus circulis, quorum diametri, vel circumferentiæ co-<lb/>gnitæ ſint: </s> <s xml:id="echoid-s9012" xml:space="preserve">Item pluribus figuris ſimilibus ſimiliterque poſitis, quarũ <lb/>latera homologa ſint nota: </s> <s xml:id="echoid-s9013" xml:space="preserve">Inuenire diametrum, vel circumferenti-<lb/>am, cuius circulus omnibus circulis propoſitis æqualis ſit. </s> <s xml:id="echoid-s9014" xml:space="preserve">Item latus <lb/>reperire, cuius figura ſimilis, ſimiliterque poſita æqualis ſit omnibus <lb/>propoſitis figuris.</s> <s xml:id="echoid-s9015" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9016" xml:space="preserve"><emph style="sc">Mvltiplicentvr</emph> diametri, vel circumferentiæ, aut latera homologa <lb/>in ſe, & </s> <s xml:id="echoid-s9017" xml:space="preserve">numeri producti in vnam ſummam colligantur. </s> <s xml:id="echoid-s9018" xml:space="preserve">Radix enim quadrata <lb/>huius ſummæ erit diameter, circumferentiaue, aut latus homologum quæſitum. <lb/></s> <s xml:id="echoid-s9019" xml:space="preserve">Verbigratia, ſi ſint quatuor diametri, circumferentiæue circulorum, autlatera <lb/>homologa ſimilium figurarum, ſimiliter que poſitarum, 84. </s> <s xml:id="echoid-s9020" xml:space="preserve">3. </s> <s xml:id="echoid-s9021" xml:space="preserve">4. </s> <s xml:id="echoid-s9022" xml:space="preserve">12. </s> <s xml:id="echoid-s9023" xml:space="preserve">atq; </s> <s xml:id="echoid-s9024" xml:space="preserve">in ſe mul-<lb/>tiplicentur, gignentur numeri 7056. </s> <s xml:id="echoid-s9025" xml:space="preserve">9. </s> <s xml:id="echoid-s9026" xml:space="preserve">16. </s> <s xml:id="echoid-s9027" xml:space="preserve">144. </s> <s xml:id="echoid-s9028" xml:space="preserve">quorum ſumma 7225. </s> <s xml:id="echoid-s9029" xml:space="preserve">Radix er-<lb/>go quadrata huius ſummæ 85. </s> <s xml:id="echoid-s9030" xml:space="preserve">erit diameter, circumferentiaue circuli, aut latus <lb/>homologum, quod quæritur: </s> <s xml:id="echoid-s9031" xml:space="preserve">ita vt circulus, cuius diameter, vel circumferen-<lb/>tia eſt 85. </s> <s xml:id="echoid-s9032" xml:space="preserve">autfigura ſupra rectam 85. </s> <s xml:id="echoid-s9033" xml:space="preserve">ſimilis, ſimiliterq; </s> <s xml:id="echoid-s9034" xml:space="preserve">poſita figuris datis, æqua-<lb/>lis ſit quatuor circulis, aut figuris propoſitis. </s> <s xml:id="echoid-s9035" xml:space="preserve">Nam cum quadratum 7225. </s> <s xml:id="echoid-s9036" xml:space="preserve">radicis <lb/> <anchor type="note" xlink:label="note-232-01a" xlink:href="note-232-01"/> 85. </s> <s xml:id="echoid-s9037" xml:space="preserve">æquale ſit quatuor quadratis 7056. </s> <s xml:id="echoid-s9038" xml:space="preserve">9. </s> <s xml:id="echoid-s9039" xml:space="preserve">16. </s> <s xml:id="echoid-s9040" xml:space="preserve">144. </s> <s xml:id="echoid-s9041" xml:space="preserve">radicũ 84. </s> <s xml:id="echoid-s9042" xml:space="preserve">3. </s> <s xml:id="echoid-s9043" xml:space="preserve">4. </s> <s xml:id="echoid-s9044" xml:space="preserve">12. </s> <s xml:id="echoid-s9045" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Circuli au- tem eandem habeant proportionem, quam quadrata diametrorum: </s> <s xml:id="echoid-s9046" xml:space="preserve">ac proin-<lb/>de quam quadrata circumferẽtiarum; </s> <s xml:id="echoid-s9047" xml:space="preserve">quod circumferentiæ diametris ſint pro-<lb/>portionales. </s> <s xml:id="echoid-s9048" xml:space="preserve">Item figuræ ſimiles, ſimiliter que poſitæ inter ſe ſint, vt quadrata <lb/>laterum homologorum, <anchor type="note" xlink:href="" symbol="b"/> propterea quod tam quadrata, quam figuræ habent <anchor type="note" xlink:label="note-232-02a" xlink:href="note-232-02"/> duplicatam proportionem laterum: </s> <s xml:id="echoid-s9049" xml:space="preserve">erit quo que tam circulus, cuius diameter, <lb/>circumferentiauẽ 85. </s> <s xml:id="echoid-s9050" xml:space="preserve">æqualis quatuor circulis, quorum diametri circumferen-<lb/>tiæue 84. </s> <s xml:id="echoid-s9051" xml:space="preserve">3. </s> <s xml:id="echoid-s9052" xml:space="preserve">4. </s> <s xml:id="echoid-s9053" xml:space="preserve">12. </s> <s xml:id="echoid-s9054" xml:space="preserve">quam figura ſupra latus 85. </s> <s xml:id="echoid-s9055" xml:space="preserve">ſimilis ſimiliter que poſita, quatuor<unsure/> <lb/>figuris, quarum lateta 84. </s> <s xml:id="echoid-s9056" xml:space="preserve">3. </s> <s xml:id="echoid-s9057" xml:space="preserve">4. </s> <s xml:id="echoid-s9058" xml:space="preserve">12. </s> <s xml:id="echoid-s9059" xml:space="preserve">æqualis, quod eſt propoſitum.</s> <s xml:id="echoid-s9060" xml:space="preserve"/> </p> <div xml:id="echoid-div549" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-232-01" xlink:href="note-232-01a" xml:space="preserve">2. duodec.</note> <note symbol="b" position="left" xlink:label="note-232-02" xlink:href="note-232-02a" xml:space="preserve">20. ſexti.</note> </div> </div> <div xml:id="echoid-div551" type="section" level="1" n="199"> <head xml:id="echoid-head212" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s9061" xml:space="preserve">AREAM propoſitæ Elipſis indagare.</s> <s xml:id="echoid-s9062" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9063" xml:space="preserve"><emph style="sc">Lvbet</emph> denique librum hunc quartum duobus problematibus terminare, <lb/>quæ ab Archimede Syracuſano accutiſsimè inuenta ſunt, ac demonſtrata. </s> <s xml:id="echoid-s9064" xml:space="preserve">Vnũ <lb/>eſt de area Ellipſis; </s> <s xml:id="echoid-s9065" xml:space="preserve">alterum de area Parabolæ. </s> <s xml:id="echoid-s9066" xml:space="preserve">Sit ergo Ellipſis A B C D, cuius <lb/>maior diameter BD, & </s> <s xml:id="echoid-s9067" xml:space="preserve">minor AC, ſecans maiorem in E, bifariam. </s> <s xml:id="echoid-s9068" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Inueniatur <anchor type="note" xlink:label="note-232-03a" xlink:href="note-232-03"/> HI, media proportionalis inter BD, & </s> <s xml:id="echoid-s9069" xml:space="preserve">AC: </s> <s xml:id="echoid-s9070" xml:space="preserve">& </s> <s xml:id="echoid-s9071" xml:space="preserve">circuli circa diametrum HI, deſcri-<lb/>pti area in quiratur, per ea, quæ c. </s> <s xml:id="echoid-s9072" xml:space="preserve">7. </s> <s xml:id="echoid-s9073" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s9074" xml:space="preserve">ſcripſimus. </s> <s xml:id="echoid-s9075" xml:space="preserve">Dico hanc aream areæ <pb o="203" file="233" n="233" rhead="LIBER QVARTVS."/> Ellipſis ABCD, eſſeæqualem.</s> <s xml:id="echoid-s9076" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim eſt, vt BD, ad AC, ita quadratum <anchor type="note" xlink:label="note-233-01a" xlink:href="note-233-01"/> ex BD, ad quadratũ ex ex HI. </s> <s xml:id="echoid-s9077" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Vt autẽ qua- <anchor type="figure" xlink:label="fig-233-01a" xlink:href="fig-233-01"/> dratum ex B D, ad quadratum ex HI, ita eſt <lb/> <anchor type="note" xlink:label="note-233-02a" xlink:href="note-233-02"/> circulus diametri B D, ad circulum diametri <lb/>HI. </s> <s xml:id="echoid-s9078" xml:space="preserve">Igitur erit quoq;</s> <s xml:id="echoid-s9079" xml:space="preserve">, vt BD, ad AC, ita circu <lb/>lus diametri B D, ad circulum diametri HI. <lb/></s> <s xml:id="echoid-s9080" xml:space="preserve">Cũ ergo per propoſitionem 5. </s> <s xml:id="echoid-s9081" xml:space="preserve">Archimedis <lb/>de Conoidibus, & </s> <s xml:id="echoid-s9082" xml:space="preserve">ſphæroidib. </s> <s xml:id="echoid-s9083" xml:space="preserve">ſit quoq;</s> <s xml:id="echoid-s9084" xml:space="preserve">, vt <lb/>maior diameter BD, ad minorem AC, ita cir-<lb/>culus diametri BD, ad Ellipſim ABCD; </s> <s xml:id="echoid-s9085" xml:space="preserve">cha-<lb/> <anchor type="note" xlink:label="note-233-03a" xlink:href="note-233-03"/> bebit circulus diametri BD, eandem propor-<lb/>tionem ad circulum diametri HI, & </s> <s xml:id="echoid-s9086" xml:space="preserve">ad Ellipſim ABCD. </s> <s xml:id="echoid-s9087" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Ideoque area circuli <anchor type="note" xlink:label="note-233-04a" xlink:href="note-233-04"/> diametri HI, areæ Ellipſis ABCD, æqualis erit. </s> <s xml:id="echoid-s9088" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s9089" xml:space="preserve"/> </p> <div xml:id="echoid-div551" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-232-03" xlink:href="note-232-03a" xml:space="preserve">13. ſexti.</note> <note symbol="a" position="right" xlink:label="note-233-01" xlink:href="note-233-01a" xml:space="preserve">coroll. 20. <lb/>ſexti.</note> <figure xlink:label="fig-233-01" xlink:href="fig-233-01a"> <image file="233-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/233-01"/> </figure> <note symbol="b" position="right" xlink:label="note-233-02" xlink:href="note-233-02a" xml:space="preserve">2. duodec.</note> <note symbol="c" position="right" xlink:label="note-233-03" xlink:href="note-233-03a" xml:space="preserve">11. quinti.</note> <note symbol="d" position="right" xlink:label="note-233-04" xlink:href="note-233-04a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div553" type="section" level="1" n="200"> <head xml:id="echoid-head213" xml:space="preserve">VI.</head> <p> <s xml:id="echoid-s9090" xml:space="preserve">AREAM propoſitæ parabolæ inueſtigare.</s> <s xml:id="echoid-s9091" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9092" xml:space="preserve"><emph style="sc">Sit</emph> data parabola ABC, cuius baſis AC, & </s> <s xml:id="echoid-s9093" xml:space="preserve">axis B D, diuidens baſem bifari-<lb/>amin D, & </s> <s xml:id="echoid-s9094" xml:space="preserve">vertex B. </s> <s xml:id="echoid-s9095" xml:space="preserve">Inſcribatur parabolæ triangulum A B C, eandem habens <lb/>baſem, ac verticem cum parabola. </s> <s xml:id="echoid-s9096" xml:space="preserve">Producta autem baſe A C, ſumatur CE, ter-<lb/>tia pars ipſius A C: </s> <s xml:id="echoid-s9097" xml:space="preserve">ita vt AE, ipſius A C, ſit ſeſquitertia. </s> <s xml:id="echoid-s9098" xml:space="preserve">Iungatur que recta E B. <lb/></s> <s xml:id="echoid-s9099" xml:space="preserve">Inquiratur denique per cap. </s> <s xml:id="echoid-s9100" xml:space="preserve">2. </s> <s xml:id="echoid-s9101" xml:space="preserve">huius libr. </s> <s xml:id="echoid-s9102" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-233-02a" xlink:href="fig-233-02"/> area triãguli ABE. </s> <s xml:id="echoid-s9103" xml:space="preserve">quam dico eſſe æqua-<lb/>lem areæ parabolæ A B C. </s> <s xml:id="echoid-s9104" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quoniã enim eſt, vt A E, ad A C, ita triangulum ABE, ad <lb/> <anchor type="note" xlink:label="note-233-05a" xlink:href="note-233-05"/> triangulum A B C: </s> <s xml:id="echoid-s9105" xml:space="preserve">Eſt autem A E, ipſius <lb/>A C, ſeſquitertia, ex conſtructione; </s> <s xml:id="echoid-s9106" xml:space="preserve">erit <lb/>quo que triangulum ABE, trianguli ABC, <lb/>ſeſquitertium. </s> <s xml:id="echoid-s9107" xml:space="preserve">Cum ergo, vt Archimedes <lb/>in lib. </s> <s xml:id="echoid-s9108" xml:space="preserve">de Quadratura paraboles demõſtra <lb/>uit, parabola quo que ABC, trianguli A-<lb/> <anchor type="note" xlink:label="note-233-06a" xlink:href="note-233-06"/> BC, ſit ſeſquitertia: </s> <s xml:id="echoid-s9109" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> habebunt triangulum A B E, & </s> <s xml:id="echoid-s9110" xml:space="preserve">parabola ABC, ad trian- <anchor type="note" xlink:label="note-233-07a" xlink:href="note-233-07"/> gulum A B C, eandem proportionem. </s> <s xml:id="echoid-s9111" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Ideoque area trianguli A B E, areæ paraboles ABC, æqualis erit. </s> <s xml:id="echoid-s9112" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s9113" xml:space="preserve"/> </p> <div xml:id="echoid-div553" type="float" level="2" n="1"> <figure xlink:label="fig-233-02" xlink:href="fig-233-02a"> <image file="233-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/233-02"/> </figure> <note symbol="e" position="right" xlink:label="note-233-05" xlink:href="note-233-05a" xml:space="preserve">1. ſexti.</note> <note symbol="f" position="right" xlink:label="note-233-06" xlink:href="note-233-06a" xml:space="preserve">11. quinti.</note> <note symbol="g" position="right" xlink:label="note-233-07" xlink:href="note-233-07a" xml:space="preserve">11. quinti.</note> </div> </div> <div xml:id="echoid-div555" type="section" level="1" n="201"> <head xml:id="echoid-head214" xml:space="preserve">FINIS LIBRI QVARTI.</head> <pb o="204" file="234" n="234"/> <figure> <image file="234-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/234-01"/> </figure> </div> <div xml:id="echoid-div556" type="section" level="1" n="202"> <head xml:id="echoid-head215" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ <lb/>LIBER QVINTVS.</head> <figure> <image file="234-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/234-02"/> </figure> </div> <div xml:id="echoid-div557" type="section" level="1" n="203"> <head xml:id="echoid-head216" xml:space="preserve">AREAS <lb/>Solidorum, corporumue perſcrutans.</head> <p style="it"> <s xml:id="echoid-s9114" xml:space="preserve">SVPEREST tertia magnitudinis ſpecies, quæ corpora ſolidaue cõ-<lb/>plectitur. </s> <s xml:id="echoid-s9115" xml:space="preserve">Et quia initio lib. </s> <s xml:id="echoid-s9116" xml:space="preserve">4. </s> <s xml:id="echoid-s9117" xml:space="preserve">dixim{us}, corpora metienda eſſe per <lb/>corpuſcula cubica: </s> <s xml:id="echoid-s9118" xml:space="preserve">ita vt quando dicitur corp{us} aliquod continere <lb/>1000. </s> <s xml:id="echoid-s9119" xml:space="preserve">plam@s, intelligendum ſit, 1000. </s> <s xml:id="echoid-s9120" xml:space="preserve">cubos æquales, quorum ſinguli <lb/>latera habent vni palmo æqualia, aream illi{us} corporis explere: </s> <s xml:id="echoid-s9121" xml:space="preserve">docendum iam <lb/>erit hoc lib. </s> <s xml:id="echoid-s9122" xml:space="preserve">qua ratione cuiuſcunque corporis areainueſtigetur, hoc eſt, numer{us} <lb/>corpuſculorum cubicorum in eo contentorum. </s> <s xml:id="echoid-s9123" xml:space="preserve">Præcipua autem corpora, de qui-<lb/>b{us} acturi ſum{us}, ſunt Parallelepipeda, Priſmata, cubi, Pyramides, Fruſtapyra-<lb/>midum, Cylindri, Coni, Fruſta Conorum, ſphæræ, ſphærarum portiones, quin<unsure/> <lb/>corpora regularia, videlicet Tetraedrum, Hexaedrum ſiue cub{us}, Octaedrum, I-<lb/>coſaedrum, Dodecaedrum, quæ omnia lib 11. </s> <s xml:id="echoid-s9124" xml:space="preserve">ab Euclid. </s> <s xml:id="echoid-s9125" xml:space="preserve">definita ſunt. </s> <s xml:id="echoid-s9126" xml:space="preserve">His ad-<lb/>iungem{us} nonnulla corpora vacua, & </s> <s xml:id="echoid-s9127" xml:space="preserve">alia quædam irregularia.</s> <s xml:id="echoid-s9128" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div558" type="section" level="1" n="204"> <head xml:id="echoid-head217" xml:space="preserve">DE AREA PARALLELEPIP EDO-<lb/>rum, Priſmatum, & Cylindrorum.</head> <head xml:id="echoid-head218" xml:space="preserve"><emph style="sc">Capvt</emph> I.</head> <p> <s xml:id="echoid-s9129" xml:space="preserve"><emph style="sc">PArallelepipedvm</emph> <anchor type="note" xlink:href="" symbol="a"/> eſt figura ſolida ſex figuris quadrilateris, qua- <anchor type="note" xlink:label="note-234-01a" xlink:href="note-234-01"/> rum, quæ ex aduerſo, parallelæ ſunt, contenta. </s> <s xml:id="echoid-s9130" xml:space="preserve">Huiuſmodi figuram ſoli-<lb/>dam exprimit columna aliqua quadrilatera vniformis craſsitiei. </s> <s xml:id="echoid-s9131" xml:space="preserve">Vt figura <lb/>ſolida A B C D E F G H, in qua tam duo plana oppoſita ABCD, EFGH, quam <pb o="205" file="235" n="235" rhead="LIBER QVINTVS."/> duo A D E H, B C F G, & </s> <s xml:id="echoid-s9132" xml:space="preserve">duo ABGH, DCFE, parallelogramma ſunt inter <lb/>ſe parallela, & </s> <s xml:id="echoid-s9133" xml:space="preserve">æqualia, dicitur parallelepipedum, Huius area ita inueſtigabitur. <lb/></s> <s xml:id="echoid-s9134" xml:space="preserve"> <anchor type="note" xlink:label="note-235-01a" xlink:href="note-235-01"/> Sit primò propoſitum parallelepipedum rectangulum habens omnia ſex pa-<lb/>rallelo gramma rectangula, ac proinde omnes eius angulos ſolidos rectos: </s> <s xml:id="echoid-s9135" xml:space="preserve">ſit-<lb/>que longitudo baſis AB, palm. </s> <s xml:id="echoid-s9136" xml:space="preserve">3. </s> <s xml:id="echoid-s9137" xml:space="preserve">latitudo AD, palm. </s> <s xml:id="echoid-s9138" xml:space="preserve">2. </s> <s xml:id="echoid-s9139" xml:space="preserve">& </s> <s xml:id="echoid-s9140" xml:space="preserve">altitudo AH, palm. </s> <s xml:id="echoid-s9141" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s9142" xml:space="preserve">Ducatur ergo latitudo 2. </s> <s xml:id="echoid-s9143" xml:space="preserve">in longitudinem 3. </s> <s xml:id="echoid-s9144" xml:space="preserve">vt producatur baſis palmorum <lb/> <anchor type="handwritten" xlink:label="hd-235-1a" xlink:href="hd-235-1"/> 6. </s> <s xml:id="echoid-s9145" xml:space="preserve">quadratorum, vt lib. </s> <s xml:id="echoid-s9146" xml:space="preserve">4. </s> <s xml:id="echoid-s9147" xml:space="preserve">cap. </s> <s xml:id="echoid-s9148" xml:space="preserve">1. </s> <s xml:id="echoid-s9149" xml:space="preserve">traditum eſt. </s> <s xml:id="echoid-s9150" xml:space="preserve">Deinde baſis hæc 6. </s> <s xml:id="echoid-s9151" xml:space="preserve">palmorum <lb/>ducatur in altitudinem 4. </s> <s xml:id="echoid-s9152" xml:space="preserve">Numerus enim productus 24. </s> <s xml:id="echoid-s9153" xml:space="preserve">indicabit in parallele-<lb/>pipedo contineri 24 cubos, quorũ ſingula latera ſingulos palmos complectun-<lb/>tur, quod ita planum faciemus. </s> <s xml:id="echoid-s9154" xml:space="preserve">Exponatur ſeorſum rectangulum I K L M, æ-<lb/>quale baſi ABCD, intelligaturque altitudo per-<lb/> <anchor type="figure" xlink:label="fig-235-01a" xlink:href="fig-235-01"/> pendicularis L N, 4. </s> <s xml:id="echoid-s9155" xml:space="preserve">palm. </s> <s xml:id="echoid-s9156" xml:space="preserve">Si igitur ducatur la-<lb/>tus I M, palm. </s> <s xml:id="echoid-s9157" xml:space="preserve">2. </s> <s xml:id="echoid-s9158" xml:space="preserve">in IK, plam. </s> <s xml:id="echoid-s9159" xml:space="preserve">3. </s> <s xml:id="echoid-s9160" xml:space="preserve">producetur area <lb/>baſis palmorum quadratorum 6. </s> <s xml:id="echoid-s9161" xml:space="preserve">ſupra quæ ſi <lb/>concipiantur extructi 6. </s> <s xml:id="echoid-s9162" xml:space="preserve">cubiæquales, imple-<lb/>bunt ij parallelepipedum vſq; </s> <s xml:id="echoid-s9163" xml:space="preserve">ad primum pal-<lb/>mum L Q, altitudinis. </s> <s xml:id="echoid-s9164" xml:space="preserve">Si deinde alij 6. </s> <s xml:id="echoid-s9165" xml:space="preserve">cubi æ-<lb/>quales prioribus ſuperimp onantur, implebit@r parallelepipedum vſq; </s> <s xml:id="echoid-s9166" xml:space="preserve">ad ſecun-<lb/>dum palmum altitudinis QP. </s> <s xml:id="echoid-s9167" xml:space="preserve">Et alij 6. </s> <s xml:id="echoid-s9168" xml:space="preserve">@ubi æquales parallelepipedum vſq; <lb/></s> <s xml:id="echoid-s9169" xml:space="preserve">tertium palmum P O, altitudinis implebunt. </s> <s xml:id="echoid-s9170" xml:space="preserve">Denique alij 6. </s> <s xml:id="echoid-s9171" xml:space="preserve">cubi appoſiti to-<lb/>tum parallelepipedum explebunt vſque ad quartũ altitudinis palmum O N. </s> <s xml:id="echoid-s9172" xml:space="preserve"><lb/>Conſtat ergo in toto parallelepipedo exiſtere toties 6. </s> <s xml:id="echoid-s9173" xml:space="preserve">cubospalmares, quoties <lb/>palmus in altitudine continetur, hoc eſt cubos 24.</s> <s xml:id="echoid-s9174" xml:space="preserve"/> </p> <div xml:id="echoid-div558" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-234-01" xlink:href="note-234-01a" xml:space="preserve">30. defin. vn-<lb/>dec.</note> <note position="right" xlink:label="note-235-01" xlink:href="note-235-01a" xml:space="preserve">Area paralle-<lb/>lepipedirectã-<lb/>guli.</note> <handwritten xlink:label="hd-235-1" xlink:href="hd-235-1a"/> <figure xlink:label="fig-235-01" xlink:href="fig-235-01a"> <image file="235-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/235-01"/> </figure> </div> <p> <s xml:id="echoid-s9175" xml:space="preserve">2. </s> <s xml:id="echoid-s9176" xml:space="preserve"><emph style="sc">Intelligatvr</emph> deinde parallelepipedum ABCE, cuius baſes ABCD, <lb/> <anchor type="note" xlink:label="note-235-02a" xlink:href="note-235-02"/> EFGH, ſint Rhombi, vel Rhomboides, ac latera AH, DE, DE, BG, CF, ad ba-<lb/>ſem A B C D, recta, ita vt altitudo ſit A H. </s> <s xml:id="echoid-s9177" xml:space="preserve">Primum ergo inquiratur area baſis <lb/>ABCD, vt lib. </s> <s xml:id="echoid-s9178" xml:space="preserve">4. </s> <s xml:id="echoid-s9179" xml:space="preserve">cap. </s> <s xml:id="echoid-s9180" xml:space="preserve">3. </s> <s xml:id="echoid-s9181" xml:space="preserve">Num. </s> <s xml:id="echoid-s9182" xml:space="preserve">1. </s> <s xml:id="echoid-s9183" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s9184" xml:space="preserve">Hæc deinde in altitudinem AH, du-<lb/>catur. </s> <s xml:id="echoid-s9185" xml:space="preserve">Productus namq; </s> <s xml:id="echoid-s9186" xml:space="preserve">numerus erit parallelepipedi area. </s> <s xml:id="echoid-s9187" xml:space="preserve">Nam ſi fiat rectan-<lb/>gulum IL, baſi AC, æquale, & </s> <s xml:id="echoid-s9188" xml:space="preserve">ſupra illud concipiatur parallelepipedũ rectan-<lb/>gulũ, cuius altitudo LN, altitudini AH, ſit æqualis <anchor type="note" xlink:href="" symbol="a"/> erit hoc parallelepipedum <anchor type="note" xlink:label="note-235-03a" xlink:href="note-235-03"/> parallelepipedo ACE, æquale. </s> <s xml:id="echoid-s9189" xml:space="preserve">Cum ergo parallelepipedum, cuius baſis rectan-<lb/>gulum IL, & </s> <s xml:id="echoid-s9190" xml:space="preserve">altitudo LN, producatur ex altitudine LN, in baſem IL, vt oſten-<lb/>ſem eſt, producetur quoque parallelepipedum ACE, ex altitudine AH, in ba-<lb/>ſem AC, baſi IL, æqualem.</s> <s xml:id="echoid-s9191" xml:space="preserve"/> </p> <div xml:id="echoid-div559" type="float" level="2" n="2"> <note position="right" xlink:label="note-235-02" xlink:href="note-235-02a" xml:space="preserve">Area paralle<unsure/>-<lb/>lepipedi nõre<unsure/>-<lb/>ctanguli.</note> <note symbol="a" position="right" xlink:label="note-235-03" xlink:href="note-235-03a" xml:space="preserve">31. duodec.</note> </div> <p> <s xml:id="echoid-s9192" xml:space="preserve"><emph style="sc">Si</emph> nullum latus parallelepipedi rectum eſt ad baſem, demittenda erit ex ali-<lb/>quo angulo ſupremi parallogrammi ad planum, in quo baſis, linea perpendi-<lb/>cularis, pro altitudine parallelepipedi, eaque diligenter metienda. </s> <s xml:id="echoid-s9193" xml:space="preserve">Sinamq; </s> <s xml:id="echoid-s9194" xml:space="preserve">area <lb/>baſis inueſtigetur vel per cap. </s> <s xml:id="echoid-s9195" xml:space="preserve">1. </s> <s xml:id="echoid-s9196" xml:space="preserve">lib. </s> <s xml:id="echoid-s9197" xml:space="preserve">4. </s> <s xml:id="echoid-s9198" xml:space="preserve">quando eſt rectangula, vel per cap. </s> <s xml:id="echoid-s9199" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s9200" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s9201" xml:space="preserve">quando non eſtrectangula, eaque in altitudinem inuentam duca-<lb/>tur, producetur area propoſiti parallelepipedi. </s> <s xml:id="echoid-s9202" xml:space="preserve">Nam ſi ſupra baſem intelligatur <lb/>parallelepipedum rectum eiuſdem altitudinis cum propoſito parallelepipedo, <lb/> <anchor type="note" xlink:label="note-235-04a" xlink:href="note-235-04"/> <anchor type="note" xlink:href="" symbol="b"/> erunt duo hæc parallelepipeda inter ſeæqualia. </s> <s xml:id="echoid-s9203" xml:space="preserve">Conſtat autẽ ex Num. </s> <s xml:id="echoid-s9204" xml:space="preserve">1. </s> <s xml:id="echoid-s9205" xml:space="preserve">& </s> <s xml:id="echoid-s9206" xml:space="preserve">2.</s> <s xml:id="echoid-s9207" xml:space="preserve"> parallelepipedum rectum gigni ex ductu bafis in altitudinem.</s> <s xml:id="echoid-s9208" xml:space="preserve"/> </p> <div xml:id="echoid-div560" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-235-04" xlink:href="note-235-04a" xml:space="preserve">29. vel 30. <lb/>vndec.</note> </div> <p> <s xml:id="echoid-s9209" xml:space="preserve">3. </s> <s xml:id="echoid-s9210" xml:space="preserve"><emph style="sc">Cvbvs</emph>, qui etiam parallelepipedum quoddam eſt rectangulum, eo dem <lb/> <anchor type="note" xlink:label="note-235-05a" xlink:href="note-235-05"/> modo producitur, nimirum ex latere in ſe, & </s> <s xml:id="echoid-s9211" xml:space="preserve">iterum in productum. </s> <s xml:id="echoid-s9212" xml:space="preserve">Vt ſi latus <lb/>cubi ſit 10. </s> <s xml:id="echoid-s9213" xml:space="preserve">erit eius area 1000. </s> <s xml:id="echoid-s9214" xml:space="preserve">quod decies decem decies procreent 1000.</s> <s xml:id="echoid-s9215" xml:space="preserve"/> </p> <div xml:id="echoid-div561" type="float" level="2" n="4"> <note position="right" xlink:label="note-235-05" xlink:href="note-235-05a" xml:space="preserve">Area c ubi.</note> </div> <note symbol="c" position="right" xml:space="preserve">13. defin. <lb/>vndec.</note> <p> <s xml:id="echoid-s9216" xml:space="preserve">4. </s> <s xml:id="echoid-s9217" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <emph style="sc">Prisma</emph> eſt figura ſolida, quæplanis continetur, quorum aduerſa duo <pb o="206" file="236" n="236" rhead="GEOMETR. PRACT."/> ſunt & </s> <s xml:id="echoid-s9218" xml:space="preserve">æqualia, & </s> <s xml:id="echoid-s9219" xml:space="preserve">ſimilia, & </s> <s xml:id="echoid-s9220" xml:space="preserve">parallela; </s> <s xml:id="echoid-s9221" xml:space="preserve">alia verò parallelogramma. </s> <s xml:id="echoid-s9222" xml:space="preserve">Vt eſt ſo-<lb/>lidum ADF, cuius baſes ſunt pentagona ABCDE, FGHIK, parallela, & </s> <s xml:id="echoid-s9223" xml:space="preserve">æqua-<lb/>lia. </s> <s xml:id="echoid-s9224" xml:space="preserve">Hanc figuram ſolidam repræſentat columna aliqua laterata æqualis craſsi-<lb/>tudinis, cuiu, baſes oppoſitæ ſunt æquales, ſimiles, ac parallelę, ſiue hæ triangu-<lb/>la ſint, ſiue quadrangula, ſiue pentagona, &</s> <s xml:id="echoid-s9225" xml:space="preserve">c. </s> <s xml:id="echoid-s9226" xml:space="preserve">Ex quo fit, vt priſma quodcun-<lb/>que ambiant tot parallelo gramma, quot latera, vel anguli in vnoquo que op-<lb/>poſitorum planorum reperiuntur. </s> <s xml:id="echoid-s9227" xml:space="preserve">Vt propoſitum priſma ambiunt quinque <lb/>parallelogramma ABGF, BCHG, CDIH, DEKI, EAFK. </s> <s xml:id="echoid-s9228" xml:space="preserve">Area porro cuiusli-<lb/> <anchor type="note" xlink:label="note-236-01a" xlink:href="note-236-01"/> bet priſmatis inuenietur, ſi area baſis inquiratur, atque in altitudinem ducatur. <lb/></s> <s xml:id="echoid-s9229" xml:space="preserve">Nam ſi concipiatur parallelepipedum eiuſdem <lb/> <anchor type="figure" xlink:label="fig-236-01a" xlink:href="fig-236-01"/> altitudinis cum priſmate, habens baſem, rectan-<lb/>gulũ baſi priſmatis æquale; </s> <s xml:id="echoid-s9230" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit hoc parallele- <anchor type="note" xlink:label="note-236-02a" xlink:href="note-236-02"/> pipedum priſmati ęquale. </s> <s xml:id="echoid-s9231" xml:space="preserve">Cũ ergo parallelepi-<lb/>pedũ producatur ex ſua baſe in altitudinem, <lb/>procreabitur quoque priſma ex multiplicatio-<lb/>ne ſuę baſis in altitudinem. </s> <s xml:id="echoid-s9232" xml:space="preserve">Area porro baſis <lb/>cognoſcetur ex iis, quæ lib. </s> <s xml:id="echoid-s9233" xml:space="preserve">4. </s> <s xml:id="echoid-s9234" xml:space="preserve">ſcrip ſimus, & </s> <s xml:id="echoid-s9235" xml:space="preserve">altitudo priſmatis, ſi eius latera re-<lb/>cta non ſint ad baſem, exploranda<unsure/> erit, vt cap. </s> <s xml:id="echoid-s9236" xml:space="preserve">præcedente Num. </s> <s xml:id="echoid-s9237" xml:space="preserve">2. </s> <s xml:id="echoid-s9238" xml:space="preserve">altitudinem <lb/>parallelepidi inueſtigandam eſſe præ@p<unsure/>imus.</s> <s xml:id="echoid-s9239" xml:space="preserve"/> </p> <div xml:id="echoid-div562" type="float" level="2" n="5"> <note position="left" xlink:label="note-236-01" xlink:href="note-236-01a" xml:space="preserve">Area priſma-<lb/>tis, tam recti, <lb/>quam obliqui.</note> <figure xlink:label="fig-236-01" xlink:href="fig-236-01a"> <image file="236-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/236-01"/> </figure> <note symbol="a" position="left" xlink:label="note-236-02" xlink:href="note-236-02a" xml:space="preserve">2. coroll. 7. <lb/>duodec.</note> </div> <p> <s xml:id="echoid-s9240" xml:space="preserve">5. </s> <s xml:id="echoid-s9241" xml:space="preserve"><emph style="sc">Cylindrvs</emph> eſt figura ſolida æqualis craſsitiei, quæ duobus circulis <lb/>æqualibus, & </s> <s xml:id="echoid-s9242" xml:space="preserve">æquidiſtantibus, & </s> <s xml:id="echoid-s9243" xml:space="preserve">rotunda ſuperficie inter ipſos interiecta con-<lb/>tinetur, inſtar columnę cuiuſpiam rotundæ. </s> <s xml:id="echoid-s9244" xml:space="preserve">Vt eſt ſolidum A C H, cuius baſes <lb/>ſunt duo circuli ABCD, EFGH, paralleli, & </s> <s xml:id="echoid-s9245" xml:space="preserve">æquales. </s> <s xml:id="echoid-s9246" xml:space="preserve">Huius quo que area pro-<lb/>creabitur ex multiplicatione baſis, ex cap. </s> <s xml:id="echoid-s9247" xml:space="preserve">7. </s> <s xml:id="echoid-s9248" xml:space="preserve">lib. </s> <s xml:id="echoid-s9249" xml:space="preserve">4. </s> <s xml:id="echoid-s9250" xml:space="preserve">inuentę in altitudinem. <lb/></s> <s xml:id="echoid-s9251" xml:space="preserve">quod in Cylindro recto explicabitur, vt Num. </s> <s xml:id="echoid-s9252" xml:space="preserve">1. </s> <s xml:id="echoid-s9253" xml:space="preserve">in parallelepipedo recto fa-<lb/>ctum eſt. </s> <s xml:id="echoid-s9254" xml:space="preserve">Nam ſi verbi gratia baſis Cylindri circularis ABCD, continet 10. </s> <s xml:id="echoid-s9255" xml:space="preserve">pal-<lb/>mos quadratos, explebunt 10. </s> <s xml:id="echoid-s9256" xml:space="preserve">cubi palmares ſupra illos 10. </s> <s xml:id="echoid-s9257" xml:space="preserve">palmos quadratos <lb/>extructi, Cylindrum vſque ad primum palmum altitudinis; </s> <s xml:id="echoid-s9258" xml:space="preserve">at 20. </s> <s xml:id="echoid-s9259" xml:space="preserve">cubi eundem <lb/>explebunt vſque ad ſecundum palmum, &</s> <s xml:id="echoid-s9260" xml:space="preserve">c. </s> <s xml:id="echoid-s9261" xml:space="preserve">Quod ſi Cylindrus obliquus ſit, <lb/>exquirenda erit eius altitudo per lineam perpendicularem ex ſuperiore baſe de-<lb/>miſſam ad planum, in quo inferior baſis exiſtit, atque in hanc altitudinem area <lb/>baſis ex cap. </s> <s xml:id="echoid-s9262" xml:space="preserve">7. </s> <s xml:id="echoid-s9263" xml:space="preserve">lib. </s> <s xml:id="echoid-s9264" xml:space="preserve">4. </s> <s xml:id="echoid-s9265" xml:space="preserve">inuenta multiplicanda. </s> <s xml:id="echoid-s9266" xml:space="preserve">Productus enim numerus dabit <lb/>aream Cylindri propoſiti, <anchor type="note" xlink:href="" symbol="b"/> cum æqualis ſit Cylindro recto eandem cum illo <anchor type="note" xlink:label="note-236-03a" xlink:href="note-236-03"/> baſem, & </s> <s xml:id="echoid-s9267" xml:space="preserve">altitudinem habenti.</s> <s xml:id="echoid-s9268" xml:space="preserve"/> </p> <div xml:id="echoid-div563" type="float" level="2" n="6"> <note symbol="b" position="left" xlink:label="note-236-03" xlink:href="note-236-03a" xml:space="preserve">coroll. 11. <lb/>duodec.</note> </div> </div> <div xml:id="echoid-div565" type="section" level="1" n="205"> <head xml:id="echoid-head219" xml:space="preserve">DE AREA PYRAMIDVM <lb/>& Conorum.</head> <head xml:id="echoid-head220" xml:space="preserve"><emph style="sc">Capvt</emph> II.</head> <p> <s xml:id="echoid-s9269" xml:space="preserve">1. </s> <s xml:id="echoid-s9270" xml:space="preserve"><emph style="sc">PYramis</emph> <anchor type="note" xlink:href="" symbol="c"/> eſt figura ſolida, quę planis continetur ab vno plano ad v- <anchor type="note" xlink:label="note-236-04a" xlink:href="note-236-04"/> num punctum conſtituta. </s> <s xml:id="echoid-s9271" xml:space="preserve">Vt figura ſolida A B C D E F, ad punctum <lb/>F, conſtituta ſupra baſem pentagonam A B C D E, & </s> <s xml:id="echoid-s9272" xml:space="preserve">quam ambiunt <pb o="207" file="237" n="237" rhead="LIBER QVINTVS."/> quinque triangula ABF, BCF, CDF, DEF, AEF, tot nimirum, quot in baſe ſunt <lb/>latera, dicitur pyramis.</s> <s xml:id="echoid-s9273" xml:space="preserve"/> </p> <div xml:id="echoid-div565" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-236-04" xlink:href="note-236-04a" xml:space="preserve">defin. 12. <lb/>vndec.</note> </div> <p> <s xml:id="echoid-s9274" xml:space="preserve"><emph style="sc">Convs</emph> autem eſt figura ſolida rotunda ad vnum punctum conſtituta, ſu-<lb/> <anchor type="note" xlink:label="note-237-01a" xlink:href="note-237-01"/> pra baſem circularem, inſtar pyramidis rotundæ, <lb/> <anchor type="figure" xlink:label="fig-237-01a" xlink:href="fig-237-01"/> qualis eſt figura ABCDE.</s> <s xml:id="echoid-s9275" xml:space="preserve"/> </p> <div xml:id="echoid-div566" type="float" level="2" n="2"> <note position="right" xlink:label="note-237-01" xlink:href="note-237-01a" xml:space="preserve">Area pyra-<lb/>midis, & co-<lb/>ni.</note> <figure xlink:label="fig-237-01" xlink:href="fig-237-01a"> <image file="237-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/237-01"/> </figure> </div> <p> <s xml:id="echoid-s9276" xml:space="preserve"><emph style="sc">Tam</emph> autem pyramidis, quam Coni area pro-<lb/>ducitur ex multiplicatione baſis in tertiã partem <lb/>altitudinis. </s> <s xml:id="echoid-s9277" xml:space="preserve">Cũ enim, vt in præcedenti cap. </s> <s xml:id="echoid-s9278" xml:space="preserve">oſten-<lb/>dimus, ex baſe in totam altitudinem gignatur <lb/>priſma vel Cylindrus eandem habens cum pyra-<lb/>mide, & </s> <s xml:id="echoid-s9279" xml:space="preserve">cono altitudinem; </s> <s xml:id="echoid-s9280" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> producetur ex eadem baſe in tertiam partem al- <anchor type="note" xlink:label="note-237-02a" xlink:href="note-237-02"/> titudinis tertia pars illius priſmatis, vel Cylindri. </s> <s xml:id="echoid-s9281" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/>, Cum ergo pyramisſit tertia <anchor type="handwritten" xlink:label="hd-237-2a" xlink:href="hd-237-2"/> pars illius priſmatis, <anchor type="note" xlink:href="" symbol="c"/> & </s> <s xml:id="echoid-s9282" xml:space="preserve">Conus tertia pars Cylindri, liquet tam pyramidẽ, quam <anchor type="note" xlink:label="note-237-03a" xlink:href="note-237-03"/> Conum produci ex baſe in tertiam partem altitudinis. </s> <s xml:id="echoid-s9283" xml:space="preserve">Ex quo fit, ſi baſis duca-<lb/>tur in totam altitudinem, tertiam partem numeri producti, eſſe quoque aream <lb/> <anchor type="note" xlink:label="note-237-04a" xlink:href="note-237-04"/> pyramidis, vel Coni. </s> <s xml:id="echoid-s9284" xml:space="preserve">Item eandem produci ex tota altitudine in tertiam par-<lb/>tem baſis: </s> <s xml:id="echoid-s9285" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> quod hac ratione tertia pars priſmatis, vel Conigignatur.</s> <s xml:id="echoid-s9286" xml:space="preserve"/> </p> <div xml:id="echoid-div567" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-237-02" xlink:href="note-237-02a" xml:space="preserve">ſchol. 14. <lb/>duodec.</note> <handwritten xlink:label="hd-237-2" xlink:href="hd-237-2a"/> <note symbol="b" position="right" xlink:label="note-237-03" xlink:href="note-237-03a" xml:space="preserve">corol. 7. <lb/>duodec.</note> <note symbol="c" position="right" xlink:label="note-237-04" xlink:href="note-237-04a" xml:space="preserve">10. duodec. <lb/>Areapyrami-<lb/>dis, & coni <lb/>aliter.</note> </div> <p> <s xml:id="echoid-s9287" xml:space="preserve">2. </s> <s xml:id="echoid-s9288" xml:space="preserve"><emph style="sc">Basis</emph> porro pyramidis, ſi triangularis eſt, cognoſcetur, vt lib. </s> <s xml:id="echoid-s9289" xml:space="preserve">4. </s> <s xml:id="echoid-s9290" xml:space="preserve">2. </s> <s xml:id="echoid-s9291" xml:space="preserve">tra-<lb/>ditum eſt: </s> <s xml:id="echoid-s9292" xml:space="preserve">ſi multilatera, reperietur, per ea, quæ eodem lib. </s> <s xml:id="echoid-s9293" xml:space="preserve">cap. </s> <s xml:id="echoid-s9294" xml:space="preserve">3. </s> <s xml:id="echoid-s9295" xml:space="preserve">4. </s> <s xml:id="echoid-s9296" xml:space="preserve">& </s> <s xml:id="echoid-s9297" xml:space="preserve">5. </s> <s xml:id="echoid-s9298" xml:space="preserve">ſcripſi-<lb/> <anchor type="note" xlink:label="note-237-05a" xlink:href="note-237-05"/> mus. </s> <s xml:id="echoid-s9299" xml:space="preserve">Baſis autem Coni inueſtigabitur ex cap. </s> <s xml:id="echoid-s9300" xml:space="preserve">7. </s> <s xml:id="echoid-s9301" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s9302" xml:space="preserve">At verò altitudo <lb/>tam pyramidis, quam Coni, obtinebitur, ſi in vertice ſtatuatur planum baſi æ-<lb/>quidiſtans, ab eoque ad planum, in quo baſis, perpendicularis demittatur, ea-<lb/> <anchor type="note" xlink:label="note-237-06a" xlink:href="note-237-06"/> <anchor type="handwritten" xlink:label="hd-237-2a" xlink:href="hd-237-2"/> que exquiſitè menſuretur. </s> <s xml:id="echoid-s9303" xml:space="preserve">Quamuis enim eadem hæc altitudo inda<unsure/>gari poſsit <lb/>Geo metrice, ſi inclinatio vnius lateris ad baſem, & </s> <s xml:id="echoid-s9304" xml:space="preserve">magnitudo quoque eiuſdem <lb/>lateris cognoſcatur: </s> <s xml:id="echoid-s9305" xml:space="preserve">quia tamen hęc ipſa exploranda ſunt materialiter per ali-<lb/>quodinſtrumentum, præſtatipſam quoque altitudinem ſtatim per inſtrumen-<lb/>tum exquirere, pręſertim per inſtrumentum partium, quod lib. </s> <s xml:id="echoid-s9306" xml:space="preserve">1. </s> <s xml:id="echoid-s9307" xml:space="preserve">cap. </s> <s xml:id="echoid-s9308" xml:space="preserve">1. </s> <s xml:id="echoid-s9309" xml:space="preserve">deſcri <lb/>pſimus: </s> <s xml:id="echoid-s9310" xml:space="preserve">cum inuentio illa Geometrica difficilior ſit, procedatq; </s> <s xml:id="echoid-s9311" xml:space="preserve">ex inclinatio-<lb/>ne, ac latere per inſtrumentum cognitis.</s> <s xml:id="echoid-s9312" xml:space="preserve"/> </p> <div xml:id="echoid-div568" type="float" level="2" n="4"> <note symbol="d" position="right" xlink:label="note-237-05" xlink:href="note-237-05a" xml:space="preserve">1. ſchol. 7. <lb/>duodec. & 11. <lb/>duodec.</note> <note position="right" xlink:label="note-237-06" xlink:href="note-237-06a" xml:space="preserve">Altitudo py-<lb/>ramidis, & <lb/>coni.</note> <handwritten xlink:label="hd-237-2" xlink:href="hd-237-2a"/> </div> <p> <s xml:id="echoid-s9313" xml:space="preserve">3. </s> <s xml:id="echoid-s9314" xml:space="preserve"><emph style="sc">Atqve</emph> hæc, quę diximus, intelligivolumus tam de pyramidibus, C@-<lb/>niſque rectis, quam de obliquis, & </s> <s xml:id="echoid-s9315" xml:space="preserve">Scalenis.</s> <s xml:id="echoid-s9316" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div570" type="section" level="1" n="206"> <head xml:id="echoid-head221" xml:space="preserve">DL AREA FRVSTI PYRA-<lb/>midis, & Coni.</head> <head xml:id="echoid-head222" xml:space="preserve"><emph style="sc">Capvt</emph> III.</head> <p> <s xml:id="echoid-s9317" xml:space="preserve">1. </s> <s xml:id="echoid-s9318" xml:space="preserve"><emph style="sc">FRvstvm</emph> pyramidis, & </s> <s xml:id="echoid-s9319" xml:space="preserve">Coni appello id, quod alij pyramidem decur-<lb/>tatam, & </s> <s xml:id="echoid-s9320" xml:space="preserve">Conum decurtatum dicunt. </s> <s xml:id="echoid-s9321" xml:space="preserve">Sit ergo fruſtum pyramidis <lb/>ABCDEF, cuius baſes ABC, DEF, ſint parallelæ, & </s> <s xml:id="echoid-s9322" xml:space="preserve">ſimiles, & </s> <s xml:id="echoid-s9323" xml:space="preserve">cuius <lb/>area inueſtiganda ſit. </s> <s xml:id="echoid-s9324" xml:space="preserve">Quod duobus modis fieri poteſt. </s> <s xml:id="echoid-s9325" xml:space="preserve">Primũ cogitetur integra <lb/>pyramis ABCH, cuius altitudinẽ HG, perpendicularem ad baſem (licet pyramis <lb/>actu nõ ſit integrata) ita inueniem<emph style="sub">9</emph>. </s> <s xml:id="echoid-s9326" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Quoniã eſt, vt ab AB, ad AH, ita DE, ad DH, <anchor type="note" xlink:label="note-237-07a" xlink:href="note-237-07"/> & </s> <s xml:id="echoid-s9327" xml:space="preserve">permutando, vt AB, ad DE, ita AH, ad DH; </s> <s xml:id="echoid-s9328" xml:space="preserve">erit quoq; </s> <s xml:id="echoid-s9329" xml:space="preserve">diuidẽdo (ſumpta AS, <pb o="208" file="238" n="238" rhead="GEOMETR. PRACT."/> æquali ipſi DE) vt SB, ad DE, ita AD, ad DH. </s> <s xml:id="echoid-s9330" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quia vero plana parallela ABC, <anchor type="note" xlink:label="note-238-01a" xlink:href="note-238-01"/> DEF, ſecant rectas AH, GH, proportionaliter in I, G; </s> <s xml:id="echoid-s9331" xml:space="preserve">erit quoque vt SB, ad <lb/>DE, ita GI, ad IH. </s> <s xml:id="echoid-s9332" xml:space="preserve">Siigitur fiat, vt SB, differentia inter latera homologa AB, <lb/>DE, baſium ad DE, ita GI, altitudo Fruſti pyramidis (quæ cognoſcetur per li-<lb/>neam perpendicularem demiſſam ad baſem ex aliquo puncto plani DEF, et-<lb/>iam producti, ſi opus eſt,) ad aliud, prodibit recta IH, altitudo nimirum pyra-<lb/>midis DEFH: </s> <s xml:id="echoid-s9333" xml:space="preserve">qua addita ad GI, tota altitudo GH, cognita erit. </s> <s xml:id="echoid-s9334" xml:space="preserve">Quocirca ſi per <lb/> <anchor type="note" xlink:label="note-238-02a" xlink:href="note-238-02"/> caput præcedens inueniatur area tam integræ pyramidis ABCH, quam abſciſ-<lb/>ſæ pyramidis D E F H, & </s> <s xml:id="echoid-s9335" xml:space="preserve">hęc ab illa dematur, reliquum fiet Fruſtum <lb/>A B C D E F.</s> <s xml:id="echoid-s9336" xml:space="preserve"/> </p> <div xml:id="echoid-div570" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-237-07" xlink:href="note-237-07a" xml:space="preserve">4. ſexti.</note> <note symbol="a" position="left" xlink:label="note-238-01" xlink:href="note-238-01a" xml:space="preserve">17. vnde.</note> <note position="left" xlink:label="note-238-02" xlink:href="note-238-02a" xml:space="preserve">Area fruſti <lb/>pyramidis.</note> </div> <p> <s xml:id="echoid-s9337" xml:space="preserve">2. </s> <s xml:id="echoid-s9338" xml:space="preserve"><emph style="sc">Non</emph> aliter fruſtum Coni ABCD, inueſtigabitur, vt patet, ſi integer Co-<lb/> <anchor type="note" xlink:label="note-238-03a" xlink:href="note-238-03"/> nus intelligatur ABH, &</s> <s xml:id="echoid-s9339" xml:space="preserve">c.</s> <s xml:id="echoid-s9340" xml:space="preserve"/> </p> <div xml:id="echoid-div571" type="float" level="2" n="2"> <note position="left" xlink:label="note-238-03" xlink:href="note-238-03a" xml:space="preserve">Areafruſti <lb/>coni.</note> </div> <p> <s xml:id="echoid-s9341" xml:space="preserve">3. </s> <s xml:id="echoid-s9342" xml:space="preserve"><emph style="sc">Alio</emph> modo idem fruſtum tam Pyramidis, quam Coni cognoſcemus, <lb/>etiamſi neque pyramis, neque conus integretur. </s> <s xml:id="echoid-s9343" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Fiant quadrata K L M N, NOPQ, baſibus ABC, DEF, notis æqualia, inueniaturque inter quadrata KM, <lb/> <anchor type="note" xlink:label="note-238-04a" xlink:href="note-238-04"/> NP, ſuperficies media proportionalis, qua-<lb/> <anchor type="figure" xlink:label="fig-238-01a" xlink:href="fig-238-01"/> lis eſt rectangula figura OK, producto la-<lb/>tere O P, ad R. </s> <s xml:id="echoid-s9344" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quoniam enim eſt, vt <anchor type="note" xlink:label="note-238-05a" xlink:href="note-238-05"/> MN, ad NO, ita KM, ad NR. </s> <s xml:id="echoid-s9345" xml:space="preserve">Item vt KN, <lb/>ad NQ, ita NR, ad NP. </s> <s xml:id="echoid-s9346" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> eſtque M N, ad <anchor type="note" xlink:label="note-238-06a" xlink:href="note-238-06"/> NO, vt KN, ad N Q: </s> <s xml:id="echoid-s9347" xml:space="preserve">erit quadratum K M, <lb/>ad rectangulum NR, vt rectangulum N R, <lb/>ad quadratum N P: </s> <s xml:id="echoid-s9348" xml:space="preserve">ideoque N R, medio <lb/>loco proportionale eſt inter quadrata <lb/>KM, NP. </s> <s xml:id="echoid-s9349" xml:space="preserve">Quamobrem, ſi radix quadrata <lb/>baſis ABC, notæ, id eſt, latus KN, quadrati KM, ducatur in radicem quadratam <lb/>baſis DEF, notę, hoc eſt, in latus NO, quadrati N P, producetur area rectangu-<lb/>li N R.</s> <s xml:id="echoid-s9350" xml:space="preserve"/> </p> <div xml:id="echoid-div572" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-238-04" xlink:href="note-238-04a" xml:space="preserve">14. ſecundi.</note> <figure xlink:label="fig-238-01" xlink:href="fig-238-01a"> <image file="238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/238-01"/> </figure> <note symbol="c" position="left" xlink:label="note-238-05" xlink:href="note-238-05a" xml:space="preserve">1. ſexti.</note> <note symbol="d" position="left" xlink:label="note-238-06" xlink:href="note-238-06a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s9351" xml:space="preserve"><emph style="sc">Iam</emph> vero ducatur GI, altitudo fruſti in ſummam ex quadrato KM, hoc eſt, <lb/>ex baſe ABC, & </s> <s xml:id="echoid-s9352" xml:space="preserve">quadrato NP, ſiue baſe DEF, & </s> <s xml:id="echoid-s9353" xml:space="preserve">ſuperficie NR, media propor-<lb/>tionali inter baſes, vel dicta quadrata collectam. </s> <s xml:id="echoid-s9354" xml:space="preserve">Productus enim numerus <lb/>triplus erit fruſti pyramidis A B C D E F; </s> <s xml:id="echoid-s9355" xml:space="preserve">ideoque tertia producti pars area erit <lb/>prædicti fruſti. </s> <s xml:id="echoid-s9356" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Quoniam enim priſma, quod fit ex GH, altitudine pyramidis <anchor type="note" xlink:label="note-238-07a" xlink:href="note-238-07"/> in baſem ABC, ſiue quadratum KM, triplum eſt pyramidis ABCDEFH: </s> <s xml:id="echoid-s9357" xml:space="preserve">erit <lb/>quoque parallelepipedum factum ex GI, in quadratum KM, vna cum paralle-<lb/>lepipedo facto ex IH, in idem quadratum KM, triplum pyramidis eiuſdem. </s> <s xml:id="echoid-s9358" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Eſt <anchor type="note" xlink:label="note-238-08a" xlink:href="note-238-08"/> aũt & </s> <s xml:id="echoid-s9359" xml:space="preserve">ablatum parallelepipedum factum ex IH, in baſem DEF, hoc eſt, in qua-<lb/>dratum NP, triplum ablatæ pyramidis DEFH, <anchor type="note" xlink:href="" symbol="g"/> Igitur & </s> <s xml:id="echoid-s9360" xml:space="preserve">reliquum, quod fit ex <anchor type="note" xlink:label="note-238-09a" xlink:href="note-238-09"/> GI, in quadratum KM, vna cum iis, quæfiunt ex IH, in KP, & </s> <s xml:id="echoid-s9361" xml:space="preserve">in RM, triplum <lb/>erit fruſti reliqui ABCDEF.</s> <s xml:id="echoid-s9362" xml:space="preserve"/> </p> <div xml:id="echoid-div573" type="float" level="2" n="4"> <note symbol="e" position="left" xlink:label="note-238-07" xlink:href="note-238-07a" xml:space="preserve">7. duodec.</note> <note symbol="f" position="left" xlink:label="note-238-08" xlink:href="note-238-08a" xml:space="preserve">7. duodec.</note> <note symbol="g" position="left" xlink:label="note-238-09" xlink:href="note-238-09a" xml:space="preserve">5. quinti.</note> </div> <p> <s xml:id="echoid-s9363" xml:space="preserve">4. </s> <s xml:id="echoid-s9364" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> <emph style="sc">Qvia</emph> verò æquales ſuperficies A B C, K M, ad ſuperficies æquales <anchor type="note" xlink:label="note-238-10a" xlink:href="note-238-10"/> DEF, NR, eandem habent proportionem, erit permutando ABC, ad DEF, vt <lb/>KM, ad NP. </s> <s xml:id="echoid-s9365" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> ideoque latus AB, ad latus DE, erit, vt latus KN, ad latus NQ, &</s> <s xml:id="echoid-s9366" xml:space="preserve"> <anchor type="note" xlink:label="note-238-11a" xlink:href="note-238-11"/> diuidendo (ſubtracta recta AS, æquali ipſi DE, ex AB,) SB, ad DE, vt KQ, ad <lb/>QN. </s> <s xml:id="echoid-s9367" xml:space="preserve">Eſt autem, vt Num. </s> <s xml:id="echoid-s9368" xml:space="preserve">1. </s> <s xml:id="echoid-s9369" xml:space="preserve">demonſtrauimus, vt SB, ad DE, ita GI, ad IH. </s> <s xml:id="echoid-s9370" xml:space="preserve">Igi-<lb/>tur erit etiam vt KQ, ad QN, ideoque vt MO, ad ON, ita GI, ad IH. </s> <s xml:id="echoid-s9371" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Sed vt KQ, ad QN, ita eſt KP, ad PN: </s> <s xml:id="echoid-s9372" xml:space="preserve">Et vt MO, ad ON, ita MR, ad RN. </s> <s xml:id="echoid-s9373" xml:space="preserve">Igitur erit <lb/> <anchor type="note" xlink:label="note-238-12a" xlink:href="note-238-12"/> <pb o="209" file="239" n="239" rhead="LIBER QVINTVS."/> quoque, vt GI, ad IH, ita tam KP, ad PN, quam MR, ad RN. </s> <s xml:id="echoid-s9374" xml:space="preserve">Contractisergo <lb/>hiſce magnitudinibus ad numeros, <anchor type="note" xlink:href="" symbol="a"/> erit numerus factus ex GI, primò in PN, <anchor type="note" xlink:label="note-239-01a" xlink:href="note-239-01"/> quartum, æqualis ei, quifit ex IH, ſecundò in KP, tertium. </s> <s xml:id="echoid-s9375" xml:space="preserve">Item numerus fa-<lb/>ctus ex GI, primò in RN, quartum æqualis ei, qui fit ex IH, ſecundò in MR, ter-<lb/>tium: </s> <s xml:id="echoid-s9376" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Ac propterea duo, qui fiunt ex GI, in PN, & </s> <s xml:id="echoid-s9377" xml:space="preserve">ex GI, in RN, æquales <anchor type="note" xlink:label="note-239-02a" xlink:href="note-239-02"/> erunt duobus, qui fiunt ex IH, in KP, & </s> <s xml:id="echoid-s9378" xml:space="preserve">IH, in MR. </s> <s xml:id="echoid-s9379" xml:space="preserve">Adiecto ergo communi, qui <lb/>fit ex GI, in KM, erunt tres, qui fiunt ex GI, in KM, & </s> <s xml:id="echoid-s9380" xml:space="preserve">in PN, & </s> <s xml:id="echoid-s9381" xml:space="preserve">in RN, æquales <lb/>tribus, qui fiunt ex GI, in KM, & </s> <s xml:id="echoid-s9382" xml:space="preserve">ex IH, in KP, & </s> <s xml:id="echoid-s9383" xml:space="preserve">in MR. </s> <s xml:id="echoid-s9384" xml:space="preserve">Sed hi poſteriores tres <lb/>tripli ſunt fruſti pyramidis ABCDEF, vt ad fin. </s> <s xml:id="echoid-s9385" xml:space="preserve">Num. </s> <s xml:id="echoid-s9386" xml:space="preserve">3. </s> <s xml:id="echoid-s9387" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s9388" xml:space="preserve">Ergo <lb/>& </s> <s xml:id="echoid-s9389" xml:space="preserve">priores tres, qui nimirum fiunt ex GI, altitudine fruſti in KM, & </s> <s xml:id="echoid-s9390" xml:space="preserve">in PN, & </s> <s xml:id="echoid-s9391" xml:space="preserve">in <lb/>RN, hoc eſt, in ſummam ex KM, baſi ABC, æquali, & </s> <s xml:id="echoid-s9392" xml:space="preserve">ex PN, baſi DEF, æqua-<lb/>li, & </s> <s xml:id="echoid-s9393" xml:space="preserve">ex RN, media proportionali inter baſes collectam, tripli erunt eiuſdem fru-<lb/>ſti: </s> <s xml:id="echoid-s9394" xml:space="preserve">ideoque tertia eorum pars æqualis erit areæ fruſti. </s> <s xml:id="echoid-s9395" xml:space="preserve">quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s9396" xml:space="preserve"/> </p> <div xml:id="echoid-div574" type="float" level="2" n="5"> <note symbol="h" position="left" xlink:label="note-238-10" xlink:href="note-238-10a" xml:space="preserve">7. quinti.</note> <note symbol="i" position="left" xlink:label="note-238-11" xlink:href="note-238-11a" xml:space="preserve">22. ſexti.</note> <note symbol="k" position="left" xlink:label="note-238-12" xlink:href="note-238-12a" xml:space="preserve">1. ſexti.</note> <note symbol="a" position="right" xlink:label="note-239-01" xlink:href="note-239-01a" xml:space="preserve">19. ſept.</note> <note symbol="b" position="right" xlink:label="note-239-02" xlink:href="note-239-02a" xml:space="preserve">2. pronunc.</note> </div> <p> <s xml:id="echoid-s9397" xml:space="preserve">5. </s> <s xml:id="echoid-s9398" xml:space="preserve"><emph style="sc">Eadem</emph> ratione fruſtũ Coni ABDC, producetur ex altitudine GI, in ſum-<lb/>mam ex baſe AB, & </s> <s xml:id="echoid-s9399" xml:space="preserve">baſe CD, & </s> <s xml:id="echoid-s9400" xml:space="preserve">ſuperficie media proportionali inter baſes colle-<lb/>ctam: </s> <s xml:id="echoid-s9401" xml:space="preserve">vt conſtat, ſi concipiantur quadrata KM, NP, baſibus æqualia, & </s> <s xml:id="echoid-s9402" xml:space="preserve">proinde ſu-<lb/>perficies RN, media proportionalis inter baſes, &</s> <s xml:id="echoid-s9403" xml:space="preserve">c.</s> <s xml:id="echoid-s9404" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div576" type="section" level="1" n="207"> <head xml:id="echoid-head223" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s9405" xml:space="preserve">1. </s> <s xml:id="echoid-s9406" xml:space="preserve"><emph style="sc">Si</emph> ea, quæ hactenus dicta ſunt, rebus materialibus accommodentur, licebit <lb/> <anchor type="note" xlink:label="note-239-03a" xlink:href="note-239-03"/> nobis per cap. </s> <s xml:id="echoid-s9407" xml:space="preserve">1. </s> <s xml:id="echoid-s9408" xml:space="preserve">metiri murum quemcunque vniformis craſſitiei, tanquam pa-<lb/>rallelepipedum quoddam, cuius longitudo eadem ſit, quæ longitudo muri: </s> <s xml:id="echoid-s9409" xml:space="preserve">lati-<lb/>tudo verò eadem quæ muri latitudo: </s> <s xml:id="echoid-s9410" xml:space="preserve">altitudo denique, ſiue profunditas eadem, <lb/>quæ altitudo muri.</s> <s xml:id="echoid-s9411" xml:space="preserve"/> </p> <div xml:id="echoid-div576" type="float" level="2" n="1"> <note position="right" xlink:label="note-239-03" xlink:href="note-239-03a" xml:space="preserve">Solidit{as} mu-<lb/>ri.</note> </div> <p> <s xml:id="echoid-s9412" xml:space="preserve"><emph style="sc">Eademqve</emph> ratione metiemur fruſtum alicuius marmoris, vel alterius ſaxi, <lb/>tanquam priſma quodpiam, ſi vniformem habeat craſſitiem, lateraque ad baſes <lb/>ſint recta.</s> <s xml:id="echoid-s9413" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9414" xml:space="preserve"><emph style="sc">Non</emph> aliter ſaccum tritici, tanquam Cylindrum quendam, dimetiemur, plus <lb/> <anchor type="note" xlink:label="note-239-04a" xlink:href="note-239-04"/> minus. </s> <s xml:id="echoid-s9415" xml:space="preserve">Nam cum ſaccus non ſit accuratus Cylindrus, vera eius menſura haberi <lb/>non poteſt.</s> <s xml:id="echoid-s9416" xml:space="preserve"/> </p> <div xml:id="echoid-div577" type="float" level="2" n="2"> <note position="right" xlink:label="note-239-04" xlink:href="note-239-04a" xml:space="preserve">Solidit{as} ali-<lb/>cui{us} fruſti <lb/>marmoris.</note> </div> <p> <s xml:id="echoid-s9417" xml:space="preserve">2. </s> <s xml:id="echoid-s9418" xml:space="preserve"><emph style="sc">Rvrsvs</emph> per caput 2. </s> <s xml:id="echoid-s9419" xml:space="preserve">aceruum tritici, tanquam conum aliquem, eodem <lb/>modo, plus minus, metiri licebit. </s> <s xml:id="echoid-s9420" xml:space="preserve">Itaque ſi cognoſcemus, quot grana, vel libræ <lb/>tritici in cubo, verbi gratia, vnius palmi contineantur, multiplicenturque grana, <lb/> <anchor type="note" xlink:label="note-239-05a" xlink:href="note-239-05"/> vel libræ vnius cubiti in numerum cuborum, qui in toto ſacco, vel aceruo reperti <lb/>ſunt, producetur numerus granorum, vellibrarum in eodem ſacco, vel aceruo exi-<lb/> <anchor type="note" xlink:label="note-239-06a" xlink:href="note-239-06"/> ſtentium.</s> <s xml:id="echoid-s9421" xml:space="preserve"/> </p> <div xml:id="echoid-div578" type="float" level="2" n="3"> <note position="right" xlink:label="note-239-05" xlink:href="note-239-05a" xml:space="preserve">Capacit{as} ſac-<lb/>citritici.</note> <note position="right" xlink:label="note-239-06" xlink:href="note-239-06a" xml:space="preserve">Capacit{as} a-<lb/>ceruitritici.</note> </div> <p> <s xml:id="echoid-s9422" xml:space="preserve"><emph style="sc">Sic</emph> etiam, ſi detur vas aliquod excauatum in modum parallelepipedi, aut Cy-<lb/>lindri, ſciemus eius capacitatem, ſi eius parallelepipedum interius aut Cylindrum, <lb/>non ſecus, ac ſi ſolida figura eſſet, metiemur: </s> <s xml:id="echoid-s9423" xml:space="preserve">ita vt ſi cognitum fuerit, quot men-<lb/>ſuræ aquæ alteriusue liquoris in cubo, verbi gratia, vnius palmi contineantur, <lb/>ignorari non poſsit, quot menſurę in cubis in toto vaſe contentis comprehen-<lb/> <anchor type="note" xlink:label="note-239-07a" xlink:href="note-239-07"/> dantur, ſi nimirum menſuræ vnius cubi ducantur in numerum cuborum, quos <lb/>in vaſe comprehendi inuenimus.</s> <s xml:id="echoid-s9424" xml:space="preserve"/> </p> <div xml:id="echoid-div579" type="float" level="2" n="4"> <note position="right" xlink:label="note-239-07" xlink:href="note-239-07a" xml:space="preserve">Capacit{as} va-<lb/>ſis excauati.</note> </div> <p> <s xml:id="echoid-s9425" xml:space="preserve">4. </s> <s xml:id="echoid-s9426" xml:space="preserve"><emph style="sc">Deniqve</emph> ſi deſideretur ſoliditas alicuius vaſis, quod tam interius, <lb/>quam exterius formam habeat parallelepipedi, ſeu priſmatis, Cylindriue, me- <pb o="210" file="240" n="240" rhead="GEOMETR. PRACT."/> tienda erit vtraque figura tuminterior, tum exterior. </s> <s xml:id="echoid-s9427" xml:space="preserve">Si namque illa ex hac de-<lb/> <anchor type="note" xlink:label="note-240-01a" xlink:href="note-240-01"/> trahetur, reliqua fiet ſoliditas vaſis excauati.</s> <s xml:id="echoid-s9428" xml:space="preserve"/> </p> <div xml:id="echoid-div580" type="float" level="2" n="5"> <note position="left" xlink:label="note-240-01" xlink:href="note-240-01a" xml:space="preserve">Solidit{as} vaſis <lb/>excauati.</note> </div> </div> <div xml:id="echoid-div582" type="section" level="1" n="208"> <head xml:id="echoid-head224" xml:space="preserve">DE AREA QVINQVE COR-<lb/>porum regularium.</head> <head xml:id="echoid-head225" xml:space="preserve"><emph style="sc">Capvt</emph> IV.</head> <p> <s xml:id="echoid-s9429" xml:space="preserve">1. </s> <s xml:id="echoid-s9430" xml:space="preserve"><emph style="sc">QVinqve</emph> tantum ſunt corpora regularia, Tetraedrum, Hexaedrum, <lb/>Octaedrum, Dodecaedrum, & </s> <s xml:id="echoid-s9431" xml:space="preserve">Ico ſaedrum, vt in ſcholio propoſ. </s> <s xml:id="echoid-s9432" xml:space="preserve">18. <lb/></s> <s xml:id="echoid-s9433" xml:space="preserve">libr. </s> <s xml:id="echoid-s9434" xml:space="preserve">13. </s> <s xml:id="echoid-s9435" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s9436" xml:space="preserve">demonſtrauimus: </s> <s xml:id="echoid-s9437" xml:space="preserve">quæ ſic ab Euclide libr. </s> <s xml:id="echoid-s9438" xml:space="preserve">11. </s> <s xml:id="echoid-s9439" xml:space="preserve">defi-<lb/>niuntur.</s> <s xml:id="echoid-s9440" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9441" xml:space="preserve"><emph style="sc">Tetraedrvm</emph> eſt figura ſolida ſub quatuor triangulis æqualibus, & </s> <s xml:id="echoid-s9442" xml:space="preserve">æ-<lb/>quilateris contenta. </s> <s xml:id="echoid-s9443" xml:space="preserve">qualem figuram exprimit pyramis triangularis æquila-<lb/>tera.</s> <s xml:id="echoid-s9444" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9445" xml:space="preserve"><emph style="sc">Hexaedrvm</emph> eſt figura ſolida ſub ſex quadratis æqualibus contenta. </s> <s xml:id="echoid-s9446" xml:space="preserve">qua-<lb/>lem refert cubus, ſeu parallelepipedum baſium quadratarum, in quo omnes tres <lb/>dimenſiones ſunt æquales.</s> <s xml:id="echoid-s9447" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9448" xml:space="preserve"><emph style="sc">Octaedrvm</emph> eſt figura ſolida ſub octo triangulis æqualibus, & </s> <s xml:id="echoid-s9449" xml:space="preserve">æquilate-<lb/>ris contenta.</s> <s xml:id="echoid-s9450" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9451" xml:space="preserve"><emph style="sc">Dodecaedrvm</emph> eſt figura ſolida ſub duodecim pentagonis æqualibus, <lb/>& </s> <s xml:id="echoid-s9452" xml:space="preserve">æquilateris, & </s> <s xml:id="echoid-s9453" xml:space="preserve">æquiangulis contenta.</s> <s xml:id="echoid-s9454" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9455" xml:space="preserve"><emph style="sc">Icosaedrvm</emph> eſt figura ſolida ſub 20. </s> <s xml:id="echoid-s9456" xml:space="preserve">triangulis æqualibus, & </s> <s xml:id="echoid-s9457" xml:space="preserve">æquilate-<lb/>ris contenta.</s> <s xml:id="echoid-s9458" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9459" xml:space="preserve">2. </s> <s xml:id="echoid-s9460" xml:space="preserve"><emph style="sc">Cvbi</emph> ſiue Hexaedri aream gigni ex multiplicatione lateris in ſe, & </s> <s xml:id="echoid-s9461" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-240-02a" xlink:href="note-240-02"/> iterum in productum, cap. </s> <s xml:id="echoid-s9462" xml:space="preserve">1. </s> <s xml:id="echoid-s9463" xml:space="preserve">Num. </s> <s xml:id="echoid-s9464" xml:space="preserve">3. </s> <s xml:id="echoid-s9465" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s9466" xml:space="preserve">Item pyramidem, ſeu Tetrae-<lb/>drum produci ex eius altitudine (quæ mechanicè cognoſcetur, vt c. </s> <s xml:id="echoid-s9467" xml:space="preserve">2. </s> <s xml:id="echoid-s9468" xml:space="preserve">Num. </s> <s xml:id="echoid-s9469" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s9470" xml:space="preserve">traditum eſt) in tertiam baſis partem: </s> <s xml:id="echoid-s9471" xml:space="preserve">vel ex eius baſe in tertiam partem altitu-<lb/>dinis, declara uimus cap. </s> <s xml:id="echoid-s9472" xml:space="preserve">2. </s> <s xml:id="echoid-s9473" xml:space="preserve">Num. </s> <s xml:id="echoid-s9474" xml:space="preserve">1. </s> <s xml:id="echoid-s9475" xml:space="preserve">Quod ſi Geometricè inuenire lubeat altitu-<lb/> <anchor type="note" xlink:label="note-240-03a" xlink:href="note-240-03"/> dinem Tetraedri, ita faciemus. </s> <s xml:id="echoid-s9476" xml:space="preserve">Quoniam quadratum diametri ſphæræ Tetrae-<lb/>drum ambientis eſt, vt 2. </s> <s xml:id="echoid-s9477" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s9478" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quod diameter ſit potentia ſeſquialtera lateris <anchor type="note" xlink:label="note-240-04a" xlink:href="note-240-04"/> pyramidis: </s> <s xml:id="echoid-s9479" xml:space="preserve">Sifiat, vt 2. </s> <s xml:id="echoid-s9480" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s9481" xml:space="preserve">ita quadratũ lateris Tetraedri ad aliud, pro dibit qua-<lb/>dratum diametri ſphærę, eiuſque quadrati quadrata radix diametrum ipſam ex-<lb/> <anchor type="note" xlink:label="note-240-05a" xlink:href="note-240-05"/> hibebit, <anchor type="note" xlink:href="" symbol="b"/> cuius duæ tertiæ partes altitudinem Tetraedri offerent.</s> <s xml:id="echoid-s9482" xml:space="preserve"/> </p> <div xml:id="echoid-div582" type="float" level="2" n="1"> <note position="left" xlink:label="note-240-02" xlink:href="note-240-02a" xml:space="preserve">Area cubi, & <lb/>Tetraedri.</note> <note position="left" xlink:label="note-240-03" xlink:href="note-240-03a" xml:space="preserve">Altitudo Te-<lb/>traedri.</note> <note symbol="a" position="left" xlink:label="note-240-04" xlink:href="note-240-04a" xml:space="preserve">13 tertiide-<lb/>cimi.</note> <note symbol="b" position="left" xlink:label="note-240-05" xlink:href="note-240-05a" xml:space="preserve">2. coroll. 13. <lb/>tertiidec.</note> </div> <p> <s xml:id="echoid-s9483" xml:space="preserve">3. </s> <s xml:id="echoid-s9484" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <emph style="sc">Qvoniam</emph> verò Octaedrum diuiditur in duas pyramides ſimiles, &</s> <s xml:id="echoid-s9485" xml:space="preserve"> <anchor type="note" xlink:label="note-240-06a" xlink:href="note-240-06"/> æquales, quarum baſis communis eſt quadratum à latere deſcriptum: </s> <s xml:id="echoid-s9486" xml:space="preserve">ſi vtriuſ-<lb/>que pyramidis inueſtigetur area, ignorari non poterit area Octaedri, cum ex <lb/> <anchor type="note" xlink:label="note-240-07a" xlink:href="note-240-07"/> areis illarum pyramidum conflata ſit. </s> <s xml:id="echoid-s9487" xml:space="preserve">Producetur autem area illarum duarum <lb/>pyramidum, ſi quadratum lateris Octaedri ducatur in diametrum Octaedri, & </s> <s xml:id="echoid-s9488" xml:space="preserve"><lb/>producti numeri tertia pars capiatur. </s> <s xml:id="echoid-s9489" xml:space="preserve">quia pro ductus ille numerus ex quadrato <lb/>lateris Octaedri in eiuſdem diametrum, eſt parallelepipedum duarum illarum <lb/>pyramidum triplum: </s> <s xml:id="echoid-s9490" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> propterea quod ſemiſsis illius parallelepipedi eandem <anchor type="note" xlink:label="note-240-08a" xlink:href="note-240-08"/> habens baſem, & </s> <s xml:id="echoid-s9491" xml:space="preserve">altitudinem, cum vtralibet pyramidum, tripla eſt vnius pyra-<lb/>midis. </s> <s xml:id="echoid-s9492" xml:space="preserve">Diameter porro Octaedri, quę à diametro ſphærę, vel quadrati lateris. </s> <s xml:id="echoid-s9493" xml:space="preserve">O-<lb/> <anchor type="note" xlink:label="note-240-09a" xlink:href="note-240-09"/> ctaedri non differt, inuenietur, ſi ex duplo quadrati lateris radix quadrata erua- <pb o="211" file="241" n="241" rhead="LIBER QVINTVS."/> tur; </s> <s xml:id="echoid-s9494" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quod tam quadratum ex diametro quadrati deſcriptum duplum ſit qua- <anchor type="note" xlink:label="note-241-01a" xlink:href="note-241-01"/> drati lateris, <anchor type="note" xlink:href="" symbol="b"/> quam quadratum diametri ſphæræ quadratilateris Octaedri. </s> <s xml:id="echoid-s9495" xml:space="preserve">Se- miſsis verò huius diametri, altitudo erit vtriuſuis Pyramidis. </s> <s xml:id="echoid-s9496" xml:space="preserve">Quare ſi hęcalti-<lb/> <anchor type="note" xlink:label="note-241-02a" xlink:href="note-241-02"/> tudo ducatur in tertiam partem quadrati lateris, producetur area vnius pyrami-<lb/>dis, id eſt, ſemiſsis Octaedri: </s> <s xml:id="echoid-s9497" xml:space="preserve">ac proinde duplum huius pyramidis aream totius <lb/>Octaedri indicabit.</s> <s xml:id="echoid-s9498" xml:space="preserve"/> </p> <div xml:id="echoid-div583" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-240-06" xlink:href="note-240-06a" xml:space="preserve">2. coroll. 14. <lb/>tertiidec.</note> <note position="left" xlink:label="note-240-07" xlink:href="note-240-07a" xml:space="preserve">Area Octae-<lb/>dri.</note> <note symbol="d" position="left" xlink:label="note-240-08" xlink:href="note-240-08a" xml:space="preserve">coroll. 7. <lb/>duodec.</note> <note position="left" xlink:label="note-240-09" xlink:href="note-240-09a" xml:space="preserve">Diameter <lb/>Octaedri.</note> <note symbol="a" position="right" xlink:label="note-241-01" xlink:href="note-241-01a" xml:space="preserve">ſchol. 47. <lb/>primi.</note> <note symbol="b" position="right" xlink:label="note-241-02" xlink:href="note-241-02a" xml:space="preserve">14. tertii-<lb/>dec.</note> </div> <p> <s xml:id="echoid-s9499" xml:space="preserve">4. </s> <s xml:id="echoid-s9500" xml:space="preserve"><emph style="sc">Deinde</emph> quia ductis ex centro Dodecaedri ad omnes eius angulos re-<lb/> <anchor type="note" xlink:label="note-241-03a" xlink:href="note-241-03"/> ctis lineis, Dodecaedrum in 12. </s> <s xml:id="echoid-s9501" xml:space="preserve">pyramides pentagonas æquales diuiditur; </s> <s xml:id="echoid-s9502" xml:space="preserve">ſi area <lb/>vnius pyramidis per cap. </s> <s xml:id="echoid-s9503" xml:space="preserve">2. </s> <s xml:id="echoid-s9504" xml:space="preserve">inuenta multip licetur per 12. </s> <s xml:id="echoid-s9505" xml:space="preserve">procreabitur area to-<lb/>tius Dodecaedri. </s> <s xml:id="echoid-s9506" xml:space="preserve">Vtautem vnius pyramidis area habeatur, neceſſe eſt, & </s> <s xml:id="echoid-s9507" xml:space="preserve">aream <lb/>baſis pentagonæ inueſtigare ex latere dato, per ea, quę lib. </s> <s xml:id="echoid-s9508" xml:space="preserve">4. </s> <s xml:id="echoid-s9509" xml:space="preserve">cap. </s> <s xml:id="echoid-s9510" xml:space="preserve">5. </s> <s xml:id="echoid-s9511" xml:space="preserve">ſcrip ſimus, <lb/>& </s> <s xml:id="echoid-s9512" xml:space="preserve">pyramidis altitu dinem, vtiam docebo. </s> <s xml:id="echoid-s9513" xml:space="preserve">Ex ſuperiori plano producto demit-<lb/>tatur ad planum baſis oppoſitæ linea perpendicularis: </s> <s xml:id="echoid-s9514" xml:space="preserve">Huius enim ſemiſsis di-<lb/>ligenter inquiſita in partibus lateris Dodecaedri, per inſtrumentum partium lib. <lb/></s> <s xml:id="echoid-s9515" xml:space="preserve">1. </s> <s xml:id="echoid-s9516" xml:space="preserve">cap. </s> <s xml:id="echoid-s9517" xml:space="preserve">1. </s> <s xml:id="echoid-s9518" xml:space="preserve">conſtructum, dabit pyramidis altitudinem quęſitam, quemadmodum <lb/>& </s> <s xml:id="echoid-s9519" xml:space="preserve">tota perpendicularis altitudinem Dodecaedri exhibet. </s> <s xml:id="echoid-s9520" xml:space="preserve">Quamtamen Geo-<lb/>metricè ita quoq; </s> <s xml:id="echoid-s9521" xml:space="preserve">deprehendemus. </s> <s xml:id="echoid-s9522" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quia cubus in Dodecaedro deſcriptus ea- <anchor type="note" xlink:label="note-241-04a" xlink:href="note-241-04"/> dem ſphęra, qua Dodecaedrum, comprehenditur, eiuſquelatus vnum angulum <lb/>Pentagoni Dodecaedri ſubtendit; </s> <s xml:id="echoid-s9523" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s9524" xml:space="preserve">eadem diameter eſt ſphærę, Dodecae-<lb/>dri, & </s> <s xml:id="echoid-s9525" xml:space="preserve">cubi: </s> <s xml:id="echoid-s9526" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Sirecta ſubtendens angulum pentagoni inueſtigetur, habebitur <anchor type="note" xlink:label="note-241-05a" xlink:href="note-241-05"/> latus cubi: </s> <s xml:id="echoid-s9527" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Et quia diameter ſphærę potentia eſt tripla lateris cubi, ſi quadra- tumlateris cubi inuenti triplicetur, habebitur quadratum diametri ſphærę, vel <lb/> <anchor type="note" xlink:label="note-241-06a" xlink:href="note-241-06"/> cubi, cuius radix quadrata ipſam diametrum dabit. </s> <s xml:id="echoid-s9528" xml:space="preserve">Cum ergo diameter Dode-<lb/>caedri, & </s> <s xml:id="echoid-s9529" xml:space="preserve">altitudo eiuſdem centra baſium oppoſitarum coniungens ſe in centro <lb/>ſecentbifariam, venabimur ſemiſſem huius altitudinis, nimirum altitudinem py-<lb/>ramidis quęſitam, hacratione. </s> <s xml:id="echoid-s9530" xml:space="preserve">Concipiatur triangulum rectangulum, cuius ba-<lb/> <anchor type="note" xlink:label="note-241-07a" xlink:href="note-241-07"/> ſis eſt ſemidiameter Dodecaedri nota, cum tota diameter proximè cognita ſit, <lb/>latera verò circa angulum rectum, altitudo pyramidis, & </s> <s xml:id="echoid-s9531" xml:space="preserve">ſemidiameter circuli <lb/>baſem Dodecaedri circumſcribentis. </s> <s xml:id="echoid-s9532" xml:space="preserve">Cum ergo ſemidiameter hæc cognoſci <lb/>poſsit, ex iis, quę lib. </s> <s xml:id="echoid-s9533" xml:space="preserve">4. </s> <s xml:id="echoid-s9534" xml:space="preserve">cap. </s> <s xml:id="echoid-s9535" xml:space="preserve">5. </s> <s xml:id="echoid-s9536" xml:space="preserve">docuimus, <anchor type="note" xlink:href="" symbol="f"/> cognoſcetur quoque reliquum la- tus, altitudo videlicet pyramidis, quam quærimus. </s> <s xml:id="echoid-s9537" xml:space="preserve">Porrò ſemidiameter prædi-<lb/> <anchor type="note" xlink:label="note-241-08a" xlink:href="note-241-08"/> cti circuli pentagonum Dodecaed@i circumſcribentis ita quo quered detur no-<lb/>ta. </s> <s xml:id="echoid-s9538" xml:space="preserve">Quoniam latus pentagoni ſubtenditin<unsure/> eo circulo grad. </s> <s xml:id="echoid-s9539" xml:space="preserve">72. </s> <s xml:id="echoid-s9540" xml:space="preserve">& </s> <s xml:id="echoid-s9541" xml:space="preserve">latus Decago-<lb/> <anchor type="note" xlink:label="note-241-09a" xlink:href="note-241-09"/> nigrad. </s> <s xml:id="echoid-s9542" xml:space="preserve">36. </s> <s xml:id="echoid-s9543" xml:space="preserve">cognita erunt hæc latera in partibus ſinus totius. </s> <s xml:id="echoid-s9544" xml:space="preserve">Si ergo fiat, vt latus <lb/>pentagoni in partibus ſinus totius cognitum ad idem latus notum ex hypothe-<lb/>ſi, ita latus Decagoni in iiſdem partibus ſinus totius cogniti ad aliud, prodibit <lb/>Decagoni latus in menſura lateris pentagoni cognitum. </s> <s xml:id="echoid-s9545" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Et quia latus penta- goni poteſt latera decagoni, & </s> <s xml:id="echoid-s9546" xml:space="preserve">Hexagoni eiuſdem circuli, ſi quadratum lateris <lb/>decagoni proximè cogniti detrahatur ex quadrato lateris pentagoni, reliquum <lb/> <anchor type="note" xlink:label="note-241-10a" xlink:href="note-241-10"/> fiet quadratum lateris Hexagoni, id eſt, ſemidiametri, ideo que eius radix qua-<lb/>drata ſemidiametrum exhibebit notam.</s> <s xml:id="echoid-s9547" xml:space="preserve"/> </p> <div xml:id="echoid-div584" type="float" level="2" n="3"> <note position="right" xlink:label="note-241-03" xlink:href="note-241-03a" xml:space="preserve">Area Dode-<lb/>caedri.</note> <note symbol="c" position="right" xlink:label="note-241-04" xlink:href="note-241-04a" xml:space="preserve">8. quinti <lb/>dec.</note> <note symbol="d" position="right" xlink:label="note-241-05" xlink:href="note-241-05a" xml:space="preserve">12. triang. <lb/>rectil.</note> <note symbol="e" position="right" xlink:label="note-241-06" xlink:href="note-241-06a" xml:space="preserve">15. tertiidec.</note> <note position="right" xlink:label="note-241-07" xlink:href="note-241-07a" xml:space="preserve">Perpendicu-<lb/>laris è centro <lb/>ſphæræ ad ba-<lb/>ſem Dodecae-<lb/>dri.</note> <note symbol="f" position="right" xlink:label="note-241-08" xlink:href="note-241-08a" xml:space="preserve">3. triang. <lb/>rectil.</note> <note position="right" xlink:label="note-241-09" xlink:href="note-241-09a" xml:space="preserve">Semidiame-<lb/>ter circuls <lb/>pentagonum <lb/>Dodesaedri <lb/>circumſcri-<lb/>bentis.</note> <note symbol="g" position="right" xlink:label="note-241-10" xlink:href="note-241-10a" xml:space="preserve">10. tertiidec.</note> </div> <p> <s xml:id="echoid-s9548" xml:space="preserve">5. </s> <s xml:id="echoid-s9549" xml:space="preserve"><emph style="sc">Postremo</emph> quia ductis ex centro Icoſaedri ad omnes eius angulos <lb/> <anchor type="note" xlink:label="note-241-11a" xlink:href="note-241-11"/> rectis lineis, Icoſaedrum in 20. </s> <s xml:id="echoid-s9550" xml:space="preserve">pyramides triangulares ęquales diuiditur; </s> <s xml:id="echoid-s9551" xml:space="preserve">ſi <lb/>area vnius pyramidis per cap. </s> <s xml:id="echoid-s9552" xml:space="preserve">2. </s> <s xml:id="echoid-s9553" xml:space="preserve">inuenta multiplicetur per 20. </s> <s xml:id="echoid-s9554" xml:space="preserve">gignetur to-<lb/>tius Icoſaedri area ex illis 20. </s> <s xml:id="echoid-s9555" xml:space="preserve">pyramidibus conflata. </s> <s xml:id="echoid-s9556" xml:space="preserve">Vtautem vnius pyramidis <lb/>area obtineatur, inueſtiganda primum erit area baſis triangularis, ex iis, quę <lb/>lib. </s> <s xml:id="echoid-s9557" xml:space="preserve">4. </s> <s xml:id="echoid-s9558" xml:space="preserve">cap. </s> <s xml:id="echoid-s9559" xml:space="preserve">2. </s> <s xml:id="echoid-s9560" xml:space="preserve">Num. </s> <s xml:id="echoid-s9561" xml:space="preserve">4. </s> <s xml:id="echoid-s9562" xml:space="preserve">& </s> <s xml:id="echoid-s9563" xml:space="preserve">5. </s> <s xml:id="echoid-s9564" xml:space="preserve">ſcripſimus, Deinde altitudo pyramidis mechanicę, <pb o="212" file="242" n="242" rhead="GEOMETR. PRACT."/> hoc pacto. </s> <s xml:id="echoid-s9565" xml:space="preserve">Ex ſuperiori plano producto, hoc eſt, ex inferiori ſuperficie alicu-<lb/>ius plani, quod corporis ſupremæ baſi imponeretur, ad planum baſis oppoſitæ <lb/>perpendicularis demittatur. </s> <s xml:id="echoid-s9566" xml:space="preserve">Hæc enim accuratè dimenſa altitudinem Icoſaedri <lb/>dabit, eiuſque ſemiſsis altitudinem pyramidis, quæ quęritur. </s> <s xml:id="echoid-s9567" xml:space="preserve">Quam Geome-<lb/>tricè ita etiam explorabimus. </s> <s xml:id="echoid-s9568" xml:space="preserve">Fiat pentagonum ex 5. </s> <s xml:id="echoid-s9569" xml:space="preserve">lateribus Icoſaedri, inue-<lb/>ſtigetur que eius ſemidiameter, & </s> <s xml:id="echoid-s9570" xml:space="preserve">latus Decagoniin circulo illud pentagonum <lb/>circumſcribente, in partibus, in quibus latus Icoſaedri datum eſt, hac ſcilicet ra-<lb/>tione. </s> <s xml:id="echoid-s9571" xml:space="preserve">Concipiatur triangulum rectangulum, cuius baſis ſemidiameter dicti cir-<lb/>culi, latera verò ſemiſsis lateris pentagoni, hoc eſt, Icoſaedri, & </s> <s xml:id="echoid-s9572" xml:space="preserve">perpendicularis <lb/>è centro ad punctum medium dictilateris demiſſa. </s> <s xml:id="echoid-s9573" xml:space="preserve">Ita namque cognoſcetur ſe-<lb/>midiameter, ex iis, quæ lib. </s> <s xml:id="echoid-s9574" xml:space="preserve">4. </s> <s xml:id="echoid-s9575" xml:space="preserve">cap. </s> <s xml:id="echoid-s9576" xml:space="preserve">5. </s> <s xml:id="echoid-s9577" xml:space="preserve">Num. </s> <s xml:id="echoid-s9578" xml:space="preserve">2. </s> <s xml:id="echoid-s9579" xml:space="preserve">tradita ſunt. </s> <s xml:id="echoid-s9580" xml:space="preserve">Latus verò Decagoni <lb/>in eodem circulo deſcripti reperietur, vt paulo ante circa finem Num. </s> <s xml:id="echoid-s9581" xml:space="preserve">4. </s> <s xml:id="echoid-s9582" xml:space="preserve">di-<lb/>ctum eſt. </s> <s xml:id="echoid-s9583" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quia diameter ſphærę, ſiue Ico ſaedri potentia eſt quintuplainuen- <anchor type="note" xlink:label="note-242-01a" xlink:href="note-242-01"/> tæ ſemidiametri; </s> <s xml:id="echoid-s9584" xml:space="preserve">ſi quadratum ſemidiametri inuentę quintupletur, procreabi-<lb/>tur quadratum diametri Icoſaedri, cuius radix quadrata diametrum offeret, <lb/>ideoque & </s> <s xml:id="echoid-s9585" xml:space="preserve">ſemidiameter Icoſaedri nota erit. </s> <s xml:id="echoid-s9586" xml:space="preserve">Vel aliter. </s> <s xml:id="echoid-s9587" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam diameter <anchor type="note" xlink:label="note-242-02a" xlink:href="note-242-02"/> ſphærę, id eſt, Icoſaedri, componitur ex latere Hexagoni, & </s> <s xml:id="echoid-s9588" xml:space="preserve">duobus lateribus <lb/>decagoni in circulo pentagonum ex quinque lateribus Icoſaedri compoſitum <lb/>circumſcribente: </s> <s xml:id="echoid-s9589" xml:space="preserve">erit ſumma collecta ex ſemidiametro illius circuli, & </s> <s xml:id="echoid-s9590" xml:space="preserve">duobus <lb/>lateribus decagoni, diametro Icoſaedriæqualis: </s> <s xml:id="echoid-s9591" xml:space="preserve">ideo que rurſus ſemidiameter <lb/>Icoſaedrinota erit.</s> <s xml:id="echoid-s9592" xml:space="preserve"/> </p> <div xml:id="echoid-div585" type="float" level="2" n="4"> <note position="right" xlink:label="note-241-11" xlink:href="note-241-11a" xml:space="preserve">Area Icoſae-<lb/>dri.</note> <note symbol="a" position="left" xlink:label="note-242-01" xlink:href="note-242-01a" xml:space="preserve">1. corol. 16. <lb/>tertiidec.</note> <note symbol="b" position="left" xlink:label="note-242-02" xlink:href="note-242-02a" xml:space="preserve">2. corol. 16. <lb/>tertiidec.</note> </div> <p> <s xml:id="echoid-s9593" xml:space="preserve"><emph style="sc">Hinc</emph> patet, Orontium cum illis, qui ipſum ſequuntur, decipi, qui putat, <lb/> <anchor type="note" xlink:label="note-242-03a" xlink:href="note-242-03"/> ex ſemiſſe ſemidiametri illius circuli, & </s> <s xml:id="echoid-s9594" xml:space="preserve">ex latere decagoni componi ſemiaxem <lb/>Icoſaedri, hoceſt, axem, vel altitudinem pyramidis, cuius baſis triangulum Ico-<lb/>ſaedri, & </s> <s xml:id="echoid-s9595" xml:space="preserve">vertex centrum ſp hæræ. </s> <s xml:id="echoid-s9596" xml:space="preserve">Nam vt ex iis conſtat, quę proximè ſcripſi-<lb/>mus, eo modo componitur ſemidiameter ſphærę, vel Icoſaedri, quæ maior <lb/>eſt prędicto axe. </s> <s xml:id="echoid-s9597" xml:space="preserve">Semidiameter porro circuli prædictum pentagonum circum-<lb/>ſcribentis reperiri quo que poterit, vt ad finem Num. </s> <s xml:id="echoid-s9598" xml:space="preserve">4. </s> <s xml:id="echoid-s9599" xml:space="preserve">diximus, ſi nimirum <lb/>quadratum lateris decagoni ex quadrato lateris dicti pentagoni, quod à late-<lb/>re Icoſaedrinon differt, tollatur, & </s> <s xml:id="echoid-s9600" xml:space="preserve">reliqui numeri radix quadrata extrahatur: <lb/></s> <s xml:id="echoid-s9601" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="c"/> propterea quodlatus pentagoni poteſt latera decagoni, & </s> <s xml:id="echoid-s9602" xml:space="preserve">hexagoni eiuſdem <anchor type="note" xlink:label="note-242-04a" xlink:href="note-242-04"/> circuli.</s> <s xml:id="echoid-s9603" xml:space="preserve"/> </p> <div xml:id="echoid-div586" type="float" level="2" n="5"> <note position="left" xlink:label="note-242-03" xlink:href="note-242-03a" xml:space="preserve">Error Oron-<lb/>tii.</note> <note symbol="c" position="left" xlink:label="note-242-04" xlink:href="note-242-04a" xml:space="preserve">10. tertiidec.</note> </div> <p> <s xml:id="echoid-s9604" xml:space="preserve"><emph style="sc">Iam</emph> verò cognita ſemidiametro Icoſaedri, inueniemus altitudinem pyra-<lb/> <anchor type="note" xlink:label="note-242-05a" xlink:href="note-242-05"/> midis, cuius baſis eſt triangulum Icoſaedri, & </s> <s xml:id="echoid-s9605" xml:space="preserve">vertex eiuſdem centrum, hoc mo-<lb/>do. </s> <s xml:id="echoid-s9606" xml:space="preserve">Quoniam diameter Icoſaedri, eiuſdemque altitudo ſeſein centro ſecant bi-<lb/>fariam, concipiatur triangulum rectangulum, cuius baſis eſt diameter Icoſaedri <lb/>proximè cognita, latera verò circa angulum rectum, altitudo pyramidis, & </s> <s xml:id="echoid-s9607" xml:space="preserve">ſe-<lb/>midiameter circuli baſem Icoſaedri circumſcribentis. </s> <s xml:id="echoid-s9608" xml:space="preserve">Cum ergo hęc ſemidia-<lb/>meter cognoſci poſsit ex iis, quæ lib. </s> <s xml:id="echoid-s9609" xml:space="preserve">4. </s> <s xml:id="echoid-s9610" xml:space="preserve">cap. </s> <s xml:id="echoid-s9611" xml:space="preserve">5. </s> <s xml:id="echoid-s9612" xml:space="preserve">docuimus, <anchor type="note" xlink:href="" symbol="d"/> cognoſcetur quoque <anchor type="note" xlink:label="note-242-06a" xlink:href="note-242-06"/> latus reliquum pyramidis, videlicet altitudo, quæ inquiritur. </s> <s xml:id="echoid-s9613" xml:space="preserve">Semidiameter <lb/>porro circuli baſem triangularem Icoſaedri circumſcribentis effi cietur hoc et-<lb/>iam pacto cognita. </s> <s xml:id="echoid-s9614" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Quoniam trianguli æquilaterilatus potentia triplum eſt <anchor type="note" xlink:label="note-242-07a" xlink:href="note-242-07"/> ſemidiametri illius circuli; </s> <s xml:id="echoid-s9615" xml:space="preserve">ſi quadratum lateris Icoſaedri diuidatur per 3. </s> <s xml:id="echoid-s9616" xml:space="preserve">erit <lb/>Quotientis radix quadrata ſemidiameter quæſita.</s> <s xml:id="echoid-s9617" xml:space="preserve"/> </p> <div xml:id="echoid-div587" type="float" level="2" n="6"> <note position="left" xlink:label="note-242-05" xlink:href="note-242-05a" xml:space="preserve">Perpendicu-<lb/>laris è centro <lb/>ſphæræ ad ba-<lb/>ſem Icoſaedri.</note> <note symbol="d" position="left" xlink:label="note-242-06" xlink:href="note-242-06a" xml:space="preserve">3. triang. re-<lb/>ctil.</note> <note symbol="e" position="left" xlink:label="note-242-07" xlink:href="note-242-07a" xml:space="preserve">12. tertiidec. <lb/>Semidiame-<lb/>ter circuli tri-<lb/>angulum Ico-<lb/>ſaedricircum <lb/>ſcribentis.</note> </div> <p> <s xml:id="echoid-s9618" xml:space="preserve">6. </s> <s xml:id="echoid-s9619" xml:space="preserve"><emph style="sc">Eadem</emph> hacarte, quæ in Dodecaedro, & </s> <s xml:id="echoid-s9620" xml:space="preserve">Icoſaedro expoſita eſt, areas <lb/>Tetraedri, cubi, & </s> <s xml:id="echoid-s9621" xml:space="preserve">Octaedriinueſtigare licebit, ſi, lineis ex eorum centris ad o-<lb/>mnes angulos ductis, in pyramides æquales diſtribuantur. </s> <s xml:id="echoid-s9622" xml:space="preserve">Tetraedrum nimi- <pb o="213" file="243" n="243" rhead="LIBER QVINTVS."/> rum in 4. </s> <s xml:id="echoid-s9623" xml:space="preserve">pyramides triangulares; </s> <s xml:id="echoid-s9624" xml:space="preserve">cubus in 6. </s> <s xml:id="echoid-s9625" xml:space="preserve">quadrangulares; </s> <s xml:id="echoid-s9626" xml:space="preserve">& </s> <s xml:id="echoid-s9627" xml:space="preserve">Octaedrum in <lb/> <anchor type="note" xlink:label="note-243-01a" xlink:href="note-243-01"/> 8. </s> <s xml:id="echoid-s9628" xml:space="preserve">triangulares. </s> <s xml:id="echoid-s9629" xml:space="preserve">Inuenta namque per cap. </s> <s xml:id="echoid-s9630" xml:space="preserve">2. </s> <s xml:id="echoid-s9631" xml:space="preserve">in quolibet area vnius pyramidis, ſi <lb/>ea in numerum baſiũ corporis regularis ducatur, inſurget totius corporis area. <lb/></s> <s xml:id="echoid-s9632" xml:space="preserve">Vt autem area vnius pyramidis habeatur, inquirenda prius erit altitudo ipſius, <lb/>vt ducta in tertiam baſis partem pyramidis aream producat. </s> <s xml:id="echoid-s9633" xml:space="preserve">Altitudo ergo hæc <lb/>in Tetraedro ſic reperietur. </s> <s xml:id="echoid-s9634" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam diameter ſphæræ Tetraedrum ambien- <anchor type="note" xlink:label="note-243-02a" xlink:href="note-243-02"/> tis potentia eſt ſeſquialtera lateris Tetraedri: </s> <s xml:id="echoid-s9635" xml:space="preserve">ſi Fiat, vt 2. </s> <s xml:id="echoid-s9636" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s9637" xml:space="preserve">ita quadratum <lb/>lateris Tetraedri dati ad aliud, prodibit quadratum diametri ſphæræ, cuius radix <lb/> <anchor type="note" xlink:label="note-243-03a" xlink:href="note-243-03"/> quadrata ipſam diametrum oſtendet. </s> <s xml:id="echoid-s9638" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Huius autem diametri ſexta pars, alti- tudo erit pyramidis quæſita, recta videlicet perpendicularis è centro ſphæræ in <lb/>baſem Tetraedri demiſſa.</s> <s xml:id="echoid-s9639" xml:space="preserve"/> </p> <div xml:id="echoid-div588" type="float" level="2" n="7"> <note position="right" xlink:label="note-243-01" xlink:href="note-243-01a" xml:space="preserve">Area Tetra-<lb/>edri, cubi & <lb/>Octaedri ali-<lb/>ter inuentæ.</note> <note symbol="a" position="right" xlink:label="note-243-02" xlink:href="note-243-02a" xml:space="preserve">13. tertij-<lb/>dec.</note> <note symbol="b" position="right" xlink:label="note-243-03" xlink:href="note-243-03a" xml:space="preserve">2. corol. 13. <lb/>tertiidec. <lb/>Perpendicu-<lb/>laris è centro <lb/>ſphæræ @d ba-<lb/>ſem Tetrae-<lb/>dri, cubi, & <lb/>Octaedri.</note> </div> <p> <s xml:id="echoid-s9640" xml:space="preserve"><emph style="sc">In</emph> cubo verò dicta altitudo ſemiſsi lateris cubiæqualis eſt, quod perpendi-<lb/>cularis è centro ſphæræ in baſem cubi demiſſa æqualis ſit ſemiſsi lateris cubi, vt <lb/>liquet.</s> <s xml:id="echoid-s9641" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9642" xml:space="preserve"><emph style="sc">In</emph> Octaedro denique eadem altitudo ſic deprehendetur. </s> <s xml:id="echoid-s9643" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quoniam dia- meter ſphæræ eſt potentia dupla lateris Octaedri: </s> <s xml:id="echoid-s9644" xml:space="preserve">ſi fiat, vt 1. </s> <s xml:id="echoid-s9645" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s9646" xml:space="preserve">ita quadratum <lb/>lateris Octaedri dati ad aliud, procreabitur quadratum diametri ſphæræ, vel O-<lb/> <anchor type="note" xlink:label="note-243-04a" xlink:href="note-243-04"/> ctaedri. </s> <s xml:id="echoid-s9647" xml:space="preserve">Huius ergo radix quadrata dabit diametrum, ideoq; </s> <s xml:id="echoid-s9648" xml:space="preserve">ſemidiameter non <lb/>ignorabitur. </s> <s xml:id="echoid-s9649" xml:space="preserve">Hinc altitudo pyramidis quæſita, hoc eſt perpendicularis è centro <lb/>ſphæræ in baſem Octaedri demiſſa, elicietur ea ratione, quã in Icoſaedro ad ſi-<lb/>nem Num. </s> <s xml:id="echoid-s9650" xml:space="preserve">5. </s> <s xml:id="echoid-s9651" xml:space="preserve">explicauimus.</s> <s xml:id="echoid-s9652" xml:space="preserve"/> </p> <div xml:id="echoid-div589" type="float" level="2" n="8"> <note symbol="c" position="right" xlink:label="note-243-04" xlink:href="note-243-04a" xml:space="preserve">14. tertii-<lb/>dec.</note> </div> <p> <s xml:id="echoid-s9653" xml:space="preserve">7. </s> <s xml:id="echoid-s9654" xml:space="preserve"><emph style="sc">Non</emph> videtur autem omittenda alia ratio dimetiendi omnia quinq; </s> <s xml:id="echoid-s9655" xml:space="preserve">cor-<lb/>pora regularia, quæ quidem in lib. </s> <s xml:id="echoid-s9656" xml:space="preserve">14. </s> <s xml:id="echoid-s9657" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s9658" xml:space="preserve">demonſtrata eſt, & </s> <s xml:id="echoid-s9659" xml:space="preserve">eſt eiuſmodi. <lb/></s> <s xml:id="echoid-s9660" xml:space="preserve"> <anchor type="note" xlink:label="note-243-05a" xlink:href="note-243-05"/> Primum quæratur ſuperficies conuexa cuiuſque corporis, ex eius latere cogni-<lb/>to, etiamſi nullius baſis area inueſtigetur: </s> <s xml:id="echoid-s9661" xml:space="preserve">hoc videlicet pacto. </s> <s xml:id="echoid-s9662" xml:space="preserve">Quoniam quæ-<lb/>libet baſis cuiuſuis corporis diuiditur per rectas ex centro baſis ad omnes angu-<lb/>los ductas in tottriangula æqualia, quot anguli, vellatera in baſe continentur: </s> <s xml:id="echoid-s9663" xml:space="preserve">ſi <lb/>ducatur hic numerus triangulorum in numerum baſium corpus regulare, quod <lb/>propoſitum eſt, ambientium, habebitur numerus omnium huiuſmo di triangu-<lb/> <anchor type="figure" xlink:label="fig-243-01a" xlink:href="fig-243-01"/> @orũ in tota ſuperficie conuexa contentorũ. </s> <s xml:id="echoid-s9664" xml:space="preserve">Vt quia baſis quadrata cubi ABCD, <lb/>diuiſa eſt in quatuor triangula ex centro E, continebuntur 24. </s> <s xml:id="echoid-s9665" xml:space="preserve">eiuſmodi trian-<lb/>gulain 6. </s> <s xml:id="echoid-s9666" xml:space="preserve">baſibus. </s> <s xml:id="echoid-s9667" xml:space="preserve">Item quia baſis triangularis Tetraedri, Octaedri, & </s> <s xml:id="echoid-s9668" xml:space="preserve">Icoſae-<lb/>dri ABC, ex centro D, diſtributa eſt in 3. </s> <s xml:id="echoid-s9669" xml:space="preserve">triangula, exiſtent in 4. </s> <s xml:id="echoid-s9670" xml:space="preserve">baſibus Tetrae-<lb/>dri 12. </s> <s xml:id="echoid-s9671" xml:space="preserve">eiuſmo ditriangula, & </s> <s xml:id="echoid-s9672" xml:space="preserve">24. </s> <s xml:id="echoid-s9673" xml:space="preserve">in 8. </s> <s xml:id="echoid-s9674" xml:space="preserve">baſibus Octaedri, & </s> <s xml:id="echoid-s9675" xml:space="preserve">60. </s> <s xml:id="echoid-s9676" xml:space="preserve">in 20. </s> <s xml:id="echoid-s9677" xml:space="preserve">baſibus Ico-<lb/>ſcedri. </s> <s xml:id="echoid-s9678" xml:space="preserve">Denique quia baſis pentagona Dodecaedri ABCDE, reſoluta eſt ex cẽ-<lb/>tro F, in 5. </s> <s xml:id="echoid-s9679" xml:space="preserve">triangula, cõplectentur 12. </s> <s xml:id="echoid-s9680" xml:space="preserve">baſes Dodecaedri 60. </s> <s xml:id="echoid-s9681" xml:space="preserve">eiuſmoditriãgula.</s> <s xml:id="echoid-s9682" xml:space="preserve"/> </p> <div xml:id="echoid-div590" type="float" level="2" n="9"> <note position="right" xlink:label="note-243-05" xlink:href="note-243-05a" xml:space="preserve">In quot trian-<lb/>gula diuidan-<lb/>tur omn{es} ba-<lb/>ſ{es} cuiuſuis <lb/>corporis regu-<lb/>laris ex earũ <lb/>centris.</note> <figure xlink:label="fig-243-01" xlink:href="fig-243-01a"> <image file="243-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/243-01"/> </figure> </div> <p> <s xml:id="echoid-s9683" xml:space="preserve"><emph style="sc">Deinde</emph> quia rectangulum contentum ſub perpendiculari è centro baſis <lb/>in latus demiſſa, & </s> <s xml:id="echoid-s9684" xml:space="preserve">ſub vno latere, æquale eſt duobus eiuſmodi triangulis, <anchor type="note" xlink:href="" symbol="d"/> <anchor type="note" xlink:label="note-243-06a" xlink:href="note-243-06"/> propterea quod vnius duplum eſt: </s> <s xml:id="echoid-s9685" xml:space="preserve">erit in cubo rectangulum illud duodecies <lb/>ſumptum toti ſup erficiei cubi æquale. </s> <s xml:id="echoid-s9686" xml:space="preserve">In Tetraedro verò ſexies ſumptum to- <pb o="214" file="244" n="244" rhead="GEOMETR. PRACT."/> tam Tetraedri ſuperficiem conficiet: </s> <s xml:id="echoid-s9687" xml:space="preserve">In Octaedro deinde duodecies acceptum <lb/> <anchor type="note" xlink:label="note-244-01a" xlink:href="note-244-01"/> toti ſuperficiei Octaedri adæquabitur: </s> <s xml:id="echoid-s9688" xml:space="preserve">Atin Dodecaedro, & </s> <s xml:id="echoid-s9689" xml:space="preserve">Icoſaedro tricies <lb/>ſumptum ſuperficiei totitam Dodecaedri, quam Icoſaedri æquale erit. </s> <s xml:id="echoid-s9690" xml:space="preserve">Dicta au-<lb/>tem perpendicularis EF, in baſe cubi æqualis eſt ſemiſsilateris cubi AB, <anchor type="note" xlink:href="" symbol="a"/> Quo- niam enim perpendicularis EF, ſecat latus AB, bifariam, <anchor type="note" xlink:href="" symbol="b"/> eſt que ipſi AF, æqua- lis, quod anguli FAE, FEA, ſemirecti ſint; </s> <s xml:id="echoid-s9691" xml:space="preserve">conſtat EF, ſemiſsi lateris cubi eſſe ę-<lb/> <anchor type="note" xlink:label="note-244-02a" xlink:href="note-244-02"/> qualem. </s> <s xml:id="echoid-s9692" xml:space="preserve">Perpendicularis autem DE, in baſe Tetraedri; </s> <s xml:id="echoid-s9693" xml:space="preserve">Octaedri, & </s> <s xml:id="echoid-s9694" xml:space="preserve">Icoſaedri, <lb/> <anchor type="note" xlink:href="" symbol="c"/> ſemiſsis eſt ſemidiametri C D. </s> <s xml:id="echoid-s9695" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Cum ergo latus A C, ſit potentia triplum ſe- <anchor type="note" xlink:label="note-244-03a" xlink:href="note-244-03"/> midiametri CD: </s> <s xml:id="echoid-s9696" xml:space="preserve">Si fiat, vt 3. </s> <s xml:id="echoid-s9697" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s9698" xml:space="preserve">ita quadratum lateris dati AC, ad aliud, prodi-<lb/> <anchor type="note" xlink:label="note-244-04a" xlink:href="note-244-04"/> bit quadratum ſemidiametri C D, cuius radix quadrata ipſam C D, indicabit, e-<lb/>iuſque ſemiſsis perpendicularem DE, exhibebit. </s> <s xml:id="echoid-s9699" xml:space="preserve">Perpendicularis denique FG, <lb/> <anchor type="note" xlink:label="note-244-05a" xlink:href="note-244-05"/> in baſe Dodecaedrie ſemiſsis eſt ſummæ ex ſemidiametro AF, & </s> <s xml:id="echoid-s9700" xml:space="preserve">latere decago-<lb/> <anchor type="note" xlink:label="note-244-06a" xlink:href="note-244-06"/> ni circuli ABD, collectæ, quodlatus decagoni cognoſcetur, vt ad finem Nume. <lb/></s> <s xml:id="echoid-s9701" xml:space="preserve">4. </s> <s xml:id="echoid-s9702" xml:space="preserve">traditum eſt.</s> <s xml:id="echoid-s9703" xml:space="preserve"/> </p> <div xml:id="echoid-div591" type="float" level="2" n="10"> <note symbol="d" position="right" xlink:label="note-243-06" xlink:href="note-243-06a" xml:space="preserve">41. primi.</note> <note position="left" xlink:label="note-244-01" xlink:href="note-244-01a" xml:space="preserve">Superficies re <lb/>gularium cor <lb/>porum & per-<lb/>pendicular{es} <lb/>baſium.</note> <note symbol="a" position="left" xlink:label="note-244-02" xlink:href="note-244-02a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note symbol="b" position="left" xlink:label="note-244-03" xlink:href="note-244-03a" xml:space="preserve">6. primi.</note> <note symbol="c" position="left" xlink:label="note-244-04" xlink:href="note-244-04a" xml:space="preserve">2. coroll. 12. <lb/>tertijdec.</note> <note symbol="d" position="left" xlink:label="note-244-05" xlink:href="note-244-05a" xml:space="preserve">12. tertijdec.</note> <note symbol="e" position="left" xlink:label="note-244-06" xlink:href="note-244-06a" xml:space="preserve">1. quartidec.</note> </div> <p> <s xml:id="echoid-s9704" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> <emph style="sc">Qvia</emph> verò ſolidum, quod fit ex perpendiculari è centro cuius cumque <anchor type="note" xlink:label="note-244-07a" xlink:href="note-244-07"/> corporis regularis ad aliquam eius baſem ducta in tertiam partem ſuperficiei i-<lb/>pſius corporis, ipſi corpori æquale eſt; </s> <s xml:id="echoid-s9705" xml:space="preserve">ſi inueſtigetur ſuperficies conuexa dati <lb/> <anchor type="note" xlink:label="note-244-08a" xlink:href="note-244-08"/> corporis regularis, vt proximè docuimus, atque in tertiam eius partem ducatur <lb/>altitudo vnius pyramidum, in quas corpus ipſum per rectas è centro ipſius du-<lb/>ctas diuiditur, (quæ altitudo reperietur, vt ſupra tradidimus) hoc eſt, perpendi-<lb/>cularis è centro corporis in eius baſem demiſſa, procreabitur area, ſiue ſoliditas <lb/>ipſius corporis. </s> <s xml:id="echoid-s9706" xml:space="preserve">Quæ etiam obtinebitur, ſi dicta altitudo ducatur in totam ſu-<lb/>perficiem conuexam, & </s> <s xml:id="echoid-s9707" xml:space="preserve">producti tertia pars capiatur.</s> <s xml:id="echoid-s9708" xml:space="preserve"/> </p> <div xml:id="echoid-div592" type="float" level="2" n="11"> <note symbol="f" position="left" xlink:label="note-244-07" xlink:href="note-244-07a" xml:space="preserve">ſchol. 20. ter-<lb/>tijdec.</note> <note position="left" xlink:label="note-244-08" xlink:href="note-244-08a" xml:space="preserve">Areæ corpo-<lb/>rumregulari-<lb/>um aliter in-<lb/>uentæ.</note> </div> <p> <s xml:id="echoid-s9709" xml:space="preserve">8. </s> <s xml:id="echoid-s9710" xml:space="preserve"><emph style="sc">Itaqve</emph> vt vides, tota difficultas in corporibus regularibus dimetien-<lb/>dis conſiſtit fermè totain altitudine pyramidis baſem habentis eandem cũ cor-<lb/>pore, verticem autem in centro ſphæræ, exquirenda: </s> <s xml:id="echoid-s9711" xml:space="preserve">cuius quideminuentio Ge-<lb/>ometrica pernumeros moleſtiſsima eſt, propter radices ſurdas, & </s> <s xml:id="echoid-s9712" xml:space="preserve">numeros fra-<lb/> <anchor type="figure" xlink:label="fig-244-01a" xlink:href="fig-244-01"/> ctos, quorum numeratores, denominatoreſq; </s> <s xml:id="echoid-s9713" xml:space="preserve">nimis magniſunt, adeo vt ope-<lb/>ræ pretium videatur eſſe eandem mechanicè explorare, vt adinitium Num. </s> <s xml:id="echoid-s9714" xml:space="preserve">2. </s> <s xml:id="echoid-s9715" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s9716" xml:space="preserve">& </s> <s xml:id="echoid-s9717" xml:space="preserve">5. </s> <s xml:id="echoid-s9718" xml:space="preserve">diximus, præſertim ſi ex quiſita diligentia in ea per inſtrumentum partium <lb/>dimetienda adhibeatur. </s> <s xml:id="echoid-s9719" xml:space="preserve">Sed quia non ſemper in promptu habemus corpora re-<lb/>gularia, vt mechanicè eam altitudinem conſequi poſsimus, libetrationem quã-<lb/>dam nouam, eamque facillimam hic præſcribere, qua ſine moleſtia illa numero-<lb/>rum, eadem illa altitudo per lineas inueniatur, etiamſi corpus regulare non <pb o="215" file="245" n="245" rhead="LIBER QVINTVS."/> adſit, ſed ſolum eius latus datum ſitac cognitum. </s> <s xml:id="echoid-s9720" xml:space="preserve">Sit ergo primo datum latus <lb/>Tetraedri A B, quotcunque palmorum, <lb/> <anchor type="figure" xlink:label="fig-245-01a" xlink:href="fig-245-01"/> conſtruaturque triangulum æquilaterum <lb/>A B C, pro baſe Tetraedri: </s> <s xml:id="echoid-s9721" xml:space="preserve">Diuiſo autem <lb/>latere A B, bifariam in D, iungatur recta <lb/> <anchor type="note" xlink:label="note-245-01a" xlink:href="note-245-01"/> C D, <anchor type="note" xlink:href="" symbol="a"/> quæ ad AB, perpendicularis erit.</s> <s xml:id="echoid-s9722" xml:space="preserve"> Conſtructo quo que Iſoſcele ABE, cuius <lb/>vtrumque latus rectæ CD, æqualeſit, de-<lb/>mittatur ad AE, perpendicularis BF, cuius <lb/>quarta pars ſit F G. </s> <s xml:id="echoid-s9723" xml:space="preserve">Dico FG, altitudinem <lb/> <anchor type="note" xlink:label="note-245-02a" xlink:href="note-245-02"/> eſſe vnius pyramidis, hoc eſt, æqualem eſ-<lb/>ſe perpendiculari ex centro ſphæræ Tetra-<lb/>edro circumſcriptæ ad vnam baſem deductæ. </s> <s xml:id="echoid-s9724" xml:space="preserve">Quoniam enim, vt ad finem Eucli-<lb/>dis ex Hypſicle demonſtrauimus, E, angulus eſt inclinationis vnius baſis Tetra-<lb/>edriad alteram, eſt que EB, perpendiculari CD, æqualis: </s> <s xml:id="echoid-s9725" xml:space="preserve">ſi triangulum B E F, <lb/>concipiatur circa EF, moueri, donec rectum ſit ad baſem Tetraedri, cadet pun-<lb/>ctum B, in verticem Tetraedri; </s> <s xml:id="echoid-s9726" xml:space="preserve">ac proinde perpendicularis BF, altitudo erit Te-<lb/>traedri. </s> <s xml:id="echoid-s9727" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et quia altitudo Tetraedri duas partes tertias diamet@i ſphæræ conti- <anchor type="note" xlink:label="note-245-03a" xlink:href="note-245-03"/> net: </s> <s xml:id="echoid-s9728" xml:space="preserve">ſi ſemidiameter ponatur 6. </s> <s xml:id="echoid-s9729" xml:space="preserve">erit altitudo B F, 4. </s> <s xml:id="echoid-s9730" xml:space="preserve">& </s> <s xml:id="echoid-s9731" xml:space="preserve">ſemidiameter 3. </s> <s xml:id="echoid-s9732" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Cum ergo altitudo vnius pyramidis ſit tertia pars ſemidiametri, erit BG, ſemidiame-<lb/> <anchor type="note" xlink:label="note-245-04a" xlink:href="note-245-04"/> ter, & </s> <s xml:id="echoid-s9733" xml:space="preserve">G F, altitudo vnius pyramidis. </s> <s xml:id="echoid-s9734" xml:space="preserve">Quam etiam inueniemus, licet Iſoſceles <lb/>AEB, non extruatur, hoc modo. </s> <s xml:id="echoid-s9735" xml:space="preserve">Sumpta dH, tertia parte perpendicularis CD, <lb/>exciteturad CD, perpendicularis HK, quæ ex D, adinteruallum CD, ſecetur in <lb/>K. </s> <s xml:id="echoid-s9736" xml:space="preserve">Dico HI, quartam partem ipſius HK, eſſe altitudinem vnius pyramidis Ere-<lb/>cto enim triangulo DHK, ſupra baſem Tetraedri ABC, cadet punctũ K, in ver-<lb/>ticem Tetraedri, quod D K, ducta æqualis ſit perpendiculari ex medio latere ad <lb/>angulum baſis oppoſitum ductæ. </s> <s xml:id="echoid-s9737" xml:space="preserve">Ergo vt prius, HK, altitudo erit Tetraedri, & </s> <s xml:id="echoid-s9738" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-245-05a" xlink:href="note-245-05"/> HI, perpendicularis ex centro ſphæræ in H, centrum baſis cadens. </s> <s xml:id="echoid-s9739" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Nam D H, tertia pars perpendicularis CD, in centrum trianguli cadit.</s> <s xml:id="echoid-s9740" xml:space="preserve"/> </p> <div xml:id="echoid-div593" type="float" level="2" n="12"> <figure xlink:label="fig-244-01" xlink:href="fig-244-01a"> <image file="244-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/244-01"/> </figure> <figure xlink:label="fig-245-01" xlink:href="fig-245-01a"> <image file="245-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/245-01"/> </figure> <note symbol="a" position="right" xlink:label="note-245-01" xlink:href="note-245-01a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note position="right" xlink:label="note-245-02" xlink:href="note-245-02a" xml:space="preserve">Altitudo py-<lb/>ramidis Te-<lb/>traedri.</note> <note symbol="b" position="right" xlink:label="note-245-03" xlink:href="note-245-03a" xml:space="preserve">2. corol. 13. <lb/>tertijdec.</note> <note symbol="c" position="right" xlink:label="note-245-04" xlink:href="note-245-04a" xml:space="preserve">2. corol. 13. <lb/>tertijdec.</note> <note symbol="d" position="right" xlink:label="note-245-05" xlink:href="note-245-05a" xml:space="preserve">2. corol. 13. <lb/>tertijdec.</note> </div> <p> <s xml:id="echoid-s9741" xml:space="preserve"><emph style="sc">Sit</emph> deinde datum latus Octaedri L M, ſupra quod conſtruatur triangulum <lb/>æquilaterum L M N, pro baſe Octaedri. </s> <s xml:id="echoid-s9742" xml:space="preserve">Diuiſo autem latere L M, bifariam in <lb/>O, iungaturrecta N O, <anchor type="note" xlink:href="" symbol="e"/> quæ ad L M, erit perpendicularis. </s> <s xml:id="echoid-s9743" xml:space="preserve">Conſtructio iam <anchor type="note" xlink:label="note-245-06a" xlink:href="note-245-06"/> Iſoſcele QRS, ſupra baſem QR, æqualem diametro ſphæræ, vel quadrati ex la-<lb/>tere Octaedri deſcripti, (quæ habebitur, ſi educatur perpendicularis MP, lateri <lb/>L M, æqualis. </s> <s xml:id="echoid-s9744" xml:space="preserve">Iuncta enim recta L P, diameter erit illius quadrati, vel ſphæræ.) <lb/></s> <s xml:id="echoid-s9745" xml:space="preserve">vtrum que laterum QS, RS, æquale habens perpendiculari N O; </s> <s xml:id="echoid-s9746" xml:space="preserve">ducatur ex R, <lb/>ad QS, perpendicularis RT, quæbifariam ſecetur in V. </s> <s xml:id="echoid-s9747" xml:space="preserve">Dico T V, eſſe altitudi-<lb/> <anchor type="note" xlink:label="note-245-07a" xlink:href="note-245-07"/> nem pyramidis quæſitam, hoc eſt, æqualem eſſe perpendiculari ex centro ſphę-<lb/>ræ ad vnam baſem Octaedri cadenti. </s> <s xml:id="echoid-s9748" xml:space="preserve">Quoniam enim, vt ad finem Euclidis ex <lb/>Hypſicle demonſtrauimus, augulus QSR, in clinationem vnius baſis ad alteram <lb/>indicat, eſt que obtuſus, erit perpendicularis R T, cadens ad partes anguli acuti <lb/>R S T, æqualis altitudini Octaed@i, id eſt, perpendicularibaſium Octaedri oppo-<lb/>ſitarum centra connectenti, vt ex Octaedro materiali perſpicuum eſt: </s> <s xml:id="echoid-s9749" xml:space="preserve">Ac pro-<lb/>pterea eius ſemiſsis T V, altitudo erit pyramidis q̃ſita, quod altitudo Octaedri <lb/>bifariam ſecetur in centro.</s> <s xml:id="echoid-s9750" xml:space="preserve"/> </p> <div xml:id="echoid-div594" type="float" level="2" n="13"> <note symbol="e" position="right" xlink:label="note-245-06" xlink:href="note-245-06a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note position="right" xlink:label="note-245-07" xlink:href="note-245-07a" xml:space="preserve">Altitudo py-<lb/>ramidis Octa-<lb/>edri.</note> </div> <p> <s xml:id="echoid-s9751" xml:space="preserve"><emph style="sc">Si</emph> deturlatus cubi, ſiue hexaedri, erit eius ſemiſsis altitudo pyramidis quæ-<lb/> <anchor type="note" xlink:label="note-245-08a" xlink:href="note-245-08"/> fita: </s> <s xml:id="echoid-s9752" xml:space="preserve">propterea quod cubialtitudo eiuſdem lateriſit æqualis.</s> <s xml:id="echoid-s9753" xml:space="preserve"/> </p> <div xml:id="echoid-div595" type="float" level="2" n="14"> <note position="right" xlink:label="note-245-08" xlink:href="note-245-08a" xml:space="preserve">Altitudo py-<lb/>ramidis cubi.</note> </div> <pb o="216" file="246" n="246" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s9754" xml:space="preserve"><emph style="sc">Datvm</emph> iam ſit AB, latus Dodecaedri, ſupra quod extruatur pentagonum <lb/>æquilaterum, & </s> <s xml:id="echoid-s9755" xml:space="preserve">æquiangulum ABCDE, pro baſe Dodecaedri. </s> <s xml:id="echoid-s9756" xml:space="preserve">Iuncta autem re-<lb/>cta CE, <anchor type="note" xlink:href="" symbol="a"/> quæ latus erit cubi in Dodecaedro, & </s> <s xml:id="echoid-s9757" xml:space="preserve">in eadem cumip ſo ſphæra deſcri- <anchor type="note" xlink:label="note-246-01a" xlink:href="note-246-01"/> pti, <anchor type="note" xlink:href="" symbol="b"/> atque lateri AB, parallela: </s> <s xml:id="echoid-s9758" xml:space="preserve">ſecetur AB, bifariam in S, connectatur que recta S D. </s> <s xml:id="echoid-s9759" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> quæ angulum CDE, bifariam ſecabit: </s> <s xml:id="echoid-s9760" xml:space="preserve">d ac proinde & </s> <s xml:id="echoid-s9761" xml:space="preserve">rectam CE, bifariã, <anchor type="note" xlink:label="note-246-02a" xlink:href="note-246-02"/> & </s> <s xml:id="echoid-s9762" xml:space="preserve">ad angulos rectos diuidet: </s> <s xml:id="echoid-s9763" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s9764" xml:space="preserve">anguli ad S, recti erunt. </s> <s xml:id="echoid-s9765" xml:space="preserve">Fiat ſupra <lb/>CE, Iſoſceles CGE, cuius vtrum que laterum CG, EG, perpendiculari SF, ſitæ-<lb/> <anchor type="note" xlink:label="note-246-03a" xlink:href="note-246-03"/> <anchor type="note" xlink:label="note-246-04a" xlink:href="note-246-04"/> <anchor type="figure" xlink:label="fig-246-01a" xlink:href="fig-246-01"/> quale. </s> <s xml:id="echoid-s9766" xml:space="preserve">Sumptis quo que FH, FI, ſemiſsi lateris AB, æqualibus, erigantur ad EC, <lb/>perpendiculares HK, IL, quæ ex C, E, ad interuallum FD, ſecentur in K, L, iun-<lb/>ganturq; </s> <s xml:id="echoid-s9767" xml:space="preserve">rectæ EL, CK. </s> <s xml:id="echoid-s9768" xml:space="preserve">His paratis, fiat angulo CGE, æqualis angulus MNO, <lb/>ponatur que N O, ipſi SD, æqualis: </s> <s xml:id="echoid-s9769" xml:space="preserve">Item angulo ELK, fiat æqualis angulus N-<lb/>OP, ponaturque OP, lateri AB, æqualis: </s> <s xml:id="echoid-s9770" xml:space="preserve">ac tandem demittatur ex P, ad MN, <lb/>perpendicularis PQ, quæ bifariam ſecetur in R. </s> <s xml:id="echoid-s9771" xml:space="preserve">Dico RQ, altitudinem eſſe py-<lb/>ramidis vnius in Dodecaedro. </s> <s xml:id="echoid-s9772" xml:space="preserve">Nam quia, vt ad finem Euclidis ex Hypſicle de-<lb/>monſtrauimus, angulus C G E, in clinationem vnius baſis ad alteram metitur, ſi <lb/>MN, concipiatur eſſe perpendicularis, quæ in baſe infima ex angulo pentagoni <lb/>ad medium punctum lateris oppoſiti ducitur, reſpondebit NO, perpendiculari, <lb/>quæin pentagono ad illam baſem inclinato ex eodem medio puncto ad oppo-<lb/>ſitum angulum ducitur: </s> <s xml:id="echoid-s9773" xml:space="preserve">propterea quod angulum M N O, angulo inclinatio-<lb/>nis CGE, & </s> <s xml:id="echoid-s9774" xml:space="preserve">rectam NO, perpendiculari S D, æqualem poſuimus. </s> <s xml:id="echoid-s9775" xml:space="preserve">Recta autem <lb/>OP, refert latus Dodecaedri inter angulum dicti pentagoni inclinati, & </s> <s xml:id="echoid-s9776" xml:space="preserve">angu-<lb/>lũ ſupremæ baſis poſitũ: </s> <s xml:id="echoid-s9777" xml:space="preserve">ꝓpterea ꝙ recta OP, poſita eſt æqualis lateri Dodeca-<lb/>edri, & </s> <s xml:id="echoid-s9778" xml:space="preserve">angulus NOP, angulo ELK, qui quidem æqualis eſt illi, quem dictum la-<lb/>tus efficit cum perpendiculari ex angulo ſupradicti pentagoni inclinatiad ba-<lb/>ſemin medium punctum lateris oppoſiti ductæ, vt conſtat, ſi vna baſis cubi Do-<lb/>decaedro inſcripti intelligatur dicto lateri Dodecaedri ſubſtrata, ita vt duo late-<lb/>ra baſis cubi ſubtendant duos angulos duorum pentagonorũ, quorum vnum <lb/>ad baſem Dodecaedri inclinatum eſt, alterum vero ò ſupremum in Dodecaedro. <lb/></s> <s xml:id="echoid-s9779" xml:space="preserve">Erit enim tuncrecta CE, æqualis rectæ duo puncta media duorum laterum di-<lb/>ctorum baſis cubi connectenti. </s> <s xml:id="echoid-s9780" xml:space="preserve">Rectæ autem EL, CK, reſpondebunt rectis ex <lb/>eiſdem punctis medijs laterum illorum baſis cubi, ad angulos prædictorũ pen-<lb/>tagonorum ductis: </s> <s xml:id="echoid-s9781" xml:space="preserve">Ac proinde angulus ELK, æqualis erit ei, quem perpendi- <pb o="217" file="247" n="247" rhead="LIBER QVINTVS."/> cularis in pentagono inclinato cum prædicto latere Dodecaedri efficit. </s> <s xml:id="echoid-s9782" xml:space="preserve">Ex quo <lb/>fit punctum P, in plano ſupremæ baſis exiſtere, atque idcirco perpendicularem <lb/> <anchor type="figure" xlink:label="fig-247-01a" xlink:href="fig-247-01"/> P Q, ad planum baſis per M N, ductum demiſſam, æqualem eſſe altitudini Do-<lb/>decaedri; </s> <s xml:id="echoid-s9783" xml:space="preserve">eiuſque ſemiſſem R Q, altitudini vnius pyramidis pentagonæ eſſe æ-<lb/>qualem. </s> <s xml:id="echoid-s9784" xml:space="preserve">Quæ omnia facil@ intelligentur, ſi Dodecaedrum aliquod materiale ad-<lb/>hibeatur.</s> <s xml:id="echoid-s9785" xml:space="preserve"/> </p> <div xml:id="echoid-div596" type="float" level="2" n="15"> <note symbol="a" position="left" xlink:label="note-246-01" xlink:href="note-246-01a" xml:space="preserve">2. coroll. 17. <lb/>@rtijdec.</note> <note symbol="b" position="left" xlink:label="note-246-02" xlink:href="note-246-02a" xml:space="preserve">coroll. 8. <lb/>quintidec.</note> <note symbol="c" position="left" xlink:label="note-246-03" xlink:href="note-246-03a" xml:space="preserve">ſchol. 12. <lb/>quarti.</note> <note symbol="d" position="left" xlink:label="note-246-04" xlink:href="note-246-04a" xml:space="preserve">4. primi.</note> <figure xlink:label="fig-246-01" xlink:href="fig-246-01a"> <image file="246-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/246-01"/> </figure> <figure xlink:label="fig-247-01" xlink:href="fig-247-01a"> <image file="247-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/247-01"/> </figure> </div> <p> <s xml:id="echoid-s9786" xml:space="preserve"><emph style="sc">Deniqve</emph> datum ſit Ico ſaedri latus a b, ſupra quod extruatur pentagonum <lb/>æquilaterum, & </s> <s xml:id="echoid-s9787" xml:space="preserve">æquiangulum a b c d e, pro baſe pyramidis ex quin que baſibus <lb/>Icoſaedri conflatæ. </s> <s xml:id="echoid-s9788" xml:space="preserve">Iuncta autem recta c e, ſeceturlatus a b, in ſ, bifariam, & </s> <s xml:id="echoid-s9789" xml:space="preserve">re-<lb/>cta ducaturſd, quæ vt in Dodecaedro oſtendimus proximè, perpendicularis e-<lb/>rit ad vtramque a b, c e. </s> <s xml:id="echoid-s9790" xml:space="preserve">Fiat ſupra latus Icoſaedri c d, triangulum æquilaterum <lb/>c d h, probaſe vna Icoſaedri; </s> <s xml:id="echoid-s9791" xml:space="preserve">& </s> <s xml:id="echoid-s9792" xml:space="preserve">diuiſo latere c d, bifariam in k, iungatur recta h-<lb/>k, <anchor type="note" xlink:href="" symbol="a"/> quæ ad c d, erit perpendicularis. </s> <s xml:id="echoid-s9793" xml:space="preserve">Præterea ſupra c e, fiat Iſoſceles c g e, cu- <anchor type="note" xlink:label="note-247-01a" xlink:href="note-247-01"/> ius vtrum que laterum c g, e g, perpendicularihk, ſit æquale. </s> <s xml:id="echoid-s9794" xml:space="preserve">Poſt hæc ſupra ſ d, <lb/>conſtituatur triangulum ſdl, cuius latus ſl, perpendiculari h k, & </s> <s xml:id="echoid-s9795" xml:space="preserve">latus dl, lateri <lb/>Icoſaedri a b, fit æquale. </s> <s xml:id="echoid-s9796" xml:space="preserve">Denique angulo c g e, fiat æqualis angulus m n o, & </s> <s xml:id="echoid-s9797" xml:space="preserve"><lb/>recta n o, perpendiculari h k, æqualis: </s> <s xml:id="echoid-s9798" xml:space="preserve">Item angulus n o p, angulo d l s, rectaque <lb/>o p, lateri Icoſaedri a b, æqualis. </s> <s xml:id="echoid-s9799" xml:space="preserve">Dico perpendicularem p q, ad m n, demiſſam, <lb/>eſſe altitudinem Icoſaedri, eiuſque ſemiſſem r q, altitudinem vnius pyramidis in <lb/> <anchor type="note" xlink:label="note-247-02a" xlink:href="note-247-02"/> Icoſaedro. </s> <s xml:id="echoid-s9800" xml:space="preserve">Quia enim, vt ex Hypſicle ad finem Euclidis demonſtrauimus, an-<lb/>gulus c g e, metitur in clinationem vnius baſis ad alteram, ſi m n, concipiatur eſ-<lb/>ſe perpendicularis, quæ in baſe infima Icoſaedri ex angulo trianguli ad medium <lb/>punctum lateris oppoſiti ducitur, reſpondebit n o, per pendiculari, quæ in trian-<lb/>gulo ad illam baſem inclinato ex eodem medio pũcto ad angulum oppoſitum <lb/>ducitur: </s> <s xml:id="echoid-s9801" xml:space="preserve">propterea quod angulum m n o, angulo inclinationis c g e, & </s> <s xml:id="echoid-s9802" xml:space="preserve">rectam <lb/>n o, perpendiculari h k, æqualem fecimus: </s> <s xml:id="echoid-s9803" xml:space="preserve">Recta verò o p, referet latus Icoſae-<lb/>dri inter angulum dicti trianguli inclinati, & </s> <s xml:id="echoid-s9804" xml:space="preserve">angulum ſupremæ baſis poſitum; <lb/></s> <s xml:id="echoid-s9805" xml:space="preserve">propterea quod recta o p, poſita eſt æqualis lateri Icoſaedri, & </s> <s xml:id="echoid-s9806" xml:space="preserve">angulus n o p, <lb/>angulo d l s: </s> <s xml:id="echoid-s9807" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> qui quidem æqualis eſt illi, quẽ dictum latus efficit cum perpen- <anchor type="note" xlink:label="note-247-03a" xlink:href="note-247-03"/> diculari ex angulo ſupradicti trianguli inclinati ad baſem, in medium punctum <lb/>lateris oppoſiti ducitur. </s> <s xml:id="echoid-s9808" xml:space="preserve">Eſt enim recta d s, æqualis perpendiculari ex angulo <lb/>pentagoni ad latus oppoſitum ductæ, & </s> <s xml:id="echoid-s9809" xml:space="preserve">latera sl, dl, æqualia perpendiculari <pb o="218" file="248" n="248" rhead="GEOMETR. PRACT."/> in triangulo inclinato, & </s> <s xml:id="echoid-s9810" xml:space="preserve">lateri @coſaedri inter angulum ſupremum pentagoni <lb/>prædicti, & </s> <s xml:id="echoid-s9811" xml:space="preserve">angulum trianguli inclinati. </s> <s xml:id="echoid-s9812" xml:space="preserve">Ex quo fit, punctump, in plano ſupre@ <lb/> <anchor type="figure" xlink:label="fig-248-01a" xlink:href="fig-248-01"/> mæ baſis exiſtere: </s> <s xml:id="echoid-s9813" xml:space="preserve">ac ꝓinde perpendicularem p q, ad planũ baſis per m n, du-<lb/>ctũ demiſſam, æqualẽ eſſe altitudini Icoſaedri, eiuſq; </s> <s xml:id="echoid-s9814" xml:space="preserve">ſemiſſem r q, altitudini v-<lb/>nius pyramidis trigoni eſſe æqualem. </s> <s xml:id="echoid-s9815" xml:space="preserve">Quæ omnia facilè percipientur, ſi adhibe-<lb/>atur materiale aliquod Icoſaedrum. </s> <s xml:id="echoid-s9816" xml:space="preserve">Inuenta porrò hocmodo altitudine pyra-<lb/>midis, cognoſcenda eadem ſumma diligentia e@it, beneficio inſtrumentiparti-<lb/>um, in partibus lateris corporis regularis propoſiti.</s> <s xml:id="echoid-s9817" xml:space="preserve"/> </p> <div xml:id="echoid-div597" type="float" level="2" n="16"> <note symbol="a" position="right" xlink:label="note-247-01" xlink:href="note-247-01a" xml:space="preserve">ſchol. 26. <lb/>Primi.</note> <note position="right" xlink:label="note-247-02" xlink:href="note-247-02a" xml:space="preserve">Altitudo py-<lb/>ramidis Ico-<lb/>ſaedri.</note> <note symbol="b" position="right" xlink:label="note-247-03" xlink:href="note-247-03a" xml:space="preserve">8. primi.</note> <figure xlink:label="fig-248-01" xlink:href="fig-248-01a"> <image file="248-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/248-01"/> </figure> </div> <p> <s xml:id="echoid-s9818" xml:space="preserve">DE AREA SPHÆRÆ, INVENTIONE-<lb/>que ſuperficiei conuexæ eiuſdem ſphæræ.</s> <s xml:id="echoid-s9819" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div599" type="section" level="1" n="209"> <head xml:id="echoid-head226" xml:space="preserve"><emph style="sc">Capvt</emph> V.</head> <p> <s xml:id="echoid-s9820" xml:space="preserve">1. </s> <s xml:id="echoid-s9821" xml:space="preserve">VT ſphæræ aream, ſoliditatemue pluribus poſsimus vijs aſſequi, demõ-<lb/>ſtranda prius erunt nonnulla ad eamrem valdè neceſſaria, atq; </s> <s xml:id="echoid-s9822" xml:space="preserve">vtilia. <lb/></s> <s xml:id="echoid-s9823" xml:space="preserve">quodſequentibus 7. </s> <s xml:id="echoid-s9824" xml:space="preserve">propoſitionibus effi ciemus.</s> <s xml:id="echoid-s9825" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div600" type="section" level="1" n="210"> <head xml:id="echoid-head227" xml:space="preserve">PROPOSITIO I.</head> <p> <s xml:id="echoid-s9826" xml:space="preserve">QVAM proportionem habent duæ quælibet partes aliquotæ magni-<lb/>tudinis cuiuſcunque, eandem habent duæ ſimiles partes alterius cu-<lb/>iuſuis magnitudinis.</s> <s xml:id="echoid-s9827" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9828" xml:space="preserve"><emph style="sc">Sit</emph> enim A, eadem pars magnitudinis B, quæ C, magnitudinis D: </s> <s xml:id="echoid-s9829" xml:space="preserve">Item E, <lb/> <anchor type="figure" xlink:label="fig-248-02a" xlink:href="fig-248-02"/> eadẽ pars magnitudinis B, quæ F, ma-<lb/>g@itudinis D. </s> <s xml:id="echoid-s9830" xml:space="preserve">Dico eſſe, vt A, ad E, ita <lb/>C, ad F, Quoniam enim eſt, vt A, ad B, <lb/>ita C, ad D, quod vtrobiq@ eadem pro-<lb/>portio ſubmultiplex poſita ſit. </s> <s xml:id="echoid-s9831" xml:space="preserve">Item <pb o="219" file="249" n="249" rhead="LIBER QVINTVS."/> vt B, ad E, ita D, ad F; </s> <s xml:id="echoid-s9832" xml:space="preserve">quod vtrobique poſita ſit eadem proportio multiplex: </s> <s xml:id="echoid-s9833" xml:space="preserve">e-<lb/>rit ex æquo, vt A ad E, ita C, ad F. </s> <s xml:id="echoid-s9834" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s9835" xml:space="preserve"/> </p> <div xml:id="echoid-div600" type="float" level="2" n="1"> <figure xlink:label="fig-248-02" xlink:href="fig-248-02a"> <image file="248-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/248-02"/> </figure> </div> <p> <s xml:id="echoid-s9836" xml:space="preserve"><emph style="sc">Idem</emph> ſequitur ſi A, & </s> <s xml:id="echoid-s9837" xml:space="preserve">C, ſintipſarum B, D, eædem partes plures non facien-<lb/>tes vnam: </s> <s xml:id="echoid-s9838" xml:space="preserve">Item, ſi E, & </s> <s xml:id="echoid-s9839" xml:space="preserve">F, earundem B, D, ſint eædem partes plures non facien-<lb/>tes vnam, vt {2/3}. </s> <s xml:id="echoid-s9840" xml:space="preserve">vel {3/5}. </s> <s xml:id="echoid-s9841" xml:space="preserve">&</s> <s xml:id="echoid-s9842" xml:space="preserve">c. </s> <s xml:id="echoid-s9843" xml:space="preserve">Nam ſi verbi gratia A, C, ſint {3/4}. </s> <s xml:id="echoid-s9844" xml:space="preserve">ipſarum B, D, erit <lb/>{1/4}. </s> <s xml:id="echoid-s9845" xml:space="preserve">ipſius B, ad B, vt {1/4}. </s> <s xml:id="echoid-s9846" xml:space="preserve">ipſius D, ad D. </s> <s xml:id="echoid-s9847" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur erunt quoq;</s> <s xml:id="echoid-s9848" xml:space="preserve">, vt {3/4}. </s> <s xml:id="echoid-s9849" xml:space="preserve">ipſius B, hoc <anchor type="note" xlink:label="note-249-01a" xlink:href="note-249-01"/> eſt, ipſa A, ad B, ita {3/4}. </s> <s xml:id="echoid-s9850" xml:space="preserve">ipſius D, hoc eſt, ipſa C, ad D. </s> <s xml:id="echoid-s9851" xml:space="preserve">Rurſus ſi verbi gratia E, F, <lb/>ſint {2/3}. </s> <s xml:id="echoid-s9852" xml:space="preserve">ipſarum B, D, erit, vt B, ad {1/3}. </s> <s xml:id="echoid-s9853" xml:space="preserve">eiuſdem B, ita D, ad {1/3}. </s> <s xml:id="echoid-s9854" xml:space="preserve">eiuſdem D. </s> <s xml:id="echoid-s9855" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Acpro- <anchor type="note" xlink:label="note-249-02a" xlink:href="note-249-02"/> inde vt B, ad {2/3}. </s> <s xml:id="echoid-s9856" xml:space="preserve">id eſt, ad E, ita D, ad {2/3}. </s> <s xml:id="echoid-s9857" xml:space="preserve">id eſt, ad F. </s> <s xml:id="echoid-s9858" xml:space="preserve">Quare, vt prius, erit ex æquo, <lb/>vt A, ad E, ita C. </s> <s xml:id="echoid-s9859" xml:space="preserve">ad F.</s> <s xml:id="echoid-s9860" xml:space="preserve"/> </p> <div xml:id="echoid-div601" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-249-01" xlink:href="note-249-01a" xml:space="preserve">ſchol. 22. <lb/>quinti.</note> <note symbol="b" position="right" xlink:label="note-249-02" xlink:href="note-249-02a" xml:space="preserve">ſchol. 22. <lb/>quinti.</note> </div> </div> <div xml:id="echoid-div603" type="section" level="1" n="211"> <head xml:id="echoid-head228" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s9861" xml:space="preserve"><emph style="sc">Seqvitvr</emph> hinc, ita eſſe {1/4}. </s> <s xml:id="echoid-s9862" xml:space="preserve">cuiuſuis magnitudinis ad {1/3}. </s> <s xml:id="echoid-s9863" xml:space="preserve">eiuſdem, vt eſt {1/2}. <lb/></s> <s xml:id="echoid-s9864" xml:space="preserve">cuiuſuis alteri<emph style="sub">9</emph> magnitudinis ad {2/3}. </s> <s xml:id="echoid-s9865" xml:space="preserve">eiuſdẽ. </s> <s xml:id="echoid-s9866" xml:space="preserve">Quoniã. </s> <s xml:id="echoid-s9867" xml:space="preserve">n. </s> <s xml:id="echoid-s9868" xml:space="preserve">vt oſtendim<emph style="sub">9</emph>, ita eſt {1/4}. </s> <s xml:id="echoid-s9869" xml:space="preserve">prio-<lb/>ris magnitudinis ad {1/3}. </s> <s xml:id="echoid-s9870" xml:space="preserve">eiuſdem, vt {1/4}. </s> <s xml:id="echoid-s9871" xml:space="preserve">poſterioris ad {1/3}. </s> <s xml:id="echoid-s9872" xml:space="preserve">eiuſdem. </s> <s xml:id="echoid-s9873" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Vtautem {1/4}.</s> <s xml:id="echoid-s9874" xml:space="preserve"> <anchor type="note" xlink:label="note-249-03a" xlink:href="note-249-03"/> poſterioris ad {1/3}. </s> <s xml:id="echoid-s9875" xml:space="preserve">ita ſunt {2/4}. </s> <s xml:id="echoid-s9876" xml:space="preserve">ad {2/3}. </s> <s xml:id="echoid-s9877" xml:space="preserve">hoc eſt, {1/2}. </s> <s xml:id="echoid-s9878" xml:space="preserve">ad {2/3}. </s> <s xml:id="echoid-s9879" xml:space="preserve">Igitur erit vt {1/4}. </s> <s xml:id="echoid-s9880" xml:space="preserve">prioris <lb/>magnitudinis ad {1/3}. </s> <s xml:id="echoid-s9881" xml:space="preserve">eiuſdem, ita {1/2}. </s> <s xml:id="echoid-s9882" xml:space="preserve">poſterioris magnitudinis ad {2/3}. </s> <s xml:id="echoid-s9883" xml:space="preserve">eiuſdem.</s> <s xml:id="echoid-s9884" xml:space="preserve"/> </p> <div xml:id="echoid-div603" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-249-03" xlink:href="note-249-03a" xml:space="preserve">1. quiuti.</note> </div> </div> <div xml:id="echoid-div605" type="section" level="1" n="212"> <head xml:id="echoid-head229" xml:space="preserve">PROPOSITIO II.</head> <p> <s xml:id="echoid-s9885" xml:space="preserve">RECTANGVLVM ſub diametro, & </s> <s xml:id="echoid-s9886" xml:space="preserve">circumferentia maximi circuli <lb/>in ſphæra comprehenſum, quadruplum eſt circuli maximi, & </s> <s xml:id="echoid-s9887" xml:space="preserve">ſuper-<lb/>ficiei conuexæ eiuſdem ſphęræ ęquale.</s> <s xml:id="echoid-s9888" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9889" xml:space="preserve"><emph style="sc">Sit</emph> rectangulum AB, comprehenſum ſub diametro AC, & </s> <s xml:id="echoid-s9890" xml:space="preserve">circumferentia <lb/>CB, maximi in ſphæra circuli. </s> <s xml:id="echoid-s9891" xml:space="preserve">Dico rectangulum AB, <lb/> <anchor type="figure" xlink:label="fig-249-01a" xlink:href="fig-249-01"/> quadruplum eſſe circuli maximi in ſphæra, & </s> <s xml:id="echoid-s9892" xml:space="preserve">ſuperficiei <lb/>conuexæ eiuſdem ſphæræ ęquale. </s> <s xml:id="echoid-s9893" xml:space="preserve">Sectis enim omnibus <lb/>lateribus bifariam in E, F, G, H, iunctiſquerectis EG, FH, <lb/>ſecantibus ſeſe in I, diuiſum erit totum rectangulum in <lb/>quatuor æqualia A I, C I, B I, D I, <anchor type="note" xlink:href="" symbol="d"/> quodrectæ E G, F H, <anchor type="note" xlink:label="note-249-04a" xlink:href="note-249-04"/> rectis A D, A C, parallelæ ſint. </s> <s xml:id="echoid-s9894" xml:space="preserve">Ac proinderectangulum A B, rectanguli C I, <lb/>quadruplum erit. </s> <s xml:id="echoid-s9895" xml:space="preserve">Eſt autem rectangulum C I, contentum ſub C E, ſemidia-<lb/>metro, & </s> <s xml:id="echoid-s9896" xml:space="preserve">ſemicircumferentia C F, circulo maximo, cuius nimirum diameter <lb/>A C, ęquale, vt lib. </s> <s xml:id="echoid-s9897" xml:space="preserve">4. </s> <s xml:id="echoid-s9898" xml:space="preserve">capit. </s> <s xml:id="echoid-s9899" xml:space="preserve">7. </s> <s xml:id="echoid-s9900" xml:space="preserve">Nume. </s> <s xml:id="echoid-s9901" xml:space="preserve">1. </s> <s xml:id="echoid-s9902" xml:space="preserve">demonſtratum eſt. </s> <s xml:id="echoid-s9903" xml:space="preserve">lgitur rectangu-<lb/>lum A B, circulimaximi quadruplum eſt. </s> <s xml:id="echoid-s9904" xml:space="preserve">Et quia eiuſdem circuli maximi qua-<lb/>drupla eſt ſuperficies conuexa ſphærę, per propoſ. </s> <s xml:id="echoid-s9905" xml:space="preserve">31. </s> <s xml:id="echoid-s9906" xml:space="preserve">lib. </s> <s xml:id="echoid-s9907" xml:space="preserve">1. </s> <s xml:id="echoid-s9908" xml:space="preserve">Archimedis de ſphę-<lb/>ra, & </s> <s xml:id="echoid-s9909" xml:space="preserve">Cylindro: </s> <s xml:id="echoid-s9910" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ęquale erit rectangulum A B, conuexæ ſuperficiei, quod e- <anchor type="note" xlink:label="note-249-05a" xlink:href="note-249-05"/> rat demonſtrandum.</s> <s xml:id="echoid-s9911" xml:space="preserve"/> </p> <div xml:id="echoid-div605" type="float" level="2" n="1"> <figure xlink:label="fig-249-01" xlink:href="fig-249-01a"> <image file="249-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/249-01"/> </figure> <note symbol="d" position="right" xlink:label="note-249-04" xlink:href="note-249-04a" xml:space="preserve">33. primi.</note> <note symbol="e" position="right" xlink:label="note-249-05" xlink:href="note-249-05a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div607" type="section" level="1" n="213"> <head xml:id="echoid-head230" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s9912" xml:space="preserve">EX demonſtratione liquet, rectangulum ſub diametro cuiuſuis circuli, (et-<lb/>iamſi non ſit maximus in ſphęra,) & </s> <s xml:id="echoid-s9913" xml:space="preserve">circumferentia eiuſdem, quadruplum eſſe <lb/>ipſius circuli. </s> <s xml:id="echoid-s9914" xml:space="preserve">Eadem enim ſemper demonſtratio adhibebitur.</s> <s xml:id="echoid-s9915" xml:space="preserve"/> </p> <pb o="220" file="250" n="250" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div608" type="section" level="1" n="214"> <head xml:id="echoid-head231" xml:space="preserve">PROPOSITIO III.</head> <p> <s xml:id="echoid-s9916" xml:space="preserve">EADEM eſt proportio quadrati circumferentiæ circuli maximi in <lb/>ſphęra ad ſuperficiem ſphęræ, quę circumferentię circuli maximi ad <lb/>diametrum. </s> <s xml:id="echoid-s9917" xml:space="preserve">Item eadem eſt proportio quadrati diametri maximi cir-<lb/>culi in ſphęra ad ſuperficiem ſphęrę, quę diametri ad circumferenti-<lb/>am eiuſdem circuli maximi.</s> <s xml:id="echoid-s9918" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9919" xml:space="preserve"><emph style="sc">Sit</emph> circulus ſphæræ maximus ABCD, eiuſque diameter AC. </s> <s xml:id="echoid-s9920" xml:space="preserve">Dico ita eſſe <lb/>quadratum ex circumferentia ABCD, deſcriptum ad ſuperficiem ſphęræ, cuius <lb/>diameter A C, vt eſt circumferentia ABCD, ad diametrum AC. </s> <s xml:id="echoid-s9921" xml:space="preserve">Itemita eſſe qua <lb/>dratum diametri AC, circulimaximi in ſphæra, ad ſuperficiem ſphęrę, vt eſt di-<lb/>ameter A C, ad circumferentiam A B C D. </s> <s xml:id="echoid-s9922" xml:space="preserve">Sit enim E F, diametro A C, & </s> <s xml:id="echoid-s9923" xml:space="preserve">recta <lb/>FG, circumferentię ABCD, æqualis, & </s> <s xml:id="echoid-s9924" xml:space="preserve">ſuper FG, conſtruatur quadratum GH, <lb/>capiaturque F I, ipſi E F, ęqualis, eritque E I, quadratum diametri E F, vel A C. <lb/></s> <s xml:id="echoid-s9925" xml:space="preserve"> <anchor type="figure" xlink:label="fig-250-01a" xlink:href="fig-250-01"/> Perfecta autem figura, vt vides, erit tam rectangulum G I, ſub ſemidiametro FI, <lb/>maximi circuli, & </s> <s xml:id="echoid-s9926" xml:space="preserve">circumfentia FG, quam rectangulum EH, ſub diametro EF, <lb/>eiuſdem circuli maximi, & </s> <s xml:id="echoid-s9927" xml:space="preserve">circumferentia FH, ęquale, per pręcedentem, ſuper-<lb/>ficiei conuexę ſphęrę. </s> <s xml:id="echoid-s9928" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum ergo ſit, vt GH, quadratum ex circumferentia FG, <anchor type="note" xlink:label="note-250-01a" xlink:href="note-250-01"/> deſcriptum adrectangulum EH, ſuperficiei conuexę ſphęrę ęquale, ita GF, cir-<lb/>cumferentia ad EF, diametrum circulimaximi, conſtat primum.</s> <s xml:id="echoid-s9929" xml:space="preserve"/> </p> <div xml:id="echoid-div608" type="float" level="2" n="1"> <figure xlink:label="fig-250-01" xlink:href="fig-250-01a"> <image file="250-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/250-01"/> </figure> <note symbol="a" position="left" xlink:label="note-250-01" xlink:href="note-250-01a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s9930" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/><emph style="sc">Item</emph> cum ſit, vt E I, quadratum diametri EF, maximi circuli, ad I G, re- <anchor type="note" xlink:label="note-250-02a" xlink:href="note-250-02"/> ctangulum ſuperficiei conuexæ ſphærę ęquale, ita E F, diameter maximi circuli <lb/>ad FG, circumferentiam, patetid, quod ſecundo loco proponitur.</s> <s xml:id="echoid-s9931" xml:space="preserve"/> </p> <div xml:id="echoid-div609" type="float" level="2" n="2"> <note symbol="b" position="left" xlink:label="note-250-02" xlink:href="note-250-02a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div611" type="section" level="1" n="215"> <head xml:id="echoid-head232" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s9932" xml:space="preserve"><emph style="sc">Hinc</emph> manifeſtum eſt (id quod lib. </s> <s xml:id="echoid-s9933" xml:space="preserve">4. </s> <s xml:id="echoid-s9934" xml:space="preserve">capit. </s> <s xml:id="echoid-s9935" xml:space="preserve">7. </s> <s xml:id="echoid-s9936" xml:space="preserve">Nume. </s> <s xml:id="echoid-s9937" xml:space="preserve">1. </s> <s xml:id="echoid-s9938" xml:space="preserve">etiam demonſtra-<lb/> <anchor type="note" xlink:label="note-250-03a" xlink:href="note-250-03"/> uimus) circuli aream gignitam ex {1/4}. </s> <s xml:id="echoid-s9939" xml:space="preserve">diametri in totam circumferentiam, quam <lb/>ex {1/4}. </s> <s xml:id="echoid-s9940" xml:space="preserve">circumferentię in totam diametrum. </s> <s xml:id="echoid-s9941" xml:space="preserve">Cum enim circulus A B C D, ſit <lb/>quarta pars rectanguli GI, quòd hoc illius quadruplum ſit oſtenſum propoſ. </s> <s xml:id="echoid-s9942" xml:space="preserve">2.</s> <s xml:id="echoid-s9943" xml:space="preserve"> <pb o="221" file="251" n="251" rhead="LIBER QVINTVS."/> <anchor type="note" xlink:href="" symbol="a"/>Contineatur autem quarta pars rectanguli G I, tam ſub {1/4}. </s> <s xml:id="echoid-s9944" xml:space="preserve">diametri F I, & </s> <s xml:id="echoid-s9945" xml:space="preserve">cir- <anchor type="note" xlink:label="note-251-01a" xlink:href="note-251-01"/> cumferentia F G, quam ſub {1/4}. </s> <s xml:id="echoid-s9946" xml:space="preserve">circumferentiæ F G, & </s> <s xml:id="echoid-s9947" xml:space="preserve">diametro F I; </s> <s xml:id="echoid-s9948" xml:space="preserve">liquido con-<lb/>ſtat, quod proponitur.</s> <s xml:id="echoid-s9949" xml:space="preserve"/> </p> <div xml:id="echoid-div611" type="float" level="2" n="1"> <note position="left" xlink:label="note-250-03" xlink:href="note-250-03a" xml:space="preserve">Area circuli.</note> <note symbol="a" position="right" xlink:label="note-251-01" xlink:href="note-251-01a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div613" type="section" level="1" n="216"> <head xml:id="echoid-head233" xml:space="preserve">PROPOSITIO IV.</head> <p> <s xml:id="echoid-s9950" xml:space="preserve">QVADRATVM circumferentiæ circuli maximi in ſphæra ad ſuper-<lb/>ficiem ſphæræ conuexam, maiorem proportionem habet, quam 223. <lb/></s> <s xml:id="echoid-s9951" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s9952" xml:space="preserve">minorem verò, quam 22. </s> <s xml:id="echoid-s9953" xml:space="preserve">ad 7.</s> <s xml:id="echoid-s9954" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9955" xml:space="preserve"><emph style="sc">Cvm</emph> enim per præcedentem ſit, vt quadratum circumferentiæ maximi cir-<lb/>culi ad ſuperficiem conuexam ſphærę, ita circumferentia eiuſdem circuli ad <lb/>diametrum: </s> <s xml:id="echoid-s9956" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ſit autem maior proportio circumferentię ad diametrum, quam <anchor type="note" xlink:label="note-251-02a" xlink:href="note-251-02"/> 3 {10/71}. </s> <s xml:id="echoid-s9957" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s9958" xml:space="preserve">hoc eſt, quam 223. </s> <s xml:id="echoid-s9959" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s9960" xml:space="preserve">minor verò, quam 3 {1/7}. </s> <s xml:id="echoid-s9961" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s9962" xml:space="preserve">hoc eſt, quam <lb/>22. </s> <s xml:id="echoid-s9963" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s9964" xml:space="preserve">liquetid, quod propoſitum eſt.</s> <s xml:id="echoid-s9965" xml:space="preserve"/> </p> <div xml:id="echoid-div613" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-251-02" xlink:href="note-251-02a" xml:space="preserve">2. de Dimẽſ@ <lb/>circuli lib. 4.</note> </div> </div> <div xml:id="echoid-div615" type="section" level="1" n="217"> <head xml:id="echoid-head234" xml:space="preserve">PROPOSITIO V.</head> <p> <s xml:id="echoid-s9966" xml:space="preserve">QVADRATVM diametri circuli in ſphæra maximi ad ſuperficiem <lb/>ſphæræ conuexam, maiorem proportionem habet, quam 7. </s> <s xml:id="echoid-s9967" xml:space="preserve">ad 22. <lb/></s> <s xml:id="echoid-s9968" xml:space="preserve">minorem verò, quam 71. </s> <s xml:id="echoid-s9969" xml:space="preserve">ad 223.</s> <s xml:id="echoid-s9970" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9971" xml:space="preserve"><emph style="sc">Cvm</emph> enim per propoſ. </s> <s xml:id="echoid-s9972" xml:space="preserve">3. </s> <s xml:id="echoid-s9973" xml:space="preserve">ſit, vt quadratum diametri ad ſuperficiem ſphærę, <lb/>ita diameter ad circumferentiam: </s> <s xml:id="echoid-s9974" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Sit autem maior proportio diametri ad cir- <anchor type="note" xlink:label="note-251-03a" xlink:href="note-251-03"/> cumferentiam, quam 7. </s> <s xml:id="echoid-s9975" xml:space="preserve">ad 22. </s> <s xml:id="echoid-s9976" xml:space="preserve">(<anchor type="note" xlink:href="" symbol="d"/> quod minor ſit proportio circumferentię ad <anchor type="note" xlink:label="note-251-04a" xlink:href="note-251-04"/> diametrum, quam 22. </s> <s xml:id="echoid-s9977" xml:space="preserve">ad 7.) </s> <s xml:id="echoid-s9978" xml:space="preserve">minor verò, quam 71. </s> <s xml:id="echoid-s9979" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s9980" xml:space="preserve">(<anchor type="note" xlink:href="" symbol="e"/> quod maior ſit proportio circumferentię ad diametrum, quam 223. </s> <s xml:id="echoid-s9981" xml:space="preserve">ad 71.) </s> <s xml:id="echoid-s9982" xml:space="preserve">patet verum eſſe, <lb/>quod proponitur.</s> <s xml:id="echoid-s9983" xml:space="preserve"/> </p> <div xml:id="echoid-div615" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-251-03" xlink:href="note-251-03a" xml:space="preserve">26. quinti.</note> <note symbol="d" position="right" xlink:label="note-251-04" xlink:href="note-251-04a" xml:space="preserve">2. de Dimẽſ. <lb/>circuli lib. 4.</note> </div> </div> <div xml:id="echoid-div617" type="section" level="1" n="218"> <head xml:id="echoid-head235" xml:space="preserve">PROPOSITIO VI.</head> <p> <s xml:id="echoid-s9984" xml:space="preserve">PROPORTIO cubi ex circumferentia maximi in ſphæra circuli de-<lb/>ſcripti, ad ſphæram maior eſt, quam 298374. </s> <s xml:id="echoid-s9985" xml:space="preserve">ad 5041. </s> <s xml:id="echoid-s9986" xml:space="preserve">minor autem, <lb/>quam 2904. </s> <s xml:id="echoid-s9987" xml:space="preserve">ad 49.</s> <s xml:id="echoid-s9988" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s9989" xml:space="preserve"><emph style="sc">Cvm</emph> enim ſit per propoſ. </s> <s xml:id="echoid-s9990" xml:space="preserve">1. </s> <s xml:id="echoid-s9991" xml:space="preserve">cap. </s> <s xml:id="echoid-s9992" xml:space="preserve">7. </s> <s xml:id="echoid-s9993" xml:space="preserve">lib. </s> <s xml:id="echoid-s9994" xml:space="preserve">4. </s> <s xml:id="echoid-s9995" xml:space="preserve">vt circumferentia maximi circuli in <lb/>quauis ſphæra, ad circumferentiam maximi circuli in quauis alia ſphęra, ita dia-<lb/>meter ad diametrum: </s> <s xml:id="echoid-s9996" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> habeat autem cubus circumferentię prioris ſphærę ad cubum circumferentię ſphærę poſterioris, proportionem triplicatam circum-<lb/> <anchor type="note" xlink:label="note-251-05a" xlink:href="note-251-05"/> ferentię ad circumferentiam; </s> <s xml:id="echoid-s9997" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Item ſphæra prior ad poſteriorem ſphęram, proportionem triplicatam diametri ad diametrum: </s> <s xml:id="echoid-s9998" xml:space="preserve">erit vt cubus ex circumfe-<lb/> <anchor type="note" xlink:label="note-251-06a" xlink:href="note-251-06"/> rentia prioris ſphærę deſcriptus ad cubum ex poſterioris ſphærę circumferen-<lb/>tia deſcriptum, ita ſphæra prior ad poſteriorem ſphęram; </s> <s xml:id="echoid-s9999" xml:space="preserve">Et permutando, vt cu-<lb/>bus circumferentię ſphærę prioris ad priorem ſphęram, ita cubus circumferen-<lb/>tiæ poſterioris ſphęrę ad ſphęram poſteriorem.</s> <s xml:id="echoid-s10000" xml:space="preserve"/> </p> <div xml:id="echoid-div617" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-251-05" xlink:href="note-251-05a" xml:space="preserve">33. vndec.</note> <note position="right" xlink:label="note-251-06" xlink:href="note-251-06a" xml:space="preserve">18. duodec.</note> </div> <pb o="222" file="252" n="252" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s10001" xml:space="preserve"><emph style="sc">Iam</emph> verè ſi circumferentia circuli maximi alicuius ſphærę ſit 1. </s> <s xml:id="echoid-s10002" xml:space="preserve">diuidatur que <lb/>per 3 {10/71}. </s> <s xml:id="echoid-s10003" xml:space="preserve">producetur diameter {71/223}. </s> <s xml:id="echoid-s10004" xml:space="preserve">maior quam vera, ex coroll. </s> <s xml:id="echoid-s10005" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s10006" xml:space="preserve">2. </s> <s xml:id="echoid-s10007" xml:space="preserve">de <lb/>dimenſione circuli. </s> <s xml:id="echoid-s10008" xml:space="preserve">Si igitur eius ſemiſsis {71/446}. </s> <s xml:id="echoid-s10009" xml:space="preserve">in {1/2}. </s> <s xml:id="echoid-s10010" xml:space="preserve">ſemiſſem circumferentiæ <lb/>ducatur, procreabitur area circuli maximi {71/892}. </s> <s xml:id="echoid-s10011" xml:space="preserve">maior, quẽ vera: </s> <s xml:id="echoid-s10012" xml:space="preserve">quæ rurſus du-<lb/>cta in {2/3}. </s> <s xml:id="echoid-s10013" xml:space="preserve">diametri inuentę, (quæ etiam maior eſt, quam vera) nimirum in <lb/>{142/669}. </s> <s xml:id="echoid-s10014" xml:space="preserve">gignet, vtinſra in regula 2. </s> <s xml:id="echoid-s10015" xml:space="preserve">oſtendam, ſoliditatem ſphæræ {10082/596748}. </s> <s xml:id="echoid-s10016" xml:space="preserve">hoc eſt <lb/>{5041/298374}. </s> <s xml:id="echoid-s10017" xml:space="preserve">maiorem tamen, quam veram. </s> <s xml:id="echoid-s10018" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Maior ergo erit proportio cubi ex <anchor type="note" xlink:label="note-252-01a" xlink:href="note-252-01"/> circumferentia 1. </s> <s xml:id="echoid-s10019" xml:space="preserve">deſcripti, qui cubus eſt 1. </s> <s xml:id="echoid-s10020" xml:space="preserve">ad ſphęram, quam ad {5041/298374}. </s> <s xml:id="echoid-s10021" xml:space="preserve">Cum <lb/>ergo ſit 1. </s> <s xml:id="echoid-s10022" xml:space="preserve">ad {5041/298374}. </s> <s xml:id="echoid-s10023" xml:space="preserve">vt 298374. </s> <s xml:id="echoid-s10024" xml:space="preserve">ad 5041. </s> <s xml:id="echoid-s10025" xml:space="preserve">(Quoniam enim ex propoſitione <lb/>2. </s> <s xml:id="echoid-s10026" xml:space="preserve">Minutiarum ad finem libr. </s> <s xml:id="echoid-s10027" xml:space="preserve">9. </s> <s xml:id="echoid-s10028" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s10029" xml:space="preserve">eadem eſt proportio Numeratoris <lb/>5041. </s> <s xml:id="echoid-s10030" xml:space="preserve">ad denominatorem 298374. </s> <s xml:id="echoid-s10031" xml:space="preserve">quæ minutię {5041/298374}. </s> <s xml:id="echoid-s10032" xml:space="preserve">ad ſuum integrum 1. <lb/></s> <s xml:id="echoid-s10033" xml:space="preserve">erit conuertendo, vt 298374. </s> <s xml:id="echoid-s10034" xml:space="preserve">ad 5041. </s> <s xml:id="echoid-s10035" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s10036" xml:space="preserve">ad {5042/298374}.) </s> <s xml:id="echoid-s10037" xml:space="preserve">maior erit proportio <lb/>cubi ex circumferentia 1. </s> <s xml:id="echoid-s10038" xml:space="preserve">deſcripti, qui cubus eſt 1. </s> <s xml:id="echoid-s10039" xml:space="preserve">ad ſphæram, qaam 298374. </s> <s xml:id="echoid-s10040" xml:space="preserve"><lb/>ad 5041. </s> <s xml:id="echoid-s10041" xml:space="preserve">Cum igitur ita ſe habeat cubus ex circumferentia maximi circuli <lb/>in ſphæra ad ſphæram qualibet deſcriptus, vt cubus circumferentiæ 1. </s> <s xml:id="echoid-s10042" xml:space="preserve">ad ſuam <lb/>ſphæram, vtinitio huius propoſ. </s> <s xml:id="echoid-s10043" xml:space="preserve">oſtendimus; </s> <s xml:id="echoid-s10044" xml:space="preserve">Conſtatid, quod primo loco <lb/>proponitur.</s> <s xml:id="echoid-s10045" xml:space="preserve"/> </p> <div xml:id="echoid-div618" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-252-01" xlink:href="note-252-01a" xml:space="preserve">8. quinti.</note> </div> <p> <s xml:id="echoid-s10046" xml:space="preserve"><emph style="sc">Rvrsvs</emph> ſi circumferentia 1. </s> <s xml:id="echoid-s10047" xml:space="preserve">diuidatur per 3 {1/7}. </s> <s xml:id="echoid-s10048" xml:space="preserve">producetur diameter {7/22}. <lb/></s> <s xml:id="echoid-s10049" xml:space="preserve">minor quam vera, ex coroll. </s> <s xml:id="echoid-s10050" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s10051" xml:space="preserve">2. </s> <s xml:id="echoid-s10052" xml:space="preserve">de dimenſione circuli. </s> <s xml:id="echoid-s10053" xml:space="preserve">Si igitur eius ſe-<lb/>miſsis {7/44}. </s> <s xml:id="echoid-s10054" xml:space="preserve">ducatur in {1/2}. </s> <s xml:id="echoid-s10055" xml:space="preserve">ſemiſſem circumferentiæ, procreabitur area circuli ma-<lb/>ximi {7/88}: </s> <s xml:id="echoid-s10056" xml:space="preserve">minor, quam vera: </s> <s xml:id="echoid-s10057" xml:space="preserve">quę rurſus ducta in {2/3}. </s> <s xml:id="echoid-s10058" xml:space="preserve">diametri inuentę (quæ etiam <lb/>minor eſt, quam vera) nimirum in {14/66}. </s> <s xml:id="echoid-s10059" xml:space="preserve">hoc eſt in {7/33}. </s> <s xml:id="echoid-s10060" xml:space="preserve">producet, vt infra in regula <lb/>2. </s> <s xml:id="echoid-s10061" xml:space="preserve">docebo, ſoliditatem ſphærę {49/2904}. </s> <s xml:id="echoid-s10062" xml:space="preserve">minoremtamen, quam veram. </s> <s xml:id="echoid-s10063" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Minor er- <anchor type="note" xlink:label="note-252-02a" xlink:href="note-252-02"/> go erit proportio cubi ex circumferentia 1. </s> <s xml:id="echoid-s10064" xml:space="preserve">deſcripti, qui cubus eſt 1. </s> <s xml:id="echoid-s10065" xml:space="preserve">ad ſphæ-<lb/>ram, quam ad {49/2904}. </s> <s xml:id="echoid-s10066" xml:space="preserve">Cum ergo ſit 1. </s> <s xml:id="echoid-s10067" xml:space="preserve">ad {49/2904}. </s> <s xml:id="echoid-s10068" xml:space="preserve">vt 2904. </s> <s xml:id="echoid-s10069" xml:space="preserve">ad 49. </s> <s xml:id="echoid-s10070" xml:space="preserve">(Quoniam enim ex <lb/>propoſ. </s> <s xml:id="echoid-s10071" xml:space="preserve">2. </s> <s xml:id="echoid-s10072" xml:space="preserve">Minutiarum ad finem libr. </s> <s xml:id="echoid-s10073" xml:space="preserve">9. </s> <s xml:id="echoid-s10074" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s10075" xml:space="preserve">eadem proportio eſt Numerato-<lb/>ris 49. </s> <s xml:id="echoid-s10076" xml:space="preserve">ad denominatorem 2904. </s> <s xml:id="echoid-s10077" xml:space="preserve">quæ minutię {49/2904}. </s> <s xml:id="echoid-s10078" xml:space="preserve">ad ſuum integrum 1. </s> <s xml:id="echoid-s10079" xml:space="preserve">erit <lb/>conuertendo, vt 2904. </s> <s xml:id="echoid-s10080" xml:space="preserve">ad 49. </s> <s xml:id="echoid-s10081" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s10082" xml:space="preserve">ad {49/2904}.) </s> <s xml:id="echoid-s10083" xml:space="preserve">minor erit proportio cubi deſcripti <lb/>ex circumferentia 1. </s> <s xml:id="echoid-s10084" xml:space="preserve">qui cubus eſt 1. </s> <s xml:id="echoid-s10085" xml:space="preserve">ad ſphæram, quam 2904. </s> <s xml:id="echoid-s10086" xml:space="preserve">ad 49. </s> <s xml:id="echoid-s10087" xml:space="preserve">Cum igi-<lb/>tur ita ſe habeat cubus ex circumferentia maximi circuli ſphærę cuiuslibet de-<lb/>ſcriptus, ad ſphæram, vt cubus circumferentiæ 1. </s> <s xml:id="echoid-s10088" xml:space="preserve">ad ſuam ſphęram, vtinitio hu-<lb/>ius propoſitionis demonſtrauimus, patet etiam id, quod ſecundo loco propo-<lb/>ſitum erat.</s> <s xml:id="echoid-s10089" xml:space="preserve"/> </p> <div xml:id="echoid-div619" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-252-02" xlink:href="note-252-02a" xml:space="preserve">@. quinti.</note> </div> </div> <div xml:id="echoid-div621" type="section" level="1" n="219"> <head xml:id="echoid-head236" xml:space="preserve">PROPOSITIO VII.</head> <p> <s xml:id="echoid-s10090" xml:space="preserve">CVBVS diametri ſphæræ ad ſphæram, maiorem proportionem habet, <lb/>quam 21. </s> <s xml:id="echoid-s10091" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10092" xml:space="preserve">minorem verò, quam 426. </s> <s xml:id="echoid-s10093" xml:space="preserve">ad 223.</s> <s xml:id="echoid-s10094" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10095" xml:space="preserve"><emph style="sc">Cvm</emph> enim ſit, vt cubus diametri cuiuslibet ſphærę ad cubum diametri al-<lb/>terius ſphærę, ita ſphæra ad ſphęram <anchor type="note" xlink:href="" symbol="c"/>; </s> <s xml:id="echoid-s10096" xml:space="preserve">quod vtra que proportio ſit trip licata <anchor type="note" xlink:label="note-252-03a" xlink:href="note-252-03"/> proportionis diametrorum: </s> <s xml:id="echoid-s10097" xml:space="preserve">erit permutando, vt cubus diametri cuiuslibet <lb/>ſphærę ad ipſam ſphęram, ita cubus diametri alterius ſphærę ad ipſam ſphęram.</s> <s xml:id="echoid-s10098" xml:space="preserve"/> </p> <div xml:id="echoid-div621" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-252-03" xlink:href="note-252-03a" xml:space="preserve">33. vndec. & <lb/>18. duodec.</note> </div> <p> <s xml:id="echoid-s10099" xml:space="preserve"><emph style="sc">Qvod</emph> ſi diameter alicuius ſphærę ponatur 1. </s> <s xml:id="echoid-s10100" xml:space="preserve">multipliceturq; </s> <s xml:id="echoid-s10101" xml:space="preserve">per 3 {1/7}. </s> <s xml:id="echoid-s10102" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> pro- <anchor type="note" xlink:label="note-252-04a" xlink:href="note-252-04"/> ueniet circuli maximi circumferentia {@@/7}. </s> <s xml:id="echoid-s10103" xml:space="preserve">maior quam vera. </s> <s xml:id="echoid-s10104" xml:space="preserve">Eius ergo ſemiſ@is <lb/>{11/7}. </s> <s xml:id="echoid-s10105" xml:space="preserve">in {1/2}. </s> <s xml:id="echoid-s10106" xml:space="preserve">ſemiſſem diametri ducta efficiet {11/@@}. </s> <s xml:id="echoid-s10107" xml:space="preserve">aream ipſius maximi circuli vera ma- <pb o="223" file="253" n="253" rhead="LIBER QVINTVS."/> iorem: </s> <s xml:id="echoid-s10108" xml:space="preserve">ac proinde ſi hęc area maior, quam vera, ducatur in {2/3}. </s> <s xml:id="echoid-s10109" xml:space="preserve">diametri, gigne-<lb/>tur, vt infra in regula 2. </s> <s xml:id="echoid-s10110" xml:space="preserve">dicetur, ſoliditas ſphærę {22/42}. </s> <s xml:id="echoid-s10111" xml:space="preserve">hoc eſt {11/21}. </s> <s xml:id="echoid-s10112" xml:space="preserve">maior quam ve-<lb/> <anchor type="note" xlink:label="note-253-01a" xlink:href="note-253-01"/> ra. </s> <s xml:id="echoid-s10113" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur cubus diametri 1. </s> <s xml:id="echoid-s10114" xml:space="preserve">qui eſt 1. </s> <s xml:id="echoid-s10115" xml:space="preserve">ad ſphæram, proportionem habebit maio- rem, quam ad {11/2@}. </s> <s xml:id="echoid-s10116" xml:space="preserve">Cum ergo ſit 1. </s> <s xml:id="echoid-s10117" xml:space="preserve">ad {11/21}. </s> <s xml:id="echoid-s10118" xml:space="preserve">vt 21. </s> <s xml:id="echoid-s10119" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10120" xml:space="preserve">(Quia enim ex propoſ. </s> <s xml:id="echoid-s10121" xml:space="preserve">2. </s> <s xml:id="echoid-s10122" xml:space="preserve">Mi-<lb/>nutiarum ad finem lib. </s> <s xml:id="echoid-s10123" xml:space="preserve">9. </s> <s xml:id="echoid-s10124" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s10125" xml:space="preserve">eadem eſt proportio Numeratoris 11. </s> <s xml:id="echoid-s10126" xml:space="preserve">ad Deno-<lb/>minatorem 21. </s> <s xml:id="echoid-s10127" xml:space="preserve">quæ Minutiæ {11/21}. </s> <s xml:id="echoid-s10128" xml:space="preserve">ad ſuum integrum 1. </s> <s xml:id="echoid-s10129" xml:space="preserve">erit conuertendo vt 21. </s> <s xml:id="echoid-s10130" xml:space="preserve">ad <lb/>11. </s> <s xml:id="echoid-s10131" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s10132" xml:space="preserve">ad {11/21}.) </s> <s xml:id="echoid-s10133" xml:space="preserve">maior erit proportio cubi 1. </s> <s xml:id="echoid-s10134" xml:space="preserve">ex diametro 1. </s> <s xml:id="echoid-s10135" xml:space="preserve">deſcripti ad ſphæram, <lb/>quam 21. </s> <s xml:id="echoid-s10136" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10137" xml:space="preserve">Et quia, vt initio huius propoſitionis oſtendimus, ita eſt cubus <lb/>diametri cuiuſuis alterius ſphæræ ad ipſam ſphærã, vt cubus diametri 1. </s> <s xml:id="echoid-s10138" xml:space="preserve">ad ſuam <lb/>ſphæram, verum eſt, quod primo loco eſt propoſitum.</s> <s xml:id="echoid-s10139" xml:space="preserve"/> </p> <div xml:id="echoid-div622" type="float" level="2" n="2"> <note symbol="d" position="left" xlink:label="note-252-04" xlink:href="note-252-04a" xml:space="preserve">corol 2. de <lb/>Dimenſ. cir-<lb/>culi.</note> <note symbol="a" position="right" xlink:label="note-253-01" xlink:href="note-253-01a" xml:space="preserve">8. quinti.</note> </div> <p> <s xml:id="echoid-s10140" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> <emph style="sc">Item</emph> ſi diameter 1. </s> <s xml:id="echoid-s10141" xml:space="preserve">ducatur in 3 {10/71}. </s> <s xml:id="echoid-s10142" xml:space="preserve">producetur circumferentia maximi <anchor type="note" xlink:label="note-253-02a" xlink:href="note-253-02"/> circuli {223/71}. </s> <s xml:id="echoid-s10143" xml:space="preserve">minor quam vera. </s> <s xml:id="echoid-s10144" xml:space="preserve">Eius ergo ſemiſsis {223/142}. </s> <s xml:id="echoid-s10145" xml:space="preserve">ducta in {1/2}. </s> <s xml:id="echoid-s10146" xml:space="preserve">ſemiſſem diame-<lb/>tri@. </s> <s xml:id="echoid-s10147" xml:space="preserve">faciet {223/284}. </s> <s xml:id="echoid-s10148" xml:space="preserve">aream circuli maximi vera minorem; </s> <s xml:id="echoid-s10149" xml:space="preserve">ideo que ſi ea ducatur in <lb/>{2/3}. </s> <s xml:id="echoid-s10150" xml:space="preserve">diametri 1. </s> <s xml:id="echoid-s10151" xml:space="preserve">pro creabitur, vt infra in regula 2. </s> <s xml:id="echoid-s10152" xml:space="preserve">dicetur, ſoliditas ſphæræ {449/852}. </s> <s xml:id="echoid-s10153" xml:space="preserve">hoc <lb/>eſt, {223/476}. </s> <s xml:id="echoid-s10154" xml:space="preserve">minor, quam vera. </s> <s xml:id="echoid-s10155" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur cubus 1 diametri 1. </s> <s xml:id="echoid-s10156" xml:space="preserve">ad ſphæram habebit pro- portionem minorem, quam ad {223/426}. </s> <s xml:id="echoid-s10157" xml:space="preserve">Cum ergo ſit 1. </s> <s xml:id="echoid-s10158" xml:space="preserve">ad {223/426}. </s> <s xml:id="echoid-s10159" xml:space="preserve">vt 426. </s> <s xml:id="echoid-s10160" xml:space="preserve">ad 223. <lb/></s> <s xml:id="echoid-s10161" xml:space="preserve"> <anchor type="note" xlink:label="note-253-03a" xlink:href="note-253-03"/> (Quia enim ex propoſ. </s> <s xml:id="echoid-s10162" xml:space="preserve">2. </s> <s xml:id="echoid-s10163" xml:space="preserve">Minutiarum ad ſinem lib. </s> <s xml:id="echoid-s10164" xml:space="preserve">9. </s> <s xml:id="echoid-s10165" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s10166" xml:space="preserve">eadem proportio <lb/>eſt Numeratoris 223. </s> <s xml:id="echoid-s10167" xml:space="preserve">ad denominatorem 426. </s> <s xml:id="echoid-s10168" xml:space="preserve">quæ Minutiæ {223/426}. </s> <s xml:id="echoid-s10169" xml:space="preserve">ad ſuum <lb/>integrum 1. </s> <s xml:id="echoid-s10170" xml:space="preserve">erit conuertendo, vt 426. </s> <s xml:id="echoid-s10171" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s10172" xml:space="preserve">ita 1. </s> <s xml:id="echoid-s10173" xml:space="preserve">ad {223/4@6}.) </s> <s xml:id="echoid-s10174" xml:space="preserve">minor erit pro-<lb/>portio cubi diametri 1. </s> <s xml:id="echoid-s10175" xml:space="preserve">ad ſuam ſphæram, quam 426. </s> <s xml:id="echoid-s10176" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s10177" xml:space="preserve">Quoniam ve-<lb/>rò, vt ad initium huius propoſitionis oſtendimus, ita eſt cubus diametri cuiusli-<lb/>bet ſphæræ alterius ad ipſam ſphæram, vt cubus diametr 1. </s> <s xml:id="echoid-s10178" xml:space="preserve">ad ſuam ſphæram, li-<lb/>quet etiam id, quod ſecundo loco propoſitum eſt.</s> <s xml:id="echoid-s10179" xml:space="preserve"/> </p> <div xml:id="echoid-div623" type="float" level="2" n="3"> <note symbol="b" position="right" xlink:label="note-253-02" xlink:href="note-253-02a" xml:space="preserve">corol. 2. d@ <lb/>Dimenſ. cir-<lb/>culi.</note> <note symbol="c" position="right" xlink:label="note-253-03" xlink:href="note-253-03a" xml:space="preserve">8. quinti.</note> </div> <p> <s xml:id="echoid-s10180" xml:space="preserve">2. </s> <s xml:id="echoid-s10181" xml:space="preserve"><emph style="sc">His</emph> præmiſsis, ſequuntur regulę ad inueſtigandam tam ſuperficiem <lb/>conuexam cuiuslibet ſphæræ, quam eiuſdem ſoliditatem.</s> <s xml:id="echoid-s10182" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div625" type="section" level="1" n="220"> <head xml:id="echoid-head237" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s10183" xml:space="preserve">SVPERFICIEM conuexam propoſitæ ſphæræ adinuenire.</s> <s xml:id="echoid-s10184" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10185" xml:space="preserve"><emph style="sc">Area</emph> maximi circuli datæ ſphærę quadruplicetur. </s> <s xml:id="echoid-s10186" xml:space="preserve">Productus enim nu-<lb/> <anchor type="note" xlink:label="note-253-04a" xlink:href="note-253-04"/> merus conuexam ſphærę ſuperficiem exhibebit: </s> <s xml:id="echoid-s10187" xml:space="preserve">propterea quod per propoſ. <lb/></s> <s xml:id="echoid-s10188" xml:space="preserve">31. </s> <s xml:id="echoid-s10189" xml:space="preserve">lib. </s> <s xml:id="echoid-s10190" xml:space="preserve">1. </s> <s xml:id="echoid-s10191" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s10192" xml:space="preserve">Cylindro, ſuperficies ſphærę quadrupla eſt <lb/>circuli maximi.</s> <s xml:id="echoid-s10193" xml:space="preserve"/> </p> <div xml:id="echoid-div625" type="float" level="2" n="1"> <note position="right" xlink:label="note-253-04" xlink:href="note-253-04a" xml:space="preserve">Superfici@s <lb/>conuexa ſpa-<lb/>ræ.</note> </div> <p> <s xml:id="echoid-s10194" xml:space="preserve"><emph style="sc">Eadem</emph> ſuperficies procreabitur, ſi diameter ſphęrę in circumferentiam <lb/>circuli maximi ducatur: </s> <s xml:id="echoid-s10195" xml:space="preserve">propterea quod per propoſ. </s> <s xml:id="echoid-s10196" xml:space="preserve">2. </s> <s xml:id="echoid-s10197" xml:space="preserve">Num. </s> <s xml:id="echoid-s10198" xml:space="preserve">2. </s> <s xml:id="echoid-s10199" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10200" xml:space="preserve">re-<lb/>ctangulum ſub diametro, & </s> <s xml:id="echoid-s10201" xml:space="preserve">circumferentia maximi circuli comprehenſum ſu-<lb/>perficiei conuexę ſphęrę eſt ęquale.</s> <s xml:id="echoid-s10202" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div627" type="section" level="1" n="221"> <head xml:id="echoid-head238" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s10203" xml:space="preserve">SOLIDITATEM propoſitæ ſphæræ exquirere.</s> <s xml:id="echoid-s10204" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10205" xml:space="preserve">1. </s> <s xml:id="echoid-s10206" xml:space="preserve">SPH AER AE ſolidit{as} producitur ex ei{us} ſemidiametro in tertiam partem <lb/> <anchor type="note" xlink:label="note-253-05a" xlink:href="note-253-05"/> ſuperficiei conuex@. </s> <s xml:id="echoid-s10207" xml:space="preserve">Velex {@/4} toti{us} diametri in {2/3}. </s> <s xml:id="echoid-s10208" xml:space="preserve">couuexæ ſuperficiei.</s> <s xml:id="echoid-s10209" xml:space="preserve"/> </p> <div xml:id="echoid-div627" type="float" level="2" n="1"> <note position="right" xlink:label="note-253-05" xlink:href="note-253-05a" xml:space="preserve">Solidit{as}<unsure/> <lb/>ſphæræ.</note> </div> <p style="it"> <s xml:id="echoid-s10210" xml:space="preserve">2. </s> <s xml:id="echoid-s10211" xml:space="preserve">ITEM ex duab{us} tertiis par@ib{us} diametri in aream circuli maximi.</s> <s xml:id="echoid-s10212" xml:space="preserve"/> </p> <pb o="224" file="254" n="254" rhead="GEOMETR. PRACT."/> <p style="it"> <s xml:id="echoid-s10213" xml:space="preserve">3. </s> <s xml:id="echoid-s10214" xml:space="preserve">VEL ex duab{us} tertiis partib{us} ar@æ circuli maximi in totam diametrum.</s> <s xml:id="echoid-s10215" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10216" xml:space="preserve">4. </s> <s xml:id="echoid-s10217" xml:space="preserve">VEL ex ſemidiametro in quatuor terti{as} part{es} areæ circuli maximi.</s> <s xml:id="echoid-s10218" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10219" xml:space="preserve">5. </s> <s xml:id="echoid-s10220" xml:space="preserve">VEL ex ſemiſſe areæ circuli maximi in quatuor terti{as} partes diametri.</s> <s xml:id="echoid-s10221" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10222" xml:space="preserve">6. </s> <s xml:id="echoid-s10223" xml:space="preserve">VEL ex dupla diametro in tertiam partem areæ circuli maximi.</s> <s xml:id="echoid-s10224" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10225" xml:space="preserve">7. </s> <s xml:id="echoid-s10226" xml:space="preserve">VEL ex diametro in 6. </s> <s xml:id="echoid-s10227" xml:space="preserve">partem ſuperficiei ſphæræ.</s> <s xml:id="echoid-s10228" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s10229" xml:space="preserve">8. </s> <s xml:id="echoid-s10230" xml:space="preserve">VEL denique ex tertia parte diametriin ſemiſſem ſuperficiei conuexæ ſphæræ.</s> <s xml:id="echoid-s10231" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10232" xml:space="preserve"><emph style="sc">Primvm</emph> demonſtatum à nobis eſt in commentariis in ſphęram, quam de-<lb/> <anchor type="note" xlink:label="note-254-01a" xlink:href="note-254-01"/> monſtrationem repetemus lib. </s> <s xml:id="echoid-s10233" xml:space="preserve">7. </s> <s xml:id="echoid-s10234" xml:space="preserve">de Iſoperimetris. </s> <s xml:id="echoid-s10235" xml:space="preserve">Idem tamen aliter hac ratio-<lb/>ne demonſtrabimus. </s> <s xml:id="echoid-s10236" xml:space="preserve">Concipiatur conus, cuius baſis maximus circulus ſphę-<lb/>rę, & </s> <s xml:id="echoid-s10237" xml:space="preserve">altitudo ſemidiameter eiuſdem. </s> <s xml:id="echoid-s10238" xml:space="preserve">Item alius conus, cuius baſis quadrupla <lb/>ſit maximi circuli, & </s> <s xml:id="echoid-s10239" xml:space="preserve">altitudo ſemidiameter eadem. </s> <s xml:id="echoid-s10240" xml:space="preserve">Et quia prioris coni tam <lb/>ſphęra, per propoſ. </s> <s xml:id="echoid-s10241" xml:space="preserve">32. </s> <s xml:id="echoid-s10242" xml:space="preserve">lib. </s> <s xml:id="echoid-s10243" xml:space="preserve">1. </s> <s xml:id="echoid-s10244" xml:space="preserve">Archimedis de ſphęra, & </s> <s xml:id="echoid-s10245" xml:space="preserve">cylindro, quadrupla <lb/>eſt, <anchor type="note" xlink:href="" symbol="a"/> quam poſterior conus: </s> <s xml:id="echoid-s10246" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erunt poſterior conus, & </s> <s xml:id="echoid-s10247" xml:space="preserve">ſphęra inter ſe æ- <anchor type="note" xlink:label="note-254-02a" xlink:href="note-254-02"/> quales.</s> <s xml:id="echoid-s10248" xml:space="preserve"/> </p> <div xml:id="echoid-div628" type="float" level="2" n="2"> <note position="left" xlink:label="note-254-01" xlink:href="note-254-01a" xml:space="preserve">Demonſtra-<lb/>tio primæ par-<lb/>tis.</note> <note symbol="a" position="left" xlink:label="note-254-02" xlink:href="note-254-02a" xml:space="preserve">11. duodec.</note> </div> <note symbol="b" position="left" xml:space="preserve">9. quinti.</note> <p> <s xml:id="echoid-s10249" xml:space="preserve"><emph style="sc">Rvrsvs</emph> quia circulus, cuius ſemidiameter ęqualis eſt toti diametro ſpę-<lb/>rę, quadruplus eſt circuli maximi. </s> <s xml:id="echoid-s10250" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> (cum enim ſit circulus ad circulum, vt qua- <anchor type="note" xlink:label="note-254-04a" xlink:href="note-254-04"/> dratum diametri ad quadratum diametri: </s> <s xml:id="echoid-s10251" xml:space="preserve">quadratum autem prioris diametri <lb/>quadruplum ſit quadrati diametri poſterioris, ex ſcholio propoſ. </s> <s xml:id="echoid-s10252" xml:space="preserve">4. </s> <s xml:id="echoid-s10253" xml:space="preserve">lib. </s> <s xml:id="echoid-s10254" xml:space="preserve">2. </s> <s xml:id="echoid-s10255" xml:space="preserve">Eucl. <lb/></s> <s xml:id="echoid-s10256" xml:space="preserve">quòd illa diameter ſit huius dupla; </s> <s xml:id="echoid-s10257" xml:space="preserve">quando quidem ſemiſsis prioris diametri <lb/>ſumpta eſt poſteriori diametro ęqualis; </s> <s xml:id="echoid-s10258" xml:space="preserve">erit quoque circulus circuli quadru-<lb/>plus.) </s> <s xml:id="echoid-s10259" xml:space="preserve">eritidem circulus, cuius ſemidiameter diametro ſphęrę ęqualis eſt, ęqua-<lb/>lis baſi poſterioris coni, cum huius baſis quadrupla etiam poſita ſit maximi circu-<lb/>li. </s> <s xml:id="echoid-s10260" xml:space="preserve">Quia verò etiam ſuperficies ſphæræ quadrupla eſt circuli maximi, ex propoſ. </s> <s xml:id="echoid-s10261" xml:space="preserve"><lb/>31. </s> <s xml:id="echoid-s10262" xml:space="preserve">lib. </s> <s xml:id="echoid-s10263" xml:space="preserve">1. </s> <s xml:id="echoid-s10264" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s10265" xml:space="preserve">cylindro: </s> <s xml:id="echoid-s10266" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Erunt ſuperficies ſphæræ, baſis po- <anchor type="note" xlink:label="note-254-05a" xlink:href="note-254-05"/> ſterioris coni, & </s> <s xml:id="echoid-s10267" xml:space="preserve">circulus ſemidiametrum habens æqualem diametro ſphæræ, inter <lb/>ſe æquales.</s> <s xml:id="echoid-s10268" xml:space="preserve"/> </p> <div xml:id="echoid-div629" type="float" level="2" n="3"> <note symbol="c" position="left" xlink:label="note-254-04" xlink:href="note-254-04a" xml:space="preserve">2. duodec.</note> <note symbol="d" position="left" xlink:label="note-254-05" xlink:href="note-254-05a" xml:space="preserve">9. quinti.</note> </div> <p> <s xml:id="echoid-s10269" xml:space="preserve"><emph style="sc">Postremo</emph> concipiatur cylindrus, cuius baſis ſit prædictus circulus ſemidia-<lb/>metrum diametro ſphæræ habens æqualem, altitudo verò ſemidiameter ſphæræ. <lb/></s> <s xml:id="echoid-s10270" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="e"/> Erit hic cylindrus triplus poſterioris coni prædicti: </s> <s xml:id="echoid-s10271" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s10272" xml:space="preserve">ſphæræ, quæ <anchor type="note" xlink:label="note-254-06a" xlink:href="note-254-06"/> ei cono eſt oſtenſa æqualis. </s> <s xml:id="echoid-s10273" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Idem autem cylindrus triplus quoque eſt cylindri, <anchor type="note" xlink:label="note-254-07a" xlink:href="note-254-07"/> qui eandem habeat altitudinem, & </s> <s xml:id="echoid-s10274" xml:space="preserve">baſem terriæ parti illius cylindri, hoc <lb/>eſt, tertiæ parti ſuperficiei ſphæræ, æqualem. </s> <s xml:id="echoid-s10275" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Ergo poſterior cylindrus, (ba- <anchor type="note" xlink:label="note-254-08a" xlink:href="note-254-08"/> ſem habenstertiæ parti ſuperficiei ſphæræ æqualem, altitudinem verò ſemidia-<lb/>metro eiuſdem ſphæræ æqualem,) & </s> <s xml:id="echoid-s10276" xml:space="preserve">ſphæra æquales ſunt. </s> <s xml:id="echoid-s10277" xml:space="preserve">Cum ergo cylindrus <lb/>hic poſterior contineatur ſub ſemidiametro ſphæræ, & </s> <s xml:id="echoid-s10278" xml:space="preserve">tertia parte ſuperficiei <lb/>ſphæricæ: </s> <s xml:id="echoid-s10279" xml:space="preserve">liquidò conſtat, ſphæræ ſoliditatem gigni ex ſemidiametro in partem <lb/>tertiam ſuperficiei ſphæræ. </s> <s xml:id="echoid-s10280" xml:space="preserve">Velex @. </s> <s xml:id="echoid-s10281" xml:space="preserve">totius diametri in {2/@}. </s> <s xml:id="echoid-s10282" xml:space="preserve">ſuperficiei ſphæræ: </s> <s xml:id="echoid-s10283" xml:space="preserve">cum <lb/>hic numerus illi ſit æqualis. </s> <s xml:id="echoid-s10284" xml:space="preserve">quod eſt primum.</s> <s xml:id="echoid-s10285" xml:space="preserve"/> </p> <div xml:id="echoid-div630" type="float" level="2" n="4"> <note symbol="e" position="left" xlink:label="note-254-06" xlink:href="note-254-06a" xml:space="preserve">10. duodec.</note> <note symbol="f" position="left" xlink:label="note-254-07" xlink:href="note-254-07a" xml:space="preserve">11. duodec.</note> <note symbol="g" position="left" xlink:label="note-254-08" xlink:href="note-254-08a" xml:space="preserve">9. quinti.</note> </div> <p> <s xml:id="echoid-s10286" xml:space="preserve"><emph style="sc">Concipiatvr</emph> rurſum cylindrus, cuius baſis maximus circulus ſphæræ, & </s> <s xml:id="echoid-s10287" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-254-09a" xlink:href="note-254-09"/> altitudo diameter ſphæræ. </s> <s xml:id="echoid-s10288" xml:space="preserve">Erit hic cylindrus ſeſquialter ſphæræ, ex coroll. </s> <s xml:id="echoid-s10289" xml:space="preserve">pro-<lb/>poſ. </s> <s xml:id="echoid-s10290" xml:space="preserve">32. </s> <s xml:id="echoid-s10291" xml:space="preserve">libr. </s> <s xml:id="echoid-s10292" xml:space="preserve">1. </s> <s xml:id="echoid-s10293" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s10294" xml:space="preserve">cylindro. </s> <s xml:id="echoid-s10295" xml:space="preserve">Quod ſi ex parte ſuperiori per <lb/>tettiam par<unsure/>tem diametri ſphæræ, vel axis cylindri, ducatur baſibus cylindri pla-<lb/>num parallelum: </s> <s xml:id="echoid-s10296" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> erit totus cylindrus ad cylindrum abſciſſum, cums axis duæ <anchor type="note" xlink:label="note-254-10a" xlink:href="note-254-10"/> tertiæ partes ſunt totius axis, ſeſquialter; </s> <s xml:id="echoid-s10297" xml:space="preserve">Ac proinde poſterior hic cylindrus <lb/> <anchor type="note" xlink:label="note-254-11a" xlink:href="note-254-11"/> abſciſſus, qui quidem continetur ſub maximo circulo, nempe ſub ſua baſi, & </s> <s xml:id="echoid-s10298" xml:space="preserve">dua-<lb/>bustertiis partibus diametriſphæræ, ſphæræ æqualis erit. </s> <s xml:id="echoid-s10299" xml:space="preserve">Pater igitur etiam ſe-<lb/>cundum.</s> <s xml:id="echoid-s10300" xml:space="preserve"/> </p> <div xml:id="echoid-div631" type="float" level="2" n="5"> <note position="left" xlink:label="note-254-09" xlink:href="note-254-09a" xml:space="preserve">Demonſtra-<lb/>tio ſecundæ <lb/>partis.</note> <note symbol="h" position="left" xlink:label="note-254-10" xlink:href="note-254-10a" xml:space="preserve">13. duodec.</note> <note symbol="i" position="left" xlink:label="note-254-11" xlink:href="note-254-11a" xml:space="preserve">9. quinti.</note> </div> <pb o="225" file="255" n="255" rhead="LIBER QVINTVS."/> </div> <div xml:id="echoid-div633" type="section" level="1" n="222"> <head xml:id="echoid-head239" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s10301" xml:space="preserve"><emph style="sc">Sit</emph> parallelepipedum A B C, comprehenſum ſub A C, duabus tertiis parti-<lb/>bus diametri ſphærę, & </s> <s xml:id="echoid-s10302" xml:space="preserve">ſub baſi AB, quę circulo maximo eiuſdem ſphærę ſit æ-<lb/>qualis. </s> <s xml:id="echoid-s10303" xml:space="preserve">Dico parallelepipedum A B C, ſphærę æquale eſſe. </s> <s xml:id="echoid-s10304" xml:space="preserve">Sit enim aliud pa-<lb/>rallelepip edum D E F, contentum ſub DF, ſemidiametro ſphærę, & </s> <s xml:id="echoid-s10305" xml:space="preserve">ſub baſe <lb/>DE, quę tertię parti ſuperficiei ſphærę ſit æqualis. </s> <s xml:id="echoid-s10306" xml:space="preserve">quod vt in prima parte huius <lb/>2. </s> <s xml:id="echoid-s10307" xml:space="preserve">regulę demonſtrauimus, æquale erit ſphærę propoſitæ. </s> <s xml:id="echoid-s10308" xml:space="preserve">Quia ergo baſis A B, <lb/>circulo maximo ſpærę æqualis, eſt {1/4}. </s> <s xml:id="echoid-s10309" xml:space="preserve">ſuperficiei ſphærę; </s> <s xml:id="echoid-s10310" xml:space="preserve">erit ex coroll. </s> <s xml:id="echoid-s10311" xml:space="preserve">propoſ@ <lb/>1. </s> <s xml:id="echoid-s10312" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10313" xml:space="preserve">vt AB, hoceſt, vt {1/4}. </s> <s xml:id="echoid-s10314" xml:space="preserve">ſuperficiei ſphærę ad DE, id eſt, ad {1/3}. </s> <s xml:id="echoid-s10315" xml:space="preserve">eiuſdem <lb/>ſuperficiei, ita DF, hoc eſt, ita {1/2}. </s> <s xml:id="echoid-s10316" xml:space="preserve">diametri ſphærę, ad A C, id <lb/>eſt, ad {2/3}. </s> <s xml:id="echoid-s10317" xml:space="preserve">eiuſdem diametri. </s> <s xml:id="echoid-s10318" xml:space="preserve">Ac proinde cum baſes A B, D E, <lb/> <anchor type="figure" xlink:label="fig-255-01a" xlink:href="fig-255-01"/> cumaltitudinibus D F, A C, reciprocentur, <anchor type="note" xlink:href="" symbol="a"/> parallelepipeda <anchor type="note" xlink:label="note-255-01a" xlink:href="note-255-01"/> ABC, DEF, æqualia inter ſe erunt. </s> <s xml:id="echoid-s10319" xml:space="preserve">Cum ergo DEF, ſphærę <lb/>æquale ſit, vt dictum eſt, erit quoque ABC, eidem ſphærę æ-<lb/>quale. </s> <s xml:id="echoid-s10320" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s10321" xml:space="preserve"/> </p> <div xml:id="echoid-div633" type="float" level="2" n="1"> <figure xlink:label="fig-255-01" xlink:href="fig-255-01a"> <image file="255-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/255-01"/> </figure> <note symbol="a" position="right" xlink:label="note-255-01" xlink:href="note-255-01a" xml:space="preserve">34. vndec.</note> </div> </div> <div xml:id="echoid-div635" type="section" level="1" n="223"> <head xml:id="echoid-head240" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s10322" xml:space="preserve"><emph style="sc">Qvoniam</emph> ex coroll. </s> <s xml:id="echoid-s10323" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s10324" xml:space="preserve">1. </s> <s xml:id="echoid-s10325" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10326" xml:space="preserve">eſt, vt {1/4}. </s> <s xml:id="echoid-s10327" xml:space="preserve">ſu-<lb/>perficiei ſphærę ad {1/@}. </s> <s xml:id="echoid-s10328" xml:space="preserve">eiuſdem ſuperficiei, ita {1/2}. </s> <s xml:id="echoid-s10329" xml:space="preserve">diametriad {2/3}. <lb/></s> <s xml:id="echoid-s10330" xml:space="preserve">eiuſdem diametri: </s> <s xml:id="echoid-s10331" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> idem numerus efficietur ex primo nu- <anchor type="note" xlink:label="note-255-02a" xlink:href="note-255-02"/> mero, nimirum ex {1/4}. </s> <s xml:id="echoid-s10332" xml:space="preserve">ſuperficiei, id eſt, ex circulo maximo ſphærę, in quartum, <lb/>nimirum in {2/3}. </s> <s xml:id="echoid-s10333" xml:space="preserve">diametri, quiex ſecundo, id eſt, ex {1/3}. </s> <s xml:id="echoid-s10334" xml:space="preserve">ſuperficiei, in tertium, hoc <lb/>eſt, in {1/2}. </s> <s xml:id="echoid-s10335" xml:space="preserve">diametri. </s> <s xml:id="echoid-s10336" xml:space="preserve">Sed ex {1/3}. </s> <s xml:id="echoid-s10337" xml:space="preserve">ſuperficiei in {1/2}. </s> <s xml:id="echoid-s10338" xml:space="preserve">diametri ſoliditas ſphærę procrea-<lb/>tur, vt in primaparte huius 2. </s> <s xml:id="echoid-s10339" xml:space="preserve">regulæoſtenſum eſt. </s> <s xml:id="echoid-s10340" xml:space="preserve">Igitur eadem ſoliditas ex cir-<lb/>culo maximo in {2/3}. </s> <s xml:id="echoid-s10341" xml:space="preserve">diametri gignetur. </s> <s xml:id="echoid-s10342" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s10343" xml:space="preserve"/> </p> <div xml:id="echoid-div635" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-255-02" xlink:href="note-255-02a" xml:space="preserve">19. ſept.</note> </div> <p> <s xml:id="echoid-s10344" xml:space="preserve"><emph style="sc">Rvrsvs</emph> quia cylindrus, cuius baſis circulus maximus ſphærę, & </s> <s xml:id="echoid-s10345" xml:space="preserve">altitudo <lb/> <anchor type="note" xlink:label="note-255-03a" xlink:href="note-255-03"/> diameter eiuſdem, ſeſquialter eſtipſius ſphærę, ex coroll. </s> <s xml:id="echoid-s10346" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s10347" xml:space="preserve">32. </s> <s xml:id="echoid-s10348" xml:space="preserve">lib. </s> <s xml:id="echoid-s10349" xml:space="preserve">1. </s> <s xml:id="echoid-s10350" xml:space="preserve">Ar-<lb/>chimedis de ſphæra, & </s> <s xml:id="echoid-s10351" xml:space="preserve">cylindro: </s> <s xml:id="echoid-s10352" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Idemque cylindrus ſeſquialter etiam cylin- <anchor type="note" xlink:label="note-255-04a" xlink:href="note-255-04"/> dri, cuius baſis æqualis ſit duabus tertiis partibus circulimaximi, & </s> <s xml:id="echoid-s10353" xml:space="preserve">altitudo ea-<lb/> <anchor type="note" xlink:label="note-255-05a" xlink:href="note-255-05"/> dem diameter; </s> <s xml:id="echoid-s10354" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erunt poſterior hic cylindrus, & </s> <s xml:id="echoid-s10355" xml:space="preserve">ſphæra æquales: </s> <s xml:id="echoid-s10356" xml:space="preserve">hoc eſt, ſphæra producetur ex {2/3}. </s> <s xml:id="echoid-s10357" xml:space="preserve">areæ maximi circuli in diametrum ſphærę. </s> <s xml:id="echoid-s10358" xml:space="preserve">quod eſt <lb/>tertium.</s> <s xml:id="echoid-s10359" xml:space="preserve"/> </p> <div xml:id="echoid-div636" type="float" level="2" n="2"> <note position="right" xlink:label="note-255-03" xlink:href="note-255-03a" xml:space="preserve">Demonſtratio <lb/>tertiæ partis.</note> <note symbol="c" position="right" xlink:label="note-255-04" xlink:href="note-255-04a" xml:space="preserve">11. duodec.</note> <note symbol="d" position="right" xlink:label="note-255-05" xlink:href="note-255-05a" xml:space="preserve">9. quinti.</note> </div> <p> <s xml:id="echoid-s10360" xml:space="preserve"><emph style="sc">Concipiantvr</emph> quoque duo parallelepipeda, quorum vnius baſis ſit {2/3}. <lb/></s> <s xml:id="echoid-s10361" xml:space="preserve"> <anchor type="note" xlink:label="note-255-06a" xlink:href="note-255-06"/> areæ maximi circuli in ſphæra æqualis, & </s> <s xml:id="echoid-s10362" xml:space="preserve">altitudo toti diametro: </s> <s xml:id="echoid-s10363" xml:space="preserve">alterius verò <lb/>baſis æqualis ſit {4/3}. </s> <s xml:id="echoid-s10364" xml:space="preserve">areę circuli maximi, & </s> <s xml:id="echoid-s10365" xml:space="preserve">altitudo ſemidiametro. </s> <s xml:id="echoid-s10366" xml:space="preserve">Et quia horum <lb/>parallelepipedorum baſes cum altitudinibus reciprocantur: </s> <s xml:id="echoid-s10367" xml:space="preserve">quod tam prioris <lb/>baſis ſubdupla ſit baſis poſterioris, quam altitudo poſterioris altitudinis prio-<lb/>ris: </s> <s xml:id="echoid-s10368" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erunt ipſa parallelepipeda æqualia. </s> <s xml:id="echoid-s10369" xml:space="preserve">Sed prius, per tertiam partem huius <anchor type="note" xlink:label="note-255-07a" xlink:href="note-255-07"/> 2. </s> <s xml:id="echoid-s10370" xml:space="preserve">regulę, ęquale eſt ſoliditati ſphęrę. </s> <s xml:id="echoid-s10371" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s10372" xml:space="preserve">poſterius. </s> <s xml:id="echoid-s10373" xml:space="preserve">Ideoque ſphęra <lb/>producetur ex ſemidiametro in {4/@}. </s> <s xml:id="echoid-s10374" xml:space="preserve">areę circuli maximi. </s> <s xml:id="echoid-s10375" xml:space="preserve">quod quarto loco pro-<lb/>ponitur.</s> <s xml:id="echoid-s10376" xml:space="preserve"/> </p> <div xml:id="echoid-div637" type="float" level="2" n="3"> <note position="right" xlink:label="note-255-06" xlink:href="note-255-06a" xml:space="preserve">Demonſtratio <lb/>quartæ partis.</note> <note symbol="e" position="right" xlink:label="note-255-07" xlink:href="note-255-07a" xml:space="preserve">34. vndec.</note> </div> <p> <s xml:id="echoid-s10377" xml:space="preserve"><emph style="sc">Præterea</emph> concipiantur duo parallelepipeda, quorum vnius baſis ęqua-<lb/> <anchor type="note" xlink:label="note-255-08a" xlink:href="note-255-08"/> lis ſit areę circuli in ſphęra maximi, & </s> <s xml:id="echoid-s10378" xml:space="preserve">altitudo ęqualis {2/3}. </s> <s xml:id="echoid-s10379" xml:space="preserve">diametri: </s> <s xml:id="echoid-s10380" xml:space="preserve">alterius ve-<lb/>rò baſis ęqualis ſit @ areę circuli maximi, & </s> <s xml:id="echoid-s10381" xml:space="preserve">altitudo {4/3}. </s> <s xml:id="echoid-s10382" xml:space="preserve">diametri. </s> <s xml:id="echoid-s10383" xml:space="preserve">Et quia horum <lb/>parallelepipedorum baſes reciprocantur cum altitudinibus; </s> <s xml:id="echoid-s10384" xml:space="preserve">quod tam baſis <lb/>in priori dupla ſit baſis in poſteriori, quam altitudo in poſteriori altitudinis in <pb o="226" file="256" n="256" rhead="GEOMETR. PRACT."/> priori: </s> <s xml:id="echoid-s10385" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> æqualia erunt ipſa parallelepipeda: </s> <s xml:id="echoid-s10386" xml:space="preserve">Sed prius eſt, per 2. </s> <s xml:id="echoid-s10387" xml:space="preserve">partem huius <anchor type="note" xlink:label="note-256-01a" xlink:href="note-256-01"/> 2. </s> <s xml:id="echoid-s10388" xml:space="preserve">regulæ, æquale ſphærę. </s> <s xml:id="echoid-s10389" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s10390" xml:space="preserve">poſterius: </s> <s xml:id="echoid-s10391" xml:space="preserve">atque idcirco ſphæra gignetur ex <lb/>{1/2}. </s> <s xml:id="echoid-s10392" xml:space="preserve">areæ circuli maximi in {4/5}. </s> <s xml:id="echoid-s10393" xml:space="preserve">diametri. </s> <s xml:id="echoid-s10394" xml:space="preserve">quod eſt quintum.</s> <s xml:id="echoid-s10395" xml:space="preserve"/> </p> <div xml:id="echoid-div638" type="float" level="2" n="4"> <note position="right" xlink:label="note-255-08" xlink:href="note-255-08a" xml:space="preserve">Demonſtratio <lb/>quintæ partis.</note> <note symbol="a" position="left" xlink:label="note-256-01" xlink:href="note-256-01a" xml:space="preserve">34. vndec.</note> </div> <p> <s xml:id="echoid-s10396" xml:space="preserve"><emph style="sc">Item</emph> intelligantur duo parallelepipeda, quorum vnius baſis æqualis ſit {2/3}. <lb/></s> <s xml:id="echoid-s10397" xml:space="preserve"> <anchor type="note" xlink:label="note-256-02a" xlink:href="note-256-02"/> areæ circuli maximi in ſphæra, & </s> <s xml:id="echoid-s10398" xml:space="preserve">altitudo diametro: </s> <s xml:id="echoid-s10399" xml:space="preserve">alterius verò baſis æqualis <lb/>ſit {1/3}. </s> <s xml:id="echoid-s10400" xml:space="preserve">areæ maximi circuli, & </s> <s xml:id="echoid-s10401" xml:space="preserve">altitudo duplæ diametro. </s> <s xml:id="echoid-s10402" xml:space="preserve">Et quia horum paralle-<lb/>lepipedorum baſes cum altitudinibus reciprocantur: </s> <s xml:id="echoid-s10403" xml:space="preserve">quod tam baſis in priori <lb/>ſit dupla baſis in poſteriori, quam altitudo in poſteriori altitudinis in priori: <lb/></s> <s xml:id="echoid-s10404" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/> eruntipſa parallelepipeda æqualia: </s> <s xml:id="echoid-s10405" xml:space="preserve">Sed prius per 3. </s> <s xml:id="echoid-s10406" xml:space="preserve">partem huius 2. </s> <s xml:id="echoid-s10407" xml:space="preserve">regulę, æ- <anchor type="note" xlink:label="note-256-03a" xlink:href="note-256-03"/> quale eſt ipſi ſphærę. </s> <s xml:id="echoid-s10408" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s10409" xml:space="preserve">poſterius: </s> <s xml:id="echoid-s10410" xml:space="preserve">Ac proinde ſphæra ex dupla dia-<lb/>metro in {1/3}. </s> <s xml:id="echoid-s10411" xml:space="preserve">areæ circuli maximi procreabitur. </s> <s xml:id="echoid-s10412" xml:space="preserve">quod ſexto loco eſt propo-<lb/>ſitum.</s> <s xml:id="echoid-s10413" xml:space="preserve"/> </p> <div xml:id="echoid-div639" type="float" level="2" n="5"> <note position="left" xlink:label="note-256-02" xlink:href="note-256-02a" xml:space="preserve">Demonſtratio <lb/>ſextæpartis.</note> <note symbol="b" position="left" xlink:label="note-256-03" xlink:href="note-256-03a" xml:space="preserve">34. vndec.</note> </div> <p> <s xml:id="echoid-s10414" xml:space="preserve"><emph style="sc">Intelligantvr</emph> quoque duo parallelepipeda, quorum vnius baſis con-<lb/> <anchor type="note" xlink:label="note-256-04a" xlink:href="note-256-04"/> tineat {1/3}. </s> <s xml:id="echoid-s10415" xml:space="preserve">ſuperficiei ſphærę, & </s> <s xml:id="echoid-s10416" xml:space="preserve">altitudo {1/3}. </s> <s xml:id="echoid-s10417" xml:space="preserve">diametri: </s> <s xml:id="echoid-s10418" xml:space="preserve">alterius verò baſis compre-<lb/>hendat {1/6}. </s> <s xml:id="echoid-s10419" xml:space="preserve">ſuperficiei, & </s> <s xml:id="echoid-s10420" xml:space="preserve">altitudo æqualis ſit diametro. </s> <s xml:id="echoid-s10421" xml:space="preserve">Et quoniam baſes cum <lb/>altitudinibus ſunt reciprocę, quod ita ſit {1/3}. </s> <s xml:id="echoid-s10422" xml:space="preserve">ſuperficiei baſis videlicet prioris <lb/>parallelepipedi, ad {1/6}. </s> <s xml:id="echoid-s10423" xml:space="preserve">ſuperficiei, id eſt, ad baſem poſterioris, vt altitudo poſte-<lb/>rioris, nempe diameter, ad prioris altitudinem, nimirum ad {1/2}. </s> <s xml:id="echoid-s10424" xml:space="preserve">diametri, cum v-<lb/>traque proportio ſit dupla: </s> <s xml:id="echoid-s10425" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ipſa parallelepipeda æqualia erunt: </s> <s xml:id="echoid-s10426" xml:space="preserve">Sed prius ipſi <anchor type="note" xlink:label="note-256-05a" xlink:href="note-256-05"/> ſphærę, per 1. </s> <s xml:id="echoid-s10427" xml:space="preserve">partem huius 2. </s> <s xml:id="echoid-s10428" xml:space="preserve">regulæ æquale eſt. </s> <s xml:id="echoid-s10429" xml:space="preserve">Igitur, & </s> <s xml:id="echoid-s10430" xml:space="preserve">poſterius: </s> <s xml:id="echoid-s10431" xml:space="preserve">hoc eſt, <lb/>ſphærę ſoliditas producetur ex diametro in ſextam partem ſuperficiei, quod eſt <lb/>ſeptimum.</s> <s xml:id="echoid-s10432" xml:space="preserve"/> </p> <div xml:id="echoid-div640" type="float" level="2" n="6"> <note position="left" xlink:label="note-256-04" xlink:href="note-256-04a" xml:space="preserve">Demonſtratio <lb/>ſeptimæ partis.</note> <note symbol="c" position="left" xlink:label="note-256-05" xlink:href="note-256-05a" xml:space="preserve">34. vndec.</note> </div> <p> <s xml:id="echoid-s10433" xml:space="preserve"><emph style="sc">Deniqve</emph> concipiantur duo parallelepipeda, quorum vnius baſis ſit {1/3}. <lb/></s> <s xml:id="echoid-s10434" xml:space="preserve"> <anchor type="note" xlink:label="note-256-06a" xlink:href="note-256-06"/> ſuperficiei ſphæræ, & </s> <s xml:id="echoid-s10435" xml:space="preserve">altitudo ſemidiameter: </s> <s xml:id="echoid-s10436" xml:space="preserve">alterius autem baſis ſit {1/2}. </s> <s xml:id="echoid-s10437" xml:space="preserve">ſuperfi-<lb/>ciei, & </s> <s xml:id="echoid-s10438" xml:space="preserve">altitudo {1/3}. </s> <s xml:id="echoid-s10439" xml:space="preserve">diametri. </s> <s xml:id="echoid-s10440" xml:space="preserve">Quia verò baſes, & </s> <s xml:id="echoid-s10441" xml:space="preserve">altitudines recipro cantur, quod <lb/>ita ſit {1/3}. </s> <s xml:id="echoid-s10442" xml:space="preserve">ſuperficiei ad {1/2}. </s> <s xml:id="echoid-s10443" xml:space="preserve">ſuperficiei, nimirũ baſis prioris parallelepipedi ad ba-<lb/>ſem poſterioris, vt {1/3}. </s> <s xml:id="echoid-s10444" xml:space="preserve">diametriad {1/2}. </s> <s xml:id="echoid-s10445" xml:space="preserve">diametri, altitudo videlicet poſterioris pa-<lb/>rallelepipedi ad altitudinem prioris; </s> <s xml:id="echoid-s10446" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> æqualia eruntipſa parallelepipeda. </s> <s xml:id="echoid-s10447" xml:space="preserve">Cum <anchor type="note" xlink:label="note-256-07a" xlink:href="note-256-07"/> ergo, per 1. </s> <s xml:id="echoid-s10448" xml:space="preserve">partem huius 2. </s> <s xml:id="echoid-s10449" xml:space="preserve">regulæ, prius ſit ſphæræ æquale, eidem quo que po-<lb/>ſterius æquale erit: </s> <s xml:id="echoid-s10450" xml:space="preserve">Ac propterea ſphæræ ſoliditas pro ducetur ex tertia part@ <lb/>diametri in ſemiſſem conuexæ ſuperficiei. </s> <s xml:id="echoid-s10451" xml:space="preserve">quod eſt o ctauum.</s> <s xml:id="echoid-s10452" xml:space="preserve"/> </p> <div xml:id="echoid-div641" type="float" level="2" n="7"> <note position="left" xlink:label="note-256-06" xlink:href="note-256-06a" xml:space="preserve">Demonſtratio <lb/>octauæ partis.</note> <note symbol="d" position="left" xlink:label="note-256-07" xlink:href="note-256-07a" xml:space="preserve">34. vndec.</note> </div> <p> <s xml:id="echoid-s10453" xml:space="preserve">3. </s> <s xml:id="echoid-s10454" xml:space="preserve"><emph style="sc">Iam</emph> vero ex propoſ. </s> <s xml:id="echoid-s10455" xml:space="preserve">4. </s> <s xml:id="echoid-s10456" xml:space="preserve">& </s> <s xml:id="echoid-s10457" xml:space="preserve">5. </s> <s xml:id="echoid-s10458" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10459" xml:space="preserve">Num. </s> <s xml:id="echoid-s10460" xml:space="preserve">1. </s> <s xml:id="echoid-s10461" xml:space="preserve">colliguntur quatuor ſe-<lb/>quentes regulæ, per quas ſuperficies ſphæræ conuexa inuenitur tum maior <lb/>quam vera, tum minor, tam ex circumferentia, quam ex diametro circuli ma-<lb/>ximi.</s> <s xml:id="echoid-s10462" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div643" type="section" level="1" n="224"> <head xml:id="echoid-head241" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s10463" xml:space="preserve">EX circumferentia circuli in ſphæra maximi ſuperficiem conuexam <lb/>ſphęrę procreare vera maiorem.</s> <s xml:id="echoid-s10464" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10465" xml:space="preserve"><emph style="sc">Fiat</emph> vt 223. </s> <s xml:id="echoid-s10466" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s10467" xml:space="preserve">ita quadratum ex circumferentia maximi circuli data de-<lb/> <anchor type="note" xlink:label="note-256-08a" xlink:href="note-256-08"/> ſcriptum ad aliud, pro dibitque ſphæræ ſuperficies maior quam vera. </s> <s xml:id="echoid-s10468" xml:space="preserve">Cum e-<lb/>nim per propoſ. </s> <s xml:id="echoid-s10469" xml:space="preserve">4. </s> <s xml:id="echoid-s10470" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10471" xml:space="preserve">Num. </s> <s xml:id="echoid-s10472" xml:space="preserve">1. </s> <s xml:id="echoid-s10473" xml:space="preserve">maior ſit proportio qua drati circumfe-<lb/>rentiæ circuli maximi ad ſuperficiem ſphæræ, quam 223. </s> <s xml:id="echoid-s10474" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s10475" xml:space="preserve">ſit autem qua-<lb/>dratum datæ circumferentiæ ad numerum procreatum, vt 223. </s> <s xml:id="echoid-s10476" xml:space="preserve">ad 71. </s> <s xml:id="echoid-s10477" xml:space="preserve">habebit <lb/>quo que quadratum circumferentiæ datæ ad ſuperficiem ſphæræ ver@m, maio- <pb o="227" file="257" n="257" rhead="LIBER QVINTVS."/> @em proportionem, quàm idem quadratum ad numerum productum: </s> <s xml:id="echoid-s10478" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/>ideo- que productus numerus maior erit ſuperficie vera.</s> <s xml:id="echoid-s10479" xml:space="preserve"/> </p> <div xml:id="echoid-div643" type="float" level="2" n="1"> <note position="left" xlink:label="note-256-08" xlink:href="note-256-08a" xml:space="preserve">Superfici{es} <lb/>ſphæræmaior, <lb/>quam vera.</note> </div> <note symbol="a" position="right" xml:space="preserve">10. quinti.</note> </div> <div xml:id="echoid-div645" type="section" level="1" n="225"> <head xml:id="echoid-head242" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s10480" xml:space="preserve">EX circumferentia circuli in ſphæra maximi ſuperficiem ſphæræ con-<lb/>uexam vera minorem eruere.</s> <s xml:id="echoid-s10481" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10482" xml:space="preserve"><emph style="sc">Fiat</emph> vt 22. </s> <s xml:id="echoid-s10483" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s10484" xml:space="preserve">ita quadratum ex circumferentia maximi circuli data de-<lb/> <anchor type="note" xlink:label="note-257-02a" xlink:href="note-257-02"/> ſcriptum ad aliud. </s> <s xml:id="echoid-s10485" xml:space="preserve">Numerus enim genitus minor erit vera ſuperficie ſphæræ. <lb/></s> <s xml:id="echoid-s10486" xml:space="preserve">Cum enim per pro poſ. </s> <s xml:id="echoid-s10487" xml:space="preserve">4. </s> <s xml:id="echoid-s10488" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10489" xml:space="preserve">Num. </s> <s xml:id="echoid-s10490" xml:space="preserve">1. </s> <s xml:id="echoid-s10491" xml:space="preserve">minor ſit proportio quadrati circum-<lb/>ferentiæ circuli maximi ad ſuperficiem ſphæræ, quam 22. </s> <s xml:id="echoid-s10492" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s10493" xml:space="preserve">Sit autem qua-<lb/>dratum datæ circumferentiæ ad numerum genitum, vt 22. </s> <s xml:id="echoid-s10494" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s10495" xml:space="preserve">habebit quo que <lb/>quadratum datæ circumferentiæ ad ſuperficiem veram ſphæræ, minorem pro-<lb/>portionem, quam idem quadratum ad numerum productum. </s> <s xml:id="echoid-s10496" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quam obrem <anchor type="note" xlink:label="note-257-03a" xlink:href="note-257-03"/> numerus productus vera ſuperficie ſphæræ min or erit.</s> <s xml:id="echoid-s10497" xml:space="preserve"/> </p> <div xml:id="echoid-div645" type="float" level="2" n="1"> <note position="right" xlink:label="note-257-02" xlink:href="note-257-02a" xml:space="preserve">Superficies <lb/>ſphæræ minor. <lb/>quam vera.</note> <note symbol="b" position="right" xlink:label="note-257-03" xlink:href="note-257-03a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div647" type="section" level="1" n="226"> <head xml:id="echoid-head243" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s10498" xml:space="preserve">EX diametro circuli in ſphęra maximi ſuperficiem ſphærę vera maio-<lb/>rem elicere.</s> <s xml:id="echoid-s10499" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10500" xml:space="preserve"><emph style="sc">Fiat</emph> vt 7. </s> <s xml:id="echoid-s10501" xml:space="preserve">ad 22. </s> <s xml:id="echoid-s10502" xml:space="preserve">ita quadratum diametri datæ circuli maximi ad aliud, gi-<lb/> <anchor type="note" xlink:label="note-257-04a" xlink:href="note-257-04"/> gnetur que ſuperficies ſphæræ maior, quam vera. </s> <s xml:id="echoid-s10503" xml:space="preserve">Nam cum, per propoſ. </s> <s xml:id="echoid-s10504" xml:space="preserve">5. </s> <s xml:id="echoid-s10505" xml:space="preserve">huius <lb/>cap. </s> <s xml:id="echoid-s10506" xml:space="preserve">Num. </s> <s xml:id="echoid-s10507" xml:space="preserve">1. </s> <s xml:id="echoid-s10508" xml:space="preserve">maior ſit proportio quadrati diametri circuli maximi ad ſuperfi-<lb/>ciem ſphæræ veram, quam 7. </s> <s xml:id="echoid-s10509" xml:space="preserve">ad 22. </s> <s xml:id="echoid-s10510" xml:space="preserve">Sit autem quadratum diametri datæ ad pro-<lb/>ductum numerum, vt 7. </s> <s xml:id="echoid-s10511" xml:space="preserve">ad 22. </s> <s xml:id="echoid-s10512" xml:space="preserve">habebit quo que quadratum datæ diametri ad ſu-<lb/>perficiem veram ſphærę, maiorem proportionem, quamidem quadratum ad <lb/>productum numerum. </s> <s xml:id="echoid-s10513" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quo circa procreatus numerus maior er@t, quam ve- <anchor type="note" xlink:label="note-257-05a" xlink:href="note-257-05"/> ra ſuperficies ſphærę.</s> <s xml:id="echoid-s10514" xml:space="preserve"/> </p> <div xml:id="echoid-div647" type="float" level="2" n="1"> <note position="right" xlink:label="note-257-04" xlink:href="note-257-04a" xml:space="preserve">Superfici{es} <lb/>ſphæræ minor, <lb/>quam vera.</note> <note symbol="c" position="right" xlink:label="note-257-05" xlink:href="note-257-05a" xml:space="preserve">10. quint.</note> </div> </div> <div xml:id="echoid-div649" type="section" level="1" n="227"> <head xml:id="echoid-head244" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s10515" xml:space="preserve">EX diametro circuli in ſphæra maximi ſuperficiem ſphærę vera mino-<lb/>rem colligere.</s> <s xml:id="echoid-s10516" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10517" xml:space="preserve"><emph style="sc">Fiat</emph> vt 71. </s> <s xml:id="echoid-s10518" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s10519" xml:space="preserve">ita quadratum datæ diametri circuli maximi ad aliud. <lb/></s> <s xml:id="echoid-s10520" xml:space="preserve"> <anchor type="note" xlink:label="note-257-06a" xlink:href="note-257-06"/> Numerus namque genitus ſuperficiem ſphæræ vera minorem exhibebit. </s> <s xml:id="echoid-s10521" xml:space="preserve">Quo <lb/>niam enim per propoſ. </s> <s xml:id="echoid-s10522" xml:space="preserve">5. </s> <s xml:id="echoid-s10523" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10524" xml:space="preserve">Num. </s> <s xml:id="echoid-s10525" xml:space="preserve">1. </s> <s xml:id="echoid-s10526" xml:space="preserve">minor eſt proportio quad ati dia-<lb/>metri ad ſuperfi ciem ſphærę, quam 71. </s> <s xml:id="echoid-s10527" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s10528" xml:space="preserve">Eſt autẽ quadratum datæ diametri <lb/>ad pro ductum numerum, vt 71. </s> <s xml:id="echoid-s10529" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s10530" xml:space="preserve">Erit quoq; </s> <s xml:id="echoid-s10531" xml:space="preserve">min or proportio quadrati da-<lb/>tæ diametriad veram ſuperficiem ſphærę quam eiuſdem quadrati ad numerum <lb/>productum. </s> <s xml:id="echoid-s10532" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quapropter minor erit productus numerus, quam vera ſuper- <anchor type="note" xlink:label="note-257-07a" xlink:href="note-257-07"/> ficies ſphærę.</s> <s xml:id="echoid-s10533" xml:space="preserve"/> </p> <div xml:id="echoid-div649" type="float" level="2" n="1"> <note position="right" xlink:label="note-257-06" xlink:href="note-257-06a" xml:space="preserve">Superfici{es} <lb/>ſphæræ mi or, <lb/>quam vera.</note> <note symbol="d" position="right" xlink:label="note-257-07" xlink:href="note-257-07a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s10534" xml:space="preserve">4. </s> <s xml:id="echoid-s10535" xml:space="preserve"><emph style="sc">Pari</emph> ratione ex propoſ. </s> <s xml:id="echoid-s10536" xml:space="preserve">6. </s> <s xml:id="echoid-s10537" xml:space="preserve">& </s> <s xml:id="echoid-s10538" xml:space="preserve">7. </s> <s xml:id="echoid-s10539" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10540" xml:space="preserve">Nũ. </s> <s xml:id="echoid-s10541" xml:space="preserve">1. </s> <s xml:id="echoid-s10542" xml:space="preserve">eliciuntur quatuor aliæ <lb/>ſequentes regulę, per quas tam ex circũferentia, quam ex diametro maximi cir-<lb/>cul inſp ra, e@uitur ſoliditas ſphærę tummai@r, quam vera, t@m minor.</s> <s xml:id="echoid-s10543" xml:space="preserve"/> </p> <pb o="228" file="258" n="258" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div651" type="section" level="1" n="228"> <head xml:id="echoid-head245" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s10544" xml:space="preserve">EX circumferentia circuli maximi in ſphæra ſoliditatem ſphæræ produ-<lb/>cere vera maiorem.</s> <s xml:id="echoid-s10545" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10546" xml:space="preserve"><emph style="sc">Fiat</emph> vt 298374. </s> <s xml:id="echoid-s10547" xml:space="preserve">ad 5041. </s> <s xml:id="echoid-s10548" xml:space="preserve">ita cubus ex data circumferentia maximi circuli <lb/> <anchor type="note" xlink:label="note-258-01a" xlink:href="note-258-01"/> deſcriptus ad aliud. </s> <s xml:id="echoid-s10549" xml:space="preserve">Numerus enim genitus dabit ſphærę ſoliditatem vera ma-<lb/>iorem. </s> <s xml:id="echoid-s10550" xml:space="preserve">Nam cum, per propoſ. </s> <s xml:id="echoid-s10551" xml:space="preserve">6. </s> <s xml:id="echoid-s10552" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10553" xml:space="preserve">Num. </s> <s xml:id="echoid-s10554" xml:space="preserve">1. </s> <s xml:id="echoid-s10555" xml:space="preserve">maior ſit proportio cubi <lb/>datæ circumferentiæ ad ſphæram, quam 298374. </s> <s xml:id="echoid-s10556" xml:space="preserve">ad 5041. </s> <s xml:id="echoid-s10557" xml:space="preserve">Sit autem cubus datę <lb/>circumferentię ad numerum procreatum, vt 298374. </s> <s xml:id="echoid-s10558" xml:space="preserve">ad 5041. </s> <s xml:id="echoid-s10559" xml:space="preserve">habebit quo que <lb/>cubus datę circumferentię ad ſphærę ſoliditatem veram, maiorem proportio-<lb/>nem, quam idem cubus ad productum numerum: </s> <s xml:id="echoid-s10560" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Ideoque productus nume- <anchor type="note" xlink:label="note-258-02a" xlink:href="note-258-02"/> rus maior erit, quam vera ſoliditas ſphærę.</s> <s xml:id="echoid-s10561" xml:space="preserve"/> </p> <div xml:id="echoid-div651" type="float" level="2" n="1"> <note position="left" xlink:label="note-258-01" xlink:href="note-258-01a" xml:space="preserve">Solidit{as} ſphæ-<lb/>ræmaior, quã <lb/>vera.</note> <note symbol="a" position="left" xlink:label="note-258-02" xlink:href="note-258-02a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div653" type="section" level="1" n="229"> <head xml:id="echoid-head246" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s10562" xml:space="preserve">EX circumferentia circuli maximi in ſphæra ſoliditatem ſphærę vera <lb/>minorem procreare.</s> <s xml:id="echoid-s10563" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10564" xml:space="preserve"><emph style="sc">Fiat</emph>, vt 2904. </s> <s xml:id="echoid-s10565" xml:space="preserve">ad 49. </s> <s xml:id="echoid-s10566" xml:space="preserve">ita cubus datę circumferentiæ ad aliud. </s> <s xml:id="echoid-s10567" xml:space="preserve">Nam pro-<lb/> <anchor type="note" xlink:label="note-258-03a" xlink:href="note-258-03"/> ductus numerus offeret ſphærę ſoliditatem vera minorem. </s> <s xml:id="echoid-s10568" xml:space="preserve">Cum enim per pro-<lb/>poſ. </s> <s xml:id="echoid-s10569" xml:space="preserve">6. </s> <s xml:id="echoid-s10570" xml:space="preserve">huius cap. </s> <s xml:id="echoid-s10571" xml:space="preserve">Num. </s> <s xml:id="echoid-s10572" xml:space="preserve">1. </s> <s xml:id="echoid-s10573" xml:space="preserve">minor ſit pro portio cubi circumferentiæ datę ad ſphę-<lb/>ram, quam 2904. </s> <s xml:id="echoid-s10574" xml:space="preserve">ad 49. </s> <s xml:id="echoid-s10575" xml:space="preserve">ſit autem cubus datę circumferentię ad numerum pro-<lb/>creatum, vt 2904. </s> <s xml:id="echoid-s10576" xml:space="preserve">ad 49. </s> <s xml:id="echoid-s10577" xml:space="preserve">habebit quoque cubus datę circumferentiæ ad veram <lb/>ſoliditatem ſphæræ, minorem proportionem, quàm idem cubus ad numerum <lb/>productum. </s> <s xml:id="echoid-s10578" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Ideo que numerus productus minor erit vera ſoliditate ſphæræ.</s> <s xml:id="echoid-s10579" xml:space="preserve"/> </p> <div xml:id="echoid-div653" type="float" level="2" n="1"> <note position="left" xlink:label="note-258-03" xlink:href="note-258-03a" xml:space="preserve">Solidit{as} ſphæ-<lb/>ræminor, quã <lb/>vera.</note> </div> <note symbol="b" position="left" xml:space="preserve">10. quinti.</note> </div> <div xml:id="echoid-div655" type="section" level="1" n="230"> <head xml:id="echoid-head247" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s10580" xml:space="preserve">EX diametro maximi circuli in ſphęra ſoliditatem ſphærę colligere ve-<lb/>ra maiorem.</s> <s xml:id="echoid-s10581" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10582" xml:space="preserve"><emph style="sc">Fiat</emph> vt 21. </s> <s xml:id="echoid-s10583" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10584" xml:space="preserve">ita cubus diametri datæ ad aliud. </s> <s xml:id="echoid-s10585" xml:space="preserve">Procreatus namque nu-<lb/> <anchor type="note" xlink:label="note-258-05a" xlink:href="note-258-05"/> merus ſoliditatem ſphæræ dabit vera maiorem. </s> <s xml:id="echoid-s10586" xml:space="preserve">Cum enim per propoſ. </s> <s xml:id="echoid-s10587" xml:space="preserve">7. </s> <s xml:id="echoid-s10588" xml:space="preserve">huius <lb/>cap. </s> <s xml:id="echoid-s10589" xml:space="preserve">Num. </s> <s xml:id="echoid-s10590" xml:space="preserve">1. </s> <s xml:id="echoid-s10591" xml:space="preserve">maior ſit proportio cubi diametriſphæræ ad ſphæram, quàm 21. </s> <s xml:id="echoid-s10592" xml:space="preserve">ad <lb/>11. </s> <s xml:id="echoid-s10593" xml:space="preserve">Sit autem cubus datæ diametri ad numerum productum, vt 21. </s> <s xml:id="echoid-s10594" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s10595" xml:space="preserve">habe-<lb/>bit quo que maiorem proportionem cubus datæ diametri ad veram ſoliditatem <lb/>ſphæræ, quamidem cubus ad numerum genitum. </s> <s xml:id="echoid-s10596" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quapropter numerus pro- <anchor type="note" xlink:label="note-258-06a" xlink:href="note-258-06"/> creatus maior erit vera ſoliditate ſphæræ.</s> <s xml:id="echoid-s10597" xml:space="preserve"/> </p> <div xml:id="echoid-div655" type="float" level="2" n="1"> <note position="left" xlink:label="note-258-05" xlink:href="note-258-05a" xml:space="preserve">Solidit{as} ſphæ-<lb/>ræ maior, quã <lb/>vera.</note> <note symbol="c" position="left" xlink:label="note-258-06" xlink:href="note-258-06a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div657" type="section" level="1" n="231"> <head xml:id="echoid-head248" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s10598" xml:space="preserve">EX diametro maximi circuli in ſphæra ſoliditatem ſphæræ concludere <lb/>vera minorem.</s> <s xml:id="echoid-s10599" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Solidit{as} ſphæ-<lb/>ræminor, quã <lb/>vera.</note> <p> <s xml:id="echoid-s10600" xml:space="preserve"><emph style="sc">Fiat</emph> vt 426. </s> <s xml:id="echoid-s10601" xml:space="preserve">ad 223. </s> <s xml:id="echoid-s10602" xml:space="preserve">ita cubus datę diametri ad aliud. </s> <s xml:id="echoid-s10603" xml:space="preserve">Numerus enim pro-<lb/>ueniens minor erit, quàm vera ſoliditas ſphærę. </s> <s xml:id="echoid-s10604" xml:space="preserve">Nam cum per propoſ. </s> <s xml:id="echoid-s10605" xml:space="preserve">7. </s> <s xml:id="echoid-s10606" xml:space="preserve">huius <pb o="229" file="259" n="259" rhead="LIBER QVINTVS."/> cap Num. </s> <s xml:id="echoid-s10607" xml:space="preserve">1. </s> <s xml:id="echoid-s10608" xml:space="preserve">minor ſit proportio cubi datæ diametri ad ſphæram, quàm 426. </s> <s xml:id="echoid-s10609" xml:space="preserve">ad <lb/>223. </s> <s xml:id="echoid-s10610" xml:space="preserve">Sit autem cubus diametri datæ ad procreatum numerum, vt 426. </s> <s xml:id="echoid-s10611" xml:space="preserve">ad 223. <lb/></s> <s xml:id="echoid-s10612" xml:space="preserve">habebit quo que cubus datæ diametri ad ſphæram, proportionem minorem, <lb/> <anchor type="note" xlink:label="note-259-01a" xlink:href="note-259-01"/> quàm idem cubus ad numerum genitum. </s> <s xml:id="echoid-s10613" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quare minor erit numerus produ- ctus, quàm vera ſoliditas ſphęræ.</s> <s xml:id="echoid-s10614" xml:space="preserve"/> </p> <div xml:id="echoid-div657" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-259-01" xlink:href="note-259-01a" xml:space="preserve">10. quinti.</note> </div> </div> <div xml:id="echoid-div659" type="section" level="1" n="232"> <head xml:id="echoid-head249" xml:space="preserve">DE AREA SEGMENTO-<lb/>rum ſphæræ.</head> <head xml:id="echoid-head250" xml:space="preserve"><emph style="sc">Capvt</emph> VI.</head> <p style="it"> <s xml:id="echoid-s10615" xml:space="preserve">I. </s> <s xml:id="echoid-s10616" xml:space="preserve">HEMISPHER II ſuperfici{es} conuexa, excluſa baſe, gignitur ex area ma-<lb/> <anchor type="note" xlink:label="note-259-02a" xlink:href="note-259-02"/> ximi circuli per 2. </s> <s xml:id="echoid-s10617" xml:space="preserve">multiplicata. </s> <s xml:id="echoid-s10618" xml:space="preserve">Vel ex ſemidiametro in circumferentiã ma-<lb/>ximi circuli. </s> <s xml:id="echoid-s10619" xml:space="preserve">Vel denique ex tota diametro in ſemiſſem circumferentiæ ma-<lb/>ximi circuli. </s> <s xml:id="echoid-s10620" xml:space="preserve">Quæ omnia perſpicua ſunt ex 1. </s> <s xml:id="echoid-s10621" xml:space="preserve">regula Num. </s> <s xml:id="echoid-s10622" xml:space="preserve">2. </s> <s xml:id="echoid-s10623" xml:space="preserve">capitis 5. </s> <s xml:id="echoid-s10624" xml:space="preserve">propterea <lb/>quod hi numeri producti ſunt ſemiſſes illorum, qui ſuperficiem conuexam to-<lb/>tius ſphęræ in earegula exhibuerunt.</s> <s xml:id="echoid-s10625" xml:space="preserve"/> </p> <div xml:id="echoid-div659" type="float" level="2" n="1"> <note position="right" xlink:label="note-259-02" xlink:href="note-259-02a" xml:space="preserve">Superfici{es} cõ-<lb/>uexa Hemi-<lb/>ſphærii.</note> </div> <p> <s xml:id="echoid-s10626" xml:space="preserve">2. </s> <s xml:id="echoid-s10627" xml:space="preserve">SVPERFICIES conuexa cuiuſlibet portionis ſphæræ hemiſphærio minoris, <lb/> <anchor type="note" xlink:label="note-259-03a" xlink:href="note-259-03"/> velmaioris, dempta baſe, æqualis eſt circulo, cui{us} ſemidiameter æqualis eſt rectæ lineæ, <lb/>quæ à vertice portionis ad circumferentiam baſis ducitur. </s> <s xml:id="echoid-s10628" xml:space="preserve">ex propoſ. </s> <s xml:id="echoid-s10629" xml:space="preserve">40. </s> <s xml:id="echoid-s10630" xml:space="preserve">lib. </s> <s xml:id="echoid-s10631" xml:space="preserve">1. </s> <s xml:id="echoid-s10632" xml:space="preserve">Archime-<lb/>dis de ſphæra, & </s> <s xml:id="echoid-s10633" xml:space="preserve">cylindro. </s> <s xml:id="echoid-s10634" xml:space="preserve">Sit enim maximus in ſphęra circulus ABCD, cuius dia-<lb/>meter AC, quàm in E, ad angulos rectos ſecet B D, recta, per quam intelligatur <lb/>duciplanum diametro ad angulos rectos, ſecans ſphæram in duas portiones, <lb/>quarum baſis communis circulus diametri B D, & </s> <s xml:id="echoid-s10635" xml:space="preserve">A, vertex minoris portionis, <lb/>maioris autem vertex C. </s> <s xml:id="echoid-s10636" xml:space="preserve">Iunctis autem rectis AB, CB; </s> <s xml:id="echoid-s10637" xml:space="preserve">erit circulus ſemidiametri <lb/>A B, ſuperficiei conuexæ minoris portionis, & </s> <s xml:id="echoid-s10638" xml:space="preserve">circulus ſemidiametri C B, con-<lb/>uexę ſuperficiei maioris portionis ęqualis, ex dicta propoſ. </s> <s xml:id="echoid-s10639" xml:space="preserve">Ar-<lb/> <anchor type="figure" xlink:label="fig-259-01a" xlink:href="fig-259-01"/> chimedis. </s> <s xml:id="echoid-s10640" xml:space="preserve">Quare ſi vtraque AB, CB, in partibus diametri A C, <lb/>fiat nota, præſertim ope inſtrumenti partium cap. </s> <s xml:id="echoid-s10641" xml:space="preserve">1. </s> <s xml:id="echoid-s10642" xml:space="preserve">lib. </s> <s xml:id="echoid-s10643" xml:space="preserve">1. </s> <s xml:id="echoid-s10644" xml:space="preserve">con-<lb/>ſtructi, & </s> <s xml:id="echoid-s10645" xml:space="preserve">areę circulorum ad interualla ſemidiametrorum AB, <lb/>CB, deſciptorum inueſtigentur, per ea, quę lib. </s> <s xml:id="echoid-s10646" xml:space="preserve">4. </s> <s xml:id="echoid-s10647" xml:space="preserve">capit. </s> <s xml:id="echoid-s10648" xml:space="preserve">7. </s> <s xml:id="echoid-s10649" xml:space="preserve">ſcri-<lb/>pſimus; </s> <s xml:id="echoid-s10650" xml:space="preserve">notæ erunt ſuperficies conuexæ dictarum portionum <lb/>ſphęræ.</s> <s xml:id="echoid-s10651" xml:space="preserve"/> </p> <div xml:id="echoid-div660" type="float" level="2" n="2"> <note position="right" xlink:label="note-259-03" xlink:href="note-259-03a" xml:space="preserve">Superfici{es} cõ-<lb/>uexæ portio-<lb/>nis ſphæræ.</note> <figure xlink:label="fig-259-01" xlink:href="fig-259-01a"> <image file="259-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/259-01"/> </figure> </div> <p> <s xml:id="echoid-s10652" xml:space="preserve"><emph style="sc">Eadem</emph> ſuperficies conuexa portionis ſphærę hemiſphęrio minoris, vel ma-<lb/>ioris, ita quoque cognoſcetur. </s> <s xml:id="echoid-s10653" xml:space="preserve">Ex demonſtratis ab Archimede propoſ. </s> <s xml:id="echoid-s10654" xml:space="preserve">3. </s> <s xml:id="echoid-s10655" xml:space="preserve">lib. </s> <s xml:id="echoid-s10656" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s10657" xml:space="preserve">de ſphęra, & </s> <s xml:id="echoid-s10658" xml:space="preserve">cylindro, eandem proportionem habet EC, ad EA, quam ſuperfi-<lb/>cies conuexa portionis ſphęræ baſem habentis circulum diametri BD, & </s> <s xml:id="echoid-s10659" xml:space="preserve">verticẽ <lb/>C, ad ſuperficiem conuexam portionis baſem habentis eundem circulum dia-<lb/>metri BD, & </s> <s xml:id="echoid-s10660" xml:space="preserve">verticem A. </s> <s xml:id="echoid-s10661" xml:space="preserve">Igitur componendo quoq; </s> <s xml:id="echoid-s10662" xml:space="preserve">erit, vt tota diameter A C, <lb/>ad AE, ita ſuperficies conuexa totius ſphęræ ad ſuperficiem cõuexam portionis <lb/>B A D. </s> <s xml:id="echoid-s10663" xml:space="preserve">Eademq; </s> <s xml:id="echoid-s10664" xml:space="preserve">ratione erit, vt tota diameter A C, ad E C, ita conuexa ſuperfici-<lb/>es totius ſphęræ ad ſuperficiem conuexam portionis BCD. </s> <s xml:id="echoid-s10665" xml:space="preserve">Quo circainueſti-<lb/>gata proportione diametri A C, ad ſegmenta AE, EC, per inſtrumentum partium <lb/>cap. </s> <s xml:id="echoid-s10666" xml:space="preserve">1. </s> <s xml:id="echoid-s10667" xml:space="preserve">lib. </s> <s xml:id="echoid-s10668" xml:space="preserve">1. </s> <s xml:id="echoid-s10669" xml:space="preserve">conſtructum; </s> <s xml:id="echoid-s10670" xml:space="preserve">ſi fiat, vt diameter AC, ad AE, ita ſuperficies conuexa <lb/>totius ſphęræ, (quæ ex regula 1. </s> <s xml:id="echoid-s10671" xml:space="preserve">Nume. </s> <s xml:id="echoid-s10672" xml:space="preserve">2. </s> <s xml:id="echoid-s10673" xml:space="preserve">cap. </s> <s xml:id="echoid-s10674" xml:space="preserve">5. </s> <s xml:id="echoid-s10675" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10676" xml:space="preserve">cognita fiet) ad alind, <pb o="230" file="260" n="260" rhead="GEOMETR. PRACT."/> proueniet conuexa ſuperficies portionis minoris BAD. </s> <s xml:id="echoid-s10677" xml:space="preserve">Similiq; </s> <s xml:id="echoid-s10678" xml:space="preserve">modò ſuperfi-<lb/>cies conuexa maioris portionis B C D, cognoſcetur; </s> <s xml:id="echoid-s10679" xml:space="preserve">ſi fiat, vt diameter A C, ad <lb/>EC, ita ſuperficies conuexa totius ſphæræ ad aliud.</s> <s xml:id="echoid-s10680" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10681" xml:space="preserve"><emph style="sc">Et</emph> quia, vt ex Archimede oſtendimus, ita eſt diameter A C, ad A E, vel ad <lb/>EC, vt tota ſuperficies ſphæræ ad ſuperficiẽ portionis BAD, vel BCD: </s> <s xml:id="echoid-s10682" xml:space="preserve">erit quo-<lb/>que ita AF, ſemiſsis diametri ad AE, vel EC, vt hemiſphęrij ſuperficies GAH, ad <lb/>ſuperficiem portionis B A D, vel B C D, quod oſtendetur eodem modo, quo <lb/>ſcholium propoſ. </s> <s xml:id="echoid-s10683" xml:space="preserve">22. </s> <s xml:id="echoid-s10684" xml:space="preserve">lib. </s> <s xml:id="echoid-s10685" xml:space="preserve">5. </s> <s xml:id="echoid-s10686" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s10687" xml:space="preserve">eſt demonſtratum: </s> <s xml:id="echoid-s10688" xml:space="preserve">Si fiat, vt ſemidiameter <lb/>ſphæræ A F, ad A E, vel E C, altitudinem portionis, ita hemiſphęrij G A H, ſu-<lb/>perficies ad aliud; </s> <s xml:id="echoid-s10689" xml:space="preserve">producetur rurſus conuexa ſuperficies portionis minoris B-<lb/>AD, vel maioris BCD.</s> <s xml:id="echoid-s10690" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10691" xml:space="preserve"><emph style="sc">Immo</emph> cum ſit vt AF, ſemidiameter ad AE, ita hemiſphęrij GAH, ſuperfici-<lb/> <anchor type="note" xlink:label="note-260-01a" xlink:href="note-260-01"/> es ad ſuperficiem portionis BAD, <anchor type="note" xlink:href="" symbol="a"/>erit per conuerſionem rationis, vt AF, ſemi- diameter ad EF, ita ſuperficies hemiſphęrij GAH, ad ſuperficiem fruſti GBDH, <lb/>demptis baſibus. </s> <s xml:id="echoid-s10692" xml:space="preserve">Ergo EF, eadem pars erit, vel partes diametri AC, vel ſemidia-<lb/>metri AF, quæ pars eſt, vel partes ſuperficies fruſtri GBDH, demptis baſibus, ſu-<lb/>perficiei totius ſphęræ, vel hemiſphærij GAH. </s> <s xml:id="echoid-s10693" xml:space="preserve">Quam obrem cognito, quæ pars <lb/>ſit EF, vel partes ſemidiametri AF, ſi ex ſuperficie hemiſphærij G A H, eadẽ pars <lb/>auferatur, vel partes, reliqua fiet ſuperficies conuexa minoris portionis B A D. <lb/></s> <s xml:id="echoid-s10694" xml:space="preserve">Et ſi ad hemiſphærij GCH, ſuperficiem adij ciatur eadem pars, vel partes, con-<lb/>flabitur conuexa ſuperficies portionis maioris BCD. </s> <s xml:id="echoid-s10695" xml:space="preserve">Verbi gratia, ſi E F, conti-<lb/>neat {3/5}. </s> <s xml:id="echoid-s10696" xml:space="preserve">ſemidiametri AF, & </s> <s xml:id="echoid-s10697" xml:space="preserve">ex ſuperficie hemiſphærij G A H, tollantur {3/5}. </s> <s xml:id="echoid-s10698" xml:space="preserve">reli-<lb/>qua fiet ſuperficies conuexa portionis minoris BAD: </s> <s xml:id="echoid-s10699" xml:space="preserve">Et ſi {3/5}. </s> <s xml:id="echoid-s10700" xml:space="preserve">ſuperficiei hemi-<lb/>ſphærij adij ciantur ad ſuperficiẽ hemiſpherij GCH, cõficietur ſuperficies cõue-<lb/>xa maioris portionis BCD. </s> <s xml:id="echoid-s10701" xml:space="preserve">Sic ſi EF, eſſet ſemiſsis ſemidiametri, auferenda eſ@et <lb/>ex hemiſphærij ſuperficie ſemiſsis ipſius, vel adijcenda: </s> <s xml:id="echoid-s10702" xml:space="preserve">Et ſic de cæteris.</s> <s xml:id="echoid-s10703" xml:space="preserve"/> </p> <div xml:id="echoid-div661" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-260-01" xlink:href="note-260-01a" xml:space="preserve">corol. 19. <lb/>quinti.</note> </div> <p> <s xml:id="echoid-s10704" xml:space="preserve">3. </s> <s xml:id="echoid-s10705" xml:space="preserve"><emph style="sc">Hemisphærii</emph> ſoliditas producitur ex ſemidiametro in tertiam par-<lb/> <anchor type="note" xlink:label="note-260-02a" xlink:href="note-260-02"/> tem ſuperficiei hemiſphærij: </s> <s xml:id="echoid-s10706" xml:space="preserve">Vel in ſextam partem ſuperficiei totius ſphæræ. </s> <s xml:id="echoid-s10707" xml:space="preserve">Vel <lb/>ex {1/4}. </s> <s xml:id="echoid-s10708" xml:space="preserve">totius diametri in {2/3}. </s> <s xml:id="echoid-s10709" xml:space="preserve">ſuperficiei hemiſphærij.</s> <s xml:id="echoid-s10710" xml:space="preserve"/> </p> <div xml:id="echoid-div662" type="float" level="2" n="4"> <note position="left" xlink:label="note-260-02" xlink:href="note-260-02a" xml:space="preserve">Solidit{as} he-<lb/>miſphærij.</note> </div> <p> <s xml:id="echoid-s10711" xml:space="preserve"><emph style="sc">Item</emph> ex duabus tertijs partibus diametri in ſemiſſem areæ circuli maximi.</s> <s xml:id="echoid-s10712" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10713" xml:space="preserve"><emph style="sc">Vel</emph> ex duabus tertijs partibus areæ circuli maximi in ſemidiametrum: </s> <s xml:id="echoid-s10714" xml:space="preserve">Aut <lb/>ex tertia parte areæ circuli maximi in totam diametrum.</s> <s xml:id="echoid-s10715" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10716" xml:space="preserve"><emph style="sc">Vel</emph> ex {1/4}. </s> <s xml:id="echoid-s10717" xml:space="preserve">totius diametri in {4/5<unsure/>}. </s> <s xml:id="echoid-s10718" xml:space="preserve">areæ circuli maximi.</s> <s xml:id="echoid-s10719" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10720" xml:space="preserve"><emph style="sc">Vel</emph> ex ſemiſſe areæ circuli maximi in {2/3<unsure/>}. </s> <s xml:id="echoid-s10721" xml:space="preserve">diametri.</s> <s xml:id="echoid-s10722" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10723" xml:space="preserve"><emph style="sc">Vel</emph> ex dupla diametro in {1/6}. </s> <s xml:id="echoid-s10724" xml:space="preserve">areæ circuli maximi.</s> <s xml:id="echoid-s10725" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10726" xml:space="preserve"><emph style="sc">Vel</emph> ex ſemidiametro in ſextam partem ſuperficiei ſphęræ.</s> <s xml:id="echoid-s10727" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10728" xml:space="preserve"><emph style="sc">Vel</emph> denique ex {1/6}. </s> <s xml:id="echoid-s10729" xml:space="preserve">diametri in ſuperficiem hemiſphærij conuexam. </s> <s xml:id="echoid-s10730" xml:space="preserve">quæ <lb/>omnia ex 2. </s> <s xml:id="echoid-s10731" xml:space="preserve">regula Num. </s> <s xml:id="echoid-s10732" xml:space="preserve">2. </s> <s xml:id="echoid-s10733" xml:space="preserve">cap. </s> <s xml:id="echoid-s10734" xml:space="preserve">5. </s> <s xml:id="echoid-s10735" xml:space="preserve">colliguntur: </s> <s xml:id="echoid-s10736" xml:space="preserve">cum omnes hi numeri producti <lb/>ſint ſemiſſes illorum, qui ſoliditatem totius ſphæræ in ea regula indicant.</s> <s xml:id="echoid-s10737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10738" xml:space="preserve">4. </s> <s xml:id="echoid-s10739" xml:space="preserve"><emph style="sc">Soliditas</emph> ſectoris ſphęræ (quinimirum componitur ex minore por-<lb/> <anchor type="note" xlink:label="note-260-03a" xlink:href="note-260-03"/> tione ſphęræ, & </s> <s xml:id="echoid-s10740" xml:space="preserve">cono baſem habente eandem cum portione, & </s> <s xml:id="echoid-s10741" xml:space="preserve">altitudinem æ-<lb/>qualem perpendiculari ex centro in baſem portionis deductæ; </s> <s xml:id="echoid-s10742" xml:space="preserve">Vel quirelin qui-<lb/>tur, ſi idem conus ex portione maiore ſubtrahitur. </s> <s xml:id="echoid-s10743" xml:space="preserve">Vt in proxima figura, ſolidũ <lb/>compoſitum ex portione ſphęræ B A D, baſem habente circulum diametri B D, <lb/>& </s> <s xml:id="echoid-s10744" xml:space="preserve">ex cono habente eandem baſem, & </s> <s xml:id="echoid-s10745" xml:space="preserve">verticemin centro F: </s> <s xml:id="echoid-s10746" xml:space="preserve">Item ſolidũ, quod <lb/>relinquitur, ſi conus idem ex portione maiore B C D, dematur, appellamus ſe- <pb o="231" file="261" n="261" rhead="LIBER QVINTVS."/> ctorem ſphæræ.) </s> <s xml:id="echoid-s10747" xml:space="preserve">hac ratione inueſtigabitur. </s> <s xml:id="echoid-s10748" xml:space="preserve">Quoniam per propoſ. </s> <s xml:id="echoid-s10749" xml:space="preserve">42. </s> <s xml:id="echoid-s10750" xml:space="preserve">lib. </s> <s xml:id="echoid-s10751" xml:space="preserve">1. </s> <s xml:id="echoid-s10752" xml:space="preserve">Ar-<lb/>chimedis de ſphęra & </s> <s xml:id="echoid-s10753" xml:space="preserve">cylindro, ſectoriſphęræ ęqualis eſt conus baſem habens <lb/>circulum ęqualem ſuperficiei conuexæ portionis ſphęræ, altitudinem verò ſe-<lb/>midiametro ſphęræ ęqualem: </s> <s xml:id="echoid-s10754" xml:space="preserve">Conus autem pro ducitur, vt c. </s> <s xml:id="echoid-s10755" xml:space="preserve">2. </s> <s xml:id="echoid-s10756" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10757" xml:space="preserve">Nu. <lb/></s> <s xml:id="echoid-s10758" xml:space="preserve">1. </s> <s xml:id="echoid-s10759" xml:space="preserve">declarauimus, vel ex baſe in {1/3}. </s> <s xml:id="echoid-s10760" xml:space="preserve">altitudinis: </s> <s xml:id="echoid-s10761" xml:space="preserve">Vel ex tota altitudine in {1/3}. </s> <s xml:id="echoid-s10762" xml:space="preserve">baſis; </s> <s xml:id="echoid-s10763" xml:space="preserve"><lb/>fit vt ſector ſphęræ gignatur vel ex ſuperficie conuexa portionis ſphęræ in {1/3}. </s> <s xml:id="echoid-s10764" xml:space="preserve">ſe-<lb/>midiametri, hoc eſt, in {1/6}. </s> <s xml:id="echoid-s10765" xml:space="preserve">totius diametri: </s> <s xml:id="echoid-s10766" xml:space="preserve">Vel ex ſemidiametro in {1/3}. </s> <s xml:id="echoid-s10767" xml:space="preserve">ſuperfi-<lb/>ciei conuexæ portionis ſphæræ.</s> <s xml:id="echoid-s10768" xml:space="preserve"/> </p> <div xml:id="echoid-div663" type="float" level="2" n="5"> <note position="left" xlink:label="note-260-03" xlink:href="note-260-03a" xml:space="preserve">Solidit{as} ſe-<lb/>ctoris ſphæræ.</note> </div> <note position="right" xml:space="preserve">Soliditas cæ<unsure/>-<lb/>iuslibet portio <lb/>nis ſphæræ.</note> <p> <s xml:id="echoid-s10769" xml:space="preserve">5. </s> <s xml:id="echoid-s10770" xml:space="preserve"><emph style="sc">Soliditas</emph> verò cuiuſcunque portionis ſphęræ hoc modo procrea-<lb/>bitur. </s> <s xml:id="echoid-s10771" xml:space="preserve">Inueſtigetur ſoliditas ſectoris ſphæræ, vt proximè tra ditum eſt. </s> <s xml:id="echoid-s10772" xml:space="preserve">Nam ſi, <lb/>quando portio propoſita minor eſt hemiſphærio, ex hoc ſectore dematur co-<lb/>nus eandem habens cum portione baſem, altitudinem verò perpendicularem <lb/>ex centro ſphęræ in baſem portionis cadentem, reliqua fiet ſoliditas portionis <lb/>minoris: </s> <s xml:id="echoid-s10773" xml:space="preserve">At verò ſi, quando portio propof<unsure/>ita hemiſphęrio maior eſt, idem co-<lb/>nus ad ſectorem adijciatur, conflabitur ſoliditas portionis maioris. </s> <s xml:id="echoid-s10774" xml:space="preserve">Id quod <lb/>perſpicuum eſt in ſuperiorifigura, cum conus BFD, ablatus ex ſectore ABFDA, <lb/>reliquam faciat portionem minorem BAD: </s> <s xml:id="echoid-s10775" xml:space="preserve">Idem vero conus BFD, ad ditus ſe-<lb/>ctori CBFDC, conſtituat maiorem portionem BCD. </s> <s xml:id="echoid-s10776" xml:space="preserve">Conus porrò prædictus <lb/>B F D, cognitus fiet ex baſe, nimirum ex circulo diametri B D, & </s> <s xml:id="echoid-s10777" xml:space="preserve">altitudine E F, <lb/>cognitis, vt cap. </s> <s xml:id="echoid-s10778" xml:space="preserve">2. </s> <s xml:id="echoid-s10779" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10780" xml:space="preserve">Num. </s> <s xml:id="echoid-s10781" xml:space="preserve">1. </s> <s xml:id="echoid-s10782" xml:space="preserve">docuimus.</s> <s xml:id="echoid-s10783" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div665" type="section" level="1" n="233"> <head xml:id="echoid-head251" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s10784" xml:space="preserve"><emph style="sc">Sit</emph> in ſphæra circulus maximus ABCD, & </s> <s xml:id="echoid-s10785" xml:space="preserve">portiones ſphęræ, quarum ba-<lb/>ſis communis circulus diametri B D, & </s> <s xml:id="echoid-s10786" xml:space="preserve">vertices A, C, quarum ſoliditates ex-<lb/>quirendæ ſunt. </s> <s xml:id="echoid-s10787" xml:space="preserve">Ex centro H, ducatur ad B D, perpendicularis H E, <anchor type="note" xlink:href="" symbol="a"/> quæ rectã <anchor type="note" xlink:label="note-261-02a" xlink:href="note-261-02"/> B D, ſecabit bifariam, <anchor type="note" xlink:href="" symbol="b"/> ac proinde & </s> <s xml:id="echoid-s10788" xml:space="preserve">vtrum que ar- <anchor type="figure" xlink:label="fig-261-01a" xlink:href="fig-261-01"/> <anchor type="note" xlink:label="note-261-03a" xlink:href="note-261-03"/> cum BAD, B C D, bifariam, hoc eſt, per vertices A, <lb/> <anchor type="note" xlink:label="note-261-04a" xlink:href="note-261-04"/> C, tranſibit. </s> <s xml:id="echoid-s10789" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Fiat, vt C E, ad ſummam rectarum C H, C E, ita A E, ad E F: </s> <s xml:id="echoid-s10790" xml:space="preserve">Item, vt A E, ad ſummam <lb/>rectarum A H, A E, ita EC, ad E G. </s> <s xml:id="echoid-s10791" xml:space="preserve">Intelligantur que <lb/>duo coni, quorum baſis communis circulus diametri BD, & </s> <s xml:id="echoid-s10792" xml:space="preserve">vertices F, G. </s> <s xml:id="echoid-s10793" xml:space="preserve">Erit <lb/>per propoſ. </s> <s xml:id="echoid-s10794" xml:space="preserve">2. </s> <s xml:id="echoid-s10795" xml:space="preserve">lib. </s> <s xml:id="echoid-s10796" xml:space="preserve">2. </s> <s xml:id="echoid-s10797" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s10798" xml:space="preserve">cylindro, conus B F D, portioni <lb/>minori B A D, & </s> <s xml:id="echoid-s10799" xml:space="preserve">conus BGD, portioni maiori BCD, æqualis. </s> <s xml:id="echoid-s10800" xml:space="preserve">Quocirca, in-<lb/>uentis horum conorum ſoliditatibus, vt cap. </s> <s xml:id="echoid-s10801" xml:space="preserve">2. </s> <s xml:id="echoid-s10802" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10803" xml:space="preserve">Numer. </s> <s xml:id="echoid-s10804" xml:space="preserve">1. </s> <s xml:id="echoid-s10805" xml:space="preserve">traditum eſt <lb/>inuentę quoque erunt ſoliditates portionum B A D, B C D. </s> <s xml:id="echoid-s10806" xml:space="preserve">quod eſt propoſi-<lb/>tum.</s> <s xml:id="echoid-s10807" xml:space="preserve"/> </p> <div xml:id="echoid-div665" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-261-02" xlink:href="note-261-02a" xml:space="preserve">3. tertij.</note> <figure xlink:label="fig-261-01" xlink:href="fig-261-01a"> <image file="261-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/261-01"/> </figure> <note symbol="b" position="right" xlink:label="note-261-03" xlink:href="note-261-03a" xml:space="preserve">ſchol. 27. <lb/>tertij.</note> <note symbol="c" position="right" xlink:label="note-261-04" xlink:href="note-261-04a" xml:space="preserve">12. ſexti.</note> </div> <note position="right" xml:space="preserve">Soliditas c@-<lb/>iuslibet fru-<lb/>ſti ſphæræ.</note> <p> <s xml:id="echoid-s10808" xml:space="preserve">6. </s> <s xml:id="echoid-s10809" xml:space="preserve"><emph style="sc">Soliditas</emph> denique cuiuſcunque fruſti ſphęrę, ſiue baſes ſint paralle-<lb/>Ię, cuiuſmodi eſt in 1. </s> <s xml:id="echoid-s10810" xml:space="preserve">figura huius cap. </s> <s xml:id="echoid-s10811" xml:space="preserve">fruſtum BDHG, inter circulos diametro-<lb/>rum BD, GH, incluſum, ſiue non parallelę, quale eſt fruſtum B D L K, hoc pa-<lb/>cto inuenietur. </s> <s xml:id="echoid-s10812" xml:space="preserve">Inueſtigetur, vt Num. </s> <s xml:id="echoid-s10813" xml:space="preserve">5. </s> <s xml:id="echoid-s10814" xml:space="preserve">diximus, vtriuſque portionis ABD, A-<lb/>GH, ſoliditas. </s> <s xml:id="echoid-s10815" xml:space="preserve">Minori enim detra cta ex maiore, reliqua erit ſoliditas fruſti BD-<lb/>HG. </s> <s xml:id="echoid-s10816" xml:space="preserve">Sic etiam, inuento I, vertice portionis KIL, ſi inueniatur ſoliditas v-<lb/>triuſque portio nis BCD, KLI, minorque ex maiore tol-<lb/>latur, remanebit ſoliditas fruſti B D L K, nota.</s> <s xml:id="echoid-s10817" xml:space="preserve"/> </p> <pb o="232" file="262" n="262" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div667" type="section" level="1" n="234"> <head xml:id="echoid-head252" xml:space="preserve">DE AREA SPHÆROIDIS, EIVSDEM-<lb/>que portionum.</head> <head xml:id="echoid-head253" xml:space="preserve"><emph style="sc">Capvt</emph> VII.</head> <p> <s xml:id="echoid-s10818" xml:space="preserve">1. </s> <s xml:id="echoid-s10819" xml:space="preserve"><emph style="sc">SIt</emph> Ellipſis ABCD, cuius maior axis AC, minor B D, priorem ad angulos <lb/>rectos ſecans. </s> <s xml:id="echoid-s10820" xml:space="preserve">Soliditatem ergo Sphæroidis, id eſt, ſolidi ex circumuolu-<lb/>tione Ellipſis circa axem effecti, ita nanciſcemur. </s> <s xml:id="echoid-s10821" xml:space="preserve">Quoniam planum per <lb/>BD, ductum, & </s> <s xml:id="echoid-s10822" xml:space="preserve">rectum ad axem AC, circulum facit, vt à Federico Commandi-<lb/>no ad propoſ. </s> <s xml:id="echoid-s10823" xml:space="preserve">12. </s> <s xml:id="echoid-s10824" xml:space="preserve">lib. </s> <s xml:id="echoid-s10825" xml:space="preserve">Archimedis de Conoidibus, & </s> <s xml:id="echoid-s10826" xml:space="preserve">Sphæroidib. </s> <s xml:id="echoid-s10827" xml:space="preserve">demonſtratur. <lb/></s> <s xml:id="echoid-s10828" xml:space="preserve">cuius diameter BD, & </s> <s xml:id="echoid-s10829" xml:space="preserve">centrum E; </s> <s xml:id="echoid-s10830" xml:space="preserve">erit per propoſ. </s> <s xml:id="echoid-s10831" xml:space="preserve">29. </s> <s xml:id="echoid-s10832" xml:space="preserve">lib. </s> <s xml:id="echoid-s10833" xml:space="preserve">Archimedis de Cono-<lb/>id. </s> <s xml:id="echoid-s10834" xml:space="preserve">& </s> <s xml:id="echoid-s10835" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10836" xml:space="preserve">ſemiſsis Sphæroidis A B D, dupla coni ean dem baſem cum illa <lb/>ſemiſſe, circulum videlicet diametri B D, habentis, & </s> <s xml:id="echoid-s10837" xml:space="preserve">altitudinem eandem E A. </s> <s xml:id="echoid-s10838" xml:space="preserve"><lb/>Igitur ſi huius coni ſoliditas per capit. </s> <s xml:id="echoid-s10839" xml:space="preserve">2. </s> <s xml:id="echoid-s10840" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10841" xml:space="preserve">inueſtigetur, & </s> <s xml:id="echoid-s10842" xml:space="preserve">duplicetur, <lb/> <anchor type="note" xlink:label="note-262-01a" xlink:href="note-262-01"/> exurget ſoliditas ſemiſsis Sphæroidis: </s> <s xml:id="echoid-s10843" xml:space="preserve">quæ duplicata ſoliditatem totius Sphę-<lb/>roidis exhibebit.</s> <s xml:id="echoid-s10844" xml:space="preserve"/> </p> <div xml:id="echoid-div667" type="float" level="2" n="1"> <note position="left" xlink:label="note-262-01" xlink:href="note-262-01a" xml:space="preserve">Soliditas Sphæ <lb/>roidis.</note> </div> <p> <s xml:id="echoid-s10845" xml:space="preserve">2. </s> <s xml:id="echoid-s10846" xml:space="preserve"><emph style="sc">Dvcatvr</emph> minori axi B D, parallela F G, ſecans maiorem axemin H, <lb/>ad rectos angulos. </s> <s xml:id="echoid-s10847" xml:space="preserve">Si igitur per F G, ducatur planum rectum ad axem, fiet cir-<lb/>culus in Sphæroide diametrum habens F G, & </s> <s xml:id="echoid-s10848" xml:space="preserve">centrum H, vt Federicus Com-<lb/>mandinus ad propoſ. </s> <s xml:id="echoid-s10849" xml:space="preserve">12. </s> <s xml:id="echoid-s10850" xml:space="preserve">lib. </s> <s xml:id="echoid-s10851" xml:space="preserve">Archim. </s> <s xml:id="echoid-s10852" xml:space="preserve">de Conoid. </s> <s xml:id="echoid-s10853" xml:space="preserve">& </s> <s xml:id="echoid-s10854" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10855" xml:space="preserve">demonſtrauit; </s> <s xml:id="echoid-s10856" xml:space="preserve">ab-<lb/> <anchor type="figure" xlink:label="fig-262-01a" xlink:href="fig-262-01"/> ſcindentur que portiones Sphæroidis F A G, minor & </s> <s xml:id="echoid-s10857" xml:space="preserve"><lb/>FCG, maior. </s> <s xml:id="echoid-s10858" xml:space="preserve">Vtriuſq; </s> <s xml:id="echoid-s10859" xml:space="preserve">ſoliditas ita fiet cognita. </s> <s xml:id="echoid-s10860" xml:space="preserve">Quo-<lb/>niam per propoſ. </s> <s xml:id="echoid-s10861" xml:space="preserve">31. </s> <s xml:id="echoid-s10862" xml:space="preserve">libri Archimedis de Conoid. </s> <s xml:id="echoid-s10863" xml:space="preserve">& </s> <s xml:id="echoid-s10864" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-262-02a" xlink:href="note-262-02"/> Sphęroid. </s> <s xml:id="echoid-s10865" xml:space="preserve">Conus, cuius baſis circulus diametri F G, & </s> <s xml:id="echoid-s10866" xml:space="preserve"><lb/>axis H A, ad minorem portionem ſphęroidis F A G, <lb/>proportionẽ habet, quam maioris portionis axis HC, <lb/>ad ſummam rectarum EC, HC: </s> <s xml:id="echoid-s10867" xml:space="preserve">Si fiat, vt HC, maioris <lb/>portionis axis ad ſummam rectarum E C, H C, ita co-<lb/>nus prædictus ad aliud, (qui quidem conus ex cap. </s> <s xml:id="echoid-s10868" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s10869" xml:space="preserve">huius libri cognitus erit.) </s> <s xml:id="echoid-s10870" xml:space="preserve">prodibit ſoliditas minoris <lb/>portionis ſphęroidis F A G.</s> <s xml:id="echoid-s10871" xml:space="preserve"/> </p> <div xml:id="echoid-div668" type="float" level="2" n="2"> <figure xlink:label="fig-262-01" xlink:href="fig-262-01a"> <image file="262-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/262-01"/> </figure> <note position="left" xlink:label="note-262-02" xlink:href="note-262-02a" xml:space="preserve">Solidit{as} por-<lb/>tionum Sphæ-<lb/>roidis.</note> </div> <p> <s xml:id="echoid-s10872" xml:space="preserve"><emph style="sc">Rvrsvs</emph> quia per propoſ. </s> <s xml:id="echoid-s10873" xml:space="preserve">33. </s> <s xml:id="echoid-s10874" xml:space="preserve">libri Archim. </s> <s xml:id="echoid-s10875" xml:space="preserve">de Conoid. </s> <s xml:id="echoid-s10876" xml:space="preserve">& </s> <s xml:id="echoid-s10877" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10878" xml:space="preserve">conus, <lb/>cuius baſis circulus diametri F G, & </s> <s xml:id="echoid-s10879" xml:space="preserve">axis H C, ad maiorem portionem Sphęro-<lb/>idis FCG, proportionem habet, quam minoris portionis axis HA, ad ſummam <lb/>rectarum E A, H A: </s> <s xml:id="echoid-s10880" xml:space="preserve">ſi fiat, vt H A, minoris portionis axis ad ſummam rectarum <lb/>EA, HA, ita prædictus conus (quem per cap. </s> <s xml:id="echoid-s10881" xml:space="preserve">2. </s> <s xml:id="echoid-s10882" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10883" xml:space="preserve">metieris) ad aliud, pro-<lb/>creabitur ſoliditas maioris portionis ſphęroidis FCG.</s> <s xml:id="echoid-s10884" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div670" type="section" level="1" n="235"> <head xml:id="echoid-head254" xml:space="preserve">DE AREA CONOIDIS <lb/>parabolici.</head> <head xml:id="echoid-head255" xml:space="preserve"><emph style="sc">Capvt</emph> VIII.</head> <p> <s xml:id="echoid-s10885" xml:space="preserve">1. </s> <s xml:id="echoid-s10886" xml:space="preserve"><emph style="sc">SIt</emph> Parabola A B C, cuius axis B D, ad baſem A C, rectus. </s> <s xml:id="echoid-s10887" xml:space="preserve">Solidita-<lb/> <anchor type="note" xlink:label="note-262-03a" xlink:href="note-262-03"/> tem igitur Conoidis parabolici, quod parabola circa axem circumducta <lb/>effi cit, ita metiemur. </s> <s xml:id="echoid-s10888" xml:space="preserve">Quo niam per ea, quæ ad prop oſ. </s> <s xml:id="echoid-s10889" xml:space="preserve">12. </s> <s xml:id="echoid-s10890" xml:space="preserve">libri Archim.</s> <s xml:id="echoid-s10891" xml:space="preserve"> <pb o="233" file="263" n="263" rhead="LIBER QVINTVS."/> de Conoid. </s> <s xml:id="echoid-s10892" xml:space="preserve">& </s> <s xml:id="echoid-s10893" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10894" xml:space="preserve">Federicus Commandinus <lb/> <anchor type="figure" xlink:label="fig-263-01a" xlink:href="fig-263-01"/> demonſtrauit, planum per AC, ductum, & </s> <s xml:id="echoid-s10895" xml:space="preserve">rectum <lb/>ad axem BD, circulum facit, cuius diameter A C, & </s> <s xml:id="echoid-s10896" xml:space="preserve"><lb/>centrum D: </s> <s xml:id="echoid-s10897" xml:space="preserve">erit per propoſ. </s> <s xml:id="echoid-s10898" xml:space="preserve">23. </s> <s xml:id="echoid-s10899" xml:space="preserve">libri Archim. </s> <s xml:id="echoid-s10900" xml:space="preserve">de <lb/>Conoid. </s> <s xml:id="echoid-s10901" xml:space="preserve">& </s> <s xml:id="echoid-s10902" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10903" xml:space="preserve">Parabolicum Conoides A-<lb/>BC, ſeſquialterum coni, cuius baſis circulus diame-<lb/>tri AC, & </s> <s xml:id="echoid-s10904" xml:space="preserve">axis BD. </s> <s xml:id="echoid-s10905" xml:space="preserve">Igitur ſi fiat, vt 2. </s> <s xml:id="echoid-s10906" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s10907" xml:space="preserve">ita prædi-<lb/>ctus conus (quem ex cap. </s> <s xml:id="echoid-s10908" xml:space="preserve">2. </s> <s xml:id="echoid-s10909" xml:space="preserve">huius libri metiemur) <lb/>ad aliud; </s> <s xml:id="echoid-s10910" xml:space="preserve">proſiliet ſoliditas Conoidis Parabolici A-<lb/>BC.</s> <s xml:id="echoid-s10911" xml:space="preserve"/> </p> <div xml:id="echoid-div670" type="float" level="2" n="1"> <note position="left" xlink:label="note-262-03" xlink:href="note-262-03a" xml:space="preserve">Soliditas Co-<lb/>noidis Para-<lb/>bolici.</note> <figure xlink:label="fig-263-01" xlink:href="fig-263-01a"> <image file="263-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/263-01"/> </figure> </div> </div> <div xml:id="echoid-div672" type="section" level="1" n="236"> <head xml:id="echoid-head256" xml:space="preserve">DE AREA CONOIDIS <lb/>Hyperbolici.</head> <head xml:id="echoid-head257" xml:space="preserve"><emph style="sc">Capvt</emph> IX.</head> <p> <s xml:id="echoid-s10912" xml:space="preserve"><emph style="sc">COncipiatvr</emph> ſuperior figura eſſe Hyperbola, & </s> <s xml:id="echoid-s10913" xml:space="preserve">recta E, æqualis ſe-<lb/> <anchor type="note" xlink:label="note-263-01a" xlink:href="note-263-01"/> miſsi diametri tranſuerſæ inter duas hyperbolas oppoſitas, hoc eſt, rectæ <lb/>ex centro hyperbolarum ad verticem B, ductæ. </s> <s xml:id="echoid-s10914" xml:space="preserve">Fietque rurſus circu-<lb/>lus, cuius diameter A C, à plano per AC, ducto, & </s> <s xml:id="echoid-s10915" xml:space="preserve">ad axem recto, vt Federicus <lb/>Commandinus ad propoſ. </s> <s xml:id="echoid-s10916" xml:space="preserve">12. </s> <s xml:id="echoid-s10917" xml:space="preserve">libri Archim. </s> <s xml:id="echoid-s10918" xml:space="preserve">de Conoid. </s> <s xml:id="echoid-s10919" xml:space="preserve">& </s> <s xml:id="echoid-s10920" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10921" xml:space="preserve">demon-<lb/>ſtrauit. </s> <s xml:id="echoid-s10922" xml:space="preserve">Soliditatem igitur Conoidis Hyperbolici, quod ab hyperbola ABC, <lb/>circa axem BD, circumuoluta effi citur, ita venabimur. </s> <s xml:id="echoid-s10923" xml:space="preserve">Quoniam per pro-<lb/>poſ. </s> <s xml:id="echoid-s10924" xml:space="preserve">27. </s> <s xml:id="echoid-s10925" xml:space="preserve">lib. </s> <s xml:id="echoid-s10926" xml:space="preserve">Archimedis de Conoid. </s> <s xml:id="echoid-s10927" xml:space="preserve">& </s> <s xml:id="echoid-s10928" xml:space="preserve">Sphæroid. </s> <s xml:id="echoid-s10929" xml:space="preserve">Conoides Hyperbolicum A-<lb/>BC, ad conum, cuius baſis eadem cum baſe Conoidis, circulus videlicet diame-<lb/>tri A C, & </s> <s xml:id="echoid-s10930" xml:space="preserve">axis idem B D, proportionem habet eandem, quam linea conflata ex <lb/>axe B D, & </s> <s xml:id="echoid-s10931" xml:space="preserve">tripla ipſius E, ad lineam conflatam ex axe BD, & </s> <s xml:id="echoid-s10932" xml:space="preserve">dupla ipſius E. </s> <s xml:id="echoid-s10933" xml:space="preserve">Si <lb/>fiat, vt linea conflata ex axe B D, & </s> <s xml:id="echoid-s10934" xml:space="preserve">duplaipſius E, ad lineam conflatam ex <lb/>axe BD, & </s> <s xml:id="echoid-s10935" xml:space="preserve">tripla ipſius E, ita prædictus conus (quem ex cap. </s> <s xml:id="echoid-s10936" xml:space="preserve">2. </s> <s xml:id="echoid-s10937" xml:space="preserve">huius libri dime-<lb/>tiemur) ad aliud; </s> <s xml:id="echoid-s10938" xml:space="preserve">gignetur ſoliditas Conoidis Hyperbolici ABC.</s> <s xml:id="echoid-s10939" xml:space="preserve"/> </p> <div xml:id="echoid-div672" type="float" level="2" n="1"> <note position="right" xlink:label="note-263-01" xlink:href="note-263-01a" xml:space="preserve">Soliditas Co-<lb/>noidis Hyper-<lb/>bolici.</note> </div> </div> <div xml:id="echoid-div674" type="section" level="1" n="237"> <head xml:id="echoid-head258" xml:space="preserve">DE AREA DOLIORVM.</head> <head xml:id="echoid-head259" xml:space="preserve"><emph style="sc">Capvt</emph> X.</head> <p> <s xml:id="echoid-s10940" xml:space="preserve"><emph style="sc">QVoniam</emph> dolia non eandem formam vbiq; </s> <s xml:id="echoid-s10941" xml:space="preserve">ſeruant, vix præſcribi po-<lb/>teſt ratio, qua dolij propoſiti capacitas accurate inueniatur. </s> <s xml:id="echoid-s10942" xml:space="preserve">Argumen-<lb/> <anchor type="note" xlink:label="note-263-02a" xlink:href="note-263-02"/> to eſt, quod ſcriptores variè de eius Dimenſione ſcripſerunt. </s> <s xml:id="echoid-s10943" xml:space="preserve">Dicam er-<lb/>go etiam ego id, quod mihi veriſimile videtur. </s> <s xml:id="echoid-s10944" xml:space="preserve">Sit dolium ABCDEF, in extre-<lb/>mitatibus habens circulos AF, CD, orificium B, per quod cogitetur planum du-<lb/>ctum rectum ad lineam KL, centra circulorum AF, CD, coniungentem, ſecans <lb/>dolium bifariam. </s> <s xml:id="echoid-s10945" xml:space="preserve">Si igitur aſſeres dolij in B, & </s> <s xml:id="echoid-s10946" xml:space="preserve">E, curuentur, & </s> <s xml:id="echoid-s10947" xml:space="preserve">deinde ſecun-<lb/>dum lineas quaſi rectas extendantur, cuiuſmo di dolia non pauca Romæ vidi:</s> <s xml:id="echoid-s10948" xml:space="preserve"> <pb o="234" file="264" n="264" rhead="GEOMETR. PRACT."/> referent ſemiſſes dolij ABEF, CDEB, conos decuratos: </s> <s xml:id="echoid-s10949" xml:space="preserve">quos ſi per ea, quæ c-<lb/>3. </s> <s xml:id="echoid-s10950" xml:space="preserve">huius libri ſcripta ſunt, metieris, dabit eorum ſumma dolij propoſiti capa cita-<lb/>tem. </s> <s xml:id="echoid-s10951" xml:space="preserve">Memortamen eſto, profunditatem dolij B E, & </s> <s xml:id="echoid-s10952" xml:space="preserve">diametrum circuli A F, <lb/>menſurandam eſſe intra aſſeres, ita, vt eorum craſsities excludantur; </s> <s xml:id="echoid-s10953" xml:space="preserve">vt habea-<lb/>tur decurtatus conus, cuius baſes ſint circuli BE, AF, & </s> <s xml:id="echoid-s10954" xml:space="preserve">c.</s> <s xml:id="echoid-s10955" xml:space="preserve"/> </p> <div xml:id="echoid-div674" type="float" level="2" n="1"> <note position="right" xlink:label="note-263-02" xlink:href="note-263-02a" xml:space="preserve">Capacit{as} do-<lb/>lii.</note> </div> <figure> <image file="264-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/264-01"/> </figure> <p> <s xml:id="echoid-s10956" xml:space="preserve">2. </s> <s xml:id="echoid-s10957" xml:space="preserve"><emph style="sc">Si</emph> vero aſſeres dolij verè ſint ali-<lb/>quo modo circulares, quod nonuulli aſ-<lb/>ſerunt, concipiendum erit dolium, tan-<lb/>quam fruſtum quoddam Sphæroidis. </s> <s xml:id="echoid-s10958" xml:space="preserve">Nã <lb/>curuitas aſſerum ſenſibiliter à curuitate <lb/>Sphæroidis, cuius axes ſunt rectæ KL, BE, <lb/>non different. </s> <s xml:id="echoid-s10959" xml:space="preserve">Sed quia ſolum axis minor <lb/>nimirum profunditas dolij, data eſt, re-<lb/>periemus ex puncto A, in Ellipſi dato, <lb/>maiorem axem KL, hoc modo. </s> <s xml:id="echoid-s10960" xml:space="preserve">Interual-<lb/>lo ſemiſsis minoris axis B G, deſcribatur <lb/>ex A, arcus ſecans in H, rectam G H, ad angulos rectos ipſi minori axi per eius <lb/>punctum medium G, ductam: </s> <s xml:id="echoid-s10961" xml:space="preserve">& </s> <s xml:id="echoid-s10962" xml:space="preserve">ex A, per H, recta emittatur ſecans minorem <lb/>axem in I. </s> <s xml:id="echoid-s10963" xml:space="preserve">Recta enim A I, dabit ſemiſſes G K, G L, maioris axis, vt lib. </s> <s xml:id="echoid-s10964" xml:space="preserve">1. </s> <s xml:id="echoid-s10965" xml:space="preserve">noſtri <lb/>Aſtrolabij in ſcholio Lemmatis 50. </s> <s xml:id="echoid-s10966" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s10967" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s10968" xml:space="preserve">ſi tam ſemiſsis ſphę-<lb/> <anchor type="note" xlink:label="note-264-01a" xlink:href="note-264-01"/> roidis BKE, quam portio minor AKF, menſuretur, vt cap. </s> <s xml:id="echoid-s10969" xml:space="preserve">7. </s> <s xml:id="echoid-s10970" xml:space="preserve">traditum eſt, & </s> <s xml:id="echoid-s10971" xml:space="preserve">ſo-<lb/>liditas portionis AKF, ex ſoliditate ſemiſsis BKE, detrahatur, remanebit capaci-<lb/>tas ſemiſsis dolij ABEF, quæ duplicata totius dolij capa citatem exhibebit. </s> <s xml:id="echoid-s10972" xml:space="preserve">V-<lb/>tram que hanc rationem dolij dimetiendi à vera dolij capacitate non longè ab-<lb/>eſſe arbitror. </s> <s xml:id="echoid-s10973" xml:space="preserve">Paratus tamen interim ſum, ſi quis accuratiorem inuenerit, eam li-<lb/>benti animo, & </s> <s xml:id="echoid-s10974" xml:space="preserve">grato acceptare.</s> <s xml:id="echoid-s10975" xml:space="preserve"/> </p> <div xml:id="echoid-div675" type="float" level="2" n="2"> <note position="left" xlink:label="note-264-01" xlink:href="note-264-01a" xml:space="preserve">Capacitas do-<lb/>lij alio modo.</note> </div> </div> <div xml:id="echoid-div677" type="section" level="1" n="238"> <head xml:id="echoid-head260" xml:space="preserve">DE AREA CORPORVM. <lb/>omnino irregularium.</head> <head xml:id="echoid-head261" xml:space="preserve"><emph style="sc">Capvt</emph> XI.</head> <p> <s xml:id="echoid-s10976" xml:space="preserve">1. </s> <s xml:id="echoid-s10977" xml:space="preserve"><emph style="sc">TRadvnt</emph> ſcriptores nonnulli regulam quandã mechanicam ad cor-<lb/>pora dimetienda, quæ omnino ſunt irregularia, ita vt ſub regulas Geo-<lb/>metricas, quæ hactenus explicatæ ſunt, cadere non poſsint: </s> <s xml:id="echoid-s10978" xml:space="preserve">cuiuſ-<lb/>modi ſunt ſtatuæ, vrnæ, amphoræ, fruſta ſaxorum, quæ neque vniformis ſunt <lb/>craſsitiei, neque latera habent prorſus recta, aut ad baſes perpendicularia, & </s> <s xml:id="echoid-s10979" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s10980" xml:space="preserve">Hæc ergo regula, quæ nullo modo videtur aſpernanda, ita ſe habet.</s> <s xml:id="echoid-s10981" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s10982" xml:space="preserve"><emph style="sc">Paretvr</emph> arca lignea ex aſſeribus leuigatis, inſtar parallelepipedi cuiuſdã, <lb/> <anchor type="note" xlink:label="note-264-02a" xlink:href="note-264-02"/> quæ pice ita oblinatur, vt aquam continere poſsit. </s> <s xml:id="echoid-s10983" xml:space="preserve">Arca hæc tantæ debet eſſe <lb/>longitudinis, latitudinis, atq; </s> <s xml:id="echoid-s10984" xml:space="preserve">altitu dinis, vt corpus metiendum intra ipſam po-<lb/>ſitum, a qua totum po ſsit operiri. </s> <s xml:id="echoid-s10985" xml:space="preserve">Poſita autem hac arca Horizonti æquidiſtan-<lb/>te, beneficio libellæ, aut perpendiculi, infun datur in eam tantũ aquæ, quantum <lb/>ſatis eſt, vt corpus imp oſitum omnino tegat, notentur que diligenter ſuprema <pb o="235" file="265" n="265" rhead="LIBER QVINTVS."/> latera aquæ in aſſeribus arcæ, vt habeatur altitudo aquæ vſque ad arcæ fundũ: <lb/></s> <s xml:id="echoid-s10986" xml:space="preserve">Extracto deinde corpore, ita tamen, vt nihil aquæ extra arcam cadat, notentur <lb/>rurſum latera aquæ, poſtquam quieuerit. </s> <s xml:id="echoid-s10987" xml:space="preserve">Quod ſi per cap. </s> <s xml:id="echoid-s10988" xml:space="preserve">1. </s> <s xml:id="echoid-s10989" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s10990" xml:space="preserve">metia-<lb/>mur duo parallelepipeda, quorũ baſis communis eſt arcæ fundus, ſiue baſis, al-<lb/>titudines vero rectæ à lateribus aquæ notatis vſque ad baſem, & </s> <s xml:id="echoid-s10991" xml:space="preserve">minus à maio-<lb/>re ſubtrahamus, relinquetur parallelepipedũ ſoliditati corporis propoſiti o-<lb/>mnino æquale. </s> <s xml:id="echoid-s10992" xml:space="preserve">quod parallelepipedũ etiam conſequeris, ſi altitu dinem inter <lb/>latera aquæ bis notata duces in baſem arcæ. </s> <s xml:id="echoid-s10993" xml:space="preserve">Sunt, qui infuſa a qua in arcam,@la-<lb/>tera eius in aſſeribus primo loco notent. </s> <s xml:id="echoid-s10994" xml:space="preserve">Deinde impoſito corpore, eiuſdem a-<lb/>quæ latera ſignent. </s> <s xml:id="echoid-s10995" xml:space="preserve">Si enim altitudo inter poſteriora latera, ac priora ducatur in <lb/>baſem arcæ, pro ducetur ſoliditas corporis impoſiti.</s> <s xml:id="echoid-s10996" xml:space="preserve"/> </p> <div xml:id="echoid-div677" type="float" level="2" n="1"> <note position="left" xlink:label="note-264-02" xlink:href="note-264-02a" xml:space="preserve">Soliditas cu-<lb/>i{us}libet cor-<lb/>poris irregu-<lb/>laris.</note> </div> <p> <s xml:id="echoid-s10997" xml:space="preserve">2. </s> <s xml:id="echoid-s10998" xml:space="preserve"><emph style="sc">Pro</emph> vrnis, at que amphoris, ſiue eæ lapideæ ſint, ſiue cretaceæ, ita fa cie-<lb/>mus. </s> <s xml:id="echoid-s10999" xml:space="preserve">Impleatur vas arena, & </s> <s xml:id="echoid-s11000" xml:space="preserve">eius orificiumita obturetur, vt a qua ingredi nul-<lb/>lo modo poſsit. </s> <s xml:id="echoid-s11001" xml:space="preserve">Impoſito deinde vaſe in aqua intra arcam contenta, ac ſi eſſet <lb/>corpus quod piam irregulare, inueſtigetur eius ſoliditas, vt Num. </s> <s xml:id="echoid-s11002" xml:space="preserve">1. </s> <s xml:id="echoid-s11003" xml:space="preserve">diximus. </s> <s xml:id="echoid-s11004" xml:space="preserve">De-<lb/>inde extra cta arena, notentur latera aquæ, antequam vas vacuum impo natur. <lb/></s> <s xml:id="echoid-s11005" xml:space="preserve">Impoſito denique vaſe vacuo, ſignentur iterum latera a quæ. </s> <s xml:id="echoid-s11006" xml:space="preserve">Si namque altitu-<lb/>do inter poſteriora, ac priora latera multiplicetur per baſem arcæ: </s> <s xml:id="echoid-s11007" xml:space="preserve">pro creabitur <lb/>ſoliditas ſolius vaſis: </s> <s xml:id="echoid-s11008" xml:space="preserve">quæ detracta ex priori ſoliditate, notamrelin quet vaſis <lb/>@apacitatem.</s> <s xml:id="echoid-s11009" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div679" type="section" level="1" n="239"> <head xml:id="echoid-head262" xml:space="preserve">DE SVPERFICIE CONVEXA <lb/>coni & cylindri recti.</head> <head xml:id="echoid-head263" xml:space="preserve"><emph style="sc">Capvt</emph> XII.</head> <p> <s xml:id="echoid-s11010" xml:space="preserve">1. </s> <s xml:id="echoid-s11011" xml:space="preserve"><emph style="sc">QVoniam</emph> ex Archimede demonſtrauimus, qua ratione ſuperficies <lb/> <anchor type="note" xlink:label="note-265-01a" xlink:href="note-265-01"/> conuexa, ſphæræ eiuſque portionum inueſtiganda ſit: </s> <s xml:id="echoid-s11012" xml:space="preserve">non deerit for-<lb/>taſſe, qui idem deſi deret in cono, ac cylindro recto. </s> <s xml:id="echoid-s11013" xml:space="preserve">quod ex ijs, quæ <lb/>ab eo dem Archimede in lib. </s> <s xml:id="echoid-s11014" xml:space="preserve">1. </s> <s xml:id="echoid-s11015" xml:space="preserve">de ſphęra, & </s> <s xml:id="echoid-s11016" xml:space="preserve">cylindro demonſtrata ſunt, obtine-<lb/>bit hoc modo. </s> <s xml:id="echoid-s11017" xml:space="preserve">Propoſito cono recto quo cunque, erit eius ſuperficies conue-<lb/>xa conica, ſecluſa baſe, æqualis circulo, cuius ſemidiameter eſt linea media pro-<lb/>portionalis inter latus coni, & </s> <s xml:id="echoid-s11018" xml:space="preserve">ſemidiametrum baſis eiuſdem coni, ex propoſ. <lb/></s> <s xml:id="echoid-s11019" xml:space="preserve"> <anchor type="note" xlink:label="note-265-02a" xlink:href="note-265-02"/> 14. </s> <s xml:id="echoid-s11020" xml:space="preserve">lib. </s> <s xml:id="echoid-s11021" xml:space="preserve">1. </s> <s xml:id="echoid-s11022" xml:space="preserve">Archimedis de ſphęra, & </s> <s xml:id="echoid-s11023" xml:space="preserve">cylindro.</s> <s xml:id="echoid-s11024" xml:space="preserve"/> </p> <div xml:id="echoid-div679" type="float" level="2" n="1"> <note position="right" xlink:label="note-265-01" xlink:href="note-265-01a" xml:space="preserve">Superficies co-<lb/>nica, dempta <lb/>baſe, cui cir-<lb/>culo ſit æqua-<lb/>lis.</note> <note position="right" xlink:label="note-265-02" xlink:href="note-265-02a" xml:space="preserve">Superficies <lb/>fruſti coni, <lb/>demptis baſi-<lb/>bus, cui circu-<lb/>lo æqualis ſit.</note> </div> <p> <s xml:id="echoid-s11025" xml:space="preserve">2. </s> <s xml:id="echoid-s11026" xml:space="preserve"><emph style="sc">Qvod</emph> ſi conus rectus ſecetur plano, quod baſi æquidiſtet, erit ſuperfi-<lb/>cies conuexa fruſti coni, demptis baſibus, æqualis circulo, cuius ſemidiameter <lb/>eſt linea media proportionalis inter latus conicum fruſti, & </s> <s xml:id="echoid-s11027" xml:space="preserve">rectam ex ſemidia-<lb/>metris duarũ baſum cõflatã, ex ꝓpoſ. </s> <s xml:id="echoid-s11028" xml:space="preserve">16. </s> <s xml:id="echoid-s11029" xml:space="preserve">lib. </s> <s xml:id="echoid-s11030" xml:space="preserve">1. </s> <s xml:id="echoid-s11031" xml:space="preserve">Archime. </s> <s xml:id="echoid-s11032" xml:space="preserve">de ſphęra, & </s> <s xml:id="echoid-s11033" xml:space="preserve">cylindro. <lb/></s> <s xml:id="echoid-s11034" xml:space="preserve"> <anchor type="note" xlink:label="note-265-03a" xlink:href="note-265-03"/> </s> </p> <div xml:id="echoid-div680" type="float" level="2" n="2"> <note position="right" xlink:label="note-265-03" xlink:href="note-265-03a" xml:space="preserve">Propo tio co-<lb/>nicæ ſuperfi-<lb/>ciei ad ſuam <lb/>baſem.</note> </div> <p> <s xml:id="echoid-s11035" xml:space="preserve"><emph style="sc">Item</emph> ſuperficies conica coni recti ad ſuam baſem, proportionẽ habet ean-<lb/>dem, quam latus coni ad ſemidiametrum baſis coni eiuſdem, ex propoſ. </s> <s xml:id="echoid-s11036" xml:space="preserve">15. </s> <s xml:id="echoid-s11037" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s11038" xml:space="preserve">1. </s> <s xml:id="echoid-s11039" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s11040" xml:space="preserve">cylindro.</s> <s xml:id="echoid-s11041" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11042" xml:space="preserve">4. </s> <s xml:id="echoid-s11043" xml:space="preserve"><emph style="sc">Deniqve</emph> ſuperficies conuexa cylindrirecti, demptis baſibus, æqualis <lb/> <anchor type="note" xlink:label="note-265-04a" xlink:href="note-265-04"/> eſt circulo, cuius ſemidia meter eſt linea media proportio nalis inter latus cylin-<lb/>dri, & </s> <s xml:id="echoid-s11044" xml:space="preserve">diametrũ baſis cylin dri eiuſdem, ex propoſ. </s> <s xml:id="echoid-s11045" xml:space="preserve">13. </s> <s xml:id="echoid-s11046" xml:space="preserve">lib. </s> <s xml:id="echoid-s11047" xml:space="preserve">1. </s> <s xml:id="echoid-s11048" xml:space="preserve">Archimedis de ſphę-<lb/>ra & </s> <s xml:id="echoid-s11049" xml:space="preserve">cylindro.</s> <s xml:id="echoid-s11050" xml:space="preserve"/> </p> <div xml:id="echoid-div681" type="float" level="2" n="3"> <note position="right" xlink:label="note-265-04" xlink:href="note-265-04a" xml:space="preserve">Superficies cy <lb/>lindrica dem <lb/>ptis baſibus, <lb/>cui circulo ſit <lb/>æqualis.</note> </div> </div> <div xml:id="echoid-div683" type="section" level="1" n="240"> <head xml:id="echoid-head264" xml:space="preserve">FINIS LIBRI QVINTI.</head> <pb o="236" file="266" n="266"/> <figure> <image file="266-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/266-01"/> </figure> </div> <div xml:id="echoid-div684" type="section" level="1" n="241"> <head xml:id="echoid-head265" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ</head> <head xml:id="echoid-head266" xml:space="preserve">LIBER SEXTVS.</head> <figure> <image file="266-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/266-02"/> </figure> <p> <s xml:id="echoid-s11051" xml:space="preserve">In quo de Geodæſia, & </s> <s xml:id="echoid-s11052" xml:space="preserve">de figuris augendis, minuen-<lb/>diſque in data proportione: </s> <s xml:id="echoid-s11053" xml:space="preserve">Item de duarum me-<lb/>diarum proportionalium inter duas datas rectas <lb/>inuentione: </s> <s xml:id="echoid-s11054" xml:space="preserve">Ac denique deradicum extractione <lb/>agitur.</s> <s xml:id="echoid-s11055" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11056" xml:space="preserve">EXPEDITIS iis, quæ de magnitudinum dimenſio-<lb/>nibus initio propoſuimus, reſtat vt de rectilinearũ ſu-<lb/>perficierum etiam diuiſione agamus: </s> <s xml:id="echoid-s11057" xml:space="preserve">quæ Geometrię <lb/>practicæ pars proprio nomine Geodæſia vocatur. </s> <s xml:id="echoid-s11058" xml:space="preserve">Nã <lb/>δ{αί}ω idem ſignificat, quod partior, vel diuido. </s> <s xml:id="echoid-s11059" xml:space="preserve">Scio <lb/>pleroſque partem etiam illam, quæ magnitudines, & </s> <s xml:id="echoid-s11060" xml:space="preserve"><lb/>terram metitur, appellare Geodæſiam; </s> <s xml:id="echoid-s11061" xml:space="preserve">Sed hi, auctore Pediaſimo de <lb/>menſuratione & </s> <s xml:id="echoid-s11062" xml:space="preserve">partitione terræ, longè à veritate aberrant. </s> <s xml:id="echoid-s11063" xml:space="preserve">Nam, in-<lb/>quit, terræ menſuratio du{as} in partes diuiditur, Geometriam ſcilicet, & </s> <s xml:id="echoid-s11064" xml:space="preserve">Geo-<lb/>deaſiam. </s> <s xml:id="echoid-s11065" xml:space="preserve">Areænamq{ue} ſecundum artem menſuratio, & </s> <s xml:id="echoid-s11066" xml:space="preserve">terræ menſuratio <lb/> <anchor type="note" xlink:label="note-266-01a" xlink:href="note-266-01"/> eſt, & </s> <s xml:id="echoid-s11067" xml:space="preserve">merito Geometria vocatur. </s> <s xml:id="echoid-s11068" xml:space="preserve">vni{us} vero, & </s> <s xml:id="echoid-s11069" xml:space="preserve">eiuſdem areæ, ſeu loci diuiſio <lb/>inter diuerſ{as} perſon{as}, partitio quædam eſt terræ, & </s> <s xml:id="echoid-s11070" xml:space="preserve">iure optimo Geodaeſia ap-<lb/>pellatur. </s> <s xml:id="echoid-s11071" xml:space="preserve">HæcPediaſimus.</s> <s xml:id="echoid-s11072" xml:space="preserve"/> </p> <div xml:id="echoid-div684" type="float" level="2" n="1"> <note position="left" xlink:label="note-266-01" xlink:href="note-266-01a" xml:space="preserve">Geometria <lb/>& Geodęſia <lb/>quid.</note> </div> <p> <s xml:id="echoid-s11073" xml:space="preserve">EDIDIT quidem Federicus Commandimus anno 1570. </s> <s xml:id="echoid-s11074" xml:space="preserve">libellũ <lb/>deſuperficierum diuiſionibus Machometo cuidam Bagdedino Arabi <lb/>adſcriptum: </s> <s xml:id="echoid-s11075" xml:space="preserve">ipſeque eadem de re alium breuiorem, & </s> <s xml:id="echoid-s11076" xml:space="preserve">magis vniuerſa-<lb/>lem conſcripſit: </s> <s xml:id="echoid-s11077" xml:space="preserve">eſtque ſanèlibellus vterque acutiſſimus, & </s> <s xml:id="echoid-s11078" xml:space="preserve">eruditio- <pb o="237" file="267" n="267" rhead="LIBER SEXTVS."/> nerefertiſſimus. </s> <s xml:id="echoid-s11079" xml:space="preserve">Idem verò poſtea argumentum alia via aggreſſus eſt, <lb/>& </s> <s xml:id="echoid-s11080" xml:space="preserve">meo certè iudicio, faciliori & </s> <s xml:id="echoid-s11081" xml:space="preserve">magis generali, Simon Steuinius <lb/>Brugenſis: </s> <s xml:id="echoid-s11082" xml:space="preserve">ſed in qua aliquid deſiderari videatur, vt omnibus ſuper-<lb/>ficiebus rectilineis (quodipſe velle videtur) conuenire poſſit. </s> <s xml:id="echoid-s11083" xml:space="preserve">quod <lb/>facilè iudicabunt, qui illius problemata Geometrica attentè perlege-<lb/>rint. </s> <s xml:id="echoid-s11084" xml:space="preserve">Res enim propoſita nulla ratione confici poteſt, niſi prius duæ <lb/>propoſitiones demonſtrentur, quarum priorem ipſe ſine demonſtra-<lb/>tione aſſumit pro principio, poſterioris verò ne meminit quidem, <lb/>cum tamen admodum ſit neceſſaria, & </s> <s xml:id="echoid-s11085" xml:space="preserve">quam Machometus Bagdedi-<lb/>nus demonſtrauit paulò aliter, quam nos. </s> <s xml:id="echoid-s11086" xml:space="preserve">Has ergo duas propoſitiones <lb/>ad initium huius lib. </s> <s xml:id="echoid-s11087" xml:space="preserve">demonſtrabimus, & </s> <s xml:id="echoid-s11088" xml:space="preserve">poſteriorem quidem longè <lb/>generalius, quam à Machometo factum eſt. </s> <s xml:id="echoid-s11089" xml:space="preserve">quod beneuolo Lecto-<lb/>ri iudicandum relinquo. </s> <s xml:id="echoid-s11090" xml:space="preserve">Deindè<unsure/> ſuperficierum rectilinearum diui-<lb/>ſionem aggrediemur, inſiſtentes eiuſdem Steuinii veſtigiis, niſi quan-<lb/>do generalius rem oportebit demonſtrare. </s> <s xml:id="echoid-s11091" xml:space="preserve">Nihil autem deratione Ma-<lb/>chometi, & </s> <s xml:id="echoid-s11092" xml:space="preserve">Federici Commandini dicemus: </s> <s xml:id="echoid-s11093" xml:space="preserve">tum quia libellus ipſo-<lb/>rum in manibus omnium eſt, ac propterea eum, quicunque vo-<lb/>let, legere poterit: </s> <s xml:id="echoid-s11094" xml:space="preserve">tum quia propoſita aliqua figura multorum an-<lb/>gulorum, non ſine difficultate, ac labore eam ſtudioſus diuidet, ni-<lb/>ſi diuiſionis omnium præcedentium figurarum memor ſit, quod in <lb/>noſtra ratione non accidit: </s> <s xml:id="echoid-s11095" xml:space="preserve">tum denique quia illorum ratio ſolum fi-<lb/>guris ordinariis conuenit, quæ videlicet omnes angulos habent intror-<lb/>ſum, tot nimirũ, quotlatera figura ipſa continet, noſtra autem via figuras <lb/>etiamillas complectitur, quæ angulos habent partim introrſum, & </s> <s xml:id="echoid-s11096" xml:space="preserve">par-<lb/>tim extrorſum vergentes.</s> <s xml:id="echoid-s11097" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div686" type="section" level="1" n="242"> <head xml:id="echoid-head267" xml:space="preserve">THOREMA 1. PROPOSITIO 1.</head> <p> <s xml:id="echoid-s11098" xml:space="preserve">SI magnitudo in quotuis partes ſecetur vtcunque, & </s> <s xml:id="echoid-s11099" xml:space="preserve">alia quæpiam ma-<lb/>gnitudo in totidem partes ordine illis proportionales: </s> <s xml:id="echoid-s11100" xml:space="preserve">habebunt <lb/>quotlibet partes prioris magnitudinis ſimul ad reliquas omnes par-<lb/>tes ſimul eandem proportionem, quam totidem partes poſterioris <lb/>magnitudinis ſimul ad reliquas omnes partes ſimul. </s> <s xml:id="echoid-s11101" xml:space="preserve">Et ſi quælibet <lb/>pars prioris magnitudinis ſecetur in duas partes vtcunque, ſecetur <lb/>autem & </s> <s xml:id="echoid-s11102" xml:space="preserve">pars poſterioris magnitudinis illi parti reſpondens in alias <lb/>duas partes<unsure/> duabus illis proportionales: </s> <s xml:id="echoid-s11103" xml:space="preserve">erunt quoque ibidem to-<lb/>tæ magnitudines ſectæ proportionaliter.</s> <s xml:id="echoid-s11104" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11105" xml:space="preserve"><emph style="sc">Sit</emph> magnitudo A B, ſecta in quotuis partes vtcunque A C, C D, D E, EF, <lb/>F B: </s> <s xml:id="echoid-s11106" xml:space="preserve">& </s> <s xml:id="echoid-s11107" xml:space="preserve">alia magnitudo qualiſcunque G H, etiamſi diuerſi ſit generis, ſecta in <pb o="238" file="268" n="268" rhead="GEOMETR. PRACT."/> totidem partes GI, IK, KL, LM, MH, illis ordine proportionales. </s> <s xml:id="echoid-s11108" xml:space="preserve">Dico ita eſ-<lb/>ſe, verbi gratia, duas partes AC, CD, ſimul ad reliquastres DE, EF, FB, ſimul, vt <lb/>ſunt duæ GI, IK, ſimulad reliquas tres KL, LM, MH, ſimul, &</s> <s xml:id="echoid-s11109" xml:space="preserve">c. </s> <s xml:id="echoid-s11110" xml:space="preserve">Quoniam enim <lb/>eſt, vt AC, ad CD, ita GI, ad IK, erit componendo etiam, vt AD, ad CD, ita GK, <lb/>ad IK: </s> <s xml:id="echoid-s11111" xml:space="preserve">Vtautem CD, ad DE, ita eſt IK, ad KL. </s> <s xml:id="echoid-s11112" xml:space="preserve">Igitur ex æqualitate erit, vt AD, <lb/>ad DE, ita GK, ad KL.</s> <s xml:id="echoid-s11113" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11114" xml:space="preserve"><emph style="sc">Rvrsvs</emph> quia conuertendo eſt, vt BF, ad F E, ita HM, ad ML; </s> <s xml:id="echoid-s11115" xml:space="preserve">erit quo-<lb/>que componendo, vt BE, ad FE, ita HL, ad ML: </s> <s xml:id="echoid-s11116" xml:space="preserve">Vtautem FE, ad ED, ita eſt <lb/>ML; </s> <s xml:id="echoid-s11117" xml:space="preserve">ad LK. </s> <s xml:id="echoid-s11118" xml:space="preserve">Igitur ex æqualitate erit, vt B E, ad ED, ita HL, ad L K; </s> <s xml:id="echoid-s11119" xml:space="preserve">& </s> <s xml:id="echoid-s11120" xml:space="preserve">com-<lb/> <anchor type="figure" xlink:label="fig-268-01a" xlink:href="fig-268-01"/> ponendo, vt B D, ad E D, ita H K, ad L K; </s> <s xml:id="echoid-s11121" xml:space="preserve">& </s> <s xml:id="echoid-s11122" xml:space="preserve">conuertendo, vt D E, ad D B, ita <lb/>KL, ad KH. </s> <s xml:id="echoid-s11123" xml:space="preserve">Itaque cumoſtenſum ſit, eſſe vt AD, ad DE, ita vt GK, ad KL, & </s> <s xml:id="echoid-s11124" xml:space="preserve">vt <lb/>D E, ad D B, ita K L, ad K H; </s> <s xml:id="echoid-s11125" xml:space="preserve">erit ex æqualitate, vt A D, ad D B, ita G K, <lb/>ad K H.</s> <s xml:id="echoid-s11126" xml:space="preserve"/> </p> <div xml:id="echoid-div686" type="float" level="2" n="1"> <figure xlink:label="fig-268-01" xlink:href="fig-268-01a"> <image file="268-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/268-01"/> </figure> </div> <p> <s xml:id="echoid-s11127" xml:space="preserve"><emph style="sc">Non</emph> aliter oſtendemus eſſe, vt AC, ad CB, ita GI, ad IH. </s> <s xml:id="echoid-s11128" xml:space="preserve">Nam rurſus con-<lb/>uertendo, componendo, & </s> <s xml:id="echoid-s11129" xml:space="preserve">ex æqualitate erit vt B C, ad D C, ita HI, ad K I; </s> <s xml:id="echoid-s11130" xml:space="preserve">& </s> <s xml:id="echoid-s11131" xml:space="preserve"><lb/>conuertendo, vt CD, ad CB, ita IK, ad IH. </s> <s xml:id="echoid-s11132" xml:space="preserve">Cum ergo ſit, vt AC, ad CD, ita GI, <lb/>ad IK, & </s> <s xml:id="echoid-s11133" xml:space="preserve">vt CD, ad CB, ita IK, ad I H; </s> <s xml:id="echoid-s11134" xml:space="preserve">erit ex æqualitate, vt AC, ad CB, ita GI, <lb/>ad I H.</s> <s xml:id="echoid-s11135" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11136" xml:space="preserve"><emph style="sc">Pari</emph> ratione erit, vt AF, ad FB, ita GM, ad MH. </s> <s xml:id="echoid-s11137" xml:space="preserve">Erit namquerurſus com-<lb/>ponendo, & </s> <s xml:id="echoid-s11138" xml:space="preserve">ex æqualitate, vt AF, ad EF, ita GM, ad LM. </s> <s xml:id="echoid-s11139" xml:space="preserve">Cum ergo ſit quo-<lb/>que, vt EF, ad FB, ita LM, ad MH: </s> <s xml:id="echoid-s11140" xml:space="preserve">erit ex æqualitate, vt AF, ad FB, ita GM, ad <lb/>MH; </s> <s xml:id="echoid-s11141" xml:space="preserve">& </s> <s xml:id="echoid-s11142" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s11143" xml:space="preserve">Conſtatigitur primum.</s> <s xml:id="echoid-s11144" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11145" xml:space="preserve"><emph style="sc">Deinde</emph> pars v. </s> <s xml:id="echoid-s11146" xml:space="preserve">g. </s> <s xml:id="echoid-s11147" xml:space="preserve">tertia DE, ſecta ſit vtcunque in partes duas D N, N E; <lb/></s> <s xml:id="echoid-s11148" xml:space="preserve">& </s> <s xml:id="echoid-s11149" xml:space="preserve">tertia quo que pars KL, in duas KO, OL, illis proportionales. </s> <s xml:id="echoid-s11150" xml:space="preserve">Dico eſſe quo-<lb/>que vt AN, ad NB, ita GO, ad OH. </s> <s xml:id="echoid-s11151" xml:space="preserve">Erit enim conuertendo, vt EN, ad N D, ita <lb/>L O, ad O K: </s> <s xml:id="echoid-s11152" xml:space="preserve">& </s> <s xml:id="echoid-s11153" xml:space="preserve">componendo, vt ED, ad DN, ita LK, ad K O. </s> <s xml:id="echoid-s11154" xml:space="preserve">Quare cum ſit, <lb/>vt CD, ad DE, ita IK, ad KL, & </s> <s xml:id="echoid-s11155" xml:space="preserve">vt DE, ad DN, ita KL, ad KO: </s> <s xml:id="echoid-s11156" xml:space="preserve">erit ex æqualita-<lb/>te, vt CD, ad DN, ita IK, ad KO, atque ita partes AC, CD, DN, partibus GI, IK, <lb/>KO, proportionales ſunt.</s> <s xml:id="echoid-s11157" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11158" xml:space="preserve"><emph style="sc">Rvrsvs</emph> quia eſt conuertendo, vt FE, ad E D, ita ML, ad LK; </s> <s xml:id="echoid-s11159" xml:space="preserve">& </s> <s xml:id="echoid-s11160" xml:space="preserve">compo-<lb/>nendo, vt DE, ad NE, ita KL, ad OL; </s> <s xml:id="echoid-s11161" xml:space="preserve">erit exæqualitate, vt FE, ad E N, ita ML, <lb/>ad LO; </s> <s xml:id="echoid-s11162" xml:space="preserve">& </s> <s xml:id="echoid-s11163" xml:space="preserve">conuertendo, vt NE, ad EF, ita OL, ad LM; </s> <s xml:id="echoid-s11164" xml:space="preserve">ac proinde omnes par-<lb/>tes AC, CD, DN, NE, EF, FB, omnibus partibus GI, IK, KO, OL, LM, <lb/>M H, proportionales ſunt. </s> <s xml:id="echoid-s11165" xml:space="preserve">Igitur vt in prima parte demonſtra-<lb/>tumeſt, erit vt AN, ad NB, ita GO, ad OH. </s> <s xml:id="echoid-s11166" xml:space="preserve">Conſtat <lb/>ergo etiam ſecundum.</s> <s xml:id="echoid-s11167" xml:space="preserve"/> </p> <pb o="239" file="269" n="269" rhead="LIBER SEXTVS."/> </div> <div xml:id="echoid-div688" type="section" level="1" n="243"> <head xml:id="echoid-head268" xml:space="preserve">PROBLEMA 1. PROPOSITIO 2.</head> <p> <s xml:id="echoid-s11168" xml:space="preserve">DATO rectilineo ſuper datam rectam inter alias duas rectas interce-<lb/>ptam, conſtituere quadrilaterum æquale, cuius latus oppoſitum in-<lb/>ter duas eaſdem rectas, interceptum datæ rectæ ſit parallelum. </s> <s xml:id="echoid-s11169" xml:space="preserve">Et <lb/>datis duobus rectilineis inæqualibus quibuſcunque, ex maiore per <lb/>lineam vni lateri parallelam detrahere rectilineum minori æquale, <lb/>quando id fieri poteſt, quod ex ipſa problematis ſolutione cogno-<lb/>ſcetur.</s> <s xml:id="echoid-s11170" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11171" xml:space="preserve"><emph style="sc">Sit</emph> rectilineum datum A, & </s> <s xml:id="echoid-s11172" xml:space="preserve">recta data B C, inter duas rectas B D, C E, <lb/>intercepta: </s> <s xml:id="echoid-s11173" xml:space="preserve">oporteatque primum conſtitue-<lb/> <anchor type="figure" xlink:label="fig-269-01a" xlink:href="fig-269-01"/> rerectilineo A, æquale quadrilaterum ſuper da-<lb/>tam rectam B C, cuius latus oppoſitum inter <lb/>eaſdem rectas BD, C E, interceptum datærectæ <lb/>BC, ſit parallelum. </s> <s xml:id="echoid-s11174" xml:space="preserve">Et ſi quidem duæ rectæ BD, <lb/>C E, ſint parallelæ (quodtum demum eueniet, <lb/>cum duo anguli B, C, æquales ſunt, duobus <lb/>rectis) efficietur problema, <anchor type="note" xlink:href="" symbol="a"/> ſi ſuper rectam B C, conſtituetur parallelo- <anchor type="note" xlink:label="note-269-01a" xlink:href="note-269-01"/> grammum B E, ſiue in angulo B C E, ſiue in angulo C B D, rectilineo A, æ-<lb/>quale.</s> <s xml:id="echoid-s11175" xml:space="preserve"/> </p> <div xml:id="echoid-div688" type="float" level="2" n="1"> <figure xlink:label="fig-269-01" xlink:href="fig-269-01a"> <image file="269-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/269-01"/> </figure> <note symbol="a" position="right" xlink:label="note-269-01" xlink:href="note-269-01a" xml:space="preserve">45. primi.</note> </div> <p> <s xml:id="echoid-s11176" xml:space="preserve">2. </s> <s xml:id="echoid-s11177" xml:space="preserve"><emph style="sc">Qvando</emph> anguli B, C, rectiſunt, facilius problema effi ciemus hac ra-<lb/>tione. </s> <s xml:id="echoid-s11178" xml:space="preserve">Rectilineo dato A, conſtituatur per ea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s11179" xml:space="preserve">14. </s> <s xml:id="echoid-s11180" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s11181" xml:space="preserve">2. </s> <s xml:id="echoid-s11182" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11183" xml:space="preserve">vel potius per ea, quæ Num. </s> <s xml:id="echoid-s11184" xml:space="preserve">4. </s> <s xml:id="echoid-s11185" xml:space="preserve">cap. </s> <s xml:id="echoid-s11186" xml:space="preserve">4. </s> <s xml:id="echoid-s11187" xml:space="preserve">lib. </s> <s xml:id="echoid-s11188" xml:space="preserve">4. </s> <s xml:id="echoid-s11189" xml:space="preserve">huius Geometriæ pra-<lb/>cticæ ſcripſimus, quadratum F G H, æquale: </s> <s xml:id="echoid-s11190" xml:space="preserve">reſoluendo videlicet rectili-<lb/>n@um in triangula, vel trapezia, & </s> <s xml:id="echoid-s11191" xml:space="preserve">cuilibet triangulo, vel trapezio æquale qua-<lb/> <anchor type="figure" xlink:label="fig-269-02a" xlink:href="fig-269-02"/> dratum conſtituendo, ac tandem omnia illa quadrata ad vnum redigendo, vt <lb/>locis citatis fusè explicauimus. </s> <s xml:id="echoid-s11192" xml:space="preserve">Deinde duabus rectis B C, F G, inueniatur <pb o="240" file="270" n="270" rhead="GEOMETR. PRACT."/> tertia proportionalis B D, acper D, ipſi B C, parallela agatur D E. </s> <s xml:id="echoid-s11193" xml:space="preserve">Rectan-<lb/>gulumenim B E, rectilineo dato A, æquale erit; </s> <s xml:id="echoid-s11194" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cum quadrato F G H, æ- <anchor type="note" xlink:label="note-270-01a" xlink:href="note-270-01"/> quale ſit; </s> <s xml:id="echoid-s11195" xml:space="preserve">propterea quod tres rectæ B C, F G, B D, continuè proportiona-<lb/>les ſunt.</s> <s xml:id="echoid-s11196" xml:space="preserve"/> </p> <div xml:id="echoid-div689" type="float" level="2" n="2"> <figure xlink:label="fig-269-02" xlink:href="fig-269-02a"> <image file="269-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/269-02"/> </figure> <note symbol="a" position="left" xlink:label="note-270-01" xlink:href="note-270-01a" xml:space="preserve">17. ſexti.</note> </div> <p> <s xml:id="echoid-s11197" xml:space="preserve"><emph style="sc">Immo</emph> eodem hoc artificio vti poterimus, quando parallelæ BI, CK, non <lb/>faciunt angulosrectos ad B, & </s> <s xml:id="echoid-s11198" xml:space="preserve">C. </s> <s xml:id="echoid-s11199" xml:space="preserve">Nam conſtituto rectangulo CD, æquali qua-<lb/>drato FH, id eſt, rectilineo A, vt dictum eſt: </s> <s xml:id="echoid-s11200" xml:space="preserve">ſi producaturlatus DE, donec ſecet <lb/>rectas B I, C K, in I, K, <anchor type="note" xlink:href="" symbol="b"/> erit parallelogrammum BK, rectangulo CD, æquale, <anchor type="note" xlink:label="note-270-02a" xlink:href="note-270-02"/> hoceſt, rectilineo A.</s> <s xml:id="echoid-s11201" xml:space="preserve"/> </p> <div xml:id="echoid-div690" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-270-02" xlink:href="note-270-02a" xml:space="preserve">35. primi.</note> </div> <p> <s xml:id="echoid-s11202" xml:space="preserve">3. </s> <s xml:id="echoid-s11203" xml:space="preserve"><emph style="sc">Si</emph> verò duæ rectæ BD, CE, non ſint parallelæ, conueniant ad partes D, <lb/>E, in F, (quod tumfiet, cum duo anguli DBC, ECB, minores ſunt duobus re-<lb/>ctis) ſitque primum propoſitum ſuper B C, verſus F, conſtituere trapezium <lb/>rectilineo A, æquale, habenslatus rectæ B C<unsure/>, oppoſitum eidem B C, paralle-<lb/>lum. </s> <s xml:id="echoid-s11204" xml:space="preserve">quodvtfieri poſsit, neceſſe eſt, rectilineum A, minus eſſe triangulo BCF, <lb/>vt patet.</s> <s xml:id="echoid-s11205" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11206" xml:space="preserve"><emph style="sc">Rectilineo</emph> ergo A, conſtruatur perea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s11207" xml:space="preserve">14. </s> <s xml:id="echoid-s11208" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s11209" xml:space="preserve"> <anchor type="figure" xlink:label="fig-270-01a" xlink:href="fig-270-01"/> 2. </s> <s xml:id="echoid-s11210" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11211" xml:space="preserve">vel potius perea, quæ Num. </s> <s xml:id="echoid-s11212" xml:space="preserve">4. </s> <s xml:id="echoid-s11213" xml:space="preserve">cap. </s> <s xml:id="echoid-s11214" xml:space="preserve">4. </s> <s xml:id="echoid-s11215" xml:space="preserve">lib. </s> <s xml:id="echoid-s11216" xml:space="preserve">4. </s> <s xml:id="echoid-s11217" xml:space="preserve">huius, vtproximè dixi-<lb/> <anchor type="note" xlink:label="note-270-03a" xlink:href="note-270-03"/> mus, æquale quadratum G, cuius latus H I. </s> <s xml:id="echoid-s11218" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Item triangulo B C F, æquale quadratum K, cuius latus L M. </s> <s xml:id="echoid-s11219" xml:space="preserve">Deinde duabus rectis L M, H I: </s> <s xml:id="echoid-s11220" xml:space="preserve">reperiatur <lb/>tertia proportionalis L N. </s> <s xml:id="echoid-s11221" xml:space="preserve">Inuenta autem recta O, media proportionali in-<lb/>ter L M, & </s> <s xml:id="echoid-s11222" xml:space="preserve">M N; </s> <s xml:id="echoid-s11223" xml:space="preserve">fiat vt L M, ad O, ita B F, ad F D; </s> <s xml:id="echoid-s11224" xml:space="preserve">actandem per D, ipſi <lb/>B C, parallela agatur D E. </s> <s xml:id="echoid-s11225" xml:space="preserve">Dico Trapezium B E, rectilineo A, æquale eſſe <pb o="241" file="271" n="271" rhead="LIBER SEXTVS."/> <anchor type="note" xlink:href="" symbol="a"/> Quoniam enim triangulum B C F, ad triangulum D E F, duplicatam pro- <anchor type="note" xlink:label="note-271-01a" xlink:href="note-271-01"/> portionem habet lateris B F, ad latus D F, hoc eſt, rectæ L M, ad rectam O: <lb/></s> <s xml:id="echoid-s11226" xml:space="preserve">Habet autem & </s> <s xml:id="echoid-s11227" xml:space="preserve">L M, ad M N, duplicatam proportionem eius, quam habet <lb/>L M, ad O, quod L M, O, M N, ſint continuè proportionales. </s> <s xml:id="echoid-s11228" xml:space="preserve">Igitur erit vt <lb/>triangulum B C F, ad triangulum D E F, ita L M, ad M N; </s> <s xml:id="echoid-s11229" xml:space="preserve">Et per conuerſio-<lb/>nem rationis, vt triangulum B C F, ad Trapezium B E, ita L M, ad L N. </s> <s xml:id="echoid-s11230" xml:space="preserve"><lb/> <anchor type="note" xlink:href="" symbol="b"/> Cum ergo ſit, vt LM, ad LN, ita quadratum K, ad quadratum G, quod LM, <anchor type="note" xlink:label="note-271-02a" xlink:href="note-271-02"/> H I, L N, continuè ſint proportionales, erit quo que vt triangulum B C F, ad <lb/>trapezium B E, ita quadratum K, ad quadratum G, hoc eſt, ita triangulum <lb/>BCF, quod ipſi K, æquale eſt, ad rectilineum A, ipſi G, æquale. </s> <s xml:id="echoid-s11231" xml:space="preserve">Quo circa cum <lb/>triangulum B C F, ad trapezium B E, & </s> <s xml:id="echoid-s11232" xml:space="preserve">ad rectilineum A, eandem habeat pro-<lb/>portionem; </s> <s xml:id="echoid-s11233" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> æqualia erunt trapezium BE, & </s> <s xml:id="echoid-s11234" xml:space="preserve">rectilineum A, quod eſt propo- <anchor type="note" xlink:label="note-271-03a" xlink:href="note-271-03"/> ſitum.</s> <s xml:id="echoid-s11235" xml:space="preserve"/> </p> <div xml:id="echoid-div691" type="float" level="2" n="4"> <figure xlink:label="fig-270-01" xlink:href="fig-270-01a"> <image file="270-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/270-01"/> </figure> <note symbol="c" position="left" xlink:label="note-270-03" xlink:href="note-270-03a" xml:space="preserve">14. ſecundi.</note> <note symbol="a" position="right" xlink:label="note-271-01" xlink:href="note-271-01a" xml:space="preserve">19. ſexti.</note> <note symbol="b" position="right" xlink:label="note-271-02" xlink:href="note-271-02a" xml:space="preserve">coroll. 2@. <lb/>ſexti.</note> <note symbol="c" position="right" xlink:label="note-271-03" xlink:href="note-271-03a" xml:space="preserve">9. quinti.</note> </div> <p> <s xml:id="echoid-s11236" xml:space="preserve">4. </s> <s xml:id="echoid-s11237" xml:space="preserve"><emph style="sc">Sit</emph> deinde ſuper B C, verſus R, S, vbianguli R B C, SCB, duobus re-<lb/>ctis ſunt maiores, non autem verſus punctum concurſus F, conſtruendum trape-<lb/>zium rectilineo A, cuiuſcunque magnitudinis ſit, æquale, habens latus oppo-<lb/>ſitum rectæ B C, parallelum. </s> <s xml:id="echoid-s11238" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Fiat rurſus triangulo B C F, æquale quadra- <anchor type="note" xlink:label="note-271-04a" xlink:href="note-271-04"/> tum K, cuius latus L M; </s> <s xml:id="echoid-s11239" xml:space="preserve">& </s> <s xml:id="echoid-s11240" xml:space="preserve">rectilineo A, aliud quadratum G, æquale, cuius <lb/>latus HI, per ea, quæ ad propoſitionem 14. </s> <s xml:id="echoid-s11241" xml:space="preserve">lib. </s> <s xml:id="echoid-s11242" xml:space="preserve">2. </s> <s xml:id="echoid-s11243" xml:space="preserve">Euclìd. </s> <s xml:id="echoid-s11244" xml:space="preserve">vel potius per ea, quæ <lb/>Num. </s> <s xml:id="echoid-s11245" xml:space="preserve">4. </s> <s xml:id="echoid-s11246" xml:space="preserve">cap. </s> <s xml:id="echoid-s11247" xml:space="preserve">4. </s> <s xml:id="echoid-s11248" xml:space="preserve">lib. </s> <s xml:id="echoid-s11249" xml:space="preserve">4. </s> <s xml:id="echoid-s11250" xml:space="preserve">huius docuimus. </s> <s xml:id="echoid-s11251" xml:space="preserve">Dein de lateribus L M, HI, inueniatur <lb/>tertia proportionalis MP, quæ ipſi LM, in continuum & </s> <s xml:id="echoid-s11252" xml:space="preserve">directum ſit poſi-<lb/>ta: </s> <s xml:id="echoid-s11253" xml:space="preserve">at que inter totam L P, & </s> <s xml:id="echoid-s11254" xml:space="preserve">L M, reperta ſit media proportionalis Q: </s> <s xml:id="echoid-s11255" xml:space="preserve">ac <lb/>poſtremo, vt Q, ad L P, ita fiat FB, ad F R: </s> <s xml:id="echoid-s11256" xml:space="preserve">ipſique BC, parallela agatur R S. <lb/></s> <s xml:id="echoid-s11257" xml:space="preserve">Dico trapezium B S, rectilineo A, eſſe æquale. </s> <s xml:id="echoid-s11258" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Quoniam enim triangulum <anchor type="note" xlink:label="note-271-05a" xlink:href="note-271-05"/> BCF, ad triangulum R S F, proportionem habet duplicatam lateris F B, ad <lb/>latus F R, hoc eſt, proportionis Q, ad L P; </s> <s xml:id="echoid-s11259" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s11260" xml:space="preserve">proportio L M, ad <lb/>L P, duplicata proportionis L M, ad Q, vel Q, ad L P, quod tres rectæ L M, <lb/>Q, L P, ſint continuè proportionales. </s> <s xml:id="echoid-s11261" xml:space="preserve">Igitur erit vt triangulum B C F, ad <lb/>triangulum R S F, ita L M, ad L P; </s> <s xml:id="echoid-s11262" xml:space="preserve">ideo que etiam per diuiſionem rationis con-<lb/>trariam in ſcholio propoſ. </s> <s xml:id="echoid-s11263" xml:space="preserve">17. </s> <s xml:id="echoid-s11264" xml:space="preserve">libr. </s> <s xml:id="echoid-s11265" xml:space="preserve">5. </s> <s xml:id="echoid-s11266" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11267" xml:space="preserve">à nobis demonſtratam, vt trian-<lb/>gulum BCF, ad trapezium BS, ita L M, ad M P. </s> <s xml:id="echoid-s11268" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Vtautem L M, ad M P, ita <anchor type="note" xlink:label="note-271-06a" xlink:href="note-271-06"/> eſt quadratum K, ad quadratum G, quod tres L M, H I, M P, ſint continuè <lb/>proportionales. </s> <s xml:id="echoid-s11269" xml:space="preserve">Igitur erit quoque, vt triangulum B C F, ad trapezium B S, <lb/>ita quadratum K, ad quadratum G. </s> <s xml:id="echoid-s11270" xml:space="preserve">Cum ergo triangulo B C F, conſtructum <lb/>ſit æquale quadratum K: </s> <s xml:id="echoid-s11271" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit quoque trapezium B S, quadrato G, æquale, <anchor type="note" xlink:label="note-271-07a" xlink:href="note-271-07"/> hoc eſt, rectilineo A, cui quadratum G, conſtructum eſt æquale. </s> <s xml:id="echoid-s11272" xml:space="preserve">quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s11273" xml:space="preserve"/> </p> <div xml:id="echoid-div692" type="float" level="2" n="5"> <note symbol="d" position="right" xlink:label="note-271-04" xlink:href="note-271-04a" xml:space="preserve">14. ſecundi.</note> <note symbol="e" position="right" xlink:label="note-271-05" xlink:href="note-271-05a" xml:space="preserve">19. ſexti.</note> <note symbol="f" position="right" xlink:label="note-271-06" xlink:href="note-271-06a" xml:space="preserve">coroll. 2@. <lb/>ſexti.</note> <note symbol="g" position="right" xlink:label="note-271-07" xlink:href="note-271-07a" xml:space="preserve">14. quinti.</note> </div> <p> <s xml:id="echoid-s11274" xml:space="preserve"><emph style="sc">Qvod</emph> ſi quando duæ rectæ B F, C F, in tam remoto puncto concurrant, <lb/>vt vix haberi poſsit, (quod quidem tunc accidet, cum ipſæ rectæ ferè pa-<lb/>rallelæ ſunt) abſoluemus problema, etiamſi punctum concurſus F, non habea-<lb/>mus, huncin modum. </s> <s xml:id="echoid-s11275" xml:space="preserve">Sumpto vtcunque puncto T, in altera earum, nimirum <lb/>in C F, agatur T V, alteri BF, parallela; </s> <s xml:id="echoid-s11276" xml:space="preserve">& </s> <s xml:id="echoid-s11277" xml:space="preserve">duabus B C, C V, inueniatur tertia <lb/>proportionalis X. </s> <s xml:id="echoid-s11278" xml:space="preserve">Conſtructo deinde ex ſcholio propoſ. </s> <s xml:id="echoid-s11279" xml:space="preserve">14. </s> <s xml:id="echoid-s11280" xml:space="preserve">lib. </s> <s xml:id="echoid-s11281" xml:space="preserve">2. </s> <s xml:id="echoid-s11282" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11283" xml:space="preserve">vel <lb/>potius, vt Num. </s> <s xml:id="echoid-s11284" xml:space="preserve">4. </s> <s xml:id="echoid-s11285" xml:space="preserve">cap. </s> <s xml:id="echoid-s11286" xml:space="preserve">4. </s> <s xml:id="echoid-s11287" xml:space="preserve">libr. </s> <s xml:id="echoid-s11288" xml:space="preserve">4. </s> <s xml:id="echoid-s11289" xml:space="preserve">huius docuimus, quadrato G, æquali rectili-<lb/>neo A, inueniatur tribus BC, X, H I, quarta proportionalis IY, agatur que Y Z, <lb/>lateribus qua drati parallela. </s> <s xml:id="echoid-s11290" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Et quoniam eſt, vt triangulum B C F, (ſi per- <anchor type="note" xlink:label="note-271-08a" xlink:href="note-271-08"/> ficeretur) ad triangulum V C T, ita recta B C, ad rectam X, hoc eſt, ita H I, <pb o="242" file="272" n="272" rhead="GEOMETR. PRACT."/> ad I Y, <anchor type="note" xlink:href="" symbol="a"/> hoc eſt, ita quadratum G, ad rectangulum I Z: </s> <s xml:id="echoid-s11291" xml:space="preserve">Eſt autem triangulum <anchor type="note" xlink:label="note-272-01a" xlink:href="note-272-01"/> B C F, maius quadrato G, ſiue rectilineo A: </s> <s xml:id="echoid-s11292" xml:space="preserve">(quando enim ad partes angulo-<lb/>rum, qui duo bus rectis minores ſunt, conſtruendum eſt trapezium dato recti-<lb/> <anchor type="figure" xlink:label="fig-272-01a" xlink:href="fig-272-01"/> lineo æquale, debet eſſe triangulum maius rectilineo) <anchor type="note" xlink:href="" symbol="b"/> erit quo que triang@- <anchor type="note" xlink:label="note-272-02a" xlink:href="note-272-02"/> lum V C T, maius rectangulo I Z. </s> <s xml:id="echoid-s11293" xml:space="preserve">Igitur vt Num. </s> <s xml:id="echoid-s11294" xml:space="preserve">3. </s> <s xml:id="echoid-s11295" xml:space="preserve">traditum eſt, conſtruatur <lb/>trapezium V b, rectangulo I Z, æquale: </s> <s xml:id="echoid-s11296" xml:space="preserve">& </s> <s xml:id="echoid-s11297" xml:space="preserve">tribus rectis C V, V a, C B, inuen-<lb/>ta quarta proportionali B D, <anchor type="note" xlink:href="" symbol="c"/> (tranſibit autem recta ducta C a, per D, ſi <anchor type="note" xlink:label="note-272-03a" xlink:href="note-272-03"/> quarta B D, ritè eſt inuenta, <anchor type="note" xlink:href="" symbol="d"/> ita vt viciſsim recta C a, ſi ex quiſitè ducatur, ex- hibeat quartam proportionalem quæſitam B D,) demittatur D E, ipſi B C, pa-<lb/> <anchor type="note" xlink:label="note-272-04a" xlink:href="note-272-04"/> rallela. </s> <s xml:id="echoid-s11298" xml:space="preserve">Dico trapezium B E, dato rectilineo A, æquale eſſe. </s> <s xml:id="echoid-s11299" xml:space="preserve">Quoniam enim <lb/>trapezium B E, trapezio V b, ſimile eſt, per ea, quæ ad propoſ. </s> <s xml:id="echoid-s11300" xml:space="preserve">18. </s> <s xml:id="echoid-s11301" xml:space="preserve">libr. </s> <s xml:id="echoid-s11302" xml:space="preserve">6. </s> <s xml:id="echoid-s11303" xml:space="preserve">Eu-<lb/>clid. </s> <s xml:id="echoid-s11304" xml:space="preserve">demonſtrauimus; </s> <s xml:id="echoid-s11305" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit trapezium B E, ad trapezium V b, vt recta B C, ad <anchor type="note" xlink:label="note-272-05a" xlink:href="note-272-05"/> rectam X, hoc eſt, vt recta HI, ad IY, <anchor type="note" xlink:href="" symbol="f"/> hoc eſt, vt quadratum G, ad rectangu- lum I Z. </s> <s xml:id="echoid-s11306" xml:space="preserve">Cum ergo trapezium V b, rectangulo I Z, æquale ſit; </s> <s xml:id="echoid-s11307" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit quo- <anchor type="note" xlink:label="note-272-06a" xlink:href="note-272-06"/> que trapezium BE, quadrato G, hoc eſt, rectilineo A, æquale. </s> <s xml:id="echoid-s11308" xml:space="preserve">quod eſt propo-<lb/> <anchor type="note" xlink:label="note-272-07a" xlink:href="note-272-07"/> ſitum.</s> <s xml:id="echoid-s11309" xml:space="preserve"/> </p> <div xml:id="echoid-div693" type="float" level="2" n="6"> <note symbol="h" position="right" xlink:label="note-271-08" xlink:href="note-271-08a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> <note symbol="a" position="left" xlink:label="note-272-01" xlink:href="note-272-01a" xml:space="preserve">1. ſexti.</note> <figure xlink:label="fig-272-01" xlink:href="fig-272-01a"> <image file="272-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/272-01"/> </figure> <note symbol="b" position="left" xlink:label="note-272-02" xlink:href="note-272-02a" xml:space="preserve">14. quinti.</note> <note symbol="c" position="left" xlink:label="note-272-03" xlink:href="note-272-03a" xml:space="preserve">ſ@hol. 4. <lb/>ſexti.</note> <note symbol="d" position="left" xlink:label="note-272-04" xlink:href="note-272-04a" xml:space="preserve">4. ſexti.</note> <note symbol="e" position="left" xlink:label="note-272-05" xlink:href="note-272-05a" xml:space="preserve">coroll. 20. <lb/>ſexti.</note> <note symbol="f" position="left" xlink:label="note-272-06" xlink:href="note-272-06a" xml:space="preserve">1. ſexti.</note> <note symbol="g" position="left" xlink:label="note-272-07" xlink:href="note-272-07a" xml:space="preserve">14. quinti.</note> </div> <p> <s xml:id="echoid-s11310" xml:space="preserve">6. </s> <s xml:id="echoid-s11311" xml:space="preserve"><emph style="sc">Non</emph> aliter ex alia parte angulorum R B C, S C B, qui duo bus rectis ſunt <lb/>maiores, etiamſi punctum concurſus F, non habeatur, trapezium conſtituemus <lb/>rectilineo T, æquale, cuiuſcunque magnitudinis illud ponatur. </s> <s xml:id="echoid-s11312" xml:space="preserve">Neque enim in <lb/>hoc caſu neceſſe eſt, ipſum eſſe minus triangulo B C F, ſi perficeretur. </s> <s xml:id="echoid-s11313" xml:space="preserve">Ducta <lb/>namque ex quolibet puncto T, rectæ C F, ipſi BF, parallella T V, eaque pro-<lb/>ducta, inueniatur duabus rectis B C, C V, tertia proportionalis X. </s> <s xml:id="echoid-s11314" xml:space="preserve">Conſtructo <pb o="243" file="273" n="273" rhead="LIBER SEXTVS."/> deinde quadrato G, quod rectilineo A, ſit æquale, reperiatur tribus rectis BC, X, <lb/>& </s> <s xml:id="echoid-s11315" xml:space="preserve">HI, quarta proportionalis IY, agaturque Y Z, lateribus quadrati parallela, ita <lb/>vt rurſus ſit triangulum BCF, ad triangulum V C T, ſicut quadratum G, ad re-<lb/>ctangulum I Z. </s> <s xml:id="echoid-s11316" xml:space="preserve">Poſt hæc, vt Num. </s> <s xml:id="echoid-s11317" xml:space="preserve">4. </s> <s xml:id="echoid-s11318" xml:space="preserve">præcepimus, rectangulo IZ, conſtruatur <lb/>trapezium æquale V e, & </s> <s xml:id="echoid-s11319" xml:space="preserve">tribus rectis C V, V d, CB, inuenta quarta proportionali <lb/>B R, <anchor type="note" xlink:href="" symbol="a"/> (tranſibit autem recta C d, ducta per R: </s> <s xml:id="echoid-s11320" xml:space="preserve">ſi quarta BR, rectè inuenta eſt:</s> <s xml:id="echoid-s11321" xml:space="preserve"> <anchor type="note" xlink:label="note-273-01a" xlink:href="note-273-01"/> <anchor type="note" xlink:href="" symbol="b"/> ita vt viciſsim recta C d, ſi accuratè ducatur, abſcindat quartam proportiona- lem quæſitam BR,) demittatur R S, ipſi BC, parallela. </s> <s xml:id="echoid-s11322" xml:space="preserve">Dico trapezium B S, recti-<lb/> <anchor type="note" xlink:label="note-273-02a" xlink:href="note-273-02"/> lineo A, æquale eſſe. </s> <s xml:id="echoid-s11323" xml:space="preserve">Quoniam enim trapezium B S, trapezio V e, ſimile eſt, per <lb/>ea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s11324" xml:space="preserve">18. </s> <s xml:id="echoid-s11325" xml:space="preserve">lib. </s> <s xml:id="echoid-s11326" xml:space="preserve">6. </s> <s xml:id="echoid-s11327" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11328" xml:space="preserve">monſtrata ſunt à nobis; </s> <s xml:id="echoid-s11329" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit trape- <anchor type="note" xlink:label="note-273-03a" xlink:href="note-273-03"/> zium B S, ad trapezium V e, vt recta BC, ad rectam X, hoc eſt, vt recta H I, ad <lb/>rectam IY, <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, vt quadratum G, ad rectangulum IZ. </s> <s xml:id="echoid-s11330" xml:space="preserve">Cum ergo trapezium <anchor type="note" xlink:label="note-273-04a" xlink:href="note-273-04"/> V e, rectangulo I Z, æqualeſit; </s> <s xml:id="echoid-s11331" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit quo que trapezium B S, quadrato G, hoc <anchor type="note" xlink:label="note-273-05a" xlink:href="note-273-05"/> eſt, rectilineo A, æquale. </s> <s xml:id="echoid-s11332" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s11333" xml:space="preserve"/> </p> <div xml:id="echoid-div694" type="float" level="2" n="7"> <note symbol="a" position="right" xlink:label="note-273-01" xlink:href="note-273-01a" xml:space="preserve">ſchol. 4. <lb/>ſexti.</note> <note symbol="b" position="right" xlink:label="note-273-02" xlink:href="note-273-02a" xml:space="preserve">4. ſexti.</note> <note symbol="c" position="right" xlink:label="note-273-03" xlink:href="note-273-03a" xml:space="preserve">coroll. 20. <lb/>ſexti.</note> <note symbol="d" position="right" xlink:label="note-273-04" xlink:href="note-273-04a" xml:space="preserve">1. ſexti.</note> <note symbol="e" position="right" xlink:label="note-273-05" xlink:href="note-273-05a" xml:space="preserve">14. quinti.</note> </div> <p> <s xml:id="echoid-s11334" xml:space="preserve">7. </s> <s xml:id="echoid-s11335" xml:space="preserve"><emph style="sc">Iam</emph> verò datis duobus rectilineis quibuſcunque A, & </s> <s xml:id="echoid-s11336" xml:space="preserve">BCDEFGHI, ſit <lb/>ex poſteriore: </s> <s xml:id="echoid-s11337" xml:space="preserve">quod maius ponatur, auferendum rectilineum habens latus la-<lb/>teri BI, parallelum, æquale priori A, quod minus ſtatuatur, ſi fieri quidem id po-<lb/>terit. </s> <s xml:id="echoid-s11338" xml:space="preserve">Fieri autem poterit ſemper in figuris omnes angulos habentibus intror-<lb/>ſum, in aliis verò non ſemper. </s> <s xml:id="echoid-s11339" xml:space="preserve">Per ea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s11340" xml:space="preserve">14 lib. </s> <s xml:id="echoid-s11341" xml:space="preserve">2. </s> <s xml:id="echoid-s11342" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11343" xml:space="preserve">vel <lb/>potius per ea, quæ Num. </s> <s xml:id="echoid-s11344" xml:space="preserve">4. </s> <s xml:id="echoid-s11345" xml:space="preserve">cap. </s> <s xml:id="echoid-s11346" xml:space="preserve">4. </s> <s xml:id="echoid-s11347" xml:space="preserve">lib. </s> <s xml:id="echoid-s11348" xml:space="preserve">4. </s> <s xml:id="echoid-s11349" xml:space="preserve">huius tradidimus, conſtruatur quadra-<lb/>tum K M, rectilineo minori A, æquale. </s> <s xml:id="echoid-s11350" xml:space="preserve">Deinde ex angulo C, quilateri BI, pro-<lb/>ximus eſt, ducta lateri B I, parallela C O, conſtituatur rectilineo B O, ea-<lb/>dem ratione quadratum æquale P Q R; </s> <s xml:id="echoid-s11351" xml:space="preserve">& </s> <s xml:id="echoid-s11352" xml:space="preserve">duabus rectis K N, PQ, inuenta ter-<lb/>tia proportionali K S, ducatur S T, ipſi K L, parallela: </s> <s xml:id="echoid-s11353" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Erit que rectangulum <anchor type="note" xlink:label="note-273-06a" xlink:href="note-273-06"/> K T, contentum ſub prima linea KL, & </s> <s xml:id="echoid-s11354" xml:space="preserve">tertia K S, quadrato mediæ PQ, hoc eſt, <lb/>rectilineo B O, æquale. </s> <s xml:id="echoid-s11355" xml:space="preserve">Et quoniam KS, inuenta eſt in hoc exemplo minor late-<lb/>re KN: </s> <s xml:id="echoid-s11356" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s11357" xml:space="preserve">rectangulum K T, minus quadrato K M, hoc eſt, rectilineo <lb/>A; </s> <s xml:id="echoid-s11358" xml:space="preserve">erit etiam rectilineum B O, minus rectilineo A. </s> <s xml:id="echoid-s11359" xml:space="preserve">Ex propinquiore ergo an-<lb/>gulo H, ducta rurſum ipſi CO, vel BI, parallela H V, fiat iterum rectilineo C H, <lb/>æquale quadratum, cuius latus X: </s> <s xml:id="echoid-s11360" xml:space="preserve">Et duabus rectis K N, & </s> <s xml:id="echoid-s11361" xml:space="preserve">X, inuenta tertia <lb/>proportionali S N, quæ in hoc exemplo terminatur in extremo lateris K N; <lb/></s> <s xml:id="echoid-s11362" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="g"/> erit rurſum rectangulum SM, ſub prima linea ST, & </s> <s xml:id="echoid-s11363" xml:space="preserve">tertia SN, comprehenſum <anchor type="note" xlink:label="note-273-07a" xlink:href="note-273-07"/> æquale quadrato mediæ X, hoc eſt, rectilineo C H. </s> <s xml:id="echoid-s11364" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s11365" xml:space="preserve">K T, ipſi <lb/>B O, ſit oſtenſum æquale: </s> <s xml:id="echoid-s11366" xml:space="preserve">erit totum quadratum K M, hoc eſt, rectilineum A, <lb/>toti rectilineo BCVHOI, æquale; </s> <s xml:id="echoid-s11367" xml:space="preserve">ac proinde ex maiori rectilineo per rectam <lb/>HV, lateri B I, parallelam rectilineum detraximus minori rectilineo A, æquale. <lb/></s> <s xml:id="echoid-s11368" xml:space="preserve">quod faciendum erat.</s> <s xml:id="echoid-s11369" xml:space="preserve"/> </p> <div xml:id="echoid-div695" type="float" level="2" n="8"> <note symbol="f" position="right" xlink:label="note-273-06" xlink:href="note-273-06a" xml:space="preserve">17. ſexti.</note> <note symbol="g" position="right" xlink:label="note-273-07" xlink:href="note-273-07a" xml:space="preserve">17. ſexti.</note> </div> <p> <s xml:id="echoid-s11370" xml:space="preserve">8. </s> <s xml:id="echoid-s11371" xml:space="preserve"><emph style="sc">Qvod</emph> ſi duabus rectis K N, & </s> <s xml:id="echoid-s11372" xml:space="preserve">X, inuenta tertia proportionalis fuiſ-<lb/>ſet minor, quam SN, nimirum æqualis ipſi S Y, ita vt rectangulum S Z, quadra-<lb/>to rectæ X, vel rectilineo C H, foret æquale: </s> <s xml:id="echoid-s11373" xml:space="preserve">ducenda eſſet ex proximo an-<lb/>gulo D, alia parallela D a, & </s> <s xml:id="echoid-s11374" xml:space="preserve">rectilineo D H, conſtituendum quadratum æ-<lb/>quale; </s> <s xml:id="echoid-s11375" xml:space="preserve">at que rectæ K N, & </s> <s xml:id="echoid-s11376" xml:space="preserve">lateri poſtremi huius quadrati inuenti adiungen-<lb/>da tertia proportionalis, & </s> <s xml:id="echoid-s11377" xml:space="preserve">ei abſcindenda æqualis Y b. </s> <s xml:id="echoid-s11378" xml:space="preserve">Et ſi Y B, foret minor, <lb/>quam Y N, ducenda adhuc eſſet ex proximo angulo G, parallela lateri B I, & </s> <s xml:id="echoid-s11379" xml:space="preserve"><lb/>rectilineo inter hanc parallelam, & </s> <s xml:id="echoid-s11380" xml:space="preserve">D a, comprehenſo effi ciendum quadratum <lb/>æquale: </s> <s xml:id="echoid-s11381" xml:space="preserve">ac rectæ K N, & </s> <s xml:id="echoid-s11382" xml:space="preserve">lateri huius quadrati adiungenda tertia proportiona-<lb/>lis, &</s> <s xml:id="echoid-s11383" xml:space="preserve">c. </s> <s xml:id="echoid-s11384" xml:space="preserve">Atque ita progrediendum deinceps, donec inuenta ſit tertia propor- <pb o="244" file="274" n="274" rhead="GEOMETR. PRACT."/> tionalis S N, quæ terminetur in N, (qualis fuit tertia proportionalis S N, dua-<lb/>bus K N, & </s> <s xml:id="echoid-s11385" xml:space="preserve">X, inuenta) vel cuius terminus vltra N, cadat. </s> <s xml:id="echoid-s11386" xml:space="preserve">Quando enim ter-<lb/>minatur in N, erit rectilineum inter B I, & </s> <s xml:id="echoid-s11387" xml:space="preserve">vltimam parallelam comprehenſum <lb/> <anchor type="figure" xlink:label="fig-274-01a" xlink:href="fig-274-01"/> æquale quadrato K M, hoc eſt, rectilineo A, vt oſtenſum fuit de rectilineo BH, <lb/>paulo ante hunc Num. </s> <s xml:id="echoid-s11388" xml:space="preserve">8. </s> <s xml:id="echoid-s11389" xml:space="preserve">Quando autem terminus tertiæ proportionalis ca-<lb/>dit vltra N, videlicet in b. </s> <s xml:id="echoid-s11390" xml:space="preserve">ita vt rectangulum Y d, ſit æquale vltimo quadrato <lb/>inuento, hoc eſt, rectilineo D H; </s> <s xml:id="echoid-s11391" xml:space="preserve">ac proinde totum rectilineuminter BI, & </s> <s xml:id="echoid-s11392" xml:space="preserve">vlti-<lb/>mam parallelam, hoc eſt, rectangulum K d, maius quadrato K M, vel rectilineo <lb/>A; </s> <s xml:id="echoid-s11393" xml:space="preserve">erit vltimum rectilineum D H, maius rectangulo N d, propterea quod K Z, <lb/>ipſi BH, æquale eſt, & </s> <s xml:id="echoid-s11394" xml:space="preserve">Y d, ipſi D H. </s> <s xml:id="echoid-s11395" xml:space="preserve">Quare ſi ſuper rectam D a, inter rectas D V, <lb/>a H, conſtituatur trapezium D f, per parallelam e f, rectangulo N d, æquale, vt <lb/>ſupra Num. </s> <s xml:id="echoid-s11396" xml:space="preserve">3. </s> <s xml:id="echoid-s11397" xml:space="preserve">traditum eſt: </s> <s xml:id="echoid-s11398" xml:space="preserve">erit rectilineum inter B I, & </s> <s xml:id="echoid-s11399" xml:space="preserve">e f, parallelas conten-<lb/>tum æquale quadrato K M, hoc eſt, rectilineo A.</s> <s xml:id="echoid-s11400" xml:space="preserve"/> </p> <div xml:id="echoid-div696" type="float" level="2" n="9"> <figure xlink:label="fig-274-01" xlink:href="fig-274-01a"> <image file="274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/274-01"/> </figure> </div> <p> <s xml:id="echoid-s11401" xml:space="preserve"><emph style="sc">Cætervm</emph> non eſt neceſſe, vt ſemper à proximo angulo parallela duca-<lb/>tur in figura B F. </s> <s xml:id="echoid-s11402" xml:space="preserve">Quando namque ſenſus iudicaret plus minus, parallelam ex <lb/>aliquo angulo non proximo ductam auferre rectilineum minus quadrato <lb/>K M, vel non multò maius, ducenda eſſet parallela ex eo angulo, & </s> <s xml:id="echoid-s11403" xml:space="preserve">recti-<lb/>lineo abſciſſo conſtruendum æquale quadratum, & </s> <s xml:id="echoid-s11404" xml:space="preserve">reliqua perficienda, vt <lb/>prius.</s> <s xml:id="echoid-s11405" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11406" xml:space="preserve"><emph style="sc">Deniqve</emph> ſi aliquando deprehenderetur, rectilineum abſciſſum non eſſe <lb/>multo minus quadrato, conſtituendum eſſet, vt Num. </s> <s xml:id="echoid-s11407" xml:space="preserve">3. </s> <s xml:id="echoid-s11408" xml:space="preserve">docuimus, ſuper paral-<lb/>lelam illam trapezium æquale rectangulo, quo quadratum KM, rectilineum ab-<lb/>ſciſſum ſuperat.</s> <s xml:id="echoid-s11409" xml:space="preserve"/> </p> <pb o="245" file="275" n="275" rhead="LIBER SEXTVS."/> <p> <s xml:id="echoid-s11410" xml:space="preserve">9. </s> <s xml:id="echoid-s11411" xml:space="preserve"><emph style="sc">Ex</emph> his puto ſatis ſtudio ſum Lectorem intelligere, quo pacto in alijs e-<lb/>xemplis ſe gerere debeat. </s> <s xml:id="echoid-s11412" xml:space="preserve">Nam ſi verbi gratia ex hoc propoſito rectilineo irre-<lb/>gulariſsimo per lineam lateri AM, parallelam abſcindenda ſit portio æqualis al-<lb/>teri cuipiam rectilineo minori, producemus MA, vſ-<lb/> <anchor type="figure" xlink:label="fig-275-01a" xlink:href="fig-275-01"/> que ad N. </s> <s xml:id="echoid-s11413" xml:space="preserve">Et ſi quidem deprehenſum fuerit trian-<lb/>gulum A B N, eſſe æquale dato rectilineo minori, <lb/>(quod ſcietur, ſi quadratum triangulo æquale con-<lb/>ſtructum, fuerit æquale quadrato, quod dato recti-<lb/>lineo minori conſtruitur æquale) recta AN, proble-<lb/>ma efficiet. </s> <s xml:id="echoid-s11414" xml:space="preserve">Siverò maius, conſtruemus ſuper AN, <lb/>verſus B, trapezium per parallelam ipſi A N, æquale exceſſui: </s> <s xml:id="echoid-s11415" xml:space="preserve">At ſi minus, du-<lb/>cemus L O, parallelam. </s> <s xml:id="echoid-s11416" xml:space="preserve">Nam ſi fuerit deprehenſum rectilineum NL, æquale <lb/>defectui, problema efficiet parallela L O: </s> <s xml:id="echoid-s11417" xml:space="preserve">Si verò maius, conſtituemus ſuper <lb/>L O, verſus MN, per parallelam ipſi MN, trapezium exceſſui æquale. </s> <s xml:id="echoid-s11418" xml:space="preserve">Ea enim <lb/>parallela problema ſoluet: </s> <s xml:id="echoid-s11419" xml:space="preserve">At ſi minus, producemus OL, ad P: </s> <s xml:id="echoid-s11420" xml:space="preserve">Etſi quidem <lb/>triangulum KLP, fuerit æquale defectui, tota parallela O P, quæſtioni ſatisfa-<lb/>ciet: </s> <s xml:id="echoid-s11421" xml:space="preserve">Si verò maius, <anchor type="note" xlink:href="" symbol="a"/> conſtituemus in angulo K, triangulum ſimile triangulo <anchor type="note" xlink:label="note-275-01a" xlink:href="note-275-01"/> KLP, & </s> <s xml:id="echoid-s11422" xml:space="preserve">exceſſuiæquale; </s> <s xml:id="echoid-s11423" xml:space="preserve">ita vt hoc triangulum vna cum rectilineo per paralle-<lb/>lam L O, abſciſſo ſit dato rectilineo minori æquale. </s> <s xml:id="echoid-s11424" xml:space="preserve">Ex quo colliges, proble-<lb/>ma in hoc caſu ſolui non poſſe, cum duæ parallelæ, nimirum L O, & </s> <s xml:id="echoid-s11425" xml:space="preserve">illa, quæ <lb/>triangulum ipſi KLP, ſimile aufert, reſecent ex toto rectilineo BG, partem dato <lb/>rectilineo minori æqualem. </s> <s xml:id="echoid-s11426" xml:space="preserve">At ſi triangulum KLP, fuerit minus defectu præ-<lb/>dicto, ita vt hoc triangulum vna cum rectilineo per parallelam L O, abſciſſo ſit <lb/>minus dato rectilineo minore, ducemus per D, parallelam Q R. </s> <s xml:id="echoid-s11427" xml:space="preserve">Et ſi quidem <lb/>rectilineum P R, æquale fuerit defectui, quo figura KPLMABNO, à dato re-<lb/>ctilineo minore deficit, factum erit per parallelam QR, quod iubetur: </s> <s xml:id="echoid-s11428" xml:space="preserve">Siverò <lb/>maius, parallela, quæ cum QR, verſus OP, auferet rectilineum huic exceſſuiæ-<lb/>quale, ſatisfaciet problemati: </s> <s xml:id="echoid-s11429" xml:space="preserve">At ſi rectilineum PR, fuerit minus prædicto de-<lb/>fectu, & </s> <s xml:id="echoid-s11430" xml:space="preserve">triangulum C D R, inuentũ fuerit vltimo huic defectui, quo rectiline-<lb/>um PR, à prædicto defectu deficit, æquale, parallela DQ@ quæſtionem diſſoluet: <lb/></s> <s xml:id="echoid-s11431" xml:space="preserve">Si autem triangulum CDR, fuerit maius hoc vltimo defectu, <anchor type="note" xlink:href="" symbol="b"/> ſi ad C, conſtiru- <anchor type="note" xlink:label="note-275-02a" xlink:href="note-275-02"/> atur triangulum exceſſui æquale, & </s> <s xml:id="echoid-s11432" xml:space="preserve">ſimile triangulo CDR, ſatisfacient quæſtio-<lb/>ni duæ parallelæ, videlicet D Q. </s> <s xml:id="echoid-s11433" xml:space="preserve">& </s> <s xml:id="echoid-s11434" xml:space="preserve">baſis prædictitrianguli conſtituti; </s> <s xml:id="echoid-s11435" xml:space="preserve">atque in <lb/>hoc caſu per vnicam parallelam ſatisfieri problemati nequit: </s> <s xml:id="echoid-s11436" xml:space="preserve">Si denique trian-<lb/>gulum CDR, minus extiterit eo dem illo vltimo defectu, ducemus parallelam <lb/>IS. </s> <s xml:id="echoid-s11437" xml:space="preserve">Et ſi quidẽ rectilineum DI, æquale fuerit illi, quo triangulum CDR, minus <lb/>eſt vltimo illo defectu, erit totum rectilineum ISDCBAMLKI, dato minori re-<lb/>ctilineo æquale: </s> <s xml:id="echoid-s11438" xml:space="preserve">Si autem rectilineum DI, inæquale fuerit, progrediemur vlte-<lb/>rius, vt iam ſæpius dictum eſt, donec rectilineum inueniamus dato minori recti-<lb/>lineo æquale; </s> <s xml:id="echoid-s11439" xml:space="preserve">Inuenietur autem omnino vnum æquale, cum totũ rectilineum <lb/>BG, maius ponatur. </s> <s xml:id="echoid-s11440" xml:space="preserve">Vides igitur, facilè conijci poſſe, quando problema per v-<lb/>nicam parallelam ſolui poſsit, & </s> <s xml:id="echoid-s11441" xml:space="preserve">quando non, ſed per duas: </s> <s xml:id="echoid-s11442" xml:space="preserve">Quotieſcunque <lb/>enimincidemus in eiuſmo ditriangulum in ipſa conſtructione, qualia fu-<lb/>erunt K L P, & </s> <s xml:id="echoid-s11443" xml:space="preserve">C D R, ex quo auferendum ſit triangulum ſimile, & </s> <s xml:id="echoid-s11444" xml:space="preserve"><lb/>æquale exceſſui alicui, ſolui problema nequit, niſi per <lb/>duas parallelas.</s> <s xml:id="echoid-s11445" xml:space="preserve"/> </p> <div xml:id="echoid-div697" type="float" level="2" n="10"> <figure xlink:label="fig-275-01" xlink:href="fig-275-01a"> <image file="275-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/275-01"/> </figure> <note symbol="a" position="right" xlink:label="note-275-01" xlink:href="note-275-01a" xml:space="preserve">25. ſexti.</note> <note symbol="b" position="right" xlink:label="note-275-02" xlink:href="note-275-02a" xml:space="preserve">25. ſexti.</note> </div> <pb o="246" file="276" n="276" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div699" type="section" level="1" n="244"> <head xml:id="echoid-head269" xml:space="preserve">PROBL. 2. PROPOS. 3.</head> <p> <s xml:id="echoid-s11446" xml:space="preserve">DIVISO rectilineo quolibet in triangula ex vno aliquo puncto, rectas <lb/>lineas ipſis triangulis ordine proportionales inuenire.</s> <s xml:id="echoid-s11447" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11448" xml:space="preserve"><emph style="sc">Sit</emph> rectilineum quo dlibet A B C D E F, diuiſum in triangula A B C, A C D, <lb/>ADE, AEF, per rectas ex angulo A, (vel aliquo puncto aſsignato in vno latere) <lb/>ad omnes angulos oppoſitos ductas: </s> <s xml:id="echoid-s11449" xml:space="preserve">atque hiſce triangulis inueniendæ ſint or-<lb/>dine totidem rectæ proportionales. </s> <s xml:id="echoid-s11450" xml:space="preserve">Ex omnibus angulis dempto angulo A, ad <lb/> <anchor type="figure" xlink:label="fig-276-01a" xlink:href="fig-276-01"/> rectas ex A, egredientes ducantur perpendiculares B I, <lb/>CL, DK, DN, EM, FO, pro altitudinibus triangulorũ. <lb/></s> <s xml:id="echoid-s11451" xml:space="preserve">(Nihil autem refert, ſi interdum perpendiculares cadant <lb/>in rectas extra triangula productas, cuiuſmodi hic ſunt <lb/>DK, DN,) ita vt ſingula triangula binas habeant altitudi-<lb/>nes, præter duo extrema, quæ ſingulas duntaxat habent. </s> <s xml:id="echoid-s11452" xml:space="preserve"><lb/>Deinde in recta quacunque GH, accipiatur GP, æ qualis <lb/>altitudini BI, primi trianguli ABC; </s> <s xml:id="echoid-s11453" xml:space="preserve">& </s> <s xml:id="echoid-s11454" xml:space="preserve">P Q, æqualis altitu-<lb/>dini DK, ſecundi trianguli A C D, reſpectu eiuſdem baſis <lb/>AC. </s> <s xml:id="echoid-s11455" xml:space="preserve">Poſt hæcſiat, <anchor type="note" xlink:href="" symbol="a"/> vt CL altitudo ſecunditriangulire- <anchor type="note" xlink:label="note-276-01a" xlink:href="note-276-01"/> ſpectu baſis AD, ad EM, altitudinẽ tertij trianguli ADE, reſpectu eiuſdem baſis <lb/>AD, ita PQ. </s> <s xml:id="echoid-s11456" xml:space="preserve">ad QR; </s> <s xml:id="echoid-s11457" xml:space="preserve">Et vt DN, altitudo tertij trianguli ADE, reſpectu baſis AE, <lb/>ad FO, altitudinem quartitrianguli AEF, reſpectu eiuſdem baſis AE, ita QR, ad <lb/>RH, atque ita deinceps, ſi plura fuerint triangula, ſumendo ſemper duas alti-<lb/>tudines ad communem baſem demiſſas, &</s> <s xml:id="echoid-s11458" xml:space="preserve">c. </s> <s xml:id="echoid-s11459" xml:space="preserve">Dico quatuor rectas G P, PQ, <lb/>QR, R H, eſſe quatuor triangulis ordine proportionales. </s> <s xml:id="echoid-s11460" xml:space="preserve">Nam vt in ſcholio <lb/>propoſ. </s> <s xml:id="echoid-s11461" xml:space="preserve">1. </s> <s xml:id="echoid-s11462" xml:space="preserve">lib. </s> <s xml:id="echoid-s11463" xml:space="preserve">6. </s> <s xml:id="echoid-s11464" xml:space="preserve">Euclid demonſtratum eſt, à nobis, ita eſt triangulum A B C, ad <lb/>triangulum ACD, vt altitudo BI, ad altitudinem D K, propter baſem commu-<lb/>nem AC, hoc eſt, vt GP, ad PQ, cum hæ ſumptæ ſint illis altitudinibus æquales. <lb/></s> <s xml:id="echoid-s11465" xml:space="preserve">Eadem de cauſa ita eſt triangulum ACD, ad triangulum ADE, vt altitudo CL, <lb/>ad altitudinem EM, hoc eſt, vt PQ. </s> <s xml:id="echoid-s11466" xml:space="preserve">ad QR, cum ex conſtructione ſit, vt CL, ad <lb/>EM, ita PQ. </s> <s xml:id="echoid-s11467" xml:space="preserve">ad QR. </s> <s xml:id="echoid-s11468" xml:space="preserve">Pari denique ratione ita eſt triangulum ADE, ad triangu-<lb/>lum AEF, vt altitudo DN, ad altitudinem, FO, hoc eſt, vt QR, ad R H, cum ſit <lb/>per conſtructionem, vt DM, ad FO, ita QR, ad RH. </s> <s xml:id="echoid-s11469" xml:space="preserve">Conſtat ergo id, quod pro-<lb/>poſitum fuit.</s> <s xml:id="echoid-s11470" xml:space="preserve"/> </p> <div xml:id="echoid-div699" type="float" level="2" n="1"> <figure xlink:label="fig-276-01" xlink:href="fig-276-01a"> <image file="276-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/276-01"/> </figure> <note symbol="a" position="left" xlink:label="note-276-01" xlink:href="note-276-01a" xml:space="preserve">12. ſexti.</note> </div> </div> <div xml:id="echoid-div701" type="section" level="1" n="245"> <head xml:id="echoid-head270" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s11471" xml:space="preserve"><emph style="sc">Sit</emph> rurſus rectilineum ABCDEF, diuiſum in triangula ABC, ACD, ADE, <lb/>AEF, ex puncto A. </s> <s xml:id="echoid-s11472" xml:space="preserve">Quoniam bina proxima triangula conſtituunt qua drila-<lb/>terum, cuius diameter eſt latus vtrique triangulo commune, cuiuſmodi eſt A-<lb/>BCD, ducemus diametro AC, ex D, parallelam D O, quæ ſecet latus BC, pro-<lb/>ductum in O. </s> <s xml:id="echoid-s11473" xml:space="preserve">Sic in quadrilatero ACDE, diametro AD, parallelam ducemus <lb/>EP, quæ ſecet latus CD, protractum in P. </s> <s xml:id="echoid-s11474" xml:space="preserve">Itemque in quadrilatero ADEF, dia-<lb/>metro AE, parallelam ducemus FQ. </s> <s xml:id="echoid-s11475" xml:space="preserve">quæ latus DE, productum ſecet in Q. </s> <s xml:id="echoid-s11476" xml:space="preserve">De-<lb/>inde in recta quauis GN, ſumantur G H, HK, ipſis BC, CO, æquales, Et tribus <pb o="247" file="277" n="277" rhead="LIBER SEXTVS."/> CD, DP, HK, reperiatur quarta proportionalis KL. </s> <s xml:id="echoid-s11477" xml:space="preserve">Ac tandem tribus DE, EQ, <lb/>KL, quarta proportionalis inueniatur L N. </s> <s xml:id="echoid-s11478" xml:space="preserve">Dico inuentas eſſe quatuor rectas <lb/>GH, HK, KL, LN, quatuor triangulis proportionales. </s> <s xml:id="echoid-s11479" xml:space="preserve">Ductis enim ex A, ad <lb/>O, P, Q, puncta concurſuum rectis AO, AP, AQ,<anchor type="note" xlink:href="" symbol="a"/> erit triangulum ACD, trian- <anchor type="note" xlink:label="note-277-01a" xlink:href="note-277-01"/> <anchor type="figure" xlink:label="fig-277-01a" xlink:href="fig-277-01"/> gulo ACO; </s> <s xml:id="echoid-s11480" xml:space="preserve">& </s> <s xml:id="echoid-s11481" xml:space="preserve">triangulum ADE, triangulo ADP; </s> <s xml:id="echoid-s11482" xml:space="preserve">& </s> <s xml:id="echoid-s11483" xml:space="preserve">triangulum AEF, trian-<lb/>gulo AEQ, æquale. </s> <s xml:id="echoid-s11484" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/>Cum ergo ſit, vt B C, ad C O, hoc eſt, vt G H, ad H K, ita <anchor type="note" xlink:label="note-277-02a" xlink:href="note-277-02"/> triangulum ABC, ad triangulum ACO, hoc eſt, ad triangulum A C D: </s> <s xml:id="echoid-s11485" xml:space="preserve">Item vt <lb/>CD, ad DP, hoc eſt, vt HK, ad KL, ita triangulum A C D, ad triangulum A D P, <lb/>hoc eſt, ad triangulum ADE: </s> <s xml:id="echoid-s11486" xml:space="preserve">Et vt DE, ad EQ, hoc eſt, vt KL, ad LN, ita tri-<lb/>angulum ADE, ad triangulum AEQ, hoc eſt, ad triangulum AEF: </s> <s xml:id="echoid-s11487" xml:space="preserve">perſpicuum <lb/>eſt id, quod proponitur.</s> <s xml:id="echoid-s11488" xml:space="preserve"/> </p> <div xml:id="echoid-div701" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-277-01" xlink:href="note-277-01a" xml:space="preserve">37. primi.</note> <figure xlink:label="fig-277-01" xlink:href="fig-277-01a"> <image file="277-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/277-01"/> </figure> <note symbol="b" position="right" xlink:label="note-277-02" xlink:href="note-277-02a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div703" type="section" level="1" n="246"> <head xml:id="echoid-head271" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s11489" xml:space="preserve"><emph style="sc">Rationes</emph> duæ expoſitæ, quæ expeditiſsimè ſunt, propria eſt triangulo-<lb/>rum, in quæ diuiditur figura per rectas ab vno aliquo puncto in quouis latere <lb/>dato, vel ab aliquo angulo emiſſas: </s> <s xml:id="echoid-s11490" xml:space="preserve">poteſt tamen idem hoc problema abſolui <lb/>alio modo, qui in quaslibet figuras conuenit, licet non ſit tam expeditus. </s> <s xml:id="echoid-s11491" xml:space="preserve">Ita er-<lb/>go agemus: </s> <s xml:id="echoid-s11492" xml:space="preserve">Sit eadem figura proxima diuiſa in triangula, vel etiam in plurium <lb/>laterum figuras. </s> <s xml:id="echoid-s11493" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>Et primo triangulo ABC, vel primæ figuræ, rectangulum, vel <anchor type="note" xlink:label="note-277-03a" xlink:href="note-277-03"/> quoduis aliud parallelogrammum non rectangulum æquale conſtruatur IS: </s> <s xml:id="echoid-s11494" xml:space="preserve">Et <lb/>ſuper rectam RS, aliud parallelogrammum S T, ſecundo triangulo ACD, vel ſe-<lb/>cundę figuræ æquale, habens angulum SR T, angulo I, æqualem. </s> <s xml:id="echoid-s11495" xml:space="preserve">Itẽ ſuper rectã <lb/>TV, aliud VX, tertio triãgulo ADE, vel tertiæ figuræ æquale, angulũ habẽs VT-<lb/>X, æqualẽ eidem angulo I: </s> <s xml:id="echoid-s11496" xml:space="preserve">Ac deniq; </s> <s xml:id="echoid-s11497" xml:space="preserve">ſuper rectam XY, aliud YM, quarto trian-<lb/>gulo A E F, vel quartæ figuræ æquale, angulum Y X M, habens æqualem eidem <lb/>angulo I: </s> <s xml:id="echoid-s11498" xml:space="preserve">atque ita deinceps, ſi plura fuerint triangula, vel figuræ. </s> <s xml:id="echoid-s11499" xml:space="preserve">Dico re-<lb/>ctas IR, RT, TX, XM, triangulis, vel figuris eſſe proportionales. </s> <s xml:id="echoid-s11500" xml:space="preserve">Nam ex qua- <pb o="248" file="278" n="278" rhead="GEOMETR. PRACT."/> tuor rectangulis, vel parallelogrammis conſtituitur vnum totum, vt ex demõ-<lb/>ſtratione propoſ 45. </s> <s xml:id="echoid-s11501" xml:space="preserve">lib. </s> <s xml:id="echoid-s11502" xml:space="preserve">1. </s> <s xml:id="echoid-s11503" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11504" xml:space="preserve">manifeſtum eſt, propter angulos I, S R T, V-<lb/> <anchor type="note" xlink:label="note-278-01a" xlink:href="note-278-01"/> TX, YXM, æquales: </s> <s xml:id="echoid-s11505" xml:space="preserve">ac proinde omnia quatuor eandem habent altitudinem.</s> <s xml:id="echoid-s11506" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur rectæ IR, RT, TX, XM, proportionales ſunt parallelogrammis, ideo que <lb/>& </s> <s xml:id="echoid-s11507" xml:space="preserve">triangulis, ſiue figuris, quod eſt propoſitum.</s> <s xml:id="echoid-s11508" xml:space="preserve"/> </p> <div xml:id="echoid-div703" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-277-03" xlink:href="note-277-03a" xml:space="preserve">44. vel 45-<lb/>primi.</note> <note symbol="a" position="left" xlink:label="note-278-01" xlink:href="note-278-01a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div705" type="section" level="1" n="247"> <head xml:id="echoid-head272" xml:space="preserve">PROBL. 3. PROPOS. 4.</head> <p> <s xml:id="echoid-s11509" xml:space="preserve">DATVM rectilineum per rectam à quouis angulo, vel puncto in ali-<lb/>quo latere ductam in proportionem datam diuidere: </s> <s xml:id="echoid-s11510" xml:space="preserve">ita vt antece-<lb/>dens proportionis, in quam malueris partem, vergat.</s> <s xml:id="echoid-s11511" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11512" xml:space="preserve"><emph style="sc">Sit</emph> primum triangulum quodcunque ABC, per rectam ex angulo A, diui-<lb/>dendum in duas partes: </s> <s xml:id="echoid-s11513" xml:space="preserve">ita vt pars ad B, vergẽs ad reliquam partẽ habeat<unsure/> pro-<lb/> <anchor type="figure" xlink:label="fig-278-01a" xlink:href="fig-278-01"/> portionem datam D, ad E. </s> <s xml:id="echoid-s11514" xml:space="preserve">Secetur latus B C, <lb/>dato angulo oppoſitum, per ea, quæin ſcholio <lb/>propoſ. </s> <s xml:id="echoid-s11515" xml:space="preserve">10. </s> <s xml:id="echoid-s11516" xml:space="preserve">lib. </s> <s xml:id="echoid-s11517" xml:space="preserve">6. </s> <s xml:id="echoid-s11518" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11519" xml:space="preserve">docuimus, in F, ita vt <lb/>eadem ſit proportio BF, ad FC, quæ D, ad E, du-<lb/>caturque recta A F. </s> <s xml:id="echoid-s11520" xml:space="preserve">Dico eſſe vt D, ad E, ita tri-<lb/> <anchor type="note" xlink:label="note-278-02a" xlink:href="note-278-02"/> angulum ABF, ad triangulum AFC. </s> <s xml:id="echoid-s11521" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/>Eſt enim triangulum ABF, ad triangulum AFC, vt BF, ad <lb/>FC, hoc eſT, vt D, ad E.</s> <s xml:id="echoid-s11522" xml:space="preserve"/> </p> <div xml:id="echoid-div705" type="float" level="2" n="1"> <figure xlink:label="fig-278-01" xlink:href="fig-278-01a"> <image file="278-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/278-01"/> </figure> <note symbol="b" position="left" xlink:label="note-278-02" xlink:href="note-278-02a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11523" xml:space="preserve"><emph style="sc">Deinde</emph> ſit idem triangulum ABC, diuidendum in duas partes, per rectam <lb/>ex puncto F, dato in latere BC, ita vt pars verſus B, ad reliquam partem habeat <lb/>proportionem datam D, ad E. </s> <s xml:id="echoid-s11524" xml:space="preserve">Ducta ex dato puncto F, ad angulum oppo-<lb/>ſitum A, recta FA, vt totum triangulum in duo triangula ſit ſectum: </s> <s xml:id="echoid-s11525" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>reperian- <anchor type="note" xlink:label="note-278-03a" xlink:href="note-278-03"/> tur duæ rectæ GH, HI, habentes eandem proportionem, quam triangulum A-<lb/>BF, ad triangulum AFC: </s> <s xml:id="echoid-s11526" xml:space="preserve">tota que GI, ſecetur in H, vt eadem ſit proportio GH, <lb/>ad HI, quæ D, ad E. </s> <s xml:id="echoid-s11527" xml:space="preserve">Et quia punctum H, cadit in extremum primæ lineæ GH; <lb/></s> <s xml:id="echoid-s11528" xml:space="preserve">eſtque vt GH, ad HI, hoc eſt, vt D, ad E, ita triangulum ABF, ad triangulum A-<lb/>FC: </s> <s xml:id="echoid-s11529" xml:space="preserve">diuidet recta F A, ex dato puncto F, ad oppoſitum angulum A, ducta tri-<lb/>angulum ABC, in duas partes in data proportione D, ad E.</s> <s xml:id="echoid-s11530" xml:space="preserve"/> </p> <div xml:id="echoid-div706" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-278-03" xlink:href="note-278-03a" xml:space="preserve">3. hui{us}.</note> </div> <p> <s xml:id="echoid-s11531" xml:space="preserve"><emph style="sc">Sit</emph> rurſus data proportio K, ad L, diuidendumque ſit triangulum ABC, ex <lb/>puncto F, in duas partes eiuſdem proportionis. </s> <s xml:id="echoid-s11532" xml:space="preserve">Diuidatur tota GI, in M, ita vt <lb/>eadem ſit proportio GM, ad MI, quæ K, ad L. </s> <s xml:id="echoid-s11533" xml:space="preserve">Et quoniam diuiſionis punctum <lb/>M, cadit in primam partem GH, totius lineæ GI, ſecabimus BA, baſem primitri-<lb/>anguli dato puncto F, oppoſitam, in N, vt eadem ſit proportio BN, ad NA, quę <lb/>GM, ad MH. </s> <s xml:id="echoid-s11534" xml:space="preserve">Dico ductam rectam FN, problema efficere, hoc eſt, ita eſſe tri-<lb/>angulum BFN, ad trapezium FNAC, vt K, ad L. </s> <s xml:id="echoid-s11535" xml:space="preserve">Quoniam enim rectæ GH, HI, <lb/>triangulis ABF, AFC, proportionales inuentæ ſunt: </s> <s xml:id="echoid-s11536" xml:space="preserve">& </s> <s xml:id="echoid-s11537" xml:space="preserve">tam primam partem GH, <lb/>in M, quam primum triangulum ABF, per rectam FN, ſecuimus proportionali-<lb/> <anchor type="note" xlink:label="note-278-04a" xlink:href="note-278-04"/> ter, <anchor type="note" xlink:href="" symbol="d"/> cum ſit triangulum BFN, ad triangulum NFA, vt BN, ad NA, hoc eſt, vt <anchor type="note" xlink:label="note-278-05a" xlink:href="note-278-05"/> GM, ad MH,<anchor type="note" xlink:href="" symbol="e"/> erit vt GM, ad MI, id eſt, vt K, ad L, ita BFN, triangulum ad tra- pezium FNAC. </s> <s xml:id="echoid-s11538" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s11539" xml:space="preserve"/> </p> <div xml:id="echoid-div707" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-278-04" xlink:href="note-278-04a" xml:space="preserve">1. ſexti.</note> <note symbol="e" position="left" xlink:label="note-278-05" xlink:href="note-278-05a" xml:space="preserve">1. hui{us}.</note> </div> <p> <s xml:id="echoid-s11540" xml:space="preserve"><emph style="sc">Deniqve</emph> data ſit proportio O, ad P, ſecandum que ſit triangulum ABC, <lb/>in duas partes eiuſdem proportionis. </s> <s xml:id="echoid-s11541" xml:space="preserve">Diuiſatota G I, in Q, ita vt eadem ſit <pb o="249" file="279" n="279" rhead="LIBER SEXTVS."/> proportio GQ.</s> <s xml:id="echoid-s11542" xml:space="preserve">ad QI@quæ O, ad P: </s> <s xml:id="echoid-s11543" xml:space="preserve">quoniam punctum diuiſionis Q. </s> <s xml:id="echoid-s11544" xml:space="preserve">eadit in ſe-<lb/>cundam partem HI, totius lineę GI, diuidemus AC, baſem ſecundi trianguli da-<lb/>to puncto F; </s> <s xml:id="echoid-s11545" xml:space="preserve">oppoſitam in R, vt eadem ſit proportio AR, ad R C, quæ H Q, ad <lb/>QI. </s> <s xml:id="echoid-s11546" xml:space="preserve">Dico ductam rectam ER, problema efficere, hoc eſt, ita eſſe trapezium AB-<lb/>FR, ad triangulum RFC, vt O, ad P. </s> <s xml:id="echoid-s11547" xml:space="preserve">Quoniam enim rectæ GH, HI, repertæ ſunt <lb/>triangulis ABF, AFC, proportionales; </s> <s xml:id="echoid-s11548" xml:space="preserve">& </s> <s xml:id="echoid-s11549" xml:space="preserve">tam ſecundam partem HI, in Q. </s> <s xml:id="echoid-s11550" xml:space="preserve">quam <lb/>ſecundum triangulum AFC, per rectam FR, ſecuimus proportionaliter: </s> <s xml:id="echoid-s11551" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/>cum <anchor type="note" xlink:label="note-279-01a" xlink:href="note-279-01"/> ſit triangulum AFR, ad triangulum CFR, vt AR, ad RC, hoc eſt, vt HQ. </s> <s xml:id="echoid-s11552" xml:space="preserve">ad QI: <lb/></s> <s xml:id="echoid-s11553" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/>Erit vt GQ, ad QI. </s> <s xml:id="echoid-s11554" xml:space="preserve">hoc eſt, vt O, ad P, ita trapezium ABFR, ad triangulũ RFC.</s> <s xml:id="echoid-s11555" xml:space="preserve"> <anchor type="note" xlink:label="note-279-02a" xlink:href="note-279-02"/> quod eſt propoſitum.</s> <s xml:id="echoid-s11556" xml:space="preserve"/> </p> <div xml:id="echoid-div708" type="float" level="2" n="4"> <note symbol="a" position="right" xlink:label="note-279-01" xlink:href="note-279-01a" xml:space="preserve">1. ſexti.</note> <note symbol="b" position="right" xlink:label="note-279-02" xlink:href="note-279-02a" xml:space="preserve">1. hui{us}.</note> </div> <p> <s xml:id="echoid-s11557" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi antecedens proportionis vergere debeat verſus C, proportio-<lb/>que data ſit O, ad P; </s> <s xml:id="echoid-s11558" xml:space="preserve">fiet id commodiſsimè, ſi triangulum ex F, diuidatur ſecũ-<lb/>dum proportionem P, ad O, ita vt antecedens vergat verſus B, ſicut docuimus. <lb/></s> <s xml:id="echoid-s11559" xml:space="preserve">Nam tunc pars verſus C, ad reliquam habebit proportionem, quam O, ad P, per <lb/>conuerſam proportionalitatem. </s> <s xml:id="echoid-s11560" xml:space="preserve">Quod etiam in alijs figuris intelligi volo.</s> <s xml:id="echoid-s11561" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11562" xml:space="preserve"><emph style="sc">Sit</emph> deinde multilatera figura quæ cunque ABCDEF, per rectam ex angu-<lb/>lo A, ductam ſecanda in duas partes, ita vt pars ad B, vergens ad reliquam par-<lb/>tem proportionem habeat datam M, ad N. </s> <s xml:id="echoid-s11563" xml:space="preserve">Ductis ex dato angulo A, ad omnes <lb/> <anchor type="note" xlink:label="note-279-03a" xlink:href="note-279-03"/> angulos oppoſitos rectis partientibus figuram in quatuortriangula; </s> <s xml:id="echoid-s11564" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>inueni- anturipſis quatu or rectæ proportionales GH, HI, IK, KL. </s> <s xml:id="echoid-s11565" xml:space="preserve">Tota deinde GL, ſe-<lb/>ceturin O, vt eadem ſit proportio G O, ad O L, quæ M, ad N. </s> <s xml:id="echoid-s11566" xml:space="preserve">Et quoniam di-<lb/>uiſionis punctum O, cadit in tertiam lineam IK, ſecabimus tertij trianguli baſem <lb/>DE, dato angulo A, oppoſitam in P, vt ſecta eſt IK, in O, ducemuſquerectam <lb/> <anchor type="figure" xlink:label="fig-279-01a" xlink:href="fig-279-01"/> AP. </s> <s xml:id="echoid-s11567" xml:space="preserve">Dico eſſerectilineum ABCDPA, ad rectilineum APEFA, vt GO, ad OL, <lb/>hoc eſt vt M, ad N. </s> <s xml:id="echoid-s11568" xml:space="preserve">Quoniam enim triangula rectilinei dati proportionalia ſunt <lb/>ordine rectis GH, HI, IK, KL, per conſtructionem, tertiumq; </s> <s xml:id="echoid-s11569" xml:space="preserve">triangulum ADE, <lb/>& </s> <s xml:id="echoid-s11570" xml:space="preserve">lineam tertiam IK, diuiſimus proportionaliter per rectam A P, & </s> <s xml:id="echoid-s11571" xml:space="preserve">in puncto <lb/>O; </s> <s xml:id="echoid-s11572" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> cum ſit vt DP, ad PE, hoc eſt, vt IO, ad OK, ita triangulum ADP, ad trian- gulum A P E; </s> <s xml:id="echoid-s11573" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Erunt totæ quoque magnitudines ſectæ proportionaliter, hoc <anchor type="note" xlink:label="note-279-04a" xlink:href="note-279-04"/> eſt, erit ABCDPA, ad APEFA, vt G O, ad O L, hoc eſt, vt M, ad N. </s> <s xml:id="echoid-s11574" xml:space="preserve">quod eſt <lb/> <anchor type="note" xlink:label="note-279-05a" xlink:href="note-279-05"/> propoſitum.</s> <s xml:id="echoid-s11575" xml:space="preserve"/> </p> <div xml:id="echoid-div709" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-279-03" xlink:href="note-279-03a" xml:space="preserve">3. hui{us}.</note> <figure xlink:label="fig-279-01" xlink:href="fig-279-01a"> <image file="279-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/279-01"/> </figure> <note symbol="d" position="right" xlink:label="note-279-04" xlink:href="note-279-04a" xml:space="preserve">1. ſexti.</note> <note symbol="e" position="right" xlink:label="note-279-05" xlink:href="note-279-05a" xml:space="preserve">1. hui{us}.</note> </div> <p> <s xml:id="echoid-s11576" xml:space="preserve"><emph style="sc">Qvod</emph> ſi punctum O, diuidens rectam GL, in duas partes proportionis da-<lb/>tæ M, ad N, caderet in aliquod punctorum H, I, K, vt in I, terminum ſecundæ li-<lb/>neæ HI, diuideret recta AD, terminans ſecundum triangulum A C D, totum re-<lb/>ctilineum in datam proportionem. </s> <s xml:id="echoid-s11577" xml:space="preserve">Nam tunc per primam partem propoſ. </s> <s xml:id="echoid-s11578" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s11579" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s11580" xml:space="preserve">eſſet ABCDA, ad ADEFA, vt M, ad N, vt peſpicuum eſt.</s> <s xml:id="echoid-s11581" xml:space="preserve"/> </p> <pb o="250" file="280" n="280" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s11582" xml:space="preserve"><emph style="sc">Datvm</emph> præterea ſit rectilineum ABCD, ex puncto E, dato in latere AB, di-<lb/>uidendum in duas partes, quarum prior ad B, vergens ad poſteriorem partem <lb/>habeat proportionem datam K, ad L. </s> <s xml:id="echoid-s11583" xml:space="preserve">Ductis ex dato puncto E, ad omnes an-<lb/> <anchor type="figure" xlink:label="fig-280-01a" xlink:href="fig-280-01"/> gulos oppoſitos rectis diuidentibus rectilineum in tot triangula vno minus, <lb/>quotlatera figura habet; </s> <s xml:id="echoid-s11584" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/>inueniantur rectæ FH, HI, IG, triangulis EBC, ECD, EDA, proportionales. </s> <s xml:id="echoid-s11585" xml:space="preserve">Secta deindetota FG, in M, ſecundum datam propor-<lb/> <anchor type="note" xlink:label="note-280-01a" xlink:href="note-280-01"/> tionem K, ad L: </s> <s xml:id="echoid-s11586" xml:space="preserve">quoniam diuiſionis punctum M, incidit in primam linea F H, <lb/>diuidemus primi trianguli E B C, baſem BC, dato puncto E, oppoſitam in X, vt <lb/>FH, ſecta eſt in M. </s> <s xml:id="echoid-s11587" xml:space="preserve">Iuncta namq; </s> <s xml:id="echoid-s11588" xml:space="preserve">recta EX,<anchor type="note" xlink:href="" symbol="b"/> erit triangulum EBX, ad figuram <anchor type="note" xlink:label="note-280-02a" xlink:href="note-280-02"/> EXCDAE, vt FM, ad MG, hoc eſt, vt K, ad L: </s> <s xml:id="echoid-s11589" xml:space="preserve">propterea quod triangula E B C, <lb/>ECD, EDA, rectis FH, HI, IG, proportionalia ſunt ex conſtructione; </s> <s xml:id="echoid-s11590" xml:space="preserve">& </s> <s xml:id="echoid-s11591" xml:space="preserve">primæ <lb/>partes EBC, FH, ſectæ ſunt per rectam EX, & </s> <s xml:id="echoid-s11592" xml:space="preserve">in M, proportionaliter; </s> <s xml:id="echoid-s11593" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>cum ſit <anchor type="note" xlink:label="note-280-03a" xlink:href="note-280-03"/> EBX, ad EXC, vt BX, ad XC, hoc eſt, vt FM, ad MH. </s> <s xml:id="echoid-s11594" xml:space="preserve">Conſtat ergo propoſitũ.</s> <s xml:id="echoid-s11595" xml:space="preserve"/> </p> <div xml:id="echoid-div710" type="float" level="2" n="6"> <figure xlink:label="fig-280-01" xlink:href="fig-280-01a"> <image file="280-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/280-01"/> </figure> <note symbol="a" position="left" xlink:label="note-280-01" xlink:href="note-280-01a" xml:space="preserve">3. hui{us}.</note> <note symbol="b" position="left" xlink:label="note-280-02" xlink:href="note-280-02a" xml:space="preserve">1. hui{us}.</note> <note symbol="c" position="left" xlink:label="note-280-03" xlink:href="note-280-03a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11596" xml:space="preserve"><emph style="sc">Si</emph> proportio data ſit N, ad O; </s> <s xml:id="echoid-s11597" xml:space="preserve">ſecta FG, in P, ſecundum proportionem N, <lb/>ad O, cadet diuiſionis punctum P, in ſecundam lineam HI. </s> <s xml:id="echoid-s11598" xml:space="preserve">Igitur ſi ſecundi tri-<lb/>anguli ECD, baſis CD, dato puncto E, oppoſita ſecetur in Q, vt ſecta eſt HI, in <lb/>P, nectatur que recta EQ, <anchor type="note" xlink:href="" symbol="d"/> erit rurſus figura EBCQE, ad figuram EQDA, vt FP, <anchor type="note" xlink:label="note-280-04a" xlink:href="note-280-04"/> ad PG, hoc eſt, vt N, ad O.</s> <s xml:id="echoid-s11599" xml:space="preserve"/> </p> <div xml:id="echoid-div711" type="float" level="2" n="7"> <note symbol="d" position="left" xlink:label="note-280-04" xlink:href="note-280-04a" xml:space="preserve">1. hui{us}.</note> </div> <p> <s xml:id="echoid-s11600" xml:space="preserve"><emph style="sc">Si</emph> denique data ſit proportio R, ad S; </s> <s xml:id="echoid-s11601" xml:space="preserve">ſecta FG, in V, ſecundum proportio-<lb/>nem R, ad S, cadet diuiſionis punctum V, in tertiam lineam I G. </s> <s xml:id="echoid-s11602" xml:space="preserve">Quamobrem <lb/>ſi tertij trianguli EDA, baſis DA, dato puncto E, oppoſita ſecetur in T, vt ſecta <lb/>eſt IG, in V, iungaturque recta ET; </s> <s xml:id="echoid-s11603" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit rurſus figura EBCDTE, ad triangulum <anchor type="note" xlink:label="note-280-05a" xlink:href="note-280-05"/> ETA, vt FV, ad VG, hoc eſt, vt R, ad S.</s> <s xml:id="echoid-s11604" xml:space="preserve"/> </p> <div xml:id="echoid-div712" type="float" level="2" n="8"> <note symbol="e" position="left" xlink:label="note-280-05" xlink:href="note-280-05a" xml:space="preserve">hui{us}.</note> </div> <p> <s xml:id="echoid-s11605" xml:space="preserve"><emph style="sc">Atqve</emph> hac via procedendum eſt in omnibus alijs figuris, quæ latera toti-<lb/>dem habeant, quot angulos, id eſt, in quibus omnes anguli introrſum vergant.</s> <s xml:id="echoid-s11606" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11607" xml:space="preserve"><emph style="sc">Idem</emph> hoc problema efficiemus in rectilineo, cuius anguli partim extror-<lb/>ſum vergant, & </s> <s xml:id="echoid-s11608" xml:space="preserve">partim introrſum, dummodo ab angulo, vel puncto dato in la-<lb/>tere ducipoſsint lineæ rectæ diuidentes rectilineum in triangula quæ nullũ ipſi-<lb/>us latus ſecent. </s> <s xml:id="echoid-s11609" xml:space="preserve">Vt in hac figura octo laterum ABCDEFGH, cuius quinque an-<lb/>guli B, C, D, F, H, introrſum vergunt, & </s> <s xml:id="echoid-s11610" xml:space="preserve">reliquitres BAH, DEF, FGH, extrorſum. <lb/></s> <s xml:id="echoid-s11611" xml:space="preserve">ductæ ſunt rectæ ex angulo A, ad omnes angulos, præter quam ad duos proxi- <pb o="251" file="281" n="281" rhead="LIBER SEXTVS."/> mos B, H, nullum figuræ latus interſecantes. </s> <s xml:id="echoid-s11612" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Siigitur reperiantur ſexrectæ IK, <anchor type="note" xlink:label="note-281-01a" xlink:href="note-281-01"/> K L, L M, M N, N O, O P, ſex triangulis <lb/> <anchor type="figure" xlink:label="fig-281-01a" xlink:href="fig-281-01"/> ABC, ACD, ADE, AEF, AFG, AGH, <lb/>proportionales; </s> <s xml:id="echoid-s11613" xml:space="preserve">& </s> <s xml:id="echoid-s11614" xml:space="preserve">tota linea L P, ſece-<lb/>tur in S, ſecundnm datam proportionem <lb/>Q, ad R; </s> <s xml:id="echoid-s11615" xml:space="preserve">atque baſis D E, tertij trianguli <lb/>(Nam diuiſio nis punctum S, in tertiamli-<lb/>neam LM, incidit) dato puncto A, oppo-<lb/>ſita diuidatur in T, vt linea L M, in S, di-<lb/>uiſa eſt, ducaturque recta A T: </s> <s xml:id="echoid-s11616" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Erit fi- <anchor type="note" xlink:label="note-281-02a" xlink:href="note-281-02"/> gura ABCDTA, ad figurã ATEFGHA, <lb/>vt IS, ad SP, hoc eſt, vt Q, ad R.</s> <s xml:id="echoid-s11617" xml:space="preserve"/> </p> <div xml:id="echoid-div713" type="float" level="2" n="9"> <note symbol="a" position="right" xlink:label="note-281-01" xlink:href="note-281-01a" xml:space="preserve">3. hui{us}.</note> <figure xlink:label="fig-281-01" xlink:href="fig-281-01a"> <image file="281-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/281-01"/> </figure> <note symbol="b" position="right" xlink:label="note-281-02" xlink:href="note-281-02a" xml:space="preserve">1. hui{us}.</note> </div> <p> <s xml:id="echoid-s11618" xml:space="preserve"><emph style="sc">Ex</emph> angulo B, vel H, non poterit propoſita figura in quamcunque propor-<lb/>tionem diuidi: </s> <s xml:id="echoid-s11619" xml:space="preserve">quia lineæ ex eorum vtrolibet ad oppoſitos angulos emiſſæ par-<lb/>tim ſecant latera, & </s> <s xml:id="echoid-s11620" xml:space="preserve">partim cadunt extra figuram. </s> <s xml:id="echoid-s11621" xml:space="preserve">Quod ſi data proportio mi-<lb/>nor eſſet, quam figuræ BCDEB, (ſi nimirum intelligatur ducta recta BE,) ad figu-<lb/>ram BEFGHAB, tum demum diuidi poſſet ex B, tota figura in datam propor-<lb/>tionem: </s> <s xml:id="echoid-s11622" xml:space="preserve">propterea quod fierent duo triangula B C D, (ducta videlicet recta <lb/>B D,) B D E, ad punctum B, quorum baſes ſunt latera figuræ C D, D E; </s> <s xml:id="echoid-s11623" xml:space="preserve">alia vero <lb/>quatuor ABE, AEF, AFG, AGH, ad punctum A, quorum etiam baſes ſunt figu-<lb/>rælatera AB, EF, FG, GH, &</s> <s xml:id="echoid-s11624" xml:space="preserve">c.</s> <s xml:id="echoid-s11625" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11626" xml:space="preserve"><emph style="sc">Eodem</emph> modo quamcunque figuram rectilineam, etiam irregulariſsimam, <lb/>partiemur in datam proportionem, non quidem ex quolibet angulo, vel pun-<lb/>cto dato, (niſi ex eo duci poſsint rectæ ad omnes angulos oppoſitos, ex-<lb/>ceptis duobus proximis, quæ nullum figuræ latus interſecent: </s> <s xml:id="echoid-s11627" xml:space="preserve">cuiuſmo-<lb/> <anchor type="figure" xlink:label="fig-281-02a" xlink:href="fig-281-02"/> di eſſet punctum V, in antecedentifigura) ſed ex aliquo puncto particulari; </s> <s xml:id="echoid-s11628" xml:space="preserve">ſi <lb/>prius figura diuidatur in triangula ex pluribus punctis, ita vt quodlibet <lb/>triangulum habeat ſaltem vnum latus, quod etiam ſit latus figuræ. </s> <s xml:id="echoid-s11629" xml:space="preserve">Vt ſi fi-<lb/>gura hæc A B C D E F G H I K A, diuidatur in octo triangula, & </s> <s xml:id="echoid-s11630" xml:space="preserve">illis in recta <lb/>L T, <anchor type="note" xlink:href="" symbol="c"/> inueniantur totidem lineæ proportionales, totaq; </s> <s xml:id="echoid-s11631" xml:space="preserve">linea L T, ſecetur in Y, <anchor type="note" xlink:label="note-281-03a" xlink:href="note-281-03"/> <pb o="252" file="282" n="282" rhead="GEOMETR. PRACT."/> ſecundum datam proportionem V, ad X; </s> <s xml:id="echoid-s11632" xml:space="preserve">& </s> <s xml:id="echoid-s11633" xml:space="preserve">baſis IK, ſexti trianguli (quod pun-<lb/>ctum Y, cadat in ſextam lineam QR,) ſecetur in Z, vt ſecta eſt QR, in Y; </s> <s xml:id="echoid-s11634" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ita vt <anchor type="note" xlink:label="note-282-01a" xlink:href="note-282-01"/> ductarecta F Z, triangulum FIK, ſectum ſit, vt ſecta eſt baſis IK, hoc eſt recta QR: <lb/></s> <s xml:id="echoid-s11635" xml:space="preserve"> <anchor type="note" xlink:label="note-282-02a" xlink:href="note-282-02"/> <anchor type="note" xlink:href="" symbol="b"/> Erit figura ABCDEFZKA, ad figuram FZIHGF, vt LY, ad YT, hoc eſt, vt V, ad X, Et ſic de cæteris.</s> <s xml:id="echoid-s11636" xml:space="preserve"/> </p> <div xml:id="echoid-div714" type="float" level="2" n="10"> <figure xlink:label="fig-281-02" xlink:href="fig-281-02a"> <image file="281-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/281-02"/> </figure> <note symbol="c" position="right" xlink:label="note-281-03" xlink:href="note-281-03a" xml:space="preserve">3. hui{us}.</note> <note symbol="a" position="left" xlink:label="note-282-01" xlink:href="note-282-01a" xml:space="preserve">1. ſexti.</note> <note symbol="b" position="left" xlink:label="note-282-02" xlink:href="note-282-02a" xml:space="preserve">1. hui{us}.</note> </div> </div> <div xml:id="echoid-div716" type="section" level="1" n="248"> <head xml:id="echoid-head273" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s11637" xml:space="preserve"><emph style="sc">His</emph> ritè intellectis, licebit nobis quamlibet figuram ſecare in quotuis par-<lb/> <anchor type="note" xlink:label="note-282-03a" xlink:href="note-282-03"/> tes æquales ex dato angulo, vel pũcto in latere, ex quo duci poſsint ad omnes <lb/>angulos, exceptis proximis duobus, rectæ lineæ, ita vt nullum figuræ latus ſe-<lb/>cent. </s> <s xml:id="echoid-s11638" xml:space="preserve">Nam ſi propoſita figura ſit ſecanda, verbi gratia in 7. </s> <s xml:id="echoid-s11639" xml:space="preserve">partes æquales, di-<lb/>uidemus eam ſecundum proportionem 1. </s> <s xml:id="echoid-s11640" xml:space="preserve">ad 6. </s> <s xml:id="echoid-s11641" xml:space="preserve">nimirum ſecundum ſub multi-<lb/>plicem denominatam à denominatore partium, minus vno, in quas figura diui-<lb/>denda eſt. </s> <s xml:id="echoid-s11642" xml:space="preserve">Ita enim prior pars erit {1/7}. </s> <s xml:id="echoid-s11643" xml:space="preserve">totius figuræ, cum poſterior 6. </s> <s xml:id="echoid-s11644" xml:space="preserve">eiuſmodi <lb/>partes complectatur. </s> <s xml:id="echoid-s11645" xml:space="preserve">Hanc deinde poſteriorem partem ſecabimus ſecundum <lb/>proportionem 1. </s> <s xml:id="echoid-s11646" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s11647" xml:space="preserve">ita vt prior pars contineat {1/6}: </s> <s xml:id="echoid-s11648" xml:space="preserve">ipſius, hoc eſt, {1/7}. </s> <s xml:id="echoid-s11649" xml:space="preserve">totius fi-<lb/>guræ. </s> <s xml:id="echoid-s11650" xml:space="preserve">Poſt hæc poſteriorem huius ſecundæ diuiſionis partem partiemur ſecũ-<lb/>dum proportionem 1. </s> <s xml:id="echoid-s11651" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s11652" xml:space="preserve">Ac rurſus partem huius tertiæ diuiſionis poſterio-<lb/>rem diuidemus ſecundum proportionem 1. </s> <s xml:id="echoid-s11653" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s11654" xml:space="preserve">Atq; </s> <s xml:id="echoid-s11655" xml:space="preserve">poſteriorem huius quar-<lb/>tæ diuiſionis partem ſecabimus ſecundum proportionem 1. </s> <s xml:id="echoid-s11656" xml:space="preserve">ad 2. </s> <s xml:id="echoid-s11657" xml:space="preserve">Ac poſtremo <lb/>partem poſteriorem huius quintæ diuiſionis partiemur in duas partes æquales, <lb/>nimirum ſecundum proportionem 1. </s> <s xml:id="echoid-s11658" xml:space="preserve">ad 1.</s> <s xml:id="echoid-s11659" xml:space="preserve"/> </p> <div xml:id="echoid-div716" type="float" level="2" n="1"> <note position="left" xlink:label="note-282-03" xlink:href="note-282-03a" xml:space="preserve">Quo pacto <lb/>figura data <lb/>ſecetur ex da-<lb/>to angulo vel <lb/>puncto in late <lb/>re, in quotuis <lb/>partes æqua-<lb/>l{es}.</note> </div> <p> <s xml:id="echoid-s11660" xml:space="preserve"><emph style="sc">Non</emph> aliter figuram irregularem, in qua à nullo angulo, vel puncto in late-<lb/>re, duci poſſunt rectæ ad angulos oppoſitos, quin aliqua figuræ latera ſecentur, <lb/>diuidere licebit in partes quotuis æquales, ex diuerſis angulis, vel punctis. </s> <s xml:id="echoid-s11661" xml:space="preserve">Nam <lb/>ſi verbi gratia vltima figura huius propoſitionis diuidenda ſit in 5. </s> <s xml:id="echoid-s11662" xml:space="preserve">partes æqua-<lb/>les; </s> <s xml:id="echoid-s11663" xml:space="preserve">reſecabimus ex ea, ab aliquo angulo, vel puncto, quintã partem. </s> <s xml:id="echoid-s11664" xml:space="preserve">Deinde <lb/>ex maiore parte complectente {4/5}. </s> <s xml:id="echoid-s11665" xml:space="preserve">totius figuræ, ab aliquo eius angulo, vel pun-<lb/>cto, detrahemus quartam partem: </s> <s xml:id="echoid-s11666" xml:space="preserve">Item tertiam partem ex maiore parte huius <lb/>diuiſionis: </s> <s xml:id="echoid-s11667" xml:space="preserve">Ac tandem ſemiſſem ex vltima parte poſtremæ huius diuiſionis. </s> <s xml:id="echoid-s11668" xml:space="preserve">Hac <lb/>enim ratione diuiſa erit tota figura in 5. </s> <s xml:id="echoid-s11669" xml:space="preserve">partes: </s> <s xml:id="echoid-s11670" xml:space="preserve">non ſecus atq; </s> <s xml:id="echoid-s11671" xml:space="preserve">in linea recta A B, <lb/>contingit. </s> <s xml:id="echoid-s11672" xml:space="preserve">Si namque eam partiri iubeamur in 7. </s> <s xml:id="echoid-s11673" xml:space="preserve">partes æquales, efficiemus <lb/> <anchor type="figure" xlink:label="fig-282-01a" xlink:href="fig-282-01"/> id, ſi primo loco ſeptimam partem AC, detrahemus; </s> <s xml:id="echoid-s11674" xml:space="preserve">deinde {1/6}. </s> <s xml:id="echoid-s11675" xml:space="preserve">C D, ex reliqua <lb/>linea CB; </s> <s xml:id="echoid-s11676" xml:space="preserve">& </s> <s xml:id="echoid-s11677" xml:space="preserve">ex reliqua DB, quintam partem DE: </s> <s xml:id="echoid-s11678" xml:space="preserve">& </s> <s xml:id="echoid-s11679" xml:space="preserve">ex reliqua EB, quartam par-<lb/>tem DE: </s> <s xml:id="echoid-s11680" xml:space="preserve">& </s> <s xml:id="echoid-s11681" xml:space="preserve">ex reliqua FB, tertiam partem F G: </s> <s xml:id="echoid-s11682" xml:space="preserve">Ac deniq; </s> <s xml:id="echoid-s11683" xml:space="preserve">relin quam lineam GB, <lb/>bifariam ſecabimus in H, hoc eſt, ſemiſſem GH, ex ea abſcindemus.</s> <s xml:id="echoid-s11684" xml:space="preserve"/> </p> <div xml:id="echoid-div717" type="float" level="2" n="2"> <figure xlink:label="fig-282-01" xlink:href="fig-282-01a"> <image file="282-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/282-01"/> </figure> </div> <p> <s xml:id="echoid-s11685" xml:space="preserve"><emph style="sc">Facilivs</emph> idem exequemur, quando ex dato angulo, vel puncto, duci <lb/>poſſunt rectæ ad omnes angulos, duo bus proximis exceptis, nullum figuræ la-<lb/>tus ſecantes, hacratione. </s> <s xml:id="echoid-s11686" xml:space="preserve">Lineam ex rectis, quę triangulis figuræ proportiona-<lb/>les ſunt, cõflatã ſecabim<emph style="sub">9</emph> in tot partes æquales, in quot figurã partiriiubemur. </s> <s xml:id="echoid-s11687" xml:space="preserve">Si <pb o="253" file="283" n="283" rhead="LIBER SEXTVS."/> enim baſes triangulorum dato angulo, vel puncto oppoſitas, quæ lineis, in <lb/>quas puncta diuiſionum cadunt, reſpondent, ita diuidemus, vt ſectæ ſunt re-<lb/>ſpondentes lineæ, atque ex dato angulo, vel puncto, ad diuiſionum puncta re-<lb/>ctas ducemus, factum erit, quod proponitur. </s> <s xml:id="echoid-s11688" xml:space="preserve">Vt ſi ſecunda figura huius pro-<lb/>poſ. </s> <s xml:id="echoid-s11689" xml:space="preserve">ſecanda ſit in 5. </s> <s xml:id="echoid-s11690" xml:space="preserve">partes æquales, partiemur lineam G I, in 5. </s> <s xml:id="echoid-s11691" xml:space="preserve">æquales partes <lb/>G H, Hb, b a, a K, KL. </s> <s xml:id="echoid-s11692" xml:space="preserve">Et quoniam primum punctum H, cadit in H, erit trian-<lb/>gulum ABC, quintæ figuræ pars; </s> <s xml:id="echoid-s11693" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> cum ſit vt GH, ad HL, ita triangulum ABC, <anchor type="note" xlink:label="note-283-01a" xlink:href="note-283-01"/> ad reliquam partem figuræ. </s> <s xml:id="echoid-s11694" xml:space="preserve">Deinde ſecabimus baſem ſecundi trianguli in d, vt <lb/>ſecunda linea HI, ſecta eſt in b: </s> <s xml:id="echoid-s11695" xml:space="preserve">Et baſem tertii trianguli in e, vt tertia linea <lb/>IK, ſecta eſt in a; </s> <s xml:id="echoid-s11696" xml:space="preserve">rectaſque ducemus A d, A e. </s> <s xml:id="echoid-s11697" xml:space="preserve">Quia verò quartum punctum K, <lb/>cadit in K, terminum quartæ lineæ IK, erit figura diuiſa in 5. </s> <s xml:id="echoid-s11698" xml:space="preserve">partes æquales ABC, <lb/> <anchor type="figure" xlink:label="fig-283-01a" xlink:href="fig-283-01"/> A C d, A d D e, A e E, AEF: </s> <s xml:id="echoid-s11699" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> propterea quod hæ partes partibus GH, Hb, b a, <anchor type="note" xlink:label="note-283-02a" xlink:href="note-283-02"/> a K, K L, proportionales ſunt.</s> <s xml:id="echoid-s11700" xml:space="preserve"/> </p> <div xml:id="echoid-div718" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-283-01" xlink:href="note-283-01a" xml:space="preserve">1. hui{us}.</note> <figure xlink:label="fig-283-01" xlink:href="fig-283-01a"> <image file="283-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/283-01"/> </figure> <note symbol="b" position="right" xlink:label="note-283-02" xlink:href="note-283-02a" xml:space="preserve">1. hui{us}.</note> </div> <p> <s xml:id="echoid-s11701" xml:space="preserve"><emph style="sc">Neqve</emph> verò difficile erit hanc eandem rationem figuris irregularibus, qua-<lb/>lis eſt vltima huius propoſ. </s> <s xml:id="echoid-s11702" xml:space="preserve">accommodare. </s> <s xml:id="echoid-s11703" xml:space="preserve">Si enim ea diuidenda ſit, verbi gra-<lb/>tia in tres partes æquales, ſecanda erit linea L T, in tres æquales partes L a, a b, <lb/>b T. </s> <s xml:id="echoid-s11704" xml:space="preserve">Et quarti trianguli baſis DE, diuidenda in d, vt quarta linea O P, diuiſa eſt <lb/>in a: </s> <s xml:id="echoid-s11705" xml:space="preserve">Item ſexti trianguli baſis K I, ſecanda in f, vt ſexta linea Q R, in b, ſecta <lb/>eſt. </s> <s xml:id="echoid-s11706" xml:space="preserve">Nam ſi ex K, angulo baſi D E, oppoſito recta ducatur K d: </s> <s xml:id="echoid-s11707" xml:space="preserve">Item ex angulo <lb/>F, baſi KI, oppoſito recta F f, erunt tres partes figuræ ABCD d KA, K d E F f K, <lb/>f FGHI f, inter ſe æquales: </s> <s xml:id="echoid-s11708" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> cum ſint rectis La, a b, b T, proportionales. </s> <s xml:id="echoid-s11709" xml:space="preserve">Ea- <anchor type="note" xlink:label="note-283-03a" xlink:href="note-283-03"/> demque de cæteris ratio eſt.</s> <s xml:id="echoid-s11710" xml:space="preserve"/> </p> <div xml:id="echoid-div719" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-283-03" xlink:href="note-283-03a" xml:space="preserve">1. hui{us}.</note> </div> </div> <div xml:id="echoid-div721" type="section" level="1" n="249"> <head xml:id="echoid-head274" xml:space="preserve">PROBLEMA 4. PROPOSITIO 5.</head> <p> <s xml:id="echoid-s11711" xml:space="preserve">DATVM rectilineum per rectam lineam datæ rectæ parallelam in dà<unsure/>-<lb/>tam proportionem diuidere, ita vt antecedens proportionis in quam <lb/>elegeris partem vergat.</s> <s xml:id="echoid-s11712" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s11713" xml:space="preserve"><emph style="sc">Sit</emph> primo triangulum A B C, diuidendum in duas partes per lineam lateri <lb/>B C, parallelam, vt pars verſus A, ad reliquam habeat proportionem datam D, <lb/>ad E. </s> <s xml:id="echoid-s11714" xml:space="preserve">Alterutro laterũ, cui linea diuidens æquidiſtare non debet, videlicet A C, <lb/>diuiſo in F, vt eadem ſit proportio AF, ad FC, quæ D, ad E, initio facto ab angu-<lb/>lo A, verſus quem antecedens proportionis vergere debet, reperiatur inter to-<lb/>tum latus AC, & </s> <s xml:id="echoid-s11715" xml:space="preserve">eius partem AF, quæ terminatur in angulo A, qui lateri oppo- <pb o="254" file="284" n="284" rhead="GEOMETR. PRACT."/> nitur, cuiæ quidiſtans ducenda eſt, media proportionalis A G, agaturque per G, <lb/>ipſi B C, parallela G H. </s> <s xml:id="echoid-s11716" xml:space="preserve">Dico hanc parallelam problema efficere, id eſt, eandem <lb/>eſſe proportionem trianguli A G H, ad trapezium B C G H, quæ eſt D, ad E. <lb/></s> <s xml:id="echoid-s11717" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/> Quoniam enim triangulum ABC, ad triangulum A G H, eſt vt latus A C, ad re- <anchor type="note" xlink:label="note-284-01a" xlink:href="note-284-01"/> ctam AF, quod tres rectæ AC, AG, AF, continuè proportionales ſint, & </s> <s xml:id="echoid-s11718" xml:space="preserve">triangu-<lb/> <anchor type="figure" xlink:label="fig-284-01a" xlink:href="fig-284-01"/> la ABC, AGH, ſuper A C, A G, ſimilia ſimiliter que poſita; </s> <s xml:id="echoid-s11719" xml:space="preserve">Erit per conuerſio-<lb/>nem rationis triangulum ABC, ad trapezium BCGH, vt AC, ad FC. </s> <s xml:id="echoid-s11720" xml:space="preserve">Ergo diui-<lb/>dendo erit triangulum AGH, ad trapezium BCGH, vt AF, ad FC, hoc eſt, vt D, <lb/>ad E. </s> <s xml:id="echoid-s11721" xml:space="preserve">quod eſt prop oſitum. </s> <s xml:id="echoid-s11722" xml:space="preserve">Quod etiam ita colligemus. </s> <s xml:id="echoid-s11723" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam eſt trian- <anchor type="note" xlink:label="note-284-02a" xlink:href="note-284-02"/> gulum A B C, ad triangulum A G H, vt recta A C, ad AF; </s> <s xml:id="echoid-s11724" xml:space="preserve">erit diuidendo trape-<lb/>zium B G, ad triangulum A G H, vt F C, ad AF: </s> <s xml:id="echoid-s11725" xml:space="preserve">Et conuertendo triangulum <lb/>AGH, ad trapezium BG, vt AF ad FC, hoc eſt, vt D, ad E.</s> <s xml:id="echoid-s11726" xml:space="preserve"/> </p> <div xml:id="echoid-div721" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-284-01" xlink:href="note-284-01a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> <figure xlink:label="fig-284-01" xlink:href="fig-284-01a"> <image file="284-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/284-01"/> </figure> <note symbol="b" position="left" xlink:label="note-284-02" xlink:href="note-284-02a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> </div> </div> <div xml:id="echoid-div723" type="section" level="1" n="250"> <head xml:id="echoid-head275" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s11727" xml:space="preserve"><emph style="sc">Diviso</emph> latere quo cunque, nimirum B C, in I, ſecundum datam proportio-<lb/>nem D, ad E, iuncta que recta AI: </s> <s xml:id="echoid-s11728" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit triangulum ABI, ad triangulum AIC, vt <anchor type="note" xlink:label="note-284-03a" xlink:href="note-284-03"/> BI, ad IC, id eſt, vt D, ad E. </s> <s xml:id="echoid-s11729" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Siigitur ſuper B C, inter rectas BA, CA, conſtitua- <anchor type="note" xlink:label="note-284-04a" xlink:href="note-284-04"/> tur trapezium BG, per parallelam G H, æquale triangulo AIC, quod eſt conſe-<lb/>quens; </s> <s xml:id="echoid-s11730" xml:space="preserve">erit reliquum triangulum A G H, reliquo triangulo ABI, æquale. </s> <s xml:id="echoid-s11731" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Qua- <anchor type="note" xlink:label="note-284-05a" xlink:href="note-284-05"/> re erit triangulum AGH, ad trapezium BG, vt triangulum ABI, ad triangulum <lb/>AIC, hoc eſt, vt BI, ad IC, vel vt D, ad E. </s> <s xml:id="echoid-s11732" xml:space="preserve">Sed prior via expeditior eſt: </s> <s xml:id="echoid-s11733" xml:space="preserve">placuit ta-<lb/>men hanc alteram etiam proponere; </s> <s xml:id="echoid-s11734" xml:space="preserve">quia generalis fermè eſt, & </s> <s xml:id="echoid-s11735" xml:space="preserve">in omnes figu-<lb/>ras multilateras quadrat, vt infra patebit.</s> <s xml:id="echoid-s11736" xml:space="preserve"/> </p> <div xml:id="echoid-div723" type="float" level="2" n="1"> <note symbol="c" position="left" xlink:label="note-284-03" xlink:href="note-284-03a" xml:space="preserve">1. ſexti.</note> <note symbol="d" position="left" xlink:label="note-284-04" xlink:href="note-284-04a" xml:space="preserve">2. hui{us}.</note> <note symbol="e" position="left" xlink:label="note-284-05" xlink:href="note-284-05a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s11737" xml:space="preserve"><emph style="sc">Non</emph> ſecus id, quod propoſitum eſt, exequemur, ſi antecedens proportio-<lb/>nis vergere debeat ad latus BC, cui recta ducenda eſt parallela. </s> <s xml:id="echoid-s11738" xml:space="preserve">Diuiſo enim rur-<lb/>ſus latere A C, in K, ſecundum datam proportionem D, ad E, ita vt antecedens <lb/>proportionis incipiat à latere B C; </s> <s xml:id="echoid-s11739" xml:space="preserve">inueniatur inter totam A C, & </s> <s xml:id="echoid-s11740" xml:space="preserve">eius partem <lb/>A K, quæ eſt conſequens proportionis, media proportionalis A L, agaturque <lb/>per L, lateri B C, parallela L M. </s> <s xml:id="echoid-s11741" xml:space="preserve">Dico hanc parallelam problema efficere, hoc eſt, <lb/>eandem eſſe proportionem trapezij BL, ad triangulum A L M, quæ eſt D, ad E. <lb/></s> <s xml:id="echoid-s11742" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="f"/> Quia enim eſt, vt triangulum ABC, ad triangulum ALM, ita CA, ad AK: </s> <s xml:id="echoid-s11743" xml:space="preserve">quod <anchor type="note" xlink:label="note-284-06a" xlink:href="note-284-06"/> tres rectæ AC, AL, AK, continuè ſint proportionales, & </s> <s xml:id="echoid-s11744" xml:space="preserve">triangula ABC, ALM, <lb/>ſuper A C, AL, ſimilia ſimiliter que poſita; </s> <s xml:id="echoid-s11745" xml:space="preserve">Erit diuidendo trapezium BL, ad <lb/>triangulum ALM, vt CK, ad AK, hoc eſt, vt D, ad E, quod eſt propoſitum. </s> <s xml:id="echoid-s11746" xml:space="preserve">Hoc <lb/>idem effici etiam poteſt via illa altera generali, quamuis non ita expeditè. </s> <s xml:id="echoid-s11747" xml:space="preserve">Di- <pb o="255" file="285" n="285" rhead="LIBER SEXTVS."/> uiſo enim quo cunque latere, nimirum B C, in I, ſecundum datam proportio-<lb/>nem D, ad E, iunctaque recta AI, <anchor type="note" xlink:href="" symbol="a"/> erit triangulum ABI, ad triangulum AIC, vt <anchor type="note" xlink:label="note-285-01a" xlink:href="note-285-01"/> BI, ad IC, id eſt, vt D, ad E. </s> <s xml:id="echoid-s11748" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Siigitur ſuper B C, inter rectas BA, CA, conſtitua- <anchor type="note" xlink:label="note-285-02a" xlink:href="note-285-02"/> tur trapezium BL, per parallelam LM, æquale triangulo ABI, quod eſt antece-<lb/>dens: </s> <s xml:id="echoid-s11749" xml:space="preserve">erit reliquum triangulum ALM, reliquo triangulo AIC, æquale. </s> <s xml:id="echoid-s11750" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Qua- <anchor type="note" xlink:label="note-285-03a" xlink:href="note-285-03"/> re erit trapezium BL, ad triangulum A L M, vt triangulum ABI, ad triangulum <lb/>AIC, hoc eſt, vt BI, ad IC, vel vt D, ad E.</s> <s xml:id="echoid-s11751" xml:space="preserve"/> </p> <div xml:id="echoid-div724" type="float" level="2" n="2"> <note symbol="f" position="left" xlink:label="note-284-06" xlink:href="note-284-06a" xml:space="preserve">coroll 19. <lb/>ſexti.</note> <note symbol="a" position="right" xlink:label="note-285-01" xlink:href="note-285-01a" xml:space="preserve">1. ſexti.</note> <note symbol="b" position="right" xlink:label="note-285-02" xlink:href="note-285-02a" xml:space="preserve">2. hui{us}.</note> <note symbol="c" position="right" xlink:label="note-285-03" xlink:href="note-285-03a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s11752" xml:space="preserve"><emph style="sc">Sed</emph> diuidendum iam ſit triangulum ABC, in datam proportionem D, ad E, <lb/>per lineam parallelam cuicunque lineæ F: </s> <s xml:id="echoid-s11753" xml:space="preserve">quæ ſi æquidiſtet vni laterum, par-<lb/>tiemur triangulum in datam proportionem, per rectam illi lateri parallelam, <lb/> <anchor type="figure" xlink:label="fig-285-01a" xlink:href="fig-285-01"/> vt iam tradidimus, ſiue antecedens vergere debeat ad angulum lateri illi oppo-<lb/> <anchor type="note" xlink:label="note-285-04a" xlink:href="note-285-04"/> ſitum, ſiue ad ipſummet latus: </s> <s xml:id="echoid-s11754" xml:space="preserve">factumque erit, quod iubetur; </s> <s xml:id="echoid-s11755" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> cum parallela illa æquidiſtet etiam datæ rectæ lineæ.</s> <s xml:id="echoid-s11756" xml:space="preserve"/> </p> <div xml:id="echoid-div725" type="float" level="2" n="3"> <figure xlink:label="fig-285-01" xlink:href="fig-285-01a"> <image file="285-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/285-01"/> </figure> <note symbol="d" position="right" xlink:label="note-285-04" xlink:href="note-285-04a" xml:space="preserve">30. primi.</note> </div> <p> <s xml:id="echoid-s11757" xml:space="preserve"><emph style="sc">Si</emph> verò data recta F, nulli laterum æquidiſtet, ducatur illi ex aliquo angulo <lb/>parallela intra triangulum cadens, qualis eſt B G. </s> <s xml:id="echoid-s11758" xml:space="preserve">Si igitur antecedens propor-<lb/>tionis ſtatuendum ſit ad partes A, ſecabimus latus A C, in H, in datam propor-<lb/>tionem D, ad E. </s> <s xml:id="echoid-s11759" xml:space="preserve">Cadat autem primo punctum H, inter G, & </s> <s xml:id="echoid-s11760" xml:space="preserve">C; </s> <s xml:id="echoid-s11761" xml:space="preserve">inueniatur-<lb/>que inter GC, & </s> <s xml:id="echoid-s11762" xml:space="preserve">eius partem CH, terminatam in angulo C, parallelæ BG, oppo-<lb/>ſito media proportionalis C I; </s> <s xml:id="echoid-s11763" xml:space="preserve">& </s> <s xml:id="echoid-s11764" xml:space="preserve">per I, ipſi B G, vel ipſi F, parallela agatur I K; <lb/></s> <s xml:id="echoid-s11765" xml:space="preserve">quam dico, problema efficere: </s> <s xml:id="echoid-s11766" xml:space="preserve">hoc eſt, eſſe trapezium A B K I, ad triangulum <lb/>IKC, vt D, ad E. </s> <s xml:id="echoid-s11767" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Quoniam enim eſt (ducta recta B H,) vt triangulum GBC, ad <anchor type="note" xlink:label="note-285-05a" xlink:href="note-285-05"/> triangulum IKC, ita GC, ad CH; </s> <s xml:id="echoid-s11768" xml:space="preserve">quod tres GC, CI, CH, ſint continuè pro-<lb/>portionales, & </s> <s xml:id="echoid-s11769" xml:space="preserve">triangula ſimilia ſimiliter que poſita: </s> <s xml:id="echoid-s11770" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Vt autem GC, ad CH, ita <anchor type="note" xlink:label="note-285-06a" xlink:href="note-285-06"/> eſt quo que triangulum G B C, ad triangulum H B C; </s> <s xml:id="echoid-s11771" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erunt triangula I K C, <anchor type="note" xlink:label="note-285-07a" xlink:href="note-285-07"/> HBC, æqualia: </s> <s xml:id="echoid-s11772" xml:space="preserve">Ac proinde & </s> <s xml:id="echoid-s11773" xml:space="preserve">reliquum trapezium A B K I, reliquo triangulo <lb/>ABH, æquale erit. </s> <s xml:id="echoid-s11774" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Igitur erit vt trapezium ABKI, ad triangulum ICK, ita trian- <anchor type="note" xlink:label="note-285-08a" xlink:href="note-285-08"/> gulum ABH, ad triangulum HBC: </s> <s xml:id="echoid-s11775" xml:space="preserve">hoc eſt, ita AH, ad HC, velita D, ad E. </s> <s xml:id="echoid-s11776" xml:space="preserve">quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s11777" xml:space="preserve"/> </p> <div xml:id="echoid-div726" type="float" level="2" n="4"> <note symbol="e" position="right" xlink:label="note-285-05" xlink:href="note-285-05a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> <note symbol="f" position="right" xlink:label="note-285-06" xlink:href="note-285-06a" xml:space="preserve">1. ſexti.</note> <note symbol="g" position="right" xlink:label="note-285-07" xlink:href="note-285-07a" xml:space="preserve">9. quinti.</note> <note symbol="h" position="right" xlink:label="note-285-08" xlink:href="note-285-08a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s11778" xml:space="preserve"><emph style="sc">Cadat</emph> deinde punctum L, (Ponimus iam datam proportionem E, ad D, <lb/>diuiſamque eſſe A C, in L, ſecundum datam proportionem) inter A, & </s> <s xml:id="echoid-s11779" xml:space="preserve">G. </s> <s xml:id="echoid-s11780" xml:space="preserve">In-<lb/>uenta ergo inter G A, & </s> <s xml:id="echoid-s11781" xml:space="preserve">eius partem AL, terminatã in angulo A, parallelæ B G, <lb/>oppoſito, media proportionali AM, agatur per M, ipſi GB, ideo que & </s> <s xml:id="echoid-s11782" xml:space="preserve">ipſi F, pa-<lb/>rallela MN: </s> <s xml:id="echoid-s11783" xml:space="preserve">quam dico problema efficere, hoc eſt, eſſe triangulum A M N, ad <lb/>trapezium M N B C, vt E, ad D. </s> <s xml:id="echoid-s11784" xml:space="preserve">Iuncta namque recta B L, <anchor type="note" xlink:href="" symbol="i"/> quoniam eſt, vt <anchor type="note" xlink:label="note-285-09a" xlink:href="note-285-09"/> triangulum A B G, ad triangulum A M N, ita AG, ad AL: </s> <s xml:id="echoid-s11785" xml:space="preserve">quod tres AG, AM, <lb/>AL, continuè proportionales ſint, & </s> <s xml:id="echoid-s11786" xml:space="preserve">triangula ſimilia ſimiliter que poſita: </s> <s xml:id="echoid-s11787" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Vt <anchor type="note" xlink:label="note-285-10a" xlink:href="note-285-10"/> <pb o="256" file="286" n="286" rhead="GEOMETR. PRACT."/> autem A G, ad A L, ita eſt idẽ quoq; </s> <s xml:id="echoid-s11788" xml:space="preserve">triangulũ A B G, ad triangulum A B L; </s> <s xml:id="echoid-s11789" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> @runt <anchor type="note" xlink:label="note-286-01a" xlink:href="note-286-01"/> triangula A M N, A B L, æqualia: </s> <s xml:id="echoid-s11790" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s11791" xml:space="preserve">reliquum trapezium M N B C, reli-<lb/>quo triangulo L B C, æquale erit. </s> <s xml:id="echoid-s11792" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quapropter erit triangulum A M N, ad tra- <anchor type="note" xlink:label="note-286-02a" xlink:href="note-286-02"/> pezium M N B C, vt triangulum A B L, ad triangulum L B C, <anchor type="note" xlink:href="" symbol="c"/>hoc eſt, vt A L, ad <anchor type="note" xlink:label="note-286-03a" xlink:href="note-286-03"/> L C, vel E, ad D. </s> <s xml:id="echoid-s11793" xml:space="preserve">quod eſt propoſitum,</s> </p> <div xml:id="echoid-div727" type="float" level="2" n="5"> <note symbol="i" position="right" xlink:label="note-285-09" xlink:href="note-285-09a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> <note symbol="k" position="right" xlink:label="note-285-10" xlink:href="note-285-10a" xml:space="preserve">1. ſexti.</note> <note symbol="a" position="left" xlink:label="note-286-01" xlink:href="note-286-01a" xml:space="preserve">9. quinti.</note> <note symbol="b" position="left" xlink:label="note-286-02" xlink:href="note-286-02a" xml:space="preserve">7. quinti.</note> <note symbol="c" position="left" xlink:label="note-286-03" xlink:href="note-286-03a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11794" xml:space="preserve"><emph style="sc">Qvod</emph> ſi punctum diuiſionis caderetin G, quod contingeret, ſi data eſſet <lb/>proportio O, ad P; </s> <s xml:id="echoid-s11795" xml:space="preserve">Ipſamet parallela B G, problema effi ceret; </s> <s xml:id="echoid-s11796" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> cum ſit triangu- <anchor type="note" xlink:label="note-286-04a" xlink:href="note-286-04"/> lum A B G, ad triangulum G B C, vt A G, G C, hoc eſt, vt O, ad P. </s> <s xml:id="echoid-s11797" xml:space="preserve">Et ſi antecedens <lb/>ſtatui debet verſus C, & </s> <s xml:id="echoid-s11798" xml:space="preserve">data proportio eſſet P, ad O; </s> <s xml:id="echoid-s11799" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit quo que triangulum <anchor type="note" xlink:label="note-286-05a" xlink:href="note-286-05"/> C B G, ad triangulum A B G, vt C G, ad G A, vel vt P, ad O.</s> <s xml:id="echoid-s11800" xml:space="preserve"/> </p> <div xml:id="echoid-div728" type="float" level="2" n="6"> <note symbol="d" position="left" xlink:label="note-286-04" xlink:href="note-286-04a" xml:space="preserve">1. ſexti.</note> <note symbol="e" position="left" xlink:label="note-286-05" xlink:href="note-286-05a" xml:space="preserve">1. ſext.</note> </div> </div> <div xml:id="echoid-div730" type="section" level="1" n="251"> <head xml:id="echoid-head276" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s11801" xml:space="preserve"><emph style="sc">Diviso</emph> quouis latere, videlicet in A C, in H, ſecundum datam proportio-<lb/>nem D, ad E, iuncta que recta B H; </s> <s xml:id="echoid-s11802" xml:space="preserve">quia punctum H, cadit inter G, & </s> <s xml:id="echoid-s11803" xml:space="preserve">C, fiat ſuper <lb/> <anchor type="note" xlink:label="note-286-06a" xlink:href="note-286-06"/> B G, inter rectas B C, G C, trapezium B I, per parallelam I K, æquale triangulo <lb/>B G H: </s> <s xml:id="echoid-s11804" xml:space="preserve">Erit que propterea totum trapezium A B K I, to titriangulo A B H, æquale, <lb/>at que id circo & </s> <s xml:id="echoid-s11805" xml:space="preserve">reliquum triangulum I K C, reliquo triangulo B C H. </s> <s xml:id="echoid-s11806" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Quocir- <anchor type="note" xlink:label="note-286-07a" xlink:href="note-286-07"/> circa erit trapezium A B K I, ad triangulum I K C, vt triangulum A B H, ad triangu-<lb/>lum B C H, <anchor type="note" xlink:href="" symbol="h"/> hoc eſt, vt A H, ad H C, vel vt D, ad E.</s> <s xml:id="echoid-s11807" xml:space="preserve"/> </p> <div xml:id="echoid-div730" type="float" level="2" n="1"> <note symbol="f" position="left" xlink:label="note-286-06" xlink:href="note-286-06a" xml:space="preserve">2. hui{us}.</note> <note symbol="g" position="left" xlink:label="note-286-07" xlink:href="note-286-07a" xml:space="preserve">7. quinti.</note> </div> <note symbol="h" position="left" xml:space="preserve">1. ſexti.</note> <p> <s xml:id="echoid-s11808" xml:space="preserve"><emph style="sc">Diviso</emph> rurſus latere A C, in L, ſecundum datam proportionem E, ad D, <lb/>iunctaque recta B L, quoniam punctum L, cadit inter A, & </s> <s xml:id="echoid-s11809" xml:space="preserve">G. </s> <s xml:id="echoid-s11810" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> fiat ſuper B G, in- <anchor type="note" xlink:label="note-286-09a" xlink:href="note-286-09"/> ter rectas B A, G A, trapezium B M, per parallelam M N, æquale triangulo B G L: <lb/></s> <s xml:id="echoid-s11811" xml:space="preserve">Erit que propterea & </s> <s xml:id="echoid-s11812" xml:space="preserve">totum trapezium M N B C, toti triangulo L B C, & </s> <s xml:id="echoid-s11813" xml:space="preserve">reli-<lb/>quum triangulum A M N, reliquo triangulo A B L, æquale, <anchor type="note" xlink:href="" symbol="k"/> Quam obrem erit <anchor type="note" xlink:label="note-286-10a" xlink:href="note-286-10"/> triangulum A M N, ad trapezium M N B C, vt triangulum A B L, ad triangulum <lb/>L B C, <anchor type="note" xlink:href="" symbol="l"/> hoc eſt, vt A L, ad L C, vel vt E, ad D. </s> <s xml:id="echoid-s11814" xml:space="preserve">quod erat faciendum.</s> <s xml:id="echoid-s11815" xml:space="preserve"/> </p> <div xml:id="echoid-div731" type="float" level="2" n="2"> <note symbol="i" position="left" xlink:label="note-286-09" xlink:href="note-286-09a" xml:space="preserve">2. hui{us}.</note> <note symbol="k" position="left" xlink:label="note-286-10" xlink:href="note-286-10a" xml:space="preserve">7. quinti.</note> </div> <note symbol="l" position="left" xml:space="preserve">1. ſexti.</note> <p> <s xml:id="echoid-s11816" xml:space="preserve"><emph style="sc">Non</emph> aliter problema abſoluemus, ſi antecedens proportionis vergere de-<lb/>beat ad partes C. </s> <s xml:id="echoid-s11817" xml:space="preserve">Diuiſo namque latere C A, in L, in proportionem datam D, <lb/>ad E: </s> <s xml:id="echoid-s11818" xml:space="preserve">Quoniam punctum L, caditinter A, & </s> <s xml:id="echoid-s11819" xml:space="preserve">G: </s> <s xml:id="echoid-s11820" xml:space="preserve">inueniemus inter G A, A L, me-<lb/>diam proportionalem A M, & </s> <s xml:id="echoid-s11821" xml:space="preserve">per M, ipſi B G, parallelam ducemus M N, quam <lb/>dico problema efficere, id eſt, ita eſſe trapezium M N B C, ad triangulum A M N, <lb/>vt D, ad E. </s> <s xml:id="echoid-s11822" xml:space="preserve">Iuncta namque recta B L: </s> <s xml:id="echoid-s11823" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> quoniam triangulum A B G, ad trian- <anchor type="note" xlink:label="note-286-12a" xlink:href="note-286-12"/> gulum A M N, eſt, vt A G, ad A L, quod tres A G, A M, A L, continuè ſint propor-<lb/>tionales, & </s> <s xml:id="echoid-s11824" xml:space="preserve">triangula ſimilia ſimiliterque poſita: </s> <s xml:id="echoid-s11825" xml:space="preserve"><anchor type="note" xlink:href="" symbol="n"/> Vt autem A G, ad A L, ita <anchor type="note" xlink:label="note-286-13a" xlink:href="note-286-13"/> eſt quo que idem triangulum A B G, ad triangulum A B L; </s> <s xml:id="echoid-s11826" xml:space="preserve"><anchor type="note" xlink:href="" symbol="o"/> æqualia erunt trian- <anchor type="note" xlink:label="note-286-14a" xlink:href="note-286-14"/> gula A M N, A B L; </s> <s xml:id="echoid-s11827" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s11828" xml:space="preserve">reliquum trapezium M N B C, reliquo trian-<lb/>gulo L B C, æquale erit. </s> <s xml:id="echoid-s11829" xml:space="preserve">Igitur erit trapezium M N B C, ad triangulum A M N, <lb/> <anchor type="note" xlink:label="note-286-15a" xlink:href="note-286-15"/> vt triangulum L B C, ad triangulum A B L, <anchor type="note" xlink:href="" symbol="p"/> hoc eſt, vt C L, ad L A, vel vt D, ad E.</s> <s xml:id="echoid-s11830" xml:space="preserve"/> </p> <div xml:id="echoid-div732" type="float" level="2" n="3"> <note symbol="m" position="left" xlink:label="note-286-12" xlink:href="note-286-12a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> <note symbol="n" position="left" xlink:label="note-286-13" xlink:href="note-286-13a" xml:space="preserve">1. ſexti.</note> <note symbol="o" position="left" xlink:label="note-286-14" xlink:href="note-286-14a" xml:space="preserve">9. quinti.</note> <note symbol="p" position="left" xlink:label="note-286-15" xlink:href="note-286-15a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11831" xml:space="preserve"><emph style="sc">Qvod</emph> ſi proportio data ſit E, ad D; </s> <s xml:id="echoid-s11832" xml:space="preserve">diuiſo eodem latere C A, in H, in pro-<lb/>portionem datam E, ad D, ductaquerecta B H; </s> <s xml:id="echoid-s11833" xml:space="preserve">quoniam punctum H, cadit in-<lb/>ter G, & </s> <s xml:id="echoid-s11834" xml:space="preserve">C, reperiemus inter G C, C H, mediam proportionalem C I, & </s> <s xml:id="echoid-s11835" xml:space="preserve">per I, pa-<lb/>rallelam ipſi B G, agemus I K: </s> <s xml:id="echoid-s11836" xml:space="preserve">quam dico problema effi cere, hoc eſt, eſſe trian-<lb/>gulum C I K, ad trapezium I K B A, vt C H, ad H A, vel vt E, ad D. </s> <s xml:id="echoid-s11837" xml:space="preserve">Iuncta enim re-<lb/>cta B H; </s> <s xml:id="echoid-s11838" xml:space="preserve"><anchor type="note" xlink:href="" symbol="q"/> quoniam eſt triangulum C B G, ad triangulum C I K, vt C G, ad C H;</s> <s xml:id="echoid-s11839" xml:space="preserve"> <anchor type="note" xlink:label="note-286-16a" xlink:href="note-286-16"/> quod tres C G, C I, C H, ſint continuè proportionales, & </s> <s xml:id="echoid-s11840" xml:space="preserve">triangula ſimilia ſimi- <pb o="257" file="287" n="287" rhead="LIBER SEXTVS."/> literque poſita: </s> <s xml:id="echoid-s11841" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Vt autem C G, ad C H, ita quoque eſt idem triangulum C B G, <anchor type="note" xlink:label="note-287-01a" xlink:href="note-287-01"/> ad triangulum C B H; </s> <s xml:id="echoid-s11842" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> æqualia erunt triangula C I K, C B H: </s> <s xml:id="echoid-s11843" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s11844" xml:space="preserve">reli- <anchor type="note" xlink:label="note-287-02a" xlink:href="note-287-02"/> <anchor type="figure" xlink:label="fig-287-01a" xlink:href="fig-287-01"/> quum trapezium I K B A, reliquo triangulo H B A, æquale erit. </s> <s xml:id="echoid-s11845" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quocirca erit <anchor type="note" xlink:label="note-287-03a" xlink:href="note-287-03"/> triangulum C I K, ad trapezium I K B A, vt triangulum C B H, ad triangulum H B A, <lb/> <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, vt C H, ad H A, vel vt E, ad D. </s> <s xml:id="echoid-s11846" xml:space="preserve">quod faciendum erat.</s> <s xml:id="echoid-s11847" xml:space="preserve"/> </p> <div xml:id="echoid-div733" type="float" level="2" n="4"> <note symbol="q" position="left" xlink:label="note-286-16" xlink:href="note-286-16a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> <note symbol="a" position="right" xlink:label="note-287-01" xlink:href="note-287-01a" xml:space="preserve">1. ſexti.</note> <note symbol="b" position="right" xlink:label="note-287-02" xlink:href="note-287-02a" xml:space="preserve">9. quinti.</note> <figure xlink:label="fig-287-01" xlink:href="fig-287-01a"> <image file="287-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/287-01"/> </figure> <note symbol="c" position="right" xlink:label="note-287-03" xlink:href="note-287-03a" xml:space="preserve">7. quinti.</note> </div> <note symbol="d" position="right" xml:space="preserve">1. ſexti.</note> <p> <s xml:id="echoid-s11848" xml:space="preserve"><emph style="sc">Fiet</emph> idem quoque aliter, ſi cadente puncto diuiſionis L, inter A, & </s> <s xml:id="echoid-s11849" xml:space="preserve">G, <lb/> <anchor type="note" xlink:href="" symbol="e"/> conſtruatur trapezium G N, per parallelam M N, æquale triangulo B G L; </s> <s xml:id="echoid-s11850" xml:space="preserve">Ca- <anchor type="note" xlink:label="note-287-05a" xlink:href="note-287-05"/> dente verò puncto diuiſionis H, inter G, & </s> <s xml:id="echoid-s11851" xml:space="preserve">C, trapezium conſtituatur B I, æqua-<lb/>le triangulo B G H, per parallelam I K, &</s> <s xml:id="echoid-s11852" xml:space="preserve">c.</s> <s xml:id="echoid-s11853" xml:space="preserve"/> </p> <div xml:id="echoid-div734" type="float" level="2" n="5"> <note symbol="e" position="right" xlink:label="note-287-05" xlink:href="note-287-05a" xml:space="preserve">2. hui{us}.</note> </div> <p> <s xml:id="echoid-s11854" xml:space="preserve"><emph style="sc">Præterea</emph> quadrilaterum A B C D, diuidendum ſit per lineam lateri <lb/>C D, parallelam in duas partes, vt pars verſus A, ad reliquam habeat datam pro-<lb/>portionem E, ad F. </s> <s xml:id="echoid-s11855" xml:space="preserve">Per ea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s11856" xml:space="preserve">14. </s> <s xml:id="echoid-s11857" xml:space="preserve">lib. </s> <s xml:id="echoid-s11858" xml:space="preserve">2. </s> <s xml:id="echoid-s11859" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11860" xml:space="preserve">vel potius per <lb/>ea, quæ Num. </s> <s xml:id="echoid-s11861" xml:space="preserve">4. </s> <s xml:id="echoid-s11862" xml:space="preserve">cap. </s> <s xml:id="echoid-s11863" xml:space="preserve">4. </s> <s xml:id="echoid-s11864" xml:space="preserve">lib. </s> <s xml:id="echoid-s11865" xml:space="preserve">4. </s> <s xml:id="echoid-s11866" xml:space="preserve">huius tradidimus, quadrilatero A B C D, conſtrua-<lb/>tur quadratum æquale G H I K; </s> <s xml:id="echoid-s11867" xml:space="preserve">ſecetur que latus G K, in L, in proportionem E, <lb/>ad F, datam, & </s> <s xml:id="echoid-s11868" xml:space="preserve">per L, ipſi G H, parallela agatur L M. </s> <s xml:id="echoid-s11869" xml:space="preserve">Deinde ſuper C D, inter re-<lb/>ctas C B, D A, <anchor type="note" xlink:href="" symbol="f"/> fiat rectilineo L I, æquale quadrilaterum D O, habenslatus N O, <anchor type="note" xlink:label="note-287-06a" xlink:href="note-287-06"/> lateri C D, parallelum: </s> <s xml:id="echoid-s11870" xml:space="preserve">Et ſi quidem punctum O, cadit in latus C B, vel in ipſum <lb/>punctum B, recta N O, problema efficiet. </s> <s xml:id="echoid-s11871" xml:space="preserve">Cum enim quadratum G I, toti qua-<lb/>drilatero A C, æquale ſit, & </s> <s xml:id="echoid-s11872" xml:space="preserve">rectangulum ablatum L I, quadril atero ablato D O; <lb/></s> <s xml:id="echoid-s11873" xml:space="preserve">erit quoque reliquum rectangulum G M, reliquo rectilineo A N O, æquale. </s> <s xml:id="echoid-s11874" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-287-07a" xlink:href="note-287-07"/> <anchor type="note" xlink:href="" symbol="g"/> Quapropter erit A N O, ad D O, vt G M, ad M K; </s> <s xml:id="echoid-s11875" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> hoc eſt, vt G L, ad L K, vel <anchor type="note" xlink:label="note-287-08a" xlink:href="note-287-08"/> vt E, ad F. </s> <s xml:id="echoid-s11876" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s11877" xml:space="preserve"/> </p> <div xml:id="echoid-div735" type="float" level="2" n="6"> <note symbol="f" position="right" xlink:label="note-287-06" xlink:href="note-287-06a" xml:space="preserve">2. hui{us}.</note> <note symbol="g" position="right" xlink:label="note-287-07" xlink:href="note-287-07a" xml:space="preserve">7. quinti.</note> <note symbol="h" position="right" xlink:label="note-287-08" xlink:href="note-287-08a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11878" xml:space="preserve"><emph style="sc">Si</emph> verò punctum O, cadit in C B, latus productum, (quod hic fiet, ſi propor-<unsure/> <lb/>tio data ſit Q, ad R. </s> <s xml:id="echoid-s11879" xml:space="preserve">Diuiſo enim latere G K, in S, in datam proportionem Q. </s> <s xml:id="echoid-s11880" xml:space="preserve">ad <lb/>R; </s> <s xml:id="echoid-s11881" xml:space="preserve">ductaque S T, parallela lateri G H; </s> <s xml:id="echoid-s11882" xml:space="preserve">ſi ſuper C D, inter rectas D A, C B, <anchor type="note" xlink:href="" symbol="i"/>fiatre- <anchor type="note" xlink:label="note-287-09a" xlink:href="note-287-09"/> ctangulo K T, æquale quadrilaterum D O, cadet O, vltra B & </s> <s xml:id="echoid-s11883" xml:space="preserve">parallela N O, ſe-<lb/>cabit latus A B, in P.) </s> <s xml:id="echoid-s11884" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> conſtituemus ſuper N P, inter rectas N A, P A, per paralle- <anchor type="note" xlink:label="note-287-10a" xlink:href="note-287-10"/> lam V X, trapezium N V, triangulo B O P, æquale, factum erit, quod iubetur. </s> <s xml:id="echoid-s11885" xml:space="preserve">Ad-<lb/>dito enim communi rectilineo C D N P B; </s> <s xml:id="echoid-s11886" xml:space="preserve">erit totum rectilineum C D X V B, toti <lb/>rectilineo D O, hoc eſt, rectangulo K T, æquale; </s> <s xml:id="echoid-s11887" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s11888" xml:space="preserve">reliquum triangu-<lb/>lum A V X, reliquo rectangulo G T, æquale. </s> <s xml:id="echoid-s11889" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Quare erit triangulum A V X, ad <anchor type="note" xlink:label="note-287-11a" xlink:href="note-287-11"/> rectilineum C D X V B, vt G T, ad K T, <anchor type="note" xlink:href="" symbol="m"/> hoc eſt, vt G S, ad S K, vel vt Q, ad R.</s> <s xml:id="echoid-s11890" xml:space="preserve"> <anchor type="note" xlink:label="note-287-12a" xlink:href="note-287-12"/> quod erat faciendum.</s> <s xml:id="echoid-s11891" xml:space="preserve"/> </p> <div xml:id="echoid-div736" type="float" level="2" n="7"> <note symbol="i" position="right" xlink:label="note-287-09" xlink:href="note-287-09a" xml:space="preserve">2. hui{us}.</note> <note symbol="k" position="right" xlink:label="note-287-10" xlink:href="note-287-10a" xml:space="preserve">2 hui{us}.</note> <note symbol="l" position="right" xlink:label="note-287-11" xlink:href="note-287-11a" xml:space="preserve">7. quinti.</note> <note symbol="m" position="right" xlink:label="note-287-12" xlink:href="note-287-12a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11892" xml:space="preserve"><emph style="sc">Non</emph> alia ratione problema abſoluemus, ſi antecedens proportionis ſtatuen- <pb o="258" file="288" n="288" rhead="GEOMETR. PRACT."/> dum ſit ad partes C D. </s> <s xml:id="echoid-s11893" xml:space="preserve">Sit namque data proportio F, ad E, vel R, ad Q. </s> <s xml:id="echoid-s11894" xml:space="preserve">Diuiſo <lb/>ergo latere K G, quadrati G I, quod qua drilatero A C, factum eſt æquale, in L, ſe-<lb/> <anchor type="figure" xlink:label="fig-288-01a" xlink:href="fig-288-01"/> cundum propottionem datam F, ad E; </s> <s xml:id="echoid-s11895" xml:space="preserve">vel in S, ſecundum datam proportionem <lb/>R, ad Q, ductaque parallela L M, vel S T: </s> <s xml:id="echoid-s11896" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ſi ſuper C D, inter rectas D A, C B, <anchor type="note" xlink:label="note-288-01a" xlink:href="note-288-01"/> conſtituemus rectilineum D O, rectangulo K M, vel rectilineum C D X V B, re-<lb/>ctangulo K T, æquale per parallelam N O, vel X V, vt paulò ante dictum eſt, ſo-<lb/>lutum erit problema, vt liquet.</s> <s xml:id="echoid-s11897" xml:space="preserve"/> </p> <div xml:id="echoid-div737" type="float" level="2" n="8"> <figure xlink:label="fig-288-01" xlink:href="fig-288-01a"> <image file="288-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/288-01"/> </figure> <note symbol="a" position="left" xlink:label="note-288-01" xlink:href="note-288-01a" xml:space="preserve">2. hui{us}.</note> </div> <p> <s xml:id="echoid-s11898" xml:space="preserve"><emph style="sc">Vervm</emph> idem quadrilaterum A B C D, diuidendum ſit per parallelam cuili-<lb/>bet alteri lineæ Y, in proportionem E, ad F, ita vt antecedens proportionis ver-<lb/>gat ad A; </s> <s xml:id="echoid-s11899" xml:space="preserve">Si igitur data recta Y, æquidiſtet vni lateri, partiemur datum quadri-<lb/>laterum in datam proportionem per lineam illi lateri, proindeque & </s> <s xml:id="echoid-s11900" xml:space="preserve">rectæ Y, <lb/>parallelam, vt proximè ſcripſimus. </s> <s xml:id="echoid-s11901" xml:space="preserve">Si verò nulli lateri recta Y, æquidiſter, du-<lb/>cemus ei ex aliquo angulo, vt ex C, parallelam C Z, quæ intra quadrilaterum ca-<lb/>dat; </s> <s xml:id="echoid-s11902" xml:space="preserve">& </s> <s xml:id="echoid-s11903" xml:space="preserve">ablato triangulo C D Z, vel rectilineo ablato, auferemus ex quadrato <lb/>G I, quod toti quadrilatero ſit æquale, rectangulum æquale K d. </s> <s xml:id="echoid-s11904" xml:space="preserve">(quod fiet, ſi <lb/>triangulo, vel rectilineo C D Z, fiat æquale quadratum, cuius latus a; </s> <s xml:id="echoid-s11905" xml:space="preserve">& </s> <s xml:id="echoid-s11906" xml:space="preserve">duabus <lb/>K I, & </s> <s xml:id="echoid-s11907" xml:space="preserve">a, tertia proportionalis reperiatur K b. </s> <s xml:id="echoid-s11908" xml:space="preserve">Ducta enim b d, ipſi K I, paralle-<lb/>la, <anchor type="note" xlink:href="" symbol="b"/> erit rectangulum K d, quadrato lateris a, hoc eſt, rectilineo C D Z, æqua- <anchor type="note" xlink:label="note-288-02a" xlink:href="note-288-02"/> le.) </s> <s xml:id="echoid-s11909" xml:space="preserve">Et ſi quidem parallela b d, cadit inter K I, & </s> <s xml:id="echoid-s11910" xml:space="preserve">L M, <anchor type="note" xlink:href="" symbol="c"/> conſtituemus ſuper C Z, <anchor type="note" xlink:label="note-288-03a" xlink:href="note-288-03"/> inter rectas C B, Z A, rectilineum C a e Z, rectangulo b M, æquale, cuius latus a e, <lb/>lateri C Z, hoc eſt, datæ rectæ Y, aquidiſtet; </s> <s xml:id="echoid-s11911" xml:space="preserve">factum que erit, quod præcipitur. <lb/></s> <s xml:id="echoid-s11912" xml:space="preserve">Erit enim totum rectilineum C D e a, toti rectangulo K M; </s> <s xml:id="echoid-s11913" xml:space="preserve">ac propterea & </s> <s xml:id="echoid-s11914" xml:space="preserve">reli-<lb/>quum A B a e, reliquo G M, æquale. </s> <s xml:id="echoid-s11915" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quapropter erit A B a e, ad e a c D, vt <anchor type="note" xlink:label="note-288-04a" xlink:href="note-288-04"/> G M, ad K M; </s> <s xml:id="echoid-s11916" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> hoc eſt, vt G L, ad L K, vel vt E, ad F. </s> <s xml:id="echoid-s11917" xml:space="preserve">Si ver ò parallela b d, coin- <anchor type="note" xlink:label="note-288-05a" xlink:href="note-288-05"/> cidit cum recta L M, efficiet problema recta C Z: </s> <s xml:id="echoid-s11918" xml:space="preserve">quia cum K M, rectilineo <lb/>C D Z, ſit æquale, erit reliquum rectangulum G M, reliquo rectilineo A B C Z, æ-<lb/>quale; </s> <s xml:id="echoid-s11919" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> atque idcirco erit A B C Z, ad C D Z, vt G M, ad K M, <anchor type="note" xlink:href="" symbol="g"/> hoc eſt, vt G L, <anchor type="note" xlink:label="note-288-06a" xlink:href="note-288-06"/> ad L K, vel vt E, ad F. </s> <s xml:id="echoid-s11920" xml:space="preserve">Si denique parallela b d, ca dit inter L M, & </s> <s xml:id="echoid-s11921" xml:space="preserve">G H; </s> <s xml:id="echoid-s11922" xml:space="preserve">erit <lb/> <anchor type="note" xlink:label="note-288-07a" xlink:href="note-288-07"/> rectilineum, vel triangulum ablatum C D Z, maius rectangulo K M. </s> <s xml:id="echoid-s11923" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Si igi- <anchor type="note" xlink:label="note-288-08a" xlink:href="note-288-08"/> tur ſuper C Z, inter rectas C D, Z D, conſtruetur rectilineum C m, rectangulo <lb/>L d, æquale; </s> <s xml:id="echoid-s11924" xml:space="preserve">erit reliquum D l m, reliquo rectangulo G d, æquale: </s> <s xml:id="echoid-s11925" xml:space="preserve">ac pro- <pb o="259" file="289" n="289" rhead="LIBER SEXTVS."/> pterea ad reliquum ml C B A, reliquo rectangulo K M. </s> <s xml:id="echoid-s11926" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur erit A B C l m, <anchor type="note" xlink:label="note-289-01a" xlink:href="note-289-01"/> a d l m D, vt G M, ad K M: </s> <s xml:id="echoid-s11927" xml:space="preserve">hoc eſt vt G L, ad L K, vel vt E, ad F. </s> <s xml:id="echoid-s11928" xml:space="preserve">quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s11929" xml:space="preserve"/> </p> <div xml:id="echoid-div738" type="float" level="2" n="9"> <note symbol="b" position="left" xlink:label="note-288-02" xlink:href="note-288-02a" xml:space="preserve">17. ſexti.</note> <note symbol="c" position="left" xlink:label="note-288-03" xlink:href="note-288-03a" xml:space="preserve">2. hui{us}.</note> <note symbol="d" position="left" xlink:label="note-288-04" xlink:href="note-288-04a" xml:space="preserve">7. quinti.</note> <note symbol="e" position="left" xlink:label="note-288-05" xlink:href="note-288-05a" xml:space="preserve">1. ſexti.</note> <note symbol="f" position="left" xlink:label="note-288-06" xlink:href="note-288-06a" xml:space="preserve">7. quinti.</note> <note symbol="g" position="left" xlink:label="note-288-07" xlink:href="note-288-07a" xml:space="preserve">1. ſexti.</note> <note symbol="h" position="left" xlink:label="note-288-08" xlink:href="note-288-08a" xml:space="preserve">2. hui{us}.</note> <note symbol="a" position="right" xlink:label="note-289-01" xlink:href="note-289-01a" xml:space="preserve">7. quinti.</note> </div> <p> <s xml:id="echoid-s11930" xml:space="preserve"><emph style="sc">Eodem</emph> prorſus modo quamlibet aliam figuram, quotquot habeat latera, <lb/>in datam prop ortionem ſecabimus per lineam, quæ vni lateri vel cuiuis alij re-<lb/>ctæ lineæ æquidiſtet. </s> <s xml:id="echoid-s11931" xml:space="preserve">Sit enim datum heptagonum qualecunque A B C D E F G, <lb/>ſecandum per lineam lateri A G, parallelam, in duas partes, vt ea, quæ ad D, ver-<lb/>git, ad reliquam habeat proportionem eandem, quam M, ad N habet. </s> <s xml:id="echoid-s11932" xml:space="preserve">Conſtitu-<lb/>to quadrato H I K L, æquali ipſi heptagono, per ea, quæin ſchol. </s> <s xml:id="echoid-s11933" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s11934" xml:space="preserve">14. <lb/></s> <s xml:id="echoid-s11935" xml:space="preserve">lib. </s> <s xml:id="echoid-s11936" xml:space="preserve">2. </s> <s xml:id="echoid-s11937" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s11938" xml:space="preserve">vel potius per ea, quæ Num. </s> <s xml:id="echoid-s11939" xml:space="preserve">4. </s> <s xml:id="echoid-s11940" xml:space="preserve">cap. </s> <s xml:id="echoid-s11941" xml:space="preserve">4. </s> <s xml:id="echoid-s11942" xml:space="preserve">lib. </s> <s xml:id="echoid-s11943" xml:space="preserve">4. </s> <s xml:id="echoid-s11944" xml:space="preserve">huius ſcripſimus; </s> <s xml:id="echoid-s11945" xml:space="preserve">& </s> <s xml:id="echoid-s11946" xml:space="preserve"><lb/>diuiſo latere H I, in O, in proportionem M, ad N, ductaque O P, lateri H L, pa-<lb/>rellela: </s> <s xml:id="echoid-s11947" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> fiat ſuper rectam A G, inter rectas A B, G F, rectangulo I P, æqua- <anchor type="note" xlink:label="note-289-02a" xlink:href="note-289-02"/> le rectilineum A G Q R, habens latus Q R, lateri A G, parallelum. </s> <s xml:id="echoid-s11948" xml:space="preserve">Et quo-<lb/>niam Q R, cadit vltra F, B, <anchor type="note" xlink:href="" symbol="c"/> conſtituemus rur- <anchor type="figure" xlink:label="fig-289-01a" xlink:href="fig-289-01"/> <anchor type="note" xlink:label="note-289-03a" xlink:href="note-289-03"/> ſum ſuper rectam S T, inter rectas S C, T E, per <lb/>rectam V X, ipſi S T, parallelam, recti-<lb/>lineum æquale triangulis F Q T, B R S, extra <lb/>heptagonum exiſtentibus; </s> <s xml:id="echoid-s11949" xml:space="preserve">factumque erit <lb/>quod proponitur. </s> <s xml:id="echoid-s11950" xml:space="preserve">Cum enim rectilineum <lb/>A G Q R, ac proinde & </s> <s xml:id="echoid-s11951" xml:space="preserve">rectilineum A B V X-<lb/>F G, rectangulo I P, ſit æquale, erit quo que re-<lb/>liquum D E X V C, reliquo rectangulo O L, æ-<lb/>quale. </s> <s xml:id="echoid-s11952" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Igitur erit D E X V C, ad A B V X F G, <anchor type="note" xlink:label="note-289-04a" xlink:href="note-289-04"/> vt O L, ad I P, <anchor type="note" xlink:href="" symbol="e"/> hoc eſt, vt H O, ad O I, vel vt <anchor type="note" xlink:label="note-289-05a" xlink:href="note-289-05"/> M, ad N, quo derat faciendum.</s> <s xml:id="echoid-s11953" xml:space="preserve"/> </p> <div xml:id="echoid-div739" type="float" level="2" n="10"> <note symbol="b" position="right" xlink:label="note-289-02" xlink:href="note-289-02a" xml:space="preserve">2. hui{us}.</note> <figure xlink:label="fig-289-01" xlink:href="fig-289-01a"> <image file="289-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/289-01"/> </figure> <note symbol="c" position="right" xlink:label="note-289-03" xlink:href="note-289-03a" xml:space="preserve">2. hui{us}.</note> <note symbol="d" position="right" xlink:label="note-289-04" xlink:href="note-289-04a" xml:space="preserve">7. quinti.</note> <note symbol="e" position="right" xlink:label="note-289-05" xlink:href="note-289-05a" xml:space="preserve">1. ſexti.</note> </div> <p> <s xml:id="echoid-s11954" xml:space="preserve"><emph style="sc">Qvod</emph> ſi latus rectilinei ex heptagono ab-<lb/>ſciſsi æquidiſtare debeat rectæ Y, q̃ nulli lateri <lb/>heptagoni æꝗdiſter (ſinãq; </s> <s xml:id="echoid-s11955" xml:space="preserve">æquidiſtaret, vni <lb/>lateri, abſolueretur problema, vt proximè tra-<lb/>ditũ eſt) ducta ex angulo G, rectæ Y, parallela <lb/>G Z, quæ intra figurã cadat; </s> <s xml:id="echoid-s11956" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> cõſtruemus recti- <anchor type="note" xlink:label="note-289-06a" xlink:href="note-289-06"/> lineo A G Z, ſuper rectã I K, æquale rectangulũ h K; </s> <s xml:id="echoid-s11957" xml:space="preserve">quod fiet, ſi rectilineo A G Z, <lb/>fiat æquale quadratum, cuius latus l, & </s> <s xml:id="echoid-s11958" xml:space="preserve">duabus I K, & </s> <s xml:id="echoid-s11959" xml:space="preserve">l, tertia proportiona-<lb/>lis reperiatur I h. </s> <s xml:id="echoid-s11960" xml:space="preserve">Ducta enim h i, ipſi I K, parallela, <anchor type="note" xlink:href="" symbol="g"/> erit rectangulum h K, qua- <anchor type="note" xlink:label="note-289-07a" xlink:href="note-289-07"/> drato lateris l, hoc eſt, rectilineo A G Z, æquale. </s> <s xml:id="echoid-s11961" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Deinde ſuper rectam G Z, <anchor type="note" xlink:label="note-289-08a" xlink:href="note-289-08"/> inter rectas G F, Z B, conſtituemus rectangulo h P, per parallelam b a, æqua-<lb/>le rectilineum G Z b a. </s> <s xml:id="echoid-s11962" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Nam ſi triangulis B b e, F a d, ſuper rectam d e, in- <anchor type="note" xlink:label="note-289-09a" xlink:href="note-289-09"/> ter rectas e C, d E, fiat per parallelam f g, rectilineum d e g f, æquale; </s> <s xml:id="echoid-s11963" xml:space="preserve">fa-<lb/>ctum erit, quod in problemate proponitur, vt ex dictis perſpicuum eſt. </s> <s xml:id="echoid-s11964" xml:space="preserve">Ea-<lb/>demque omninò ratio eſt in omnibus aliis rectilineis quamuis irregularibus, <lb/>dummodo in iis ducipoſsit vna linea parallela datærectæ, quæ rectilineum au-<lb/>ferat dato rectilineo æquale. </s> <s xml:id="echoid-s11965" xml:space="preserve">Non enim ſemper hoc fieri poſſe in figuris, cu-<lb/>ius anguli partim introrſum, & </s> <s xml:id="echoid-s11966" xml:space="preserve">partim extrorſum vergant, ad finem propoſ. <lb/></s> <s xml:id="echoid-s11967" xml:space="preserve">2. </s> <s xml:id="echoid-s11968" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s11969" xml:space="preserve">declarauimus. </s> <s xml:id="echoid-s11970" xml:space="preserve">Id quod conſtructio ipſa problematis perſpicuè <lb/>nos docebit.</s> <s xml:id="echoid-s11971" xml:space="preserve"/> </p> <div xml:id="echoid-div740" type="float" level="2" n="11"> <note symbol="f" position="right" xlink:label="note-289-06" xlink:href="note-289-06a" xml:space="preserve">2. hui{us}.</note> <note symbol="g" position="right" xlink:label="note-289-07" xlink:href="note-289-07a" xml:space="preserve">17. ſexti.</note> <note symbol="h" position="right" xlink:label="note-289-08" xlink:href="note-289-08a" xml:space="preserve">2. hui{us}.</note> <note symbol="i" position="right" xlink:label="note-289-09" xlink:href="note-289-09a" xml:space="preserve">2. hui{us}.</note> </div> <pb o="260" file="290" n="290" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div742" type="section" level="1" n="252"> <head xml:id="echoid-head277" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s11972" xml:space="preserve"><emph style="sc">Dividi</emph> ergo poterit quælibet figura rectilinea in quotuis partes æquales <lb/> <anchor type="note" xlink:label="note-290-01a" xlink:href="note-290-01"/> per lineas, quæ datæ cuiuis rectæ lineæ æquidiſtent. </s> <s xml:id="echoid-s11973" xml:space="preserve">Nam ſi verbi gratia data fi-<lb/>gura ſecanda ſit in 8. </s> <s xml:id="echoid-s11974" xml:space="preserve">partes æquales per lineas datæ rectæ parallelas, diuidemus <lb/>eam primum in duas partes inter ſe proportionem habentes 1. </s> <s xml:id="echoid-s11975" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s11976" xml:space="preserve">Ita namque <lb/>prior pars erit {1/8}. </s> <s xml:id="echoid-s11977" xml:space="preserve">totius figuræ. </s> <s xml:id="echoid-s11978" xml:space="preserve">Deinde poſteriorem partem ſecabimus in pro-<lb/>portionem 1. </s> <s xml:id="echoid-s11979" xml:space="preserve">ad 6. </s> <s xml:id="echoid-s11980" xml:space="preserve">ita vt prior pars huius diuiſionis ſit {1/7}. </s> <s xml:id="echoid-s11981" xml:space="preserve">illius partis diuiſæ, hoc <lb/>eſt, {1/7}. </s> <s xml:id="echoid-s11982" xml:space="preserve">totius figuræ, cum pars illa diuiſa complectatur {7/8}. </s> <s xml:id="echoid-s11983" xml:space="preserve">totius figuræ. </s> <s xml:id="echoid-s11984" xml:space="preserve">Poſtea <lb/>partem poſt eriorem proximæ diuiſionis partiemur in proportionem 1. </s> <s xml:id="echoid-s11985" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s11986" xml:space="preserve">Et <lb/>poſteriorem huius diuiſionis partem in proportionem 1. </s> <s xml:id="echoid-s11987" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s11988" xml:space="preserve">Atqueita dein-<lb/>ceps, minuendo ſemper, don@c ad partem deueniamus, quæ ſecanda ſit in pro-<lb/>portionem 1. </s> <s xml:id="echoid-s11989" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s11990" xml:space="preserve">hoc eſt, in partes æquales.</s> <s xml:id="echoid-s11991" xml:space="preserve"/> </p> <div xml:id="echoid-div742" type="float" level="2" n="1"> <note position="left" xlink:label="note-290-01" xlink:href="note-290-01a" xml:space="preserve">Quo pacto fi-<lb/>gura data ſe-<lb/>c{et}ur per li-<lb/>ne{as} parallel{as} <lb/>in quotuis par <lb/>t{es} æqual{es}.</note> </div> <p> <s xml:id="echoid-s11992" xml:space="preserve"><emph style="sc">Hoc</emph> idem effici poterit ea ratione, quam ad finem ſcholij propoſ. </s> <s xml:id="echoid-s11993" xml:space="preserve">4. </s> <s xml:id="echoid-s11994" xml:space="preserve">expo-<lb/>ſuimus: </s> <s xml:id="echoid-s11995" xml:space="preserve">ſi videlicetlatus quadrati H I, quod rectilineo dato conſtructum eſt <lb/>æquale, in tot æquales partes ſecetur, in quot partes datum rectilineum diuiden-<lb/>dum eſt, & </s> <s xml:id="echoid-s11996" xml:space="preserve">primo rectilineum diuidatur in proportionem primæ partis ad reli-<lb/>quas: </s> <s xml:id="echoid-s11997" xml:space="preserve">Deinde poſterior pars rectilinei in proportionem ſecundæ partis lateris <lb/>H I, ad reliquas: </s> <s xml:id="echoid-s11998" xml:space="preserve">at que ita deinceps, &</s> <s xml:id="echoid-s11999" xml:space="preserve">c.</s> <s xml:id="echoid-s12000" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12001" xml:space="preserve"><emph style="sc">Atqve</emph> hic finem habet noſtra Geodæſia complectens diuiſionem omnium <lb/>figurarum rectilinearum: </s> <s xml:id="echoid-s12002" xml:space="preserve">ſequuntur iam particulares nonnullæ diuiſiones qua-<lb/>rundam figurarum, quæ tum, quia ſubtiles acutaſque demonſtrationes conti-<lb/>nent, tum quia pleraſque earum eruditi quo que Geometræ, vt Leonardus Pi-<lb/>ſanus, Frater Lucas Pacciolus, & </s> <s xml:id="echoid-s12003" xml:space="preserve">Nicolaus Tartalea tradiderunt, omittendæ <lb/>nullo modo viſæ ſunt: </s> <s xml:id="echoid-s12004" xml:space="preserve">Vt autem Geometricè eas demonſtremus, præmittenda <lb/>ſunt Theoremata nonnulla, quorum primum ſit hoc.</s> <s xml:id="echoid-s12005" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div744" type="section" level="1" n="253"> <head xml:id="echoid-head278" xml:space="preserve">THEOREMA 2. PROPOS. 6.</head> <p> <s xml:id="echoid-s12006" xml:space="preserve">SI duo triangula æqualia habeant vnum latus commune, & </s> <s xml:id="echoid-s12007" xml:space="preserve">in diuerſas <lb/>partes vergant: </s> <s xml:id="echoid-s12008" xml:space="preserve">Recta oppoſitos angulos connectens à latere illo <lb/>communi bifariam ſecatur.</s> <s xml:id="echoid-s12009" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12010" xml:space="preserve"><emph style="sc">Sint</emph> æqualia duo triangula A B C, A B D, habentia latus A B, commune, & </s> <s xml:id="echoid-s12011" xml:space="preserve">in <lb/>diuerſas partes vergentia. </s> <s xml:id="echoid-s12012" xml:space="preserve">Dico rectam C D, oppoſitos angulos C, D, iungentem <lb/>ſecari in E, bifariam à latere com̃uni A B. </s> <s xml:id="echoid-s12013" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniã enim eſt tá<unsure/>m <anchor type="figure" xlink:label="fig-290-01a" xlink:href="fig-290-01"/> <anchor type="note" xlink:label="note-290-02a" xlink:href="note-290-02"/> triangulum A C E, ad triangulum A D E, quàm triangulum B C E, <lb/>ad triangulum B D E, vt C E, ad E D; </s> <s xml:id="echoid-s12014" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit triangulum A C E, ad <anchor type="note" xlink:label="note-290-03a" xlink:href="note-290-03"/> triangulum A D E, vt triangulum B C E, ad triangulum B D E. <lb/></s> <s xml:id="echoid-s12015" xml:space="preserve"> <anchor type="note" xlink:label="note-290-04a" xlink:href="note-290-04"/> <anchor type="note" xlink:href="" symbol="c"/> Igitur erunt quo que duo triangula ſimul A C E, B C E, hoc eſt, totum triangulum A B C, ad duo triangula ſimul A D E, B D E, id <lb/>eſt, ad totum triangulum A B D, vt A C E, ad A D E, hoc eſt, vt <lb/>C E, ad E D. </s> <s xml:id="echoid-s12016" xml:space="preserve">Cum ergo triangula A B C, A B D, ponantur æqualia; </s> <s xml:id="echoid-s12017" xml:space="preserve">erunt quo-<lb/>que rectæ C E, E D, æquales, ac proinde C D, in E, ſecta eſt bifariam. </s> <s xml:id="echoid-s12018" xml:space="preserve">quod erat <lb/>oſtendendum.</s> <s xml:id="echoid-s12019" xml:space="preserve"/> </p> <div xml:id="echoid-div744" type="float" level="2" n="1"> <figure xlink:label="fig-290-01" xlink:href="fig-290-01a"> <image file="290-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/290-01"/> </figure> <note symbol="a" position="left" xlink:label="note-290-02" xlink:href="note-290-02a" xml:space="preserve">1. ſexti.</note> <note symbol="b" position="left" xlink:label="note-290-03" xlink:href="note-290-03a" xml:space="preserve">11. quinti.</note> <note symbol="c" position="left" xlink:label="note-290-04" xlink:href="note-290-04a" xml:space="preserve">12. quinti.</note> </div> <pb o="261" file="291" n="291" rhead="LIBER SEXTVS."/> </div> <div xml:id="echoid-div746" type="section" level="1" n="254"> <head xml:id="echoid-head279" xml:space="preserve">THEOR. 3. PROPOS. 7.</head> <p> <s xml:id="echoid-s12020" xml:space="preserve">SI in triangulo baſi parallela ducatur, & </s> <s xml:id="echoid-s12021" xml:space="preserve">extrema parallelarum rectis <lb/>iungantur ſe ſeinterſecantibus: </s> <s xml:id="echoid-s12022" xml:space="preserve">habebit vtriuſuis harum rectarum ſe-<lb/>gmentur ab angulo incipiens ad reliquum in latere terminatum ean-<lb/>dem proportionem, quam latus ab illa recta diuiſum ad partem eius <lb/>ſuperiorem. </s> <s xml:id="echoid-s12023" xml:space="preserve">Recta autem ex tertio angulo per interſectionem dicta-<lb/>rum rectarum extenſa ſecabit vtramque parallelam bifariam.</s> <s xml:id="echoid-s12024" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12025" xml:space="preserve"><emph style="sc">In</emph> triangulo ABC, ducta ſit DE, baſi BC, parallela, & </s> <s xml:id="echoid-s12026" xml:space="preserve">iunctæ rectæ BE, CD, <lb/>ſeinterſecent in F. </s> <s xml:id="echoid-s12027" xml:space="preserve">Dico eſſe BF, ad FE, vt AC, ad AE: </s> <s xml:id="echoid-s12028" xml:space="preserve">Item CF, ad FD, vt AB, <lb/>ad AD. </s> <s xml:id="echoid-s12029" xml:space="preserve">Et iunctam rectam AF, ſecare parallelas DE, BC, bi-<lb/>fariam in G, & </s> <s xml:id="echoid-s12030" xml:space="preserve">H. </s> <s xml:id="echoid-s12031" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim triangula B D C, C E B, æ- <anchor type="figure" xlink:label="fig-291-01a" xlink:href="fig-291-01"/> <anchor type="note" xlink:label="note-291-01a" xlink:href="note-291-01"/> qualia ſunt; </s> <s xml:id="echoid-s12032" xml:space="preserve">ablato communi BFC, reliqua BDF, CEF, æqua-<lb/>lia quoque erunt. </s> <s xml:id="echoid-s12033" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quia verò eſt, vt B D, ad D A, ita C E, ad <anchor type="note" xlink:label="note-291-02a" xlink:href="note-291-02"/> EA: </s> <s xml:id="echoid-s12034" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Vt autem BD, ad DA, ita eſt triangulum BFD, ad trian- <anchor type="note" xlink:label="note-291-03a" xlink:href="note-291-03"/> gulum AFD: </s> <s xml:id="echoid-s12035" xml:space="preserve">Et vt CE, ad EA, ita triangulum CFE, ad trian-<lb/>gulum AFE; </s> <s xml:id="echoid-s12036" xml:space="preserve">erit quoque triangulum BFD, ad triangulum AFD, vt triangulum <lb/>CFE, ad triangulum AFE. </s> <s xml:id="echoid-s12037" xml:space="preserve">Cum ergo triangulum BFD, triangulo CFE, oſten-<lb/>ſum ſit æquale; </s> <s xml:id="echoid-s12038" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erit quoque triangulum AFD, triangulo AFE, æquale. </s> <s xml:id="echoid-s12039" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igi- <anchor type="note" xlink:label="note-291-04a" xlink:href="note-291-04"/> tur DE, in G, ſecta eſt bifariam: </s> <s xml:id="echoid-s12040" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> ac proinde & </s> <s xml:id="echoid-s12041" xml:space="preserve">parallela BC, ſecta erit bifariam <anchor type="note" xlink:label="note-291-05a" xlink:href="note-291-05"/> in H. </s> <s xml:id="echoid-s12042" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Et quoniam triangulum AFB, ad triangula æqualia AFD, AFE, eandem <anchor type="note" xlink:label="note-291-06a" xlink:href="note-291-06"/> habetproportionem; </s> <s xml:id="echoid-s12043" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> eſt que vt AFB, ad AFD, ita AB, ad AD: </s> <s xml:id="echoid-s12044" xml:space="preserve">Et vt AFB, ad <anchor type="note" xlink:label="note-291-07a" xlink:href="note-291-07"/> AFE, ita BF, ad FE: </s> <s xml:id="echoid-s12045" xml:space="preserve">erit quoque BA, ad AD, ideoque AC, ad AE, vt BF, ad FE: <lb/></s> <s xml:id="echoid-s12046" xml:space="preserve"> <anchor type="note" xlink:label="note-291-08a" xlink:href="note-291-08"/> Eademque ratione erit A B, ad A D, vel A C, ad A E, vt C F, ad F D. </s> <s xml:id="echoid-s12047" xml:space="preserve">quod <lb/> <anchor type="note" xlink:label="note-291-09a" xlink:href="note-291-09"/> etiam inde patet; </s> <s xml:id="echoid-s12048" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> cum ſit vt C F, ad F D, ita C F E, ad D E F, hoc eſt, ita B F D, <anchor type="note" xlink:label="note-291-10a" xlink:href="note-291-10"/> ipſi CFE, æquale ad idem DEF, <anchor type="note" xlink:href="" symbol="k"/> hoc eſt, ita BF, ad FE. </s> <s xml:id="echoid-s12049" xml:space="preserve">quod erat demonſtran- dum.</s> <s xml:id="echoid-s12050" xml:space="preserve"/> </p> <div xml:id="echoid-div746" type="float" level="2" n="1"> <figure xlink:label="fig-291-01" xlink:href="fig-291-01a"> <image file="291-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/291-01"/> </figure> <note symbol="a" position="right" xlink:label="note-291-01" xlink:href="note-291-01a" xml:space="preserve">37. primi.</note> <note symbol="b" position="right" xlink:label="note-291-02" xlink:href="note-291-02a" xml:space="preserve">2. ſexti.</note> <note symbol="c" position="right" xlink:label="note-291-03" xlink:href="note-291-03a" xml:space="preserve">1. ſexti.</note> <note symbol="d" position="right" xlink:label="note-291-04" xlink:href="note-291-04a" xml:space="preserve">14. quinti.</note> <note symbol="e" position="right" xlink:label="note-291-05" xlink:href="note-291-05a" xml:space="preserve">6. hui{us}.</note> <note symbol="f" position="right" xlink:label="note-291-06" xlink:href="note-291-06a" xml:space="preserve">ſchol. 4. ſexti.</note> <note symbol="g" position="right" xlink:label="note-291-07" xlink:href="note-291-07a" xml:space="preserve">7. quinti.</note> <note symbol="h" position="right" xlink:label="note-291-08" xlink:href="note-291-08a" xml:space="preserve">1. ſexti.</note> <note symbol="i" position="right" xlink:label="note-291-09" xlink:href="note-291-09a" xml:space="preserve">1. ſexti.</note> <note symbol="k" position="right" xlink:label="note-291-10" xlink:href="note-291-10a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div748" type="section" level="1" n="255"> <head xml:id="echoid-head280" xml:space="preserve">THEOR. 4. PROPOS. 8.</head> <p> <s xml:id="echoid-s12051" xml:space="preserve">SI in triangulo à duobus angulis duæ rectæ ducantur ad media puncta <lb/>oppoſitorum laterum: </s> <s xml:id="echoid-s12052" xml:space="preserve">Recta ex angulo reliquo perinterſectionem <lb/>earum deducta ſecat quoque reliquum latus bifariam. </s> <s xml:id="echoid-s12053" xml:space="preserve">Cuiuslibet au-<lb/>tem illarum trium linearum ſegmentum prope angulum adreliquum <lb/>ſegmentum duplam habet proportionem. </s> <s xml:id="echoid-s12054" xml:space="preserve">Triangulum denique per <lb/>rectas ab interſectione ad angulos ductas in tria triangula æqualia di-<lb/>uiditur.</s> <s xml:id="echoid-s12055" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12056" xml:space="preserve"><emph style="sc">In</emph> triangulo præcedentis propoſ. </s> <s xml:id="echoid-s12057" xml:space="preserve">ABC, duærectæ BE, CD, <lb/>ſecent latera AC, AB, bifariamin E, D, ſe autem mutuo interſe-<lb/> <anchor type="figure" xlink:label="fig-291-02a" xlink:href="fig-291-02"/> cet in F. </s> <s xml:id="echoid-s12058" xml:space="preserve">Dico rectam ductam AF, ſecare quoque latus BC, bi-<lb/>fariamin H, &</s> <s xml:id="echoid-s12059" xml:space="preserve">c. </s> <s xml:id="echoid-s12060" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Iuncta enim recta D E, parallela erit ipſi B C, <anchor type="note" xlink:label="note-291-11a" xlink:href="note-291-11"/> cum ſecet latera A B, A C, proportionaliter, in partes videlicet <lb/>æquales: </s> <s xml:id="echoid-s12061" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> Quamobrem A F, vtramque parallelam D E, B C, <anchor type="note" xlink:label="note-291-12a" xlink:href="note-291-12"/> bifariam ſecabit. </s> <s xml:id="echoid-s12062" xml:space="preserve">quod eſt primum.</s> <s xml:id="echoid-s12063" xml:space="preserve"/> </p> <div xml:id="echoid-div748" type="float" level="2" n="1"> <figure xlink:label="fig-291-02" xlink:href="fig-291-02a"> <image file="291-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/291-02"/> </figure> <note symbol="l" position="right" xlink:label="note-291-11" xlink:href="note-291-11a" xml:space="preserve">2. ſexti.</note> <note symbol="m" position="right" xlink:label="note-291-12" xlink:href="note-291-12a" xml:space="preserve">7. hui{us}.</note> </div> <pb o="262" file="292" n="292" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s12064" xml:space="preserve"><emph style="sc">Deinde</emph> <anchor type="note" xlink:href="" symbol="a"/> quia eſt, vt AB, ad AD, ita CF, ad FD: </s> <s xml:id="echoid-s12065" xml:space="preserve">Eſt autem AB, ipſius AD, <anchor type="note" xlink:label="note-292-01a" xlink:href="note-292-01"/> dupla; </s> <s xml:id="echoid-s12066" xml:space="preserve">erit quo que CF, ipſius FD, dupla. </s> <s xml:id="echoid-s12067" xml:space="preserve">Eademqueratione & </s> <s xml:id="echoid-s12068" xml:space="preserve">BF, ipſius FE; <lb/></s> <s xml:id="echoid-s12069" xml:space="preserve">& </s> <s xml:id="echoid-s12070" xml:space="preserve">AF, ipſius FH, dupla erit. </s> <s xml:id="echoid-s12071" xml:space="preserve">quod eſt ſecundum.</s> <s xml:id="echoid-s12072" xml:space="preserve"/> </p> <div xml:id="echoid-div749" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-292-01" xlink:href="note-292-01a" xml:space="preserve">7. hui{us}.</note> </div> <p> <s xml:id="echoid-s12073" xml:space="preserve"><emph style="sc">Postremo</emph> <anchor type="note" xlink:href="" symbol="b"/> quia eſt vt AF, ad FH, ita triangulum A F B, ad triangulum <anchor type="note" xlink:label="note-292-02a" xlink:href="note-292-02"/> BFH: </s> <s xml:id="echoid-s12074" xml:space="preserve">Eſt autem AF, ipſius F H, oſtenſa dupla; </s> <s xml:id="echoid-s12075" xml:space="preserve">erit quoque triangulum A F B, <lb/>trianguli B F H, duplum. </s> <s xml:id="echoid-s12076" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s12077" xml:space="preserve">triangulum B F C, eiuſdem trianguli B-<lb/> <anchor type="note" xlink:label="note-292-03a" xlink:href="note-292-03"/> FH, duplum; </s> <s xml:id="echoid-s12078" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> quod triangula B F H, C F H, æqualia ſint. </s> <s xml:id="echoid-s12079" xml:space="preserve">Igitur æqualia erunt triangula AFB, BFC. </s> <s xml:id="echoid-s12080" xml:space="preserve">Eodemq; </s> <s xml:id="echoid-s12081" xml:space="preserve">modo triangulum AFC, eidem triangulo BFC, <lb/>æquale erit: </s> <s xml:id="echoid-s12082" xml:space="preserve">ac proinde omnia tria AFB, BFC, CFA, æqualia erunt. </s> <s xml:id="echoid-s12083" xml:space="preserve">quod eſt <lb/>tertium.</s> <s xml:id="echoid-s12084" xml:space="preserve"/> </p> <div xml:id="echoid-div750" type="float" level="2" n="3"> <note symbol="b" position="left" xlink:label="note-292-02" xlink:href="note-292-02a" xml:space="preserve">1. ſexti.</note> <note symbol="c" position="left" xlink:label="note-292-03" xlink:href="note-292-03a" xml:space="preserve">38. primi.</note> </div> </div> <div xml:id="echoid-div752" type="section" level="1" n="256"> <head xml:id="echoid-head281" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s12085" xml:space="preserve"><emph style="sc">Itaqve</emph> facilè inueniri poteſt punctum intra triangulum, à quo tres rectæ <lb/>ad tres angulos ductæ ipſum triungulum in tria æqualia triangula partiantur. <lb/></s> <s xml:id="echoid-s12086" xml:space="preserve">Huiuſmodi enim punctum in propoſito triangulo eſt F, vbi duæ rectæ ex duo-<lb/>bus quibuſuis angulis ad media puncta oppoſitorum laterum ductæ ſe interſe-<lb/>cant, vt in tertia parte huius propoſ. </s> <s xml:id="echoid-s12087" xml:space="preserve">oſtendimus.</s> <s xml:id="echoid-s12088" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div753" type="section" level="1" n="257"> <head xml:id="echoid-head282" xml:space="preserve">THEOR. 5. PROPOS. 9.</head> <p> <s xml:id="echoid-s12089" xml:space="preserve">SI in triangulo ducatur recta vtcunque duo latera ſecans: </s> <s xml:id="echoid-s12090" xml:space="preserve">Erit totum <lb/>triangulum ad abſciſſum triangulum, vt rectangulum ſub duobus la-<lb/>teribus ſectis totius trianguli comprehenſum, ad rectangulum ſub <lb/>duobus lateribus trianguli abſciſſi, quæ priorum ſegmenta ſunt, com-<lb/>prehenſum.</s> <s xml:id="echoid-s12091" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12092" xml:space="preserve"><emph style="sc">In</emph> triangulo ABC, recta D E, ſecet latera A B, A C, in D, E. </s> <s xml:id="echoid-s12093" xml:space="preserve">Dico eſſe vt re-<lb/>ctangulum ſub AB, AC, adrectangulum ſub AD, AE, ita tri-<lb/>angulum ABC, ad triangulum ADE. </s> <s xml:id="echoid-s12094" xml:space="preserve">Quoniam enim triangu-<lb/> <anchor type="figure" xlink:label="fig-292-01a" xlink:href="fig-292-01"/> la ABC, ADE, angulum habent communem A; </s> <s xml:id="echoid-s12095" xml:space="preserve">habebunt per <lb/>propoſ. </s> <s xml:id="echoid-s12096" xml:space="preserve">4. </s> <s xml:id="echoid-s12097" xml:space="preserve">ſchol. </s> <s xml:id="echoid-s12098" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12099" xml:space="preserve">23. </s> <s xml:id="echoid-s12100" xml:space="preserve">lib. </s> <s xml:id="echoid-s12101" xml:space="preserve">6. </s> <s xml:id="echoid-s12102" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12103" xml:space="preserve">eandem propor-<lb/>tionem, quamrectangula ſub lateribus AB, A C, & </s> <s xml:id="echoid-s12104" xml:space="preserve">ſub A D, <lb/>AE, comprehenſa. </s> <s xml:id="echoid-s12105" xml:space="preserve">quod oſtendendum erat.</s> <s xml:id="echoid-s12106" xml:space="preserve"/> </p> <div xml:id="echoid-div753" type="float" level="2" n="1"> <figure xlink:label="fig-292-01" xlink:href="fig-292-01a"> <image file="292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/292-01"/> </figure> </div> </div> <div xml:id="echoid-div755" type="section" level="1" n="258"> <head xml:id="echoid-head283" xml:space="preserve">PROBL. 5. PROPOS. 10.</head> <p> <s xml:id="echoid-s12107" xml:space="preserve">DATVM triangulum ex dato puncto in eius latere in quotlibet par-<lb/>tes æquales diuidere.</s> <s xml:id="echoid-s12108" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12109" xml:space="preserve"><emph style="sc">Propositione</emph> quartadecima ſcholij propoſ 33. </s> <s xml:id="echoid-s12110" xml:space="preserve">lib. </s> <s xml:id="echoid-s12111" xml:space="preserve">6. </s> <s xml:id="echoid-s12112" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12113" xml:space="preserve">tradidi-<lb/>mus regulam, qua triangulum in duas partes ſecundum datam proportionem <lb/>diuidendum ſit: </s> <s xml:id="echoid-s12114" xml:space="preserve">Etquo pacto ex triangulo pars imperata ſit auferenda. </s> <s xml:id="echoid-s12115" xml:space="preserve">Si igi-<lb/>tur triangulum ex dato puncto in eius latere quouis ſecandum ſit in quorlibet <pb o="263" file="293" n="293" rhead="LIBER SEXTVS."/> part<unsure/>es æquales, detrahenda primum erit per lineam rectam ex dato puncto du-<lb/>ctam pars denominata à numero partium, in quas diuidendum eſt triangulum. <lb/></s> <s xml:id="echoid-s12116" xml:space="preserve"> <anchor type="figure" xlink:label="fig-293-01a" xlink:href="fig-293-01"/> Deinde duæ tales partes: </s> <s xml:id="echoid-s12117" xml:space="preserve">poſtea tres, atque ita deinceps, donec tot partes, vna <lb/>minus, detractæ ſint, in quot partes diuidendum proponitur triangulum. </s> <s xml:id="echoid-s12118" xml:space="preserve">Vt ſi <lb/>triangulum ABC, ex puncto D, diuidendum ſit in quinque partes æquales, diui-<lb/>demus latus BC, in quo datum punctum eſt, in quinque partes æquales, in pun-<lb/>ctis E, F, G, H. </s> <s xml:id="echoid-s12119" xml:space="preserve">Iuncta deinderecta DA, ducemus ei parallelas EI, FK, GL, HM. <lb/></s> <s xml:id="echoid-s12120" xml:space="preserve">Sinamque connectantur rectæ D I, D K, D L, D M, diuiſum erit triangulum in <lb/>quinque partes æquales. </s> <s xml:id="echoid-s12121" xml:space="preserve">Nam vt in dicta propoſ. </s> <s xml:id="echoid-s12122" xml:space="preserve">14. </s> <s xml:id="echoid-s12123" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s12124" xml:space="preserve">33. </s> <s xml:id="echoid-s12125" xml:space="preserve">lib. </s> <s xml:id="echoid-s12126" xml:space="preserve">6. </s> <s xml:id="echoid-s12127" xml:space="preserve"><lb/>Euclid. </s> <s xml:id="echoid-s12128" xml:space="preserve">oſtenſum eſt, triangulum DBI, eſt {@/5}. </s> <s xml:id="echoid-s12129" xml:space="preserve">totius trianguli, hoc eſt, ita ſe ha-<lb/>bet DBI, ad ABC, vt BE, ad BC. </s> <s xml:id="echoid-s12130" xml:space="preserve">Triangulum autem DBK, continet {2/3}. </s> <s xml:id="echoid-s12131" xml:space="preserve">totius <lb/>trianguli, id eſt, ita ſe habet D B K, ad A B C, vt B F, ad B C. </s> <s xml:id="echoid-s12132" xml:space="preserve">At vero triangulum <lb/>DBL, complectitur {3/5}. </s> <s xml:id="echoid-s12133" xml:space="preserve">totius trianguli, id eſt, ita ſe habet DBL, ad ABC, vt BG, <lb/>ad BC. </s> <s xml:id="echoid-s12134" xml:space="preserve">Quadrilaterum denique ABDM, comprehendit {4/5}. </s> <s xml:id="echoid-s12135" xml:space="preserve">totius trianguli, hoc <lb/>eſt, ita ſe habet ABDM, ad ABC, vt B H, ad B C. </s> <s xml:id="echoid-s12136" xml:space="preserve">Ex quo fit, reliquum triangu-<lb/>lum DMC, eſſe {1/5}. </s> <s xml:id="echoid-s12137" xml:space="preserve">eiuſdem trianguli ABC.</s> <s xml:id="echoid-s12138" xml:space="preserve"/> </p> <div xml:id="echoid-div755" type="float" level="2" n="1"> <figure xlink:label="fig-293-01" xlink:href="fig-293-01a"> <image file="293-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/293-01"/> </figure> </div> <p> <s xml:id="echoid-s12139" xml:space="preserve"><emph style="sc">Qvando</emph> punctum datum eſt in vno angulo, manifeſtum eſt, ſi latus op-<lb/>poſitum in tot partes ſecetur, in quot triangulum diuidendum eſt, <anchor type="note" xlink:href="" symbol="a"/> rectas ex <anchor type="note" xlink:label="note-293-01a" xlink:href="note-293-01"/> eo angulo ad puncta diuiſionum eductas ſecare triangulum in propoſitas par-<lb/>tes æquales.</s> <s xml:id="echoid-s12140" xml:space="preserve"/> </p> <div xml:id="echoid-div756" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-293-01" xlink:href="note-293-01a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div758" type="section" level="1" n="259"> <head xml:id="echoid-head284" xml:space="preserve">PROBL. 6. PROPOS. 11.</head> <p> <s xml:id="echoid-s12141" xml:space="preserve">DATVM triangulum per lineas vni lateri parallelas in quotlibet æ-<lb/>quales partes diuidere,</s> </p> <p> <s xml:id="echoid-s12142" xml:space="preserve"><emph style="sc">Sit</emph> triangulum A B C, diuidendum verbi gratia in quatuor partes æquales <lb/>perlineas lateri BC, æquidiſtantes. </s> <s xml:id="echoid-s12143" xml:space="preserve">Secetur vtrumuis reliquorum laterum ni-<lb/> <anchor type="figure" xlink:label="fig-293-02a" xlink:href="fig-293-02"/> mirum A B, in 4. </s> <s xml:id="echoid-s12144" xml:space="preserve">partes æquales, in tot videlicet, in quot triangulũ diuidendum <lb/>eſt, in punctis D, E, F. </s> <s xml:id="echoid-s12145" xml:space="preserve">Etinter A B, A D, inuenta media proportionali A E;</s> <s xml:id="echoid-s12146" xml:space="preserve"> <pb o="264" file="294" n="294" rhead="GEOMETR. PRACT."/> atque inter AB, AE, media proportionali A G; </s> <s xml:id="echoid-s12147" xml:space="preserve">ac deniqueinter AB, AF, media <lb/>proportionali AH; </s> <s xml:id="echoid-s12148" xml:space="preserve">ducantur EI, GK, HL, lateri BC, parallelæ@ quas dico trian-<lb/>gulum partiriin 4. </s> <s xml:id="echoid-s12149" xml:space="preserve">partes æquales. </s> <s xml:id="echoid-s12150" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim triangulum ABC, triangu- <anchor type="note" xlink:label="note-294-01a" xlink:href="note-294-01"/> lo AEI, ſimile eſt; </s> <s xml:id="echoid-s12151" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit triangulum ABC, ad triangulum AEI, vt A B, ad A D, quod tres A B, A E, A D, ſint continuè pro portionales. </s> <s xml:id="echoid-s12152" xml:space="preserve">Eſt autem A D, quarta <lb/> <anchor type="note" xlink:label="note-294-02a" xlink:href="note-294-02"/> parsipſius AB. </s> <s xml:id="echoid-s12153" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12154" xml:space="preserve">triangulum AEI, quarta pars eſt trianguli ABC.</s> <s xml:id="echoid-s12155" xml:space="preserve"/> </p> <div xml:id="echoid-div758" type="float" level="2" n="1"> <figure xlink:label="fig-293-02" xlink:href="fig-293-02a"> <image file="293-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/293-02"/> </figure> <note symbol="a" position="left" xlink:label="note-294-01" xlink:href="note-294-01a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> <note symbol="b" position="left" xlink:label="note-294-02" xlink:href="note-294-02a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> </div> <p> <s xml:id="echoid-s12156" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <emph style="sc">Non</emph> aliter oſtendemus, eſſetriangulum A B C, ad triangulum A G K, vt <anchor type="note" xlink:label="note-294-03a" xlink:href="note-294-03"/> AB, ad AE, quod etiam tres AB, AG, AE, ſint continue proportionales. </s> <s xml:id="echoid-s12157" xml:space="preserve">Quare <lb/>cum AE, contineat {2/4}. </s> <s xml:id="echoid-s12158" xml:space="preserve">rectæ AB, continebit etiam AGK, triangulum {2/4}. </s> <s xml:id="echoid-s12159" xml:space="preserve">trian-<lb/>guli A B C: </s> <s xml:id="echoid-s12160" xml:space="preserve">Ideoq; </s> <s xml:id="echoid-s12161" xml:space="preserve">cum AEI, ſit {1/4}. </s> <s xml:id="echoid-s12162" xml:space="preserve">trianguli A B C, vt oſtendimus, erit EIK G, {1/4}. <lb/></s> <s xml:id="echoid-s12163" xml:space="preserve">eiuſdem trianguli A B C. </s> <s xml:id="echoid-s12164" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s12165" xml:space="preserve">eadem ratione erit triangulum A B C, ad trian-<lb/>gulum AHL, vt AB, ad AF, quod etiam tres AB, AH, AF, ſint continue propor-<lb/>tionales: </s> <s xml:id="echoid-s12166" xml:space="preserve">ac proinde triangulum AHL, complectetur {3/4}. </s> <s xml:id="echoid-s12167" xml:space="preserve">trianguli ABC; </s> <s xml:id="echoid-s12168" xml:space="preserve">quem-<lb/>admodum AF, continet {3/4}. </s> <s xml:id="echoid-s12169" xml:space="preserve">ipſius AB: </s> <s xml:id="echoid-s12170" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s12171" xml:space="preserve">BHLC, erit {1/4}. </s> <s xml:id="echoid-s12172" xml:space="preserve">trianguli ABC, &</s> <s xml:id="echoid-s12173" xml:space="preserve">c.</s> <s xml:id="echoid-s12174" xml:space="preserve"/> </p> <div xml:id="echoid-div759" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-294-03" xlink:href="note-294-03a" xml:space="preserve">coroll. 19. <lb/>ſexti.</note> </div> </div> <div xml:id="echoid-div761" type="section" level="1" n="260"> <head xml:id="echoid-head285" xml:space="preserve">PROBL. 7. PROPOS. 12.</head> <p> <s xml:id="echoid-s12175" xml:space="preserve">DATVM triangulum per rectam ex puncto extra triangulum dato <lb/>ductam in duas partes æquales diuidere.</s> <s xml:id="echoid-s12176" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12177" xml:space="preserve">Ex puncto D, extra triangulum AB C, dato ducenda ſit linea diuidens tri-<lb/>angulum bifariam. </s> <s xml:id="echoid-s12178" xml:space="preserve">Ducta recta D A, ad angulum oppoſitum ſecante latus B C, <lb/>in E: </s> <s xml:id="echoid-s12179" xml:space="preserve">ſi quidem B C, in E, diuiditur bifariam, factum erit, quod diubetur: </s> <s xml:id="echoid-s12180" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> quod tunc triangula ABE, ACE, ſint æqualia. </s> <s xml:id="echoid-s12181" xml:space="preserve">Si vero B C, non bifariam diuiditur in <lb/> <anchor type="note" xlink:label="note-294-04a" xlink:href="note-294-04"/> E, ſit ſegmentum CE, maius, cui ducatur parallelaDF, occurrens lateri AC, pro-<lb/>ducto in F. </s> <s xml:id="echoid-s12182" xml:space="preserve">Secto latere AC, bifariam in G, inueniatur tribus DF. </s> <s xml:id="echoid-s12183" xml:space="preserve">BC, CG, quar-<lb/>ta proportionalis CH; </s> <s xml:id="echoid-s12184" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Eritque rectangulum ſub DF, CH, æquale rectangulo ſub B C, C G; </s> <s xml:id="echoid-s12185" xml:space="preserve">hoc eſt ſemiſsirectanguli ſub B C, C A: </s> <s xml:id="echoid-s12186" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> cum rectangulum ſub <anchor type="note" xlink:label="note-294-05a" xlink:href="note-294-05"/> B C, C A, duplum ſit rectanguli ſub B C, C G. </s> <s xml:id="echoid-s12187" xml:space="preserve">Deinde <lb/> <anchor type="note" xlink:label="note-294-06a" xlink:href="note-294-06"/> <anchor type="figure" xlink:label="fig-294-01a" xlink:href="fig-294-01"/> inuenta L, media proportionaliinter F C, C H, <anchor type="note" xlink:href="" symbol="g"/> vt qua- dratum ex L, æquale ſitrectangulo ſub FC, CH, adiũ-<lb/> <anchor type="note" xlink:label="note-294-07a" xlink:href="note-294-07"/> gaturipſi CH, recta H@, vtrectangulum ſub tota CI, & </s> <s xml:id="echoid-s12188" xml:space="preserve"><lb/>adiuncta HI, ęquale ſit quadrato exL, ſiue rectangulo <lb/>ſub F C, C H, quemadmodum ad finem ſcholij pro-<lb/>poſ. </s> <s xml:id="echoid-s12189" xml:space="preserve">36. </s> <s xml:id="echoid-s12190" xml:space="preserve">lib. </s> <s xml:id="echoid-s12191" xml:space="preserve">3. </s> <s xml:id="echoid-s12192" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12193" xml:space="preserve">ſcripſimus: </s> <s xml:id="echoid-s12194" xml:space="preserve">ducaturque recta DI, <lb/>ſecans BC, in K. </s> <s xml:id="echoid-s12195" xml:space="preserve">Dico rectam DI, ſecare triangulum ABC, in duas partes AB-<lb/>KI, IKC, æquales. </s> <s xml:id="echoid-s12196" xml:space="preserve">Quoniam enim per conſtructionem rectangulũ ſub CI, IH, <lb/>æquale eſt rectangulo ſub CF, CH, <anchor type="note" xlink:href="" symbol="h"/> erit vt CI, ad CF, ita CH, ad IH; </s> <s xml:id="echoid-s12197" xml:space="preserve">Et cõuer- tendo, vt CF, ad CI, ita IH, ad CH: </s> <s xml:id="echoid-s12198" xml:space="preserve">& </s> <s xml:id="echoid-s12199" xml:space="preserve">cõponẽdo vt IF, ad CI, ad CH. </s> <s xml:id="echoid-s12200" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Vt <anchor type="note" xlink:label="note-294-08a" xlink:href="note-294-08"/> aũt IF, ad CI, ita eſt FD, ad CK. </s> <s xml:id="echoid-s12201" xml:space="preserve">Igitur erit quoq; </s> <s xml:id="echoid-s12202" xml:space="preserve">FD, ad CK, vt CI, ad CH: </s> <s xml:id="echoid-s12203" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Ac <anchor type="note" xlink:label="note-294-09a" xlink:href="note-294-09"/> {pue}inde rectangulũ ſub FD, CH æquale erit rectangulo ſub CK, CI: </s> <s xml:id="echoid-s12204" xml:space="preserve">Erat aũtre-<lb/>ctangulũ ſub FD, CH, per conſtructionẽ æquale ſemiſsirectanguli ſub BC, CA. <lb/></s> <s xml:id="echoid-s12205" xml:space="preserve"> <anchor type="note" xlink:label="note-294-10a" xlink:href="note-294-10"/> Igitur & </s> <s xml:id="echoid-s12206" xml:space="preserve">rectangulum ſub C K, C I, æquale erit ſemiſsi rectanguli ſub B C, <lb/>C A. </s> <s xml:id="echoid-s12207" xml:space="preserve">Vt autem rectangulum ſub CK, CI, ad rectangulum ſub BC, CA, <anchor type="note" xlink:href="" symbol="l"/> ita eſt tri- <anchor type="note" xlink:label="note-294-11a" xlink:href="note-294-11"/> angulum CKI, ad triangulum ABC. </s> <s xml:id="echoid-s12208" xml:space="preserve">Igitur triangulum CKI, æquale quoque <pb o="265" file="295" n="295" rhead="LIBER SEXTVS."/> erit ſemiſsi trianguli ABC: </s> <s xml:id="echoid-s12209" xml:space="preserve">ac proinde quadrilaterum ABKI, reliquæ ſemiſsitri-<lb/>anguli ABC, æquale erit. </s> <s xml:id="echoid-s12210" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12211" xml:space="preserve"/> </p> <div xml:id="echoid-div761" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-294-04" xlink:href="note-294-04a" xml:space="preserve">38. primi.</note> <note symbol="e" position="left" xlink:label="note-294-05" xlink:href="note-294-05a" xml:space="preserve">16. ſexti.</note> <note symbol="f" position="left" xlink:label="note-294-06" xlink:href="note-294-06a" xml:space="preserve">1. ſexti.</note> <figure xlink:label="fig-294-01" xlink:href="fig-294-01a"> <image file="294-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/294-01"/> </figure> <note symbol="g" position="left" xlink:label="note-294-07" xlink:href="note-294-07a" xml:space="preserve">17. ſexti.</note> <note symbol="h" position="left" xlink:label="note-294-08" xlink:href="note-294-08a" xml:space="preserve">16. ſexti.</note> <note symbol="i" position="left" xlink:label="note-294-09" xlink:href="note-294-09a" xml:space="preserve">4. ſexti & <lb/>permutando.</note> <note symbol="k" position="left" xlink:label="note-294-10" xlink:href="note-294-10a" xml:space="preserve">16. ſexti.</note> <note symbol="l" position="left" xlink:label="note-294-11" xlink:href="note-294-11a" xml:space="preserve">9. hui{us}.</note> </div> <p> <s xml:id="echoid-s12212" xml:space="preserve"><emph style="sc">Eadem</emph> ratione, ſi pro CG, ſumamus {1/3}. </s> <s xml:id="echoid-s12213" xml:space="preserve">vel {1/4}. </s> <s xml:id="echoid-s12214" xml:space="preserve">vel quamcumque partem <lb/>lateris AC, & </s> <s xml:id="echoid-s12215" xml:space="preserve">reliqua fiant, vt ſupra, auferemus perrectam ex D, ductam {1/3}. </s> <s xml:id="echoid-s12216" xml:space="preserve">vel <lb/>{1/4}. </s> <s xml:id="echoid-s12217" xml:space="preserve">vel deniq; </s> <s xml:id="echoid-s12218" xml:space="preserve">talẽ partem ex triangulo ABC, qualis ſumpta eſt C G, ipſius A C, <lb/>vt perſpicuum eſt.</s> <s xml:id="echoid-s12219" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12220" xml:space="preserve"><emph style="sc">Leonardvs</emph> Piſanus, & </s> <s xml:id="echoid-s12221" xml:space="preserve">Nicolaus Tartalea, idem hoc problema ſoluũt, <lb/>quando datum punctum eſt extra triangulum in tali loco, vt vnum latus trian-<lb/>guli pro ductum, in illud incidat, cuiuſmodi eſſet punctum F, datum. </s> <s xml:id="echoid-s12222" xml:space="preserve">Item <lb/>quando eſt inter duo latera producta: </s> <s xml:id="echoid-s12223" xml:space="preserve">Vt ſi triangulum foret AEC, punctũ au-<lb/>teminter B, & </s> <s xml:id="echoid-s12224" xml:space="preserve">D, exiſteret, ita vt ab eo ſolum per angulum E, duci poſſet linea <lb/>ſecans latus A C: </s> <s xml:id="echoid-s12225" xml:space="preserve">quippe cumrectæ ab eo ad angulos A, C, ductæ nullum latus <lb/>interſecarent. </s> <s xml:id="echoid-s12226" xml:space="preserve">Verum quia hęc curio ſa magis, quam vtilia ſunt, dedita opera à <lb/>nobis omittuntur. </s> <s xml:id="echoid-s12227" xml:space="preserve">Quiautem ea deſiderat, auctores prædictos legere pote-<lb/>rit. </s> <s xml:id="echoid-s12228" xml:space="preserve">Pariratione abſtinemus ab eo problemate, quando punctum datum eſt in-<lb/>tra triangulum (quod tamen ijdem auctores ſoluere conantur) quia non ſem-<lb/>per per punctum interius duci poteſt linea, quæ triangulũ bifariam ſecet, vt ex-<lb/>perientia conſtat.</s> <s xml:id="echoid-s12229" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div763" type="section" level="1" n="261"> <head xml:id="echoid-head286" xml:space="preserve">PROBL. 8. PROPOS. 13.</head> <p> <s xml:id="echoid-s12230" xml:space="preserve">DATVM parallelogrammum in quotcunq; </s> <s xml:id="echoid-s12231" xml:space="preserve">partes æquales perlineas <lb/>duobus lateribus oppoſitis æquidiſtantes diuidere.</s> <s xml:id="echoid-s12232" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12233" xml:space="preserve"><emph style="sc">Sit</emph> parallelogrammum ABCD, diuidendum verbi gratia in tres partes æ-<lb/>quales per lineas lateribus AB, DC, æquidiſtantes. </s> <s xml:id="echoid-s12234" xml:space="preserve">Diuiſo <lb/> <anchor type="figure" xlink:label="fig-295-01a" xlink:href="fig-295-01"/> alterutro reliquorum duorum laterum, nimirum B C, in <lb/>tres partes æquales, in quot videlicet parallelogrammum <lb/>proponitur diuidendum, in E, & </s> <s xml:id="echoid-s12235" xml:space="preserve">F, punctis, ducantur EG, <lb/>FH, ipſis AB, DC, parallelæ: </s> <s xml:id="echoid-s12236" xml:space="preserve">factumque erit, quodiube-<lb/>tur; </s> <s xml:id="echoid-s12237" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quod parallelogramma A E, E H, H C, ęqualia ſint, <anchor type="note" xlink:label="note-295-01a" xlink:href="note-295-01"/> propter æquales baſes BE, EF, FC.</s> <s xml:id="echoid-s12238" xml:space="preserve"/> </p> <div xml:id="echoid-div763" type="float" level="2" n="1"> <figure xlink:label="fig-295-01" xlink:href="fig-295-01a"> <image file="295-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/295-01"/> </figure> <note symbol="a" position="right" xlink:label="note-295-01" xlink:href="note-295-01a" xml:space="preserve">1. ſexti vel <lb/>38. primi.</note> </div> </div> <div xml:id="echoid-div765" type="section" level="1" n="262"> <head xml:id="echoid-head287" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s12239" xml:space="preserve"><emph style="sc">Itaqve</emph> ſi ex latere auferatur {1/2}. </s> <s xml:id="echoid-s12240" xml:space="preserve">vel {1/3}. </s> <s xml:id="echoid-s12241" xml:space="preserve">vel {3/4}. </s> <s xml:id="echoid-s12242" xml:space="preserve">vel denique qualiſcunque <lb/>pars, vel partes, & </s> <s xml:id="echoid-s12243" xml:space="preserve">per extremum eius punctum parallela lateri AB, ducatur, ab-<lb/>lata erit ex toto parallelo grammo eadem pars, vel eædem partes. </s> <s xml:id="echoid-s12244" xml:space="preserve">Ita vides A E, <lb/>eſſe partem tertiam parallelogrammi A C, quemadmodum & </s> <s xml:id="echoid-s12245" xml:space="preserve">B E, tertia pars eſt <lb/>lateris B C, &</s> <s xml:id="echoid-s12246" xml:space="preserve">c.</s> <s xml:id="echoid-s12247" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div766" type="section" level="1" n="263"> <head xml:id="echoid-head288" xml:space="preserve">PROBL. 9. PROPOS. 14.</head> <p> <s xml:id="echoid-s12248" xml:space="preserve">DATVM parallelogrammum per rectam ex puncto ſiue extra, ſiue <lb/>intra ipſum, ſiue in aliquo latere dato ductam, bifariam diuidere.</s> <s xml:id="echoid-s12249" xml:space="preserve"/> </p> <pb o="266" file="296" n="296" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s12250" xml:space="preserve"><emph style="sc">Sit</emph> primò parallelo grammum A B C D, per rectam ex puncto E, exteriori <lb/>ductam ſecundum bifariam. </s> <s xml:id="echoid-s12251" xml:space="preserve">Ducta diametro B D, eaque ſecta bifariam in F, <lb/>ducatur ex E, per F, recta EH, quam dico parallelogrammum <lb/> <anchor type="figure" xlink:label="fig-296-01a" xlink:href="fig-296-01"/> partiri bifariam. </s> <s xml:id="echoid-s12252" xml:space="preserve">Nam vt in ſcholio Propoſ. </s> <s xml:id="echoid-s12253" xml:space="preserve">34. </s> <s xml:id="echoid-s12254" xml:space="preserve">lib. </s> <s xml:id="echoid-s12255" xml:space="preserve">1. </s> <s xml:id="echoid-s12256" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12257" xml:space="preserve">de-<lb/>monſtrauimus, recta G H, diuidens diametrum B D, in F, bifa-<lb/>riam, ſecat parallelo grammum bifariam. </s> <s xml:id="echoid-s12258" xml:space="preserve">Idem fiet, ſi recta IK, <lb/>latera AD, BC, ſecans bifariam, diuidatur bifariam in F, & </s> <s xml:id="echoid-s12259" xml:space="preserve">per <lb/>F, extendatur recta E F, propterea quod I K, diametrum ſecat <lb/>bifariam, ac proinde per F, punctum medium diametritranſit. <lb/></s> <s xml:id="echoid-s12260" xml:space="preserve"> <anchor type="note" xlink:label="note-296-01a" xlink:href="note-296-01"/> <anchor type="note" xlink:href="" symbol="a"/>Cum enim anguli IDF, F I D, angulis alternis KBF, FKB, æquales ſint, & </s> <s xml:id="echoid-s12261" xml:space="preserve">latera <anchor type="note" xlink:label="note-296-02a" xlink:href="note-296-02"/> I D, KB, quibus adiacent, æqualia; </s> <s xml:id="echoid-s12262" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erunt tam latera D F, quam I<unsure/> F, K F, inter ſe æqualia.</s> <s xml:id="echoid-s12263" xml:space="preserve"/> </p> <div xml:id="echoid-div766" type="float" level="2" n="1"> <figure xlink:label="fig-296-01" xlink:href="fig-296-01a"> <image file="296-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/296-01"/> </figure> <note symbol="a" position="left" xlink:label="note-296-01" xlink:href="note-296-01a" xml:space="preserve">29. primi.</note> <note symbol="b" position="left" xlink:label="note-296-02" xlink:href="note-296-02a" xml:space="preserve">26. primi.</note> </div> <p> <s xml:id="echoid-s12264" xml:space="preserve"><emph style="sc">Eodem</emph> modo ex puncto interioriL; </s> <s xml:id="echoid-s12265" xml:space="preserve">Item ex puncto G, in latere BC, recta <lb/>ducta LF, vel GF, parallelogrammum bifariam diuidet.</s> <s xml:id="echoid-s12266" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div768" type="section" level="1" n="264"> <head xml:id="echoid-head289" xml:space="preserve">PROBL. 10. PROPOS. 15.</head> <p> <s xml:id="echoid-s12267" xml:space="preserve">INTER datas duas rectas, duas medias proportionales prope verum <lb/>inuenire.</s> <s xml:id="echoid-s12268" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12269" xml:space="preserve"><emph style="sc">Exposita</emph> Geodæſia noſtra prioribus quinq; </s> <s xml:id="echoid-s12270" xml:space="preserve">propoſitionibus huius lib. <lb/></s> <s xml:id="echoid-s12271" xml:space="preserve">& </s> <s xml:id="echoid-s12272" xml:space="preserve">nouem alijs propoſitionibus, ijs demonſtratis, quæ addenda eſſe cenſuimus <lb/>adidem argumentum ſpectantia: </s> <s xml:id="echoid-s12273" xml:space="preserve">agendumiam eſſet de augendis minuendiſq; </s> <s xml:id="echoid-s12274" xml:space="preserve"><lb/>figuris in data proportione, vt in titulo huius lib. </s> <s xml:id="echoid-s12275" xml:space="preserve">6. </s> <s xml:id="echoid-s12276" xml:space="preserve">propoſuimus. </s> <s xml:id="echoid-s12277" xml:space="preserve">Verum quia <lb/>ſicutid in planis figuris effi ci non poteſt ſine inuentione medię proportionalis <lb/>inter duasrectas propoſitas, quam inuentionem Euclid. </s> <s xml:id="echoid-s12278" xml:space="preserve">lib. </s> <s xml:id="echoid-s12279" xml:space="preserve">6. </s> <s xml:id="echoid-s12280" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12281" xml:space="preserve">13. </s> <s xml:id="echoid-s12282" xml:space="preserve">nobis <lb/>tradidit: </s> <s xml:id="echoid-s12283" xml:space="preserve">ita idem abſolui in figuris ſolidis nulla ratione poteſt, niſi inter duas <lb/>rectas datas duæ mediæ reperiantur proportionales. </s> <s xml:id="echoid-s12284" xml:space="preserve">Quo circa prius in hac pro-<lb/>poſ. </s> <s xml:id="echoid-s12285" xml:space="preserve">in medium afferemus, quæ antiqui Geometræ nobis hac de reſcripta relin-<lb/>querunt. </s> <s xml:id="echoid-s12286" xml:space="preserve">Multorum enim ingenia res hęc exercuit, at qu<unsure/>e torſit, quamuis ne-<lb/>mo ad hanc vſque diem, verè, ac Geometricè duas medias proportionales inter <lb/>duas rectas datas inuenerit. </s> <s xml:id="echoid-s12287" xml:space="preserve">Prætermi@sis autem modis Eratoſthenis; </s> <s xml:id="echoid-s12288" xml:space="preserve">Plato-<lb/>nis; </s> <s xml:id="echoid-s12289" xml:space="preserve">Pappi Alexandrini; </s> <s xml:id="echoid-s12290" xml:space="preserve">Spori; </s> <s xml:id="echoid-s12291" xml:space="preserve">menechmi tum beneficio Hyperbolę, ac para-<lb/>bolę, tum ope duarum parabolarum; </s> <s xml:id="echoid-s12292" xml:space="preserve">& </s> <s xml:id="echoid-s12293" xml:space="preserve">Architæ Tarentini, quamuis acutiſsi-<lb/>mis ſubtiliſsimiſque: </s> <s xml:id="echoid-s12294" xml:space="preserve">ſolum quatu or ab Herone, Apollonio Pergæo, Philone <lb/>Byſantio, Philoppono, Diocle, & </s> <s xml:id="echoid-s12295" xml:space="preserve">Nicomede traditos explicabimus, quos cõ-<lb/>modiores, facilioreſque, & </s> <s xml:id="echoid-s12296" xml:space="preserve">errori minus obnoxiosiudicauimus. </s> <s xml:id="echoid-s12297" xml:space="preserve">Qui aliorum <lb/>rationes deſiderat, legere eas poterit in Commentarijs Euto cij Aſcalonitæ in li-<lb/>brum 2. </s> <s xml:id="echoid-s12298" xml:space="preserve">Archimedis de Sphęra, & </s> <s xml:id="echoid-s12299" xml:space="preserve">Cylindro: </s> <s xml:id="echoid-s12300" xml:space="preserve">Item in libello Ioannis Verneri <lb/>Norimbergenſis de ſectionibus Conicis. </s> <s xml:id="echoid-s12301" xml:space="preserve">Hinc ita que exor diamur.</s> <s xml:id="echoid-s12302" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div769" type="section" level="1" n="265"> <head xml:id="echoid-head290" xml:space="preserve">MODVS HERONIS IN MECHANICIS <lb/>introductionibus, & telis fabricandis: qui etiam Apollo-<lb/>nio Pergæo aſcribitur.</head> <p> <s xml:id="echoid-s12303" xml:space="preserve"><emph style="sc">Sint</emph> duæ lineęrectę AB, BC, inter quas oporteat duas medias proportio-<lb/>nales in quirere. </s> <s xml:id="echoid-s12304" xml:space="preserve">Conſtituautur ad angulumrectum B, & </s> <s xml:id="echoid-s12305" xml:space="preserve">perficiatur rectangulũ <pb o="267" file="297" n="297" rhead="LIBER SEXTVS."/> ABCD, cum diametris AC, BD, <anchor type="note" xlink:href="" symbol="a"/> quæ ſe mutuo bifariam diuidentin E. </s> <s xml:id="echoid-s12306" xml:space="preserve">Satis eſ- <anchor type="note" xlink:label="note-297-01a" xlink:href="note-297-01"/> ſet vnam tantum diametrum ducere, eamquein E, ſecare bifariam. </s> <s xml:id="echoid-s12307" xml:space="preserve">Protractis <lb/>autemlateribus DA, DC, intelligatur circa punctum B, moueriregula hincinde, <lb/>donecita ſecet D A, D C, productas in F, & </s> <s xml:id="echoid-s12308" xml:space="preserve">G, vtrectæ emiſſæ E F, E G, ęquales <lb/>ſint. </s> <s xml:id="echoid-s12309" xml:space="preserve">Vel certè, vt vult Apollonius, ex E, plures circulideſcribantur LI, GF, <lb/>MN, donec chorda arcus vnius pręciſè per punctum B, incedat, qualis eſt GF. <lb/></s> <s xml:id="echoid-s12310" xml:space="preserve">Quod ſi chorda ſupra B, tranſeat, cuiuſmodi eſt chorda LI, deſcribendus erit cir-<lb/>culus j<unsure/>. </s> <s xml:id="echoid-s12311" xml:space="preserve">L; </s> <s xml:id="echoid-s12312" xml:space="preserve">Si verò infra punctũ B, tranſeat, qualis <lb/> <anchor type="figure" xlink:label="fig-297-01a" xlink:href="fig-297-01"/> eſt chorda MN, deſcribẽdus erit circul<emph style="sub">9</emph> s<unsure/>.</s> <s xml:id="echoid-s12313" xml:space="preserve">M. </s> <s xml:id="echoid-s12314" xml:space="preserve">At-<lb/>que hoc opus toties iterandum, donec aliqua <lb/>chorda, qualis eſt GF, per B, incedat. </s> <s xml:id="echoid-s12315" xml:space="preserve">Erunt enim <lb/>hacratione EF, EG, ex centro E, ad circumferen-<lb/>tiam GF, interſe ęquales. </s> <s xml:id="echoid-s12316" xml:space="preserve">Quibus ita conſtructis. <lb/></s> <s xml:id="echoid-s12317" xml:space="preserve">Dico A F, C G, eſſe medio loco proportionales <lb/>inter AB, BC: </s> <s xml:id="echoid-s12318" xml:space="preserve">hoc eſt, ita eſſe AB, ad AF, vt AF, <lb/>ad CG, & </s> <s xml:id="echoid-s12319" xml:space="preserve">CG, ad CB.</s> <s xml:id="echoid-s12320" xml:space="preserve"><unsure/> Diuiſis enim AD, CD, bi-<lb/> <anchor type="note" xlink:label="note-297-02a" xlink:href="note-297-02"/> fariam in K, & </s> <s xml:id="echoid-s12321" xml:space="preserve">H; </s> <s xml:id="echoid-s12322" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erunt ductę E K, E H, ad A D, C D, perpendiculares. </s> <s xml:id="echoid-s12323" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quoniam verò rectan- <anchor type="note" xlink:label="note-297-03a" xlink:href="note-297-03"/> gulum ſub D F, A F, vna cum quadrato ex A K, <lb/>quadrato ex K F, ęquale eſt; </s> <s xml:id="echoid-s12324" xml:space="preserve">addito communi <lb/>quadrato ex E K, eritrectangulum ſub D F, A F, <lb/>vna cum quadratis ex A K, E K, <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, vna <anchor type="note" xlink:label="note-297-04a" xlink:href="note-297-04"/> cũ quadrato ex EA, ęquale quadratis ex KF, EF, <anchor type="note" xlink:href="" symbol="e"/> hoceſt, quadrato ex EF, hoc <anchor type="note" xlink:label="note-297-05a" xlink:href="note-297-05"/> eſt, quadrato ex EG, quę ipſi EF, eſt æqualis. </s> <s xml:id="echoid-s12325" xml:space="preserve">Eademratione oſtendemus, re-<lb/>ctangulum ſub DG, GC, vna cum quadrato ex CE, id eſt, ex EA, ęquale eſſe ei-<lb/>dem quadrato ex E G. </s> <s xml:id="echoid-s12326" xml:space="preserve">Igitur rectangulum ſub D F, A F, vna cum quadrato <lb/>ex EA, ęquale erit rectangulo ſub DG, GC, vna cum quadrato ex EA: </s> <s xml:id="echoid-s12327" xml:space="preserve">dempto-<lb/>que communi quadrato EA; </s> <s xml:id="echoid-s12328" xml:space="preserve">remanebitrectangulum ſub DG, GC, rectangu-<lb/>lo ſub DF, AF, ęquale. </s> <s xml:id="echoid-s12329" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Quo circa erit DG, ad DF, vt AF, ad CG: </s> <s xml:id="echoid-s12330" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Vt autem <anchor type="note" xlink:label="note-297-06a" xlink:href="note-297-06"/> DG, ad DF, ita eſt AB, ad AF. </s> <s xml:id="echoid-s12331" xml:space="preserve">Ergo erit vt AB, ad AF, ita A F, ad C G: </s> <s xml:id="echoid-s12332" xml:space="preserve">hoc eſt, <lb/> <anchor type="note" xlink:label="note-297-07a" xlink:href="note-297-07"/> tres AB, AF, CG, continuè proportionales erunt. </s> <s xml:id="echoid-s12333" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Sed rurſuseſt, vt D G, ad <anchor type="note" xlink:label="note-297-08a" xlink:href="note-297-08"/> DF, ita CG, ad CB. </s> <s xml:id="echoid-s12334" xml:space="preserve">Igitur erit quoque CG, ad CB, vt AB, ad A F; </s> <s xml:id="echoid-s12335" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s12336" xml:space="preserve">vt AF <lb/>ad CG. </s> <s xml:id="echoid-s12337" xml:space="preserve">Quare erunt quatuor AB, AF, CG, CB, continuè proportionales. </s> <s xml:id="echoid-s12338" xml:space="preserve">quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s12339" xml:space="preserve"/> </p> <div xml:id="echoid-div769" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-297-01" xlink:href="note-297-01a" xml:space="preserve">ſchol. 34. <lb/>primi.</note> <figure xlink:label="fig-297-01" xlink:href="fig-297-01a"> <image file="297-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/297-01"/> </figure> <note symbol="b" position="right" xlink:label="note-297-02" xlink:href="note-297-02a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note symbol="c" position="right" xlink:label="note-297-03" xlink:href="note-297-03a" xml:space="preserve">6. ſecundi.</note> <note symbol="d" position="right" xlink:label="note-297-04" xlink:href="note-297-04a" xml:space="preserve">47. primi.</note> <note symbol="e" position="right" xlink:label="note-297-05" xlink:href="note-297-05a" xml:space="preserve">47. primi.</note> <note symbol="f" position="right" xlink:label="note-297-06" xlink:href="note-297-06a" xml:space="preserve">16. ſexti.</note> <note symbol="g" position="right" xlink:label="note-297-07" xlink:href="note-297-07a" xml:space="preserve">4. ſexti.</note> <note symbol="h" position="right" xlink:label="note-297-08" xlink:href="note-297-08a" xml:space="preserve">4. ſexti.</note> </div> </div> <div xml:id="echoid-div771" type="section" level="1" n="266"> <head xml:id="echoid-head291" xml:space="preserve">MODVS PHILONIS BYSANTII, <lb/>qui Philoppono quoque tribuitur.</head> <p> <s xml:id="echoid-s12340" xml:space="preserve"><emph style="sc">Sint</emph> rurſus in eadem figura inter rectas A B, B C, inueniendę duę medię <lb/>proportionales. </s> <s xml:id="echoid-s12341" xml:space="preserve">Conſtituto rectangulo ABCD, vna cum diametro CA, produ-<lb/>ctiſq; </s> <s xml:id="echoid-s12342" xml:space="preserve">lateribus D A, D C, vt ſupra; </s> <s xml:id="echoid-s12343" xml:space="preserve">deſcribatur ex E, medio puncto diametricir-<lb/>culus CBA, ad interuallum E C, vel EA, <anchor type="note" xlink:href="" symbol="i"/> qui neceſſario per angulum rectum B, <anchor type="note" xlink:label="note-297-09a" xlink:href="note-297-09"/> tranſibit. </s> <s xml:id="echoid-s12344" xml:space="preserve">Deinde circa punctum B, regula hincinde moueatur, ſecans DA, DC, <lb/>protractas in F, & </s> <s xml:id="echoid-s12345" xml:space="preserve">G, & </s> <s xml:id="echoid-s12346" xml:space="preserve">circumferentiamin O, donec B G, O F, ęquales ſint. <lb/></s> <s xml:id="echoid-s12347" xml:space="preserve">Quod fiet, ſi per B, plurimę lineę occultę ducantur. </s> <s xml:id="echoid-s12348" xml:space="preserve">Vna enim earum habebit <lb/>ſegmentum inter rectam DG, & </s> <s xml:id="echoid-s12349" xml:space="preserve">circulum æquale ſegmento inter DF, & </s> <s xml:id="echoid-s12350" xml:space="preserve">eundẽ <pb o="268" file="298" n="298" rhead="GEOMETR. PRACT."/> circulum. </s> <s xml:id="echoid-s12351" xml:space="preserve">Quibus peractis, dico AF, CG, medias proportionales eſſe inter AB, <lb/>CB. </s> <s xml:id="echoid-s12352" xml:space="preserve">Quoniam enim ęquales ſunt GB, FO; </s> <s xml:id="echoid-s12353" xml:space="preserve">addita communi BO, æquales quo-<lb/>que erunt GO, FB: </s> <s xml:id="echoid-s12354" xml:space="preserve">ideo querectangulum ſub GO, GB, rectangulo ſub FB, FO, <lb/>æquale erit. </s> <s xml:id="echoid-s12355" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Sedillud rectangulo ſub DG, GC, & </s> <s xml:id="echoid-s12356" xml:space="preserve">hocrectangulo ſub DF, AF, <anchor type="note" xlink:label="note-298-01a" xlink:href="note-298-01"/> eſt æquale. </s> <s xml:id="echoid-s12357" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s12358" xml:space="preserve">rectangulum ſub DG, GC, rectangulo ſub DF, AF, æqua-<lb/>le erit. </s> <s xml:id="echoid-s12359" xml:space="preserve">Quamobrem, vt in præcedentimodo, oſtendemus, AB, AF, CG, CB, eſſe <lb/>continue proportionales. </s> <s xml:id="echoid-s12360" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12361" xml:space="preserve"/> </p> <div xml:id="echoid-div771" type="float" level="2" n="1"> <note symbol="i" position="right" xlink:label="note-297-09" xlink:href="note-297-09a" xml:space="preserve">ſchol. 31. <lb/>tertij.</note> <note symbol="a" position="left" xlink:label="note-298-01" xlink:href="note-298-01a" xml:space="preserve">1. coroll. 36. <lb/>tertij.</note> </div> </div> <div xml:id="echoid-div773" type="section" level="1" n="267"> <head xml:id="echoid-head292" xml:space="preserve">MODIS DIOCLIS IN LIBRO DE <lb/>Piriis pulcherrimus.</head> <p> <s xml:id="echoid-s12362" xml:space="preserve"><emph style="sc">Præmittit</emph> prius Diocles Lemma tale. </s> <s xml:id="echoid-s12363" xml:space="preserve">Deſcribatur circulus A B C D, <lb/>cuius centrum E, cum diametris A C, B D, ſe ſe ad angulos rectos ſecantibus in <lb/>centro E. </s> <s xml:id="echoid-s12364" xml:space="preserve">Sumptis deinde duobus arcubus æqualibus DF, DG, iungaturrecta <lb/>CG, & </s> <s xml:id="echoid-s12365" xml:space="preserve">per F, ipſi B D, parallela agatur F K, ſecans C G, in H. </s> <s xml:id="echoid-s12366" xml:space="preserve">Hoc facto, erunt <lb/> <anchor type="figure" xlink:label="fig-298-01a" xlink:href="fig-298-01"/> FK, K C, mediæ proportionales inter AK, K H. </s> <s xml:id="echoid-s12367" xml:space="preserve">Ducta namque G L, parallela <lb/> <anchor type="note" xlink:label="note-298-02a" xlink:href="note-298-02"/> ipſi B D, iunctiſquerectis EF, EG, <anchor type="note" xlink:href="" symbol="b"/> quoniam anguli LEG, KEF, inſiſtentes ar- <anchor type="note" xlink:label="note-298-03a" xlink:href="note-298-03"/> cubus æqualibus AG, CF, æquales ſunt, <anchor type="note" xlink:href="" symbol="c"/> & </s> <s xml:id="echoid-s12368" xml:space="preserve">anguli L, K, recti, lateraque EG, EF, <anchor type="note" xlink:label="note-298-04a" xlink:href="note-298-04"/> æqualia; </s> <s xml:id="echoid-s12369" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erunt & </s> <s xml:id="echoid-s12370" xml:space="preserve">GL, FK, & </s> <s xml:id="echoid-s12371" xml:space="preserve">E L, E K, inter ſe æquales: </s> <s xml:id="echoid-s12372" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s12373" xml:space="preserve">reliquæ AL, CK; </s> <s xml:id="echoid-s12374" xml:space="preserve">Immo addita communi L K, & </s> <s xml:id="echoid-s12375" xml:space="preserve">totæ A K, C L, æquales inter ſe erunt. <lb/></s> <s xml:id="echoid-s12376" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="e"/> Quoniam igitur eſt CL, ad LG, vt CK, ad KH: </s> <s xml:id="echoid-s12377" xml:space="preserve">eſt que vt CL, ad LG, ita AK, <anchor type="note" xlink:label="note-298-05a" xlink:href="note-298-05"/> ad K F, quod hæ illis æquales ſint: </s> <s xml:id="echoid-s12378" xml:space="preserve">erit quoque AK, ad KF, vt CK, ad KH. </s> <s xml:id="echoid-s12379" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Vt <anchor type="note" xlink:label="note-298-06a" xlink:href="note-298-06"/> autem A K, ad KF, ita eſt KF, ad CK. </s> <s xml:id="echoid-s12380" xml:space="preserve">Igitur erit A K, ad K F, vt K F, ad C K, & </s> <s xml:id="echoid-s12381" xml:space="preserve"><lb/>CK, ad KH, hoc eſt, KF, CK, mediæ proportionales erunt inter AK, KH. </s> <s xml:id="echoid-s12382" xml:space="preserve">quod <lb/>eſt propoſitum. </s> <s xml:id="echoid-s12383" xml:space="preserve">Pari ratione, ſi, ſumptis arcubus ęqualibus DM, DN, iunctaq; <lb/></s> <s xml:id="echoid-s12384" xml:space="preserve">recta CM, ducatur NP, ipſi BD, parallela ſecans C M, in O; </s> <s xml:id="echoid-s12385" xml:space="preserve">erunt PN, CP, inter <lb/>A P, P O, mediæ proportionales, &</s> <s xml:id="echoid-s12386" xml:space="preserve">c.</s> <s xml:id="echoid-s12387" xml:space="preserve"/> </p> <div xml:id="echoid-div773" type="float" level="2" n="1"> <figure xlink:label="fig-298-01" xlink:href="fig-298-01a"> <image file="298-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/298-01"/> </figure> <note symbol="b" position="left" xlink:label="note-298-02" xlink:href="note-298-02a" xml:space="preserve">27. tertij.</note> <note symbol="c" position="left" xlink:label="note-298-03" xlink:href="note-298-03a" xml:space="preserve">27. primi.</note> <note symbol="d" position="left" xlink:label="note-298-04" xlink:href="note-298-04a" xml:space="preserve">26. primi.</note> <note symbol="e" position="left" xlink:label="note-298-05" xlink:href="note-298-05a" xml:space="preserve">4. ſexti.</note> <note symbol="f" position="left" xlink:label="note-298-06" xlink:href="note-298-06a" xml:space="preserve">ſchol. 13. <lb/>ſexti.</note> </div> <p> <s xml:id="echoid-s12388" xml:space="preserve"><emph style="sc">Hoc</emph> lemmate præmiſſo, ſint inter rectas AB, BC, reperiendæ duæ mediæ ꝓ-<lb/>portionales. </s> <s xml:id="echoid-s12389" xml:space="preserve">Conſtituantur in altera figura ad angulum rectum B, & </s> <s xml:id="echoid-s12390" xml:space="preserve">centro B, <lb/>ad interuallum maioris BA, deſcribatur circulus AFDE, ad cuius circumferen-<lb/>tiam vſque protendantur AB, BC. </s> <s xml:id="echoid-s12391" xml:space="preserve">Deinde ex A, per C, ducta recta ACG, ſuma-<lb/>tur in quadrante DE, punctum H, in tali ſitu, vt ducta H K, ipſi E F, parallela ſe-<lb/>canteipſam AG, in L; </s> <s xml:id="echoid-s12392" xml:space="preserve">recta ex D, per L, emiſſa auferat arcũ EM, arcui EH, æqua-<lb/>lem. </s> <s xml:id="echoid-s12393" xml:space="preserve">Namhacratione, per lemma præmiſſum, erunt KH, DK, mediæ propor-<lb/>tionales inter AK, KL. </s> <s xml:id="echoid-s12394" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Et quoniam eſt vt AK, ad KL, ita AB, ad BC: </s> <s xml:id="echoid-s12395" xml:space="preserve">ſi fiat, vt <anchor type="note" xlink:label="note-298-07a" xlink:href="note-298-07"/> A K, ad K H, ita A B, ad N; </s> <s xml:id="echoid-s12396" xml:space="preserve">& </s> <s xml:id="echoid-s12397" xml:space="preserve">vt K H, ad D K, ita N, ad O; </s> <s xml:id="echoid-s12398" xml:space="preserve">erunt quoque <lb/>N, O, mediæ proportionales inter A B, B C.</s> <s xml:id="echoid-s12399" xml:space="preserve"><unsure/> Neque enim dubitandum <lb/>eſt, eſſe & </s> <s xml:id="echoid-s12400" xml:space="preserve">O, ad BC, vt DK, ad K L. </s> <s xml:id="echoid-s12401" xml:space="preserve">Cum enim ſit, vt AK, ad KL, ita AB, ad BC:</s> <s xml:id="echoid-s12402" xml:space="preserve"> <pb o="269" file="299" n="299" rhead="LIBER SEXTVS."/> habeatautem AK, ad KL, proportionem triplicatã AK, ad HK, hoc eſt, AB, ad <lb/>N; </s> <s xml:id="echoid-s12403" xml:space="preserve">habebit etiam AB, ad BC, triplicatam proportionẽ AB, ad N. </s> <s xml:id="echoid-s12404" xml:space="preserve">Cum ergo pro-<lb/>portio AB, ad N, ſit æqualis proportioni N, ad O; </s> <s xml:id="echoid-s12405" xml:space="preserve">erit eidem æqualis proportio <lb/>O, ad B C, vt tres æquales proportiones exiſtant inter A B, & </s> <s xml:id="echoid-s12406" xml:space="preserve">B C. </s> <s xml:id="echoid-s12407" xml:space="preserve">Igitur qua-<lb/>tuor AB, N, O, BC, continuè proportionales ſunt, quemadmodum quatuor <lb/>AK, KH, DK, KL. </s> <s xml:id="echoid-s12408" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12409" xml:space="preserve"/> </p> <div xml:id="echoid-div774" type="float" level="2" n="2"> <note symbol="g" position="left" xlink:label="note-298-07" xlink:href="note-298-07a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s12410" xml:space="preserve"><emph style="sc">Vervm</emph>, quia diffi cile viſum fuit Diocli accipere in poſteriori figura pun-<lb/>ctum H, in tali ſitu, vtrecta DM, ſecans AG, & </s> <s xml:id="echoid-s12411" xml:space="preserve">parallelam HK, in L, auferat ar-<lb/>cum EM, arcui EH, æqualem: </s> <s xml:id="echoid-s12412" xml:space="preserve">deſcripſit lineam quandam inflexam ad hancrem <lb/>aptiſsimam, hac ratione. </s> <s xml:id="echoid-s12413" xml:space="preserve">Deſcribatur circulus A B C D, cuius centrum E, cum <lb/>diametris AC, BD, ſeſead angulosrectos in E, ſecantibus. </s> <s xml:id="echoid-s12414" xml:space="preserve">Deinde in quadrante <lb/>CD, capiantur quotcunque puncta parum interſe diſtantia, quæ ex D, & </s> <s xml:id="echoid-s12415" xml:space="preserve">B, or-<lb/>dine in quadrantes D A, B C, transferantur. </s> <s xml:id="echoid-s12416" xml:space="preserve">Poſt hæc applicata regula ad bina <lb/>puncta quadrantum DC, BC, æqualiter à B, D, diſtantia, ducantur rectæ occul-<lb/> <anchor type="figure" xlink:label="fig-299-01a" xlink:href="fig-299-01"/> tæ, <anchor type="note" xlink:href="" symbol="a"/> quæ ipſi BD, parallelę erunt. </s> <s xml:id="echoid-s12417" xml:space="preserve">Et ex C, ad ſingula puncta quadrantis D A, rectę occultæ emittantur, notentur que harum interſectiones cum prædictis pa-<lb/> <anchor type="note" xlink:label="note-299-01a" xlink:href="note-299-01"/> rallelis occultis; </s> <s xml:id="echoid-s12418" xml:space="preserve">nimirum punctum T, vbi recta ex C, ad proximum punctum <lb/>ipſi D, ducta interſecat proximam parallelam ipſi B D, & </s> <s xml:id="echoid-s12419" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s12420" xml:space="preserve">Nam ſi <lb/>omnia hæc interſectionum puncta ritè per lineam inflexam coniungantur, qua-<lb/>lis eſt CK TD, conſtructa erit figura mediis duabus proportionalibus inuenien-<lb/>dis aptiſsima. </s> <s xml:id="echoid-s12421" xml:space="preserve">Sint enim inter duas F, G, duæ mediæ proportinales inuenien-<lb/>dæ. </s> <s xml:id="echoid-s12422" xml:space="preserve">In diametro AC, etiam producta, ſi opus eſt, ſumatur AH, maiori F, æqua-<lb/>lis. </s> <s xml:id="echoid-s12423" xml:space="preserve">Ducta deinde perpendiculari H P, abſcindatur H I, minori G, ęqua-<lb/>lis. </s> <s xml:id="echoid-s12424" xml:space="preserve">Ducta autem AI, ſecante lineam in flexam in K, agatur per K, ipſi BD, paral-<lb/>lela LM. </s> <s xml:id="echoid-s12425" xml:space="preserve">Denique ſumpta L N, ipſi L C, æquali, ducantur per N, & </s> <s xml:id="echoid-s12426" xml:space="preserve">M, rectæ <lb/>AN, AM, ſecantes HP, in O, P. </s> <s xml:id="echoid-s12427" xml:space="preserve">Dico HP, HO, eſſe medias proportionales in-<lb/>ter AH, HI, hoc eſt, inter F, & </s> <s xml:id="echoid-s12428" xml:space="preserve">G. </s> <s xml:id="echoid-s12429" xml:space="preserve">Quoniam enim punctum K, lineæ inflexæ in-<lb/>uentum eſt per rectam ad punctum quadrantis DA, ductã, quod tanto interual-<lb/>lo à puncto D, abeſt, quanto punctum M, ab eodem diſtat, vt ex deſcriptione <lb/>lineæ inflexæ liquet; </s> <s xml:id="echoid-s12430" xml:space="preserve">erunt ex lemmate Dioclis quatuor rectæ AL, L M, LC, <lb/>vel LN, & </s> <s xml:id="echoid-s12431" xml:space="preserve">LK, continuè proportionales.</s> <s xml:id="echoid-s12432" xml:space="preserve"/> </p> <div xml:id="echoid-div775" type="float" level="2" n="3"> <figure xlink:label="fig-299-01" xlink:href="fig-299-01a"> <image file="299-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/299-01"/> </figure> <note symbol="a" position="right" xlink:label="note-299-01" xlink:href="note-299-01a" xml:space="preserve">ſchol. 27. <lb/>tertii.</note> </div> <pb o="270" file="300" n="300" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s12433" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum ergo hiſce quatuor rectis proportionales ſint quatuor rectæ AH, HP, <anchor type="note" xlink:label="note-300-01a" xlink:href="note-300-01"/> HO, HI; </s> <s xml:id="echoid-s12434" xml:space="preserve">erunt hæ quoque continuè proportionales. </s> <s xml:id="echoid-s12435" xml:space="preserve">quod eſt propoſirum.</s> <s xml:id="echoid-s12436" xml:space="preserve"/> </p> <div xml:id="echoid-div776" type="float" level="2" n="4"> <note symbol="a" position="left" xlink:label="note-300-01" xlink:href="note-300-01a" xml:space="preserve">4. ſexti.</note> </div> </div> <div xml:id="echoid-div778" type="section" level="1" n="268"> <head xml:id="echoid-head293" xml:space="preserve">MODVS NICOMEDIS IN <lb/>libro de lineis Conchoidibus.</head> <p> <s xml:id="echoid-s12437" xml:space="preserve"><emph style="sc">Nicomedes</emph> conſtruit prius inſtrumentum quoddam, quo lineaminfle-<lb/>xam deſcribit, quam Conchilem, vel Conchoideos appellat. </s> <s xml:id="echoid-s12438" xml:space="preserve">Sed nos omiſſo eo <lb/>inſtrumento, eandem, (quod ad noſtrum inſtitutum ſatis eſt) per puncta deli-<lb/>neabimus, hac ratione. </s> <s xml:id="echoid-s12439" xml:space="preserve">Sit recta linea A B, & </s> <s xml:id="echoid-s12440" xml:space="preserve">ad eam perpendicularis C D, in <lb/>puncto E. </s> <s xml:id="echoid-s12441" xml:space="preserve">Sumatur deinde infra E, punctum D, pro polo lineæ deſcribendæ, <lb/>& </s> <s xml:id="echoid-s12442" xml:space="preserve">ſupra E, aliud punctum C, vt libet. </s> <s xml:id="echoid-s12443" xml:space="preserve">In vſu lineæ deſcriptæ conſtabit, quan-<lb/>tum tam punctum D, quam punctum C, à puncto E, abeſſe debeat. </s> <s xml:id="echoid-s12444" xml:space="preserve">Si igitur ex <lb/>D, ducantur plurimæ lineæ occultę parum inter ſe diſtantes, & </s> <s xml:id="echoid-s12445" xml:space="preserve">ex ſingulis ab-<lb/> <anchor type="figure" xlink:label="fig-300-01a" xlink:href="fig-300-01"/> ſcindantur portiones rectæ E C, æquales, initio ſemper facto à recta AB; </s> <s xml:id="echoid-s12446" xml:space="preserve">ex-<lb/>trema autem harum portionum puncta per lineam inflexam coniungantur de-<lb/>ſcripta erit linea conchilis. </s> <s xml:id="echoid-s12447" xml:space="preserve">Exemplum habes in quatuor lineis D H, D G, D F, <lb/>DN, in quibus ſumptæ ſunt L H, K G, SF, BN, ipſi EC, æquales, per quarum ex-<lb/>trema puncta H, G, F, N, inflexa linea incedit. </s> <s xml:id="echoid-s12448" xml:space="preserve">Et quo plures lineæ occultę ex <lb/>D, educentur, eo crebriora puncta inuenientur, per quæ tranſire debet linea in-<lb/>flexa.</s> <s xml:id="echoid-s12449" xml:space="preserve"/> </p> <div xml:id="echoid-div778" type="float" level="2" n="1"> <figure xlink:label="fig-300-01" xlink:href="fig-300-01a"> <image file="300-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/300-01"/> </figure> </div> <p> <s xml:id="echoid-s12450" xml:space="preserve"><emph style="sc">Seqvitvr</emph> ex deſcriptione huius lineæ, eam nunquã poſſe cum recta AB, <lb/>conuenire, licet vtra que in infinitum producatur: </s> <s xml:id="echoid-s12451" xml:space="preserve">quia puncta, per quæ in cedit, <lb/>ſunt omnia ſupra rectam A B, terminantia nimirum ſegmenta rectarum ex D, <lb/>prodeuntium (<anchor type="note" xlink:href="" symbol="b"/> quæ quidem omnes rectam AB, interſecant) ipſi EC, æqualia.</s> <s xml:id="echoid-s12452" xml:space="preserve"/> </p> <note symbol="b" position="left" xml:space="preserve">pronuncia-<lb/>tum 11. lib. 1.</note> <p> <s xml:id="echoid-s12453" xml:space="preserve"><emph style="sc">Demonstrat</emph> deinde Nicomedes duas proprietates huius lineæ inſignes. <lb/></s> <s xml:id="echoid-s12454" xml:space="preserve">Prima eſt. </s> <s xml:id="echoid-s12455" xml:space="preserve">Quodlibet eius punctum à puncto C, diuerſum minus diſtat à recta <lb/>AB, quampunctum C: </s> <s xml:id="echoid-s12456" xml:space="preserve">Aliorum autem punctorum, quod remotius eſt à C, <lb/>minus diſtat ab eadem recta A B, quam quod minus remotum eſt. </s> <s xml:id="echoid-s12457" xml:space="preserve">Ducta enim <lb/>recta quacunque D G, demittatur perpendicularis GI. </s> <s xml:id="echoid-s12458" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Et quia K G, maior<unsure/> <anchor type="note" xlink:label="note-300-03a" xlink:href="note-300-03"/> eſt quam GI; </s> <s xml:id="echoid-s12459" xml:space="preserve">erit quoque perpendicularis E C, (ipſi K G, æqualis) maior quam <lb/>perpendicularis IG, hoc eſt, punctum C, magis diſtabit à recta AB, quam pun-<lb/>ctum G. </s> <s xml:id="echoid-s12460" xml:space="preserve">Eademq; </s> <s xml:id="echoid-s12461" xml:space="preserve">ratione magis à recta AB, diſtabit punctum C, quam quod-<lb/>uis aliud. </s> <s xml:id="echoid-s12462" xml:space="preserve">Sumatur deinde aliud punctum H, remotius à C, quam punctum G, <lb/>demittatur que perpendicularis HA. </s> <s xml:id="echoid-s12463" xml:space="preserve">Dico punctum H, minus diſtare à recta <lb/>AB, quampunctum G, hoc eſt, perpendicularem HA, minorem eſſe perpendi- <pb o="271" file="301" n="301" rhead="LIBER SEXTVS."/> culari G I. </s> <s xml:id="echoid-s12464" xml:space="preserve">Ducta namque recta D H, <anchor type="note" xlink:href="" symbol="a"/> erit angulus DKE, maio@ angulo DLE.</s> <s xml:id="echoid-s12465" xml:space="preserve"> <anchor type="note" xlink:label="note-301-01a" xlink:href="note-301-01"/> <anchor type="note" xlink:href="" symbol="b"/> hoc eſt angulus GKI, angulo HLA. </s> <s xml:id="echoid-s12466" xml:space="preserve">Cum ergo recti I, A, æquales ſint; </s> <s xml:id="echoid-s12467" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit <anchor type="note" xlink:label="note-301-02a" xlink:href="note-301-02"/> reliquus G, reliquo AHL, minor. </s> <s xml:id="echoid-s12468" xml:space="preserve">Siigitur ipſi G, fiat æqualis AHM; </s> <s xml:id="echoid-s12469" xml:space="preserve">erunt trian-<lb/> <anchor type="note" xlink:label="note-301-03a" xlink:href="note-301-03"/> gula KGI, MHA, æquiangula; </s> <s xml:id="echoid-s12470" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> ideoque erit, vt MH, ad HA, ita KG, ad GI.</s> <s xml:id="echoid-s12471" xml:space="preserve"> <anchor type="note" xlink:label="note-301-04a" xlink:href="note-301-04"/> <anchor type="note" xlink:href="" symbol="e"/> Et quia L H, maior eſt, quam M H, <anchor type="note" xlink:href="" symbol="f"/> (quod angulus HML, maior ſit recto A, <anchor type="note" xlink:label="note-301-05a" xlink:href="note-301-05"/> <anchor type="note" xlink:href="" symbol="g"/> & </s> <s xml:id="echoid-s12472" xml:space="preserve">HLM, minor) <anchor type="note" xlink:href="" symbol="h"/> erit maior proportio LH, ad HA, quam HM, ad HA, hoc <anchor type="note" xlink:label="note-301-06a" xlink:href="note-301-06"/> eſt, quam GK, ad GI: </s> <s xml:id="echoid-s12473" xml:space="preserve">ac proinde cum GK, HL, æquales ſint, erit quo que ma-<lb/> <anchor type="note" xlink:label="note-301-07a" xlink:href="note-301-07"/> ior proportio HL, ad HA, quam HL, ad GI; </s> <s xml:id="echoid-s12474" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> ideo que HA, minor erit quam <anchor type="note" xlink:label="note-301-08a" xlink:href="note-301-08"/> GI. </s> <s xml:id="echoid-s12475" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12476" xml:space="preserve"/> </p> <div xml:id="echoid-div779" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-300-03" xlink:href="note-300-03a" xml:space="preserve">19. primi.</note> <note symbol="a" position="right" xlink:label="note-301-01" xlink:href="note-301-01a" xml:space="preserve">16. primi.</note> <note symbol="b" position="right" xlink:label="note-301-02" xlink:href="note-301-02a" xml:space="preserve">15. primi.</note> <note symbol="c" position="right" xlink:label="note-301-03" xlink:href="note-301-03a" xml:space="preserve">32. primi.</note> <note symbol="d" position="right" xlink:label="note-301-04" xlink:href="note-301-04a" xml:space="preserve">4. ſexti.</note> <note symbol="e" position="right" xlink:label="note-301-05" xlink:href="note-301-05a" xml:space="preserve">19. primi.</note> <note symbol="f" position="right" xlink:label="note-301-06" xlink:href="note-301-06a" xml:space="preserve">16. primi.</note> <note symbol="g" position="right" xlink:label="note-301-07" xlink:href="note-301-07a" xml:space="preserve">17. primi.</note> <note symbol="h" position="right" xlink:label="note-301-08" xlink:href="note-301-08a" xml:space="preserve">8. quinti.</note> </div> <note symbol="i" position="right" xml:space="preserve">10. quinti.</note> <p> <s xml:id="echoid-s12477" xml:space="preserve"><emph style="sc">Altera</emph> proprietas eſt. </s> <s xml:id="echoid-s12478" xml:space="preserve">Quamuis Conchilis CF, nunquam conueniat cum <lb/>recta EB, tamen cum qualibet alia recta, etiam ipſi EB, propinquiſsima conue-<lb/>nit. </s> <s xml:id="echoid-s12479" xml:space="preserve">Sit enim primum recta NO, ipſi EB, parallela, ſecans EC, in O. </s> <s xml:id="echoid-s12480" xml:space="preserve">Fiatvt EO, <lb/>ad OD, ita E C, ad P. </s> <s xml:id="echoid-s12481" xml:space="preserve">Et quoniam E O, minor eſt quam E C;</s> <s xml:id="echoid-s12482" xml:space="preserve"><unsure/> <anchor type="note" xlink:href="" symbol="k"/> erit quoque <anchor type="note" xlink:label="note-301-10a" xlink:href="note-301-10"/> OD, minor quam P. </s> <s xml:id="echoid-s12483" xml:space="preserve">Si igitur ex D, ad interuallum rectę P, deſcribatur arcus cir-<lb/>culi, ſecabit is rectam ON, in aliquo puncto, vt in N. </s> <s xml:id="echoid-s12484" xml:space="preserve">Dico Conchilem CF, pro-<lb/>longatam coire cum O N, in N. </s> <s xml:id="echoid-s12485" xml:space="preserve">Ducta enim recta D N, ſecante E B, in B, quæ <lb/>ipſi P, æqualis erit; </s> <s xml:id="echoid-s12486" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> quoniam eſt vt EO, ad OD, ita BN, ad ND; </s> <s xml:id="echoid-s12487" xml:space="preserve">hoc eſt, ad ſi- <anchor type="note" xlink:label="note-301-11a" xlink:href="note-301-11"/> bi æqualem P. </s> <s xml:id="echoid-s12488" xml:space="preserve">Fuit autem etiam, vt EO, ad OD, ita EC,<unsure/> ad P. </s> <s xml:id="echoid-s12489" xml:space="preserve">Igitur BN, EC, <lb/>ad P, eandem proportionem habebunt: </s> <s xml:id="echoid-s12490" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> ac proinde inter ſe æquales erunt;</s> <s xml:id="echoid-s12491" xml:space="preserve"> <anchor type="note" xlink:label="note-301-12a" xlink:href="note-301-12"/> ideoque Conchilis per N, tranſibit.</s> <s xml:id="echoid-s12492" xml:space="preserve"/> </p> <div xml:id="echoid-div780" type="float" level="2" n="3"> <note symbol="k" position="right" xlink:label="note-301-10" xlink:href="note-301-10a" xml:space="preserve">14. quinti.</note> <note symbol="l" position="right" xlink:label="note-301-11" xlink:href="note-301-11a" xml:space="preserve">4. ſexti.</note> <note symbol="m" position="right" xlink:label="note-301-12" xlink:href="note-301-12a" xml:space="preserve">9. quinti.</note> </div> <p> <s xml:id="echoid-s12493" xml:space="preserve"><emph style="sc">Sit</emph> deinde recta Q F, non parallela ipſi E B, ſed eam ſecet in E, vergatque <lb/>verſus Conchilem. </s> <s xml:id="echoid-s12494" xml:space="preserve">Quia igitur Conchilis cum recta ON, conuenit, conueniet <lb/>prius cum ipſa QF, in F, vt perſpicuum eſt.</s> <s xml:id="echoid-s12495" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12496" xml:space="preserve"><emph style="sc">Post</emph> hæc Nicomedes diſſoluit huiuſmodi problema. </s> <s xml:id="echoid-s12497" xml:space="preserve">Dato quouis angu-<lb/>lo rectilineo, & </s> <s xml:id="echoid-s12498" xml:space="preserve">puncto extra lineas angulum datum comprehendentes: </s> <s xml:id="echoid-s12499" xml:space="preserve">Ab <lb/>illo puncto educere rectam ſecantem rectas datum continentes angulum, ita vt <lb/>eius portio inter illas rectas intercepta æqualis ſit datæ rectę. </s> <s xml:id="echoid-s12500" xml:space="preserve">In eadem nam-<lb/>que figura rectæ EB, EF, angulum contineant BEF, ducendaque ſit ex D, linea, <lb/>ita vt eius portio inter E B, E F, æqualis ſit datæ rectæ, R. </s> <s xml:id="echoid-s12501" xml:space="preserve">Ex O, ad inferiorem <lb/>lineam E B, ducatur perpendicularis DE, ſumatur que EC, datæ rectæ R, æqua-<lb/>lis: </s> <s xml:id="echoid-s12502" xml:space="preserve">& </s> <s xml:id="echoid-s12503" xml:space="preserve">polo D, interuallo verò EG, Conchilis deſcribatur, quæ per ſecun-<lb/>dam proprietatem rectam E F, ſecabit in F. </s> <s xml:id="echoid-s12504" xml:space="preserve">Ducta ergo recta D F, ſecante <lb/>E B, in S; </s> <s xml:id="echoid-s12505" xml:space="preserve">erit S F, ipſi EC, hoc eſt, ipſi R, æqualis, vt ex deſcriptione Conchi-<lb/>lis liquet.</s> <s xml:id="echoid-s12506" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12507" xml:space="preserve"><emph style="sc">His</emph> præmiſsis, ſint duæ rectæ AB BC, ad angulum rectum B, coniunctæ, in-<lb/>ter quas reperiendæ ſint duę lineæ medię proportionales. </s> <s xml:id="echoid-s12508" xml:space="preserve">Compleatur rectan-<lb/>gulum AC, cuius duo latera A D, CD, bifariam ſecentur in F, E. </s> <s xml:id="echoid-s12509" xml:space="preserve">Ducta autem <lb/>ex B, per E, recta ſecante A D, productam in G; </s> <s xml:id="echoid-s12510" xml:space="preserve"><anchor type="note" xlink:href="" symbol="n"/> erit DG, ipſi CB, hoc eſt, ipſi <anchor type="note" xlink:label="note-301-13a" xlink:href="note-301-13"/> D A, æqualis; </s> <s xml:id="echoid-s12511" xml:space="preserve">propterea quod anguli D, E, trianguli D E G, angulis CE, trian-<lb/>guli CEB, æquales ſunt, & </s> <s xml:id="echoid-s12512" xml:space="preserve">latera quoque DE, CE, quibus adiacent, æqualia. <lb/></s> <s xml:id="echoid-s12513" xml:space="preserve">Rurſus ductam perpendicularem FH, ſecet AH, ipſi CE, æqualis, quod fiet, ſi ex <lb/>A, ad interuallum C E, arcus delineetur ſecans F H, in H. </s> <s xml:id="echoid-s12514" xml:space="preserve">Deinde iuncta recta <lb/>GH, ducatur ei parallela AI: </s> <s xml:id="echoid-s12515" xml:space="preserve">atq; </s> <s xml:id="echoid-s12516" xml:space="preserve">producta DA; </s> <s xml:id="echoid-s12517" xml:space="preserve">ex H, per problema præcedens, <lb/>ducatur recta HK, vtramque AI, AK, ita ſecans, vt inter cepta IK, ipſi AH, vel <lb/>CE, æqualis ſit. </s> <s xml:id="echoid-s12518" xml:space="preserve">quod fiet, ſi ex H, plurimæ rectæ ducentur occultæ, donec v-<lb/>nius portio inter cepta æqualis ſit ipſi AH, vel CE. </s> <s xml:id="echoid-s12519" xml:space="preserve">Poſtremò ex K, per B, recta <pb o="272" file="302" n="302" rhead="GEOMETR. PRACT."/> extendatur ſecans DC, productamin L. </s> <s xml:id="echoid-s12520" xml:space="preserve">Dico duas AK, CL, medias propor-<lb/>tionales eſſe inter AB, BC. </s> <s xml:id="echoid-s12521" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim <anchor type="note" xlink:label="note-302-01a" xlink:href="note-302-01"/> eſt LC, ad CD, vt LB, ad BK, hoc eſt, vt DA, <lb/>ad AK; </s> <s xml:id="echoid-s12522" xml:space="preserve">Et vt CD, ad C E, ita eſt GA, ad DA, <lb/>quod vtraque CD, GA, ſecta ſit bifariam in <lb/>E, D: </s> <s xml:id="echoid-s12523" xml:space="preserve">erit ex proportione perturbata LC, ad <lb/> <anchor type="figure" xlink:label="fig-302-01a" xlink:href="fig-302-01"/> CE, vt GA ad AK, vt in hac formula apparet: <lb/></s> <s xml:id="echoid-s12524" xml:space="preserve">hoc eſt, vt HI, ad IK. </s> <s xml:id="echoid-s12525" xml:space="preserve">Cum ergo C E, ipſi I K, ſit æqualis per conſtructionem, <lb/> <anchor type="note" xlink:label="note-302-02a" xlink:href="note-302-02"/> <anchor type="note" xlink:href="" symbol="b"/> erit quoque LC, ipſi HI, æqualis, & </s> <s xml:id="echoid-s12526" xml:space="preserve">tota LE, toti HK. </s> <s xml:id="echoid-s12527" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Deinde quia rectan- <anchor type="note" xlink:label="note-302-03a" xlink:href="note-302-03"/> gulum ſub DK, KA, vna cum quadrato ex AF, æquale eſt quadrato FK; </s> <s xml:id="echoid-s12528" xml:space="preserve">addi-<lb/>to communi quadrato ex FH, eritrectangulum ſub DK, KA, vnà cum quadra-<lb/> <anchor type="note" xlink:label="note-302-04a" xlink:href="note-302-04"/> tis ex AF, FH, <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, vna cum quadrato ex AH, vel ex CE, æquale quadratis <anchor type="note" xlink:label="note-302-05a" xlink:href="note-302-05"/> ex KF, FH, hoc eſt, quadrato ex HK, id eſt, ex LE, ipſi HK, æquali. </s> <s xml:id="echoid-s12529" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Sed & </s> <s xml:id="echoid-s12530" xml:space="preserve">rectã- gulum ſub DL, LC, vna cum eodem quadrato ex CE, æquale quoq; </s> <s xml:id="echoid-s12531" xml:space="preserve">eſt eidem <lb/>quadrato ex LE. </s> <s xml:id="echoid-s12532" xml:space="preserve">Igitur rectangulum ſub D K, A K, vna cum quadrato ex C E, <lb/>æquale erit rectangulo ſub DL, LC, vna cum eodem quadrato ex CE; </s> <s xml:id="echoid-s12533" xml:space="preserve">Et dem-<lb/>pto communi quadrato C E, reliquum rectangulum ſub D L, L C, reliquo re-<lb/>ctangulo ſub DK, A K, æquale erit. </s> <s xml:id="echoid-s12534" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Igitur erit D L, ad D K, <anchor type="note" xlink:href="" symbol="g"/> hoc eſt, AB, ad <anchor type="note" xlink:label="note-302-06a" xlink:href="note-302-06"/> AK, vt AK, ad LC. </s> <s xml:id="echoid-s12535" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Vtautem AB, ad A K, ita eſt quo que LC, ad CB. </s> <s xml:id="echoid-s12536" xml:space="preserve">Igitur <anchor type="note" xlink:label="note-302-07a" xlink:href="note-302-07"/> erit AB, ad AK, vt AK, ad LC, & </s> <s xml:id="echoid-s12537" xml:space="preserve">vt LC, ad CB: </s> <s xml:id="echoid-s12538" xml:space="preserve">ac proinde AK, LC, medię pro-<lb/> <anchor type="note" xlink:label="note-302-08a" xlink:href="note-302-08"/> portionales erunt inter datas AB, BC, quod eſt propoſitum.</s> <s xml:id="echoid-s12539" xml:space="preserve"/> </p> <div xml:id="echoid-div781" type="float" level="2" n="4"> <note symbol="n" position="right" xlink:label="note-301-13" xlink:href="note-301-13a" xml:space="preserve">26. primi.</note> <note symbol="a" position="left" xlink:label="note-302-01" xlink:href="note-302-01a" xml:space="preserve">2. ſexti.</note> <figure xlink:label="fig-302-01" xlink:href="fig-302-01a"> <image file="302-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/302-01"/> </figure> <note symbol="b" position="left" xlink:label="note-302-02" xlink:href="note-302-02a" xml:space="preserve">14. quinti.</note> <note symbol="c" position="left" xlink:label="note-302-03" xlink:href="note-302-03a" xml:space="preserve">6. ſecundi.</note> <note symbol="d" position="left" xlink:label="note-302-04" xlink:href="note-302-04a" xml:space="preserve">47. primi.</note> <note symbol="e" position="left" xlink:label="note-302-05" xlink:href="note-302-05a" xml:space="preserve">6. ſecundi.</note> <note symbol="f" position="left" xlink:label="note-302-06" xlink:href="note-302-06a" xml:space="preserve">16. ſexti.</note> <note symbol="g" position="left" xlink:label="note-302-07" xlink:href="note-302-07a" xml:space="preserve">4. ſexti.</note> <note symbol="h" position="left" xlink:label="note-302-08" xlink:href="note-302-08a" xml:space="preserve">4. ſexti.</note> </div> <p> <s xml:id="echoid-s12540" xml:space="preserve"><emph style="sc">Qvod</emph> ſi datę duę rectæ ſint nimis longę, accipi poterunt earum ſemiſſes, <lb/>vel tertię partes, &</s> <s xml:id="echoid-s12541" xml:space="preserve">c. </s> <s xml:id="echoid-s12542" xml:space="preserve">atque inter eas duę medię inquirendę. </s> <s xml:id="echoid-s12543" xml:space="preserve">Nam ſi inuentę du-<lb/>plicentur, veltriplicentur, &</s> <s xml:id="echoid-s12544" xml:space="preserve">c. </s> <s xml:id="echoid-s12545" xml:space="preserve">habebuntur duę medię inter datas duas. </s> <s xml:id="echoid-s12546" xml:space="preserve">Quod <lb/>etiam in aliis modis intelligendum eſt.</s> <s xml:id="echoid-s12547" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div783" type="section" level="1" n="269"> <head xml:id="echoid-head294" xml:space="preserve">PROBL. 11. PROPOS. 16.</head> <p> <s xml:id="echoid-s12548" xml:space="preserve">DATAM figuram planam, vel circulum augere, vel minuere in data <lb/>proportione.</s> <s xml:id="echoid-s12549" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12550" xml:space="preserve"><emph style="sc">Hoc</emph> problema, quod ad figuras planas rectilineas attinet, explicauimus <lb/>propoſ. </s> <s xml:id="echoid-s12551" xml:space="preserve">15. </s> <s xml:id="echoid-s12552" xml:space="preserve">ſcholij propoſ. </s> <s xml:id="echoid-s12553" xml:space="preserve">33. </s> <s xml:id="echoid-s12554" xml:space="preserve">lib. </s> <s xml:id="echoid-s12555" xml:space="preserve">6. </s> <s xml:id="echoid-s12556" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12557" xml:space="preserve">Nuncidem ad circulos quoque ex-<lb/>tendemus. </s> <s xml:id="echoid-s12558" xml:space="preserve">Sit ergo rectilineum, cuius latus A B, vel circulus, cuius diameter <lb/>AB, oporteatque conſtituere maius rectilineum, vel circulum maiorem in pro-<lb/>portione, C, ad D, nimirum ſub tripla. </s> <s xml:id="echoid-s12559" xml:space="preserve">Tribus lineis C, D, AB, inueniatur quar- <pb o="273" file="303" n="303" rhead="LIBER SEXTVS."/> ta proportionalis E atque inter AB, & </s> <s xml:id="echoid-s12560" xml:space="preserve">E, reperiatur media proportionalis FG, <lb/>& </s> <s xml:id="echoid-s12561" xml:space="preserve">ſupra FG, figura conſtruatur ſimilis datæ figuræ AB, ſimiliterque poſita. </s> <s xml:id="echoid-s12562" xml:space="preserve">Item <lb/>circulus deſcribatur circa diametrnm FG. </s> <s xml:id="echoid-s12563" xml:space="preserve">Dico tam rectilineum A B, eſſe ter-<lb/>tiam partem rectilinei FG, quam circulum AB, circuli FG, nimirum eandem ha-<lb/>bere proportionem AB, ad FG, quam habet C, ad D. </s> <s xml:id="echoid-s12564" xml:space="preserve">Quoniam enim tres rectę <lb/>AB, FG, & </s> <s xml:id="echoid-s12565" xml:space="preserve">E, continuè proportionales ſunt, <anchor type="note" xlink:href="" symbol="a"/> erit figura A B, ad figuram FG, <anchor type="note" xlink:label="note-303-01a" xlink:href="note-303-01"/> vt AB, ad E, hoc eſt, vt C, ad D. </s> <s xml:id="echoid-s12566" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quia verò eſt, vt quadratum ex A B, ad qua- dratum FG, ita circulus AB, ad circulum FG; </s> <s xml:id="echoid-s12567" xml:space="preserve">eſtque quadratum AB, ad quadra-<lb/> <anchor type="note" xlink:label="note-303-02a" xlink:href="note-303-02"/> tum FG, vt AB, ad E; </s> <s xml:id="echoid-s12568" xml:space="preserve">erit quoque circulus AB, ad circulum FG, vt AB, ad E, vel <lb/>vt C, ad D.</s> <s xml:id="echoid-s12569" xml:space="preserve"/> </p> <div xml:id="echoid-div783" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-303-01" xlink:href="note-303-01a" xml:space="preserve">coroll. 19. <lb/>vel 20. ſexti.</note> <note symbol="b" position="right" xlink:label="note-303-02" xlink:href="note-303-02a" xml:space="preserve">2. duodec.</note> </div> <p> <s xml:id="echoid-s12570" xml:space="preserve"><emph style="sc">Sit</emph> deinde figura, vel circulus H I, oporteatque conſtruere minorem figu-<lb/>ram, vel circulum in proportione K, ad L, nimirum tripla. </s> <s xml:id="echoid-s12571" xml:space="preserve">Tribus rectis K, L, H <lb/>I, inueniatur quarta proportionalis M: </s> <s xml:id="echoid-s12572" xml:space="preserve">atque inter HI, & </s> <s xml:id="echoid-s12573" xml:space="preserve">M, media proportio-<lb/>nalis inueniatur N O, ſupra quam conſtruatur figura ſimilis ſimiliter que poſita <lb/>figuræ HI: </s> <s xml:id="echoid-s12574" xml:space="preserve">Item circulus deſcribatur circa diametrum NO. </s> <s xml:id="echoid-s12575" xml:space="preserve">Dico tam figuram <lb/>HI, ad figuram NO, quam circulum HI, ad circulum NO, habereproportionem <lb/>triplam, eandem videlicet, quam habet K, ad L. </s> <s xml:id="echoid-s12576" xml:space="preserve">Quoniam enim tres rectæ HI, <lb/> <anchor type="note" xlink:label="note-303-03a" xlink:href="note-303-03"/> NO, & </s> <s xml:id="echoid-s12577" xml:space="preserve">M, continuè ſunt proportionales; </s> <s xml:id="echoid-s12578" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit figura HI, ad figuram NO, vt recta HI, ad M, hoc eſt, vt K, ad L. </s> <s xml:id="echoid-s12579" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et quia eſt, vt quadratum HI, ad quadra- <anchor type="note" xlink:label="note-303-04a" xlink:href="note-303-04"/> tum NO, ita circulus HI, ad circulum NO, eſt que quadratum HI, ad quadra-<lb/>tum NO, vt recta HI, ad M; </s> <s xml:id="echoid-s12580" xml:space="preserve">erit quoque circulus HI, ad circulum NO, vtre-<lb/>cta HI, ad M, vel vt K, ad L.</s> <s xml:id="echoid-s12581" xml:space="preserve"/> </p> <div xml:id="echoid-div784" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-303-03" xlink:href="note-303-03a" xml:space="preserve">coroll. 19. vel <lb/>20. ſexti.</note> <note symbol="d" position="right" xlink:label="note-303-04" xlink:href="note-303-04a" xml:space="preserve">2. duodec.</note> </div> <p> <s xml:id="echoid-s12582" xml:space="preserve">Ex his conſtat, qua ratione, dato foramine rotundo, vel etiam quadrato ali-<lb/>cuius fontis, aliud foramen rotundum, vel quadratum maius, vel minus in qua-<lb/>cunque proportione conſtruendum ſit.</s> <s xml:id="echoid-s12583" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div786" type="section" level="1" n="270"> <head xml:id="echoid-head295" xml:space="preserve">PROBL. 12. PROPOS. 17.</head> <p> <s xml:id="echoid-s12584" xml:space="preserve">DATAM figuram ſolidam qualemcunque ex iis, de quibus Eucl. </s> <s xml:id="echoid-s12585" xml:space="preserve">in <lb/>lib. </s> <s xml:id="echoid-s12586" xml:space="preserve">Stereometriæ agit, augere vel minuere in proportione data.</s> <s xml:id="echoid-s12587" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12588" xml:space="preserve"><emph style="sc">Hvivsmodi</emph> figuræ ſolidæ ſunt parallelepipedum, Pyra-<lb/> <anchor type="figure" xlink:label="fig-303-01a" xlink:href="fig-303-01"/> mis, Priſma, ſphæra, Conus, Cylindrus, & </s> <s xml:id="echoid-s12589" xml:space="preserve">quinque corporare-<lb/>gularia.</s> <s xml:id="echoid-s12590" xml:space="preserve"/> </p> <div xml:id="echoid-div786" type="float" level="2" n="1"> <figure xlink:label="fig-303-01" xlink:href="fig-303-01a"> <image file="303-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/303-01"/> </figure> </div> <p> <s xml:id="echoid-s12591" xml:space="preserve"><emph style="sc">Sit</emph> ergo figura ſolida, cuius latus A, vel ſphæra, cuius dia-<lb/>meter A, augenda primum in proportione B, ad C. </s> <s xml:id="echoid-s12592" xml:space="preserve">Tribus re-<lb/>ctis B, C, A, inueniatur quarta proportionalis D: </s> <s xml:id="echoid-s12593" xml:space="preserve">atque inter A, <lb/>& </s> <s xml:id="echoid-s12594" xml:space="preserve">D, reperiantur duæ mediæ proportionales E, F. </s> <s xml:id="echoid-s12595" xml:space="preserve">Dico ſoli-<lb/>dum lateris A, ad ſolidum ſupra latus E, nimirum ſupra me-<lb/>diam proportionalem, quæ propinquior eſt lateri dato A, <lb/>conſtructum ſimile, ſimiliter que poſitum ſolido ſupra latus A, <lb/>conſtituto, habere proportionem, quam B, habet ad C. </s> <s xml:id="echoid-s12596" xml:space="preserve">Item <lb/>ſphæram datam diametri A, ad ſphæram diametri E, eſſe, vt B, <lb/>ad C. </s> <s xml:id="echoid-s12597" xml:space="preserve">Quoniam enim figura ſolida lateris A, ad figuram ſoli-<lb/>dam lateris E, ſimilem ſimiliter que poſitam habet proportio-<lb/>nem triplicatam lateris A, ad latus E, vt lib. </s> <s xml:id="echoid-s12598" xml:space="preserve">11. </s> <s xml:id="echoid-s12599" xml:space="preserve">& </s> <s xml:id="echoid-s12600" xml:space="preserve">12. </s> <s xml:id="echoid-s12601" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s12602" xml:space="preserve">de- <pb o="274" file="304" n="304" rhead="GEOMETR. PRACT."/> monſtratum eſt: </s> <s xml:id="echoid-s12603" xml:space="preserve">Similiter Conus & </s> <s xml:id="echoid-s12604" xml:space="preserve">Cylindrus, cuius baſis diameter A, ad Co-<lb/>num & </s> <s xml:id="echoid-s12605" xml:space="preserve">Cylindrum ſimilem, cuius diameter E: </s> <s xml:id="echoid-s12606" xml:space="preserve">Necnon ſphæra diametri A, ad <lb/>ſphæram diametri E; </s> <s xml:id="echoid-s12607" xml:space="preserve">eſt autem, ex defin. </s> <s xml:id="echoid-s12608" xml:space="preserve">10. </s> <s xml:id="echoid-s12609" xml:space="preserve">lib. </s> <s xml:id="echoid-s12610" xml:space="preserve">5. </s> <s xml:id="echoid-s12611" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12612" xml:space="preserve">proportio quoque <lb/>A, ad D, triplicata proportionis A, ad E: </s> <s xml:id="echoid-s12613" xml:space="preserve">Erit ſolidum A, ad ſolidum E, vt A, ad <lb/>D, hoc eſt, vt B, ad C. </s> <s xml:id="echoid-s12614" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12615" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12616" xml:space="preserve"><emph style="sc">Sit</emph> deinde ſolidum lateris, vel diametri D, minuendum in proportione da-<lb/>ta C, ad B. </s> <s xml:id="echoid-s12617" xml:space="preserve">Tribus C, B, D, inueniatur quarta proportionalis A; </s> <s xml:id="echoid-s12618" xml:space="preserve">atque inter D, <lb/>& </s> <s xml:id="echoid-s12619" xml:space="preserve">A, reperiantur duæ mediæ proportionales F, E. </s> <s xml:id="echoid-s12620" xml:space="preserve">Dico ſolidum lateris, vel dia-<lb/>metri D, ad ſolidum ſimile ſimiliter que deſcriptum ſupra F, nimirum ſupra me-<lb/>diam proportionalem, quæ lateri dato D, propinquior eſt, proportio nem habe-<lb/>re, quam C, ad B. </s> <s xml:id="echoid-s12621" xml:space="preserve">Quoniam enim ſolidum D, ad ſimile ſimiliter que deſcriptum <lb/>ſolidum F, proportionem habet triplicatam lateris D, ad latus F: </s> <s xml:id="echoid-s12622" xml:space="preserve">qualem etiam <lb/>habet ex defin. </s> <s xml:id="echoid-s12623" xml:space="preserve">10. </s> <s xml:id="echoid-s12624" xml:space="preserve">lib. </s> <s xml:id="echoid-s12625" xml:space="preserve">5. </s> <s xml:id="echoid-s12626" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s12627" xml:space="preserve">recta D, ad rectam A: </s> <s xml:id="echoid-s12628" xml:space="preserve">erit ſolidum D, ad ſolidum <lb/>F, vt D, ad A, id eſt, vt C, ad B. </s> <s xml:id="echoid-s12629" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s12630" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12631" xml:space="preserve"><emph style="sc">Constat</emph> ex his, qua ratione Cubus non ſolum duplicandus ſit (quod <lb/>veteres inquirebant) ſed etiam augendus minuenduſue in quacunque propor-<lb/>tione: </s> <s xml:id="echoid-s12632" xml:space="preserve">Item quo pacto pylæ bombardarum maiores, aut minores fieri debeant <lb/>ſecundum proportionem datam.</s> <s xml:id="echoid-s12633" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div788" type="section" level="1" n="271"> <head xml:id="echoid-head296" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s12634" xml:space="preserve"><emph style="sc">Figvras</emph> ſolidas ſimiliterque poſitas habere proportionem triplicatam <lb/>homologorum laterum, demonſtratum eſt de parallelepipedis quidem lib. </s> <s xml:id="echoid-s12635" xml:space="preserve">11. <lb/></s> <s xml:id="echoid-s12636" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s12637" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12638" xml:space="preserve">33. </s> <s xml:id="echoid-s12639" xml:space="preserve">Depyramidibus verò lib. </s> <s xml:id="echoid-s12640" xml:space="preserve">12. </s> <s xml:id="echoid-s12641" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12642" xml:space="preserve">8. </s> <s xml:id="echoid-s12643" xml:space="preserve">eiuſque coroll. </s> <s xml:id="echoid-s12644" xml:space="preserve">& </s> <s xml:id="echoid-s12645" xml:space="preserve">de <lb/>Priſmatis, in eiuſdem ſcholio. </s> <s xml:id="echoid-s12646" xml:space="preserve">Deſphæra autem lib. </s> <s xml:id="echoid-s12647" xml:space="preserve">eodem 12. </s> <s xml:id="echoid-s12648" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12649" xml:space="preserve">18. </s> <s xml:id="echoid-s12650" xml:space="preserve">De <lb/>Conis deinde & </s> <s xml:id="echoid-s12651" xml:space="preserve">Cylindris eodem lib. </s> <s xml:id="echoid-s12652" xml:space="preserve">12. </s> <s xml:id="echoid-s12653" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12654" xml:space="preserve">12. </s> <s xml:id="echoid-s12655" xml:space="preserve">ſi pro lateribus homolo-<lb/>gis ſumantur diametri ſphærarum, & </s> <s xml:id="echoid-s12656" xml:space="preserve">diametribaſium Conorum, & </s> <s xml:id="echoid-s12657" xml:space="preserve">Cylindro-<lb/>rum. </s> <s xml:id="echoid-s12658" xml:space="preserve">Actandem de quinque corporibus regularibus in coroll. </s> <s xml:id="echoid-s12659" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s12660" xml:space="preserve">17. </s> <s xml:id="echoid-s12661" xml:space="preserve">lib. </s> <s xml:id="echoid-s12662" xml:space="preserve"><lb/>12. </s> <s xml:id="echoid-s12663" xml:space="preserve">quippe cum omnia hæc corpora in ſphæris deſcribi poſsint.</s> <s xml:id="echoid-s12664" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12665" xml:space="preserve"><emph style="sc">Qvamvis</emph> autem problema hoc deſupradictis corporibus duntaxat pro-<lb/>poſuerimus, idem tamen etiam locum habet in aliis cuiuſque generis corpori-<lb/>bus ſimilibus, ſimiliter que poſitis, vt perſpicuum eſt; </s> <s xml:id="echoid-s12666" xml:space="preserve">propterea quod diuidi <lb/>poſſunt in pyramides ſimiles, æquales numero; </s> <s xml:id="echoid-s12667" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quæ quidem proportionem <anchor type="note" xlink:label="note-304-01a" xlink:href="note-304-01"/> habent laterum homologorum triplicatam. </s> <s xml:id="echoid-s12668" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cum ergo ſit, vt vna pyramis ad vnam pyramidem, ita omnes ad omnes, id eſt, ita ſolidum ad ſolidum; </s> <s xml:id="echoid-s12669" xml:space="preserve">ſint-<lb/> <anchor type="note" xlink:label="note-304-02a" xlink:href="note-304-02"/> que eadem latera homologa ſolidorum, quæ pyramidum ſimilium, conſtat <lb/>propoſitum.</s> <s xml:id="echoid-s12670" xml:space="preserve"/> </p> <div xml:id="echoid-div788" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-304-01" xlink:href="note-304-01a" xml:space="preserve">8. duodec. <lb/>eiuſ coroll.</note> <note symbol="b" position="left" xlink:label="note-304-02" xlink:href="note-304-02a" xml:space="preserve">12. quinti.</note> </div> </div> <div xml:id="echoid-div790" type="section" level="1" n="272"> <head xml:id="echoid-head297" xml:space="preserve">PROBL. 13. PROPOS. 18.</head> <p> <s xml:id="echoid-s12671" xml:space="preserve">INTER duos numeros datos tum vnum, tum duos medios propor-<lb/>tionales reperire.</s> <s xml:id="echoid-s12672" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12673" xml:space="preserve"><emph style="sc">Non</emph> rarò figura ſiue plana, ſiue ſolida augenda, vel minuenda eſt per nume-<lb/>ros, quod quidem ſine inuentione vnius medij proportionalis, vel duorum <lb/>mediorum inter datos duos numeros perfici nonpoteſt: </s> <s xml:id="echoid-s12674" xml:space="preserve">idcirco artem præ- <pb o="275" file="305" n="305" rhead="LIBER SEXTVS."/> ſcribemus, qua huiuſmodi medias inuenire poſsimus. </s> <s xml:id="echoid-s12675" xml:space="preserve">Propoſitis igitur primum <lb/>duobus numeris quibuſcunque 9. </s> <s xml:id="echoid-s12676" xml:space="preserve">& </s> <s xml:id="echoid-s12677" xml:space="preserve">25. </s> <s xml:id="echoid-s12678" xml:space="preserve">inter quos reperiendus ſit vnus medius <lb/>proportionalis; </s> <s xml:id="echoid-s12679" xml:space="preserve">ſi multiplicentur inter ſe, & </s> <s xml:id="echoid-s12680" xml:space="preserve">producti numeri 225. </s> <s xml:id="echoid-s12681" xml:space="preserve">radix qua-<lb/>drata eruatur 15. </s> <s xml:id="echoid-s12682" xml:space="preserve">vt in Arithmetica practica cap. </s> <s xml:id="echoid-s12683" xml:space="preserve">26. </s> <s xml:id="echoid-s12684" xml:space="preserve">docuimus: </s> <s xml:id="echoid-s12685" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit radix hæc <anchor type="note" xlink:label="note-305-01a" xlink:href="note-305-01"/> quadrata medio loco proportionalis inter datos numeros, vt hic 9. </s> <s xml:id="echoid-s12686" xml:space="preserve">15. </s> <s xml:id="echoid-s12687" xml:space="preserve">25. </s> <s xml:id="echoid-s12688" xml:space="preserve">quip-<lb/>pe cum quadratum medij numeri æquale ſit rectangulo ſub extremis compre-<lb/>henſo. </s> <s xml:id="echoid-s12689" xml:space="preserve">Sic inter 5. </s> <s xml:id="echoid-s12690" xml:space="preserve">& </s> <s xml:id="echoid-s12691" xml:space="preserve">13. </s> <s xml:id="echoid-s12692" xml:space="preserve">medius proportionalis erit radix quadrata numeri 65. <lb/></s> <s xml:id="echoid-s12693" xml:space="preserve">qui ex multiplicatione datorum numerorum gignitur, quæ radix paulò maior <lb/>eſt, quam 8 {1/17}. </s> <s xml:id="echoid-s12694" xml:space="preserve">& </s> <s xml:id="echoid-s12695" xml:space="preserve">paulò minor, quam 8 {1/16}.</s> <s xml:id="echoid-s12696" xml:space="preserve"/> </p> <div xml:id="echoid-div790" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-305-01" xlink:href="note-305-01a" xml:space="preserve">17. ſexti, vel <lb/>20 ſept.</note> </div> <p> <s xml:id="echoid-s12697" xml:space="preserve"><emph style="sc">Sint</emph> deinde duo numeri 2. </s> <s xml:id="echoid-s12698" xml:space="preserve">& </s> <s xml:id="echoid-s12699" xml:space="preserve">54. </s> <s xml:id="echoid-s12700" xml:space="preserve">inter quos inueniendi ſint duo medij <lb/>proportionales. </s> <s xml:id="echoid-s12701" xml:space="preserve">Multiplicetur quadratus minoris in maiorem. </s> <s xml:id="echoid-s12702" xml:space="preserve">Producti nam-<lb/>que numeri 216. </s> <s xml:id="echoid-s12703" xml:space="preserve">radix cubica 6. </s> <s xml:id="echoid-s12704" xml:space="preserve">erit primus medius iuxta minorem collocan-<lb/>dus. </s> <s xml:id="echoid-s12705" xml:space="preserve">Et ſi maioris quadratus ducatur in minorem, erit producti numeri 5832. <lb/></s> <s xml:id="echoid-s12706" xml:space="preserve">radix cubica 18. </s> <s xml:id="echoid-s12707" xml:space="preserve">alter medius iuxta maiorem ſtatuendus, vt hic 2. </s> <s xml:id="echoid-s12708" xml:space="preserve">6. </s> <s xml:id="echoid-s12709" xml:space="preserve">18. </s> <s xml:id="echoid-s12710" xml:space="preserve">54. </s> <s xml:id="echoid-s12711" xml:space="preserve">Ra-<lb/>tio huius rei eſt, quod datis quatuor lineis continuè proportionalibus, paralle-<lb/>lepipedum ſub quadrato alterutrius extremarum, & </s> <s xml:id="echoid-s12712" xml:space="preserve">ſub altera extrema com-<lb/>prehenſum, æquale eſt cubo mediæ proportionalis, quæ priori extremo aſſum-<lb/>pto propinquior eſt, vt in ſequenti Lemmate demonſtrabimus. </s> <s xml:id="echoid-s12713" xml:space="preserve">Quoniam ve-<lb/>rò, vt in ſcholio propoſ. </s> <s xml:id="echoid-s12714" xml:space="preserve">19. </s> <s xml:id="echoid-s12715" xml:space="preserve">lib. </s> <s xml:id="echoid-s12716" xml:space="preserve">8. </s> <s xml:id="echoid-s12717" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s12718" xml:space="preserve">oſtendimus, propoſitis hiſce tribus <lb/>numeris 2. </s> <s xml:id="echoid-s12719" xml:space="preserve">2. </s> <s xml:id="echoid-s12720" xml:space="preserve">54. </s> <s xml:id="echoid-s12721" xml:space="preserve">idem procreatur numerus, ſiue prius ducantur 2. </s> <s xml:id="echoid-s12722" xml:space="preserve">in 2. </s> <s xml:id="echoid-s12723" xml:space="preserve">deinde <lb/>productus 4. </s> <s xml:id="echoid-s12724" xml:space="preserve">in 54. </s> <s xml:id="echoid-s12725" xml:space="preserve">ſiue prius 2. </s> <s xml:id="echoid-s12726" xml:space="preserve">in 54. </s> <s xml:id="echoid-s12727" xml:space="preserve">deinde productus 108. </s> <s xml:id="echoid-s12728" xml:space="preserve">in 2. </s> <s xml:id="echoid-s12729" xml:space="preserve">Item datis hiſ-<lb/>ce tribus numeris 54. </s> <s xml:id="echoid-s12730" xml:space="preserve">54. </s> <s xml:id="echoid-s12731" xml:space="preserve">2. </s> <s xml:id="echoid-s12732" xml:space="preserve">idem numerus gignitur, ſiue prius ducantur 54. </s> <s xml:id="echoid-s12733" xml:space="preserve">in <lb/>54. </s> <s xml:id="echoid-s12734" xml:space="preserve">deinde productus 2916. </s> <s xml:id="echoid-s12735" xml:space="preserve">in 2. </s> <s xml:id="echoid-s12736" xml:space="preserve">ſiue prius 54. </s> <s xml:id="echoid-s12737" xml:space="preserve">in 2. </s> <s xml:id="echoid-s12738" xml:space="preserve">deinde productus 108. </s> <s xml:id="echoid-s12739" xml:space="preserve">in <lb/>54. </s> <s xml:id="echoid-s12740" xml:space="preserve">manifeſto colligitur, ſi minor 2. </s> <s xml:id="echoid-s12741" xml:space="preserve">ducatur in maiorem 54. </s> <s xml:id="echoid-s12742" xml:space="preserve">& </s> <s xml:id="echoid-s12743" xml:space="preserve">productus 108. </s> <s xml:id="echoid-s12744" xml:space="preserve"><lb/>in minorem 2. </s> <s xml:id="echoid-s12745" xml:space="preserve">produci quoque cubum medij proportionalis iuxta minorem <lb/>conſtituendi: </s> <s xml:id="echoid-s12746" xml:space="preserve">Item ſi maior 54. </s> <s xml:id="echoid-s12747" xml:space="preserve">ducatur in minorem 2. </s> <s xml:id="echoid-s12748" xml:space="preserve">& </s> <s xml:id="echoid-s12749" xml:space="preserve">productus 108. </s> <s xml:id="echoid-s12750" xml:space="preserve">in ma-<lb/>iorem 54. </s> <s xml:id="echoid-s12751" xml:space="preserve">pro creari cubum medij proportionalis iuxta maiorem ſcribendi. </s> <s xml:id="echoid-s12752" xml:space="preserve">Sic <lb/>inter 4 & </s> <s xml:id="echoid-s12753" xml:space="preserve">100. </s> <s xml:id="echoid-s12754" xml:space="preserve">erunt duo medij proportionales, Radix cubica numeri 1600. </s> <s xml:id="echoid-s12755" xml:space="preserve">& </s> <s xml:id="echoid-s12756" xml:space="preserve"><lb/>Radix cubica numeri 40000. </s> <s xml:id="echoid-s12757" xml:space="preserve">Cæterum inuento altero mediorum numero-<lb/>rum, reperietur alter etiam, ſi inuentus per extremum remotiorem multiplice-<lb/>tur & </s> <s xml:id="echoid-s12758" xml:space="preserve">producti numeri radix quadrata capiatur. </s> <s xml:id="echoid-s12759" xml:space="preserve">Vt in dato exemplo 2. </s> <s xml:id="echoid-s12760" xml:space="preserve">6. </s> <s xml:id="echoid-s12761" xml:space="preserve">18. </s> <s xml:id="echoid-s12762" xml:space="preserve"><lb/>54. </s> <s xml:id="echoid-s12763" xml:space="preserve">ſi medius inuentus 6. </s> <s xml:id="echoid-s12764" xml:space="preserve">ducatur in 54. </s> <s xml:id="echoid-s12765" xml:space="preserve">erit producti numeri 324. </s> <s xml:id="echoid-s12766" xml:space="preserve">radix quadra-<lb/>ta 18. </s> <s xml:id="echoid-s12767" xml:space="preserve">alter medius: </s> <s xml:id="echoid-s12768" xml:space="preserve">Item inuentus medius 18. </s> <s xml:id="echoid-s12769" xml:space="preserve">ſi multiplicetur per 2. </s> <s xml:id="echoid-s12770" xml:space="preserve">erit pro ducti <lb/>numeri 36. </s> <s xml:id="echoid-s12771" xml:space="preserve">radix quadrata 6. </s> <s xml:id="echoid-s12772" xml:space="preserve">alter medius: </s> <s xml:id="echoid-s12773" xml:space="preserve">propterea quod tam 2. </s> <s xml:id="echoid-s12774" xml:space="preserve">6. </s> <s xml:id="echoid-s12775" xml:space="preserve">18. </s> <s xml:id="echoid-s12776" xml:space="preserve">quam 6. </s> <s xml:id="echoid-s12777" xml:space="preserve"><lb/>18. </s> <s xml:id="echoid-s12778" xml:space="preserve">54. </s> <s xml:id="echoid-s12779" xml:space="preserve">ſunt tres continuè proportionales.</s> <s xml:id="echoid-s12780" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div792" type="section" level="1" n="273"> <head xml:id="echoid-head298" xml:space="preserve">LEMMA.</head> <p> <s xml:id="echoid-s12781" xml:space="preserve">SI ſint quatuor lineæ continuè proportionales: </s> <s xml:id="echoid-s12782" xml:space="preserve">parallelepipedum ſub <lb/>quadrato alterutrius extremarum, & </s> <s xml:id="echoid-s12783" xml:space="preserve">altera extrema comprehenſum, <lb/>æquale eſt cubo mediæ proportionalis, quæ priori extremæ aſſumptæ <lb/>propinquior eſt.</s> <s xml:id="echoid-s12784" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12785" xml:space="preserve"><emph style="sc">Repetatvr</emph> figura propoſ. </s> <s xml:id="echoid-s12786" xml:space="preserve">17. </s> <s xml:id="echoid-s12787" xml:space="preserve">in qua lineæ quatuor continuè propor-<lb/>tionales ſunt A, E, F, D. </s> <s xml:id="echoid-s12788" xml:space="preserve">Dico parallelepipedum ſub quadrato extremæ A, &</s> <s xml:id="echoid-s12789" xml:space="preserve"> <pb o="276" file="306" n="306" rhead="GEOMETR. PRACT."/> altera D, contentum, cubo rectæ E, æquale eſſe. </s> <s xml:id="echoid-s12790" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim quadratum <anchor type="note" xlink:label="note-306-01a" xlink:href="note-306-01"/> rectæ A, ad quadratum rectæ E proportionem habet, quam A, ad F, id eſt, quam <lb/>E, ad D, recipro cabuntur baſes cum altitudinibus, cum baſis parallelepipedi <lb/>ſit quadratum rectæ A, & </s> <s xml:id="echoid-s12791" xml:space="preserve">eiuſdem altitudo recta D: </s> <s xml:id="echoid-s12792" xml:space="preserve">cubi autem baſis quadra-<lb/>tum rectæ E, & </s> <s xml:id="echoid-s12793" xml:space="preserve">altitudo ipſamet recta E. </s> <s xml:id="echoid-s12794" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Igitur æqualia erunt parallelepipe- <anchor type="note" xlink:label="note-306-02a" xlink:href="note-306-02"/> dum, & </s> <s xml:id="echoid-s12795" xml:space="preserve">cubus. </s> <s xml:id="echoid-s12796" xml:space="preserve">Eadem ratione erit parallelepipedum ſub quadrato extremæ <lb/>D, & </s> <s xml:id="echoid-s12797" xml:space="preserve">ſub altera extrema A, contentum æquale cubo rectæ F. </s> <s xml:id="echoid-s12798" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Nam cum ſit, vt <anchor type="note" xlink:label="note-306-03a" xlink:href="note-306-03"/> quadratum rectæ D, ad quadratum rectæ F, id eſt, vt baſis dicti parallelepipe-<lb/>di ad baſem dicti cubi, ita D, ad E, hoc eſt, ita F, ad A, hoc eſt, ita altitudo cubi, <lb/>ad altitudinem parallelepipedi; </s> <s xml:id="echoid-s12799" xml:space="preserve">reciprocabuntur quo que baſes cum altitudi-<lb/>nibus: </s> <s xml:id="echoid-s12800" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> ideo que æqualia erunt parallelepipedum, & </s> <s xml:id="echoid-s12801" xml:space="preserve">cubus. </s> <s xml:id="echoid-s12802" xml:space="preserve">quod eſt <anchor type="note" xlink:label="note-306-04a" xlink:href="note-306-04"/> propoſitum.</s> <s xml:id="echoid-s12803" xml:space="preserve"/> </p> <div xml:id="echoid-div792" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-306-01" xlink:href="note-306-01a" xml:space="preserve">coroll 20. <lb/>ſexti.</note> <note symbol="b" position="left" xlink:label="note-306-02" xlink:href="note-306-02a" xml:space="preserve">34. vnde-<lb/>cimi.</note> <note symbol="c" position="left" xlink:label="note-306-03" xlink:href="note-306-03a" xml:space="preserve">coroll. 20. <lb/>ſexti.</note> <note symbol="d" position="left" xlink:label="note-306-04" xlink:href="note-306-04a" xml:space="preserve">34. vnde-<lb/>cimi.</note> </div> <p> <s xml:id="echoid-s12804" xml:space="preserve"><emph style="sc">Qvia</emph> verò in noſtra Arithmetica practica ſolum radicis quadratæ extra-<lb/>ctionem explicauimus, operæ me pretium facturum puto, radicis cubicæ extra-<lb/>ctionem hoc loco, quamuis fortaſſe alieno, inſerere: </s> <s xml:id="echoid-s12805" xml:space="preserve">quando quidem ea neceſ-<lb/>ſaria omninò eſt, vt problema hoc 13. </s> <s xml:id="echoid-s12806" xml:space="preserve">ad opus poſsit deduci. </s> <s xml:id="echoid-s12807" xml:space="preserve">Hoc autem ef-<lb/>ficiam, ſi præſcribam artem quandam generalem, qua cuiuſcunque generis ra-<lb/>dicem extrahere poſsimus, ex libro eximij cuiuſdam Arithmetici Germani de-<lb/>promptam fermè totam: </s> <s xml:id="echoid-s12808" xml:space="preserve">quod quidem ſtudioſo Lectorinon iniucundum, aut <lb/>ingratum fore confido.</s> <s xml:id="echoid-s12809" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div794" type="section" level="1" n="274"> <head xml:id="echoid-head299" xml:space="preserve">PROBL. 14. PROPOS. 19.</head> <head xml:id="echoid-head300" xml:space="preserve">RADICEM cuiuslibet generis extrahere.</head> <p> <s xml:id="echoid-s12810" xml:space="preserve"><emph style="sc">Extractio</emph> radicis eſt inuentio numeri ex propoſito numero, qui mul-<lb/> <anchor type="note" xlink:label="note-306-05a" xlink:href="note-306-05"/> tiplicatione aliqua in ſe numerum propoſitum producat. </s> <s xml:id="echoid-s12811" xml:space="preserve">Vt extractio qua-<lb/>dratæ radicis eſt inuentio numeri ex numero quadrato, qui quadratè mul-<lb/>tiplicatus ipſum producat: </s> <s xml:id="echoid-s12812" xml:space="preserve">Et extractio radicis cubicæ, eſt inuentio nume-<lb/>ri, qui in ſe ductus cubicè producat cubum propoſitum, &</s> <s xml:id="echoid-s12813" xml:space="preserve">c. </s> <s xml:id="echoid-s12814" xml:space="preserve">Quid autem <lb/>ſit multiplicare numerum quadratè, aut cubicè, aut alio modo, mox expli-<lb/>cabo.</s> <s xml:id="echoid-s12815" xml:space="preserve"/> </p> <div xml:id="echoid-div794" type="float" level="2" n="1"> <note position="left" xlink:label="note-306-05" xlink:href="note-306-05a" xml:space="preserve">Extractio ra-<lb/>dicis quid.</note> </div> <p> <s xml:id="echoid-s12816" xml:space="preserve"><emph style="sc">Qvemadmodvm</emph> igitur infinitæ ſunt ſpecies multiplicationum nume-<lb/> <anchor type="note" xlink:label="note-306-06a" xlink:href="note-306-06"/> rorumin ſe, vt ſtatim dicam, ex quibus oriuntur numeri quadrati; </s> <s xml:id="echoid-s12817" xml:space="preserve">& </s> <s xml:id="echoid-s12818" xml:space="preserve">ſolidi, vt <lb/>cubi, Zenficenſi, Surdeſolidi, &</s> <s xml:id="echoid-s12819" xml:space="preserve">c. </s> <s xml:id="echoid-s12820" xml:space="preserve">qui à Iunioribus nonnullis in Algebra ex-<lb/>plicari ſolent: </s> <s xml:id="echoid-s12821" xml:space="preserve">ſic etiam infinitæ ſunt radicum ſpecies iuxta varias numerorum <lb/>appellationes, qui conſurgunt ex varia radicum multiplicatione. </s> <s xml:id="echoid-s12822" xml:space="preserve">Quæ omnia <lb/>pulchrè nobis repræſentat naturalis numerorum progreſsio, inſeruiens <lb/>progreſsionibus Geometricis ab vnitate incipienti-<lb/>bus: </s> <s xml:id="echoid-s12823" xml:space="preserve">vt hic.</s> <s xml:id="echoid-s12824" xml:space="preserve"/> </p> <div xml:id="echoid-div795" type="float" level="2" n="2"> <note position="left" xlink:label="note-306-06" xlink:href="note-306-06a" xml:space="preserve">Infinitæ ſpe-<lb/>ci{es}<unsure/> radicum.</note> </div> <pb o="277" file="307" n="307" rhead="LIBER SEXTVS."/> <note position="right" xml:space="preserve"> <lb/>0. # 1. # 2. # 3. # 4. # 5. # 6. # 7. # 8. # 9. # 10.&c. <lb/>1. # 2. # 4. # 8. # 16. # 32. # 64. # 128. # 256. # 512. # 1024. &c. <lb/># Radix # Quadratus # Cubus. # Zenſizenſus # Surdeſoli- \\ dus. # Zenſicubus # B, ſurdeſo- \\ lidus. # Zenſizen- \\ zenſus # Cubicubus # Zenſurde- \\ ſolidus. <lb/></note> <p> <s xml:id="echoid-s12825" xml:space="preserve"><emph style="sc">Primvm</emph> numeri ſuperioris progreſsionis ſignificant ſpecies multiplicatio-<lb/>num. </s> <s xml:id="echoid-s12826" xml:space="preserve">Vt 2. </s> <s xml:id="echoid-s12827" xml:space="preserve">ſupra quadratum ſignificat, multiplicationem quadratam fieri, dum <lb/>radix bis ponitur, & </s> <s xml:id="echoid-s12828" xml:space="preserve">ſic multiplicatur, vt 2. </s> <s xml:id="echoid-s12829" xml:space="preserve">2. </s> <s xml:id="echoid-s12830" xml:space="preserve">facit 4. </s> <s xml:id="echoid-s12831" xml:space="preserve">Sic 3. </s> <s xml:id="echoid-s12832" xml:space="preserve">ſignificat, multipli-<lb/>cationem cubicam fieri, dum radix ter ponitur, atque ita multiplicatur, vt 2. </s> <s xml:id="echoid-s12833" xml:space="preserve">2. </s> <s xml:id="echoid-s12834" xml:space="preserve">2. <lb/></s> <s xml:id="echoid-s12835" xml:space="preserve">facit 8. </s> <s xml:id="echoid-s12836" xml:space="preserve">Pari ratione 4. </s> <s xml:id="echoid-s12837" xml:space="preserve">oſtendit multiplicationem Zenſizenſicam: </s> <s xml:id="echoid-s12838" xml:space="preserve">Et 5. </s> <s xml:id="echoid-s12839" xml:space="preserve">ſurde-<lb/>ſolidam, &</s> <s xml:id="echoid-s12840" xml:space="preserve">c.</s> <s xml:id="echoid-s12841" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12842" xml:space="preserve"><emph style="sc">Deinde</emph> ijdem @numeri ſignificant radicum ſpecies. </s> <s xml:id="echoid-s12843" xml:space="preserve">Vt 2. </s> <s xml:id="echoid-s12844" xml:space="preserve">ſignificat, radi-<lb/>cem quadratam producere quadratum per multiplicationem quadratam: </s> <s xml:id="echoid-s12845" xml:space="preserve">Et <lb/>3. </s> <s xml:id="echoid-s12846" xml:space="preserve">denotat, radicem Cubicam procreare Cubum per multiplicationem cubicam: <lb/></s> <s xml:id="echoid-s12847" xml:space="preserve">Et ſic deinceps.</s> <s xml:id="echoid-s12848" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12849" xml:space="preserve"><emph style="sc">In</emph> extra ctionibus igitur radicum obſeruanda eſt ſignatio figurarum per pũ-<lb/>cta in numero, ex quo radix aliqua extrahenda eſt, hoc modo.</s> <s xml:id="echoid-s12850" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12851" xml:space="preserve"><emph style="sc">In</emph> extractione radicis quadratæ ſignantur omnes figuræ in Iocis imparibus, <lb/>incipiendo à dextris: </s> <s xml:id="echoid-s12852" xml:space="preserve">ita vt alternatim ſemper vna figura omittatur, quæ non <lb/>ſignetur.</s> <s xml:id="echoid-s12853" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12854" xml:space="preserve"><emph style="sc">In</emph> extractione cubica omittuntur ſemper <lb/> <anchor type="note" xlink:label="note-307-02a" xlink:href="note-307-02"/> duæ figuræ. </s> <s xml:id="echoid-s12855" xml:space="preserve">In Zenſizenſica tres. </s> <s xml:id="echoid-s12856" xml:space="preserve">In ſurdeſolida <lb/> <anchor type="note" xlink:label="note-307-03a" xlink:href="note-307-03"/> quatuor. </s> <s xml:id="echoid-s12857" xml:space="preserve">Et ſic deinceps in infinitum. </s> <s xml:id="echoid-s12858" xml:space="preserve">Vtin ap-<lb/>poſitis exemplis vides.</s> <s xml:id="echoid-s12859" xml:space="preserve"/> </p> <div xml:id="echoid-div796" type="float" level="2" n="3"> <note position="right" xlink:label="note-307-02" xlink:href="note-307-02a" xml:space="preserve"> <lb/>Pro quadrata. <lb/>68719476736 <lb/>. . . . . . <lb/>Pro cubica <lb/>68719476736 <lb/>. . . . <lb/>Zenſizenſica. <lb/>68719476736 <lb/>. . . <lb/>Pro ſurdeſolida. <lb/>68719476736 <lb/>. . . <lb/></note> <note position="right" xlink:label="note-307-03" xlink:href="note-307-03a" xml:space="preserve">Quo modo fi-<lb/>guræ per pun-<lb/>cta ſignentur.</note> </div> <p> <s xml:id="echoid-s12860" xml:space="preserve"><emph style="sc">Respondent</emph> autem hæ ſignationes me-<lb/>dijs proportionalibus. </s> <s xml:id="echoid-s12861" xml:space="preserve">Vt quoniam inter duos <lb/>quadratos cadit vnus medius, ideo in extractio-<lb/>ne ra dicis quadratę omittitur ſemper vnafigu-<lb/>ra: </s> <s xml:id="echoid-s12862" xml:space="preserve">Inter duos verò Cubos cadunt duo medij, <lb/>idcirco omittuntur ſemper duæ figuræ, & </s> <s xml:id="echoid-s12863" xml:space="preserve">ſic de <lb/>cæteris.</s> <s xml:id="echoid-s12864" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12865" xml:space="preserve"><emph style="sc">Pro</emph> qualibet autem ſpecie radicis extrahẽ-<lb/>dę inſeruiunt quidam numeri peculiares: </s> <s xml:id="echoid-s12866" xml:space="preserve">qui per <lb/>ſequentem tabulam inueniuntur, quæ hoc mo-<lb/>dò conſtruitur. </s> <s xml:id="echoid-s12867" xml:space="preserve">Prima columna continet ſeriem <lb/>naturalẽ numerorũ. </s> <s xml:id="echoid-s12868" xml:space="preserve">Ex hac colũna naſcit ſecũ-<lb/>da: </s> <s xml:id="echoid-s12869" xml:space="preserve">tertia ex ſecũda: </s> <s xml:id="echoid-s12870" xml:space="preserve">& </s> <s xml:id="echoid-s12871" xml:space="preserve">quarta ex tertia, hoc mo-<lb/>do. </s> <s xml:id="echoid-s12872" xml:space="preserve">Relictis duab. </s> <s xml:id="echoid-s12873" xml:space="preserve">cellulis primæ colũnæ, repetit <lb/>numerus tertiæ cellulæ in ſecunda columna. </s> <s xml:id="echoid-s12874" xml:space="preserve">De-<lb/>inde ex additione duorum numerorum, id eſt, ex <lb/>tertio primæ columnæ, & </s> <s xml:id="echoid-s12875" xml:space="preserve">primo ſecundæ columnæ, fit ſecundus numerus ſe-<lb/>cundæ columnæ. </s> <s xml:id="echoid-s12876" xml:space="preserve">Eodem modo ex ſecundo numero ſecundæ columnæ, & </s> <s xml:id="echoid-s12877" xml:space="preserve">ex <lb/>eius collaterali conficitur tertius ſecundæ columnæ: </s> <s xml:id="echoid-s12878" xml:space="preserve">Atque ex tertio numero <lb/>ſecundæ columnę, & </s> <s xml:id="echoid-s12879" xml:space="preserve">ex eius collaterali fit quartus eiuſdem ſecundę columnæ, <pb o="278" file="308" n="308" rhead="GEOMETR. PRACT."/> <anchor type="note" xlink:label="note-308-01a" xlink:href="note-308-01"/> <anchor type="note" xlink:label="note-308-02a" xlink:href="note-308-02"/> 2 <lb/>3 # 3 <lb/>4 # 6 <lb/>5 # 10 # 10 <lb/>6 # 15 # 20 <lb/>7 # 21 # 35 # 35 <lb/>8 # 28 # 56 # 70 <lb/>9 # 36 # 84 # 126 # 126 <lb/>10 # 45 # 120 # 210 # 252 <lb/>11 # 55 # 165 # 330 # 462 # 462 <lb/>12 # 66 # 220 # 495 # 792 # 924 <lb/>13 # 78 # 286 # 715 # 1287 # 1716 # 1716 <lb/>14 # 91 # 364 # 100 # 2002 # 3003 # 3432 <lb/>15 # 105 # 455 # 1365 # 3003 # 5005 # 6435 # 6435 <lb/>16 # 120 # 560 # 1820 # 4368 # 8008 # 11440 # 12870 <lb/>17 # 136 # 680 # 2380 # 6188 # 12376 # 16448 # 24310 # 24310 <lb/>& </s> <s xml:id="echoid-s12880" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s12881" xml:space="preserve">Continentur autem in ſecunda columna omnes numeri trian-<lb/>gulares. </s> <s xml:id="echoid-s12882" xml:space="preserve">Non aliter tertia columna ex ſecunda oritur, & </s> <s xml:id="echoid-s12883" xml:space="preserve">quarta ex tertia, <lb/>&</s> <s xml:id="echoid-s12884" xml:space="preserve">c. </s> <s xml:id="echoid-s12885" xml:space="preserve">Semper enim in qualibet columna relinquũtur duæ primæ cellu@æ, & </s> <s xml:id="echoid-s12886" xml:space="preserve">nu-<lb/>merus tertiæ cellulæ repetitur pro primo numero ſequentis columnæ: </s> <s xml:id="echoid-s12887" xml:space="preserve">atque ex <lb/>additione eius numeri cum collaterali præcedentis columnæ conflatur ſecun-<lb/>dus numerus, &</s> <s xml:id="echoid-s12888" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s12889" xml:space="preserve"> <anchor type="note" xlink:label="note-308-03a" xlink:href="note-308-03"/> </s> </p> <div xml:id="echoid-div797" type="float" level="2" n="4"> <note position="left" xlink:label="note-308-02" xlink:href="note-308-02a" xml:space="preserve">Conſtructio <lb/>tabulæ miri-<lb/>ficæ.</note> <note position="left" xlink:label="note-308-02" xlink:href="note-308-02a" xml:space="preserve">Conſtructio <lb/>tabulæ miri-<lb/>ficæ.</note> <note position="left" xlink:label="note-308-03" xlink:href="note-308-03a" xml:space="preserve">Quo pacto <lb/>ex ſuperiori <lb/>tabula deſu-<lb/>mantur nu-<lb/>meripro ſin-<lb/>gulis radicum <lb/>ſpecieb{us}.</note> </div> <p> <s xml:id="echoid-s12890" xml:space="preserve"><emph style="sc">Hac</emph> extructa tabula, deſumuntur numeri peculiares ex ordinibus trans-<lb/>uerſalibus hoc ordine. </s> <s xml:id="echoid-s12891" xml:space="preserve">Numeri cuiuslibet ordinis tranſuerſalis ordine ſcrib un-<lb/>tur, ijdemque ordineretrogrado repetuntur, vltimo ſemper ex cepto, & </s> <s xml:id="echoid-s12892" xml:space="preserve">penul-<lb/>timo etiam tuncſolum, quando vltimo æqualis eſt. </s> <s xml:id="echoid-s12893" xml:space="preserve">Vt in ſecundo ordine ſu-<lb/>mitur tantum numerus 2. </s> <s xml:id="echoid-s12894" xml:space="preserve">quia cum ſit vltimus, non repetitur. </s> <s xml:id="echoid-s12895" xml:space="preserve">In tertio ſumun-<lb/>tur quo que hi duo tantum 3.</s> <s xml:id="echoid-s12896" xml:space="preserve">3. </s> <s xml:id="echoid-s12897" xml:space="preserve">quia penultimus non repetitur, cum ab vltimo <lb/>non differat. </s> <s xml:id="echoid-s12898" xml:space="preserve">In quarto autem ſumendi ſunt hi tres. </s> <s xml:id="echoid-s12899" xml:space="preserve">4. </s> <s xml:id="echoid-s12900" xml:space="preserve">6. </s> <s xml:id="echoid-s12901" xml:space="preserve">4. </s> <s xml:id="echoid-s12902" xml:space="preserve">In nono hi octo 9. <lb/></s> <s xml:id="echoid-s12903" xml:space="preserve">36. </s> <s xml:id="echoid-s12904" xml:space="preserve">84. </s> <s xml:id="echoid-s12905" xml:space="preserve">126. </s> <s xml:id="echoid-s12906" xml:space="preserve">126. </s> <s xml:id="echoid-s12907" xml:space="preserve">84. </s> <s xml:id="echoid-s12908" xml:space="preserve">36. </s> <s xml:id="echoid-s12909" xml:space="preserve">9. </s> <s xml:id="echoid-s12910" xml:space="preserve">Et in decimo hinouem 10. </s> <s xml:id="echoid-s12911" xml:space="preserve">45. </s> <s xml:id="echoid-s12912" xml:space="preserve">120. </s> <s xml:id="echoid-s12913" xml:space="preserve">210. </s> <s xml:id="echoid-s12914" xml:space="preserve">252. </s> <s xml:id="echoid-s12915" xml:space="preserve">210. </s> <s xml:id="echoid-s12916" xml:space="preserve">120. </s> <s xml:id="echoid-s12917" xml:space="preserve"><lb/>45. </s> <s xml:id="echoid-s12918" xml:space="preserve">10. </s> <s xml:id="echoid-s12919" xml:space="preserve">&</s> <s xml:id="echoid-s12920" xml:space="preserve">c. </s> <s xml:id="echoid-s12921" xml:space="preserve">Vbivides, ſemper tot numeros aſſumi, quot ſunt vnitates in primo <lb/>numero tranſuerſali, minus vno. </s> <s xml:id="echoid-s12922" xml:space="preserve">Vtin ordine ſeptimodecimo aſſumendi erunt <lb/>hiſexdecim 17. </s> <s xml:id="echoid-s12923" xml:space="preserve">136. </s> <s xml:id="echoid-s12924" xml:space="preserve">680. </s> <s xml:id="echoid-s12925" xml:space="preserve">2380. </s> <s xml:id="echoid-s12926" xml:space="preserve">6188. </s> <s xml:id="echoid-s12927" xml:space="preserve">12376. </s> <s xml:id="echoid-s12928" xml:space="preserve">19448. </s> <s xml:id="echoid-s12929" xml:space="preserve">24310. </s> <s xml:id="echoid-s12930" xml:space="preserve">24310. </s> <s xml:id="echoid-s12931" xml:space="preserve">19448. </s> <s xml:id="echoid-s12932" xml:space="preserve">12376. </s> <s xml:id="echoid-s12933" xml:space="preserve"><lb/>6188. </s> <s xml:id="echoid-s12934" xml:space="preserve">2380. </s> <s xml:id="echoid-s12935" xml:space="preserve">680. </s> <s xml:id="echoid-s12936" xml:space="preserve">136. </s> <s xml:id="echoid-s12937" xml:space="preserve">17. </s> <s xml:id="echoid-s12938" xml:space="preserve">& </s> <s xml:id="echoid-s12939" xml:space="preserve">ſic de cæteris.</s> <s xml:id="echoid-s12940" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12941" xml:space="preserve"><emph style="sc">Cvilibet</emph> deinde numero præponendæ ſunttot cifræ, quot numeriab eo <lb/>incluſiue numerantur vſque ad vltimum aſſumptum. </s> <s xml:id="echoid-s12942" xml:space="preserve">Vt numero 2. </s> <s xml:id="echoid-s12943" xml:space="preserve">ſecundi or-<lb/>dinis præponenda eſt vna cifra, hoc modo, 20. </s> <s xml:id="echoid-s12944" xml:space="preserve">Sed duo tertij ordinis habe-<lb/>bunt has cifras 300. </s> <s xml:id="echoid-s12945" xml:space="preserve">30. </s> <s xml:id="echoid-s12946" xml:space="preserve">Sic in nono ordinè, vbiaſſumuntur octo numeri, pri-<lb/>mus habebit octo cifras, ſecundus ſeptem, tertius ſex, & </s> <s xml:id="echoid-s12947" xml:space="preserve">ſic ſemper minuendo <lb/>vnam. </s> <s xml:id="echoid-s12948" xml:space="preserve">Cuius autem radicis extra ctioni inſeruiant prædicti nu neri in quolibet <lb/>ordine tranſuerſali accepti, pulchrè indicat primus numerus o@dinis tranſuerſa- <pb o="279" file="309" n="309" rhead="LIBER SEXTVS."/> lis. </s> <s xml:id="echoid-s12949" xml:space="preserve">Vt quia in ſuperiori progreſsione numerus 2. </s> <s xml:id="echoid-s12950" xml:space="preserve">notat quadratum, & </s> <s xml:id="echoid-s12951" xml:space="preserve">3. </s> <s xml:id="echoid-s12952" xml:space="preserve">cubũ: <lb/></s> <s xml:id="echoid-s12953" xml:space="preserve">& </s> <s xml:id="echoid-s12954" xml:space="preserve">4. </s> <s xml:id="echoid-s12955" xml:space="preserve">Zenſizenſum, &</s> <s xml:id="echoid-s12956" xml:space="preserve">c. </s> <s xml:id="echoid-s12957" xml:space="preserve">ideo numerus 2. </s> <s xml:id="echoid-s12958" xml:space="preserve">ſecundi ordinis cum ſua cifra, hoc mo-<lb/>do. </s> <s xml:id="echoid-s12959" xml:space="preserve">20. </s> <s xml:id="echoid-s12960" xml:space="preserve">inſeruit radici quadratæ: </s> <s xml:id="echoid-s12961" xml:space="preserve">Et duo numeri ex tertio ordine aſſumpti cum <lb/>ſuis cifris, hoc modo 300. </s> <s xml:id="echoid-s12962" xml:space="preserve">30. </s> <s xml:id="echoid-s12963" xml:space="preserve">radici cubicæ: </s> <s xml:id="echoid-s12964" xml:space="preserve">Et tres quarti ordinis, hocmodo <lb/>4000. </s> <s xml:id="echoid-s12965" xml:space="preserve">600. </s> <s xml:id="echoid-s12966" xml:space="preserve">40. </s> <s xml:id="echoid-s12967" xml:space="preserve">radici Zenſizenſicæ: </s> <s xml:id="echoid-s12968" xml:space="preserve">& </s> <s xml:id="echoid-s12969" xml:space="preserve">quatuor quinti ordinis, hoc modo, <lb/>50000. </s> <s xml:id="echoid-s12970" xml:space="preserve">10000. </s> <s xml:id="echoid-s12971" xml:space="preserve">1000. </s> <s xml:id="echoid-s12972" xml:space="preserve">50. </s> <s xml:id="echoid-s12973" xml:space="preserve">radici ſurdeſolidæ, &</s> <s xml:id="echoid-s12974" xml:space="preserve">c.</s> <s xml:id="echoid-s12975" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12976" xml:space="preserve"><emph style="sc">Iam</emph> verò vt propius ad extra ctionem radicum accedamus, ſciendum eſt, <lb/> <anchor type="note" xlink:label="note-309-01a" xlink:href="note-309-01"/> radicẽ cuiuslibet numeri habere tot figuras, quot puncta ſub ipſo ſignata ſunt, <lb/>ſecundum do ctrinam ſuperiorem. </s> <s xml:id="echoid-s12977" xml:space="preserve">Item ad punctum vltimum verſus ſiniſtram <lb/>pertinere ipſam figuram ſupra punctum poſitam, cum omnibus alijs, quæipſam <lb/>verſus ſiniſtram præcedunt. </s> <s xml:id="echoid-s12978" xml:space="preserve">Ex quo puncto ſi ſubtrahatur numerus, vt mox <lb/>dicemus, ſpectabit ad penultimum punctum figura ſupra ipſum punctum cum <lb/>reliquis ad ſiniſtram, & </s> <s xml:id="echoid-s12979" xml:space="preserve">ſic de cæteris.</s> <s xml:id="echoid-s12980" xml:space="preserve"/> </p> <div xml:id="echoid-div798" type="float" level="2" n="5"> <note position="right" xlink:label="note-309-01" xlink:href="note-309-01a" xml:space="preserve">Quot figur{as} <lb/>quælib{et} ra-<lb/>dix habeat.</note> </div> <p> <s xml:id="echoid-s12981" xml:space="preserve"><emph style="sc">Vt</emph> autem ritè incipiat extractio cuiuslibet radicis, conſtruenda erit tabella <lb/>quadratorum, cuborum, ſurdeſolidorum, B, ſurdeſolidorum, & </s> <s xml:id="echoid-s12982" xml:space="preserve">aliorum nu-<lb/>merorum, qui ex nouem figuris Arithmeticis pro ducuntur, cum ſuisradicibus. <lb/></s> <s xml:id="echoid-s12983" xml:space="preserve">Vthic vides.</s> <s xml:id="echoid-s12984" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>Radices. # Quadrati. # Radices. # Cubi. # Radices. # Surdeſoli- \\ di. # Radices. # B, Surdeſo- \\ lidi. <lb/>1 # 1 # 1 # 1 # 1 # 1 # 1 # 1 <lb/>2 # 4 # 2 # 8 # 2 # 32 # 2 # 128 <lb/>3 # 9 # 3 # 27 # 3 # 243 # 3 # 2187 <lb/>4 # 16 # 4 # 64 # 4 # 1024 # 4 # 16384 <lb/>5 # 25 # 5 # 125 # 5 # 3125 # 5 # 78125 <lb/>6 # 36 # 6 # 216 # 6 # 7776 # 6 # 279936 <lb/>7 # 49 # 7 # 343 # 7 # 16807 # 7 # 823543 <lb/>8 # 64 # 8 # 512 # 8 # 32768 # 8 # 2097152 <lb/>9 # 81 # 9 # 729 # 9 # 59049 # 9 # 4782969 <lb/></note> <p> <s xml:id="echoid-s12985" xml:space="preserve">Cur autem non appoſuerim tabellas Zenſizenſorum. </s> <s xml:id="echoid-s12986" xml:space="preserve">& </s> <s xml:id="echoid-s12987" xml:space="preserve">zenſicuborũ, cau-<lb/>ſa eſt, quod eiuſmodi numerorum radices non egent nouis præceptis, vt poſtea <lb/>dicetur. </s> <s xml:id="echoid-s12988" xml:space="preserve">Sequuntur ergo iam extractionum exempla aliquot.</s> <s xml:id="echoid-s12989" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div800" type="section" level="1" n="275"> <head xml:id="echoid-head301" xml:space="preserve">EXTRACTIO RADICIS <lb/>Quadratæ. <lb/></head> <note position="right" xml:space="preserve">Quadr atæ <lb/>radicis ex-<lb/>trastio.</note> <p> <s xml:id="echoid-s12990" xml:space="preserve"><emph style="sc">Sit</emph> numerus propoſitus 6765201. </s> <s xml:id="echoid-s12991" xml:space="preserve">ex quo erui debet radix quadrata. <lb/></s> <s xml:id="echoid-s12992" xml:space="preserve">. </s> <s xml:id="echoid-s12993" xml:space="preserve">. </s> <s xml:id="echoid-s12994" xml:space="preserve">. </s> <s xml:id="echoid-s12995" xml:space="preserve">.</s> <s xml:id="echoid-s12996" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s12997" xml:space="preserve"><emph style="sc">Primvm</emph> ab vltimo puncto, id eſt, à figura 6. </s> <s xml:id="echoid-s12998" xml:space="preserve">(Hoc enim punctum vnã tan-<lb/>tum habet figuram, cum eam nulla alia præcedat) ſubtrahitur maximus quadra-<lb/> <anchor type="note" xlink:label="note-309-04a" xlink:href="note-309-04"/> tus, quiſubtrahipoteſt, nimirum 4. </s> <s xml:id="echoid-s12999" xml:space="preserve">& </s> <s xml:id="echoid-s13000" xml:space="preserve">in Quotiente ad marginem ponitur eius <lb/>radix 2. </s> <s xml:id="echoid-s13001" xml:space="preserve">Inreſiduo autem manent 2. </s> <s xml:id="echoid-s13002" xml:space="preserve">pro ſequenti puncto. </s> <s xml:id="echoid-s13003" xml:space="preserve">quod erit 276. </s> <s xml:id="echoid-s13004" xml:space="preserve">atque <lb/>ita abſolutum eſt vltimum punctum, quod eſt in operatione primum.</s> <s xml:id="echoid-s13005" xml:space="preserve"/> </p> <div xml:id="echoid-div800" type="float" level="2" n="1"> <note position="right" xlink:label="note-309-04" xlink:href="note-309-04a" xml:space="preserve">Radix 2601.</note> </div> <pb o="280" file="310" n="310" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s13006" xml:space="preserve"><emph style="sc">Deinde</emph> paro diuiſorem ex figura 2. </s> <s xml:id="echoid-s13007" xml:space="preserve">inuenta in Quotiente multiplicata per <lb/>20. </s> <s xml:id="echoid-s13008" xml:space="preserve">nimirum per numerum radici quadratę inſeruientem: </s> <s xml:id="echoid-s13009" xml:space="preserve">quem inuenio 40. <lb/></s> <s xml:id="echoid-s13010" xml:space="preserve">Per hunc diuido punctum ſequens 276. </s> <s xml:id="echoid-s13011" xml:space="preserve">& </s> <s xml:id="echoid-s13012" xml:space="preserve">pro Quotiente reperio 6. </s> <s xml:id="echoid-s13013" xml:space="preserve">Pono er-<lb/>go primum Quotientem 2. </s> <s xml:id="echoid-s13014" xml:space="preserve">ad ſiniſtram, numerum peculiarem <lb/> <anchor type="note" xlink:label="note-310-01a" xlink:href="note-310-01"/> 20. </s> <s xml:id="echoid-s13015" xml:space="preserve">in medio, & </s> <s xml:id="echoid-s13016" xml:space="preserve">nouum Quotientem 6. </s> <s xml:id="echoid-s13017" xml:space="preserve">ad dextrã, ſub quo ſcri-<lb/>bo eius quadratum 36. </s> <s xml:id="echoid-s13018" xml:space="preserve">vt hic vides.</s> <s xml:id="echoid-s13019" xml:space="preserve"/> </p> <div xml:id="echoid-div801" type="float" level="2" n="2"> <note position="right" xlink:label="note-310-01" xlink:href="note-310-01a" xml:space="preserve"> <lb/>2--20--6. <lb/>36. <lb/></note> </div> <p> <s xml:id="echoid-s13020" xml:space="preserve"><emph style="sc">Post</emph> hęc multiplico tres numeros 2. </s> <s xml:id="echoid-s13021" xml:space="preserve">20. </s> <s xml:id="echoid-s13022" xml:space="preserve">6. </s> <s xml:id="echoid-s13023" xml:space="preserve">inter ſe, & </s> <s xml:id="echoid-s13024" xml:space="preserve">producto 240. </s> <s xml:id="echoid-s13025" xml:space="preserve">addo <lb/>quadratum 36. </s> <s xml:id="echoid-s13026" xml:space="preserve">& </s> <s xml:id="echoid-s13027" xml:space="preserve">ſummam 276. </s> <s xml:id="echoid-s13028" xml:space="preserve">ex puncto 276. </s> <s xml:id="echoid-s13029" xml:space="preserve">detraho, nihil que relinquitur, <lb/>atque ita abſolutum eſt ſequens punctum: </s> <s xml:id="echoid-s13030" xml:space="preserve">aliudque ſequens punctum eſt 52.</s> <s xml:id="echoid-s13031" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13032" xml:space="preserve"><emph style="sc">Paro</emph> iam diuiſorem ex toto Quotiente inuento 26, ducto in 20. </s> <s xml:id="echoid-s13033" xml:space="preserve">id eſt, in <lb/>numerum peculiarem quadratæ radicis, quem inuenio 520. </s> <s xml:id="echoid-s13034" xml:space="preserve">Et quia perhunc <lb/>diuidi non poteſt punctum 52. </s> <s xml:id="echoid-s13035" xml:space="preserve">pono in Quotienteo. </s> <s xml:id="echoid-s13036" xml:space="preserve">neque opus eſt multipli-<lb/>care, vtreperiatur numerus ſubtrahendus, quia nihil ſubrahitur, cumo. </s> <s xml:id="echoid-s13037" xml:space="preserve">multi-<lb/>plicans producato. </s> <s xml:id="echoid-s13038" xml:space="preserve">Et ſic fit in omnibus alijs extractionibus, quando diuiſor <lb/>inuentus in puncto propoſito ne ſemel quidem continetur: </s> <s xml:id="echoid-s13039" xml:space="preserve">atque ita abſolu-<lb/>tum eſt punctum 52. </s> <s xml:id="echoid-s13040" xml:space="preserve">punctum queinſequens eſt 5201.</s> <s xml:id="echoid-s13041" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13042" xml:space="preserve"><emph style="sc">Paro</emph> iterũ diuiſorẽ ex toto Quotiẽte inuẽto 260. </s> <s xml:id="echoid-s13043" xml:space="preserve">ducto in numerũ pecu-<lb/>liarem 20. </s> <s xml:id="echoid-s13044" xml:space="preserve">quem reperio 5200. </s> <s xml:id="echoid-s13045" xml:space="preserve">qui ſemel in puncto 5201. </s> <s xml:id="echoid-s13046" xml:space="preserve">continetur. </s> <s xml:id="echoid-s13047" xml:space="preserve">Pono er-<lb/>go totum Quotientem prius inuentum 260. </s> <s xml:id="echoid-s13048" xml:space="preserve">ad ſiniſtram, & </s> <s xml:id="echoid-s13049" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-310-02a" xlink:href="note-310-02"/> numerum peculiarem 20. </s> <s xml:id="echoid-s13050" xml:space="preserve">in medio, & </s> <s xml:id="echoid-s13051" xml:space="preserve">quotientẽ 1. </s> <s xml:id="echoid-s13052" xml:space="preserve">nunc inuen-<lb/>tumad dexteram, eiuſque quadratum 1. </s> <s xml:id="echoid-s13053" xml:space="preserve">ſub illo, vt in exemplo patet.</s> <s xml:id="echoid-s13054" xml:space="preserve"/> </p> <div xml:id="echoid-div802" type="float" level="2" n="3"> <note position="right" xlink:label="note-310-02" xlink:href="note-310-02a" xml:space="preserve"> <lb/>260--20--1 <lb/>1 <lb/></note> </div> <p> <s xml:id="echoid-s13055" xml:space="preserve"><emph style="sc">Mvltiplicatio</emph> trium ſuperiorum numerorum facit 5200. </s> <s xml:id="echoid-s13056" xml:space="preserve">addo qua-<lb/>dratum 1. </s> <s xml:id="echoid-s13057" xml:space="preserve">figuræ inuentæ 1. </s> <s xml:id="echoid-s13058" xml:space="preserve">fit numerus 5201. </s> <s xml:id="echoid-s13059" xml:space="preserve">qui ex puncto 5201. </s> <s xml:id="echoid-s13060" xml:space="preserve">detractus nil <lb/>relinquit. </s> <s xml:id="echoid-s13061" xml:space="preserve">Eſt ergo abſoluta extractio, radixque inuenta eſt 2601. </s> <s xml:id="echoid-s13062" xml:space="preserve">quæ quadra-<lb/>tè, id eſt, in ſe multiplicata producit propoſitum numerum 6765201.</s> <s xml:id="echoid-s13063" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13064" xml:space="preserve"><emph style="sc">Atqve</emph> hæc eſt probatio, vel examen cuiuſuis extra ctionis, vt videlicet ra-<lb/>dix inuenta in ſemultip licetur vel quadratè, vel cubice, vel ſurdeſolide, &</s> <s xml:id="echoid-s13065" xml:space="preserve">c.</s> <s xml:id="echoid-s13066" xml:space="preserve">ꝓ <lb/>qualitateradicis. </s> <s xml:id="echoid-s13067" xml:space="preserve">Si enim in extractione nihil fuit relictum, veluti in noſtro exẽ-<lb/>plo, neceſſe eſt, numerum productum ęqualem eſſe propoſitio numero, ex quo <lb/>fa cta eſt extractio: </s> <s xml:id="echoid-s13068" xml:space="preserve">Si autem in extractione aliquid fuit relictum, illud additũ <lb/>producto numero conficiet numerum propoſitum. </s> <s xml:id="echoid-s13069" xml:space="preserve">Inſtitui quoq; </s> <s xml:id="echoid-s13070" xml:space="preserve">poteſt exa-<lb/>men per 9. </s> <s xml:id="echoid-s13071" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s13072" xml:space="preserve">vt in Diuiſione. </s> <s xml:id="echoid-s13073" xml:space="preserve">Namſi ex inuenta radice abijciantur 9. </s> <s xml:id="echoid-s13074" xml:space="preserve">vel 7. <lb/></s> <s xml:id="echoid-s13075" xml:space="preserve">quoties fieri poteſt, & </s> <s xml:id="echoid-s13076" xml:space="preserve">reſiduum collocetur tum in ſiniſtra parte crucis, tum in <lb/>dextra, quod Quotiens, vel radix inuenta ſit etiam inſtar Diuiſoris. </s> <s xml:id="echoid-s13077" xml:space="preserve">Hoc enim <lb/>reſiduo in ſe multiplicato quadrate, vel cubice, &</s> <s xml:id="echoid-s13078" xml:space="preserve">c. </s> <s xml:id="echoid-s13079" xml:space="preserve">& </s> <s xml:id="echoid-s13080" xml:space="preserve">ex producto abiectis 9. </s> <s xml:id="echoid-s13081" xml:space="preserve"><lb/>vel 7. </s> <s xml:id="echoid-s13082" xml:space="preserve">neceſſe eſt, reſiduum hoc æquale eſſe reſiduo numeri propoſiti, ſi abijciã-<lb/>tur ex eo omnia 9. </s> <s xml:id="echoid-s13083" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s13084" xml:space="preserve">& </s> <s xml:id="echoid-s13085" xml:space="preserve">nihil in extra ctione relictum ſit. </s> <s xml:id="echoid-s13086" xml:space="preserve">Nam alio-<lb/> <anchor type="figure" xlink:label="fig-310-01a" xlink:href="fig-310-01"/> quin ex reſiduo extractionis, & </s> <s xml:id="echoid-s13087" xml:space="preserve">ex producto radicis inuẽtæin ſe mul-<lb/>tiplicatæ abijcienda erunt omnia 9. </s> <s xml:id="echoid-s13088" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s13089" xml:space="preserve">Hoc enim reſiduum ęquale <lb/>eſſe debetreſi duo numeri propoſiti, ſi omnia 9. </s> <s xml:id="echoid-s13090" xml:space="preserve">vel 7. </s> <s xml:id="echoid-s13091" xml:space="preserve">abijciantur. </s> <s xml:id="echoid-s13092" xml:space="preserve">In <lb/>noſtro exemplo, ſi probatio inſtituatur per 9. </s> <s xml:id="echoid-s13093" xml:space="preserve">reſiduum ſemper eſto. <lb/></s> <s xml:id="echoid-s13094" xml:space="preserve">Siverò fiat per 7. </s> <s xml:id="echoid-s13095" xml:space="preserve">ſtabit exemplum examinis vt hic apparet.</s> <s xml:id="echoid-s13096" xml:space="preserve"/> </p> <div xml:id="echoid-div803" 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/xxxxxxxx/figures/310-01"/> </figure> </div> </div> <div xml:id="echoid-div805" type="section" level="1" n="276"> <head xml:id="echoid-head302" xml:space="preserve">EXTRACTIO RADICIS CVBICE.</head> <p> <s xml:id="echoid-s13097" xml:space="preserve"><emph style="sc">Sit</emph> ex numero 239483190. </s> <s xml:id="echoid-s13098" xml:space="preserve">extrahenda radix cubica.</s> <s xml:id="echoid-s13099" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13100" xml:space="preserve">. </s> <s xml:id="echoid-s13101" xml:space="preserve">. </s> <s xml:id="echoid-s13102" xml:space="preserve">.</s> <s xml:id="echoid-s13103" xml:space="preserve"/> </p> <pb o="281" file="311" n="311" rhead="LIBER SEXTVS."/> <p> <s xml:id="echoid-s13104" xml:space="preserve"><emph style="sc">Primvm</emph> expuncto 239. </s> <s xml:id="echoid-s13105" xml:space="preserve">ſubtraho cubum 216. </s> <s xml:id="echoid-s13106" xml:space="preserve">qui eſt maximus in eo con-<lb/> <anchor type="note" xlink:label="note-311-01a" xlink:href="note-311-01"/> tentus, cuius radicem 6. </s> <s xml:id="echoid-s13107" xml:space="preserve">ſcribo in Quotiente ad marginem. </s> <s xml:id="echoid-s13108" xml:space="preserve">Et quia relinquitur <lb/>numerus 23. </s> <s xml:id="echoid-s13109" xml:space="preserve">erit ſequens punctum 23483.</s> <s xml:id="echoid-s13110" xml:space="preserve"/> </p> <div xml:id="echoid-div805" type="float" level="2" n="1"> <note position="right" xlink:label="note-311-01" xlink:href="note-311-01a" xml:space="preserve">Cubicæ radi-<lb/>cis extractio. <lb/>radix 621.</note> </div> <p> <s xml:id="echoid-s13111" xml:space="preserve"><emph style="sc">Deinde</emph> paro diuiſorem hocmodo. </s> <s xml:id="echoid-s13112" xml:space="preserve">Supra radicem inuen-<lb/>tam 6. </s> <s xml:id="echoid-s13113" xml:space="preserve">pono eius quadratum 36. </s> <s xml:id="echoid-s13114" xml:space="preserve">Et ad dextram colloco duos nu-<lb/> <anchor type="note" xlink:label="note-311-02a" xlink:href="note-311-02"/> meros peculiares radicis cubicæ, nimirum 300. </s> <s xml:id="echoid-s13115" xml:space="preserve">& </s> <s xml:id="echoid-s13116" xml:space="preserve">30. </s> <s xml:id="echoid-s13117" xml:space="preserve">vt hic vides. <lb/></s> <s xml:id="echoid-s13118" xml:space="preserve">Multiplico ſuperiores duos numeros 36. </s> <s xml:id="echoid-s13119" xml:space="preserve">& </s> <s xml:id="echoid-s13120" xml:space="preserve">300. </s> <s xml:id="echoid-s13121" xml:space="preserve">inter ſe, & </s> <s xml:id="echoid-s13122" xml:space="preserve">pro-<lb/>ducto 10800. </s> <s xml:id="echoid-s13123" xml:space="preserve">addo productum 180. </s> <s xml:id="echoid-s13124" xml:space="preserve">ex multiplicatione numerorum inferiorũ <lb/>6. </s> <s xml:id="echoid-s13125" xml:space="preserve">& </s> <s xml:id="echoid-s13126" xml:space="preserve">30. </s> <s xml:id="echoid-s13127" xml:space="preserve">inter ſe. </s> <s xml:id="echoid-s13128" xml:space="preserve">Nam ſumma 10980. </s> <s xml:id="echoid-s13129" xml:space="preserve">erit Diuiſor. </s> <s xml:id="echoid-s13130" xml:space="preserve">Satis etiam eſſet productus <lb/>ex duobus ſuperioribus inter ſe multiplicatis, nimirum 10800. </s> <s xml:id="echoid-s13131" xml:space="preserve">pro Diuiſore. </s> <s xml:id="echoid-s13132" xml:space="preserve"><lb/>quodin alijs extractionibus intelligendum quoque eſt. </s> <s xml:id="echoid-s13133" xml:space="preserve">Diuido ergo punctum <lb/>meum 23483. </s> <s xml:id="echoid-s13134" xml:space="preserve">per diuiſoreminuentum 10980. </s> <s xml:id="echoid-s13135" xml:space="preserve">& </s> <s xml:id="echoid-s13136" xml:space="preserve">Quotientem 2. </s> <s xml:id="echoid-s13137" xml:space="preserve">ſcribo poſt fi-<lb/>guram 6. </s> <s xml:id="echoid-s13138" xml:space="preserve">prius inuentam. </s> <s xml:id="echoid-s13139" xml:space="preserve">Pingo poſt hæc figuram huiuſmodi. </s> <s xml:id="echoid-s13140" xml:space="preserve">Ad dextram <lb/>numerorum 36. </s> <s xml:id="echoid-s13141" xml:space="preserve">& </s> <s xml:id="echoid-s13142" xml:space="preserve">300. </s> <s xml:id="echoid-s13143" xml:space="preserve">colloco inuentam figuram 2. </s> <s xml:id="echoid-s13144" xml:space="preserve">& </s> <s xml:id="echoid-s13145" xml:space="preserve">infra <lb/>eam eius quadratum 4. </s> <s xml:id="echoid-s13146" xml:space="preserve">& </s> <s xml:id="echoid-s13147" xml:space="preserve">ſub hoc cubum eiuſdem 8. </s> <s xml:id="echoid-s13148" xml:space="preserve">Nam ſi <lb/> <anchor type="note" xlink:label="note-311-03a" xlink:href="note-311-03"/> tam ſuperiores tres numeri 36. </s> <s xml:id="echoid-s13149" xml:space="preserve">300. </s> <s xml:id="echoid-s13150" xml:space="preserve">& </s> <s xml:id="echoid-s13151" xml:space="preserve">2. </s> <s xml:id="echoid-s13152" xml:space="preserve">quam inferiores tres <lb/>6. </s> <s xml:id="echoid-s13153" xml:space="preserve">30. </s> <s xml:id="echoid-s13154" xml:space="preserve">& </s> <s xml:id="echoid-s13155" xml:space="preserve">4. </s> <s xml:id="echoid-s13156" xml:space="preserve">inter ſe multiplicentur, & </s> <s xml:id="echoid-s13157" xml:space="preserve">productis 21600. </s> <s xml:id="echoid-s13158" xml:space="preserve">& </s> <s xml:id="echoid-s13159" xml:space="preserve">720. <lb/></s> <s xml:id="echoid-s13160" xml:space="preserve">addatur cubus 8. </s> <s xml:id="echoid-s13161" xml:space="preserve">fiet numerus 22328. </s> <s xml:id="echoid-s13162" xml:space="preserve">quem ſi ex meo puncto <lb/>23483. </s> <s xml:id="echoid-s13163" xml:space="preserve">ſubtraham, remanent 1155. </s> <s xml:id="echoid-s13164" xml:space="preserve">atque adeo punctum ſequens erit 1155190.</s> <s xml:id="echoid-s13165" xml:space="preserve"/> </p> <div xml:id="echoid-div806" type="float" level="2" n="2"> <note position="right" xlink:label="note-311-02" xlink:href="note-311-02a" xml:space="preserve"> <lb/>36--300 <lb/>6-- 30 <lb/></note> <note position="right" xlink:label="note-311-03" xlink:href="note-311-03a" xml:space="preserve"> <lb/>36--300--2. <lb/>6-- 30--4. <lb/>8. <lb/></note> </div> <p> <s xml:id="echoid-s13166" xml:space="preserve"><emph style="sc">Paro</emph> iam alium diuiſorem hocpacto. </s> <s xml:id="echoid-s13167" xml:space="preserve">Supra 62. </s> <s xml:id="echoid-s13168" xml:space="preserve">radicem hactenus inuen-<lb/>tam ſcribo eius quadratum 3844. </s> <s xml:id="echoid-s13169" xml:space="preserve">& </s> <s xml:id="echoid-s13170" xml:space="preserve">ad dextram eoſdem nu-<lb/>meros peculiares 300. </s> <s xml:id="echoid-s13171" xml:space="preserve">& </s> <s xml:id="echoid-s13172" xml:space="preserve">30. </s> <s xml:id="echoid-s13173" xml:space="preserve">Multiplicatio duorum ſuperio-<lb/> <anchor type="note" xlink:label="note-311-04a" xlink:href="note-311-04"/> rum inter ſefacit diuiſorem 1153200. </s> <s xml:id="echoid-s13174" xml:space="preserve">per quem ſi diuidatur pũ-<lb/>ctum 1155190. </s> <s xml:id="echoid-s13175" xml:space="preserve">fit Quotiens 1. </s> <s xml:id="echoid-s13176" xml:space="preserve">poſt alias duas figuras 62. </s> <s xml:id="echoid-s13177" xml:space="preserve">ſcri-<lb/>bendus. </s> <s xml:id="echoid-s13178" xml:space="preserve">Pingo iam figuram talem. </s> <s xml:id="echoid-s13179" xml:space="preserve">Ad dextram numero-<lb/>rum 3844. </s> <s xml:id="echoid-s13180" xml:space="preserve">& </s> <s xml:id="echoid-s13181" xml:space="preserve">300. </s> <s xml:id="echoid-s13182" xml:space="preserve">pono figuram 1. </s> <s xml:id="echoid-s13183" xml:space="preserve">proximè inuentam, & </s> <s xml:id="echoid-s13184" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-311-05a" xlink:href="note-311-05"/> infra eam eius quadratum 1. </s> <s xml:id="echoid-s13185" xml:space="preserve">& </s> <s xml:id="echoid-s13186" xml:space="preserve">ſub hoc eiuſdem cubum. </s> <s xml:id="echoid-s13187" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s13188" xml:space="preserve">Nam ſi tam tres ſuperiores numeri 3844.</s> <s xml:id="echoid-s13189" xml:space="preserve">300. </s> <s xml:id="echoid-s13190" xml:space="preserve">& </s> <s xml:id="echoid-s13191" xml:space="preserve">1. </s> <s xml:id="echoid-s13192" xml:space="preserve">quam <lb/>inferiores tres 62. </s> <s xml:id="echoid-s13193" xml:space="preserve">30. </s> <s xml:id="echoid-s13194" xml:space="preserve">& </s> <s xml:id="echoid-s13195" xml:space="preserve">1. </s> <s xml:id="echoid-s13196" xml:space="preserve">inter ſe multiplicentur, & </s> <s xml:id="echoid-s13197" xml:space="preserve">pro-<lb/>ductis 1153200. </s> <s xml:id="echoid-s13198" xml:space="preserve">& </s> <s xml:id="echoid-s13199" xml:space="preserve">1860. </s> <s xml:id="echoid-s13200" xml:space="preserve">addatur cubus 1. </s> <s xml:id="echoid-s13201" xml:space="preserve">fiet numerus 1155061. </s> <s xml:id="echoid-s13202" xml:space="preserve">quem ſi ex meo <lb/>puncto 1155190. </s> <s xml:id="echoid-s13203" xml:space="preserve">demam, remanent 129. </s> <s xml:id="echoid-s13204" xml:space="preserve">Eſt ergo abſoluta extractio, radixque <lb/>inuenta eſt 621. </s> <s xml:id="echoid-s13205" xml:space="preserve">quæ in ſe cubice multiplicata procreatnumerum 239483061. </s> <s xml:id="echoid-s13206" xml:space="preserve">cui <lb/>ſi adijciatur reſiduum 129. </s> <s xml:id="echoid-s13207" xml:space="preserve">coflabitur numerus propoſitus 239483190.</s> <s xml:id="echoid-s13208" xml:space="preserve"/> </p> <div xml:id="echoid-div807" type="float" level="2" n="3"> <note position="right" xlink:label="note-311-04" xlink:href="note-311-04a" xml:space="preserve"> <lb/>3844--300. <lb/>62-- 30. <lb/></note> <note position="right" xlink:label="note-311-05" xlink:href="note-311-05a" xml:space="preserve"> <lb/>3844--300--1. <lb/>62-- 30--1. <lb/>1. <lb/></note> </div> </div> <div xml:id="echoid-div809" type="section" level="1" n="277"> <head xml:id="echoid-head303" xml:space="preserve">EXTRACTIO RADICIS <lb/>Surdeſolidæ.</head> <p> <s xml:id="echoid-s13209" xml:space="preserve"><emph style="sc">Sit</emph> numerus 1039589621. </s> <s xml:id="echoid-s13210" xml:space="preserve">cuius radix ſurdeſolida quæritur.</s> <s xml:id="echoid-s13211" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13212" xml:space="preserve">. </s> <s xml:id="echoid-s13213" xml:space="preserve">.</s> <s xml:id="echoid-s13214" xml:space="preserve"/> </p> <note position="right" xml:space="preserve">Surdeſolidæ <lb/>ràdicis extra-<lb/>ctio <lb/>radix 63.</note> <p> <s xml:id="echoid-s13215" xml:space="preserve"><emph style="sc">Primvm</emph> à puncto 10395. </s> <s xml:id="echoid-s13216" xml:space="preserve">ſubtraho maximum ſurdeſolidum 7776. </s> <s xml:id="echoid-s13217" xml:space="preserve">in eo in-<lb/>cluſum, cuius radicem 6. </s> <s xml:id="echoid-s13218" xml:space="preserve">ſcribo in margine ꝓ Quotiente: </s> <s xml:id="echoid-s13219" xml:space="preserve">& </s> <s xml:id="echoid-s13220" xml:space="preserve">quia numerus relin-<lb/>quitur 2619. </s> <s xml:id="echoid-s13221" xml:space="preserve">ideo punctum ſequens erit 261989621.</s> <s xml:id="echoid-s13222" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13223" xml:space="preserve"><emph style="sc">Deinde</emph> paro diuiſorem hoc modo. </s> <s xml:id="echoid-s13224" xml:space="preserve">Supra radicem inuentam 6. </s> <s xml:id="echoid-s13225" xml:space="preserve">ſcribo e-<lb/>ius quadratum 36. </s> <s xml:id="echoid-s13226" xml:space="preserve">& </s> <s xml:id="echoid-s13227" xml:space="preserve">ſupra hunc eiuſdem cubum 216. </s> <s xml:id="echoid-s13228" xml:space="preserve">& </s> <s xml:id="echoid-s13229" xml:space="preserve">ſupra hunc eiuſdem <lb/>Zenſizenſum, vel quadrati quadratum 1296: </s> <s xml:id="echoid-s13230" xml:space="preserve">ita vt conſtituatur progreſsio <pb o="282" file="312" n="312" rhead="GEOMETR. PRACT."/> Geometrica aſcendens à radice 6. </s> <s xml:id="echoid-s13231" xml:space="preserve">denominata totterminorum, quot numeri <lb/>peculiares requiruntur in extra ctione ſurdeſolida, Et ad <lb/> <anchor type="note" xlink:label="note-312-01a" xlink:href="note-312-01"/> dextram colloco quatuor numeros peculiares requiſi-<lb/>tos, vt exemplum monſtrat. </s> <s xml:id="echoid-s13232" xml:space="preserve">Multiplico duos ſuperio-<lb/>res numeros (quod ſatis eſt) 1296. </s> <s xml:id="echoid-s13233" xml:space="preserve">& </s> <s xml:id="echoid-s13234" xml:space="preserve">50000. </s> <s xml:id="echoid-s13235" xml:space="preserve">inter ſe. </s> <s xml:id="echoid-s13236" xml:space="preserve">Pro-<lb/>ductus namque numerus 64800000. </s> <s xml:id="echoid-s13237" xml:space="preserve">erit diuiſor, per quem ſi diuidatur pun-<lb/>ctum relictum 261989621. </s> <s xml:id="echoid-s13238" xml:space="preserve">poſſet eſſe Quotiens vel 4. </s> <s xml:id="echoid-s13239" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s13240" xml:space="preserve">vel 2. </s> <s xml:id="echoid-s13241" xml:space="preserve">Accipio au-<lb/>tem 3. </s> <s xml:id="echoid-s13242" xml:space="preserve">quia figura 2. </s> <s xml:id="echoid-s13243" xml:space="preserve">eſt nimis parua, & </s> <s xml:id="echoid-s13244" xml:space="preserve">4. </s> <s xml:id="echoid-s13245" xml:space="preserve">nimis magna, vt ex ſequentibus pate-<lb/>bit; </s> <s xml:id="echoid-s13246" xml:space="preserve">quem Quotientam 3. </s> <s xml:id="echoid-s13247" xml:space="preserve">in margine ſcribo poſt inuentam figuram 6. </s> <s xml:id="echoid-s13248" xml:space="preserve">Pingo er-<lb/>ergo talem figuram. </s> <s xml:id="echoid-s13249" xml:space="preserve">Ad dextram numerorum 1296. </s> <s xml:id="echoid-s13250" xml:space="preserve">& </s> <s xml:id="echoid-s13251" xml:space="preserve">50000. </s> <s xml:id="echoid-s13252" xml:space="preserve">pono figuram <lb/>Quotientis acceptam 3. </s> <s xml:id="echoid-s13253" xml:space="preserve">& </s> <s xml:id="echoid-s13254" xml:space="preserve">infra eam eius quadratum 9. </s> <s xml:id="echoid-s13255" xml:space="preserve">& </s> <s xml:id="echoid-s13256" xml:space="preserve">ſub hoc eiuſdem cu-<lb/>bum 27. </s> <s xml:id="echoid-s13257" xml:space="preserve">& </s> <s xml:id="echoid-s13258" xml:space="preserve">ſub hoc eiuſdem Zenſizenſum, vel quadrati quadratum 81. </s> <s xml:id="echoid-s13259" xml:space="preserve">& </s> <s xml:id="echoid-s13260" xml:space="preserve">ſub <lb/>hoc eiuſdem ſurdeſolidum 243. </s> <s xml:id="echoid-s13261" xml:space="preserve">ita vt ad dextram conſtituatur progreſsio Geo-<lb/>metrica deſcendens denominata à figura Quotientis 3. </s> <s xml:id="echoid-s13262" xml:space="preserve">inuenta tot terminorum <lb/>vno amplius, quot numeri peculiares requirun-<lb/>tur: </s> <s xml:id="echoid-s13263" xml:space="preserve">adeo vt vltimus terminus ſit numerus ſurde-<lb/> <anchor type="note" xlink:label="note-312-02a" xlink:href="note-312-02"/> ſolidus figuræ inuentæ, quemadmodum in cubi-<lb/>ca extractione fuit cubus, & </s> <s xml:id="echoid-s13264" xml:space="preserve">in quadrata qua-<lb/>dratus. </s> <s xml:id="echoid-s13265" xml:space="preserve">Nam ſi terni numeritranſuerſales inter ſe <lb/>multiplicentur, & </s> <s xml:id="echoid-s13266" xml:space="preserve">ad productos 194400000. <lb/></s> <s xml:id="echoid-s13267" xml:space="preserve">19440000. </s> <s xml:id="echoid-s13268" xml:space="preserve">972000. </s> <s xml:id="echoid-s13269" xml:space="preserve">24300. </s> <s xml:id="echoid-s13270" xml:space="preserve">adijciatur ſurdeſoli-<lb/>dus 243. </s> <s xml:id="echoid-s13271" xml:space="preserve">efficietur numerus 214836543. </s> <s xml:id="echoid-s13272" xml:space="preserve">qui ex puncto 261989621. </s> <s xml:id="echoid-s13273" xml:space="preserve">detractus re-<lb/>linquit 47153078. </s> <s xml:id="echoid-s13274" xml:space="preserve">Eſt ergo radix ſurdeſolida inuenta 63. </s> <s xml:id="echoid-s13275" xml:space="preserve">quę in ſe ſurdeſolidè <lb/>multiplicata, ſi nimirum quinquies ponatur hoc modo, 63. </s> <s xml:id="echoid-s13276" xml:space="preserve">63. </s> <s xml:id="echoid-s13277" xml:space="preserve">63. </s> <s xml:id="echoid-s13278" xml:space="preserve">63. </s> <s xml:id="echoid-s13279" xml:space="preserve">63. </s> <s xml:id="echoid-s13280" xml:space="preserve">pro du-<lb/>cit numerum 992436543, cui ſi addatur reſiduum 47153078. </s> <s xml:id="echoid-s13281" xml:space="preserve">conflabitur propo-<lb/>fitus numerus 1039589621.</s> <s xml:id="echoid-s13282" xml:space="preserve"/> </p> <div xml:id="echoid-div809" type="float" level="2" n="1"> <note position="right" xlink:label="note-312-01" xlink:href="note-312-01a" xml:space="preserve"> <lb/>1296--50000 <lb/>216--10000 <lb/>36-- 100 <lb/>6-- 50 <lb/></note> <note position="right" xlink:label="note-312-02" xlink:href="note-312-02a" xml:space="preserve"> <lb/>1296--50000-- 3. <lb/>216--10000-- 9. <lb/>36-- 1000-- 27. <lb/>6-- 50-- 81. <lb/>243 <lb/></note> </div> <p> <s xml:id="echoid-s13283" xml:space="preserve"><emph style="sc">Qvod</emph> ſi ſupereſſet aliud punctum, conſtituenda eſſet progreſsio aſcendẽs <lb/>denominata à tota radice hactenus inuenta 63. <lb/></s> <s xml:id="echoid-s13284" xml:space="preserve">quatuor terminorum, vt hic vides. </s> <s xml:id="echoid-s13285" xml:space="preserve">Nam produ-<lb/> <anchor type="note" xlink:label="note-312-03a" xlink:href="note-312-03"/> ctus 78764805. </s> <s xml:id="echoid-s13286" xml:space="preserve">ex ſuperioribus duobus numeris <lb/>inter ſe multiplicatis eſſet nouus diuiſor. </s> <s xml:id="echoid-s13287" xml:space="preserve">Deinde <lb/>ex noua figura inuenta eſſet conſtituen da pro-<lb/>greſsio deſcendens vſque ad ſurdeſolidum illius <lb/>figuræ, quemadmodum ſupra cum figura 3. </s> <s xml:id="echoid-s13288" xml:space="preserve">factum eſt.</s> <s xml:id="echoid-s13289" xml:space="preserve"/> </p> <div xml:id="echoid-div810" type="float" level="2" n="2"> <note position="right" xlink:label="note-312-03" xlink:href="note-312-03a" xml:space="preserve"> <lb/>15752961--50000 <lb/>250047--10000 <lb/>3969-- 1000 <lb/>63-- 50 <lb/></note> </div> <p> <s xml:id="echoid-s13290" xml:space="preserve"><emph style="sc">Atqve</emph> in hunc modum radicem cuiuſcunque ſpeciei extrahes, ſi diligen-<lb/>ter inquires numeros propoſitæ radiciinſeruientes, vt ſupra docuimus. </s> <s xml:id="echoid-s13291" xml:space="preserve">Quæ <lb/>ſanè ratio mihi ſemper præclara eſt viſa. </s> <s xml:id="echoid-s13292" xml:space="preserve">Nam etiamſi operatio videatur ali-<lb/>quanto longior eſſe, quam par ſit, difficilis tamen non eſt, quippe cum ignorari <lb/>in ea non poſsit, quid faciendum ſit: </s> <s xml:id="echoid-s13293" xml:space="preserve">cum tamen in extractionibus ab alijs Ari-<lb/>thmeticis traditis (quadrata excepta) tanta ſit operationis difficultas, vt infini-<lb/>ta ferememoria opus ſit ad retinendũ ea, quę ad extrahendas radices adhiben-<lb/>da ſunt, vt in radice cubica extrahenda per aliorum regulam, ſi adhibeatur, pa-<lb/>tebit: </s> <s xml:id="echoid-s13294" xml:space="preserve">cum tamen cubica extractio ſit longè facilior extractione ſurdeſolida, & </s> <s xml:id="echoid-s13295" xml:space="preserve"><lb/>alijs inſequentibus, quę ferè inextricabiles ſunt.</s> <s xml:id="echoid-s13296" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13297" xml:space="preserve"><emph style="sc">Sola</emph> vna difficultas tam in noſtra, quam in aliorum extra ctione exiſtit, <lb/>quod nimirum dubium interdum ſit, num figuram nimis paruam in Quotien-<lb/>te alicuius puncti accep erimus. </s> <s xml:id="echoid-s13298" xml:space="preserve">Vt in ſecundo puncto extractionis ſurdeſolidæ <pb o="283" file="313" n="313" rhead="LIBER SEXTVS."/> potuit eſſe Quotiens vel 4. </s> <s xml:id="echoid-s13299" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s13300" xml:space="preserve">vel 2. </s> <s xml:id="echoid-s13301" xml:space="preserve">Nos autem accepimus 3. </s> <s xml:id="echoid-s13302" xml:space="preserve">Si ergo certi <lb/> <anchor type="note" xlink:label="note-313-01a" xlink:href="note-313-01"/> eſſe velimus, an accipi potuiſſet figura 4. </s> <s xml:id="echoid-s13303" xml:space="preserve">quando quidem ſuperfuit numerus <lb/>47153078. </s> <s xml:id="echoid-s13304" xml:space="preserve">valdè magnus, faciendum periculum erit cum figura 4. </s> <s xml:id="echoid-s13305" xml:space="preserve">conſtituen-<lb/>do ſcilicet progreſsionem deſcendentem Geo-<lb/>metricam à 4. </s> <s xml:id="echoid-s13306" xml:space="preserve">denominatam. </s> <s xml:id="echoid-s13307" xml:space="preserve">Et quia qua-<lb/> <anchor type="note" xlink:label="note-313-02a" xlink:href="note-313-02"/> tuor numeri producti tranſuerſales cum ſurde-<lb/>ſolido 1024. </s> <s xml:id="echoid-s13308" xml:space="preserve">faciunt numerum 297141824. </s> <s xml:id="echoid-s13309" xml:space="preserve">qui <lb/>ex puncto ſecundo 261989621. </s> <s xml:id="echoid-s13310" xml:space="preserve">ſubtrahi ne-<lb/>quit; </s> <s xml:id="echoid-s13311" xml:space="preserve">argumento eſt, figuram 4. </s> <s xml:id="echoid-s13312" xml:space="preserve">nimis magnam <lb/>eſſe, ac proinde figuram 2. </s> <s xml:id="echoid-s13313" xml:space="preserve">nimis paruam; </s> <s xml:id="echoid-s13314" xml:space="preserve">quã-<lb/>do quidem cum figura 3. </s> <s xml:id="echoid-s13315" xml:space="preserve">tales numeri pro creati ſunt, qui ex propoſito puncto <lb/>potuerunt ſubtrahi. </s> <s xml:id="echoid-s13316" xml:space="preserve">Hoc ergo remedium ſi adhibeatur, quamuis longiuſcu-<lb/>lum, tutiſsima erit noſtra ratio extrahendarum radicum.</s> <s xml:id="echoid-s13317" xml:space="preserve"/> </p> <div xml:id="echoid-div811" type="float" level="2" n="3"> <note position="right" xlink:label="note-313-01" xlink:href="note-313-01a" xml:space="preserve">Difficult{as} in <lb/>extractioni-<lb/>b{us} quo pact@ <lb/>ſuperetur.</note> <note position="right" xlink:label="note-313-02" xlink:href="note-313-02a" xml:space="preserve"> <lb/>1296--50000-- 4 <lb/>216--10000-- 16 <lb/>36-- 1000-- 64 <lb/>6-- 50-- 256 <lb/>1024 <lb/></note> </div> <note position="right" xml:space="preserve">Cur exemplũ <lb/>non ponatur <lb/>de radice Zen-<lb/>ſiZ<unsure/>enſica, & c.</note> <p> <s xml:id="echoid-s13318" xml:space="preserve"><emph style="sc">Iam</emph> verò cur in ſuperiori<unsure/> tabella qua dratorum, cuborum, ſurdeſolidorum, <lb/>&</s> <s xml:id="echoid-s13319" xml:space="preserve">c. </s> <s xml:id="echoid-s13320" xml:space="preserve">omiſerim Zenſizenſos, ſiue quadrati quadratos, atque adeo radicis Zen-<lb/>ſizenſicæ extractionem præterierim; </s> <s xml:id="echoid-s13321" xml:space="preserve">ratio eſt, quod radices Zenſizenſica, Zenſi-<lb/>cubica, Zenſizenzenſica, cubicubica, Zenſurdeſolida, &</s> <s xml:id="echoid-s13322" xml:space="preserve">c. </s> <s xml:id="echoid-s13323" xml:space="preserve">quamuis erui poſ-<lb/>ſint, ſicut aliæ, habent tamen aliam etiam extractionis regulã, quam vel exipſis <lb/>nominibus colligere licet. </s> <s xml:id="echoid-s13324" xml:space="preserve">Videlicet.</s> <s xml:id="echoid-s13325" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13326" xml:space="preserve"><emph style="sc">Extractvrvs</emph> radicem Zenſizenſicam, ſiue quadrati quadratam, ex-<lb/>trahe primo radicem quadratam: </s> <s xml:id="echoid-s13327" xml:space="preserve">Deinde ex hac radice erue iterum quadratam <lb/>radicem. </s> <s xml:id="echoid-s13328" xml:space="preserve">Hæc enim erit radix Zenſizenſica, quæ quæritur.</s> <s xml:id="echoid-s13329" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13330" xml:space="preserve"><emph style="sc">Extractvrvs</emph> verò radicem Zenſicubicam, id eſt, quadrati cubicam, <lb/>vel cubi quadratam, extrahe primo radicem quadratam, & </s> <s xml:id="echoid-s13331" xml:space="preserve">ex hac deinderadi-<lb/>cem cubicam. </s> <s xml:id="echoid-s13332" xml:space="preserve">Vel primo erue radicem cubicam, & </s> <s xml:id="echoid-s13333" xml:space="preserve">ex hac quadratam. </s> <s xml:id="echoid-s13334" xml:space="preserve">Vlti-<lb/>ma enim radix eruta erit ea, quam quæris. </s> <s xml:id="echoid-s13335" xml:space="preserve">Idem iudicium habeto de alijs ra-<lb/>dicibus numerorum compo ſitorum, vt de radice Zenſizenzenſica, cubicubi-<lb/>ca, Zenſurdeſolida, &</s> <s xml:id="echoid-s13336" xml:space="preserve">c.</s> <s xml:id="echoid-s13337" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div813" type="section" level="1" n="278"> <head xml:id="echoid-head304" xml:space="preserve">REGVLA PROPRIA EXTRA-<lb/>ctionis radicis cubicæ.</head> <p> <s xml:id="echoid-s13338" xml:space="preserve"><emph style="sc">Qvoniam</emph> frequentior vſus eſt radicis quadratæ, & </s> <s xml:id="echoid-s13339" xml:space="preserve">cubicæ apud Mathe-<lb/>maticos, quam aliarum radicum, lubet in ſtudio ſorũ gratiam præſcribere hoc <lb/>loco regulam propriam ad cubicam radicem extrahendam: </s> <s xml:id="echoid-s13340" xml:space="preserve">quemadmodum <lb/>idem de quadrata radice fecimus in noſtra Arithmetica practica. </s> <s xml:id="echoid-s13341" xml:space="preserve">Relictis au-<lb/>tem aliorum regulis, quod minus faciles, minuſque expeditæ ſint, excerpam <lb/>vnam quaſi nouam ex ſuperiori extractione radicis cubicæ, quæ ſic ſehabet.</s> <s xml:id="echoid-s13342" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13343" xml:space="preserve"><emph style="sc">Sit</emph> eruendaradix cubica ex numero 1860867.</s> <s xml:id="echoid-s13344" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13345" xml:space="preserve">. </s> <s xml:id="echoid-s13346" xml:space="preserve">. </s> <s xml:id="echoid-s13347" xml:space="preserve">.</s> <s xml:id="echoid-s13348" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13349" xml:space="preserve"><emph style="sc">Ex</emph> primo puncto 1. </s> <s xml:id="echoid-s13350" xml:space="preserve">ad ſiniſtram ſubtraho cubum 1. </s> <s xml:id="echoid-s13351" xml:space="preserve">maximũ in eo contentũ, <lb/> <anchor type="note" xlink:label="note-313-04a" xlink:href="note-313-04"/> nihilq; </s> <s xml:id="echoid-s13352" xml:space="preserve">remanet. </s> <s xml:id="echoid-s13353" xml:space="preserve">Erit ergo ſequẽs punctũ 860. </s> <s xml:id="echoid-s13354" xml:space="preserve">& </s> <s xml:id="echoid-s13355" xml:space="preserve">pro radice inuenta eſt figura 1.</s> <s xml:id="echoid-s13356" xml:space="preserve"/> </p> <div xml:id="echoid-div813" type="float" level="2" n="1"> <note position="right" xlink:label="note-313-04" xlink:href="note-313-04a" xml:space="preserve">Regula pro<unsure/> <lb/>pria rad@cis <lb/>cubicæ. <lb/>radix 123.</note> </div> <p> <s xml:id="echoid-s13357" xml:space="preserve"><emph style="sc">Paro</emph> diuiſorem, multiplicando quadratum figuræ inuentæ, nimirum 1. </s> <s xml:id="echoid-s13358" xml:space="preserve">per <lb/>300. </s> <s xml:id="echoid-s13359" xml:space="preserve">qui erit 300. </s> <s xml:id="echoid-s13360" xml:space="preserve">per quem ſi diuidam punctum 860 inuenio Quotientẽ 2. </s> <s xml:id="echoid-s13361" xml:space="preserve">pro <lb/>ſecunda figura radicis. </s> <s xml:id="echoid-s13362" xml:space="preserve">Hanc duco in diuiſorem inuentum 300. </s> <s xml:id="echoid-s13363" xml:space="preserve">facio que 600. <lb/></s> <s xml:id="echoid-s13364" xml:space="preserve">Deinde duco quadratũ nouæ figuræ 2. </s> <s xml:id="echoid-s13365" xml:space="preserve">inuentæ, nimirũ 4. </s> <s xml:id="echoid-s13366" xml:space="preserve">in productũ ex priori <pb o="284" file="314" n="314" rhead="GEOMETR. PRACT."/> figura 1. </s> <s xml:id="echoid-s13367" xml:space="preserve">multiplicata per 30. </s> <s xml:id="echoid-s13368" xml:space="preserve">hoc eſt, in 30. </s> <s xml:id="echoid-s13369" xml:space="preserve">facio que 120. </s> <s xml:id="echoid-s13370" xml:space="preserve">Poſtremò ad ſummã <lb/>duorum horum productorum 600. </s> <s xml:id="echoid-s13371" xml:space="preserve">& </s> <s xml:id="echoid-s13372" xml:space="preserve">120. </s> <s xml:id="echoid-s13373" xml:space="preserve">id eſt, ad 720. </s> <s xml:id="echoid-s13374" xml:space="preserve">adijcio cubum inuen-<lb/>tę nouæ figuræ 2. </s> <s xml:id="echoid-s13375" xml:space="preserve">nimirum 8. </s> <s xml:id="echoid-s13376" xml:space="preserve">totamque ſummam 728. </s> <s xml:id="echoid-s13377" xml:space="preserve">ex meo puncto 860. </s> <s xml:id="echoid-s13378" xml:space="preserve">ſub-<lb/>traho. </s> <s xml:id="echoid-s13379" xml:space="preserve">Et quia remanent 132. </s> <s xml:id="echoid-s13380" xml:space="preserve">erit vltimum punctum 132867. </s> <s xml:id="echoid-s13381" xml:space="preserve">Scribo ergo inuen-<lb/>tam figuram 2. </s> <s xml:id="echoid-s13382" xml:space="preserve">poſt priorem 1.</s> <s xml:id="echoid-s13383" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13384" xml:space="preserve"><emph style="sc">Deinde</emph> paro eodem modo diuiſorem nouum pro vltimo puncto. </s> <s xml:id="echoid-s13385" xml:space="preserve">Nimi-<lb/>rum quadratum totius radicis 12. </s> <s xml:id="echoid-s13386" xml:space="preserve">hactenus inuentæ, id eſt, 144. </s> <s xml:id="echoid-s13387" xml:space="preserve">duco iterum in <lb/>300. </s> <s xml:id="echoid-s13388" xml:space="preserve">Productus enim numerus 43200. </s> <s xml:id="echoid-s13389" xml:space="preserve">erit diuiſor, per quem ſi diuidam meum <lb/>punctum 132867. </s> <s xml:id="echoid-s13390" xml:space="preserve">reperio Quotientem 3. </s> <s xml:id="echoid-s13391" xml:space="preserve">ſcribendum poſt radicem 12. </s> <s xml:id="echoid-s13392" xml:space="preserve">hacte-<lb/>nusinuentam. </s> <s xml:id="echoid-s13393" xml:space="preserve">Hanc figuram inuentam 3. </s> <s xml:id="echoid-s13394" xml:space="preserve">ſimiliter duco in diuiſorem inuentum <lb/>43200. </s> <s xml:id="echoid-s13395" xml:space="preserve">facioque 129600. </s> <s xml:id="echoid-s13396" xml:space="preserve">Deinde quadratum eiuſdem nouæ figuræ 3. </s> <s xml:id="echoid-s13397" xml:space="preserve">nimirum <lb/>9. </s> <s xml:id="echoid-s13398" xml:space="preserve">duco in productum ex radice 12 prius inuenta, multiplicata per 30. </s> <s xml:id="echoid-s13399" xml:space="preserve">hoc eſt, <lb/>in 360. </s> <s xml:id="echoid-s13400" xml:space="preserve">efficioque 3240. </s> <s xml:id="echoid-s13401" xml:space="preserve">Poſtremò ad ſummam horum duorum productorum <lb/>129600. </s> <s xml:id="echoid-s13402" xml:space="preserve">& </s> <s xml:id="echoid-s13403" xml:space="preserve">3240. </s> <s xml:id="echoid-s13404" xml:space="preserve">hoc eſt, ad 132840. </s> <s xml:id="echoid-s13405" xml:space="preserve">adijcio cubum 27. </s> <s xml:id="echoid-s13406" xml:space="preserve">ex eadem noua figura <lb/>genitum, fitque numerus 132867: </s> <s xml:id="echoid-s13407" xml:space="preserve">qui ex puncto 123867. </s> <s xml:id="echoid-s13408" xml:space="preserve">detractus nihil relin-<lb/>quit. </s> <s xml:id="echoid-s13409" xml:space="preserve">Atque ita inuenta eſt radix 123. </s> <s xml:id="echoid-s13410" xml:space="preserve">numeripropoſiti 1860867.</s> <s xml:id="echoid-s13411" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13412" xml:space="preserve"><emph style="sc">Qvod</emph> ſi ſupereſſet aliud punctum, ducendus eſſet quadratus totius radi-<lb/>cis 123. </s> <s xml:id="echoid-s13413" xml:space="preserve">hactenus inuentæ in 300. </s> <s xml:id="echoid-s13414" xml:space="preserve">vt nouus diuiſor exurgeret, &</s> <s xml:id="echoid-s13415" xml:space="preserve">c. </s> <s xml:id="echoid-s13416" xml:space="preserve">Vides ergo in <lb/>hacregula plus memoriæ requiri, quam in ſuperiori, quamuis facilior ſit, quam <lb/>aliorum regulæ. </s> <s xml:id="echoid-s13417" xml:space="preserve">Memor tamen eſto, quando dubitas, an nimis paruam figuram <lb/>in Quotiente acceperis, vt facias periculum de maiori figura, vt ſupra dictum eſt.</s> <s xml:id="echoid-s13418" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div815" type="section" level="1" n="279"> <head xml:id="echoid-head305" xml:space="preserve">PROBL. 15. PROPOS. 20.</head> <p> <s xml:id="echoid-s13419" xml:space="preserve">IN numeris non quadratis, non cubis, non Zenſizenſicis, non ſurde-<lb/>ſolidis, &</s> <s xml:id="echoid-s13420" xml:space="preserve">c. </s> <s xml:id="echoid-s13421" xml:space="preserve">radicem, veræ propinquam inuenire.</s> <s xml:id="echoid-s13422" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13423" xml:space="preserve"><emph style="sc">Qvando</emph> numerus propoſitus non eſt quadratus, aut cubus, aut Zenſizẽ-<lb/>ſus, aut ſurdeſ<unsure/>olidus, &</s> <s xml:id="echoid-s13424" xml:space="preserve">c. </s> <s xml:id="echoid-s13425" xml:space="preserve">non poteſt habere veramradicem, ſed per regulas ſu-<lb/>periores inueniturradix maximi quadrati, vel cubi, vel Zenſizenſi vel ſurdeſoli-<lb/>diin dato numero contenti. </s> <s xml:id="echoid-s13426" xml:space="preserve">Vt igitur ſciamus, quænam fractio ad inuentam ra-<lb/>dicem addenda ſit, vt habeaturradix propinquior veræ, agendum erit hoc mo-<lb/>do.</s> <s xml:id="echoid-s13427" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13428" xml:space="preserve"><emph style="sc">Nvmero</emph> propoſito apponantur aliquot binarij cifrarum, ſi quadrata ra-<lb/> <anchor type="note" xlink:label="note-314-01a" xlink:href="note-314-01"/> dix propinqua inquiritur: </s> <s xml:id="echoid-s13429" xml:space="preserve">vel ſi cubica, aliquot cifrarum ternarij; </s> <s xml:id="echoid-s13430" xml:space="preserve">vel aliquot <lb/>quaternarij, ſi radix Zenſizenſica deſideratur, vel aliquot quinarij cifrarum, ſi <lb/>de ſurdeſolida radice agitur, &</s> <s xml:id="echoid-s13431" xml:space="preserve">c. </s> <s xml:id="echoid-s13432" xml:space="preserve">Reſpondet autem in qualibet ſpecie radicis <lb/>numerus cifrarum aliquoties repetendus numero, qui in progreſsione ad initi-<lb/>um præcedentis problematis poſita ſcribitur ſupra numerum, à quo radix no-<lb/>men ſumit. </s> <s xml:id="echoid-s13433" xml:space="preserve">Vt quia ſupra quadratum ponitur 2. </s> <s xml:id="echoid-s13434" xml:space="preserve">ideo pro quadrata radice ap-<lb/>ponuntur binę cifræ aliquoties, at pro cubica ternæ, quod ſupra cubum ſcri-<lb/>ptus ſit numerus 3. </s> <s xml:id="echoid-s13435" xml:space="preserve">&</s> <s xml:id="echoid-s13436" xml:space="preserve">c. </s> <s xml:id="echoid-s13437" xml:space="preserve">ita vt pro radice cubicuba propinqua apponendi ſinta-<lb/>liquot nouenarij cifrarũ, quippe cũ ſupra cubicubũ numerus 9. </s> <s xml:id="echoid-s13438" xml:space="preserve">reperiat<unsure/> deſcri-<lb/>ptus. </s> <s xml:id="echoid-s13439" xml:space="preserve">Idem numerus cifrarum aliquoties repetendus reſpondet quoq; </s> <s xml:id="echoid-s13440" xml:space="preserve">ſignatio-<lb/>nibus punctorum, quæ faciendæ ſunt, vtr adix extrahatur. </s> <s xml:id="echoid-s13441" xml:space="preserve">Vt quia in quadrata <pb o="285" file="315" n="315" rhead="LIBER SEXTVS."/> radice extrahenda ſignatur ſecunda quæuis figura, propterea aliquot cifrarum <lb/>binarij aſcribendi ſunt; </s> <s xml:id="echoid-s13442" xml:space="preserve">in cubica verò radice aliquot ternarij cifrarum adiun-<lb/>gendi ſunt, quia in ea extrahenda tertia quæque figura ſignatur, &</s> <s xml:id="echoid-s13443" xml:space="preserve">c.</s> <s xml:id="echoid-s13444" xml:space="preserve"/> </p> <div xml:id="echoid-div815" type="float" level="2" n="1"> <note position="left" xlink:label="note-314-01" xlink:href="note-314-01a" xml:space="preserve">Quot binarij <lb/>cifrarum vel <lb/>ternarij vel <lb/>quaternarij, <lb/>&c. ad propin <lb/>quam radicẽ <lb/>eruendam ap-<lb/>ponendi ſint.</note> </div> <p> <s xml:id="echoid-s13445" xml:space="preserve"><emph style="sc">Qvo</emph> autem plures binarij, vel ternarij cifrarum, &</s> <s xml:id="echoid-s13446" xml:space="preserve">c. </s> <s xml:id="echoid-s13447" xml:space="preserve">numero propoſito ap-<lb/>ponetur, eo propinquiorradix eruetur.</s> <s xml:id="echoid-s13448" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13449" xml:space="preserve"><emph style="sc">Appositis</emph> hoc modo cifris ad numerum, ex quo radix eruenda eſt, ex-<lb/> <anchor type="note" xlink:label="note-315-01a" xlink:href="note-315-01"/> trahenda eſt ex toto illo numero radix, vt ſupra traditum eſt. </s> <s xml:id="echoid-s13450" xml:space="preserve">Deinde ex ea ra-<lb/>dice abiectis ad dexteram tot figuris, quot cifrarum binarij, vel ternarij, vel <lb/>quaternarij, &</s> <s xml:id="echoid-s13451" xml:space="preserve">c. </s> <s xml:id="echoid-s13452" xml:space="preserve">appoſitifuere, reliquæ figuræradicem integram dabunt, cui <lb/>addenda eſt fractio numeratorem habens figuras abiectas, denominatorem ve-<lb/>rò vnitatem, cum totidem cifris, quot binarij cifrarum, vel ternarij, &</s> <s xml:id="echoid-s13453" xml:space="preserve">c. </s> <s xml:id="echoid-s13454" xml:space="preserve">addi-<lb/>ti fuerunt, nimirum vel 10. </s> <s xml:id="echoid-s13455" xml:space="preserve">ſi vnus binarius, vel ternarius, &</s> <s xml:id="echoid-s13456" xml:space="preserve">c. </s> <s xml:id="echoid-s13457" xml:space="preserve">additus fuit, <lb/>vel 100. </s> <s xml:id="echoid-s13458" xml:space="preserve">ſi duo binarij: </s> <s xml:id="echoid-s13459" xml:space="preserve">vel ternarij, &</s> <s xml:id="echoid-s13460" xml:space="preserve">c. </s> <s xml:id="echoid-s13461" xml:space="preserve">additi fuerunt: </s> <s xml:id="echoid-s13462" xml:space="preserve">vel 1000. </s> <s xml:id="echoid-s13463" xml:space="preserve">ſi tres, & </s> <s xml:id="echoid-s13464" xml:space="preserve">ſic <lb/>deinceps ita vt fractio illa contineat vel decimas, vel centeſimas, vel milleſi-<lb/>mas, &</s> <s xml:id="echoid-s13465" xml:space="preserve">c.</s> <s xml:id="echoid-s13466" xml:space="preserve"/> </p> <div xml:id="echoid-div816" type="float" level="2" n="2"> <note position="right" xlink:label="note-315-01" xlink:href="note-315-01a" xml:space="preserve">Quæ fractio <lb/>addenda ſit <lb/>radici, vt pro-<lb/>pinquior ra-<lb/>dix gignatur.</note> </div> <p> <s xml:id="echoid-s13467" xml:space="preserve"><emph style="sc">Exempli</emph> cauſa. </s> <s xml:id="echoid-s13468" xml:space="preserve">Ex numero 29. </s> <s xml:id="echoid-s13469" xml:space="preserve">extrahenda ſitradix quadrata. </s> <s xml:id="echoid-s13470" xml:space="preserve">Appoſitis <lb/>tribus binariis cifrarum, hoc modo 29000000. </s> <s xml:id="echoid-s13471" xml:space="preserve">inuenietur huius numeri qua-<lb/>drata radix 5385. </s> <s xml:id="echoid-s13472" xml:space="preserve">minor quam vera, quippe cum in extra ctione aliquid remanſe-<lb/>rit: </s> <s xml:id="echoid-s13473" xml:space="preserve">addita verò vnitate, fietradix 5386. </s> <s xml:id="echoid-s13474" xml:space="preserve">maior, quam vera. </s> <s xml:id="echoid-s13475" xml:space="preserve">Abiectis igitur tribus <lb/>figuris ad dexteram, propter tres cifrarum binarios additos, erit propinqua ra-<lb/>dix 5 {385/1000}. </s> <s xml:id="echoid-s13476" xml:space="preserve">minor tamen quam vera: </s> <s xml:id="echoid-s13477" xml:space="preserve">at 5 {386/1000}. </s> <s xml:id="echoid-s13478" xml:space="preserve">maior quam vera. </s> <s xml:id="echoid-s13479" xml:space="preserve">Illius enim <lb/>quadratus numerus eſt 28 {998225/1000000}. </s> <s xml:id="echoid-s13480" xml:space="preserve">minor quam propoſitus numerus 29. <lb/></s> <s xml:id="echoid-s13481" xml:space="preserve">Huius verò numerus quadratus eſt 29 {8996/1000000}. </s> <s xml:id="echoid-s13482" xml:space="preserve">maior eodem numero propo-<lb/>ſito 29.</s> <s xml:id="echoid-s13483" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13484" xml:space="preserve"><emph style="sc">Item</emph> ex numero 29160. </s> <s xml:id="echoid-s13485" xml:space="preserve">eruenda ſit radix cubica. </s> <s xml:id="echoid-s13486" xml:space="preserve">Apponantur tres terna-<lb/>rij cifrarum, vt rurſus habeantur in fractione partes denominatæ à 1000. </s> <s xml:id="echoid-s13487" xml:space="preserve">atque <lb/>ex toto numero 29160000000000. </s> <s xml:id="echoid-s13488" xml:space="preserve">extrahatur radix cubica, quæ reperietur <lb/>30779. </s> <s xml:id="echoid-s13489" xml:space="preserve">minor quam vera, quod in extractione fuerit aliquis numerus reſiduus: <lb/></s> <s xml:id="echoid-s13490" xml:space="preserve">atque adeò maior quàm vera, erit 30780. </s> <s xml:id="echoid-s13491" xml:space="preserve">Abiectis tribus figuris ad dexteram, <lb/>propter tres cifrarum ternarios appoſitos, erit propinqua radix cubica 30 {779/1000}. </s> <s xml:id="echoid-s13492" xml:space="preserve"><lb/>minor quam vera, cum eius cubus ſit tantum 29158 {388419@39/1000000000}. </s> <s xml:id="echoid-s13493" xml:space="preserve">maior autem <lb/>propinqua radix, quam vera, erit 30 {780/1000}. </s> <s xml:id="echoid-s13494" xml:space="preserve">quippe cum eius cubus ſit <lb/>29161 {230552000/1000000000}.</s> <s xml:id="echoid-s13495" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13496" xml:space="preserve"><emph style="sc">Demonstratio</emph> huius inuentionis radicis propinquæ hæc eſt. </s> <s xml:id="echoid-s13497" xml:space="preserve">Quando <lb/>pro radice quadrata apponuntur 00. </s> <s xml:id="echoid-s13498" xml:space="preserve">ad numerum propoſitum, verbi gratia ad <lb/>5. </s> <s xml:id="echoid-s13499" xml:space="preserve">multiplicatur propoſitus numerus per 100. </s> <s xml:id="echoid-s13500" xml:space="preserve">hoc eſt, per quadratum radicis <lb/>10. </s> <s xml:id="echoid-s13501" xml:space="preserve">Et quia quadrati 500. </s> <s xml:id="echoid-s13502" xml:space="preserve">& </s> <s xml:id="echoid-s13503" xml:space="preserve">5. </s> <s xml:id="echoid-s13504" xml:space="preserve">(Nam datus numerus, & </s> <s xml:id="echoid-s13505" xml:space="preserve">conflatus ex additio-<lb/>ne 00. </s> <s xml:id="echoid-s13506" xml:space="preserve">ſumendi ſunt tanquam quadrati, cum eorum radices quærantur) <anchor type="note" xlink:href="" symbol="a"/> ha- <anchor type="note" xlink:label="note-315-02a" xlink:href="note-315-02"/> bent proportionem ſuarum radicum duplicatam: </s> <s xml:id="echoid-s13507" xml:space="preserve">Eſt autem 500. </s> <s xml:id="echoid-s13508" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s13509" xml:space="preserve">vt 100. <lb/></s> <s xml:id="echoid-s13510" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13511" xml:space="preserve">propterea quod 5. </s> <s xml:id="echoid-s13512" xml:space="preserve">multiplicatus per 100. </s> <s xml:id="echoid-s13513" xml:space="preserve">fecit 500. </s> <s xml:id="echoid-s13514" xml:space="preserve">Centupla verò pro-<lb/>portio decuplę duplicata eſt, vt in hoc appoſito exemplo <lb/> <anchor type="note" xlink:label="note-315-03a" xlink:href="note-315-03"/> patet; </s> <s xml:id="echoid-s13515" xml:space="preserve">erit proportio radicis numeri 500. </s> <s xml:id="echoid-s13516" xml:space="preserve">ad radicem nu-<lb/>meri 5. </s> <s xml:id="echoid-s13517" xml:space="preserve">decupla. </s> <s xml:id="echoid-s13518" xml:space="preserve">Cum ergo radix 500. </s> <s xml:id="echoid-s13519" xml:space="preserve">ſit 22. </s> <s xml:id="echoid-s13520" xml:space="preserve">minor quam <lb/>vera, erit eius {1/10}. </s> <s xml:id="echoid-s13521" xml:space="preserve">nimirum 2 {2/10}. </s> <s xml:id="echoid-s13522" xml:space="preserve">radix numeri 5. </s> <s xml:id="echoid-s13523" xml:space="preserve">minor <lb/>quam vera: </s> <s xml:id="echoid-s13524" xml:space="preserve">ac proinde 2 {3/10}. </s> <s xml:id="echoid-s13525" xml:space="preserve">erit maior quam vera. </s> <s xml:id="echoid-s13526" xml:space="preserve">Rectè <lb/>ergo præcepimus, quando apponuntur 00. </s> <s xml:id="echoid-s13527" xml:space="preserve">abiiciendam eſſe ex radice 22. </s> <s xml:id="echoid-s13528" xml:space="preserve">vnam <lb/>figuram, vt relinquatur radix 2 {2/10}.</s> <s xml:id="echoid-s13529" xml:space="preserve"/> </p> <div xml:id="echoid-div817" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-315-02" xlink:href="note-315-02a" xml:space="preserve">11. octaui.</note> <note position="right" xlink:label="note-315-03" xlink:href="note-315-03a" xml:space="preserve"> <lb/>1. # 10. # 100. <lb/># 5. # 500. <lb/></note> </div> <pb o="286" file="316" n="316" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s13530" xml:space="preserve"><emph style="sc">Qvando</emph> autem apponuntur 0000. </s> <s xml:id="echoid-s13531" xml:space="preserve">multiplicatur numerus propoſitus, <lb/>verbi gratia numerus 5. </s> <s xml:id="echoid-s13532" xml:space="preserve">per 10000. </s> <s xml:id="echoid-s13533" xml:space="preserve">id eſt, per quadratum radicis 100. </s> <s xml:id="echoid-s13534" xml:space="preserve">Et quia <lb/>quadrati 50000. </s> <s xml:id="echoid-s13535" xml:space="preserve">5. </s> <s xml:id="echoid-s13536" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> habent duplicatam proportionem ſuarum radicum. </s> <s xml:id="echoid-s13537" xml:space="preserve">Eſt <anchor type="note" xlink:label="note-316-01a" xlink:href="note-316-01"/> autem 50000. </s> <s xml:id="echoid-s13538" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s13539" xml:space="preserve">vt 10000. </s> <s xml:id="echoid-s13540" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13541" xml:space="preserve">propterea quod| 5. </s> <s xml:id="echoid-s13542" xml:space="preserve">multiplicatus per 10000. <lb/></s> <s xml:id="echoid-s13543" xml:space="preserve">fecit 50000. </s> <s xml:id="echoid-s13544" xml:space="preserve">Proportio autem 10000. </s> <s xml:id="echoid-s13545" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13546" xml:space="preserve">duplicata eſt proportionis 1000. </s> <s xml:id="echoid-s13547" xml:space="preserve"><lb/>ad 1. </s> <s xml:id="echoid-s13548" xml:space="preserve">vt in hoc exemplo patet; </s> <s xml:id="echoid-s13549" xml:space="preserve">erit proportio ra-<lb/> <anchor type="note" xlink:label="note-316-02a" xlink:href="note-316-02"/> dicis numeri 50000. </s> <s xml:id="echoid-s13550" xml:space="preserve">nimirum 223. </s> <s xml:id="echoid-s13551" xml:space="preserve">ad radicem nu-<lb/>meri 5. </s> <s xml:id="echoid-s13552" xml:space="preserve">vt 100. </s> <s xml:id="echoid-s13553" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13554" xml:space="preserve">Quare ſi radix 223. </s> <s xml:id="echoid-s13555" xml:space="preserve">diuidatur <lb/>per 100. </s> <s xml:id="echoid-s13556" xml:space="preserve">procreabitur radix quadrata propinqua <lb/>2 {23/100}. </s> <s xml:id="echoid-s13557" xml:space="preserve">minor quam vera, at 2 {24/100}. </s> <s xml:id="echoid-s13558" xml:space="preserve">erit maior quam <lb/>vera. </s> <s xml:id="echoid-s13559" xml:space="preserve">Rectè ergo præcepimus, cum apponuntur 0000. </s> <s xml:id="echoid-s13560" xml:space="preserve">abiiciendas eſſe ex radi-<lb/>ce 223. </s> <s xml:id="echoid-s13561" xml:space="preserve">duas figuras, vt reliqua fiatradix 2 {23/100}.</s> <s xml:id="echoid-s13562" xml:space="preserve"/> </p> <div xml:id="echoid-div818" type="float" level="2" n="4"> <note symbol="a" position="left" xlink:label="note-316-01" xlink:href="note-316-01a" xml:space="preserve">11. octaui.</note> <note position="right" xlink:label="note-316-02" xlink:href="note-316-02a" xml:space="preserve"> <lb/>1. # 100. # 10000. <lb/># 5. # 50000. <lb/></note> </div> <p> <s xml:id="echoid-s13563" xml:space="preserve"><emph style="sc">Pari</emph> ratione, ſi apponantur 000000. </s> <s xml:id="echoid-s13564" xml:space="preserve">procreabitur radix propinqua in <lb/>milleſimis, at queita deinceps.</s> <s xml:id="echoid-s13565" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13566" xml:space="preserve"><emph style="sc">Rvrsvs</emph> quando pro radice cubica ad numerum propoſitum, vt ad 5. </s> <s xml:id="echoid-s13567" xml:space="preserve">ad-<lb/>duntur 000. </s> <s xml:id="echoid-s13568" xml:space="preserve">multiplicatur datus numerus per 1000. </s> <s xml:id="echoid-s13569" xml:space="preserve">id eſt, per cubum radicis <lb/>10. </s> <s xml:id="echoid-s13570" xml:space="preserve">Et quia cubi 5000. </s> <s xml:id="echoid-s13571" xml:space="preserve">5. </s> <s xml:id="echoid-s13572" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> habentproportionem ſuarum radicum triplicatam:</s> <s xml:id="echoid-s13573" xml:space="preserve"> <anchor type="note" xlink:label="note-316-03a" xlink:href="note-316-03"/> Eſt autem 5000. </s> <s xml:id="echoid-s13574" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s13575" xml:space="preserve">vt 1000. </s> <s xml:id="echoid-s13576" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13577" xml:space="preserve">quod 1000. </s> <s xml:id="echoid-s13578" xml:space="preserve">mul-<lb/>tiplicans 5. </s> <s xml:id="echoid-s13579" xml:space="preserve">fecit 5000. </s> <s xml:id="echoid-s13580" xml:space="preserve">Proportio autem 1000. </s> <s xml:id="echoid-s13581" xml:space="preserve">ad 1. <lb/></s> <s xml:id="echoid-s13582" xml:space="preserve"> <anchor type="note" xlink:label="note-316-04a" xlink:href="note-316-04"/> triplicata eſt proportionis 10. </s> <s xml:id="echoid-s13583" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13584" xml:space="preserve">vt in hoc appoſito <lb/>exemplo apparet; </s> <s xml:id="echoid-s13585" xml:space="preserve">erit proportio radicis numeri 5000. <lb/></s> <s xml:id="echoid-s13586" xml:space="preserve">nimirum 17. </s> <s xml:id="echoid-s13587" xml:space="preserve">ad radicem numeri 5. </s> <s xml:id="echoid-s13588" xml:space="preserve">vt 10. </s> <s xml:id="echoid-s13589" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13590" xml:space="preserve">Quo-<lb/>circa ſi radix 17. </s> <s xml:id="echoid-s13591" xml:space="preserve">diuidatur per 10. </s> <s xml:id="echoid-s13592" xml:space="preserve">fit radix cubica propinqua 1 {7/10}. </s> <s xml:id="echoid-s13593" xml:space="preserve">minor quam <lb/>vera, at 1 {8/10}. </s> <s xml:id="echoid-s13594" xml:space="preserve">erit maior quam vera. </s> <s xml:id="echoid-s13595" xml:space="preserve">Rectè ergo præcepimus, quando pro radice <lb/>cubica apponuntur 000. </s> <s xml:id="echoid-s13596" xml:space="preserve">abiiciendam eſſe ex radice inuenta 17. </s> <s xml:id="echoid-s13597" xml:space="preserve">vnam figuram, <lb/>vt reliqua fiatradix 1 {7/10}.</s> <s xml:id="echoid-s13598" xml:space="preserve"/> </p> <div xml:id="echoid-div819" type="float" level="2" n="5"> <note symbol="b" position="left" xlink:label="note-316-03" xlink:href="note-316-03a" xml:space="preserve">12. octaui.</note> <note position="right" xlink:label="note-316-04" xlink:href="note-316-04a" xml:space="preserve"> <lb/>1. # 10. # 100. # 1000. <lb/># # 5. # 5000. <lb/></note> </div> <p> <s xml:id="echoid-s13599" xml:space="preserve"><emph style="sc">Simili</emph> modo ſi apponantur 000000. </s> <s xml:id="echoid-s13600" xml:space="preserve">inuenietur propinqua radix in cen-<lb/>teſimis: </s> <s xml:id="echoid-s13601" xml:space="preserve">Et ſi apponantur 000000000. </s> <s xml:id="echoid-s13602" xml:space="preserve">in milleſimis, &</s> <s xml:id="echoid-s13603" xml:space="preserve">c. </s> <s xml:id="echoid-s13604" xml:space="preserve">Nam ibi multipli-<lb/>catur numerus per 1000000. </s> <s xml:id="echoid-s13605" xml:space="preserve">id eſt, per cubum radicis 100. </s> <s xml:id="echoid-s13606" xml:space="preserve">hic verò per <lb/>1000000000. </s> <s xml:id="echoid-s13607" xml:space="preserve">nimirum per cubum radicis 1000. </s> <s xml:id="echoid-s13608" xml:space="preserve">Cætera eodem modo demon-<lb/>ſtrabuntur.</s> <s xml:id="echoid-s13609" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13610" xml:space="preserve"><emph style="sc">Neqve</emph> verò diuerſa ratio eſt in aliis radicibus. </s> <s xml:id="echoid-s13611" xml:space="preserve">Nam in ſurdeſolida verbi <lb/>gratia, quando adduntur 00000. </s> <s xml:id="echoid-s13612" xml:space="preserve">fit <lb/>multiplicatio per 100000. </s> <s xml:id="echoid-s13613" xml:space="preserve">id eſt, per <lb/> <anchor type="note" xlink:label="note-316-05a" xlink:href="note-316-05"/> ſurdeſolidum radicis 10. </s> <s xml:id="echoid-s13614" xml:space="preserve">Et propor-<lb/>tio 100000. </s> <s xml:id="echoid-s13615" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13616" xml:space="preserve">eſt quintuplicata proportionis 10. </s> <s xml:id="echoid-s13617" xml:space="preserve">ad 1. </s> <s xml:id="echoid-s13618" xml:space="preserve">vt in appoſito exem-<lb/>plo apparet, &</s> <s xml:id="echoid-s13619" xml:space="preserve">c.</s> <s xml:id="echoid-s13620" xml:space="preserve"/> </p> <div xml:id="echoid-div820" type="float" level="2" n="6"> <note position="right" xlink:label="note-316-05" xlink:href="note-316-05a" xml:space="preserve"> <lb/>1. # 10. # 100. # 1000. # 10000. # 100000. <lb/></note> </div> </div> <div xml:id="echoid-div822" type="section" level="1" n="280"> <head xml:id="echoid-head306" xml:space="preserve">PROBL. 16. PROPOS. 21.</head> <p> <s xml:id="echoid-s13621" xml:space="preserve">RADICEM cuiuſque generis ex data minutia extrahere.</s> <s xml:id="echoid-s13622" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13623" xml:space="preserve"><emph style="sc">In</emph> minutiis extrahenda eſt radix eiuſdem appellationis cum radice, quæ <lb/> <anchor type="note" xlink:label="note-316-06a" xlink:href="note-316-06"/> quæritur, tum ex numeratore, tum ex denominatore. </s> <s xml:id="echoid-s13624" xml:space="preserve">Ita enim fiet fractio, quæ <lb/>radix eſt propoſitæ minutiæ. </s> <s xml:id="echoid-s13625" xml:space="preserve">Vtradix quadrata minutiæ {4/9}. </s> <s xml:id="echoid-s13626" xml:space="preserve">eſt {2/3}. </s> <s xml:id="echoid-s13627" xml:space="preserve">Et radix cubica <lb/>minutiæ {8/27}. </s> <s xml:id="echoid-s13628" xml:space="preserve">eſt ſimiliter {2/3<unsure/>}. </s> <s xml:id="echoid-s13629" xml:space="preserve">Et radix Zenſizenſica minutiæ {30/81}. </s> <s xml:id="echoid-s13630" xml:space="preserve">eſt quo que {@/@}. </s> <s xml:id="echoid-s13631" xml:space="preserve">Et <lb/>radix ſurdeſolida minutiæ {32/243}. </s> <s xml:id="echoid-s13632" xml:space="preserve">eſt pari ratione {@/@}. </s> <s xml:id="echoid-s13633" xml:space="preserve">& </s> <s xml:id="echoid-s13634" xml:space="preserve">ſic de aliis.</s> <s xml:id="echoid-s13635" xml:space="preserve"/> </p> <div xml:id="echoid-div822" type="float" level="2" n="1"> <note position="left" xlink:label="note-316-06" xlink:href="note-316-06a" xml:space="preserve">Extractio ra-<lb/>dicum ex mi-<lb/>nutiis.</note> </div> <pb o="287" file="317" n="317" rhead="LIBER SEXTVS."/> <p> <s xml:id="echoid-s13636" xml:space="preserve"><emph style="sc">Qvod</emph> ſi data minutia fuerit fractio, vel minutia alterius minutiæ, reducen-<lb/>da prius erit ad minutiam ſimplicem. </s> <s xml:id="echoid-s13637" xml:space="preserve">Vt ſi quærenda ſit radix quadrata ex hac <lb/>minutia minutiæ<unsure/> {2/4}. </s> <s xml:id="echoid-s13638" xml:space="preserve">{8/9}, reducenda erit ad hanc ſimplicem minutiam {16/36}. </s> <s xml:id="echoid-s13639" xml:space="preserve">cuius ra-<lb/>dix quadrata eſt {4/6}. </s> <s xml:id="echoid-s13640" xml:space="preserve">vel {2/3}. </s> <s xml:id="echoid-s13641" xml:space="preserve">&</s> <s xml:id="echoid-s13642" xml:space="preserve">c.</s> <s xml:id="echoid-s13643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13644" xml:space="preserve"><emph style="sc">Similiter</emph> ſi fractio adhæreat integris, erunt integra prius reducenda ad <lb/>fractionem eiuſdem denominationis. </s> <s xml:id="echoid-s13645" xml:space="preserve">Vt ſi quærenda ſitradix cubica numeri <lb/>2 {10/27}. </s> <s xml:id="echoid-s13646" xml:space="preserve">reducendus erit ad hanc fractionem {64/27}. </s> <s xml:id="echoid-s13647" xml:space="preserve">cuius radix cubica eſt {4/3}. </s> <s xml:id="echoid-s13648" xml:space="preserve">hoc eſt, <lb/>1 {1/3}. </s> <s xml:id="echoid-s13649" xml:space="preserve">&</s> <s xml:id="echoid-s13650" xml:space="preserve">c.</s> <s xml:id="echoid-s13651" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13652" xml:space="preserve"><emph style="sc">Si</emph> vel numerator, vel denominator minutiæ, vel vterque numerus careat <lb/>radice eius appellationis, quæ deſideratur, non habebitilla minutia radicem, quę <lb/>quæritur. </s> <s xml:id="echoid-s13653" xml:space="preserve">Vt neque {4/7}. </s> <s xml:id="echoid-s13654" xml:space="preserve">neque {6/9}. </s> <s xml:id="echoid-s13655" xml:space="preserve">neque {5/12}. </s> <s xml:id="echoid-s13656" xml:space="preserve">habentradicem quadratam præcisè, <lb/>propterea quod denominator in prima numerator verò in ſecunda, & </s> <s xml:id="echoid-s13657" xml:space="preserve">vterque <lb/>numerus in tertia quadratam radicem non habet.</s> <s xml:id="echoid-s13658" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13659" xml:space="preserve"><emph style="sc">Cognosces</emph> autem, an data fractio habeat radicem quæſitam, nec ne, ſi <lb/>eam, ad minimos terminos reduces. </s> <s xml:id="echoid-s13660" xml:space="preserve">Si namque ita reducta habuerit radicem, <lb/>dicetur quoque data minutia eandem radicem habere: </s> <s xml:id="echoid-s13661" xml:space="preserve">Si verò reducta ad mi-<lb/>nimos terminos radicem non habuerit, neque propoſita minutia radicem habe-<lb/>bit. </s> <s xml:id="echoid-s13662" xml:space="preserve">Vt ſi proponatur minutia {20/45}. </s> <s xml:id="echoid-s13663" xml:space="preserve">volo ſcire, an habeat radicem quadratam: <lb/></s> <s xml:id="echoid-s13664" xml:space="preserve">Ea redacta ad minimos terminos eſt {4/9}. </s> <s xml:id="echoid-s13665" xml:space="preserve">quæ radicem quadratam habet {2/3}. </s> <s xml:id="echoid-s13666" xml:space="preserve"><lb/>Hanc ergo eandem radicem quadratam dicetur habere minutia propoſita {20/45}. </s> <s xml:id="echoid-s13667" xml:space="preserve"><lb/>Atverò minutia {6/9}. </s> <s xml:id="echoid-s13668" xml:space="preserve">non habebit radicem quadratam: </s> <s xml:id="echoid-s13669" xml:space="preserve">quia neque {2/3}. </s> <s xml:id="echoid-s13670" xml:space="preserve">in mini-<lb/>mis terminis, ad quam reducitur, eam habet. </s> <s xml:id="echoid-s13671" xml:space="preserve">Pari ratione minutia {24/81}. </s> <s xml:id="echoid-s13672" xml:space="preserve">habe-<lb/>bit radicem cubicam {2/3}. </s> <s xml:id="echoid-s13673" xml:space="preserve">eandem nimirum, quam habet minutia {8/27}. </s> <s xml:id="echoid-s13674" xml:space="preserve">in minimis <lb/>terminis, ad quam illa reducitur. </s> <s xml:id="echoid-s13675" xml:space="preserve">Minutia autem {13/20}. </s> <s xml:id="echoid-s13676" xml:space="preserve">radice cubica carebit, <lb/>quod minutia {3/5}. </s> <s xml:id="echoid-s13677" xml:space="preserve">ad quam in minimis terminis reuocatur, eadem careat. </s> <s xml:id="echoid-s13678" xml:space="preserve">Et ſic <lb/>de aliis.</s> <s xml:id="echoid-s13679" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13680" xml:space="preserve"><emph style="sc">Qvando</emph> ergo minutia ad minimos reuocata terminos radicem quæſitam <lb/>non habuerit, ex quirenda erit radix propinqua tam numeratoris, quam deno-<lb/>minatoris, apponendo videlicet vtrique prius numero aliquot cifrarum bina-<lb/>rios, vel ternarios, quaternarioſue, &</s> <s xml:id="echoid-s13681" xml:space="preserve">c. </s> <s xml:id="echoid-s13682" xml:space="preserve">prout quadrata radix, aut cubica, aut <lb/>Zenſizenſica, &</s> <s xml:id="echoid-s13683" xml:space="preserve">c. </s> <s xml:id="echoid-s13684" xml:space="preserve">inquiritur. </s> <s xml:id="echoid-s13685" xml:space="preserve">Si namque radix propinqua numeratoris per pro-<lb/>pinquam denominatoris radicem diuidatur, prodibit radix propinqua, quam <lb/>quærimus. </s> <s xml:id="echoid-s13686" xml:space="preserve">Verbi gratia, ſi proponatur minutia {6/7}. </s> <s xml:id="echoid-s13687" xml:space="preserve">cuius radix quadrata inqui-<lb/>renda ſit, appoſitis tribus binariis cifrarum reperietur numeratoris radix propin-<lb/>qua 2 {449/100@}. </s> <s xml:id="echoid-s13688" xml:space="preserve">denominatoris verò 2 {645/1000}. </s> <s xml:id="echoid-s13689" xml:space="preserve">Si @gitur illa per hanc diuidatur, pro-<lb/>ueniet radix quæſita {2449/264@}. </s> <s xml:id="echoid-s13690" xml:space="preserve">ſatis propinqua. </s> <s xml:id="echoid-s13691" xml:space="preserve">Idem iudicium de aliis radicibus <lb/>habeatur, ſi memineris tamen, in cubica tam numeratori, quam denominatori <lb/>apponendos eſſe aliquot ternarios cifrarum, vt propinquæ eorum radices e-<lb/>ruantur: </s> <s xml:id="echoid-s13692" xml:space="preserve">In Zenſizenſica verò aliquot quaternarios, & </s> <s xml:id="echoid-s13693" xml:space="preserve">in ſurdeſolida aliquot <lb/>quinarios, &</s> <s xml:id="echoid-s13694" xml:space="preserve">c.</s> <s xml:id="echoid-s13695" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13696" xml:space="preserve"><emph style="sc">Qvia</emph> verò moleſtum eſt inquirere duas radices propinquas, vnam pro <lb/>numeratore propoſitæ fractionis, & </s> <s xml:id="echoid-s13697" xml:space="preserve">pro denominatore alteram, traduntur à <lb/>Cardano, & </s> <s xml:id="echoid-s13698" xml:space="preserve">Tartalea pro radice quadrata, & </s> <s xml:id="echoid-s13699" xml:space="preserve">cubica, quæ nimirum magis in v-<lb/>ſu ſunt, peculiares quædamregulæ, quas hic explicare lubet: </s> <s xml:id="echoid-s13700" xml:space="preserve">quippe cum in eis <lb/>tantummodo radicis propinquæ inuentione opus ſit.</s> <s xml:id="echoid-s13701" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13702" xml:space="preserve"><emph style="sc">Pro</emph> quadrata igitur radice, duc numeratorem in denominatorẽ, & </s> <s xml:id="echoid-s13703" xml:space="preserve">produ-<lb/> <anchor type="note" xlink:label="note-317-01a" xlink:href="note-317-01"/> cti numeri radicẽ quadratam propinquã diuide per denominatorẽ: </s> <s xml:id="echoid-s13704" xml:space="preserve">Vel nume- <pb o="288" file="318" n="318" rhead="GEOMETR. PRACT."/> ratorem per radicem illam propinquam partire. </s> <s xml:id="echoid-s13705" xml:space="preserve">Vtroque enim modo radix pro-<lb/> <anchor type="note" xlink:label="note-318-01a" xlink:href="note-318-01"/> pinqua fractionis propoſitæ gignetur. </s> <s xml:id="echoid-s13706" xml:space="preserve">Et ſi quidem propinqua illa radix nume-<lb/>ri producti ex numeratore in denominatorem fuerit minor quam vera, reperie-<lb/>tur priori modo radix fractionis propinqua minor quo que quam vera; </s> <s xml:id="echoid-s13707" xml:space="preserve">pro-<lb/>pterea quod numerus verò minor diuiditur: </s> <s xml:id="echoid-s13708" xml:space="preserve">poſteriori verò modo inuenietur <lb/>radix propinqua fractionis maior quam vera, quod tunc diuiſio fiat per nume-<lb/>rum vero minorem. </s> <s xml:id="echoid-s13709" xml:space="preserve">Contrarium eueniet, ſi radix illa propinqua numeri ex nu-<lb/>meratore in denominatorem producti fuerit maior quam vera. </s> <s xml:id="echoid-s13710" xml:space="preserve">Nam priori mo-<lb/>do gignetur radix fractionis propinqua maior, quam vera, poſteriori vero mo-<lb/>do minor, quam vera, vt perſpicuum eſt. </s> <s xml:id="echoid-s13711" xml:space="preserve">Hanc regulam propoſui quo que libr. <lb/></s> <s xml:id="echoid-s13712" xml:space="preserve">4. </s> <s xml:id="echoid-s13713" xml:space="preserve">cap. </s> <s xml:id="echoid-s13714" xml:space="preserve">2. </s> <s xml:id="echoid-s13715" xml:space="preserve">Num. </s> <s xml:id="echoid-s13716" xml:space="preserve">5. </s> <s xml:id="echoid-s13717" xml:space="preserve">ibique eandem demonſtraui. </s> <s xml:id="echoid-s13718" xml:space="preserve">Exemplum huius etiam regulæ <lb/>ibidem habes.</s> <s xml:id="echoid-s13719" xml:space="preserve"/> </p> <div xml:id="echoid-div823" type="float" level="2" n="2"> <note position="right" xlink:label="note-317-01" xlink:href="note-317-01a" xml:space="preserve">Alia extr actio <lb/>radicis qua-</note> <note position="left" xlink:label="note-318-01" xlink:href="note-318-01a" xml:space="preserve">dratæ & cu-<lb/>bicæ ex data <lb/>minutia.</note> </div> <p> <s xml:id="echoid-s13720" xml:space="preserve"><emph style="sc">Pro</emph> radice verò cubica: </s> <s xml:id="echoid-s13721" xml:space="preserve">duc numeratorem in quadratum denominatoris, <lb/>& </s> <s xml:id="echoid-s13722" xml:space="preserve">producti numeri radicem cubicam propinquam diuide per denominatorem: <lb/></s> <s xml:id="echoid-s13723" xml:space="preserve">Vel duc denominatorem in quadratum numeratoris, & </s> <s xml:id="echoid-s13724" xml:space="preserve">per numeri producti <lb/>radicem cubicam propinquam partire numeratorem. </s> <s xml:id="echoid-s13725" xml:space="preserve">Vtroque enim modo <lb/>propinqua radix propoſitæ minutiæ proueniet. </s> <s xml:id="echoid-s13726" xml:space="preserve">Et priori quidem modo, ſi il-<lb/>la radix cubica propinqua fuerit minor quam vera, reperietur radix propinqua <lb/>fractionis minor quo que quam vera, propterea quod diuiſio fit numeri ve-<lb/>ro minoris per denominatorem fractionis: </s> <s xml:id="echoid-s13727" xml:space="preserve">Si autemradix illa propinqua fue-<lb/>rit maior quam vera, gignetur quoque radix propinqua fractionis maior quam <lb/>vera, quod tunc numerus vero maior per denominatorem fractionis diuidatur. </s> <s xml:id="echoid-s13728" xml:space="preserve"><lb/>Poſteriorivero modo, ſi radix illa cubica propinquior fuerit minor quam vera, <lb/>producetur radix propinqua fractionis maior quam vera, quod tunc diuiſio <lb/>fiat per numerum vero minorem: </s> <s xml:id="echoid-s13729" xml:space="preserve">At ſi illa radix cubica propinqua fuerit ma-<lb/>ior quam vera, erit inuenta radix fractionis propinqua minor quam vera, quan-<lb/>doquidem tunc diuiditur numerator per numerum vero maiorẽ. </s> <s xml:id="echoid-s13730" xml:space="preserve">Exemplum <lb/>in fractione {8/27}. </s> <s xml:id="echoid-s13731" xml:space="preserve">habente verã radicem cubicam {2/3}. </s> <s xml:id="echoid-s13732" xml:space="preserve">Ducto numeratore 8. </s> <s xml:id="echoid-s13733" xml:space="preserve">in 729. </s> <s xml:id="echoid-s13734" xml:space="preserve"><lb/>quadratum denominatoris 27. </s> <s xml:id="echoid-s13735" xml:space="preserve">fit numerus 5832. </s> <s xml:id="echoid-s13736" xml:space="preserve">cuius radix cubica eſt 18. </s> <s xml:id="echoid-s13737" xml:space="preserve">Hæc <lb/>diuiſa per denominatorem 27. </s> <s xml:id="echoid-s13738" xml:space="preserve">facit {18/27}. </s> <s xml:id="echoid-s13739" xml:space="preserve">id eſt, {2/3}. </s> <s xml:id="echoid-s13740" xml:space="preserve">pro radice cubica fractionis {8/27}. </s> <s xml:id="echoid-s13741" xml:space="preserve"><lb/>Item ducto denominatore 27. </s> <s xml:id="echoid-s13742" xml:space="preserve">in 64. </s> <s xml:id="echoid-s13743" xml:space="preserve">quadratum numeratoris 8. </s> <s xml:id="echoid-s13744" xml:space="preserve">gignitur nume-<lb/>rus 1728. </s> <s xml:id="echoid-s13745" xml:space="preserve">cuius radix cubica eſt 12. </s> <s xml:id="echoid-s13746" xml:space="preserve">per quam ſi diuidatur numerator 8. </s> <s xml:id="echoid-s13747" xml:space="preserve">fit Quo-<lb/>tiens {8/12}. </s> <s xml:id="echoid-s13748" xml:space="preserve">hoc eſt, {2/3}. </s> <s xml:id="echoid-s13749" xml:space="preserve">vt prius, pro radice cubica fractionis {8/27}. </s> <s xml:id="echoid-s13750" xml:space="preserve">propoſitæ. </s> <s xml:id="echoid-s13751" xml:space="preserve">Alte-<lb/>rum exemplum in fractione {5/7}. </s> <s xml:id="echoid-s13752" xml:space="preserve">non habente veram radicem cubicam. </s> <s xml:id="echoid-s13753" xml:space="preserve">Ducto <lb/>numeratore 5. </s> <s xml:id="echoid-s13754" xml:space="preserve">in 49. </s> <s xml:id="echoid-s13755" xml:space="preserve">quadratum denominatoris 7. </s> <s xml:id="echoid-s13756" xml:space="preserve">fit numerus 245. </s> <s xml:id="echoid-s13757" xml:space="preserve">cuius radix <lb/>cubica propinqua 6 {25/100}. </s> <s xml:id="echoid-s13758" xml:space="preserve">(inuenta per appoſitionum duorum ternariorum ci-<lb/>frarum) diuiſa per denominatorẽ 7. </s> <s xml:id="echoid-s13759" xml:space="preserve">facit Quotientem {625/700}. </s> <s xml:id="echoid-s13760" xml:space="preserve">hoc eſt, {25/28}. </s> <s xml:id="echoid-s13761" xml:space="preserve">. </s> <s xml:id="echoid-s13762" xml:space="preserve">pro ra-<lb/>dice ſractionis {5/7}. </s> <s xml:id="echoid-s13763" xml:space="preserve">Item ducto denominatore 7. </s> <s xml:id="echoid-s13764" xml:space="preserve">in 25. </s> <s xml:id="echoid-s13765" xml:space="preserve">quadratum numeratoris 5. </s> <s xml:id="echoid-s13766" xml:space="preserve"><lb/>fit numerus 175. </s> <s xml:id="echoid-s13767" xml:space="preserve">per cuius radicem cubicam 5 {59/100}. </s> <s xml:id="echoid-s13768" xml:space="preserve">propinquam inuentam per ap-<lb/>poſitionẽ 000000. </s> <s xml:id="echoid-s13769" xml:space="preserve">ad 175. </s> <s xml:id="echoid-s13770" xml:space="preserve">ſi partiamur numeratorem 5. </s> <s xml:id="echoid-s13771" xml:space="preserve">inueniemus Quotien-<lb/>tem {500/559}. </s> <s xml:id="echoid-s13772" xml:space="preserve">pro radice cubica propinqua datæ fractionis {5/7}. </s> <s xml:id="echoid-s13773" xml:space="preserve">atque ita de aliis. </s> <s xml:id="echoid-s13774" xml:space="preserve">Por-<lb/>ro hoc modo reperitur radix fractionis propinquior, quam per ſuperiorem re-<lb/>gulam: </s> <s xml:id="echoid-s13775" xml:space="preserve">quia hic ſolum vnus error irrepit propter radicem cubicam propin-<lb/>quam, quæ vera non eſt, manente tam denominatore in priori modo, quam nu-<lb/>meratore in poſteriori, in propria ſua quantitate; </s> <s xml:id="echoid-s13776" xml:space="preserve">at in ſuperioriregula duo <lb/>interueniunt errores, propter duas radices cubicas propinquas, quæ veræ <lb/>non ſunt.</s> <s xml:id="echoid-s13777" xml:space="preserve"/> </p> <pb o="289" file="319" n="319" rhead="LIBER SEXTVS."/> <p> <s xml:id="echoid-s13778" xml:space="preserve"><emph style="sc">Demonstro</emph> vtrumque hunc modum hac ratione. </s> <s xml:id="echoid-s13779" xml:space="preserve">Quando numerator <lb/>in quadratum denominatoris ducitur, erit producti numeriradix cubica vnus <lb/>duorum mediorum proportionalium inter numeratorem ac denominatorem <lb/>collo candus iuxta denominatorem, vt conſtat ex iis, quæ propoſ. </s> <s xml:id="echoid-s13780" xml:space="preserve">18. </s> <s xml:id="echoid-s13781" xml:space="preserve">huius lib. <lb/></s> <s xml:id="echoid-s13782" xml:space="preserve">demonſtrata ſunt. </s> <s xml:id="echoid-s13783" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Erit igitur proportio numeratoris ad denominatorem tri- <anchor type="note" xlink:label="note-319-01a" xlink:href="note-319-01"/> plicata proportionis radicis cubicæ inuentæ ad denominatorem: </s> <s xml:id="echoid-s13784" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Eſt autem eadem proportio numeratoris ad denominatorem, tanquam cubi ad cubum, <lb/> <anchor type="note" xlink:label="note-319-02a" xlink:href="note-319-02"/> triplicata quoque proportionis radicis cubicæ numeratoris ad radicem cubi-<lb/>cam denominatoris. </s> <s xml:id="echoid-s13785" xml:space="preserve">Igitur erit radix cubica inuenta ad denominatorem, vt <lb/>radix cubica numeratoris ad radicem cubicam denominatoris: </s> <s xml:id="echoid-s13786" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ac proinde <anchor type="note" xlink:label="note-319-03a" xlink:href="note-319-03"/> minutia, cuius numerator radix cubica inuenta, ac denominator ipſe denomi-<lb/>nator, æqualis erit minutiæ, cuius numerator radix cubica numeratoris, ac <lb/>denominator radix cubica denominatoris. </s> <s xml:id="echoid-s13787" xml:space="preserve">Quam ob rem ſicut hæc minu-<lb/>tia eſt radix cubica fractionis propoſitæ, ita quoque illa eritradix cubica eiuſ-<lb/>dem fractionis. </s> <s xml:id="echoid-s13788" xml:space="preserve">Diuiſa ergo radice illa cubica inuenta per denominatorem <lb/>fractionis propoſitæ (quæ diuiſio fit, quia illa radix cubica inuenta eſt fractio, <lb/>ac proinde, vt cognoſcatur val or minutiæ, cuius numerator radix illa inuen-<lb/>ta, ac denominator ipſemet denominator fractionis propoſitæ, diuidendus eſt <lb/>numerator huius minutiæ per eius denominatorem: </s> <s xml:id="echoid-s13789" xml:space="preserve">alio quin ſi illa radix cu-<lb/>bica inuenta foret numerus integer, diuiſio facienda non eſſet) producetur ra-<lb/>dix cubica fractionis propoſitæ: </s> <s xml:id="echoid-s13790" xml:space="preserve">quemadmodum ex diuiſione radicis cubicæ <lb/>numeratoris per radicem cubicam denominatoris procreatur radix cubica fra-<lb/>ctionis propoſitæ: </s> <s xml:id="echoid-s13791" xml:space="preserve">quippe cum minutia nil aliud ſit, niſi numerator per deno-<lb/>minatorem diuiſus.</s> <s xml:id="echoid-s13792" xml:space="preserve"/> </p> <div xml:id="echoid-div824" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-319-01" xlink:href="note-319-01a" xml:space="preserve">10. defiu. <lb/>quinti.</note> <note symbol="b" position="right" xlink:label="note-319-02" xlink:href="note-319-02a" xml:space="preserve">12. octaui.</note> <note symbol="c" position="right" xlink:label="note-319-03" xlink:href="note-319-03a" xml:space="preserve">7. minutia-<lb/>rum ad finem <lb/>lib. 9.</note> </div> <p> <s xml:id="echoid-s13793" xml:space="preserve"><emph style="sc">Qvando</emph> verò denominator in quadratum numeratoris ducitur, erit <lb/>numeri producti radix cubica vnus duorum mediorum proportionalium in-<lb/>ter numeratorem, ac denominatorem, collocandus iuxta numeratorem, vt <lb/>ex demonſtratis propoſ. </s> <s xml:id="echoid-s13794" xml:space="preserve">18. </s> <s xml:id="echoid-s13795" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s13796" xml:space="preserve">manifeſtum eſt. </s> <s xml:id="echoid-s13797" xml:space="preserve">Ergo rurſus erit propor-<lb/>tio numeratoris ad denominatorem triplicata proportionis numeratoris ad il-<lb/>lam radicem cubicam inuentam, quemadmodum & </s> <s xml:id="echoid-s13798" xml:space="preserve">proportionis radicis cu-<lb/>bicæ numeratoris ad radicem cubicam denominatoris, eadem illa proportio <lb/>numeratoris ad denominatorem, tanquam cubi ad cubum, triplicata eſt: </s> <s xml:id="echoid-s13799" xml:space="preserve">Ac <lb/>propterea, vt ſupra, minutia, cuius numerator ipſemet numerator fractio-<lb/>nis propoſitæ, denominator verò radix illa cubica inuenta, æqualis erit mi-<lb/>nutiæ, cuius numerator, radix cubica numeratoris fractionis propoſitæ, & </s> <s xml:id="echoid-s13800" xml:space="preserve">de-<lb/>nominator radix cubica denominatoris eiuſdem fractionis. </s> <s xml:id="echoid-s13801" xml:space="preserve">Quocirca diuiſo <lb/>numeratore per illam inuentam radicem cubicam, prodibit radix cubica fra-<lb/>ctionis propoſitæ.</s> <s xml:id="echoid-s13802" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div826" type="section" level="1" n="281"> <head xml:id="echoid-head307" xml:space="preserve">PROBL. 17. PROPOS. 22.</head> <p> <s xml:id="echoid-s13803" xml:space="preserve">RADICEM quadratam & </s> <s xml:id="echoid-s13804" xml:space="preserve">cubicam in numeris non quadratis, & </s> <s xml:id="echoid-s13805" xml:space="preserve"><lb/>non cubicis per lineas Geometrice inuenire.</s> <s xml:id="echoid-s13806" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13807" xml:space="preserve"><emph style="sc">Sit</emph> datus numerus 10. </s> <s xml:id="echoid-s13808" xml:space="preserve">repræſentans quadratum 10. </s> <s xml:id="echoid-s13809" xml:space="preserve">palmorum, vel pedum, <pb o="290" file="320" n="320" rhead="GEOMETR. PRACT. LIBER SEXT."/> vel aliarum menſurarum. </s> <s xml:id="echoid-s13810" xml:space="preserve">Capiatur linea A, palmorum, vel pedum 10. </s> <s xml:id="echoid-s13811" xml:space="preserve">&</s> <s xml:id="echoid-s13812" xml:space="preserve">c. </s> <s xml:id="echoid-s13813" xml:space="preserve">& </s> <s xml:id="echoid-s13814" xml:space="preserve">li-<lb/>nea B, palmi, vel pedis 1. </s> <s xml:id="echoid-s13815" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Inuenta C, media proportionali inter A, & </s> <s xml:id="echoid-s13816" xml:space="preserve">B. </s> <s xml:id="echoid-s13817" xml:space="preserve">Dico <anchor type="note" xlink:label="note-320-01a" xlink:href="note-320-01"/> C, eſſe radicem quadratam, ſiue latus quadra-<lb/> <anchor type="figure" xlink:label="fig-320-01a" xlink:href="fig-320-01"/> tum, numeri A, 10. </s> <s xml:id="echoid-s13818" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam enim quadra- <anchor type="note" xlink:label="note-320-02a" xlink:href="note-320-02"/> tum rectæ C, æquale eſt rectangulo ſub A, & </s> <s xml:id="echoid-s13819" xml:space="preserve"><lb/>B, comprehenſo: </s> <s xml:id="echoid-s13820" xml:space="preserve">Eſt autem hocrectangulum <lb/>10. </s> <s xml:id="echoid-s13821" xml:space="preserve">quod vnitas B, numerum A, 10. </s> <s xml:id="echoid-s13822" xml:space="preserve">multipli-<lb/>cans procreet 10. </s> <s xml:id="echoid-s13823" xml:space="preserve">perſpicuum eſt, rectam C, <lb/>eſſe latus quadratum 10. </s> <s xml:id="echoid-s13824" xml:space="preserve">palmorum, vel pe-<lb/>dum, &</s> <s xml:id="echoid-s13825" xml:space="preserve">c. </s> <s xml:id="echoid-s13826" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s13827" xml:space="preserve"/> </p> <div xml:id="echoid-div826" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-320-01" xlink:href="note-320-01a" xml:space="preserve">13. ſexti.</note> <figure xlink:label="fig-320-01" xlink:href="fig-320-01a"> <image file="320-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/320-01"/> </figure> <note symbol="b" position="left" xlink:label="note-320-02" xlink:href="note-320-02a" xml:space="preserve">17. ſexti.</note> </div> <p> <s xml:id="echoid-s13828" xml:space="preserve"><emph style="sc">Rvrsvs</emph> <anchor type="note" xlink:href="" symbol="c"/> inuentis inter A, & </s> <s xml:id="echoid-s13829" xml:space="preserve">B, duabus mediis proportionalibus D, <anchor type="note" xlink:label="note-320-03a" xlink:href="note-320-03"/> & </s> <s xml:id="echoid-s13830" xml:space="preserve">E. </s> <s xml:id="echoid-s13831" xml:space="preserve">Dico E, quæ minori B, propinquior eſt, latus eſſe cubicum, ſiue radi-<lb/>cem cubicam, numeri 10. </s> <s xml:id="echoid-s13832" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam enim cubus rectæ E, æqualis eſt parallele- <anchor type="note" xlink:label="note-320-04a" xlink:href="note-320-04"/> pipedo ſub quadrato rectæ B, & </s> <s xml:id="echoid-s13833" xml:space="preserve">ſub A, recta comprehenſo. </s> <s xml:id="echoid-s13834" xml:space="preserve">Eſt autem hoc pa-<lb/>rallelepipedum 10. </s> <s xml:id="echoid-s13835" xml:space="preserve">quod vnitas B, ſe multiplicans faciat quadratum 1. </s> <s xml:id="echoid-s13836" xml:space="preserve">& </s> <s xml:id="echoid-s13837" xml:space="preserve"><lb/>quadratum 1. </s> <s xml:id="echoid-s13838" xml:space="preserve">multiplicans numerum A, 10. </s> <s xml:id="echoid-s13839" xml:space="preserve">gignat 10. </s> <s xml:id="echoid-s13840" xml:space="preserve">liquidò con-<lb/>ſtat, rectam E, eſſe latus cubicum 10. </s> <s xml:id="echoid-s13841" xml:space="preserve">palmorum, vel <lb/>pedum, &</s> <s xml:id="echoid-s13842" xml:space="preserve">c. </s> <s xml:id="echoid-s13843" xml:space="preserve">quod eſt propoſi-<lb/>tum.</s> <s xml:id="echoid-s13844" xml:space="preserve"/> </p> <div xml:id="echoid-div827" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-320-03" xlink:href="note-320-03a" xml:space="preserve">15. hui{us}.</note> <note symbol="d" position="left" xlink:label="note-320-04" xlink:href="note-320-04a" xml:space="preserve">Lemma 18. <lb/>hui{us}.</note> </div> </div> <div xml:id="echoid-div829" type="section" level="1" n="282"> <head xml:id="echoid-head308" xml:space="preserve">FINIS LIBRI SEXTI.</head> <figure> <image file="320-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/320-02"/> </figure> <pb o="291" file="321" n="321"/> <figure> <image file="321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/321-01"/> </figure> </div> <div xml:id="echoid-div830" type="section" level="1" n="283"> <head xml:id="echoid-head309" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ</head> <head xml:id="echoid-head310" xml:space="preserve">LIBER SEPTIMVS.</head> <figure> <image file="321-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/321-02"/> </figure> </div> <div xml:id="echoid-div831" type="section" level="1" n="284"> <head xml:id="echoid-head311" xml:space="preserve">De figuris Iſoperimetris diſputans: cui Appendicis <lb/>loco annectitur breuis de circulo per lineas <lb/>quadrando tractatiuncula.</head> <p style="it"> <s xml:id="echoid-s13845" xml:space="preserve">NOBILIS ac præſtans de Iſoperimetris figuris ſemper apud <lb/>omnes habita est diſputatio@ in qua videlicet inquiritur, vtra <lb/>duarum ſit altera maior, & </s> <s xml:id="echoid-s13846" xml:space="preserve">capacior, & </s> <s xml:id="echoid-s13847" xml:space="preserve">quæ ſit omnium <lb/>maxima, & </s> <s xml:id="echoid-s13848" xml:space="preserve">capaciſſima. </s> <s xml:id="echoid-s13849" xml:space="preserve">Non pauci enim rerum Geometri-<lb/>carum ignari hac in re hallucinati ſunt, putantes figur {as} Iſo-<lb/>perimetr{as}, quæ nimirum æquales ambit{us} continent, eſſe <lb/>inter ſe æquales, ſiue æquè capaces. </s> <s xml:id="echoid-s13850" xml:space="preserve">Immoè contrario, quod mirum est, nonnul-<lb/>li, qui ſe Geometr {as} appellant, adduci vix poſſunt, vt credant, dari poſſe du{as} fi-<lb/>gur{as} Iſoperimetr {as} inter ſe omnino æquales, quod tamen fieri poſſe, clariſſimè <lb/>propoſ. </s> <s xml:id="echoid-s13851" xml:space="preserve">20. </s> <s xml:id="echoid-s13852" xml:space="preserve">21. </s> <s xml:id="echoid-s13853" xml:space="preserve">22. </s> <s xml:id="echoid-s13854" xml:space="preserve">hui{us} demonſtrabim{us}. </s> <s xml:id="echoid-s13855" xml:space="preserve">Quamuis autem de @ſoperimetris fi-<lb/>guris tractionem benè longam, & </s> <s xml:id="echoid-s13856" xml:space="preserve">copioſam in commentariis noſtris in ſphæ-<lb/>ram inſtituerim{us}, exemplum in hoc ſecuti Theonis Alexandrini, qui idem ar-<lb/>gumentum in commentariis in Almageſtum Ptolomei perſecut{us} eſt: </s> <s xml:id="echoid-s13857" xml:space="preserve">tamen <lb/>quia id in alieno fortaſſis loco factum eſſe ſuſpicari quis poſſet; </s> <s xml:id="echoid-s13858" xml:space="preserve">transferem{us} eam <lb/>tractationem ex @oſtris commentariis in ſphæram in hanc noſtram Geometriam <lb/>practicam, tanquam in magis proprium locum, additis trib{us}, aut quatuor pro-<lb/>poſitionib{us}, quæ in illa tractatione deſiderantur, & </s> <s xml:id="echoid-s13859" xml:space="preserve">tamen maximè ad hanc <lb/>materiam ſpectare videntur.</s> <s xml:id="echoid-s13860" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div832" type="section" level="1" n="285"> <head xml:id="echoid-head312" xml:space="preserve">DEFINITIONES.</head> <note position="right" xml:space="preserve">Definition{es} <lb/>ad tractationẽ <lb/>Iſoperim{et}ra-<lb/>rũ figurarum <lb/>pertinent{es}.</note> </div> <div xml:id="echoid-div833" type="section" level="1" n="286"> <head xml:id="echoid-head313" xml:space="preserve">I.</head> <p> <s xml:id="echoid-s13861" xml:space="preserve">ISOPERIMETRÆ figuræ ſunt, quæ æquales ambitus continent.</s> <s xml:id="echoid-s13862" xml:space="preserve"/> </p> <pb o="292" file="322" n="322" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div834" type="section" level="1" n="287"> <head xml:id="echoid-head314" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s13863" xml:space="preserve">REGVLARIS figura dicitur ea, quæ & </s> <s xml:id="echoid-s13864" xml:space="preserve">æquilatera, & </s> <s xml:id="echoid-s13865" xml:space="preserve">æquiangu-<lb/>la eſt.</s> <s xml:id="echoid-s13866" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div835" type="section" level="1" n="288"> <head xml:id="echoid-head315" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s13867" xml:space="preserve">CENTRVM figuræ regularis dicitur punctum illud, quod centrum <lb/>eſt circuli figuræ inſcripti, vel circumſcripti.</s> <s xml:id="echoid-s13868" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div836" type="section" level="1" n="289"> <head xml:id="echoid-head316" xml:space="preserve">IIII.</head> <p> <s xml:id="echoid-s13869" xml:space="preserve">AREA cuiuslibet figuræ dicitur capacitas, ſpatium ſiue ſuperficies in-<lb/>tra latera ipſius comprehenſa.</s> <s xml:id="echoid-s13870" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div837" type="section" level="1" n="290"> <head xml:id="echoid-head317" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s13871" xml:space="preserve">OMNE ſolidum rectangulum (cuius nimirum baſes æquidiſtantes <lb/>ſunt, & </s> <s xml:id="echoid-s13872" xml:space="preserve">æquales, lateraque ad baſes recta, quale eſt Parallelepipedum) <lb/>contineri dicitur ſub altera baſium, ac perpendiculari ab illa baſi ad <lb/>alteram protracta.</s> <s xml:id="echoid-s13873" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13874" xml:space="preserve"><emph style="sc">Qvia</emph> nimirum alterutra baſium indicat longitudinem ac latitudinem fi-<lb/>guræ, perpendicularis verò altitudinem, ſiue profunditatem eiuſdem de-<lb/>monſtrat.</s> <s xml:id="echoid-s13875" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div838" type="section" level="1" n="291"> <head xml:id="echoid-head318" xml:space="preserve">THEOR. 1. PROPOS. 1.</head> <p> <s xml:id="echoid-s13876" xml:space="preserve">AREA cuiuslibet trianguli æqualis eſt rectangulo comprehenſo ſub <lb/> <anchor type="note" xlink:label="note-322-01a" xlink:href="note-322-01"/> perpendiculari â vertice ad baſim protracta, & </s> <s xml:id="echoid-s13877" xml:space="preserve">dimidia parte baſis. <lb/></s> <s xml:id="echoid-s13878" xml:space="preserve">Item rectangulo comprehenſo ſub ſemiſſe perpendicularis, & </s> <s xml:id="echoid-s13879" xml:space="preserve">tota <lb/>baſe. </s> <s xml:id="echoid-s13880" xml:space="preserve">Vel denique ſemiſsi rectanguli ſub tota perpendiculari, & </s> <s xml:id="echoid-s13881" xml:space="preserve">tota <lb/>baſe comprehenſi.</s> <s xml:id="echoid-s13882" xml:space="preserve"/> </p> <div xml:id="echoid-div838" type="float" level="2" n="1"> <note position="left" xlink:label="note-322-01" xlink:href="note-322-01a" xml:space="preserve">Triangulum <lb/>quodcunque <lb/>cuirectangulo <lb/>aquale ſit.</note> </div> <p> <s xml:id="echoid-s13883" xml:space="preserve"><emph style="sc">Sit</emph> triangulum A B C, ex cuius vertice A, ad baſim BC, ducatur perpendi-<lb/>cularis A D, diuidatque primò baſim BC, bifa-<lb/> <anchor type="figure" xlink:label="fig-322-01a" xlink:href="fig-322-01"/> riam, vt in prima figura. </s> <s xml:id="echoid-s13884" xml:space="preserve">Per A, ducatur E A F, <lb/>in vtramque partem æquidiſtans rectæ B C, <lb/>compleatur que rectangulum B E F C, <anchor type="note" xlink:href="" symbol="a"/> quod <anchor type="note" xlink:label="note-322-02a" xlink:href="note-322-02"/> erit duplum trianguli A B C; </s> <s xml:id="echoid-s13885" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Item duplum <anchor type="note" xlink:label="note-322-03a" xlink:href="note-322-03"/> rectanguli ADBE. </s> <s xml:id="echoid-s13886" xml:space="preserve">Quare rectangulum ADBE, <lb/>quod nimirum continetur ſub perpendiculari <lb/>AD, & </s> <s xml:id="echoid-s13887" xml:space="preserve">dimidio baſis BD, æquale eſt triangulo ABC, diuidat ſecundo perpendi-<lb/>cularis AD, baſim BC, non bifariam, vel etiã cadat in baſim CB, protra ctam, vt in <lb/>2. </s> <s xml:id="echoid-s13888" xml:space="preserve">& </s> <s xml:id="echoid-s13889" xml:space="preserve">3. </s> <s xml:id="echoid-s13890" xml:space="preserve">figura; </s> <s xml:id="echoid-s13891" xml:space="preserve">Et per A, ducatur rurſus AF, in vtramq; </s> <s xml:id="echoid-s13892" xml:space="preserve">partẽ æquidiſtãs rectę BC, <lb/>compleaturq; </s> <s xml:id="echoid-s13893" xml:space="preserve">rectangulũ ADCF. </s> <s xml:id="echoid-s13894" xml:space="preserve">Diuiſa deinde baſe BC, bifariã in G, ducantur <pb o="293" file="323" n="323" rhead="LIBER SEPTIMVS."/> rectæ B E, G H, ipſi A D, æquidiſtantes, <anchor type="note" xlink:href="" symbol="a"/> eritque G H, ęqualis perpendiculari <anchor type="note" xlink:label="note-323-01a" xlink:href="note-323-01"/> A D. </s> <s xml:id="echoid-s13895" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniamigitur rectangulum BCFE, duplum eſt trianguli ABC; </s> <s xml:id="echoid-s13896" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Item <anchor type="note" xlink:label="note-323-02a" xlink:href="note-323-02"/> duplum rectanguli BEHG: </s> <s xml:id="echoid-s13897" xml:space="preserve">erit rectangulum BEHG, quod continetur ſub per-<lb/> <anchor type="note" xlink:label="note-323-03a" xlink:href="note-323-03"/> pendiculari GH, vel AD, & </s> <s xml:id="echoid-s13898" xml:space="preserve">dimidio baſis BG, æquale triangulo ABC.</s> <s xml:id="echoid-s13899" xml:space="preserve"/> </p> <div xml:id="echoid-div839" 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/xxxxxxxx/figures/322-01"/> </figure> <note symbol="a" position="left" xlink:label="note-322-02" xlink:href="note-322-02a" xml:space="preserve">41. primi.</note> <note symbol="b" position="left" xlink:label="note-322-03" xlink:href="note-322-03a" xml:space="preserve">36. primi.</note> <note symbol="a" position="right" xlink:label="note-323-01" xlink:href="note-323-01a" xml:space="preserve">34. primi.</note> <note symbol="b" position="right" xlink:label="note-323-02" xlink:href="note-323-02a" xml:space="preserve">41. primi.</note> <note symbol="c" position="right" xlink:label="note-323-03" xlink:href="note-323-03a" xml:space="preserve">36. primi.</note> </div> <p> <s xml:id="echoid-s13900" xml:space="preserve"><emph style="sc">Secetvr</emph> iam perpendicularis AD, vel G H, bifariam in I, agaturque per I, <lb/>ipſi BC, parallela KL. </s> <s xml:id="echoid-s13901" xml:space="preserve">Dico triangulum idem ABC, æquale quoque eſſe rectã-<lb/>gulo BCLK, in 1. </s> <s xml:id="echoid-s13902" xml:space="preserve">& </s> <s xml:id="echoid-s13903" xml:space="preserve">2. </s> <s xml:id="echoid-s13904" xml:space="preserve">figura, Item rectangulo BCLM, in 3. </s> <s xml:id="echoid-s13905" xml:space="preserve">figura, comprehen-<lb/>ſo nimirum ſub ID, vel IG, ſemiſſe perpendicularis AD, vel HG. </s> <s xml:id="echoid-s13906" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam <anchor type="note" xlink:label="note-323-04a" xlink:href="note-323-04"/> enim triangulum ABC, dimidium eſt rectanguli E C, eiuſdemque dimidium et-<lb/>iam eſt rectangulum BL; </s> <s xml:id="echoid-s13907" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> quod rectangula BL, LE, ſuper æquales baſes æqua- <anchor type="note" xlink:label="note-323-05a" xlink:href="note-323-05"/> lia ſint: </s> <s xml:id="echoid-s13908" xml:space="preserve">æqualia inter ſe erunt triangulum A B C, & </s> <s xml:id="echoid-s13909" xml:space="preserve">rectangulum B L. </s> <s xml:id="echoid-s13910" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Et quia <anchor type="note" xlink:label="note-323-06a" xlink:href="note-323-06"/> rectangulum B F, contentum ſub perpendiculari A D, vel B E, & </s> <s xml:id="echoid-s13911" xml:space="preserve">baſe trianguli <lb/>BC, duplum eſt trianguli ABC; </s> <s xml:id="echoid-s13912" xml:space="preserve">erit triangulum ſemiſsiillius rectanguli ęquale. <lb/></s> <s xml:id="echoid-s13913" xml:space="preserve">Area igitur cuiuslibet trianguli æqualis eſt, &</s> <s xml:id="echoid-s13914" xml:space="preserve">c. </s> <s xml:id="echoid-s13915" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s13916" xml:space="preserve"/> </p> <div xml:id="echoid-div840" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-323-04" xlink:href="note-323-04a" xml:space="preserve">41. primi.</note> <note symbol="e" position="right" xlink:label="note-323-05" xlink:href="note-323-05a" xml:space="preserve">36. primi.</note> <note symbol="f" position="right" xlink:label="note-323-06" xlink:href="note-323-06a" xml:space="preserve">41. primi.</note> </div> </div> <div xml:id="echoid-div842" type="section" level="1" n="292"> <head xml:id="echoid-head319" xml:space="preserve">PROBL. 2. PROPOS. 2.</head> <note position="right" xml:space="preserve">Regularis fi-<lb/>gura quæcun-<lb/>que cuirectã-<lb/>gulo @qualis <lb/>ſit.</note> <p> <s xml:id="echoid-s13917" xml:space="preserve">AREA cuiuslibet figuræ regularis æqualis eſt rectangulo contento ſub <lb/>perpendiculari à centro figuræ ad vnum latus ducta, & </s> <s xml:id="echoid-s13918" xml:space="preserve">ſub dimidia-<lb/>to ambitu eiuſdem figuræ.</s> <s xml:id="echoid-s13919" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13920" xml:space="preserve"><emph style="sc">Sit</emph> figura regularis quæcunque ABCDEF, & </s> <s xml:id="echoid-s13921" xml:space="preserve">centrum eius punctum G, à <lb/>quo ducatur GH, perpendicularis ad vnum latus, nempe ad AB: </s> <s xml:id="echoid-s13922" xml:space="preserve">Sit quoq; </s> <s xml:id="echoid-s13923" xml:space="preserve">re-<lb/>ctangulum I K L M, contentum ſub I K, quæ æqualis ſit perpendiculari G H, & </s> <s xml:id="echoid-s13924" xml:space="preserve"><lb/>ſub KL, recta, quæ æqualis ponatur dimidiæ parti ambitus figuræ ABCDEF. </s> <s xml:id="echoid-s13925" xml:space="preserve">Di-<lb/> <anchor type="figure" xlink:label="fig-323-01a" xlink:href="fig-323-01"/> co huic rectangulo æqualem eſſe figuram regularem ABCDEF. </s> <s xml:id="echoid-s13926" xml:space="preserve">Ducantur enim <lb/>ex G, ad ſingulos angulos lineæ rectæ, vt tota figura in triangula reſoluatur, quæ <lb/>omnia æqualia inter ſe erunt, vt in corollario propoſ. </s> <s xml:id="echoid-s13927" xml:space="preserve">8. </s> <s xml:id="echoid-s13928" xml:space="preserve">lib. </s> <s xml:id="echoid-s13929" xml:space="preserve">1. </s> <s xml:id="echoid-s13930" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s13931" xml:space="preserve">demonſtra-<lb/>tum eſt à nobis: </s> <s xml:id="echoid-s13932" xml:space="preserve">propterea quòd omnia latera triangulorum à puncto G, ex-<lb/>euntia ſint inter ſe æqualia, habeantq; </s> <s xml:id="echoid-s13933" xml:space="preserve">baſes æquales, nempè latera figuræ regu-<lb/>laris. </s> <s xml:id="echoid-s13934" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Hinc enim effi citur, omnes angulos ad G, æquales eſſe, ac proinde, ex di- <anchor type="note" xlink:label="note-323-08a" xlink:href="note-323-08"/> cto corollario, triangula ipſa inter ſe quo que eſſe æqualia. </s> <s xml:id="echoid-s13935" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Quoniam igitur re- <anchor type="note" xlink:label="note-323-09a" xlink:href="note-323-09"/> ctangulum contentum ſub GH, perpendiculari, & </s> <s xml:id="echoid-s13936" xml:space="preserve">medietate baſis AB, æquale <lb/>eſt triangulo ABG, ſi ſumantur tot huiuſmodi rectangula, in quot triangula di-<lb/>uiſa eſt figura regularis, erunt omnia ſimul figuræ ABCDEF, ęqualia; </s> <s xml:id="echoid-s13937" xml:space="preserve">propterea <pb o="294" file="324" n="324" rhead="GEOMETR. PRACT."/> quod omnia triangula oſtenſa ſint æqualia triangulo ABG. </s> <s xml:id="echoid-s13938" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cum igitur eadem <anchor type="note" xlink:label="note-324-01a" xlink:href="note-324-01"/> ſimul æqualia ſint rectangulo IKLM; </s> <s xml:id="echoid-s13939" xml:space="preserve">propterea quòd K L, æqualis ponitur di-<lb/>midio ambitus ABCDEF, hoc eſt omnibus medietatibus baſium ſimul; </s> <s xml:id="echoid-s13940" xml:space="preserve">& </s> <s xml:id="echoid-s13941" xml:space="preserve">recta <lb/>IK, perpendiculari G H; </s> <s xml:id="echoid-s13942" xml:space="preserve">erit figura regularis A B C D E F, æqualis rectangulo <lb/>IKLM. </s> <s xml:id="echoid-s13943" xml:space="preserve">Area igitur cuiuslibet figuræ regularis æqualis eſt, &</s> <s xml:id="echoid-s13944" xml:space="preserve">c. </s> <s xml:id="echoid-s13945" xml:space="preserve">quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s13946" xml:space="preserve"/> </p> <div xml:id="echoid-div842" type="float" level="2" n="1"> <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/xxxxxxxx/figures/323-01"/> </figure> <note symbol="g" position="right" xlink:label="note-323-08" xlink:href="note-323-08a" xml:space="preserve">8. primi.</note> <note symbol="h" position="right" xlink:label="note-323-09" xlink:href="note-323-09a" xml:space="preserve">1. hui{us}.</note> <note symbol="a" position="left" xlink:label="note-324-01" xlink:href="note-324-01a" xml:space="preserve">1. ſecundi.</note> </div> </div> <div xml:id="echoid-div844" type="section" level="1" n="293"> <head xml:id="echoid-head320" xml:space="preserve">THEOR. 3. PROPOS. 3.</head> <p> <s xml:id="echoid-s13947" xml:space="preserve">AREA cuiuslibet figuræ regularis æqualis eſt triangulo rectangulo, <lb/> <anchor type="note" xlink:label="note-324-02a" xlink:href="note-324-02"/> cuius vnum latus circa angulum rectum æquale eſt perpendiculari à <lb/>centro figuræ ad vnum latus ductæ, alterum verò æquale ambitui e-<lb/>iuſdem figuræ.</s> <s xml:id="echoid-s13948" xml:space="preserve"/> </p> <div xml:id="echoid-div844" type="float" level="2" n="1"> <note position="left" xlink:label="note-324-02" xlink:href="note-324-02a" xml:space="preserve">Regularis fi-<lb/>gura quæcũ-<lb/>que cui trian-<lb/>gulo rectan-<lb/>gulo æqualis <lb/>ſit.</note> </div> <p> <s xml:id="echoid-s13949" xml:space="preserve"><emph style="sc">Sit</emph> rurſus figura regularis A B C, cuius centrum D, à quo perpendicularis <lb/>ad latus AB, ducta ſit D E; </s> <s xml:id="echoid-s13950" xml:space="preserve">triangulum verò rectangulum DEF, habens angulũ <lb/> <anchor type="figure" xlink:label="fig-324-01a" xlink:href="fig-324-01"/> E, rectum, & </s> <s xml:id="echoid-s13951" xml:space="preserve">latus DE, æquale perpendiculari DE, latus autem EF, æquale am-<lb/>bitui figuræ ABC. </s> <s xml:id="echoid-s13952" xml:space="preserve">Dico triangulum DEF, figuræ ABC, æquale eſſe. </s> <s xml:id="echoid-s13953" xml:space="preserve">Complea-<lb/>tur enim rectangulum DEFG; </s> <s xml:id="echoid-s13954" xml:space="preserve">& </s> <s xml:id="echoid-s13955" xml:space="preserve">diuiſa E F, bifariam in puncto H, ducatur HI, <lb/>æquidiſtans rectæ D E. </s> <s xml:id="echoid-s13956" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Erit igitur rectangulum D E H I, contentum ſub D E, <anchor type="note" xlink:label="note-324-03a" xlink:href="note-324-03"/> perpendiculari, & </s> <s xml:id="echoid-s13957" xml:space="preserve">ſub EH, dimidio ambitus figuræ, æquale figuræ ABC: </s> <s xml:id="echoid-s13958" xml:space="preserve">Atre-<lb/>ctangulo DEHI, æquale eſt triangulum D E F. </s> <s xml:id="echoid-s13959" xml:space="preserve">Nam rectangulum D E H I, eſt <lb/> <anchor type="note" xlink:label="note-324-04a" xlink:href="note-324-04"/> dimidium rectanguli DEFG; </s> <s xml:id="echoid-s13960" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> propterea quod ęqualia ſunt rectangula DEHI, <anchor type="note" xlink:label="note-324-05a" xlink:href="note-324-05"/> IHFG; </s> <s xml:id="echoid-s13961" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Triangulum quoque DEF, dimidium eſt eiuſdem rectanguli DEFG.</s> <s xml:id="echoid-s13962" xml:space="preserve"> Igitur & </s> <s xml:id="echoid-s13963" xml:space="preserve">triangulum DEF, æquale erit figuræ A B C. </s> <s xml:id="echoid-s13964" xml:space="preserve">Area ergo cuiuslibet figu-<lb/>ræ regularis æqualis eſt triangulo rectangulo, &</s> <s xml:id="echoid-s13965" xml:space="preserve">c. </s> <s xml:id="echoid-s13966" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s13967" xml:space="preserve"/> </p> <div xml:id="echoid-div845" type="float" level="2" n="2"> <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/xxxxxxxx/figures/324-01"/> </figure> <note symbol="b" position="left" xlink:label="note-324-03" xlink:href="note-324-03a" xml:space="preserve">2. hui{us}.</note> <note symbol="c" position="left" xlink:label="note-324-04" xlink:href="note-324-04a" xml:space="preserve">36 primi.</note> <note symbol="d" position="left" xlink:label="note-324-05" xlink:href="note-324-05a" xml:space="preserve">41. primi.</note> </div> </div> <div xml:id="echoid-div847" type="section" level="1" n="294"> <head xml:id="echoid-head321" xml:space="preserve">THEOR. 4. PROPOS. 4.</head> <p> <s xml:id="echoid-s13968" xml:space="preserve">AREA cuiuslibet circuli æqualis eſt rectangulo comprehenſo ſub ſe-<lb/> <anchor type="note" xlink:label="note-324-06a" xlink:href="note-324-06"/> midiametro, & </s> <s xml:id="echoid-s13969" xml:space="preserve">dimidiata circumferentia circuli.</s> <s xml:id="echoid-s13970" xml:space="preserve"/> </p> <div xml:id="echoid-div847" type="float" level="2" n="1"> <note position="left" xlink:label="note-324-06" xlink:href="note-324-06a" xml:space="preserve">Circul{us} qui-<lb/>cunque cui <lb/>rectangulo æ-<lb/>qualis ſit.</note> </div> <p> <s xml:id="echoid-s13971" xml:space="preserve"><emph style="sc">Esto</emph> circulus ABC, cuius ſemidiameter D B: </s> <s xml:id="echoid-s13972" xml:space="preserve">Rectangulum autem DBEF, <lb/>comprehenſum ſub D B, ſemidiametro circuli, & </s> <s xml:id="echoid-s13973" xml:space="preserve">B E, recta, quę æqualis ſit di-<lb/>midiatæ circumferentiæ circuli. </s> <s xml:id="echoid-s13974" xml:space="preserve">Dico aream circuli ABC, æqualem eſſe rectan-<lb/>gulo DBEF. </s> <s xml:id="echoid-s13975" xml:space="preserve">Producatur enim BE, in continuum, ponatur que EG, æqualis i-<lb/>pſi BE, vt ſit BG, recta æqualis toti circumferentiæ circuli. </s> <s xml:id="echoid-s13976" xml:space="preserve">Coniungantur deniq;</s> <s xml:id="echoid-s13977" xml:space="preserve"> <pb o="295" file="325" n="325" rhead="LIBER SEPTIMVS."/> puncta D, G, recta D G. </s> <s xml:id="echoid-s13978" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam igitur circulus A B C, æqualis eſt triangulo <anchor type="note" xlink:label="note-325-01a" xlink:href="note-325-01"/> DBG: </s> <s xml:id="echoid-s13979" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Eſt autem triangulum DBG, rectangulo D B E F, æquale; </s> <s xml:id="echoid-s13980" xml:space="preserve">quod baſis <anchor type="figure" xlink:label="fig-325-01a" xlink:href="fig-325-01"/> <anchor type="note" xlink:label="note-325-02a" xlink:href="note-325-02"/> trianguli dupla ſit baſis rectanguli; </s> <s xml:id="echoid-s13981" xml:space="preserve">(Id quod etiam ex demonſtratione antece-<lb/>dentis propoſ. </s> <s xml:id="echoid-s13982" xml:space="preserve">liquet, vbi oſtendimus, triangulum DEF, æquale eſſe rectangu-<lb/>lo DEHI:) </s> <s xml:id="echoid-s13983" xml:space="preserve">erit quoque circulus ABC, rectangulo DBEF, æqualis. </s> <s xml:id="echoid-s13984" xml:space="preserve">Area ergo <lb/>cuiuslibet circuli æqualis eſt rectangulo, &</s> <s xml:id="echoid-s13985" xml:space="preserve">c. </s> <s xml:id="echoid-s13986" xml:space="preserve">quod oſtendendum erat.</s> <s xml:id="echoid-s13987" xml:space="preserve"/> </p> <div xml:id="echoid-div848" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-325-01" xlink:href="note-325-01a" xml:space="preserve">1. de Dimẽs. <lb/>circuli Ar-<lb/>chim.</note> <figure xlink:label="fig-325-01" xlink:href="fig-325-01a"> <image file="325-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/325-01"/> </figure> <note symbol="b" position="right" xlink:label="note-325-02" xlink:href="note-325-02a" xml:space="preserve">ſchol. 41. <lb/>primi.</note> </div> </div> <div xml:id="echoid-div850" type="section" level="1" n="295"> <head xml:id="echoid-head322" xml:space="preserve">THEOR. 5. PROPOS. 5.</head> <note position="right" xml:space="preserve">Propriet{as} <lb/>quædam tri-<lb/>anguli rectan-<lb/>guli.</note> <p> <s xml:id="echoid-s13988" xml:space="preserve">IN omnitriangulo rectangulo, ſi ab vno acutorum angulorum vtcun-<lb/>que ad latus oppoſitum linea recta ducatur, erit maior proportio hu-<lb/>ius lateris ad eius ſegmentum, quod prope angulum rectum exiſtit, <lb/>quam anguli acuti prædicti, ad eius partem dicto ſegmento lateris op-<lb/>poſitum.</s> <s xml:id="echoid-s13989" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s13990" xml:space="preserve"><emph style="sc">Sit</emph> triangulum rectangulum ABC, cuius angulus C, ſitrectus; </s> <s xml:id="echoid-s13991" xml:space="preserve">ducaturque <lb/>ab acuto angulo A, ad latus oppoſitum BC, recta AD, vt-<lb/> <anchor type="figure" xlink:label="fig-325-02a" xlink:href="fig-325-02"/> cunque; </s> <s xml:id="echoid-s13992" xml:space="preserve">Dico maiorem eſſe proportionem rectæ B C, ad <lb/>rectam CD, quam anguli BAC, ad angulum CAD. </s> <s xml:id="echoid-s13993" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quo- <anchor type="note" xlink:label="note-325-04a" xlink:href="note-325-04"/> niam enim recta AD, maior quidem eſt, quam A C; </s> <s xml:id="echoid-s13994" xml:space="preserve">minor <lb/>verò, quam AB; </s> <s xml:id="echoid-s13995" xml:space="preserve">ſi centro A, interuallo autem A D, circu-<lb/>lus deſcribatur, ſecabit is rectam A C, protractam infra <lb/>punctum C, vtin E, at verò rectam AB, ſupra punctum B, <lb/>vtin F. </s> <s xml:id="echoid-s13996" xml:space="preserve">Et quia maior eſt proportio trianguli BAD, ad ſectorem FAD, quã tri-<lb/>anguli DAC, ad ſectorem DAE, (propterea quod ibi eſt proportio maioris in-<lb/>æqualitatis, hic autem minoris inæqualitatis) <anchor type="note" xlink:href="" symbol="d"/> erit quoque permutando maior <anchor type="note" xlink:label="note-325-05a" xlink:href="note-325-05"/> proportio trianguli BAD, ad triangulum D A C, quam ſectoris FAD, ad ſectorẽ <lb/>D A E. </s> <s xml:id="echoid-s13997" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Componendo igitur maior quoque erit proportio trianguli B A C, ad <anchor type="note" xlink:label="note-325-06a" xlink:href="note-325-06"/> triangulum D A C, hoc eſt, rectæ BC, ad rectam CD, <anchor type="note" xlink:href="" symbol="f"/> (habent enim triangula B- <anchor type="note" xlink:label="note-325-07a" xlink:href="note-325-07"/> AC, DAC, eandem proportionem, quam baſes BC, CD.) </s> <s xml:id="echoid-s13998" xml:space="preserve">quam ſectoris F A E, <lb/>ad ſectorẽ DAE, hoc eſt, quam anguli BAC, ad angulum CAD; </s> <s xml:id="echoid-s13999" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> quod eandem <anchor type="note" xlink:label="note-325-08a" xlink:href="note-325-08"/> habeant proportionem ſectores, quam anguli. </s> <s xml:id="echoid-s14000" xml:space="preserve">Quo circa in omnitriangulo re-<lb/>ctangulo, &</s> <s xml:id="echoid-s14001" xml:space="preserve">c. </s> <s xml:id="echoid-s14002" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s14003" xml:space="preserve"/> </p> <div xml:id="echoid-div850" type="float" level="2" n="1"> <figure xlink:label="fig-325-02" xlink:href="fig-325-02a"> <image file="325-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/325-02"/> </figure> <note symbol="c" position="right" xlink:label="note-325-04" xlink:href="note-325-04a" xml:space="preserve">19. primi.</note> <note symbol="d" position="right" xlink:label="note-325-05" xlink:href="note-325-05a" xml:space="preserve">27. quinti.</note> <note symbol="e" position="right" xlink:label="note-325-06" xlink:href="note-325-06a" xml:space="preserve">28. quinti.</note> <note symbol="f" position="right" xlink:label="note-325-07" xlink:href="note-325-07a" xml:space="preserve">1. ſexti.</note> <note symbol="g" position="right" xlink:label="note-325-08" xlink:href="note-325-08a" xml:space="preserve">coroll. 33. <lb/>ſexti.</note> </div> <p> <s xml:id="echoid-s14004" xml:space="preserve"><emph style="sc">Hæc</emph> propoſitio vera quoque eſt in triangulo non rectangulo, dummodo <lb/>angulus C, maior ſit angulo A D C, vt patet; </s> <s xml:id="echoid-s14005" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> quod tunc etiam A D, maior ſit, <anchor type="note" xlink:label="note-325-09a" xlink:href="note-325-09"/> quam AC, minor vero, quam AB, &</s> <s xml:id="echoid-s14006" xml:space="preserve">c.</s> <s xml:id="echoid-s14007" xml:space="preserve"/> </p> <div xml:id="echoid-div851" type="float" level="2" n="2"> <note symbol="h" position="right" xlink:label="note-325-09" xlink:href="note-325-09a" xml:space="preserve">19. primi.</note> </div> <pb o="296" file="326" n="326" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div853" type="section" level="1" n="296"> <head xml:id="echoid-head323" xml:space="preserve">THEOR. 6. PROPOS. 6.</head> <p> <s xml:id="echoid-s14008" xml:space="preserve">ISOPERIMETRARVM figurarum regularium maior eſt illa, quæ <lb/> <anchor type="note" xlink:label="note-326-01a" xlink:href="note-326-01"/> plures continet angulos, pluraue latera.</s> <s xml:id="echoid-s14009" xml:space="preserve"/> </p> <div xml:id="echoid-div853" type="float" level="2" n="1"> <note position="left" xlink:label="note-326-01" xlink:href="note-326-01a" xml:space="preserve">Inter figur{as} <lb/>Iſoperimetr{as}, <lb/>quæ plur{es} an-<lb/>gulos, ſeu late-<lb/>ra continet, il-<lb/>la maior eſt.</note> </div> <p> <s xml:id="echoid-s14010" xml:space="preserve"><emph style="sc">Sint</emph> duæ figuræ regulares iſoperimetræ ABC, DEF, habeatque plura late-<lb/>ra, ſiue angulos figura ABC, quam DEF. </s> <s xml:id="echoid-s14011" xml:space="preserve">Dico ABC, maiorem eſſe, quam DEF. <lb/></s> <s xml:id="echoid-s14012" xml:space="preserve"> <anchor type="figure" xlink:label="fig-326-01a" xlink:href="fig-326-01"/> Deſcribatur enim circa figuras circuli, à quorum centris G, H, ducantur ad BC, <lb/>EF, perpendiculares GI, HK, <anchor type="note" xlink:href="" symbol="a"/> quæ diuident rectas BC, EF, bifariam. </s> <s xml:id="echoid-s14013" xml:space="preserve">Quoniam <anchor type="note" xlink:label="note-326-02a" xlink:href="note-326-02"/> igitur figura ABC, plura habet latera, quam DEF, ſibi iſoperimetra, efficitur, vt <lb/>latus BC, ſæpius repetitum metiatur ambitum figuræ ABC, quam latus EF, am-<lb/>bitum figuræ DEF. </s> <s xml:id="echoid-s14014" xml:space="preserve">Quare latus B C, minus erit latere EF, ideo que BI, medietas <lb/>lateris B C, minor, quàm EK, medietas lateris EF. </s> <s xml:id="echoid-s14015" xml:space="preserve">Ponatur KL, ęqualis ipſi BI, & </s> <s xml:id="echoid-s14016" xml:space="preserve"><lb/>ducantur rectæ LH, HE, HF, GB, GC. </s> <s xml:id="echoid-s14017" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et quia omnes arcus circuli D E F, ſunt <anchor type="note" xlink:label="note-326-03a" xlink:href="note-326-03"/> æquales, quòd & </s> <s xml:id="echoid-s14018" xml:space="preserve">rectæ ſubtenſę æquales ponantur; </s> <s xml:id="echoid-s14019" xml:space="preserve">erit recta E F, ita ſubmul-<lb/>tiplex ambitus figuræ D E F, vt arcus E F, ſubmultiplex eſt circumferen-<lb/>tiæ circuli D E F: </s> <s xml:id="echoid-s14020" xml:space="preserve">Eademque ratione ita multiplex ambitus figuræ A B C, <lb/>rectæ B C, ſicut multiplex eſt circumferentia A B C, arcus B C. </s> <s xml:id="echoid-s14021" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Vt autem <anchor type="note" xlink:label="note-326-04a" xlink:href="note-326-04"/> arcus EF, ad circumferentiam circuli DEF, ita eſt angulus EHF, ad quatuor re-<lb/>ctos; </s> <s xml:id="echoid-s14022" xml:space="preserve">Igitur erit quoque vt recta EF, ad ambitum figuræ DEF, hoc eſt, ad ambi-<lb/>tum figuræ ABC, illi æqualem, ita angulus EHF, ad quatuor rectos: </s> <s xml:id="echoid-s14023" xml:space="preserve">Vt autem <lb/>ambitus figuræ ABC, ad rectam BC, ita eſt circumferentia circuli ABC, ad arcum <lb/>BC, <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, ita quatuor rectiad angulum B G C. </s> <s xml:id="echoid-s14024" xml:space="preserve">Ex æquo igitur, vt recta E F, <anchor type="note" xlink:label="note-326-05a" xlink:href="note-326-05"/> ad rectam BC, <anchor type="note" xlink:href="" symbol="e"/> hoc eſt, vt recta EK, ad rectam BI, hoc eſt, ad rectam KL, ita an- gulus EHF, ad angulum BGC, <anchor type="note" xlink:href="" symbol="f"/> hoc eſt, ita angulus EHK, ad angulum B G I. </s> <s xml:id="echoid-s14025" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> <anchor type="note" xlink:label="note-326-06a" xlink:href="note-326-06"/> Eſt autem maior proportio rectæ EK, ad rectam KL, quam anguli EHK, ad an-<lb/> <anchor type="note" xlink:label="note-326-07a" xlink:href="note-326-07"/> gulum KHL. </s> <s xml:id="echoid-s14026" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Quare maior erit proportio quoque anguli EHK, ad angulum <anchor type="note" xlink:label="note-326-08a" xlink:href="note-326-08"/> BGI, quam eiuſdem anguli EHK, ad angulum KHL; </s> <s xml:id="echoid-s14027" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> ideoq; </s> <s xml:id="echoid-s14028" xml:space="preserve">maior erit angu- <anchor type="note" xlink:label="note-326-09a" xlink:href="note-326-09"/> lus KHL, quam angulus B G I. </s> <s xml:id="echoid-s14029" xml:space="preserve">Cumigitur anguli H K L, G I B, ſint ęquales, vt <lb/> <anchor type="note" xlink:label="note-326-10a" xlink:href="note-326-10"/> pote recti; </s> <s xml:id="echoid-s14030" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> erit reliquus angulus HLK, minor reliquo angulo GBI. </s> <s xml:id="echoid-s14031" xml:space="preserve">Fiatigitur <anchor type="note" xlink:label="note-326-11a" xlink:href="note-326-11"/> angulus K L M, æqualis angulo G B I; </s> <s xml:id="echoid-s14032" xml:space="preserve">cadetque LM, extra L H; </s> <s xml:id="echoid-s14033" xml:space="preserve">conuenietque <lb/>cum KH, producta vltra H, in puncto M. </s> <s xml:id="echoid-s14034" xml:space="preserve">Quoniam igitur duo anguli B, I, trian-<lb/>guli GBI, ęquales ſunt duobus angulis L, K, trianguli MLK, & </s> <s xml:id="echoid-s14035" xml:space="preserve">latera BI, LK, æ-<lb/>qualia, <anchor type="note" xlink:href="" symbol="l"/> erunt rectæ GI, MK, ęquales. </s> <s xml:id="echoid-s14036" xml:space="preserve">Recta ergo GI, maior eſt, quam recta HK.</s> <s xml:id="echoid-s14037" xml:space="preserve"> <anchor type="note" xlink:label="note-326-12a" xlink:href="note-326-12"/> <pb o="297" file="327" n="327" rhead="LIBER SEPTIMVS."/> Quamobrem rectangulum ſub GI, & </s> <s xml:id="echoid-s14038" xml:space="preserve">dimidio ambitu figuræ ABC, contentum, <lb/>maius erit rectangulo contento ſub HK, & </s> <s xml:id="echoid-s14039" xml:space="preserve">dimidio ambitu figuræ DEC, quiæ-<lb/>qualis ponitur dimidio ambitus figuræ A B C, <anchor type="note" xlink:href="" symbol="a"/> Quocirca cumillud rectangulũ <anchor type="note" xlink:label="note-327-01a" xlink:href="note-327-01"/> oſtenſum ſit æquale figuræ ABC, hoc autem figuræ DEF, æquale; </s> <s xml:id="echoid-s14040" xml:space="preserve">maior quoq; <lb/></s> <s xml:id="echoid-s14041" xml:space="preserve">erit figura ABC, quàm figura DEF. </s> <s xml:id="echoid-s14042" xml:space="preserve">Iſoperimetrarum ergo figurarum regularium <lb/>maior eſt illa &</s> <s xml:id="echoid-s14043" xml:space="preserve">c. </s> <s xml:id="echoid-s14044" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s14045" xml:space="preserve"/> </p> <div xml:id="echoid-div854" type="float" level="2" n="2"> <figure xlink:label="fig-326-01" xlink:href="fig-326-01a"> <image file="326-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/326-01"/> </figure> <note symbol="a" position="left" xlink:label="note-326-02" xlink:href="note-326-02a" xml:space="preserve">3. tertij.</note> <note symbol="b" position="left" xlink:label="note-326-03" xlink:href="note-326-03a" xml:space="preserve">28. tertij.</note> <note symbol="c" position="left" xlink:label="note-326-04" xlink:href="note-326-04a" xml:space="preserve">2. coroll. 33. <lb/>ſexti.</note> <note symbol="d" position="left" xlink:label="note-326-05" xlink:href="note-326-05a" xml:space="preserve">2. coroll. 33. <lb/>ſexti.</note> <note symbol="c" position="left" xlink:label="note-326-06" xlink:href="note-326-06a" xml:space="preserve">15. quinti.</note> <note symbol="f" position="left" xlink:label="note-326-07" xlink:href="note-326-07a" xml:space="preserve">15. quinti.</note> <note symbol="g" position="left" xlink:label="note-326-08" xlink:href="note-326-08a" xml:space="preserve">5. hui{us}.</note> <note symbol="h" position="left" xlink:label="note-326-09" xlink:href="note-326-09a" xml:space="preserve">13. quinti.</note> <note symbol="i" position="left" xlink:label="note-326-10" xlink:href="note-326-10a" xml:space="preserve">10. quinti.</note> <note symbol="k" position="left" xlink:label="note-326-11" xlink:href="note-326-11a" xml:space="preserve">32. primi.</note> <note symbol="l" position="left" xlink:label="note-326-12" xlink:href="note-326-12a" xml:space="preserve">26. primi.</note> <note symbol="a" position="right" xlink:label="note-327-01" xlink:href="note-327-01a" xml:space="preserve">2. hui{us}.</note> </div> <note position="right" xml:space="preserve">Qua arte <lb/>triangulum <lb/>Iſoſcel{es} con-<lb/>ſtituatur Iſo-<lb/>perimetrum <lb/>cuiuis trian-<lb/>gulo non Iſo-<lb/>ſceli.</note> </div> <div xml:id="echoid-div856" type="section" level="1" n="297"> <head xml:id="echoid-head324" xml:space="preserve">PROBL. 1. PROPOS. 7.</head> <p> <s xml:id="echoid-s14046" xml:space="preserve">PROPOSITO triangulo, cuius duo latera ſint inæqualia, ſupra re-<lb/>liquum latus triangulum priori Iſoperimetrum, ac duo habens latera <lb/>æqualia, deſcribere.</s> <s xml:id="echoid-s14047" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14048" xml:space="preserve"><emph style="sc">Sit</emph> triangulum ABC, cuius duo latera AB, BC, ſintinæqualia, nempe AB, <lb/>maius, quam BC; </s> <s xml:id="echoid-s14049" xml:space="preserve">oporteat que ſupra AC, conſtruere triangulum Iſoſceles, atq; <lb/></s> <s xml:id="echoid-s14050" xml:space="preserve">Iſoperimetrum triangulo ABC. </s> <s xml:id="echoid-s14051" xml:space="preserve">Sumatur recta D E, æqualis duobus lateribus <lb/>AB, BC, ſimul, diuidaturque bifariam in F. </s> <s xml:id="echoid-s14052" xml:space="preserve">Et quoniam latera AB, BC, ſimul ma-<lb/>iora ſunt latere AC, erunt quoque DF, FE, ſimul, maiores quam linea A C. </s> <s xml:id="echoid-s14053" xml:space="preserve">At-<lb/> <anchor type="note" xlink:label="note-327-03a" xlink:href="note-327-03"/> que ob id tres lineæ AC, DF, FE, ita ſeſe habebunt, vt quęlibet duæ ſintreliqua <lb/>maiores. </s> <s xml:id="echoid-s14054" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/>Si igitur ex ipſis conficiatur triangulum <anchor type="note" xlink:label="note-327-04a" xlink:href="note-327-04"/> <anchor type="figure" xlink:label="fig-327-01a" xlink:href="fig-327-01"/> AGC, effectum erit, quod proponitur. </s> <s xml:id="echoid-s14055" xml:space="preserve">Erunt enim <lb/>latera A G, G C, & </s> <s xml:id="echoid-s14056" xml:space="preserve">inter ſe ęqualia, & </s> <s xml:id="echoid-s14057" xml:space="preserve">ſimul ſumpta <lb/>æqualia lateribus AB, BC, ſimul ſumptis: </s> <s xml:id="echoid-s14058" xml:space="preserve">Addito igi-<lb/>tur communi A C, erunt triangula ABC, AGC, Iſo-<lb/>perimetra. </s> <s xml:id="echoid-s14059" xml:space="preserve">Propoſito igitur triangulo, cuius duo la-<lb/>tera ſint inæqualia, ſupra reliquum latus triangulum, <lb/>&</s> <s xml:id="echoid-s14060" xml:space="preserve">c. </s> <s xml:id="echoid-s14061" xml:space="preserve">deſcripſimus. </s> <s xml:id="echoid-s14062" xml:space="preserve">quod faciendum erat.</s> <s xml:id="echoid-s14063" xml:space="preserve"/> </p> <div xml:id="echoid-div856" type="float" level="2" n="1"> <note position="right" xlink:label="note-327-03" xlink:href="note-327-03a" xml:space="preserve">16. primi.</note> <note symbol="b" position="right" xlink:label="note-327-04" xlink:href="note-327-04a" xml:space="preserve">22. primi.</note> <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/xxxxxxxx/figures/327-01"/> </figure> </div> </div> <div xml:id="echoid-div858" type="section" level="1" n="298"> <head xml:id="echoid-head325" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s14064" xml:space="preserve"><emph style="sc">Cadet</emph> autem neceſſario punctum G, extra triangulum ABC: </s> <s xml:id="echoid-s14065" xml:space="preserve">Sinamque <lb/>caderet in latus AB, vtad punctum H, <anchor type="note" xlink:href="" symbol="c"/> eſſet ducta recta HC, minor, quam HB, <anchor type="note" xlink:label="note-327-05a" xlink:href="note-327-05"/> BC, ſimul, & </s> <s xml:id="echoid-s14066" xml:space="preserve">obid triangulum AHC, non eſſet Iſoperimetrum triangulo ABC, <lb/>cuius contrarium ex conſtructione eſt demonſtratum. </s> <s xml:id="echoid-s14067" xml:space="preserve">Multo minus cadet pũ-<lb/>ctum G, intra triangulum ABC. </s> <s xml:id="echoid-s14068" xml:space="preserve">Quare extra cadet, quod eſt propoſitum.</s> <s xml:id="echoid-s14069" xml:space="preserve"/> </p> <div xml:id="echoid-div858" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-327-05" xlink:href="note-327-05a" xml:space="preserve">20. primi.</note> </div> </div> <div xml:id="echoid-div860" type="section" level="1" n="299"> <head xml:id="echoid-head326" xml:space="preserve">THEOR. 7. PROPOS. 8.</head> <note position="right" xml:space="preserve">Iſoſcel{es} tri-<lb/>angulum ma-<lb/>i{us} eſt trian-<lb/>gulo ſibi Iſo-<lb/>perimetro non <lb/>Iſoſcele.</note> <p> <s xml:id="echoid-s14070" xml:space="preserve">DVORVM triangulorum Iſoperimetrorum eandem habentium ba-<lb/>ſim, quorum vnius duo latera ſint æqualia, alterius verò inæqualia; <lb/></s> <s xml:id="echoid-s14071" xml:space="preserve">maius erit illud, cuius duo latera æqualia ſunt.</s> <s xml:id="echoid-s14072" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14073" xml:space="preserve"><emph style="sc">Esto</emph> triangulum ABC, cuius latus AB, maius ſit latere BC, <anchor type="note" xlink:href="" symbol="d"/> conſtituatur- <anchor type="note" xlink:label="note-327-07a" xlink:href="note-327-07"/> que ſuper baſim AC, triangulo ABC, triangulum Iſoperimetrum ADC, habens <lb/>latera AD, DC, æqualia & </s> <s xml:id="echoid-s14074" xml:space="preserve">inter ſe, & </s> <s xml:id="echoid-s14075" xml:space="preserve">lateribus AB, BC, ſimul ſumptis. </s> <s xml:id="echoid-s14076" xml:space="preserve">Dico tri- <pb o="298" file="328" n="328" rhead="GEOMETR. PRACT."/> angulum ADC, maius eſſe triangulo ABC. </s> <s xml:id="echoid-s14077" xml:space="preserve">Producatur enim AD, ad partes D, <lb/> <anchor type="figure" xlink:label="fig-328-01a" xlink:href="fig-328-01"/> ſitque D E, æqualis ipſi A D, ſiue ipſi D C. </s> <s xml:id="echoid-s14078" xml:space="preserve">Ducantur <lb/> <anchor type="note" xlink:label="note-328-01a" xlink:href="note-328-01"/> quoque rectæ DB, BE. </s> <s xml:id="echoid-s14079" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam igitur AB, BE, ma- iores ſunt, quam A E, hoc eſt, quam A D, D C, ſimul, <lb/>hoc eſt, quam A B, B C, ſimul; </s> <s xml:id="echoid-s14080" xml:space="preserve">ablata communi A B, <lb/>erit B E, maior quam B C. </s> <s xml:id="echoid-s14081" xml:space="preserve">Et quia latera E D, D B, tri-<lb/>anguli EDB, æqualia ſunt lateribus CD, DB, trianguli <lb/>CDB. </s> <s xml:id="echoid-s14082" xml:space="preserve">At verò baſis BE, baſe BC, maior, <anchor type="note" xlink:href="" symbol="b"/> erit angulus <anchor type="note" xlink:label="note-328-02a" xlink:href="note-328-02"/> EDB, maior angulo C D B. </s> <s xml:id="echoid-s14083" xml:space="preserve">Quare angulus EDB, <lb/>maior eſt, quam dimidium anguli EDC: </s> <s xml:id="echoid-s14084" xml:space="preserve">Eſt autem an-<lb/>gulus DAC, dimidium anguli EDC; </s> <s xml:id="echoid-s14085" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> propterea quod <anchor type="note" xlink:label="note-328-03a" xlink:href="note-328-03"/> anguli DAC, DCA, æquales ſunt, <anchor type="note" xlink:href="" symbol="d"/> & </s> <s xml:id="echoid-s14086" xml:space="preserve">his ſimul ſum- <anchor type="note" xlink:label="note-328-04a" xlink:href="note-328-04"/> ptis ęqualis quo que externus angulus E D C. </s> <s xml:id="echoid-s14087" xml:space="preserve">Maior <lb/>igitur erit angulus EDB, angulo DAC. </s> <s xml:id="echoid-s14088" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Fiat angulus EDF, ęqualis angulo in- <anchor type="note" xlink:label="note-328-05a" xlink:href="note-328-05"/> terno DAC; </s> <s xml:id="echoid-s14089" xml:space="preserve">cadetque DF, recta ſupra rectam DB, <anchor type="note" xlink:href="" symbol="f"/> æquidiſtabit que rectæ A C.</s> <s xml:id="echoid-s14090" xml:space="preserve"> <anchor type="note" xlink:label="note-328-06a" xlink:href="note-328-06"/> Producatur DF, donec cum AB, protracta conueniat in F, ducaturq; </s> <s xml:id="echoid-s14091" xml:space="preserve">recta F C. <lb/></s> <s xml:id="echoid-s14092" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="g"/> Quoniam igitur triangula ADC, AFC, æqualia ſunt, triangulum autem AFC, <anchor type="note" xlink:label="note-328-07a" xlink:href="note-328-07"/> maius eſt triangulo ABC; </s> <s xml:id="echoid-s14093" xml:space="preserve">maius quoque erit triangulum ADC, triangulo ABC. <lb/></s> <s xml:id="echoid-s14094" xml:space="preserve">Quam ob rem duorum triangulorum Iſoperimetrorum eandem habentium <lb/>baſim, &</s> <s xml:id="echoid-s14095" xml:space="preserve">c. </s> <s xml:id="echoid-s14096" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s14097" xml:space="preserve"/> </p> <div xml:id="echoid-div860" type="float" level="2" n="1"> <note symbol="d" position="right" xlink:label="note-327-07" xlink:href="note-327-07a" xml:space="preserve">7. hui{us}.</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/xxxxxxxx/figures/328-01"/> </figure> <note symbol="a" position="left" xlink:label="note-328-01" xlink:href="note-328-01a" xml:space="preserve">20. primi.</note> <note symbol="b" position="left" xlink:label="note-328-02" xlink:href="note-328-02a" xml:space="preserve">25. primi.</note> <note symbol="c" position="left" xlink:label="note-328-03" xlink:href="note-328-03a" xml:space="preserve">5. primi.</note> <note symbol="d" position="left" xlink:label="note-328-04" xlink:href="note-328-04a" xml:space="preserve">32. primi.</note> <note symbol="e" position="left" xlink:label="note-328-05" xlink:href="note-328-05a" xml:space="preserve">23. primi.</note> <note symbol="f" position="left" xlink:label="note-328-06" xlink:href="note-328-06a" xml:space="preserve">28. primi.</note> <note symbol="g" position="left" xlink:label="note-328-07" xlink:href="note-328-07a" xml:space="preserve">37. primi.</note> </div> </div> <div xml:id="echoid-div862" type="section" level="1" n="300"> <head xml:id="echoid-head327" xml:space="preserve">THEOR. 8. PROPOS. 9.</head> <note position="left" xml:space="preserve">Proprietas <lb/>duorum tri-<lb/>angulorum <lb/>rectangulorũ <lb/>ſimilium.</note> <p> <s xml:id="echoid-s14098" xml:space="preserve">IN ſimilibus triangulis rectangulis quadratum à lateribus, quæ angulis <lb/>rectis ſubtenduntur, tanquam ab vna linea, deſcriptum, æquale eſt <lb/>quadratis duobus ſimul, quæ à reliquis homologis lateribus tanquam <lb/>ex duabus lineis, ita vt quælibet duo latera homologa conficiant vnã <lb/>lineam rectam, deſcribuntur.</s> <s xml:id="echoid-s14099" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14100" xml:space="preserve"><emph style="sc">Sint</emph> triangula rectangula ſimilia ABC, DEF, ita vt anguli B, & </s> <s xml:id="echoid-s14101" xml:space="preserve">E, ſint <lb/> <anchor type="figure" xlink:label="fig-328-02a" xlink:href="fig-328-02"/> recti, anguli verò C, & </s> <s xml:id="echoid-s14102" xml:space="preserve">F, inter ſe æquales; </s> <s xml:id="echoid-s14103" xml:space="preserve">itemque <lb/>anguli A, & </s> <s xml:id="echoid-s14104" xml:space="preserve">D, inter ſe æquales; </s> <s xml:id="echoid-s14105" xml:space="preserve">homologaque late-<lb/>ra AB, DE; </s> <s xml:id="echoid-s14106" xml:space="preserve">Item BC, EF, & </s> <s xml:id="echoid-s14107" xml:space="preserve">AC, DF. </s> <s xml:id="echoid-s14108" xml:space="preserve">Dico quadratũ <lb/>ex AC, DF, tanquam ex linea vna, deſcriptum, æqua-<lb/>le eſſe duobus quadratis, quorumvnum ex AB, DE, <lb/>tanquam exvna linea, alterum verò ex BC, EF, tan-<lb/>quam exvna quoque linea, deſcribitur. </s> <s xml:id="echoid-s14109" xml:space="preserve">Producta <lb/>namque DE, ad partes E, ſumatur E G, æqualis rectæ <lb/>A B, & </s> <s xml:id="echoid-s14110" xml:space="preserve">ducatur G H, recta æquidiſtans rectæ E F, do-<lb/>nec cum DF, producta conueniat in puncto H; </s> <s xml:id="echoid-s14111" xml:space="preserve">Dein-<lb/>de per F, ducatur recta F I, æquidiſtans rectæ EG. </s> <s xml:id="echoid-s14112" xml:space="preserve">Erit <lb/>igitur triangulum FIH, æquiangulum triangulo DEF, <lb/>hoc eſt, triangulo ABC. </s> <s xml:id="echoid-s14113" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Nam angulus FIH, æqua- <anchor type="note" xlink:label="note-328-09a" xlink:href="note-328-09"/> lis eſt angulo G, <anchor type="note" xlink:href="" symbol="i"/> & </s> <s xml:id="echoid-s14114" xml:space="preserve">hic æqualis angulo D E F, hoc <anchor type="note" xlink:label="note-328-10a" xlink:href="note-328-10"/> eſt, angulo B; </s> <s xml:id="echoid-s14115" xml:space="preserve">angulus verò H, æqualis eſt angulo <pb o="299" file="329" n="329" rhead="LIBER SEPTIMVS."/> DFE, hoc eſt, angulo C; </s> <s xml:id="echoid-s14116" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ac proinde & </s> <s xml:id="echoid-s14117" xml:space="preserve">angulus IFH, angulo A: </s> <s xml:id="echoid-s14118" xml:space="preserve">Sunt autem <anchor type="note" xlink:label="note-329-01a" xlink:href="note-329-01"/> & </s> <s xml:id="echoid-s14119" xml:space="preserve">latera AB, FI, æqualia; </s> <s xml:id="echoid-s14120" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Nam recta FI, eſt æqualis rectæ E G, hæc autem rectæ <anchor type="note" xlink:label="note-329-02a" xlink:href="note-329-02"/> AB, ſumpta fuit ęqualis. </s> <s xml:id="echoid-s14121" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur & </s> <s xml:id="echoid-s14122" xml:space="preserve">latera BC, IH; </s> <s xml:id="echoid-s14123" xml:space="preserve">item AC, FH, æqualia inter ſe <anchor type="note" xlink:label="note-329-03a" xlink:href="note-329-03"/> erunt. </s> <s xml:id="echoid-s14124" xml:space="preserve">Quare recta DH, compoſita erit ex AC, & </s> <s xml:id="echoid-s14125" xml:space="preserve">DF, Recta verò DG, ex AB, <lb/>& </s> <s xml:id="echoid-s14126" xml:space="preserve">DE, Recta denique GH, ex BC, & </s> <s xml:id="echoid-s14127" xml:space="preserve">EF; </s> <s xml:id="echoid-s14128" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> quod GI, recta æqualis ſit rectæ EF.</s> <s xml:id="echoid-s14129" xml:space="preserve"> <anchor type="note" xlink:label="note-329-04a" xlink:href="note-329-04"/> <anchor type="note" xlink:href="" symbol="e"/> Et quoniam quadratum rectæ DH, æquale eſt quadratis rectarum DG, GH, <anchor type="note" xlink:label="note-329-05a" xlink:href="note-329-05"/> ſimul, conſtat verum eſſe, quod proponitur. </s> <s xml:id="echoid-s14130" xml:space="preserve">In ſimilibus igitur triangulis re-<lb/>ctangulis quadratum à lateribus, quæ angulis rectis ſubtenduntur, &</s> <s xml:id="echoid-s14131" xml:space="preserve">c. </s> <s xml:id="echoid-s14132" xml:space="preserve">quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s14133" xml:space="preserve"/> </p> <div xml:id="echoid-div862" type="float" level="2" n="1"> <figure xlink:label="fig-328-02" xlink:href="fig-328-02a"> <image file="328-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/328-02"/> </figure> <note symbol="h" position="left" xlink:label="note-328-09" xlink:href="note-328-09a" xml:space="preserve">29. primi.</note> <note symbol="i" position="left" xlink:label="note-328-10" xlink:href="note-328-10a" xml:space="preserve">29. primi.</note> <note symbol="a" position="right" xlink:label="note-329-01" xlink:href="note-329-01a" xml:space="preserve">32. primi.</note> <note symbol="b" position="right" xlink:label="note-329-02" xlink:href="note-329-02a" xml:space="preserve">34. primi.</note> <note symbol="c" position="right" xlink:label="note-329-03" xlink:href="note-329-03a" xml:space="preserve">26. primi.</note> <note symbol="d" position="right" xlink:label="note-329-04" xlink:href="note-329-04a" xml:space="preserve">34. primi.</note> <note symbol="e" position="right" xlink:label="note-329-05" xlink:href="note-329-05a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div864" type="section" level="1" n="301"> <head xml:id="echoid-head328" xml:space="preserve">PROBL. 2. PROPOS. 10.</head> <note position="right" xml:space="preserve">Qua arte cõ-<lb/>ſtituantur <lb/>duo triangula <lb/>Iſoſcelia ſimi-<lb/>lia quidem <lb/>interſe Iſope-<lb/>rimetra vero <lb/>aliis duob{us} <lb/>Iſoſcel b{us}.</note> <p> <s xml:id="echoid-s14134" xml:space="preserve">DATIS duobus triangulis Iſoſcelibus; </s> <s xml:id="echoid-s14135" xml:space="preserve">quorum baſes inæquales exi-<lb/>ſtant, duoque latera vnius æqualia ſint duobus lateribus alterius; </s> <s xml:id="echoid-s14136" xml:space="preserve">Su-<lb/>per eiſdem baſibus duo alia triangula Iſoſcelia inter ſe quidem ſimi-<lb/>lia, prioribus verò ſimul ſumptis Iſoperimetra ſimul ſumpta, conſti-<lb/>tuere.</s> <s xml:id="echoid-s14137" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14138" xml:space="preserve"><emph style="sc">Sint</emph> ſuper baſes inæquales A B, C D, duo triangula Iſoſcelia A E B, C F D, <lb/>ſintque quatuor lineę AE, EB, C F, F D, inter ſe æquales; </s> <s xml:id="echoid-s14139" xml:space="preserve">maior autem ſit baſis <lb/>AB, baſe C D. </s> <s xml:id="echoid-s14140" xml:space="preserve">quibus poſitis, <anchor type="note" xlink:href="" symbol="f"/> erit angulus E, maior angulo F, ideoque trian- <anchor type="note" xlink:label="note-329-07a" xlink:href="note-329-07"/> <anchor type="figure" xlink:label="fig-329-01a" xlink:href="fig-329-01"/> gula non ſimilia, cum nec æquiangula. </s> <s xml:id="echoid-s14141" xml:space="preserve">Oporteat iam ſuper baſes eaſdem AB, <lb/>CD, conſtituere alia duo triangula Iſoſcelia inter ſe quidem ſimilia, Iſoperime-<lb/>tra verò ſimul ſumpta prioribus triangulis ſimul ſumptis. </s> <s xml:id="echoid-s14142" xml:space="preserve">Ponatur recta GH, æ-<lb/>qualis quatu or rectis AE, EB, CF, FD, <anchor type="note" xlink:href="" symbol="g"/> diuidatur que in puncto K, vt eſſet recta <anchor type="note" xlink:label="note-329-08a" xlink:href="note-329-08"/> compoſita ex AB, & </s> <s xml:id="echoid-s14143" xml:space="preserve">C D, diuiſa in puncto B, hoc eſt, ſit ea proportio GK, ad <lb/>KH, quæ eſt AB, ad CD. </s> <s xml:id="echoid-s14144" xml:space="preserve">Et quia maior eſt recta AB, quam recta CD, maior quo-<lb/>que erit recta G K, quam recta K H, cum vtrobique ſit proportio maioris inæ-<lb/>qualitatis. </s> <s xml:id="echoid-s14145" xml:space="preserve">Diuidaturvtraque GK, KH, bifariam in punctis L, & </s> <s xml:id="echoid-s14146" xml:space="preserve">M. </s> <s xml:id="echoid-s14147" xml:space="preserve">Itaque cum <lb/>ſit vt GK, ad KH, ita AB, ad CD, erit componendo, vt GH, ad KH, ita AB, CD, <lb/>ſimul ad C D: </s> <s xml:id="echoid-s14148" xml:space="preserve">Eſt autem G H, maior, quàm A B, C D, ſimul, <anchor type="note" xlink:href="" symbol="h"/> quod & </s> <s xml:id="echoid-s14149" xml:space="preserve">quatuor <anchor type="note" xlink:label="note-329-09a" xlink:href="note-329-09"/> rectæ AE, EB, CF, FD, quæ æquales ſunt rectæ GH, maiores ſint, quam AB, CD. <lb/></s> <s xml:id="echoid-s14150" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="i"/> Igitur & </s> <s xml:id="echoid-s14151" xml:space="preserve">K H, maior erit quam C D: </s> <s xml:id="echoid-s14152" xml:space="preserve">Eademque ratione maior erit G K, quam <anchor type="note" xlink:label="note-329-10a" xlink:href="note-329-10"/> A B. </s> <s xml:id="echoid-s14153" xml:space="preserve">Quoniam igitur trium rectarum A B, G L, L K, duæ reliqua ſunt maiores <lb/>omnifariam ſumptæ; </s> <s xml:id="echoid-s14154" xml:space="preserve">(Duæ enim G L, L K, maiores ſunt quam A B, quod <lb/>tota G K, maior ſit, quam A B, vt modo fuit oſtenſum; </s> <s xml:id="echoid-s14155" xml:space="preserve">Manifeſtum <lb/>autem eſt, A B, G L, maiores eſſe reliqua L K; </s> <s xml:id="echoid-s14156" xml:space="preserve">Itemque A B, L K, re-<lb/>liqua G L, eſſe maiores, propterea quod G K, diuiſa eſt bifariam in puncto <lb/>L. </s> <s xml:id="echoid-s14157" xml:space="preserve">Idem quoq; </s> <s xml:id="echoid-s14158" xml:space="preserve">dices de tribus rectis C D, K M, M H.) </s> <s xml:id="echoid-s14159" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> conſtituatur ex tribus <anchor type="note" xlink:label="note-329-11a" xlink:href="note-329-11"/> <pb o="300" file="330" n="330" rhead="GEOMETR. PRACT."/> rectis AB, GL, LK, triangulum ANB, quod erit Iſoſceles, cadetque punctum N, <lb/>extra triangulum AEB, cum AE, EB, ſimul dimidium conſtituant rectæ G, H; </s> <s xml:id="echoid-s14160" xml:space="preserve">at <lb/>verò A N, N@B, ſimul maius efficiant, quam dimidium rectæ G H. </s> <s xml:id="echoid-s14161" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Rurſus ex <anchor type="note" xlink:label="note-330-01a" xlink:href="note-330-01"/> tribus rectis CD, KM, MH, conſtituatur quoque triangulum C O D, quod Iſo-<lb/>ſceles erit, cadetque punctum O, intra triangulum CFD, eo quod CF, FD, ſimul <lb/>æquales ſint dimidio rectæ GH; </s> <s xml:id="echoid-s14162" xml:space="preserve">at CO, OD, ſimul minores ſint dimidio rectæ <lb/>GH. </s> <s xml:id="echoid-s14163" xml:space="preserve">Et quoniam quatuor latera AE, EB, CF, FD, ſimul; </s> <s xml:id="echoid-s14164" xml:space="preserve">Item AN, NB, C O, <lb/>O D, ſimul æqualia ſunt rectæ G H, erunt priora quatuor ſimul, poſterioribus <lb/>quatuor ſimul æqualia; </s> <s xml:id="echoid-s14165" xml:space="preserve">additis ergo communibus AB, CD, fient ſex latera AE, <lb/>EB, BA, CF, FD, DC, ſimul ęqualia ſex lateribus AN, NB, BA, CO, OD, DC, ſi-<lb/>mul; </s> <s xml:id="echoid-s14166" xml:space="preserve">ideo que triangula ANB, COD, ſimul Iſoperimetra erunt triangulis AEB, <lb/>CFD, ſimul. </s> <s xml:id="echoid-s14167" xml:space="preserve">Dico iam, quod & </s> <s xml:id="echoid-s14168" xml:space="preserve">ſimilia inter ſe ſunt triangula ANB, COD. </s> <s xml:id="echoid-s14169" xml:space="preserve">Nam <lb/>quoniam eſt, vt AB, ad CD, ita GK, ad KH, <anchor type="note" xlink:href="" symbol="b"/> hoc eſt, ita GL, ad KM, hoc eſt ita <anchor type="note" xlink:label="note-330-02a" xlink:href="note-330-02"/> A N, ad C O, & </s> <s xml:id="echoid-s14170" xml:space="preserve">N B, ad O D; </s> <s xml:id="echoid-s14171" xml:space="preserve">erit permutando, vt A B, ad AN, <lb/>ita C D, ad C O; </s> <s xml:id="echoid-s14172" xml:space="preserve">& </s> <s xml:id="echoid-s14173" xml:space="preserve">vt A N, ad N B, ita C O, ad O D. </s> <s xml:id="echoid-s14174" xml:space="preserve">Proportionalia ergo <lb/>ſunt latera triangulorum ANB, COD; </s> <s xml:id="echoid-s14175" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ac proinde æquiangula inter ſe erunt, &</s> <s xml:id="echoid-s14176" xml:space="preserve"> <anchor type="note" xlink:label="note-330-03a" xlink:href="note-330-03"/> idcirco ſimilia. </s> <s xml:id="echoid-s14177" xml:space="preserve">Quare datis duobus triangulis Iſoſcelibus, quorum baſes inæ-<lb/>quales exiſtant &</s> <s xml:id="echoid-s14178" xml:space="preserve">c. </s> <s xml:id="echoid-s14179" xml:space="preserve">conſtituimus. </s> <s xml:id="echoid-s14180" xml:space="preserve">quod faciendum erat.</s> <s xml:id="echoid-s14181" xml:space="preserve"/> </p> <div xml:id="echoid-div864" type="float" level="2" n="1"> <note symbol="f" position="right" xlink:label="note-329-07" xlink:href="note-329-07a" xml:space="preserve">25. primi.</note> <figure xlink:label="fig-329-01" xlink:href="fig-329-01a"> <image file="329-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/329-01"/> </figure> <note symbol="g" position="right" xlink:label="note-329-08" xlink:href="note-329-08a" xml:space="preserve">10. ſexti.</note> <note symbol="h" position="right" xlink:label="note-329-09" xlink:href="note-329-09a" xml:space="preserve">20. primi.</note> <note symbol="i" position="right" xlink:label="note-329-10" xlink:href="note-329-10a" xml:space="preserve">14. quinti.</note> <note symbol="k" position="right" xlink:label="note-329-11" xlink:href="note-329-11a" xml:space="preserve">22. primi.</note> <note symbol="a" position="left" xlink:label="note-330-01" xlink:href="note-330-01a" xml:space="preserve">22. primi.</note> <note symbol="b" position="left" xlink:label="note-330-02" xlink:href="note-330-02a" xml:space="preserve">15. quinti.</note> <note symbol="c" position="left" xlink:label="note-330-03" xlink:href="note-330-03a" xml:space="preserve">5. ſexti.</note> </div> </div> <div xml:id="echoid-div866" type="section" level="1" n="302"> <head xml:id="echoid-head329" xml:space="preserve">THEOR. 9. PROPOS. 11.</head> <note position="left" xml:space="preserve">Triangula <lb/>duo Iſoſcelia <lb/>ſimilia maio-<lb/>ra ſunt duo-<lb/>bus Iſoſcelib{us} <lb/>non ſi milib{us}, <lb/>quæillis ſint <lb/>Iſoperimetr@, <lb/>baſeſque ha-<lb/>beant eaſdem.</note> <p> <s xml:id="echoid-s14182" xml:space="preserve">DVO triangula Iſoſcelia ſimilia ſuper inæqualibus baſibus conſtituta, <lb/>vtraque ſimul maiora ſunt duobus triangulis Iſoſcelibus, vtriſque ſi-<lb/>mul, quæ habeant eaſdem baſes cum prioribus, ſintq; </s> <s xml:id="echoid-s14183" xml:space="preserve">diſſimilia qui-<lb/>dem inter ſe, at Iſoperimetra prioribus duobus, nec non quatuor la-<lb/>tera inter ſe habeant æqualia.</s> <s xml:id="echoid-s14184" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14185" xml:space="preserve"><emph style="sc">Svper</emph> baſibus inæqualibus A C, C E, ſint duo triangula Iſoſcelia inter ſe <lb/>nonſimilia ABC, CDE, ita vt quatuor latera AB, BC, C D, D E, inter ſe ſint æ-<lb/>qualia. </s> <s xml:id="echoid-s14186" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Atque ſuper eiſdem baſibus A C, C E, conſtituantur alia duo triangu- <anchor type="note" xlink:label="note-330-05a" xlink:href="note-330-05"/> <anchor type="figure" xlink:label="fig-330-01a" xlink:href="fig-330-01"/> la Iſoſcelia AFC, CGE, ſimilia inter ſe, & </s> <s xml:id="echoid-s14187" xml:space="preserve">Iſoperimetra ſimul prioribus triangulis <lb/>ſimul. </s> <s xml:id="echoid-s14188" xml:space="preserve">Dico duo triãgula AFC, CGE, ſimul maiora eſſe duobus triangulis ABC, <lb/>CDE, ſimul. </s> <s xml:id="echoid-s14189" xml:space="preserve">Ponantur enim A C, C E, ſecundum lineam rectam vnam; </s> <s xml:id="echoid-s14190" xml:space="preserve">ſitque <pb o="301" file="331" n="331" rhead="LIBER SEPTIMVS."/> AC, baſis maior baſe CE. </s> <s xml:id="echoid-s14191" xml:space="preserve">Deinde ex F, per B, ducatur recta FBK, ſecans rectam <lb/>AC, in puncto K; </s> <s xml:id="echoid-s14192" xml:space="preserve">Item ex D, per G, punctum ducaturrecta DGH, ſecansre-<lb/>ctam CE, in H. </s> <s xml:id="echoid-s14193" xml:space="preserve">Et quia latera AF, FB, trianguli AFB, æqualia ſunt lateribus CF, <lb/> <anchor type="note" xlink:label="note-331-01a" xlink:href="note-331-01"/> FB, triãguli CFB, & </s> <s xml:id="echoid-s14194" xml:space="preserve">baſis AB, baſi BC, æqualis, <anchor type="note" xlink:href="" symbol="a"/> erit angulus AFB, angulo CFB, æqualis. </s> <s xml:id="echoid-s14195" xml:space="preserve">Rurſus quia latera AF, FK, trianguli AFK, æqualia ſunt lateribus CF, <lb/>FK, trianguli CFK, & </s> <s xml:id="echoid-s14196" xml:space="preserve">angulus AFK, angulo CFK, æqualis, vt probatum eſt; <lb/></s> <s xml:id="echoid-s14197" xml:space="preserve"> <anchor type="note" xlink:label="note-331-02a" xlink:href="note-331-02"/> <anchor type="note" xlink:href="" symbol="b"/> erunt baſes AK, KC, æquales, & </s> <s xml:id="echoid-s14198" xml:space="preserve">anguli ad K, æquales quoque, hoc eſt, recti.</s> <s xml:id="echoid-s14199" xml:space="preserve"> Eadem ratio cinatione concludemus rectam CE, in puncto H, diuidi bifariam, <lb/>anguloſque ad H, eſſerectos. </s> <s xml:id="echoid-s14200" xml:space="preserve">Producatur recta DH, ad partes H, ſumatur que <lb/>HL, æqualis rectæ DH, & </s> <s xml:id="echoid-s14201" xml:space="preserve">extendatur à puncto L, per punctum C, recta LCN. <lb/></s> <s xml:id="echoid-s14202" xml:space="preserve">Quoniam verò latera DH, HC, trianguli DCH, æqualia ſunt lateribus LH, <lb/>HC, trianguli LCH, & </s> <s xml:id="echoid-s14203" xml:space="preserve">anguli ad H, æquales, vtpote recti; </s> <s xml:id="echoid-s14204" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt baſes DC, LC, <anchor type="note" xlink:label="note-331-03a" xlink:href="note-331-03"/> æquales, & </s> <s xml:id="echoid-s14205" xml:space="preserve">anguli DCH, LCH, æquales etiam: </s> <s xml:id="echoid-s14206" xml:space="preserve">Atqui angulus DCH, maior <lb/>eſt angulo GCH, & </s> <s xml:id="echoid-s14207" xml:space="preserve">angulus GCH, æqualis eſt angulo FAK, propter ſimilitu-<lb/>dinem triangulorum GCE, & </s> <s xml:id="echoid-s14208" xml:space="preserve">FAC, hoc eſt, angulo FCA, <anchor type="note" xlink:href="" symbol="d"/> qui angulo FAC, æqualis eſt. </s> <s xml:id="echoid-s14209" xml:space="preserve">Erit igitur angulus DCH, hoc eſt, angulus LCH, qui illi oſtenſus <lb/> <anchor type="note" xlink:label="note-331-04a" xlink:href="note-331-04"/> eſt æqualis, hoc eſt, angulus NCK, <anchor type="note" xlink:href="" symbol="e"/> qui angulo LCH, ad verticem eſt æqua- <anchor type="note" xlink:label="note-331-05a" xlink:href="note-331-05"/> lis, maior etiam angulo FCA: </s> <s xml:id="echoid-s14210" xml:space="preserve">& </s> <s xml:id="echoid-s14211" xml:space="preserve">obid CN, recta extra rectam CF, cadet ne-<lb/>ceſſariò; </s> <s xml:id="echoid-s14212" xml:space="preserve">& </s> <s xml:id="echoid-s14213" xml:space="preserve">rectæ LC, CB, propterea comprehendent ad partes K, angulum <lb/>BCL. </s> <s xml:id="echoid-s14214" xml:space="preserve">Quare ſi ducatur recta B L, ſecabit ea lineam C K, in aliquo puncto in-<lb/>ter puncta C, & </s> <s xml:id="echoid-s14215" xml:space="preserve">K, quod ſit M. </s> <s xml:id="echoid-s14216" xml:space="preserve">Quoniam verò rectæ AB, BC, CD, DE, ſimul <lb/>æquales ſunt rectis AF, FC, CG, GE, ſimul, propter triangula iſoperimetra, <lb/>erunt quoque dimidia earum æqualia inter ſe, nimirum rectę BC, CD, hoc eſt, <lb/>BC, CL, ſimul æquales ipſis FC, CG, ſimul: </s> <s xml:id="echoid-s14217" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Sunt autem rectæ BC, CL, ſimul <anchor type="note" xlink:label="note-331-06a" xlink:href="note-331-06"/> maiores recta BL. </s> <s xml:id="echoid-s14218" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14219" xml:space="preserve">FC, CG, ſimul maiores erunt eadem recta BL: <lb/></s> <s xml:id="echoid-s14220" xml:space="preserve"> <anchor type="figure" xlink:label="fig-331-01a" xlink:href="fig-331-01"/> ideo que quadratum ex F C, C G, tanquam ex vna linea, deſcriptum maius erit <lb/> <anchor type="note" xlink:label="note-331-07a" xlink:href="note-331-07"/> quadrato BL. </s> <s xml:id="echoid-s14221" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Quod autem ex F C, CG, tanquam ex vna linea, deſcribitur quadratum, æquale eſt quadrato ex F K, G H, tanquam ex vna linea deſcripto, <lb/>vna cum quadrato, quod ex K C, C H, tanquam ex vna linea deſcribitur. <lb/></s> <s xml:id="echoid-s14222" xml:space="preserve"> <anchor type="note" xlink:label="note-331-08a" xlink:href="note-331-08"/> <anchor type="note" xlink:href="" symbol="h"/> Quadratum verò ex L B, deſcriptum æquale eſt quadrato ex B K, L H, hoc eſt, ex B K, D H, tanquam ex vna linea, deſcripto, vna cum quadrato, quod ex <lb/>KM, MH, tanquam ex vna linea, deſcribitur; </s> <s xml:id="echoid-s14223" xml:space="preserve">quod triangula rectangula BKM, <lb/> <anchor type="note" xlink:label="note-331-09a" xlink:href="note-331-09"/> LHM, ſint ſimilia inter ſe. </s> <s xml:id="echoid-s14224" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Sunt enim anguli M, ad verticemæquales, <pb o="302" file="332" n="332" rhead="GEOMETR. PRACT."/> & </s> <s xml:id="echoid-s14225" xml:space="preserve">anguli K, H, recti, <anchor type="note" xlink:href="" symbol="a"/> ideoque & </s> <s xml:id="echoid-s14226" xml:space="preserve">reliqui KBM, HLM, æquales. </s> <s xml:id="echoid-s14227" xml:space="preserve">Igitur qua- <anchor type="note" xlink:label="note-332-01a" xlink:href="note-332-01"/> dratum ex FK, GH, tanquam ex vna linea, deſcriptum, & </s> <s xml:id="echoid-s14228" xml:space="preserve">quadratum ex KC, <lb/>CH, tanquam ex vna linea deſcriptum, hoc eſt, quadratum K H, vtraque ſi-<lb/>mul, maiora ſunt quadrato ex BK, D H, tanquam ex vna linea, deſcripto, & </s> <s xml:id="echoid-s14229" xml:space="preserve"><lb/>quadrato ex K M, M H, tanquam ex vna linea deſcripto, hoc eſt, quadrato <lb/>KH, vtriſque ſimul. </s> <s xml:id="echoid-s14230" xml:space="preserve">Ablato ergo communi quadrato KH, erit quadratum ex <lb/>FK, GH, tanquam ex vna linea, deſcriptum maius quadrato ex BK, DH, tan-<lb/>quam ex vna linea, deſcripto; </s> <s xml:id="echoid-s14231" xml:space="preserve">ideoque maiores erunt rectæ lineæ FK, GH, ſi-<lb/>mul rectis BK, DH, ſimul; </s> <s xml:id="echoid-s14232" xml:space="preserve">Acpropterea, demptis communibus BK, GH, erit <lb/>FB, reliqua maior, quam reliqua D G. </s> <s xml:id="echoid-s14233" xml:space="preserve">Eſt autem & </s> <s xml:id="echoid-s14234" xml:space="preserve">K C, maior quam H C, <lb/>quod tota A C, cuius dimidium eſt K C, maior ponitur quam tota C E, cuius <lb/>dimidium eſt HC. </s> <s xml:id="echoid-s14235" xml:space="preserve">Quapropter rectangulum ſub FB, KC, contentum, maius <lb/>erit rectangulo ſub DG, HC, contento. </s> <s xml:id="echoid-s14236" xml:space="preserve">Et quoniam triangulum FBC, dimi-<lb/>dium eſt rectanguli ſub FB, KC, contenti; </s> <s xml:id="echoid-s14237" xml:space="preserve">(Nam ſi ſuper F B, conſtituatur re-<lb/>ctangulum altitudinem habens K C, ita vt triangulum, & </s> <s xml:id="echoid-s14238" xml:space="preserve">rectangulum inter <lb/> <anchor type="figure" xlink:label="fig-332-01a" xlink:href="fig-332-01"/> <anchor type="note" xlink:label="note-332-02a" xlink:href="note-332-02"/> eaſdem ſint parallelas; </s> <s xml:id="echoid-s14239" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit triangulum parallelo grammi dimidium. </s> <s xml:id="echoid-s14240" xml:space="preserve">quod quidem parallelo grammumidem eſt, quod rectangulum ſub FB, KC, conten-<lb/>tum, vt conſtat.) </s> <s xml:id="echoid-s14241" xml:space="preserve">Triangulum verò DGC, dimidium eſt rectanguli contenti <lb/>ſub DG, HC; </s> <s xml:id="echoid-s14242" xml:space="preserve">(Sienim ſuper D G, conſtituatur rectangulum altitudinem ha-<lb/> <anchor type="note" xlink:label="note-332-03a" xlink:href="note-332-03"/> bens H C, ita vt triangulum, & </s> <s xml:id="echoid-s14243" xml:space="preserve">rectangulum inter eaſdem ſint parallelas: </s> <s xml:id="echoid-s14244" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit triangulum parallelogrammi dimidium. </s> <s xml:id="echoid-s14245" xml:space="preserve">quod quidem parallelo grammum <lb/>idem eſt, quod rectangulum ſub DG, HC, contentum, vt conſtat.) </s> <s xml:id="echoid-s14246" xml:space="preserve">erit quo-<lb/>que triangulum F B C, maius triangulo D G C; </s> <s xml:id="echoid-s14247" xml:space="preserve">ac propterea duplum trianguli <lb/>F B C, nimirum rectilineum AFCBA, maius erit duplo trianguli D G C, vtpote <lb/>rectilineo CDEGC. </s> <s xml:id="echoid-s14248" xml:space="preserve">Quocirca, addito communi compoſito ex triangulis <lb/>ABC, CGE; </s> <s xml:id="echoid-s14249" xml:space="preserve">erunt triangula AFC, CGE, vtra que ſimul maiora triangulis ABC, <lb/>C D E, vtriſque ſimul. </s> <s xml:id="echoid-s14250" xml:space="preserve">Duo ergo triangula Iſoſcelia ſimilia ſuper inæ-<lb/>qualibus baſibus conſtituta, &</s> <s xml:id="echoid-s14251" xml:space="preserve">c. </s> <s xml:id="echoid-s14252" xml:space="preserve">quod oſtenden-<lb/>dum eſt.</s> <s xml:id="echoid-s14253" xml:space="preserve"/> </p> <div xml:id="echoid-div866" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-330-05" xlink:href="note-330-05a" xml:space="preserve">10. hui{us}.</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/xxxxxxxx/figures/330-01"/> </figure> <note symbol="a" position="right" xlink:label="note-331-01" xlink:href="note-331-01a" xml:space="preserve">8. primi.</note> <note symbol="b" position="right" xlink:label="note-331-02" xlink:href="note-331-02a" xml:space="preserve">4. primi.</note> <note symbol="c" position="right" xlink:label="note-331-03" xlink:href="note-331-03a" xml:space="preserve">4. primi.</note> <note symbol="d" position="right" xlink:label="note-331-04" xlink:href="note-331-04a" xml:space="preserve">5. primi.</note> <note symbol="e" position="right" xlink:label="note-331-05" xlink:href="note-331-05a" xml:space="preserve">15. primi.</note> <note symbol="f" position="right" xlink:label="note-331-06" xlink:href="note-331-06a" xml:space="preserve">20. primi.</note> <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/xxxxxxxx/figures/331-01"/> </figure> <note symbol="g" position="right" xlink:label="note-331-07" xlink:href="note-331-07a" xml:space="preserve">9. hui{us}.</note> <note symbol="h" position="right" xlink:label="note-331-08" xlink:href="note-331-08a" xml:space="preserve">9. hui{us}.</note> <note symbol="i" position="right" xlink:label="note-331-09" xlink:href="note-331-09a" xml:space="preserve">15. primi.</note> <note symbol="a" position="left" xlink:label="note-332-01" xlink:href="note-332-01a" xml:space="preserve">32. 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/xxxxxxxx/figures/332-01"/> </figure> <note symbol="b" position="left" xlink:label="note-332-02" xlink:href="note-332-02a" xml:space="preserve">41. primi.</note> <note symbol="c" position="left" xlink:label="note-332-03" xlink:href="note-332-03a" xml:space="preserve">41. primi.</note> </div> <pb o="303" file="333" n="333" rhead="LIBER SEPTIMVS."/> </div> <div xml:id="echoid-div868" type="section" level="1" n="303"> <head xml:id="echoid-head330" xml:space="preserve">THEOR. 10. PROPOS. 12.</head> <note position="right" xml:space="preserve">Inter Iſoperi-<lb/>metr{as} figur{as} <lb/>æqualia nu-<lb/>mero habent{es} <lb/>latera maxi-<lb/>ma, & æqui-<lb/>latera eſt, & <lb/>æquiangula.</note> <p> <s xml:id="echoid-s14254" xml:space="preserve">ISOPERIMETRARVM figurarum latera numero æqualia haben-<lb/>tium maxima & </s> <s xml:id="echoid-s14255" xml:space="preserve">æquilatera eſt, & </s> <s xml:id="echoid-s14256" xml:space="preserve">æquiangula.</s> <s xml:id="echoid-s14257" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14258" xml:space="preserve"><emph style="sc">Esto</emph> figura quotcunque laterum A B C D E F, maxima inter omnes toti-<lb/>dem laterum ſibi Iſoperimetras, ita vt maior dari non poſsit. </s> <s xml:id="echoid-s14259" xml:space="preserve">Dico eam eſſe æ-<lb/>quilateram, & </s> <s xml:id="echoid-s14260" xml:space="preserve">æquiangulam. </s> <s xml:id="echoid-s14261" xml:space="preserve">Sit enim ſi fieri poteſt, primum non æquilatera, <lb/>ſed ſint latera AB, BC, proximain æqualia. </s> <s xml:id="echoid-s14262" xml:space="preserve">Ducta igitur recta AC, <anchor type="note" xlink:href="" symbol="a"/> ſi conſtitua- <anchor type="note" xlink:label="note-333-02a" xlink:href="note-333-02"/> tur ſuper AC, triangulũ Iſoſceles AGC, quod <lb/> <anchor type="figure" xlink:label="fig-333-01a" xlink:href="fig-333-01"/> ſit iſoperimetrum triangulo ABC; </s> <s xml:id="echoid-s14263" xml:space="preserve">erit tota fi-<lb/> <anchor type="note" xlink:label="note-333-03a" xlink:href="note-333-03"/> gura AGCDEF. </s> <s xml:id="echoid-s14264" xml:space="preserve">Iſoperimetra figurę ABCD-<lb/>EF. </s> <s xml:id="echoid-s14265" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et quia triangulum AGC, maius eſt tri- angulo ABC; </s> <s xml:id="echoid-s14266" xml:space="preserve">ſi addatur commune polygo-<lb/>num ACDEF, erit ſigura AGCDEF, maior <lb/>quam figura ABCDEF. </s> <s xml:id="echoid-s14267" xml:space="preserve">quod eſt contrarium <lb/>hypotheſi. </s> <s xml:id="echoid-s14268" xml:space="preserve">Non ergo inæqualia ſunt latera <lb/>AB, BC, ſed æqualia. </s> <s xml:id="echoid-s14269" xml:space="preserve">Eademq; </s> <s xml:id="echoid-s14270" xml:space="preserve">ratione oſten-<lb/>demus, latera proxima BC, CD; </s> <s xml:id="echoid-s14271" xml:space="preserve">Item proxima <lb/>deinceps æqualia eſſe. </s> <s xml:id="echoid-s14272" xml:space="preserve">Maxima igitur figura <lb/>inter ſibi iſoperimetras æqualia numero late-<lb/>ra habentes æquilatera eſt, quod eſt primum.</s> <s xml:id="echoid-s14273" xml:space="preserve"/> </p> <div xml:id="echoid-div868" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-333-02" xlink:href="note-333-02a" xml:space="preserve">7. hui{us}.</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/xxxxxxxx/figures/333-01"/> </figure> <note symbol="b" position="right" xlink:label="note-333-03" xlink:href="note-333-03a" xml:space="preserve">8. hui{us}.</note> </div> <p> <s xml:id="echoid-s14274" xml:space="preserve"><emph style="sc">Sit</emph> deinde, ſi fieri poteſt, figura ABCDEF, <lb/>æquilatera quidem, vt iam demonſtratum eſt, <lb/>at non æquiangula, ſed anguli B, D, non pro-<lb/>ximi inæquales ſint, maiorque angulus B, <lb/>quam angulus D. </s> <s xml:id="echoid-s14275" xml:space="preserve">Quo niamigitur demonſtra-<lb/>tum eſt, figuram maximam eſſe æquilateram, <lb/>erunt duo triangula ABC, CDE, Iſoſcelia, ita <lb/>vt duo latera AB, BC, æqualia ſint duobus la-<lb/>teribus CD, DE: </s> <s xml:id="echoid-s14276" xml:space="preserve">Ponitur autem angulus B, <lb/>maior angulo D; </s> <s xml:id="echoid-s14277" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit recta AC, maior quam <anchor type="note" xlink:label="note-333-04a" xlink:href="note-333-04"/> recta CE. </s> <s xml:id="echoid-s14278" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Si igitur conſtituantur ſuper baſes <anchor type="note" xlink:label="note-333-05a" xlink:href="note-333-05"/> AC, CE, alia duo triangula Iſoſcelia AGC, CHE, ſimilia inter ſe, & </s> <s xml:id="echoid-s14279" xml:space="preserve">Iſoperime-<lb/>tra triangulis ABC, CDE; </s> <s xml:id="echoid-s14280" xml:space="preserve">erunt triangula AGC, CHE, vtra que ſimul maiora <lb/> <anchor type="note" xlink:label="note-333-06a" xlink:href="note-333-06"/> triangulis ABC, CDE, vtriſque ſimul. </s> <s xml:id="echoid-s14281" xml:space="preserve">Si igitur addatur commune polygonum <lb/>ACEF: </s> <s xml:id="echoid-s14282" xml:space="preserve">erit figura AGCHEF, maior, quam figura ABCDEF, quod cum hypo-<lb/>theſi pugnat, quod hæc omnium maxima ponatur. </s> <s xml:id="echoid-s14283" xml:space="preserve">Non ergo inæquales ſunt <lb/>anguli B, D, ſed æquales. </s> <s xml:id="echoid-s14284" xml:space="preserve">Eademque ratione oſtendemus, angulos non pro-<lb/>ximos C, E, æquales eſſe, & </s> <s xml:id="echoid-s14285" xml:space="preserve">binos alios quo ſuis non proximos. </s> <s xml:id="echoid-s14286" xml:space="preserve">Ex quo effici-<lb/>tur, totam figuram æquiangulam eſſe, nempe proximos etiam angulos inter <lb/>fe eſſe æquales. </s> <s xml:id="echoid-s14287" xml:space="preserve">Si enim verbi gratia angulus B, non dicatur æqualis eſſe an-<lb/>gulo C; </s> <s xml:id="echoid-s14288" xml:space="preserve">cum angulus C, æqualis ſit non proximo angulo E; </s> <s xml:id="echoid-s14289" xml:space="preserve">erit quo que an-<lb/>gulus B, angulo E, non æqualis, quod abſurdum eſt. </s> <s xml:id="echoid-s14290" xml:space="preserve">Bini enim anguli non pro-<lb/>ximi inter ſe æquales ſunt, vt oſtendimus. </s> <s xml:id="echoid-s14291" xml:space="preserve">Maxima ergo figura inter ſibi Iſope- <pb o="304" file="334" n="334" rhead="GEOMETR. PRACT."/> rimetras æqualia numero latera habentes non ſolum æquilatera, ſed & </s> <s xml:id="echoid-s14292" xml:space="preserve">æ-<lb/>quiangula eſt. </s> <s xml:id="echoid-s14293" xml:space="preserve">Quocirca Iſoperimetrarum figurarum latera numero æqua-<lb/>lia habentium maxima & </s> <s xml:id="echoid-s14294" xml:space="preserve">æquilatera eſt, & </s> <s xml:id="echoid-s14295" xml:space="preserve">æquiangula. </s> <s xml:id="echoid-s14296" xml:space="preserve">quod demonſtran-<lb/>dum erat.</s> <s xml:id="echoid-s14297" xml:space="preserve"/> </p> <div xml:id="echoid-div869" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-333-04" xlink:href="note-333-04a" xml:space="preserve">24. prim.</note> <note symbol="d" position="right" xlink:label="note-333-05" xlink:href="note-333-05a" xml:space="preserve">10. hui{us}.</note> <note symbol="e" position="right" xlink:label="note-333-06" xlink:href="note-333-06a" xml:space="preserve">11. hui{us}.</note> </div> </div> <div xml:id="echoid-div871" type="section" level="1" n="304"> <head xml:id="echoid-head331" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s14298" xml:space="preserve"><emph style="sc">Circa</emph> demonſtrationem prioris partis huius propoſ. </s> <s xml:id="echoid-s14299" xml:space="preserve">obſeruandum eſt, <lb/> <anchor type="note" xlink:label="note-334-01a" xlink:href="note-334-01"/> accipienda eſſe duo latera inæqualia proxima inter ſe, ita vt angulum conſti-<lb/>tuant, nullumque aliud inter ea interponatur, qualia ſunt latera accepta AB, <lb/>BC, angulum B, efficientia. </s> <s xml:id="echoid-s14300" xml:space="preserve">Hac enim ratione, ducta recta AC, factum erit <lb/>triangulum ABC, cuius duo latera AB, BC, inæqualia ſunt, vtin demonſtra-<lb/>ne aſſumebatur. </s> <s xml:id="echoid-s14301" xml:space="preserve">Neque verò dubitare quis poterit, in figura non æquilatera, <lb/>qualis ponitur ABCDEF, accipi poſſe duo latera proxima inæqualia. </s> <s xml:id="echoid-s14302" xml:space="preserve">Nam ſi <lb/>quis dicat latera AB, BC, eſſe æqualia, ſumemus latera AB, AF, quæ ſi dican-<lb/>tur etiam æqualia eſſe, accipiemus AF, FE: </s> <s xml:id="echoid-s14303" xml:space="preserve">Et ſi hæc adhuc æqualia eſſe di-<lb/>cantur, capiemus EF, ED; </s> <s xml:id="echoid-s14304" xml:space="preserve">& </s> <s xml:id="echoid-s14305" xml:space="preserve">ſic deinceps progrediemur; </s> <s xml:id="echoid-s14306" xml:space="preserve">donec ad duo la-<lb/>tera proxima inæqualia veniamus, quæ angulum conſtituant. </s> <s xml:id="echoid-s14307" xml:space="preserve">Neceſſariò au-<lb/>tem ad duo huiuſmo di latera perueniemus: </s> <s xml:id="echoid-s14308" xml:space="preserve">alias figura eſſet æquilatera, quod <lb/>non conceditur</s> </p> <div xml:id="echoid-div871" type="float" level="2" n="1"> <note position="left" xlink:label="note-334-01" xlink:href="note-334-01a" xml:space="preserve">Quæ obſer-<lb/>uanda ſint in <lb/>demonſtratio-<lb/>ne hui{us} pro-<lb/>poſ.</note> </div> <p> <s xml:id="echoid-s14309" xml:space="preserve"><emph style="sc">Qvod</emph> verò ad poſterioris partis demonſtrationem attinet, aduertendum <lb/>eſt, in figuris multilateris accipiendos eſſe duos angulos inæquales non proxi-<lb/>mos inter ſe, ita vt inter ipſos vnus, vel plures anguli interponantur, quales ſunt <lb/>anguli accepti B, D, inter quos ponitur angulus C. </s> <s xml:id="echoid-s14310" xml:space="preserve">Hac enim ratione duæ rectę <lb/>AC, CE, dictos angulos ſubtendentes ſe mutuò non interſecabunt, conſtituen-<lb/>turque duæ figuræ ABCDEF, AGCHEF, ex additione communis figuræ <lb/>ACEF, ad triangula ſupra baſes AC, CE, conſtructa: </s> <s xml:id="echoid-s14311" xml:space="preserve">quod non contingeret, ſi <lb/>duo anguliinæquales proximi inter ſe ſumerentur, vt conſtat. </s> <s xml:id="echoid-s14312" xml:space="preserve">Non eſt autem in <lb/>dubitum vertendũ, an tales duo anguli poſsint accipi. </s> <s xml:id="echoid-s14313" xml:space="preserve">In omni enim figura mul-<lb/>tilatera non æquiangula neceſſariò erunt aliqui duo anguli non proximi inter <lb/>ſe inæquales. </s> <s xml:id="echoid-s14314" xml:space="preserve">Nam in propoſita figura ABCDEF, comparabimus angulum B, <lb/>cum omnibus non proximis angulis D, E, F, quineceſſariò duo erunt in penta-<lb/>gono, in hexagono verò tres, & </s> <s xml:id="echoid-s14315" xml:space="preserve">ita deinceps. </s> <s xml:id="echoid-s14316" xml:space="preserve">Quod ſi vni alicui eorum fue-<lb/>rit inæqualis, habebimus iam duos angulos non proximos inter ſe inæquales, <lb/>nempe angulum B, & </s> <s xml:id="echoid-s14317" xml:space="preserve">illum, cui inæqualis eſt: </s> <s xml:id="echoid-s14318" xml:space="preserve">Si verò omnibus dicatur æqua-<lb/>lis, erit tunc angulus B, ſaltem alteri proximorum inæqualis, alias figura eſſet <lb/>æquiangula. </s> <s xml:id="echoid-s14319" xml:space="preserve">Siergo inæqualis fuerit angulo A, erit angulus A, tam angulo E, <lb/>quam angulo D; </s> <s xml:id="echoid-s14320" xml:space="preserve">non proximo inæqualis, cum vtriuis horum æqualis ponatur <lb/>angulus B: </s> <s xml:id="echoid-s14321" xml:space="preserve">Si verò inæqualis fuerit angulo C, erit angulus C, tam angulo E, <lb/>quam angulo F, non proximo inæqualis, quod vtriuis horum angulus B, pona-<lb/>tur æqualis.</s> <s xml:id="echoid-s14322" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14323" xml:space="preserve"><emph style="sc">Sed</emph> quoniam propoſitio hæc demonſtrata tantum eſt in figuris multilate-<lb/>ris, vtex iis conſtat, quæ proximè de duobus angulis non proximis inæquali-<lb/>bus diximus: </s> <s xml:id="echoid-s14324" xml:space="preserve">In triangulis enim, & </s> <s xml:id="echoid-s14325" xml:space="preserve">quadrilateris figuris anguli eiuſmo di repe-<lb/>riri non poſſunt, cum in triangulis æquilateris omnes anguli ſint æquales, vt ex <lb/>coroll. </s> <s xml:id="echoid-s14326" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14327" xml:space="preserve">5. </s> <s xml:id="echoid-s14328" xml:space="preserve">lib. </s> <s xml:id="echoid-s14329" xml:space="preserve">1. </s> <s xml:id="echoid-s14330" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s14331" xml:space="preserve">patet; </s> <s xml:id="echoid-s14332" xml:space="preserve">in quadrilateris autem figuris omnia late-<lb/>ra habentibus æqualia, (quoniam neceſſario ſunt parallelo gramma, vt in ſcho- <pb o="305" file="335" n="335" rhead="LIBER SEPTIMVS."/> lio propoſ. </s> <s xml:id="echoid-s14333" xml:space="preserve">34. </s> <s xml:id="echoid-s14334" xml:space="preserve">lib. </s> <s xml:id="echoid-s14335" xml:space="preserve">1. </s> <s xml:id="echoid-s14336" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s14337" xml:space="preserve">oſtendimus) <anchor type="note" xlink:href="" symbol="a"/> ſinguli oppoſiti inter ſeſint æqua- <anchor type="note" xlink:label="note-335-01a" xlink:href="note-335-01"/> les: </s> <s xml:id="echoid-s14338" xml:space="preserve">Idcirco totam hanc propoſitionem in triangulis, & </s> <s xml:id="echoid-s14339" xml:space="preserve">quadrilateris figuris ita <lb/> <anchor type="figure" xlink:label="fig-335-01a" xlink:href="fig-335-01"/> demonſtrabimus. </s> <s xml:id="echoid-s14340" xml:space="preserve">Sit primum triangulum ABC, inter ſibi Iſoperimetra triangu-<lb/>la maximum. </s> <s xml:id="echoid-s14341" xml:space="preserve">Dico illud æquilaterum eſſe & </s> <s xml:id="echoid-s14342" xml:space="preserve">æquiangulum. </s> <s xml:id="echoid-s14343" xml:space="preserve">Si enim non eſt <lb/> <anchor type="note" xlink:label="note-335-02a" xlink:href="note-335-02"/> æquilaterum, ſed latera AB, BC, ſuntinæqualia, <anchor type="note" xlink:href="" symbol="b"/> ſi ſuper baſem AC, conſtitua- tur triangulum Iſoſceles ADC, ita vt latera AD, DC, ſimul æqualia ſint lateri-<lb/>bus AB, BC, ſimul; </s> <s xml:id="echoid-s14344" xml:space="preserve">erunt triangula ABC, ADC, Iſoperimetra; </s> <s xml:id="echoid-s14345" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> atque adeo <anchor type="note" xlink:label="note-335-03a" xlink:href="note-335-03"/> ADC, maius, quam ABC, quod eſt contra hypotheſim. </s> <s xml:id="echoid-s14346" xml:space="preserve">Non ergo inæqualia <lb/>ſunt latera AB, BC, ſed æqualia. </s> <s xml:id="echoid-s14347" xml:space="preserve">Eademque ratio eſt de cæteris. </s> <s xml:id="echoid-s14348" xml:space="preserve">Æquilaterum <lb/>ergo eſt triangulum ABC. </s> <s xml:id="echoid-s14349" xml:space="preserve">Igitur, ex coroll. </s> <s xml:id="echoid-s14350" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s14351" xml:space="preserve">5. </s> <s xml:id="echoid-s14352" xml:space="preserve">lib. </s> <s xml:id="echoid-s14353" xml:space="preserve">1. </s> <s xml:id="echoid-s14354" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s14355" xml:space="preserve">& </s> <s xml:id="echoid-s14356" xml:space="preserve">æquian-<lb/>gulum eſt. </s> <s xml:id="echoid-s14357" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s14358" xml:space="preserve"/> </p> <div xml:id="echoid-div872" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-335-01" xlink:href="note-335-01a" xml:space="preserve">34. primi.</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/xxxxxxxx/figures/335-01"/> </figure> <note symbol="b" position="right" xlink:label="note-335-02" xlink:href="note-335-02a" xml:space="preserve">7. hui{us}.</note> <note symbol="c" position="right" xlink:label="note-335-03" xlink:href="note-335-03a" xml:space="preserve">8. hui{us}.</note> </div> <p> <s xml:id="echoid-s14359" xml:space="preserve"><emph style="sc">Deinde</emph> ſit quadrilaterum ABCD, inter omnia ſibi Iſoperimetra maxi-<lb/>mum. </s> <s xml:id="echoid-s14360" xml:space="preserve">Dico illud eſſe, & </s> <s xml:id="echoid-s14361" xml:space="preserve">æquilaterum, & </s> <s xml:id="echoid-s14362" xml:space="preserve">æquiangulum. </s> <s xml:id="echoid-s14363" xml:space="preserve">Si enim non eſt æqui-<lb/>laterum, ſint latera AB, BC, ſi fieri poteſt, inæqualia, ducaturq; </s> <s xml:id="echoid-s14364" xml:space="preserve">recta AC. </s> <s xml:id="echoid-s14365" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Si <anchor type="note" xlink:label="note-335-04a" xlink:href="note-335-04"/> igitur ſuper AC, conſtituatur triangulum Iſoſceles, AEC, Iſoperimetrum trian-<lb/>gulo ABC; </s> <s xml:id="echoid-s14366" xml:space="preserve">erit triangulum AEC, maius triangulo ABC. </s> <s xml:id="echoid-s14367" xml:space="preserve">Addito ergo com-<lb/>muni triangulo ACD, <anchor type="note" xlink:href="" symbol="e"/> erit quadrilaterum AECD, maius quadrilatero ABCD.</s> <s xml:id="echoid-s14368" xml:space="preserve"> <anchor type="note" xlink:label="note-335-05a" xlink:href="note-335-05"/> quod eſt contra hypotheſim, cum ABCD, maximum ponatur. </s> <s xml:id="echoid-s14369" xml:space="preserve">Non ergo inæ-<lb/>qualia ſunt latera AB, BC, ſed æqualia. </s> <s xml:id="echoid-s14370" xml:space="preserve">Eademque ratio eſt de cæteris. </s> <s xml:id="echoid-s14371" xml:space="preserve">Æqui-<lb/>latera ergo eſt figura ABCD.</s> <s xml:id="echoid-s14372" xml:space="preserve"/> </p> <div xml:id="echoid-div873" type="float" level="2" n="3"> <note symbol="d" position="right" xlink:label="note-335-04" xlink:href="note-335-04a" xml:space="preserve">7. hui{us}.</note> <note symbol="e" position="right" xlink:label="note-335-05" xlink:href="note-335-05a" xml:space="preserve">8. hui{us}.</note> </div> <p> <s xml:id="echoid-s14373" xml:space="preserve"><emph style="sc">Sit</emph>iam quadrilatera figura ABCD, omnium iſoperimetrarum maxima, æ-<lb/>quilatera, vt oſtenſum eſt, at non æquiangula, ſed anguli BAD, CDA, inæqua-<lb/>les ſint. </s> <s xml:id="echoid-s14374" xml:space="preserve">Quoniam igitur figura ABCD, cum ſit æquilatera, parallelogrammum <lb/>eſt, vt in ſcholio propoſ. </s> <s xml:id="echoid-s14375" xml:space="preserve">34. </s> <s xml:id="echoid-s14376" xml:space="preserve">lib. </s> <s xml:id="echoid-s14377" xml:space="preserve">1. </s> <s xml:id="echoid-s14378" xml:space="preserve">demonſtrauimus; </s> <s xml:id="echoid-s14379" xml:space="preserve">neuter que angulorum A, <lb/>D, rectus eſt; </s> <s xml:id="echoid-s14380" xml:space="preserve">(alias, <anchor type="note" xlink:href="" symbol="f"/> cum ambo duobus rectis ſint æquales, eſſent ambo recti) <anchor type="note" xlink:label="note-335-06a" xlink:href="note-335-06"/> ſed vnus acutus, & </s> <s xml:id="echoid-s14381" xml:space="preserve">obtuſus alter: </s> <s xml:id="echoid-s14382" xml:space="preserve">ſi educantur ex A, & </s> <s xml:id="echoid-s14383" xml:space="preserve">D, duæ lineæ perpen-<lb/>diculares AH, DG, occurrentes lateri BC, in H, & </s> <s xml:id="echoid-s14384" xml:space="preserve">G; </s> <s xml:id="echoid-s14385" xml:space="preserve">erit quo que AHGD, pa-<lb/>rallelogrammum. </s> <s xml:id="echoid-s14386" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Quia verò latera AB, DC, maiora ſunt lateribus AH, DG;</s> <s xml:id="echoid-s14387" xml:space="preserve"> <anchor type="note" xlink:label="note-335-07a" xlink:href="note-335-07"/> producantur hæc, vt fiant rectæ AE, DF, lateribus AB, DC, æquales, iungatur-<lb/>que recta EF. </s> <s xml:id="echoid-s14388" xml:space="preserve">Quo facto, erit figura AEFD, iſoperimetra parallelogrammo <lb/>ABCD; </s> <s xml:id="echoid-s14389" xml:space="preserve">cum latera AE, DF, lateribus AB, DC, æqualia ſint, latus verò AD, <lb/>commune, & </s> <s xml:id="echoid-s14390" xml:space="preserve">latus EF, lateri B C, æquale, <anchor type="note" xlink:href="" symbol="h"/> quod vtrumque æquale ſit lateri <anchor type="note" xlink:label="note-335-08a" xlink:href="note-335-08"/> oppoſito AD. </s> <s xml:id="echoid-s14391" xml:space="preserve">Cum ergo figura AEFD, maior ſit parallelogrammo AHGD; <lb/></s> <s xml:id="echoid-s14392" xml:space="preserve"> <anchor type="note" xlink:label="note-335-09a" xlink:href="note-335-09"/> <anchor type="note" xlink:href="" symbol="i"/> hoc autem æquale ſit parallelogrammo ABCD; </s> <s xml:id="echoid-s14393" xml:space="preserve">erit quoque figura AEFD, maior parallelogrammo ABCD. </s> <s xml:id="echoid-s14394" xml:space="preserve">Quare cum eidem ſit iſoperimetra, non erit <lb/>ABCD, figura quadrilatera inter ſibi Iſoperimetras maxima. </s> <s xml:id="echoid-s14395" xml:space="preserve">quod eſt contra <lb/>hypotheſim. </s> <s xml:id="echoid-s14396" xml:space="preserve">Non ergo inæquales ſunt anguli BAD, CDA, ſed æquales: </s> <s xml:id="echoid-s14397" xml:space="preserve">at-<lb/> <anchor type="note" xlink:label="note-335-10a" xlink:href="note-335-10"/> que adeò cum ABCD, ſit parallelogrammum, <anchor type="note" xlink:href="" symbol="k"/> erunt anguli oppoſiti B, C, angulis D, A, æquales, proptereaque tota figura æquiangula erit, quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s14398" xml:space="preserve"/> </p> <div xml:id="echoid-div874" type="float" level="2" n="4"> <note symbol="f" position="right" xlink:label="note-335-06" xlink:href="note-335-06a" xml:space="preserve">29. primi.</note> <note symbol="g" position="right" xlink:label="note-335-07" xlink:href="note-335-07a" xml:space="preserve">19. primi.</note> <note symbol="h" position="right" xlink:label="note-335-08" xlink:href="note-335-08a" xml:space="preserve">34. primi.</note> <note symbol="i" position="right" xlink:label="note-335-09" xlink:href="note-335-09a" xml:space="preserve">35. primi.</note> <note symbol="k" position="right" xlink:label="note-335-10" xlink:href="note-335-10a" xml:space="preserve">34. primi.</note> </div> <pb o="306" file="336" n="336" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div876" type="section" level="1" n="305"> <head xml:id="echoid-head332" xml:space="preserve">THEOR. 11. PROPOS. 13.</head> <p> <s xml:id="echoid-s14399" xml:space="preserve">CIRCVLVS omnibus figuris rectilineis regularibus ſibi iſoperime-<lb/> <anchor type="note" xlink:label="note-336-01a" xlink:href="note-336-01"/> tris maior eſt.</s> <s xml:id="echoid-s14400" xml:space="preserve"/> </p> <div xml:id="echoid-div876" type="float" level="2" n="1"> <note position="left" xlink:label="note-336-01" xlink:href="note-336-01a" xml:space="preserve">Circul{us} o-<lb/>mnium ſigu-<lb/>varum recti-<lb/>linearum re-<lb/>gularium ſibi <lb/>iſoperimetra-<lb/>rum maxi-<lb/>m{us} est.</note> </div> <p> <s xml:id="echoid-s14401" xml:space="preserve"><emph style="sc">Esto</emph> circulus ABC, figura autem regularis quotcunque laterum ei iſoperi-<lb/>metra DEF. </s> <s xml:id="echoid-s14402" xml:space="preserve">Dico circulum ABC, eſſe maiorem figura DEF. </s> <s xml:id="echoid-s14403" xml:space="preserve">Sit enim G, cen-<lb/>trum circuli A B C, & </s> <s xml:id="echoid-s14404" xml:space="preserve">H, centrum figuræ D E F; </s> <s xml:id="echoid-s14405" xml:space="preserve">Deſcribaturque circa circulum <lb/>ABC, figura BIKC, tot laterum, & </s> <s xml:id="echoid-s14406" xml:space="preserve">angulorum æqualium, quot continet figu-<lb/>ra DEF, per ea, quæ in ſcholio propoſ. </s> <s xml:id="echoid-s14407" xml:space="preserve">16. </s> <s xml:id="echoid-s14408" xml:space="preserve">lib. </s> <s xml:id="echoid-s14409" xml:space="preserve">4. </s> <s xml:id="echoid-s14410" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s14411" xml:space="preserve">docuimus. </s> <s xml:id="echoid-s14412" xml:space="preserve">Deinde ex <lb/> <anchor type="figure" xlink:label="fig-336-01a" xlink:href="fig-336-01"/> puncto contactus A, ad centrum G, ducatur recta A G, <anchor type="note" xlink:href="" symbol="a"/> quæ perpendicularis <anchor type="note" xlink:label="note-336-02a" xlink:href="note-336-02"/> erit ad IK. </s> <s xml:id="echoid-s14413" xml:space="preserve">Ducatur rurſus HD, ad LM, perpendicularis; </s> <s xml:id="echoid-s14414" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Diuidentque rectæ <anchor type="note" xlink:label="note-336-03a" xlink:href="note-336-03"/> GA, HD, rectas IK, LM, bifariã, vt conſtat, ſi figuris BIKC, DEF, circumſcriban-<lb/>tur circuli. </s> <s xml:id="echoid-s14415" xml:space="preserve">Ducantur quoque rectæ GI, HL, quæ diuident angulos I, & </s> <s xml:id="echoid-s14416" xml:space="preserve">L, bi-<lb/>fariam, vt manifeſtum eſt ex demonſtratione propoſ. </s> <s xml:id="echoid-s14417" xml:space="preserve">12. </s> <s xml:id="echoid-s14418" xml:space="preserve">lib. </s> <s xml:id="echoid-s14419" xml:space="preserve">4. </s> <s xml:id="echoid-s14420" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s14421" xml:space="preserve">Quo-<lb/>niam igitur toti anguli I, & </s> <s xml:id="echoid-s14422" xml:space="preserve">L, ſunt æquales, propter ſimilitudinem figurarum, <lb/>erunt etiamip ſorum dimidia (videlicet anguli AIG, DLH,) æqualia. </s> <s xml:id="echoid-s14423" xml:space="preserve">Cum er-<lb/>go & </s> <s xml:id="echoid-s14424" xml:space="preserve">anguli I A G, L D H, ſintæ quales, vtpote recti; </s> <s xml:id="echoid-s14425" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt triangula A I H, <anchor type="note" xlink:label="note-336-04a" xlink:href="note-336-04"/> DLH, æquiangula. </s> <s xml:id="echoid-s14426" xml:space="preserve">Quia verò ambitus figuræ B I K C, maior eſt (per 1. </s> <s xml:id="echoid-s14427" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s14428" xml:space="preserve">libr. </s> <s xml:id="echoid-s14429" xml:space="preserve">1. </s> <s xml:id="echoid-s14430" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s14431" xml:space="preserve">cylindro) ambitu circuli A B C; </s> <s xml:id="echoid-s14432" xml:space="preserve">Ambitus au-<lb/>tem circuli æqualis ponitur ambitui figuræ D E F; </s> <s xml:id="echoid-s14433" xml:space="preserve">erit quo que ambitus figuræ <lb/>BIKC, maior ambitus figuræ DEF. </s> <s xml:id="echoid-s14434" xml:space="preserve">Cum igitur figuræ ſint regulares, & </s> <s xml:id="echoid-s14435" xml:space="preserve">ſimiles, <lb/>erit etiam latus IK, latere LM, maius; </s> <s xml:id="echoid-s14436" xml:space="preserve">& </s> <s xml:id="echoid-s14437" xml:space="preserve">ideò I A, dimidium lateris IK, maius, <lb/>quam LD, dimidium lateris LM. </s> <s xml:id="echoid-s14438" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Rurſus quoniam eſt, vt IA, ad G A, ita L D, <anchor type="note" xlink:label="note-336-05a" xlink:href="note-336-05"/> ad D H; </s> <s xml:id="echoid-s14439" xml:space="preserve">Et eſt I A, maior quam L D; </s> <s xml:id="echoid-s14440" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit quo que A G, maior, quam D H.</s> <s xml:id="echoid-s14441" xml:space="preserve"> <anchor type="note" xlink:label="note-336-06a" xlink:href="note-336-06"/> Quamobrem rectangulum contentum ſub A G, & </s> <s xml:id="echoid-s14442" xml:space="preserve">dimidio ambitu circuli <lb/>ABC, <anchor type="note" xlink:href="" symbol="f"/> quod circulo ABC, eſt æquale, maius eſt, quam rectangulum conten- <anchor type="note" xlink:label="note-336-07a" xlink:href="note-336-07"/> tum ſub DH, & </s> <s xml:id="echoid-s14443" xml:space="preserve">dimidio ambitu figuræ DEF, hoc eſt, <anchor type="note" xlink:href="" symbol="g"/> quam area figuræ DEF.</s> <s xml:id="echoid-s14444" xml:space="preserve"> <anchor type="note" xlink:label="note-336-08a" xlink:href="note-336-08"/> Circulus igitur omnibus figuris rectilineis regularibus ſibi iſoperimetris maior <lb/>eſt. </s> <s xml:id="echoid-s14445" xml:space="preserve">quod oſtendendum erat.</s> <s xml:id="echoid-s14446" xml:space="preserve"/> </p> <div xml:id="echoid-div877" 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/xxxxxxxx/figures/336-01"/> </figure> <note symbol="a" position="left" xlink:label="note-336-02" xlink:href="note-336-02a" xml:space="preserve">18. tertii.</note> <note symbol="b" position="left" xlink:label="note-336-03" xlink:href="note-336-03a" xml:space="preserve">3. tertii.</note> <note symbol="c" position="left" xlink:label="note-336-04" xlink:href="note-336-04a" xml:space="preserve">32. primi.</note> <note symbol="d" position="left" xlink:label="note-336-05" xlink:href="note-336-05a" xml:space="preserve">4. ſexti.</note> <note symbol="e" position="left" xlink:label="note-336-06" xlink:href="note-336-06a" xml:space="preserve">14. quinti.</note> <note symbol="f" position="left" xlink:label="note-336-07" xlink:href="note-336-07a" xml:space="preserve">4. hui{us}.</note> <note symbol="g" position="left" xlink:label="note-336-08" xlink:href="note-336-08a" xml:space="preserve">2. hui{us}.</note> </div> </div> <div xml:id="echoid-div879" type="section" level="1" n="306"> <head xml:id="echoid-head333" xml:space="preserve">COROLLARIVM.</head> <note position="left" xml:space="preserve">Circul{us} o-<lb/>mnib{us} figu-<lb/>ris rectilin{eis} <lb/>ſibi iſoperime-<lb/>tris maior eſt.</note> <p> <s xml:id="echoid-s14447" xml:space="preserve">EX omnibus iis, quæ demonſtrata ſunt, perſpicuum eſt, circulum ab-<lb/>ſolutè omnium figurarum rectilinearum ſibi iſoperimetrarum maxi-<lb/>mum eſſe.</s> <s xml:id="echoid-s14448" xml:space="preserve"/> </p> <pb o="307" file="337" n="337" rhead="LIBER SEPTIMVS."/> <p> <s xml:id="echoid-s14449" xml:space="preserve"><emph style="sc">Qvoniam</emph> enim ex propoſitione 5. </s> <s xml:id="echoid-s14450" xml:space="preserve">habetur, regularium figurarum iſope-<lb/>rimetrarum eam, quæ plura latera continet, eſſe maiorem: </s> <s xml:id="echoid-s14451" xml:space="preserve">Rurſus ex propoſi-<lb/>tione 12. </s> <s xml:id="echoid-s14452" xml:space="preserve">conſtat, inter omnes figuras iſoperimetras æqualia numero latera ha-<lb/>bentes, eam maximam eſſe, quæ regularis eſt: </s> <s xml:id="echoid-s14453" xml:space="preserve">Ex hac denique 13. </s> <s xml:id="echoid-s14454" xml:space="preserve">propoſitio-<lb/>ne perſpicuum eſt, circulum omnium figurarum iſoperimetrarum regularium <lb/>eſſe maximum: </s> <s xml:id="echoid-s14455" xml:space="preserve">Manifeſtè concluditur, circulum abſolutè ac ſimpliciter o-<lb/>mnium figurarum rectilinearum ſibi iſoperimetrarum maximum eſſe. </s> <s xml:id="echoid-s14456" xml:space="preserve">quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s14457" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div880" type="section" level="1" n="307"> <head xml:id="echoid-head334" xml:space="preserve">THEOR. 12. PROPOS. 14.</head> <p> <s xml:id="echoid-s14458" xml:space="preserve">AREA cuiuslibet pyramidis æqualis eſt ſolido rectangulo contento <lb/> <anchor type="note" xlink:label="note-337-01a" xlink:href="note-337-01"/> ſub perpendiculari à vertice ad baſim protracta, & </s> <s xml:id="echoid-s14459" xml:space="preserve">tertia parte baſis.</s> <s xml:id="echoid-s14460" xml:space="preserve"/> </p> <div xml:id="echoid-div880" type="float" level="2" n="1"> <note position="right" xlink:label="note-337-01" xlink:href="note-337-01a" xml:space="preserve">Pyramis quæ-<lb/>lib{et} cui pa-<lb/>rallelepipedo <lb/>ſit æqualis.</note> </div> <p> <s xml:id="echoid-s14461" xml:space="preserve"><emph style="sc">Sit</emph> pyramis, cuius baſis quotcunque laterum ABCDE, & </s> <s xml:id="echoid-s14462" xml:space="preserve">vertex F. </s> <s xml:id="echoid-s14463" xml:space="preserve">Soli-<lb/>dum autem rectangulum G N, cuius baſis G H I K, æqualis ſit tertiæ parti baſis <lb/>A B C D E; </s> <s xml:id="echoid-s14464" xml:space="preserve">altitudo verò ſiue perpendicularis <lb/> <anchor type="figure" xlink:label="fig-337-01a" xlink:href="fig-337-01"/> GL, æqualis altitudini pyramidis, ſiue perpen-<lb/>diculari à vertice pyramidis ad eius baſim pro-<lb/>ductæ. </s> <s xml:id="echoid-s14465" xml:space="preserve">Dico ſolidum rectangulum GN, æqua-<lb/>le eſſe, pyramidi A B C D E F. </s> <s xml:id="echoid-s14466" xml:space="preserve">Ducantur enim <lb/>ab omnibus angulis baſis G H I K, ad aliquod <lb/>punctum baſis oppoſitæ, nimirum ad L, lineæ <lb/>rectæ, ita vt conſtituatur pyramis G H I K L, <lb/>eandem habens baſim cum ſolido G N, ean-<lb/>demque altitudinem & </s> <s xml:id="echoid-s14467" xml:space="preserve">cum eodem ſolido <lb/>G N, & </s> <s xml:id="echoid-s14468" xml:space="preserve">cum pyramide A B C D E F. </s> <s xml:id="echoid-s14469" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quo- <anchor type="note" xlink:label="note-337-02a" xlink:href="note-337-02"/> niam igitur pyramis A B C D E F, tripla eſt py-<lb/>ramidis GHIKL; </s> <s xml:id="echoid-s14470" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et ſolidum G N, triplum <anchor type="note" xlink:label="note-337-03a" xlink:href="note-337-03"/> quo que eſt eiuſdem pyramidis GHIKL: </s> <s xml:id="echoid-s14471" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit <anchor type="note" xlink:label="note-337-04a" xlink:href="note-337-04"/> ſolidum G N, pyramidi A B C D E F, æquale. <lb/></s> <s xml:id="echoid-s14472" xml:space="preserve">Quapropter area cuiuslibet pyramidis æqua-<lb/>lis eſt ſolido rectangulo, &</s> <s xml:id="echoid-s14473" xml:space="preserve">c. </s> <s xml:id="echoid-s14474" xml:space="preserve">quod erat oſten-<lb/>dendum.</s> <s xml:id="echoid-s14475" xml:space="preserve"/> </p> <div xml:id="echoid-div881" 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/xxxxxxxx/figures/337-01"/> </figure> <note symbol="a" position="right" xlink:label="note-337-02" xlink:href="note-337-02a" xml:space="preserve">ſchol. 6. <lb/>duodec.</note> <note symbol="b" position="right" xlink:label="note-337-03" xlink:href="note-337-03a" xml:space="preserve">coroll. 7. <lb/>duodec.</note> <note symbol="c" position="right" xlink:label="note-337-04" xlink:href="note-337-04a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div883" type="section" level="1" n="308"> <head xml:id="echoid-head335" xml:space="preserve">THEOR. 13. PROPOS. 15.</head> <note position="right" xml:space="preserve">Corp{us} quod-<lb/>libet, in qua <lb/>ſphæra deſcri-<lb/>bipotest, cui <lb/>parallelepipe-<lb/>do æquale ſit.</note> <p> <s xml:id="echoid-s14476" xml:space="preserve">AREA cuiuslibet corporis planis ſuperficiebus contenti, & </s> <s xml:id="echoid-s14477" xml:space="preserve">circa <lb/>ſphæram aliquam circumſcriptibilis, hoc eſt, à cuius puncto aliquo <lb/>medio omnes perpendiculares ad baſes eius productæ ſunt æquales, <lb/>æqualis eſt ſolido rectangulo contento ſub vna perpendicularium, <lb/>& </s> <s xml:id="echoid-s14478" xml:space="preserve">tertia parte ambitus corporis.</s> <s xml:id="echoid-s14479" xml:space="preserve"/> </p> <pb o="308" file="338" n="338" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s14480" xml:space="preserve"><emph style="sc">Esto</emph> corpus planis ſuperficiebus contentum A B C D, circa ſphæram <lb/>EFGH, cuius centrum I, deſcriptum, in quo ducantur ex I, ad puncta conta-<lb/>ctuum lineæ rectæ IE, IF, IG, IH, quæ ad baſes ſolidi erunt perpendiculares. </s> <s xml:id="echoid-s14481" xml:space="preserve">Nam <lb/>ſi verbi gratia per rectam IE, ducatur planum faciens in ſphæra, per propoſ. </s> <s xml:id="echoid-s14482" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s14483" xml:space="preserve"> <anchor type="note" xlink:label="note-338-01a" xlink:href="note-338-01"/> lib. </s> <s xml:id="echoid-s14484" xml:space="preserve">1. </s> <s xml:id="echoid-s14485" xml:space="preserve">Theod. </s> <s xml:id="echoid-s14486" xml:space="preserve">circulum EFGH, <anchor type="note" xlink:href="" symbol="a"/> & </s> <s xml:id="echoid-s14487" xml:space="preserve">in baſi rectam AB; </s> <s xml:id="echoid-s14488" xml:space="preserve">tanget circulus EFGH, rectam A B, in puncto E, propterea quod ſphæra baſim non ſecat, ſed tangit. <lb/></s> <s xml:id="echoid-s14489" xml:space="preserve"> <anchor type="note" xlink:label="note-338-02a" xlink:href="note-338-02"/> <anchor type="note" xlink:href="" symbol="b"/>Igitur IE, ad rectam AB, perpendicularis erit. </s> <s xml:id="echoid-s14490" xml:space="preserve">Eadem ratione, ſi per I E, duca- tur aliud planum, à priori differens, fiet alius circulus in ſphæra, & </s> <s xml:id="echoid-s14491" xml:space="preserve">alia linea recta <lb/>in eadem baſi ſecans rectam A B, in E; </s> <s xml:id="echoid-s14492" xml:space="preserve">ad quam et-<lb/> <anchor type="figure" xlink:label="fig-338-01a" xlink:href="fig-338-01"/> <anchor type="note" xlink:label="note-338-03a" xlink:href="note-338-03"/> iam I E, perpendicularis erit: </s> <s xml:id="echoid-s14493" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Ac propterea IE, ad baſim ſolidi per illas rectas ductam perpendicularis <lb/>erit. </s> <s xml:id="echoid-s14494" xml:space="preserve">Non aliter oſtendemus, rectas IF, IG, IH, ad alias <lb/>baſes eſſe perpendiculares. </s> <s xml:id="echoid-s14495" xml:space="preserve">Sit quo que ſolidum re-<lb/>ctangulum L R, cuius baſes KLMN, ſit æqualis ter-<lb/>tiæ parti ambitus corporis ABCD; </s> <s xml:id="echoid-s14496" xml:space="preserve">altitudo verò ſi-<lb/>ue perpendicularis L P, æqualis vni perpendicula-<lb/>rium ex centro I, ad baſes corporis A B C D, caden-<lb/>tium; </s> <s xml:id="echoid-s14497" xml:space="preserve">quæ omnes inter ſe æquales ſunt ex defin. <lb/></s> <s xml:id="echoid-s14498" xml:space="preserve">ſphæræ. </s> <s xml:id="echoid-s14499" xml:space="preserve">Dico ſolidum LR, corpori ABCD, æquale <lb/>eſſe. </s> <s xml:id="echoid-s14500" xml:space="preserve">Ducantur enim ex centro I, ad omnes angulos <lb/>corporis ABCD, rectælineæ, vt totum corpus in py-<lb/>ramides, ex quibus componitur, diuidatur: </s> <s xml:id="echoid-s14501" xml:space="preserve">quarum <lb/>quidem pyramidum baſes eædem ſunt quæ corpo-<lb/> <anchor type="note" xlink:label="note-338-04a" xlink:href="note-338-04"/> ris, vertex autem communis centrum I. </s> <s xml:id="echoid-s14502" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam igitur quælibet harum pyramidum æqualis eſt ſolido <lb/>rectangulo ſub perpendiculari L P, quæ ſingulis perpendicularibus corporis <lb/>ABCD, æqualis ponitur, & </s> <s xml:id="echoid-s14503" xml:space="preserve">tertia parte ſuæ baſis contento; </s> <s xml:id="echoid-s14504" xml:space="preserve">Si fiant tot ſolida <lb/>rectangula, quot ſunt pyramides, erunt omnia hæc ſimul æqualia ſolido rectan-<lb/>gulo LR. </s> <s xml:id="echoid-s14505" xml:space="preserve">(Si enim rectangulum K L M N, diuidatur in tot rectangula, quot <lb/>baſes ſunt in ſolido propoſito, ita vt primum æquale ſit tertiæ parti vnius baſis, <lb/>& </s> <s xml:id="echoid-s14506" xml:space="preserve">ſecundum tertiæ parti alterius, & </s> <s xml:id="echoid-s14507" xml:space="preserve">ita deinceps, quando quidem totum rectan-<lb/>gulum K L M N, æquale ponitur tertiæ parti totius ambitus ſolidi; </s> <s xml:id="echoid-s14508" xml:space="preserve">intelligan-<lb/>tur autem ſuper illa rectangula conſtitui parallelepipeda; </s> <s xml:id="echoid-s14509" xml:space="preserve">erunt omnia ſimul <lb/> <anchor type="note" xlink:label="note-338-05a" xlink:href="note-338-05"/> æqualia parallelepipedo L R.) </s> <s xml:id="echoid-s14510" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Cum ergo ſingula parallelepipeda ſingulis pyramidibus ſint æqualia; </s> <s xml:id="echoid-s14511" xml:space="preserve">erunt quo que omnes pyramides, nempe corpus <lb/>A B C D, ex illis compoſitum, æquale ſolido rectangulo L R. </s> <s xml:id="echoid-s14512" xml:space="preserve">Quamobrem <lb/>area cuiuslibet corporis planis ſuperficiebus contenti, &</s> <s xml:id="echoid-s14513" xml:space="preserve">c. </s> <s xml:id="echoid-s14514" xml:space="preserve">quod demonſtran-<lb/>dum erat.</s> <s xml:id="echoid-s14515" xml:space="preserve"/> </p> <div xml:id="echoid-div883" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-338-01" xlink:href="note-338-01a" xml:space="preserve">3. vndec.</note> <note symbol="b" position="left" xlink:label="note-338-02" xlink:href="note-338-02a" xml:space="preserve">18. tertii.</note> <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/xxxxxxxx/figures/338-01"/> </figure> <note symbol="c" position="left" xlink:label="note-338-03" xlink:href="note-338-03a" xml:space="preserve">4. vndec.</note> <note symbol="d" position="left" xlink:label="note-338-04" xlink:href="note-338-04a" xml:space="preserve">14. hui{us}.</note> <note symbol="e" position="left" xlink:label="note-338-05" xlink:href="note-338-05a" xml:space="preserve">14. hui{us}.</note> </div> </div> <div xml:id="echoid-div885" type="section" level="1" n="309"> <head xml:id="echoid-head336" xml:space="preserve">THEOR. 14. PROPOS. 16.</head> <p> <s xml:id="echoid-s14516" xml:space="preserve">AREA cuiuslibet ſphæræ æqualis eſt ſolido rectangulo comprehenſo <lb/> <anchor type="note" xlink:label="note-338-06a" xlink:href="note-338-06"/> ſub ſemidiametro ſphæræ, & </s> <s xml:id="echoid-s14517" xml:space="preserve">tertia parte ambitus ſphæræ.</s> <s xml:id="echoid-s14518" xml:space="preserve"/> </p> <div xml:id="echoid-div885" type="float" level="2" n="1"> <note position="left" xlink:label="note-338-06" xlink:href="note-338-06a" xml:space="preserve">Sphæræ quæ-<lb/>libet cui pa-<lb/>rallelepipedo <lb/>ſit æqualis.</note> </div> <p> <s xml:id="echoid-s14519" xml:space="preserve"><emph style="sc">Esto</emph> ſphæra ABC, cuius centrum D, ſemidiameter AD: </s> <s xml:id="echoid-s14520" xml:space="preserve">Solidum autem <lb/>rectangulum E, contentũ ſub ſemidiametro AD, & </s> <s xml:id="echoid-s14521" xml:space="preserve">tertia parte ambitus ſphærę, <pb o="309" file="339" n="339" rhead="LIBER SEPTIMVS."/> ABC. </s> <s xml:id="echoid-s14522" xml:space="preserve">Dico corpus E, ſphæræ ABC, eſſe æquale. </s> <s xml:id="echoid-s14523" xml:space="preserve">Nam ſi non eſt æquale: </s> <s xml:id="echoid-s14524" xml:space="preserve">ſit, ſi <lb/>fieri poteſt, primum maius, ſitque exceſſus corporis E, ſupra ſphæram A B C, <lb/>quantitas F. </s> <s xml:id="echoid-s14525" xml:space="preserve">Intelligatur circa centrum D, deſcripta ſphæra GHK, maior quam <lb/>ſphæra ABC, ita tamen, vt exceſſus ſphæræ GHK, ſupra ſphęram ABC, non ſit <lb/>maior quantitate F, ſed vel æqualis, vel minor, hoc eſt, vt ſphæra GHK, ſit vel ę-<lb/>qualis ſolido E, quando nimirum ipſa excedit ſphærã A B C, præciſè quantitate <lb/>F; </s> <s xml:id="echoid-s14526" xml:space="preserve">vel minor, ſi nimirum ipſa excedit ſphæram A B C, minori quãtitate, quam F. <lb/></s> <s xml:id="echoid-s14527" xml:space="preserve">Neceſſario enim aliqua ſphæra erit, quæ vel æqualis ſit magnitudini E, atq; </s> <s xml:id="echoid-s14528" xml:space="preserve">ad-<lb/>eo maior quam ſphæra ABC; </s> <s xml:id="echoid-s14529" xml:space="preserve">vel maior quidem quam ſphæra ABC, minor verò <lb/> <anchor type="note" xlink:label="note-339-01a" xlink:href="note-339-01"/> quam magnitudo E, quæ maior ponitur, quam ſphæra ABC. </s> <s xml:id="echoid-s14530" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Inſcribatur de- inde intra ſphæram GHK, corpus, quod non tangat ſphæram A B C, ita vt vna-<lb/>quæque perpendicularium ex centro D, ad baſes iſtius corporis eductarum ma-<lb/>ior ſit ſemidiametro A D. </s> <s xml:id="echoid-s14531" xml:space="preserve">Siigitur à centro D, ad omnes angulos dicti corporis <lb/>ducantur lineæ rectæ, vt to tum corpus in pyramides diuidatur, quarum baſes <lb/>ſunt eædem, quæ corporis G H K, vertex autem communis centrum D; </s> <s xml:id="echoid-s14532" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit <anchor type="note" xlink:label="note-339-02a" xlink:href="note-339-02"/> quælibet pyramis æqualis ſolido rectangulo contento ſub eius perpendiculari, <lb/>& </s> <s xml:id="echoid-s14533" xml:space="preserve">tertia parte baſis; </s> <s xml:id="echoid-s14534" xml:space="preserve">Atq; </s> <s xml:id="echoid-s14535" xml:space="preserve">idcirco ſolidum <lb/> <anchor type="figure" xlink:label="fig-339-01a" xlink:href="fig-339-01"/> rectangulum contentum ſub ſemidiame-<lb/>tro AD, & </s> <s xml:id="echoid-s14536" xml:space="preserve">tertia parte baſis cuiuslibet py-<lb/>ramidis, minus ipſa pyramide erit. </s> <s xml:id="echoid-s14537" xml:space="preserve">Et quo-<lb/>niam omnia ſolida rectangula cõtenta ſub <lb/>ſingulis perpendicularibus ex centro D, ad <lb/>baſes corporis dicti protractis, & </s> <s xml:id="echoid-s14538" xml:space="preserve">ſingulis <lb/>tertijs partibus baſium, ſimul æqualia ſunt <lb/>toti corpori; </s> <s xml:id="echoid-s14539" xml:space="preserve">efficiunt autem omnes tertiæ <lb/>partes baſium ſimul tertiam partem ambi-<lb/>tus corporis; </s> <s xml:id="echoid-s14540" xml:space="preserve">erit ſolidum rectangulũ con-<lb/>tẽtum ſub ſemidiametro AD, & </s> <s xml:id="echoid-s14541" xml:space="preserve">tertia par-<lb/>te ambitus præfati corporis inſcripti intra <lb/>ſphæram G H K, minus corpore inſcripto. <lb/></s> <s xml:id="echoid-s14542" xml:space="preserve">Quoniam verò ambitus corporis inſcripti <lb/>maior eſt ambitu ſphęræ ABC, vt demon-<lb/>ſtrat Archimedes lib. </s> <s xml:id="echoid-s14543" xml:space="preserve">1. </s> <s xml:id="echoid-s14544" xml:space="preserve">de ſphęra & </s> <s xml:id="echoid-s14545" xml:space="preserve">cylin-<lb/>dro propoſ. </s> <s xml:id="echoid-s14546" xml:space="preserve">27. </s> <s xml:id="echoid-s14547" xml:space="preserve">atque adeo & </s> <s xml:id="echoid-s14548" xml:space="preserve">tertia pars <lb/>ambitus dicti corporis maior tertia parte ambitus ſphęrę ABC: </s> <s xml:id="echoid-s14549" xml:space="preserve">erit ſolidum re-<lb/>ctangulum contentum ſub ſemidiametro AD, & </s> <s xml:id="echoid-s14550" xml:space="preserve">tertia parte ambitus ſphęrę A-<lb/>B C, hoc eſt, ſolidum E, multo minus corpore inſcripto intra ſphęram G H K: </s> <s xml:id="echoid-s14551" xml:space="preserve"><lb/>Poſita eſt autem ſphæra GHK, vel æqualis ſolido E, vel minor. </s> <s xml:id="echoid-s14552" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14553" xml:space="preserve">ſphę-<lb/>ra G H K, minor erit corpore intra ipſam deſcripto, totum parte quod eſt ab-<lb/>ſurdum. </s> <s xml:id="echoid-s14554" xml:space="preserve">Quo circa ſolidum E, maius non erit ſphęra ABC.</s> <s xml:id="echoid-s14555" xml:space="preserve"/> </p> <div xml:id="echoid-div886" type="float" level="2" n="2"> <note symbol="a" position="right" xlink:label="note-339-01" xlink:href="note-339-01a" xml:space="preserve">17. duodec.</note> <note symbol="b" position="right" xlink:label="note-339-02" xlink:href="note-339-02a" xml:space="preserve">14. hui{us}.</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/xxxxxxxx/figures/339-01"/> </figure> </div> <p> <s xml:id="echoid-s14556" xml:space="preserve"><emph style="sc">Sit</emph> <emph style="sc">Deinde</emph>, ſi fieri poteſt, ſolidum E, minus, quam ſphęra ABC, exce-<lb/>datur que à ſphęra ABC, quantitate F. </s> <s xml:id="echoid-s14557" xml:space="preserve">Intelligatur circa centrum D, ſphęra de-<lb/>ſcripta L M N, minor quàm ſphęra A B C, ita tamen vt exceſſus, quo ſphęra <lb/>LMN, ſuperatur à ſphęra ABC, non ſit maior quantitate F, ſed vel æqualis, vel <lb/>minor, hoc eſt, vt ſphęra LMN, ſit vel æqualis ſolido E, ſi nimirum ipſa exceda-<lb/>tur à ſphęra ABC, quantitate F, vel maior ſolido E, ſi videlicet ſphęra L M N, à <lb/>ſphęra ABC, ſuperetur minori quantitate, quam F. </s> <s xml:id="echoid-s14558" xml:space="preserve">Neceſſario enim aliqua ſphę- <pb o="310" file="340" n="340" rhead="GEOMETR. PRACT."/> ra erit, quæ vel ęqualis ſit ſolido E, atque adeo minor quam ſphęra ABC, vel <lb/>minor quidem quàm ſphæra ABC, maior verò quam magnitudo E, quæ minor <lb/>ponitur, quam ſphęra ABC. </s> <s xml:id="echoid-s14559" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Deſcribatur deinde intra ſphæram ABC, corpus, <anchor type="note" xlink:label="note-340-01a" xlink:href="note-340-01"/> quod minimè tangat ſphæram L M N; </s> <s xml:id="echoid-s14560" xml:space="preserve">ita vt vnaquæque perpendicularium ex <lb/>centro D, ad baſes huius corporis inſcripti cadentium minor ſit ſemidiametro <lb/>AD. </s> <s xml:id="echoid-s14561" xml:space="preserve">Siigitur à centro D, ad omnes eius angulos lineæ extendantur, vt totum <lb/>corpus in pyramides reſoluatur, quarũ baſes ſunt eædem, quæ corporis ABC, <lb/>vertex autem communis centrum D; </s> <s xml:id="echoid-s14562" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit quęlibet pyramis ęqualis ſolido re- <anchor type="note" xlink:label="note-340-02a" xlink:href="note-340-02"/> ctangulo contento ſub eius perpendiculari, & </s> <s xml:id="echoid-s14563" xml:space="preserve">tertia parte baſis; </s> <s xml:id="echoid-s14564" xml:space="preserve">Et ideo ſoli-<lb/>dum rectangulum contentum ſub ſemidiametro A D, & </s> <s xml:id="echoid-s14565" xml:space="preserve">tertia parte baſis cu-<lb/>iuſuis pyramidis, maius erit pyramide ipſa. </s> <s xml:id="echoid-s14566" xml:space="preserve">Et quoniam omnia ſolida rectan-<lb/>gula contenta ſunt ſingulis perpendicularibus ex centro D, ad baſes corporis <lb/>dicti protractis, & </s> <s xml:id="echoid-s14567" xml:space="preserve">ſingulis tertijs partibus baſium, ſimul æqualia ſunt toti cor-<lb/>pori; </s> <s xml:id="echoid-s14568" xml:space="preserve">efficiunt autem omnes tertiæ partes baſium ſimul tertiam partem ambi-<lb/>tus corporis, erit ſolidum rectangulum contentum ſub ſemidiametro A D, & </s> <s xml:id="echoid-s14569" xml:space="preserve"><lb/>tertia parte ambitus dicti corporis ſphęræ ABC, inſcripti, maius corpore inſcri-<lb/>pto. </s> <s xml:id="echoid-s14570" xml:space="preserve">Cum igitur ambitus ſphęræ A B C, maior ſit ambitu corporis ſibi inſcri-<lb/>pti, atque adeo & </s> <s xml:id="echoid-s14571" xml:space="preserve">tertia pars ambitus ſphęræ maior tertia parte ambitus dicti <lb/>corporis; </s> <s xml:id="echoid-s14572" xml:space="preserve">erit ſolidum rectangulum contentum ſub AD, ſemidiametro, & </s> <s xml:id="echoid-s14573" xml:space="preserve">ter-<lb/>tia parte ambitus ſphęræ ABC, hoc eſt, ſolidum E, multo maius corpore inſcri-<lb/>pto intra ſphęram ABC: </s> <s xml:id="echoid-s14574" xml:space="preserve">Ponebatur autem ſphęra L M N, vel ęqualis ſolido <lb/>E, vel maior. </s> <s xml:id="echoid-s14575" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s14576" xml:space="preserve">ſphęra L M N, maior erit corpore intra ſphęram A B C, <lb/>deſcripto, pars toto, quod eſt abſurdum. </s> <s xml:id="echoid-s14577" xml:space="preserve">Non igitur ſolidum E, m@nu<unsure/>s erit ſphę-<lb/>ra ABC. </s> <s xml:id="echoid-s14578" xml:space="preserve">Cum ergo neque maius ſit oſtenſum, æquale omnino erit: </s> <s xml:id="echoid-s14579" xml:space="preserve">Ac propte-<lb/>rea area cuiuslibet ſphęræ ęqualis eſt ſolido rectangulo comprehenſo ſub ſemi-<lb/>diametro ſphęræ, & </s> <s xml:id="echoid-s14580" xml:space="preserve">tertia parte ambitus ſphęræ. </s> <s xml:id="echoid-s14581" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s14582" xml:space="preserve"/> </p> <div xml:id="echoid-div887" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-340-01" xlink:href="note-340-01a" xml:space="preserve">17. duodec.</note> <note symbol="b" position="left" xlink:label="note-340-02" xlink:href="note-340-02a" xml:space="preserve">14. hui{us}.</note> </div> </div> <div xml:id="echoid-div889" type="section" level="1" n="310"> <head xml:id="echoid-head337" xml:space="preserve">THEOR. 15. PROPOS. 17.</head> <note position="left" xml:space="preserve">Sphæra ma-<lb/>ior eſt omni-<lb/>b{us} corpori-<lb/>b{us} ſibi Iſope-<lb/>rimetris, & <lb/>circa ali{as} <lb/>ſphær{as} cir-<lb/>cumſcriptibi-<lb/>lib{us}, quæ pla <lb/>nis ſuperficie-<lb/>b{us} continen-<lb/>tur.</note> <p> <s xml:id="echoid-s14583" xml:space="preserve">SPHÆRA omnibus corporibus ſibi Iſoperimetris, quæ planis <lb/>ſuperficiebus contineantur, circaque alias ſphæras circumſcriptibilia <lb/>ſint, hoc eſt, quorum omnes perpendiculares ad baſes productæ ab <lb/>aliquo puncto medio ſint æquales, maior eſt.</s> <s xml:id="echoid-s14584" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14585" xml:space="preserve"><emph style="sc">Esto</emph> ſphæra A, cuius centrum A, & </s> <s xml:id="echoid-s14586" xml:space="preserve">ſemidiameter AB: </s> <s xml:id="echoid-s14587" xml:space="preserve">Solidum autem cir-<lb/>ca aliquam ſphæram circumſcriptibile ſibi Iſoperimetrum C, cuius vna perpen-<lb/>dicularium C D. </s> <s xml:id="echoid-s14588" xml:space="preserve">Dico ſphæram A, maior<unsure/>em eſſe ſolido C. </s> <s xml:id="echoid-s14589" xml:space="preserve">Intelligatur enim <lb/>circa ſphęram A, corpus deſcriptum ſimile prorſus ſolido C, ita vt ſingula quo-<lb/>que latera contingant ſphęram A, hoc eſt, eius perpendiculares, quarum vna <lb/>ſit AB, ſint quo que æquales, nempe ſemidiametri ſphęræ A, exiſtentes. </s> <s xml:id="echoid-s14590" xml:space="preserve">Ita que <lb/>quoniam ambitus corporis circa ſphęram A, maior eſt ambitu ſphęræ A, (per ea, <lb/>quę ab Archimede ſunt demonſtrata lib 1. </s> <s xml:id="echoid-s14591" xml:space="preserve">de ſphæra & </s> <s xml:id="echoid-s14592" xml:space="preserve">cylindro, propoſ. </s> <s xml:id="echoid-s14593" xml:space="preserve">27.) <lb/></s> <s xml:id="echoid-s14594" xml:space="preserve">erit quoque eiuſdem corporis ambitus maior ambitu corpori C. </s> <s xml:id="echoid-s14595" xml:space="preserve">Quare per-<lb/>pendicularis AB, hoc eſt ſemidiameter ſphærę A, maior erit perpendiculari CD.</s> <s xml:id="echoid-s14596" xml:space="preserve"> <pb o="311" file="341" n="341" rhead="LIBER SEPTIMVS."/> Quamobrẽ rectangulũ ſolidũ contentũſub ſemidia-<lb/> <anchor type="note" xlink:label="note-341-01a" xlink:href="note-341-01"/> metro A B, & </s> <s xml:id="echoid-s14597" xml:space="preserve">tetria parte ambitus ſphęræ A, <anchor type="note" xlink:href="" symbol="a"/> quod <anchor type="figure" xlink:label="fig-341-01a" xlink:href="fig-341-01"/> ſphęræ A, ęquale eſt, maius erit, quàm rectangulum <lb/>ſolidum contentum ſub perpendiculari CD, & </s> <s xml:id="echoid-s14598" xml:space="preserve">tertia <lb/> <anchor type="note" xlink:label="note-341-02a" xlink:href="note-341-02"/> parte ambitus corporis C, <anchor type="note" xlink:href="" symbol="b"/> hoc eſt, quam corpus C.</s> <s xml:id="echoid-s14599" xml:space="preserve"> Sphęra igitur omnibus corporibus ſibi Iſoperime-<lb/>tris, quę planis ſuperficiebus contineantur, &</s> <s xml:id="echoid-s14600" xml:space="preserve">c. </s> <s xml:id="echoid-s14601" xml:space="preserve">ma-<lb/>ior eſt. </s> <s xml:id="echoid-s14602" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s14603" xml:space="preserve"/> </p> <div xml:id="echoid-div889" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-341-01" xlink:href="note-341-01a" xml:space="preserve">16. hui{us}.</note> <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/xxxxxxxx/figures/341-01"/> </figure> <note symbol="b" position="right" xlink:label="note-341-02" xlink:href="note-341-02a" xml:space="preserve">15. hui{us}.</note> </div> <note position="right" xml:space="preserve">Sphæra ma-<lb/>ior eſt quouis <lb/>corpore regu-<lb/>lari ſibi Iſope-<lb/>rimetro.</note> </div> <div xml:id="echoid-div891" type="section" level="1" n="311"> <head xml:id="echoid-head338" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s14604" xml:space="preserve"><emph style="sc">Constat</emph> hinc, ſphęram maiorem eſſe quoli-<lb/> <anchor type="figure" xlink:label="fig-341-02a" xlink:href="fig-341-02"/> bet corpore regulariſibi Iſoperimetro: </s> <s xml:id="echoid-s14605" xml:space="preserve">quip pe cum <lb/>omnes perpendiculares è centro ad baſes corporis <lb/>regularis inter ſe ęquales ſint; </s> <s xml:id="echoid-s14606" xml:space="preserve">propterea quod ęqua-<lb/> <anchor type="note" xlink:label="note-341-04a" xlink:href="note-341-04"/> les ſunt ſemidiametro ſphęrę, <anchor type="note" xlink:href="" symbol="c"/> quæintra corpus de- ſcribipoteſt.</s> <s xml:id="echoid-s14607" xml:space="preserve"/> </p> <div xml:id="echoid-div891" type="float" level="2" n="1"> <figure xlink:label="fig-341-02" xlink:href="fig-341-02a"> <image file="341-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/341-02"/> </figure> <note symbol="c" position="right" xlink:label="note-341-04" xlink:href="note-341-04a" xml:space="preserve">21. quinti-<lb/>dec.</note> </div> </div> <div xml:id="echoid-div893" type="section" level="1" n="312"> <head xml:id="echoid-head339" xml:space="preserve">THEOR. 16. PROPOS. 18.</head> <note position="right" xml:space="preserve">Sphæramaior <lb/>eſt omnib{us} <lb/>corporib{us} ſi-<lb/>bi Iſoperime-<lb/>tris, & circa <lb/>ali{as} ſphær{as} <lb/>circumſcripti <lb/>bilib<emph style="sub">9</emph>, quæ co-<lb/>nicis ſuperfi-<lb/>cieb{us} conti-<lb/>nentur.</note> <p> <s xml:id="echoid-s14608" xml:space="preserve">SPHÆRA omnibus corporibus ſibi Iſoperimetris, & </s> <s xml:id="echoid-s14609" xml:space="preserve">circa alias ſphę-<lb/>ras circumſcriptibilibus, quæ ſuperficiebus conicis contineantur, ita <lb/>vt latera omnia conica ſint æqualia, maior eſt.</s> <s xml:id="echoid-s14610" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14611" xml:space="preserve"><emph style="sc">Esto</emph> circulus A B C D, cui circumſcribatur figura regularis E F G-<lb/>HIKLM, ita vt numerus laterum à quaternario menſuretur, cuiuſmodi <lb/>eſt quadratum, figura 8. </s> <s xml:id="echoid-s14612" xml:space="preserve">12. </s> <s xml:id="echoid-s14613" xml:space="preserve">16. </s> <s xml:id="echoid-s14614" xml:space="preserve">20. </s> <s xml:id="echoid-s14615" xml:space="preserve">24. </s> <s xml:id="echoid-s14616" xml:space="preserve">vel 28. </s> <s xml:id="echoid-s14617" xml:space="preserve">laterum, angulorumque <lb/> <anchor type="figure" xlink:label="fig-341-03a" xlink:href="fig-341-03"/> æqualium, &</s> <s xml:id="echoid-s14618" xml:space="preserve">c. </s> <s xml:id="echoid-s14619" xml:space="preserve">Ducaturque ex angulo E, per centrum ad angulum I, recta <lb/>EI. </s> <s xml:id="echoid-s14620" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s14621" xml:space="preserve">ſi circa manentem rectam EI, immobilem circumagatur planũ, in quo <lb/>eſt circulus ABCD, & </s> <s xml:id="echoid-s14622" xml:space="preserve">figura EFGHIKLM, deſcribet circul<emph style="sub">9</emph> ſphærã, figura vero <pb o="312" file="342" n="342" rhead="GEOMETR. PRACT."/> corpus circa ſphęram conicis ſuperficiebus contentum, quarum ſuperficierum <lb/>latera æqualia ſunt, nemp è eadem, quę figuræ, vt ab Archimede demonſtra-<lb/>tur propoſ. </s> <s xml:id="echoid-s14623" xml:space="preserve">22. </s> <s xml:id="echoid-s14624" xml:space="preserve">& </s> <s xml:id="echoid-s14625" xml:space="preserve">27. </s> <s xml:id="echoid-s14626" xml:space="preserve">lib. </s> <s xml:id="echoid-s14627" xml:space="preserve">1. </s> <s xml:id="echoid-s14628" xml:space="preserve">de ſphęra & </s> <s xml:id="echoid-s14629" xml:space="preserve">cylindro. </s> <s xml:id="echoid-s14630" xml:space="preserve">Sit iam ſphæra N, Iſoperi-<lb/>metra corpori EFGHIKLM, circa ſphęram A B C D, deſcripto. </s> <s xml:id="echoid-s14631" xml:space="preserve">Dico ſphęram <lb/>N, dicto corpore eſſe maiorem. </s> <s xml:id="echoid-s14632" xml:space="preserve">Quoniam enim ambitus ſolidi EF GHIKLM, <lb/>maior eſt (per propoſ. </s> <s xml:id="echoid-s14633" xml:space="preserve">27. </s> <s xml:id="echoid-s14634" xml:space="preserve">lib. </s> <s xml:id="echoid-s14635" xml:space="preserve">1. </s> <s xml:id="echoid-s14636" xml:space="preserve">Archimedis deſphęra & </s> <s xml:id="echoid-s14637" xml:space="preserve">cylindro) ambitu ſphę-<lb/>rę ABCD, erit quoque ambitus ſphęrę N, maior ambitu ſphęrę ABCD; </s> <s xml:id="echoid-s14638" xml:space="preserve">ideoq; <lb/></s> <s xml:id="echoid-s14639" xml:space="preserve">ſemidiameter ſphęrę N, maior erit ſemidiametro ſphęrę ABCD. </s> <s xml:id="echoid-s14640" xml:space="preserve">Et quia ſuper-<lb/>ficies ſphęræ quadrupla eſt (per propoſ. </s> <s xml:id="echoid-s14641" xml:space="preserve">31. </s> <s xml:id="echoid-s14642" xml:space="preserve">lib. </s> <s xml:id="echoid-s14643" xml:space="preserve">1. </s> <s xml:id="echoid-s14644" xml:space="preserve">Archimedis de ſphęra & </s> <s xml:id="echoid-s14645" xml:space="preserve">cy-<lb/>lindro) maximi circuli in ſphęra; </s> <s xml:id="echoid-s14646" xml:space="preserve">ſi ſumat circul<emph style="sub">9</emph> O P, quadrupl<emph style="sub">9</emph> circuli maximi <lb/>in ſphęra N; </s> <s xml:id="echoid-s14647" xml:space="preserve">(quod quidem facilè fiet, ſi diameter O P, dupla ſumatur diametri <lb/> <anchor type="note" xlink:label="note-342-01a" xlink:href="note-342-01"/> maximi circuli in ſphęra N. </s> <s xml:id="echoid-s14648" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim vt circulus O P, ad circulum ma- ximum in ſphęra N, ita quadratum diametri O P, ad quadratum diametri circuli <lb/> <anchor type="note" xlink:label="note-342-02a" xlink:href="note-342-02"/> maximi in ſphęra N; </s> <s xml:id="echoid-s14649" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Eſt autem quadrati ad quadratum proportio duplica- ta proportionis laterum homologorum, erit quo que circulus O P, ad circulum <lb/>maximum in ſphęra N, in proportione duplicata proportionis diametri O P, ad <lb/>diametrum circuli maximi in ſphęra N. </s> <s xml:id="echoid-s14650" xml:space="preserve">Cum igitur diametri ponantur habere <lb/>proportionem duplam, habebunt circuli proportionem quadruplam; </s> <s xml:id="echoid-s14651" xml:space="preserve">qua-<lb/>drupla enim proportio duplicata eſt ꝓportionis duplæ, vt in his numeris appa-<lb/>ret. </s> <s xml:id="echoid-s14652" xml:space="preserve">1. </s> <s xml:id="echoid-s14653" xml:space="preserve">2. </s> <s xml:id="echoid-s14654" xml:space="preserve">4.) </s> <s xml:id="echoid-s14655" xml:space="preserve">erit circulus OP, ęqualis ſuperficiei ſphæræ N. </s> <s xml:id="echoid-s14656" xml:space="preserve">Accipiatur rurſus cir-<lb/>culus S T, æqualis circulo O P. </s> <s xml:id="echoid-s14657" xml:space="preserve">Statuatur deinde ſupra circulum S T, conus re-<lb/>ctus S T V, axem V X, æqualem habens ſemidiametro ſphæræ N: </s> <s xml:id="echoid-s14658" xml:space="preserve">item ſupra cir-<lb/>culum O P, alter conus N P Q, conſtruatur habens axem Q R, ęqualem ſemidia-<lb/> <anchor type="figure" xlink:label="fig-342-01a" xlink:href="fig-342-01"/> metro ſphęrę ABCD; </s> <s xml:id="echoid-s14659" xml:space="preserve">eritque maior altitudo coni S T V, quam coni O P Q. </s> <s xml:id="echoid-s14660" xml:space="preserve">at <lb/> <anchor type="note" xlink:label="note-342-03a" xlink:href="note-342-03"/> baſes ęquales erunt. </s> <s xml:id="echoid-s14661" xml:space="preserve">Quare conus S T V, maior erit cono O P Q; </s> <s xml:id="echoid-s14662" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> propterea quod coni æqualium baſium eaminter ſe habent proportionem, quam altitudi-<lb/>nes. </s> <s xml:id="echoid-s14663" xml:space="preserve">Quoniam verò ſphęra N, quadrupla eſt eius coni, qui baſem habet ęqua-<lb/>lem maximo in ſphęra N, circulo, & </s> <s xml:id="echoid-s14664" xml:space="preserve">altitudinem ęqualem ſemidiametro ſphęrę <lb/>N, vt demonſtrauit Archimedes lib. </s> <s xml:id="echoid-s14665" xml:space="preserve">1. </s> <s xml:id="echoid-s14666" xml:space="preserve">de ſphęra & </s> <s xml:id="echoid-s14667" xml:space="preserve">cylindro propoſ. </s> <s xml:id="echoid-s14668" xml:space="preserve">32. </s> <s xml:id="echoid-s14669" xml:space="preserve">Huius <lb/>autem eiuſdem coni quadruplus eſt conus S T V, <anchor type="note" xlink:href="" symbol="d"/> eo quod coni eandem habẽ- <anchor type="note" xlink:label="note-342-04a" xlink:href="note-342-04"/> tes altitu dinem proportionem habent quam baſes; </s> <s xml:id="echoid-s14670" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit conus S T V, ſphęrę <anchor type="note" xlink:label="note-342-05a" xlink:href="note-342-05"/> <pb o="313" file="343" n="343" rhead="LIBER SEPTIMVS."/> N, æqualis. </s> <s xml:id="echoid-s14671" xml:space="preserve">Eodem pacto, quia baſis coni O P Q. </s> <s xml:id="echoid-s14672" xml:space="preserve">æqualis eſt ambitui corporis <lb/>EFGHIKLM; </s> <s xml:id="echoid-s14673" xml:space="preserve">quia & </s> <s xml:id="echoid-s14674" xml:space="preserve">æqualis ſuperficiei ſphæræ N, quæ corpori illi Iſoperi-<lb/>metra eſt: </s> <s xml:id="echoid-s14675" xml:space="preserve">altitudo vero æqualis ſemidiametro ſphęræ ABCD; </s> <s xml:id="echoid-s14676" xml:space="preserve">erit ſolido EFG-<lb/>HIKLM, æqualis conus O P Q, per ea, quæ Archimedes lib 1. </s> <s xml:id="echoid-s14677" xml:space="preserve">de ſphæra & </s> <s xml:id="echoid-s14678" xml:space="preserve">cy-<lb/>lindro propoſ. </s> <s xml:id="echoid-s14679" xml:space="preserve">29. </s> <s xml:id="echoid-s14680" xml:space="preserve">demonſtrauit. </s> <s xml:id="echoid-s14681" xml:space="preserve">Quamobrem & </s> <s xml:id="echoid-s14682" xml:space="preserve">ſphæra N, maiorerit ſolido <lb/>EFGHIKLM, conicis ſuperficiebus contento. </s> <s xml:id="echoid-s14683" xml:space="preserve">Sphæra igitur omnibus cor-<lb/>poribus ſibi Iſoperimetris, & </s> <s xml:id="echoid-s14684" xml:space="preserve">circa alias ſphæras circumſcrip tibilibus, &</s> <s xml:id="echoid-s14685" xml:space="preserve">c. </s> <s xml:id="echoid-s14686" xml:space="preserve">maior <lb/>eſt. </s> <s xml:id="echoid-s14687" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s14688" xml:space="preserve"/> </p> <div xml:id="echoid-div893" type="float" level="2" n="1"> <figure xlink:label="fig-341-03" xlink:href="fig-341-03a"> <image file="341-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/341-03"/> </figure> <note symbol="a" position="left" xlink:label="note-342-01" xlink:href="note-342-01a" xml:space="preserve">2. duodec.</note> <note symbol="b" position="left" xlink:label="note-342-02" xlink:href="note-342-02a" xml:space="preserve">20. ſexti.</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/xxxxxxxx/figures/342-01"/> </figure> <note symbol="c" position="left" xlink:label="note-342-03" xlink:href="note-342-03a" xml:space="preserve">14. duodec.</note> <note symbol="d" position="left" xlink:label="note-342-04" xlink:href="note-342-04a" xml:space="preserve">11. duodec.</note> <note symbol="e" position="left" xlink:label="note-342-05" xlink:href="note-342-05a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div895" type="section" level="1" n="313"> <head xml:id="echoid-head340" xml:space="preserve">THEOR. 17. PROPOS. 19.</head> <note position="right" xml:space="preserve">Sphæramaior <lb/>eſt quolibet <lb/>cono & cy-<lb/>lindro ſibi Iſo-<lb/>perimetro.</note> <p> <s xml:id="echoid-s14689" xml:space="preserve">SPHÆRA quolibet cono, & </s> <s xml:id="echoid-s14690" xml:space="preserve">cylindro ſibi Iſoperimetro maior eſt.</s> <s xml:id="echoid-s14691" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14692" xml:space="preserve"><emph style="sc">Proposita</emph> enim quacunque ſphæra, ſi fiat conus baſem habens æqua-<lb/>lem ſuperficiei ſphærę, id eſt, quadruplam maximi in ſphæra circuli, altitudinem <lb/>verò ſemidiametro ſphæræ æqualem: </s> <s xml:id="echoid-s14693" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit ſphæra huic cono æqualis; </s> <s xml:id="echoid-s14694" xml:space="preserve">propte- <anchor type="note" xlink:label="note-343-02a" xlink:href="note-343-02"/> rea quod ad conum, cuius baſis eſt maximus in ſphæra circulus, & </s> <s xml:id="echoid-s14695" xml:space="preserve">altitudo ſe-<lb/>midiameter ſphæræ, tam ſphæra, ex propoſ. </s> <s xml:id="echoid-s14696" xml:space="preserve">32. </s> <s xml:id="echoid-s14697" xml:space="preserve">libri 1. </s> <s xml:id="echoid-s14698" xml:space="preserve">Archimedis de ſphæra & </s> <s xml:id="echoid-s14699" xml:space="preserve"><lb/>cylindro, <anchor type="note" xlink:href="" symbol="b"/> quam prior conus baſem habens quadruplã maximi circuli in ſphæ- <anchor type="note" xlink:label="note-343-03a" xlink:href="note-343-03"/> ra, hoc eſt, ſuperficiei ſphærę æqualem, & </s> <s xml:id="echoid-s14700" xml:space="preserve">altitudinem ſemidiametrum ſphæræ, <lb/>proportionem habet quadruplam. </s> <s xml:id="echoid-s14701" xml:space="preserve">Cum ergo ambitus conibaſem habentis ſu-<lb/>perficiei ſphæræ æqualem maior ſit ambitu ſphæræ, quippe cumille hunc exce-<lb/>dattota ſuperficie coni, ſecluſa baſi, quæ ambitui ſphæræ ponitur æqualis, li-<lb/>quido conſtat, ſi fiat conus ſphærę Iſoperimeter, hunc eſſe illo cono, ac proin-<lb/>de & </s> <s xml:id="echoid-s14702" xml:space="preserve">ſphęra minorem.</s> <s xml:id="echoid-s14703" xml:space="preserve"/> </p> <div xml:id="echoid-div895" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-343-02" xlink:href="note-343-02a" xml:space="preserve">9. quinti.</note> <note symbol="b" position="right" xlink:label="note-343-03" xlink:href="note-343-03a" xml:space="preserve">11. duodec.</note> </div> <p> <s xml:id="echoid-s14704" xml:space="preserve"><emph style="sc">Rvrsvs</emph> ſi fiat cylindrus baſem habens æqualem ſuperficiei ſphęræ, & </s> <s xml:id="echoid-s14705" xml:space="preserve">al-<lb/>titudinem ſemidiametrum ſphærę; </s> <s xml:id="echoid-s14706" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit hic cylindrus triplus illius coni baſem <anchor type="note" xlink:label="note-343-04a" xlink:href="note-343-04"/> habentis æqualem eidem ſuperficiei ſphęræ, & </s> <s xml:id="echoid-s14707" xml:space="preserve">altitudinem ſemidiametrum ean-<lb/>dem ſphærę, quem ſphęræ æqualem eſſe proximè oſtendimus: </s> <s xml:id="echoid-s14708" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s14709" xml:space="preserve">tri-<lb/>plusipſius ſphæræ. </s> <s xml:id="echoid-s14710" xml:space="preserve">Tertia ergo pars illius cylindri (cylindrus videlicet eandem <lb/>habens baſem, altitudinem vero tertiam partem altitudinis cylindri illius: </s> <s xml:id="echoid-s14711" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> cũ <anchor type="note" xlink:label="note-343-05a" xlink:href="note-343-05"/> ille cylindrus ſit huius triplus) æqualis erit ſphæræ. </s> <s xml:id="echoid-s14712" xml:space="preserve">Cum ergo poſterior hic <lb/>cylindrus habeat ambitum maiorẽ ambitu ſphęræ, quod ille hunc excedat am-<lb/>bitu totius cylindri, ſecluſa vna baſe; </s> <s xml:id="echoid-s14713" xml:space="preserve">quis non videt, ſi fiat cylindrus ſphęræ I-<lb/>ſoperimeter, hunc eſſe priore illo cylindro, acproinde & </s> <s xml:id="echoid-s14714" xml:space="preserve">ſp hæra maiorẽ? </s> <s xml:id="echoid-s14715" xml:space="preserve">Sphę-<lb/>ra ergo quolibet cono, & </s> <s xml:id="echoid-s14716" xml:space="preserve">cylindro ſibi Iſoperimetro maioreſt. </s> <s xml:id="echoid-s14717" xml:space="preserve">quod demon-<lb/>ſtrandum erat.</s> <s xml:id="echoid-s14718" xml:space="preserve"/> </p> <div xml:id="echoid-div896" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-343-04" xlink:href="note-343-04a" xml:space="preserve">10. duodec.</note> <note symbol="d" position="right" xlink:label="note-343-05" xlink:href="note-343-05a" xml:space="preserve">14. duode.</note> </div> </div> <div xml:id="echoid-div898" type="section" level="1" n="314"> <head xml:id="echoid-head341" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s14719" xml:space="preserve"><emph style="sc">Hæc</emph> omnia ferè ex Theone Alexandrino in commentarijs in Almageſtũ <lb/>Ptolemaei, & </s> <s xml:id="echoid-s14720" xml:space="preserve">ex Pappo Alexandrino in Mathematicis collectionibus, licet ple-<lb/>raque eorum clarius & </s> <s xml:id="echoid-s14721" xml:space="preserve">facilius demonſtrauerimus, excerpta ſunt: </s> <s xml:id="echoid-s14722" xml:space="preserve">quæ verò ſe-<lb/>quntur, à nobis inuenta ſunt, ac demonſtrata.</s> <s xml:id="echoid-s14723" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div899" type="section" level="1" n="315"> <head xml:id="echoid-head342" xml:space="preserve">PROBL. 3. PROPOS. 20.</head> <p> <s xml:id="echoid-s14724" xml:space="preserve">DATO ſemicirculo vel quadranti, vel octauæ parti circuli, aut deci- <pb o="314" file="344" n="344" rhead="GEOMETR. PRACT."/> męſextæ, &</s> <s xml:id="echoid-s14725" xml:space="preserve">c. </s> <s xml:id="echoid-s14726" xml:space="preserve">rectangulum conſtituere Iſoperimetrum & </s> <s xml:id="echoid-s14727" xml:space="preserve">æquale; </s> <s xml:id="echoid-s14728" xml:space="preserve">ſi li-<lb/> <anchor type="note" xlink:label="note-344-01a" xlink:href="note-344-01"/> nea recta periphæriæ detur æqualis.</s> <s xml:id="echoid-s14729" xml:space="preserve"/> </p> <div xml:id="echoid-div899" type="float" level="2" n="1"> <note position="left" xlink:label="note-344-01" xlink:href="note-344-01a" xml:space="preserve">Semicirculo <lb/>& aliis parti-<lb/>b{us} ſubduplis <lb/>circuli æqua-<lb/>lia rectangu-<lb/>la & Iſoperi-<lb/>metra conſti-<lb/>tuere.</note> </div> <p> <s xml:id="echoid-s14730" xml:space="preserve"><emph style="sc">Sit</emph> ſemicirculus A B C: </s> <s xml:id="echoid-s14731" xml:space="preserve">Quadrans A B D, octaua pars circuli A N D, pars <lb/>ſextadecima AOD, &</s> <s xml:id="echoid-s14732" xml:space="preserve">c. </s> <s xml:id="echoid-s14733" xml:space="preserve">conſtruatur rectangulum AE, contentum ſub ſemidia-<lb/>metro A D, & </s> <s xml:id="echoid-s14734" xml:space="preserve">ſub recta A F, quæ quartæ parti peripheriæ æqualis ſit; </s> <s xml:id="echoid-s14735" xml:space="preserve">Item re-<lb/>ctangulum A G, ſub ſemidiametro AD, & </s> <s xml:id="echoid-s14736" xml:space="preserve">ſub AH, octaua parte peripheriæ. </s> <s xml:id="echoid-s14737" xml:space="preserve">Itẽ <lb/>rectangulum A I, ſub ſemidiametro AD, & </s> <s xml:id="echoid-s14738" xml:space="preserve">ſub AK, <lb/> <anchor type="figure" xlink:label="fig-344-01a" xlink:href="fig-344-01"/> parte decimaſexta peripherię. </s> <s xml:id="echoid-s14739" xml:space="preserve">Item rectangulum AL, <lb/>ſub ſemidiametro AD, & </s> <s xml:id="echoid-s14740" xml:space="preserve">ſub AM, parte trigeſima ſe-<lb/>cunda peripheriæ. </s> <s xml:id="echoid-s14741" xml:space="preserve">Erit igitur ex ijs, quę lib. </s> <s xml:id="echoid-s14742" xml:space="preserve">4. </s> <s xml:id="echoid-s14743" xml:space="preserve">cap. <lb/></s> <s xml:id="echoid-s14744" xml:space="preserve">7. </s> <s xml:id="echoid-s14745" xml:space="preserve">ad finem Num. </s> <s xml:id="echoid-s14746" xml:space="preserve">1. </s> <s xml:id="echoid-s14747" xml:space="preserve">oſtendimus, AE, ſemicir culo ABC; </s> <s xml:id="echoid-s14748" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s14749" xml:space="preserve">AG. </s> <s xml:id="echoid-s14750" xml:space="preserve">Quadranti ABD; </s> <s xml:id="echoid-s14751" xml:space="preserve">& </s> <s xml:id="echoid-s14752" xml:space="preserve">AI, octauę parti AND: </s> <s xml:id="echoid-s14753" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s14754" xml:space="preserve">AL, parti ſextæ decimæ A O D, æquale, &</s> <s xml:id="echoid-s14755" xml:space="preserve">c. </s> <s xml:id="echoid-s14756" xml:space="preserve">Dico <lb/>hæc eadem rectangula eſſe Iſoperimetra prædictis cir-<lb/>culi partibus, ſingula ſingulis. </s> <s xml:id="echoid-s14757" xml:space="preserve">quod quidem perſpi-<lb/>cuum eſt ex conſtructione. </s> <s xml:id="echoid-s14758" xml:space="preserve">Nam AD, EF, æqualia <lb/>ſunt diametro AC, & </s> <s xml:id="echoid-s14759" xml:space="preserve">AF, DE, ſemicircumferentiæ A-<lb/>BC, nimirum duabus quartis partibus circumferentiæ. </s> <s xml:id="echoid-s14760" xml:space="preserve">Item AD, GH, æqualia <lb/>ſunt ſemidiametris A D, D B, & </s> <s xml:id="echoid-s14761" xml:space="preserve">A H, D G, duabus partibus octauis, hoc eſt, <lb/>quartæ parti cir cumferentiæ AB. </s> <s xml:id="echoid-s14762" xml:space="preserve">Item AD, IK, duabus ſemidiametris AD, DN, <lb/>& </s> <s xml:id="echoid-s14763" xml:space="preserve">AK, DI, duabus ſextis decimis, id eſt, octauæ parti circumferentiæ AN. </s> <s xml:id="echoid-s14764" xml:space="preserve">Item <lb/>AD, LM, duabus ſemidiametris AD, DO, & </s> <s xml:id="echoid-s14765" xml:space="preserve">AM, DL, duabus partibus trigeſi-<lb/>mis ſecundis, hoc eſt, parti decimæſextæ AO; </s> <s xml:id="echoid-s14766" xml:space="preserve">Atque ita deinceps, ſi tam peri-<lb/>pheria AO, quam recta AM, continue ſubdiuidatur. </s> <s xml:id="echoid-s14767" xml:space="preserve">Dato ergo ſemicirculo, <lb/>vel Quadranti, &</s> <s xml:id="echoid-s14768" xml:space="preserve">c. </s> <s xml:id="echoid-s14769" xml:space="preserve">rectangulum Iſoperimetrum, & </s> <s xml:id="echoid-s14770" xml:space="preserve">æquale conſtituimus. </s> <s xml:id="echoid-s14771" xml:space="preserve">quod <lb/>faciendum erat.</s> <s xml:id="echoid-s14772" xml:space="preserve"/> </p> <div xml:id="echoid-div900" 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/xxxxxxxx/figures/344-01"/> </figure> </div> <p> <s xml:id="echoid-s14773" xml:space="preserve"><emph style="sc">Hoc</emph> problema, quod ad ſemicirculum, ac Quadrantem attinet, aduertit <lb/>etiam nuper R. </s> <s xml:id="echoid-s14774" xml:space="preserve">P. </s> <s xml:id="echoid-s14775" xml:space="preserve">Odo Malcotius Mathematicus ingenioſus, cum problema <lb/>Mathematicum per ſuos auditores exhiberet in Collegio Romano: </s> <s xml:id="echoid-s14776" xml:space="preserve">quamuis <lb/>illud inſtar Theorematis propoſuerit.</s> <s xml:id="echoid-s14777" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div902" type="section" level="1" n="316"> <head xml:id="echoid-head343" xml:space="preserve">PROBL. 4. PROPOS. 21.</head> <note position="left" xml:space="preserve">Parallelogrã-<lb/>mum dato <lb/>triangulo æ-<lb/>quale & Iſo-<lb/>perimetrum <lb/>conſtituere.</note> <p> <s xml:id="echoid-s14778" xml:space="preserve">DATO triangulo cuicunque parallelogrammum æquale, atque Iſo-<lb/>perimetrum conſtituere.</s> <s xml:id="echoid-s14779" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14780" xml:space="preserve"><emph style="sc">Sit</emph> datum triangulum qualecunque ABC. </s> <s xml:id="echoid-s14781" xml:space="preserve">Per A, ducatur AM, baſi B C, <lb/>parallela. </s> <s xml:id="echoid-s14782" xml:space="preserve">Et quia, ſi neuter angulorum B, C, rectus eſt, <anchor type="note" xlink:href="" symbol="a"/> vtrumquelatus AB, @ maius eſt perpendiculariex A, vel B, C, D, in oppoſitam parallelam demiſ-<lb/> <anchor type="note" xlink:label="note-344-03a" xlink:href="note-344-03"/> ſa: </s> <s xml:id="echoid-s14783" xml:space="preserve">ſi verò alter angulorumrectus eſt, hoc eſt, alterutrum laterum perpendicu-<lb/>lare eſt ad dictas parallelas; </s> <s xml:id="echoid-s14784" xml:space="preserve">vtrumque latus AB, AC, ſimulmaius eſt, quam du-<lb/>plum prædictæ perpendicularis; </s> <s xml:id="echoid-s14785" xml:space="preserve">ideo que ſemiſsis aggregati ex vtro que latere <lb/>maior perpendiculari eadem; </s> <s xml:id="echoid-s14786" xml:space="preserve">id eſt, ſi accipiatur GH, lateri AB, & </s> <s xml:id="echoid-s14787" xml:space="preserve">HI, lateri AC, <lb/>ęqualis, vttota GI, ſummæ laterum AB, AC, æqualis ſit, diuidatur que GI, bifa-<lb/>riam in K, ſemiſsis GK, maior erit perpendiculari DE. </s> <s xml:id="echoid-s14788" xml:space="preserve">Si igitur ex D, medio pun- <pb o="315" file="345" n="345" rhead="LIBER SEPTIMVS."/> cto baſis B C, adinteruallũ GK, arcus circuli deſcribatur, ſecabitis rectam AM, <lb/>in aliquo puncto, vtin L. </s> <s xml:id="echoid-s14789" xml:space="preserve">Sumpta autem LM, ipſi B D, æquali, ducantur rectæ <lb/> <anchor type="figure" xlink:label="fig-345-01a" xlink:href="fig-345-01"/> DL, BM, <anchor type="note" xlink:href="" symbol="a"/> quæ parallelæ inter ſe erunt; </s> <s xml:id="echoid-s14790" xml:space="preserve">ideo que parallelo grammum erit D M, <anchor type="note" xlink:href="" symbol="b"/> <anchor type="note" xlink:label="note-345-01a" xlink:href="note-345-01"/> triangulo ABC, æquale. </s> <s xml:id="echoid-s14791" xml:space="preserve">Dico hoc idem triangulo eſſe Iſoperimetrum, quod <lb/> <anchor type="note" xlink:label="note-345-02a" xlink:href="note-345-02"/> perſpicuum eſt ex conſtructione: </s> <s xml:id="echoid-s14792" xml:space="preserve">quippe cum D L, B M, vtraque æqualis ſit <lb/>ipſi G K, hoc eſt, ſemiſsi laterum A B, A C, ac proinde ambæ D L, B M, ſimul æ-<lb/>quales ambobus lateribus AB, AC, ſimul; </s> <s xml:id="echoid-s14793" xml:space="preserve">rectæ autem BD, LM, ſimul æquales <lb/>baſi BC. </s> <s xml:id="echoid-s14794" xml:space="preserve">Conſtructum ergo eſt parallelogrammum D M, non rectangulum æ-<lb/>quale, & </s> <s xml:id="echoid-s14795" xml:space="preserve">Iſoperimetrum triangulo ABC.</s> <s xml:id="echoid-s14796" xml:space="preserve"/> </p> <div xml:id="echoid-div902" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-344-03" xlink:href="note-344-03a" xml:space="preserve">coroll. 19. <lb/>primi.</note> <figure xlink:label="fig-345-01" xlink:href="fig-345-01a"> <image file="345-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/345-01"/> </figure> <note symbol="a" position="right" xlink:label="note-345-01" xlink:href="note-345-01a" xml:space="preserve">33. primi.</note> <note symbol="b" position="right" xlink:label="note-345-02" xlink:href="note-345-02a" xml:space="preserve">ſchol. 41. <lb/>primi.</note> </div> <p> <s xml:id="echoid-s14797" xml:space="preserve"><emph style="sc">Qvod</emph> ſi optes rectangulum eidem triangulo ABC, æquale, & </s> <s xml:id="echoid-s14798" xml:space="preserve">Iſoperime-<lb/>trum, ita agendum erit. </s> <s xml:id="echoid-s14799" xml:space="preserve">Erectis perpendicularibus B F, D F, <anchor type="note" xlink:href="" symbol="c"/> erit rectangulum <anchor type="note" xlink:label="note-345-03a" xlink:href="note-345-03"/> BE, triangulo æquale, ſed non Iſoperimetrum; </s> <s xml:id="echoid-s14800" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> quod BF, DE, minores ſintla- teribus AB, AC, ſed BD, EF, baſi BC, æquales: </s> <s xml:id="echoid-s14801" xml:space="preserve">ac proinde ambitus rectanguli <lb/> <anchor type="note" xlink:label="note-345-04a" xlink:href="note-345-04"/> BE, ambitu trianguli ABC, minor; </s> <s xml:id="echoid-s14802" xml:space="preserve">ideoque ſi pro ducantur BF, DE, ad æqualita-<lb/>tem ſemiſsis laterum AB, AC, fiet quidem rectangulum triangulo ABC, Iſope-<lb/>rimetrum, ſed triangulo maius, cum ſuperet rectangulum BE. </s> <s xml:id="echoid-s14803" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/>Inuenta au- <anchor type="note" xlink:label="note-345-05a" xlink:href="note-345-05"/> tem inter BF, BD, media proportionali N; </s> <s xml:id="echoid-s14804" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> erit quadratumrectæ N, rectangu- <anchor type="note" xlink:label="note-345-06a" xlink:href="note-345-06"/> lo BE, ideo que & </s> <s xml:id="echoid-s14805" xml:space="preserve">triangulo ABC, ęquale. </s> <s xml:id="echoid-s14806" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Quia vero BF, BD, ſimul maio- <anchor type="note" xlink:label="note-345-07a" xlink:href="note-345-07"/> res ſunt, quam duplarectæ N; </s> <s xml:id="echoid-s14807" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> eſt que BM, maior quam BF, erunt BM, BD, ſi- mul multo maiores, quam dupla rectæ N. </s> <s xml:id="echoid-s14808" xml:space="preserve">Sumpta ergo QP, ipſi B D, & </s> <s xml:id="echoid-s14809" xml:space="preserve">P O, <lb/> <anchor type="note" xlink:label="note-345-08a" xlink:href="note-345-08"/> ipſi BM, æquali, vt tota QO, duabus BD, BM, ſimul ſit æqualis; </s> <s xml:id="echoid-s14810" xml:space="preserve">erit quoque <lb/>QO, maior quam dupla rectæ N. </s> <s xml:id="echoid-s14811" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/>Secetur ergo QO, in R, ita vt N, ſit inter ſe- <anchor type="note" xlink:label="note-345-09a" xlink:href="note-345-09"/> gmenta Q R, RO, media proportionalis, perficiatur que rectangulum QS, <lb/>ſub ſegmentis QR, RO, comprehenſum. </s> <s xml:id="echoid-s14812" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> quod quadrato rectæ N, hoc eſt, <anchor type="note" xlink:label="note-345-10a" xlink:href="note-345-10"/> rectangulo BE, vel triangulo ABC, æquale erit. </s> <s xml:id="echoid-s14813" xml:space="preserve">Dico idem eſſe triangulo ABC, <lb/>Iſoperimetrum. </s> <s xml:id="echoid-s14814" xml:space="preserve">Quoniam enim QR, RS, ſimul, id eſt, recta QO, æquales ſunt <lb/>rectis B D, B M, ſimul, ex conſtru ctione; </s> <s xml:id="echoid-s14815" xml:space="preserve">eruntreliquæ S T, T Q. </s> <s xml:id="echoid-s14816" xml:space="preserve">reliquis LM, <lb/>L D, æquales: </s> <s xml:id="echoid-s14817" xml:space="preserve">ideoque rectangulum QS, parallelogrammo BL, ac proinde & </s> <s xml:id="echoid-s14818" xml:space="preserve"><lb/>triangulo ABC, (cui parallelogrammum BK, Iſoperimetrum eſt oſtenſum) erit <lb/>Iſoperimetrum. </s> <s xml:id="echoid-s14819" xml:space="preserve">Dato igitur triangulo cuicunq; </s> <s xml:id="echoid-s14820" xml:space="preserve">parallelogrammum, &</s> <s xml:id="echoid-s14821" xml:space="preserve">c. </s> <s xml:id="echoid-s14822" xml:space="preserve">con-<lb/>ſtituimus. </s> <s xml:id="echoid-s14823" xml:space="preserve">quod faciendum erat.</s> <s xml:id="echoid-s14824" xml:space="preserve"/> </p> <div xml:id="echoid-div903" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-345-03" xlink:href="note-345-03a" xml:space="preserve">ſchol. 41. <lb/>primi.</note> <note symbol="d" position="right" xlink:label="note-345-04" xlink:href="note-345-04a" xml:space="preserve">coroll. 19. <lb/>primi.</note> <note symbol="e" position="right" xlink:label="note-345-05" xlink:href="note-345-05a" xml:space="preserve">13. ſexti.</note> <note symbol="f" position="right" xlink:label="note-345-06" xlink:href="note-345-06a" xml:space="preserve">17. ſexti.</note> <note symbol="g" position="right" xlink:label="note-345-07" xlink:href="note-345-07a" xml:space="preserve">ſchol. 25. <lb/>quinti.</note> <note symbol="h" position="right" xlink:label="note-345-08" xlink:href="note-345-08a" xml:space="preserve">19. primi.</note> <note symbol="i" position="right" xlink:label="note-345-09" xlink:href="note-345-09a" xml:space="preserve">ſchol. 13. <lb/>ſexti.</note> <note symbol="k" position="right" xlink:label="note-345-10" xlink:href="note-345-10a" xml:space="preserve">17 ſexti.</note> </div> <pb o="316" file="346" n="346" rhead="GEOMETR. PRACT."/> <figure> <image file="346-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/346-01"/> </figure> </div> <div xml:id="echoid-div905" type="section" level="1" n="317"> <head xml:id="echoid-head344" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s14825" xml:space="preserve"><emph style="sc">Qvod</emph> ſi ſumatur punctum V, vtc<unsure/>unque inter R, & </s> <s xml:id="echoid-s14826" xml:space="preserve">O, erit rectangulum <lb/>ſub QV, VO, adhuc Iſoperimetrum triangulo ABC, ſed minus. </s> <s xml:id="echoid-s14827" xml:space="preserve">Si vero capia-<lb/>tur punctum X, vtcunque inter R, & </s> <s xml:id="echoid-s14828" xml:space="preserve">Y, (diuiſa Q O, bifariam in Y,) erit adhuc <lb/>rectangulum ſub QX, XO, Iſoperimetrum triangulo A B C, ſed maius. </s> <s xml:id="echoid-s14829" xml:space="preserve">Qua-<lb/>dratum denique rectæ QV, Iſoperimetrum quo que eſt triangulo A B C, & </s> <s xml:id="echoid-s14830" xml:space="preserve">ma-<lb/>ius, quæ omnia ita demonſtrabimus. </s> <s xml:id="echoid-s14831" xml:space="preserve">Prædicta rectangula, & </s> <s xml:id="echoid-s14832" xml:space="preserve">quadratum rectæ <lb/>QY, Iſoperimetra eſſe triangulo ABC, hoc eſt, rectangulo QS, patet: </s> <s xml:id="echoid-s14833" xml:space="preserve">cum bi-<lb/>na latera circa angulumrectum æqualia ſemper ſintrectæ QO, hoc eſt, binis la-<lb/>teribus rectanguli QS. </s> <s xml:id="echoid-s14834" xml:space="preserve">Eadem verò eſſe inæ qualia triangulo ABC, ſic oſtendo. <lb/></s> <s xml:id="echoid-s14835" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="a"/>Quoniam quadrata QV, VO, maiora ſunt quadratis QR, RO, ſimul: </s> <s xml:id="echoid-s14836" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Sunt <anchor type="note" xlink:label="note-346-01a" xlink:href="note-346-01"/> autem tam illa duo, vna cum rectangulo ſub QV, VO, bis, quam hæc duo, vna <lb/> <anchor type="note" xlink:label="note-346-02a" xlink:href="note-346-02"/> cumrectangulo ſub QR, RO, bis, quadrato QO, æqualia; </s> <s xml:id="echoid-s14837" xml:space="preserve">erit rectangulum ſub <lb/>QV, VO, bis minus rectangulo ſub QR, R O, bis; </s> <s xml:id="echoid-s14838" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s14839" xml:space="preserve">rectangulum ſub <lb/>QV, VO, ſemel, rectangulo ſub QR, RO, ſemel minus erit. </s> <s xml:id="echoid-s14840" xml:space="preserve">Non aliter oſten-<lb/>demus, rectangulum ſub QR, R O, minus eſſe rectangulo ſub QX, XO, hoc <lb/>eſt rectangulum ſub QX, XO, maius eſſe rectangulo ſub QR, R O. </s> <s xml:id="echoid-s14841" xml:space="preserve">Denique <lb/> <anchor type="note" xlink:href="" symbol="c"/> quoniam rectangulum ſub QX, XO, vna cum quadrato XY, æquale eſt qua- <anchor type="note" xlink:label="note-346-03a" xlink:href="note-346-03"/> drato Y O, vel QY; </s> <s xml:id="echoid-s14842" xml:space="preserve">erit quadratum QY, maius rectangulo ſub QX, XO, ideo-<lb/>que multo maius rectangulo ſub QR, R O, id eſt, triangulo A B C. </s> <s xml:id="echoid-s14843" xml:space="preserve">quæ omnia <lb/>demonſtranda erant.</s> <s xml:id="echoid-s14844" xml:space="preserve"/> </p> <div xml:id="echoid-div905" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-346-01" xlink:href="note-346-01a" xml:space="preserve">lemma 42. <lb/>decimi.</note> <note symbol="b" position="left" xlink:label="note-346-02" xlink:href="note-346-02a" xml:space="preserve">4. ſecundi.</note> <note symbol="c" position="left" xlink:label="note-346-03" xlink:href="note-346-03a" xml:space="preserve">6. ſecundi.</note> </div> <p> <s xml:id="echoid-s14845" xml:space="preserve">EX quo conſtat, quadratum QY, ex ſemiſſæ rectæ QO, deſcriptum maxi-<lb/>mum eſſe omnium rectangulorum ſub quibuſcunque ſegmentis rectæ QO, cõ-<lb/>prehenſorum, quod etiam ex propoſ. </s> <s xml:id="echoid-s14846" xml:space="preserve">12. </s> <s xml:id="echoid-s14847" xml:space="preserve">huius lib. </s> <s xml:id="echoid-s14848" xml:space="preserve">liquet.</s> <s xml:id="echoid-s14849" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div907" type="section" level="1" n="318"> <head xml:id="echoid-head345" xml:space="preserve">PROBL. 5. PROPOS. 22.</head> <note position="left" xml:space="preserve">Rectangulũ <lb/>datæfiguræ <lb/>Iſoperimetrũ <lb/>& æquale cõ-<lb/>ſtituere.</note> <p> <s xml:id="echoid-s14850" xml:space="preserve">DATO rectilineo parallelogrammum rectangulum æquale, & </s> <s xml:id="echoid-s14851" xml:space="preserve">Iſope-<lb/>rimetrum conſtituere. </s> <s xml:id="echoid-s14852" xml:space="preserve">Oportet autem latus quadrati rectilineo æ-<lb/>qualis, maius non eſſe ſemiſſe dimidiati ambitus dati rectilinei.</s> <s xml:id="echoid-s14853" xml:space="preserve"/> </p> <pb o="317" file="347" n="347" rhead="LIBER SEPTIMVS."/> <p> <s xml:id="echoid-s14854" xml:space="preserve"><emph style="sc">Sit</emph> hexagonum datum A, æquilaterum quidem, ſed non æquiangulum, ita <lb/>vt B, ad latus quadrati illi æqualis <anchor type="note" xlink:href="" symbol="a"/> inuentum maius non ſit ſemiſſe, dimidiati <anchor type="note" xlink:label="note-347-01a" xlink:href="note-347-01"/> ambitus hexagoni. </s> <s xml:id="echoid-s14855" xml:space="preserve">Sumpta ergo recta C D, æquali ſemiſsi ambitus hexagoni; <lb/></s> <s xml:id="echoid-s14856" xml:space="preserve">erit B, recta non maior ſemiſſe ipſius C D, ſed vel æqualis, vel minor. </s> <s xml:id="echoid-s14857" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Secta au- <anchor type="note" xlink:label="note-347-02a" xlink:href="note-347-02"/> tem CD, in E, ita vt B, ſit media proportionalis inter ſegmenta DE, EC, fiatre-<lb/>ctangulum E G, contentum ſub ſegmentis D E, E C. </s> <s xml:id="echoid-s14858" xml:space="preserve">Dico rectangulum E G, <lb/>æquale eſſe, & </s> <s xml:id="echoid-s14859" xml:space="preserve">iſoperimetrum hexagono A. </s> <s xml:id="echoid-s14860" xml:space="preserve">Quoniam enim tres D E, B, E C, <lb/>continuè proportionales ſunt; </s> <s xml:id="echoid-s14861" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit rectangulum E G, quadrato B, id eſt, he- <anchor type="note" xlink:label="note-347-03a" xlink:href="note-347-03"/> xagono A, æquale. </s> <s xml:id="echoid-s14862" xml:space="preserve">Et quia duo latera DE, EF, æqualia ſunt rectæ CD, hoc eſt, <lb/>ſemiſsi ambitus hexagoni A, ideo que reliquæ duæ FG, GD, alteri ſemiſsi: </s> <s xml:id="echoid-s14863" xml:space="preserve">@erit <lb/>totum rectangulum E G, hexagono A, iſoperimetrum. </s> <s xml:id="echoid-s14864" xml:space="preserve">Dato ergo rectilineo <lb/>parallelogrammum rectangulum ęquale, & </s> <s xml:id="echoid-s14865" xml:space="preserve">iſo perimetrum conſtituimus: </s> <s xml:id="echoid-s14866" xml:space="preserve">quod <lb/>erat faciendum.</s> <s xml:id="echoid-s14867" xml:space="preserve"/> </p> <div xml:id="echoid-div907" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-347-01" xlink:href="note-347-01a" xml:space="preserve">14. ſecundi.</note> <note symbol="b" position="right" xlink:label="note-347-02" xlink:href="note-347-02a" xml:space="preserve">ſchol. 13. <lb/>ſexti.</note> <note symbol="c" position="right" xlink:label="note-347-03" xlink:href="note-347-03a" xml:space="preserve">17. ſexti.</note> </div> </div> <div xml:id="echoid-div909" type="section" level="1" n="319"> <head xml:id="echoid-head346" xml:space="preserve">SCHOLIVM.</head> <figure> <image file="347-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/347-01"/> </figure> <p> <s xml:id="echoid-s14868" xml:space="preserve"><emph style="sc">Qvod</emph> ſi B, latus quadrati foret maius ſemiſſe di-<lb/>midij ambitus rectilinei A, hoc eſt, maius recta CD, <lb/>non poſſet C D, ita ſecari, vt B, eſſet medio loco pro-<lb/>portionalis inter ſegmenta, vt liquidò conſtat.</s> <s xml:id="echoid-s14869" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14870" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi ſumatur punctum H, inter C, & </s> <s xml:id="echoid-s14871" xml:space="preserve">E, <lb/>vtcunque; </s> <s xml:id="echoid-s14872" xml:space="preserve">erit rectangulum ſub D H, H C, adhuc <lb/>iſoperimetrum figuræ A, ſed tamen minus. </s> <s xml:id="echoid-s14873" xml:space="preserve">Si verò ac-<lb/>cipiatur punctum I, vtcunque inter E, & </s> <s xml:id="echoid-s14874" xml:space="preserve">L, punctum <lb/>medium rectæ C D; </s> <s xml:id="echoid-s14875" xml:space="preserve">erit adhuc rectangulum ſub D I, <lb/>I C, figuræ A, iſoperimetrum, maius tamen. </s> <s xml:id="echoid-s14876" xml:space="preserve">Sic et-<lb/>iam quadratum ſemiſsis D L, erit iſo perimetrum ei-<lb/>dem figuræ & </s> <s xml:id="echoid-s14877" xml:space="preserve">maius; </s> <s xml:id="echoid-s14878" xml:space="preserve">quæ omnia demonſtrabun-<lb/>tur, vt in ſcholio præcedentis problematis dictum <lb/>eſt.</s> <s xml:id="echoid-s14879" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div910" type="section" level="1" n="320"> <head xml:id="echoid-head347" xml:space="preserve">APPENDIX.</head> <p> <s xml:id="echoid-s14880" xml:space="preserve">De circulo per lineas quadrando.</s> <s xml:id="echoid-s14881" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s14882" xml:space="preserve">1. </s> <s xml:id="echoid-s14883" xml:space="preserve"><emph style="sc">Locvs</emph> hic me admonet, vt quoniam hoc libro demonſtratum eſt, cir-<lb/>culum figurarum omnium ſibi iſoperimetrarum eſſe maximum, breuiter do-<lb/>ceam, quaratione dato circulo quadratum conſtrui poſsit æquale, & </s> <s xml:id="echoid-s14884" xml:space="preserve">viciſsim <lb/>dato quadrato circulus æqualis; </s> <s xml:id="echoid-s14885" xml:space="preserve">atqueid per lineas: </s> <s xml:id="echoid-s14886" xml:space="preserve">cum lib. </s> <s xml:id="echoid-s14887" xml:space="preserve">4. </s> <s xml:id="echoid-s14888" xml:space="preserve">cap 7. </s> <s xml:id="echoid-s14889" xml:space="preserve">copiosè <lb/> <anchor type="note" xlink:label="note-347-04a" xlink:href="note-347-04"/> traditum ſit, quo pacto ex inuentis ab Archimede, per numeros circulus qua-<lb/>drandus ſit, hoc eſt, qua ratione area circuli, ſiue capacitas tum ex diametro, tum <lb/>ex circumferentia cognita ſit inuenienda: </s> <s xml:id="echoid-s14890" xml:space="preserve">Huius enim areæ radix quadrata, la-<lb/>tus eſt quadrati, quod circulo æquale eſt. </s> <s xml:id="echoid-s14891" xml:space="preserve">Sic è contrario cap. </s> <s xml:id="echoid-s14892" xml:space="preserve">8. </s> <s xml:id="echoid-s14893" xml:space="preserve">eiuſdem lib. </s> <s xml:id="echoid-s14894" xml:space="preserve">re-<lb/>gula 1. </s> <s xml:id="echoid-s14895" xml:space="preserve">Num. </s> <s xml:id="echoid-s14896" xml:space="preserve">1. </s> <s xml:id="echoid-s14897" xml:space="preserve">docuimus qua via ex data circuli area indaganda ſit tam circum-<lb/>ferentia, quam diameter illius circuli: </s> <s xml:id="echoid-s14898" xml:space="preserve">hoc eſt, propoſito quadrato, inſtar areæ <lb/>circuli alicuius, quomodo circulus deſcribendus ſit illi quadrato æqualis. </s> <s xml:id="echoid-s14899" xml:space="preserve">In- <pb o="318" file="348" n="348" rhead="GEOMETR. PRACT."/> uenta enim diametro per prædictamregulam 1. </s> <s xml:id="echoid-s14900" xml:space="preserve">Num. </s> <s xml:id="echoid-s14901" xml:space="preserve">1. </s> <s xml:id="echoid-s14902" xml:space="preserve">cap. </s> <s xml:id="echoid-s14903" xml:space="preserve">8. </s> <s xml:id="echoid-s14904" xml:space="preserve">lib. </s> <s xml:id="echoid-s14905" xml:space="preserve">7. </s> <s xml:id="echoid-s14906" xml:space="preserve">circulus il-<lb/>lius diametri erit is, qui quæritur. </s> <s xml:id="echoid-s14907" xml:space="preserve">Viſum eſt autem appendicem hanc libro huic <lb/>ſeptimo adiungere, quod tractatio de circuli Tetragoniſmo, ſiue quadratura, <lb/>non parum affinis ſit de iſoperimetris figuris diſputationi.</s> <s xml:id="echoid-s14908" xml:space="preserve"/> </p> <div xml:id="echoid-div910" type="float" level="2" n="1"> <note position="right" xlink:label="note-347-04" xlink:href="note-347-04a" xml:space="preserve">Quo pacto re-<lb/>periatur per <lb/>numeros qua-<lb/>dratum cir-<lb/>culo æquale, <lb/>& contra ex <lb/>doctrina Ar-<lb/>chimedis.</note> </div> <p> <s xml:id="echoid-s14909" xml:space="preserve"><emph style="sc">Qvadratvra</emph> autem circuli per numeros, quam Arabes tradiderunt, & </s> <s xml:id="echoid-s14910" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-348-01a" xlink:href="note-348-01"/> quam Ioſephus Scaliger in ſuis Cyclometricis elementis veram eſſe credit, o-<lb/>mnino reiiciẽda eſt, cum ſit extra limites Archimedis, per quos conſtat, propor-<lb/>tionem circumferentiæ ad diametrum minorem debere eſſe tripla ſeſquiſepti-<lb/>ma, maiorem verò tripla ſuperdecupartiẽte ſeptuageſimas primas. </s> <s xml:id="echoid-s14911" xml:space="preserve">quod in nu-<lb/>meris Arabum non cernitur. </s> <s xml:id="echoid-s14912" xml:space="preserve">Dicunt enim proportionẽ circumferentiæ ad dia-<lb/>metrum eſſe potentia decuplam: </s> <s xml:id="echoid-s14913" xml:space="preserve">adeò vt ſi quadratum circumferentiæ pona-<lb/>tur 10. </s> <s xml:id="echoid-s14914" xml:space="preserve">quadratum diametrum ſit 1. </s> <s xml:id="echoid-s14915" xml:space="preserve">quod falſum eſt. </s> <s xml:id="echoid-s14916" xml:space="preserve">Nam cum radix quadrata <lb/>numeri 10. </s> <s xml:id="echoid-s14917" xml:space="preserve">ſit maior quam 3 {1/7}. </s> <s xml:id="echoid-s14918" xml:space="preserve">quod huius radicis quadratum ſit tantum 9 {43/49}. <lb/></s> <s xml:id="echoid-s14919" xml:space="preserve">Radix autem vnitatis ſit 1. </s> <s xml:id="echoid-s14920" xml:space="preserve">eſſet maior proportio circumferentiæ ad diametrum, <lb/>quam tripla ſeſquiſeptima: </s> <s xml:id="echoid-s14921" xml:space="preserve">cum tamen ſecundum Archimedem ſit minor. </s> <s xml:id="echoid-s14922" xml:space="preserve"><lb/>Item quia poſita diametro 7. </s> <s xml:id="echoid-s14923" xml:space="preserve">circumferentia minor eſt, quam 22. </s> <s xml:id="echoid-s14924" xml:space="preserve">ex Archimede; </s> <s xml:id="echoid-s14925" xml:space="preserve"><lb/>erit quadratum circumferentiæ minus, quam 484. </s> <s xml:id="echoid-s14926" xml:space="preserve">quod ad 49. </s> <s xml:id="echoid-s14927" xml:space="preserve">quadratum dia <lb/>metri minorem proportionem habet, quam decuplam; </s> <s xml:id="echoid-s14928" xml:space="preserve">quippe cum 490. </s> <s xml:id="echoid-s14929" xml:space="preserve">ad <lb/>49. </s> <s xml:id="echoid-s14930" xml:space="preserve">proportionem habeant decuplam. </s> <s xml:id="echoid-s14931" xml:space="preserve">Minor igitur eſt proportio quadrati cir-<lb/>cumferentiæ ad qua dratum diametri, quam decupla.</s> <s xml:id="echoid-s14932" xml:space="preserve"/> </p> <div xml:id="echoid-div911" type="float" level="2" n="2"> <note position="left" xlink:label="note-348-01" xlink:href="note-348-01a" xml:space="preserve">Circuli qua-<lb/>dratura per <lb/>numeros ſe-<lb/>cundum Ara-<lb/>b{es} falſa.</note> </div> <p> <s xml:id="echoid-s14933" xml:space="preserve"><emph style="sc">Simili</emph> modo reiicienda eſt ratio quadrandi circuli per numeros Alberti <lb/> <anchor type="note" xlink:label="note-348-02a" xlink:href="note-348-02"/> Dureri, qui exiſtimat, di<unsure/>uiſa diametro circuliin octo partes æquales, diametrum <lb/>quadrati circulo æqualis eſſe 10. </s> <s xml:id="echoid-s14934" xml:space="preserve">adeò vt diameter quadrari circulo æqualis ad <lb/>diametrum circuli proportionem habeat, quam 10. </s> <s xml:id="echoid-s14935" xml:space="preserve">ad 8. </s> <s xml:id="echoid-s14936" xml:space="preserve">quod etiam falſum eſt. <lb/></s> <s xml:id="echoid-s14937" xml:space="preserve">Nam cum quadratum diametri 10. </s> <s xml:id="echoid-s14938" xml:space="preserve">ſit 100. </s> <s xml:id="echoid-s14939" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/>duplumque quadrati, cuius diame- ter eſt 10. </s> <s xml:id="echoid-s14940" xml:space="preserve">& </s> <s xml:id="echoid-s14941" xml:space="preserve">quod circulo diametri 8. </s> <s xml:id="echoid-s14942" xml:space="preserve">dicitur æquale: </s> <s xml:id="echoid-s14943" xml:space="preserve">erit quadratum circulo æ-<lb/> <anchor type="note" xlink:label="note-348-03a" xlink:href="note-348-03"/> quale 50. </s> <s xml:id="echoid-s14944" xml:space="preserve">Sed ex diametro 8. </s> <s xml:id="echoid-s14945" xml:space="preserve">reperitur area circuli maior, quam vera, 50 {2/7}. </s> <s xml:id="echoid-s14946" xml:space="preserve">vt <lb/>cap. </s> <s xml:id="echoid-s14947" xml:space="preserve">7. </s> <s xml:id="echoid-s14948" xml:space="preserve">lib. </s> <s xml:id="echoid-s14949" xml:space="preserve">4. </s> <s xml:id="echoid-s14950" xml:space="preserve">Num. </s> <s xml:id="echoid-s14951" xml:space="preserve">4. </s> <s xml:id="echoid-s14952" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s14953" xml:space="preserve">Vera ergo circuli area maior erit, quam 50 {2/7}. <lb/></s> <s xml:id="echoid-s14954" xml:space="preserve">atque adeò multò maior, quam 50. </s> <s xml:id="echoid-s14955" xml:space="preserve">Eſt igitur quadratum Alberti minus area <lb/>circuli, non autem æquale.</s> <s xml:id="echoid-s14956" xml:space="preserve"/> </p> <div xml:id="echoid-div912" type="float" level="2" n="3"> <note position="left" xlink:label="note-348-02" xlink:href="note-348-02a" xml:space="preserve">Quadr atura <lb/>circuli per nu <lb/>meros ex Al-<lb/>berto Durero <lb/>falſa.</note> <note symbol="a" position="left" xlink:label="note-348-03" xlink:href="note-348-03a" xml:space="preserve">ſchol. 47, <lb/>primi.</note> </div> <p> <s xml:id="echoid-s14957" xml:space="preserve">2. </s> <s xml:id="echoid-s14958" xml:space="preserve"><emph style="sc">Iam</emph> verò, vt ad quadraturam circuli per lineas aggrediamur, pudet me <lb/> <anchor type="note" xlink:label="note-348-04a" xlink:href="note-348-04"/> refellere illam, quæ imperitis vera eſſe videtur, & </s> <s xml:id="echoid-s14959" xml:space="preserve">quam ſciolus, neſcio quis, <lb/>Campano Mathematico non indo cto affinxit, typiſque mandauit. </s> <s xml:id="echoid-s14960" xml:space="preserve">Eſt autem <lb/>talis. </s> <s xml:id="echoid-s14961" xml:space="preserve">Linea recta circumferentiæ circuli æqualis (quo pacto autem eiuſmo di <lb/>linea inueniatur, non docet) ſecetur in 4. </s> <s xml:id="echoid-s14962" xml:space="preserve">partes æquales, ex quibus quadratum <lb/>conſtituatur. </s> <s xml:id="echoid-s14963" xml:space="preserve">quod ſciolus ille circulo dicit eſſe æquale. </s> <s xml:id="echoid-s14964" xml:space="preserve">quæ res omnin ò Geo-<lb/>metraindigna eſt, & </s> <s xml:id="echoid-s14965" xml:space="preserve">planè ridicula. </s> <s xml:id="echoid-s14966" xml:space="preserve">Si enim quadratum illud circulo eſt Iſope-<lb/>rimetrum; </s> <s xml:id="echoid-s14967" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> circulus autem omnium figurarum rectilinearum ſibi iſo perime- <anchor type="note" xlink:label="note-348-05a" xlink:href="note-348-05"/> trarum capaciſsimus eſt, quis non videt, quadratum illud circulo minus eſſe?</s> <s xml:id="echoid-s14968" xml:space="preserve"/> </p> <div xml:id="echoid-div913" type="float" level="2" n="4"> <note position="left" xlink:label="note-348-04" xlink:href="note-348-04a" xml:space="preserve">Falſa qua-<lb/>dratura cir-<lb/>culi per line{as} <lb/>Campano a-<lb/>ſcripta.</note> <note symbol="b" position="left" xlink:label="note-348-05" xlink:href="note-348-05a" xml:space="preserve">13. hui{us}. <lb/>Quadratura <lb/>Hyppocratis <lb/>Chii.</note> </div> <p> <s xml:id="echoid-s14969" xml:space="preserve">3. </s> <s xml:id="echoid-s14970" xml:space="preserve"><emph style="sc">De</emph> Tetragoniſmo etiam Hippo cratis Chij nihil dicerem, niſi in eius de-<lb/>monſtrationè acumen ingenij lateret, quamuis metam propoſitam non attin-<lb/>gat. </s> <s xml:id="echoid-s14971" xml:space="preserve">Ita enim progreditur. </s> <s xml:id="echoid-s14972" xml:space="preserve">Sit quadrandus circulus AFBE, cuius diameter AB, <lb/> <anchor type="note" xlink:label="note-348-06a" xlink:href="note-348-06"/> ex qua deſcribatur quadratum ABCD, <anchor type="note" xlink:href="" symbol="c"/> circa quod circulus deſcribatur AB- <anchor type="note" xlink:label="note-348-07a" xlink:href="note-348-07"/> CD, cuius diameter BD, datum circulum AFBE, ſecet in E. </s> <s xml:id="echoid-s14973" xml:space="preserve">Ducta ergo recta AE, <lb/> <anchor type="note" xlink:label="note-348-08a" xlink:href="note-348-08"/> <anchor type="note" xlink:href="" symbol="d"/> erit angulus AEB, in ſemicirculo rectus, <anchor type="note" xlink:href="" symbol="e"/> ideoque perpendicularis AE, diui- det baſem B D, trianguli Iſoſcelis ABD, bifariam: </s> <s xml:id="echoid-s14974" xml:space="preserve">ac proinde E, centrum erit <lb/> <anchor type="note" xlink:label="note-348-09a" xlink:href="note-348-09"/> circuli ABCD. </s> <s xml:id="echoid-s14975" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Et quia quadratum diametri BD, duplum eſt quadrati diame- <pb o="319" file="349" n="349" rhead="LIBER SEPTIMVS."/> tri AB; </s> <s xml:id="echoid-s14976" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> eſt que vt quadratum BD, ad quadratum AB, ita circulus ABCD, ad <anchor type="note" xlink:label="note-349-01a" xlink:href="note-349-01"/> circulum AFBE: </s> <s xml:id="echoid-s14977" xml:space="preserve">erit quo que circulus circuli duplus; </s> <s xml:id="echoid-s14978" xml:space="preserve">& </s> <s xml:id="echoid-s14979" xml:space="preserve">ſemicirculus BAD, <lb/>ſemicirculi AFB; </s> <s xml:id="echoid-s14980" xml:space="preserve">ideo que ſemiſsis ſemicir culi BAD: </s> <s xml:id="echoid-s14981" xml:space="preserve">id eſt, quadrãs ABE, <anchor type="note" xlink:href="" symbol="b"/> (eſt enim ABE, quadrans, ob angulum rectum in centro E,) ſemicirculo AFB, æqua-<lb/>lis. </s> <s xml:id="echoid-s14982" xml:space="preserve">Dempto igitur communi ſegmento AGB, reliquum triangulum AFB, reli-<lb/>quæ Lunulæ A F B G A, æquale erit: </s> <s xml:id="echoid-s14983" xml:space="preserve">ac proinde ſi triangulo fiat quadratum æ-<lb/>quale erit idem hoc quadratum Lunulæ AFBGA, æquale. </s> <s xml:id="echoid-s14984" xml:space="preserve">Atque ita quadrata <lb/>eſt Lunula AFBGA.</s> <s xml:id="echoid-s14985" xml:space="preserve"/> </p> <div xml:id="echoid-div914" type="float" level="2" n="5"> <note symbol="c" position="left" xlink:label="note-348-06" xlink:href="note-348-06a" xml:space="preserve">9. quinti.</note> <note symbol="d" position="left" xlink:label="note-348-07" xlink:href="note-348-07a" xml:space="preserve">31. tertii.</note> <note symbol="e" position="left" xlink:label="note-348-08" xlink:href="note-348-08a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note symbol="f" position="left" xlink:label="note-348-09" xlink:href="note-348-09a" xml:space="preserve">ſchol. 27. <lb/>primi.</note> <note symbol="a" position="right" xlink:label="note-349-01" xlink:href="note-349-01a" xml:space="preserve">2. duodec.</note> </div> <figure> <image file="349-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/349-01"/> </figure> <p> <s xml:id="echoid-s14986" xml:space="preserve"><emph style="sc">Deinde</emph> ſitrecta HI, diametri AB, dupla, circa quam ſemicirculo deſcripto, <lb/>aptentur in eo tresrectæ ſemidiametro huius circuli, hoc eſt, diametro A B, æ-<lb/>quales HK, KL, LI, continentes ſemiſſem hexagoni: </s> <s xml:id="echoid-s14987" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> cum latus hexagoni ſit <anchor type="note" xlink:label="note-349-02a" xlink:href="note-349-02"/> ſemidiametro æquale. </s> <s xml:id="echoid-s14988" xml:space="preserve">Deſcriptis autem circa illas tres rectas ſemicirculis HMK, <lb/>KOL, LQI, qui ſemicirculo AFB, æquales ſunt, propter diametros æquales; <lb/></s> <s xml:id="echoid-s14989" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="c"/> quoniam quadratum rectæ HI, quadrati rectæ HK, quadruplum eſt. </s> <s xml:id="echoid-s14990" xml:space="preserve">quod la- <anchor type="note" xlink:label="note-349-03a" xlink:href="note-349-03"/> tus lateris ſit duplum: </s> <s xml:id="echoid-s14991" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erit quo que circulus diametri H I, circuli diametri HK, quadruplus, & </s> <s xml:id="echoid-s14992" xml:space="preserve">ſemicirculus HKLI, ſemicirculis HMK, KOL, LQI, AFB, æ-<lb/> <anchor type="note" xlink:label="note-349-04a" xlink:href="note-349-04"/> qualis erit: </s> <s xml:id="echoid-s14993" xml:space="preserve">demptiſque ſegmentis communibus HNK, KPL, LRI, reliquum <lb/>trapezium HKLI, æquale erit tribus Lunulis HNKM, KPLO, LRIQ, vna cum <lb/>ſemicirculo AFB. </s> <s xml:id="echoid-s14994" xml:space="preserve">Si igitur tres illæ Lunulæ quadrentur, vt traditum eſt, & </s> <s xml:id="echoid-s14995" xml:space="preserve">tri-<lb/>bus illis quadratis auferatur ex trapezio rectilineum æquale, hoc eſt, <anchor type="note" xlink:href="" symbol="e"/> inqui- <anchor type="note" xlink:label="note-349-05a" xlink:href="note-349-05"/> ratur exceſſus trapezii ſuper tria illa quadrata; </s> <s xml:id="echoid-s14996" xml:space="preserve">erit exceſſus hic rectilinea figura <lb/>ſemicirculo AFB, æqualis. </s> <s xml:id="echoid-s14997" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Si igitur huic figuræ quadratum fiat æquale, erit <anchor type="note" xlink:label="note-349-06a" xlink:href="note-349-06"/> idem hoc quadratum ſemicirculo A F B, æquale, & </s> <s xml:id="echoid-s14998" xml:space="preserve">quadratum ex illius qua-<lb/>drati diametro deſcriptum toti circulo AFBE, æquale. </s> <s xml:id="echoid-s14999" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> quod tam quadratum <anchor type="note" xlink:label="note-349-07a" xlink:href="note-349-07"/> quadrati duplum ſit, quam circulus ſemicirculi. </s> <s xml:id="echoid-s15000" xml:space="preserve">Quadratus ergo circulus eſt.</s> <s xml:id="echoid-s15001" xml:space="preserve"/> </p> <div xml:id="echoid-div915" type="float" level="2" n="6"> <note symbol="b" position="right" xlink:label="note-349-02" xlink:href="note-349-02a" xml:space="preserve">coroll. 15. <lb/>quarti.</note> <note symbol="c" position="right" xlink:label="note-349-03" xlink:href="note-349-03a" xml:space="preserve">ſchol. 4. ſe-<lb/>cundi.</note> <note symbol="d" position="right" xlink:label="note-349-04" xlink:href="note-349-04a" xml:space="preserve">2. duodec.</note> <note symbol="e" position="right" xlink:label="note-349-05" xlink:href="note-349-05a" xml:space="preserve">ſchol. 45. <lb/>primi.</note> <note symbol="f" position="right" xlink:label="note-349-06" xlink:href="note-349-06a" xml:space="preserve">14. ſecundi.</note> <note symbol="g" position="right" xlink:label="note-349-07" xlink:href="note-349-07a" xml:space="preserve">ſchol. 45. <lb/>primi.</note> </div> <p> <s xml:id="echoid-s15002" xml:space="preserve"><emph style="sc">Hæc</emph> eſt quadratura Hyppocratis, acuta quidem, quod Lunulam AGBF, <lb/> <anchor type="note" xlink:label="note-349-08a" xlink:href="note-349-08"/> verè quadrauerit, vitio ſa autem, quod tres Lunulas HNKM, KPLO, LRIQ, <lb/>quadratas à ſe eſſe arbitratur, quod verum non eſt. </s> <s xml:id="echoid-s15003" xml:space="preserve">Solum enim ex eius demon-<lb/>ſtratione Lunula ea quadratur, cuius inferior peripheria eſt quarta pars peri-<lb/>pheriæ alicuius circuli, ſuperior autem ſemicirculus alterius circuli, qualis fuit <lb/>Lunula AGBF. </s> <s xml:id="echoid-s15004" xml:space="preserve">Nam AGB, quarta pars eſt circumferentiæ ABCD, & </s> <s xml:id="echoid-s15005" xml:space="preserve">AFB, ſe-<lb/>miſsis peripheriæ AFBE. </s> <s xml:id="echoid-s15006" xml:space="preserve">At eiuſmodi non ſunttres aliæ Lunulæ, quippe cum <lb/>earum peripheriæ inferiores HNK, KPL, LRI, ſint ſextæ partes totius circumfe-<lb/> <anchor type="note" xlink:label="note-349-09a" xlink:href="note-349-09"/> rentiæ, quamuis peripheriæ ſuperiores ſint ſemicirculi, vt in illa: </s> <s xml:id="echoid-s15007" xml:space="preserve">quæ nondum <lb/>ſunt quadratæ. </s> <s xml:id="echoid-s15008" xml:space="preserve">Quod ſi inuenta eſſet ars quadran di huiuſmodi Lunulas, veriſ-<lb/>ſimè quo que quadraretur circulus, ſine inuentione lineæ rectæ circuli periphe-<lb/>riæ æqualis. </s> <s xml:id="echoid-s15009" xml:space="preserve">quæ ſanè res foret præclara.</s> <s xml:id="echoid-s15010" xml:space="preserve"/> </p> <div xml:id="echoid-div916" type="float" level="2" n="7"> <note position="right" xlink:label="note-349-08" xlink:href="note-349-08a" xml:space="preserve">Fallacia qua-<lb/>draturæ Hip-<lb/>pocratis.</note> <note position="right" xlink:label="note-349-09" xlink:href="note-349-09a" xml:space="preserve">Quid deſide-<lb/>retur in Hip-<lb/>pocratis qua-<lb/>dratura.</note> </div> <pb o="320" file="350" n="350" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s15011" xml:space="preserve"><emph style="sc">Colligitvr</emph> ergo ex hac ratione Hippocratis, quadraturam circuli eſſe <lb/>poſsibilem, cum ſicut Lunula A G B F, quadrata eſt, ita quo que Lunulam <lb/>HNKM, quadrari poſſe, nihil obſtet, quamuis adhuc non ſit à quo quam qua-<lb/>drata. </s> <s xml:id="echoid-s15012" xml:space="preserve">Et certè, vt quidam rectè affirmat, quod hic oſtenditur ab Hippocrate <lb/>de Lunula AGBF, quæ pars eſt circuli AFBE, nihil idem prohibet de circulo to-<lb/>to ſciri poſſe, etiam non inueſtigata quantitate peripheriæ circuli, cum ſolum <lb/>deſit ars quadrandi Lunulam HNKM. </s> <s xml:id="echoid-s15013" xml:space="preserve">Immo plus aliquando dubitationis in-<lb/>ferretinuentio quadraturæ Lunulæ AGBF, non cognita, quam circuli.</s> <s xml:id="echoid-s15014" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15015" xml:space="preserve">4. </s> <s xml:id="echoid-s15016" xml:space="preserve"><emph style="sc">Mvlta</emph> quoque hic dicenda eſſent de falſis aliorum quadraturis, ſed <lb/> <anchor type="note" xlink:label="note-350-01a" xlink:href="note-350-01"/> quia hæ vel ſe ipſas produnt, cum in progreſſu earum facilè appareat, aliquid <lb/>deeſſe ad conſtituendum circulo æquale quadratum, cuiuſmo di eſt quadratura <lb/>Iacobi Falconis Equitis Hiſpani, qui ſine inuẽtione lineæ rectæ, quæ peripheriæ <lb/>ſit æqualis, circulum quadrare conatur: </s> <s xml:id="echoid-s15017" xml:space="preserve">vel ab aliis iam dudum ſunt confuta-<lb/>tæ, nimirum Nicolai Cuſani Cardinalis quadratura à Ioanne Regiomontano, ac <lb/>Ioanne Buteone, & </s> <s xml:id="echoid-s15018" xml:space="preserve">Orontij Finaei Tetragoniſmus tum ab eodem Buteone, <lb/>tum à Petro Nonio Luſitano in libello de Erratis Orontij: </s> <s xml:id="echoid-s15019" xml:space="preserve">quorum vterque <lb/>variis viis lineam rectam circumferentiæ æqualem ſeinueniſſe putauit, nihil o-<lb/>mniò dicendum mihi eſſe ſtatuo, ne fruſtra tempus terere inutiliter videar. <lb/></s> <s xml:id="echoid-s15020" xml:space="preserve">Quamobrem ſolum hoc loco eam quadraturam ſubiiciam, & </s> <s xml:id="echoid-s15021" xml:space="preserve">plenius aliquan-<lb/>to exponam, quam ad finem libr. </s> <s xml:id="echoid-s15022" xml:space="preserve">6. </s> <s xml:id="echoid-s15023" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s15024" xml:space="preserve">conſcripſi, quæ videlicet per li-<lb/> <anchor type="note" xlink:label="note-350-02a" xlink:href="note-350-02"/> neam Quadratricem (ſic enim eam appellare lubet, lineam rectam inuenit cir-<lb/>culari æqualem. </s> <s xml:id="echoid-s15025" xml:space="preserve">Hæc enim via licet ad Geometricè inueniendum punctum <lb/>quoddam, nonnihil in ea deſideretur, accuratior tamen eſt omnibus aliis, quas <lb/>hactenus videre potui; </s> <s xml:id="echoid-s15026" xml:space="preserve">ita vt practicè à ſcopo aberrare non poſsimus. </s> <s xml:id="echoid-s15027" xml:space="preserve">Vt au-<lb/>tem clarè atque ordinatè procedam, abſoluam totum negotium paucis quibuſ-<lb/>dam propoſitionibus.</s> <s xml:id="echoid-s15028" xml:space="preserve"/> </p> <div xml:id="echoid-div917" type="float" level="2" n="8"> <note position="left" xlink:label="note-350-01" xlink:href="note-350-01a" xml:space="preserve">Cur defalſis <lb/>aliorum qua-<lb/>draturis hic <lb/>nihil dicatur.</note> <note position="left" xlink:label="note-350-02" xlink:href="note-350-02a" xml:space="preserve">Quæ nõ qua-<lb/>dratura per <lb/>line{as} hic ex-<lb/>plicetur.</note> </div> </div> <div xml:id="echoid-div919" type="section" level="1" n="321"> <head xml:id="echoid-head348" xml:space="preserve">I. <lb/>QVADRA TRICEM lineam deſcribere.</head> <p> <s xml:id="echoid-s15029" xml:space="preserve"><emph style="sc">Dinostratvs</emph>, & </s> <s xml:id="echoid-s15030" xml:space="preserve">Nicomedes, vt auctor eſt Pappus Alexandrinus in <lb/>4. </s> <s xml:id="echoid-s15031" xml:space="preserve">libr. </s> <s xml:id="echoid-s15032" xml:space="preserve">Mathematicarum collectionum, lineam quandam inflexam excogita-<lb/>runt ad circuli quadraturam, ideo que ab officio {τε}{τρ}α{γο}νίζ{ου}{σα} ab iiſdem eſt ap-<lb/>pellata; </s> <s xml:id="echoid-s15033" xml:space="preserve">à nobis verò eadem de cauſa quadratrix dicetur. </s> <s xml:id="echoid-s15034" xml:space="preserve">Quanquam autem <lb/>prædicti auctores conentur huiuſmodi lineam deſcribere per duos motus ima-<lb/>ginarios duarum rectarum ſeſe interſecantium, qua in re principium (vt philo-<lb/>ſophilo quuntur) petunt, vt propterea à Pappo reiiciatur, tanquam inutilis, & </s> <s xml:id="echoid-s15035" xml:space="preserve"><lb/>quæ deſcribi non poſsit: </s> <s xml:id="echoid-s15036" xml:space="preserve">nostamen eam ſineillis motibus Geometricè delinea-<lb/>bimus per inuentionem quotuis punctorum, per quæ duci debeat; </s> <s xml:id="echoid-s15037" xml:space="preserve">quemadmo-<lb/>dum in deſcriptionibus conicarum ſectio<unsure/>num fieri ſolet.</s> <s xml:id="echoid-s15038" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15039" xml:space="preserve">5. </s> <s xml:id="echoid-s15040" xml:space="preserve"><emph style="sc">Sit</emph> ergo in quadrato ABCD, deſcriptus Quadrans BD. </s> <s xml:id="echoid-s15041" xml:space="preserve">Si igitur, vt vo-<lb/> <anchor type="note" xlink:label="note-350-03a" xlink:href="note-350-03"/> lunt inuentores lineæ Quadratricis, tam ſemidiameter A D, æquabiliter ferri in-<lb/>telligatur circa centrum A, quam latus quadrati ſupremum CD, deorſum verſus <lb/>æquabiliter quo que: </s> <s xml:id="echoid-s15042" xml:space="preserve">ita vt quo tempore punctum D, circumferentiam DB, vni-<lb/>formi ſemper motu percurrit, eodemrecta DC, vniformi etiam motu deſcendens <lb/>adlatus AB, perueniat, ſic tamen, vt perpetuo ſit lateri AB, parallela. </s> <s xml:id="echoid-s15043" xml:space="preserve">& </s> <s xml:id="echoid-s15044" xml:space="preserve">cumlate- <pb o="321" file="351" n="351" rhead="LIBER SEPTIMVS."/> ribus AD, BC, angulos rectos effi ciat, ſecabunt ſe mutuò continuè ſemidiame-<lb/>ter in circumferentia D B, circumacta, & </s> <s xml:id="echoid-s15045" xml:space="preserve">recta D C, deorſum lata, in punctis, <lb/>quæ lineam Quadratricem deſcribent: </s> <s xml:id="echoid-s15046" xml:space="preserve">hoc eſt, per quæ linea Quadratrix <lb/>tranſibit, cuiuſmodi eſt linea inflexa DE. </s> <s xml:id="echoid-s15047" xml:space="preserve">Sed quia duo iſti motus vniformes, <lb/>quorum vnus per circumferentiam D B, fit, & </s> <s xml:id="echoid-s15048" xml:space="preserve">alter per lineas rectas D A, C B, <lb/>effici non poſſunt, niſi proportio habeatur cir-<lb/> <anchor type="figure" xlink:label="fig-351-01a" xlink:href="fig-351-01"/> cularis lineæ ad rectam, meritò à Pappo deſcri-<lb/>ptio hæc repræhenditur: </s> <s xml:id="echoid-s15049" xml:space="preserve">quippe cum ignota <lb/>adhuc ſit ea proportio, & </s> <s xml:id="echoid-s15050" xml:space="preserve">quæ per hanc lineam <lb/>inueſtiganda proponatur. </s> <s xml:id="echoid-s15051" xml:space="preserve">Quare nos Geome-<lb/>tricè eandem lineam Quadratricem deſcribe-<lb/>mus hoc modo. </s> <s xml:id="echoid-s15052" xml:space="preserve">Arcus B D, in quotuis partes <lb/>æquales diuidatur, & </s> <s xml:id="echoid-s15053" xml:space="preserve">latus vtrumque AD, BC, <lb/>in totidem æquales partes. </s> <s xml:id="echoid-s15054" xml:space="preserve">Facillima diuiſio <lb/>erit, ſi & </s> <s xml:id="echoid-s15055" xml:space="preserve">arcus D B, & </s> <s xml:id="echoid-s15056" xml:space="preserve">vtrumque latus AD, BC, <lb/>ſecetur primum bifariam, deinde vtraque ſe-<lb/>miſsis iterum bifariam, atq; </s> <s xml:id="echoid-s15057" xml:space="preserve">ita deinceps, quan-<lb/>tum libuerit. </s> <s xml:id="echoid-s15058" xml:space="preserve">Quo autem plures extiterint diuiſiones, eo accuratius Quadratrix <lb/>linea deſcribetur. </s> <s xml:id="echoid-s15059" xml:space="preserve">Nos ad confuſionem vitandam ſecuimus tam arcum D B, <lb/>quam duo latera AD, BC, in octo tantum partes æquales.</s> <s xml:id="echoid-s15060" xml:space="preserve"/> </p> <div xml:id="echoid-div919" type="float" level="2" n="1"> <note position="left" xlink:label="note-350-03" xlink:href="note-350-03a" xml:space="preserve">Quadratricis <lb/>deſcriptio.</note> <figure xlink:label="fig-351-01" xlink:href="fig-351-01a"> <image file="351-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/351-01"/> </figure> </div> <p> <s xml:id="echoid-s15061" xml:space="preserve"><emph style="sc">Deinde</emph> bina puncta laterum AD, BC, æqualiter diſtantia à latere DC, vel <lb/>AB, coniungantur lineis rectis occultis, atque ex centro A, aliæ rectæ occultæ ad <lb/>ſingula diuiſionũ puncta Quadrantis DB, extendantur. </s> <s xml:id="echoid-s15062" xml:space="preserve">Vbi enim hærectæ prio-<lb/>res rectas interſecabunt, prima primam, ſecunda ſecundam, &</s> <s xml:id="echoid-s15063" xml:space="preserve">c. </s> <s xml:id="echoid-s15064" xml:space="preserve">perea puncta <lb/>Quadratrix linea congruenter ducenda eſt, ita vtnon ſit ſinuoſa, ſed æquabili-<lb/>liter ſemper progrediatur nullum effi ciens gibbum, autangulum alicubi: </s> <s xml:id="echoid-s15065" xml:space="preserve">qua-<lb/>lis eſt linea inflexa D E, ſecans ſemidiametrum AB, in E.</s> <s xml:id="echoid-s15066" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15067" xml:space="preserve">6. </s> <s xml:id="echoid-s15068" xml:space="preserve"><emph style="sc">Sed</emph> quia punctum E, in latere A B, inuenire Geometricè non poteſt, <lb/>cum ibi omnis ſectio rectarum ceſſet: </s> <s xml:id="echoid-s15069" xml:space="preserve">vt illud ſine notabili errore, quiſcilicet <lb/>ſub ſenſum cadat, reperiamus: </s> <s xml:id="echoid-s15070" xml:space="preserve">vtemur hoc artificio: </s> <s xml:id="echoid-s15071" xml:space="preserve">Infimam partem A F, la-<lb/>teris AD, ſi ſatis exigua non ſit, ſecabimus bifariam continuè, donec infima par-<lb/>ticula ſit perexigua: </s> <s xml:id="echoid-s15072" xml:space="preserve">Eodemque modo infimam partem B I, arcus D B, bifariam <lb/>continuè ſecabimus, donectot fiant ſub diuiſiones, quot in parte A F, factæ ſunt, <lb/>vt pa<unsure/>rticula B K, talis pars ſit totius arcus D B, qualis pars eſt A G, totius lateris <lb/>AD. </s> <s xml:id="echoid-s15073" xml:space="preserve">Particulæ deinde A G, æquales abſcindemus BL, BN, AM, ducemuſque <lb/>rectas occultas GL, MN. </s> <s xml:id="echoid-s15074" xml:space="preserve">Ducta verò ex A, centro recta occulta AK, quæ ſecet <lb/>GL, in H, puncto, quod accuratiſsimè notetur (adhibito videlicet Lemmate <lb/>Probl. </s> <s xml:id="echoid-s15075" xml:space="preserve">1. </s> <s xml:id="echoid-s15076" xml:space="preserve">lib. </s> <s xml:id="echoid-s15077" xml:space="preserve">2. </s> <s xml:id="echoid-s15078" xml:space="preserve">vt concurſus H, quam ex quiſitiſsimè reperiatur) ſumemus ipſi <lb/>GH, æqualem M P. </s> <s xml:id="echoid-s15079" xml:space="preserve">Si enim Quadratricem vſque ad H, deſcriptam continua-<lb/>bimus æquabili, atque vniformi extenſione vſq; </s> <s xml:id="echoid-s15080" xml:space="preserve">ad P, ſecabit Quadratrix li-<lb/>nea latus AB, in E, puncto, quod quæritur. </s> <s xml:id="echoid-s15081" xml:space="preserve">Nam propter paruam rectarum GH, <lb/>A E, M P, inter ſe diſtantiam efficitur, vt fermè ſint æquales, licet Geometri-<lb/>cèloquendo recta A E, ſemper maior ſit aliquanto, quantumuis parum eæ re-<lb/>ctæ inter ſe diſtent: </s> <s xml:id="echoid-s15082" xml:space="preserve">ſed exceſlus ille circino deprehendi non poteſt: </s> <s xml:id="echoid-s15083" xml:space="preserve">adeò vt <lb/>arcus circuli ex A, per H, P, deſcriptus verum punctum E, quod ad ſenſum at-<lb/>tinet, indicare videatur. </s> <s xml:id="echoid-s15084" xml:space="preserve">Id quod etiam in circumferentia circuli contingit.</s> <s xml:id="echoid-s15085" xml:space="preserve"> <pb o="322" file="352" n="352" rhead="GEOMETR. PRACT."/> Rectæ namque GL, AB, MN, ſi parum inter ſe diſtent, in circulo omnino æqua-<lb/>les iudicabuntur, quamuis verè AB, aliquanto maior ſit. </s> <s xml:id="echoid-s15086" xml:space="preserve">Itaque ſi res illæ rectæ <lb/>GH, AE, MP, perexiguam habeant diſtantiam inter ſe, dubitari non poteſt, pun-<lb/>ctum E, in quo quadratrix linea ſemidiametrum AB, ſecat, ab eo, quod verè in <lb/>Quadratriceibi exiſtit, non differre notabiliter: </s> <s xml:id="echoid-s15087" xml:space="preserve">dummodo puncta H, P, exqui-<lb/>ſitè & </s> <s xml:id="echoid-s15088" xml:space="preserve">ſumma adhibita diligentia, inuenta ſint.</s> <s xml:id="echoid-s15089" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15090" xml:space="preserve"><emph style="sc">Rectam</emph> porrò AD, vocamus latus Quadratricis: </s> <s xml:id="echoid-s15091" xml:space="preserve">& </s> <s xml:id="echoid-s15092" xml:space="preserve">rectam AE, eiuſdem <lb/> <anchor type="note" xlink:label="note-352-01a" xlink:href="note-352-01"/> baſem: </s> <s xml:id="echoid-s15093" xml:space="preserve">ac denique punctum A, centrum eiuſdem.</s> <s xml:id="echoid-s15094" xml:space="preserve"/> </p> <div xml:id="echoid-div920" type="float" level="2" n="2"> <note position="left" xlink:label="note-352-01" xlink:href="note-352-01a" xml:space="preserve">Lat{us} baſis & <lb/>@entrũ Qua-<lb/>dratricis.</note> </div> <p> <s xml:id="echoid-s15095" xml:space="preserve">7. </s> <s xml:id="echoid-s15096" xml:space="preserve"><emph style="sc">Vervm</emph> puncta Quadratricis prope baſem certius inueniemus (ſine in-<lb/>terſectionibus linearum, quæ ibi valdè obliquæ ſunt) per lineas perpendicula-<lb/>res: </s> <s xml:id="echoid-s15097" xml:space="preserve">hocmodo. </s> <s xml:id="echoid-s15098" xml:space="preserve">Ducta chorda Quadrantis B Q, ſecetur in D, bifariam. </s> <s xml:id="echoid-s15099" xml:space="preserve">quod <lb/>fiet, ſi ex A, ad C, punctum medium Quadrantis recta ducatur. </s> <s xml:id="echoid-s15100" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Hæc enim re- <anchor type="note" xlink:label="note-352-02a" xlink:href="note-352-02"/> ctam BQ, ſecabit bifariam. </s> <s xml:id="echoid-s15101" xml:space="preserve">Deinde rectæ AD, ſumatur æqualis AE; </s> <s xml:id="echoid-s15102" xml:space="preserve">iuncta que <lb/>recta D E, ſecetur bifariam in F. </s> <s xml:id="echoid-s15103" xml:space="preserve">quod etiam fiet, ſi ex A, ad I, punctum medium <lb/> <anchor type="note" xlink:label="note-352-03a" xlink:href="note-352-03"/> arcus B C, ducatur AI. </s> <s xml:id="echoid-s15104" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Hæc enim chordam B C, <anchor type="figure" xlink:label="fig-352-01a" xlink:href="fig-352-01"/> (ſi duceretur) ſecaret bifariam: </s> <s xml:id="echoid-s15105" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ac proinde &</s> <s xml:id="echoid-s15106" xml:space="preserve"> <anchor type="note" xlink:label="note-352-04a" xlink:href="note-352-04"/> rectam DE, <anchor type="note" xlink:href="" symbol="d"/> quæ chordæ BC, eſt parallela: </s> <s xml:id="echoid-s15107" xml:space="preserve">pro- pterea quod latera A B, A C, in triangulo A B C, <lb/> <anchor type="note" xlink:label="note-352-05a" xlink:href="note-352-05"/> proportionaliter ſecantur, in D, E: </s> <s xml:id="echoid-s15108" xml:space="preserve">quippe cum <lb/>tam AB, AC, quam AE, AD, æquales ſint. </s> <s xml:id="echoid-s15109" xml:space="preserve">Rurſus <lb/>rectæ AF, capiatur æqualis AG, iunctaque FG, ſe-<lb/>cetur bifariam in H. </s> <s xml:id="echoid-s15110" xml:space="preserve">quod etiam fiet per rectam <lb/>AK, ductam ad K, punctum medium arcus BI. </s> <s xml:id="echoid-s15111" xml:space="preserve">At-<lb/>que hoc modo, ſi rectæ A H, æqualis accipiatur, <lb/>& </s> <s xml:id="echoid-s15112" xml:space="preserve">reliqua fiant, vt prius, inuenietur aliud punctum <lb/>inter H, & </s> <s xml:id="echoid-s15113" xml:space="preserve">G. </s> <s xml:id="echoid-s15114" xml:space="preserve">Et ſic deinceps quotuis alia puncta reperiemus viciniora ipſi <lb/>A B, per lineas perpendiculares, non autem per obliquas ſectiones, vt in prio-<lb/> <anchor type="note" xlink:label="note-352-06a" xlink:href="note-352-06"/> rifigura. </s> <s xml:id="echoid-s15115" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Eſt enim A D, ad D B, perpendicularis, <anchor type="note" xlink:href="" symbol="f"/>& </s> <s xml:id="echoid-s15116" xml:space="preserve">A F, ad DE, & </s> <s xml:id="echoid-s15117" xml:space="preserve">DH, ad <anchor type="note" xlink:label="note-352-07a" xlink:href="note-352-07"/> FG, &</s> <s xml:id="echoid-s15118" xml:space="preserve">c.</s> <s xml:id="echoid-s15119" xml:space="preserve"/> </p> <div xml:id="echoid-div921" type="float" level="2" n="3"> <note symbol="a" position="left" xlink:label="note-352-02" xlink:href="note-352-02a" xml:space="preserve">ſchol. 27. <lb/>tertii.</note> <note symbol="b" position="left" xlink:label="note-352-03" xlink:href="note-352-03a" xml:space="preserve">ſchol. 27. <lb/>tertii.</note> <figure xlink:label="fig-352-01" xlink:href="fig-352-01a"> <image file="352-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/352-01"/> </figure> <note symbol="c" position="left" xlink:label="note-352-04" xlink:href="note-352-04a" xml:space="preserve">ſchol. 4. ſe-<lb/>xti.</note> <note symbol="d" position="left" xlink:label="note-352-05" xlink:href="note-352-05a" xml:space="preserve">2. ſexti.</note> <note symbol="e" position="left" xlink:label="note-352-06" xlink:href="note-352-06a" xml:space="preserve">3. tertii.</note> <note symbol="f" position="left" xlink:label="note-352-07" xlink:href="note-352-07a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> </div> <p> <s xml:id="echoid-s15120" xml:space="preserve"><emph style="sc">Omnia</emph> verò hæc puncta inuenta D, F, H, &</s> <s xml:id="echoid-s15121" xml:space="preserve">c. </s> <s xml:id="echoid-s15122" xml:space="preserve">eſſe in Quadratrice, ita oſten-<lb/> <anchor type="note" xlink:label="note-352-08a" xlink:href="note-352-08"/> do. </s> <s xml:id="echoid-s15123" xml:space="preserve">Ductis DL, FM, NH, ipſi AB, parallelis ſecantibus BD, in O, P; </s> <s xml:id="echoid-s15124" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit vt BD, ad D Q, ita AL, ad L Q, ideoque & </s> <s xml:id="echoid-s15125" xml:space="preserve">A Q, ſectaerit in L, bifariam. </s> <s xml:id="echoid-s15126" xml:space="preserve">Sectus autem <lb/>eſt & </s> <s xml:id="echoid-s15127" xml:space="preserve">arcus B Q, in bifariam. </s> <s xml:id="echoid-s15128" xml:space="preserve">Igitur, vt oſtenſum eſt, punctum D, eſt in Qua-<lb/> <anchor type="note" xlink:label="note-352-09a" xlink:href="note-352-09"/> dratrice. </s> <s xml:id="echoid-s15129" xml:space="preserve">Rurſus quia D E, ſecta eſt bifariam in F, <anchor type="note" xlink:href="" symbol="h"/> erit quoque DB, ſecta bifa- <anchor type="note" xlink:label="note-352-10a" xlink:href="note-352-10"/> riam in O, <anchor type="note" xlink:href="" symbol="i"/>ideoque erit vt EF, ad FD, ita BO, ad OD. </s> <s xml:id="echoid-s15130" xml:space="preserve">Sed vt BO, ad OD; </s> <s xml:id="echoid-s15131" xml:space="preserve">ita eſt AM, ad ML, & </s> <s xml:id="echoid-s15132" xml:space="preserve">vt EF, ad FD, ita arcus BI, ad IC. </s> <s xml:id="echoid-s15133" xml:space="preserve">Ergo vt oſten dimus, ſeca-<lb/>bunt ſeſe AI, MO, in puncto Quadratricis. </s> <s xml:id="echoid-s15134" xml:space="preserve">Eademque ratio eſt de aliis punctis <lb/>hac arte inuentis.</s> <s xml:id="echoid-s15135" xml:space="preserve"/> </p> <div xml:id="echoid-div922" type="float" level="2" n="4"> <note symbol="g" position="left" xlink:label="note-352-08" xlink:href="note-352-08a" xml:space="preserve">2. ſexti.</note> <note symbol="h" position="left" xlink:label="note-352-09" xlink:href="note-352-09a" xml:space="preserve">2. ſexti.</note> <note symbol="i" position="left" xlink:label="note-352-10" xlink:href="note-352-10a" xml:space="preserve">2. ſexti.</note> </div> <p> <s xml:id="echoid-s15136" xml:space="preserve">8. </s> <s xml:id="echoid-s15137" xml:space="preserve"><emph style="sc">Esse</emph> porrò lineam hanc inflexam DE, à nobis Geometricè deſcriptam, <lb/>eandem, quam Dinoſtratus, & </s> <s xml:id="echoid-s15138" xml:space="preserve">Nicomedes per duos illos motus imaginarios <lb/>deſcribi concipiebant, perſpicuũ eſt. </s> <s xml:id="echoid-s15139" xml:space="preserve">Nam ſi ſemidiameter A D, in priori figura <lb/>circa centrum A, per arcum D B, eodem tempore moueatur motu vniformi, <lb/>quo latus DC, deorſum fertur motu quo que vniformi: </s> <s xml:id="echoid-s15140" xml:space="preserve">fit vt quando ſemidia-<lb/>meter AD, pertranſiuit quamcunque partem arcus DB, tunc latus DC, ſimilem <lb/>partem laterum DA, CB, percurrerit. </s> <s xml:id="echoid-s15141" xml:space="preserve">Alias aut duo illi motus non eſſent vni-<lb/>formes, aut non eodem tempore ad latus A B, tam ſemidiameter AD, quam la- <pb o="323" file="353" n="353" rhead="LIBER SEPTIMVS."/> tus DC perueniret. </s> <s xml:id="echoid-s15142" xml:space="preserve">Cum ergo rectæ ex centro A, per partes arcus DB, emiſſæ, & </s> <s xml:id="echoid-s15143" xml:space="preserve"><lb/>lineæ parallelæ per partes laterum D A, C B, ductæ abſcindant ſemper ex arcu <lb/>DB, & </s> <s xml:id="echoid-s15144" xml:space="preserve">ex lateribus DA, CB, partes ſimiles, ex conſtructione: </s> <s xml:id="echoid-s15145" xml:space="preserve">liquidò conſtat, <lb/>puncta lineæ inflexæ DE, à nobis Geometricè inuenta, à punctis, quæ à duobus <lb/>illis motibus reperirentur non differre.</s> <s xml:id="echoid-s15146" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15147" xml:space="preserve"><emph style="sc">Hæc</emph> igitur eſt deſcriptio lineæ Quadratricis Geometrica quo dammodo, <lb/>quemadmo dum & </s> <s xml:id="echoid-s15148" xml:space="preserve">conicarum ſectionum deſcriptiones, quæ per puncta et-<lb/>iam fiunt, vt ab Apollonio traditur, Geometricæ dicuntur, cum tamen errori <lb/>magis ſint obnoxiæ, quam noſtra deſcriptio, propterinuentionem plurimarum <lb/>linearum mediarum proportionalium, quæ ad earum deſcriptiones ſunt neceſ-<lb/>ſariæ, quibus in Quadratricis deſcriptione opus non eſt. </s> <s xml:id="echoid-s15149" xml:space="preserve">Quare niſi quis to-<lb/>tam conicarum ſectionum do ctrinam, quam tanto ingenij acumine Appollo-<lb/>nius Pergaeus perſecutus eſt, vt propterea Magnus Geometra appellatus ſit, <lb/>reiicere velit, tan quam inutilem, & </s> <s xml:id="echoid-s15150" xml:space="preserve">non Geometricam, (quod neminem in Geo-<lb/>metria peritum facturum exiſtimo, cum ſectiones conicas ad demonſtrationes <lb/>adhibuerint præſtantiſsimi Geometræ. </s> <s xml:id="echoid-s15151" xml:space="preserve">Nam Menechmus Hyperbola, ac Pa-<lb/>rabola vſus eſt in duarum linearum mediarum prop ortionalium inter quaſuis <lb/>duas rectas inuentione; </s> <s xml:id="echoid-s15152" xml:space="preserve">Et Archimedes ipſe multa præclarè de iiſdem ſectioni-<lb/>bus conicis demonſtrauit: </s> <s xml:id="echoid-s15153" xml:space="preserve">ac denique eiuſmodi ſectiones inſignem vſum ha-<lb/>bẽt in re Gnomonica, vt ex noſtra Gnomonica apparet) admittere omninò co-<lb/>getur, hanc deſcriptionem noſtram Quadratricis lineæ eſſe quodammodo Geo-<lb/>metricam. </s> <s xml:id="echoid-s15154" xml:space="preserve">Adde quod linea conchilis, qua Nicomedes duas medias lineas <lb/>proportionales acutiſsimè inueſtigat, per puncta etiam deſcribitur, vt lib. </s> <s xml:id="echoid-s15155" xml:space="preserve">6. </s> <s xml:id="echoid-s15156" xml:space="preserve">pro-<lb/>poſ. </s> <s xml:id="echoid-s15157" xml:space="preserve">15. </s> <s xml:id="echoid-s15158" xml:space="preserve">diximus.</s> <s xml:id="echoid-s15159" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15160" xml:space="preserve"><emph style="sc">Habet</emph> linea hæc quadratrix multas, & </s> <s xml:id="echoid-s15161" xml:space="preserve">inſignes vtilitates, quarum nonnul-<lb/>las ad finem lib. </s> <s xml:id="echoid-s15162" xml:space="preserve">6. </s> <s xml:id="echoid-s15163" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s15164" xml:space="preserve">demonſtrauimus, quas hoc loco repetere ſuperuaca-<lb/>neum eſt. </s> <s xml:id="echoid-s15165" xml:space="preserve">Solum igitur eius vſum in quadrandis circulis hic exponemus. </s> <s xml:id="echoid-s15166" xml:space="preserve">Qua <lb/>in re indigemus tantummodo vltimo puncto E, in priori figura, etiamſi nullum <lb/>aliud Quadratricis punctum inuentum eſſet. </s> <s xml:id="echoid-s15167" xml:space="preserve">quod quidem vltimum punctum <lb/>licet Geometricè, ac præcisè non reperiatur: </s> <s xml:id="echoid-s15168" xml:space="preserve">tamen ſi artificium poſterioris fi-<lb/>guræ adhibeatur, non aberrabimus à vero puncto notabiliter, vt ſupra diximus. <lb/></s> <s xml:id="echoid-s15169" xml:space="preserve">Quando namque deprehenſum fuerit, vltimam perpendicularem A H, æqua-<lb/>lem eſſe præcedenti vltimæ lineæ translatæ A G, ita vt nulla differentia inter illas <lb/>per circinum diſcernatur: </s> <s xml:id="echoid-s15170" xml:space="preserve">ſumi poterit citra errorem notabilem vltimum illud <lb/>punctum G, pro puncto extremo Quadratricis: </s> <s xml:id="echoid-s15171" xml:space="preserve">Sin minus, ducendæ erunt aliæ <lb/>perpendiculares eo artificio, quo AF, AH, ductæ ſunt, donecinter vltimam, & </s> <s xml:id="echoid-s15172" xml:space="preserve"><lb/>poſtremo loco inuentam rectam in ſemidiametro AB, nullum appareat diſcri-<lb/>men. </s> <s xml:id="echoid-s15173" xml:space="preserve">cuius quidem rei operatio ipſa optimus erit magiſter.</s> <s xml:id="echoid-s15174" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div924" type="section" level="1" n="322"> <head xml:id="echoid-head349" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s15175" xml:space="preserve">9. </s> <s xml:id="echoid-s15176" xml:space="preserve">Ex deſcriptione Quadratricis colligitur, ſi ex centro A, ducatur recta vt-<lb/>cunque A Q, ſecans arcum Quadrantis in Q, & </s> <s xml:id="echoid-s15177" xml:space="preserve">Quadratricem in O; </s> <s xml:id="echoid-s15178" xml:space="preserve">ita eſſe <lb/>arcum BD, ad arcum B Q, vt eſt ſemidiameter A D, ad rectam A R, ducta prius <lb/>O R, ipſi A B, parallela: </s> <s xml:id="echoid-s15179" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s15180" xml:space="preserve">ad rectam, quæ ex O, ad A B, demit-<lb/>titur perpendicularis. </s> <s xml:id="echoid-s15181" xml:space="preserve">Quia enim eadem pars eſt arcus D Q, totius arcus DB, quę <lb/> <anchor type="note" xlink:label="note-353-01a" xlink:href="note-353-01"/> <pb o="324" file="354" n="354" rhead="GEOMETR. PRACT."/> pars eſt recta DR, totius ſemidiametri DA, quippe cum in deſcriptione Quadra-<lb/>tricis arcus D Q, totius arcus DB, tot particulas complectatur, quot partes re-<lb/> <anchor type="figure" xlink:label="fig-354-01a" xlink:href="fig-354-01"/> cta DR, totius DA, continet: </s> <s xml:id="echoid-s15182" xml:space="preserve">quando quidem <lb/>rectæ A Q, R O, ſeſe interſecant in O, puncto <lb/>Quadratricis. </s> <s xml:id="echoid-s15183" xml:space="preserve">Neque hæc ſimilitudo impedi-<lb/>tur, etiamſi tam arcus DQ, toti arcui DB, quã <lb/>recta D R, totilateri D A, ſit incommenſura-<lb/>bilis, cum perpetuò Quadratrix eadem vni-<lb/>formitate progrediatur per omnia ſua puncta. <lb/></s> <s xml:id="echoid-s15184" xml:space="preserve">Si enim recta DR, non eſt talis pars, ſiue com-<lb/>menſurabilis, ſiue incommenſurabilis totius <lb/>lateris D A, qualis pars eſt arcus D Q, totius <lb/>arcus DB; </s> <s xml:id="echoid-s15185" xml:space="preserve">ſi cogitetur pars lateris D A, minor <lb/>quam DR, vel maior, ſecabit parallela ex eius <lb/>puncto exrremo ducta rectam A Q, vel ſupra O, velinfra, in puncto, per quod <lb/>Quadratrix deſcribenda eſt: </s> <s xml:id="echoid-s15186" xml:space="preserve">ac proinde ea nõ tranſibit per O, quod eſt abſurdũ, <lb/>& </s> <s xml:id="echoid-s15187" xml:space="preserve">contra hypotheſim. </s> <s xml:id="echoid-s15188" xml:space="preserve">Quia inquam eadẽ pars eſt arcus D Q, totius arcus DB, <lb/> <anchor type="note" xlink:label="note-354-01a" xlink:href="note-354-01"/> quæ pars eſt recta DR, totius lateris DA; </s> <s xml:id="echoid-s15189" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit quoque reliquus arcus QB, ea- dem pars totius arcus D B, quæ pars eſt reliqua recta R A, totius lateris D A, <lb/>quod eadem ſit proportio totius D B, ad D Q, quæ totius D A, ad DR: </s> <s xml:id="echoid-s15190" xml:space="preserve">Et per-<lb/>mutando eadem totius DB, ad totam D A, quæ ablati arcus D Q, ad ablatam <lb/>rectam D R. </s> <s xml:id="echoid-s15191" xml:space="preserve">Quocirca erit, vt totus arcus D B, ad arcum Q B, ita totum latus <lb/>D A, ad rectam R A, hoc eſt, ad rectam perpendicularem ex O, ad AB, demiſ-<lb/>ſam, <anchor type="note" xlink:href="" symbol="b"/> quæ ipſi R A, æqualis eſt.</s> <s xml:id="echoid-s15192" xml:space="preserve"/> </p> <div xml:id="echoid-div924" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-353-01" xlink:href="note-353-01a" xml:space="preserve">34. primi.</note> <figure xlink:label="fig-354-01" xlink:href="fig-354-01a"> <image file="354-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/354-01"/> </figure> <note symbol="a" position="left" xlink:label="note-354-01" xlink:href="note-354-01a" xml:space="preserve">19. quinti.</note> </div> <note symbol="b" position="left" xml:space="preserve">34. primi.</note> </div> <div xml:id="echoid-div926" type="section" level="1" n="323"> <head xml:id="echoid-head350" xml:space="preserve">II.</head> <p> <s xml:id="echoid-s15193" xml:space="preserve">SI Quadrantis, & </s> <s xml:id="echoid-s15194" xml:space="preserve">Quadratricis idem centrum ſit; </s> <s xml:id="echoid-s15195" xml:space="preserve">erunt arcus Qua-<lb/>drantis, ſemidiameter, & </s> <s xml:id="echoid-s15196" xml:space="preserve">baſis quadratricis continué proportionales.</s> <s xml:id="echoid-s15197" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15198" xml:space="preserve"><emph style="sc">Hæc</emph> eſt eximia, atque inſignis proprietas Quadratricis. </s> <s xml:id="echoid-s15199" xml:space="preserve">Sit Quadrans, <lb/>& </s> <s xml:id="echoid-s15200" xml:space="preserve">Quadratrix ex eo deſcripta, vt ſupra. </s> <s xml:id="echoid-s15201" xml:space="preserve">Dico arcum BD, ſemidiametrum AD, <lb/>& </s> <s xml:id="echoid-s15202" xml:space="preserve">Quadratricis baſem A E, continuè eſſe proportionales, hoc eſt, eſſe B D, <lb/>ad AD, vt AD, ad AE. </s> <s xml:id="echoid-s15203" xml:space="preserve">Sin minus, ſit vt BD, ad AD, ita AD, ad AF, maiorem ipſa <lb/> <anchor type="figure" xlink:label="fig-354-02a" xlink:href="fig-354-02"/> AE, minoremue: </s> <s xml:id="echoid-s15204" xml:space="preserve">ſitque primum AF, maior, quam AE. </s> <s xml:id="echoid-s15205" xml:space="preserve">Deſcripto ex centro A, <lb/>Quadrante FG, per F, ſecante Quadratricem in H, ducatur per H, ſemidiameter <pb o="325" file="355" n="355" rhead="LIBER SEPTIMVS."/> AHK, demittaturque perpendicularis HI. </s> <s xml:id="echoid-s15206" xml:space="preserve">Quoniam igitur ponitur arcus BD, <lb/>ad rectam AD, vt AD, hoc eſt, vt A B, ad AF; </s> <s xml:id="echoid-s15207" xml:space="preserve">eſt que vt A B, ſemidiameter ad ſe-<lb/>midiametrum A F, ita arcus B D, ad arcum F G; </s> <s xml:id="echoid-s15208" xml:space="preserve">(Cum enim ſit, vt lib. </s> <s xml:id="echoid-s15209" xml:space="preserve">4. </s> <s xml:id="echoid-s15210" xml:space="preserve">capit. </s> <s xml:id="echoid-s15211" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s15212" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15213" xml:space="preserve">1. </s> <s xml:id="echoid-s15214" xml:space="preserve">demonſtrauimus, diameter ad diametrum, vt circumferentia ad circũ-<lb/> <anchor type="note" xlink:label="note-355-01a" xlink:href="note-355-01"/> ferentiam; </s> <s xml:id="echoid-s15215" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit quo que ſemidiameter AB, ad ſemidiametrum AF, vt eadem cir- <anchor type="note" xlink:label="note-355-02a" xlink:href="note-355-02"/> cumferentia ad eandem circumferentiam: </s> <s xml:id="echoid-s15216" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ac proinde etiam, vt quarta pars circumferentiæ ad quartam partem circumferentiæ, hoc eſt, vt arcus BD, ad ar-<lb/> <anchor type="note" xlink:label="note-355-03a" xlink:href="note-355-03"/> cum F G.) </s> <s xml:id="echoid-s15217" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Erit quoque arcus B D, ad rectam A D, vtidem arcus B D, ad arcum <anchor type="note" xlink:label="note-355-04a" xlink:href="note-355-04"/> F G; </s> <s xml:id="echoid-s15218" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> ac proptera æquales erunt recta A D, & </s> <s xml:id="echoid-s15219" xml:space="preserve">arcus F G. </s> <s xml:id="echoid-s15220" xml:space="preserve">Quia verò ex præce- denti coroll. </s> <s xml:id="echoid-s15221" xml:space="preserve">eſt, vt arcus B D, ad arcum BK, ita recta AD, ad rectam HI, & </s> <s xml:id="echoid-s15222" xml:space="preserve">vt ar-<lb/> <anchor type="note" xlink:label="note-355-05a" xlink:href="note-355-05"/> cus BD, ad arcum BK, ita eſt arcus FG, ad arcum FH, <anchor type="note" xlink:href="" symbol="e"/> quod arcus B D, B K, ar- cubus FG, FH, ſimiles ſint; </s> <s xml:id="echoid-s15223" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> erit quoque recta AD, ad rectam HI, vt arcus F G, <anchor type="note" xlink:label="note-355-06a" xlink:href="note-355-06"/> ad arcum FH. </s> <s xml:id="echoid-s15224" xml:space="preserve">Cum ergo oſtenſa ſit recta A D, arcui F G, æqualis: </s> <s xml:id="echoid-s15225" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/>erit quo- <anchor type="note" xlink:label="note-355-07a" xlink:href="note-355-07"/> que recta HI, arcui F H, ęqualis quod eſt abſurdum. </s> <s xml:id="echoid-s15226" xml:space="preserve">Eſt enim recta H I, minor <lb/> <anchor type="note" xlink:label="note-355-08a" xlink:href="note-355-08"/> arcu F H, cum ea ſit ſemiſsis chordæ ſubten dentis arcum duplum arcus F H: </s> <s xml:id="echoid-s15227" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> <anchor type="note" xlink:label="note-355-09a" xlink:href="note-355-09"/> (Nam recta A F, ſecat eam chordam bifariam; </s> <s xml:id="echoid-s15228" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> ac proinde & </s> <s xml:id="echoid-s15229" xml:space="preserve">arcum) chorda autem ſemper ſuo arcu minor ſit. </s> <s xml:id="echoid-s15230" xml:space="preserve">Non ergo eſt arcus B D, ad ſemidiametrum <lb/>AD, vt AD, ad rectam maiorem baſe AE, Quadratricis.</s> <s xml:id="echoid-s15231" xml:space="preserve"/> </p> <div xml:id="echoid-div926" type="float" level="2" n="1"> <figure xlink:label="fig-354-02" xlink:href="fig-354-02a"> <image file="354-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/354-02"/> </figure> <note symbol="a" position="right" xlink:label="note-355-01" xlink:href="note-355-01a" xml:space="preserve">15. quinti.</note> <note symbol="b" position="right" xlink:label="note-355-02" xlink:href="note-355-02a" xml:space="preserve">15. quinti.</note> <note symbol="c" position="right" xlink:label="note-355-03" xlink:href="note-355-03a" xml:space="preserve">11. quinti.</note> <note symbol="d" position="right" xlink:label="note-355-04" xlink:href="note-355-04a" xml:space="preserve">9. quinti.</note> <note symbol="e" position="right" xlink:label="note-355-05" xlink:href="note-355-05a" xml:space="preserve">ſchol. 33. <lb/>ſexti.</note> <note symbol="f" position="right" xlink:label="note-355-06" xlink:href="note-355-06a" xml:space="preserve">11. quinti.</note> <note symbol="g" position="right" xlink:label="note-355-07" xlink:href="note-355-07a" xml:space="preserve">14. quinti.</note> <note symbol="h" position="right" xlink:label="note-355-08" xlink:href="note-355-08a" xml:space="preserve">3. tertij.</note> <note symbol="i" position="right" xlink:label="note-355-09" xlink:href="note-355-09a" xml:space="preserve">ſchol. 27. <lb/>tertij.</note> </div> <p> <s xml:id="echoid-s15232" xml:space="preserve"><emph style="sc">Sit</emph> deinde, ſi fieri poteſt, vt arcus BD, ad AD, ita A D, ad A I, min orem baſe <lb/>AE. </s> <s xml:id="echoid-s15233" xml:space="preserve">Deſcripto igitur ex centro A, per I, Quadrante IL, erigatur ex I, ad AE, per-<lb/>pendicularis I H, ſecans Quadratricem in H, puncto, per quod ſemidia-<lb/>meter ducatur AK, ſecans arcum IL, in M. </s> <s xml:id="echoid-s15234" xml:space="preserve">Oſtendemus ergo, vt prius, arcum <lb/>IL, rectæ AD, æqualem eſſe. </s> <s xml:id="echoid-s15235" xml:space="preserve">Item ita eſſe arcum BD, ad arcum BK, hoc eſt, arcum <lb/>I L, ad arcum I M, vt eſt recta A D; </s> <s xml:id="echoid-s15236" xml:space="preserve">ad rectam H I. </s> <s xml:id="echoid-s15237" xml:space="preserve">Quare cum arcus <lb/> <anchor type="note" xlink:label="note-355-10a" xlink:href="note-355-10"/> IL, oſtenſus ſit æqualis rectæ A D, <anchor type="note" xlink:href="" symbol="k"/> erit quoq; </s> <s xml:id="echoid-s15238" xml:space="preserve">arcus I M, æqualis rectæ HI. </s> <s xml:id="echoid-s15239" xml:space="preserve">quod <anchor type="note" xlink:label="note-355-11a" xlink:href="note-355-11"/> eſt abſurdum. </s> <s xml:id="echoid-s15240" xml:space="preserve">Eſt enimrecta H I, maior arcu I M. </s> <s xml:id="echoid-s15241" xml:space="preserve">Nam ſi ex H, duceretur ver-<lb/>ſus G, alia recta tangens circulum IL, ſicut H I, eundẽ tangit in I, <anchor type="note" xlink:href="" symbol="l"/> eſſent hę duæ <anchor type="note" xlink:label="note-355-12a" xlink:href="note-355-12"/> tangentes æquales, arcuſq; </s> <s xml:id="echoid-s15242" xml:space="preserve">inter eas interceptus ſecaretur bifariam in M, <anchor type="note" xlink:href="" symbol="m"/> pro- pterea quod angulus ab eis comprehenſus bifariam diuideretur à recta AH, <anchor type="note" xlink:href="" symbol="n"/> ac <anchor type="note" xlink:label="note-355-13a" xlink:href="note-355-13"/> proinde & </s> <s xml:id="echoid-s15243" xml:space="preserve">angulus in centro A, ſi ad alterum punctum conta ctus recta adiun-<lb/>geretur: </s> <s xml:id="echoid-s15244" xml:space="preserve"><anchor type="note" xlink:href="" symbol="o"/> ideoque arcus, quibus inſiſtunt, æquales forent. </s> <s xml:id="echoid-s15245" xml:space="preserve">Igitur cum, vt lib.</s> <s xml:id="echoid-s15246" xml:space="preserve"> <anchor type="note" xlink:label="note-355-14a" xlink:href="note-355-14"/> 8. </s> <s xml:id="echoid-s15247" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15248" xml:space="preserve">1. </s> <s xml:id="echoid-s15249" xml:space="preserve">probabimus cum Archimede, duæ illætangentes ſimul maiores <lb/>ſint arcu ab eis comprehenſo, erit & </s> <s xml:id="echoid-s15250" xml:space="preserve">earum ſemiſsis HI, maior ſemiſſe IM, illius <lb/>arcus. </s> <s xml:id="echoid-s15251" xml:space="preserve">Non eſt ergo arcus BD, ad ſemidiametrum AD, vt AD, ad rectam minorẽ <lb/>baſe AE, Quadratricis; </s> <s xml:id="echoid-s15252" xml:space="preserve">Sed neque vt AD, ad maiorem, ſicut oſtenſum eſt. </s> <s xml:id="echoid-s15253" xml:space="preserve">Igitur <lb/>vt AD, ad ipſam baſem AE. </s> <s xml:id="echoid-s15254" xml:space="preserve">quod demonſtrandum erat.</s> <s xml:id="echoid-s15255" xml:space="preserve"/> </p> <div xml:id="echoid-div927" type="float" level="2" n="2"> <note symbol="k" position="right" xlink:label="note-355-10" xlink:href="note-355-10a" xml:space="preserve">14. quinti.</note> <note symbol="l" position="right" xlink:label="note-355-11" xlink:href="note-355-11a" xml:space="preserve">2. coroll. 36. <lb/>tertij.</note> <note symbol="m" position="right" xlink:label="note-355-12" xlink:href="note-355-12a" xml:space="preserve">ſchol. 27. <lb/>tertij.</note> <note symbol="n" position="right" xlink:label="note-355-13" xlink:href="note-355-13a" xml:space="preserve">4. vel 8. <lb/>primi.</note> <note symbol="o" position="right" xlink:label="note-355-14" xlink:href="note-355-14a" xml:space="preserve">26. tertij.</note> </div> </div> <div xml:id="echoid-div929" type="section" level="1" n="324"> <head xml:id="echoid-head351" xml:space="preserve">COROLLARIVM I.</head> <note position="right" xml:space="preserve">Rectam cir-<lb/>cunferentiæ <lb/>circuli æqua-<lb/>lem reperire.</note> <p> <s xml:id="echoid-s15256" xml:space="preserve">HINC facilè rectam reperiemus arcui Quadrantis, ex quo Quadratrix <lb/>deſcripta eſt, ac proinde & </s> <s xml:id="echoid-s15257" xml:space="preserve">ſemicircumferentiæ, immo & </s> <s xml:id="echoid-s15258" xml:space="preserve">toti circũ-<lb/>ferentiæ æqualem.</s> <s xml:id="echoid-s15259" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15260" xml:space="preserve"><emph style="sc">Qvoniam</emph> eſt arcus B D, ad ſemidiametrum A D, vt A D, ad ba-<lb/>ſem Quadratricis A E; </s> <s xml:id="echoid-s15261" xml:space="preserve">erit conuertendo quoque A E, ad A D, vt A D, ad <lb/> <anchor type="note" xlink:label="note-355-16a" xlink:href="note-355-16"/> arcum B D. </s> <s xml:id="echoid-s15262" xml:space="preserve">Si igitur duabus rectis A E, A D, inueniatur tertia proportionalis; </s> <s xml:id="echoid-s15263" xml:space="preserve"><anchor type="note" xlink:href="" symbol="p"/> erit AD, ad eam tertiam, vt ad arcum BD, cum vtraq; </s> <s xml:id="echoid-s15264" xml:space="preserve">proportio ſit eadem, quæ <pb o="326" file="356" n="356" rhead="GEOMETR. PRACT."/> AE. </s> <s xml:id="echoid-s15265" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quare tertia illa proportionalis arcui Quadrantis B D, æqualis erit: </s> <s xml:id="echoid-s15266" xml:space="preserve">Et ſi <anchor type="note" xlink:label="note-356-01a" xlink:href="note-356-01"/> duplicetur, fiet recta æqualis ſemicircumferentiæ eiuſdem circuli: </s> <s xml:id="echoid-s15267" xml:space="preserve">Si verò qua-<lb/>druplicetur, fiet recta toti circumferentiæ æqualis.</s> <s xml:id="echoid-s15268" xml:space="preserve"/> </p> <div xml:id="echoid-div929" type="float" level="2" n="1"> <note symbol="p" position="right" xlink:label="note-355-16" xlink:href="note-355-16a" xml:space="preserve">11. quinti.</note> <note symbol="a" position="left" xlink:label="note-356-01" xlink:href="note-356-01a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div931" type="section" level="1" n="325"> <head xml:id="echoid-head352" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s15269" xml:space="preserve">SEQVITVR quoque exhis, ſi baſis Quadratricis AE, ſtatuatur ſemi-<lb/>diameter alicuius circuli, eius latus A D, quartæ parti circumferentię <lb/>illius circuli eſſe æquale: </s> <s xml:id="echoid-s15270" xml:space="preserve">Et lineam lateris A D, duplam æqualem eſſe <lb/>ſemicircumferentiæ eiuſdem circuli: </s> <s xml:id="echoid-s15271" xml:space="preserve">Et lineam quadruplam lateris <lb/>toti circumferentiæ eſſe æqualem.</s> <s xml:id="echoid-s15272" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15273" xml:space="preserve"><emph style="sc">Cvm</emph> enim, vt lib. </s> <s xml:id="echoid-s15274" xml:space="preserve">4. </s> <s xml:id="echoid-s15275" xml:space="preserve">cap. </s> <s xml:id="echoid-s15276" xml:space="preserve">7. </s> <s xml:id="echoid-s15277" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15278" xml:space="preserve">1. </s> <s xml:id="echoid-s15279" xml:space="preserve">oſtendimus, diametri circulorum cir-<lb/>cumferentijs ſint proportionales, <anchor type="note" xlink:href="" symbol="b"/> erunt quo que ſemidiametri ſemicircumfe- <anchor type="note" xlink:label="note-356-02a" xlink:href="note-356-02"/> <anchor type="figure" xlink:label="fig-356-01a" xlink:href="fig-356-01"/> rentijs, & </s> <s xml:id="echoid-s15280" xml:space="preserve">quadrantibus proportionales. </s> <s xml:id="echoid-s15281" xml:space="preserve">Igitur erit, vt A D, ad A E, hoc eſt, vt <lb/>ſupradicta tertia proportionalis ad AD, ita Quadrans BD, ſemidiametri AD, ad <lb/>Quadrantem ſemidiametri A E. </s> <s xml:id="echoid-s15282" xml:space="preserve">Cum ergo tertia illa proportionalis æqualis ſit <lb/> <anchor type="note" xlink:label="note-356-03a" xlink:href="note-356-03"/> oſtenſa Quadranti BD; </s> <s xml:id="echoid-s15283" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit quo que recta A D, quadranti ſemidiametri A E, æ- qualis. </s> <s xml:id="echoid-s15284" xml:space="preserve">Dupla ergo linea ipſius AD, ſemicircumferentiæ circuli, cuius ſemidia-<lb/>meter AE; </s> <s xml:id="echoid-s15285" xml:space="preserve">& </s> <s xml:id="echoid-s15286" xml:space="preserve">quadrupla toti circumferentiæ erit ęqualis.</s> <s xml:id="echoid-s15287" xml:space="preserve"/> </p> <div xml:id="echoid-div931" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-356-02" xlink:href="note-356-02a" xml:space="preserve">15. quinti.</note> <figure xlink:label="fig-356-01" xlink:href="fig-356-01a"> <image file="356-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/356-01"/> </figure> <note symbol="c" position="left" xlink:label="note-356-03" xlink:href="note-356-03a" xml:space="preserve">14. quinti.</note> </div> </div> <div xml:id="echoid-div933" type="section" level="1" n="326"> <head xml:id="echoid-head353" xml:space="preserve">COROLLARIVM III.</head> <p> <s xml:id="echoid-s15288" xml:space="preserve">EX his quoque infertur, ſi duæ rectę N, O, in præcedenti figura eandem <lb/>proportionem habeant, quam AD, AE, minor autem O, ſtatuatur ſe-<lb/>midiameter circuli alicuius, maiorem N, æqualem eſſe arcui Quadrã-<lb/>tis illius circuli.</s> <s xml:id="echoid-s15289" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15290" xml:space="preserve"><emph style="sc">Cvm</emph> enim ſit AD, ad AE, vt N, ad O; </s> <s xml:id="echoid-s15291" xml:space="preserve">erit permutando AD, ad N, vt AE, ad <lb/>O. </s> <s xml:id="echoid-s15292" xml:space="preserve">Vt autem AE, ad O, ita eſt Quadrans ſemidiametri A E, ad Quadrantem ſe-<lb/>midiametri O, vt lib. </s> <s xml:id="echoid-s15293" xml:space="preserve">4. </s> <s xml:id="echoid-s15294" xml:space="preserve">cap. </s> <s xml:id="echoid-s15295" xml:space="preserve">7. </s> <s xml:id="echoid-s15296" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15297" xml:space="preserve">1. </s> <s xml:id="echoid-s15298" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s15299" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Igitur @rit quoque <anchor type="note" xlink:label="note-356-04a" xlink:href="note-356-04"/> AD, ad N, vt Quadrans ſemidiametri AE, ad Quadrantem ſemidiametri O. </s> <s xml:id="echoid-s15300" xml:space="preserve">Cũ <pb o="327" file="357" n="357" rhead="LIBER SEPTIMVS."/> ergo AD, æqualis ſit oſtenſa Quadranti ſemidiametri AE; </s> <s xml:id="echoid-s15301" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit quoqe N, æqua- <anchor type="note" xlink:label="note-357-01a" xlink:href="note-357-01"/> lis Quadranti ſemidiametri O. </s> <s xml:id="echoid-s15302" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s15303" xml:space="preserve"/> </p> <div xml:id="echoid-div933" type="float" level="2" n="1"> <note symbol="d" position="left" xlink:label="note-356-04" xlink:href="note-356-04a" xml:space="preserve">11. quinti.</note> <note symbol="a" position="right" xlink:label="note-357-01" xlink:href="note-357-01a" xml:space="preserve">14. quinti.</note> </div> </div> <div xml:id="echoid-div935" type="section" level="1" n="327"> <head xml:id="echoid-head354" xml:space="preserve">III.</head> <p> <s xml:id="echoid-s15304" xml:space="preserve">DATO circulo quadratum æquale conſtituere.</s> <s xml:id="echoid-s15305" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15306" xml:space="preserve"><emph style="sc">Sit</emph> quadrandus circulus ad interuallum ſemidiametri B C, deſcriptus. </s> <s xml:id="echoid-s15307" xml:space="preserve">Tri-<lb/> <anchor type="note" xlink:label="note-357-02a" xlink:href="note-357-02"/> bus rectis A E, baſi Quadratricis; </s> <s xml:id="echoid-s15308" xml:space="preserve">A D, lateri eiuſdem pręcedentis figuræ, & </s> <s xml:id="echoid-s15309" xml:space="preserve">rectę <lb/> <anchor type="figure" xlink:label="fig-357-01a" xlink:href="fig-357-01"/> BC, inuenta quarta proportionali F; </s> <s xml:id="echoid-s15310" xml:space="preserve">erit ex coroll. </s> <s xml:id="echoid-s15311" xml:space="preserve">3. <lb/></s> <s xml:id="echoid-s15312" xml:space="preserve">antecedenti recta F, quadranti circuli dati æqualis, <lb/>atq; </s> <s xml:id="echoid-s15313" xml:space="preserve">eius dupla ſemicircumferentiæ æqualis erit. </s> <s xml:id="echoid-s15314" xml:space="preserve">In-<lb/>uenta autem inter ſemidiametrum B C, & </s> <s xml:id="echoid-s15315" xml:space="preserve">duplami-<lb/>pſius F, media proportionali GH: </s> <s xml:id="echoid-s15316" xml:space="preserve">Dico quadratum <lb/>ex G H, deſcriptum æquale eſſe circulo ad interual-<lb/> <anchor type="note" xlink:label="note-357-03a" xlink:href="note-357-03"/> lum B C, deſcripto. </s> <s xml:id="echoid-s15317" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam enim rectangulum ſub B C, ſemidiametro, & </s> <s xml:id="echoid-s15318" xml:space="preserve">ſub ſemicircumferẽtia cir-<lb/>culi, id eſt, ſub dupla rectæ F, inuentæ, æquale eſt cir-<lb/> <anchor type="note" xlink:label="note-357-04a" xlink:href="note-357-04"/> culo: </s> <s xml:id="echoid-s15319" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Prędicto autem rectangulo æquale eſt qua- dratum lateris G H; </s> <s xml:id="echoid-s15320" xml:space="preserve">erit quoque quadratum lateris <lb/>G H, circulo ſemidiametri B C, æquale.</s> <s xml:id="echoid-s15321" xml:space="preserve"/> </p> <div xml:id="echoid-div935" type="float" level="2" n="1"> <note position="right" xlink:label="note-357-02" xlink:href="note-357-02a" xml:space="preserve">Quadratum <lb/>circulo æqua-<lb/>le exhibere.</note> <figure xlink:label="fig-357-01" xlink:href="fig-357-01a"> <image file="357-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/357-01"/> </figure> <note symbol="b" position="right" xlink:label="note-357-03" xlink:href="note-357-03a" xml:space="preserve">4. Iſoperi-<lb/>metrorum.</note> <note symbol="c" position="right" xlink:label="note-357-04" xlink:href="note-357-04a" xml:space="preserve">17. ſexti.</note> </div> <p> <s xml:id="echoid-s15322" xml:space="preserve"><emph style="sc">Vervm</emph> vt expedite linea recta inueniatur æqualis quartæ parti circumfe-<lb/> <anchor type="note" xlink:label="note-357-05a" xlink:href="note-357-05"/> rentiæ dati circuli, atque id circo & </s> <s xml:id="echoid-s15323" xml:space="preserve">ſemicircumferentiæ, vel toti circumferentiæ, <lb/>conſtruenda erit figura eiuſmodi. </s> <s xml:id="echoid-s15324" xml:space="preserve">Fiat angulus rectus D A E, recta que AD, ęqua-<lb/>lis ſit ſemidiametro Quadrantis, ex quo Quadratrix deſcripta eſt; </s> <s xml:id="echoid-s15325" xml:space="preserve">& </s> <s xml:id="echoid-s15326" xml:space="preserve">AE, baſi <lb/>eiuſdem Quadratricis æqualis. </s> <s xml:id="echoid-s15327" xml:space="preserve">Vel certe ex centro <lb/> <anchor type="figure" xlink:label="fig-357-02a" xlink:href="fig-357-02"/> A, noua Quadratrix deſcribatur DE, cuius latus AD, <lb/>& </s> <s xml:id="echoid-s15328" xml:space="preserve">baſis A E. </s> <s xml:id="echoid-s15329" xml:space="preserve">Ducta namquerecta D E, conſtru-<lb/>cta erit figura aptiſsima ad rectam circumferen-<lb/>tię dati circuli æqualem inueniendam. </s> <s xml:id="echoid-s15330" xml:space="preserve">Si <lb/>enim circuli quadrandi ſemidiametro abſcindatur æ-<lb/>qualis AF, ducanturque F G, ipſi D E, parallela; </s> <s xml:id="echoid-s15331" xml:space="preserve">erit <lb/>ex coroll. </s> <s xml:id="echoid-s15332" xml:space="preserve">3. </s> <s xml:id="echoid-s15333" xml:space="preserve">antecedenti A G, æqualis quartæ parti <lb/>circumferentiæ dati circuli, cuius ſemidiameter nimi-<lb/>rum eſt AF, (quemadmodum A D, quartæ parti cir-<lb/>cumferentiæ circuli ſemidiametri AE, æqualis eſt, vt <lb/>ex coroll. </s> <s xml:id="echoid-s15334" xml:space="preserve">2. </s> <s xml:id="echoid-s15335" xml:space="preserve">præcedenti conſtat) <anchor type="note" xlink:href="" symbol="d"/> propterea quod <anchor type="note" xlink:label="note-357-06a" xlink:href="note-357-06"/> A F, A G, eandem habent proportionem, quam A E, <lb/>AD. </s> <s xml:id="echoid-s15336" xml:space="preserve">Eadem ratione, ductis HI, KL, MN, eidem DE, <lb/>parallelis, erunt AI, AL, AN, æquales quartis parti-<lb/>bus circumferentiarum circulorum ex ſemidiame-<lb/>tris AH, AK, AM, deſcriptorum. </s> <s xml:id="echoid-s15337" xml:space="preserve">Hæautem rectæ <lb/>duplicatę ſemicir cumferentijs æquales erunt, & </s> <s xml:id="echoid-s15338" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s15339" xml:space="preserve">Atque hac arte inuenietur recta ęqualis quartæ parti <lb/>circumferentię cuiuſuis circuli, ſi eius ſemidiametro ex recta A E, æqualem <lb/>lineam abſcindemus, ab eiuſque extremo rectæ DE, parallelã ducemus, &</s> <s xml:id="echoid-s15340" xml:space="preserve">c.</s> <s xml:id="echoid-s15341" xml:space="preserve"/> </p> <div xml:id="echoid-div936" type="float" level="2" n="2"> <note position="right" xlink:label="note-357-05" xlink:href="note-357-05a" xml:space="preserve">Facilis inuen-<lb/>tio rectæ æqu@ <lb/>lis circumfe-<lb/>rentiæ.</note> <figure xlink:label="fig-357-02" xlink:href="fig-357-02a"> <image file="357-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/357-02"/> </figure> <note symbol="d" position="right" xlink:label="note-357-06" xlink:href="note-357-06a" xml:space="preserve">4. ſexti.</note> </div> <pb o="328" file="358" n="358" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s15342" xml:space="preserve"><emph style="sc">Vt</emph> quoque ſine vllo labore dato cuicunq; </s> <s xml:id="echoid-s15343" xml:space="preserve">circulo quadratum æquale ex-<lb/>hibeamus, vtendum erit hoc artificio. </s> <s xml:id="echoid-s15344" xml:space="preserve">Inuento ſemellatere quadrati alicui cir-<lb/> <anchor type="note" xlink:label="note-358-01a" xlink:href="note-358-01"/> culo æqualis, vt paulò ante docuimus, conſtruemus figuram ad quadrandos <lb/>alios circulos quo ſcunque accommodatiſsimam, hoc modo. </s> <s xml:id="echoid-s15345" xml:space="preserve">Detur circulus <lb/>A B C, diametri A C, ſitque A B, media proportionalis inter ſemidiametrum, <lb/>& </s> <s xml:id="echoid-s15346" xml:space="preserve">rectam ſemicircumferentiæ æqualem inuentam ex præcedenti figura, ita vt <lb/>quadratum rectæ AB, circulo diametri A C, ſit æquale: </s> <s xml:id="echoid-s15347" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> accommodetur AB, in <anchor type="note" xlink:label="note-358-02a" xlink:href="note-358-02"/> circulo, quæ certius applicabitur, ſi fortè circinus ex A, ad interuallũ datæ AB, <lb/>deſcriptus nimis oblique peripheriam A B C, ſecet in B, hoc modo. </s> <s xml:id="echoid-s15348" xml:space="preserve">Duabus <lb/>rectis, nimirum diametro AC, & </s> <s xml:id="echoid-s15349" xml:space="preserve">lateri AB, quadrati inuento reperiatur tertia ꝓ-<lb/>portionalis AD. </s> <s xml:id="echoid-s15350" xml:space="preserve">Perpendicularis namque DB, cadet in punctum B, in quod la-<lb/> <anchor type="note" xlink:label="note-358-03a" xlink:href="note-358-03"/> tus inuentum duci debet: </s> <s xml:id="echoid-s15351" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> propterea quod tres rectæ AC, AB, AD, ſunt conti- nuè proportionales, quemadmodum recta A C, latus quadrati inuentum, & </s> <s xml:id="echoid-s15352" xml:space="preserve"><lb/>AD, continuam ſeruant proportionem, ex conſtructione. </s> <s xml:id="echoid-s15353" xml:space="preserve">Liquet autem inter <lb/>AC, AD, vnam tantum poſſe eſſe mediam proportionalem. </s> <s xml:id="echoid-s15354" xml:space="preserve">Hac figura extru-<lb/>cta, dicto citius quemcunque circulum quadrabimus. </s> <s xml:id="echoid-s15355" xml:space="preserve">Sinamque diametro da-<lb/>ti circulirectam æqualem abſcindemus A F, circa quam ſemicirculus deſcriba-<lb/>tur, reſecabit is ex recta AB, latus AE, cuius quadratum circulo dato eſt æqua-<lb/> <anchor type="figure" xlink:label="fig-358-01a" xlink:href="fig-358-01"/> le. </s> <s xml:id="echoid-s15356" xml:space="preserve">Quia enim angulus externus AEF, inter-<lb/>no ABC, æqualis eſt: </s> <s xml:id="echoid-s15357" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> quod vterque in ſemi- <anchor type="note" xlink:label="note-358-04a" xlink:href="note-358-04"/> circulo rectus ſit; </s> <s xml:id="echoid-s15358" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erunt E F, B C, parallelæ;</s> <s xml:id="echoid-s15359" xml:space="preserve"> <anchor type="note" xlink:label="note-358-05a" xlink:href="note-358-05"/> ideoque triangula AEF, ABC, æquiangula. </s> <s xml:id="echoid-s15360" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <anchor type="note" xlink:label="note-358-06a" xlink:href="note-358-06"/> Igitur erit CA, ad AB, vt FA, ad AE; </s> <s xml:id="echoid-s15361" xml:space="preserve">Et permu-<lb/>tando CA, ad FA, vt AB, ad AE. </s> <s xml:id="echoid-s15362" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Ideoque e- <anchor type="note" xlink:label="note-358-07a" xlink:href="note-358-07"/> rit quoque quadratum ex AC, ad quadratum <lb/>ex A F: </s> <s xml:id="echoid-s15363" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> hoc eſt, vt circulus diametri A C, ad <anchor type="note" xlink:label="note-358-08a" xlink:href="note-358-08"/> circulum diametri A F, vt quadratum ex A B, <lb/>ad quadratum ex AE. </s> <s xml:id="echoid-s15364" xml:space="preserve">Eſt autem circulus dia-<lb/>metri A C, quadrato ex A B, per conſtru ctio-<lb/>nem, ęquale. </s> <s xml:id="echoid-s15365" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Igitur & </s> <s xml:id="echoid-s15366" xml:space="preserve">circulus diametri AF, <anchor type="note" xlink:label="note-358-09a" xlink:href="note-358-09"/> quadrato ex AE, æquale erit. </s> <s xml:id="echoid-s15367" xml:space="preserve">Ita quo que qua-<lb/>dratum rectæ A G, circulo diametri A H, erit <lb/>ęquale. </s> <s xml:id="echoid-s15368" xml:space="preserve">Et ſic de cęteris.</s> <s xml:id="echoid-s15369" xml:space="preserve"/> </p> <div xml:id="echoid-div937" type="float" level="2" n="3"> <note position="left" xlink:label="note-358-01" xlink:href="note-358-01a" xml:space="preserve">Facilis inuen-<lb/>tio quadrati <lb/>circulo æqua-<lb/>lis.</note> <note symbol="a" position="left" xlink:label="note-358-02" xlink:href="note-358-02a" xml:space="preserve">1. quinti.</note> <note symbol="b" position="left" xlink:label="note-358-03" xlink:href="note-358-03a" xml:space="preserve">coroll. 8. <lb/>ſexti.</note> <figure xlink:label="fig-358-01" xlink:href="fig-358-01a"> <image file="358-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/358-01"/> </figure> <note symbol="c" position="left" xlink:label="note-358-04" xlink:href="note-358-04a" xml:space="preserve">31. tertii.</note> <note symbol="d" position="left" xlink:label="note-358-05" xlink:href="note-358-05a" xml:space="preserve">28. primi.</note> <note symbol="e" position="left" xlink:label="note-358-06" xlink:href="note-358-06a" xml:space="preserve">4. ſexti.</note> <note symbol="f" position="left" xlink:label="note-358-07" xlink:href="note-358-07a" xml:space="preserve">22. ſexti.</note> <note symbol="g" position="left" xlink:label="note-358-08" xlink:href="note-358-08a" xml:space="preserve">2. duodec.</note> <note symbol="h" position="left" xlink:label="note-358-09" xlink:href="note-358-09a" xml:space="preserve">14. quinti.</note> </div> <p> <s xml:id="echoid-s15370" xml:space="preserve"><emph style="sc">Iam</emph> verò quoniam lib. </s> <s xml:id="echoid-s15371" xml:space="preserve">4. </s> <s xml:id="echoid-s15372" xml:space="preserve">cap. </s> <s xml:id="echoid-s15373" xml:space="preserve">6. </s> <s xml:id="echoid-s15374" xml:space="preserve">propoſ. <lb/></s> <s xml:id="echoid-s15375" xml:space="preserve"> <anchor type="note" xlink:label="note-358-10a" xlink:href="note-358-10"/> 3. </s> <s xml:id="echoid-s15376" xml:space="preserve">ex Archimede demonſtrauimus, quadratũ <lb/>diametri ad circulum habere ferme propor-<lb/>tionem, quam 14. </s> <s xml:id="echoid-s15377" xml:space="preserve">ad 11. </s> <s xml:id="echoid-s15378" xml:space="preserve">ſi quis volet ſecundũ <lb/>hanc proportionem reperire quadratum cir-<lb/>culo æquale; </s> <s xml:id="echoid-s15379" xml:space="preserve">diuidenda erit recta A C, in 14. <lb/></s> <s xml:id="echoid-s15380" xml:space="preserve">partes æquales, & </s> <s xml:id="echoid-s15381" xml:space="preserve">ex vndecima parte D, (ita vt AD, contineat partes 11. </s> <s xml:id="echoid-s15382" xml:space="preserve">& </s> <s xml:id="echoid-s15383" xml:space="preserve">DC, <lb/>3.) </s> <s xml:id="echoid-s15384" xml:space="preserve">ex citanda perpendicularis DB, vſque ad circumferentiam circa A C, deſcri-<lb/>ptam. </s> <s xml:id="echoid-s15385" xml:space="preserve">Recta enim enim ducta A B, latus erit quadrati circulo diametri A C, æ-<lb/> <anchor type="note" xlink:label="note-358-11a" xlink:href="note-358-11"/> qualis. </s> <s xml:id="echoid-s15386" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Cum enim tres rectæ AC, AB, AD, ſint continue proportionales; </s> <s xml:id="echoid-s15387" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> erit quadratum ex A C, ad quadratum ex A B, vt A C, ad A D, videlicet vt 14. </s> <s xml:id="echoid-s15388" xml:space="preserve">ad 11. <lb/></s> <s xml:id="echoid-s15389" xml:space="preserve"> <anchor type="note" xlink:label="note-358-12a" xlink:href="note-358-12"/> Cum ergo etiam ſit, vt diximus, quadratum diametriad circulum, vt 14. </s> <s xml:id="echoid-s15390" xml:space="preserve">ad 11. <lb/></s> <s xml:id="echoid-s15391" xml:space="preserve">ferme: </s> <s xml:id="echoid-s15392" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> erit quadratum ex AC, ad quadratum ex AB, vt ad circulum diametri <anchor type="note" xlink:label="note-358-13a" xlink:href="note-358-13"/> AC. </s> <s xml:id="echoid-s15393" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> Igitur quadratum ex AB, circulo diametri A C, æquale erit. </s> <s xml:id="echoid-s15394" xml:space="preserve">Quod ſi ſe- <anchor type="note" xlink:label="note-358-14a" xlink:href="note-358-14"/> <pb o="329" file="359" n="359" rhead="LIBER SEPTIMVS."/> cundum varias diametros deſcribantur circuli per A, tranſeuntes, abſcindent <lb/>quoq; </s> <s xml:id="echoid-s15395" xml:space="preserve">ij circuli ex recta AB, latera quadratorum illis circulis æqualium. </s> <s xml:id="echoid-s15396" xml:space="preserve">Habes <lb/>ergo viam facilem inueniendi quadratum circulo dato æquale, ſiue quadratri-<lb/>cemn oſtram adhibeas, ſiue demonſtrata ab Archimede ſequaris.</s> <s xml:id="echoid-s15397" xml:space="preserve"/> </p> <div xml:id="echoid-div938" type="float" level="2" n="4"> <note position="left" xlink:label="note-358-10" xlink:href="note-358-10a" xml:space="preserve">Facilis inuen-<lb/>tio quadrati-<lb/>circulo æqua-<lb/>lis ex Archi-<lb/>mede.</note> <note symbol="i" position="left" xlink:label="note-358-11" xlink:href="note-358-11a" xml:space="preserve">coroll. 2. <lb/>ſexti.</note> <note symbol="k" position="left" xlink:label="note-358-12" xlink:href="note-358-12a" xml:space="preserve">coroll. 20. <lb/>ſexti.</note> <note symbol="l" position="left" xlink:label="note-358-13" xlink:href="note-358-13a" xml:space="preserve">11. quinti.</note> <note symbol="m" position="left" xlink:label="note-358-14" xlink:href="note-358-14a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div940" type="section" level="1" n="328"> <head xml:id="echoid-head355" xml:space="preserve">IV.</head> <p> <s xml:id="echoid-s15398" xml:space="preserve">DATO quadrato circulum æqualem deſcribere.</s> <s xml:id="echoid-s15399" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15400" xml:space="preserve"><emph style="sc">Sit</emph> datum quadratum lateris AE, cui circulus æqualis eſt deſcribendus. </s> <s xml:id="echoid-s15401" xml:space="preserve">In <lb/>proxima figura ex recta AB, abſcindatur recta A E, dato lateri quadrati æqualis: <lb/></s> <s xml:id="echoid-s15402" xml:space="preserve">Et ex E, ducatur ad AB, perpendicularis E F, ſecans A C, in F. </s> <s xml:id="echoid-s15403" xml:space="preserve">Eritque circulus <lb/>diametri AF, quadrato lateris AE, æqualis, vt ex proximè demonſtratis liquet.</s> <s xml:id="echoid-s15404" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div941" type="section" level="1" n="329"> <head xml:id="echoid-head356" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s15405" xml:space="preserve"><emph style="sc">Ex</emph> his, quæ demonſtrata ſunt, conſtruemus circulum cuicunq; </s> <s xml:id="echoid-s15406" xml:space="preserve">figuræ re-<lb/>ctilineæ æqualem. </s> <s xml:id="echoid-s15407" xml:space="preserve">Et contra cuicunque circulo figuram rectilineam æqualem <lb/>conſtituemus, quæ alteri datæ figuræ rectilineæ cuicunque ſimilis ſit. </s> <s xml:id="echoid-s15408" xml:space="preserve">Nam ſi da-<lb/>tæ figuræ rectilineæ <anchor type="note" xlink:href="" symbol="a"/> deſcribamus quadratum æquale, & </s> <s xml:id="echoid-s15409" xml:space="preserve">huic quadrato circu- <anchor type="note" xlink:label="note-359-01a" xlink:href="note-359-01"/> lum æqualem per hanc 4. </s> <s xml:id="echoid-s15410" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15411" xml:space="preserve">conſtituamus; </s> <s xml:id="echoid-s15412" xml:space="preserve">erit idem hic circulus datæ fi-<lb/>guræ rectilineæ æqualis.</s> <s xml:id="echoid-s15413" xml:space="preserve"/> </p> <div xml:id="echoid-div941" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-359-01" xlink:href="note-359-01a" xml:space="preserve">14. ſecundi.</note> </div> <p> <s xml:id="echoid-s15414" xml:space="preserve"><emph style="sc">Rvrsvs</emph> ſi per propoſitionem 3. </s> <s xml:id="echoid-s15415" xml:space="preserve">dato circulo quadratum ęquale conſtru-<lb/>amus, huic autem quadrato <anchor type="note" xlink:href="" symbol="b"/> conſtituamus figuram rectilineam æqualem, & </s> <s xml:id="echoid-s15416" xml:space="preserve">ſi- <anchor type="note" xlink:label="note-359-02a" xlink:href="note-359-02"/> milem alteri datę figuræ rectilineæ; </s> <s xml:id="echoid-s15417" xml:space="preserve">erit eadem hæc figura rectilinea conſtituta, <lb/>dato circulo æqualis. </s> <s xml:id="echoid-s15418" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s15419" xml:space="preserve"/> </p> <div xml:id="echoid-div942" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-359-02" xlink:href="note-359-02a" xml:space="preserve">25. ſexti.</note> </div> </div> <div xml:id="echoid-div944" type="section" level="1" n="330"> <head xml:id="echoid-head357" xml:space="preserve">V.</head> <p> <s xml:id="echoid-s15420" xml:space="preserve">DATÆ rectæ lineæ circumferentiam circuli reperire æqualem.</s> <s xml:id="echoid-s15421" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15422" xml:space="preserve"><emph style="sc">In</emph> ſecunda figura propoſ. </s> <s xml:id="echoid-s15423" xml:space="preserve">3. </s> <s xml:id="echoid-s15424" xml:space="preserve">ſit rectæ O, exhibenda æqualis circumferentia. <lb/></s> <s xml:id="echoid-s15425" xml:space="preserve">Eius quartæ parti capiatur in latere Quadratricis A D, recta æqualis A I, ac per I, <lb/>ipſi D E, agatur parallela I H. </s> <s xml:id="echoid-s15426" xml:space="preserve">Eritque circumferentia circuli ex diametro A H, <lb/>deſcripti æqualis datę rectæ O, propterea quod quarta pars eius circum-<lb/>ferentiæ æqualis eſt rectæ AI, vt oſtenſum eſt ac proinde tota circũ-<lb/>ferentia æqualis erit quadruplæ rectæ AI, hoc eſt, æqua-<lb/>lis rectæ O, cuius quarta pars poſita eſt recta <lb/>AI. </s> <s xml:id="echoid-s15427" xml:space="preserve">Datæ ergo rectæ circumfentiã æ-<lb/>qualem reperim<emph style="sub">9</emph>. </s> <s xml:id="echoid-s15428" xml:space="preserve">Quod faci-<lb/>endum erat.</s> <s xml:id="echoid-s15429" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div945" type="section" level="1" n="331"> <head xml:id="echoid-head358" xml:space="preserve">FINIS LIBRI SEPTIMI.</head> <pb o="330" file="360" n="360"/> <figure> <image file="360-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/360-01"/> </figure> </div> <div xml:id="echoid-div946" type="section" level="1" n="332"> <head xml:id="echoid-head359" xml:space="preserve">GEOMETRIÆ <lb/>PRACTICÆ <lb/>LIBER OCTAVVS.</head> <figure> <image file="360-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/360-02"/> </figure> </div> <div xml:id="echoid-div947" type="section" level="1" n="333"> <head xml:id="echoid-head360" xml:space="preserve">Varia Theoremata, ac problemata Geometrica <lb/>demonſtrans.</head> <p style="it"> <s xml:id="echoid-s15430" xml:space="preserve">VT extremam manum Geometriæ huic noſtræ practicæ impona-<lb/>m{us}, concludem{us} eam variis nonnullis Theorematib{us}, at que <lb/>problematib{us} Geometricis, tum collectis ex Geometris aliis, tum <lb/>proprio, vt aiunt, Marte excogitatis, ac demonſtratis. </s> <s xml:id="echoid-s15431" xml:space="preserve">Qua in <lb/>re exemplum illuſtre habem{us} in Pappo Alexandrino, qui octo totos libros con-<lb/>ſcripſit de Mathematicis collectionib{us}. </s> <s xml:id="echoid-s15432" xml:space="preserve">Neque vero hoc præter inſtitutum no-<lb/>ſtrum exiſtimare quis debet: </s> <s xml:id="echoid-s15433" xml:space="preserve">cum per eiuſmodi demonſtrationes Geometric{as} ſtu-<lb/>dioſo Lectori via multiplex aperiatur ad inueſtigand{as} ſimiles ſpeculationes in <lb/>reb{us} Geometricis: </s> <s xml:id="echoid-s15434" xml:space="preserve">quippe cum in iis ad exercendum ingenium ampliſſimum <lb/>campum habeat. </s> <s xml:id="echoid-s15435" xml:space="preserve">Eſt & </s> <s xml:id="echoid-s15436" xml:space="preserve">alia cauſa, quæ me ad hunc librum octauum conſcri-<lb/>bendum permouit, ne videlicet tot Theoremata, ac problemata non ſi{ne} magno <lb/>labore perueſtigata pereant, cum ad nullam Geometriæ partem magis propriè <lb/>pertineant, quam ad hanc Geometriam practicam: </s> <s xml:id="echoid-s15437" xml:space="preserve">præſertim quod pleraque <lb/>c<unsure/>orum praxes Geometric{as} pertractent. </s> <s xml:id="echoid-s15438" xml:space="preserve">Adde quod non pauci viri docti & </s> <s xml:id="echoid-s15439" xml:space="preserve">gra-<lb/>@es ad hunc librum perſcribendum auctores mihi, atque ſuaſores fuerunt.</s> <s xml:id="echoid-s15440" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div948" type="section" level="1" n="334"> <head xml:id="echoid-head361" xml:space="preserve">THEOR. 1. PROPOS. 1.</head> <p> <s xml:id="echoid-s15441" xml:space="preserve">FIGVRA regularis circulo circumſcripta maiorem ambitum habet, <lb/>quam circulus.</s> <s xml:id="echoid-s15442" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15443" xml:space="preserve"><emph style="sc">Hæc</emph> eſt prima propoſitio Archimedis in lib. </s> <s xml:id="echoid-s15444" xml:space="preserve">1. </s> <s xml:id="echoid-s15445" xml:space="preserve">de ſphęra & </s> <s xml:id="echoid-s15446" xml:space="preserve">Cylindro: </s> <s xml:id="echoid-s15447" xml:space="preserve">quã <lb/>demonſtrat, hoc aſſumpto principio.</s> <s xml:id="echoid-s15448" xml:space="preserve"/> </p> <pb o="331" file="361" n="361" rhead="LIBER OCTAVVS."/> <p> <s xml:id="echoid-s15449" xml:space="preserve">Si duæ lineæ in plano eoſdem habeant terminos, & </s> <s xml:id="echoid-s15450" xml:space="preserve">in eaſdem partes cauæ ſint, compre-<lb/>hendens comprehenſa maior eſt. </s> <s xml:id="echoid-s15451" xml:space="preserve">quod quidem principium eſſe verum, ex eo euidẽ-<lb/>ter intelligi poteſt quod ex eo, non ſolum Archimedes, verum etiam plurimi a-<lb/>lij Geometræ tum veteres, tum recentiores, innumera propemodum, atque ad-<lb/>miranda Theoremata, problemataque demonſtrarint, quæ vt veriſsima, ab o-<lb/>mnibus recepta ſunt; </s> <s xml:id="echoid-s15452" xml:space="preserve">neque vnquam ex illo abſurdi aliquid conſecutum eſt, <lb/>aut contra id quiſquam hactenus à du obus ferme millibus annorum, noui quid <lb/>commentus eſt. </s> <s xml:id="echoid-s15453" xml:space="preserve">Hoc ergo poſito principio, facilis eſt de-<lb/> <anchor type="figure" xlink:label="fig-361-01a" xlink:href="fig-361-01"/> monſtratio Archimedis. </s> <s xml:id="echoid-s15454" xml:space="preserve">Sit namque figura regularis ABC-<lb/>DEF, deſcripta circa circulũ, cuius centrũ N, tangens eũ in <lb/>punctis G, H, I, K, L, M. </s> <s xml:id="echoid-s15455" xml:space="preserve">Quoniã igitur per præmiſlum prin-<lb/>cipium rectæ A G, A M, maiores ſunt arcu G M: </s> <s xml:id="echoid-s15456" xml:space="preserve">Item B G, <lb/>B H, maiores arcu G H, & </s> <s xml:id="echoid-s15457" xml:space="preserve">ſic de reliquis; </s> <s xml:id="echoid-s15458" xml:space="preserve">erunt omnes rectæ <lb/>ſimul conficientes totum ambitum figuræ, maiores omni-<lb/>bus arcubus ſimul conficientibus totam circuli perip heriam. </s> <s xml:id="echoid-s15459" xml:space="preserve">quod erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s15460" xml:space="preserve"/> </p> <div xml:id="echoid-div948" type="float" level="2" n="1"> <figure xlink:label="fig-361-01" xlink:href="fig-361-01a"> <image file="361-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/361-01"/> </figure> </div> </div> <div xml:id="echoid-div950" type="section" level="1" n="335"> <head xml:id="echoid-head362" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15461" xml:space="preserve"><emph style="sc">Cardanvs</emph> in libro quinto de proportionibus propoſ. </s> <s xml:id="echoid-s15462" xml:space="preserve">201. </s> <s xml:id="echoid-s15463" xml:space="preserve">conatur de-<lb/>monſtrare, duas rectas circulum contingentes, cuiuſmo di ſunt A G, A M, maio-<lb/>res eſſe arcu intercepto GM, (quod Archimedes ex ſuo aſſumpto principio de-<lb/>duxit (præmiſsis tribus Lemmatibus, & </s> <s xml:id="echoid-s15464" xml:space="preserve">vno principio. </s> <s xml:id="echoid-s15465" xml:space="preserve">quorum primum eſt <lb/>hoc.</s> <s xml:id="echoid-s15466" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div951" type="section" level="1" n="336"> <head xml:id="echoid-head363" xml:space="preserve">LEMMA I.</head> <p> <s xml:id="echoid-s15467" xml:space="preserve">SI fuerint quatuor quantitates, & </s> <s xml:id="echoid-s15468" xml:space="preserve">minor ſit exceſſus inter primã & </s> <s xml:id="echoid-s15469" xml:space="preserve">ſecũ-<lb/>dã, quam inter tertiã & </s> <s xml:id="echoid-s15470" xml:space="preserve">quartam; </s> <s xml:id="echoid-s15471" xml:space="preserve">ſitq; </s> <s xml:id="echoid-s15472" xml:space="preserve">prima non minor, quam tertia, <lb/>maior verò quam ſecunda, Item tertia maior quam quarta: </s> <s xml:id="echoid-s15473" xml:space="preserve">Erit mi-<lb/>nor proportio primæ ad ſecundam, quam tertiæ ad quartam.</s> <s xml:id="echoid-s15474" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15475" xml:space="preserve"><emph style="sc">Sint</emph> quatuor quantitates A, BC, D, EF; </s> <s xml:id="echoid-s15476" xml:space="preserve">ſitque GB, exceſſus inter primam <lb/>A, & </s> <s xml:id="echoid-s15477" xml:space="preserve">ſecundam BC, minor exceſſu H E, inter tertiam <lb/> <anchor type="figure" xlink:label="fig-361-02a" xlink:href="fig-361-02"/> D, & </s> <s xml:id="echoid-s15478" xml:space="preserve">quartam E F; </s> <s xml:id="echoid-s15479" xml:space="preserve">Item prima A, non ſit minor, quã <lb/>tertia D: </s> <s xml:id="echoid-s15480" xml:space="preserve">maior verò quam ſecunda B C: </s> <s xml:id="echoid-s15481" xml:space="preserve">Ac deni-<lb/>que tertia D; </s> <s xml:id="echoid-s15482" xml:space="preserve">maiorſit quam quarta E F; </s> <s xml:id="echoid-s15483" xml:space="preserve">Dico mino-<lb/>rem eſſe proportionem primæ A, ad ſecundam B C, <lb/>quam tertiæ D, ad quartam E F. </s> <s xml:id="echoid-s15484" xml:space="preserve">Cum enim A, non <lb/>minor ſit, ꝗ̃ D: </s> <s xml:id="echoid-s15485" xml:space="preserve">at GB, minor, ꝗ̃ HE, <anchor type="note" xlink:href="" symbol="a"/> erit maior ꝓ por- <anchor type="note" xlink:label="note-361-01a" xlink:href="note-361-01"/> tio A, ad GB, quam ad HE: </s> <s xml:id="echoid-s15486" xml:space="preserve">Eſt autem A, (ſi eſt æqua-<lb/>lis ip ſi D,) ad H E, vt D, ad H E; </s> <s xml:id="echoid-s15487" xml:space="preserve">vel maior eſt proportio A, (ſi maior eſt, quam <lb/>D,) ad HE. </s> <s xml:id="echoid-s15488" xml:space="preserve">quam D, ad HE. </s> <s xml:id="echoid-s15489" xml:space="preserve">Igitur maior erit proportio A, ad GB, quam D, ad <lb/>H E. </s> <s xml:id="echoid-s15490" xml:space="preserve">Si igitur fiat vt D, ad HE, ita A, ad G I, habebit quo que A, ad GB, ma-<lb/>iorem proportionem, quam ad G I; </s> <s xml:id="echoid-s15491" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ac proinde erit GI, maior quam G B; </s> <s xml:id="echoid-s15492" xml:space="preserve">id- <anchor type="note" xlink:label="note-361-02a" xlink:href="note-361-02"/> eo que I C, minor, quam B C. </s> <s xml:id="echoid-s15493" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Maior ergo erit proportio A, ad I C. </s> <s xml:id="echoid-s15494" xml:space="preserve">quam <anchor type="note" xlink:label="note-361-03a" xlink:href="note-361-03"/> <pb o="332" file="362" n="362" rhead="GEOMETR. PRACT."/> ad B C. </s> <s xml:id="echoid-s15495" xml:space="preserve">Et quoniam G C, ipſi A, æqualis, eſt ad G I, vt H F, ipſi D, æqualis, ad <lb/>HE; </s> <s xml:id="echoid-s15496" xml:space="preserve">Erit quo que per conuerſionem rationis GC, hoc eſt, A, ad IC, vt HF, hoc <lb/>eſt, vt D, ad E F. </s> <s xml:id="echoid-s15497" xml:space="preserve">Cum ergo oſtenſum ſit, maiorem eſſe proportionem A, ad <lb/>IC, quam ad B C; </s> <s xml:id="echoid-s15498" xml:space="preserve">erit quo que maior proportio D, ad E F, quam A, ad B C, hoc <lb/>eſt, A, ad B C, minorem proportionem habebit, quam D, ad E F. </s> <s xml:id="echoid-s15499" xml:space="preserve">quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s15500" xml:space="preserve"/> </p> <div xml:id="echoid-div951" type="float" level="2" n="1"> <figure xlink:label="fig-361-02" xlink:href="fig-361-02a"> <image file="361-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/361-02"/> </figure> <note symbol="a" position="right" xlink:label="note-361-01" xlink:href="note-361-01a" xml:space="preserve">8. quinti.</note> <note symbol="b" position="right" xlink:label="note-361-02" xlink:href="note-361-02a" xml:space="preserve">10. quinti.</note> <note symbol="c" position="right" xlink:label="note-361-03" xlink:href="note-361-03a" xml:space="preserve">8. quinti.</note> </div> </div> <div xml:id="echoid-div953" type="section" level="1" n="337"> <head xml:id="echoid-head364" xml:space="preserve">LEMMA II.</head> <p> <s xml:id="echoid-s15501" xml:space="preserve">SI circuli arcum duæ rectæ tangant in vno puncto coeuntes; </s> <s xml:id="echoid-s15502" xml:space="preserve">& </s> <s xml:id="echoid-s15503" xml:space="preserve">in eo-<lb/>dem arcu aptentur quotlibet rectæ æquales diuidentes ipſum in par-<lb/>tes totidem æquales. </s> <s xml:id="echoid-s15504" xml:space="preserve">Erunt duæ illæ tangentes omnibus hiſce chor-<lb/>dis ſimul maiores.</s> <s xml:id="echoid-s15505" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15506" xml:space="preserve"><emph style="sc">Tangant</emph> arcum AB, duæ rectæ AK, BK, coeuntes in K, aptenturq; </s> <s xml:id="echoid-s15507" xml:space="preserve">quot-<lb/>libet rectæ in eo æquales AC, CD, DE, EF, FG, GB, diuidentes arcum in totidem <lb/>partes æquales. </s> <s xml:id="echoid-s15508" xml:space="preserve">Dico rectas AK, BK, ſimul maiores eſſe omnibus illis rectis ſub-<lb/>tenſis ſimul. </s> <s xml:id="echoid-s15509" xml:space="preserve">Productis enim rectis AC, BG, donec coeant in H; </s> <s xml:id="echoid-s15510" xml:space="preserve">Item pro-<lb/>ductis rectis CD, GF, donec concurrantin I, & </s> <s xml:id="echoid-s15511" xml:space="preserve">ſic deinceps, ſi plures rectæ fu-<lb/>erint: </s> <s xml:id="echoid-s15512" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Erunt rectæ DI, FI, maiores rectis <anchor type="figure" xlink:label="fig-362-01a" xlink:href="fig-362-01"/> <anchor type="note" xlink:label="note-362-01a" xlink:href="note-362-01"/> DE, FE. </s> <s xml:id="echoid-s15513" xml:space="preserve">Additis ergo æqualibus DC, FG; <lb/></s> <s xml:id="echoid-s15514" xml:space="preserve">erunt etiam rectæ C I, G I; </s> <s xml:id="echoid-s15515" xml:space="preserve">maiores rectis <lb/> <anchor type="note" xlink:label="note-362-02a" xlink:href="note-362-02"/> CD, DE, EF, FG, ſimul. </s> <s xml:id="echoid-s15516" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Sed C H, G H, maiores ſunt rectis C I, G I. </s> <s xml:id="echoid-s15517" xml:space="preserve">Igitur multo <lb/>maiores erunt CH, GH, rectis C D, D E, <lb/>EF, F G; </s> <s xml:id="echoid-s15518" xml:space="preserve">additiſque æqualibus A C, B G, <lb/>maiores erunt AH, B H, ſimul quam A C, <lb/>CD, DE, EF, FG, GB, ſimul. </s> <s xml:id="echoid-s15519" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Sunt autem <anchor type="note" xlink:label="note-362-03a" xlink:href="note-362-03"/> & </s> <s xml:id="echoid-s15520" xml:space="preserve">AK, BK, maiores, quam AH, BH. </s> <s xml:id="echoid-s15521" xml:space="preserve">Igi-<lb/>tur multo maiores erunt A K, B K, ſimul <lb/>quam AC, CD, DE, EF, FG, GB, ſimul. </s> <s xml:id="echoid-s15522" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s15523" xml:space="preserve"/> </p> <div xml:id="echoid-div953" type="float" level="2" n="1"> <figure xlink:label="fig-362-01" xlink:href="fig-362-01a"> <image file="362-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/362-01"/> </figure> <note symbol="a" position="left" xlink:label="note-362-01" xlink:href="note-362-01a" xml:space="preserve">21. primi.</note> <note symbol="b" position="left" xlink:label="note-362-02" xlink:href="note-362-02a" xml:space="preserve">21. primi.</note> <note symbol="c" position="left" xlink:label="note-362-03" xlink:href="note-362-03a" xml:space="preserve">21. primi.</note> </div> </div> <div xml:id="echoid-div955" type="section" level="1" n="338"> <head xml:id="echoid-head365" xml:space="preserve">EEMMA III.</head> <p> <s xml:id="echoid-s15524" xml:space="preserve">SI circuli arcum tres rectæ tangant in duobus punctis coeuntes, ita vt <lb/>contactuum punctum medium diuidat arcum bifariam: </s> <s xml:id="echoid-s15525" xml:space="preserve">In eodem <lb/>autem arcu accommodentur quotlibet rectæ numero pares, & </s> <s xml:id="echoid-s15526" xml:space="preserve">inter <lb/>ſe æquales, Erunt tres illæ tangẽtes omnib. </s> <s xml:id="echoid-s15527" xml:space="preserve">his ſimul ſumptis maiores.</s> <s xml:id="echoid-s15528" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15529" xml:space="preserve"><emph style="sc">In</emph> antecedente figura arcum AB, tangant tres rectæ AC, CD, DB, conueni-<lb/>entes in duobus punctis C, D, ſecantes ipſum bifariam in E. </s> <s xml:id="echoid-s15530" xml:space="preserve">Accommo den-<lb/>turque in eo dem arcu quotlibet rectæ æquales, & </s> <s xml:id="echoid-s15531" xml:space="preserve">numero pares AF, F E, EG, <lb/>GB. </s> <s xml:id="echoid-s15532" xml:space="preserve">Dico tres AC, CD, DB, ſimul ſumptas eſſe maiores rectis AF, FE, EG, GB, <lb/>ſimul ſumptis. </s> <s xml:id="echoid-s15533" xml:space="preserve">Quoniam enim per Lemma præcedens AC, C E, maiores ſunt, <lb/>rectis AF, FE: </s> <s xml:id="echoid-s15534" xml:space="preserve">Item BD, DE, maiores rectis B G, G E, Erunt quo que A C, C D, <lb/>D B, ſimul maiores rectis AF, FE, EG, GB, ſimul. </s> <s xml:id="echoid-s15535" xml:space="preserve">quod oſtendendum erat.</s> <s xml:id="echoid-s15536" xml:space="preserve"/> </p> <pb o="333" file="363" n="363" rhead="LIBER OCTAVVS."/> <p> <s xml:id="echoid-s15537" xml:space="preserve"><emph style="sc">Hæc</emph> ergo ſunt tria lemmata, quæ Cardanus præmittit: </s> <s xml:id="echoid-s15538" xml:space="preserve">quibus adiungit <lb/> <anchor type="note" xlink:label="note-363-01a" xlink:href="note-363-01"/> hoc poſtulatum, ſine principium. </s> <s xml:id="echoid-s15539" xml:space="preserve">Cum arcus quilibet maior ſit quotcunque <lb/>rectis in eo ſubtenſis ſimul ſumptis, & </s> <s xml:id="echoid-s15540" xml:space="preserve">quo plures ſubtenſæ fuerint, eo minori <lb/>exceſſu arcus illas ſuperet: </s> <s xml:id="echoid-s15541" xml:space="preserve">fieri poteſt, vt tot ſubtenſæ duci poſsint, ita vt ex-<lb/>ceſſus, quo arcus illas ſuperet, minor ſit quauis recta propoſita. </s> <s xml:id="echoid-s15542" xml:space="preserve">Hoc princi-<lb/>pium videtur eſſe manifeſtum, cumtam paruus arcus poſsit accipi, vt eius chor-<lb/>da illi ferè æqualis ſit; </s> <s xml:id="echoid-s15543" xml:space="preserve">adeò vt ſenſus nullam percipere poſsit inter arcum, & </s> <s xml:id="echoid-s15544" xml:space="preserve"><lb/>chordam differentiam. </s> <s xml:id="echoid-s15545" xml:space="preserve">A poſteriori tamen illud confirmari poteſt per nume-<lb/>ros. </s> <s xml:id="echoid-s15546" xml:space="preserve">Poſita enim proportione circumferentiæ ad diametrum fermè 31415926. <lb/></s> <s xml:id="echoid-s15547" xml:space="preserve">ad 10000000. </s> <s xml:id="echoid-s15548" xml:space="preserve">vt lib. </s> <s xml:id="echoid-s15549" xml:space="preserve">4. </s> <s xml:id="echoid-s15550" xml:space="preserve">cap. </s> <s xml:id="echoid-s15551" xml:space="preserve">7. </s> <s xml:id="echoid-s15552" xml:space="preserve">Num. </s> <s xml:id="echoid-s15553" xml:space="preserve">5. </s> <s xml:id="echoid-s15554" xml:space="preserve">ex probatis auctoribus retulimus, depre-<lb/>hendemus, ſi arcus propoſitus continuè ſecetur bifariam per rectas ſubtenſas, <lb/>ſemper à præ cedenti exceſſu plus dimidio auferri; </s> <s xml:id="echoid-s15555" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ac proinde tandem relin- <anchor type="note" xlink:label="note-363-02a" xlink:href="note-363-02"/> qui exceſſum omni quantitate minorem. </s> <s xml:id="echoid-s15556" xml:space="preserve">Nam arcus verbi gratia graduum 4. <lb/></s> <s xml:id="echoid-s15557" xml:space="preserve">erit 698131. </s> <s xml:id="echoid-s15558" xml:space="preserve">& </s> <s xml:id="echoid-s15559" xml:space="preserve">eius chorda ex tabula ſinuum eruta 697990. </s> <s xml:id="echoid-s15560" xml:space="preserve">ita vt arcus chor-<lb/>dam ſuperet hoc numero 141. </s> <s xml:id="echoid-s15561" xml:space="preserve">Summa deinde duarum chordarum graduum 2. </s> <s xml:id="echoid-s15562" xml:space="preserve"><lb/>erit 698096. </s> <s xml:id="echoid-s15563" xml:space="preserve">quæ ſuperatur ab eodem arcu grad. </s> <s xml:id="echoid-s15564" xml:space="preserve">4. </s> <s xml:id="echoid-s15565" xml:space="preserve">numero hoc 35. </s> <s xml:id="echoid-s15566" xml:space="preserve">qui minor <lb/>eſt ſemiſſe præcedentis exceſſus 141. </s> <s xml:id="echoid-s15567" xml:space="preserve">ac proinde plus dimidio ab eo ablatum <lb/>erit. </s> <s xml:id="echoid-s15568" xml:space="preserve">Rurſus ſumma quatuor chordarum gradus 1. </s> <s xml:id="echoid-s15569" xml:space="preserve">erit 698120. </s> <s xml:id="echoid-s15570" xml:space="preserve">quam idem ar-<lb/>cus 4. </s> <s xml:id="echoid-s15571" xml:space="preserve">graduum ſuperat hoc numero 11. </s> <s xml:id="echoid-s15572" xml:space="preserve">qui etiam minor eſt ſemiſſe proximi ex-<lb/>ceſſus 35. </s> <s xml:id="echoid-s15573" xml:space="preserve">Item ſumma 8. </s> <s xml:id="echoid-s15574" xml:space="preserve">chordarum, quarum quælibet 30. </s> <s xml:id="echoid-s15575" xml:space="preserve">minutis debetur, erit <lb/>698128. </s> <s xml:id="echoid-s15576" xml:space="preserve">exceſſus autem inter eam, & </s> <s xml:id="echoid-s15577" xml:space="preserve">eundem arcum 4. </s> <s xml:id="echoid-s15578" xml:space="preserve">graduum, numerus 3. </s> <s xml:id="echoid-s15579" xml:space="preserve"><lb/>qui minor quoque eſt, quam ſemiſsis proximi exceſſus 11. </s> <s xml:id="echoid-s15580" xml:space="preserve">& </s> <s xml:id="echoid-s15581" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s15582" xml:space="preserve"><lb/>Scio confirmationem hanc propoſiti principii non eſſe demonſtratiuam, cum <lb/>prop ortio circumferentiæ ad diametrum colligatur ex eo, quod demonſtrare <lb/>conamur, nimirum figuram circulo circumſcriptam habere maiorem ambitum <lb/>ambitu circuli: </s> <s xml:id="echoid-s15583" xml:space="preserve">eam tamen probabilem eſſe, nemo dubitabit, cum vix credi@-<lb/>le videatur, (ſi illa proportio longè à vero abeſſet) exceſſus illos paulatim ita <lb/>minui, vt ſemper minor numerus ſemiſſe præcedentis exceſſus relin quatur; </s> <s xml:id="echoid-s15584" xml:space="preserve">adeò <lb/>vt tandem nulla ferè differentia inter arcum, & </s> <s xml:id="echoid-s15585" xml:space="preserve">ſummam chordarum ſubtenſa-<lb/>rum rep eriatur.</s> <s xml:id="echoid-s15586" xml:space="preserve"/> </p> <div xml:id="echoid-div955" type="float" level="2" n="1"> <note position="right" xlink:label="note-363-01" xlink:href="note-363-01a" xml:space="preserve">Principium <lb/>Cardani.</note> <note symbol="a" position="right" xlink:label="note-363-02" xlink:href="note-363-02a" xml:space="preserve">1. decimi.</note> </div> <p> <s xml:id="echoid-s15587" xml:space="preserve"><emph style="sc">His</emph> præmiſsis, tangant duæ rectæ AB, AL, arcum BCL. </s> <s xml:id="echoid-s15588" xml:space="preserve">Dico eas eſſe ma-<lb/> <anchor type="note" xlink:label="note-363-03a" xlink:href="note-363-03"/> iores arcu. </s> <s xml:id="echoid-s15589" xml:space="preserve">Sint enim, ſi fieri poteſt, non maiores, ac proinde arcus BCL, ſit vel <lb/> <anchor type="figure" xlink:label="fig-363-01a" xlink:href="fig-363-01"/> æqualis rectis AB, AL, vel maior. </s> <s xml:id="echoid-s15590" xml:space="preserve">Secto ergo arcu bifariam <lb/>in C, ducatur DCE, tangens arcum in C. </s> <s xml:id="echoid-s15591" xml:space="preserve">Diuiſis quo que ar-<lb/>cubus CB, CL, bifariam in G, F, iungantur rectæ B G, G C, <lb/>CF, FL. </s> <s xml:id="echoid-s15592" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et quia AD, AE, maiores ſunt quam DE; </s> <s xml:id="echoid-s15593" xml:space="preserve">addi- <anchor type="note" xlink:label="note-363-04a" xlink:href="note-363-04"/> tis DL, EB, communibus, quæ æquales ſunt; </s> <s xml:id="echoid-s15594" xml:space="preserve">(Namiunctis <lb/>rectis NA, NB, NL, ex centro N; </s> <s xml:id="echoid-s15595" xml:space="preserve">quoniã tria latera trianguli <lb/>ABN, tribus lateribus triãguli ALN, æqualia ſunt; </s> <s xml:id="echoid-s15596" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt tã <anchor type="note" xlink:label="note-363-05a" xlink:href="note-363-05"/> anguli ad N, quam ad A, æquales; </s> <s xml:id="echoid-s15597" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> ideoq; </s> <s xml:id="echoid-s15598" xml:space="preserve">arcus CB, CL, <anchor type="note" xlink:label="note-363-06a" xlink:href="note-363-06"/> æquales erunt; </s> <s xml:id="echoid-s15599" xml:space="preserve">ac proinde recta N A, per contactum C, <lb/> <anchor type="note" xlink:label="note-363-07a" xlink:href="note-363-07"/> tranſibit, <anchor type="note" xlink:href="" symbol="e"/> eritque ad DE, perpendicularis. </s> <s xml:id="echoid-s15600" xml:space="preserve">Cum igitur duo anguli DAC, DCA, <anchor type="note" xlink:label="note-363-08a" xlink:href="note-363-08"/> duobus angulis E A C, E C A, æquales ſint, & </s> <s xml:id="echoid-s15601" xml:space="preserve">latus adiacens A C, commune; <lb/></s> <s xml:id="echoid-s15602" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="f"/> erunt latera AD, AE, æqualia: </s> <s xml:id="echoid-s15603" xml:space="preserve">proptereaque & </s> <s xml:id="echoid-s15604" xml:space="preserve">reliquæ DL, EB, æquales erunt, cum tangentes AL, AB, æquales ſint) erunt A L, A B, maiores tribus L D, DE, <lb/>EB. </s> <s xml:id="echoid-s15605" xml:space="preserve">Sit ergo exceſſus H. </s> <s xml:id="echoid-s15606" xml:space="preserve">Rurſus quia arcus BL, maior eſt rectis BG, GC, CF, FL, <lb/>ſit exceſſus I, qui minor ſit exceſſu H. </s> <s xml:id="echoid-s15607" xml:space="preserve">Si nam que minor non eſt, diuidemus ar- <pb o="334" file="364" n="364" rhead="GEOMETR. PRACT."/> cus LF, FC, CG, GB, bifariam, & </s> <s xml:id="echoid-s15608" xml:space="preserve">hos rurſus bifariã, & </s> <s xml:id="echoid-s15609" xml:space="preserve">c. </s> <s xml:id="echoid-s15610" xml:space="preserve">connectemuſq; </s> <s xml:id="echoid-s15611" xml:space="preserve">rectas, <lb/>donec fiat exceſſus minor ex ceſſu H, per ſuperius principium Cardani. </s> <s xml:id="echoid-s15612" xml:space="preserve">Quo-<lb/>niam igitur arcus L B, prima quantitas ſuperat ſecundam, videlicet rectas L F, <lb/>FC, CG, GB, ſimul exceſſu I; </s> <s xml:id="echoid-s15613" xml:space="preserve">Et tertia quantitas, nimirum ſumma rectarum AL, <lb/>AB, ſuperat quartam, id eſt, ſummam rectarum LD, DE, EB, exceſſu H: </s> <s xml:id="echoid-s15614" xml:space="preserve">Eſt que <lb/>exceſſus I, minor exceſſu H; </s> <s xml:id="echoid-s15615" xml:space="preserve">Et prima quantitas, hoc eſt, arcus BL, ponitur non <lb/>minor, quam tertia ex AB, AL, conflata; </s> <s xml:id="echoid-s15616" xml:space="preserve">item tertia AB, AI, maior, quam quar-<lb/>ta LD, DE, EB: </s> <s xml:id="echoid-s15617" xml:space="preserve">erit per 1. </s> <s xml:id="echoid-s15618" xml:space="preserve">Lemma, minor proportio arcus BL, primæ quantita-<lb/>tis ad ſecundam LF, FC, CG, GB q@iam tertiæ quantitatis AL, AB, ad quartam <lb/>L D, D E, E B; </s> <s xml:id="echoid-s15619" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et permutando minor erit proportio arcus L B, ad A L, A B, <anchor type="note" xlink:label="note-364-01a" xlink:href="note-364-01"/> ſimul, quam rectarum LF, FC, CG, GB, ſimul ad rectas LD, DE, EB, ſimul. </s> <s xml:id="echoid-s15620" xml:space="preserve">Sit <lb/>ergo vt compoſita ex LF, FC, CG, GB, ad compoſitam ex LD, DE, EB, ita ar-<lb/>cus BK, ad rectas AL, AB, ſimul: </s> <s xml:id="echoid-s15621" xml:space="preserve">Eritque propterea minor etiam proportio ar-<lb/>cus B L, ad AL, AB, ſimul, quam arcus B L, ad arcum BK; </s> <s xml:id="echoid-s15622" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> ideo que arcus BK, <anchor type="note" xlink:label="note-364-02a" xlink:href="note-364-02"/> maior erit arcu BL. </s> <s xml:id="echoid-s15623" xml:space="preserve">Cum ergo eadem ſit proportio rectarum LF, FC, CG, GB, <lb/>ſimul ad LD, DE, EB, ſimul, quæ arcus BK, ad AL, AB, ſimul: </s> <s xml:id="echoid-s15624" xml:space="preserve">ſintque per 3. </s> <s xml:id="echoid-s15625" xml:space="preserve">Lem-<lb/>ma, rectæ LF, FC, CG, GB, ſimul minores, quam LD, DE, EB, ſimul; </s> <s xml:id="echoid-s15626" xml:space="preserve">erit quo-<lb/>que arcus B K, minor, quam AL, AB, ſimul. </s> <s xml:id="echoid-s15627" xml:space="preserve">Multò ergo minor erit arcus BL, <lb/>duabus AL, AB, ſimul. </s> <s xml:id="echoid-s15628" xml:space="preserve">Quare rectæ tangentes AL, AB, ſimul maiores ſunt ar-<lb/>cu BL, quod erat oſtendendum.</s> <s xml:id="echoid-s15629" xml:space="preserve"/> </p> <div xml:id="echoid-div956" type="float" level="2" n="2"> <note position="right" xlink:label="note-363-03" xlink:href="note-363-03a" xml:space="preserve">Demonſtra-<lb/>tio Cardani</note> <figure xlink:label="fig-363-01" xlink:href="fig-363-01a"> <image file="363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/363-01"/> </figure> <note symbol="b" position="right" xlink:label="note-363-04" xlink:href="note-363-04a" xml:space="preserve">20. primi.</note> <note symbol="c" position="right" xlink:label="note-363-05" xlink:href="note-363-05a" xml:space="preserve">8. primi.</note> <note symbol="d" position="right" xlink:label="note-363-06" xlink:href="note-363-06a" xml:space="preserve">26. tertii.</note> <note symbol="e" position="right" xlink:label="note-363-07" xlink:href="note-363-07a" xml:space="preserve">18. tertii.</note> <note symbol="f" position="right" xlink:label="note-363-08" xlink:href="note-363-08a" xml:space="preserve">26. primi.</note> <note symbol="a" position="left" xlink:label="note-364-01" xlink:href="note-364-01a" xml:space="preserve">ſchol. 27. <lb/>quinti.</note> <note symbol="b" position="left" xlink:label="note-364-02" xlink:href="note-364-02a" xml:space="preserve">10. quinti.</note> </div> <p> <s xml:id="echoid-s15630" xml:space="preserve"><emph style="sc">Est</emph> autem hæc demonſtratio Cardani admirabilis, & </s> <s xml:id="echoid-s15631" xml:space="preserve">non abſimilis illi, qua <lb/>Eucl, in propoſ. </s> <s xml:id="echoid-s15632" xml:space="preserve">12. </s> <s xml:id="echoid-s15633" xml:space="preserve">lib. </s> <s xml:id="echoid-s15634" xml:space="preserve">9. </s> <s xml:id="echoid-s15635" xml:space="preserve">vtitur. </s> <s xml:id="echoid-s15636" xml:space="preserve">In vtraque enim infertur concluſio demonſtra-<lb/>tione affirmatiua ex eius oppoſito, vt patet.</s> <s xml:id="echoid-s15637" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15638" xml:space="preserve"><emph style="sc">Attvli</emph> hanc demonſtrationem Cardani, non quòd verè Geometrica ſit, <lb/>niſi principium illud ſuum admittatur, ſed quod ingenioſa ſit & </s> <s xml:id="echoid-s15639" xml:space="preserve">acuta. </s> <s xml:id="echoid-s15640" xml:space="preserve">Sine ta-<lb/>men hac demonſtratione concedendum erit, ambitum figuræ circumſcriptæ eſ-<lb/>ſe inaiorem peripheria circuli propter demonſtrationem Archimedis, cumnihil <lb/>vnquam in contrarium à quo quam ſit allatum, vt ſupra diximus.</s> <s xml:id="echoid-s15641" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div958" type="section" level="1" n="339"> <head xml:id="echoid-head366" xml:space="preserve">THEOR. 2. PROPOS. 2.</head> <p> <s xml:id="echoid-s15642" xml:space="preserve">CIRCVLORVM diametri inter ſe ſunt, vt circumferentiæ.</s> <s xml:id="echoid-s15643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15644" xml:space="preserve"><emph style="sc">Hoc</emph> demonſtrauimus nos in libr. </s> <s xml:id="echoid-s15645" xml:space="preserve">4. </s> <s xml:id="echoid-s15646" xml:space="preserve">cap. </s> <s xml:id="echoid-s15647" xml:space="preserve">7. </s> <s xml:id="echoid-s15648" xml:space="preserve">num. </s> <s xml:id="echoid-s15649" xml:space="preserve">3. </s> <s xml:id="echoid-s15650" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15651" xml:space="preserve">1. </s> <s xml:id="echoid-s15652" xml:space="preserve">idem autem <lb/>hic aliter demonſtrabimus ex Pappo, hoc modo. </s> <s xml:id="echoid-s15653" xml:space="preserve">Sint duo circuli A B C D, <lb/>EFGH, quorum diametri AC, EG. </s> <s xml:id="echoid-s15654" xml:space="preserve">Dico eſſe cir-<lb/> <anchor type="figure" xlink:label="fig-364-01a" xlink:href="fig-364-01"/> cumferentiam ad circumferentiam, vt eſt diameter <lb/>ad diametrum. </s> <s xml:id="echoid-s15655" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Quoniam enim eſt circulus ad cir- <anchor type="note" xlink:label="note-364-03a" xlink:href="note-364-03"/> culum, vt quadratum diametri ad quadratum dia-<lb/>metri. </s> <s xml:id="echoid-s15656" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Vt autem circulus A B C D, ad circulum <anchor type="note" xlink:label="note-364-04a" xlink:href="note-364-04"/> E F G H, ita eſt quadruplum circuli ad quadru-<lb/>plam circuli. </s> <s xml:id="echoid-s15657" xml:space="preserve">Igitur erit quoque quadruplum <lb/>circuli A B C D, ad quadruplum circuli E F-<lb/>G H, vt quadratum diametri A C, ad quad atum <lb/>diametri EG. </s> <s xml:id="echoid-s15658" xml:space="preserve">Sed rectangulum ſub diametro AC, & </s> <s xml:id="echoid-s15659" xml:space="preserve">recta, quæ circumferentiæ <lb/>ABCD, ſit æqualis, comprehenſum, quadruplũ eſt circuli ABCD; </s> <s xml:id="echoid-s15660" xml:space="preserve">& </s> <s xml:id="echoid-s15661" xml:space="preserve">rectangu-<lb/>lum ſub diametro E G, & </s> <s xml:id="echoid-s15662" xml:space="preserve">circumferentia EFGH, quadruplum circuli E F G H, <pb o="335" file="365" n="365" rhead="LIBER OCTAVVS."/> ex coroll. </s> <s xml:id="echoid-s15663" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s15664" xml:space="preserve">2. </s> <s xml:id="echoid-s15665" xml:space="preserve">cap. </s> <s xml:id="echoid-s15666" xml:space="preserve">5. </s> <s xml:id="echoid-s15667" xml:space="preserve">Num. </s> <s xml:id="echoid-s15668" xml:space="preserve">1. </s> <s xml:id="echoid-s15669" xml:space="preserve">lib. </s> <s xml:id="echoid-s15670" xml:space="preserve">5. </s> <s xml:id="echoid-s15671" xml:space="preserve">Igitur erit rectangulum ſub diametro <lb/>AC, & </s> <s xml:id="echoid-s15672" xml:space="preserve">circumferentia ABCD, contentum, ad rectangulum ſub diametro E G, <lb/>& </s> <s xml:id="echoid-s15673" xml:space="preserve">circumferentia EFGH, comprehenſum, vt quadratum ex A C, ad quadra-<lb/>tum ex E G; </s> <s xml:id="echoid-s15674" xml:space="preserve">Et permutan do erit rectangulum ſub diametro A C, & </s> <s xml:id="echoid-s15675" xml:space="preserve">circumfe-<lb/>rentia ABCD, ad quadratũ ex AC, vt rectangulum ſub diametro EG, & </s> <s xml:id="echoid-s15676" xml:space="preserve">circum-<lb/>ferentia EF GH, ad quadratũ ex E G. </s> <s xml:id="echoid-s15677" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Eſt autem rectangulum ſub A C, & </s> <s xml:id="echoid-s15678" xml:space="preserve">re- <anchor type="note" xlink:label="note-365-01a" xlink:href="note-365-01"/> cta, quæ circumferentiæ ABCD, ſit æqualis, ad quadratum ex AC, vt recta cir-<lb/>cumferentiæ æqualis ad A C: </s> <s xml:id="echoid-s15679" xml:space="preserve">propterea quod rectangulum, & </s> <s xml:id="echoid-s15680" xml:space="preserve">quadratum <lb/>eandem habent altitu dinem A C. </s> <s xml:id="echoid-s15681" xml:space="preserve">Eodemque modo eſt rectangulum ſub E G, <lb/>& </s> <s xml:id="echoid-s15682" xml:space="preserve">recta, quæ circum ferentiæ EFGH, ſit æqualis, ad quadratum ex EG, vt recta <lb/>circumferentiæ æqualis ad EG. </s> <s xml:id="echoid-s15683" xml:space="preserve">Igitur erit, vt circumferentia A B C D, ad dia-<lb/>metrum A C, ita circumferentia EFGH, ad diametrum EG: </s> <s xml:id="echoid-s15684" xml:space="preserve">Et permutando cir-<lb/>cumferentia ad circumferentiam, vt diameter ad diametrum, quod demon-<lb/>ſtrandum erat.</s> <s xml:id="echoid-s15685" xml:space="preserve"/> </p> <div xml:id="echoid-div958" type="float" level="2" n="1"> <figure xlink:label="fig-364-01" xlink:href="fig-364-01a"> <image file="364-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/364-01"/> </figure> <note symbol="c" position="left" xlink:label="note-364-03" xlink:href="note-364-03a" xml:space="preserve">2. duodec.</note> <note symbol="d" position="left" xlink:label="note-364-04" xlink:href="note-364-04a" xml:space="preserve">15. quinti.</note> <note symbol="a" position="right" xlink:label="note-365-01" xlink:href="note-365-01a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div960" type="section" level="1" n="340"> <head xml:id="echoid-head367" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15686" xml:space="preserve"><emph style="sc">Svnt</emph> qui putent, fruſtrà à Pappo hoc theorema demonſtrari, cum videatur <lb/>eſſe per ſe notum, ita eſſe circumferentiam cuiuſuis circuli ad ſuam diametrum, <lb/>vt eſt circumferentia alterius circuli ad ſuam diametrum. </s> <s xml:id="echoid-s15687" xml:space="preserve">ac proinde permutan-<lb/>do eſſe circumferentiam ad circumferentiam, vt eſt diameter ad diametrum. <lb/></s> <s xml:id="echoid-s15688" xml:space="preserve">Qua in re mirum in modum decipiuntur. </s> <s xml:id="echoid-s15689" xml:space="preserve">Cum enim à Ptolomæo (quod & </s> <s xml:id="echoid-s15690" xml:space="preserve">à <lb/>nobis propoſ. </s> <s xml:id="echoid-s15691" xml:space="preserve">10. </s> <s xml:id="echoid-s15692" xml:space="preserve">Sinuum factum eſt) demonſtretur, maiorem eſſe proportio-<lb/>nem maioris arcus ad minorem eiuſdem circuli, quam chordæ ad chordam, <lb/>(quod etiam de arcubus, & </s> <s xml:id="echoid-s15693" xml:space="preserve">chordis in circulis inæqualibus verum eſt, niſi ar-<lb/>cus illi ſimiles ſint, vt in ſequenti Theoremate oſtendemus) quis ſine demon-<lb/>ſtratione concederet, eandem eſſe proportionem circumferentiæ ad circumfe-<lb/>rentiã, quæ eſt diametri ad diametrum? </s> <s xml:id="echoid-s15694" xml:space="preserve">Quod ſi demonſtratum eſſet, ita eſſe ar-<lb/>cum cuiuſuis circuli ad ſimilem arcum alterius circuli, vt eſt corda ad chordam, <lb/>tum demum conſtaret, ita eſſe circumferentiam ad circumferentiam, <anchor type="note" xlink:href="" symbol="b"/> ac proin- <anchor type="note" xlink:label="note-365-02a" xlink:href="note-365-02"/> de & </s> <s xml:id="echoid-s15695" xml:space="preserve">ſemir cumferentiam ad ſemicircumferentiam, vt eſt diameter ad diame-<lb/>trum: </s> <s xml:id="echoid-s15696" xml:space="preserve">propterea quod arcus ſemicirculorum ſimiles ſunt, quorum chordæ ſunt <lb/>diametri. </s> <s xml:id="echoid-s15697" xml:space="preserve">Verum hoc demonſtrari non poteſt, niſi prius demonſtretur, ita eſ-<lb/>ſe circumferentiam ad circumferentiam, vt eſt diameter ad diametrum, vt in <lb/>Theoremate ſequenti conſtabit. </s> <s xml:id="echoid-s15698" xml:space="preserve">Meritò ergo, & </s> <s xml:id="echoid-s15699" xml:space="preserve">non ſine cauſa, theorema <lb/>præcedens à Pappo fuit demonſtratum.</s> <s xml:id="echoid-s15700" xml:space="preserve"/> </p> <div xml:id="echoid-div960" type="float" level="2" n="1"> <note symbol="b" position="right" xlink:label="note-365-02" xlink:href="note-365-02a" xml:space="preserve">15. quinti.</note> </div> </div> <div xml:id="echoid-div962" type="section" level="1" n="341"> <head xml:id="echoid-head368" xml:space="preserve">THEOR. 3. PROPOS. 3.</head> <p> <s xml:id="echoid-s15701" xml:space="preserve">ARCVS cuiuſuis circuli ad arcum ſimilem alterius circuli eandem <lb/>habet proportionem, quam chorda ad chordam. </s> <s xml:id="echoid-s15702" xml:space="preserve">Et contra arcus <lb/>candem habentes proportionem, quam chordæ, ſimiles ſunt.</s> <s xml:id="echoid-s15703" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15704" xml:space="preserve"><emph style="sc">In</emph> figura præ cedentis propoſ. </s> <s xml:id="echoid-s15705" xml:space="preserve">ducantur ad diametros perpen diculares P B, <lb/>QF, ex centris P, Q, diuidentes ſemicirculos in binos quadrantes: </s> <s xml:id="echoid-s15706" xml:space="preserve">ſintque ar-<lb/>cus BI, BK, æquales, quibus ſimiles capiantur FL, FM; </s> <s xml:id="echoid-s15707" xml:space="preserve">adeò vt toti arcus IK, <pb o="336" file="366" n="366" rhead="GEOMETR. PRACT."/> LM, ſimiles ſint, quorum chordæ IK, LM, bifariam ſectæ ſint à ſemidiametris in <lb/>N, O. </s> <s xml:id="echoid-s15708" xml:space="preserve">Dico eandem eſſe proportionem arcus IBK, ad arcum L F M, quæ eſt <lb/>chordæ IK, ad chordam LM, &</s> <s xml:id="echoid-s15709" xml:space="preserve">c. </s> <s xml:id="echoid-s15710" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim eſt circumferentia ABD, ad <anchor type="note" xlink:label="note-366-01a" xlink:href="note-366-01"/> circumferentiam EFH, <anchor type="note" xlink:href="" symbol="b"/> hoc eſt, quadrans AB, ad quadrantem EF, vt diame- <anchor type="note" xlink:label="note-366-02a" xlink:href="note-366-02"/> ter AC, ad diametrum EG; </s> <s xml:id="echoid-s15711" xml:space="preserve">hoc eſt, vt ſemidiame-<lb/> <anchor type="figure" xlink:label="fig-366-01a" xlink:href="fig-366-01"/> ter PB, ad ſemidiametrum QF. </s> <s xml:id="echoid-s15712" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Eſt autem, vt qua- <anchor type="note" xlink:label="note-366-03a" xlink:href="note-366-03"/> drans AB, ad quadrantem EF, ita arcus IB, ad arcum <lb/>LF, cum ſimiles ponantur. </s> <s xml:id="echoid-s15713" xml:space="preserve">Igitur erit quoque ar-<lb/>cus IB, ad arcum L F, vt ſemidiameter P B, ad ſemi-<lb/>diametrum QF. </s> <s xml:id="echoid-s15714" xml:space="preserve">Cum ergo ex Lemmate propoſ. </s> <s xml:id="echoid-s15715" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s15716" xml:space="preserve">lib. </s> <s xml:id="echoid-s15717" xml:space="preserve">1. </s> <s xml:id="echoid-s15718" xml:space="preserve">noſtræ Gnomonicæ, vel ex Lemmate 5. </s> <s xml:id="echoid-s15719" xml:space="preserve">libr. </s> <s xml:id="echoid-s15720" xml:space="preserve">1. </s> <s xml:id="echoid-s15721" xml:space="preserve"><lb/>noſtri Aſtrolabilij, ita ſe habeat P B, ſinus totus ad <lb/>QF, ſinum totum, quemadmo dum ſinus IN, ad ſinum L O. </s> <s xml:id="echoid-s15722" xml:space="preserve">erit quo que arcus <lb/>IB, ad arcum LF, vt ſinus IN, ad ſinum LO, <anchor type="note" xlink:href="" symbol="d"/> id eſt, duplus arcus IK, ad duplum <anchor type="note" xlink:label="note-366-04a" xlink:href="note-366-04"/> arcum LM, vt chorda IK, ipſius IN, dupla, ad chordam LM, ipſius LO, duplam. <lb/></s> <s xml:id="echoid-s15723" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s15724" xml:space="preserve"/> </p> <div xml:id="echoid-div962" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-366-01" xlink:href="note-366-01a" xml:space="preserve">2. hui{us}.</note> <note symbol="b" position="left" xlink:label="note-366-02" xlink:href="note-366-02a" xml:space="preserve">15. quinti.</note> <figure xlink:label="fig-366-01" xlink:href="fig-366-01a"> <image file="366-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/366-01"/> </figure> <note symbol="c" position="left" xlink:label="note-366-03" xlink:href="note-366-03a" xml:space="preserve">15. quinti.</note> <note symbol="d" position="left" xlink:label="note-366-04" xlink:href="note-366-04a" xml:space="preserve">15. quinti.</note> </div> <p> <s xml:id="echoid-s15725" xml:space="preserve"><emph style="sc">Vervm</emph> ſit iamarcus IK, ad arcum LM, vt chorda IK, ad chordam LM. </s> <s xml:id="echoid-s15726" xml:space="preserve">Di-<lb/>co arcus ſimiles eſſe. </s> <s xml:id="echoid-s15727" xml:space="preserve">Facta enim eadem conſtructione, e erit quoque arcus IB, ad <lb/> <anchor type="note" xlink:label="note-366-05a" xlink:href="note-366-05"/> arcum LF, ſemiſsis ad ſemiſſem, vt IN, ad LO, ſemiſsis ad ſemiſſem. </s> <s xml:id="echoid-s15728" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Vt autem <anchor type="note" xlink:label="note-366-06a" xlink:href="note-366-06"/> IB, ad LF, ita eſt quadrans AB, ad quadrantem EF; </s> <s xml:id="echoid-s15729" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Et vt quadrans ad quadran- tem, ita eſt ſemidiameter PB, ad ſemidiametrũ. </s> <s xml:id="echoid-s15730" xml:space="preserve">QF. </s> <s xml:id="echoid-s15731" xml:space="preserve">Igitur eri@ quo que ſinus IN, <lb/> <anchor type="note" xlink:label="note-366-07a" xlink:href="note-366-07"/> ad ſinum LO, vt ſinus totus PB, ad ſinum totum QF: </s> <s xml:id="echoid-s15732" xml:space="preserve">Atque idcirco ex Lem-<lb/>mate propoſ. </s> <s xml:id="echoid-s15733" xml:space="preserve">1. </s> <s xml:id="echoid-s15734" xml:space="preserve">lib. </s> <s xml:id="echoid-s15735" xml:space="preserve">1. </s> <s xml:id="echoid-s15736" xml:space="preserve">noſtræ Gnomonicæ, vel ex Lemmate 5. </s> <s xml:id="echoid-s15737" xml:space="preserve">lib. </s> <s xml:id="echoid-s15738" xml:space="preserve">1. </s> <s xml:id="echoid-s15739" xml:space="preserve">noſtri Aſtro-<lb/>labij, arcus IB, LF, ſimiles erunt; </s> <s xml:id="echoid-s15740" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s15741" xml:space="preserve">eorum dupli IBK, LFM, ſimiles <lb/>erunt. </s> <s xml:id="echoid-s15742" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s15743" xml:space="preserve"/> </p> <div xml:id="echoid-div963" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-366-05" xlink:href="note-366-05a" xml:space="preserve">15. quinti.</note> <note symbol="f" position="left" xlink:label="note-366-06" xlink:href="note-366-06a" xml:space="preserve">ſchol. 33. <lb/>ſexti.</note> <note symbol="g" position="left" xlink:label="note-366-07" xlink:href="note-366-07a" xml:space="preserve">2. hui{us}.</note> </div> </div> <div xml:id="echoid-div965" type="section" level="1" n="342"> <head xml:id="echoid-head369" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s15744" xml:space="preserve"><emph style="sc">Seqvitvr</emph> hinc, ſi arcus IBK, LFM, non ſint ſimiles, eos non habere ean-<lb/>dem cum chordis proportionem, Sinamque eandem haberent, ipſi ſimiles eſ-<lb/>ſent, vt in ſecunda parte huius propoſ. </s> <s xml:id="echoid-s15745" xml:space="preserve">fuit oſtenſum, quod eſt abſurdum, cum <lb/>ponantur non ſimiles.</s> <s xml:id="echoid-s15746" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div966" type="section" level="1" n="343"> <head xml:id="echoid-head370" xml:space="preserve">PROBL. 1. PROPOS. 4.</head> <p> <s xml:id="echoid-s15747" xml:space="preserve">DATO quadrilatero æquale parallelogrammum in dato angulo faci-<lb/>lius, quam per propoſ. </s> <s xml:id="echoid-s15748" xml:space="preserve">45. </s> <s xml:id="echoid-s15749" xml:space="preserve">lib. </s> <s xml:id="echoid-s15750" xml:space="preserve">1. </s> <s xml:id="echoid-s15751" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s15752" xml:space="preserve">conſtituere.</s> <s xml:id="echoid-s15753" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15754" xml:space="preserve"><emph style="sc">Sit</emph> quadrilaterum quodcunque ABCD, & </s> <s xml:id="echoid-s15755" xml:space="preserve">datus angulus K. </s> <s xml:id="echoid-s15756" xml:space="preserve">Ducta dia-<lb/>metro BD, eaque diuiſa bifariam in E, ducatur per E, <lb/> <anchor type="figure" xlink:label="fig-366-02a" xlink:href="fig-366-02"/> recta FG, faciens in E, angulum FED, dato angulo K, <lb/>æqualem. </s> <s xml:id="echoid-s15757" xml:space="preserve">Deinde ducta per D, ipſi FG, parallela HI, <lb/>& </s> <s xml:id="echoid-s15758" xml:space="preserve">per A, C, duabus AH, CI, ipſi BD, parallelis ſecanti-<lb/>bus FG, HI, in F, H, G, I, conſtitutum erit parallelo-<lb/> <anchor type="note" xlink:label="note-366-08a" xlink:href="note-366-08"/> grammum FI, in dato angulo G, <anchor type="note" xlink:href="" symbol="h"/> qui æqualis eſt an- gulo, FED, internus externo, hoc eſt, angulo K. </s> <s xml:id="echoid-s15759" xml:space="preserve">Dico <pb o="339" file="367" n="367" rhead="LIBER OCTAVVS."/> idem parallelogrammum quadrilatero dato ABCD, æquale eſſe. </s> <s xml:id="echoid-s15760" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quia enim <anchor type="note" xlink:label="note-367-01a" xlink:href="note-367-01"/> parallelogrammum FD, triangulo ABD, & </s> <s xml:id="echoid-s15761" xml:space="preserve">parallelogrammum EI, triangulo <lb/>CBD, æquale eſt; </s> <s xml:id="echoid-s15762" xml:space="preserve">erit totum parallelogrammum FI, toti quadrilatero ABCD, <lb/>æquale.</s> <s xml:id="echoid-s15763" xml:space="preserve"><unsure/> quod eſt propoſitum.</s> <s xml:id="echoid-s15764" xml:space="preserve"/> </p> <div xml:id="echoid-div966" type="float" level="2" n="1"> <figure xlink:label="fig-366-02" xlink:href="fig-366-02a"> <image file="366-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/366-02"/> </figure> <note symbol="h" position="left" xlink:label="note-366-08" xlink:href="note-366-08a" xml:space="preserve">29. primi.</note> <note symbol="a" position="right" xlink:label="note-367-01" xlink:href="note-367-01a" xml:space="preserve">ſchol. 41. <lb/>primi.</note> </div> </div> <div xml:id="echoid-div968" type="section" level="1" n="344"> <head xml:id="echoid-head371" xml:space="preserve">PROBL. 2. PROPOS. 5.</head> <p> <s xml:id="echoid-s15765" xml:space="preserve">DATO rectangulo ſupra datam rectam æquale rectangulum facilius, <lb/>quam per propoſ. </s> <s xml:id="echoid-s15766" xml:space="preserve">45. </s> <s xml:id="echoid-s15767" xml:space="preserve">lib. </s> <s xml:id="echoid-s15768" xml:space="preserve">1. </s> <s xml:id="echoid-s15769" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s15770" xml:space="preserve">conſtituere.</s> <s xml:id="echoid-s15771" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15772" xml:space="preserve"><emph style="sc">Sit</emph> rectangulum A B C D, cui ſupra datam rectam conſtituendum eſt re-<lb/>ctangulum æquale. </s> <s xml:id="echoid-s15773" xml:space="preserve">Producto quolibet latere, ni-<lb/> <anchor type="figure" xlink:label="fig-367-01a" xlink:href="fig-367-01"/> mirum A B, capiatur B E, æqualis datærectæ, ſiue ea <lb/>ſit maior latere AB, ſiue minor: </s> <s xml:id="echoid-s15774" xml:space="preserve">atque ex E, per C, <lb/>recta ducatur ſecans AD, productam in F, complea-<lb/>turque rectangulum AH, & </s> <s xml:id="echoid-s15775" xml:space="preserve">rectæ BC, DC. </s> <s xml:id="echoid-s15776" xml:space="preserve">produ-<lb/>cantur vſque ad G, I. </s> <s xml:id="echoid-s15777" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Eritigitur complementum <anchor type="note" xlink:label="note-367-02a" xlink:href="note-367-02"/> CH, complemento CA, ęquale. </s> <s xml:id="echoid-s15778" xml:space="preserve">Cumigitur latus CI, <lb/>ſit lateri BE, id eſt, datærectæ æquale: </s> <s xml:id="echoid-s15779" xml:space="preserve">factum erit, quod proponitur.</s> <s xml:id="echoid-s15780" xml:space="preserve"/> </p> <div xml:id="echoid-div968" type="float" level="2" n="1"> <figure xlink:label="fig-367-01" xlink:href="fig-367-01a"> <image file="367-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/367-01"/> </figure> <note symbol="b" position="right" xlink:label="note-367-02" xlink:href="note-367-02a" xml:space="preserve">43. primi.</note> </div> </div> <div xml:id="echoid-div970" type="section" level="1" n="345"> <head xml:id="echoid-head372" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s15781" xml:space="preserve"><emph style="sc">Datæ</emph> rectæ BE, & </s> <s xml:id="echoid-s15782" xml:space="preserve">duobus lateribus AB, BC, dati rectanguli, <anchor type="note" xlink:href="" symbol="c"/> inueniatur <anchor type="note" xlink:label="note-367-03a" xlink:href="note-367-03"/> quarta proportionalis C G. </s> <s xml:id="echoid-s15783" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Nam rectangulum ſub BE, prima, & </s> <s xml:id="echoid-s15784" xml:space="preserve">CG, quarta <anchor type="note" xlink:label="note-367-04a" xlink:href="note-367-04"/> comprehenſum, rectangulum videlicet C H, æquale erit dato rectangulo AC, <lb/>ſub ſecunda AB, & </s> <s xml:id="echoid-s15785" xml:space="preserve">tertia BC, comprehenſo. </s> <s xml:id="echoid-s15786" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s15787" xml:space="preserve"/> </p> <div xml:id="echoid-div970" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-367-03" xlink:href="note-367-03a" xml:space="preserve">17. ſexti.</note> <note symbol="d" position="right" xlink:label="note-367-04" xlink:href="note-367-04a" xml:space="preserve">16. ſexti.</note> </div> </div> <div xml:id="echoid-div972" type="section" level="1" n="346"> <head xml:id="echoid-head373" xml:space="preserve">PROBL. 3. PROPOS. 6.</head> <p> <s xml:id="echoid-s15788" xml:space="preserve">DATO rectilineo æquale rectangulum facilius, quam per propoſ. </s> <s xml:id="echoid-s15789" xml:space="preserve">45. <lb/></s> <s xml:id="echoid-s15790" xml:space="preserve">lib. </s> <s xml:id="echoid-s15791" xml:space="preserve">1. </s> <s xml:id="echoid-s15792" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s15793" xml:space="preserve">conſtituere.</s> <s xml:id="echoid-s15794" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15795" xml:space="preserve"><emph style="sc">Hoc</emph> per duas præcedentes propoſ. </s> <s xml:id="echoid-s15796" xml:space="preserve">facilè expedietur. </s> <s xml:id="echoid-s15797" xml:space="preserve">Nam ſi figura recti-<lb/>linea in triangula reſoluatur, conſtituent quælibet duo commune latus haben-<lb/>tia trapezium, cuius latus commune eſt diameter. </s> <s xml:id="echoid-s15798" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Igitur ſi ſingulis trapeziis ſingula rectangula fiant æqualia, atque etiam vltimo triangulo, ſi fortè nume-<lb/>rus triangulorum eſt impar. </s> <s xml:id="echoid-s15799" xml:space="preserve">Deinde, ſi, vt in præcedenti propoſ. </s> <s xml:id="echoid-s15800" xml:space="preserve">dictum eſt, <lb/>vni lateri primi rectanguli, & </s> <s xml:id="echoid-s15801" xml:space="preserve">duobus lateribus ſecundi, <anchor type="note" xlink:href="" symbol="e"/> inueniatur quarta <anchor type="note" xlink:label="note-367-05a" xlink:href="note-367-05"/> proportionalis; </s> <s xml:id="echoid-s15802" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> erit rectangulum ſub aſſumpto latere in primo rectangulo, <anchor type="note" xlink:label="note-367-06a" xlink:href="note-367-06"/> & </s> <s xml:id="echoid-s15803" xml:space="preserve">quarta proportionali, ſecundo rectangulo æquale. </s> <s xml:id="echoid-s15804" xml:space="preserve">Quocirca ſi alterum la-<lb/> <anchor type="note" xlink:label="note-367-07a" xlink:href="note-367-07"/> tus prope aſſumptum in primo rectangulo producatur, & </s> <s xml:id="echoid-s15805" xml:space="preserve">ex producto abſcin-<lb/>datur recta æqualis quartæ proportionali, compleatur que totum rectangulum, <lb/>habebitur rectangulum ex duobus compoſitum æquale duobus primis tra-<lb/>peziis. </s> <s xml:id="echoid-s15806" xml:space="preserve">Et ſi eidem lateri, ac duobus tertij rectanguli reperiatur rurſum quar-<lb/>ta proportionalis, & </s> <s xml:id="echoid-s15807" xml:space="preserve">huic quartæ ſumatur in priori latere producto recta æqua- <pb o="340" file="368" n="368" rhead="GEOMETR. PRACT."/> lis, conficietur eodem pacto rectangulum ex tribus conflatum æquale tribus <lb/>trapeziis, &</s> <s xml:id="echoid-s15808" xml:space="preserve">c.</s> <s xml:id="echoid-s15809" xml:space="preserve"/> </p> <div xml:id="echoid-div972" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-367-05" xlink:href="note-367-05a" xml:space="preserve">4. hui{us}.</note> <note symbol="f" position="right" xlink:label="note-367-06" xlink:href="note-367-06a" xml:space="preserve">12. ſexti.</note> <note symbol="g" position="right" xlink:label="note-367-07" xlink:href="note-367-07a" xml:space="preserve">16. ſexti.</note> </div> <p> <s xml:id="echoid-s15810" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> <emph style="sc">Vltimo</emph> porrò triangulo, ſi quod fuerit, conſtituetur rectangulum æqua- <anchor type="note" xlink:label="note-368-01a" xlink:href="note-368-01"/> I<unsure/>e ſupra ſemiſſem baſis, in eadem altitudine cum triangulo.</s> <s xml:id="echoid-s15811" xml:space="preserve"/> </p> <div xml:id="echoid-div973" type="float" level="2" n="2"> <note symbol="a" position="left" xlink:label="note-368-01" xlink:href="note-368-01a" xml:space="preserve">ſchol. 41. <lb/>primi.</note> </div> </div> <div xml:id="echoid-div975" type="section" level="1" n="347"> <head xml:id="echoid-head374" xml:space="preserve">THEOR. 4. PROPOS. 7.</head> <p> <s xml:id="echoid-s15812" xml:space="preserve">SI ex duobus punctis ad vnum punctum cuiuſuis lineæ rectæ, quæ <lb/>communis ſectio ſit plani per duo puncta ducti cum alio quopiam <lb/>plano, duæ rectæ ducantur, facientes cum illa duos angulos æquales: <lb/></s> <s xml:id="echoid-s15813" xml:space="preserve">erunt duæ hæ rectæ breuiores quibuſcunque aliis duabus rectis, quæ <lb/>ex eiſdem duobus punctis ad aliud punctum eiuſdem lineæ rectæ <lb/>ducuntur.</s> <s xml:id="echoid-s15814" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15815" xml:space="preserve"><emph style="sc">Ex</emph> duobus punctis A, B, ad C, punctum in recta CD, ita vt planum per CD, <lb/>du ctum tranſeat reuolutum per A, B, ducantur duæ rectæ AC, BC, facientes an-<lb/>gulos A C F, B C D, æquales: </s> <s xml:id="echoid-s15816" xml:space="preserve">& </s> <s xml:id="echoid-s15817" xml:space="preserve">ex eiſdem punctis A, B, ducantur primum ad <lb/>aliud punctum D, ad dextram ipſius C, aliæ duæ rectæ AD, BD. </s> <s xml:id="echoid-s15818" xml:space="preserve">Dico AC, BC, <lb/>eſſe breuiores, quam AD, BD. </s> <s xml:id="echoid-s15819" xml:space="preserve">Producta enim AC, verſus C, fiat CE, ipſi CB, <lb/>æqualis, iungaturque DE. </s> <s xml:id="echoid-s15820" xml:space="preserve">Et quia angulus ACF, angulo BCD, ponitur æqua-<lb/>lis, <anchor type="note" xlink:href="" symbol="b"/> eſtque angulus ACF, angulo ECD, ad verticem æqualis, erit quoque an- <anchor type="note" xlink:label="note-368-02a" xlink:href="note-368-02"/> gulus BCD, angulo ECD, æqualis. </s> <s xml:id="echoid-s15821" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s15822" xml:space="preserve">duo latera BC, CD, duobus <lb/>lateribus EC, CD, æqualia ſint: </s> <s xml:id="echoid-s15823" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit baſis <anchor type="note" xlink:label="note-368-03a" xlink:href="note-368-03"/> <anchor type="figure" xlink:label="fig-368-01a" xlink:href="fig-368-01"/> D B, baſi D E, æqualis; </s> <s xml:id="echoid-s15824" xml:space="preserve">ac proinde A D, <lb/>D B, ſimul ipſis A D, D E. </s> <s xml:id="echoid-s15825" xml:space="preserve">ſimul æquales <lb/>erunt. </s> <s xml:id="echoid-s15826" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Sunt autem A D, D E, maiores <anchor type="note" xlink:label="note-368-04a" xlink:href="note-368-04"/> quam AE, hoc eſt, quam AC, CB; </s> <s xml:id="echoid-s15827" xml:space="preserve">quod <lb/>CB, CE, poſitæ ſint æquale@. </s> <s xml:id="echoid-s15828" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s15829" xml:space="preserve">AD, <lb/>BD, maiores erunt, quam AC, BC. </s> <s xml:id="echoid-s15830" xml:space="preserve">quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s15831" xml:space="preserve"/> </p> <div xml:id="echoid-div975" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-368-02" xlink:href="note-368-02a" xml:space="preserve">15. primi.</note> <note symbol="c" position="left" xlink:label="note-368-03" xlink:href="note-368-03a" xml:space="preserve">4. primi.</note> <figure xlink:label="fig-368-01" xlink:href="fig-368-01a"> <image file="368-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/368-01"/> </figure> <note symbol="d" position="left" xlink:label="note-368-04" xlink:href="note-368-04a" xml:space="preserve">@primi.</note> </div> <p> <s xml:id="echoid-s15832" xml:space="preserve"><emph style="sc">Dvcantvr</emph> deinde ex punctis A, B, <lb/>ad aliud punctum F, ad ſiniſtram ipſius C, <lb/>aliæ duæ rectæ AF, BF. </s> <s xml:id="echoid-s15833" xml:space="preserve">Dico rurſus A C, <lb/>BC, breuiores eſſe, quam AF, BF. </s> <s xml:id="echoid-s15834" xml:space="preserve">Producta enim rurſum AC, ſumptaque CE, <lb/>ipſi CB, æquali, iungatur EF. </s> <s xml:id="echoid-s15835" xml:space="preserve">Et quoniam anguli ACF, BCD, æquales ponun-<lb/>tur; </s> <s xml:id="echoid-s15836" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> eſtque ACF, angulo ECD, ad verticem æqualis; </s> <s xml:id="echoid-s15837" xml:space="preserve">erunt qu@que anguli <anchor type="note" xlink:label="note-368-05a" xlink:href="note-368-05"/> BCD, ECD, æquales: </s> <s xml:id="echoid-s15838" xml:space="preserve">ac proinde & </s> <s xml:id="echoid-s15839" xml:space="preserve">ex duobus rectis reliqui BCF, ECF, æqua-<lb/>les erunt. </s> <s xml:id="echoid-s15840" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s15841" xml:space="preserve">duo latera BC, CF, duobus lateribus EC, CF, æqualia <lb/>ſint; </s> <s xml:id="echoid-s15842" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> erit quo que baſis BF, baſi EF, æqualis: </s> <s xml:id="echoid-s15843" xml:space="preserve">Ac proinde AF, FE, ipſis AF, BF, <anchor type="note" xlink:label="note-368-06a" xlink:href="note-368-06"/> æquales erunt. </s> <s xml:id="echoid-s15844" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Sunt autem AF, FE, maiores, quam AE, hoc eſt, quam AC, BC, <anchor type="note" xlink:label="note-368-07a" xlink:href="note-368-07"/> quod BC, CE, poſitæ ſint æquales. </s> <s xml:id="echoid-s15845" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s15846" xml:space="preserve">AF, BF, maiores erunt, quam A C, <lb/>CB. </s> <s xml:id="echoid-s15847" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s15848" xml:space="preserve"/> </p> <div xml:id="echoid-div976" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-368-05" xlink:href="note-368-05a" xml:space="preserve">15. primi.</note> <note symbol="f" position="left" xlink:label="note-368-06" xlink:href="note-368-06a" xml:space="preserve">4. primi.</note> <note symbol="g" position="left" xlink:label="note-368-07" xlink:href="note-368-07a" xml:space="preserve">20. primi.</note> </div> </div> <div xml:id="echoid-div978" type="section" level="1" n="348"> <head xml:id="echoid-head375" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s15849" xml:space="preserve"><emph style="sc">Qvia</emph> ergo Natura non impedita agit per lineas breuiſsimas; </s> <s xml:id="echoid-s15850" xml:space="preserve">fit, vtradius <lb/>Solis, vel viſualis cadens ex A, in planum terſum D F, ita vt reflectatur ad pun- <pb o="341" file="369" n="369" rhead="LIBER OCTAVVS."/> ctum B, cadat neceſſariò in punctum C, vbiangulus ACF, (quem Perſpectiui <lb/>angulum incidentiæ dicunt.) </s> <s xml:id="echoid-s15851" xml:space="preserve">æqualis efficitur angulo BCD, quem reflexionis <lb/>appellant. </s> <s xml:id="echoid-s15852" xml:space="preserve">Nam ſi radius caderet in D, vel F, reflecteretur que ad B, non ageret <lb/>Natura per lineas breuiſsimas; </s> <s xml:id="echoid-s15853" xml:space="preserve">cum tam AD, BD, quam AF, BF, longiores ſint, <lb/>quam AC, BC vt demonſtrauimus. </s> <s xml:id="echoid-s15854" xml:space="preserve">quod eſt abſurdum. </s> <s xml:id="echoid-s15855" xml:space="preserve">Atque ita demonſtra-<lb/>tum eſt, quod Perſpectiui aſſumunt, angulum ſcilicet incidentiæ æqualem eſſe <lb/>angulo reflexionis.</s> <s xml:id="echoid-s15856" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div979" type="section" level="1" n="349"> <head xml:id="echoid-head376" xml:space="preserve">PROBL. 4. PROPOS. 8.</head> <p> <s xml:id="echoid-s15857" xml:space="preserve">SI quis numerum mente conceperit, quot ei vnitates poſttres operatio-<lb/>nes imperatas reliquæ ſint, coniicere.</s> <s xml:id="echoid-s15858" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15859" xml:space="preserve"><emph style="sc">Ivbe</emph> conceptum numerum per quemcunque numerum, vt per 2. </s> <s xml:id="echoid-s15860" xml:space="preserve">vel 3. </s> <s xml:id="echoid-s15861" xml:space="preserve">vel <lb/>4. </s> <s xml:id="echoid-s15862" xml:space="preserve">vel 10. </s> <s xml:id="echoid-s15863" xml:space="preserve">&</s> <s xml:id="echoid-s15864" xml:space="preserve">c. </s> <s xml:id="echoid-s15865" xml:space="preserve">multiplicari, & </s> <s xml:id="echoid-s15866" xml:space="preserve">producto adde tu quemlibet numerum à numero <lb/>multiplicante numeratum. </s> <s xml:id="echoid-s15867" xml:space="preserve">Deinde iube ex parte aliquota ſummę totius à mul-<lb/>tiplicante numero denominata auferri ſimilem partem aliquotam numeri pro-<lb/>ducti ex multiplicante in numerum conceptum, hoc eſt, ipſum numerum <lb/>conceptum. </s> <s xml:id="echoid-s15868" xml:space="preserve">Ita enim reliquus numerus erit ſimilis pars aliquota numeri, quem <lb/>adiunxiſti. </s> <s xml:id="echoid-s15869" xml:space="preserve">Cum ergo numerus adiunctus tibi notus ſit, habeatque partem <lb/>aliquotam à numero multiplicante denominatam, ac proinde tibi cognitam: <lb/></s> <s xml:id="echoid-s15870" xml:space="preserve">dices reliquas vnitates illam partem aliquotam conficere. </s> <s xml:id="echoid-s15871" xml:space="preserve">Exempli gratia. </s> <s xml:id="echoid-s15872" xml:space="preserve"><lb/>Concipiataliquis numerum 4. </s> <s xml:id="echoid-s15873" xml:space="preserve">iube multiplicari per 6. </s> <s xml:id="echoid-s15874" xml:space="preserve">fiunt 24. </s> <s xml:id="echoid-s15875" xml:space="preserve">iube addi nu-<lb/>merum 30. </s> <s xml:id="echoid-s15876" xml:space="preserve">à multiplicante 6. </s> <s xml:id="echoid-s15877" xml:space="preserve">numeratum, fiunt 54. </s> <s xml:id="echoid-s15878" xml:space="preserve">Ex ſexta parte huius ſummæ <lb/>denominata à multiplicante numero 6. </s> <s xml:id="echoid-s15879" xml:space="preserve">id eſt, ex 9. </s> <s xml:id="echoid-s15880" xml:space="preserve">fac detrahi partem quoque <lb/>ſextam prioris numeri producti 24. </s> <s xml:id="echoid-s15881" xml:space="preserve">nimirum 4. </s> <s xml:id="echoid-s15882" xml:space="preserve">videlicetipſum numerum con-<lb/>ceptum. </s> <s xml:id="echoid-s15883" xml:space="preserve">Ita namque remanet ſexta pars numeri adiuncti 30. </s> <s xml:id="echoid-s15884" xml:space="preserve">nimirum 5. </s> <s xml:id="echoid-s15885" xml:space="preserve">quod <lb/>in hunc modum demonſtratur.</s> <s xml:id="echoid-s15886" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15887" xml:space="preserve"><emph style="sc">Sit</emph> conceptus numerus A, quo ducto verbi gratia in 6. </s> <s xml:id="echoid-s15888" xml:space="preserve">gignatur B, & </s> <s xml:id="echoid-s15889" xml:space="preserve">ad-<lb/>dito F, fiat ſumma C. </s> <s xml:id="echoid-s15890" xml:space="preserve">Ex E, parte aliquota ſum-<lb/>mæ C, denominata à 6. </s> <s xml:id="echoid-s15891" xml:space="preserve">detrahatur D, pars ali-<lb/> <anchor type="note" xlink:label="note-369-01a" xlink:href="note-369-01"/> quota prioris producti B, denominata quo-<lb/>que à 6. </s> <s xml:id="echoid-s15892" xml:space="preserve">hoc eſt, ipſemet numerus conceptus <lb/>A, reliquuſque fiat numerus G, quem dico par-<lb/>tem eſſe aliquotam numeri a diuncti F, à nume-<lb/>ro quoque 6. </s> <s xml:id="echoid-s15893" xml:space="preserve">denominatam. </s> <s xml:id="echoid-s15894" xml:space="preserve">Quoniam enim <lb/>ita eſt multiplex totus C, totius E, vt ablatus <lb/>B, ex C, ipſius D, ex E, ablati, quippe cum po-<lb/>natur E, talis pars ipſius C, qualis D, ipſius B, <lb/>denominata videlicet à 6. </s> <s xml:id="echoid-s15895" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur erit quoquereliquus F, (detracto nimirum <anchor type="note" xlink:label="note-369-02a" xlink:href="note-369-02"/> producto B, ex ſumma C,) ita multiplex reliqui G, (dempto ſcilicet D, ex E,) <lb/>vttotus C, totius E. </s> <s xml:id="echoid-s15896" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s15897" xml:space="preserve"/> </p> <div xml:id="echoid-div979" type="float" level="2" n="1"> <note position="right" xlink:label="note-369-01" xlink:href="note-369-01a" xml:space="preserve"> <lb/>A # B # C # F <lb/>4 # 24 # 54 # 30 <lb/># D # E # G <lb/># 4. # 9. # 5. <lb/></note> <note symbol="a" position="right" xlink:label="note-369-02" xlink:href="note-369-02a" xml:space="preserve">ſchol. 7. <lb/>ſeptimi.</note> </div> <p> <s xml:id="echoid-s15898" xml:space="preserve"><emph style="sc">Est</emph> autem iucundum, hoc idem coniici poſſe inter plures. </s> <s xml:id="echoid-s15899" xml:space="preserve">Nam ſi plures <lb/>concipiant mente numeros, ſinguli videlicet ſingulos, nullo eorum conſcio, <lb/>quem quiſq; </s> <s xml:id="echoid-s15900" xml:space="preserve">numerum conceperit; </s> <s xml:id="echoid-s15901" xml:space="preserve">& </s> <s xml:id="echoid-s15902" xml:space="preserve">iubeas quemlibet ſuum numerum multi-<lb/>pl@care per quemuis numerum à te electum; </s> <s xml:id="echoid-s15903" xml:space="preserve">deinde addere numerũ à tuo electo <pb o="342" file="370" n="370" rhead="GEOMETR. PRACT."/> numeratum, quicunqueille ſit; </s> <s xml:id="echoid-s15904" xml:space="preserve">ac poſtremo ex parte aliquota ſummæ, cuius de-<lb/>nominator eſt numerus à te electus, auferre ſimilem partem ex productis ſingu-<lb/>lorum, hoc eſt, ipſos conceptos numeros: </s> <s xml:id="echoid-s15905" xml:space="preserve">reliquus numerus cuiuſque erit ſi-<lb/>milis pars numeri adiecti.</s> <s xml:id="echoid-s15906" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15907" xml:space="preserve"><emph style="sc">Qvod</emph> ſi malueris diuerſos numeros, dic vt ſecundus ſuum reſiduum du-<lb/>plicet, & </s> <s xml:id="echoid-s15908" xml:space="preserve">tertius triplicet, &</s> <s xml:id="echoid-s15909" xml:space="preserve">c. </s> <s xml:id="echoid-s15910" xml:space="preserve">Ita enim coniicies, primi reſiduum eſſe illam par-<lb/>tem aliquotam numeriadiecti: </s> <s xml:id="echoid-s15911" xml:space="preserve">ſecundum verò habere duplum illius, & </s> <s xml:id="echoid-s15912" xml:space="preserve">ter-<lb/>tium triplum, &</s> <s xml:id="echoid-s15913" xml:space="preserve">c. </s> <s xml:id="echoid-s15914" xml:space="preserve">Vbivides eos reſiduum illud per quoſcunque numeros poſ-<lb/>ſe multiplicare, dummodo memor ſis in coniiciendis numeris, per quos nume-<lb/>ros factæ ſunt multiplicationes.</s> <s xml:id="echoid-s15915" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div981" type="section" level="1" n="350"> <head xml:id="echoid-head377" xml:space="preserve">PROBL. 5. PROPOS. 9.</head> <p> <s xml:id="echoid-s15916" xml:space="preserve">DATVM numerum quadratum in quotuis quadratos numeros par-<lb/>tiri.</s> <s xml:id="echoid-s15917" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s15918" xml:space="preserve"><emph style="sc">Qvamvis</emph> problema hoc videatur ferè impoſsibile: </s> <s xml:id="echoid-s15919" xml:space="preserve">(qui enim fieri po-<lb/>teſt, dicet aliquis, vt quilibet numerus quadratus diuidi poſsit in quotlibet nu-<lb/>meros, qui omnes ſint quadrati?) </s> <s xml:id="echoid-s15920" xml:space="preserve">ſolutio tamen eius non eſt difficilis. </s> <s xml:id="echoid-s15921" xml:space="preserve">Sit igitur <lb/>quadratus numerus datus 36. </s> <s xml:id="echoid-s15922" xml:space="preserve">diuidendus in 5. </s> <s xml:id="echoid-s15923" xml:space="preserve">numeros quadratos. </s> <s xml:id="echoid-s15924" xml:space="preserve">Per ea, quæ <lb/>ad propoſ. </s> <s xml:id="echoid-s15925" xml:space="preserve">47. </s> <s xml:id="echoid-s15926" xml:space="preserve">lib. </s> <s xml:id="echoid-s15927" xml:space="preserve">1. </s> <s xml:id="echoid-s15928" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s15929" xml:space="preserve">ſcripſimus, reperiantur tres numeri, quorummaio-<lb/>ris quadratus reliquorum quadratis ſit æqualis, nimirum 5. </s> <s xml:id="echoid-s15930" xml:space="preserve">4. </s> <s xml:id="echoid-s15931" xml:space="preserve">3. </s> <s xml:id="echoid-s15932" xml:space="preserve">Deinde dic: </s> <s xml:id="echoid-s15933" xml:space="preserve">ſi <lb/>5. </s> <s xml:id="echoid-s15934" xml:space="preserve">dant 4. </s> <s xml:id="echoid-s15935" xml:space="preserve">quid dabunt 6. </s> <s xml:id="echoid-s15936" xml:space="preserve">quadrata videlicet radix dati quadrati? </s> <s xml:id="echoid-s15937" xml:space="preserve">Item ſi 5. </s> <s xml:id="echoid-s15938" xml:space="preserve">dant <lb/>3. </s> <s xml:id="echoid-s15939" xml:space="preserve">quid dabunt 6? </s> <s xml:id="echoid-s15940" xml:space="preserve">Inuenieſq; </s> <s xml:id="echoid-s15941" xml:space="preserve">{24/5}. </s> <s xml:id="echoid-s15942" xml:space="preserve">& </s> <s xml:id="echoid-s15943" xml:space="preserve">{18/5}. </s> <s xml:id="echoid-s15944" xml:space="preserve">hoc eſt, 4 {4/5}. </s> <s xml:id="echoid-s15945" xml:space="preserve">& </s> <s xml:id="echoid-s15946" xml:space="preserve">3 {3/5}. </s> <s xml:id="echoid-s15947" xml:space="preserve">radices duorum qua-<lb/>dratorum quadrato 36. </s> <s xml:id="echoid-s15948" xml:space="preserve">dato æqualium. </s> <s xml:id="echoid-s15949" xml:space="preserve">Nam cum ita ſe habeat radix 6. </s> <s xml:id="echoid-s15950" xml:space="preserve">ad in-<lb/>uentos duos numeros, vt 5. </s> <s xml:id="echoid-s15951" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s15952" xml:space="preserve">& </s> <s xml:id="echoid-s15953" xml:space="preserve">3. </s> <s xml:id="echoid-s15954" xml:space="preserve">ex conſtructione: </s> <s xml:id="echoid-s15955" xml:space="preserve">fiet ex lateribus 6. </s> <s xml:id="echoid-s15956" xml:space="preserve">4 {4/5}. <lb/></s> <s xml:id="echoid-s15957" xml:space="preserve">3 {3/5}. </s> <s xml:id="echoid-s15958" xml:space="preserve">triangulum rectangulum, ſimile nimirum triangulo rectangulo ex lateribus <lb/>5. </s> <s xml:id="echoid-s15959" xml:space="preserve">4. </s> <s xml:id="echoid-s15960" xml:space="preserve">3. </s> <s xml:id="echoid-s15961" xml:space="preserve">conſtructo. </s> <s xml:id="echoid-s15962" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Igitur quadrati ex 4 {4/5}. </s> <s xml:id="echoid-s15963" xml:space="preserve">& </s> <s xml:id="echoid-s15964" xml:space="preserve">{3/5}. </s> <s xml:id="echoid-s15965" xml:space="preserve">æquales erunt quadrato radi- <anchor type="note" xlink:label="note-370-01a" xlink:href="note-370-01"/> cis 6. </s> <s xml:id="echoid-s15966" xml:space="preserve">dato. </s> <s xml:id="echoid-s15967" xml:space="preserve">Rurſus ſi fiat, vt 5. </s> <s xml:id="echoid-s15968" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s15969" xml:space="preserve">& </s> <s xml:id="echoid-s15970" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s15971" xml:space="preserve">ita 3 {3/5}. </s> <s xml:id="echoid-s15972" xml:space="preserve">ad aliud; </s> <s xml:id="echoid-s15973" xml:space="preserve">(ſumendo mino-<lb/>rem radicem inuentam, ne coincidamus cum aliqua præcedente radiceiam in-<lb/>uenta) inueniẽtur alij duo numeri, quorũ quadrati æquales ſint quadrato radi-<lb/>cis 3 {3/5}. </s> <s xml:id="echoid-s15974" xml:space="preserve">nimirum 2 {22/25}. </s> <s xml:id="echoid-s15975" xml:space="preserve">& </s> <s xml:id="echoid-s15976" xml:space="preserve">2 {4/25}. </s> <s xml:id="echoid-s15977" xml:space="preserve">Atqueita iam (relicta radice 3 {3/5}.) </s> <s xml:id="echoid-s15978" xml:space="preserve">inuentæ erunt <lb/>tres radices 4 {4/5}. </s> <s xml:id="echoid-s15979" xml:space="preserve">2 {22/25}. </s> <s xml:id="echoid-s15980" xml:space="preserve">2 {4/25}. </s> <s xml:id="echoid-s15981" xml:space="preserve">quarum quadrati æquales erunt quadrato 36. </s> <s xml:id="echoid-s15982" xml:space="preserve">pro-<lb/>poſito. </s> <s xml:id="echoid-s15983" xml:space="preserve">Eodemmodo, ſi fiat, vt 5. </s> <s xml:id="echoid-s15984" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s15985" xml:space="preserve">& </s> <s xml:id="echoid-s15986" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s15987" xml:space="preserve">ita 2 {4/25}. </s> <s xml:id="echoid-s15988" xml:space="preserve">ad aliud, reperientur <lb/>duæ aliæ radices 1 {91/125}. </s> <s xml:id="echoid-s15989" xml:space="preserve">& </s> <s xml:id="echoid-s15990" xml:space="preserve">1 {37/125}. </s> <s xml:id="echoid-s15991" xml:space="preserve">Quare (relicta radice 2 {4/25}. </s> <s xml:id="echoid-s15992" xml:space="preserve">cuius loco duas inue-<lb/>nimus) inuentæ iam erunt quatuor radices 4 {4/5}. </s> <s xml:id="echoid-s15993" xml:space="preserve">2 {22/25}. </s> <s xml:id="echoid-s15994" xml:space="preserve">1 {91/125}. </s> <s xml:id="echoid-s15995" xml:space="preserve">& </s> <s xml:id="echoid-s15996" xml:space="preserve">1 {37/125}. </s> <s xml:id="echoid-s15997" xml:space="preserve">quarumnu-<lb/>meri quadrati quadrato 36. </s> <s xml:id="echoid-s15998" xml:space="preserve">æquales erunt. </s> <s xml:id="echoid-s15999" xml:space="preserve">Denique ſi rurſus fiat vt 5. </s> <s xml:id="echoid-s16000" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s16001" xml:space="preserve">& </s> <s xml:id="echoid-s16002" xml:space="preserve"><lb/>ad 3. </s> <s xml:id="echoid-s16003" xml:space="preserve">ita 1 {37/125}. </s> <s xml:id="echoid-s16004" xml:space="preserve">minor radix inuenta ad aliud, reperientur duæ aliæ radices 1 {23/625}. <lb/></s> <s xml:id="echoid-s16005" xml:space="preserve">& </s> <s xml:id="echoid-s16006" xml:space="preserve">{486/625}. </s> <s xml:id="echoid-s16007" xml:space="preserve">Quocirca (relicta radice 1 {37/@25}. </s> <s xml:id="echoid-s16008" xml:space="preserve">pro qua duas proximas inuenimus) in-<lb/>uentæ erunt quinque radices 4 {4/5}. </s> <s xml:id="echoid-s16009" xml:space="preserve">2 {22/25}. </s> <s xml:id="echoid-s16010" xml:space="preserve">1 {91/125}. </s> <s xml:id="echoid-s16011" xml:space="preserve">1 {23/625}. </s> <s xml:id="echoid-s16012" xml:space="preserve">& </s> <s xml:id="echoid-s16013" xml:space="preserve">{486/625}. </s> <s xml:id="echoid-s16014" xml:space="preserve">quarum quadrati nu-<lb/>meri 23 {1/25}. </s> <s xml:id="echoid-s16015" xml:space="preserve">8 {184/625}. </s> <s xml:id="echoid-s16016" xml:space="preserve">2 {1@406/35625}. </s> <s xml:id="echoid-s16017" xml:space="preserve">1 {29279/190625}. </s> <s xml:id="echoid-s16018" xml:space="preserve">& </s> <s xml:id="echoid-s16019" xml:space="preserve">{236196/39@625}. </s> <s xml:id="echoid-s16020" xml:space="preserve">conficiunt datum quadratum 36. </s> <s xml:id="echoid-s16021" xml:space="preserve">At-<lb/>que in hunc modum plures quadrati inueniri poterunt æquales numero <lb/>36. </s> <s xml:id="echoid-s16022" xml:space="preserve">ſi nimirum fiat;</s> <s xml:id="echoid-s16023" xml:space="preserve"><unsure/> vt 5. </s> <s xml:id="echoid-s16024" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s16025" xml:space="preserve">& </s> <s xml:id="echoid-s16026" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s16027" xml:space="preserve">ita vltima radix inuenta {486/625}. </s> <s xml:id="echoid-s16028" xml:space="preserve"><lb/>quæ minima eſt, ad aliud, &</s> <s xml:id="echoid-s16029" xml:space="preserve">c.</s> <s xml:id="echoid-s16030" xml:space="preserve"/> </p> <div xml:id="echoid-div981" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-370-01" xlink:href="note-370-01a" xml:space="preserve">47. primi.</note> </div> <pb o="343" file="371" n="371" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div983" type="section" level="1" n="351"> <head xml:id="echoid-head378" xml:space="preserve">THEOR. 5. PROPOS. 10.</head> <p> <s xml:id="echoid-s16031" xml:space="preserve">PROPOSITIS duabus minutiis inæqualibus; </s> <s xml:id="echoid-s16032" xml:space="preserve">minutia, cuius nume-<lb/>rator ex illarum numeratoribus, denominator autem ex denomina-<lb/>toribus conflatur, maior quidem eſt minore, minor vero maiore.</s> <s xml:id="echoid-s16033" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16034" xml:space="preserve"><emph style="sc">Sint</emph> duæ minutiæ inæquales, maior {3/5}. </s> <s xml:id="echoid-s16035" xml:space="preserve">& </s> <s xml:id="echoid-s16036" xml:space="preserve">minor {4/7}. </s> <s xml:id="echoid-s16037" xml:space="preserve">Iungantur tam nume-<lb/>tatores, quam denominatores, vt fiat minutia {7/12}. </s> <s xml:id="echoid-s16038" xml:space="preserve">Dico hanc maiorem eſſe, <lb/>quam {4/7}. </s> <s xml:id="echoid-s16039" xml:space="preserve">& </s> <s xml:id="echoid-s16040" xml:space="preserve">minorem quam {3/5}. </s> <s xml:id="echoid-s16041" xml:space="preserve">Quoniam enim maior eſt minutia {3/5}. </s> <s xml:id="echoid-s16042" xml:space="preserve">quam {4/7}. <lb/></s> <s xml:id="echoid-s16043" xml:space="preserve">erit per propoſ. </s> <s xml:id="echoid-s16044" xml:space="preserve">8. </s> <s xml:id="echoid-s16045" xml:space="preserve">Minutiarum lib. </s> <s xml:id="echoid-s16046" xml:space="preserve">9. </s> <s xml:id="echoid-s16047" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s16048" xml:space="preserve">maior proportio 3. </s> <s xml:id="echoid-s16049" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s16050" xml:space="preserve">quam 4. </s> <s xml:id="echoid-s16051" xml:space="preserve">ad <lb/>7. </s> <s xml:id="echoid-s16052" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et permutando, maior 3. </s> <s xml:id="echoid-s16053" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s16054" xml:space="preserve">quam 5. </s> <s xml:id="echoid-s16055" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s16056" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Igitur &</s> <s xml:id="echoid-s16057" xml:space="preserve"> <anchor type="note" xlink:label="note-371-01a" xlink:href="note-371-01"/> <anchor type="note" xlink:label="note-371-02a" xlink:href="note-371-02"/> componendo, maior 3. </s> <s xml:id="echoid-s16058" xml:space="preserve">4. </s> <s xml:id="echoid-s16059" xml:space="preserve">ſimul, hoc eſt, 7. </s> <s xml:id="echoid-s16060" xml:space="preserve">ad 4. </s> <s xml:id="echoid-s16061" xml:space="preserve">quam 5. </s> <s xml:id="echoid-s16062" xml:space="preserve">7. <lb/></s> <s xml:id="echoid-s16063" xml:space="preserve"> <anchor type="note" xlink:label="note-371-03a" xlink:href="note-371-03"/> ſimul, id eſt, quam 12. </s> <s xml:id="echoid-s16064" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s16065" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Et permutando, maior 7. </s> <s xml:id="echoid-s16066" xml:space="preserve">ad 12.</s> <s xml:id="echoid-s16067" xml:space="preserve"> <anchor type="note" xlink:label="note-371-04a" xlink:href="note-371-04"/> quam 4. </s> <s xml:id="echoid-s16068" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s16069" xml:space="preserve">Ac proinde per propoſ. </s> <s xml:id="echoid-s16070" xml:space="preserve">8. </s> <s xml:id="echoid-s16071" xml:space="preserve">Minutiarum libri 9. </s> <s xml:id="echoid-s16072" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16073" xml:space="preserve">maior erit <lb/>minutia {7/12}. </s> <s xml:id="echoid-s16074" xml:space="preserve">quam {4/7}. </s> <s xml:id="echoid-s16075" xml:space="preserve">quod eſt primum.</s> <s xml:id="echoid-s16076" xml:space="preserve"/> </p> <div xml:id="echoid-div983" type="float" level="2" n="1"> <note position="right" xlink:label="note-371-01" xlink:href="note-371-01a" xml:space="preserve"> <lb/>{3/5}. # {4/7}. # {7/12}. <lb/></note> <note symbol="a" position="right" xlink:label="note-371-02" xlink:href="note-371-02a" xml:space="preserve">27 quinti.</note> <note symbol="b" position="right" xlink:label="note-371-03" xlink:href="note-371-03a" xml:space="preserve">28. quinti.</note> <note symbol="c" position="right" xlink:label="note-371-04" xlink:href="note-371-04a" xml:space="preserve">27. quinti.</note> </div> <p> <s xml:id="echoid-s16077" xml:space="preserve"><emph style="sc">Deinde</emph> quia minor eſt {4/7}. </s> <s xml:id="echoid-s16078" xml:space="preserve">quam {3/5}. </s> <s xml:id="echoid-s16079" xml:space="preserve">erit per propoſ. </s> <s xml:id="echoid-s16080" xml:space="preserve">8. </s> <s xml:id="echoid-s16081" xml:space="preserve">Minutiarum libri <lb/>9. </s> <s xml:id="echoid-s16082" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16083" xml:space="preserve">minor proportio 4. </s> <s xml:id="echoid-s16084" xml:space="preserve">ad 7. </s> <s xml:id="echoid-s16085" xml:space="preserve">quam 3. </s> <s xml:id="echoid-s16086" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s16087" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> & </s> <s xml:id="echoid-s16088" xml:space="preserve">permutando, minor 4. </s> <s xml:id="echoid-s16089" xml:space="preserve">ad <anchor type="note" xlink:label="note-371-05a" xlink:href="note-371-05"/> 3. </s> <s xml:id="echoid-s16090" xml:space="preserve">quam 7. </s> <s xml:id="echoid-s16091" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s16092" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igitur & </s> <s xml:id="echoid-s16093" xml:space="preserve">componendo minor 4. </s> <s xml:id="echoid-s16094" xml:space="preserve">3. </s> <s xml:id="echoid-s16095" xml:space="preserve">ſimul, id eſt, 7. </s> <s xml:id="echoid-s16096" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s16097" xml:space="preserve">quam <anchor type="note" xlink:label="note-371-06a" xlink:href="note-371-06"/> 7. </s> <s xml:id="echoid-s16098" xml:space="preserve">5. </s> <s xml:id="echoid-s16099" xml:space="preserve">ſimul, hoc eſt. </s> <s xml:id="echoid-s16100" xml:space="preserve">quam 12. </s> <s xml:id="echoid-s16101" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s16102" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Et permutando, minor 7. </s> <s xml:id="echoid-s16103" xml:space="preserve">ad 12. </s> <s xml:id="echoid-s16104" xml:space="preserve">quam 3. </s> <s xml:id="echoid-s16105" xml:space="preserve">ad <anchor type="note" xlink:label="note-371-07a" xlink:href="note-371-07"/> 5. </s> <s xml:id="echoid-s16106" xml:space="preserve">Ac proinde per propoſ. </s> <s xml:id="echoid-s16107" xml:space="preserve">8. </s> <s xml:id="echoid-s16108" xml:space="preserve">Minutiarum lib. </s> <s xml:id="echoid-s16109" xml:space="preserve">9. </s> <s xml:id="echoid-s16110" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s16111" xml:space="preserve">minor erit minutia {7/12}. </s> <s xml:id="echoid-s16112" xml:space="preserve">quã <lb/>{3/5}. </s> <s xml:id="echoid-s16113" xml:space="preserve">quod eſt ſecundum.</s> <s xml:id="echoid-s16114" xml:space="preserve"/> </p> <div xml:id="echoid-div984" type="float" level="2" n="2"> <note symbol="d" position="right" xlink:label="note-371-05" xlink:href="note-371-05a" xml:space="preserve">27. quinti.</note> <note symbol="e" position="right" xlink:label="note-371-06" xlink:href="note-371-06a" xml:space="preserve">28. quinti.</note> <note symbol="f" position="right" xlink:label="note-371-07" xlink:href="note-371-07a" xml:space="preserve">27. quinti.</note> </div> </div> <div xml:id="echoid-div986" type="section" level="1" n="352"> <head xml:id="echoid-head379" xml:space="preserve">THEOR. 6. PROPOS. 11.</head> <p> <s xml:id="echoid-s16115" xml:space="preserve">SI duonumeri inter ſe primi non ſint ambo quadrati aut cubi; </s> <s xml:id="echoid-s16116" xml:space="preserve">neque <lb/>eorum æquè multiplices vlli, quadrati erunt, aut cubi. </s> <s xml:id="echoid-s16117" xml:space="preserve">Et ſi eorum æ-<lb/>què multiplices aliqui ſint ambo quadrati, aut cubi, etiam ipſi erunt <lb/>quadrati aut cubi.</s> <s xml:id="echoid-s16118" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16119" xml:space="preserve"><emph style="sc">Sint</emph> enim A, B, numeri inter ſe primi, & </s> <s xml:id="echoid-s16120" xml:space="preserve">non ambo quadrati, vel cubi, quã-<lb/>uis vnus eorum qua dratus ſit, vel cubus. </s> <s xml:id="echoid-s16121" xml:space="preserve">ſintque eorum æ-<lb/> <anchor type="note" xlink:label="note-371-08a" xlink:href="note-371-08"/> què multiplices C, D. </s> <s xml:id="echoid-s16122" xml:space="preserve">Dico neq; </s> <s xml:id="echoid-s16123" xml:space="preserve">hos eſſe ambos quadra-<lb/>tos, aut cubos. </s> <s xml:id="echoid-s16124" xml:space="preserve">Sint enim, ſi fieri poteſt, ambo quadrati, <lb/>vel cubi. </s> <s xml:id="echoid-s16125" xml:space="preserve">Et quoniam idem numerus multiplicans A, & </s> <s xml:id="echoid-s16126" xml:space="preserve">B, <lb/>fecit C, & </s> <s xml:id="echoid-s16127" xml:space="preserve">D, quod hi illorum ſint æque multiplices: </s> <s xml:id="echoid-s16128" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit <anchor type="note" xlink:label="note-371-09a" xlink:href="note-371-09"/> A, ad B, vt C, ad D. </s> <s xml:id="echoid-s16129" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Cadit autem inter C, & </s> <s xml:id="echoid-s16130" xml:space="preserve">D, vnus me- <anchor type="note" xlink:label="note-371-10a" xlink:href="note-371-10"/> dius proportionalis, aut duo. </s> <s xml:id="echoid-s16131" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Igitur & </s> <s xml:id="echoid-s16132" xml:space="preserve">inter A, B, vnus ca- <anchor type="note" xlink:label="note-371-11a" xlink:href="note-371-11"/> det medius proportionalis, aut duo. </s> <s xml:id="echoid-s16133" xml:space="preserve">Cum ergo extremi A, <lb/>B, ponantur inter ſe primi; </s> <s xml:id="echoid-s16134" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> erunt omnes tres, vel quatuor <anchor type="note" xlink:label="note-371-12a" xlink:href="note-371-12"/> proportionales, minimi in ſua proportione: </s> <s xml:id="echoid-s16135" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Ac proinde <anchor type="note" xlink:label="note-371-13a" xlink:href="note-371-13"/> A, B, ambo qua drati erunt, vel cubi. </s> <s xml:id="echoid-s16136" xml:space="preserve">quod eſt contra hy-<lb/>potheſim. </s> <s xml:id="echoid-s16137" xml:space="preserve">Non ergo C, D, ambo qua drati ſunt, aut cubi. </s> <s xml:id="echoid-s16138" xml:space="preserve">quod erat oſtenden-<lb/>dum.</s> <s xml:id="echoid-s16139" xml:space="preserve"/> </p> <div xml:id="echoid-div986" type="float" level="2" n="1"> <note position="right" xlink:label="note-371-08" xlink:href="note-371-08a" xml:space="preserve"> <lb/>A # B <lb/>4. # 11. <lb/>C # D <lb/>12. # 33. <lb/></note> <note symbol="g" position="right" xlink:label="note-371-09" xlink:href="note-371-09a" xml:space="preserve">17. ſeptimi.</note> <note symbol="h" position="right" xlink:label="note-371-10" xlink:href="note-371-10a" xml:space="preserve">11. & 12. <lb/>octaui.</note> <note symbol="i" position="right" xlink:label="note-371-11" xlink:href="note-371-11a" xml:space="preserve">8. octaui.</note> <note symbol="k" position="right" xlink:label="note-371-12" xlink:href="note-371-12a" xml:space="preserve">1. octaui.</note> <note symbol="l" position="right" xlink:label="note-371-13" xlink:href="note-371-13a" xml:space="preserve">coroll. 2. <lb/>octaui.</note> </div> <p> <s xml:id="echoid-s16140" xml:space="preserve"><emph style="sc">Sed</emph> ſintiam C, D, ipſorum A, B, inter ſe primorum æquè multiplices, & </s> <s xml:id="echoid-s16141" xml:space="preserve">am-<lb/>bo quadrati, vel cubi. </s> <s xml:id="echoid-s16142" xml:space="preserve">Dico etiam A, B, ambos eſſe quadratos, vel cubos. </s> <s xml:id="echoid-s16143" xml:space="preserve">Si. </s> <s xml:id="echoid-s16144" xml:space="preserve">n.</s> <s xml:id="echoid-s16145" xml:space="preserve"> <pb o="344" file="372" n="372" rhead="GEOMETR. PRACT."/> non ſunt; </s> <s xml:id="echoid-s16146" xml:space="preserve">neque ipſi C, D, erunt ambo quadrati, vel cubi, vt demonſtratum <lb/>eſt. </s> <s xml:id="echoid-s16147" xml:space="preserve">quod cum hypotheſi pugnat.</s> <s xml:id="echoid-s16148" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div988" type="section" level="1" n="353"> <head xml:id="echoid-head380" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s16149" xml:space="preserve"><emph style="sc">Hinc</emph> fit, ſi tam Numerator, quam Denominator alicuius minutiæ fuerit <lb/>quadratus aut cubus: </s> <s xml:id="echoid-s16150" xml:space="preserve">tam Numeratorem quoque, quam Denominatoreme-<lb/>iuſdem minutiæ ad minimos reductę terminos, eſſe quadratum, vel cubum; </s> <s xml:id="echoid-s16151" xml:space="preserve">cum <lb/>minimi termini ſint numeri inter ſe primi, habeantque eandem proportionem, <lb/>quam Numerator, ac Denominator prioris minutiæ: </s> <s xml:id="echoid-s16152" xml:space="preserve">quippe cum minutiæ ſint <lb/>æquales. </s> <s xml:id="echoid-s16153" xml:space="preserve">Item ſi vterque numerus minutiæ cuiuſpiam in minimis terminis non <lb/>ſit quadratus, aut cubus, neque vtrum que numerum alterius minutiæ æquiua-<lb/>lentis eſſe quadratum, aut cubum.</s> <s xml:id="echoid-s16154" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div989" type="section" level="1" n="354"> <head xml:id="echoid-head381" xml:space="preserve">THEOR. 7. PROPOS. 12.</head> <p> <s xml:id="echoid-s16155" xml:space="preserve">IN omni quadrilatera figura rectilinea, tria latera, vt libet, aſſumpta, ma-<lb/>iora ſunt reliquo latere.</s> <s xml:id="echoid-s16156" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16157" xml:space="preserve"><emph style="sc">Sit</emph> quadrilaterum ABCD. </s> <s xml:id="echoid-s16158" xml:space="preserve">Dico quælibet tria latera, nimirum DA, AB, <lb/>BC, ſimul ſumpta eſſe maiora reliquo latere DC. <lb/></s> <s xml:id="echoid-s16159" xml:space="preserve"> <anchor type="note" xlink:label="note-372-01a" xlink:href="note-372-01"/> <anchor type="figure" xlink:label="fig-372-01a" xlink:href="fig-372-01"/> Ducta enim diametro BD; </s> <s xml:id="echoid-s16160" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> eruntrectæ BD, BC, maiores quam DC; </s> <s xml:id="echoid-s16161" xml:space="preserve">Sed eadem ratione AB, AD, <lb/>maiores ſunt quam BD. </s> <s xml:id="echoid-s16162" xml:space="preserve">Maiores erunt ergo tres <lb/>AD, AB, BC, quam duæ B D, B C; </s> <s xml:id="echoid-s16163" xml:space="preserve">ac proinde <lb/>multo maiores, quam D C. </s> <s xml:id="echoid-s16164" xml:space="preserve">Idemque demon-<lb/>ſtrabitur ſimili modo de quibuſcunque alijs tri-<lb/>bus lateribus, vt conſtat. </s> <s xml:id="echoid-s16165" xml:space="preserve">In omni ergo quadrilatera figura rectilinea, tria latera, <lb/>vt libet, aſſumpta. </s> <s xml:id="echoid-s16166" xml:space="preserve">maiora ſunt reliquo latere. </s> <s xml:id="echoid-s16167" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s16168" xml:space="preserve"/> </p> <div xml:id="echoid-div989" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-372-01" xlink:href="note-372-01a" xml:space="preserve">20. primi.</note> <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/xxxxxxxx/figures/372-01"/> </figure> </div> </div> <div xml:id="echoid-div991" type="section" level="1" n="355"> <head xml:id="echoid-head382" xml:space="preserve">PROBL. 6. PROPOS. 13.</head> <p> <s xml:id="echoid-s16169" xml:space="preserve">DATIS tribus punctis, per quæ circulis deſcribendus ſit, inuenire alia <lb/>puncta, per quæ idem circulus tranſire debeat.</s> <s xml:id="echoid-s16170" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16171" xml:space="preserve"><emph style="sc">Solent</emph> interdum tria data puncta tam parum inter ſe diſtare, aut fere in <lb/>recta linea iacere, vt non facilè eorum centrum inueniri poſsit, propterea quod <lb/>rectæ ſecantes lineas illa puncta connectentes bifariam, & </s> <s xml:id="echoid-s16172" xml:space="preserve">ad angulos rectos, <lb/>nimis obliquè ſe in centro interſecant. </s> <s xml:id="echoid-s16173" xml:space="preserve">Vtigitur magis ex quiſitè centrum re-<lb/>periatur, inueſtiganda erunt alia duo puncta, vel plura, per quæ idem circulus <lb/>incedere debeat, hoc modo. </s> <s xml:id="echoid-s16174" xml:space="preserve">Sint data tria puncta A, B, C. </s> <s xml:id="echoid-s16175" xml:space="preserve">Iunctis rectis A B, <lb/>A C, BC, conſtituatur ſuper baſem BC, triangulum BCD, t@iangulo ABC, æqui-<lb/>laterum, ita vt angulus D, vergat in eampartem, verſus quam circumferentia <lb/>deſcribenda tranſire debet, lateraque æqualia non ab eodem puncto exeant, <lb/>hoceſt, latus C D, lateri B A, & </s> <s xml:id="echoid-s16176" xml:space="preserve">latus B D, lateri C A, ſit æquale. </s> <s xml:id="echoid-s16177" xml:space="preserve">Quod quidem <lb/>fiet, ſiex C, arcus delineetur ad interuallum BA, quem alius arcus ex B, ad inter-<lb/> <anchor type="note" xlink:label="note-372-02a" xlink:href="note-372-02"/> uallum CA, delineatus ſecet in D. </s> <s xml:id="echoid-s16178" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Erit enim angulus D, angulo A, æqualis: </s> <s xml:id="echoid-s16179" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <anchor type="note" xlink:label="note-372-03a" xlink:href="note-372-03"/> ac proinde circulus per tria puncta A, B, C, deſcriptus tranſibit quoq; </s> <s xml:id="echoid-s16180" xml:space="preserve">per quar- <pb o="345" file="373" n="373" rhead="LIBER OCTAVVS."/> tum punctum D. </s> <s xml:id="echoid-s16181" xml:space="preserve">Eadem ratione, ſi ſuper baſem C D, triangulum conſtruatur <lb/>C D E, triangulo B C D, æquilaterum ordine prædicto, ita vt latus D E, lateri C B, <lb/> <anchor type="figure" xlink:label="fig-373-01a" xlink:href="fig-373-01"/> & </s> <s xml:id="echoid-s16182" xml:space="preserve">latus CE, lateri BD, æquale ſit, inuentum erit aliud punctum E, per quod cir-<lb/>cumferentia incedat. </s> <s xml:id="echoid-s16183" xml:space="preserve">Atque eadem arte reperietur aliud punctum F, per trian-<lb/>gulum DEF, triangulo E D C, æquilaterum, &</s> <s xml:id="echoid-s16184" xml:space="preserve">c. </s> <s xml:id="echoid-s16185" xml:space="preserve">Eodem modo ex altera parte <lb/>reperietur aliud punctum G, per triangulum A B G, triangulo B A C, æquilate-<lb/>rum, & </s> <s xml:id="echoid-s16186" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s16187" xml:space="preserve">Si igitur eligantur tria puncta, ita vt rectæ ea connecten-<lb/>tes conſtituant quaſi angulum rectum, qualia ſunt tria puncta A, C, D, & </s> <s xml:id="echoid-s16188" xml:space="preserve">ex <lb/>proximis A, C, ad quo dcunque idem interuallum bini arcus deſcribantur, & </s> <s xml:id="echoid-s16189" xml:space="preserve"><lb/>ex proximis C, D, binialij; </s> <s xml:id="echoid-s16190" xml:space="preserve">ac per interſectiones horum arcuum rectæ lineæ <lb/>emittantur, ſecabunt ſeſe in centro H, &</s> <s xml:id="echoid-s16191" xml:space="preserve">c. </s> <s xml:id="echoid-s16192" xml:space="preserve">Aptiſsima quo que eſſent tria pun-<lb/>cta G, B, D, quamuis angulus D B G, acutus ſit. </s> <s xml:id="echoid-s16193" xml:space="preserve">Item tria puncta G, B, E, & </s> <s xml:id="echoid-s16194" xml:space="preserve">C, <lb/>E, F, &</s> <s xml:id="echoid-s16195" xml:space="preserve">c.</s> <s xml:id="echoid-s16196" xml:space="preserve"/> </p> <div xml:id="echoid-div991" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-372-02" xlink:href="note-372-02a" xml:space="preserve">8 primi.</note> <note symbol="c" position="left" xlink:label="note-372-03" xlink:href="note-372-03a" xml:space="preserve">ſchol. 21. <lb/>tertij.</note> <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/xxxxxxxx/figures/373-01"/> </figure> </div> </div> <div xml:id="echoid-div993" type="section" level="1" n="356"> <head xml:id="echoid-head383" xml:space="preserve">PROBL. 7. PROPOS. 14.</head> <p> <s xml:id="echoid-s16197" xml:space="preserve">DATO exceſſu diametri Quadrati ſupra latus: </s> <s xml:id="echoid-s16198" xml:space="preserve">Item dato exceſſu dia-<lb/>metri Rhombi ſupra latus, vellateris ſupra diametrum (quando illud <lb/>maius eſt) vna cum vno Rhombi angulo: </s> <s xml:id="echoid-s16199" xml:space="preserve">Dato præterea exceſſu dia-<lb/>metri rectanguli ſupra vtr umlibet laterum inæqualium, vna cum an-<lb/>gulo, quem diameter cum eo latere facit, vel vna cum proportione <lb/>eorundem inæqualium laterum: </s> <s xml:id="echoid-s16200" xml:space="preserve">Dato deniq; </s> <s xml:id="echoid-s16201" xml:space="preserve">exceſſu diametri Rhõ-<lb/>boidis ſupra vtrumuis laterum inæqualium, vel vtriuſuis inæqualium <lb/>laterũ ſupra diametrum (quando illud maius eſt) vna cũ vno angulo <lb/>Rhomboidis, & </s> <s xml:id="echoid-s16202" xml:space="preserve">inſuper cum angulo, quem diameter cum eo latere <lb/>facit, vel inſuper cum proportione duortum laterum in æqualiũ; </s> <s xml:id="echoid-s16203" xml:space="preserve">Qua-<lb/>dratum ipſum, Rhombũ, Rectangulũ, & </s> <s xml:id="echoid-s16204" xml:space="preserve">Rhomboides conſtituere.</s> <s xml:id="echoid-s16205" xml:space="preserve"/> </p> <pb o="346" file="374" n="374" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s16206" xml:space="preserve"><emph style="sc">Hoc</emph> problema, quod ad quadratum attinet, alio modo ad finem lib. </s> <s xml:id="echoid-s16207" xml:space="preserve">2. </s> <s xml:id="echoid-s16208" xml:space="preserve">Eu-<lb/>clid. </s> <s xml:id="echoid-s16209" xml:space="preserve">abſoluimus. </s> <s xml:id="echoid-s16210" xml:space="preserve">Sit A, datus exceſſus diametri quadrati cuiuſpiam ſupra la-<lb/>tus. </s> <s xml:id="echoid-s16211" xml:space="preserve">Fiat quodcunque quadratum B C D E, cuius diameter B D, excedat latus <lb/> <anchor type="figure" xlink:label="fig-374-01a" xlink:href="fig-374-01"/> exceſſu D F; </s> <s xml:id="echoid-s16212" xml:space="preserve">qui ſi æqualis fuerit dato exceſſui A; </s> <s xml:id="echoid-s16213" xml:space="preserve">factum erit, quod iubetur: </s> <s xml:id="echoid-s16214" xml:space="preserve">Si <lb/>vero inæqualis, <anchor type="note" xlink:href="" symbol="a"/> fiat vt DF, ad datum exceſſum A, ita diameter BD, ad BG, per- <anchor type="note" xlink:label="note-374-01a" xlink:href="note-374-01"/> ficiatur que quadratum H I; </s> <s xml:id="echoid-s16215" xml:space="preserve">quod dico eſſeid, quod quæritur. </s> <s xml:id="echoid-s16216" xml:space="preserve">Sumpta enim <lb/>recta GK, ipſi A, æquali, quoniam eſt per conſtructionem, vt tota BD, ad totam <lb/>BG, ita DF, ablata ad A, hoc eſt, ad GK, ablatam; </s> <s xml:id="echoid-s16217" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit quoq; </s> <s xml:id="echoid-s16218" xml:space="preserve">vt tota BD, adto- <anchor type="note" xlink:label="note-374-02a" xlink:href="note-374-02"/> tam BG, ita reliqua BF, ad reliquam BK. </s> <s xml:id="echoid-s16219" xml:space="preserve">Et permutando, vt BD, ad BF, ita BG, <lb/>ad BK,. </s> <s xml:id="echoid-s16220" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Eſt autem vt BD, ad BF, ita BD, ad BC, (quod BF, BC, æquales ſint;</s> <s xml:id="echoid-s16221" xml:space="preserve"> <anchor type="note" xlink:label="note-374-03a" xlink:href="note-374-03"/> cum DF, ponatur ex ceſſus diametri BD, ſupra latus BC.) </s> <s xml:id="echoid-s16222" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et vt BD, ad BC, ita <anchor type="note" xlink:label="note-374-04a" xlink:href="note-374-04"/> BG, ad BH. </s> <s xml:id="echoid-s16223" xml:space="preserve">Igitur erit quo que vt BG, ad BK, ita BG, ad BH; </s> <s xml:id="echoid-s16224" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Acproinde BK, <anchor type="note" xlink:label="note-374-05a" xlink:href="note-374-05"/> BH, æquales erunt. </s> <s xml:id="echoid-s16225" xml:space="preserve">Diameter ergo BG, ſuperatlatus BH, hoc eſt, BK, recta GK, <lb/>quæ dato ex ceſſui A, æqualis eſt. </s> <s xml:id="echoid-s16226" xml:space="preserve">Quod eſt propoſitum.</s> <s xml:id="echoid-s16227" xml:space="preserve"/> </p> <div xml:id="echoid-div993" 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/xxxxxxxx/figures/374-01"/> </figure> <note symbol="a" position="left" xlink:label="note-374-01" xlink:href="note-374-01a" xml:space="preserve">12. ſexti.</note> <note symbol="b" position="left" xlink:label="note-374-02" xlink:href="note-374-02a" xml:space="preserve">19. quinti.</note> <note symbol="c" position="left" xlink:label="note-374-03" xlink:href="note-374-03a" xml:space="preserve">7. quinti.</note> <note symbol="d" position="left" xlink:label="note-374-04" xlink:href="note-374-04a" xml:space="preserve">4. ſexti.</note> <note symbol="e" position="left" xlink:label="note-374-05" xlink:href="note-374-05a" xml:space="preserve">9. quinti.</note> </div> <p> <s xml:id="echoid-s16228" xml:space="preserve"><emph style="sc">Sit</emph> deinde A, exceſſus diametri in Rhombo aliquo ſupra latus, vna cum <lb/>angulo L, datus. </s> <s xml:id="echoid-s16229" xml:space="preserve">Fiat Rhombus quicunque BCDE, habens angulum C, æqua-<lb/>lem dato angulo L, vt in primo Rhombo, vel angulum B, vt in ſecundo. </s> <s xml:id="echoid-s16230" xml:space="preserve">Siue <lb/>ergo diameter opponi debeat dato angulo C, vt in primo Rhombo, ſiue datum <lb/>angulum B, ſecare, vt in ſecundo, ducatur diameter B D, excedens latus B C, <lb/>recta DF, quæ ſi æqualis fuerit dato exceſſui A, factum erit, quo diubetur: </s> <s xml:id="echoid-s16231" xml:space="preserve">Si ve-<lb/> <anchor type="figure" xlink:label="fig-374-02a" xlink:href="fig-374-02"/> ro inæqualis, <anchor type="note" xlink:href="" symbol="f"/> fiat vt D F, ad exceſſum datum A, ita diameter B D, ad B G, com- <anchor type="note" xlink:label="note-374-06a" xlink:href="note-374-06"/> pleaturque Rhombus HI, quem dico eſſe, eum, qui quæritur. </s> <s xml:id="echoid-s16232" xml:space="preserve">Abſciſſa enimre-<lb/>cta GK, exceſſui dato A, ęquali, adhibenda eſt, eadem omnino demonſtratio, <lb/>quæ in quadrato facta eſt.</s> <s xml:id="echoid-s16233" xml:space="preserve"/> </p> <div xml:id="echoid-div994" type="float" level="2" n="2"> <figure xlink:label="fig-374-02" xlink:href="fig-374-02a"> <image file="374-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/374-02"/> </figure> <note symbol="f" position="left" xlink:label="note-374-06" xlink:href="note-374-06a" xml:space="preserve">12. ſexti.</note> </div> <pb o="347" file="375" n="375" rhead="LIBER OCTAVVS."/> <p> <s xml:id="echoid-s16234" xml:space="preserve"><emph style="sc">Qvod</emph> ſi diameter latere Rhombi minor fuerit, ſit datus exceſſus A, lateris <lb/>inaliquo Rhombo ſupra diametrum, vna cum angulo L. </s> <s xml:id="echoid-s16235" xml:space="preserve">Conſtruatur Rhom-<lb/>bus quantuſcunque BDCE, habens angulum D, æqualem dato angulo L, vt in <lb/>tertio Rhombo. </s> <s xml:id="echoid-s16236" xml:space="preserve">Et quia vt latus ſuperet diametrum, ducenda eſt diameter per <lb/>angulos obtuſos, <anchor type="note" xlink:href="" symbol="a"/> (quod diameter per acutos angulos ducta ſemper maior eſt <anchor type="note" xlink:label="note-375-01a" xlink:href="note-375-01"/> Rhombilatere) ducatur diameter BC, quamlatus BD, excedatrecta DF, quę ſi <lb/>æqualis fuerit exceſſui dato A, factum erit, quod jubetur: </s> <s xml:id="echoid-s16237" xml:space="preserve">Siver ò inæqualis: <lb/></s> <s xml:id="echoid-s16238" xml:space="preserve">fiat vt DF, ad exceſſum A, ita BD, latus ad B G, compleaturque Rhombus G I, <lb/>circa eandem diametrum, quem dico eſſe quæſitum. </s> <s xml:id="echoid-s16239" xml:space="preserve">Abſciſſa namq; </s> <s xml:id="echoid-s16240" xml:space="preserve">recta GK, <lb/>æquali exceſſui A: </s> <s xml:id="echoid-s16241" xml:space="preserve">fiet demonſtratio, vt in quadrato, vt perſpicuum eſt, ſi loco <lb/>diametrorum BD, BG, in quadrato, ſumantur hic latera BD, BG.</s> <s xml:id="echoid-s16242" xml:space="preserve"/> </p> <div xml:id="echoid-div995" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-375-01" xlink:href="note-375-01a" xml:space="preserve">19. primi.</note> </div> <p> <s xml:id="echoid-s16243" xml:space="preserve"><emph style="sc">Tvnc</emph> autem latus Rhombimaius erit diametro (vt hoc etiam obiter mo-<lb/>neamus) cum ſemiſsis anguli obtuſi maior fuerit angulo acuto eiuſdẽ Rhom-<lb/>bi. </s> <s xml:id="echoid-s16244" xml:space="preserve">Nam ſi in tertio Rhombo angulus CBD, quiſemiſsis eſt anguli obtuſi B, vt <lb/>in ſchol. </s> <s xml:id="echoid-s16245" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16246" xml:space="preserve">34. </s> <s xml:id="echoid-s16247" xml:space="preserve">lib. </s> <s xml:id="echoid-s16248" xml:space="preserve">1. </s> <s xml:id="echoid-s16249" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16250" xml:space="preserve">oſtendimus, maiorſit angulo acuto D; </s> <s xml:id="echoid-s16251" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit <anchor type="note" xlink:label="note-375-02a" xlink:href="note-375-02"/> latus BD, hoc eſt, CD, maius diametro BC, in triangulo BCD. </s> <s xml:id="echoid-s16252" xml:space="preserve">Quando autem <lb/>ſemiſsis anguli obtuſi fuerit minor angulo acuto, vt in Rhombo ſecundo; </s> <s xml:id="echoid-s16253" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit <anchor type="note" xlink:label="note-375-03a" xlink:href="note-375-03"/> diameter latere maior in triangulo B C D.</s> <s xml:id="echoid-s16254" xml:space="preserve"/> </p> <div xml:id="echoid-div996" type="float" level="2" n="4"> <note symbol="b" position="right" xlink:label="note-375-02" xlink:href="note-375-02a" xml:space="preserve">19. primi.</note> <note symbol="c" position="right" xlink:label="note-375-03" xlink:href="note-375-03a" xml:space="preserve">19. primi.</note> </div> <p> <s xml:id="echoid-s16255" xml:space="preserve"><emph style="sc">Tertio</emph> ſit datus exceſſus A, diametri rectanguli alicuius ſupra alterum <lb/>laterum inæqualium, vna cum angulo L, quem diameter cum eo latere conſti-<lb/>tuit, velvna cum proportione M, ad N, quam illud latus ad alterum habet. </s> <s xml:id="echoid-s16256" xml:space="preserve">Si <lb/>ergo angulus L, eſt ſemirecto minor, vel certè proportio M, ad N, maiors in-<lb/>æqualitatis, vt in priori rectangulo, erit A, exceſſus diametri ſupra maius latus: <lb/></s> <s xml:id="echoid-s16257" xml:space="preserve">Si verò angulus L, eſt maior ſemirecto, vel proportio M, ad N, minoris inæqua-<lb/> <anchor type="figure" xlink:label="fig-375-01a" xlink:href="fig-375-01"/> litatis, erit A, exceſſus diametri ſupra minus latus. </s> <s xml:id="echoid-s16258" xml:space="preserve">Conſtituatur ergo angulus <lb/>CBD, angulo L, æqualis, fiatque rectangulum BCDE, circa aſſumptam diame-<lb/>trum BD. </s> <s xml:id="echoid-s16259" xml:space="preserve">Vel fiat B C, quanta cunque ad CD, perpendicularem, vt M, ad N: <lb/></s> <s xml:id="echoid-s16260" xml:space="preserve">completo que rectangulo CE, ducatur diameter B D, excedens latus B G, recta <lb/>D F, quæ ſi fuerit æqualis dato exceſſui A, conſtructum erit rectangulum C E, <lb/>quod quæritur: </s> <s xml:id="echoid-s16261" xml:space="preserve">Si verò inæqualis, fiat vt D F, ad exceſſum datum A, ita B D, <lb/>ad B G, compleaturque rectangulum H I, quod erit quæſitum. </s> <s xml:id="echoid-s16262" xml:space="preserve">Ab-<lb/>ſciſſa enim recta G K, æquali exceſſui A, demonſtrabitur propoſitum, vt <lb/>in quadrato.</s> <s xml:id="echoid-s16263" xml:space="preserve"/> </p> <div xml:id="echoid-div997" type="float" level="2" n="5"> <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/xxxxxxxx/figures/375-01"/> </figure> </div> <p> <s xml:id="echoid-s16264" xml:space="preserve"><emph style="sc">Qvarto</emph> & </s> <s xml:id="echoid-s16265" xml:space="preserve">vltimo ſit in aliquo Rhomboide datus exceſſus A, diametri <lb/>ſupra vtrumuis inæqualium laterum, vna cum angulo Rhomboidis O, & </s> <s xml:id="echoid-s16266" xml:space="preserve">inſu-<lb/>per cum angulo L, quem diameter cum latere, cuius exceſſus ſumptus eſt, effi- <pb o="348" file="376" n="376" rhead="GEOMETR. PRACT."/> cit, velinſuper cum proportione M, ad N, quam latus illud ad alterum latus ha-<lb/>bet. </s> <s xml:id="echoid-s16267" xml:space="preserve">Conſtituatur angulus BCD, in prima figura, vel CBE, in ſecunda dato an-<lb/>gulo O, æqualis. </s> <s xml:id="echoid-s16268" xml:space="preserve">Deinde ſiue diameter dato angulo C, opponi debeat, vt in pri-<lb/>ma figura, ſiue datum angulũ CBE, ſecare, vt in ſecũda, fiat ad B, angulus CBD, <lb/>angulo L, dato æqualis, ſecetque CD, rectam BD, in D: </s> <s xml:id="echoid-s16269" xml:space="preserve">vel fiat vt M, ad N, ita <lb/>BC, ad CD; </s> <s xml:id="echoid-s16270" xml:space="preserve">ac Rhomboides compleatur CE, cuius diameter latus BC, excedat <lb/>recta DF. </s> <s xml:id="echoid-s16271" xml:space="preserve">quæ ſi æqualis fuerit dato exceſſui A, factum erit, quodiubetur. </s> <s xml:id="echoid-s16272" xml:space="preserve">Sive-<lb/>rò inæqualis, fiat vt DF, ad exceſſum A, ita BD, ad BG, compleaturque Rhom-<lb/>boides HI, circa eandem diametrum BD, quod dico eſſe quæſitum. </s> <s xml:id="echoid-s16273" xml:space="preserve">Nam ſi reſe-<lb/>cetur GK, exceſſui A, æqualis, adhibenda eſt eadem demonſtratio, quæin præ-<lb/>cedentibus.</s> <s xml:id="echoid-s16274" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16275" xml:space="preserve"><emph style="sc">Qvod</emph> ſi diameter Rhomboidis cuiuſpiam minor fuerit latere maiore, vt in <lb/>tertia figura. </s> <s xml:id="echoid-s16276" xml:space="preserve">Sit datus exceſſus A, lateris maioris in aliquo Rhomboide ſupra <lb/>diametrum, vna cum angulo O, Rhomboidis, & </s> <s xml:id="echoid-s16277" xml:space="preserve">inſuper cum angulo L, quem <lb/> <anchor type="figure" xlink:label="fig-376-01a" xlink:href="fig-376-01"/> diameter cum illo latere maiore efficere debet, vel inſuper cum proportione M, <lb/>ad N, quam maius latus ad minus habet. </s> <s xml:id="echoid-s16278" xml:space="preserve">Conſtituatur angulus BDC, dato an-<lb/>gulo O, æqualis: </s> <s xml:id="echoid-s16279" xml:space="preserve">Etſi eſt acutus, fiat in B, angulus D B C, angulo L, æqualis, <lb/>(ſi datus angulus Rhomboidis foret obtuſus, nimirum DBE, conſtituendus eſ-<lb/>ſet angulus DBC, in ipſo angulo dato) ſecetque recta BC, rectam D C, in C; </s> <s xml:id="echoid-s16280" xml:space="preserve">vel <lb/>fiat vt M, ad N, ita BD, ad DC; </s> <s xml:id="echoid-s16281" xml:space="preserve">ac Rhomboides cõpleatur D E, cuius latus B D, <lb/>diametrum BC, ſuperetrecta DF, quæ ſi æqualis fuerit dato exceſſui, factum e-<lb/>rit, quodiubetur: </s> <s xml:id="echoid-s16282" xml:space="preserve">Si verò inæqualis, fiat vt DF, ad A, ita BD, ad B G, perficia-<lb/>turque Rhomboides GI, quod dico eſſe quæſitum. </s> <s xml:id="echoid-s16283" xml:space="preserve">Nam ſi capiatur GK, æqua-<lb/>lis ipſi A, demonſtrabitur propoſitum, vt ſupra in quadrato, ſi loco diametrorũ <lb/>BD, BG, quadrati, accipiantur hic latera BD, BG, vt perſpicuum eſt.</s> <s xml:id="echoid-s16284" xml:space="preserve"/> </p> <div xml:id="echoid-div998" type="float" level="2" n="6"> <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/xxxxxxxx/figures/376-01"/> </figure> </div> <p> <s xml:id="echoid-s16285" xml:space="preserve"><emph style="sc">Tvnc</emph> autemlatus maius diametrum excedet, quando angulus, quem dia-<lb/>meter cum minore latere efficit, maior eſt acuto angulo Rhomboidis. </s> <s xml:id="echoid-s16286" xml:space="preserve">Nam ſi <lb/>in tertia figura angulus BCD, maior eſt angulo D; </s> <s xml:id="echoid-s16287" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erit recta B D, maior, ꝗ̃ BC.</s> <s xml:id="echoid-s16288" xml:space="preserve"> <anchor type="note" xlink:label="note-376-01a" xlink:href="note-376-01"/> </s> </p> <div xml:id="echoid-div999" type="float" level="2" n="7"> <note symbol="a" position="left" xlink:label="note-376-01" xlink:href="note-376-01a" xml:space="preserve">19. primi.</note> </div> </div> <div xml:id="echoid-div1001" type="section" level="1" n="357"> <head xml:id="echoid-head384" xml:space="preserve">THEOR. 8. PROPOS. 15.</head> <p> <s xml:id="echoid-s16289" xml:space="preserve">IN rectangulo parallelogrammo, ſumptis exceſſibus, quibus diameter <lb/>duo latera ſuperat; </s> <s xml:id="echoid-s16290" xml:space="preserve">Rectangulum ſub differentia exceſſuum, & </s> <s xml:id="echoid-s16291" xml:space="preserve">mino- <pb o="349" file="377" n="377" rhead="LIBER OCTAVVS."/> re exceſſu bis ſumptum, vna cum quadrato minoris exceſſus bis ſum-<lb/>pto, æquale eſt quadrato rectæ, qua minus latus minorem exceſſum <lb/>ſuperat.</s> <s xml:id="echoid-s16292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16293" xml:space="preserve"><emph style="sc">Sit</emph> rectangulum A C, cuius diametro B D, æqualis ſit recta B E, vt exceſſus <lb/>minor, quo diameter maius latus BC, ſuperat, ſit CE; </s> <s xml:id="echoid-s16294" xml:space="preserve">Sumpta autem BF, æqua-<lb/>liminorilateri CD, vt EF, exceſſus ſit, quo diameter BD, velilli æqualis BE, mi-<lb/>nus latus CD, velilli æqualem BF, ſuperat: </s> <s xml:id="echoid-s16295" xml:space="preserve">ac proinde CF, ſit differentia exceſ-<lb/>ſuum EC, EF. </s> <s xml:id="echoid-s16296" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quia latera BC, CD, maiora ſunt latere BD hoc eſt, recta BE, <anchor type="note" xlink:label="note-377-01a" xlink:href="note-377-01"/> dempta communi B C, erit reliqua C D, maior, quam reli-<lb/> <anchor type="figure" xlink:label="fig-377-01a" xlink:href="fig-377-01"/> qua CE; </s> <s xml:id="echoid-s16297" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s16298" xml:space="preserve">BF, æqualis ip ſi CD, maior erit, quam <lb/>C E. </s> <s xml:id="echoid-s16299" xml:space="preserve">Abſc@ſſa ergo F G, ipſi C E, æquali, erit B G, exceſſus <lb/>quo minus latus BF, minorem exceſſum FG, ſuperat. </s> <s xml:id="echoid-s16300" xml:space="preserve">Di-<lb/>corectangulum bis ſumptum ſub F C, differentia exceſſu-<lb/>um, vna cum quadrato minoris exceſſus CE, bis ſumpto, <lb/>æquale eſſe quadrato rectæ BG, quaminus latus BF, mino-<lb/>rem exceſſum F G, ſuperat. </s> <s xml:id="echoid-s16301" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam enim quadratum <anchor type="note" xlink:label="note-377-02a" xlink:href="note-377-02"/> rectæ B E, æquale eſt quadratis rectarum B C, C E, vna cumrectangulo bis ſub <lb/>BC, CE, hoc eſt, rectangulo ſemel ſumpto ſub B C, & </s> <s xml:id="echoid-s16302" xml:space="preserve">recta ipſius C E, dupla: </s> <s xml:id="echoid-s16303" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <anchor type="note" xlink:label="note-377-03a" xlink:href="note-377-03"/> Eſt autem rectangulum ſub B C, & </s> <s xml:id="echoid-s16304" xml:space="preserve">dupla ipſius C E, æquale rectangulis ſub <lb/>B F, & </s> <s xml:id="echoid-s16305" xml:space="preserve">dupla ipſius C E, æquale rectangulis ſub B F, & </s> <s xml:id="echoid-s16306" xml:space="preserve">duplaipſius C E, & </s> <s xml:id="echoid-s16307" xml:space="preserve">ſub <lb/>FC, & </s> <s xml:id="echoid-s16308" xml:space="preserve">dupla ipſius CE; </s> <s xml:id="echoid-s16309" xml:space="preserve">hoc eſt, rectangulo ſub BF, & </s> <s xml:id="echoid-s16310" xml:space="preserve">CE, bis vna cum rectan-<lb/>gulo ſub FC, & </s> <s xml:id="echoid-s16311" xml:space="preserve">CE, bis; </s> <s xml:id="echoid-s16312" xml:space="preserve">Erit quadratum rectæ BE, ſiue rectæ BD, æquale quo-<lb/>que quadratis rectarum BC, CE, vna cum rectangulis ſub BF, CE, bis, & </s> <s xml:id="echoid-s16313" xml:space="preserve">ſub FC, <lb/>CE, bis; </s> <s xml:id="echoid-s16314" xml:space="preserve">Ac proinde & </s> <s xml:id="echoid-s16315" xml:space="preserve">quadrata rectarum BC, CD, <anchor type="note" xlink:href="" symbol="d"/> quæ quadrato rectæ BD, <anchor type="note" xlink:label="note-377-04a" xlink:href="note-377-04"/> æqualia ſunt, æqualia erunt quadratis rectarum B C, C E, vna cum rectangulis <lb/>ſub BF, CE, bis, & </s> <s xml:id="echoid-s16316" xml:space="preserve">ſub FC, CE, bis. </s> <s xml:id="echoid-s16317" xml:space="preserve">Ablato ergo communi quadrato rectæ BC, <lb/>erit reliquum quadratum rectæ CD, hoc eſt, rectæ BF, æqualereliquis rectangu-<lb/>lis ſub BF, CE, bis, & </s> <s xml:id="echoid-s16318" xml:space="preserve">ſub FC, CE, bis, vna cum quadrato rectæ C E. </s> <s xml:id="echoid-s16319" xml:space="preserve">Addito <lb/>igitur communi quadrato rectæ FG, erunt quadrata rectarum B F, F G, æqualia <lb/>rectangulis ſub BF, CE, bis, & </s> <s xml:id="echoid-s16320" xml:space="preserve">ſub FC, CE, bis, vna cum quadratis rectarum CE, <lb/> <anchor type="note" xlink:label="note-377-05a" xlink:href="note-377-05"/> FG. </s> <s xml:id="echoid-s16321" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Sed quadratarectarum BF, FG, æqualia ſunt rectangulo ſub BF, FG, bis, vna cum quadrato rectæ BG. </s> <s xml:id="echoid-s16322" xml:space="preserve">Igitur rectangulum quoque ſub BF, FG, hoc eſt, <lb/>ſub BF, CE, bis, vna cum quadrato rectæ BG, æquale erit rectangulis ſub BF, CE, <lb/>bis, & </s> <s xml:id="echoid-s16323" xml:space="preserve">ſub FC, CE, bis, vna cum quadratis rectarũ CE, FG. </s> <s xml:id="echoid-s16324" xml:space="preserve">Ablato ergo cõmu-<lb/>ni rectangulo ſub BF, CE, bis ſumpto; </s> <s xml:id="echoid-s16325" xml:space="preserve">erit reliquum quadratum BG, æquale re-<lb/>liquo rectangulo ſub F C, C E, bis, vna cum quadratis rectarum C E, F G: </s> <s xml:id="echoid-s16326" xml:space="preserve">hoc <lb/>eſt, rectangulum ſub F C, diſferentia exceſſuum, & </s> <s xml:id="echoid-s16327" xml:space="preserve">CE, minore exceſſu bis ſum-<lb/>ptum, vna cum quadrato minoris exceſſus CE, bis ſumpto, æquale eſt qua dra-<lb/>to rectæ B G, qua minus latus B F, minorem exceſſum F G, ſuperat, quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s16328" xml:space="preserve"/> </p> <div xml:id="echoid-div1001" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-377-01" xlink:href="note-377-01a" xml:space="preserve">20. primi.</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/xxxxxxxx/figures/377-01"/> </figure> <note symbol="b" position="right" xlink:label="note-377-02" xlink:href="note-377-02a" xml:space="preserve">4. ſecundi.</note> <note symbol="c" position="right" xlink:label="note-377-03" xlink:href="note-377-03a" xml:space="preserve">1. ſecundi.</note> <note symbol="d" position="right" xlink:label="note-377-04" xlink:href="note-377-04a" xml:space="preserve">47. primi.</note> <note symbol="e" position="right" xlink:label="note-377-05" xlink:href="note-377-05a" xml:space="preserve">7 ſecundi.</note> </div> </div> <div xml:id="echoid-div1003" type="section" level="1" n="358"> <head xml:id="echoid-head385" xml:space="preserve">PROBL. 8. PROPOS. 16.</head> <p> <s xml:id="echoid-s16329" xml:space="preserve">DATIS duobus exceſſibus, quibus diameter rectanguli vtrumque la-<lb/>tus ſuperat, vtrumque latus, & </s> <s xml:id="echoid-s16330" xml:space="preserve">diametrum inuenire.</s> <s xml:id="echoid-s16331" xml:space="preserve"/> </p> <pb o="350" file="378" n="378" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s16332" xml:space="preserve"><emph style="sc">Sit</emph> datus exceſſus FE, diametri ſupra latus minus, & </s> <s xml:id="echoid-s16333" xml:space="preserve">C E, ſupra maius, ita <lb/>vt differentia exceſſuum ſit F C. </s> <s xml:id="echoid-s16334" xml:space="preserve">Ex C, educatur ad FE, perpendicularis CL, ca-<lb/>piantur que CH, HI, EK, minori exceſſui CE, æquales, ita vt totæ CI, CK, æqua-<lb/>les ſint, vt pote ipſius CE, duplæ, perficiaturque parallelogrammum FI. </s> <s xml:id="echoid-s16335" xml:space="preserve">Diuiſa <lb/>deinde FK, bifariam in N, deſcribatur ex N, per F, & </s> <s xml:id="echoid-s16336" xml:space="preserve">K, ſemicirculus FLK, ſecans <lb/>CL; </s> <s xml:id="echoid-s16337" xml:space="preserve">in L. </s> <s xml:id="echoid-s16338" xml:space="preserve">Ducta denique HE, ſumatur illi æqualis CM, iungaturque recta LM. <lb/></s> <s xml:id="echoid-s16339" xml:space="preserve">Dico L M, differentiam eſſe inter minus latus quæſitum, & </s> <s xml:id="echoid-s16340" xml:space="preserve">minorem exceſſum <lb/>datum CE, ita vt CE, addita ad LM, efficiat minus latus; </s> <s xml:id="echoid-s16341" xml:space="preserve">cui ſi addatur F C, dif-<lb/>ferentia datorum exceſſum, fiat maius la-<lb/> <anchor type="figure" xlink:label="fig-378-01a" xlink:href="fig-378-01"/> tus. </s> <s xml:id="echoid-s16342" xml:space="preserve">(Eſt enim differentia exceſſuum dia-<lb/>metri ſupra vtrumque latus rectanguli æ-<lb/>qualis exceſſui maioris lateris ſupra mi-<lb/>nus: </s> <s xml:id="echoid-s16343" xml:space="preserve">vt in figura præcedentis propoſ. </s> <s xml:id="echoid-s16344" xml:space="preserve">pa-<lb/>tet; </s> <s xml:id="echoid-s16345" xml:space="preserve">vbidiameter eſt BD, vel BE; </s> <s xml:id="echoid-s16346" xml:space="preserve">exceſſus <lb/>maior F E, quo diameter minus latus B F, <lb/>ſuperat; </s> <s xml:id="echoid-s16347" xml:space="preserve">exceſſus minor CE, quo eadem <lb/>diameter maius latus B C, ſuperat: </s> <s xml:id="echoid-s16348" xml:space="preserve">eſt que <lb/>FC, differentia exceſſuum, exceſſus, quo <lb/>maius latus B C, ſuperat minus B F,) Ac <lb/>tandem maiori lateriinuẽto adijciatur minor exceſſus CE, vt diameter habeatur. <lb/></s> <s xml:id="echoid-s16349" xml:space="preserve">quæ omnia ita demonſtrabuntur. </s> <s xml:id="echoid-s16350" xml:space="preserve">Per præcedentem, rectangulum ſub FC, dif-<lb/>ferentia exceſſuum, & </s> <s xml:id="echoid-s16351" xml:space="preserve">CE, minori exceſſu bis ſumptum, hoc eſt, rectangulum <lb/>FI, vna cum quadrato rectæ CE, bis etiam ſumpto, hoc eſt, vna cum quadrato <lb/>rectæ HE, vel CM, æquale eſt quadrato rectæ, qua minus latus quæſitum, mi-<lb/>norem exceſſum CE, ſuperat. </s> <s xml:id="echoid-s16352" xml:space="preserve">Cum ergo quadratum rectæ CL, æquale ſit re-<lb/>ctangulo FI, vt ex demonſtratione vltimæ propoſ. </s> <s xml:id="echoid-s16353" xml:space="preserve">lib. </s> <s xml:id="echoid-s16354" xml:space="preserve">2. </s> <s xml:id="echoid-s16355" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16356" xml:space="preserve">conſtat; </s> <s xml:id="echoid-s16357" xml:space="preserve">erunt <lb/>quo que quadrata rectarum CL, CM, æqualia quadrato eiuſdem rectæ, qua mi-<lb/>nuslatus quæſitum ſuperat minorem exceſſum CE, <anchor type="note" xlink:href="" symbol="a"/> Ac proinde cum quadra- <anchor type="note" xlink:label="note-378-01a" xlink:href="note-378-01"/> tis rectarum CL, CM, ſit æquale quadratum rectæ LM: </s> <s xml:id="echoid-s16358" xml:space="preserve">erit quo que quadratum <lb/>rectæ LM, æquale quadrato rectæ, qua minus latus quæſitum minorem exceſ-<lb/>ſum CE, ſuperat. </s> <s xml:id="echoid-s16359" xml:space="preserve">Eſt ergo LM, exceſſus minoris lateris quæſiti ſupra minorem <lb/>exceſſum CE. </s> <s xml:id="echoid-s16360" xml:space="preserve">Ideo que recta ex LM, CE, conflata erit minus latus quæſitum: <lb/></s> <s xml:id="echoid-s16361" xml:space="preserve">cui ſi addatur FC, differentia exceſſuum, fiet maius latus quæſitum: </s> <s xml:id="echoid-s16362" xml:space="preserve">cui ſi tan-<lb/>dem minor exceſſus C E, adijciatur, conflabitur diameter quæſita. </s> <s xml:id="echoid-s16363" xml:space="preserve">quæ omnia <lb/>demonſtranda erant.</s> <s xml:id="echoid-s16364" xml:space="preserve"/> </p> <div xml:id="echoid-div1003" type="float" level="2" n="1"> <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/xxxxxxxx/figures/378-01"/> </figure> <note symbol="a" position="left" xlink:label="note-378-01" xlink:href="note-378-01a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div1005" type="section" level="1" n="359"> <head xml:id="echoid-head386" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s16365" xml:space="preserve"><emph style="sc">Itaqve</emph> recta LM, cuius quadratum æquale eſt rectangulo FI, ſub FC, dif-<lb/>ferentia exceſſuum, & </s> <s xml:id="echoid-s16366" xml:space="preserve">dupla minoris exceſſus CE, comprehenſo vna cum du-<lb/>plo quadrati exceſſus minoris CE, addita minori exceſſui CE, efficit minus latus <lb/>quæſitum, &</s> <s xml:id="echoid-s16367" xml:space="preserve">c.</s> <s xml:id="echoid-s16368" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16369" xml:space="preserve"><emph style="sc">Immo</emph> quia quadratum rectæ, CL, rectangulo FI, ſub FC, differentia exceſ-<lb/>ſuum, & </s> <s xml:id="echoid-s16370" xml:space="preserve">C K, duplo minoris exceſſus C E, comprehenſo æquale eſt, vt in de-<lb/>monſtratione dictum eſt; </s> <s xml:id="echoid-s16371" xml:space="preserve">& </s> <s xml:id="echoid-s16372" xml:space="preserve">rectangulum CP, duplum eſt quadrati exceſſus mi-<lb/>noris CE, hoc eſt, quadrato rectæ CM, æquale: </s> <s xml:id="echoid-s16373" xml:space="preserve">erit quadratũ rectæ LM, toti re-<lb/>ctãgulo FP, ſub maiori exceſſu FE, & </s> <s xml:id="echoid-s16374" xml:space="preserve">EP, dupla minoris exceſſus CE, contento <pb o="351" file="379" n="379" rhead="LIBER OCTAVVS."/> æquale; </s> <s xml:id="echoid-s16375" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ideoque L M, media proportionalis eritinter maiorem exceſſum, ac <anchor type="note" xlink:label="note-379-01a" xlink:href="note-379-01"/> duplum minoris exceſſus. </s> <s xml:id="echoid-s16376" xml:space="preserve">Quo circaſi inter maiorem exceſſum, & </s> <s xml:id="echoid-s16377" xml:space="preserve">duplum mi-<lb/>noris exceſſus, ſumatur media proportionalis LM, habebitur rurſus differentia <lb/>inter minus latus, & </s> <s xml:id="echoid-s16378" xml:space="preserve">minorem exceſſum, &</s> <s xml:id="echoid-s16379" xml:space="preserve">c,</s> </p> <div xml:id="echoid-div1005" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-379-01" xlink:href="note-379-01a" xml:space="preserve">17. ſexti.</note> </div> </div> <div xml:id="echoid-div1007" type="section" level="1" n="360"> <head xml:id="echoid-head387" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s16380" xml:space="preserve"><emph style="sc">Hoc</emph> problema, vna cum antecedente Theoremate in Gallia, vnde mihi <lb/>tranſmiſſum eſt, abingenioſo quodam Geometra demonſtratum fuit, cuius no-<lb/>men, ſi mihi eſſet cognitum, hic libenter aſſcriberem. </s> <s xml:id="echoid-s16381" xml:space="preserve">Idem tamen problemaad <lb/>finem lib. </s> <s xml:id="echoid-s16382" xml:space="preserve">2. </s> <s xml:id="echoid-s16383" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16384" xml:space="preserve">ex Marino Gheraldo Patritio Raguſino aliter quo que de-<lb/>monſtrauimus non infeliciter.</s> <s xml:id="echoid-s16385" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1008" type="section" level="1" n="361"> <head xml:id="echoid-head388" xml:space="preserve">PROBL. 9. PROPOS. 17.</head> <p> <s xml:id="echoid-s16386" xml:space="preserve">DATO exceſſu diametri rectanguli ſupra maius latus, & </s> <s xml:id="echoid-s16387" xml:space="preserve">exceſſu ma-<lb/>ioris lateris ſupra minus: </s> <s xml:id="echoid-s16388" xml:space="preserve">vtrumque latus, ac diametrum inuenire.</s> <s xml:id="echoid-s16389" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16390" xml:space="preserve"><emph style="sc">Qvoniam</emph>, vt in præcedenti problem. </s> <s xml:id="echoid-s16391" xml:space="preserve">dictum eſt, exceſſus maioris lateris <lb/>ſupra minus, æqualis eſt differentiæ inter exceſſus diametri ſupra vtrumque la-<lb/>tus: </s> <s xml:id="echoid-s16392" xml:space="preserve">fit vt exceſſus diametri ſupra maius latus, additus ad exceſſum maioris la-<lb/>teris ſupra minus, conficiat exceſſum diametri ſupra minus latus. </s> <s xml:id="echoid-s16393" xml:space="preserve">Quare cum <lb/>cogniti ſint exceſſus diametri ſupra vtrumque latus, reliqua cognoſcentur, vt in <lb/>præmiſſo problemate traditum eſt.</s> <s xml:id="echoid-s16394" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1009" type="section" level="1" n="362"> <head xml:id="echoid-head389" xml:space="preserve">PROBL. 10. PROPOS. 18.</head> <p> <s xml:id="echoid-s16395" xml:space="preserve">SECTA linea recta vtcunque, adiungere ei verſus vtramuis partem li-<lb/>neam rectam, ita vt quadratum totius rectæ compoſitæ æquale ſit qua-<lb/>drato rectæ adiunctæ; </s> <s xml:id="echoid-s16396" xml:space="preserve">vna cum quadrato rectæ, quæ ex adiuncta, & </s> <s xml:id="echoid-s16397" xml:space="preserve"><lb/>proximo ſegmento prioris lineæ conflatur.</s> <s xml:id="echoid-s16398" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16399" xml:space="preserve"><emph style="sc">In</emph> figura propoſ. </s> <s xml:id="echoid-s16400" xml:space="preserve">15. </s> <s xml:id="echoid-s16401" xml:space="preserve">ſit recta EF, ſecta in C, vtcunque, oporteatque ei verſus <lb/>F, adiungere rectam, ita vt quadratum totius comp oſitæ ſit æquale, quadrato <lb/>adiunctæ, vna cum quadrato rectæ ex ſegmento F C, & </s> <s xml:id="echoid-s16402" xml:space="preserve">adiuncta compoſitæ. <lb/></s> <s xml:id="echoid-s16403" xml:space="preserve"> <anchor type="figure" xlink:label="fig-379-01a" xlink:href="fig-379-01"/> Statuantur EF, EC, exceſſus, quibus diameter alicuius re-<lb/>ctanguli vtrumque latus ſuperat. </s> <s xml:id="echoid-s16404" xml:space="preserve">Atque ex propoſ. </s> <s xml:id="echoid-s16405" xml:space="preserve">16. </s> <s xml:id="echoid-s16406" xml:space="preserve">in-<lb/>ueniatur minus latus BF. </s> <s xml:id="echoid-s16407" xml:space="preserve">Dico rectam BF, ipſi EF, adiun-<lb/>ctam efficere, quod proponitur. </s> <s xml:id="echoid-s16408" xml:space="preserve">Fiat enim rectangulum <lb/>AC, ſub BC, & </s> <s xml:id="echoid-s16409" xml:space="preserve">CD, ipſi BF, æquali comprehẽſum. </s> <s xml:id="echoid-s16410" xml:space="preserve">Et quia <lb/>FC, differentia exceſſuum addita minori lateri inuẽto BF, <lb/>facit maius latus, vt propoſ. </s> <s xml:id="echoid-s16411" xml:space="preserve">16. </s> <s xml:id="echoid-s16412" xml:space="preserve">dictũ eſt, erit BE, diametro <lb/>BD æqualis, quando quidem excedit minus latus BF, vel CD, recta EF, & </s> <s xml:id="echoid-s16413" xml:space="preserve">maius <lb/>recta EC. </s> <s xml:id="echoid-s16414" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam verò quadratum rectæ BE, hoc eſt, diametri BD, æquale <anchor type="note" xlink:label="note-379-02a" xlink:href="note-379-02"/> eſt quadrato rectæ CD, id eſt, adiunctæ BF, vna cum quadrato rectæ B C, com-<lb/>poſitæ ex adiuncta BF, & </s> <s xml:id="echoid-s16415" xml:space="preserve">proximo ſegmento F C, liquidò conſtatid, quod pro-<lb/>ponitur.</s> <s xml:id="echoid-s16416" xml:space="preserve"/> </p> <div xml:id="echoid-div1009" 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/xxxxxxxx/figures/379-01"/> </figure> <note symbol="b" position="right" xlink:label="note-379-02" xlink:href="note-379-02a" xml:space="preserve">47. primi.</note> </div> <pb o="352" file="380" n="380" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1011" type="section" level="1" n="363"> <head xml:id="echoid-head390" xml:space="preserve">PROBL. 11. PROPOS. 19.</head> <p> <s xml:id="echoid-s16417" xml:space="preserve">DATIS duabus rectis inæqualibus, quarum maior diametrum qua-<lb/>drati ex minore deſcripti non ſuperet: </s> <s xml:id="echoid-s16418" xml:space="preserve">maiorem ita ſecare in duas <lb/>partes inæquales, vt earum quadrata ſimul ſumpta quadrato minoris <lb/>lineæ ſint æqualia,</s> </p> <p> <s xml:id="echoid-s16419" xml:space="preserve"><emph style="sc">Sint</emph> datæ duæ rectæ AB, maior, & </s> <s xml:id="echoid-s16420" xml:space="preserve">AC, minor, ita vt AB, non ſit maior dia-<lb/>metro quadrati ex AC, deſcripti. </s> <s xml:id="echoid-s16421" xml:space="preserve">Erigatur perpendicularis AD, maiori AB, æ-<lb/>qualis: </s> <s xml:id="echoid-s16422" xml:space="preserve">Et ducta recta B D, ſecetur bifariam in E, iungatur que recta AE, <anchor type="note" xlink:href="" symbol="a"/> quæ <anchor type="note" xlink:label="note-380-01a" xlink:href="note-380-01"/> ad B D, perpendicularis erit: </s> <s xml:id="echoid-s16423" xml:space="preserve">diuidetque angulum rectum A, bifariam in duos <lb/>ſemirectos: </s> <s xml:id="echoid-s16424" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Suntautem & </s> <s xml:id="echoid-s16425" xml:space="preserve">B, D, ſemirecti. </s> <s xml:id="echoid-s16426" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur latera E A, E B, æqualia <anchor type="note" xlink:label="note-380-02a" xlink:href="note-380-02"/> ſunt; </s> <s xml:id="echoid-s16427" xml:space="preserve">ac proinde AB, diameter erit quadrati rectæ AE. </s> <s xml:id="echoid-s16428" xml:space="preserve">Et quoniam A B, poni-<lb/> <anchor type="note" xlink:label="note-380-03a" xlink:href="note-380-03"/> tur non maior diametro quadrati minoris A C, non erit A C, minor quam A E, <lb/>ſed vel maior, vel æqualis. </s> <s xml:id="echoid-s16429" xml:space="preserve">Si namque minor eſſet A C, quam A E, ſumpta ipſi <lb/>æquali AL, ductaque LM, ipſi EB, parallela, eſſet AM, diameter quadratimino-<lb/>ris A L, ideoque maior AB, ſuperaret diametrum quadrati ex minore deſcripti. <lb/></s> <s xml:id="echoid-s16430" xml:space="preserve">quod non ponitur. </s> <s xml:id="echoid-s16431" xml:space="preserve">Sit ergo primum A C, maior quam A E, productaque AE, <lb/>vt A F, ipſi A C, ſit æqualis, deſcribatur ex A, per <lb/> <anchor type="figure" xlink:label="fig-380-01a" xlink:href="fig-380-01"/> C, F, circulus ſecans BD, in H, N, demittaturque <lb/>HI, ad AB, perpendicularis. </s> <s xml:id="echoid-s16432" xml:space="preserve">Dico maiorem AB, ita <lb/>eſſe ſectam in I, vt quadrata rectarum AI, IB, æqua-<lb/>lia ſint quadrato minoris A O. </s> <s xml:id="echoid-s16433" xml:space="preserve">Quoniam enim an-<lb/>gulus I, rectus eſt, & </s> <s xml:id="echoid-s16434" xml:space="preserve">B, ſemirectus: </s> <s xml:id="echoid-s16435" xml:space="preserve">erit quoque H, <lb/>in triangulo B H I, ſemirectus; </s> <s xml:id="echoid-s16436" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> ideoque late- <anchor type="note" xlink:label="note-380-04a" xlink:href="note-380-04"/> ra B I, H I, æqualia erunt. </s> <s xml:id="echoid-s16437" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Cum ergo, ducta re- <anchor type="note" xlink:label="note-380-05a" xlink:href="note-380-05"/> cta A H, quadrata rectarum A I, I H, quadrato re-<lb/>ctæ A H, æqualia ſint; </s> <s xml:id="echoid-s16438" xml:space="preserve">erunt quoque quadrata ſe-<lb/>gmentorum A I, I B, æqualia quadrato rectæ A H, <lb/>hoc eſt, quadrato rectæ AC. </s> <s xml:id="echoid-s16439" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s16440" xml:space="preserve"/> </p> <div xml:id="echoid-div1011" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-380-01" xlink:href="note-380-01a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note symbol="b" position="left" xlink:label="note-380-02" xlink:href="note-380-02a" xml:space="preserve">2. coroll. <lb/>32. primi.</note> <note symbol="c" position="left" xlink:label="note-380-03" xlink:href="note-380-03a" xml:space="preserve">6. primi.</note> <figure xlink:label="fig-380-01" xlink:href="fig-380-01a"> <image file="380-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/380-01"/> </figure> <note symbol="d" position="left" xlink:label="note-380-04" xlink:href="note-380-04a" xml:space="preserve">6. primi.</note> <note symbol="e" position="left" xlink:label="note-380-05" xlink:href="note-380-05a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s16441" xml:space="preserve"><emph style="sc">Sit</emph> deinde minor linea data A K, æqualis ipſi <lb/>AE, ita vt maior AB, diametro quadrati ex minore <lb/>A E, deſcripti æqualis ſit. </s> <s xml:id="echoid-s16442" xml:space="preserve">Demittatur perpendicu-<lb/>laris EG, <anchor type="note" xlink:href="" symbol="f"/> quæ & </s> <s xml:id="echoid-s16443" xml:space="preserve">baſem AB, & </s> <s xml:id="echoid-s16444" xml:space="preserve">angulum rectum E, diuidet bifariam in ſemire- <anchor type="note" xlink:label="note-380-06a" xlink:href="note-380-06"/> ctos. </s> <s xml:id="echoid-s16445" xml:space="preserve">Dico AB, ita eſſe ſectam in G, vt quadrata ſegmentorum æqualium A G, <lb/>GB, æqualia ſint quadrato minoris AK, vel AE. </s> <s xml:id="echoid-s16446" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Erunt enim rurſus latera EG, <anchor type="note" xlink:label="note-380-07a" xlink:href="note-380-07"/> GB, æqualia, ob ſemirectos angulos æquales B, & </s> <s xml:id="echoid-s16447" xml:space="preserve">E, in triangulo BEG. </s> <s xml:id="echoid-s16448" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Cum <anchor type="note" xlink:label="note-380-08a" xlink:href="note-380-08"/> ergo quadrata rectarum AG, GE, æqualia ſint quadrato rectæ AE, erunt quo-<lb/>que quadrata ſegmentorum AG, GB, æqualia quadrato AE, vel minoris lineæ <lb/>A K. </s> <s xml:id="echoid-s16449" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s16450" xml:space="preserve"/> </p> <div xml:id="echoid-div1012" type="float" level="2" n="2"> <note symbol="f" position="left" xlink:label="note-380-06" xlink:href="note-380-06a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> <note symbol="g" position="left" xlink:label="note-380-07" xlink:href="note-380-07a" xml:space="preserve">6. primi.</note> <note symbol="h" position="left" xlink:label="note-380-08" xlink:href="note-380-08a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s16451" xml:space="preserve"><emph style="sc">Si</emph> ex altera ſectione N, demittatur perpendicularis NO, erit AB, ſecta in O, <lb/>vt in I, ita vt etiam quadrata ſegmentorum BO, OA, æqualia ſint quadrato mi-<lb/>noris AC, vel AH. </s> <s xml:id="echoid-s16452" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Cum enim rectæ BD, BA, ſectæ ſint proportionaliter in H, <anchor type="note" xlink:label="note-380-09a" xlink:href="note-380-09"/> I, & </s> <s xml:id="echoid-s16453" xml:space="preserve">N, O, ſintque æquales B H, N D; </s> <s xml:id="echoid-s16454" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> (propterea quod perpendicularis AE, <anchor type="note" xlink:label="note-380-10a" xlink:href="note-380-10"/> ſecat H N, bifariam. </s> <s xml:id="echoid-s16455" xml:space="preserve">Ablatis ergo æqualibus E H, E N, ex æqualibus E B, E D, <pb o="353" file="381" n="381" rhead="LIBER OCTAVVS."/> reliquæ BH, DN, æquales erunt) erunt quo que BI, AO, æquales; </s> <s xml:id="echoid-s16456" xml:space="preserve">necnon <lb/>BO, AC, &</s> <s xml:id="echoid-s16457" xml:space="preserve">c.</s> <s xml:id="echoid-s16458" xml:space="preserve"/> </p> <div xml:id="echoid-div1013" type="float" level="2" n="3"> <note symbol="i" position="left" xlink:label="note-380-09" xlink:href="note-380-09a" xml:space="preserve">2. ſexti.</note> <note symbol="k" position="left" xlink:label="note-380-10" xlink:href="note-380-10a" xml:space="preserve">3. tertii.</note> </div> </div> <div xml:id="echoid-div1015" type="section" level="1" n="364"> <head xml:id="echoid-head391" xml:space="preserve">PROBL. 12. PROPOS. 20.</head> <p> <s xml:id="echoid-s16459" xml:space="preserve">DATA chorda alicuius arcus, vna cum perpendiculari, quæ ex medio <lb/>puncto chordæ ad arcum vſque educitur: </s> <s xml:id="echoid-s16460" xml:space="preserve">quot gradus, vel palmos <lb/>tam arcus, quam ſemidiameter circuli complectitur, inuenire.</s> <s xml:id="echoid-s16461" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16462" xml:space="preserve"><emph style="sc">Sit</emph> data chorda AB, palmorum 74. </s> <s xml:id="echoid-s16463" xml:space="preserve">& </s> <s xml:id="echoid-s16464" xml:space="preserve">perpendicularis CD, ex medio pun-<lb/>cto C, educta palmorum 10. </s> <s xml:id="echoid-s16465" xml:space="preserve">Iuncta recta AD; </s> <s xml:id="echoid-s16466" xml:space="preserve">quoniam in triangulo rectan-<lb/>gulo ACD, latera AC, CD, nota ſunt, quod CD, ſit 10. </s> <s xml:id="echoid-s16467" xml:space="preserve">& </s> <s xml:id="echoid-s16468" xml:space="preserve">AC, 37. </s> <s xml:id="echoid-s16469" xml:space="preserve">ſemiſsis nimi-<lb/>rum chordæ AB, 74. </s> <s xml:id="echoid-s16470" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Si fiat vt CD, 10. </s> <s xml:id="echoid-s16471" xml:space="preserve">ad ſinum totum <anchor type="figure" xlink:label="fig-381-01a" xlink:href="fig-381-01"/> <anchor type="note" xlink:label="note-381-01a" xlink:href="note-381-01"/> 100000. </s> <s xml:id="echoid-s16472" xml:space="preserve">ita AC, 37. </s> <s xml:id="echoid-s16473" xml:space="preserve">ad aliud; </s> <s xml:id="echoid-s16474" xml:space="preserve">inuenietur AC, 370000, tangẽs <lb/>anguli ADC; </s> <s xml:id="echoid-s16475" xml:space="preserve">ac proinde ipſe angulus erit gr. </s> <s xml:id="echoid-s16476" xml:space="preserve">74. </s> <s xml:id="echoid-s16477" xml:space="preserve">min. </s> <s xml:id="echoid-s16478" xml:space="preserve">53. </s> <s xml:id="echoid-s16479" xml:space="preserve">fe-<lb/>re. </s> <s xml:id="echoid-s16480" xml:space="preserve">Et ſi concipiatur duci AE, ad centrum E, erit quo que an-<lb/>gulus DAE, gr. </s> <s xml:id="echoid-s16481" xml:space="preserve">74. </s> <s xml:id="echoid-s16482" xml:space="preserve">min. </s> <s xml:id="echoid-s16483" xml:space="preserve">53. </s> <s xml:id="echoid-s16484" xml:space="preserve">fere; </s> <s xml:id="echoid-s16485" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> quippe cum anguli ADE, <anchor type="note" xlink:label="note-381-02a" xlink:href="note-381-02"/> DAE, æquales ſint. </s> <s xml:id="echoid-s16486" xml:space="preserve">Quod ſi vtriuſque ſumma gr. </s> <s xml:id="echoid-s16487" xml:space="preserve">149. </s> <s xml:id="echoid-s16488" xml:space="preserve">min. <lb/></s> <s xml:id="echoid-s16489" xml:space="preserve">46. </s> <s xml:id="echoid-s16490" xml:space="preserve">fere, detrahatur ex duobusrectis, hoc eſt, ex gr. </s> <s xml:id="echoid-s16491" xml:space="preserve">180. </s> <s xml:id="echoid-s16492" xml:space="preserve">re-<lb/>liquus fiet tertius angulus E, in centro gr. </s> <s xml:id="echoid-s16493" xml:space="preserve">30. </s> <s xml:id="echoid-s16494" xml:space="preserve">min. </s> <s xml:id="echoid-s16495" xml:space="preserve">14. </s> <s xml:id="echoid-s16496" xml:space="preserve">fere, ac <lb/>totidem grad. </s> <s xml:id="echoid-s16497" xml:space="preserve">erit arcus AD; </s> <s xml:id="echoid-s16498" xml:space="preserve">ideoque eius duplus ADB, grad. </s> <s xml:id="echoid-s16499" xml:space="preserve">60. </s> <s xml:id="echoid-s16500" xml:space="preserve">min. </s> <s xml:id="echoid-s16501" xml:space="preserve">28. </s> <s xml:id="echoid-s16502" xml:space="preserve"><lb/>fermè.</s> <s xml:id="echoid-s16503" xml:space="preserve"/> </p> <div xml:id="echoid-div1015" type="float" level="2" n="1"> <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/xxxxxxxx/figures/381-01"/> </figure> <note symbol="a" position="right" xlink:label="note-381-01" xlink:href="note-381-01a" xml:space="preserve">8. triang. <lb/>rectil.</note> <note symbol="b" position="right" xlink:label="note-381-02" xlink:href="note-381-02a" xml:space="preserve">5. primi.</note> </div> <p> <s xml:id="echoid-s16504" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/>Quia verò quadratum ex AC, æquale eſt rectangulo ſub CD, & </s> <s xml:id="echoid-s16505" xml:space="preserve">reliqua <anchor type="note" xlink:label="note-381-03a" xlink:href="note-381-03"/> parte diametri: </s> <s xml:id="echoid-s16506" xml:space="preserve">ſi AC, 37, palm. </s> <s xml:id="echoid-s16507" xml:space="preserve">ducatur in ſe, & </s> <s xml:id="echoid-s16508" xml:space="preserve">productus numerus 1369. </s> <s xml:id="echoid-s16509" xml:space="preserve">diui-<lb/>datur per CD, palm. </s> <s xml:id="echoid-s16510" xml:space="preserve">10. </s> <s xml:id="echoid-s16511" xml:space="preserve">prodibit reliqua pars diametri palm. </s> <s xml:id="echoid-s16512" xml:space="preserve">136 {9/10}. </s> <s xml:id="echoid-s16513" xml:space="preserve">ac proinde <lb/>addita CD, palm. </s> <s xml:id="echoid-s16514" xml:space="preserve">10. </s> <s xml:id="echoid-s16515" xml:space="preserve">tota diameter erit palm. </s> <s xml:id="echoid-s16516" xml:space="preserve">146 {9/10}. </s> <s xml:id="echoid-s16517" xml:space="preserve">& </s> <s xml:id="echoid-s16518" xml:space="preserve">ſemidiameter palm. <lb/></s> <s xml:id="echoid-s16519" xml:space="preserve">73. </s> <s xml:id="echoid-s16520" xml:space="preserve">{9/20}. </s> <s xml:id="echoid-s16521" xml:space="preserve">Si igitur fiat vt 7. </s> <s xml:id="echoid-s16522" xml:space="preserve">ad 22. </s> <s xml:id="echoid-s16523" xml:space="preserve">ita 146 {9/10}. </s> <s xml:id="echoid-s16524" xml:space="preserve">palmi ad aliud; </s> <s xml:id="echoid-s16525" xml:space="preserve">reperietur per 1. </s> <s xml:id="echoid-s16526" xml:space="preserve">re-<lb/>gulam Num. </s> <s xml:id="echoid-s16527" xml:space="preserve">2. </s> <s xml:id="echoid-s16528" xml:space="preserve">cap. </s> <s xml:id="echoid-s16529" xml:space="preserve">7. </s> <s xml:id="echoid-s16530" xml:space="preserve">lib. </s> <s xml:id="echoid-s16531" xml:space="preserve">4. </s> <s xml:id="echoid-s16532" xml:space="preserve">huius, circumferentia circuli palm. </s> <s xml:id="echoid-s16533" xml:space="preserve">430 {9/35}. </s> <s xml:id="echoid-s16534" xml:space="preserve">Ergo <lb/>ſi rurſus fiat, vt tota circumferentia grad. </s> <s xml:id="echoid-s16535" xml:space="preserve">360. </s> <s xml:id="echoid-s16536" xml:space="preserve">ad palmos 430 {9/35}. </s> <s xml:id="echoid-s16537" xml:space="preserve">ita arcus ADB, <lb/>grad. </s> <s xml:id="echoid-s16538" xml:space="preserve">60. </s> <s xml:id="echoid-s16539" xml:space="preserve">min. </s> <s xml:id="echoid-s16540" xml:space="preserve">28. </s> <s xml:id="echoid-s16541" xml:space="preserve">ad aliud, inuenietur hic arcus palm. </s> <s xml:id="echoid-s16542" xml:space="preserve">72 {54513/189000}. </s> <s xml:id="echoid-s16543" xml:space="preserve">hoc eſt, 72 {10/39}. </s> <s xml:id="echoid-s16544" xml:space="preserve"><lb/>paulò amplius.</s> <s xml:id="echoid-s16545" xml:space="preserve"/> </p> <div xml:id="echoid-div1016" type="float" level="2" n="2"> <note symbol="c" position="right" xlink:label="note-381-03" xlink:href="note-381-03a" xml:space="preserve">35. tertii.</note> </div> </div> <div xml:id="echoid-div1018" type="section" level="1" n="365"> <head xml:id="echoid-head392" xml:space="preserve">THEOR. 9. ROPOS. 21.</head> <p> <s xml:id="echoid-s16546" xml:space="preserve">IN omni triangulo quadratum maximi lateris minus eſt, quam duplum <lb/>ſummæ quadratorum ex reliquis duobus lateribus deſcriptorum.</s> <s xml:id="echoid-s16547" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16548" xml:space="preserve"><emph style="sc">In</emph> triangulo ABC, maximum latus ſit AC, <lb/> <anchor type="figure" xlink:label="fig-381-02a" xlink:href="fig-381-02"/> & </s> <s xml:id="echoid-s16549" xml:space="preserve">angulus oppoſitus B, obtuſus. </s> <s xml:id="echoid-s16550" xml:space="preserve">Si namque re-<lb/>ctus eſſet, vel acutus; </s> <s xml:id="echoid-s16551" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> eſſet quadratum rectæ <anchor type="note" xlink:label="note-381-04a" xlink:href="note-381-04"/> AC, vel æquale duobus quadratis rectarum <lb/>AB, BC; </s> <s xml:id="echoid-s16552" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> vel minus: </s> <s xml:id="echoid-s16553" xml:space="preserve">ac proinde multo <anchor type="note" xlink:label="note-381-05a" xlink:href="note-381-05"/> minus duplo ſummæ quadratorum AB, BC. <lb/></s> <s xml:id="echoid-s16554" xml:space="preserve">Ex maiore latere AC, dematur AD, recta æ-<lb/>qualis lateri AB. </s> <s xml:id="echoid-s16555" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Et quia duo@ latera AB, BC, <anchor type="note" xlink:label="note-381-06a" xlink:href="note-381-06"/> <pb o="354" file="382" n="382" rhead="GEOMETR. PRACT."/> maiora ſuntlatere AC; </s> <s xml:id="echoid-s16556" xml:space="preserve">erit reliqua CD, minor latere BC; </s> <s xml:id="echoid-s16557" xml:space="preserve">ac proinde duo qua-<lb/>drata AD, DC, minora duobus quadratis AB, BC. </s> <s xml:id="echoid-s16558" xml:space="preserve">Eſt autem quadratum AC, <lb/>minus duplo quadratorum AD, DC: </s> <s xml:id="echoid-s16559" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> propterea quod æquale eſt duobus <anchor type="note" xlink:label="note-382-01a" xlink:href="note-382-01"/> quadratis AD, DC, vna cum rectangulo bis ſub AD, DC; </s> <s xml:id="echoid-s16560" xml:space="preserve">quod quidem rectan-<lb/>gulum bis, minus eſt duobus quadratis AD, DC, ex Lemmate propoſ. </s> <s xml:id="echoid-s16561" xml:space="preserve">39. </s> <s xml:id="echoid-s16562" xml:space="preserve">lib. <lb/></s> <s xml:id="echoid-s16563" xml:space="preserve">10. </s> <s xml:id="echoid-s16564" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16565" xml:space="preserve">Multo ergo minus erit quadratum AC, duplo quadratorum AB, BC. </s> <s xml:id="echoid-s16566" xml:space="preserve"><lb/>quod erat demonſtrandum.</s> <s xml:id="echoid-s16567" xml:space="preserve"/> </p> <div xml:id="echoid-div1018" type="float" level="2" n="1"> <figure xlink:label="fig-381-02" xlink:href="fig-381-02a"> <image file="381-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/381-02"/> </figure> <note symbol="d" position="right" xlink:label="note-381-04" xlink:href="note-381-04a" xml:space="preserve">47. primi.</note> <note symbol="e" position="right" xlink:label="note-381-05" xlink:href="note-381-05a" xml:space="preserve">13. ſecund.</note> <note symbol="f" position="right" xlink:label="note-381-06" xlink:href="note-381-06a" xml:space="preserve">20. primi.</note> <note symbol="a" position="left" xlink:label="note-382-01" xlink:href="note-382-01a" xml:space="preserve">4. ſecundi.</note> </div> </div> <div xml:id="echoid-div1020" type="section" level="1" n="366"> <head xml:id="echoid-head393" xml:space="preserve">PROBL. 13. PROPOS. 22.</head> <p> <s xml:id="echoid-s16568" xml:space="preserve">DATIS tribus rectis vtcunque in plano non parallelis, niſi quando ex-<lb/>tremæ à media æqualiter diſtant, rectam lineam ducere, & </s> <s xml:id="echoid-s16569" xml:space="preserve">quidem <lb/>per datum punctum in media, ſi omnes tres in vno puncto conue-<lb/>niant; </s> <s xml:id="echoid-s16570" xml:space="preserve">ita vt eius ſegmenta inter mediam, & </s> <s xml:id="echoid-s16571" xml:space="preserve">extremas ſint inter ſe æ-<lb/>qualia, vel datam habeant proportionem.</s> <s xml:id="echoid-s16572" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16573" xml:space="preserve"><emph style="sc">Sint</emph> tres rectæ AB, CD, EF. </s> <s xml:id="echoid-s16574" xml:space="preserve">Ducatur vt cunque recta BF, ſecans omnes <lb/>tres. </s> <s xml:id="echoid-s16575" xml:space="preserve">qua ſecta bifariamin G, ſi quidem punctum G, cadet in mediam, factum <lb/>erit, quodiubetur. </s> <s xml:id="echoid-s16576" xml:space="preserve">Si verò G, cadet extra lineam mediam CD, agatur per G, <lb/>alteri extremarum, nimirum ipſi EF, (dummodo einon æquidiſtet media CD) <lb/>parallela AG, ſecans mediam in C. </s> <s xml:id="echoid-s16577" xml:space="preserve">Cum <lb/> <anchor type="figure" xlink:label="fig-382-01a" xlink:href="fig-382-01"/> enim CD, ponatur non æquidiſtareipſi <lb/>EF, ſecabit vtique productam EF, & </s> <s xml:id="echoid-s16578" xml:space="preserve">pro-<lb/>inde eius quoque parallelam AG. </s> <s xml:id="echoid-s16579" xml:space="preserve">Quod <lb/>ſi CD, ipſi EF, æquidiſtaret, ducenda eſſet <lb/>per G, ipſi AB, parallela. </s> <s xml:id="echoid-s16580" xml:space="preserve">Poſtremò ex B, <lb/>puncto ductæ rectæ BF, ſumpto in extre-<lb/>ma AB, cui non æquidiſtat AG, ducatur <lb/>per C, vbi parallela AG, mediam CD, ſe-<lb/>cat, recta BC, ſecans EF, in E. </s> <s xml:id="echoid-s16581" xml:space="preserve">Dico re-<lb/>ctas BC, CE, æquales eſſe. </s> <s xml:id="echoid-s16582" xml:space="preserve">Cum enim in <lb/>triangulo BEF, recta CG, ſit baſi EF, parallela: </s> <s xml:id="echoid-s16583" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit vt BG, ad GF, ita BC, ad <anchor type="note" xlink:label="note-382-02a" xlink:href="note-382-02"/> ad CE; </s> <s xml:id="echoid-s16584" xml:space="preserve">Sed BG, ipſi GF, per conſtructionem, æqualis eſt. </s> <s xml:id="echoid-s16585" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s16586" xml:space="preserve">BC, ipſi CE, <lb/>æqualis erit.</s> <s xml:id="echoid-s16587" xml:space="preserve"/> </p> <div xml:id="echoid-div1020" 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/xxxxxxxx/figures/382-01"/> </figure> <note symbol="b" position="left" xlink:label="note-382-02" xlink:href="note-382-02a" xml:space="preserve">2. ſexti.</note> </div> <p> <s xml:id="echoid-s16588" xml:space="preserve"><emph style="sc">Qvod</emph> ſi per G, agatur ipſi AB, parallela GI, ſecans mediam CD, in I; </s> <s xml:id="echoid-s16589" xml:space="preserve">erit <lb/>ducta FIK, ſecta quoque bifariam in I: </s> <s xml:id="echoid-s16590" xml:space="preserve">propterea quod in triangulo FBK, recta <lb/>GI, baſi BK, æquidiſtat; </s> <s xml:id="echoid-s16591" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ideoque ſicut BF, in G, diuiſa eſt bifariam, ita quo- <anchor type="note" xlink:label="note-382-03a" xlink:href="note-382-03"/> que FK, in I, bifariam ſecabitur.</s> <s xml:id="echoid-s16592" xml:space="preserve"/> </p> <div xml:id="echoid-div1021" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-382-03" xlink:href="note-382-03a" xml:space="preserve">2. ſexti.</note> </div> <p> <s xml:id="echoid-s16593" xml:space="preserve"><emph style="sc">Ex</emph> his patet, cur media linea non debeat vtrique extremæ æquidiſtare, ſed <lb/>vel neutri, vel alteri tantum. </s> <s xml:id="echoid-s16594" xml:space="preserve">Nam ſi vtrique æquidiſtaret, <anchor type="note" xlink:href="" symbol="d"/> recta per G, ducta <anchor type="note" xlink:label="note-382-04a" xlink:href="note-382-04"/> parallela alterutra extremarum, æquidiſtaret quoque mediæ, ac proinde eam <lb/>non ſecaret. </s> <s xml:id="echoid-s16595" xml:space="preserve">Quod ſi neutri æquidiſtet, duci poterit per G, parallela vtrilibet <lb/>extremarum. </s> <s xml:id="echoid-s16596" xml:space="preserve">Si verò vni extremarum æquidiſtet, ducenda erit per G, alteripa-<lb/>rallela.</s> <s xml:id="echoid-s16597" xml:space="preserve"/> </p> <div xml:id="echoid-div1022" type="float" level="2" n="3"> <note symbol="d" position="left" xlink:label="note-382-04" xlink:href="note-382-04a" xml:space="preserve">30. primi.</note> </div> <p> <s xml:id="echoid-s16598" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi tres datæ rectæ coeant in eodem puncto productæ, vt in H, <lb/>facilius erit problema, etiamſi in media linea detur punctum C, per quod duci <pb o="355" file="383" n="383" rhead="LIBER OCTAVVS."/> debeat linea. </s> <s xml:id="echoid-s16599" xml:space="preserve">Ducta enim per datum punctum C, in media, alterutri extrema-<lb/>rum, vt ipſi HF, parallela CA, ſecante alte-<lb/> <anchor type="figure" xlink:label="fig-383-01a" xlink:href="fig-383-01"/> ram extremam in A; </s> <s xml:id="echoid-s16600" xml:space="preserve">& </s> <s xml:id="echoid-s16601" xml:space="preserve">ipſi HA, æqualis <lb/>capiatur AB, ſecabitur ducta BCE, in C, <lb/>bifariã: </s> <s xml:id="echoid-s16602" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quippe cũ ſit BG, ad CE, vt recta <anchor type="note" xlink:label="note-383-01a" xlink:href="note-383-01"/> BA, ad rectam AH, &</s> <s xml:id="echoid-s16603" xml:space="preserve">c. </s> <s xml:id="echoid-s16604" xml:space="preserve">Sic etiam ſi detur <lb/>punctum I, in media; </s> <s xml:id="echoid-s16605" xml:space="preserve">ducta per I, alteru-<lb/>triextremarum, vtipſi HB, parallela LI, <lb/>ſumptaque ipſi HL, æquali LF, ſeca-<lb/>bitur ducta FIK, in I, bifariam; </s> <s xml:id="echoid-s16606" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> propter- <anchor type="note" xlink:label="note-383-02a" xlink:href="note-383-02"/> ea quod eſt FI, ad IK, vt FL, ad LH. <lb/></s> <s xml:id="echoid-s16607" xml:space="preserve">&</s> <s xml:id="echoid-s16608" xml:space="preserve">c.</s> <s xml:id="echoid-s16609" xml:space="preserve"/> </p> <div xml:id="echoid-div1023" type="float" level="2" n="4"> <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/xxxxxxxx/figures/383-01"/> </figure> <note symbol="a" position="right" xlink:label="note-383-01" xlink:href="note-383-01a" xml:space="preserve">2. ſexti.</note> <note symbol="b" position="right" xlink:label="note-383-02" xlink:href="note-383-02a" xml:space="preserve">2. ſexti.</note> </div> <p> <s xml:id="echoid-s16610" xml:space="preserve"><emph style="sc">Eadem</emph> ratione ducemus lineam, quæ à media ſecetur in duas partes da-<lb/>tam habentes proportionem. </s> <s xml:id="echoid-s16611" xml:space="preserve">Si namque ducta BF, vtcunque ſecetur in datam <lb/>proportionẽ in G, & </s> <s xml:id="echoid-s16612" xml:space="preserve">reliqua fiant, vt ſupra; </s> <s xml:id="echoid-s16613" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit rurſus vt FG, ad GB, ita EC, <anchor type="note" xlink:label="note-383-03a" xlink:href="note-383-03"/> ad CB; </s> <s xml:id="echoid-s16614" xml:space="preserve">Vel vt BG, ad GF, ita BC, ad CE, prout videlicet proportio data eſt FG, <lb/>ad GB, vel BG, ad GF. </s> <s xml:id="echoid-s16615" xml:space="preserve">Sic etiam ſi tres rectæ datæ coeantin H; </s> <s xml:id="echoid-s16616" xml:space="preserve">ducta GA, ipſi <lb/>EF, parallela, fiatque HA, ad AB, vt antecedens datæ proportionis ad conſe-<lb/>quens: </s> <s xml:id="echoid-s16617" xml:space="preserve">ſi ducatur BCE, <anchor type="note" xlink:href="" symbol="d"/> erit EC, ad CB, vt HA, ad AB. </s> <s xml:id="echoid-s16618" xml:space="preserve">Et ſi fiat HA, ad AB, <anchor type="note" xlink:label="note-383-04a" xlink:href="note-383-04"/> vt conſequens datæ proportionis ad antecedens, erit rurſus BC, ad CE, vt BA, <lb/>antecedens ad conſequens AH.</s> <s xml:id="echoid-s16619" xml:space="preserve"/> </p> <div xml:id="echoid-div1024" type="float" level="2" n="5"> <note symbol="c" position="right" xlink:label="note-383-03" xlink:href="note-383-03a" xml:space="preserve">2. ſexti.</note> <note symbol="d" position="right" xlink:label="note-383-04" xlink:href="note-383-04a" xml:space="preserve">1. ſexti.</note> </div> </div> <div xml:id="echoid-div1026" type="section" level="1" n="367"> <head xml:id="echoid-head394" xml:space="preserve">PROBL. 14. PROPOS. 23.</head> <p> <s xml:id="echoid-s16620" xml:space="preserve">CVIVSLIBET lineæ, quamuis minimæ, exhibere multiplicem quam-<lb/>cunque, etiamſi circino ipſa non accipiatur. <lb/></s> <s xml:id="echoid-s16621" xml:space="preserve"> <anchor type="figure" xlink:label="fig-383-02a" xlink:href="fig-383-02"/> </s> </p> <div xml:id="echoid-div1026" type="float" level="2" n="1"> <figure xlink:label="fig-383-02" xlink:href="fig-383-02a"> <image file="383-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/383-02"/> </figure> </div> <p> <s xml:id="echoid-s16622" xml:space="preserve"><emph style="sc">Sit</emph> ſumenda verbigratia lineolæ AB, tripla. </s> <s xml:id="echoid-s16623" xml:space="preserve">Extenſo circino quantumli-<lb/>bet ex A, ad C, ſumantur ipſi AC, duæ æ-<lb/> <anchor type="figure" xlink:label="fig-383-03a" xlink:href="fig-383-03"/> quales CF, FD, vt tota AD, ipſius AC, ſit <lb/>tripla. </s> <s xml:id="echoid-s16624" xml:space="preserve">Item ipſi CB, ſumantur tres æ qua-<lb/> <anchor type="handwritten" xlink:label="hd-383-4a" xlink:href="hd-383-4"/> les DG, GH, HE. </s> <s xml:id="echoid-s16625" xml:space="preserve">Dico AE, eſſe ipſius AB, <lb/>triplam. </s> <s xml:id="echoid-s16626" xml:space="preserve">Quoniam enim tam multiplex eſt <lb/>AD, totius AC, quam multiplex eſt ablata <lb/>DE, ablatæ CB, nimirum tripla: </s> <s xml:id="echoid-s16627" xml:space="preserve">erit quoque ita multiplex reliqua EA, reliquæ <lb/>AB, vt tota totius, videlicet tripla. </s> <s xml:id="echoid-s16628" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s16629" xml:space="preserve"/> </p> <div xml:id="echoid-div1027" type="float" level="2" n="2"> <figure xlink:label="fig-383-03" xlink:href="fig-383-03a"> <image file="383-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/383-03"/> </figure> <handwritten xlink:label="hd-383-4" xlink:href="hd-383-4a"/> </div> </div> <div xml:id="echoid-div1029" type="section" level="1" n="368"> <head xml:id="echoid-head395" xml:space="preserve">PROBL. 15. PROPOS. 24.</head> <p> <s xml:id="echoid-s16630" xml:space="preserve">EX qualibet lineola, quamuis minima, auferre partem, vel partes impe-<lb/>ratas,</s> </p> <p> <s xml:id="echoid-s16631" xml:space="preserve"><emph style="sc">In</emph> figura præcedentis@ propoſ. </s> <s xml:id="echoid-s16632" xml:space="preserve">ſit ex lineola AB, detrahenda tertia pars. </s> <s xml:id="echoid-s16633" xml:space="preserve">Per <lb/>præcedentem ſumatur ipſius AB, tripla AE, quæ, ſi videbitur nimis exigua, mul-<lb/>tiplicetur, vt libet. </s> <s xml:id="echoid-s16634" xml:space="preserve">In exemplo quadruplicata eſt vſque ad D; </s> <s xml:id="echoid-s16635" xml:space="preserve">ita vt AD, ſit <lb/>ipſius AB, duodecupla: </s> <s xml:id="echoid-s16636" xml:space="preserve">(quod ſcietur, ſi numerus partium AE, nimirum 3 du- <pb o="356" file="384" n="384" rhead="GEOMETR. PRACT."/> catur in numerũ partium ipſius AD, ipſi AE, æqualium, nimirum in 4.) </s> <s xml:id="echoid-s16637" xml:space="preserve">ac pro-<lb/>inde ſi AB, diuiſa eſſe intelligatur in 3. </s> <s xml:id="echoid-s16638" xml:space="preserve">partes, tota AD, continebit tales partes <lb/>36. </s> <s xml:id="echoid-s16639" xml:space="preserve">Quo circa ſi in inſtrumento partiũ lib. </s> <s xml:id="echoid-s16640" xml:space="preserve">1. </s> <s xml:id="echoid-s16641" xml:space="preserve">cap. </s> <s xml:id="echoid-s16642" xml:space="preserve">1. </s> <s xml:id="echoid-s16643" xml:space="preserve">conſtructo interuallum AD, <lb/>ſtatuatur inter partes 36. </s> <s xml:id="echoid-s16644" xml:space="preserve">36. </s> <s xml:id="echoid-s16645" xml:space="preserve">Deinde interuallũ inter 35. </s> <s xml:id="echoid-s16646" xml:space="preserve">35. </s> <s xml:id="echoid-s16647" xml:space="preserve">(nimirum tota AD, <lb/>vna parte minus) tranferatur ex D, ad I, erit AI, tertia pars ipſius AB, hoc eſt, <lb/>pars trigeſima ſexta totius AD. </s> <s xml:id="echoid-s16648" xml:space="preserve">Cum ergo AB, contineat tres trigeſimas ſextas <lb/>partes totius AD, erit AG, ipſius AB, pars tertia. </s> <s xml:id="echoid-s16649" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s16650" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1030" type="section" level="1" n="369"> <head xml:id="echoid-head396" xml:space="preserve">PROBL. 16. PROPOS. 25.</head> <p> <s xml:id="echoid-s16651" xml:space="preserve">ANGVLVM datum rectilineum in tres æquales partes partiri.</s> <s xml:id="echoid-s16652" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16653" xml:space="preserve"><emph style="sc">Problema</emph> hoc veteres Geometras diu, multumque exagitauit, neque <lb/>ab vllo ad hanc vſque diem Geometrice eſt ſolutum. </s> <s xml:id="echoid-s16654" xml:space="preserve">Pappus Alexandrinus <lb/>inter alios illud ſoluere conatus eſt per deſcriptionem hyperboles. </s> <s xml:id="echoid-s16655" xml:space="preserve">Nos idem <lb/> <anchor type="figure" xlink:label="fig-384-01a" xlink:href="fig-384-01"/> abſoluemus per lineam Conchoideos, quam lib. </s> <s xml:id="echoid-s16656" xml:space="preserve">6. </s> <s xml:id="echoid-s16657" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s16658" xml:space="preserve">15. </s> <s xml:id="echoid-s16659" xml:space="preserve">huius ex Nico-<lb/>mede deſcripſimus, hoc modo. </s> <s xml:id="echoid-s16660" xml:space="preserve">Sit datus angulus acutus ABC: </s> <s xml:id="echoid-s16661" xml:space="preserve">Demiſſa au-<lb/>tem ex quouis puncto A, ad BC, perpendiculari AD, fumatur ipſius AB, dupla <lb/>DC; </s> <s xml:id="echoid-s16662" xml:space="preserve">Et polo B, interuallo autem DC, deſcribatur linea Conchoideos CE, ſe-<lb/>cans rectam AE, ipſi BC, ductam parallelam in E, ducatur que recta BE. </s> <s xml:id="echoid-s16663" xml:space="preserve">Di-<lb/>co angulum CBE, eſſe tertiam partem dati anguli ABC; </s> <s xml:id="echoid-s16664" xml:space="preserve">hoc eſt, angulum ABE, <lb/>duplum eſſe anguli CBE, adeò vt diuiſo angulo ABE, bifariam, totus angulus <lb/>ABC, ſectus ſit in tres partes æquales. </s> <s xml:id="echoid-s16665" xml:space="preserve">Quoniam enim ex deſo<unsure/>riptione Con-<lb/>choideos, recta GE, ipſi DC, æqualis eſt; </s> <s xml:id="echoid-s16666" xml:space="preserve">ac proinde ipſius AB, dupla: </s> <s xml:id="echoid-s16667" xml:space="preserve">ſi ſecetur <lb/>bifariamin F, erit vtra que ſemiſsis ipſi AB, æqualis. </s> <s xml:id="echoid-s16668" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quia verò circulus ex F, <anchor type="note" xlink:label="note-384-01a" xlink:href="note-384-01"/> circa GE, deſcriptus tranſit per angulum rectum GAE, erit quoque ducta FA, <lb/>vtrique ſemiſsi FE, FG, ideo que & </s> <s xml:id="echoid-s16669" xml:space="preserve">ipſi AB, æqualis. </s> <s xml:id="echoid-s16670" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Igitur tam anguli FAE, <anchor type="note" xlink:label="note-384-02a" xlink:href="note-384-02"/> FEA, quam AFB, ABF, æquales erunt. </s> <s xml:id="echoid-s16671" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Eſt autem externus AFB, duobus in- <anchor type="note" xlink:label="note-384-03a" xlink:href="note-384-03"/> ternis FAE, FEA. </s> <s xml:id="echoid-s16672" xml:space="preserve">æqualis: </s> <s xml:id="echoid-s16673" xml:space="preserve">ideoque ipſius FEA, duplus. </s> <s xml:id="echoid-s16674" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s16675" xml:space="preserve">ABF, eiuſ-<lb/> <anchor type="note" xlink:label="note-384-04a" xlink:href="note-384-04"/> dem FEA, <anchor type="note" xlink:href="" symbol="d"/> hoc eſt, alterni CBE, duplus erit.</s> <s xml:id="echoid-s16676" xml:space="preserve"/> </p> <div xml:id="echoid-div1030" 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/xxxxxxxx/figures/384-01"/> </figure> <note symbol="a" position="left" xlink:label="note-384-01" xlink:href="note-384-01a" xml:space="preserve">ſchol. 31. <lb/>tertii.</note> <note symbol="b" position="left" xlink:label="note-384-02" xlink:href="note-384-02a" xml:space="preserve">5. primi.</note> <note symbol="c" position="left" xlink:label="note-384-03" xlink:href="note-384-03a" xml:space="preserve">32. primi.</note> <note symbol="d" position="left" xlink:label="note-384-04" xlink:href="note-384-04a" xml:space="preserve">29. primi.</note> </div> <p> <s xml:id="echoid-s16677" xml:space="preserve"><emph style="sc">Si</emph> angulus datus rectus eſt, diuidetur in tres æquales angulos, vt in ſcholio <lb/>propoſ. </s> <s xml:id="echoid-s16678" xml:space="preserve">32. </s> <s xml:id="echoid-s16679" xml:space="preserve">lib. </s> <s xml:id="echoid-s16680" xml:space="preserve">1. </s> <s xml:id="echoid-s16681" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16682" xml:space="preserve">tradidimus.</s> <s xml:id="echoid-s16683" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16684" xml:space="preserve"><emph style="sc">Si</emph> verò eſt obtuſus, ſecabimus eum bifariam, & </s> <s xml:id="echoid-s16685" xml:space="preserve">ſemiſſem alterutram intres <lb/>partes æquales, vt docuimus hoc loco. </s> <s xml:id="echoid-s16686" xml:space="preserve">Nam duæ partes tertiæ illius ſemiſsis ef-<lb/>ficient propoſiti anguli obtuſi tertiam partem, vt perſpicuum eſt.</s> <s xml:id="echoid-s16687" xml:space="preserve"/> </p> <pb o="357" file="385" n="385" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1032" type="section" level="1" n="370"> <head xml:id="echoid-head397" xml:space="preserve">PROBL. 17. PROPOS. 26.</head> <p> <s xml:id="echoid-s16688" xml:space="preserve">SI per idem punctum diametri in rectangulo duæ lineæ ducantur late-<lb/>ribus parallelæ: </s> <s xml:id="echoid-s16689" xml:space="preserve">Erit rectangulum ſub ſegmentis diametri compre-<lb/>henſum æquale duobus rectangulis ſub ſegmentis duorum laterum <lb/>comprehenſis.</s> <s xml:id="echoid-s16690" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16691" xml:space="preserve"><emph style="sc">In</emph> rectangulo BD, per E, punctum diametri AC, ductæ ſint FG, HI, lateri-<lb/>bus parallelæ. </s> <s xml:id="echoid-s16692" xml:space="preserve">Dico rectangulum ſub AE, EC, æquale eſſe rectangulis ſub AF, <lb/> <anchor type="note" xlink:label="note-385-01a" xlink:href="note-385-01"/> FB, & </s> <s xml:id="echoid-s16693" xml:space="preserve">ſub BI, IC. </s> <s xml:id="echoid-s16694" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim quadratum ex AC, æquale eſt quadratis ex <anchor type="note" xlink:label="note-385-02a" xlink:href="note-385-02"/> AE, EC, vna cum rectangulo bis ſub AE, EC. </s> <s xml:id="echoid-s16695" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Suntautem quadrata ex AB, BC, quadrato ex AC, æqualia; </s> <s xml:id="echoid-s16696" xml:space="preserve">erunt quoque duo quadrata ex AE, EC, vna <lb/>cum rectangulo bis ſub AE, EC, æqualia quadratis ex <lb/> <anchor type="figure" xlink:label="fig-385-01a" xlink:href="fig-385-01"/> <anchor type="note" xlink:label="note-385-03a" xlink:href="note-385-03"/> AB, BC. </s> <s xml:id="echoid-s16697" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Sed quadratum ex AB, æquale eſt quadra- tis duobus ex AF, FB, vna cum rectangulo bis ſub AF, <lb/>FB; </s> <s xml:id="echoid-s16698" xml:space="preserve">Et quadratum ex BC, æquale eſt duobus quadra-<lb/>tis ex BI, IC, vna cum rectangulo bis ſub BI, IC. </s> <s xml:id="echoid-s16699" xml:space="preserve">Igi-<lb/>tur duo quadrata ex AE, EC, vna cum rectangulo bis <lb/>ſub AE, EC, æqualia erunt quatu or quadratis ex AF, <lb/>FB, BI, IC, vna cum rectangulis bis ſub AF, FB, & </s> <s xml:id="echoid-s16700" xml:space="preserve">ſub BI, IC. </s> <s xml:id="echoid-s16701" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Cum ergo qua- <anchor type="note" xlink:label="note-385-04a" xlink:href="note-385-04"/> dratum ex AE, quadratis ex AF, FE, hoc eſt, ex AF, BI: </s> <s xml:id="echoid-s16702" xml:space="preserve">& </s> <s xml:id="echoid-s16703" xml:space="preserve">quadratum ex EC, <lb/>quadratis ex EI, IC, hoc eſt ex FB, IC: </s> <s xml:id="echoid-s16704" xml:space="preserve">æquale ſit; </s> <s xml:id="echoid-s16705" xml:space="preserve">Erunt quatuor quadrata <lb/>ex AF, BI, FB, IC, vna cum rectangulo bis ſub AE, EC, æqualia quatuor qua-<lb/>dratis ex AF, FB, BI, IC, vna cum rectangulis bis ſub AF, FB, & </s> <s xml:id="echoid-s16706" xml:space="preserve">ſub BI, IC; </s> <s xml:id="echoid-s16707" xml:space="preserve">Ab-<lb/>latiſque vtrobique prædictis quatuor quadratis communibus, eritreliquum re-<lb/>ctangulum bis ſub AE, EC, reliquis rectangulis bis ſub AF, FB, & </s> <s xml:id="echoid-s16708" xml:space="preserve">ſub BI, IC, æ-<lb/>quale. </s> <s xml:id="echoid-s16709" xml:space="preserve">Ideo que rectangulum ſemel ſub AE, EC, rectangulis ſemel ſub AF. <lb/></s> <s xml:id="echoid-s16710" xml:space="preserve">FB, & </s> <s xml:id="echoid-s16711" xml:space="preserve">ſub BI, IC, erit æquale. </s> <s xml:id="echoid-s16712" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s16713" xml:space="preserve"/> </p> <div xml:id="echoid-div1032" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-385-01" xlink:href="note-385-01a" xml:space="preserve">4. ſecundi.</note> <note symbol="b" position="right" xlink:label="note-385-02" xlink:href="note-385-02a" xml:space="preserve">47. primi.</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/xxxxxxxx/figures/385-01"/> </figure> <note symbol="c" position="right" xlink:label="note-385-03" xlink:href="note-385-03a" xml:space="preserve">4. ſecundi.</note> <note symbol="d" position="right" xlink:label="note-385-04" xlink:href="note-385-04a" xml:space="preserve">47. primi.</note> </div> </div> <div xml:id="echoid-div1034" type="section" level="1" n="371"> <head xml:id="echoid-head398" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s16714" xml:space="preserve"><emph style="sc">Itaqve</emph> in quadrato, vt in poſteriori figura, rectangulum ſub ſegmentis <lb/>diametri AE, EC, comprehenſum æquale eſt duobus complementis DE, BE, <lb/>quippe cum complementa ſub ſegmentis laterum comprehendantur.</s> <s xml:id="echoid-s16715" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1035" type="section" level="1" n="372"> <head xml:id="echoid-head399" xml:space="preserve">PROBL. 18. PROPOS. 27.</head> <p> <s xml:id="echoid-s16716" xml:space="preserve">DATO centro Ellipſis in linea in infinitum producta, vna cum duo-<lb/>bus punctis ad eaſdem partes axis, vel centri, per quæ tranſire dicatur <lb/>Ellipſis: </s> <s xml:id="echoid-s16717" xml:space="preserve">vtrumque axis vtriuſque extremum inuenire.</s> <s xml:id="echoid-s16718" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16719" xml:space="preserve"><emph style="sc">Hoc</emph> problema conicum eſt, & </s> <s xml:id="echoid-s16720" xml:space="preserve">acutum. </s> <s xml:id="echoid-s16721" xml:space="preserve">Sit A, centrum, id eſt, punctum me-<lb/>dium alicuius Ellipſis in linea axis maioris BC, quantacunq;</s> <s xml:id="echoid-s16722" xml:space="preserve">: Et duo puncta in <lb/>eadem Ellipſi D, E, verſus eandem partem, hoc eſt, ſiue ſupra centrum A, ſiue <lb/>infra; </s> <s xml:id="echoid-s16723" xml:space="preserve">è quibus ad BC, perpendiculares ducantur DF, EG: </s> <s xml:id="echoid-s16724" xml:space="preserve">Eritque DF, minor <pb o="358" file="386" n="386" rhead="GEOMETR. PRACT."/> quam EG, quod perpendicularis remotior à centro minor ſemper ſit, quam pro-<lb/>pinquior. </s> <s xml:id="echoid-s16725" xml:space="preserve">Ducta igitur recta ED, ſecabit lineam axis in B. </s> <s xml:id="echoid-s16726" xml:space="preserve">Secta autem A B, bi-<lb/>fariam in H, deſcribatur ex H, circa AB, ſemicirculus AKB. </s> <s xml:id="echoid-s16727" xml:space="preserve">Diuiſa quoque F G, <lb/>inter ductas perpendiculares bifariam in I, ducatur IK, ad AB, perpendicularis <lb/>ſecans circumferentiam in K, & </s> <s xml:id="echoid-s16728" xml:space="preserve">rectam B E, in L. </s> <s xml:id="echoid-s16729" xml:space="preserve">Ducta quoque recta B K, ſe-<lb/> <anchor type="figure" xlink:label="fig-386-01a" xlink:href="fig-386-01"/> cante perpendiculares DF, EG, in M, N, iungantur re-<lb/>ctæ AM, AK, AN. </s> <s xml:id="echoid-s16730" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quoniam eſt, vt FI, ad IG, æqua- <anchor type="note" xlink:label="note-386-01a" xlink:href="note-386-01"/> lem, ita MK, ad KN, erit quoque MK, ipſi KN, æqua-<lb/>lis. </s> <s xml:id="echoid-s16731" xml:space="preserve">Igitur duo latera AK, KM, duobus lateribus AK, <lb/>KN, æqualia erunt. </s> <s xml:id="echoid-s16732" xml:space="preserve">Cum ergo & </s> <s xml:id="echoid-s16733" xml:space="preserve">angulos æquales <lb/>comprehendant, vtpoterectos, <anchor type="note" xlink:href="" symbol="b"/> quod angulus AKB, <anchor type="note" xlink:label="note-386-02a" xlink:href="note-386-02"/> in ſemicirculo rectus ſit; </s> <s xml:id="echoid-s16734" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt baſes AM, AN, æqua- <anchor type="note" xlink:label="note-386-03a" xlink:href="note-386-03"/> les. </s> <s xml:id="echoid-s16735" xml:space="preserve">Circulus igitur ex A, per N, deſcriptus tranſibit <lb/>per M, ſecabitque BC, in O, & </s> <s xml:id="echoid-s16736" xml:space="preserve">C. </s> <s xml:id="echoid-s16737" xml:space="preserve">Dico OC, eſſe axem <lb/>Ellipſis maiorem. </s> <s xml:id="echoid-s16738" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Cum enim ſit, vt MD, ad DF, ita <anchor type="note" xlink:label="note-386-04a" xlink:href="note-386-04"/> NE, ad EG, tranſibit neceſſariò Ellipſis, quæ per O, D, <lb/>C, deſcribitur, (poſſe autem Ellipſim deſcribi circa <lb/>O C, tanquam axem maiorem, per punctum D, con-<lb/>ſtat ex iis, quæ ad finem ſcholij propoſ. </s> <s xml:id="echoid-s16739" xml:space="preserve">8. </s> <s xml:id="echoid-s16740" xml:space="preserve">lib. </s> <s xml:id="echoid-s16741" xml:space="preserve">1. </s> <s xml:id="echoid-s16742" xml:space="preserve">Gno-<lb/>monices, & </s> <s xml:id="echoid-s16743" xml:space="preserve">in ſcholio Lemmatis 50. </s> <s xml:id="echoid-s16744" xml:space="preserve">lib. </s> <s xml:id="echoid-s16745" xml:space="preserve">1. </s> <s xml:id="echoid-s16746" xml:space="preserve">Aſtrolabij <lb/>ſcripſimus) per punctum E, ex ſcholio Lemmatis 51. <lb/></s> <s xml:id="echoid-s16747" xml:space="preserve">Aſtrolabij: </s> <s xml:id="echoid-s16748" xml:space="preserve">Acproinde Ellipſis per data puncta D, E, <lb/>circa centrum A, deſcripta tranſibit per O, C, ita vt ab Ellipſi per O, D, C, deſcri-<lb/>pta non differat. </s> <s xml:id="echoid-s16749" xml:space="preserve">Alioquin Ellipſis Ellipſim in 8. </s> <s xml:id="echoid-s16750" xml:space="preserve">punctis ſecaret, nimirum in D, <lb/>E, & </s> <s xml:id="echoid-s16751" xml:space="preserve">aliis duobus reſpondentibus ex altera parte axis: </s> <s xml:id="echoid-s16752" xml:space="preserve">deinde in aliis 4. </s> <s xml:id="echoid-s16753" xml:space="preserve">infra <lb/>centrum reſpondentibus. </s> <s xml:id="echoid-s16754" xml:space="preserve">quod eſt abſurdum <anchor type="note" xlink:href="" symbol="e"/> quippe cum Ellipſis Ellipſim in 4. </s> <s xml:id="echoid-s16755" xml:space="preserve">tantum punctis ſecet.</s> <s xml:id="echoid-s16756" xml:space="preserve"/> </p> <div xml:id="echoid-div1035" type="float" level="2" n="1"> <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/xxxxxxxx/figures/386-01"/> </figure> <note symbol="a" position="left" xlink:label="note-386-01" xlink:href="note-386-01a" xml:space="preserve">2. ſexti.</note> <note symbol="b" position="left" xlink:label="note-386-02" xlink:href="note-386-02a" xml:space="preserve">31. tertii.</note> <note symbol="c" position="left" xlink:label="note-386-03" xlink:href="note-386-03a" xml:space="preserve">4. primi.</note> <note symbol="d" position="left" xlink:label="note-386-04" xlink:href="note-386-04a" xml:space="preserve">ſchol 4.</note> </div> <note symbol="e" position="left" xml:space="preserve">25. quinti <lb/>Apollonii.</note> </div> <div xml:id="echoid-div1037" type="section" level="1" n="373"> <head xml:id="echoid-head400" xml:space="preserve">THEOR. 10. PROPOS. 28.</head> <p> <s xml:id="echoid-s16757" xml:space="preserve">SI in circuli diametro producta punctum ſumatur, ab eoque recta cir-<lb/>culum tangens ducatur; </s> <s xml:id="echoid-s16758" xml:space="preserve">à puncto autem contactus chorda ducatur <lb/>ad diametrum perpendicularis: </s> <s xml:id="echoid-s16759" xml:space="preserve">Recta ex eodem contactus puncto <lb/>ad vtrumlibet extremum diametri ducta diuidet angulum à tangen-<lb/>te, & </s> <s xml:id="echoid-s16760" xml:space="preserve">prædicta chorda perpendiculari comprehenſum bifariam. <lb/></s> <s xml:id="echoid-s16761" xml:space="preserve">Item ſi ab eodem puncto in diametro producta aſſumpto recta du-<lb/>catur circulum ſecans, & </s> <s xml:id="echoid-s16762" xml:space="preserve">ab alterutro ſectionis puncto ad interſectio-<lb/>nem diametri cum prædicta chorda perpendiculari recta iungatur: </s> <s xml:id="echoid-s16763" xml:space="preserve"><lb/>Recta ex eodem ſectionis puncto ad vtrumlibet diametri extremum <lb/>ducta ſecabit quoque angulum à linea ſecante, & </s> <s xml:id="echoid-s16764" xml:space="preserve">illa alia, quæ per in-<lb/>terſectionem diametri cum prædicta chorda perpendiculari ducitur, <lb/>bifariam.</s> <s xml:id="echoid-s16765" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16766" xml:space="preserve"><emph style="sc">In</emph> circulo ABCD, cuius centrum E, pro ducta ſit diameter AC, & </s> <s xml:id="echoid-s16767" xml:space="preserve">ex F, duca-<lb/>tur primum recta FH, tangens circulum in B, atque ex B, ducatur chorda B D, <pb o="359" file="387" n="387" rhead="LIBER OCTAVVS."/> ſecans diametrum in G, ad rectos angulos, iunganturque ad extrema diametri <lb/> <anchor type="figure" xlink:label="fig-387-01a" xlink:href="fig-387-01"/> rectæ BC, BA. </s> <s xml:id="echoid-s16768" xml:space="preserve">Dico tamangulos CBF, CBG, <lb/>quam (producta F B, ad H,) angulos ABH, <lb/>ABG, eſſe æquales. </s> <s xml:id="echoid-s16769" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam enim angulus <anchor type="note" xlink:label="note-387-01a" xlink:href="note-387-01"/> C B F, angulo B A C, in alterno ſegmento æ-<lb/>qualis eſt. </s> <s xml:id="echoid-s16770" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> & </s> <s xml:id="echoid-s16771" xml:space="preserve">angulus C B D, eidem angulo <anchor type="note" xlink:label="note-387-02a" xlink:href="note-387-02"/> B A C, æqualis, ob arcus æquales CB, CD; </s> <s xml:id="echoid-s16772" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> <anchor type="note" xlink:label="note-387-03a" xlink:href="note-387-03"/> (vel etiam angulus CBG, angulo BAC, æqua-<lb/>lis eſt; </s> <s xml:id="echoid-s16773" xml:space="preserve">quod BG, in triangulo rectangulo AB-<lb/>C, ad baſem AC, perpendicularis ſit) erunt an-<lb/>guli CBF, CBG, inter ſe quoque æquales. <lb/></s> <s xml:id="echoid-s16774" xml:space="preserve">Producta autem CB, ad I, ſi ex rectis angulis <lb/>ABC, ABI, tollantur æquales CBG, HBI, <anchor type="note" xlink:href="" symbol="d"/> <anchor type="note" xlink:label="note-387-04a" xlink:href="note-387-04"/> (cum enim CBF, æqualis ſit angulo H B I, ad <lb/>verticem: </s> <s xml:id="echoid-s16775" xml:space="preserve">& </s> <s xml:id="echoid-s16776" xml:space="preserve">angulus CBG, angulo CBF, o-<lb/>ſtenſus æqualis; </s> <s xml:id="echoid-s16777" xml:space="preserve">erit quo que angulus C B G, <lb/>angulo H B I, æqualis.) </s> <s xml:id="echoid-s16778" xml:space="preserve">erunt quo que reliqui <lb/>anguli ABG, ABH, inter ſe æquales. </s> <s xml:id="echoid-s16779" xml:space="preserve">quod eſt <lb/>primum.</s> <s xml:id="echoid-s16780" xml:space="preserve"/> </p> <div xml:id="echoid-div1037" 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/xxxxxxxx/figures/387-01"/> </figure> <note symbol="a" position="right" xlink:label="note-387-01" xlink:href="note-387-01a" xml:space="preserve">32. tertij.</note> <note symbol="b" position="right" xlink:label="note-387-02" xlink:href="note-387-02a" xml:space="preserve">27. tertij.</note> <note symbol="c" position="right" xlink:label="note-387-03" xlink:href="note-387-03a" xml:space="preserve">8. ſexti.</note> <note symbol="d" position="right" xlink:label="note-387-04" xlink:href="note-387-04a" xml:space="preserve">15. primi.</note> </div> <p> <s xml:id="echoid-s16781" xml:space="preserve"><emph style="sc">Deinde</emph> ducatur recta FN, ſecans circu-<lb/>lum in K, L, ductiſque rectis KGO, LGM, per <lb/>G, iungantur tam rectæ K C, K A, quam L C, <lb/>L A, ad extrema diametri. </s> <s xml:id="echoid-s16782" xml:space="preserve">Dico rurſus, tam angulos CLF, CLG, quam ALG, <lb/>ALN: </s> <s xml:id="echoid-s16783" xml:space="preserve">Item tam CKF, CKG, quam AKG, AKL, eſſe æquales. </s> <s xml:id="echoid-s16784" xml:space="preserve">Ductis enim ex <lb/>centro rectis EB, EK; </s> <s xml:id="echoid-s16785" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> erit angulus EBF, rectus: </s> <s xml:id="echoid-s16786" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Igitur erit FB, media propor- <anchor type="note" xlink:label="note-387-05a" xlink:href="note-387-05"/> tionalis inter EF, FG: </s> <s xml:id="echoid-s16787" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Ideoque rectangulum ſub EF, FG, quadrato ex FB, æ- <anchor type="note" xlink:label="note-387-06a" xlink:href="note-387-06"/> quale erit; </s> <s xml:id="echoid-s16788" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Eſt autem eidem quadrato æquale quo que rectangulum ſub LF, FK. </s> <s xml:id="echoid-s16789" xml:space="preserve">Igitur rectangulum ſub EF, FG, rectangulo ſub L F, F K, æquale erit: </s> <s xml:id="echoid-s16790" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Ac <anchor type="note" xlink:label="note-387-07a" xlink:href="note-387-07"/> pro inde erit vt EF, prima ad FK, ſecundam, ita LF, tertia ad FG, quartam. </s> <s xml:id="echoid-s16791" xml:space="preserve">Qua-<lb/> <anchor type="note" xlink:label="note-387-08a" xlink:href="note-387-08"/> re cum triangula EFK, LFG, habeant latera circa communem angulum F, pro-<lb/> <anchor type="note" xlink:label="note-387-09a" xlink:href="note-387-09"/> portionalia; </s> <s xml:id="echoid-s16792" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> erunt anguli FEK, FLG, homologis lateribus FK, FG, oppoſiti <anchor type="note" xlink:label="note-387-10a" xlink:href="note-387-10"/> æquales. </s> <s xml:id="echoid-s16793" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Eſt autem angulus FEK, in centro anguli CIK, ad circumferentiam, <anchor type="note" xlink:label="note-387-11a" xlink:href="note-387-11"/> (cum habeãt eandem baſem C K,) duplus. </s> <s xml:id="echoid-s16794" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s16795" xml:space="preserve">angulus FLG, eiuſdem an-<lb/>guli C L K, duplus erit; </s> <s xml:id="echoid-s16796" xml:space="preserve">ac proinde angulus F L G, ſectus erit bifariam à recta <lb/>LC, hoc eſt, anguli CLF, CLG, æquales erunt. </s> <s xml:id="echoid-s16797" xml:space="preserve">Producta autem CL, ad P, ſi ex <lb/>rectis angulis ALC, ALP, demantur æquales anguli CLG, P L N, (Cum enim <lb/>CLF, oſtenſus ſit æqualis angulo CLG, <anchor type="note" xlink:href="" symbol="m"/> & </s> <s xml:id="echoid-s16798" xml:space="preserve">CLF, & </s> <s xml:id="echoid-s16799" xml:space="preserve">æqualis ſitangulo PLN, <anchor type="note" xlink:label="note-387-12a" xlink:href="note-387-12"/> ad verticem, erit quoque CLG, eidem angulo PLN, æqualis.) </s> <s xml:id="echoid-s16800" xml:space="preserve">erunt quoq; </s> <s xml:id="echoid-s16801" xml:space="preserve">re-<lb/>liquianguli ALG, ALN, æquales. </s> <s xml:id="echoid-s16802" xml:space="preserve">Rurſus quia anguli CLK, CLM, oſtenſi ſunt <lb/> <anchor type="note" xlink:label="note-387-13a" xlink:href="note-387-13"/> æquales; </s> <s xml:id="echoid-s16803" xml:space="preserve"><anchor type="note" xlink:href="" symbol="n"/> erunt arcus CK, CM, æquales. </s> <s xml:id="echoid-s16804" xml:space="preserve"><anchor type="note" xlink:href="" symbol="o"/> Igitur anguli CGK, CGM: </s> <s xml:id="echoid-s16805" xml:space="preserve"><anchor type="note" xlink:href="" symbol="p"/> Ideoq;</s> <s xml:id="echoid-s16806" xml:space="preserve"> <anchor type="note" xlink:label="note-387-14a" xlink:href="note-387-14"/> & </s> <s xml:id="echoid-s16807" xml:space="preserve">anguli A G O, AGL, ad verticem æquales erunt: </s> <s xml:id="echoid-s16808" xml:space="preserve"><anchor type="note" xlink:href="" symbol="q"/> Ac proinde arcus etiam AO, AL, æquales erunt: </s> <s xml:id="echoid-s16809" xml:space="preserve"><anchor type="note" xlink:href="" symbol="r"/> ideoque & </s> <s xml:id="echoid-s16810" xml:space="preserve">anguli AKO, AKL, erunt æquales. </s> <s xml:id="echoid-s16811" xml:space="preserve">Pro- <anchor type="note" xlink:label="note-387-15a" xlink:href="note-387-15"/> ducta autem AK, ad Q, ſi ex rectis angulis CKA, CKQ, auferantur æquales AKG, <lb/> <anchor type="note" xlink:label="note-387-16a" xlink:href="note-387-16"/> FKQ, (Cum enim angulus AKG, angulo AKL, oſtenſus ſit æqualis: </s> <s xml:id="echoid-s16812" xml:space="preserve"><anchor type="note" xlink:href="" symbol="s"/> hic autem angulo FKQ, ad verticem ſit æqualis; </s> <s xml:id="echoid-s16813" xml:space="preserve">erit quoque angulus AKG, angulo FKQ. <lb/></s> <s xml:id="echoid-s16814" xml:space="preserve"> <anchor type="note" xlink:label="note-387-17a" xlink:href="note-387-17"/> æqualis.) </s> <s xml:id="echoid-s16815" xml:space="preserve">erunt etiam reliqui anguli CKG, CKF, inter ſe æquales. </s> <s xml:id="echoid-s16816" xml:space="preserve">Quæ omnia <lb/> <anchor type="note" xlink:label="note-387-18a" xlink:href="note-387-18"/> demonſtranda erant.</s> <s xml:id="echoid-s16817" xml:space="preserve"/> </p> <div xml:id="echoid-div1038" type="float" level="2" n="2"> <note symbol="e" position="right" xlink:label="note-387-05" xlink:href="note-387-05a" xml:space="preserve">18. tertij.</note> <note symbol="f" position="right" xlink:label="note-387-06" xlink:href="note-387-06a" xml:space="preserve">coroll. 8. <lb/>ſexti.</note> <note symbol="g" position="right" xlink:label="note-387-07" xlink:href="note-387-07a" xml:space="preserve">17. ſexti.</note> <note symbol="h" position="right" xlink:label="note-387-08" xlink:href="note-387-08a" xml:space="preserve">36. tertij.</note> <note symbol="i" position="right" xlink:label="note-387-09" xlink:href="note-387-09a" xml:space="preserve">16. ſexti.</note> <note symbol="k" position="right" xlink:label="note-387-10" xlink:href="note-387-10a" xml:space="preserve">6. ſexti.</note> <note symbol="l" position="right" xlink:label="note-387-11" xlink:href="note-387-11a" xml:space="preserve">20. tertij.</note> <note symbol="m" position="right" xlink:label="note-387-12" xlink:href="note-387-12a" xml:space="preserve">15. primi.</note> <note symbol="n" position="right" xlink:label="note-387-13" xlink:href="note-387-13a" xml:space="preserve">26. tertij.</note> <note symbol="o" position="right" xlink:label="note-387-14" xlink:href="note-387-14a" xml:space="preserve">ſchol. 29. <lb/>tertij.</note> <note symbol="p" position="right" xlink:label="note-387-15" xlink:href="note-387-15a" xml:space="preserve">15 primi</note> <note symbol="q" position="right" xlink:label="note-387-16" xlink:href="note-387-16a" xml:space="preserve">ſchol. 29. <lb/>tertij.</note> <note symbol="r" position="right" xlink:label="note-387-17" xlink:href="note-387-17a" xml:space="preserve">27. tertij.</note> <note symbol="s" position="right" xlink:label="note-387-18" xlink:href="note-387-18a" xml:space="preserve">15. primi.</note> </div> <pb o="360" file="388" n="388" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1040" type="section" level="1" n="374"> <head xml:id="echoid-head401" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s16818" xml:space="preserve"><emph style="sc">Hoc</emph> Theorema valdè vtile eſt ad deſcriptionem paralleli cuiuſuis circuli <lb/>maximi per datum punctum in Aſtrolabio, vt ex propoſ. </s> <s xml:id="echoid-s16819" xml:space="preserve">18. </s> <s xml:id="echoid-s16820" xml:space="preserve">lib. </s> <s xml:id="echoid-s16821" xml:space="preserve">2. </s> <s xml:id="echoid-s16822" xml:space="preserve">Aſtrolabij per-<lb/>ſpicuum eſt: </s> <s xml:id="echoid-s16823" xml:space="preserve">cum multa ibi demonſtrari poſsint per hoc Theorema, ſine ijs, <lb/>quæ ex Aſtrolabij deſcriptione pendent.</s> <s xml:id="echoid-s16824" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1041" type="section" level="1" n="375"> <head xml:id="echoid-head402" xml:space="preserve">THEOR. 11. PROPOS. 29.</head> <p> <s xml:id="echoid-s16825" xml:space="preserve">DESCRIPTIONEM Pentagoni æquilateri, & </s> <s xml:id="echoid-s16826" xml:space="preserve">æquianguli ſupra <lb/>datam rectam ab Alberto Durero traditam, & </s> <s xml:id="echoid-s16827" xml:space="preserve">quam omnes fere Ar-<lb/>chitecti, atq; </s> <s xml:id="echoid-s16828" xml:space="preserve">artifices approbant, falſam eſſe, demonſtrare.</s> <s xml:id="echoid-s16829" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16830" xml:space="preserve"><emph style="sc">Praxis</emph> hæc eſt. </s> <s xml:id="echoid-s16831" xml:space="preserve">Sit data recta A B. </s> <s xml:id="echoid-s16832" xml:space="preserve">Ex centris A, B, & </s> <s xml:id="echoid-s16833" xml:space="preserve">interuallo eodem <lb/>AB, deſcribantur duo circuli ſe ſe interſecantes in C, D. </s> <s xml:id="echoid-s16834" xml:space="preserve">Ducta autem CD, quã-<lb/>tacunque, deſcribatur eodem interuallo AB, ex C, per A, B, circulus rectam CD, <lb/>in E, & </s> <s xml:id="echoid-s16835" xml:space="preserve">priores circulos ſecans in F, G. </s> <s xml:id="echoid-s16836" xml:space="preserve">Item ducantur ex F, G, per E, rectæ ſe-<lb/>cantes priores circulos in H, I. </s> <s xml:id="echoid-s16837" xml:space="preserve">Denique eodem interuallo ex HI, duo arcus de-<lb/>ſcripti ſe ſe interſecent in K@ iunganturque rectæ AI, IK, KH, HB. </s> <s xml:id="echoid-s16838" xml:space="preserve">Putat ergo Du-<lb/>rerus, pentagonum ABHKI, eſſe æquilaterum, & </s> <s xml:id="echoid-s16839" xml:space="preserve">æquiangulum. </s> <s xml:id="echoid-s16840" xml:space="preserve">quod falſum <lb/>eſt. </s> <s xml:id="echoid-s16841" xml:space="preserve">Nam æquilaterum quidem eſt, ex deſcriptione, non autem æquiangulum. <lb/></s> <s xml:id="echoid-s16842" xml:space="preserve">quod vt manifeſtum fiat, demonſtranda ſunt prius nonnulla.</s> <s xml:id="echoid-s16843" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16844" xml:space="preserve">1. </s> <s xml:id="echoid-s16845" xml:space="preserve"><emph style="sc">Arcvs</emph> tres FA, AB, BG, ſextæ partes circuliſunt, quod rectæ eos ſubtẽ-<lb/>dentes ſemidiametri ſint circuli FABG, ex conſtructione. </s> <s xml:id="echoid-s16846" xml:space="preserve">Igitur FABG, ſemi-<lb/> <anchor type="figure" xlink:label="fig-388-01a" xlink:href="fig-388-01"/> <anchor type="note" xlink:label="note-388-01a" xlink:href="note-388-01"/> circulus eſt, cuius diameter FG; </s> <s xml:id="echoid-s16847" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> ideoque angulus <anchor type="note" xlink:label="note-388-02a" xlink:href="note-388-02"/> FEG, in ſemicirculo rectus: </s> <s xml:id="echoid-s16848" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Et diameter F G, re- ctæ AB, parallela, ob arcus A F, B G, æquales. </s> <s xml:id="echoid-s16849" xml:space="preserve">Et <lb/>quoniã, vt conſtat ex demonſtratione praxis ſcho-<lb/>lij propoſ. </s> <s xml:id="echoid-s16850" xml:space="preserve">@0. </s> <s xml:id="echoid-s16851" xml:space="preserve">& </s> <s xml:id="echoid-s16852" xml:space="preserve">11. </s> <s xml:id="echoid-s16853" xml:space="preserve">lib. </s> <s xml:id="echoid-s16854" xml:space="preserve">1. </s> <s xml:id="echoid-s16855" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16856" xml:space="preserve">recta C D, ſecat re-<lb/> <anchor type="note" xlink:label="note-388-03a" xlink:href="note-388-03"/> ctam AB, bifariam in M, & </s> <s xml:id="echoid-s16857" xml:space="preserve">ad angulos rectos; </s> <s xml:id="echoid-s16858" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ſe- cabit eadem parallela quoque F G, ad angulosre-<lb/> <anchor type="note" xlink:label="note-388-04a" xlink:href="note-388-04"/> ctos in C. </s> <s xml:id="echoid-s16859" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Eadem quoq; </s> <s xml:id="echoid-s16860" xml:space="preserve">CD, ſecabit arcum A B, bifariamin E, ac propterea toti arcus EF, EG, ęqua <lb/> <anchor type="note" xlink:label="note-388-05a" xlink:href="note-388-05"/> les erunt, videlicet quadrantes; </s> <s xml:id="echoid-s16861" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ideoq; </s> <s xml:id="echoid-s16862" xml:space="preserve">rectæ EF, EG, latera ſunt quadrati in circulo F A B G, deſcri-<lb/> <anchor type="note" xlink:label="note-388-06a" xlink:href="note-388-06"/> pti, eiuſque diameter FG. </s> <s xml:id="echoid-s16863" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Igitur anguli F, G, ſemi- recti erunt: </s> <s xml:id="echoid-s16864" xml:space="preserve">ac proinde cum anguliad C, recti ſint, <lb/> <anchor type="note" xlink:label="note-388-07a" xlink:href="note-388-07"/> <anchor type="note" xlink:href="" symbol="g"/> erunt quo que O E M, NEM, ſemirecti; </s> <s xml:id="echoid-s16865" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s16866" xml:space="preserve">&</s> <s xml:id="echoid-s16867" xml:space="preserve"> EOM, ENM, ſemirecti. </s> <s xml:id="echoid-s16868" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Ac proinde tam latera E M, N O, quam E M, M N, æ- <anchor type="note" xlink:label="note-388-08a" xlink:href="note-388-08"/> qualia: </s> <s xml:id="echoid-s16869" xml:space="preserve">atqueidcirco & </s> <s xml:id="echoid-s16870" xml:space="preserve">OM, NM, inter ſe æqualia erunt; </s> <s xml:id="echoid-s16871" xml:space="preserve">nec non & </s> <s xml:id="echoid-s16872" xml:space="preserve">totæ OB, <lb/>NA, æquales erunt. </s> <s xml:id="echoid-s16873" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Immo & </s> <s xml:id="echoid-s16874" xml:space="preserve">EO, EN, erunt æquales, quod latera EM, MO, <anchor type="note" xlink:label="note-388-09a" xlink:href="note-388-09"/> laterib. </s> <s xml:id="echoid-s16875" xml:space="preserve">EM, MN, æqualia ſint, cõprehendãtq; </s> <s xml:id="echoid-s16876" xml:space="preserve">angulos æquales, vt pote rectos.</s> <s xml:id="echoid-s16877" xml:space="preserve"/> </p> <div xml:id="echoid-div1041" 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/xxxxxxxx/figures/388-01"/> </figure> <note symbol="a" position="left" xlink:label="note-388-01" xlink:href="note-388-01a" xml:space="preserve">31. tertij.</note> <note symbol="b" position="left" xlink:label="note-388-02" xlink:href="note-388-02a" xml:space="preserve">ſchol. 27. <lb/>tertij.</note> <note symbol="c" position="left" xlink:label="note-388-03" xlink:href="note-388-03a" xml:space="preserve">29. primi.</note> <note symbol="d" position="left" xlink:label="note-388-04" xlink:href="note-388-04a" xml:space="preserve">ſchol. 27. <lb/>tertij.</note> <note symbol="e" position="left" xlink:label="note-388-05" xlink:href="note-388-05a" xml:space="preserve">6. quanti.</note> <note symbol="f" position="left" xlink:label="note-388-06" xlink:href="note-388-06a" xml:space="preserve">ſchol. 34. <lb/>primi.</note> <note symbol="g" position="left" xlink:label="note-388-07" xlink:href="note-388-07a" xml:space="preserve">32. primi.</note> <note symbol="h" position="left" xlink:label="note-388-08" xlink:href="note-388-08a" xml:space="preserve">6. primi.</note> <note symbol="i" position="left" xlink:label="note-388-09" xlink:href="note-388-09a" xml:space="preserve">4. primi.</note> </div> <p> <s xml:id="echoid-s16878" xml:space="preserve">2. </s> <s xml:id="echoid-s16879" xml:space="preserve"><emph style="sc">Deinde</emph> quialatera AN, AI, lateribus BO, BH, æqualia ſunt; </s> <s xml:id="echoid-s16880" xml:space="preserve">ſuntq; </s> <s xml:id="echoid-s16881" xml:space="preserve">an-<lb/>guli N, O, ſemirecti æquales, & </s> <s xml:id="echoid-s16882" xml:space="preserve">vterque reliquorum angulorum I, H, minorre-<lb/> <anchor type="note" xlink:label="note-388-10a" xlink:href="note-388-10"/> cto; </s> <s xml:id="echoid-s16883" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> quod vterque minor ſit ſemirecto ad O, & </s> <s xml:id="echoid-s16884" xml:space="preserve">N; </s> <s xml:id="echoid-s16885" xml:space="preserve">propterea quod tam latus AN, minus eſt latere AI, ꝗ̃ latus B O, latere BH: </s> <s xml:id="echoid-s16886" xml:space="preserve">eruntper ea, quæ ad finem lib.</s> <s xml:id="echoid-s16887" xml:space="preserve"> <pb o="361" file="389" n="389" rhead="LIBER OCTAVVS."/> 1. </s> <s xml:id="echoid-s16888" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s16889" xml:space="preserve">demonſtrauimus, tam baſes NI, OH, quam anguli A, B, & </s> <s xml:id="echoid-s16890" xml:space="preserve">I, H, æqua-<lb/>les. </s> <s xml:id="echoid-s16891" xml:space="preserve">Igitur duo anguli A, B, in pentagono æqualesinter ſe ſunt.</s> <s xml:id="echoid-s16892" xml:space="preserve"/> </p> <div xml:id="echoid-div1042" type="float" level="2" n="2"> <note symbol="k" position="left" xlink:label="note-388-10" xlink:href="note-388-10a" xml:space="preserve">19. primi.</note> </div> <p> <s xml:id="echoid-s16893" xml:space="preserve">3. </s> <s xml:id="echoid-s16894" xml:space="preserve"><emph style="sc">Rvrsvs</emph> demptis OE, NE, æqualibus ex æqualibus OH, NI, reliquæ re-<lb/> <anchor type="note" xlink:label="note-389-01a" xlink:href="note-389-01"/> ctæ EH, EI, æquales ſunt: </s> <s xml:id="echoid-s16895" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erunt anguli EIH, EHI, æquales, ac proinde ſemire- <anchor type="figure" xlink:label="fig-389-01a" xlink:href="fig-389-01"/> cti, cum HEI, ſit rectus; </s> <s xml:id="echoid-s16896" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Suntautem & </s> <s xml:id="echoid-s16897" xml:space="preserve">anguli H, <anchor type="note" xlink:label="note-389-02a" xlink:href="note-389-02"/> I, in Iſoſcele KIH, æquales. </s> <s xml:id="echoid-s16898" xml:space="preserve">Igitur toti anguli H, I, <lb/>in pentagono æquales ſunt.</s> <s xml:id="echoid-s16899" xml:space="preserve"/> </p> <div xml:id="echoid-div1043" type="float" level="2" n="3"> <note symbol="a" position="right" xlink:label="note-389-01" xlink:href="note-389-01a" xml:space="preserve">5. primi.</note> <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/xxxxxxxx/figures/389-01"/> </figure> <note symbol="b" position="right" xlink:label="note-389-02" xlink:href="note-389-02a" xml:space="preserve">5. primi.</note> </div> <p> <s xml:id="echoid-s16900" xml:space="preserve">4. </s> <s xml:id="echoid-s16901" xml:space="preserve"><emph style="sc">Postremo</emph> cũ latera EL, EI, lateribus EL, <lb/>E H, ſintæqualia, contineantque angulos æquales <lb/>ſemirectos; </s> <s xml:id="echoid-s16902" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erunt & </s> <s xml:id="echoid-s16903" xml:space="preserve">baſes LI, LH, æquales, & </s> <s xml:id="echoid-s16904" xml:space="preserve">an- <anchor type="note" xlink:label="note-389-03a" xlink:href="note-389-03"/> guli ad L, ideoque recti. </s> <s xml:id="echoid-s16905" xml:space="preserve">Ex quo efficitur, <anchor type="note" xlink:href="" symbol="d"/> rectas AB, HI, eſſe parallelas: </s> <s xml:id="echoid-s16906" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> quod etiam conſtat ex eo, <anchor type="note" xlink:label="note-389-04a" xlink:href="note-389-04"/> quod alterni anguli INA, NIH, æquales ſint, nimi-<lb/> <anchor type="note" xlink:label="note-389-05a" xlink:href="note-389-05"/> rum ſemirecti. </s> <s xml:id="echoid-s16907" xml:space="preserve">Hinc etiam ſequitur, rectam C D L, <lb/>productam cadere in angulum K, diuidereque eum <lb/>bifariam. </s> <s xml:id="echoid-s16908" xml:space="preserve">Si enim dicatur non cadere in K, <anchor type="note" xlink:href="" symbol="f"/> diuidet perpendicularis ex K, ad HI, demiſſa baſem Iſoſce-<lb/> <anchor type="note" xlink:label="note-389-06a" xlink:href="note-389-06"/> lis KIH, bifariam in alio puncto, quam in L, quod eſt abſurdum. </s> <s xml:id="echoid-s16909" xml:space="preserve">Quia ergo la-<lb/>tera KI, KL, lateribus KH, KL, æqualia ſunt, & </s> <s xml:id="echoid-s16910" xml:space="preserve">baſis IL, baſi HL, oſtenſa æqua-<lb/>lis: </s> <s xml:id="echoid-s16911" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erunt anguliad K, æquales.</s> <s xml:id="echoid-s16912" xml:space="preserve"/> </p> <div xml:id="echoid-div1044" type="float" level="2" n="4"> <note symbol="c" position="right" xlink:label="note-389-03" xlink:href="note-389-03a" xml:space="preserve">4. primi.</note> <note symbol="d" position="right" xlink:label="note-389-04" xlink:href="note-389-04a" xml:space="preserve">28. primi.</note> <note symbol="e" position="right" xlink:label="note-389-05" xlink:href="note-389-05a" xml:space="preserve">27. primi.</note> <note symbol="f" position="right" xlink:label="note-389-06" xlink:href="note-389-06a" xml:space="preserve">ſchol. 26. <lb/>primi.</note> </div> <note symbol="g" position="right" xml:space="preserve">8. primi.</note> <p> <s xml:id="echoid-s16913" xml:space="preserve"><emph style="sc">His</emph> ita præmiſsis, demonſtrabimusiam pentagonum ABHKI, non eſſe æ-<lb/> <anchor type="note" xlink:label="note-389-08a" xlink:href="note-389-08"/> quiangulum, hocmodo. </s> <s xml:id="echoid-s16914" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Omnes 5. </s> <s xml:id="echoid-s16915" xml:space="preserve">anguli in pentagono quolibet, ſiue ſit ę- quilaterum, & </s> <s xml:id="echoid-s16916" xml:space="preserve">æquiangulum, ſiue non, æquales ſunt 6. </s> <s xml:id="echoid-s16917" xml:space="preserve">rectis, hoc eſt, gra. </s> <s xml:id="echoid-s16918" xml:space="preserve">540. <lb/></s> <s xml:id="echoid-s16919" xml:space="preserve">quibus diuiſis per 5. </s> <s xml:id="echoid-s16920" xml:space="preserve">efficitur vnus angulus pentagoni æquilateri, & </s> <s xml:id="echoid-s16921" xml:space="preserve">quianguli <lb/>grad. </s> <s xml:id="echoid-s16922" xml:space="preserve">108. </s> <s xml:id="echoid-s16923" xml:space="preserve">At vterlibet duorum angulorum A, B, in pentagono Dureri maior eſt, <lb/>quam grad. </s> <s xml:id="echoid-s16924" xml:space="preserve">108. </s> <s xml:id="echoid-s16925" xml:space="preserve">& </s> <s xml:id="echoid-s16926" xml:space="preserve">vterlibet duorum H, I, minor, & </s> <s xml:id="echoid-s16927" xml:space="preserve">angulus K, maior quolibet <lb/>reliquorum quatuor. </s> <s xml:id="echoid-s16928" xml:space="preserve">vt oſtendemus. </s> <s xml:id="echoid-s16929" xml:space="preserve">Igitur pentagonũ Durerinon eſt æquian-<lb/>gulum. </s> <s xml:id="echoid-s16930" xml:space="preserve">Hoc autem ita fiet perſpicuum.</s> <s xml:id="echoid-s16931" xml:space="preserve"/> </p> <div xml:id="echoid-div1045" type="float" level="2" n="5"> <note symbol="h" position="right" xlink:label="note-389-08" xlink:href="note-389-08a" xml:space="preserve">ſchol. 32. <lb/>primi.</note> </div> <p> <s xml:id="echoid-s16932" xml:space="preserve"><emph style="sc">Qvoniam</emph> poſito ſinu toto AB, 10000000. </s> <s xml:id="echoid-s16933" xml:space="preserve">eius ſemiſsis BM, ſinus vide-<lb/>licet grad. </s> <s xml:id="echoid-s16934" xml:space="preserve">30. </s> <s xml:id="echoid-s16935" xml:space="preserve">eſt 5000000. </s> <s xml:id="echoid-s16936" xml:space="preserve">cui ſi addatur MO, id eſt, ME, ſinus verſus grad. </s> <s xml:id="echoid-s16937" xml:space="preserve">30. <lb/></s> <s xml:id="echoid-s16938" xml:space="preserve">nimirum 1339746. </s> <s xml:id="echoid-s16939" xml:space="preserve">fiet tota BO, 6339746. </s> <s xml:id="echoid-s16940" xml:space="preserve">Quia ergo in triangulo BHO, duo la-<lb/>tera dantur BH, 10000000. </s> <s xml:id="echoid-s16941" xml:space="preserve">& </s> <s xml:id="echoid-s16942" xml:space="preserve">BO, 6339746. </s> <s xml:id="echoid-s16943" xml:space="preserve">vna cum angulo O, grad. </s> <s xml:id="echoid-s16944" xml:space="preserve">45. </s> <s xml:id="echoid-s16945" xml:space="preserve">nec <lb/> <anchor type="note" xlink:label="note-389-09a" xlink:href="note-389-09"/> non cum ſpecie anguli H, qui ſupra oſtenſus fuit recto minor: </s> <s xml:id="echoid-s16946" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Si fiat,</s> </p> <div xml:id="echoid-div1046" type="float" level="2" n="6"> <note symbol="i" position="right" xlink:label="note-389-09" xlink:href="note-389-09a" xml:space="preserve">15. triang. re-<lb/>ctil.</note> </div> <note style="it" position="right" xml:space="preserve"> <lb/>Vt lat{us} B H, \\ 10000000. # ad 707 068. ſinum an- \\ guli O, g ad. 45. # Italat{us} B O, \\ 6339746. # ad aliud, <lb/></note> <p> <s xml:id="echoid-s16947" xml:space="preserve">inuenietur ſinus anguli BHO, 4482877 {1/2}. </s> <s xml:id="echoid-s16948" xml:space="preserve">ferme, qui in tabula ſinuum offeret <lb/>ipſum angulum grad. </s> <s xml:id="echoid-s16949" xml:space="preserve">26. </s> <s xml:id="echoid-s16950" xml:space="preserve">min. </s> <s xml:id="echoid-s16951" xml:space="preserve">38. </s> <s xml:id="echoid-s16952" xml:space="preserve">cui ſi addatur angulus BOH, grad. </s> <s xml:id="echoid-s16953" xml:space="preserve">45. </s> <s xml:id="echoid-s16954" xml:space="preserve">fiet ſum-<lb/>ma angulorum H, O, grad. </s> <s xml:id="echoid-s16955" xml:space="preserve">71. </s> <s xml:id="echoid-s16956" xml:space="preserve">min. </s> <s xml:id="echoid-s16957" xml:space="preserve">38. </s> <s xml:id="echoid-s16958" xml:space="preserve">quæ ſumma dempta ex duobusrectis, id <lb/>eſt, ex grad. </s> <s xml:id="echoid-s16959" xml:space="preserve">180. </s> <s xml:id="echoid-s16960" xml:space="preserve">relinquet angulum OBH, grad. </s> <s xml:id="echoid-s16961" xml:space="preserve">108. </s> <s xml:id="echoid-s16962" xml:space="preserve">min. </s> <s xml:id="echoid-s16963" xml:space="preserve">22. </s> <s xml:id="echoid-s16964" xml:space="preserve">Igitur vterque an-<lb/>gulus A, B, in pentagono maior eſt verò angulo pentagoni grad. </s> <s xml:id="echoid-s16965" xml:space="preserve">108.</s> <s xml:id="echoid-s16966" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s16967" xml:space="preserve"><emph style="sc">Deinde</emph> ducta BP, ad HI, perpendiculari, ſi iterum ſtatuatur BH, finus to-<lb/>tus 10000000. </s> <s xml:id="echoid-s16968" xml:space="preserve">erit HP, 3150970. </s> <s xml:id="echoid-s16969" xml:space="preserve">anguli HBP, grad 18. </s> <s xml:id="echoid-s16970" xml:space="preserve">min. </s> <s xml:id="echoid-s16971" xml:space="preserve">22. </s> <s xml:id="echoid-s16972" xml:space="preserve">quirelin quitur, ſi <lb/>rectus angulus A B P, grad 90. </s> <s xml:id="echoid-s16973" xml:space="preserve">detrahatur ex angulo A B H, inuento grad. </s> <s xml:id="echoid-s16974" xml:space="preserve">108. <lb/></s> <s xml:id="echoid-s16975" xml:space="preserve">min. </s> <s xml:id="echoid-s16976" xml:space="preserve">22. </s> <s xml:id="echoid-s16977" xml:space="preserve">Si igitur addatur PL. </s> <s xml:id="echoid-s16978" xml:space="preserve">5000000. </s> <s xml:id="echoid-s16979" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> cum ſit æqualis ipſi BM, ſemiſsi ſinus <anchor type="note" xlink:label="note-389-11a" xlink:href="note-389-11"/> totius, fiet tota HL, ſinus anguli HKL, 8150970. </s> <s xml:id="echoid-s16980" xml:space="preserve">Ac propterea angulus ipſe erit <lb/>grad. </s> <s xml:id="echoid-s16981" xml:space="preserve">54. </s> <s xml:id="echoid-s16982" xml:space="preserve">min. </s> <s xml:id="echoid-s16983" xml:space="preserve">36. </s> <s xml:id="echoid-s16984" xml:space="preserve">qui duplicatus dabit totum angulum H K I, grad. </s> <s xml:id="echoid-s16985" xml:space="preserve">109. </s> <s xml:id="echoid-s16986" xml:space="preserve">min. </s> <s xml:id="echoid-s16987" xml:space="preserve">12. <lb/></s> <s xml:id="echoid-s16988" xml:space="preserve">maiore verò angulo pentagoni grad. </s> <s xml:id="echoid-s16989" xml:space="preserve">108.</s> <s xml:id="echoid-s16990" xml:space="preserve"/> </p> <div xml:id="echoid-div1047" type="float" level="2" n="7"> <note symbol="k" position="right" xlink:label="note-389-11" xlink:href="note-389-11a" xml:space="preserve">34. primi.</note> </div> <pb o="362" file="390" n="390" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s16991" xml:space="preserve"><emph style="sc">Iam</emph> verò ſi ſumma trium angulorum A, B, K, inuentorum, nimirum grad. <lb/></s> <s xml:id="echoid-s16992" xml:space="preserve"> <anchor type="note" xlink:label="note-390-01a" xlink:href="note-390-01"/> 325. </s> <s xml:id="echoid-s16993" xml:space="preserve">min. </s> <s xml:id="echoid-s16994" xml:space="preserve">56. </s> <s xml:id="echoid-s16995" xml:space="preserve">auferatur ex grad. <lb/></s> <s xml:id="echoid-s16996" xml:space="preserve">540. </s> <s xml:id="echoid-s16997" xml:space="preserve">ſumma omnium 5. </s> <s xml:id="echoid-s16998" xml:space="preserve">angu-<lb/>lorum pentagoni, reliqua fiet <lb/>ſumma angulorum H, I, grad. </s> <s xml:id="echoid-s16999" xml:space="preserve"><lb/>214. </s> <s xml:id="echoid-s17000" xml:space="preserve">min. </s> <s xml:id="echoid-s17001" xml:space="preserve">4. </s> <s xml:id="echoid-s17002" xml:space="preserve">Acproinde vterque <lb/>erit grad. </s> <s xml:id="echoid-s17003" xml:space="preserve">107. </s> <s xml:id="echoid-s17004" xml:space="preserve">min. </s> <s xml:id="echoid-s17005" xml:space="preserve">2. </s> <s xml:id="echoid-s17006" xml:space="preserve">minor ve-<lb/>rò angulo pentagoni grad. </s> <s xml:id="echoid-s17007" xml:space="preserve">180. </s> <s xml:id="echoid-s17008" xml:space="preserve"><lb/>Nonergo æquiangulum eſt Du-<lb/>reri pentagonum, ſed ſolum æ-<lb/>quilaterum. </s> <s xml:id="echoid-s17009" xml:space="preserve">Omnes tamen 5. </s> <s xml:id="echoid-s17010" xml:space="preserve"><lb/>anguli cõficiunt ſummam grad. </s> <s xml:id="echoid-s17011" xml:space="preserve"><lb/>540. </s> <s xml:id="echoid-s17012" xml:space="preserve">ſicut in pentagono æquila-<lb/>tero, atque æquiangulo, vt hæ <lb/>formula indicat.</s> <s xml:id="echoid-s17013" xml:space="preserve"/> </p> <div xml:id="echoid-div1048" type="float" level="2" n="8"> <note position="right" xlink:label="note-390-01" xlink:href="note-390-01a" xml:space="preserve"> <lb/>Angulus # A # Grad. # 108 # min. # 22 <lb/># B # Grad. # 108 # min. # 22 <lb/># H # Grad. # 107 # min. # 2 <lb/># I # Grad. # 107 # min. # 2 <lb/># K # Grad. # 109 # min. # 12 <lb/>## Summa # # 540 # min. # 0 <lb/></note> </div> </div> <div xml:id="echoid-div1050" type="section" level="1" n="376"> <head xml:id="echoid-head403" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s17014" xml:space="preserve"><emph style="sc">Svnt</emph> alij nonnulli, qui ad interuallum cuiuſuis rectæ AB, deſcriptis ex cẽ-<lb/>tris A, B, duobus circulis ſe interſecantibus in C, D, vt in ſuperiori figura, ducunt <lb/>rectam AD, affirmantque AD, latus eſſe pentagoni in circulo, cuius ſemidiame-<lb/>ter DM, inſcripti. </s> <s xml:id="echoid-s17015" xml:space="preserve">ſed toto cœlo aberrant. </s> <s xml:id="echoid-s17016" xml:space="preserve">Eſt enim AD, minus latere pentago-<lb/> <anchor type="note" xlink:label="note-390-02a" xlink:href="note-390-02"/> ni circuli prædicti. </s> <s xml:id="echoid-s17017" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Nam quia latus pentagonipoteſt & </s> <s xml:id="echoid-s17018" xml:space="preserve">latus hexagoni, & </s> <s xml:id="echoid-s17019" xml:space="preserve">la- tus decagoni circuli eiuſdem: </s> <s xml:id="echoid-s17020" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Poteſt autem AD, rectas DM, MA; </s> <s xml:id="echoid-s17021" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> & </s> <s xml:id="echoid-s17022" xml:space="preserve">DM, la- <anchor type="note" xlink:label="note-390-03a" xlink:href="note-390-03"/> tus eſt hexagoni in circulo, cuius ſemidiameter DM, eſſet AM, latus decagoni in <lb/> <anchor type="note" xlink:label="note-390-04a" xlink:href="note-390-04"/> eodem circulo. </s> <s xml:id="echoid-s17023" xml:space="preserve">quod falſum eſt. </s> <s xml:id="echoid-s17024" xml:space="preserve">Quoniam enim latus decagoni maius eſt ſe-<lb/>miſſe lateris pentagoni, quod duo latera decagoniſupra latus Pentagoni con-<lb/> <anchor type="note" xlink:label="note-390-05a" xlink:href="note-390-05"/> ſtituant Iſoſceles <anchor type="note" xlink:href="" symbol="d"/> in quo duo latera maiora ſunt latere pentagoni: </s> <s xml:id="echoid-s17025" xml:space="preserve">Erit AM, ſe- <anchor type="note" xlink:label="note-390-06a" xlink:href="note-390-06"/> miſsis ipſius AB, vel AD, minor latere decagoni. </s> <s xml:id="echoid-s17026" xml:space="preserve">Igitur AD, minor eſt latere pẽ-<lb/>tagoni; </s> <s xml:id="echoid-s17027" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> quando quidem latus pentagoni poteſt & </s> <s xml:id="echoid-s17028" xml:space="preserve">latus hexagoni DM, &</s> <s xml:id="echoid-s17029" xml:space="preserve"> latus decagoni, quod maius eſt, quam AM, ſemiſsis ipſius AD, vt diximus.</s> <s xml:id="echoid-s17030" xml:space="preserve"/> </p> <div xml:id="echoid-div1050" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-390-02" xlink:href="note-390-02a" xml:space="preserve">10. tertijde-<lb/>cimi.</note> <note symbol="b" position="left" xlink:label="note-390-03" xlink:href="note-390-03a" xml:space="preserve">47. primi.</note> <note symbol="c" position="left" xlink:label="note-390-04" xlink:href="note-390-04a" xml:space="preserve">coroll. 15. <lb/>quanti.</note> <note symbol="d" position="left" xlink:label="note-390-05" xlink:href="note-390-05a" xml:space="preserve">20. primi.</note> <note symbol="e" position="left" xlink:label="note-390-06" xlink:href="note-390-06a" xml:space="preserve">10. tertijde-<lb/>cimi.</note> </div> </div> <div xml:id="echoid-div1052" type="section" level="1" n="377"> <head xml:id="echoid-head404" xml:space="preserve">THEOR. 12. PROPOS. 30.</head> <p> <s xml:id="echoid-s17031" xml:space="preserve">INVENTIONEM lateris heptagoni in dato circulo non rectè à qui-<lb/>buſdam tradi, demonſtrare.</s> <s xml:id="echoid-s17032" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17033" xml:space="preserve"><emph style="sc">Carolvs</emph> Marianus Cremonenſis totum vnum libellũ <lb/> <anchor type="figure" xlink:label="fig-390-01a" xlink:href="fig-390-01"/> edidit de inuentione lateris heptagoni in circulo dato, in <lb/>quo probare conatur, latus heptagoni reperiri hac ratione. <lb/></s> <s xml:id="echoid-s17034" xml:space="preserve">Sit circulus ABC, cuius centrum D, diameter CA, in qua pro-<lb/>ducta capiatur AE, æqualis quartæ parti ſemidiametri AD, ita <lb/>vt AE, quinta pars ſit rectæ DE. </s> <s xml:id="echoid-s17035" xml:space="preserve">Deſcripto autem ex E, ad in-<lb/>teruallum ſemidiametri AD, circulo ſecante datum circulum <lb/>in B, iũgatur recta AB, quam dicit eſſe latus heptagoni, quod <lb/>falſum eſſe, ita oſtendemus. </s> <s xml:id="echoid-s17036" xml:space="preserve">Si AB, eſſet verum latus hepta-<lb/>goni, & </s> <s xml:id="echoid-s17037" xml:space="preserve">ducta BE, æquali ſemidiametro DB, (quod fiet, ſi ex <pb o="363" file="391" n="391" rhead="LIBER OCTAVVS."/> B, ad interuallum ſemidiametri recta D E, ſecetur in E,) ſecante ar@um AB, in F, <lb/>diuideretur arcus AB, in F, vel angulus ADB, bifariam. </s> <s xml:id="echoid-s17038" xml:space="preserve">quod tamen in eius de-<lb/>ſcriptione non contingit, vt demonſtrabitur. </s> <s xml:id="echoid-s17039" xml:space="preserve">Non ergo eius linea A B, verum <lb/>latus eſt heptagoni. </s> <s xml:id="echoid-s17040" xml:space="preserve">Ductis enim rectis DB, DF, ſi AB, eſt ſeptima pars circum-<lb/>ferentiæ, continebit tam angulus ADB, quam DEB, (<anchor type="note" xlink:href="" symbol="a"/> quiæquales ſunt) {2/7}. </s> <s xml:id="echoid-s17041" xml:space="preserve">vel <anchor type="note" xlink:label="note-391-01a" xlink:href="note-391-01"/> {4/14}. </s> <s xml:id="echoid-s17042" xml:space="preserve">duorum rectorum. </s> <s xml:id="echoid-s17043" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Ergo reliqui DAB, DBA, ſimul continebunt {5/7}. </s> <s xml:id="echoid-s17044" xml:space="preserve">vel <anchor type="note" xlink:label="note-391-02a" xlink:href="note-391-02"/> {10/14}. </s> <s xml:id="echoid-s17045" xml:space="preserve">duorum rectorum. </s> <s xml:id="echoid-s17046" xml:space="preserve">Ac proinde vterqueipſorum continebit {5/14}. </s> <s xml:id="echoid-s17047" xml:space="preserve">duorum <lb/>rectorum. </s> <s xml:id="echoid-s17048" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Cum ergo DAB, æqualis ſit duobus E, & </s> <s xml:id="echoid-s17049" xml:space="preserve">ABE, continebunt etiam <anchor type="note" xlink:label="note-391-03a" xlink:href="note-391-03"/> hi ſimul {5/14}. </s> <s xml:id="echoid-s17050" xml:space="preserve">duorum rectorum. </s> <s xml:id="echoid-s17051" xml:space="preserve">Continet autem E, ſolus {4/14}. </s> <s xml:id="echoid-s17052" xml:space="preserve">duorum recto-<lb/>rum. </s> <s xml:id="echoid-s17053" xml:space="preserve">Igitur ABE, continebit {1/14}. </s> <s xml:id="echoid-s17054" xml:space="preserve">duorum rectorum.</s> <s xml:id="echoid-s17055" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Et quia A D F, duplus eſt <anchor type="note" xlink:label="note-391-04a" xlink:href="note-391-04"/> ipſius ABE, propter eandem baſem AF, continebit angulus ADF, {2/14}. </s> <s xml:id="echoid-s17056" xml:space="preserve">id eſt, {1/7}. <lb/></s> <s xml:id="echoid-s17057" xml:space="preserve">duorum rectorum. </s> <s xml:id="echoid-s17058" xml:space="preserve">Cum ergo totus ADB, complectatur {2/7}. </s> <s xml:id="echoid-s17059" xml:space="preserve">vt dictum eſt, con-<lb/>tinebit quoq; </s> <s xml:id="echoid-s17060" xml:space="preserve">BDF, {1/7}. </s> <s xml:id="echoid-s17061" xml:space="preserve">duorum rectorum; </s> <s xml:id="echoid-s17062" xml:space="preserve">ideoque æquales erunt ADF, BDF.</s> <s xml:id="echoid-s17063" xml:space="preserve"/> </p> <div xml:id="echoid-div1052" type="float" level="2" n="1"> <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/xxxxxxxx/figures/390-01"/> </figure> <note symbol="a" position="right" xlink:label="note-391-01" xlink:href="note-391-01a" xml:space="preserve">5. primi.</note> <note symbol="b" position="right" xlink:label="note-391-02" xlink:href="note-391-02a" xml:space="preserve">32. primi.</note> <note symbol="c" position="right" xlink:label="note-391-03" xlink:href="note-391-03a" xml:space="preserve">32. primi.</note> <note symbol="d" position="right" xlink:label="note-391-04" xlink:href="note-391-04a" xml:space="preserve">20. tertij.</note> </div> <p> <s xml:id="echoid-s17064" xml:space="preserve"><emph style="sc">Sed</emph> iam AB, ſit inuenta per conſtructionem prædicti auctoris; </s> <s xml:id="echoid-s17065" xml:space="preserve">eritque EB, <lb/>æqualis ipſi DB. </s> <s xml:id="echoid-s17066" xml:space="preserve">Si ergo AB, eſſet verum latus heptagoni, caderet DI, perpendi-<lb/>cularis, diuidens nimirum angulum A D B, bifariam, in F, quod verum non eſt. <lb/></s> <s xml:id="echoid-s17067" xml:space="preserve">Poſita enim BE, 4. </s> <s xml:id="echoid-s17068" xml:space="preserve">erit tota CE, 9. </s> <s xml:id="echoid-s17069" xml:space="preserve">& </s> <s xml:id="echoid-s17070" xml:space="preserve">DE, 5.</s> <s xml:id="echoid-s17071" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Cum ergo ſit, vt BD, ad DE, ita BF, <anchor type="note" xlink:label="note-391-05a" xlink:href="note-391-05"/> ad F E; </s> <s xml:id="echoid-s17072" xml:space="preserve">(quod angulus A D B, ſectus ſit bifariam) erit componendo, ſumma ex <lb/>BD, DE, nimirum 9. </s> <s xml:id="echoid-s17073" xml:space="preserve">ad DE, 5. </s> <s xml:id="echoid-s17074" xml:space="preserve">vt BE, ad FE. </s> <s xml:id="echoid-s17075" xml:space="preserve">Si igitur fiat, vt 9. </s> <s xml:id="echoid-s17076" xml:space="preserve">ad 5. </s> <s xml:id="echoid-s17077" xml:space="preserve">ita BE, 4. </s> <s xml:id="echoid-s17078" xml:space="preserve">ad <lb/>aliud, inuenietur FE, 2 {2/9}. </s> <s xml:id="echoid-s17079" xml:space="preserve">ac propterea rectangulum ſub BE, 4. </s> <s xml:id="echoid-s17080" xml:space="preserve">& </s> <s xml:id="echoid-s17081" xml:space="preserve">EF, 2 {2/9}. </s> <s xml:id="echoid-s17082" xml:space="preserve">erit <lb/>8 {8/9}. </s> <s xml:id="echoid-s17083" xml:space="preserve">& </s> <s xml:id="echoid-s17084" xml:space="preserve">rectangulum ſub CE, 9. </s> <s xml:id="echoid-s17085" xml:space="preserve">& </s> <s xml:id="echoid-s17086" xml:space="preserve">EA, 1. </s> <s xml:id="echoid-s17087" xml:space="preserve">erit 9. </s> <s xml:id="echoid-s17088" xml:space="preserve">quod eſt abſurdum; </s> <s xml:id="echoid-s17089" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> cum hæcre- <anchor type="note" xlink:label="note-391-06a" xlink:href="note-391-06"/> ctangula ſint æqualia. </s> <s xml:id="echoid-s17090" xml:space="preserve">Non ergo recta DI, cadit in punctum F, interſectionis re-<lb/>ctæ BE, cum arcu AB, quando quidem rectangulum ſub BE, EF, æquale non eſt <lb/>rectangulo ſub CE, EA, ſed minus: </s> <s xml:id="echoid-s17091" xml:space="preserve">Ac proindenonrectè illa ratione latus he-<lb/>ptagoni inuenitur.</s> <s xml:id="echoid-s17092" xml:space="preserve"/> </p> <div xml:id="echoid-div1053" type="float" level="2" n="2"> <note symbol="e" position="right" xlink:label="note-391-05" xlink:href="note-391-05a" xml:space="preserve">3. ſexti.</note> <note symbol="f" position="right" xlink:label="note-391-06" xlink:href="note-391-06a" xml:space="preserve">1. coroll. 36. <lb/>tertij.</note> </div> <p> <s xml:id="echoid-s17093" xml:space="preserve"><emph style="sc">Albertvs</emph> Durerus ad KL, latus trianguli æquilateri (ſumptis videlicet ar-<lb/>cubus AK, AL, quorum vterque ſextam partẽ circumferentiæ contineat) per-<lb/>pendicularem ducit A H, dicitque K H, ſemiſſem illius lateris eſſe latus hepta-<lb/>goni. </s> <s xml:id="echoid-s17094" xml:space="preserve">quod ſimiliter falſum eſt. </s> <s xml:id="echoid-s17095" xml:space="preserve">Nam KH, omnino æqualis eſt rectæ AB, quam <lb/>proximè demonſtrauimus non eſſelatus heptagoni. </s> <s xml:id="echoid-s17096" xml:space="preserve">Si enim iungeretur recta <lb/>AK, fieret triangulum æquilaterum AKD. </s> <s xml:id="echoid-s17097" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Igitur perpendicularis K H, diuidet <anchor type="note" xlink:label="note-391-07a" xlink:href="note-391-07"/> AD, bifariam: </s> <s xml:id="echoid-s17098" xml:space="preserve">Acproinde poſita DK, vel DA, 4. </s> <s xml:id="echoid-s17099" xml:space="preserve">erit DH, 2. </s> <s xml:id="echoid-s17100" xml:space="preserve">Quocirca ſi detrahe-<lb/>mus 4. </s> <s xml:id="echoid-s17101" xml:space="preserve">quadratum DH, ex 16. </s> <s xml:id="echoid-s17102" xml:space="preserve">quadrato DK, <anchor type="note" xlink:href="" symbol="h"/> reliquum erit quadratum KH 12.</s> <s xml:id="echoid-s17103" xml:space="preserve"> <anchor type="note" xlink:label="note-391-08a" xlink:href="note-391-08"/> At tantum etiam deprehendemus eſſe quadratum AB. </s> <s xml:id="echoid-s17104" xml:space="preserve">Quoniam enim quadra-<lb/>tum BE, eſt 16. </s> <s xml:id="echoid-s17105" xml:space="preserve">hoc eſt, {64/4}. </s> <s xml:id="echoid-s17106" xml:space="preserve">& </s> <s xml:id="echoid-s17107" xml:space="preserve">quadratum EG, {25/4}. </s> <s xml:id="echoid-s17108" xml:space="preserve">(Nam perpendicularis BG, <lb/>ſecat in Iſoſcele EBD, baſem ED, bifariam. </s> <s xml:id="echoid-s17109" xml:space="preserve">Cum ergo ED, ſit 5. </s> <s xml:id="echoid-s17110" xml:space="preserve">erit DG, 2 {1/2}. <lb/></s> <s xml:id="echoid-s17111" xml:space="preserve">cuius quadratum eſt {25/4})<anchor type="note" xlink:href="" symbol="i"/> erit quadratum BG, {39/4}. </s> <s xml:id="echoid-s17112" xml:space="preserve">Sed quadratum A G, eſt {9/4}.</s> <s xml:id="echoid-s17113" xml:space="preserve"> <anchor type="note" xlink:label="note-391-09a" xlink:href="note-391-09"/> quodrecta A G, @@t 1 {1/2}. </s> <s xml:id="echoid-s17114" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> Igitur quadratum AB, erit {48/4}. </s> <s xml:id="echoid-s17115" xml:space="preserve">id eſt, 12. </s> <s xml:id="echoid-s17116" xml:space="preserve">quod eſt pro- <anchor type="note" xlink:label="note-391-10a" xlink:href="note-391-10"/> poſitum.</s> <s xml:id="echoid-s17117" xml:space="preserve"/> </p> <div xml:id="echoid-div1054" type="float" level="2" n="3"> <note symbol="g" position="right" xlink:label="note-391-07" xlink:href="note-391-07a" xml:space="preserve">ſchol 26. <lb/>primi.</note> <note symbol="h" position="right" xlink:label="note-391-08" xlink:href="note-391-08a" xml:space="preserve">47 primi.</note> <note symbol="i" position="right" xlink:label="note-391-09" xlink:href="note-391-09a" xml:space="preserve">47. primi.</note> <note symbol="k" position="right" xlink:label="note-391-10" xlink:href="note-391-10a" xml:space="preserve">47. primi.</note> </div> <p> <s xml:id="echoid-s17118" xml:space="preserve"><emph style="sc">Franciscvs</emph> Fluſlas Candalla vir nobiliſsimus, ac <lb/> <anchor type="figure" xlink:label="fig-391-01a" xlink:href="fig-391-01"/> do ctiſsimus conatus eſt conſtruere triangulum Iſoſceles <lb/>habens vtrumuis angulorum æqualiũ ad baſem triplum re-<lb/>liqui anguli, vt beneficio ipſius in dato circulo heptagonum <lb/>inſcribatur, vt in ſcholio propoſ. </s> <s xml:id="echoid-s17119" xml:space="preserve">15. </s> <s xml:id="echoid-s17120" xml:space="preserve">lib. </s> <s xml:id="echoid-s17121" xml:space="preserve">4. </s> <s xml:id="echoid-s17122" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s17123" xml:space="preserve">tradidi-<lb/>mus. </s> <s xml:id="echoid-s17124" xml:space="preserve">Ita ergo ſcribit. </s> <s xml:id="echoid-s17125" xml:space="preserve">Sit triangulum æquilaterum DMN, <lb/>in quo perpendicularis DO, ad baſem ſecetur bifariam in P. <lb/></s> <s xml:id="echoid-s17126" xml:space="preserve">Deſcripto deinde ex M, per N, D, circulo, quem ſecet per- <pb o="364" file="392" n="392" rhead="GEOMETR. PRACT."/> pendicularis PQ, in Q, iungantur rectæ QN, MQ. </s> <s xml:id="echoid-s17127" xml:space="preserve">Dicit igitur, in Iſoſcele MNQ, <lb/>vtrumlibet angulorum N, Q, triplum eſſe anguli M. </s> <s xml:id="echoid-s17128" xml:space="preserve">quod falſum eſſe, hincin-<lb/>telligi poteſt. </s> <s xml:id="echoid-s17129" xml:space="preserve">Demiſſa perpendiculari QR, pro ſinu arcus QN, vel anguli N, <lb/>poſito ſinu toto MQ, vel MN, 10000000. </s> <s xml:id="echoid-s17130" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Quoniam latus DN, potentia ſe- <anchor type="note" xlink:label="note-392-01a" xlink:href="note-392-01"/> ſquitertium eſt perpendicularis D O, ſi fiat vt 4. </s> <s xml:id="echoid-s17131" xml:space="preserve">ad 3. </s> <s xml:id="echoid-s17132" xml:space="preserve">ita 100000000000000. <lb/></s> <s xml:id="echoid-s17133" xml:space="preserve">quadratum lateris DN, ad aliud reperietur quadratum DO, 75000000000000. </s> <s xml:id="echoid-s17134" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-392-02a" xlink:href="note-392-02"/> <anchor type="note" xlink:href="" symbol="b"/> quod cum ſit quadruplum quadrati P O, vel Q R, erit quadratum Q R, 18750000000000. </s> <s xml:id="echoid-s17135" xml:space="preserve">ipſumque latus QR, erit 4330127. </s> <s xml:id="echoid-s17136" xml:space="preserve">verò minus, vel 4330128. <lb/></s> <s xml:id="echoid-s17137" xml:space="preserve">vero maius, cui in tabula ſinuum (adhibita parte proportionali) reſpondent <lb/>grad. </s> <s xml:id="echoid-s17138" xml:space="preserve">25. </s> <s xml:id="echoid-s17139" xml:space="preserve">min. </s> <s xml:id="echoid-s17140" xml:space="preserve">39. </s> <s xml:id="echoid-s17141" xml:space="preserve">ſec. </s> <s xml:id="echoid-s17142" xml:space="preserve">32. </s> <s xml:id="echoid-s17143" xml:space="preserve">pro arcu QN, vel angulo MNQ, quo ablato ex duobus <lb/>rectis, ſiue ex grad. </s> <s xml:id="echoid-s17144" xml:space="preserve">180. </s> <s xml:id="echoid-s17145" xml:space="preserve">reliqua erit ſumma angulorum æqualium ad baſem QN, <lb/>grad. </s> <s xml:id="echoid-s17146" xml:space="preserve">154. </s> <s xml:id="echoid-s17147" xml:space="preserve">min. </s> <s xml:id="echoid-s17148" xml:space="preserve">20. </s> <s xml:id="echoid-s17149" xml:space="preserve">ſec. </s> <s xml:id="echoid-s17150" xml:space="preserve">28. </s> <s xml:id="echoid-s17151" xml:space="preserve">atque idcirco vterque complectetur grad. </s> <s xml:id="echoid-s17152" xml:space="preserve">77. </s> <s xml:id="echoid-s17153" xml:space="preserve">min. </s> <s xml:id="echoid-s17154" xml:space="preserve">10. </s> <s xml:id="echoid-s17155" xml:space="preserve"><lb/>ſec. </s> <s xml:id="echoid-s17156" xml:space="preserve">14. </s> <s xml:id="echoid-s17157" xml:space="preserve">qui maior eſt, quam triplus anguli M N Q. </s> <s xml:id="echoid-s17158" xml:space="preserve">grad. </s> <s xml:id="echoid-s17159" xml:space="preserve">25. </s> <s xml:id="echoid-s17160" xml:space="preserve">min. </s> <s xml:id="echoid-s17161" xml:space="preserve">39. </s> <s xml:id="echoid-s17162" xml:space="preserve">ſec. </s> <s xml:id="echoid-s17163" xml:space="preserve">32. </s> <s xml:id="echoid-s17164" xml:space="preserve">cum <lb/>hic angulus triplicatus efficiat tantummodo grad. </s> <s xml:id="echoid-s17165" xml:space="preserve">76. </s> <s xml:id="echoid-s17166" xml:space="preserve">min. </s> <s xml:id="echoid-s17167" xml:space="preserve">58. </s> <s xml:id="echoid-s17168" xml:space="preserve">ſec. </s> <s xml:id="echoid-s17169" xml:space="preserve">36. </s> <s xml:id="echoid-s17170" xml:space="preserve">Falſum <lb/>ergo eſt, quod Candalla nititur probare. </s> <s xml:id="echoid-s17171" xml:space="preserve">Paralogiſmos tum Caroli Mariani, <lb/>tum Candallæ, quos committunt, non eſt huius loci manifeſtare: </s> <s xml:id="echoid-s17172" xml:space="preserve">ſatis nobis <lb/>eſt, indicaſſe eos non rectè deſcripſiſſe heptagonũ æquilaterũ, & </s> <s xml:id="echoid-s17173" xml:space="preserve">æquiangulũ.</s> <s xml:id="echoid-s17174" xml:space="preserve"/> </p> <div xml:id="echoid-div1055" type="float" level="2" n="4"> <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/xxxxxxxx/figures/391-01"/> </figure> <note symbol="a" position="left" xlink:label="note-392-01" xlink:href="note-392-01a" xml:space="preserve">12. quarti-<lb/>dectmi.</note> <note symbol="b" position="left" xlink:label="note-392-02" xlink:href="note-392-02a" xml:space="preserve">ſchol. 4. <lb/>ſecundi.</note> </div> </div> <div xml:id="echoid-div1057" type="section" level="1" n="378"> <head xml:id="echoid-head405" xml:space="preserve">THEOR. 13. PROPOS. 31.</head> <p> <s xml:id="echoid-s17175" xml:space="preserve">OCTOGONVM æquilaterum & </s> <s xml:id="echoid-s17176" xml:space="preserve">æquiangulum circulo inſcriptum <lb/>medio loco proportionale eſt inter quadratum eidem circulo circũ-<lb/>ſcriptum, & </s> <s xml:id="echoid-s17177" xml:space="preserve">quadratum inſcriptum.</s> <s xml:id="echoid-s17178" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17179" xml:space="preserve"><emph style="sc">Hoc</emph> Theorema eſt Orontij, quod facilè ita demonſtrabitur. </s> <s xml:id="echoid-s17180" xml:space="preserve">Sit circu-<lb/>lus ABCD, cuius centrum E; </s> <s xml:id="echoid-s17181" xml:space="preserve">duæ diametri AC, BD, ſecantes ſe ſe in E, ad angu-<lb/>los rectos. </s> <s xml:id="echoid-s17182" xml:space="preserve">Iunctis ergo rectis AB, B C, C D, D A, erit quadratum circulo inſcri-<lb/>ptum ABCD, vt ex demonſtratione propoſ. </s> <s xml:id="echoid-s17183" xml:space="preserve">6. </s> <s xml:id="echoid-s17184" xml:space="preserve">lib. </s> <s xml:id="echoid-s17185" xml:space="preserve">4. </s> <s xml:id="echoid-s17186" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s17187" xml:space="preserve">conſtat. </s> <s xml:id="echoid-s17188" xml:space="preserve">Ducantur <lb/> <anchor type="figure" xlink:label="fig-392-01a" xlink:href="fig-392-01"/> quoque per A, B, C, D, perpendiculares ad diame-<lb/>tros coeuntes in F, G, H, I, eritque quadratum cir-<lb/>cumſcriptum F G H I, vt patet ex demonſtratione <lb/>propoſ. </s> <s xml:id="echoid-s17189" xml:space="preserve">7. </s> <s xml:id="echoid-s17190" xml:space="preserve">lib. </s> <s xml:id="echoid-s17191" xml:space="preserve">4. </s> <s xml:id="echoid-s17192" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s17193" xml:space="preserve">Ductis autem diametris <lb/>FH, GI, <anchor type="note" xlink:href="" symbol="c"/> ſecabuntur quadrantes AB, BC, CD, DA, <anchor type="note" xlink:label="note-392-03a" xlink:href="note-392-03"/> bifariam; </s> <s xml:id="echoid-s17194" xml:space="preserve">propterea quod anguli in centro ſunt o-<lb/> <anchor type="note" xlink:label="note-392-04a" xlink:href="note-392-04"/> mnes æquales, <anchor type="note" xlink:href="" symbol="d"/> nimirum ſemirecti: </s> <s xml:id="echoid-s17195" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ac proinde &</s> <s xml:id="echoid-s17196" xml:space="preserve"> latera quadrati inſcripti diuiſa erunt bifariam, & </s> <s xml:id="echoid-s17197" xml:space="preserve">ad <lb/> <anchor type="note" xlink:label="note-392-05a" xlink:href="note-392-05"/> angulos rectos. </s> <s xml:id="echoid-s17198" xml:space="preserve">Et ſi iungantur rectæ, AK, KB, &</s> <s xml:id="echoid-s17199" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s17200" xml:space="preserve">deſcriptum erit octogonum intra circulum. </s> <s xml:id="echoid-s17201" xml:space="preserve">Dico <lb/>ita eſſe quadratum exterius ad octogonum, vt o-<lb/>ctogonum ad quadratum interius. </s> <s xml:id="echoid-s17202" xml:space="preserve">Quoniam enim <lb/>triangula AEF, EAL, ęquiangula ſunt, quod rectos <lb/>habeant angulos, & </s> <s xml:id="echoid-s17203" xml:space="preserve">ſemirectos: </s> <s xml:id="echoid-s17204" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Erit E F, ad F A, hoc eſt, ad EK, (eſt namque EK, ipſi EA, hoc eſt, ipſi AF, æqualis) vt EA, hoc eſt, <lb/> <anchor type="note" xlink:label="note-392-06a" xlink:href="note-392-06"/> vt EK, ad AL, hoc eſt, ad EL,<anchor type="note" xlink:href="" symbol="g"/> quod AL, EL, ſint æquales, propter angulos ſe- <anchor type="note" xlink:label="note-392-07a" xlink:href="note-392-07"/> mirectos A, E, intriangulo AEL. </s> <s xml:id="echoid-s17205" xml:space="preserve">Sunt ergo tres rectę EF, EK, EL, continue pro-<lb/>portio les. </s> <s xml:id="echoid-s17206" xml:space="preserve">Igitur & </s> <s xml:id="echoid-s17207" xml:space="preserve">triangula AEF, AEK, AEL, continue erunt proportiona-<lb/> <anchor type="note" xlink:label="note-392-08a" xlink:href="note-392-08"/> lia: </s> <s xml:id="echoid-s17208" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> na baſibus EF; </s> <s xml:id="echoid-s17209" xml:space="preserve">EK, EL, ſint proportionalia; </s> <s xml:id="echoid-s17210" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Ac proinde & </s> <s xml:id="echoid-s17211" xml:space="preserve">eorũ octu- <anchor type="note" xlink:label="note-392-09a" xlink:href="note-392-09"/> <pb o="365" file="393" n="393" rhead="LIBER OCTAVVS."/> pla continuerunt proportionalia, quadratum videlicet FGHI, octogonum A-<lb/>KBCDA, & </s> <s xml:id="echoid-s17212" xml:space="preserve">quadratum ABCD; </s> <s xml:id="echoid-s17213" xml:space="preserve">quippe cum prædicta triangula ſint harum fi-<lb/>gurarum octauæ partes, vtliquet. </s> <s xml:id="echoid-s17214" xml:space="preserve">Octogonum igitur medio loco proportiona-<lb/>le eſt inter quadrata FGHI, ABCD, quod demonſtrandum erat.</s> <s xml:id="echoid-s17215" xml:space="preserve"/> </p> <div xml:id="echoid-div1057" type="float" level="2" n="1"> <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/xxxxxxxx/figures/392-01"/> </figure> <note symbol="c" position="left" xlink:label="note-392-03" xlink:href="note-392-03a" xml:space="preserve">27. tertij.</note> <note symbol="d" position="left" xlink:label="note-392-04" xlink:href="note-392-04a" xml:space="preserve">ſchol. 34. <lb/>primi.</note> <note symbol="e" position="left" xlink:label="note-392-05" xlink:href="note-392-05a" xml:space="preserve">ſchol. 27. <lb/>tertij.</note> <note symbol="f" position="left" xlink:label="note-392-06" xlink:href="note-392-06a" xml:space="preserve">4. ſexti.</note> <note symbol="g" position="left" xlink:label="note-392-07" xlink:href="note-392-07a" xml:space="preserve">6. primi.</note> <note symbol="h" position="left" xlink:label="note-392-08" xlink:href="note-392-08a" xml:space="preserve">1. ſexti.</note> <note symbol="i" position="left" xlink:label="note-392-09" xlink:href="note-392-09a" xml:space="preserve">15 quinti@</note> </div> </div> <div xml:id="echoid-div1059" type="section" level="1" n="379"> <head xml:id="echoid-head406" xml:space="preserve">THEOR. 14. PROPOS. 32.</head> <p> <s xml:id="echoid-s17216" xml:space="preserve">SI ex diametro quadrati detrahatur ipſius latus: </s> <s xml:id="echoid-s17217" xml:space="preserve">Reliqua linea erit latus <lb/>alterius quadrati, cuius diameter eſt linea, quæ relinquitur, ſi latus in-<lb/>uentum bis ex diametro prioris quadrati auferatur; </s> <s xml:id="echoid-s17218" xml:space="preserve">vel ſi idem latus <lb/>inuentum ex prioris quadrati latere tollatur.</s> <s xml:id="echoid-s17219" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17220" xml:space="preserve"><emph style="sc">Ex</emph> diametro BD, quadrati ABCD, abſcindatur recta BE, lateri AB, æqualis, <lb/>& </s> <s xml:id="echoid-s17221" xml:space="preserve">ex eadem diametro dematur reliqua DE, bis vſque ad F, ita vt EF, ſit ipſi DE, <lb/>æqualis: </s> <s xml:id="echoid-s17222" xml:space="preserve">vel (quod idem eſt) reliqua DE, ſiue EF, illi æqualis ex latere AB, hoc <lb/>eſt, ex BE, auferatur. </s> <s xml:id="echoid-s17223" xml:space="preserve">Dico DE, vel EF, latus eſſe quadrati, cuius diameter BF. <lb/></s> <s xml:id="echoid-s17224" xml:space="preserve">Ductis enim per F, rectis GH, IK, parallelis ipſis AD, AB; </s> <s xml:id="echoid-s17225" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> erunt G K, H I, circa <anchor type="note" xlink:label="note-393-01a" xlink:href="note-393-01"/> diametrum quadrata. </s> <s xml:id="echoid-s17226" xml:space="preserve">Dico rectam EF, vel DE, æqualem eſ-<lb/> <anchor type="figure" xlink:label="fig-393-01a" xlink:href="fig-393-01"/> ſe lateri B G, quadrati G K, cuius diameter B F, reliqua fuit <lb/>poſt detractionem D E, bis ex diametro B D. </s> <s xml:id="echoid-s17227" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Quoniam e- <anchor type="note" xlink:label="note-393-02a" xlink:href="note-393-02"/> nim quadratum ex D F, duplum eſt quadrati ex I F, ſiue ex <lb/>A G, <anchor type="note" xlink:href="" symbol="c"/> & </s> <s xml:id="echoid-s17228" xml:space="preserve">quadruplum quadrati ex E F; </s> <s xml:id="echoid-s17229" xml:space="preserve">ſi quadratum ex D F, <anchor type="note" xlink:label="note-393-03a" xlink:href="note-393-03"/> ponatur 4. </s> <s xml:id="echoid-s17230" xml:space="preserve">erit quadratum ex A G, 2. </s> <s xml:id="echoid-s17231" xml:space="preserve">& </s> <s xml:id="echoid-s17232" xml:space="preserve">quadratum ex EF, 1. <lb/></s> <s xml:id="echoid-s17233" xml:space="preserve">Ac proinde quadratum ex AG, duplum erit quadrati ex EF. </s> <s xml:id="echoid-s17234" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-393-04a" xlink:href="note-393-04"/> <anchor type="note" xlink:href="" symbol="d"/> Eſt autem & </s> <s xml:id="echoid-s17235" xml:space="preserve">quadratum ex B F, quadrati ex B G, duplum. </s> <s xml:id="echoid-s17236" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Igitur erit vt re- cta A G, ad rectam EF, ita recta BF, ad rectam BG; </s> <s xml:id="echoid-s17237" xml:space="preserve">quando quidem quadrata ea-<lb/> <anchor type="note" xlink:label="note-393-05a" xlink:href="note-393-05"/> rum proportionalia ſunt, habentia nimirum proportionem duplam. </s> <s xml:id="echoid-s17238" xml:space="preserve">Capiatur <lb/>B L, ipſi BF, æqualis ita vt reliqua LA, reliquæ FE, æqualis ſit. </s> <s xml:id="echoid-s17239" xml:space="preserve">Erit igitur quoq; <lb/></s> <s xml:id="echoid-s17240" xml:space="preserve">AG, ad AL, vt BF, ſiue BL, ad BG; </s> <s xml:id="echoid-s17241" xml:space="preserve">Et diuidendo GL, ad LA, vt GL, ad BG. </s> <s xml:id="echoid-s17242" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Quo <anchor type="note" xlink:label="note-393-06a" xlink:href="note-393-06"/> circa AL, ſiue EF, & </s> <s xml:id="echoid-s17243" xml:space="preserve">BG, æquales ſunt. </s> <s xml:id="echoid-s17244" xml:space="preserve">quod erat oſtendendum.</s> <s xml:id="echoid-s17245" xml:space="preserve"/> </p> <div xml:id="echoid-div1059" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-393-01" xlink:href="note-393-01a" xml:space="preserve">coroll. 4. <lb/>ſecundi.</note> <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/xxxxxxxx/figures/393-01"/> </figure> <note symbol="b" position="right" xlink:label="note-393-02" xlink:href="note-393-02a" xml:space="preserve">ſchol. 47. <lb/>primi.</note> <note symbol="c" position="right" xlink:label="note-393-03" xlink:href="note-393-03a" xml:space="preserve">ſchol. 4. ſe-<lb/>cundi.</note> <note symbol="d" position="right" xlink:label="note-393-04" xlink:href="note-393-04a" xml:space="preserve">ſchol. 47. <lb/>primi.</note> <note symbol="e" position="right" xlink:label="note-393-05" xlink:href="note-393-05a" xml:space="preserve">22. ſexti.</note> <note symbol="f" position="right" xlink:label="note-393-06" xlink:href="note-393-06a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div1061" type="section" level="1" n="380"> <head xml:id="echoid-head407" xml:space="preserve">PROBL. 19. PROPOS. 33.</head> <p> <s xml:id="echoid-s17246" xml:space="preserve">OCTOGONVM æquilaterum, & </s> <s xml:id="echoid-s17247" xml:space="preserve">æquiangulum ad datam altitudi-<lb/>nem, latitudinemue conſtituere.</s> <s xml:id="echoid-s17248" xml:space="preserve"/> </p> <figure> <image file="393-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/393-02"/> </figure> <p> <s xml:id="echoid-s17249" xml:space="preserve"><emph style="sc">Sit</emph> ad altitudinem datam AB, conſtruendum <lb/>octogonum æquilaterum, & </s> <s xml:id="echoid-s17250" xml:space="preserve">æquiangulum. </s> <s xml:id="echoid-s17251" xml:space="preserve">De-<lb/>ſcripto ex AB, quadrato ABCD, ductiſque diame-<lb/>tris AC, BD, ſe in E, ſecantibus bifariam, & </s> <s xml:id="echoid-s17252" xml:space="preserve">ad an-<lb/>gulos rectos; </s> <s xml:id="echoid-s17253" xml:space="preserve">abſcindantur ad interuallum EA, ex <lb/>quatuor angulis quadratirectæ æquales A F, A G; <lb/></s> <s xml:id="echoid-s17254" xml:space="preserve">DH, DI; </s> <s xml:id="echoid-s17255" xml:space="preserve">BK, BL; </s> <s xml:id="echoid-s17256" xml:space="preserve">CM, CN, iunganturq; </s> <s xml:id="echoid-s17257" xml:space="preserve">rectæ HK, <lb/>G M, L I, N F. </s> <s xml:id="echoid-s17258" xml:space="preserve">Dico octogonum FHKGMLIN, <lb/>eſſe æquilaterum, & </s> <s xml:id="echoid-s17259" xml:space="preserve">æquiangulum. </s> <s xml:id="echoid-s17260" xml:space="preserve">Quoniam <lb/>enim rectæ A H, A K; </s> <s xml:id="echoid-s17261" xml:space="preserve">B G, B M; </s> <s xml:id="echoid-s17262" xml:space="preserve">C L, C I; </s> <s xml:id="echoid-s17263" xml:space="preserve"><lb/>D F, D N, relinquuntur poſt detractionem <pb o="366" file="394" n="394" rhead="GEOMETR. PRACT."/> <anchor type="figure" xlink:label="fig-394-01a" xlink:href="fig-394-01"/> rectæ A B, ex lateribus æqualibus quadrati ABCD, <lb/>ipſæ inter ſe æquales erunt; </s> <s xml:id="echoid-s17264" xml:space="preserve">ideoq; </s> <s xml:id="echoid-s17265" xml:space="preserve">& </s> <s xml:id="echoid-s17266" xml:space="preserve">earũ, quadra-<lb/>ta erunt æqualia.</s> <s xml:id="echoid-s17267" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Cũ ergo quadratũ ex KH, ęquale <anchor type="note" xlink:label="note-394-01a" xlink:href="note-394-01"/> ſit quadratis ex AH, AK, ipſum duplũ erit tã qua-<lb/>drati ex AH, ꝗ̃ quadrati ex AK. </s> <s xml:id="echoid-s17268" xml:space="preserve">Quia verò A B, dia-<lb/>meter eſt quadratiex AE, deſcripti, abſciſſaq; </s> <s xml:id="echoid-s17269" xml:space="preserve">eſt re-<lb/> <anchor type="note" xlink:label="note-394-02a" xlink:href="note-394-02"/> cta BK, lateri AE, æqualis; </s> <s xml:id="echoid-s17270" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit reliqua AK, lat<emph style="sub">9</emph> qua- drati, cuius diameter GK, quæ relin quitur poſt de-<lb/>tractionem ipſius A K, bis ex diametro A B, vel ex <lb/>latere A G, ſemel. </s> <s xml:id="echoid-s17271" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Igitur & </s> <s xml:id="echoid-s17272" xml:space="preserve">quadratum ex G K, duplum erit quadrati ex K A: </s> <s xml:id="echoid-s17273" xml:space="preserve">ac proinde quadrata <lb/> <anchor type="note" xlink:label="note-394-03a" xlink:href="note-394-03"/> ex K H, K G, æqualia inter ſe erunt; </s> <s xml:id="echoid-s17274" xml:space="preserve">ideo que & </s> <s xml:id="echoid-s17275" xml:space="preserve">re-<lb/>ctæ KH, KG, æquales erunt. </s> <s xml:id="echoid-s17276" xml:space="preserve">Eadem ratione oſten-<lb/>demus, eandem GK, æqualem eſſe rectæ G M; </s> <s xml:id="echoid-s17277" xml:space="preserve">& </s> <s xml:id="echoid-s17278" xml:space="preserve">G M, æqualem rectæ M L, & </s> <s xml:id="echoid-s17279" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-394-04a" xlink:href="note-394-04"/> ſic de cæteris. </s> <s xml:id="echoid-s17280" xml:space="preserve">Æquilaterum ergo eſt octogonum. </s> <s xml:id="echoid-s17281" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Quoniam autem bini <anchor type="note" xlink:label="note-394-05a" xlink:href="note-394-05"/> anguliad H, K, G, M, L, I, N, F, æquales ſunt duobus rectis; </s> <s xml:id="echoid-s17282" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> ſuntque angulia- <anchor type="note" xlink:label="note-394-06a" xlink:href="note-394-06"/> cuti verſus angulos quadrati omnes inter ſe æquales: </s> <s xml:id="echoid-s17283" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> immo ſemirecti, quod KH, GM, &</s> <s xml:id="echoid-s17284" xml:space="preserve">c. </s> <s xml:id="echoid-s17285" xml:space="preserve">ſint diametri quadratorum exlateribus AH, GB, &</s> <s xml:id="echoid-s17286" xml:space="preserve">c. </s> <s xml:id="echoid-s17287" xml:space="preserve">deſcripto-<lb/>rum: </s> <s xml:id="echoid-s17288" xml:space="preserve">Erunt reliqui anguli obtuſi in octogono æquales; </s> <s xml:id="echoid-s17289" xml:space="preserve">ideoque octogonum <lb/>æquiangulum etiam eſt. </s> <s xml:id="echoid-s17290" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17291" xml:space="preserve"/> </p> <div xml:id="echoid-div1061" type="float" level="2" n="1"> <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/xxxxxxxx/figures/394-01"/> </figure> <note symbol="a" position="left" xlink:label="note-394-01" xlink:href="note-394-01a" xml:space="preserve">47. primi.</note> <note symbol="b" position="left" xlink:label="note-394-02" xlink:href="note-394-02a" xml:space="preserve">32. hui{us}.</note> <note symbol="c" position="left" xlink:label="note-394-03" xlink:href="note-394-03a" xml:space="preserve">ſchol. 47. <lb/>primi.</note> <note symbol="d" position="left" xlink:label="note-394-04" xlink:href="note-394-04a" xml:space="preserve">13 primi.</note> <note symbol="e" position="left" xlink:label="note-394-05" xlink:href="note-394-05a" xml:space="preserve">4. primi.</note> <note symbol="f" position="left" xlink:label="note-394-06" xlink:href="note-394-06a" xml:space="preserve">ſchol. 34. <lb/>primi.</note> </div> </div> <div xml:id="echoid-div1063" type="section" level="1" n="381"> <head xml:id="echoid-head408" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s17292" xml:space="preserve"><emph style="sc">Hæc</emph> praxis, quam antè aliquot annos à quo dam Architecto ſine demon-<lb/>ſtrationetamen accepi, pulcherrima eſt: </s> <s xml:id="echoid-s17293" xml:space="preserve">quippe quæ non requirat diuiſionem <lb/>circuli in octo partes æquales, & </s> <s xml:id="echoid-s17294" xml:space="preserve">deſcribat octogonum ad datam altitudinem, <lb/>latitudinemuè, vt patet. </s> <s xml:id="echoid-s17295" xml:space="preserve">Quam praxem vt demonſtrarem, oportuit prius de-<lb/>monſtrare præcedens theorema. </s> <s xml:id="echoid-s17296" xml:space="preserve">Ex eo enim facile problema propoſitum con-<lb/>ficitur, vt patuit.</s> <s xml:id="echoid-s17297" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1064" type="section" level="1" n="382"> <head xml:id="echoid-head409" xml:space="preserve">PROBL. 20. PROPOS. 34.</head> <p> <s xml:id="echoid-s17298" xml:space="preserve">AMBITVM terræ ex edito aliquo monte metiri.</s> <s xml:id="echoid-s17299" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17300" xml:space="preserve"><emph style="sc">Circa</emph> finem cap. </s> <s xml:id="echoid-s17301" xml:space="preserve">1. </s> <s xml:id="echoid-s17302" xml:space="preserve">ſphęræ Ioan. </s> <s xml:id="echoid-s17303" xml:space="preserve">de Sacro boſco propoſui rationem, qua <lb/>Franciſcus Maurolycus ambitum terræ ex edito aliquo monte inueſtigare do-<lb/>cuit, quæ talis eſt. </s> <s xml:id="echoid-s17304" xml:space="preserve">Sit circulus terræ B C D, in quo eligatur editiſsimus aliquis <lb/>mons, (ipſe in Sicilia montem Ætnam ad hoc negotium cenſuit eligendum) <lb/>cuius altitudo AB, inquiratur vel per Quadrantem, vt lib. </s> <s xml:id="echoid-s17305" xml:space="preserve">2. </s> <s xml:id="echoid-s17306" xml:space="preserve">problem. </s> <s xml:id="echoid-s17307" xml:space="preserve">2. </s> <s xml:id="echoid-s17308" xml:space="preserve">3. </s> <s xml:id="echoid-s17309" xml:space="preserve">& </s> <s xml:id="echoid-s17310" xml:space="preserve">4. <lb/></s> <s xml:id="echoid-s17311" xml:space="preserve"> <anchor type="figure" xlink:label="fig-394-02a" xlink:href="fig-394-02"/> docuimus, vel per Quadratum Geometricum, vt lib. </s> <s xml:id="echoid-s17312" xml:space="preserve">3. </s> <s xml:id="echoid-s17313" xml:space="preserve">pro-<lb/>blem. </s> <s xml:id="echoid-s17314" xml:space="preserve">6. </s> <s xml:id="echoid-s17315" xml:space="preserve">7. </s> <s xml:id="echoid-s17316" xml:space="preserve">8. </s> <s xml:id="echoid-s17317" xml:space="preserve">& </s> <s xml:id="echoid-s17318" xml:space="preserve">9. </s> <s xml:id="echoid-s17319" xml:space="preserve">vel potius vt in ſcholio problem. </s> <s xml:id="echoid-s17320" xml:space="preserve">7. </s> <s xml:id="echoid-s17321" xml:space="preserve">ac 9. <lb/></s> <s xml:id="echoid-s17322" xml:space="preserve">tradidimus. </s> <s xml:id="echoid-s17323" xml:space="preserve">Deinde ex A, vertice montis menſuretur totum <lb/>illud ſpacium pelagi, ſeu terrę, (vbi tamen montes nonſint) <lb/>quod inde conſpicitur, ita vt radius AC, maris vel terræ ſu-<lb/>perficiẽ contingat in C. </s> <s xml:id="echoid-s17324" xml:space="preserve">Hoc autẽ fiet per ea, quæ in proble-<lb/>matib. </s> <s xml:id="echoid-s17325" xml:space="preserve">citatis tradita ſunt. </s> <s xml:id="echoid-s17326" xml:space="preserve">Ex his poſtea explorat magnitudi-<lb/>nẽ lineæ tangẽtis A C,<anchor type="note" xlink:href="" symbol="g"/> ꝓ pterea ꝙ ei<emph style="sub">9</emph> quadrato æqualia ſunt <anchor type="note" xlink:label="note-394-07a" xlink:href="note-394-07"/> quadrata AB, BC, (ſumpto ſpacio BC, ꝓ linea recta) <anchor type="note" xlink:href="" symbol="h"/> cuius <anchor type="note" xlink:label="note-394-08a" xlink:href="note-394-08"/> <pb o="367" file="395" n="395" rhead="LIBER OCTAVVS."/> quadratum æquale eſt rectangulo ſub AD, AB, quo diuiſo per AB, altitudinem <lb/>montis, prodibit in Quotiente recta A D; </s> <s xml:id="echoid-s17327" xml:space="preserve">ex qua ſi dematur altitudo montis <lb/>A B, nota relinquetur diameter terræ BD. </s> <s xml:id="echoid-s17328" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Ac proinde circumferentia B C D, <anchor type="note" xlink:label="note-395-01a" xlink:href="note-395-01"/> cognita fiet.</s> <s xml:id="echoid-s17329" xml:space="preserve"/> </p> <div xml:id="echoid-div1064" type="float" level="2" n="1"> <figure xlink:label="fig-394-02" xlink:href="fig-394-02a"> <image file="394-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/394-02"/> </figure> <note symbol="g" position="left" xlink:label="note-394-07" xlink:href="note-394-07a" xml:space="preserve">47. primi.</note> <note symbol="h" position="left" xlink:label="note-394-08" xlink:href="note-394-08a" xml:space="preserve">36. tertij.</note> <note symbol="a" position="right" xlink:label="note-395-01" xlink:href="note-395-01a" xml:space="preserve">coroll. 2. de <lb/>Dimens. cir-<lb/>culi lib. 4. hu-<lb/>i{us}.</note> </div> <p> <s xml:id="echoid-s17330" xml:space="preserve"><emph style="sc">Sed</emph> quia in hac ratione metiendi ambitus terreſtris aſſumitur, arcum B C, <lb/>à linea recta non diſferre. </s> <s xml:id="echoid-s17331" xml:space="preserve">quod verum non eſt, quando mons tam altus eſt, vt <lb/>ſpacium 200. </s> <s xml:id="echoid-s17332" xml:space="preserve">vel 300. </s> <s xml:id="echoid-s17333" xml:space="preserve">milliariorum cerni poſsit, quod tunc arcus BC, iuxta am-<lb/>bitum à Ptolomæo poſitum contineat grad. </s> <s xml:id="echoid-s17334" xml:space="preserve">3. </s> <s xml:id="echoid-s17335" xml:space="preserve">min. </s> <s xml:id="echoid-s17336" xml:space="preserve">11. </s> <s xml:id="echoid-s17337" xml:space="preserve">vel grad. </s> <s xml:id="echoid-s17338" xml:space="preserve">4. </s> <s xml:id="echoid-s17339" xml:space="preserve">min. </s> <s xml:id="echoid-s17340" xml:space="preserve">48. </s> <s xml:id="echoid-s17341" xml:space="preserve">Ac <lb/>proinde non rectè linea tangens A C, ex lateribus A B, BC, colligitur. </s> <s xml:id="echoid-s17342" xml:space="preserve">Adde <lb/>quod per ptoblemata lib. </s> <s xml:id="echoid-s17343" xml:space="preserve">2. </s> <s xml:id="echoid-s17344" xml:space="preserve">& </s> <s xml:id="echoid-s17345" xml:space="preserve">3. </s> <s xml:id="echoid-s17346" xml:space="preserve">citata inuenitur perpendicularis BE, in plano, <lb/>ad quod mons eſt ad angulos rectos: </s> <s xml:id="echoid-s17347" xml:space="preserve">Redigemus rationem hanc ad meliorem <lb/>formam multis viis hoc modo. </s> <s xml:id="echoid-s17348" xml:space="preserve">Deprehenſo angulo A, per Quadrantem, vel <lb/>Quadratum, quando radius viſualis per dioptram circulum terræ tangit. </s> <s xml:id="echoid-s17349" xml:space="preserve">Quod <lb/>tum denique certiſsimè fiet, cum per dioptram conſpicitur Sol, aut alia ſtella, <lb/>quando oritur, vel occidit. </s> <s xml:id="echoid-s17350" xml:space="preserve">Deprehenſo, inquam, angulo A, inuenienda erit <lb/>perpendicularis BE, per problemata paulò ante citata. </s> <s xml:id="echoid-s17351" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Etrecta AE, ex duabus <anchor type="note" xlink:label="note-395-02a" xlink:href="note-395-02"/> A B, B E. </s> <s xml:id="echoid-s17352" xml:space="preserve">Si enim ad A E, adiicietur BE, hoc eſt, EC, <anchor type="note" xlink:href="" symbol="c"/> quæ ipſi BE, æqualis eſt, <anchor type="note" xlink:label="note-395-03a" xlink:href="note-395-03"/> nota fiet tota tangens AC, ex qua, vt ſupra dictum eſt, & </s> <s xml:id="echoid-s17353" xml:space="preserve">diameter terræ BD, & </s> <s xml:id="echoid-s17354" xml:space="preserve"><lb/>circumferentia inueſtigabitur. </s> <s xml:id="echoid-s17355" xml:space="preserve">Quin etiam cognito angulo A, ac proinde & </s> <s xml:id="echoid-s17356" xml:space="preserve">eius <lb/>complemento E, <anchor type="note" xlink:href="" symbol="d"/> reperietur tam latus B E, <anchor type="note" xlink:href="" symbol="e"/> quam baſis A E, ſineproblemati- <anchor type="note" xlink:label="note-395-04a" xlink:href="note-395-04"/> bus ex lib. </s> <s xml:id="echoid-s17357" xml:space="preserve">2. </s> <s xml:id="echoid-s17358" xml:space="preserve">& </s> <s xml:id="echoid-s17359" xml:space="preserve">3. </s> <s xml:id="echoid-s17360" xml:space="preserve">citatis, &</s> <s xml:id="echoid-s17361" xml:space="preserve">c.</s> <s xml:id="echoid-s17362" xml:space="preserve"/> </p> <div xml:id="echoid-div1065" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-395-02" xlink:href="note-395-02a" xml:space="preserve">47. primi.</note> <note symbol="c" position="right" xlink:label="note-395-03" xlink:href="note-395-03a" xml:space="preserve">2. coroll. 36. <lb/>tertii.</note> <note symbol="d" position="right" xlink:label="note-395-04" xlink:href="note-395-04a" xml:space="preserve">4. triang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s17363" xml:space="preserve"><emph style="sc">Vel</emph> ſic agemus. </s> <s xml:id="echoid-s17364" xml:space="preserve">Cognito per dioptram angulo A, cognitus etiam erit (du-<lb/> <anchor type="note" xlink:label="note-395-05a" xlink:href="note-395-05"/> cta recta F C, <anchor type="note" xlink:href="" symbol="f"/> quæ ad A C, perpendicularis erit) angulus F, eius complemen- tum in centro. </s> <s xml:id="echoid-s17365" xml:space="preserve">Quia verò ducta recta FE, duo latera EC, CF, duobus lateribus <lb/> <anchor type="note" xlink:label="note-395-06a" xlink:href="note-395-06"/> E B, BF, æqualia ſunt, comprehenduntque angulos æquales, nemperectos: <lb/></s> <s xml:id="echoid-s17366" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="g"/> erunt anguli ad F, æquales. </s> <s xml:id="echoid-s17367" xml:space="preserve">Cum ergo totus angulus B F C, cognitus ſit, vt <anchor type="note" xlink:label="note-395-07a" xlink:href="note-395-07"/> proximè diximus; </s> <s xml:id="echoid-s17368" xml:space="preserve">cognitus etiam erit BFE, tanquam ſemiſsis ipſius: </s> <s xml:id="echoid-s17369" xml:space="preserve">ac proin-<lb/>de & </s> <s xml:id="echoid-s17370" xml:space="preserve">eius complementum B E F, notum erit. </s> <s xml:id="echoid-s17371" xml:space="preserve">Igitur in triangulo ABE, ex an-<lb/>gulis A, E, & </s> <s xml:id="echoid-s17372" xml:space="preserve">latere AE, <anchor type="note" xlink:href="" symbol="h"/> reperietur BE, in partibus altitudinis montis A B, no- <anchor type="note" xlink:label="note-395-08a" xlink:href="note-395-08"/> tæ. </s> <s xml:id="echoid-s17373" xml:space="preserve">Atque eodem modo in triangulo B E F, ex angulis E, F, & </s> <s xml:id="echoid-s17374" xml:space="preserve">latere BE, cog-<lb/>noſcetur ſemidiameter B F, in partibus lateris BE, hoc eſt, in partibus altitudinis <lb/>montis A B; </s> <s xml:id="echoid-s17375" xml:space="preserve">ideoque & </s> <s xml:id="echoid-s17376" xml:space="preserve">tota diameter B D, nota fiet, & </s> <s xml:id="echoid-s17377" xml:space="preserve">ex hac ambitus terræ. <lb/></s> <s xml:id="echoid-s17378" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17379" xml:space="preserve"/> </p> <div xml:id="echoid-div1066" type="float" level="2" n="3"> <note symbol="e" position="right" xlink:label="note-395-05" xlink:href="note-395-05a" xml:space="preserve">5. triang. re-<lb/>ctil.</note> <note symbol="f" position="right" xlink:label="note-395-06" xlink:href="note-395-06a" xml:space="preserve">18. tertii.</note> <note symbol="g" position="right" xlink:label="note-395-07" xlink:href="note-395-07a" xml:space="preserve">4. primi.</note> <note symbol="h" position="right" xlink:label="note-395-08" xlink:href="note-395-08a" xml:space="preserve">4. rectang. <lb/>rectil.</note> </div> <p> <s xml:id="echoid-s17380" xml:space="preserve"><emph style="sc">Deniqve</emph> hoc etiam modo idem aſſequemur. </s> <s xml:id="echoid-s17381" xml:space="preserve">Cognito per dioptram an-<lb/>gulo A, quando radius viſualis terram contingit, cognitus etiam erit angulus <lb/>A F C, eius complementum. </s> <s xml:id="echoid-s17382" xml:space="preserve">Ergo huius anguli ſecans AF, cognita erit in parti-<lb/>busſinus totius FC. </s> <s xml:id="echoid-s17383" xml:space="preserve">Ex qua ſecante, ſi dematur ſinus BF, nota relinquetur alti-<lb/>tudo montis AB, in partibus ſinus totius BF. </s> <s xml:id="echoid-s17384" xml:space="preserve">Si igitur fiat, vt altitudo montis <lb/>AB, nota in partibus ſinus totius ad eandem A B, notam in data menſura, ita ſi-<lb/>nus totus BF, ad aliud; </s> <s xml:id="echoid-s17385" xml:space="preserve">proueniet ſemidiameter BF, nota in partibus altitudinis <lb/>montis, &</s> <s xml:id="echoid-s17386" xml:space="preserve">c.</s> <s xml:id="echoid-s17387" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1068" type="section" level="1" n="383"> <head xml:id="echoid-head410" xml:space="preserve">PROBL. 21. PROPOS. 35.</head> <p> <s xml:id="echoid-s17388" xml:space="preserve">PRISMATI cuicunque Cylindrum æqualem, & </s> <s xml:id="echoid-s17389" xml:space="preserve">Pyramidi Conum <lb/>æqualem: </s> <s xml:id="echoid-s17390" xml:space="preserve">Ac viciſsim Cylindro Priſma æquale, & </s> <s xml:id="echoid-s17391" xml:space="preserve">Cono æqualem <lb/>Pyramidem conſtituere.</s> <s xml:id="echoid-s17392" xml:space="preserve"/> </p> <pb o="368" file="396" n="396" rhead="GEOMETR. PRACT."/> <p> <s xml:id="echoid-s17393" xml:space="preserve"><emph style="sc">Si</emph> baſi priſmatis, vel pyramidis conſtruatur circulus æqualis, per ea, quæ ad <lb/>finem lib. </s> <s xml:id="echoid-s17394" xml:space="preserve">7. </s> <s xml:id="echoid-s17395" xml:space="preserve">ſcripſimus: </s> <s xml:id="echoid-s17396" xml:space="preserve">Et ſuper hunc circulum extruatur cylindrus, vel conus <lb/>eiuſdem altitu dinis cum priſmate, vel pyramide; </s> <s xml:id="echoid-s17397" xml:space="preserve">erit cylindrus priſmati, & </s> <s xml:id="echoid-s17398" xml:space="preserve">co-<lb/>nus pyramidi æqualis. </s> <s xml:id="echoid-s17399" xml:space="preserve">Cum enim tam baſes, quam altitudines æquales ſint: <lb/></s> <s xml:id="echoid-s17400" xml:space="preserve">pro ducatur autem priſma, & </s> <s xml:id="echoid-s17401" xml:space="preserve">Cylindrus ex baſe in altitudinem multiplicata, & </s> <s xml:id="echoid-s17402" xml:space="preserve"><lb/>pyramis, atque conus ex tertia parte baſis in altitudinem multiplicata, vt lib. </s> <s xml:id="echoid-s17403" xml:space="preserve">5. </s> <s xml:id="echoid-s17404" xml:space="preserve"><lb/>cap. </s> <s xml:id="echoid-s17405" xml:space="preserve">1. </s> <s xml:id="echoid-s17406" xml:space="preserve">declarauimus; </s> <s xml:id="echoid-s17407" xml:space="preserve">manifeſtum eſt, cylindrum priſmati, & </s> <s xml:id="echoid-s17408" xml:space="preserve">conum pyramidi <lb/>eſſe æqualem.</s> <s xml:id="echoid-s17409" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17410" xml:space="preserve"><emph style="sc">Si</emph> viciſsim baſi cylindri, vel coni conſtituatur quadratum, aut alia quæuis <lb/>rectilinea figura æqualis, per ea, quæ ad finem lib. </s> <s xml:id="echoid-s17411" xml:space="preserve">7. </s> <s xml:id="echoid-s17412" xml:space="preserve">diximus, & </s> <s xml:id="echoid-s17413" xml:space="preserve">ſuper hoc qua-<lb/>dratum, aut figuram rectilineam fiat priſma, vel pyramis eiuſdem altitudinis <lb/>cum cylindro, vel cono, erit priſma cylindro, & </s> <s xml:id="echoid-s17414" xml:space="preserve">pyramis cono æqualis. </s> <s xml:id="echoid-s17415" xml:space="preserve">quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s17416" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1069" type="section" level="1" n="384"> <head xml:id="echoid-head411" xml:space="preserve">PROBL. 22. PROPOS. 36.</head> <p> <s xml:id="echoid-s17417" xml:space="preserve">DATO Cylindro, aut priſmati æqualem conum, vel pyramidem ſub <lb/>eadem altitudine. </s> <s xml:id="echoid-s17418" xml:space="preserve">Et viciſsim dato cono, vel pyramidi æqualem cy-<lb/>lindrum, aut priſma eiuſdem altitudinis conſtituere.</s> <s xml:id="echoid-s17419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17420" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> <emph style="sc">Si</emph> tam baſis cylindri, quam priſmatis tripletur, & </s> <s xml:id="echoid-s17421" xml:space="preserve">ſuper triplicatam extrua- <anchor type="note" xlink:label="note-396-01a" xlink:href="note-396-01"/> tur conus, vel pyramis eiuſdem altitudinis, factum erit, quod in prima paite <lb/>proponitur. </s> <s xml:id="echoid-s17422" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Cumenim cylindrus ttiplus ſit coni eandem cumillo baſem, &</s> <s xml:id="echoid-s17423" xml:space="preserve"> <anchor type="note" xlink:label="note-396-02a" xlink:href="note-396-02"/> altitudinem habentis: </s> <s xml:id="echoid-s17424" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> Item priſma triplum pyramidis eandem cum illo ba- ſem, atque altitudinem habentis: </s> <s xml:id="echoid-s17425" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Sit autem & </s> <s xml:id="echoid-s17426" xml:space="preserve">conus extructus eiuſdem @l- <anchor type="note" xlink:label="note-396-03a" xlink:href="note-396-03"/> lius coni triplus; </s> <s xml:id="echoid-s17427" xml:space="preserve">necnon & </s> <s xml:id="echoid-s17428" xml:space="preserve">pyramis conſtructa eiuſdem illius pyramidis tri-<lb/>pla. </s> <s xml:id="echoid-s17429" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Erit tam conus extructus cylindro æqualis, quam pyramis conſtructa <anchor type="note" xlink:label="note-396-04a" xlink:href="note-396-04"/> priſmati. </s> <s xml:id="echoid-s17430" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17431" xml:space="preserve"/> </p> <div xml:id="echoid-div1069" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-396-01" xlink:href="note-396-01a" xml:space="preserve">16. ſexti. <lb/>hui{us}.</note> <note symbol="b" position="left" xlink:label="note-396-02" xlink:href="note-396-02a" xml:space="preserve">10. duode-<lb/>cimi.</note> <note symbol="c" position="left" xlink:label="note-396-03" xlink:href="note-396-03a" xml:space="preserve">coroll. 7. <lb/>duodec.</note> <note symbol="d" position="left" xlink:label="note-396-04" xlink:href="note-396-04a" xml:space="preserve">11. & 6. duo-<lb/>dec.</note> </div> <p> <s xml:id="echoid-s17432" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> <emph style="sc">Si</emph> viciſsim baſes coni, & </s> <s xml:id="echoid-s17433" xml:space="preserve">pyramidis in tripla proportione minuantur, &</s> <s xml:id="echoid-s17434" xml:space="preserve"> <anchor type="note" xlink:label="note-396-05a" xlink:href="note-396-05"/> <anchor type="note" xlink:label="note-396-06a" xlink:href="note-396-06"/> ſuper tertias has partes cylindrus erigatur, & </s> <s xml:id="echoid-s17435" xml:space="preserve">priſma: </s> <s xml:id="echoid-s17436" xml:space="preserve">Erit tam cylindrus dato <lb/>cono, quam priſma datæ pyramidi æquale. </s> <s xml:id="echoid-s17437" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Quoniam enim tam conus datus, <anchor type="note" xlink:label="note-396-07a" xlink:href="note-396-07"/> quam cylindrus extructus, triplus eſt coni eandem baſem, altitudinem que ha-<lb/>bentis cum cylindro extructo. </s> <s xml:id="echoid-s17438" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> Item tam data pyramis, quam priſma extru- <anchor type="note" xlink:label="note-396-08a" xlink:href="note-396-08"/> ctum, triplum eſt pyramidis eandem habentis baſem, atque altitudinem cum <lb/>priſmate extructo. </s> <s xml:id="echoid-s17439" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Erit tam cylindrus extructus dato cono æqualis, quam <anchor type="note" xlink:label="note-396-09a" xlink:href="note-396-09"/> priſma conſtructum datæ pyramidis æquale. </s> <s xml:id="echoid-s17440" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17441" xml:space="preserve"/> </p> <div xml:id="echoid-div1070" type="float" level="2" n="2"> <note symbol="e" position="left" xlink:label="note-396-05" xlink:href="note-396-05a" xml:space="preserve">9. quinti.</note> <note symbol="f" position="left" xlink:label="note-396-06" xlink:href="note-396-06a" xml:space="preserve">16. ſexti. <lb/>hui{us}.</note> <note symbol="g" position="left" xlink:label="note-396-07" xlink:href="note-396-07a" xml:space="preserve">11. & 10. <lb/>duodec.</note> <note symbol="h" position="left" xlink:label="note-396-08" xlink:href="note-396-08a" xml:space="preserve">6. & 7. duo-<lb/>dec.</note> <note symbol="i" position="left" xlink:label="note-396-09" xlink:href="note-396-09a" xml:space="preserve">9. quinti.</note> </div> </div> <div xml:id="echoid-div1072" type="section" level="1" n="385"> <head xml:id="echoid-head412" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s17442" xml:space="preserve"><emph style="sc">Qvia</emph> igitur <anchor type="note" xlink:href="" symbol="k"/> omne priſma in cylindrum, & </s> <s xml:id="echoid-s17443" xml:space="preserve">pyramis in conum conuerti- <anchor type="note" xlink:label="note-396-10a" xlink:href="note-396-10"/> tur: </s> <s xml:id="echoid-s17444" xml:space="preserve">Et contra cylindrus in priſma, & </s> <s xml:id="echoid-s17445" xml:space="preserve">conus in pyramidem: </s> <s xml:id="echoid-s17446" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Item cylindrus <anchor type="note" xlink:label="note-396-11a" xlink:href="note-396-11"/> in conum, & </s> <s xml:id="echoid-s17447" xml:space="preserve">priſma in pyramidem: </s> <s xml:id="echoid-s17448" xml:space="preserve">Et contra conus in cylindrum, & </s> <s xml:id="echoid-s17449" xml:space="preserve">pyramis <lb/>in priſma conuerti poteſt; </s> <s xml:id="echoid-s17450" xml:space="preserve">fit vt indifferenter tam cylindrus, quam priſma tranſ-<lb/>mutari poſsit in pyramidem, aut conum, ac pyramis in cylindrum, aut priſma <lb/>æquale.</s> <s xml:id="echoid-s17451" xml:space="preserve"/> </p> <div xml:id="echoid-div1072" type="float" level="2" n="1"> <note symbol="k" position="left" xlink:label="note-396-10" xlink:href="note-396-10a" xml:space="preserve">35. hui{us}.</note> <note symbol="l" position="left" xlink:label="note-396-11" xlink:href="note-396-11a" xml:space="preserve">36. hui{us}.</note> </div> <pb o="369" file="397" n="397" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1074" type="section" level="1" n="386"> <head xml:id="echoid-head413" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s17452" xml:space="preserve">Ex his etiam manifeſtè colligitur, omnem cylindrum, ac priſma; </s> <s xml:id="echoid-s17453" xml:space="preserve">ſimiliter & </s> <s xml:id="echoid-s17454" xml:space="preserve"><lb/>conum, ac pyramidem conuerti poſſe in parallelepipedum rectangulum, cuius <lb/>baſis ſit quadrata. </s> <s xml:id="echoid-s17455" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Nam conuerſo cylindro, aut cono, vel pyramidein priſma <anchor type="note" xlink:label="note-397-01a" xlink:href="note-397-01"/> qualecunque, ſi baſi priſmatis fiat quadratum æquale, & </s> <s xml:id="echoid-s17456" xml:space="preserve">ſupra illud erigatur pa-<lb/> <anchor type="note" xlink:label="note-397-02a" xlink:href="note-397-02"/> rallelepipedum eiuſdem altitudinis; </s> <s xml:id="echoid-s17457" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit hoc parallelepipedum priori æquale, ac proinde & </s> <s xml:id="echoid-s17458" xml:space="preserve">propoſito cylindro, vel cono, aut pyramidi.</s> <s xml:id="echoid-s17459" xml:space="preserve"/> </p> <div xml:id="echoid-div1074" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-397-01" xlink:href="note-397-01a" xml:space="preserve">35. & 36. <lb/>hui{us}.</note> <note symbol="b" position="right" xlink:label="note-397-02" xlink:href="note-397-02a" xml:space="preserve">2. coroll. 7. <lb/>duodec.</note> </div> </div> <div xml:id="echoid-div1076" type="section" level="1" n="387"> <head xml:id="echoid-head414" xml:space="preserve">PROBL. 23. PROPOS. 37.</head> <p> <s xml:id="echoid-s17460" xml:space="preserve">DATVM cylindrum, vel priſma: </s> <s xml:id="echoid-s17461" xml:space="preserve">ſimiliter datum conum, vel pyrami-<lb/>dem cuiuſcunque altitudinis, in æqualem ſub data qualibet alia altitu-<lb/>dine, & </s> <s xml:id="echoid-s17462" xml:space="preserve">ſupra baſem quotcunque angulorum, reuocare.</s> <s xml:id="echoid-s17463" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17464" xml:space="preserve"><emph style="sc">In</emph> proportione, quam data altitudo ad altitudinem propoſiti ſolidi habet, <lb/> <anchor type="note" xlink:href="" symbol="c"/> augeatur vel minuatur baſis eiuſdem ſolidi dati. </s> <s xml:id="echoid-s17465" xml:space="preserve">Nam ſolidum ſupra hanc ba- <anchor type="note" xlink:label="note-397-03a" xlink:href="note-397-03"/> ſem auctam, vel diminutam ſecundum datam altitudinem conſtructum, erit id, <lb/>quod quæritur. </s> <s xml:id="echoid-s17466" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Erit enim æquale dato ſolido: </s> <s xml:id="echoid-s17467" xml:space="preserve">quippe cum altitudines cum <anchor type="note" xlink:label="note-397-04a" xlink:href="note-397-04"/> baſibus reciprocæ ſint. </s> <s xml:id="echoid-s17468" xml:space="preserve">Quod ſi baſi conſtructi ſolidi fiat æqualis baſis quot-<lb/>cunque angulorum & </s> <s xml:id="echoid-s17469" xml:space="preserve">ſupra eam conſtituatur ſolidum ſub data altitudine; </s> <s xml:id="echoid-s17470" xml:space="preserve">erit <lb/>hoc etiam ſolidum ſolido propoſito æquale.</s> <s xml:id="echoid-s17471" xml:space="preserve"/> </p> <div xml:id="echoid-div1076" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-397-03" xlink:href="note-397-03a" xml:space="preserve">16. ſexti. <lb/>hui{us}.</note> <note symbol="d" position="right" xlink:label="note-397-04" xlink:href="note-397-04a" xml:space="preserve">15. & 9. <lb/>duodec.</note> </div> </div> <div xml:id="echoid-div1078" type="section" level="1" n="388"> <head xml:id="echoid-head415" xml:space="preserve">PROBL. 24. PROPOS. 38.</head> <p> <s xml:id="echoid-s17472" xml:space="preserve">DATO parallelepipedo rectangulo cubum æqualem deſcribere.</s> <s xml:id="echoid-s17473" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17474" xml:space="preserve"><emph style="sc">Si</emph> parallelepipedum non habet baſem quadratam, <anchor type="note" xlink:href="" symbol="e"/> fiat eius baſi quadra- <anchor type="note" xlink:label="note-397-05a" xlink:href="note-397-05"/> <anchor type="figure" xlink:label="fig-397-01a" xlink:href="fig-397-01"/> tum æquale BCD, ſupra quod erigatur parallelepi-<lb/>pedum rectangulum eiuſdem altitudinis A B, cum <lb/>parallelepipedo dato, <anchor type="note" xlink:href="" symbol="f"/> quod æquale erit dato pa- <anchor type="note" xlink:label="note-397-06a" xlink:href="note-397-06"/> rallelepipedo. </s> <s xml:id="echoid-s17475" xml:space="preserve">Huic ergo cubum æqualem con-<lb/>ſtruemus hacarte. </s> <s xml:id="echoid-s17476" xml:space="preserve">Inter BC, latus quadrati BD, & </s> <s xml:id="echoid-s17477" xml:space="preserve"><lb/>A B, altitudinem parallelepipedi, inueniantur duæ <lb/>med@æ prop@rtionales EF, H: </s> <s xml:id="echoid-s17478" xml:space="preserve">ita vt ſit BC, ad EF, <lb/>quemadmodum EF, ad H, & </s> <s xml:id="echoid-s17479" xml:space="preserve">H, ad A B. </s> <s xml:id="echoid-s17480" xml:space="preserve">Et ſuper <lb/>EF, propinquiorem lateri B C, conſtruatur cubus EFG. </s> <s xml:id="echoid-s17481" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> qui parallelepipedo <anchor type="note" xlink:label="note-397-07a" xlink:href="note-397-07"/> ABCD, æqualis erit.</s> <s xml:id="echoid-s17482" xml:space="preserve"/> </p> <div xml:id="echoid-div1078" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-397-05" xlink:href="note-397-05a" xml:space="preserve">14. ſecundi.</note> <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/xxxxxxxx/figures/397-01"/> </figure> <note symbol="f" position="right" xlink:label="note-397-06" xlink:href="note-397-06a" xml:space="preserve">2. coroll. 7. <lb/>duodec.</note> <note symbol="g" position="right" xlink:label="note-397-07" xlink:href="note-397-07a" xml:space="preserve">Lemma 18. <lb/>ſexti hui{us}.</note> </div> <p> <s xml:id="echoid-s17483" xml:space="preserve"><emph style="sc">Qvod</emph> ſi fortè accidat, tres dimenſiones parallelepipedi dati, cuius baſis <lb/>quadrata non ſit eſſe continuè proportionales; </s> <s xml:id="echoid-s17484" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit cubus ex media deſcriptus <anchor type="note" xlink:label="note-397-08a" xlink:href="note-397-08"/> parallelepipedo æqualis.</s> <s xml:id="echoid-s17485" xml:space="preserve"/> </p> <div xml:id="echoid-div1079" type="float" level="2" n="2"> <note symbol="h" position="right" xlink:label="note-397-08" xlink:href="note-397-08a" xml:space="preserve">36. vndec.</note> </div> </div> <div xml:id="echoid-div1081" type="section" level="1" n="389"> <head xml:id="echoid-head416" xml:space="preserve">COROLLARIVM.</head> <p> <s xml:id="echoid-s17486" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> <emph style="sc">Cvm</emph> igitur omnis cylindrus, omne priſma, conus ac pyramis in rectangu- <anchor type="note" xlink:label="note-397-09a" xlink:href="note-397-09"/> lum parallelepipedum poſsit commutari, liquidò conſtat, cuilibet ſolido eiuſ-<lb/>modi, cubum poſſe conſtrui æqualem.</s> <s xml:id="echoid-s17487" xml:space="preserve"/> </p> <div xml:id="echoid-div1081" type="float" level="2" n="1"> <note symbol="i" position="right" xlink:label="note-397-09" xlink:href="note-397-09a" xml:space="preserve">2. coroll. <lb/>36. hui{us}.</note> </div> <pb o="370" file="398" n="398" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1083" type="section" level="1" n="390"> <head xml:id="echoid-head417" xml:space="preserve">PROBL. 25. PROPOS. 39.</head> <p> <s xml:id="echoid-s17488" xml:space="preserve">DATO cubo æquale parallelepipedum rectangulum ſub data altitu-<lb/>dine, vel ſupra datam baſem conſtruere.</s> <s xml:id="echoid-s17489" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17490" xml:space="preserve"><emph style="sc">Sit</emph> in præcedenti figura datus cubus E F G, & </s> <s xml:id="echoid-s17491" xml:space="preserve">primum data altitudo A B, <lb/> <anchor type="note" xlink:label="note-398-01a" xlink:href="note-398-01"/> ſub qua conſtruendum ſit parallelepipedum rectangulum cubo æquale. </s> <s xml:id="echoid-s17492" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Alti- tudini AB, & </s> <s xml:id="echoid-s17493" xml:space="preserve">lateri cubi E F, reperiatur tertia proportionalis BC. </s> <s xml:id="echoid-s17494" xml:space="preserve">Et fiat rectan-<lb/>gulum BD, comprehenſum ſub tertia proportionali BC, & </s> <s xml:id="echoid-s17495" xml:space="preserve">recta CD, lateri cu-<lb/>bi E F, æquali; </s> <s xml:id="echoid-s17496" xml:space="preserve">erigatur que ſupra B D, parallelepipedum rectangulum ſub data <lb/>altitudine AB. </s> <s xml:id="echoid-s17497" xml:space="preserve">quod dico cubo eſſe æquale. </s> <s xml:id="echoid-s17498" xml:space="preserve">Quoniam enim parallelepipedum <lb/>rectangulum A B D, continetur ſub tribus rectis AB, CD, BC, hoc eſt, ſub A B, <lb/>E F, B C, continuè proportionalibus; </s> <s xml:id="echoid-s17499" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit parallelepipedum æquale cubo ex <anchor type="note" xlink:label="note-398-02a" xlink:href="note-398-02"/> media E F, deſcripto. </s> <s xml:id="echoid-s17500" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17501" xml:space="preserve"/> </p> <div xml:id="echoid-div1083" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-398-01" xlink:href="note-398-01a" xml:space="preserve">11. ſexti.</note> <note symbol="b" position="left" xlink:label="note-398-02" xlink:href="note-398-02a" xml:space="preserve">36. vndec.</note> </div> <p> <s xml:id="echoid-s17502" xml:space="preserve"><emph style="sc">Sit</emph> deinde data baſis BD, quæ ſi non eſt parallelogrammum, <anchor type="note" xlink:href="" symbol="c"/> r@uocetur ad <anchor type="note" xlink:label="note-398-03a" xlink:href="note-398-03"/> parallelogrammum æquale. </s> <s xml:id="echoid-s17503" xml:space="preserve">Et quam proportio-<lb/> <anchor type="figure" xlink:label="fig-398-01a" xlink:href="fig-398-01"/> nem habeat baſis data BD, ad baſem cubi dati, eam <lb/>habet latus cubi E F, ad rectam A B. </s> <s xml:id="echoid-s17504" xml:space="preserve">(quod fiet, ſi <lb/>ſupra latus cubi E F, fiat rectangulum æquale baſi <lb/>B D, & </s> <s xml:id="echoid-s17505" xml:space="preserve">ſuper alterum latus huius rectangulialiud <lb/>rectangulum æquale quadrato lateris cubi E F. </s> <s xml:id="echoid-s17506" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> <anchor type="note" xlink:label="note-398-04a" xlink:href="note-398-04"/> Namtunc erit, vt primum rectangulum, id eſt, ba-<lb/>ſis BD, ad ſecundum rectangulum, id eſt, ad quadratum, vel baſem cubi, ita primi <lb/>rectanguli baſis, videlicet EF, ad baſem ſecundirectanguli.) </s> <s xml:id="echoid-s17507" xml:space="preserve">Nam ſi ſupra baſem <lb/>BD, erigatur parallelepipedum in altitu dine inuenta AB, <anchor type="note" xlink:href="" symbol="e"/> erunt parallelepipe- <anchor type="note" xlink:label="note-398-05a" xlink:href="note-398-05"/> dum, & </s> <s xml:id="echoid-s17508" xml:space="preserve">cubus æqualia: </s> <s xml:id="echoid-s17509" xml:space="preserve">quippe cum baſes, & </s> <s xml:id="echoid-s17510" xml:space="preserve">altitudines ſint reciprocæ, ex con-<lb/>ſtructione. </s> <s xml:id="echoid-s17511" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17512" xml:space="preserve"/> </p> <div xml:id="echoid-div1084" type="float" level="2" n="2"> <note symbol="c" position="left" xlink:label="note-398-03" xlink:href="note-398-03a" xml:space="preserve">45. primi.</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/xxxxxxxx/figures/398-01"/> </figure> <note symbol="d" position="left" xlink:label="note-398-04" xlink:href="note-398-04a" xml:space="preserve">1. ſexti.</note> <note symbol="e" position="left" xlink:label="note-398-05" xlink:href="note-398-05a" xml:space="preserve">34. vndec.</note> </div> </div> <div xml:id="echoid-div1086" type="section" level="1" n="391"> <head xml:id="echoid-head418" xml:space="preserve">COR OLLARIVM.</head> <p> <s xml:id="echoid-s17513" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> <emph style="sc">Qvoniam</emph> igitur cuilibet cylindro, priſmati, cono, ac pyramidi parallele- <anchor type="note" xlink:label="note-398-06a" xlink:href="note-398-06"/> pipedum rectangulum conſtrui poteſt æquale: </s> <s xml:id="echoid-s17514" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> ſi huic parallelepipedo fiat cu- bus æqualis; </s> <s xml:id="echoid-s17515" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/> & </s> <s xml:id="echoid-s17516" xml:space="preserve">huic cubo parallelepipedum rectangulum ſub data altitudi- <anchor type="note" xlink:label="note-398-07a" xlink:href="note-398-07"/> ne, vel baſe data æquale: </s> <s xml:id="echoid-s17517" xml:space="preserve">commutatus erit cylindrus, priſma, conus, ac pyramis <lb/> <anchor type="note" xlink:label="note-398-08a" xlink:href="note-398-08"/> in parallelepipedum rectangulum æquale datæ altitudinis, vel baſis.</s> <s xml:id="echoid-s17518" xml:space="preserve"/> </p> <div xml:id="echoid-div1086" type="float" level="2" n="1"> <note symbol="f" position="left" xlink:label="note-398-06" xlink:href="note-398-06a" xml:space="preserve">coroll. 38. <lb/>hui{us}.</note> <note symbol="g" position="left" xlink:label="note-398-07" xlink:href="note-398-07a" xml:space="preserve">38. hui{us}.</note> <note symbol="h" position="left" xlink:label="note-398-08" xlink:href="note-398-08a" xml:space="preserve">39. hui{us}.</note> </div> </div> <div xml:id="echoid-div1088" type="section" level="1" n="392"> <head xml:id="echoid-head419" xml:space="preserve">PROBL. 26. PROPOS. 40.</head> <p> <s xml:id="echoid-s17519" xml:space="preserve">SPHÆRÆ datæ cubum æqualem: </s> <s xml:id="echoid-s17520" xml:space="preserve">Et dato cubo æqualem ſphæram <lb/>conſtituere.</s> <s xml:id="echoid-s17521" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17522" xml:space="preserve"><emph style="sc">Qvoniam</emph> per propoſ. </s> <s xml:id="echoid-s17523" xml:space="preserve">32. </s> <s xml:id="echoid-s17524" xml:space="preserve">lib. </s> <s xml:id="echoid-s17525" xml:space="preserve">1. </s> <s xml:id="echoid-s17526" xml:space="preserve">Archimedis de ſphæra, & </s> <s xml:id="echoid-s17527" xml:space="preserve">cylindro, Cylin-<lb/>drus rectus, cuius baſis eſt maximus ſphæræ circulus, & </s> <s xml:id="echoid-s17528" xml:space="preserve">altitudo diametro eiuſ-<lb/>dem ſphæræ æqualis, ſeſquialteram habet proportionem ad ſphæram: </s> <s xml:id="echoid-s17529" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Hab@t <anchor type="note" xlink:label="note-398-09a" xlink:href="note-398-09"/> autem idem cylind@us ad cylindrum eiuſdem baſis, cuius altitudo contineat <lb/>{2/3}. </s> <s xml:id="echoid-s17530" xml:space="preserve">diametri ſphæræ, proportionem quo que ſeſquialteram; </s> <s xml:id="echoid-s17531" xml:space="preserve"><anchor type="note" xlink:href="" symbol="k"/> erit poſterior hic <anchor type="note" xlink:label="note-398-10a" xlink:href="note-398-10"/> <pb o="371" file="399" n="399" rhead="LIBER OCTAVVS."/> cylindrus ſphæræ æqualis. </s> <s xml:id="echoid-s17532" xml:space="preserve">Si igitur huic cylindro <anchor type="note" xlink:href="" symbol="a"/> fiat cubus æqualis; </s> <s xml:id="echoid-s17533" xml:space="preserve">eritidem <anchor type="note" xlink:label="note-399-01a" xlink:href="note-399-01"/> hic cubus datæ ſphæræ æqualis. </s> <s xml:id="echoid-s17534" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17535" xml:space="preserve"/> </p> <div xml:id="echoid-div1088" type="float" level="2" n="1"> <note symbol="i" position="left" xlink:label="note-398-09" xlink:href="note-398-09a" xml:space="preserve">14. vndec.</note> <note symbol="k" position="left" xlink:label="note-398-10" xlink:href="note-398-10a" xml:space="preserve">9. quinti.</note> <note symbol="a" position="right" xlink:label="note-399-01" xlink:href="note-399-01a" xml:space="preserve">coroll. 38. <lb/>hui{us}.</note> </div> <p> <s xml:id="echoid-s17536" xml:space="preserve"><emph style="sc">Vel</emph> quia per eandẽ propoſ. </s> <s xml:id="echoid-s17537" xml:space="preserve">32. </s> <s xml:id="echoid-s17538" xml:space="preserve">Archimedis, ſphæra quadrupla eſt coni, cu-<lb/>ius baſis eſt maximus ſphærę circulus, & </s> <s xml:id="echoid-s17539" xml:space="preserve">altitudo ſemidiameter ſphæræ æqualis: <lb/></s> <s xml:id="echoid-s17540" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="b"/> Eſt autem eiuſdem coni quadruplus etiam conus eiuſdem altitudinis, baſem <anchor type="note" xlink:label="note-399-02a" xlink:href="note-399-02"/> habens circuli maximi in ſphæra quadruplam, hoc eſt, baſem habens circulum, <lb/>cuius ſemidiameter æqualis diametro maximi circuli; </s> <s xml:id="echoid-s17541" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> erit poſterior hic conus <anchor type="note" xlink:label="note-399-03a" xlink:href="note-399-03"/> ſphæræ æqualis. </s> <s xml:id="echoid-s17542" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> Si igitur huic cono fiat cubus æqualis, erit hic idem cubus <anchor type="note" xlink:label="note-399-04a" xlink:href="note-399-04"/> ſphæræ datæ æqualis. </s> <s xml:id="echoid-s17543" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17544" xml:space="preserve"/> </p> <div xml:id="echoid-div1089" type="float" level="2" n="2"> <note symbol="b" position="right" xlink:label="note-399-02" xlink:href="note-399-02a" xml:space="preserve">11. duodec.</note> <note symbol="c" position="right" xlink:label="note-399-03" xlink:href="note-399-03a" xml:space="preserve">9. quinti.</note> <note symbol="d" position="right" xlink:label="note-399-04" xlink:href="note-399-04a" xml:space="preserve">coroll. 38. <lb/>hui{us}.</note> </div> <p> <s xml:id="echoid-s17545" xml:space="preserve"><emph style="sc">Sit</emph> viciſsim dato cubo fabricanda ſphæra æqualis. </s> <s xml:id="echoid-s17546" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Fiat cubo, tanquam <anchor type="note" xlink:label="note-399-05a" xlink:href="note-399-05"/> Priſmati, cylindrus æqualis. </s> <s xml:id="echoid-s17547" xml:space="preserve">Deinde ſphæra fabricetur, habens diametrum ſeſ-<lb/>quialteram aititu dinis cylindri. </s> <s xml:id="echoid-s17548" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Hæc enim ſphæra cylindro, ac proinde cubo <anchor type="note" xlink:label="note-399-06a" xlink:href="note-399-06"/> dato æqualis erit: </s> <s xml:id="echoid-s17549" xml:space="preserve">propterea quod cylindrus eiuſdem baſis altitudinem habens <lb/>æqualem diametro ſphæræ, <anchor type="note" xlink:href="" symbol="g"/> ſeſquialter eſt tam prioris cylindri, <anchor type="note" xlink:href="" symbol="h"/> quam datæ <anchor type="note" xlink:label="note-399-07a" xlink:href="note-399-07"/> <anchor type="note" xlink:label="note-399-08a" xlink:href="note-399-08"/> ſphæræ. </s> <s xml:id="echoid-s17550" xml:space="preserve">quod eſt propoſitum.</s> <s xml:id="echoid-s17551" xml:space="preserve"/> </p> <div xml:id="echoid-div1090" type="float" level="2" n="3"> <note symbol="e" position="right" xlink:label="note-399-05" xlink:href="note-399-05a" xml:space="preserve">35. hui{us}.</note> <note symbol="f" position="right" xlink:label="note-399-06" xlink:href="note-399-06a" xml:space="preserve">9. quinti.</note> <note symbol="g" position="right" xlink:label="note-399-07" xlink:href="note-399-07a" xml:space="preserve">14. duodec.</note> <note symbol="h" position="right" xlink:label="note-399-08" xlink:href="note-399-08a" xml:space="preserve">32. lib. 1. de <lb/>ſph. & cyl.</note> </div> </div> <div xml:id="echoid-div1092" type="section" level="1" n="393"> <head xml:id="echoid-head420" xml:space="preserve">COROLLARIVM I.</head> <p> <s xml:id="echoid-s17552" xml:space="preserve"><emph style="sc">Qvia</emph> verò <anchor type="note" xlink:href="" symbol="i"/> ſi baſi cubi fiat æqualis figura quotcunq; </s> <s xml:id="echoid-s17553" xml:space="preserve">laterum, ſiue eare- <anchor type="note" xlink:label="note-399-09a" xlink:href="note-399-09"/> gularis ſit, ſiue non; </s> <s xml:id="echoid-s17554" xml:space="preserve">& </s> <s xml:id="echoid-s17555" xml:space="preserve">ſupra hanc figuram erigatur ſolidum rectangulum ad al-<lb/> <anchor type="note" xlink:label="note-399-10a" xlink:href="note-399-10"/> titudinẽ cubi <anchor type="note" xlink:href="" symbol="k"/> ſolidum hoc cubo eſt æquale: </s> <s xml:id="echoid-s17556" xml:space="preserve">fit vt ſphæræ datæ conſtrui poſsit æquale ſolidum rectangulum ſupra baſem quotlibet angulorum; </s> <s xml:id="echoid-s17557" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> ſi nimirum <anchor type="note" xlink:label="note-399-11a" xlink:href="note-399-11"/> prius conſtruatur cubus æqualis: </s> <s xml:id="echoid-s17558" xml:space="preserve">deinde huic cubo ſolidum rectangulum æ-<lb/>quale, vt proximè dictum eſt. </s> <s xml:id="echoid-s17559" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> Item quia cuicunque priſmati pyramis conſtrui <anchor type="note" xlink:label="note-399-12a" xlink:href="note-399-12"/> poteſt æqualis: </s> <s xml:id="echoid-s17560" xml:space="preserve">ſi cubo, qui ſphærę eſt æqualis, tanquam priſmati, fiat pyramis <lb/>æqualis; </s> <s xml:id="echoid-s17561" xml:space="preserve">erit quo que eadem pyramis ſp hærę æqualis. </s> <s xml:id="echoid-s17562" xml:space="preserve"><anchor type="note" xlink:href="" symbol="n"/> Immo quoniam cuili- <anchor type="note" xlink:label="note-399-13a" xlink:href="note-399-13"/> bet cylindro conus fieri poteſt æqualis: </s> <s xml:id="echoid-s17563" xml:space="preserve">ſi cylindrus extruatur ſphærę æqualis, <lb/>ſupra baſem videlicet maximo circulo in ſphæra æqualẽ, & </s> <s xml:id="echoid-s17564" xml:space="preserve">cuius altitudo con-<lb/>tineat {2/3}. </s> <s xml:id="echoid-s17565" xml:space="preserve">diametri, vt ad initium huius propoſ. </s> <s xml:id="echoid-s17566" xml:space="preserve">oſtendimus: </s> <s xml:id="echoid-s17567" xml:space="preserve">Deinde huic cylin-<lb/>dro conus æqualis; </s> <s xml:id="echoid-s17568" xml:space="preserve">conſtitutus erit conus quo que datæ ſphærę æqualis.</s> <s xml:id="echoid-s17569" xml:space="preserve"/> </p> <div xml:id="echoid-div1092" type="float" level="2" n="1"> <note symbol="i" position="right" xlink:label="note-399-09" xlink:href="note-399-09a" xml:space="preserve">25. ſexti.</note> <note symbol="k" position="right" xlink:label="note-399-10" xlink:href="note-399-10a" xml:space="preserve">2. eoroll. 7. <lb/>duodec.</note> <note symbol="l" position="right" xlink:label="note-399-11" xlink:href="note-399-11a" xml:space="preserve">40. hui{us}.</note> <note symbol="m" position="right" xlink:label="note-399-12" xlink:href="note-399-12a" xml:space="preserve">36 hui{us}.</note> <note symbol="n" position="right" xlink:label="note-399-13" xlink:href="note-399-13a" xml:space="preserve">36. hui{us}.</note> </div> <p> <s xml:id="echoid-s17570" xml:space="preserve"><emph style="sc">Vicissim</emph> <anchor type="note" xlink:href="" symbol="o"/> quia cuilibet priſmati conſtrui poteſt cubus æqualis: </s> <s xml:id="echoid-s17571" xml:space="preserve"><anchor type="note" xlink:href="" symbol="p"/> Si huic <anchor type="note" xlink:label="note-399-14a" xlink:href="note-399-14"/> cubo fiat æqualis ſphæra, erit eadem hæc ſphæra conſtituta æqualis dato priſma-<lb/> <anchor type="note" xlink:label="note-399-15a" xlink:href="note-399-15"/> tiſupra baſem quotcunque angulorum.</s> <s xml:id="echoid-s17572" xml:space="preserve"/> </p> <div xml:id="echoid-div1093" type="float" level="2" n="2"> <note symbol="o" position="right" xlink:label="note-399-14" xlink:href="note-399-14a" xml:space="preserve">37. hui{us}.</note> <note symbol="p" position="right" xlink:label="note-399-15" xlink:href="note-399-15a" xml:space="preserve">40. hui{us}.</note> </div> </div> <div xml:id="echoid-div1095" type="section" level="1" n="394"> <head xml:id="echoid-head421" xml:space="preserve">COROLLARIVM II.</head> <p> <s xml:id="echoid-s17573" xml:space="preserve"><emph style="sc">Qvin</emph> etiam colligitur, poſſe ſphęram conſtrui æqualem cuilibet corporire-<lb/>gulari. </s> <s xml:id="echoid-s17574" xml:space="preserve">Nam de cubo quidem oſtenſum eſt hac propoſ. </s> <s xml:id="echoid-s17575" xml:space="preserve">40. </s> <s xml:id="echoid-s17576" xml:space="preserve">De Tetraedro ve-<lb/>ro, ſiue Pyramide regulari patet. </s> <s xml:id="echoid-s17577" xml:space="preserve">Nam ſi Pyramidi <anchor type="note" xlink:href="" symbol="q"/> fiat Parallelepipedum æqua- <anchor type="note" xlink:label="note-399-16a" xlink:href="note-399-16"/> le: </s> <s xml:id="echoid-s17578" xml:space="preserve"><anchor type="note" xlink:href="" symbol="r"/> Et huic parallelepipedo cubus æqualis; </s> <s xml:id="echoid-s17579" xml:space="preserve">Ac tandem huic cubo fabrice- tur ſphęra æqualis; </s> <s xml:id="echoid-s17580" xml:space="preserve">erit eadem hæc ſphæra Tetraedro, ſiue pyramidi regulariæ-<lb/> <anchor type="note" xlink:label="note-399-17a" xlink:href="note-399-17"/> qualis. </s> <s xml:id="echoid-s17581" xml:space="preserve">De Octaedro autem, Icoſaedro, & </s> <s xml:id="echoid-s17582" xml:space="preserve">Dodecaedro ita res peragetur. </s> <s xml:id="echoid-s17583" xml:space="preserve">Si o-<lb/>mnibus baſibus corporis regularis fiat quadratum æquale, per ea, quæ ad finem <lb/>lib. </s> <s xml:id="echoid-s17584" xml:space="preserve">2. </s> <s xml:id="echoid-s17585" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s17586" xml:space="preserve">vel potius per ea, quę lib. </s> <s xml:id="echoid-s17587" xml:space="preserve">4. </s> <s xml:id="echoid-s17588" xml:space="preserve">huius Geometrię cap. </s> <s xml:id="echoid-s17589" xml:space="preserve">4. </s> <s xml:id="echoid-s17590" xml:space="preserve">Num. </s> <s xml:id="echoid-s17591" xml:space="preserve">4. </s> <s xml:id="echoid-s17592" xml:space="preserve">tra-<lb/>didimus; </s> <s xml:id="echoid-s17593" xml:space="preserve">& </s> <s xml:id="echoid-s17594" xml:space="preserve">ſuper hoc quadratum fiat pyramis habens altitudinem æqualem <lb/>perpendiculari è centro corporis ad quamlibet baſem ductę, hoc eſt, altitudini <lb/>vnius pyramidis ex iis, in quas corpus diuiditur è centro: </s> <s xml:id="echoid-s17595" xml:space="preserve"><anchor type="note" xlink:href="" symbol="s"/> Erit hæc pyramis <anchor type="note" xlink:label="note-399-18a" xlink:href="note-399-18"/> <pb o="372" file="400" n="400" rhead="GEOMETR. PRACT."/> corpori regulari æqualis; </s> <s xml:id="echoid-s17596" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> quippe cum ita ſe habeat tam pyramis hæc quadrila- <anchor type="note" xlink:label="note-400-01a" xlink:href="note-400-01"/> tera ad vnam pyramidem corporis regularis, quam omnes pyramides corporis <lb/>regularis ad vnam pyramidem, vt baſis illius vel baſes omnium pyramidum <lb/>corporis, ad vnam baſem; </s> <s xml:id="echoid-s17597" xml:space="preserve">propterea quod in Octaedro proportio eſt vtro-<lb/>bique octupla: </s> <s xml:id="echoid-s17598" xml:space="preserve">In Icoſaedro, vigecupla: </s> <s xml:id="echoid-s17599" xml:space="preserve">Et in Dodecaedro, duo decupla. </s> <s xml:id="echoid-s17600" xml:space="preserve">Qua-<lb/>re ſi totiilli pyramidi cubus conſtruatur æqualis, vt paulò ante de Tetraedro di-<lb/>ctum eſt: </s> <s xml:id="echoid-s17601" xml:space="preserve">atque huic tandem cubo ſphæra æqualis fabricetur; </s> <s xml:id="echoid-s17602" xml:space="preserve">erit eadem ſphæ-<lb/>ra illi pyramidi, hoc eſt, corpori regulari æqualis.</s> <s xml:id="echoid-s17603" xml:space="preserve"/> </p> <div xml:id="echoid-div1095" type="float" level="2" n="1"> <note symbol="q" position="right" xlink:label="note-399-16" xlink:href="note-399-16a" xml:space="preserve">2. coroll. 36. <lb/>hui{us}.</note> <note symbol="r" position="right" xlink:label="note-399-17" xlink:href="note-399-17a" xml:space="preserve">38. cui{us}.</note> <note symbol="s" position="right" xlink:label="note-399-18" xlink:href="note-399-18a" xml:space="preserve">9. quinti.</note> <note symbol="a" position="left" xlink:label="note-400-01" xlink:href="note-400-01a" xml:space="preserve">6. duodec.</note> </div> </div> <div xml:id="echoid-div1097" type="section" level="1" n="395"> <head xml:id="echoid-head422" xml:space="preserve">PROBL. 27. PROPOS. 41.</head> <p> <s xml:id="echoid-s17604" xml:space="preserve">DVOBVS aut pluribus cubis vnum cubum æqualem efficere.</s> <s xml:id="echoid-s17605" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17606" xml:space="preserve"><emph style="sc">Si</emph> ſupra baſem ſuperiorem primi cubi, <anchor type="note" xlink:href="" symbol="b"/> conſtruatur parallelepipedum re- <anchor type="note" xlink:label="note-400-02a" xlink:href="note-400-02"/> ctangulum ſecundo cubo æquale, vt fiat vnum parallelepipedum duo bus cu-<lb/>bis æquale: </s> <s xml:id="echoid-s17607" xml:space="preserve">Et ſupra huius parallelepipedi baſem ſuperiorem aliud parallelepi-<lb/>pedum æquale tertio cubo, & </s> <s xml:id="echoid-s17608" xml:space="preserve">ſic deinceps, ſi plures adſint cubi, conſtructum <lb/>erit parallelepipedum propoſitis cubis æquale. </s> <s xml:id="echoid-s17609" xml:space="preserve">Huic ergo <anchor type="note" xlink:href="" symbol="c"/> ſi fiat cubus æqua- <anchor type="note" xlink:label="note-400-03a" xlink:href="note-400-03"/> lis, factum erit, quod proponitur.</s> <s xml:id="echoid-s17610" xml:space="preserve"/> </p> <div xml:id="echoid-div1097" type="float" level="2" n="1"> <note symbol="b" position="left" xlink:label="note-400-02" xlink:href="note-400-02a" xml:space="preserve">39 hui{us}.</note> <note symbol="c" position="left" xlink:label="note-400-03" xlink:href="note-400-03a" xml:space="preserve">38. hui{us}.</note> </div> </div> <div xml:id="echoid-div1099" type="section" level="1" n="396"> <head xml:id="echoid-head423" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s17611" xml:space="preserve"><emph style="sc">Eadem</emph> arte quotlibet figuris ſolidis non cubis, conſtruetur cubus æqualis: <lb/></s> <s xml:id="echoid-s17612" xml:space="preserve"> <anchor type="note" xlink:href="" symbol="d"/> ſi nimirum reuo centur ad vnum parallelepipedum, &</s> <s xml:id="echoid-s17613" xml:space="preserve">c.</s> <s xml:id="echoid-s17614" xml:space="preserve"/> </p> <note symbol="d" position="left" xml:space="preserve">37. hui{us}.</note> </div> <div xml:id="echoid-div1100" type="section" level="1" n="397"> <head xml:id="echoid-head424" xml:space="preserve">PROBL. 28. PROPOS. 42.</head> <p> <s xml:id="echoid-s17615" xml:space="preserve">DATO cubo, corpus regulare, quod ex quinque elegeris, æquale con-<lb/>ſtruere.</s> <s xml:id="echoid-s17616" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17617" xml:space="preserve"><emph style="sc">Sit</emph> datus cubus, cuius latus A, cui verbi gratia conſtruendum ſit æquale <lb/>Dodecaedrum. </s> <s xml:id="echoid-s17618" xml:space="preserve"><anchor type="note" xlink:href="" symbol="e"/> Fiat quodcunque Dodecaedrum, <anchor type="note" xlink:label="note-400-05a" xlink:href="note-400-05"/> cuius latus B: </s> <s xml:id="echoid-s17619" xml:space="preserve">cui per ea, quæ in 2. </s> <s xml:id="echoid-s17620" xml:space="preserve">coroll. </s> <s xml:id="echoid-s17621" xml:space="preserve">præceden-<lb/> <anchor type="note" xlink:label="note-400-06a" xlink:href="note-400-06"/> tis propoſ. </s> <s xml:id="echoid-s17622" xml:space="preserve">dicta ſunt, fiat æqualis cubus, cuius latus <lb/>C. </s> <s xml:id="echoid-s17623" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> Et tribus lateribus C, A, B, reperiatur quarta proportionalis D. </s> <s xml:id="echoid-s17624" xml:space="preserve">Dico Do- <anchor type="note" xlink:label="note-400-07a" xlink:href="note-400-07"/> decaedrum ſupra latus D, conſtructum, æquale eſſe dato cubo lateris A. </s> <s xml:id="echoid-s17625" xml:space="preserve">Quo-<lb/>niam enim, vt ex demonſtratione propoſ. </s> <s xml:id="echoid-s17626" xml:space="preserve">37. </s> <s xml:id="echoid-s17627" xml:space="preserve">lib. </s> <s xml:id="echoid-s17628" xml:space="preserve">11. </s> <s xml:id="echoid-s17629" xml:space="preserve">Eucl. </s> <s xml:id="echoid-s17630" xml:space="preserve">patet. </s> <s xml:id="echoid-s17631" xml:space="preserve">ita eſt cubus la-<lb/>teris C, ad cubum lateris A, vt Dodecaedrum lateris B, ad Dodecaedrum late-<lb/>ris D: </s> <s xml:id="echoid-s17632" xml:space="preserve">Eſt autem per conſtructionem, cubus lateris C, æqualis Dodecaedro <lb/>lateris B; </s> <s xml:id="echoid-s17633" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> erit quo que cubus lateris A, Dodecaedro lateris D, æqualis, quod eſt <anchor type="note" xlink:label="note-400-08a" xlink:href="note-400-08"/> propoſitum.</s> <s xml:id="echoid-s17634" xml:space="preserve"/> </p> <div xml:id="echoid-div1100" type="float" level="2" n="1"> <note symbol="e" position="left" xlink:label="note-400-05" xlink:href="note-400-05a" xml:space="preserve">17. tertii-<lb/>decimi.</note> <note position="right" xlink:label="note-400-06" xlink:href="note-400-06a" xml:space="preserve"> <lb/>C. # A. # B. # D. <lb/></note> <note symbol="f" position="left" xlink:label="note-400-07" xlink:href="note-400-07a" xml:space="preserve">12. ſexti.</note> <note symbol="g" position="left" xlink:label="note-400-08" xlink:href="note-400-08a" xml:space="preserve">14. quinti.</note> </div> </div> <div xml:id="echoid-div1102" type="section" level="1" n="398"> <head xml:id="echoid-head425" xml:space="preserve">PROBL. 29. PROPOS. 43.</head> <p> <s xml:id="echoid-s17635" xml:space="preserve">EX maiori cubo detrahere minorem, reſiduoque cubum æqualem ex-<lb/>hibere.</s> <s xml:id="echoid-s17636" xml:space="preserve"/> </p> <pb o="373" file="401" n="401" rhead="LIBER OCTAVVS."/> <p> <s xml:id="echoid-s17637" xml:space="preserve"><emph style="sc">Svpra</emph> baſem maioris cubi <anchor type="note" xlink:href="" symbol="a"/> conſtruatur parallelepipedum cubo minori <anchor type="note" xlink:label="note-401-01a" xlink:href="note-401-01"/> æquale. </s> <s xml:id="echoid-s17638" xml:space="preserve">Et ex latere cubi maioris ab ſcindatur recta æqualis altitudini conſtructi <lb/>parallelepipedi. </s> <s xml:id="echoid-s17639" xml:space="preserve">Si enim per punctum abſciſsionis ducatur planum baſibus cu-<lb/>bi parallelum, detractum erit parallelepip edum parallelepipedo conſtructo æ-<lb/>quale, cum habeat eandem baſem & </s> <s xml:id="echoid-s17640" xml:space="preserve">altitudinem cumillo, hoc eſt, minori cu-<lb/>bo æquale. </s> <s xml:id="echoid-s17641" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> Si igitur reliquo parallelepipedo fiat cubus æqualis, factum erit, <anchor type="note" xlink:label="note-401-02a" xlink:href="note-401-02"/> quod proponitur.</s> <s xml:id="echoid-s17642" xml:space="preserve"/> </p> <div xml:id="echoid-div1102" type="float" level="2" n="1"> <note symbol="a" position="right" xlink:label="note-401-01" xlink:href="note-401-01a" xml:space="preserve">39. hui{us}.</note> <note symbol="b" position="right" xlink:label="note-401-02" xlink:href="note-401-02a" xml:space="preserve">38. hui{us}.</note> </div> </div> <div xml:id="echoid-div1104" type="section" level="1" n="399"> <head xml:id="echoid-head426" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s17643" xml:space="preserve"><emph style="sc">Idem</emph> fieri poteſt in aliis figuris ſolidis; </s> <s xml:id="echoid-s17644" xml:space="preserve"><anchor type="note" xlink:href="" symbol="c"/> ſi prius reducantur ad parallelepi- <anchor type="note" xlink:label="note-401-03a" xlink:href="note-401-03"/> peda rectangula, quando non ſunt parallelepipeda. </s> <s xml:id="echoid-s17645" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> & </s> <s xml:id="echoid-s17646" xml:space="preserve">deinde parallelepi- peda ad cubos, &</s> <s xml:id="echoid-s17647" xml:space="preserve">c.</s> <s xml:id="echoid-s17648" xml:space="preserve"/> </p> <div xml:id="echoid-div1104" type="float" level="2" n="1"> <note symbol="c" position="right" xlink:label="note-401-03" xlink:href="note-401-03a" xml:space="preserve">1. & 2. coroll. <lb/>36. hui{us}.</note> </div> <note symbol="d" position="right" xml:space="preserve">38. hui{us}.</note> </div> <div xml:id="echoid-div1106" type="section" level="1" n="400"> <head xml:id="echoid-head427" xml:space="preserve">PROBL. 30. PROPOS. 44.</head> <p> <s xml:id="echoid-s17649" xml:space="preserve">DATIS duabus, aut pluribus ſphæris, ſphæram vnam æqualem con-<lb/>ſtituere.</s> <s xml:id="echoid-s17650" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17651" xml:space="preserve"><emph style="sc">Sphæris</emph> propoſitis <anchor type="note" xlink:href="" symbol="e"/> conſtruantur cubi æquales: </s> <s xml:id="echoid-s17652" xml:space="preserve"><anchor type="note" xlink:href="" symbol="f"/> His deinde vnus <anchor type="note" xlink:label="note-401-05a" xlink:href="note-401-05"/> <anchor type="note" xlink:label="note-401-06a" xlink:href="note-401-06"/> cubus æqualis fiat, qui etiam ſphæris datis erit æqualis. </s> <s xml:id="echoid-s17653" xml:space="preserve"><anchor type="note" xlink:href="" symbol="g"/> Si igitur huic cubo <anchor type="note" xlink:label="note-401-07a" xlink:href="note-401-07"/> extruatur ſphæra æqualis; </s> <s xml:id="echoid-s17654" xml:space="preserve">factum erit, quod iubetur.</s> <s xml:id="echoid-s17655" xml:space="preserve"/> </p> <div xml:id="echoid-div1106" type="float" level="2" n="1"> <note symbol="e" position="right" xlink:label="note-401-05" xlink:href="note-401-05a" xml:space="preserve">40. hui{us}.</note> <note symbol="f" position="right" xlink:label="note-401-06" xlink:href="note-401-06a" xml:space="preserve">41. hui{us}.</note> <note symbol="g" position="right" xlink:label="note-401-07" xlink:href="note-401-07a" xml:space="preserve">40. hui{us}.</note> </div> </div> <div xml:id="echoid-div1108" type="section" level="1" n="401"> <head xml:id="echoid-head428" xml:space="preserve">PROBL. 31. PROPOS. 45.</head> <p> <s xml:id="echoid-s17656" xml:space="preserve">EX maiori ſphæra minorem ſphæram detrahere, reſiduoque ſphæram <lb/>æqualem exhibere.</s> <s xml:id="echoid-s17657" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17658" xml:space="preserve"><anchor type="note" xlink:href="" symbol="h"/><emph style="sc">Vtraqve</emph> ſphæra in cubum reuocetur. </s> <s xml:id="echoid-s17659" xml:space="preserve"><anchor type="note" xlink:href="" symbol="i"/> Detracto deinde minore ex <anchor type="note" xlink:label="note-401-08a" xlink:href="note-401-08"/> maiore, <anchor type="note" xlink:href="" symbol="k"/> ſi reſiduo ſphæra fiat æqualis; </s> <s xml:id="echoid-s17660" xml:space="preserve">factum erit, quod proponitur.</s> <s xml:id="echoid-s17661" xml:space="preserve"> <anchor type="note" xlink:label="note-401-09a" xlink:href="note-401-09"/> <anchor type="note" xlink:label="note-401-10a" xlink:href="note-401-10"/> </s> </p> <div xml:id="echoid-div1108" type="float" level="2" n="1"> <note symbol="h" position="right" xlink:label="note-401-08" xlink:href="note-401-08a" xml:space="preserve">40. hui{us}.</note> <note symbol="i" position="right" xlink:label="note-401-09" xlink:href="note-401-09a" xml:space="preserve">43. hui{us}.</note> <note symbol="k" position="right" xlink:label="note-401-10" xlink:href="note-401-10a" xml:space="preserve">40. hui{us}.</note> </div> </div> <div xml:id="echoid-div1110" type="section" level="1" n="402"> <head xml:id="echoid-head429" xml:space="preserve">PROBL. 32. PROPOS. 46.</head> <p> <s xml:id="echoid-s17662" xml:space="preserve">DATVM cubum aut parallelepipedum, ſecundum proportionem da-<lb/>tam ſecare.</s> <s xml:id="echoid-s17663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17664" xml:space="preserve"><emph style="sc">Si</emph> namque vnum latus in baſe cubi, aut parallelepipedi ſecetur ſecundum <lb/>datam proportionem, & </s> <s xml:id="echoid-s17665" xml:space="preserve">per punctum ſectionis ducatur planum duabus baſi-<lb/>ſibus erectis ſolidi parallelum, diuidens ipſum ſolidum in duo parallelepipeda: <lb/></s> <s xml:id="echoid-s17666" xml:space="preserve">habebunt hæc parallelepipeda datam proportionem. </s> <s xml:id="echoid-s17667" xml:space="preserve"><anchor type="note" xlink:href="" symbol="l"/> Habent enim propor- <anchor type="note" xlink:label="note-401-11a" xlink:href="note-401-11"/> tionem inter ſe eandem, quam baſes. </s> <s xml:id="echoid-s17668" xml:space="preserve"><anchor type="note" xlink:href="" symbol="m"/> Cum ergo baſes habeant eandem pro- <anchor type="note" xlink:label="note-401-12a" xlink:href="note-401-12"/> portionem, quam ſegmenta lateris ſecundum datam proportionem diuiſi; </s> <s xml:id="echoid-s17669" xml:space="preserve">con-<lb/>ſtat id, quod propoſitum eſt.</s> <s xml:id="echoid-s17670" xml:space="preserve"/> </p> <div xml:id="echoid-div1110" type="float" level="2" n="1"> <note symbol="l" position="right" xlink:label="note-401-11" xlink:href="note-401-11a" xml:space="preserve">32. vndec.</note> <note symbol="m" position="right" xlink:label="note-401-12" xlink:href="note-401-12a" xml:space="preserve">@. ſexti.</note> </div> </div> <div xml:id="echoid-div1112" type="section" level="1" n="403"> <head xml:id="echoid-head430" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s17671" xml:space="preserve"><emph style="sc">Non</emph> aliter priſma quo dlibet, aut cylindrus ſecundum datam proportio-<lb/>nem ſecabitur, ſi altitudo in datã ſecetur proportionem, & </s> <s xml:id="echoid-s17672" xml:space="preserve">per punctũ ſectionis <pb o="374" file="402" n="402" rhead="GEOMETR. PRACT."/> planum ducatur baſibus parallelum. </s> <s xml:id="echoid-s17673" xml:space="preserve">Hoc enim ſecabit <anchor type="note" xlink:href="" symbol="a"/> tam priſma, <anchor type="note" xlink:href="" symbol="b"/> quam cy- <anchor type="note" xlink:label="note-402-01a" xlink:href="note-402-01"/> lindrum in datam proportionem. <lb/></s> <s xml:id="echoid-s17674" xml:space="preserve"> <anchor type="note" xlink:label="note-402-02a" xlink:href="note-402-02"/> </s> </p> <div xml:id="echoid-div1112" type="float" level="2" n="1"> <note symbol="a" position="left" xlink:label="note-402-01" xlink:href="note-402-01a" xml:space="preserve">ſchol. 14. <lb/>duodec.</note> <note symbol="b" position="left" xlink:label="note-402-02" xlink:href="note-402-02a" xml:space="preserve">14. duodec.</note> </div> </div> <div xml:id="echoid-div1114" type="section" level="1" n="404"> <head xml:id="echoid-head431" xml:space="preserve">PROBL. 33. PROPOS. 47.</head> <p> <s xml:id="echoid-s17675" xml:space="preserve">FIGVRAM Ellipſi ſimilem, quam ouatam dicunt, circino deſcri-<lb/>bere.</s> <s xml:id="echoid-s17676" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17677" xml:space="preserve"><emph style="sc">Libet</emph> miſcellaneorum hunc librum peruulgato illo problemate conclu-<lb/>dere, quo artifices ope circini deſcribere ſolent ſiguram ouatam Ellipſi ſimilem, <lb/>ita vt nulli anguli appareant: </s> <s xml:id="echoid-s17678" xml:space="preserve">cum non rarò eiuſmo di figura à Geometris in ſuis <lb/>delineationibus adhibeatur. </s> <s xml:id="echoid-s17679" xml:space="preserve">Docui quidem in lib. </s> <s xml:id="echoid-s17680" xml:space="preserve">1 noſtræ Gnomonicę in ſcho-<lb/>lio propoſ. </s> <s xml:id="echoid-s17681" xml:space="preserve">8. </s> <s xml:id="echoid-s17682" xml:space="preserve">qua ratione vera Ellipſis, quę coni-<lb/> <anchor type="figure" xlink:label="fig-402-01a" xlink:href="fig-402-01"/> ca ſectio eſt, deſcribenda ſit: </s> <s xml:id="echoid-s17683" xml:space="preserve">Sed hic ſimilem figu-<lb/> <anchor type="note" xlink:label="note-402-03a" xlink:href="note-402-03"/> ram ex ſegmentis circulorum conſtantem deſcri-<lb/>bendam proponimus. </s> <s xml:id="echoid-s17684" xml:space="preserve">Ita ergo, vt ex variis ſcri-<lb/>ptoribus colligitur, agemus. </s> <s xml:id="echoid-s17685" xml:space="preserve">Conſtruantur duo <lb/>triangula æquilatera, vel Iſoſcelia ſupra ba-<lb/>ſem communem A C, in diuerſas partes A B C, <lb/>ADC. </s> <s xml:id="echoid-s17686" xml:space="preserve">(Æquilatera venuſtiorem faciunt figuram, <lb/>vt experientia te docebit) productiſque lateribus, <lb/>deſcribantur ex A, C, duo arcus EFG, HIK, vſque <lb/>ad latera producta. </s> <s xml:id="echoid-s17687" xml:space="preserve">Si namque ex B, D, per E, K, <lb/>G, H, alij arcus deſcribantur, <anchor type="note" xlink:href="" symbol="c"/> tangent hipriores <anchor type="note" xlink:label="note-402-04a" xlink:href="note-402-04"/> arcus in punctis E, K, G, H: </s> <s xml:id="echoid-s17688" xml:space="preserve">ac proinde illos non <lb/>ſecabunt, conſtituta que erit figura ouata.</s> <s xml:id="echoid-s17689" xml:space="preserve"/> </p> <div xml:id="echoid-div1114" 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/xxxxxxxx/figures/402-01"/> </figure> <note symbol="c" position="left" xlink:label="note-402-03" xlink:href="note-402-03a" xml:space="preserve">15. primi.</note> <note symbol="d" position="left" xlink:label="note-402-04" xlink:href="note-402-04a" xml:space="preserve">ſchol. 13. <lb/>tertii.</note> </div> <p> <s xml:id="echoid-s17690" xml:space="preserve"><emph style="sc">Bene</emph> autem vides, ex eiſdem centris A, C, B, D, deſcribi poſſe varias figuras, <lb/>prout arcus EFG, HIK, maiores fuerint, autminores, vt in figura apparet.</s> <s xml:id="echoid-s17691" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17692" xml:space="preserve"><emph style="sc">Qvod</emph> ſi triangula conſtituta ſint Iſoſcelia, poterunt latera AB, CB, &</s> <s xml:id="echoid-s17693" xml:space="preserve">c. </s> <s xml:id="echoid-s17694" xml:space="preserve">vel <lb/>maiora fieri baſe AC, vel minora. </s> <s xml:id="echoid-s17695" xml:space="preserve">In figura noſtra ſunt minora.</s> <s xml:id="echoid-s17696" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17697" xml:space="preserve"><emph style="sc">Potes</emph> etiam, ſi placet, primo loco ex centris B, D, deſcribere arcus EMK, <lb/>GLH, ad quodcunque interuallum, pro latitudine figuræ deſcribendę: </s> <s xml:id="echoid-s17698" xml:space="preserve">deinde <lb/>ex centris A, C, minores arcus delineare EFG, HIK.</s> <s xml:id="echoid-s17699" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17700" xml:space="preserve"><emph style="sc">Qvin</emph> etiam ſine conſtructione triangulorum idem effi ciemus hoc modo. <lb/></s> <s xml:id="echoid-s17701" xml:space="preserve">Ductis duabus rectis AC, BD, ad angulos rectos ſe ſecantibus in N; </s> <s xml:id="echoid-s17702" xml:space="preserve">ſumptiſque <lb/>æqualibus NA, NC, quantiſcunque pro longitudine figurę, deſcribantur ex A, <lb/>C, arcus circulorum EFG, HIK, parui, aut magni, prout deſideras extremitates <lb/>figurę ſecundum longitudinem habere anguſtiores, latioreſue. </s> <s xml:id="echoid-s17703" xml:space="preserve">Deinde acce-<lb/>ptis aliis duabus rectis æqualibus NB, ND, quantiſcunque, (quo autem puncta <lb/>B, D, remotiora fuerint ab N, eo anguſtior figura euadet: </s> <s xml:id="echoid-s17704" xml:space="preserve">& </s> <s xml:id="echoid-s17705" xml:space="preserve">quo minus remota, <lb/>eo latior. </s> <s xml:id="echoid-s17706" xml:space="preserve">Sed vſus magiſter optimus facilè docebit, quantæ debeant eſſe re-<lb/>ctæ NB, ND,) ducantur ex B, D, per centra A, C, rectæ ſecantes priores arcus in <lb/>E, K, &</s> <s xml:id="echoid-s17707" xml:space="preserve">c. </s> <s xml:id="echoid-s17708" xml:space="preserve">Nam ſi ex B, D, per puncta E, K, &</s> <s xml:id="echoid-s17709" xml:space="preserve">c. </s> <s xml:id="echoid-s17710" xml:space="preserve">alij duo arcus deſcribantur, per-<lb/>fecta erit figura Ellipſi ſimilis.</s> <s xml:id="echoid-s17711" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17712" xml:space="preserve"><emph style="sc">Vt</emph> autem videas, venuſtiores figuras deſcribi, ſi triangula ABC, ADC, ſint <pb o="375" file="403" n="403" rhead="LIBER OCTAVVS."/> æquilatera, quæ fere ratio ab artificibus ſeruari ſolet, deſcrip ſimus hic duas fi-<lb/>guras. </s> <s xml:id="echoid-s17713" xml:space="preserve">In minori eſt latus BA, rectæ AE, duplum, in maiorivero æquale, &</s> <s xml:id="echoid-s17714" xml:space="preserve">c.</s> <s xml:id="echoid-s17715" xml:space="preserve"/> </p> <figure> <image file="403-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/403-01"/> </figure> <p> <s xml:id="echoid-s17716" xml:space="preserve"><emph style="sc">Iam</emph> verò area figuræ ouatæ, beneficio trianguli æquilateri deſcriptæ, quam <lb/> <anchor type="note" xlink:label="note-403-01a" xlink:href="note-403-01"/> artifices non raro expetunt, facilè inuenietur, hoc modo. </s> <s xml:id="echoid-s17717" xml:space="preserve">Sector BEK, eſt ſe-<lb/>xta pars circuli, cuius ſemidiameter BE, nota, nimirum latus trianguli ęquilate-<lb/>ri BEK, quod eſt in maiorifigura duplum lateris AB, aſſumpti ad libitum: </s> <s xml:id="echoid-s17718" xml:space="preserve">in mi-<lb/>nori vero ſeſquialterum eſt eiuſdem lateris AB. </s> <s xml:id="echoid-s17719" xml:space="preserve">Inuenta ergo area illius circuli, <lb/>vt lib. </s> <s xml:id="echoid-s17720" xml:space="preserve">4. </s> <s xml:id="echoid-s17721" xml:space="preserve">cap. </s> <s xml:id="echoid-s17722" xml:space="preserve">7. </s> <s xml:id="echoid-s17723" xml:space="preserve">docuimus, ſi ex eius ſexta parte dematur triangulum æquilate-<lb/>rum BEK, cuius area reperietur per ea, quæ in eodem lib. </s> <s xml:id="echoid-s17724" xml:space="preserve">4. </s> <s xml:id="echoid-s17725" xml:space="preserve">cap. </s> <s xml:id="echoid-s17726" xml:space="preserve">2. </s> <s xml:id="echoid-s17727" xml:space="preserve">Num. </s> <s xml:id="echoid-s17728" xml:space="preserve">5. </s> <s xml:id="echoid-s17729" xml:space="preserve">tradi-<lb/>ta ſunt: </s> <s xml:id="echoid-s17730" xml:space="preserve">reliquum fiet ſegmentum EK, ac proinde & </s> <s xml:id="echoid-s17731" xml:space="preserve">GH, notum. </s> <s xml:id="echoid-s17732" xml:space="preserve">Item ſector <lb/>AGE, eſt pars duo decima circuli, cuius ſemidiameter AG, nota, nimirum vel æ-<lb/>qualis lateri aſſumpto AB, vel ſemiſsis ipſius. </s> <s xml:id="echoid-s17733" xml:space="preserve">Si igitur ex duo decima parte areæ <lb/>illius circuli auferatur area Iſoſcelis AGE, quæ reperietur per ea, quæ lib. </s> <s xml:id="echoid-s17734" xml:space="preserve">4. </s> <s xml:id="echoid-s17735" xml:space="preserve">cap. <lb/></s> <s xml:id="echoid-s17736" xml:space="preserve">2. </s> <s xml:id="echoid-s17737" xml:space="preserve">Num. </s> <s xml:id="echoid-s17738" xml:space="preserve">4. </s> <s xml:id="echoid-s17739" xml:space="preserve">ſcrip ſimus: </s> <s xml:id="echoid-s17740" xml:space="preserve">remanebit ſegmentum GFE, notum, ideoque & </s> <s xml:id="echoid-s17741" xml:space="preserve">ſegmen-<lb/>tum H I K. </s> <s xml:id="echoid-s17742" xml:space="preserve">Quocirca ſi quatuor ſegmentis cognitis adij ciatur area rectanguli <lb/>EGHK, cognita erit area totius figuræ. </s> <s xml:id="echoid-s17743" xml:space="preserve">Cognoſcetur autem area huius rectan-<lb/>guli ex doctrina cap. </s> <s xml:id="echoid-s17744" xml:space="preserve">1. </s> <s xml:id="echoid-s17745" xml:space="preserve">lib. </s> <s xml:id="echoid-s17746" xml:space="preserve">4. </s> <s xml:id="echoid-s17747" xml:space="preserve">cum latus E K, chorda ſit ſextæ partis circuli, hoc <lb/>eſt, ſemidiametro æquale: </s> <s xml:id="echoid-s17748" xml:space="preserve">at E F, ſit ſinus grad. </s> <s xml:id="echoid-s17749" xml:space="preserve">60. </s> <s xml:id="echoid-s17750" xml:space="preserve">id eſt, ſemiſsis chordæ <lb/>grad. </s> <s xml:id="echoid-s17751" xml:space="preserve">120.</s> <s xml:id="echoid-s17752" xml:space="preserve"/> </p> <div xml:id="echoid-div1115" type="float" level="2" n="2"> <note position="right" xlink:label="note-403-01" xlink:href="note-403-01a" xml:space="preserve">Area figura <lb/>ouatæ hic de <lb/>ſcripta.</note> </div> <p> <s xml:id="echoid-s17753" xml:space="preserve"><emph style="sc">Vervm</emph> quia hac ratione deſcribi nequit figura ad datam longitudinem, <lb/>latitudinem que; </s> <s xml:id="echoid-s17754" xml:space="preserve">(quoniam ſi longitudo eligatur FI, ignotum erit, quanta ſit fu-<lb/>tura latitudo: </s> <s xml:id="echoid-s17755" xml:space="preserve">propterea quod arcus ex B, D, deſcripti raro tranſeunt per electa <lb/>puncta latitudinis: </s> <s xml:id="echoid-s17756" xml:space="preserve">Siverò eligatur latitudo L M, in prima figura, ignorabitur fu-<lb/>tura longitudo: </s> <s xml:id="echoid-s17757" xml:space="preserve">quip pe cum arcus ex A, C, deſcriptiraro etiam per electa pun-<lb/>cto longitudinis tranſeant: </s> <s xml:id="echoid-s17758" xml:space="preserve">vt perſpicuum eſt.) </s> <s xml:id="echoid-s17759" xml:space="preserve">do cebimus cum Ioan. </s> <s xml:id="echoid-s17760" xml:space="preserve">Baptiſta <lb/>Benedicto, quo pacto, data tam longitudine, quam latiudine figura Ellip ſi ſimi-<lb/>lis deſcribenda ſit. </s> <s xml:id="echoid-s17761" xml:space="preserve">Sitergo data longitudo AB, & </s> <s xml:id="echoid-s17762" xml:space="preserve">latitudo CD, quæ ſe bifa- <pb o="376" file="404" n="404" rhead="GEOMETR. PRACT"/> riam, & </s> <s xml:id="echoid-s17763" xml:space="preserve">ad rectos angulos ſecent in E. </s> <s xml:id="echoid-s17764" xml:space="preserve">Ex latitu dine CD, abſcindatur recta DF@ <lb/> <anchor type="figure" xlink:label="fig-404-01a" xlink:href="fig-404-01"/> quantacunoue vltra E, maior tamen interual-<lb/>lo inter punctum F, & </s> <s xml:id="echoid-s17765" xml:space="preserve">A, extremum longitu-<lb/>dinis. </s> <s xml:id="echoid-s17766" xml:space="preserve">Hoc enim niſi fiat, deſcribi non pote-<lb/>rit figura ouata. </s> <s xml:id="echoid-s17767" xml:space="preserve">Deinde centro F, & </s> <s xml:id="echoid-s17768" xml:space="preserve">interual-<lb/>lo FD, deſcribatur circulus DG, quineceſſa-<lb/>rio vltra punctum A, tranſibit, quippe cum ſe-<lb/>midiameter F D, maior poſita ſit interuallo <lb/>F A. </s> <s xml:id="echoid-s17769" xml:space="preserve">Ducta autem FG, longitudini AB, paral-<lb/>lela ſecante circulum deſcriptum in G; </s> <s xml:id="echoid-s17770" xml:space="preserve">duca-<lb/>tur ex G, per A, recta ſecans eundem circulum <lb/>in H, puncto, è quo ducatur HIK, latitudini <lb/>CD, parallela, iungatur que HF, ſecans AB, <lb/>longitudin emin L; </s> <s xml:id="echoid-s17771" xml:space="preserve">eruntque FG, FH, æqua-<lb/> <anchor type="note" xlink:label="note-404-01a" xlink:href="note-404-01"/> les è centro F, ad circumferentiam. </s> <s xml:id="echoid-s17772" xml:space="preserve"><anchor type="note" xlink:href="" symbol="a"/> Et quia triangula H G F, H A L, ſimilia ſunt; </s> <s xml:id="echoid-s17773" xml:space="preserve"><anchor type="note" xlink:href="" symbol="b"/> erit vt <anchor type="note" xlink:label="note-404-02a" xlink:href="note-404-02"/> GF, ad FH, ita AL, ad LH. </s> <s xml:id="echoid-s17774" xml:space="preserve">Cum ergo GF, i-<lb/>pſi FH. </s> <s xml:id="echoid-s17775" xml:space="preserve">ſit æqualis; </s> <s xml:id="echoid-s17776" xml:space="preserve">erit quoque AL, ipſi LH, æqualis. </s> <s xml:id="echoid-s17777" xml:space="preserve">Circulus ergo AH, ex L, <lb/> <anchor type="note" xlink:label="note-404-03a" xlink:href="note-404-03"/> per A, deſcriptus tranſibit per H, <anchor type="note" xlink:href="" symbol="c"/> ibique priorem circulum D, G, tanget. </s> <s xml:id="echoid-s17778" xml:space="preserve">Si igitur capiatur EO, æqualis ipſi EI, & </s> <s xml:id="echoid-s17779" xml:space="preserve">EM, ipſi EL, ducatur que POQ, per O, re-<lb/>ctæ CD, parallela, atque ex M, centro, interuallo autem LH, vel MP, circulus <lb/>PBQ, deſcribatur, tanget hic quo que priorem circulum in P. </s> <s xml:id="echoid-s17780" xml:space="preserve">Si denique ſum-<lb/>pta EN, ipſi EF, æquali, deſcribatur ex N, ad interuallum FD, prioris circuli cir-<lb/>culus KCQ tanget hic circulos HAK, PBQ, in K, Q, perfecta que erit figura <lb/>ouata.</s> <s xml:id="echoid-s17781" xml:space="preserve"/> </p> <div xml:id="echoid-div1116" type="float" level="2" n="3"> <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/xxxxxxxx/figures/404-01"/> </figure> <note symbol="a" position="left" xlink:label="note-404-01" xlink:href="note-404-01a" xml:space="preserve">coroll. 4. <lb/>ſexti.</note> <note symbol="b" position="left" xlink:label="note-404-02" xlink:href="note-404-02a" xml:space="preserve">4. ſexti.</note> <note symbol="c" position="left" xlink:label="note-404-03" xlink:href="note-404-03a" xml:space="preserve">ſchol. 13. <lb/>tertij.</note> </div> <p> <s xml:id="echoid-s17782" xml:space="preserve"><emph style="sc">Sed</emph> quia, vt dictum eſt, conſtat que ex deſcrip tione, niſi latitudo CD, tanta <lb/>ſit, vt ex ea abſcindi poſsit recta DF, maior interuallo FA, figura hac ratione de-<lb/>ſcribinequit: </s> <s xml:id="echoid-s17783" xml:space="preserve">adeo vt longitudo, ac latitudo ad libitum aſſumi non poſsint; </s> <s xml:id="echoid-s17784" xml:space="preserve">in-<lb/>ſtituetur operatio alio modo, ſumpta quacun que longitudine AB, & </s> <s xml:id="echoid-s17785" xml:space="preserve">latitudi-<lb/>ne CD. </s> <s xml:id="echoid-s17786" xml:space="preserve">Secent ſe in prima figura longitudo, latitudo que FI, LM, datæ mutuo <lb/>bifariam in N, & </s> <s xml:id="echoid-s17787" xml:space="preserve">ad angulos rectos, & </s> <s xml:id="echoid-s17788" xml:space="preserve">ſumantur rectæ FA, IC, æquales, & </s> <s xml:id="echoid-s17789" xml:space="preserve">mi-<lb/>nores ſemiſſe latitudinis LN; </s> <s xml:id="echoid-s17790" xml:space="preserve">deſcribantur que ex A, & </s> <s xml:id="echoid-s17791" xml:space="preserve">C, per F, & </s> <s xml:id="echoid-s17792" xml:space="preserve">I, circelli E-<lb/>FG, HIK, Sumpta deinde MO, ſemidiametro AF, æquali, iungatur OA, ex O, <lb/>ad centrum A, quam bifariam, & </s> <s xml:id="echoid-s17793" xml:space="preserve">adangulos recto sin P, ſecet recta PB, ſecans <lb/>LM, etiam pro ductum, ſi opus eſt, in B, ducatur que BA, vſque ad circulum G-<lb/>FE. </s> <s xml:id="echoid-s17794" xml:space="preserve">Et quoniam duo latera OP, PB, duobus lateribus AP, PB, æqualia ſunt, an-<lb/> <anchor type="note" xlink:label="note-404-04a" xlink:href="note-404-04"/> guloſque continent rectos, id eſt, æquales: </s> <s xml:id="echoid-s17795" xml:space="preserve"><anchor type="note" xlink:href="" symbol="d"/> erunt baſes OB, AB, æquales; </s> <s xml:id="echoid-s17796" xml:space="preserve">ad- ditiſque æqualibus OM, AE, (ſumpta nam que fuit MO, æqualis ſemidiametro <lb/>FA, vel IC,) totæ BE, BM, æquales erunt. </s> <s xml:id="echoid-s17797" xml:space="preserve">Deſcriptus ergo circulus ex B, per M, <lb/> <anchor type="note" xlink:label="note-404-05a" xlink:href="note-404-05"/> tranſibit per E, <anchor type="note" xlink:href="" symbol="e"/> ibique circellum GFE, tanget. </s> <s xml:id="echoid-s17798" xml:space="preserve">Eodem que modo circellum H- IK, tanget. </s> <s xml:id="echoid-s17799" xml:space="preserve">Siigitur ſumpta ND, ipſi NB, æquali, deſcribatur ex D, per L, circulus <lb/>tangens eoſdem priores circellos in G, H, abſoluta erit figura.</s> <s xml:id="echoid-s17800" xml:space="preserve"/> </p> <div xml:id="echoid-div1117" type="float" level="2" n="4"> <note symbol="d" position="left" xlink:label="note-404-04" xlink:href="note-404-04a" xml:space="preserve">4. primi.</note> <note symbol="e" position="left" xlink:label="note-404-05" xlink:href="note-404-05a" xml:space="preserve">ſchol. 13. <lb/>tertij.</note> </div> </div> <div xml:id="echoid-div1119" type="section" level="1" n="405"> <head xml:id="echoid-head432" xml:space="preserve">SCHOLIVM.</head> <p> <s xml:id="echoid-s17801" xml:space="preserve"><emph style="sc">Qvo</emph> autem ſemidiameter FA, ſumpta fuerit minor, quam MN, co certius <lb/>centrum B, reperietur vt liquet,</s> </p> <pb o="377" file="405" n="405" rhead="LIBER OCTAVVS."/> <p> <s xml:id="echoid-s17802" xml:space="preserve"><emph style="sc">Rogatv</emph> multorum placet Epilogi loco apponere tabulam qua dratorũ, <lb/>& </s> <s xml:id="echoid-s17803" xml:space="preserve">cuborum, qui ex numeris ab 1. </s> <s xml:id="echoid-s17804" xml:space="preserve">vſque ad 1000. </s> <s xml:id="echoid-s17805" xml:space="preserve">pro ducuntur: </s> <s xml:id="echoid-s17806" xml:space="preserve">propterea quod <lb/>huiuſce tabulæ multiplex, & </s> <s xml:id="echoid-s17807" xml:space="preserve">inſignis eſt vſus cum in alijs rebus Mathematicis, <lb/>tum verò maxime in radicibus quadratis, & </s> <s xml:id="echoid-s17808" xml:space="preserve">cubicis ex magnis numeris extra-<lb/>hendis, vt poſt tabulam paucis exponam. </s> <s xml:id="echoid-s17809" xml:space="preserve">Non extendi autem tabulam vltra <lb/>radicem 1000. </s> <s xml:id="echoid-s17810" xml:space="preserve">contentus radicibus tres figuras non ſuperantibus: </s> <s xml:id="echoid-s17811" xml:space="preserve">propterea <lb/>quod ſi extenderetur vſque ad radicem 10000. </s> <s xml:id="echoid-s17812" xml:space="preserve">vt radices haberentur quatuor <lb/>figurarum, decies maior tabula conficienda eſſſet. </s> <s xml:id="echoid-s17813" xml:space="preserve">Si quis tamen eam extende-<lb/>re volet, inueniet ad finem tabulæ regulas, quibus id facile poſsit exe-<lb/>qui. </s> <s xml:id="echoid-s17814" xml:space="preserve">Quamuis quadratorum tabulam Doctiſsimus Maginus in ſua <lb/>tabula Tetragonica vſque ad radicem 10000. </s> <s xml:id="echoid-s17815" xml:space="preserve">pro-<lb/>mouerit.</s> <s xml:id="echoid-s17816" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1120" type="section" level="1" n="406"> <head xml:id="echoid-head433" xml:space="preserve">SEQVITVR TABVLA QVADRATO-<lb/>rum, & Cuborum, quorum radices maio-<lb/>res non ſunt, quam 1000.</head> <pb o="378" file="406" n="406" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1121" type="section" level="1" n="407"> <head xml:id="echoid-head434" xml:space="preserve">Tabula Quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>1 # 1 # 1 <lb/>2 # 4 # 8 <lb/>3 # 9 # 27 <lb/>4 # 16 # 64 <lb/>5 # 25 # 125 <lb/>6 # 36 # 216 <lb/>7 # 49 # 343 <lb/>8 # 64 # 512 <lb/>9 # 81 # 729 <lb/>10 # 100 # 1000 <lb/>11 # 121 # 1331 <lb/>12 # 144 # 1728 <lb/>13 # 169 # 2197 <lb/>14 # 196 # 2744 <lb/>15 # 225 # 3375 <lb/>16 # 256 # 4096 <lb/>17 # 289 # 4913 <lb/>18 # 324 # 5832 <lb/>19 # 361 # 6859 <lb/>20 # 400 # 8000 <lb/>21 # 441 # 9261 <lb/>22 # 484 # 10648 <lb/>23 # 529 # 12167 <lb/>24 # 576 # 13824 <lb/>25 # 625 # 15625 <lb/>26 # 676 # 17576 <lb/>27 # 729 # 19683 <lb/>28 # 784 # 21952 <lb/>29 # 841 # 24389 <lb/>30 # 900 # 27000 <lb/>31 # 961 # 29791 <lb/>32 # 1024 # 32768 <lb/>33 # 1089 # 35937 <lb/>34 # 1156 # 39304 <lb/>35 # 1225 # 42875 <lb/>36 # 1296 # 46656 <lb/>37 # 1369 # 50653 # 77 <lb/>38 # 1444 # 54872 <lb/>39 # 1521 # 59319 <lb/>40 # 1600 # 64000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/># 41 # 1681 # 68921 <lb/># 42 # 1764 # 74088 <lb/># 43 # 1849 # 79507 <lb/># 44 # 1936 # 85184 <lb/># 45 # 2025 # 91125 <lb/># 46 # 2116 # 97336 <lb/># 47 # 2209 # 103823 <lb/># 48 # 2304 # 110592 <lb/># 49 # 2401 # 117649 <lb/># 50 # 2500 # 125000 <lb/># 51 # 2601 # 132651 <lb/># 52 # 2704 # 140608 <lb/># 53 # 2809 # 148877 <lb/># 54 # 2916 # 157464 <lb/># 55 # 3025 # 166375 <lb/># 56 # 3136 # 175616 <lb/># 57 # 3249 # 185193 <lb/># 58 # 3364 # 195112 <lb/># 59 # 3481 # 205379 <lb/># 60 # 3600 # 216000 <lb/># 61 # 3721 # 226981 <lb/># 62 # 3844 # 238328 <lb/># 63 # 3969 # 250047 <lb/># 64 # 4096 # 262144 <lb/># 65 # 4225 # 274625 <lb/># 66 # 4356 # 287496 <lb/># 67 # 4489 # 300763 <lb/># 68 # 4624 # 314432 <lb/># 69 # 4761 # 328509 <lb/># 70 # 4900 # 343000 <lb/># 71 # 5041 # 357911 <lb/># 72 # 5184 # 373248 <lb/># 73 # 5329 # 389017 <lb/># 74 # 5476 # 405224 <lb/># 75 # 5625 # 421875 <lb/># 76 # 5776 # 438976 <lb/># 77 # 5929 # 456533 <lb/># 78 # 6084 # 474552 <lb/># 79 # 6241 # 493039 <lb/># 80 # 6400 # 512000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>81 # 1656 # 531441 <lb/>82 # 4672 # 551368 <lb/>83 # 6889 # 571787 <lb/>84 # 9705 # 592704 <lb/>85 # 7225 # 614125 <lb/>86 # 7396 # 636056 <lb/>87 # 7569 # 658503 <lb/>88 # 7744 # 681472 <lb/>89 # 7921 # 704968 <lb/>90 # 8100 # 729000 <lb/>91 # 8281 # 753571 <lb/>92 # 8464 # 778688 <lb/>93 # 8649 # 804357 <lb/>94 # 8836 # 830584 <lb/>95 # 9025 # 857375 <lb/>96 # 9216 # 884736 <lb/>97 # 9409 # 912673 <lb/>98 # 9604 # 941192 <lb/>99 # 9801 # 970299 <lb/>100 # 10000 # 1000000 <lb/>101 # 10201 # 1030301 <lb/>102 # 10404 # 1061208 <lb/>103 # 10609 # 1092727 <lb/>104 # 10816 # 1124864 <lb/>105 # 11025 # 1157625 <lb/>106 # 11236 # 1191016 <lb/>107 # 11449 # 1225043 <lb/>108 # 11664 # 1259712 <lb/>109 # 11881 # 1295029 <lb/>110 # 12100 # 1331000 <lb/>111 # 12321 # 1367631 <lb/>112 # 12544 # 1404928 <lb/>113 # 12769 # 1442897 <lb/>114 # 12996 # 1481544 <lb/>115 # 13225 # 1520875 <lb/>116 # 13456 # 1560896 <lb/>117 # 13685 # 1601613 <lb/>118 # 13924 # 1643032 <lb/>119 # 14161 # 1685159 <lb/>120 # 14400 # 1728000 <lb/></note> <pb o="379" file="407" n="407" rhead="LIBER OCTAVS."/> </div> <div xml:id="echoid-div1122" type="section" level="1" n="408"> <head xml:id="echoid-head435" xml:space="preserve">Tabula Quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>121 # 14641 # 1771561 <lb/>122 # 14884 # 1815848 <lb/>123 # 15129 # 1860867 <lb/>124 # 15376 # 1906624 <lb/>125 # 15625 # 1953125 <lb/>126 # 15876 # 2000376 <lb/>127 # 16129 # 2048383 <lb/>128 # 16384 # 2097152 <lb/>129 # 16641 # 2146689 <lb/>130 # 16900 # 2197000 <lb/>131 # 17161 # 2248091 <lb/>132 # 17424 # 2299968 <lb/>133 # 17689 # 2352637 <lb/>134 # 17956 # 2406104 <lb/>135 # 18225 # 2460375 <lb/>136 # 18496 # 2515456 <lb/>137 # 18769 # 2571353 <lb/>138 # 19044 # 2628027 <lb/>139 # 19321 # 2685619 <lb/>140 # 19600 # 2744000 <lb/>141 # 19881 # 2803221 <lb/>142 # 20164 # 2863288 <lb/>143 # 20449 # 2924207 <lb/>144 # 20736 # 2985984 <lb/>145 # 21025 # 3048625 <lb/>146 # 21316 # 3112136 <lb/>147 # 21609 # 3176523 <lb/>148 # 21904 # 3241792 <lb/>149 # 22201 # 3307949 <lb/>150 # 22500 # 3375000 <lb/>151 # 22801 # 3442951 <lb/>152 # 23104 # 3511808 <lb/>153 # 23409 # 3581577 <lb/>154 # 23716 # 3652264 <lb/>155 # 24025 # 3723875 <lb/>156 # 24336 # 3796416 <lb/>157 # 24649 # 3869893 <lb/>158 # 24964 # 3944312 <lb/>159 # 2528@ # 4019679 <lb/>160 # 25600 # 4096000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>161 # 25921 # 4173281 <lb/>162 # 26244 # 4251528 <lb/>163 # 26569 # 4330747 <lb/>164 # 26896 # 4410944 <lb/>165 # 27225 # 4492125 <lb/>166 # 27556 # 4574296 <lb/>167 # 27889 # 4657463 <lb/>168 # 28224 # 4741632 <lb/>169 # 28561 # 4826809 <lb/>170 # 28900 # 4913000 <lb/>171 # 29241 # 5000211 <lb/>172 # 29584 # 5088448 <lb/>173 # 29929 # 5177717 <lb/>174 # 30276 # 5268024 <lb/>175 # 30625 # 5359375 <lb/>176 # 30976 # 5451776 <lb/>177 # 31329 # 5545233 <lb/>178 # 31684 # 5639752 <lb/>179 # 32041 # 5735339 <lb/>180 # 32400 # 5832000 <lb/>181 # 32761 # 5929741 <lb/>182 # 33124 # 6028568 <lb/>183 # 33489 # 6128487 <lb/>184 # 33856 # 6229504 <lb/>185 # 34225 # 6331625 <lb/>186 # 34596 # 6434856 <lb/>187 # 34969 # 6539203 <lb/>188 # 35344 # 6644672 <lb/>189 # 35721 # 6751269 <lb/>190 # 36100 # 6859000 <lb/>191 # 36481 # 6967871 <lb/>192 # 36864 # 7077888 <lb/>193 # 37249 # 7189057 <lb/>194 # 37636 # 7301384 <lb/>195 # 38025 # 7414875 <lb/>196 # 38416 # 7529536 <lb/>197 # 38809 # 7645373 <lb/>198 # 39204 # 7762392 <lb/>199 # 39601 # 7880599 <lb/>200 # 40000 # 8000000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>201 # 40401 # 8120601 <lb/>202 # 40804 # 8242408 <lb/>203 # 41209 # 8365427 <lb/>204 # 41616 # 8489664 <lb/>205 # 42025 # 8615125 <lb/>206 # 42436 # 8741816 <lb/>207 # 42849 # 8869743 <lb/>208 # 43264 # 8998912 <lb/>209 # 43681 # 9129329 <lb/>210 # 44100 # 9261000 <lb/>211 # 44521 # 9393931 <lb/>212 # 44944 # 9528128 <lb/>213 # 45369 # 9663597 <lb/>214 # 45796 # 9800344 <lb/>215 # 46225 # 9938375 <lb/>216 # 46656 # 10077696 <lb/>217 # 47089 # 10218313 <lb/>218 # 47524 # 10360232 <lb/>219 # 47961 # 10503459 <lb/>220 # 48400 # 10648080 <lb/>221 # 48841 # 10793861 <lb/>222 # 49284 # 10941048 <lb/>223 # 49729 # 11089567 <lb/>224 # 50176 # 11239414 <lb/>225 # 50625 # 11390625 <lb/>226 # 51076 # 11543176 <lb/>227 # 51529 # 11697083 <lb/>228 # 51984 # 11852352 <lb/>229 # 52441 # 12008989 <lb/>230 # 52900 # 12167000 <lb/>231 # 53361 # 12326391 <lb/>232 # 53824 # 12487168 <lb/>233 # 54289 # 12649337 <lb/>234 # 54756 # 12812904 <lb/>235 # 55225 # 12977875 <lb/>236 # 55696 # 13144256 <lb/>237 # 56169 # 13312053 <lb/>238 # 56644 # 13481272 <lb/>239 # 57121 # 13651919 <lb/>240 # 57600 # 13824000 <lb/></note> <pb o="380" file="408" n="408" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1123" type="section" level="1" n="409"> <head xml:id="echoid-head436" xml:space="preserve">Tabula Quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>241 # 58081 # 13997521 <lb/>242 # 58564 # 14172488 <lb/>243 # 59049 # 14348907 <lb/>244 # 59536 # 14526784 <lb/>245 # 60025 # 14706125 <lb/>246 # 60516 # 14886936 <lb/>247 # 61009 # 15069223 <lb/>248 # 61504 # 15252992 <lb/>249 # 62001 # 15438249 <lb/>250 # 62500 # 15625000 <lb/>251 # 63001 # 15813251 <lb/>252 # 63504 # 16003008 <lb/>253 # 64009 # 16194277 <lb/>254 # 64516 # 16387064 <lb/>255 # 65025 # 16581375 <lb/>256 # 65536 # 16777216 <lb/>257 # 66049 # 16974593 <lb/>258 # 66564 # 17173512 <lb/>259 # 67081 # 17373979 <lb/>260 # 67600 # 17576000 <lb/>261 # 68121 # 17779581 <lb/>262 # 68644 # 17984728 <lb/>263 # 69169 # 18191447 <lb/>264 # 69696 # 18399744 <lb/>265 # 70225 # 18609625 <lb/>266 # 70756 # 18821096 <lb/>267 # 71289 # 19034163 <lb/>268 # 71824 # 19248832 <lb/>269 # 72361 # 19465109 <lb/>270 # 72900 # 19683000 <lb/>271 # 73441 # 19902511 <lb/>272 # 73984 # 20123648 <lb/>273 # 74529 # 20346417 <lb/>274 # 75076 # 20570824 <lb/>275 # 75625 # 20796875 <lb/>276 # 76176 # 21024576 <lb/>277 # 76729 # 21253933 <lb/>278 # 77284 # 21484952 <lb/>279 # 77841 # 21717639 <lb/>280 # 78400 # 21952000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>281 # 78961 # 22188041 <lb/>282 # 79524 # 22425768 <lb/>283 # 80089 # 22665187 <lb/>284 # 80656 # 22909304 <lb/>285 # 81225 # 23149125 <lb/>286 # 81796 # 23393656 <lb/>287 # 82369 # 23639903 <lb/>288 # 82944 # 23887872 <lb/>289 # 83521 # 24137569 <lb/>290 # 84100 # 24389000 <lb/>291 # 84681 # 24642171 <lb/>292 # 85264 # 24897088 <lb/>293 # 85849 # 25153757 <lb/>294 # 86436 # 25412184 <lb/>295 # 87025 # 25672375 <lb/>296 # 87616 # 25934336 <lb/>297 # 88209 # 26198073 <lb/>298 # 88804 # 26463592 <lb/>299 # 89401 # 26730899 <lb/>300 # 90000 # 27000000 <lb/>301 # 90601 # 27270901 <lb/>302 # 91204 # 27543608 <lb/>303 # 91809 # 27818127 <lb/>304 # 92416 # 28094464 <lb/>305 # 93025 # 28372625 <lb/>306 # 93636 # 28652616 <lb/>307 # 94249 # 28934443 <lb/>308 # 94864 # 29218112 <lb/>309 # 95481 # 29503629 <lb/>310 # 96100 # 29791000 <lb/>311 # 96721 # 30080231 <lb/>312 # 97344 # 30371328 <lb/>313 # 97969 # 30664297 <lb/>314 # 98596 # 30956144 <lb/>315 # 99225 # 31255875 <lb/>316 # 99856 # 31554496 <lb/>317 # 100489 # 31855013 <lb/>318 # 101124 # 32157432 <lb/>319 # 101761 # 32461759 <lb/>320 # 102400 # 32768000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Qua- \\ drati # Cubi <lb/>321 # 103041 # 33076161 <lb/>322 # 103684 # 33386248 <lb/>323 # 104329 # 33698267 <lb/>324 # 104976 # 34012224 <lb/>325 # 105625 # 34328125 <lb/>326 # 106276 # 34645976 <lb/>327 # 106929 # 34965783 <lb/>328 # 107584 # 35287552 <lb/>329 # 108241 # 35611289 <lb/>330 # 108900 # 35937000 <lb/>331 # 106561 # 36264691 <lb/>332 # 110224 # 36594368 <lb/>333 # 110889 # 36926037 <lb/>334 # 111556 # 37259704 <lb/>335 # 112225 # 37595375 <lb/>336 # 112896 # 37933056 <lb/>337 # 113569 # 38272753 <lb/>338 # 114244 # 38614472 <lb/>339 # 114921 # 38958219 <lb/>340 # 115600 # 39304000 <lb/>341 # 116281 # 39651821 <lb/>342 # 116964 # 40001688 <lb/>343 # 117649 # 40353607 <lb/>344 # 118336 # 40707584 <lb/>345 # 119025 # 41063625 <lb/>346 # 119716 # 41421736 <lb/>347 # 120409 # 41781923 <lb/>348 # 121104 # 42144192 <lb/>349 # 121801 # 42508549 <lb/>350 # 122500 # 42875000 <lb/>351 # 123201 # 43243551 <lb/>352 # 123904 # 43614208 <lb/>353 # 124609 # 43986977 <lb/>354 # 125316 # 44361864 <lb/>355 # 126025 # 44738875 <lb/>356 # 126736 # 45118016 <lb/>357 # 127449 # 45499293 <lb/>358 # 128164 # 45882712 <lb/>359 # 128881 # 46268279 <lb/>360 # 129600 # 46656000 <lb/></note> <pb o="381" file="409" n="409" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1124" type="section" level="1" n="410"> <head xml:id="echoid-head437" xml:space="preserve">Tabula Quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>361 # 130321 # 47045881 <lb/>362 # 131044 # 47437928 <lb/>363 # 131769 # 47832147 <lb/>364 # 132496 # 48238544 <lb/>365 # 133225 # 48627125 <lb/>366 # 133956 # 49027896 <lb/>367 # 134689 # 49430863 <lb/>368 # 135424 # 49836032 <lb/>369 # 136161 # 50243409 <lb/>370 # 136900 # 50653000 <lb/>371 # 137641 # 51064811 <lb/>372 # 138384 # 51478848 <lb/>373 # 139129 # 51895117 <lb/>374 # 139876 # 52313624 <lb/>375 # 140625 # 52734375 <lb/>376 # 141376 # 53157376 <lb/>377 # 142129 # 53582633 <lb/>378 # 142884 # 54010152 <lb/>379 # 143641 # 54439939 <lb/>380 # 144400 # 54872000 <lb/>381 # 145161 # 55306341 <lb/>382 # 145924 # 55742968 <lb/>383 # 146689 # 56181887 <lb/>384 # 147456 # 56623104 <lb/>385 # 148225 # 57066625 <lb/>386 # 148996 # 57512456 <lb/>387 # 149769 # 57960603 <lb/>388 # 150544 # 58411072 <lb/>389 # 151321 # 58863869 <lb/>390 # 152100 # 59319000 <lb/>391 # 152881 # 59776471 <lb/>392 # 153664 # 60236288 <lb/>393 # 154449 # 60698457 <lb/>394 # 155236 # 61162984 <lb/>395 # 156025 # 61629875 <lb/>396 # 156816 # 62099136 <lb/>397 # 157609 # 62570773 <lb/>398 # 158404 # 63044792 <lb/>399 # 159201 # 63521199 <lb/>400 # 160000 # 64000000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>401 # 160801 # 64481201 <lb/>402 # 161604 # 64964808 <lb/>403 # 162409 # 65450827 <lb/>404 # 163216 # 65939264 <lb/>405 # 164025 # 66430125 <lb/>406 # 164836 # 66923416 <lb/>407 # 165649 # 67419143 <lb/>408 # 166464 # 67917312 <lb/>409 # 167281 # 68417929 <lb/>410 # 168100 # 68921000 <lb/>411 # 168921 # 69426531 <lb/>412 # 169744 # 69934528 <lb/>413 # 170569 # 70444997 <lb/>414 # 171396 # 70957944 <lb/>415 # 172225 # 71473375 <lb/>416 # 173056 # 71991296 <lb/>417 # 173889 # 72511713 <lb/>418 # 174724 # 73034632 <lb/>419 # 175561 # 73560059 <lb/>420 # 176400 # 74088000 <lb/>421 # 177241 # 74618461 <lb/>422 # 178084 # 75151448 <lb/>423 # 178929 # 75686967 <lb/>424 # 179776 # 76225024 <lb/>425 # 180625 # 76765625 <lb/>426 # 181476 # 77308776 <lb/>427 # 182329 # 77854483 <lb/>428 # 183184 # 78402752 <lb/>429 # 184041 # 78953589 <lb/>430 # 184900 # 79507000 <lb/>431 # 185761 # 80062991 <lb/>432 # 186624 # 80621668 <lb/>433 # 187489 # 81182737 <lb/>434 # 188356 # 81746504 <lb/>435 # 189225 # 82312875 <lb/>436 # 190096 # 82881856 <lb/>437 # 190969 # 83453453 <lb/>438 # 191844 # 84027672 <lb/>439 # 192721 # 84604519 <lb/>440 # 193600 # 85184000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>441 # 194481 # 85766121 <lb/>442 # 195364 # 86350888 <lb/>443 # 196249 # 86938307 <lb/>444 # 197136 # 87528384 <lb/>445 # 198025 # 88121125 <lb/>446 # 198916 # 88716536 <lb/>447 # 199809 # 89314623 <lb/>448 # 200704 # 89915392 <lb/>449 # 201601 # 90518849 <lb/>450 # 202500 # 91125000 <lb/>451 # 203401 # 91733851 <lb/>452 # 204304 # 92345408 <lb/>453 # 205209 # 92959677 <lb/>454 # 206116 # 93576664 <lb/>455 # 207025 # 94196375 <lb/>456 # 207936 # 94818816 <lb/>457 # 208849 # 95443993 <lb/>458 # 209764 # 96071912 <lb/>459 # 210681 # 96702579 <lb/>460 # 211600 # 97336000 <lb/>461 # 212521 # 97972181 <lb/>462 # 213444 # 98611128 <lb/>463 # 214369 # 99252847 <lb/>464 # 215296 # 99897344 <lb/>465 # 216225 # 100544625 <lb/>466 # 217156 # 101194696 <lb/>467 # 218089 # 101847563 <lb/>468 # 219024 # 102503232 <lb/>469 # 219961 # 103161709 <lb/>470 # 220900 # 103823000 <lb/>471 # 221841 # 104487111 <lb/>472 # 222784 # 105154048 <lb/>473 # 223729 # 105823817 <lb/>474 # 224676 # 106496424 <lb/>475 # 225625 # 107171875 <lb/>476 # 226576 # 107850176 <lb/>477 # 227529 # 108531333 <lb/>478 # 228484 # 109215352 <lb/>479 # 229441 # 109902239 <lb/>480 # 230400 # 110592000 <lb/></note> <pb o="382" file="410" n="410" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1125" type="section" level="1" n="411"> <head xml:id="echoid-head438" xml:space="preserve">Tabula quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>481 # 231361 # 111284641 <lb/>482 # 232324 # 111980168 <lb/>483 # 233289 # 112678587 <lb/>484 # 234256 # 113379904 <lb/>485 # 235225 # 114084125 <lb/>486 # 236196 # 114791256 <lb/>487 # 237169 # 115501303 <lb/>488 # 238144 # 116214272 <lb/>489 # 23912@ # 116930169 <lb/>490 # 240100 # 117649000 <lb/>491 # 241081 # 118370771 <lb/>492 # 242064 # 119095488 <lb/>493 # 243049 # 119823157 <lb/>494 # 244036 # 120553784 <lb/>495 # 245025 # 121287375 <lb/>496 # 246016 # 122023936 <lb/>497 # 247009 # 122763473 <lb/>498 # 248004 # 123505992 <lb/>499 # 249001 # 124251499 <lb/>500 # 250000 # 125000000 <lb/>501 # 251001 # 125751501 <lb/>502 # 252004 # 126506008 <lb/>503 # 253009 # 127263527 <lb/>504 # 254016 # 128024064 <lb/>505 # 255025 # 128787625 <lb/>506 # 256036 # 129554216 <lb/>507 # 257049 # 130323843 <lb/>508 # 258064 # 131096512 <lb/>509 # 259081 # 131872229 <lb/>510 # 260100 # 132651000 <lb/>511 # 261121 # 133432831 <lb/>512 # 262144 # 134217728 <lb/>513 # 263169 # 135005697 <lb/>514 # 264196 # 135796744 <lb/>515 # 265225 # 136590875 <lb/>516 # 266256 # 137388096 <lb/>517 # 267289 # 138188413 <lb/>518 # 268324 # 138991832 <lb/>519 # 269361 # 139798359 <lb/>520 # 270400 # 140608000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>521 # 271441 # 141420761 <lb/>522 # 272484 # 142236648 <lb/>523 # 273529 # 143055667 <lb/>524 # 274576 # 143877824 <lb/>525 # 275625 # 144703125 <lb/>526 # 276676 # 145531576 <lb/>527 # 277729 # 146363183 <lb/>528 # 278784 # 147197952 <lb/>529 # 279841 # 148035889 <lb/>530 # 280900 # 148877000 <lb/>531 # 281961 # 149721291 <lb/>532 # 283024 # 150568768 <lb/>533 # 284089 # 151419437 <lb/>534 # 285156 # 152273304 <lb/>535 # 286225 # 153130375 <lb/>536 # 287296 # 153990656 <lb/>537 # 288369 # 154854153 <lb/>538 # 289444 # 155720872 <lb/>539 # 290521 # 156590819 <lb/>540 # 291600 # 157464000 <lb/>541 # 292681 # 158340421 <lb/>542 # 293764 # 159220088 <lb/>543 # 294849 # 160103007 <lb/>544 # 295936 # 160989184 <lb/>545 # 297025 # 161878625 <lb/>546 # 298116 # 162771336 <lb/>547 # 299209 # 163667323 <lb/>548 # 300304 # 164566592 <lb/>549 # 301401 # 165469149 <lb/>550 # 302500 # 166375000 <lb/>551 # 303601 # 167284151 <lb/>552 # 304704 # 168196608 <lb/>553 # 305809 # 169112377 <lb/>554 # 306916 # 170031464 <lb/>555 # 308025 # 170953875 <lb/>556 # 309136 # 171879616 <lb/>557 # 310249 # 172808693 <lb/>558 # 311364 # 173741112 <lb/>559 # 312481 # 174676879 <lb/>560 # 313600 # 175616000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti # Cubi <lb/>561 # 314721 # 176558481 <lb/>562 # 315844 # 177504328 <lb/>563 # 316969 # 178453547 <lb/>564 # 318096 # 179406144 <lb/>565 # 319225 # 180362125 <lb/>566 # 320356 # 181321496 <lb/>567 # 321489 # 182284263 <lb/>568 # 322624 # 183250432 <lb/>569 # 323761 # 184220009 <lb/>570 # 324900 # 185193000 <lb/>571 # 326041 # 186169411 <lb/>572 # 327184 # 187149248 <lb/>573 # 328329 # 188131518 <lb/>574 # 329476 # 189119224 <lb/>575 # 330625 # 190109375 <lb/>576 # 331776 # 191102976 <lb/>577 # 332929 # 192100033 <lb/>578 # 334084 # 193100552 <lb/>579 # 335241 # 194104539 <lb/>580 # 336400 # 195112000 <lb/>581 # 337561 # 196122941 <lb/>582 # 338724 # 197137368 <lb/>583 # 339889 # 198155287 <lb/>584 # 341056 # 199176704 <lb/>585 # 342225 # 200201625 <lb/>586 # 343396 # 201230056 <lb/>587 # 344569 # 202262003 <lb/>588 # 345744 # 203297472 <lb/>589 # 346921 # 204336469 <lb/>590 # 348100 # 205379000 <lb/>591 # 349281 # 206425071 <lb/>592 # 350464 # 207474688 <lb/>593 # 351649 # 208527857 <lb/>594 # 352836 # 209584584 <lb/>595 # 354025 # 210644875 <lb/>596 # 355216 # 211708736 <lb/>597 # 356409 # 212776173 <lb/>598 # 357604 # 213847192 <lb/>599 # 358801 # 214921799 <lb/>600 # 360000 # 216000000 <lb/></note> <pb o="383" file="411" n="411" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1126" type="section" level="1" n="412"> <head xml:id="echoid-head439" xml:space="preserve">Tabula Quadratorum, & cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>601 # 361201 # 217081801 <lb/>602 # 362404 # 218767208 <lb/>603 # 363609 # 219256227 <lb/>604 # 364816 # 220348864 <lb/>605 # 366025 # 221445125 <lb/>606 # 367236 # 222545016 <lb/>607 # 368449 # 223648543 <lb/>608 # 369664 # 224755712 <lb/>609 # 370881 # 225866529 <lb/>610 # 372100 # 226981000 <lb/>611 # 373321 # 228099131 <lb/>612 # 374544 # 229220928 <lb/>613 # 375769 # 230346397 <lb/>614 # 376996 # 231475544 <lb/>615 # 378225 # 232608375 <lb/>616 # 379456 # 233744896 <lb/>617 # 380689 # 234885113 <lb/>618 # 381924 # 236029032 <lb/>619 # 383161 # 237176659 <lb/>620 # 384400 # 238328000 <lb/>621 # 385641 # 239483061 <lb/>622 # 386884 # 240641848 <lb/>623 # 388129 # 241804367 <lb/>624 # 389376 # 242970624 <lb/>625 # 390625 # 244140625 <lb/>626 # 391876 # 245314376 <lb/>627 # 393129 # 246491883 <lb/>628 # 394384 # 247673152 <lb/>629 # 395641 # 248858189 <lb/>630 # 396900 # 250047000 <lb/>631 # 398161 # 251239591 <lb/>632 # 399424 # 252435968 <lb/>633 # 400689 # 253636137 <lb/>634 # 401956 # 254840104 <lb/>635 # 403225 # 256047875 <lb/>636 # 404496 # 257259456 <lb/>637 # 405769 # 258474853 <lb/>638 # 407044 # 259694072 <lb/>639 # 408321 # 260917119 <lb/>640 # 409600 # 262144000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>641 # 410881 # 263374721 <lb/>642 # 412164 # 264609288 <lb/>643 # 413449 # 265847707 <lb/>644 # 414736 # 267089984 <lb/>645 # 416025 # 268336125 <lb/>646 # 417316 # 269586136 <lb/>647 # 418609 # 270840023 <lb/>648 # 419904 # 272097792 <lb/>649 # 421201 # 273359449 <lb/>650 # 422500 # 274625000 <lb/>651 # 423801 # 275894451 <lb/>652 # 425104 # 277167808 <lb/>653 # 426409 # 278445077 <lb/>654 # 427716 # 279726264 <lb/>655 # 429025 # 281011375 <lb/>656 # 430336 # 282300416 <lb/>657 # 431649 # 283593393 <lb/>658 # 432964 # 284890312 <lb/>659 # 434281 # 286191179 <lb/>660 # 435600 # 287496000 <lb/>661 # 436921 # 288804781 <lb/>662 # 438244 # 290117528 <lb/>663 # 439569 # 291434247 <lb/>664 # 440896 # 292754944 <lb/>665 # 442225 # 294079625 <lb/>666 # 443556 # 295408296 <lb/>667 # 444889 # 296740963 <lb/>668 # 446224 # 298077632 <lb/>669 # 447561 # 299418309 <lb/>670 # 448900 # 300763000 <lb/>671 # 450241 # 302111711 <lb/>672 # 451584 # 303464448 <lb/>673 # 452929 # 304821217 <lb/>674 # 454276 # 306182024 <lb/>675 # 455625 # 307546875 <lb/>676 # 456976 # 308915776 <lb/>677 # 458329 # 310288733 <lb/>678 # 459684 # 311665752 <lb/>679 # 461041 # 313046839 <lb/>680 # 462400 # 314432000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>681 # 463761 # 315821241 <lb/>682 # 465124 # 317214568 <lb/>683 # 466489 # 318611987 <lb/>684 # 467856 # 320013504 <lb/>685 # 469225 # 321419125 <lb/>686 # 470596 # 322828856 <lb/>687 # 471969 # 324242703 <lb/>688 # 473344 # 325660672 <lb/>689 # 474721 # 327082769 <lb/>690 # 476100 # 328509000 <lb/>691 # 477481 # 329939371 <lb/>692 # 478864 # 331373888 <lb/>693 # 480249 # 332812557 <lb/>694 # 481636 # 334255384 <lb/>695 # 483025 # 335702375 <lb/>696 # 484416 # 337153536 <lb/>697 # 485809 # 338608873 <lb/>698 # 487204 # 340068392 <lb/>699 # 488601 # 341532099 <lb/>700 # 490000 # 343000000 <lb/>701 # 491401 # 344472101 <lb/>702 # 492804 # 345948408 <lb/>703 # 494209 # 347428927 <lb/>704 # 495616 # 348913664 <lb/>705 # 497025 # 350402625 <lb/>706 # 498436 # 351895816 <lb/>707 # 499849 # 353393243 <lb/>708 # 501264 # 354894912 <lb/>709 # 502681 # 356400829 <lb/>710 # 504100 # 357911000 <lb/>711 # 505521 # 359425431 <lb/>712 # 506944 # 360944128 <lb/>713 # 508369 # 362467097 <lb/>714 # 508796 # 363994344 <lb/>715 # 511225 # 365525875 <lb/>716 # 512656 # 367061696 <lb/>717 # 514089 # 368601813 <lb/>718 # 515524 # 370146232 <lb/>719 # 516961 # 371694959 <lb/>720 # 518400 # 373248000 <lb/></note> <pb o="384" file="412" n="412" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1127" type="section" level="1" n="413"> <head xml:id="echoid-head440" xml:space="preserve">Tabula Quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>721 # 519841 # 374805361 <lb/>722 # 521284 # 376367048 <lb/>723 # 522729 # 377933067 <lb/>724 # 524176 # 379503424 <lb/>725 # 525625 # 381078125 <lb/>726 # 527076 # 382657176 <lb/>727 # 528529 # 384240583 <lb/>728 # 529984 # 385828352 <lb/>729 # 531441 # 387420489 <lb/>730 # 532900 # 389017000 <lb/>731 # 534361 # 390617891 <lb/>732 # 535824 # 392223168 <lb/>733 # 537289 # 393832837 <lb/>734 # 538756 # 395446904 <lb/>735 # 940225 # 397065375 <lb/>736 # 541696 # 398688256 <lb/>737 # 543169 # 400315553 <lb/>738 # 544644 # 401947272 <lb/>739 # 546121 # 403583419 <lb/>740 # 547600 # 405224000 <lb/>741 # 549081 # 406869021 <lb/>742 # 550564 # 408518488 <lb/>743 # 552049 # 410172407 <lb/>744 # 553536 # 411830784 <lb/>745 # 555025 # 413493625 <lb/>746 # 556516 # 415160936 <lb/>747 # 558009 # 416832723 <lb/>748 # 559504 # 418508992 <lb/>749 # 561001 # 420189749 <lb/>750 # 562500 # 421875000 <lb/>751 # 564001 # 423564751 <lb/>752 # 565504 # 425259008 <lb/>753 # 567009 # 426957777 <lb/>754 # 568516 # 428661064 <lb/>755 # 570025 # 430368875 <lb/>756 # 571536 # 432081216 <lb/>757 # 573049 # 433798093 <lb/>758 # 574564 # 435519512 <lb/>759 # 576081 # 437245479 <lb/>760 # 577600 # 438976000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>761 # 579121 # 440711081 <lb/>762 # 580644 # 442450728 <lb/>763 # 582169 # 444194947 <lb/>764 # 583696 # 445943744 <lb/>765 # 585225 # 447697125 <lb/>766 # 586756 # 449455096 <lb/>767 # 588289 # 451217663 <lb/>768 # 589824 # 452984832 <lb/>769 # 591361 # 454756609 <lb/>770 # 592900 # 456533000 <lb/>771 # 594441 # 458314011 <lb/>772 # 595984 # 460099648 <lb/>773 # 597529 # 461889917 <lb/>774 # 599076 # 463684824 <lb/>775 # 600625 # 465484375 <lb/>776 # 602176 # 467288576 <lb/>777 # 603729 # 469097433 <lb/>778 # 605284 # 470910952 <lb/>779 # 606841 # 472729139 <lb/>780 # 608400 # 474552000 <lb/>781 # 609961 # 476379541 <lb/>782 # 611524 # 478211768 <lb/>783 # 613089 # 480048687 <lb/>784 # 614656 # 481860304 <lb/>785 # 616225 # 483736625 <lb/>786 # 617796 # 485587656 <lb/>787 # 619369 # 487443403 <lb/>788 # 620944 # 489303872 <lb/>789 # 622521 # 491169069 <lb/>790 # 624100 # 493039000 <lb/>791 # 625681 # 494913671 <lb/>792 # 627264 # 496793088 <lb/>793 # 628849 # 498677257 <lb/>794 # 630436 # 500566184 <lb/>795 # 632025 # 502459875 <lb/>796 # 633616 # 504358336 <lb/>797 # 635209 # 506261573 <lb/>798 # 636804 # 508169592 <lb/>799 # 638401 # 510082399 <lb/>800 # 640000 # 512000000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>801 # 641601 # 513922401 <lb/>802 # 643204 # 515849608 <lb/>803 # 644809 # 517781627 <lb/>804 # 646416 # 519718464 <lb/>805 # 648025 # 521660125 <lb/>806 # 949636 # 523606616 <lb/>807 # 651249 # 525557943 <lb/>808 # 652864 # 527514112 <lb/>809 # 654481 # 529475129 <lb/>810 # 656100 # 531441000 <lb/>811 # 657721 # 533411731 <lb/>812 # 659344 # 535387328 <lb/>813 # 660969 # 537367797 <lb/>814 # 662596 # 539353144 <lb/>815 # 664225 # 541343375 <lb/>816 # 665856 # 543338496 <lb/>817 # 667489 # 545338513 <lb/>818 # 669124 # 547343432 <lb/>819 # 670761 # 549353259 <lb/>820 # 672400 # 551368000 <lb/>821 # 674041 # 553387661 <lb/>822 # 675684 # 555412248 <lb/>823 # 677329 # 557441767 <lb/>824 # 678976 # 559476224 <lb/>825 # 680625 # 561515625 <lb/>826 # 682276 # 563559976 <lb/>827 # 683929 # 565609283 <lb/>828 # 685584 # 567663552 <lb/>829 # 687241 # 569722789 <lb/>830 # 688900 # 571787000 <lb/>831 # 690561 # 573856191 <lb/>832 # 692224 # 575930368 <lb/>833 # 693889 # 578009537 <lb/>834 # 695556 # 580093704 <lb/>835 # 697225 # 582182875 <lb/>836 # 698896 # 584277056 <lb/>837 # 700569 # 586376253 <lb/>838 # 70224@ # 588480472 <lb/>839 # 703921 # 590589719 <lb/>840 # 705600 # 592704000 <lb/></note> <pb o="385" file="413" n="413" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1128" type="section" level="1" n="414"> <head xml:id="echoid-head441" xml:space="preserve">Tabula Quadratorum, & cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>841 # 707281 # 594823321 <lb/>842 # 708964 # 596947688 <lb/>843 # 710649 # 599077107 <lb/>844 # 712336 # 601211584 <lb/>845 # 714025 # 603351125 <lb/>846 # 715716 # 605495736 <lb/>847 # 717409 # 607645423 <lb/>848 # 719104 # 609800192 <lb/>849 # 720801 # 611960049 <lb/>850 # 722500 # 614125000 <lb/>851 # 724201 # 616295051 <lb/>852 # 725904 # 618470208 <lb/>853 # 727609 # 620650477 <lb/>854 # 729316 # 622835864 <lb/>855 # 731025 # 625026375 <lb/>856 # 732736 # 627222016 <lb/>857 # 734449 # 629422793 <lb/>858 # 736164 # 631628712 <lb/>859 # 737881 # 633839779 <lb/>860 # 739600 # 636056000 <lb/>861 # 741321 # 638277381 <lb/>862 # 743044 # 640503928 <lb/>863 # 744769 # 642735647 <lb/>864 # 746496 # 644972544 <lb/>865 # 748225 # 647214625 <lb/>866 # 749956 # 649461896 <lb/>867 # 751689 # 651714363 <lb/>868 # 753424 # 653972032 <lb/>869 # 755161 # 656234909 <lb/>870 # 756900 # 658503000 <lb/>871 # 758641 # 660776311 <lb/>872 # 760384 # 663054848 <lb/>873 # 762129 # 665338617 <lb/>874 # 763876 # 667627624 <lb/>875 # 765625 # 669921875 <lb/>876 # 767376 # 672221376 <lb/>877 # 769129 # 674526133 <lb/>878 # 770884 # 676836152 <lb/>879 # 772641 # 679151439 <lb/>880 # 774400 # 681472000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>881 # 776161 # 683797841 <lb/>882 # 777924 # 686128968 <lb/>883 # 779689 # 688465387 <lb/>884 # 781456 # 690807504 <lb/>885 # 783225 # 693154125 <lb/>886 # 784996 # 695506456 <lb/>887 # 786769 # 697864103 <lb/>888 # 788544 # 700227072 <lb/>889 # 790321 # 702595369 <lb/>890 # 792100 # 704969000 <lb/>891 # 793881 # 707347971 <lb/>892 # 795664 # 709732288 <lb/>893 # 797449 # 712121957 <lb/>894 # 799236 # 714516984 <lb/>895 # 801025 # 716917375 <lb/>896 # 802816 # 719323136 <lb/>897 # 804609 # 721734273 <lb/>898 # 806404 # 724150792 <lb/>899 # 808201 # 726572699 <lb/>900 # 810000 # 729000000 <lb/>901 # 811801 # 731432701 <lb/>902 # 813604 # 733870808 <lb/>903 # 815409 # 736314327 <lb/>904 # 817216 # 738763264 <lb/>905 # 819025 # 741217625 <lb/>906 # 820836 # 743677416 <lb/>907 # 822649 # 746142643 <lb/>908 # 824464 # 748613312 <lb/>909 # 826281 # 751085429 <lb/>910 # 828100 # 753571000 <lb/>911 # 829921 # 756058031 <lb/>912 # 831744 # 758550528 <lb/>913 # 833569 # 761048497 <lb/>914 # 835396 # 763551944 <lb/>915 # 837225 # 766060875 <lb/>916 # 839056 # 768575296 <lb/>917 # 840889 # 771095213 <lb/>918 # 842724 # 773620632 <lb/>919 # 844561 # 776151559 <lb/>920 # 846400 # 778688000 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>921 # 848241 # 781229961 <lb/>922 # 850084 # 783777448 <lb/>923 # 851929 # 786330467 <lb/>924 # 853776 # 788889024 <lb/>925 # 855625 # 791453125 <lb/>926 # 857476 # 794022776 <lb/>927 # 859329 # 796597983 <lb/>928 # 861184 # 799178752 <lb/>929 # 863041 # 801767089 <lb/>930 # 864900 # 804357000 <lb/>931 # 866761 # 806954491 <lb/>932 # 868624 # 809557568 <lb/>933 # 870489 # 812166237 <lb/>934 # 872356 # 814780504 <lb/>935 # 874225 # 817400375 <lb/>936 # 876096 # 820025856 <lb/>937 # 877969 # 822656953 <lb/>938 # 879844 # 825293672 <lb/>939 # 881721 # 827936019 <lb/>940 # 883600 # 830584000 <lb/>941 # 885481 # 833237621 <lb/>942 # 887364 # 835896888 <lb/>943 # 889249 # 838561807 <lb/>944 # 891136 # 841232384 <lb/>945 # 893025 # 843908625 <lb/>946 # 794916 # 846590536 <lb/>947 # 896809 # 849278123 <lb/>948 # 898704 # 851971392 <lb/>949 # 900601 # 854670349 <lb/>950 # 902500 # 857375000 <lb/>951 # 904401 # 860085351 <lb/>952 # 906304 # 862801408 <lb/>953 # 908209 # 865523177 <lb/>954 # 910116 # 868250664 <lb/>955 # 912025 # 870983875 <lb/>956 # 913936 # 873722816 <lb/>957 # 915849 # 876467493 <lb/>958 # 917764 # 879217912 <lb/>959 # 919681 # 881974079 <lb/>960 # 921600 # 884736000 <lb/></note> <pb o="386" file="414" n="414" rhead="GEOMETR. PRACT."/> </div> <div xml:id="echoid-div1129" type="section" level="1" n="415"> <head xml:id="echoid-head442" xml:space="preserve">Tabula Quadratorum, & Cuborum.</head> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>961 # 923521 # 887503681 <lb/>962 # 925444 # 890277128 <lb/>963 # 927369 # 893056347 <lb/>964 # 929296 # 895841344 <lb/>965 # 931225 # 898632125 <lb/>966 # 933156 # 901428696 <lb/>967 # 935089 # 904231063 <lb/>968 # 937024 # 907039232 <lb/>969 # 938961 # 909853209 <lb/>970 # 940900 # 912673000 <lb/>971 # 942841 # 915498611 <lb/>972 # 944784 # 918330048 <lb/>973 # 946729 # 921167317 <lb/>974 # 948676 # 924010424 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>975 # 950625 # 926859375 <lb/>976 # 952576 # 929714176 <lb/>977 # 954529 # 932574833 <lb/>978 # 956484 # 935441352 <lb/>979 # 958441 # 938313739 <lb/>980 # 960400 # 941192000 <lb/>981 # 962361 # 944076141 <lb/>982 # 964324 # 946966168 <lb/>983 # 966289 # 949862087 <lb/>984 # 968256 # 952763904 <lb/>985 # 970225 # 955671625 <lb/>986 # 972196 # 958585256 <lb/>987 # 974169 # 961504803 <lb/>988 # 976144 # 964430272 <lb/></note> <note position="right" xml:space="preserve"> <lb/>Radices # Quadra \\ ti: # Cubi <lb/>989 # 978121 # 967361669 <lb/>990 # 980100 # 970299000 <lb/>991 # 982081 # 973242271 <lb/>992 # 984064 # 976191488 <lb/>993 # 986049 # 979146657 <lb/>994 # 988036 # 982107784 <lb/>995 # 990025 # 985074875 <lb/>996 # 992016 # 988047936 <lb/>997 # 994009 # 991026973 <lb/>998 # 996004 # 994011992 <lb/>999 # 998001 # 997002999 <lb/>1000 # 1000000 # 1000000000 <lb/></note> <pb o="387" file="415" n="415" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1130" type="section" level="1" n="416"> <head xml:id="echoid-head443" xml:space="preserve">DE DIFFERENTHS QVADRATO-<lb/>rum & cuborum, & de continuationeta-<lb/>bulæ eorundem.</head> <p> <s xml:id="echoid-s17817" xml:space="preserve"><emph style="sc">Qvoniam</emph> quadrati numeri creantur per continuam additionem nume-<lb/> <anchor type="note" xlink:label="note-415-01a" xlink:href="note-415-01"/> rorum imparium, vt Arithmetici demonſtrant: </s> <s xml:id="echoid-s17818" xml:space="preserve">fit vt differentia inter quemlibet <lb/>quadratum, & </s> <s xml:id="echoid-s17819" xml:space="preserve">proximè inſequen-<lb/> <anchor type="note" xlink:label="note-415-02a" xlink:href="note-415-02"/> tem ſit duplaradicis minoris, addi-<lb/>ta inſuper vnitate. </s> <s xml:id="echoid-s17820" xml:space="preserve">Itaque duobus <lb/>modis tabula quadratorum com-<lb/>poni poteſt, & </s> <s xml:id="echoid-s17821" xml:space="preserve">continuari. </s> <s xml:id="echoid-s17822" xml:space="preserve">Vno <lb/> <anchor type="note" xlink:label="note-415-03a" xlink:href="note-415-03"/> modo, ſi omnes numeri impares <lb/>ordine ponantur, initio ſumpto ab <lb/>1. </s> <s xml:id="echoid-s17823" xml:space="preserve">Nam 1. </s> <s xml:id="echoid-s17824" xml:space="preserve">dat primum quadratum <lb/>1. </s> <s xml:id="echoid-s17825" xml:space="preserve">Et ex 1. </s> <s xml:id="echoid-s17826" xml:space="preserve">& </s> <s xml:id="echoid-s17827" xml:space="preserve">3. </s> <s xml:id="echoid-s17828" xml:space="preserve">fit ſecundus 4. </s> <s xml:id="echoid-s17829" xml:space="preserve">cui ſi <lb/>addatur ſequens impar 5. </s> <s xml:id="echoid-s17830" xml:space="preserve">fit tertius <lb/>9. </s> <s xml:id="echoid-s17831" xml:space="preserve">& </s> <s xml:id="echoid-s17832" xml:space="preserve">ſi addatur impar ſequens 7. </s> <s xml:id="echoid-s17833" xml:space="preserve">fit <lb/>quartus quadratus 16. </s> <s xml:id="echoid-s17834" xml:space="preserve">at que ita de-<lb/>inceps. </s> <s xml:id="echoid-s17835" xml:space="preserve">Habet autem quilibet qua-<lb/>dratusra dicem tot vnitatum, quot <lb/>numeri impares ipſum conficiunt. <lb/></s> <s xml:id="echoid-s17836" xml:space="preserve">Vt quia ſolus impar 1. </s> <s xml:id="echoid-s17837" xml:space="preserve">dat primum <lb/>quadratum 1. </s> <s xml:id="echoid-s17838" xml:space="preserve">propterea eius radix <lb/>eſt 1. </s> <s xml:id="echoid-s17839" xml:space="preserve">Deinde quia duo impares 1. </s> <s xml:id="echoid-s17840" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s17841" xml:space="preserve">3. </s> <s xml:id="echoid-s17842" xml:space="preserve">conficiunt ſecundum quadra-<lb/>tum 4. </s> <s xml:id="echoid-s17843" xml:space="preserve">erit eius radix 2. </s> <s xml:id="echoid-s17844" xml:space="preserve">Sic quia <lb/>duo decim numeri impares 1. </s> <s xml:id="echoid-s17845" xml:space="preserve">3. </s> <s xml:id="echoid-s17846" xml:space="preserve">5. </s> <s xml:id="echoid-s17847" xml:space="preserve">7. </s> <s xml:id="echoid-s17848" xml:space="preserve"><lb/>9. </s> <s xml:id="echoid-s17849" xml:space="preserve">11. </s> <s xml:id="echoid-s17850" xml:space="preserve">13. </s> <s xml:id="echoid-s17851" xml:space="preserve">15. </s> <s xml:id="echoid-s17852" xml:space="preserve">17. </s> <s xml:id="echoid-s17853" xml:space="preserve">19. </s> <s xml:id="echoid-s17854" xml:space="preserve">21. </s> <s xml:id="echoid-s17855" xml:space="preserve">23. </s> <s xml:id="echoid-s17856" xml:space="preserve">compo-<lb/>nunt quadratum 144. </s> <s xml:id="echoid-s17857" xml:space="preserve">erit eius ra-<lb/>dix 12. </s> <s xml:id="echoid-s17858" xml:space="preserve">& </s> <s xml:id="echoid-s17859" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s17860" xml:space="preserve">Atq; </s> <s xml:id="echoid-s17861" xml:space="preserve">in hũc <lb/>modum ſi ſemper ſequens nume-<lb/>us impar adiiciatur ad quadratum præcedentem, conflatur ſequens numerus <lb/>quadratus, continuabiturque tabula in infinitum: </s> <s xml:id="echoid-s17862" xml:space="preserve">ſitamen prius ſeries nume-<lb/>rorum imparium continuetur. </s> <s xml:id="echoid-s17863" xml:space="preserve">Radices ſerie numerorum naturali progre-<lb/>diuntur.</s> <s xml:id="echoid-s17864" xml:space="preserve"/> </p> <div xml:id="echoid-div1130" type="float" level="2" n="1"> <note position="right" xlink:label="note-415-01" xlink:href="note-415-01a" xml:space="preserve">Differentiæ <lb/>quadratorum</note> <note position="right" xlink:label="note-415-02" xlink:href="note-415-02a" xml:space="preserve"> <lb/>Numeri \\ impares. # Quadra- \\ ti. # Radices. <lb/>1 # 1 # 1 <lb/>3 # 4 # 2 <lb/>5 # 9 # 3 <lb/>7 # 16 # 4 <lb/>9 # 25 # 5 <lb/>11 # 36 # 6 <lb/>13 # 49 # 7 <lb/>15 # 64 # 8 <lb/>17 # 81 # 9 <lb/>19 # 100 # 10 <lb/>21 # 121 # 11 <lb/>23 # 144 # 12 <lb/></note> <note position="right" xlink:label="note-415-03" xlink:href="note-415-03a" xml:space="preserve">Compoſitio ta<unsure/>-<lb/>bulæ quadra-<lb/>torum.</note> </div> <p> <s xml:id="echoid-s17865" xml:space="preserve"><emph style="sc">Alio</emph> modo condi poterit tabula quadratorum, & </s> <s xml:id="echoid-s17866" xml:space="preserve">in infinitum continua-<lb/>ri, ſine numerorum imparium ſerie, ſi omnes radices ponantur ordine, vt in tabu-<lb/>la vides. </s> <s xml:id="echoid-s17867" xml:space="preserve">Cum enim primus quadratus ſit 1. </s> <s xml:id="echoid-s17868" xml:space="preserve">cuius radix 1. </s> <s xml:id="echoid-s17869" xml:space="preserve">ſi hæc radix dupli-<lb/>cata, addita inſuper 1. </s> <s xml:id="echoid-s17870" xml:space="preserve">addatur primo quadrato 1. </s> <s xml:id="echoid-s17871" xml:space="preserve">fit ſecundus 4. </s> <s xml:id="echoid-s17872" xml:space="preserve">cuius ra-<lb/>dix 2. </s> <s xml:id="echoid-s17873" xml:space="preserve">Hæc duplicata, & </s> <s xml:id="echoid-s17874" xml:space="preserve">inſuper addita 1. </s> <s xml:id="echoid-s17875" xml:space="preserve">ſi adiiciatur ſecundo quadra-<lb/>to 4. </s> <s xml:id="echoid-s17876" xml:space="preserve">fit tertius 9. </s> <s xml:id="echoid-s17877" xml:space="preserve">cuius radix 3. </s> <s xml:id="echoid-s17878" xml:space="preserve">quæ duplicata, & </s> <s xml:id="echoid-s17879" xml:space="preserve">inſuper addita 1. </s> <s xml:id="echoid-s17880" xml:space="preserve">facit 7. <lb/></s> <s xml:id="echoid-s17881" xml:space="preserve">Si igitur addantur 7. </s> <s xml:id="echoid-s17882" xml:space="preserve">ad quadratum 9. </s> <s xml:id="echoid-s17883" xml:space="preserve">fit quartus quadratus 16. </s> <s xml:id="echoid-s17884" xml:space="preserve">& </s> <s xml:id="echoid-s17885" xml:space="preserve">ſic in infi-<lb/>nitum.</s> <s xml:id="echoid-s17886" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17887" xml:space="preserve"><emph style="sc">Nvmeri</emph> autem cubi gignuntur quoque ex additione numerorum impa-<lb/>rium, hoc modo. </s> <s xml:id="echoid-s17888" xml:space="preserve">Deſcripta ſerie imparium numerorum ab 1. </s> <s xml:id="echoid-s17889" xml:space="preserve">incipientium, pri- <pb o="388" file="416" n="416" rhead="GEOMETR. PRACT."/> mus 1. </s> <s xml:id="echoid-s17890" xml:space="preserve">dat primum cubum 1. </s> <s xml:id="echoid-s17891" xml:space="preserve">cuius radix 1. </s> <s xml:id="echoid-s17892" xml:space="preserve">Duo deinde ſequentes 3. </s> <s xml:id="echoid-s17893" xml:space="preserve">5. </s> <s xml:id="echoid-s17894" xml:space="preserve">coacer-<lb/> <anchor type="note" xlink:label="note-416-01a" xlink:href="note-416-01"/> uati præbent ſecundum cubum <lb/> <anchor type="note" xlink:label="note-416-02a" xlink:href="note-416-02"/> 8. </s> <s xml:id="echoid-s17895" xml:space="preserve">cuius radix 2. </s> <s xml:id="echoid-s17896" xml:space="preserve">Tres inſequentes <lb/>7. </s> <s xml:id="echoid-s17897" xml:space="preserve">9. </s> <s xml:id="echoid-s17898" xml:space="preserve">11. </s> <s xml:id="echoid-s17899" xml:space="preserve">exhibent tertium cubum <lb/>27. </s> <s xml:id="echoid-s17900" xml:space="preserve">cuius radix 3. </s> <s xml:id="echoid-s17901" xml:space="preserve">Atque eundem <lb/>in modũ ſequentes quatuor im-<lb/>pares conficiẽt quartum cubum, <lb/>& </s> <s xml:id="echoid-s17902" xml:space="preserve">inſequentes quinq; </s> <s xml:id="echoid-s17903" xml:space="preserve">quintum, <lb/>& </s> <s xml:id="echoid-s17904" xml:space="preserve">ſic deinceps in infinitum. </s> <s xml:id="echoid-s17905" xml:space="preserve">Qui-<lb/>libet autem cubus radicem habet <lb/>tot vnitatum quot impares nu-<lb/>meri coaceruati ipſum compo-<lb/>nunt.</s> <s xml:id="echoid-s17906" xml:space="preserve"/> </p> <div xml:id="echoid-div1131" type="float" level="2" n="2"> <note position="left" xlink:label="note-416-01" xlink:href="note-416-01a" xml:space="preserve">Generatio cu-<lb/>borum.</note> <note position="right" xlink:label="note-416-02" xlink:href="note-416-02a" xml:space="preserve"> <lb/>Numeri \\ impares. # Cubi # Radices. <lb/>1 # 1 # 1 <lb/>3 <lb/>5 # 8 # 2 <lb/>7 <lb/>9 <lb/>11 # 27 # 3 <lb/>13 <lb/>15 <lb/>17 <lb/>19 # 64 # 4 <lb/>21 <lb/>23 <lb/>25 <lb/>27 <lb/>29 # 125 # 5 <lb/></note> </div> <p> <s xml:id="echoid-s17907" xml:space="preserve"><emph style="sc">Prodvcitvr</emph> quoq; </s> <s xml:id="echoid-s17908" xml:space="preserve">cu-<lb/>bus cuiuſcunque radicis, ſi ea ra-<lb/>dix in ſuum quadratum ducatur. <lb/></s> <s xml:id="echoid-s17909" xml:space="preserve">Vt cubus radicis 16. </s> <s xml:id="echoid-s17910" xml:space="preserve">eſt 4096. </s> <s xml:id="echoid-s17911" xml:space="preserve">ge-<lb/>nitus ex radice 16. </s> <s xml:id="echoid-s17912" xml:space="preserve">multiplicata in <lb/>256. </s> <s xml:id="echoid-s17913" xml:space="preserve">quadratum eiuſdem radicis.</s> <s xml:id="echoid-s17914" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17915" xml:space="preserve"><emph style="sc">Vervm</emph> quia permoleſtum <lb/>eſt, tam ſeriem numerorum impa-<lb/>rium in tot terminis continuare, <lb/>vt ex eorum additione omnes <lb/>cubi generentur, vt dictum eſt, <lb/>quã radices omnes per ſuos qua-<lb/>dratos multiplicare, obſeruauit <lb/>Ioan. </s> <s xml:id="echoid-s17916" xml:space="preserve">Baptiſta Villalpandus no-<lb/>ſtræ Societatis ſacerdos Theolo-<lb/>gus, ac Mathematicus egregius, <lb/>cuius eru ditionis in rebus Mathe-<lb/>maticis ſpecimẽ præclarum extat <lb/>in apparatu Vrbis, ac Templi Hiero ſolymitani, præſertim verò in multiplici <lb/> <anchor type="note" xlink:label="note-416-03a" xlink:href="note-416-03"/> duarum mediarum proportionalium inter duas rectas inuentione, obſeruauit <lb/>inquam pulcherrimam proprietatem cuborum numer orum, per quam differen-<lb/>tiæ cub orum ordine ſine magna difficultate reperiuntur, quæ poſtea ordine ad <lb/>cubos præcedentes additæ conficiunt cubos inſequentes. </s> <s xml:id="echoid-s17917" xml:space="preserve">Res autem ſic ſe ha-<lb/> <anchor type="note" xlink:label="note-416-04a" xlink:href="note-416-04"/> bet. </s> <s xml:id="echoid-s17918" xml:space="preserve">In columna ſiniſtra ſcribatur progreſsio Arithmetica, quæ à 6. </s> <s xml:id="echoid-s17919" xml:space="preserve">incipiat, & </s> <s xml:id="echoid-s17920" xml:space="preserve"><lb/>per 6. </s> <s xml:id="echoid-s17921" xml:space="preserve">progrediatur. </s> <s xml:id="echoid-s17922" xml:space="preserve">In ſecunda columna reponantur numeri, qui ex nume-<lb/>ris primæ columnæ componuntur hac arte. </s> <s xml:id="echoid-s17923" xml:space="preserve">Iuxta 6. </s> <s xml:id="echoid-s17924" xml:space="preserve">ponatur 1. </s> <s xml:id="echoid-s17925" xml:space="preserve">Deinde 7. </s> <s xml:id="echoid-s17926" xml:space="preserve">qui <lb/>numerus ex 1. </s> <s xml:id="echoid-s17927" xml:space="preserve">& </s> <s xml:id="echoid-s17928" xml:space="preserve">6. </s> <s xml:id="echoid-s17929" xml:space="preserve">conflatur. </s> <s xml:id="echoid-s17930" xml:space="preserve">Poſt hæc numerus 19. </s> <s xml:id="echoid-s17931" xml:space="preserve">ex 7. </s> <s xml:id="echoid-s17932" xml:space="preserve">& </s> <s xml:id="echoid-s17933" xml:space="preserve">12. </s> <s xml:id="echoid-s17934" xml:space="preserve">coaceruatus. <lb/></s> <s xml:id="echoid-s17935" xml:space="preserve">Atque ita deinceps quilibet bini numeri primæ, ac ſecundæ columnæ ſimul ad-<lb/>diti conficient numerum inferiorem in ſecunda columna. </s> <s xml:id="echoid-s17936" xml:space="preserve">In tertia deinde co-<lb/>lumna collo centur omnes cubi, qui per continuam additionem numerorum ſe-<lb/>cundæ columnæ, quæ quidem differentias cuborum continet, colliguntur hac <lb/>ratione. </s> <s xml:id="echoid-s17937" xml:space="preserve">Primus cubus eſt @ cui ſi addas ſequentem differentiam 7. </s> <s xml:id="echoid-s17938" xml:space="preserve">facies ſecun-<lb/>dum cubum 8. </s> <s xml:id="echoid-s17939" xml:space="preserve">& </s> <s xml:id="echoid-s17940" xml:space="preserve">addendo differentiam ſequentem 19. </s> <s xml:id="echoid-s17941" xml:space="preserve">conflabis tertium cu-<lb/>bum 27. </s> <s xml:id="echoid-s17942" xml:space="preserve">& </s> <s xml:id="echoid-s17943" xml:space="preserve">ita deinceps. </s> <s xml:id="echoid-s17944" xml:space="preserve">Semper enim in tabella appoſita iuxta quemlibet cu- <pb o="389" file="417" n="417" rhead="LIBER OCTAVVS."/> bum ponitur differentia, qua à præcedenti cubo differt. </s> <s xml:id="echoid-s17945" xml:space="preserve">In quarta denique co-<lb/>lumna ſcribantur ordine cuborum radices.</s> <s xml:id="echoid-s17946" xml:space="preserve"/> </p> <div xml:id="echoid-div1132" type="float" level="2" n="3"> <note position="left" xlink:label="note-416-03" xlink:href="note-416-03a" xml:space="preserve">Differentiæ <lb/>cuborum quo <lb/>modo repe-<lb/>riantur.</note> <note position="left" xlink:label="note-416-04" xlink:href="note-416-04a" xml:space="preserve">Conſtructio <lb/>tabulæ cubo-<lb/>rum.</note> </div> <note position="right" xml:space="preserve"> <lb/>Progreſſio \\ ſenarij. # Differẽtiæ \\ cuborum. # Cubi. # Radices. <lb/>6 # 1 # 1 # 1 <lb/>12 # 7 # 8 # 2 <lb/>18 # 19 # 27 # 3 <lb/>24 # 37 # 64 # 4 <lb/>30 # 61 # 125 # 5 <lb/>36 # 91 # 216 # 6 <lb/>42 # 127 # 343 # 7 <lb/>48 # 169 # 512 # 8 <lb/>54 # 217 # 729 # 9 <lb/>60 # 271 # 1000 # 10 <lb/>66 # 331 # 1331 # 11 <lb/>72 # 397 # 1728 # 12 <lb/>78 # 469 # 2197 # 13 <lb/>84 # 547 # 2744 # 14 <lb/>90 # 631 # 3375 # 15 <lb/>96 # 721 # 4096 # 16 <lb/></note> <p> <s xml:id="echoid-s17947" xml:space="preserve"><emph style="sc">Eædem</emph> differentiæ cuborum in ſecunda columna deſcriptæ inuenientur <lb/>quo que hoc modo. </s> <s xml:id="echoid-s17948" xml:space="preserve">Radicis propoſitæ quadratum triplicetur, addaturq; </s> <s xml:id="echoid-s17949" xml:space="preserve">radix <lb/>triplicata, atque inſuper 1. </s> <s xml:id="echoid-s17950" xml:space="preserve">Conflatus enim numerus erit differentia, qua cubus <lb/>propoſitæ radicis ab inſequenti cubo differt. </s> <s xml:id="echoid-s17951" xml:space="preserve">Vt ſi deſideretur differentia inter <lb/>cubum 216. </s> <s xml:id="echoid-s17952" xml:space="preserve">radicis 6. </s> <s xml:id="echoid-s17953" xml:space="preserve">& </s> <s xml:id="echoid-s17954" xml:space="preserve">cubum proximè maiorem. </s> <s xml:id="echoid-s17955" xml:space="preserve">Quadratum radicis 6. </s> <s xml:id="echoid-s17956" xml:space="preserve">eſt 36. <lb/></s> <s xml:id="echoid-s17957" xml:space="preserve">triplum eius eſt 108. </s> <s xml:id="echoid-s17958" xml:space="preserve">cui ſi addatur triplum radicis, videlicet 18. </s> <s xml:id="echoid-s17959" xml:space="preserve">& </s> <s xml:id="echoid-s17960" xml:space="preserve">inſuper vnitas, <lb/>conflabitur differentia 127. </s> <s xml:id="echoid-s17961" xml:space="preserve">quæſita. </s> <s xml:id="echoid-s17962" xml:space="preserve">Atque hoc modo, ſi continuentur differen-<lb/>tiæ cuborum ope progreſsionis ſenarij, extendetur tabula cuborum, quantum <lb/>libuerit. </s> <s xml:id="echoid-s17963" xml:space="preserve">Suntautem, vt vides, numeri progreſsionis ſenarij ſextupli radicum cu-<lb/>borum, ſinguli ſingularum. </s> <s xml:id="echoid-s17964" xml:space="preserve">Vt 60. </s> <s xml:id="echoid-s17965" xml:space="preserve">ſextuplus eſt radicis 10.</s> <s xml:id="echoid-s17966" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s17967" xml:space="preserve"><emph style="sc">Qvia</emph> verò diximus, cubos gigni ex ad ditione numerorum imparium, nimi-<lb/>rum primum eſſe 1. </s> <s xml:id="echoid-s17968" xml:space="preserve">ſecundum ex duobus ſequentibus 3. </s> <s xml:id="echoid-s17969" xml:space="preserve">5. </s> <s xml:id="echoid-s17970" xml:space="preserve">confici, & </s> <s xml:id="echoid-s17971" xml:space="preserve">tertium ex <lb/>tribus ſequentibus 7. </s> <s xml:id="echoid-s17972" xml:space="preserve">9. </s> <s xml:id="echoid-s17973" xml:space="preserve">11. </s> <s xml:id="echoid-s17974" xml:space="preserve">&</s> <s xml:id="echoid-s17975" xml:space="preserve">c. </s> <s xml:id="echoid-s17976" xml:space="preserve">ita vt quilibet ex tot imparibus coaceruetur, <lb/>quotvnitates in eius radice continentur; </s> <s xml:id="echoid-s17977" xml:space="preserve">ſi curioſus quiſpiam noſſe deſideret, <pb o="390" file="418" n="418" rhead="GEOMETR. PRACT."/> (quod ſcire non iniucundum eſt) quotnam ſint illi numeri impares propoſi-<lb/> <anchor type="note" xlink:label="note-418-01a" xlink:href="note-418-01"/> tum cubum componentes, hoc eſt, à quonam impari illi impares incipiant, & </s> <s xml:id="echoid-s17978" xml:space="preserve">in <lb/>quo deſinant, conſequetur id hoc artificio. </s> <s xml:id="echoid-s17979" xml:space="preserve">Radix propoſiti cubi, ſi impar eſt, <lb/>multiplicetur per ſemiſſem numeri proximè minoris, & </s> <s xml:id="echoid-s17980" xml:space="preserve">duplo producti ad-<lb/>datur 1. </s> <s xml:id="echoid-s17981" xml:space="preserve">Numerus enim conflatus erit primus imparium quæſitorum. </s> <s xml:id="echoid-s17982" xml:space="preserve">Quod ſi <lb/>radix fuerit par, ducatur eius ſemiſsis in numerum proximè minorem radice, & </s> <s xml:id="echoid-s17983" xml:space="preserve"><lb/>duplo producti adiiciatur 1. </s> <s xml:id="echoid-s17984" xml:space="preserve">Rurſus enim primus imparium, qui quæruntur, <lb/>conficietur. </s> <s xml:id="echoid-s17985" xml:space="preserve">Exempli gratia. </s> <s xml:id="echoid-s17986" xml:space="preserve">Propoſitus ſit cubus 125. </s> <s xml:id="echoid-s17987" xml:space="preserve">cuius radix 5. </s> <s xml:id="echoid-s17988" xml:space="preserve">ac proin-<lb/>de ex quin que numeris imparibus coaceruatus. </s> <s xml:id="echoid-s17989" xml:space="preserve">Duco radicem 5. </s> <s xml:id="echoid-s17990" xml:space="preserve">quia eſt im-<lb/>par in 2. </s> <s xml:id="echoid-s17991" xml:space="preserve">ſemiſſem numeri 4. </s> <s xml:id="echoid-s17992" xml:space="preserve">proximè minoris, efficio que 10. </s> <s xml:id="echoid-s17993" xml:space="preserve">duplico. </s> <s xml:id="echoid-s17994" xml:space="preserve">fiunt 20. <lb/></s> <s xml:id="echoid-s17995" xml:space="preserve">& </s> <s xml:id="echoid-s17996" xml:space="preserve">addita 1. </s> <s xml:id="echoid-s17997" xml:space="preserve">fit primus imparium 21. </s> <s xml:id="echoid-s17998" xml:space="preserve">Ergo quinque impares quæſiti ſunt. </s> <s xml:id="echoid-s17999" xml:space="preserve">21. </s> <s xml:id="echoid-s18000" xml:space="preserve">23. </s> <s xml:id="echoid-s18001" xml:space="preserve"><lb/>25. </s> <s xml:id="echoid-s18002" xml:space="preserve">27. </s> <s xml:id="echoid-s18003" xml:space="preserve">29. </s> <s xml:id="echoid-s18004" xml:space="preserve">quiin vnam ſummam collecti conſtituunt cubum 125. </s> <s xml:id="echoid-s18005" xml:space="preserve">Rurſus datus <lb/>ſit cubus 1728. </s> <s xml:id="echoid-s18006" xml:space="preserve">cuiusradix 12. </s> <s xml:id="echoid-s18007" xml:space="preserve">ideo que ex 12. </s> <s xml:id="echoid-s18008" xml:space="preserve">imparibus numeris coagmenta-<lb/>tus, Semiſſem radicis, quia par eſt, nimirum 6. </s> <s xml:id="echoid-s18009" xml:space="preserve">duco in 11. </s> <s xml:id="echoid-s18010" xml:space="preserve">numerum proximè <lb/>minorem radice numerumq; </s> <s xml:id="echoid-s18011" xml:space="preserve">productum 66. </s> <s xml:id="echoid-s18012" xml:space="preserve">duplico, addoq; </s> <s xml:id="echoid-s18013" xml:space="preserve">1. </s> <s xml:id="echoid-s18014" xml:space="preserve">Numerus enim <lb/>compoſitus 133. </s> <s xml:id="echoid-s18015" xml:space="preserve">erit primus 12. </s> <s xml:id="echoid-s18016" xml:space="preserve">imparium, quos quærimus. </s> <s xml:id="echoid-s18017" xml:space="preserve">Ergo omnes 12. </s> <s xml:id="echoid-s18018" xml:space="preserve"><lb/>erunt hi. </s> <s xml:id="echoid-s18019" xml:space="preserve">133. </s> <s xml:id="echoid-s18020" xml:space="preserve">135. </s> <s xml:id="echoid-s18021" xml:space="preserve">137. </s> <s xml:id="echoid-s18022" xml:space="preserve">139. </s> <s xml:id="echoid-s18023" xml:space="preserve">141. </s> <s xml:id="echoid-s18024" xml:space="preserve">143. </s> <s xml:id="echoid-s18025" xml:space="preserve">145. </s> <s xml:id="echoid-s18026" xml:space="preserve">147. </s> <s xml:id="echoid-s18027" xml:space="preserve">149. </s> <s xml:id="echoid-s18028" xml:space="preserve">151. </s> <s xml:id="echoid-s18029" xml:space="preserve">153. </s> <s xml:id="echoid-s18030" xml:space="preserve">155. </s> <s xml:id="echoid-s18031" xml:space="preserve">conficientes <lb/>cubum 1728.</s> <s xml:id="echoid-s18032" xml:space="preserve"/> </p> <div xml:id="echoid-div1133" type="float" level="2" n="4"> <note position="left" xlink:label="note-418-01" xlink:href="note-418-01a" xml:space="preserve">Qui numeri <lb/>impar{es} da-<lb/>tum cubum <lb/>@omponant.</note> </div> <p> <s xml:id="echoid-s18033" xml:space="preserve"><emph style="sc">Regvla</emph> hæc ſic demonſtratur. </s> <s xml:id="echoid-s18034" xml:space="preserve">Quando radix impar per ſemiſſem numeri <lb/>proximè minoris multiplicatur, vel ſemiſsis radicis paris per numerum proximè <lb/>minorem radice, producitur ſumma terminorum ſeriei naturalis numerorum ab <lb/>1. </s> <s xml:id="echoid-s18035" xml:space="preserve">vſque ad numerum proximè minorem radice, vt in progreſsione Arithmetica <lb/>diximus, hoc eſt, numerus terminorum imparium, qui primum imparem quæ-<lb/>ſitum præcedunt in ſerie numerorum imparium; </s> <s xml:id="echoid-s18036" xml:space="preserve">cum primus ſit vnus, deinde <lb/>ſequantur duo, poſtea tres, &</s> <s xml:id="echoid-s18037" xml:space="preserve">c. </s> <s xml:id="echoid-s18038" xml:space="preserve">Igitur ea ſumma indicat, quotum locum oc-<lb/>cupet impar numerus imparem quæſitum antecedens: </s> <s xml:id="echoid-s18039" xml:space="preserve">quia verò is locus du-<lb/>plicatus, dempta 1. </s> <s xml:id="echoid-s18040" xml:space="preserve">ex duplo, exhibet vltimum illum imparem, ſi ad eundem du-<lb/>plicatum addatur 1. </s> <s xml:id="echoid-s18041" xml:space="preserve">effi cietur primus imparium quæſitorum. </s> <s xml:id="echoid-s18042" xml:space="preserve">Verbi gratia, ſi <lb/>quæratur primus ſeptem imparium, qui conficiunt cubum 343. </s> <s xml:id="echoid-s18043" xml:space="preserve">cuius radix 7. <lb/></s> <s xml:id="echoid-s18044" xml:space="preserve">quærimus prius ſummam 6. </s> <s xml:id="echoid-s18045" xml:space="preserve">terminorum ſeriei naturalis, quæ eſt 21. </s> <s xml:id="echoid-s18046" xml:space="preserve">quod fit, <lb/>ſi ad 6. </s> <s xml:id="echoid-s18047" xml:space="preserve">addo 1. </s> <s xml:id="echoid-s18048" xml:space="preserve">& </s> <s xml:id="echoid-s18049" xml:space="preserve">ſummam 7. </s> <s xml:id="echoid-s18050" xml:space="preserve">id eſt, radicem imparem, ducam in 3. </s> <s xml:id="echoid-s18051" xml:space="preserve">ſemiſſem 6. </s> <s xml:id="echoid-s18052" xml:space="preserve"><lb/>terminorum. </s> <s xml:id="echoid-s18053" xml:space="preserve">Igitur impar numerus præcedens primum quæſitum ponitur in 21. </s> <s xml:id="echoid-s18054" xml:space="preserve"><lb/>loco. </s> <s xml:id="echoid-s18055" xml:space="preserve">Duplico, addoq; </s> <s xml:id="echoid-s18056" xml:space="preserve">1. </s> <s xml:id="echoid-s18057" xml:space="preserve">fit primus impar quæſitus 43. </s> <s xml:id="echoid-s18058" xml:space="preserve">Sunt ergo ſeptem quæ-<lb/>ſiti 43. </s> <s xml:id="echoid-s18059" xml:space="preserve">45. </s> <s xml:id="echoid-s18060" xml:space="preserve">47. </s> <s xml:id="echoid-s18061" xml:space="preserve">49. </s> <s xml:id="echoid-s18062" xml:space="preserve">51. </s> <s xml:id="echoid-s18063" xml:space="preserve">53. </s> <s xml:id="echoid-s18064" xml:space="preserve">55. </s> <s xml:id="echoid-s18065" xml:space="preserve">qui conſtituunt cubum 343.</s> <s xml:id="echoid-s18066" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18067" xml:space="preserve"><emph style="sc">Sed</emph> vt certi ſimus, quando radix eſt magna, num primus imparium rectè ſit <lb/>inuentus, ne cogamur omnes impares cõtinuare, quod laborio ſum eſt, inueſti-<lb/>gabimus vltimum ex primo, deinde ſummam omnium, hoc modo. </s> <s xml:id="echoid-s18068" xml:space="preserve">Differen-<lb/>tiam progreſsionis numerorum imparium, nimirum 2. </s> <s xml:id="echoid-s18069" xml:space="preserve">ducemus in numerum ter-<lb/>minorum, minus vno, pro ductumq; </s> <s xml:id="echoid-s18070" xml:space="preserve">primo inuento addemus. </s> <s xml:id="echoid-s18071" xml:space="preserve">Conflatus enim <lb/>numerus erit poſtremus imparium quæſitorum. </s> <s xml:id="echoid-s18072" xml:space="preserve">Si igitur ad hunc apponamus <lb/>primum, & </s> <s xml:id="echoid-s18073" xml:space="preserve">ſummæ ſemiſſem in numerum terminorum, id eſt, in radicem cubi <lb/>propoſiti ducamus, producetur propoſitus cubus, ſi primus imparium rectè in-<lb/>uentuseſt. </s> <s xml:id="echoid-s18074" xml:space="preserve">Vt in proximo cubo 343. </s> <s xml:id="echoid-s18075" xml:space="preserve">cuius ra dix 7. </s> <s xml:id="echoid-s18076" xml:space="preserve">ſi ad 43. </s> <s xml:id="echoid-s18077" xml:space="preserve">primum imparem in-<lb/>uentum addamus 12. </s> <s xml:id="echoid-s18078" xml:space="preserve">numerum ſcilicet pro ductum ex 2. </s> <s xml:id="echoid-s18079" xml:space="preserve">in radicẽ 7. </s> <s xml:id="echoid-s18080" xml:space="preserve">minus vno, <lb/>fit ſeptimus numerus impar quæſitus 55. </s> <s xml:id="echoid-s18081" xml:space="preserve">ad quem ſi apponatur primus 43. </s> <s xml:id="echoid-s18082" xml:space="preserve">fiunt <lb/>98. </s> <s xml:id="echoid-s18083" xml:space="preserve">Et ſi huius numeri ſemiſsis 49. </s> <s xml:id="echoid-s18084" xml:space="preserve">ducatur in radicẽ 7. </s> <s xml:id="echoid-s18085" xml:space="preserve">producetur cub<emph style="sub">9</emph> 343. </s> <s xml:id="echoid-s18086" xml:space="preserve">&</s> <s xml:id="echoid-s18087" xml:space="preserve">c.</s> <s xml:id="echoid-s18088" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18089" xml:space="preserve"><emph style="sc">Alia</emph> etiamratione perfacili eoſdem numero simpares datum cubum com- <pb o="391" file="419" n="419" rhead="LIBER OCTAVVS."/> ponentes reperiemus hoc modo. </s> <s xml:id="echoid-s18090" xml:space="preserve">Sit datus cubus 117649. </s> <s xml:id="echoid-s18091" xml:space="preserve">cuius radix 49. </s> <s xml:id="echoid-s18092" xml:space="preserve">ac <lb/>proinde reperiendi 49. </s> <s xml:id="echoid-s18093" xml:space="preserve">numeri impares, quorum ſumma ſit 117649. </s> <s xml:id="echoid-s18094" xml:space="preserve">Ponamus <lb/>primum imparem eſſe 1. </s> <s xml:id="echoid-s18095" xml:space="preserve">℞. </s> <s xml:id="echoid-s18096" xml:space="preserve">id eſt, 1. </s> <s xml:id="echoid-s18097" xml:space="preserve">Radicem (vtin Algebra fieri ſolet) Et quia <lb/>differentia numerorum imparium eſt 2. </s> <s xml:id="echoid-s18098" xml:space="preserve">erit ſecundus impar 1. </s> <s xml:id="echoid-s18099" xml:space="preserve">℞. </s> <s xml:id="echoid-s18100" xml:space="preserve">† 2. </s> <s xml:id="echoid-s18101" xml:space="preserve">hoc eſt, 1. <lb/></s> <s xml:id="echoid-s18102" xml:space="preserve">℞. </s> <s xml:id="echoid-s18103" xml:space="preserve">plus 2. </s> <s xml:id="echoid-s18104" xml:space="preserve">tertius 1. </s> <s xml:id="echoid-s18105" xml:space="preserve">℞. </s> <s xml:id="echoid-s18106" xml:space="preserve">† 4. </s> <s xml:id="echoid-s18107" xml:space="preserve">Et ne cogamur continuare omnes 49. </s> <s xml:id="echoid-s18108" xml:space="preserve">terminos, du-<lb/>cemus differentiam 2. </s> <s xml:id="echoid-s18109" xml:space="preserve">in 48. </s> <s xml:id="echoid-s18110" xml:space="preserve">numerum terminorum, minusvno, & </s> <s xml:id="echoid-s18111" xml:space="preserve">ad produ-<lb/>ctum 96. </s> <s xml:id="echoid-s18112" xml:space="preserve">addemus primum, quem poſuimus eſſe 1. </s> <s xml:id="echoid-s18113" xml:space="preserve">℞. </s> <s xml:id="echoid-s18114" xml:space="preserve">Ita enim fiet vltimus ter-<lb/>minus quęſitus 1. </s> <s xml:id="echoid-s18115" xml:space="preserve">℞. </s> <s xml:id="echoid-s18116" xml:space="preserve">† 96. </s> <s xml:id="echoid-s18117" xml:space="preserve">cui ſi addamus primum, videlicet 1. </s> <s xml:id="echoid-s18118" xml:space="preserve">℞.</s> <s xml:id="echoid-s18119" xml:space="preserve">. fit ſumma 2. </s> <s xml:id="echoid-s18120" xml:space="preserve"><lb/>℞. </s> <s xml:id="echoid-s18121" xml:space="preserve">† 96. </s> <s xml:id="echoid-s18122" xml:space="preserve">cuius ſemiſsis 1. </s> <s xml:id="echoid-s18123" xml:space="preserve">℞. </s> <s xml:id="echoid-s18124" xml:space="preserve">† 48. </s> <s xml:id="echoid-s18125" xml:space="preserve">ducta in 49. </s> <s xml:id="echoid-s18126" xml:space="preserve">numerum terminorum, facit nu-<lb/>merum 49. </s> <s xml:id="echoid-s18127" xml:space="preserve">℞. </s> <s xml:id="echoid-s18128" xml:space="preserve">† 2352. </s> <s xml:id="echoid-s18129" xml:space="preserve">quiæquatur cubo propoſito 117649. </s> <s xml:id="echoid-s18130" xml:space="preserve">Ablatis ergo vtrin-<lb/>que 2352. </s> <s xml:id="echoid-s18131" xml:space="preserve">remanebunt 49. </s> <s xml:id="echoid-s18132" xml:space="preserve">℞. </s> <s xml:id="echoid-s18133" xml:space="preserve">æquales numero 115297. </s> <s xml:id="echoid-s18134" xml:space="preserve">quo diuiſo per 49. </s> <s xml:id="echoid-s18135" xml:space="preserve">fit <lb/>Quotiens 2353. </s> <s xml:id="echoid-s18136" xml:space="preserve">pro primo impari quæſito. </s> <s xml:id="echoid-s18137" xml:space="preserve">quod vt probemus, ducemus diffe-<lb/>rentiam imparium, nimirum 2. </s> <s xml:id="echoid-s18138" xml:space="preserve">in 48. </s> <s xml:id="echoid-s18139" xml:space="preserve">numerum terminorum, minus vno, pro-<lb/>ductum que 96. </s> <s xml:id="echoid-s18140" xml:space="preserve">primo 2353. </s> <s xml:id="echoid-s18141" xml:space="preserve">addemus, vt conficiamus vltimũ terminum 2449. </s> <s xml:id="echoid-s18142" xml:space="preserve"><lb/>Deinde primum huic adij ciemus, & </s> <s xml:id="echoid-s18143" xml:space="preserve">ſummæ 4802. </s> <s xml:id="echoid-s18144" xml:space="preserve">ſemiſſem 2401. </s> <s xml:id="echoid-s18145" xml:space="preserve">in 49. </s> <s xml:id="echoid-s18146" xml:space="preserve">nume-<lb/>rum terminorum ducemus, procreabitur que numerus 117649. </s> <s xml:id="echoid-s18147" xml:space="preserve">qui dato cubo <lb/>æqualis eſt. </s> <s xml:id="echoid-s18148" xml:space="preserve">Recte ergo primus impar inuentus eſt.</s> <s xml:id="echoid-s18149" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1135" type="section" level="1" n="417"> <head xml:id="echoid-head444" xml:space="preserve">VSVS PRÆCEDENTIS TABVLÆ <lb/>quadratorum, & cuborum in radicibus <lb/>extrahendis.</head> <p> <s xml:id="echoid-s18150" xml:space="preserve"><emph style="sc">Inter</emph> alias vtilitates habet ſuperior tabula quadratorum, & </s> <s xml:id="echoid-s18151" xml:space="preserve">cuborume-<lb/>gregium vſum in radicibus quadratis, & </s> <s xml:id="echoid-s18152" xml:space="preserve">cubicis extrahendis; </s> <s xml:id="echoid-s18153" xml:space="preserve">quippe cum per <lb/>eam ſtatim expediantur tria prima puncta ad ſiniſtram, ſimul que tres figuræ ra-<lb/>dicis inueniantur: </s> <s xml:id="echoid-s18154" xml:space="preserve">quod vno, aut altero exemplo planum fiet.</s> <s xml:id="echoid-s18155" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18156" xml:space="preserve"><emph style="sc">Sit</emph> primum extrahenda radix quadrata ex appoſito numero. </s> <s xml:id="echoid-s18157" xml:space="preserve">Primum <lb/>1 1 7 6 8 9 0 1 4 5 (3 4 3 0 5. <lb/></s> <s xml:id="echoid-s18158" xml:space="preserve">. </s> <s xml:id="echoid-s18159" xml:space="preserve">. </s> <s xml:id="echoid-s18160" xml:space="preserve">. </s> <s xml:id="echoid-s18161" xml:space="preserve">. </s> <s xml:id="echoid-s18162" xml:space="preserve">. </s> <s xml:id="echoid-s18163" xml:space="preserve"><lb/>quæro inter quadratos numerum 117689. </s> <s xml:id="echoid-s18164" xml:space="preserve">trium punctorum, quem quia non in-<lb/>uenio, capio proximè minorem quadratum 117649. </s> <s xml:id="echoid-s18165" xml:space="preserve">eiuſque radicem 343. </s> <s xml:id="echoid-s18166" xml:space="preserve">in <lb/>Quotiente pono. </s> <s xml:id="echoid-s18167" xml:space="preserve">Inuento autem quadrato ſubtracto ex numero 117689. </s> <s xml:id="echoid-s18168" xml:space="preserve">trium <lb/>punctorum, remanent 40. </s> <s xml:id="echoid-s18169" xml:space="preserve">Ergo ſequens punctum erit 4001. </s> <s xml:id="echoid-s18170" xml:space="preserve">quo (relicta figu-<lb/>ra 1.) </s> <s xml:id="echoid-s18171" xml:space="preserve">diuiſo per 686. </s> <s xml:id="echoid-s18172" xml:space="preserve">duplumradicis inuentæ, reperitur Quotiens 0. </s> <s xml:id="echoid-s18173" xml:space="preserve">erit que vlti-<lb/>mum punctum 400145. </s> <s xml:id="echoid-s18174" xml:space="preserve">quo (relicta etiam figura 5.) </s> <s xml:id="echoid-s18175" xml:space="preserve">diuiſo per 6860. </s> <s xml:id="echoid-s18176" xml:space="preserve">duplum <lb/>radicis inuentæ, inuenitur Quotiens 5. </s> <s xml:id="echoid-s18177" xml:space="preserve">Eſt ergo tota radix 34305 {57120/68611}.</s> <s xml:id="echoid-s18178" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18179" xml:space="preserve"><emph style="sc">Sit</emph> deinde ex appoſito numero extrahenda radix cubica. </s> <s xml:id="echoid-s18180" xml:space="preserve">Primum inter <lb/>4 2 5 0 9 5 4 9 6 1 3 0 7 (1 6 1 9 9 <lb/>. </s> <s xml:id="echoid-s18181" xml:space="preserve">. </s> <s xml:id="echoid-s18182" xml:space="preserve">. </s> <s xml:id="echoid-s18183" xml:space="preserve">. </s> <s xml:id="echoid-s18184" xml:space="preserve">. <lb/></s> <s xml:id="echoid-s18185" xml:space="preserve">cubos quæro numerum 4250954. </s> <s xml:id="echoid-s18186" xml:space="preserve">trium primorũ punctorum, quem quia non <lb/>reperio, accipio cubum proxime minorem 4173281. </s> <s xml:id="echoid-s18187" xml:space="preserve">cum ſua radice 161. </s> <s xml:id="echoid-s18188" xml:space="preserve">qui cu-<lb/>bus ex tribus punctis ſubtractus relin quit 77673. </s> <s xml:id="echoid-s18189" xml:space="preserve">ita vt ſequens punctum ſit <lb/>77673961.</s> <s xml:id="echoid-s18190" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18191" xml:space="preserve"><emph style="sc">Paro</emph> ergo diuiſorem, vt lib. </s> <s xml:id="echoid-s18192" xml:space="preserve">6. </s> <s xml:id="echoid-s18193" xml:space="preserve">ad finem propoſ. </s> <s xml:id="echoid-s18194" xml:space="preserve">19. </s> <s xml:id="echoid-s18195" xml:space="preserve">docuimus, multipli- <pb o="392" file="420" n="420" rhead="GEOME. PRACT. LIB. OCTAVVS."/> cando radicis inuentæ quadratum ex eadem tabula excerptum, nimirum 25921. <lb/></s> <s xml:id="echoid-s18196" xml:space="preserve">per 300. </s> <s xml:id="echoid-s18197" xml:space="preserve">qui erit 7776300. </s> <s xml:id="echoid-s18198" xml:space="preserve">per quem ſi meum punctum diuido, reperio Quotiẽ-<lb/>tem 9. </s> <s xml:id="echoid-s18199" xml:space="preserve">& </s> <s xml:id="echoid-s18200" xml:space="preserve">facta operatione remanent 7295302. </s> <s xml:id="echoid-s18201" xml:space="preserve">adeo vt vltimum punctum ſit <lb/>7295302307</s> </p> <p> <s xml:id="echoid-s18202" xml:space="preserve"><emph style="sc">Deniqve</emph> paro nouum diuiſorem, multiplicando radicis 1619. </s> <s xml:id="echoid-s18203" xml:space="preserve">inuentæ <lb/>quadratum 2621161. </s> <s xml:id="echoid-s18204" xml:space="preserve">per 300. </s> <s xml:id="echoid-s18205" xml:space="preserve">qui erit 786348300. </s> <s xml:id="echoid-s18206" xml:space="preserve">per quem ſi vltimum meum <lb/>punctum diuido, inuenio Quotientem 9. </s> <s xml:id="echoid-s18207" xml:space="preserve">Facta autem operatione, remanent <lb/>214232708. </s> <s xml:id="echoid-s18208" xml:space="preserve">abſolutaque eſt extractio.</s> <s xml:id="echoid-s18209" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s18210" xml:space="preserve"><emph style="sc">Neqve</emph> vero negligendum videtur non inutile compendium in extractio-<lb/>ne cubica. </s> <s xml:id="echoid-s18211" xml:space="preserve">quod eſt eiuſmodi. </s> <s xml:id="echoid-s18212" xml:space="preserve">Quando nouus diuiſor parandus eſt, ne coga-<lb/>mur totius radicis inuentæ quadratum ſupputare, diuidemus radicem inuentam <lb/>in duas partes, ita vt vna pars ſit vltima figura Quotientis inuenta, nimirum in-<lb/>uenta quarta figura 9. </s> <s xml:id="echoid-s18213" xml:space="preserve">in dato exemplo, partiemur radicem inuentam 1619. </s> <s xml:id="echoid-s18214" xml:space="preserve">in <lb/>1610. </s> <s xml:id="echoid-s18215" xml:space="preserve">& </s> <s xml:id="echoid-s18216" xml:space="preserve">9. </s> <s xml:id="echoid-s18217" xml:space="preserve">quas partes bis ſcribemus, vt in appoſita formula <lb/>vides. </s> <s xml:id="echoid-s18218" xml:space="preserve">Si igitur ſingulas partes in ſingulas partes ducemus, <lb/> <anchor type="note" xlink:label="note-420-01a" xlink:href="note-420-01"/> producetur quadratus quæſitus radicis 1619. </s> <s xml:id="echoid-s18219" xml:space="preserve">vt ad propoſ. </s> <s xml:id="echoid-s18220" xml:space="preserve">1. <lb/></s> <s xml:id="echoid-s18221" xml:space="preserve">lib. </s> <s xml:id="echoid-s18222" xml:space="preserve">2. </s> <s xml:id="echoid-s18223" xml:space="preserve">Euclid. </s> <s xml:id="echoid-s18224" xml:space="preserve">demonſtrauimus. </s> <s xml:id="echoid-s18225" xml:space="preserve">Verbi gratia quia præcedens <lb/>quadratum numeri 161. </s> <s xml:id="echoid-s18226" xml:space="preserve">fuit 25921. </s> <s xml:id="echoid-s18227" xml:space="preserve">appoſitis duabus cifris, habebimus produ-<lb/>ctum 2592100. </s> <s xml:id="echoid-s18228" xml:space="preserve">ex 1610 in 1610. </s> <s xml:id="echoid-s18229" xml:space="preserve">cui ſi addemus 14490. </s> <s xml:id="echoid-s18230" xml:space="preserve">bis numerum videlicet <lb/>productum ex 1610. </s> <s xml:id="echoid-s18231" xml:space="preserve">in 9. </s> <s xml:id="echoid-s18232" xml:space="preserve">vel ex 9. </s> <s xml:id="echoid-s18233" xml:space="preserve">in 1610. </s> <s xml:id="echoid-s18234" xml:space="preserve">& </s> <s xml:id="echoid-s18235" xml:space="preserve">inſuper 81. </s> <s xml:id="echoid-s18236" xml:space="preserve">productum ex 9. </s> <s xml:id="echoid-s18237" xml:space="preserve">in 9. </s> <s xml:id="echoid-s18238" xml:space="preserve"><lb/>conficiemus 2621161. </s> <s xml:id="echoid-s18239" xml:space="preserve">quadratum radicis inuentæ 1619. </s> <s xml:id="echoid-s18240" xml:space="preserve">Eadem ratione ſi huius <lb/>numeri 16199. </s> <s xml:id="echoid-s18241" xml:space="preserve">quadratum deſideremus, diuidemus eum in 16190. </s> <s xml:id="echoid-s18242" xml:space="preserve">9. </s> <s xml:id="echoid-s18243" xml:space="preserve">bis, vt in <lb/>hac formula vides. </s> <s xml:id="echoid-s18244" xml:space="preserve">Deinde quia iam quadratum habuimus <lb/>2621161. </s> <s xml:id="echoid-s18245" xml:space="preserve">numeri 1619. </s> <s xml:id="echoid-s18246" xml:space="preserve">appoſitis duabus cifris, habebimus 26-<lb/> <anchor type="note" xlink:label="note-420-02a" xlink:href="note-420-02"/> 2116100. </s> <s xml:id="echoid-s18247" xml:space="preserve">quadratum numeri 16190. </s> <s xml:id="echoid-s18248" xml:space="preserve">cui ſi addemus 145710. <lb/></s> <s xml:id="echoid-s18249" xml:space="preserve">bis, productum videlicet ex 16190. </s> <s xml:id="echoid-s18250" xml:space="preserve">in 9. </s> <s xml:id="echoid-s18251" xml:space="preserve">vel ex 9. </s> <s xml:id="echoid-s18252" xml:space="preserve">in 16190. </s> <s xml:id="echoid-s18253" xml:space="preserve">& </s> <s xml:id="echoid-s18254" xml:space="preserve"><lb/>inſuper 81 productum ex 9. </s> <s xml:id="echoid-s18255" xml:space="preserve">in 9. </s> <s xml:id="echoid-s18256" xml:space="preserve">efficiemus 262407601. </s> <s xml:id="echoid-s18257" xml:space="preserve">quadratum numeri 161-<lb/>99. </s> <s xml:id="echoid-s18258" xml:space="preserve">& </s> <s xml:id="echoid-s18259" xml:space="preserve">ſic de cæteris. </s> <s xml:id="echoid-s18260" xml:space="preserve">Hoc ergo compendium reddet faciliorem cubicæ radicis <lb/>extractionem, cum ſemper præcedentis radicis inuentæ quadratum habeamus, <lb/>& </s> <s xml:id="echoid-s18261" xml:space="preserve">appoſitis duabus cifris, quadratum conficiamus eiuſdem radicis, appoſita ei <lb/>vna cifra, &</s> <s xml:id="echoid-s18262" xml:space="preserve">c.</s> <s xml:id="echoid-s18263" xml:space="preserve"/> </p> <div xml:id="echoid-div1135" type="float" level="2" n="1"> <note position="right" xlink:label="note-420-01" xlink:href="note-420-01a" xml:space="preserve"> <lb/>1610. # 9. <lb/>1610. # 9. <lb/></note> <note position="right" xlink:label="note-420-02" xlink:href="note-420-02a" xml:space="preserve"> <lb/>16190. # 9. <lb/>16190. # 9. <lb/></note> </div> <p> <s xml:id="echoid-s18264" xml:space="preserve"><emph style="sc">Bene</emph> autem vides, ſi tabula ſuperior extenderetur, vt haberentur quadra-<lb/>@i, & </s> <s xml:id="echoid-s18265" xml:space="preserve">cubiradicum 4. </s> <s xml:id="echoid-s18266" xml:space="preserve">aut 5. </s> <s xml:id="echoid-s18267" xml:space="preserve">figurarum, multo faciliorem effi ci extra ctionem ra-<lb/>dicum. </s> <s xml:id="echoid-s18268" xml:space="preserve">Sed quia tabula excreſceret hac ratione in immenſum, conten-<lb/>ti fuimus tabula, in quaradices habent 3. </s> <s xml:id="echoid-s18269" xml:space="preserve">figuras ad ſum-<lb/>mum, cum eã quilibet ex ijs, quæ diximus, <lb/>extendere poſsit, & </s> <s xml:id="echoid-s18270" xml:space="preserve">continuare.</s> <s xml:id="echoid-s18271" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1137" type="section" level="1" n="418"> <head xml:id="echoid-head445" xml:space="preserve">FINIS LIBRI OCTAVI.</head> <pb file="421" n="421" rhead="INDEX."/> </div> <div xml:id="echoid-div1138" type="section" level="1" n="419"> <head xml:id="echoid-head446" xml:space="preserve">ALPHABETICVS RERVM. <lb/>AC VERBORVM.</head> <note style="it" position="right" xml:space="preserve"> <lb/>## A. <lb/><emph style="sc">ACclivem</emph> diſtantiã mõ-<lb/># tis à loco menſoris vſque ad <lb/># baſem altitudinis monti im-<lb/># poſitæ, etiam non viſam, vnà <lb/># cum ipſa altitudine, quando <lb/># menſor in aſcenſu montis conſiſtit, pro-<lb/># pè verum, beneficio quadrati efficere <lb/># cognitam. # 132 <lb/>Accliuem montis aſcenſum à loco menſo-<lb/># ris vſ ad baſem altitudinis monti im-<lb/># poſitæ, {et}iam non viſam, vna cum ipſa <lb/># altitudine, quando menſor in aſcenſu <lb/># montis conſiſtit, prope verum, beneficio <lb/># quadrantis notum efficere. # 79 <lb/>Aceru{us} tritici, quo pacto menſuretur. # 209 <lb/>Ædificij cuiuſcum ad perpendiculum <lb/># erecti, vel putei profunditatem, ſi modo <lb/># angul{us} fundi, vel ſignum aliquod in <lb/># fuisdo poſitum conſpiciatur, per qua-<lb/># drantem reperire. # 80 <lb/>Ædificij cuiuſuis ad perpendiculum ere-<lb/># cti, vel putei profunditatem ſi modo an-<lb/># gul{us} fundi, vel ſignum aliquod infun-<lb/># ao poſitum conſpiciatur, per quadratũ <lb/># efficere cognitam. # 134 <lb/>Æquilateri trianguli area. # 166 <lb/>Agri, campiue, aut vrbis, vel regionis ſitũ <lb/># in plano deſcribere. # 148 <lb/>Agro propoſito figur am ſimilem in charta <lb/># deſcribere. # 172 <lb/>Alberti Dureri quadratura circuli per <lb/># numeros falſa eſt. # 318 <lb/>Aliquotæ part{es} ſimiles plurium magnitu-<lb/># dinũ eandem habent proportionem. # 218 <lb/>Altera parte longioris area. # 158 <lb/>Altim{et}ra ſcala quid. # 85 <lb/>Altitudinem, ad cui{us} baſem pateat acceſ-<lb/># ſ{us} beneficio ſpeculi plani, vnà cum di-<lb/># ſtantia ſpeculi à cacumine altitudinis <lb/># deprehendere. # 144 <lb/>Altitudinem, cui{us} baſis impoſita ſit mon-<lb/># @i, vel altericuipiã altitudini, & vtra-<lb/># queilli{us} extremit{as} cerni poſſit, {et}iam-<lb/># ſi infimum punctum alteri{us}, cui impe-<lb/># nitur, lateat, & eiuſdem puncti infimi <lb/># diſt antia à loco menſoris cognita nõ ſit, <lb/># per quadratum ex valle, aut ex plano <lb/># Horizontis explorare. # 131 <lb/>Altitudinem cui{us}lib{et} rei erectæ, per e-<lb/># i{us} diſtantiam ab oculo menſoris, bene-<lb/># ficio quadrati conijcere. # 106 <lb/>Altitudinẽ cui{us} libet rei erectæ, ẽt ſi ei<emph style="sub">9</emph> di-<lb/># ſtantia ab oculo menſoris ne data ſit, <lb/># ne inuenta, per du{as} ſtation{es} in plano <lb/># fact{as}, auxilio quadratipatefacere. # 107 <lb/>Altitudinem cui{us} baſis impoſita ſit alteri <lb/># altitudini, & vtraque illi{us} extremit{as} <lb/># cer ni poſſit, {et}iamſi infimum punctum <lb/># alteri{us}, cui imponitur, lateat, & eiuſdẽ <lb/># puncti infimi diſt antia à loco menſoris <lb/># cognita non ſit, per quadrantẽ ex valle, <lb/># aut ex plano Horizontis explorare. # 77 <lb/>Altitudinẽ cuiuſ rei erectæ, ex ei{us} vm-<lb/># bra, quam Sole lucente proijcit, ſi nota <lb/># fuerit, per quadratũ deprehendere. # 140 <lb/>Altitudinem inacceſſibilem beneficio ſpe-<lb/># culi plani, vna cum ſpeculi diſtantia <lb/># tam à baſe {et}iam non viſa, quam a ca-<lb/># cumine altitudinis, cognoſcere. # 145 <lb/>Altitudinem in acceſſibilem, quando ne <lb/># diſt antia à loco menſoris ad ei{us} baſem <lb/># nota eſt, ne{q́ue} è directo ipſi{us} duæ ſtatio-<lb/># n@@ in plano fieri poſſunt, neque denique <lb/># baſis appar{et}, per quadrantem notam <lb/># reddere. Atque hinc obiter ipſam quo-<lb/># que diſtantiam elicere. # 72 <lb/>Altitudinem inacceſſibilem, cui{us} baſis nõ <lb/># videatur, & ad quam per nullum ſpa-<lb/># tium ſecundum lineam rectam acce-<lb/># dere poſſim{us}, aut recedere, vt duæ ſta-<lb/># tion{es} fieri poſſint, ſed ſolũ ad dextram, <lb/># ſiniſtramuc ad locum, è quo ei{us} baſis <lb/># appareat, per quadrantẽ explorare. # 72 <lb/>Altitudinem inacceſſibilem, cui{us} baſis <lb/># non videatur, & ad quã per nullũ ſpa-<lb/># tiũ ſecundũ rectã lineã accedere poſſit <pb file="422" n="422" rhead="INDEX"/> # menſor, aut recedere, vt duæ ſtation{es} <lb/># fieri poſſint, ſed ſolum ad dextram, ſini-<lb/># ſtramue ad locum, è quo ei{us} baſis cer-<lb/># natur, per quadratum explorare. # 129 <lb/>Altitudinem inacceſſibilem, quando di-<lb/># ſtantia à loco menſoris ad baſem altitu-<lb/># dinis ignota eſt, per du{as} ſtation{es} in pla-<lb/># no fact{as} per quadrantem dim{et}iri. At-<lb/># que hinc diſtantiam quo ipſam erue-<lb/># re, {et}iamſi extrem{us} ei{us} termin{us} non <lb/># cernatur. # 57 <lb/>Altitudinem maiorem ex minori incogni-<lb/># ta, ſi tamen baſis maioris cerni poſſit, per <lb/># quadratum ven ari. # 130 <lb/>Altitudinem maiorem ex minori cognita, <lb/># {et}iamſi ſolum maioris vertex cernatur, <lb/># per quadratum efficere notam. # 129 <lb/>Altitudinem maiorem ex minori cognita, <lb/># per du{as} ſt ation{es} in ſummitate, vel in <lb/># duab{us} feneſtris fact{as}, {et}iamſi ſolum <lb/># maioris altitudinis vertex cernatur, <lb/># per quadrantem adinuenire. At hinc <lb/># diſtantiam quo inter altitudin{es} colli-<lb/># gere. # 74 <lb/>Altitudinem maiorem ex minori incogni-<lb/># ta, dummodo baſis maioris cerni poſſit, <lb/># per quadrantem perſcrutari. # 75 <lb/>Altitudinem minorem ex maiori cognita, <lb/># lic{et} baſis minoris cerni non poſſit, per <lb/># quadratum ſcrutari. # 130 <lb/>Altitudinem minorem ex maiori incogni@ <lb/># ta, dummodo baſis minoris appare{at}, <lb/># per quadratum elicere. # 130 <lb/>Altitudinem minorem ex maiori cogni-<lb/># ta, licet baſis minoris non cerni poſſit, <lb/># ope quadrantis perueſtigare. At hinc <lb/># diſtantiam quo inter altitudin{es} du{as} <lb/># eruere. # 76 <lb/>Altitudinem minorem ex maiori incogni-<lb/># ta, dummodo baſis minoris videri poſſit, <lb/># per quadrantem explorare. Atque hinc <lb/># diſt antiam quoque inter du{as} altitudi-<lb/># nes conijcere. # 76 <lb/>Altitudinem montis, vel turris ex ei{us} fa-<lb/># ſtigio, quando è directo menſoris inter-<lb/># uallum aliquod inter duo ſigna vel {et}-<lb/># iam inter ſignũ quodpiam ac turrim co <lb/># gnitum est per quadratũ conijcere # 123 <lb/>Altitudinẽ montis, aut turris ex ei{us} ver-<lb/># tice per du{as} ſtation{es} in eiuſdem ſum-<lb/># mitate fact{as}, è quib{us} ſignum aliquod <lb/># in Horiz@nte appareat, per quadrantẽ <lb/># dimetiri. At hinc ipſam qu@ diſtan-<lb/># tiam à turris baſe, vel perpendiculo <lb/># montis ad ſignum illud inueſtigare. # 59 <lb/>Altitudinem monti impoſitam, ſi modo al-<lb/># titudinis baſis poſſit conſpici: vel portio-<lb/># nem ſuperiorem alicui{us} turris, benefi-<lb/># cio ſpeculi plani efficere notam. # 146 <lb/>Altitudinem montis m{et}iri per quadran-<lb/># tem. # 57. & 59 <lb/>Altitudinem montis, aut turris ex ei{us} ver <lb/># tice per du{as} ſt ation{es} in haſta aliqua e-<lb/># recta, vel in duab{us} feneſtris turris, qua <lb/># rum vna ſupra aliam exiſtat, fact{as}, è <lb/># quib{us} ſignũ aliquod in Horizonte vi-<lb/># deri poſſit, per quadrantem m{et}iri. At <lb/># hinc diſtãtiã quo à ꝑpẽdiculo mõtisvel <lb/># turris, vſ ad ſignũ visũ cognoſcere. # 62 <lb/>Altitudinẽ montis, aut turris ex ei{us} ver-<lb/># tice per quadrantem metiri, ſi in plano, <lb/># cui in ſiſtit, ſpatiũ aliquod è directo mẽ-<lb/># ſoris notum ſit, deprehendere. # 64 <lb/>Altitudinẽ per vnicã ſtationẽ m{et}iri ꝑ qua <lb/># drantẽ, quando diſtantia nota est. # 56 <lb/>Altitudo pyramidum, in qu{as} corporare-<lb/># gularia è centris reſoluuntur. # 214 <lb/>Altitudinem propoſitã ſingulari quodam <lb/># modo inueſtigare. # 108. & 109 <lb/>Altitudinem propoſitam, eiuſ diſtantiam <lb/># ab oculo menſoris, vnà cum hypotenuſa <lb/># ab oculo ad faſtigium altitudinis extẽ-<lb/># ſa, venariope quadrati ſt abilis per vni-<lb/># cam ſtationem, etiamſi ſolum faſtigiũ <lb/># rei erectæ cernatur. # 112 <lb/>Altitudinem propoſitam, quando diſtan-<lb/># tia ab oculo menſoris ne data eſt, ne <lb/># inuenta, neque è dir ecto altitudinis duæ <lb/># ſtation{es} in aliqua haſta erecta fact{as}, <lb/># per quadratum indagare. # 111 <lb/>Altitudo quãdo maior ſit, quã diſtãtia, & <lb/># quando æqualis, & quando minor. # 140 <lb/>Altitudinem Solis, vel ſtellæ cuiuſuis per <lb/># quadratum obſeruare. # 87 <pb file="423" n="423" rhead="INDEX."/> Altitudinem turris, aut alteri{us} rei per <lb/># baculum indagare. # 137 <lb/>Altitud@nem turris vel mõtis ex ei{us} ſum-<lb/># mitate per vnicam ſtationem, ope qua <lb/># d@ati ſtabilis m{et}iri, vna cum diſtantia <lb/># ſigni in Horizonte viſi vſ ad turrem, <lb/># vel montis perpendiculum. # 118 <lb/>Altitudinem turris ex ei{us} vertice per v-<lb/># nicam ſtationem per quadrantem me-<lb/># tiri, ſi diſt antia ſigni in Horizonte viſi <lb/># vſ ad baſem turris nota ſit. # 64 <lb/>Altitudinẽ<unsure/> turris, aut montis, ex ei{us} ſum-<lb/># mitate per quadratum dim{et}iri, quan-<lb/># do in plano ſummitatis Horizonti æqui-<lb/># diſtante duæ ſtation{es} fieri poſſunt, & ſi-<lb/># gnũ aliquod in Horizonte cernitur. # 114 <lb/>Altitudinẽ turris, vel montis ex ei{us} ſum-<lb/># mitate per du{as} ſtation{es} in haſta ali@ <lb/># qua erecta fact{as}, inueſtigare per qua-<lb/># dratum, quando ſignum aliquod in <lb/># Horizonte videri poteſt. # 116 <lb/>Altitudinem turris, aut alteri{us} rei, per <lb/># Normam inueſtigare. # 138 <lb/>Altitudinis portionem ex minore altitudi-<lb/># ne, & minoris portionem ex maiore, per <lb/># quadratum percipere. # 131 <lb/>Altitudinis maioris portionem ex minori <lb/># altitudine, & minoris portionem ex ma <lb/># iori per quadrantem cognoſcere. # 76 <lb/>Altitudinis minoris portionem ex minore <lb/># altitudine, & maioris portionem ex mi-<lb/># nore, per quadratum elicere. # 131 <lb/>Ambitum terræ ex edito aliquo monte me-<lb/># tiri. # 366 <lb/>Anguli quantit{as}, quem latera inſtrumẽ-<lb/># ti partium continent, quo pacto cogno-<lb/># ſcatur. # 11 <lb/>Angulorum, & linearum quarundam <lb/># mechanicam menſur ationem admit-<lb/># tendam eſſe. # 169 <lb/>Angulos duos trianguli obliquanguli ex <lb/># duob{us} laterib{us}, & angulo ab ipſis cõ-<lb/># prehenſo, reperire. # 48 <lb/>Angulos duos trianguli obliq@ anguli ex <lb/># duob{us} laterib{us}, & angulo vni eorum <lb/># oppoſito, (ſi modo conſt{et} ſpeci{es} anguli <lb/># alteri lateri oppoſiti, quando dat{us} an-<lb/># gul{us} acut{us} eſt) expiſcari. # 48 <lb/>Angulos omn{es} tr{es} trianguli obliquanguli <lb/># ex omnib{us} trib{us} laterib{us} perueſtiga-<lb/># re. # 49 <lb/>Angulum acutum trianguli rectanguli <lb/># ex baſe, & vno latere inquirere. # 46 <lb/>Angulum acutum trianguli rectanguli <lb/># ex vtro{q́ue} latere reddere cognitum. # 46 <lb/>Angulum rectilineum datum in tr{es} æquæ <lb/># l{es} angulos diuide e. # 356 <lb/>Angul{us} incidentiæ cur angulo r@flexionis <lb/># ſit æqualis. # 341 <lb/>Angul{us} obſeruationis quis. # 52 <lb/>Angul{us}, quem filum cum proximo qua-<lb/># drati latere facit, quando offerat alt@tu-<lb/># dinem Solis, vel ſtellæ, & quando com-<lb/># plementum altitudinis. # 88 <lb/>Arabum quadratura circuli per numeros <lb/># falſa eſt. # 318 <lb/>Arc{us} circuli ad arcum ſimilẽ alteri{us} cir <lb/># culi eſt, vt chorda ad chordã & cõtra # 397 <lb/>Arcus datorum grad. ac Min. quo pacto <lb/># ex circulo quouis ope inſtrumenti par-<lb/># tium abſcindantur. # 9. & 10 <lb/>Area alter a parte longioris. # 158 <lb/>Area campi, intra quem lac{us}, vel ſylua <lb/># exiſtat. # 172 <lb/>Area campi, quando in triangula reſolui <lb/># non poteſt, # 171 <lb/>Area circuli accuratior. # 198 <lb/>Area circuli cui triangulo rectangulo ſit <lb/># æqualis, ſecundum Archimedem. # 182 <lb/>Area circuli æqualis eſt rectangulo compre <lb/># henſo ſub ſemid ametro, & ſemiſſe cir-<lb/># cumferentiæ circuli. # 294 <lb/>Area circulitrib{us} viis, ex cognita diame-<lb/># tro, & circumferentia. # 192 <lb/>Area Conoidis Hyperbolici. # 233 <lb/>Area Conoidis parabolici. # 233 <lb/>Area corporis planis ſuperficieb{us} conten-<lb/># ti, & circa ſphæram circum ſcriptibilis, <lb/># cui ſolido rectangulo ſit æqualis. # 307 <lb/>Area corporum omnino irregularium, <lb/># quæ. # 234 <lb/>Area cui{us} libet portionis ſphæræ. # 231 <lb/>Areæ cuiuſuis figuræ pulchra inuentio. <lb/># 173 <pb file="424" n="424" rhead="INDEX."/> Area cui{us}lib{et} trianguli cui rectangulo <lb/># ſit æqualis. # 292 <lb/>Areæ datæ Ellipſis. # 203 <lb/>Area datæ parabolæ. # 203 <lb/>Area doliorum. # 233 <lb/>Area figuræ lenticularis. # 200 <lb/>Area figuræ quadrilateræ omnino irregu-<lb/># laris. # 170 <lb/>Area figuræ ex variis circulorum ſegmen-<lb/># tis cogmentatæ. # 200 <lb/>Area figuræ regularis, cui{us} lat{us} eſt vni-<lb/># t{as}, quo pacto inueniatur. # 180 <lb/>Area figuræregularis, cui rectãgulo æqua-<lb/># lis ſit. # 293 <lb/>Area figuræ regularis, quo pacto ex area <lb/># alteri{us} figuræ ſimilis cognita eruatur. <lb/># 179 <lb/>Area figuræ regularis, cuitriangulo re-<lb/># ctangulo ſit æqualis. # 294 <lb/>Areæ figurarũ regulariũ à triãgulo vſ ad <lb/># Dodecagonũ, quãdo lat{us} eſt vnit{as}. # 180 <lb/>Area fruſti pyramidis, & coni. # 208 <lb/>Area fruſti ſphæræ. # 231 <lb/>Aream circuli vera maiorem ex diame-<lb/># tro inueſtigare. # 197 <lb/>Aream circuli ver a minorem, ex circum-<lb/># ferentia concludere. # 197 <lb/>Aream circuli vera maiorem, ex diame-<lb/># tro colligere. # 197 <lb/>Aream circuli vera maiorem, ex circum-<lb/># ferentia inferre. # 197 <lb/>Areã figuræ Ellipſi ſimilis, quæ circino de-<lb/># ſcribitur, inquirere. # 375 <lb/>Area multilateræ figuræ irregularis q̃. # 171 <lb/>Area par allelogrammorum. # 170 <lb/>Area par allelepipedorum, Priſmatum, & <lb/># cylindrorum. # 204 <lb/>Area portionum ſphæroidis. # 232 <lb/>Area pyramidis cui ſolido rectangulo ſit æ-<lb/># qualis. # 307 <lb/>Area pyramidum, & conorum. # 206 <lb/>Area quadrati. # 158 <lb/>Area quadrilaterorum non rectangulo-<lb/># rum. # 169 <lb/>Area quinque corporum regularium quæ. <lb/># 210. & 214 <lb/>Area rectangulorum. # 158 <lb/>Area regularium figurarum. # 175 <lb/>Area Rhombi, & Rhomboidis. # 169 <lb/>Area ſectoris circuli. # 199 <lb/>Area ſemicirculi, quadrantis, octauæ par-<lb/># tis, & c. # 193 <lb/>Area ſegmentorum circuli. # 199 <lb/>Area ſegmentorum ſphæræ. # 229 <lb/>Area ſphæræ vera minor, ex diametro cir-<lb/># culi maximi. # 228 <lb/>Area ſphæræ vera maior, ex circumferen-<lb/># tia maximi circuli. # 228 <lb/>Area ſphæræ vera minor, ex circumferen-<lb/># tia circuli maximi. # 228 <lb/>Area ſphæræ vera maior, ex diametro cir-<lb/># culi maximi. # 228 <lb/>Area ſphæræ æqualis eſt ſolido rectangulo <lb/># comprehenſo ſub ſemidiam{et}ro, & ter-<lb/># tia parte ſuperficiei conuexæ. # 229 <lb/>Area ſphæræ & ſuperfici{es} eiuſdem conue-<lb/># xa. # 218. & 223 <lb/>Area, vel ſolidit{as} ſectoris ſphæræ. # 230 <lb/>Area, vel ſolidit{as} hemiſphærij. # 230 <lb/>Area ſphæroidis. # 232 <lb/>Area trapezij nulla habentis latera paral-<lb/># lela. # 170 <lb/>Area trapezij habentis duo latera paralle-<lb/># la. # 170 <lb/>Area triangulorum. # 158. & 161 <lb/>Area trianguli rectanguli. # 165 <lb/>Area trianguli rectanguli, ex vno latere <lb/># circa angulum rectum, & vno angulo <lb/># acuto. # 168 <lb/>Area trianguli rectanguli, ex vno latere <lb/># circa angulum rectum, & latere quod <lb/># recto angulo opponitur. # 168 <lb/>Area trianguli rectanguli, ex latere, quod <lb/># recto angulo opponitur, & vno angulo <lb/># acuto. # 167 <lb/>Area trianguli Iſoſeelis. # 165 <lb/>Area trianguli æquilateri. # 166 <lb/>Area trianguli obliquanguli, ex duob{us} <lb/># laterib{us}, & angulo ab ipſis comprehen-<lb/># ſo. # 168 <lb/>Area trianguli obliquanguli, ex vno late-<lb/># re, ac duob{us} angulis. # 168 <lb/>Area vaſis excauati. # 209 <lb/>Arundinis, vel baculi beneficio diſtantiã <pb file="425" n="425" rhead="INDEX."/> # propoſitam m{et}iri. # 99 <lb/>Aſcenſum accliuem montis à loco menſo-<lb/># ris vſ ad baſem altitudinis monti im-<lb/># poſitæ, {et}iam non viſam, vna cum ipſa <lb/># altitud@ne, quando menſor in aſcenſu <lb/># montis conſiſtit, prope verum, beneficio <lb/># quadrantis efficere cognitum. # 79 <lb/>B <lb/>BAculi beneficio diſtantiam interped{es} <lb/># menſoris, & ſignum aliquod in plano <lb/># Horizontis m{et}@ri, quando extrem{us} <lb/># termin{us} diſtantiæ videri petest. # 137 <lb/>Baculi beneficio turrim, aut alteri{us} rei <lb/># altitudinem m{et}iri. # 137 <lb/>Baculi, aut arundinis beneficio diſtanti-<lb/># am propoſitam metiri. # 99 <lb/>Baſem trianguli rectanguli ex vno latere, <lb/># & vno angulo acuto inueſtigare. # 45 <lb/>Baſem trianguli rectanguli ex vtroquela-<lb/># tere perſcrutari. # 45 <lb/>Baſis quadratricis, ſemidiameter quadrã-<lb/># tis, & quadrans ſunt continue propor-<lb/># tional{es}. # 324 <lb/>C <lb/>CAmpano aſſcripta circuli quadratura <lb/># per line{as} falſa eſt. # 318 <lb/>Campi, agriu@, autvrbis, vel regionis ſitum <lb/># in plano deſcribere. # 148 <lb/>Campi area, quando in triangula reſolui <lb/># non poteſt. # 172 <lb/>Campi, intra quem lac{us}, vel ſylua exi-<lb/># ſtat, area. # 172 <lb/>Campo propoſito figuram in charta ſimilẽ <lb/># deſcribere. # 172 <lb/>Centeſimæ, vel milleſimæ part{es} in quauis <lb/># recta linea quo p@cto capiantur, ope in-<lb/># ſtrumenti partium. # 6 <lb/>Chorda alicui{us} arc{us} data, vna cum per-<lb/># pendiculari ex medio ei{us} puncto ad <lb/># arcum educta; quotgrad{us}, vel palmos <lb/># tam arc{us}, quam ſemidiameter com-<lb/># plectitur, inue@ire. # 253 <lb/>Circuli pulcherrim apropriet{as}. # 358 <lb/>Circuli quadraturam eſſe poſſibilem. # 320 <lb/>Circuli quadratura per Hyppoc atem <lb/># Ch@um falſa est. # 319 <lb/>Circuli quadratura per line{as} Campano <lb/># aſſcripta falſa est. # 318 <lb/>Circuli quadratura per numeros ſecundũ <lb/># Arab{es} falſa eſt. # 318 <lb/>Circuli quadratura per numeros ex Al-<lb/># berto Durero falſa est. # 318 <lb/>Circuli quadratura per line{as}. # 317 <lb/>Circuli area trib{us} viis, ex cognita dia-<lb/># metro, & circumferentia. # 192 <lb/>Circuli area accuratior. # 198 <lb/>Circuli aream vera maiorem, ex diame-<lb/># tro inueſtigare. # 197 <lb/>Circuli aream vera minorem, ex diame-<lb/># tro colligere. # 197 <lb/>Circuli aream vera minorem, ex circum-<lb/># ferentia inferre. # 197 <lb/>Circuli aream vera maiorem, ex circum-<lb/># ferentia concludere. # 197 <lb/>Circuli area cui triangulo rectangulo ſit <lb/># æqualis, ſecundum Archimedis doctri-<lb/># nam. # 182 <lb/>Circuli area æqualis eſt rectangulo com-<lb/># prehenſo ſub ſemidiametro, & ſemiſſe <lb/># circum ferentiæ circuli. # 294 <lb/>Circuli dimenſio ex Archimede. # 181 <lb/>Circuli diameter quam proportionem ha-<lb/># beat ad peripheriam, ſecundum Archi-<lb/># medem. # 185 <lb/>Circuli diameter in numeris ex dato ar-<lb/># cu. # 201 <lb/>Circuli diameter ex data periphæria, & <lb/># periphæria ex data diametro accura-<lb/># tior. # 198 <lb/>Circuli diametrum vera maiorem ex da-<lb/># ta circumferentia indagare. # 194 <lb/>Circuli diametrum vera minorem ex da-<lb/># ta periphæria inueſtigare. # 194 <lb/>Circuli diameter ducta in 3 {1/7}. gignit nu-<lb/># merum maiorem circumferentia. # 191 <lb/>Circuli peripheria, ac diameter, ex ei{us} <lb/># area. # 201 <lb/>Circuli peripheria ex data diametro, & <lb/># diameter ex data peripheria accurati-<lb/># or. # 198 <lb/>Circuli periphæria quam proportionẽ ha-<lb/># beat ad diametrum, ſecundum Archi-<lb/># medem. # 185 <lb/>Circuli periphæria diuiſa per 3 {1/7}. facit nu- <pb file="426" n="426" rhead="INDEX."/> # merum minorem diametro. # 191 <lb/>Circuli peripheriam veramaiorem, ex da <lb/># ta diametro reperire. # 193 <lb/>Circuli peripheriam vera minorem, ex da <lb/># ta diam{et}ro elicere. # 193 <lb/>Circuli parti octauæ, decimæſextæ & c. re-<lb/># ctangulum conſtituere Iſoperimetrum, <lb/># & æquale. # 214 <lb/>Circuli ſectoris area. # 199 <lb/>Circulo figuram rectilineam æqualem, & <lb/># alteri ſimilem conſtituere. # 329 <lb/>Circulo dato quadratum æquale conſtitu-<lb/># ere. # 327 <lb/>Circulo quadratũ æquale quo pacto facile <lb/># exhibeatur ex propriafigura. # 327 <lb/>Circulorum peripheriæ inter ſe ſunt, vt <lb/># diametri. # 195. & 336 <lb/>Circulorũ duorũ, vel figurarum ſimilium <lb/># ꝓportio, ex datis diametris circũferen-<lb/># tiisue, vel duob. laterib homologis. # 201 <lb/>Circulorum diam{et}ri inter ſe ſunt, vt cir-<lb/># cumferentiæ. # 194. & 336 <lb/>Circulũ quadrato æqualẽ deſcribere. # 329 <lb/>Circulum, velſiguram planam rectiline-<lb/># am, in data proportione augere velmi-<lb/># nuere. # 272 <lb/>Circulum cuilib{et} figuræ rectilineæ æqua-<lb/># lem deſcribere. # 329 <lb/>Circulum per tria puncta deſcribere, inuẽ-<lb/># tis nimirum aliis punctis, per quæ tran-<lb/># ſire deb{et}. # 344 <lb/>Circulum plurib{us} circulis, quorum dia-<lb/># m{et}ri, vel circumferentiæ datæ ſint, æ-<lb/># qualem: Et figuram ſimilem plurib{us} <lb/># figuris ſimilib{us}, quarum latera homo-<lb/># loga data ſint, æqualem deſcribere. # 202 <lb/>Circul{us} ad quadratum diametri propor-<lb/># tionem hab{et} quam 11. ad 14 proxime, <lb/># ſecundum Archimedem. # 191 <lb/>Circul{us} ad quadratum circumferentiæ <lb/># maiorẽ proportionẽ habet, quam 7. ad <lb/># 88. minorẽ vero, quam 71. ad 892. # 196 <lb/>Circul{us} omnib{us} figuris rectilineis regu-<lb/># larib{us} ſibi Iſoperimetris maior eſt. # 306 <lb/>Circul{us} omnium figurarum rectilinea-<lb/># rũ ſibi Iſoperimetrarũ maxim<emph style="sub">9</emph> eſt. # 306 <lb/>Circul{us} ad quadratum diametri maiorẽ <lb/># proportionem hab{et}, quam 223. ad 284 <lb/># minorẽ vero, quam 11. ad 14 # 196 <lb/>Circumferentiæ circuli ad diametrũ pro-<lb/># portio accuratior, quæ. # 198 <lb/>Compendium pulchrum in longitudini-<lb/># b{us} metiendis ꝑ quadratum ſtabile. # 98 <lb/>Conchoideos lineæ deſcriptio, eiuſque duæ <lb/># propri{et}at{es} inſign{es}. # 270 <lb/>Coni, & cylindri ſuperfici{es} conuexa. # 235 <lb/>Conicæ ſuperficiei proportio ad ſuam ba-<lb/># ſem. # 235 <lb/>Conoidis Hyperboliciſolidit{as}. # 233 <lb/>Cono, cylindro priſmati, ac pyramidi cubũ <lb/># æqualem efficere. # 369 <lb/>Cono, vel pyramidi æqualem cylindrum, <lb/># aut priſma ſub eadem altitudine: ct <lb/># contra cylindro, vel priſmati æqualem <lb/># conum, aut pyramidem ſub eadem al-<lb/># titudine conſtituere. # 368 <lb/>Conoidis parabolici ſolidit{as}. # 232 <lb/>Conorum, ac pyramidum area. # 206 <lb/>Conſtructio tabulæ Gnomonicæ facillima, <lb/># eiuſque vſ{us}. # 89 <lb/>Conſtructio & vſ{us} tabellæ pro minutis, & <lb/># ſecundis. # 18 <lb/>Conſtructio & vſ{us} tabulæ ꝓ minutis, & <lb/># ſec. cognoſcendis ex quadrante. # 20 <lb/>Conſtructio quadrantis ad min. & ſec. co-<lb/># gnoſcenda. # 15 <lb/>Conſtructio quadrati Geometrici. # 84 <lb/>Conſtructio pinnacidiorum pro radio vi-<lb/># ſuali. # 17 <lb/>Conſtructio regulæ loco filij cum perpendi-<lb/># culo. # 17 <lb/>Conſtructio inſtrumenti partium. # 4. & 13 <lb/>Conum, ac pyramidem in cylindrum, & <lb/># priſma: Item priſma, & cylindrum in <lb/># conũ, ac pyramidem tranſmutare. # 368 <lb/>Conum, cylindrũ, priſma, ac pyramidem <lb/># inæqualẽ ſub data altitudine, & ſupra <lb/># baſem quotuis angulorũ reuocare. # 369 <lb/>Conum, cylindrum, priſma, acpyramidem <lb/># in parallelepipedum æquale datæ altitu-<lb/># dinis vel baſis commutare. # 368 <lb/>Conũ datæ ſphæræ cõſtituere æqualem. # 371 <lb/>Conũ pyramidi, & cylindrum priſmati æ-<lb/># qualem: Ac viciſſim pyramidem cono <pb file="427" n="427" rhead="INDEX."/> # æqualem, & priſmæ cylindro æquale <lb/># conſtituere. # 368 <lb/>Conum, pyramidem, priſma, & cylindrum <lb/># in parallelepipedum ſupra baſem qua-<lb/># dratam conuertere. # 369 <lb/>Corpori regulari ſphæram æqualem exhi-<lb/># bere. # 371 <lb/>Corporum quinque regularium ſuperfici{es} <lb/># conuexæ. # 214 <lb/>Corporum quinque regularium area quæ. <lb/># 210. & 214 <lb/>Corporum omnino irregularium area. # 234 <lb/>Corp{us} regulare quoduis dato cubo æquale <lb/># conſtituere. # 372 <lb/>Corp{us} planis ſuperficieb{us} contentum, & <lb/># circa ſphæram circumſcriptibile, cui ſo-<lb/># lido rectangulo ſit æquale. # 307 <lb/>Cubicam, & quadratam radicem in nu-<lb/># meris non quadratis, & non cubis per <lb/># line{as} Geom{et}ricè inuenir@. # 290 <lb/>Cubicæ radicis extractio. # 281 <lb/>Cubicæ & quadratæ radicis extractio ex <lb/># data minutia. # 287 <lb/>Cubo dato corp{us} regulare quodcunque æ-<lb/># quale conſtruere. # 372 <lb/>Cubo dato parallelepipedum rectangulum <lb/># ſub data altitudine, vel ſupra datam ba-<lb/># ſem, æquale conſtituere. # 270 <lb/>Cuborum differentiæ quo pacto reperian-<lb/># tur. # 388 <lb/>Cuborum, & quadratorum tabula vſque <lb/># ad radicem 1000. # 378 <lb/>Cuborum generatio. # 388 <lb/>Cubum cylindro priſmati, cono, ac pyrami-<lb/># di æqualem conſtruere. # 370 <lb/>Cubum datæ ſphæræ æqualem: Et ſphæram <lb/># dato cubo æqualem efficere. # 370 <lb/>Cubum datum aut parallelepipedum in <lb/># datam proportionem ſecare. # 373 <lb/>Cubum duob{us}, aut plurib{us} cubis æqua. <lb/># lem conſtruere. # 372 <lb/>Cubum minorem ex maiori d{et}rahere re-<lb/># ſiduum{q́ue} in cubum conuertere. # 373 <lb/>Cubum parallelepipedo rectangulo æqua-<lb/># lem conſtruere. # 369 <lb/>Cubum ſolidis quotlib{et} æqualem conſti-<lb/># tuere. # 372 <lb/>Cub{us} alterutri{us} mediarum proportiona-<lb/># lium inter du{as} rect{as}, æqualis eſt paral-<lb/># lelepipedo ſub quadrato extremæ prope <lb/># mediam aſſumptam & altera extrema <lb/># comprehenſo. # 275 <lb/>Cub{us} diam{et}ri ſphæræ ad ſphæram, maio-<lb/># rem proportionem habet, quam 21. ad <lb/># 11. minorem vero, quã 426. ad 223. # 222 <lb/>Cub{us} circumferentiæ maximi circuli in <lb/># ſphæra ad ſphæram, maiorem proportio-<lb/># nem habet, quam 298374. ad 5041. mi-<lb/># rem autem, quam 2904. ad 49. # 221 <lb/>Cylindri, & coni ſuperfici{es} conuexa. # 235 <lb/>Cylindrica ſuperfici{es}, demptis baſib{us} # 235 <lb/>Cylindrorum, priſmatum, ac parallelepipe-<lb/># dorum area. # 205 <lb/>Cylindro, aut priſmati æqualem conum, vel <lb/># pyramidem ſub eadem altitudine; Et <lb/># viciſſim cono, vel pyramidi æqualem <lb/># cylindrum, aut priſma eiuſdem altitu-<lb/># dinis conſtituere. # 368 <lb/>Cylindro priſmati, cono, ac pyramidi cubũ <lb/># æqualem conſtituere. # 369 <lb/>Cylindrum datæ ſphæræ conſtituere æqua-<lb/># lem. # 370 <lb/>Cylindrum, ac priſma in pyramidem, & co-<lb/># num: Item conum, ac pyramidem in cy-<lb/># lindrum, vel priſma æquale tranſmuta-<lb/># re. # 368 <lb/>Cylindrum, aut priſma datum in propor-<lb/># tionem datam diuidere. # 373 <lb/>Cylindrum, conum, priſma, ac pyramidem <lb/># in parallelepipedũ rectangulum æquale <lb/># datæ altitudinis, vel baſis cõmutare. # 370 <lb/>Cylindrum, priſma, conum, ac pyramidem <lb/># cuiuſcunque altitudinis, in æqualem <lb/># ſub data qualib{et} alia altitudine, & ſu-<lb/># pra baſem quotcunque angulorum con-<lb/># uertere. # 368 <lb/>Cylindrum priſmati, & conum pyramidi <lb/># æqualem: Et viciſſim priſma cylindro <lb/># æquale, & pyramidem cono æqualem <lb/># conſlruere. # 368 <lb/>Cylindrum priſma, conum, & pyramidem <lb/># in parallelepipedum ſupra baſem qua-<lb/># dratam conuertere. # 369 <pb file="428" n="428" rhead="INDEX."/> D. <lb/>DEcimæ part{es} milleſimarum, quo pacto <lb/># ſumantur, {et}iamſi inſtrumentum par-<lb/># tium diuiſum ſit in 100. part{es} duntaxat. # 8 <lb/>Decimæ, vel centeſimæ part{es} quotcunque, <lb/># quo pacto ex quauis parte rectæ in par-<lb/># t{es} æqual{es} diuiſæ per circinum auferan-<lb/># tur. # 44 <lb/>Declinationem cuiuslibet paralleli in dia-<lb/># metro Aſtrolabii, per inſtrumentum <lb/># partium inuenire. # 11 <lb/>Diameter circuli, ac peripheriæ, ex ei{us} <lb/># area. # 201 <lb/>Diameter circuli in numeris, ex dato ar-<lb/># cu. # 201 <lb/>Diam{et}ri circuli ad circumferentiam pro-<lb/># portio accuratior, quæ. # 198 <lb/>Diameter circuli ducta in 3 {1/7}. facit nume-<lb/># rum maiorem circumferentia. # 191 <lb/>Diametrum circuli vera maiorem, ex da-<lb/># ta circumferentia indagare. # 194 <lb/>Diametrum circuli vera minorem, ex data <lb/># circumferentia inueſtigare. # 194 <lb/>Diameter circuli quam proportionem ha-<lb/># beat ad peripheriam, ſecundum Archi-<lb/># medem. # 185 <lb/>Diameter circuli ex data peripheria, & pe-<lb/># ripheriã ex data diametro accuratior. # 198 <lb/>Diametri circulorum inter ſe ſunt, vt cir-<lb/># cumferentiæ. # 194. & 336 <lb/>Differẽtiæ cuborũ quo pacto reperiãtur # 388 <lb/>Differentiæ quadratorum. # 387 <lb/>Differentia ſt ationum quid. # 52 <lb/>Difficult{as} in extractionib{us} radicum quæ <lb/># ſit, & quo pacto ſuperetur. # 283 <lb/>Dimenſion{es} quo modo ſine numerorum <lb/># ſupputatione fiant. # 55. 58. 61. 64 <lb/>Dimenſio altitudinis quo pacto fiat, ſinere-<lb/># ductione vmbræ verſæ ad rectam, quan@ <lb/># do in vna ſtatione vmbra recta, & verſa <lb/># in altera ſecatur. # 110 <lb/>Dimenſion{es} diſtantiarum eodẽ modo fiunt <lb/># in quadrato ſtabili, ac pendulo. # 102 <lb/>Dimenſio diſtãtiæ quo modo fiat ſine redu-<lb/># ctione vmbrarum rectarum ad verſ{as}, <lb/># quando in vtraque ſtatione lat{us} vm-<lb/># bræ rectæ ſecatur. # 104 <lb/>Dimenſio diſtantiæ quo modo fiat ſine re-<lb/># ductione vmbræ rectæ ad verſam, quan-<lb/># do in vna ſtatione vmbra recta & in al-<lb/># tera verſa ſecatur. # 105 <lb/>Dimenſio circuli ex Archimede. # 181 <lb/>Dioptræ portio intra quadratum ſtabile <lb/># quo pacto reperiatur. # 113 <lb/>Diſtantiam ab oculo vel pede menſoris ad <lb/># quoduis punctum in Horizonte nota-<lb/># tum, per vnicam ſtationem, per qua-<lb/># drantem metiri. # 67 <lb/>Diſtantiam accliuem montis à loco men-<lb/># ſoris vſque ad baſem altitudinis monti <lb/># impoſitæ, {et}iam non viſam, vna cum ipſa <lb/># altitudine<unsure/>, quando menſor in aſcenſu <lb/># montis conſiſtit, prope verum efficere <lb/># cognitam, beneficio quadrantis. # 78 <lb/>Diſtantiam accliuem montis à loco men-<lb/># ſoris vſque ad baſem altitudinis monti <lb/># impoſitæ, {et}iam non viſam, vna cum ipſa <lb/># altitudine, quando menſor in aſcenſis <lb/># montis conſiſtit, prope verum, beneficio <lb/># quadrati efficere cognitam. # 132 <lb/>Diſtantiam a ſummitate turris, vel muri <lb/># vſque ad ſignum aliquod in Horizonte <lb/># poſitum, licet ad illud acceſſ{us} non pa-<lb/># teat, per quadratum eruere, vbicunque <lb/># menſor exiſtat. # 128 <lb/>Diſtantiam horizontalẽ inter turrim ali-<lb/># quam, & al@ud quodpiã ſignum, ex tur-<lb/># ri per du{as} ſtation{es} in faſtigio fact{as}, vel <lb/># in duab{us} feneſtris, quarum vna ad per-<lb/># pendiculum ſit ſub alia, quando ſpa@ium <lb/># inter ill{as} feneſtr{as} notum est, etiamſi <lb/># toti{us} turris altitudo ignota ſit. per qua-<lb/># drantem dimetiri. Atque hinc obiter <lb/># altitudinem turris patefacere. # 70 <lb/>Diſtantiã ab oculo, vel pede menſoris (vbi-<lb/># cun exiſtat) ad quoduis punctum in <lb/># aliqua altitudine, vel etiam in Horizon-<lb/># te notatum per quadratum exquirere, <lb/># per vnicam etiam ſtationem. # 123. & 126 <lb/>Diſta tiam inter duo puncta in quolibet <lb/># plano eleuato, ſiue illud ad Horizontem <lb/># ſit rectum, ſiue inclinatum per quadran-<lb/># tem metiri. # 67 <lb/>Diſtantiam in plano, ſiue acceſſibilis ea ſit, <pb file="429" n="429" rhead="INDEX."/> # ſiue inacceſſibilis, per du{as} ſtation{es} in <lb/># eodem plano fact{as} per quadrantem tam <lb/># pendulum, quam ſtabilem m{et}iri, quan-<lb/># do in ei{us} extremo erecta est altitudo <lb/># aliqua perpendicularis, {et}iamſi infi-<lb/># mum ei{us} extremum non cernatur. At-<lb/># que hinc altitudinem quoque ipſam eli-<lb/># cere. # 52, & 55 <lb/>Diſtantiam à baſe turris ad ſignum propo-<lb/># ſitum in Horizonte, ex ſummitate tur-<lb/># ris, vel ex aliqua ei{us} feneſtra, per qua-<lb/># dratum cognoſcere. # 119 <lb/>Diſtantiam ab oculo, vel pede menſoris ad <lb/># quoduis punctum in aliqua al@itudine <lb/># notatum, per du{as} ſtation{es} in plano fa-<lb/># ct{as}, per quadrantem m{et}iri. # 65 <lb/>Diſtantiam inter ſignum quodpiam in Ho-<lb/># rizonte poſitum, & ſummitatem turris, <lb/># vel muri alicui{us} lic{et} ad ipſum ſignum <lb/># acceſſ{us} non pateat, per quadrantem <lb/># colligere. # 72 <lb/>Diſtantiam inter te, & ſignum quodcun <lb/># in plano Horizontis poſitum, per qua-<lb/># dratum perue@tigare. # 96 & 97 <lb/>Diſtantiam in Horizonte inter menſorem, <lb/># & ſignum aliquod viſum, per ſimplicißi-<lb/># mũ quoddã inſtrumentũ indagare. # 142 <lb/>Diſtantiam in plano per du{as} ſtation@ in <lb/># eodem plano fact{as}, per quadratum me <lb/># tiri, quando in ei{us} extr@@o erecta est <lb/># altitudo aliqua perpendicularis, etiam-<lb/># ſi infimum ei{us} extremum non cerna-<lb/># tur. # 100 <lb/>Diſtantiam inter duo montium, aut tur-<lb/># rium cacumina, per ſimpliciſſimum <lb/># quoddam inſtrumentum reperire. # 142 <lb/>Diſtantiam in plano Horizõtis inter men-<lb/># ſorem, & ſignum quoduis, beneficio <lb/># Normæ adinuenire. # 138 <lb/>Diſtantiam inter duo ſigna in plano, cui al-<lb/># titudo inſiſtit, ſi ea diſtantia è directo <lb/># menſoris iaceat, & vtrum ei{us} extre-<lb/># mum cerni poſſit, ex altitudinis faſtigio, <lb/># {et}iamſi altitudo ſit menſoris ſt atura, per <lb/># quadratum comprehendere. # 121 <lb/>Diſtantiam in plano Horizontis, quæ non <lb/># ſit valdè magna, facillimo quodam mo-<lb/># do dim{et}iri. # 139 <lb/>Diſtantiam inter ped{es} menſoris, & ſignum <lb/># aliquod in plano Horizontis, beneficio <lb/># baculi m{et}iri, quando extrem{us} termi-<lb/># n{us} diſtantiæ videri potest. # 140 <lb/>Diſtantiam inter duo ſigna, vel punctain <lb/># quolib{et} plano ſiue recto ad Horizontẽ, <lb/># ſiue inclinato, per quadratũ m{et}iri. # 126 <lb/>Diſtantiam per vnicam ſtationem m{et}iri <lb/># per quadrantem, quando altitudo nota <lb/># est. # 59 <lb/>Diſtantiam, quando menſor in vno ei{us} ex-<lb/># tremo, vel in aliqua altitudine nota ad <lb/># planum, in quo eſt diſtantia, perpendi-<lb/># culari exiſtens alterum extremum vi-<lb/># dere potest, per quadrantem metiri. # 68 <lb/>Doliorum capacit{as}. # 233 <lb/>E. <lb/>ELlipſis centro dato in linea axis, vna <lb/># cum duob{us} punctis Ellipſis, vtrum <lb/># axis vtriuſ extremum reperire. # 357 <lb/>Ellipſis datæ area. # 203 <lb/>Ellipſi ſimilem figuram, quam ouatam di-<lb/># cunt, circino deſcribere. # 374 <lb/>Examen extractionis radicum. # 280 <lb/>Extrahere radicem cuiuſuis generis ex da-<lb/># to numero. # 276 <lb/>F. <lb/>FAcilis inuentio lineæ rectæ cuiuis cir-<lb/># cũferentiæ æqualis, ex ꝓpria figura. # 327 <lb/>Facilis inuentio quadrati circulo æqualis. <lb/># 328 <lb/>Figura regularis circulo circumſcripta ma-<lb/># iorem ambitum hab{et}, quam circul{us}. <lb/># 330. & 335 <lb/>Figuræ numeri, ex quo radix extr ahitur, <lb/># quo modo per puncta ſignentur. # 277 <lb/>Figuræ rectilineæ cuilibet circulum æqua-<lb/># lem deſcribere. # 329 <lb/>Figuræ regularis area, cui triãgulo rectan-<lb/># gulo ſit æqualis. # 294 <lb/>Figuræ regularis area, cui rectangulo æqua-<lb/># lis ſit. # 293 <lb/>Figuram Ellipſi ſimilem, quam ouatam di-<lb/># cunt, circino deſcribere. # 374 <lb/>Figuram rectilineam circulo æqualem, & <lb/># alteri ſimilem conſtituere. # 329 <lb/>Figuram rectilineã in quotuis partes æqua- <pb file="430" n="430" rhead="INDEX."/> # l{es} per rectam datæ rectæ parallelam di-<lb/># ſtribuere. # 260 <lb/>Figuram ſolidam quamcun{q́ue} ex iis, de qui-<lb/># b{us} Eucidi. in Stereom{et}ria agit augere <lb/># vel minuere in data proportione. # 273 <lb/>Figuram rectilineam ex dato angulo, vel <lb/># puncto in latere, in quotuis part{es} æqua-<lb/># l{es} partiri. # 252 <lb/>Figuram planam rectilineam, vel circu-<lb/># lum, in data proportione augere, vel mi-<lb/># nuere. # 272 <lb/>Figuram ſimilem plurib{us} figuris ſimili-<lb/># b{us}, quarum latera homologa data ſint, <lb/># æqualem. Et circulum plurib{us} circulis, <lb/># quorum diam{et}ri, circumferẽtiæue da-<lb/># tæ ſint, æqualem deſcribere. # 202 <lb/>Figurarum duarum ſimilium, aut circulo-<lb/># rum proportio, ex datis duob{us} laterib{us} <lb/># homologis, vel diam{et}ris, circumferen-<lb/># tiiſue. # 201 <lb/>Figurarum iſoperim{et}rarum latera nu-<lb/># mero æqualia habentium, maxima & <lb/># æquilatera eſt, & æquiangula. # 303 <lb/>Figurarum regularium Iſoperim{et}rarum <lb/># maior eſt illa, quæ plur{es} contin{et} angu-<lb/># los pluraue latera. # 296 <lb/>Figuris rectilin{eis} regularib{us} circul{us}, cui <lb/># iſoperim{et}ræ ſunt, maior eſt. # 306 <lb/>Filum perpendiculi ſecãs vmbram rectam <lb/># facit angulum complementi altitudinis: <lb/># ſecans verò vmbram verſam, angulum <lb/># conſtituit ipſi{us} altitudinis. # 89 <lb/>Fractionem magnam ad minorem ferè æ-<lb/># quiualentem reducere. # 178 <lb/>Fractionis inter du{as} mediæ facilis inuen-<lb/># tio. # 178 <lb/>Fruſti marmoris regularis ſ@lidit{as}. # 209 <lb/>Fruſti pyramidis, & coni area. # 208 <lb/>Fruſti ſphæræ ſolidit{as}. # 231 <lb/>G. <lb/>GEner aradicum innumera. # 276 <lb/># Generatio cuborum. # 388 <lb/>Generatio quadratorum. # 387 <lb/>Geodæſia, ac Geom{et}ria quid. # 236 <lb/>Geom{et}rici quadrati conſtructio. # 85 <lb/>Gnomonica tabula. # 91 <lb/>Gnomonicæ tabulæ facillima conſtructio, <lb/># eiuſ{q́ue} vſ{us}. # 89 <lb/>Gnomon, ſeu lat{us} quadrati medio loco <lb/># proportionale eſt inter vmbram rectam, <lb/># ac verſam. # 87 <lb/>Grad{us} ac Min. in dato arcu quot conti-<lb/># neantur, per inſtrumentum partium <lb/># cognoſcere. # 10 <lb/>Grad{us}, ac Min. quotlib{et}, quo pacto ex <lb/># circulo quouis, ope inſtrumẽti partium, <lb/># abſcindantur. # 9. & 10 <lb/>H. <lb/>HEmiſpherii conuexa ſuperfici{es}. # 229 <lb/># Hemiſphærii ſolidit{as}. # 230 <lb/>Heptagoni lat{us} non rectè à Carolo Maria-<lb/># no, Alberto Durero, & Franciſco Fluſ-<lb/># ſate inueniri. # 362 <lb/>Hyperbolici Conoidis ſolidit{as}. # 233 <lb/>Hypotenuſæ inuentio per quadrantem. # 53. <lb/># 58. 61. 63 <lb/>I. <lb/>IMpar{es} numeros quemlib{et} cubũ com-<lb/># ponent{es} inuenire. # 390 <lb/>Imperata pars quo pacto ex data recta ab-<lb/># ſcindatur, per inſtrumentũ partium. # 10 <lb/>Inacceßibilem altitudinem, cui{us} baſis non <lb/># videatur, & ad quam per nullũ ſpatium <lb/># ſecundum rectam lineam accedere poßit <lb/># menſor, aut recedere, vt duæ ſtation{es} fie-<lb/># ri poſſint, ſed ſolum ad dextram, ſini-<lb/># ſtramue ad locum, è quo ei{us} baſis cer-<lb/># natur, per quadratum explorare. # 128 <lb/>Inacceſſibilem altitudinem, cui{us} baſis non <lb/># videatur, & ad quam per nullũ ſpatium <lb/># ſecundum lineam rectam accedere poſ-<lb/># ſim{us}, aut recedere, vt duæ ſtation{es} fieri <lb/># poſſint, ſed ſolum ad dextram, ſiniſtram-<lb/># ue ad locum, è quo ei{us} baſis appare{at}, <lb/># per quadrantem explorare, # 72 <lb/>Inacceſſibilem altitudinem, quando diſtan-<lb/># tia a loco menſoris ad baſem altitudinis <lb/># ignota eſt, per du{as} ſtation{es} in plano fa-<lb/># ct{as}, per quadrantẽ dim{et}iri. At hinc <lb/># diſtantiam quo ipſam eruere, {et}iamſi <lb/># extrem{us} ei{us} termin{us} nõ cernatur. # 57 <lb/>Inacceſſibilem altitudinem, quando neque <lb/>diſtantia à loco menſoris ad ei{us} baſem <pb file="431" n="431" rhead="INDEX"/> # nota eſt, ne{q́ue} è directo ipſi{us} duæ ſtatio-<lb/># n{es} in plano fieri poſſunt, neque denique <lb/># baſis appar{et}, per quadrantem notam <lb/># reddere. Atque hinc obiter ipſam quo <lb/># diſtantiam elicere. # 73 <lb/>Inacceſſibilem altitudinem beneficio ſpecu-<lb/># li plani, vnà cum ſpeculi diſtantia tam <lb/># à baſe, {et}iam non viſa, quam à cacumi-<lb/># ne altitudinis cognoſcere. # 145 <lb/>Inacceſſibilem diſtantiã per quadrantem <lb/># tam pendulum, quam ſtabilem m{et}iri, <lb/># quando in ei{us} extremo erecta eſt alti-<lb/># tudo perpẽdicularis, etiamſi infimũ ei{us} <lb/># extremum non cernatur. At hinc alti-<lb/># tudinem quo ipſam elicere. # 52. & 55 <lb/>Incidentiæ angul{us} cur angulo reflexionis <lb/># ſit æqualis. # 341 <lb/>Inſtrumenta menſurandi varia. # 51 <lb/>Inſtrumenti, quod Italis Squadra zoppa <lb/># dicitur, conſtructio, & vſ{us}. # 150 <lb/>Inſtrumentum partium quid, & quo pacto <lb/># conſtruatur. # 3. & 4 <lb/>Inſtrumentum partium quo pacto aliter <lb/># conſtruatur. # 13 <lb/>Inſtrumentum pro librationib{us} aptiſſi-<lb/># mum. # 153 <lb/>Interuallum, ad cui{us} extrema accedere <lb/># non liceat, dummodo ea appareant & <lb/># ipſum interuallum productum ad ped{es} <lb/># menſoris pertingat, ex altitudine aliqua <lb/># nota, per quadrantem m{et}iri. # 68 <lb/>Interuallum è directo menſoris poſitum cu-<lb/># i{us} vtrumque extremum, vel alterum <lb/># non appareat, niſi menſor ad dextram, <lb/># vel ſiniſtram accedat, per quadrantem <lb/># comprehendere. # 71 <lb/>Interuallũ in Horizonte inter turrim ali-<lb/># quam, & aliud quodpiam ſignum, ex <lb/># turri per du{as} ſtation{es} in faſtigio fa-<lb/># ct{as}, vel in duab{us} feneſtris, quarũ vna <lb/># ſit ad perpendiculum ſub alia, quando <lb/># ſpatium inter ill{as} feneſtr{as} notum est, <lb/># {et}iamſi toti{us} turris altitudo ignota ſit, <lb/># per quadrantem dim{et}iri. Atque hinc <lb/># obiter altitudinem turris patefacere. # 70 <lb/>Interuallũ in Horizonte, inter menſorem, <lb/># & ſignũ aliquod viſum, per ſimpliciſſi-<lb/># mũ quoddam inſtrumentũ indagare. # 142 <lb/>Interuallũ in plano Horizontis inter men-<lb/># ſorem, & ſignum quoduis beneficio <lb/># Normæ adinuenire. # 138 <lb/>Interuallum inter duo puncta in quolib{et} <lb/># plano eleuato, ſiue illud ad Horizontem <lb/># ſit rectum, ſiue inclinatũ, per quadran-<lb/># tem metiri. # 67 <lb/>Interuallum inter duo ſigna, vel puncta in <lb/># quolibet plano ſiue recto ad Horizontẽ, <lb/># ſiue inclinato, per quadratũ metiri. # 126 <lb/>Interuallum, quãdo menſor in vno ei{us} ex-<lb/># tremo, vel in aliqua altitudine nota ad <lb/># planum, in quo interuallũ eſt, perpendi-<lb/># culari exiſtens alterum extremum vi-<lb/># dere poteſt, per quadrantem metiri. # 68 <lb/>Interuallum tranſuerſum in Horizonte, <lb/># cui{us} vtrum{q́ue} extremum videripoteſt, <lb/># per quadratum metiri. # 128 <lb/>Interuallum inter pedes menſoris, & ſignũ <lb/># aliquod in plano Horizontis, beneficio <lb/># baculi metiri, quando extrem{us} termi-<lb/># n{us} interualli videri potest. # 137 <lb/>Interuallum tranſuerſum in Horizonte, <lb/># cui{us} vtrum extremum inſpicipoteſt, <lb/># per quadrantem efficere notum. # 69 <lb/>Ioſeph{us} Scaliger perperam Archimedem <lb/># de Dimenſione circuli reprehendit. # 184 <lb/>Irregularium omnino corporum area. # 334 <lb/>Iſoperimetra figuræ quæ, & tractatio de {eis} <lb/># inſtituta. # 291 <lb/>Iſoperimetrarum figurarum regularium <lb/># maior eſt illa, quæ plur{es} continet angu-<lb/># los, pluraue latera. # 296 <lb/>Iſoperimetrarum figurarum latera nume-<lb/># ro habentium æqualia, maxima & æ-<lb/># quilatera eſt, & æquiangula. # 303 <lb/>Iſoperimetrorum triangulorum eandem <lb/># habentium baſem, mai{us} eſt illud, quod <lb/># duo latera habet æqualia. # 297 <lb/>Iſoſcelis trianguli area. # 165 <lb/>Iſoſcelia duo triangula ſimilia baſium inæ-<lb/># qualium, ſimulmaiora ſunt duob{us} Iſo-<lb/># ſcelib{us} ſimul ſuper eaſdem baſ{es}, quæ <lb/># quidem inter ſe ſint diſſimilia, priorib{us} <lb/># verò Iſoperimetra, habeant quatuor <lb/># latera inter ſe æqualia. # 360 <pb file="432" n="432" rhead="INDEX"/> Iſoſcelib{us} duob{us} triangulis datis, quorum <lb/># baſ{es} inæqual{es} ſint, & duo latera vni{us} <lb/># duob{us} alteri{us} æqualia: ſuper eiſdem <lb/># baſib{us} triangula Iſoſcelia ſimilia, & <lb/># priorib{us} ſimul ſumptis Iſoperim{et}ra <lb/># conſtituere. # 299 <lb/>L. <lb/>LAtera duo trianguli obliquanguli, ex <lb/># tertio latere, & duob{us} quibuſuis an-<lb/># gulis, inuenire. # 46 <lb/>Latera tria in quadrilatero maiora ſunt <lb/># quarto latere. # 344 <lb/>Lateris trianguli obliquanguli ſegmenta <lb/># à perpendicularifacta, ex datis trib{us} <lb/># laterib{us} cognoſcere. # 46 <lb/>Laterum proportion{es} ex datis angulis cu-<lb/># iuſuis trianguli patefacere. # 44 <lb/>Lat{us} figuræ regularis, quo pacto ex ei{us} <lb/># area deprehendatur. # 181 <lb/>Lat{us} figuræ regularis quo pacto ex ſemi-<lb/># diam{et}ro circuli circumſcripti cogno-<lb/># ſcatur. # 178 <lb/>Lat{us} polygoni propoſiti quo pacto in dato <lb/># circulo per inſtrumentum partium in-<lb/># ueniatur. # 11 <lb/>Lat{us} quadratricis æquale eſt quadranti <lb/># circuli, cui{us} ſemidiameter est baſis <lb/># quadratricis. # 326 <lb/>Lat{us} trianguli rectanguli, ex baſe, & al-<lb/># terutro angulorum acutorum, notum <lb/># efficere. # 45 <lb/>Lat{us} trianguli rectanguli, ex baſe, & alte-<lb/># ro cognoſcere. # 45 <lb/>Lat{us} trianguli rectanguli ex altero latere <lb/># & alterutro angulo acuto eruere. # 45 <lb/>Lat{us} trianguli obliquanguli ex duob{us} <lb/># laterib{us}, & angulo ab ipſis comprehen-<lb/># ſo colligere. # 47 <lb/>Lat{us} trianguli obliquanguli ex duob{us} <lb/># reliquis laterib{us}, & duob{us} quibuſuis <lb/># angulis, addiſcere. # 47 <lb/>Lat{us} trianguli obliquanguli ex duob{us} <lb/># laterib{us}, & angulo vni eorum oppoſito, <lb/># (ſi modo conſtet ſpeci{es} anguli alteri la-<lb/># teri dato oppoſiti, quando dat{us} angul{us} <lb/># acut{us} est) exquirere. # 48 <lb/>Lenticularis figuræ area. # 200 <lb/>Librare ſpatium terræ inæquale, pro ducen-<lb/># dis aquis: aut {et}iam, ſi lubet, Horizonti <lb/># æquidiſtans efficere. # 153 <lb/>Linea recta diuiſa in quotuis part{es} æqua-<lb/># l{es}, quot eiuſmodi part{es} in quauis alia <lb/># recta contineantur, ope inſtrumenti par-<lb/># tium cognoſcere. # 6 <lb/>Linea recta in quotuis part{es} æqual{es} diui-<lb/># ſa, quot decimæ, vel centeſimæ, & c. in <lb/># quauis particula vni{us} partis contineã-<lb/># tur, per circinum deprehendere. # 42 <lb/>Lineæ ſuperfici{es}, ac ſolida, pen{es} quid men-<lb/># ſurentur. # 157 <lb/>Lineæ rectæ ſub dimenſionem cadent{es} quæ <lb/># ſint. # 51 <lb/>Lineæ duæ, vna recta, & altera inflexa, <lb/># nunquam concurrent{es}, lic{et} in infini-<lb/># tum producantur, & ſemper magis vna <lb/># ad alteram acced{at}. # 270 <lb/>Lineam quadratricem deſcribere. # 320 <lb/>Lineam rectam, ad cui{us} extrema accede-<lb/># re non liceat, dummodo ea appareant, <lb/># & ipſa linea recta producta ad ped{es} <lb/># menſoris pertingat, ex altitudine aliquæ <lb/># nota, per quadrantem metiri. # 68 <lb/>Lineam rectam in Horizonte inter turrim <lb/># aliquam, & aliud quodpiam ſignum, ex <lb/># turri per du{as} ſtation{es} in faſtigio fact{as}, <lb/># vel in duab{us} feneſtris, quarum vna ad <lb/># perpendiculum ſit ſub alia, quando ſpæ<unsure/>-<lb/># tium inter ill{as} feneſtr{as} notum eſt, et-<lb/># iam ſi toti{us} turris altitudo ſit ignota, per <lb/># quadrantem dimetiri. Atque hinc obi-<lb/># ter altitudinem turris patefacere. # 70 <lb/>Lineam rectam datam per inſtrumentum <lb/># partiũ diuidere, vt alia recta diuiſa eſt. # 12 <lb/>Lineam rectam è directo menſoris poſitam <lb/># cui{us} vtrum extremum, vel alterum, <lb/># non appareat, niſi ad dextram vel ſini-<lb/># ſtram menſor accedat, per quadrantem <lb/># comprehendere. # 71 <lb/>Lineam rectam in Horizonte per quadra-<lb/># tum metiri, quando menſor in vno ei{us} <lb/># extremo exiſtens alterum extremum <lb/># videre non potest, neque altitudo in <lb/># promptu est, ſed ſolum ad dextram, vel <lb/># ſiniſtram per lineam perpendicularem <pb file="433" n="433" rhead="INDEX"/> # dere poteſt ad locum, è quo alterum ex-<lb/># tremum appare{at}. # 121 <lb/>Lineam rectam arcui quadrantis æqua-<lb/># lem reperire. # 325 <lb/>Lineam rectã in Horizonte è directo men-<lb/># ſoris iacentem, per quadratum cognoſce-<lb/># re, ad cui{us} extrema neque accedere li-<lb/># ceat, neque è loco menſoris eam dim{et}i-<lb/># ri: dummodo ad dextram, vel ſiniſtram <lb/># per lineam perpendicularem ad locum <lb/># aliquẽire poſſit menſor, ex quo vtrum <lb/># extremum appare{at}. # 121 <lb/>Lineam rectã, quando menſor in vno ei{us} <lb/># extremo, vel in aliqua altitudine nota <lb/># ad planum, in quo eſt linea, perpendicu-<lb/># lari exiſtens alterum extremum videre <lb/># potest, per quadrantem m{et}iri. # 68 <lb/>Lineam rectam tranſuerſam in Horizon-<lb/># te, cui{us} vtrum{q́ue} extremum videri po-<lb/># test, per quadrantem metiri. # 127 <lb/>Lineam rectam tranſuerſam in Horizon-<lb/># te, cui{us} vtrum extremum inſpici po-<lb/># test, per quadrantem notam efficere. # 69 <lb/>Linearum quarundã, & angulorũ mecha-<lb/># nicã menſurationẽ admittendã eſſe. # 169 <lb/>Longitudinem tranſuerſam in Horizonte, <lb/># cui{us} vtrum extremum inſpici poteſt, <lb/># per quadrantem notam efficere. # 69 <lb/>Longitudinem trabis ad Horizontem in-<lb/># clinatæ, cui{us} portio ſuperior tantum <lb/># conſpiciatur, vnà cum angulo inclina-<lb/># tionis, diſtantia baſis à menſore, & alti-<lb/># tudine faſtigii ſupra Horizontem, per <lb/># quadratum metiri. # 151 <lb/>Longitudinem rectæ è diam{et}ro menſoris <lb/># poſitæ, cui{us} vtrum{q́ue} extremum vel al-<lb/># terum non appareat, niſi ad dextram, <lb/># vel ſiniſtram accedat menſor, per qua-<lb/># drantem comprehendere. # 71 <lb/>Longitudinem vmbræ ab altitudine, Sole <lb/># lucente, quando proiectæ altitudo eſt co-<lb/># gnita, ope quadrati adipiſci. # 141 <lb/>Longitudinem lineæ rectæ, quando menſor <lb/># in vno ei{us} extremo, vel in altitudine <lb/># aliqua nota, quæ perpendicularis ſit in <lb/># eo extremo ad planum, in quo linea ia-<lb/># c{et}, exiſtens alterum extemũ videre po-<lb/># teſt, per quadrantem comprehendere. # 68 <lb/>Longitudinem in Horizonte inter turrim <lb/># aliquam, & aliud quodpiam ſignum, ex <lb/># turri per du{as} ſtation{es} in faſtigio fact{as}, <lb/># vel in duab{us} feneſtris quarum vna ſit <lb/># ſub alia ad perpendiculum, quando ſpa-<lb/># tium inter ill{as} feneſtr{as} notum eſt, {et}-<lb/># iamſitoti{us} turris altitudo ignota ſit, per <lb/># quadrantem dimetiri. At hinc obiter <lb/># altitudinem turris patefacere. # 70 <lb/>Longitudinem in Horizonte extenſam per <lb/># quadratum m{et}iri, quando menſor in <lb/># vno ei{us} extremo exiſtens alterũ videre <lb/># non poteſt, propter tumorem aliquem in-<lb/># teriectũ, ne altitudo in promptu eſt, ſed <lb/># ſolum ad dextrã, vel ſiniſtram per lineã <lb/># perpendicularẽ recedere poteſt ad locũ, è <lb/># quo alterum extremum appare{at}. # 121 <lb/>Longitudinem in Horizonte è directo men-<lb/># ſoris iacentem, per quadratũ cognoſce-<lb/># re, ad cui{us} extrema neque accedere li-<lb/># ceat, ne è loco menſoris eam dimetiri, <lb/># ne vlla adſit altitudo: dũmodo ad dex-<lb/># trã, vel ſiniſtrã per lineam perpendicula-<lb/># rem ad locũ aliquemire poſſit menſor, ex <lb/># quo vtrum extremum appare{at}. # 122 <lb/>Longitndinem, ad cui{us} extrema accedere <lb/># non liceat, dummodo ea appareant, & <lb/># ipſa longitudo producta ad ped{es} menſo-<lb/># ris pertingat, ex altitudine aliqua nota, <lb/># per quadrantem metiri. # 68 <lb/>Longitudinem aſcenſ{us} alicui{us} montis, ſi <lb/># ei{us} cacumen ab oculo in radice conſti-<lb/># tuto videatur, per inſtrumentum ſim-<lb/># plicißimum cognoſcere. # 143 <lb/>M. <lb/>MAgnitudinum quatuor propriet{as} <lb/># quædam. # 331 <lb/>Magnitudinib{us} in part{es} proportional{es} <lb/># ſectis, ſi in ſingulis vna pars itreum ſece-<lb/># tur proportionaliter, erunt ibidem totæ <lb/># etiam ſectæ proportionaliter. # 237 <lb/>Marmoris regularis ſolidit{as}. # 209 <lb/>Mechanica menſuratio in nonnullis lin{eis}, <lb/># & angulis admittenda est. # 169 <pb file="434" n="434" rhead="INDEX"/> Medi{as} du{as} proportional{es} inter du{as} da-<lb/># d{as}, ex Nicomede, prope verum, adin-<lb/># uenire. # 270 <lb/>Medi{as} du{as} ꝓportional{es} inter du{as} datas, <lb/># ex Diocle, prope verum, inquirere. # 268 <lb/>Medi{as} du{as} proportional{es} inter dat{as} <lb/># du{as}, ex Herone, & Apollonio Pergæo, <lb/># prope verum, inuenire. # 266. & 267 <lb/>Medi{as} du{as} proportional{es} inter du{as} da-<lb/># t{as}, ex Philone Byſantio, & Philopono, <lb/># prope verum inquirere. # 268 <lb/>Medium numerum proportionalem, vel <lb/># duos medios inter duos datos numeros <lb/># comperire. # 274 <lb/>Menſorum ratio communis in area cuiuſ-<lb/># uis figuræ, vel agri inueſtiganda. # 173 <lb/>Menſurandi varia inſtrumenta. # 51 <lb/>Menſuræ linearum, ſuperficierum, ac ſoli-<lb/># dorum pen{es} quid ſumantur. # 157 <lb/>Milleſimarum decimæ part{es} quo modo ſu-<lb/># mantur, {et}iam ſi inſtrumentũ partium <lb/># diuiſum ſit in 100. part{es} duntaxat. # 8 <lb/>Milleſimæ part{es} quomodo capiantur, {et}-<lb/># iamſi in inſtrumento partium conti-<lb/># neantur tantum 100. part{es}. # 7 <lb/>Milleſimæ, vel centeſimæ part{es} in quauis <lb/># recta linea quo pacto capiantur, ope in-<lb/># ſtrumenti partium. # 6 <lb/>Minuta quotlib{et} quo pacto ex gradu quo-<lb/># uis abſcindantur per circinum. # 41 <lb/>Minuta ac ſecũda quo pacto per circinum <lb/># in quauis particula grad{us} deprehen-<lb/># dantur. # 39 <lb/>Minuta, & ſecũda quo pacto ex quadran-<lb/># te conſtructo reperiantur. # 18 <lb/>Minutia, cui{us} Numerator ex duarum <lb/># minutiarũ Numeratorib{us}, & Deno-<lb/># minator ex denominatorib{us} conflatur, <lb/># maior eſt minore, & maiore minor. # 178 <lb/>Minutiæ inter du{as} mediæ facilis inuentio. <lb/># 778 <lb/>Minutiam magnam ad minorem ferè æ-<lb/># quiualentemreducere. # 178 <lb/>Montis altitudinem m{et}iri per quadran-<lb/># tem. # 57. & 59 <lb/>Montis vel turris altitudinem ex ei{us} ſum-<lb/># mitate per vnicam ſtationem, ope qua-<lb/># drati ſtabilis m{et}iri, vna cum diſtantia <lb/># ſigni in Horizonte viſi vſ ad perpendi-<lb/># culum montis, aut turris. # 117 <lb/>Montis, aut turris altitudinem ex ei{us} ver-<lb/># tice per quadrantem m{et}iri, ſi in plano, <lb/># cui inſiſtit, ſpatium aliquod è directo <lb/># menſoris notum ſit, deprehendere. # 64 <lb/>Montis, aut turris altitudinẽ ex ei{us} ſum-<lb/># mitate per quadratum dim{et}iri, quãdo <lb/># in plano ſummitatis Horizonti æquidi-<lb/># ſtante duæ ſtation{es} fieri poſſunt, & ſignũ <lb/># aliquod in Horizonte cernitur. # 114 <lb/>Montis aut turris altitudinem ex ei{us} ver-<lb/># tice per du{as} ſtation{es} in haſta aliqua e-<lb/># recta, vel in duab{us} feneſtris turris, qua-<lb/># rum vna ſit ſupra aliam fact{as}, è quib{us} <lb/># ſignũ aliquod in Horizõte videri poßit, <lb/># per quadrantẽ m{et}iri. At hinc diſtan-<lb/># tiã quo à perpẽdiculo mõtis, velturris, <lb/># vſ ad ſignum viſum cognoſcere. # 62 <lb/>Montis, aut turris altitudinem ex ei{us}ver-<lb/># tice per du{as} ſtation{es} in eiuſdem ſummi-<lb/># tate fact{as}, è quib{us} ſignum aliquod in <lb/># Horizonte appareat, per quadrantem <lb/># dim{et}iri. At hinc ipſam quo diſtan-<lb/># tiam à montis perpendiculo, vel turris <lb/># baſe ad ſignum illud inueſtigare. # 59 <lb/>Montis vel turris altitudinem ex ei{us} faſti-<lb/># gio, quando è directo menſoris interual-<lb/># lum aliquod inter duo ſigna vel {et}iam <lb/># inter ſignum quodpiam ac turrim cog-<lb/># nitum eſt, per quadratum coniicere. # 122 <lb/>Montis, vel turris altitudinẽ ex ei{us} ſum-<lb/># mitate per du{as} ſt ation{es} in haſta aliqua <lb/># erecta fact{as}, inueſtigare per quadra-<lb/># tum, quando ſignum aliquod in Hori-<lb/># zonte videri potest. # 116 <lb/>Multilateræ figuræ irregularis area quæ. # 171 <lb/>Muri cuiuſque ſolidit{as}. # 209 <lb/>N. <lb/>NOrmæ beneficio altitudinem turris, <lb/># aut alteri{us} rei inueſtigare. # 139 <lb/>Normæ beneficio diſtantiam in plano Ho-<lb/># rizontis inter menſorem, & ſignũ quod-<lb/># uis percipere. # 139 <lb/>Numeri particular{es} pro ſingulis radicum <lb/># ſpecieb{us}, quo modo reperiantur. # 278 <pb file="435" n="435" rhead="INDEX"/> Numeros impar{es} datum cubum compo-<lb/># nent{es} reperire. # 390 <lb/>Numerum aliquo concipiente, quot ei v-<lb/># nitat{es} remaneant poſt tr{es} operation{es} <lb/># imper at{as}, conijcere. # 341 <lb/>O <lb/>OBliquanguli trianguli area, exvno la-<lb/># tere, ac duob{us} angulis. # 168 <lb/>Obliquanguli trianguli area, ex duob{us} <lb/># laterib{us}, & angulo ab ipſis cõprehẽſo. # 168 <lb/>Obliquangulorũ quadrilaterorũ area. # 169 <lb/>Obliquangulorum triangulorum rectili-<lb/># neorum problemata. # 46 <lb/>Obſeruationis angul{us} quis. # 52 <lb/>Ouatam figuram Ellipſi ſimilem circino <lb/># deſcribere. # 374 <lb/>Octogonum regulare ad datam altitudinẽ <lb/># latitudinemue conſtituere. # 365 <lb/>Octogonum regulare circulo inſcriptum <lb/># medio proportionale eſt inter quadra-<lb/># tum circulo circumſcriptum, & qua-<lb/># dratum eidem inſcriptum. # 364 <lb/>P <lb/>PArabolici Conoidis ſolidit{as}. # 232 <lb/># Parabolæ datæ area. # 203 <lb/>Parallelepipedũ ſub quadrato alterutri{us} <lb/># extremarum, (ſi ſint quatuor lineæ cõ-<lb/># tinue proportional{es}) & altera extrema <lb/># comprehenſum, æquale est cubo mediæ <lb/># proportionalis, quæ priori extremæ aſ-<lb/># ſumptæ eſt propinquior. # 275 <lb/>Parallelepipedorum, Priſmatum, & cylin-<lb/># drorum area. # 204. & 205 <lb/>Parallelepipedo rectangulo cubum æqua-<lb/># lem exhibere. # 369 <lb/>Parallelepipedum, aut cubum in datam <lb/># proportionem diuidere. # 373 <lb/>Parallelepipedum rectangulum ſub data <lb/># altitudine, vel ſupra datam baſem dato <lb/># cubo æquale conſtituere. # 370 <lb/>Parallelogrammum datum in quotcun <lb/># part{es} æquales diuidere per rect{as} duo-<lb/># b{us} laterib{us} oppoſitis parallel{as}. # 206 <lb/>Parallelogrammum datum per rectam ex <lb/># puncto ſiue extra, ſiue intra ipſum, ſiue <lb/># in aliquo latere dato ductam bifariã ſe-<lb/># care. # 266 <lb/>Parallelogrammum in dato angulo æqua-<lb/># le dato quadrilatero conſtituere. # 338 <lb/>Pars imperata quo pacto ex data recta ab-<lb/># ſcindatur, per inſtrumentum partiũ. # 10 <lb/>Part{es} aliquotæ ſimil{es} plurium magnitu-<lb/># dinum eandem habent proportione. # 218 <lb/>Part{es} decimæ milleſimarum quo pacto ſu-<lb/># mantur, {et}iamſi inſtrumentũ partium <lb/># diuiſum ſit in 100. part{es} duntax{at}. # 8 <lb/>Part{es} centeſimæ, vel mille ſimæ in quauis <lb/># recta linea, quo pacto ope inſtrumenti <lb/># partium capiantur. # 6 <lb/>Part{es} quotcun decimæ, vel cẽteſimæ, & c. <lb/># quo pacto ex quauis parte rectæ in par-<lb/># t{es} æqual{es} diuiſæꝑcircinũ auferãtur. # 45 <lb/>Part{es} milleſimæ, quo modo capiantur, {et}-<lb/># iamſi in inſtrumento partium contine-<lb/># antur tantum 100. part{es}. # 7 <lb/>Particula quælib{et} vni{us} partis centenſi-<lb/># mæ inſtrumenti partium, quot part{es} <lb/># decim{as} vni{us} centeſimæ, vel quot mil-<lb/># leſim{as} toti{us} lateris inſtrumenti cõple-<lb/># ctatur, quo pacto cognoſcatur. # 9 <lb/>Partium inſtrumentum quid, & quo pa-<lb/># cto conſtruatur. # 3. & 4 <lb/>Partium inſtrumentum, quo pacto aliter <lb/># conſtruatur. # 13 <lb/>Pendul{us} quadrans quid. # 18 <lb/>Pentagonum regulare non rectè conſtrui <lb/># ab Alberto Durero, & aliis. # 360 <lb/>Peripheriæ circuli ex data diametro; & di-<lb/># am{et}er ex data peripheria, accuratior. # 198 <lb/>Peripheria circuli, ac diam{et}er, ex ei{us} a-<lb/># rea. # 201 <lb/>Peripheria circuli quã proportionẽ habeat <lb/># ad diametrũ, ſecundũ Archimedẽ. # 185 <lb/>Peripheria circuli diuiſa per 3 {1/7}. gignit nu-<lb/># merum maiorem diam{et}ro. # 191 <lb/>Peripheriæ circuli ad diametrum propor-<lb/># tio accuratior, quæ. # 198 <lb/>Peripheriæ circulorum inter ſe ſunt, vt di-<lb/># ametri. # 194. & 336 <lb/>Peripheriam circuli vera maiorem, ex da-<lb/># ta diam{et}ro reperire. # 193 <lb/>Peripheriã datærectæ æqualẽ reperire. # 329 <lb/>Peripheriam circuli vera minorem. ex da-<lb/># ta diam{et}ro elicere. # 193 <pb file="436" n="436" rhead="INDEX."/> Perpendicularem in lat{us} quodcun tri-<lb/># anguli ex angulo oppoſito cadentẽ, ex o-<lb/># mnib. trib{us} laterib{us} efficere notã. # 49 <lb/>Perpendicularisquæ ſegmenta faciat in la-<lb/># tere trianguli obliquanguli. # 46 <lb/>Perpendicularis ex quolibet gradu qua-<lb/># drantis demiſſa, in quodnam punctum <lb/># ſemidiam{et}ri cadat, per inſtrumentum <lb/># partium cognoſcere. # 11 <lb/>Perpendicularis in triangulo quando est <lb/># numer{us} ſurd{us}, quid agendum. # 165 <lb/>Perpendicularis in triangulo æquilatero <lb/># quo pacto cognoſcatur. # 175 <lb/>Perpendicularis è centro figuræ regularis, <lb/># quo pacto cognoſcatur. # 175 <lb/>Perpendiculi filum ſecans vmbrãrectam <lb/># facit angulum complementi altitudi-<lb/># nis: ſecans vero vmbram verſam, an-<lb/># gulum conſtituit ipſi{us} altitudinis. # 89 <lb/>Pinnacidia pro radio viſuali quo pacto cõ-<lb/># ſtruenda ſint. # 17 <lb/>Pinnacidia quo pacto in quadrante ſint <lb/># affigenda. # 17 <lb/>Polygoni propoſiti lat{us} quo pacto in dato <lb/># circulo per inſtrumentum partium in-<lb/># ueniatur. # 11 <lb/>Portionem altitudinis maioris ex minori <lb/># altitudine, & minoris portionẽ, ex ma-<lb/># iori per quadrantem cognoſcere. # 76 <lb/>Portionem altitudinis maioris ex minore <lb/># altitudine, & minoris portionẽ ex ma-<lb/># iore, per quadratum percipere. # 131 <lb/>Portionis ſphæræ ſuperfici{es} conuexa. # 229 <lb/>Portionis ſphæræ ſolidit{as}. # 231 <lb/>Portionum ſphæroidis ſolidit{as}. # 232 <lb/>Primi inter ſe numeri ſi ambo nõ ſint qua@ <lb/># drati, aut cubi; neque vlli eorum æque <lb/># multiplic{es} quadrati erunt aut cubi. Et <lb/># ſi duorum numerorum æ<unsure/>que multipli-<lb/># c{es} ſint ambo quadrati, aut cubi, etiam <lb/># ipſi quadrati erunt, aut cubi. # 343 <lb/>Priſma, ac c<unsure/>yl@ndrum in conum, & pyra-<lb/># midem: Item, conum, ac pyramidem in <lb/># cylindrum, vel priſma æquale tranſmu-<lb/># tare. # 369 <lb/>Priſma conum, cylindrũ ac pyramidẽ in æ-<lb/># quale ſub data altitudine, & ſupra basẽ <lb/># quotuis angulorum conuertere. # 369 <lb/>Priſma, cylindrum, conum, ac pyramidem <lb/># in parallelepipedum ſupra baſem qua-<lb/># dratam conuertere. # 369 <lb/>Priſmati, cono, cyl@ndro, ac pyramidi cubũ <lb/># æqualem conſtruere. # 370 <lb/>Priſmati cuicun cylindrum æqualem, & <lb/># pyramidi conum æqualem: Ac viciſſim <lb/># cylindro priſma æquale, & cono æqua-<lb/># lem pyramidem conſtruere. # 368 <lb/>Priſmati dato ſphæram æqualem conſtru-<lb/># ere. # 371 <lb/>Pryſma pyramidem, cylindrum, & conum <lb/># in parallelepipedum rect angulũ aquale <lb/># datæ altitudinis, vel baſis comutare. # 370 <lb/>Priſma, vel cylindrum datum in datam <lb/># proportionem diuidere. # 373 <lb/>Priſmati, aut cyli<unsure/>ndro æqualem pyramidẽ <lb/># vel conum ſub eadem altitudine: Et <lb/># cõtra pyramidi, vel conoæquale priſma, <lb/># vel cylindrum conſtituere eiuſdem al-<lb/># titudinis. # 368 <lb/>Priſmatum, parallelepipedorum, & cylin-<lb/># drorum area. # 204. & 205 <lb/>Probatio extractionis radicum. # 280 <lb/>Problemata 3. 4. 5. 6. & 7. libri 3. quo pa-<lb/># cto per vnicam ſt ationem, ope quadrati <lb/># ſtabilis abſoluantur. # 112 <lb/>Problemeta triangulorum rectilineorum <lb/># rectangulorum. # 44 <lb/>Profunditatem putei, vel adificij cuiuſuis <lb/># ad perpendiculum erecti, ſi modo angu-<lb/># l{us} fundi, vel ſignum aliquod in fundo <lb/># poſitum conſpiciatur, per quadratum <lb/># efficere cognitam # 134 <lb/>Profunditatem putei, vel a dificij cuiuſcũ-<lb/># que ad perpendiculum erecti, ſi modo <lb/># angul{us} fundi, vel ſignum aliquod in-<lb/># fundo poſitum conſpiciatur, per qua-<lb/># drantem reperire. # 80 <lb/>Profunditatem vallis, eiuſdemque deſcen-<lb/># ſum obliquum, ſi non ſit valde in qua-<lb/># lis, etuſ termin{us}, vel aliquod in valle <lb/># ſignum conſpici poſſit, per quadr antem <lb/># ſcrutari. # 82 <lb/>Profundit atem vallis, eiuſdem que deſcen-<lb/># ſum obliquum, ſi non ſit valde inæ qua-<lb/># lis, & ei{us} termi<unsure/>n{us}, vel aliquod i<unsure/>n ea <pb file="437" n="437" rhead="INDEX."/> # ſignum conſpici poſſit, per quadratum <lb/># cognoſcere. # 136 <lb/>Propinquam radicem veræ, in numeris nõ <lb/># quadratis non cubis, nõ Zenſizenſis, non <lb/># ſurdeſolidis, & c. inuenire. # 284 <lb/>Proportionalem numerum medium, vel <lb/># duos medios, inter datos duos numeros <lb/># reperire. # 274 <lb/>Proportional{es} du{as} medi{as} inter du{as} da-<lb/># t{as}, ex Diocle; ꝓpe verũ inueſtigare. # 268 <lb/>Proportional{es} du{as} medi{as} inter du{as} da-<lb/># t{as}, ex Nicomede, prope verum, adinue-<lb/># nire. # 270 <lb/>Proportional{es} du{as} medi{as} inter du{as} da-<lb/># t{as}, ex Philone Byſantio, & Philopono, <lb/># prope verum, inquirere. # 267 <lb/>Proportional{es} du{as} medi{as} inter dat{as} <lb/># du{as} rect{as}, prope verum, ex Herone, & <lb/># Apollonio Pergæo inuenire. # 267 <lb/>Proportion{es} laterum ex datis angulis cu-<lb/># iuſuis trianguli patefacere. # 44 <lb/>Proportionalis tertia, & quarta quo pacto <lb/># ꝑ inſtrumentum partium reperiatur. # 13 <lb/>Propriet{as} pulchra quadrati. # 365 <lb/>Propriet{as} circuli pulcherrima. # 358 <lb/>Propriet{as} quædam quatuor magnitudi-<lb/># num. # 331 <lb/>Propriet{as} pulchra Quadratricis. # 323 <lb/>Punctum, in quo duæ rectæ ad in uicem in-<lb/># clinatæ concurrant, inuenire. # 55 <lb/>Punctum declinationis cui{us} libet paralle-<lb/># li in diametro Aſtrolabij per inſtrumẽ-<lb/># tum partium inuenire. # 11 <lb/>Putei, vel æd@fici<unsure/>j cuiuſcun<unsure/>que ad perpen-<lb/># diculum erecti profunditatem, ſi modo <lb/># angul{us} fundi, vel ſignum aliquod in <lb/># fundo poſitum conſp@ciatur, per qua-<lb/># drantem reperire. # 80 <lb/>Putei, vel ædificij<unsure/> cuiuſuis ad perpendicu-<lb/># lum erecti profunditatem, ſi modo an-<lb/># gul{us} fundi, vel ſignum aliquod in <lb/># fundo poſitum conſpiciatur, per qua-<lb/># dratum efficere cognitam. # 134 <lb/>Pyramidẽ, conũ, cylindrũ, ac priſma in æ-<lb/># qualẽ ſub data altitudine, & ſupra ba-<lb/># ſem quotuis angulorum reuocare. # 369 <lb/>Pyramidem, priſma, conum, & cylindrũ in <lb/># parallepipedum rectangulũ æquale da-<lb/># tæ altitudinis, vel baſis commutare # 370 <lb/>Pyramidem datæ ſphæræ æqualem extrue-<lb/># re. # 371 <lb/>Pyramidem, priſma, cylindrum, & conum <lb/># in parallelepipedum ſupra baſem qua-<lb/># dratam conuertere. # 369 <lb/>Pyramidi, cono, cylindro, ac priſmati cubũ <lb/># æqualem efficere. # 369 <lb/>Pyramidi conum æqualem, & priſm ati cy-<lb/># lindrum æqualem: Et contra cono py-<lb/># ramidem æqualem, & cylindro priſma <lb/># æquale conſtruere. # 368 <lb/>Pyramidi, vel cono æquale priſma, vel cy-<lb/># lindrum eiuſdem altitudinis: Et viciſ-<lb/># ſim cylindro, vel priſmati æqualem co-<lb/># num, vel pyramidem ſub eadem altitis-<lb/># dine conſtituere. # 367 <lb/>Pyramidum, & conorum area. # 206 <lb/>Pyramis cui ſoliào rectangulo æqualis ſit. <lb/># 307 <lb/>Q <lb/>QVadrans pendul{us} quid. # 18 <lb/>Quadrans ſtabilis quid # 18 <lb/>Quadrans, ſemidiameter, & baſis Qua-<lb/># dratricis, continue ſunt proportional{es}. <lb/># 324. <lb/>Quadrantis lib. 1. cap. 2. conſtructi vſ{us} in <lb/># minutis, & ſecundis ex quirendis. # 18 <lb/>Quadranti circuli rectangulum conſtitu-<lb/># ere Iſoperimetrum & æquale. # 214 <lb/>Quadrantis conſtructio ad Min. & Sec. <lb/># cognoſcenda. # 15 <lb/>Quadr ati area. # 158 <lb/>Quadratæ radicis extractio. # 279 <lb/>Quadratæ, & cubicæ radicis extractio ex <lb/># data minutia. # 288 <lb/>Quadratam & cubicam radicem in nu-<lb/># meris non quadratis, & non cubis per <lb/># line{as} Geometricè inuenire. # 289 <lb/>Quadrati Geometrici conſtructio. # 84 <lb/>Quadrati pulchra propriet{as}. # 365 <lb/>Quadratorum differentiæ. # 387 <lb/>Quadratorum, & cuborum tabula vſque <lb/># adradicem 1000. # 378 <lb/>Quadratorum generatio. # 387 <lb/>Quadratum, altera parte longi{us}, Rhom-<lb/># bum ac Rhomboid{es}, ex exceſſu diame-<lb/># tri ſupra lat{us}, & c. deſcribere. # 345 <pb file="438" n="438" rhead="INDEX."/> Quadratum numerum in quotlibet qua-<lb/># dratos deſtribuere. # 342 <lb/>Quadrilateri tria latera maiora ſunt <lb/># quarto latere. # 344 <lb/>Quadrato circulũ æqualẽ deſcribere. # 329 <lb/>Quadratricis lat{us} æquale eſt quadr anti <lb/># circuli, cui{us} ſemidiameter eſt baſisqua <lb/># dratricis. # 326 <lb/>Quadratricis baſis, ſemidiam{et}er qua-<lb/># drantis, & quadrans ſunt continue ꝓ-<lb/># portional{es}. # 324 <lb/>Quadratricis pulchra propri{et}{as}. # 363 <lb/>Quadratricem lineam deſcribere. # 324 <lb/>Quadratum pendulum, ac ſt abile, quo-<lb/># modo vſurpetur. # 86 <lb/>Quadratum circumferentiæ circuli ma-<lb/># ximi in ſphæra ita eſt ad ſuperficiem <lb/># ſphæræ, vt circumferentia circuli ma-<lb/># ximi ad diametrum. # 219 <lb/>Quadratum circulo æquale, quo pacto fa-<lb/># cile exhibeatur ex propria figura. # 328 <lb/>Quadratum dato circulo æquale conſti-<lb/># tuere. # 327 <lb/>Quadratum diam{et}ri circuli maximi in <lb/># ſphæra ad ſuperficiẽ ſphæræ, maiorẽ pro-<lb/># portionem habet quam 7. ad 22. minorẽ <lb/># vero, quam 71. ad 223. # 220 <lb/>Quadratum datæ figuræ æquale, quo pa-<lb/># cto facile conſtruatur. # 173 <lb/>Quadratum diametri circuli ad circulũ <lb/># proportionem habet, quam 14. ad 11. ꝓ-<lb/># ximè, ſecundum Archimedem. # 191 <lb/>Quadratũ diametri circuli ad circulum <lb/># habet maiorẽ proportionẽ, quã 14. ad 11. <lb/># minorem vero, quam 284. ad 223. # 195 <lb/>Quadratũ circũferentiæ circuli ad circu-<lb/># lũ habet maiorẽ ꝓportionẽ, ꝗ̃ 892. ad 71 <lb/># minorẽ vero, quam 88. ad 7. # 196 <lb/>Qua dratũ circũferentiæ circuli maximi <lb/># in ſphæra ad ſuperficiẽ conuexã ſphæræ, <lb/># maiorẽ proportionẽ habet, ꝗ̃ 223. ad 71. <lb/># minorem vero, quam 22. ad 7. # 221 <lb/>Quadratũ maximi lateris triãguli min{us} <lb/># eſt, ꝗ̃ duplũ ſummæ quadratorũ ex reli-<lb/># quis duob{us} laterib{us} deſcriptorum. # 353 <lb/>Quadratura circuli per numeros ſecundũ <lb/># Arab{es} falſa est. # 318 <lb/>Quadratura circuli per line{as} Campano <lb/># aſſcripta falſa est. # 318 <lb/>Quadratura circuli per numeros ex Al-<lb/># berto Durero falſa eſt. # 318 <lb/>Quadratura circuli per line{as}. # 317 <lb/>Quadratura circuli per Hyppocratem <lb/># Chium falſa eſt. # 318. & 319 <lb/>Quadraturam circuli eſſe poſſibilem. # 320 <lb/>Quadrilateri omnino irregularis area. # 171 <lb/>Quadrilatero æquale par allelogr ammum <lb/># facile conſtruere in dato angulo. # 338 <lb/>Quadrilaterorũ nõ rectangulorũ area. # 169 <lb/>Quantit{as} anguli, quẽ latera in ſtrumenti <lb/># partiũ cõtinẽt, quo pacto cognoſcatur. # 12 <lb/>Quarta & tertia proportionalis, quo pacto <lb/># ꝑ inſtrumentum partium reperiatur. # 13 <lb/>R <lb/>RAdicis cubicæ extrahendæregula pro-<lb/># pria. # 283 <lb/>Radicis cubicæ extractio. # 281 <lb/>Radicis quadratæ, & cubicæ extractio ex <lb/># data minutia. # 288 <lb/>Radicis quadratæ extractio. # 279 <lb/>Radicis ſurdeſolidæ extractio. # 281 <lb/>Radicem cui{us}lib{et} generis extrahere ex <lb/># dato numero. # 276 <lb/>Radicem cuiuſque generis ex data minu-<lb/># tia extrahere. # 287 <lb/>Radicem quadratam, & cubicam in nu-<lb/># meris non quadratis, & non cubis per <lb/># line{as} Geometricè inuenire. # 290 <lb/>Radicem veræ propinquam in numeris nõ <lb/># quadratis, non cubis, non Zenſizenſis, <lb/># non ſurdeſolidis, & c. inuenire. # 244 <lb/>Radicum infinitæ ſpeci{es}. # 276 <lb/>Rædix quadrata numeri fracti quo pacto <lb/># eruatur. # 166 <lb/>Radix quælib{et} extrahenda quot figur{as} <lb/># habere poſſit. # 279 <lb/>Recta linea in quotuis part{es} æqual{es} diui-<lb/># ſa quot decimæ, vel centeſimæ, & c. in <lb/># quauis particula vni{us} partis contine-<lb/># antur, per circinum deprehe@dere. # 42 <lb/>Recta ducta ex angulo acuto trianguli re-<lb/># ctãguli in oppoſitũ lat<emph style="sub">9</emph>, maior eſt ꝓportio <lb/># hui{us} lateris ad ei{us} ſegmentũ prope re-<lb/># ctum angulum, quam illi{us} anguli a- <pb file="439" n="439" rhead="INDEX."/> # cuti ad ei{us} partem dicto ſegmento la-<lb/># teris oppoſitam. # 295 <lb/>Recta linea diuiſa in quotuis part{es} æqua-<lb/># l{es}, quot eiuſmodi part{es} in quauis alia <lb/># recta contineantur, ope inſtrumẽti par-<lb/># tium, cognoſcere. # 6 <lb/>Recta data, quamuis minima, partem, vel <lb/># part{es} imperat{as} ex ea auferre. # 355 <lb/>Recta connectens duos angulos oppoſitos in <lb/># duob{us} triangulis æqualib{us} lat{us} com-<lb/># mune habentib{us}, & in diuer ſ{as} part{es} <lb/># vergentib{us}, à communi latere bifari-<lb/># am diuiditur. # 260 <lb/>Recta cuiuis circũferentiæ æqualis, quo pa-<lb/># cto facile reperiatur ex ꝓpria figura. # 327 <lb/>Rectæ duæ tangent{es} circulum, & in vno <lb/># puncto coeunt{es}, maior{es} ſunt omnib{us} <lb/># chordis interceptum arcum diuidenti-<lb/># b{us} in quotcunque part{es} æqual{es}. # 332 <lb/>Rectæ lineæ adiungere rectam, ita vt qua-<lb/># dratum toti{us} compoſitæ æquale ſit qua-<lb/># dato rectæ adiunctæ, vna cum quadra-<lb/># torectæ, quæ ex adiuncta, & proxi@o ſe-<lb/># gmento prioris lineæ conflatur. # 351 <lb/>Rectæ tr{es} circulum tangent{es}, & in duo-<lb/># b{us} punctis coeunt{es}, maior{es} ſuntomni-<lb/># b{us} chordis arc{us} duos interceptos in <lb/># part{es} æqual{es} ſecantib{us}. # 332 <lb/>Rectæ lineæ circumferentiam æqualem re-<lb/># perire. # 329 <lb/>Rectæ quamuis minimæ exhibere multi-<lb/># plicem quamcunque, etiamſi circino i-<lb/># pſa non accipiatur. # 355 <lb/>Rectæ lineæ ſub dimenſionem cadent{es} quæ <lb/># ſint. # 51 <lb/>Rectam lineã tranſuerſam in Horizonte, <lb/># cui{us} vtrum extremũ inſpici poteſt, <lb/># per quadrantem notam efficere. # 69 <lb/>Rectam linem tranſuerſam in Horizon-<lb/># te, cui{us} vtrum extremum videri po-<lb/># teſt, per quadratum metiri. # 127 <lb/>Rectam lineam ad cui{us} extrema accede-<lb/># re non liceat, dummodo ea appareant, <lb/># & ipſarecta linea producta ad ped{es} mẽ-<lb/># ſoris perting at, ex altitudine aliqua no-<lb/># ta, per quadratem metiri. # 69 <lb/>Rectam lineam in Horizonte per quadra-<lb/># tum m{et}iri, quando menſor in vno ei{us} <lb/># extremo exiſtens alterum extremũ vi@ <lb/># dere non poteſt neque altitudo in prom-<lb/># ptu eſt, ſed ſolum ad dextram, vel ſi-<lb/># niſtram per lineam perpendicularem <lb/># recedere poteſt ad locum, è quo alterum <lb/># extremum appareat. # 121 <lb/>Rectam lineam, quando menſor in vno e-<lb/># i{us}extremo, vel in aliquo altitudine no-<lb/># ta ad planum, in quo eſt recta, perpen-<lb/># diculari exiſtens alterum extremũ vi-<lb/># dere poteſt, per quadrantem metiri. # 68 <lb/>Rectam lineam è directo menſoris poſitam, <lb/># cui{us} vtrum extremum, vel alterum <lb/># non appareat, niſi ad dextram, vel ſini-<lb/># ſtram menſor accedat, per quadrantem <lb/># comprehendere. # 71 <lb/>Rectam lineam in Horizonte è directo mẽ-<lb/># ſoris iacentem, per quadratum cogno-<lb/># ſcere, ad cui{us} extremane accedere li-<lb/># ceat, neque è loco menſoris eam dim{et}i-<lb/># ri: dummodo ad dextram, vel ſin@ſtrã <lb/># per lineam perpendicularem ad locum <lb/># aliquemire poſſit menſor, ex quo vtrũ-<lb/># que extremum appareat. # 122 <lb/>Rectam lineam in Horizonte inter turrim <lb/># aliquam, & aliud quodpiam ſignum, <lb/># ex turri per du{as} ſtation{es}in faſtigio fa-<lb/># ct{as}, vel in duab{us} feneſtris, quarum <lb/># vna ſit ad perpendiculũ ſub alia, quan-<lb/># do ſpatium inter ill{as} feneſtr{as} notum <lb/># eſt, etiamſi toti{us} turris altitudo ſit igno <lb/># ta, per quadr antem dimetiri. Atque <lb/># hinc obiter altitudinem turris patefa-<lb/># cere. # 70 <lb/>Rectã arcui quadrãtisæ qualẽreperire. # 325 <lb/>Rectam lineam datã per inſtrum entũ par-<lb/># tium diuidere, vt alia recta diuiſa eſt. # 11 <lb/>Rectanguli trianguli area, ex latere, quod <lb/># recto angulo opponitur, & vno angulo <lb/># acuto. # 167 <lb/>Rectanguli trianguli area, ex vno latere <lb/># circa angulum rectum, & vno angulo <lb/># acuto. # 168 <lb/>Rectangulorum duorum triangulorum ſi-<lb/># m@lium propriet{as} qu@dam. # 398 <lb/>Rectanguli trianguli area, ex vno latere <pb file="440" n="440" rhead="INDEX."/> # @irca angulum rectum & latere, quod <lb/># recto angulo opponitur. # 168 <lb/>Rectanguli trianguli area. # 165 <lb/>Rectangulo dato ſupra datã rectã æquale <lb/># rectangulum facile conſtruere. # 162 <lb/>Rectangulum, cui{us} duo exceſſ{us} dantur, <lb/># quib{us} diameter vtrumque lat{us} ſupe-<lb/># rat, conſtituere. # 350 <lb/>Rectangulum dato rectilineo æquale faci-<lb/># lè conſtruere. # 339 <lb/>Rectangulum, in quo exceſſ{us} diametriſu-<lb/># pra mai{us} lat{us}, & maioris lateris ſu-<lb/># pramin{us} datur, conſtituere. # 351 <lb/>Rectangulum ſub differentia exceſſuum, <lb/># quib{us} diameter alicui{us} rectanguli v-<lb/># trumque lat{us} ſuper at, & minore exceſ-<lb/># ſu bis ſumptum, vnà cum quadrato mi-<lb/># noris exceſſ{us}bis ſurn<unsure/>pto æquale eſt qua <lb/># drato rectæ, qua min{us} lat{us} minorem <lb/># exceſſum ſuperat. # 349 <lb/>Rectangulum ſub ſegmentis diametri ali-<lb/># cui{us} rectanguli (ductis per idem pun-<lb/># ctum diametri parallelis) comprehen-<lb/># ſum, æquale eſt duob{us} rectangulis ſub <lb/># ſegmentis duorum laterum inæquali-<lb/># um comprehenſis. # 357 <lb/>Rectangulum ſub diametro, & circumfe-<lb/># rentia maximi circuli in ſphæra, qua-<lb/># druplum eſt circuli maximi, & ſuperfi-<lb/># ciei conuexæ eiuſdẽſphæræ æquale. # 219 <lb/>Rectangulorum area. # 157 <lb/>Rectangulorum triangulorum rectilineo-<lb/># rum problemata. # 44 <lb/>Rectarum duarum proportionem haben-<lb/># tiũ, quãlat{us} quadratricis ad basẽ, ma-<lb/># ior æqualis eſt quadranti circuli, cui{us} <lb/># ſemidiameter eſt minor recta. # 326 <lb/>Rectilineã figurã planã, vel circulũ in da-<lb/># ta proportione augere, vel minuere. # 272 <lb/>Rectilineum angulum in tr{es} æqual{es} an-<lb/># gulos diuidere. # 356 <lb/>Rectis trib{us} datis in vno plano non par al-<lb/># lelis, rectam ducere, {et}iam per datum <lb/># interdum punctum in media, ita vt ei{us} <lb/># ſegmenta inter mediã, & extrem{as} ſint <lb/># æqual{es}, vel datã habeãt ꝓportionẽ. # 354 <lb/>Re@tilineis duob{us} inæqualib{us} datis, ex <lb/># maiore ꝑ lineam vni lateri parallelam <lb/># detrahere rectilineũ minori æquale. # 243 <lb/>Rectili<unsure/>neo cuilibet æquale rectangulum fa <lb/># cilè conſtruere. # 339 <lb/>Rectilineo dato æquale quadrilaterũ inter <lb/># du{as} rect{as} ſuper datam rectam per li-<lb/># neam parallelam conſtituere. # 239 <lb/>Rectilineo in triangulo reſoluto ex vno ali-<lb/># quo puncto, rect{as} ipſis triangulis or di-<lb/># ne proportional{es} inuenire. # 246 <lb/>Rectilineo dato parallelogrãmũ rectangu-<lb/># lũ æquale, & Iſoperimetrũ cõſtituere. # 216 <lb/>Rectilineum datum per rectam à quouis <lb/># angulo, vel puncto lateris, in datam ꝓ-<lb/># portionem ſecare. # 248 <lb/>Rectilineum datum ex angulo, vel puncto <lb/># dato in latere, in quotuis part{es} æqual{es} <lb/># diſtribuere. # 252 <lb/>Rectilineũ datum in quotuis part{es} æqua-<lb/># l{es} diſtribuere. # 260 <lb/>Rectilineũ datũ rectã datæ rectæ paralle-<lb/># lã in datam proportionem diu dere. # 253 <lb/>Rectilineorum triangulorum obliquan-<lb/># gulorum problemata. # 47 <lb/>Rectilineorum triangulorum rectangulo-<lb/># rum problemata. # 44 <lb/>Regionis, aut vrbis, vel campi ſitum in pla-<lb/># no deſcribere. # 147 <lb/>Rectis duab{us} datis, quarum maior dia-<lb/># metrum quadrati minoris non ſuperet: <lb/># maiorem it a ſecare, vt partium quadræ <lb/># ta ſimul ſumpta quadrato minoris li-<lb/># neæ ſint æqualia. # 352 <lb/>Reflexionis angul{us} cur angulo incidentiæ <lb/># ſit æqualis. # 341 <lb/>Regula communis menſorum in area cu-<lb/># iuſuis figuræ, vel agri inueſtiganda. # 173 <lb/>Regula propria extractionis radicis cubi-<lb/># cæ. # 283 <lb/>Regula vnica adomn{es} rect{as} dimetiẽd{as}, <lb/># quando rerum extrema videntur. # 152 <lb/>Regulæ conſtructio loco fili cum perpendi-<lb/># culo. # 17 <lb/>Regulare corp{us} quodlibet dato cubo æ-<lb/># quale conſtituere. # 37@ <lb/>Regulari corpori ſphæram æqualem exhi-<lb/># ber@. # 37@ <pb file="441" n="441" rhead="INDEX."/> Regularis figuræ ærea, cui rectangulo æ-<lb/># qualis ſit. # 293 <lb/>Regularis figuræ area, cui triangulo rectã-<lb/># gulo ſit æqualis. # 294 <lb/>Regularis figura circulo circumſcripta <lb/># maiorem ambitum habet, quam circu-<lb/># l{us}. # 330. & 331 <lb/>Regulærib{us} figuris rectilineis circul{us}, cui <lb/># Iſoperimetræ ſunt@ maior eſt. # 306 <lb/>Regularium figurarum Iſoperimetrarum <lb/># maior eſt illa, quæplur{es} continet angu-<lb/># los, pluraue latera. # 296 <lb/>Regularium figurarum areæ à triangulo <lb/># vſque ad Dodecagonism, quando lat{us} <lb/># eſt vnit{as}. # 180 <lb/>Regularium figurarum area. # 175 <lb/>Regularium quinque corporum area quæ. <lb/># 210. & 214. <lb/>Regularium quinque corporum ſuperfici{es} <lb/># conuexa # 214 <lb/>Rhomboidis, ac Rhombi area. # 170 <lb/>Rhombi, ac Rhomboidis area. # 170 <lb/>Rhombum, ac Rhomboid{es}, ex exceſſu dia-<lb/># metri ſupra lat{us}, &c. deſcribere. # 346 <lb/>S <lb/>SAcc{us} tritici, quo pacto menſuretur. # 209 <lb/>Saxiregularis ſolidit{as}. # 209 <lb/>Scala ali<unsure/>imetra quid. # 85 <lb/>Secant{es} ac ſin{us} quo pacto in inſtrumento <lb/># partium capiantur. # 8 <lb/>Sectoris circuli area. # 199 <lb/>Sectoris ſphæræ ſolidit{as}. # 230 <lb/>Segmenta lateris tria<unsure/>nguli obliquanguli à <lb/># perpendiculari facta, ex datis trib{us} la-<lb/># terib{us} cognoſcere. # 46 <lb/>Semicirculo, quadranti, octauæ parti cir-<lb/># culi, &c. rectangulum conſtituere Iſo-<lb/># perimetrum, & æ<unsure/>quale. # 214 <lb/>Semidiametri circuli inuentio ex latere fi-<lb/># guræ regularis inſcriptæ. # 178 <lb/>Similem figuram plurib{us} figuris ſimilib<emph style="sub">9</emph>, <lb/># quarum latera homologa data ſint, æ-<lb/># qualem: ct circulum plurib{us} circu-<lb/># lis, quorum diametri, circumferentiæ-<lb/># ue datæ ſint, æqualem deſcribere. # 202 <lb/>Similium duorum triangulorum rec@an-<lb/># gulorum propriet {as} quædam. # 298 <lb/>Similium duarũ figurarũ, aut circulorũ <lb/># ꝓportio, ex datis duob. laterib. homolo-<lb/># gis, vel diametris, circũferentiisue. # 201 <lb/>Simil{es} pa<gap/>{es} aliquotæ pluriũ magnitudi-<lb/># num eandem habent proportionem. # 218 <lb/>Sin{us} tot{us} quando tam paru{us} eſt, vt in <lb/># inſtrumentum partium transferri ne-<lb/># queat, quid agendum. # 12 <lb/>Sinu toto poſito 10000. quo pacto in inſtru-<lb/># mento partium Tangent{es} ſumantur. # 8 <lb/>Sinu toto poſito 100. quo pacto Tangent{es} ꝑ <lb/># inſtrumentum partium capiantur. # 6 <lb/>Sin{us} ac ſecant{es} quo pacto in inſtrument@ <lb/># partium capiantur. # 7 <lb/>Solidã figurã quamcun ex iis, de quib{us} <lb/># Eucl. in Stereometria agit ſecundũ pro-<lb/># portionẽ datã augere, vel minuere. # 273 <lb/>Solida, vel corpora præcipua, quorum areæ <lb/># inueſtigantur, quæ. # 204 <lb/>Solida, ſuperfici{es}, ac lineæ pen{es} quid men-<lb/># ſurentur. # 157 <lb/>Solidit{as} cuiuſuis portionis ſphæræ. # 231 <lb/>Solidit{as} ſphæræ. # 223 <lb/>Solidit{as} ſphæræ vera minor, ex circumfe-<lb/># rentia maximi circuli. # 228 <lb/>Solidit{as} ſphæræ vera maior, ex diametro <lb/># circuli maximi. # 228 <lb/>Solidit{as} ſphæræ vera minor, ex diametro <lb/># maximi circuli. # 228 <lb/>Solidit{as} ſphæræ vera maior, ex circumfe-<lb/># rentia circuli maximi. # 228 <lb/>Solidit{as} ſphæroidis. # 232 <lb/>Solidit{as}, vel area hemiſphærij. # 230 <lb/>Solidit{as} portionum ſphæroidis. # 232 <lb/>Solidit{as} ſectoris ſphæræ. # 230 <lb/>Solidit{as} muri cuiuſque. # 209 <lb/>Solidit{as} fruſtiregularis marmoris. # 209 <lb/>Solidit{as} fruſti ſphæræ. # 231 <lb/>Solidit{as} Conoidis Hyperbolici. # 233 <lb/>Solidit{as} Conoidis parabolici. # 232 <lb/>Solidit{as} vaſis excauati. # 209 <lb/>Solidum planis ſuperficieb{us} contentum, <lb/># & circa ſphæram circumſcriptibile, cui <lb/># ſolido rectangulo ſit æquale. # 307 <lb/>Solidorum quinque regularium area quæ. <lb/># 210. & 214. <lb/>Solidorum quinque regularium ſuper <pb file="442" n="442" rhead="INDEX."/> # fici{es} conuexa. # 214 <lb/>Solidorum omnino irregulariũ area. # 334 <lb/>Solidum min{us} ex maiori detrahere, reſi-<lb/># duum in cubum transformare. # 373 <lb/>Solidum rectangulum ſupra baſem quot-<lb/># cunque angulorum, datæ ſphæræ æqua-<lb/># lem conſtruere. # 371 <lb/>Solis, velſtellæ cuiuſuis altitudinem per <lb/># quadratum obſeruare. # 88 <lb/>Spatium terræinæquale pro ducẽdis aquis <lb/># librare: aut etiam, ſi lub{et}, Horizonti <lb/># æquidiſtans efficere. # 153 <lb/>Spatiũ inter duo pũcta in quolib{et} plano e-<lb/># leuato, fiue illud ad Horizõ. ſit rectũ, ſi-<lb/># ue inclinatũ per quadrantẽ m{et}iri. # 67 <lb/>Speci{es} radicum infinitæ. # 276 <lb/>Speculi plani beneficio altitudinem monti <lb/># impoſitam, ſi modo altitudinis baſis poſ-<lb/># ſit conſpici: Velportionem ſuperiorem <lb/># alicui{us} turris, metiri. # 147 <lb/>Speculi plani beneficio altitudinem, ad cu-<lb/># i{us} baſem pateat acceſſ{us}, vnà cum di-<lb/># ſtantia<unsure/> ſpeculi à cacumine altitudinis, <lb/># deprehendere. # 144 <lb/>Speculi plani beneficio altitudinem in ac-<lb/># ceſſibilem, vna cum ſpeculi diſt antia tã <lb/># à baſe, {et}iam nonviſa, quam a cacumi-<lb/># altitudinis, cognoſcere. # 145 <lb/>Sphæra ad cubum circumferentiæ maxi-<lb/># mi circuli, maiorem proportionem ha-<lb/># bet, quam 49. ad 2904. minorem vero, <lb/># quam 5041. ad 298374. # 221 <lb/>Sphæra quolib{et} cono, & cylindro ſibi Iſo-<lb/># perimetro maior est. # 313 <lb/>Sphæraæqualis eſt ſolido rectangulo com-<lb/># prehenſo ſub ſemidiametro, & terti@ <lb/># parte ſuperfictei conuexæ. # 309 <lb/>Sphæra maior eſt quouis corpore regulari <lb/># ſibi Iſoperimetro. # 311 <lb/>Sphæra omnib{us} corporib{us} ſibi Iſoperime-<lb/># tris, quæ planis ſuperficieb{us} contineæn-<lb/># tur, circa ali{as} ſphær{as} circumſcripti@ <lb/># bilia ſint, maior eſt. # 311 <lb/>Sphæra omnib{us} corporib{us} ſibi Iſoperime-<lb/># tris, & circa ali{as} ſphær{as} circumſcri-<lb/># ptibilib{us}, quæ ſuperficieb{us} conicis cõ-<lb/># tineantur, maior est. # 311 <lb/>Sphæræ area, vel ſolidit{as}, ex diametro mæ<unsure/> <lb/># ximi circuli. # 228 <lb/>Sphæræ datæ conum efficere æqualem. # 371 <lb/>Sphæræ datæ cylindrũ æqualẽ exhibere. # 371 <lb/>Sphæræ datæ cubum æqualem: ct dato cu-<lb/># bo æqualem ſphæram efficere. # 370 <lb/>Sphæræ datæ piramidem conſtituere æqua-<lb/># lem. # 371 <lb/>Sphæræ datæ ſolidum rectangulum æquale <lb/># ſupra baſem quotcunque angulorum <lb/># conſtituere. # 371 <lb/>Sphæræ ſuperfici{es} vera minor, ex circum-<lb/># ferentia maximi circuli. # 227 <lb/>Sphæræ ſuperfici{es} vera minor, ex diame-<lb/># tro circuli maximi. # 227 <lb/>Sphæræ ſuperfici{es} vera maior, ex diame-<lb/># tro circuli maximi. # 227 <lb/>Sphæræ area, ſeu ſolidit{as} vera minor, ex <lb/># circumferentia circuli maximi. # 228 <lb/>Sphæræ area, ſeu ſolidit{as} maior, quã vera, <lb/># ex circumferentia maximi circuli. # 228 <lb/>Sphæræ area, ſiue ſolidit{as}, ex diametro <lb/># maximi circuli. # 228 <lb/>Sphæræ ſolidit{as}. # 223 <lb/>Sphæra ad cubum diametri ſphæræ maio-<lb/># rem proportionem habet, quam 223. ad <lb/># 426. minorẽ verò, quàm 11. ad 21. # 222 <lb/>Sphæræ ſuperfici{es} conuexa cui rectangulo <lb/># ſit æqualis. # 219 <lb/>Sphæræ ſuperfici{es} conuexa ad quadratum <lb/># circumferentiæ circuli maximi, maio-<lb/># rem proportionem habet, quam 7. ad 22 <lb/># minorem verò, quàm 71. ad 223. # 221 <lb/>Sphæræ ſegmentorum area. # 229 <lb/>Sphæræ ſuperfici{es} ad quadratũ circũferẽ-<lb/># tie<unsure/> circuli maximi eſt, vt diameter ad <lb/># circumferentiam circuli maximi. # 219 <lb/>Sphæræ ſuperfici{es} vera maior, ex circum-<lb/># ferentia circuli maximi. # 226 <lb/>Sphæræ portionis ſuperfici{es} conuexa. # 229 <lb/>Sphæræ portionis ſolidit{as}. # 231 <lb/>Sphæræ area, eiuſdemque ſuperfici{es} con-<lb/># uexa. # 218. & 223 <lb/>Sphæræ ſuperfici{es} ad quadratum diame-<lb/># tri ſphæræ, vel circuli maximi, maiorẽ <lb/># proportionem habet quam 223. ad 71. <lb/># minorem vero quam 22. ad 7. # 221 <pb file="443" n="443" rhead="INDEX."/> Sphæram duab{us}, aut plurib{us} ſphæris æ-<lb/># qualem deſcribere. # 373 <lb/>Sphæram dato corpori regulari æqualem <lb/># conſtituere. # 371 <lb/>Sphærã dato priſmati æqualẽ conſtruere. # 371 <lb/>Sphæram minorẽ ex maiori detrahere, reſi-<lb/># duo æqualem ſphæram exhibere. # 373 <lb/>Spheroidis portionum ſolidit{as}. # 232 <lb/>Sphæroidis area, vel ſolidit{as}. # 232 <lb/>Sphæram, vel figuram ſolidam in data pro-<lb/># portione augere, vel minuere. # 273 <lb/>Squadra Zoppa apud Italos quid, eiuſque <lb/># vſ{us}. # 150 <lb/>Stabilis quadrans quid. # 17 <lb/>Stationum differentia quid. # 52 <lb/>Stellæ cuiuſuis, vel Solis altitudinem per <lb/># quadratum obſeruare. # 87 <lb/>Superfici{es} conuexa quinque corporum re-<lb/># gularium. # 214 <lb/>Superficiei conicæ ꝓportio ad ſuã baſem. # 235 <lb/>Superfici{es} conuexa ſphæræ, eiuſdemque <lb/># area. # 218. & 223 <lb/>Superfici{es} conuexa hemiſphærii. # 229 <lb/>Superfici{es} conuexa ſphæræ ad quadratum <lb/># circumferentiæ circuli maximi, maiorẽ <lb/># proportionem habet, quam 7. ad 22. mi-<lb/># norem verò, quam 71. ad 223. # 221 <lb/>Superfici{es}, lineæ, ac ſolida, pen{es} quid men-<lb/># ſurentur. # 157 <lb/>Superfici{es} conuexa ſphæræ cui rectangulo <lb/># ſit æqualis. # 219 <lb/>Superfici{es} ſphæræ vera maior, ex circum-<lb/># ferentia circuli maximi. # 226 <lb/>Superfici{es} ſphæræ vera maior, ex diam{et}ro <lb/># maximi circuli. # 227 <lb/>Superfici{es} ſphæræ vera minor, ex circum-<lb/># ferentia circuli maximi. # 227 <lb/>Superfici{es} ſphæræ ad quadratũ circumfe-<lb/># rentiæ circuli maximi eſt, vt diam{et}er <lb/># ad circumferentiã circuli maximi. # 219 <lb/>Superfici{es} conuexa portionis ſphæræ. # 229 <lb/>Superfici{es} ſphæræ ad quadratum diam{et}ri <lb/># circuli maximi, maiorem proportionem <lb/># hab{et}, quam 223. ad 71. minorem verò <lb/># quam 22. ad 7. # 221 <lb/>Superfici{es} co@uexa coni, & cylindri recti. <lb/># 235 <lb/>Superfici{es} cylindrica, demptis baſib{us}. # 235 <lb/>Superfici{es} fruſti coni, demptis baſib{us}. # 235 <lb/>Surdeſolidæ ræ<unsure/>dicis extractio. # 281 <lb/>T. <lb/>TAbula continens latera figurarum re-<lb/># gularium à triangulo vſque ad figu-<lb/># rã 80. laterũ, poſita diam{et}ro 20000000. <lb/># vel ſinu toto 10000000. # 177 <lb/>Tabula mirifica ad depromendos numeros <lb/># pro ſingulis radicum ſpecieb{us}. # 277 <lb/>Tabulæ conſtructio & vſ{us} pro minutis, & <lb/># ſec. ex quadr ante cognoſcendis. # 20 <lb/>Tabula gnomonica. # 91 <lb/>Tabulæ gnomonicæ facillima conſtructio, <lb/># eiuſ{q́ue} vſ{us}. # 89 <lb/>Tabulæ pro minutis ad plur{es} quadrant{es} <lb/># extenſio ſine regula aurea. # 20 <lb/>Tabula pro minutis, & ſecundis ex qua-<lb/># drante inueſtigandis. # 22 <lb/>Tabula quadratorum, & cuborum vſque <lb/># ad radicem 1000. # 378 <lb/>Tabella pro min. & ſec. cognoſcendis ex <lb/># quadrante conſtructo. # 19 <lb/>Tangens quando ſuperat ſinum totũ, quid <lb/># agendum, vt per vnicam tranſlationem <lb/># per inſtrumentum partium punctum <lb/># quæſitum reperiatur. # 12 <lb/>Tangent{es}, poſito ſinu toto 10000. quo pacto <lb/># in inſtrumento partium ſumantur. # 8 <lb/>Tangent{es} quo modo inueniantur ope in-<lb/># ſtrumenti partiũ, poſito ſinu toto 1000. # 7 <lb/>Tangent{es} tr{es} in duob{us} punctis coeunt{es}, <lb/># maior{es} ſunt omnib<emph style="sub">9</emph> chordis duos arcus <lb/># interceptos in part{es} æqual{es} ſecantib. # 332 <lb/>Tangẽt{es} duæ in vnopuncto coeũt{es}, maio-<lb/># r{es} ſunt omnib{us} chordis arcũ interceptũ <lb/># in quotcũ part{es} æqual{es} diuidẽtib. # 332 <lb/>Tangent{es} quo modo accipiãtur reſpectu ſi-<lb/># n{us} toti{us} 100. ope inſtrumenti partium. # 6 <lb/>Tangent{es} quo pacto ſumantur, quando in-<lb/># ſtrumentũ partium cõtin{et} 1000. part{es}. # 6 <lb/>Tangente aliqua inuenta in latere inſtru-<lb/># menti partium, quo pacto eadem repe-<lb/># riatur reſpectu dati ſin{us} toti{us}. # 8 <lb/>Tangent{es} perexiguæ quo pacto in inſt<unsure/>ru-<lb/># mento partium ſumantur. # 8 <lb/>Terræ ambitũ ex edito aliquo mõte metiri. <lb/># 366<unsure/> <pb file="444" n="444" rhead="INDEX."/> Tertia & quarta proportionalis, quo pacto <lb/># per inſtrumentum partium reperiatur. # 12 <lb/>Trabis longitudinem ad Horizontẽ incli-<lb/># natæ, cui{us} portio ſuperior tantum con-<lb/># ſpiciatur, vnà cum angulo inclinatio-<lb/># nis, diſt antia baſis à menſore, & altitu-<lb/># dine faſtigii ſupra Horizontem, per qua-<lb/># dratum m{et}iri. # 151 <lb/>Trapezii habentis duo latera parallela, <lb/># area. # 170 <lb/>Trapezii irregularis facilis dimenſio. # 175 <lb/>Triangis<unsure/>la duo Iſoſcelia ſimilia baſium in-<lb/># æqualium, ſimul maiora ſunt duob{us} <lb/># Iſoſcelib{us} ſimul ſuper eaſdem baſ{es}, quæ <lb/># quidem inter ſe ſint diſſimilia, priorib{us} <lb/># verò Iſoperimetra. habeant quatuor <lb/># latera inter ſe æqualia. # 300 <lb/>Trianguli Iſoſcelis area. # 165 <lb/>Trianguli obliquanguli area. # 168 <lb/>Trianguli rectanguli area, ex vno latere <lb/># circa angulum rectum, & latere, quod <lb/># recto angulo opponitur. # 167 <lb/>Trianguli pulchra propriet{as}, ſi in eo du-<lb/># catur vni lateri parallela, &c. # 261 <lb/>Triangulis duob{us} Iſoſcelib{us} datis, quo-<lb/># rum baſ{es} ſint inæqual{es}, & duo latera <lb/># vni{us} duob{us} alteri{us} æqualia: ſuper eiſ-<lb/># dem baſib{us} duo triangula Iſoſcelia ſi-<lb/># milia, & priorib{us} ſimul ſumptis Iſope-<lb/># rimetra conſtituere. # 299 <lb/>Trianguli æquilateri area. # 166 <lb/>Trianguli cui{us}libet area cui rectangulo <lb/># ſit æqualis. # 292 <lb/>Trianguli rectanguli area. # 165 <lb/>Trianguli rectãguli area, ex vno latere cir-<lb/># ca angulũ rectũ, & vno angulo acuto. # 169 <lb/>Trianguli rectãguli area, ex latere, ꝙ recto <lb/># angulo opponitur, & vno angulo acuto. # 167 <lb/>Trianguli inſignis propriet{as}, ſi in eo à duo-<lb/># b{us} angulis ad media puncta oppoſitorum <lb/># laterum rectæ ducantur, &c. # 252 <lb/>Triangulo duorum laterum inæqualium <lb/># ſupra tertium lat{us} triangulum conſti-<lb/># tuere priori Iſoperimetrum duorum æ-<lb/># qualium laterum. # 297 <lb/>Triangulo parallelogr ammum æquale, & <lb/># Iſoperimetrum conſtituere. # 214 <lb/>Triangulorum rectilineorum rectangulo-<lb/># rum problemata. # 44 <lb/>Triangulorum duorum rectangulorum ſi-<lb/># milium propriet{as} quædam. # 298 <lb/>Triangulorum Iſoperimetrorum eandem <lb/># habentium baſem, mai{us} erit illud, quod <lb/># duo latera habet æqualia. # 297 <lb/>Triangulorum rectilineorum obliquangu-<lb/># lorum problemata. # 46 <lb/>Triangulum datũ ex dato puncto in latere <lb/># in quotlibet part{es} æqual{es} diuidere. # 262 <lb/>Triangulum datum per line{as} vni lateri <lb/># parallel{as} in quotlibet part{es} æqual{es} di-<lb/># ſtribuere. # 263 <lb/>Triangulum datum per rectum ex puncto <lb/># extra triangulum dato in du{as} part{es} æ-<lb/># qual{es} partiri. # 264 <lb/>Triangulum totum ad triangulum abſciſ-<lb/># ſum per rectam, eſt, vt rectangulum ſub <lb/># duob{us} laterib{us} ſectis ad rectangulum <lb/># ſub duob{us} laterib{us} trianguli abſcißi <lb/># comprehenſum. # 262 <lb/>Tritici aceru{us}, quo pacto menſuretur. # 209 <lb/>Tritici ſacc{us}, quo pacto menſuretur. # 209 <lb/>Turris, aut montis altitudinẽ, ex ei{us} ſum-<lb/># mitate per quadratũ dimetiri, quando in <lb/># plano ſummitatis Horizonti æquidiſt ante <lb/># duæ ſtation{es} fieri poſſunt, & ſignum ali-<lb/># quod in Horizonte cernitur. # 114 <lb/>Turris, aut alteri{us} rei altitudinem per ba-<lb/># culum indagare. # 137 <lb/>Turris aut alteri{us} rei altitudinem, per <lb/># Normam inueſtigare. # 139 <lb/>Turris, vel montis altitudinẽ ex ei{us} ſum-<lb/># mitate per du{as} ſtation{es} in haſta aliqua <lb/># erecta fact{as} inueſtigare per quadratũ, <lb/># quando ſignum aliquod in Horizonte <lb/># videri poteſt. # 116 <lb/>Turris altitudinem ex ei{us} vertice per vn@-<lb/># cam ſt ationem per quadrantem metiri, <lb/># ſi diſtantia ſigni in Horizonte viſi vſque <lb/># ad baſem turris nota ſit. # 64 <lb/>Turris, vel montis altitudinẽ, ex ei{us} ſum-<lb/># mitate per vnicam ſtationem, ope qua-<lb/># drati ſtabilis m{et}iri, vnà cum diſtantia, <lb/># ſigni viſi in Horizonte vſ ad turrem <lb/># vel @@ontis perpendiculum. # 119 <pb file="445" n="445" rhead="INDEX."/> Turris, aut montis altitudinem ex ei{us} ver-<lb/># tice per quadrantem m{et}iri, ſi in plano, <lb/># cui inſiſtit, ſpatiũ aliquod è directo men-<lb/># ſoris poſitum notum ſit. # 64 <lb/>Turris, vel mõtis altitudinẽ ex ei{us} faſtigio, <lb/># quando è directo menſoris interuallum <lb/># aliquod inter duo ſigna, vel {et}iam inter <lb/># <gap/>m quodpiam, acturrim cognitum <lb/># <gap/>er quadratum coniicere. # 123 <lb/># <gap/>ut montis altitudinem ex ei{us} ver-<lb/># <gap/>er du{as} ſtation{es} in eiuſdẽ ſummi-<lb/># <gap/>act{as}, è quib{us} ſignum aliquod in <lb/># <gap/>Zonte appareat, per quadrantem <lb/># aim@tiri. At hinc ipſam quo diſtan-<lb/># tiam à turris baſe, vel perpẽdiculo mon-<lb/># tis ad ſignum illud inueſtigare. # 79 <lb/>Turris, aut montis altitudine ex ei{us} verti-<lb/># ce per du{as} ſtation{es} in haſta aliqua ere-<lb/># cta, velin duab{us} feneſtris turris, quarũ <lb/># vna ſit ſupra aliã, fact{as}, è quib{us} ſignũ <lb/># aliquod in Horizonte videri poſſit, per <lb/># quadrantẽ m{et}iri. At hinc diſtantiam <lb/># quo{q́ue} à perpendiculo turris, vel montis, <lb/># vſ ad ſignum viſum cognoſcere. # 62 <lb/>Turrium duarũ ſummitatib{us} viſis, etiãſi <lb/># baſ{es} propter ædificia interiecta occul-<lb/># tentur, diſtanti<unsure/>ã tam inter earum baſ{es}, <lb/># quàm inter earundem faſtigia, vnà cum <lb/># ipſarum al@itudinib{us}, ac diſtantiis à <lb/># menſore, per quadratum coniicere. # 152 <lb/>V. <lb/>Vallis profunditatem, eiuſdemque de-<lb/># ſcenſum obliquum, ſi non ſit valdè <lb/># inæqualis, eiuſ termin{us}, vel aliquod <lb/># in valle ſignum conſpici poſſit, per qua-<lb/># drantem ſcrutari. # 82 <lb/>Vallis profunditatẽ, eiuſdem deſcenſum <lb/># obliquũ, ſi nõ ſit valdè inæqualis, & ei{us} <lb/># termin{us}, vel aliquod in ea ſignũ conſpi-<lb/># ci poſſit, per quadratum cognoſcere. # 136 <lb/>Varia inſtrumenta menſurandi. # 51 <lb/>Vaſis excauati ſolidit{as}. # 209 <lb/>Vaſis excauati capaeit{as}. # 209 <lb/>Vmbrarecta, ac verſa quo pacto in qua-<lb/># drato Geom{et}rico cognoſcenda ſit. # 86 <lb/>Vmbræ longitudinem ab altitudine, Sole <lb/># lucente, proiectæ, quando altitudo est <lb/># cognita, ope quadrati adipiſci. # 141 <lb/>Vmbra recta ac verſa in quadrato quæ & <lb/># in quot part{es} à Geometris vtraque ſe-<lb/># cetur. # 85 <lb/>Vmbrarecta ac verſa in quet partes in hoc <lb/># opere diuidatur. # 85 <lb/>Vmbræ rectæ ad verſam reductio, & con-<lb/># tra. # 87 <lb/>Vrbis, vel campi, aut regionis ſitum in pla-<lb/># no deſcribere. # 147 <lb/>Vſ{us} & conſtructio tabulæ pro minutis & <lb/># ſec. cognoſcendis ex quadrante. # 19 <lb/>Vſ{us} quadrantis conſtructi in minutis, & <lb/># ſecundis exquirendis. # 17 <lb/>Vſ{us} tabulæ quadr atorum & cuborum in <lb/># extrahendis radicib{us} quadratis & cu-<lb/># bi<unsure/>cis. # 391 <lb/></note> </div> <div xml:id="echoid-div1139" type="section" level="1" n="420"> <head xml:id="echoid-head447" xml:space="preserve">ERRATA.</head> <p> <s xml:id="echoid-s18272" xml:space="preserve">In tabula quadratorum & </s> <s xml:id="echoid-s18273" xml:space="preserve">cuborum. </s> <s xml:id="echoid-s18274" xml:space="preserve">pag. </s> <s xml:id="echoid-s18275" xml:space="preserve">378. </s> <s xml:id="echoid-s18276" xml:space="preserve">in quadrato Radicis 117. </s> <s xml:id="echoid-s18277" xml:space="preserve">loco quinto à dextera-<lb/>pro 5. </s> <s xml:id="echoid-s18278" xml:space="preserve">reſtitue 9. </s> <s xml:id="echoid-s18279" xml:space="preserve">Pag. </s> <s xml:id="echoid-s18280" xml:space="preserve">381. </s> <s xml:id="echoid-s18281" xml:space="preserve">in cubo Rad. </s> <s xml:id="echoid-s18282" xml:space="preserve">364. </s> <s xml:id="echoid-s18283" xml:space="preserve">loc. </s> <s xml:id="echoid-s18284" xml:space="preserve">quarto pro 3. </s> <s xml:id="echoid-s18285" xml:space="preserve">r. </s> <s xml:id="echoid-s18286" xml:space="preserve">2. </s> <s xml:id="echoid-s18287" xml:space="preserve">P. </s> <s xml:id="echoid-s18288" xml:space="preserve">381. </s> <s xml:id="echoid-s18289" xml:space="preserve">in cubo Rad. </s> <s xml:id="echoid-s18290" xml:space="preserve">432. </s> <s xml:id="echoid-s18291" xml:space="preserve">loco ſexto <lb/>pro 6. </s> <s xml:id="echoid-s18292" xml:space="preserve">r. </s> <s xml:id="echoid-s18293" xml:space="preserve">5. </s> <s xml:id="echoid-s18294" xml:space="preserve">P. </s> <s xml:id="echoid-s18295" xml:space="preserve">382. </s> <s xml:id="echoid-s18296" xml:space="preserve">in cubo Rad. </s> <s xml:id="echoid-s18297" xml:space="preserve">573. </s> <s xml:id="echoid-s18298" xml:space="preserve">loco vlt. </s> <s xml:id="echoid-s18299" xml:space="preserve">pro 8.</s> <s xml:id="echoid-s18300" xml:space="preserve">r.</s> <s xml:id="echoid-s18301" xml:space="preserve">7. </s> <s xml:id="echoid-s18302" xml:space="preserve">P 383. </s> <s xml:id="echoid-s18303" xml:space="preserve">in quadrato Rad 714. </s> <s xml:id="echoid-s18304" xml:space="preserve">loco tertio pro 8. <lb/></s> <s xml:id="echoid-s18305" xml:space="preserve">r. </s> <s xml:id="echoid-s18306" xml:space="preserve">9. </s> <s xml:id="echoid-s18307" xml:space="preserve">P 384. </s> <s xml:id="echoid-s18308" xml:space="preserve">in quadrato Rad 735. </s> <s xml:id="echoid-s18309" xml:space="preserve">loco primo pro 9 r. </s> <s xml:id="echoid-s18310" xml:space="preserve">5. </s> <s xml:id="echoid-s18311" xml:space="preserve">P. </s> <s xml:id="echoid-s18312" xml:space="preserve">385. </s> <s xml:id="echoid-s18313" xml:space="preserve">in cubo Rad. </s> <s xml:id="echoid-s18314" xml:space="preserve">884. </s> <s xml:id="echoid-s18315" xml:space="preserve">loco antepen. </s> <s xml:id="echoid-s18316" xml:space="preserve">pro 5.</s> <s xml:id="echoid-s18317" xml:space="preserve">r.</s> <s xml:id="echoid-s18318" xml:space="preserve">1. </s> <s xml:id="echoid-s18319" xml:space="preserve"><lb/>In cubo Rad.</s> <s xml:id="echoid-s18320" xml:space="preserve">929. </s> <s xml:id="echoid-s18321" xml:space="preserve">loco ſexto pro 7. </s> <s xml:id="echoid-s18322" xml:space="preserve">r. </s> <s xml:id="echoid-s18323" xml:space="preserve">5. </s> <s xml:id="echoid-s18324" xml:space="preserve">In cubo Rad. </s> <s xml:id="echoid-s18325" xml:space="preserve">864. </s> <s xml:id="echoid-s18326" xml:space="preserve">loco pen. </s> <s xml:id="echoid-s18327" xml:space="preserve">pro 5. </s> <s xml:id="echoid-s18328" xml:space="preserve">r. </s> <s xml:id="echoid-s18329" xml:space="preserve">4. </s> <s xml:id="echoid-s18330" xml:space="preserve">In quadr. </s> <s xml:id="echoid-s18331" xml:space="preserve">Rad. </s> <s xml:id="echoid-s18332" xml:space="preserve">946. </s> <s xml:id="echoid-s18333" xml:space="preserve">pro <lb/><gap/>8. </s> <s xml:id="echoid-s18334" xml:space="preserve">In cubo Rad. </s> <s xml:id="echoid-s18335" xml:space="preserve">909. </s> <s xml:id="echoid-s18336" xml:space="preserve">pro 5. </s> <s xml:id="echoid-s18337" xml:space="preserve">r. </s> <s xml:id="echoid-s18338" xml:space="preserve">9. </s> <s xml:id="echoid-s18339" xml:space="preserve">P. </s> <s xml:id="echoid-s18340" xml:space="preserve">272@ in propoſitione 16. </s> <s xml:id="echoid-s18341" xml:space="preserve">d@ſideratur hic ſubiecta figura.</s> <s xml:id="echoid-s18342" xml:space="preserve"/> </p> <figure> <image file="445-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/445-01"/> </figure> <pb file="446" n="446"/> <pb file="447" n="447"/> <pb file="448" n="448"/> <pb file="449" n="449"/> <pb file="450" n="450"/> </div></text> </echo>