texts/archimedes/xml/bianc_locam_009_la_1615.xml - Colored diff

Main History Search Repository tree

Return to bianc_locam_009_la_1615.xml CVS log

Up to [CVSROOT] / texts / archimedes / xml

Colored diff for /texts/archimedes/xml/bianc_locam_009_la_1615.xml between version 1.2 and 1.3

version 1.2, 2003/11/20 17:53:50

version 1.3, 2003/12/05 02:02:22

Line 2079

<s id="id.000679">Quilibet circulus æqualis e&longs;t triangulo rectangulo, cuius

<lb/>quidem &longs;emidiameter vni laterum, quæ circa rectum angulum &longs;unt, ambi

<lb/>tus verò ba&longs;i eius e&longs;t æqualis.</s><figure id="id.009.01.034.1.jpg" xlink:href="009/01/034/1.jpg"lace="text"/>

<lb/>tus verò ba&longs;i eius e&longs;t æqualis.</s><figure id="id.009.01.034.1.jpg" xlink:href="009/01/034/1.jpg" place="text"/>

<s id="id.000680">Sit, v.g. datus circulus, cuius &longs;emidiameter A B; & fit trian gulum rectangu

<lb/>lum A B C, cuius angulus B, &longs;it rectus, & latus B A, <expan abbr="con&longs;titu&etilde;s">con&longs;tituens</expan> angulum re

Line 2185

de motu <emph type="italics"/>(Quadratum augetur Gnomone circumpo&longs;ito)<emph.end type="italics"/> Gnomon vox

<lb/>græca inter alia &longs;ignificat in&longs;trumentum illud, quod Latini tum amu&longs;&longs;im,

<lb/>tum normam appellant, Itali verò, Squadra, ad

<lb/>cuius &longs;imilitudinem Geometræ denominarunt fi

<lb/>guram quandam, &longs;eu portionem cuiu&longs;uis paralle

Line 2358

<s id="id.000740">&longs;ic; commen&longs;. </s>

<s id="id.000741">magnitudines dicuntur, quas

<lb/>eadem men&longs;ura metitur, vt &longs;i fuerint duæ magnitu

<lb/>dines, A, & B, quas eadem men&longs;ura C, ide&longs;t quan

<lb/>titas C, metiatur, ide&longs;t <expan abbr="quãtitas">quantitas</expan> C, applicata quan

Line 2369

<lb/>petita ip&longs;am con&longs;umeret, diceretur <expan abbr="vtranq;">vtranque</expan> metiri, & proinde duas lineas

<lb/>A, & B, e&longs;&longs;e comm. definit po&longs;tea <expan abbr="incomm&etilde;&longs;">incommen&longs;</expan> hoc modo, incomm, autem, qua

<lb/>rum nullam contingit communem men&longs;uram reperiri; vt &longs;i duarum linea

<lb/>rum, A, B, nunquam po&longs;&longs;et reperiri aliqua men&longs;u

<lb/>ra, quæ <expan abbr="vtranq;">vtranque</expan> adæquatè metiretur, v. g. &longs;i linea

<lb/>C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta

Line 2408

<s id="id.000747">&longs;unt diameter, & latus eiu&longs;

<lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti

<lb/>e&longs;t lineola E, in præ&longs;enti quadrato, etiam&longs;i illam in

<lb/>infinitum &longs;ubdiuidas, quæ <expan abbr="vtranq;">vtranque</expan> lineam, diame

<lb/>trum &longs;cilicet A C, & latus quoduis ex quatuor, v. g.

Line 2490

<lb/>ex quodam græco codice, qui characteres hoc modo appo&longs;uerat. </s>

<s id="id.000761">&longs;it I&longs;o&longs;ce

<lb/>les C A B, cuius ba&longs;is C B, Dico angulos &longs;upra ba&longs;im,

<lb/>in quibus literæ E F, e&longs;&longs;e inuicem æquales. </s>

Line 2604

<lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt

<lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="ill&utilde;">illum</expan>, quem

<lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C,

<lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro,

<lb/>e&longs;t ratio anguli. </s>

Line 2631

<s id="id.000786">Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita

<lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g.

<lb/>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu

<lb/>li partiales B A D, D A C, erunt æquales totali angulo

<lb/>B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua

Line 2643

<lb/>li &longs;imul ad vnum punctum con&longs;tituti <expan abbr="adæquar&etilde;tur">adæquarentur</expan> mucro

<lb/>niilli, quem con&longs;tituerent alij duo anguli, quibus illi tres

<lb/>&longs;unt pares, v.g. &longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti,

<lb/>quos linea perpendicularis D E, facit cum li

<lb/>nea F G; &longs;it <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G,

<lb/>tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dic&etilde;tur">dicentur</expan> æqua

<lb/>les duobus hi&longs;ce rectis, &longs;i tres illi mucrones

<lb/>trianguli fimul &longs;umpti, & vniti ad punctum

<lb/>E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum

<lb/>rectorum coeunt, congruent omnino duobus

<lb/>prædictis angulis rectis, &longs;iue duobus illis mu

<lb/>cronibus angulorum rectorum, &longs;iue con&longs;ti

Line 2792

<s id="id.000813">quoad

<lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens.

<lb/>probat Euclides in 28. primi Elem.

quod &longs;i

Line 2896

<s id="id.000830">Præterea &longs;i &longs;upponamus aliam fal&longs;itatem, &longs;cilicet triangulum habere tres

<lb/>angulos maiores duobus rectis, &longs;equetur eadem iterum fal&longs;itas, &longs;cilicet pa

<lb/>rallelas coincidere, & probatur &longs;ic; &longs;int enim

<lb/><expan abbr="triãguli">trianguli</expan> A B C, tres anguli maiores, quam duo

<lb/>recti anguli, & per punctum C, ducta &longs;it recta

Line 3048

<lb/>quod patet ducta &longs;emidiametro H G, perpendiculari ex B C, quæ &longs;uo extre

<lb/>mo puncto G, &longs;ecat bifariam, & <expan abbr="diametr&utilde;">diametrum</expan> B D, & circumferentiam B G C.

<lb/>facto ergo centro G, de&longs;cribatur alter circulus per puncta B C D F, <expan abbr="conne-ctatur&qacute;">conne

<lb/>ctaturque</expan>; recta G C. in triangulo orthogonio B C D, latus B D, &longs;ubtenditur <pb xlink:href="009/01/045.jpg" pagenum="45"/><figure id="id.009.01.045.1.jpg" xlink:href="009/01/045/1.jpg"lace="text"/>

<lb/>angulo recto C, ergo quadratum eius ex eorol

<lb/>lario 47. primi, duplum erit quadrati B C, quare

<lb/>etiam circulus B C D F, duplus erit circuli A B

Line 3070

<s id="id.000859">&longs;ed vt reliquum circu

<lb/>li quadret, &longs;ic pergit, ponatur recta L M, dupla

<lb/>ip&longs;ius B C, &longs;upra quam &longs;emicirculus de&longs;cribatur

<lb/>L O M, cui in&longs;cribatur hexagoni

<lb/>æquilateri dimidium L Q S M, & &longs;u

<lb/>per tribus hexagoni lateribus, &longs;int

Line 3180

Euclidis 31. in qua talis fi

<lb/>gura proponitur qualis e&longs;t præ&longs;ens, in qua vides triangulum A B C. in &longs;e

<lb/>micirculo. </s>

<s id="id.000881">tunc autem dicitur triangulum in

Line 3333

<s id="id.000912">Per æquilaterum, intelligit quadratum, quadratus autem numerus defi

<lb/>nitione 18. &longs;eptimi &longs;ic explicatur: Quadratus numerus e&longs;t, qui &longs;ub duobus

<lb/>æqualibus numeris continetur, ide&longs;t, qui fit ex ductu vnius numeri in &longs;e ip

<lb/>&longs;um, vt &longs;i ducantur 3. in 3. fient 9. qui continetur &longs;ub duobus

<lb/>ternarijs; omnes autem ternarij &longs;unt æquales. </s>

Line 3355

<lb/>ex multiplicatione numeri alicuius in &longs;e ip&longs;um.</s>

<s id="id.000916">Per altera parte longius, intelligit numerum, qui producitur à duobus

<lb/>numeris inæqualibus inuicem multiplicatis, qualis e&longs;t

<lb/>duodenarius, qui ex ductu trium in quatuor produci

<lb/>tur, & refert figuram altera parte longiorem, &longs;iue, vt

Line 3421

<lb/>ex 28. primi Elem.

de&longs;umitur, quam propterea primo loco exponendam

<lb/>cen&longs;ui. </s>

<s id="id.000932">Quando igitur duæ rectæ con&longs;titu

Line 3439

<lb/>quia linea E F, pote&longs;t facere aliquando prædictos angulos non <expan abbr="tant&utilde;">tantum</expan> æqua

<lb/>les duobus rectis, verum etiam rectos, quo etiam modo <expan abbr="probar&etilde;tur">probarentur</expan> cædem

<lb/>lineæ e&longs;&longs;e parallelæ, vt in &longs;equenti figura, cum &longs;int anguli A G I, C I G, re

<lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan

Line 3535

<s id="id.000950">Alternam igitur proportionem definit Eu

<lb/>clides definitione 12. quinti, &longs;ic, e&longs;t &longs;umptio antecedentis ad <expan abbr="anteced&etilde;tem">antecedentem</expan>,

<lb/>& con&longs;equentis ad con&longs;equentem. </s>

<s id="id.000951">Explico, exponantur qua

Line 3686

<s id="id.000979">8. qui e&longs;t ad in&longs;tar cubi geometrici, & continetur&longs;ub tribus binarijs

<lb/>multiplicatis inuicem, quæ multiplicatio &longs;ic in&longs;tituitur, exponuntur tres bi

<lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur.

<s id="id.000980">4. qui e&longs;t numerus quadratus huius figuræ, <figure id="id.009.01.052.2.jpg" xlink:href="009/01/052/2.jpg"lace="text"/>, deinde

<s id="id.000980">4. qui e&longs;t numerus quadratus huius figuræ, <figure id="id.009.01.052.2.jpg" xlink:href="009/01/052/2.jpg" place="text"/>, deinde

<lb/>tertius binarius ducitur in prædictum quadratum 4. & pro

<lb/>ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua

<lb/>ternarij, vnus &longs;upra alterum, vt in præ&longs;enti figura refe

<lb/>runt cubicam figuram, cuius tam longitudo, quam latitudo, <pb xlink:href="009/01/053.jpg" pagenum="53"/>& altitudo, e&longs;t 2. Similiter cubus numerus e&longs;t 27. quia &longs;it ex tribus terna

<lb/>rijs inuicem modo prædicto multiplicatis, 3. 3. 3. nam 3. in 3. ductis &longs;it 9.

<lb/>qui e&longs;t quadratus. </s>

<s id="id.000981">quo deinde ducto in tertium ter

Line 3741

<lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden

<lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora

<lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat

<lb/>per 21. primi Elem.

&longs;it enim ma

Line 3764

<s id="id.000990">Exemplum &longs;ubalternationis Muficæ &longs;it, <expan abbr="con&longs;onãtia">con&longs;onantia</expan> Diapa&longs;on, quam

<lb/>vulgò octauam appellant in data chorda collocare, hoc e&longs;t, vocem grauio

<lb/>rem facere duplam vocis acutioris &longs;umatur chorda A B, & diuidatur bifa

<lb/>riam, &longs;ine in æqualia in C; tota igitur chorda A B, ad dimidium A C, haber <pb xlink:href="009/01/054.jpg" pagenum="54"/><figure id="id.009.01.054.1.jpg" xlink:href="009/01/054/1.jpg"lace="text"/>

<lb/>proportionem, quam 2. ad 1.

<lb/>&longs;iue duplam, ergo etiam &longs;o

<lb/>nus totius chordæ A B, ad <expan abbr="&longs;o-n&utilde;">&longs;o

Line 3806

<lb/>drandus circulus A B C D, cui circum&longs;eribatur quadratum E F G H. per

<lb/>7 quarti, & alterum quadratum I L M N, eidem in&longs;cribatur per 6. quarti,

<lb/>quid autem &longs;it circum&longs;cribere, & in&longs;cribere figuram circulo, ex definitione

<lb/>3. & 4. eiu&longs;dem libri petatur, quamuis

<lb/>ex in&longs;pectione figuræ <expan abbr="pr&ecedil;s&etilde;tis">pr&ecedil;sentis</expan> &longs;atis per

<lb/>cipi po&longs;&longs;it; deinde aliud <expan abbr="quadrat&utilde;">quadratum</expan> me

Line 3852

<lb/>teles innuit, hoc enim modo ip&longs;ius Ari&longs;t.

mentem probè penetrare poteri

<s id="id.001008">&longs;it ergo <expan abbr="triãgulum">triangulum</expan> A B C. </s>

Line 4030

<s id="id.001041">Quoniam enim in 13.

<lb/>Aequep. probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo

<lb/>quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D, <expan abbr="centr&utilde;">centrum</expan> grauitatis. <pb xlink:href="009/01/057.jpg" pagenum="57"/><figure id="id.009.01.057.1.jpg" xlink:href="009/01/057/1.jpg"lace="text"/>

<lb/>&longs;ed eadem ratione erit etiam in linea B E, er

<lb/>go non ni&longs;i in puncto F, quod <expan abbr="&longs;ol&utilde;">&longs;olum</expan> e&longs;t in vtra

<lb/>que, quod erat demon&longs;trandum. </s>

Line 4447

<lb/>Ari&longs;t.

con&longs;idera&longs;&longs;e, Minam re&longs;pectu Talenti, re&longs;pectu enim illius dici pote&longs;t

<lb/>principium, cum &longs;ex millia minarum in Attico talento continerentur.</s><figure id="id.009.01.061.1.jpg" xlink:href="009/01/061/1.jpg"lace="text"/>

<lb/>principium, cum &longs;ex millia minarum in Attico talento continerentur.</s><figure id="id.009.01.061.1.jpg" xlink:href="009/01/061/1.jpg" place="text"/>

Line 4578

<s id="id.001170">Et paulo po&longs;t <emph type="italics"/>(Qutmadmod&ubreve; &longs;t vi

<lb/>trum perforatum videremus, & lumen

Line 4717

<lb/>cebantnon licere Mu&longs;ico vltra quaternarium pertran&longs;ire, eò quod &longs;olæ pro

<lb/>portiones, vt diximus, numerorum quaternario contentorum, concordem,

<lb/>ac con&longs;onantem concentum efficere poterant: quod vt adhuc melius per

<lb/>cipiamus, accipe exemplum. </s>

<s id="id.001198">Sint duæ chordæ

Line 4840

<s id="id.001227">per angulum in &longs;emicirculo intelligas eum, qui

<lb/>fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum

<lb/>circumferentiæ coeuntibus, vt in figura

<lb/>præ&longs;enti vides lineas A C, B C, ad C, pun

<lb/>ctum conuenire, <expan abbr="ibi&qacute;">ibique</expan>; facere angulum,

Line 4989

<s id="id.001261">vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angul&utilde;">angulum</expan> A, angulo D:

<lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb xlink:href="009/01/067.jpg" pagenum="67"/><figure id="id.009.01.067.1.jpg" xlink:href="009/01/067/1.jpg"lace="text"/>

<lb/>circa angulos æquales, v. g. cirea an

<lb/>gulos A, & D, habeant proportiona

<lb/>lia, hoc e&longs;t, vt latus A B, ad latus A C;

Line 5185

<lb/>allatum &longs;ic videtur explicandum: Conetur aliquis demon&longs;trare hanc pro

<lb/>po&longs;itionem; &longs;i linea ducta fuerit æquidi&longs;tans lateri vnius plani trianguli, &longs;e

<lb/>cabit & latera, & locum, ideft &longs;uperficiem illam triangularem &longs;imiliter, ide &longs;t

<lb/>in eadem proportione, vt in triangulo A B C,

<lb/>linea D E, parallela ba&longs;i B C, &longs;ecat latera A B,

<lb/>& A C, in punctis D, & E, in eadem ratione,

Line 5333

<s id="id.001335">Ad &longs;inem cap. <emph type="italics"/>(Aut vt Antiphon quadra

<lb/>uit)<emph.end type="italics"/> &longs;imile peccatum pecca&longs;&longs;e Antiphon

Line 5516

<s id="id.001368">pro medio autem

<lb/>huius pa&longs;&longs;ionis accipit lineam perpendicularem, quam innuit verbis illis

<lb/><emph type="italics"/>(quoniam enim hoc rectum e&longs;t<emph.end type="italics"/>) vt in figura &longs;it triangulum A B C, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; vt latus

<lb/>A C, &longs;it perpendiculare <expan abbr="c&utilde;">cum</expan> latere B C, & pro

<lb/>ducatur B C, in D; tunc triangulum A B C,

<lb/>habere tres angulos, A, B, & A C B, æquales

Line 5566

numeros impares &longs;olos appellabant Gnomones, eò quod in for

<lb/>mam normæ æquilateræ, &longs;iue Gnomonis con&longs;titui po&longs;&longs;int, vt patet in his

<lb/>nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de

<lb/>reliquis imparibus. </s>

Line 5574

<lb/>queunt in figuram normæ æquilateræ di&longs;poni, cum

<lb/>non habeant vnitatem pro angulo, & paria po&longs;tea la

<lb/>tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di

<lb/>&longs;ponatur &longs;ic <figure id="id.009.01.074.2.jpg" xlink:href="009/01/074/2.jpg"lace="text"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con

<lb/>&longs;ponatur &longs;ic <figure id="id.009.01.074.2.jpg" xlink:href="009/01/074/2.jpg" place="text"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con

<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="id.009.01.074.3.jpg" xlink:href="009/01/074/3.jpg"lace="text"/> quia dee&longs;t huic figuræ angularis vnitas, quæ

<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="id.009.01.074.3.jpg" xlink:href="009/01/074/3.jpg" place="text"/> quia dee&longs;t huic figuræ angularis vnitas, quæ

<lb/>illi nece&longs;iaria e&longs;t. </s>

<s id="id.001377">Pythagorici igitur dicebant, numerum parem ideò e&longs;&longs;e

Line 5606

<s id="id.001382">&longs;it vnitas, & &longs;uo or

<lb/>dine &longs;equantur impares, vt in &longs;equenti hac &longs;erie apparet 1. 3. 5. 7. 9. & c.

<lb/>&longs;i igitur vnitati addatur ternarius in Gnomo

<lb/>nis modum, vt vides in prima figura, produ

<lb/>cetur quaternarius numerus, qui e&longs;t numerus

Line 5636

<s id="id.001387">&longs;i igitur vnitati binarius coaceruetur, fit numerus

<lb/>triangularis, vt in prima figura. </s>

<s id="id.001388">&longs;i huic ternario

Line 5781

<s id="id.001425">Tex. 36. <emph type="italics"/>(Sit <expan abbr="itaq;">itaque</expan> linea, in qua A G E, infinita ad partes E; & alia vtrinque

<lb/>infinita, in qua <foreign lang="greek">b</foreign> B; &longs;i <expan abbr="itaq;">itaque</expan> de&longs;cribat circulum linea A G E, cir ca centrum G, fe-<emph.end type="italics"/>

<lb/><emph type="italics"/>retur circulariter linea A G E, &longs;ecans ali

<lb/>quando lineam <foreign lang="greek">b</foreign> B, tempore finito; totum

<lb/>enim tempus, in quo circulariter latum

Line 5950

<lb/>gis autem concauum e&longs;t, quod centro propinquius est. </s>

<s id="id.001457">ducantur ergo ex centro A,<emph.end type="italics"/>

<lb/><emph type="italics"/>linea A B, & linea A C, & producatur, in qua B C,

<lb/>ducta igitur ad ba&longs;im linea, in qua A D, minor e&longs;t eis,

<lb/>quæ ex centro. </s>

Line 6009

<lb/>alijs maius corpus velocius fertur propria latione, &longs;ic, & in circularibus: maius

<lb/>enim e&longs;t eorum, quæ auferuntur ab eis, quæ ex centro, maioris circuli &longs;egmentum)<emph.end type="italics"/>

<lb/>ex intellectione vltimæ periodi textus totius intelligentia pendet: &longs;it igitur

<lb/>figura præ&longs;ens, in qua cum &longs;int duo circuli concen

<lb/>trici, vnus altero maior, <expan abbr="eductæ&qacute;">eductæque</expan>; &longs;int ex <expan abbr="c&etilde;tro">centro</expan> duæ

<lb/>&longs;emidiametri A D, A E, quæ <expan abbr="vtrunq;">vtrunque</expan> circulum &longs;e

Line 6109

<s id="id.001488">intellige tamen partem illam Solis, quæ non eclyp&longs;atur, habere figu

<lb/>ram lunulæ, pars enim à Luna obumbrata non videtur, et&longs;i videretur oua

<lb/>lem quandam &longs;peciem, præ&longs;eferret: pars igitur, illa e&longs;t corniculata, quia

<lb/>cum Solis defectio ex interpo&longs;itione Lunæ inter nos, &

<lb/>Solem contingat, & Luna &longs;it &longs;phærica, nece&longs;&longs;ariò &longs;phæ

<lb/>ricè, & circulariter Solem obumbrabit; quare pars illa

Line 6266

<lb/>Vr&longs;am dicit) magnopere mutatur horizon: quod apparet primo ex varia

<lb/>tionc a&longs;trorum, nam quæ in primo loco &longs;upra no&longs;trum verticem <expan abbr="trã&longs;ibant">tran&longs;ibant</expan>,

<lb/>in &longs;ecundo loco non amplius, &longs;ed alia, <expan abbr="atq;">atque</expan> alia valde ab inuicem &longs;eiuncta

<lb/>exfacto quamuis paruo itinere tran&longs;eunt. </s>

<s id="id.001518">&longs;it in

Line 6415

<s id="id.001549">huiu&longs;modi &longs;unt triangula æquilatera (de his enim

<lb/>intelligendus e&longs;t textus) quadrata, & hexagona, &longs;iue &longs;exilatera regularia;

<lb/>nam &longs;ex triangula æquilatera &longs;imul iuncta in plano paui

<lb/>re po&longs;&longs;unt, vt patet in figura præ&longs;enti; ratio huius e&longs;t,

<lb/>quia omnes anguli circa idem punctum (y. </s>

Line 6432

<lb/>A, totum locum circa illud implere po&longs;&longs;unt. </s>

<s id="id.001552">Quadratum etiam replere lo

<lb/>cum manife&longs;tum e&longs;t, cum enim ip&longs;ius anguli &longs;intrecti, &longs;i

<lb/>quatuor quadrata ad idem punctum A, copulentur, vt in

<lb/>figura apparet, replebunt eadem de cau&longs;a vacuum.</s>

Line 6444

<lb/>hil vacui inter ip&longs;a relinquetur, vt in figura hac o&longs;tenditur. </s>

<s id="id.001554">præter has tres

<lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol

<lb/>&longs;it. </s>

Line 6725

<s id="id.001606">boc autem vtrum accidit ad ip&longs;um tcrræ medium,

<lb/>an ad vniuer&longs;i, quoniam idem ip&longs;orum &longs;it, alius &longs;ermo e&longs;t)<emph.end type="italics"/> cum vellet

<lb/>probare Ari&longs;toteles dari <expan abbr="p&utilde;ctum">punctum</expan> quoddam in medio

<lb/>mundi, ad quod grauia de&longs;cendant, & concurrent:

<lb/>& à quo leuia a&longs;cendat; vtitur, præter alias, etiam

Line 6862

<lb/>quà &longs;it illuminatum, vmbram proijci fa&longs;tigiatam: cum deinde Solem val

<lb/>de a terra di&longs;tare certum &longs;it, optimè infertur, eum re&longs;pectu terræ e&longs;&longs;e maxi

<lb/>mum: quanto enim duæ lineæ, &longs;iue radij B A, B C. à terra ad partes Solis

<lb/>magis elongantur, tan

<lb/>to maius corpus <expan abbr="illu-minãs">illu

<lb/>minans</expan> intercipiunt. </s>

Line 7121

<lb/>queretur nece&longs;&longs;ariò eum tanta velocitate videri à nobis circumferri, vt po

<lb/>tius fulgor quidam, &longs;eu radius pertran&longs;iens ab oriente in occidentem appa

<lb/>reret, quam &longs;tella qu&ecedil;dam: <expan abbr="id&qacute;">idque</expan>; propter propinquitatem; a&longs;tra enim ob ni

<lb/>miam di&longs;tantiam videntur tardè moueri, quamuis veloci&longs;&longs;imè moueantur.</s><figure id="id.009.01.092.1.jpg" xlink:href="009/01/092/1.jpg"lace="text"/>

<lb/>miam di&longs;tantiam videntur tardè moueri, quamuis veloci&longs;&longs;imè moueantur.</s><figure id="id.009.01.092.1.jpg" xlink:href="009/01/092/1.jpg" place="text"/>

<s id="id.001684">Quod melius ex &longs;equenti figura

<lb/><expan abbr="cõuincitur">conuincitur</expan>, vbi circulus interior e&longs;t

Line 7762

<lb/>eminere pote&longs;t, quod &longs;ic geometricè demon&longs;trabo. </s>

<s id="id.001805">de&longs;cribatur enim figura

<lb/><figure id="id.009.01.100.1.jpg" xlink:href="009/01/100/1.jpg"lace="text"/><pb xlink:href="009/01/101.jpg" pagenum="101"/>illa, qua ad vaporum altitudines indagandas vtuntur Alhazenus, Vitellio,

<lb/><figure id="id.009.01.100.1.jpg" xlink:href="009/01/100/1.jpg" place="text"/><pb xlink:href="009/01/101.jpg" pagenum="101"/>illa, qua ad vaporum altitudines indagandas vtuntur Alhazenus, Vitellio,

<lb/>& Clauius, in qua terræ globus e&longs;t F L G E, regiò vaporum, & exhalatio

<lb/>num M X N T. horizon a&longs;tronomicus O P. phy&longs;icus Q R, tangens terram

<lb/>in puncto F, vbi etiam ponendus e&longs;t huius horizontis habitator, vnà cum.

Line 8494

<s id="id.001956">eodem autem modo ad inferiorem polum alij duo coni terræ &longs;egmenta fa

<lb/>ciunt)<emph.end type="italics"/> vt benè duas ha&longs;ce terræ portiones, quas &longs;olas habitabiles putat Ari

<lb/>&longs;tot. concipias, <expan abbr="reliquaq;">reliquaque</expan> huius loci intelligas, in&longs;pice &longs;equentem figuram.

<lb/>Maior circulus &longs;it c&oelig;lum, in quo polus L, articus; M, antarticus, ille eleua

<lb/>tus &longs;upra no&longs;trum horizontem S N, 45. gradibus, i&longs;te verò totidem infra

<lb/>depre&longs;&longs;us. </s>

Line 8639

<s id="id.001992"><emph type="italics"/>De&longs;criptus &longs;it igitur, vt clarior res euadat horizontis circulus quapropter, &

<lb/>rotundus)<emph.end type="italics"/> vt in &longs;equenti figura circulus A G B H, de&longs;criptus horizontem

<lb/>referret,</s><figure id="id.009.01.111.1.jpg" xlink:href="009/01/111/1.jpg"lace="text"/><pb xlink:href="009/01/112.jpg" pagenum="112"/>

<lb/>referret,</s><figure id="id.009.01.111.1.jpg" xlink:href="009/01/111/1.jpg" place="text"/><pb xlink:href="009/01/112.jpg" pagenum="112"/>

<s id="id.001993"><emph type="italics"/>Oportet autem ip&longs;ius alteram portionem intelligere, quæ nobis habitatur; quæ

<lb/>eodem modo diuidi poterit)<emph.end type="italics"/> ide&longs;t oportet intelligere ip&longs;ius horizontis, vel ter

Line 8808

<lb/>dem enim &longs;igno ad idem &longs;ignum æquales frangentur &longs;uper circuli lineam &longs;emper. </s>

<s id="id.002030">&longs;it<emph.end type="italics"/>

<lb/><emph type="italics"/>enim à puncto, in quo A, ad B, fracta, & ea, quæ est

<lb/>A C B, & quæ A F B, & quæ A D B, æquales autem

<lb/>& hæ A C, A F, A D, inuiccm. </s>

Line 8937

<lb/>ferentiam incident lineæ, quæ à K, & &longs;i quidem in ortu, aut in occa&longs;u a&longs;tri reflexio

<lb/>fiat, &longs;emicirculus ab <expan abbr="horizõte">horizonte</expan> a&longs;&longs;umetur &longs;uper terram factus. </s>

<s id="id.002050">&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb xlink:href="009/01/116.jpg" pagenum="116"/><figure id="id.009.01.116.1.jpg" xlink:href="009/01/116/1.jpg"lace="text"/>

<s id="id.002050">&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb xlink:href="009/01/116.jpg" pagenum="116"/><figure id="id.009.01.116.1.jpg" xlink:href="009/01/116/1.jpg" place="text"/>

<lb/><emph type="italics"/>&longs;emper &longs;emicirculo, minus autem,

<lb/>cum in meridie fuerit a&longs;trum)<emph.end type="italics"/> quod

<lb/>&longs;upra monui, iterum moneo, <expan abbr="re-tin&etilde;dam">re

Line 8991

<s id="id.002063">Ibidem <emph type="italics"/>(Sit enim in <expan abbr="ori&etilde;te">oriente</expan> pri

<lb/>mum vbi G, & refracta &longs;it K M,

Line 9036

<lb/>tum; ea &longs;cilicet, quæ per reflexionem (de quorum numero e&longs;t Iris) viden

<lb/>tur, videri, tunc &longs;olum, quando angulus incidentiæ fuerit æqualis angulo

<lb/>reflexionis, quia tunc breui&longs;&longs;imis lineis fit vi&longs;io; quibus &longs;oli, natura (&longs;i fieri

<lb/>pote&longs;t) vtitur. </s>

<s id="id.002070">v. g. in figura præ&longs;enti &longs;it &longs;pe

Line 9149

<s id="id.002087">Iam &longs;i <expan abbr="triãgulum">triangulum</expan> G M K, prioris figuræ circumuoluatur circa axem G K O,

<lb/>punctum ip&longs;ius M, de&longs;cribit prædictum ambitum L M N. hunc ambitum

<lb/>inquit Ariltot. linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t.

<lb/>Erit præterea &longs;ectio circunferentiarum ho

<lb/>rizontis, & huius amb tus data, cuius extre

<lb/>ma puncta e&longs;&longs;ent L, & N. &longs;i enim <expan abbr="cõcipiamus">concipiamus</expan>

Line 9210

<s id="id.002095">Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta,

<lb/>nam &longs;ecundum veritatem po&longs;&longs;unt in eadem nube con&longs;titui plures anguli

<lb/>æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra <expan abbr="alter&utilde;">alterum</expan>; vt in figura præ

<lb/>&longs;enti, &longs;i nubes e&longs;&longs;et vbi B D.

<lb/>oculus in C, Sol in A. e&longs;&longs;ent

<lb/>duo anguli A B C, A D C, æ

Line 9271

cum data &longs;it proportio linearum K M, & M G, in &longs;upe

<lb/>riori &longs;ecunda figura numeri 164. quam nunc iterum in&longs;picere opertet; ex

<lb/>ponatur alia linea recta B D. quæ diui

<lb/>datur in partes B, & D. proportionales

<lb/>cum lineis K M, G M, per 10. 6. cum

Line 9416

<lb/>erat primo loco demon&longs;trandum.</s>

<s id="id.002132">Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum

<lb/>ad M, ducatur M P, faciens angulum K P M, æqua

<lb/>lem angulo G M K, per 23. primi, erunt enim duo

<lb/>triangula <expan abbr="æquiãgula">æquiangula</expan> G P M, K P M, angulus enim

Line 9452

<lb/>e&longs;t enim i&longs;te, in quo G P. </s>

<s id="id.002140">Quoniam autem &longs;upra diametrum, quæ A C, quod K G,

<lb/>centrum vtique erit &longs;ub horizonte priori eius, in quo A C, in linea K P, in quo <foreign lang="greek">w,</foreign><emph.end type="italics"/><pb xlink:href="009/01/123.jpg" pagenum="123"/><figure id="id.009.01.123.1.jpg" xlink:href="009/01/123/1.jpg"lace="text"/>

<lb/><emph type="italics"/>Quare minor erit &longs;uperior &longs;ectio &longs;emicir

<lb/>culo, in qua S T, (nam Q S T, &longs;emicir

<lb/>culus est, nunc autem inter&longs;ectus e&longs;t ab

Line 9621

<s id="id.002168">non in hoc &longs;itu, & ouali figura, &longs;ed

<lb/>e&longs;&longs;e perfectè &longs;emicircularem, <expan abbr="habere&qacute;">habereque</expan>; talem po&longs;itionem, vt pars ip&longs;ius B F,

<lb/>&longs;it citra chartam eleuata, <expan abbr="ip&longs;i&qacute;">ip&longs;ique</expan>; perpendicularis, pars verò E B, vltra pagi<pb xlink:href="009/01/126.jpg" pagenum="126"/>nam rectà recedat, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; diameter Iridis E F, faciat angulos rectos cum

<lb/>linea horizontali A C L, in quo &longs;itu oculo C, totus ex oppo&longs;ito directè &longs;pe

Line 9784

<s id="id.002202">Maior quidem, imò etiam integer circulus, &longs;ed ab oculo in &longs;ummitate

<lb/>montis con&longs;tituto, <expan abbr="Sole&qacute;">Soleque</expan>; iam multum eleuato videri pote&longs;t, vt in

<lb/>hac &longs;ecunda figura cernitur, vbi euecto Sole ad locum E, &longs;upra horizontem

<lb/>A B, poterit oculus in vertice montis C, po&longs;itus Iridem F G H I, comple

<lb/>tam videre, quia infra lineam E C D, in qua exi&longs;tunt Sol, oculus, & Iridis

<lb/>centrum, nihil e&longs;t ad partes D, vbi nubes irrorat, quod Iridis apparitioni

Line 9797

<s id="id.002205">Qvia eleuato Sole ad E, vt in hac tertia figura, nece&longs;&longs;ario centrum Iri

<lb/>dis D, infra horizontem A B, deprimetur, cum in eadem recta E C D.

<lb/>Sol E, oculus C, <expan abbr="centr&utilde;&qacute;">centrunque</expan>; Iridis D, exi&longs;tant: vnde nece&longs;&longs;ariò &longs;equitur Iridis

<lb/>portionem F G H, &longs;upra horizontem extantem, &longs;emicirculo minorem e&longs;&longs;e.</s>

Line 9947

<lb/>&longs;i quidem cum corpore quodam &longs;emper e&longs;t)<emph.end type="italics"/> Propo&longs;itio 2. tertij Elem.

&pacute;robat li

<lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro

<lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem

<lb/>cadere intra circulum. </s>

Line 9982

<lb/>in vnico puncto, vt hoc loco innuit Philo&longs;ophus. </s>

<s id="id.002246">probat autem hac ferè ra

<lb/>tione. </s>

<s id="id.002247">&longs;it &longs;phæra A B C, quæ tangat quodpiam planum

Line 10057

<lb/>&longs;ens figura A B C D, cuius quadrandæ ratio e&longs;t huiu&longs;modi. </s>

<s id="id.002262">per 13. 6. inue

<lb/>niatur recta linea media proportionalis inter

<lb/>duo latera figuræ A B, B C, <expan abbr="ea&qacute;">eaque</expan>; &longs;it B D, in &longs;e

<lb/>quenti figura. </s>

Line 10070

<s id="id.002264">quadratum autem li

<lb/>neæ B D, æquale e&longs;t rectangulo dato A B C D, per 17.6. Inuentio porrò hu

<lb/>ius mediæ proportionalis, quia facilis e&longs;t, & &longs;citu iucunda, eam &longs;ic habeto.

<lb/>accipe duo latera A B, & B C, <expan abbr="quadrãdi">quadrandi</expan> rectan

<lb/>guli, <expan abbr="ea&qacute;">eaque</expan>; in directum con&longs;titue, vt vnicam re

<lb/>ctam con&longs;tituant A C, vt apparet in figura; de

Line 10084

<s id="id.002265">Quadratum igitur huius B D, hoc e&longs;t quadratum,

<lb/>cuius quatuor latera &longs;iut æqualia lineæ B D, quale e&longs;t præ&longs;ens, æquale erit

<lb/>dato &longs;uperiori rectangulo A B C D, <expan abbr="atq;">atque</expan> hoc modo per

<lb/>acta erit quadratio, &longs;eu tetragoni&longs;mus dati quadrilateri

Line 10241

<s id="id.002305">Eodem cap. <emph type="italics"/>(Velut Diapa&longs;on, & Diapente)<emph.end type="italics"/> quid &longs;it con&longs;onantia Diapa

<lb/>&longs;on, explicaui in primo Po&longs;ter. tex. 1. Diapente verò e&longs;t con&longs;onantia ex duo

<lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt

<lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

Line 10743

<s id="id.002431">Tex. 17. <emph type="italics"/>(Veluti diametrum commen&longs;urabilem e&longs;&longs;e impo&longs;&longs;ibile est)<emph.end type="italics"/> huius expo

<lb/>&longs;itionem inuenies 1. Priorum, &longs;ecto 1. cap.

23.</s><figure id="id.009.01.141.1.jpg" xlink:href="009/01/141/1.jpg"lace="text"/>

23.</s><figure id="id.009.01.141.1.jpg" xlink:href="009/01/141/1.jpg" place="text"/>

Line 10836

<lb/>con&longs;cripta &longs;unt (<emph type="italics"/>Cur triangulus duo recti?<emph.end type="italics"/>) ide&longs;t, cur triangulus habet tres

<lb/>angulos æquales duobus rectis angulis (<emph type="italics"/>Quia qui circa vnum punctum anguli

<lb/>duobus rectis angulis æquales &longs;unt<emph.end type="italics"/>) ni&longs;i hoc dictum ad bonum trahatur &longs;en&longs;um,

<lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum

<lb/>punctum, v. g. A, &longs;unt con&longs;tituti, æquales &longs;unt

<lb/>non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis,

Line 10862

<s id="id.002457">tales &longs;unt quatuor anguli

<lb/>ad &longs;uperiores partes lineæ B C, & ad punctum A, con&longs;tituti, qui, vt patet,

<lb/>&longs;unt æquales duobus rectis B A D, D A C,

<lb/>tales etiam &longs;unt in hac &longs;ecunda figura tres

<lb/>anguli B C A, A C D, D C E, qui quidem

Line 10903

<s id="id.002465">nunc &longs;olùm <pb xlink:href="009/01/143.jpg" pagenum="143"/><figure id="id.009.01.143.1.jpg" xlink:href="009/01/143/1.jpg"lace="text"/>

<s id="id.002465">nunc &longs;olùm <pb xlink:href="009/01/143.jpg" pagenum="143"/><figure id="id.009.01.143.1.jpg" xlink:href="009/01/143/1.jpg" place="text"/>

<lb/>hæc addenda &longs;unt. </s>

<s id="id.002466">Re&longs;pondet Ari&longs;t.

Line 11525

<s id="id.002620">Quoniam autem &longs;ecundum contrarias &longs;imul motiones mouetur circulus,

<lb/>& alterum quidem diametri extremum vbi A, in figura præ&longs;enti antror&longs;um

<lb/>mouetur; alterum verò vbi B, retror

<lb/>&longs;um, efficiunt nonnulli, vt ab vnica mo

<lb/>tione multi contrariò &longs;imul mouean

Line 11578

<s id="id.002628">&longs;it enim in figura proportio lateris A B, ad latus A C, quam

<lb/>ctiam habent duo motus, &longs;ecundum quos latum quodpiam feratur, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; la

<lb/>tum illud A, & feratur motu vno ver&longs;us B, per lineam A B, altero verò mo

<lb/>tu feratur ver&longs;us C. quod fiet &longs;i cogitemus latus A B, <expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> ver&longs;us M C, <pb xlink:href="009/01/151.jpg" pagenum="151"/><figure id="id.009.01.151.1.jpg" xlink:href="009/01/151/1.jpg"lace="text"/>

<lb/>ip&longs;i æquidi&longs;tanter, dum punctum A, mouetnr

<lb/>ad B. his duabus lationibus A, latum. </s>

Line 11624

<lb/>extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus

<lb/>orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, &

<lb/>retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta <expan abbr="de&longs;crib&etilde;s">de&longs;cribens</expan> cir

<lb/>culum, nunquam ad diametri perpendiculum

<lb/>perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a

<lb/>à centro perpendicularis diametro. </s>

Line 11690

<lb/>uer&longs;um agitur, &longs;eu &longs;ecus centrum, ob quam cogitur in gyrum duci, minor

<lb/>autem linea &longs;ecundum hanc motionem innaturalem plus fertur, quàm ma

<lb/>ior, ide&longs;t plus ip&longs;ius progre&longs;&longs;io inflectitur in orbem; quia enim e&longs;t centro

<lb/>vicinior, quod quodammodo retra

<lb/>hit à motu naturali, propterea ma

<lb/>gis vincitur, quàm remotior. </s>

Line 11800

<lb/>bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ

<lb/>brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun

<lb/>ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri

<lb/>à centro exeuntes, vt in figura cerne

<lb/>re e&longs;t, in qua centrum, &longs;iue &longs;partum

<lb/>e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. </s>

Line 11861

<lb/>te plus libræ extra perpendiculum fit, &longs;partum enim e&longs;t perpendiculum,

<lb/>quare nece&longs;&longs;e est deor&longs;um ferri id, quod plus est, donec a&longs;cendat, quæ bifariam li

<lb/>bram diuidit ad ip&longs;um perpendicul&ubreve;, cum onus incumbat ad libræ partem tractam.<emph.end type="italics"/>

<lb/><emph type="italics"/>&longs;it libra vbi recta B C, &longs;partum autem

<lb/>A D: hoc igitur &longs;ur&longs;um erecto, perpendi

<lb/>culum erit vbi A D M. &longs;i igitur in ip&longs;o B,

Line 11880

<s id="id.002681">&longs;i autem deor&longs;um fuerit, id quod &longs;ub

<lb/>stat, contrarium facit; plus enim dimidio fit libræ, quæ deor&longs;um e&longs;t, pars, quàm

<lb/>quod perpendiculum &longs;ecet; quapropter non a&longs;cendit, pars enim eleuata leuior e&longs;t.<emph.end type="italics"/>

<lb/><emph type="italics"/>&longs;it libra vbi N G, perpendiculum autem

<lb/>K L M, bifariam igitur &longs;ecatur N G. im

<lb/>po&longs;ito autem onere in ip&longs;o N, erit quidem

Line 11906

<s id="id.002686">Ari&longs;t

<lb/>igitur ponit duas libræ &longs;pecies, &longs;iue potius duas eiu&longs;dem libræ po&longs;itiones,

<lb/>vnam, quæ haber &longs;partum, &longs;iue perpendiculum &longs;upra; alteram, quæ infra.

<lb/>vt in præ&longs;enti figura, &longs;it libra B C, cuius

<lb/>&longs;partum, &longs;iue perpendiculum A D, &longs;it &longs;ur

<lb/>&longs;um, ita vt in puncto A, &longs;it affixum perpen

Line 11917

<s id="id.002687">hæc e&longs;t prima libræ collocatio. </s>

<s id="id.002688">&longs;it deinde libra B C, cuius &longs;partum,

<lb/>&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; circa pun<pb xlink:href="009/01/156.jpg" pagenum="156"/><figure id="id.009.01.156.1.jpg" xlink:href="009/01/156/1.jpg"lace="text"/>

<lb/>ctum A, tanquam circa <expan abbr="centr&utilde;">centrum</expan>, aut axem

<lb/>ita fixum, vt ip&longs;i libræ conuer&longs;io innita

<lb/>tur, quæ e&longs;t altera libræ po&longs;itio. </s>

Line 12067

<s id="id.002715">vt in &longs;ubiectis figuris apparet. </s>

<s id="id.002716">In prima, vectis e&longs;t A B, fultu

<lb/>ra E, onus C. potentia autem &longs;eu vis,

<lb/>&longs;eu aliud pondus <expan abbr="mou&etilde;s">mouens</expan> &longs;it vbi D. quæ

<lb/>deor&longs;um in D, præmens eleuabit &longs;ur

Line 12187

<s id="id.002743">E&longs;t enim propriè Temo, &longs;iue gu

<lb/>bernaculum nauis, vectis &longs;ecundi generis, vt mox explicabo, e&longs;t enim temo

<lb/>in&longs;trumentum in extrema nanis par

<lb/>te, &longs;eu puppi affixum, vt in figura prç

<lb/>&longs;enti vides tabellam, in qua B C D,

Line 12311

<lb/>eiu&longs;dem rationem, quia nimirum parua motione facta in puppi multo ma

<lb/>ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ

<lb/>&longs;unt longiores, eò maiorem &longs;ubten&longs;am &longs;ibi lineam re&longs;picit, quod facilè in

<lb/>ad&longs;cripta figura intueri licet; in qua duæ

<lb/>lineæ A B, A C, continent angulum A, cui

<lb/>angulo &longs;ubtenduntur tres lineæ parallelæ

Line 12357

<lb/>quo po&longs;ito &longs;unt etiam latera vnius proportionalia lateribus alterius, vt pa

<lb/>tet ex prima definitione 6. necnon ex quarta eiu&longs;dem demon&longs;tratione. </s>

<lb/>quidem duo triangula &longs;unt &longs;imi

<lb/>lia, & rectè concluditur F Y, mi

<lb/>nus e&longs;&longs;e quàm D Y, &longs;ed tamen

Line 12442

<s id="id.002790">Cum igitur A, remi ca

<lb/>put in fine ip&longs;ius remigationis eò tran&longs;latum fuerit D, non erit B, vbi E; &longs;i

<lb/>enim ibi fuerit; remus igitur po&longs;itionem

<lb/>habebit rectam lineam D E; & quoniam

<lb/>contrapo&longs;iti anguli, qui ad C, æquales &longs;unt,

Line 12543

<s id="id.002812">Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij

<lb/>motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i

<lb/>tum rectitudinis habeat E F. </s>

<s id="id.002813">Spatium

Line 12590

<s id="id.002822">Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad

<lb/>rectos angulos &longs;uper B G, & à <expan abbr="p&utilde;cto">puncto</expan> A, recta A H,

<lb/>&longs;uper D F: itemque à puncto E, recta C E, &longs;uper

<lb/>E R; ip&longs;arum verò rectarum linearum E R, &

Line 12627

<s id="id.002828">Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan>

<lb/>partem perpendicularis E Z, in quam veniant a punctis A, & C,

<lb/>ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio <pb xlink:href="009/01/166.jpg" pagenum="166"/><figure id="id.009.01.166.1.jpg" xlink:href="009/01/166/1.jpg"lace="text"/>

<lb/>decur&longs;um motu proprio &longs;patij B D, duplum

<lb/>&longs;it: recta verò linea C H, curuæ re&longs;pondeat

<lb/>C G, quæ à remi palmula de&longs;cripta e&longs;t. </s>

Line 12716

<lb/>motum manubrij &longs;uperat.</s>

<s id="id.002850">Habeat enim remus incipiente motu po&longs;itionem A C: de&longs;inente

<lb/>verò <expan abbr="&longs;it&utilde;">&longs;itum</expan> rectitudinis F G. &longs;cal

<lb/>mus igitur B, propter nauigij

<lb/>motum tran&longs;latus, erit in D, &longs;it

Line 12825

<lb/>vires. </s>

<s id="id.002877">dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte

<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb xlink:href="009/01/169.jpg" pagenum="169"/><figure id="id.009.01.169.1.jpg" xlink:href="009/01/169/1.jpg"lace="text"/>

<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb xlink:href="009/01/169.jpg" pagenum="169"/><figure id="id.009.01.169.1.jpg" xlink:href="009/01/169/1.jpg" place="text"/>

<lb/>veluti centro parte E. quia ve

<lb/>rò ob maris liquiditatem na

<lb/>uis minimè obfi&longs;tit, &longs;ed facilè

Line 12919

<s id="id.002895">quod dictum e&longs;t de nutu anguli

<lb/>maioris circuli ad angulum minoris ex appo&longs;ita figura facilè patebit, vbi

<lb/>pro minore angulo intelligendus e&longs;t arcus C B,

<lb/>pro maiore autem arcus D E, quorum <expan abbr="vterq;">vterque</expan> vo

<lb/>catur angulus, quoniam angulo A, qui e&longs;t in cen

Line 13016

<s id="id.002912">vt autem clarè appareat

<lb/>vectis in trochlea, & hypomoclion, & æquales di&longs;tantiæ, &longs;it figura, in qua

<lb/>pondus D, ductario funi D C B E, alligatum. </s>

<s id="id.002913">poten

Line 13136

<lb/>vt Geometræ dicunt, Cylindrum; in cuius tamen extremitatibus e&longs;&longs;ent

<lb/>margines duo aliquantulum prominentes, ceu binæ rotæ, cum ip&longs;o tamen

<lb/>continuæ, & connexæ, vt cum ip&longs;o &longs;imul conuoluerentur; non tamen tan

<lb/>quam circa axem. </s>

<s id="id.002939">cuius hanc accipe fi

Line 13347

<lb/>cur quanto gracilius fucrit corpus &longs;ucculæ A B, tanto facilius vertitur.

<s id="id.002991">Ratio e&longs;t, quia collops, quemadmodum etiam iugum, reducitur ad vectem, <pb xlink:href="009/01/176.jpg" pagenum="176"/><figure id="id.009.01.176.1.jpg" xlink:href="009/01/176/1.jpg"lace="text"/>

<lb/>cuius hypomoclion e&longs;t in medio

<lb/>&longs;ucculæ, &longs;iue in axe ip&longs;ius &longs;ucculæ;

<lb/>potentia verò e&longs;t in &longs;ummitatibus

Line 13420

<lb/>premit, quod nunc e&longs;t in&longs;tar potentiæ motricis, quare A, & C, &longs;unt &longs;ibi in

<lb/>uicem, & potentiæ, & pondera. </s>

<s id="id.003012">In minori autem ligno, onus ligni in D, <pb xlink:href="009/01/177.jpg" pagenum="177"/><figure id="id.009.01.177.1.jpg" xlink:href="009/01/177/1.jpg"lace="text"/>

<s id="id.003012">In minori autem ligno, onus ligni in D, <pb xlink:href="009/01/177.jpg" pagenum="177"/><figure id="id.009.01.177.1.jpg" xlink:href="009/01/177/1.jpg" place="text"/>

<lb/>fultura manus in E, potentia alterius ma

<lb/>nus in F. iam inquir Ari&longs;t.

Line 13474

<lb/>magna vi percutit, quia motus mouet, &longs;eu quia mouens malleum, mouet

<lb/>ip&longs;um etiam dum e&longs;t in ip&longs;a latione, vnde ip&longs;a lationis celeritate malleus

<lb/>fit valentior: <expan abbr="hoc&qacute;">hocque</expan>; modo paruos cunci vectes maiores con&longs;equuntur vires,

<lb/>quàm ip&longs;a vectium magnitudo po&longs;tulet.

Line 13525

<lb/>res parua vi attollunt ingentia pondera. </s>

<s id="id.003037">communiter autem con&longs;tat ex plu

<lb/>ribus orbiculis, qui partim &longs;uperiori loculamento,

<lb/>partim infetiori inditi &longs;unt, per quos orbiculos cer

<lb/>ta lege circumductus e&longs;t ductarius funis, qui deinde

Line 13700

<lb/>nem antiquæ &longs;ecuris, &longs;ic puto eruendam. </s>

<s id="id.003078">Ducatur primo recta A C, quæ

<lb/>erit in&longs;tar manubrij &longs;ecuris. </s>

<s id="id.003079">de

Line 13728

<s id="id.003082">De&longs;cribatur iterum &longs;ecuralis figura prædicto modo,

<lb/>&longs;<expan abbr="itq;">itque</expan> ea A B C. ducantur præterea ad &longs;ingulos angulos tres rectæ A B, B C,

<lb/>C A, quæ con&longs;tituunt trianguium æquilaterum A B C, tria enim ip&longs;ius late

<lb/>ra &longs;ubtendunt tres arcus æquales A B, B C, C A,

<lb/>&longs;unt enim tres &longs;extantes æqualium circulorum,

<lb/>ut facilè colligi pote&longs;t ex 15. 4. ex quo etiam &longs;e

Line 13761

<s id="id.003088">Re&longs;tat, vt de antiquæ bipennis etiam figura di&longs;&longs;eramus; quæ nihil aliud

<lb/>erat, quàm duplex &longs;ecuris, &longs;iue &longs;ecuris anceps, qualis e&longs;t præ&longs;ens figura, vt

<lb/>propterea etiam &longs;æpius <expan abbr="bip&etilde;nis">bipennis</expan> ip

<lb/>&longs;a &longs;ecuris appelletur. </s>

Line 13829

<lb/>tere nolui.</s><pb xlink:href="009/01/182.jpg" pagenum="182"/>

<s id="id.003103"><emph type="italics"/>Simmiæ Rhodij

<lb/>Bipennis.<emph.end type="italics"/></s><figure id="id.009.01.182.1.jpg" xlink:href="009/01/182/1.jpg"lace="text"/><pb xlink:href="009/01/183.jpg" pagenum="183"/>

<lb/>Bipennis.<emph.end type="italics"/></s><figure id="id.009.01.182.1.jpg" xlink:href="009/01/182/1.jpg" place="text"/><pb xlink:href="009/01/183.jpg" pagenum="183"/>

<s id="id.003104"><emph type="italics"/>QVÆSTIO XX.<emph.end type="italics"/></s>

Line 13849

<lb/>adulterinas reijcere; <expan abbr="in&qacute;">inque</expan>; earum locum veras re&longs;tituere licebit. </s>

<s id="id.003109">erat igitur

<lb/>&longs;tatera, quantum ex Ari&longs;t.

conijcio

Line 13986

<s id="id.003140">Aut tandem, quia forceps hæc duos

<lb/>in &longs;e continet contrarios vectes; quorum, vnum tantum e&longs;t hypomoclion,

<lb/>corum &longs;cilicet connexio; Virtute igitur

<lb/>vectis arctius dentem per&longs;tringunt, <expan abbr="atq;">atque</expan>

<lb/>adeò obtinent, <expan abbr="atq;">atque</expan> hinc etiam facilius

Line 14053

<s id="id.003158">qui duo vectes vnicum habent hypomoclion ip&longs;am &longs;cilicet connexionem

<lb/>A. vectes &longs;unt binæ in&longs;trumenti ha&longs;tæ, F A D,

<lb/>E A C. <expan abbr="dilatãdo">dilatando</expan> igitur extrema C D, deducun

<lb/>tur etiam alia extrema F, E, & impo&longs;ita nuce in

Line 14083

<s id="id.003165">Rhombus ex definitione 23. primi Elem.

e&longs;t figura æquilatera qui

<lb/>dem, &longs;ed non æquiangula, habet enim

<lb/>binos oppo&longs;itos angulos acutos, & alies

<lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ

Line 14242

<lb/>cundum æqualem conuoluuntur, quando circa idem po&longs;iti fuerint centrum,

<lb/>manife&longs;tum e&longs;t, ita tamen, vt aliquando ambæ æquales &longs;int ei, &longs;ecundum

<lb/>quam &longs;olus maior conuolueretur; aliquando verò &longs;ecundum quam minor.

<lb/>&longs;it enim circulus maior quidem vbi

<lb/>D F C, minor verò vbi E G B, <expan abbr="vtriq;">vtrique</expan>

<lb/>autem centrum A, linea, &longs;ecundum

Line 14385

<lb/>diametraliter &longs;ubtendantur.</s>

<s id="id.003222">Tertia demum ratio e&longs;t, quia hac ratione minus re&longs;tium ab&longs;umitur, quæ

<lb/>vt benè intelligatur, de&longs;eriba

<lb/>tur lectuli figura A F G K, &

<lb/>bifariam diuidatur latus F G,

Line 14444

<s id="id.003230">&longs;i autem &longs;eeundum diame

<lb/>trum extendantur re&longs;tes, vti factum e&longs;t in lectulo A B C D, neutiquam re

<lb/>&longs;tes omnes &longs;imul &longs;uperiori quantitati adæquabuntur, &longs;ed illam longè &longs;upe

<lb/>rabunt. </s>

<s id="id.003231">Sit igitur lectus A B

Line 14503

<lb/>i&longs;tud æquilibrandum, opus e&longs;t alia potentia in altero extremo. </s>

<s id="id.003243">&longs;it lignum

<lb/>A B, &longs;u&longs;pen&longs;um ex medio C.

<lb/>hoc modo lignum ponderi

<lb/>bus libratum &longs;uis manet in

<lb/>æquilibrio, pote&longs;tque à &longs;ola

<lb/>potentia illud eleuante etiam deferri: quia A, & B, extrema &longs;e mutuò &longs;u&longs;ti

<s id="id.003244">quod &longs;i non ex medio eleuaretur,

Line 14568

<lb/>con&longs;truunt, quale à figura &longs;equenti refertur. </s>

<s id="id.003260">vbi puteus F, tolleno con&longs;tat

<lb/>erecto tigno D C, & tran&longs;

<lb/>uer&longs;a ha&longs;ta A C B, vnà cum

<lb/>fune B E, & hydria E. ap

Line 14646

<s id="id.003278">Cvm &longs;edemus, præcipuè &longs;i commodè &longs;edeamus, &longs;olemus duos angu

<lb/>los rectos facere, vnum quidem, quem facit thorax cum femore;

<lb/>alterum quem facit femur cum crure, vt in figura thorax &longs;it A B,

<lb/>femur B C, crus C D, anguli duo recti &longs;unt B,

Line 14799

<s id="id.003311">Lunam &longs;cilicet eam habere vim in mare, vt pars il

<lb/>la, quæ Lunæ &longs;irbiacet, &longs;iue quam Luna radijs ferit, æftuet, & turgeat; non

<lb/>aliter pars maris huic antipoda, & auer&longs;a, quamuis tota terræ moles inter

<lb/>ip&longs;am, & Lunam interpona

<lb/>tur, æ&longs;tuat, fluxumque, ac re

<lb/>fluxum quamuis priori mi

Line 14983

<s id="id.003352">Ex &longs;tellis igitur circa A, & C, reflecti pote&longs;t ex quarto fundamen

<lb/>to lumen <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> luminaris ad D, quod &longs;atis e&longs;t efficax, cùm ferè perpendi<pb xlink:href="009/01/199.jpg" pagenum="199"/><figure id="id.009.01.199.1.jpg" xlink:href="009/01/199/1.jpg"lace="text"/>

<lb/>culariter terræ D, incidat. </s>

<s id="id.003353">quamuis autem ex &longs;tellis F, E, lumen aliquod ad

Line 15251

<s id="id.003418">Sciendum e&longs;t omnes lineas <expan abbr="cõmen&longs;urabiles">commen&longs;urabiles</expan> longitudine, e&longs;&longs;e etiam com

<lb/>men&longs;urabiles (vt aiunt Geometræ) potentia, ide&longs;t &longs;ecundum quadrata ea<pb xlink:href="009/01/203.jpg" pagenum="203"/>rum, &longs;iue dicas quadrata <expan abbr="quoq;">quoque</expan> earum e&longs;&longs;e commen&longs;urabilia, v. g. linea dua

<lb/>rum vnciarum, & linea trium vnciarum &longs;unt

<lb/>commen&longs;urabiles longitudine, & potentia,

<lb/>quia potentia lineæ duarum vnciarum, &longs;iue

Line 15290

<s id="id.003425">Propo&longs;ita igitur linea quapiam, v. g. trium palmorum qualis e&longs;t linea A,

<lb/>po&longs;&longs;unt inueniri quamplurimæ lineæ, quarum aliæ &longs;int illi longitudine com

<lb/>men&longs;urabiles, &longs;iue quæ cum expo&longs;ita A, ha

<lb/>beant communem men&longs;uram. </s>

Line 15307

<s id="id.003427">Cæterum lineam primò expo&longs;itam, vt e&longs;t in præ

<lb/>&longs;entia A, quod e&longs;&longs;et notæ quantitatis, Græci appella

Line 15331

<s id="id.003431">Quæ igitur rationali propo&longs;itæ &longs;unt commen&longs;urabiles aliquo modo, &longs;iue

<lb/>longitudine, & potentia (<expan abbr="quæcunq;">quæcunque</expan> enim commen&longs;urabilis e&longs;t longitudine,

<lb/>e&longs;t etiam potentia) &longs;iue potentia &longs;olùm, rationales ip&longs;æ quoque dicuntur. <pb xlink:href="009/01/204.jpg" pagenum="204"/><figure id="id.009.01.204.1.jpg" xlink:href="009/01/204/1.jpg"lace="text"/>

<lb/>Aliæ verò (quarum permultæ in decimo reperiun

<lb/>tur) quæ nec longitudine, nec potentia illi &longs;unt

<lb/>commen&longs;urabiles, irrationales appellantur, qua

Line 15343

<s id="id.003433">v. g. A, com

<lb/>men&longs;urabilibus inuicem tantum potentia, componitur linea, quæ cum ea

<lb/>dem expo&longs;ita e&longs;t irrationalis, <expan abbr="vocatur&qacute;">vocaturque</expan>; ex

<lb/>duobus nominibus, &longs;iue Binomium, vt &longs;i ex

<lb/>latere A, & diametro C D, componatur li

Line 15430

<lb/>contra mathematicorum dogmata, non enim Geometræ hoc a&longs;&longs;erunt, cùm

<lb/>ip&longs;orum demon&longs;trationibus a duer&longs;etur; &longs;ed <expan abbr="tant&utilde;">tantum</expan> dicunt omnes lineas, quæ

<lb/>ad inuicem &longs;unt commen&longs;urabiles, commen&longs;urari, vna <expan abbr="eadem&qacute;">eademque</expan>; men&longs;ura,

<lb/>&longs;ed non tamen vnica, ide&longs;t non vnica, ac determi

Line 15460

<lb/>ac &longs;ine ratione, imò contra rationem addidit: tum quia in Græco textu non

<lb/>extant, tum quia &longs;en&longs;us totius &longs;ententiæ is e&longs;t, vt potius debui&longs;&longs;et affirmati

<lb/>uè dicere <emph type="italics"/>(Diuiduas credere)<emph.end type="italics"/> nam Ari&longs;toteles videtur &longs;ic <expan abbr="argum&etilde;tari">argumentari</expan>, quan

<lb/>do recta linea A B, vt in appo&longs;ita figura mo

<lb/>uetur intrando in &longs;emicirculum C A D B, ita

<lb/>vt primò &longs;it in &longs;itu A B, &longs;ecundò in E F, tertiò

Line 15491

<lb/>illorum dicta.<emph.end type="italics"/></s>

<s id="id.003461">Hæc e&longs;t alia ratio, qua probat totam circuli periphæriam e&longs;&longs;e diuiduam.

<lb/>&longs;int enim duo circuli æquales primum in eo

<lb/>dem loco, <expan abbr="vocentur&qacute;">vocenturque</expan>; A, & B, deinde circu

<lb/>lus B, moueatur, & di&longs;cedat à circulo A, ma

Line 15597

<lb/>rem lstitudinem facit applicata, æquale ei, quod ab indiuidua, & pedali copulatis

<lb/>circa bipedalem, minorem faciet latitudinem, quàm &longs;it indiuidua: erit minus, quod

<lb/>circa indiuiduam)<emph.end type="italics"/> ide&longs;t cùm minor linea applicata cum maiore, latitudinem

<lb/>faciat. </s>

<s id="id.003486">v. g. linea minor A B, applicata cum ma

Line 15615

<lb/>aduer&longs;arij velint extare huiu&longs;modi lineas ato

<lb/>mas, con&longs;tituatur figura &longs;ub vna ex illis, quæ &longs;it v. g. A B, & altera maiori,

<lb/>quæ &longs;it pedalis, v. g. B C, vt in præcedenti figura, &longs;umatur deinde linea bi

<lb/>pedalis E F, cui per 45. primi ap

<lb/>plicetur &longs;patium E F G H, æquale

<lb/>&longs;patio &longs;uperiori A B C D, nece&longs;&longs;a

Line 15650

ex tribus datis lineis, quarum quælibet duæ &longs;int, re

<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui

<lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in

<lb/>diuiduæ lineæ &longs;int æquales. </s>

Line 15669

<s id="id.003499">neque duplum erit &longs;patium à diame

<lb/>tro con&longs;urgens illius, quod ab indiuidua procreatur: nans æquali ablato, reliquum

<lb/>erit minus indiutdua, nam &longs;i æqualis, diameter quadruplum de&longs;criberet, &c.)<emph.end type="italics"/>

<lb/>ide&longs;t &longs;i per 46 primi quadratum. </s>

<s id="id.003500">v.g. A B C D, ex qua

Line 15807

<s id="id.003534">iam &longs;i linea con&longs;taret ex punctis indiuiduis

<lb/>tanquam partibus, po&longs;&longs;et circulus <expan abbr="tãgere">tangere</expan> rectam lineam in duobus punctis.

<lb/>Sit circulus, cuius centrum A, tangens lineam

<lb/>rectam B C, con&longs;tantem ex punctis, quorum vnus

<lb/>&longs;it in extremo D, lineæ B D, alterum verò in E,

Line 15962

<s id="id.003566">Ex qna repetita &longs;æpius ob&longs;eruatione lu

<lb/>ce clarius comperi Araneum non &longs;implex filum, &longs;ed ramo&longs;um, ac multiplex

<lb/>emittere, <expan abbr="atq;">atque</expan> aliquando ex &longs;ece&longs;&longs;u etiam ip&longs;o duo &longs;imul eijcere, <expan abbr="alter&utilde;">alterum</expan> quo

<lb/>&longs;u&longs;pendatur, alterum

<lb/>verò, quod &longs;orte hac,

<lb/><expan abbr="atq;">atque</expan> illac volitans, ali

Line 16046

<lb/>&longs;ciendum, quod lineam hypotenu&longs;am, quemadmodum etiam Athenæus lib.

<lb/>10. te&longs;tatur, eam appellant geometræ, quæ in triangulo rectangulo recto

<lb/>angulo &longs;ubtenditur, vnde & denominata e&longs;t hypotenu&longs;a, ide&longs;t &longs;ubten&longs;a, vt

<lb/>in triangulo A B C, cuius angulus B, rectus &longs;it, recta

<lb/>A C, angulo recto B, &longs;ubten&longs;a, hypotenu&longs;a dicitur.

Line 16069

<lb/>A D, fiet hypotenu&longs;a trianguli A B C, & quia crus hoc A D, factum hypo

<lb/>tenu&longs;a æquale e&longs;t alteri manenti A B, nequit totius veræ hypotenufæ A C,

<lb/>officio fungi, quæ æquiualet toti A D, & præterea interiacenti D C, vrea au

<lb/>tem hypotenu&longs;a debet e&longs;&longs;e maior, quia opponitun

<lb/>maiori angulo nimirum recto B, quam latus A B,

<lb/>quod angulo acuto C, opponitur per 19. primi, &

Line 16141

<lb/>e&longs;t pa&longs;&longs;us. </s>

<s id="id.003596">& tunc caput ambulantis fit inferius, quàm antequam gradere

<lb/>tur; quia tunc ambo crura erant horizonti perpen

<lb/>dicularia. </s>

Line 16176

<s id="id.003603">cum autem flectitur, & mouetur, hoc quidem &longs;ignum mo-<emph.end type="italics"/>

<lb/><emph type="italics"/>etur, illud autem manet in flexibus, quemadmodum <expan abbr="vtiq;">vtique</expan> &longs;i dia

<lb/>metri, quæ quidem A D, maneat, quæ cutem B, moueatur, &

<lb/>fiat A C, &longs;ed hic quidem videtur, &longs;ecundum omnem modum in

Line 16306

<lb/><expan abbr="vtriq;">vtrique</expan> con&longs;iderant: ille quatenus <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ad opus vtile e&longs;t, hic verò cum ve

<lb/>ritatis &longs;peculator &longs;it, quid, & qualis &longs;it, indagat)<emph.end type="italics"/> Id quod dicit Ari&longs;t.

<lb/>confirmatur ex eo, quod Fabri omnes vtuntur amu&longs;&longs;i, &longs;eu norma,

<lb/>quæ nihil aliud e&longs;t quàm angulus rectus, quæ vulgò

<lb/>&longs;quadra dicitur, vt eius auxilio angulum ip&longs;um re

<lb/>ctum in opus conferant, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; normæ, aut amu&longs;&longs;is du

Line 16344

<s id="id.003641">Porrò apud Mathematicos tria &longs;unt genera pro

<lb/>portionum, &longs;iue medietatum, Arithmetica quam modo &longs;uppo&longs;ui; Geome

<lb/>trica, & Harmonica, quas inferius oblata occa&longs;ione opportunius explicabo.</s><figure id="id.009.01.217.1.jpg" xlink:href="009/01/217/1.jpg"lace="text"/>

<lb/>trica, & Harmonica, quas inferius oblata occa&longs;ione opportunius explicabo.</s><figure id="id.009.01.217.1.jpg" xlink:href="009/01/217/1.jpg" place="text"/>

Line 17459

<lb/>rumpunt: committitur enim rectilmea figura in argulis, vel &longs;ecundum angulos)<emph.end type="italics"/>

<lb/>Vt rectè problema hoc percipiamus, proponenda e&longs;t figura rectilinea, &

<lb/>vna ex ijs, quæ paralle logramma dicuntur, vt &longs;unt Quadratum, Quadrila

<lb/>terum, Rhombus, Rhomboides, cuiu&longs;mo

<lb/>di e&longs;t præ&longs;ens, aliter verba Ari&longs;t.

Line 17742

<s id="id.003956">pro cuius intelligentia, ac &longs;olutione in&longs;picienda e&longs;t figura &longs;equens:

<lb/>in qua &longs;emicirculus, &longs;iue arcus A B C, &longs;it is, per quem Sol incedit, dum ele

<lb/>uatur &longs;upra horizontem H I; & quia Sol vniformiter &longs;candit hunc arcum, <pb xlink:href="009/01/231.jpg" pagenum="231"/><figure id="id.009.01.231.1.jpg" xlink:href="009/01/231/1.jpg"lace="text"/>

<lb/>&longs;int duo arcus æquales A B, B C.

<lb/>in plano autem horizontis ere

<lb/>ctum &longs;it corpus G D, quod à So

Line 17955

<s id="id.003999">Secundò, quò magis duorum vicinorum circulorum peripheriæ augen

<lb/>tur, eò magis ad vnius circuli &longs;imilitudinem accedere; vt in hac figura cer

<lb/>nere licet, vbi &longs;unt primò duo parui

<lb/>circuli circa centra A, & B, de&longs;cripti,

<lb/>quorum circunferentiæ cre&longs;cant <expan abbr="v&longs;q;">v&longs;que</expan>

Line 17980

<s id="id.004003">Tertiò, lumen Solis per foramen tam <expan abbr="exigu&utilde;">exiguum</expan>, quod &longs;it in&longs;tar puncti tran&longs;

<lb/>mi&longs;&longs;um, <expan abbr="figurã">figuram</expan> Solis rotundam videlicet, quamuis conuer&longs;am referre; quod

<lb/>hac de&longs;criptione patefiet. </s>

<s id="id.004004">&longs;it Sol vbi A B, fora

Line 18051

<s id="id.004017">ex tertia igitur pr&ecedil;

<lb/>notatione per punctum C, procedit rotunda illuminatio, cuius diameter

<lb/>F H, &longs;imiliter per punctum D, illuminatio

<lb/>rotunda emanat, cuius diameter K E, & pa

<lb/>riter ex omnibus alijs rimulæ punctis ro

Line 18263

<lb/>cùm Luna &longs;emiplena e&longs;t, linea illa, quæ terminus e&longs;t partis illuminatæ, &

<lb/>partis ob&longs;curæ, <expan abbr="quæ&qacute;">quæque</expan>; Lunam bifariam diuidit, videtur linea recta, cùm ta

<lb/>men non &longs;it; cùm enim fit in globo&longs;a &longs;uperficie Lunæ, nece&longs;&longs;ariò circularis

<lb/>e&longs;t? </s>

<s id="id.004062">vt autem rem hanc melius intelli

Line 18408

<lb/>norem &longs;ubinde? </s>

<s id="id.004088">quia linea recta, quæ de &longs;uperiori puncto elicitur, interius cadit.<emph.end type="italics"/>

<lb/><emph type="italics"/>&longs;it Gnomon A B. Sol, vbi C, & vbi D.

<lb/>radius igitur ex C, prefici&longs;cens, e&longs;t C F,

<lb/>& exterius proced t, quàm radius D E.

Line 18445

<s id="id.004096"><expan abbr="itaq;">itaque</expan> nece&longs;&longs;e e&longs;t radium à &longs;upe

<lb/>riore procedentem intus cadere. </s>

<s id="id.004097">&longs;it Gnomon A D, Luna B Sol C, Lunæ radius B F.<emph.end type="italics"/><pb xlink:href="009/01/240.jpg" pagenum="240"/><figure id="id.009.01.240.1.jpg" xlink:href="009/01/240/1.jpg"lace="text"/>

<lb/><emph type="italics"/>ergò vmbra Lunæ D F, Solis radius

<lb/>C E; vmbra igitur nece&longs;&longs;ariò minor

<lb/>e&longs;t, e&longs;t enim D E.)<emph.end type="italics"/> de hac re vide

Line 18548

<lb/>c&oelig;lo contingit: hoc e&longs;t, &longs;i in c&oelig;lo &longs;uperior pars deliquium patiatur, in ra

<lb/>dijs apparebit inferior deficere, vt ratio exigit optica) <expan abbr="hucu&longs;q;">hucu&longs;que</expan> Gemma Fri

<lb/>&longs;ius; ex quo etiam placuit accipere totius huius <expan abbr="experi&etilde;tiæ">experientiæ</expan> figuram, quam

<lb/>ip&longs;e <expan abbr="cuiu&longs;dã">cuiu&longs;dam</expan> eclyp&longs;is an

<lb/>ni 1544. apponit. </s>

Line 18661

<s id="id.004138">quod qui

<lb/>dem antiquitus demon&longs;trauit Ari&longs;tarchus Samius in libello de di&longs;tantijs So

<lb/>lis, & Lunæ: &

<lb/>po&longs;tea Vitellio

<lb/>lib.

Line 18739

<s id="id.004156">Cur ea, quæ in

<lb/>terram cadunt, <expan abbr="atq;">atque</expan> re&longs;iliunt angulos ad planitiem, faciunt &longs;imiles vtraque

<lb/>ex parte, qua planum tetigerint? </s>

<s id="id.004157">v. g. &longs;i

Line 18786

<lb/>&longs;eu ad angulos rectos, & ideo nece&longs;&longs;ariò per candein lineam, qua decide

<lb/>rant, repercutiuntur; cuius cau&longs;a e&longs;&longs;e pote&longs;t, quia diameter &longs;cilicet mundi,

<lb/>per quam delap&longs;a &longs;unt, ea bifariam diuidit, vt in figura, graue E, per D C,

<lb/>plano A B, perpendicularem de&longs;cendat;

<lb/>quæ <expan abbr="perp&etilde;dicularis">perpendicularis</expan> coincidit cum mun

<lb/>di diametro, perpendicula enim omnia

Line 18986

<s id="id.004205">In 2. problema. </s>

<s id="id.004206">In verba illa <emph type="italics"/>(Sed quemadmodum linea bipedalis non<emph.end type="italics"/>

<lb/><emph type="italics"/>duplum, &longs;ed quadruplum quoddam de&longs;cribit,

<lb/>&longs;ic, &c.)<emph.end type="italics"/> ide&longs;t, quemadmodum linea bi

<lb/>pedalis, quæ quamuis &longs;it dupla lineæ pedalis non

Line 19042

<lb/>tationi obnoxiæ. </s>

<s id="id.004218">&longs;int hæ duæ chordæ æquales omninò, atque æquè inten&longs;æ,

<lb/>ita vt &longs;int vni&longs;onæ, hoc e&longs;t, vna <expan abbr="tan-t&utilde;">tan

<lb/>tum</expan> vox &longs;it. </s>

Line 19095

<lb/>plam, aut octauam reddunt.</s>

<s id="id.004228">Rur&longs;us eadem chorda C D, diuidatur in tres partes æquales in punctis

<s id="id.004229">F D igitur erit duæ ter

Line 19122

<lb/>& quinta, Sol, con&longs;tat.</s>

<s id="id.004234">Eadem iterum chorda in quatuor æquas partes &longs;ecetur in punctis H, E, I,

<lb/>ita vt chorda H D, &longs;it tres quartæ

<lb/>totius A D. facto deinde tactu in

<lb/>H, pul&longs;entur &longs;imul A B, H D, & au

Line 19238

<s id="id.004255">vete

<lb/>res illos Græcos denomina&longs;&longs;e, nam primam, ide&longs;t graui&longs;&longs;imam vocem, &longs;iue

<lb/>chordam, quam modò Vt, dicunt, eam ip&longs;i Hypaten vocarunt, & reliquas

<lb/>ordine &longs;equenti.</s><figure id="id.009.01.250.1.jpg" xlink:href="009/01/250/1.jpg"lace="text"/>

<lb/>ordine &longs;equenti.</s><figure id="id.009.01.250.1.jpg" xlink:href="009/01/250/1.jpg" place="text"/>

<s id="id.004256">Vt, Hypate — ide&longs;t Principalis.</s>

Line 19294

<s id="id.004274">vbi per cantandi principium puto ip&longs;um intellige

<lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vtip&longs;e ait primo

<lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vt ip&longs;e ait primo

<lb/>Po&longs;ter. cap.

Line 19432

<lb/>neten non hypatèn omittebant? </s>

<s id="id.004311">&longs;upra octo voces, &longs;eu chordas recen&longs;ui, <expan abbr="ra-tionem&qacute;">ra

<lb/>tionemque</expan>; ip&longs;arum, vnà cum antiquis appellationibus explicani, quarum <pb xlink:href="009/01/252.jpg" pagenum="252"/><figure id="id.009.01.252.1.jpg" xlink:href="009/01/252/1.jpg"lace="text"/>

<lb/>prima e&longs;t hypate, vltima verò note, quibus re

<lb/>petitis facilè e&longs;t intelligere, quod re&longs;pondet

<lb/>Ari&longs;t. </s>

Line 19785

<lb/>bus, vt, re, mi, fa; <expan abbr="quaru&utilde;">quarum</expan> vocum interualla vnius tetrachordi generis, vnius,

<lb/>differebant ab interuallis alterius tetrachordi alterius generis, v.g. in ge

<lb/>nere Diatonico erat huiu&longs;modi tetrachordium, cuius primum interualllum

<lb/><figure id="id.009.01.255.1.jpg" xlink:href="009/01/255/1.jpg"lace="text"/>erat &longs;omitonium, reliqua verò duo erant to

<lb/><figure id="id.009.01.255.1.jpg" xlink:href="009/01/255/1.jpg" place="text"/>erat &longs;omitonium, reliqua verò duo erant to

<s id="id.004403">& à prima voce Hypate, ad vltimam Me

Line 19801

<lb/><expan abbr="chromatic&utilde;">chromaticum</expan> inter chordas &longs;ui tetrachor

<lb/>di <expan abbr="&longs;equ&etilde;tia">&longs;equentia</expan> interualla &longs;eruabat. </s>

Line 19815

<s id="id.004406">Enharmo

<lb/>nicum tandem genus tetrachordo vtebatur, cuius interualla erant ea, quæ

<lb/>&longs;equuntur.</s><figure id="id.009.01.256.2.jpg" xlink:href="009/01/256/2.jpg"lace="text"/>

<lb/>&longs;equuntur.</s><figure id="id.009.01.256.2.jpg" xlink:href="009/01/256/2.jpg" place="text"/>

<s id="id.004407">Erat hic etiam inter hypate, & Me&longs;e Diate&longs;&longs;aron; huic aliud tetrachor

<lb/>dum pariter addebatur, vt in alijs generibus, ex quibus tota Diapa&longs;on <expan abbr="cõ-flabatur">con

Line 20365

<lb/>termedios, alijs grauiores fui&longs;&longs;e, vt in &longs;equenti figura,

<lb/>in qua, tanquam in tabella, omnia, quæ de hi&longs;ce

<lb/>modis dici &longs;olent, per&longs;picuè licet

<lb/>intucri.</s><pb xlink:href="009/01/262.jpg" pagenum="262"/><figure id="id.009.01.262.1.jpg" xlink:href="009/01/262/1.jpg"lace="text"/>

<lb/>intucri.</s><pb xlink:href="009/01/262.jpg" pagenum="262"/><figure id="id.009.01.262.1.jpg" xlink:href="009/01/262/1.jpg" place="text"/>

<s id="id.004546"><emph type="italics"/>ORDO ANTIQVORVM MODORVM.<emph.end type="italics"/>

<lb/><arrow.to.target n="table6"/></s><pb xlink:href="009/01/263.jpg" pagenum="263"/><table><table.target id="table6"/><row><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell>Nete.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell>T.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell>S.</cell><cell>S.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell/><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell></row><row><cell/><cell/><cell/><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell></row><row><cell/><cell/><cell/><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell></row><row><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos</cell><cell>Parhypate</cell></row><row><cell/><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypate.</cell></row><row><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypatc.</cell><cell>Pro&longs;lamb.</cell></row><row><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>— —</cell></row><row><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Hypermix</cell></row><row><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>— —</cell><cell>tolydius.</cell></row><row><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Mixtolyd.</cell><cell/></row><row><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>— —</cell><cell/><cell/></row><row><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Lydius.</cell><cell/><cell/></row><row><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>— —</cell><cell/><cell/><cell/></row><row><cell>Lychanos</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Phrygius.</cell><cell/><cell/><cell/></row><row><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>— —</cell><cell/><cell/><cell/><cell/></row><row><cell>Parhypate</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Dorius.</cell><cell/><cell/><cell/><cell/></row><row><cell>T.</cell><cell>T.</cell><cell>— —</cell><cell/><cell/><cell/><cell/><cell/></row><row><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Hypolyd.</cell><cell/><cell/><cell/><cell/><cell/></row><row><cell>T.</cell><cell>— —</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>Pro&longs;lamb.</cell><cell>Hypophry</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>— —</cell><cell>gius.</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>Hypodor.</cell><cell/><cell/><cell/><cell/><cell/><cell/><cell/></row></table>

Line 21160

<lb/>raphra&longs;im percipies. </s>

<s id="id.004760">Verum ante omnia antiquæ lyræ ex antiquis monu

<lb/>mentis figuram oculis &longs;ubijciam.</s><figure id="id.009.01.270.1.jpg" xlink:href="009/01/270/1.jpg"lace="text"/>

<lb/>mentis figuram oculis &longs;ubijciam.</s><figure id="id.009.01.270.1.jpg" xlink:href="009/01/270/1.jpg" place="text"/>

<s id="id.004761">Porrò, vt tradit Vincentius Galilæus in &longs;uis Dialogis, erat eius figura,

<lb/>qua&longs;i ex caprino capite con&longs;tructa, cuius duo brachia erant capræ cornua;

Line 21745

<s id="id.004915">vt autem &longs;pecies <expan abbr="vtrumq;">vtrumque</expan> oculum con&longs;imiliter <expan abbr="inform&etilde;t">informent</expan>, ne

<lb/>ce&longs;&longs;e e&longs;t, vt axes vi&longs;uales, quales

<lb/>&longs;unt in appo&longs;ita figura, C B, D B,

<lb/>ab oculis C, D, ad obiectum B,

Line 21873

<s id="id.004947">Vt ea, quæ dicenda &longs;unt, facilius percipi po&longs;&longs;int, nece&longs;&longs;e e&longs;t breuiter

<lb/>oculi fabricam, in qua mirum totius naturæ Opificis artificium elu

<lb/>cet, auxilio &longs;equentium figurarum præmittere.</s><figure id="id.009.01.278.1.jpg" xlink:href="009/01/278/1.jpg"lace="text"/>

<lb/>cet, auxilio &longs;equentium figurarum præmittere.</s><figure id="id.009.01.278.1.jpg" xlink:href="009/01/278/1.jpg" place="text"/>

<s id="id.004948">A B C, cornea. </s>

Legend:

Removed from v.1.2
changed lines
	Added in v.1.3