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<archimedes xmlns:xlink="http://www.w3.org/1999/xlink" >
  <info>
    <author>Cardano, Girolamo</author>
    <title>Opus novum de proportionibus</title>
    <date>1570</date>
    <place>Basel</place>
    <translator></translator>
    <lang>la</lang>
    <cvs_file>carda_propo_015_la_1570.xml</cvs_file>
    <cvs_version></cvs_version>
    <locator>015.xml</locator>
  </info>
  <text>
    <front>
      <section>
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        <p type="head">
          <s id="id000001">HIERONYMI <lb></lb>CARDANI MEDIO<lb></lb>LANENSIS, CIVISQVE BONO­<lb></lb>NIENSIS, PHILOSOPHI, MEDICI ET <lb></lb>Mathematici clariſsimi,</s>
        </p>
        <p type="head">
          <s id="id000002">OPVS NOVVM DE <lb></lb>PROPORTIONIBVS NVMERORVM, MO<lb></lb>TVVM, PONDERVM, SONORVM, ALIARVMQVE RERVM <lb></lb>menſurandarum, non ſolùm Geometrico  more ſtabilitum, ſed etiam <lb></lb>uarijs experimentis &amp; obſeruationibus rerum in natura, ſolerti <lb></lb>demonſtratione illuſtratum, ad multiplices uſus ac­<lb></lb>commodatum, &amp; in &lt;var&gt;V&lt;/var&gt; libros digeſtum.</s>
        </p>
        <p type="head">
          <s id="id000003">PRAETEREA.</s>
        </p>
        <p type="head">
          <s id="id000004">ARTIS MAGNÆ, SIVE DE REGVLIS <lb></lb>ALGEBRAICIS, LIBER VNVS, ABSTRVSISSIMVS <lb></lb>&amp; inexhauſtus plane totius Arithmeticæ theſaurus, ab <lb></lb>authore recens multis in locis recogni­<lb></lb>tus &amp; auctus.</s>
        </p>
        <p type="head">
          <s id="id000005">ITEM.</s>
        </p>
        <p type="head">
          <s id="id000006">DE ALIZA REGVLA LIBER, HOC EST, ALGEBRAICAE <lb></lb>logiſticæ ſuæ, numeros recondita numerandi ſubtilitate, ſecundum Geo­<lb></lb>metricas quantitates inquirentis, neceſſaria Coronis, <lb></lb>nunc demum in lucem edita.</s>
        </p>
        <p type="head">
          <s id="id000007">O<emph type="italics"></emph>pus<emph.end type="italics"></emph.end> P<emph type="italics"></emph>hyſicis &amp;<emph.end type="italics"></emph.end> M<emph type="italics"></emph>athematicis imprimis <lb></lb>utile &amp; neceſſarium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id000008">Cum Cæſ. </s>
          <s id="id000009">Maieſt. </s>
          <s id="id000010">Gratia &amp; Priuilegio.</s>
        </p>
        <p type="head">
          <s id="id000011">BASILEÆ.</s>
        </p>
      </section>
      <section>
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        <p type="head">
          <s id="id000012">IN LIBRVM DE <lb></lb>PROPORTIONIBVS HIERONYMI <lb></lb>CARDANI MEDIOLANENSIS, CIVISQVE <lb></lb>Bononienſis, Medici, Præfatio ad M. A. </s>
          <s id="id000013">Amulium <lb></lb>Venetum Card. </s>
          <s id="id000014">Illuſtriſsimum.</s>
        </p>
        <p type="main">
          <s id="id000015">Bene Dictum eſt  meo iudicio à Platone M.  <lb></lb>A. </s>
          <s id="id000016">Amuli optime, beatas fore Reſpub. </s>
          <s id="id000017">ſi uel <lb></lb>illarum domini ſapientiæ amatores eſſent, <lb></lb>aut qui ſapientiæ eſſent amatores domina­<lb></lb>rentur, hoc ipſum clarè intelligens, ſtudio ſa<lb></lb>pientiæ nihil eſſe utilius humano generi: <lb></lb>quo ſimul &amp; pietas, &amp; iuſtitia, &amp; mutuus <lb></lb>amor hominum inter ſe &amp; eorum commo­<lb></lb>da continerentur. </s>
          <s id="id000018">Nempe hiſce quatuor tota noſtra felicitas com­<lb></lb>prehenditur. </s>
          <s id="id000019">Si quidem pietate in Deos nihil  niſi ſanctum, &amp; pu­<lb></lb>rum, &amp; illuſtre ſapimus: hoc ipſo primum quod ſupra nos eſt, intel­<lb></lb>ligimus, Deos ueneramur, gratias agimus, timor cum ueneratione <lb></lb>noſtros animos ſubit, &amp; de futura uita cogitamus, hæc ipſa morta­<lb></lb>lia ſi non negligentes ſaltem paruifacientes. </s>
          <s id="id000020">Iuſtitiam autem adeò <lb></lb>neceſſariam humano generi eſſe ſcimus, ut ſine illa neque eſſe, nedum <lb></lb>benè eſſe poſsímus, ut neque latronum cœtus abſque ea diu ſtare poſ­<lb></lb>ſint. </s>
          <s id="id000021">Porrò quid dicam de concordia, &amp; mutua hominum beneuo­<lb></lb>lentia, in quibus omnis uitę humanę dulcedo repoſita eſt: nec quis <lb></lb>ſuſtineat uiuere, qui ſe omnibus odioſum eſſe ſentiat. </s>
          <s id="id000022">His ipſis fi­<lb></lb>lios in ſpem alimus, parentes fouemus, fratres tuemur, &amp; adiuua­<lb></lb>mus, amicis opitulamur, cum hominibus hilarem &amp; iucundam ui­<lb></lb>tam ducimus. </s>
          <s id="id000023">Si quis ſerpentem in lecto haberet, nunquam ſom­<lb></lb>num caperet: ita nihil moleſtius eſt in hac uita, quam eſſe cum quo <lb></lb>nolis, &amp; priuari conſuetudine eorum cum quibus maximè uiuere <lb></lb>cupias. </s>
          <s id="id000024">Quid enim habent Principes præcipuum cum tota illa po­<lb></lb>tentia quam habent, niſi hoc unum, quod ſuis quos amant bene fa­<lb></lb>cere poſsint: nam reliqua omnia exerceri, uenari, edere, bibere, dor­<lb></lb>mire, iter agere, loca amæna inuiſere multis alijs conceſſum eſt, ma­<lb></lb>ioreque commodo qui in uita priuata degunt. </s>
          <s id="id000025">Si ergo principatum <lb></lb>cum tot laboribus, curis, periculis, &amp; meritò omnes appetunt: nec <lb></lb>eſt in eo quicquam præcipuum præter hoc, cui dubium eſt quin <lb></lb>hoc non ſit ſummum huius uitæ hominibus bonum? </s>
          <s id="id000026">propter cu­<lb></lb>ius uel dubiam ſpem eorum, quæ habent obliti mortales pericli­<lb></lb>tantur. </s>
          <s id="id000027">Succedunt inde tot commoda, non ſolum utilia, ſed pleraque<pb xlink:href="015/01/007.jpg"></pb>etiam neceſſaria, quæ nos ſapientia docet: huiuſmodi ergo omnia <lb></lb>cùm libris contineantur, meritò optimus quiſque librorum bono­<lb></lb>rum perpetuitati atque in columitati fauere debet. </s>
          <s id="id000028">C. </s>
          <s id="id000029">Caligulam exe­<lb></lb>cramur ſolum ob id quod Vergilij, &amp; T. </s>
          <s id="id000030">Liuij ſcripta delere cogi­<lb></lb>tauerit. </s>
          <s id="id000031">Quid facturi eſſemus, ſi feciſſet quod cogitauerat? </s>
          <s id="id000032">Eſt in ſa­<lb></lb>pientum monumentis bonum ſine malo, mens ſine corporea labe: <lb></lb>Virtutes abſque uitijs, gratiæ &amp; iucunditas ſine ſorde, &amp; immundi­<lb></lb>tia, uoluptas ſine dolore, conuerſatio abſque tædio, delitiæ abſque miſe<lb></lb>ria nuda, omnia bona præſtant, atque laudabilia ab omnibus morta­<lb></lb>litatis exuuijs libera, tantum commodi afferunt libri. </s>
          <s id="id000033">Sed &amp; in eo­<lb></lb>rum electione ac ſtudijs modus, ac medio critas quædam ſeruanda <lb></lb>eſt, quæ ſi quis neglexerit non leui incommodo afficietur: eam an­<lb></lb>tiqui rationem alij proportionem appellarunt, non equidem etiam <lb></lb>in pertritis tam <expan abbr="facillimã">facillimam</expan>, ut rentur homines: nam in alijs rebus per­<lb></lb>obſcuram eſſe fatentur, ego difficillimam puto undique, &amp; magis for<lb></lb>ſan ubi non exiſtimamus. </s>
          <s id="id000034">Vnde plures decidere uidemus magnis <lb></lb>cum auxilijs, &amp; euidenti ſpe: quid aliud eſt in cauſa quàm ignota <lb></lb>menſura rerum? </s>
          <s id="id000035">quam tamen plerique tenere ſe putant. </s>
          <s id="id000036">Ergo, cùm <lb></lb>ſummum bonum in hac menſura ſitum eſſe cernerem, ut clarè oſten<lb></lb>dunt muſicæ uoces, quæ non niſi indiuiduo (ut ita dicam) ſpatio <lb></lb>ſeu loco ſtare poſſunt, ita &amp; in figuris picturarum &amp; ſtatuarum, &amp; <lb></lb>diebus decretorijs, &amp; negotijs ciuilibus operę pretium me factu­<lb></lb>rum exiſtimaui, ſi omnia hæc quæ latè patebant breuiter in unum <lb></lb>redegiſſem, <expan abbr="nõ">non</expan> tantum ne lectorem tædio afficerem, quàm ut quòd <lb></lb>aliàs do cui, breuibus tractationibus, &amp; plura continerentur, &amp; faci<lb></lb>lius docerentur. </s>
          <s id="id000037">Cum uerò bona fortuna quædam effeciſſet, ut tibi <lb></lb>libellum dedicaſſem de Prouidentia ex conſtitutione temporum, <lb></lb>longe meliore occaſione nominis tui typographi obliti ſint, indi­<lb></lb>gnum fore putaui, ut non ærea (quemadmodum cum Glauco Dio<lb></lb>medes) cum aureis commutarem. </s>
          <s id="id000038">Itaque infinitis licet circumuentus <lb></lb>negotijs totus huic operæ in cubui, atque adeò ut præter ſpem unius <lb></lb>anni penè ſpatio liber abſolueretur. </s>
          <s id="id000039">Qui cum tibi (ut dixi) iam iurè <lb></lb>deberetur, eò tamen magis dedicandum putaui, quod non ego ſo­<lb></lb>lum quanquam id maximè, ſed communis conſenſus ho­<lb></lb>minum exiſtimet, te ſingulari uirtute omnibus <lb></lb>ſtudioſis plurimum fauere, <lb></lb>Vale.</s>
        </p>
      </section>
      <section>
        <pb xlink:href="015/01/008.jpg"></pb>
        <p type="head">
          <s id="id000040">TABVLA PRO­<lb></lb>POSITIONVM DE <lb></lb>PROPORTIONIBVS.<lb></lb><arrow.to.target n="table1"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table1"></table.target>
          <row>
            <cell>I.</cell>
            <cell>Proportionem <emph type="italics"></emph>in proportionem duci, eſt ſuperiores numeros atque inferiores inuicem ducere.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>pagina<emph.end type="italics"></emph.end> 6</cell>
          </row>
          <row>
            <cell>II.</cell>
            <cell>P<emph type="italics"></emph>roportio extremorum producitur ex intermedijs.<emph.end type="italics"></emph.end></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell>III.</cell>
            <cell>S<emph type="italics"></emph>i proportio ex duabus proportionibus in quatuor terminis producatur,   ipſa uerò proportio inter duas alias quantitates fuerit conſtituta: conſurgent trecen­ti ſexaginta modi productionis proportionis.<emph.end type="italics"></emph.end></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell>IIII.</cell>
            <cell>S<emph type="italics"></emph>i fuerit proportio primi ad ſecundum, producta ex proportionibus tertij ad quartum,   &amp; quinti ad ſextum, producetur etiam ex proportione tertij ad ſextum, &amp; quinti ad   quartum.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell>V.</cell>
            <cell>S<emph type="italics"></emph>i fuerit proportio primi ad ſecundum, producta ex proportione tertij ad quartum, &amp;   quinti ad ſextum: erit proportio tertij ad ſextum, producta ex proportionibus primi   ad ſecundum, &amp; quarti ad quintum.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell>VI.</cell>
            <cell>E<emph type="italics"></emph>x trecentis ſexaginta modis producendarum proportionum triginta ſex tantum eſſe   neceſſarios.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell>VII.</cell>
            <cell>I<emph type="italics"></emph>n modis qui neceſſariò producuntur ex duabus proportionibus, cum duæ quantitates ex   illis quæ modos conficiunt, æquales fuerint: proportio producta ad quatuor quanti­tates omiologas reducetur.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell>VIII.</cell>
            <cell>S<emph type="italics"></emph>i duarum proportionum ſuperiores numeri alternatim cum inferioribus multiplicen­tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in­uicem proportio, ex primis proportionibus compoſita.<emph.end type="italics"></emph.end></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>IX.</cell>
            <cell>S<emph type="italics"></emph>i duarum proportionum ſuperiores numeri alternatim cum inferioribus multiplicen­tur, minusque productum ex maiore detrahatur, erit reſidui ad productum ex inſe­rioribus proportio uelut illa, quæ relinquitur detracta minore proportione ex ma­iore.<emph.end type="italics"></emph.end></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>X.</cell>
            <cell>S<emph type="italics"></emph>i fuerit alicuius quantitatis ad unam partem proportio, uelut alterius partis ad ſecun­dam quantitatem, erit proportio cuiuſuis quantitatis eiuſdem generis ad ſecundam   compoſita proportio, ex proportionibus eiuſdem quantitatis, aſſumptæ ad utranque   partem primæ quantitatis ſeorſum.<emph.end type="italics"></emph.end></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>XI.</cell>
            <cell>P<emph type="italics"></emph>roportio aggregati quarumlibet duarum quantitatum ad aggregatum duarum æqua­lium <expan abbr="quantitatũ">quantitatum</expan> eſt, compoſita ex proportionibus primis, &amp; diuiſa per duplam.<emph.end type="italics"></emph.end></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell>XII.</cell>
            <cell>P<emph type="italics"></emph>ropoſitis duabus proportionibus unam alteri iungere abſque multiplicatione.<emph.end type="italics"></emph.end></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell>XIII.</cell>
            <cell>P<emph type="italics"></emph>roportio confuſa aggregata primæ &amp; tertiæ quatuor quantitatum omiologarum ad   aggregatum ſecundæ &amp; quartæ, eſt uelut compoſita ex eiſdem diuiſa per du­plam.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell>XIIII.</cell>
            <cell>P<emph type="italics"></emph>roportiones confuſæ &amp; coniunctæ in tribus quantitatibus inuicem commutantur.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell>XV.</cell>
            <cell>S<emph type="italics"></emph>i fuerint quatuor quantitates proportio confuſa, aggregati primæ &amp; tertiæ, ad aggre­gatum ſecundæ &amp; quartæ, erit ut monadis addito prouentu, qui fit diuiſa differentia,   differentiarum primæ &amp; ſecundæ, atque quartæ &amp; tertiæ, per aggregatum tertiæ &amp;   quartæ ad ipſam monadem.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell>XVI.</cell>
            <cell>O<emph type="italics"></emph>mnium quatuor quantitatum propoſita prima, quæ non minorem habet proportio­nem ad ſuam correſpondentem quàm alia ad aliam, erit proportio confuſa illarum,<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb xlink:href="015/01/009.jpg"></pb>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>ut producti ex aggregato primæ &amp; tertiæ, in tertiam ad productum ex aggre gato tertiæ &amp; omiotatæ ad ſecundam in ipſam quartam.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell>XVII.</cell>
            <cell>O<emph type="italics"></emph>mnes duæ proportiones conuerſæ producunt æqualem proportionem.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell>XVIII.</cell>
            <cell>S<emph type="italics"></emph>i fuerint quotlibet quantitates in continua proportione multiplici præter, <expan abbr="ultimã">ultimam</expan>   proportio uerò penultimæ ad ultimam, qualis reſidui primæ ad ſecundam,   erit primæ ad aggregatum reliquarum, uelut penultimæ ad ultimam.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell>XIX.</cell>
            <cell>S<emph type="italics"></emph>i fuerint aliquot quantitates arithmeticæ omiologæ, quarum exceſſus ſit æqualis   minimè, omnibus autem deficientibus ſupplementa ad æqualitatem maximè   adiungantur, erunt quadrata omnium quantitatum æqualium, adiecto rurſus   quadrato primæ cum eo quod fit ex minima primi ordinis in aggregatum o­mnium quantitatum eiuſdem, tripla aggregato quadratorum omnium quanti tatum primi ordinis pariter acceptis.<emph.end type="italics"></emph.end></cell>
            <cell>17</cell>
          </row>
          <row>
            <cell>XX.</cell>
            <cell>C<emph type="italics"></emph>um fuerint quatuor quantitates, fueritque <expan abbr="ſecũda">ſecunda</expan> æqualis tertiæ, aut prima æqualis   quartæ, erit proportio primæ ad quartam, aut tertiæ ad ſecundam, producta   ex proportionibus primæ ad ſecundam &amp; tertiæ ad quartam.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell>XXI.</cell>
            <cell>C<emph type="italics"></emph>um decuſſatim ducta fuerit prima in quartam, &amp; ſecunda in tertiam, produ­ctumque primæ in quartam, diuiſum fuerit per productum ſecundæ in tertiam,   erit proportio primæ ad ſecundam, diuiſa per proportíonem tertiæ ad quar­tam.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t ſimiliter interpoſita omiologa.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell>XXII.</cell>
            <cell>C<emph type="italics"></emph>um fuerit proportio primæ ad ſecundam maior quàm tertiæ ad quartam, erit   confuſa ex his maior quàm tertiæ ad quartam, minor autem quàm primæ ad   ſecundam.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>XXIII.</cell>
            <cell>O<emph type="italics"></emph>mnis motus naturalis ad locum ſuum eſt: ideò per rectam lineam fit.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>XXIIII.</cell>
            <cell>O<emph type="italics"></emph>mnis motus circularis uoluntarius eſt.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>XXV.</cell>
            <cell>T<emph type="italics"></emph>res ſunt motus omnino ſimplices naturalis, uoluntarius, &amp; uiolentus.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell>XXVI.</cell>
            <cell>M<emph type="italics"></emph>otus ergo compoſiti quatuor neceſſariò ſunt ſpecies.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell>XXVII.</cell>
            <cell>M<emph type="italics"></emph>otus uoluntarius eſt in loco: naturalis ad locum: uiolentus ex loco.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell>XXVIII.</cell>
            <cell>M<emph type="italics"></emph>otus quilibet uoluntarius aut uiolentus in aliquo medio fit.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell>XXIX.</cell>
            <cell>O<emph type="italics"></emph>mnis motus uoluntarius æqualis eſt ſemper: ſimpliciter etiam quilibet alius mo­tus.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell>XXX.</cell>
            <cell>I<emph type="italics"></emph>n omni corpore mobili in medio partes medij reſiſtunt obuiæ, aliæ impel­lunt.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell>XXXI.</cell>
            <cell>O<emph type="italics"></emph>mnis motus naturalis in æquali medio ualidior eſt in fine quàm in principio.<emph.end type="italics"></emph.end>V<emph type="italics"></emph>iolentus contrà.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell>XXXII.</cell>
            <cell>O<emph type="italics"></emph>mne mobile naturaliter motum ſeu uiolenter uelocius mouetur in medio rariore   quàm denſiore.<emph.end type="italics"></emph.end> M<emph type="italics"></emph>aior quoque eſt proportio finis motus in corpore rariore ad   finem motus in corpore denſiore quàm principij.<emph.end type="italics"></emph.end> I<emph type="italics"></emph>n uiolento autem celerius   perueniret ad finem motus in corpore denſiore.<emph.end type="italics"></emph.end></cell>
            <cell>27</cell>
          </row>
          <row>
            <cell>XXXIII.</cell>
            <cell>O<emph type="italics"></emph>mnia duo mobilia æqualis undique magnitudinis quæ æquali in tempore æqualia   ſpacia pertranſeunt in diuerſis ſubſtantia medijs neceſſe eſt, ut ſit ponderis ad   pondus, quem ad modum medij ad medium proportio duplicata.<emph.end type="italics"></emph.end></cell>
            <cell>27</cell>
          </row>
          <row>
            <cell>XXXIIII.</cell>
            <cell>P<emph type="italics"></emph>roportio corporis cubi ad ſuam ſuperficiem quadratam, eſt uelut eiuſdem ſuperfi   ciei, ad latus eiuſdem uerò ad monadem.<emph.end type="italics"></emph.end></cell>
            <cell>28</cell>
          </row>
          <row>
            <cell>XXXV.</cell>
            <cell>V<emph type="italics"></emph>ocum magnitudines excreſcunt in acumine, non in grauitate, finis autem eſt in   utroque extremo.<emph.end type="italics"></emph.end> P<emph type="italics"></emph>ropter hoc minima facta uariatione in hypate acutæ uix   ferunt.<emph.end type="italics"></emph.end></cell>
            <cell>29</cell>
          </row>
          <row>
            <cell>XXXVI.</cell>
            <cell>S<emph type="italics"></emph>i proportio per proportionem minorem æquali ducatur, proportio minor pro­<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb xlink:href="015/01/010.jpg"></pb>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>ducetur.<emph.end type="italics"></emph.end> V<emph type="italics"></emph>nde manifeſtum eſt duas proportiones minores æqualitate <expan abbr="inuicẽ">inuicem</expan> du ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"></emph.end></cell>
            <cell>30</cell>
          </row>
          <row>
            <cell>XXXVII.</cell>
            <cell>S<emph type="italics"></emph>i plures homines, quorum per ſe nauim mouere poßint, aut pondus ferre ſimul   iuncti eam moueant, aut pondus ferant, erunt illæ proportiones coniunctæ non productæ.<emph.end type="italics"></emph.end></cell>
            <cell>30</cell>
          </row>
          <row>
            <cell>XXXVIII.</cell>
            <cell>O<emph type="italics"></emph>mne corpus tantum reſiſtit motui contrario ſuo natúrali, quantum mouetur oc­culto motu quieſcendo.<emph.end type="italics"></emph.end></cell>
            <cell>31</cell>
          </row>
          <row>
            <cell>XXXIX.</cell>
            <cell>A<emph type="italics"></emph>b æquali aut minore ui quàm ſit impedimentum non fit motus.<emph.end type="italics"></emph.end></cell>
            <cell>31</cell>
          </row>
          <row>
            <cell>XL.</cell>
            <cell>O<emph type="italics"></emph>mne corpus ſphæricum tangens planum in puncto mouetur ad latus per quam­cunque uim, quæ medium diuidere poteſt.<emph.end type="italics"></emph.end></cell>
            <cell>31</cell>
          </row>
          <row>
            <cell>XLI.</cell>
            <cell>S<emph type="italics"></emph>i fuerint duæ quantitates ſumaturque toties <expan abbr="aggregatũ">aggregatum</expan> maioris &amp; minoris, quo­ties aggregatum minoris &amp; maioris, erit proportio confuſa maioris aggregati   ad minus, minor quam multiplicis maioris ad multiplex minoris.<emph.end type="italics"></emph.end></cell>
            <cell>32</cell>
          </row>
          <row>
            <cell>XLII.</cell>
            <cell>T<emph type="italics"></emph>rahentium nauim, aut ferentium pondera proportiones in ſe inuicem, quomodo   ducere oporteat conſiderare.<emph.end type="italics"></emph.end></cell>
            <cell>32</cell>
          </row>
          <row>
            <cell>XLIII.</cell>
            <cell>P<emph type="italics"></emph>roductionem ad additionem retrahere.<emph.end type="italics"></emph.end></cell>
            <cell>33</cell>
          </row>
          <row>
            <cell>XLIIII.</cell>
            <cell>S<emph type="italics"></emph>i fuerit proportio motoris ad id quod eſt maximum non mouens, &amp; ſpatium &amp;   tempus, nota erit etiam reliquorum nota.<emph.end type="italics"></emph.end></cell>
            <cell>33</cell>
          </row>
          <row>
            <cell>XLV.</cell>
            <cell>R<emph type="italics"></emph>ationem ſtateræ oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>34</cell>
          </row>
          <row>
            <cell>XLVI.</cell>
            <cell>A<emph type="italics"></emph>n ſit aliqua proportio &amp; qualis inter animam &amp; uitas, &amp; ſua corpora conſide­rare.<emph.end type="italics"></emph.end></cell>
            <cell>35</cell>
          </row>
          <row>
            <cell>XLVII.</cell>
            <cell>S<emph type="italics"></emph>i duo mobilia æqualister in eodem circulo iuxta proprios motus moueantur, pro­ductum temporis circuituum inuicem, erit æquale producto differentiæ tempo   rum circuitus ductæ in tempus coniunctionis primæ.<emph.end type="italics"></emph.end></cell>
            <cell>36</cell>
          </row>
          <row>
            <cell>XLVIII.</cell>
            <cell>S<emph type="italics"></emph>i tria mobilia ex eodem puncto diſcedant, fuerintque duorum ac duorum coniun­ctiones in temporibus commenſis, illa tria mobilia denuo coniungentur in tem   pore producto ex denominatore diuiſionis temporis maioris per minus in mi­nus aut numeratore in maius.<emph.end type="italics"></emph.end></cell>
            <cell>37</cell>
          </row>
          <row>
            <cell>XLIX.</cell>
            <cell>P<emph type="italics"></emph>ropofitio mobilis in circulo circuitus tempore dataque ratione diſtantiæ ab illo mo   bilis circuitum inuenire, quod ex <expan abbr="eodẽ">eodem</expan> puncto diſcedens <expan abbr="cũalio">cunalio</expan> mobili in dato   puncto <expan abbr="cõueniat">conueniat</expan> ſub <expan abbr="quocũque">quocunque</expan> numero <expan abbr="circuituũ">circuituum</expan> <expan abbr="tẽpus">tempus</expan> quoque <expan abbr="cõiunctionis">coniunctionis</expan>.<emph.end type="italics"></emph.end></cell>
            <cell>39</cell>
          </row>
          <row>
            <cell>L.</cell>
            <cell>O<emph type="italics"></emph>mnes circuituum portiones in eiſdem temporibus repetuntur.<emph.end type="italics"></emph.end></cell>
            <cell>40</cell>
          </row>
          <row>
            <cell>LI.</cell>
            <cell>O<emph type="italics"></emph>perationes dictas exemplo declarare.<emph.end type="italics"></emph.end></cell>
            <cell>41</cell>
          </row>
          <row>
            <cell>LII.</cell>
            <cell>T<emph type="italics"></emph>ria mobilia coniuncta in <expan abbr="eodẽ">eodem</expan> puncto, quorum duo &amp; duo conueniant in partib.   incommenſis inter ſe, in perpetuum in nullo unquam puncto conuenient.<emph.end type="italics"></emph.end></cell>
            <cell>42</cell>
          </row>
          <row>
            <cell>LIII.</cell>
            <cell>C<emph type="italics"></emph>irculorum ſe in aduerſum mouentium proportionem declarare.<emph.end type="italics"></emph.end></cell>
            <cell>43</cell>
          </row>
          <row>
            <cell>LIIII.</cell>
            <cell>P<emph type="italics"></emph>roportio circuli ad ſuum diametrum per ſimilitudinem eſt quarta pars periphe­riæ.<emph.end type="italics"></emph.end> R<emph type="italics"></emph>urſusque eiuſdem circuli ad peripheriam diametri quarta pars.<emph.end type="italics"></emph.end></cell>
            <cell>44</cell>
          </row>
          <row>
            <cell>LV.</cell>
            <cell>P<emph type="italics"></emph>roportionem medicamentorum per ordines ſup poſita æquali proportione in or­dinibus per quantitates &amp; proportiones demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>44</cell>
          </row>
          <row>
            <cell>LVI.</cell>
            <cell>P<emph type="italics"></emph>roportio cuiuſuis binomij ad ſuum reciſum, uel ei commenſum eſt duplicata ei   quæ ad numeri latus.<emph.end type="italics"></emph.end></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell>LVII.</cell>
            <cell>M<emph type="italics"></emph>otus rationem ad pondus inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell>LVIII.</cell>
            <cell>Q<emph type="italics"></emph>uæ ex alto deſcendunt, cur non eandem pro diſtantia motus rationem in libero   aëre ſeruent conſiderare.<emph.end type="italics"></emph.end></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell>LIX.</cell>
            <cell>O<emph type="italics"></emph>mne mobile motum duobus motibus non ad idem tendentibus utroque ſeorſum tar   dius mouetur ſimili motu.<emph.end type="italics"></emph.end></cell>
            <cell>50</cell>
          </row>
          <row>
            <cell>LX.</cell>
            <cell>O<emph type="italics"></emph>mne mobile motu naturali deſcendentis parte, deſcendit grauiore ſecundum gra­<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb xlink:href="015/01/011.jpg"></pb>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>uitatis centrum.<emph.end type="italics"></emph.end></cell>
            <cell>51</cell>
          </row>
          <row>
            <cell>LXI.</cell>
            <cell>P<emph type="italics"></emph>roportionum ictus ad pondus rei &amp; diſtantiam generaliter conſiderare.<emph.end type="italics"></emph.end></cell>
            <cell>52</cell>
          </row>
          <row>
            <cell>LXII.</cell>
            <cell>P<emph type="italics"></emph>roportionem motoris in plano ad motorem, qui eleuat pondus iuxta id quod   mouet, inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>53</cell>
          </row>
          <row>
            <cell>LXIII.</cell>
            <cell>O<emph type="italics"></emph>mne graue quanto proximius alligatum plano, tantò facilius trabitur.<emph.end type="italics"></emph.end></cell>
            <cell>53</cell>
          </row>
          <row>
            <cell>LXIIII.</cell>
            <cell>O<emph type="italics"></emph>mne mobile quantò latius tanto tardius moustur in plano.<emph.end type="italics"></emph.end></cell>
            <cell>54</cell>
          </row>
          <row>
            <cell>LXV.</cell>
            <cell>P<emph type="italics"></emph>roportionem duorum mobilium inter ſe cum auxilio medij inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>54</cell>
          </row>
          <row>
            <cell>LXVI.</cell>
            <cell>P<emph type="italics"></emph>roportionem laterum eptagoni, &amp; ſubtenſarum conſiderare, &amp; quæ à reflexa   proportione pendent.<emph.end type="italics"></emph.end></cell>
            <cell>55</cell>
          </row>
          <row>
            <cell>LXVII.</cell>
            <cell>S<emph type="italics"></emph>i fuerint aliquot quantitates ab una quantitate aliæque totidem ab eadem analo­gæ, erit proportio tertiæ unius ordinis ad tertiam alterius, ut ſecundæ ad ſe­cundum duplicata, &amp; quartæ ad quartam triplicata, quintæ ad quintam   quadruplicata, atque ſic de alijs.<emph.end type="italics"></emph.end></cell>
            <cell>57</cell>
          </row>
          <row>
            <cell>LXVIII.</cell>
            <cell>P<emph type="italics"></emph>ropoſitio collectorum ab<emph.end type="italics"></emph.end> E<emph type="italics"></emph>uclide &amp;<emph.end type="italics"></emph.end> A<emph type="italics"></emph>rchimede.<emph.end type="italics"></emph.end></cell>
            <cell>57</cell>
          </row>
          <row>
            <cell>LXIX.</cell>
            <cell>P<emph type="italics"></emph>ropoſitio collectorum ex quatuor libris<emph.end type="italics"></emph.end> A<emph type="italics"></emph>pollonij<emph.end type="italics"></emph.end> P<emph type="italics"></emph>ergei &amp;<emph.end type="italics"></emph.end> <expan abbr="q.">que</expan> S<emph type="italics"></emph>ereni.<emph.end type="italics"></emph.end></cell>
            <cell>59</cell>
          </row>
          <row>
            <cell>LXX.</cell>
            <cell>S<emph type="italics"></emph>i fuerint tres quantitates in continua proportione, aliæque totidem in continua   proportione poterunt conſtituere tres quantitates in æquali differentia per­uerſim copulatæ.<emph.end type="italics"></emph.end></cell>
            <cell>62</cell>
          </row>
          <row>
            <cell>LXXI.</cell>
            <cell>P<emph type="italics"></emph>roportionem leuitatis ponderis per uirgam torcularem attracti ad rectam ſu­ſpenſionem inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>63</cell>
          </row>
          <row>
            <cell>LXXII.</cell>
            <cell>P<emph type="italics"></emph>roportionem ponderis ſphæræ pendentis ad aſcendentem per accliue planum   inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>63</cell>
          </row>
          <row>
            <cell>LXXIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem ponderum attractorum penes figuram in plano inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>64</cell>
          </row>
          <row>
            <cell>LXXIIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem concutientis ad concuſſum inſtabili inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>64</cell>
          </row>
          <row>
            <cell>LXXV.</cell>
            <cell>P<emph type="italics"></emph><expan abbr="roportionẽ">roportionem</expan> immoti in aqua, ad <expan abbr="immotũ">immotum</expan> in terra in excipiendo <expan abbr="ictũ">ictum</expan> inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>65</cell>
          </row>
          <row>
            <cell>LXXVI.</cell>
            <cell>P<emph type="italics"></emph>roportionem <expan abbr="duorũ">duorum</expan> mobilium ſibi <expan abbr="inuicẽ">inuicem</expan> <expan abbr="concurrentiũ">concurrentium</expan> per <expan abbr="rectã">rectam</expan> inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>66</cell>
          </row>
          <row>
            <cell>LXXVII.</cell>
            <cell>P<emph type="italics"></emph>roportionem motus obliqui ad motum rectum in nauibus inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>66</cell>
          </row>
          <row>
            <cell>LXXVIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem nauis ad triremes quotuis concurrentes demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>67</cell>
          </row>
          <row>
            <cell>LXXIX.</cell>
            <cell>P<emph type="italics"></emph>roportionem medicamentorum purgantium inuicem declarare<emph.end type="italics"></emph.end></cell>
            <cell>68</cell>
          </row>
          <row>
            <cell>LXXX.</cell>
            <cell>P<emph type="italics"></emph>roportionem motus ſecundum obliquum ad rectum in ſpacio declarare.<emph.end type="italics"></emph.end></cell>
            <cell>69</cell>
          </row>
          <row>
            <cell>LXXXI.</cell>
            <cell>Q<emph type="italics"></emph>ualis ſit angulus, per quem poteſt moueri nauis ad rectum explorare.<emph.end type="italics"></emph.end></cell>
            <cell>70</cell>
          </row>
          <row>
            <cell>LXXXII.</cell>
            <cell>P<emph type="italics"></emph>roportionem uelorum indagare.<emph.end type="italics"></emph.end></cell>
            <cell>70</cell>
          </row>
          <row>
            <cell>LXXXIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem receſſus à recta uia ad obliquitatem inueſtigare.<emph.end type="italics"></emph.end></cell>
            <cell>72</cell>
          </row>
          <row>
            <cell>LXXXIIII.</cell>
            <cell>D<emph type="italics"></emph><expan abbr="iſtantiã">iſtantiam</expan> centri terræ à centro mundi per motum lapidis<emph.end type="italics"></emph.end> H<emph type="italics"></emph>erculei declarare.<emph.end type="italics"></emph.end></cell>
            <cell>73</cell>
          </row>
          <row>
            <cell>LXXXV.</cell>
            <cell>P<emph type="italics"></emph>roportio ponderis unius grauis ad aliud ſub eadem menſura eſt ueluti eiuſdem   ad differentiam ponderis uaſis repleti ex altero graui, &amp; ex ambobus de­tracto priore.<emph.end type="italics"></emph.end></cell>
            <cell>74</cell>
          </row>
          <row>
            <cell>LXXXVI.</cell>
            <cell>S<emph type="italics"></emph>i circuli in æ quales ſeu in ſphæra ſeu in plano ſe ſecuerint, nunquàm oppoſitos   angulos æquales habent.<emph.end type="italics"></emph.end></cell>
            <cell>77</cell>
          </row>
          <row>
            <cell>LXXXVII.</cell>
            <cell>P<emph type="italics"></emph>roportiones craßitiei aquæ ad <expan abbr="aẽrẽ">aerrem</expan> in <expan abbr="cõparatione">comparatione</expan> ad radios demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>78</cell>
          </row>
          <row>
            <cell>LXXXVIII.</cell>
            <cell>I<emph type="italics"></emph><expan abbr="nſtrumentũ">nſtrumentum</expan><emph.end type="italics"></emph.end> A<emph type="italics"></emph>colingen, quo momenta temporum <expan abbr="deprehendãtur">deprehendantur</expan> fabricare.<emph.end type="italics"></emph.end></cell>
            <cell>79</cell>
          </row>
          <row>
            <cell>LXXXIX.</cell>
            <cell>P<emph type="italics"></emph>roportionem denſitatis aquæ ad aërem per pondera inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>82</cell>
          </row>
          <row>
            <cell>XC.</cell>
            <cell>R<emph type="italics"></emph>ationem impetus uiolenti extra mißi ponderis ad æqualitatem reducere.<emph.end type="italics"></emph.end></cell>
            <cell>82</cell>
          </row>
          <row>
            <cell>XCI.</cell>
            <cell>P<emph type="italics"></emph>roportionem grauis cubi, &amp; ſphærici æqualium in accliui, &amp; deſcenſus eorum   demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>83</cell>
          </row>
          <row>
            <cell>XCII.</cell>
            <cell>P<emph type="italics"></emph><expan abbr="roportionẽ">roportionem</expan> ponderis æqualis iuxta longitudinis <expan abbr="cõparationẽ">comparationem</expan> demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>85</cell>
          </row>
          <row>
            <cell>XCIII.</cell>
            <cell>P<emph type="italics"></emph>ropter qd in <expan abbr="cõcußione">concußione</expan> <expan abbr="etiã">etiam</expan> leui nauis loco moueatar <expan abbr="oſtẽdere">oſtendere</expan>.<emph.end type="italics"></emph.end> V<emph type="italics"></emph>nde manifi <expan abbr="ſiũ">ſium</expan>   eſt duas naues ſibi <expan abbr="inuicẽ">inuicem</expan> occurſantes retrocedere, &amp; <expan abbr="quãtũ">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"></emph.end></cell>
            <cell>86</cell>
          </row>
          <pb xlink:href="015/01/012.jpg"></pb>
          <row>
            <cell>XCIIII.</cell>
            <cell>S<emph type="italics"></emph>i <expan abbr="quãtitas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="quãtitas">quantitas</expan> nota ſimiliter.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t ſi duæ   proportiones notæ fuerint, erit producta ex his atque diuiſa coniunctaque atque detra­cta nota.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t ſi fuerit totius ad partem proportio nota, erit et ad aliam partem nota:   &amp; alterius partis ad <expan abbr="alterã">alteram</expan> uno minor.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t ſi fuerit partis ad partem, erit ad totum   monade minor atque nota.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t ſi fuerit unius <expan abbr="quãtitatis">quantitatis</expan> ad duas <expan abbr="quãtitates">quantitates</expan> proportio   nota, erit &amp; <expan abbr="cõfuſa">confuſa</expan> ex eis nota.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t ſi fuerint trium quantitatum omiologarum, aut   quatuor analogarum omnes præter unam cognitæ, erunt &amp; illa alia cognita.<emph.end type="italics"></emph.end></cell>
            <cell>87</cell>
          </row>
          <row>
            <cell>XCV.</cell>
            <cell>C<emph type="italics"></emph>uiuſuis trigoni rectanguli, aut cuius duo auguli ſint in dupla proportione, aut qui   circulo inſcriptus ſit cognita quantitate unius lateris in comparatione ad dimetien   <expan abbr="tẽ">tem</expan>, ſi proportio duorum laterum cognita fuerit, <expan abbr="erũt">erunt</expan> omnia eius latera cognita.<emph.end type="italics"></emph.end></cell>
            <cell>88</cell>
          </row>
          <row>
            <cell>XCVI.</cell>
            <cell>C<emph type="italics"></emph>um in <expan abbr="perſpicuũ">perſpicuum</expan> denſum radij luminoſi inciderint, quatuor fiunt luminis genera.<emph.end type="italics"></emph.end></cell>
            <cell>89</cell>
          </row>
          <row>
            <cell>XCVII.</cell>
            <cell>M<emph type="italics"></emph><expan abbr="otũ">otum</expan> inuerſionis in figuris in <expan abbr="cõparatione">comparatione</expan> ad <expan abbr="motũ">motum</expan> ſphæræ in plano inueſtigare.<emph.end type="italics"></emph.end></cell>
            <cell>91</cell>
          </row>
          <row>
            <cell>XCVIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem ponderum æqualium per differentiam angulorum inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>92</cell>
          </row>
          <row>
            <cell>XCIX.</cell>
            <cell>P<emph type="italics"></emph>roportionem grauitatum per multitudinem ſuppoſitorum orbium oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>93</cell>
          </row>
          <row>
            <cell>C.</cell>
            <cell>P<emph type="italics"></emph><expan abbr="roportionẽ">roportionem</expan> grauitatis <expan abbr="ponderũ">ponderum</expan> attractorum per <expan abbr="trochlearũ">trochlearum</expan> <expan abbr="numerũ">numerum</expan> inueſtigare.<emph.end type="italics"></emph.end></cell>
            <cell>93</cell>
          </row>
          <row>
            <cell>CI.</cell>
            <cell>P<emph type="italics"></emph>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>94</cell>
          </row>
          <row>
            <cell>CII.</cell>
            <cell>P<emph type="italics"></emph>roportionem motuum inuerſionis, &amp; attractionis in plano inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>CIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem eorundem in accliui demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>CIIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem motus attractionis in decliui ad motum in plano determinare.<emph.end type="italics"></emph.end></cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>CV.</cell>
            <cell>P<emph type="italics"></emph>roportionem ferentium pondus in pertica inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>96</cell>
          </row>
          <row>
            <cell>CVI.</cell>
            <cell>Q<emph type="italics"></emph>uales proportiones angulorum doceant laterum proportiones.<emph.end type="italics"></emph.end> A<emph type="italics"></emph>tque uicißim deter­minare.<emph.end type="italics"></emph.end></cell>
            <cell>97</cell>
          </row>
          <row>
            <cell>CVII.</cell>
            <cell>S<emph type="italics"></emph>i in circulo duæ diametri ad rectum angulum ſe ſecauerint: aliæ uerò ad perpendicu­lum ex diametro exicrint ad circum ferentiam, ſingulæ ſupra diametrum erunt ma   iores portionibus reliquis diametri ſuperioribus, infra autem minores.<emph.end type="italics"></emph.end> D<emph type="italics"></emph>imidium   autem portionis ſuperioris reſiduum ad centrum maius ſagitta habebit.<emph.end type="italics"></emph.end> I<emph type="italics"></emph>n aliqua   præterea portionis ſuperioris parte, quæ uerſus diametrum tranſuerſum poſita   eſt, maior eſt differentia partis diametri ei <expan abbr="correſpõdentis">correſpondentis</expan>, <expan abbr="q̃">quae</expan> line æ tranſuerſæ.<emph.end type="italics"></emph.end></cell>
            <cell>100</cell>
          </row>
          <row>
            <cell>CVIII.</cell>
            <cell>P<emph type="italics"></emph>unctum æqualitatis differentiæ deſcenſus &amp; remotionis à centro inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>100</cell>
          </row>
          <row>
            <cell>CIX.</cell>
            <cell>R<emph type="italics"></emph>ationem libræ expendere.<emph.end type="italics"></emph.end></cell>
            <cell>101</cell>
          </row>
          <row>
            <cell>CX.</cell>
            <cell>S<emph type="italics"></emph>i duæ ſphæræ ex eadem materia deſcendant in aëre, eodem temporis momento ad   planum ueniunt.<emph.end type="italics"></emph.end></cell>
            <cell>104</cell>
          </row>
          <row>
            <cell>CXI.</cell>
            <cell>C<emph type="italics"></emph>ur ex medio tela ualidiorem ictum, &amp; naues in ſcalmo à remo ac malo recipiant in­de ex puppi explorare.<emph.end type="italics"></emph.end></cell>
            <cell>105</cell>
          </row>
          <row>
            <cell>CXII.</cell>
            <cell>C<emph type="italics"></emph>ur ex imo leuia longiùs ferantur declarare,<emph.end type="italics"></emph.end></cell>
            <cell>106</cell>
          </row>
          <row>
            <cell>CXIII.</cell>
            <cell>C<emph type="italics"></emph>ur uirga longius mittatur à puero quam à uiro inueftigare.<emph.end type="italics"></emph.end></cell>
            <cell>107</cell>
          </row>
          <row>
            <cell>CXIIII.</cell>
            <cell>C<emph type="italics"></emph>ircularis motus differentias quatuor eſſe, earumque rationem contemplari.<emph.end type="italics"></emph.end></cell>
            <cell>108</cell>
          </row>
          <row>
            <cell>CXV.</cell>
            <cell>P<emph type="italics"></emph>roportionem motuum impulſionis, &amp; attractionis inter ſe, ab eadem ui decla­rare.<emph.end type="italics"></emph.end></cell>
            <cell>110</cell>
          </row>
          <row>
            <cell>CXVI.</cell>
            <cell>C<emph type="italics"></emph>ur machinæ oblongæ igneæ longius emittant ſphæram explorare.<emph.end type="italics"></emph.end></cell>
            <cell>111</cell>
          </row>
          <row>
            <cell>CXVII.</cell>
            <cell>I<emph type="italics"></emph>n curriculis maior eſt uis pulueris copioſioris ampliore in ſpacio, quàm paucioris in   minore iuxta proportionem eandem.<emph.end type="italics"></emph.end></cell>
            <cell>112</cell>
          </row>
          <row>
            <cell>CXVIII.</cell>
            <cell>Q<emph type="italics"></emph>uanta proportione decedat ictus in obliquum parietem ab eo qui eſt ad perpendi­culum declarare.<emph.end type="italics"></emph.end></cell>
            <cell>114</cell>
          </row>
          <row>
            <cell>CXIX.</cell>
            <cell>Q<emph type="italics"></emph>uantum ictus machinæ procliuis ad angulum minuatur explorare.<emph.end type="italics"></emph.end></cell>
            <cell>115</cell>
          </row>
          <row>
            <cell>CXX</cell>
            <cell>P<emph type="italics"></emph>roportionem partium nauis ad eundem obliquum uentum explorare.<emph.end type="italics"></emph.end></cell>
            <cell>118</cell>
          </row>
          <row>
            <cell>CXXI.</cell>
            <cell>F<emph type="italics"></emph>labelli uires atque naturam declarare.<emph.end type="italics"></emph.end></cell>
            <cell>219</cell>
          </row>
          <row>
            <cell>CXXII.</cell>
            <cell>C<emph type="italics"></emph>ontemptus circa<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olis rationem in umbris declarare.<emph.end type="italics"></emph.end></cell>
            <cell>120</cell>
          </row>
          <pb xlink:href="015/01/013.jpg"></pb>
          <row>
            <cell>CXXIII.</cell>
            <cell>C<emph type="italics"></emph>ognita ratione umbræ ad gnomonem ſinum, &amp; arcum altitudinis ab horizon­te, quouis tempore dignoſcere.<emph.end type="italics"></emph.end></cell>
            <cell>121</cell>
          </row>
          <row>
            <cell>CXXIIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem umbræ uerſæ eſſe ad gnomonem, uelut gnomonis ad umbram   uerſam.<emph.end type="italics"></emph.end></cell>
            <cell>122</cell>
          </row>
          <row>
            <cell>CXXV.</cell>
            <cell>P<emph type="italics"></emph>roportionem dimetientis, &amp; peripheriæ cuiuslibet circuli paralleli æquino­ctiali per cognitam partem magni circuli demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>123</cell>
          </row>
          <row>
            <cell>CXXVI.</cell>
            <cell>C<emph type="italics"></emph>irculi horarij naturam declarare.<emph.end type="italics"></emph.end></cell>
            <cell>123</cell>
          </row>
          <row>
            <cell>CXXVII.</cell>
            <cell>D<emph type="italics"></emph>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"></emph.end></cell>
            <cell>124</cell>
          </row>
          <row>
            <cell>CXXVIII.</cell>
            <cell>N<emph type="italics"></emph>ota amplitudine ortus, cuiuſque puncti arcum ſemidiurnum inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>124</cell>
          </row>
          <row>
            <cell>CXXIX.</cell>
            <cell>D<emph type="italics"></emph>ata altitudine<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olis in quacunque regione, quacunque die diſtantiam<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olis à meri­diano cognoſcere.<emph.end type="italics"></emph.end></cell>
            <cell>124</cell>
          </row>
          <row>
            <cell>CXXX.</cell>
            <cell>D<emph type="italics"></emph>ata regionis altitudine, &amp; loco<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olis proportionem gnomonis, tam ad um­bram rectam quàm uerſam, uel etiam in cylindro determinare.<emph.end type="italics"></emph.end></cell>
            <cell>125</cell>
          </row>
          <row>
            <cell>CXXXI.</cell>
            <cell>S<emph type="italics"></emph>i lineæ alicui duplum alterius adiungatur, erit proportio duarum ad primam   maior quàm dupli cum prima ad primam cum una adiecta.<emph.end type="italics"></emph.end></cell>
            <cell>126</cell>
          </row>
          <row>
            <cell>CXXXII.</cell>
            <cell>S<emph type="italics"></emph>i ad duas lineas quarum una alteri dupla ſit eadem linea addatur, erit aggrega­ti ex minore, &amp; adiecta ad ipſam minorem, minor proportio quàm aggre­gati ex maiore, &amp; adiecta ad ipſam maiorem duplicata.<emph.end type="italics"></emph.end></cell>
            <cell>126</cell>
          </row>
          <row>
            <cell>CXXXIII.</cell>
            <cell>S<emph type="italics"></emph>i fuerint duæ quantitates, <expan abbr="quarũ">quarum</expan> una alteri dupla ſit: minuatur à minore quæ­dam quantitas, <expan abbr="eadẽque">eadenque</expan> maiori addatur, erit minoris ad reſiduum maior pro­portio, quàm aggregati ad maiorem duplicata.<emph.end type="italics"></emph.end> S<emph type="italics"></emph>i uerò minori addatur, &amp;   à maiore detrabatur, erit aggregati ad minorem minor proportio quàm   maioris ad reſiduum duplicata.<emph.end type="italics"></emph.end></cell>
            <cell>127</cell>
          </row>
          <row>
            <cell>CXXXIIII.</cell>
            <cell>S<emph type="italics"></emph>i rectangula ſuperficies ſit, cuius pars tertia quadrata ſit corpus, quod ex la­tere quadratæ in reſiduum ſuperficiei conſtat, maius eſt quouis corpore ex   eadem ſuperficies, aliter diuiſa conſtituto.<emph.end type="italics"></emph.end></cell>
            <cell>127</cell>
          </row>
          <row>
            <cell>CXXXV.</cell>
            <cell>S<emph type="italics"></emph>i linea in duas partes, quarum una fit alteri dupla diuidatur, erit quod fit ex   tertia parte in quadratum reſidui parallelipedum maius omni pararalleli­pedo, quod ex diuiſione eiuſdem lineæ creari poßit.<emph.end type="italics"></emph.end></cell>
            <cell>128</cell>
          </row>
          <row>
            <cell>CXXXVI.</cell>
            <cell>D<emph type="italics"></emph>enominationes in infinitum extendere.<emph.end type="italics"></emph.end></cell>
            <cell>129</cell>
          </row>
          <row>
            <cell>CXXXVII.</cell>
            <cell>R<emph type="italics"></emph>ationem numerorum ex progreßione declarare.<emph.end type="italics"></emph.end></cell>
            <cell>131</cell>
          </row>
          <row>
            <cell>CXXXVIII.</cell>
            <cell>M<emph type="italics"></emph>odos uſus horum numerorum declarare.<emph.end type="italics"></emph.end></cell>
            <cell>131</cell>
          </row>
          <row>
            <cell>CXXXIX.</cell>
            <cell>R<emph type="italics"></emph>adices omnes à propoſitis numeris extrahere.<emph.end type="italics"></emph.end></cell>
            <cell>132</cell>
          </row>
          <row>
            <cell>CXL.</cell>
            <cell>R<emph type="italics"></emph>adices per numeros fractos determinare.<emph.end type="italics"></emph.end></cell>
            <cell>133</cell>
          </row>
          <row>
            <cell>CXLI.</cell>
            <cell>N<emph type="italics"></emph>umeros fractos ad minores in ea <expan abbr="iẽ">iem</expan> proportione ualde propinqud deducere<emph.end type="italics"></emph.end></cell>
            <cell>136</cell>
          </row>
          <row>
            <cell>CXLII.</cell>
            <cell>D<emph type="italics"></emph><expan abbr="enominationũ">enominationum</expan> in <expan abbr="cremẽta">crementa</expan> ex extrema cognita inuenire.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t <expan abbr="cõuerſo">conuerſo</expan> modo.<emph.end type="italics"></emph.end></cell>
            <cell>137</cell>
          </row>
          <row>
            <cell>CXLIII.</cell>
            <cell>S<emph type="italics"></emph>i linea in duas partes diuidatur, corpora quæ fiunt ex una parte in alterius   quadratum mutuo æqualia ſunt corpori, quod fit ex tota linea in ſuperfi­ciem unius partis in alteram.<emph.end type="italics"></emph.end></cell>
            <cell>138</cell>
          </row>
          <row>
            <cell>CXLIIII.</cell>
            <cell>D<emph type="italics"></emph>uplum cubi medietatis maius eſt aggregato corporum mutuorum, cuiuslibet   diuiſionis quantum eſt, quod fit ex tota in quadratum differentiæ.<emph.end type="italics"></emph.end></cell>
            <cell>139</cell>
          </row>
          <row>
            <cell>CXLV.</cell>
            <cell>S<emph type="italics"></emph>i linea in duas partes diuidatur quadrata ambarum partium detracto eo, quod   fit ex una parte in alteram, æqualia ſunt producto unius in alteram cum   quadrato differentiæ.<emph.end type="italics"></emph.end></cell>
            <cell>139</cell>
          </row>
          <row>
            <cell>CXLVI.</cell>
            <cell>C<emph type="italics"></emph>orpus quod fit ex linea diuiſa in ſuperficiem æqualem quadratis ambarum par   tium detracta ſuperficie unius partis in alteram, eſt æquale aggregato cubo­rum ambarum partium.<emph.end type="italics"></emph.end></cell>
            <cell>139</cell>
          </row>
          <row>
            <cell>CXLVII.</cell>
            <cell>P<emph type="italics"></emph>ropoſita linea diuiſa duas ei line as adijcere, ut proportio <expan abbr="additarũ">additarum</expan> ſingularium<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb xlink:href="015/01/014.jpg"></pb>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>&amp; partium ſimul iunctarum ad additas ſit mutua.<emph.end type="italics"></emph.end></cell>
            <cell>148</cell>
          </row>
          <row>
            <cell>CXLVIII.</cell>
            <cell>P<emph type="italics"></emph>ropoſitis tribus lineis primam ſic diuidere, ut adiectis duabus alijs lineis, ſecun­dum <expan abbr="rationẽ">rationem</expan> mutuam ſingularum ſingulis, <expan abbr="aggregatũ">aggregatum</expan> ex una <expan abbr="adiectarũ">adiectarum</expan>, &amp; par   te ad <expan abbr="aggregatũ">aggregatum</expan> ex alia parte, &amp; adiecta ſe habeat, ut ſecunda ad <expan abbr="tertiã">tertiam</expan>.<emph.end type="italics"></emph.end></cell>
            <cell>140</cell>
          </row>
          <row>
            <cell>CXLIX.</cell>
            <cell>D<emph type="italics"></emph>atam lineam ſic diuidere, ut proportio quadratorum ad dupium unius partis in   alteram ſit, ut lineæ datæ ad lineam datam.<emph.end type="italics"></emph.end></cell>
            <cell>141</cell>
          </row>
          <row>
            <cell>CL.</cell>
            <cell>P<emph type="italics"></emph>ropoſitis duabus lineis, lineam communem utrique adiungere, ut ſit maioris ad ad­ditam proportio, uelut quadratorum minoris, &amp; adiectæ ad duplum unius in   alteram.<emph.end type="italics"></emph.end></cell>
            <cell>141</cell>
          </row>
          <row>
            <cell>CLI.</cell>
            <cell>P<emph type="italics"></emph>roportio differentiæ quadratorum partium cuiuſuis lineæ, ad quadratum diffe­rentiæ illarum eſt, uelut totius lineæ ad differentiam.<emph.end type="italics"></emph.end></cell>
            <cell>142</cell>
          </row>
          <row>
            <cell>CLII.</cell>
            <cell>S<emph type="italics"></emph>i linea in duas partes æquales, duasque inæquales diuidatur, fueritque proportio ag­gregati ex maiore, &amp; dimidio ad ipſam maiorem, uelut ex minore, &amp; ali­qua linea ad ipſam minorem, &amp; rurſus aggregati ex minore, &amp; dimidio ad   ipſam minorem, uelut aggregati ex maiore, &amp; alia addita ad ipſam maiorem,   erit proportio dimidij ad partem unam inæqualem, uelut alterius partis inæ­qualis ad ſuam additam mutuò, &amp; etiam proportio additarum inuicem, uelut   proportio <expan abbr="partiũ">partium</expan> <expan abbr="inæqualiũ">inæqualium</expan> duplicata, &amp; rurſus ipſum <expan abbr="dimidiũ">dimidium</expan> lineæ aſſum­ptæ <expan abbr="mediũ">medium</expan>, erit proportione inter additas.<emph.end type="italics"></emph.end> D<emph type="italics"></emph><expan abbr="emũ">emum</expan> proportio dimidij <expan abbr="cũ">cum</expan> addita   maiore ad <expan abbr="dimidiũ">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minorẽ">minorem</expan>.<emph.end type="italics"></emph.end></cell>
            <cell>142</cell>
          </row>
          <row>
            <cell>CLIII.</cell>
            <cell>V<emph type="italics"></emph>im quamcunque manus multiplicare.<emph.end type="italics"></emph.end></cell>
            <cell>144</cell>
          </row>
          <row>
            <cell>CLIIII.</cell>
            <cell>S<emph type="italics"></emph>i lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ   rectæ in unum punctum concurrant proportionem habentes, quam mediam   inter tota m &amp; adiectam, &amp; adiectam erit punctus, concurſus à puncto extre­mo lineæ adiectæ diſtans per lineam mediam.<emph.end type="italics"></emph.end> Q<emph type="italics"></emph>uod ſi ab extremo alicuius li­neæ æqua&#039;is mediæ, ſeu peripheria circuli, cuius ſemidiameter ſit media linea   duæ lineæ ad prædicta puncta producantur, ipſæ erunt in proportione mediæ   ad adiectam.<emph.end type="italics"></emph.end></cell>
            <cell>145</cell>
          </row>
          <row>
            <cell>CLV.</cell>
            <cell>Q<emph type="italics"></emph>uadr atorum numerum proportionem &amp; inuentionem conſiderare.<emph.end type="italics"></emph.end></cell>
            <cell>147</cell>
          </row>
          <row>
            <cell>CLVI.</cell>
            <cell>H<emph type="italics"></emph>orologiorum tempus multiplicare.<emph.end type="italics"></emph.end></cell>
            <cell>152</cell>
          </row>
          <row>
            <cell>CLVII.</cell>
            <cell>H<emph type="italics"></emph>orologiorum molarium rationem oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>154</cell>
          </row>
          <row>
            <cell>CLVIII.</cell>
            <cell>R<emph type="italics"></emph>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur ex­plicare.<emph.end type="italics"></emph.end></cell>
            <cell>156</cell>
          </row>
          <row>
            <cell>CLIX.</cell>
            <cell>N<emph type="italics"></emph>ullus angulus rectilineus æqualis eſſe poteſt alicui angulo contento recta, &amp; cir   culi portione.<emph.end type="italics"></emph.end></cell>
            <cell>158</cell>
          </row>
          <row>
            <cell>CLX.</cell>
            <cell>P<emph type="italics"></emph>ropoſita linea tribusque in ea ſignis punctum inuenire, ex quo ductæ tres lineæ ad   ſigna ſint in proportionibus datis.<emph.end type="italics"></emph.end></cell>
            <cell>162</cell>
          </row>
          <row>
            <cell>CLXI.</cell>
            <cell>S<emph type="italics"></emph>i fuerint duo trianguli, quorum baſes in eadem linea ſint conſtituti, &amp; æquales   ad unum punctum terminati, &amp; latus unum commune inter reliqua quantita­te medium neceſſe eſt angulum à maioribus lineis <expan abbr="contentũ">contentum</expan> minorem eſſe.<emph.end type="italics"></emph.end></cell>
            <cell>162</cell>
          </row>
          <row>
            <cell>CLXII.</cell>
            <cell>P<emph type="italics"></emph>roportionem duorum orbium, quorum diametrorum conuexæ partis, &amp; conca­uæ proportiones datæ ſint inueſtigare.<emph.end type="italics"></emph.end></cell>
            <cell>164</cell>
          </row>
          <row>
            <cell>CLXIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem uirium ſtellarum per motus ſuos indagare.<emph.end type="italics"></emph.end></cell>
            <cell>165</cell>
          </row>
          <row>
            <cell>CLXIIII.</cell>
            <cell>S<emph type="italics"></emph>yderum proportionem in magnitudine oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>166</cell>
          </row>
          <row>
            <cell>CLXV.</cell>
            <cell>P<emph type="italics"></emph>roportionem motuum omnium ſtellarum ad<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olem conſiderare.<emph.end type="italics"></emph.end></cell>
            <cell>167</cell>
          </row>
          <row>
            <cell>CLXVI.</cell>
            <cell>P<emph type="italics"></emph>roportiones muſicas ſuperpartientes in eas, quæ particulá una tantum abundant   reducere.<emph.end type="italics"></emph.end></cell>
            <cell>168</cell>
          </row>
          <pb xlink:href="015/01/015.jpg"></pb>
          <row>
            <cell>CLXVII.</cell>
            <cell>P<emph type="italics"></emph>roportionem muſicam ad ſapores &amp; odores coaptare.<emph.end type="italics"></emph.end></cell>
            <cell>176</cell>
          </row>
          <row>
            <cell>CLXVIII.</cell>
            <cell>P<emph type="italics"></emph>icturarum proportiones explicare.<emph.end type="italics"></emph.end></cell>
            <cell>179</cell>
          </row>
          <row>
            <cell>CLXIX.</cell>
            <cell>P<emph type="italics"></emph>roportionem muſicam in inſtrumentis declarare iuxta compoſitionis ra­tionem.<emph.end type="italics"></emph.end></cell>
            <cell>182</cell>
          </row>
          <row>
            <cell>CLXX.</cell>
            <cell>C<emph type="italics"></emph>oniugationes cuiuſuis numeri breuiter inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>185</cell>
          </row>
          <row>
            <cell>CLXXI.</cell>
            <cell>P<emph type="italics"></emph>ropoſitis duobus quibuslibet numeris, quotuis alios ſeu in continuum ſeu   medios in continua proportione arithmetica, geometrica &amp; muſica in­uenire.<emph.end type="italics"></emph.end></cell>
            <cell>187</cell>
          </row>
          <row>
            <cell>CLXXII.</cell>
            <cell>P<emph type="italics"></emph>roportiones<emph.end type="italics"></emph.end> S<emph type="italics"></emph>tiphelij deſcribere.<emph.end type="italics"></emph.end></cell>
            <cell>191</cell>
          </row>
          <row>
            <cell>CLXXIII.</cell>
            <cell>C<emph type="italics"></emph>irculum ſuper centro ſuo mouere æqualiter, ita quod omnia illius puncta   per rectam lineam moueantur ultro citroque.<emph.end type="italics"></emph.end></cell>
            <cell>192</cell>
          </row>
          <row>
            <cell>CLXXIIII.</cell>
            <cell>P<emph type="italics"></emph>rogreſſus &amp; regreſſus, tam ſine latitudine quàm cum latitudine in planetis   per ſolos concentricos circulos æqualiter motos demonſtrare.<emph.end type="italics"></emph.end></cell>
            <cell>194</cell>
          </row>
          <row>
            <cell>CLXXV.</cell>
            <cell>C<emph type="italics"></emph>auſam uarietatis diametrorum ex ſuppoſitis concentricis demonſtra­re.<emph.end type="italics"></emph.end></cell>
            <cell>195</cell>
          </row>
          <row>
            <cell>CLXXVI.</cell>
            <cell>R<emph type="italics"></emph>ationem centri grauitatis declarare.<emph.end type="italics"></emph.end></cell>
            <cell>197</cell>
          </row>
          <row>
            <cell>CLXXVII.</cell>
            <cell>S<emph type="italics"></emph>i proportio aliqua ex duabus proportionibus eiuſdem quantitatis ad alias   duas componatur, erit proportio illarum duarum eadem proportioni   producti ex proportione in primam duarum quantitatum, detracta prio­re illa quantitate, quæ ad duas comparatur, ad eandem priorem quanti­tatem.<emph.end type="italics"></emph.end></cell>
            <cell>198</cell>
          </row>
          <row>
            <cell>CLXXVIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem miſtionis metallorum, maximè auri &amp; argenti declara­re.<emph.end type="italics"></emph.end></cell>
            <cell>199</cell>
          </row>
          <row>
            <cell>CLXXIX.</cell>
            <cell>S<emph type="italics"></emph>i duobus totis duæ portiones ſimiles abſcindantur ab eiſdem denuò, &amp; ab­ſcißis portionibus partes eædem auferantur, denuoque ac denuò quoties   libuerit à portionibus, &amp; ù reſiduis ipſarum quantitatum partes eædem   auferantur, erit reſiduí ad reſiduum, ueluti totius ad totum.<emph.end type="italics"></emph.end></cell>
            <cell>200</cell>
          </row>
          <row>
            <cell>CLXXX.</cell>
            <cell>S<emph type="italics"></emph>i aliqua quantitas in duas partes diuidatur, fueritque alicuius quantitatis ad   partes illas compoſita proportio, non poterit eiuſdem quantitatis ad par­tes alias quantitatis diuiſa, aliter proportio eadem componi.<emph.end type="italics"></emph.end></cell>
            <cell>202</cell>
          </row>
          <row>
            <cell>CLXXXI.</cell>
            <cell>C<emph type="italics"></emph>um fuerit aliqua proportio, compoſita ex proportionibus primæ ad ſecun­dam &amp; tertiam, &amp; rurſus quartæ ad quintam &amp; ſextam: ita ſe habebit   proportio ſecundæ ad tertiam, ad proportionem quintæ ad ſextam, uelut   producti ex proportione in ſecundam detracta prima ad primam ad pro­ductum ex proportione in quintam, detracta quarta ad quartam.<emph.end type="italics"></emph.end></cell>
            <cell>203</cell>
          </row>
          <row>
            <cell>CLXXXII.</cell>
            <cell>P<emph type="italics"></emph>ropoſita differentia proportionum partium ſimilium ad partes aſſumptas,   propoſitaque proportione totius ad reſidua eadem, differentiam propor­tionum totius ad reliquum reſidui inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>203</cell>
          </row>
          <row>
            <cell>CLXXXIII.</cell>
            <cell>S<emph type="italics"></emph>pacium uitæ naturalis per ſpacium uitæ fortuitum declarare.<emph.end type="italics"></emph.end></cell>
            <cell>204</cell>
          </row>
          <row>
            <cell>CLXXXIIII.</cell>
            <cell>Q<emph type="italics"></emph>uæcunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri­mum uerſa mergantur.<emph.end type="italics"></emph.end></cell>
            <cell>211</cell>
          </row>
          <row>
            <cell>CLXXXV.</cell>
            <cell>C<emph type="italics"></emph>ur homo ſedens quanto altius ſedet, &amp; quanto magis crura ad fœmora, &amp;   fœmora ad pectus reclinata habet, facilius conſurgat, cum tamen hæc op­poſito modo inuicem ſe habeant, declarare.<emph.end type="italics"></emph.end></cell>
            <cell>213</cell>
          </row>
          <row>
            <cell>CLXXXVI.</cell>
            <cell>S<emph type="italics"></emph>i fuerit proportio primæ &amp; ſecundæ quantitatis ad tertiam, ut primæ &amp;   quartæ ad quintam, fueritque quarta ſecunda maior, erit proportio quar­tæ ad quintam maior quàm ſecundæ ad tertiam.<emph.end type="italics"></emph.end> Q<emph type="italics"></emph>uod ſi fuerit maior<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb xlink:href="015/01/016.jpg"></pb>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>quartæ ad quintam quàm ſecundæ ad tertiam, neceſſe eſt quartam ſecunda eſſe   maiorem.<emph.end type="italics"></emph.end></cell>
            <cell>214</cell>
          </row>
          <row>
            <cell>CLXXXVII.</cell>
            <cell>S<emph type="italics"></emph>i eiſdem uiribus &amp; ‘eadem’ proportione cum auxilio ponderis tertij quar­tum pondus moueatur quibus ſecundum, auxilio primi neceſſe eſt <expan abbr="quartũ">quartum</expan> pon   dus tardius &amp; maiore cum difficultate moueri quàm ſecundum.<emph.end type="italics"></emph.end></cell>
            <cell>214</cell>
          </row>
          <row>
            <cell>CLXXXVIII.</cell>
            <cell>S<emph type="italics"></emph>i uires aliquæ moueant cum ponderibus aliqua pondera, ut compoſita pro­portio ſit eadem proportioni uirium &amp; duorum ponderum mouentium ag­gregatum æquale duorum ponderum, ubi maior fuerit partium in æqualitas,   ibi erit maior difficultas.<emph.end type="italics"></emph.end></cell>
            <cell>214</cell>
          </row>
          <row>
            <cell>CLXXXIX.</cell>
            <cell>S<emph type="italics"></emph>i pondus minus ad longitudinem minorem ſub æquali proportione coapte­tar, facilius deorſum trahetur quàm quod maius eſt &amp; propius.<emph.end type="italics"></emph.end></cell>
            <cell>215</cell>
          </row>
          <row>
            <cell>CXC.</cell>
            <cell>S<emph type="italics"></emph>i fuerit primum graue minus ſecundo, &amp; ſecundum minus tertio, proportio   autem primi ad ſecundum multo maior quàm ſecundi ad tertium, poſibile erit   propoſitis uiribus eiſdem addere pondus <expan abbr="ſecũdo">ſecundo</expan>, ut ipſum &amp; tertium mouea­tur faciliùs ab eiſdem uiribus, &amp; primo uel ſecundo quàm antea.<emph.end type="italics"></emph.end></cell>
            <cell>215</cell>
          </row>
          <row>
            <cell>CXCL.</cell>
            <cell>C<emph type="italics"></emph>um fuerint duo pondera &amp; uires, duxerisque aggregatum ex uiribus &amp; mi­nore pondere in maius, addiderisque inſuper quantum eſt productum dimidij ui   rium in ſe latus aggregati detracto dimidio uirium, dicetur pondus auxiliare   æqualis proportionis.<emph.end type="italics"></emph.end></cell>
            <cell>215</cell>
          </row>
          <row>
            <cell>CXCII.</cell>
            <cell>S<emph type="italics"></emph>i ex medio diametri linea ad perpendiculum erigatur ad circuli peripheri­am, ex eo puncto autem quotlibet lineæ ducantur ſeu intus ad circun ferentiam uſque, ſeu extra ad diametrum, erit proportio totius lineæ ad totam uelut mu­tuo partis ad partem.<emph.end type="italics"></emph.end></cell>
            <cell>217</cell>
          </row>
          <row>
            <cell>CXCIII.</cell>
            <cell>R<emph type="italics"></emph>ationem ponderis triplicem explicare.<emph.end type="italics"></emph.end></cell>
            <cell>218</cell>
          </row>
          <row>
            <cell>CXCIIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem ponderis longioris in medio ſuſpenſi, ad breuius illi æquale &amp; in   medio ſuſpenſum declarare.<emph.end type="italics"></emph.end></cell>
            <cell>219</cell>
          </row>
          <row>
            <cell>CXCV.</cell>
            <cell>S<emph type="italics"></emph>i lectus fiat dupla longitudine ad latitudinem, melius ſuffulcietur reſtibus   ex medio ad angulos &amp; eius æquidiſtantibus quàm ſecundum longitudinem   &amp; latitudinem.<emph.end type="italics"></emph.end></cell>
            <cell>220</cell>
          </row>
          <row>
            <cell>CXCVI.</cell>
            <cell>S<emph type="italics"></emph>i duo circuli ſuper eodem centro eodem motu trans feruntur, æquale ſpacium   ſuperant.<emph.end type="italics"></emph.end></cell>
            <cell>221</cell>
          </row>
          <row>
            <cell>CXCVII.</cell>
            <cell>C<emph type="italics"></emph>ur lances ad locum ſuum ſuſpenſi redeant, impendentes <expan abbr="nõ">non</expan>, <expan abbr="demõſtrare">demonſtrare</expan>.<emph.end type="italics"></emph.end></cell>
            <cell>224</cell>
          </row>
          <row>
            <cell>CXCVIII.</cell>
            <cell>C<emph type="italics"></emph>ur ſolidum quod cubus uocatur<emph.end type="italics"></emph.end> P<emph type="italics"></emph>yramide ſtabilius ſit oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>225</cell>
          </row>
          <row>
            <cell>CXCIX.</cell>
            <cell>R<emph type="italics"></emph>ationem remorum nauim impellentium inuenire.<emph.end type="italics"></emph.end></cell>
            <cell>227</cell>
          </row>
          <row>
            <cell>CC.</cell>
            <cell>C<emph type="italics"></emph>ur temo cum paruus ſit, magnam nauim agere poteſt, &amp; cur cùm uarietas ſit   in prora, ipſe conſtituatur in puppi.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>t cum transuerſim ab aqua prematur   rectà nauim dirigat.<emph.end type="italics"></emph.end></cell>
            <cell>228</cell>
          </row>
          <row>
            <cell>CCI.</cell>
            <cell>S<emph type="italics"></emph>i duæ lineæ non ſecantes circuli peripheriam in unum punctum ex ea coe­ant exterius, neceſſe eſt illas peripheria contenta eſſe maiores.<emph.end type="italics"></emph.end></cell>
            <cell>229</cell>
          </row>
          <row>
            <cell>CCII.</cell>
            <cell>R<emph type="italics"></emph>ationem ſtrepitus oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>232</cell>
          </row>
          <row>
            <cell>CCIII.</cell>
            <cell>C<emph type="italics"></emph>ur ſcytalis onera portentur faciliùs, explorare.<emph.end type="italics"></emph.end></cell>
            <cell>233</cell>
          </row>
          <row>
            <cell>CCIIII.</cell>
            <cell>C<emph type="italics"></emph>ur pluribus trochleis, pondera facilius eleuentur oſtendere.<emph.end type="italics"></emph.end></cell>
            <cell>233</cell>
          </row>
          <row>
            <cell>CCV.</cell>
            <cell>S<emph type="italics"></emph>uper uerbis<emph.end type="italics"></emph.end> P<emph type="italics"></emph>latonis de fine<emph.end type="italics"></emph.end> R<emph type="italics"></emph>eipublicæ.<emph.end type="italics"></emph.end></cell>
            <cell>234</cell>
          </row>
          <row>
            <cell>CCVI.</cell>
            <cell>R<emph type="italics"></emph>hombi paßiones quaſdam declarare.<emph.end type="italics"></emph.end></cell>
            <cell>235</cell>
          </row>
          <row>
            <cell>CCVII.</cell>
            <cell>P<emph type="italics"></emph>roportionem agentium naturalium in tranſmutatione conſiderare.<emph.end type="italics"></emph.end></cell>
            <cell>238</cell>
          </row>
          <row>
            <cell>CCVIII.</cell>
            <cell>M<emph type="italics"></emph>ota res à centro grauitatis per <expan abbr="priorẽ">priorem</expan> motum, in reditu uelocius mouetur   quam ſi quieuerit.<emph.end type="italics"></emph.end></cell>
            <cell>238</cell>
          </row>
          <pb xlink:href="015/01/017.jpg"></pb>
          <row>
            <cell>CCIX.</cell>
            <cell>S<emph type="italics"></emph>i ſuperficies rectangula in duas partes æquales diuiſa intelligatur, quæ am­bæ quadratæ ſint, itemque in duas inæquales, erit parallelipedum ex latere   mediæ partis in totam ſuperficiem maius aggregato parallelipedorum ex   partibus inæqualibus in latera alterius partis mutuo, in eo, quod fit ex dif   ferentia lateris minoris partis à mediæ latere in differentiam maioris par­tis ſuperficiei à media ſuperficie bis, &amp; ex differentia amborum laterum   inæqualium iunctorum ad ambo latera, æqualia iuncta in minorem par­tem ſuperficiei.<emph.end type="italics"></emph.end></cell>
            <cell>241</cell>
          </row>
          <row>
            <cell>CCX.</cell>
            <cell>S<emph type="italics"></emph>i duæ lineæ ad æquales angulos ab eodem puncto peripheriæ circuli refle­ctantur, neceſſe eſt angulos cum dimetiente factos æquales eſſe.<emph.end type="italics"></emph.end> V<emph type="italics"></emph>nde ma­nifeſtum eſt, protractam diametrum angulum ſuppoſitum per æqualia di­uidere.<emph.end type="italics"></emph.end></cell>
            <cell>242</cell>
          </row>
          <row>
            <cell>CCXI.</cell>
            <cell>S<emph type="italics"></emph>i duæ lineæ ex duobus punctis peripheriam contingentes, in eandem par­tem protrahantur, ſemper magis diſtabunt inuicem ea ex parte, &amp; nun­quam concurrent.<emph.end type="italics"></emph.end></cell>
            <cell>243</cell>
          </row>
          <row>
            <cell>CCXII.</cell>
            <cell>S<emph type="italics"></emph>i ab eodem puncto ad circuli peripheriam lineæ quotuis ducantur, tres inue­nire lineas, quæ non in alium punctum reflectentur.<emph.end type="italics"></emph.end></cell>
            <cell>244</cell>
          </row>
          <row>
            <cell>CCXIII.</cell>
            <cell>P<emph type="italics"></emph>ropoſito circulo, atque in eius peripheria puncto ſignato, lineas contingentes   ultra cítraque, &amp; eam ab ipſomet deducere.<emph.end type="italics"></emph.end></cell>
            <cell>245</cell>
          </row>
          <row>
            <cell>CCXIIII.</cell>
            <cell>S<emph type="italics"></emph>i extra circulum duo puncta æqualiter à centro diſtantia ſignentur, erit pun­ctum reflexionis æqualis in medio arcus intercepti inter lineas, quæ à cen   tro ducuntur ad illa puncta.<emph.end type="italics"></emph.end> S<emph type="italics"></emph>i uerò unum centro proximius fuerit altero,   punctum æqualitatis in peripheria tantò longius, uerſus breuiorem line­am, quantò punctum aliud à centro magis diſteterit.<emph.end type="italics"></emph.end></cell>
            <cell>245</cell>
          </row>
          <row>
            <cell>CCXV.</cell>
            <cell>P<emph type="italics"></emph>unctum reflexionis punctorum inæqualiter diſtantium à centro, æqualiter   diſtat à lineis, ductis à centro ad puncta æqualiter diſtantia alterutrin­que.<emph.end type="italics"></emph.end></cell>
            <cell>246</cell>
          </row>
          <row>
            <cell>CCXVI.</cell>
            <cell>S<emph type="italics"></emph>i fuerint circuli duo inæquales, &amp; extra utrunqúe punctum ad illud ex mi­nore reflexè per magnam partem minoris à maiore perueuire pote­runt.<emph.end type="italics"></emph.end></cell>
            <cell>247</cell>
          </row>
          <row>
            <cell>CCXVII.</cell>
            <cell>O<emph type="italics"></emph>culus uidet partem ſuperficiei<emph.end type="italics"></emph.end> L<emph type="italics"></emph>unæ illuminatam à<emph.end type="italics"></emph.end> S<emph type="italics"></emph>ole per radios reflexos   à<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olis corpore: nec tamen poteſt uidere imaginem ipſius in<emph.end type="italics"></emph.end> L<emph type="italics"></emph>una tan   quam in ſpeculo.<emph.end type="italics"></emph.end></cell>
            <cell>248</cell>
          </row>
          <row>
            <cell>CCXVIII.</cell>
            <cell>R<emph type="italics"></emph>ationem maculæ<emph.end type="italics"></emph.end> L<emph type="italics"></emph>unæ indagare.<emph.end type="italics"></emph.end></cell>
            <cell>248</cell>
          </row>
          <row>
            <cell>CCXIX.</cell>
            <cell>R<emph type="italics"></emph>ationem eorum quæ apparent circa<emph.end type="italics"></emph.end> S<emph type="italics"></emph>olem ſpeculo in aqua poſito decla­rare.<emph.end type="italics"></emph.end></cell>
            <cell>150</cell>
          </row>
          <row>
            <cell>CCXX.</cell>
            <cell>C<emph type="italics"></emph>auſam cur<emph.end type="italics"></emph.end> S<emph type="italics"></emph>ol æſtiuis diebus exoriens umbram ad meridiem, cum in meridie   ad boream mittat, explorare.<emph.end type="italics"></emph.end></cell>
            <cell>252</cell>
          </row>
          <row>
            <cell>CCXXI.</cell>
            <cell>M<emph type="italics"></emph>agnitudo<emph.end type="italics"></emph.end> L<emph type="italics"></emph>unæ &amp; cæterorum aſtrorum dignoſcitur ex proportione alio­rum ad eam iuxta diſtantiam: ipſius uerò iuxta rationem pupillæ ad<emph.end type="italics"></emph.end> L<emph type="italics"></emph>u­nam diſtantiæ ratione.<emph.end type="italics"></emph.end></cell>
            <cell>354</cell>
          </row>
          <row>
            <cell>CCXXII.</cell>
            <cell>Q<emph type="italics"></emph>uantitates quæ æquales eſſe non poſſunt in eodem genere, maius tamen &amp;   minus recipiunt, ſunt in proportione poteſtatis.<emph.end type="italics"></emph.end></cell>
            <cell>255</cell>
          </row>
          <row>
            <cell>CCXXIII.</cell>
            <cell>Q<emph type="italics"></emph>uantitates quæ actu æquales eſſe non poſſunt, in nulla proportione actu   eſſe poſſunt.<emph.end type="italics"></emph.end></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>CCXXIIII.</cell>
            <cell>N<emph type="italics"></emph>eque temporis totius, ut imaginamur, ipſum eſſe infinitum, neque æui ui­tarum proportio ulla eſt ad tempus, quod poteſtate eſt, utpotè diem<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb xlink:href="015/01/018.jpg"></pb>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>uel menſem.<emph.end type="italics"></emph.end></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>CCXXV.</cell>
            <cell>P<emph type="italics"></emph>roportio media non eſt ex ratione agentis, ſed patientis.<emph.end type="italics"></emph.end></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>CCXXVI.</cell>
            <cell>P<emph type="italics"></emph>roportio ſublimis non conſiſtit in magnitudine, ſed ordine, iuxta quem diffe­rentia eſt eius quod eſt ante &amp; poſt.<emph.end type="italics"></emph.end></cell>
            <cell>257</cell>
          </row>
          <row>
            <cell>CCXXVII.</cell>
            <cell>V<emph type="italics"></emph>itæ iuxta numerum perfectionum in comparatione ad cogitationem no­ſtram proportionem quand am habent.<emph.end type="italics"></emph.end></cell>
            <cell>259</cell>
          </row>
          <row>
            <cell>CCXXVIII.</cell>
            <cell>P<emph type="italics"></emph>roportionem ſcientiæ futurorum &amp; cæterorum occultorum conſidera­re.<emph.end type="italics"></emph.end></cell>
            <cell>260</cell>
          </row>
          <row>
            <cell>CCXXIX.</cell>
            <cell>I<emph type="italics"></emph>ncorporea omnia unum ſunt, neque numerus eſt eorum.<emph.end type="italics"></emph.end></cell>
            <cell>261</cell>
          </row>
          <row>
            <cell>CCXXX.</cell>
            <cell>P<emph type="italics"></emph>roportio incorporeorum aſcendentium ſemper maior eſt.<emph.end type="italics"></emph.end></cell>
            <cell>262</cell>
          </row>
          <row>
            <cell>CCXXXI.</cell>
            <cell>T<emph type="italics"></emph>res eſſe mundos atque inter ipſos nullam eſſe proportionem: nec numero cos   definiri.<emph.end type="italics"></emph.end></cell>
            <cell>263</cell>
          </row>
          <row>
            <cell>CCXXXII.</cell>
            <cell>O<emph type="italics"></emph>mnis motus naturalis quanto uelocior eſt tanto propior eſt &amp; magis ſimil   limus quieti.<emph.end type="italics"></emph.end></cell>
            <cell>264</cell>
          </row>
          <row>
            <cell>CCXXXIII.</cell>
            <cell>Q<emph type="italics"></emph>uod eſt in mundo incorporeo æternum eſt, beatum, ſecurum, immutabile   ſecundum locum, ſolum iuxta eſſentiam fit: iuxta quod uelut à leui ſu­ſurro aquæ &amp; aura æſtiua demulcetur.<emph.end type="italics"></emph.end></cell>
            <cell>270</cell>
          </row>
        </table>
        <p type="head">
          <s id="id000041">FINIS.</s>
        </p>
        <pb xlink:href="015/01/019.jpg"></pb>
      </section>
    </front>
    <body>
      <chap>
        <pb pagenum="1" xlink:href="015/01/020.jpg"></pb>
        <p type="head">
          <s id="id000042">HIERONYMI CAR<lb></lb>DANI MEDIOLANENSIS, CI­<lb></lb>VISQVE BONONIENSIS, MEDICI <lb></lb>de Proportionibus, ſeu Ope­<lb></lb>ris Perfecti <lb></lb>LIBER QVINTVS.</s>
        </p>
        <p type="main">
          <s id="id000043">Prima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000044">Proportio ab Euclide ſic deſcribitur, Quòd <lb></lb>ſit duarum quantitatum eiuſdem generis, <lb></lb>quod ad magnitudinem attinet, compara­<lb></lb>tio certa.</s>
        </p>
        <p type="main">
          <s id="id000045">Secunda diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000046">Proportiones per ſimilitudinem <expan abbr="dicũtur">dicuntur</expan>, <lb></lb>cùm quantitas quantitati <expan abbr="comparat̃">comparatur</expan> alterius <lb></lb>generis, cui fingitur æqualis eſſe poteſtate.</s>
        </p>
        <p type="main">
          <s id="id000047">Velut ſi a b fingatur monas in comparatione <lb></lb>ad b c erit rectangulum a c æquale lineæ b c.</s>
        </p>
        <figure id="id.015.01.020.1.jpg" xlink:href="015/01/020/1.jpg"></figure>
        <p type="main">
          <s id="id000048">Tertia diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000049">Proportio æqualis proportioni eſt, cùm eodem modo termini <lb></lb>ſe habent inuicem in utraque</s>
        </p>
        <p type="main">
          <s id="id000050">Quarta diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000051">Proportiones ſecundum genus notæ dicuntur, cùm nouimus, <lb></lb>quòd ſint maiores, aut minores. </s>
          <s id="id000052">Nam cùm æquales ſunt, ſimul ne<lb></lb>ceffe eſt, ut cognoſcamus genus, &amp; ſpeciem.</s>
        </p>
        <p type="main">
          <s id="id000053">Quinta diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000054">Datum poſitione eſt: quod neceſſariò ex poſitis certam habet <lb></lb>quantitatem.</s>
        </p>
        <p type="main">
          <s id="id000055">Sexta diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000056">Datum ſimpliciter dicitur, quod ex propoſitis cognoſci poteſt, <lb></lb>quantum ſit.</s>
        </p>
        <p type="main">
          <s id="id000057">Septima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000058">Proportiones poteſtate <expan abbr="dicunt̃">dicuntur</expan>, quæ ſub comparatione aliarum <lb></lb><expan abbr="quantitatũ">quantitatum</expan> neceſſariam habentium <expan abbr="cõnexionem">connexionem</expan> <expan abbr="ſolũ">ſolum</expan> <expan abbr="cognoſcunt̃">cognoſcuntur</expan>.</s>
        </p>
        <p type="main">
          <s id="id000059">Hæ autem ſunt aliquando eiuſdem generis, cum primis ut nu­<lb></lb>meri: aliquandò alterius, ut linearum &amp; ſuperficierum, angulorum, <lb></lb>&amp; arcuum: aliquando eiuſdem generis, &amp; diuerſarum ſpecierum, <lb></lb>ut arcuum per ſinus, qua utuntur Aſtronomi.</s>
        </p>
        <p type="main">
          <s id="id000060">Octaua diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000061">Proportio homonyma dicitur duarum quantitatum diuerſi ge</s>
        </p>
        <p type="main">
          <s id="id000062"><arrow.to.target n="marg1"></arrow.to.target><lb></lb>neris, ſed alterius a b altero dependentium, uelut motus ad tem­





<pb pagenum="2" xlink:href="015/01/021.jpg"></pb>pus. </s>
          <s id="id000063">Dicimus enim motum tardum, uel uelocem in comparatione <lb></lb>ad tempus.</s>
        </p>
        <p type="margin">
          <s id="id000064"><margin.target id="marg1"></margin.target>C<emph type="italics"></emph>ar<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000065">Nona diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000066">Proportionum aliæ dicuntur rhete, aliæ alogæ, rhetæ quæ ſunt <lb></lb>ut numeri ad numerum, alogæ quæ non ſunt numeri ad numerum.</s>
        </p>
        <p type="main">
          <s id="id000067">Decima diffinitio</s>
        </p>
        <p type="main">
          <s id="id000068">Proportio rhete alia æqualis, alia multiplex, uel ſubmultiplex: <lb></lb>alia unius partis exceſſus, aut defectus, alia plurium, quam ſuper­<lb></lb>partientem, aut ſupartientem uocant.</s>
        </p>
        <p type="main">
          <s id="id000069">Vndecima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000070">Cum diuiſo denominatore per numeratorem exit quantitas alo<lb></lb>ga, proportio dicitur aloga: ſi autem numerus integer, aut pars nu­<lb></lb>meri nota dicitur rhete.</s>
        </p>
        <p type="main">
          <s id="id000071">Duodecima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000072">Proportionem in proportionem duci eſt, quoties recto ordine <lb></lb>tres quantitates in eiſdem collo<expan abbr="cant̃">cantur</expan>: ut ſint tres quan<lb></lb><figure id="id.015.01.021.1.jpg" xlink:href="015/01/021/1.jpg"></figure><lb></lb>titates a b c dicetur proportio a ad c producta ex pro <lb></lb>portione a ad b &amp; b ad c, &amp; ſimiliter proportio c ad <lb></lb>a producitur ex proportione b ad a, &amp; c ad b.</s>
        </p>
        <p type="main">
          <s id="id000073">Tertia decima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000074">Proportionem per proportionem diuidi eſt, quoties ad eandem <lb></lb>quantitatem duæ quantitates comparantur, tunc illarum propor­<lb></lb>tio eſt, quæ prodit una per alteram diuiſa.</s>
        </p>
        <p type="main">
          <s id="id000075">Sint proportiones a &amp; b ad c &amp; interponatur b inter a &amp; c, dico <lb></lb>proportionem a ad c diuiſam per proportionem a ad b, &amp; prodire <lb></lb>proportionem b ad c, conſtat ex conuerſa præcedentis.</s>
        </p>
        <p type="main">
          <s id="id000076">Quarta decima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000077">Additio proportionum intelligitur quotiens duarum quanti­<lb></lb>tatum ad unam tertiam, proportiones per aggregatum ipſarum <lb></lb>quantitatum ad eandem coniunguntur.</s>
        </p>
        <p type="main">
          <s id="id000078">Velut ſi comparentur a b &amp; b c ad d, inde tota <lb></lb><figure id="id.015.01.021.2.jpg" xlink:href="015/01/021/2.jpg"></figure><lb></lb>a c ad d dicemus proportionem, ac ad d eſſe con<lb></lb><expan abbr="iunctã">iunctam</expan> ex duabus proportionibus a b ad d &amp; b c <lb></lb>ad <expan abbr="eandẽ">eandem</expan> d. </s>
          <s id="id000079">Hoc &amp; duo ſequentes ſicut &amp; duę <expan abbr="antecedẽtes">antecedentes</expan> demon­<lb></lb>ſtrabitur eſſe. </s>
          <s id="id000080">nunc ſolum quomodo <expan abbr="intelligendũ">intelligendum</expan> ſit proponimus.</s>
        </p>
        <p type="main">
          <s id="id000081">Quinta decima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000082">Detractionem proportionis à proportione intelligimus fieri <lb></lb>per <expan abbr="detractionẽ">detractionem</expan> minoris quantitatis à maiore, comparatam ad ean­<lb></lb>dem quantitatem.</s>
        </p>
        <p type="main">
          <s id="id000083">Velut in exemplo ſuperiore detracta proportione b c ad d ex 


<pb pagenum="3" xlink:href="015/01/022.jpg"></pb>proportione a c ad d, relinquetur proportio a b ad d. </s>
          <s id="id000084">&amp; probatur <lb></lb>ex conuerſione præcedentis.</s>
        </p>
        <p type="main">
          <s id="id000085">Sexta decima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000086">Extractio radicum alicuius proportionis fit per extractionem <lb></lb>radicum quantitatum illius iuxta unam, &amp; eandem rationem.</s>
        </p>
        <p type="main">
          <s id="id000087">Velut quadratæ, uel cubæ, uel pronicæ, uel uninerſalis, uel alte­<lb></lb>rius modi.</s>
        </p>
        <p type="main">
          <s id="id000088">Decima ſeptima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000089">Cùm fuerint duæ proportiones ſimiles in tribus terminis con­<lb></lb>tinuatæ, dicetur proportio primæ quantitatis ad tertiam ueluti <lb></lb>primæ ad ſecundam duplicata. </s>
          <s id="id000090">Et ſi ſint tres proportiones ſimiles <lb></lb>in quatuor terminis, dicetur proportio primæ quantitatis ad quar­<lb></lb>tam triplicatà ei, quæ eſt primæ ad ſecundam,</s>
        </p>
        <p type="main">
          <s id="id000091">Decima octaua diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000092">Confuſa proportio dicitur ſimplicis, aut compoſitæ quantitatis <lb></lb>ad compoſitam in comparatione ad proportiones ad partes.</s>
        </p>
        <p type="main">
          <s id="id000093">Decimanona diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000094">Quantitates quę in continua ſunt proportione Analogæ <expan abbr="uocant̃">uocantur</expan>.</s>
        </p>
        <p type="main">
          <s id="id000095">Dictum eſt hoc ad fugiendum nomen barbarum, etiam ut bre­<lb></lb>uiter tamen poſſemus ſententiam explicare.</s>
        </p>
        <p type="main">
          <s id="id000096">Vigeſima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000097">Reflexa proportio dicitur cum trium quantitatum aggregatum <lb></lb>primæ, &amp; tertiæ ſe habet ad ſecundam uelut ſecunda ad tertiam,</s>
        </p>
        <p type="main">
          <s id="id000098">Vigeſima prima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000099">Trium quantitatum analogarum aliæ quidem Geometricæ, <lb></lb>cùm proportio ſimilis eſt: Aliæ Arithmeticæ, cum fuerit æqualis <lb></lb>exceſſus huc indè: Aliæ muſicæ cum fuerit proportio primæ ad ter<lb></lb>tiam multiplex, aut ſimplex, aut compoſita exceſſus quæ ſimplici <lb></lb>iuncta ſit ad multiplicis perfectionem: eadem autem ſit proportio <lb></lb>exceſſus primæ, &amp; ſecundæ ad exceſſum ſecundæ ſupra tertiam.</s>
        </p>
        <p type="main">
          <s id="id000100">Velut proportio 6. 4. 3. dupla eſt utrinque, &amp; 6. 3. 2 tripla. </s>
          <s id="id000101">&amp; 28. 24. <lb></lb>21. &amp; 45. 40. 36. Geometrica uerò &amp; arithmetica facilius continuan­<lb></lb>tur in quotquot quantitatibus, ſed &amp; muſica uelut 12. 8. 6. 4. 3. &amp; <lb></lb>proportio 8 ad 5 muſica eſt: quia proportio 5 ad 4 muſica eſt, &amp; <lb></lb>bene ſonans, igitur conſtitutis 8. 5. 4. cum 8 ad 4 benè ſonet, &amp; 5 <lb></lb>ad 4, &amp; 4 ſit extrema non media inde 8. &amp; 5 benè <expan abbr="ſonãt">ſonant</expan>. </s>
          <s id="id000102">nam in me­<lb></lb>dijs <expan abbr="nõ">non</expan> eſt <expan abbr="uerũ">uerum</expan>, ut in 9. 6. 4 bis diapente, &amp; 16. 12. 9 bis diateſſaron.</s>
        </p>
        <p type="main">
          <s id="id000103">Vigeſima ſecunda diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000104">Quantitates quæ ſimilem habent proportionem non continua­<lb></lb>tam, omiologæ appellantur.</s>
        </p>
        <p type="main">
          <s id="id000105">Vigeſima tertia diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000106">Prima operatione conſiſtere dicuntur proportiones, cùm inter <lb></lb>primo conflatas quantitates conſtiterint.</s>
        </p>
        <pb pagenum="4" xlink:href="015/01/023.jpg"></pb>
        <p type="main">
          <s id="id000107">PRIMA Animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000108">Omnis Proportio eſt, aut æqualitatis, aut maior inæqualis, <lb></lb>aut minor.</s>
        </p>
        <p type="main">
          <s id="id000109">Secunda animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000110">Quilibet numerus tantus dicitur, quanta eſt illius proportio ad <lb></lb>monadem.</s>
        </p>
        <p type="main">
          <s id="id000111">Dicimus enim quatuor, quod monadem quater contineat. </s>
          <s id="id000112">Et <lb></lb>duo cum dimidio cùm monadem bis &amp; ſemis contineat.</s>
        </p>
        <p type="main">
          <s id="id000113">Tertia animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000114">Proportionem defectus, ſeu detractæ quantitatis ad defectum <lb></lb>eſſe poſſe, ut quantitatis ad quantitatem dicuntur communes ani­<lb></lb>mi ſententiæ, quæ ex intellectu ſolo terminorum, quod ueræ ſint, <lb></lb>cognoſcuntur. </s>
          <s id="id000115">Si ergo defectus eſt quantitas, &amp; quantitas eiuſdem <lb></lb>ſpeciei, quia detrahitur, &amp; defectus non eſt ſimplicitur, ſed detra­<lb></lb>cto ergo per quartam petitionem: uel primam diffinitionem erit <lb></lb>proportio inter illas. </s>
          <s id="id000116">Sunt enim ambæ detractæ.</s>
        </p>
        <p type="main">
          <s id="id000117">Quarta animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000118">Inter quantitatem, &amp; defectum minorem quantitate, cuius eſt de<lb></lb>fectus, eſt proportio, quatenus eſt quantitas. </s>
          <s id="id000119">Sit a b linea, &amp; detra­<lb></lb>cta quantitas b c, non maior a b &amp; d ſit alia quæuis quantitas eiuſ­<lb></lb><figure id="id.015.01.023.1.jpg" xlink:href="015/01/023/1.jpg"></figure><lb></lb><expan abbr="dẽ">dem</expan> generis, dico quòd inter d &amp; b c eſt propor­<lb></lb>tio quatenus b c eſt quantitas, quia ſunt eiuſ­<lb></lb>dem generis ideo ſunt in aliqua proportione <lb></lb>per primam diffinitionem. </s>
          <s id="id000120">Sed ut b c eſt defectus, nulla eſt propor­<lb></lb>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, &amp; <lb></lb>hoc eſt contra demonſtrata ab Euclide.</s>
        </p>
        <p type="main">
          <s id="id000121">Quinta animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000122">Cum proportio producitur ex proportionibus quælibet illa­<lb></lb>rum dicetur producta diuiſa per alteram.</s>
        </p>
        <p type="main">
          <s id="id000123">Sexta animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000124">Æqualium quantitatum ſeu proportionum ad tertiam compa­<lb></lb>rabilium eadem eſt proportio atque uiciſsim. </s>
          <s id="id000125">Hæc etſi demonſtre­<lb></lb>tur ab Euclide, eſt tamen hic generalior: &amp; ſatis per ſe nota. </s>
          <s id="id000126">Vt ſit <lb></lb>propior animi communi ſententiæ, quàm rei demonſtrandæ.</s>
        </p>
        <p type="main">
          <s id="id000127">Septima animi communis ſententia.</s>
        </p>
        <p type="main">
          <s id="id000128">Ad quod quantitas proportionem habet infinitam, id in genere <lb></lb>illius quantitatis non comprehenditur.</s>
        </p>
        <p type="main">
          <s id="id000129">Nam proportio eſt duarum quantitatum eiuſdem generis com­<lb></lb>paratio certa: at hæc comparatio certa non eſt: non igitur quantita­<lb></lb>tes ambæ ſunt, aut non eiuſdem generis.</s>
        </p>
        <pb pagenum="5" xlink:href="015/01/024.jpg"></pb>
        <p type="main">
          <s id="id000130">PRIMA Petitio.</s>
        </p>
        <p type="main">
          <s id="id000131">Si fuerit primi ad ſecundum, ut tertij ad quartum, &amp; ex primo in <lb></lb>ſecundum producatur æquale, aut maius, aut minus primo, uel <lb></lb>ſecundo, producetur eodem modo ex tertio in quartum ęquale aut <lb></lb>maius, aut minus tertio, uel quarto eadem ratione &amp; ordine.</s>
        </p>
        <p type="main">
          <s id="id000132">Secunda petitio.</s>
        </p>
        <p type="main">
          <s id="id000133">Proportiones poſſunt duci, diuidi, iungi, &amp; auferri, &amp; ſumi radix <lb></lb>in eis cuiuſcunque generis, atque earum quantitatis, ut libet, poſſe <lb></lb>tranſponere.</s>
        </p>
        <p type="main">
          <s id="id000134">Tertia petitio.</s>
        </p>
        <p type="main">
          <s id="id000135">Proportionis cuiuſuis nomen à denominatore ſuprà ſcripto, &amp; <lb></lb>numeratore infrà ſcripto ſumitur.</s>
        </p>
        <p type="main">
          <s id="id000136">Quarta petitio.</s>
        </p>
        <p type="main">
          <s id="id000137">Diuiſa quauis quantitate per aliam eiuſdem generis, quod exit <lb></lb>proportio dicitur.</s>
        </p>
        <p type="main">
          <s id="id000138">Quinta petitio.</s>
        </p>
        <p type="main">
          <s id="id000139">Quęlibet proportio eſt uel inter duas quantitates, uel per unam <lb></lb>ſignificatur.</s>
        </p>
        <p type="main">
          <s id="id000140">Nam per tertiam petitionem ſi ſint duæ quantitates, quæ non ha<lb></lb>beant unius rationem, nomen ſumit proportio à duobus numeris, <lb></lb>ſin autem ſit altera monas, erit per ſecundam animi communem ſen<lb></lb>tentiam, proportio numerus ipſe Ideò patet, quod dicitur.</s>
        </p>
        <p type="main">
          <s id="id000141">Sexta petitio.</s>
        </p>
        <p type="main">
          <s id="id000142">Propoſita proportione quacunque, &amp; monade quantitatem inue<lb></lb>nire, quæ ſe habeat ad monadem in proportione propoſita.</s>
        </p>
        <p type="main">
          <s id="id000143">Nam cùm per quartam petitionem diuiſa quantitate per quan­<lb></lb>titatem exeat proportio, &amp; numerus ad <expan abbr="monadẽ">monadem</expan> ſe habeat, ut pro­<lb></lb>portio, ideo ſumpta monade ſecundum illum numerum, ille nume <lb></lb>rus eſt quantitas quæſita.</s>
        </p>
        <p type="main">
          <s id="id000144">Septima petitio.</s>
        </p>
        <p type="main">
          <s id="id000145">Quamlibet quantitatem per aliam eiuſdem generis diuidere <lb></lb>poſſe.</s>
        </p>
        <p type="main">
          <s id="id000146">Octaua petitio.</s>
        </p>
        <p type="main">
          <s id="id000147">Proportionem in proportionem ducere poſſe: quamuis ſint in­<lb></lb>ter quantitates diuerſi generis.</s>
        </p>
        <p type="main">
          <s id="id000148">Quod dicitur de multiplicatione intelligendum eſt de alijs ope­<lb></lb>rationibus ſuprà enumeratis.</s>
        </p>
        <p type="main">
          <s id="id000149">Nona petitio.</s>
        </p>
        <p type="main">
          <s id="id000150">Monadem ſemper ſumere in quo cunque genere poſſe propoſi­<lb></lb>ta proportione.</s>
        </p>
        <pb pagenum="6" xlink:href="015/01/025.jpg"></pb>
        <p type="main">
          <s id="id000151">Nam licet diuidere per ſeptimam petitionem quantitatem per <lb></lb>quantitatem proportionis: &amp; quod exit, eſt proportio per quar­<lb></lb>tam petitionem, &amp; per ſecundam animi communem ſententiam <lb></lb>illa proportio eſt numero æqualis: ergo diuiſa proportione, per ſi­<lb></lb>milem numerum ſtatuetur monas.</s>
        </p>
        <p type="main">
          <s id="id000152">Decima petitio.</s>
        </p>
        <p type="main">
          <s id="id000153">In quouis genere quantitatum ſumere poſſe quantitatem, quæ <lb></lb><arrow.to.target n="marg2"></arrow.to.target><lb></lb>ſe habeat ad monadem in proportione data. </s>
          <s id="id000154">Similem huic propo­<lb></lb>nit Euclides in lineis generaliter: nos autem contrà generaliter in <lb></lb>omnibus quantitatibus, ſed de monade tantum.</s>
        </p>
        <p type="margin">
          <s id="id000155"><margin.target id="marg2"></margin.target>D<emph type="italics"></emph>uodecima <lb></lb>ſexti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end>Vndecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000156">Monadem in quancunque quantitatem ductam æquale ipſi pro­<lb></lb>ducere. </s>
          <s id="id000157">Similiter &amp; proportionem æqualem.</s>
        </p>
        <p type="main">
          <s id="id000158">Nam cum aliqua quantitas augeat ducta aliqua minuat, neceſſe <lb></lb>eſt aliquam eſſe, quæ nec augeat, nec minuat, &amp; hæc eſt monas. <lb></lb></s>
          <s id="id000159">Idem dico de diuiſione. </s>
          <s id="id000160">Aequalitas etiam ducta, uel diuidens non <lb></lb><arrow.to.target n="marg3"></arrow.to.target><lb></lb>mutat proportionem: nec quantitatem ipſam, igitur monas æqua­<lb></lb>litatem refert. </s>
          <s id="id000161">Quod etiam eſt perſpicuum ex ſupradictis.</s>
        </p>
        <p type="margin">
          <s id="id000162"><margin.target id="marg3"></margin.target>S<emph type="italics"></emph>ecunda ani <lb></lb>mi <expan abbr="cõmunis">communis</expan> <lb></lb>ſententia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000163">Duodecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000164">Cum fuerint quatuor quantitates &amp; ad primam, &amp; tertiam æquè <lb></lb>multiplicibus aſſumptis, item que ad ſecundam &amp; quartam, &amp; ſi mul­<lb></lb>tiplex primæ maius eſt multiplici ſecundæ, multiplex tertiæ ſit ma­<lb></lb>ius multiplici quartæ, &amp; ſi minus minus, &amp; ſi æquale æquale, idque<lb></lb> ſemper quouis modo aſſumptis his proportionibus ad primam &amp; <lb></lb>tertiam, &amp; ad ſecundam &amp; quartam erit proportio primæ ad ſecun<lb></lb>dam, ut tertiæ ad quartam. </s>
          <s id="id000165">Hæc etiam aſſumitur ab Euclide. </s>
          <s id="id000166">Et per <lb></lb><arrow.to.target n="marg4"></arrow.to.target><lb></lb>hanc intelligimus etiam conuerſam.</s>
        </p>
        <p type="margin">
          <s id="id000167"><margin.target id="marg4"></margin.target>Q<emph type="italics"></emph>uinto<emph.end type="italics"></emph.end> E<emph type="italics"></emph>le. <lb></lb>diff.<emph.end type="italics"></emph.end> 6.</s>
        </p>
        <p type="main">
          <s id="id000168">Tertiadecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000169">Quantitates æquales, atque proportiones in quaſuis quanti­<lb></lb>tates ductæ eandem ſeruant rationem. </s>
          <s id="id000170">Euclides hanc demonſtrat, <lb></lb>nos autem ad uitandum tædium petimus concedi, ſub qua in­<lb></lb><arrow.to.target n="marg5"></arrow.to.target><lb></lb>cluduntur diuiſio etiam additio, detractio, laterum omnium in­<lb></lb>uentio.</s>
        </p>
        <p type="margin">
          <s id="id000171"><margin.target id="marg5"></margin.target>Q<emph type="italics"></emph>uarta quin<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000172">Quartadecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000173">Cùm termini alicuius quantitatis eandem ſeruant rationem in <lb></lb>omnibus, &amp; firmi ſunt ac ſtabiles eiuſdem rationis comparatione <lb></lb>contentæ partes æqualem ſeruant exceſſum, ſeu proportionem.</s>
        </p>
        <p type="main">
          <s id="id000174">PROPOSITIO prima.</s>
        </p>
        <p type="main">
          <s id="id000175">Proportionem in proportionem duci eſt ſuperiores nume­<lb></lb>ros atque inferiores inuicem ducere.</s>
        </p>
        <pb pagenum="7" xlink:href="015/01/026.jpg"></pb>
        <p type="main">
          <s id="id000176">Sit proportio lineæ a ad lineam b, ut anguli c ad angulum d, ſta­<lb></lb><arrow.to.target n="marg6"></arrow.to.target><lb></lb>tuatur e monas in genere a <lb></lb><figure id="id.015.01.026.1.jpg" xlink:href="015/01/026/1.jpg"></figure><lb></lb>b, &amp; fiat f ad e, ut c ad d, &amp; du<lb></lb><arrow.to.target n="marg7"></arrow.to.target><lb></lb>catur a in f &amp; b in e, &amp; pro­<lb></lb>ducantur g &amp; h. </s>
          <s id="id000177">Quia ergo <lb></lb><arrow.to.target n="marg8"></arrow.to.target><lb></lb>f eſt proportio ipſa, erit g ad <lb></lb><arrow.to.target n="marg9"></arrow.to.target><lb></lb>a ut c ad d, ſed h eſt æqualis <lb></lb>b, igitur a ad h ut ad b. </s>
          <s id="id000178">Du­<lb></lb>cta ergo dicetur proportio a <lb></lb><arrow.to.target n="marg10"></arrow.to.target><lb></lb>ad b in proportionem c ad d <lb></lb>ducendo terminos proportionis, ſeu quantitatis recta ſcilicet ſu­<lb></lb>periores cum ſuperioribus, &amp; inferiores cum inferioribus. </s>
          <s id="id000179">Nam ſi <lb></lb><arrow.to.target n="marg11"></arrow.to.target><lb></lb>rurſum conſtituantur f ad e ut a ad b cùm f ſit proportio, &amp; k ad f ut <lb></lb><arrow.to.target n="marg12"></arrow.to.target><lb></lb>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb></lb>quæ eſt fin proportionem c ad d, liquet igitur propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id000180"><margin.target id="marg6"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000181"><margin.target id="marg7"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 9. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000182"><margin.target id="marg8"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000183"><margin.target id="marg9"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000184"><margin.target id="marg10"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. A<emph type="italics"></emph>ni­<lb></lb>mi ſentent.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000185"><margin.target id="marg11"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000186"><margin.target id="marg12"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000187">Propoſitio <expan abbr="ſecũnda">ſecunda</expan>.</s>
        </p>
        <p type="main">
          <s id="id000188">Proportio extremorum producitur ex intermedijs.<lb></lb><arrow.to.target n="marg13"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000189"><margin.target id="marg13"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000190">Sint a b c quantitates dico proportio­<lb></lb><figure id="id.015.01.026.2.jpg" xlink:href="015/01/026/2.jpg"></figure><lb></lb>nem a ad c, produci ex proportione a ad b </s>
        </p>
        <p type="main">
          <s id="id000191"><arrow.to.target n="marg14"></arrow.to.target><lb></lb>&amp; b ad c, ſtatuantur totidem à monade d e <lb></lb>f, erúntque ex demonſtrantis ab Euclide in <lb></lb>quinto <expan abbr="Elemẽtorum">Elementorum</expan> in eadem proportio­<lb></lb>ne, ſtatuatur ergo d prima quantitas e ſe­<lb></lb>cunda &amp; tertia f quarta. </s>
          <s id="id000192">eritqúe per præce­<lb></lb><arrow.to.target n="marg15"></arrow.to.target><lb></lb>dentem proportio productorum ex d in e <lb></lb>&amp; ſit g, &amp; in f &amp; ſit h, producta ex propor­<lb></lb>tionibus d ad e &amp; e ad f, quare ex propor­<lb></lb>tionibus a ad b &amp; b ad e, ſed ex dictis cum <lb></lb>e ſit eadem, erit proportio d ad f, ut g ad h &amp; proportio, d ad f per <lb></lb>æquam proportionem ab Euclide demonſtratam, ut a ad c, igitur <lb></lb><arrow.to.target n="marg16"></arrow.to.target><lb></lb>proportio a ad c producitur ex proportionibus a ad b &amp; b ad c, &amp; <lb></lb>eſt proportio ipſa a ad c d numerus, ut oſtenſum eſt.</s>
        </p>
        <p type="margin">
          <s id="id000193"><margin.target id="marg14"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end> 9. <lb></lb>P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000194"><margin.target id="marg15"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000195"><margin.target id="marg16"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000196">Ex hoc ſequitur, quòd cùm fuerit quantitas tertia monas ex pro­<lb></lb><arrow.to.target n="marg17"></arrow.to.target><lb></lb>portionibus inuicem ductis producetur prima quantitas.<lb></lb><arrow.to.target n="marg18"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000197"><margin.target id="marg17"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="margin">
          <s id="id000198"><margin.target id="marg18"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3</s>
        </p>
        <p type="main">
          <s id="id000199">Ex hoc ſequitur, quòd conuerſa proportio producitur ex con­<lb></lb>uerſis proportionibus.</s>
        </p>
        <p type="main">
          <s id="id000200">Propoſitio tertia.</s>
        </p>
        <p type="main">
          <s id="id000201">Si proportio ex duabus proportionibus in quatuor terminis <lb></lb>producatur, ipſa uerò proportio inter duas alias quantitates fue­


<pb pagenum="8" xlink:href="015/01/027.jpg"></pb>rit conſtituta: conſurgent trecenti ſexaginta modi productionis <lb></lb>proportionis.</s>
        </p>
        <p type="main">
          <s id="id000202"><arrow.to.target n="marg19"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000203"><margin.target id="marg19"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000204">Hęc propoſitio ut præcedens &amp; <expan abbr="ſequẽtes">ſequentes</expan> tres ab Alchindo ſum­<lb></lb>ptæ ſunt, &amp; ab eo demonſtrantur. </s>
          <s id="id000205">Sit ergo proportio a ad b, pro­<lb></lb><arrow.to.target n="table2"></arrow.to.target><lb></lb><figure id="id.015.01.027.1.jpg" xlink:href="015/01/027/1.jpg"></figure>ducta ex proportione c ad d &amp; e ad f, conſtat <lb></lb>quòd cum ſint ſex quantitates, quòd fieri pote­<lb></lb>runt quindecim coniugationes, quas poſui à la­<lb></lb>tere facilitatis gratia, quibus reſpondent totidem <lb></lb><arrow.to.target n="table3"></arrow.to.target><lb></lb>conuerſæ: erunt ergo triginta. </s>
          <s id="id000206">Singulæ autem ha <lb></lb>rum produci poſſunt duodecim modis: ductis <lb></lb><figure id="id.015.01.027.2.jpg" xlink:href="015/01/027/2.jpg"></figure>duodecim in triginta, fiunt trecenti ſexaginta mo <lb></lb>di. </s>
          <s id="id000207">Et hoc eſt clarum perſe, modo <expan abbr="demõſtremus">demonſtremus</expan>, <lb></lb>quod ſinguli horum modorum poſsint produ­<lb></lb>ci duodecim modis, &amp; capiamus ab primam quę <lb></lb>poteſt produci ex c d &amp; e f: Item ambabus con­<lb></lb>uerſis d c &amp; fe: &amp; rurſus altera recta altera con­<lb></lb>uerſa: &amp; hoc bifariam c d &amp; f e, &amp; d c &amp; e f, ſunt er­<lb></lb>go iam quatuor modi. </s>
          <s id="id000208">Totidem ex c e &amp; d f, toti­<lb></lb>demque ex c f &amp; d e, igitur erunt duodecim mo­<lb></lb>di, quibus produci poſſe intelligitur propor­<lb></lb>tio a ad b.</s>
        </p>
        <table>
          <table.target id="table2"></table.target>
          <row>
            <cell>a</cell>
            <cell>b</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell>d</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>e</cell>
            <cell>f</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
        </table>
        <table>
          <table.target id="table3"></table.target>
          <row>
            <cell>a b</cell>
            <cell>b a</cell>
          </row>
          <row>
            <cell>a c</cell>
            <cell>c a</cell>
          </row>
          <row>
            <cell>a d</cell>
            <cell>d a</cell>
          </row>
          <row>
            <cell>a e</cell>
            <cell>e a</cell>
          </row>
          <row>
            <cell>a f</cell>
            <cell>f a</cell>
          </row>
          <row>
            <cell>b c</cell>
            <cell>c b</cell>
          </row>
          <row>
            <cell>b d</cell>
            <cell>d b</cell>
          </row>
          <row>
            <cell>b e</cell>
            <cell>e b</cell>
          </row>
          <row>
            <cell>b f</cell>
            <cell>f b</cell>
          </row>
          <row>
            <cell>c d</cell>
            <cell>d c</cell>
          </row>
          <row>
            <cell>c e</cell>
            <cell>e c</cell>
          </row>
          <row>
            <cell>c f</cell>
            <cell>f c</cell>
          </row>
          <row>
            <cell>d e</cell>
            <cell>e d</cell>
          </row>
          <row>
            <cell>d f</cell>
            <cell>f d</cell>
          </row>
          <row>
            <cell>e f</cell>
            <cell>f e</cell>
          </row>
          <row>
            <cell>direc.</cell>
            <cell>conuer.</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000209">Propoſitio quarta.</s>
        </p>
        <p type="main">
          <s id="id000210">Si fuerit proportio primi ad ſecundum produ­<lb></lb>cta ex proportionibus tertij ad quartum, &amp; quin <lb></lb>ti ad ſextum, producetur etiam ex proportione <lb></lb>tertij ad ſextum, &amp; quinti ad quartum.</s>
        </p>
        <p type="main">
          <s id="id000211">Sit proportio a b producta ex proportioni­<lb></lb><arrow.to.target n="table4"></arrow.to.target><lb></lb><figure id="id.015.01.027.3.jpg" xlink:href="015/01/027/3.jpg"></figure>bus c ad d, &amp; e ad f, dico quod etiam erit produ­</s>
        </p>
        <table>
          <table.target id="table4"></table.target>
          <row>
            <cell>a</cell>
            <cell>b</cell>
            <cell></cell>
          </row>
          <row>
            <cell>c</cell>
            <cell>e</cell>
            <cell>g</cell>
          </row>
          <row>
            <cell>d</cell>
            <cell>f</cell>
            <cell>h</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell>e</cell>
            <cell>g</cell>
          </row>
          <row>
            <cell>f</cell>
            <cell>d</cell>
            <cell>h</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000212"><arrow.to.target n="marg20"></arrow.to.target><lb></lb>cta ex proportionibus c ad f, &amp; e ad d, diſponan­<lb></lb>tur ut in figura &amp; fiat ex c in e g, &amp; ex d in fh, ergo <lb></lb><arrow.to.target n="marg21"></arrow.to.target><lb></lb>per primam harum g ad h ut a ad b, ſed per præ­<lb></lb>ſuppoſita in ſecunda productione etiam prode­<lb></lb>unt g &amp; h, igitur per primam propoſitionem ha­<lb></lb>rum a ad b proportio producitur ex proportionibus c ad f tertiæ <lb></lb>ſcilicet ad ſextam, &amp; e ad d quintę ad quartam, quod fuit <expan abbr="propoſitũ">propoſitum</expan>.</s>
        </p>
        <p type="margin">
          <s id="id000213"><margin.target id="marg20"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>petit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000214"><margin.target id="marg21"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>petit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000215">Propoſitio quinta.</s>
        </p>
        <p type="main">
          <s id="id000216">Si fuerit proportio primi ad ſecundum producta ex proportio­<lb></lb>ne tertij ad quartum, &amp; quinta ad ſextum: erit proportio tertij ad <lb></lb>ſextum producta ex proportionibus primi ad ſecundum, &amp; quar­<lb></lb>ti ad quintum.</s>
        </p>
        <pb pagenum="9" xlink:href="015/01/028.jpg"></pb>
        <p type="main">
          <figure id="id.015.01.028.1.jpg" xlink:href="015/01/028/1.jpg"></figure>
          <s id="id000217">Sit proportio a ad b producta ex proportio­<lb></lb><arrow.to.target n="marg22"></arrow.to.target><lb></lb><arrow.to.target n="table5"></arrow.to.target><lb></lb>nibus c ad d, &amp; e ad f, dico quod proportio c ad <lb></lb>f producitur ex proportione a ad b &amp; d ad e. </s>
          <s id="id000218">In­<lb></lb>terponam d e inter c &amp; f, eritque ex ſecunda pro­<lb></lb>poſitione repetita proportio c ad f producta ex <lb></lb>tribus proportionibus c ad d, d ad e, e ad f, ſed <lb></lb>proportiones c ad d, &amp; e ad f producunt pro­<lb></lb><figure id="id.015.01.028.2.jpg" xlink:href="015/01/028/2.jpg"></figure>portionem a ad b, igitur proportio c ad f produ<lb></lb>citur ex proportionibus a ad b, &amp; e ad f.<lb></lb><arrow.to.target n="table6"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000219"><margin.target id="marg22"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <table>
          <table.target id="table5"></table.target>
          <row>
            <cell>a</cell>
            <cell>b</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>d</cell>
            <cell>f</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
        </table>
        <table>
          <table.target id="table6"></table.target>
          <row>
            <cell>c</cell>
          </row>
          <row>
            <cell>-----</cell>
          </row>
          <row>
            <cell>d</cell>
          </row>
          <row>
            <cell>-----</cell>
          </row>
          <row>
            <cell>e</cell>
          </row>
          <row>
            <cell>-----</cell>
          </row>
          <row>
            <cell>f</cell>
          </row>
          <row>
            <cell>-----</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000220">Propoſitio ſexta.</s>
        </p>
        <p type="main">
          <s id="id000221">Ex trecentis ſexaginta modis producenda­<lb></lb>rum proportionum triginta ſex tantum eſſe ne­<lb></lb>ceſſarios.<lb></lb><arrow.to.target n="table7"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table7"></table.target>
          <row>
            <cell>c</cell>
            <cell>p</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>a</cell>
            <cell>d</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>b</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
        </table>
        <p type="main">
          <figure id="id.015.01.028.3.jpg" xlink:href="015/01/028/3.jpg"></figure>
          <s id="id000222">Per quartam enim proportio a ad b produ­<lb></lb><arrow.to.target n="marg23"></arrow.to.target><lb></lb>citur bifariam, &amp; ex c ad d, &amp; e ad f, &amp; ex c ad f, &amp; <lb></lb>e ad d. </s>
          <s id="id000223">&amp; per præcedentem c ad f producitur ex <lb></lb>a ad b, &amp; d ad e, &amp; per quartam rurſus ex a ad e, <lb></lb>&amp; d ad b. </s>
          <s id="id000224">Et per præcedentem rurſus a ad e ex c <lb></lb>ad f &amp; b ad d, igitur per quartam eadem produ­<lb></lb>cetur ex c ad d &amp; b ad f. </s>
          <s id="id000225">Quare per præceden­<lb></lb>tem c ad f ex a ad e, &amp; d ad b, &amp; ita diſponemus <lb></lb>hos modos in tabula. </s>
          <s id="id000226">Vides etiam <lb></lb><arrow.to.target n="table8"></arrow.to.target><lb></lb><figure id="id.015.01.028.4.jpg" xlink:href="015/01/028/4.jpg"></figure>aliquos modos non produci, ut pri­<lb></lb>mi ad quartum nec ad ſextum, &amp; li­<lb></lb>quet, quòd cùm ſint quindecim o­<lb></lb>mnes modi qui produci poſſe intelli­<lb></lb>guntur, &amp; nouem tantum producan­<lb></lb>tur ſex eſſe, qui non producuntur, quos <lb></lb>ſeorſum in tabula coniunxi. </s>
          <s id="id000227">Et con­<lb></lb>ſtat etiam, quod totidem conuerſi ſci­<lb></lb>licet decem octo <expan abbr="producũtur">producuntur</expan>, de qui­<lb></lb>bus diximus, ut ſint omnes triginta <lb></lb>ſex, qui conſtat ex duabus propoſi­<lb></lb>tionibus præmiſsis, &amp; hac tertia, <expan abbr="quã">quam</expan> <lb></lb>adiungemus ſcilicet, quòd propor­<lb></lb>tio primi ad tertium producatur ex <lb></lb>proportionibus <expan abbr="ſecũdi">ſecundi</expan> ad quartum, <lb></lb>&amp; quinti ad <expan abbr="ſextũ">ſextum</expan>. </s>
          <s id="id000228">Hoc enim ex præ­<lb></lb>cedentibus non liquet: benè liquet <lb></lb>permutatis ordinibus, quod ſi pro­<lb></lb>portio primi ad tertium producitur, 


<pb pagenum="9 [=10]" xlink:href="015/01/029.jpg"></pb>quod etiam propor­<lb></lb><figure id="id.015.01.029.1.jpg" xlink:href="015/01/029/1.jpg"></figure><arrow.to.target n="marg24"></arrow.to.target><lb></lb>tio primi ad <expan abbr="quintũ">quintum</expan>. <lb></lb></s>
          <s id="id000229">Nam tertium, &amp; quin <lb></lb>tum, item que quartum, <lb></lb>&amp; ſextum non <expan abbr="diffe­rũt">diffe­<lb></lb>runt</expan> niſi ordine uolun<lb></lb>tario. </s>
          <s id="id000230">Ergo interpoſi­<lb></lb>to e inter a, &amp; c per ſe­<lb></lb>cundam propoſitio­<lb></lb>nem proportio a ad c <lb></lb>producitur ex proportionibus a ad <lb></lb>e, &amp; e ad c, ut ex demonſtratis in præ­<lb></lb>ſenti proportio a ad c producitur ex <lb></lb>c ad f &amp; b ad d. </s>
          <s id="id000231">Proportio ergo a ad <lb></lb>c producitur ex proportionibus e <lb></lb>ad c &amp; c ad f &amp; b ad d, at e ad c &amp; c ad <lb></lb>f producunt eam, quæ eſt e ad f per <lb></lb><expan abbr="ſecũdam">ſecundam</expan> propoſitionem. </s>
          <s id="id000232">Igitur pro­<lb></lb>portio a ad c producitur ex propor­<lb></lb>tionibus b ad d ſecundi ad quartum, <lb></lb>&amp; e ad f quinti ad ſextum. </s>
          <s id="id000233">Hæc Al­<lb></lb>chindus in ſuo libello: ſed licet inge­<lb></lb>nio ſa ualde: parum <expan abbr="tamẽ">tamen</expan> utilia olim <lb></lb><expan abbr="erãt">erant</expan> neceſſaria ad intelligendum ma­<lb></lb>gnam <expan abbr="cõpoſitionem">compoſitionem</expan> Ptolemęi, nunc <lb></lb>poſtquam Heber has ſex quantita­<lb></lb>tes traduxit ad quatuor, prorſus hæc <lb></lb>ſcientia ulli uſui eſſe deſijt.<lb></lb><arrow.to.target n="table9"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000234"><margin.target id="marg23"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000235"><margin.target id="marg24"></margin.target>Modi qui <expan abbr="nõ">non</expan> <lb></lb>producuntur <lb></lb>pri. ad quartu <lb></lb>pri. ad ſextum <lb></lb>ſec. ad <expan abbr="tertiũ">tertium</expan> <lb></lb>ſec. ad <expan abbr="quintũ">quintum</expan> <lb></lb>tert. </s>
          <s id="id000236">ad quint. <lb></lb></s>
          <s id="id000237">quart. </s>
          <s id="id000238">ad ſext.</s>
        </p>
        <table>
          <table.target id="table8"></table.target>
          <row>
            <cell></cell>
            <cell>Primi ad ſecundum.</cell>
          </row>
          <row>
            <cell>1</cell>
            <cell>tertij ad <expan abbr="quartũ">quartum</expan>, &amp; quin</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad ſextum.</cell>
          </row>
          <row>
            <cell>2</cell>
            <cell>tertij ad ſextum, &amp; quin</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad quartum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Primi ad tertium.</cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>ſecundi ad quartum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>quinti ad ſextum.</cell>
          </row>
          <row>
            <cell>4</cell>
            <cell>ſecundi ad ſextum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>quinti ad quartum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Primi ad quintum.</cell>
          </row>
          <row>
            <cell>5</cell>
            <cell>ſecundi ad <expan abbr="ſextũ">ſextum</expan>, &amp; ter­</cell>
          </row>
          <row>
            <cell></cell>
            <cell>tij ad quartum.</cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>ſecundi ad quartum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>tertij ad ſextum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Secundi ad quartum.</cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>primi ad tertium, &amp; ſex</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad quintum.</cell>
          </row>
          <row>
            <cell>8</cell>
            <cell>primi ad quintum, et ſex</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad tertium.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Secundi ad ſextum.</cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>primi ad <expan abbr="quintũ">quintum</expan>, &amp; quar</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad tertium.</cell>
          </row>
          <row>
            <cell>10</cell>
            <cell>primi ad <expan abbr="tertiũ">tertium</expan>, &amp; quar­</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad quintum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Tertij ad quartum.</cell>
          </row>
          <row>
            <cell>11</cell>
            <cell>primi ad ſecundum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ſexti ad quintum.</cell>
          </row>
          <row>
            <cell>12</cell>
            <cell>primi ad quintum, &amp; ſex</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad ſecundum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Tertij ad ſextum.</cell>
          </row>
          <row>
            <cell>13</cell>
            <cell>primi ad ſecundum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>quarti ad quintum.</cell>
          </row>
          <row>
            <cell>14</cell>
            <cell>primi ad quintum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>quarti ad ſecundum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Quarti ad quintum.</cell>
          </row>
          <row>
            <cell>15</cell>
            <cell>ſecundi ad primum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>tertij ad ſextum.</cell>
          </row>
          <row>
            <cell>16</cell>
            <cell>ſecundi ad ſextum, &amp; ter</cell>
          </row>
          <row>
            <cell></cell>
            <cell>tij ad primum.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Quinti ad ſextum.</cell>
          </row>
          <row>
            <cell>17</cell>
            <cell>primi ad ſecundum, &amp;</cell>
          </row>
          <row>
            <cell></cell>
            <cell>quarti ad tertium.</cell>
          </row>
          <row>
            <cell>18</cell>
            <cell>primi ad <expan abbr="tertiũ">tertium</expan>, &amp; quar­</cell>
          </row>
          <row>
            <cell></cell>
            <cell>ti ad ſecundum.</cell>
          </row>
        </table>
        <table>
          <table.target id="table9"></table.target>
          <row>
            <cell>a</cell>
            <cell>e c</cell>
            <cell>a e</cell>
            <cell>e c</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell>c b</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell>f d</cell>
            <cell>c</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>f</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000239">Propoſitio ſeptima.</s>
        </p>
        <figure id="id.015.01.029.2.jpg" xlink:href="015/01/029/2.jpg"></figure>
        <p type="main">
          <s id="id000240">In modis qui neceſſariò produ­<lb></lb>cuntur ex duabus proportionibus, <lb></lb>cum duę quantitates ex illis, quę mo <lb></lb><figure id="id.015.01.029.3.jpg" xlink:href="015/01/029/3.jpg"></figure>dos conficiunt, æquales fuerint: pro­<lb></lb><arrow.to.target n="table10"></arrow.to.target><lb></lb>portio producta ad quatuor quanti­<lb></lb>tates omiologas reducetur.<lb></lb><arrow.to.target n="marg25"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000241"><margin.target id="marg25"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <table>
          <table.target id="table10"></table.target>
          <row>
            <cell>a</cell>
            <cell>b</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
          <row>
            <cell>d</cell>
            <cell>f</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000242">Sint ſex quantitates a b c d e f, &amp; <lb></lb>producatur proportio a ad b ex pro­<lb></lb>portione c ad d, &amp; e ad f, tu ſcis, quòd <lb></lb>modi recepti ſunt prima cum ſecunda, tertia uel quinta, &amp; ſecunda <lb></lb>cum quarta, &amp; ſexta, &amp; tertia ſimiliter cum eiſdem, &amp; quinta eodem <lb></lb>modo cum eiſdem: ſi igitur duę quantitates ex his, quę faciunt pro­


<pb pagenum="11" xlink:href="015/01/030.jpg"></pb>portionem productam inter ſe fuerint æquales reducetur hæc pro­<lb></lb>portio ad quatuor quantitates omologas, ſciliter abiectis amba­<lb></lb>bus æqualibus. </s>
          <s id="id000243">Sit gratia exempli prima æqualis quintæ: &amp; quia <lb></lb>in octauo modo proportio <expan abbr="ſecũdi">ſecundi</expan> ad quartum producitur ex pro­<lb></lb>portione primi ad quintum, &amp; ſexti ad tertium, ergo per expoſita <lb></lb>proportio ſecundi ad quartum, ut ſexti ad tertium, &amp; ita permutan­<lb></lb>do, &amp; conuertendo ſecundi ad ſextum, ut quarti ad tertium, &amp; tertij </s>
        </p>
        <p type="main">
          <s id="id000244"><arrow.to.target n="marg26"></arrow.to.target><lb></lb>ad quartum, ut ſexti ad ſecundum.</s>
        </p>
        <p type="margin">
          <s id="id000245"><margin.target id="marg26"></margin.target>V<emph type="italics"></emph>ndecima <lb></lb>petitione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000246">Propoſitio octaua.</s>
        </p>
        <p type="main">
          <s id="id000247">Si duarum <expan abbr="proportionũ">proportionum</expan> ſuperiores numeri alternatim cum infe<lb></lb>rioribus multiplicentur, atque coniungantur: erit proportio aggre­<lb></lb>gati ad productum ex inferioribus inuicem proportio ex primis <lb></lb>proportionibus compoſita.</s>
        </p>
        <figure id="id.015.01.030.1.jpg" xlink:href="015/01/030/1.jpg"></figure>
        <p type="main">
          <s id="id000248">Sit proportio una a ad b, alia c ad d, ducatur b in <lb></lb><arrow.to.target n="marg27"></arrow.to.target><lb></lb>c, fiatque e &amp; a in d, &amp; fiat f, iunganturque e &amp; f &amp; fiat h, <lb></lb>&amp; ducatur b in d et fiat g: dico <expan abbr="proportionẽ">proportionem</expan> h g com­<lb></lb>poſitam eſſe ex proportione a ad b, &amp; c ad d. </s>
          <s id="id000249">Quia <lb></lb><arrow.to.target n="marg28"></arrow.to.target><lb></lb>enim ex b in c fit e, &amp; ex b in d fit g, erit proportio e <lb></lb>ad g, ut c ad d, &amp; ſimiliter, quia ex d in a fit f, &amp; ex d in b fit g, erit f ad <lb></lb>g ut a ad b. </s>
          <s id="id000250">Sed e &amp; f componunt h, igitur proportio h ad g eſt com<lb></lb>poſita ex proportionibus e &amp; f ad g, igitur per communem animi <lb></lb>ſententiam, &amp; diffinitionem compoſitæ proportionis, proportio h <lb></lb><arrow.to.target n="marg29"></arrow.to.target><lb></lb>ad g compoſita eſt ex proportionibus a ad b, &amp; c ad d, quod eſt <lb></lb>propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id000251"><margin.target id="marg27"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000252"><margin.target id="marg28"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 13 <emph type="italics"></emph>peti­<lb></lb>tione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000253"><margin.target id="marg29"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 14 <emph type="italics"></emph>diffi <lb></lb>nitionem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000254">Propoſitio nona.</s>
        </p>
        <p type="main">
          <s id="id000255">Si duarum proportionum ſuperiores numeri alternatim cum <lb></lb>inferioribus multiplicentur, minusque productum ex maiore detra­<lb></lb>hatur, erit reſidui ad productum ex inferioribus proportio uelut <lb></lb>illa, quæ relinquitur detracta minore proportione ex maiore.</s>
        </p>
        <p type="main">
          <s id="id000256">Hæc eodem modo probatur, ut præcedens, niſi quod h fit de­<lb></lb><arrow.to.target n="marg30"></arrow.to.target><lb></lb>tracto è minore: gratia exempli ex f, &amp; ita ex diffinitione patet pro­<lb></lb>poſitum.</s>
        </p>
        <p type="margin">
          <s id="id000257"><margin.target id="marg30"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. <lb></lb>152.</s>
        </p>
        <p type="main">
          <s id="id000258">Propoſitio decima.</s>
        </p>
        <p type="main">
          <s id="id000259">Si fuerit alicuius quantitatis ad unam partem proportio uelut al<lb></lb>terius partis ad <expan abbr="ſecũdam">ſecundam</expan> quantitatem erit proportio cuiuſuis quan<lb></lb>titatis eiuſdem generis ad ſecundam compoſita proportio ex pro­<lb></lb>portionibus eiuſdem quantitatis aſſumptæ ad utran que partem pri­<lb></lb>mæ quantitatis ſeorſum.<lb></lb><arrow.to.target n="marg31"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000260"><margin.target id="marg31"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.030.2.jpg" xlink:href="015/01/030/2.jpg"></figure>
        <p type="main">
          <s id="id000261">Sit a b quantitas diuiſa in c, &amp; ſi c ut a b ad a c, <lb></lb>ita b c ad d: eritque iterum permutando a b ad b c, <lb></lb>ut a c ad d, &amp; ſumatur quædam quantitas e eiuſ­


<pb pagenum="12" xlink:href="015/01/031.jpg"></pb>dem tamen generis, cum illis dico quòd proportio e ad d eſt com­<lb></lb>poſita ex proportionibus e ad a c, &amp; e ad b c. </s>
          <s id="id000262">Poſita ergo e tan&lt;08&gt; ſu­<lb></lb>periore numero, &amp; a c &amp; c b inferioribus, erit ex octaua propoſitio­<lb></lb>ne huius proportio productorum ex e in a c, &amp; coniunctorum, &amp; <lb></lb>ex conſequenti per primam ſecundi Elementorum producti ex e in <lb></lb>a b ad productum ex a c in c b compoſita ex proportionibus e ad <lb></lb>a c, &amp; e ad c b: at quod fit ex a c in c b, eſt æquale ei quod fit ex a b in <lb></lb>d, eo quòd a b, a c, c b &amp; d ſunt omiologæ per decimam ſextam ſexti <lb></lb><expan abbr="Elemẽtorum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb></lb>ex d in a b eſt compoſita ex proportionibus e ad a c, &amp; e ad e b: At <lb></lb>proportio producti ex e in a b ad productum ex d in a b, eſt uelut e <lb></lb><arrow.to.target n="marg32"></arrow.to.target><lb></lb>ad d. </s>
          <s id="id000263">per ſuppoſita igitur proportio e ad d eſt compoſita ex propor<lb></lb>tionibus e ad a c, &amp; e ad b c, quod fuit demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id000264"><margin.target id="marg32"></margin.target>13. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000265">Propoſitio undecima.</s>
        </p>
        <p type="main">
          <s id="id000266">Proportio aggregati quarumlibet duarum quantitatum ad ag­<lb></lb>gregatum duarum æqualium quantitatum eſt compoſita ex pro­<lb></lb>portionibus primis, &amp; diuiſa per duplam.<lb></lb><arrow.to.target n="marg33"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000267"><margin.target id="marg33"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000268">Sit proportio a ad c, &amp; b ad d, &amp; ſint c &amp; d <lb></lb><figure id="id.015.01.031.1.jpg" xlink:href="015/01/031/1.jpg"></figure><lb></lb>æquales, dico quòd proportio a b ad c d eſt <lb></lb>compoſita ex proportionibus a ad c, &amp; b ad <lb></lb>d diuiſo compoſito per duplam. </s>
          <s id="id000269">Quia enim </s>
        </p>
        <p type="main">
          <s id="id000270"><arrow.to.target n="marg34"></arrow.to.target><lb></lb>c &amp; d ſunt æquales, erit b ad c, ut b ad d, qua­<lb></lb>re ex diffinitione cùm proportio a b ad c d <lb></lb><arrow.to.target n="marg35"></arrow.to.target><lb></lb>ſit compoſita ex proportionibus a ad c, &amp; b <lb></lb>ad c, erit etiam compoſita ex dictis ex propoſitione a ad c, &amp; b ad d, <lb></lb><arrow.to.target n="marg36"></arrow.to.target><lb></lb>ſtatuatur ergo e æqualis c d media inter a b &amp; c. </s>
          <s id="id000271">Et erit per ſecun­<lb></lb>dam propoſitionem proportio aggregati a b ad c producta ex <lb></lb><arrow.to.target n="marg37"></arrow.to.target><lb></lb>proportione aggregati a b ad c, &amp; e ad c, igitur proportio a b ad e <lb></lb>erit proportio a b ad c, diuiſa per proportionem e ad c, ſed e ad c eſt <lb></lb><arrow.to.target n="marg38"></arrow.to.target><lb></lb>dupla: igitur proportio a b ad c d eſt proportio a b ad c diuiſa per <lb></lb>duplam.</s>
        </p>
        <p type="margin">
          <s id="id000272"><margin.target id="marg34"></margin.target>E<emph type="italics"></emph>x ſexta<emph.end type="italics"></emph.end> A<emph type="italics"></emph>nim. <lb></lb>com. </s>
          <s id="id000273">ſententia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000274"><margin.target id="marg35"></margin.target>D<emph type="italics"></emph>ecimaquarta<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000275"><margin.target id="marg36"></margin.target>13. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000276"><margin.target id="marg37"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000277"><margin.target id="marg38"></margin.target>P<emph type="italics"></emph>er quintam <emph.end type="italics"></emph.end><lb></lb>A<emph type="italics"></emph>nim. </s>
          <s id="id000278">com. </s>
          <s id="id000279">ſen <lb></lb>tentiam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000280">Propoſitio duodecima.</s>
        </p>
        <p type="main">
          <s id="id000281">Propoſitis duabus proportionibus unam alteri iungere abſque <lb></lb>multiplicatione.<lb></lb><arrow.to.target n="marg39"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000282"><margin.target id="marg39"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb>10. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000283">Sint propoſitæ proportiones a ad c &amp; <lb></lb><figure id="id.015.01.031.2.jpg" xlink:href="015/01/031/2.jpg"></figure><lb></lb>b ad d, &amp; aſſumo e ad c, iuxta ea quæ Eu­<lb></lb>clides demonſtrauit, ut b ad d, erit igitur </s>
        </p>
        <p type="main">
          <s id="id000284"><arrow.to.target n="marg40"></arrow.to.target><lb></lb>proportio a e ad c, compoſita ex proportionibus a ad c, &amp; e ad c, <lb></lb>ſed proportio e ad c eſt, ut b ad d, igitur proportio a e ad c compo­<lb></lb>ſita eſt ex proportionibus a ad c, &amp; b ad d.</s>
        </p>
        <p type="margin">
          <s id="id000285"><margin.target id="marg40"></margin.target>E<emph type="italics"></emph>x generali <lb></lb>com.<emph.end type="italics"></emph.end> A<emph type="italics"></emph>nim. ſen <lb></lb>tentia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000286">Aliter ex b in c fiat f ex a in d, g ex c in d h coniunctum ex f g, k.</s>
        </p>
        <pb pagenum="13" xlink:href="015/01/032.jpg"></pb>
        <figure id="id.015.01.032.1.jpg" xlink:href="015/01/032/1.jpg"></figure>
        <p type="main">
          <s id="id000287">Quia ergo ex c in b fit f, ex c in d h, erit f ad h, <lb></lb>ut b ad d, igitur ut e ad c, ſed a ad c, ut g ad h igi<lb></lb><arrow.to.target n="marg41"></arrow.to.target><lb></lb>tur a e ad c, ut k ad h, ſed k ad h cómponitur ex <lb></lb>proportionibus a ad c, &amp; b ad d. </s>
          <s id="id000288">Ex octaua ha <lb></lb>rum igitur proportio a c ad c compoſita eſt ex <lb></lb>eiſdem. </s>
          <s id="id000289">Forſan quis dicat hanc eandem eſſe <lb></lb>octauæ ſed <expan abbr="nõ">non</expan> eſt, in illa enim proportio com­<lb></lb>paratur ad productum, in hac ad unam ex <lb></lb>quantitatibus.</s>
        </p>
        <p type="margin">
          <s id="id000290"><margin.target id="marg41"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000291">Ex hoc ſequitur quòd: Quælibet duæ quantitates quarum ag­<lb></lb><arrow.to.target n="marg42"></arrow.to.target><lb></lb>gregatum eſt idem ad eam quantitatem, componunt eandem pro­<lb></lb>portionem.</s>
        </p>
        <p type="margin">
          <s id="id000292"><margin.target id="marg42"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000293">Propoſitio tertia decima.</s>
        </p>
        <p type="main">
          <s id="id000294">Proportio confuſa aggregati primæ &amp; tertiæ quatuor quantita­<lb></lb>tum omiologarum ad <expan abbr="aggregatũ">aggregatum</expan> ſecundæ &amp; quartæ, eſt uelut com<lb></lb>poſita ex eiſdem diuiſa per duplam.<lb></lb><arrow.to.target n="marg43"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000295"><margin.target id="marg43"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000296">Sint a ad b, ut c ad d, dico, quòd erit confuſa <lb></lb><figure id="id.015.01.032.2.jpg" xlink:href="015/01/032/2.jpg"></figure><arrow.to.target n="table11"></arrow.to.target><lb></lb>proportio a c aggregati ad <expan abbr="aggregatũ">aggregatum</expan> b d, com<lb></lb>poſitæ ex his proportionibus diuiſæ per du­<lb></lb>plam æqualis. </s>
          <s id="id000297">Erit enim aggregati ex a c ad aggregatum ex b d, ue­<lb></lb>lut a ad b per 18 quinti Elementorum. </s>
          <s id="id000298">Sed proportiones a ad b, <lb></lb>&amp; c ad d componunt proportionem producti a in d, &amp; c in b per <lb></lb>octauam harum, ad <expan abbr="productũ">productum</expan> ex b in d, productum uerò ex a in d <lb></lb>eſt æquale producto ex b in c per decimam ſextam ſexti Elemento­<lb></lb>rum, &amp; proportio producti ex b in c ad productum ex b in d eſt ue <lb></lb>lut c ad d, quare ut aggregati a c ad aggregatum b d, igitur propor­<lb></lb>tio compoſita ex a ad b, &amp; c ad d, eſt uelut confuſa bis ſumpta. </s>
          <s id="id000299">Igi­<lb></lb>tur confuſa eſt uelut compoſita diuiſa per duplam per modum un­<lb></lb>decimæ huius.</s>
        </p>
        <table>
          <table.target id="table11"></table.target>
          <row>
            <cell>a</cell>
            <cell>c</cell>
          </row>
          <row>
            <cell>-----</cell>
            <cell>-----</cell>
          </row>
          <row>
            <cell>b</cell>
            <cell>d</cell>
          </row>
          <row>
            <cell>---</cell>
            <cell>---</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000300">Propoſitio quarta decima.</s>
        </p>
        <p type="main">
          <s id="id000301">Proportiones confuſæ, &amp; coniunctæ in tribus quantitatibus in­<lb></lb>uicem commutantur.</s>
        </p>
        <figure id="id.015.01.032.3.jpg" xlink:href="015/01/032/3.jpg"></figure>
        <p type="main">
          <s id="id000302">Sint tres quantitates, dico, quod proportio c </s>
        </p>
        <p type="main">
          <s id="id000303"><arrow.to.target n="marg44"></arrow.to.target><lb></lb>ad a b confuſa eſt, conuerſa coniunctæ a &amp; b ad <lb></lb><arrow.to.target n="marg45"></arrow.to.target><lb></lb>c. </s>
          <s id="id000304">Nam per dicta proportio a b ad c efficit con­<lb></lb>iunctam ex a b ad c: ſed c ad a b conuerſa eſt eius, quæ eſt a b ad c, &amp; <lb></lb>proportio c ad a b eſt confuſa eius, quæ eſt c ad a &amp; b. </s>
          <s id="id000305">Igitur pro­<lb></lb>portio confuſa in tribus quantitatibus eſt contraria coniunctæ in <lb></lb>eiſdem.</s>
        </p>
        <p type="margin">
          <s id="id000306"><margin.target id="marg44"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000307"><margin.target id="marg45"></margin.target>14. <emph type="italics"></emph>diff.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000308">Ex quauis ergo illarum data, data erit &amp; reliqua.<lb></lb><arrow.to.target n="marg46"></arrow.to.target></s>
        </p>
        <pb pagenum="14" xlink:href="015/01/033.jpg"></pb>
        <p type="margin">
          <s id="id000309"><margin.target id="marg46"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>diff.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000310">Propoſitio quinta decima.</s>
        </p>
        <p type="main">
          <s id="id000311">Si fuerint quatuor quantitas proportio confuſa aggregati pri­<lb></lb>mæ &amp; tertiæ ad aggregatum ſecundæ, &amp; quartæ erit ut monadis <lb></lb>addito prouentu, qui fit diuiſa differentia differentiarum primæ &amp; <lb></lb>ſecundæ, atque quartæ &amp; tertiæ per aggregatum tertiæ, &amp; quartæ ad <lb></lb>ipſam monadem.</s>
        </p>
        <figure id="id.015.01.033.1.jpg" xlink:href="015/01/033/1.jpg"></figure>
        <p type="main">
          <s id="id000312">Sint quatuor quantitates a b, c, d, e f, &amp; <lb></lb><arrow.to.target n="marg47"></arrow.to.target><lb></lb>ſit a b maior cin a h, &amp; e f maior d in f g, &amp; <lb></lb>differentia f g &amp; a h ſit a k: dico proportio­<lb></lb>nem a b, &amp; d confuſam ad c &amp; e f, eſſe ut mo<lb></lb>nadis addito prouentu, uel detracto a k diuiſæ per aggregatum c. <lb></lb>&amp; e f ad ipſam monadem, &amp; manifeſtum eſt, quòd poteſt continge­<lb></lb>re pluribus modis: Primus ut a b ſit maior c &amp; e f minor d, &amp; tunc <lb></lb>differentiæ coniungentur, &amp; prouentus, addetur monadi. </s>
          <s id="id000313">Idem fa­<lb></lb>ciendum erit ſi a b ſit maior c, &amp; e f ſit minor d, ſed exceſſus ſuperet <lb></lb>defectum. </s>
          <s id="id000314">At ſi uel a b ſit minor c, &amp; e f maior d, uel ita minor, ut c <lb></lb>exceſſus ſupra b a ſit maior defectu, detrahemus prouentum à mo­<lb></lb>nade. </s>
          <s id="id000315">Alia cautio eſt quòd ſi fuerint utrinque exceſſus, aut defectus, <lb></lb>minuemus minorem de maiore: ſi autem unus ſit exceſſus alter de­<lb></lb>fectus, iungemus illos, &amp; poſt diuidemus. </s>
          <s id="id000316">uno ergo demonſtrato <lb></lb>ut pote primo intelligentur reliqui. </s>
          <s id="id000317">Quia ergo b h eſt æqualis c &amp; <lb></lb>e g æqualis d &amp; h k æqualis g f, erit ex communi animi ſententia ag<lb></lb>gregatum ex d &amp; k b æquale aggregato ex c &amp; e f, igitur per dicta <lb></lb>proportio aggregati ad aggregatum eſt unum. </s>
          <s id="id000318">at uerò diuiſa k a <lb></lb>per c &amp; e f fit quantum diuiſa eadem per b k, &amp; d, ſed diuiſa k a per b <lb></lb>k, &amp; d iunctas, exit proportio a k ad aggregatum b k &amp; d: igitur di­<lb></lb>uiſa a k per aggregatum e f &amp; c, exibit eadem proportio, igitur a b <lb></lb>&amp; d ad aggregatum c &amp; e f eſt coniuncta ex monade &amp; proportio­<lb></lb>ne a k ad aggregatum c &amp; e f, quod erat demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id000319"><margin.target id="marg47"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.033.2.jpg" xlink:href="015/01/033/2.jpg"></figure>
        <p type="main">
          <s id="id000320">Ex hoc patet quod proportionum confuſio <lb></lb><arrow.to.target n="marg48"></arrow.to.target><lb></lb>fit iunctis denominatoribus numeratoris: mul­<lb></lb>tiplicatio multiplicatis: additio multiplicatis <lb></lb>decuſſatim in numeratores ad productum ex <lb></lb>denominatoribus, ut in exemplis.</s>
        </p>
        <p type="margin">
          <s id="id000321"><margin.target id="marg48"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000322">Propoſitio ſexta decima.</s>
        </p>
        <p type="main">
          <s id="id000323">Omnium quatuor quantitatum propoſita <lb></lb>prima, quæ non minorem habet proportionem <lb></lb>ad ſuam correſpondentem, quàm alia ad aliam <lb></lb><figure id="id.015.01.033.3.jpg" xlink:href="015/01/033/3.jpg"></figure><lb></lb>erit proportio confuſa illarum, ut pro­<lb></lb>ducti ex aggregato primæ &amp; tertiæ in 


<pb pagenum="15" xlink:href="015/01/034.jpg"></pb>tertiam, ad productum ex aggregato tertiæ &amp; omiotatæ ad ſecun­<lb></lb>dam in ipſam quartam.</s>
        </p>
        <p type="main">
          <s id="id000324">Hæc magis reducit confuſam proportionem ad notitiam, quàm, <lb></lb>præcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, quę operatio <lb></lb>eſt ſimpliciſsima, ſiue per multiplicationem quantitatum fiat, duæ <lb></lb>ſunt tantum multiplicationes, ſiue per eundem terminum ſufficit <lb></lb>alium addere. </s>
          <s id="id000325">Summatur ergo a b, c, d &amp; e, &amp; non ſit maior propor­<lb></lb>tio d ad e, quàm a b ad c, &amp; ſtatuatur tunc prima a b, ſecunda c, ter­<lb></lb>tia d, quarta e, &amp; poſtquam non eſt minor ratio a b ad c, quàm d ad <lb></lb>e, ſumatur a f ad c, ut d ad e. </s>
          <s id="id000326">licet enim hoc facere. </s>
          <s id="id000327">Dico quod pro­<lb></lb>portio confuſa a b &amp; d ad c &amp; e eſt uelut producti ex aggregato a b <lb></lb>&amp; d in d ad productum ex aggregato a f &amp; d in e. </s>
          <s id="id000328">Statuatur aggre­<lb></lb><arrow.to.target n="marg49"></arrow.to.target><lb></lb>gatum a b &amp; d linea a d prima quantitas, &amp; aggregatum a f &amp; d, <lb></lb><figure id="id.015.01.034.1.jpg" xlink:href="015/01/034/1.jpg"></figure><lb></lb>a d ſecunda quantitas, &amp; d tertia, <lb></lb>&amp; c quarta, &amp; ex a b in d fiat g, ex <lb></lb>a d in e fiat h, erit ergo per pri­<lb></lb>mam propoſitionem g ad h pro­<lb></lb><arrow.to.target n="marg50"></arrow.to.target><lb></lb>ducta ex proportionibus a b d ad <lb></lb>a f d, &amp; d ad c. </s>
          <s id="id000329">Sed proportio a f d <lb></lb>ad aggregatum c e, eſt uelut d ad <lb></lb>e. </s>
          <s id="id000330">Proportio uerò a b d ad a f d, &amp; <lb></lb>a f d ad e c producunt proportio­<lb></lb>nem a b d ad c &amp; e per ſecundam propoſitionem, harum igitur con­<lb></lb>fuſa a b ad c, &amp; d ad e, &amp; eſt proportio a b d ad c &amp; e, producuntur <lb></lb>ex proportionibus a b d ad a f d, &amp; d ad e. </s>
          <s id="id000331">Ergo proportio g ad h <lb></lb>eſt confuſa ex a b ad e, &amp; d ad e, quod erat demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id000332"><margin.target id="marg49"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000333"><margin.target id="marg50"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000334">Propoſitio decima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id000335">Omnes duę proportiones conuerſæ producunt æqualem pro­<lb></lb>portionem.<lb></lb><arrow.to.target n="table12"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table12"></table.target>
          <row>
            <cell>a</cell>
          </row>
          <row>
            <cell>-----</cell>
          </row>
          <row>
            <cell>b</cell>
          </row>
          <row>
            <cell>---</cell>
          </row>
          <row>
            <cell>c</cell>
          </row>
          <row>
            <cell>----</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000336">Sint duæ proportiones a ad b &amp; b ad a conuerſa, <lb></lb><figure id="id.015.01.034.2.jpg" xlink:href="015/01/034/2.jpg"></figure><arrow.to.target n="marg51"></arrow.to.target><lb></lb>dico, quòd producunt proportionem æqualem. </s>
          <s id="id000337">fiat <lb></lb>enim b ad c, ut b ad a, erit igitur a æqualis c &amp; b c con<lb></lb><arrow.to.target n="marg52"></arrow.to.target><lb></lb>uerſa eius quæ eſt a ad b, ſed per ſecundam harum <lb></lb>proportiones a ad b, &amp; b ad c producunt propor­<lb></lb>tionem a ad c, igitur proportiones etiam a ad b &amp; b ad a produ­<lb></lb>cunt eandem.</s>
        </p>
        <p type="margin">
          <s id="id000338"><margin.target id="marg51"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000339"><margin.target id="marg52"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. A<emph type="italics"></emph>ni­<lb></lb>mi <expan abbr="communẽ">communem</expan> <lb></lb>ſententiam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000340">Propoſitio decima octaua.</s>
        </p>
        <p type="main">
          <s id="id000341">Si fuerint quotlibet quantitates in continua proportione multi­<lb></lb>plici præter ultimam: proportio uerò penultimæ ad ultimam qua­<lb></lb>lis reſidui primæ ad ſecundam, erit primæ ad aggregatum reliqua­<lb></lb>rum uelut penultimæ ad ultimam.


<pb pagenum="16" xlink:href="015/01/035.jpg"></pb><arrow.to.target n="marg53"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000342"><margin.target id="marg53"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000343">Sint quantitates a b c d in continua proportione multiplici, ſed <lb></lb>d ad e ſit uelut reſidui a &amp; b ad b, dico proportionem a ad b c d e <lb></lb>eſſe ut d ad e. </s>
          <s id="id000344">Quia enim eſt gnomonis e ad quadratum d, ut d ad e <lb></lb>ex ſuppoſito erit per coniunctam proportionem c &amp; d ad d &amp; e, ut</s>
        </p>
        <p type="main">
          <s id="id000345"><arrow.to.target n="marg54"></arrow.to.target><lb></lb>d ad e, ſed e gnomo cum quadrato d efficit qua­<lb></lb><figure id="id.015.01.035.1.jpg" xlink:href="015/01/035/1.jpg"></figure><lb></lb>dratum e, igitur ut c quadrati ad d &amp; eiuncta, ita <lb></lb>d ad e. </s>
          <s id="id000346">Rurſus, quia b quadrati ad c quadratum, <lb></lb><arrow.to.target n="marg55"></arrow.to.target><lb></lb>ut c ad d erit gnomonis b ad quadratum c, ut <lb></lb>gnomonis c ad quadratum d, &amp; ita d ad e, igitur <lb></lb><arrow.to.target n="marg56"></arrow.to.target><lb></lb>gnomonum b c cum quadrato d ad aggrega­<lb></lb>tum c d e quadratorum, ut d ad e, ſed c gno­<lb></lb>mo cum d quadrato perficit c quadratum, <lb></lb>&amp; c quadratum cum gnomone b perficit <lb></lb>quadratum b, igitur proportio quadrati b <lb></lb>ad quadrata c d e, ut d quadrati a d e. </s>
          <s id="id000347">Et ita <lb></lb>repetendo de quotuis quantitatibus in infi<lb></lb>nitum uſque. </s>
          <s id="id000348">Hæc proponitur ab Archimede in libro de quadrato <lb></lb>æquali parabolæ, &amp; minus generaliter &amp; pluribus demonſtratur. <lb></lb></s>
          <s id="id000349">Ego tamen quia eſt generalis, deſcribam illam per corrolarium: ad­<lb></lb>damque aliud quod ex hoc ſequitur.<lb></lb><arrow.to.target n="marg57"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000350"><margin.target id="marg54"></margin.target>13. P<emph type="italics"></emph>ropoſ. <lb></lb>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000351"><margin.target id="marg55"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 19. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000352"><margin.target id="marg56"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000353"><margin.target id="marg57"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000354">Si fuerint quotlibet <expan abbr="quãtitates">quantitates</expan> omnes analogæ præter ultimam, <lb></lb>ſit autem penultima ad ultimam qualis reſidui primæ &amp; ſecundæ <lb></lb>ad ſecundam, erit proportio primæ ad aggregatum omnium alia­<lb></lb>rum ueluti penultimæ ad ultimam.</s>
        </p>
        <p type="main">
          <s id="id000355"><arrow.to.target n="marg58"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000356"><margin.target id="marg58"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000357">Hæc enim eſt euidens, quia conuenit ei demonſtratio propoſita. <lb></lb><figure id="id.015.01.035.2.jpg" xlink:href="015/01/035/2.jpg"></figure><lb></lb>exemplo autem in numeris à latere <lb></lb>poſito uides declarationem. </s>
          <s id="id000358">nam <lb></lb>proportio 16 ad 32 eſt uelut 27 reſi<lb></lb>dui primæ &amp; ſecundæ ad ipſam ſe­<lb></lb>cundam ſcilicet ad 54.</s>
        </p>
        <p type="main">
          <s id="id000359"><arrow.to.target n="marg59"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000360"><margin.target id="marg59"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000361">Ex hoc patet etiam quòd aſſumptis omnibus, ſub multiplicibus <lb></lb>analogiæ uſque in infinitum prima quantitas eſt multiplex aggre­<lb></lb>gati omnium reliquarum numero 1 m: quo prima eſt multiplex <lb></lb>ſecundæ.</s>
        </p>
        <p type="main">
          <s id="id000362"><arrow.to.target n="marg60"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000363"><margin.target id="marg60"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id000364">Si fuerint quotlibet quantitates in ſuper particulari proportio­<lb></lb>ne analogæ, erit proportio primæ ad aggregatum omnium in infi­<lb></lb>nitum iuxta proportionem multiplicem conuerſam illius partis.</s>
        </p>
        <p type="main">
          <s id="id000365"><arrow.to.target n="marg61"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000366"><margin.target id="marg61"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000367">Velut collectæ in ſeſquialtera duplæ in ſexquitertia triplæ in <lb></lb>ſexquiſeptima ſeptuplæ. </s>
          <s id="id000368">Vt capio 512 448 392 343, &amp; ita deinceps <lb></lb>uſque in infinitum aggregatum omnium earum erit 3584. Septu­


<pb pagenum="17" xlink:href="015/01/036.jpg"></pb>plum 512, &amp; aggregatum 18. 12. 8. 5 2/3, &amp; ita deinceps in ſexquialtera <lb></lb>erit 54 duplum 27 primæ in eo ordine.</s>
        </p>
        <p type="head">
          <s id="id000369">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id000370">Ex quo patet genus demonſtrandi nouun &amp; pulchrum: nam <lb></lb>ſupponatur 54, aggregatum duplum 27, primæ igitur addito 27 <lb></lb>ad 54, cum ſit dimidium, &amp; addito 13 1/2, dimidio 27 ad 27, nam ex <lb></lb>ſuppoſito quantitas ſequens eſt ſexquialtera ad 27, igitur 81 eſt du­</s>
        </p>
        <p type="main">
          <s id="id000371"><arrow.to.target n="marg62"></arrow.to.target><lb></lb>plum ad 40 1/2. Igitur conuertendo eſt proportio aggregati prioris <lb></lb>ad 27 eſt dupla, ergo aggregatum eſt 54.<lb></lb><arrow.to.target n="marg63"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000372"><margin.target id="marg62"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000373"><margin.target id="marg63"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id000374">Ex hoc patet eandem generaliter quod proportio maioris quan<lb></lb>titatis ad aggregatum reliquarum analogarum eſt, uelut eius quod <lb></lb>prouenit diuiſo quadrato maioris termini per differentiam eius, &amp; <lb></lb>ſequentis maioris in eadem proportione ad ipſum maiorem.</s>
        </p>
        <p type="main">
          <s id="id000375"><arrow.to.target n="marg64"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000376"><margin.target id="marg64"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000377">Exemplum ſit proportio augens 25 &amp; 35 duarum quintarum, uo <lb></lb>lo ſcire quantum ſit aggregatum omnium citra 25, maximam acci­<lb></lb>pio 35, ulteriorem ad 25, cuius differentia a 25 eſt 10, cum quo diui­<lb></lb>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. </s>
          <s id="id000378">Et facile po­</s>
        </p>
        <p type="main">
          <s id="id000379"><arrow.to.target n="marg65"></arrow.to.target><lb></lb>reſt demonſtrari. </s>
          <s id="id000380">Si quis dicat in qua proportione ſunt infinitæ <lb></lb>quantitates analogæ cum 12, quæ iunctæ efficiunt 10, iunge 10 cum <lb></lb>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, &amp; in ea pro­<lb></lb>portione <expan abbr="erũt">erunt</expan> illæ quantitates, in qua ſunt 26 2/3 ad 12: duc per 5 fiunt <lb></lb>60, &amp; 132 diuide per 12, exeunt 11 &amp; 5, &amp; ita erunt in proportione 11 <lb></lb>ad 5 experiaris, &amp; inuenies, &amp; demonſtratur ex prioribus.</s>
        </p>
        <p type="margin">
          <s id="id000381"><margin.target id="marg65"></margin.target>Q<emph type="italics"></emph>uæſtio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000382">Propoſitio decimanona.</s>
        </p>
        <p type="main">
          <s id="id000383">Si fuerint aliquot quantitates arithmeticæ omiologæ, quarum <lb></lb>exceſſus ſit æqualis minimè, omnibus autem deficientibus ſupple­<lb></lb>menta ad ęqualitatem maximè adiungantur, erunt quadrata omni­<lb></lb>um quantitatum æqualium adiecto rurſus quadrato primæ cum <lb></lb>eo quod fit ex minima primi ordinis in <expan abbr="aggregatũ">aggregatum</expan> omnium quan­<lb></lb>titatum eiuſdem tripla aggregato quadra­<lb></lb><figure id="id.015.01.036.1.jpg" xlink:href="015/01/036/1.jpg"></figure><lb></lb>torum omnium quantitatum primi ordinis <lb></lb><arrow.to.target n="marg66"></arrow.to.target><lb></lb>pariter acceptis.</s>
        </p>
        <p type="margin">
          <s id="id000384"><margin.target id="marg66"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000385">Sint aliquot quantitates a b c d e f g h in <lb></lb>continua proportione. </s>
          <s id="id000386">Arithmetica diſpoſitę <lb></lb>ita ut minima <expan abbr="earũ">earum</expan> quę ſit h, ſit ęqualis diffe­<lb></lb>rentię quantitatum <expan abbr="ſecundũ">ſecundum</expan> ordinem diſpo<lb></lb><expan abbr="ſitarũ">ſitarum</expan>, uelut differentia a &amp; b, &amp; b &amp; c, &amp; c &amp; <lb></lb>d, et ita de alijs, addantur <expan abbr="aũt">aut</expan> <expan abbr="ſupplemẽta">ſupplementa</expan> ſin <lb></lb>gulis harum, quæ ſint i k l m n o p, ita ut <expan abbr="oẽs">oes</expan> <lb></lb>fiant ęquales <expan abbr="cũ">cum</expan> ſuis ſupplementis ipſi lineę <lb></lb>à maiori. </s>
          <s id="id000387">Eſtque <expan abbr="idẽ">idem</expan> ac ſi eſſent aliquot quanti<pb pagenum="18" xlink:href="015/01/037.jpg"></pb>tates, &amp; <expan abbr="diuiderent̃">diuiderentur</expan> ſingulę <expan abbr="ſecundũ">ſecundum</expan> numerum <expan abbr="illarũ">illarum</expan>, ſi quatuor in <lb></lb>quatuor partes æquales, ſi quinque in quinque, ſi decem in decem, ea ra<lb></lb>tione ut ultima <expan abbr="diuideret̃">diuideretur</expan>, ubi eſt finis primæ partis, penultima ubi <lb></lb>eſt finis ſecundæ partis, ante penultima ubi eſt finis tertiæ, &amp; ſic de <lb></lb>alijs. </s>
          <s id="id000388">Vocabo ergo primas <expan abbr="quãtitates">quantitates</expan> propoſitas a b c d e f g h quan­<lb></lb>titates primi ordinis, ſed quantitates æquales quæ <expan abbr="conſtãt">conſtant</expan> ex quan <lb></lb>titatis. </s>
          <s id="id000389">primi ordinis, &amp; ſupplementis, appellabo quantitates ſecun<lb></lb>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb></lb>quia non eſt diuiſa, reliquæ omnes differunt, quantitates uerò quas <lb></lb>adiunxi nominabo <expan abbr="ſupplemẽta">ſupplementa</expan>, &amp; ſunt una minus <expan abbr="quã">quam</expan> quantitates <lb></lb>ordinum: ut ſi <expan abbr="quãtitates">quantitates</expan> ordinum ſint octo, erunt ſupplementa ſe­<lb></lb>ptem, &amp; ſi quantitates <expan abbr="ordinũ">ordinum</expan>, eſſent ſeptem eſſent <expan abbr="ſupplemẽta">ſupplementa</expan> ſex, <lb></lb>quia inter ſupplementa <expan abbr="nõ">non</expan> <expan abbr="adnumerat̃">adnumeratur</expan> quantitas indiuiſa. </s>
          <s id="id000390">Erunt er<lb></lb>go ſupplementa i k l m n o p, quæ tanto erunt maiora quanto quan<lb></lb>titates primi ordinis ſunt minores, &amp; contrà tanto maiora, quanto <lb></lb><expan abbr="quãtitates">quantitates</expan> primi ordinis ſunt maiores. </s>
          <s id="id000391">quantitates <expan abbr="aũt">aut</expan> ſecundi ordi<lb></lb>nis <expan abbr="appellabunt̃">appellabuntur</expan> a, b i, ck, dl, em, fn, go, &amp; hp. </s>
          <s id="id000392">Hæc uolui pluribus <lb></lb>agere, ut dilucidior eſſet propoſitio. </s>
          <s id="id000393">quæ licet <expan abbr="nõ">non</expan> ſit difficilis, eſt <expan abbr="tamẽ">tamen</expan> <lb></lb>confuſa ualde propter multitudinem <expan abbr="quantitatũ">quantitatum</expan> &amp; ordinum. </s>
          <s id="id000394">Dico <lb></lb>ergo q̊d aggregatum <expan abbr="quadratorũ">quadratorum</expan> quantitatum ſecundi ordinis pri<lb></lb>mo quadrato bis repetito, ſeu uno addito <expan abbr="cũ">cum</expan> eo quod fit ex minima <lb></lb>in aggregatum quantitatum primi ordinis eſt <expan abbr="triplũ">triplum</expan> aggregato ex <lb></lb>quadratis omnibus <expan abbr="quantitatũ">quantitatum</expan> <expan abbr="eiuſdẽ">eiuſdem</expan> primi ordinis, &amp; utres exem <lb></lb>plo facilius innoteſcat, ſint <expan abbr="quãtitates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb></lb>quorum quadrata ſint 64. 49. 36. 25. 16. &amp; 9.4 &amp; 1. quæ iuncta <expan abbr="faciũt">faciunt</expan> <lb></lb>204, dico quod ſi ſumamus quadrata omnium <expan abbr="quãtitatum">quantitatum</expan> ſecundi <lb></lb>ordinis, quæ ſunt octies 64, &amp; eis addiderimus unum <expan abbr="quadratũ">quadratum</expan> ex <lb></lb>his, ut fiant nouies 64, &amp; erunt 556, ſimul iuncta &amp; eis addamus, q̊d <lb></lb>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti­<lb></lb>tatum omnium primi ordinis, &amp; eſt tale <expan abbr="productũ">productum</expan> 36, ut fiat totum <lb></lb>612, quod tale 612 eſt triplum 204, aggregati <expan abbr="quadratorũ">quadratorum</expan> primi or­<lb></lb>dinis unius demonſtratio hęc eſt. </s>
          <s id="id000395">Quia ex quarta ſecundi Element. <lb></lb>Euclidis ſingula quadrata <expan abbr="quantitatũ">quantitatum</expan> <expan abbr="diuiſarũ">diuiſarum</expan> ſecundi ordinis con<lb></lb>ſtant ex quatuor partibus quarum duę ſunt quadrata partium, reli­<lb></lb>quæ duæ ſunt producta ex partibus <expan abbr="inuicẽ">inuicem</expan> bis, &amp; quia h fuit æqua­<lb></lb>lis 1, &amp; p ęqualis b, quia ſupplementa <expan abbr="fuerũtęqualia">fuerunt ęqualia</expan> mutuò quanti<lb></lb>tatibus, &amp; ita c æqualis o &amp; k æqualis g &amp; d, æqualis n &amp; l, æqualis <lb></lb>f, e <expan abbr="aũt">aut</expan> ęqualis m. </s>
          <s id="id000396"><expan abbr="Sequit̃">Sequitur</expan> ergo quod ſumptis duabus quantitatibus <lb></lb>ſecundi ordinis habentibus <expan abbr="ſupplemẽta">ſupplementa</expan> mutuò æqualia ipſis quan<lb></lb>titatibus quod quadrata partium <expan abbr="erũt">erunt</expan> dupla quadratis primarum <lb></lb>quantitatum: ueluti capio b i ſecundam &amp; h p ultimam, <expan abbr="quarũ">quarum</expan> qua­


<pb pagenum="19" xlink:href="015/01/038.jpg"></pb>drata partium ſunt quadrata b &amp; i, &amp; h &amp; p, ſed b eſt æqualis p, &amp; h <lb></lb>æqualis i. </s>
          <s id="id000397">Ergo quatuor quadrata b i &amp; h p ſunt dupla quadratis b <lb></lb>&amp; h, &amp; ita <expan abbr="concludã">concludam</expan> de omnibus ubi duæ quantitates duabus com<lb></lb>parantur: ſed in e m quia eſt ſola una quantitas, iſtud eſt etiam cla­<lb></lb>rius, quia quadrata e &amp; m ſunt dupla quadrato e ſoli eo, quod &amp; m <lb></lb><arrow.to.target n="marg67"></arrow.to.target><lb></lb>ſunt æquales. </s>
          <s id="id000398">Igitur per demonſtrata ab Euclide erit proportio o­<lb></lb>mnium quadratorum b i, c k, d l, e m, f n, g o, h p, ad quadrata b c d e <lb></lb>f g h, pariter accepta proportio dupla. </s>
          <s id="id000399">at uerò addito quadrato a <lb></lb>quadratis b c d e f g h, &amp; erunt quadrata omnium quantitatum, &amp; <lb></lb>quadratis b i, c k, d l, e m, f n, g o, h p, duplo quadrati a ſcilicet ſemel, <lb></lb>quia a eſt ex ſecundo ordine quantitatum, &amp; ſemel, quia hoc fuit aſ­<lb></lb>ſumptum in Problemate. </s>
          <s id="id000400">Sequitur ut quadrata omnia <expan abbr="quãtitatum">quantitatum</expan> <lb></lb>ſecundi ordinis, pro ut ſunt diuiſa in partes addito quadrato a, ſint <lb></lb>dupla quadratis primarum quantítatum, ſimul pariter acceptis. </s>
          <s id="id000401">Re<lb></lb>liquum eſt modo ut oſtendamus dupla <expan abbr="illorũ">illorum</expan> productorum, cum <lb></lb>eo quod fit ex minima quantitate, ſcilicet h in aggregatum ipſarum <lb></lb>quantitatum primi ordinis eſſe æquale quadratis, <expan abbr="quantitatũ">quantitatum</expan> eiuſ­<lb></lb>dem primi ordinis pariter acceptis. </s>
          <s id="id000402">Conſtat igitur, quod duplum i<lb></lb>in b eſt æquale duplo h in ipſum b, quia h &amp; i ſunt æquales, &amp; du­<lb></lb>plum k in ipſum c, eſt æquale quadruplo h in idem c, quia k eſt du­<lb></lb>pla h, &amp; ſimiliter duplum l in ipſum d eſt æquale ſexcuplo, h in d, <lb></lb>quia l eſt tripla h, &amp; ita procedendo erunt illa dupla producta æ­<lb></lb>qualia productis ex h in ipſas quantitates toties ſumptis quantus <lb></lb>eſt numerus, qui prouenit duplicato numero, ſecundum <expan abbr="quẽ">quem</expan> h con<lb></lb>tinetur in illo ſupplemento, exemplum uolo duplum producti lin <lb></lb>d bis, ſcio quòd ſupplementum l continet h ter, duplicabo tria &amp; fi­<lb></lb>ent ſex, <expan abbr="igit̃">igitur</expan> <expan abbr="duplũ">duplum</expan> lin d æquale eſt ſexcuplo h in ipſum d. </s>
          <s id="id000403">Quo con­<lb></lb>ſtituto, cum ſuppoſitum ſit producta illa duplicata cum producto h <lb></lb>in aggregatum primarum <expan abbr="quãtitatum">quantitatum</expan> eſſe æqualia quadratis ipſa­<lb></lb>rum quantitatum, igitur addemus <expan abbr="productũ">productum</expan> ex h in ſingulas quan­<lb></lb>titates productis illis prioribus, &amp; fiet productum h in a ſemel, in b <lb></lb>ter, in c quinquies, in d ſepties, in e nouies, in f undecies, in g trede­<lb></lb>cies, &amp; in h quindecies æquale duplo producti uniuſcuiuſque quan­<lb></lb>titatis in ſuum ſupplementum cum producto h in <expan abbr="aggregatũ">aggregatum</expan> ipſa­<lb></lb>rum quantitatum, at quadratum a eſt ęquale producto ex h in eam, <lb></lb>quę talem habet proportionem ad ipſum a, <expan abbr="qualẽ">qualem</expan> habet a ad ipſum <lb></lb><arrow.to.target n="marg68"></arrow.to.target><lb></lb>h per demonſtrata ab Euclide, &amp; pariter de quadrato b, quod eſt ę­<lb></lb>quale ei quod fit ex h in eam quæ toties continet b, quotiens b con<lb></lb>tinet h, &amp; ita quadratum c æquale eſt ei, quod continetur ſub h, &amp; <lb></lb>habente proportionem ad b eandem, quam b ad h, &amp; ſimiliter de <lb></lb>quadrato c &amp; omnibus reliquis, uſque ad h ipſum. </s>
          <s id="id000404">Gratia ergo exem<pb pagenum="20" xlink:href="015/01/039.jpg"></pb>pli quadratum a, erit æquale producto ex h in omnes quantitates ſe­<lb></lb>cundas, quia quotus eſt numerus quantitatum, totus eſt numerus <lb></lb>ſecundum quem a continet h, &amp; ſimiliter quotus eſt numerus quan <lb></lb>títatum incipiendo à b, &amp; quotus eſt numerus quantitatum incipi­<lb></lb>endo à c, toties b uel c <expan abbr="continẽt">continent</expan> h, &amp; ita de alijs, quadrata ergo om­<lb></lb>nium quantitatum ſimul iuncta ſunt æqualia productis ex h in ſin­<lb></lb>gulas illarum toties ſumptis, quoties illæ <expan abbr="cõtinent">continent</expan> h, ſeu quotus eſt <lb></lb>numerus illius quantitatis, incipiendo ab h, &amp; <expan abbr="numerãdo">numerando</expan> uerſus a. <lb></lb></s>
          <s id="id000405">Rurſus dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb></lb>minimam, ſeu quadratum eiuſdem quantitatis ęquale eſt producto <lb></lb>eiuſdem quantitatis, &amp; dupli omnium ſequentium primi ordinis in <lb></lb>ipſam minimam quantitatem, uelut quadratum a eſt æquale produ<lb></lb>cto ex h in a, &amp; in duplum b c d e f g h, hoc <expan abbr="autẽ">autem</expan> facile eſt probare in <lb></lb>his quantitatibus, quia ſi quadratum a eſt æquale producto h in o­<lb></lb>mnes quantitates ſecundi ordinis, &amp; omnes quantitates ſecundi or <lb></lb>dinis ſimul ſumptæ ſunt ęquales ipſi a, &amp; duplo <expan abbr="reliquarũ">reliquarum</expan> primi or <lb></lb>dinis, quia tales quantitates ſunt æquales ſuis ſupplementis uiciſ­<lb></lb>ſim, ut h cum i, k cum g, f cum l, e <expan abbr="cũ">cum</expan> m, ergo tam ſupplementa, quàm <lb></lb>quantitates primi ordinis ſunt dimidium quantitatum ſecundi or­<lb></lb>dinis, ergo duplum quantitatum primi ordinis eſt dimidium quan<lb></lb>titatum ſecundi ordinis, uerùm de b dico idem accidere, quia qua­<lb></lb>dratum b eſt ęquale producto ex h in b, &amp; in duplum reliquarum à <lb></lb>b, ſcilicet duplum c d e f g h, &amp; hoc eſt oſtendere, quod iſtę quantita<lb></lb>tes ſunt dimidium totidem quantitatum æqualium b, nam c eſt mi­<lb></lb>nor b in h, &amp; ſupplementum p quod eſt æquale ipſi b, ſi tota h p fiat <lb></lb>æqualis ipſi b, ut pote h q erit ipſa q dempta h æqualis ipſi c, ergo <lb></lb>quantitates primi ordinis ſemper ſunt æquales ſupplementis non <lb></lb>ueris, ſed prioris quantitatis aſſumptæ, ſeu in comparatione ad il­<lb></lb>lam, quadratum igitur b eſt æquale producto ex h in b, &amp; in duplum <lb></lb>c d e f g h, &amp; ſimiliter per eadem, quadratum c eſt æquale producto <lb></lb>ex h in c, &amp; in duplum d e f g h, &amp; ſic de alijs. </s>
          <s id="id000406">Habemus ergo, quod <lb></lb>quadrata a b c d e f g h ſimul iuncta ſunt æqualia producto ex h in <lb></lb>a, &amp; in duplum reliquarum, &amp; ex h in b, &amp; in duplum reliquarum <lb></lb>ſequentium, &amp; producto ex h in c ſemel, &amp; in duplum ſequentium <lb></lb>uſque ad h, &amp; ita de reliquis. </s>
          <s id="id000407">hoc enim eſt, quod nuper demonſtraui­<lb></lb>mus. </s>
          <s id="id000408">Antea quo que <expan abbr="demõſtratum">demonſtratum</expan> eſt, quod duplum b in i, c in k, d in <lb></lb>l, e in m, f in n, g in o, h in p, <expan abbr="cũ">cum</expan> producto h in <expan abbr="aggregatũ">aggregatum</expan> a b c d e f g h <lb></lb>erat ęquale productis ex h in a ſemel, &amp; in b ter, &amp; in c quinquies, in <lb></lb>d ſepties, in e nouies, in fundecies, in g tredecies, in ſe ipſam h quin­<lb></lb>decies, detractis ergo p <expan abbr="ordinẽ">ordinem</expan>, q̊d fit ex h in a ab utro que aggregato, <lb></lb>&amp; ex h in b c d e f g h bis <expan abbr="relinquet̃">relinquetur</expan> ex una parte, quae fit ex h in b ſemel 


<pb pagenum="21" xlink:href="015/01/040.jpg"></pb>cum ſuis duplicatis ſequentibus, &amp; in c, &amp; in d, &amp; in reliquis pa­<lb></lb>riter conduplicatis ſuis ſequentibus ex altera, quod fit ex h in b ſe­<lb></lb>mel, in c ter, in d quinquies, in e ſepties, in f nouies, in g undecies, <lb></lb>in h tredecies, detractis ergo rurſus quod fit ex h in b ſemel, &amp; ex <lb></lb>h in c d e f g h bis relinquetur, quod fit ex h in c, &amp; duplo ſequen­<lb></lb>tium, &amp; d &amp; duplo ſequentium, &amp; e &amp; aliarum pariter: &amp; ex alia <lb></lb>parte, quod fit ex h in c ſemel, &amp; in d ter, &amp; in e quinquies, in f ſe­<lb></lb>pties, in g nouies, in h undecies. </s>
          <s id="id000409">Ab his rurſus detractis, quòd fit <lb></lb>ex h in c ſemel, &amp; in ſequentes bis, relinquetur h in d ſemel cum ſuis <lb></lb>ſequentibus bis, &amp; in e ſemel cum ſuis ſequentibus &amp; in f, &amp; in g &amp; <lb></lb>in h pariter, &amp; ex alia parte, quod fit ex h in d ſemel, in e ter, f quin­<lb></lb>quies, g ſepties, h nouies, ab his rurſus detraho, quod fit ex h in d <lb></lb>ſemel, &amp; in ſequentes bis, relinquetur ex una parte, quod fit ex h <lb></lb>in e f g h cum duplo ſequentium ex alia, quod fit ex h in e ſe­<lb></lb>mel, f ter, g quinquies, h ſepties, &amp; ſimiliter ab his detractis, quod <lb></lb>fit ex h in e ſemel, &amp; bis in ſequentes, relinquetur ex una par­<lb></lb>te; quod fit ex h in f ſemel, &amp; in g h bis, &amp; in g ſemel, &amp; in h bis, <lb></lb>&amp; in h ſemel, &amp; ex alia, quod fit ex h in f ſemel, in g ter, in h quin­<lb></lb>quies. </s>
          <s id="id000410">Iterum detractis, quod fit ex h in f ſemel, &amp; in g h bis com­<lb></lb>muniter relinquetur, quod fit ex h in g ſemel, &amp; in h bis, &amp; in h ſe­<lb></lb>mel, &amp; ex alia parte quod fit ex h in g ſemel, &amp; ex h in h ter. </s>
          <s id="id000411">Sed <lb></lb>iſta, quæ relicta ſunt iam, ſunt manifeſtè æqualia, ergo etiam pri­<lb></lb>ma aggregata ab initio fuere æqualia, ergo &amp; æqualia illis qua­<lb></lb>drata a b c d e f g h his, quæ fiunt, ex h in eaſdem quantita­<lb></lb>tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o, <lb></lb>h in p, ſed iam his quadratis a b c d e f g h demonſtrata ſunt eſſe du­<lb></lb>pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra­<lb></lb>ti a, ergo quadrata omnium quantitatum ſecundi ordinis cum <lb></lb>quadrato a rurſus repetito, &amp; producto h in aggregatum quanti­<lb></lb>tatum primi ordinis ſunt tripla quadratis quantitatum primi ordi­<lb></lb>nis pariter acceptis, quod fuit propoſitum, &amp; fuit Archimedis in li <lb></lb>bro de lineis ſpiralibus, &amp; ego adieci hic propter modum demon<lb></lb>ſtrandi, qui eſt elegantiſsimus, &amp; procedit ex principijs arithmeti­<lb></lb>cis, &amp; diuerſis à communibus, &amp; ideo non reuoluitur, ut ſolent re­<lb></lb>liquæ quæſtiones.</s>
        </p>
        <p type="margin">
          <s id="id000412"><margin.target id="marg67"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> 5. E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 12.</s>
        </p>
        <p type="margin">
          <s id="id000413"><margin.target id="marg68"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 6. E<emph type="italics"></emph>le.<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 17.</s>
        </p>
        <p type="main">
          <s id="id000414">Propoſitio uigeſima.</s>
        </p>
        <p type="main">
          <s id="id000415">Cùm fuerint quatuor quantitates, fueritque ſecunda æqualis ter­<lb></lb>tiæ, aut primæ æqualis quartæ, erit proportio primæ ad quartam, <lb></lb>aut tertiæ ad ſecundam producta ex proportionibus primæ ad ſe­<lb></lb>cundam, &amp; tertiæ ad quartam.<lb></lb><arrow.to.target n="marg69"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000416"><margin.target id="marg69"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000417">Cùm enim quantitates hæ non fuerint ęquales, <expan abbr="cõſtat">conſtat</expan> per ſecun­


<pb pagenum="22" xlink:href="015/01/041.jpg"></pb>dam harum, quod proportio primæ ad <expan abbr="quartã">quartam</expan> producitur ex pro­<lb></lb>portione primæ ad ſecundam, ſecundę ad tertiam, &amp; tertię ad quar<lb></lb>tam: ergo non ex ſolis proportionibus primæ ad ſecundam, &amp; ter­<lb></lb>tiæ ad quartam, &amp; ſimiliter ex prima harum proportio primę ad ſe­<lb></lb>cundam, &amp; tertiæ ad quartam producunt proportionem producti <lb></lb>primæ in ſecundam ad productum tertiæ in quartam. </s>
          <s id="id000418">Et in multi­<lb></lb>plicatione proportio, quæ ſolet eſſe inter producta illa, &amp; eſt quaſi <lb></lb>duplicata eſt inter ipſas quantitates. </s>
          <s id="id000419">Sint igitur quantitates a b c d, <lb></lb>&amp; ſit b æqualis c, ponantur ergo recto ordine a b c d, eritque propor<lb></lb><figure id="id.015.01.041.1.jpg" xlink:href="015/01/041/1.jpg"></figure><lb></lb>tio a ad d producta ex proportioni­<lb></lb>bus a ad b, b ad c, &amp; c ad d, producan­<lb></lb>tur igitur ex proportionibus a ad b, c <lb></lb>ad d. </s>
          <s id="id000420">proportio c ad f, erit igitur pro­<lb></lb>portio e ad f, ſi multiplicetur per pro­<lb></lb>portionem b ad c eadem quæ prius, &amp; </s>
        </p>
        <p type="main">
          <s id="id000421"><arrow.to.target n="marg70"></arrow.to.target><lb></lb>producta iam eſt eadem ei, quæ eſt a <lb></lb>ad d, ergo proportio a ad d erit producta ex proportionibus a ad <lb></lb>b, c ad d per primam propoſitionem. </s>
          <s id="id000422">Quod uerò diximus de pri­<lb></lb>ma &amp; quarta ſi ſint æquales, manifeſtum eſt, quòd res redit ad idem <lb></lb>ſolum tranſmutato ordine, ut tertia, &amp; quarta præmittantur primę, <lb></lb>&amp; ſecundæ. </s>
          <s id="id000423">Hæc igitur propoſitio nihil aliud innuit, quàm quod <lb></lb>in hoc caſu productio, quæ ſolet fieri ex tribus proportionibus fiat <lb></lb>ex duabus tantum.</s>
        </p>
        <p type="margin">
          <s id="id000424"><margin.target id="marg70"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000425">Propoſitio uigeſima prima.</s>
        </p>
        <p type="main">
          <s id="id000426">Cùm decuſſatim ducta fuerit prima in quartam, &amp; ſecunda in ter<lb></lb>tiam; productumque primæ in quartam diuiſum fuerit per produ­<lb></lb>ctum ſecundæ in tertiam erit proportio primæ ad ſecundam diui­<lb></lb>ſa per proportionem tertiæ ad quartam. </s>
          <s id="id000427">Et ſimiliter interpoſita <lb></lb>omiologa.<lb></lb><arrow.to.target n="marg71"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000428"><margin.target id="marg71"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.041.2.jpg" xlink:href="015/01/041/2.jpg"></figure>
        <p type="main">
          <s id="id000429">Primum exponamus ſecundam partem, ſit <lb></lb>proportio a ad b, quam uolo diuidere per <lb></lb>proportionem c ad d, facio e ad b, ut c ad d, erit <lb></lb><arrow.to.target n="marg72"></arrow.to.target><lb></lb>ergo per <expan abbr="ſecũdam">ſecundam</expan> harum proportio ad b pro­<lb></lb>ducta ex proportione a ad e, &amp; e ad b, quare ex a ad e, &amp; c ad d, ergo <lb></lb>diuiſa proportione a ad b per proportionem c ad d exit proportio <lb></lb>a ad e, &amp; hic eſt ſecundus modus. </s>
          <s id="id000430">Primus autem modus ducatur a <lb></lb>in d &amp; fiat f, &amp; b in c &amp; fiat g, dico proportione f ad g eſſe prouen­<lb></lb>tum proportionis a ad b, diuide per proportionem c ad d, ducatur <lb></lb>igitur c in f &amp; fiat h, &amp; d in g &amp; fiat k, quia igitur h producitur ex c <lb></lb>in f, &amp; f producitur ex a in d, ergo h producetur ex producto c in d, <lb></lb>in a, &amp; ſimiliter quia k producitur ex d in g, &amp; g producitur ex b in 


<pb pagenum="23" xlink:href="015/01/042.jpg"></pb>c, ergo k producetur ex c d in b, ergo ex c d in a fit h, ex c d in b fit k. <lb></lb></s>
          <s id="id000431">erit a ad b ut h ad k, igitur ex prima harum cum ex c in f producatur <lb></lb>h, &amp; ex d in g k, &amp; dicatur produci proportio h ad k ex proportio­<lb></lb>ne c ad d, &amp; f ad g, &amp; proportio h ad k ſit eadem, quæ a ad b, ergo <lb></lb>proportio a ad b producitur ex c ad d, &amp; f ad g, ergo diuiſa propor­<lb></lb>tione a ad b prodibit proportio f ad g, quod fuit propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id000432"><margin.target id="marg72"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000433">Propoſitio uigeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id000434">Cùm fuerit proportio primæ ad ſecundam maior, quàm tertiæ <lb></lb>ad quartam, erit confuſa ex his maior quàm tertiæ ad quartam, mi­<lb></lb>nor autem quàm primæ ad ſecundam.</s>
        </p>
        <figure id="id.015.01.042.1.jpg" xlink:href="015/01/042/1.jpg"></figure>
        <p type="main">
          <s id="id000435">Sit proportio a ad b maior quàm c <lb></lb><arrow.to.target n="marg73"></arrow.to.target><lb></lb>ad d, dico, quod confuſa ex a c ad b d <lb></lb>eſt maior, quàm c ad d, et minor quàm <lb></lb>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiam decimam ha­<lb></lb><arrow.to.target n="marg74"></arrow.to.target><lb></lb>rum e c ad b d confuſa minor quàm a c ad b d, nam e eſt minor a, <lb></lb>quia proportionem habent minorem ad b quam a eo quòd e ha­<lb></lb>bet proportionem ad b, quam c ad d, quæ <expan abbr="autẽ">autem</expan> c ad d minor, quám <lb></lb>a ad b, ut ſuppoſitum eſt, igitur e c ad b d minor, quàm a b ad c d, e b <lb></lb>autem ad c d eſt, ut demonſtratum eſt qualis c ad d, ergo c ad d mi­<lb></lb>nor, quàm confuſa a b ad c d, quod eſt ſecundum per idem proba­<lb></lb>bitur, &amp; primum poſita f ad d, ut a ad b, eritque a maior c, igitur ma­<lb></lb>ior proportio a f ad b d, quàm a c ad b d, ſed a f ad b d, ut a ad b per <lb></lb>eandem tertiam decimam huius ergo proportio confuſa a b ad c d <lb></lb>eſt minor, quàm a ad b.</s>
        </p>
        <p type="margin">
          <s id="id000436"><margin.target id="marg73"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000437"><margin.target id="marg74"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000438">Propoſitio uigeſima tertia.</s>
        </p>
        <p type="main">
          <s id="id000439">Omnis motus naturalis ad locum ſuum eſt: ideo per rectam li­<lb></lb>neam fit.<lb></lb><arrow.to.target n="marg75"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000440"><margin.target id="marg75"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000441">Motus naturalis eſt, ut conſeruetur corpus, &amp; conueniat locus <lb></lb>corpori, igitur fit ad ſuum locum. </s>
          <s id="id000442">Locus autem dicitur in compara<lb></lb>tione ad uniuerſum. </s>
          <s id="id000443">ideo omnis motus naturalis eſt à centro mun­<lb></lb>di ſurſum, uel ad centrum deorſum. </s>
          <s id="id000444">Et quia quanto natura celerius <lb></lb>ſuum finem poteſt aſſequi (quia finis bonus eſt aliter non illum ap­<lb></lb>peteret) eum quærit, cùm ſit ſapientiſsimæ uitæ miniſtra: at linea re­</s>
        </p>
        <p type="main">
          <s id="id000445"><arrow.to.target n="marg76"></arrow.to.target><lb></lb>cta breuiſsima eſt Euclide teſte à puncto ad punctum, igitur omnis <lb></lb>motus naturalis eſt ſurſum aut deorſum per rectam lineam.</s>
        </p>
        <p type="margin">
          <s id="id000446"><margin.target id="marg76"></margin.target>D<emph type="italics"></emph>iſt. tertia <lb></lb>primi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000447">Propoſitio uigeſima quarta.</s>
        </p>
        <p type="main">
          <s id="id000448">Omnis motus circularis uoluntarius eſt.</s>
        </p>
        <p type="main">
          <s id="id000449">Sit motus in circulo ſeu per circulum in orbe cuius ſit centrum, <lb></lb>ſit c mundi centrum: igitur ex diffinitione circuli tantum diſtabit a, <lb></lb>quantum b ab ipſo c: ſed in motu naturali per pręcedentem neceſſe <lb></lb>eſt, ut recta feratur ad c, uel recedat, igitur motus a eſt uoluntarius, 


<pb pagenum="24" xlink:href="015/01/043.jpg"></pb><figure id="id.015.01.043.1.jpg" xlink:href="015/01/043/1.jpg"></figure><lb></lb>non naturalis. </s>
          <s id="id000450">nam ſi uiolentus eſſet, non <lb></lb>eſſet perpetuus. </s>
          <s id="id000451">Omnia ergo aſtra feruntur <lb></lb>circa centrum mundi. </s>
          <s id="id000452">Sit modo rota e f g, di<lb></lb>co e non moueri motu circulari nam linea <lb></lb>e c <expan abbr="lõgior">longior</expan> eſt g c, ergo recta mouetur ad cen<lb></lb>trum non circa centrum. </s>
          <s id="id000453">Indicio etiam id <lb></lb>eſt: quòd ſi in e ponatur fruſtum aliquod <lb></lb>inſigne plumbi in motu ad g per f deſcen­<lb></lb>det raptim: at dum ex g in e magna cum dif­<lb></lb>ficultate, igitur motus hic non eſt naturalis, <lb></lb>nec circularis. </s>
          <s id="id000454">nihil etiam hoc modo ſponte mouetur. </s>
          <s id="id000455">Sed cum non <lb></lb>moueatur per rectam naturaliter, nec æquidiſtans à centro per cir­<lb></lb>culum relinquitur, ut moueatur motu uiolento, aut miſto, ſed non <lb></lb>ex uoluntario, cum nullo modo moueatur æquidiſtans à centro, <lb></lb>ſed ſemper ab e lineæ ad centrum fiant breuiores, liquet eſſe mo­<lb></lb>tum uiolentum: aut miſtum ex naturali, &amp; uiolento.</s>
        </p>
        <p type="main">
          <s id="id000456">Propoſitio uigeſima quinta.</s>
        </p>
        <p type="main">
          <s id="id000457">Tres ſunt motus omnino ſimplices naturalis, uoluntarius &amp; <lb></lb>uiolentus.<lb></lb><arrow.to.target n="marg77"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000458"><margin.target id="marg77"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000459">Tres ſunt modi, quibus poſſunt moueri in comparatione ad cen<lb></lb>trum ſcilicet uel recta cum centro, uel æquidiſtando à centro, uel <lb></lb>neutro modo, igitur tres motus. </s>
          <s id="id000460">Rurſus uel à principio interiore <lb></lb>non intelligente, &amp; eſt naturalis, uel intelligente &amp; eſt uoluntarius: <lb></lb>uel exteriore &amp; eſt uiolentus. </s>
          <s id="id000461">Hæc autem diuiſio eſt ſolum propria <lb></lb>non prima. </s>
          <s id="id000462">Nam eſt uiolentus in recta ad centrum: ideo omnis, qui <lb></lb>non eſt in recta ad centrum, nec æquidiſtat, uiolentus eſt: non ta­<lb></lb>men omnis uiolentus eſt extra rectam. </s>
          <s id="id000463">Attractio autem, quæ fit ob <lb></lb>raritatem corporum, ſeu, ut dicunt, à uacuo, uiolenta eſt non natu­<lb></lb>ralis niſi ratione finis, non agentis. </s>
          <s id="id000464">Sunt enim quatuor genera mo­</s>
        </p>
        <p type="main">
          <s id="id000465"><arrow.to.target n="marg78"></arrow.to.target><lb></lb>tus uiolenti ab Ariſtotele poſita, uectio, tractio, pulſio, &amp; uolutio: <lb></lb>quanquam his non opus ſit in demonſtratiua ſcientia. </s>
          <s id="id000466"><expan abbr="cõſtat">conſtat</expan> enim <lb></lb>uolutionem ex tractione, &amp; pulſione apud illum conſiſtere.</s>
        </p>
        <p type="margin">
          <s id="id000467"><margin.target id="marg78"></margin.target>7. P<emph type="italics"></emph>hyſ. <lb></lb>cap.<emph.end type="italics"></emph.end> 2.</s>
        </p>
        <p type="main">
          <s id="id000468">Propoſitio uigeſima.</s>
        </p>
        <p type="main">
          <s id="id000469">Motus ergo compoſiti quatuor neceſſariò ſunt ſpecies.</s>
        </p>
        <p type="main">
          <s id="id000470">Si tantum ſunt tres ſpecies ſimplicium, conſtat ratione arithme­<lb></lb>tica quatuor eſſe compoſitorum. </s>
          <s id="id000471">Diſquiramus ergo an ſint natura­<lb></lb>liter tot ſpecies, forſan enim repugnabit aliquis alicui. </s>
          <s id="id000472">Porrò uidea­<lb></lb>mus primò, quot ſint uiolentorum ſpecies: Prima erit cum non ſe­<lb></lb>cundum rectam lineam fuerit: nec à centro æquidiſtantem. </s>
          <s id="id000473">Secun­<lb></lb>da cum fuerit ſecundum rectam, ſed non ad centrum. </s>
          <s id="id000474">Tertia cum <lb></lb>fuerit in recta ad centrum, ſed contrario modo, uelut terræ ſurſum. 


<pb pagenum="25" xlink:href="015/01/044.jpg"></pb>Quarta cùm in recta ad centrum, ſecundum naturam, ſed <expan abbr="nõ">non</expan> à prin<lb></lb>cipio naturali. </s>
          <s id="id000475">Velut cum quis proijcit lapidem rectà in terram è <lb></lb>turri uiolentius, quàm ille ſua grauitate deſcenſurus eſſet. </s>
          <s id="id000476">Hic igi­<lb></lb>tur motus eſt compoſitus ex naturali, &amp; uiolento. </s>
          <s id="id000477">Animalium au­<lb></lb>tem motus uoluntarius eſt, cum ſit à principio interiore cognoſcen <lb></lb>te: &amp; ſit quatenus à principio in linea circulari æqualiter diſtante à <lb></lb>centro: ſed quia obſtat grauitas, ideò miſtus eſt ex naturali, &amp; uo­<lb></lb>luntario. </s>
          <s id="id000478">Sed circularis, &amp; uiolentus ſoli eſſe non poſſunt: nam uio <lb></lb>lentus eſt neceſſariò in corpore graui aut leui: ſed omne corpus gra<lb></lb>ue aut leue, cùm mouetur, naturaliter mouetur ſaltem in fine: &amp; per <lb></lb>totum motum, motu ócculto, qui maximè in hoc libro dignus eſt <lb></lb>conſideratione, igitur motus uoluntarius, &amp; uiolentus non poſ­<lb></lb>ſunt eſſe ſimul ſoli. </s>
          <s id="id000479">Erunt ergo ſecundum naturam tantùm tres ſpe­<lb></lb>cies. </s>
          <s id="id000480">Velut cùm quis ſcandit, autſ alit: Eſt enim motus naturalis ſal­<lb></lb>tem in fine, &amp; uoluntarius, &amp; uiolentus. </s>
          <s id="id000481">Si quis autem uelit uiolen­<lb></lb>tum cum uoluntario copulare dicemus conſtare eam compoſitio­<lb></lb>nem in initio ſaliendi. </s>
          <s id="id000482">Motum autem occultum uocamus grauita­<lb></lb>tem aut leuitatem.</s>
        </p>
        <p type="main">
          <s id="id000483">Propoſitio uigeſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id000484">Motus uoluntarius eſt in loco: naturalis ad locum: uiolentus <lb></lb>exloco.</s>
        </p>
        <p type="main">
          <s id="id000485">Hæc eſt tertia differentia primarum ſpecierum motuum uolun­<lb></lb>tarius fit manente corpore toto in eodem loco, ideo proprius eſt <lb></lb>cœlo, corpora autem animalium in eodem loco feruntur: quia in <lb></lb>eodem orbe nata redire ad proprium locum. </s>
          <s id="id000486">Et ideò, ut dixi, eſt mo<lb></lb>tus miſtus ex naturali, &amp; uoluntario, qui ſi per ſe fieret, non fatiga­<lb></lb>ret mobile, cùm ex utroque principio ab interiore ui procedat. </s>
          <s id="id000487">Sed <lb></lb>quia fit per muſculos, qui trahuntur: hic autem motus eſt uiolen­<lb></lb>tus, ideò per conſequentiam fatigat. </s>
          <s id="id000488">Qui uerò naturalis, eſt ut re­<lb></lb>deat corpus ad ſuum locum, igitur naturalis eſt ad locum. </s>
          <s id="id000489">Sed <lb></lb>uiolenti finis eſt, ut protrudatur ex loco in quo eſt, non habens cer­<lb></lb>tum finem. </s>
          <s id="id000490">licet enim qui trahit, ad ſuum locum trahat, non tamen <lb></lb>ad locum mobilis.</s>
        </p>
        <p type="main">
          <s id="id000491">Propoſitio uigeſimaoctaua.</s>
        </p>
        <p type="main">
          <s id="id000492">Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb></lb><arrow.to.target n="marg79"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000493"><margin.target id="marg79"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000494">Cùm uacuum non detur, &amp; omnis motus naturalis ſit ad locum, <lb></lb>et uiolentus ex loco per præcedentem, igitur cùm non ſit in medio, <lb></lb>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s>
        </p>
        <p type="main">
          <s id="id000495">Propoſitio uigeſima nona.</s>
        </p>
        <p type="main">
          <s id="id000496">Omnis motus uoluntarius æqualis eſt ſemper: ſimpliciter etiam <lb></lb>quilibet alius motus.</s>
        </p>
        <pb pagenum="26" xlink:href="015/01/045.jpg"></pb>
        <p type="main">
          <s id="id000497"><arrow.to.target n="marg80"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000498"><margin.target id="marg80"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id000499">Motus uoluntarius non habet, quòd fatiget, &amp; ſumma perfectio <lb></lb>eſt æqualitas, &amp; natura quæ mouet non debilitatur, igitur perpe­<lb></lb>tuo perſeuerat æqualis. </s>
          <s id="id000500">neque enim eſt, ut dixi, per medium corpus. <lb></lb></s>
          <s id="id000501">Naturalis quoque, &amp; uiolentus cum ratione proportionis mouentis <lb></lb>ſupra mobile perſe non uarientur, &amp; ab ęquali proportione ęqua­<lb></lb>lis uelo citas proueniat, igitur natura tales motus ſunt ęquales, nam <lb></lb>in utroque mouens, mouet ſecundum ultimam ſuam uim.</s>
        </p>
        <p type="main">
          <s id="id000502">Propoſitio trigeſima.</s>
        </p>
        <p type="main">
          <s id="id000503">In omni corpore mobili in medio, partes medij reſiſtunt obuiæ, <lb></lb>aliæ impellunt.</s>
        </p>
        <p type="main">
          <s id="id000504"><arrow.to.target n="marg81"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000505"><margin.target id="marg81"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000506">Sit mobile a cui partes ſubiaceant directæ b, &amp; ſit graue. </s>
          <s id="id000507">Et pa­<lb></lb>tet ne diuidatur b reſiſtere, cum autem ſuperauerit, partes b deſcen­<lb></lb>dunt ante a, &amp; trahunt partes c &amp; d adhęrentes ſecum, atque ita e c d f <lb></lb><figure id="id.015.01.045.1.jpg" xlink:href="015/01/045/1.jpg"></figure><lb></lb>adiuuant ad deſcenſum partes etiam laterales <lb></lb>g &amp; h cum a tranſit in b, ne detur uacuum, tran­<lb></lb>ſeunt in k uelo ci motu, ergo propellunt a maio<lb></lb>re impetu inferius.</s>
        </p>
        <p type="main">
          <s id="id000508"><arrow.to.target n="marg82"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000509"><margin.target id="marg82"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000510">Ex quo patet, quod in omni motu naturali, <lb></lb>uel uiolento fit augumentum uelocitatis ab initio ſaltem uſque <lb></lb>ad aliquid.</s>
        </p>
        <p type="main">
          <s id="id000511"><arrow.to.target n="marg83"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000512"><margin.target id="marg83"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000513">Et ideò etiam bellicæ machinæ cuiuſcunque generis certam exi­<lb></lb>gunt diſtantiam, ut uiolentius feriant.</s>
        </p>
        <p type="main">
          <s id="id000514">Propoſitio trigeſima prima.</s>
        </p>
        <p type="main">
          <s id="id000515">Omnis motus naturalis in æquali medio ualidior eſt in fine, <lb></lb>quàm in principio: uiolentus contrà.</s>
        </p>
        <p type="main">
          <s id="id000516"><arrow.to.target n="marg84"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000517"><margin.target id="marg84"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000518">Cùm enim ex præcedenti augeantur ſemper ob medium, &amp; cau­<lb></lb>ſa, quæ mouet, ſit perpetua, &amp; à principio æterno, quod per dictæ <lb></lb>æqualiter mouet, igitur motus ille fiet uelocior in fine quàm in alia <lb></lb>parte temporis. </s>
          <s id="id000519">In uiolento autem, cùm perueniat ad finem deſinit </s>
        </p>
        <p type="main">
          <s id="id000520"><arrow.to.target n="marg85"></arrow.to.target><lb></lb>uis illa neceſſariò, quæ mouet, &amp; ſuperatur à ui naturali, quæ mo­<lb></lb>uet in contrarium, igitur antequam ceſſet motus fiet tardiſsimus <lb></lb>in fine.</s>
        </p>
        <p type="margin">
          <s id="id000521"><margin.target id="marg85"></margin.target> 29. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000522">Ex quo patet, quòd motus quadrifariam miſti dicuntur, aut ſpe­<lb></lb><arrow.to.target n="marg86"></arrow.to.target><lb></lb>cie, ut cùm quis iacit lapidem è turri: uel ex occulto naturali, &amp; uio­<lb></lb>lento manifeſto: uelut cùm quis iacit lapidem, &amp; deſcendit poſt mo<lb></lb><figure id="id.015.01.045.2.jpg" xlink:href="015/01/045/2.jpg"></figure><lb></lb>dum ex b in c motu utroque manifeſto, ſed ex a <lb></lb>in b motu uiolento manifeſto, &amp; naturali oc­<lb></lb>culto: uel ratione medij, &amp; hoc modo omnis <lb></lb>motus naturalis etiam non ſolum uiolentus eſt <lb></lb>miſtus ex proportione uirtutis mouentis, cum motu medij, ad me­<lb></lb>dium ipſum, uel ſi uiolentus ſit ex proportione uirtutis mouentis, 


<pb pagenum="27" xlink:href="015/01/046.jpg"></pb>&amp; medij ad mobile, ac medium, quod reſiſtit. </s>
          <s id="id000523">Quarto ex motibus <lb></lb>imperfectis natura ſua, &amp; non eſt uera miſtio, &amp; hoc apparet in mo­<lb></lb>tibus uoluntarijs animalium, qui non ſunt neque æquales, neque perfe <lb></lb>ctè circa medium: ſed ſunt potius ſimiles uoluntarijs. </s>
          <s id="id000524">Et ideo de­<lb></lb>monſtrationes illæ Ariſtotelis quoad uſum nihil iuuant nos.</s>
        </p>
        <p type="margin">
          <s id="id000525"><margin.target id="marg86"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000526">Propoſitio trigeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id000527">Omne mobile naturaliter motum, ſeu uiolenter uelocius moue­<lb></lb>tur in medio rariore, quàm denſiore. </s>
          <s id="id000528">Maior quoque eſt proportio fi­<lb></lb>nis motus in corpore rariore ad finem motus in corpore denſiore, <lb></lb>quàm principij. </s>
          <s id="id000529">In uiolento autem celeriùs perueniet ad finem mo<lb></lb>tus in corpore denſiore.</s>
        </p>
        <figure id="id.015.01.046.1.jpg" xlink:href="015/01/046/1.jpg"></figure>
        <p type="main">
          <s id="id000530">A mobile moueatur in b medio rariore, &amp; in c denſio­<lb></lb><arrow.to.target n="marg87"></arrow.to.target><lb></lb>re, igitur b minus reſiſtit, quàm c &amp; magis adiuuat, quia <lb></lb>uelociùs mouetur: igitur duplici de cauſa a mouebitur <lb></lb>uelociùs in b quàm in c: &amp; quia per corrolarium trigeſi­<lb></lb>mæ, &amp; præcedentis proportio finis (ubi æqualiter moueantur) ad <lb></lb>ſua principia maior erit in d, quàm in e: ergo per <expan abbr="demõſtrata">demonſtrata</expan> à Cam <lb></lb>pano poſita d prima, b ſecunda, e tertia, c quarta, maior erit propor­<lb></lb>tio d ad e, quàm b ad c quod fuit propoſitum in naturali.</s>
        </p>
        <p type="margin">
          <s id="id000531"><margin.target id="marg87"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000532">Propoſitio trigeſima ertia.</s>
        </p>
        <p type="main">
          <s id="id000533">Omnia duo mobilia æqualis undique magnitudinis, quæ æquali <lb></lb>in tempore æqualia ſpatia pertranſeunt in diuerſis ſubſtantia me­<lb></lb>dijs, neceſſe eſt, ut ſit ponderis ad pondus, quemadmodum medij <lb></lb>ad medium, proportio duplicata.<lb></lb><arrow.to.target n="marg88"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000534"><margin.target id="marg88"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000535">Sint duo mobilia a &amp; b magnitudine, &amp; forma omnino paria, <lb></lb>&amp; ſint media c &amp; d, exempli gratia: &amp; pertranſeant æquale ſpatium <lb></lb>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb></lb>pondus a eſſe duplicatam ei quæ eſt raritatis c ad raritatem d. </s>
          <s id="id000536">Quia <lb></lb>enim feruntur æqualiter, nam in æquali tem­<lb></lb><figure id="id.015.01.046.2.jpg" xlink:href="015/01/046/2.jpg"></figure><lb></lb>pore, ſeu eodem æqualia ſpatia pertranſe­<lb></lb>unt, erit proportio potentiæ a cum ſuo auxi­<lb></lb>lio ad id, quod reſiſtit ex c ut b cum ſuo au­<lb></lb>xilio ad id, quod reſiſtit ex d, permutando igi <lb></lb>tur d ad c, ut b ad a, ſed c ad d proportio rari­<lb></lb>tatis duplicat actionem, tum minus reſiſten­<lb></lb>do, tum adiuuando motum a, igitur proportio differentiæ motus <lb></lb>eſt duplicata proportioni raritatis: ſed proportio motus eſt æqua­<lb></lb>lis proportioni ponderis uiciſsim per uigeſimam ſextam ſexti Ele­<lb></lb>mentorum b ad a, igitur proportio b ad a ponderis eſt duplicata ei, <lb></lb>quæ eſt raritatis c ad raritatem d.</s>
        </p>
        <pb pagenum="28" xlink:href="015/01/047.jpg"></pb>
        <p type="head">
          <s id="id000537">SCHOLIVM PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id000538">Ne tamen ſine exemplo intelligas hanc duplicatam rationem, <lb></lb>proponatur c raritas quatuor, d unum, a pondus duodecim libra­<lb></lb><figure id="id.015.01.047.1.jpg" xlink:href="015/01/047/1.jpg"></figure><lb></lb>rum, tunc c reſiſtit ſolum ex quarta parte, &amp; effi­<lb></lb>cit a quadruplo maioris actionis, ſcilicet ut qua­<lb></lb>draginta octo, tota igitur proportio, qua mo­<lb></lb>uebitur a in c, erit centum nonaginta duorum, &amp; hoc diuidemus <lb></lb>per d, quod eſt unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. </s>
          <s id="id000539">Pro­<lb></lb>portio igitur b ad a eſt ſex de cupla, &amp; hæc eſt duplicata quadruplæ <lb></lb>raritatis c ad raritatem d.</s>
        </p>
        <p type="main">
          <s id="id000540">Quòd ſi quis neget tantundem augere c actionem a, quanto mi­<lb></lb>nus reſiſtit, ſed aut magis aut minus, &amp; ſit proportio b ad a dupli­<lb></lb>cata ipſi f, dico feſſe proportionem c ad d, nam proportio b ad a <lb></lb>eſt uelut actionis c ad d per decimam ſextam ſexti Elementorum, <lb></lb>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, ſed ex <lb></lb>fin ſe fit proportio b ad a ex diffinitione proportionis duplicatæ. <lb></lb></s>
          <s id="id000541">Sed ex duabus proportionibus a ad c, &amp; actionis ex c ad a produ­<lb></lb>citur proportio b ad a, igitur per <expan abbr="decimamſeptimã">decimam ſeptimam</expan> ſexti Elemento­<lb></lb>rum proportio c ad d eſt media inter proportiones a ad c, &amp; actio­<lb></lb>nis a in c, quare æqualis f, igitur proportio b ad a duplicata ei, quæ <lb></lb>eſt c ad d quod erat demonſtrandum.</s>
        </p>
        <p type="head">
          <s id="id000542">SCHOLIVM SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id000543">Si autem media fuerint diuerſarum rationum, ut aqua, &amp; aër non <lb></lb>demonſtrat argumentum, quia pondera inter ſe non ſeruant ratio­<lb></lb>nem. </s>
          <s id="id000544">Nam lignum centum librarum ex ſalicis arbore, non magis <lb></lb>deſcendit, quàm lignum libræ unius. </s>
          <s id="id000545">Ideò nec in comparatione ad <lb></lb>medium aëris.</s>
        </p>
        <p type="main">
          <s id="id000546">Propoſitio trigeſima quarta.</s>
        </p>
        <p type="main">
          <s id="id000547">Proportio corporis cubi ad ſuam ſuperficiem quadratam, eſt ue­<lb></lb>lut eiuſdem ſuperficiei ad latus, eiuſdem uerò ad monadem.</s>
        </p>
        <p type="main">
          <s id="id000548"><arrow.to.target n="marg89"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000549"><margin.target id="marg89"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000550">Sit cubus a b c eius quadrata, ſuperficies a <lb></lb><figure id="id.015.01.047.2.jpg" xlink:href="015/01/047/2.jpg"></figure><lb></lb>c, latus a b, monas d, dico eas eſſe inuicem <lb></lb>analogas. </s>
          <s id="id000551">Quia enim proportio a b c ad a c <lb></lb>eſt, ut quoties aſſumitur a c in a b c, &amp; toties <lb></lb>etiam aſſumitur a b in a c ex diffinitione Eucli </s>
        </p>
        <p type="main">
          <s id="id000552"><arrow.to.target n="marg90"></arrow.to.target><lb></lb>dis ſecundo Elementorum, ſi ergo monas eſt <lb></lb>in continua proportione, habeo intentum: ſi <lb></lb>non ponatur e media inter a e &amp; d, erit ergo <lb></lb>per decimam noni Elementorum elatus a c, <lb></lb>ergo æqualis a b, igitur cum a c, e &amp; d ſint analogæ, erunt &amp; a b c, <lb></lb>a b, &amp; d analogæ, quod fuit demonſtrandum.</s>
        </p>
        <pb pagenum="39 [=29]" xlink:href="015/01/048.jpg"></pb>
        <p type="margin">
          <s id="id000553"><margin.target id="marg90"></margin.target>P<emph type="italics"></emph>rima ex<emph.end type="italics"></emph.end><lb></lb>C<emph type="italics"></emph>ampano.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000554">Propoſitio trigeſima quinta.</s>
        </p>
        <p type="main">
          <s id="id000555">Vocum magnitudines excreſcunt in acumine non in grauitate, <lb></lb>finis autem eſt in utroque extremo, propter hoc minima facta uaria­<lb></lb>tione in hypate acutæ uix ferunt.<lb></lb><arrow.to.target n="marg91"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000556"><margin.target id="marg91"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id000557">Quoniam facta uariatione in hypate, quæ eſt <lb></lb>in Diapaſon, uel bis Díapaſon maiore interual­<lb></lb><figure id="id.015.01.048.1.jpg" xlink:href="015/01/048/1.jpg"></figure><lb></lb>lo diſtat, uelut ex a in b in grauiore, maius eſt in­<lb></lb>teruallum ex c in d, igitur maior eſt b d, quàm a c <lb></lb>ergo ſingulæ uoces inter b &amp; d magis diſtant, <lb></lb>quàm inter a &amp; c, &amp; quanto magis appropin­<lb></lb>quant ad d, igitur d maius eſt quàm b. </s>
          <s id="id000558">Ergo magnitudo eſt ratione <lb></lb>acuitatis, non grauitatis, cum ſuppoſuerimus d eſſe acutiorem b &amp; <lb></lb>c ipſo a. </s>
          <s id="id000559">Oſtenditur etiam idem quia uox grauis fit ex priuatione <lb></lb>motus ſicut acuta ex uehementia. </s>
          <s id="id000560">Motus autem eſt res, quies, <lb></lb>priuatio.</s>
        </p>
        <p type="main">
          <s id="id000561">Secundum ſic: nam remiſsio mota non feriet aurem, ideò ſonum <lb></lb>non pariet ob nimiam tarditatem. </s>
          <s id="id000562">At in uelociſsimo motu oportet <lb></lb>uel fidem uel arteriam contrahi, &amp; non contrahitur niſi per muſcu­<lb></lb>los, igitur contentio illa finem habet. </s>
          <s id="id000563">Si autem non ſit neceſſarium <lb></lb>habere, uel ualde procul poſsit extendi contentio, ut in machinis <lb></lb>igneis ſtrepitus fit maximus, nam motus, ut motus eſt etiam in aëre <lb></lb>nullum finem per ſe habet niſi ratione inſtrumenti, ergo ſtrepitus <lb></lb>tantus eſſe poteſt, ut fermè obſurdeſcant, qui audierint, ut ferunt de <lb></lb>Nili cataractis.</s>
        </p>
        <figure id="id.015.01.048.2.jpg" xlink:href="015/01/048/2.jpg"></figure>
        <p type="main">
          <s id="id000564">Tertium ſic ſit a b humi­<lb></lb>lior uox, quæ excreſcat ſe­<lb></lb>mitonio minore ſolum in <lb></lb>c, &amp; ſit d e dupla ad ab ſe­<lb></lb>cundum naturam, ut in uo­<lb></lb>cibus medijs fiet, ut ſi e debeat excreſcere ſemitonio minore per de­<lb></lb>cimam nonam quinti <expan abbr="Elemẽtorum">Elementorum</expan> f e dupla c b, &amp; in acutis ubi ex­<lb></lb>creuerit ad diapaſon quadrupla: pueri autem uox, quæ iam diapa­<lb></lb>ſon altior eſt d e, erit bis diapaſon, &amp; ideò quadrupla b c, ſed in acu­<lb></lb>tioribus erit dupla, nullus enim puer eſt adeo fractæ uocis, quiſu­<lb></lb>pra humillimam non aſcendat per diapaſon, igitur interuallum uo­<lb></lb>cum erit octuplum a d, b c, ſed communiter aſcendunt ad bis diapa<lb></lb>ſon, igitur interuallum unius uocis etiam cum ſemitonio propor­<lb></lb>tionem habentis eſt æquale fermè toti a b, cum autem in diapaſon <lb></lb>ſint duodecim ſemitonia, &amp; duo comata, manifeſtum eſt, quod ex­<lb></lb>tenſio illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. </s>
          <s id="id000565">Et ideò <lb></lb>minimum in crementum in humilioribus uocibus, ubi quis coga­


<pb pagenum="40 [=30]" xlink:href="015/01/049.jpg"></pb>tur aſcendere, maximum eſſe uidetur, adeò ut ægrè à pluribus fera­<lb></lb>tur, à quibuſdam non omnino feratur.</s>
        </p>
        <p type="head">
          <s id="id000566">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id000567">Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb></lb>breuitate intenderentur, ſed ex conſtrictione ligulæ, ut dicunt, ſu­<lb></lb>per aſperam arteriam uox ad diapaſon acueretur addito impetu <lb></lb>proportione, ut ex conſtrictione, &amp; impetu <expan abbr="cõſurgeret">conſurgeret</expan> dupla pro­<lb></lb>portio. </s>
          <s id="id000568">Hoc autem manifeſtè experimur in elymis in quibus nullæ <lb></lb>prorſus facta mutatione inſtrumenti conſtantibus digitis omni­<lb></lb>bus præter pollicem ſiniſtræ uocem exacuimus ad diapaſon, inde <lb></lb>etiam ad bis diapaſon: ſicut declarauimus in commentarijs Epi­<lb></lb>demiorum.</s>
        </p>
        <p type="main">
          <s id="id000569">Propoſitio trigeſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id000570">Si proportio per proportionem minorem æquali ducatur, pro­<lb></lb>portio minor producetur. </s>
          <s id="id000571">Vnde manifeſtum eſt duas proportio­<lb></lb>nes minores æqualitate inuicem ductas proportionem minorem <lb></lb>unaquaque illarum producere.</s>
        </p>
        <p type="main">
          <s id="id000572"><arrow.to.target n="marg92"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000573"><margin.target id="marg92"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.049.1.jpg" xlink:href="015/01/049/1.jpg"></figure>
        <p type="main">
          <s id="id000574">Proportio a b ad c, qualiſcunque ſit, duca­<lb></lb>tur in proportionem minorem æqualitate <lb></lb>f ad g, dico quod producta proportio erit <lb></lb>minor ea, quæ eſt a b ad c fiat d ad a b, ut f <lb></lb>ad g, et erit per ſecundam huius d ad c pro­<lb></lb>ducta ex proportionibus a b ad c, &amp; f g. </s>
          <s id="id000575">Itemque per decimam quar­<lb></lb><arrow.to.target n="marg93"></arrow.to.target><lb></lb>tam quinti <expan abbr="Elementorũ">Elementorum</expan> erit d minor a b, igitur maior a b ad c, quàm <lb></lb>d ad c. igitur quàm proportio a b ad c in proportionem f ad g. </s>
          <s id="id000576">Sit <lb></lb>autem utraque minor æqualitate ea, quæ a b ad c, &amp; ea quæ f ad g, di­<lb></lb>co productam unaquaque earum eſſe minorem. </s>
          <s id="id000577">Quod enim (manen<lb></lb>tibus his, quæ dicta ſunt) minor ſit d ad c, quam a b ad c ex prima <lb></lb>parte oſtenſum eſt. </s>
          <s id="id000578">Quòd uerò etiam minor ſit d ad c, quàm d ad <lb></lb>a b, &amp; ex conſequenti quàm f ad g demonſtratur ſic. </s>
          <s id="id000579">Quia enim mi­<lb></lb>nor eſt a b ad c, æqualitate erit a b minor c, fiat ergo h æqualis a b, <lb></lb>erit ergo d ad h, ut d ad a b per ſeptimam quinti Elementorum, at d <lb></lb>ad c minor quàm d ad h per octauam eiuſdem, igitur minor d ad c, <lb></lb>quàm d ad a b, igitur patet propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id000580"><margin.target id="marg93"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000581">Propoſitio trigeſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id000582">Si plures homines, quorum nulli per ſe nauim mouere poſsint, <lb></lb>aut pondus ferre ſimul iuncti eam moueant, aut pondus ferant, <lb></lb>erunt illæ proportiones coniunctæ non productæ.<lb></lb><arrow.to.target n="marg94"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000583"><margin.target id="marg94"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000584">Cùm enim primus non poſsit mouere nec ſecundus, erunt pro­<lb></lb>portiones minores æqualitate, Ideò per ſecundam partem præce­<lb></lb>dentis multo minus mouerent duo, quàm unus. </s>
          <s id="id000585">Et ſi quatuor mo­


<pb pagenum="41 [=31]" xlink:href="015/01/050.jpg"></pb>uerent unusque per ſe mouere non poſſet, adderetur ſi proportio <lb></lb>produceretur, fieret minor, ergo minus mouerent quinque quàm <lb></lb>quatuor ex ijſdem, quod eſt abſurdum.</s>
        </p>
        <p type="main">
          <s id="id000586">Propoſitio trigeſima octaua.</s>
        </p>
        <p type="main">
          <s id="id000587">Omne corpus tantùm reſiſtit motui contrario ſuo naturali quan <lb></lb>cum mouetur occulto motu quieſcendo.</s>
        </p>
        <p type="main">
          <s id="id000588"><arrow.to.target n="marg95"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000589"><margin.target id="marg95"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id000590">Sit a corpus quieſcens in pauimento b, &amp; mouetur in eo occul­</s>
        </p>
        <p type="main">
          <s id="id000591"><arrow.to.target n="marg96"></arrow.to.target><lb></lb>to motu uerſus centrum, ut ſuprà uiſum eſt, contra­<lb></lb><figure id="id.015.01.050.1.jpg" xlink:href="015/01/050/1.jpg"></figure><lb></lb>rius illi ſit motus ad c, ſi ergo a quieſceret in c moue­<lb></lb>retur ad b occulto motu certa ui, ergo eadem reſtitit, <lb></lb>ne traheretur ad c. </s>
          <s id="id000592">Manifeſtum eſt autem, quod hic <lb></lb><arrow.to.target n="marg97"></arrow.to.target><lb></lb>motus occultus eſt minor manifeſto.<lb></lb><arrow.to.target n="marg98"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000593"><margin.target id="marg96"></margin.target>I<emph type="italics"></emph>n commen.<emph.end type="italics"></emph.end><lb></lb>26. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000594"><margin.target id="marg97"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 30. P<emph type="italics"></emph>ro <lb></lb>poſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000595"><margin.target id="marg98"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000596">Ex hoc patet cur naues &amp; currus ab initio tardè &amp; difficulter mo<lb></lb>ueantur, ubi moueri cœperint motus augetur: quoniam reſiſtunt </s>
        </p>
        <p type="main">
          <s id="id000597"><arrow.to.target n="marg99"></arrow.to.target><lb></lb>per motum occultum naturalem qui maximus eſt dum quieſcunt, <lb></lb>ut etiam docebat philoſophus in mechanicis, nam motus ille natu­<lb></lb>ralis eſt, &amp; ideò contrarius uiolento: Ergo cum iam mouetur uio­<lb></lb>lenter minus, mouetur naturaliter, igitur minus reſiſtit. </s>
          <s id="id000598">Declarabi­<lb></lb>tur enim infrà quòd omne quod mouetur duobus motibus tanto <lb></lb><arrow.to.target n="marg100"></arrow.to.target><lb></lb>minus uno mouetur quanto magis altero.</s>
        </p>
        <p type="margin">
          <s id="id000599"><margin.target id="marg99"></margin.target>Q<emph type="italics"></emph>ueſt.<emph.end type="italics"></emph.end> 31.</s>
        </p>
        <p type="margin">
          <s id="id000600"><margin.target id="marg100"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 59.</s>
        </p>
        <p type="main">
          <s id="id000601">Propoſitio trigeſima nona.</s>
        </p>
        <p type="main">
          <s id="id000602">Ab æquali aut minore ui, quàm ſit <expan abbr="impedimentũ">impedimentum</expan>, non fit motus.</s>
        </p>
        <p type="main">
          <s id="id000603">Sit a quod reſiſtat, ne ſurſum trahatur per decem, dico, quod <expan abbr="nõ">non</expan> <lb></lb><arrow.to.target n="marg101"></arrow.to.target><lb></lb>ſurſum trahetur neque à decem, neque minore: nam ſi impedimen­<lb></lb>tum non eſſet, moueretur infra ut decem, ergo ſi traheretur ſurſum <lb></lb>per decem tantum moueretur ſurſum, <expan abbr="quantũ">quantum</expan> deorſum, ergo quie­<lb></lb>ſceret. </s>
          <s id="id000604">Si uerò à minore moueretur à maiore ui deorſum, quam ſur­<lb></lb>ſum, ergo deorſum ſimpliciter non ſurſum.</s>
        </p>
        <p type="margin">
          <s id="id000605"><margin.target id="marg101"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000606">Propoſitio quadrageſima.</s>
        </p>
        <p type="main">
          <s id="id000607">Omne corpus ſphæricum tangens planum in puncto mouetur <lb></lb>ad latus per quancunque uim, quæ medium diuidere poteſt.</s>
        </p>
        <figure id="id.015.01.050.2.jpg" xlink:href="015/01/050/2.jpg"></figure>
        <p type="main">
          <s id="id000608">Sit corpus ad unguem ſphæricum a tan­<lb></lb><arrow.to.target n="marg102"></arrow.to.target><lb></lb>gens planum b in puncto c (eſt enim hoc <lb></lb>neceſſarium ex demonſtratis ab Euclide in <lb></lb>decimaſexta Propoſitione tertij Elemento­<lb></lb>rum) dico, quod mouebitur à ui, quæ poteſt <lb></lb>ſcindere aërem. </s>
          <s id="id000609">Nam cum non aſcendat, nec <lb></lb>deſcendat, ſed quaſi in circulo ad centrum <lb></lb>mundi moueatur, pondus non affert. </s>
          <s id="id000610">Neque<lb></lb> ratione magnitudinis contactus, cum ſit in <lb></lb>puncto ſolo, igitur remanet ſolum aëris impedimentum.


<pb pagenum="42 [=32]" xlink:href="015/01/051.jpg"></pb><arrow.to.target n="marg103"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000611"><margin.target id="marg102"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000612"><margin.target id="marg103"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000613">Ex hoc liquet, quod oportet b planum eſſe ex duriſsima mate­<lb></lb>ria, quæ nullo modo cedat, aliter tanget pluſquàm in puncto.</s>
        </p>
        <p type="main">
          <s id="id000614"><arrow.to.target n="marg104"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000615"><margin.target id="marg104"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000616">Vix fieri poteſt, utin elementaribus ſphæra tangat planum in <lb></lb>puncto. </s>
          <s id="id000617">Vel quia planum non erit exactè rectum, uel non durum, <lb></lb>ut prorſus non cedat, uel non ad æquilibrium poſitum, uel ſphæra <lb></lb>non erit exactè rotunda.</s>
        </p>
        <p type="main">
          <s id="id000618">Propoſitio quadrageſima prima.</s>
        </p>
        <p type="main">
          <s id="id000619">Si fuerint duæ quantitates ſumaturque totius aggregatum maio­<lb></lb>ris &amp; minoris, quoties aggregatum minoris, &amp; maioris, erit pro­<lb></lb>portio confuſa maioris aggregati ad minus, minor quàm multipli­<lb></lb>cis maioris ad multiplex minoris.</s>
        </p>
        <p type="main">
          <s id="id000620"><arrow.to.target n="marg105"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000621"><margin.target id="marg105"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000622">Sint duæ magnitudines a &amp; b, &amp; ſit a maior <lb></lb><figure id="id.015.01.051.1.jpg" xlink:href="015/01/051/1.jpg"></figure><lb></lb>b, &amp; ſumatur exempli gratia a quater cum b ſe­<lb></lb>mel, &amp; b quater cum a ſemel, dico, quod propor<lb></lb>tio (quam confuſam eſſe liquet) aggregati primi ad ſecundum, eſt </s>
        </p>
        <p type="main">
          <s id="id000623"><arrow.to.target n="marg106"></arrow.to.target><lb></lb>minor quàm quadrupla. </s>
          <s id="id000624">Conſtat enim quod proportio quadru­<lb></lb>pli a ad a eſt maior, quam b ad quadruplum b, cum una ſit quadru­<lb></lb>pla, alia ſub quadrupla, igitur per uigeſimam ſecundam huius ag­<lb></lb>gregati quadrupli a cum b ſemel, ad quadruplum b cum a ſemel mi <lb></lb><arrow.to.target n="marg107"></arrow.to.target><lb></lb>nor, quàm quadrupli a ad a, &amp; maior quàm b ad quadruplum b, &amp; <lb></lb>eſt pro intellectu Archimedis.</s>
        </p>
        <p type="margin">
          <s id="id000625"><margin.target id="marg106"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>diff.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000626"><margin.target id="marg107"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>lib. 

de<emph.end type="italics"></emph.end><lb></lb>A<emph type="italics"></emph>tqui pon­<lb></lb>deran.<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 10.</s>
        </p>
        <p type="main">
          <s id="id000627">Propoſitio quadrageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id000628">Trahentium nauim, ut ferentium pondera proportiones in ſe in­<lb></lb>uicem, quomodo ducere oporteat conſiderare.<lb></lb><arrow.to.target n="marg108"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000629"><margin.target id="marg108"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000630">Hoc quomodo non poſsit fieri ſuprà docuimus, nunc etiam ge­</s>
        </p>
        <p type="main">
          <s id="id000631"><arrow.to.target n="marg109"></arrow.to.target><lb></lb>neraliter dicam, cum conſiſtant hæc in duobus terminis, productio <lb></lb>uerò præſupponit quatuor terminos, ut in prima propoſitione, aut <lb></lb>ſaltem tres, atque in his medius habet rationem mouentis, &amp; moti, <lb></lb>ergo cum in huiuſmodi <expan abbr="nõ">non</expan> ſint quatuor termini, nec tres, è quibus <lb></lb>unus ſit mouens, &amp; motum proportio non poterit produci. </s>
          <s id="id000632">Illud <lb></lb>etiam patet exemplo, nam ſi eſſet lapis, aut nauis obſiſtens ut ſex, &amp; <lb></lb>eſſent homines uiribus ſinguli, ut quatuor cum dimidio, tres mo­<lb></lb>uerent in proportione dupla ſexquiquarta perdicta ſuperius eo­<lb></lb>dem loco, at ſi proportio duci poſſet aliquorum hominum nume­<lb></lb>rus poſſet mouere in duplicata proportione ad unguem ſcilicet <lb></lb>5 1/16 ut eſſet uix hominum collectorum 30 3/8 at nullus eſt numerus ho<lb></lb>minum qui collectus faciat hunc numerum, nam ſex homines ex­<lb></lb>plent numerum 27, &amp; ſeptem 31 1/2, &amp; ideò non poteſt duci propor­<lb></lb>tio. </s>
          <s id="id000633">Et ideò maximus eſt error dicendo decem homines mouent na <lb></lb>uim proportione tripla, ergo triginta alij additis illis ſimiles robo­<lb></lb>re mouebunt à proportione uiginti ſeptupla ſcilicet ducta nonu­


<pb pagenum="33" xlink:href="015/01/052.jpg"></pb>pla in triplam. </s>
          <s id="id000634">Sed ſumpta proportione alio modo producitur. </s>
          <s id="id000635">Ve<lb></lb>lut ſi dicam, homines decem mouent nauim, aut <expan abbr="ferũt">ferunt</expan> pondus pro­<lb></lb>portione tripla, igitur quadraginta homines idem facient propor­<lb></lb>tione duodecupla ſcilicet quadrupla in triplam ducta. </s>
          <s id="id000636">Cum ergo <lb></lb>addo triginta homines, qui mouent in proportione nonupla, non <lb></lb>oportet ducere nonuplam in triplam, ſed totum numerum accipe­<lb></lb>re, &amp; quam proportionem habet ad partem, tandem habet uis mo­<lb></lb>uens ad uim <expan abbr="mouẽtem">mouentem</expan>. </s>
          <s id="id000637">Vnde ſi duo moueant in proportione ſex­<lb></lb>quialtera, &amp; ſex in proportione quadrupla cum dimidia, &amp; iungan <lb></lb>tur, ut fiant octo, non oportebit ducere ſexquialteram, in quadru­<lb></lb>plam ſexquialteram, ſed cum octo ad duo ſit in proportione qua­<lb></lb>drupla, ſumemus quadruplam ad ſexquialteram, quę erit ſexcupla, <lb></lb>&amp; octo mouebunt, aut pondus gerentin proportione ſexcupla.</s>
        </p>
        <p type="margin">
          <s id="id000638"><margin.target id="marg109"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 37.</s>
        </p>
        <p type="main">
          <s id="id000639">Propoſitio quadrageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id000640">Productionem ad additionem retrahere.<lb></lb><arrow.to.target n="marg110"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000641"><margin.target id="marg110"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.052.1.jpg" xlink:href="015/01/052/1.jpg"></figure>
        <p type="main">
          <s id="id000642">Sit proportio a ad b dupla poteſtate li­<lb></lb>cet ſint quinque homines, &amp; ſint quindecim <lb></lb>homines c, &amp; habebunt ad b ſexcuplam <lb></lb>proportionem per præcedentem. </s>
          <s id="id000643">Iuncta <lb></lb>ergo a, &amp; c per octauam huius <expan abbr="mouebũt">mouebunt</expan> <lb></lb>b proportione octupla, dico, quod ſi du­<lb></lb>xeris <expan abbr="proportionẽ">proportionem</expan> c ad a plus uno. </s>
          <s id="id000644">i. </s>
          <s id="id000645">qua­<lb></lb>druplam in proportionem a ad b, quæ eſt dupla, proueniet eadem <lb></lb>octupla. </s>
          <s id="id000646">Nam quia in coniunctione ſufficit iungere c cum a, &amp; ſu­<lb></lb>mitur ſecundum proportionem a ad b, igitur cum proportio a ad <lb></lb>b comparata ad proportionem c &amp; a ad b ſit, ſicut proportio c &amp; a <lb></lb>ad a, &amp; proportio c &amp; a ad a ſit, ſicut proportio c ad a, &amp; a ad a, &amp; <lb></lb>proportio a ad a habet rationem unius, igitur proportio aggregati <lb></lb>c a ad b eſt producta ex proportione c ad a plus monade in propor<lb></lb>tionem a ad b, quod erat demonſtrandum.</s>
        </p>
        <p type="main">
          <s id="id000647">Propoſitio quadrageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id000648">Si fuerit proportio motoris ad id, quod eſt maximum non mo­<lb></lb>uens &amp; ſpatium, &amp; tempus, nota erit etiam reliquorum nota.</s>
        </p>
        <p type="main">
          <s id="id000649">Sæpe contingit, ut quinque homines moueant nauim, &amp; ſpatium <lb></lb>ad tempus notum, &amp; etiam cognitum maximum, quod mouere <lb></lb>non poteſt. </s>
          <s id="id000650">Sit ergo a numerus hominum, b na­<lb></lb><figure id="id.015.01.052.2.jpg" xlink:href="015/01/052/2.jpg"></figure><lb></lb>uis, c maximum, quod non mouere poteſt, d <lb></lb>tempus, e ſpatium, f motor alius ſiue numerus <lb></lb>hominum notus, &amp; g tempus, dico, quod h ſpatium notum erit, ſeu <lb></lb><expan abbr="notũ">notum</expan> g tempus, &amp; h ſpatium, dico, quod erit f motor, ſeu numerus 


<pb pagenum="34" xlink:href="015/01/053.jpg"></pb>hominum notus. </s>
          <s id="id000651">Quoniam ergo notum eſt a &amp; c, quia eſt æquale <lb></lb>b, igitur proportio a ad b nota eſt: ſed iuxta illam a mouet b in d <lb></lb>tempore per e ſpatium, igitur per præcedentem, ut f ad a ita ſpatij <lb></lb>ad e in d tempore. </s>
          <s id="id000652">Sed per eadem ut temporis d ad ſpatium illud, <lb></lb>ita g ad h, ergo cum nota ſint d e f g erit etiam h, &amp; ita conuertendo.</s>
        </p>
        <p type="main">
          <s id="id000653">Propoſitio quadrageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id000654">Rationem ſtateræ oſtendere.<lb></lb><arrow.to.target n="marg111"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000655"><margin.target id="marg111"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000656">Archimedes nititur huic fundamento, quod pondera, quæ pro­<lb></lb>portionem mutuam habent, ut diſtantiæ à libella a, quæ ſuſpen­<lb></lb>duntur, æqualiter ponderant, ſit ergo libella a b, &amp; ſuſpenſa in a cen<lb></lb>trum mundi c, ad quod dirigitur pondus, &amp; liquet, quod ipſum <lb></lb>non ſe inclinabit ex uigeſima tertia propoſitione. </s>
          <s id="id000657">Si ergo ponantur <lb></lb>lo co lineæ b d in e &amp; f, &amp; ſit proportio e b <lb></lb><figure id="id.015.01.053.1.jpg" xlink:href="015/01/053/1.jpg"></figure><lb></lb>ad b f, ut g ad h, dico, quòd erit æquili­<lb></lb>brium, per eandem enim h mouebitur in k, <lb></lb>ſcilicet ut perueniat in rectam a d, ſi enim <lb></lb>non eſſet | ſuſpenſum h, moueretur in re­<lb></lb>cta e h per eandem, quia ergo retinetur, mo­<lb></lb>uetur per obliquam h k, &amp; ſumatur in pro­<lb></lb>pin quum punctum in b e, &amp; n in æquali di­<lb></lb>ſtantia in e f, quia ergo e b totum mouetur <lb></lb>eadem ui in ſingulis partibus, quia a pon­<lb></lb>dere h, &amp; in h mouetur per h k in m per m <lb></lb>p, ergo qualis eſt proportio magnitudinis h k ad m p, talis eſt uis <lb></lb>in m p ad uim in h k, &amp; ita in b erit penè infinita: quia quanta ui ex­<lb></lb>tenditur ex h in k tanta puncta b, ſe circumuertit ergo propor­<lb></lb>tio hypomochlij ad ſpatium, uelut roboris ad robur, at eadem n o <lb></lb>ad h k, eſt enim n o æqualis m p, &amp; n b, &amp; b m æquales, ut uerò g ad <lb></lb>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb></lb>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb></lb></s>
          <s id="id000658">nam idem pondus ſcilicet g mouet totam b f, igitur ut g ſe habet </s>
        </p>
        <p type="main">
          <s id="id000659"><arrow.to.target n="marg112"></arrow.to.target><lb></lb>ad n o, ita h ad m p, ſed m p &amp; n o ſunt æquales, ergo tanta eſt uis g <lb></lb>in f, quanta h in e.<lb></lb><arrow.to.target n="marg113"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000660"><margin.target id="marg112"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000661"><margin.target id="marg113"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000662">Ex quo patet, quod hypomochlion moueretur infinita ui, ſi poſ­<lb></lb>ſet eſſe punctus: ſed quia in extrema ſuperficie cylindri, ideò poteſt <lb></lb>aliqua ui retineri.</s>
        </p>
        <p type="main">
          <s id="id000663"><arrow.to.target n="marg114"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000664"><margin.target id="marg114"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000665">Et ſi quis poſſet capere haſtam in extremo puncto, non poſſet <lb></lb>eam mouere, etiam quod haberet robur infinitum, quia ab æquali <lb></lb>non fit motus per trigeſimamnonam propoſitionem.</s>
        </p>
        <p type="main">
          <s id="id000666"><arrow.to.target n="marg115"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000667"><margin.target id="marg115"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id000668">Et libella nihil retinet niſi quantum eſt pondus eius quod cu­


<pb pagenum="35" xlink:href="015/01/054.jpg"></pb>pit ad centrum peruenire, &amp; pondus ei appenſum non prohi­<lb></lb>bet motum, etiam ſi eſſet infinitum, niſi quatenus non uult recede­<lb></lb>re ex directo centri mundi: &amp; ut grauat hypomochlion faciens im­<lb></lb>preſsionem.</s>
        </p>
        <p type="main">
          <s id="id000669"><arrow.to.target n="marg116"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000670"><margin.target id="marg116"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id000671">Et ſi terra tota eſſet appenſa polo, moueretur magna ui: quoni­<lb></lb>am uis eadem eſt in polo, quæ in circulo toto æquinoctij.</s>
        </p>
        <p type="main">
          <s id="id000672"><arrow.to.target n="marg117"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000673"><margin.target id="marg117"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 5.</s>
        </p>
        <p type="main">
          <s id="id000674">Et rota, quanto uelocius mouetur in ambitu, tanto mi<lb></lb>norem habet uim: ſed propter aërem, qui ſecum circum­<lb></lb><figure id="id.015.01.054.1.jpg" xlink:href="015/01/054/1.jpg"></figure><lb></lb>fertur, mouetur magno impetu, &amp; magnas facit læſiones. <lb></lb></s>
          <s id="id000675">Ideò hoc in cono non accidit.</s>
        </p>
        <p type="main">
          <s id="id000676"><arrow.to.target n="marg118"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000677"><margin.target id="marg118"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 6.</s>
        </p>
        <p type="main">
          <s id="id000678">Ex quo patet ratio eleuandi pondera magna per tra­<lb></lb>bem, ut à latere uides.</s>
        </p>
        <p type="main">
          <s id="id000679">Propoſitio quadrageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id000680">An ſit aliqua proportio, &amp; qualis inter animam, &amp; ui­<lb></lb>tas, &amp; ſua corpora conſiderare.</s>
        </p>
        <p type="main">
          <s id="id000681"><arrow.to.target n="marg119"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000682"><margin.target id="marg119"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000683">Declarauimus motum cœli eſſe uoluntarium, obſequente cœ­<lb></lb>lo per uirtutem in eo infuſam. </s>
          <s id="id000684">In animalibus autem, &amp; præcipuè <lb></lb>in homine notius eſt hoc experientibus nobis in ipſis: ſed motus <lb></lb>hic, ut dixi ſupra, miſtus eſt, ille uerò cœleſtis ignotior eſt. </s>
          <s id="id000685">Certum </s>
        </p>
        <p type="main">
          <s id="id000686"><arrow.to.target n="marg120"></arrow.to.target><lb></lb>tamen eſt plenè obſequi cœlum uitæ, nec prorſus repugnare. </s>
          <s id="id000687">So­<lb></lb>let Ariſtoteli imponi, quòd ſi adderetur aſtrum cœlo, quòd cœlum <lb></lb>aut quieſceret, aut tardius moueretur: quod eſt, ac ſi diceremus, <lb></lb>quòd homo paruus ſi fieret maior, non eſſet adeò agilis, tanquam <lb></lb>motus ille eſſet ab externa cauſa. </s>
          <s id="id000688">Imò perinde eſſet, ac ſi quis dice­<lb></lb>ret, quod lapides magni minus uelociter deſcenderent, quam par­<lb></lb>ui. </s>
          <s id="id000689">Quin potius ut lapis magnus uelociùs mouetur: quàm par­<lb></lb>uus naturali motu, &amp; tardius præternaturali, ita cœlum motu uo­<lb></lb>luntario, ſi ita dici poſſet æqualius &amp; maiore cum efficacia, quan­<lb></lb>to denſius. </s>
          <s id="id000690">Et ita ſi Ariſtoteles illud dixiſſet, oſtendiſſet magnam <lb></lb>imperitiam. </s>
          <s id="id000691">Ideò quale iudicium debemus facere de Alexandro, &amp; <lb></lb><arrow.to.target n="marg121"></arrow.to.target><lb></lb>Aueroe, qui hoc ei tribuunt. </s>
          <s id="id000692"><expan abbr="legit̃">legitur</expan> enim in textu Arabico tale quip­<lb></lb>piam. </s>
          <s id="id000693">De Animalibus forſan poſſet hoc dici, <expan abbr="quoniã">quoniam</expan>, ut ſuprà dixi­<lb></lb>mus, motus ille miſtus eſt. </s>
          <s id="id000694">Remanet ergo difficultas, <expan abbr="quoniã">quoniam</expan> ſi mo­<lb></lb>tus iſte non à proportione fit, quare non eſt infinitus? </s>
          <s id="id000695">&amp; dico quae in <lb></lb>animalibus tres ſunt cauſæ, una, quia eſt miſtus, &amp; habet repugnan<lb></lb>tiam: ſecunda, quia eſt de loco ad locum, motus autem cœli eſt in lo<lb></lb>co: tertia eſt communis etiam cœlo, et eſt, <expan abbr="quoniã">quoniam</expan> non eſt ratio finis. <lb></lb></s>
          <s id="id000696">Natura enim diuina non appetit mouere <expan abbr="tã">tam</expan> celeriter. </s>
          <s id="id000697">Quid eſt ergo <lb></lb>proportio, <expan abbr="cũ">cum</expan> ſit <expan abbr="ultimũ">ultimum</expan> uoluntatis uitę, ut obtemperet primæ cauſæ, <lb></lb>ideo illud eſt <expan abbr="ultimũ">ultimum</expan>, q̊ mouet. </s>
          <s id="id000698">Eſt <expan abbr="aũt">aut</expan> idem uelle, &amp; poſſe. </s>
          <s id="id000699">In natura 


<pb pagenum="46 [=36]" xlink:href="015/01/055.jpg"></pb>enim cœli eſt ille appetitus, cuius principium eſt uita: &amp; eíus uolun<lb></lb>tatis bonum ipſum. </s>
          <s id="id000700">Et ideo hæc proportio <expan abbr="nõ">non</expan> diuiditur. </s>
          <s id="id000701">In anima­<lb></lb>libus autem non eſt uis illa niſi, cum proportione, quia primum in­<lb></lb>ſtrumentum, quod recipit, &amp; eſt ſpiritus uim habet determinatam, <lb></lb>cum ſit uirtus in materia: ideo <expan abbr="nõ">non</expan> mouet niſi cum certa proportio­<lb></lb>ne, uelut lumen in medio in ſe non habet proportionem niſi ad lu­<lb></lb>cem, ſed ut eſt in illo, poteſt eſſe remiſſum, <expan abbr="obſcurũ">obſcurum</expan> &amp; hebes. </s>
          <s id="id000702">Quæ­<lb></lb>ritur ergo quantitas illius? </s>
          <s id="id000703">ſi dicas, quòd eſt à luce: quæro quanti­<lb></lb>tas lucis, unde ſit? </s>
          <s id="id000704">forſan dicendum, quòd uelutin motibus, quanto <lb></lb>denſiora ſunt corpora tanto <expan abbr="mouent̃">mouentur</expan> maiore nixu, &amp; robore. </s>
          <s id="id000705">Nam <lb></lb>calor in materia augetur iuxta illius quantitatem: idem in luce, &amp; <lb></lb>reliquis. </s>
          <s id="id000706">Dico ergo proportionem eſſe infinitam: nam ſi corpus eſ­<lb></lb>ſet infinitum &amp; optimè diſpoſitum infinita ui moueretur &amp; agili­<lb></lb>tate, ut enim maius eſt eo maiores uires habet.</s>
        </p>
        <p type="margin">
          <s id="id000707"><margin.target id="marg120"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 27.</s>
        </p>
        <p type="margin">
          <s id="id000708"><margin.target id="marg121"></margin.target>T<emph type="italics"></emph>ex.<emph.end type="italics"></emph.end> 71. <lb></lb>2. <emph type="italics"></emph>de<emph.end type="italics"></emph.end> C<emph type="italics"></emph>œlo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000709">Propoſitio quadrageſimaſeptima.</s>
        </p>
        <p type="main">
          <s id="id000710">Si duo mobilia æqualiter in eodem circulo iuxta proprios mo­<lb></lb>tus moueantur, productum temporis circuituum inuicem erit æ­<lb></lb>quale producto differentiæ temporum circuitus ductæ in tempus <lb></lb>coniunctionis primæ.<lb></lb><arrow.to.target n="marg122"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000711"><margin.target id="marg122"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000712">Sint duo mobilia a &amp; b in eodem pun­<lb></lb><figure id="id.015.01.055.1.jpg" xlink:href="015/01/055/1.jpg"></figure><lb></lb>cto, quæ æqualiter uerſus eandem partem <lb></lb>moueantur æqualibus in temporibus, inui<lb></lb>cem tamen in æqualiter, ita quod a in f &amp; b <lb></lb>in g temporibus abſoluant circulum, &amp; ho <lb></lb>rum differentia ſit h. </s>
          <s id="id000713">Dum itaque a perficit <lb></lb>circulum b perueniat in c, igitur c d b eſt dif<lb></lb>ferentia, quæ ſuperanda eſt, &amp; proportio <lb></lb>circuli ad b c ut g ad f, quare reliqui ad reli­<lb></lb>quum, ut reſidui ad reſiduum, ſcilicet circu­<lb></lb>li ad c d b, ut g ad h, &amp; b c ad c d b ut f ad h, coniungantur igitur in k <lb></lb>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb></lb>diuiſo per certam quantitatem &amp; cum circulo &amp; b c &amp; c d b diffe­<lb></lb>rentia, &amp; ſit ſ productum ex f in g, dico quod diuiſa ſ per h exibit k <lb></lb>tempus coniunctionis primæ, ſit itaque d locus coniunctionis, dico <lb></lb>igitur quod differentia ſpatij pertranſiti a b, a &amp; a, b in reditu ex con<lb></lb>iunctione prima ad d eſt unus circulus completus, non enim poſ­<lb></lb>ſunt eſſe plures, nam ſequeretur, quòd a aliquando pertranſiſſet b, <lb></lb>et ſic non eſſet prima coniunctio, nec poteſt eſſe minus, nam ſic cum <lb></lb>a &amp; b ſint in d ultra perfectas circulationes uterque eorum pertran<lb></lb>ſiuit arcum b c, igitur nullo modo differentia poteſt eſſe minor cir­<lb></lb>culo, neque maior, ut declaratum eſt, igitur eſt unus circulus ad un­


<pb pagenum="37" xlink:href="015/01/056.jpg"></pb>guem. </s>
          <s id="id000714">Hoch declarato ponatur m spatium compositum ex circulis <lb></lb>pertranſitis a b a cum ſpatio b d, etenim ſpatium, quod pertranſit <lb></lb>b a coniunctione in a, ad coniunctionem primam in d, &amp; erit ex de­<lb></lb>monſtratis horum differentia circulus qui uocetur o, &amp; ſit p ſpa­<lb></lb>tium, quod pertranſit b in tempore eodem, in quo a pertranſit o, &amp; <lb></lb>ſit q differentia o, &amp; p quę in circulo eſt c d l b, quia igitur in eodem <lb></lb>tempore a pertranſit m &amp; b, n, erit m ad n, ut a ad b, &amp; eadem ratio­<lb></lb>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb></lb>ad p, quare cum o ſit differentia m &amp; n, &amp; q, differentia o &amp; p erit ex <lb></lb>decimanona quinti Euclidis, m ad o, ut o ad q, &amp; ita circulus eſt ana<lb></lb>logus inter ſpatium pertranſitum à motore uelociori, &amp; inter diffe­<lb></lb>rentiam ſpatij quæ accidit, dum uelocior motor pertranſit circu­<lb></lb>lum, id eſt quòd circulus a c d eſt analogus inter c d l b, &amp; circulos <lb></lb>pertranſitos a b a cum portione b d. </s>
          <s id="id000715">Reuertor igitur ad propoſi­<lb></lb>tum, cum ſit m ad o, ut o ad q, &amp; m ad o, ut n ad p, ex ſexta decima <lb></lb>quinti Euclidis, erit ex undecima eiuſdem n ad p, ut o ad q, quare ex <lb></lb>ſexta decima ſexti Elementorum ducto o, id eſt circulo, ſeu maiore <lb></lb>numero in p ſpatium pertranſitum a b, ſeu ducto fin g, &amp; diuiſo per <lb></lb>q differentiam ſpatiorum, ſeu per h exibit n, ſeu ſpatium quod <lb></lb>pertranſit b ab una coniunctione ad aliam quod erat demon­<lb></lb>ſtrandum.</s>
        </p>
        <p type="main">
          <s id="id000716"><arrow.to.target n="marg123"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000717"><margin.target id="marg123"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000718">Ex hoc patet, quod proportio temporis coniunctionis ad tem­<lb></lb>pus tardioris motus circuitionis eſt ueluti temporis circuitus uelo<lb></lb>cioris motoris ad differentiam temporis motus tardioris, &amp; uelo­<lb></lb>cioris motoris in uno circuitu.</s>
        </p>
        <p type="main">
          <s id="id000719">Propoſitio quadrageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id000720">Si tria mobilia ex eodem puncto diſcedant, fuerintque duorum, ac <lb></lb>duorum coniunctiones in temporibus commenſis illa tria mobi­<lb></lb>lia denuò coniungentur in tempore producto ex denominatore di <lb></lb>uiſionis temporis maioris per minus in minus, aut numeratore <lb></lb>in maius.</s>
        </p>
        <p type="main">
          <s id="id000721"><arrow.to.target n="marg124"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000722"><margin.target id="marg124"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000723">Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb></lb>c in ſeptem. </s>
          <s id="id000724">Dico quod primum redibunt in numero producto ex <lb></lb>ſeptem quinque &amp; duobus, qui ſunt numeri primi, &amp; erit ille nume­<lb></lb>rus ſeptuaginta annorum. </s>
          <s id="id000725">Nam in ſeptuaginta annis a perficiet tri­<lb></lb>ginta quinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redibũt">redibunt</expan> <lb></lb>per perfectos circuitus ad idem punctum. </s>
          <s id="id000726">Oſtendo modo quod <lb></lb>non ante: nam ſi ſic: ſit, ut in triginta quinque annis igitur b &amp; c per­<lb></lb>ficient perfectos circuitus, ergo <expan abbr="redibũt">redibunt</expan> ad idem punctum, a autem <lb></lb>non redibit, quoniam eius circuitus non numerat trigintaquinque <lb></lb>aliter non fuiſſet ſeptuaginta minimus numeratus ab a b c, cum 


<pb pagenum="38" xlink:href="015/01/057.jpg"></pb>ergo iam ſupponatur numerari a b &amp; c non numerabitur a b a, er­<lb></lb>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="locũ">locum</expan>, ergo <lb></lb>non erit iunctus cum b &amp; c. </s>
          <s id="id000727">Quod ſi dicas a b c coniungi in decem <lb></lb>ſeptem annis numero non numerato ab ali <lb></lb><figure id="id.015.01.057.1.jpg" xlink:href="015/01/057/1.jpg"></figure><lb></lb>quo illorum temporum, auferantur perfe­<lb></lb>ctæ circulationes, &amp; <expan abbr="remanebũt">remanebunt</expan> dimidium <lb></lb>ex a, duæ quintæ ex b, tres ſeptimæ ex c, igi­<lb></lb>tur oportebit ut hæ portiones ſint æqua­<lb></lb>les, ut poſt perfectas circulationes in idem <lb></lb>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 &amp; 2/5 &amp; 3/7 æqui­<lb></lb>ualebunt, quare proportio 7 ad 3 &amp; 5 ad 2 <lb></lb>&amp; 2 ad 1, eſt una, quare permutando 3 ad 2 <lb></lb>ut 7 ad 5, ſed 7 &amp; 5 ſunt contra ſe primi, ergo in ſua proportione mi <lb></lb>nimi per dicta in ſeptimo Elementorum: ergo tria, &amp; duo non ſunt <lb></lb>in eadem proportione. </s>
          <s id="id000728">Rurſus dicantur conuenire in annis qua­</s>
        </p>
        <p type="main">
          <s id="id000729"><arrow.to.target n="marg125"></arrow.to.target><lb></lb>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite­<lb></lb>rum: ergo per ſecundam partem erit ſeptem ad unum, ut duo ad <lb></lb>unum, igitur permutando unius ad unum, ut ſeptem ad duo, ſed <lb></lb>unum eſt æquale uni, ergo duo erunt æqualia ſeptem. </s>
          <s id="id000730">Rurſus dica­<lb></lb>mus, quod in tempore annorum &lt;02&gt; quadrata decem ſimiliter aufe­<lb></lb>ram integras reuolutiones, quas potero, &amp; erunt &lt;02&gt; 2 1/2 m: 1, &amp; &lt;02&gt; 2/5 &amp; <lb></lb>&lt;02&gt; 10/49 æqualia. </s>
          <s id="id000731">Hic uides infinita ſequi in conuenientia, quæ longum <lb></lb>eſſet numerare, nam ſeptem eſſet æquale quinque, &amp; proportio reciſi <lb></lb>ad potentia rethe, ut numeri ad numerum. </s>
          <s id="id000732">Igitur non conueniunt <lb></lb>ante ſeptuaginta annos.<lb></lb><arrow.to.target n="marg126"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000733"><margin.target id="marg125"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 23</s>
        </p>
        <p type="margin">
          <s id="id000734"><margin.target id="marg126"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000735">Ex hoc ſequitur, quòd nullibi conuenient præterquàm in eo­<lb></lb>dem puncto, ſcilicet in quo ab initio coniuncti fuerunt.</s>
        </p>
        <p type="main">
          <s id="id000736"><arrow.to.target n="marg127"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000737"><margin.target id="marg127"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>m. </s>
          <s id="id000738">2.</s>
        </p>
        <p type="main">
          <s id="id000739">Sequitur denuo ex propoſitione ipſa repetita, &amp; primo corrola­<lb></lb>rio, quod nullibi alibi conuenient quàm in dato primo puncto, in <lb></lb>quo coniuncti fuerant ab initio etiam uſque in æternum.</s>
        </p>
        <p type="main">
          <s id="id000740">Sit rurſus ut a circuat in annis duobus cum dimidio, b in tribus <lb></lb>cum tertia parte, cin quatuor cum quarta parte ducam per ſuos <lb></lb>denominatores, &amp; erit ut a in quinque annis. </s>
          <s id="id000741">b in decem, c in decem­<lb></lb>ſeptem circuant, &amp; redeant ad idem punctum, &amp; quia quin que nu­<lb></lb>merat decem, &amp; decem, &amp; decemſeptem ſunt numeri inuicem pri­<lb></lb>mi, ducam decem in decemſeptem fiunt centum ſeptuaginta. </s>
          <s id="id000742">Con­<lb></lb>ſtat igitur c quadragíes, b quinquagies ſemel, a ſexagies octies cir­<lb></lb>cumuerti, &amp; redire ad idem punctum: ergo rurſus coibunt poſt tot <lb></lb>annos in eo, dico modo, quod non ante: nam ſi non ſit, ut in trigin­<lb></lb>ta tribus annis. </s>
          <s id="id000743">gratia exempli, aufero <expan abbr="decemſeptẽ">decemſeptem</expan>, decem, &amp; quin­<lb></lb>que, &amp; relinquentur ſexdecim tria &amp; tria, &amp; rurſus ex ſexdecim tres <pb pagenum="39" xlink:href="015/01/058.jpg"></pb>circuitus c, &amp; relinquentur 3 3/4 ſequetur igitur, ut ſit proportio 17 ad <lb></lb>13, &amp; 2 1/2 ad 1/2 &amp; 3 1/3 ad 3 eadem, &amp; ita 17/13, 5/2 &amp; 10/9 eadem ſi iam ſupponi<lb></lb>mus 17 &amp; 10 eſſe primos inuicem, ut in ſecunda demonſtratione./&gt;<lb></lb></s>
          <s id="id000744">Igitur ſequuntur eadem corrolaria, quæ dicta ſunt.</s>
        </p>
        <p type="main">
          <s id="id000745">Propoſitio quadrageſima nona.</s>
        </p>
        <p type="main">
          <s id="id000746">Propoſito mobilis in circulo circuitus tempore, dataque ratione <lb></lb>diſtantiæ ab illo mobilis circuitum inuenire, quod ex eodem pun­<lb></lb>cto diſcedens cum alio mobili in dato puncto conueniat ſub quo­<lb></lb>cunque numero circuituum tempus quoque coniunctionis.</s>
        </p>
        <p type="main">
          <s id="id000747"><arrow.to.target n="marg128"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000748"><margin.target id="marg128"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.058.1.jpg" xlink:href="015/01/058/1.jpg"></figure>
        <p type="main">
          <s id="id000749">Sit in circuli peripheria a <expan abbr="pũctus">punctus</expan>, qui cir <lb></lb>cuat æquali motu (hoc enim ſemper intel­<lb></lb>ligitur) in b tempore: &amp; ſit datus punctus c <lb></lb>in quo diſcedens e mobile ex coniunctio­<lb></lb>ne cum a poſt certos circuitus proprios, <lb></lb>aut etiam. </s>
          <s id="id000750">ſine ulla circuitione perfecta de­<lb></lb>beat conuenire. </s>
          <s id="id000751">Volo ſcire tempus circui­<lb></lb>tionis e: &amp; etiam tempus coniunctionis. <lb></lb></s>
          <s id="id000752">Sit ergo primum ut abſque circuitione ulla e, a debeat comprehen­<lb></lb>dere e in c poſt numerum circuitionum ipſius a, qui ſit f. </s>
          <s id="id000753">nam ſi a o c <lb></lb>currit e in prima circuitione ipſius e, igitur a mouetur uelocius <lb></lb>quàm e, cum ergo debeat attingere ipſum e, neceſſe eſt ut a pertran­<lb></lb>ſeat prius per punctum ex quo diſceſsit antequam redeat ad con­<lb></lb>iunctionem e: ergo perficiet ſaltem unam circuitionem. </s>
          <s id="id000754">Ducemus <lb></lb>ergo f in b, &amp; fiet g tempus circuitus aut circuituum a, &amp; quia ſpa­<lb></lb>tium a c datum eſt, ſit b temporis circuitus a ad h, uelut circuli to­<lb></lb><arrow.to.target n="marg129"></arrow.to.target><lb></lb>tius ad a c, &amp; iungatur g cum h &amp; fiat k. </s>
          <s id="id000755">Fiat quoque, ut monadis <lb></lb>ad h, ita l ad monadem, &amp; ducatur l in k, &amp; fiat m: dico m eſſe tem­<lb></lb>pus circuitus e. </s>
          <s id="id000756">Conſtat enim ex ſuppoſito, quod k eſt tempus to­<lb></lb>tum in quo a peruenit poſt b circuitiones in c, ſi ergo e moueretur <lb></lb>per m tempus totum ex ſuppoſito perficeret circuitum, at quia cir­<lb></lb>cuitus ad a c, ut monadis ad h, igitur etiam ut l ad monadem, ergo <lb></lb>proportio circuitus ad a c, ut m ad monadem: ergo ſi in m tranſit to <lb></lb>tum circuitum in monade tranſit a c: ſed monas ducta in k facit k, <lb></lb>igitur e in tempore k perueniet in c, quod erat demonſtrandum. <lb></lb></s>
          <s id="id000757">Proponatur modo tempus reuolutionum e ipſum d: eodem mo­<lb></lb><arrow.to.target n="marg130"></arrow.to.target><lb></lb>do agemus ducendo fin b fit g, addatur h &amp; fiat k, diuidatur k per <lb></lb>aggregatum d &amp; a e, &amp; exeat m, (idem enim eſt diuidere per aggre­<lb></lb>gatum d &amp; h, &amp; multiplicare per l) dico ergo ut in demonſtratione <lb></lb>priore, quod m eſt tempus circuitus e. </s>
          <s id="id000758">Nam cum k ſit tempus, in <lb></lb>quo a poſt circuitus f peruenit ad c, ergo diuiſo ipſo toto tempore 


<pb pagenum="40" xlink:href="015/01/059.jpg"></pb>per numerum reuolutionum d, &amp; partem reuolutionis exibit tem­<lb></lb>pus unius reuolutionis.</s>
        </p>
        <p type="margin">
          <s id="id000759"><margin.target id="marg129"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000760"><margin.target id="marg130"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000761">Exemplum primi in re paulò obſcuriore: ſit f 4 &amp; b 2 1/2 &amp; a c 4/5, du<lb></lb>cemus 4 in 2 1/2 fit 10, adde 4/5 6 quod eſt 2 fit 12, diuide per 4/5 ſeu mul­<lb></lb>tiplica per 5/4 quod idem eſt, fit 15 circuitus e, in quatuor ergo circui­<lb></lb>tibus, &amp; 4/5 qui ſunt duodecim anni perueniet a ad c, &amp; in duodecim <lb></lb>annis e perueniet ad c, nam 12 ſunt 4/5 ipſius 15. Similiter in ſecundo <lb></lb>caſu ſit f 4 ut prius b 2 1/3 a c 1/7, ducemus 4 in 2 1/3 fit 9 1/3, addemusque h <lb></lb>portionem b qualis a c eſt totius circuitus, id eſt 1/7, eſt autem 1/7 2 1/3, 1/3 <lb></lb>fient 9 1/3, ſimiliter ponatur d 5, &amp; quia a c eſt 1/7 erunt 36/7, diuide ergo <lb></lb>9 2/3 id eſt 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s>
          <s id="id000762">Quin que ergo <lb></lb>reuolutiones e erunt 1015/108 addita ſeptima parte, quæ eſt 29/108 fient 2044/108 <lb></lb>ſeu 261/27, &amp; ſunt anni 9 18/27 ſeu 9 2/3, ergo in tanto tempore a faciet qua­<lb></lb>tuor circuitus, &amp; ſeptimam partem, &amp; e quinque circuitus, &amp; ſe­<lb></lb>ptimam.<lb></lb><arrow.to.target n="marg131"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000763"><margin.target id="marg131"></margin.target>C<emph type="italics"></emph>om./&gt;<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000764">Ex hoc patet, quod non coniungentur in alio loco, neque alio tem<lb></lb>pore ante prædictum tempus.</s>
        </p>
        <p type="main">
          <s id="id000765">Propoſitio quinquageſima.</s>
        </p>
        <p type="main">
          <s id="id000766">Omnes circuituum portiones in eiuſdem temporibus <expan abbr="repetunt̃">repetuntur</expan>.</s>
        </p>
        <p type="main">
          <s id="id000767">Sint in circulo a b c d e f g: a &amp; b iuncta, &amp; in primo congreſſu <lb></lb>iungantur in c, in ſecundo in d, in tertio in e, in quarto in f, in quinto <lb></lb>in g, in ſexto in h, in ſeptimo in k, in octauo in l. </s>
          <s id="id000768">Et ſic deinceps <expan abbr="cũquetempora">cuique<lb></lb>tempora</expan> ſint æqualia, erunt &amp; circuitus totidem numero, &amp; exceſ­<lb></lb>ſus æquales etiam a c, c d, d e, e f, f g, g h, h k, <lb></lb><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"></figure><lb></lb>k l. </s>
          <s id="id000769">Et ſi aggregatum a ſcilicet circulorum, <lb></lb>&amp; portionis fuerit commenſum circulo, &amp; <lb></lb>ita de b erunt omnia <expan abbr="cõmenſa">commenſa</expan> ad circulum, </s>
        </p>
        <p type="main">
          <s id="id000770"><arrow.to.target n="marg132"></arrow.to.target><lb></lb>&amp; etiam inter ſe. </s>
          <s id="id000771">Et ſi inter ſe aggregata, uel <lb></lb>portiones erunt, &amp; eodem modo reliqua. <lb></lb></s>
          <s id="id000772">Et quoniam circuli circulis commenſi ſunt: <lb></lb>ſi portiones erunt inuicem commenſæ <expan abbr="erũt">erunt</expan>, <lb></lb>&amp; toti circuitus cum partibus commenſi, &amp; <lb></lb>ſi non commenſi, neque erunt inter ſe, neque ad circulum. </s>
          <s id="id000773">Et ſi totum <lb></lb>ſpatium cum circuitibus erit unius generis, erunt duplicata, &amp; tri­<lb></lb>plicata, &amp; quadruplicata eiuſdem generis: quare cum ſpatia ipſa <lb></lb>detractis circuitibus uelut rhete habeant naturam reciſi, &amp; ſpatia <lb></lb>ipſa tota ſint eiuſdem generis, erunt ſpatia, quæ relinquuntur eiuſ­<lb></lb>dem generis. </s>
          <s id="id000774">Erunt tamen incommenſa neceſſariò, ſi partes fuerint <lb></lb>incommenſæ toti. </s>
          <s id="id000775">Ponatur a c incommenſa toti circulo dico, quod <lb></lb>a k <expan abbr="etiã">etiam</expan> eſt incommenſa toti circulo: &amp; <expan abbr="etiã">etiam</expan> a k, &amp; k c. </s>
          <s id="id000776">Quia enim a c <lb></lb>eſt incommenſa circulo, &amp; k a cum toto circulo ſemel eſt commen­


<pb pagenum="41" xlink:href="015/01/060.jpg"></pb>ſa a c, quia multiplex ei. </s>
          <s id="id000777">igitur cum circulus, &amp; a k diuidantur in cir­<lb></lb><arrow.to.target n="marg133"></arrow.to.target><lb></lb>culum et a k, &amp; circulus ſit incommenſus circulo, cum a k erit aggre<lb></lb></s>
          <s id="id000778">gatum ex circulo, &amp; a k incommenſum ipſi a k, &amp; a k pariter incom<lb></lb><arrow.to.target n="marg134"></arrow.to.target><lb></lb>menſa circulo. </s>
          <s id="id000779">Rurſus quia a k eſt incommenſa circulo cum a k, &amp; <lb></lb>circulus cum a k ſit multiplex ad a c, erit a k incommenſa a c, quare <lb></lb><arrow.to.target n="marg135"></arrow.to.target><lb></lb>erit c k incommenſa a k &amp; a c, &amp; circulo ad dita a k. </s>
          <s id="id000780">Si ergo a c ſit <lb></lb>commenſa circulo, erunt omnes portiones e genere numeri, &amp; ſi <lb></lb><arrow.to.target n="marg136"></arrow.to.target><lb></lb>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra­<lb></lb>ctis, ut a k &amp; a l reciſa: &amp; a c ſit potentia ſecunda rhete, id eſt radix cu<lb></lb>bica erunt omnes c d, d e, e f, potentia ſecunda rhete, et radices cubi­<lb></lb>cæ numeri, ſeu latera corporum rhete, a k uero &amp; a l, &amp; huiuſmodi <lb></lb>in infinitum reciſa potentia rhete.<lb></lb><arrow.to.target n="marg137"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000781"><margin.target id="marg132"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb><emph type="italics"></emph>præcedentis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000782"><margin.target id="marg133"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 14. <emph type="italics"></emph>deci <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000783"><margin.target id="marg134"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <lb></lb><emph type="italics"></emph>eiuſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000784"><margin.target id="marg135"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 14. <lb></lb><emph type="italics"></emph>rurſus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000785"><margin.target id="marg136"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <lb></lb><emph type="italics"></emph>rurſus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000786"><margin.target id="marg137"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000787">Ex hoc patet, quod cum circulus poſsit diuidi in infinita gene­</s>
        </p>
        <p type="main">
          <s id="id000788"><arrow.to.target n="marg138"></arrow.to.target><lb></lb>ra quantitatum, quæ non ſunt inuicem commenſæ cumque coniun­<lb></lb>ctiones hæ ſemper in eodem genere maneant, quod infinita pun­<lb></lb>cta, &amp; infinitis in ſpeciebus quantitatum remanebunt in quibus a <lb></lb>&amp; b in perpetuum nunquam conuenient. </s>
          <s id="id000789">Velut ſi coniunctio pri­<lb></lb>ma fiat in &lt;02&gt; cu. </s>
          <s id="id000790">1/2 alicuius circuli, nunquam conuenient, neque in me­<lb></lb>dietate, neque in quarta parte, nec octaua, nec tertia, nec ſexta, nec no­<lb></lb>na, nec quinta, nec decima, &amp; ſic de ſingulis in genere commenſa­<lb></lb>rum toti circulo. </s>
          <s id="id000791">Neque in &lt;02&gt; quadrata 1/2 uel 1/3 uel 1/5 neque &lt;02&gt; 1/6 uel 1/20, <lb></lb>neque in &lt;02&gt; 3 m: 1, nec 2 m: &lt;02&gt; 3 nec in &lt;02&gt; &lt;02&gt; 2 aut 3 aut 7 nec in &lt;02&gt; rela­<lb></lb>ta alicuius numeri, nec in 2 m: &lt;02&gt; &lt;02&gt; cub. </s>
          <s id="id000792">3 nec 2 m: &lt;02&gt; cub. </s>
          <s id="id000793">4, &amp; ſic <lb></lb>de alijs.</s>
        </p>
        <p type="margin">
          <s id="id000794"><margin.target id="marg138"></margin.target>P<emph type="italics"></emph>er penulti­<lb></lb>mam uigeſi­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000795">Propoſitio quinquageſima prima.</s>
        </p>
        <p type="main">
          <s id="id000796">Operationes dictas exemplo declarare.<lb></lb><arrow.to.target n="marg139"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000797"><margin.target id="marg139"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000798">Supponamus in circulo prædicto a c &lt;02&gt; 7 conſtat, quod eſſe non <lb></lb>poteſt, quia &lt;02&gt; 7 eſt maior monade, ideo toto circulo, quare non po<lb></lb>terit eſſe pars circuli, ſed referetur ad <expan abbr="quantitatẽ">quantitatem</expan> certam, uelut quod <lb></lb>circulus ſit 10. ſemper ergo diuidemus &lt;02&gt; 7, ſeu eam portionem per <lb></lb>10 quantitatem circuli &amp; exibit &lt;02&gt; 7/100, &amp; hæc erit portio circuli, &amp; ita <lb></lb>ſi portio ſit &lt;02&gt; cub. </s>
          <s id="id000799">16, diuidemus &lt;02&gt; cub. </s>
          <s id="id000800">16 per 10 exibit &lt;02&gt; cu 2/125, &amp; <lb></lb>ita de alijs.</s>
        </p>
        <p type="main">
          <s id="id000801">Sed cum ex repetitione creſcat portio illa, donec exuperet mo­<lb></lb>nadem, aut aliquem quemuis numerum detracta monade aut nu­<lb></lb>mero circuituum habebit rationem reciſi. </s>
          <s id="id000802">Velut &lt;02&gt; 7/100 quater ſum­<lb></lb>pta efficit &lt;02&gt; 112/100. Et hoc eſt potentia rhete, ſed ſi quis auferat mona­<lb></lb>dem fiet &lt;02&gt; 112/100 m: 1, &amp; hoc eſt reciſum 1, ſcilicet 1 p: &lt;02&gt; v: 23/25 m: &lt;02&gt; 28/25, ſed ta<lb></lb>men uerè eſt linea media.</s>
        </p>
        <p type="main">
          <s id="id000803">Quod uerò non contingat coniungi in alio loco, neque tem­<lb></lb>pore ſit, ut a b iungantur in c, &amp; ſit reuolutio a triplex integra, &amp; b 


<pb pagenum="42" xlink:href="015/01/061.jpg"></pb>ſexcuplex, &amp; tempus totum decem annorum: ita ut a c ſit tertia <lb></lb>pars circuitus, &amp; a circuitus tres anni, &amp; quia circuitus b ſunt ſex <lb></lb>cum tertia, diuidemus decem per 6 1/3 exit <lb></lb>1 11/29, dico quod non prius, neque in alio <lb></lb><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"></figure><lb></lb>puncto. </s>
          <s id="id000804">Si enim primùm in eodem pun­<lb></lb>cto, &amp;, gratia exempli, in quatuor annis <lb></lb>congruit enim, &amp; b dicamus quod per­<lb></lb>egerit duas reuolutiones cum tertia, hoc <lb></lb>enim eſt neceſſarium, ſi debet perueni­<lb></lb>re ad c, &amp; erunt anni tres, &amp; 23/19, non ergo <lb></lb>anni quatuor. </s>
          <s id="id000805">Cum enim tempora di­<lb></lb>uerſa diuiduntur per numeros haben­<lb></lb>tes proportionem erunt, qui prodeunt <lb></lb><arrow.to.target n="table13"></arrow.to.target><lb></lb>numeri in eadem ratione. </s>
          <s id="id000806">Diuiſo ergo <lb></lb>10 per 1 11/19 exit 6 2/3, &amp; diuiſo 4 per 1 11/19 exit <lb></lb>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb></lb>non poteſt eſſe æquale 1/3. Si enim per <lb></lb>præcedentem repetuntur, ergo non poſ­<lb></lb>ſunt redire, donec iterum coniungantur in ipſo a. </s>
          <s id="id000807">Si enim aliter ſit <lb></lb>ut ex e, igitur e c eſt æqualis a c pars toti, quod contingere non po­<lb></lb>teſt. </s>
          <s id="id000808">Sin uerò coniunctio fiat in d, igitur per præcedentem d e eſt <lb></lb>pars a c ſubmultiplex quomodolibet, quare non fuerunt aſſum­<lb></lb>pti primi numeri. </s>
          <s id="id000809">Veluti in exemplo conſtituimus, quod a, &amp; b <lb></lb>conueniunt in c in decem annis, &amp; a c eſt tertia pars circuitus: er­<lb></lb>go in triginta annis conueniunt in a, &amp; in quadraginta rurſus in c. <lb></lb>ſi ergo quis aſſumpſiſſet quadraginta annos ab initio pro con­<lb></lb>greſſu, &amp; diuiſiſſet per 1 12/19 exiret 25 1/3, &amp; ſi per 3 exiret 13 1/3, &amp; mani­<lb></lb>feſtum eſt, quod uterque numerus poteſt diuidi per eundem nu­<lb></lb>merum, utpote 4 &amp; exit numerus cum eadem parte ſcilicet 6 1/3 &amp; <lb></lb>3 1/3 ergo conuenient ante, non ergo aſſumpſiſti minimos in ea pro­<lb></lb>portione. </s>
          <s id="id000810">Illi autem nequaquam amplius diuidi non poſſunt eo­<lb></lb>dem modo.</s>
        </p>
        <table>
          <table.target id="table13"></table.target>
          <row>
            <cell>Decem</cell>
            <cell></cell>
            <cell>Quatuor</cell>
            <cell></cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>3 1/3</cell>
            <cell>1 11/19</cell>
            <cell>2 8/15)</cell>
          </row>
          <row>
            <cell>1 11/19</cell>
            <cell>6 1/3</cell>
            <cell></cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="id000811">Propoſitio quinquageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id000812">Tria mobilia coniuncta in eodem puncto, quorum duo, &amp; duo <lb></lb>conueniant in partibus in commenſis inter ſe, in perpetuum in nul­<lb></lb>lo unquam puncto conuenient.</s>
        </p>
        <p type="main">
          <s id="id000813"><arrow.to.target n="marg140"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000814"><margin.target id="marg140"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000815">Sint a b c iuncta, &amp; primo iungantur a &amp; b, iterum in d &amp; b, &amp; <lb></lb>c in e, &amp; ſint a d, a e incommenſæ, dico quòd a b c nunquam con­<lb></lb>uenient in aliquo puncto, ſeu primo, ſeu alio à primo: ſi non con­<pb pagenum="43" xlink:href="015/01/062.jpg"></pb><figure id="id.015.01.062.1.jpg" xlink:href="015/01/062/1.jpg"></figure><lb></lb>ueniant in f, erunt ergo in g tempore re­<lb></lb>uolutiones integræ, &amp; portio a f inſuper. <lb></lb></s>
          <s id="id000816">Et quia hæ conſtituuntur per congreſſus <lb></lb>b cum a, &amp; ſunt ſpatia a d, &amp; b cum c, &amp; <lb></lb>ſunt ſpatia e f, igitur ſpatium a f erit ex ge­<lb></lb>nere quantitatis a d, &amp; a e per quinqua­<lb></lb>geſimam, harum ergo erunt commenſæ: <lb></lb>quod eſt contra ſuppoſitum. </s>
          <s id="id000817">Et harum <lb></lb>propoſitionum principium eſt traditum <lb></lb>à Campano Nouarienſi Euclidis expoſitore, in quodam libello <lb></lb>non edito qui diligentia patris mei Facij ad me peruenit.</s>
        </p>
        <p type="main">
          <s id="id000818">Propoſitio quinquageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id000819"><expan abbr="Circulorũ">Circulorum</expan> ſe in aduerſum mouentium proportionem declarare.</s>
        </p>
        <p type="main">
          <s id="id000820"><arrow.to.target n="marg141"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000821"><margin.target id="marg141"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000822">Sit orbis a b cuius cen­<lb></lb><figure id="id.015.01.062.2.jpg" xlink:href="015/01/062/2.jpg"></figure><lb></lb>centrum c, manubrium c <lb></lb>d f e, ſeu uero tangat circu<lb></lb>lum g, ſeu more gemmas <lb></lb>ſculpentium aligetur al­<lb></lb>teri orbi funiculo a l b, &amp; <lb></lb>ſit in uertice axis k m or­<lb></lb>biculus ſolidus aut ſemi­<lb></lb>circulari forma m, dico <lb></lb>quod proportio motus a <lb></lb>b ad motum m eſt produ<lb></lb>cta ex duabus proportio­<lb></lb>nibus c n <expan abbr="ſemidimetiẽtis">ſemidimetientis</expan>, <lb></lb>&amp; ſemidimetientis m ad k <lb></lb>o, quare ut rectanguli c n <lb></lb>in dimidium dimetientis <lb></lb>m ad quadratum o, ut enim a b ad ol orbem, id eſt <expan abbr="peripheriarũ">peripheriarum</expan> ita <lb></lb>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo­<lb></lb>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="continet̃">continetur</expan> in peripheria a b, ergo quoties o k con­<lb></lb>tinetur in c n toties in una circuitione a b o l circumuertitur, ſed <lb></lb>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo­<lb></lb>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb></lb>b toties o k continetur in c n, ergo ſi fiat comparatio ſemidiametri <lb></lb>m ad c n, erit producta proportio circuitus a b ad circuitum m ex <lb></lb>proportione c n ad o k, et ſemidimetientis m ad <expan abbr="idẽ">idem</expan> o k, ergo per 26 <lb></lb>proportio numeri circuitus unius p <expan abbr="alterũ">alterum</expan> eſt, ut rectanguli ſub c n, <lb></lb>&amp; ſemidimetiente m ad quadratum k o, quod erat <expan abbr="demonſtrandũ">demonſtrandum</expan>.</s>
        </p>
        <p type="main">
          <s id="id000823">Manifeſtum eſt autem ex ipſa ſola conſtitutione, quod ſi a b mo­</s>
        </p>
        <p type="main">
          <s id="id000824"><arrow.to.target n="marg142"></arrow.to.target>


<pb pagenum="44" xlink:href="015/01/063.jpg"></pb>uetur ſurſum à dextro in ſiniſtrum in inferiore parte, mouebitur à <lb></lb>ſiniſtro in dextrum, &amp; uterque circulorum g &amp; k in ſuperiore parte, <lb></lb>&amp; in inferiore mouebitur contrario motu, ſcilicet in ſuperiore à ſini<lb></lb>ſtro in dextrum, &amp; inferiore à dextro in ſiniſtrum, illi uerò duo or­<lb></lb>bes ſimili motu mouebuntur tam in parte ſuperiore, quàm inferio­<lb></lb>re, &amp; proportio motuum eorum inter ſe erit uelut dimetientium <lb></lb>eorundem.<lb></lb><arrow.to.target n="marg143"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000825"><margin.target id="marg142"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="margin">
          <s id="id000826"><margin.target id="marg143"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000827">Rurſus cum a b circumuertatur cum manubrio c d f e, tanto uelo<lb></lb>cius circumuertetur, &amp; in ea proportione, qua d f continetur in c n, <lb></lb>&amp; in eodem tempore, in quo manubrium circumuertitur in eodem <lb></lb>axis circumuertitur, &amp; orbis, ut dictum eſt, ergo in eodem tempo­<lb></lb>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb></lb>uidebitur moueri axis ipſo orbe, quanta eſt proportio minoris in <lb></lb>æqualitatis ipſius axis, ſeu ambitus, ſeu ſemidimetientis ad ambi­<lb></lb>tum, ſeu ſemidimetientem orbis.</s>
        </p>
        <p type="main">
          <s id="id000828">Propoſitio quinquageſimaquarta.</s>
        </p>
        <p type="main">
          <s id="id000829">Proportio circuli ad ſuum diametrum per <expan abbr="ſimilitudinẽ">ſimilitudinem</expan> eſt quar­<lb></lb>ta pars peripheriæ. </s>
          <s id="id000830">Rurſusque eiuſdem circuli ad peripheriam diame<lb></lb>tri quarta pars.</s>
        </p>
        <p type="main">
          <s id="id000831"><arrow.to.target n="marg144"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000832"><margin.target id="marg144"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000833">Quoniam enim ſuperficies circuli, ut ab <lb></lb><figure id="id.015.01.063.1.jpg" xlink:href="015/01/063/1.jpg"></figure><lb></lb>Archimede demonſtratum eſt, fit ex dimi­</s>
        </p>
        <p type="main">
          <s id="id000834"><arrow.to.target n="marg145"></arrow.to.target><lb></lb>dio diametri in <expan abbr="dimidiũ">dimidium</expan> peripheriæ erit, ut <lb></lb>eadem fiat ex tota peripheria in <expan abbr="quartã">quartam</expan> par<lb></lb>tem diametri, &amp; ex tota diametro in quar­<lb></lb>tam <expan abbr="partẽ">partem</expan> peripherię. </s>
          <s id="id000835">ergo proportio areę <lb></lb>circuli ad diametrum per ſimilitudinem <lb></lb><arrow.to.target n="marg146"></arrow.to.target><lb></lb>eſt quarta pars peripherię, &amp; proportio areę <lb></lb>ad <expan abbr="peripheriã">peripheriam</expan> eſt quarta pars dimetientis, quod erat probandum.</s>
        </p>
        <p type="margin">
          <s id="id000836"><margin.target id="marg145"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000837"><margin.target id="marg146"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>diff.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000838">Propoſitio quinquageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id000839">Proportionem medicamentorum per ordines ſuppoſita æquali <lb></lb>proportione in ordinibus per quantitates, &amp; proportiones de­<lb></lb>monſtrare.<lb></lb><arrow.to.target n="marg147"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000840"><margin.target id="marg147"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000841">Galenus libro quinto de Simplicibus medicamentis, quem ſe­</s>
        </p>
        <p type="main">
          <s id="id000842"><arrow.to.target n="marg148"></arrow.to.target><lb></lb>quuti ſunt alij medici, ponit quatuor ordines <expan abbr="medicamentorũ">medicamentorum</expan> iux­<lb></lb>ta qualitates calidi, frigidi, ſicci, &amp; humidi, &amp; primus eſt cum <expan abbr="medi­camentũ">medi­<lb></lb>camentum</expan> non ſentitur quale ſit licet operetur, uelut camęmelon, ab­<lb></lb>ſynthium, &amp; oriza: ſecundus eſt, cum ſentitur, ſed non lædit, ut nux <lb></lb>myriſtica, ſaluia, ozimum: tertius eſt cum ſentitur, &amp; lædit, ſed <lb></lb>non deſtruit, neque corrumpit corpus, uelut aſſarum apium ſta­<lb></lb>phiſagria, cappares, myrrha, ruta: quartus eſt, cum deſtruit ue­<lb></lb>lut pyretrum, piper, euphorbium cæpe aggreſte, &amp; ſinapis, cina­


<pb pagenum="45" xlink:href="015/01/064.jpg"></pb>momum autem, &amp; gingiber numerantur inter medicinas calídas <lb></lb>tertij gradus, &amp; hoc opus comparatur ad corpus ſicut dicit Gale­<lb></lb>nus, &amp; Serapio non ad linguam, ut medici noſtri temporis interpre<lb></lb>tantur. </s>
          <s id="id000843">Ex quo patet, quod aliqua medicina poterit eſſe quarti ordi<lb></lb>nis, &amp; non lædere linguam in guſtu, &amp; alia tertij ordinis, quæ non <lb></lb>ſolum lædet linguam, ſed ſenſum eius corrumpet, et deſtruet, quod <lb></lb>contingit propter ſubſtantiam tenuem craſſæ miſtam cum ſiccitate <lb></lb>pari ipſi calori. </s>
          <s id="id000844">Sed non oportet hęc nunc tractar, enon ſolum quia <lb></lb>non ſit locus, ſed etiam quòd confuſa ſit per ſe ipſa materia abſque <lb></lb>eo, quod difficultatem difficultati addamus, ſolum ergo eas dubita<lb></lb>tiones adiungemus, quas <expan abbr="uolẽtes">uolentes</expan> declarare propoſitionem præſen<lb></lb>tem, neque ſuperfugere, neque declinare poſſumus. </s>
          <s id="id000845">Nam de ſicco, <lb></lb>&amp; humido, cum ſint longè minoris actionis, quàm calidum, &amp; fri­<lb></lb>gidum, &amp; præcipuè humidum, non uideo quomodo poſsit Gale­<lb></lb>nus ſtatuere medicinam humidam tertij gradus, nedum quarti, <lb></lb>cum non poſsit inueniri medicina, quæ deſtruat corpus noſtrum <lb></lb>propter humidam qualitatem. </s>
          <s id="id000846">Et licet Serapio poſuerit gingiber <lb></lb><arrow.to.target n="marg149"></arrow.to.target><lb></lb>&amp; enulam &amp; zelim in tertio ordine calidorum &amp; humidorum: &amp; <lb></lb>inter frigidas, &amp; humidas in tertio portulacam, aizoum, &amp; uirgam <lb></lb>paſtoris, &amp; fungos. </s>
          <s id="id000847">Primum non auſus eſt ponere medicinas ullas <lb></lb>calidas, aut frigidas in quarto ordine, quę ſint humidæ. </s>
          <s id="id000848">ſecundum, <lb></lb>quando dicit medicinas calídas, aut frigidas, atque humídas in ter­<lb></lb>tio ordine, intelligit ſolum de qualitate actiua ſcilicet caliditate, uel <lb></lb>frigiditate, &amp; non de humida qualitate, quod oſtendit de gingibe­<lb></lb>re, &amp; enula, dicens, quod ſunt calidæ in tertio ordine, &amp; humidæ <lb></lb>humido crudo, non auſus addere ordinem, quia non uídit ratio­<lb></lb>nem, qua poſſent dici humidæ in tertio. </s>
          <s id="id000849">Et clarius in capite de zei­<lb></lb>len, quem ſtatuerat inter medicinas calidas, &amp; humidas in tertio, di<lb></lb>cit quod eſt calida in tertio, &amp; humida in primo, ergo non intelligit <lb></lb>per medicinas calidas &amp; humidas in tertio ordine, quod ſint humi­<lb></lb>dæ in tertio ordine. </s>
          <s id="id000850">Clarius etiam de frigidis &amp; humidis, nam por­<lb></lb>tula cam dicit eſſe frigidam in tertio, humidam in ſecundo, &amp; quod <lb></lb>maius, eſt cum collo caſſet aizoum inter medicinas frigidas, &amp; hu­<lb></lb>midas in tertio ordine, dicit, quod eſt frigidum in tertio ordine, ad­<lb></lb>ijcit, quod eſt ſiccum parum, &amp; de uirga paſtoris nihil dicit de hu­<lb></lb>mido, ſed dicit, quod aſtringit, ex quo concludo, quod ſecun­<lb></lb>dum mentem Serapionis nulla eſt medicina humidior portulaca, <lb></lb>etiam uidetur innuere de fungis, ſatis eſt quod non excedunt ſecun<lb></lb>dum ordinem in humido neque calida neque frigida, ſed frigida ſunt <lb></lb>humidiora, ut fungi, &amp; portulaca, quia frigiditas in generatione <lb></lb>humidum magis admittit, quàm caliditas, &amp; calida magis hu<pb pagenum="46" xlink:href="015/01/065.jpg"></pb>mectant, quia magis penetrat uis medicamenti, &amp; hæc regula de <lb></lb>humido, &amp; ſicco eſt generalis apud Serapionem, quod non intelli­<lb></lb>gitur ordo in paſsiuis, niſi ſpecialiter exprimatur, nam de ſiccitate <lb></lb>non nego, quin inueniantur medicinæ ſiccæ in tertio, &amp; forſan in <lb></lb>quarto ordine, ſed de hac Galeni oſcitantia, quæ in illo peculiaris <lb></lb>eſt dum uult ſequi ſuas methodos ſine alio diſcrimine, medicis con<lb></lb>ſiderandum relinquo.</s>
        </p>
        <p type="margin">
          <s id="id000851"><margin.target id="marg148"></margin.target>C<emph type="italics"></emph>ap. </s>
          <s id="id000852">ult.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000853"><margin.target id="marg149"></margin.target>C<emph type="italics"></emph>ap.<emph.end type="italics"></emph.end> 336. <lb></lb>337. &amp; <lb></lb>338.</s>
        </p>
        <p type="main">
          <s id="id000854">Secunda difficultas eſt maior, &amp; magis pertinet ad nos, &amp; eſt, <lb></lb>quòd non declarauit an iſti ordines inter ſe <expan abbr="aliquã">aliquam</expan> proportionem <lb></lb>ſeruarent, an omnino nullam, ſi enim nulla proportio ſeruatur, fieri <lb></lb>nullo modo poteſt, ut per cognitionem temperaturæ ſimplicium <lb></lb>medicamentorum cognoſcamus temperaturam compoſitorum ex <lb></lb>illis ratione ulla, ſed oportebit ſolum experiri. </s>
          <s id="id000855">Sed ſi ordines ſer­<lb></lb>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb></lb>ſit Arithmetica, uel Geometrica, uel Muſica, &amp; nihil mirum eſſet, <lb></lb>quod eſſet Muſica, ut aliâs docuimus, ubi tractauimus de differen­<lb></lb>tia inter ſenſum auditus, et uiſus. </s>
          <s id="id000856">Sed quia de hac nullus medicus ui <lb></lb>detur intellexiſſe, omittam hanc tractationem. </s>
          <s id="id000857">Et quanquàm Gale­<lb></lb>nus poſsit uideri non exiſtimaſſe, quòd hi ordines non ſeruent <lb></lb>proportionem ullam, quia non auſus eſt tractare de temperamen­<lb></lb>to medicamentorum compoſitorum per rationem temperamen­<lb></lb>ti ſimplicium, nihilominus ſuppoſito quod ita eſſet, quod ſeruetur <lb></lb>altera proportionum, uolo oſtendere rationem componendi in <lb></lb>utraque proportione &amp; Arithmetica, &amp; Geometrica. </s>
          <s id="id000858">Ex quo ſe­<lb></lb>quitur, quod Aueroes quàm oſcitanter tractauerit in quinto ſuo­<lb></lb>rum collectaneorum de hoc, &amp; non diſtinguit, neque docet pri­<lb></lb>mum an ſit aliqua proportio, deinde ſi qua ſit, cuius generis ſit, &amp; <lb></lb>cum in re tam clara pugnet prorſus, ut cœcus ictus maximos eden­<lb></lb>do, ſed in caſſum pleroſque, quàm malè agant qui ei in arduis tan­<lb></lb>tum tribuunt fidei, &amp; authoritatis, ſed hæc eſt infelicitas noſtra, &amp; <lb></lb>ira Deorum. </s>
          <s id="id000859">Suppoſito ergo quod primò ordines diſtinguantur <lb></lb>per proportionem arithmeticam, ſit ſuperficies a b pro quantitate, <lb></lb><figure id="id.015.01.065.1.jpg" xlink:href="015/01/065/1.jpg"></figure><lb></lb>&amp; a ſit calida in primo gradu, &amp; b in ter­<lb></lb>tio, erit ergo perinde ac ſi duo corpora <lb></lb>eſſent unum altitudinis unius cum baſi <lb></lb>quadrilatera rectangula a, aliud altitu­<lb></lb>dinis trium, baſi autem quadrilatera ſu­<lb></lb>perficie rectangula b, hoc igitur erit to­<lb></lb>tum miſtum, &amp; quia quantitas medicamenti non mutatur quæ eſt <lb></lb>a, b, ergo talia corpora æquantur uni corpori, cuius baſis eſt a b, <lb></lb>cum ergo talia corpora producantur ex a in unum, &amp; b in tria, ergo 


<pb pagenum="47" xlink:href="015/01/066.jpg"></pb>diuiſo aggregato per a b prodibit altitudo, ſeu ordo qualitatis to­<lb></lb>tius medicamenti, iuxta quod conſtituitur regula prima libri artis <lb></lb>medendi paruæ huiuſmodi, &amp; reliquæ, traduxi autem illas ad hunc <lb></lb>locum, “quia pendent ex demonſtratione hac: “duc numerum ordi­<lb></lb>nis ſingulorum medicamentorum in numerum quantitatis, ſimilia <lb></lb>iunge, diſsimilia detrahe, quod fit, diuide per aggregatum, quanti­<lb></lb>tatum, exibit numerus ordinis compoſiti. </s>
          <s id="id000860">Sic miſcendo calidum in <lb></lb>ſecundo ordine cum duplo pondere temperati conflabit calidum <lb></lb>in beſſe. </s>
          <s id="id000861">Secunda ſi ex pluribus diuerſarum, qualitatum, &amp; ordi­<lb></lb>num temperatum efficere uelis, duc quæ ſunt eiuſdem qualitatis in <lb></lb>ſuas quantitates, &amp; iunge, quod fit, diuide per numerum ordinis <lb></lb>medicamenti contrarij, exibit quantitas illius, ſub qua ſi iungatur, <lb></lb>fiet medicamentum temperatum. </s>
          <s id="id000862">Tertia cum nolueris ex tempera­<lb></lb>to, &amp; alio cuiuſcunque ordinis medicamen conficere ordinis re­<lb></lb>miſsionis, detrahe numerum ordinis eius, quod conficere uis ex nu<lb></lb>mero ordinis eius, quod habes, &amp; cum reſiduo diuide numerum <lb></lb>medicaminis, quod conficere uis, quod exit eſt numerus quantita­<lb></lb>tis medicamenti non temperati in comparatione ad temperatum.” <lb></lb>Ex his potes propoſitis quibuſcunque medicamentis conficere <lb></lb>antidotum ſub quo cunque ordine remiſsiore potentiſsimo ex il­<lb></lb>lis. </s>
          <s id="id000863">Quarta in compoſitione, quæ non fermenteſcit calida, calidis <lb></lb>iuncta ſemper opus augent, ut mel cum pipere. </s>
          <s id="id000864">Quæ autem ſub mi<lb></lb>nore quantitate exhibentur non ſub remiſsiore ordine agant, ſed <lb></lb>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb></lb>immutant.</s>
        </p>
        <p type="main">
          <s id="id000865">Quod ſi ſtatuamus proportionem eſſe Geometricam, modus <lb></lb>erit idem in omnibus, &amp; quo ad numerum etiam in primo, &amp; ſecun<lb></lb>do ordine, quia in proportione dupla Geometrica ſecundus ordo <lb></lb>tantundem diſtat à primo, quantum primus ab æqualitate, quia <lb></lb>unum &amp; duo ſeruant proportionem, &amp; æqualem diſtantiam, ſed in <lb></lb>cæteris ordinibus non ita erit, quia qui eſſet trium in Arithmetica, <lb></lb>ſcilicet totius ordo eſt, quatuor in Geometrica, &amp; quartus ordo, <lb></lb>qui eſſet quatuor in Arithmetica, eſſet octo in Geometrica, ideo <lb></lb><figure id="id.015.01.066.1.jpg" xlink:href="015/01/066/1.jpg"></figure><lb></lb>ſcribemus ordines hoc modo, &amp; operabimur cum <lb></lb>numeris loco ordinum, exemplum ergo primum <lb></lb>ſit medicina calida in tertio ordine quatuor uncia­<lb></lb>rum, &amp; medicina frigida in <expan abbr="ſecũdo">ſecundo</expan> ordine duarum <lb></lb>unciarum, duco quatuor in tria, ſi proportio ſit Arithmetica, fit <lb></lb>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo­<lb></lb>decim, quia omnis medicina tantum retondit de contrario, ſeu mi­<lb></lb>nuit relinquuntur octo ſcilicet caliditatis, diuido per ſex ag­<pb pagenum="48" xlink:href="015/01/067.jpg"></pb>gregatum unciarum exit unum, &amp; tertia, ergo erit calida in princi­<lb></lb>pio ſecundi ordinis. </s>
          <s id="id000866">Secundum exemplum ſint eædem medicinæ, <lb></lb>&amp; ſit proportio Geometrica, ducemus ergo quatuor in quatuor, &amp; <lb></lb>fiunt ſexdecim, &amp; duo in duo fiunt quatuor, detrahe quatuor ex ſex<lb></lb>decim, &amp; remanent duodecim, diuide per ſex, ut prius, exeunt duo, <lb></lb>ergo erit calida in fine ſecundi gradus uides ergo diſcrimen. </s>
          <s id="id000867">rurſus <lb></lb>ſint ambæ medicinæ calidæ, &amp; ducemus, ut prius in tertio exem­<lb></lb>plo, ubi proportio ſit Arithmetica iungendo duodecim cum qua­<lb></lb>tuor, &amp; fient ſexdecim, diuide per ſex, exeunt duo, &amp; duæ tertiæ, er­<lb></lb>go erit calida in medio tertij gradus, rurſus in quarto exemplo iun<lb></lb>gemus ſedecim cum quatuor, &amp; fient uiginti, diuide per ſex exi­<lb></lb>bunt tria &amp; tertia, &amp; ita erit in medio tertij gradus, ut prius, ſed ſi <lb></lb>ille quatuor unciæ eſſent calidæ in quarto gradu, &amp; illæ duæ unciæ <lb></lb>in ſecundo gradu, ut prius ducendo quatuor in quatuor fiunt ſex­<lb></lb>decim, &amp; duo in duo fiunt quatuor, iunge, &amp; fient uiginti, diuide <lb></lb>per ſex exeunt tria cum tertia, ergo erit calida in principio quarti <lb></lb>gradus ſecundum proportionem Arithmeticam, ſed ſecundum <lb></lb>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua­<lb></lb>tuor ut prius, ſcilicet productum duorum in duo fiunt triginta ſex, <lb></lb>diuide per ſex, exeunt ſex, &amp; quia ſex ad quatuor maiorem habent <lb></lb>proportionem, quàm octo ad ſex ideo hæc medicina erit calida ul­<lb></lb>tra medium quarti gradus, iam ergo uides rationem, &amp; differen­<lb></lb>tiam horum.</s>
        </p>
        <p type="main">
          <s id="id000868">Quod ſi quis dicat, an debeat attendi Geometrica proportio in <lb></lb>medicamentis, an Arithmetica, reſpondeo, quòd ueriſimilius eſt de <lb></lb>Arithmetica, quia illa proportio etiam quod ſit minor quatuor ad <lb></lb>trium, quàm trium ad duo, &amp; multò minor quàm duo ad unum ni­<lb></lb>hilominus longè plus operatur, quia tertius ordo iam incipit eſſe <lb></lb>præter naturam, &amp; uidemus, quod læſio facta in uulnerato, etiam <lb></lb>quòd ſit quadruplo minor, plus nocet longè, quàm in ſano qua­<lb></lb>druplo maior: quia termini præter naturam ſunt ualdè anguſti in <lb></lb>comparatione ad latitudinem naturalem, ſicut etiam uidemus in­<lb></lb>tendendis chordis ſcorpionum, quod ultima pars eſt breuis &amp; ta­<lb></lb>men homini tantam difficultatem adijcit. </s>
          <s id="id000869">Notandum eſt etiam, <lb></lb>quòd ob hoc diuiſerunt ordines in tres partes, uelut gingiber eſt <lb></lb>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb></lb>principio, &amp; ita euphorbium eſt calidum in principio quarti gra­<lb></lb>dus, ſed in fine principij piper, in principio principij aqua ſepara­<lb></lb>tionis in medio quarti ordinis, ſed oleum chalcanthi factum ea ar­<lb></lb>te, ut exurat paleas, ſicut ignis eſt calidum in fine quarti ordinis, &amp; <lb></lb>ita ſufficiet diuidere propter eandem cauſam primum, &amp; ſecun­


<pb pagenum="49" xlink:href="015/01/068.jpg"></pb>dum ordinem in duas tantum partes non ratione latitudinis, quæ <lb></lb>eſt æqualis, uel etiam forſan maior, ſed ratione uarietatis operatio­<lb></lb>nis quæ minus ſentitur, &amp; maximè in primo ordine.</s>
        </p>
        <p type="main">
          <s id="id000870">Propoſitio quinquageſimaſexta.</s>
        </p>
        <p type="main">
          <s id="id000871">Proportio cuiuſuis binomij ad ſuum reciſum, uel ei commen­<lb></lb>ſum eſt duplicata ei, quæ ad numeri latus.<lb></lb><arrow.to.target n="marg150"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000872"><margin.target id="marg150"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id000873">Cum enim proportionis medium ſit latus numeri eo quod ex bi<lb></lb>nomio in reciſum ſuum fit numerus ex his, quæ demonſtrata ſunt <lb></lb>generaliter in tertio Arithmeticæ de omnibus binomijs cum ſuis </s>
        </p>
        <p type="main">
          <s id="id000874"><arrow.to.target n="marg151"></arrow.to.target><lb></lb>reciſis, uel in quadratis lateribus erit &lt;02&gt; numeri media proportione <lb></lb>inter binomium, &amp; ſuum reciſum, igitur cum proportio producto­<lb></lb>rum ex binomio in commenſa reciſo ſit, ut commenſorum ad reci­<lb></lb><arrow.to.target n="marg152"></arrow.to.target><lb></lb>ſa erunt omnia producta ex binomio in commenſa reciſo ſuo &lt;02&gt; nu<lb></lb><arrow.to.target n="marg153"></arrow.to.target><lb></lb>meri, igitur proportio binomij ad reciſum ſuum, &amp; omnia com­<lb></lb>menſa illi, eſt duplicata ei quæ ad &lt;02&gt; numeri.<lb></lb><arrow.to.target n="marg154"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000875"><margin.target id="marg151"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. P<emph type="italics"></emph>ro­<lb></lb>poſ. </s>
          <s id="id000876">lib. 

de<emph.end type="italics"></emph.end><lb></lb>A<emph type="italics"></emph>liza.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000877"><margin.target id="marg152"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000878"><margin.target id="marg153"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <lb></lb><emph type="italics"></emph>ſeptimi <lb></lb>eiuſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000879"><margin.target id="marg154"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>deci­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement:<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000880">Propoſitio quinquageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id000881">Motus rationem ad pondus inuenire.</s>
        </p>
        <p type="main">
          <s id="id000882"><arrow.to.target n="marg155"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000883"><margin.target id="marg155"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000884">Oſtenſum eſt antea, quod motus naturalis uelocior fit in fine, ac <lb></lb>magis augetur ob aëris motum, ubi uerò hæret eſt ac ſi quieſcat. <lb></lb></s>
          <s id="id000885">Eadem autem eſt ratio in motis uiolenter, &amp; naturaliter dum ęqua­<lb></lb>li impetu feruntur. </s>
          <s id="id000886">Sed ſubitò poſt etiam, quod motus æqualiter <lb></lb>augerentur minus tamen creſcit proportio uiolenti ſcilicet ob im­<lb></lb><figure id="id.015.01.068.1.jpg" xlink:href="015/01/068/1.jpg"></figure><lb></lb>pedimentum naturale. </s>
          <s id="id000887">Sed ſi uis mouens fuerit <lb></lb>adeò ualida ut proportio incrementi ex aëre ſit <lb></lb>maior, quàm impedimentum, &amp; in crementum al<lb></lb>terius mobilis naturaliter moti, motus ille uelo­<lb></lb>cior fiet naturali, ut in ſphæris ferreis ex machina <lb></lb>igne excuſsis, quod ergo attinet ad præſentem <lb></lb>motum ratio eſt eadem. </s>
          <s id="id000888">Quicunque ergo motus <lb></lb>minoris grauis cogit deſcendere lancem ex ad­<lb></lb>uerſo proportionem habet eandem ad ſuum mo <lb></lb>bile quam habet graue æquiponderans. </s>
          <s id="id000889">Sit ergo <lb></lb>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb></lb>g, h, erit ergo ponderum h, g, f, e, ad ſe inuicem, &amp; ad a qualis mo­<lb></lb>tuum ob diſtantiam intentorum. </s>
          <s id="id000890">Experimentum ergo docet, quòd <lb></lb>dimidium ponderis æquilibrium facit ex palmo minoris dimidio <lb></lb>motum manifeſtum, &amp; ex palmo quarta pars ponderis, ergo ſe ha­<lb></lb>bent prope portionem.</s>
        </p>
        <p type="main">
          <s id="id000891">Propoſitio quinquageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id000892">Quę ex alto deſcendunt cur non eandem pro diſtantia motus ra<lb></lb>tionem in libero aëre ſeruent conſiderare.</s>
        </p>
        <pb pagenum="50" xlink:href="015/01/069.jpg"></pb>
        <p type="main">
          <s id="id000893">Aër in ſublimiore eius regione ſemper naturali motu fertur ex <lb></lb>Oriente in Occidentem, ſed &amp; infra uerum minus manifeſtè. </s>
          <s id="id000894">At ca­<lb></lb>ſu plerun que contingit, ut moueatur longè uehementius, ſeu ad ean­<lb></lb>dem partem, ſeu aliam. </s>
          <s id="id000895">Qui uerò naturalis eſt, debilis <lb></lb><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"></figure><lb></lb>eſt, quoniam in tenui ualde ſubſtantia eſt: nec <expan abbr="cõtinuus">continuus</expan> <lb></lb>ſed inſtar motus aquæ maris fluit ac refluit: aliter ne­<lb></lb>ceſſe eſſet, ut ſingulis horis per mille milliaria procede­<lb></lb>ret, ut ſic ne que latere poſſet, quandoquidem fortuiti mo<lb></lb>tus, qui ſunt multo tardiores non latent nos. </s>
          <s id="id000896">Nam tardiores illos <lb></lb>eſſe <expan abbr="cõſtat">conſtat</expan>, cum in hora ſint pulſus arteriarum, quatuor millia <expan abbr="ictuũ">ictuum</expan> <lb></lb>in homine prope temperamentum: ſi igitur motus naturalis aëris <lb></lb>eſſet continuus, in hora aër procederet ob ambitum terræ millies <lb></lb>mille paſſus, <expan abbr="igit̃">igitur</expan> in ictu pulſus ſuperaret paſſus 250. At experimur <lb></lb>nullum uentum aut procellam ſuperare quinquaginta paſſus, cum <lb></lb>etiam continuus eſſe nunquam ſoleat, imò ne poſsit quidem, itaque <lb></lb>cum hic multo tardior etiam in ſublimi, dum eſt, nos latere non <lb></lb>queat, multo minus poſſet naturalis latere, ſi adeò uelox &amp; in ea­<lb></lb>dem parte <expan abbr="aẽris">aeris</expan> eſſet at que continuus. </s>
          <s id="id000897">Præterea tantus impetus nun­<lb></lb>quam à minore motu, aut cauſa ſuperaretur, adeò ut ſemper flatum <lb></lb>aëris orientalem ſentiremus. </s>
          <s id="id000898">Quotidie etiam aduenire ad nos aë­<lb></lb>rem ex Illyrico, Macedonia, Myſia, Ponto, Bythínia, Capadocia, Sy <lb></lb>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to­<lb></lb>to præterea Oceano orientali tam uaſto, &amp; Gallica noua, terraque flo<lb></lb>rida non ſolum res eſt admirabilis&#039;, &amp; incredibilis, ſed etiam aliena <lb></lb>à ſenſu, &amp; ab his, quæ eueniunt. </s>
          <s id="id000899">A&#039;ſenſu quidem, quoniam nebulę, <lb></lb>quæ in aëre mouentur, primùm non in eandem partem ſemper mo<lb></lb>uentur: nun quam autem adeò celeriter: at ſi aër ſic circumuoluere­<lb></lb>tur, mouerentur &amp; illa, quę in eo continentur, quotidieque aërem ex­<lb></lb>periremur &amp; nubiloſum, &amp; madidum propter mare. </s>
          <s id="id000900">Nechis, quæ <lb></lb>eueniunt hoc ſatis reſpondet, nec nobis id contingeret, ut ſi peſti <lb></lb>aliqua in regione noſtra directa ſæuiret, ut aër ſingulis diebus la­<lb></lb>be ea infectus ad nos deferretur. </s>
          <s id="id000901">Moueri uerò aërem ſemper mani­<lb></lb>feſtiſsimum eſt tum experimento, tum ratione: ratione ſiquidem, <lb></lb>quod aqua &amp; cœlum naturaliter perpetuò mouentur, quare etiam <lb></lb>aër. </s>
          <s id="id000902">Experimento, quòd ubi hiant oſtia, &amp; ianuæ, ibi perpetuus ſen­<lb></lb>titur flatus. </s>
          <s id="id000903">Ergo ſi a pondus deſcendat in c, ex alto fertur rectà, ſed <lb></lb>ſi ex ſublimi transferetur in b, &amp; indirecta, &amp; ad latus, unde ex <lb></lb>hoc ſequitur.</s>
        </p>
        <p type="main">
          <s id="id000904">Propoſitio quin quageſima nona.</s>
        </p>
        <p type="main">
          <s id="id000905"><arrow.to.target n="marg156"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000906"><margin.target id="marg156"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id000907">Omne mobile motum duobus motibus non ad idem tendenti­<lb></lb>bus, utro que ſeorſum tardius mouetur ſimili motu.</s>
        </p>
        <pb pagenum="51" xlink:href="015/01/070.jpg"></pb>
        <p type="main">
          <s id="id000908">Sit a mobile, quod moueatur per a b c impulſu uenti aut uiolen­</s>
        </p>
        <p type="main">
          <s id="id000909"><arrow.to.target n="marg157"></arrow.to.target><lb></lb><figure id="id.015.01.070.1.jpg" xlink:href="015/01/070/1.jpg"></figure><lb></lb>to cum naturali coniuncto: &amp; ſit terminus naturalis e, <lb></lb><arrow.to.target n="marg158"></arrow.to.target><lb></lb>&amp; uiolenti d: uter que in directo c, dico, quod tardius per­<lb></lb>ueniet ad c quam d, uel e. </s>
          <s id="id000910">De e manifeſtum eſt, quoniam <lb></lb>motus aëris, qui intendit motum a, diuíditur in partem, <lb></lb>quæ iuuat motum ad d, &amp; partem, quæ mouetur ad e, <lb></lb>igitur fit minor adiectio. </s>
          <s id="id000911">Et etiam quia a c eſt longior <lb></lb>a e ex diffinitione rectæ: quare tardius perueniet ad c quàm ad e du<lb></lb>plici ratione. </s>
          <s id="id000912">Dico etiam, quod tardius ad c quàm d. </s>
          <s id="id000913">Quia enim <lb></lb>uis, quæ fert ad d repugnat ei, quæ fert ad e, &amp; uis, quæ fert ad e, re­<lb></lb>pugnat ei quæ fert ad d, igitur tardius perueniet ad c, quàm d. </s>
          <s id="id000914">Nec <lb></lb>potes dicere, quòd uis, quæ fert ad c adiuuet ad motum è regione <lb></lb>d, nam cum unus motus non poſsit perfici ſine altero, igitur quan­<lb></lb>tum motus ad e retardabit motum ad d, tanto motus a c erit tardí­<lb></lb>or abſolutè motu ad d. </s>
          <s id="id000915">Verum etiam eſt, quod c e breuior erit a d, <lb></lb>quia motus ad e ſemper contrahit motum ad d naturalis uiolen­<lb></lb>tum ob cauſam dictam. </s>
          <s id="id000916">Vtrùm uerò motus ad c abſolutè ſit tardi­<lb></lb>or, quàm ad d, non ſuppoſito, quod c e ſit æqualis a d, ſed minor, <lb></lb>nunc non eſt locus determinandi.</s>
        </p>
        <p type="margin">
          <s id="id000917"><margin.target id="marg157"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000918"><margin.target id="marg158"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>bu­ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000919">Ex hoc patet, quod motus æquidiſtantis mobilis, finis eſt mini­<lb></lb><arrow.to.target n="marg159"></arrow.to.target><lb></lb>mus omnium: quoniam mobile quaſi quieſcit in illo. </s>
          <s id="id000920">Velut ſi a mo<lb></lb>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb></lb>naturalis: nam cum incipiat, erit debiliſsimus, quia non <lb></lb><figure id="id.015.01.070.2.jpg" xlink:href="015/01/070/2.jpg"></figure><lb></lb>eſt motus actu: uiolentus autem æqualis eſt naturali, <lb></lb>dum minimus eſt: ergo cum ex diſtantia medij palmi <lb></lb>duplicetur, naturalis erit motus in b minimus, niſi b c <lb></lb><arrow.to.target n="marg160"></arrow.to.target><lb></lb>eſſet minor dimidio palmi. </s>
          <s id="id000921">Et etiam quòd eſſet minor, quia ut di­<lb></lb>ctum eſt, uter que ſimul iunctus eſt æqualis uni eorum non impedito <lb></lb>uel minor.</s>
        </p>
        <p type="margin">
          <s id="id000922"><margin.target id="marg159"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000923"><margin.target id="marg160"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 57. <emph type="italics"></emph>bu­ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000924">Propoſitio ſexageſima.</s>
        </p>
        <p type="main">
          <s id="id000925">Omne mobile motu naturali deſcendens parte, deſcendit gra­<lb></lb>uiore ſecundum grauitatis centrum.</s>
        </p>
        <p type="main">
          <s id="id000926">Sit a mobile, grauitatis centrum b, cuius pars ei pro­<lb></lb><arrow.to.target n="marg161"></arrow.to.target><lb></lb><figure id="id.015.01.070.3.jpg" xlink:href="015/01/070/3.jpg"></figure><lb></lb>ximior ſit c a, dico quod deſcendat motu naturali c a, <lb></lb>parte tangendo terram, quia enim totum a non poteſt <lb></lb>deſcendere ad centrum deſcendit b, quia eadem eſt na­<lb></lb>tura partis, &amp; totius: totius autem terræ natura eſt ut <lb></lb>centrum, totius ſit centrum grauitatis, quare b breuiore uia fertur <lb></lb><arrow.to.target n="marg162"></arrow.to.target><lb></lb>ad centrum, ergo per c d proximiorem partem ipſi b. </s>
          <s id="id000927">Sed pars pro­<lb></lb>ximior neceſſariò eſt grauior, quia centrum eſt in medio grauita­


<pb pagenum="52" xlink:href="015/01/071.jpg"></pb>tis, ergo omne mobile deſcendit motu naturali per ſui grauio­<lb></lb>rem partem.<lb></lb><arrow.to.target n="marg163"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000928"><margin.target id="marg161"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id000929"><margin.target id="marg162"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>bu­ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id000930"><margin.target id="marg163"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000931">Ex hoc ſequitur, quòd graue habens partes inæquales, ſeu ſub­<lb></lb>ſtantia, ſeu forma, ſi ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> ſit, infrà opor­<lb></lb>tet, ut circumuoluatur.</s>
        </p>
        <p type="main">
          <s id="id000932">Propoſitio ſexageſima prima.</s>
        </p>
        <p type="main">
          <s id="id000933">Proportionem ictus ad pondus rei, &amp; diſtantiam generaliter <lb></lb>conſiderare.</s>
        </p>
        <p type="main">
          <s id="id000934"><arrow.to.target n="marg164"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000935"><margin.target id="marg164"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000936">Dictum eſt ſuperius de proportione deſcenſus ad grauitatem: </s>
        </p>
        <p type="main">
          <s id="id000937"><arrow.to.target n="marg165"></arrow.to.target><lb></lb>&amp; quòd ſi graue deſcendat ex alto impeditur à motu aëris: &amp; quòd <lb></lb><arrow.to.target n="marg166"></arrow.to.target><lb></lb>res, quæ mouetur duobus motibus non ad idem tendentibus tar­<lb></lb><arrow.to.target n="marg167"></arrow.to.target><lb></lb>dius mouetur, quam motus ſit unuſquiſque. </s>
          <s id="id000938">Demùm quòd graue <lb></lb><arrow.to.target n="marg168"></arrow.to.target><lb></lb>deſcendens circumuoluitur, ſi pars grauior non ſit, deorſum: &amp; an­<lb></lb>tea ubi egimus de proportione motus ad grauitatem, quod hęc in­<lb></lb>telligenda ſunt pro ut poſſunt intelligi de motu etiam uiolento. <lb></lb></s>
          <s id="id000939">Cum ergo uideamus duo hæc, quod res acuta frangit caput, ſi ex <lb></lb>alto incidat, ſed non concutit, lata concutit, ſed non diuidit, premit <lb></lb>tamen carnem ſubiectam: nec hoc accidit merito ponderis: nam ut <lb></lb>uiſum eſt ſemilibra lapidis, uel ferri cadens ex alto contundit caput, <lb></lb>&amp; uulnerat, &amp; non eleuat in æquilibrio, ut potè ex alto cadens loco <lb></lb>per ſpatium octo palmorum pondus ſexdecim librarum, &amp; a pon­<lb></lb>dere ſexdecim librarum homo non læditur, nec uulneratur, ergo id <lb></lb>accidit ex alia cauſa, &amp; eſt, quod aër interceptus inter graue, &amp; cor­<lb></lb>pus noſtrum non poteſt dilabi tam citò, ergo ne corpus penetret, <lb></lb>cogitur ingredi locum, cui eſt obuius, at que ita concutere, &amp; diuide­<lb></lb>re. </s>
          <s id="id000940">Ex quibus ſequuntur omnia hæc.<lb></lb><arrow.to.target n="marg169"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000941"><margin.target id="marg165"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 57.</s>
        </p>
        <p type="margin">
          <s id="id000942"><margin.target id="marg166"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 58.</s>
        </p>
        <p type="margin">
          <s id="id000943"><margin.target id="marg167"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 59.</s>
        </p>
        <p type="margin">
          <s id="id000944"><margin.target id="marg168"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 60.</s>
        </p>
        <p type="margin">
          <s id="id000945"><margin.target id="marg169"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000946">Primùm ſi quod incidit, molle fuerit, non uulneratur caput, uel <lb></lb>pars ſubiecta, quia reſilit in corpus molle: nec à molli, quia retundi­<lb></lb>tur, poteſt uulnerari: ergo nullo modo. </s>
          <s id="id000947">Sed neque adeò concutit, <lb></lb>quia aër rediens, &amp; receptus in molli corpore pro parte, non uer­<lb></lb>berat locum.</s>
        </p>
        <p type="main">
          <s id="id000948"><arrow.to.target n="marg170"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000949"><margin.target id="marg170"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000950">Secundum in omni colliſione ſeu duri, ſeu mollis, ſed magis du­<lb></lb>ri, dilabuntur partes aëris ad latera, ideo quod partes mediæ pre­<lb></lb>muntur. </s>
          <s id="id000951">Et quanto motus eſt tardior.</s>
        </p>
        <p type="main">
          <s id="id000952"><arrow.to.target n="marg171"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000953"><margin.target id="marg171"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000954">Tertium in motu uelo ci fit maior ictus &amp; læſio, &amp; maiora omnia <lb></lb>quam pro proportione motus: quoniam ob uelo<expan abbr="citatẽ">citatem</expan> minus diffu<lb></lb>git aëris. </s>
          <s id="id000955">Et ideò fiunt grauia uulnera ex modico incremento uelo­<lb></lb>citatis motus.</s>
        </p>
        <p type="main">
          <s id="id000956"><arrow.to.target n="marg172"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000957"><margin.target id="marg172"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000958">Quartum res latæ, duræ concutiunt, &amp; non uulnerant niſi ſint <lb></lb>cum magno impetu, aut ualde graues: acutæ autem uulnerant, ſed <lb></lb>non concutiunt, niſi parti acutæ lata ſuccedat.</s>
        </p>
        <pb pagenum="53" xlink:href="015/01/072.jpg"></pb>
        <p type="main">
          <s id="id000959">Quintum, corpora dura magis læduntur à latis, quia ſcindun­</s>
        </p>
        <p type="main">
          <s id="id000960"><arrow.to.target n="marg173"></arrow.to.target><lb></lb>tur, mollia autem à tenuibus, quia diuiduntur: nam mollitie excipi­<lb></lb>unt aërem, &amp; ita à latis non adeò patiuntur, &amp; etiam, quoniam nec <lb></lb>franguntur, nec ſponte ſcinduntur.</s>
        </p>
        <p type="margin">
          <s id="id000961"><margin.target id="marg173"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000962">Sextum, etiam in duris penetrat aliquid aëris, aliter tota frange­<lb></lb><arrow.to.target n="marg174"></arrow.to.target><lb></lb>rentur. </s>
          <s id="id000963">Conſtat etiam omnem lapidem marmoreum, aut ſiliceum <lb></lb>eſſe poroſum, ut dicunt. </s>
          <s id="id000964">Et etiam quia recipitur in mollioribus, er­<lb></lb>go etiam in durioribus &amp; in duriſsimis: quod ſi non recipiant ut ui<lb></lb>trum, &amp; gemmæ tota franguntur. </s>
          <s id="id000965">Hoc etiam uidetur ſenſiſſe Philo<lb></lb>ſophus, qui uult, quòd res franguntur ob poros.</s>
        </p>
        <p type="margin">
          <s id="id000966"><margin.target id="marg174"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000967">Propoſitio ſexageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id000968">Proportionem motoris in plano ad motorem, qui eleuat pon­<lb></lb>dus iuxta id, quod mouet inuenire.</s>
        </p>
        <p type="main">
          <s id="id000969">Conſtitutum eſt inuenire proportionem uirium, quæ eleuant <lb></lb><arrow.to.target n="marg175"></arrow.to.target><lb></lb>pondus ad uires, quæ ipſum in plano leui trahere poſ­<lb></lb><figure id="id.015.01.072.1.jpg" xlink:href="015/01/072/1.jpg"></figure><lb></lb>ſunt. </s>
          <s id="id000970">Vires enim, quæ eleuant pondus a ſunt eædem <lb></lb>puta b, quæ uero trahunt c, ſed hæ poſſunt uariari, nam <lb></lb>quanto uinculum altius, aut decliuis locus magis, aut <lb></lb>aſpera ſuperficies ſeu ponderis ſeu plani, tanto difficilius trahitur, <lb></lb>&amp; maiores expoſcit uires: hoc enim experimento deprehenditur. <lb></lb></s>
          <s id="id000971">Duæ uerò poſtremæ cauſæ etiam per ſe perſpicuæ ſunt, nec demon <lb></lb>ſtratione indigent: niſi quod ſi planum ſit duriſsimum, ac leuiſsi­<lb></lb>mum, quod eſt aſperum facilius trahitur, quia minore ſui parte pla­<lb></lb>num tangit. </s>
          <s id="id000972">Nos præterea ſupponimus planum æquale undique <lb></lb>leue durum, &amp; corpus undique ſibi ſimile, id eſt cubi formam refe­<lb></lb>rens, &amp; uinculum in imo: Demonſtrare igitur expedit primum, <lb></lb>quòd in hoc caſu b eſt duplum ad c. </s>
          <s id="id000973">Quia enim cum a eleuatur b ui <lb></lb>res ſuperant motum obſcurum ſeu occultum, ſeu pondus a, &amp; ſi <lb></lb>permitteretur ſine eo, quod ſuſtineret, deſcenderet iuxta pondus <lb></lb>ſuum, quod ſit d: nititur ergo per pondus d, at quia trahendo duci­<lb></lb>tur circa medium, nam plana ſuperficies parum differt à rotunda <lb></lb>terræ ob terræ magnitudinem, media erit repugnantia: in eo enim <lb></lb>quod mouetur, grauitatem habet d in eo, quod <expan abbr="nõ">non</expan> remouetur nul­<lb></lb>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb></lb>ad d, ita c ad dimidium, grauitatis a, at b eſt primum, quod poteſt <lb></lb>mouere d, igitur c eſt primum, quod poteſt mouere dimidium a, ut <lb></lb>ergo dimidium a ad d, ita c ad b, eſt igitur c dimidium b.</s>
        </p>
        <p type="margin">
          <s id="id000974"><margin.target id="marg175"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000975">Propoſitio ſexageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id000976">Omne graue quanto proximius alligatum plano, tanto faci­<lb></lb>lius trahitur.


<pb pagenum="54" xlink:href="015/01/073.jpg"></pb><arrow.to.target n="marg176"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000977"><margin.target id="marg176"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000978">Sit graue a b c alligatum funibus in d ef, dico, <lb></lb><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"></figure><lb></lb>quòd facilius trahetur per fe quàm c b &amp; e b, quàm <lb></lb>d a, quia ſi debet trahi ex a uel b, aut cadet, aut uis ex <lb></lb>a &amp; b communicabitur c, igitur erit minor quàm in <lb></lb>c, &amp; hoc naturaliter. </s>
          <s id="id000979">Mathematica autem ratione quoniam ex a tra­<lb></lb>hetur c, quaſi per lineam d c: at attractio recta eſt ualidior obliqua <lb></lb>igitur attractio c per d eſt debilior, quàm per f. </s>
          <s id="id000980">Rurſus ſi e trahitur <lb></lb>per d cùm a peruenerit in d, erit perinde ac, ſi attractum eſſet per li­<lb></lb>neam c d, ſed linea c d mouet duobus motibus, uno ad ſuperiora, al </s>
        </p>
        <p type="main">
          <s id="id000981"><arrow.to.target n="marg177"></arrow.to.target><lb></lb>tero ad latus, ergo lentius ad f per d c quàm f c, quod erat demon­<lb></lb>ſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id000982"><margin.target id="marg177"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 59. <emph type="italics"></emph>bu­ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id000983">Propoſitio ſexageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id000984">Omne mobile quanto latius tanto tardius mouetur in plano.<lb></lb><arrow.to.target n="marg178"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000985"><margin.target id="marg178"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000986">Demonſtratum eſt ſuperius quòd ſi mobile ſit ſphęricum, &amp; tan</s>
        </p>
        <p type="main">
          <s id="id000987"><arrow.to.target n="marg179"></arrow.to.target><lb></lb>gat planum in puncto, quòd mouetur per quancunque uim aptam <lb></lb>diuidere medium. </s>
          <s id="id000988">Quia ergo ſi tangat in puncto facillime moue­<lb></lb>tur, ſi in linea paulò difficilius, ſi per ſuperficiem adhuc difficilius, <lb></lb>igitur cum fiat attritio in motu quanto latius eſt mobile eo diffici­<lb></lb>lius mouetur. </s>
          <s id="id000989">Sit ergo mobile a b, quod moueatur uerſus c, &amp; quia <lb></lb>pars b ſeu dimidium mouetur iuxta rationem me­<lb></lb><figure id="id.015.01.073.2.jpg" xlink:href="015/01/073/2.jpg"></figure><lb></lb>dietatis, &amp; pars a eodem modo, ergo conduplicata <lb></lb>difficultate, quia medietas b impedit medietatem, a <lb></lb>quanto latius eſt, &amp; longius a b, tanto difficilius <lb></lb><arrow.to.target n="marg180"></arrow.to.target><lb></lb>mouetur. </s>
          <s id="id000990">Et hoc intelligitur de corporibus ualde <lb></lb>latis propter dicta ſuperius.</s>
        </p>
        <p type="margin">
          <s id="id000991"><margin.target id="marg179"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 40.</s>
        </p>
        <p type="margin">
          <s id="id000992"><margin.target id="marg180"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 62</s>
        </p>
        <p type="main">
          <s id="id000993">Propoſitio ſexageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id000994">Proportionem duorum mobilium inter ſe cum auxilio medij <lb></lb>inuenire.<lb></lb><arrow.to.target n="marg181"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id000995"><margin.target id="marg181"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id000996">Graue deſcendit naturaliter quatuor cauſis: prima eſt ponderis <lb></lb>magnitudo, unde quod grauius eſt celerius deſcendit. </s>
          <s id="id000997">Secundò ob <lb></lb>paruam medij repugnantiam, ideo quanto medium eſt rarius &amp; <lb></lb>mobile tenuius, tanto celerius deſcendit: contrà uerò tardius. </s>
          <s id="id000998">Ter­<lb></lb>tiò ob impetum aëris ſub ſequentis: &amp; ideo mobile quòd ex eadem </s>
        </p>
        <p type="main">
          <s id="id000999"><arrow.to.target n="marg182"></arrow.to.target><lb></lb>materia conſtat, ſemper deſcendit parte acutiore ſuprapoſita, ne aër <lb></lb>cogatur celerius ferri: &amp; quanto diutius deſcendit, tanto magis in­<lb></lb>tenditur motus, at que augetur, ut ſuprà de claratum eſt. </s>
          <s id="id001000">Quarta cauſa <lb></lb>eſt, quod non impediatur ab aëre tranſuerſim moto, et à latere: ideo <lb></lb>leuia mobilia &amp; magna non ſolum lentius deſcendunt, quoniam <lb></lb><arrow.to.target n="marg183"></arrow.to.target><lb></lb>paruam uim habeant, &amp; magnam repugnantiam, ſed quia tranſuer<lb></lb><arrow.to.target n="marg184"></arrow.to.target><lb></lb>ſim impulſa minus mouentur motu recto, ut ſupra uiſum eſt. </s>
          <s id="id001001">Por­


<pb pagenum="55" xlink:href="015/01/074.jpg"></pb>rò proportio ratione deſcenſus aucta, declarata eſt paulo antè, <lb></lb>quare cum medium ſupponatur eiuſdem generis, &amp; figura non <lb></lb>eiuſmodi, nec leuitas, ut prorſus non impellat, nedum ut moueat la<lb></lb>tus: figura quo que eadem ambobus relinquetur proportio motus <lb></lb>ad motum producta ex proportionibus incrementi in proportio­<lb></lb><arrow.to.target n="marg185"></arrow.to.target><lb></lb>nem ponderum, &amp; iam habuimus proportionem incrementi ex <lb></lb><arrow.to.target n="marg186"></arrow.to.target><lb></lb>motu aëris, ergo proportio unius motus producti ad alteram no­<lb></lb>ta erit.</s>
        </p>
        <p type="margin">
          <s id="id001002"><margin.target id="marg182"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 30.</s>
        </p>
        <p type="margin">
          <s id="id001003"><margin.target id="marg183"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 59.</s>
        </p>
        <p type="margin">
          <s id="id001004"><margin.target id="marg184"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 62.</s>
        </p>
        <p type="margin">
          <s id="id001005"><margin.target id="marg185"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 42. <emph type="italics"></emph>ha­rum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001006"><margin.target id="marg186"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> 61. <emph type="italics"></emph>ha­rum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001007">Propoſitio ſexageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id001008">Proportionem laterum eptagoni, &amp; ſubtenſarum conſiderare, <lb></lb>&amp; quæ à reflexa proportione pendent.<lb></lb><arrow.to.target n="marg187"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001009"><margin.target id="marg187"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id001010">Sit eptagonus a b d f g e c, &amp; ſubtenſæ b <lb></lb><figure id="id.015.01.074.1.jpg" xlink:href="015/01/074/1.jpg"></figure><lb></lb>c, &amp; f e duobus lateribus, tribus autem d c <lb></lb>d e, &amp; erunt (quia intelligitur eptagono æ­<lb></lb>quilatero, &amp; æquiangulo) b c &amp; e f inuicem <lb></lb>æquales: &amp; item d c, &amp; d e æquales: &amp; ſi du­<lb></lb>cerentur b e &amp; c f inuicem æquales: &amp; ad a c <lb></lb>&amp; d g: quare cum angulus cb d conſiſtatin </s>
        </p>
        <p type="main">
          <s id="id001011"><arrow.to.target n="marg188"></arrow.to.target><lb></lb>arcu c e g f d, &amp; angulus b d c in arcu b a c, <lb></lb>&amp; angulus b c d in arcu b d; &amp; ſit arcus c e g <lb></lb>f d duplus arcus b a c, quia c e g f d ſubtendit quatuor latera epta­<lb></lb>goni, &amp; arcus b a c duo, &amp; ita arcus etiam b a c duplus arcui b d <lb></lb>erit angulus d b e duplus angulo c d b, &amp; angulus c d b duplus an­<lb></lb><arrow.to.target n="marg189"></arrow.to.target><lb></lb>gulo b c d, quare per demonſtrata à nobis proportio laterum b d, <lb></lb>b c, c d, eſt reflexa, igitur proportio d b &amp; b c, ad d c, ut d e ad b c, &amp; <lb></lb><arrow.to.target n="marg190"></arrow.to.target><lb></lb>rurſus proportio b d &amp; d e ad b e, ut b e ad b d. </s>
          <s id="id001012">Quare ſuppoſita <lb></lb>d b 1, b c 1 poſitione, erit d c latus 1 quad. </s>
          <s id="id001013">p: 1 poſitione. </s>
          <s id="id001014">Proportio <lb></lb><arrow.to.target n="marg191"></arrow.to.target><lb></lb>uerò, ut dictum eſt b d &amp; d c ad b c, id eſt p: &lt;02&gt; 1 quad. </s>
          <s id="id001015">p: 1 pos, ad 1 <lb></lb>pos eſt, ut b c ad b d, id eſt 1 pos ad 1, igitur 1 p: &lt;02&gt; v: 1 quad. </s>
          <s id="id001016">p: 1 pos <lb></lb>æquatur quadrato b c, quod eſt 1 quad. </s>
          <s id="id001017">igitur 1 quad. </s>
          <s id="id001018">m: 1 æquatur <lb></lb>&lt;02&gt; v: 1 quad. </s>
          <s id="id001019">p: 1 pos quare 1 quad. </s>
          <s id="id001020">quad. </s>
          <s id="id001021">m: 2, quad. </s>
          <s id="id001022">p: 1 æquatur 1 <lb></lb>quad. </s>
          <s id="id001023">p: 1 pos. </s>
          <s id="id001024">Additis igitur communiter quatuor quadratis fient <lb></lb>1 quad. </s>
          <s id="id001025">quad. </s>
          <s id="id001026">p: 2 quad. </s>
          <s id="id001027">p: 1 æqualia 5 quad. </s>
          <s id="id001028">p: 1 pos. </s>
          <s id="id001029">Et reducitur ad <lb></lb>1 cu. </s>
          <s id="id001030">æqualem 1 3/4 pos p: 7/8.</s>
        </p>
        <p type="margin">
          <s id="id001031"><margin.target id="marg188"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 28. &amp; 29. <emph type="italics"></emph>tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001032"><margin.target id="marg189"></margin.target>P<emph type="italics"></emph>er ult. </s>
          <s id="id001033">ſexti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001034"><margin.target id="marg190"></margin.target>D<emph type="italics"></emph>e<emph.end type="italics"></emph.end> S<emph type="italics"></emph>uh. lib.<emph.end type="italics"></emph.end> 16.</s>
        </p>
        <p type="margin">
          <s id="id001035"><margin.target id="marg191"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>diff.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001036">Aliter ſtante ſuppoſitione ut Ludouicus Ferrarius ex demon­<lb></lb>ſtratis à Ptolemæo quadratum b c, &amp; eſt 1 quad eſt æquale produ­<lb></lb>cto ex b d in c e, quod eſt 1, &amp; a b in d c, igitur detracto 1, produ­<lb></lb>cto b d in c e ex 1 quad. </s>
          <s id="id001037">quadrato c b, relinquitur productum ex <lb></lb>a b in c d 1 quad. </s>
          <s id="id001038">m: 1, ergo diuiſo co per a b, quæ eſt 1, relinquitur <lb></lb>c d 1 quad. </s>
          <s id="id001039">m: 1 huius uerò quadratum per <expan abbr="eadẽ">eadem</expan> demonſtrata à Pto­


<pb pagenum="56" xlink:href="015/01/075.jpg"></pb>lemæo, ęquale eſt rectangulis ex b c in de, &amp; b d in c e, igitur 1 quad. <lb></lb></s>
          <s id="id001040">quad. </s>
          <s id="id001041">m: 2 quad. </s>
          <s id="id001042">p: 1 eſt æquale 1 producto b d in c e, &amp; producto b <lb></lb>cin d e detracto 1 communi, relinquetur productum ex b c in d e 1 <lb></lb>quad. </s>
          <s id="id001043">quad. </s>
          <s id="id001044">m: 2 quad. </s>
          <s id="id001045">igitur diuiſo 1 quad. </s>
          <s id="id001046">quad. </s>
          <s id="id001047">m: 2 quad. </s>
          <s id="id001048">per 1 <lb></lb>pos, exit 1 cu. </s>
          <s id="id001049">m: 2 pos æqualia d e, &amp; d e eſt æqualis d c, ut ab initio <lb></lb>demonſtrauimus, &amp; d c fuit 1 quad. </s>
          <s id="id001050">m: 1, igitur 1 cu. </s>
          <s id="id001051">m: 2 æquantur 1 <lb></lb>quad. </s>
          <s id="id001052">m: 1, igitur 1 cu. </s>
          <s id="id001053">p: 1 æquantur 1 quad. </s>
          <s id="id001054">p: 2 pos.</s>
        </p>
        <p type="main">
          <s id="id001055">Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, &amp; du<lb></lb>cantur perpendiculares a f, d g &amp; c d, &amp; ſit c e i ca 1 pos, &amp; quia ut <lb></lb><arrow.to.target n="marg192"></arrow.to.target><lb></lb>a e ad a c, ita d e ad b c, erit ergo b c (1 posp: 1)/(1 pos) quare b f (1/2 pos 1/2,)/(2 pos) &amp; <lb></lb>quia d h eſt dimidium d e, erit d h, &amp; g f <lb></lb><figure id="id.015.01.075.1.jpg" xlink:href="015/01/075/1.jpg"></figure><lb></lb>1/2, cum ergo b f ſit (1/2 pos p: 1/2)/pos erit ergo di­<lb></lb>uiſa 1/2 pos per 1 pos, &amp; exit 1/2, b f 1/2p: 1/2/pos <lb></lb>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb></lb>&amp; eius quadratum 1/4/(1 quad). igitur cum qua­<lb></lb>dratum b d ſit 1, erit quadratum g d 1 m: <lb></lb>2/4/(2 quad)g c autem eſt compoſita ex e f, quæ <lb></lb>eſt 1/2p: 1/2/(1 pos) &amp; f g quæ eſt 1/2, erit igitur c <lb></lb>g 1 p: 1/2/(1 pos), &amp; <expan abbr="quadratũ">quadratum</expan> eius 1 p: 1/pos eſt 1/4/(1 quad.) quare <expan abbr="q̃dratũ">quadratum</expan> e d q̊d eſt <lb></lb><arrow.to.target n="marg193"></arrow.to.target><lb></lb>compoſitum ex quadratis c g &amp; g d erit 2 p: 1/pos c a uerò eſt æqua­<lb></lb>lis c d, quia, ut demonſtratum eſt angulus d c e eſt ſeptima pars <lb></lb>duorum rectorum, &amp; angulus b c e ei duplus, quare cum c f a ſit re­<lb></lb>ctus erit ex trigeſima ſecunda primi Elementorum f a c tres ſepti­<lb></lb>mæ unius recti, ergo d a c 6/7 unius recti, d c a uerò 2/7 unius recti, quia <lb></lb><arrow.to.target n="marg194"></arrow.to.target><lb></lb>eſt ſeptima pars duorum rectorum, ígitur a d c eſt 6/7 unius recti: igi­<lb></lb>tur c d eſt æqualis c a, ergo quadratum quadrato: igitur 1 quad. </s>
          <s id="id001056">p: 2 <lb></lb>pos p: 1, æquatur 2 p: 1/(1 pos) igitur 1 quad. </s>
          <s id="id001057">p: 2 pos, æquantur 1 p: 1/(1 pos). <lb></lb>Quare 1 cub. </s>
          <s id="id001058">p: 2 quad. </s>
          <s id="id001059">æquatur 1 pos p: 1. <lb></lb><figure id="id.015.01.075.2.jpg" xlink:href="015/01/075/2.jpg"></figure><lb></lb>Sit etiam angulus a duplus b, &amp; b c dupla <lb></lb>b a: &amp; erit per eadem proportio a c, &amp; a b <lb></lb>ad c b, ut c b ad c a. </s>
          <s id="id001060">Ponamus ergo ab 1, erit <lb></lb>b c 2, &amp; a c 1 pos, &amp; a c, a b 1 pos p: 1, &amp; du­<lb></lb>cta in a c fit 1 quad. </s>
          <s id="id001061">p: 1 pos, &amp; hoc eſt æquale 4 quadrato b c per re­<lb></lb>flexæ proportionis diffinitionem. </s>
          <s id="id001062">Igitur a c eſt &lt;02&gt; 4 1/4 m: 1/2, &amp; ita <lb></lb>de alijs.</s>
        </p>
        <p type="margin">
          <s id="id001063"><margin.target id="marg192"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 42. <emph type="italics"></emph>pri mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001064"><margin.target id="marg193"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>pri mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001065"><margin.target id="marg194"></margin.target>P<emph type="italics"></emph>er ſextam eiuſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001066">Propoſitio ſexageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id001067">Si fuerint aliquot quantitates ab una quantitate, aliæque totidem 


<pb pagenum="57" xlink:href="015/01/076.jpg"></pb>ab eadem analogæ, erit proportio tertiæ unius ordinis ad tertiam <lb></lb>alterius, ut ſecundæ ad ſecundam duplicata, &amp; quartæ ad quartam <lb></lb>triplicata, quintæ ad quintam quadruplicata, at que ſic de alijs.<lb></lb><arrow.to.target n="marg195"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001068"><margin.target id="marg195"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id001069">Sint quantitates b c d e f, ab a in continua proportio­<lb></lb><figure id="id.015.01.076.1.jpg" xlink:href="015/01/076/1.jpg"></figure><arrow.to.target n="table14"></arrow.to.target><lb></lb>ne, &amp; aliæ totidem g h k l m, dico quod proportio h c eſt <lb></lb>duplicata ei, quæ eſt g ad b, &amp; k ad d triplicata, &amp; l ad e <lb></lb>quadruplicata, &amp; ſic deinceps, ſumatur enim unum, &amp; ab </s>
        </p>
        <table>
          <table.target id="table14"></table.target>
          <row>
            <cell></cell>
            <cell>a</cell>
            <cell></cell>
          </row>
          <row>
            <cell>b</cell>
            <cell></cell>
            <cell>g</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell></cell>
            <cell>h</cell>
          </row>
          <row>
            <cell>d</cell>
            <cell></cell>
            <cell>k</cell>
          </row>
          <row>
            <cell>e</cell>
            <cell></cell>
            <cell>l</cell>
          </row>
          <row>
            <cell>f</cell>
            <cell></cell>
            <cell>m</cell>
          </row>
          <row>
            <cell></cell>
            <cell>n</cell>
            <cell></cell>
          </row>
          <row>
            <cell>o</cell>
            <cell></cell>
            <cell>t</cell>
          </row>
          <row>
            <cell>p</cell>
            <cell><foreign lang="grc">α</foreign></cell>
            <cell>u</cell>
          </row>
          <row>
            <cell>q</cell>
            <cell><foreign lang="grc">β γ</foreign></cell>
            <cell>x</cell>
          </row>
          <row>
            <cell>z</cell>
            <cell></cell>
            <cell>y</cell>
          </row>
          <row>
            <cell>s</cell>
            <cell></cell>
            <cell>z</cell>
          </row>
        </table>
        <p type="main">
          <s id="id001070"><arrow.to.target n="marg196"></arrow.to.target><lb></lb>eo o p q r s in proportione b ad a, &amp; t u x y z in propor­<lb></lb>tione g ad a, erit igitur p quadratum o, &amp; u quadratum t, <lb></lb>&amp; q cubus o, &amp; x cubus t, &amp; ita de alijs: ergo proportio <lb></lb><arrow.to.target n="marg197"></arrow.to.target><lb></lb>n ad p duplicata ei, quæ t ad o, &amp; x ad q triplicata ei, quæ t <lb></lb>ad o, &amp; poteſt etiam demonſtrari generaliter ultra qua­<lb></lb><arrow.to.target n="marg198"></arrow.to.target><lb></lb>dratum, &amp; cubum: nam ſi ducatur t in o, fiat que <foreign lang="grc">α</foreign> erit, pro­<lb></lb>portio enim ad <foreign lang="grc">α</foreign> eadem quæ t ad o, &amp; proportio a ad p, <lb></lb>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb></lb><arrow.to.target n="marg199"></arrow.to.target><lb></lb>poſitam in quinto libro ab Euclide u ad p duplicata ei, <lb></lb>quæ t ad o, &amp; ſimiliter ex t in p fit <foreign lang="grc">β</foreign> ex o in u, <foreign lang="grc">γ</foreign> eruntque<lb></lb><arrow.to.target n="marg200"></arrow.to.target><lb></lb>q <foreign lang="grc">β γ</foreign> x in continua proportione per eandem. </s>
          <s id="id001071">Quia ergo propor­<lb></lb>tio q ad <foreign lang="grc">β</foreign> eſt ut o ad t, patet, quod x ad q eſt triplicata ei, quæ eſt t ad <lb></lb>o, &amp; ita de reliquis, cum ergo proportio p ad o ſit, ut e ad b, &amp; o ad <lb></lb><arrow.to.target n="marg201"></arrow.to.target><lb></lb>n, ut b ad a, &amp; n ad t, ut a ad g, &amp; t ad u, ut g ad h, ſequitur ut ſit t ad a, <lb></lb>ut g ad b, &amp; u ad p, ut h ad c, igitur cum ſit ut u ad p duplicata ei, quę <lb></lb>eſt t ad o erit h ad e, duplicata ei quæ eſt g ad b, &amp; ita de reliquis, &amp; <lb></lb>noǹ refert, ſeu dicas u ad p duplicatam ei, quæ eſt t ad o, ſeu dicas p <lb></lb><arrow.to.target n="marg202"></arrow.to.target><lb></lb>ad u duplicatam ei, quæ eſt o ad t. </s>
          <s id="id001072">Aliter &amp; euidentius in duabus <lb></lb>ſoleo demonſtrare: cum enim ſit e &amp; h duplicata ei quæ eſt b &amp; g <lb></lb>ad a, ut ſupra, &amp; quadrati b ad quadratum a, &amp; quadrati g ad qua­<lb></lb><arrow.to.target n="marg203"></arrow.to.target><lb></lb>dratum a duplicata his quæ b &amp; g ad a erunt b &amp; g quadratorum <lb></lb>ad quadratum a, uelut c &amp; h ad a. </s>
          <s id="id001073">Et conuertendo qua­<lb></lb><arrow.to.target n="table15"></arrow.to.target><lb></lb>drati a ad quadratum g, ut a ad h, conſtituantur ergo <lb></lb><figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg"></figure>hic &amp; erit quadrati b ad <expan abbr="quadratũ">quadratum</expan> g, ita c ad h: ſed qua­<lb></lb>drati b ad quadratum g, ut b ad g proportio duplicata <lb></lb>igitur e ad h, ut b ad g duplicata.</s>
        </p>
        <p type="margin">
          <s id="id001074"><margin.target id="marg196"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>noni<emph.end type="italics"></emph.end> E<emph type="italics"></emph>le.<emph.end type="italics"></emph.end> &amp; 22. &amp; 23. <emph type="italics"></emph>octaui.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001075"><margin.target id="marg197"></margin.target>V<emph type="italics"></emph>ide per<emph.end type="italics"></emph.end> 23. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001076"><margin.target id="marg198"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>ſex ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end> &amp; 33. <emph type="italics"></emph>undeci­mi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001077"><margin.target id="marg199"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſe­ptimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001078"><margin.target id="marg200"></margin.target>D<emph type="italics"></emph>iff.<emph.end type="italics"></emph.end> 10.</s>
        </p>
        <p type="margin">
          <s id="id001079"><margin.target id="marg201"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001080"><margin.target id="marg202"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10 <emph type="italics"></emph>diff. </s>
          <s id="id001081">quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001082"><margin.target id="marg203"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ſex ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <table>
          <table.target id="table15"></table.target>
          <row>
            <cell><expan abbr="q̃d">quad</expan>.</cell>
            <cell>b</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell><expan abbr="q̃d">quad</expan>.</cell>
            <cell>a</cell>
            <cell>a</cell>
          </row>
          <row>
            <cell><expan abbr="q̃d">quad</expan>.</cell>
            <cell>g</cell>
            <cell>h</cell>
          </row>
        </table>
        <p type="main">
          <s id="id001083">Propoſitio ſexageſimaoctaua, collectorum ab Euclide <lb></lb>&amp; Archimede.</s>
        </p>
        <p type="main">
          <s id="id001084">Omnis cylindrus cono habenti baſim, &amp; altitudinem eandem <lb></lb><arrow.to.target n="marg204"></arrow.to.target><lb></lb>triplus eſt. </s>
          <s id="id001085">Omnis cylindrus ſphæræ habenti eundem magnum <lb></lb><arrow.to.target n="marg205"></arrow.to.target><lb></lb>circulum, &amp; altitudinem ſexquialter eſt. </s>
          <s id="id001086">Omnis ſphæra dupla eſt <lb></lb><arrow.to.target n="marg206"></arrow.to.target><lb></lb>cono, cuius baſis eſt eius circulus magnus, &amp; altitudo eadem, quæ <lb></lb>ſphæræ ipſius. </s>
          <s id="id001087">Omnis ſuperficies ſphæræ quadrupla eſt maiori <lb></lb><arrow.to.target n="marg207"></arrow.to.target><lb></lb>ſuo circulo. </s>
          <s id="id001088">Superficies portionis ſphæræ eſt æqualis circulo, cu <lb></lb><arrow.to.target n="marg208"></arrow.to.target>


<pb pagenum="58" xlink:href="015/01/077.jpg"></pb>ius ſemidiameter eſt linea ducta à uertice portionis ad finem illius.</s>
        </p>
        <p type="margin">
          <s id="id001089"><margin.target id="marg204"></margin.target>1</s>
        </p>
        <p type="margin">
          <s id="id001090"><margin.target id="marg205"></margin.target>2</s>
        </p>
        <p type="margin">
          <s id="id001091"><margin.target id="marg206"></margin.target>3</s>
        </p>
        <p type="margin">
          <s id="id001092"><margin.target id="marg207"></margin.target>4</s>
        </p>
        <p type="margin">
          <s id="id001093"><margin.target id="marg208"></margin.target>5</s>
        </p>
        <p type="main">
          <s id="id001094">Quilibet ſector ſphæræ æqualis eſt cono, cuius baſis eſt circu­<lb></lb>lus æqualis ſuperficiei eiuſdem portionis, altitudo uerò ſphæræ ſe­<lb></lb>midiameter. </s>
          <s id="id001095">Proportio ſphæræ ad ſectorem datum, eſt duplica­<lb></lb>ta ei, quę eſt dimetientis ad lineam, quæ à uertice portionis ad lim­<lb></lb>bum. </s>
          <s id="id001096">Cum enim ſphæra ſit æqualis cono, cuius baſis eſt maior cir­<lb></lb>culus, altitudo uerò dupla dimetienti per tertiam harum, quæ hic <lb></lb><arrow.to.target n="marg209"></arrow.to.target><lb></lb>proponuntur: erit ſphæra æqualis cono baſim habenti circulum, <lb></lb>cuius ſemidiameter ſit æqualis diametro ſphæræ, altitudo uerò ſe­<lb></lb>midiameter ſphæræ. </s>
          <s id="id001097">At per ſextam harum ſector ſphæræ eſt æqua­<lb></lb>lis cono habenti altitudinem ſemidiametrum ſphærę, baſim autem <lb></lb><arrow.to.target n="marg210"></arrow.to.target><lb></lb>ipſam portionis ſuperficiem: igitur proportio ſphæræ ad ſecto­<lb></lb>rem, uelut circuli cuius diameter eſt dupla dimetienti ſphæræ ad <lb></lb>círculum æqualem ſuperficiei portionis: at ſuperficies portionis <lb></lb>per quintam harum eſt æqualis circulo, cuius ſemidiameter eſt li­<lb></lb>nea à uertice portionis ad limbum eiuſdem: ergo proportio ſphæ­<lb></lb>ræ ad ſuum ſectorem eſt uelut circuli, cuius dimetiens eſt duplus di <lb></lb>metienti ſphæræ, aut ſemidimetiens eſt æqualis dimetienti ſphæræ <lb></lb>ad circulum, cuius ſemidimetiens eſt linea à uertice portionis ad <lb></lb>limbum. </s>
          <s id="id001098">Sed proportio talium circulorum eſt duplicata propor­<lb></lb><arrow.to.target n="marg211"></arrow.to.target><lb></lb>tioni ſemidimetientium, igitur proportio ſphæræ ad ſuum ſecto­<lb></lb>rem eſt ueluti dimetientis ſphæræ ad lineam, quæ á uertice portio­<lb></lb><arrow.to.target n="marg212"></arrow.to.target><lb></lb>nis ad limbum duplicata. </s>
          <s id="id001099">Cuicunque portioni ſphæræ conus ille <lb></lb>habetur æqualis, qui baſim habeat eandem cum portione, altitudi­<lb></lb>nem uerò lineam rectam, quæ ad altitudinem portionis eandem <lb></lb>habeat proportionem, quam ſemidiametros ſphæræ unà cum alti­<lb></lb>tudine reliquæ portionis habet ad eandem reliquæ portionis alti­<lb></lb><arrow.to.target n="marg213"></arrow.to.target><lb></lb>tudinem. </s>
          <s id="id001100">Earum ſphæræ portionum, quæ æqualibus ſuperfi­<lb></lb><arrow.to.target n="marg214"></arrow.to.target><lb></lb>ciebus continentur medietas ſphæræ maxima exiſtit. </s>
          <s id="id001101">Proportio <lb></lb>ſuperficiei ſphæræ plano diuiſæ ad reliquæ portionis ſuperficiem, <lb></lb>&amp; reſidui ſectoris ad ſectorem, eſt uelut quadratorum duarum li­<lb></lb>nearum quæ à uerticulis ſectionum ad communem ſuperficiem <lb></lb>plani portiones ſecantis deſcendunt: nam ſectorem ſphæræ, dico <lb></lb><arrow.to.target n="marg215"></arrow.to.target><lb></lb>corpus compoſitum ex portione, &amp; cono illo. </s>
          <s id="id001102">Ille idem etiam defi­<lb></lb>nit Ellipſim coni a cuti anguli ſectionem, quam dicit etiam fieri ſe­<lb></lb><arrow.to.target n="marg216"></arrow.to.target><lb></lb>cto cylindro per planum non ad angulos rectos ſtante ſuper cylin­<lb></lb>dri axem. </s>
          <s id="id001103">Ab hac igitur coni acuti anguli ſectione ſeu ellipſi cir­<lb></lb><arrow.to.target n="marg217"></arrow.to.target><lb></lb>cumacta figura ſphæroides corpus quod baſim rotundam habet, <lb></lb>uocat: id que duplex ob longum, quod fit diametro longiore quie­<lb></lb>ſcente, &amp; prolatum quod fit quieſcente breuiore: ſicut reliquam ſci <lb></lb>licet parabolen aut hyperbolen, quia inferius non eſt terminata, 


<pb pagenum="59" xlink:href="015/01/078.jpg"></pb>in cono rectangulo uocat rectanguli coni ſectionem: ex qua cir­<lb></lb>cumacta fit conoidale, quia planam habet baſim. </s>
          <s id="id001104">Si ergo in ea­<lb></lb><arrow.to.target n="marg218"></arrow.to.target><lb></lb>dem rectanguli coni ſectione à plano portiones æquales habentes <lb></lb>diametros abſcindantur, illæ portiones erunt æquales. </s>
          <s id="id001105">Et triangu­<lb></lb>li in eiſdem portionibus inſcripti æquales erunt. </s>
          <s id="id001106">Diametrum uo­<lb></lb>cat in <expan abbr="quacunqũe">quacunqune</expan> portione lineam, quæ omnes lineas baſi æquidi­<lb></lb>ſtantes per æqualia diuidit. </s>
          <s id="id001107">Omnis circuli cuius diameter eſt ma<lb></lb><arrow.to.target n="marg219"></arrow.to.target><lb></lb>ior diameter ellipſis proportio ad ellipſim eſt uelut directè diame­<lb></lb>tri ellipſis ad diametrum tranſuerſam. </s>
          <s id="id001108">Ex quo patet quod pro­<lb></lb><arrow.to.target n="marg220"></arrow.to.target><lb></lb>portio cuiuslibet circuli ad ellipſim eſt uelut quadrati ſuæ diame­<lb></lb>tri ad rectangulum recta, &amp; tranſuerſa diametro ellipſis compre­<lb></lb>henſum. </s>
          <s id="id001109">Ex hoc rurſus ſequitur quod ellipſis ad ellipſim, ut re­<lb></lb><arrow.to.target n="marg221"></arrow.to.target><lb></lb>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s>
        </p>
        <p type="margin">
          <s id="id001110"><margin.target id="marg209"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 14. &amp; 15. <emph type="italics"></emph>duodeci mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>le.<emph.end type="italics"></emph.end> E<emph type="italics"></emph>ucl.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001111"><margin.target id="marg210"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>duodecimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>le.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001112"><margin.target id="marg211"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>duodecimi<emph.end type="italics"></emph.end>, &amp; 20. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001113"><margin.target id="marg212"></margin.target>8</s>
        </p>
        <p type="margin">
          <s id="id001114"><margin.target id="marg213"></margin.target>9</s>
        </p>
        <p type="margin">
          <s id="id001115"><margin.target id="marg214"></margin.target>10</s>
        </p>
        <p type="margin">
          <s id="id001116"><margin.target id="marg215"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 22. <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001117"><margin.target id="marg216"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ſex ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001118"><margin.target id="marg217"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001119"><margin.target id="marg218"></margin.target>11</s>
        </p>
        <p type="margin">
          <s id="id001120"><margin.target id="marg219"></margin.target>12</s>
        </p>
        <p type="margin">
          <s id="id001121"><margin.target id="marg220"></margin.target>13</s>
        </p>
        <p type="margin">
          <s id="id001122"><margin.target id="marg221"></margin.target>14</s>
        </p>
        <p type="main">
          <s id="id001123">Si conoides &amp; ſphæroides ſecet plano æquidiſtanti axi fiet ſe­<lb></lb><arrow.to.target n="marg222"></arrow.to.target><lb></lb>ctio conoidalis ſimilis ei à qua conoides ſeu ſphæroides deſcri­<lb></lb>ptum eſt. </s>
          <s id="id001124">Sin autem ſupra axem plano ad perpendiculum erecto <lb></lb>ſectio circulus erit. </s>
          <s id="id001125">Et ſi ſecentur obliquè fiet ellipſis, modo omnia <lb></lb>latera comprehendat. </s>
          <s id="id001126">Omnis portio conoidalis rectanguli, quam <lb></lb><arrow.to.target n="marg223"></arrow.to.target><lb></lb>planum ſecat, ſexquialtera eſt, cono qui baſim &amp; axem eandem ha­<lb></lb>bet. </s>
          <s id="id001127">Ex quo patet, quod ſi portio conoidalis rectanguli &amp; ſphæ­<lb></lb><arrow.to.target n="marg224"></arrow.to.target><lb></lb>ræ medietas eandem baſim habeant &amp; axem eundem, medietas <lb></lb>ſphæræ ſexquitertia erit conoidali portioni. </s>
          <s id="id001128">Et ſi eiuſdem rectan<lb></lb><arrow.to.target n="marg225"></arrow.to.target><lb></lb>guli conoidalis portiones abſcin dantur erit portionum propor­<lb></lb>tio uelut quadratorum axium. </s>
          <s id="id001129">Cuiuslibet ſphæroidis pars pla­<lb></lb><arrow.to.target n="marg226"></arrow.to.target><lb></lb>no per centrum abſciſſa dupla eſt cono baſim &amp; axem eadem ha­<lb></lb>benti. </s>
          <s id="id001130">Si autem non ſuper centrum erit proportio earum ad co­<lb></lb><arrow.to.target n="marg227"></arrow.to.target><lb></lb>num baſim, &amp; axem eandem habentem uelut coniunctæ ex axe al­<lb></lb>terius partis &amp; dimidio axis ſphæroidis ad axem alterius partis.</s>
        </p>
        <p type="margin">
          <s id="id001131"><margin.target id="marg222"></margin.target>15</s>
        </p>
        <p type="margin">
          <s id="id001132"><margin.target id="marg223"></margin.target>16</s>
        </p>
        <p type="margin">
          <s id="id001133"><margin.target id="marg224"></margin.target>17</s>
        </p>
        <p type="margin">
          <s id="id001134"><margin.target id="marg225"></margin.target>18</s>
        </p>
        <p type="margin">
          <s id="id001135"><margin.target id="marg226"></margin.target>19</s>
        </p>
        <p type="margin">
          <s id="id001136"><margin.target id="marg227"></margin.target>20</s>
        </p>
        <p type="main">
          <s id="id001137">Demum proportio partis conoidis obtuſi anguli plano abſciſ­<lb></lb><arrow.to.target n="marg228"></arrow.to.target><lb></lb>ſæ ad conum, baſim &amp; axem eadem habentem eſt ueluti lineæ, com<lb></lb>poſitæ ex axe portionis &amp; triplo adiectæ ad compoſitum ex axe <lb></lb>portionis &amp; duplo eiuſdem adiectæ. </s>
          <s id="id001138">Adiectam uocat hyperbolis <lb></lb>tranſuerſam. </s>
          <s id="id001139">Omnis cylindrus cono triplus eſt habenti eandem <lb></lb><arrow.to.target n="marg229"></arrow.to.target><lb></lb>baſim &amp; altitudinem. </s>
          <s id="id001140">Omnes cylindri coni ſphæræ ſunt in pro­<lb></lb><arrow.to.target n="marg230"></arrow.to.target><lb></lb>portione corporum ſimilium planis ſuperficiebus contentarum.</s>
        </p>
        <p type="margin">
          <s id="id001141"><margin.target id="marg228"></margin.target>21</s>
        </p>
        <p type="margin">
          <s id="id001142"><margin.target id="marg229"></margin.target>22</s>
        </p>
        <p type="margin">
          <s id="id001143"><margin.target id="marg230"></margin.target>23</s>
        </p>
        <p type="main">
          <s id="id001144">Propoſitio ſexageſima nona, collectorum ex quatuor libris <lb></lb>Apollonij Pergei &amp; <expan abbr="q.">que</expan> Sereni.</s>
        </p>
        <p type="main">
          <s id="id001145">Si fuerit linea bifariam diuiſa, eique in longum alia addita, &amp; rur­<lb></lb><arrow.to.target n="marg231"></arrow.to.target><lb></lb>ſus alia detracta, fueritque totius cum addita ad eam, quæ addita eſt <lb></lb>ueluti reſidui ad detractam erit lineæ com­<lb></lb><figure id="id.015.01.078.1.jpg" xlink:href="015/01/078/1.jpg"></figure><lb></lb>poſitæ ex addita, &amp; dimidia ad dimidiam 


<pb pagenum="60" xlink:href="015/01/079.jpg"></pb>ipſam uelut dimidiæ ad differentiam eius, &amp; detractæ. </s>
          <s id="id001146">Rurſusque li­<lb></lb>neæ compoſitæ ex dimidio &amp; reſiduo dimidiæ ac detractæ ad li­<lb></lb>neam compoſitam ex addita &amp; detracta ut reſidui dimidiæ, &amp; de­<lb></lb>tractæ ad partem detractam. </s>
          <s id="id001147">Et rurſus totius compoſitæ ad com­<lb></lb>poſitam ex dimidia &amp; addita, uelut compoſitæ ex addita, &amp; diffe­<lb></lb>rentia ad ipſam additam. </s>
          <s id="id001148">Velut ſit propoſita a b per æqualia diuiſa <lb></lb>in c, addita b d, &amp; detracta b e, ſit proportio a d ad d b, ut a e ad e b, <lb></lb>dico eſſe, ut c d ad cb, ita ab ad c e. </s>
          <s id="id001149">Et ut a e ad e d ut c e ad e b. </s>
          <s id="id001150">Et ite­<lb></lb><arrow.to.target n="marg232"></arrow.to.target><lb></lb>rum ut a d ad c d uelut e d ad d b. </s>
          <s id="id001151">In parabole proportio partium <lb></lb>diametri ad uerticem terminantium duplicata eſt proportioni li­<lb></lb>nearum ab eiſdem punctis ordinatim ductarum ad ipſam ſectio­<lb></lb><arrow.to.target n="marg233"></arrow.to.target><lb></lb>nem. </s>
          <s id="id001152">In hyperbole autem &amp; ellipſi &amp; circuli circumferentia erit <lb></lb>quadratorum linearum ordinatim ductarum inter ſe uelut rectan­<lb></lb><arrow.to.target n="marg234"></arrow.to.target><lb></lb>gulorum partium diametri ad eadem puncta terminantium. </s>
          <s id="id001153">Et in <lb></lb>eiſdem ſi à puncto peripheriæ contingens ad diametrum ducatur, <lb></lb>&amp; ab eodem ordinata, erit ut partis diametri interceptę inter extre­<lb></lb>mum, &amp; ordinatam ad partem inter ordinatam &amp; peripheriam, ue­<lb></lb>lut interceptæ inter extremum &amp; contingentem ad interceptam <lb></lb><arrow.to.target n="marg235"></arrow.to.target><lb></lb>exterius inter finem contingentis &amp; peripheriam. </s>
          <s id="id001154">Et in eiſdem <lb></lb>quadratum ſemidiametri æquale eſſe rectangulo ex intercepta in­<lb></lb>ter centrum &amp; caſum contingentis in interceptam inter centrum &amp; <lb></lb><arrow.to.target n="marg236"></arrow.to.target><lb></lb>caſum ordinatæ à loco contactus productæ. </s>
          <s id="id001155">Si parabolen recta <lb></lb>linea contingens ad diametrum perueniat, ſumptoque puncto alio <lb></lb>in ſectione æquidiſtans ab eo ducatur contingenti: &amp; ab utroque <lb></lb>etiam ad diametrum ordinatæ, demum à uertice æquidiſtans illis, <lb></lb>&amp; à priore puncto diametro æquidiſtans donec concurrant, erit <lb></lb>triangulus ex ordinata, &amp; æquidiſtante à ſecundo puncto, &amp; dia­<lb></lb>metri parte contentus rectangulo ex prima ordinata &amp; parte dia­<lb></lb>metri inter uerticem &amp; ſecundam ordinatam contento æqualis.<lb></lb><arrow.to.target n="marg237"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001156"><margin.target id="marg231"></margin.target>1</s>
        </p>
        <p type="margin">
          <s id="id001157"><margin.target id="marg232"></margin.target>2</s>
        </p>
        <p type="margin">
          <s id="id001158"><margin.target id="marg233"></margin.target>3</s>
        </p>
        <p type="margin">
          <s id="id001159"><margin.target id="marg234"></margin.target>4</s>
        </p>
        <p type="margin">
          <s id="id001160"><margin.target id="marg235"></margin.target>5</s>
        </p>
        <p type="margin">
          <s id="id001161"><margin.target id="marg236"></margin.target>6</s>
        </p>
        <p type="margin">
          <s id="id001162"><margin.target id="marg237"></margin.target>7</s>
        </p>
        <p type="main">
          <s id="id001163">Si in parabole contingente ad diametrum ducta ex alio puncto <lb></lb>ei æquidiſtans ducatur ex ipſa ſectione, ubi iterum ſecat ſectionem <lb></lb>intercepta per æqualia diuidetur linea à puncto contingentis dia­</s>
        </p>
        <p type="main">
          <s id="id001164"><arrow.to.target n="marg238"></arrow.to.target><lb></lb>metro æquidiſtanti ducta. </s>
          <s id="id001165">Idem uerò fermè continget ducta li­<lb></lb>nea à centro in locum contactus, ſecabit enim omnes contingenti <lb></lb><arrow.to.target n="marg239"></arrow.to.target><lb></lb>æquidiſtantes in hyperbole, ellipſi at que circulo. </s>
          <s id="id001166">Eſt autem omne <lb></lb>centrum in medio diametri: diameter autem in circulo &amp; ellipſi il­<lb></lb>las per æqualia diuidit intus enim eſt: in contrapoſitis inter uerti­<lb></lb>cem, &amp; uerticem poſita eſt exterius utriuſque contingenti ad per­<lb></lb>pendiculum inſiſtens. </s>
          <s id="id001167">In hyperbole autem exterius etiam adiacet, <lb></lb>ut in contrapoſitis eadem &amp; tranſuerſa uocatur: cuius terminus eſt <lb></lb>punctus concurſus cum latere trianguli, qui conum per axem diui­


<pb pagenum="61" xlink:href="015/01/080.jpg"></pb>dit: linea uerò tangens uerticem hyperbolis ad quam ordinatæ <lb></lb><arrow.to.target n="marg240"></arrow.to.target><lb></lb>poſſunt, Recta appellabitur. </s>
          <s id="id001168">Data recta linea poſitione, aliaque ma<lb></lb>gnitudine data &amp; angülo parabolen, &amp; hyperbolen, &amp; ellipſim, <lb></lb>&amp; contra poſitas circa datam poſitione tanquàm diametrum de­<lb></lb>ſcribere tanquàm cono erecto, ut angulus ad uerticem ſectionis <lb></lb>comprehenſus ſit, &amp; per rectam rectangulum æquale comprehen­<lb></lb>datur quadrato datæ lineæ magnitudine. </s>
          <s id="id001169">Si linea in duas partes <lb></lb><arrow.to.target n="marg241"></arrow.to.target><lb></lb>diuidatur, eique utrinque æquales lineæ adiun­<lb></lb><figure id="id.015.01.080.1.jpg" xlink:href="015/01/080/1.jpg"></figure><lb></lb>gantur erit rectangulum ex partibus totius æ­<lb></lb>quale rectangulis partium prioris lineæ, &amp; ex <lb></lb>priore linea cum una adiecta in eam, quæ adiecta eſt. </s>
          <s id="id001170">Si hyperbo<lb></lb><arrow.to.target n="marg242"></arrow.to.target><lb></lb>len recta linea in uertice contingat, &amp; utrinque abſcindatur, quan­<lb></lb>tum eſt, quod poteſt in quartam partem rectanguli ex diametro <lb></lb>tranſuerſa hyperbolis, quæ exterius adiacetin eam, quæ recta dici­<lb></lb>tur, ad quam, quæ ordinatim ducuntur, ſunt æquidiſtantes lineæ, <lb></lb>quæ à ſectionis centro ad terminos contingentis ducuntur ſemper <lb></lb>ipſi ſectioni magis appropinquabunt, nec unquam conuenient: &amp; <lb></lb>ob id aſymptoton appellantur. </s>
          <s id="id001171">Nec ullæ aliæ intra <expan abbr="angulũ">angulum</expan> illum <lb></lb><arrow.to.target n="marg243"></arrow.to.target><lb></lb>inueniri poterunt. </s>
          <s id="id001172">Vnde etiam intra <expan abbr="datũ">datum</expan> angulum deſcribere do­<lb></lb>cemur hyperbolen cuius anguli latera ſint aſymptota. </s>
          <s id="id001173">Aſymptotis <lb></lb><arrow.to.target n="marg244"></arrow.to.target><lb></lb>duabus propoſitis uni hyperboli, in finitas alías eidem aſymptotas <lb></lb>inuenire. </s>
          <s id="id001174">Duabus rectis aſymptotis infinitas ſubijci poſſe hyperbo<lb></lb>les illis rectis, &amp; inter ſe aſymptotas. </s>
          <s id="id001175">Cum in duabus ſuperficie­<lb></lb><arrow.to.target n="marg245"></arrow.to.target><lb></lb>bus æquidiſtantibus duo circuli æquales, quorum linea per cen­<lb></lb>tra non eſt ad perpendiculum earum infinitis planis ſecantur, fiunt <lb></lb>in ipſis lineæ à peripheria in peripheriam rectæ quæ corpus cylin­<lb></lb>dricum claudunt quod ſcalenus cylindrus appellatur: longè alius <lb></lb>ab eo, qui fit recto cylindro per duo plana æquidiſtantia, ſed non <lb></lb>ad perpendiculum poſita diſſecto. </s>
          <s id="id001176">nam eius extremæ ſuperficies <lb></lb>non circuli, ſed ellipſes ſunt. </s>
          <s id="id001177">Si ſcalenus cylindrus plano non æ­<lb></lb><arrow.to.target n="marg246"></arrow.to.target><lb></lb>quidiſtanti baſi, ſed ita ut angulos interiores æquales faciat angu­<lb></lb>lis baſis ſectio circulus erit: uocaturque hæc ſectio ſub contraria: nec <lb></lb>ulla præter hanc &amp; baſi æquidiſtantem ſectio circulus eſſe poteſt: <lb></lb>ſed ſunt ellipſes. </s>
          <s id="id001178">Super eundem circulum, &amp; ſub eadem altitudi­<lb></lb><arrow.to.target n="marg247"></arrow.to.target><lb></lb>ne ellipſes ſimiles in cono &amp; cylindro eſſe poſſunt, quæ ab eodem <lb></lb>plano fiant, docetque uel baſi uel cono uel cylindro, aut cono pro­<lb></lb>poſito reliqua facere, quod eſt ualde admirabile: cum ellipſis cylin­<lb></lb>drica ſemper æqualis ſit in utraque parte à diametro tranſuerſa <lb></lb>utrinque æqualiter diſtante, conica uerò minor neceſſariò ſit in ſu­<lb></lb>periore parte uerſus coni uerticem latior in inferiore, ubi partes a <lb></lb>diametro tranſuerſa æqualiter diſteterint: ipſę autem non ſolum ſi­


<pb pagenum="62" xlink:href="015/01/081.jpg"></pb><arrow.to.target n="marg248"></arrow.to.target><lb></lb>miles, ſed unam perſæpe in utriſ que eſſe uult. </s>
          <s id="id001179">Sed &amp; hoc Archime­<lb></lb>des dicere uidetur: lineæ ductæ à uertice coniſcaleni ad perpendi­<lb></lb>culum ſuper baſes ſingulas omnium triangulorum per axem coni <lb></lb>tranſeuntium in peripheriam unius circuli cadunt.</s>
        </p>
        <p type="margin">
          <s id="id001180"><margin.target id="marg238"></margin.target>8</s>
        </p>
        <p type="margin">
          <s id="id001181"><margin.target id="marg239"></margin.target>9</s>
        </p>
        <p type="margin">
          <s id="id001182"><margin.target id="marg240"></margin.target>10</s>
        </p>
        <p type="margin">
          <s id="id001183"><margin.target id="marg241"></margin.target>11</s>
        </p>
        <p type="margin">
          <s id="id001184"><margin.target id="marg242"></margin.target>12</s>
        </p>
        <p type="margin">
          <s id="id001185"><margin.target id="marg243"></margin.target>13</s>
        </p>
        <p type="margin">
          <s id="id001186"><margin.target id="marg244"></margin.target>14</s>
        </p>
        <p type="margin">
          <s id="id001187"><margin.target id="marg245"></margin.target>15</s>
        </p>
        <p type="margin">
          <s id="id001188"><margin.target id="marg246"></margin.target>16</s>
        </p>
        <p type="margin">
          <s id="id001189"><margin.target id="marg247"></margin.target>17</s>
        </p>
        <p type="margin">
          <s id="id001190"><margin.target id="marg248"></margin.target>18</s>
        </p>
        <p type="main">
          <s id="id001191">Propoſitio ſeptuageſima.</s>
        </p>
        <p type="main">
          <s id="id001192">Si fuerint tres quantitates in continua proportione, aliæque toti­<lb></lb>dem in continua proportione, poterunt conſtituere tres quantita­<lb></lb>tes in æquali differentia peruerſim copulatæ.<lb></lb><arrow.to.target n="marg249"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001193"><margin.target id="marg249"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id001194">Velut ſint a b c primi ordi­<lb></lb><figure id="id.015.01.081.1.jpg" xlink:href="015/01/081/1.jpg"></figure><lb></lb>nis, &amp; d ef ſecundi, &amp; ſit 28, </s>
        </p>
        <p type="main">
          <s id="id001195"><arrow.to.target n="marg250"></arrow.to.target><lb></lb>b 4, c 2, &amp; d 2 1/4, e 1 1/2, f 1, tunc <lb></lb>iunctis a &amp; e fit 9 1/2, &amp; b &amp; d b <lb></lb>1/4, &amp; e cum f 3, at 3 &amp; 6 1/4 &amp; 9 1/2 <lb></lb>æqualiter diſtant, nam diffe­<lb></lb>rentia eſt 3 1/4. At ſi iungatur <lb></lb>cum e, &amp; b cum f, &amp; c cum d <lb></lb>idem poterit contingere: ut in <lb></lb>figura uides, nam a e eſt 8 1/2, <lb></lb>p: &lt;02&gt; 1 1/4, &amp; b f 7, &amp; c d 5 1/2, m: &lt;02&gt; 1 1/4, &amp; differentia b f ab utro que com­<lb></lb>poſito, eſt 1 1/2 p: &lt;02&gt; 1 1/4, qua excedit &amp; exceditur. </s>
          <s id="id001196">Dico modo, quaſi <lb></lb>ex ordine coniungantur qualeſcunque proportiones fuerint, modo <lb></lb>non ſint ambæ æqualitatis 1, ut b iungatur cum c, &amp; reliquæ ut li­<lb></lb>bet, uelut a cum d, &amp; c cum f, uel a cum f, &amp; e cum d, nunquam fient <lb></lb><arrow.to.target n="marg251"></arrow.to.target><lb></lb>æquales exceſſus, nam de primo eſt clarum: nam ſi a cum d iun­<lb></lb>gatur, &amp; ambæ fuerint maximæ, maior eſt differentia a ad b, quàm <lb></lb>b ad c, &amp; maior etiam d ad e quàm e ad f, ideo maior erit differentia <lb></lb>a &amp; d ad b e quàm b e ad c f, quod erat probandum. </s>
          <s id="id001197">Eodem modo <lb></lb>ſed laborioſius demonſtratur reliquus modus ſcilicet, quod con­<lb></lb>iunctio a f ad b e eſt maior aut minor quàm b e ad c d, ex hoc ſe­<lb></lb>quuntur corrolaria.</s>
        </p>
        <p type="margin">
          <s id="id001198"><margin.target id="marg250"></margin.target>16</s>
        </p>
        <p type="margin">
          <s id="id001199"><margin.target id="marg251"></margin.target>17</s>
        </p>
        <p type="main">
          <s id="id001200">Primum, tres æquales quantitates non poſſunt diuidi in tres, &amp; <lb></lb>tres quantitates in continua proportione ordinatè, ut dixi, niſi u­<lb></lb>triuſque ordinis tres, ac tres inuicem ſint æquales.</s>
        </p>
        <p type="main">
          <s id="id001201">Secundum, tres quantitates in æquali exceſſu ordinate, ut dixi, <lb></lb>non poſſunt diuidi in tres, &amp; tres quantitates, quæ ſint in eadem <lb></lb>proportione quantumcunque proportiones illæ duorum ordinum <lb></lb>fint diuerſæ.</s>
        </p>
        <p type="main">
          <s id="id001202">Tertium, tres quantitates, quæ ſint in eadem proportione non <lb></lb>poſſunt diuidi ordinate in tres ac tres, quæ ſint in continua propor<lb></lb>tione niſi ſint ambæ proportiones eædem cum proportione ipſa­<lb></lb>rum quantitatum.</s>
        </p>
        <pb pagenum="63" xlink:href="015/01/082.jpg"></pb>
        <p type="main">
          <s id="id001203">Propoſitio ſeptuageſima prima.</s>
        </p>
        <p type="main">
          <s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra­<lb></lb>cti ad rectam ſuſpenſionem inuenire.</s>
        </p>
        <figure id="id.015.01.082.1.jpg" xlink:href="015/01/082/1.jpg"></figure>
        <p type="main">
          <s id="id001205">Sit torcularis uirga, cuius ſpiræ a b per circui­<lb></lb><arrow.to.target n="marg252"></arrow.to.target><lb></lb>tum ſint centuplæ ad altitudinem a b, &amp; axis d c <lb></lb><arrow.to.target n="marg253"></arrow.to.target><lb></lb>ſemidiametro b c centupla, &amp; quoniam per ſupe­<lb></lb>rius aſſumpta, qualis eſt proportio ſpatij ad ſpa­<lb></lb>tium, talis leuitatis ad <expan abbr="leuitatẽ">leuitatem</expan>, <expan abbr="igit̃">igitur</expan> e pondus aſcen<lb></lb>dens per a b leuius quam per b <expan abbr="crectã">c rectam</expan> centuplo, et <lb></lb>ſimiliter cum circuitus b c, &amp; d c ſint in eodem tem<lb></lb>pore, &amp; circuitus d c, ſit centuplus ad ſpiralem b c <lb></lb>per demonſtrata ab Euclide, ergo e erit centuplo <lb></lb>leuius circum ductum per d quàm b, ſed per b circumductum cen­<lb></lb>tuplo leuius eſt, quàm per rectam, igitur e ponderat ſolum particu­<lb></lb>lam ex decem millibus recti ponderis.</s>
        </p>
        <p type="margin">
          <s id="id001206"><margin.target id="marg252"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="margin">
          <s id="id001207"><margin.target id="marg253"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 45.</s>
        </p>
        <p type="main">
          <s id="id001208">Propoſitio ſeptuageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id001209">Proportionem ponderis ſphęræ pendentis ad aſcendentem per <lb></lb>accliue planum inuenire</s>
        </p>
        <figure id="id.015.01.082.2.jpg" xlink:href="015/01/082/2.jpg"></figure>
        <p type="main">
          <s id="id001210">Sit ſphæra æqualis ponderi g in pun­<lb></lb><arrow.to.target n="marg254"></arrow.to.target><lb></lb>cto b, quæ debeat trahi ſuper b c accli­<lb></lb>ue planum b e ad perpendiculum pla­<lb></lb><arrow.to.target n="marg255"></arrow.to.target><lb></lb>ni b f. </s>
          <s id="id001211">Quia ergo in b e mouetur a, qua­<lb></lb>uis modica ui per dicta ſuperius, erit per <lb></lb>communem animi ſententiam uis, quæ <lb></lb>mouebit a per e b nulla: per dicta uerò <lb></lb>a mouebitur ad f ſemper, a conſtanti ui <lb></lb>æquali g, &amp; per b c a conſtanti ui æqua­<lb></lb>li k, ſicut per b d a conſtanti æquali h, ergo per ultimam petitio­<lb></lb>nem, cum termini ſeruent, quo ad partes eandem rationem ſin­<lb></lb>guli per ſe, &amp; motus per b e ſit a nulla ui, erit proportio g ad k, ue­<lb></lb>lut proportio uis, quæ mouet per b f ad uim, quæ mouet per <lb></lb>b c, &amp; uelut anguli per e b f recti ad angulum e b c, &amp; ita uis, <lb></lb>quæ mouet a per b f, &amp; eſt, ut dictum eſt, g ad uim, quæ mouet <lb></lb>per b d, &amp; eſt h ex ſuppoſito, ut c b f ad e b d, igitur proportio dif­<lb></lb>ficultatis motus a per b d ad idem a per b c, eſt uelut h ad k, quod <lb></lb>erat demonſtrandum.</s>
        </p>
        <pb pagenum="64" xlink:href="015/01/083.jpg"></pb>
        <p type="margin">
          <s id="id001212"><margin.target id="marg254"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="margin">
          <s id="id001213"><margin.target id="marg255"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 40. 7</s>
        </p>
        <p type="main">
          <s id="id001214">Propoſitio ſeptuageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id001215">Proportionem ponderum attractorum penes figuram in pla­<lb></lb>no inuenire.<lb></lb><arrow.to.target n="marg256"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001216"><margin.target id="marg256"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001217">Sint duo pondera æqualia in plano a &amp; b, &amp; ſit <lb></lb><figure id="id.015.01.083.1.jpg" xlink:href="015/01/083/1.jpg"></figure><lb></lb>a ſuperficies qua planum tangit dupla b ſuperfi­<lb></lb>ciei, qua planum tangit: dico quod ſi trahantur ab <lb></lb>imo, quod erunt æqualia: ſuſpendantur, &amp; erunt <lb></lb>æqualia ex ſuppoſito, ſed a quieſcens in plano eſt <lb></lb>dimidium a ſuſpenſi, &amp; b quieſcens in plano eſt di<lb></lb>midium b ſuſpenſi ex demonſtratis ſuperius, igi­<lb></lb>tur per communem animi ſententiam a &amp; b in pla­<lb></lb>no ſunt æqualia.</s>
        </p>
        <p type="main">
          <s id="id001218"><arrow.to.target n="marg257"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001219"><margin.target id="marg257"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001220">Ex hoc manifeſtum eſt, quod proportio uirium trahentium pon<lb></lb>dera in plano eadem eſt, quæ ipſorum ponderum dum ſuſpendun­<lb></lb>tur. </s>
          <s id="id001221">Vbi planum æquale ſit, &amp; ſolidum.</s>
        </p>
        <p type="main">
          <s id="id001222"><arrow.to.target n="marg258"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001223"><margin.target id="marg258"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 62.</s>
        </p>
        <p type="main">
          <s id="id001224">Propoſitio ſeptuageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id001225">Proportionem concutientis ad concuſſum ſtabili inuenire.</s>
        </p>
        <p type="main">
          <s id="id001226"><arrow.to.target n="marg259"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001227"><margin.target id="marg259"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001228">Intelligo concutiens eſſe ſolidum, quod non frangitur, idque gra­<lb></lb>uitate, &amp; impetu concutere, nam de duritie ſupponitur, &amp; grauitas, <lb></lb>ut demonſtrabitur in corrolario eſt iuxta ſuperficiem inferiorem <lb></lb>ponderi comparatam. </s>
          <s id="id001229">Cum ergo motus concuſsionis magnitudo <lb></lb>conſtet ex grauitate, impetu &amp; figura, concuſsi autem ex pondere <lb></lb>&amp; connexione: multiplicatis inuicem partibus productorum pro­<lb></lb>portio, erit proportio concuſsionis: ut ſit grauitas decem, impetus <lb></lb>quadraginta: pondus icti centum connexio ut duo, ducemus qua­<lb></lb>draginta in decem, &amp; fient quadringenta, et duo in centum, fient du<lb></lb>centa, igitur concuſsio erit dupla.</s>
        </p>
        <p type="main">
          <s id="id001230"><arrow.to.target n="marg260"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001231"><margin.target id="marg260"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001232">Cum fuerit figura rotunda, concuſsio erit integra in puncto: <lb></lb>quia ſphæra iacens in plano totum pondus in punctum cogit.</s>
        </p>
        <p type="main">
          <s id="id001233"><arrow.to.target n="marg261"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001234"><margin.target id="marg261"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001235">Si autem planum eſt, quod ijcitur, proportio totius ad totum eſt <lb></lb>minor, quàm partis ad partem pro ratione quantitatis latitudinis. </s>
        </p>
        <p type="main">
          <s id="id001236"><arrow.to.target n="marg262"></arrow.to.target><lb></lb>ſed maior ratione aëris comprehenſi, de quo infrà.<lb></lb><arrow.to.target n="marg263"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001237"><margin.target id="marg262"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 84.</s>
        </p>
        <p type="margin">
          <s id="id001238"><margin.target id="marg263"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001239">Cum proportio minor fuerit ſtabile, non poterit in ſolido plano <lb></lb>moueri: aliter fieret motus à debiliore, &amp; per præcedentem etiam <lb></lb>poſſet pari ratione eleuari.</s>
        </p>
        <p type="main">
          <s id="id001240"><arrow.to.target n="marg264"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001241"><margin.target id="marg264"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id001242">Cumque ſtabile non mouetur, &amp; omne agens agat aliquid neceſſe <lb></lb>eſt, ut ſtabilis partes cedant, aut diſſoluantur. </s>
          <s id="id001243">Quanto ergo magis <lb></lb>cedit, tanto minus diſſoluitur.</s>
        </p>
        <pb pagenum="65" xlink:href="015/01/084.jpg"></pb>
        <p type="main">
          <s id="id001244">Cauſæ igitur quæ alleuiant ictum, ne diſſoluatur, ſunt ſeptem le­</s>
        </p>
        <p type="main">
          <s id="id001245"><arrow.to.target n="marg265"></arrow.to.target><lb></lb>uitas ictus, ponderis, fractura, mollities eius, quod icitur, mollities <lb></lb>eius, quod excipit ictum, motus eiuſdem, &amp; figura lata, &amp; inæqua­<lb></lb>lis. </s>
          <s id="id001246">Durities ergo, quatenus fracturæ opponitur, aliud eſt, quam ut <lb></lb>mollitiei: &amp; utra que eſt cauſa, quæ auget ictum, ut reliquæ <lb></lb> oppoſitæ minuunt, dicemus autem de his inferius.</s>
        </p>
        <p type="margin">
          <s id="id001247"><margin.target id="marg265"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 9.</s>
        </p>
        <p type="main">
          <s id="id001248">Propoſitio ſeptuageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id001249">Proportionem immoti in aqua ad immotum in terra in excipien <lb></lb>do ictum inuenire.</s>
        </p>
        <p type="main">
          <s id="id001250">Sit pondus a in terra æquale b eiuſdem naturæ magnitudinis fi­<lb></lb><arrow.to.target n="marg266"></arrow.to.target><lb></lb>guræ, &amp; eodem in ſitu, quod ſit in aqua porrò a, ſi eſſet affixum ter­<lb></lb>ræ oportet, ut conuellatur, aut diſſoluatur aut frangatur. </s>
          <s id="id001251">Et clarum <lb></lb><figure id="id.015.01.084.1.jpg" xlink:href="015/01/084/1.jpg"></figure><lb></lb>eſt, quod totum ictum excipit. </s>
          <s id="id001252">Si uerò <lb></lb>affixum non ſit, euertitur, &amp; tanto mino­<lb></lb>rem partem excipit ictus, quanto faci­<lb></lb>lior eſt ad euerſionem. </s>
          <s id="id001253">Vnde nata fabu­<lb></lb>la de quercu, quæ cum immobilis eſſet, <lb></lb>&amp; ſtaret uento euerſa eſt, arundo flecten­<lb></lb>do ſe, cecidit quidem, ſed non eſt eradi­<lb></lb>cata. </s>
          <s id="id001254">Sermo igitur eſt de b inſidenti aquę <lb></lb>in comparatione ad a, quando excipit <lb></lb>plenum ictum. </s>
          <s id="id001255">Cum ergo b tangitur, ex­<lb></lb>cipit plenum ictum illo inſtanti, ſed quia <lb></lb>non excipitur ictus cedente materia, &amp; <lb></lb>antequam materia cedat b mouetur loco, quia inſidet aquæ, ergo <lb></lb>non excipit ictum. </s>
          <s id="id001256">Proponatur ergo, quod moueatur b per c ſpa­<lb></lb>tium in d tempore, &amp; ſit, ut idem b ab e ui trahatur per idem ſpa­<lb></lb>tium in eodem tempore ex loco directo ad eandem partem: qua­<lb></lb>lis ergo proportio e ad b, &amp; aërem, qui cum eo reſiſtit, talis propor­<lb></lb>tio ictus f grauis puta in a ad ictum Y in b. </s>
          <s id="id001257">Quia per demonſtra­<lb></lb><arrow.to.target n="marg267"></arrow.to.target><lb></lb>ta ſuperius proportio f ad a producitur ex proportionibus e ad b, <lb></lb><arrow.to.target n="marg268"></arrow.to.target><lb></lb>&amp; a ad e, ergo diuiſa proportione f ad a per proportionem c ad b <lb></lb>exibit proportio ictus Y in a ad ictum Y in b quod erat demon­<lb></lb>ſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id001258"><margin.target id="marg266"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001259"><margin.target id="marg267"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 2.</s>
        </p>
        <p type="margin">
          <s id="id001260"><margin.target id="marg268"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 42. &amp; 43. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001261">Ex hoc patet, quod b quanto mollius, leuius, &amp; ſtrictius in imo, <lb></lb><arrow.to.target n="marg269"></arrow.to.target><lb></lb>&amp; in tenuiore aqua, eo minus lædetur. </s>
          <s id="id001262">Et quanto ictus lentior fue­<lb></lb>rit etiam quod ſit grauius Y.</s>
        </p>
        <pb pagenum="66" xlink:href="015/01/085.jpg"></pb>
        <p type="margin">
          <s id="id001263"><margin.target id="marg269"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001264">Propoſitio ſeptuageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id001265">Proportionem duorum mobilium ſibi inuicem concurrentium <lb></lb>per rectam inuenire.<lb></lb><arrow.to.target n="marg270"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001266"><margin.target id="marg270"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001267">Iam cognito, quod mobilia, quæ loco mouentur per præceden­<lb></lb>tes, ſed omnino quieſcunt integros excipiunt ictus: alia quidem, <lb></lb>quæ concurrunt, non omnino reſiliunt, alia uero reſiliunt, &amp; quæ <lb></lb>reſiliunt minores excipiunt ictus, ſequitur ut diuerſa ſit compara­<lb></lb>tio: nam erunt, quæ ſtando excipient ictus, &amp; hæc integros ut mu­<lb></lb>ri, &amp; quæ concurrendo, nec reſiliendo, ut equi curſu incitati: &amp; quæ <lb></lb>ſtando, ſed reſiliendo, ut naues ſtantes: &amp; quæ concurrendo, reſi­<lb></lb>liendo qúe ut naues uentis, &amp; triremes ab impulſu: bifariam ergo <lb></lb>contingit intelligi, quod proponitur. </s>
          <s id="id001268">Sed in utroque etiam ſenſu <lb></lb>uarietas eſt: nam ut concurrit pars altera celerius, ita etiam magis <lb></lb>concutitur. </s>
          <s id="id001269">Et ideo ſit, ut proportio ictùs ſit in comparatione ad <lb></lb>grauitatem duplá, &amp; concurrant æqualiter, &amp; ſint æquè grauia, &amp; <lb></lb>neutrum reſiliat, erunt in proportione quadrupla, &amp; eodem mo­<lb></lb>do ſi utrunque reſiliat. </s>
          <s id="id001270">At ſi diuerſo impetu ferantur, ut dixi, tria <lb></lb>erunt præcipuè conſideranda grauitas ſeu pondus, impetus, &amp; an <lb></lb>reſiliat. </s>
          <s id="id001271">Quanto enim grauiora fuerint, &amp; maiore impetu agen­<lb></lb>tur, &amp; non reſilierint eo maiorem ictum recipient: quanto leuio­<lb></lb>ra, &amp; minore impetu, &amp; magis reſilierint, minus lædentur. </s>
          <s id="id001272">Sed &amp; <lb></lb>in debilitando ictum conſiderare oportet tria, quod reſiliat, quod <lb></lb>diffugiat, quod circumuertatur: reſiliunt naues, ſi roſtris concur­<lb></lb>rant pleno ictu: ſi uerò non pleno ictu concurrant, ſed diffugiant <lb></lb>hoc experimento compertum eſt minimum eſſe ictum: ſi roſtro <lb></lb>tranſuerſum nauis feriatur medium, eſt hoc.</s>
        </p>
        <figure id="id.015.01.085.1.jpg" xlink:href="015/01/085/1.jpg"></figure>
        <p type="main">
          <s id="id001273">Sit ergo ut a b nauis tangat roſtro b c ſic ut <lb></lb>diffugiat, erit hypomochlium c, &amp; ſi tangat <lb></lb>e f hypomochlium eſt in d dupla, ergo eſt c b <lb></lb>ipſi d e, igitur ictus duplo minor excipitur à <lb></lb>c b quàm ef. </s>
          <s id="id001274">Eſt etiam tempus longè maius, <lb></lb>quo excipit ictum ef, quàm b c: ſtatim enim diſcedit b c occurritque<lb></lb>aliis partibus, in c f autem impingit, &amp; angulus a d c eſt longè ma­<lb></lb>ior recto, quàm a b f: ob hæc igitur longè maior eſt ictus c f quàm <lb></lb>b c: uocant autem hoc declinationem.</s>
        </p>
        <p type="main">
          <s id="id001275">Propoſitio ſeptuageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id001276">Proportionem motus obliqui ad motum rectum in nauibus <lb></lb>inuenire.</s>
        </p>
        <p type="main">
          <s id="id001277"><arrow.to.target n="marg271"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001278"><margin.target id="marg271"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001279">Cùm uentus fertur ad puppim rectà, nauiſqúe gubernaculum di<pb pagenum="67" xlink:href="015/01/086.jpg"></pb>rigitur, tendunturqúe uela ac expanduntur ſumma in parte mali, <lb></lb>tunc motus eſt uelociſsimus: fingamus autem, quod omnia ad <lb></lb>idem tendant præter uentum, qui non directus ſit ad puppim, ſed <lb></lb>à latere, ut uides, &amp; temo ſitin contrarium tantundem directus, &amp; <lb></lb>ſupponamus pro nunc, quod uelum ſit ſolum in anteriore parte <lb></lb>nauis, nam ſecus eſſet nimis magna differentia, <lb></lb><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"></figure><lb></lb>quod nauis una ageretur tribus malis alia una: <lb></lb>Quæritur igitur proportio motus b c ad mo­<lb></lb>tum d e: fiat ergo c f æqualis e g, ita ut f angulus <lb></lb>rectus ſit, &amp; manifeſtum eſt, quod h c maior eſt <lb></lb>c f, cum ergo angulus f rectus ſit, quanto maior <lb></lb>erit angulus h c f, tanto maior erit proportio h c <lb></lb>ad c f, quod eſt primum a, ińde noto angulo h c f <lb></lb>per ea, quæ tradita ſunt ab Aſtrologis de ſinu &amp; <lb></lb>arcu erit nota proportio c h ad c f, ideo ad e g <lb></lb>fiat ergo c k æqualis c h, igitur c k erit maior e g, ſi ergo perambula­<lb></lb>bit æqualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb></lb>ad c f, igitur cum nota ſit c k, eſt enim æqualis c h, erit temporis ad <lb></lb>tempus proportio nota. </s>
          <s id="id001280">Quod autem in æquali tempore mouebi­<lb></lb>tur nauis per c k &amp; h c patet ex aſſumpto inferius declarando.</s>
        </p>
        <p type="main">
          <s id="id001281"><arrow.to.target n="marg272"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001282"><margin.target id="marg272"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 99.</s>
        </p>
        <p type="main">
          <s id="id001283">Propoſitio ſeptuageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id001284">Propoſitionem nauis ad triremes quotuis concurrentes de­<lb></lb>monſtrare.</s>
        </p>
        <p type="main">
          <s id="id001285">Sit nauis deferens pondus decuplo maius triremi, &amp; conſtat, </s>
        </p>
        <p type="main">
          <s id="id001286"><arrow.to.target n="marg273"></arrow.to.target><lb></lb>quod impulſu æquabitur decem triremibus, ubi flante uento e <lb></lb>puppi æqualiter feratur in aduerſum, quantum triremes ui homi­<lb></lb>num. </s>
          <s id="id001287">Sed quoniam triremes impediuntur à uento licet ſine uelis <lb></lb>ſint, habent enim &amp; ipſę malum, &amp; uelum, ſed exigua comparatio­<lb></lb><arrow.to.target n="marg274"></arrow.to.target><lb></lb>ne nauium, ideo ictus ille multo ualidior eſt ex demonſtratis. </s>
          <s id="id001288">Cum <lb></lb>uero uis illa ſimul ſit, liquet, &#039;quòd hoc in caſu niſi machinæ obſta­<lb></lb>rent una nauis mille poſſet obruere triremes diſiunctas per tantum <lb></lb>ſpatium inter ſe, quantum eſt id, in quo nauis poteſt uenti impul­<lb></lb>ſum recipere. </s>
          <s id="id001289">At impedimentorum maximum ſunt machinæ, quæ <lb></lb>in nauim collimant à lateribus, cum triremes quaquâ uerſum ſe a­<lb></lb>gant, &amp; ob id proram ſolam exponunt ictibus, in quam difficile <lb></lb>eſt collimare, &amp; ſi tangatur pars ea robuſtior eſt, nec periculum <lb></lb>euerſionis adeò in currit, ut à lateribus: nec enim adeò anguſta eſt a <lb></lb>prora ad puppim nauis, quam à latere ad latus: his tot cauſis mi­<lb></lb>nus eſt obnoxia machinis triremis, quám nauis. </s>
          <s id="id001290">Sed &amp; alia cauſa <lb></lb>eſt, quoniam neceſſe eſt ut ob angulum laterum ad proram 


<pb pagenum="68" xlink:href="015/01/087.jpg"></pb>ictus dilabatur ſępius ſolum traiecta ſuperficie. </s>
          <s id="id001291">Secundum impe­<lb></lb>dimentum eſt à uento, ſi ualde obliquus ſit, nam ad rectum impul­<lb></lb>ſum, multum debilitatur: aut ſi inconſtans ſit, uiribusque remittatur. <lb></lb></s>
          <s id="id001292">Tertium uerò ſi triremes inuicem connexæ ſint, ac ſe tangant, in <lb></lb>quas nauis dirigitur. </s>
          <s id="id001293">Sed &amp; hoc infrà demonſtrabitur nauim, ut le­<lb></lb><arrow.to.target n="marg275"></arrow.to.target><lb></lb>uior fuerit facilius elabi, ſed ut pondere magis onerata grauiores <lb></lb>ictus inferre: ob hoc triremem inuenerunt mediam maximi uſus <lb></lb><foreign lang="grc">ἀμφήρην. </foreign></s>
          <s id="id001294">Galeonum uulgò uocant.</s>
        </p>
        <p type="margin">
          <s id="id001295"><margin.target id="marg273"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001296"><margin.target id="marg274"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 74.</s>
        </p>
        <p type="margin">
          <s id="id001297"><margin.target id="marg275"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 109.</s>
        </p>
        <p type="main">
          <s id="id001298">Propoſitio ſeptuageſima nona.</s>
        </p>
        <p type="main">
          <s id="id001299">Proportionem medicamentorum purgantium inuicem de­<lb></lb>clarare.<lb></lb><arrow.to.target n="marg276"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001300"><margin.target id="marg276"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001301">Scio, quàm multa concurrant, etiam per ſe ad purgationem mul <lb></lb>titudo humorum præparatio locus propinquus, ſed nobis ſer­<lb></lb>mo eſt pari ſub conditione, ut ſit dimidia uncia Caſsiæ nigræ in tri­<lb></lb>bus uicibus expurget libram humorum, &amp; uelim ſcire ab una un­<lb></lb>cia, quoties expurgabitur, &amp; quantum. </s>
          <s id="id001302">Dico, quod in ſcamonio, &amp; <lb></lb>agarico hæc ratio deprehendi poteſt: in his autem medicamentis, <lb></lb>quæ magis leniunt, quàm à proprietate educant, ut eſt caſsia nigra, <lb></lb>ratio hæc non ualet, quoniam feces quando que pro maiore par­<lb></lb>te educuntur, ita ut etiam multiplicato medicamento deſit, quod <lb></lb>educatur. </s>
          <s id="id001303">Et quamuis humores iuxta proportionem trahat, cum <lb></lb>tamen feces proportionem non ſeruent, ſequitur: ut aggregati ad </s>
        </p>
        <p type="main">
          <s id="id001304"><arrow.to.target n="marg277"></arrow.to.target><lb></lb>aggregatum proportio non ſeruetur. </s>
          <s id="id001305">At non eſt facile poſtmo­<lb></lb>dum internoſcere feces ab humoribus, quocirca uidetur propor­<lb></lb>tio illa confundi. </s>
          <s id="id001306">Quod ſi medicamentum leniens, fiat ob quanti­<lb></lb>tatem purgans humores, ut de multa caſsia nigra, tunc non poteſt <lb></lb>aſsignari illa comparatio niſi ut eſt medicamentum purgans. </s>
          <s id="id001307">Et ſit <lb></lb>gratia exempli, primum ut grana ſex ſcamonij purgent aliquem <lb></lb>ter, &amp; uncias decem bilis, dico iuxta rationem ſupra poſitam, quod <lb></lb><arrow.to.target n="marg278"></arrow.to.target><lb></lb>grana duodecim purgabunt iuxta proportionem duplam ſexqui­<lb></lb>alteram, ſi duo grana nil purgant, ſed commouent. </s>
          <s id="id001308">æqualia enim <lb></lb><arrow.to.target n="marg279"></arrow.to.target><lb></lb>ſunt: ut quatuor ſint dupla, &amp; ſex tripla, &amp; mouent ter, quia ſexqui­<lb></lb>alteram habent proportionem ad exceſſum, igitur duodecim du­<lb></lb>plam, &amp; ſexquialteram ad quatuor, nam decem ad quatuor eſt du­<lb></lb>pla ſexquialtera, &amp; purgabit ſepties cum nixu libras duas fer­<lb></lb>me bilis. </s>
          <s id="id001309">Vt comparatio fiat exceſſus ad uim, quæ reſiſtit eodem <lb></lb>modo. </s>
          <s id="id001310">In caſsia ergo nigra ſi uncia <expan abbr="unanõ">una non</expan> purga, ſed lenit tantum, <lb></lb>&amp; duæ unciæ purgant ter, &amp; libram unam bilis, tres unciæ duplam 


<pb pagenum="69" xlink:href="015/01/088.jpg"></pb>habent proportionem iuxta exceſſum ad unam, exceſſus igitur <lb></lb>duplum purgabunt, &amp; duplo magis, id eſt præter feces libras <lb></lb>duas bilis in ſex uicibus.</s>
        </p>
        <p type="margin">
          <s id="id001311"><margin.target id="marg277"></margin.target>E<emph type="italics"></emph>x conuerſa<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>quint.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001312"><margin.target id="marg278"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 37.</s>
        </p>
        <p type="margin">
          <s id="id001313"><margin.target id="marg279"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 42.</s>
        </p>
        <p type="main">
          <s id="id001314">Propoſitio octuageſima.</s>
        </p>
        <p type="main">
          <s id="id001315">Proportionem motus ſecundum obliquum ad rectum in ſpa­<lb></lb>tio declarare.</s>
        </p>
        <p type="main">
          <s id="id001316">Hæc uídetur ſimilis ſuperiori cuidam propoſitioni, ſed tamen in <lb></lb><arrow.to.target n="marg280"></arrow.to.target><lb></lb>hoc differt, quoniam in c a ſupponimus nauim moueri, ut concu­<lb></lb>tiat, hic autem iuxta motum ſolum: ut proponamus b nauim ferri <lb></lb><figure id="id.015.01.088.1.jpg" xlink:href="015/01/088/1.jpg"></figure><lb></lb>uerſus a uento recto ex b in a: ſit autem uentus ex <lb></lb>cin a mouens nauim ex b in a: nòn enim moue­<lb></lb>bit ut quidam putant in ratione c a ad b a: ut ſi ca <lb></lb>ſit ſexquiquarta ad b a, ut æquali impetu ex b &amp; <lb></lb>c flante uento moueretur tardius per c a, quam <lb></lb>per b a, quia æqualiter ex ſuppoſito: ergo tanto <lb></lb>tardius c fertur in a, quam b in idem quanto lon­<lb></lb>gior eſt c a, b a igitur ſi b perueniet in a in qua­<lb></lb>tuor diebus c perueniet in idem a in quinque <lb></lb>diebus. </s>
          <s id="id001317">Hoc enim eſt per ſe manifeſtum: ſed non quærimus id, ſed <lb></lb>ut uento c a æquali per c a ei, qui eſt b a per b a, ubi b moueatur uen <lb></lb>to c a per b a, quanto tardius mouebitur. </s>
          <s id="id001318">Mouebitur. </s>
          <s id="id001319">n. </s>
          <s id="id001320">tardius ad <lb></lb>a per b a, quam per c a, at per c a tardius, quam ex b in a per æqua­<lb></lb>lem uim, ergo multo tardius ex b in a per c a uentum, quam per uen <lb></lb>tum ex b in a. </s>
          <s id="id001321">Quærimus ergo compoſitionem horum, ut ſit c <lb></lb>nauis, quæ debeat transferri ad a per uentum ex b, &amp; ſequitur, <lb></lb>quod tardius, quam ex c per uentum ex c in a, &amp; tardius ex b per <lb></lb>uentum ex cin a. </s>
          <s id="id001322">Ergo malus, qui in prora eſt conuoluto eo, qui <lb></lb>eſt in puppi, ut etiam Ariſtoteles docet tantundem nititur ad re­<lb></lb><arrow.to.target n="marg281"></arrow.to.target><lb></lb>ctum ex cin æquidiſtantem locum ab a quantum c diſtat ab con­<lb></lb>tra temo, qui in puppi eſt dirigitur ad h, &amp; ſi ualidius ſit uentus e­<lb></lb>tiam adiuuante temonem, ſeu contra nitente, quantum licet mo­<lb></lb>bili pondere nauis ad id latus, premitur enim nauis, quaſi ſubmer­<lb></lb>gi debeat, uento in aduerſum premente, ut ſi uentus repente huic <lb></lb>contrarius exoriatur, <expan abbr="periculũ">periculum</expan> ſubeat, ne obruatur. </s>
          <s id="id001323">Cum ergo uen­<lb></lb>tus ex b feratur, æquidiſtans c h, &amp; c feratur per temonem in k, &amp; ab <lb></lb>oppoſitis æqualis actio ſequatur, imò tota impeditur, ex c in h fere­<lb></lb>tur iuxta proportionem anguli, quem conſtituit h c cum a c ad to­<lb></lb>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob 


<pb pagenum="70" xlink:href="015/01/089.jpg"></pb>uim uenti, &amp; uiæ longitudinem, angulus uerò h c a ſit ſexta re­<lb></lb>cti pars, feretur ex c uerſus a ad quantitatem b a in quatuorde­<lb></lb>cim horis: igitur rurſus quanta eſt proportio c a ad b a tan­<lb></lb>tum eſt temporis, in quo fertur ex c ad a ad quatuor decim horas <lb></lb>per uentum b a.</s>
        </p>
        <p type="margin">
          <s id="id001324"><margin.target id="marg280"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001325"><margin.target id="marg281"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 7. M<emph type="italics"></emph>echanica.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001326">Propoſitio octuageſima prima.</s>
        </p>
        <p type="main">
          <s id="id001327">Qualis ſit angulus, per quem poteſt moueri nauis ad rectum <lb></lb>explorare.<lb></lb><arrow.to.target n="marg282"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001328"><margin.target id="marg282"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001329">Cum in præcedenti propoſitione oſtenſum ſit angulum k c a <lb></lb>oportere eſſe æqualem angulo h c a, ut feratur, c in a uento c h, nec <lb></lb>tamen prorſus, ſed temo magis inflectit uerſus k quam uentus co­<lb></lb>git uerſus h: ſicut contra maiori ui uentus dirigit ad h, quàm temo <lb></lb>ad k, ut neceſſe ſit nauim flecti ad k pondere, ideo ſi uentus eſſet <lb></lb>tranſuerſus periclitaretur, neceſſe eſt, ut per omnes uentos, qui fe­<lb></lb>runt ab ea, quæ ad perpendiculum ſuper c a, &amp; ſunt quatuor decim: <lb></lb>ſed quoniam, ut dixi, pondere adiuuante uis uenti minor fit, neceſ­<lb></lb>ſe eſt, ut per uentos debiliores feratur magis ab extremis, qui pro­<lb></lb>pe perpendiculum ſunt: ita ut numerus omnium ſit, cum leuiſsimi <lb></lb>fuerint, quatuor decim, cum uiolentiſsimi, tres tantum proprius, &amp; <lb></lb>qui diſtant trigeſima ſecunda parte totius circuli, id eſt partibus un<lb></lb>decimi, cum quarta reliqui undecim, medij ſunt: ut tanto plures aſ­<lb></lb>ſumi poſsint à Nauclero, quanto molliores ſunt uenti, tanto pau­<lb></lb>ciores, quo uiolentiores. </s>
          <s id="id001330">Tutius autem fuerit in ualidis uentis diri­<lb></lb>gere nauim per uentum proximiorem, quam per ipſummet, qui re­</s>
        </p>
        <p type="main">
          <s id="id001331"><arrow.to.target n="marg283"></arrow.to.target><lb></lb>ctè tendit ad locum. </s>
          <s id="id001332">Veluti tendat nauis ex a in b, uentus tendat in <lb></lb>c ualidior, cumque magnus fuerit angulus c a b, ut potè dodrans to­<lb></lb>tius recti, ut eſſet temo dirigendus ad ſextum uentum altrinſecus di <lb></lb>rigemus ſolum ad quintum, ut feratur in d, &amp; hoc erit tanto cele­<lb></lb>rius, &amp; celerius feratur per a d &amp; d b, quàm ſi nauis recta lata eſſet <lb></lb>ex a in b. </s>
          <s id="id001333">inſuper tutius.</s>
        </p>
        <p type="margin">
          <s id="id001334"><margin.target id="marg283"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 83</s>
        </p>
        <p type="main">
          <s id="id001335">Propoſitio octuageſimaſecunda.</s>
        </p>
        <p type="main">
          <s id="id001336">Proportionem uelorum indagare.<lb></lb><arrow.to.target n="marg284"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001337"><margin.target id="marg284"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001338">Vela tribus in locis diſponi ſolent dolo b, quod in prora con­<lb></lb>ſtituitur, &amp; in malo, qui ponitur in medio ratione, quæ inferius <lb></lb>oſtendetur, ſed non ad unguem, quia cum malus in anteriorem <lb></lb>partem à uento impellatur, ſi eſſet in medio, ſemper præmeretur <lb></lb>nauis in anteriorem partem, ex quo duo magna incommoda ſeque <lb></lb>rentur: primùm ut periculum ſubiret, ne inuerſa in anteriorem par­


<pb pagenum="71" xlink:href="015/01/090.jpg"></pb>tem ſubmergeretur. </s>
          <s id="id001339">Secundum ne preſſa in parte anteriore dif­<lb></lb>ficilius aquas diſſecaret, &amp; ob id longe tardius, moueretur. </s>
          <s id="id001340">Pro­<lb></lb>pter hæc duo incommoda igitur malus etiam ſi unicus eſſet <lb></lb>(quod uulgatiſsimum maioribus noſtris |fuit) in parte magis <lb></lb>proræ proxima locabatur à gubernatoribus, ut eſſet quaſi in trien<lb></lb>te à roſtro in beſſe à puppi: Rarum fuit, &amp; memorabile, quod nunc <lb></lb>paſsim habet olim Antigoni <foreign lang="grc">τριαμέου&amp;</foreign> 1, uelorum trium: quorum <lb></lb>poſtremum Epidromus ut ipſa uoce intelligamus non fuiſſe ue­<lb></lb>lum in malo ipſo medio, ſed in puppi conſtitutum. </s>
          <s id="id001341">Cauſa Dolonis <lb></lb>inferius exponetur: quod autem eſſet paruum, &amp; omnium mini­<lb></lb>mum, ut nauis facile ab eo inuerteretur. </s>
          <s id="id001342">Vnde etiam nunc minus <lb></lb>minime habent tam quantitate, quam etiam altitudine, quod uo­<lb></lb>cant Trinehetum, ſolum enim ſuſtinet nauim, quæ à uentis, uel un­<lb></lb>dis mergi ſolet: ab undis ubi humilior eſt, à uentis à lateribus, et an­<lb></lb>teriore parte. </s>
          <s id="id001343">Vnde humile, &amp; exiguum uelum efficit, ut nauis ante­<lb></lb>riore parte leuis, nec mergatur prona à uentis, nec aquas ea exci­<lb></lb>piat, nec tamen impelli poteſt nauis in ſcopulos, nec euerti ob cau­<lb></lb>ſas dictas: ob quæ in magnis tempeſtatibus hoc ipſo duntaxat uti <lb></lb>ſolent. </s>
          <s id="id001344">Quod etſi nimium ſæuierint, etiam illud demittunt, &amp; ſi <lb></lb>fieri poteſt, etiam malum ipſam quamuis ſine uelo ſit. </s>
          <s id="id001345">Sed plerun­<lb></lb>que circumuolutam, &amp; implicatam ſolet antennam annexam, at­<lb></lb>que ſuſpenſam habere. </s>
          <s id="id001346">Sed &amp; ne nauis prorſum obruatur, quo­<lb></lb>niam ea pars omnem uentorum uim excipere ſolet, &amp; ut leuiſsima <lb></lb>ſit ijdem Gubernatores puppim multa arena, lapillis qúe onerant. <lb></lb></s>
          <s id="id001347">Ergo uelocitas nauis à uentorum impetu, eorumqúe rectitudi­<lb></lb>ne à uelorum magnitudine, &amp; loco humiliore, aut ſublimiore ha­<lb></lb>betur: tum nauis leuitate, &amp; forma. </s>
          <s id="id001348">Quæ enim non merguntur ut <lb></lb><foreign lang="grc">δρομάδες</foreign> (ſic enim uocat Ariſtophanes) eas, quas nunc uulgus fre­<lb></lb>gatas appellat) quaſi aquas innatantes curſu ſunt uelociſsimæ. </s>
          <s id="id001349">Et <lb></lb>longiores latis. </s>
          <s id="id001350">Poſt has ſunt, quæ carinam habent tenuem, ut fa­<lb></lb>cile aquas diuidant. </s>
          <s id="id001351">Vltimo loco, quæ quaſi mediæ, ante quidem <lb></lb>tenues, pòſt latiores ad uelocem curſum, &amp; ferendum onera aptæ, <lb></lb>&amp; humiles altis: &amp; leui ex ligno. </s>
          <s id="id001352">Sed nos de uelorum uarieta­<lb></lb>te loquimur, non ea&#039;, quæ ad malos pertinet. </s>
          <s id="id001353">Conſtat enim me­<lb></lb>dio loco plus mouere, quam in extremis, ut infrà docebi­<lb></lb>mus. </s>
          <s id="id001354">Antiquo enim tempore opus non fuit malorum mul­<lb></lb>titudine, quoniam ſijderibus uias dirigebant ob id non ad <lb></lb>amuſsim, quoniam linea dirigi non poterat maximè ob mo­<lb></lb>tus obliquitatem in circulo uiſus: ideò mali multi confu­<lb></lb>ſionem in curſu, &amp; impedimentum in naui, maiuſqúe pericu­<lb></lb>lum attuliſſent. </s>
          <s id="id001355">At nunc inuenta pyxide, &amp; lapidis Her­


<pb pagenum="72" xlink:href="015/01/091.jpg"></pb>culei auxilio pluribus locis uela diſpoſita melius dirigunt iter, ut <lb></lb>quaſi craſſa minerua depictum, &amp; poteſtate deformatum, ad amuſ­<lb></lb>ſim contrahant. </s>
          <s id="id001356">Motus ergo magnitudo non ſimpliciter conſtat, <lb></lb>ſed comparatione ſuperficiei ueli ad uelum longitudine quidem, </s>
        </p>
        <p type="main">
          <s id="id001357"><arrow.to.target n="marg285"></arrow.to.target><lb></lb>ac latitudine conflata per multiplicationem. </s>
          <s id="id001358">Altitudinis quo que ut <lb></lb><arrow.to.target n="marg286"></arrow.to.target><lb></lb>infrà exponetur. </s>
          <s id="id001359">Ex quorum omnium ductu, quaſi cubica, uel tri­<lb></lb>plicata ratione, ut ſuperius oſtenſum eſt, ratio uelocitatis motus na <lb></lb>uium conflatur.</s>
        </p>
        <p type="margin">
          <s id="id001360"><margin.target id="marg285"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 86.</s>
        </p>
        <p type="margin">
          <s id="id001361"><margin.target id="marg286"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 42.</s>
        </p>
        <p type="main">
          <s id="id001362">Propoſitio octuageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id001363">Proportionem receſſus à recta uia ad obliquitatem inueſtigare.<lb></lb><arrow.to.target n="marg287"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001364"><margin.target id="marg287"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001365">Sit nauis in a itura in b (uentus rectus ad c, medius ad e) per <expan abbr="ob­liquũ">ob­<lb></lb>liquum</expan>, cum ergo tardius moueatur per a e quàm a c &amp; per a b, quam <lb></lb>per a d, &amp; ſint ad perpendiculum b e, b d quas conſtat eſſe breuiſsi­<lb></lb>mas earum, quæ ad a c &amp; ad a d. </s>
          <s id="id001366">Queritur igitur quando uelocius <lb></lb><figure id="id.015.01.091.1.jpg" xlink:href="015/01/091/1.jpg"></figure><lb></lb>ferretur ad b, an cum per a c, c b, an cum per a d, d b, <lb></lb>an cum per a b ſimpliciter. </s>
          <s id="id001367">Et conſtat quod a d &amp; d b <lb></lb>longiores ſunt a b, iſtud enim demonſtratum eſt ab <lb></lb>Euclide in primo Elementorum, dico modo a c, &amp; </s>
        </p>
        <p type="main">
          <s id="id001368"><arrow.to.target n="marg288"></arrow.to.target><lb></lb>c b eſſe longiores a d &amp; d b, nam quadrata a d &amp; d b <lb></lb>&amp; a c &amp; c b ſunt æqualia quadrato a b per dicta ibi­<lb></lb><arrow.to.target n="marg289"></arrow.to.target><lb></lb>dem, &amp; ideo quadrata a c &amp; c b ęqualia quadratis a d <lb></lb>&amp; d b, ſed a d eſt longior a c, quia ducta c d angulus <lb></lb>d c a eſt obtuſus, igitur ad maiorem a c per decimam <lb></lb>nonam primi Elementorum: quare per communem <lb></lb>animi ſententiam quadratum a d maius eſt quadrato a c, quare rur­<lb></lb>ſus per communem animi ſententiam quadratum c b maius eſt <lb></lb>quadrato d b. </s>
          <s id="id001369">Cum ergo quadrata a d &amp; d b æqualia ſint quadra­<lb></lb>tis a c &amp; c b, &amp; a d ſit maior a c &amp; c b maior d b, ſequitur per nonam <lb></lb>ſecundi Elementorum, quod a c &amp; c d ſint maiores a d &amp; d b pari­<lb></lb>ter acceptis. </s>
          <s id="id001370">Si ergo maior fuerit exceſſus quàm proportio motus <lb></lb>per temonem cohibiti, ut ſupra uiſum eſt, tardius mouebitur per <lb></lb>a d, d b quàm a b per a c, c b quàm per a d, d b, ſed ſi contrà maior ſit <lb></lb>proportio motus cohibiti à temone ad motum liberum quàm ex­<lb></lb><arrow.to.target n="marg290"></arrow.to.target><lb></lb>ceſſus ad exceſſum uelocius mouebitur per a d d b, quàm per a b, <lb></lb>&amp; per a c quàm per a b. </s>
          <s id="id001371">Accedit huc e incommodo longioris uiæ, <lb></lb>quod uento a c non poterit ferri nauis ex c d in b, quoniam antea <lb></lb>ægre ferebatur: &amp; nunc ægrius per c b quàm a b, plus enim diſtat <lb></lb>uentus a c ab itinere c a quàm à uento a b, ut uiſum eſt ſuperius, igi­<lb></lb>tur multo melius eſt (ni quid obſtet) ire per a b quàm per <expan abbr="ullã">ullam</expan> aliam <lb></lb><arrow.to.target n="marg291"></arrow.to.target><lb></lb>uiam: niſi ſtationes ſint in c d, uel periculum immineat in a b. </s>
          <s id="id001372">Vbi ta<lb></lb>men uenti ſecundarent, tantum eſt uirium in recto curſu, &amp; æquali 


<pb pagenum="73" xlink:href="015/01/092.jpg"></pb>uelocitate ferretur citius ex a in b per a d d b, &amp; etiam citius per a c, <lb></lb>c b in b quam per ipſam a b, quod fuit propoſitum declarare.</s>
        </p>
        <p type="margin">
          <s id="id001373"><margin.target id="marg288"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 20.</s>
        </p>
        <p type="margin">
          <s id="id001374"><margin.target id="marg289"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 47.</s>
        </p>
        <p type="margin">
          <s id="id001375"><margin.target id="marg290"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 80.</s>
        </p>
        <p type="margin">
          <s id="id001376"><margin.target id="marg291"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 81. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001377">Propoſitio octuageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id001378">Diſtantiam centri terræ à centro mundi per motum lapidis Her<lb></lb>culei declarare.<lb></lb><arrow.to.target n="marg292"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001379"><margin.target id="marg292"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id001380">Non me later Ariſtotelem exiſtimare centrum mundi eſſe cen­<lb></lb>trum terræ illudque probaſſe, quod tamen ex demonſtratione noſtra <lb></lb>mathematica apparet nunc ſubijciam, &amp; quid ad illius rationes di­<lb></lb>cendum ſit, aliâs etiam dicendum erit: nam liber hic, ut mathemati­<lb></lb>ca decet, eſſe debet ab omnibus contentionibus abſolutus. </s>
          <s id="id001381">Con­<lb></lb>ſtat ſanè non eſſe propriam uim lapidis illius, ut qui non ſit circum­<lb></lb>ſcriptus ſed fruſtulum quoduis id poteſt, neque per ſe, ſed in ferro &amp; <lb></lb>pendulo, nec fieri poteſt, ut ſit illius <expan abbr="tãquam">tanquam</expan> ſpeciei unius lapidum, <lb></lb>ſed quaſi perfectæ portionis cuiuſdam generis terræ, quæ abſolu­<lb></lb>ta ſit, cuius indicium eſt illius copia, neque enim ullibi non inuenitur, <lb></lb>&amp; ubi ferrum effoditur, ut in Ilua Inſula Tyrrheno mari, eſt ergo fer <lb></lb><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"></figure><lb></lb>ri uis terræ maritæ, quæ perfecta in ſuo ge­<lb></lb>nere, ubi uim fecundam acceperit à maſcu­<lb></lb>lo ſcilicet Herculeo lapide, quærit primum <lb></lb>ut deſcendat, ubi hoc non poſsit <expan abbr="ſaltẽ">ſaltem</expan> quæ­<lb></lb>rit, ut quieſcere poſsit. </s>
          <s id="id001382">Vt ergo quieſcat à <lb></lb>motu cœli qui eſt ab Oriente in Occiden­<lb></lb>tem iuxta axis cœli ſitum ſe dirigit, quod <lb></lb>ille ſolus quieſcat in ſuo motu, uel ſaltem <lb></lb>tardiſsimè moueatur: indicio eſt quod ſi <lb></lb>extra ſitum illum acus ferrea imbuta eo lapide ponatur, ſtatim tre­<lb></lb>mit uehementer, adeò ut nec momento ullo conſiſtat, ſed miſerè &amp; <lb></lb>grauiter torqueri uideatur, non ergo quod ſentiat polorum locum <lb></lb>qui tantum abeſt ab illa, ut nec ab homine perito mathematicarum, <lb></lb>ſed quod uix illa cœli ſentiatur circa centrum mundi. </s>
          <s id="id001383">Cuius indi­<lb></lb>cio eſt Oceani maris, aquarum fluxus &amp; refluxus. </s>
          <s id="id001384">Duos ergo ha­<lb></lb>bet motus terra perfecta, ſeu ferrum lapide Herculeo <expan abbr="imbutũ">imbutum</expan> ſub­<lb></lb>ordinatos imperfectum perfecto: perfectus eſt, ut deſcendat ad cen<lb></lb>trum terræ, ut ibi quieſcat: imperfectum, cum à perfecto prohibe­<lb></lb>tur, ut quieſcat ſaltem extra centrum cum in clinatione ad centrum, <lb></lb>et hoc fiet ſi ſecundum longitudinem acus dirigatur per axem mun <lb></lb>di, cum ſitu tamen deſcenſui ad terræ centrum proximiore, ut ſæpi­<lb></lb>us ſuperius declarauimus, dum de motu grauium &amp; præcipuè li­<lb></lb>bræ, &amp; centro grauitatis loqueremur. </s>
          <s id="id001385">Quibus demonſtratis tum <lb></lb>experimento tum ratione à Fortunio Affaytato Cremonenſi Me­<lb></lb>dico, cum per hæc poſtmodum cogeretur fateri acum ad polum 


<pb pagenum="74" xlink:href="015/01/093.jpg"></pb>tendere, cum tamen tendat à dextro latere ſcilicet ab Oriente no­<lb></lb>uem partibus, ſeu decima parte unius recti in centro terræ, quæ eſt <lb></lb>quadrageſima totius ambitus cœli. </s>
          <s id="id001386">Statuatur centrum mundi a, &amp; <lb></lb>b a c axis, ſecundum quam mouetur motu diurno, ita l a dextra exit <lb></lb>oriens, k a ſiniſtra occidens, &amp; ſtatuatur d centrum terræ, ſeu ſuprà <lb></lb>ſeu infrà, non tamen in linea b c, ſed uel ſuprà in dextra parte, uel in­<lb></lb>frà in ſiniſtra, ita ut ducta linea per illud punctum arcus b g ſit no­<lb></lb>uem partium. </s>
          <s id="id001387">Conſtituta ergo acu in e puncto, ubi linea h ad g ſecat <lb></lb>peripheriam terrę dico, quod acus dirigetur per h g, &amp; non per b c, <lb></lb>nam acus mouetur ad centrum per eam, &amp; in eo ſitu tota dirigitur, <lb></lb>quia omnes partes grauis conſentiunt in motu principij grauitatis <lb></lb>ad centrum, hoc enim demonſtratum: nixus ergo eſt ut moueatur <lb></lb>per c d, &amp; in eo nixu qui eſt quies cuſtodit lineam axis, quæ eſt a b, <lb></lb>ut quieſcat, ergo non quieſcet, niſi in linea d g, quod erat demon­<lb></lb>ſtrandum. </s>
          <s id="id001388">Quæ autem ſequuntur ex his corrolaria omnia concor­<lb></lb>dant cum experimentis. </s>
          <s id="id001389">Ergo hic ſermo eſt demonſtratiuus, ut e­<lb></lb>nim bene dixit Auerroes: Sermo demonſtratiuus ſatisfacit omni­<lb></lb>bus problematibus quæ <expan abbr="cõtingunt">contingunt</expan> circa principale quæſitum. </s>
          <s id="id001390">Ex <lb></lb>hoc ergo patet, quod angulus diſtantia d ab a in latitudine eſt deci­<lb></lb>ma pars recti, et quod quanto magis diſtatin longitudine centrum <lb></lb>terræ à centro mundi, tanto etiam minus diſtatin latitudine. </s>
          <s id="id001391">Hæc <lb></lb>enim ſunt demonſtrata clarè in mathematicis. </s>
          <s id="id001392">Vnde fieri poſſet <lb></lb>quod hæc quantitas diſtantiæ eſſet res, per quam exigua etiam ſi <lb></lb>non eſſet maior quatuor digitis ſufficeret, modo etiam per ualde <lb></lb>paruum ſpatium diſtaret ab eodem in longitudine. </s>
          <s id="id001393">De cauſa au­<lb></lb>tem huius differentiæ aliâs dicendum erit, hic locus non eſt, ſed ſuf­<lb></lb>ficit ſcire quod ita ſit, quod ſi mobilis ſit punctus d, clarum eſt ali­<lb></lb>quando futurum ut minus diſtet g à b, aliquando ut ſit idem. </s>
          <s id="id001394">Et <lb></lb>qualiſcunque motus ſit, neceſſe eſt eam diſtantiam uariari.</s>
        </p>
        <p type="main">
          <s id="id001395">Propoſitio octuageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id001396">Proportio ponderis unius grauis ad aliud ſub eadem menſura <lb></lb>eſt, ueluti eiuſdem ad differentiam ponderis uaſis repleti ex altero <lb></lb>graui, &amp; ex ambobus detracto priore.</s>
        </p>
        <p type="main">
          <s id="id001397"><arrow.to.target n="marg293"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001398"><margin.target id="marg293"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001399">Sit aurum a, &amp; liquor b, quæ repleant uas c, &amp; <lb></lb>pondus amborum ſit librarum quadraginta, &amp; <lb></lb><figure id="id.015.01.093.1.jpg" xlink:href="015/01/093/1.jpg"></figure><lb></lb>uas repletum liquore ſolo ſit librarum xxix, au­<lb></lb>rum autem ſit ponderis librarum xij, igitur reli­<lb></lb>quum erit ponderis xxviij, differentia ergo ua­<lb></lb>ſis pleni, &amp; non pleni liquore eſt libra una, pon­<lb></lb>dus auri eſt librarum duodecim: dico quod au­<lb></lb>ri pondus eſt duodecuplum ponderi liquoris, &amp; <pb pagenum="75" xlink:href="015/01/094.jpg"></pb>ſi fuiſſet pondus amborum libræ xxxix, manentibus reliquis, ſeque <lb></lb>retur quod pondus liquoris eſſet xxvij, &amp; quia plenum uas ſuppo­<lb></lb>nitur eſſe librarum xxix, eſſet differentia libræ ij, at auri pondus eſt <lb></lb>libræ xij, igitur proportio ponderis auri ad liquorem eſſet ſexcu­<lb></lb>pla. </s>
          <s id="id001400">Nam ſi uas plenum liquore ex ſuppoſito eſt librarum xxix, &amp; <lb></lb>cum auro xl, gratia exempli, &amp; auri pondus eſt xij, igitur liquoris <lb></lb>pondus eſt xxviij librarum: ſed cum liquor ſit corpus ſimilium par­<lb></lb>tium, igitur loci ad lo cum, ut ponderis ad pondus, ergo dum adeſt <lb></lb>aurum, liquor occupat xxviij partes cxxxix, totius uaſis igitur au­<lb></lb>rum continet unam partem tantum, &amp; cum aurum pondus habeat <lb></lb>librarum xij, &amp; liquor unius: quia totum uas cxxxix librarum dum <lb></lb>eſt plenum, &amp; eſt diuiſum in xxix partes, igitur pondus unius par­<lb></lb>tis liquoris eſt una libra, igitur pondus auri eſt duodecuplum ad <lb></lb>pondus liquoris quod fuit propoſitum.</s>
        </p>
        <p type="main">
          <s id="id001401"><arrow.to.target n="marg294"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001402"><margin.target id="marg294"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001403">Ex quo ſequitur quòd ſi ducatur pondus illud partis per pon­<lb></lb>dus repleti uaſis ex alio graui, &amp; productum diuidatur per differen<lb></lb>tiam illam, prodibit pondus uaſis repleti liquore graui.</s>
        </p>
        <p type="main">
          <s id="id001404"><arrow.to.target n="marg295"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001405"><margin.target id="marg295"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001406">Exemplum, ſi pondus auri fuerit librarum xij, pondus uaſis re­<lb></lb>pleti liquore xxix librarum, pondus auri &amp; liquoris replentium <lb></lb>uas xxxix librarum, ducemus xij in xxix fit cccxlviij, diuido perij <lb></lb>differentiam xxvij ponderis uaſis, repleti ex ambobus detracto au­<lb></lb>ri pondere, &amp; xxix ponderis uaſis repleti liquore exit clxxiiij, &amp; tan <lb></lb>tum auri uas illud continebit, nam cum duæ partes quas occupa­<lb></lb>bat aurum eſſent ponderis librarum xij, totum quod erat partium <lb></lb>xxix, continebit decies &amp; quater cum dimidio illud aurum xij, aut <lb></lb>ductum in xiiij cum dimidio, efficit cclxxiiij ut prius.</s>
        </p>
        <p type="head">
          <s id="id001407">EXEMPLVM.</s>
        </p>
        <p type="main">
          <s id="id001408">Quia ergo in ſuperiore propoſitione docui, quod ferrum eſt ue­<lb></lb>ra terra: uolui ſcire qualis eſſet proportio ferri ad aquam. </s>
          <s id="id001409">Accepi ur<lb></lb>ceum cuius aqua dum plenus eſſet ponderis, fuit unciarum ſex, &amp; <lb></lb>ſeptuncis unciæ, &amp; ſeptuncis duodecimæ partis unciæ &amp; pondus <lb></lb>ferri unciæ ſeptem, &amp; triens unciæ &amp; triens duodecimæ partis un­<lb></lb>ciæ: &amp; uaſis aquę &amp; ferro eodem repleti unciæ tredecim, &amp; duode­<lb></lb>cima &amp; ſeptunx duodecimæ partis unciæ. </s>
          <s id="id001410">Detrahemus ergo vij &amp; <lb></lb>trientem &amp; trientem duodecimæ. </s>
          <s id="id001411">i. </s>
          <s id="id001412">7 &amp; 64/144 pondus ferri ex 13 19/144, &amp; <lb></lb>relinquentur 5 99/144, detrahe ex 6 81/144, pondere aquæ totius uaſis relin<lb></lb>quuntur 17/18, diuide 7 64/144 per 17/18 exit proportio ponderis ferri ad pon<lb></lb>dus aquæ 7 15/17. Et hoc eſt proximum ei quod dixit Philoſophus de <lb></lb>proportione ponderis terræ &amp; aquæ.</s>
        </p>
        <p type="main">
          <s id="id001413"><arrow.to.target n="marg296"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001414"><margin.target id="marg296"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001415">Ex hoc patet ſolutio problematis cuiuſdam propoſiti aliasque mi <lb></lb>nus bene ſoluti cùm cauſam habeat manifeſtiſsimam, ſcilicet quod 


<pb pagenum="76" xlink:href="015/01/095.jpg"></pb>uaſe aqua pleno impoſitis ſenſim centum aureis coronatis nihil ef­<lb></lb>funditur, non quod quicquam abſumatur in metallo, ſed cauſa eſt <lb></lb>quod cum aurum ſit duplum pondere ferro, erit ex demonſtratis <lb></lb>ſex decuplum ad pondus aquæ. </s>
          <s id="id001416">Igitur cum ſit proportio ponderis <lb></lb>auri ad differentiam ſpatij eadem, ſi ſit uas aquæ ponderis libræ <lb></lb>unius &amp; mediæ, erit pondus totum xxiij unciarum, igitur aqua de­<lb></lb>ficiet ſolum ex decimaoctaua parte ſeu creſcet ex impoſitione auri, <lb></lb>ſed illa pars in tumore aquæ abſumitur, <expan abbr="nõ">non</expan> ſolum, quia <lb></lb><figure id="id.015.01.095.1.jpg" xlink:href="015/01/095/1.jpg"></figure><lb></lb>dum aureos imponimus plana ſolum ſit, ſed quia non ex <lb></lb>quauis rotunditate defluit, aliter in urceo tam exiguo <lb></lb>non poſſet apparere rotunda: quod enim rotunditas to­<lb></lb>tius terræ, quæ etiam planam oſtendit totam unam re­<lb></lb>gionem ad rotunditatem quæ apparet in exiguo urceo <lb></lb>aquæ. </s>
          <s id="id001417">Eſt igitur rotunditas illa potius ob lentorem aquę qui auge­<lb></lb>tur à lentore argenti, &amp; etiam magis auri, cum ſenſu digitorum per­<lb></lb>cipiatur.</s>
        </p>
        <p type="main">
          <s id="id001418"><arrow.to.target n="marg297"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001419"><margin.target id="marg297"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001420">Ex hoc apparet ratio quomodo Archimedes potuerit deprehen<lb></lb>dere coronam à Hierone propoſitam quantum auri &amp; argenti con<lb></lb>tineret. </s>
          <s id="id001421">Sit ergo uas a b aqua <expan abbr="plenũ">plenum</expan> ponderis unciarum triginta, &amp; <lb></lb>cum libra auri ſit ponderis unciarum quadraginta unius, &amp; cum li­<lb></lb>bra argenti ponderis unciarum quadraginta cum dimidio, igitur <lb></lb>erit auri pondus ad aquæ pondus duodecuplum, argenti autem <lb></lb>ad idem octuplum, quare auri ad <expan abbr="argẽtum">argentum</expan> pondus ſexquialterum. <lb></lb></s>
          <s id="id001422">Ponamus ergo quod corona impoſita ex auro &amp; argento ſolo fa­<lb></lb>bricata (hoc enim ſupponere oportet) fuerit unciarum ſexaginta, <lb></lb>pondus autem aquæ contentę cum corona in uaſe unciarum uigin<lb></lb>ti quatuor cum dimidio, ſcilicet totum octuaginta quatuor cum di­<lb></lb>midia, erit ergo proportio ponderis coronæ ad pondus aquæ, ut <lb></lb>cxx ad xi, aurum igitur eſt proportione duodecuplum, argentum <lb></lb>autem octuplum, corona ut cxx ad xi. </s>
          <s id="id001423">Conſtituantur ſub eiſdem ra­<lb></lb>tionibus ducen do lxxxviij. </s>
          <s id="id001424">cxx. </s>
          <s id="id001425">cxxxij. </s>
          <s id="id001426">hoc eſt ac ſi dicamus, accipe <lb></lb>partes ex cxxxij &amp; lxxxviij, tot ut faciant integrum &amp; componant <lb></lb>cxx. </s>
          <s id="id001427">Et ideò reduces ad minores numeros, ſcilicet xxxiij. </s>
          <s id="id001428">xxij. </s>
          <s id="id001429">et xxx. </s>
        </p>
        <p type="main">
          <s id="id001430"><arrow.to.target n="marg298"></arrow.to.target><lb></lb>&amp; operaberis per regulam de conſolatione monetarum, quas po­<lb></lb>nemus infrà, &amp; fient auri partes octo &amp; argen<lb></lb><figure id="id.015.01.095.2.jpg" xlink:href="015/01/095/2.jpg"></figure><lb></lb>ti partes iij, nam cum duxeris iij in octo pon­<lb></lb>dus argenti fiet xxiiij, &amp; cum duxeris viij in <lb></lb>xij, pondus auri fiet xcvi, igitur totum pon­<lb></lb>dus erit cxx, diuidendum per xi, aggregatum <lb></lb>partium auri &amp; argenti, ita uero uncia ad unciam, ut tota corona mi <lb></lb>ſta ad coronam puram auri &amp; argenti.</s>
        </p>
        <pb pagenum="77" xlink:href="015/01/096.jpg"></pb>
        <p type="margin">
          <s id="id001431"><margin.target id="marg298"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 178.</s>
        </p>
        <p type="main">
          <s id="id001432">Ex hoc etiam patet modus <expan abbr="cognoſcẽdi">cognoſcendi</expan> proportionem grauium <lb></lb><arrow.to.target n="marg299"></arrow.to.target><lb></lb>inuicem per ſolam aquam, uelut auri ad plumbum, ad lapides uel <lb></lb>æs, aut æris ad lapidem &amp; ſimilia, ut in præcedenti operatione de­<lb></lb>prehendiſti: nam cum ſit nota proportio auri ad aquam &amp; æris uel <lb></lb>lapidis ad eandem, erit auri ad æs uel lapidem nota.</s>
        </p>
        <p type="margin">
          <s id="id001433"><margin.target id="marg299"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id001434">Et ſimiliter ſciemus per hoc accipere partes diuerſorum, quę iun<lb></lb><arrow.to.target n="marg300"></arrow.to.target><lb></lb>ctæ faciant conſtitutum pondus. </s>
          <s id="id001435">Velut uolo facere maſſam ex mel­<lb></lb><figure id="id.015.01.096.1.jpg" xlink:href="015/01/096/1.jpg"></figure><lb></lb>le &amp; aqua, quæ impleat uas, quod mellis contineat <lb></lb>quindecim, aquæ duodecim, uolo ut contentum ſit <lb></lb>ponderis quatuordecim, operabor, ut in <expan abbr="cõſolatio­nibus">conſolatio­<lb></lb>nibus</expan>, ponam duas partes mellis &amp; unam aquæ, ut <lb></lb>uides in operatione à latere.</s>
        </p>
        <p type="margin">
          <s id="id001436"><margin.target id="marg300"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 5.</s>
        </p>
        <p type="main">
          <s id="id001437">Propoſitio octuageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id001438">Si circuli in æquales, ſeu in ſphæra, ſeu in plano ſe ſecuerint nun­<lb></lb>quam oppoſitos angulos æquales habent.</s>
        </p>
        <p type="main">
          <s id="id001439">Capiantur tres quartæ circulorum magnorum a b, a c, b c, &amp; alia <lb></lb><arrow.to.target n="marg301"></arrow.to.target><lb></lb>b d ad rectos angulos <expan abbr="erũtque">eruntque</expan> uiciſsim poli, &amp; ducatur per medium <lb></lb>parallelus, erit ergo e f æqualis e g, &amp; f e æqualis f g, ſed baſis c g eſt <lb></lb><figure id="id.015.01.096.2.jpg" xlink:href="015/01/096/2.jpg"></figure><lb></lb>quarta circuli, &amp; baſis c b dimidium quartæ <lb></lb>circuli eo quod tota b a eſt quarta circuli, igi­<lb></lb>tur per modum 25 primi Elementorum quæ <lb></lb>tenet, erit angulus c f g maior oppoſito c f b. <lb></lb></s>
          <s id="id001440">Hoc autem tenet in eiuſdem rationis ſuperfi­<lb></lb>ciebus, quales ſunt hæ, quæ ſunt ſuperficies eiuſdem ſphęræ. </s>
          <s id="id001441">poſſet <lb></lb>etiam demonſtrari per modum quartæ primi Elementorum. </s>
          <s id="id001442">Et eti­<lb></lb>am conſtituta ſphæra e f g, cuius hic circulus eſſet maior circulus, &amp; <lb></lb>non tangeret niſi in illa linea ſphæra maiorem, &amp; utrin que ſecaret eo­<lb></lb>dem circulo. </s>
          <s id="id001443">Et etiam per cordas &amp; trigonos rectilineos, auxilio <lb></lb><expan abbr="tamẽ">tamen</expan> regulæ dialecticæ. </s>
          <s id="id001444">Ex hoc ſequitur auxilio regulæ dialecticæ, <lb></lb><figure id="id.015.01.096.3.jpg" xlink:href="015/01/096/3.jpg"></figure><lb></lb>quod in omnibus parallelis a c d &amp; e f g cum b c circulo <lb></lb>maiore, &amp; per aliam regulam dialecticam in omnibus cira<lb></lb>culis inæqualibus inter ſe ad æquales angulos ſecanti­<lb></lb>bus &amp; ex tertia demum regula dialectica, ſequitur in o­<lb></lb>mnibus circulis in æqualibus ſe ſecantibus ad quemuis <lb></lb>angulum in ſphæræ ſuperficie. </s>
          <s id="id001445">Sunt autem hæ regulæ mediæ inter <lb></lb>axiomata &amp; demonſtrata. </s>
          <s id="id001446">Et ex logica propria illi arti. </s>
          <s id="id001447">In plano au­<lb></lb><arrow.to.target n="marg302"></arrow.to.target><lb></lb>tem ſpatium d b c minus eſt a b c, ſed ſpatium c b d eſt unum, ergo <lb></lb>per communem animi ſententiam ſpatium a b d, maius eſt ſpatio <lb></lb>c b c, quod fuit probandum.</s>
        </p>
        <pb pagenum="78" xlink:href="015/01/097.jpg"></pb>
        <p type="margin">
          <s id="id001448"><margin.target id="marg301"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001449"><margin.target id="marg302"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>terd <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001450">Propoſitio octuageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id001451">Proportionem craſsitiei aquæ ad aërem in comparatione ad ra­<lb></lb>dios demonſtrare.<lb></lb><arrow.to.target n="marg303"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001452"><margin.target id="marg303"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001453">Sit in aheno a b c d in imo e dena<lb></lb><figure id="id.015.01.097.1.jpg" xlink:href="015/01/097/1.jpg"></figure><lb></lb>rius argenteus cera affixus uel cla­<lb></lb>uo, quem uideat ex h impoſita aqua <lb></lb>clara uſque ad f, uideat ex k, igitur per <lb></lb>aquam deflectitur à perpendiculo <lb></lb>per angulum k f n, &amp; in l, per angu­<lb></lb>lum l g o creſcente aqua demum in <lb></lb>labro m a p, &amp; ſit e annexus, &amp; tabu<lb></lb>la h k l m ſit affixa ſolo uel pondere <lb></lb>firma foraminibus obliquis infrà <lb></lb>ſpectantibus, &amp; per a aſpicientibus extremitatem e. </s>
          <s id="id001454">Poſſumus ergo <lb></lb>imaginari primum, quòd omnes inclinationes ſint à perpendicu­<lb></lb>lari, dum exit aqua, &amp; ita denarius uideretur, uel in ſuperficie aquæ <lb></lb>in directo e, uel in recta ex oculo in imo, quorum neutrum uerum <lb></lb>eſt. </s>
          <s id="id001455">Secundus modus eſt, ut radius delatus e a flectatur ad k uel l, &amp; <lb></lb>hoc non quia in a non eſt mutatio medij. </s>
          <s id="id001456">Tertius eſt, ut linea ex ocu<lb></lb>lo ducta perueniat per punctum a ad ſuperficiem aquæ, &amp; ex ea <lb></lb>per directum ad denarium, &amp; tunc quia oculus iudicat ſe uidere <lb></lb>per rectam, ideo iudicabit ſe uidere per l a g in q, eo quod ſemper in <lb></lb>directo loci in quo eſt e. </s>
          <s id="id001457">At quoniam non ex qua cunque diſtantia ui­<lb></lb>detur e, ſed ex longinquiore loco, ubi uas fuerit humilius quod li­<lb></lb>neæ ad a ex oculo, quanto a fuerit humilius, tanto propius ipſi e <lb></lb>procedunt. </s>
          <s id="id001458">Et uerſa uice lineæ ex e ad a, quanto e eſt humilius ad <lb></lb>quencunque locum inflectuntur, tanto inferius <expan abbr="cadũt">cadunt</expan>. </s>
          <s id="id001459">Ergo cum fue<lb></lb>rint ad æquilibrium h, magis diſtabunt ab e, &amp; ita e magis procul <lb></lb>uidebitur. </s>
          <s id="id001460">Cauſa ergo triplex eſt humilitas, uel altitudo uaſis: humi <lb></lb>litas uel altitudo aquæ: &amp; labri uaſis altitudo. </s>
          <s id="id001461">Sed hanc relinquere <lb></lb>poſſumus. </s>
          <s id="id001462">Difficultas ergo experimenti etiam rectè facti eſt, quo­<lb></lb>niam poſito uaſe n c d ſolum, ut altitudo ſit tantum n e, procul ma­<lb></lb>gis uidebitur e, quàm ſi uas ſit a b c d, &amp; totum plenum. </s>
          <s id="id001463">Vbi autem <lb></lb>uas fit a b c d, magis procul uidebitur e cum ſuerit totum plenum, <lb></lb>quam cum fuerit plena ſola pars n c d. </s>
          <s id="id001464">Sic difficile eſt conſiderare <lb></lb>an altitudo aquæ faciat ad uiſionem procul, cum in humiliore, ſed <lb></lb>diſsipari uaſe longius uideatur in pauca, quia labrum non obſtat: <lb></lb>in eodem autem longius in pluri aqua, quia labrum etiam non ob­<lb></lb>ſtat, ſed alia ratione. </s>
          <s id="id001465">Vt ergo uideamus hoc experimentum, capie­


<pb pagenum="79" xlink:href="015/01/098.jpg"></pb>mus duo uaſa a b c d duplum h k l m ſub eadem proportione alti­<lb></lb>tudinis &amp; latitudinis, &amp; collocabimus ita ut p n radius æquidiſtet <lb></lb>f e, &amp; collocabimus tabulas cum foraminibus, ut prius, &amp; g f p q <lb></lb><figure id="id.015.01.098.1.jpg" xlink:href="015/01/098/1.jpg"></figure><lb></lb>in æquilibrio, in de uidebimus, an q p ſit æqualis aut breuior, nam <lb></lb>longior eſſe non poteſt, quoniam inflectitur a minore aqua, ideo <lb></lb>angulus p h q non poteſt eſſe maior f a g, ſuppoſita p h æquali a f: <lb></lb>quod ſi non eſſet, ſufficeret, ut q &amp; p eſſent in æquilibrio uno, &amp; f g <lb></lb>alio. </s>
          <s id="id001466">Sed ueritas eſt quod à maiore aqua maior fit reflexio: tum <lb></lb>quia in his, quæ ſunt ſecundum naturam corpoream, &amp; ſubſtan­<lb></lb>tiam denſam, aut tenuem uarietas quantitatis uariat uires: tum <lb></lb>quia uidemus, quod in altiore aqua denarius uidetur magis cum <lb></lb>fundo elatus. </s>
          <s id="id001467">Igitur his cognitis experimentum fiat cum uaſe ple­<lb></lb>no. </s>
          <s id="id001468">Et (ut dixi) conſiderabimus proportionem anguli f a g ad far, <lb></lb>ſeu f e c quæ ſanè eſt notabilis: adeò ut ſit maior proportio aquæ ad <lb></lb>aërem comparatione grauium quàm lucis.</s>
        </p>
        <p type="main">
          <s id="id001469"><arrow.to.target n="marg304"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001470"><margin.target id="marg304"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001471">Ex his cognoſcemus comparatione eiuſdem aquæ tenuitatem <lb></lb>aëris unius regionis in comparatione ad aërem alterius: nam ubi <lb></lb>remotius uidebitur denarius, ibi aër erit tenuior.</s>
        </p>
        <p type="main">
          <s id="id001472"><arrow.to.target n="marg305"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001473"><margin.target id="marg305"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001474">Et per idem in eadem regione comparationem aquarum. </s>
          <s id="id001475">Nam <lb></lb>cum ſit idem aër, &amp; uas, ac reliqua paria, ubi magis procul uidebi­<lb></lb>tur denarius, aqua erit craſsior ideò deterior.</s>
        </p>
        <p type="main">
          <s id="id001476"><arrow.to.target n="marg306"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001477"><margin.target id="marg306"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001478">Sequitur etiam quòd omnes res propiores in aqua uidentur, <lb></lb>quam ſint, &amp; ideò maiores: &amp; ob id etiam omnis aqua profundior <lb></lb>eſt, quam uideatur. </s>
          <s id="id001479">Vt ingredi perſæpè ſit periculoſum.</s>
        </p>
        <p type="main">
          <s id="id001480">Propoſitio octuageſima octaua. </s>
          <s id="id001481">De inſtrumento <lb></lb>momentorum.</s>
        </p>
        <p type="main">
          <s id="id001482">Inſtrumentum Acolingen, quo momenta temporum deprehen<lb></lb>dantur fabricare.</s>
        </p>
        <pb pagenum="80" xlink:href="015/01/099.jpg"></pb>
        <p type="main">
          <s id="id001483"><arrow.to.target n="marg307"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001484"><margin.target id="marg307"></margin.target>C<emph type="italics"></emph>om.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001485">Et quoniam motus naturales fiunt in tempore: &amp; dicuntur ue­<lb></lb>lociores, uel ob ſpatium loci magnum, quod ſuperatur, uel ob tem<lb></lb>poris breuitatem in uelociſsimis motibus, quod ad ſpatia attinet, <lb></lb>facilius dignoſcuntur uelociores, quoniam ſpatium maius &amp; ma­<lb></lb>net, ut menſurari commodè poſsit: ſed quòd ad tempus, quanto tar<lb></lb>diores, quoniam in uelo cibus quantitas temporis eſt exigua: &amp; e­<lb></lb>tiam tempus ipſum perpetuò diffluit: ideò difficillimè deprehendi <lb></lb>poteſt. </s>
          <s id="id001486">Huius cauſa excogitauimus inſtrumentum, quod uo caui­<lb></lb>mus Acolingen: quod conſtat tribus rotis: prima eſt pedum duo­<lb></lb>decim diametri, in ambitu autem habet denticulos ccclx æqua­<lb></lb>les, &amp; æqualiter inter ſe diſtantes, huius peripheriæ funis cum pon­<lb></lb>deribus inſeritur, ita ut cum alijs duabus rotis renitentibus in una <lb></lb>hora circumagatur æqualiter. </s>
          <s id="id001487">Duodecim ex his denticulis curru­<lb></lb>lis duodecim denticulorum axis ſecundæ rotæ inſeritur: ſic ut cum <lb></lb>rota magna duodecim conuerſa fuerit partibus, ſecunda rota cu­<lb></lb>ius axis ſit pedum duorum, ſcilicet ſexcuplo maior circumuerta­<lb></lb>tur. </s>
          <s id="id001488">Huius minoris ambitus diuiſus ſit in cxx partes æquales, &amp; <lb></lb>unicuique parti denticulus inſertus ſit: ita hæc rota tricies in una <lb></lb>hora conuertetur. </s>
          <s id="id001489">Singulis uerò denticulis currulis axis rotæ ha­<lb></lb>bentis denticulos quatuor inſeratur, ita ut dum ſecunda rota uer­<lb></lb>titur ſemel minima circumuertatur tricies: nam pro ſingulis qua­<lb></lb>tuor denticulis, quibus media rota circumagetur, minima tota cir­<lb></lb>cumuertetur, ideoqúe nongenties in una hora. </s>
          <s id="id001490">Hæc minima ro­<lb></lb>tula beſſem pedis in dimetiente habebit, ut ſit ſexta pars illius, in <lb></lb>ambitu autem diuiſa erit in xl partes, ut cum circumuerſa fue­<lb></lb>rit nongenties in una hora pertranſierit partes xxxvi. </s>
          <s id="id001491">Et cum <lb></lb>pulſus hominis communis ſint in hora &lt;23&gt;, uel circa nouem partes <lb></lb>ex his rotę minoris perficient circiter unam pulſationem ex diaſto­<lb></lb>le &amp; ſiſtole, ſeu ex diſtentione &amp; contractione perfectam: ut partis <lb></lb>unius conuerſio fiat in nona parte, uel circa unius pulſationis pul­<lb></lb>ſus humani: &amp; hoc eſt minimum fermè, quod ab humano ſen­<lb></lb>ſu percipi poſsit. </s>
          <s id="id001492">Erit etiam proportio rotarum eadem tam in dia­<lb></lb>metris, quàm circuitibus ſcilicet ſexcupla, neque motus diffor­<lb></lb>mis, quoniam maior tanto tardius mouebitur, quanto quod ue­<lb></lb>locius mouetur etiam minus erit, tamen proportio uelo citatis ma­<lb></lb>ioris ad minorem in æqualibus ſpatijs uigintiquincupla, ut ma­<lb></lb>ioris ad mediam quintupla, nam cum ſit ſexcupla in ambitu, <lb></lb>&amp; tricies moueatur uelocius comparatione totius, ſequitur, ut <lb></lb>proportio ſpatij, quod ſuperabit media ad ſpatium, quod ſu­<lb></lb>perabit maior in eiſdem temporibus, erit quintupla, ſemper ad un­<lb></lb>guem. </s>
          <s id="id001493">Et ita mediæ ad minorem quintupla, &amp; ideò maioris ad 


<pb pagenum="81" xlink:href="015/01/100.jpg"></pb>minorem uelo citas uiginti quincupla, ut non ſit difformis, neque <lb></lb>periculoſa, ut in rotis moletrinis, &amp; ſit diuiſa per medium iuxta <lb></lb>proportionem, cum ſit tanto uelocior minor media, quanto media <lb></lb>maiore. </s>
          <s id="id001494">Rurſus proportio partium maioris ad mediæ partes tripla <lb></lb>eſt ſcilicet ccclx ad cxx, &amp; mediæ ad <expan abbr="minorẽ">minorem</expan> tripla cxx ad xl, &amp; pro­<lb></lb>portio eſt ſexcupla, iterum igitur partes maioris ad mediam, &amp; me­<lb></lb>diæ ad minorem erunt in dupla proportione, utrobique, &amp; eſt pul­<lb></lb>chrum. </s>
          <s id="id001495">Ideò partes etiam minimæ rotæ erunt ſatis magnæ: nam <lb></lb>cum diameter ſit bes pedis, ambitus peripheriæ erit duorum pe­<lb></lb>dum. </s>
          <s id="id001496">1. unciarum uiginti quatuor: igitur diuiſa peripheria in xl par­<lb></lb>te r, unaquæque pars erit maior dimidia uncia.</s>
        </p>
        <p type="head">
          <s id="id001497">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001498">Et cum defuerit inſtrumentum, utemur menſura expulſu homi­<lb></lb>nis deſumpta, ſed non eſt adeò exacta. </s>
          <s id="id001499">Accedit aliud commodum, <lb></lb>quòd cum in una hora circumuertantur partes xxxvi, id eſt triginta <lb></lb>ſex mille: &amp; octauus orbis circumuertatur in totidem annis, tot <lb></lb>erunt momenta ex his in una hora, quot anni in uno circuitu ſtella­<lb></lb>rum fixarum. </s>
          <s id="id001500">Vt intelligamus, quàm breui tranſit una hora apud <lb></lb>nos, ita apud Deum, ut ita dicam (nam nulla in infinito proportio) <lb></lb>unus annus magnus, &amp; reditus rerum omnium. </s>
          <s id="id001501">Comparata etiam <lb></lb>rota minima ad rotam moletrini ſic ſe habet, quòd cùm modica ad­<lb></lb>eſt, uerſatur rota in una pulſatione: cum ſatis abundans quinquies, <lb></lb>aut ſexies cum immodica duo decies.</s>
        </p>
        <figure id="id.015.01.100.1.jpg" xlink:href="015/01/100/1.jpg"></figure>
        <pb pagenum="82" xlink:href="015/01/101.jpg"></pb>
        <p type="main">
          <s id="id001502"><arrow.to.target n="marg308"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001503"><margin.target id="marg308"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001504">Ex hoc ſequitur, quod homo ſi moueretur uelo citate motus ro­<lb></lb>tæ moletrinæ in ſex ebdomadibus perueniret ad ſydus Lunæ, nam <lb></lb>rotarum earum, quibus ferrum acuitur ſemidimetiens communi­<lb></lb>ter eſt bes unius paſſus, ideò dimetiens paſſus cum triente: ambi­<lb></lb>tus ergo quatuor paſſus, &amp; xxi pars, colligamus nunc integra, in <lb></lb>uno ictu pulſus circumagitur decies, id eſt paſſus xl, in hora ſunt <lb></lb>&lt;23&gt; pulſationes: in hora igitur ſpatium pertranſitum eſt cxl paſſuum <lb></lb>in M. horis, ergo erunt clx M. paſſuum addita parte xxi, erunt clxviij <lb></lb>M. paſſuum, &amp; tantum diſtat luna à terra: &amp; M. horæ ſunt dies penè <lb></lb>xlij, ebdomadæ ſcilicet ſex.</s>
        </p>
        <p type="main">
          <s id="id001505">Propoſitio octuageſima nona.</s>
        </p>
        <p type="main">
          <s id="id001506">Proportionem denſitatis aquæ ad aërem per pondera inuenire.</s>
        </p>
        <p type="main">
          <s id="id001507"><arrow.to.target n="marg309"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001508"><margin.target id="marg309"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001509">Contingit hoc multis modis: primum acceptis duabus ſphæru­<lb></lb>lis æqualibus ex cryſtali ſubſtantia unaque demiſſa ab altiſsima turri, <lb></lb>&amp; menſurato ictu per inſtrumentum præcedens, &amp; ſub totidem <lb></lb>momentis alia demiſſa in aquam, in de ſub eodem tempore dimen­<lb></lb>ſa altitudine, erit proportio ſpatij ad ſpatium, ut denſitatis aquæ, ad <lb></lb>denſitatem aëris. </s>
          <s id="id001510">Item emiſſa ſphærula per inſtrumentum in aërem, <lb></lb>in de in aquam: &amp; ſumpta proportione. </s>
          <s id="id001511">Et uidimus ſcorpionem, <lb></lb>qui <expan abbr="ſphærulã">ſphærulam</expan> creteam emittebat pedibus lxx, &amp; in aqua per unum <lb></lb>&amp; dimidium adeò, ut proportio fuerit, ut quinquaginta ad unum: <lb></lb>ideò eſt fallax experimentum in uiolento motu: nam cum emitte­<lb></lb>batur in aquam erat propè, &amp; ob id in ſummo robore: cùm in aë­<lb></lb>rem, emittitur ſenſim uis. </s>
          <s id="id001512">De hoc ergo loquar.</s>
        </p>
        <p type="main">
          <s id="id001513"><arrow.to.target n="marg310"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001514"><margin.target id="marg310"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001515">Et erumpentia ob id magis quàm è terra, et minus quàm ex aëre: <lb></lb>diuiditur enim aqua cum graue petit fundum, &amp; aqua feruet: &amp; eſt <lb></lb>mirabilius, quàm utile.</s>
        </p>
        <p type="main">
          <s id="id001516">Propoſitio nonageſima.</s>
        </p>
        <p type="main">
          <s id="id001517">Rationem impetus uiolenti extra miſsi ponderis ad æqualita­<lb></lb>tem reducere.</s>
        </p>
        <p type="main">
          <s id="id001518"><arrow.to.target n="marg311"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001519"><margin.target id="marg311"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001520">Sit uiolentum a quod moueatur per b c d e, e ſpatium, &amp; quia <lb></lb>uiolentum contrà nititur naturali, cadat ergo in planum in e: ſunt <lb></lb>ergo tria conſideran da, primum quod, ut dixi aliâs, motus uiolen­<lb></lb>tus pro certa diſtantia augetur, &amp; cauſam ibi reddidi, ut potè uſque <lb></lb>ad c, ſed hoc eſſet difficile cognitu. </s>
          <s id="id001521">Secundum, quod ubi incipit de­<lb></lb>creſcere, ſemper magis ac magis decreſcit propter naturalem ni­<lb></lb>xum contra operantem. </s>
          <s id="id001522">Tertium quod ubi deſcendere incipit, ibi <lb></lb>eſt æqualis uis uiolentum motum agens cum naturali. </s>
          <s id="id001523">Certum eſt <lb></lb>etiam quod motus æqualis intelligitur erecta ad perpendiculum <lb></lb>e f, donec occurrat a d: &amp; diuiſa tota b f per tempus, locus ergo, in <lb></lb>quo mouetur per tantum ſpatium, dicitur locus motus æqualis: 


<pb pagenum="83" xlink:href="015/01/102.jpg"></pb>qui ſit gratia exempli g h, cuius medium proportione ſit k, di­<lb></lb>co k conſiſtere propiorem f, quàm b, etiamſi æqualiter mouere­<lb></lb>tur. </s>
          <s id="id001524">Primum quòd in tota g f declinat, &amp; totus motus eſt lentior, <lb></lb>quàm in tota b g, &amp; tamen tardatur tantundem, ergo per commu­<lb></lb>nem animi ſententiam, k eſt propior f, quàm b. </s>
          <s id="id001525">Secundò, quia per <lb></lb>ſecundum ſuppo ſitum motus a uerſus f, continuè fit lentior, igitur <lb></lb>per communem animi ſententiam multò longius eſt tempus mo­<lb></lb>tus a k, quam f, &amp; tanto maius ſpatium. </s>
          <s id="id001526">Tertiò, quia motus ex b uer<lb></lb>ſus c augetur, &amp; ſi eſſet æqualis adhuc multò eſſet breuior k f quam <lb></lb>a k, igitur multò magis hoc modo, &amp; triplicata ratione. </s>
          <s id="id001527">Si ergo b k <lb></lb><figure id="id.015.01.102.1.jpg" xlink:href="015/01/102/1.jpg"></figure><lb></lb>eſſet ſexquiquarta ſolum ipſi k f, <lb></lb>erit b k dupla: fermè ex triplicata <lb></lb>ratione ipſi k f, &amp; iuxta eundem <lb></lb>modum ponemus mediam uim <lb></lb>xlvi paſsibus à ſcorpione a quam <lb></lb>&amp; hoc modo erit propè id quod eſt.</s>
        </p>
        <p type="head">
          <s id="id001528">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001529">Dubitat autem Philoſophus in mechanicis quæ nam uis ſit, quę <lb></lb>moueat lapidem iam excuſſum? </s>
          <s id="id001530">&amp; dubium non eſt quin ex parte ſit <lb></lb>aër motus tum ratione, quia mouetur ergo mouet, tum experimen <lb></lb>to, ut in fulminibus, &amp; his quæ uento impelluntur, ut hypophyſis, <lb></lb>ſed in ſcorpionibus &amp; arcubus &amp; pilis id non ſufficere uidetur. </s>
          <s id="id001531">Ita­<lb></lb>que uelut &amp; caliditas &amp; frigiditas in corporibus natura contrarijs <lb></lb>aliquandiu manent, &amp; agunt ita &amp; uiolentos motus, idque Alexan­<lb></lb>der &amp; Simplicius uolunt. </s>
          <s id="id001532">Inditio ſunt quòd mota &amp; emiſſa ex lon­<lb></lb>gioribus machinis quan quam non aërem continentibus, nec in­<lb></lb>anibus tamen, longius eijciunt ſagittas &amp; miſsilia, quoniam uis <lb></lb>illa firmius imprimitur, uelut etiam de lapidibus &amp; ferro, quod di­<lb></lb>utius in igne moram traxit, aut continuè follibus ignitum eſt, nam <lb></lb>etiam tanto tardius refrigeratur unum quod que horum, &amp; alia urit <lb></lb>&amp; accendit calore illo externo, quanquam natura frigidum ſit: di­<lb></lb>cemus autem &amp; de hoc ſuo loco.</s>
        </p>
        <p type="main">
          <s id="id001533">Propoſitio nonageſima prima.</s>
        </p>
        <p type="main">
          <s id="id001534">Proportionem grauis cubi, &amp; ſphærici æqualium in accliui, &amp; <lb></lb>deſcenſus eorum demonſtrare.</s>
        </p>
        <p type="main">
          <s id="id001535">Hic non pauca ſunt <expan abbr="cõſideranda">conſideranda</expan>: Primum <lb></lb><figure id="id.015.01.102.2.jpg" xlink:href="015/01/102/2.jpg"></figure><lb></lb>quòd hoc intelligi poteſt, uel de motibus at­<lb></lb>tractionis, uel impulſionis, uel inuerſionis. <lb></lb></s>
          <s id="id001536">Secundum quod omne, quod impellitur ſuperiùs, tantundem gra­<lb></lb>uat attractum, quantum ad deſcenſum, ſi ſit rotundum, nam qua­<lb></lb>drata, <expan abbr="etiã">etiam</expan> alia non deſcendunt ſponte in decliui, &amp; ſi ſit locus ualdè 


<pb pagenum="84" xlink:href="015/01/103.jpg"></pb>decliuis, tanto minus deſcendunt, quanto ſunt latiora. </s>
          <s id="id001537">Quia tamen <lb></lb>omnia difficiliùs deſcendunt ſphæricis, &amp; facilius quàm in plano, <lb></lb>ubi ponderant niſi per dimidium grauitatis, ideò proportio hæc <lb></lb>conſtat ex proportione anguli deſcenſus ad totum rectum, &amp; ma­<lb></lb>gnitudine ſuperficiei, qua incumbit ad pondus comparata. </s>
          <s id="id001538">Omne <lb></lb>enim graue, quanto grauius tam ad quietem, quàm ad motum na­<lb></lb>turalem potentius eſt: hoc enim perſpicuum eſt, quia quieti natu­<lb></lb>rali motus uiolentus, &amp; motui naturali quies uiolenta opponitur: <lb></lb>quia ergo maiore ui opus eſt ad motum præter naturam, ergo ſe­<lb></lb>cundum naturam etiam maiore ui quieſcit. </s>
          <s id="id001539">Aſſumpſimus ergo cu­<lb></lb>bum, ut magis notum. </s>
          <s id="id001540">Sphæra igitur in omni decliui deſcendit, <lb></lb>quia ut dictum eſt, nil habet quod reſiſtat ad motum: &amp; ipſa gra­<lb></lb>uior eſt in decliui, quàm in plano, quia c pun­<lb></lb>ctus cadit ultra e, ergo punctus contactus, &amp; <lb></lb><figure id="id.015.01.103.1.jpg" xlink:href="015/01/103/1.jpg"></figure><lb></lb>centrum grauitatis, &amp; centrum mundi, non ſunt <lb></lb>in una linea. </s>
          <s id="id001541">Si enim b c contangeretur, eſſet b c <lb></lb>plana. </s>
          <s id="id001542">Si uerò tangit, angulus eſt maior angulo <lb></lb>contactus, ergo cum neceſſarium ſit, æquidiſta­<lb></lb>re aliter non eſſet ſphæricum, oportet, ut eleue­<lb></lb>tur ex parte c, &amp; deſcendat uerſus b, &amp; ideò ut <lb></lb>continuetur motus. </s>
          <s id="id001543">Si uerò ſit in linea conta­<lb></lb>ctus b c f, &amp; æquidiſtet non erit, ut dixi punctus <lb></lb>contactus in linea centrorum, ſed in a c, cum ſuppoſitum ſit lineam <lb></lb>a d eſſe lineam centrorum: maior eſt ergo portio g c e, quàm reſi­<lb></lb>duum, ergo deſcendet in b. </s>
          <s id="id001544">Cubus uerò non deſcendet, niſi cum di­<lb></lb>midium d addito, quod intercipitur inter lineam mediam, &amp; quæ à <lb></lb>centro mundi ad punctum medium contactus uſque quò perueniat <lb></lb>ad oppoſitam partem, eam habuerit proportionem ad idem me­<lb></lb>dium eadem portione detracta, quem iuncta proportioni anguli <lb></lb>declinationis ad reſiduum recti dimidiam proportionem efficiat. <lb></lb></s>
          <s id="id001545">Eademque ratio aliorum planorum. </s>
          <s id="id001546">Dico præterea quòd motus <lb></lb>ſphæræ, &amp; etiam corporum rectarum ſuperficierum in deſcenſu <lb></lb>alius eſt æqualis, &amp; alius inæqualis, &amp; quaſi à latere, uelut ſi angu­<lb></lb>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb></lb>hoc, &amp; maximè ſi non retineatur æqualiter, &amp; difficile ſit in medio <lb></lb>retinere, propterea prolapſus hi melius <expan abbr="retinẽtur">retinentur</expan> duobus uinculis, <lb></lb>quàm in medio, non ſolum ob hanc æqualitatem, &amp; complexum <lb></lb>meliorem, ſed <expan abbr="etiã">etiam</expan>, quod omnes motus, omnes ponderum nixus fa<lb></lb>ciliùs cohibentur, &amp; <expan abbr="deducunt̃">deducuntur</expan> diuiſi in partes, &lt;08&gt; ſi toti contin <expan abbr="eant̃">eantur</expan>, <lb></lb>aut ui <expan abbr="trahãtur">trahantur</expan>. </s>
          <s id="id001547">Et ideo uincula in rami cibus duplicia dextra, &amp; ſini<lb></lb>ſtra ſcilicet in <expan abbr="eadẽ">eadem</expan> parte tamën longe ſunt meliora etiam ferreis, quæ <lb></lb>ſolum in medio nectantur.</s>
        </p>
        <pb pagenum="85" xlink:href="015/01/104.jpg"></pb>
        <p type="main">
          <s id="id001548"><arrow.to.target n="marg312"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001549"><margin.target id="marg312"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001550">Ex hoc etiam ſequitur, <lb></lb><figure id="id.015.01.104.1.jpg" xlink:href="015/01/104/1.jpg"></figure><lb></lb>quod cùm omne graue <lb></lb>ſpontè ſemper appropin­<lb></lb>quet centro mundi, &amp; a ſi <lb></lb>moueretur per planum e, <lb></lb>magis remoueretur à cen­<lb></lb>tro mundi, ut per e c per ea <lb></lb>quæ diximus, &amp; quoniam <lb></lb>linea ex centro mundi ad <lb></lb>c longior eſt, quàm ad e, <lb></lb>multò poteſt enim eſſe, ut <lb></lb>in proportione diametri <lb></lb>quadrati ad latus eius, &amp; <lb></lb>etiam maior. </s>
          <s id="id001551">ergo poterit <lb></lb>eſſe adeò parum decliuis <lb></lb>linea c d, ut c punctus ma­<lb></lb>gis diſter à centro mundi, <lb></lb>quàm d, &amp; tamen feretur <lb></lb>ex d in c motu naturali, ut demonſtratum eſt, ergo per purum mo­<lb></lb>tum naturalem poterit a remoueri à centro mundi. </s>
          <s id="id001552">Hoc uolui pro­<lb></lb>ponere, ut intelligeres in plano uero c e non moueri a ſponte, quia <lb></lb>c neceſſariò altior eſt d: ſi ergo mouebitur, non erit c e recta, ſed <lb></lb>pars proportionis circuli ſuperficiei terræ, quæ ſenſu à recta diſtin­<lb></lb>gui non poterit. </s>
          <s id="id001553">Hoc ergo eſt primum, ex quo ſequitur.</s>
        </p>
        <p type="main">
          <s id="id001554"><arrow.to.target n="marg313"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001555"><margin.target id="marg313"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001556">Quod aliquid poterit uideri decliue, in quo non deſcendet imò <lb></lb>erit, ut potè ſi aliqua linea obliqua eſſet inter c e, &amp; f e, illa eſſet decli­<lb></lb>uis ſpecie, &amp; re, &amp; tamen graue in illa non deſcenderet, quia à cen­<lb></lb>tro mundi magis remoueretur: hoc tamen eſt perdifficile factu, &amp; <lb></lb>maximè in parua diſtantia, uel etiam unius miliaris. </s>
          <s id="id001557">Atque hæc <lb></lb>in leuigatis.</s>
        </p>
        <p type="main">
          <s id="id001558">Propoſitio nonageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id001559">Proportionem ponderis æqualis iuxta longitudinis compara­<lb></lb>tionem demonſtrare.</s>
        </p>
        <figure id="id.015.01.104.2.jpg" xlink:href="015/01/104/2.jpg"></figure>
        <p type="main">
          <s id="id001560">Hoc eſt, quod Archimedes reliquit </s>
        </p>
        <p type="main">
          <s id="id001561"><arrow.to.target n="marg314"></arrow.to.target><lb></lb>intactum, cum eſſet maximè neceſſa­<lb></lb>rium, &amp; oſtendit magis abſtruſa, ſed <lb></lb>pace illius dixerim minus utilia. </s>
          <s id="id001562">Cum <lb></lb>ergo ſumpſiſſem uirgam b f ponderis <lb></lb>unciarum xxiij, fuiſſet b a uigeſima quarta pars, b f fuit pondus æ­<lb></lb>quilibrij in b appenſum librarum uiginti ſex cum dimidia: fuit igi­<lb></lb>tur proportio ponderis e f ad pondus f b, ut tredecim ferme ad 


<pb pagenum="86" xlink:href="015/01/105.jpg"></pb>unum. </s>
          <s id="id001563">Et rurſus feci a b quintam partem a f, &amp; fuit a b unciarum <lb></lb>quatuor, &amp; pondus quod æquauit librarum quatuor, ideò du­<lb></lb>plum ad pondus b f, ſicut c f ad c b: conſtat enim quòd pondus ap­<lb></lb>penſum eſt æquale ponderi cf. </s>
          <s id="id001564">Et rurſus poſui b a quartam partem <lb></lb>b f, &amp; fuit pondus, quod æquauit in b duæ libræ: ex quo manife­<lb></lb>ſtum eſt, quòd proportio c f ad c b eſt ſemper uelut ponderis c f ad <lb></lb>totam b f. </s>
          <s id="id001565">Et hoc eſt, ac ſi dicamus, quòd proportio ponderis c f ad <lb></lb>totam eſt confuſa ex proportione e f ad c b, &amp; c f, quod eſt 1 p. </s>
          <s id="id001566">Id <lb></lb><arrow.to.target n="marg315"></arrow.to.target><lb></lb>etiam declaratum eſt in primo de Subtilitate. </s>
          <s id="id001567">Proponatur ergo <lb></lb>lemma, iam ſic proportio ponderis cf ad pondus b c, eſt primum <lb></lb>ut longitudinis cf, ſi eſſet ſuſpenſa in medio ad longitudinem b c, <lb></lb>quia ſupponuntur proportione ſimiles ſuis longitudinibus ma­<lb></lb>gnitudines, &amp; pondera. </s>
          <s id="id001568">At c f ſuſpenſa in c, tanto eſt grauior pon­<lb></lb>dere proprio, quanto proportionis longitudinis cf ad cb quadra­<lb></lb>tum, quia in ſe ducitur proportio: igitur proportio ponderis c f in <lb></lb>loco ſuo ad b c pondus eſt confuſa ex proportione longitudinis <lb></lb>cf ad c b, &amp; quadratis eiuſdem proportionis longitudinis cf ad c <lb></lb>b. </s>
          <s id="id001569">Sed quadratum proportionis longitudinis cf ad cb eſt æquale <lb></lb>producto proportionis longitudinis c f in ipſam c f, propterea <lb></lb>quòd ex proportione longitudinis cf ad cb in ipſam c b fit c f, igi­<lb></lb>tur proportio ponderis c f ad pondus c b eſt confuſa ex propor­<lb></lb>tione ponderis c f ad pondus c b, &amp; proportione ponderis cf alicu<lb></lb>ius ſe habentis ad pondus cf, ut cf longitudo ad longitudinem <lb></lb>c b, igitur proportio ponderis cf ad pondus b f, ut cf ad c b in lon­<lb></lb>gitudine, quod erat probandum.</s>
        </p>
        <p type="margin">
          <s id="id001570"><margin.target id="marg314"></margin.target>C<emph type="italics"></emph>om.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001571"><margin.target id="marg315"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>diff.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001572">Propoſitio nonageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id001573">Propter quid in concuſsione etiam leui nauis loco moueatur <lb></lb>oſtendere. </s>
          <s id="id001574">Vnde manifeſtum eſt, duas naues ſibi inuicem occurſan <lb></lb>tes retrocedere, &amp; quantum retrocedant ambæ.<lb></lb><arrow.to.target n="marg316"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001575"><margin.target id="marg316"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001576">Proponatur, quod proportio motus grauis in a d graue in aqua <lb></lb>ſit, uelut proportio ponderis attracti in terra ad denſitatem aquæ <lb></lb>cum profunditate, nam ubi pondus ſupernataret aquæ, quia aqua <lb></lb>eſt rotunda, eſt ac ſi tangeret in puncto. </s>
          <s id="id001577">Quare per demonſtrata ſu­<lb></lb>periùs mouebitur à quacunque ui, ergo nixus contrarius aduenit ob </s>
        </p>
        <p type="main">
          <s id="id001578"><arrow.to.target n="marg317"></arrow.to.target><lb></lb>profunditatem, &amp; aquæ denſitatem, ſed quanto aqua denſior eſt, <lb></lb>tanto minus nauis deſcendit, &amp; quanto minus denſa, tanto magis: <lb></lb>ergo pari modo fermè redduntur mobiles, &amp; in aqua dulci &amp; ſalſa, <lb></lb>ubi naues ſint ſimiles forma, pondere, magnitudine. </s>
          <s id="id001579">Quia ergo ne­<lb></lb>ceſſe eſt tabulam nauis eſſe duriorem, quam aqua ad reſiſtendum, <lb></lb>ergo pars maior ictus mouebit primo nauim, quam tabulam pe­<lb></lb>netret, cum ergo quod facilius eſt, præcedat, difficilius ergo naues 


<pb pagenum="87" xlink:href="015/01/106.jpg"></pb>utrinque mouebuntur, &amp; quia inter duos quoſcunque motus contra­<lb></lb>rios <expan abbr="nõ">non</expan> eſſe eos, ut utar uocabulo Auerrois quinto Phyſicorum, ne­<lb></lb>ceſſe eſt, ut intercedat quies media, &amp; in quiete ab ictu, ut uiſum eſt <lb></lb>ſuperius, oportet, ut quod excipit ictum uel loco moueatur, uel ce­<lb></lb><arrow.to.target n="marg318"></arrow.to.target><lb></lb>dat, &amp; ictus penetret, uel aër non condenſetur ob tarditatem ultra <lb></lb>metam, nec retro cedere poteſt ex ſuppoſito, &amp; ictus eſt magnus, <lb></lb>clarum eſt, quod oportet, ut cedat, &amp; ſi durum ſit confringatur. <lb></lb></s>
          <s id="id001580">Proportio ergo receſſus ad ictum eſt ut temporis, &amp; magnitudinis <lb></lb>partis, quæ cedit, &amp; retro ceſſus poſito ictu tanquam monade.</s>
        </p>
        <p type="margin">
          <s id="id001581"><margin.target id="marg317"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 40.</s>
        </p>
        <p type="margin">
          <s id="id001582"><margin.target id="marg318"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 74.</s>
        </p>
        <p type="main">
          <s id="id001583">Propoſitio nonageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id001584">Si quantitas aliqua nota atque proportio erit producta quantitas <lb></lb>nota ſimiliter. </s>
          <s id="id001585">Et ſi duæ proportiones notæ fuerint, erit producta <lb></lb>ex his atque diuiſa, coniunctaque, atque detracta nota. </s>
          <s id="id001586">Et ſi fuerit totius <lb></lb>ad partem proportio nota erit, &amp; ad aliam partem nota, &amp; alterius <lb></lb>partis ad alteram uno minor. </s>
          <s id="id001587">Et ſi fuerit partis ad partem, erit ad to<lb></lb>tum monade minor atque nota. </s>
          <s id="id001588">Et ſi fuerit unius quantitatis ad duas <lb></lb>quantitates proportio nota, erit &amp; confuſa ex eis nota. </s>
          <s id="id001589">Et ſi fuerint <lb></lb>trium quantitatum omiologarum, aut quatuor analogarum, o­<lb></lb>mnes præter unam cognitæ erunt, &amp; illa alia cognita.</s>
        </p>
        <figure id="id.015.01.106.1.jpg" xlink:href="015/01/106/1.jpg"></figure>
        <p type="main">
          <s id="id001590">Sit quantitas a b &amp; ducta in d proportionem, <lb></lb><arrow.to.target n="marg319"></arrow.to.target><lb></lb>producat b c: dico quod duobus quibuslibet ex <lb></lb>his cognitis, erit cognitum tertium: nam cogni­<lb></lb>tum quodlibet dicitur in comparatione ad ſimpliciter cognitum, <lb></lb>quod eſt unum per ſe omnibus cognitum. </s>
          <s id="id001591">Ob id Arithmetica eſt <lb></lb>prima omnium diſciplinarum, quia habet principium cognitum, <lb></lb>&amp; id, quod eſt, ad principium comparatum cognitum in illius com<lb></lb>paratione: neque aliter cognitum dici poteſt. </s>
          <s id="id001592">Quia ergo d cognita <lb></lb>eſt, erunt monades, &amp; partes cognitæ in ea: aliter non eſſet cognita <lb></lb>b a, igitur cum cognita ſit, erit cognita per ſingulas monades, quan<lb></lb>ta ſit. </s>
          <s id="id001593">Et ſi diceres quòd b a non eſt cognita per partem monadis: <lb></lb>dico quod pars monadis non eſt incognita, quia cum monades <lb></lb>ſunt cognitæ, eſſet d incognita. </s>
          <s id="id001594">Omnes enim, quod componitur ex <lb></lb>cognito &amp; incognito, eſt incognitum, quia cognitum ſolum ratio­<lb></lb>ne partis cognitæ. </s>
          <s id="id001595">Si ergo pars monadis eſt cognita, erit pars a b <lb></lb>quælibet prout ex monade componitur ſimpliciter cognita. </s>
          <s id="id001596">Su­<lb></lb><arrow.to.target n="marg320"></arrow.to.target><lb></lb>pereſt, ut ſolum pars partis: &amp; dico quod illa etiam eſt cognita: <lb></lb>quia ſi pars ab eſſet, monas eſſet cognita: eſſet enim pars ipſa.</s>
        </p>
        <p type="margin">
          <s id="id001597"><margin.target id="marg319"></margin.target>C<emph type="italics"></emph>om.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001598"><margin.target id="marg320"></margin.target>E<emph type="italics"></emph>x ſecunda <lb></lb>animi com­<lb></lb>muni ſenten<lb></lb>tia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001599">Sed ſi ſit pars, erit ſumpta ſecundum partem monadis ipſius, <lb></lb>ideò erit cognita iuxta nomen, uelut dimidium eſt dimidium mo­<lb></lb>nadis, dimidium tertiæ partis monadis eſt cognitum, quia tertia <lb></lb>pars eſt cognita, &amp; ſcimus, quanta pars aſſumatur illius. </s>
          <s id="id001600">Ergo ſi a b, 


<pb pagenum="88" xlink:href="015/01/107.jpg"></pb>&amp; d cognitæ ſunt erit &amp; b c, quod eſt primum. </s>
          <s id="id001601">Per hæc eadem pro­<lb></lb>bantur quatuor ſequentes partes eodem modo. </s>
          <s id="id001602">Sexta ſic: ſit pro­<lb></lb>portio a c ad c b, nota igitur in comparatione ad monadem, ſed pro <lb></lb>portio a c ad c b b a eſt monas, igitur proportio a c ad a b nota eſt, <lb></lb>quoniam aliter non poſſet dici proportio a c ad b c nota. </s>
          <s id="id001603">Aliter, ſit <lb></lb>proportio a c ad c b e nota, ex ſuppoſito igitur conuerſa nota quæ <lb></lb>ſit f ex f, igitur in a c fit b c ex g in a c, fiat a b ergo ex a c in f g fit a, c igi<lb></lb>tur f g eſt monas, f autem nota eſt, igitur in comparatione ad mona­<lb></lb><arrow.to.target n="marg321"></arrow.to.target><lb></lb>dem, ergo reſiduum g notum. </s>
          <s id="id001604">Cum uerò proportio a c ad c b com­<lb></lb>ponatur ex proportione a b b c ad b c, &amp; proportio b c ad b c ſit <lb></lb>monas, &amp; proportio a c ad b c nota, erit proportio a b ad b c cogni<lb></lb><arrow.to.target n="marg322"></arrow.to.target><lb></lb>ta, &amp; monade minor proportione a c ad b c. </s>
          <s id="id001605">Per idem octaua pars <lb></lb><figure id="id.015.01.107.1.jpg" xlink:href="015/01/107/1.jpg"></figure><lb></lb>demonſtrabitur. </s>
          <s id="id001606">Inde ſit proportio a ad b, &amp; ad c no­<lb></lb>ta, erit ergo b, &amp; c ad a nota, quare b c ad a nota, ſed <lb></lb><arrow.to.target n="marg323"></arrow.to.target><lb></lb>hæc eſt conuerſa ad b c confuſa, igitur proportio a <lb></lb>ad b confuſa nota eſt. </s>
          <s id="id001607">Vltimum ſit, ſint a b c omiologæ, &amp; ſint a &amp; b <lb></lb><arrow.to.target n="marg324"></arrow.to.target><lb></lb>notæ duo, quod c nota eſt, nam a b, ſi notæ ſunt, nota eſt proportio <lb></lb>earum. </s>
          <s id="id001608">Ergo &amp; proportio b ad c ergo per primam partem huius <lb></lb><arrow.to.target n="marg325"></arrow.to.target><lb></lb>cum ſit b nota, exit &amp; c. </s>
          <s id="id001609">Et ſi ponantur a c notæ, dico, quòd b nota <lb></lb>erit: nam proportio a c ad c nota eſt, quæ ſit d, igitur d ad monadem <lb></lb>ut a ad c, ergo latus notum erit, quod ductum in c producit b, b igi­<lb></lb><arrow.to.target n="marg326"></arrow.to.target><lb></lb>tur nota. </s>
          <s id="id001610">Et ſimiliter in analogis ſint a b c notæ: &amp; ideò erit propor­<lb></lb>tio a ad b nota ergo c ad d. </s>
          <s id="id001611">cumque c nota ſit, ergo per primam par­<lb></lb>tem huius erit d nota, quod fuit demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id001612"><margin.target id="marg321"></margin.target>P<emph type="italics"></emph>er demon­<lb></lb>ſtrat.<emph.end type="italics"></emph.end> 12. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001613"><margin.target id="marg322"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. P<emph type="italics"></emph>et.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001614"><margin.target id="marg323"></margin.target>E<emph type="italics"></emph>x demonſt.<emph.end type="italics"></emph.end><lb></lb>12. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001615"><margin.target id="marg324"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 14. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001616"><margin.target id="marg325"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001617"><margin.target id="marg326"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 2. A<emph type="italics"></emph>nimi <lb></lb>ſententia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001618">Propoſitio nonageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id001619">Cuiuſuis trigoni rectanguli, aut cuius duo anguli ſint in dupla <lb></lb>proportione, aut qui circulo inſcriptus ſit cognita quantitate uni­<lb></lb>us lateris in comparatione ad dimetientem ſi proportio <expan abbr="duorũ">duorum</expan> la­<lb></lb>terum cognita fuerit, erunt omnia eius latera cognita.<lb></lb><arrow.to.target n="marg327"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001620"><margin.target id="marg327"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001621">Non de cognitione propinqua <expan abbr="aſtronomorũ">aſtronomorum</expan>, de qua abundè ab <lb></lb>Heber tractatum eſt, ſed de exacta, de qua ſuperius egi nunc ſermo </s>
        </p>
        <p type="main">
          <s id="id001622"><arrow.to.target n="marg328"></arrow.to.target><lb></lb>eſt: ſit igitur primum a b c trigonus orthogonius: &amp; ſit a rectus, &amp; <lb></lb>proportio <expan abbr="duorũ">duorum</expan> laterum cognita, dico, quod omnia latera cognita <lb></lb><arrow.to.target n="marg329"></arrow.to.target><lb></lb><figure id="id.015.01.107.2.jpg" xlink:href="015/01/107/2.jpg"></figure><lb></lb>erunt: nam ſit proportio, gratia exempli, <lb></lb>a b ad b c, erit ergo quadrati a b ad qua­<lb></lb>dratum b c cognita, quia duplicata: at <lb></lb>quadrata a b, &amp; a c perficiunt quadratum <lb></lb>b c, igitur proportio quadrati a b ad a c et <lb></lb>eſt 1 p: cognita erit, quare &amp; a b ad a c, &amp; <expan abbr="eodẽ">eodem</expan> modo a c ad b c: quod <lb></lb>eſt primum. </s>
          <s id="id001623">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb></lb>ſub quadruplum quadrato a b quare ſubtriplum quadrato a c igi­<pb pagenum="89" xlink:href="015/01/108.jpg"></pb>tur ſi a b ponatur 1 b c erit 2, &amp; a c &lt;02&gt; 3. Rurſus ponatur angulus b <lb></lb>duplus angulo c qualiſcunque ſit, erit per demonſtrata ſuperius pro­<lb></lb>portio a b b c ad a c, ut a c ad a b, ſi igitur nota ſit proportio a c ad <lb></lb>a b, erit nota proportio a b b c ad a b per præcedentem. </s>
          <s id="id001624">Ergo per <lb></lb>eandem omnia nota ſcilicet b c ad b a, &amp; b c ad c a. </s>
          <s id="id001625">Et ſi eſſet nota <lb></lb>proportio a b ad b c, dico, quod eſſent nota omnia, nam nota eſſet <lb></lb>a b, &amp; b c, &amp; quod fit ex a b in ipſum aggregatum. </s>
          <s id="id001626">Sed hoc eſt æ­<lb></lb><arrow.to.target n="marg330"></arrow.to.target><lb></lb>quale quadrato a c, igitur notum eſt quadratum a c ergo a c: igitur <lb></lb>proportio a b b c ad a c, &amp; a c ad a b. </s>
          <s id="id001627">Vt ſi a b eſſet 4 b c 5, eſſet a b b c <lb></lb>9 ducta in a b, quæ eſt, fit 36, cuius latus eſt b a c ſcilicet. </s>
          <s id="id001628">Et ſi eſſet <lb></lb>trigonus aliquis in circulo, cuius proportio duorum laterum ſit co<lb></lb>gnita ad dimetientem relata, ſequitur per demonſtrata ſupe­<lb></lb>rius, quod etiam tertium latus erit cognitum in comparatione ad <lb></lb>eadem, &amp; ideo etiam proportio illorum laterum ad unguem co­<lb></lb>gnita erit.</s>
        </p>
        <p type="margin">
          <s id="id001629"><margin.target id="marg328"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 97.</s>
        </p>
        <p type="margin">
          <s id="id001630"><margin.target id="marg329"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001631"><margin.target id="marg330"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 17.</s>
        </p>
        <p type="main">
          <s id="id001632">Multa præterea cognita eſſent in hoc genere, quæ nunc præter­<lb></lb><arrow.to.target n="marg331"></arrow.to.target><lb></lb>mitto, quia non ſunt ad finem neceſſaria. </s>
          <s id="id001633">Alia præterea per diligen­<lb></lb>tem inquiſitionem maioris artis quàm alias edidimus. </s>
          <s id="id001634">tum uerò <lb></lb>etiam per nouas demonſtrationes.</s>
        </p>
        <p type="margin">
          <s id="id001635"><margin.target id="marg331"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001636">Propoſitio nonageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id001637">Cum in perſpicuum denſum radij luminoſi inciderint, quatuor <lb></lb>fiunt luminis genera.<lb></lb><arrow.to.target n="marg332"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001638"><margin.target id="marg332"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001639">Sit ſol a, &amp; perſpicuum denſum, exempli gratia, ut ampula <lb></lb>magna aqua plena b c d, &amp; ſi ſit rotunda accendit ignem ex ad­<lb></lb>uerſo ut in e. </s>
          <s id="id001640">Dico ergo in b c d eſſe quatuor genera luminis. </s>
          <s id="id001641">Pri­<lb></lb>mum quod eſt ualidius, &amp; rectà tranſit, ualidius enim eſt, quod <lb></lb>tranſit quàm quod tranſire non poteſt, &amp; etiam quia, ut dixi, <lb></lb>ignem accendit. </s>
          <s id="id001642">Secundum eſt quod colligitur in ampula, &amp; dein­<lb></lb>de ſpargitur <expan abbr="circũcircà">circuncircà</expan>, nam id ualidius eſt, quia penetrat, &amp; reſilit <lb></lb>quàm quod non penetrat, aut ſi penetrat, non ſpargitur, &amp; hoc dif­<lb></lb>funditur circa uas, nec reflectitur rectè, ſed quaſi intro colligitur, &amp; <lb></lb>diuerſa ratione diffunditur, eſt tamen imbecillius primo, ut dictum <lb></lb>eſt. </s>
          <s id="id001643">Tertium genus eſt, quod illuminat intus ingrediendo, ſed non <lb></lb>ſpargitur, &amp; hoc eſt debilius ſecundo, quia <expan abbr="nõ">non</expan> poteſt ſpargi. </s>
          <s id="id001644">Quar­<lb></lb><figure id="id.015.01.108.1.jpg" xlink:href="015/01/108/1.jpg"></figure><lb></lb>tum eſt, quod non ingreditur omnino, ſed refle­<lb></lb>ctitur, iſtud eſt abſque dubio imbecillimum, quo­<lb></lb>niam penetrare non poteſt. </s>
          <s id="id001645">Et licet in ſpeculis <lb></lb>concauis radius reflexus uideatur eſſe ualidior, <lb></lb>ſtatim enim accendit ignem, hoc non contin­<lb></lb>git, niſi quia in ſpeculo cauo radij omnes col­


<pb pagenum="90" xlink:href="015/01/109.jpg"></pb><expan abbr="ligunt̃">liguntur</expan> ob <expan abbr="opacũ">opacum</expan>, quod à tergo eſt, neque <expan abbr="ſpargunt̃">ſparguntur</expan>, neque <expan abbr="tranſeũt">tranſeunt</expan>, neque<lb></lb> combibuntur, ut ita dicam ſed omnes <expan abbr="reflectũtur">reflectuntur</expan>. </s>
          <s id="id001646">Ex quo colligitur <lb></lb>quincuplex ordo radiorum iuxta rationem uirium, primus eſt refle<lb></lb><expan abbr="xorũ">xorum</expan> à ſpeculo <expan abbr="cõcauo">concauo</expan>, &amp; hi ſunt <expan abbr="potẽtiſsimi">potentiſsimi</expan> ob <expan abbr="rationẽ">rationem</expan> <expan abbr="dictã">dictam</expan>, poſt <lb></lb>quos ſunt radij, qui tranſeunt per perſpicuum maximè rotundum, <lb></lb>qui &amp; ipſi generant ignem, &amp; debiliorem primo, deinde reliqui <lb></lb>tres ſequentes ſupra dicti. </s>
          <s id="id001647">Sextus eſt radiorum, qui reflectuntur à <lb></lb>rebus non nitidis, ut à muris, &amp; tabulis, nam omnia dura reflectunt <lb></lb>&amp; etiam mollium pleraque, &amp; hæc reflexio eſt fermè infinita, &amp; ob id <lb></lb>cubicula etiam in angulis illuminantur.</s>
        </p>
        <p type="main">
          <s id="id001648"><arrow.to.target n="marg333"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001649"><margin.target id="marg333"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001650">Ex hoc ſequitur, quòd Luna remittit lumen, non reflectit, nam <lb></lb>ſecus non illuminaret to tum orbem, ſed ſolum portionem oppo­<lb></lb>ſitam Soli, &amp; hoc etiam rarò, ergo combibitur, &amp; illuſtrat circun­<lb></lb>circa ubique.</s>
        </p>
        <p type="main">
          <s id="id001651"><arrow.to.target n="marg334"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001652"><margin.target id="marg334"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001653">In ſtellis lumen Solis pertranſit aliter, ſi reflecteretur, non illumi­<lb></lb>naret nos, aut apparerent, uelut cometæ, quia pars una eſſet clarior <lb></lb>reliqua, &amp; ſi conbiberent lumen, non uiderentur æquè claræ, cum <lb></lb>Sol eſſet propinquus, aut remotus.</s>
        </p>
        <p type="main">
          <s id="id001654"><arrow.to.target n="marg335"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001655"><margin.target id="marg335"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001656">Luna tota intus illuminatur à Sole, quoniam ſi ante coniun­<lb></lb>ctionem illuminatur à ſiniſtra parte, &amp; combibit lumen per cor­<lb></lb>rolarium primum, &amp; poſt coniunctionem illuminatur à dex­<lb></lb>tra, &amp; combibit pariter lumen, ergo eſt tota naturæ perſpicuæ, ſed <lb></lb>uidetur obſcura ex aduerſo, propterea quòd radij ualidiores refle­<lb></lb>xi illuſtrant illam ex parte Solis, diffugiunt à contraria, quod ma­<lb></lb>nifeſtè apparet in ampula expoſita Soli. </s>
          <s id="id001657">Pars enim clarior uerſus <lb></lb>Solem uidetur, quam ex aduerſo, hoc autem longè magis in Luna <lb></lb>ob diſtantiam.</s>
        </p>
        <p type="main">
          <s id="id001658"><arrow.to.target n="marg336"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001659"><margin.target id="marg336"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id001660">In omni Solis eclipſi fit colectio radiorum ad aſpectum, &amp; <lb></lb>ideo in regione illa, in qua centrum Solis integitur à centro Lunæ, <lb></lb>&amp; ubicunque fit, fit incendium per tertium corrolarium. </s>
          <s id="id001661">Hoc autem <lb></lb>fit ſemper in quauis coniunctione, &amp; dum Luna ſilet in regione ae­<lb></lb>ris, ſed terris non ſecundùm centrum, uerùm ad latitudinem, &amp; ad <lb></lb>Orientem ante coniunctionem cum Sole, &amp; ad Occidentem poſt: <lb></lb>ſed centra non ſunt in linea uiſus.</s>
        </p>
        <p type="main">
          <s id="id001662"><arrow.to.target n="marg337"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001663"><margin.target id="marg337"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 5.</s>
        </p>
        <p type="main">
          <s id="id001664">Ex hoc ſequitur, quod oportet ſubſtantiam Lunæ eſſe ualde cla­<lb></lb>ram, cum uideamus ab ampula tam paruum lumen diffundi, &amp; ra­<lb></lb>rum, à Luna uerò in uniuerſum orbem, &amp; tam copioſum, ut neceſ­<lb></lb>ſarium ſit ſubſtantiam Lunæ eſſe denſam, &amp; lucidam ualde.</s>
        </p>
        <p type="head">
          <s id="id001665">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001666">Et ſi quis dicat, quòd ſi incendium illud fieri poſſet in hora ecli­<lb></lb>pſis, ſequeretur, quòd ut in ampula in medio Lunæ uideretur ma­


<pb pagenum="91" xlink:href="015/01/110.jpg"></pb>gnus ſplendor, referens corpus Solis. </s>
          <s id="id001667">Propterea dico, quòd uel ac­<lb></lb>cidit, quia homo non poteſt ea hora intueri Solem, &amp; etiam eſt im­<lb></lb>peditus à radijs circumſtantibus, cuius indicio eſt, quod in ſpe­<lb></lb>culo poſito in aqua, ſimile uidetur ſtellulæ in centro Lunę: &amp; hic eſt <lb></lb>ſplendor Solis collectus in centro Lunæ. </s>
          <s id="id001668">poſſet etiam dici, quòd <lb></lb>Luna circa medium propter maculam non admitteret lumen, &amp; ita <lb></lb>eſſet inæqualium partium.</s>
        </p>
        <p type="main">
          <s id="id001669">Propoſitio nonageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id001670">Motum inuerſionis in figuris in comparatione ad motum ſphæ<lb></lb>ræ in plano inueſtigare.</s>
        </p>
        <p type="main">
          <s id="id001671"><arrow.to.target n="marg338"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001672"><margin.target id="marg338"></margin.target>C<emph type="italics"></emph>om.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001673">Voco motum inuerſionis, qui ſimilis eſt motui ſphæræ, ſcili­<lb></lb>cet circumuertendo graue à uertice, &amp; manifeſtum eſt, quòd in <lb></lb>quacunque figura, qua graue inſidet plano per punctum ue­</s>
        </p>
        <p type="main">
          <s id="id001674"><arrow.to.target n="marg339"></arrow.to.target><lb></lb>lut ouata ipſum mouetur à quauis ui, ſed ſi inſideat per ſuperfi­<lb></lb>ciem, quanto maior eſt, &amp; humilior, tanto difficilius mouetur, <lb></lb>ideò in corpore uiginti baſium, quòd inter regularia uocata, plu­<lb></lb>res habet, ſuperficies pro ratione æqualis ponderis, motus erit <lb></lb>longe facilior. </s>
          <s id="id001675">Alia cauſa eſt inæqualitas partium, unde quæ ro­<lb></lb>tunda ſunt, quia prominent, facile mouentur, &amp; cum partes me­<lb></lb>diæ inſiſtant plano, quanto minores erunt tanto facilius moue­<lb></lb>buntur ratione ponderis. </s>
          <s id="id001676">Vnde patet, quòd corpora ouata faci­<lb></lb>lius mouentur, etiam quàm ſphærica, habent enim partem me­<lb></lb>diam minorem, &amp; paria ſunt ratione inceſſus plani, ſed aëris mul­<lb></lb>titudine tardius, quoniam enim ſphæra ſub æquali ambitu plus <lb></lb>continet corporis, ergo ouatum æquale ſphæræ habet maio­<lb></lb>rem ambitum ipſa ſphæra. </s>
          <s id="id001677">Hæc autem à Theone partim de­<lb></lb>monſtrata ſunt, partim ab Archimede, &amp; partim à nobis, ergo <lb></lb>motus ouati eſt fermè æqualis motui ſphæræ, &amp; tardior eſt con­<lb></lb><figure id="id.015.01.110.1.jpg" xlink:href="015/01/110/1.jpg"></figure><lb></lb>citatus, quàm ſphæræ, quia à ma­<lb></lb>iore excipitur aëre, &amp; partes exte­<lb></lb>riores non ita incumbunt in me­<lb></lb>dium ſecundum longitudinem. </s>
          <s id="id001678">Cu­<lb></lb>bus uero tardior eſt propter æqua­<lb></lb>litatem, &amp; latitudinem ſuperficiei in­<lb></lb>ferioris, omnium <expan abbr="autẽ">autem</expan> minime pro­<lb></lb>pter has cauſas conus ambligonius, <lb></lb>&amp; quanto magis fuerit, ratio uero <lb></lb>eleuationis eſt, ut ſit cubus b c, cuius <lb></lb>medium grauitatis ſit b ſuper pla­


<pb pagenum="92" xlink:href="015/01/111.jpg"></pb>no de, &amp; eleuetur ex a, &amp; manifeſtum eſt, quod inſidebit per totam <lb></lb>lineam c f ipſi plano, &amp; proportio grauitatis totius ſuſpenſi in com<lb></lb>paratione ad grauitatem eius, qui inuertit, eſt, uelut proportio par­<lb></lb>tis terminatæ ad lineam c f uerſus eum, qui eleuat ad partem, quæ <lb></lb>ultra eſt, cum uerò hæ partes notæ ſint iuxta perpendiculum ex <lb></lb>centro grauitatis, manifeſtum eſt, quod ſciemus pondus corporis <lb></lb>a b cf, dum inuertitur in quo cunque ſitu ad pondus eius, dum ſu­<lb></lb>ſpenditur, &amp; clarum eſt, quòd cùm centrum, &amp; medium grauitatis <lb></lb>fuerint in una linea per c f, tunc nulla erit grauitas.</s>
        </p>
        <p type="margin">
          <s id="id001679"><margin.target id="marg339"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 40.</s>
        </p>
        <p type="main">
          <s id="id001680">Propoſitio nonageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id001681">Proportionem ponderum æqualium per differentiam angulo­<lb></lb>rum inuenire.<lb></lb><arrow.to.target n="marg340"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001682"><margin.target id="marg340"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001683">Sit a b, quæ ſi appenſa eſſet ad æquidi­<lb></lb><figure id="id.015.01.111.1.jpg" xlink:href="015/01/111/1.jpg"></figure><lb></lb>ſtantem terræ ſuperficiei, nulla ui poſſet ele</s>
        </p>
        <p type="main">
          <s id="id001684"><arrow.to.target n="marg341"></arrow.to.target><lb></lb>uari, inflectatur ergo ad c punctum, omiſſa <lb></lb>c g, &amp; manifeſtum eſt, quod ſi b c inſiſteret <lb></lb><arrow.to.target n="marg342"></arrow.to.target><lb></lb>ad perpendiculum, ponderaret a c ſi eſſet in <lb></lb>æquilibrio, ponatur ergo accliuis in c d per <lb></lb>notum angulum. </s>
          <s id="id001685">Quia igitur b c ad c a no­<lb></lb>ta eſt, erit dicta ſuperiùs notum pondus <lb></lb>b h, poſita h c æquali c a, quare totius a b, <lb></lb>&amp; iam fuit e k notum, &amp; punctus d notus: <lb></lb>hoc enim infrà demonſtrabitur, qualis igitur proportio lineæ <lb></lb><arrow.to.target n="marg343"></arrow.to.target><lb></lb>tranſuerſæ dl ad lineam deſcendentem d m, talis differentiæ pon­<lb></lb>derum c m, &amp; c e, id eſt partis ad partem. </s>
          <s id="id001686">hæc autem inferiùs de­<lb></lb>monſtrabuntur. </s>
          <s id="id001687">Neque enim abſurdum eſt in materijs miſtis, ali­<lb></lb><arrow.to.target n="marg344"></arrow.to.target><lb></lb>quando uti nondum demonſtratis cum fuerint mathematica, quia <lb></lb>obtinent principij rationem, quod etiam facit Archimedes. </s>
          <s id="id001688">Ma­<lb></lb>nifeſtum eſt autem, quod in angulo m c d recti dimidio, propor­<lb></lb>tio media erit. </s>
          <s id="id001689">Sed hoc bifariam contingere poteſt ſcilicet, ut ſit <lb></lb>media, per quantitatem, &amp; per proportionem, eſt autem media, ut <lb></lb><arrow.to.target n="marg345"></arrow.to.target><lb></lb>demonſtrabitur infrà ſecundum proportionem l d ad l e, propo­<lb></lb>natur ergo c e b, erit latus quadrati &lt;02&gt; 72, igitur latus octogoni eſt <lb></lb>&lt;02&gt; v: 72 m: &lt;02&gt; 2592, &amp; latus reſidui &lt;02&gt; v: 72 p: &lt;02&gt; 2592. quadrata er­<lb></lb>go partium baſis differunt in &lt;02&gt; 10368. Quare partes baſis ſunt <lb></lb>6 p: &lt;02&gt; 18, &amp; 6 m: &lt;02&gt; 18 ſcilicet l e, l d autem eſt &lt;02&gt; 18, igitur differen­<lb></lb>tia, &amp; proportio eſt, qualis &lt;02&gt; 18 ad 6 m: &lt;02&gt; 18 fermê, ut 17 ad 7, &amp; ta­<lb></lb>lis eſt proportio ponderis c d ad pondus c e ratione in crementi, <lb></lb>ſeu differentiæ. </s>
          <s id="id001690">Vt ſi pondus in c e eſſet decem librarum in c in 


<pb pagenum="93" xlink:href="015/01/112.jpg"></pb>quadraginta erit in c d triginta unius cum quarta, ſed proportionis <lb></lb>ratione eſſet uiginti octo cum tertia.</s>
        </p>
        <p type="margin">
          <s id="id001691"><margin.target id="marg341"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2. <lb></lb>45. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001692"><margin.target id="marg342"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 86. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001693"><margin.target id="marg343"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 99.</s>
        </p>
        <p type="margin">
          <s id="id001694"><margin.target id="marg344"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 97.</s>
        </p>
        <p type="margin">
          <s id="id001695"><margin.target id="marg345"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 98.</s>
        </p>
        <p type="main">
          <s id="id001696">Propoſitio nonageſima nona.</s>
        </p>
        <p type="main">
          <s id="id001697">Proportionem grauitatum per multitudinem ſuppoſitorum or <lb></lb>bium oſtendere.<lb></lb><arrow.to.target n="marg346"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001698"><margin.target id="marg346"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001699">Omne, quod mouetur, mouetur ſecundum naturam ponderis, <lb></lb>quæ in attractione, ut demonſtratum eſt, æqualis eſt dimidio ſu­<lb></lb>ſpenſi, cum ergo diuidatur in multiplices partes motus uniuſcuiuſ­<lb></lb>que, eſt ſecundum dimidium illius partis, ut, ſi ſint ſex rotæ in cur­<lb></lb>ru det, quod uehitur, ſit pondus ſexaginta librarum, unaquæque </s>
        </p>
        <p type="main">
          <s id="id001700"><arrow.to.target n="marg347"></arrow.to.target><lb></lb>rota habet pondus quinque librarum, ſcilicet diuiſo triginta per <lb></lb>ſex, &amp; quia quod cunque mouetur ſphæricè non habet pondus, <lb></lb>niſi quantum premitur axis, ideò pondus ſexaginta librarum in <lb></lb>uehendo redditur læſus, quanto proportio producta minor eſt <lb></lb>additione. </s>
          <s id="id001701">Exemplum, ſit deductum pondus ſexaginta librarum <lb></lb>per ſex rotas ad uiginti quatuor, quia ſi rotæ poſſent circumduci, <lb></lb>ut in inuerſione dictum eſt, &amp; eſſent æquales, &amp; in ſolido æquali, <lb></lb>ac duro, nulla ui mouerentur, ſed quaſi per ſe, ergo ſuppoſito pon­<lb></lb>dere uiginti quatuor librarum aſſumemus unamquamque partem, <lb></lb>&amp; ducemus eam in ſe ipſam, ſcilicet detraham quintam partem ex <lb></lb>toto 30, fit 24, duc 30 in ſe, fit 900, duc 24 in ſe, fit 576, proportio ut <lb></lb>25 ad 16, at diuiſo 30 in ſex partes, fit 5, detrahe quintam partem, fit <lb></lb>4, duc in ſe, fit 16, duc in ſex, fit 96, igitur proportio 900 ad 96 eſt ut <lb></lb>25 ad 2 2/3, quod ergo erat 16 factum eſt 2 2/3, proportio ergo de­<lb></lb>creſcentis maior eſt diuiſo per plura. </s>
          <s id="id001702">Sed plerunque additis ro­<lb></lb>tis creſcit pondus nihilo ſecius, redditur etiam leuius. </s>
          <s id="id001703">Sed &amp; de <lb></lb>hoc in ſequenti.</s>
        </p>
        <p type="margin">
          <s id="id001704"><margin.target id="marg347"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 40.</s>
        </p>
        <p type="main">
          <s id="id001705">Propoſitio centeſima.</s>
        </p>
        <p type="main">
          <s id="id001706">Proportionem grauitatis ponderum attractorum per trochlea­<lb></lb>rum numerum inueſtigare.<lb></lb><arrow.to.target n="marg348"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001707"><margin.target id="marg348"></margin.target>C<emph type="italics"></emph>om.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001708">Ariſtoteles in Mechanicis cenſet cauſam leuitatis trochlearum </s>
        </p>
        <p type="main">
          <s id="id001709"><arrow.to.target n="marg349"></arrow.to.target><lb></lb>eſſe in pondere eleuando, quòd pondera auxilio uectium facilius <lb></lb>mouentur, quàm manibus. </s>
          <s id="id001710">Rotulæ uerò in trochleis uectes ſunt, <lb></lb>&amp; axis miſta hypomochlij, ergo facilius pondus trahitur per u­<lb></lb>nam rotulam, quàm ſi manu traheretur, at uerò per duas tres, <lb></lb>unde tris paſſus longe facilius, &amp; etiam facilius per quinque, unde <lb></lb>pentas paſſus, nam quinque orbiculis, quaſi totidem uectibus <lb></lb>diuiſum pondus manifeſtè fit leuius, &amp; ut dictum eſt, tanquam <lb></lb>totidem uectibus pondus eleuatur, eſtqúe proportio produ­


<pb pagenum="94" xlink:href="015/01/113.jpg"></pb>cta, ſemperque prior hypomochlij locum habet, ueruntamen minus <lb></lb>aſſumit laboris, poſterior uerò uectis maiorem partem ſibi ponde­<lb></lb>ris ſeruat, uelut in ſuccula etiam iugum traiectum per plures colo­<lb></lb>pes facilius uertitur. </s>
          <s id="id001711">Et ſi quis dicat nónne totum pondus inſidet <lb></lb>primę trochleæ per trochleam, intelligo nunc ſolùm rotulam cum <lb></lb>ipſo axe, ſeu axiculo (ut dicunt) non autem in proprio ſignificato, <lb></lb>in quo etiam funis traiectus, &amp; inſidens rotulæ, ſeu rotulis, nam <lb></lb>una trochlea plures continere&#039;poteſt orbiculos, &amp; axes. </s>
          <s id="id001712">Licet ergo <lb></lb>pondus inſideat primæ trochleæ, ſeu rotulæ, in eo tamen, quod tra<lb></lb>hitur, diuiditur&#039;, licet non æqualiter dico, præter id funis motum <lb></lb>intendi. </s>
          <s id="id001713">nam motus actionem auget, &amp; ideò quanto longior, eo fa­<lb></lb>cilius mouet ob concuſsionem, demum quia leuis eſt rotula circa <lb></lb>axem, ut plus uecte poſsit.</s>
        </p>
        <p type="margin">
          <s id="id001714"><margin.target id="marg349"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> M<emph type="italics"></emph>echan.<emph.end type="italics"></emph.end><lb></lb>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 18.</s>
        </p>
        <p type="main">
          <s id="id001715">Propoſitio centeſima prima.</s>
        </p>
        <p type="main">
          <s id="id001716">Proportionem precij gemmarum ex tribus in eodem genere co<lb></lb>gnitis inuenire.<lb></lb><arrow.to.target n="marg350"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001717"><margin.target id="marg350"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001718">Solent gemmarij uendere adamantem ponderis unius grani <lb></lb>uno coronato, duorum autem granorum tribus coronatis, qua­<lb></lb>tuor autem, gratia exempli, quadraginta coronatis, quęritur quan­<lb></lb>tum ualebit adamas octo granorum, quoniam ergo proportio <lb></lb>non ſeruatur. </s>
          <s id="id001719">Eſt enim in pondere utraque dupla, in precio autem <lb></lb>ex prima habetur tripla, ex ſecunda habetur proportio maior, <lb></lb>quàm tredecim ad unum, propterea utendum eſt proportione <lb></lb>propinquiori, ſi ſatis faceret. </s>
          <s id="id001720">gratia exempli, in prima ad ditione fuit <lb></lb>unum granum, &amp; acquiſiuit proportionem triplam, in ſecunda fue<lb></lb>runt duo grana, ſi ergo acquiſiſſet ſolum ſexcuplam proportio­<lb></lb>nem, haberemus intentum. </s>
          <s id="id001721">Propterea in iſto caſu oportet demon­<lb></lb>ſtrare forma Geometrica, ſuppoſito, quòd ſit figura recta ex uno la <lb></lb><figure id="id.015.01.113.1.jpg" xlink:href="015/01/113/1.jpg"></figure><lb></lb>tere a b, ita ut angulus, uel minimus capiat b c æqualem a b, &amp; ex <lb></lb>æquali b a c addito fiat b d tripla b c, &amp; ex angulo b a e duplo b a d, <lb></lb>fiat b c d e quadragintupla a b, &amp; iuxta rationem erit in infinitum. <lb></lb></s>
          <s id="id001722">Siue ſit parabole, ſiue hyperbole, ſeu ſit alia coincidentium.</s>
        </p>
        <pb pagenum="95" xlink:href="015/01/114.jpg"></pb>
        <p type="head">
          <s id="id001723">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001724">Et nota, quòd ſi res hæc eſſet naturalis, oſtenderet infinitum in <lb></lb>rebus ex regula dialectica, ſed quia ex <expan abbr="uolũtaria">uoluntaria</expan>, nullas habet uires.</s>
        </p>
        <p type="main">
          <s id="id001725">Propoſitio centeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id001726">Proportionem motuum inuerſionis, &amp; attractionis in plano <lb></lb>inuenire.</s>
        </p>
        <p type="main">
          <s id="id001727">Et ſit, ut aliquid inuertatur, declaratum autem eſt ſuprà, quid ſit </s>
        </p>
        <p type="main">
          <s id="id001728"><arrow.to.target n="marg351"></arrow.to.target><lb></lb>inuerſio, &amp; quàm diuerſa ſit rurſus, &amp; quòd attractio eſt dimidium <lb></lb><arrow.to.target n="marg352"></arrow.to.target><lb></lb>ponderis eleuati. </s>
          <s id="id001729">Cum ergo conſtet in inuerſione, quanta ſit pro­<lb></lb>portio ponderis ſuſpenſi ad pondus inuerſum, &amp; pondus ſuſpenſi <lb></lb><arrow.to.target n="marg353"></arrow.to.target><lb></lb>ſit duplum ponderi attracti, ſequitur, ut diuiſa proportione ponde<lb></lb>ris ſuſpenſi ad pondus inuerſum per medium cognoſcatur propor<lb></lb>tio attractionis ad inuerſionem.</s>
        </p>
        <p type="margin">
          <s id="id001730"><margin.target id="marg351"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id001731"><margin.target id="marg352"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 89.</s>
        </p>
        <p type="margin">
          <s id="id001732"><margin.target id="marg353"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 62.</s>
        </p>
        <p type="main">
          <s id="id001733">Ex hoc ſequitur, quod aliquod pondus trahi poteſt, quod non <lb></lb><arrow.to.target n="marg354"></arrow.to.target><lb></lb>poteſt inuerti, hoc autem indiget longa declaratione, quam doce­<lb></lb>bimus inferiùs: &amp; tamen attigit hoc rarò.</s>
        </p>
        <p type="margin">
          <s id="id001734"><margin.target id="marg354"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001735">Propoſitio centeſima tertia.</s>
        </p>
        <p type="main">
          <s id="id001736">Proportionem eorundem in accliui demonſtrare.</s>
        </p>
        <p type="main">
          <s id="id001737">Dupliciter poteſt intelligi, uel deſcendendo, uel aſcendendo. <lb></lb><arrow.to.target n="marg355"></arrow.to.target><lb></lb><arrow.to.target n="marg356"></arrow.to.target><lb></lb>Sed ego nunc loquor de aſcenſu, contraria ratione intelliges de <lb></lb>deſcenſu, &amp; circa inuerſionem demonſtrata eſt proportio eius <lb></lb>iuxta angulum aſcenſus, &amp; ſimiliter declarabitur de proportione <lb></lb><arrow.to.target n="marg357"></arrow.to.target><lb></lb>attractionis iuxta eundem angulum aſcenſus, &amp; nuper declarata <lb></lb>eſt proportio inuerſionis in plano ad attractionem, ex quibus ſe­<lb></lb>quitur per ea, quæ dicam inferius, quòd proportio cuiuſuis mobi­<lb></lb>lis inuerſi ad attractum ſub quibuſcunque angulis nota erit.</s>
        </p>
        <p type="margin">
          <s id="id001738"><margin.target id="marg355"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001739"><margin.target id="marg356"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 72.</s>
        </p>
        <p type="margin">
          <s id="id001740"><margin.target id="marg357"></margin.target>I<emph type="italics"></emph>n ſequenti.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001741">Propoſitio centeſima quarta.</s>
        </p>
        <p type="main">
          <s id="id001742">Proportionem motus attractionis in decliui ad motum in pla­<lb></lb>no determinare.</s>
        </p>
        <p type="main">
          <s id="id001743">Si ab accliue, ſeu decliue in quo d ad attra­<lb></lb><arrow.to.target n="marg358"></arrow.to.target><lb></lb><arrow.to.target n="marg359"></arrow.to.target><lb></lb><figure id="id.015.01.114.1.jpg" xlink:href="015/01/114/1.jpg"></figure><lb></lb>hendum, cuius nota eſt ex ſuperioribus dif­<lb></lb>ficultas in plano ratione figuræ conſtante, er­<lb></lb>go ea quæritur proportio aſcenſus, &amp; quo­<lb></lb>niam terminus ad perpendiculum eſt dupla <lb></lb>proportio, &amp; iam grauitas in plano eſt dimidium, ideò quicquid <lb></lb>acquiritur in eleuatione eſt in comparatione ad illud dimidium, <lb></lb>cum ergo attractio ſecundum eandem proportionem augeatur, er­<lb></lb>go ſemper maior difficultas augebitur, ergo ab initio minimum 


<pb pagenum="96" xlink:href="015/01/115.jpg"></pb>erit diſcrimen ab attractione in plano. </s>
          <s id="id001744">Exempli gratia ſit, ut graue d <lb></lb>in plano ſit, ut quin que, &amp; ſuſpenſum decem, ergo in medio angulo <lb></lb>erit penè ſeptem, ſed ſeptem minus longe <expan abbr="diſtãt">diſtant</expan> à quin que, quàm de­<lb></lb>cem ad ſeptem, ergo in ſecunda parte plus longè augebitur difficul<lb></lb>tas attractionis ſupra difficultatem in medio angulo accliui, quam <lb></lb>in prima parte à plano ad medium accliue, &amp; quoniam planum in <lb></lb>plano deſcendit, tanto uehementius, quanto difficilius attrahitur, <lb></lb>ergo planum in decliui ſublimi longe maiore impetu feretur infrà <lb></lb>quam ſit proportio anguli ad angulum. </s>
          <s id="id001745">Exempli gratia, planum in <lb></lb>medio angulo, ſi incipiat deſcendere in dodrante multo lentius, <lb></lb>quàm pro dimidio uirium deſcenſus totius anguli, imò initium de­<lb></lb>ſcenſus eſt à medio recti ad unguem, ubi omnia plana ſint, &amp; duriſ­<lb></lb>ſima, &amp; cauſa huius eſt, quia omne graue tendit ad centrum, quòd <lb></lb>maior pars ipſius grauis eſt ultra medium grauitatis in decliui <lb></lb>humiliore.</s>
        </p>
        <p type="margin">
          <s id="id001746"><margin.target id="marg358"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001747"><margin.target id="marg359"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 62. &amp; <lb></lb>64. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001748">Propoſitio centeſima quinta.</s>
        </p>
        <p type="main">
          <s id="id001749">Proportionem ferentium pondus in pertica inuenire.</s>
        </p>
        <figure id="id.015.01.115.1.jpg" xlink:href="015/01/115/1.jpg"></figure>
        <p type="main">
          <s id="id001750">Hæc proponitur etiam à Philoſo­<lb></lb><arrow.to.target n="marg360"></arrow.to.target><lb></lb>pho, &amp; ponatur ab, &amp; ſi pondus ſit in <lb></lb><arrow.to.target n="marg361"></arrow.to.target><lb></lb>medio d grauat æqualiter utrunque, <lb></lb>nam in hoc conſentit experimentum <lb></lb>cum ratione, at uerò ſi ponatur in cita, <lb></lb>ut b c ſit tripla b a uiderentur a &amp; b, tanquam hypomochlia, &amp; pon<lb></lb><arrow.to.target n="marg362"></arrow.to.target><lb></lb>dus ipſum b, ut grauior eſſet cb, quam c a. </s>
          <s id="id001751">Ariſtoteles, ſeu author <lb></lb>ille hoc uidens bifariam reſpondet: primum quòd hoc eſt inuer­<lb></lb><arrow.to.target n="marg363"></arrow.to.target><lb></lb>ſum inſtrumentum, cum in cæteris motor ſit ex aduerſo hypomo­<lb></lb>chlij, hic in ipſo, geſtans enim mouet &amp; hypomochlij inſtar eſt hu­<lb></lb>merus. </s>
          <s id="id001752">At hoc uerum non eſt: quod mouet enim eſt pondus, &amp; eſt <lb></lb>in c: nam a, &amp; contingit moueri: quia ſi ſtarent, idem ſequeretur. </s>
          <s id="id001753">Se­<lb></lb>cunda reſponſio eſt, quod utrunque premit ſcilicet ferentes &amp; pon­<lb></lb>dus, &amp; quòd qui longior eſt ab hypomochlio facilius mouet, &amp; <lb></lb>redit ad idem fermè: nam in c conſtituitur, quod moueri debet, ca­<lb></lb>pita uectium ſunt a, &amp; b: motus autem eſt ipſum ſuſtinere pondus. <lb></lb></s>
          <s id="id001754">At hoc non uidetur, quoniam ratio, qua uectis longior facilius mo<lb></lb>uet, eſt ambitus magnitudo, ob quam motus redditur tardior, &amp; <lb></lb>ideo leuior: igitur non eſt hoc uerum de motu occulto, ſicut eſt gra<lb></lb>uis prementis, ſed circumducente, cum in occulto uelut in ſtatera <lb></lb>contrarium accidere docuerimus aliâs. </s>
          <s id="id001755">Quidam dixere b premere <lb></lb>c uerſus a, a contrà uerſus b, &amp; ideò grauari magis a àb, quàm b ab <lb></lb>a, quia maiorem uim habet b e, quàm a c. </s>
          <s id="id001756">Iſtud falſum eſt bifariam. <lb></lb></s>
          <s id="id001757">Primum, quia &amp; ſi a, &amp; b ſint in æquilibrio, ut nec unus in alterum 


<pb pagenum="97" xlink:href="015/01/116.jpg"></pb>incumbat, nec impellat, ſed tantum ſuſtineat nihilo ſecius res uera <lb></lb>eſt. </s>
          <s id="id001758">Et etiam quia non eſt uerum, quòd qui longius incumbit, ma­<lb></lb>iorem uim inferat. </s>
          <s id="id001759">Propterea dicendum eſt, quòd qui ex commu­<lb></lb>nibus propria nituntur demonſtrare, omnes corrumpunt diſcipli­<lb></lb>nas. </s>
          <s id="id001760">Nihil deterius eſt his monſtris. </s>
          <s id="id001761">Nam etſi hæc ratio uera eſſet: <lb></lb>non tamen reddit cauſam, quia non eſt ex proprijs principijs. </s>
          <s id="id001762">Dico <lb></lb>ergo, quod ſi c deſcendat in e, per perpendiculum deſcendet, igitur <lb></lb>d b eſt longior d a, quare angulus e a b maior e b a: igitur pondus c <lb></lb>plus deſcendit comparatione a, quàm b, ergo plus grauat c ipſum a <lb></lb>quàm b, ſeu ex cauſa, quod magis premat, ſeu ex effectu, quòd ma­<lb></lb>gis deceſſerit. </s>
          <s id="id001763">Cauſa ergo erroris eſt, quod ſi ponatur angulus f b a <lb></lb>æqualis angulo f a b, &amp; ponatur b f ęqualis b c, tun c in eodem tem­<lb></lb>pore, in quo tranſit dimidium c in e, tranſibit aliud dimidium c in f. <lb></lb></s>
          <s id="id001764">quia ſeparatę partes grauiores ſunt in c b, quàm c a, propter diſtan­<lb></lb>tiam ab hypomochlio, ſed tunc uelocius mouentur, &amp; angulus fit <lb></lb>ęqualis. </s>
          <s id="id001765">Sed quando pondus eſt unum, &amp; c deſcendit ad e, cum de­<lb></lb>ſcendat inæquali tempore, &amp; peragat maiorem angulum compa­<lb></lb>ratione a, quam b, ſequitur, ut uelocius moueatur comparatione a <lb></lb>quàm b. </s>
          <s id="id001766">Ergo ſi non mouetur, cum omnis potentia ſit ſimilis actui, <lb></lb>tum quia ab eo producitur, &amp; effectus eſt ſimilis cauſæ: tum quia <lb></lb>eſt initium actus, igitur etiam quod a b non inclinetur, nec deſcen­<lb></lb>dat, grauius erit pondus, comparatione a quàm b, quod erat de­<lb></lb>monſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id001767"><margin.target id="marg360"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001768"><margin.target id="marg361"></margin.target>Q<emph type="italics"></emph>usſt.<emph.end type="italics"></emph.end> 59. <lb></lb>M<emph type="italics"></emph>echanic.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001769"><margin.target id="marg362"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 45.</s>
        </p>
        <p type="margin">
          <s id="id001770"><margin.target id="marg363"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 103.</s>
        </p>
        <p type="main">
          <s id="id001771">Ex hoc ſequitur, quòd aliqua iuncta erunt grauiora reſpectu u­<lb></lb>nius, quæ erunt mutato ordine diuiſa leuiora. </s>
          <s id="id001772">Quoniam diuiſa, <lb></lb>quæ longius diſtant æqualem, aut maiorem angulum faciunt, iun­<lb></lb>cta minorem.</s>
        </p>
        <p type="main">
          <s id="id001773">Propoſitio centeſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id001774">Quales proportiones angulorum doceant laterum proportio­<lb></lb>nes. </s>
          <s id="id001775">At que uiciſsim determinare.</s>
        </p>
        <p type="main">
          <s id="id001776">Sit circulus a b c, cuius dimetiens, nota b d ſit b, erit ergo latus <lb></lb><arrow.to.target n="marg364"></arrow.to.target><lb></lb><figure id="id.015.01.116.1.jpg" xlink:href="015/01/116/1.jpg"></figure><lb></lb>exagoni a b dimidium b d, id eſt 3. igitur <lb></lb>cum angulus a ſit rectus, erit a d &lt;02&gt; 27 latus <lb></lb>trianguli. </s>
          <s id="id001777">Et latus quadrati per eandem &lt;02&gt;<lb></lb>18. Vt latus exagoni ſit &lt;02&gt; 9. Quadrati &lt;02&gt; 18 <lb></lb>Trianguli &lt;02&gt; 27, &amp; ita poteſtate ſe habent <lb></lb>hæc ut 1. 2. 3. Et ſunt nota. </s>
          <s id="id001778">Et quia latus d e c <lb></lb>agoni eſt &lt;02&gt; 11 1/4 m, 1 1/2. &amp; ipſum erit notum. <lb></lb></s>
          <s id="id001779">Quare latus pentagoni eſt &lt;02&gt; v 22 1/2 m: &lt;02&gt;<lb></lb>101 1/4 notum. </s>
          <s id="id001780">Et iam notum fuit latus epta­<lb></lb>goni. </s>
          <s id="id001781">Habebimus igitur latera Trianguli 


<pb pagenum="98" xlink:href="015/01/117.jpg"></pb>quadrati pentagoni, &amp; eptagoni æquilaterorum nota: &amp; etiam <lb></lb>ſubtenſorum duobus ex his. </s>
          <s id="id001782">Sit, gratia exempli, a b 3 &amp; b c &lt;02&gt; 11 1/4m: <lb></lb>1 1/2, ut prius, &amp; ponatur b d diameter, erit ad &lt;02&gt; 27 &amp; c d &lt;02&gt; v 22 1/2 m: <lb></lb>&lt;02&gt; 101 1/4, quam ducemus in a b, &amp; fiet &lt;02&gt; v 202 1/2 m: &lt;02&gt; 8201 1/4. Duce­<lb></lb>mus itidem &lt;02&gt; 27 a d in b c &lt;02&gt; 11 1/4 m: 1 1/2 fiet &lt;02&gt; 303 3/4m: &lt;02&gt; 60 3/4, hoc to­<lb></lb>tum diuide per 66, quæ eſt b: fiet a c &lt;02&gt; 8 7/16 m: &lt;02&gt; 1 11/16 p: &lt;02&gt; v: 5 45/72 m: &lt;02&gt;<lb></lb>6 1701/5184. Nec credas te errare, quoniam latus pentagoni eſſet, ac ſi an­<lb></lb>gulus b rectus eſſet: ſed quia eſt obtuſus, ideo a c eſt alia linea, &amp; <lb></lb>maior latere pentagoni. </s>
          <s id="id001783">Et ſimiliter ſi a b, &amp; a c notæ eſſent, utpo­<lb></lb><arrow.to.target n="marg365"></arrow.to.target><lb></lb>te a b 3, ut prius a c 5 dico, quòd b c nota eſt: nam a d erit &lt;02&gt; 27, &amp; <lb></lb>quia ex b d in a c fit 30, fiet ex b c in a d pos &lt;02&gt; 27, et ex a b in c d &lt;02&gt; 324 <lb></lb>m: 9 quad. </s>
          <s id="id001784">igitur 30 m: pos &lt;02&gt; 27 æquantur &lt;02&gt; 324 m: 9 quad. </s>
          <s id="id001785">quare <lb></lb>900 p: 27 quad. </s>
          <s id="id001786">m: pos &lt;02&gt; 97200 <expan abbr="æquãtur">æquantur</expan> 324 m: 9 quad. </s>
          <s id="id001787">igitur 576 <lb></lb>p: 16 quad. </s>
          <s id="id001788">ęquantur pos &lt;02&gt; 97200. Quadratum igitur p: 36 ęquan­<lb></lb>tur pos &lt;02&gt; 379 11/16, erit ergo b c &lt;02&gt; v: &lt;02&gt; 94 59/64 p: &lt;02&gt; 58 59/64 &amp; ſimiliter ſi a c <lb></lb>ſit nota, puta 4 erit a b ſubtenſa dimidio arcus a c nota. </s>
          <s id="id001789">Erit enim a e <lb></lb>2 ergo d e 3 p: &lt;02&gt; 5 et b e 3 m: &lt;02&gt; 5, <expan abbr="igit̃">igitur</expan> a b &lt;02&gt; v: 18 m, &lt;02&gt; 180. Igitur hoc <lb></lb>modo diuidendo, iungendo, &amp; detrahendo habebimus ex quatu­<lb></lb>or illis ſimplicibus trianguli quadrati. </s>
          <s id="id001790">Pentagoni, &amp; eptagoni in <lb></lb>numeras linearum magnitudines in circulo. </s>
          <s id="id001791">Et ſimiliter quouis mo <lb></lb>do, ut dictum eſt, in quauis figura æquilatera, utpote ſuppoſito <lb></lb><figure id="id.015.01.117.1.jpg" xlink:href="015/01/117/1.jpg"></figure><lb></lb>quod deſcriptum ſit non angulum in <lb></lb>circulo æquilaterum, quod etiam erit <lb></lb>æquiangulum, &amp; ſit arcus a b duplus <lb></lb>arcui a c, erit angulus a c b duplus an­<lb></lb>gulo a b c, &amp; angulus b a c in portione <lb></lb>b d e c ſexcuplus a b c, &amp; triplus a c b. <lb></lb></s>
          <s id="id001792">Erit ergo per demonſtrata proportio <lb></lb><arrow.to.target n="marg366"></arrow.to.target><lb></lb>b a ad a c, uelut a c, &amp; c b, ad a b: pro­<lb></lb>portio autem a b arcus ad a c, ex ſup­<lb></lb>poſito maior eſt proportione rectæ a b ad a c, igitur etiam propor­<lb></lb>tione a c &amp; c b ad a b, ergo duo latera trianguli ad tertium minorem <lb></lb>habent proportionem, quam arcus ad arcum, quanto rectæ ad re­<lb></lb>ctam minor eſt. </s>
          <s id="id001793">Sit rurſus in triangulo b e d quomodolibet modo <lb></lb>ſit angulus b d e quadruplus angulo b e d, &amp; diuidatur d per ęqua­<lb></lb>lia ducta d f, erit igitur proportio f d, d e ad f e, ut e f ad f d, ſed e f ad <lb></lb><arrow.to.target n="marg367"></arrow.to.target><lb></lb>f b ut d e ad d b. </s>
          <s id="id001794">igitur proportio b d, d e ad f b <expan abbr="cõpoſita">compoſita</expan> ex propor­<lb></lb>tionibus e f ad f d, &amp; e d ad d b. </s>
          <s id="id001795">Proportio igitur b d, d e ad f b, ut <lb></lb>producti ex e f in e d ad productum ex d fin d b. </s>
          <s id="id001796">Rurſus ponamus, <lb></lb><arrow.to.target n="marg368"></arrow.to.target><lb></lb>quod in quadrangulo a b c d primæ figuræ ſit a b 4 b c 3 c d 5 ad 6 <lb></lb>dico, quòd ſpatium contentum erit notum. </s>
          <s id="id001797">Ductis rectis a c &amp; b d 


<pb pagenum="99" xlink:href="015/01/118.jpg"></pb>quomodolibet, ut ſe ſecent in e, erunt anguli d c a, &amp; d b a æquales, <lb></lb><arrow.to.target n="marg369"></arrow.to.target><lb></lb>quia in ead́em portione circuli a d, &amp; anguli a d e ęquales, quia con<lb></lb>tra ſe poſiti. </s>
          <s id="id001798">igitur trianguli a b e, &amp; c d e ſimiles, &amp; proportio d c ad <lb></lb><arrow.to.target n="marg370"></arrow.to.target><lb></lb>a b, ut c e ad b e, c d autem fuit 5 a b 4, igitur ſi b e ponatur 4 pos c e <lb></lb>erit 5 pos. </s>
          <s id="id001799">Per eaſdem, &amp; eodem modo a d ad b c ut d e ad e c. igitur <lb></lb>poſita c e 5 pos erit e d 10 pos, tota igitur d b 14 pos. </s>
          <s id="id001800">Et quoniam ea­<lb></lb><arrow.to.target n="marg371"></arrow.to.target><lb></lb>dem proportio a e ad e b per eadem, &amp; e b fuit 4 pos: igitur a e eſt 8 <lb></lb>pos, quare a e 13. poſt productum igitur ex a c in d b, eſt 182 quad. <lb></lb></s>
          <s id="id001801">&amp; hoc æquatur productis a b in c d, quod eſt 20, &amp; b c in a d quod <lb></lb>eſt 18, totum igitur eſt 38, igitur res eſt &lt;02&gt; 19/91. Quare notę erunt lineæ <lb></lb>b e, e d, a e, &amp; e c, ſed ſufficit, ut cognita ſit a c, uel b d. </s>
          <s id="id001802">Per regulam <lb></lb>enim triangulorum erunt notæ areæ a b c, &amp; a d e, quare tota ſuper­<lb></lb>ficies a b c d. </s>
          <s id="id001803">Et eſt inuentum Scipionis Ferri Bononienſis de quo <lb></lb>aliâs. </s>
          <s id="id001804">Poteſt etiam inuenta a c uel b d haberi ſuperficies facilius <lb></lb>per catheros.</s>
        </p>
        <p type="margin">
          <s id="id001805"><margin.target id="marg364"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id001806"><margin.target id="marg365"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 52. E<emph type="italics"></emph>le <lb></lb>ment.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001807"><margin.target id="marg366"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>de<emph.end type="italics"></emph.end><lb></lb>S<emph type="italics"></emph>ubtil.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001808"><margin.target id="marg367"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001809"><margin.target id="marg368"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001810"><margin.target id="marg369"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 21. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001811"><margin.target id="marg370"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001812"><margin.target id="marg371"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001813">Sit modo obtuſi angulus a b c, &amp; nota latera ſingula, &amp; angu­<lb></lb>lus a b c, &amp; producantur latera ad perpendicu­<lb></lb><figure id="id.015.01.118.1.jpg" xlink:href="015/01/118/1.jpg"></figure><lb></lb>lum, ut ſint d &amp; e recti, &amp; quia anguli ad a ſunt <lb></lb>æquales, erunt anguli e b a, &amp; d e a ſemper æ­<lb></lb><arrow.to.target n="marg372"></arrow.to.target><lb></lb>quales. </s>
          <s id="id001814">Et hoc idem contingit in acuti angulis <lb></lb>triangulis intus, &amp; eſt utile mechanicum: &amp; <lb></lb>quia a b c notus eſt, &amp; d notus, erunt anguli tri<lb></lb>goni d b c noti: &amp; ſi fuerit angulus a notus, <expan abbr="erũt">erunt</expan> anguli d a c &amp; e a b <lb></lb>noti, &amp; ideo anguli e b a, &amp; d c a: &amp; ſemper notum, quod fit ex b a <lb></lb>in a d, uel c a in a e, ſunt enim ęqualia inter ſe: etiam notæ ad &amp; a c, <lb></lb>quoniam duplum horum eſt exceſſus quadrati b c ſuper quadrata <lb></lb>a b, &amp; a c. </s>
          <s id="id001815">Quod uerò propositurà Monteregio de cognitione an­<lb></lb>gulorum in triangulis non eſt intelligendum, ut uerba ſignificant, <lb></lb><arrow.to.target n="marg373"></arrow.to.target><lb></lb>ſed ſolum de cognitione quoad uſum tabularum.</s>
        </p>
        <p type="margin">
          <s id="id001816"><margin.target id="marg372"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001817"><margin.target id="marg373"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>ſe­<lb></lb>cundi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001818">Et iterum ponamus, quòd proportio a c c b ad a b ſit qualis a b <lb></lb>ad a c, dico quòd angulus c duplus eſt angulo b. </s>
          <s id="id001819">Si non ducatur c d <lb></lb><figure id="id.015.01.118.2.jpg" xlink:href="015/01/118/2.jpg"></figure><lb></lb>faciens angulum d c b duplum b, erit igitur pro­<lb></lb>portio d c c b ad d b, ut d b ad d c. </s>
          <s id="id001820">Maior eſt <expan abbr="autẽ">autem</expan> <lb></lb>d c, quàm a c, aut æqualis, aut minor, ſi æqualis, <lb></lb>igitur maior proportio d c c b ad b d quàm b a, <lb></lb>igitur maior proportio b d ad d c quam b a ad a c <lb></lb>ad a c &amp; æquales ſunt igitur b d maior d a pars toto, quod eſſe non <lb></lb>poteſt. </s>
          <s id="id001821">Si uerò d c ponatur maior a c, magis ex hoc ſequitur b d ma­<lb></lb>iorem eſſe b a. </s>
          <s id="id001822">Quod ſi minor ſit d c quàm a c. </s>
          <s id="id001823">Ex demonſtratio­<lb></lb>ne ipſius reflexæ proportionis patet hoc contingere non poſſe. <lb></lb></s>
          <s id="id001824">Et ſimiliter patet conuerſas in reliquis etiam ueras eſſe, non ſolum 


<pb pagenum="100" xlink:href="015/01/119.jpg"></pb>in proportionibus notiſsimis angulorum ſed etiam in coniuncti­<lb></lb>one &amp; detractione. </s>
          <s id="id001825">Et eſt ex ſubtiliſsimis operationibus, quæ ho­<lb></lb>mini in hoc genere eueniant.</s>
        </p>
        <p type="main">
          <s id="id001826">Propoſitio centeſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id001827">Si in circulo duo diametri ad rectum angulum ſe ſecauer int: alię <lb></lb>uerò ad perpendiculum ex diametro exierint ad circumferentiam, <lb></lb>ſingulæ ſupra diametrum erunt maiores portionibus reliquis dia­<lb></lb>metri ſuperioribus, infra autem minores. </s>
          <s id="id001828">Dimidium autem porti­<lb></lb>onis ſuperioris reſiduum ad centrum maius ſagitta habebit. </s>
          <s id="id001829">In ali­<lb></lb>qua præterea portionis ſuperioris parte, quæ uerſus diametrum <lb></lb>tranſuerſum poſita eſt, maior eſt differentia partis diametri ei cor­<lb></lb>reſpondentis, quam lineæ tranſuerſæ.</s>
        </p>
        <figure id="id.015.01.119.1.jpg" xlink:href="015/01/119/1.jpg"></figure>
        <p type="main">
          <s id="id001830">Sint duę diametri a b, c d ad perpendi <lb></lb>culum ſecantes ſe in centro, &amp; <expan abbr="ducũtur">ducuntur</expan> <lb></lb>ſupr f g k h, &amp; infra m l ad perpendicu­<lb></lb>lum ſupra a b: dico f g eſſe maiorem f a, <lb></lb>&amp; k h k a, &amp; contrà minorem m l, quàm <lb></lb>m a. </s>
          <s id="id001831">Per octauam enim ſexti, quod fit ex <lb></lb><arrow.to.target n="marg374"></arrow.to.target><lb></lb>b f in f a æquale eſt <expan abbr="q̃drato">quadrato</expan> f g, ſed b f eſt <lb></lb>maior f g, quia b f eſt maior c b, &amp; ideo <lb></lb>e c g f, ergo f g maior eſt f a, m l <expan abbr="aũt">aut</expan> minor eſt per <expan abbr="eadẽ">eadem</expan> e c, quare e a, <lb></lb>multo igitur minor m a, quod eſt primum. </s>
          <s id="id001832">Suppoſito etiam, quòd <lb></lb><arrow.to.target n="marg375"></arrow.to.target><lb></lb>a g arcus ſit dimidium a c, dico a f <expan abbr="minorẽ">minorem</expan> eſſe f e, nam quadratum e <lb></lb><arrow.to.target n="marg376"></arrow.to.target><lb></lb>g æquale eſt quadratis f e, &amp; f g, &amp; <expan abbr="quadratũ">quadratum</expan> a g quadratis f g &amp; f a <lb></lb>&amp; e g eſt ęqualis lateri exagoni, &amp; a g latus octogoni, igitur e g ma­<lb></lb><arrow.to.target n="marg377"></arrow.to.target><lb></lb>ior g a, &amp; duo quadrata e f &amp; f g maiora duobus quadratis f g &amp; <lb></lb>f a, detracto igitur communi f g quadrato, patet propoſitum.<lb></lb><arrow.to.target n="marg378"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001833"><margin.target id="marg374"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 31. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001834"><margin.target id="marg375"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>tertij<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001835"><margin.target id="marg376"></margin.target>1. <emph type="italics"></emph>eiuſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001836"><margin.target id="marg377"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001837"><margin.target id="marg378"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb>15. <emph type="italics"></emph>quarti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001838">Cum rurſus ex prima parte huius lineę f g &amp; k h ſint maiores f a, <lb></lb>&amp; k a &amp; ea ſit æqualis e c, neceſſe eſt ut iuxta punctum c augeatur </s>
        </p>
        <p type="main">
          <s id="id001839"><arrow.to.target n="marg379"></arrow.to.target><lb></lb>magis linea in ea, quam ſit differentia lineæ tranſuerſæ ad lineam <lb></lb>tranſuerſam per communem animi ſententiam, quod eſt tertium.</s>
        </p>
        <p type="margin">
          <s id="id001840"><margin.target id="marg379"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 28. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001841">Propoſitio centeſima octaua.</s>
        </p>
        <p type="main">
          <s id="id001842">Punctum ęqualitatis differentię deſcenſus, &amp; remotionis à cen­<lb></lb>tro inuenire.</s>
        </p>
        <p type="main">
          <s id="id001843">Per præcedentem moto puncto a uerſus c ſemper uſ que ad e, c ma<lb></lb><arrow.to.target n="marg380"></arrow.to.target><lb></lb>gis diſtat <expan abbr="pũctum">punctum</expan> a linea a e, quàm à puncto a uerſus, quia linea n h <lb></lb>maior eſt n a, &amp; per eandem dum appropinquat ad c cum e c fiat <lb></lb>ęqualis ea, maius fit in crementum in a e, quàm reſpectu lineæ tranſ­<lb></lb>uerſalis. </s>
          <s id="id001844">Volo ergo inuenire punctum hoc in quo fit mutatio: &amp; <lb></lb>diuido arcum ac per æqualia in f, &amp; dico illum eſſe punctum quæ­<lb></lb>ſitum: accepto quouis puncto in e f, puta k, duco g o h p ęquidiſtan<pb pagenum="101" xlink:href="015/01/120.jpg"></pb><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"></figure><lb></lb>tes a b, &amp; c d: erunt que anguli q &amp; n recti <lb></lb><arrow.to.target n="marg381"></arrow.to.target><lb></lb>&amp; anguli f e a, &amp; f e c ęquales, igitur uter <lb></lb><arrow.to.target n="marg382"></arrow.to.target><lb></lb>que dimidium recti: igitur per dicta in <lb></lb>primo Elementorum Euclidis e n ęqua <lb></lb><arrow.to.target n="marg383"></arrow.to.target><lb></lb>lis n k, igitur c q æqualis e n, quare h p <lb></lb>æqualis g o, ſed quod fit ex o k in k g eſt <lb></lb><arrow.to.target n="marg384"></arrow.to.target><lb></lb>æquale ei, quod fit ex p k in k h, igitur <lb></lb><arrow.to.target n="marg385"></arrow.to.target><lb></lb>k h eſt æqualis k g ex eisdem oſtendi­<lb></lb>tur f l m k quadratum eſſe. </s>
          <s id="id001845">Quia ergo <lb></lb>k h eſt æqualis k g, &amp; k l æqualis k m, erit l g æqualis m h. </s>
          <s id="id001846">Er­<lb></lb>go deſcendendo ex g in f, quantum f l ſuperat l g, tantum deſcen­<lb></lb>dendo ex f in h, f m ſuperat m h per communem animi ſententi­<lb></lb>am. </s>
          <s id="id001847">At f m eſt deſcenſus f in linea a e, &amp; m h diſtantia, quæ acqui­<lb></lb>ritur in linea f r, n m enim eſt æqualis f r, igitur n h excedit f r in <lb></lb>h m, &amp; ita a n excedit a r in n r ęquali f m. </s>
          <s id="id001848">Quantum ergo in g f, <lb></lb>l f excedit l g, tantum in deſcenſu ex f in h, f m, quæ refert g l, ex­<lb></lb>cedit h m, quæ refert f l. </s>
          <s id="id001849">Arcus autem f g eſt æqualis arcui f h, <lb></lb>quod <expan abbr="cũ">cum</expan> poſſem oſtendere pluribus modis ſatis conſtat, quia chor<lb></lb><arrow.to.target n="marg386"></arrow.to.target><lb></lb>darum illorum quadrata ſunt inuicem æqualia, quia lineæ f m, &amp; <lb></lb><arrow.to.target n="marg387"></arrow.to.target><lb></lb>f l item que m h &amp; l g ſunt æquales, &amp; anguli m, &amp; l recti. </s>
          <s id="id001850">Igitur cum <lb></lb>ad quod uis punctum in linea e f ſemper linea deſcenſus in parte <lb></lb>inferiore eſt maior linea diſtantiæ tanto, quanto per æqualem ar­<lb></lb>cum in ſuperiore linea diſtantiæ eſt maior linea, deſcenſus ſequitur <lb></lb>per regulam Dialecticam quod punctus f, eſt punctus ęqualitatis. <lb></lb></s>
          <s id="id001851">Per idem diceremus in quarta parte inferiore.</s>
        </p>
        <p type="margin">
          <s id="id001852"><margin.target id="marg380"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001853"><margin.target id="marg381"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 29. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001854"><margin.target id="marg382"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001855"><margin.target id="marg383"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 32. <lb></lb>&amp; 6.</s>
        </p>
        <p type="margin">
          <s id="id001856"><margin.target id="marg384"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 34. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001857"><margin.target id="marg385"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>tertij<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001858"><margin.target id="marg386"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001859"><margin.target id="marg387"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001860">Propoſitio centeſima nona.</s>
        </p>
        <p type="main">
          <s id="id001861">Rationem libræ expendere.</s>
        </p>
        <p type="main">
          <s id="id001862">Cum libra moueatur, uelut rota circa axem, quia trutina manet, <lb></lb>ideò ſi pondus ponatur, dum iugum fuerit in linea a b nihil mo­<lb></lb>uebitur, quia appetitus deſcenſus ex puncto a maximus eſt, &amp; ni­<lb></lb>hil iuuat motum extra naturam, idem dico de graui poſito in uerti­<lb></lb>ce b a. </s>
          <s id="id001863">Nam duo ſunt motus in rota, &amp; in libra unus, per quem <lb></lb>dum fertur per arcum a f, gratia exempli deſcendit, quantum eſt <lb></lb><arrow.to.target n="marg388"></arrow.to.target><lb></lb>a r, quæ eſt minor dimidio e r, &amp; ideò minor e r, quæ eſt maior di­<lb></lb>midio, ut demonſtratum eſt, &amp; etiam minor r f, quæ æqualis eſt r e <lb></lb><arrow.to.target n="marg389"></arrow.to.target><lb></lb>per demonſtrata rurſus: &amp; hic eſt naturalis ut palam eſt: alter præ­<lb></lb>ter <expan abbr="naturã">naturam</expan>, &amp; eſt ferri ad latus, quoniam hoc eſt <expan abbr="propriũ">proprium</expan> immortali­<lb></lb>bus: cun que hic ſit ad latus eſt etiam <expan abbr="cõtra">contra</expan> naturam, quia magis diſtat <lb></lb>a centro, nam e f eſt longior c r, ſi ergo r ferretur in f, moueretur à <lb></lb>centro, &amp; contra naturam. </s>
          <s id="id001864">Dum ergo fertur ex a in f, multo lentius 


<pb pagenum="102" xlink:href="015/01/121.jpg"></pb>fertur, quàm ex f in c: uelocius autem ex c uſque ad medium: nam <lb></lb>plurimum deſcendit. </s>
          <s id="id001865">Ex h ad b autem celerrimè, quoniam deſcen­<lb></lb>dit, &amp; appropinquat lineæ a b, ut uter que motus ſit naturalis. </s>
          <s id="id001866">Non <lb></lb>ergo mouetur pręter naturam niſi quatenus longius recedit à linea <lb></lb>a b, unde in inferiore parte mouetur ad eandem, ideò de parte c b <lb></lb>tota perſpicua eſt ratio, cur facillimè deſcendat, ſimiliter &amp; tota, <lb></lb>hoc enim eſt demonſtratum. </s>
          <s id="id001867">Similiter &amp; quare difficillimè feratur <lb></lb>ex b uſ que ad p, &amp; ultra p uſ que ad directum r f: at de motu ex a in f, <lb></lb>quod debeat ferri, quia plus remouetur, quam deſcendat, nulla eſt <lb></lb>ratio: ut nec cur ex oppoſito f ad a difficilem ſe præſtet: &amp; hoc eſt, <lb></lb>quia tertiam rationem etiam ipſe Ariſtoteles, &amp; qui eum ſequuti <lb></lb>ſunt, prætermiſit. </s>
          <s id="id001868">Ea autem eſt, quod dum fertur ad g, uel f etiam li­<lb></lb>cet non deſcendat magis, quàm remoueatur, ex a <lb></lb><figure id="id.015.01.121.1.jpg" xlink:href="015/01/121/1.jpg"></figure><lb></lb>ad centrum terræ tamen magis appropinquat. <lb></lb></s>
          <s id="id001869">Quia enim e a eſt ęqualis e c, quoniam prodeunt <lb></lb>à centro circuli eiuſdem, &amp; b e, &amp; e c ſunt maio­<lb></lb>res b c, ideò b a erit maior b c, eſt autem b cen­<lb></lb><arrow.to.target n="marg390"></arrow.to.target><lb></lb>trum mundi, ergo a motum ad c, appropinqua­<lb></lb>uit ipſi b</s>
        </p>
        <p type="margin">
          <s id="id001870"><margin.target id="marg388"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 98.</s>
        </p>
        <p type="margin">
          <s id="id001871"><margin.target id="marg389"></margin.target>I<emph type="italics"></emph>n præceden <lb></lb>ti.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001872"><margin.target id="marg390"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id001873">Dico etiam quod libra ex chalybe tenuiſsimo, <lb></lb>&amp; quanto <expan abbr="leuiorũ">leuiorum</expan> concharum, &amp; longioris iugi <lb></lb>10 exactior, quoniam lances illæ minori exceſſu <lb></lb>mouentur, quia plus diſtant ab hypomochlio. <lb></lb></s>
          <s id="id001874">Sit ergo libra, cuius iugum a b trutina c: lances d &amp; e, alia libra, <lb></lb>cuius lances h, &amp; k, &amp; l m longiores, iugum f g. </s>
          <s id="id001875">Conſtat, quod <lb></lb>qualis proportio f g ad a b, talis ambitus, ad ambitum: motus er­<lb></lb>go ſi ſit æqualis utrarumque, igitur a tanto minore proportione <lb></lb><figure id="id.015.01.121.2.jpg" xlink:href="015/01/121/2.jpg"></figure>


<pb pagenum="103" xlink:href="015/01/122.jpg"></pb>mouebitur in h, quam in d, uelut ſit proportio f g ad a b dupla, ut <lb></lb>ergo æqualiter moueantur, ſi ſit dupla ſexquiquarta in d cum lan­<lb></lb>ce ad e uacuam, erit in h ſexquialtera, &amp; mouebit æquali tempore. <lb></lb></s>
          <s id="id001876">Ergo iuxta hoc fient libræ, quæ examinabunt decimam, &amp; uigeſi­<lb></lb>mam partem grani, quod eſt neceſſarium in pretioſis rebus, &amp; me­<lb></lb>dicamentis potentibus, &amp; longè magis in mechanicis experimen­<lb></lb>tis, &amp; maximè quæ ad demonſtrationem pertinent magnitudinis <lb></lb>ſuperficierum, &amp; conſtat res in tribus, in longitudine, f g iungi, in le <lb></lb>uitate materiæ illius, &amp; lancium, nam tanto maior redditur propor<lb></lb>tio ponderis exigui, &amp; in firmitate iugi ac rectitudine. </s>
          <s id="id001877">ideò debet <lb></lb>fieri ex chalybe purgato, durato ac tenuiſsimo, natura que leui, &amp; ut c <lb></lb>ſit in medio, &amp; mobilis f g.</s>
        </p>
        <p type="main">
          <s id="id001878">Conſiderandum eſt demum an f l &amp; g m ſint grauiores f h, &amp; <lb></lb>g k. </s>
          <s id="id001879">Vt enim grauiores extiterint minus facilè mouentur. </s>
          <s id="id001880">Viden­<lb></lb>tur autem mihi, qui de his conſcripſerunt perperam contempſiſſe <lb></lb>hoc, conſtat enim, quòd dum l deſcendit, remouetur a b n c tru­<lb></lb>tina, &amp; m, quæ aſcendit contra appropinquat. </s>
          <s id="id001881">Videtur autem hoc <lb></lb>bifariam contra naturam: nam ut diximus pondus applicat ſe ad <lb></lb>rectam n c, quia uerſus centrum, &amp; etiam quia facit angulum ob­<lb></lb>tuſum, cum deberet, ut ab initio ſaltem conſtituere cum iugo re­<lb></lb>ctum. </s>
          <s id="id001882">Et de m nihil mirum eſt, cum acutum, ut ſe ad lineam, quæ ad <lb></lb>centrum retrahat. </s>
          <s id="id001883">Huiuſmodi præterijſſe Ariſtotelem, demiror, <lb></lb>quæ nimis fuerunt in conſpicuo, ut dubitem ne non ſuus ſit ille li­<lb></lb>ber, qui eius penè nihil ſapiat præter obſcuritatem. </s>
          <s id="id001884">Tentan­<lb></lb>dum eſt igitur horum cauſas aſsignare. </s>
          <s id="id001885">nam quæ huiuſmodi po­<lb></lb>teſt eſſe doctrina niſi perfecta fuerit, in omnibus etenim neceſſe eſt <lb></lb>aut omnia ſcire, aut ignorare. </s>
          <s id="id001886">In hoc igitur dico, quod h f, ſeu l f, <lb></lb>ſemper æquidiſtant n c trutinæ, ergo cum angulus f c n in clina­<lb></lb>to iugo fiat obtuſus deſcendente pondere, &amp; n c g aſcendente pon­<lb></lb>dere fiat acutus, ergo angulus l f c tantundem fiet obtuſior, &amp; m g c <lb></lb>acutior, quanto anguli ad c tales ſunt. </s>
          <s id="id001887">Et cauſa eſt quia n c ratio­<lb></lb>ne ponderis eſt directa ad centrum, ergo oportet, ut pondera l, uel <lb></lb>h, &amp; m, uel k, ſi debent tendere ad centrum, ut f l, &amp; g m æquidi­<lb></lb>ſtent n c, niſi quantum eſt pro diſtantia f, à puncto c, &amp; g a b eodem, <lb></lb>quæ comparata ad <expan abbr="centrũ">centrum</expan> terrę, ſeu mundi, eſt inſenſibilis omnino. <lb></lb></s>
          <s id="id001888">Circa hæc <expan abbr="notandũ">notandum</expan> iſtud mirabile ſcilicet, quod ratio motus, quan­<lb></lb>tumuis exigua ſufficit ad motus <expan abbr="modũ">modum</expan>, licet uelo citas <expan abbr="pẽdeat">pendeat</expan> ex gra<lb></lb>uitate, &amp; alijs. </s>
          <s id="id001889">Et quae graue, quod expers eſt ſenſus, debeat ſequi ratio <lb></lb>nem Geometricam uix ſapientibus <expan abbr="cognitã">cognitam</expan>, cauſa tamen una eſt, &amp; <lb></lb>perſpicua: <expan abbr="nã">nam</expan> omne graue eſt in linea à centro <expan abbr="mũdi">mundi</expan>: ſi <expan abbr="aũt">aut</expan> medium <lb></lb>grauis ſit extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, quę eſt in eo, nam <expan abbr="centrũ">centrum</expan> ſem<pb pagenum="104" xlink:href="015/01/123.jpg"></pb>per eſt in <expan abbr="eadẽ">eadem</expan>. </s>
          <s id="id001890">Ergo ſola inclinatio ad hoc ut <expan abbr="mediũ">medium</expan> grauis ſit in li­<lb></lb>nea <expan abbr="centrorũ">centrorum</expan> grauitatis &amp; terræ, ſufficit. </s>
          <s id="id001891">Eſt ergo principium in ſe i­<lb></lb>pſo. </s>
          <s id="id001892">In appenſis ſimiliter. </s>
          <s id="id001893">Trutina enim, &amp; finis iugi, &amp; grauis <expan abbr="cen­trũ">cen­<lb></lb>trum</expan> mundi <expan abbr="centrũ">centrum</expan> ſunt in <expan abbr="eadẽ">eadem</expan> linea, ut eſſe poſſunt, cum exigua illa <lb></lb>&amp; ſola diſtantia intercedat. </s>
          <s id="id001894">&amp; hoc eſt primum. </s>
          <s id="id001895">Quia ergo <expan abbr="iugũ">iugum</expan> eſt <lb></lb>ex materia ſolida, mouetur ratione, quæ dicta eſt, lances autem <lb></lb>oportet cum filis appenſi ſint, ut puncta f &amp; h, uel l, &amp; g k, uel g m <lb></lb>ſint in una linea cum centro terræ. </s>
          <s id="id001896">Et quia l magis diſtat a b f quam <lb></lb>h, &amp; m a g magis, quam k, &amp; oportet faciant eandem inclinatio­<lb></lb>nem, quia anguli trutinæ cum iugó ſunt ijdem, &amp; linea cl eſt ma­<lb></lb>ior c h, &amp; c m, quàm c k in quouis ſitu, ergo ſpatium, quod ambitur, <lb></lb>eſt maius ergo per d e monſtrata ſuperius l eſt grauius h etiam <lb></lb>præter uinculorum additionem, &amp; m grauius k. </s>
          <s id="id001897">Quanto igi­<lb></lb>tur longiores ſunt funiculi à libræ extremitate ſeu iugi, tanto gra­<lb></lb>uius redditur pondus, quod tamen multi putant eſſe falſum: nec <lb></lb>aliquid referre, quòd ſit longum, aut breue ſuſtentaculum.</s>
        </p>
        <p type="main">
          <s id="id001898">Propoſitio centeſima decima.</s>
        </p>
        <p type="main">
          <s id="id001899">Si duæ ſphæræ ex eadem materia deſcendant in <expan abbr="aẽ">ae</expan> <lb></lb>re eodem temporis momento ad planum ueniunt.<lb></lb><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"></figure><lb></lb><arrow.to.target n="marg391"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001900"><margin.target id="marg391"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001901">Supponitur quod ex eodem loco. </s>
          <s id="id001902">Sermo enim <lb></lb>abſurda ſub interpretatione nunquam niſi ab inui­<lb></lb>dioſo, uel imperito intelligi debet. </s>
          <s id="id001903">Sit ergo a tripla <lb></lb>ad b, ſphærula ad ſphærulam ex plumbo ambæ fer­<lb></lb>ro uel lapide eiuſdem generis, dico, quòd inæquali <lb></lb>tempore peruenient ad planum c d. </s>
          <s id="id001904">Nam a propor­<lb></lb>tionem habet ad b, ut uiginti ſeptem ad unum. </s>
          <s id="id001905">pro­<lb></lb>portio autem ſpatij a ad ſpatium b nonupla eſt, &amp; <lb></lb>proportio denſitatis aëris ad aërem eſt tripla, propterea quod den­<lb></lb>ſitas illa multiplicatur propter impetus magnitudinem. </s>
          <s id="id001906">nam ſi ro­<lb></lb>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du­<lb></lb>plo, quàm ſit robur, ut quinque percutiat baculo, ut duo: propter <lb></lb>denſitatem ergo maiorem aëris in a, quam in b: &amp; quoniam ſi ſub <lb></lb>maiore impetu mouetur <expan abbr="aẽr">aer</expan> ſub a, quam ſub b, igitur proportio <lb></lb>erit comparanda longitudini à centro a ad longitudinem a centro <lb></lb>b, quæ eſt tripla. </s>
          <s id="id001907">Si ergo ſubtripla eſt ratio motus b ad a, quod <lb></lb>ad medium attinet, tripla autem propter uelocitatem diſceſſus aë­<lb></lb>ris à medio grauitatis, quod eſt in ſuperficie e regione centri graui­<lb></lb>tatis in linea ad centrum mundi, ut dictum eſt in præcedenti: mani­<lb></lb>feſtum eſt, quod a, &amp; b inæquali tempore peruenient ad ſubie­<lb></lb>ctum planum, &amp; æquidiſtans centris eorum. </s>
          <s id="id001908">Similiter &amp; in aqua: 


<pb pagenum="105" xlink:href="015/01/124.jpg"></pb>cum uerò uideatur in illa tanto celerius a deſcendere, quàm b, <lb></lb>quanto eſt ſemidiameter a longior ſemidiametro b, liquet ex hoc, <lb></lb>quod æquali uelo citate deſcendunt, ſed ob uelocitatem motus in <lb></lb>aëre latet diſcrimen anticipationis contactus ſoli a ante b, qui di­<lb></lb>gnoſcitur in aqua, ex quo patet exactam eſſe æqualitatem. </s>
          <s id="id001909">Sed reſi­<lb></lb>liunt ſemel in aqua ambæ, cum pluries in aëre a ſolo, quare etiam in <lb></lb>aqua perturbatur cognitio in parum accuratis, at que ſenſu præditis, <lb></lb>ſicut etiam in caſu, ne altera alteram perueniat, utra que comprehenſa <lb></lb>duobus digitis, altera alteram tangente, &amp; uſque ad centrum in <lb></lb>aquam demiſsis ſimul digitis dilatatis dimittendæ ſunt.</s>
        </p>
        <p type="main">
          <s id="id001910">Propoſitio centeſima undecima.</s>
        </p>
        <p type="main">
          <s id="id001911">Cur ex medio tela ualidiorem ictum, &amp; naues in ſcalmo à remo, <lb></lb>ac malo recipiant inde ex puppi explorare.</s>
        </p>
        <p type="main">
          <s id="id001912">Ariſtoteles uidetur in Mechanicis, &amp; qui eum ſequuti ſunt, ui­</s>
        </p>
        <p type="main">
          <s id="id001913"><arrow.to.target n="marg392"></arrow.to.target><lb></lb>dentur rem nauticam quòd ad remos attinet, referre in longitu­<lb></lb>dinem partis, quæ ſcalmum tanquàm hypomochlium interiacet <lb></lb>&amp; manum: ea enim circa medium nauis cum illa ibi ſit latior ma­<lb></lb>ior eſt. </s>
          <s id="id001914">Sed &amp; qui lembos ducunt, &amp; in puppe magis diſtant à <lb></lb>ſcalmo &amp; in prora, quàm in medio nauis, nec tamen uelocius il­<lb></lb>lam agunt: non quòd ratio illa falſa ſit, ſed quia uelocius ferun­<lb></lb>tur etiam ob aliam cauſam, quàm ſit hæc, &amp; magis uniuerſalem. <lb></lb></s>
          <s id="id001915">Primum igitur ſumamus, quod ſuperiùs demonſtratum eſt ſcili­<lb></lb><arrow.to.target n="marg393"></arrow.to.target><lb></lb>cet, quòd ubi pondus aliquod æquale undique tanquam in li­<lb></lb>bra ſuſpenſum fuerit, proportio ponderis partium inæqualium <lb></lb>ad duas partes æquales, eſt confuſa ex proportione longitudi­<lb></lb>nis earundem, &amp; quadrato eiuſdem proportionis. </s>
          <s id="id001916">Sit ergo diui­<lb></lb>ſa a b in c, &amp; fiat c e æqualis c a: proportio igitur ponderis b e ad <lb></lb>pondus e a eſt compoſita ex proportione b e ad e a, &amp; quadrato <lb></lb><figure id="id.015.01.124.1.jpg" xlink:href="015/01/124/1.jpg"></figure><lb></lb>eius <expan abbr="ſecũdum">ſecundum</expan> longitudinem. </s>
          <s id="id001917">at poſita agi <lb></lb>na d g in medio a b, proportio ponderis b e <lb></lb>ad pondus ea eſt, ueluti longitudinis b e <lb></lb>ad e a, igitur proportio <expan abbr="põderis">ponderis</expan> b e ad e a, <lb></lb>cum agina eſt extra medium in c, eſt tanto <lb></lb>maior proportione b c ad ea, quantum eſt quadratum illius pro­<lb></lb><arrow.to.target n="marg394"></arrow.to.target><lb></lb>portionis, ergo b e pondus maius eſt, cum agina eſt in c, quàm in d. <lb></lb></s>
          <s id="id001918">igitur per <expan abbr="communẽ">communem</expan> animi <expan abbr="ſententiã">ſententiam</expan> addito communi pondere a e, <lb></lb>erit pondus a b minus ſemper cum agina eſt in d, &lt;08&gt; in ullo alio lo­<lb></lb>co a b. </s>
          <s id="id001919">Ergo pondus a b apprehenſum in d <expan abbr="mouebit̃">mouebitur</expan> a b æquali ui <lb></lb><arrow.to.target n="marg395"></arrow.to.target><lb></lb>maiore proportione, &lt;08&gt; in ullo alio loco. </s>
          <s id="id001920">Haſtile ergo in medio ap­<lb></lb>prehenſum maiore ui mouebitur, quàm in ulla alia parte. </s>
          <s id="id001921">Et ſi gra­


<pb pagenum="106" xlink:href="015/01/125.jpg"></pb>cilius ſit in anteriore parte propinquius comprehenſum calci, &amp; ſi <lb></lb>craſsius, uel grauius propius cuſpidi. </s>
          <s id="id001922">Semper igitur ob hanc cau­<lb></lb>ſam mota ex medio grauitatis, ſeu uelo, ſeu ramo, ſeu manu uelo­<lb></lb>cius mouentur, quàm ex alijs partibus. </s>
          <s id="id001923">In remo etiam poteſt acce­<lb></lb>dere illud commodum, cuius meminit Ariſtoteles. </s>
          <s id="id001924">Propter hoc igi<lb></lb>tur, qui malum in naui collo carunt tantùm unum, in medio fermè <lb></lb>eum collocarunt, ut antiqui: &amp; qui duos aut tres, maiorem craſsio­<lb></lb><arrow.to.target n="marg396"></arrow.to.target><lb></lb>rem ſcilicet, &amp; altiorem in medio conſtituerunt.</s>
        </p>
        <p type="margin">
          <s id="id001925"><margin.target id="marg392"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001926"><margin.target id="marg393"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 86.</s>
        </p>
        <p type="margin">
          <s id="id001927"><margin.target id="marg394"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001928"><margin.target id="marg395"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id001929"><margin.target id="marg396"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 82.</s>
        </p>
        <p type="main">
          <s id="id001930">Propoſitio centeſimaduodecima.</s>
        </p>
        <p type="main">
          <s id="id001931">Cur ex imo leuia longius ferantur declarare.</s>
        </p>
        <p type="main">
          <s id="id001932">Iam uerò <expan abbr="cõſideremus">conſideremus</expan>, quòd propoſitum eſt, non ſolum in com­<lb></lb><arrow.to.target n="marg397"></arrow.to.target><lb></lb>paratione ad medium, ſed extremorum inuicem, miſſa enim ab imo <lb></lb>uelo cius feruntur, quàm à medio non ſolum manu, ſed ſcorpioni­<lb></lb>bus, &amp; arcubus. </s>
          <s id="id001933">Videmus &amp; hoc obſeruare pueros uirgam lon­<lb></lb>gius iacentes non ex medio, ſed imo apprehenſam, quoniam pars <lb></lb>ipſa anterior, &amp; quæ manu apprehenſa eſt, uehementi impetu emit­<lb></lb>titur: &amp; ut recipit impetum magis æqualem, longius fertur, nam <lb></lb>quod emittitur proportionem habet ad ſpatium. </s>
          <s id="id001934">Cum ergo appre<lb></lb>henſa in medio uirga ſolum medietate anteriore impetum recipiat <lb></lb>per ſe, ob id minus fertur: at impetus ſequitur proportionem, ut ui­<lb></lb>ſum eſt, quæ eſt circa medium ob leuitatem ponderis. </s>
          <s id="id001935">In leuibus <lb></lb>ergo maius ſpatium ſuperabunt emiſſa ex imo, quoniam propor­<lb></lb>tio ſpatij eadem eſt ad duplum, &amp; ad dimidium. </s>
          <s id="id001936">igitur ex imo fer­<lb></lb>me duplum etiam ſpatij ſuperabit: non tamen omnino quia maio­<lb></lb>rem, ut dixi proportionem habet ad id, quod ex medio comprehen<lb></lb>ſum eſt. </s>
          <s id="id001937">At in leuibus non eſt neceſſarium, ut ex medio apprehen­<lb></lb>dantur, quoniam etiam cum incremento illo ponderis iam leuia <lb></lb>ſunt: plus ergo facit longitudo eius, quod eiaculatur, quàm impe­<lb></lb><figure id="id.015.01.125.1.jpg" xlink:href="015/01/125/1.jpg"></figure><lb></lb>tus, cuius demonſtratio eſt hæc. </s>
          <s id="id001938">Sit uirga <lb></lb>a b apprehenſa in medio ponderis unciæ <lb></lb>mediæ, &amp; in a d, ut ſit d a palmus, &amp; uigeſi­<lb></lb>ma pars totius a b, erit ergo reſiduum ad duplum, a d nonuplum, <lb></lb><arrow.to.target n="marg398"></arrow.to.target><lb></lb>&amp; a b tota unciarum quin que cum dimidia, ſi igitur grauetur, quia in <lb></lb>ſitu recto eſt mediæ unciæ, in æquidiſtanti terræ, quin que unciarum <lb></lb>cum dimidio, erit in ſitu dimidij recti unciarum trium. </s>
          <s id="id001939">Eſt igitur <lb></lb>proportio ſexcupla, ſi apprehendatur in medio, &amp; ad æquidiſtan­<lb></lb>tem, ad apprehenſam in imo, &amp; ad angulum medium: at emiſſa ex <lb></lb><arrow.to.target n="marg399"></arrow.to.target><lb></lb>a d habet totum aërem a b circumdantem impulſum ex c b ſolum <lb></lb>dimidium reliqua pars ui trahitur, ergo proportio ſpatij a b, erit <lb></lb>ſexdecupla fermè ſpatio b c, quoniam eſt triplicata corporis ad cor<lb></lb>pus eius, quæ eſt longitudinis ad longitudinem, &amp; quadruplicata 


<pb pagenum="107" xlink:href="015/01/126.jpg"></pb>reſpectu aëris a c, qui reſiſtit apprehenſa a b in c. </s>
          <s id="id001940">Et iam minus fere­<lb></lb>batur quinta parte, ideo longius eiaculabitur triplo ex a, quàm ex <lb></lb>c. </s>
          <s id="id001941">Nec tamen maiore impetu, quia obliquè fertur, &amp; quæ obliquè <lb></lb><expan abbr="feriũt">feriunt</expan>, minore cum impetu feriunt: at que eo magis ſi leuia fuerint: ab <lb></lb>aëre enim circumambiente perturbantur, &amp; in incertum trudun­<lb></lb>tur. </s>
          <s id="id001942">Quæ ergo grauia ſunt ex medio emiſſa, &amp; ad æquidiſtantem <lb></lb>longius feruntur, &amp; maiore cum impetu, quia magis directè: leuia <lb></lb>autem longius ex imo, ſed minore cum impetu, ſi aliqua cauſa à re­<lb></lb>cto, &amp; æquidiſtante declinauerint. </s>
          <s id="id001943">At ſi à ſuprema parte, &amp; iuxta <lb></lb>cuſpidem, neque procul feruntur, neque cum impetu ob cauſas di­<lb></lb>ctas. </s>
          <s id="id001944">Eadem quoque ratio eſt omnium machinarum: ideò ob lon­<lb></lb>gę longius eiaculantur, quoniam proportionem ſeruant ad cana­<lb></lb><arrow.to.target n="marg400"></arrow.to.target><lb></lb>lem. </s>
          <s id="id001945">Sed de hoc inferius agetur.</s>
        </p>
        <p type="margin">
          <s id="id001946"><margin.target id="marg397"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001947"><margin.target id="marg398"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 86.</s>
        </p>
        <p type="margin">
          <s id="id001948"><margin.target id="marg399"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 89.</s>
        </p>
        <p type="margin">
          <s id="id001949"><margin.target id="marg400"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 107.</s>
        </p>
        <p type="main">
          <s id="id001950">Propoſitio centeſimatertia decima.</s>
        </p>
        <p type="main">
          <s id="id001951">Cur uirga longius mittatur à puero, quàm à uiro inueſtigare.<lb></lb><arrow.to.target n="marg401"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001952"><margin.target id="marg401"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id001953">Diligentia, &amp; uſus puerilis efficit, ut uirga feratur ſecundum me­<lb></lb>dium rectianguli: uir autem non conſtanter iacit, &amp; ſecundum re­<lb></lb>ctum, at rectus inceſſus in leuibus, quia ab aëre in obliquum defle­<lb></lb>ctitur uirga ob longitudinem efficit, ut inflectatur infrà celerius, &amp; <lb></lb>deſinat citius motus, ac finiatur. </s>
          <s id="id001954">Tertia cauſa eſt, quòd leuiſsima <lb></lb>non adeò recipiunt impetum ut grauia: nam leuiſsimam &amp; exigu­<lb></lb>am ligni portionem maximo nixu uix excutiemus è manu. </s>
          <s id="id001955">Cauſa <lb></lb>ergo eſt: quoniam uim, oportet, ut habeat, quod contra naturam <lb></lb>mouetur, ut naturaliter moueri poſsit, quæcunque igitur naturaliter <lb></lb>exiguum habent motum, ut pluma, palea, feſtucæ nulla ratione ue­<lb></lb>hementer contra naturam agi poſſunt. </s>
          <s id="id001956">Quædam ergo à pueris lon<lb></lb>gius <expan abbr="iaciũtur">iaciuntur</expan> ob ſolam peritiam, &amp; exercitationem, quædam quo­<lb></lb>niam ad angulum latiorem magis feruntur, quàm ſit rectus, quæ­<lb></lb>dam quoniam leuiſsima ſunt. </s>
          <s id="id001957">Sed ſi leuiora non feruntur ualido <lb></lb>motu uiolento, cur tamen à pueris iacta longius <expan abbr="ferũtur">feruntur</expan>? </s>
          <s id="id001958">Ratio eſt, <lb></lb>quoniam maior uis deficiente obiecto magis fatigatur, atque ideò <lb></lb>minus mouet. </s>
          <s id="id001959">Propter hæc igitur omnia non ſolùm in pueris, ſed <lb></lb>in machinis, quæ accommodata ſunt, melius impelluntur, a c lon­<lb></lb>gius feruntur, quàm leuiſsima. </s>
          <s id="id001960">nam nec palea ſcorpione iacta tam <lb></lb>procul, quàm ſagitta fertur, cum proportio maior ſit, tamen ad pa­<lb></lb>leam, quàm ad ſagittam. </s>
          <s id="id001961">Inde fit, ut quemadmodum Turca ille lite­<lb></lb>ras ſui Principis, cum timeret ad noſtros propius accedere, lapidi al<lb></lb>ligatas longius emiſit. </s>
          <s id="id001962">Cauſam autem huius docet Ariſtoteles in <lb></lb>Mechanicis dum quærit cur, &amp; grauia &amp; leuia ualde longe proijci <lb></lb>nequeunt: nam grauia nimis, moueri <expan abbr="nõ">non</expan> facilè poſſunt: leuia etiam <lb></lb>ualde ad rem mouere non ualent. </s>
          <s id="id001963">Ob hæc utra que ex his paruo cum 


<pb pagenum="108" xlink:href="015/01/127.jpg"></pb>impetu emittuntur, tametſi uehementer nitaris. </s>
          <s id="id001964">Sed &amp; leuia ferun­<lb></lb>tur hac illac, ut non poſsint retinere impetum prioris uiolentiæ: in­<lb></lb>natum enim eſt, ut duorum motuum ſimul in eadem re uigentium, <lb></lb>cum illa proprio impetu feratur, unus alterum impediat: nam ſi ro­<lb></lb>ta uehatur circulariter acta, non tamen ceſſabit, aut iminuetur impe<lb></lb>tus circulationis. </s>
          <s id="id001965">Multa ergo in huiuſmodi anomalis motibus con<lb></lb>ſideranda ſunt, ut illorum impetum robur, ac locum definiamus.</s>
        </p>
        <p type="main">
          <s id="id001966">Ex hoc liquet, cur plumbeæ ſphærulæ longius ferantur à tor­</s>
        </p>
        <p type="main">
          <s id="id001967"><arrow.to.target n="marg402"></arrow.to.target><lb></lb>mento emiſſæ, quàm ligneæ, etiam ſi non frangantur.</s>
        </p>
        <p type="margin">
          <s id="id001968"><margin.target id="marg402"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id001969">Propoſitio centeſima quarta decima.</s>
        </p>
        <p type="main">
          <s id="id001970">Circularis motus differentias quatuor eſſe, earum qúe rationem <lb></lb>contemplari.</s>
        </p>
        <p type="main">
          <s id="id001971">In motu circulari aut axis <expan abbr="progredit̃">progreditur</expan>, aut ſuo loco manet. </s>
          <s id="id001972">Vtroque <lb></lb><arrow.to.target n="marg403"></arrow.to.target><lb></lb>autem modo uel mouetur ab axe, uel circumferentia, igitur conſtat <lb></lb>quatuor eſſe motuum differentias: quas cum tres proponat author <lb></lb>libri Mechanicarum, aut Ariſtotelem illum eſſe, credendum non <lb></lb>eſt, aut illum ſtupidum dicere neceſſe eſt, nam modum diuidendi <lb></lb>eum latuiſſe quis putet. </s>
          <s id="id001973">cum rota igitur aut ſphæra in plano cir­<lb></lb>cumagitur, motus eſt ex circumferentia prægrediente axe: ut pa­<lb></lb>lam eſt: motis enim loco nobis mouentur omnia, quæ ſunt in no­<lb></lb>bis. </s>
          <s id="id001974">Cum uerò rotæ ſub curru ſunt, progreditur axis earum, &amp; rota <lb></lb>ob id cum quieſcere nequeat, quia facilius circumuertitur, quàm <lb></lb>trahatur, procedit, &amp; hic eſt ſecundus modus, quo rota ex circumfe<lb></lb>rentia mouetur, &amp; ex axe initium eſt motus. </s>
          <s id="id001975">At uerò in rota molari, <lb></lb>&amp; quibus gladij exacuuntur, cum loco non moueantur, motus eſt <lb></lb>ex axe: axis enim rotam circumagit, non rota axem, quieſcit tamen <lb></lb>in eodem loco rota, &amp; axis ſcilicet, quia non progreditur, ſed in lo­<lb></lb>co mouetur: atque hic eſt tertius modus. </s>
          <s id="id001976">Demum ſuccula putei, &amp; <lb></lb>ipſa mouetur circulari motu, &amp; trochleæ etiam, neque enim progre­<lb></lb>diuntur: ſed non ex axe mouentur, uerùm ſuccula per coloppes cir<lb></lb>cumducitur, &amp; trochlea per funes, axis que in ſuccula mouetur, in tro<lb></lb>chleis autem quieſcit prorſus: dico mouetur, id eſt circumducitur, <lb></lb>non quod progrediatur: ut non ſolum ſint quatuor modi, ſed po­<lb></lb>tius quin que, nam &amp; demonſtratione oſtenduntur, &amp; experimento <lb></lb>docente deprehenduntur. </s>
          <s id="id001977">Horum omnium liberrimus eſt, primus <lb></lb>ex circumferentia progrediente toto, ſeu attracto ſeu impulſo &amp; ue<lb></lb>lociſsimus, cuius cauſam ſuprà oſtendimus. </s>
          <s id="id001978">Proximus huic eſt mo­<lb></lb><arrow.to.target n="marg404"></arrow.to.target><lb></lb>tus rotarum per axem, quoniam axis premit rotam interius ſo­<lb></lb>lam, &amp; labitur: ideo que quod &amp; axis, &amp; rota intus ſint leuiſsima, pro­<lb></lb>deſt plurimum: &amp; aurigæ axungia inungunt, &amp; nomen ab eo traxit 


<pb pagenum="109" xlink:href="015/01/128.jpg"></pb>axungia. </s>
          <s id="id001979">Et quae rota magna ſit: quoniam cum <expan abbr="nõ">non</expan> rota, ſed axis traha­<lb></lb>tur in æquali tempore &amp; magna, &amp; parua trahitur: utra que uerò una <lb></lb>conuerſione tantam <expan abbr="lineã">lineam</expan> rectam ſuperat, quanta eſt rotæ periphe­<lb></lb>ria. </s>
          <s id="id001980">Quod ſi plures ſint rotæ celerius feruntur, quia axis minus tan­<lb></lb>to <expan abbr="rotã">rotam</expan> premit. </s>
          <s id="id001981">Et ſi rectus ſit axis, &amp; bene rotundus, &amp; foramen ro<lb></lb>tundum, &amp; latius, &amp; è duriſsimo ligno, ut non poſsit in clinari: &amp; <lb></lb>rota ipſa in ambitu æqualis, omnia hæc faciunt ad motus uelo cita<lb></lb>tem, unde Homerus.<lb></lb><arrow.to.target n="marg405"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id001982"><margin.target id="marg403"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id001983"><margin.target id="marg404"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 40.</s>
        </p>
        <p type="margin">
          <s id="id001984"><margin.target id="marg405"></margin.target>I<emph type="italics"></emph>liad.<emph.end type="italics"></emph.end> 23.</s>
        </p>
        <p type="main">
          <s id="id001985"><foreign lang="grc">ἴχνια τύπτε όδεοσι άρ &amp; κόνιν ἀμφιχυθῡναι</foreign>.</s>
        </p>
        <p type="main">
          <s id="id001986">Id eſt, ueſtigia per cuſsit pedibus, ante que illa puluis pedibus ex­<lb></lb>cuſſus (ueſtigia ſcilicet relinquentibus) ingrederetur. </s>
          <s id="id001987">Principalis <lb></lb>autem cauſa uelo citatis eſt agens, uelut equi. </s>
          <s id="id001988">Sed inter <expan abbr="hũc">hunc</expan> motum <lb></lb>&amp; priorem medius eſt Scitalæ uocatæ, nam ut in primo axis proci­<lb></lb>dit &amp; rotundum à ſuperficie circumagitur, licet axis etiam circum­<lb></lb>ducatur, ut axis, &amp; rota, aut ſphæra duplici motu moueantur, ſci­<lb></lb>licet antrorſum, &amp; circumcirca, in rota currus duo ijdem motus <lb></lb>ſint, axis quo que antrorſum moueatur, ſed non circumagatur: unde <lb></lb>impeditior eſt hic motus: ita in Scytala utrunque utro que motu mo­<lb></lb>uetur, &amp; circumcirca, &amp; antrorſum, at que id commune eſt, cum pri­<lb></lb>mo ita axis mouet rotas, non rotæ axem, quòd ſecundo motui ro­<lb></lb>tarum in curru proprium eſt, ut tantum degenerent à primo motu, <lb></lb>quanto leuius uertuntur, quàm in ſecundo motu. </s>
          <s id="id001989">Trahitur ergo <lb></lb><figure id="id.015.01.128.1.jpg" xlink:href="015/01/128/1.jpg"></figure><lb></lb>iugum in ſcitala, uelut in rotis currus, <lb></lb>ſed eſt annexum rotis non in curri­<lb></lb>bus. </s>
          <s id="id001990">Propterea in primo motu trahi­<lb></lb>tur, uel impellitur à ſuperficie: in ſe­<lb></lb>cundo a b axe, ſed non affixo rotis, unde ægrè trahuntur in ſcyta­<lb></lb>la ab axe affixo rotę. </s>
          <s id="id001991">Quare leuius quàm in curru, difficilius quàm <lb></lb>in rota uel ſphæra à ſuperficie extima circumacta. </s>
          <s id="id001992">Quartus modus <lb></lb>eſt, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb></lb>moletrinis, &amp; rotis, quibus ferrum exacuitur. </s>
          <s id="id001993">Eſt enim hic ſimilior <lb></lb>primo, quia contrarius, in primo enim procedit rota, &amp; uertitur à <lb></lb>circumferentia, hic quieſcit rota, &amp; mouetur ab axe. </s>
          <s id="id001994">Proximus huic <lb></lb>eſt, qui fit in ſucculis ob firmitatem axis: nam axis eſt coniunctus <lb></lb>rotæ. </s>
          <s id="id001995">Vltimus eſt trochlearum, qui &amp; difficillimus: ſit enim à cir­<lb></lb>cumferentia, &amp; axis diſiunctus eſt à trochlea: quod ad dit difficulta­<lb></lb>tem. </s>
          <s id="id001996">Sed &amp; trochlea caret colloppibus. </s>
          <s id="id001997">Ergo uerum eſt, quod o­<lb></lb>mnia rotunda facilius circumaguntur, ſed uaria ratione: nam plus <lb></lb>mota ſuper aliquo plano, ut in plauſtris &amp; ſcytalis: minus in ſuccu­<lb></lb>lis, &amp; rotis acuentibus ferrum, &amp; molis: nam &amp; ſi rotunditatem iu­<lb></lb>uet ob æqualitatem ad conuerſionem, non tamen in his eſt ad eò 


<pb pagenum="110" xlink:href="015/01/129.jpg"></pb>utilis. </s>
          <s id="id001998">Vtilitas ergo prima eſt, cum circumuertitur in plano, uelut <lb></lb>in rotis ſcytalis, &amp; ſphæris. </s>
          <s id="id001999">Secunda quæ minor eſt, cum à ſuperfi­<lb></lb>cie circumuertitur, ut in trochleis. </s>
          <s id="id002000">Tertia cum à coloppis, quæ mi­<lb></lb>nima eſt omnium, ut in ſucculis. </s>
          <s id="id002001">Motus autem cœli non eſt ex tri­<lb></lb>plici primo genere, cum ſit in loco, &amp; non ad locum, neque ut rotæ <lb></lb>molaris: nam ille eſt ex axe: nec ut in trochlea: nam in ea axis quieſ­<lb></lb>cit ipſum autem cœlum circa axem non uertitur, ſed cum axe, ſi ta­<lb></lb>men inſecabilis linea circumagi poteſt dici. </s>
          <s id="id002002">Relinquitur ergo, ut <lb></lb>Cœli motus propior ſit motui ſucculæ, quàm alij motui. </s>
          <s id="id002003">Differt <lb></lb>ab eo in hoc, quod in ſuccula mouetur axis ab orbe: at in cœlo <lb></lb>ut non mouetur ab axe, ita nec axis ab orbe: cun que ſit motus ſim­<lb></lb>pliciſsimus, in alio genere collocandus eſt: quando quidem in illo <lb></lb>nulla pars poſsit dici primo, quod <expan abbr="neceſſariũ">neceſſarium</expan> eſt in uno quo que <expan abbr="horũ">horum</expan>.</s>
        </p>
        <p type="main">
          <s id="id002004">Propoſitio centeſima quinta decima.</s>
        </p>
        <p type="main">
          <s id="id002005">Proportionem motuum impulſionis, &amp; attractionis inter&#039;ſe ab <lb></lb>eadem ui declarare.</s>
        </p>
        <p type="main">
          <s id="id002006">Conſtat, quòd attractio cum fune longiore ualidior eſt, quam </s>
        </p>
        <p type="main">
          <s id="id002007"><arrow.to.target n="marg406"></arrow.to.target><lb></lb>cum manibus, quoniam eſt cum motu quodam: motus autem au­<lb></lb>get actionem, ideo attractio ualidior eſt hac de cauſa, ſed &amp; impul­<lb></lb>ſio cum baculo ualidior eſt, quam cum manibus, quoniam licet col<lb></lb>ligere omnes uires in illo baculo, &amp; ipſum applicare loco, unde fa­<lb></lb>cilius impelli poteſt. </s>
          <s id="id002008">Velut ſphæra ex medio latere: nam ibi magis <lb></lb>colliguntur uires, &amp; ad impellendum facilius eſt, quodcunque leui­<lb></lb>us eſt. </s>
          <s id="id002009">Pars autem magis remota à centro grauitatis eſt leuior, his <lb></lb>duabus cauſis, ſphæra ex medio latere facilius ac magis impellitur. <lb></lb></s>
          <s id="id002010">Sed nos ſupponimus nunc applicationem æqualem eſſe, nam ſe­<lb></lb>cus ad impellendum facilius eſt applicare totum corpus, quàm at­<lb></lb>tractionem. </s>
          <s id="id002011">Pectore enim magna ui impellimus, nihil eſt compar, <lb></lb>quo trahere poſsimus. </s>
          <s id="id002012">Sed, ut dixi, ſit baculus applicatus alicui la­<lb></lb>pidi ea parte, qua facilius poteſt impelli &amp; trahi, &amp; quæritur, quæ <lb></lb>maior ſit uis, an attrahendi? </s>
          <s id="id002013">&amp; dico quòd homo, uel conatur trahe­<lb></lb>re toto corpore, &amp; impellere, at que hoc modo magis trahit, quàm <lb></lb>impellet, quoniam corporis pondus melius adhibetur in tractione <lb></lb>quàm impulſu: uel citra corporis pondus, ſed ſola ui membrorum: <lb></lb>&amp; tunc magis impellit, quoniam impulſus fit corpore prono in <expan abbr="an­teriorẽ">an­<lb></lb>teriorem</expan> partem, quæ inclinatio, &amp; motus eſt naturalis magis, quàm <lb></lb>in attractione in partem poſteriorem. </s>
          <s id="id002014">Sed ubi nulla ſit diuerſitas <lb></lb>neque horum, neque figurarum æqualis uis æqualem efficit motum: <lb></lb>quia impulſus impellentis comparatione eſt attractio reſpectu al­<lb></lb>terius. </s>
          <s id="id002015">Verùm non eſt eadem uis nec propè par impellendi, at que <lb></lb>attrahendi hominibus, cum attractio fiat per muſculos ad origi­


<pb pagenum="111" xlink:href="015/01/130.jpg"></pb>nem ſuam naturaliter ſe retrahentibus impulſui nullum inſtrumen<lb></lb>tum à natura delegatum inuenio, nam ad extenſionem muſculi ſa­<lb></lb>nè ex aduerſo ſunt fabricati: cum ergo duo ſint tantum motus mu­<lb></lb>ſculorum tenſio, dum <expan abbr="retrahũtur">retrahuntur</expan> ad principium ſuum, &amp; remiſsio, <lb></lb>dum membrum quieſcit in naturali nullus erit locus impulſioni, <lb></lb>niſi ex conſequentia non per ſe, quamobrem multo infirmiorem il­<lb></lb>lum attractione in brachijs eſſe, neceſſe eſt.</s>
        </p>
        <p type="margin">
          <s id="id002016"><margin.target id="marg406"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002017">Propoſitio centeſima ſexta decima.</s>
        </p>
        <p type="main">
          <s id="id002018">Cur machinæ ablongæ igneæ longius emittant ſphæram ex­<lb></lb>plorare.<lb></lb><arrow.to.target n="marg407"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002019"><margin.target id="marg407"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002020">Quoniam ratio ſuperius adducta, neque in his, neque in hypophy­</s>
        </p>
        <p type="main">
          <s id="id002021"><arrow.to.target n="marg408"></arrow.to.target><lb></lb>ſis (uocant cerbatanas) non poteſt ſatisfacere, cum tamen idem ſe­<lb></lb>quatur in his, ut in illis uidetur, quaſi uis eſſe in ſphærula ſic emiſ­<lb></lb>ſa, &amp; non in aëre, quemadmodum dicebamus, coniuncto eſſe. </s>
          <s id="id002022">Ex <lb></lb>quo neceſſe eſſet, ut quod longius ferretur, etiam ualidiores ictus <lb></lb><figure id="id.015.01.130.1.jpg" xlink:href="015/01/130/1.jpg"></figure><lb></lb>inferret, hoc autem <lb></lb>non ita ſe habet, ſed <lb></lb>ictus magnitudo <lb></lb>ex robore machi­<lb></lb>narum tam ignea­<lb></lb>rum, quam ſcorpio <lb></lb>num pendet, nam <lb></lb>ſit a ſcorpio ma­<lb></lb>gnus, ſed tenuis, ex <lb></lb>hòc palam eſt lon­<lb></lb>gius mittere ſagit­<lb></lb>tam, quòd à parua, <lb></lb>&amp; breui, quantun­<lb></lb>uis craſſa non lon­<lb></lb>ge mittitur: at uerò <lb></lb>quod b craſſus &amp; paruus maiore cum impetu mittat oſtenditur <lb></lb>nam ea pondera ſagittæ mouet, quæ non poteſt mouere a, igitur b <lb></lb>ualidiore robore mouet, quam a. </s>
          <s id="id002023">Præterea illud oſtendit iugum fu­<lb></lb>nis arcus craſsiora duriora, quæ maioribus uiribus <expan abbr="indigẽt">indigent</expan>, quam <lb></lb>a, qui à puero tendi poterit. </s>
          <s id="id002024">Non eſt ergo eadem ratio mittendi <lb></lb>longius, &amp; ualidiore cum robore. </s>
          <s id="id002025">Eadem ergo cum ratio ſit in <lb></lb>machinis igneis, craſsiores enim, &amp; latiores ac breuiores magis <lb></lb>concutiunt, quam longiores tenuiores minoris ſphæræ capaces: <lb></lb>non ſolum ob magnitudinem ſphæræ magis illæ concutiunt, ſed, <lb></lb>ut dixi, ob maiorem impetus uim: cauſa ergo eſt manifeſta in his, <lb></lb>ſed non cauſa, qua longius ferantur in longiore canali. </s>
          <s id="id002026">Sed uide­


<pb pagenum="112" xlink:href="015/01/131.jpg"></pb>tur una, eadem que eſſe ratio in utriſque. </s>
          <s id="id002027">Conſtituatur can alis a b <lb></lb>lońgior, &amp; c d breuior, ut ſit ſexquialter a b ad c d, &amp; ſit rurſus <lb></lb><figure id="id.015.01.131.1.jpg" xlink:href="015/01/131/1.jpg"></figure><lb></lb>ſphærulæ locus e in longiore, <lb></lb>ſexquialter in diſtantia a b, qua <lb></lb>lis eſt in f a d, &amp; erit per dicta <lb></lb>ab Euclide in quinto, ac ſexqui<lb></lb>altera c f. </s>
          <s id="id002028">Poſſemus igitur di­<lb></lb>cere, quod uelut ab hypomo­<lb></lb>chlio longiore ſpatio circuma­<lb></lb>gitur pondus: ita &amp; a b c, &amp; f. <lb></lb></s>
          <s id="id002029">Sed rurſus incidimus in id, ut <lb></lb>maiore impetu feratur e quàm f. </s>
          <s id="id002030">Ideo ſi concedatur maiore ferri ex <lb></lb>e, quam ex f non ſequitur, ut celerius, aut maiore impetu. </s>
          <s id="id002031">Percutit <lb></lb>puer pugno quanta ui poteſt ac celerrimè, uir robuſtus lentè, &amp; mi­<lb></lb>nore impetu, ſed tamen ictus longè maior eſt. </s>
          <s id="id002032">Eſt enim ictus robur <lb></lb>non à uelo citate ſolum, ſed maiore ex ponderis grauitate, quæ ſola <lb></lb>premit, urget, &amp; frangit etiam ſine motu. </s>
          <s id="id002033">Solum ergo id reſtat du­<lb></lb>bium, cur ſi grauius eſt, moueatur eodem fermé impetu: nam quo <lb></lb>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con<lb></lb>cutit, aut quaſſat, ſed grauitas ad hoc plus facit impetu. </s>
          <s id="id002034">Palea maxi­<lb></lb>mo impetu demiſſa non ferit, non ledit, &amp; celerius deſcendit, fer­<lb></lb>rum ſola grauitate actum, imò etiam temperato ictu lædit graui­<lb></lb>ter, quaſſat, &amp; frangit: itaque f maiore indiget quantitate pyrij pulue­<lb></lb>ris, quàm e: ſiquidem tertia parte ponderis ſuæ ſphæræ: at maius <lb></lb>eſt pondus f quam e, ergo maius pondus pulueris f quàm e, ergo <lb></lb>maior uehementia ictus, ſiquidem ea ſequitur, robur cauſæ mouen<lb></lb>tis ſimpliciter: ut concludamus longitudinem ictus ſequi propor­<lb></lb>tionem motoris ad motum, ſed uehementia robur motoris: nam ſi <lb></lb>ex portione mouet æquale pondus maiore cum impetu mouet, <lb></lb>quoniam maior eſt proportio: ſi minore igitur pondus maius eſt, <lb></lb>&amp;, ut dixi plus facit magnitudo ponderis cum leui ictu, quàm ma­<lb></lb>gnitudo ictus cum leui pondere. </s>
          <s id="id002035">Quæ ergo feruntur per longio­<lb></lb>res canales maiore impetu feruntur, &amp; ſocietatem <expan abbr="habẽt">habent</expan> aëris moti <lb></lb>per longius <expan abbr="ſpatiũ">ſpatium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="uĩs">uis</expan><lb></lb>motus confirmata eſt, &amp; proportio eius, quòd mouet, maior eſt ad id, <lb></lb>quod <expan abbr="mouet̃">mouetur</expan>, quia minus extenditur, at uerò f <expan abbr="motũ">motum</expan> minore propor­<lb></lb>tione <expan abbr="ictũ">ictum</expan> facit <expan abbr="maiorẽ">maiorem</expan>, quia, ut dixi, <expan abbr="tãto">tanto</expan> grauius, eſt quod ferit. </s>
          <s id="id002036">Quod <lb></lb><expan abbr="autẽ">autem</expan> minus <expan abbr="extẽdatur">extendatur</expan> machina a b quam c d, <expan abbr="nũc">nunc</expan> <expan abbr="oſtẽdere">oſtendere</expan> oportet.</s>
        </p>
        <p type="margin">
          <s id="id002037"><margin.target id="marg408"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 103.</s>
        </p>
        <p type="main">
          <s id="id002038">Propoſitio centeſima decima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id002039">In cuniculis maior eſt uis pulueris copioſioris ampliore in ſpa­<lb></lb>tio, quàm paucioris in minore iuxta proportionem eandem.</s>
        </p>
        <pb pagenum="113" xlink:href="015/01/132.jpg"></pb>
        <p type="main">
          <s id="id002040">Sit ſpatium f d ſexqui tertium b e, puluis quo que in f d ſpatio ſi­<lb></lb><arrow.to.target n="marg409"></arrow.to.target><lb></lb>militer ſexqui tertius pulueri b e pondere, &amp; manifeſtum eſt, quod <lb></lb>dum conuertitur in ignem qualiſcunque ſit proportio (modo eadem <lb></lb>ignis ad puluerem) erit ignis in f d pariter ſexqui tertius igni in b e, <lb></lb>dico quòd ſi craſsities f d ſit etiam ſexqui tertia craſsitiei b e, quod <lb></lb>poterit frangi, &amp; moueri f d quieſcente b e. </s>
          <s id="id002041">Vnde idem in cuniculis <lb></lb>ut magnus cuniculus cum multo puluere poſsit mouere montem <lb></lb>paruus cum puluere proportione reſpondente priori non poſsit. <lb></lb></s>
          <s id="id002042">Nam cùm æqualia ſint omnia iuxta que rationem eandem, neceſſe eſt <lb></lb>ut pro ratione extendantur, at in paruo ſpatio minor fit denſitas cę­<lb></lb>tera paria ſunt, ergo à paruo ſpatio non tantus fit impetus, quantus <lb></lb>à magno. </s>
          <s id="id002043">Impetus etiam proportionem habet ad <expan abbr="põdus">pondus</expan>, &amp; ad con­<lb></lb>iunctionem, à maiore igitur impetu plura, &amp; maiora mouentur, &amp; <lb></lb>conuelluntur, quam à minore, ob hæc igitur minores cuniculi ſuc­<lb></lb>cutiunt, maiores euertunt, maximi exturbant, &amp; proijciunt. </s>
          <s id="id002044">Nam <lb></lb>qui ſuccutiunt, ubi pondus, aut coniunctio maior ſit, quàm ut di­<lb></lb>ſtrahere poſsint, condenſant partes proximiores, &amp; rimas faciunt, <lb></lb>per quas exhalat ignis aut omnino extinguitur, aut condenſatur. <lb></lb></s>
          <s id="id002045">At ergo in bellicis machinis, minus dilatat puluis, cum fuerit in lon <lb></lb>go canali, ob id ergo maiore impetu feruntur per illas, quàm per <lb></lb>breuiores, etiam quòd minor ſit puluis, minor ſit ignis. </s>
          <s id="id002046">Experimen <lb></lb>tum facies in canali, ubi ſambuci medulla pro globulo flatu impel­<lb></lb>lente expellitur abſ que periculo: nam quanto minor fuerit canalis <lb></lb>ambitu ac longior eo maiore impetu pellitur. </s>
          <s id="id002047">Forſan quiſpiam nos <lb></lb>meritò poterit uideri <expan abbr="reprehẽdiſſe">reprehendiſſe</expan>, quòd inanis gloriæ ſtudio per­<lb></lb>nicioſa humano generi doceam. </s>
          <s id="id002048">Quibus reſpondeo, me nihil do cu <lb></lb>iſſe, quod ín humani generis detrimentum cedat, huiuſmodi que prę­<lb></lb>cepta iam obſcuraſſe, ut ne quid mali accidere poſſet hominibus ex <lb></lb>his: <expan abbr="nã">nam</expan> quòd ad ea, quæ declarata, ſunt, cauſas ſolùm retuli, effectus <lb></lb>ipſi modi artis <expan abbr="nimiũ">nimium</expan> feruntur, ac nimio pluſquam <expan abbr="uellẽ">uellem</expan> in telligun­<lb></lb>tur. </s>
          <s id="id002049">Vt cum ad copiam, ad magnitudinem, ad coacta imperia miſe­<lb></lb>rorum reſpicio, nihil plus poſsit addi. </s>
          <s id="id002050">Omnia enim huiuſque <expan abbr="ſpectãt">ſpectant</expan> <lb></lb>ad potentiorum in crementa. </s>
          <s id="id002051">An ergo ſuccurrere afflictis, obſeſsis, <lb></lb>cinctis, æquare <expan abbr="conditionẽ">conditionem</expan>, liberare à ſeruitute etiam rebelles <expan abbr="nõ">non</expan> li­<lb></lb>cebit? </s>
          <s id="id002052">Ab initio fuimus omnes liberi: excogitata fuit regni ratio ad <lb></lb>commodum hominum, ea uerſa eſt per uim in <expan abbr="Tyrannidẽ">Tyrannidem</expan>. </s>
          <s id="id002053">Subtili <lb></lb>ergo ratione <expan abbr="occurrendũ">occurrendum</expan> eſt imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni­<lb></lb>mis, ut dixi, quę ad cuniculos ad <expan abbr="magnitudinẽ">magnitudinem</expan> <expan abbr="machinarũ">machinarum</expan> ad rectos <lb></lb>ictus ad <expan abbr="libramẽta">libramenta</expan> ad longitudinem ſpatij, per quos globus ille de­<lb></lb>fertur, nota ſunt improbis illis artificibus, nec noſtrum eſt ſpectare, <lb></lb>cur id licuerit, poſtquam Deus hanc uiolentiam eſſe uoluit. </s>
          <s id="id002054">Multa <lb></lb>damnamus, <expan abbr="q̃">quae</expan> Deus eſſe uult: boni uiri eſt <expan abbr="nõ">non</expan> niſi opitulari homini­<lb></lb>bus, <expan abbr="etiã">etiam</expan> malis modo bonis futuri <expan abbr="nõ">non</expan> ſint <expan abbr="impedimẽto">impedimento</expan>: <expan abbr="quamobrẽ">quamobrem</expan> 


<pb pagenum="114" xlink:href="015/01/133.jpg"></pb>ea tradenda ſunt, quæ oppreſsis ſint auxilio: ea ſunt, quę ſubtilibus <lb></lb><expan abbr="conſtãt">conſtant</expan> rationibus, et multiplicata <expan abbr="amittũt">amittunt</expan> uim ut quaſi <expan abbr="pręſtẽt">pręſtent</expan> pau<lb></lb>ca multis, &amp; exigua magnis. </s>
          <s id="id002055">In cęteris obſcurare ita decet cuncta, <expan abbr="q̃">quae</expan> <lb></lb>obeſſe poſſunt, aut quouis modo puerti ad malos uſus <expan abbr="queãt">queant</expan>, ut di­<lb></lb>cta <expan abbr="nõ">non</expan> dicta eſſe <expan abbr="putẽt">putent</expan>, hoc eſt <expan abbr="officiũ">officium</expan> <expan abbr="nõ">non</expan> ſolum probi, ſed <expan abbr="etiã">etiam</expan> pruden<lb></lb>tis uiri.</s>
        </p>
        <p type="margin">
          <s id="id002056"><margin.target id="marg409"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id002057">Propoſitio centeſima decima octaua.</s>
        </p>
        <p type="main">
          <s id="id002058">Quanta proportione decedat ictus in obliquum parietem ab eo, <lb></lb>qui eſt ad perpendiculum declarare.</s>
        </p>
        <figure id="id.015.01.133.1.jpg" xlink:href="015/01/133/1.jpg"></figure>
        <p type="main">
          <s id="id002059">Sit paries b d e, ex a <expan abbr="ferat̃">feratur</expan> in dictus, qui ſi <lb></lb><arrow.to.target n="marg410"></arrow.to.target><lb></lb>eſſet in c d <expan abbr="parietẽ">parietem</expan> eſſe ad perpendiculum, &amp; <lb></lb>ualidiſsimus, ſin uero in f g abraderet, &amp; <expan abbr="nõ">non</expan> <lb></lb><expan abbr="cõquaſſaret">conquaſſaret</expan>. </s>
          <s id="id002060">Quæritur ergo ex b d e muro <lb></lb>qualis excipietur? </s>
          <s id="id002061">erit ergo proportio anguli c d a ad <expan abbr="angulũ">angulum</expan> b d a, <lb></lb>ueluti ictus a d in d c ad <expan abbr="ictũ">ictum</expan> in b d, <expan abbr="manifeſtũ">manifeſtum</expan> eſt <expan abbr="aũt">aut</expan> ſequi proportio­<lb></lb>nem, <expan abbr="quoniã">quoniam</expan> maxima uarietate <expan abbr="cõſtat">conſtat</expan> dum ex angulo b d a acuto fit <lb></lb>acutior, <expan abbr="quoniã">quoniam</expan> ſi b d c ſit <expan abbr="q̃druplus">quadruplus</expan> b d a erit reſiduus ad <expan abbr="dimidiũ">dimidium</expan> b <lb></lb>d a nonuplus ipſi dimidio, &amp; ad <expan abbr="quartã">quartam</expan> <expan abbr="partẽ">partem</expan> habebit proportionem <lb></lb><expan abbr="decemnouẽ">decem nouem</expan> ad <expan abbr="unũ">unum</expan>. </s>
          <s id="id002062">Si ergo <expan abbr="etiã">etiam</expan> in <expan abbr="idẽ">idem</expan> tenderent, <expan abbr="nõ">non</expan> efficerent mille <lb></lb>ictus q̊d tres, cuius demonſtratio hęc eſt. </s>
          <s id="id002063">Supponamus <expan abbr="proportionẽ">proportionem</expan> <lb></lb>b d c ad <expan abbr="q̃rtam">quartam</expan> <expan abbr="partẽ">partem</expan> a d b ad dito reſiduo ad b d c eſſe <expan abbr="ſolũ">ſolum</expan> <expan abbr="decuplã">decuplam</expan>: <lb></lb><expan abbr="tũc">tunc</expan> ex duob. </s>
          <s id="id002064">ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, quę in b e, <expan abbr="etiã">etiam</expan> tribus <lb></lb>millecupla: nam <expan abbr="cõquaſſata">conquaſſata</expan> turri in primo ictu, id d decuplo magis <lb></lb>ad perpendiculum &lt;08&gt; in b d e <expan abbr="ſumat̃">ſumatur</expan> decima pars in ambitu d, &amp; illa <lb></lb>erit ergo <expan abbr="tã">tam</expan> diſſoluta, &amp; infirma ex ſuppoſito, &lt;08&gt; eſt tota b e: ſed ex ſe<lb></lb>cundo ictu decuplo magis <expan abbr="cõquaſſabit̃">conquaſſabitur</expan> illa pars, &lt;08&gt; b e ergo tota d c <lb></lb>centuplo magis <expan abbr="quaſſabit̃">quaſſabitur</expan> ex duob. </s>
          <s id="id002065">ictibus c d turris, &lt;08&gt; b e, &amp; ita in <lb></lb>tribus: ex <expan abbr="decẽ">decem</expan> millibus ergo ictibus <expan abbr="etiã">etiam</expan> ad amuſsim directis, <expan abbr="cũ">cum</expan> ta<lb></lb><expan abbr="mẽid">men id</expan> uix fieri poſsit in <expan abbr="tãta">tanta</expan> multitudine <expan abbr="nõ">non</expan> plus <expan abbr="cõminuet̃">comminuetur</expan> b d e, &lt;08&gt;<lb></lb>ex decë c d <expan abbr="p̃ter">pnter</expan> <expan abbr="quã">quam</expan> <expan abbr="exiguũ">exiguum</expan> <expan abbr="quippiã">quippiam</expan> in ſuperficie. </s>
          <s id="id002066">Imo ut <expan abbr="declaratũ">declaratum</expan> <lb></lb>eſt multo minus repetita ratione multiplicis. </s>
          <s id="id002067">Ob id in arce <expan abbr="Medio­lanẽſi">Medio­<lb></lb>lanenſi</expan> exterius lapidibus uiuis in <expan abbr="rotundũ">rotundum</expan> diducta ſuperficie inter­<lb></lb><figure id="id.015.01.133.2.jpg" xlink:href="015/01/133/2.jpg"></figure><lb></lb>uallo que <expan abbr="q̃">quae</expan>drato hunc in <expan abbr="modũ">modum</expan> munitę ſunt altiores tur<lb></lb>res. </s>
          <s id="id002068">Fiat ergo murus cuius proportio a d c ad b d a ſit ſex<lb></lb>quitertia, erit que angulus b d c <expan abbr="dodrãs">dodrans</expan> recti, &amp; <expan abbr="parũ">parum</expan> incli <lb></lb>natis, <expan abbr="ſiquidẽ">ſiquidem</expan> b d c erit quarta pars recti, &amp; ſit tantę ma­<lb></lb>gnitudinis, at que duritiei, ac adeò benè coniunctus fer­<lb></lb><arrow.to.target n="table16"></arrow.to.target><lb></lb>reis cathenis, ac ſtolonibus, ut poſsit reſiſtere <expan abbr="machinarũ">machinarum</expan> <expan abbr="fe­rentiũ">fe­<lb></lb>rentium</expan> <expan abbr="ſphęrã">ſphęram</expan> <expan abbr="librarũ">librarum</expan> ducentarum (quæ ſanè maximæ ſunt) <lb></lb><figure id="id.015.01.133.3.jpg" xlink:href="015/01/133/3.jpg"></figure>quinquaginta: <expan abbr="tũc">tunc</expan> cum proportio ſexquitertia nouies repeti­<lb></lb>ta, ut in numeris uides, efficiat quinquies replicatis nouem <lb></lb>ictibus, fiet proportio decupla quinquies producta, quę eſt cen <lb></lb><expan abbr="tũ">tum</expan> millium ad <expan abbr="unũ">unum</expan> in quadraginta quin que ictibus. </s>
          <s id="id002069"><expan abbr="Antequã">Antequam</expan> <lb></lb>ergo peruenit ad quinquaginta ictus rectos neceſſe erit, ut 


<pb pagenum="115" xlink:href="015/01/134.jpg"></pb>multo plures centum millibus ictus excipiat ante &lt;08&gt; euertatur, quæ <lb></lb>recta ſi eſſet quinquaginta ſolùm potuiſſet ſuſtinere. </s>
          <s id="id002070">Quæ ergo hu<lb></lb>mana potentia ſufficeret. </s>
          <s id="id002071">In arce Medio<expan abbr="lanẽſi">lanenſi</expan> uidimus uix attactas <lb></lb>in illis extuberationibus lapideis. </s>
          <s id="id002072">Sed quoniam hic occurritur per <lb></lb>inclinationem machinarum, ideò de hoc <expan abbr="ſermonẽ">ſermonem</expan> ſum habiturus.</s>
        </p>
        <p type="margin">
          <s id="id002073"><margin.target id="marg410"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <table>
          <table.target id="table16"></table.target>
          <row>
            <cell>729</cell>
          </row>
          <row>
            <cell>972</cell>
          </row>
          <row>
            <cell>1296</cell>
          </row>
          <row>
            <cell>1728</cell>
          </row>
          <row>
            <cell>2304</cell>
          </row>
          <row>
            <cell>3072</cell>
          </row>
          <row>
            <cell>4096</cell>
          </row>
          <row>
            <cell>5461 1/3</cell>
          </row>
          <row>
            <cell>7281 7/9</cell>
          </row>
        </table>
        <p type="main">
          <s id="id002074">Propoſitio centeſima decima nona.</s>
        </p>
        <p type="main">
          <s id="id002075">Quantum ictus machinę procliuis ad <expan abbr="angulũ">angulum</expan> <expan abbr="minuat̃">minuatur</expan> explorare.</s>
        </p>
        <p type="main">
          <s id="id002076">Huiuſce cauſa <expan abbr="excogitarũt">excogitarunt</expan>, ut ictus ad <expan abbr="perpendiculũ">perpendiculum</expan> <expan abbr="dirigeret̃">dirigeretur</expan>, &amp; <lb></lb><arrow.to.target n="marg411"></arrow.to.target><lb></lb><expan abbr="quanquã">quanquam</expan> angulus d e f ſit ęquali angulo a b c, longè <expan abbr="tñ">tamen</expan> maior eſt uis <lb></lb>a b &lt;08&gt; d e duplici cauſa, &amp; <expan abbr="quoniã">quoniam</expan> a b eſt <expan abbr="ſecundũ">ſecundum</expan> nat uram impetus <lb></lb><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"></figure><lb></lb>ignis, &amp; <expan abbr="etiã">etiam</expan> <expan abbr="eorũ">eorum</expan>, quę <expan abbr="emittunt̃">emittuntur</expan> in altum: &amp; q̊d pars <lb></lb>ſuperior in b retineat <expan abbr="ictũ">ictum</expan>, in e non retineat. </s>
          <s id="id002077">Sed caui<lb></lb>tas fiat maior in inferiore parte: cuius <expan abbr="experimẽtum">experimentum</expan> <lb></lb>quiliber facere poteſt <expan abbr="cũ">cum</expan> haſta. </s>
          <s id="id002078">Huic ergo ſolertiæ, <expan abbr="q̃">quae</expan> <lb></lb>tormenta iubet altius collocare obſtat <expan abbr="primũ">primum</expan>, quod <lb></lb>ictus ex decliui ſitu periculoſior eſt pro machina, &amp; ma<lb></lb>ximè q̊d retro impellit, quae ex retro ceſſa, poſt &lt;08&gt; exone<lb></lb>rata eſt, <expan abbr="dignoſcit̃">dignoſcitur</expan>, &amp; ad <expan abbr="collimandũ">collimandum</expan> decedit parte <expan abbr="ui­riũ">ui­<lb></lb>rium</expan> ſuarum, q̊d etſi <expan abbr="paruũ">paruum</expan> ſit in ductu <expan abbr="tñ">tamen</expan>, &amp; <expan abbr="ictuũ">ictuum</expan> mul<lb></lb>tiplicatione <expan abbr="magnũ">magnum</expan> affert diſcrimen. </s>
          <s id="id002079">Habet &amp; <expan abbr="cõmo">commo</expan> <lb></lb>dum ſitus muri accliuis <expan abbr="terrã">terram</expan> <expan abbr="ſuppoſitã">ſuppoſitam</expan> ad perpendiculum, <expan abbr="q̃">quae</expan> ictum <lb></lb>ſuſtinet: adeò ut omnib. </s>
          <s id="id002080"><expan abbr="inuicẽ">inuicem</expan> collectis, perinde ſit ac ſi ex perpen­<lb></lb>diculo, et ęquidiſtanti ad <expan abbr="ſolũ">ſolum</expan> <expan abbr="feriat̃">feriatur</expan>. </s>
          <s id="id002081">Venetus. </s>
          <s id="id002082">S. aliter Patauij cauit, <lb></lb>uidetur que, quae ſapientiſsimus ſit, &amp; eandem ſequatur ubi que normam, <lb></lb>poſt &lt;08&gt; in <expan abbr="rotundã">rotundam</expan> figuram <expan abbr="totũ">totum</expan> urbis ambitum formauit, &amp; foſſa la<lb></lb>ta, ac pro fundiſsima aqua que perenni muniuit, &amp; <expan abbr="ſummã">ſummam</expan> muri partem <lb></lb><expan abbr="rotundã">rotundam</expan> in hunc <expan abbr="modũ">modum</expan> effecit <expan abbr="cauã">cauam</expan> que interius undique, ne cuniculis <lb></lb><figure id="id.015.01.134.2.jpg" xlink:href="015/01/134/2.jpg"></figure><lb></lb>poſſet euerti, à lateribus uerò humiles, ac craſsiſsimas turres, ut nul<lb></lb>la ui poſſent dirui, eas que tormentis bellicis, undique latera luſtrantib. <lb></lb></s>
          <s id="id002083">repleſſet, illud diligentiſsime cauit, ne murus humilior eſſet aduerſa <lb></lb>ripa, ſed ad <expan abbr="libellã">libellam</expan> tamen depreſſus, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb></lb>ſis ſphęrulæ non tangerent <expan abbr="murũ">murum</expan>: nam <expan abbr="cũ">cum</expan> foſſa ſit quadraginta paſ­<lb></lb>ſuum, excedat <expan abbr="aũt">aut</expan> murus <expan abbr="exteriorẽ">exteriorem</expan> aggerem uno paſſu, ut quicquid <lb></lb>in ambitu eſt uno ictu oculi cognoſci poſsit, &amp; aggeris angulus ma<lb></lb>ior ſit uno paſſu, <expan abbr="tũ">tum</expan> magis adiecta craſsitie machinę fieri non poteſt, <lb></lb>ut ictus in <expan abbr="murũ">murum</expan> dirigatur. </s>
          <s id="id002084">Eam ob cauſam <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="ędificiũ">ędificium</expan> ul­<lb></lb><figure id="id.015.01.134.3.jpg" xlink:href="015/01/134/3.jpg"></figure><lb></lb>lum, aut planta, uel colliculus eſſet cir­<lb></lb>cum circa <expan abbr="urbẽ">urbem</expan> ad tria M. P. laborat hoc <lb></lb>periculo hęc urbs, ne tota ędificijs euer­<lb></lb>ſis concidat. </s>
          <s id="id002085"><expan abbr="Turcarũ">Turcarum</expan> enim Princeps di­<lb></lb>dicit, ut in Nouo caſtro in Melitę Inſulę <lb></lb>arce S. </s>
          <s id="id002086">Elmi appellata pluſ &lt;08&gt; mille icti­<lb></lb>bus in ſingulos dies imo M D obtundere 


<pb pagenum="116" xlink:href="015/01/135.jpg"></pb>munitiones. </s>
          <s id="id002087">Eum que impetum producere ad quindecim dies, &amp; ui­<lb></lb>ginti tum etiam longius, ut facilè domos omnes euertat, homines <lb></lb>occidat: ſi qui ſuperſunt tot in commodis obruuntur uigilijs, fame, <lb></lb>ſiti, puluere, ut inutiles reddantur. </s>
          <s id="id002088">Ideò huic <expan abbr="incõmodo">incommodo</expan> occurrunt <lb></lb>aggeribus intra mœnia erectis, in quos uis <expan abbr="tormẽtorum">tormentorum</expan> igneorum <lb></lb>emoritur. </s>
          <s id="id002089">Sed dices, cur ergo non pro muris erigere eos præſtat, &amp; <lb></lb>minore ſumptu ſatis? </s>
          <s id="id002090">quoniam ſubruuntur à foſſoribus facillimè, ſi <lb></lb>ad illos peruenire poſsit hoſtis. </s>
          <s id="id002091">Ideò intra mœnia utiliſsimi ſunt, pro <lb></lb>mœnijs parum proſunt. </s>
          <s id="id002092">Quod uerò ad teſtudines attinet, ſub qui­<lb></lb>bus <expan abbr="latẽt">latent</expan> foſſores machinæ laterales, &amp; à fronte &amp; ignes, &amp; aqua al­<lb></lb>tior prohibent omnino iniuriam, quę ab his imminet. </s>
          <s id="id002093">Cæterum hu­<lb></lb>iuſmodi cum in longum <expan abbr="differunt̃">differuntur</expan> morbis, illuuie, <expan abbr="incõmodis">incommodis</expan>, plu­<lb></lb>uijs, frigoribus omnino <expan abbr="diſſoluũtur">diſſoluuntur</expan>, ut nulla multitudo huic operi <lb></lb>ſufficere poſsit. </s>
          <s id="id002094">Rhodus, Alba regia, Melita, Caſtrum <expan abbr="nouũ">nouum</expan>, Byzan<lb></lb>tium, ſi diferri potuiſſent tempora, non ceſsiſſent uictori quantum­<lb></lb>uis ſuperbo. </s>
          <s id="id002095">Vicit pertinacia, audacia que ſumma, <expan abbr="Corcyrã">Corcyram</expan>, Viennam <lb></lb>capere <expan abbr="nõ">non</expan> potuit, quoniam in <expan abbr="longũ">longum</expan> trahebatur oppugnatio. </s>
          <s id="id002096">Mul<lb></lb>tæ machinæ, &amp; pauci homines prædæ obſeſſorum expoſitæ ſunt: <lb></lb>paucę, &amp; pauci homines obſidebuntur potius, quam obſidebunt. <lb></lb></s>
          <s id="id002097">Exercitus magnus diſſoluitur, &amp; ſemet ipſum conſumit, ſi nulla fiat <lb></lb>acceſsio aut exigua quomodo ſtabit: ſi magna auxilia omnia cor­<lb></lb>rumpuntur. </s>
          <s id="id002098">Contrà obſeſsis auxilia ſi ueniant luſtrata, &amp; munita, et <lb></lb>omnibus neceſſarijs ornata uiri integri <expan abbr="cõtra">contra</expan> fatigatos, &amp; feſſos cor <lb></lb>pore, armati contra inermes, alacres contra torpidos ſuperueniunt. <lb></lb></s>
          <s id="id002099">Ob id præcipuum eſt auxilium pręter hęc his, qui oppugnantur co<lb></lb>pia militum, qui per initia nun &lt;08&gt; quieſcant diu noctu que, <expan abbr="uerũ">uerum</expan> noctu <lb></lb>duo tubicines perſæpe <expan abbr="exercitũ">exercitum</expan> <expan abbr="inſomnẽ">inſomnem</expan> in armis tota nocte <expan abbr="cõtine">contine</expan> <lb></lb><expan abbr="bũt">bunt</expan>. </s>
          <s id="id002100">Serio <expan abbr="aũt">aut</expan> die pugnare, &amp; noctu <expan abbr="cũ">cum</expan> minimè id <expan abbr="ſperãt">ſperant</expan>, &amp; fatigati <lb></lb>ſunt: mira euenire ſolent in his inſperatis, ac audacibus eruptionib. <lb></lb></s>
          <s id="id002101">perſępe <expan abbr="etiã">etiam</expan> omnino ſupra <expan abbr="fidẽ">fidem</expan>. </s>
          <s id="id002102">Ita <expan abbr="nõ">non</expan> conquieſcere oportet donec, <lb></lb>uel omnino à capto deſinat hoſtis, aut <expan abbr="locũ">locum</expan> occupet ſibi <expan abbr="relictũ">relictum</expan> po­<lb></lb>tius &lt;08&gt; <expan abbr="quẽ">quem</expan> elegerit. </s>
          <s id="id002103">nam <expan abbr="experimentũ">experimentum</expan> frequens docuit, ubi illæ ma<lb></lb>gnę uires ſuo arbitrio <expan abbr="locũ">locum</expan>, <expan abbr="quẽ">quem</expan> <expan abbr="elegerũt">elegerunt</expan> obtinere potuerint, <expan abbr="tandẽ">tandem</expan> <lb></lb>potiri locis <expan abbr="quãtumuis">quantumuis</expan> munitis in hoc q̊d diximus <expan abbr="cõtra">contra</expan> <expan abbr="opponat̃">opponatur</expan>. <lb></lb></s>
          <s id="id002104">Etenim <expan abbr="ſeptẽ">ſeptem</expan> modis <expan abbr="cũ">cum</expan> urbes, at que arces <expan abbr="capiant̃">capiantur</expan>, <expan abbr="quorũ">quorum</expan> duo ſunt ex<lb></lb>tra <expan abbr="p̃ſentẽ">pnſentem</expan> <expan abbr="conſiderationẽ">conſiderationem</expan> obſidio, <expan abbr="q̃">quae</expan> magnitudine ambitus loci <expan abbr="tol­lit̃">tol­<lb></lb>litur</expan>, &amp; proditio, <expan abbr="q̃">quae</expan> cuſto<expan abbr="dũ">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euerſio ſuperioris muri, <lb></lb>euerſio ab imo per machinas, cuniculi, ſeu ſuffoſsio, urbis euerſio, ſeu <lb></lb><expan abbr="ędificiorũ">ędificiorum</expan>: &amp; <expan abbr="q̃uo">qua uo</expan>cant aggreſsio, ſeu oppugnatio per ſcalas, &amp; crates <lb></lb><expan abbr="cũ">cum</expan> ſagittarijs: his omnib. </s>
          <s id="id002105"><expan abbr="ſatisfactũ">ſatisfactum</expan> puto, pręter &lt;08&gt; oppugnationi pro­<lb></lb>pter <expan abbr="humilitatẽ">humilitatem</expan> <expan abbr="murorũ">murorum</expan>: <expan abbr="nã">nam</expan> lignis <expan abbr="opplent̃">opplentur</expan>, at que faſciculis, terra que foſ<lb></lb>ſę: nihil. </s>
          <s id="id002106">n. </s>
          <s id="id002107">reſiſtit immenſę illi poteſtati, &amp; crudelitati <expan abbr="ſęuiſsimorũ">ſęuiſsimorum</expan> ty<lb></lb><expan abbr="rãnorũ">rannorum</expan>. </s>
          <s id="id002108"><expan abbr="Verũ">Verum</expan>, ut dixi, terra noctu <expan abbr="effodit̃">effoditur</expan>, ligna artificioſis ignib. </s>
          <s id="id002109">eru<pb pagenum="117" xlink:href="015/01/136.jpg"></pb>untur. </s>
          <s id="id002110">Et longum eſt opus ſiue per paucos, ſiue per multos quis ef­<lb></lb>ficere conetur: ut non minus exigat temporis, quàm obſidio: nam <lb></lb>multitudine unus alterum impedit, &amp; mortui uiuos, ut omnino res <lb></lb>ſit non ſperanda niſi aduerſus inertiſsimos. </s>
          <s id="id002111">Pontes euertunt machi<lb></lb>næ, ignes que. </s>
          <s id="id002112">Sed ubi etiam muros obtinuerint ob rotunditatem in <lb></lb>illis conſiſtere non poſſunt. </s>
          <s id="id002113">Inde à defenſoribus propulſantur ſariſ­<lb></lb>ſis, telis, ignibus, tranſuerſis trabibus, machinis: illudque accedit com<lb></lb>modi, ut quanto plures eo facilius excutiantur. </s>
          <s id="id002114">Dixi non debere <lb></lb>uereri maxima etiam præter id, quoniam &amp; iſtę ipſę tanto ſanguine <lb></lb>acquiſitę tanto deorum &amp; hominum iniuria modica ſcintilla ignis <lb></lb>ſine munitionibus, exercitibus, ſiue machinis, abſque terræ <expan abbr="cõcuſsio­ne">concuſsio­<lb></lb>ne</expan>, aut inundatione, uel peſte euertuntur. </s>
          <s id="id002115">In illam miſeram lachry­<lb></lb>mam patris ſcintilla ignis inferni, cùm Deo placuerit, <expan abbr="mittit̃">mittitur</expan>, ex qua, <lb></lb>quod <expan abbr="coalitũ">coalitum</expan> eſt, multis ſeculis imperium luxu, crudelitate, ſtultitia <lb></lb>unius filij, uix uno luſtro toto diſſoluitur. </s>
          <s id="id002116">Hanc <expan abbr="ſcintillã">ſcintillam</expan> cum felici <lb></lb>etiam genio ſecum ex utero detulit Alexander Magnus. </s>
          <s id="id002117">In alijs alij <lb></lb>genium ſortiti ſunt, alij <expan abbr="ſcintillã">ſcintillam</expan> detulere ab Orco. </s>
          <s id="id002118">Ex imperio Aſſy<lb></lb>riorum per luxum Sardanapalus: ex Medorum per <expan abbr="ſcintillã">ſcintillam</expan> Aſtya­<lb></lb>ges: ex <expan abbr="Perſarũ">Perſarum</expan> per ſtultitiam Darius: ex <expan abbr="Romanorũ">Romanorum</expan> Honorius. </s>
          <s id="id002119">Di <lb></lb>ces, hęc quid ad proportionem? </s>
          <s id="id002120">Imò uelut machina ad <expan abbr="perpendiculũ">perpendiculum</expan> <lb></lb>librata pauculo illo puluere Pyrio <expan abbr="urbẽ">urbem</expan> euertit, ita ſcintilla illa infer <lb></lb>ni ignis ſemini magni tyranni indita euertit at que diſſoluit totum re­<lb></lb>gnum ſine machinis, ut dixi, uel exercitibus ullis, &amp; quod maius eſt <lb></lb>remedio nullo. </s>
          <s id="id002121">Sed puerulo indito luxus, ignauiæ, crudelitatis atque <lb></lb>ſtultitię fontibus, mirabile dictu ſanè, &amp; ad proportionem diuino­<lb></lb>rum <expan abbr="inſtrumentorũ">inſtrumentorum</expan> pertinens. </s>
          <s id="id002122">Sed redeamus ad inſtitutum: Video <lb></lb>enim, quid poſsit obijci, ſcilicet muros craſſos, et altiores tueri <expan abbr="urbẽ">urbem</expan> <lb></lb>&amp; ædificia illius poſſe abſque aggeris erectione, &amp; ſi <expan abbr="diruant̃">diruantur</expan> manere <lb></lb>etiam nihilominus imo magis, quod eſt terram, uſque <expan abbr="quoniã">quoniam</expan> eadem <lb></lb>ratione manet, quia concuti non poſsit à machinis: nec hoſtes id cu<lb></lb>raturos, ſperantes hoc <expan abbr="ſolũ">ſolum</expan> ſufficere, quod mœnia ſolo <expan abbr="æquent̃">æquentur</expan>, at que id <lb></lb><expan abbr="factũ">factum</expan> eſt Mediolani, &amp; in arce eius, <expan abbr="tũ">tum</expan> Papię &amp; in Cremonenſi arce. <lb></lb></s>
          <s id="id002123">Verùm ni fallor, ut paruis arcibus à tanta ui tormentorum nullum <lb></lb>eſt <expan abbr="præſidiũ">præſidium</expan>, aut ſalutis ſpes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag<lb></lb>geri confidant, nam &amp; pauci homines tanto labori non ſufficerent, <lb></lb>&amp; agger cum foſſa effoſſa ſcilicet terra defenſores nimis in <expan abbr="anguſtũ">anguſtum</expan> <lb></lb>cogeret. </s>
          <s id="id002124">At in urbibus contra eueniet: muris enim erectis altius ma<lb></lb>chinæ lapidum fruſtis hominem <expan abbr="occidẽt">occident</expan>: an percuſſa ſuperiore par <lb></lb>te ob coniunctionem inferior concutitur, &amp; in de <expan abbr="totũ">totum</expan> ſimul cadit, <lb></lb>ut uidimus Papię, quo <expan abbr="cadẽte">cadente</expan>, &amp; foſſa impletur, &amp; <foreign lang="grc">τEIκολέτοις</foreign> facilior <lb></lb>aditus ad ſubruendum reliquas partes <expan abbr="prębet̃">prębetur</expan>: imò perculſi defen­


<pb pagenum="118" xlink:href="015/01/137.jpg"></pb>ſores ſæpe muneris ſui obliuiſcuntur, deſertaque ea parte liberum <lb></lb>ingreſſum hoſtibus exhibent. </s>
          <s id="id002125">Tum uerò magis, quod non confi­<lb></lb>dunt animo <expan abbr="nõ">non</expan> ad id parato, poſſe aggerem ſufficientem, &amp; in tam <lb></lb>breui tempore exſtruere, &amp; etiam intelligunt, antequam erigatur, <lb></lb>patere à lateribus introitum hoſtibus.</s>
        </p>
        <p type="margin">
          <s id="id002126"><margin.target id="marg411"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002127">Propoſitio centeſimauigeſima.</s>
        </p>
        <p type="main">
          <s id="id002128">Proportionem partium nauis ad eundem obliquum uentum <lb></lb>explorare.<lb></lb><arrow.to.target n="marg412"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002129"><margin.target id="marg412"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002130">Sint mali in naui a b c, ad b e, c f uentus è regione g h k etiam ad <lb></lb>perpendiculum feratur, ut anguli g d a, h e b, k f c ſint æquales, dico <lb></lb>tamen diuerſo modo affici: nam cum premitur a uerſus l, c premi­<lb></lb>tur uerſus f: at ſi prematur c uerſus n a, premitur uerſus d, at ſi pre­<lb></lb><figure id="id.015.01.137.1.jpg" xlink:href="015/01/137/1.jpg"></figure><lb></lb>matur b uerſus m, &amp; a uer­<lb></lb>ſus l, ſed non quantum ex <lb></lb>g d, &amp; c uerſus n, ſed non <lb></lb>quantum ex k f, ab eodem <lb></lb>ergo uento contrarij mo­<lb></lb>tus efficiuntur ex uelorum <lb></lb>diuerſitate, etenim per uen <lb></lb>tum d feretur ad meridiem <lb></lb>nauis, &amp; per uelum f ad Se<lb></lb>ptentrionem etiam didu­<lb></lb>cto auxilio e l a ui, quanto <lb></lb>magis cum illo: &amp; ſi uen­<lb></lb>tus excipiatur in f uelo, <lb></lb>non iuuabit clauus, &amp; ſi in <lb></lb>d dirigetur, &amp; temperabitur motus, &amp; ſi in e medio modo. </s>
          <s id="id002131">Ergo ſi <lb></lb>uentus feratur rectè iuuabit, ut dici ſolet omnibus, &amp; plenis uelis <lb></lb>excipere, ſi ex obliquo demittere antennam puppis, ſin autem ual­<lb></lb>de obliquus ſit, ſolo proræ uelo utemur. </s>
          <s id="id002132">Si ualidior quàm oportet <lb></lb>humiliore. </s>
          <s id="id002133">Atque hæc poſtmodum ſunt diligenter numeranda, ac <lb></lb>metienda: nunc ſufficiat cauſam reddidiſſe, &amp; admonuiſſe diuerſi­<lb></lb>tatis motuum, quæ ex uelis contingit: nam eò fertur nauis, quò <lb></lb>prora dirigitur. </s>
          <s id="id002134">Ergo cum puppis tanto feratur uerſus meridiem <lb></lb>a b, quanto prora uerſus meridiem a d, &amp; quanto puppis fertur uer<lb></lb>ſus <expan abbr="meridiẽ">meridiem</expan>, tanto prora fertur uerſus boream, igitur quanto prora <lb></lb>fertur uerſus meridiem a d, tanto uerſus boream a b f, ſed ſitus claui <lb></lb>poteſt multo plus in comparatione ueli d, quam f ſcilicet, quia di­<lb></lb>ſtantia a b a eſt o a, &amp; diſtantia e c eſt o c, tanto plus ergo poteſt cla­<lb></lb>ui ſitus in comparatione ad uelum d, quam f, quanta eſt proportio 


<pb pagenum="119" xlink:href="015/01/138.jpg"></pb>o a, ad o c, igitur clauus eſt longè potentior in comparatione ueli <lb></lb>d, quam f, ergo uelum d minus agit nauim, quam f. </s>
          <s id="id002135">Sed ut extrema <lb></lb>ſe habent, ita medium eorum comparatione, igitur malus b e uali­<lb></lb>dior eſt, multo d a, &amp; infirmior c f. </s>
          <s id="id002136">Verùm, ut dixi, ob ſitum ſimpli­<lb></lb>citer ualidius eſt, uelum e quam f, &amp; etiam quia, ut dixi, altior &amp; <lb></lb>craſsior ſolet eſſe, ideo multo ualidior tribus his cauſis, quàm e f: <lb></lb>adde quartam quòd uelum habet maius, antiquo tempore uoca­<lb></lb>tum acatius. </s>
          <s id="id002137">At ut etiam docui c b non eſt in medio, nec æquidiſtat <lb></lb>ab a d &amp; c f, ſed inclinatur ad proram ideoque imbecillior: cum ergo <lb></lb>ſit æqualium, &amp; paulo maiorum uirium, quàm c f, &amp; tutior, &amp; me­<lb></lb>lius agatur per <expan abbr="clauũ">clauum</expan> quàm c f, &amp; ſit a d nimis iuſto imbecillis, pro­<lb></lb>pterea b e mali, &amp; ueli maximus eſt uſus: adeò mali nomen per an­<lb></lb>tonomaſiam de ipſo ſimpliciter intelligatur.</s>
        </p>
        <p type="main">
          <s id="id002138">Propoſitio centeſima uigeſima prima.</s>
        </p>
        <p type="main">
          <s id="id002139">Flabelli uires, at que naturam declarare.</s>
        </p>
        <p type="main">
          <s id="id002140">Sit flabellum a b c appenſum, ut ſolet, in a, &amp; moueatur motu </s>
        </p>
        <p type="main">
          <s id="id002141"><arrow.to.target n="marg413"></arrow.to.target><lb></lb>quaſi circa axem p a q in parte inferiore, &amp; aër comprehenſus ſub <lb></lb>b h k, &amp; ſpatium ſit 1 m figuræ nauicularis, quæ conſtat eſſe par­<lb></lb>tem cylindri inanis ex formatione ab Euclide ſcripta: nam ſi pro­<lb></lb>poneretur p a q ad perpendiculum ſuperſtans plano, fieret circum­<lb></lb>ducta a b c ſuperficie, quæ eſſet lata ſuperius, ſicut etiam inferius <lb></lb><arrow.to.target n="marg414"></arrow.to.target><lb></lb>cylindrus: at ſuperius a b tenuis eſt, &amp; anguſta, ergo fiet pars cy­<lb></lb>lindri inanis: quia non circumuoluitur, donec redeat. </s>
          <s id="id002142">Ergo per di­<lb></lb>cta ſuperius ſectio illius p r q s per axem eſt pars cuiuſdam elly­<lb></lb><arrow.to.target n="marg415"></arrow.to.target><lb></lb>pſis. </s>
          <s id="id002143">Et ſectio quæuis planæ ſuperficiei æquidiſtans a b c uelut tu, <lb></lb>item que æquidiſtans axi p a q eſt ſuperficies rectangula, quarum <lb></lb>una eſt ſimilis, &amp; æqualis b h k, eſt in una ſuperficie cum axe p a q <lb></lb>alia uerò eſt æquidiſtans eidem axi maior aut minor æquidiſtanti­<lb></lb>um, &amp; ipſa laterum, at que rectangula ac ſi cylindrus ſtans axi plano <lb></lb>æquidiſtanti ſecaretur iuxta longitudinem ſeu altitudinem ſuam: <lb></lb>&amp; manifeſtum eſt, quod iſta duo plana, &amp; eorum ſuperficies ſecant <lb></lb>ſe mutuò ad rectos angulos.</s>
        </p>
        <p type="margin">
          <s id="id002144"><margin.target id="marg413"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="margin">
          <s id="id002145"><margin.target id="marg414"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 11. <lb></lb><emph type="italics"></emph>diff.<emph.end type="italics"></emph.end> 21.</s>
        </p>
        <p type="margin">
          <s id="id002146"><margin.target id="marg415"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 69.</s>
        </p>
        <p type="main">
          <s id="id002147">Quibus conſtitutis, qui ſtabunt iuxta l, &amp; m longitudines aëris <lb></lb>moti, &amp; loci, per quem tranſit flabellum, ſentient magnum uentum, <lb></lb>quoniam cum corpus m x l ab extremis partibus ſit elatius a b ex­<lb></lb>tremis, ſtantes, &amp; alti tangentur à uento agitato. </s>
          <s id="id002148">Si uero ſedeant, aer <lb></lb>primum non attinget illos, ut etiam quia ſurſum pellitur non per­<lb></lb>ueniet ad illos, imò diffugiet, ergo non refrigerabuntur. </s>
          <s id="id002149">Qui uerò <lb></lb>à lateribus l x m <expan abbr="ſtabũt">ſtabunt</expan> hiccinde, uelut in f g, ſi ſteterint, <expan abbr="nõ">non</expan> refrigera<lb></lb><expan abbr="bũtur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer deſcendet, ergo fugi <lb></lb>et ab illis, cum autem fuerit in x, erit in loco humiliori, &amp; mouebi­


<pb pagenum="120" xlink:href="015/01/139.jpg"></pb>tur diuerſa ratione, quippe ab f in h, &amp; non ad latera, ergo ne que <lb></lb><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"></figure><lb></lb>contactu, neque motu, qui <lb></lb>fiet per æquidiſtantem f, <lb></lb>&amp; g non poterunt refrige­<lb></lb>rari. </s>
          <s id="id002150">Sed ſi humili loco ſe­<lb></lb>deant, quoniam aër deſcen <lb></lb>dit, ex l &amp; m uerſus x, &amp; <lb></lb>etiam, quia erunt proximi <lb></lb>h k, <expan abbr="quãdo">quando</expan> fuerit in x, <expan abbr="refri­gerabunt̃">refri­<lb></lb>gerabuntur</expan> ualde. </s>
          <s id="id002151">Qui <expan abbr="autẽ">autem</expan> <lb></lb><expan abbr="erũt">erunt</expan> iuxta h &amp; k minus <expan abbr="re­frigerabunt̃">re­<lb></lb>frigerabuntur</expan> utriſque, ſed pau<lb></lb>lulum in reditibus propin <lb></lb>quis, &amp; neque ſtantes, <expan abbr="nequeſedẽtes">neque <lb></lb>ſedentes</expan>, ſed ſi altius attolla­<lb></lb>tur h k. </s>
          <s id="id002152">Rurſus ſi b h k fue­<lb></lb>rit grauior eodem, ut de­<lb></lb>ſcendat tanto impetu, <expan abbr="quã­to">quan­<lb></lb>to</expan> aſcendit attractum, ut <lb></lb>pote ex ligno tenui nucis, <lb></lb>tunc multo magis refrige­<lb></lb>rabit, &amp; procul, <expan abbr="nõ">non</expan> ob uim <lb></lb>ualidiorem, ſed quoniam <lb></lb>celerius occurſantes ſibi <lb></lb>contrarijs motibus, ac <expan abbr="ue­hemẽtibus">ue­<lb></lb>hementibus</expan> fiet colliſio par<lb></lb>tium aëris, &amp; ideo in ambitum impelletur, &amp; undique cubiculum <lb></lb>refrigerabit, quod non faciet maius longè flabellum lento motu <lb></lb>agitatum, aut ex materia leui. </s>
          <s id="id002153">Idem multo magis contingeret, ubi <lb></lb>duo eſſent flabella laquearibus appenſa, quæ ad perpendiculum <lb></lb><expan abbr="aẽrem">aerrem</expan> mouerent, ſeu quod ſuperficies eo modo ſe haberent: &amp; ſi <lb></lb>flabella rotunda eſſent, tunc maiorem ambitum aëris occuparent, <lb></lb>&amp; uelocius deficientibus angulis mouebuntur.</s>
        </p>
        <p type="main">
          <s id="id002154">Propoſitio centeſima uigeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id002155">Contemptus circa ſolis rationem in umbris declarare.</s>
        </p>
        <p type="main">
          <s id="id002156">Conſtat primùm ſolem, &amp; ex centro, &amp; toto eius ambitu illumi­<lb></lb>nare hanc primùm diuerſitatem, quæ aliquando tota diametro <lb></lb>computata dimidium unius partis totius cœli excedit: ſcioterici <lb></lb>negligunt, ut exiguam. </s>
          <s id="id002157">Secundò etiam diuerſitatis illius, qua mo­<lb></lb>dò à terra uerſus abſidem defertur, modò ad terram deſcendere to­<lb></lb>tidem uariata altitudine, non parum nullam habent rationem, ſeu 


<pb pagenum="121" xlink:href="015/01/140.jpg"></pb>quòd tanta ne ſit, ut euidentem in gnomonibus faciat uarietatem, <lb></lb>ſeu quòd incertum adhuc ſit, an id uerè ſoli accidat. </s>
          <s id="id002158">Tertium eſt fi­<lb></lb>nis umbræ ipſius gnomonis, qui incertus eſt, ut pars non contem­<lb></lb>nenda in dubium uertatur, quoniam ſenſim ex obſcuro in illumi­<lb></lb>natum feratur, at tamen contemnitur etiam. </s>
          <s id="id002159">Quartum quòd cum <lb></lb>ſol moueatur in ſpira, fingitur quaſi in parallelo æquinoctiali circu<lb></lb>lo circumagatur ab his, qui horologia deſcribunt. </s>
          <s id="id002160">Quintum quòd <lb></lb>cum inæqualiter in orbe ſuo moueatur quanuis exigua ſit hæc dif­<lb></lb>ferentia, æqualiter <expan abbr="tamẽ">tamen</expan> moueri præſupponitur. </s>
          <s id="id002161">Sextum eſt, quòd <lb></lb>dies æquales ſupponuntur, qui tamen tum ex ratione partis pera­<lb></lb>gratæ, tum ratione aſcenſus <expan abbr="eiuſdẽ">eiuſdem</expan> ſunt inæquales, &amp; <expan abbr="tamẽ">tamen</expan> hæc in­<lb></lb>qualitas <expan abbr="etiã">etiam</expan> in <expan abbr="horarũ">horarum</expan> computatione prætermittitur. </s>
          <s id="id002162">Sed &amp; hęc ut <lb></lb>prior ratione magis, <expan abbr="quã">quam</expan> ſenſu <expan abbr="deprehendit̃">deprehenditur</expan>. </s>
          <s id="id002163"><expan abbr="Septimũ">Septimum</expan> eſt diſcrimen, <lb></lb>q̊d oritur ex uiſus circulo ſeu horizonte, &amp; circulo tranſeunte p cen<lb></lb><expan abbr="trũ">trum</expan> mundi, nam horizon uere <expan abbr="tãto">tanto</expan> minor eſt circulo magno, quan­<lb></lb>tum eſt ſemidiameter terrę, <expan abbr="cõparatus">comparatus</expan> ad <expan abbr="ſemidiametrũ">ſemidiametrum</expan> orbis cœle<lb></lb>ſtis, ſed eſt inſenſilis quantitatis. </s>
          <s id="id002164"><expan abbr="Octauũ">Octauum</expan> eſt, quod trianguli ex gno­<lb></lb>mone umbra, &amp; radijs ſolis latera non mutant lineas, quæ à ſole ad <lb></lb>centrum terræ deueniunt, nec quòd maius eſt, radius ſolis ad uerti­<lb></lb>cem hominis breuior habetur femidimetiente. </s>
          <s id="id002165">Hæc <expan abbr="igit̃">igitur</expan> omnia <expan abbr="ſci­otericorũ">ſci­<lb></lb>otericorum</expan> opifices non obſeruant, ſed negligunt. </s>
          <s id="id002166">Verum quatuor <lb></lb>tantùm altitudinem poli regionis locum ſolis in eclyptica locum <lb></lb>ſolis in circulo æquinoctialis, uel æquinoctiali parallelo, ex qui­<lb></lb>bus tribus fit altitudo ſolis, una in circulo ſcilicet uerticali ab hori­<lb></lb>zonte, &amp; differentia lineæ meridianæ à linea uerſus polum, quam <lb></lb><arrow.to.target n="marg416"></arrow.to.target><lb></lb>oſtendit lapis Herculeus, de qua dictum eſt ſuperius.</s>
        </p>
        <p type="margin">
          <s id="id002167"><margin.target id="marg416"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 84.</s>
        </p>
        <p type="main">
          <s id="id002168">Propoſitio centeſima uigeſima tertia.</s>
        </p>
        <p type="main">
          <s id="id002169">Cognita ratione umbrę ad gno<lb></lb>monem ſinum, &amp; arcum altitudi­<lb></lb>nis ab horizonte quouis tempo­<lb></lb>re dignoſcere.</s>
        </p>
        <figure id="id.015.01.140.1.jpg" xlink:href="015/01/140/1.jpg"></figure>
        <p type="main">
          <s id="id002170">Sit circulus magnus, in quo ſol <lb></lb><arrow.to.target n="marg417"></arrow.to.target><lb></lb>a f g ſuperſtans ad perpendicu­<lb></lb>lum circulo uiſus f e g, quos mani <lb></lb>feſtum eſt tranſire per idem cen­<lb></lb>trum mundi c, quia magni ſunt, &amp; <lb></lb>ſit c d erecta ad perpendiculum <lb></lb>ſuper f g, nam perinde eſt per ſe­<lb></lb>ptimum contemptum, ac ſi ſuper­<lb></lb><arrow.to.target n="marg418"></arrow.to.target><lb></lb>ficies horizontis tranſeat per terrę centrum, &amp; pedes per octauum, <lb></lb><arrow.to.target n="marg419"></arrow.to.target><lb></lb>ideo proportio e c ad c d umbræ ad gnomonem, ut b e ad b a, ergo 


<pb pagenum="123 [=122]" xlink:href="015/01/141.jpg"></pb>per demonſtrata b a cognita in comparatione a d e a, e a autem per <lb></lb>octauum contemptum eſt dimetiens circuli, ergo a b ſinus notus, <lb></lb>&amp; arcus f a, quod eſt primum cognitum. </s>
          <s id="id002171">Et hic quidem circulus <lb></lb>uerticalis dicitur, quia per illum tranſit, aliter non eſſet ad perpen­<lb></lb>diculum horizonti.<lb></lb><arrow.to.target n="marg420"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002172"><margin.target id="marg417"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id002173"><margin.target id="marg418"></margin.target>P<emph type="italics"></emph>ræced.<emph.end type="italics"></emph.end> P<emph type="italics"></emph>ro <lb></lb>poſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002174"><margin.target id="marg419"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 113.</s>
        </p>
        <p type="margin">
          <s id="id002175"><margin.target id="marg420"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002176">Ex hoc ſequitur, quod altitudines ſolis æquales omnes in uno <lb></lb>ſunt circulo horizonti parallelo. </s>
          <s id="id002177">Et ſi ſol fuerit in uno circulo ho­<lb></lb>rizonti parallelo, altitudines ſolis, &amp; umbræ magnitudines æqua­<lb></lb>les erunt.</s>
        </p>
        <p type="main">
          <s id="id002178"><arrow.to.target n="marg421"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002179"><margin.target id="marg421"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002180">Sol niſi bis in una die poteſt eſſe in circulo horizonti parallelo, <lb></lb>ſemel ante meridiem, &amp; ſemel poſt, tantundem ab eodem diſtans.</s>
        </p>
        <p type="main">
          <s id="id002181"><arrow.to.target n="marg422"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002182"><margin.target id="marg422"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id002183">Cum ergo ita ſit, neceſſe eſt umbras æquales, &amp; circulum hori­<lb></lb>zonti <expan abbr="parallelũ">parallelum</expan> fieri ſub in æqualibus horis in diuerſis ſemper die­<lb></lb>bus, præterquam cum in punctis fuerit æqualis ab ęquinoctiali, &amp; <lb></lb>in eandem partem declinationis, &amp; hoc bis <expan abbr="cõtingit">contingit</expan> ſolum in anno <lb></lb>pro quolibet circulo parallelo, ſicut in eodem die etiam bis <expan abbr="tãtum">tantum</expan>, <lb></lb>ut dictum eſt.</s>
        </p>
        <p type="main">
          <s id="id002184"><arrow.to.target n="marg423"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002185"><margin.target id="marg423"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002186">Nam exempli gratia, cum ſol eſt in initio Capricorni, &amp; in Cœli <lb></lb>medio, minima eſt umbra eius diei, &amp; totius anni. </s>
          <s id="id002187">Cum ergo fuerit <lb></lb>ante meridiem, uel poſt, erit umbra maior ex ſuppoſito ſecudo um­<lb></lb>bra meridiei: at ei æqualis poterit eſſe umbra meridiei alterius diei <lb></lb>ex primo ſuppoſito, ergo umbræ æquales diuerſorum dierum fi­<lb></lb>unt ſub diuerſo ſitu ſolis, quo ad circulum meridiei, quod erat de­<lb></lb>monſtrandum.</s>
        </p>
        <p type="main">
          <s id="id002188"><arrow.to.target n="marg424"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002189"><margin.target id="marg424"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id002190">Ex hoc ſequitur, quod horarum determinatio fit ſecundum line­<lb></lb>am in æqualem obliquam, quæ toti anno ſeruiat, ut æqualium um­<lb></lb>brarum determinatio hararum &amp; partium eius numerum.</s>
        </p>
        <p type="main">
          <s id="id002191"><arrow.to.target n="marg425"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002192"><margin.target id="marg425"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 5.</s>
        </p>
        <p type="main">
          <s id="id002193">Ex quo colligitur modus faciendi gnomonem, ſeu per umbras <lb></lb>rectas, ſeu per uerſas, qui docebit toto anno non <expan abbr="ſolũ">ſolum</expan> horas, ſed mo <lb></lb>menta <expan abbr="pulſuũ">pulſuum</expan>, de quibus <expan abbr="dictũ">dictum</expan> eſt quod MMMDC horam <expan abbr="perficiũt">perficiunt</expan>.</s>
        </p>
        <p type="main">
          <s id="id002194">Propoſitio centeſima uigeſima quarta.</s>
        </p>
        <p type="main">
          <s id="id002195">Proportionem umbræ uerſæ eſſe ad gnomonem, uelut gnomo­<lb></lb>nis ad umbram uerſam.</s>
        </p>
        <p type="main">
          <s id="id002196"><arrow.to.target n="marg426"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002197"><margin.target id="marg426"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002198">Vmbra uerſa dicitur, quoties gnomo in pariete ad perpendicu­<lb></lb>lum figitur, ſic ut gnomo æquidiſtet circulo horizontis. </s>
          <s id="id002199">Sit ergo <lb></lb>paries c k ad perpendiculum f g, &amp; h k a d gnomo ad perpendicu­<lb></lb>lum parietis &amp; ſol, ut prius in a, &amp; ſit primo k h tantæ longitudinis </s>
        </p>
        <p type="main">
          <s id="id002200"><arrow.to.target n="marg427"></arrow.to.target><lb></lb>ut umbræ locus ſit <expan abbr="pũctus">punctus</expan> d, ut ſit radius a h d e, eritque angulus d u­<lb></lb>trin que æqualis, &amp; propterea triangulus k h d ſimilis d c e. </s>
          <s id="id002201">Sit modo <lb></lb><arrow.to.target n="marg428"></arrow.to.target><lb></lb>gnomo maior m l ipſo h k &amp; c l maior c k ſeu æqualis, &amp; quam an­<lb></lb>guli k &amp; l recti ſunt, &amp; anguli l m n, &amp; k h d æqualis, quia a n, &amp; a c 


<pb pagenum="113 [=123]" xlink:href="015/01/142.jpg"></pb>ſunt æquidiſtantes per octauum contemptum, erunt per dicta tri­<lb></lb>anguli ſimiles, igitur proportio l m gnomonis ad l n umbram <lb></lb>ut k h gnomonis ad k d umbram, ſed k h, ad k d, ut c e umbræ ad c d <lb></lb>gnomonem: igitur proportio l m gnomonis ad l n <expan abbr="umbrã">umbram</expan>, ut um­<lb></lb>bræ c e ad c d gnomonem, quod fuit demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id002202"><margin.target id="marg427"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002203"><margin.target id="marg428"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002204">Ex hoc primùm patet &amp; pręcedenti, quod cognita proportione <lb></lb><arrow.to.target n="marg429"></arrow.to.target><lb></lb>umbrę uerſę ad gnomonem cognoſcitur ſinus ſolis, &amp; arcus altitu­<lb></lb>dinis in circulo magno, &amp; eſt altitudo ab horizontis parte, quæ <lb></lb>proximior eſt loco ſolis, ut demonſtratum à nobis in Geometricis.</s>
        </p>
        <p type="margin">
          <s id="id002205"><margin.target id="marg429"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002206">Sequitur etiam, quòd cùm umbra fuerit æqualis gnomoni, ſeu <lb></lb><arrow.to.target n="marg430"></arrow.to.target><lb></lb>recta, ſeu uerſa ſolis, uel Lunæ, uel ſtellæ, altitudo erit partium qua­<lb></lb>draginta quin que: nam anguli d &amp; e, uel d &amp; h erunt æquales: igitur <lb></lb>arcus f a medietas quartæ ideò partium xlv. </s>
          <s id="id002207">Et ſi gnomo fuerit ma­<lb></lb>ior umbra uerſa, uel minor recta, erit arcus f a minor xlv partibus, ſi <lb></lb>contrà maior. </s>
          <s id="id002208">Et hoc ubique terrarum. </s>
          <s id="id002209">Et ubi non poſsit tantundem <lb></lb>eleuari, ut quando ſol eſt ſub circulo capricorni, nunquam nobis <lb></lb><arrow.to.target n="marg431"></arrow.to.target><lb></lb>gnomo æquabitur umbræ rectæ ſed ſemper erit minor, &amp; ſemper <lb></lb><arrow.to.target n="marg432"></arrow.to.target><lb></lb>maior umbra uerſa pari ratione.</s>
        </p>
        <p type="margin">
          <s id="id002210"><margin.target id="marg430"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="margin">
          <s id="id002211"><margin.target id="marg431"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>primi<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002212"><margin.target id="marg432"></margin.target>P<emph type="italics"></emph>er ult. </s>
          <s id="id002213">ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002214">Propoſitio centeſima uigeſima quinta.</s>
        </p>
        <p type="main">
          <s id="id002215">Proportionem dimetientis, &amp; peripherię cuiuslibet circuli paral <lb></lb>leli æquinoctiali per cognitam partem magni circuli demonſtrare.</s>
        </p>
        <p type="main">
          <s id="id002216">Hæc erat tam clara, ut hic locum non mereretur: tam neceſſaria <lb></lb><arrow.to.target n="marg433"></arrow.to.target><lb></lb>huic propoſito, ut non potuerit omitti. </s>
          <s id="id002217">Sit ergo Aequinoctij circu­<lb></lb>lus a b portio circuli magni nota, a c parallelus circulus, ęquinoctij <lb></lb>circulo c d, erit igitur ſinus c d notus. </s>
          <s id="id002218">Et ideò <expan abbr="quadratũ">quadratum</expan> c d notum, <lb></lb><arrow.to.target n="marg434"></arrow.to.target><lb></lb>ergo &amp; pars utraque b d d a nota. </s>
          <s id="id002219">Quare detracta a d ex d b relinqui­<lb></lb>tur d g æqualis f c diametro paralleli aſsignari. </s>
          <s id="id002220">Quare proportio <lb></lb><arrow.to.target n="marg435"></arrow.to.target><lb></lb>a b ad e f nota ex obiter ſuprà demonſtratis, &amp; pariter ambi­<lb></lb>tus circuli a b ad ambitum circuli c d, eſt enim ut dimetientis ad di­<lb></lb><arrow.to.target n="marg436"></arrow.to.target><lb></lb>metientem.</s>
        </p>
        <p type="margin">
          <s id="id002221"><margin.target id="marg433"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002222"><margin.target id="marg434"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>tertij,<emph.end type="italics"></emph.end><lb></lb>&amp; 8. &amp; 17. <lb></lb><emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002223"><margin.target id="marg435"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>ſecun­<lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002224"><margin.target id="marg436"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 113. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002225">Propoſitio centeſima uigeſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id002226">Circuli horarij naturam declarare.<lb></lb><arrow.to.target n="marg437"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002227"><margin.target id="marg437"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <figure id="id.015.01.142.1.jpg" xlink:href="015/01/142/1.jpg"></figure>
        <p type="main">
          <s id="id002228">Circulus horarius eſt circulus magnus <lb></lb>tranſiens per <expan abbr="ſolẽ">ſolem</expan>, aut lunam, aut quoduis <lb></lb>ſydus, de quo agitur, &amp; per polos mundi, <lb></lb>ideò differt à circulo priore altitudinis So­<lb></lb>lis, quia ille ſtat ad perpendiculum ſuper <lb></lb>horizontem, niſi cum tangitur uice meridi­<lb></lb>ani, uterque tamen tranſit per <expan abbr="centrũ">centrum</expan> mundi, <lb></lb>ac ſolis. </s>
          <s id="id002229">Hic etiam ad ſimiles partes æqui­<lb></lb>noctij circulum, &amp; omnes parallelos ſecat. 


<pb pagenum="124" xlink:href="015/01/143.jpg"></pb>Et principalis eſt meridianus, ideò ab illo Aſtrologi horas utrinque<lb></lb> ante, &amp; poſt numerant. </s>
          <s id="id002230">Ideò <expan abbr="clarũ">clarum</expan> eſt, quòd horæ à meridie com­<lb></lb>putatæ ſunt <expan abbr="cõmunes">communes</expan>, habitantibus ſub quauis altitudine poli, &amp; <lb></lb>ubiuis ſit, ſol modò regiones æqualiter diſtent à fortunatis, ſeu ſint <lb></lb>in eadem longitudine.</s>
        </p>
        <p type="main">
          <s id="id002231">Propoſitio centeſima uigeſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id002232">Data Poli altitudine ortus amplitudinem demonſtrare.<lb></lb><arrow.to.target n="marg438"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002233"><margin.target id="marg438"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002234">Sit horizon a d b æquinoctij circulus <lb></lb><figure id="id.015.01.143.1.jpg" xlink:href="015/01/143/1.jpg"></figure><lb></lb>a k f eclyptica c g, &amp; punctus ortus in ea g. <lb></lb></s>
          <s id="id002235">&amp; c initium arietis, &amp; g b amplitudo ortiua <lb></lb>&amp; c e, c f quartæ circulorum, ut ſit e f maxi­<lb></lb>ma ſolis declinatio, &amp; polus mundi borea­<lb></lb>lis l, quia igitur l d nota eſt ex ſuppoſito, &amp; <lb></lb>l k quadrans erit k h <expan abbr="reſiduũ">reſiduum</expan> ad dimidium <lb></lb>circuli notum. </s>
          <s id="id002236">Quia uerò æquinoctium, &amp; <lb></lb>Meridianus ſecant ſe ad angulos rectos, &amp; <lb></lb>b a æquidiſtat ab utro que polo, erit b polus <lb></lb>h d, quare b k, quarta circuli, &amp; angulus k <lb></lb>rectus. </s>
          <s id="id002237">Igitur ſumus in diſpoſitione tabula­<lb></lb>rum primi mobilis, ergo etiam oppoſitus <lb></lb>triangulus, qui ei eſt æqualis, &amp; ęquiangu­<lb></lb>lus in eadem diſpoſitione b m d, quare cum <lb></lb>data ſit g n declinatio <expan abbr="pũcti">puncti</expan> g dati, datus erit, &amp; arcus g b quæſitus.</s>
        </p>
        <p type="main">
          <s id="id002238">Propoſitio centeſima uigeſima octaua.</s>
        </p>
        <p type="main">
          <s id="id002239">Nota amplitudine ortus cuiuſque <expan abbr="pũcti">puncti</expan> <expan abbr="arcũ">arcum</expan> <expan abbr="ſemidiurnũ">ſemidiurnum</expan> inuenire.</s>
        </p>
        <p type="main">
          <s id="id002240"><arrow.to.target n="marg439"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002241"><margin.target id="marg439"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002242">Sit in eadem figura nota g b, uolo illius <expan abbr="arcũ">arcum</expan> ſemidiurnum. </s>
          <s id="id002243">Cum <lb></lb>ergo g n ſit declinatio, erit pars arcus Meridiani horarij per polos <lb></lb>tranſeuntis, compleatur ergo l g n o, &amp; quia g n nota eſt, quia de­<lb></lb>clinatio puncti dati, &amp; g b nota ex ſuppoſito, &amp; f angulus rectus, <lb></lb>quia e f eſt portio meridiani, erit b n nota differentia aſcenſionis a <lb></lb>quarta circuli k b, <expan abbr="igit̃">igitur</expan> tota k n arcus ſemidiurnus. </s>
          <s id="id002244"><expan abbr="Quoniã">Quoniam</expan> g p paral <lb></lb>lelus ſimilis eſt k n, &amp; in eo <expan abbr="reuoluit̃">reuoluitur</expan> Sol: ergo quando enim perue­<lb></lb>niet ad p. </s>
          <s id="id002245">Poſſumus etiam ſine inuentione arcus ortus amplitudi­<lb></lb>nis per triangulum k m d ex notitia g n cognoſcere eandem n b.</s>
        </p>
        <p type="main">
          <s id="id002246"><arrow.to.target n="marg440"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002247"><margin.target id="marg440"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002248">Ex his duabus ſequitur <expan abbr="cõuerſa">conuerſa</expan> ſcilicet, quae data magnitudine diei <lb></lb><expan abbr="cuiuſcũque">cuiuſcunque</expan> in quauis regione nota erit poli altitudo <expan abbr="eiuſdẽ">eiuſdem</expan> regionis.</s>
        </p>
        <p type="main">
          <s id="id002249">Propoſitio centeſima uigeſima nona.</s>
        </p>
        <p type="main">
          <s id="id002250">Data altitudine ſolis in quacunque regione quacunque die diſtan­<lb></lb>tiam ſolis à Meridiano cognoſcere.</s>
        </p>
        <p type="main">
          <s id="id002251"><arrow.to.target n="marg441"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002252"><margin.target id="marg441"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002253">Sit Horizon a b c d æquinoctij circulus b e d. </s>
          <s id="id002254">Meridianus a e c <lb></lb>Polus mundi Borealis f uertex, g, <expan abbr="pũctus">punctus</expan> in eclyptica h ducatur ex 


<pb pagenum="125" xlink:href="015/01/144.jpg"></pb>polo mundi circulus horarius f h k ad æquinoctij circulum, &amp; uer­<lb></lb>ticalis circulus p h l uſque ad Horizontem, &amp; circulus parallelus æ­<lb></lb>quinoctij circulo h m, ſit ergo h l altitudo ſolis nota, igitur h g nota </s>
        </p>
        <p type="main">
          <s id="id002255"><arrow.to.target n="marg442"></arrow.to.target><lb></lb>erit reſiduum quartę circuli, &amp; ſimiliter h k <lb></lb><figure id="id.015.01.144.1.jpg" xlink:href="015/01/144/1.jpg"></figure><lb></lb>nota, quia declinatio puncti dati in eclypti<lb></lb>ca eſt n nota dies, &amp; locus ſolis ex ſuppoſi­<lb></lb>to ergo nota fh <expan abbr="reſiduũ">reſiduum</expan> quartę circuli no­<lb></lb>ta eſt <expan abbr="etiã">etiam</expan> g e, quæ eſt ęqualis altitudini po­<lb></lb>li ex ſuppoſito, ergo reſiduum quadrantis <lb></lb>f g, ergo triangulus f g h notorum laterum <lb></lb>ergo notus angulus f, ergo arcus k e diſtan <lb></lb><arrow.to.target n="marg443"></arrow.to.target><lb></lb>tia ſumpta in æquinoctij circulo puncti h, <lb></lb>cui ſimilis eſt arcus h m ex parallelo h m, nam quando k perueniet <lb></lb><arrow.to.target n="marg444"></arrow.to.target><lb></lb>in e h perueniet in m, &amp; in æquali tempore, qua diuiſa per quinde­<lb></lb>cim gradus, habebimus horas <expan abbr="diſtãtię">diſtantię</expan> ſolis à Meridie ante, uel poſt, <lb></lb>&amp; minuta horarum dando quibuslibet gradibus quatuor minuta <lb></lb>horæ, &amp; quibuslibet minutis graduum quatuor ſecunda horæ, &amp; <lb></lb>ita habebimus tempus exactiſsimum à Meridie in quacunque regi­<lb></lb>one, &amp; in quacunque hora diei.</s>
        </p>
        <p type="margin">
          <s id="id002256"><margin.target id="marg442"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 123. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002257"><margin.target id="marg443"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 34. <lb></lb><emph type="italics"></emph>lib.<emph.end type="italics"></emph.end> 4.</s>
        </p>
        <p type="margin">
          <s id="id002258"><margin.target id="marg444"></margin.target>D<emph type="italics"></emph>e<emph.end type="italics"></emph.end> T<emph type="italics"></emph>riang.<emph.end type="italics"></emph.end><lb></lb>M<emph type="italics"></emph>onteregij.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002259">Propoſitio centeſima trigeſima.</s>
        </p>
        <p type="main">
          <s id="id002260">Data regionis altitudine, &amp; loco ſolis proportionem gnomo­<lb></lb>nis tam ad umbram rectam, quam uerſam, uel etiam in cylindro de­<lb></lb>terminare.</s>
        </p>
        <p type="main">
          <s id="id002261">Hęc eſt propoſitio illa pulcherrima, quam tot ambagibus tradi­<lb></lb><arrow.to.target n="marg445"></arrow.to.target><lb></lb>dere antiqui cum ſuis analematibus, &amp; ſcioteris, nec tamen demon <lb></lb>ſtrationem, nec rationem exactam inſtrumenorum conſtructio­<lb></lb>nem, qua poſſemus per umbras rectas uerſas, &amp; cylindricas ſcire ad <lb></lb>unguem, qualis hora, &amp; minutum, &amp; ſecundum diei eſſet quocun­<lb></lb>que anni tempore. </s>
          <s id="id002262">Plerique autem tam laborioſè id conati ſunt de­<lb></lb>monſtrare, ut ſtudioſos deterruerint ab opere: res autem ipſa facil­<lb></lb>lima eſt. </s>
          <s id="id002263">Propoſita ergo Poli exacta altitudine ſolis in Meridie <lb></lb>declinatione addita uel detracta, habebis reſiduum eius ad qua­<lb></lb>drantem f g, &amp; ſimiliter habebis ex declinatione nota loci ſolis de­<lb></lb>tracta à quadrante f h &amp; iuxta horam tuam, &amp; minutum multi­<lb></lb><arrow.to.target n="marg446"></arrow.to.target><lb></lb>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb></lb>g h, quare reſiduum h l, igitur punctum umbrę rectę, uel uerſę ipſi­<lb></lb>us gnomonis ad unguem, &amp; ita conſtitues horologium exactiſsi­<lb></lb>mum ſecundum ea, quæ dixi in Corrolarijs ſupradictis, &amp; quia ho­<lb></lb><arrow.to.target n="marg447"></arrow.to.target><lb></lb>rizon a b c d ſecat æquinoctialem in <expan abbr="cẽtro">centro</expan> terræ ducta g h k, erunt <lb></lb>anguli b h g, &amp; k h l ęquales. </s>
          <s id="id002264">Igitur poſito g ortu puncti eclypti­<lb></lb>cæ, erit g b ortus amplitudo nota, &amp; ideò angulus b h g, &amp; k h l 


<pb pagenum="126" xlink:href="015/01/145.jpg"></pb><arrow.to.target n="marg448"></arrow.to.target><lb></lb>notus, &amp; ita extendemus per totum annum. </s>
          <s id="id002265">Cum uerò fuerit g ele­<lb></lb>uatus erit, ut <expan abbr="demõſtratum">demonſtratum</expan> eſt, in circulo magno uerticali, ergo an­<lb></lb>gulus fiet in eodem circulo, quia gnomo eſt etiam in illius ſuperfi­<lb></lb>cie. </s>
          <s id="id002266">Ergo angulus erit æqualis angulo, quem faceret ſol, ſi oriretur <lb></lb><arrow.to.target n="marg449"></arrow.to.target><lb></lb><figure id="id.015.01.145.1.jpg" xlink:href="015/01/145/1.jpg"></figure><lb></lb>in puncto horizontis, quem ſecat circulus <lb></lb>uerticalis ſub ea altitudine: ſed his eſt no­<lb></lb>tus: nam in priore figura g h f eſt notus ea­<lb></lb><arrow.to.target n="marg450"></arrow.to.target><lb></lb><expan abbr="dẽ">dem</expan> ratione, qua f, &amp; ideò ei oppoſitus k h n, <lb></lb>&amp; k rectus, eſt enim f polus b d, &amp; h k decli<lb></lb>natio nota ergo k n, &amp; h n notæ. </s>
          <s id="id002267">At e k, &amp; <lb></lb>g h fuere notæ. </s>
          <s id="id002268">Ergo e n, &amp; g n, quare reſi­<lb></lb>duæ n l &amp; n b notæ. </s>
          <s id="id002269">Eſt autem angulus l <lb></lb>rectus. </s>
          <s id="id002270">ergo ortus amplitudo puncti l nota <lb></lb>ſcilicet arcus l b, ergo in præſenti figura angulus m h b, ergo k h l. <lb></lb></s>
          <s id="id002271">igitur poterimus ſtatuere angulos umbrarum, &amp; iam poſſumus <lb></lb>determinare magnitudinem: ergo punctum ad <expan abbr="unguẽ">unguem</expan> umbrę qua­<lb></lb>libet hora, &amp; parte horæ ſingulis diebus in quacunque regione datæ <lb></lb>altitudinis poli uerſa, &amp; rects. </s>
          <s id="id002272">In cylindrica autem eodem modo ſi­<lb></lb>cut in uerſa, eſt enim ſpecies umbrę uerſę, niſi quod analema ob ob­<lb></lb>liquitatem cylindri melius aptatur, rotundum ſcilicet cum <expan abbr="rotũdo">rotundo</expan>.</s>
        </p>
        <p type="margin">
          <s id="id002273"><margin.target id="marg445"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002274"><margin.target id="marg446"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 28. <emph type="italics"></emph>li.<emph.end type="italics"></emph.end> 4. <lb></lb><emph type="italics"></emph>loan. </s>
          <s id="id002275">de<emph.end type="italics"></emph.end> M<emph type="italics"></emph>on <lb></lb>teregij de<emph.end type="italics"></emph.end><lb></lb>T<emph type="italics"></emph>riang.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002276"><margin.target id="marg447"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 123. <lb></lb><emph type="italics"></emph>uel<emph.end type="italics"></emph.end> 124. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002277"><margin.target id="marg448"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 123. <lb></lb>C<emph type="italics"></emph>orol.<emph.end type="italics"></emph.end> 1.</s>
        </p>
        <p type="margin">
          <s id="id002278"><margin.target id="marg449"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 127. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002279"><margin.target id="marg450"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002280">Propoſitio centeſima trigeſima prima.</s>
        </p>
        <p type="main">
          <s id="id002281">Si lineæ alicui dupla alterius <expan abbr="adiungat̃">adiungatur</expan>, erit proportio duarum ad <lb></lb><expan abbr="primã">primam</expan> maior, quam dupli, cum prima ad primam cum una adiecta.</s>
        </p>
        <p type="main">
          <s id="id002282">Sit a b linea, cui adiecta ſit b c, &amp; rurſus ad b c c d <expan abbr="æq́ualis">æqualis</expan> b c <lb></lb>dico, quod proportio a c ad a b eſt maior, quàm a d ad a c. </s>
          <s id="id002283">Propor<lb></lb><arrow.to.target n="marg451"></arrow.to.target><lb></lb>tio enim c d ad c a minor eſt, quàm ad a b per octauam quinti E­<lb></lb>lementorum. </s>
          <s id="id002284">Ergo minor d c ad c a quàm c b ad a b, quia b c &amp; c d <lb></lb>ſunt æquales, ideò <expan abbr="æqualẽ">æqualem</expan> habent <expan abbr="proportionẽ">proportionem</expan> <lb></lb>ad a b: <expan abbr="igit̃">igitur</expan> coniungendo per 28. Quinti propor<lb></lb><figure id="id.015.01.145.2.jpg" xlink:href="015/01/145/2.jpg"></figure><lb></lb>tio d a ad a c minor, quam c a ad a b, quod erat demonſtrandum.<lb></lb><arrow.to.target n="marg452"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002285"><margin.target id="marg451"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="margin">
          <s id="id002286"><margin.target id="marg452"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002287">Propoſitio centeſima trigeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id002288">Si ad duas lineas, quarum una alteri dupla ſit eadem linea adda­<lb></lb>tur erit aggregati ex minore, &amp; a d adiecta ad ipſam <expan abbr="minorẽ">minorem</expan> minor <lb></lb>proportio quam aggregati ex maiore, &amp; adiecta ad ipſam maio­<lb></lb>rem duplicata.</s>
        </p>
        <p type="main">
          <s id="id002289"><arrow.to.target n="marg453"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002290"><margin.target id="marg453"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002291">Sint duæ lineę a b, &amp; c d. </s>
          <s id="id002292">&amp; ſit c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb></lb><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"></figure><lb></lb>b e, &amp; uocetur iuncta c d, d f dico, <lb></lb>quod proportio e a ad a b, eſt mi­<lb></lb>nor duplicata f c ad c d, adijcia­<lb></lb>tur d f æqualis g f, quia ergo g d <lb></lb>eſt dupla ad f d, ideo ad e b c d autem eſt dupla ad a b, tota igitur <pb pagenum="127" xlink:href="015/01/146.jpg"></pb>g c dupla toti e a. </s>
          <s id="id002293">quare ut g c ad g d ut e a ad e b <expan abbr="permutãdo">permutando</expan>, &amp; per <lb></lb>euerſam ut e a ad a b, ita g c ad c d, ut g c ad c d <expan abbr="cõponitur">componitur</expan> ex g e ad <lb></lb>f e, &amp; f c ad c d, igitur e a ad c b componitur ex eiſdem. </s>
          <s id="id002294">Proportio <lb></lb>autem g c ad f c eſt minor, quam f c ad c d, igitur minor quàm du­<lb></lb>plicata f c ad c d. </s>
          <s id="id002295">conſtat uerò ex eiſdem, quod proportio c a ad a b <lb></lb>maior eſt duplicata g c ad f c.</s>
        </p>
        <p type="main">
          <s id="id002296">Propoſitio centeſima trigeſima tertia.</s>
        </p>
        <p type="main">
          <s id="id002297">Si fuerint duæ quantitates, quarum una alteri dupla ſit: minua­<lb></lb>tur à minore quædam <expan abbr="quãtitas">quantitas</expan> eademque maiori addatur, erit mino­<lb></lb>ris ad <expan abbr="reſiduũ">reſiduum</expan> maior proportio, <expan abbr="quã">quam</expan> aggregati ad <expan abbr="maiorẽ">maiorem</expan> duplicata. <lb></lb></s>
          <s id="id002298">Si uerò minori addatur et à maiore detrahatur, erit aggregati ad mi<lb></lb>nore m minor proportio quàm maioris ad reſiduum duplicata.</s>
        </p>
        <p type="main">
          <s id="id002299"><arrow.to.target n="marg454"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002300"><margin.target id="marg454"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <figure id="id.015.01.146.1.jpg" xlink:href="015/01/146/1.jpg"></figure>
        <p type="main">
          <s id="id002301">Sit a b dupla c d, &amp; addatur quæ­<lb></lb>dam ad b a, quę ſit a g, eadem detraha­<lb></lb>tur ex c d &amp; ſit c h, dico, quod propor­<lb></lb>tio e d ad d h maior eſt, quam duplica­<lb></lb>ta g b ad a b, &amp; rurſus ſi quædam ad c &amp; minuatur ex a b utpotè <lb></lb>c f addatur c d, &amp; a e minuatur ex a b, erit proportio f d ad c d mi­<lb></lb>nor duplicata a b ad g e. </s>
          <s id="id002302"><expan abbr="Primũ">Primum</expan> ſic reſecentur a n &amp; k l æquales ſin­<lb></lb>gulæ c h, igitur a l dupla eſt e h &amp; a b fuit dupla a d, c d igitur ut in <lb></lb>priore conſtitutioné præcedentis a b ad l b, ut c d ad h d &amp; a b ad <lb></lb>b l maior, quam duplicata a b ad b k ut minor quàm k b ad b l. </s>
          <s id="id002303">hoc <lb></lb>enim demonſtratum eſt in fine, igitur c d ad h d maior, quàm du­<lb></lb>plicata a k ad k b, ſed a k ad k b maior eſt per uigeſimam tertiam, hu­<lb></lb>ius ſcilicet per demonſtrationem illius, quàm g b ad b a, igitur mul­<lb></lb>to maior c d ad d h, quàm duplicata g b ad b a, quod eſt primum.</s>
        </p>
        <p type="main">
          <s id="id002304">Secundum ſic per eadem, addito enim duplo f c ipſi <lb></lb><figure id="id.015.01.146.2.jpg" xlink:href="015/01/146/2.jpg"></figure><lb></lb>a b ut in ſecunda figura, &amp; ſint a m, &amp; m n erit f d ad c d, <lb></lb>ut n a ad a b, quare cum n a ad a b ſit minor duplicata per <lb></lb>præcedentem in b ad a b, &amp; a b ad e b ſit maior, ut demon <lb></lb>ſtratum eſt in uigeſima tertia huius, quàm m b ad a b, erit <lb></lb>f d ad d c multo minor duplicata a b ad b e, quod eſt ſe­<lb></lb>cundum.</s>
        </p>
        <p type="main">
          <s id="id002305">Propoſitio centeſima trigeſima quarta.</s>
        </p>
        <p type="main">
          <s id="id002306">Si rectangula ſuperficies ſit cuius pars tertia quadrata ſit, corpus <lb></lb>quod ex latere quadratæ in reſiduum ſuperficiei conſtat maius eſt <lb></lb>quouis corpore ex eadem ſuperficies aliter diuiſa conſtituto.</s>
        </p>
        <p type="main">
          <s id="id002307">Sit rectangulum a c cuius tertia pars c e ſit quadrata, dico quod <lb></lb><arrow.to.target n="marg455"></arrow.to.target><lb></lb>corpus, quod <expan abbr="cõſtat">conſtat</expan> ex e d in a b eſt maius omni corpore, quod fue <lb></lb>rit ex latere partis ſuperficiei a b in reliquam <expan abbr="partẽ">partem</expan>. </s>
          <s id="id002308">Si non diuidatur <lb></lb>uel ſupra uel infra, &amp; primo in f erit <expan abbr="autẽ">autem</expan> proportio e d ad d f, ut e c ad 


<pb pagenum="128" xlink:href="015/01/147.jpg"></pb>e k, &amp; f a ad a e, ut ſuperficierum ipſa­<lb></lb><figure id="id.015.01.147.1.jpg" xlink:href="015/01/147/1.jpg"></figure><lb></lb>rum per primam ſexti Elementorum: at <lb></lb>per præcedentem maior eſt proportio <lb></lb>e d ad d f, quàm a f ad a e, duplicata igi­<lb></lb>tur maior eſt proportio e d ad eam, quę <lb></lb>poteſt ſuper f c ſuperficiem, quam f a ad <lb></lb>a e, igitur maior, quàm a k ad a b ex pri­<lb></lb>ma ſexti Elementorum: igitur per trige<lb></lb>ſimam quartam undecimi. </s>
          <s id="id002309">Parallelipe­<lb></lb>dum ex e d in a b maius eſt parallelipedo ex ea, quæ poteſt in f c ſu­<lb></lb>perficiem in ipſam ſuperficiem a k. </s>
          <s id="id002310">Si uerò diuiſio facta fuerit in g, <lb></lb>conſtat ex præcedenti, quod minor eſt proportio g e ad e d, quàm <lb></lb>ſit duplicata e a ad a d a g, eam igitur minor proportio eius lineæ, <lb></lb>quæ poteſt in g e ſuperficiem ad e d quam a b ad a h, igitur paralle­<lb></lb>lipedum ex e d in a b eſt maius parallelipedo ex ea, quæ poteſt g c <lb></lb>in a h cum ſit a b ad a h, ut dictum eſt, uelut a e ad a g.<lb></lb><arrow.to.target n="marg456"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002311"><margin.target id="marg455"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002312"><margin.target id="marg456"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002313">Manifeſtum eſt autem, quòd tale corpus eſt æquale duplo cubi <lb></lb>lateris partis tertiæ quadratæ.</s>
        </p>
        <p type="main">
          <s id="id002314">Propoſitio centeſima trigeſima quinta.</s>
        </p>
        <p type="main">
          <s id="id002315">Si linea in duas partes, quarum una ſit alteri dupla, diuidatur <lb></lb>erit, quod fit ex tertia parte in quadratum reſidui parallelipedum <lb></lb>maius omni parallelipedo, quod ex diuiſione eiuſdem lineæ crea­<lb></lb>ri poſsit.</s>
        </p>
        <p type="main">
          <s id="id002316"><arrow.to.target n="marg457"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002317"><margin.target id="marg457"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002318">Sit a c dupla b c, &amp; ſit quadratum ad ipſius a c, dico parallelipe­<lb></lb><figure id="id.015.01.147.2.jpg" xlink:href="015/01/147/2.jpg"></figure><lb></lb>dum ex b c in a d maius eſſe quouis alio ex <lb></lb>diuiſione lineæ a b ſimiliter creato. </s>
          <s id="id002319">Secetur <lb></lb>primo in e, &amp; fiat quadratum a f, eritque per <lb></lb>uigeſimam quintam. </s>
          <s id="id002320">Huius proportio c b <lb></lb>ad b c maior duplicata a e ad a c, quare ma­<lb></lb>ior, quam a f ad a d per uigeſimam ſexti Ele<lb></lb>mentorum, igitur per trigeſimam quartam <lb></lb>undecimi, Parallelipedum ex b c in a d maius eſt parallelipedo e b <lb></lb>in a f, quod eſt demonſtrandum. </s>
          <s id="id002321">Si uerò diuiſio cadat in g, fiat qua­<lb></lb>dratum a h, et erit per uigeſimamtertiam huius proportio g c ad c b <lb></lb>minor, quam duplicata c a ad a g: igitur minor, quàm a d ad a h, igi­<lb></lb>tur per eandem parallelipedum ex c b in a d maius eſt parallelipe­<lb></lb>do ex g b in a h.</s>
        </p>
        <p type="main">
          <s id="id002322"><arrow.to.target n="marg458"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002323"><margin.target id="marg458"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002324">Ex hoc liquet quòd parallelipedum illud erit quadruplum cu­<lb></lb>bo minoris partis, &amp; dimidium cubi maioris.</s>
        </p>
        <pb pagenum="129" xlink:href="015/01/148.jpg"></pb>
        <p type="main">
          <s id="id002325">Propoſitio centeſima trigeſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id002326">Denominationes in infinitum extendere.</s>
        </p>
        <p type="main">
          <s id="id002327">Inquit Euclides, ſi fuerint quotlibet quantitates ab uno in conti­</s>
        </p>
        <p type="main">
          <s id="id002328"><arrow.to.target n="marg459"></arrow.to.target><lb></lb><arrow.to.target n="marg460"></arrow.to.target><lb></lb>nua proportione, erit tertius numerus quadratus, &amp; omnes alij ſe­<lb></lb>quentes uno intermiſſo. </s>
          <s id="id002329">Tertia igitur in comparatione ad ſecun­<lb></lb>dam etiam, quod non ſit numerus, eſt quadratum: eſt enim tertia <lb></lb>ab uno quadratum ſecundæ, quæ eſt proportio. </s>
          <s id="id002330">Detracto igitur <lb></lb>uno omnes quantitates lo co pari ſunt quadratæ: ut ſcias ergo cu­<lb></lb>ius ſunt quadratæ diuide per medium, &amp; erit quadratum illius, er­<lb></lb>go quadrageſima erit quadratum uigeſimæ, &amp; uigeſima decimæ, <lb></lb>&amp; decima quintæ, &amp; uigeſima ſexta tertiæ decimæ, &amp; ita de alijs. <lb></lb></s>
          <s id="id002331">Iuxta hoc dicemus, quod ſecunda erit <expan abbr="quadratũ">quadratum</expan>, &amp; quarta quadra­<lb></lb>tum quadrati, &amp; octaua <expan abbr="quadratũ">quadratum</expan> quadrati quadrati. </s>
          <s id="id002332">Et ſextadeci­<lb></lb>ma quad quad quad quad. </s>
          <s id="id002333">&amp; ita trigeſima ſecunda quad quad quad <lb></lb>quad quad. </s>
          <s id="id002334">Quod autem quad. </s>
          <s id="id002335">eſt quarta in ordine, ideo &amp; octa­<lb></lb>ua &amp; duodecima &amp; decimaſexta, &amp; ſic de alijs ſunt quadrata qua­<lb></lb>drati, &amp; ſicut quarta eſt quadratum quadrati primæ, ita octaua ſe­<lb></lb>cundæ, &amp; duodecima tertiæ, &amp; ſexta decima quartæ, &amp; uigeſima <lb></lb>quintæ, &amp; ita ſemper diuidendo per quatuor.</s>
        </p>
        <p type="margin">
          <s id="id002336"><margin.target id="marg459"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002337"><margin.target id="marg460"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 9. P<emph type="italics"></emph>ro <lb></lb>poſ.<emph.end type="italics"></emph.end> 8.</s>
        </p>
        <p type="main">
          <s id="id002338">Secunda regula dicebat ibidem Euclides, ſi fuerint quotlibet <lb></lb><arrow.to.target n="marg461"></arrow.to.target><lb></lb>quantitates ab uno in continua proportione quartus, ab uno erit <lb></lb>cubus ſupple ſecundæ, &amp; ita duobus ſemper intermiſsis, uno igi­<lb></lb>tur ipſo relicto quolibet loco ternario, ut tertia, ſexta, nona, duode­<lb></lb>cima ſunt cubi, &amp; cubi eius quantitatis, quę exit diuiſo numero per <lb></lb>tria, uelut tertia primæ, ſexta ſecundæ, nona tertię, duo decima quar <lb></lb>tæ: &amp; ita tertia erit cubus nona cubus cubi, &amp; uigeſima ſeptima cu­<lb></lb>bus cubi cubi ſcilicet primæ. </s>
          <s id="id002339">Et trigeſima nona eſt cubus ter­<lb></lb>tiæ decimæ.</s>
        </p>
        <p type="margin">
          <s id="id002340"><margin.target id="marg461"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 9. P<emph type="italics"></emph>ro­<lb></lb>poſ.<emph.end type="italics"></emph.end> 8.</s>
        </p>
        <p type="main">
          <s id="id002341">Tertia regula quarta quantitas, ut uiſum eſt: eſt quad quad. </s>
          <s id="id002342">Et <lb></lb>quinta eſt relatum primum, quia 5 eſt numerus primus, &amp; 7 eſt re­<lb></lb>latum ſecundum, quia eſt ſecundus numerus primus: &amp; undecima <lb></lb>tertium: &amp; tertiadecima quartum: &amp; decimaſeptima quintum: &amp; <lb></lb>decimanona ſextum: &amp; uigeſima tertia ſeptimum &amp; uigeſima quin­<lb></lb>ta, quia eſt primus numerus præter quam ad quintam, ideò eſt rela­<lb></lb>tum quintæ, quæ eſt relatum primum primæ, omnes ergo numeri <lb></lb>primi ſunt relata, alij omnes ſunt ex natura cubi uel quadrati. </s>
          <s id="id002343">Sed <lb></lb>relata ſunt inter ſe omnia diuerſorum generum niſi <expan abbr="uigeſimũ">uigeſimum</expan> quin­<lb></lb>tum, quod eſt relatum primum primi relati, &amp; quadrageſimum no­<lb></lb>num eſt relatum ſecundum relati ſecundi. </s>
          <s id="id002344">Et ita centeſimum uigeſi­<lb></lb>mum primum eſt relatum tertium tertij relati, reliqua, ut dixi, me­<lb></lb>dia inter hæc ſunt ſui generis.</s>
        </p>
        <pb pagenum="130" xlink:href="015/01/149.jpg"></pb>
        <p type="main">
          <s id="id002345">Quarta regula propoſita quantitate ab uno in continua propor<lb></lb>tione, ſi uis ſcire cuius naturæ ſit detracto uno conſidera, an poſsit <lb></lb>diuidi per duo, eſt quadratum medietatis, &amp; ita procedes diuiden­<lb></lb>do uſque ad numerum primum, qui uel eſt 2, &amp; erit ex genere quad <lb></lb>quad. </s>
          <s id="id002346">uel 3, &amp; erit ex genere quadratorum cuborum, &amp; ſimiliter ſi <lb></lb>ſit 9, erit ex genere quadratorum cubi cubi. </s>
          <s id="id002347">Et ſi proueniat alius nu<lb></lb>merus primus, ut 5. 7. 11. 13. erit quadratum relati illius ordinis. </s>
          <s id="id002348">Et ſi <lb></lb>non poteſt diuidi numerus quantitatum per 2 uide, ſi poſsit diuidi <lb></lb>per 3, tunc erit cubus illius quantitatis, &amp; ſi illa quantitas, quæ pro­<lb></lb>uenit ex diuiſione: fuerit 3, uel potuerit diuidi per 3, erit cubus, uel <lb></lb>cubus cubi, &amp; ita deinceps. </s>
          <s id="id002349">Si uerò ſit alius numerus primus, ut 5. <lb></lb>7. 11. erit cubus relati. </s>
          <s id="id002350">Et ita ſi <expan abbr="nõ">non</expan> poſsit diuidi per 2, nec per 3, erit ex <lb></lb>genere relati. </s>
          <s id="id002351">Et tunc ſi poſsit diuidi per alium numerum, ut 35, erit <lb></lb>relatum ex eo genere. </s>
          <s id="id002352">Vtpotè trigeſima quinta quantitas eſt rela­<lb></lb>tum ſecundum relati primi, ſeu relatum primum relati ſecundi. <lb></lb></s>
          <s id="id002353">Nam quoties quantitas poteſt diuidi per duos numeros, dicetur <lb></lb>ſub utro que uiciſsim, ut duodecima poteſt diuidi per 4 &amp; 3, ideò di­<lb></lb>cetur cubus quad quad. </s>
          <s id="id002354">uel quad quad. </s>
          <s id="id002355">cub. </s>
          <s id="id002356">&amp; per 2 &amp; 6, &amp; dicetur <lb></lb>quadratum cubi quadrati, &amp; quadratum cubicum quadrati ipſius <lb></lb>proportionis, ad quam omnia referri debent.</s>
        </p>
        <p type="main">
          <s id="id002357">Quinta regula ex præcedenti pendet, &amp; eſt, quod denomina­<lb></lb>tiones, &amp; proportiones uiciſsim commutantur: uelut 256 eſt quad <lb></lb>quad quad, &amp; inter quad quad quad, &amp; quad quad ſunt quatuor ter <lb></lb>mini ipſo computato, &amp; inter quad quad, &amp; quod uiſi duo, ergo <lb></lb>quad quad quad continet plures proportiones, &amp; proportiones <lb></lb>duplicatæ non conſtituunt quad: nam 64 continet duas duplas <lb></lb>ad 16, non tamen eſt quadratum 16, ideo oportet diligenter ani­<lb></lb>maduertere.</s>
        </p>
        <p type="main">
          <s id="id002358">Sexta regula ſimiliter ex dictis pendet, &amp; eſt, quòd gratia exem­<lb></lb>pli relatum primum comparatum ad primum terminum eſt ſexta <lb></lb>quantitas, cum autem comparatur ad rem, iam præſupponit pro­<lb></lb>portionem. </s>
          <s id="id002359">Exemplum relatum primum proportionis 21/20 eſt 4084101/3200000 <lb></lb>&amp; eſt aliquanto maior ſexquiquarta, &amp; ſi colligas terminos 100. <lb></lb>105. 110 1/4 115 61/80 121 861/1600 127 19681/32000. Tu uides quòd ſunt ſex termini in <lb></lb>utra que computando primum, ſed in 21/20 ſunt duo termini, &amp; in qua­<lb></lb>drato tres, &amp; in quadrato quadrati per præcedentem, adduntur <lb></lb>duo &amp; ultimus ſcilicet ſextus fit ex relato ipſo. </s>
          <s id="id002360">Ergo ultra propor­<lb></lb>tionem ſunt tantum quatuor termini.</s>
        </p>
        <p type="main">
          <s id="id002361">Septima regula ad effugiendum omnes errores tu ſcis, quòd <lb></lb>4096 quadratum 64 eſt ſextus a 64, ad quem habet proportionem <lb></lb>quadrati, &amp; 64 eſt ſimiliter ſextus ab uno illo ſcilicet non compu­


<pb pagenum="131" xlink:href="015/01/150.jpg"></pb>tato, &amp; ita 64 habet rationem unius, &amp; licet comparetur ad 2 rem, <lb></lb>&amp; ſit ſextus ab eo, eo computato 4096 autem à 64 ſit ſeptimus, ta­<lb></lb>men non eſt eadem ratio, quia 64 non eſt quadratum 2.</s>
        </p>
        <p type="main">
          <s id="id002362">Propoſitio centeſima trigeſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id002363">Rationem numerorum ex progreſsione declarare.</s>
        </p>
        <p type="main">
          <s id="id002364">Michaël Stifelius rationem pulcherrimam tradidit ad inuentio­<lb></lb><arrow.to.target n="marg462"></arrow.to.target><lb></lb><arrow.to.target n="marg463"></arrow.to.target><lb></lb>nem numerorum, qui uocantur multiplicandi, &amp; componitur hoc <lb></lb>modo. </s>
          <s id="id002365">Ex prima componitur 1 &amp; 2, faciunt 3. 1. 2. 3 faciunt 6. 1. 2. 3. 4 <lb></lb>faciunt 10, &amp; ita prima tabula conſtituit ſecundam recta ſerie nu­<lb></lb>merorum iunctis o­<lb></lb>mnibus ab uno. </s>
          <s id="id002366">Ter<lb></lb><figure id="id.015.01.150.1.jpg" xlink:href="015/01/150/1.jpg"></figure><arrow.to.target n="table17"></arrow.to.target><lb></lb>tia fit ex ſecunda &amp; <lb></lb>tertia, primò aſſumi<lb></lb>tur 10 in tertia, ut in <lb></lb>ſecunda, &amp; ex 10 ſe­<lb></lb>cundæ, &amp; 10 tertiæ <lb></lb>fit 20, &amp; ex 15 ſecun­<lb></lb>dæ, &amp; 20 tertiæ fit <lb></lb>35, &amp; ex 21 ſecundæ, <lb></lb>&amp; 35 tertiæ fit 56, &amp; <lb></lb>ex 28, &amp; 56 fit 84. Et <lb></lb>quanta fit ex tertia, <lb></lb>&amp; ex ſe ipſa. </s>
          <s id="id002367">primum <lb></lb>aſſumendo 35 ex ter<lb></lb>tia, &amp; ponitur pro <lb></lb>primo numero quartæ, &amp; ex 35 tertiæ, &amp; 35 quartæ fit 70 numerus <lb></lb>ſecundæ quartæ: &amp; ita ex 56 &amp; 70 fit 126, &amp; ex 84, &amp; 126. 210. &amp; ita <lb></lb>quinta ex quarta &amp; ſe ipſa, &amp; ſic in infinitum.</s>
        </p>
        <p type="margin">
          <s id="id002368"><margin.target id="marg462"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id002369"><margin.target id="marg463"></margin.target>P<emph type="italics"></emph>rimæ ſuæ<emph.end type="italics"></emph.end><lb></lb>A<emph type="italics"></emph>rith.<emph.end type="italics"></emph.end></s>
        </p>
        <table>
          <table.target id="table17"></table.target>
          <row>
            <cell>1</cell>
            <cell>2</cell>
            <cell>3</cell>
            <cell>4</cell>
            <cell>5</cell>
            <cell>6</cell>
            <cell>7</cell>
            <cell>8</cell>
          </row>
          <row>
            <cell>1</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>2</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>3</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>4</cell>
            <cell>6</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>5</cell>
            <cell>10</cell>
            <cell>10</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>15</cell>
            <cell>20</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>21</cell>
            <cell>35</cell>
            <cell>35</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>8</cell>
            <cell>28</cell>
            <cell>56</cell>
            <cell>70</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>36</cell>
            <cell>84</cell>
            <cell>126</cell>
            <cell>126</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>10</cell>
            <cell>45</cell>
            <cell>120</cell>
            <cell>210</cell>
            <cell>252</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>11</cell>
            <cell>55</cell>
            <cell>165</cell>
            <cell>330</cell>
            <cell>462</cell>
            <cell>462</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>12</cell>
            <cell>66</cell>
            <cell>220</cell>
            <cell>495</cell>
            <cell>792</cell>
            <cell>924</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>13</cell>
            <cell>78</cell>
            <cell>286</cell>
            <cell>715</cell>
            <cell>1297</cell>
            <cell>1716</cell>
            <cell>1716</cell>
            <cell></cell>
          </row>
          <row>
            <cell>14</cell>
            <cell>91</cell>
            <cell>364</cell>
            <cell>1001</cell>
            <cell>2002</cell>
            <cell>3003</cell>
            <cell>3432</cell>
            <cell></cell>
          </row>
          <row>
            <cell>15</cell>
            <cell>105</cell>
            <cell>455</cell>
            <cell>1365</cell>
            <cell>3003</cell>
            <cell>5005</cell>
            <cell>6435</cell>
            <cell>6435</cell>
          </row>
          <row>
            <cell>16</cell>
            <cell>120</cell>
            <cell>560</cell>
            <cell>1820</cell>
            <cell>4368</cell>
            <cell>8008</cell>
            <cell>11440</cell>
            <cell>12870</cell>
          </row>
          <row>
            <cell>17</cell>
            <cell>136</cell>
            <cell>680</cell>
            <cell>2380</cell>
            <cell>6188</cell>
            <cell>12376</cell>
            <cell>19448</cell>
            <cell>24310</cell>
          </row>
        </table>
        <p type="main">
          <s id="id002370">Regula ergo eſt, quòd binarius ſeruit &lt;02&gt; quadratæ, &amp; quia nihil <lb></lb>eſt in eius directo, ſolus ipſe ſeruiet &lt;02&gt; quadratæ. </s>
          <s id="id002371">Ternarius autem <lb></lb>cubicæ, &amp; quia in eius directo eſt alter ternarius, ille etiam ſeruiet <lb></lb>&lt;02&gt; cubicæ. </s>
          <s id="id002372">Quaternarius autem ſeruiet quadrato quadrati, &amp; ſena­<lb></lb>rius, qui eſt in illius directo. </s>
          <s id="id002373">Ergo quinarius ſeruiet &lt;02&gt; relatę primę, <lb></lb>&amp; duo ſequentes numeri ſcilicet 10 &amp; 10, &amp; eo dem modo ſenarius <lb></lb>numeri duo ſequentes 15 &amp; 20 ſeruient cubo quadrati, &amp; ita etiam <lb></lb>ſeptenarius cum tribus ſequentibus numeris 21. 35 &amp; 35 ſeruient <lb></lb>rel. </s>
          <s id="id002374">ſecundi radici, &amp; ita deinceps in infinitum.</s>
        </p>
        <p type="main">
          <s id="id002375">Propoſitio centeſima trigeſima octaua.</s>
        </p>
        <p type="main">
          <s id="id002376">Modos uſus horum numerorum declarare.</s>
        </p>
        <p type="main">
          <s id="id002377">In quouis numero denominationis oportet tot addere o, quo­<lb></lb><arrow.to.target n="marg464"></arrow.to.target>


<pb pagenum="132" xlink:href="015/01/151.jpg"></pb>tus eſt ordo, &amp; facere tot numeros ſequentes; quotus eſt ordo, &amp; <lb></lb>ſemper minuere unam o, uelut quia quadrata &lt;02&gt; eſt prima ad 2 ad­<lb></lb>demus o, &amp; fiet 20, nec alium quęremus numerum. </s>
          <s id="id002378">Sed quia cubi­<lb></lb>ca eſt ſecundo loco, habebit prima nota 00, &amp; fiet 300, &amp; ſecundum <lb></lb>3 unam 0, &amp; fiet 30, &amp; in quadrato quadrati addemus 000 primo, <lb></lb>&amp; 00 ſecundo, &amp; o tertio, &amp; ita habebimus 4000. 600. 40. ſed quia <lb></lb>in tabula non eſt 4 ultimum, addemus ſimilem primo ſemper. </s>
          <s id="id002379">In <lb></lb>relato primo, ergo habebimus 50000. 1000. 1000. 50. &amp; in cubo <lb></lb>quadrati 600000. 150000. 20000. 1500. 60. Manifeſtum eſt, quòd <lb></lb>his uice uerſa aſſumpſimus 15 &amp; 6 ſimiles prioribus addendo ſem­<lb></lb>per ut dixi o minus, donec ad unam peruenerit. </s>
          <s id="id002380">Et ita in relato ſe­<lb></lb>cundo 7000000. 2100000. 350000. 35000. 2100. 70. &amp; ita deinceps.</s>
        </p>
        <p type="margin">
          <s id="id002381"><margin.target id="marg464"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002382">Propoſitio centeſima trigeſima nona.</s>
        </p>
        <p type="main">
          <s id="id002383">Radices omnes à propoſitis numeris extrahere.<lb></lb><arrow.to.target n="marg465"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002384"><margin.target id="marg465"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002385">Propoſitis quibuſuis numeris utpotè 916132832, uolo detrahere <lb></lb>&lt;02&gt; relatam primam, primum habebo in tabula deſcripta relata pri­<lb></lb>ma numerorum ſimplicium uſque ad 10 uelut in exemplo. </s>
          <s id="id002386">Dein de <lb></lb><figure id="id.015.01.151.1.jpg" xlink:href="015/01/151/1.jpg"></figure><lb></lb>ſubſcribam pun­<lb></lb>ctum ſub prima <lb></lb>nota à dextra, &amp; <lb></lb>quia eſt quarta in <lb></lb><figure id="id.015.01.151.2.jpg" xlink:href="015/01/151/2.jpg"></figure>ordine hoc, ſeu quinta denominatio ſecun­<lb></lb>dum noſtrum, omittam quatuor notas in­<lb></lb>ter medias, &amp; ſubſcribam punctum aliud, <lb></lb>&amp; ita facerem ſi eſſent plures quàm decem <lb></lb>notæ: relinquitur ergo ad <expan abbr="pũctum">punctum</expan> primum <lb></lb>à ſiniſtra 9161, cuius quęro &lt;02&gt; relatam pri­<lb></lb>mam in tabula, quam inuenio eſſe 6, nam <lb></lb><figure id="id.015.01.151.3.jpg" xlink:href="015/01/151/3.jpg"></figure>7776 eius relatum primum eſt <lb></lb>proximius ex minoribus ad 9161, <lb></lb>detraho igitur 7776, ex numero <lb></lb>propoſitio relinquitur. </s>
          <s id="id002387">Dein de<lb></lb>póno 6 &amp; quadratum eius, &amp; cub. </s>
          <s id="id002388">&amp; quadratum <lb></lb>quadrati, quia, ut dixi, eſt quarta denominatio a­<lb></lb><figure id="id.015.01.151.4.jpg" xlink:href="015/01/151/4.jpg"></figure>pud illum, &amp; è regione numeros præcedentes in­<lb></lb>uentos relati primi ex præcedenti propoſitione: &amp; duco ſingulos <lb></lb>cum ſuis collateralibus, ut uides etiam in figura, et cum ultimo pro­<lb></lb>ducto, ſcilicet 64800000 diuido 138532832 exit 2, huius accipio o­<lb></lb>mnes numeros ad relatum primum uſque ut uides, &amp; pono minores <lb></lb>è regione maiorum, utpotè 2 è regione 1296 &amp; 50000, &amp; 4 è regio­


<pb pagenum="133" xlink:href="015/01/152.jpg"></pb>ne 216 &amp; 10000, &amp; 8 è regione 36 &amp; 10000, &amp; 16 è regione 6, &amp; 50, <lb></lb>&amp; duco 6 in 50 fit 300, duco in 16 fit 4800, duco 36 in 1000 fit <lb></lb>36000, duco 36 in 8 fit 288000, duco etiam 216 in 10000 &amp; fit <lb></lb>2160000, &amp; duco hos per 4 fit 86400000, duco rurſus 1296 in <lb></lb>50000 fit 64800000, duco in 2 fit 129600000. Demum addo 32 re­<lb></lb>latum primum 2, &amp; fit ſumma omnium 138532832, &amp; ita habemus <lb></lb>radicem relatam primam dicti numeri eſſe 62. Et ſi numerus produ<lb></lb>ctus fuiſſet maior oportuiſſet accipere proximo minorem. </s>
          <s id="id002389">Inde per <lb></lb>regulam ſequentem addere minutias.</s>
        </p>
        <p type="main">
          <s id="id002390">Propoſitio centeſima quadrageſima.</s>
        </p>
        <p type="main">
          <s id="id002391">Radices per numeros fractos determinare.</s>
        </p>
        <p type="main">
          <s id="id002392">Duplex eſt modus, ut etiam docui in arithmeticis, ſcilicet ut pro </s>
        </p>
        <p type="main">
          <s id="id002393"><arrow.to.target n="marg466"></arrow.to.target><lb></lb>radice quadrata addatur duo o, &amp; pro cuba tria, &amp; pro quadrata <lb></lb>quadrata quatuor, &amp; pro relata prima quinque, &amp; ita deinceps, &amp; <lb></lb>prę decimis ſemel, pro centeſimis bis, pro milleſimis ter, pro millia­<lb></lb>ribus ſeu partibus earum quater, pro centeſimis milleſimis quin­<lb></lb>quies, pro milleſimis milleſimarum ſexies, &amp; ita deinceps deinde <lb></lb>per præcedentem detrahere radicem, &amp; erit ualde exacta. </s>
          <s id="id002394">Exemplo <lb></lb>non utar, niſi quòd ſi uelles radicem relatam 16 ad milleſimas, acci­<lb></lb>cipies radicem relatam numeri à latere propoſiti, &amp; ita de alijs <lb></lb>1600000, 00000, 00000, &amp; ſi uelles &lt;02&gt; cub. </s>
          <s id="id002395">5 1/5 per milleſimas, pri<lb></lb>mo addes ter 000, &amp; fiet 3000000000, inde ſume 1/5 1000000000, <lb></lb>qui eſt 200000000, &amp; adde ad 5000000000, fit 2500000000, <lb></lb>&amp; hoc quia unum refert numerum 1000000000 ex ſuppoſito &amp; 1/5 <lb></lb>eſt 1/5 unius.</s>
        </p>
        <p type="margin">
          <s id="id002396"><margin.target id="marg466"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002397">Secundus modus eſt, ut accipias proximè maiorem, &amp; multipli­<lb></lb>ca in ſe, &amp; detrahe numerum propoſitum, &amp; reſiduum diuide per <lb></lb>duplum radicis primo inuentæ, ſi fuerit quadrata, &amp; per triplum <lb></lb>quadrati eiuſdem ſi fuerit cubica, &amp; per quadruplum cubi, ſi fuerit <lb></lb>quadrata quadrata, &amp; per quincuplum quadrati quadrati, &amp; quod <lb></lb>exit detrahes ex priore radice, &amp; rurſus quod relinquitur, multipli­<lb></lb>ca in ſe, &amp; eodem modo agendo quod ſupereſt à numero propoſi­<lb></lb>to, diuide per duplum radicis prioris, ſi ſit radix quadrata, uel per <lb></lb>triplum quadrati ſi ſit cubica, &amp; quod exit rurſus detrahe, &amp; ita a­<lb></lb>gendo, peruenies ad exactiſsimam radicem, exemplum uolo radi­<lb></lb>cem quadratam 5 proxima maior eſt 3, quadratum 9, differentia 4, <lb></lb>diuide per 6 duplum 3 exit 2/3, detrahe ex 3 fit 2 1/3, quadratum eſt 49/9 <lb></lb>quod eſt 5 4/9, rurſus diuido 4/9 differentiam 5 4/9 &amp; 5 per 4 2/3 duplum <lb></lb>radicis primæ exit 2/21, detrahe ex 2 1/3, relinquitur 2 5/21, radix ſatis pro­<lb></lb>pinqua, nam eius quadratum eſt 5 4/441, in cubica ſimiliter uolo &lt;02&gt;<lb></lb>cu. </s>
          <s id="id002398">5, proxima maior eſt 2, cubus 8, differentia 3, diuide per triplum 


<pb pagenum="134" xlink:href="015/01/153.jpg"></pb>quadrati 2 quod eſt 12 exit 1/4 detrahe ex 2 fit 1 3/4 cuius cubus eſt 5 23/64 <lb></lb>differentia eſt 23/64 diuide per triplum quadrati 1 3/4 quòd eſt 9 3/16 exit <lb></lb>23/588 detrahe ex 1 3/4 <expan abbr="relinquũtur">relinquuntur</expan> 1 107/147 cuius cubus eſt 5 504449/3176523 Ita diuides <lb></lb>hunc exceſſum ſi placet per triplum quadrati 1 107/147 &amp; eſt fermè 9 exit <lb></lb>56050/3176523 quaſi detrahe ex 1 107/147 relinquuntur 323159/453789.</s>
        </p>
        <p type="main">
          <s id="id002399">Tertius modus eſt ſubtilior, tu ſcis, q̊d duo decima denominatio <lb></lb>eſt quadrata ſextę, &amp; quadrata quad, tertiæ, &amp; cuba quarti, quarta <lb></lb>autem eſt inter <expan abbr="tertiã">tertiam</expan> &amp; ſextam ſecunda quantitas in continua pro­<lb></lb>portione: ergo inuenta &lt;02&gt; numeri propoſiti &amp; &lt;02&gt; radicis inuentæ <lb></lb><expan abbr="reducã">reducam</expan> ad unam denominationem, et inter numeratores collo cabo <lb></lb>duas quantitates, quod facile erit ſenſim procedendo, &amp; habebo &lt;02&gt;<lb></lb>cu. </s>
          <s id="id002400">quæſitam, ſcilicet minorem ex duabus intermedijs. </s>
          <s id="id002401">Et ſimiliter <lb></lb>pro relata prima, capiam ſexaginta denominationes, &amp; ſcis, quòd <lb></lb>quinta decima eſt &lt;02&gt; &lt;02&gt; ſexageſimę, &amp; decima eſt &lt;02&gt; cu. </s>
          <s id="id002402">&lt;02&gt; ſexageſimę, <lb></lb>&amp; duodecima &lt;02&gt; relata prima ſexageſimæ per eandem inuenta, er­<lb></lb>go &lt;02&gt; numeri propoſiti tanquam ille ſit ſexageſima denominatio, <lb></lb>inueniam illius radicis inuentæ &lt;02&gt; quadratam, &amp; cubicam, &amp; <lb></lb>quia duodecima quantitas quæ eſt &lt;02&gt; relata prima numeri eſt <lb></lb>ſecunda, quatuor intermediarum inter ponam inter &lt;02&gt; quadra­<lb></lb>tum, quadratum, &amp; cubicam quadratam quatuor numeros in <lb></lb>continua proportione, &amp; ſecundus ex minoribus erit &lt;02&gt; relata <lb></lb>prima numeri propoſiti. </s>
          <s id="id002403">Exemplum cubicæ uolo &lt;02&gt; cu: 5 habui &lt;02&gt;<lb></lb>quadratam eius 2 5/21 ſed uolo proximiorem diuidendo 4/441 per 4, <lb></lb>quod eſt fermè duplum 2 5/21 exit 1/441 detraho ex 2 5/21 relinquitur ualde <lb></lb>proxima &lt;02&gt; 5. 2 104/441 huius igitur radix quadrata, primo inuenta eſt 1 1/2 <lb></lb>ſecunda proximior eſt 1 41/84 reduco ad eandem denominationem fi­<lb></lb>ent 284/9261 2 416/1764 &amp; 1 861/1764 inter 3944, &amp; 2625, inueniemus duos nume­<lb></lb>ros in continua proportione, ut uides, &amp; erit ſecunda quantitas <lb></lb><figure id="id.015.01.153.1.jpg" xlink:href="015/01/153/1.jpg"></figure><lb></lb>3006/7641, quod eſt 167/98 proximum ad 1 5/7, &lt;02&gt; cubica. </s>
          <s id="id002404">5. <lb></lb><expan abbr="nã">nam</expan> eius cubus eſt 5. 13/343 at exactiſsima eſt ergo 1 69/98. <lb></lb>ut liquet. </s>
          <s id="id002405">Pro relata prima ergo ponamus, ut ue­<lb></lb>lim &lt;02&gt; relatam <expan abbr="primã">primam</expan> 25, accipio 5 &lt;02&gt; 25 cuius &lt;02&gt; eſt, ut uiſum eſt, 2 104/441 <lb></lb>ſimiliter &lt;02&gt; cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb></lb>&amp; inueniam quatuor numeros in <expan abbr="cõtinua">continua</expan> proportione inter illos, <lb></lb>&amp; ſecundus poſt minimum ex illis erit &lt;02&gt; relata prima propinquiſ­<lb></lb>ſima 25. Quomodo uerò inueniantur facillimè illi termini, do­<lb></lb>cui in ſexto libro operis perfecti.</s>
        </p>
        <p type="main">
          <s id="id002406">Quarta regula eſt utilior, licet minus uideatur nobilis, &amp; eſt fun­<lb></lb>data in hoc, quod ſi a b ſit maior c &amp; eis ad dantur b e, &amp; d f æqua­<lb></lb>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, &amp; ex <lb></lb>conſequenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c f ipſius a e, quàm <pb pagenum="135" xlink:href="015/01/154.jpg"></pb>c d ipſius a f ex Euclide. </s>
          <s id="id002407">Dico ergo quod maior eſt proportio a b <lb></lb><figure id="id.015.01.154.1.jpg" xlink:href="015/01/154/1.jpg"></figure><lb></lb>ad c d, quàm a e ad e f, fiat d g ad quam ſit b c ut <lb></lb><arrow.to.target n="marg467"></arrow.to.target><lb></lb>a b ad c d, eritque a e ad c g ut a b ad c d, minor au­<lb></lb>tem eſt a e ad c f, quam ad c g, igitur minor a e ad <lb></lb>c f quàm a b ad c d quod fuit propoſitum. </s>
          <s id="id002408">Simili <lb></lb>ter ſi fuerint duæ quantitates, a b &amp; c d, quarum a b ſit maiore, c d <lb></lb>autem eadem e minor, dico, quòd dimidium aggregati a b &amp; c d <lb></lb>maiorem habebit proportionem ad e, quàm c d &amp; minor, nam iun­<lb></lb>cta b f æquali d e ad a b, ita ut f g ſit dimidium totius a f, qùia ergo <lb></lb><figure id="id.015.01.154.2.jpg" xlink:href="015/01/154/2.jpg"></figure><lb></lb>f g eſt dimidium f a &amp; fb eſt minor dimidio <lb></lb><arrow.to.target n="marg468"></arrow.to.target><lb></lb>f a cum ſit minor b a, &amp; ſimiliter f g eſt mi­<lb></lb>nor a b, quia a b eſt maior dimidio a f, quia <lb></lb>eſt maior b f, ergo proportio g f ad c eſt ma<lb></lb>ior quam b f ad e, ita quam c d ad e, &amp; mi­<lb></lb><arrow.to.target n="marg469"></arrow.to.target><lb></lb>nor quàm a b ad e, quod fuit propoſitum. </s>
          <s id="id002409">Quo uiſo uolo &lt;02&gt; 1000 <lb></lb>quadratam, &amp; quòd de quadrata dico, dico etiam de alijs radici­<lb></lb>bus &amp; erit ex ſecunda regula harum 31 39/62 &amp; quadratum erit 1000 <lb></lb>1521/3844. Iuxta ergo primam partem regulæ 31 38/61 erit minus, &amp; in ueritate <lb></lb>in eo, quod fit ducendo, ut uides, &amp; hoc eſt pro­<lb></lb><figure id="id.015.01.154.3.jpg" xlink:href="015/01/154/3.jpg"></figure><lb></lb>ximum ad 11/160, multiplico igitur duplum 31 39/62, <lb></lb>quod eſt fermè 63 1/4 in 1/160 fient 63/160 detrahe ex <lb></lb>1521/3844 hoc modo, diuide 3844 per 160 exit 24   /40 <lb></lb>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb></lb>1521/3844 ſunt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, &amp; erit &lt;02&gt; exa­<lb></lb>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita<lb></lb>tem, &amp; propinquitatem in producto differentia eſt 1/100 aut parum <lb></lb>maius quod ad radicem comparatum cum debeat diuidi per du­<lb></lb>plum eius erit paulo maius 1/6300. Vnde facilior eſt, &amp; breuior hæc <lb></lb>uia quàm per 00 additus. </s>
          <s id="id002410">Rurſus uolo aliquid <expan abbr="adim̃ere">adimere</expan> &amp; cum pro<lb></lb>pinquitate ita facio. </s>
          <s id="id002411">Conſidero quòd 31 38/61 eſt maius 1/6300 radice, di­<lb></lb>uido 6300 per 62 exit 103 fermè, neque enim curo in hoc fractiones, <lb></lb>multiplico ergo 103 in 38/61 &amp; habeo 3914/6283 hic denominator eſt proxi­<lb></lb>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam &lt;02&gt; 1000, <lb></lb>31 3913/6283 cuius quadratum eſt 1000 minus 1/1048 hoc ut dixi diuiſum <lb></lb>per duplum &lt;02&gt; quod eſt 63 eſt omnino inſenſile in radice.</s>
        </p>
        <p type="margin">
          <s id="id002412"><margin.target id="marg467"></margin.target>8. P<emph type="italics"></emph>ropoſ. <lb></lb>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002413"><margin.target id="marg468"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem. <lb></lb><expan abbr="amplificatã">amplificatam</expan>.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002414"><margin.target id="marg469"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002415">Quinta regula eſt omnium pulcherrima, &amp; eſt communis omni <lb></lb>bus &amp; fractis &amp; integris &amp; omnibus generibus radicum, &amp; ſit ex­<lb></lb>emplum, uolo &lt;02&gt; radicis ſupraſcriptæ ſcilicet 31 3913/6283 multiplico 31 <lb></lb>in 6283, &amp; fit 194793, cui addo 3913, fit 198686 manifeſtum eſt igi­<lb></lb>tur, quod 198686/6283 æquiualet 31 3913/6283 hoc facto, quod eſt commune om­


<pb pagenum="136" xlink:href="015/01/155.jpg"></pb>nibus radicibus extrahendis pro radice quadrata, multiplicabo nù<lb></lb>meratorem, qui eſt 194686 per denominatorem, qui eſt 6283, &amp; ſi <lb></lb>uoluero radicem cubicam, multiplicabo eundem numeratorem <lb></lb>per quadratum denominatoris, &amp; ſi uoluero radicem radicis, mul­<lb></lb>tiplicabo per cubum, multiplicabo per quadratum quadratum <lb></lb>6283, &amp; ita de alijs una diminutione minore, &amp; eius qui prouenit <lb></lb>numeri &lt;02&gt; ſupra poſita denominatori erit &lt;02&gt; eiuſmodi, quam ſuſce­<lb></lb>piſti, uelut in exemplo fuit numerus 198686/6283 quia ergo uolo &lt;02&gt; quad. <lb></lb></s>
          <s id="id002416">multiplico 198686 in 6283, &amp; fit 1248344138, huius accipio &lt;02&gt;<lb></lb>quad. </s>
          <s id="id002417">quæ eſt 35332, hæc autem eſt diuidenda per 6283, &amp; exeunt <lb></lb>5 3917/12566, ecce uides radicem exactam admodum, &amp; facilem. </s>
          <s id="id002418">Volo rur­<lb></lb>ſus &lt;02&gt; quadrat. </s>
          <s id="id002419">5 3917/12566, multiplico 12566 per 5 &amp; fit 62830, cui addo <lb></lb>3917, &amp; fit 66747, cui ſuppono 12566 denominatorem, fient ergo <lb></lb>66747/12566, manifeſtum eſt igitur quòd hoc æquiualet 5 3917/12566, ſi igitur mul<lb></lb>tiplicarem denominatorem per denominatorem &amp; numeratorem, <lb></lb>quod proueniret, eſſet æquale eidem numero, ergo &lt;02&gt; eius eſſet ea­<lb></lb>dem cum &lt;02&gt; prioris, ſed &lt;02&gt; denominatoris eſſet prior numerus, er­<lb></lb>go ſufficiet extrahere &lt;02&gt; producti ex denominatore in numerato­<lb></lb>rem, &amp; ita productum erit ex denominatore in numeratorem <lb></lb>838742802, cuius &lt;02&gt; eſt 28961, hæc igitur diuiſa per 12566 oſten­<lb></lb>dit &lt;02&gt; 2 3892/12566. In hac autem quadrata eſt alius modus ſine multiplica­<lb></lb>tione, ſed non eſt communis alijs, ubi ſtatueris denominatorem <lb></lb>pro denominatore &lt;02&gt;, utpote 12566, &amp; numeratorem 66747, con­<lb></lb>ſtitues medium ſenſim augendo.</s>
        </p>
        <p type="main">
          <s id="id002420">Rurſus uolo &lt;02&gt; relatam 2 3829/12566 reduco ad denominatorem, &amp; fit <lb></lb>ut prius 28961/12566, duco igitur 12566 ad quad. </s>
          <s id="id002421">quad. </s>
          <s id="id002422">ſed ſufficiet in hoc <lb></lb>caſu deducere ad minores denominationes, utpotè diuide 28961 <lb></lb>per 12566 exit 2 3829/12566 multiplico per 566 fit 1104 5862/12566, hoc detrahe <lb></lb>ex 28961 habebis 27856/12000, diuide igitur per 1000 habebis 12 &amp; 27 107/125 <lb></lb>at 108/126 ſunt 6/7, igitur habes 12 pro denominatore, &amp; 27 6/7 pro nume­<lb></lb>ratore, quare erunt numeri 195/84, erit ergo per hanc regulam, ut ducas <lb></lb>84 ad quad. </s>
          <s id="id002423">quadrati, &amp; fit 49787136, duc in 195 fit 9708491520, <lb></lb>cuius &lt;02&gt; relata prima eſt 99, igitur &lt;02&gt; relata prima 2 3829/12566 eſt 1 15/84 pau­<lb></lb>lo maior, id eſt 1 13/70. Et nota quod ſi denominator haberet &lt;02&gt; illius <lb></lb>generis, quam quæris, ſufficeret inuenire radicem eiuſdem generis <lb></lb>abſque alia numerorum multiplicatione.</s>
        </p>
        <p type="main">
          <s id="id002424">Propoſitio centeſima quadrageſima prima. (deducere.</s>
        </p>
        <p type="main">
          <s id="id002425">Numeros fractos ad minores in <expan abbr="eadẽ">eadem</expan> proportione ualde propinqua</s>
        </p>
        <p type="main">
          <s id="id002426">Cum plerunque numeri fracti habeantur per radices, ut aliquan­<lb></lb><arrow.to.target n="marg470"></arrow.to.target><lb></lb>do maiores ſint, aut minores eo fit, ut poſsint reduci ad mino­<lb></lb>res numeros, ut melius intelligi poſsint &amp; facilius tractari, &amp; 


<pb pagenum="137" xlink:href="015/01/156.jpg"></pb>cum hoc ſit exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb></lb>uolo certa ratione ad minores diuiſiones deducere. </s>
          <s id="id002427">Deduco pri­<lb></lb>mò totum ad fractiones ducendo 2 in 12566, &amp; addendo 3829, &amp; <lb></lb>fit 26961/12566, multiplico 12566 per 9, quia proportio unius ad alterum <lb></lb>eſt fermè, ut 9 ad 4, &amp; fit 113094, multiplico 4 in 28961 fit 115844, <lb></lb>hoc igitur eſt maius, igitur proportio 28961 ad 12566 eſt maior <lb></lb>quàm 9 ad 4, detraho igitur 12566 ex 28961, relinquitur 16395, de­<lb></lb>traho 113094 ex 115844, relinquitur 2750, diuido 2750 per 16395 <lb></lb>exit 55/328 addo 2 denominatori fit 55/330, quod eſt 1/6, nam iſtæ additiones <lb></lb>paruæ præter quòd parum uariant quantitatem etiam dum ad ex­<lb></lb>amen reducuntur, nihil impediunt, detrahe igitur 1/6 à 9/4, &amp; ducendo <lb></lb>per 6, &amp; detrahendo 53/23, duco igitur primos numeros ſcilicet 28961/12566 <lb></lb>mutuo in 53/23, fiunt 665998, &amp; 666107, ita uides, quod proportio <lb></lb>53 ad 23 eſt paulo minor, quàm 28961 ad 12566, &amp; æquiualent 27/23<lb></lb>&amp; 2 3829/12566.</s>
        </p>
        <p type="margin">
          <s id="id002428"><margin.target id="marg470"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id002429">Propoſitio centeſima quadrageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id002430">Denominationum incrementa ex extrema cognita inuenire, &amp; <lb></lb>conuerſo modo.</s>
        </p>
        <p type="main">
          <s id="id002431"><expan abbr="Quidã">Quidam</expan> per uſuram <expan abbr="rediuiuã">rediuiuam</expan> fecit 40000 coronatos ex 40 in 40 <lb></lb><arrow.to.target n="marg471"></arrow.to.target><lb></lb>annis. </s>
          <s id="id002432">Quęro <expan abbr="qutãa">qutana</expan> fuerit uſura, &amp; <expan abbr="quãdo">quando</expan> habuit 1000 coronatos, <lb></lb><expan abbr="quidã">quidam</expan> uellent ſoluere per regulam trium quantitatum, in qua com­<lb></lb>mitterentur maximi errores. </s>
          <s id="id002433">Et in ea multi ſunt modi, &amp; omnes fal­<lb></lb>ſi præter hanc uiam nulla eſt uera, adde quòd uellent multi per ſor­<lb></lb>tem inuentam ſoluere augendo per ſingulos annos, quod adeò <lb></lb>difficile eſſet, &amp; penè foret impoſsibile. </s>
          <s id="id002434">Ideò diuides 40000 per 40 <lb></lb>numerum ſortis exit 1000, igitur in 40 annis unum fit mille, ſunt <lb></lb>ergo 40 denominationes ab uno, quarum quadrageſima eſt 1000, <lb></lb>igitur uigeſima eſt &lt;02&gt; 1000 |ſcilicet |31 3913/6283, igitur decima eſt &lt;02&gt; eius <lb></lb><arrow.to.target n="marg472"></arrow.to.target><lb></lb>5 3917/12566 huius radix, erit quinta quantitas 2 7/23, cuius &lt;02&gt; relata prima, <lb></lb><arrow.to.target n="table18"></arrow.to.target><lb></lb>erit proportio 1 13/70, cuius quadratum eſt 1 1889/4900 ſeu <lb></lb>1 67/165 pro ſecunda quantitate, duces ergo primam, <lb></lb><figure id="id.015.01.156.1.jpg" xlink:href="015/01/156/1.jpg"></figure>quæ eſt 83/70 in quintam, quæ eſt reducta ad mino­<lb></lb>res fractiones facilitatis cauſa 53/23, &amp; habebis ſex­<lb></lb>tam quantitatem 2 118/161, duco etiam quintam quan­<lb></lb>titatem ſcilicet 53/23 in ſecundam quæ eſt 232/165, &amp; fit ſe­<lb></lb>ptimi anni quantitas, duco igitur ſeptem anno­<lb></lb>rum numerum, qui eſt 3 14/61 in 31 38/61 fit 102 992/6283. At in <lb></lb>ſex annis additis ad uiginti, fit tanto minus, quan­<lb></lb>to 31 38/61 ductum in differentiam ſeptem, &amp; ſex an­<lb></lb>norum quæ eſt 60/121, fit ergo 15 35/492. Quia ergo an­


<pb pagenum="138" xlink:href="015/01/157.jpg"></pb>nuatim ſolum uſura adijcitur ſorti, ſufficiet diuidere 2 992/6283 per 15 35/492 <lb></lb>ſcilicet multiplicando per 12 numerum menſium 2 992/6283 fit 25 5621/6283 di­<lb></lb>uide 25 5621/6283 per 15 35/492, exit menſis unus, &amp; dies 21, detrahe ex 27 an­<lb></lb>nis, remanent anni 26, menſes 10, dies 9, in quo tempore habuit <lb></lb>4000 aureos coronatos. </s>
          <s id="id002435">Vſura autem fuit ut uiſum 13/70, igitur per re­<lb></lb>gulam trium duc 13 in 100 fit 1300, diuide 1300 per 70 exit 18 4/7 &amp; <lb></lb>tanta fuit pro centum. </s>
          <s id="id002436">Et cum computaueris in tribus annis, acqui­<lb></lb>rit modico plus beſſe eius, quod habet. </s>
          <s id="id002437">Et ita in 13 annis, &amp; parua <lb></lb>illa parte perueniet ad decuplum eius, quod habet, ſcilicet 4000 au <lb></lb>reorum, &amp; habebit aureos 40000, ut propoſitum eſt.</s>
        </p>
        <p type="margin">
          <s id="id002438"><margin.target id="marg471"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002439"><margin.target id="marg472"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 136. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <table>
          <table.target id="table18"></table.target>
          <row>
            <cell>Anni</cell>
            <cell>Aurei</cell>
          </row>
          <row>
            <cell>1</cell>
            <cell>1 13/70</cell>
          </row>
          <row>
            <cell>2</cell>
            <cell>1 67/165</cell>
          </row>
          <row>
            <cell>5</cell>
            <cell>2 7/23</cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>2 118/161</cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>3 14/61</cell>
          </row>
          <row>
            <cell>10</cell>
            <cell>5 3917/12566</cell>
          </row>
          <row>
            <cell>20</cell>
            <cell>31 38/61</cell>
          </row>
          <row>
            <cell>40</cell>
            <cell>1000</cell>
          </row>
        </table>
        <p type="head">
          <s id="id002440">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002441">In propoſita proportione numero que terminorum rediuiuam u­<lb></lb>ſuram inuenire.</s>
        </p>
        <p type="main">
          <s id="id002442">Sit gratia exempli, in ſex annis uſura rediuiua uigeſimæ, erit­<lb></lb>qúe proportio 21/20, cuius numeratorem ſexies ducam in ſe primum <lb></lb>bis fit 441: ergo ducto 441 in ſe fit qúe 194481 ductum in 441 <lb></lb>fit 85766121 ſexies ductum 21, quinquies autem ducam 20 deno­<lb></lb><figure id="id.015.01.157.1.jpg" xlink:href="015/01/157/1.jpg"></figure><lb></lb>minatorem in ſe fit bis 400, ter 8000, <lb></lb>quinquies ergo 3200000, diuide nume­<lb></lb>ratorem per denominatorem abiectis <lb></lb>quinque notis erit 26 2566121/3200000. Quæ propor<lb></lb>tio eſt proxima 26 4/5 ad 20, &amp; ita ut 134 ad <lb></lb>100. Et ſi pigeret tædij aut laboris poſſes <lb></lb>pro xij annis, ducere 134 in ſe, &amp; fit 17956 <lb></lb>diuide per 100 eadem ratione, exit 179 14/25 <lb></lb>&amp; ita 100 in xij annis, fit tantundem. </s>
          <s id="id002443">Et <lb></lb>ita pro xviij &amp; xx annis.</s>
        </p>
        <p type="main">
          <s id="id002444">Propoſitio centeſima quadrageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id002445">Si linea in duas partes diuidatur, corpora, quæ fiunt ex una par­<lb></lb>te in alterius quadratum mutuò æqualia ſunt corpori, quod fit ex <lb></lb>tota linea in ſuperficiem unius partis in alteram.<lb></lb><arrow.to.target n="marg473"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002446"><margin.target id="marg473"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002447">Sit a c diuiſa in a b, b c quadratum a b ſit <lb></lb><figure id="id.015.01.157.2.jpg" xlink:href="015/01/157/2.jpg"></figure><lb></lb>a d, <expan abbr="quadratũ">quadratum</expan> b c, ſit b e <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb></lb>ex a b in b e, a f dico quòd corpora ex a b in <lb></lb>b e, &amp; b c in a d æqualia ſunt corpori ex a c <lb></lb>in a f. </s>
          <s id="id002448">Quia enim corpus ex a c in a f conſtat <lb></lb>ex a b in a f, &amp; b c in a f, per primam ſecun­</s>
        </p>
        <p type="main">
          <s id="id002449"><arrow.to.target n="marg474"></arrow.to.target><lb></lb>di Elementorum. </s>
          <s id="id002450">corpus autem ex a b in a f <lb></lb>eſt æquale corpori ex b c in a d, &amp; corpus <lb></lb>ex b c in a f eſt æquale corpori ex a b in b c <lb></lb>igitur conſtat propoſitum.</s>
        </p>
        <pb pagenum="140 [=139]" xlink:href="015/01/158.jpg"></pb>
        <p type="margin">
          <s id="id002451"><margin.target id="marg474"></margin.target>I<emph type="italics"></emph>d eſt per <lb></lb>eius demon­<lb></lb>ſtrationem.<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 29. <emph type="italics"></emph>un <lb></lb>decimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002452">Propoſitio centeſima quadrageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id002453">Duplum cubi medietatis maius eſt aggregato corporum mutu­<lb></lb>orum cuiuslibet diuiſionis, quantum eſt, quod fit ex tota in quadra <lb></lb>tum differentiæ.<lb></lb><arrow.to.target n="marg475"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002454"><margin.target id="marg475"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002455">Sit a b diuiſa per æqualia in c, &amp; per inæqua­<lb></lb>lia in d, dico, quòd duplum cubi a c eſt maius ag<lb></lb><figure id="id.015.01.158.1.jpg" xlink:href="015/01/158/1.jpg"></figure><lb></lb>gregato corporum ex a d in quadratum b d, &amp; b d in quadratum <lb></lb>a cin eo quod fit ex a b in quadratum c d, nam per <expan abbr="præcedentẽ">præcedentem</expan> du­<lb></lb>plum cubi a c eſt æquale corpori ex a b in quadratum a c: aggrega­<lb></lb>tum quo que corporum ex a d in quadratum b d, &amp; b d in quadra­<lb></lb>tum a d eſt ęquale ei, quod fit ex a b in <expan abbr="rectangulũ">rectangulum</expan> ex a d in d b. </s>
          <s id="id002456"><expan abbr="qua­dratũ">qua­<lb></lb>dratum</expan> <expan abbr="autẽ">autem</expan> a c eſt maius rectangulo a d in d b quadrato c d differen<lb></lb>tiæ, igitur duplum cubi a c excedit aggregatum <expan abbr="corporũ">corporum</expan> <expan abbr="mutuorũ">mutuorum</expan> <lb></lb>in corpore ex a b in quadratum c d differentię, quod eſt <expan abbr="propoſitũ">propoſitum</expan>.</s>
        </p>
        <p type="main">
          <s id="id002457"><arrow.to.target n="marg476"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002458"><margin.target id="marg476"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002459">Propoſitio centeſima quadrageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id002460">Si line a in duas partes diuidatur quadrata ambarum partium <lb></lb>detracto eo quod fit ex una parte in alteram, ęqualia ſunt producto <lb></lb>unius in alteram cum quadrato differentiæ.</s>
        </p>
        <p type="main">
          <s id="id002461"><arrow.to.target n="marg477"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002462"><margin.target id="marg477"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002463">Sit linea a c diuiſa in b, &amp; ſit differentia a b, <lb></lb>b c, b d, dico quod quadrata a b &amp; b c detracto <lb></lb><figure id="id.015.01.158.2.jpg" xlink:href="015/01/158/2.jpg"></figure><lb></lb>eo quod fit ex a b in b c, æqualia ſunt producto a b in b c cum qua­<lb></lb>drato b d. </s>
          <s id="id002464">Quoniam. </s>
          <s id="id002465">n. </s>
          <s id="id002466">quadrata a b, b c æqualia quadratis a d d b <lb></lb>b c &amp; productis ex a d in d b bis &amp; quod fit ex a b in b c æquale eſt <lb></lb>ei quod fit ex a d in ſe cum eo quod fit ex a d in d b, quia a d eſt ęqua </s>
        </p>
        <p type="main">
          <s id="id002467"><arrow.to.target n="marg478"></arrow.to.target><lb></lb>lis b c ideo quadrata a b &amp; b c detracto eo quod fit ex a b in b c ſunt <lb></lb>æqualia quadratis a d d b, &amp; producto a d in d b ſemel: a c quadra­<lb></lb><arrow.to.target n="marg479"></arrow.to.target><lb></lb>tum a d cum producto a d in d b eſt æquale producto a b in a d, &amp; <lb></lb>ex conſequenti in b c, igitur reſiduum quadratorum a b &amp; b c de­<lb></lb>tracto producti a b in b c eſt æquale a b in b c cum quadrato b d <lb></lb>quod fuit propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002468"><margin.target id="marg478"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002469"><margin.target id="marg479"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002470">Propoſitio centeſima quadrageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id002471">Corpus quod fit ex linea diuiſa in ſuperficiem ęqualem quadra­<lb></lb>tis ambarum partium detracta ſuperficie unius partis in <expan abbr="alterã">alteram</expan>, eſt <lb></lb>æquale aggregato cuborum <expan abbr="ambarũ">ambarum</expan> <expan abbr="partiũ">partium</expan>.</s>
        </p>
        <figure id="id.015.01.158.3.jpg" xlink:href="015/01/158/3.jpg"></figure>
        <p type="main">
          <s id="id002472">Sic a b diuiſa in e quadrata partium e f &amp; <lb></lb><arrow.to.target n="marg480"></arrow.to.target><lb></lb>b d detrahatur ex e f, f g æqualis a d, dico cor<lb></lb>pus ex a b in ſuperficies b d, d g æquale eſ­<lb></lb>ſe cubis a c &amp; c b pariter acceptis, quia. </s>
          <s id="id002473">n. <lb></lb></s>
          <s id="id002474">ex a b in b d fiunt duo corpora cubus <lb></lb>b d &amp; corpus ex a d in quadratum d b hoc <lb></lb>autem eſt æquale corpori ex b cin a d quia 


<pb pagenum="140" xlink:href="015/01/159.jpg"></pb>fíunt ex æqualibus lineis: at corpus quod fit ex a b in d g æquale eſt <lb></lb>corporibus quæ fiunt ex a c, c b in ſuperficiem d g at cubus a c con­<lb></lb>tinet duo corpora quę fiunt &amp; a c in d g &amp; g f, igitur cubus a c ſupe­<lb></lb>rat productum ex a b in d g in producto ex a c in f g &amp; ſuperatur ab <lb></lb>eo in producto ex b c in d g, ſuperabatur etiam, ut uiſum eſt, cubus <lb></lb>b c à producto b a in d b in producto b cin c f, igitur cubi a c c b ſu­<lb></lb>perantur à producto a b in ad in producto b c in c f &amp; in d g, quare <lb></lb>in producto b c in f e: ſi quidem f e &amp; f g ſunt æqualia ex ſuppoſito <lb></lb>ſuperant autem in producto ex c b in e f, igitur tantum eſt in in quo <lb></lb>ſuperantur quantum eſt id in quo ſuperant: ergo ſunt æqualia.</s>
        </p>
        <p type="margin">
          <s id="id002475"><margin.target id="marg480"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002476">Propoſitio centeſima quadrageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id002477">Propoſita linea diuiſa duas ei lineas adijcere, ut proportio addita­<lb></lb>rum ſingularum &amp; partium ſimul iunctarum ad additas ſit mutua.<lb></lb><arrow.to.target n="marg481"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002478"><margin.target id="marg481"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002479">Sit linea a b diuiſa in c uolo eius <lb></lb><figure id="id.015.01.159.1.jpg" xlink:href="015/01/159/1.jpg"></figure><lb></lb>partibus addere lineas, ut propoſi­</s>
        </p>
        <p type="main">
          <s id="id002480"><arrow.to.target n="marg482"></arrow.to.target><lb></lb>tum eſt, ſtatuo mediam c d inter a e &amp; <lb></lb><arrow.to.target n="marg483"></arrow.to.target><lb></lb>c b quæ ſit c d, &amp; facio ut c d ad c a ita <lb></lb>c a ad a e, &amp; ut d c ad c b ita c b ad b f, quia ergo d e media eſt inter <lb></lb><arrow.to.target n="marg484"></arrow.to.target><lb></lb>a c &amp; c b, &amp; ut ea ad a cita d c a c b ad c f erunt omnes in continua <lb></lb><arrow.to.target n="marg485"></arrow.to.target><lb></lb>proportione, quare proportio e c ad c a ut c f ad b f &amp; e c ad ea ut <lb></lb>c f ad c b quod eſt propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002481"><margin.target id="marg482"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002482"><margin.target id="marg483"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002483"><margin.target id="marg484"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002484"><margin.target id="marg485"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002485">Propoſitio centeſima quadrageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id002486">Propoſitis tribus lineis primam ſic diuidere, ut adiectis duabus <lb></lb>alijs lineis ſecundum rationem mutuam ſingularum ſingulis ag­<lb></lb>gregatum ex una adiectarum &amp; parte ad aggregatum ex alia parte <lb></lb>&amp; adiecta ſe habeat, ut ſecunda ad tertiam.<lb></lb><arrow.to.target n="marg486"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002487"><margin.target id="marg486"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002488">Sit a, b, c, d, propoſitæ lineę, <lb></lb><figure id="id.015.01.159.2.jpg" xlink:href="015/01/159/2.jpg"></figure><lb></lb>uolo diuidere a b ita in e ut <lb></lb>ſumpta ſecundum proportio­<lb></lb>nem alicuius quantitatis, puta <lb></lb>g ad a e ſic b f ad e b &amp; ut g ad <lb></lb>e b ſic g a ad a e ut ſit propor­<lb></lb>tio g e ad e f ut c ad d. </s>
          <s id="id002489">Sint ergo <lb></lb>omnia <expan abbr="cõſtituta">conſtituta</expan> &amp; ſit g rectan­<lb></lb>gulum ex a e in e b, cum ergo <lb></lb>g a contineat a e ut g continet e b, g autem continet e b ſecundum <lb></lb>a e, igitur g a continet a e ſecundum a c, ergo ex diffinitione qua­</s>
        </p>
        <p type="main">
          <s id="id002490"><arrow.to.target n="marg487"></arrow.to.target><lb></lb>drati a g eſt quadratum a e. </s>
          <s id="id002491">Pari ratione b f eſt quadratum b e. </s>
          <s id="id002492">pro­<lb></lb>portio igitur g e ad e f cum ſit ut c ad e ex ſuppoſito erit ut ipſi pro­<lb></lb>portioni addamus, &amp; detrahamus ex duplo a b &amp; dimidium reſi­<lb></lb>dui ducamus in ſe, &amp; addamus aggregato quadrati a b cum ipſa 


<pb pagenum="141" xlink:href="015/01/160.jpg"></pb>a b, &amp; latus eius detracto dimidio reſidui erit b c linea, quare diui­<lb></lb>ſio nota, &amp; eſt ut dicamus : uolo diuidere datam lineam, ut quantita­<lb></lb>tes adiectæ ſub mutua proportione ad unam tertiam cum parti­<lb></lb>bus obtineant inter ſe proportionem datam.</s>
        </p>
        <p type="margin">
          <s id="id002493"><margin.target id="marg487"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>ſecun<lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002494">Propoſitio centeſima quadrageſima nona.</s>
        </p>
        <p type="main">
          <s id="id002495">Datam lineam ſic diuidere, ut proportio quadratorum ad du­<lb></lb>plum unius partis in alteram ſit, ut lineę datæ ad lineam datam.</s>
        </p>
        <p type="main">
          <s id="id002496">Sit data a b quam uolo diuidere, ut proponitur ſub proportio­<lb></lb><arrow.to.target n="marg488"></arrow.to.target><lb></lb>ne c d ad e, diuido a b bifariam in f, &amp; abſcindo <lb></lb><figure id="id.015.01.160.1.jpg" xlink:href="015/01/160/1.jpg"></figure><lb></lb>g d æqualem d e, &amp; inter c g <expan abbr="reſiduũ">reſiduum</expan> &amp; c e inter­<lb></lb>pono proportione, &amp; ut h ad c g ita a f medietatis a b ad fk. </s>
          <s id="id002497">Omnia <lb></lb>iſta ſunt notiſsima ex primo &amp; ſexto Elemento­<lb></lb><figure id="id.015.01.160.2.jpg" xlink:href="015/01/160/2.jpg"></figure><lb></lb><expan abbr="rũ">rum</expan> Euclidis. </s>
          <s id="id002498">Si ergo abſcindantur fk ex fa, dico <lb></lb>quod proportio quadratorum l k &amp; k a ad du­<lb></lb>plum rectanguli a k in k b eſt ut c d ad d e. </s>
          <s id="id002499">Quia. n. </s>
          <s id="id002500">c e ad c g dupli­<lb></lb>cata eſt ei quę eſt h ad c g, duplicata eſt <expan abbr="etiã">etiam</expan> ei quæ eſt f a ad fk, qua­<lb></lb>re ut quadrati a f ad fk, ita c e ad c g, igitur diſiungendo c g ad g e ut <lb></lb>reſidui quadrati k f ad reſiduum quadrati a f, quare c g ad g d ut <lb></lb>quadrati k f ad dimidium reſidui quadrati a f, igitur coniunctim c d <lb></lb>ad d g ut quadrati k f &amp; dimidij reſidui quadrati a f ad ipſum dimi­<lb></lb>dium reſidui. </s>
          <s id="id002501">At uerò cum g d ſit æqualis d e, erit c d ad d e ut qua­<lb></lb>drati k f cum dimidio reſidui ſæpius dicti ad ipſum dimidium reſi­<lb></lb>dui. </s>
          <s id="id002502">Igitur etiam ut dupli quadrati k f cum reſiduo ad <expan abbr="reſiduũ">reſiduum</expan>, ſunt <lb></lb>enim omnia duplicata. </s>
          <s id="id002503">At <expan abbr="duplũ">duplum</expan> quadrati k f <expan abbr="cũ">cum</expan> reſiduo eſt æqua­<lb></lb>le quadratis a f &amp; f k, igitur quadratorum a f &amp; f k ad differentiam <lb></lb>eo rum proportio eſt ut c d ad d e, igitur dupli quadratorum a f &amp; <lb></lb>f k ad duplum differentiæ quadratorum a f &amp; fk ut c d ad d e. </s>
          <s id="id002504">Ve­<lb></lb><arrow.to.target n="marg489"></arrow.to.target><lb></lb>rum duplum quadratorum a f &amp; f k æquatur quadratis b k &amp; k a. <lb></lb><arrow.to.target n="marg490"></arrow.to.target><lb></lb>Et duplum differentiæ quadratorum a f &amp; fk eſt ęquale duplo pro <lb></lb>ducti b k in k a, igitur proportio quadratorum k b &amp; k a ad <expan abbr="duplũ">duplum</expan> <lb></lb>producti k b in k a eſt ueluti c d ad d e, quod eſt propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002505"><margin.target id="marg488"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002506"><margin.target id="marg489"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002507"><margin.target id="marg490"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002508">Propoſitio centeſima quinquageſima.</s>
        </p>
        <p type="main">
          <s id="id002509">Propoſitis duabus lineis <expan abbr="lineã">lineam</expan> communem <lb></lb><figure id="id.015.01.160.3.jpg" xlink:href="015/01/160/3.jpg"></figure><lb></lb>utrique adiungere, ut ſit maioris ad additam pro­<lb></lb>portio, uelut quadratorum minoris &amp; adiectæ <lb></lb>ad duplum unius in alteram.</s>
        </p>
        <p type="main">
          <s id="id002510">Hæc eſt quaſi conuerſa <expan abbr="præcedẽtis">præcedentis</expan>. </s>
          <s id="id002511">Sit a ma­<lb></lb><arrow.to.target n="marg491"></arrow.to.target><lb></lb>ior, &amp; b c minor, &amp; fiat b d dupla b c, ſuper <expan abbr="quã">quam</expan> <lb></lb>erigatur b f æqualis a; &amp; ſit rectangulum d f &amp; <lb></lb>deſcribatur quadratum b c quod ſit b g reſiduę <lb></lb>ſuperficiei ad d f latus ſit h, dico h eſſe lineam quæſitam. </s>
          <s id="id002512">Superficies 


<pb pagenum="142" xlink:href="015/01/161.jpg"></pb>enim d f cum fiat ex a in duplum b c, dupla erit ſuperficiei a in b c, ſu<lb></lb>perficies f d, tota æquatur quadratis h &amp; b c, igitur quadrata h &amp; b <lb></lb>c dupla ſunt ſuperficiei a in b c, quod uerò fit ex a in duplum b c ſe <lb></lb>habet ad id quod fit ex h in duplum b c, ut a ad h, cum per eandem <lb></lb>lineam ducantur, igitur quod fit ex a in duplum b c, &amp; ſunt quadra­<lb></lb>ta h &amp; b c, ſe habent ad duplum h in b c, ut a ad h, quod fuit de­<lb></lb>monſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id002513"><margin.target id="marg491"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002514">Propoſitio centeſima quinquageſima prima.</s>
        </p>
        <p type="main">
          <s id="id002515">Proportio differentiæ quadratorum partium, cuiuſuis lineæ ad <lb></lb>quadratum differentiæ <expan abbr="illarũ">illarum</expan> eſt uelut totius lineę ad differentiam.<lb></lb><arrow.to.target n="marg492"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002516"><margin.target id="marg492"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002517">Sit a b diuiſa in puncto c, &amp; fiat c d æqualis <lb></lb>c b, manifeſtum eſt quod differentia partium <lb></lb><figure id="id.015.01.161.1.jpg" xlink:href="015/01/161/1.jpg"></figure><lb></lb>eſt a d, dico proportionem differentiæ quadra <lb></lb>torum a c &amp; c b ad quadratum a d differentiæ partium eſſe ut a b ad </s>
        </p>
        <p type="main">
          <s id="id002518"><arrow.to.target n="marg493"></arrow.to.target><lb></lb>a d. </s>
          <s id="id002519">Quoniam differentia quadratorum a c &amp; c b eſt, quod fit ex a d <lb></lb>in d c bis cum quadrato a d, &amp; ideò quod fit ex a d in d b cum qua­<lb></lb>drato a d, &amp; ideò quod fit ex tota a b in a d. </s>
          <s id="id002520">Igitur differentia qua­<lb></lb><arrow.to.target n="marg494"></arrow.to.target><lb></lb>drato a c &amp; c b eſt quod fit ex a b in a d, quare cum quadratum a d <lb></lb>fiat ex a d in a d, erit proportio a b ad a d, uelut differentiæ quadra­<lb></lb><arrow.to.target n="marg495"></arrow.to.target><lb></lb>torum a c &amp; b c ad quadratum a d differentiæ partium. </s>
          <s id="id002521">Quod fuit <lb></lb>propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002522"><margin.target id="marg493"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002523"><margin.target id="marg494"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002524"><margin.target id="marg495"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002525">Propoſitio centeſima quinquageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue­<lb></lb>ritque proportio aggregati ex maiore &amp; dimidio ad ipſam maiorem <lb></lb>uelut ex minore, &amp; aliqua linea ad ipſam minorem, &amp; rurſus aggre­<lb></lb>gati ex minore dimidio ad ipſam minorem, uelut aggregati ex ma­<lb></lb>iore &amp; alia addita ad ipſam maiorem, erit proportio dimidij ad par<lb></lb>tem unam inæqualem, uelut alterius partis inæqualis ad ſuam ad­<lb></lb>ditam mutuò, &amp; etiam proportio additarum inuicem, uelut pro­<lb></lb>portio partium inæqualium duplicata, &amp; rurſus ipſum dimidium <lb></lb>lineæ aſſumptæ medium erit proportione inter additas. </s>
          <s id="id002527">Demum <lb></lb>proportio dimidij cum ad dita maiore ad dimidium cum addita mi<lb></lb>nore, uelut maioris partis ad minorem.<lb></lb><arrow.to.target n="marg496"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002528"><margin.target id="marg496"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002529">Sit propoſita a b diuiſa per <lb></lb><figure id="id.015.01.161.2.jpg" xlink:href="015/01/161/2.jpg"></figure><lb></lb>æqualia in c per inæqualia in <lb></lb>d, &amp; ſit ut addantur a g &amp; b f, <lb></lb>ita ut proportio c a, &amp; a d ad a d ſit ueluti f d ad d b, &amp; c b &amp; b d ad <lb></lb>b d, uelut g d ad d a, &amp; hæc eſt quarta <expan abbr="ſecũdi">ſecundi</expan> Archimedis de ſphęra, <lb></lb>&amp; Cylindro: quia ergo a c &amp; a d ad a d, ut f d ad d b erit a c ad a d, <lb></lb>fb ad b d. </s>
          <s id="id002530">Et ſimiliter quia eſt c b &amp; b d ad b d, uelut g d ad d a erit 


<pb pagenum="143" xlink:href="015/01/162.jpg"></pb>c b ad b d, uelut g a ad a d, &amp; hoc eſt primum. </s>
          <s id="id002531">Quia ergo c a eſt æ­<lb></lb>qualis c b, erit c a ad b d, uelut g a ad a d, &amp; iam fuit a d ad c a, ut b d <lb></lb>ad f b, per conuerſam igitur a d ad b d, ut g a ad a d, &amp; ut b d ad fb, <lb></lb>interpoſitis ergo a d &amp; d b inter a g &amp; b f cum compoſita ſit pro­<lb></lb>portio a g ad b f ex proportione a g ad a d, &amp; ad d b, &amp; d b <lb></lb>ad b f, &amp; proportio a d ad d b, ſit æqualis proportioni <lb></lb><figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg"></figure><lb></lb>a g ad a d, &amp; d b ad b f, igitur proportio a g ad b f. </s>
          <s id="id002532">Per de­<lb></lb>monſtrata ab Alchindo eſt duplicata proportioni a d ad <lb></lb>d b quod eſt ſecundum. </s>
          <s id="id002533">Rurſus quia ex primo demon­<lb></lb>ſtrato, uel eius conuerſo proportio a d ad a c eſt uelut b d <lb></lb>ad b f, &amp; d b ad a c, ut a d ad a g, proportiones ergo <lb></lb><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"></figure><lb></lb>a d &amp; d b ad a c componunt proportionem produ­<lb></lb>cti a d in d b, quod ſit h ad quadratum a c quod ſit <lb></lb>k, &amp; ſimiliter proportio b d ad b f &amp; a d ad a g com­<lb></lb>ponunt proportionem producti ex b d in a d, quod <lb></lb>ſit l ad productum b f in a g, quod ſit m, per demonſtrata ab Eucli­<lb></lb>de in ſexto Elementorum, igitur proportio h ad k ut l ad m, ſed h &amp; </s>
        </p>
        <p type="main">
          <s id="id002534"><arrow.to.target n="marg497"></arrow.to.target><lb></lb>l ſunt æquales, quia producuntur ex eiſdem, igitur per demonſtra­<lb></lb>ta in quinto Elementorum Euclidis, k eſt æquale m, ergo a c eſt me­<lb></lb>dia pro portione inter b f &amp; g a, quod eſt tertium. </s>
          <s id="id002535">Quia uerò ex pri­<lb></lb>mo demonſtrato eſt fb ad b d, ut a c ad a d, &amp; c b ad idem b d, ut g a <lb></lb>ad idem a d erit coniungendo fb &amp; b c ad b d, ut coniun­<lb></lb><figure id="id.015.01.162.3.jpg" xlink:href="015/01/162/3.jpg"></figure><lb></lb>gendo g a &amp; a c ad a d, ſed fb &amp; b c componunt f c &amp; g a, <lb></lb>&amp; a c componunt g c, igitur ut f c ad b d, ita g c ad a d, er­<lb></lb>go permutando g c ad f c, ut a d ad b d, quod eſt quartum.</s>
        </p>
        <p type="margin">
          <s id="id002536"><margin.target id="marg497"></margin.target>I<emph type="italics"></emph>n<emph.end type="italics"></emph.end> P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 23 <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 9.</s>
        </p>
        <p type="main">
          <s id="id002537">Cum ergo punctum d fuerit datum, licet inuenire a g &amp; b f, faci­<lb></lb>lè, ut Archimedes præſupponit proportionem g d ad d f datam &amp; <lb></lb>quærit eam, quæ eſt a d ad d b, &amp; peruenitur ad res numero triplo <lb></lb>quadrati dimidij lineæ aſſumptæ æquales cubo &amp; numero, qui ſit <lb></lb>ex duplo cubi dimidij in 1 m: ipſa proportione, &amp; quod produci­<lb></lb>tur diuiſo per 1 p: ipſa proportione. </s>
          <s id="id002538">Veluti poſita a b 10, &amp; propor­<lb></lb>tione quam uolo g d ad d f ſexcupla, duco 5 dimidium 10 in ſe fit 25, <lb></lb>&amp; triplico, fit 75 numerus rerum. </s>
          <s id="id002539">Inde duco 5 idem dimidium ad <lb></lb>cubum fit 125, duplico fit 250, duco in 5, qui eſt 1 m: proportione fit <lb></lb>1250, diuido per 7, qui eſt 1 p: proportione exit 178 4/7 numerus, qui <lb></lb>cum cubo æquatur 75 rebus. </s>
          <s id="id002540">Cum ergo conſtituta fuerit diuiſio in <lb></lb>c, non recipit proportionem g d ad f d quam uolueris, ſed ſequitur <lb></lb>una ſola ad <expan abbr="illã">illam</expan>, &amp; eſt mirabile, quoniam lineę uidentur ſumi liberè. <lb></lb></s>
          <s id="id002541">Sed non eſt ita. </s>
          <s id="id002542">Et <expan abbr="etiã">etiam</expan> quia Archimedes <expan abbr="uidet̃">uidetur</expan> aſſumere <expan abbr="aliã">aliam</expan> lineam, <lb></lb>ſed non inueſtigat eam, imò oſtendit eam ex aſſumptis. </s>
          <s id="id002543">At Eutoci­<lb></lb>us oſtendit ambas, <expan abbr="unã">unam</expan> ex propria inuentione, aliam ex Diocle, ſed 


<pb pagenum="144" xlink:href="015/01/163.jpg"></pb>una eſt ſuperflua, quia ut dixi, una ſequitur ad aliam. </s>
          <s id="id002544">Ex hoc pa­<lb></lb>tet cur Diocles aſſumpſerit lineam unam, quæ eſt a c, quæ ſe ha­<lb></lb>bet ad a d, &amp; d b, ut uiciſsim a d, &amp; d b ad additas, quod eſt pri­<lb></lb>mum demonſtratum. </s>
          <s id="id002545">Sic enim omittit primum quod proponit Ar<lb></lb>chimedes, &amp; aſſumit quod proximum eſt: &amp; ideò Archimedes non <lb></lb>probat, nec præſupponit, quod à Diocle probatur, ſcilicet datum <lb></lb>eſſe punctum d in linea a b, ſed ſolum in linea g f, ideò cogitur pro­<lb></lb>bare ſecundum quod demonſtratur ab Eutocio, &amp; à nobis demon <lb></lb>ſtratum eſt ſuprà. </s>
          <s id="id002546">Archimedes <expan abbr="aũt">aut</expan> aſſumit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb></lb>uocat b f, quæ eſt æqualis b c medietati: aliam aſſumit quam uocat <lb></lb>b h, cuius proportio ad b d eſt ſicut quadrati ad a d quadratum a b. <lb></lb></s>
          <s id="id002547">Conſtat ergo quod proportio g d ad d f eſt data. </s>
          <s id="id002548">Et ſimiliter f g ad <lb></lb>g d, &amp; eſt 1 præ proportione data. </s>
          <s id="id002549">Vnde notandum quod datum <lb></lb>dicitur, ſimpliciter cognitum alio modo, dicitur datum poſitione, <lb></lb>quod eſt certum &amp; tale, uelut ſi quis dicat, diuide 10 in duos nume­<lb></lb>ros quadratos: hoc non eſt datum, poteſt enim diuidi pluribus mo <lb></lb>dis. </s>
          <s id="id002550">At ſi dicas ut una pars ſit alterius <expan abbr="quadratũ">quadratum</expan>, iſtud antequàm ſci<lb></lb>untur partes, dicitur datum poſitione. </s>
          <s id="id002551">Ergo datum poſitione eſt du<lb></lb>plex, uel ut ratio nota ſit, non autem quantitas, ut ſi dicam a b eſt du<lb></lb>pla ad b c, utra que dicitur nota poſitione, quo­<lb></lb>niam neſcio quanta ſit a b. </s>
          <s id="id002552">Vel ſi quantitas eſt <lb></lb><figure id="id.015.01.163.1.jpg" xlink:href="015/01/163/1.jpg"></figure><lb></lb>nota proportio ignota ſit, ut ſi a c ſit 10, &amp; ſit, <lb></lb>ut b c ſit &lt;02&gt; relata, a b erit punctus b, &amp; proportio a b ad b c data po<lb></lb>ſitione, non tamen nota. </s>
          <s id="id002553">Et ſi dicas igitur omnia, quæ habent deter<lb></lb>minationem erunt data poſitione? </s>
          <s id="id002554">Dico quod non, quia oportet, <lb></lb>ut illa determinatio comprehendatur ſub una ratione, eaque ſaltem <lb></lb>generaliter cognita.</s>
        </p>
        <p type="main">
          <s id="id002555">Propoſitio centeſima quinquageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id002556">Vim quan cun que manus multiplicare.<lb></lb><arrow.to.target n="marg498"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002557"><margin.target id="marg498"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002558">Cum enim radimus aut trahimus manifeſtum eſt, </s>
        </p>
        <p type="main">
          <s id="id002559"><arrow.to.target n="marg499"></arrow.to.target><lb></lb>quod ambabus manibus uis conduplicatur, &amp; ma­<lb></lb><figure id="id.015.01.163.2.jpg" xlink:href="015/01/163/2.jpg"></figure><lb></lb>ior redditur, quanta eſt proportio totius ad exceſ­<lb></lb>ſum: uelut ſit a quod mouetur ab una manu uiribus <lb></lb>ut b, quæ ſunt exceſſus b d ſupra a, cum ergo propor<lb></lb>tio c b d ad a ſit compoſita ex proportionibus c &amp; <lb></lb>b d ad a manifeſtum eſt, quod erit producta ex pro­<lb></lb>portione c b d ad b d, &amp; b d ad a, ſed e b d eſt dupla <lb></lb>ad b d, quia e eſt æqualis, c igitur proportio c b d ad <lb></lb><arrow.to.target n="marg500"></arrow.to.target><lb></lb>a eſt maior multo quàm duorum exceſſuum, qui mo<lb></lb>uerent in proportione dupla: uelut ſi adderemus f 


<pb pagenum="145" xlink:href="015/01/164.jpg"></pb>ad d b æqualem b, multo maior eſt ex communi animi ſententia e f <lb></lb>b d <expan abbr="quã">quam</expan> f b d, quia e continet f, &amp; quantum eſt d inſuper: cum ergo <lb></lb>b cum d moueat a in proportione b d ad a &amp; f cum d mouebit a in <lb></lb>proportione eadem qua b d, ergo per uiam additionis duplo ue­<lb></lb>locius, quàm dupla proportione, uerùm dupla comparatione ad <lb></lb>proportionem b d ad a, non autem duplicata ſed dupla, ut dixi, quę <lb></lb>erit maior quàm dupla per <expan abbr="additionẽ">additionem</expan> exceſſus. </s>
          <s id="id002560">Ergo ſi addatur al­<lb></lb>ter homo, erit dupla ad illam duplam, ueluti addendo æqualem d b <lb></lb>f e, adeò ut ſi proportio d b f e eſſet quintupla, mouerent illi duo in <lb></lb>proportione decupla. </s>
          <s id="id002561">Sed annexo baculo aut lima aut ſerra annu­<lb></lb>lo h, ita ut circunuolui poſsit h æquabit uires non ſolum d b f e ſed <lb></lb>multorum hominum. </s>
          <s id="id002562">igitur multo plus aget homo ambabus ma­<lb></lb>nibus radendo aut ſecando cum g, quàm quadrupla proportione <lb></lb>unius manus, &amp; hoc incrementum eſt non ſolum magnæ <lb></lb>utilitatis, ſed ualde <expan abbr="accõmodatum">accommodatum</expan> in actionibus artificum <lb></lb>operum grauiorum. </s>
          <s id="id002563">Et huiuſmodi conduplicatio eſt ratio <lb></lb>limæ quam ſurdam uocamus.</s>
        </p>
        <p type="margin">
          <s id="id002564"><margin.target id="marg499"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 37.</s>
        </p>
        <p type="margin">
          <s id="id002565"><margin.target id="marg500"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2.</s>
        </p>
        <figure id="id.015.01.164.1.jpg" xlink:href="015/01/164/1.jpg"></figure>
        <p type="main">
          <s id="id002566">Propoſitio centeſima quadrageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id002567">Si lineę datę alia linea adiungatur, ab extremitatibus autem pri­<lb></lb>oris lineę duæ rectæ in unum punctum concurrant proportionem <lb></lb>habentes quam media inter totam &amp; adiectam, ad adiectam erit <lb></lb>punctus concurſus à puncto extremo lineæ adiectæ diſtans per li­<lb></lb>neam mediam. </s>
          <s id="id002568">Quòd ſi ab extremo alicuius lineæ æqualis mediæ <lb></lb>ſeu peripheria circuli cuius ſemidiameter ſit media linea duæ lineæ <lb></lb>ad prædicta puncta producantur, ipſę erunt in proportione medię <lb></lb>ad adiectam.<lb></lb><arrow.to.target n="marg501"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002569"><margin.target id="marg501"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id002570">Hęc propoſitio eſt admirabilis: &amp; etiam deſcripſi, ut multa ſecre­<lb></lb>ta Dialecticæ potius <expan abbr="aperirent̃">aperirentur</expan> quam quod huic propoſito <expan abbr="multũ">multum</expan> <lb></lb>congrueret. </s>
          <s id="id002571">Ideò potius ſcholij cauſa poſita eſt quam ipſius tracta­<lb></lb>tionis: ut <expan abbr="modũ">modum</expan> demonſtrandi magis quam id, q̊d <expan abbr="demonſtrat̃">demonſtratur</expan>, re­<lb></lb>ſpicere oporteat. </s>
          <s id="id002572"><expan abbr="Conſtituat̃">Conſtituatur</expan> ergo (per uiam problematis) linea a b <lb></lb>&amp; proportio c ad d, &amp; fiat d e ad c, ut c ad d, &amp; a b ad e ut b f ad d, &amp; <lb></lb>ut g ad c, eritque g media inter a f &amp; f b, quod licet ſolum ſupponatur <lb></lb>ab Appollonio, <expan abbr="tamẽ">tamen</expan> facilè demonſtratur &amp; à Commandino adie­<lb></lb>cta eſt <expan abbr="demõ">demon</expan>ſtratio. </s>
          <s id="id002573">Concurrant ergo ex a &amp; b duę lineę in aliquod </s>
        </p>
        <p type="main">
          <s id="id002574"><arrow.to.target n="marg502"></arrow.to.target><lb></lb>punctum, putat h ut ſit a h ad h b uelut c ad d, dico quod ſi ducat <lb></lb>h f quod ipſa erit æqualis g, ducatur b l æquidiſtans a h, &amp; quia <lb></lb><arrow.to.target n="marg503"></arrow.to.target><lb></lb>ex ſuppoſito a h ad h b, ut g ad b f, erit b h ad h a, ut b f ad g, &amp; quia <lb></lb>trianguli a h f &amp; b l f ſunt ſimiles erit proportio a h ad b l, ueluti a f <lb></lb><arrow.to.target n="marg504"></arrow.to.target><lb></lb>ad fb, igitur per ęquam proportionem b e h ad b l, ut a f ad g, ſed ut <lb></lb><arrow.to.target n="marg505"></arrow.to.target><lb></lb>a f ad g ita g ad b f ex ſuppoſito: &amp; ut a f ad g, it a h a ad h b, ex ſuppo 


<pb pagenum="146" xlink:href="015/01/165.jpg"></pb>ſito igitur ut a h ad h b ita h b ad b l, ſed angulus a h b eſt æqualis <lb></lb>angulo h b l, ergo triangulus a h b eſt <lb></lb>ſimilis triangulo h b l, quare angulus <lb></lb>b h l eſt ęqualis angulo h a f, igitur du <lb></lb>orum triangulorum f a h, &amp; fb h duo <lb></lb><arrow.to.target n="marg506"></arrow.to.target><lb></lb>anguli unius a &amp; f ſunt æquales duo­<lb></lb>bus angulis, alterius igitur propor­<lb></lb><figure id="id.015.01.165.1.jpg" xlink:href="015/01/165/1.jpg"></figure><lb></lb>tio a f ad fh reſpicientium angulos ę­<lb></lb><arrow.to.target n="marg507"></arrow.to.target><lb></lb>quales ut a h ad h b reſpicientium an­<lb></lb><arrow.to.target n="marg508"></arrow.to.target><lb></lb>gulum f, ſed a h ad h b ut c ad d, ex ſup <lb></lb>poſito igitur a f ad f h, ut c ad d, ſed ut c ad d ita a f ad g, ex ſuppoſito <lb></lb>ergo h f eſt æqualis g.<lb></lb><arrow.to.target n="marg509"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002575"><margin.target id="marg502"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 29. <emph type="italics"></emph>pri <lb></lb>mi, &amp;<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002576"><margin.target id="marg503"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 22. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002577"><margin.target id="marg504"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002578"><margin.target id="marg505"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002579"><margin.target id="marg506"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>pri<lb></lb>mi, &amp;<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002580"><margin.target id="marg507"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002581"><margin.target id="marg508"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002582"><margin.target id="marg509"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002583">Cum ergo hęc demonſtratio ſit ex ſenſu in uno puncto h, ideò ad <lb></lb>quælibet puncta traduci poteſt, quæ potero imaginari, &amp; ita pri­<lb></lb>ma uocabitur ſenſus, <expan abbr="ſecũda">ſecunda</expan> imaginandi: Et <expan abbr="quoniã">quoniam</expan> in demonſtran­<lb></lb>do non aſſumimus aliquid, quod ſit proprium alicui puncto, niſi <lb></lb>proportionem h a ad h b ſimilem eſſe c ad d, ideo hoc pertinet ad <lb></lb>intellectum, &amp; eſt tertium. </s>
          <s id="id002584">Et idem dico ſi k eſſet ultra h quod po­<lb></lb>teſt contingere. </s>
          <s id="id002585">modò k a ad k b ſit ut c ad d &amp; k f ſit ęqualis g idem <lb></lb>ſequetur, &amp; comprehenditur ſub tertio &amp; pertinet ad intellectum, <lb></lb>&amp; quoniam demonſtratur quod punctum k ubicunque ſumatur, eſt <lb></lb>in ęquali <expan abbr="diſtãtia">diſtantia</expan> à puncto f ſcilicet per g lineam, erit ſemper in peri­<lb></lb>pheria circuli, &amp; hoc poteſt eſſe in infinitis locis ſimpliciter &amp; extra <lb></lb>infinitum nihil eſt, igitur ſub hoc continetur conuerſum ſcilicet, <lb></lb>quod a quolibet puncto circuli ductis lineis ad a &amp; b ipſę erunt in <lb></lb>proportione c ad d. </s>
          <s id="id002586">Et ita abſque principijs Geometricis concluditur <lb></lb>propoſitio Geometrica &amp; hoc eſt <foreign lang="grc">περιλάμπουσιν</foreign> &amp; fermè ſummum in­<lb></lb>tellectus humani. </s>
          <s id="id002587">Et poteſt demonſtrari Geometricè duobus uer­<lb></lb>bis. </s>
          <s id="id002588">Quia. n. </s>
          <s id="id002589"><expan abbr="fſupponit̃">f ſupponitur</expan> æqualis g eo quòd h eſt in peripheria circu­<lb></lb>li erit media inter a f &amp; f b, quare cum angulus f ſit communis, erit <lb></lb>proportio a h ad h b, laterum reſpicientium angulum f in utroque </s>
        </p>
        <p type="main">
          <s id="id002590"><arrow.to.target n="marg510"></arrow.to.target><lb></lb>triangulo, uelut h f lateris in maiori ad f b latus in minori, quare <lb></lb><arrow.to.target n="marg511"></arrow.to.target><lb></lb>cum ex ſuppoſito h f ad fb ſit ut c ad d, erit a ad b, ut c ad d. </s>
          <s id="id002591">Et uides <lb></lb>Apollonium, &amp; Pappium quanta ſuperflua adijciant in hac ſecun­<lb></lb><arrow.to.target n="marg512"></arrow.to.target><lb></lb>da parte demonſtrationis, quæ eſt prima apud illos, &amp; ducunt <expan abbr="unã">unam</expan> <lb></lb>lineam non neceſſariam ex puncto b ad latus fh. </s>
          <s id="id002592">Vt <expan abbr="antiquorũ">antiquorum</expan> ple <lb></lb>rique non tantum potuerint Geometria &amp; ingenio, quæ ferunt excel<lb></lb>lentiſsima in illis, quantum nos ex Dialectica <foreign lang="grc">πε̣ριλάμπουσιν</foreign> inducen <lb></lb>tes. </s>
          <s id="id002593">eſt enim ſingulare hoc exemplum.<lb></lb><arrow.to.target n="marg513"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002594"><margin.target id="marg510"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002595"><margin.target id="marg511"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph><expan abbr="eiuſdẽ">eiuſdem</expan><emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002596"><margin.target id="marg512"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end><lb></lb>I<emph type="italics"></emph>n primo<emph.end type="italics"></emph.end> C<emph type="italics"></emph>o <lb></lb>nicor.<emph.end type="italics"></emph.end> A<emph type="italics"></emph>pol. <lb></lb>in<emph.end type="italics"></emph.end> P<emph type="italics"></emph>ræfat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002597"><margin.target id="marg513"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002598">Ex hoc <expan abbr="etiã">etiam</expan> patet quod ſi circulus duceretur ſecundum f k tran­<lb></lb>ſiretque per m &amp; n eſſet a m ad m b &amp; a n ad b n, ut a h ad h b.</s>
        </p>
        <pb pagenum="147" xlink:href="015/01/166.jpg"></pb>
        <p type="head">
          <s id="id002599">SCHOLIVM</s>
        </p>
        <p type="main">
          <s id="id002600">Ex hoc pater qualiter ex uera demonſtratione ſenſu oſtenſa per­<lb></lb>uenimus ad quotquot imaginando, inde intellectu abiectis condi­<lb></lb>tionibus non neceſſarijs facimus infinitum &amp; uniuerſale. </s>
          <s id="id002601">Demum <lb></lb>ſine artis ſpecialis auxilio oſtendimus theorema uniuerſale (quod <lb></lb>etiam poterat oſtendi Geometricè, ſed longè pulchrius eſt, ac ſubli­<lb></lb>mius per <foreign lang="grc">περιλαμπουσιν</foreign>, qua hoc ipſo infinita alia docemus generaliter <lb></lb>per ſimplicem <expan abbr="comprehẽſionem">comprehenſionem</expan> oſtendere) ſcilicet quod à quouis <lb></lb>puncto peripherię circuli, cuius ſemidiameter eſt media proportio­<lb></lb>ne inter totam extenſam à centro uſque exterius, &amp; partem quæ&#039; eſt à <lb></lb>centro ad punctum deſcriptum ſub proportione continua <expan abbr="datarũ">datarum</expan> <lb></lb>linearum lineæ ductæ ex eo ad punctum exterius, &amp; punctum de­<lb></lb>ſcriptum ſunt in proportione datarum linearum.</s>
        </p>
        <p type="main">
          <s id="id002602">Propoſitio centeſima quinquageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id002603"><expan abbr="Quadratorũ">Quadratorum</expan> <expan abbr="numerorũ">numerorum</expan> proportionem &amp; <expan abbr="inuentionẽ">inuentionem</expan> <expan abbr="cõſiderare">conſiderare</expan>.</s>
        </p>
        <figure id="id.015.01.166.1.jpg" xlink:href="015/01/166/1.jpg"></figure>
        <p type="main">
          <s id="id002604">Primùm oportet ſcire eſſe tres naturales <lb></lb>numerorum ſeries, primam Euclidis iuxta </s>
        </p>
        <p type="main">
          <s id="id002605"><arrow.to.target n="marg514"></arrow.to.target><lb></lb>quamuis <expan abbr="proportionẽ">proportionem</expan>, in qua unum &amp; ter­<lb></lb>tius &amp; quintus, &amp; ita uno ſemper intermiſ­<lb></lb>ſo ſunt quadrati. </s>
          <s id="id002606">Primus quo que. </s>
          <s id="id002607">1. unum &amp; <lb></lb>quartus &amp; ſeptimus &amp; ita duobus intermiſsis ſunt cubi. </s>
          <s id="id002608">In ſecun­<lb></lb>do ordine eſt naturalis ſeries numerorum, ex qua colligitur alia, &amp; <lb></lb>ex illa bini quilibet ſe ſequentes conſtituunt numerum <expan abbr="quadratũ">quadratum</expan>. <lb></lb></s>
          <s id="id002609">In tertia numeri impares, qui ſemper collati efficiunt quadratum.</s>
        </p>
        <p type="margin">
          <s id="id002610"><margin.target id="marg514"></margin.target>E<emph type="italics"></emph><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"></emph.end> 1.</s>
        </p>
        <figure id="id.015.01.166.2.jpg" xlink:href="015/01/166/2.jpg"></figure>
        <p type="main">
          <s id="id002611">Sit ergo propoſitus numerus cui uelim <lb></lb>addere quadratum numerum, ut fiat qua­<lb></lb><arrow.to.target n="marg515"></arrow.to.target><lb></lb>dratus totus, accipe numerum quadratum <lb></lb>minorem illo quem uis, &amp; detrahe à propo<lb></lb>ſito numero ſeu quadrato ſeu non reſidu­<lb></lb><arrow.to.target n="marg516"></arrow.to.target><lb></lb>um, diuide per duplum &lt;02&gt; quadrati quod <lb></lb>detraxiſti, q̊d exit duc in ſe fiet quadratus numerus, idem que additus <lb></lb>numero propoſito, faciet quadratum. </s>
          <s id="id002612">Velut capio 16 qui eſt qua­<lb></lb>dratus, aufero 9 quadratum <expan abbr="minorẽ">minorem</expan> relinquitur 7, diuido per 6 du­<lb></lb>plum &lt;02&gt; 9, exit 1 1/6 quadratum eius eſt 1 13/36 qui additus ad 16 facit 17 13/36 <lb></lb><expan abbr="quadratũ">quadratum</expan> cuius &lt;02&gt; eſt 4 1/6.</s>
        </p>
        <p type="margin">
          <s id="id002613"><margin.target id="marg515"></margin.target>E<emph type="italics"></emph><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"></emph.end> 2.</s>
        </p>
        <p type="margin">
          <s id="id002614"><margin.target id="marg516"></margin.target>E<emph type="italics"></emph><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"></emph.end> 3.</s>
        </p>
        <p type="main">
          <s id="id002615">Ex hoc patet propoſito quouis numero <expan abbr="q̃drato">quadrato</expan> modus inuenien­<lb></lb><arrow.to.target n="marg517"></arrow.to.target><lb></lb>di infinitos numeros quadratos qui <expan abbr="cũ">cum</expan> illo iuncti facient <expan abbr="quadratũ">quadratum</expan>.</s>
        </p>
        <p type="margin">
          <s id="id002616"><margin.target id="marg517"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="head">
          <s id="id002617">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002618">Poſſem adducere demonſtrationes omnium <expan abbr="horũ">horum</expan>, ſed reddere­<lb></lb>tur res longa <expan abbr="cũ">cum</expan> ſint manifeſtę ex ſeptimo octauo &amp; nono Euclidis. <lb></lb></s>
          <s id="id002619">Exemplum ſecundum capio modò 14 qui non eſt quadratus, aufe­<lb></lb>ro 9, remanet 5, diuido per 6 duplum &lt;02&gt; 9 exit 5/6 <expan abbr="quadratũ">quadratum</expan> eius eſt 25/36 


<pb pagenum="148" xlink:href="015/01/167.jpg"></pb>hic additus ad 14 conſtituit 14 25/36 quadratum 3 5/6. Et ita 14 eſt diffe­<lb></lb>rentia duorum quadratorum, ſcilicet 25/36 &amp; 14 25/36.<lb></lb><arrow.to.target n="marg518"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002620"><margin.target id="marg518"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002621">Ex hoc habebis duo quadrata in datis terminis quæ different <lb></lb>dato numero, &amp; eſt pulchrum. </s>
          <s id="id002622">Velut uolo duo quadrata quæ dif­<lb></lb>ferant in 2, &amp; &lt;02&gt; minoris ſit inter 1 &amp; 2, tunc capies per regulam i­<lb></lb>pſam 2, &amp; auferes <expan abbr="numerũ">numerum</expan> quadratum ita quòd reſiduum diuiſum <lb></lb>per duplum radicis efficiat <expan abbr="numerũ">numerum</expan> inter 1 &amp; 2. Veluti capio 4/9 qua­<lb></lb>dratum, aufero ex 2, relinquitur 1 5/9 diuido per duplum 2/13 radicis 4/9 &amp; <lb></lb>eſt 1 1/3 &amp; exit 1 1/6, &amp; hic eſt minor numerus cuius quadratum eſt 1 13/36 <lb></lb>cui ſi addantur 2, fient 3 13/36 numerus quadratus 1 5/6.</s>
        </p>
        <p type="main">
          <s id="id002623"><arrow.to.target n="marg519"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002624"><margin.target id="marg519"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id002625">Cum autem uolueris duo quadrata quæ differant in 100, tunc <lb></lb>per regulam datam ſi auferes 1, peruenires ad numeros magnos &amp; <lb></lb>fractos, &amp; ideo melius eſt quia numerus eſt par, ut detrahas nume­<lb></lb>rum parem quadratum, ita quod reſiduum poſsit diuidi per <expan abbr="duplũ">duplum</expan> <lb></lb>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb></lb>84 <expan abbr="reſiduũ">reſiduum</expan> non <expan abbr="põt">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb></lb><expan abbr="detrahã">detraham</expan> 4 &amp; <expan abbr="relinquet̃">relinquetur</expan> 96, diuido per <expan abbr="duplũ">duplum</expan> radicis quod eſt 4 exit <lb></lb>24, cuius quadratum qua eſt 576 addito 100 facit 676 <expan abbr="quadratũ">quadratum</expan> 26. <lb></lb>Et ita ex 433 non auferam ſed 9, quia relinquetur 24 qui poteſt diui­<lb></lb>di per ſe, duplum &lt;02&gt; 9 &amp; exit 4 cuius <expan abbr="quadratũ">quadratum</expan> eſt 16, addito 33 fit 49.</s>
        </p>
        <p type="main">
          <s id="id002626">Secunda regula, cum uolueris propoſito uno numero quadra­<lb></lb>to illum diuidere infinitis modis in duos numeros quadratos, cape <lb></lb>quemuis numerum quadratum per primum exemplum regulę pri<lb></lb>mæ, &amp; cum eo diuide numerum propoſitum, &amp; qui proueniet erit <lb></lb>quadratus, <expan abbr="hũc">hunc</expan> ergo duces in partes numeri quadrati quę ſunt nu­<lb></lb>meri <expan abbr="q̃drati">quadrati</expan>, &amp; fient duo quadrati numeri, &amp; illi <expan abbr="componẽt">component</expan> <expan abbr="numerũ">numerum</expan> <lb></lb><expan abbr="quadratũ">quadratum</expan> <expan abbr="priorẽ">priorem</expan> quem diuiſiſti. </s>
          <s id="id002627">quia multiplicatio fit per <expan abbr="eoſdẽ">eoſdem</expan> nu­<lb></lb>meros qui ſunt partes diuiſoris. </s>
          <s id="id002628">Velut uolo facere de 4 duas partes <lb></lb>quę ſint <expan abbr="q̃drati">quadrati</expan> numeri, capio <expan abbr="numerũ">numerum</expan> <expan abbr="q̃dratũ">quadratum</expan> qui <expan abbr="cõponat̃">componatur</expan> ex duo­<lb></lb>bus <expan abbr="q̃dratis">quadratis</expan>, uelut 25, diuido 4 per 25 exit 4/25 <expan abbr="hũc">hunc</expan> duco per 9 &amp; 16 <expan abbr="q̃dra­tos">quadra­<lb></lb>tos</expan> numeros <expan abbr="cõponentes">componentes</expan> 25 <expan abbr="fiũt">fiunt</expan> 1 11/25 &amp; 2 14/25 <expan abbr="q̃drati">quadrati</expan> 1 2/5 &amp; 1 3/5 Et hi <expan abbr="q̃drati">quadrati</expan> <lb></lb><expan abbr="cõponunt">componunt</expan> 4. Et ita poſſes diuidere infinitis modis, puta per 17 13/36 &amp; <lb></lb>per 169. Tertia regula cum unus numerus additus <lb></lb><figure id="id.015.01.167.1.jpg" xlink:href="015/01/167/1.jpg"></figure><lb></lb>primo &amp; detractis à <expan abbr="ſecũdo">ſecundo</expan> facit ambo quadrata, <expan abbr="idẽ">idem</expan> <lb></lb>numerus coniunctus cum differentia illorum nume­<lb></lb>rorum &amp; detractus à primo &amp; additus ſecundo facit <lb></lb>eoſdem numeros quadratos, ueluti capio 10 primum <lb></lb>3 ſecundum 6 additus ad 10 &amp; detractus à 7 efficit 6 <lb></lb>&amp; 1 quadratos dico quod iunctus 16 cum 3 differen­<lb></lb>tia 10 &amp; 7 fit 9, qui detractus à 10 &amp; additus ad 7 effi­<lb></lb>cit 1 &amp; 16 numeros quadratos priores.</s>
        </p>
        <pb pagenum="149" xlink:href="015/01/168.jpg"></pb>
        <p type="head">
          <s id="id002629">SCHOLIVM</s>
        </p>
        <p type="main">
          <s id="id002630">Sunt &amp; alij modi plures faciendi huiuſmodi, ſed <expan abbr="nõ">non</expan> ſunt ad eò ge<lb></lb>nerales, &amp; nihilo minus ſunt magis confuſi, &amp; non aliquid plus.</s>
        </p>
        <p type="main">
          <s id="id002631">Quarta regula, <expan abbr="cũ">cum</expan> uolueris <expan abbr="numerũ">numerum</expan> aliquem non quad. </s>
          <s id="id002632">qui bifa<lb></lb><expan abbr="riã">riam</expan> <expan abbr="componat̃">componatur</expan> ex duob. </s>
          <s id="id002633"><expan abbr="q̃d">quad</expan>. </s>
          <s id="id002634">uelut 10 ex 25, &amp; 25 &amp; 49 &amp; 1, <lb></lb><figure id="id.015.01.168.1.jpg" xlink:href="015/01/168/1.jpg"></figure><lb></lb>&amp; <expan abbr="ſumat̃">ſumatur</expan> a b numerus quad. </s>
          <s id="id002635">diuiſus in <expan abbr="ſupplemẽta">ſupplementa</expan>, ita quae c <lb></lb>d ſit portio minor eiuſmodi, ut adiecta illi <expan abbr="æq̃li">æquali</expan> c d gnomo <lb></lb>cir<expan abbr="cũſcriptus">cunſcriptus</expan> c k l <expan abbr="cũ">cum</expan> <expan abbr="fq̃drato">fquadrato</expan>, ſit <expan abbr="ęq̃lis">ęqualis</expan> a b <expan abbr="q̃drato">quadrato</expan>, detractis <lb></lb><expan abbr="igit̃">igitur</expan> c e &amp; e d, <expan abbr="æq̃libus">æqualibus</expan> erunt duo <expan abbr="ſupplemẽta">ſupplementa</expan> c k l <expan abbr="cũf">cunf</expan> qua­<lb></lb>drato ęqualia duob. </s>
          <s id="id002636"><expan abbr="ſupplemẽtis">ſupplementis</expan> a b <expan abbr="cũ">cum</expan> <expan abbr="q̃drato">quadrato</expan> h g. </s>
          <s id="id002637">Maio­<lb></lb>ra <expan abbr="aũt">aunt</expan> <expan abbr="ſupplemẽta">ſupplementa</expan> <expan abbr="excedũt">excedunt</expan> minora in duplo quad. </s>
          <s id="id002638">c d <expan abbr="igit̃">igitur</expan> detractis <lb></lb>minoribus ſupplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="duplũ">duplum</expan> quad. </s>
          <s id="id002639">c d <expan abbr="cũ">cum</expan> f qua­<lb></lb>drato ęqualia h g <expan abbr="q̃drato">quadrato</expan>. </s>
          <s id="id002640">Ergo propoſito numero, putà 3 ducam in ſe <lb></lb>fit 9, <expan abbr="ducã">ducam</expan> 2 <expan abbr="minorẽ">minorem</expan> in ſe fit 4, duplicabo fit 8, detraho ex 9, <expan abbr="relinquit̃">relinquitur</expan> <lb></lb>1 numerus <expan abbr="q̃dratus">quadratus</expan>, <expan abbr="igit̃">igitur</expan> <expan abbr="dicã">dicam</expan> q̊d 3 <expan abbr="cũ">cum</expan> duplo 2, &amp; erit <expan abbr="totũ">totum</expan> 7, eſt unus <lb></lb>numerus, alter &lt;02&gt; 1. 1. 1, &amp; <expan abbr="horũ">horum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002641"><expan abbr="cõponunt">componunt</expan> 50, <expan abbr="duplũ">duplum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002642">5. Et ſimi <lb></lb>liter capio 6 <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002643">36 <expan abbr="duplũ">duplum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002644">4. 32 differentia 4, numerus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002645">2, ideo <lb></lb>6 <expan abbr="cũ">cum</expan> duplo 4, &amp; eſt 14, eſt unus numerus, alter 2, <expan abbr="quorũ">quorum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002646">ſunt 200, <lb></lb><expan abbr="dimidiũ">dimidium</expan> eſt 100 <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002647">10 <expan abbr="cõpoſiti">compoſiti</expan> ex 6 &amp; 4. Et ita capio 9, <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002648">eius 81 du<lb></lb><expan abbr="plũ">plum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002649">6. 72 differentia 9 numerus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002650"><expan abbr="igit̃">igitur</expan> cum duplo 6, &amp; eſt 21, eſt <lb></lb>unus <expan abbr="illorũ">illorum</expan>, alter 3 <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002651">450, <expan abbr="duplũ">duplum</expan> 225 <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002652">15, qui conſtat ex 9 &amp; 6. Et <lb></lb>ita capio 11 <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002653">cuius eſt 121, <expan abbr="duplũ">duplum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002654">6 eſt 72 differentia, 72 &amp; 21 eſt <lb></lb>49 numerus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002655">7, <expan abbr="igit̃">igitur</expan> 23 qui conſtat ex 11, &amp; duplo 6 numeri mino<lb></lb>ris eſt unus numerus, alter eſt 7 <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002656"><expan abbr="quorũ">quorum</expan> ſunt 578. <expan abbr="duplũ">duplum</expan> 289, <expan abbr="q̃d">quad</expan>. <lb></lb></s>
          <s id="id002657">17, qui conſtat ex 11 &amp; 6. Quinta regula, per hoc inueniemus infini<lb></lb>tos numeros <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002658"><expan abbr="cõponentes">componentes</expan> 32, nam <expan abbr="cũ">cum</expan> 32 ſit duplus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002659"><expan abbr="diuidã">diuidam</expan> per <lb></lb>unum <expan abbr="aggregatũ">aggregatum</expan> ex inuentis puta 578, &amp; quia ambo ex ſuppoſito <lb></lb>ſunt dupli ad <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002660">qui proueniet erit <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002661">ſcilicet 16/289, duc in numeros <expan abbr="q̃­dratos">qua­<lb></lb>dratos</expan> qui componunt 578, &amp; ſunt 529 &amp; 49, &amp; fient 2 206/289 &amp; 29 83/289, <lb></lb>&amp; hi iuncti <expan abbr="fiũt">fiunt</expan> 32, quia ſunt multiplicatæ partes numeri, per quem <lb></lb>eſt diuiſus numerus. </s>
          <s id="id002662">Et ita poteris diuidere 32 in infinitos alios <expan abbr="q̃d">quad</expan>.</s>
        </p>
        <p type="main">
          <s id="id002663">Sexta regula, ponamus modò quod uelim diuidere 10, <expan abbr="cõpoſitũ">compoſitum</expan> ex <lb></lb>duob. </s>
          <s id="id002664"><expan abbr="q̃d">quad</expan>. </s>
          <s id="id002665">9 &amp; 1, &amp; non <expan abbr="duplũ">duplum</expan> numero <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002666">ita quod ſit diuiſus in alios <lb></lb>duos: <expan abbr="ducã">ducam</expan> 10 in 25 <expan abbr="cõpoſitũ">compoſitum</expan> ex duob. </s>
          <s id="id002667"><expan abbr="q̃d">quad</expan>. </s>
          <s id="id002668">fit 250/25, at 250 <expan abbr="cõponit̃">componitur</expan> aliter <lb></lb>ex duob. </s>
          <s id="id002669">quad. </s>
          <s id="id002670">&lt;08&gt; 225/25 &amp; 25/25, ſcilicet 169/25 &amp; 81/25, id eſt 6 19/25 &amp; 3 6/25, qui ſunt <expan abbr="q̃d">quad</expan>. <lb></lb></s>
          <s id="id002671">2 3/5 &amp; 1 4/5, &amp; ita uolo diuidere 13 in duo alia <expan abbr="q̃drata">quadrata</expan> &lt;08&gt; 9 &amp; 4, duco 13 in <lb></lb>25 &amp; fit 325/25, qui neceſſario <expan abbr="cõponit̃">componitur</expan> ex 225/25 &amp; 100/25, ſed ego uolo q̊d <expan abbr="cõpo">compo</expan> <lb></lb><expan abbr="nat̃">natur</expan> aliter, uelut ex 289/25 &amp; 63/25, &amp; ita ex 11 14/25 &amp; 1 11/25, qui ſunt numeri <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002672">com<lb></lb>ponentes 13, &amp; &lt;02&gt; ſunt 3 2/5 &amp; 1 1/5, &amp; in his opus eſt induſtria, ſcilicet ut <lb></lb><expan abbr="multiplicet̃">multiplicetur</expan> per numeros <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002673">ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> comp <lb></lb>ſiti ex <expan abbr="q̃dratis">quadratis</expan>. </s>
          <s id="id002674">Vt uerò uideamus <expan abbr="reſiduũ">reſiduum</expan>, proponamus quae uelim diui <lb></lb>dere 6 in duos numeros <expan abbr="q̃d">quad</expan>, <expan abbr="primũ">primum</expan> ſcire debes q̊d non poſſunt eſſe 


<pb pagenum="150" xlink:href="015/01/169.jpg"></pb>integri ex ratione dicta, quia oporteret ut eſſent ambo impares aut <lb></lb>pares, &amp; ſic <expan abbr="differrẽt">differrent</expan> numero pari, ergo oporteret ut eſſet unus me­<lb></lb>dius numerus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002675">ſunt &amp; alię rationes, ſed neque unus poſſet eſſe inte<lb></lb>ger, &amp; alius fractus, <expan abbr="nõ">non</expan> eſſet. </s>
          <s id="id002676">n. </s>
          <s id="id002677">6 numerus integer: <expan abbr="relinquit̃">relinquitur</expan> ergo ut <lb></lb>ſint duo fracti: ſed in numeris fractis <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002678">deductis ad minimas deno <lb></lb>minationes <expan abbr="operũ">operum</expan>, ut tam denominator &lt;08&gt; numerator habeat radi­<lb></lb>ces, ergo oportet q̊d hoc ſit in illis, &amp; quia iuncti debent facere inte­<lb></lb>gros 6, neceſſe eſt ut denominator ſit unus, &amp; <expan abbr="idẽ">idem</expan> in utroque, et q̊d nu<lb></lb>meratores ſimul iuncti ſint <expan abbr="ſexcuplũ">ſexcuplum</expan> denominatoris, ſi fracti <expan abbr="debẽt">debent</expan> <lb></lb>ęquipollere 6, ergo ille denominator <expan abbr="cũ">cum</expan> ſit <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002679">&amp; numeratores am­<lb></lb>bo ſint <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002680">&amp; ſint <expan abbr="ſexcuplũ">ſexcuplum</expan> denominatoris, oportebit inuenire <expan abbr="nu­merũ">nu­<lb></lb>merum</expan> <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002681">qui ductus in 6, faciat <expan abbr="numerũ">numerum</expan> qui <expan abbr="cõponit̃">componitur</expan> ex duob. </s>
          <s id="id002682"><expan abbr="q̃d">quad</expan>. <lb></lb></s>
          <s id="id002683">aut <expan abbr="cõponit̃">componitur</expan> ęqualiter, ergo proportio medietatis ad <expan abbr="medietatẽ">medietatem</expan> 6, eſt <lb></lb>ueluti totius ad 6, ſed totu continet 6 in <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002684">quia ex 6 in <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002685">fit <expan abbr="totũ">totum</expan>, <lb></lb>ergo ex medietate in <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002686">idem fit medietas, ſed medietas eſt nume­<lb></lb>rus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002687">ergo 3 eſſet numerus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002688">quod eſt falſum, oportet <expan abbr="igit̃">igitur</expan> ut nume <lb></lb>ri illi ſint inæ quales, &amp; ut 6 diuidatur in duas partes inęquales, hoc <lb></lb><expan abbr="aũt">aut</expan> fit diuidendo quemlibet <expan abbr="numerũ">numerum</expan> parem, qui <expan abbr="cõponit̃">componitur</expan> ex duob. <lb></lb></s>
          <s id="id002689">numeris <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002690">nam ſi eſſet impar, <expan abbr="nõ">non</expan> poſſet prodire numerus integer, &amp; <lb></lb><expan abbr="cũ">cum</expan> prouenerit numerus <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002691">ille erit <expan abbr="quẽ">quem</expan> quęrimus, <expan abbr="nã">nam</expan> diuiſo 6 per to­<lb></lb>tum <expan abbr="illũ">illum</expan> numerum, inde q̊d prouenit multiplicato per numeros <expan abbr="q̃d">quad</expan>, <lb></lb><expan abbr="cõponentes">componentes</expan> illum <expan abbr="numerũ">numerum</expan> productum, <expan abbr="producunt̃">producuntur</expan> partes 6, quæ <expan abbr="erũt">erunt</expan> <lb></lb>numeri <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002692">quia denominator utriuſque partis ex ſuppoſito eſt nume <lb></lb>rus <expan abbr="q̃dratus">quadratus</expan>, qui multiplicatus eſt per 6, &amp; numeratores ſunt nume <lb></lb>ri <expan abbr="q̃drati">quadrati</expan>, qui <expan abbr="cõponebant">componebant</expan> <expan abbr="numerũ">numerum</expan> <expan abbr="productũ">productum</expan>, et tales partes <expan abbr="ęquant̃">ęquantur</expan> <lb></lb>6, quia numerus productus <expan abbr="componit̃">componitur</expan> ex numeratoribus, &amp; <expan abbr="produ­cit̃">produ­<lb></lb>citur</expan> tale <expan abbr="cõpoſitum">compoſitum</expan> ex 6 in <expan abbr="denominatorẽ">denominatorem</expan>, &amp; hic eſt diuiſus per deno <lb></lb><expan abbr="minatorẽ">minatorem</expan>, ergo prouenit 6, ſi <expan abbr="em̃">emm</expan> multiplicato 3 in 4 fit 12, diuiſo 12 per <lb></lb>4, exit neceſſario idem 3. Pro colligendo ergo numeros omnes, qui <lb></lb><expan abbr="cõponuntur">componuntur</expan> ex <expan abbr="q̃dratis">quadratis</expan>, propones tibi ſeriem <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002693"><expan abbr="omniũ">omnium</expan>, &amp; inde iun­<lb></lb>ges, &amp; diuides per 6, &amp; <expan abbr="cũ">cum</expan> prodierit <expan abbr="q̃dratus">quadratus</expan>, <expan abbr="inuenit̃">inuenitur</expan> denominator, <lb></lb>&amp; numeri <expan abbr="cõponentes">componentes</expan> ipſum erunt numeratores, et ſuppoſiti deno <lb></lb>minatoribus <expan abbr="cõſtituent">conſtituent</expan> partes. </s>
          <s id="id002694">Vt uerò cognoſcas, ex quibus poſ­<lb></lb>ſit componi primum ex imparibus, non oportet aſſumere niſi 135, <lb></lb>quia 7 diuiſum per 6 relinquit 1, &amp; 9 diuiſum per 6, relinquit 3, &amp; 35 <lb></lb>diuiſum per 6 relinquit 5. ergo non poteſt componi numerus im­<lb></lb>par, qui diuidatur per 6, ut ſuperſit impar alius quàm 1. 3. 5. ſed 1 &amp; 3 <lb></lb>&amp; 5, &amp; 5 componunt 4 &amp; 1, &amp; 1 &amp; 3 &amp; 5 componunt 2, ſcilicet abie­<lb></lb>cto 6, ergo tales numeri <expan abbr="q̃drati">quadrati</expan> ſi ſint impares, uel ambo terminan­<lb></lb>tur in 3, ut 9 &amp; 81, qui faciunt 90, uel in 1 &amp; 5, ſed nullus numerus <lb></lb>quadratus diuiſus per 6 terminatur in 5, quia 1 ductum in ſe produ­<lb></lb>cit 1, &amp; 3 pro ducit 3, &amp; 5 pro ducit 1, ut 5 in 5 facit 25, &amp; 11 in 11 produ­


<pb pagenum="151" xlink:href="015/01/170.jpg"></pb>cit 121, quibus diuiſis per 6 ſupereſt 1. Quod etiam ſic demonſtratur <lb></lb>de 5, &amp; compoſitis à 5, nam diuiſo 5 in 3 &amp; 2, quadratum eius <expan abbr="cõpo­nitur">compo­<lb></lb>nitur</expan> ex duplo 3 in 2, in quo nihil ſupereſt, ſi diuidatur per 6, &amp; ex <lb></lb>quadrato 3, quòd eſt 9, in quo ſupereſt 3, &amp; ex quadrato 2 quod eſt </s>
        </p>
        <p type="main">
          <s id="id002695"><arrow.to.target n="marg520"></arrow.to.target><lb></lb>4, ſed iunctis 4 &amp; 3, &amp; abiecto 6 ſupereſt 1, ergo 5 in 5 <expan abbr="ductũ">ductum</expan>, &amp; diui<lb></lb>ſo producto relinquitur 1. Et ſimiliter capio 17, et <expan abbr="componit̃">componitur</expan> ex 12 &amp; <lb></lb>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil ſu­<lb></lb>pereſt, &amp; duplo 5 in 12, in quo <expan abbr="etiã">etiam</expan> nihil ſupereſt, ſi diuidatur per 6: <lb></lb>&amp; ex quadrato 5, in quo ſupereſt 1, ergo in nullo numero <expan abbr="cõpoſito">compoſito</expan> <lb></lb>ex 5 &amp; 6, uel compoſitis ex 6, poterit produci numerus, qui diuiſus <lb></lb>per 6 relinquat 5, igitur neque talis numerus potérit <expan abbr="cõponi">componi</expan> ex duo­<lb></lb>bus quadratis, in quib. </s>
          <s id="id002696">ſuperſit 5 &amp; 1, quia nullus eſt, in quo ſuper­<lb></lb>ſit 5 facta diuiſione per 6. Ex quo colligitur una regula: quod ſi quis <lb></lb>dicat multiplicaui 27 in ſe, et diuiſi per 13, uellem ſcire quid ſupereſt, <lb></lb>dico quod ſine multiplicatione et diuiſione poteris hoc ſcire ex de­<lb></lb>monſtratione dicta, diuide ergo 27 per 13, &amp; relinquitur 1, duc in ſe <lb></lb>fit 1: dices ergo, quod ſupererit 1, &amp; ita ſi ducerem 28 in ſe, &amp; diuide­<lb></lb>rem per 11, dico quod ſupererit 3, nam diuiſo 28 per 11, relinquitur <lb></lb>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum eſt, &amp; tantum <lb></lb><expan abbr="relinquit̃">relinquitur</expan> ducto 28 in ſe &amp; fit 784, &amp; diuiſo per 11. Reuertendo ergo <lb></lb>ad propoſitum, pater quod ex duobus tantum numeris imparibus <lb></lb>quadratis poteſt conflari ille numerus, <expan abbr="quorũ">quorum</expan> radices diuiſæ per 6 <lb></lb>relinquunt 3. Sed de paribus uel ſupereſt 2 uel 4 uel nihil, ſed <expan abbr="q̃dra­tum">quadra­<lb></lb>tum</expan> 2 eſt 4, &amp; <expan abbr="q̃dratum">quadratum</expan> 4 diuiſum per 6 etiam relinquit 4, ergo neque <lb></lb>ex duobus numeris, in quibus ſuperſint 2, neque in quibus ſuperſint <lb></lb>4, neque in quibus ſuperſint in uno 2, in altero 4 <expan abbr="poterũt">poterunt</expan> quadrata, in <lb></lb>quibus ſemper ſupererit 4, &amp; iuncta faciunt 8, in quod ̊ſupereſt 2, <expan abbr="cõ">con</expan>fla­<lb></lb>re <expan abbr="numerũ">numerum</expan> <expan abbr="dictũ">dictum</expan> ſeu <expan abbr="quæſitũ">quæſitum</expan>, qui poſsit diuidi per 6: neque ex <expan abbr="q̃d">quad</expan>. </s>
          <s id="id002697"><expan abbr="duo­rũ">duo­<lb></lb>rum</expan> <expan abbr="numẽrorũ">numerorum</expan>, in <expan abbr="quorũ">quorum</expan> altero nihil ſuperſit in reliquo ſuperſit 2 uel <lb></lb>4, quia in aggregato <expan abbr="q̃dratorũ">quadratorum</expan> ſemper ſupererit 4. Ergo relinqui­<lb></lb>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb></lb>ribus, quorum latera diuiſa per 6 relinquunt 3, uel ex duobus pari­<lb></lb>bus, quorum latera diuiſa per 6 nihil relinquant. </s>
          <s id="id002698">Oportet igitur <lb></lb>inuenire duos tales numeros quadratos numerorum imparium, in <lb></lb>quibus ſuperſit 3, ſi diuidantur per 6, aut parium in quibus nihil ſu­<lb></lb>perſit, quorum aggregato diuiſo per 6 prodeat numerus <expan abbr="q̃dratus&#039;">quadratus&#039;</expan>.</s>
        </p>
        <p type="margin">
          <s id="id002699"><margin.target id="marg520"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E <emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002700">His uiſis dico, quod conſtat radices talium numerorum opor­<lb></lb>tere eſſe in imparibus per additionem 6 incipiendo à 3, ut ſint <lb></lb>3. 9. 15. 21. 27. 33. 39. 45. 51. &amp; ſic deinceps: in paribus au­<lb></lb>tem per additionem eiuſdem 6 incipiendo à 6, uelut 6. 12. <lb></lb>18. 24. 30. 36. 42. 48. 54. 60. Dico ergo quod diui­<lb></lb>ſo numero illo compoſito per 6 in imparibus exibit numerus, 


<pb pagenum="152" xlink:href="015/01/171.jpg"></pb>qui diuiſus per 6 ſupererit 3, &amp; in paribus qui poterit diuidi per 6. <lb></lb>Quia <expan abbr="componunt̃">componuntur</expan> ex huiuſmodi: uelut 3 in ſe facit 9, &amp; 25 in ſe facit <lb></lb>225, qui <expan abbr="iũcti">iuncti</expan> <expan abbr="faciũt">faciunt</expan> 234, diuiſo 235 per 6 exit 39, qui <expan abbr="iterũ">iterum</expan> diuiſus per 6 <lb></lb>ſupereſt 3, &amp; ſimiliter capio 6 &amp; 12, <expan abbr="quorũ">quorum</expan> <expan abbr="q̃drata">quadrata</expan> ſunt 36 &amp; 144, &amp; <lb></lb><expan abbr="aggregatũ">aggregatum</expan> 180, qui diuiſus per 6 exit 30, qui <expan abbr="iterũ">iterum</expan> poteſt diuidi per <lb></lb>6. Et hoc quia <expan abbr="quilibetillorũ">quilibet illorum</expan> poteſt diuidi per <expan abbr="q̃dratũ">quadratum</expan> 6 in paribus, <lb></lb>ergo aggregato diuiſo per 6 q̊d prodit, <expan abbr="iterũ">iterum</expan> poterit diuidi per 6. <lb></lb>Et in imparibus quodlibet <expan abbr="q̃dratorũ">quadratorum</expan> exuperat ſupra ſenarios in 3, <lb></lb><expan abbr="igit̃">igitur</expan> <expan abbr="aggregatũ">aggregatum</expan> diuiſum in 2 pariet <expan abbr="numerũ">numerum</expan> qui diuiſus per 3, exibit <lb></lb>numerus impar <expan abbr="cõpoſitus">compoſitus</expan> ex ſenarijs &amp; 3. Illud ergo <expan abbr="quadratũ">quadratum</expan>, q̊d <lb></lb>prodibit, uel erit <expan abbr="cõpoſitum">compoſitum</expan> ex ſenarijs, uel ſupererit 3. Sed <expan abbr="cũ">cum</expan> 3 nume <lb></lb>ret 6, ergo tres <expan abbr="q̃drati">quadrati</expan> numeri ſcilicet duo, qui <expan abbr="cõponunt">componunt</expan> <expan abbr="numerũ">numerum</expan>, <lb></lb><arrow.to.target n="marg521"></arrow.to.target><lb></lb>&amp; qui prodit per <expan abbr="diuiſionẽ">diuiſionem</expan> 6, erunt <expan abbr="cõpoſiti">compoſiti</expan> inter ſe, ergo &amp; radices il<lb></lb>lorum. </s>
          <s id="id002701"><expan abbr="Igit̃">Igitur</expan> radix numeri <expan abbr="q̃drati">quadrati</expan>, qui prouenit diuiſo aggregato <expan abbr="qua­dratorũ">qua­<lb></lb>dratorum</expan> per 6 eſt ex <expan abbr="eodẽ">eodem</expan> ordine <expan abbr="impariũ">imparium</expan>, ſi impares numeri <expan abbr="q̃drati">quadrati</expan> <lb></lb><expan abbr="fuerũt">fuerunt</expan>, aut <expan abbr="pariũ">parium</expan> ſi pares. </s>
          <s id="id002702">At hoc eſſe <expan abbr="nõ">non</expan> poteſt, <expan abbr="nã">nam</expan> fracti illi numeri, <lb></lb>qui <expan abbr="erũt">erunt</expan> radices, <expan abbr="nõ">non</expan> <expan abbr="erũt">erunt</expan> minimi, ſed diuiſi per 3 oſtendent minores, <lb></lb>quod eſt contra ſuppoſitum, quare nullo modo 6 poteſt diuidi in <lb></lb>duos numeros quadratos, neque integros, neque fractos, quod erat <lb></lb>demonſtrandum. </s>
          <s id="id002703">Habes igitur ex hoc demonſtrationem quando <lb></lb><expan abbr="nõ">non</expan> poſsit diuidi, &amp; quando poſsit, quod poſsit, &amp; quomodo ſimul.</s>
        </p>
        <p type="margin">
          <s id="id002704"><margin.target id="marg521"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 29. <emph type="italics"></emph>ſe­<lb></lb>ptimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002705">Propoſitio centeſima quinquageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id002706">Horologiorum tempus multiplicare.<lb></lb><arrow.to.target n="marg522"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002707"><margin.target id="marg522"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002708">Contingit quandoque q̊d <expan abbr="horologiorũ">horologiorum</expan> tem<lb></lb><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"></figure><lb></lb>pus breue eſt, uolumus <expan abbr="aũt">aut</expan> maius efficere: id <lb></lb>duob. </s>
          <s id="id002709">modis poſſumus, <expan abbr="quorũ">quorum</expan> unus diffici­<lb></lb>lior eſt ſed perpetuus, &amp; longè nobilior, nam <lb></lb>grauitas ponderis uerſatilis efficit <expan abbr="quidẽ">quidem</expan> <expan abbr="tar­diorẽ">tar­<lb></lb>diorem</expan>, ſed difficilius <expan abbr="mobilẽ">mobilem</expan>, &amp; ob id grauio­<lb></lb>re <expan abbr="põdere">pondere</expan> in<expan abbr="digentẽ">digentem</expan>. </s>
          <s id="id002710">Sit ergo rota a b uerſati­<lb></lb>lis, quæ certam menſuram exigit pro quacunque funis parte correſperon<lb></lb>dentis uni denti ex centum, in quos diſtincta ſit, curriculum <expan abbr="aũt">aut</expan> c d <lb></lb>quinque <expan abbr="dentiũ">dentium</expan>, per q̊drota ſexaginta dentes <expan abbr="habẽs">habens</expan> <expan abbr="circumuoluat̃">circumuoluatur</expan> in <lb></lb><expan abbr="cõuerſione">conuerſione</expan>, <expan abbr="igit̃">igitur</expan> primę rotę uities <expan abbr="circumferet̃">circumferetur</expan>, <expan abbr="ſecũda">ſecunda</expan> <expan abbr="dẽtesque">dentesque</expan> M. CC. <lb></lb>rurſus ad <expan abbr="hãc">hanc</expan> <expan abbr="ſecundã">ſecundam</expan> tertia <expan abbr="nectat̃">nectatur</expan> cum curriculo ſex <expan abbr="dentiũ">dentium</expan>, atque in <lb></lb>ea <expan abbr="dẽtes">dentes</expan> ſeptuaginta duo, ut in una <expan abbr="cõuerſione">conuerſione</expan> ſint xiiij cccc, dentes <lb></lb><expan abbr="igit̃">igitur</expan> tot dentes in una <expan abbr="cõuerſione">conuerſione</expan> primę rotę circumuoluentur. </s>
          <s id="id002711">Iam <lb></lb>uerò tempus illud poterit duplicari ac triplicari iuxta <expan abbr="tarditatẽ">tarditatem</expan> tem<lb></lb>poris uerſatilis: <expan abbr="quãto">quanto</expan> <expan abbr="igit̃">igitur</expan> ponderoſius fuerit illud <expan abbr="tẽpus">tempus</expan>, tanto tar­<lb></lb>dius <expan abbr="mouebit̃">mouebitur</expan>, pauciores que circumuolutiones neceſſarię <expan abbr="erũt">erunt</expan> ad <expan abbr="ex­plẽdam">ex­<lb></lb>plendam</expan> unam <expan abbr="diẽ">diem</expan>: id eſt horas 24, ſed hoc in <expan abbr="cõmodi">commodi</expan> accedet, quòd <lb></lb>reuolutio indicis tanto tardior erit, ut <expan abbr="nõ">non</expan> iuſtè oſten dat horas: pro­


<pb pagenum="153" xlink:href="015/01/172.jpg"></pb>poſitum igitur eſt, ut pondera tardius ferantur, index <expan abbr="aũt">aut</expan>, &amp; quę ad <lb></lb>indicem ſequuntur horarum demonſtrationes celerius aut eodem <lb></lb>modo ferantur. </s>
          <s id="id002712">Ponamus ergo poſt&lt;08&gt; eadem eſt ratio celerioris &amp; <lb></lb>æqué uelocis, ponderis <expan abbr="aũt">aut</expan> tardius deſcendentis, aut <expan abbr="cõtrà">contrà</expan> tardio­<lb></lb>ris, aut æqualiter circumducti in dicis, celerioris <expan abbr="aũt">aut</expan> deſcenſus pon­<lb></lb>deris, quod ad nullam <expan abbr="utilitatẽ">utilitatem</expan> profuturum uideo. </s>
          <s id="id002713">Sit ergo ut pon<lb></lb>dus uelim tardius deſcendere, rotam <expan abbr="aũt">aut</expan> ęqualiter circumferri, dico <lb></lb>quod ex tempore mobili ſeu uerſatili (&amp; eſt ferrum, quod in ſum­<lb></lb>mo horologij citra ultraque <expan abbr="fert̃">fertur</expan> tam in horologijs ponderum &lt;08&gt; mo <lb></lb>læ) id fieri non poteſt: nam quantum tardabitur rota tertia ſecunda <lb></lb>&amp; prima, atque ob id deſcenſus ponderum, tantum remorabitur rota <lb></lb>prima quæ indicem oſtendit, ergo tantum index tardabitur quan­<lb></lb>tum <expan abbr="põdera">pondera</expan>, &amp; ut uno uerbo dicam, cùm <expan abbr="eadẽ">eadem</expan> rota index circumfe­<lb></lb>ratur, &amp; <expan abbr="põdus">pondus</expan> deſcendat, <expan abbr="quantũ">quantum</expan> unum tardatur tantum &amp; aliud.</s>
        </p>
        <p type="main">
          <s id="id002714">Secundus modus eſt, ut rota una totum tempus cum indice in ui<lb></lb>gintiquatuor horis circumuoluatur, &amp; currulis in quo funis minor <lb></lb>fiat: neceſſe eſt <expan abbr="igit̃">igitur</expan>, ut circumuoluta rota aut ſemel aut bis, <expan abbr="t̃er">tur</expan>, qua­<lb></lb>ter decies, &amp; <expan abbr="circumuoluat̃">circumuoluatur</expan> pleno circuitu index, et ſine errore: quo­<lb></lb>niam tempus &amp; dentes menſuræ reſpondent: igitur ſub eiſdem cir­<lb></lb>cuitibus numero eodemque tempore minus ex fune <expan abbr="deſcendẽt">deſcendent</expan> in cur<lb></lb>ruli paruo &lt;08&gt; magno: quare mutatione indiget currulis, aut ut funis <lb></lb>circumuoluens rotam curriculum habeat <expan abbr="annexũ">annexum</expan> rotæ oſten denti <lb></lb>horas, in qua pauciores ſint dentes: nam in eodem tempore, &amp; cir­<lb></lb>cuitu paucioribus uicibus circumuoluitur rota funis quæ grauita­<lb></lb>te temporis, &amp; multitudine <expan abbr="dentiũ">dentium</expan> certam <lb></lb><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"></figure><lb></lb>ſeruabit <expan abbr="menſurã">menſuram</expan>. </s>
          <s id="id002715">Sed in hoc neceſſe eſt gra<lb></lb>uius efficere pondus, aut leuius <expan abbr="tẽpus">tempus</expan> <expan abbr="quo­niã">quo­<lb></lb>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb></lb><expan abbr="tñ">tn</expan> tardè quod ſit pro paruitatis circuitus ratione.</s>
        </p>
        <p type="main">
          <s id="id002716">Tertius modus facilior eſt, &amp; magis com<lb></lb><expan abbr="pẽdioſus">pendioſus</expan>: Sit horologium a b c, in quo rota <lb></lb>d quæ funem <expan abbr="cõtinet">continet</expan> baſis horologij e f, cui <lb></lb>firmiter ſint <expan abbr="appẽſę">appenſę</expan> duę trochleę g &amp; h, &amp; fu <lb></lb>nis una parte trochleę appenſus in k, <expan abbr="ducat̃">ducatur</expan> <lb></lb>ad inferiorem aliam trochleam l inſeraturque <lb></lb>ibi orbiculo ſuo, &amp; redeat à dextra ſuperius <lb></lb><expan abbr="inſerat̃que">inſeraturque</expan> orbiculo ſuperioris trochleę, dedu<lb></lb><expan abbr="cat̃que">caturque</expan> uerſus <expan abbr="ſiniſtrã">ſiniſtram</expan>: atque ibi <expan abbr="deſcendẽs">deſcendens</expan> habe <lb></lb>at <expan abbr="põdus">pondus</expan> tractorium in m, <expan abbr="deducat̃que">deducaturque</expan> ſupra <lb></lb>ad <expan abbr="rotã">rotam</expan> horologij d, et circumuolutus exeat <lb></lb>ipſum, &amp; <expan abbr="deſcẽdat">deſcendat</expan> ad tro<expan abbr="chleãn">chlean</expan>, ſub que ea circumuolutus <expan abbr="iterũ">iterum</expan> aſcen<pb pagenum="154" xlink:href="015/01/173.jpg"></pb>dat à dextra parte, et circumuoluatur h cochleę rediens ad ſiniſtram <lb></lb>ibique deſcendens connectatur trochleæ in inferiori in o, cuius imæ <lb></lb>parti annectatur pondus remorans in imo annexum parte troch­<lb></lb>leæ p. </s>
          <s id="id002717">Cum ergo trahitur n trochlea, trahitur funis adeò ut pon­<lb></lb>dus m, tandem aſcendat cum trochlea l prope k: quia ergo in duo­<lb></lb>decim horis pondus m deſcenderet per k l funem reuolutionibus <lb></lb>circa d rotam dicamus uiginti, ergo ſi debet deſcendere à k ad l, per <lb></lb>funem duplicatam k l cum ipſam neceſſe ſit obequitantem d reuo­<lb></lb>lutionibus quadraginta circumuolui d, nam tota o h n d m g l k lon<lb></lb>gè maior eſt duplo k l, neceſſe eſt m deſcendere tardius quàm in du<lb></lb>plo temporis, quo deſcenderet per rectum funem k l, quod erat de­<lb></lb>monſtrandum. </s>
          <s id="id002718">Et hanc appendicem uidi apud Cæſarem Odonum <lb></lb>Apulum medicum, uirum elegantem lepidique ingenij. </s>
          <s id="id002719">Memento <lb></lb>uerò quod ubi orbiculi non cederent funi, uel quia duriores in cir<lb></lb>cumuolutione, uel quia latius exciperent illum reduplicato fune <lb></lb>circa illos omnin o circumducuntur, ſed difficilius ideò egent gra­<lb></lb>uiori pondere.</s>
        </p>
        <p type="main">
          <s id="id002720">Propoſitio centeſima quinquageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id002721">Horologiorum molarium rationem oſtendere.</s>
        </p>
        <p type="main">
          <s id="id002722"><arrow.to.target n="marg523"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002723"><margin.target id="marg523"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002724">Sunt horum duo genera primum, &amp; anti<lb></lb><figure id="id.015.01.173.1.jpg" xlink:href="015/01/173/1.jpg"></figure><lb></lb>quius licet multo poſterius eo quod pon­<lb></lb>deribus ducitur, quod funiculo ex inteſti­<lb></lb>nis ouium ſeu fidibus liræ agitur. </s>
          <s id="id002725">Sit igitur <lb></lb>axis f k erectus ſuper plano, cui per longum <lb></lb>coniuncta mola multiplicis ſpiræ in fine, cu<lb></lb>ius c annectatur ferreo circulo, qui habeatur loco capſulæ b c, quæ <lb></lb>circumuolui poſsit: huic <expan abbr="circũductus">circunductus</expan> funis d e multipliciter in pun<lb></lb>cto g, ſit autem e h in modum pyramidis ſenſim in acutum, ſed non <lb></lb>ualde per <expan abbr="ſpirã">ſpiram</expan> exculptam deſinentis, cui rota in uertice inſerta den<lb></lb>ſiculo, &amp; uertatur h e, colligens funiculum tractum in ſpira uerſus <lb></lb>apicem: unde funiculus circumuoluet b g d, <expan abbr="capſulã">capſulam</expan> uerſus c, trahet <lb></lb>ergo molam, &amp; conſtringet uiolenter <expan abbr="quãtum">quantum</expan> fert longitudo funis <lb></lb>quæ circumuolui poteſt a b e ad h: &amp; cum trahitur in d eremittitur, <lb></lb>non poteſt mola ſtatim retrahere reluctantibus denticulis h l rotæ, <lb></lb>&amp; alijs quæ implicantur curriculo m, a igitur mola conſtructa uio­<lb></lb>lenter mouet b g d, capſulam motu contrario à c in d &amp; in g &amp; in b, <lb></lb>quare funis d e trahitur, &amp; trahit e h illum circumuoluendo contra­<lb></lb>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb></lb>aliam <expan abbr="rotã">rotam</expan>, &amp; ſic deinceps donec tempus moueatur, &amp; rota indicis. <lb></lb></s>
          <s id="id002726">Hic adeſt capſula, &amp; quod circumuertitur à claue non eſt axis molę <lb></lb>ſed extra molam, ſcilicet e h. </s>
          <s id="id002727">Et quoniam hac ratione quanto mola a 


<pb pagenum="155" xlink:href="015/01/174.jpg"></pb>magis <expan abbr="explicabit̃">explicabitur</expan>, tanto lentius trahet, &amp; uertet e h, ideò hoc ex ſtru<lb></lb>ctura auxilium præſtatur, ut funis in inferiore parte <expan abbr="cõplexus">complexus</expan> latio­<lb></lb>res orbes, &amp; è regione tanto uehementius uertat e h: &amp; ita uis quæ <lb></lb>remittitur ob molæ laxitatem, augetur tantundem ob ſitum &amp; ma­<lb></lb>gnitudinem ſpirarum ut diſtantiorum ſua extremitate ab hypomo<lb></lb>chlio, quod eſt axis coni e h, ſeu inſtar axis.</s>
        </p>
        <p type="main">
          <s id="id002728">Alterum genus horologiorum cum mola ſine fune loco capſulę <lb></lb>habet <expan abbr="rotã">rotam</expan> plano ſub ſtratam, plenam denticulis axis, quo circum­<lb></lb>agitur uiolenter, non eſt extra molam, ſed ei annexa eſt mola intus, <lb></lb>exterius <expan abbr="aũt">aut</expan> rotę; ergo circumducto axe molę uim patitur circulus <lb></lb>exterior, ſed non <expan abbr="mouet̃">mouetur</expan>, quoniam clauo <expan abbr="impedit̃">impeditur</expan>. </s>
          <s id="id002729">Vbi mola quan­<lb></lb>tum decet conſtricta eſt ſublato clauo ſtatim ſecum trahit rotam, &amp; <lb></lb>illa <expan abbr="curriculũ">curriculum</expan> rotas que alias, &amp; tempus agitur, &amp; index uertitur. </s>
          <s id="id002730">Sed <lb></lb>in hoc idem eſt in commodum ſine remedio <lb></lb><figure id="id.015.01.174.1.jpg" xlink:href="015/01/174/1.jpg"></figure><lb></lb>quod fuit in priore. </s>
          <s id="id002731">Vbi enim cœperit laxa­<lb></lb>ri mola tanto tardius progrediuntur rotæ <lb></lb>atque index. </s>
          <s id="id002732">Veluti axis a b cui ſecun dum lon<lb></lb>gitudinem molæ caput interius annexum <lb></lb>eſt altero circulo rotæ in c d curriculum rotæ e, implexum rotæ f <lb></lb>clauus rotam retinens, donec circumducto a b mola conſtringa­<lb></lb>tur, &amp; latus eius trahat rotam ex c. </s>
          <s id="id002733">Inde ſublato clauo circulus, ſeu <lb></lb>rota trahitur ex c in g, &amp; in famola, quæ etiam ſecundum eandem <lb></lb>partem circumuoluta eſt: igitur d circumagetur à rota &amp; reliqua. <lb></lb></s>
          <s id="id002734">Sed ut dixi conſtructio hæc non ſatisfacit.</s>
        </p>
        <p type="main">
          <s id="id002735">Aliam ergo oportuit excogitare quę huiuſmodi eſt. </s>
          <s id="id002736">Sub axe a b, <lb></lb>qui circumuertitur ad molam contrahendam rotam, collocant par <lb></lb>uam quæ eſt, ut ita dicam, pars axis ima cui inſeruntur dentes in am<lb></lb>bitu ea ratione, ut dum mola tenditur, premant denticulos interio­<lb></lb>res, atque ita elabitur, totiesque circumducitur manente g f, donec <lb></lb>colligatur mola, quæ non ut in priore reliquo extremo ulli rotæ <lb></lb>affixa eſt, ſed columnæ in continenti <lb></lb>opercula horologij. </s>
          <s id="id002737">Cum ergo mola <lb></lb>tenta retrahat axem a b contrario mo­<lb></lb><figure id="id.015.01.174.2.jpg" xlink:href="015/01/174/2.jpg"></figure><lb></lb>tu, &amp; ille rotam mobilem, quæ cum <lb></lb>non poſsit regredi propter auerſos <lb></lb>dentes, mouet rotam f g contrario mo<lb></lb>tu, quæ circumacta per denticulos ſu­<lb></lb>os curriculum agit, &amp; reliqua omnia <lb></lb>neceſſaria. </s>
          <s id="id002738">Cur autem cum laxatur mo <lb></lb>la, &amp; uertit lentius c e rotam coniun­<lb></lb>ctam, ideoque g f, &amp; reliqua omnia <expan abbr="nõ">non</expan> tardetur tempus, &amp; circumuo­


<pb pagenum="156" xlink:href="015/01/175.jpg"></pb>lutio indicis cauſa eſt alia longè quàm in priore, nam mola longior <lb></lb>fit craſsior, &amp; durior adeoque robuſta, &amp; rotæ leues, ac tempus dum <lb></lb>laxata fuerit munus ſuum iuſto in tempore obeant: quare neceſſe <lb></lb>eſt, ut ab initio uehementius agat, &amp; celerius rotam cum axe qui tra<lb></lb>hitur à mola. </s>
          <s id="id002739">Ergo excogitarunt aliud genus retinaculi forma co­<lb></lb>chleæ quod ab initio moratur <expan abbr="uehemẽter">uehementer</expan> axem ne circumagatur, et <lb></lb>quanto magis mola explicatur eo minus retinet <expan abbr="impetũ">impetum</expan> illius, adeo <lb></lb>ut uehementer retineat uehementem concitationem mediocriter <lb></lb>moderatam, ſegniter lentam, nullo modo iuſtam: ita fit, ut ſemper <lb></lb>fermè æqualiter moueatur. </s>
          <s id="id002740">Difficile eſt tamen ad unguem ſeruare <lb></lb>moderationem, &amp; æqualitatem, &amp; magis etiam in his horologijs, <lb></lb>quæ uno circuitu molæ tempus <expan abbr="lõgius">longius</expan> exigunt: at difficilius etiam <lb></lb>efficere molam, quæ longo tempore duret, cum intenta ualde cele­<lb></lb>rius moueat rotas, &amp; ob id breui abſoluat circuitum, mollior au­<lb></lb>tem citò remittatur. </s>
          <s id="id002741">Et ob id longior &amp; non adeò <lb></lb>dura melior eſt. </s>
          <s id="id002742">Ratio autem cochleæ ita ſe habet. <lb></lb><figure id="id.015.01.175.1.jpg" xlink:href="015/01/175/1.jpg"></figure><lb></lb>Circa axem molæ d deducitur cochlea a b c, quæ <lb></lb>dum laxatur mola cochlea mouetur ex b in c, atque<lb></lb> ita pariter laxatur uis cochleæ retinentis axem.</s>
        </p>
        <p type="main">
          <s id="id002743">Propoſitio centeſima quinquageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id002744">Rationem indicis mobilis cum rota horarum numerus per ictus <lb></lb>indicatur explicare.</s>
        </p>
        <p type="main">
          <s id="id002745"><arrow.to.target n="marg524"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002746"><margin.target id="marg524"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id002747">Hoc fieri poteſt in ſingulo genere horologij trium <expan abbr="deſcriptorũ">deſcriptorum</expan>. <lb></lb></s>
          <s id="id002748">Propterea ſufficiat de uno oſtendiſſe. </s>
          <s id="id002749">Sed &amp; in ſingulo genere ſunt <lb></lb>multi modi, unius tamen reddidiſſe <expan abbr="rationẽ">rationem</expan> ſufficiat. </s>
          <s id="id002750">Hoc <expan abbr="aũt">aut</expan> qua­<lb></lb>tuor habet difficultates: prima ut horarum ictus conueniant cum <lb></lb>indice: ſecunda ut conuerſo indice conuertatur, &amp; rota ictuum: ter<lb></lb>tia ut ictuum numerus cum numero indicis conueniat. </s>
          <s id="id002751">Vnde mul­<lb></lb>ta ſunt horologia, in quibus ictus unus ſolum auditur ſingulis ho­<lb></lb>ris, atque hic modus facilis eſt: quarta cur in horum pleriſ que ſi non <lb></lb>pulſata ſtatim hora <expan abbr="transferat̃ur">transferatur</expan> index, non ceſſat pulſatio: imò nec <lb></lb>retineri poteſt, donec pondus illud deſcenderit. </s>
          <s id="id002752">Ergo primi &amp; ter­<lb></lb>tij ratio hæc habeatur, cum rota quę indicis rotam circumagit, per­<lb></lb>uenerit ad horæ finem, denticulo ſoluit aliam, eleuans obicem, illa <lb></lb>mouetur à pondere proprio alio, ſcilicet ab illo quod tempus agit: <lb></lb>aut ſi ſit horologium molæ à mola alia propria, quæ malleos cir­<lb></lb>cumacta perpetuò mouet, atque motura eſſet ſemper, donec pondus <lb></lb>ad terram deſcenderet: uerum dum mouetur deſcendit ferrum pro <lb></lb>quouis ictu quod in rotæ limbum incidit, &amp; donec inciderit in eam <lb></lb>partem quæ lenis eſt dilabitur, nec retinetur, &amp; ita eleuatur rurſus, 


<pb pagenum="157" xlink:href="015/01/176.jpg"></pb>at uero cum in concauam partem incidit retineri neceſſe eſt: atque ita <lb></lb>pondus non amplius deſcendit, rota ſiſtitur, malleus manet immo­<lb></lb>bilis: ſpatia ergo quæ ſunt inter cauitates ſunt ſecundum magnitu­<lb></lb>dinem proportionis numerórum <expan abbr="horarũ">horarum</expan>, uel ad ſex, uel ad duode­<lb></lb>cim, uel ad uiginti ­<lb></lb><figure id="id.015.01.176.1.jpg" xlink:href="015/01/176/1.jpg"></figure><lb></lb> quatuor terminan­<lb></lb>tium. </s>
          <s id="id002753">Ita quod, gra­<lb></lb>tia exempli, ſit iam <lb></lb>in cauitate a duode­<lb></lb>cimę horæ uncus, di<lb></lb>uidam circulum to­<lb></lb>tum in duas partes <lb></lb>æquales, quia in ſin <lb></lb>gulis medietatibus <lb></lb>propoſitum eſt, duo<lb></lb>decim facere cauita­<lb></lb>tes pro unco retinen­<lb></lb>do. </s>
          <s id="id002754">Et quia in una­<lb></lb>quaque medietate o­<lb></lb>portet, ut pulſent ho<lb></lb>ræ lxxviij, &amp; præterea ſint ibi ſex ſpatia cauitatum, quarum ſingulæ <lb></lb>contineant, gratia exempli, duo ſpatia unius ictus, ut certius retinea <lb></lb>tur uncus, <expan abbr="erũt">erunt</expan> igitur ſpatia omnia nonaginta: diuidemus ergo me­<lb></lb>dietatem circuli utranque in nonaginta partes æquales incipiendo <lb></lb>ab a, &amp; dabimus b primæ horę quod ſpatium eſt unius tantum par<lb></lb>tis ex nonaginta, poſt deſcribemus c cauitatem duarum partium, <lb></lb>ita ubi ictum unum dederit uncus, retinebitur in c, pòſt accipiemus <lb></lb>duo ſpatia, &amp; ſint ſignificata d litera, poſt quę faciemus cauitatem e: <lb></lb>&amp; ita uncus bis cadet in d, &amp; pulſabunt duo ictus, &amp; pòſt retinebi­<lb></lb>tur uncus in e. </s>
          <s id="id002755">Et poſt accipiam ſpatium trium partium, quod ſit f, <lb></lb>&amp; poſt deſcribam cauitatem g duarum partium, atque ita procedam <lb></lb>uſque ad duodecim.</s>
        </p>
        <p type="main">
          <s id="id002756">Ex quo manifeſtum eſt pondus quod agit rotam uolæ non de­</s>
        </p>
        <p type="main">
          <s id="id002757"><arrow.to.target n="marg525"></arrow.to.target><lb></lb>ſcendere, niſi dum horæ pulſant, ſecus quieſcere.</s>
        </p>
        <p type="margin">
          <s id="id002758"><margin.target id="marg525"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002759">Secundum, quòd deſcendit illud pondus plus &amp; minus, iuxta <lb></lb><arrow.to.target n="marg526"></arrow.to.target><lb></lb>proportionem numeri horarum, ita quod quando pulſabit una ho <lb></lb>ra parum ualde deſcendet, cum ſex horæ ſexcuplo magis, cum duo­<lb></lb>decim adhuc longè magis, id eſt duplo plus quàm cum pulſant <lb></lb>ſex horæ.</s>
        </p>
        <p type="margin">
          <s id="id002760"><margin.target id="marg526"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002761">Secunda conſtructio hanc habet rationem: Cum n rota indicis <lb></lb>coniuncta fuerit rotæ, quæ transfert malleum, neceſſe eſt ut unà fe­


<pb pagenum="158" xlink:href="015/01/177.jpg"></pb>rantur: qui nimò illud magis mirum de quo illi non mirantur quia <lb></lb>frequens eſt, ſcilicet cur aut quomodo ſi diuiſæ ſunt ut cir<expan abbr="çũducto">çunducto</expan> <lb></lb>indice non transferatur rota mallei, <expan abbr="põdere">pondere</expan> tamen uerſata rota in­<lb></lb>dicis in idem incidat, ut horæ quæ pulſu declarantur ad unguem <lb></lb>&amp; in eiſdem ſectionibus <expan abbr="cõueniant">conueniant</expan> cum horis quas index oſtendit.</s>
        </p>
        <p type="main">
          <s id="id002762">Verum quia multis modis contingit ordinem horologiorum <lb></lb>peruerti: in ſimilibus quidem ſi hora indicis ſimul &amp; pulſus unà <lb></lb>circumferuntur, ſed tardius ambo index traducitur ad locum debi­<lb></lb>tum, inde ponderi aliquid additur. </s>
          <s id="id002763">Si uerò antè proceſſerit quam. <lb></lb></s>
          <s id="id002764">Sol in dicet ablato pondere, ſines tempus fluere uſque ad indicis lo­<lb></lb>cum ſine motu horologij, pondus quoque ipſum minues. </s>
          <s id="id002765">At ſi pon­<lb></lb>dus pulſus in terram deuenerit uel propè, expecta donec ſuper li­<lb></lb>nea index fuerit, inde trahe, neque. </s>
          <s id="id002766">n. </s>
          <s id="id002767">excurret: nam ſi dum index eſt in <lb></lb>medio horæ aut propè, traxeris pondus pulſus, non deſinet deſcen <lb></lb>dere, pulſabuntqúe horæ donec ad terram pondus deuenerit, <lb></lb>quòd ſi iam in errorem incideris pulſentque horę &amp; deſcendat, pon­<lb></lb>dus, ſenſim deducito indicem, cum. </s>
          <s id="id002768">n. </s>
          <s id="id002769">ad finem horę peruenerit ini­<lb></lb>tiumque ſequentis, quoniam ferrum in interuallum deuenerit rota &amp; <lb></lb>pondus firmabitur. </s>
          <s id="id002770">Inde ſublato <expan abbr="põdere">pondere</expan> donec Sol ad <expan abbr="horã">horam</expan> quam <lb></lb>index monſtrat peruenerit, reddes pondus horologio. </s>
          <s id="id002771">Si ergo ho­<lb></lb>ram pulſu <expan abbr="eandẽ">eandem</expan> declarat quam index, bene eſt, ſi non, <expan abbr="paululũ">paululum</expan> <expan abbr="uir­gulã">uir­<lb></lb>gulam</expan> eleua quę eſt iuxta fores horologij pulſabitque ſequens hora, id <lb></lb>uero toties repetes immoto in dies &amp; ſublato, ſi uereris ne extra <expan abbr="in­teruallũ">in­<lb></lb>teruallum</expan> ferrum feratur, &amp; ob id excurrat rota pulſus <expan abbr="horarũ">horarum</expan>, donec <lb></lb>hora pulſet quæ cum indice conuenit, ſtatimque pondus quo horæ <lb></lb>pulſant ſurſum retrahes. </s>
          <s id="id002772">His quinque regulis uſum diſces ſimilium <lb></lb>horologiorum, unumquodque autem proprias habet: ſed duæ pri­<lb></lb>mæ omni horologiæ ſatisfaciunt. </s>
          <s id="id002773">Quòd ſi hæ non ſatisfaciunt iam <lb></lb>horologium laborat: tum uerò illud diſſoluere oportet &amp; deterge­<lb></lb>re &amp; inungere, iuuat autem uel capſula uel linteo perpetuo pul­<lb></lb>uerem ab illo arcere. </s>
          <s id="id002774">Quòd ſi nec ſic reſtituitur neceſſe eſt diſſol­<lb></lb>uere &amp; antea conſiderare impedimentum, pòſt denticulum qui la­<lb></lb>borat, plerunque. </s>
          <s id="id002775">n. </s>
          <s id="id002776">aliquem inuenies huius modi, quem lima aut alia <lb></lb>ratione reſtitues, ſemper autém hi fermè reſtituuntur: at qui mola <lb></lb>aguntur præter rotarum &amp; axium &amp; indicum labores, molæ etiam <lb></lb>inæqualitati &amp; defectibus ſubiciuntur, qui ſi nimis uelo citer agunt <lb></lb>rotas cum difficultate reſtituuntur moderationi, ſi lentius rarò uel <lb></lb>nunquam emendantur, uix etiam noua inducta mola.</s>
        </p>
        <p type="main">
          <s id="id002777">Propoſitio centeſima quinquageſima nona.</s>
        </p>
        <p type="main">
          <s id="id002778">Nullus angulus rectilineus æqualis eſſe poteſt alicui angulo con<lb></lb>tento recta &amp; circuli portione.</s>
        </p>
        <pb pagenum="159" xlink:href="015/01/178.jpg"></pb>
        <p type="main">
          <s id="id002779">Sit angulus a &amp; circulus b c, dico non poſſe aliquem angulum <lb></lb><arrow.to.target n="marg527"></arrow.to.target><lb></lb>contentum recta &amp; circuli portione eſſe illi <lb></lb><figure id="id.015.01.178.1.jpg" xlink:href="015/01/178/1.jpg"></figure><lb></lb>æqualem. </s>
          <s id="id002780">ſi enim eſſe poſsit, ſit c b e. </s>
          <s id="id002781">duca­<lb></lb>tur recta b d faciens rectilineum d b c ęqua<lb></lb><arrow.to.target n="marg528"></arrow.to.target><lb></lb>lem a, erit igitur d b c ęqualis e b c per com­<lb></lb>munem animi ſententiam, ſeu ergo b d ca­<lb></lb>dat intra circulum ſeu extra, erit pars ęqua­<lb></lb>lis toti quod eſſe non poteſt. </s>
          <s id="id002782">Sed neque po­<lb></lb>teſt cadere recta ſuper b e. </s>
          <s id="id002783">nam id eſt contra demonſtrata ab Eucli­<lb></lb><arrow.to.target n="marg529"></arrow.to.target><lb></lb>de. </s>
          <s id="id002784">At ſi ſit angulus c b e exterior ſimiliter producta b d, ſeu intus, <lb></lb>ſeu extrà cadat, pars erit æqualis toti quod eſſe non poteſt.</s>
        </p>
        <p type="margin">
          <s id="id002785"><margin.target id="marg527"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id002786"><margin.target id="marg528"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002787"><margin.target id="marg529"></margin.target>23. E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002788">Ex hoc patet quod nullus angulus peripheria circuli &amp; recta <expan abbr="cõ­">con­<lb></lb></expan><arrow.to.target n="marg530"></arrow.to.target><lb></lb>tentus poteſt eſſe æqualis recto, quia rectus etiam rectilineus eſt.</s>
        </p>
        <p type="margin">
          <s id="id002789"><margin.target id="marg530"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002790">Et rurſus nullus angulus peripheria &amp; <lb></lb><arrow.to.target n="marg531"></arrow.to.target><lb></lb><figure id="id.015.01.178.2.jpg" xlink:href="015/01/178/2.jpg"></figure><lb></lb>recta contentus à recta linea per æqualia <lb></lb>diuidi poteſt, patet quia una pars eſſet an­<lb></lb>gulus rectilineus, alia contentus recta &amp; pe<lb></lb>ripheria: iſti <expan abbr="autẽ">autem</expan> non poſſunt eſſe æquales, <lb></lb>quare nec prior potuit per æqualia diuidi.</s>
        </p>
        <p type="margin">
          <s id="id002791"><margin.target id="marg531"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002792">Ex hoc etiam patet quod ſpatium con­<lb></lb><arrow.to.target n="marg532"></arrow.to.target><lb></lb><expan abbr="tentũ">tentum</expan> à peripheria circuli nulli angulo rectilineo ęquale eſſe poteſt. <lb></lb></s>
          <s id="id002793">nam dimidium eſſet æquale dimidio, quod eſt contra demonſtrata.</s>
        </p>
        <p type="margin">
          <s id="id002794"><margin.target id="marg532"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="head">
          <s id="id002795">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id002796">Inter duos circulos qui ſe diuidant infinitæ lineæ duci poſſunt. <lb></lb></s>
          <s id="id002797">Inter circulos autem qui ſe tangant, recta linea duci non poteſt.<lb></lb><arrow.to.target n="marg533"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002798"><margin.target id="marg533"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002799">Sint duo circuli a b &amp; a c, qui ſe diuidant </s>
        </p>
        <p type="main">
          <s id="id002800"><arrow.to.target n="marg534"></arrow.to.target><lb></lb>in a, &amp; ducatur ex centro inferioris d a &amp; <lb></lb><figure id="id.015.01.178.3.jpg" xlink:href="015/01/178/3.jpg"></figure><lb></lb>a d, &amp; ad d a cathetus a e, dico quòd a e di­<lb></lb>uidet angulum b a c ducatur ex centro ſu­<lb></lb><arrow.to.target n="marg535"></arrow.to.target><lb></lb>perioris a c b quod ſit f, fa cui cathetus a g, <lb></lb>quia ergo e a cadit infra a g, &amp; inter a g &amp; <lb></lb><arrow.to.target n="marg536"></arrow.to.target><lb></lb>a b non poteſt duci recta, igitur e a cadit in­<lb></lb><figure id="id.015.01.178.4.jpg" xlink:href="015/01/178/4.jpg"></figure><lb></lb>tra a c b circulum. </s>
          <s id="id002801">Rurſus tangant ſe circuli <lb></lb>c d &amp; c e, &amp; ducatur a b per centra <expan abbr="eorũ">eorum</expan> quę <lb></lb>applicabit ad c, ex c ducatur cathetus c f &amp; <lb></lb><expan abbr="quoniã">quoniam</expan> c f contangit <expan abbr="circulũ">circulum</expan> c e, l igitur, du­<lb></lb>cta quauis linea infra c f, cadet intra <expan abbr="circulũ">circulum</expan> <lb></lb>c e. </s>
          <s id="id002802">Non ergo poterit cadere inter c d &amp; c e.</s>
        </p>
        <p type="margin">
          <s id="id002803"><margin.target id="marg534"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002804"><margin.target id="marg535"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>ter<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002805"><margin.target id="marg536"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id002806">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id002807">Dato angulo contento duabus peripherijs <expan abbr="æqualiũ">æqualium</expan> circulorum <lb></lb>ſe ſecantium æqualem rectilineum illi fabricare.</s>
        </p>
        <pb pagenum="160" xlink:href="015/01/179.jpg"></pb>
        <p type="main">
          <s id="id002808">Sit angulus a b c duabus peripherijs æqualium circulorum con<lb></lb><arrow.to.target n="marg537"></arrow.to.target><lb></lb>tentus, uolo ei æqualem rectilineum fabricare, ducantur b d &amp; b e <lb></lb><arrow.to.target n="marg538"></arrow.to.target><lb></lb>æquales, ut pote facto b centro eritque angulus d b a æqualis angu­<lb></lb>lo e b c, addito utrique communi d b e ex peri<lb></lb><figure id="id.015.01.179.1.jpg" xlink:href="015/01/179/1.jpg"></figure><lb></lb>pheria &amp; recta, fiet angulus d b e ex rectis <lb></lb>æqualis a b c ex peripherijs, quod crat de­<lb></lb>monſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id002809"><margin.target id="marg537"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002810"><margin.target id="marg538"></margin.target>P<emph type="italics"></emph>er modum<emph.end type="italics"></emph.end><lb></lb>8. <emph type="italics"></emph>primi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>l.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002811">Ex hoc patet quod reliqua duo ſpatia <lb></lb><arrow.to.target n="marg539"></arrow.to.target><lb></lb>non poſſunt eſſe æqualia rectilineo. </s>
          <s id="id002812">Nam <lb></lb>ſpatium b a c demonſtratum eſt æquale eſ­<lb></lb>ſe rectilineo, &amp; b ad non eſt æquale rectili­<lb></lb>neo, <expan abbr="igit̃">igitur</expan> <expan abbr="ſpatiũ">ſpatium</expan> c a d non poteſt eſſe æquale <lb></lb>angulo rectilineo, nam ſi ſic ſit b a c ęquale <lb></lb>f g h &amp; c a d h g k, <expan abbr="igit̃">igitur</expan> <expan abbr="totũ">totum</expan>, b a d erit ęquale <lb></lb><arrow.to.target n="marg540"></arrow.to.target><lb></lb>toti f g k quod eſt contra <expan abbr="ſuppoſitũ">ſuppoſitum</expan>, ideò neque<lb></lb>b a e quia b a c &amp; d a e ſunt <expan abbr="æq̃lia">æqualia</expan> rectilineis <lb></lb>per ſe, &amp; <expan abbr="etiã">etiam</expan> pariter accepta. </s>
          <s id="id002813">Totum <expan abbr="aũt">aunt</expan> <expan abbr="ſpatiũ">ſpatium</expan> a eſt <expan abbr="ęq̃le">ęquale</expan> quatuor, re­<lb></lb>ctis ergo <expan abbr="reſiduũ">reſiduum</expan>, ſcilicet ſpatia c a d &amp; b a c pariter accepta ſunt <expan abbr="ęq̃­lia">ęqua­<lb></lb>lia</expan> rectilineis ſpatijs, ſed <expan abbr="ſpatiũ">ſpatium</expan> e a d non eſt <expan abbr="æq̃le">æquale</expan> rectilineo, ergo per<lb></lb>demonſtrata hic, nec b a e, <expan abbr="nã">nam</expan> ſi ſit, ſit ergo b a e æquale h g k &amp; quia <lb></lb>ambo ſpatia b a e &amp; c a d ſunt <expan abbr="æq̃lia">æqualia</expan> rectilineo ex demonſtratis, ſit <lb></lb>ergo æqualia f g k, erit ergo ex communi animi ſententia ſpatium f <lb></lb>g h æquale ſpatio c a d, quod eſt contra primam partem corrolarij.</s>
        </p>
        <p type="margin">
          <s id="id002814"><margin.target id="marg539"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="margin">
          <s id="id002815"><margin.target id="marg540"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb><emph type="italics"></emph>præſentis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id002816">LEMMA TERTIVM.<lb></lb><arrow.to.target n="marg541"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002817"><margin.target id="marg541"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002818">Inter duas rectas lineas ſe tangentes circuli dati peripheriam </s>
        </p>
        <p type="main">
          <s id="id002819"><arrow.to.target n="marg542"></arrow.to.target><lb></lb>ducere. </s>
          <s id="id002820">Sit circulus datus a b rectilineus <lb></lb><figure id="id.015.01.179.2.jpg" xlink:href="015/01/179/2.jpg"></figure><lb></lb>angulus c d e, uolo illum diuidere circuli <lb></lb> periferia data b f, duco perpendicularem <lb></lb>d g ex, d ſuper d c, &amp; facio g d æqualem a b <lb></lb><arrow.to.target n="marg543"></arrow.to.target><lb></lb>&amp; duco circulum per d qui ſit d h qui cadet <lb></lb>infra d c &amp; ob id etiam ſupra d e, igitur di­<lb></lb>uidet angulum c d e, quare cum circulus d h ſit æqualis circulo b f <lb></lb><arrow.to.target n="marg544"></arrow.to.target><lb></lb>patet propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002821"><margin.target id="marg542"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph><expan abbr="eiuſdẽ">eiuſdem</expan><emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002822"><margin.target id="marg543"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002823"><margin.target id="marg544"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 6.</s>
        </p>
        <p type="main">
          <s id="id002824">Ex hoc patet quod infinitis modis poteſt diuidi angulus c d e <lb></lb><arrow.to.target n="marg545"></arrow.to.target><lb></lb>peripheria b f, nam diuiſo per rectam c d e linea d k per ęqualia &amp; di <lb></lb><arrow.to.target n="marg546"></arrow.to.target><lb></lb>uiſo k d e per præſentem peripheria b f, patet propoſitum quoniam <lb></lb>angulus c d e poteſtin infinitum recta diuidi, &amp; ita ſemper per peri­<lb></lb>pheriam, unde patet propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002825"><margin.target id="marg545"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>diff. <lb></lb></s>
          <s id="id002826">tertij <expan abbr="eiuſdẽ">eiuſdem</expan>.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002827"><margin.target id="marg546"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id002828">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002829">Atque hæc omnia ſequuntur de mente Euclidis, quæ tamen ui­<lb></lb>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe<pb pagenum="161" xlink:href="015/01/180.jpg"></pb>ria &amp; recta ſunt ex genere quantitatis continuæ, &amp; quòd detur ma­<lb></lb>ius &amp; minus &amp; nunquam detur ęquale, uidetur abſurdum ne dum <lb></lb>admirabile. </s>
          <s id="id002830">Et maximè quod etiam anguli ex peripheria &amp; recta <lb></lb>ſunt diuerſorum generum inter ſe &amp; infinitorum. </s>
          <s id="id002831">Pręterea iſtud re­<lb></lb>pugnare uidetur ipſimet Euclidi, dicenti duabus magnitudinibus <lb></lb><arrow.to.target n="marg547"></arrow.to.target><lb></lb><arrow.to.target n="marg548"></arrow.to.target><lb></lb>propoſitis inæqualibus, ſi de maiore earum plus dimidio detraha­<lb></lb>tur, atque iterum de reſiduo maius dimidio, &amp; rurſus de eo quod re­<lb></lb>linquitur plus dimidio, neceſſe erit ut tandem minor minore quan­<lb></lb>titas relinquatur. </s>
          <s id="id002832">Neque illud argumentum uidetur concludere an­<lb></lb>gulus contactus, ex recta, &amp; circuli circumferentia non poteſt recta <lb></lb>diuidi, &amp; rectilineus poteſt diuidi, ergo rectilineus ſemper eſt ma­<lb></lb>ior angulo contactus, quia hoc contingit in angulo contactus pro<lb></lb>pter modum anguli, non paruitatem: ſicut etiam non ualet de figu­<lb></lb><figure id="id.015.01.180.1.jpg" xlink:href="015/01/180/1.jpg"></figure><lb></lb>ra a lunari, &amp; quadrangulo b. </s>
          <s id="id002833">nam poteſt b diuidi <lb></lb>ab angulo ad angulum recta &amp; a non poteſt, &amp; <lb></lb>tamen a maius eſt quam b, cum contineat ipſam. <lb></lb></s>
          <s id="id002834">Proponantur ergo duo circuli a d e &amp; a f g qui ſe contingant in a, &amp; <lb></lb>eorum centra ſint b &amp; c &amp; ducantur rectæ a f d &amp; a g e &amp; conſtat <lb></lb>qui portiones a d &amp; a f ſimiles ſunt, <lb></lb><figure id="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"></figure><lb></lb>itemque a e &amp; a g, ducta enim a b c <lb></lb><arrow.to.target n="marg549"></arrow.to.target><lb></lb>per centra circulorum ex contactu <lb></lb>tranſibit per illa: quare anguli h a g <lb></lb>&amp; h a e ſunt ijdem &amp; ſimiliter h a f <lb></lb>&amp; h a d ijdem, portiones ergo af &amp; <lb></lb>a d itemque a g &amp; a e ſimiles ſunt: an­<lb></lb>gulus igitur g a e ex peripherijs &amp; <lb></lb><arrow.to.target n="marg550"></arrow.to.target><lb></lb>e a d ex rectis ſunt ijdem in puncto <lb></lb>a: ſed quod ad baſsim maior eſt ba­<lb></lb>ſis g e quam e d: hoc enim ſuppono <lb></lb>quod per ſe eſt manifeſtum toties <lb></lb><expan abbr="diuidẽdo">diuidendo</expan> arcum d e ut fiat minor recta g e. </s>
          <s id="id002835">Quia ergo ſunt duę ma­<lb></lb>gnitudines, quarum ter mini ſunt ijdem ex una parte, ſcilicet pun­<lb></lb>ctum a, ex alia autem unus eſt maior altero, ſcilicet g e quam e f &amp; <lb></lb><arrow.to.target n="marg551"></arrow.to.target><lb></lb>a d e peripheria eſt maior recta a g e. </s>
          <s id="id002836">Ergo per regulam dialecti­<lb></lb>cam ſi ſub eadem proportione procederent, maius eſſet ſpatium <lb></lb>ſemper inter peripherias quàm rectas. </s>
          <s id="id002837">igitur angulus peripheria­<lb></lb>rum eſt maior angulo à rectis contento. </s>
          <s id="id002838">Cum angulus non ſit <lb></lb>niſi quidam habitus propinquitatis linearum, ſed angulus con­<lb></lb>tactus ex recta &amp; peripheria maior eſt contento ex peripherijs cum <lb></lb>habeat rationem totius ad partem, igitur angulus contactus eſt <lb></lb>maior dato angulo rectilineo.</s>
        </p>
        <pb pagenum="162" xlink:href="015/01/181.jpg"></pb>
        <p type="margin">
          <s id="id002839"><margin.target id="marg547"></margin.target>1. P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002840"><margin.target id="marg548"></margin.target>10. E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002841"><margin.target id="marg549"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>ter<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002842"><margin.target id="marg550"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>diff. <lb></lb></s>
          <s id="id002843">tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002844"><margin.target id="marg551"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>deci­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002845">Propoſitio centeſima ſexageſima.</s>
        </p>
        <p type="main">
          <s id="id002846">Propoſita linea tribus que in ea ſignis punctum inuenire, ex que <lb></lb>ductæ tres lineæ ad ſigna ſint in proportionibus datis.<lb></lb><arrow.to.target n="marg552"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002847"><margin.target id="marg552"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id002848">Sit data linea a b c in qua puncta dicta &amp; datæ tres lineę d e f, uo­<lb></lb>lo inuenire punctum, puta g ex quo ductæ tres <lb></lb>lineæ ad a b c puncta ſint in proportione a g ad </s>
        </p>
        <p type="main">
          <s id="id002849"><arrow.to.target n="marg553"></arrow.to.target><lb></lb>g b, ut d ad e &amp; g b ad g c, ut e ad f. </s>
          <s id="id002850">Per pręceden<lb></lb><figure id="id.015.01.181.1.jpg" xlink:href="015/01/181/1.jpg"></figure><lb></lb>tia inuenio circulum ex cuius peripheria omni­<lb></lb>bus ex punctis ductæ lineæ ad a b ſint in pro­<lb></lb>portione d ad e, &amp; per idem circulum ex cuius <lb></lb>peripheria quælibet lineæ ductæ ad b c puncta <lb></lb>ſint in proportione c ad f, ſi igitur iſti duo circu­<lb></lb>li ſe ſecabunt in aliquo puncto puta g: liquet <lb></lb>quod lineæ ductæ ex g ad a b c, erunt in propor<lb></lb>tione d e f.<lb></lb><arrow.to.target n="marg554"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002851"><margin.target id="marg553"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 154.</s>
        </p>
        <p type="margin">
          <s id="id002852"><margin.target id="marg554"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}_{m}.</s>
        </p>
        <p type="main">
          <s id="id002853">Ex quo liquet quod ſi uoluero ducere ad tria puncta data, tres <lb></lb>lineas in continua proportione data d ad e, ſubijciam tertiam uel in<lb></lb>terponam, ſi uoluero mediam. </s>
          <s id="id002854">Et ſi uellem, ut eſſet a g ad g b dupli­<lb></lb>cata ei quæ eſt g b ad b c, &amp; uellem quòd proportio d ad a d f data <lb></lb>eſſet, oporteret inuenire duas medias proportione inter d &amp; f, in de <lb></lb>operari cum una earum per modum propoſitum. </s>
          <s id="id002855">Differt corrola­<lb></lb>rium hoc à propoſitione in hoc, quod in propoſitione non quæri­<lb></lb>mus niſi proportionem g a ad g b &amp; g b ad b c, non g a ad g c, neque <lb></lb>comparationem proportionum: at in corrolario quærimus tres <lb></lb>proportiones g a g b &amp; g c, &amp; comparationem proportionum in­<lb></lb>ter ſe, ſcilicet æqualitatem.</s>
        </p>
        <p type="main">
          <s id="id002856">Propoſitio centeſima ſexageſima prima.</s>
        </p>
        <p type="main">
          <s id="id002857">Si fuerint duo trianguli quorum baſes in eadem linea ſint con­<lb></lb>ſtituti &amp; æquales &amp; ad unum punctum terminati, &amp; latus unum <lb></lb>commune inter reliqua quantita­<lb></lb><figure id="id.015.01.181.2.jpg" xlink:href="015/01/181/2.jpg"></figure><lb></lb>te medium, neceſſe eſt angulum à <lb></lb>maioribus lineis contentum mi­<lb></lb>norem eſſe.</s>
        </p>
        <p type="main">
          <s id="id002858">Sint duo trianguli a b c, a c d, </s>
        </p>
        <p type="main">
          <s id="id002859"><arrow.to.target n="marg555"></arrow.to.target><lb></lb>quales proponuntur, &amp; ſit a d ma­<lb></lb><arrow.to.target n="marg556"></arrow.to.target><lb></lb>ior a b dico angulum d a c eſſe mi­<lb></lb>norem. </s>
          <s id="id002860">Si non fiat angulus d a c æ­<lb></lb>qualis ex alia parte, &amp; oportet ſi non ſit minor ut uel cadat a d ſu­<lb></lb><arrow.to.target n="marg557"></arrow.to.target><lb></lb>per a b &amp; ducta a d ad ęqualitatem cadet infra b, ducta ergo d c erit <lb></lb>trigonus a d c maior a b c, quod eſſe non poteſt cum ſint æquales. 


<pb pagenum="163" xlink:href="015/01/182.jpg"></pb>Si autem a d cadat extra a b ducatur d e: quæ ſi cadat ſupra b c uel <lb></lb>infra, cum totum ſit maius parte erit a d e, ut prius maior a b c quod <lb></lb><arrow.to.target n="marg558"></arrow.to.target><lb></lb>eſt contra Euclidem. </s>
          <s id="id002861">Reliquum eſt ut d c cadat ſupra b c: hoc au­<lb></lb><arrow.to.target n="marg559"></arrow.to.target><lb></lb>tem eſſe non poteſt, nam cum ſuppoſuerimus a b eſſe minorem a c <lb></lb>erit angulus a c b minor angulo a b c, quare a c b eſt minor recto, &amp; <lb></lb><arrow.to.target n="marg560"></arrow.to.target><lb></lb>ideò a c d maior recto, at a c d æqualis eſt a c d, alteri igitur a c d eſt <lb></lb><arrow.to.target n="marg561"></arrow.to.target><lb></lb>maior recto a c b minor, erit ergo pars maior toto.</s>
        </p>
        <p type="margin">
          <s id="id002862"><margin.target id="marg555"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002863"><margin.target id="marg556"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002864"><margin.target id="marg557"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 38. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002865"><margin.target id="marg558"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002866"><margin.target id="marg559"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002867"><margin.target id="marg560"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002868"><margin.target id="marg561"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id002869">LEMMA.</s>
        </p>
        <p type="main">
          <s id="id002870">His demonſtratis quis dicere poſſet ex ſuperius expoſitis quod <lb></lb><arrow.to.target n="marg562"></arrow.to.target><lb></lb>angulus rectilineus ſemper eſſet maior angulo contactus? </s>
          <s id="id002871">quia an­<lb></lb>gulus contactus non poteſt diuidi niſi obliqua linea, recti lineus <lb></lb>autem tam obliqua quam recta. </s>
          <s id="id002872">Propter hoc exponantur circuli <lb></lb><figure id="id.015.01.182.1.jpg" xlink:href="015/01/182/1.jpg"></figure><lb></lb>tres ſe tangentes a b, a c, a d hac rati­<lb></lb>one ut a b, b c, c d ſint æquales, erunt <lb></lb><arrow.to.target n="marg563"></arrow.to.target><lb></lb>enim centra omnia in linea conta­<lb></lb>ctus, &amp; ducatur a e f g recta quomo<lb></lb><arrow.to.target n="marg564"></arrow.to.target><lb></lb>dolibet: &amp; erunt ductis lineis b c, <lb></lb><arrow.to.target n="marg565"></arrow.to.target><lb></lb>c f, d g anguli e f g recti, quare om­<lb></lb>nes trigoni a b e, a c f, a d g, ſimiles <lb></lb><arrow.to.target n="marg566"></arrow.to.target><lb></lb>&amp; ideo a e, e f, f g æquales, atque por­<lb></lb>tiones a g, a f, a e, iuxta proportio­<lb></lb>nem circulorum, quare a g, erit ſex­<lb></lb>quialtera a f &amp; a f dupla a e, igitur <lb></lb><arrow.to.target n="marg567"></arrow.to.target><lb></lb>per præcedentem maior erit angu­<lb></lb>lus e a f, quam f a g, &amp; a d a ex recta <lb></lb><arrow.to.target n="marg568"></arrow.to.target><lb></lb>&amp; peripheria quam e a f, igitur augendo eadem ratione cum perue­<lb></lb>niamus ad angulum b a g qui fermè eſt recto æqualis cum deficiat <lb></lb>ſolo angulo contactus, liquet angulum e a g eſſe longè maiorem <lb></lb>multis rectilineis. </s>
          <s id="id002873">Iſtud poſſet etiam demonſtrari uia Archimedis <lb></lb>diuidendo arcus g a in h &amp; f a in k bifariam ducendo que lineas re­<lb></lb>ctas g h &amp; fk &amp; ita diuidendo h a in 1, &amp; k a in m bifariam, &amp; ducen­<lb></lb>do rectas atque ita ſemper appropinquando puncto a. </s>
          <s id="id002874">Concludo er­<lb></lb>go quod angulus <expan abbr="cõtactus">contactus</expan> ex recta &amp; peripheria eſt maior multis <lb></lb>rectilineis. </s>
          <s id="id002875">Cauſa autem erroris eſt quod multi exiſtimarunt corro­<lb></lb>larium illud eſſe Euclidis cum non ſit. </s>
          <s id="id002876">Nam Euclidi ſufficit hoc <lb></lb>quòd angulus contactus <expan abbr="nõ">non</expan> poſsit recta diuidi, nam eo utitur poſt <lb></lb><expan abbr="modũ">modum</expan> in demonſtrationibus. </s>
          <s id="id002877">Eo uerò quod ſit minor omnibus re­<lb></lb>ctilineis angulis non utitur, ideò etiam ſi <expan abbr="uerũ">uerum</expan> fuiſſet <expan abbr="nõ">non</expan> addidiſſet: <lb></lb>quanto minus: cum uerum non ſit, ideò fuit <expan abbr="adiectũ">adiectum</expan> ab aliquo qui <lb></lb><expan abbr="idẽ">idem</expan> fore credidit <expan abbr="nõ">non</expan> poſſe diuidi recta linea &amp; eſſe minus quocunque<lb></lb> quod recta linea diuidi poſſet, quod apertè ut dixi falſum eſt.</s>
        </p>
        <pb pagenum="164" xlink:href="015/01/183.jpg"></pb>
        <p type="margin">
          <s id="id002878"><margin.target id="marg562"></margin.target>L<emph type="italics"></emph>emmate<emph.end type="italics"></emph.end> 3. <lb></lb>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 159.</s>
        </p>
        <p type="margin">
          <s id="id002879"><margin.target id="marg563"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>ter<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002880"><margin.target id="marg564"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 31. <emph type="italics"></emph>ter<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002881"><margin.target id="marg565"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002882"><margin.target id="marg566"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002883"><margin.target id="marg567"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>diff <lb></lb>tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002884"><margin.target id="marg568"></margin.target>P<emph type="italics"></emph>er præce­<lb></lb>dentem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id002885">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002886">Ratio autem quòd omnis angulus contactus indiuiduus ſit, ſeu <lb></lb>duorum circulorum, ſeu circuli cum recta eſt, quoniam cum fuerint <lb></lb>duæ rationes contrariæ, &amp; una perpetuò minuitur, alia manet ne­<lb></lb>ceſſe eſt, ut tandem, quæ minuitur, ſuperetur ab ea quæ manet: cum <lb></lb>ergo circuli curuitas maneat, &amp; angulus tendat in punctum perpe­<lb></lb>tua diminutione neceſſe eſt, ut curuitas circuli impediat diuiſio­<lb></lb>nem rectè: ſed hoc habet duplicem obicem. </s>
          <s id="id002887">Primum, quia nullus <lb></lb>angulus ex circumferentia &amp; recta poſſet diuidi: hoc autem falſum <lb></lb>eſt manifeſtè, cum ſolus ille qui fit ex contactu lineæ, quæ non di­<lb></lb>uidit circulum, diuidi non poſsit. </s>
          <s id="id002888">Secundò, quod angulus conta­<lb></lb>ctus duorum circulorum ſe exterius tangentium multo minus <lb></lb>poſſet diuidi angulo contactus interioris duorum circulorum, <lb></lb>quod tamen falſum eſt: &amp; hoc animaduertit Campanus noſter, uir <lb></lb>acutus. </s>
          <s id="id002889">Dico ergo quòd in his qui ſe tangunt exterius, non fit diui­<lb></lb>ſio niſi ſemel: &amp; quamuis inclinentur mutuò, tamen in concurſu <lb></lb>non aptantur, ut cum obuiat rectæ aut cauæ parti circuli quia ne­<lb></lb>ceſſe eſt, ut accedat, in alio autem diſcedat: indicio eſt quod circu­<lb></lb>los ſe exterius tangentes, in puncto facilè deſcribes, interius uix fie­<lb></lb>ri poteſt, ſed uidentur coniuncti <lb></lb><figure id="id.015.01.183.1.jpg" xlink:href="015/01/183/1.jpg"></figure><lb></lb>per longum interuallum. </s>
          <s id="id002890">Ad aliud <lb></lb>dico, quòd ille angulus ex recta &amp; <lb></lb>peripheria conuexa circuli propter <lb></lb>diſceſſum ſeruat maiorem inclina­<lb></lb>tionem in quocunque puncto, quàm <lb></lb>ſit acceſſus conuexæ partis exterio­<lb></lb>ris circuli.</s>
        </p>
        <p type="main">
          <s id="id002891">Propoſitio centeſima ſexageſima <lb></lb>ſecunda.</s>
        </p>
        <p type="main">
          <s id="id002892">Proportionem duorum orbium <lb></lb>quorum diametrorum <expan abbr="cõuexæ">conuexæ</expan> par<lb></lb>tis, &amp; concauæ proportiones datæ <lb></lb>ſint, inueſtigare.</s>
        </p>
        <p type="main">
          <s id="id002893">Sint duo orbes a b c d &amp; e f g h, <lb></lb><arrow.to.target n="marg569"></arrow.to.target><lb></lb>&amp; ſit proportio a d ad b c, data &amp; e <lb></lb>h ad f g, data &amp; rurſus a d ad e h, di­<lb></lb>co orbis proportionem a b c d ad <lb></lb><expan abbr="orbẽ">orbem</expan> e f g h eſſe <expan abbr="datã">datam</expan>. </s>
          <s id="id002894">Quia. n. </s>
          <s id="id002895">propor<lb></lb>tio a d ſphærę ad b c eſt ueluti ad di <lb></lb>metientis ad b c <expan abbr="dimetientẽ">dimetientem</expan> triplicata, ideò <expan abbr="cũ">cum</expan> nota ſit a d ad b c di <lb></lb><arrow.to.target n="marg570"></arrow.to.target><lb></lb><expan abbr="metientiũ">metientium</expan>, erit nota <expan abbr="etiã">etiam</expan> a d ſphæræ ad b c <expan abbr="ſphęrã">ſphęram</expan>. </s>
          <s id="id002896">quare orbis ad ad <lb></lb><expan abbr="ſphęrã">ſphęram</expan> b c. nota eſt <expan abbr="etiã">etiam</expan> proportio b c <expan abbr="dimetiẽtis">dimetientis</expan> ad a d &amp; ad a d e h &amp; 


<pb pagenum="165" xlink:href="015/01/184.jpg"></pb>e h ad f g, igitur b c proportio dimetientis ad f g dimetientem nota. <lb></lb><arrow.to.target n="marg571"></arrow.to.target><lb></lb>Quare ſphæræ b c ad f g ſphæram. </s>
          <s id="id002897">at nota eſt proportio f g ad e h <lb></lb>dimetientium igitur &amp; ſphærarum: igitur nota eſt f g ſphæræ ad or<lb></lb>bem e h, igitur cum nota ſit proportio orbis ad a d ſphæram b c, &amp; <lb></lb>b c ſphæræ ad f g ſphæram, &amp; f g ſphæræ ad orbem e h, erit propor<lb></lb>tio orbis a d ad orbem e h nota, quod eſt propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id002898"><margin.target id="marg569"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002899"><margin.target id="marg570"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>duo <lb></lb>decimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id002900"><margin.target id="marg571"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 22. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem. <lb></lb></s>
          <s id="id002901">&amp;<emph.end type="italics"></emph.end> A<emph type="italics"></emph>lizam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id002902">Propoſitio centeſima ſexageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id002903">Proportionem uirium ſtellarum per motus ſuos indagare.</s>
        </p>
        <p type="main">
          <s id="id002904">Mouentur ſtellæ omnes ab Oriente in Occidentem die una, qui <lb></lb><arrow.to.target n="marg572"></arrow.to.target><lb></lb>motus fit à prima mente, quæ mouet: ideò quod ad hoc attinet non <lb></lb>eſt diuerſitas: uerùm in motibus ab Occidente in Orientem <expan abbr="cũ">cum</expan> ſint <lb></lb>proprij, oportet conſiderare tempus, in quo <expan abbr="circumuertũtur">circumuertuntur</expan>, &amp; ma<lb></lb>gnitudinem ambitus, &amp; inde magnitudinem orbis, qui circumagi­<lb></lb>tur, &amp; horum trium facta comparatione dignoſcitur robur uirium <lb></lb>ſtellarum &amp; uitarum quæ mouent eas. </s>
          <s id="id002905">Ponatur ergo, ut uelim pro­<lb></lb>portionem uitę Saturni ad uitam Lunæ: erit ergo (ut docet Alphra<lb></lb><arrow.to.target n="marg573"></arrow.to.target><lb></lb>ganus) Luna, cum eſt in longitudine propiore, altitudinem habens <lb></lb>109000 M.P. &amp; cum eſt in longitudine longiore 208500, tota igitur <lb></lb>dimetiens 417000 M.P. mane 218000 M.P. </s>
          <s id="id002906">Igitur proportio ſolida­<lb></lb>rum ſphærarum eſt uelut 72511713 ad 10360232, remanebit ergo <lb></lb>proportio orbis ad ſphæram elementorum, ut 62151481 ad <lb></lb>10360232, &amp; eſt ſexcuplum fermè. </s>
          <s id="id002907">Rurſus proportio dimetientis al­<lb></lb>titudinis Saturni ad contentum eſt uelut 2011 ad 1440, &amp; eſt propè <lb></lb>201 ad 114, quare 67 ad 38, quare ſphærarum ut 300000 ad 55000 <lb></lb>ferme. </s>
          <s id="id002908">Igitur ferè ut 60 ad 11. Rurſus proportio dimetientis ſphæ­<lb></lb>ræ Saturni ad dimetientem ſphæræ Lunæ eſt propè 313, &amp; ſphæra­<lb></lb>rum ſolidarum 306 317 10. Perinde eſt. </s>
          <s id="id002909">Quia ergo proportio ſphæ­<lb></lb>ræ Saturni ad ſphæram Lunæ eſt 30631710, &amp; orbis Lunæ eſt 5/6 <lb></lb>ſolum ſphæræ ſuæ diuidemus 30631710 per 5/6, &amp; exibit proportio <lb></lb>ſphæræ Saturni ad orbem Lunæ 36758052, at quia proportio ſo­<lb></lb>lidæ ſphæræ Saturni ad contentum eſt ut 60 ad 11, erit ſphæræ ad <lb></lb>orbem, ut 60 ad 49 reſiduum, diuidam ergo 36758052 per 60, exe­<lb></lb>unt 612634, &amp; ducam per 49, id eſt per 100, fit 61263400, &amp; diuiden <lb></lb>do per 2, exit 30631700, detraho 612634, relinquitur proportio or­<lb></lb>bis Saturni ad orbem Lunæ 30019066.</s>
        </p>
        <p type="margin">
          <s id="id002910"><margin.target id="marg572"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002911"><margin.target id="marg573"></margin.target>D<emph type="italics"></emph>iff.<emph.end type="italics"></emph.end> 21.</s>
        </p>
        <p type="main">
          <s id="id002912">Iam uerò circuitus Saturni ad circulum Lunæ, proportio eſt 313, <lb></lb>ut uiſum eſt, Lunæ autem tempus per ſex ductum eſt 164 dies, Sa­<lb></lb>turni 177 anni propemodum, qui ſunt dies 64649 diuide, duc <lb></lb>ergo 313 in 164, fiunt 51332. Idem ergo peragrat Luna in <lb></lb>51332 diebus, quod Saturnus in 64649, &amp; eſt quo ad hoc agi­


<pb pagenum="166" xlink:href="015/01/185.jpg"></pb>lior, ut ita dicam, quarta parte: at Saturnus, ut dictum eſt, mouet or­<lb></lb>bem 30019066, ſed lentiùs quinta parte, detrahe illam fiet robur Sa<lb></lb>turni in comparatione ad Lunam 24015253.</s>
        </p>
        <p type="main">
          <s id="id002913">Eſt tamen Luna multo agilior ob propinquitatem, &amp; ob uarie­<lb></lb>tatem luminis, &amp; magnitudinem ſuperficiei. </s>
          <s id="id002914">Et etiam quod maius <lb></lb>eſt ob id quod defert ad nos uires omnium ſyderum, nihilominus <lb></lb>quo ad uires uix eſt comparatio.</s>
        </p>
        <p type="head">
          <s id="id002915">SCHOLIVM.<lb></lb><arrow.to.target n="marg574"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002916"><margin.target id="marg574"></margin.target>46</s>
        </p>
        <p type="main">
          <s id="id002917">Multum autem differt hæc propoſitio à ſuperiore, nam in illa <lb></lb>quæſiuimus uim uitarum ex proportione ad ſua corpora, quæ <lb></lb>quodammodo eſt quodammodo, non hic autem exponimus uim <lb></lb>uitarum ex earum operatione. </s>
          <s id="id002918">Propterea ſubijciemus breuiter alti­<lb></lb>tudinem proportiones in minore longitudine &amp; maiori<lb></lb><arrow.to.target n="table19"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table19"></table.target>
          <row>
            <cell>Luna</cell>
            <cell>in minore altitudine</cell>
            <cell>51</cell>
            <cell>in maiore</cell>
            <cell>64</cell>
          </row>
          <row>
            <cell>Mercurij</cell>
            <cell>in minore</cell>
            <cell>64</cell>
            <cell>in maiore</cell>
            <cell>167</cell>
          </row>
          <row>
            <cell>Veneris</cell>
            <cell>in minore</cell>
            <cell>167</cell>
            <cell>in maiore</cell>
            <cell>1120</cell>
          </row>
          <row>
            <cell>Solis</cell>
            <cell>in minore</cell>
            <cell>1120</cell>
            <cell>in maiore</cell>
            <cell>1220</cell>
          </row>
          <row>
            <cell>Martis</cell>
            <cell>in minore</cell>
            <cell>1220</cell>
            <cell>in maiore</cell>
            <cell>8876</cell>
          </row>
          <row>
            <cell>Iouis</cell>
            <cell>in minore</cell>
            <cell>8876</cell>
            <cell>in maiore</cell>
            <cell>14405</cell>
          </row>
          <row>
            <cell>Saturni</cell>
            <cell>in minore</cell>
            <cell>14405</cell>
            <cell>in maiore</cell>
            <cell>20110</cell>
          </row>
        </table>
        <p type="main">
          <s id="id002919">Stellarum fixarum propior 20110 longior non habetur. </s>
          <s id="id002920">Et hæ <lb></lb>menſuræ ſunt in comparatione ad ſemidiametrum terræ. </s>
          <s id="id002921">Et iuxta <lb></lb>id quod potuit ſecundum rationem haberi: nam demonſtratio ſola <lb></lb>eſt de altitudinibus Solis &amp; Lunæ, &amp; eorum magnitudinibus à </s>
        </p>
        <p type="main">
          <s id="id002922"><arrow.to.target n="marg575"></arrow.to.target><lb></lb>Ptolemæo in magna compoſitione.</s>
        </p>
        <p type="margin">
          <s id="id002923"><margin.target id="marg575"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>cap.<emph.end type="italics"></emph.end><lb></lb>14. 15. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end><lb></lb>16.</s>
        </p>
        <p type="main">
          <s id="id002924">Propoſitio centeſima ſexageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id002925">Syderum proportionem in magnitudine oſtendere.<lb></lb><arrow.to.target n="table20"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table20"></table.target>
          <row>
            <cell>Luna ad terram comparata</cell>
            <cell>1/39</cell>
          </row>
          <row>
            <cell>Mercurij corpus</cell>
            <cell>1/22000</cell>
          </row>
          <row>
            <cell>Veneris</cell>
            <cell>1/29</cell>
          </row>
          <row>
            <cell>Solis corpus</cell>
            <cell>166</cell>
          </row>
          <row>
            <cell>Martis</cell>
            <cell>15/8</cell>
          </row>
          <row>
            <cell>Iouis</cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>Saturni</cell>
            <cell>91</cell>
          </row>
        </table>
        <p type="main">
          <s id="id002926">Stellarum autem fixarum inſignium unaquæque etiam minima, ſi <lb></lb><arrow.to.target n="marg576"></arrow.to.target><lb></lb>credendum eſt Alphragano, eſt centies maior tota terra, unde ca­<lb></lb>nem neceſſe eſt centies mille maiorem eſſe, eſt enim in eadem altitu<lb></lb>dine, &amp; dimetiens decuplus dimetienti ſtellarum ſecundæ magni­<lb></lb>tudinis, quas ille inſignes uocat: aliter Saturnus non tantus eſſe <lb></lb>poſſet, cum ſit minimus aſpectu.</s>
        </p>
        <pb pagenum="167" xlink:href="015/01/186.jpg"></pb>
        <p type="margin">
          <s id="id002927"><margin.target id="marg576"></margin.target>D<emph type="italics"></emph>iff.<emph.end type="italics"></emph.end> 22.</s>
        </p>
        <p type="main">
          <s id="id002928">Propoſitio centeſima ſexageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id002929">Propoſitionem motuum omnium <expan abbr="ſtellarũ">ſtellarum</expan> ad ſolem conſiderare.</s>
        </p>
        <p type="main">
          <s id="id002930">Videtur Sol quaſi Rex in Cœlo, nam omnes orbes cum illius <lb></lb><arrow.to.target n="marg577"></arrow.to.target><lb></lb>motu conueniunt, &amp; uidetur es admiratione digna his, qui non <lb></lb>nouerunt, quanta ſit concordia omnium rerum, de qua infrà dice­<lb></lb>mus. </s>
          <s id="id002931">Ergo Luna primum hoc habet, ut linea æqualis motu Solis <lb></lb>ſemper media ſit inter lineam æqualis motus Lunę &amp; loci maximè <lb></lb>inæqualitatis motus eius, ubi ſcilicet tardiſsimè mouetur, Veneris <lb></lb>autem &amp; Mercurij ut motus æquales idem ſemper ſint cum motu <lb></lb>æquali, &amp; locus cum loco ipſius Solis ad unguem præter id quod <lb></lb>infrà dicemus. </s>
          <s id="id002932">Trium uerò <expan abbr="ſuperiorũ">ſuperiorum</expan> ratio ſic <expan abbr="cõſtat">conſtat</expan> ad Solem ut à <lb></lb>Ptolemęo <expan abbr="obſeruatũ">obſeruatum</expan> eſt ex Hipparcho. </s>
          <s id="id002933">In omni reſtitutione cuiuſ­<lb></lb>libet planetę ſuperioris numerus <expan abbr="reuolutionũ">reuolutionum</expan> Solis ęqualis eſt nu­<lb></lb>mero <expan abbr="reſtitutionũ">reſtitutionum</expan> planetę <expan abbr="ſecundũ">ſecundum</expan> <expan abbr="motũ">motum</expan> æqualitatis &amp; inęqualita<lb></lb>tis pariter acceptis. </s>
          <s id="id002934">Velut Saturnus in annis quinquaginta nouem <lb></lb>die una &amp; horis decem octo quinquageſies ſepties per motum inę­<lb></lb>qualem ad <expan abbr="unguẽ">unguem</expan>, per æqualem autem duabus reuolutionibus par <lb></lb>te inſuper una &amp; quadraginta quin que minutijs, quæ reſpondent di­<lb></lb>ei uni, &amp; horis decem octo ex motu Solis, &amp; ita bis Saturnus reuol<lb></lb>uitur ſecundum motum æqualitatis &amp; quinquageſies ſepties per <lb></lb>motum inæqualem &amp; ſimiliter. </s>
          <s id="id002935">Iupiter in annis 70, diebus trecen­<lb></lb>tis ſexaginta, horis quatuor, ſexaginta quinque reuolutiones inęqua<lb></lb>les perficiet &amp; ſex ęquales, deficientibus ex ęqualibus quatuor par­<lb></lb>tibus &amp; dextante quod eſt <expan abbr="quãtum">quantum</expan> peragraret Solin quatuor die­<lb></lb>bus, &amp; dextante diei ad perfectionem ſcilicet annorum ſeptuaginta <lb></lb>atque unius. </s>
          <s id="id002936">Martis quo que ſtella in annis ſeptuaginta nouem, &amp; die­<lb></lb>bus tribus &amp; horis fermè quatuor triginta nouem facit inæquali­<lb></lb>tatis reuolutiones: æqualitatis autem quadraginta duas, &amp; inſuper <lb></lb>partes tres cum ſextante, quas manifeſtum eſt peragrari à Sole in <lb></lb>diebus tribus atque horis quatuor. </s>
          <s id="id002937">Veneris quo que ſydus in octo an­<lb></lb>nis deficientibus diebus duobus &amp; quadrante, inæqualitatis quin­<lb></lb>que perficit reuolutiones, æqualitatis autem tantundem ad un<expan abbr="guẽ">guem</expan> <lb></lb>quantum Sol deficiente eadem parte ſeu diebus duobus &amp; qua­<lb></lb>drante. </s>
          <s id="id002938">Mercurij quo que ſtella in quadraginta ſex annis &amp; una die <lb></lb>&amp; hora una fermè quadraginta ſex fermè perficit reuolutiones æ­<lb></lb>qualis motus &amp; inſuper gradum unum cum portione reſpondenti <lb></lb>portioni temporis, id eſt, horæ fermè uni: in æqualitatis autem cen­<lb></lb>ſum quadraginta quin que. </s>
          <s id="id002939">Atque hęc ſunt manifeſtiſsima et ut dixi ad­<lb></lb>miranda ſunt, præterea alia minus generalia, aut minus manifeſta <lb></lb>aut non tanti momenti quæ conſultò prætermitto, non eſt. </s>
          <s id="id002940">n. </s>
          <s id="id002941">locus <lb></lb>hic docendi artes ſingulas ſed ſolum ea tractandi quæ ad argumen<pb pagenum="168" xlink:href="015/01/187.jpg"></pb>tum pertinent. </s>
          <s id="id002942">Igitur ut ad rem redeam. </s>
          <s id="id002943">Solis cum octauo Orbe ea <lb></lb>ratio eſt, ut linea quam ille permeat eadem ſit quam quę fixę ſtellæ, <lb></lb>non. </s>
          <s id="id002944">n. </s>
          <s id="id002945">ad eandem diſtantiam &amp; mente conceptam ab æquinoctijs <lb></lb>deſcendentem ac æquidiſtantem mouetur, ſed ad eam ſecundum <lb></lb>quam ſtellę fixę in octauo orbe mouentur in comparatione ad ecli­<lb></lb>pticam ſuperioris orbis. </s>
          <s id="id002946">Porrò de his atque huiuſmodi in Paralipo­<lb></lb>menis diximus, ubi etiam docuimus quomodo ſecundum duos cir<lb></lb><arrow.to.target n="marg578"></arrow.to.target><lb></lb>culos, qui ſolum circa ſuum centrum mouentur, punctus datus per<lb></lb>petuò in recta linea feratur.</s>
        </p>
        <p type="margin">
          <s id="id002947"><margin.target id="marg577"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id002948"><margin.target id="marg578"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 14. <lb></lb><emph type="italics"></emph>cap.<emph.end type="italics"></emph.end> 7.</s>
        </p>
        <p type="main">
          <s id="id002949">Propoſitio centeſima ſexageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id002950">Proportiones muſicas ſuperpartientes in eas quæ particula una <lb></lb>tantum abundant reducere.<lb></lb><arrow.to.target n="marg579"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id002951"><margin.target id="marg579"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id002952">Ptolemęi hoc inuentum fuit, ut &amp; multa alia pręclara: itaque ſta­<lb></lb>tuendum eſt, primum uoces ęquales non concentum efficere, quia <lb></lb>diuerſæ non ſunt, quę autem diuerſę ſunt, nihilominus proportio­<lb></lb>ne conſtant ſimpliciſsima &amp; multiplici, tales optimam efficiunt ar­<lb></lb>moniam. </s>
          <s id="id002953">Eiuſmodi ſunt quæ in dupla ſunt proportione, uocatur <lb></lb>autem diapaſon. </s>
          <s id="id002954">1. quaſi omnia comprehendens non à numero uo­<lb></lb>cum uelut diapente &amp; diateſſaron à quatuor &amp; quin que uo cibus. </s>
          <s id="id002955">In <lb></lb>diapaſo. </s>
          <s id="id002956">n. </s>
          <s id="id002957">omnia <expan abbr="cõprehendi">comprehendi</expan> uidentur. </s>
          <s id="id002958">1. omnes uo <expan abbr="cũ">cum</expan> differentiæ, <lb></lb><expan abbr="quanquã">quanquam</expan> ex octo <expan abbr="tantũ">tantum</expan> uo cibus conſtet. </s>
          <s id="id002959">Pòſt ſunt quæ in <expan abbr="q̃drupla">quadrupla</expan>, <lb></lb>unde bis diapaſon, poſt quæ in tripla, nam propior eſt monadi ſeu ę­<lb></lb>qualitati: ſed non adeò ſimplex ut bis diapaſon. </s>
          <s id="id002960">Vocant <expan abbr="aũt">aut</expan> hanc <lb></lb>diapaſon diapente: inde <expan abbr="ſubſequit̃">ſubſequitur</expan> octupla quę uix in uocibus </s>
          <s id="id002961">huma­<lb></lb>nis habetur: <expan abbr="frequẽs">frequens</expan> in inſtrumentis, uo<expan abbr="cat̃que">caturque</expan> tris diapaſon inde ſex­<lb></lb>cupla, ſeu bis diapaſon diapente. </s>
          <s id="id002962">Quintupla <expan abbr="aũt">aut</expan> minus <expan abbr="cõcors">concors</expan> eſt: <lb></lb>ſed de hac inferius dicemus, atque de multiplicibus </s>
          <s id="id002963">dicta ſunto. </s>
          <s id="id002964">Sed de <lb></lb><expan abbr="cõ">com</expan>centu ex particula ſuperaddita ſexquialtera ſexquitertia atque alijs <lb></lb>nunc agendum. </s>
          <s id="id002965">Clarum eſt. </s>
          <s id="id002966">n. </s>
          <s id="id002967">has eſſe ſimpliciſsimas. </s>
          <s id="id002968">Cum ergo du<lb></lb>pla proportio non magis poſsit diuidi æqualibus interuallis atque<lb></lb> ſimplicibus proportionibus quàm in ſexquialteram &amp; ſexquiter­<lb></lb>tiam, uelut inter 4 &amp; 2 interpoſito 3. nam proportio 3 ad 2 eſt ſex­<lb></lb>quialtera, &amp; 4 ad 3 ſexquitertia: nec melius poteſt diuidi, at ſexqui­<lb></lb>alteram &amp; ſexquitertiam quantumuis magnis numeris diuidere <lb></lb>non licebat melius aut commodius quam per ſexquioctauas: uelu­<lb></lb>ti ſumpto numero 64 cui duplus eſt 128, inter medius 96 qui cum <lb></lb>64 ſexquialteram facit proportionem, quæ ſuauiſsima eſt omni­<lb></lb>um deductis multiplicibus, uocaturque diapente. </s>
          <s id="id002969">At quæ eſt 128 ad <lb></lb>96 ſexquitertia eſt minuſque benè ſonat per ſe, ſed in acutioribus uo­<lb></lb>cibus ſolum cum alijs benè ſonat, uelut cum diapente, perficiens <lb></lb>diapaſon, interuallum, ergo inter 96 &amp; 64 diuiſum per ſexquiocta­<pb pagenum="169" xlink:href="015/01/188.jpg"></pb>uas producit 72 et 81, <expan abbr="nã">nam</expan> 72 ad 64 eſt ſexquio<expan abbr="ctauũ">ctauum</expan>, ſicut 81 ad 72. uerùm <lb></lb>id accidebat in <expan abbr="cõmodi">commodi</expan> quae 81 ad 64 <expan abbr="nullã">nullam</expan> habet <expan abbr="proportionẽ">proportionem</expan> <expan abbr="commodã">commodam</expan>, <lb></lb>&amp; multo minus 96 ad 81, quare uiſum eſt Ptolemęo ut ſubtracta mona<lb></lb>de <expan abbr="fierẽt">fierent</expan> termini 64, 72, 80, &amp; 96, proportio <expan abbr="aũt">aut</expan> 80 ad 64 <expan abbr="cõſtituit">conſtituit</expan> ſexqui<lb></lb><expan abbr="quartã">quartam</expan> atque <expan abbr="ditonũ">ditonum</expan>, proportio quoque 96 ad 72 <expan abbr="ſexquitertiã">ſexquitertiam</expan> <expan abbr="ſemiditonũ">ſemiditonum</expan> que. <lb></lb></s>
          <s id="id002970">Rurſus proportio 128 ad 64 <expan abbr="cõponit̃">componitur</expan> ex proportionibus </s>
          <s id="id002971">80 ad 64, <expan abbr="q̃">quae</expan> <expan abbr="habet̃">habetur</expan> <lb></lb>pro ditono ut <expan abbr="dictũ">dictum</expan> eſt, &amp; eſt ſexquiquarta proportio. </s>
          <s id="id002972">At 128 cum 80 eſt in <lb></lb>proportione ſuperpartiente tres quintas, <expan abbr="q̃">quae</expan> <expan abbr="iterũ">iterum</expan> eſt conſona. </s>
          <s id="id002973">Regula <expan abbr="em̃">emm</expan> <lb></lb>eſt quae ubi conſonantia uo <expan abbr="cũ">cum</expan> <expan abbr="diuidat̃">diuidatur</expan> in duas partes, <expan abbr="quarũ">quarum</expan> una ſit conſo<lb></lb>nans, <expan abbr="reliquã">reliquam</expan> <expan abbr="etiã">etiam</expan> eſſe <expan abbr="conſonantẽ">conſonantem</expan>, at <expan abbr="nõ">non</expan> <expan abbr="cõuertit̃">conuertitur</expan>. </s>
          <s id="id002974">Sępe. </s>
          <s id="id002975">n. </s>
          <s id="id002976">fit ut ex duabus <lb></lb></s>
          <s id="id002977">conſonantibus diſſonans <expan abbr="cõpoſitio">compoſitio</expan> <expan abbr="oriat̃">oriatur</expan>, uelut ex duplici <expan abbr="diapẽte">diapente</expan>, aut <lb></lb><expan abbr="diapẽte">diapente</expan> <expan abbr="cũ">cum</expan> ditono, ſed ut ad <expan abbr="propoſitũ">propoſitum</expan> reuertar, alia diapaſon eſt inter 80 <lb></lb>&amp; 40, at inter 48 &amp; 40 eſt ſemiditonus ut <expan abbr="oſtẽſum">oſtenſum</expan> eſt, uelut inter 96 &amp; <lb></lb>80, nam inter 45 &amp; 40 eſt proportio ſexquioctaua, inter 48 <expan abbr="aũt">aut</expan> &amp; 45 ſex­<lb></lb>quiquinta decima, <expan abbr="igit̃">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="q̃">quae</expan> eſt ſuperbi­<lb></lb>partiens tertias ſeu ſolida <expan abbr="cũ">cum</expan> beſſe ſeu ſexta maior erit <expan abbr="cõſonans">conſonans</expan>. </s>
          <s id="id002978">Iam er<lb></lb>go uidemus detractione aut additione ſexquioctuageſimæ, concinnas <lb></lb>reddi uulgatiores armonias: <expan abbr="tertiã">tertiam</expan> utran que <expan abbr="maiorẽ">maiorem</expan> ſcilicet &amp; <expan abbr="minorẽ">minorem</expan>, ac <lb></lb>rurſus <expan abbr="ſextã">ſextam</expan> <expan abbr="maiorẽ">maiorem</expan> atque minore <expan abbr="q̃">quae</expan> in minoribus numeris ſcilicet à mo­<lb></lb>nade ad octo poſitæ ſunt. </s>
          <s id="id002979">Vides præterea <expan abbr="ſemiditonũ">ſemiditonum</expan> in ſexquiquinta <lb></lb><arrow.to.target n="table21"></arrow.to.target><lb></lb><expan abbr="cõſtare">conſtare</expan>: adeò ut à ſenario infra nihil inutile <lb></lb><figure id="id.015.01.188.1.jpg" xlink:href="015/01/188/1.jpg"></figure>reddatur. </s>
          <s id="id002980">Diateſſaron <expan abbr="aũt">aut</expan> cum primum di <lb></lb>uidi poteſt, ſi ſecus diuidatur &lt;08&gt; in <expan abbr="ditonũ">ditonum</expan> <lb></lb>&amp; <expan abbr="ſemitoniũ">ſemitonium</expan>, aut in ſemiditonum &amp; <expan abbr="tonũ">tonum</expan>, <lb></lb>ſcilicet in duo <expan abbr="tantũ">tantum</expan> interualla, non <expan abbr="cõmo­dius">commo­<lb></lb>dius</expan> <expan abbr="quã">quam</expan> inter octo &amp; ſeptem &amp; ſex diuidi <lb></lb>poteſt. </s>
          <s id="id002981">Cum ergo octo ad <expan abbr="ſeptẽ">ſeptem</expan> diſſona ſit, <lb></lb>quippe nimis remota eſt hęc proportio à ſen<lb></lb>ſu humano: <expan abbr="quamobrẽ">quamobrem</expan> ex regula data, ne­<lb></lb>que proportio <expan abbr="ſeptẽ">ſeptem</expan> ad ſex. </s>
          <s id="id002982">Sed dubitabis <lb></lb>meritò, quia <expan abbr="cũ">cum</expan> diateſſaron diuidatur <expan abbr="bifa­riã">bifa­<lb></lb>riam</expan>, in <expan abbr="ditonũ">ditonum</expan> &amp; <expan abbr="ſemitoniũ">ſemitonium</expan>, ac rurſus in <expan abbr="ſe­miditonũ">ſe­<lb></lb>miditonum</expan> &amp; <expan abbr="tonũ">tonum</expan>, quarum altera <expan abbr="cõſonans">conſonans</expan> eſt, reliqua <expan abbr="nõ">non</expan>. </s>
          <s id="id002983"><expan abbr="Videt̃">Videtur</expan> ergo <lb></lb>infirmari regula illa, quae conſonantia diuiſa ſi una pars <expan abbr="cõſonet">conſonet</expan>, alia non <lb></lb>poſsit eſſe diſſonans, <expan abbr="nã">nam</expan> conſtat <expan abbr="coniũ">conium</expan> &amp; <expan abbr="ſemitoniũ">ſemitonium</expan> tam per ſe quam in <lb></lb><expan abbr="cõpoſitione">compoſitione</expan> diſſonare: &amp; <expan abbr="nõ">non</expan> <expan abbr="parũ">parum</expan> ſed acerbè. </s>
          <s id="id002984"><expan abbr="Verũ">Verum</expan> reſpondeo diateſſa<lb></lb>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="em̃">emm</expan> per <lb></lb>ſe eſt, diſſonans eſt: at que ſic in <expan abbr="conſonantẽ">conſonantem</expan> &amp; diſſonantem diuidi poteſt: <lb></lb>quatenus <expan abbr="aũt">aut</expan> pars eſt diapaſon <expan abbr="cõſonans">conſonans</expan> in acutis: quan &lt;08&gt; <expan abbr="etiã">etiam</expan> adiecta <lb></lb>ditono aut ſemiditono ſuprà efficiat <expan abbr="ſextã">ſextam</expan> maiorem aut <expan abbr="minorẽ">minorem</expan> parum <lb></lb>benè ſonantes. </s>
          <s id="id002985">At quintupla proportio ut ab initio propoſitum eſt, <expan abbr="cõſtat">conſtat</expan> <lb></lb>bis diapaſon, &amp; ſexquiquarta, ut planè <expan abbr="manifeſtũ">manifeſtum</expan> eſt: ſexquiquarta <expan abbr="aũt">aut</expan> 


<pb pagenum="170" xlink:href="015/01/189.jpg"></pb>ditonus: bis diapaſon <expan abbr="aũt">aut</expan> quindecim uo cibus. </s>
          <s id="id002986">Omnes igitur decem, &amp; <lb></lb><expan abbr="ſeptẽ">ſeptem</expan> uoces, <expan abbr="q̃">quae</expan> ſexdecim interuallis <expan abbr="diſtinguunt̃">diſtinguuntur</expan>, conſonantes ſunt: &amp; ex <lb></lb>genere ditoni, &amp; ſexquiquartæ, ſed paulo minus benè <expan abbr="ſonãt">ſonant</expan> quod ditonus <lb></lb>ipſe. </s>
          <s id="id002987">Igitur <expan abbr="quintuplã">quintuplam</expan> multiplicem ad ſex <expan abbr="quiquartã">quiquartam</expan> reduximus. </s>
          <s id="id002988">Verum <lb></lb>ut oſtenſum eſt &amp; decimaſeptima, <expan abbr="q̃">quae</expan> bis diapaſon <expan abbr="cõſtat">conſtat</expan>, &amp; ſemiditono <lb></lb>benè ſonat, hęc <expan abbr="aũt">aut</expan> inter nonaginta ſex &amp; uiginti: quadrupla <expan abbr="igit̃">igitur</expan> eſt &amp; <lb></lb>ſuperquadripartiens quintas. </s>
          <s id="id002989">Diapaſon quo que cum ſexta maiore &amp; mi<lb></lb>nore eandem habent rationem quam 16 ad 5, &amp; 10 ad 3, triplam utranque, <lb></lb>ſed altera ſexquiquinta, altera ſexquitertia: bis diapaſon uerò <expan abbr="cũ">cum</expan> eiſdem <lb></lb>ut uiginti ad tria, &amp; 32 ad quin que ſexcupla utraque: ſed altera ſuperbipar­<lb></lb>tiens tertias, altera quintas. </s>
          <s id="id002990"><expan abbr="Manifeſtũ">Manifeſtum</expan> eſt igitur hanc diuiſionem <expan abbr="nõ">non</expan> ſo­<lb></lb>lum concinnam magis eſſe &amp; ſuauem ſed omnem <expan abbr="tonorũ">tonorum</expan> &amp; ſemitonio­<lb></lb>rum <expan abbr="neceſsitatẽ">neceſsitatem</expan> effugere. </s>
          <s id="id002991">Quòd uerò in cauſa fuit ut toni &amp; ſemitonia <lb></lb>in uſu eſſent, id eſt, quoniam in <expan abbr="diſcẽdo">diſcendo</expan> neceſſe eſt eandem ſeruari ratio­<lb></lb>nem in <expan abbr="crementorũ">crementorum</expan>, ne que arithmeticam ſed <expan abbr="geometricã">geometricam</expan>. </s>
          <s id="id002992">Ideò <expan abbr="aſcẽſus">aſcenſus</expan> per <lb></lb>tonos &amp; ſemitonia <expan abbr="cõmodus">commodus</expan> fuit, nam duplicem <expan abbr="ſolũ">ſolum</expan> differentiam pue<lb></lb>ri uſu aſſequi coguntur. </s>
          <s id="id002993">At uerò poterat &amp; per ſexquiſextam diuidi dia<lb></lb>teſſaron, ut inter triginta ſex &amp; quadraginta nouem interpoſitis 42, ue­<lb></lb>rùm triplex <expan abbr="ſequebat̃">ſequebatur</expan> in <expan abbr="cõueniens">conueniens</expan>: primum ut diateſſaron ad amuſsim <lb></lb>non ſeruaretur, ſed incidebat in cacophoniam, addita quadrageſima o­<lb></lb>ctaua parte: deficiente <expan abbr="aũt">aut</expan> in duabus ſexquiſeptimis numeris ſeu propor<lb></lb>tione ſexquitertia: ut inter 49 &amp; 64 loco 48 &amp; 64, uelut <expan abbr="etiã">etiam</expan> inter 48 ad <lb></lb>36, addita igitur monade in termino medio utrin que fit diſſonantia. </s>
          <s id="id002994">Se­<lb></lb>cundum inconueniens, eſt quae ſic diuidente non ſeruabatur ratio ſexqui­<lb></lb>quartæ &amp; ſexquiquintæ ſeu ditoni &amp; ſemiditoni, quæ uoces benè ſo­<lb></lb>nant. </s>
          <s id="id002995">Tertium inconueniens erat, quòd hæc ratio diuidendi diapentes <lb></lb>minimè ſatisfaciebat, uelut inter 324 &amp; 216. Interponere enim neceſſe <lb></lb>erat 252 &amp; 294, unde incongrua rurſus erat diuiſio. </s>
          <s id="id002996">His tot cauſis cum <lb></lb>proportiones maiores non fatisfacerent ut ſexqui quinta quæ diateſſa­<lb></lb>ron nullo modo æqualiter diuidere poteſt, &amp; in diapente deficit ſexqui<lb></lb>uigeſima quarta, ut inter 25 &amp; 36, coacti ſunt cum nec ſexquiſexta nec <lb></lb>ſexquiſeptima idoneæ eſſent ad ſexquioctauam confugere.</s>
        </p>
        <table>
          <table.target id="table21"></table.target>
          <row>
            <cell>Diapaſon</cell>
            <cell>2</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Bis diapaſon</cell>
            <cell>4</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Diapaſon diapente</cell>
            <cell>3</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Tris diapaſon</cell>
            <cell>8</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Bis diapaſon <expan abbr="diapẽte">diapente</expan></cell>
            <cell>6</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Hæmiolia</cell>
            <cell>3</cell>
            <cell>2</cell>
          </row>
          <row>
            <cell>Hæmitritæa</cell>
            <cell>4</cell>
            <cell>3</cell>
          </row>
          <row>
            <cell>Ditonus</cell>
            <cell>5</cell>
            <cell>4</cell>
          </row>
          <row>
            <cell>Semiditonus</cell>
            <cell>6</cell>
            <cell>5</cell>
          </row>
          <row>
            <cell>Sexta minor</cell>
            <cell>8</cell>
            <cell>5</cell>
          </row>
          <row>
            <cell>Sexta maior</cell>
            <cell>5</cell>
            <cell>3</cell>
          </row>
          <row>
            <cell>Bis diapaſon ditonus</cell>
            <cell>5</cell>
            <cell>1</cell>
          </row>
        </table>
        <p type="main">
          <s id="id002997">Eſt &amp; alia diuiſio toni in ſemitonia, <expan abbr="q̃">quae</expan> eſt uaria <expan abbr="ponẽdo">ponendo</expan> <expan abbr="tonũ">tonum</expan> inter 18 <lb></lb>&amp; 16, media uox eſt 17 ſemitonium maius inter 17 &amp; 16, ſed minus inter <lb></lb>18 &amp; 17, <expan abbr="quorũ">quorum</expan> differentia eſt 1/288. Hic ſubit admiratio quomodo <expan abbr="ſemi­toniũ">ſemi­<lb></lb>tonium</expan> minus <expan abbr="aptet̃">aptetur</expan> tam gratè in ſymphonijs, maius <expan abbr="aũt">aunt</expan> <expan abbr="nequaquã">nequaquam</expan>. </s>
          <s id="id002998">Ptole<lb></lb>męus hoc negaret, quia ſexquiquinta ſeu ſemiditonus <expan abbr="cõſtat">conſtat</expan> tono inte­<lb></lb>gro, qui eſt inter 90 &amp; 80, &amp; ſemitonio <expan abbr="pluſquã">pluſquam</expan> maiore quod eſt inter <lb></lb>96 &amp; 90, &amp; eſt ſexquiquinta decima: <expan abbr="q̃">quae</expan> maior eſt tono maiore 1/255. Pro­<lb></lb>pterea dicemus cauſam eſſe quae poſito ſemiditono inter 81 &amp; 96, id eſt, <lb></lb>27 &amp; 32 ſublato tono, id eſt, 234 &amp; 216, remanebit 13 differentia 256 ad <lb></lb>243, ſeu qualis eſt 96 ad 91 &amp; 1/8 quæ eſt ut 768 ad 729 et redit ad <expan abbr="idẽ">idem</expan>, ſcili<pb pagenum="171" xlink:href="015/01/190.jpg"></pb>cet, ut 256 ad 243, 13 autem eſt paulo plus decimanona, ergo multo mi­<lb></lb>nus ſemitonio minore. </s>
          <s id="id002999">ſecundum <expan abbr="mẽtem">mentem</expan> ergo Ptolemæi, poſito tono <lb></lb>inter 135, &amp; 120, &amp; ſemitonio maiore inter 128 &amp; 120 remanebit ſemito­<lb></lb>nium minus fermè inter 19 &amp; 18, id eſt, 133 &amp; 126, quę proportio differt <lb></lb>à 135 &amp; 138. Si quis autem bene animaduertat, ſexquioctuageſima illa <lb></lb>adimitur, ex tono &amp; additur ſemitonio minori, &amp; hæc eſt cauſa quòd <lb></lb>ſemitonium maius Ptolemæi ſit concinnum, quia additur tonis imper<lb></lb>fectis. </s>
          <s id="id003000">Dimidium autem ſemitonij minoris eſt inter 36 &amp; 35, &amp; uocatur <lb></lb><expan abbr="cõma">comma</expan>: &amp; eſt minus &amp; maius: maius eſt inter 35 &amp; 34, rurſus <expan abbr="cõma">comma</expan> mi­<lb></lb>nus diuiditur in duas dieſes, minorem, quæ eſt inter 72 &amp; 71, &amp; maio­<lb></lb>rem, quę eſt inter 71 &amp; 70, &amp; ideò manet difficultas quomodo intenta <lb></lb>uoce per dieſim fiat melior conſonantia? </s>
          <s id="id003001">nam de remiſsione poſſemus <lb></lb>dicere quòd accipitur loco ſexquioctuageſimæ: ſed in ſexquioctuage­<lb></lb>ſima remittitur de tono ſecundum mentem Ptolemæi, in dieſi intendi­<lb></lb>tur ſemitonium minus, ſicut oſtendit experimentum, ſed forſan conue<lb></lb>niunt quia intentio ſemitonij minoris deducit ſemiditonum ad ſexqui<lb></lb>quintam: eſt enim differentia ſemitonij minoris intenti hoc modo ad <lb></lb>ſemitonium minus, ut 136 ad 135: ſed hoc eſt longè minus ſexquioctua<lb></lb>geſima, unum ſat eſt, hanc eſſe ultimam diuiſionem toni in octo par­<lb></lb>tes, &amp; ut in diatonico toni dominantur, ita in chromatico ſemitonia in <lb></lb>enarmonico dieſes, ſed dieſes fugitando (ut ita dicam) ac aures uelli­<lb></lb>cando, mirum in modum oblectant audientes: uelut toni ſtando, un­<lb></lb>de etiam nomen, ſemitonia medium modum obtinent.</s>
        </p>
        <p type="main">
          <s id="id003002">Tertium genus proportionis (omitto modò <expan abbr="diuiſionẽ">diuiſionem</expan> temporum <lb></lb>binarij, ternarij, quinarij, qui ultimus eſt eorum quos ſenſus recipiat, <lb></lb>nam ſeptenarius propinquior eſt binarij diuiſioni ob octonarium, &amp; <lb></lb>modos illos ſatis notos Doricum, Lydium &amp; Phrigium, ac eiuſmodi) <lb></lb>eſt Ptolemæi: rurſus qui cum uideret deſpectam futuram muſicæ con­<lb></lb>templationem, conatus eſt illius aliquod ſingulare emolumentum <lb></lb>oſtendere, quemadmodum fecit &amp; in libro de Prædictionibus, exiſti­<lb></lb>mans ni illos compoſuiſſet ueluti pręmium oſtendentes tanti laboris <lb></lb>quantus neceſſarius uideretur ad intellectum librorum Magnæ com­<lb></lb>poſitionis, futurum eſſe, ut hi negligerentur, ergo &amp; hoc in muſicæ li­<lb></lb>bris oſtendere molitus eſt, ſcilicet, præclarum eſſe <expan abbr="aliquẽ">aliquem</expan> huius <expan abbr="cõtem­plationis">contem­<lb></lb>plationis</expan> finem, quod <expan abbr="utinã">utinam</expan> non feciſſet, ne illud uerè de eo dici poſſet:</s>
        </p>
        <p type="main">
          <s id="id003003">—Non omnia poſſumus omnes.</s>
        </p>
        <p type="main">
          <s id="id003004">Virum enim hunc ſupra omnem humani ingenij <expan abbr="metã">metam</expan> fuiſſe <expan abbr="nõ">non</expan> nega­<lb></lb>mus: ſed hanc partem quam hic agit, adeò infeliciter tractat, ut malim <lb></lb>credere <expan abbr="totũ">totum</expan> illum tertium <expan abbr="librũ">librum</expan> fuiſſe ab aliquo alio <expan abbr="adiectũ">adiectum</expan>. </s>
          <s id="id003005">Etenim <lb></lb>quid turpius ſapienti homini &lt;08&gt; imitari uulgares illos? </s>
          <s id="id003006"><expan abbr="ſeptẽ">ſeptem</expan> planetæ, <lb></lb>ſeptem mundi miracula, <expan abbr="ſeptẽ">ſeptem</expan> artes liberales: quid enim ſimilitudo nu<pb pagenum="172" xlink:href="015/01/191.jpg"></pb>meri iuuare poteſt, aut quàm afferre utilitatem? </s>
          <s id="id003007">nimis certè in <expan abbr="dignũ">dignum</expan> eſt <lb></lb>uti <expan abbr="argumẽto">argumento</expan> à ſimilitudine ſumpto: tum maximè adeò leui. </s>
          <s id="id003008">Sed quo­<lb></lb>niam conſtat omnia quæ in mundo ſunt ordine coniuncta eſſe, &amp; ne­<lb></lb>ceſsitate uinciri, ideò cùm finis ipſe uerus ſit, non tam debemus Ptole­<lb></lb>mæum damnare, quae non probauerit, quàm laudare, quod <expan abbr="ueritatẽ">ueritatem</expan> ſine <lb></lb>ratione ſit aſſectus. </s>
          <s id="id003009">Sæpe enim accidit huiuſmodi uiris adeò pręſtan­<lb></lb>tibus ut ueritas detegatur, quam cùm illi, ut mos eſt <expan abbr="hominũ">hominum</expan>, rationi­<lb></lb>bus adornare nituntur, tranſgredientes metam muneris, in abſurda &amp; <lb></lb>ineptias <expan abbr="incidũt">incidunt</expan>. </s>
          <s id="id003010">Ergo id modò declarare aggrediar, ſupponens quae ue­<lb></lb>rum eſt, ſcilicet hanc muſicam <expan abbr="concinnitatẽ">concinnitatem</expan> cum diuinis <expan abbr="connexã">connexam</expan> eſſe, <lb></lb>&amp; ab illis originem ducere. </s>
          <s id="id003011">Verùm dubium eſt, an ſoni propter nume <lb></lb>ros iucundi ſint, an propter aliud? </s>
          <s id="id003012">&amp; ſi propter aliud, cur ergo numeri <lb></lb>ad hoc ſunt neceſſarij? </s>
          <s id="id003013">&amp; cur obſeruare eos oportet ne ab illorum ordi <lb></lb>ne diſiungi poſsint? </s>
          <s id="id003014">Hoc <expan abbr="aũt">aut</expan> perfacilè <expan abbr="intelligit̃">intelligitur</expan>, &amp; à nobis aliâs decla­<lb></lb>ratum eſt, ſcilicet delectare nos, quæ percipiuntur quæque ratione facta <lb></lb>uidentur, <expan abbr="quoniã">quoniam</expan> in his naturæ uis relucet &amp; imago uniuerſi, ergo dele<lb></lb>ctant nos, quoniam naturę ordine nos conſtamus. </s>
          <s id="id003015">Illud difficilius lon<lb></lb>gè q̊d <expan abbr="tamẽ">tamen</expan> diligenti obſeruatione <expan abbr="dignũ">dignum</expan> uidetur, ſcilicet, quonam pa<lb></lb>cto harmonia cum rebus cœleſtibus aut humanis <expan abbr="cõiuncta">coniuncta</expan> ſit. </s>
          <s id="id003016">Forſan <lb></lb>&amp; illud ab re non eſſet intelligere, cur nullum animal pręter hominem <lb></lb>capax ſit harmoniæ? </s>
          <s id="id003017">an forſan <expan abbr="quoniã">quoniam</expan> ſolus homo ratione participet, <lb></lb>&amp; ob id ſolus gaudet ratione? </s>
          <s id="id003018">ordinata <expan abbr="aũt">aut</expan> ratione <expan abbr="cõſtant">conſtant</expan> aut ſola aut <lb></lb>maximè, numerus autem quid aliud eſt quàm ordinis <expan abbr="ſeparatorũ">ſeparatorum</expan> ima­<lb></lb>go. </s>
          <s id="id003019">Porrò hæc accipienda ſunt ex his quæ ſenſibus deprehenduntur, <lb></lb>qualia ſunt quae animus mouetur &amp; uarios affectus induit iuxta harmo­<lb></lb>niæ diuerſitatem lætitię, triſtitię, impetus, remiſsionis, timoris, ſpei, ira­<lb></lb>cundiæ, &amp; commiſerationis. </s>
          <s id="id003020">Nos enim maximè octo affectus mouent <lb></lb>muſicæ modulationes. </s>
          <s id="id003021">Secundum quid autem mouent? </s>
          <s id="id003022">uel quia con­<lb></lb>ſonæ aut diſſonæ, uel quia concitatę aut tardæ, uel quod maius eſt quae<lb></lb> tendant in acutum ad alacritatem, uel in grauem deſinant &amp; remiſſum <lb></lb>ſonum ad <expan abbr="cõmiſerationem">commiſerationem</expan>, &amp; lachrymas, aut etiam ex modo tetrachor<lb></lb>dorum. </s>
          <s id="id003023">Illud ſanè non obſcurum eſt, <expan abbr="animã">animam</expan> cum ſono maximè eſſe con<lb></lb><expan abbr="iunctã">iunctam</expan>, nam neque odoribus ut odores ſunt, neque ſaporibus, aut his quæ <lb></lb>tanguntur licet plurimum delectent, aut etiam lædant, anima mouetur <lb></lb>ad affectus, licet, ut dixi, magis homo delectetur, aut triſtitia afficiatur <lb></lb>quemadmodum ex ſonorum uaria natura, quod etiam in morſis à Ta<lb></lb>rantula (araneę genus eſt) deprehenditur. </s>
          <s id="id003024">Quinimò nec à luce nec à co<lb></lb>loribus aut pictura, niſi ut hæc ad memoriam <expan abbr="reuocãt">reuocant</expan> ea, propter quæ <lb></lb>ad hilaritatem aut triſtitiam uel iram, uel commiſerationem mouemur. <lb></lb></s>
          <s id="id003025">Vnde <expan abbr="quoſdã">quoſdam</expan> reges ferunt iniurias acceptas iuſsiſſe depingi in aula ne <lb></lb>poſſent obliuiſci, at longè plures <expan abbr="curarũt">curarunt</expan>, ut potius <expan abbr="eorũ">eorum</expan> facta egregia 


<pb pagenum="173" xlink:href="015/01/192.jpg"></pb>pingerentur continuata per memoriam uoluptate, quam dum illa àge <lb></lb>rent, <expan abbr="cõceperant">conceperant</expan>: nihilominus, neque color ipſe, nec lux aut ſpectaculum <lb></lb>uel imagines poſſunt adeò mouere animi affectus, uel ſonus. </s>
          <s id="id003026">Nam <lb></lb>duo in uniuerſum ex uiſu ad animi affectus mouendos habentur, tene<lb></lb>bræ ad triſtitiam &amp; metum, pictura regionum <expan abbr="amœnarũ">amœnarum</expan> ad iucundita<lb></lb>tem, ſed <expan abbr="irã">iram</expan> quæ moueant picturæ alacritatemúe aut <expan abbr="cõmiſerationem">commiſerationem</expan>, <lb></lb>non habemus. </s>
          <s id="id003027">Videtur ergo ob hæc ſonus ipſe magis animæ intimus <lb></lb>&lt;08&gt; ullum aliud ſenſile. </s>
          <s id="id003028">Quod ſi odoratus eſt in <expan abbr="appẽdicibus">appendicibus</expan> cerebri, ui <lb></lb>ſus in pupilla oculi, guſtus in linguæ neruis, ueri ſimile eſt magis inti­<lb></lb>mum eſſe auditum, ſcilicet in cerebro ipſo, atque ob id magis ab illo mo­<lb></lb>ueri animam. </s>
          <s id="id003029">Neque <expan abbr="em̃">emm</expan> in <expan abbr="aẽre">aere</expan> concepto à concauitatibus auris, qui no<lb></lb>ſtri pars non eſt: neque à tympano, cùm ſuperflua fuiſſet cauitas interior <lb></lb>omnis: neque enim inter pupillam &amp; cerebrum pars ulla cernitur ad ui­<lb></lb>ſum adiuuandum idonea: ſed ſolus ſufficit conſenſus pupillę cum cere<lb></lb>bro: nam ad nos per ſpiritus differtur imago, non <expan abbr="em̃">emm</expan> uiſus eſſet unus, <lb></lb>nec in uno tempore fieret, ſed ueluti è <expan abbr="ſecũdo">ſecundo</expan> ſpeculo &amp; decimo ſimul, <lb></lb>&amp; eodem tempore reflectitur imago, ut à primo ita ſenſus uiſus ex pu­<lb></lb>pilla in cerebro &amp; in corde &amp; anima ſimul relucet. </s>
          <s id="id003030">At ergo non potuit <lb></lb>in tympano uel neruo denſiore fieri auditus, ſed in cerebro ipſo, ob q̊d <lb></lb>magis moueret affectus. </s>
          <s id="id003031">Sed &amp; magis incorporeus eſt ſonus, ut qui <lb></lb>inſtrumentum proprium non afficiat, niſi cum immoderatus fuerit, at <lb></lb>omnis color, omnis lux oculum afficit, ac, ut ita dicam, tingit, neque ſuc­<lb></lb>ceſsiones illas ob id adeò minutas oculus percipere poteſt ut auris, <lb></lb>ſed coinquinatur, ut ita dicam, priorum obiectorum reliquijs atque ima<lb></lb>ginibus. </s>
          <s id="id003032">Vt in uniuerſum conſtet puriorem eſſe auditus ſenſum etiam <lb></lb>animæ noſtræ propiorem quàm uiſum.</s>
        </p>
        <p type="main">
          <s id="id003033">Quibus conſtitutis uidendum eſt, quomodo ſonus permutet affe­<lb></lb>ctus: hoc autem <expan abbr="nõ">non</expan> quia animam, quæ immortalis eſt &amp; immateriaria, <lb></lb>ſed quoniam aut corporis eam partem, quæ eſt animæ inſtrumentum, <lb></lb>id eſt, ſpiritum, aut animæ <expan abbr="principalẽ">principalem</expan> coniunctionem qua corpori an­<lb></lb>nexa eſt. </s>
          <s id="id003034">Vt enim corpus deſerit aut impeditur à corporis commercio <lb></lb>corpus immoritur: hoc præſentiens animus, fiunt illa duo præuia ad <lb></lb>mortem timor &amp; triſtitia. </s>
          <s id="id003035">Vt contrà, lætitia non eſt niſi communicatio <lb></lb>animæ corpori, &amp; quatenus communicatur ſolum de uita cogitat, atque<lb></lb> ob id quaſi immortalis, qui lætatur obliuiſcitur mortis. </s>
          <s id="id003036">Ergo animę ra<lb></lb>tio illa erit, quæ ut cognoſcit perfectè exhilaratur dulcedine uo cum, &amp; <lb></lb>hoc fit in diapaſon. </s>
          <s id="id003037">Vt uerò imperfectè diapente, ut imperfectius dia­<lb></lb>teſſaron, at cum ex diateſſaro &amp; diapente perficitur diapaſon, accidit ei <lb></lb><expan abbr="idẽ">idem</expan>, quod <expan abbr="quærẽti">quærenti</expan> gemmas in matrice dum inuenit, &amp; ei qui ex tabulis <lb></lb>arcam <expan abbr="cõficit">conficit</expan>, &amp; puero <expan abbr="cũ">cum</expan> adoleſcit, &amp; generaliter ei qui ex imperfectis <lb></lb>perfecta colligit: ex quintæ enim &amp; quartæ ſenſu <expan abbr="imperfectarũ">imperfectarum</expan> conſo­


<pb pagenum="174" xlink:href="015/01/193.jpg"></pb>nantiarum percipit perfectam diapaſon. </s>
          <s id="id003038">Videamus ergo an aliquid ſit <lb></lb>ſimile in animæ facultatibus, nec <expan abbr="dubiũ">dubium</expan> eſt quin ex ſenſibus. </s>
          <s id="id003039">exterioribus <lb></lb>atque interioribus fiat intelligentia. </s>
          <s id="id003040">Et ſenſus <expan abbr="quidẽ">quidem</expan> exteriores ſexquiter <lb></lb>tia <expan abbr="cõſtant">conſtant</expan>: eſt enim <expan abbr="illorũ">illorum</expan> imperfecta cognitio: maior longè memorię <lb></lb>unius &amp; rationis reliquarumque <expan abbr="facultatũ">facultatum</expan>, ex quibus <expan abbr="intelligẽtia">intelligentia</expan> oritur. <lb></lb></s>
          <s id="id003041">Iam uerò habemus exactam <expan abbr="ſimilitudinẽ">ſimilitudinem</expan> facultatum animę humanę, <expan abbr="q̃">quae</expan> <lb></lb>cognoſcit. </s>
          <s id="id003042">Nunc ulterius procedamus et uideamus, an ſit aliqua <expan abbr="etiã">etiam</expan> con<lb></lb>iunctio inter illas, nam ſimilitudo etſi ſit una originis cauſa, non tamen <lb></lb>ſola digna eſt ut à Philoſopho <expan abbr="numeret̃">numeretur</expan> inter cauſas ordinis &amp; natura­<lb></lb>lis uinculi. </s>
          <s id="id003043">Non eſt ut <expan abbr="tetrachordorũ">tetrachordorum</expan> genera ad partes animę <expan abbr="cõparen­tur">comparen­<lb></lb>tur</expan>, <expan abbr="cũ">cum</expan> ſint uoluntaria diuiſione, non natura conſtituta. </s>
          <s id="id003044">Sed ſi quis hoc <lb></lb>uelit, magis ad rationem proprietatis reſpiciat, ſuauitas in chromatico, <lb></lb>ſubtilitas in Enarmonico, ſtabilitas in diatonico: Vt <expan abbr="Enarmonicũ">Enarmonicum</expan> ad <lb></lb>mentem uerè referri poſsit, <expan abbr="chromaticũ">chromaticum</expan> ad ſenſus: <expan abbr="diatonicũ">diatonicum</expan> ad <expan abbr="uitã">uitam</expan> na <lb></lb>turalemque facultatem. </s>
          <s id="id003045">Sed, ut dixi, iam propius accedamus, <expan abbr="cõcitatior">concitatior</expan> ſo<lb></lb>nus, ut Doricus ad alacritatem pertinet, ad pugnam, ad uim animę ira­<lb></lb>ſcibilis: Phrygius ad <expan abbr="uoluptatẽ">uoluptatem</expan>, Lydius ad intelligentiam remiſsione <lb></lb><expan abbr="corporeorũ">corporeorum</expan> affectuum. </s>
          <s id="id003046">Sed <expan abbr="nõ">non</expan> quęrere decet aut laborare, ut malè in­<lb></lb>uenta aut diſtributa aptemus ordini naturę, ſed ut res rebus. </s>
          <s id="id003047">Diximus <lb></lb>quatuor eſſe <expan abbr="differẽtias">differentias</expan> <expan abbr="nobiliorũ">nobiliorum</expan> <expan abbr="affectuũ">affectuum</expan> animi, ſcilicet, timoris, ſpei, <lb></lb><expan abbr="iracũdię">iracundię</expan> ſeu ſęuitię &amp; <expan abbr="cõmiſerationis">commiſerationis</expan>, lętitię, triſtitię, impetus ac remiſ­<lb></lb>ſionis. </s>
          <s id="id003048">Et <expan abbr="uidet̃">uidetur</expan> muſica nec hoc ęqualiter monere, ſed <expan abbr="primũ">primum</expan> uideamus <lb></lb>an hi ſoli affectus ſint maximi, quippe deeſſe <expan abbr="uident̃">uidentur</expan> amor atque odium. <lb></lb></s>
          <s id="id003049">Et mihi dubium non eſt quin hi potentiſsimi ſint <expan abbr="omniũ">omnium</expan> præter <expan abbr="metũ">metum</expan>. <lb></lb></s>
          <s id="id003050">Sed metus <expan abbr="cũ">cum</expan> cauſa, affectus propriè <expan abbr="nõ">non</expan> eſt, ſed potius ſcientia <expan abbr="quædã">quædam</expan>. <lb></lb></s>
          <s id="id003051">Proprium enim perturbationum eſt excedere rationem: at metus mor<lb></lb>tis, proprię aut de filio, non eſt à ratione alienús, nec excedit metas, modò <lb></lb>inanis non ſit aut falſus, ob hoc metum excludemus ab hoc negocio: <lb></lb>tum maximè ob id quod nulla muſica eſt quæ <expan abbr="metũ">metum</expan> excitet cùm ea, <expan abbr="nõ">non</expan> <lb></lb>opus ſit in eo, qui ſit cum ratione coniunctus. </s>
          <s id="id003052">Indicio eſt quae potius <expan abbr="illũ">illum</expan> <lb></lb>excudit abrupta muſica, ſicut &amp; omnia alia quæ perturbant rationem, <lb></lb>ueluti <expan abbr="ſolanũ">ſolanum</expan> &amp; madrangora atque cicuta. </s>
          <s id="id003053">Amorem igitur &amp; odium <expan abbr="nõ">non</expan> <lb></lb>excitat muſica, quia amor &amp; odium alicuius ſunt amor &amp; odium, muſi <lb></lb>ca <expan abbr="aũt">aunt</expan> generales ſolum mouet animi affectus. </s>
          <s id="id003054">Et commiſeratio, licet ſit <lb></lb>Didonis aut Phillidis, tamen eſt generaliter miſerentis. </s>
          <s id="id003055">Quęramus er­<lb></lb>go rurſus qui ſint affectus generales animi. </s>
          <s id="id003056">Et ſanè <expan abbr="uident̃">uidentur</expan> eſſe lætitia <lb></lb>atque triſtitia: impetus &amp; remiſsio: ſęuitia ac miſericordia &amp; audacia. </s>
          <s id="id003057"><expan abbr="Sũt">Sunt</expan> <lb></lb>tria ferme <expan abbr="cõiũcta">coniuncta</expan> ſimul impetus &amp; ſæuitia atque audacia, <expan abbr="quoniã">quoniam</expan> <expan abbr="cũ">cum</expan> mo<lb></lb>tu perturbato animi ſunt eiecta ratione. </s>
          <s id="id003058">Ob id <expan abbr="unũquod">ununquod</expan>que <expan abbr="horũ">horum</expan> ab ira­<lb></lb>cundia <expan abbr="deriuat̃">deriuatur</expan>. </s>
          <s id="id003059">Quapropter &amp; ita <expan abbr="rationẽ">rationem</expan> expellit aut ſuppeditat. </s>
          <s id="id003060">at ra<lb></lb>tio <expan abbr="perturbat̃">perturbatur</expan>, aut ab immodicis ſonis, aut in <expan abbr="cõptis">comptis</expan> et magnas mutatio 


<pb pagenum="175" xlink:href="015/01/194.jpg"></pb>nes habentibus atque aſperis. </s>
          <s id="id003061">Hæc autem, ut ita dicam, nulla eſt muſica. <lb></lb></s>
          <s id="id003062">Sed neque muſica ulla triſtitiam gignit, cum ut dixi, triſtitia nil aliud ſit &lt;08&gt;<lb></lb>mortis imago, muſica <expan abbr="aũt">aut</expan> uitam fouet. </s>
          <s id="id003063">Vnde <expan abbr="nõ">non</expan> immeritò fertur Xeno <lb></lb>philus muſicus <expan abbr="centũ">centum</expan> quinque annis ſine aliquo <expan abbr="incõmodo">incommodo</expan> uixiſſe, quod <lb></lb>ſingulare eſſe exemplum in humana uita refert Plinius. </s>
          <s id="id003064">Relinquitur igi<lb></lb>tur tandem, ut muſica maximè moueat tres affectus lætitiam, remiſsio­<lb></lb>nem &amp; miſericordiam. </s>
          <s id="id003065">Et quod ex his poſtmodum ad labores inſurga­<lb></lb>mus intentius, hoc non eſt ex muſicę ui aut facultate, ſed <expan abbr="cõſequentibus">conſequentibus</expan> <lb></lb>ad illa alia cauſis. </s>
          <s id="id003066">Neque ergo <expan abbr="horũ">horum</expan> cauſas ex diuiſionibus atque diſtribu­<lb></lb>tionibus uoluntarijs muſicæ <expan abbr="cõſiderare">conſiderare</expan> oportet, ſed ex ipſa <expan abbr="rerũ">rerum</expan> natura <lb></lb>atque eſſentia. </s>
          <s id="id003067">Veluti intentionis et remiſsionis, aſperitatis atque ſuauitatis <lb></lb>celeritatis ac tarditatis; <expan abbr="cõſonantium">conſonantium</expan> aut diſſonantium uo <expan abbr="cũ">cum</expan> at que muta­<lb></lb>tionis: hæ enim differentię præcipuę ſunt uo cum, uel etiam teſte Ariſto <lb></lb>tele. </s>
          <s id="id003068">Verùm <expan abbr="nõ">non</expan> obſcurum eſt: quemadmodum remiſsiones fiant animi </s>
        </p>
        <p type="main">
          <s id="id003069"><arrow.to.target n="marg580"></arrow.to.target><lb></lb>affectuum, <expan abbr="cũ">cum</expan> remittuntur uoces aut intendantur ad <expan abbr="earũ">earum</expan> intentionem. <lb></lb></s>
          <s id="id003070">Sed non eſt æqualis ratio, quoniam natura noſtra ad <expan abbr="remiſsionẽ">remiſsionem</expan> natu­<lb></lb>raliter inclinata eſt, ad intentionem non ita, ſed per uim <expan abbr="quandã">quandam</expan> aut me­<lb></lb>dio uoluptatis, aut cum anima purior eſt à corporis impedimentis. </s>
          <s id="id003071">Et <lb></lb>ob id ad ſtudia nil aptius eſt pura ſobrietate: nihil ineptius crapula atque<lb></lb> temulentia. </s>
          <s id="id003072">At lętitię cauſę ſunt, &amp; <expan abbr="cõ">con</expan>cordia uo <expan abbr="cũ">cum</expan>, &amp; mutatio ex aſpera <lb></lb>in ſuauem, <expan abbr="nõ">non</expan> ſecus ac eius qui euadit è paupertate uel è moleſtia aliqua <lb></lb>aut dolore aut alio <expan abbr="incõmodo">incommodo</expan>, tum intenſio uo <expan abbr="cũ">cum</expan> ac liber ſonus. </s>
          <s id="id003073">Vnde <lb></lb>in lętitia ſolent homines exclamare. </s>
          <s id="id003074">At ad <expan abbr="cõmiſerationem">commiſerationem</expan> mouendam <lb></lb>omnia remitti oportet ex magna in parua, adeoque deficientem ex aſpera <lb></lb>in leuem, ex ueloci in tardam, ex diſſona in conſonantem. </s>
          <s id="id003075">Antiqui ergo <lb></lb>(ut author eſt Cælius Rhodiginius) Dorico ad temperantiam &amp; mode <lb></lb><arrow.to.target n="marg581"></arrow.to.target><lb></lb>rationem utebantur, ſcilicet quòd non haberet præcipites lapſus, neque<lb></lb> arduas intentiones: Phrygio ad impetum &amp; bellicum ardorem, ſcilicet <lb></lb>per aſperas intentiones: Lydio ad fletus &amp; lamentationes per caſus &amp; <lb></lb>remiſsiones longas ac ſuaues: ideo funeribus peculiaris: Mixolydio ad <lb></lb>commiſerationem, ut defectiones interponantur &amp; breues abruptæque<lb></lb> remiſsiones, iuuantque in hoc plurimum &amp; ſenſus uerborum, familiaris <lb></lb>hic tragædijs: Aeolicus qui &amp; Ionicus tranquillitatis animi author eſt <lb></lb>ſomnumque conciliat: Dorico non abſimilis ſed ſuauior &amp; mollior: ideò <lb></lb>chromatici generis. </s>
          <s id="id003076">Quę uerò ad cœli motus referuntur, diapaſon qui­<lb></lb>dem refertur ad motum diurnum, nam maximo conſtat, &amp; exactiſsimo <lb></lb>interuallo, unusque eſt in omnibus &amp; iucundiſsimus &amp; omnia continet, <lb></lb>uelut &amp; diurnus motus. </s>
          <s id="id003077">Proprius autem tàm erraticis quàm fixis, qui <lb></lb>etiam æqualitati propinquior eſt, &amp; ad maiorem diſtantiam ſcilicet de­<lb></lb>clinationis ſigniferi ab æquinoctij circulo ad diapente refertur. </s>
          <s id="id003078">Rurſus <lb></lb>diateſſaron quòd minimo <expan abbr="cõſtat">conſtat</expan> interuallo ac maximè inæquali, &amp; per <lb></lb>ſe quidem quaſi non neceſſario ad motum in latitudinem <expan abbr="refert̃">refertur</expan>, is enim 


<pb pagenum="176" xlink:href="015/01/195.jpg"></pb>exiguus eſt &amp; inæqualis. </s>
          <s id="id003079">Ex horum itaque duorum <expan abbr="cõpoſitione">compoſitione</expan> quem­<lb></lb>admodum et ex diateſſaro &amp; diapente conformatur diapaſon, pulchra <lb></lb>conſtruitur exortus &amp; occaſus ſyderum ratio, quæ primo motu <expan abbr="cõſtat">conſtat</expan>.</s>
        </p>
        <p type="margin">
          <s id="id003080"><margin.target id="marg580"></margin.target>I<emph type="italics"></emph>n lib. 

de<emph.end type="italics"></emph.end> A<emph type="italics"></emph>u<lb></lb>dibilibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003081"><margin.target id="marg581"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>ca.<emph.end type="italics"></emph.end> 3.</s>
        </p>
        <p type="main">
          <s id="id003082">Porrò de participatione diapente, quam non <expan abbr="ſolũ">ſolum</expan> uſurpamus in <expan abbr="in­ſtrumẽtis">in­<lb></lb>ſtrumentis</expan> fiſtularum organis dictis: ſed <expan abbr="etiã">etiam</expan> in fidibus <expan abbr="monachordorũ">monachordorum</expan> <lb></lb>ſeu <expan abbr="clauichordorũ">clauichordorum</expan> (ita. </s>
          <s id="id003083">n. </s>
          <s id="id003084">nunc uo<expan abbr="cant̃">cantur</expan> <expan abbr="inſtrumẽta">inſtrumenta</expan> quib. </s>
          <s id="id003085">caruerunt anti­<lb></lb>qui) <expan abbr="nõ">non</expan> alia eſt ratio, quàm <expan abbr="q̃">quae</expan> dicta eſt <expan abbr="conſtituendarũ">conſtituendarum</expan> conſonantiarum <lb></lb>in ditonis &amp; ſemiditonis ſextaque utraque. </s>
          <s id="id003086">Vt <expan abbr="em̃">emm</expan> quatuor conſonantiæ <lb></lb>ſuauiores <expan abbr="efficerent̃">efficerentur</expan>, neceſſe fuit <expan abbr="unã">unam</expan>, ſcilicet <expan abbr="diapentẽ">diapentem</expan> uariari. </s>
          <s id="id003087">Exempli <lb></lb>gratia, ſint fides expoſitę octo, &amp; ut <expan abbr="conſtituat̃">conſtituatur</expan> proportio h ad c, ut 128 <lb></lb><figure id="id.015.01.195.1.jpg" xlink:href="015/01/195/1.jpg"></figure><arrow.to.target n="table22"></arrow.to.target><lb></lb>ad 80, id eſt ut 8 ad 5, c facta eſt remiſsior octogeſima, quare <expan abbr="cũ">cum</expan> <lb></lb>81 diapente habeat ad 121 <expan abbr="cũ">cum</expan> dimidio, erit ad 80 maior 1 1/2, id eſt <lb></lb>octuageſima parte 120, quare intentior diapente. </s>
          <s id="id003088">At in diapaſo <lb></lb>omnia ad <expan abbr="idẽ">idem</expan> redeunt: <expan abbr="horũ">horum</expan> etiam cauſa ſemitonia nigra illa ad­<lb></lb>dita ſunt. </s>
          <s id="id003089">Sed hęc tractatio proprium <expan abbr="locũ">locum</expan> exigeret, ſecus eſſet ni­<lb></lb>mis curioſi illa huc traducere. </s>
          <s id="id003090">quemadmodum, &amp; ut uellemus <lb></lb>Philoſophiam naturalem, <expan abbr="moralẽ">moralem</expan>, &amp; <expan abbr="mathematicã">mathematicam</expan> ad <expan abbr="muſicã">muſicam</expan> tra<lb></lb>ducere <expan abbr="proportionẽ">proportionem</expan>. </s>
          <s id="id003091">Melius ſanè fuiſſet ſubtilioribus rationibus <lb></lb><expan abbr="hãc">hanc</expan> <expan abbr="mẽſuris">menſuris</expan> <expan abbr="motuũ">motuum</expan> <expan abbr="aſtrorũ">aſtrorum</expan> pro ut <expan abbr="cõueniũt">conueniunt</expan> (<expan abbr="quantũ">quantum</expan> fieri potuit) aptaſſe.</s>
        </p>
        <table>
          <table.target id="table22"></table.target>
          <row>
            <cell>a</cell>
            <cell>ut</cell>
          </row>
          <row>
            <cell>b</cell>
            <cell>re</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell>mi</cell>
          </row>
          <row>
            <cell>d</cell>
            <cell>fa</cell>
          </row>
          <row>
            <cell>e</cell>
            <cell>ſol</cell>
          </row>
          <row>
            <cell>f</cell>
            <cell>re</cell>
          </row>
          <row>
            <cell>g</cell>
            <cell>mi</cell>
          </row>
          <row>
            <cell>h</cell>
            <cell>fa</cell>
          </row>
        </table>
        <p type="main">
          <s id="id003092">Propoſitio centeſima ſexageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id003093">Proportionem muſicam ad ſapores &amp; odores coaptare.<lb></lb><arrow.to.target n="marg582"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003094"><margin.target id="marg582"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003095">Melius feciſſet Ptolemęus, ſi <expan abbr="hãc">hanc</expan> proportionem ad ſapores &amp; odores <lb></lb>et picturas, <expan abbr="quemadmodũ">quemadmodum</expan> inuenimus nos, applicaſſet, uel ut Vitruuius <lb></lb>ad machinas, poterat <expan abbr="em̃">emm</expan> hoc ſcire, cum Vitruuius pluſ&lt;08&gt; centum quin­<lb></lb>quaginta annis <expan abbr="Ptolemęũ">Ptolemęum</expan> anteceſſerit. </s>
          <s id="id003096">Et quan&lt;08&gt; Latinè ſcripſerit, non <lb></lb>tam turpè erat latina legiſſe, aut <expan abbr="cõuerſa">conuerſa</expan> ab alio quopiam intellexiſſe, &lt;08&gt;<lb></lb>neſciuiſſe neceſſaria pulchraque inuenta aliorum clarorum uirorum, &amp; <lb></lb>quod deterius erat, <expan abbr="rerũ">rerum</expan> memorabilium loco fabulas ſubtexuiſſe. </s>
          <s id="id003097">Ergo <lb></lb>ut ad rem ueniam: muſica proportio bifariam <expan abbr="inuenit̃">inuenitur</expan> in ſaporibus: ſim­<lb></lb>pliciter, &amp; ex comparatione, &amp; ſimpliciter quidem ſumma ſuauitas ad <lb></lb>diapaſon refertur: eſt enim ſuauiſsimus concenſus in ſaporibus, ergo <lb></lb>dulce ei <expan abbr="reſpõdet">reſpondet</expan>, ut ſimplex, quid enim ſuauius eſſe poteſt in utro que ge<lb></lb>nere. </s>
          <s id="id003098">At pinguis, qualis in carnibus &amp; ouis benè pręparatis ad <expan abbr="diapẽte">diapente</expan> <lb></lb>refertur, eſt enim &amp; ipſe ſuauiſsimus poſt dulce, at que in ſuo genere perfe<lb></lb>ctus, diateſſaron uerò optimè ſalſo <expan abbr="cõuenit">conuenit</expan>. </s>
          <s id="id003099">Hic enim per ſe improbus <lb></lb>eſt &amp; inſuauis, ſicut etiam ſapor ſalſus eſt, diateſſaron <expan abbr="aũt">aunt</expan> cum diapente <lb></lb>perficit diapaſon, &amp; cum diapaſo inutile eſt, et diſcordat, ita ſapor ſalſus <lb></lb>cum pingui ſummam delectationem affert: cum dulci adeò parum con<lb></lb>gruit, ut melius ſocietur <expan abbr="cũ">cum</expan> amaro, uelut in oliuis benè ſalſis. </s>
          <s id="id003100">Ergo ſal­<lb></lb>ſus ſapor cum diateſſaro ad <expan abbr="unguẽ">unguem</expan> congruit rurſus ſemiditonus <expan abbr="cũ">cum</expan> inſi<lb></lb>pido, &amp; aſtringens cum ditono conueniunt ad unguem, nam uterque <expan abbr="nõ">non</expan> <lb></lb>illepidus, &amp; cum dulci conuenit, ita ſemiditonus &amp; ditonus cum diapa<pb pagenum="171 [=177]" xlink:href="015/01/196.jpg"></pb>ſo conueniunt, uterque etiam horum ſaporum parum mouet ſen­<lb></lb>ſum, &amp; inter ſe ſunt quaſi ſimiles quod ditono accidit &amp; ſemidito­<lb></lb>no, ſed &amp; neuter horum cum pingui conuenit, neque ditonus aut ſe­<lb></lb>miditonus cum diapente congruit, diſcordat enim hęc compoſitio <lb></lb>non parum. </s>
          <s id="id003101">Rurſus &amp; in hoc ſimiles ſunt quod diateſſaron cum di­<lb></lb>tono &amp; ſemiditono plurimum conuenit, ita &amp; inſipidum, &amp; aſtrin­<lb></lb>gens cum ſalſo bellè <expan abbr="cõueniunt">conueniunt</expan>. </s>
          <s id="id003102">Diateſſaron enim cum ditono ſex­<lb></lb>tam efficit maiorem, &amp; cum ſemiditono minorem quę utrique conſo<lb></lb>nant, non tamen plus ſuaues per ſe ſunt, quòd dulci &amp; pingui care­<lb></lb>ant, ut nec ſexta maior aut minor, q̊d neque diapaſon perficiant neque <lb></lb>diapente: Acris <expan abbr="autẽ">autem</expan> ſapor ſexta maiori ſimilis eſt, acidus minori: <lb></lb>mutuo conueniunt cum inſipido acris, &amp; cum aſtringente acidus, <lb></lb>quemadmodum &amp; ſexta maior cum ſemiditono, &amp; minor cum di­<lb></lb>tono copulatur perficientes diapaſon: ſed minus ſuauem, quia ab­<lb></lb>eſt diapente ibi, quia abeſt pingue: auſterum uero cum acri mode­<lb></lb>rato conuenit, propterea bene uterque cum inſipido iungitur, unde <lb></lb>illud Epigrammatici:</s>
        </p>
        <p type="main">
          <s id="id003103">Vt ſapiant fatuæ fabrorum prandia betæ, <lb></lb>O quam ſæpe petet uina piperque coquus.</s>
        </p>
        <p type="main">
          <s id="id003104">Piper enim acre eſt, &amp; uinum auſterum eſt. </s>
          <s id="id003105">Et iuſta querela Cicero­<lb></lb>nis in Epiſtolis familiaribus, qui à maluis fatetur ſe uictum, ut deci­<lb></lb>derit in lienteriam: conueniunt ambo hi ſapores <expan abbr="cũ">cum</expan> dulci &amp; pingui, <lb></lb>uelut &amp; utraque ſexta maior &amp; minor cum diapaſon &amp; diapente, at <lb></lb>neuter cum ſalſo, nam neque diateſſaron cum ſextamaiore uel mino­<lb></lb>re iungi poteſt. </s>
          <s id="id003106">Amarus autem ſapor tono perſimilis eſt, diſſonus <lb></lb>enim per ſe eſt ſemper, &amp; amarus perſe odioſus tonus origo eſt o­<lb></lb>mnium <expan abbr="conſonantiarũ">conſonantiarum</expan>, ita omnes fructus, ſeu dulces ſeu aſtringen­<lb></lb>tes, ſeu acidi, ſeu acres prius amari ſunt: tonus præterea nulla cum <lb></lb>conſonantia peius coit quàm cum diapaſo, ita neque amarus ſapor <lb></lb>infelicius iungitur quàm cum dulci, amarus quo que ſapor cum nul­<lb></lb>lo magis conuenit <expan abbr="quã">quam</expan> cum ſalſo, ita tonus additus diateſſaro, perfi<lb></lb>cit diapente dulciſsimam conſonantiam, ut multi oliuas benèſalſas <lb></lb>prætulerint faſianis: tantum conuenit ſalſo cum amaro, amarus, <lb></lb>quo que ſapor leuis non abhorret à pingui, deteriorem <expan abbr="tamẽ">tamen</expan> aliquan<lb></lb>to efficit, ut intortis ex abſynthio ouis &amp; caſeo, atque in uitibus in <lb></lb>quibus coma abſynthij in cocta fuit parum, degenerat tamen ſapor <lb></lb>ille à pingui: ita tono addito ad diapente fit ſexta maior, non adeò <lb></lb>ſuauis ut diapente, at tamen <expan abbr="nõ">non</expan> prorſus inſuauis. </s>
          <s id="id003107">Similiter ſi tonus <lb></lb>addatur ad ſemiditonum aut ad ditonum ex altero fit diateſſaron, <lb></lb>qui non concordat ex reliquo tritonus omnium aſperrimus. </s>
          <s id="id003108">Ergo <lb></lb>cum idem fiat coniuncto amaro cum inſipido, ac deterius <expan abbr="cũ">cum</expan> aſtrin­


<pb pagenum="172 [=178]" xlink:href="015/01/197.jpg"></pb>gente, uelut in acerbis glandibus, quibus nihil triſtius guſtari po­<lb></lb>teſt. </s>
          <s id="id003109">Manifeſtum eſt igitur optimè conuenire hanc ſaporum diui­<lb></lb>ſionem cum muſica proportione.</s>
        </p>
        <p type="main">
          <s id="id003110">Cumque ſapores ex ſeptem planetis pendent manifeſtè, Saturnus <lb></lb><expan abbr="em̃">emm</expan> habet aſtringens, quoniam frigidus eſt &amp; ſiccus. </s>
          <s id="id003111">Iupiter pingue <lb></lb><expan abbr="cõtraria">contraria</expan> ratione, &amp; <expan abbr="quoniã">quoniam</expan> hic ſuauis eſt, ille triſtis, acre &amp; auſterum <lb></lb><expan abbr="cõueniuntſoli">conueniunt ſoli</expan>, apparetque in eis uis maxima ad <expan abbr="ſpiritũ">ſpiritum</expan> uitalem <expan abbr="cõfir">confir</expan> <lb></lb>mandum, uiresque <expan abbr="oẽs">oens</expan> adauget, uelut &amp; Sol. </s>
          <s id="id003112">Venus habet dulce: de­<lb></lb>monſtratione hoc non indiget. </s>
          <s id="id003113">Mars ſalſum &amp; <expan abbr="cũ">cum</expan> peruerſè diſpoſi­<lb></lb>tus eſt, <expan abbr="amarũ">amarum</expan>. </s>
          <s id="id003114">Luna inſipidum. </s>
          <s id="id003115">Mercurius <expan abbr="acidũ">acidum</expan>, etenim frigida eſt <lb></lb>&amp; humida Luna, &amp; Mercurius <expan abbr="tenuitatẽ">tenuitatem</expan> quan dam habet <expan abbr="cũ">cum</expan> tempe<lb></lb><expan abbr="ramẽto">ramento</expan> moderato, cuiuſmodi fermè eſt acidus ſapor, quan&lt;08&gt; ad fri­<lb></lb>giditatem declinet, <expan abbr="parũ">parum</expan> enim habet <expan abbr="uiriũ">uirium</expan> Mercurius q̊d minima ſit <lb></lb>ſtellarum, ut ſuprà docuimus. </s>
          <s id="id003116">Huiuſmodi ergo ratione conſiderata <lb></lb>Luna ad <expan abbr="ſemiditonũ">ſemiditonum</expan> pertinebit Mercurius ad <expan abbr="ſextã">ſextam</expan> minorem, Sol <lb></lb>ad ſextam maiorem, Mars ad <expan abbr="tetrachordũ">tetrachordum</expan>, Saturnus ad ditonum, <lb></lb>Iupiter ad diapente, Venus ad diapaſon, unde plena illius dona uul<lb></lb>garis felicitatis opum honoris amoris &amp; uoluptatis, poſt quem eſt <lb></lb>Iupiter, ut ſine his duobus omnino nulla poſsit eſſe felicitas.</s>
        </p>
        <p type="main">
          <s id="id003117">Sed &amp; in circulo ſigniferi aliquam muſica proportio habebit ra­<lb></lb>tionem: diapaſon <expan abbr="em̃">emm</expan> erit &amp; totius ad dimidium, &amp; beſsis ad trien­<lb></lb>tem, &amp; dimidij ad quadrantem, &amp; trientis ad <expan abbr="ſextantẽ">ſextantem</expan>, diapente <expan abbr="aũt">aut</expan> <lb></lb>totius circuli ad beſſem, &amp; dodrantis ad <expan abbr="dimidiũ">dimidium</expan>, &amp; dimidij ad tri­<lb></lb>entem, &amp; <expan abbr="quadrãtis">quadrantis</expan> ad <expan abbr="ſextantẽ">ſextantem</expan>, diateſſaron <expan abbr="aũt">aunt</expan> totius circuli ad do<lb></lb>drantem, &amp; beſsis ad <expan abbr="dimidiũ">dimidium</expan>, &amp; trientis ad <expan abbr="quadrãtem">quadrantem</expan>: itaque in hoc <lb></lb>ſolo <expan abbr="cũ">cum</expan> Ptolemęo concordamus, in reliquis duobus neſcio qua ra­<lb></lb>tione Ptolemęus omiſerit unam <expan abbr="cõiugationem">coniugationem</expan>, nam <expan abbr="cũ">cum</expan> eſſent qua­<lb></lb>tuor in diapaſon &amp; diapente, tres tantum numerauit. </s>
          <s id="id003118">Reliquas <expan abbr="aũt">aunt</expan> <lb></lb>quatuor per integra ſigna numerare licebit, ad <expan abbr="rationẽ">rationem</expan>, tamen aſpe­<lb></lb>ctuum deducere non poſſumus, propterea efficaciam quandam ha<lb></lb>bent etiam ſignorum mutationes, ſed harmoniam non perficiunt, <lb></lb>nam &amp; ſi ſumamus ſexquiquartam &amp; ſexquiquintam, ut in his ſex­<lb></lb>quialteram, ſeu diapente conſtituamus, aut tria aut ſex ſigna acci­<lb></lb>pere oportebit: utrunque fuerit, reliqua pars ad diateſſaron pertinere <lb></lb>minimè poteſt: quamobrem conuenientius eſſet meo iudicio, ut to<lb></lb>tus circulus non ad diapaſon, uelut Ptolemæus, referretur, ſed po­<lb></lb>tius ad diapaſon diapente: ita enim conſtitutis quatuor, quinque, <lb></lb>ſex, duodecimque numeris, conſtaret tota ratio harmonica, diuiſo e­<lb></lb>tiam diapente in ditonum &amp; ſemiditonum. </s>
          <s id="id003119">ſed de hoc ſatis.</s>
        </p>
        <p type="main">
          <s id="id003120">Reuertamur ad ſapores, in quibus diximus aliam eſſe rationem <lb></lb>muſicam iuxta <expan abbr="cõpoſitionem">compoſitionem</expan>: cum enim inter ſapores qui quouiſ­


<pb pagenum="173 [=179]" xlink:href="015/01/198.jpg"></pb>modo conueniunt, dupla fuerit optimi ſaporis proportío ad dete­<lb></lb>riorem, medius uerò ad deteriorem ſexquitertia, optimus ad me­<lb></lb>dium ſexquialtera, ſapor ille optimus erit. </s>
          <s id="id003121">Et primum quidem id <lb></lb>in pingui tanquàm medio dulcique &amp; ſalſo experiamur, ſimiliter in <lb></lb>ſalſo, acri, atque inſipido. </s>
          <s id="id003122"><expan abbr="Manifeſtũ">Manifeſtum</expan> eſt enim quod horum optimus <lb></lb>eſt inſipidus, quia per ſe ferri poteſt, ſalſus autem medius, acris de­<lb></lb>terrimus, ſuperabit ergo inſipidus ſalſum ſexquialtera, acrem du­<lb></lb>pla proportione, ſalſus acrem ſexquitertia. </s>
          <s id="id003123">Rurſus dulcem copule­<lb></lb>mus cum acri, &amp; cum inſipido aut cum acido, &amp; inſipido præſtabit, <lb></lb>ut dulcis dupla, aut quadrupla, aut octupla proportione inſipi­<lb></lb>dum ſuperet, id eſt, per diapaſon, uel bis diapaſon, aut ter diapa­<lb></lb>ſon: acidum uero inſipidum ſexquitertia ſuperabit. </s>
          <s id="id003124">Alia rurſus ra­<lb></lb>tio in coniunctionibus ſaporum ad ſenſum uniuſcuiuſque referenda <lb></lb>eſt, in quo enim eſt ſumma uoluptas comparatione ad illum, hic ſta <lb></lb>tuemus diapaſon, optimumque conſtituemus ſaporem, dimidium il<lb></lb>lius quod ad uires attinet ex minus iucundo ſexquitertium, ad il­<lb></lb>lum minus iucundum ex medio. </s>
          <s id="id003125">Exempli gratia, proponamus ut <lb></lb>alicui auſtera maximè iucunda ſint (nam ſalſa nemini, quòd nullum <lb></lb>animal præter hominem, imò ne plantæ quidem niſi admodum <lb></lb>paucæ, &amp; ſui generis ſalſo alantur, iucunda eſſe poſſunt: cum ſalſum <lb></lb>amari pars ſit, eoque deterius quod acutum ſit ſalſum, unde in ſale <lb></lb>nullum animal naſcitur: in abſynthio, quanquàm ualde amaro, exi­<lb></lb>guum muſcarum genus, nigrum tota æſtate oritur, &amp; in ruta uer­<lb></lb>miculi) is ergo auſteri, quantum ſatis erit ſumet, dulcis <expan abbr="tãquàm">tanquàm</expan> me­<lb></lb>dij. </s>
          <s id="id003126">gratia exempli (nam optima ad extremum oppoſitum uix tran­<lb></lb>ſire queunt) beſſem accipito huius, gratia exempli, tanquàm deter­<lb></lb>rimi aſtringentis dodrantem, ut ſit dulcis ad aſtringentem dupla <lb></lb>proportio. </s>
          <s id="id003127">Sic ergo conſtituetur iuxta naturam propriam muſica <lb></lb>proportione ſapor iucundiſsimus.</s>
        </p>
        <p type="main">
          <s id="id003128">Idem quo que in odoribus &amp; eadem ratione, ſed ex ſaporibus hoc <lb></lb>cum intellectum ſit, fruſtra fuerit conſumere tempus, eadem enim <lb></lb>in omnibus ad ſciendum proportionem intelligenda erunt.</s>
        </p>
        <p type="main">
          <s id="id003129">Propoſitio centeſima ſexageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id003130">Picturarum proportiones explicare.</s>
        </p>
        <p type="main">
          <s id="id003131">Eſt pictura imago rei corporeæ quanquàm, &amp; per illam, &amp; acti­</s>
        </p>
        <p type="main">
          <s id="id003132"><arrow.to.target n="marg583"></arrow.to.target><lb></lb>ones, &amp; cogitationes, ſed non niſi ut per corpora ſignificantur: ut <lb></lb>ergo corpora ipſa referamus. </s>
          <s id="id003133">coloribus opus eſt, nam corpora, co­<lb></lb>lorata ſunt, ſecundò ipſa rerum natura ſcientiaque illarum, unde pi­<lb></lb>ctorem multiſcium eſſe neceſſe eſt. </s>
          <s id="id003134">tertium eſt, ut minimas earum <lb></lb>differentias explicare norit. </s>
          <s id="id003135">quartum, ut affectiones, uelut in ira­


<pb pagenum="174 [=180]" xlink:href="015/01/199.jpg"></pb>to ruborem, ciliorum <expan abbr="cõtractionem">contractionem</expan>, tumorem faciei in ambulante <lb></lb>inclinationem quandam, flexionem cruris atque ſimilia. </s>
          <s id="id003136">quintum eſt <lb></lb>lux coloribus <expan abbr="exhibẽda">exhibenda</expan>, ſed de horum nullo propoſitum eſt hic lo­<lb></lb>qui, quando quidem hæc uſu magis &amp; conſideratione, quàm ratio­<lb></lb>ne conſtent proportioneúe, nec ſint adeò admiranda ut neque ſim­<lb></lb>plex magnitudo <expan abbr="quãſexto">quanſexto</expan> loco reponere poſſumus. </s>
          <s id="id003137">Tria ergo ui­<lb></lb>dentur eſſe præcipua quorum nunc ratio habenda eſſet, ut ſint in <lb></lb>totum nouem, ſed unum ex his relinquemus, tum quia alienum ab <lb></lb>hac conſideratione, tum quia alibi pertractatum atque etiam ab alijs, <lb></lb>neque adeò admiratione dignum ſcilicet magnitudo picturarum re­<lb></lb>ſpondens magnitudini corporum iuxta ſitus differentiam, nam <lb></lb>quę altiores ſunt paulo latiores atque in ſuperiori magis parte quam <lb></lb>in inferiore, multò autem longiores eſſe oportet, ſic &amp; quæ à latere <lb></lb>erunt eadem ratione iuxta aſpectus ingredientium rationem. </s>
          <s id="id003138">Ve­<lb></lb>rum hoc ut dixi omittamus, &amp; de duplici miraculo in pictura lo­<lb></lb>quamur, ſcilicet diſtantia magna quam in parua tabella referimus, <lb></lb>et corporeitate quam in plano repręſentamus. </s>
          <s id="id003139">Horum autem duo­<lb></lb>rum aliqua communia ſunt aliqua propria. </s>
          <s id="id003140">Dicemus ergo <expan abbr="primũ">primum</expan> <lb></lb>de corpore ita pingendo, ut palàm extra tabulam prominere uide<lb></lb>atur. </s>
          <s id="id003141">Hoc autem primum ex forma ſumitur, nam ſi corpus in plano <lb></lb>ſit neceſſe eſt, ut partes illius quædam prorſus abſcondantur, par­<lb></lb>tes aliæ non prorſus, aliæ prorſus ſint in conſpicuo. </s>
          <s id="id003142">Ergo pictu­<lb></lb>ram talem fingere oportebit, quæ partes ſingulas pro ratione oſten <lb></lb>dat aut occultet. </s>
          <s id="id003143"><expan abbr="Secũda">Secunda</expan> ratio eſt quod ima corporis obſcura ſunt, <lb></lb>ſummę partes lucidę &amp; claræ ac lumine quaſi dealbatæ: media, me­<lb></lb>dia quadam ratione ut in columnis, tantumque poteſt hæc ratio, ut <lb></lb>uel ſola picturas fallere nos faciat corpora eas eſſe putantes. </s>
          <s id="id003144">Opor­<lb></lb>tet autem imum eſſe ad unguem ſimile in colore colori anguli loci <lb></lb>&amp; ſummum parti quæ ſe oculis maximè ſubiectam præbet &amp; cla­<lb></lb>ram: media uerò qualia ex umbris obſcurari ſolent. </s>
          <s id="id003145">Tertia ratio eſt <lb></lb>pro modo partium iuxta <expan abbr="obliquitatẽ">obliquitatem</expan> aſpectus: nam inſpicienti a b <lb></lb>in c d ex e oculo: depingemus in c d iuxta obli­<lb></lb><figure id="id.015.01.199.1.jpg" xlink:href="015/01/199/1.jpg"></figure><lb></lb>quitatem ſuam, quia cum c d uideatur per line­<lb></lb>as e a c &amp; e b d, &amp; eleuatum in ſitu a b, neceſſe eſt <lb></lb>ut uideatur in ſitu a b, ergo eleuatum à c d. </s>
          <s id="id003146">Eſt <lb></lb>&amp; alia conſideratio proportionis ad proxima <lb></lb>remotaque, grati a exempli, ſi homo eſſet poſt co­<lb></lb>lumnam a b, lateret eius pars, quæ eſt propinquior parieti c d, ergo <lb></lb>ſi depinxerimus hominis partes tantum dextram, reliquum ſub um<lb></lb>bra, cogitur oculus iudicare columnam eleuatam a pariete. </s>
          <s id="id003147">De­<lb></lb>mum omnia hæc ita ſunt ſubijcienda oculis, &amp; per minimas diffe­


<pb pagenum="175 [=181]" xlink:href="015/01/200.jpg"></pb>rentias &amp; animaduerſiones ita dijudicanda, atque experimento ſub­<lb></lb>ijcienda, tum proprio, tum aliorum non artis in expertium, ut res <lb></lb>prorſus abſoluta uideatur, atque in hoc multum refert multiplices <lb></lb>partes ſecundum longitudinem coloribus diſtinguere ad hoc a­<lb></lb>ptis, qui ſunt obſcurus, ſub obſcurus, cinereus, qualis ſilicis candi­<lb></lb>dus ſine luce, demum etiam aliquid nigri adijciendum, nam diuiſio <lb></lb>ſecundum longitudinem multum impedit, hanc repræſentationem <lb></lb>iuuant, &amp; extrema benè coaptata, uelut ſcapi imi, &amp; capitula &amp; ſu­<lb></lb>premi, <expan abbr="tũ">tum</expan> trabeationes ex materia coronæ, zofoni, tœnia, epiſtylia, <lb></lb>plinthi, echini, hypotrachelia, aſtagali, apophyges. </s>
          <s id="id003148">Quæ etiam in <lb></lb>parte inferiore <expan abbr="cũ">cum</expan> ſpira ſeu baſi &amp; limbo &amp; toro &amp; plintho inferio­<lb></lb>re, &amp; ſtylobata, et alia tœnia ſumma diligentia, &amp; cum eleuatione ac <lb></lb>magnitudine ultra columnæ limites extendantur. </s>
          <s id="id003149">Sicin ſtylobata <lb></lb>ratio diapente conſtat, cui ſolet addi utrinque ſexta pars pro coro­<lb></lb>nice, manifeſtum eſt autem, quod in ea conſtat muſica ratio diapa­<lb></lb>ſon ex diapente &amp; diateſſaro, compoſiti nam duæ ſextæ partes, alte<lb></lb>ra utrinque adiecta tertiam conficiunt ut ſit diateſſaron ſuprà diapen<lb></lb>te. </s>
          <s id="id003150">In regionibus autem &amp; ſpatijs depingendis eadem fermè ſeruan <lb></lb>da ſunt duobus tamen adiectis, <expan abbr="quorũ">quorum</expan> unum eſt ut longinquiſsima <lb></lb>pars, <expan abbr="nõ">non</expan> per nigrum aut obſcurum, ſed cœruleum <expan abbr="colorẽ">colorem</expan>, qualis in <lb></lb>cœlo determinanda eſt (niſi nox fingatur) nam cœlum longiſsimè <lb></lb>à nobis diſtat, ita nubes coloribus proprijs, &amp; montes cum niui­<lb></lb>bus, &amp; ſpatia uelut fluminis alueus, mare, lacus, atque hæc omnia <lb></lb>per colores diſtantiæ finguntur, uelut fluminis pars propior clara <lb></lb>&amp; lympida, &amp; colore aqueo cernitur remota obſcura, quæ maxi­<lb></lb>mè procul abeſt nigra. </s>
          <s id="id003151">Sed maxima eſt confirmatio in compara­<lb></lb>tionibus: ut ſi arbores propè magnæ ſint, &amp; homines &amp; animalia, <lb></lb>in remotiore autem parte minimi, ac quaſi puncti magnitudinem <lb></lb>referentes, atque ut in his muſica non geometrica aut arithmeti­<lb></lb>ca proportio ſeruetur. </s>
          <s id="id003152">Equidem ſi quis iudicio hæc conſequa­<lb></lb>tur, ac diligentia quæ ſcribi non poſſunt, ſed contemplatione ha­<lb></lb>bentur, ſenſu quoque, quem experimentum docet, nec ipſum man­<lb></lb>dare literis, licet ex rationibus tamen, quas hic docemus intelli­<lb></lb>get parum differre repræſentationem à re ipſa corporea. </s>
          <s id="id003153">Sed de <lb></lb>his hactenus, quæ ſi diligentius quis perſequi uelit ſine <lb></lb>artis experientia, plus adimet perfectioni rei, <lb></lb>quam adijciet. </s>
          <s id="id003154">Hoc enim aliâs <lb></lb><arrow.to.target n="marg584"></arrow.to.target><lb></lb>declarauimus.</s>
        </p>
        <pb pagenum="176 [=182]" xlink:href="015/01/201.jpg"></pb>
        <p type="margin">
          <s id="id003155"><margin.target id="marg583"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003156"><margin.target id="marg584"></margin.target>I<emph type="italics"></emph>n prima <emph.end type="italics"></emph.end><lb></lb>D<emph type="italics"></emph>islcfficæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003157">Propoſitio centeſima ſexageſima nona.</s>
        </p>
        <p type="main">
          <s id="id003158">Proportionem muſicam in inſtrumentis declarare iuxta compo<lb></lb>ſitionis rationem.<lb></lb><arrow.to.target n="marg585"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003159"><margin.target id="marg585"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003160">Tria ſunt inſtrumentorum genera, in quibus maximè relucet ra­<lb></lb>tio compoſitionis muſicæ quæ à nobis nunc ſunt demonſtranda, <lb></lb>ſcilicet machinæ bellicę, ut catapultæ &amp; baliſtę &amp; ſcorpiones, &amp; hy<lb></lb>draulica inſtrumenta ad modulationes parata, quæ antiquo tem­<lb></lb>pore maximè in uſu fuerunt nunc deſita, de quibus Vitruuius agit </s>
        </p>
        <p type="main">
          <s id="id003161"><arrow.to.target n="marg586"></arrow.to.target><lb></lb>in decimo libro. </s>
          <s id="id003162">Tertium eſt æneorum inſtrumentorum, quorum <lb></lb>etiam uſus deſijt in ſcœnicis theatris, ad intendendam uocem cum <lb></lb>modulatione, ut etiam clamor audientium &amp; uulgi cum uoluptate <lb></lb><arrow.to.target n="marg587"></arrow.to.target><lb></lb>excipiatur, de quo idem in quinto libro egit. </s>
          <s id="id003163">Sed nil melius quàm <lb></lb>uerba ipſius explicare de hoc tractantis, ſunt autem hæc. </s>
          <s id="id003164">“Muſicen <lb></lb>autem ſciat oportet, uti canonicam rationem &amp; mathematicam no­<lb></lb>tam habeat: præterea baliſtarum, catapultarum, ſcorpionum tem­<lb></lb>peraturas poſsit rectè facere. </s>
          <s id="id003165">In capitulis enim dextra ac ſiniſtra <lb></lb>ſunt foramina homotonorum, per quę tenduntur ergatis aut ſucu­<lb></lb>lis &amp; uectibus è neruo torti funes, qui non præcluduntur, nec præ­<lb></lb>ligantur niſi ſonitus ad artificis aures certos &amp; ęquales fuerint. </s>
          <s id="id003166">Bra­<lb></lb>chia enim quæ in eas tentiones includuntur cum extenduntur æ­<lb></lb>qualiter &amp; parter utraque plagam emittere debent. </s>
          <s id="id003167">Quod ſi non ho­<lb></lb>motona fuerint, impedient directam telorum miſsionem. </s>
          <s id="id003168">Item the­<lb></lb>atris uaſa ærea, quę in cellis ſub gradibus. </s>
          <s id="id003169">mathematica ratione <expan abbr="collo­cant̃">collo­<lb></lb>cantur</expan>, &amp; <expan abbr="ſonitũ">ſonitum</expan> diſcrimina, quę Gręci <foreign lang="grc">̓ηχ̂εια</foreign> <expan abbr="uocãt">uocant</expan>, ad ſymphonias mu <lb></lb>ſicas ſiue concentus <expan abbr="componunt̃">componuntur</expan>, diuiſa in circinatione diateſſaron <lb></lb>&amp; diapente &amp; diapaſon, uti uox ſcœnici ſonitus <expan abbr="cõueniens">conueniens</expan> in diſpo <lb></lb>ſitionibus, tactu <expan abbr="cũ">cum</expan> oſtenderit aucta <expan abbr="cũ">cum</expan> <expan abbr="incremẽto">incremento</expan> clarior et ſuauior <lb></lb>ad <expan abbr="ſpectatorũ">ſpectatorum</expan> perueniat aures. </s>
          <s id="id003170">Hydraulicas quo que machinas &amp; cæ­<lb></lb>tera <expan abbr="q̃">quae</expan> ſunt ſimilia his organis ſine muſicis rationibus. </s>
          <s id="id003171">efficere nemo <lb></lb>poterit. </s>
          <s id="id003172">Capiamus ergo primum illud q̊d eſt manifeſtius, ſcilicet de <lb></lb>hydraulicis organis quorum meminit Suetonius in Nerone: Reli­<lb></lb>quam diei partem per organa hydraulica noui &amp; ignoti generis cir<lb></lb>cunduxit, oſtendenſque ſingula de ratione ac difficultate cuiuſque diſ­<lb></lb>ſerens iam ſe prolaturum, ut conſtet illa fuiſſe magni opificij quæ <lb></lb>noſtra ętate deſiere.” Reſtat unicum &amp; ualde leue <expan abbr="exemplũ">exemplum</expan> auiculæ <lb></lb>æneæ uelligneæ reſonantis. </s>
          <s id="id003173">Certum eſt <expan abbr="aẽre">aere</expan> effici ſonum, ſed ita mi<lb></lb>ſceri aquæ, ut dulcior &amp; mollior non ſolum euadat, ſed etiam acuti­<lb></lb>or ac modulatior. </s>
          <s id="id003174">Eadem autem ratio maris: ſed cum aquæ corpus <lb></lb>moueatur, uidetur difficile ſeruare proportionem. </s>
          <s id="id003175">ea prima diffi­<lb></lb>cultas. </s>
          <s id="id003176">ſecunda eſt, quod cùm aqua moueatur, uix fieri poſſe uide­<lb></lb>tur ut totum ſeruet uocis integrum tenorem. </s>
          <s id="id003177">tertia ob illius con­


<pb pagenum="179 [=183]" xlink:href="015/01/202.jpg"></pb>ſumptionem. </s>
          <s id="id003178">Propterea nil mirum eſt ſi Nexo de his ſubtiliter di­<lb></lb>ſputauit, mirum fuit quod in tanta animi perturbatione niſi ad <lb></lb>amentia, ut illi putant, referatur. </s>
          <s id="id003179">Sed quid iam amplius uagor, extat <lb></lb><arrow.to.target n="marg588"></arrow.to.target><lb></lb>compendioſa ratio conſtructionis illius apud eundem Vitruuium <lb></lb>ubi Philander ex Atheneo ſonus hydradis ſuauis admodum atque <lb></lb><arrow.to.target n="marg589"></arrow.to.target><lb></lb>iucundus auditu eſt: ita ut omnes concinnitate capti conuerterent, <lb></lb>fuitque Alexendrinę urbis inuentum authore Cteſibio tonſore, eſt <lb></lb>autem magnæ Clepſydræ inſtrumentum non abſimile, ſunt enim <lb></lb>fiſtulæ in aquam contortæ, quæ, cùm aqua à iuuene quopiam per­<lb></lb>cutitur, axinis per organum tranſeuntibus inflantur, <expan abbr="periucũdum­qúe">periucundum­<lb></lb>qúe</expan> ſonum emittunt. </s>
          <s id="id003180">Eſt autem aræ rotundæ hoc inſtrumentum <lb></lb>perſimile inuentumque Ptolemæi ſecundi Euergitę temporibus, de <lb></lb>quo eundem Cteſibium ſcripſiſſe ferunt. </s>
          <s id="id003181">Fiebant autem ex ære &amp; <lb></lb>baſis e ligno cum regulis dextra ac ſiniſtra ſcalari regula compactis, <lb></lb>aqua autem in ęrea arca continebatur. </s>
          <s id="id003182">Facilè autem eſt per hæc reli<lb></lb>qua inuenire: nam epiſtomijs includebatur aër atque reſerabatur, &amp; <lb></lb>modus erat per uectes: non tamen octo <expan abbr="fiſtularũ">fiſtularum</expan> &amp; exin de uocum <lb></lb>numerum inſtrumentum id ſuperabat organa noſtra ut locupleti­<lb></lb>ora ita aſperiora. </s>
          <s id="id003183">Liquet ergo ſi fabrilis omnis ars ad Architectum <lb></lb>pertinet, illum etiam hac ratione oportere eſſe peritum muſicæ.<lb></lb><arrow.to.target n="marg590"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003184"><margin.target id="marg586"></margin.target>C<emph type="italics"></emph>ap.<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>ad<emph.end type="italics"></emph.end><lb></lb>18. <emph type="italics"></emph>&amp; in <lb></lb>cap.<emph.end type="italics"></emph.end> 13.</s>
        </p>
        <p type="margin">
          <s id="id003185"><margin.target id="marg587"></margin.target>C<emph type="italics"></emph>ap.<emph.end type="italics"></emph.end> 5.</s>
        </p>
        <p type="margin">
          <s id="id003186"><margin.target id="marg588"></margin.target>L<emph type="italics"></emph>ib,<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>cd,<emph.end type="italics"></emph.end><lb></lb>16.</s>
        </p>
        <p type="margin">
          <s id="id003187"><margin.target id="marg589"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>cap.<emph.end type="italics"></emph.end><lb></lb>24.</s>
        </p>
        <p type="margin">
          <s id="id003188"><margin.target id="marg590"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>ca.<emph.end type="italics"></emph.end> 5.</s>
        </p>
        <p type="main">
          <s id="id003189">“De Vaſis uerò æneis theatri quod melius eſt quàm ut eundem <lb></lb>authorem conſulamus, dicentem uaſa ęrea pro ratione magnitudi­<lb></lb>nis theatri ita fabricentur, ut cum <expan abbr="tangũtur">tanguntur</expan>, ſonitum facere poſsint <lb></lb>inter ſe diateſſaron diapente, ex ordine addit diapaſon, poſtea inter <lb></lb>ſedes theatri conſtitutis cellis ratione muſica ibi collocentur: ita uti <lb></lb>nullum parietem tangant circaque habeant locum <expan abbr="uacuũ">uacuum</expan> et à ſummo <lb></lb>capite ſpatium, ponantque inuerſa &amp; habeant in parte quę ſpectat ad <lb></lb>ſcenam ſuppoſitos cuneos ne minus alios ſemipede, contraque eas <lb></lb>cellas relinquantur aperturę inferiorum graduum cubilibus lon­<lb></lb>gę pedes duos altæ ſemipedem. </s>
          <s id="id003190">Et ſi non erit ampla magnitudine <lb></lb>theatrum, media altitudinis tranſuerſaregio deſignetur, &amp; in ea tre<lb></lb>decim cellæ duodecim æqualibus. 

interuallis diſtantes <expan abbr="confornicent̃">confornicentur</expan> <lb></lb>uti ea echea quæ ſupra ſcripta ſunt, ad neten hyperboleon ſonan­<lb></lb>tia in cellis quæ ſuntin cornibus extremis utraque parte prima col­<lb></lb>locentur, ſecunda ab extremis diateſſaron ad <expan abbr="netẽ">netem</expan> diezeugmenon, <lb></lb>tertia diateſſaron ad neten parameſon, quarta ad neten ſynemme­<lb></lb>non, quinta diateſſaron ad meſen, ſexta diateſſaron ad hypaten me­<lb></lb>ſen in medio unum diateſſaron ad hypaten hypaton. </s>
          <s id="id003191">Quæ sequun­<lb></lb>tur &amp; ad intelligentiam prædictorum melius ex Gulielmo Philan­<lb></lb>dro emendata ſic tranſcribemus: Eas regiones in tredecim cellas <lb></lb>diuidit æqualibus interuallis: id eſt, cellas paribus uiciſsim inter­


<pb pagenum="178 [=184]" xlink:href="015/01/203.jpg"></pb>ſticijs diſpoſitas diſtribuit ſex hinc atque hinc &amp; unam mediam, quæ <lb></lb>tamen non uſus, ſed partitionis &amp; reſponſus cauſa fit in media prę­<lb></lb>cinctione. </s>
          <s id="id003192">In ima præcinctione ponuntur uaſa quę habent harmo­<lb></lb>nię <expan abbr="rationẽ">rationem</expan>, hoc modo. </s>
          <s id="id003193">In <expan abbr="cornuũ">cornuum</expan> cellis collocantur quæ <expan abbr="ſonitũ">ſonitum</expan> ha­<lb></lb>bent netes hyperboleon. </s>
          <s id="id003194">Subſequuntur utrinque quæ ſunt ad neten <lb></lb>diezeugmenon interuallo conſonantia diateſſaron. </s>
          <s id="id003195">In tertijs cel­<lb></lb>lis ſunt quæ ad neten parameſen interuallo item diateſſaron, quæ <lb></lb>ſunt in quartis tono ſolummodo diſtant &amp; ſunt netes ſynemenon. <lb></lb></s>
          <s id="id003196">In quintis cellis ſunt ad meſen interuallo diateſſaron. </s>
          <s id="id003197">In ſextis cellis <lb></lb>ad hypaten meſon, <expan abbr="itẽ">item</expan> diateſſaron ſpatio. </s>
          <s id="id003198">In media cella ſunt ad hy<lb></lb>paten hypaton interuallo diateſſaron. </s>
          <s id="id003199">In media præcinctione ſunt <lb></lb>uaſa chromatos, collocantur autem in cornibus uaſa quæ ſunt ad <lb></lb>paraneten hyperbolem. </s>
          <s id="id003200">In ſecundis cellis ad paraneten diezeugme <lb></lb><expan abbr="nõ">non</expan> ſpatio diateſſaron, in tertijs ad paraneten hynemenon ſpatio dia <lb></lb>pente. </s>
          <s id="id003201">In quartis ad lichanon meſon interuallo diateſſaron. </s>
          <s id="id003202">In quin<lb></lb>tis ad lichanon hypaton, <expan abbr="itẽ">item</expan> diateſſaron. </s>
          <s id="id003203">In ſextis ad parameſen q̊d <lb></lb>ſpatium ad paraneten hyperboleon eſt diapente ad paraneten hy­<lb></lb>nemenon diateſſaron. </s>
          <s id="id003204">In chromatis media cella nulla ſunt uaſa, <lb></lb>quod à lichano hypaton ad proslambanomenon, aut ad aliam o­<lb></lb>mnino decem &amp; octo uocum nulla ſit conſonantia, ſunt enim hæ­<lb></lb>mitonia tantum duo &amp; tonus. </s>
          <s id="id003205">In tertia præcinctione collocantur <lb></lb>uaſa diatoni. </s>
          <s id="id003206">Etin cornibus quidem ea quæ ſunt ad paraneten, hy­<lb></lb>perboleon. </s>
          <s id="id003207">In ſecundis cellis ad paraneten diezeugmenon. </s>
          <s id="id003208">ſpatio <lb></lb>diateſſaron. </s>
          <s id="id003209">In tertijs ad paraneten hynemenon diapente. </s>
          <s id="id003210">In quar­<lb></lb>tis ad lichanon meſon diateſſaron. </s>
          <s id="id003211">In quintis ad lichanon hypaton <lb></lb>diateſſaron. </s>
          <s id="id003212">In ſextis quæ ad proslambanomenon diateſſaron ſpa­<lb></lb>tio. </s>
          <s id="id003213">In media quæ ſunt ad meſen, quod ea ad proslambanomenon <lb></lb>habet conſonantiam diapaſon, &amp; ad lychanon hypaton diapente.” </s>
        </p>
        <p type="main">
          <s id="id003214"><arrow.to.target n="marg591"></arrow.to.target><lb></lb>Hæc autem ex figura patent in opere de Subtilitate deſcripta.</s>
        </p>
        <p type="margin">
          <s id="id003215"><margin.target id="marg591"></margin.target>L<emph type="italics"></emph>ib.<emph.end type="italics"></emph.end> 16.</s>
        </p>
        <p type="main">
          <s id="id003216">Porrò quod ad machinas attinet. </s>
          <s id="id003217">Sit catapulta, cuius rudens a b <lb></lb>quam oportet trahere, ſi emittere debeat lapi­<lb></lb><figure id="id.015.01.203.1.jpg" xlink:href="015/01/203/1.jpg"></figure><lb></lb>dem, aut ſcorpio ſagittam ad aliquod ſignum <lb></lb>puta c, cum ergo ſonus c a &amp; c b homotenus fue<lb></lb>rit, non ſolum æqualiter pertractæ erunt c a &amp; <lb></lb>c b, ſed etiam æquales: nam ſi æquales eſſent, &amp; <lb></lb>inęqualiter tractæ, aut inęquales &amp; inæqualiter <lb></lb>tractę <expan abbr="ſonũ">ſonum</expan> diuerſum <expan abbr="reddẽt">reddent</expan> euidenter. </s>
          <s id="id003218">At ſi in­<lb></lb>ęquales &amp; <expan abbr="ęqualẽ">ęqualem</expan> ſonum reddant, erit <expan abbr="tñ">tnm</expan> ut fidis <lb></lb>notæ quæ ſtrepitum edit duplicem, &amp; effigiem <lb></lb>oculis <expan abbr="multiplicẽ">multiplicem</expan>, unde ſagitta in partem aduer­<lb></lb>ſam dirigitur <expan abbr="rudẽtis">rudentis</expan> intentioris, atque hæc ex Vitruuio eodem dum <lb></lb>de his agit.</s>
        </p>
        <pb pagenum="185" xlink:href="015/01/204.jpg"></pb>
        <p type="main">
          <s id="id003219">Propoſitio centeſima ſeptuageſima.</s>
        </p>
        <p type="main">
          <s id="id003220">Coniugationes cuiuſuis numeri breuiter inuenire.</s>
        </p>
        <p type="main">
          <s id="id003221">Sint gratia exempli <expan abbr="decẽ">decem</expan> homines, &amp; patet quod poſſent eſſe ſin<lb></lb><arrow.to.target n="marg592"></arrow.to.target><lb></lb>guli, &amp; hoc <expan abbr="decẽ">decem</expan> modis, quia ſunt <expan abbr="decẽ">decem</expan>, ut Petrus &amp; Ioannes: item, <lb></lb>poſſunt eſſe omnes ſimul, &amp; hoc uno modo tantum, &amp; poſſunt eſſe <lb></lb>duo, &amp; hoc poteſt uariari <expan abbr="q̃">qua</expan>draginta quinque modis: &amp; poſſunt eſſe <lb></lb>octo, &amp; manifeſtum eſt, quod <expan abbr="totidẽ">totidem</expan> modis uariantur, ſcilicet qua­<lb></lb>draginta quinque, nam cum erunt octo, duo <expan abbr="quirelinquũtur">qui relinquuntur</expan>, uariari <lb></lb>poſſunt 45 modis, ergo &amp; illi octo ad <expan abbr="unguẽ">unguem</expan> totidem modis. </s>
          <s id="id003222">Et ſi­<lb></lb>militer tres quot modis uariantur tot modis <expan abbr="ſeptẽ">ſeptem</expan>, &amp; quot modis <lb></lb>quatuor tot ſex: quinque autem quia ſunt dimidium decem, pluribus <lb></lb>modis uariantur. </s>
          <s id="id003223">Et ideò pro ordine huius detrahes <expan abbr="unũ">unum</expan>, ut ſi ſint <lb></lb>undecim uiri pones decem, ſi decem pones <expan abbr="nouẽ">nouem</expan>, &amp; colliges natu­<lb></lb>ralem seriem numerorum, ut infrà uides uno ſemper termino defi­<lb></lb>ciente: &amp; ex priore ordine, ubi uidebis ſemper <expan abbr="etiã">etiam</expan> duplicari nume­<lb></lb>ros: ut 3. 6. in de ſub 6. 10. &amp; 20 àlatere, &amp; ſub 20 35. &amp; à latere 70 du­<lb></lb>plum 35, &amp; ſub <lb></lb><arrow.to.target n="table23"></arrow.to.target><lb></lb><figure id="id.015.01.204.1.jpg" xlink:href="015/01/204/1.jpg"></figure>70 126, &amp; à late­<lb></lb>re 252, &amp; hoc pro <lb></lb>cognitione q̊d <lb></lb>rectè ſis opera­<lb></lb>tus. </s>
          <s id="id003224">Secundò a­<lb></lb>nimaduertes <expan abbr="ſe­quẽtes">ſe­<lb></lb>quentes</expan> ordines <lb></lb>fieri ex recta li­<lb></lb>nea priorum, ue<lb></lb>lut ſextus ordo eſt 7. 28. 84. 210. 462. ita incipiendo in primo ordi­<lb></lb>ne à 7, &amp; tendendo ad dextram, inuenies illos eoſdem numeros ad <lb></lb>unguem, &amp; ita in ſeptimo ordine 8. 36. 120. 330. à ſiniſtra inuento 8 <lb></lb>in primo ordine, &amp; procedendo ad dextram, inuenies 36. 120. &amp; <lb></lb>330. Tertium eſt quod numeri ultimi à medio ſunt ijdem, ut 462 &amp; <lb></lb>462. 330 &amp; 330. 165 &amp; 165. 55 &amp; 55. 11 &amp; 11. Et ſeorſum, ut dixi, rema­<lb></lb>net 1. Oportet igitur colligere numeros angulares, ut à latere ui­<lb></lb>des, &amp; fit 2047 numerus coniugationum, tot enim modis poſſunt <lb></lb>uariari. </s>
          <s id="id003225">Et ſi eſſent decem tantum, ut ab initio propoſui, primus or­<lb></lb>do finitur ad 10, ſecundus ad 45, tertius ad 120, quartus ad 210, quin<lb></lb>tus ad 252, ſextus redit ad 210, ſeptimus ad 120, octauus ad 45, no­<lb></lb>nus ad 10, decimus ad 1. Et ita colligeretur ſumma ex extremis nu­<lb></lb>meris angularibus 1023. Et tot erunt coniugationes. </s>
          <s id="id003226">Hic uides quia <lb></lb>numerus 10 eſt par, et quod adempta monade, relinquitur 9, qui eſt <lb></lb>impar quòd medius qui pertinet ad quintum ordinem eſt maxi­


<pb pagenum="186" xlink:href="015/01/205.jpg"></pb>mus, &amp; eſt 252, &amp; eſt coniugatio quinarij: hoc uolui dixiſſe, <lb></lb><figure id="id.015.01.205.1.jpg" xlink:href="015/01/205/1.jpg"></figure><arrow.to.target n="table24"></arrow.to.target><lb></lb>ut intelligeres rationes colligendi ſingulos ordines ſeor­<lb></lb>ſum. </s>
          <s id="id003227">Quod ergo attinet ad collectionem maximi numeri, <lb></lb>primus ordo ſeruit ſemper ultimo <expan abbr="relinquẽdo">relinquendo</expan> monadem, <lb></lb>&amp; ſecundus penultimo, &amp; tertius antepenultimo, &amp; ita de <lb></lb><figure id="id.015.01.205.2.jpg" xlink:href="015/01/205/2.jpg"></figure>alijs, nam ſi ſecundus uariatur 55 modis, &amp;&#039;pen­<lb></lb>ultimus uariabitur 55 modis. </s>
          <s id="id003228">Et ſi tertius uaria­<lb></lb>tur 165 modis, antepenultimus uariatur 165 mo <lb></lb>dis. </s>
          <s id="id003229">Et ita de alijs.<lb></lb><arrow.to.target n="table25"></arrow.to.target><lb></lb><arrow.to.target n="marg593"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003230"><margin.target id="marg592"></margin.target>C<emph type="italics"></emph>o.<emph.end type="italics"></emph.end> ^{m}</s>
        </p>
        <p type="margin">
          <s id="id003231"><margin.target id="marg593"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <table>
          <table.target id="table23"></table.target>
          <row>
            <cell>1</cell>
            <cell>2</cell>
            <cell>3</cell>
            <cell>4</cell>
            <cell>5</cell>
            <cell>6</cell>
            <cell>7</cell>
            <cell>8</cell>
            <cell>9</cell>
            <cell>10</cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>2</cell>
            <cell>3</cell>
            <cell>4</cell>
            <cell>5</cell>
            <cell>6</cell>
            <cell>7</cell>
            <cell>8</cell>
            <cell>9</cell>
            <cell>10</cell>
            <cell>11</cell>
            <cell></cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>6</cell>
            <cell>10</cell>
            <cell>15</cell>
            <cell>21</cell>
            <cell>28</cell>
            <cell>36</cell>
            <cell>45</cell>
            <cell>55</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>4</cell>
            <cell>10</cell>
            <cell>20</cell>
            <cell>35</cell>
            <cell>56</cell>
            <cell>84</cell>
            <cell>120</cell>
            <cell>165</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>5</cell>
            <cell>15</cell>
            <cell>35</cell>
            <cell>70</cell>
            <cell>126</cell>
            <cell>210</cell>
            <cell>330</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>21</cell>
            <cell>56</cell>
            <cell>126</cell>
            <cell>252</cell>
            <cell>462</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>28</cell>
            <cell>84</cell>
            <cell>210</cell>
            <cell>462</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>8</cell>
            <cell>36</cell>
            <cell>120</cell>
            <cell>330</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>45</cell>
            <cell>165</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>10</cell>
            <cell>55</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>11</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
        </table>
        <table>
          <table.target id="table24"></table.target>
          <row>
            <cell>11</cell>
          </row>
          <row>
            <cell>55</cell>
          </row>
          <row>
            <cell>165</cell>
          </row>
          <row>
            <cell>330</cell>
          </row>
          <row>
            <cell>462</cell>
          </row>
          <row>
            <cell>462</cell>
          </row>
          <row>
            <cell>330</cell>
          </row>
          <row>
            <cell>165</cell>
          </row>
          <row>
            <cell>55</cell>
          </row>
          <row>
            <cell>11</cell>
          </row>
          <row>
            <cell>1</cell>
          </row>
          <row>
            <cell>----</cell>
          </row>
          <row>
            <cell>2047</cell>
          </row>
        </table>
        <table>
          <table.target id="table25"></table.target>
          <row>
            <cell>10</cell>
          </row>
          <row>
            <cell>45</cell>
          </row>
          <row>
            <cell>120</cell>
          </row>
          <row>
            <cell>210</cell>
          </row>
          <row>
            <cell>252</cell>
          </row>
          <row>
            <cell>210</cell>
          </row>
          <row>
            <cell>120</cell>
          </row>
          <row>
            <cell>45</cell>
          </row>
          <row>
            <cell>10</cell>
          </row>
          <row>
            <cell>1</cell>
          </row>
          <row>
            <cell>----</cell>
          </row>
          <row>
            <cell>1023</cell>
          </row>
        </table>
        <p type="main">
          <s id="id003232">Hæc autem ratio ſatisfacit multum, &amp; eſt ne­<lb></lb>ceſſaria in temperiebus corporis humani. </s>
          <s id="id003233">Vt in <lb></lb>ſecundo, De dentibus. </s>
          <s id="id003234">Et etiam ut quælibet di­<lb></lb>ſciplina quàm breuiſsimè tradi poſsit, ut gratia <lb></lb>exempli, medicina tota in una pagina, dico me­<lb></lb>dicina <expan abbr="nõ">non</expan> ſolum Græcorum, ſed etiam Arabum <lb></lb>&amp; Latinorum, &amp; etiam longè plus: nam ſi tradatur uiginti quatuor <lb></lb>regulis simplicibus, &amp; ex illis fiant coniugationes 16777215, mani <lb></lb>feſtum eſt quod erunt regulæ omnes hæ multo plures, quàm con­<lb></lb>tineantur in omnibus libris Græcorum, &amp; Arabum, &amp; Latino­<lb></lb>rum, qui extant. </s>
          <s id="id003235">Et tamen perſpicuum eſt, uiginti quatuor regulas <lb></lb>una pagina commodiſsimè contineri. </s>
          <s id="id003236">Et hoc aliâs docui, quan­<lb></lb>quàm credam me erraſſe in ſupputatione, nam locum inuenire non <lb></lb>potui. </s>
          <s id="id003237">Vnum eſt id certum, quòd hæc ratio quàm nunc explicabo, <lb></lb>eſt uera &amp; demonſtratiua, &amp; facillima.</s>
        </p>
        <p type="main">
          <s id="id003238">Cum enim ſuperior ſit uera &amp; demonſtratiua, non eſt tamen fa­<lb></lb>cilis, &amp; præcipuè in magnis numeris. </s>
          <s id="id003239">Et ideò inueni hanc, quæ (ut <lb></lb>dixi) facillima eſt: adde numero propoſito monadem, in de confla­<lb></lb>ri inuenias numerum à monade in eodem ordine, &amp; ab eo detra­<lb></lb>cta monade habes numerum coniugationum. </s>
          <s id="id003240">Exemplum, ſi ſint <lb></lb>10 adde 1 fit 11. Vndecimus ergo numerus in proportione dupla <lb></lb>eſt 1024, detrahe 1 &amp; relinquantur 1023 numerus coniugationum, <lb></lb>ut in priore ſupputatione. </s>
          <s id="id003241">Item ſi ſint 11 numeri adde 1 fit 12, duo de­<lb></lb>cimus ergo numerus in proportione dupla eſt 2048, detrahe 1 re­<lb></lb>linquuntur 2047, coniugationes 11, ut prius in ſuprà ſcripto exem­<lb></lb>plo. </s>
          <s id="id003242">Et ita pro uiginti quatuor regulis adde 1 fit 25, uigeſimus quin­<lb></lb>tus igitur numerus in ordine duplæ proportionis à monade eſt <lb></lb>16777216, ergo detracta monade relinquitur numerus (ut dixi) re­<lb></lb>gularum &amp; coniugationum uiginti quatuor regularum, quæ ta­<lb></lb>men non ſint contrariæ inuicem: nam tunc eſſent pauciores. </s>
          <s id="id003243">Et <lb></lb>quia in iſtis numeris duplicandis poſſes facile incidere in errorem, <lb></lb>diuide ultimum per 16, &amp; ſi nihil ſupereſt, rectè proceſsit opus: ſin 


<pb pagenum="187" xlink:href="015/01/206.jpg"></pb>autem aliquid ſuperſit, aberraſti. </s>
          <s id="id003244">Vt au­<lb></lb><figure id="id.015.01.206.1.jpg" xlink:href="015/01/206/1.jpg"></figure><arrow.to.target n="table26"></arrow.to.target><lb></lb>tem habeas numeros ſingulorum or­<lb></lb>dinum, in quauis multitudine, deduci­<lb></lb>to numerum ordinis à primo, &amp; diui­<lb></lb>de per numerum ordinis ipſius reli­<lb></lb>quum, &amp; illud quod prouenit, duci­<lb></lb>to in numerum maximum præceden­<lb></lb>tis ordinis, &amp; habebis numerum quæ­<lb></lb>ſitum. </s>
          <s id="id003245">Velut ſi ſint undecim, uolo ſci­<lb></lb>re breuiter numeros, qui fiunt ex ua­<lb></lb>riatione trium. </s>
          <s id="id003246">Primum deduco pro <lb></lb>ſecundo ordine 1 ex 11 fit 10, diuido per <lb></lb>2 numerum ordinis, exit 5, duco in 11 fit <lb></lb>55 numerus ſecundi ordinis. </s>
          <s id="id003247">Inde detra<lb></lb>ho 2, qui eſt numerus differentiæ ordi­<lb></lb>nis tertij à primo ex 11, relinquitur 9, di­<lb></lb>uido 9 per 3 <expan abbr="numerũ">numerum</expan> ordinis exit 3, du­<lb></lb>co 3 in 55 numerum ſecundi fit 165, nu­<lb></lb>merus tertij ordinis. </s>
          <s id="id003248">Similiter uolo nu<lb></lb>merum uariationum quatuor, deduco <lb></lb>3 differentiam 4 à primo ordine ab 11, <lb></lb>relinquitur 8. diuido 8 per 4 numerum ordinis, exit 2, duc 2 in 195 <lb></lb>fit 330. numerus quarti ordinis. </s>
          <s id="id003249">Similiter pro quinto detraho 4 dif­<lb></lb>ferentiam à primo ordine, relinquitur 7, diuido per 5 numerum or­<lb></lb>dinis exit 1 2/5, duco in 330 numerum præcedentis ordinis, fit 462 <lb></lb>numerus quinti ordinis.</s>
        </p>
        <table>
          <table.target id="table26"></table.target>
          <row>
            <cell>1</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>2</cell>
            <cell>2</cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>4</cell>
          </row>
          <row>
            <cell>4</cell>
            <cell>8</cell>
          </row>
          <row>
            <cell>5</cell>
            <cell>16</cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>32</cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>64</cell>
          </row>
          <row>
            <cell>8</cell>
            <cell>128</cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>10</cell>
            <cell>512</cell>
          </row>
          <row>
            <cell>11</cell>
            <cell>1024</cell>
          </row>
          <row>
            <cell>12</cell>
            <cell>2048</cell>
          </row>
          <row>
            <cell>13</cell>
            <cell>4096</cell>
          </row>
          <row>
            <cell>14</cell>
            <cell>8192</cell>
          </row>
          <row>
            <cell>15</cell>
            <cell>16384</cell>
          </row>
          <row>
            <cell>16</cell>
            <cell>32768</cell>
          </row>
          <row>
            <cell>17</cell>
            <cell>65536</cell>
          </row>
          <row>
            <cell>18</cell>
            <cell>131072</cell>
          </row>
          <row>
            <cell>19</cell>
            <cell>262144</cell>
          </row>
          <row>
            <cell>20</cell>
            <cell>524288</cell>
          </row>
          <row>
            <cell>21</cell>
            <cell>1048576</cell>
          </row>
          <row>
            <cell>22</cell>
            <cell>2097152</cell>
          </row>
          <row>
            <cell>23</cell>
            <cell>4194304</cell>
          </row>
          <row>
            <cell>24</cell>
            <cell>8388608</cell>
          </row>
          <row>
            <cell>25</cell>
            <cell>16777216</cell>
          </row>
        </table>
        <p type="main">
          <s id="id003250">Ex hoc colligitur manifeſtè modus conuertendi proportionem </s>
        </p>
        <p type="main">
          <s id="id003251"><arrow.to.target n="marg594"></arrow.to.target><lb></lb>arithmeticam in proportionem miſtam: dico miſtam, quia opor­<lb></lb>tet addere monadem in priore numero: dein de quia numerum <lb></lb>terminorum oportet ſumere iuxta numerum aſsignatum, ſcilicet <lb></lb>addita monade: demum, quia oportet detrahere monadem ipſam. <lb></lb></s>
          <s id="id003252">Eſt tamen ſumpta à proportione Geometrica ut liquet, ſcilicet con­<lb></lb>tinua dupla.</s>
        </p>
        <p type="margin">
          <s id="id003253"><margin.target id="marg594"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003254">Propoſitio centeſima ſeptuageſima prima.</s>
        </p>
        <p type="main">
          <s id="id003255">Propoſitis duobus quibuslibet numeris, quotuis alios, ſeu in <lb></lb>continuum, ſeu medios in continua proportione arithmetica, geo­<lb></lb>metrica &amp; muſica inuenire.</s>
        </p>
        <p type="main">
          <s id="id003256">Hæc tota propoſitio pendet ex intellectu diffinitionis earum. <lb></lb><arrow.to.target n="marg595"></arrow.to.target><lb></lb>Sint ergo propoſiti duo numeri 2 &amp; 3, &amp; uelim tertium in conti­<lb></lb><arrow.to.target n="marg596"></arrow.to.target><lb></lb>nua proportione arithmetica, duplico quemuis, ut pote 3 fit 6, de­


<pb pagenum="188" xlink:href="015/01/207.jpg"></pb>traho 2, reliquum remanet 4 tertius numerus. </s>
          <s id="id003257">Item uolo quar­<lb></lb>tum, duplico 4 fit 8, detraho 3 remanet 5 quartus numerus: item <lb></lb>uolo minorem 3 &amp; 2, duplico 2 fit 4, detraho 3 remanet 1, ſi autem <lb></lb>uellem minorem uno, non poſſet, quia eſſet nihil, ſed creſcendo <lb></lb>poteſt extendi in infinitum, ita capio 2, &amp; &lt;02&gt; 10, duplico &lt;02&gt; 10, fit &lt;02&gt;<lb></lb>40, detraho 2, remanet &lt;02&gt; 40 m: 2, &amp; ita ſi uolo quartum numerum, <lb></lb>duplico &lt;02&gt; 40 m: 2 fit &lt;02&gt; 160 m: 4, detrahe &lt;02&gt; 10 ex &lt;02&gt; 160 m: 4, re­<lb></lb>manet &lt;02&gt; 90 m:4, &amp; ita 2 &lt;02&gt; 10 &lt;02&gt; 40 m: 2, &amp; &lt;02&gt; 90 m: 4, ſunt in con­<lb></lb>tinua proportione arithmetica, &amp; ita poteſt extendi in infini­<lb></lb>tum. </s>
          <s id="id003258">Sed ſi uellem unum, aut duos, aut tres terminos, uel quouis <lb></lb>medio 5 arithmeticæ, diuido differentiam per 1 p:numero termi­<lb></lb>norum, &amp; partes addo minori numero. </s>
          <s id="id003259">Exemplum, uolo tres nu­<lb></lb>meros medios inter 2 &amp; 7 in continua proportione arithmeti­<lb></lb>ca, detraho 2 à 7 remanet 5, diuido 5 per 1 p: quam 3, id eſt per 4, <lb></lb>exit 1 1/4, adde ergo 1 1/4 ad 2 fit 3 1/4 primus terminus, cui adde iterum <lb></lb>1 1/4 fit 4 1/2 ſecundus terminus, cui adde iterum 1 1/4 fit 5 3/4 tertius <lb></lb>numerus: fient ergo quinque termini, hoc modo in continua pro­<lb></lb>portione arithmetica 23 1/4 4 1/2 5 3/4 &amp; 7. Rurſus uolo totidem, uolo <lb></lb>inter 2 &amp; &lt;02&gt; 32, detraho 2 ex &lt;02&gt; 32 remanet &lt;02&gt; 32 m: 2, diuido per 4, <lb></lb>qui eſt 1 p: numero terminorum, exit &lt;02&gt; 2 m: 1/2, addo ergo &lt;02&gt; 2 m: <lb></lb>1/2 ad 2 fit 1 1/2, p: &lt;02&gt; 2 primus terminus, cui iterum addo &lt;02&gt; 2 m: 1/2 fit <lb></lb>&lt;02&gt; 8 p:1, ſecundus terminus, cui etiam addo &lt;02&gt; 2 m: 1/2 fit &lt;02&gt; 18 m: <lb></lb>1/2, &amp; ita habes tres terminos medios in continua proportione <lb></lb>arithmetica inter 2 &amp; &lt;02&gt; 32, &amp; ita ſi uelles quatuor terminos, diui­<lb></lb>deres differentiam per 5, &amp; ſi uelles quinque, diuideres per ſex. </s>
          <s id="id003260">&amp; <lb></lb>ita de alijs quibuſcunque.</s>
        </p>
        <p type="margin">
          <s id="id003261"><margin.target id="marg595"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id003262"><margin.target id="marg596"></margin.target>D<emph type="italics"></emph>iff,<emph.end type="italics"></emph.end> 20.</s>
        </p>
        <p type="main">
          <s id="id003263">Pro Geometrica proponantur, gratia exempli, 2 &amp; 4, ſi uelim in <lb></lb>continua proportione tertium, duco 4 in ſemet fit 16, diuido per 2 <lb></lb>exit 8. &amp; ſi uelles quartum duc 8 in ſe fit 64, diuide per 4 exit 16 <lb></lb>quartus terminus, &amp; ita in infinitum, &amp; ſi uelles minorem 2, duc 2 <lb></lb>in ſe fit 4, diuide 4 per 4 exit 1 tertius terminus, &amp; ita ſi uelles mino­<lb></lb>rem. </s>
          <s id="id003264">duc 1 in ſe fit 1, diuide per 2 exit 1/2 quartus terminus, &amp; ita ha­<lb></lb>bes quoſuis terminos, &amp; eſt ſimilis arithmeticæ hæc operatio, ſed <lb></lb>in arithmetica duplicamus unum terminum, &amp; detrahimus alium: <lb></lb>in geometrica multiplicamus unum terminum ad productum, &amp; <lb></lb>diuidimus per alium. </s>
          <s id="id003265">Et ſi uelim terminum in continua proportio­<lb></lb>ne 2 &amp; &lt;02&gt; 10, duco eodem modo &lt;02&gt; 10 in ſe fit 10, diuido per 2 fit 5 <lb></lb>tertius terminus, &amp; uelim quartum, duco 5 in ſe fit 25, diuido per &lt;02&gt;<lb></lb>10 exit &lt;02&gt; 62 1/2 quartus terminus.</s>
        </p>
        <p type="main">
          <s id="id003266">Et ſi uelles plures terminos medios in proportione geometrica, de <lb></lb>ducito maius extremum in ſe <expan abbr="ſecundũ">ſecundum</expan> <expan abbr="denominationẽ">denominationem</expan> <expan abbr="inferiorẽ">inferiorem</expan>, id 


<pb pagenum="189" xlink:href="015/01/208.jpg"></pb>eſt, ſi uolo duos terminos ſemel, &amp; dein de in minorem, &amp; &lt;02&gt;<lb></lb>cubica producti eſt ſecundus terminus, idem facio de minore in <lb></lb>ſe in de in maiorem, &amp; accipio &lt;02&gt; cu. </s>
          <s id="id003267">Exemplum, uolo duos termi­<lb></lb>nos inter 2 &amp; 3, duco 3 in ſe fit 9, duco 2 in 9 fit 18, capio &lt;02&gt; cu. </s>
          <s id="id003268">18. hic <lb></lb>eſt unus terminus, &amp; ita duco 2 in ſe fit 4, duco in 3 fit 12, capio &lt;02&gt; cu. <lb></lb></s>
          <s id="id003269">12 pro ſecundo termino. </s>
          <s id="id003270">Et ſi uolo tres terminos, duco 3 in 3 fit 9, du<lb></lb>co 3 in 9 fit 27, duco 2 in 27 fit 54, &amp; &lt;02&gt; &lt;02&gt; 54 eſt primus terminus. <lb></lb></s>
          <s id="id003271">Item duco 2 in 2 fit 4, duco 3 in 3 fit 9, duco 4 in 9 fit 36, &amp; &lt;02&gt; &lt;02&gt; 36, id <lb></lb>eſt, &lt;02&gt; 36 eſt ſecundus terminus, ſimiliter duco 2 ad ſuum cubum fit <lb></lb>8, duco 3 in 8 fit 24, &amp; &lt;02&gt; &lt;02&gt; 24, eſt tertius terminus. </s>
          <s id="id003272">Similiter uolo <lb></lb>quatuor terminos medios, duco 3 in 3 fit 9, duco 9 in 9 fit 81, duco 2 <lb></lb>in 81 fit 162, &amp; &lt;02&gt; relata prima 162, eſt primus terminus, item duco 2 <lb></lb>in 2 fit 4, &amp; 4 in 4 fit 16, &amp; 3 in 16 fit 48, &amp; &lt;02&gt; relata prima 48 erit <lb></lb>quartus terminus, item ducendo 3 ad cubum fit 27, &amp; 2 ad quadra­<lb></lb>tum, &amp; fit 4, &amp; 4 in 27 fit 108, &amp; &lt;02&gt; relata prima 108, erit ſecundus <lb></lb>terminus, &amp; ſimiliter ducendo 2 ad cubum fit 8, &amp; 3 ad quadratum <lb></lb>fit 9, &amp; 9 in 8 fit 72, &amp; &lt;02&gt; relata prima 72 eſt tertius terminus. </s>
          <s id="id003273">Habe­<lb></lb>bis ergo terminos in continua proportione 2, id eſt, &lt;02&gt; relata pri­<lb></lb>ma 32, &lt;02&gt; relata prima 48, &lt;02&gt; relata prima 72, &lt;02&gt; relata prima 108, &lt;02&gt;<lb></lb>relata prima 172, &amp; &lt;02&gt; relata prima 243, quod eſt 3, &amp; ita de alijs in <lb></lb>infinitum.</s>
        </p>
        <p type="main">
          <s id="id003274">At pro muſica, ſi ſint exhibiti duo numeri minores utpotè 2 &amp; 3, <lb></lb>uelim tertium terminum, diuido 2 per 1 differentiam exit 2, detraho <lb></lb>1 pro regula remanet 1, diuido 3 maiorem terminum per 1 exit 3, ad­<lb></lb>de 3 ad 3, fit 6 maior terminus. </s>
          <s id="id003275">Similiter capio 3 &amp; 4, diuide 3 mino­<lb></lb>rem terminum per 1 differentiam exit 3, detrahe 1 pro regula, relin­<lb></lb>quitur 2, diuide 4 terminum medium per 2 exit 2, adde ad 4 fit 6 ma<lb></lb>ior terminus. </s>
          <s id="id003276">Stiphelius autem erat in ſua regula, nam ſic 12 4 &amp; 3 <lb></lb>eſſent in continua proportione muſica ex ſua regula. </s>
          <s id="id003277">Dico ergo, <lb></lb>quod ſi proponantur 5 &amp; 7, &amp; uelim muſicam proportionem con­<lb></lb>tinuare, detraho 5 de 7 relinquitur 2, diuido 5 per 2 exit 2 1/2, detra­<lb></lb>he 1 pro regula remanet 1 1/2, diuide 7 per 1 1/2 exit 4 &amp; 2/3, adde ad 7 <lb></lb>fit 11 2/3, reduc ad integra multiplicando omnia per 3, habebis <lb></lb>35, 21, &amp; 15, in continua proportione muſica, nam 35 ad 15 eſt ut 7 <lb></lb>ad 3, &amp; 14 ad 6, eſt ut 7 ad 3, eſt autem 14 differentia 21 &amp; 35, &amp; 6 dif­<lb></lb>ferentia 21 &amp; 15, &amp; ita poſſes continuare inueniendo quartum, <lb></lb>quintum, ſextum, in infinitum. </s>
          <s id="id003278">Rurſus ſint propoſiti duo termini <lb></lb>maiores, uelut 6 &amp; 4, detrahe 4 à 6 exit 2, diuide 6 per 2 exit 3, ad­<lb></lb>de 1 pro regula fit 4, diuide 4 minorem terminum per 4 exit 1, de­<lb></lb>trahe 1 ex 4, relinquitur 3 minor terminus, &amp; ita propoſitis 6 &amp; 3 


<pb pagenum="190" xlink:href="015/01/209.jpg"></pb>differentia eſt 3, diuide 6 per 3 differentiam exit 2, adde 1 pro re­<lb></lb>gula fit 3, diuide 3 per 3 exit 1, detrahe ex 3 relinquitur 2 minor ter­<lb></lb>minus, &amp; ita potes inuenire quotuis. </s>
          <s id="id003279">Gratia exempli, habeo 3 &amp; 2 <lb></lb>maiores, capio 1 differentiam, per quam diuido 3 exit 3, addo 1 <lb></lb>fit 4, diuido 2 minorem terminum per 4 exit 1/2, detrahe 1/2 ex <lb></lb>2, relinquuntur 1 1/2, erunt ergo 32 &amp; 1 1/2, 1. 6. 4. 3. duplican­<lb></lb>do 2, ut prius in continua proportione muſica, quia ergo 632 <lb></lb>ſunt in continua proportione muſica, &amp; 32, &amp; 1 1/2 ſunt in con­<lb></lb>tinua proportione muſica, erunt duplicando 3. 4. 6. 12. in con­<lb></lb>tinua proportione muſica. </s>
          <s id="id003280">Rurſus ſint propoſiti maior, &amp; mi­<lb></lb>nor terminus, ut 6 &amp; 2, diuides maiorem per minorem exit 3, <lb></lb>cui addes 1 fit 4, diuide 4 differentiam 6 à 2 per 4 iam inuentum <lb></lb>exiti, adde ad 2 fit 3 medius terminus, ſimiliter inter 6 &amp; 3, uolo me­<lb></lb>dium terminum in proportione muſica, detraho 3 à 6, relinquitur <lb></lb>3, ſimiliter diuido 6 maiorem terminum per 3 minorem terminum, <lb></lb>exit 2, addo 1 pro regula fit 3, diuido 3 differentiam iam ſeruatam <lb></lb>per hoc 3 iam inuentum exit 1, addo ad 3 minorem terminum fit 4, <lb></lb>medius terminus, ſic uolo inter 4 &amp; 6 medium terminum in con­<lb></lb>tinua proportione muſica, diuido 6 per 4: exit 1 1/2, addo ei pro re­<lb></lb>gula fit 2 1/2, diuide 2 differentiam 4 &amp; 6 per 2 1/2 exit 4/5, adde ad 4 <lb></lb>fit 4 4/5 terminus medius, duc omnes in 5, habebis integros nume­<lb></lb>ros 30, 24 &amp; 20, &amp; ſunt pulcherrimæ regulæ, quia poſſes diui­<lb></lb>dere 24 &amp; 20 interponendo medium, id eſt capiendo 6 &amp; 5, diui­<lb></lb>de 6 per 5 exit 1 1/5, adde 1 pro regula fit 2 1/5, diuide 1 differentiam <lb></lb>per 2 1/5 exit 5/11, adde ad 5 fient termini 5 5/11 &amp; 6, reduc ad integra fi­<lb></lb>ent 55. 60. 66. &amp; quia 30. 24. &amp; 20, etiam erant in continua propor­<lb></lb>tione, &amp; 30 ad 20, erat ſexquialter, ideò capiam ſexquialterum ad <lb></lb>55, &amp; eſt 82 1/2, erunt ergo 82 1/2 66. 60. &amp; 55. in continua proportio­<lb></lb>ne muſica, ergo duplicando 165 132 120 &amp; 110, erunt in continua <lb></lb>proportione.</s>
        </p>
        <p type="main">
          <s id="id003281">Adnotat Stiphelius, quod cum fuerint tres termini in continua <lb></lb>proportione geometrica, &amp; inter primum &amp; tertium interpoſitus <lb></lb>fuerit terminus in continua proportione arithmetica, quod ibi <lb></lb>erit proportio muſica, &amp; dat exemplum de 12. 9. 8 &amp; 6, ſed ita eſt in­<lb></lb>telligendum, ut aſſumpta proportione arithmetica, ut potè 12 9 &amp; <lb></lb>6, in de ut eſt 9 ad 6, ita fiat 12 ad 8, tunc iſti tres termini 128 &amp; 6 e­<lb></lb>runt in continua proportione muſica. </s>
          <s id="id003282">Et hoc eſt pulchrum, ſi ita in­<lb></lb>telligatur, ſcilicet ex proportione Geometrica &amp; Arithmetica con­<lb></lb>ſtituere proportionem muſicam.</s>
        </p>
        <pb pagenum="185 [=191]" xlink:href="015/01/210.jpg"></pb>
        <p type="main">
          <s id="id003283">Ex hoc patet q̊d in <expan abbr="proportionẽ">proportionem</expan> Arithmetica &amp; muſica ſemper, ſi <lb></lb><arrow.to.target n="marg597"></arrow.to.target><lb></lb>duo termini fuerint numeri, tertius erit numerus, &amp; in Geometrica <lb></lb>idem erit, ſi medius &amp; extremus fuerint numeri, erit alter extremus <lb></lb>numerus, ſed tamen ſi unus euariet, omnes poterunt eſſe diuerſi.</s>
        </p>
        <p type="margin">
          <s id="id003284"><margin.target id="marg597"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003285">Propoſitio centeſima ſeptuageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id003286">Proportiones Stiphelij deſcribere.</s>
        </p>
        <p type="main">
          <s id="id003287">Conſiderauit Michael Stiphelius quod ſumpſit à <expan abbr="Boẽtio">Boentio</expan>, quaſ­<lb></lb><arrow.to.target n="marg598"></arrow.to.target><lb></lb>dam inueniri proportiones tribus numeris conſtitutis, quæ in nul­<lb></lb>lo trium primorum generum continerentur, ſed quædam tamen <lb></lb>geometricis aliæ muſicis aſsimilarentur, prima ergo Geometrica­<lb></lb>rum eſt, quoties proportio ſecundæ ad primam fuerit, uelut diffe­<lb></lb>rentiæ ſecundæ &amp; primæ ad differentiam ſecundæ &amp; tertiæ. </s>
          <s id="id003288">Velut <lb></lb><arrow.to.target n="marg599"></arrow.to.target><lb></lb>capio 2, 4, 5, proportio 4 ad 2 eſt dupla talis eſt 2 differentiæ 4 &amp; 2 <lb></lb><arrow.to.target n="marg600"></arrow.to.target><lb></lb>ad 1 differentiam 5 &amp; 4, nam in uera proportione Geometrica fit <lb></lb>conuerſo modo, quia proportio ſecundæ ad primam eſt, uelut dif­<lb></lb>ferentię tertiæ &amp; ſecundæ ad differentiam ſecundæ à prima ut in 4. <lb></lb>6. &amp; 9 proportio 6 ad 4 eſt uelut 3 differentiæ 9 ad 6 ad 2 differen­<lb></lb>tiam 6 &amp; 4.</s>
        </p>
        <p type="margin">
          <s id="id003289"><margin.target id="marg598"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003290"><margin.target id="marg599"></margin.target>2 1</s>
        </p>
        <p type="margin">
          <s id="id003291"><margin.target id="marg600"></margin.target>2 4 5</s>
        </p>
        <p type="main">
          <s id="id003292"><expan abbr="Secũda">Secunda</expan> proportio quam ille appellat poſteriorem, eſt in qua pro <lb></lb>portio tertij ad ſecundum eſt uelut differentiæ primi &amp; ſecundi ad <lb></lb>differentiam ſecundi &amp; tertij: Velut capio 1, 4, 6, proportio 6 ad 4 <lb></lb><arrow.to.target n="marg601"></arrow.to.target><lb></lb>tertij ſcilicet, &amp; ſecundum eſt uelut 3 differentiæ 4 &amp; 1, ad 2, differen­<lb></lb><arrow.to.target n="marg602"></arrow.to.target><lb></lb>tiam 6 &amp; 4, &amp; hæc ſimiliter differt à Geometrica uera in eo quo in <lb></lb>Geometrica uera oporteret, ut proportio tertij ad ſecundum eſſet <lb></lb>ut differentia tertij &amp; ſecundi ad differentiam ſecundi &amp; primi. </s>
          <s id="id003293">Dif­<lb></lb>fert à priore, quoniam in illa differentiæ ſeruant eundem ordinem, <lb></lb>quanuis transferantur in hac uerò fit conuerſus modus.</s>
        </p>
        <p type="margin">
          <s id="id003294"><margin.target id="marg601"></margin.target>3 2</s>
        </p>
        <p type="margin">
          <s id="id003295"><margin.target id="marg602"></margin.target>1 4 6</s>
        </p>
        <p type="main">
          <s id="id003296">Tertia eſt ut ſit proportio differentiæ primæ &amp; tertiæ ad diffe­<lb></lb>rentiam primæ &amp; ſecundæ, uelut ſecundæ ad primam, in Geometri <lb></lb>ca autem eſſet ſicut aggregati ſecundæ &amp; primæ ad ipſam primam, <lb></lb>tales ergo quantitates erunt uelut 4, 6, 7, nam proportio 6 ad 4 eſt <lb></lb><arrow.to.target n="marg603"></arrow.to.target><lb></lb>uelut 3 differentiæ 4 &amp; 7 ad 2 differentiam 4 &amp; 6.<lb></lb><arrow.to.target n="marg604"></arrow.to.target><lb></lb><arrow.to.target n="marg605"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003297"><margin.target id="marg603"></margin.target>3</s>
        </p>
        <p type="margin">
          <s id="id003298"><margin.target id="marg604"></margin.target>4 6 7</s>
        </p>
        <p type="margin">
          <s id="id003299"><margin.target id="marg605"></margin.target>2</s>
        </p>
        <p type="main">
          <s id="id003300">Quarta proportio ſimilis Geometricæ eſt cum fuerit proportio <lb></lb>differentiæ primæ &amp; tertiæ ad differentiam tertiæ &amp; ſecundę, uelut <lb></lb>ſecundæ ad primam, uelut in 2, 3, 5 proportio differentiæ 5 &amp; 2 quæ </s>
        </p>
        <p type="main">
          <s id="id003301"><arrow.to.target n="marg606"></arrow.to.target><lb></lb><arrow.to.target n="marg607"></arrow.to.target><lb></lb>eſt 3 ad differentiam ſecundæ &amp; tertiæ, quæ eſt 2 eſt uelut 3 quantita<lb></lb><arrow.to.target n="marg608"></arrow.to.target><lb></lb>tis ſecundæ ad 2 quantitatem primam.</s>
        </p>
        <p type="margin">
          <s id="id003302"><margin.target id="marg606"></margin.target>3</s>
        </p>
        <p type="margin">
          <s id="id003303"><margin.target id="marg607"></margin.target>2 3 5</s>
        </p>
        <p type="margin">
          <s id="id003304"><margin.target id="marg608"></margin.target>2</s>
        </p>
        <p type="main">
          <s id="id003305">Prima <expan abbr="autẽ">autem</expan> <expan abbr="harmonicarũ">harmonicarum</expan> quæ notha eſt nec legitima, hoc modo <lb></lb>ſumitur: Vt ſit proportio primæ ad tertiam uelut differentiæ ſecun <lb></lb><arrow.to.target n="marg609"></arrow.to.target><lb></lb>dæ &amp; tertiæ ad differentiam ſecundæ &amp; primæ, ueluti capio 6 pri­<lb></lb><arrow.to.target n="marg610"></arrow.to.target><lb></lb>mam 5 ſecundum 3 tertiam proportio 6 ad 3 eſt dupla ſicut 2 diffe­


<pb pagenum="186 [=192]" xlink:href="015/01/211.jpg"></pb>rentiæ ſecundæ à tertia ad 1 differentiam ſecundæ à prima. </s>
          <s id="id003306">Manife­<lb></lb>ſtum eſt autem quod in uera harmonica proportio differentiarum <lb></lb>eſt primæ &amp; ſecundæ ad illam quæ ſecundæ &amp; tertiæ.</s>
        </p>
        <p type="margin">
          <s id="id003307"><margin.target id="marg609"></margin.target>1 2</s>
        </p>
        <p type="margin">
          <s id="id003308"><margin.target id="marg610"></margin.target>6 5 3</s>
        </p>
        <p type="main">
          <s id="id003309">Secunda notha harmonica eſt, ut ſit propor­<lb></lb><figure id="id.015.01.211.1.jpg" xlink:href="015/01/211/1.jpg"></figure><lb></lb>tio primæ ad tertiam, uelut differentiæ primæ à <lb></lb>tertia ad differentiam ſecundæ à tertia, ponatur <lb></lb>25, prima 21, ſecunda 15, tertia proportio 25 ad 15 <lb></lb>eſt uelut 10 differentiæ primę à tertia ad b differen<lb></lb>tiam ſecundæ à tertia.</s>
        </p>
        <p type="main">
          <s id="id003310">Tertia eſt ſimilis priori, niſi quod ſumitur dif­<lb></lb><figure id="id.015.01.211.2.jpg" xlink:href="015/01/211/2.jpg"></figure><lb></lb>ferentia primæ à ſecunda pro ultimo termino. </s>
          <s id="id003311">Ex­<lb></lb>emplum, 25 primus terminus, 19 ſecundus, 15 ter­<lb></lb>tius, proportio 25 ad 15 eſt uelut 10 differentiæ pri­<lb></lb>mæ a tertia ad b, differentiam primæ à ſecunda. <lb></lb></s>
          <s id="id003312">Has proportiones quanquàm exiguæ utilitatis, proponere uo­<lb></lb>lui, ut excogitatis aliquibus demonſtrationibus, uelut ſuperius <lb></lb>diximus, pulchra theoremata &amp; problemata tradi poſſent.</s>
        </p>
        <p type="main">
          <s id="id003313">Propoſitio centeſima ſeptuageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id003314">Circulum ſuper centro ſuo mouere æqualiter, ita quòd omnia <lb></lb>illius puncta per rectam lineam moueantur ultro citro que.<lb></lb><arrow.to.target n="marg611"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003315"><margin.target id="marg611"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003316">Sit a centrum circuli b c, &amp; æqualis ei <lb></lb><figure id="id.015.01.211.3.jpg" xlink:href="015/01/211/3.jpg"></figure><lb></lb>circulus d e, centrum eius b in circumfe­<lb></lb>rentia circuli b c, fixum ita ut ibi mouea­<lb></lb>tur ad motum circuli b c: &amp; moueatur b <lb></lb>uerſus c æqualiter, &amp; e contrario motu <lb></lb>etiam regulariter, &amp; duplo uelocius ex e <lb></lb>uerſus d, dico omnia puncta d e moue­<lb></lb>ri in linea recta, &amp; primum capio pun­<lb></lb>ctum d, quod ſit in linea recta centro­<lb></lb>rum: &amp; moueatur b ad c, &amp; ſi circulus d e <lb></lb>eſſet immobilis, palam eſt quòd pun­<lb></lb>ctum d cum ſit in una linea a b, cum b <lb></lb>perueniret in c, d eſſet in linea a c, putà in <lb></lb>h ſecundum quantitatem, ergo b d ex </s>
        </p>
        <p type="main">
          <s id="id003317"><arrow.to.target n="marg612"></arrow.to.target><lb></lb>centro c, deſcribo circuli portionem h k, <lb></lb>duco etiam c k, erit ergo angulus h c k <lb></lb>duplus a, quare arcus h k duplus b c, <lb></lb>nam conſiſtunt in centris circulorum æ­<lb></lb>qualium: igitur cum ex h motu conuerſo, &amp; duplo ueloci in codem <lb></lb>tempore feratur d perueniet in k, &amp; ita ſecundum rectam lineam <lb></lb>erit motum eadem ratione ex d in k, quod erat demonſtrandum.</s>
        </p>
        <pb pagenum="187 [=193]" xlink:href="015/01/212.jpg"></pb>
        <p type="margin">
          <s id="id003318"><margin.target id="marg612"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003319">Ex hoc patet quòd quando b <lb></lb><figure id="id.015.01.212.1.jpg" xlink:href="015/01/212/1.jpg"></figure><lb></lb><arrow.to.target n="marg613"></arrow.to.target><lb></lb>erit in c peracta quarta circuli, ut in <lb></lb>ſecunda figura erit per motum l e <lb></lb>in a: nam cum d a ſit dupla c b, igi­<lb></lb>tur in eodem tempore l perueniet <lb></lb>ad a, in quo b perueniet ad c.</s>
        </p>
        <p type="margin">
          <s id="id003320"><margin.target id="marg613"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003321">Dico etiam, quod <expan abbr="quãdo">quando</expan> b per­<lb></lb><arrow.to.target n="marg614"></arrow.to.target><lb></lb>ueniet ad fin prima figura, d perue­<lb></lb>niet ad g, quia permeabit totum cir<lb></lb>culum, &amp; a b d ſunt in una recta li­<lb></lb>nea. </s>
          <s id="id003322">Et cum b perueniet ad m in ſe­<lb></lb>cunda figura, d rurſus perueniet ad a centrum.</s>
        </p>
        <p type="margin">
          <s id="id003323"><margin.target id="marg614"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003324">Ex hoc patet, quòd punctum d permeabit lineam rectam æqua­<lb></lb><arrow.to.target n="marg615"></arrow.to.target><lb></lb>lem duplo diametri unius circuli, id eſt, quantum eſt linea a g in pri<lb></lb>ma figura.</s>
        </p>
        <p type="margin">
          <s id="id003325"><margin.target id="marg615"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id003326">Sequitur etiam, quòd d punctum meabit et remeabit per rectam <lb></lb><arrow.to.target n="marg616"></arrow.to.target><lb></lb>lineam ag, peragendo bis eam in uno circuitu circuli b c, ſeu duo­<lb></lb>bus circuitibus d e.</s>
        </p>
        <p type="margin">
          <s id="id003327"><margin.target id="marg616"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id003328">Oſten damus modo, quod pun<lb></lb><figure id="id.015.01.212.2.jpg" xlink:href="015/01/212/2.jpg"></figure><lb></lb>ctum d extra lineam centrorum, ſci <lb></lb>licet in linea d c a f tranſibit per <expan abbr="re­ctã">re­<lb></lb>ctam</expan> eandem, ut in tertia figura pro­<lb></lb>ducatur c d uſque ad k, ita ut c k ſit <lb></lb>æqualis c a, erit ergo punctus d pri<lb></lb>mæ figuræ m è regione k tertiæ, &amp; <lb></lb>dum c mouetur ad e, d perueniat <lb></lb>ad g, erit ergo e g æqualis ea, &amp; ſe­<lb></lb>cet circulus g h rectam a d in h, &amp; <lb></lb>ducatur c h. </s>
          <s id="id003329">Et erit ut prius angu­<lb></lb>lus h e g duplus h a g, ergo arcus <lb></lb>g h duplus e c, ergo g remeauit in <lb></lb>h in tempore quo c feretur in e, <lb></lb>quare d deſcendit per rectam in h.</s>
        </p>
        <p type="main">
          <s id="id003330">Dico rurſus, quòd quanto ma­<lb></lb>gis d erit propinquum lineæ d g, <lb></lb>tanto minus deſcendet in recta, <lb></lb>quanto magis propinquum longi<lb></lb>tudinibus medijs, <expan abbr="tãto">tanto</expan> celerius mo<lb></lb>uebitur, adeò ut in ſecunda figura <lb></lb>apparet motum ex d in g, non deſcendit niſi per d n, &amp; motum ex g <lb></lb>in l deſcendit ex n in a centrum fixum. </s>
          <s id="id003331">Deſcendat ergo ex e in h &amp; h 


<pb pagenum="188 [=194]" xlink:href="015/01/213.jpg"></pb>in k per arcus æquales, &amp; ducantur arcus h l &amp; k m. </s>
          <s id="id003332">Quia n m &amp; n l <lb></lb>ſunt minores quarta circuli, &amp; maiores ſunt f e &amp; fl, &amp; angulus an­<lb></lb>gulo non minor, patet propoſitum. </s>
          <s id="id003333">Ita ergo motus, ut appropin­<lb></lb>quant <expan abbr="pũctis">punctis</expan> medijs ſunt uelociores, &amp; in æquali <expan abbr="diſtãtia">diſtantia</expan> æquales.</s>
        </p>
        <p type="main">
          <s id="id003334">Et hoc inuentum fuit Ludouici Ferrarij, cuius meminimus in Ar<lb></lb>te magna, &amp; nos ei ſubtexuimus ex noſtra inuentione, cuius ille de­<lb></lb>monſtrationem inuenire nequiuit.</s>
        </p>
        <p type="main">
          <s id="id003335">Propoſitio centeſima ſeptuageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id003336">Progreſſus &amp; regreſſus tam ſine latitudine, quàm cum latitudi­<lb></lb>ne in planetis per ſolos concentricos circulos æqualiter motos de­<lb></lb>monſtrare.<lb></lb><arrow.to.target n="marg617"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003337"><margin.target id="marg617"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003338">Sit eclyptica a b c d, &amp; arcus regreſſus b c in partes <lb></lb><figure id="id.015.01.213.1.jpg" xlink:href="015/01/213/1.jpg"></figure><lb></lb>quatuor æquales diuiſus, &amp; deſcribantur circuli duo b <lb></lb>h &amp; e k ſuper e &amp; f, &amp; ſupponatur orbis ſuperior ſub <lb></lb>eclyptica tamen, cuius polus in f, qui circumagatur in du<lb></lb>plo temporis retroceſſus planetæ, &amp; in diſtantia circuli <lb></lb>e k ſub puncto e eclypticæ, polus alterius orbis concen­<lb></lb>trici inferioris, qui circumagatur in tempore retro ceſſus <lb></lb>planetæ, &amp; planeta ſit in puncto 6, liquet ergo quòd pla<lb></lb>neta ille in uno circuitu e k circuli permeabit b c &amp; re­<lb></lb>meabit, &amp; ſemper erit ſub ipſa eclyptica. </s>
          <s id="id003339">Sed enim eclyptica habet <lb></lb>rationem rectæ lineæ, ut quiuis circulus maximus. </s>
          <s id="id003340">Et ſi quis relu­<lb></lb>ctetur fingamus rectam ſubtenſam arcui b c, &amp; aliam poſtmodum <lb></lb>æquidiſtantem in eadem ſuperficie, &amp; in orbe inferiore, &amp; tunc pa­<lb></lb>tebit liquidò propoſitum. </s>
          <s id="id003341">Sed ſi uelim latitudinem deſcribam, ma­<lb></lb>ximam latitudinem à puncto b, &amp; ducam circulum magnum per <lb></lb>punctum illud: reliqua ut prius, ad unguem: nihil enim refert quod <lb></lb>ad demonſtrationem præcedentis attinet, ſeu a d ponatur eclypti­<lb></lb>ca, ſeu alius circulus magnus.</s>
        </p>
        <p type="main">
          <s id="id003342"><arrow.to.target n="marg618"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003343"><margin.target id="marg618"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003344">Ex hoc patet cauſa cur retroceſſus in initio, &amp; in fine ſint exigui, <lb></lb>in medio ſint magni imò maximi, &amp; quomodo perpetuò uarietur <lb></lb>latitudo in tempore retro ceſſus, &amp; ratio omnium, &amp; ſimiliter de in­<lb></lb>crementis &amp; uelocitate motus.</s>
        </p>
        <p type="main">
          <s id="id003345"><arrow.to.target n="marg619"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003346"><margin.target id="marg619"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003347">Ex hoc ſequitur, quod cum erratica fuerit in centro ſeu polo f, &amp; <lb></lb>tunc mouetur uelociſsímè, quòd tamen erit in oppoſito ſolis, &amp; <lb></lb>tunc etiam ibi erit ipſe polus, quare alter erit cum ipſo ſole.</s>
        </p>
        <p type="main">
          <s id="id003348"><arrow.to.target n="marg620"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003349"><margin.target id="marg620"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id003350">Et quia dum motus eſt uelociſsimi ſecundum ordinem ſigno­<lb></lb>rum, tunc erratica ſuperior eſt ſoli iuncta, eſtque in polo, oportet ut <lb></lb>polus f moueatur ſecundum ordinem ſignorum, adeò ut cum ſol <lb></lb>peruenerit ad illius oppoſitum, orbis ſuperior dimidium perfecerit 


<pb pagenum="195" xlink:href="015/01/214.jpg"></pb>circuitus, inferior autem integrum. </s>
          <s id="id003351">Ergo orbis ſuperior tanto tar­<lb></lb>diùs mouetur ſole, quantum eſt id quod peragit polus ſine æquali <lb></lb>motu in orbe ſignorum, per motum circunducentis orbis ſuperio­<lb></lb>ris in tempore dimidij circuitus. </s>
          <s id="id003352">Inferior ergo cum moueatur du­<lb></lb>plo uelociùs ſuperiore, ut dictum eſt, igitur duplo uelocius ſole, ni­<lb></lb>ſi quantum eſt duplum motus poli ſuperioris per motum orbis <lb></lb>circunducentis.</s>
        </p>
        <p type="head">
          <s id="id003353">SCHOLIVM I.</s>
        </p>
        <p type="main">
          <s id="id003354">Intelligo autem per arcum retro ceſſus non ſolum illum quo pla­<lb></lb>neta retrocedit, nam hic eſt longè minor arcu proceſſus, ſed in quo <lb></lb>motus in æqualis eſt minor æquali, palam autem eſt hunc fore æ­<lb></lb>qualem arcui uelocioris motus quàm ſit motus æqualis.</s>
        </p>
        <p type="head">
          <s id="id003355">SCHOLIVM II.</s>
        </p>
        <p type="main">
          <s id="id003356">Cum ergo, dum erratica eſt in polo orbis ſuperioris, ibi quieſcat <lb></lb>motu eius, motu autem inferioris orbis uelociſsimè moueatur ſeu <lb></lb>progrediendo ſeu regrediendo motuque circulari, &amp; tamen per re­<lb></lb>ctam lineam, igitur uideretur quòd motus circularis partes poſſet <lb></lb>tranſire in rectum. </s>
          <s id="id003357">Reſpondeo quòd ſufficit ſola inclinatio ob ma­<lb></lb>gnitudinem anguli: nam dum ſydus transfertur extra centrum mo­<lb></lb>tu orbis inferioris, mouetur uelociter quo ad angulum motu orbis <lb></lb>ſuperioris.</s>
        </p>
        <p type="main">
          <s id="id003358">Propoſitio centeſima ſeptuageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id003359">Cauſam uarietatis diametrorum ex ſuppoſitis concentricis de­<lb></lb>monſtrare.</s>
        </p>
        <p type="main">
          <s id="id003360">In tribus ſuperioribus planetis &amp; quibuſcunque ſtellis octaui or­</s>
        </p>
        <p type="main">
          <s id="id003361"><arrow.to.target n="marg621"></arrow.to.target><lb></lb>bis manifeſtum eſt, quòd pars quæ reſpicit nos quantò remotior <lb></lb>fuerit à Sole, <expan abbr="tãto">tanto</expan> magis illuminatur. </s>
          <s id="id003362">Manifeſtum eſt etiam &amp; expe­<lb></lb>rimento &amp; ratione, quòd illud quod magis lucet, &amp; eſt <expan abbr="illuminatũ">illuminatum</expan> <lb></lb>à Sole in nocte, maius uidetur, ſicut etiam de facibus nocturnis. </s>
          <s id="id003363">Et <lb></lb>rurſus, quod ſub ſtantia orbium circa loca quæ habentur pro polis <lb></lb>eſt denſior, &amp; quod res in medio denſo apparent maiores, ſicut de <lb></lb>piſcibus in aqua, denarijs &amp; baculis. </s>
          <s id="id003364">Demonſtratum <expan abbr="aũt">aunt</expan> eſt in præ­<lb></lb>cedenti, quod quando ſtella fuerit in polo orbis ſuperioris, quòd <lb></lb>tunc maximè retrocedit, &amp; ideò cum in tempore maximi retro ceſ­<lb></lb>ſus ſit in oppoſito Solis <expan abbr="dũ">dum</expan> tres ſuperiores ſunt in oppoſitu Solis, <lb></lb>multo maiores duabus ex cauſis eſſe uidentur, &amp; iuxta proportio­<lb></lb>nem propinquitatis ad Solem commutant quantitatem &amp; tanto <lb></lb>minores apparent, quia non poſſunt, commutare <expan abbr="formã">formam</expan>, uelut Lu­<lb></lb>na propter æqualitatem ſubſtantię &amp; luminis proprij copiam, quę <lb></lb>non ſinit diſcerni uarietatem figurę. </s>
          <s id="id003365">In Luna autem ſecus eſt, nam in 


<pb pagenum="196" xlink:href="015/01/215.jpg"></pb>ipſa diſcernitur ob paucitatem luminis proprij figuræ uarietas, &amp; <lb></lb>ob id non apparet maior, imò minor aut mediæ quantitatis in op­<lb></lb>poſito Solis, ſed maxima in longitudinibus medijs, quoniam ibi <lb></lb>ſunt poli motus uarietatis ut dictum eſt, quę habet locum retro ceſ­<lb></lb>ſus, ſed ob motus paruitatem Luna non poteſt retrocedere, uerùm <lb></lb>ſolùm motus tardatur. </s>
          <s id="id003366">Nam licet denſitas ſit in cœlo ſuperiore &amp; <lb></lb>motus uelox nihilominus efficit imaginem maiorem, ſicut apparet <lb></lb>de piſce in magna aqua in medio, &amp; in parua in imo, nam in parua <lb></lb>uidetur longè maior quàm in magna, licet ſit in æquali diſtantia. </s>
          <s id="id003367">In <lb></lb>Venere autem &amp; Mercurio eadem eſt ratio diſtantiæ à Sole ut di­<lb></lb>ctum eſt in præcedenti. </s>
          <s id="id003368">Cum ergo ſub Sole multum moueantur <lb></lb>motu differentiæ uel ſecundum ſucceſsionem, uel contra ſucceſ­<lb></lb>ſionem in medijs longitudinibus, parum tunc uidentur eſſe mino­<lb></lb>res, quia ſunt remotiores à polo orbis ſuperioris. </s>
          <s id="id003369">Quod autem pro<lb></lb>pinqui coniunctioni Solis, &amp; ueloces uideantur minores, iſtud <lb></lb>contingit ob primam cauſam, quia minus illuminantur, ea parte <lb></lb>quæ ad nos uergit. </s>
          <s id="id003370">Reſtat ergo ſolum oſtendere cur propinqui <lb></lb>Soli &amp; in retroceſſu <expan abbr="uideãtur">uideantur</expan> maiores, cùm utraque ratio obſtet, ſunt <lb></lb>enim remoti à polo orbis ſuperioris &amp; propinqui Soli, cauſa eſt <lb></lb>quoniam apparent ſolùm in crepuſculis quando ſunt ſic diſpoſiti, <lb></lb>&amp; tunc aër eſt craſsior. </s>
          <s id="id003371">Quæ cauſa facit, ut neque dum uelociſsimi <lb></lb>ſunt ſemper parui uideantur, ideò non poteſt conſtitui certa ratio. <lb></lb></s>
          <s id="id003372">imò iſta deducta ſunt potius ex fundamento falſo illius figmen­<lb></lb>ti, quam ex ſenſu (ita enim argumentantur) retro cedunt, ergo ſunt <lb></lb>propinquiores terræ, ergo uidentur maiores, &amp; ita fingunt ſen­<lb></lb>ſu ſehabere quod falſa ratione oſtendere uidentur. </s>
          <s id="id003373">quodque iſtud <lb></lb>ſit uerum, patet quia nullum <expan abbr="inſtrumẽtum">inſtrumentum</expan> etiam in aëre clariſsimo <lb></lb>Aegypti poteſt oſtendere differentiam minorem ſex minutis, &amp; <lb></lb>hic eſt fermè diameter Mercurij, nec tanta eſt differentia in Venere. <lb></lb></s>
          <s id="id003374">Reliquum eſt ut ſatisfaciamus obiectioni quam faciunt de diuer­<lb></lb>ſitate magnitudinis Lunæ propter eclipſim, nam uidetur eſſe ali­<lb></lb>quando maior, &amp; aliquando minor in æquali diſtantia à ſectione <lb></lb>capitis &amp; caudæ draconis, adeò ut non uideatur poſſe aſsignari. </s>
          <s id="id003375">di<lb></lb>co ergo huius cauſam eſſe umbram ipſius Lunæ dubiam, ſicut eti­<lb></lb>am in crepuſculis, quoniam Sol in diuerſo ſitu facit diuerſam um­<lb></lb>bram comparatione oculi noſtri, maior eſt enim in hyeme quàm <lb></lb>in æſtate, &amp; quæ eſt propior nobis quàm quæ procul, &amp; quæ eſt in <lb></lb>meridie quàm iuxta Ortum uel Occaſum, &amp; ideò tam parua diffe­<lb></lb>rentia &amp; incerta, &amp; quæ aliquando uariat, nullo modo uitiare po­<lb></lb>teſt rationem motuum æternorum.</s>
        </p>
        <pb pagenum="197" xlink:href="015/01/216.jpg"></pb>
        <p type="margin">
          <s id="id003376"><margin.target id="marg621"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003377">Propoſitio centeſima ſeptuageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id003378">Rationem centri grauitatis declarare.</s>
        </p>
        <p type="main">
          <s id="id003379">Duplicem rationem <expan abbr="cẽtri">centri</expan> grauitatis inuenit Archimedes, unam <lb></lb><arrow.to.target n="marg622"></arrow.to.target><lb></lb>ſuſpenſorum ponderum: alteram ſupernatantium aquæ, in qua­<lb></lb>rum utraque ſubtilitatis certè eſt quantum dignum eſt authore illo <lb></lb>ingenioſiſsimo, ſicut etiam in elica linea, fructus autem non pro ra­<lb></lb>tione laboris, neque enim ab ætate illa uſque nunc inuentus eſt quiſ­<lb></lb>quam, qui potuerit docere, nec ille idem quæ nam utilitas ex huiuſ­<lb></lb>modi contemplatione haberetur, propterea totum hoc una propo<lb></lb>ſitione concluſimus.</s>
        </p>
        <p type="margin">
          <s id="id003380"><margin.target id="marg622"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003381">Dico igitur quòd <expan abbr="cẽtrum">centrum</expan> grauitatis in appenſis æqualibus qua­<lb></lb>dratis aut quadrilateris parallelis eſt, ubi ſe interſecant duæ diame­<lb></lb>tri. </s>
          <s id="id003382">Et quod in triangulis eſt punctus in quo concurrant tres lineæ, <lb></lb>ductę ab angulis ad latera illa per æqualia ſecando. </s>
          <s id="id003383">In quadrilatero <lb></lb>autem trapezio centrum grauitatis eſt in puncto lineæ, quæ ſecat <lb></lb>ambo latera oppoſita per æqualia, ita ut proportio partis eius li­<lb></lb>neæ, quæ intercipitur à minore æquidiſtantium, ad partem quæ in­<lb></lb>tercipitur à maiore æquidiſtantium, ſit ueluti dupli maioris æqui­<lb></lb>diſtantium cum minore ad duplum minoris æquidiſtantium cum <lb></lb>maiore. </s>
          <s id="id003384">Cuiuſcunque portionis à recta linea, &amp; rectanguli coni ſecti­<lb></lb>one comprehenſæ, centrum grauitatis diuidit diametrum portio­<lb></lb>nis, ita ut pars eius ad uerticem terminata, ſit ad partem eam ſexqui­<lb></lb>altera, quæ ad baſim portionis terminatur. </s>
          <s id="id003385">Cuiuslibet fruſti à ſecti­<lb></lb>one rectanguli coni ablati, centrum grauitatis eſt in linea recta, quę <lb></lb>fruſti exiſtit diametros: qua in quinque partes æquas diuiſa, cen­<lb></lb>trum in quinta eius media exiſtit, atque in eo eius puncto quo ipſa <lb></lb>quinta ſic diuiditur, ut portio eius propinquior minori baſi fru­<lb></lb>ſti ad reliquam eius portionem eam habeat proportionem, quam <lb></lb>habet ſolidum, cuius baſis ſit quadratum lineæ illius quæ fruſti ba­<lb></lb>ſis maior extiterit.. Altitudo ueró iſtis utriſque ſimul æqualis lineæ <lb></lb>quæ dupla ſit minoris baſis fruſti, &amp; baſi maiori eiuſdem, ad ſoli­<lb></lb>dum quod baſim habeat quadratum baſis minoris fruſti, altitudi­<lb></lb>nem uero iſtis utriſque ſimul æqualem lineæ quæ dupla ſit maioris <lb></lb>baſis, &amp; baſi minori. </s>
          <s id="id003386">Et hæc de prima, multa qúe alia pulchra de­<lb></lb>clarat Federicus Comandinus, in ſuo libro de Centro grauitatis, ut <lb></lb>pote. </s>
          <s id="id003387">Quod cuiuslibet portionis conoidis rectanguli axis à cen­<lb></lb>tro grauitatis ita diuiditur ut pars, quæ determinatur ad uerticem <lb></lb>reliquæ, quæ ad baſim terminatur dupla ſit, &amp; longè ſubtiliora quę <lb></lb>quilibet uidere poterit apud illum.</s>
        </p>
        <pb pagenum="198" xlink:href="015/01/217.jpg"></pb>
        <p type="head">
          <s id="id003388">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id003389">Partes omnes conſentiunt in grauitatem medij, quoniam una <lb></lb>aliam non uult centro mundi fieri propiorem.</s>
        </p>
        <p type="main">
          <s id="id003390">De ſecunda præcipua ſunt, quod ſi magnitudo aliqua humido <lb></lb>leuior ea in grauitate proportionem habebit ad humidum ęqualis <lb></lb>molis, quam pars magnitudinis demerſa ad totam magnitudinem, <lb></lb>&amp; hoc intelligitur quando magnitudo illa fuerit è genere ſolido­<lb></lb>rum rectorum &amp; rectangulorum. </s>
          <s id="id003391">Secunda eſt, quòd quæ ſimilia <lb></lb>ſunt ſuperficiebus, ita ut axem habeant in medio, ſecundum ſitum <lb></lb>axis merguntur &amp; prominent, &amp; ſi aliter mergantur, redeunt. </s>
          <s id="id003392">Ter­<lb></lb>tia, quod quę anguſtiora ſunt, ab oppoſita parte uerò latiora, incli­<lb></lb>nantur ad partem acutiorem, quia ſic facilius deſcendunt. </s>
          <s id="id003393">Quarta <lb></lb>eſt, de corporibus non æqualibus, ipſa enim neceſſe eſt, ut ab hac ſe <lb></lb>inflectant, &amp; ratio horum diuerſa eſt iuxta rationem proportionis <lb></lb>partium quæ merguntur adinuicem. </s>
          <s id="id003394">Quinta eſt, quòd merſa in hu­<lb></lb>mido, quanto minus merſa fuerint, tanto facilius &amp; eo frequenti­<lb></lb>us commutantur.</s>
        </p>
        <p type="main">
          <s id="id003395">Propoſitio centeſima ſeptuageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id003396">Si proportio aliqua ex duabus proportionibus eiuſdem quanti <lb></lb>tatis ad alias duas componatur: erit proportio illarum duarum ea­<lb></lb>dem proportioni producti ex proportione in primam duarum <lb></lb>quantitatum detracta priore illa quantitate, quæ ad duas compara<lb></lb>tur, ad eandem priorem quantitatem.</s>
        </p>
        <p type="main">
          <s id="id003397">Sit proportio a ad compoſita ex proportionibus c <lb></lb><arrow.to.target n="marg623"></arrow.to.target><lb></lb><figure id="id.015.01.217.1.jpg" xlink:href="015/01/217/1.jpg"></figure><lb></lb>ad d &amp; c ad e, dico quòd proportio d ad e eſt, ut produ­<lb></lb>cti ex proportione in d detracto c ad ipſum c. </s>
          <s id="id003398">Et nos <lb></lb>ſuperius expoſuimus conuerſam huius. </s>
          <s id="id003399">Erit enim per <lb></lb><expan abbr="ſecundã">ſecundam</expan> demonſtrationem illius proportio a ad b, uelut producti <lb></lb>ex c in d, &amp; e ad productum d in e: at productum d in e &amp; in propor<lb></lb>tionem, eſt idem quod productum proportionis in d in ipſum e: igi<lb></lb>tur cum in uno ſit productum e in c, &amp; d in c, in alio productum a b <lb></lb>in d in de in e, quæ ſunt æqualia, detracto producto e in c ex produ­<lb></lb>cto proportionis in d &amp; inde in e, relinquetur, productum c in d æ­<lb></lb>quale producto a b .i. </s>
          <s id="id003400">proportionis in productum d in e, detracto <lb></lb>numero c in e: igitur ducto c in d, &amp; diuiſo per productum a b in d <lb></lb>numero c, exibit e, igitur cum illud productum fiat ex d, ſcilicet in c, <lb></lb>&amp; ex e in productum proportionis in d dempto numero c, erit pro <lb></lb>portio d ad e, uelut producti ex d in proportionem, detracto e ad <lb></lb>ipſum c, uelut c ſit 12, d 4, e 6, a b erit 5 proportio d ad e, uelut d in a b, <lb></lb>id eſt 20, detracto c, &amp; eſt 8 ad c 12.</s>
        </p>
        <pb pagenum="199" xlink:href="015/01/218.jpg"></pb>
        <p type="margin">
          <s id="id003401"><margin.target id="marg623"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003402">Ex demonſtratione ſequitur, quod qualis eſt proportio e ad a b, <lb></lb><arrow.to.target n="marg624"></arrow.to.target><lb></lb>talis eſt producti d in e, ad aggregatum eorum. </s>
          <s id="id003403">Si quis ergo dicat, <lb></lb>habeo 10, &amp; uolo inuenire duas quantitates, quarum differentia ſit <lb></lb>1, &amp; proportio 10, ad eas componat quintuplam, dices quintupla <lb></lb>eſt dimidium 10, igitur in uenias duas quantitates, quarum differen<lb></lb>tia ſit 1, &amp; proportio producti unius in alteram ad aggregatum ſit <lb></lb>dupla. </s>
          <s id="id003404">Et hoc eſt manifeſtum.</s>
        </p>
        <p type="margin">
          <s id="id003405"><margin.target id="marg624"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id003406">Propoſitio centeſima ſeptuageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id003407">Proportionem miſtionis metallorum, maximè auri &amp; argenti <lb></lb>declarare.</s>
        </p>
        <p type="main">
          <s id="id003408">Dubium non eſt, quod miſtio non cognoſcatur ducto ponde­<lb></lb><arrow.to.target n="marg625"></arrow.to.target><lb></lb>re totius in partem auri uel argenti, &amp; productis collectis diuiſo <lb></lb>aggregato per aggregatum ponderis, idqúe eſt per ſe manife­<lb></lb>ſtum, nam qualis eſt proportio partis ad partem, talis eſt totius ad <lb></lb>totum.</s>
        </p>
        <p type="margin">
          <s id="id003409"><margin.target id="marg625"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003410">Sed eſt genus miſtionis, quod uocant conſolationem. </s>
          <s id="id003411">Veluti, <lb></lb>uolo ex argento perfectionis decem &amp; ſeptem, &amp; quinque, confla­<lb></lb>re argenti maſſam centum librarum perfectionis nouem, ita agen­<lb></lb>dum eſt. </s>
          <s id="id003412">Detrahe 9 à 10, &amp; omni maiori 10, relinqui­<lb></lb>tur 1, hoc ſuppone 7 &amp; 5, item detrahe 7 &amp; 5, &amp; omne <lb></lb><figure id="id.015.01.218.1.jpg" xlink:href="015/01/218/1.jpg"></figure><lb></lb>minus 9 à 9, relinquitur 2 &amp; 4, iunge omnia reſidua <lb></lb>fient 8, nam 4. 2. 11. Dicemus ergo quod 8 uncię per­<lb></lb>fectionis nouem componentur ex 6 uncijs perfe­<lb></lb>ctionis decem &amp; una ſeptem alia quinque. </s>
          <s id="id003413">Poſt di­<lb></lb>ces, ſi unciæ octo fiant 100, ſex &amp; una, &amp; una, quot fient, eruntque un­<lb></lb>ciæ aut libræ, aut ut uocant marchæ perfectionis decem, &amp; duo de­<lb></lb>cim cum dimidia, ac duodecim cum dimidia perfectionis, ut ſe­<lb></lb>ptem &amp; ut quinque: licebit etiam propoſitis terminis pluribus ex <lb></lb>repetita operatione idem facere, ueluti ſint maſſæ perfectionis 10. <lb></lb>7. 5. &amp; 2. uolo maſſam perfectionis ut 8. Tu ſcis quod ex 10. 7 &amp; 5. <lb></lb>fit maſſa perfectionis nouem data lege ſub 6. 1 &amp; 1. nunc habeo iam <lb></lb>perfectam ut 9, aliam ut 2, detraho 2 ex 8, relinquitur 6 &amp; 8, x 9 re­<lb></lb>linquitur 1, iunge fient 7, erunt ergo ſeptem unciæ, in <lb></lb><figure id="id.015.01.218.2.jpg" xlink:href="015/01/218/2.jpg"></figure><lb></lb>quibus ſex erunt perfectionis, ut 9 &amp; 1 perfectionis ut <lb></lb>2, &amp; totum erit perfectionis ut octo. </s>
          <s id="id003414">Duc ergo, ut ex­<lb></lb>plores ueritatem, 6 in 9 fit 54, duc 2 in 1 fit 2, iunge fit 56 <lb></lb>diuide per 7 exit 8 perfectio quæſita.</s>
        </p>
        <p type="main">
          <s id="id003415">Per idem intelliges detractionem ex maſſa argenti perfectionis <lb></lb>7, detraxi quartam partem perfectionis 10, uolo ſcire dodrantem <pb pagenum="200" xlink:href="015/01/219.jpg"></pb>

qualis relinquatur perfectionis, duc quadrantem in 10 fit 30, duc 12 <lb></lb>in 7 fit 84, detrahe 30 ex 84, relinquitur 54, divide 54 per 9, reſidu­<lb></lb>um 12 &amp; 3, exit 6 perfectio reſidui.</s>
          <lb></lb>
        </p>
        <p type="main">
          <s id="id003416">Si quis dicat propoſitis argenti pondo 50 &amp; dodrante perfe­<lb></lb>ctionis 11/18, uolo partem aſſumere, &amp; igne perficere, ita purum ar­<lb></lb>gentum, quod relinquitur additum reſiduo, efficiat ipſum perfe­<lb></lb>ctionis dextantis &amp; beſsis unciæ pro libra, ſeu 8/9, divide 11/18 per 8/9 exit <lb></lb>11/16, duc in pondo 50 cum dodrante, fiunt pondo 34, unciæ 7 1/8, hoc <lb></lb>igitur erit aggregatum conflatum ex argento puro &amp; reſiduo. </s>
          <s id="id003417">De­<lb></lb>trahe igitur 11/18 ex integro, relinquitur 7/18, detrahe pondo 34, uncia <lb></lb>7 1/8 ex pondo 50 cum dodrante, relinquuntur pondo 15 unciæ 6 7/8 <lb></lb>(pondo enim uncias continet ſub hoc ſenſu, quia uſui ſeruimus o­<lb></lb>cto) divide per 7/8, exeunt pondo 40 unciæ 6 1/4, &amp; tanta pars debuit <lb></lb>igne purgari. </s>
          <s id="id003418">In ea enim erunt puri argenti pondo 24, unciæ 7 /78, <lb></lb>quæ addita reſiduo, ſcilicet pondo 9, uncijs 7 3/4 conficiunt pondo <lb></lb>34 uncias 7 1/8 perfectionis dictæ.</s>
          <lb></lb>
        </p>
        <p type="main">
          <s id="id003419">Quidam miſcuit uncias decem auri perfectionis dextantis, &amp; <lb></lb>partem perfectionis dextantis cum dimidio, &amp; aliud perfectionis <lb></lb>beſsis concreuit maſſa perfectionis dodrantis unciarum octuagin<lb></lb>ta, quæruntur pondera reliquarum partium, ſubtrahe 10 pondus <lb></lb>ex 80 pondere, relinquuntur 70 perfectionis 17 5/7, inde detrahe per <lb></lb>modum ſuperiorem, &amp; relinquuntur 3 2/7 &amp; 1 5/7, <lb></lb><figure id="id.015.01.219.1.jpg" xlink:href="015/01/219/1.jpg"></figure>iunge ſimul fiunt 5, dico ergo, ſi 6 producit <lb></lb>70, quid producet 3 2/7 &amp; 1 5/7, &amp; inuenies quod 1 5/7 <lb></lb>producet 24 &amp; 3 5/7 producet 46, qui iuncti faci­<lb></lb>unt 70. </s>
          <s id="id003420">Igitur aurum perfectionis dextantis <lb></lb>cum dimidio fuit unciarum 46 aurum perfe­<lb></lb>ctionis, beſsis unciarum 24. </s>
          <s id="id003421">Reliqua interro­<lb></lb>gata diſſolues per regulas Algebræ, horum <lb></lb>modo.</s>
        </p>
        <p type="main">
          <s id="id003422">Propoſitio centeſima ſeptuageſima nona.</s>
        </p>
        <p type="main">
          <s id="id003423">Si duobus totis duæ portiones ſimiles abſcindantur, ab eiſdem <lb></lb>denuo, &amp; abſciſsis proportionibus partes eædem auferantur, denuo<expan abbr="q̀">que</expan><lb></lb> ac denuo, quoties libuerit à portionibus, &amp; à reſiduis ipſarum <lb></lb>quantitatum partes eædem auferantur, erit reſidui ad reſiduum, ue <lb></lb>luti totius ad totum.</s>
        </p>
        <p type="margin">
          <s id="id003424"><margin.target id="marg906"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003425">Sint duæ quanitates a b &amp; k l, &amp; abſciſſæ duæ partes ſimiles ex <lb></lb>utraque b c &amp; l m, &amp; ſit propoſita aliqua proportio, quæ ſit h, &amp; <lb></lb>ſumatur portio b d ipſius b c, ſecundum proportionem h, &amp; ſi­<lb></lb>


<pb pagenum="201" xlink:href="015/01/220.jpg"></pb>militer l n ipſius l m, iuxta pro­<lb></lb><figure id="id.015.01.220.1.jpg" xlink:href="015/01/220/1.jpg"></figure><lb></lb>portionem h, ſumatur rurſus <lb></lb><arrow.to.target n="marg626"></arrow.to.target><lb></lb>de ipſius a b pars ſecundum h, <lb></lb><arrow.to.target n="marg627"></arrow.to.target><lb></lb>&amp; n o ipſius k l, ſecundum ean <lb></lb>dem proportionem. </s>
          <s id="id003426">Et rurſus <lb></lb><arrow.to.target n="marg628"></arrow.to.target><lb></lb>ſumatur e f æqualis d b, &amp; o p <lb></lb><arrow.to.target n="marg629"></arrow.to.target><lb></lb>æqualis n l, ut ſint portiones <lb></lb>b c &amp; l m ſecundum proportionem h, &amp; ſumatur f g ipſius a c, ſecun <lb></lb><arrow.to.target n="marg630"></arrow.to.target><lb></lb>dum proportionem h, &amp; p q ipſius k o, ſecundum eandum propor­<lb></lb><arrow.to.target n="marg631"></arrow.to.target><lb></lb>tionem, &amp; ita procedendo ſemper, dico quod erit a g reſidui ad k q <lb></lb><arrow.to.target n="marg632"></arrow.to.target><lb></lb>reſiduum, ut a b ad k l. </s>
          <s id="id003427">Quia enim a b ad b c, ut k l ad l m ex ſuppoſi­<lb></lb><arrow.to.target n="marg633"></arrow.to.target><lb></lb>to, erit a b ad b d, ut k l ad l n: eſt etiam a b ad d e, ut k l ad n o ex ſup­<lb></lb>poſito, igitur a b ad b c, ut k l ad l o. </s>
          <s id="id003428">Igitur a b ad a c, ut k l ad k o. </s>
          <s id="id003429">Rur<lb></lb><arrow.to.target n="marg634"></arrow.to.target><lb></lb>ſus quia b c ad e f, ut l m ad o p, erit a b ad e f, ut k l ad o p, at fuit a b <lb></lb>ad a e, ut k l ad k o &amp; a e ad g f, ut k o ad p q, igitur a b ad&#039; g f, ut k l ad <lb></lb>q p. </s>
          <s id="id003430">Quare a b ad g e, ut k l ad q o. </s>
          <s id="id003431">Iterum ergo a b ad b g, ut k l ad <lb></lb>l <expan abbr="q.">que</expan> Ergo a b ad a g, ut k l ad k <expan abbr="q.">que</expan> Igitur a b ad k l, ut a g ad k q, quod <lb></lb>erat demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id003432"><margin.target id="marg626"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 22. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003433"><margin.target id="marg627"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003434"><margin.target id="marg628"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 19. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end><lb></lb>22. <emph type="italics"></emph>eiuſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003435"><margin.target id="marg629"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 22. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003436"><margin.target id="marg630"></margin.target>P<emph type="italics"></emph>er eandem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003437"><margin.target id="marg631"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 19. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end><lb></lb>22 <emph type="italics"></emph>eiuſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003438"><margin.target id="marg632"></margin.target>P<emph type="italics"></emph>er eaſdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003439"><margin.target id="marg633"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 19 <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003440"><margin.target id="marg634"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003441">Ex hoc patet, quod etſi proportio non maneat eadem in parti­<lb></lb><arrow.to.target n="marg635"></arrow.to.target><lb></lb>bus totius, &amp; partis modo ſit eadem in totis ad partes aſſumptas, et <lb></lb>in partibus ad partes aſſumptas, nihilominus ſequitur idem.</s>
        </p>
        <p type="margin">
          <s id="id003442"><margin.target id="marg635"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003443">Sequitur rurſus, quod etſi proportio eadem non maneat quan­<lb></lb><arrow.to.target n="marg636"></arrow.to.target><lb></lb>titatum aſſumptarum ad partes quæ ſumuntur, nec etiam partium <lb></lb>modo ſemper pars, quæ aſſumitur ſit totius pars, &amp; alia partis idem <lb></lb>ueratur.</s>
        </p>
        <p type="margin">
          <s id="id003444"><margin.target id="marg636"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003445">Velut ſi prima uice capiam b d partem b c, ut l n partem l m ſe­<lb></lb><arrow.to.target n="marg637"></arrow.to.target><lb></lb>cundum h proportionem, &amp; deinde capiam d e partem a b &amp; n o <lb></lb>partem k l ſecundum proportionem r, quæ ſit alia ab h, &amp; ſecunda <lb></lb>uice capiam e f partem b c, &amp; o p partem l m ſecundum proportio­<lb></lb>nem h, quæ ſit alia ab h &amp; r. </s>
          <s id="id003446">Et capiam f g partem a e &amp; p q partem <lb></lb>k o, ſecundum eandem proportionem, ſed tamen quæ non ſit ali­<lb></lb>qua prædictarum, ſcilicet h r s, ſed diuerſa ab eis, &amp; uocetur t, dico <lb></lb>quod nihilominus erit proportio a g ad k q, ut a b ad k l, quæ pa­<lb></lb>tent ex ui demonſtrationum, in quibus nil plus aſſumitur ad de­<lb></lb>monſtrandum, quàm id quod proponitur in corrolarijs.</s>
        </p>
        <p type="margin">
          <s id="id003447"><margin.target id="marg637"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id003448">Ex hoc etiam ſequitur, quod ſecundum quem numerum prima <lb></lb><arrow.to.target n="marg638"></arrow.to.target><lb></lb>quantitas abſumetur, ſecundum eundem abſumetur &amp; ſecunda.</s>
        </p>
        <p type="margin">
          <s id="id003449"><margin.target id="marg638"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. .3.</s>
        </p>
        <p type="main">
          <s id="id003450">Velut ſi prima quantitas abſumatur ad unguem in quinta detra­<lb></lb><arrow.to.target n="marg639"></arrow.to.target><lb></lb>ctione, etiam ſecunda k l in quinta detractione ad unguem abſume<lb></lb>tur, quod patet per demonſtrata, nam reſidua ſemper ſunt eædem <lb></lb>partes ipſarum quantitatum.</s>
        </p>
        <pb pagenum="202" xlink:href="015/01/221.jpg"></pb>
        <p type="margin">
          <s id="id003451"><margin.target id="marg639"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003452">Quarto ſequitur, quod ſi detractio fuerit facta eodem modo, &amp; <lb></lb><arrow.to.target n="marg640"></arrow.to.target><lb></lb>fuerit proportio totius ad totum, ut reſidui ad reſiduum, erunt par <lb></lb>tes aſſumptæ ſimiles.</s>
        </p>
        <p type="margin">
          <s id="id003453"><margin.target id="marg640"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 4.</s>
        </p>
        <p type="main">
          <s id="id003454">Velut ſi fuerit facta detractio iuxta propoſitionem, aut primum <lb></lb><arrow.to.target n="marg641"></arrow.to.target><lb></lb>uel ſecundum corrolarium, &amp; fuerit proportio a g ad k g, ut a b ad <lb></lb>k l, erit a b ad b c, ut k l ad l m.</s>
        </p>
        <p type="margin">
          <s id="id003455"><margin.target id="marg641"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003456">Sequitur etiam, quod ſi fuerit aſſumpta proportio <expan abbr="primarũ">primarum</expan> par­<lb></lb><arrow.to.target n="marg642"></arrow.to.target><lb></lb>tium eadem, &amp; facta fuerit detractio in omnibus præter unam iux­<lb></lb>ta dicta, &amp; fuerit totius ad totum, ut reſidui ad reſiduum, erit ut illa <lb></lb>etiam reliqua detractio, ſeu ad tota, ſeu ad partes ſit facta, ſecundum <lb></lb>eandem proportionem.</s>
        </p>
        <p type="margin">
          <s id="id003457"><margin.target id="marg642"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 5.</s>
        </p>
        <p type="main">
          <s id="id003458">Velut ſi ſit proportio a b ad k l, ut a g ad k g, &amp; rurſus ut b c ad <lb></lb><arrow.to.target n="marg643"></arrow.to.target><lb></lb>l m, &amp; aſſumptæ ſint proportiones eædem ſemper totius, &amp; totius <lb></lb>ad partes, &amp; reſiduorum ad partes, etiam &amp; b c &amp; l m ad partes, eti­<lb></lb>am excepta una ſeu quantitatum a b &amp; k l, ſeu reſiduorum ut a c &amp; <lb></lb>k o, ſeu partium ut b c &amp; l m ad partes, dico quod hæ partes etiam <lb></lb>erunt aſſumptæ ſecundum eandem proportionem ad ipſas magni­<lb></lb>tudines, uel partes primas uel reſidua.</s>
        </p>
        <p type="margin">
          <s id="id003459"><margin.target id="marg643"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003460">Sed &amp; id ſequitur ex his, quod cuiuſcunque ſeu totius ſeu partis <lb></lb><arrow.to.target n="marg644"></arrow.to.target><lb></lb>ſeu utriuſque pars maior aſſumetur, erit maior proportio totius ad <lb></lb>totum quàm reſidui ad reſiduum.</s>
        </p>
        <p type="margin">
          <s id="id003461"><margin.target id="marg644"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 6.</s>
        </p>
        <p type="main">
          <s id="id003462">Hæc demonſtrantur à Campano, nam ſi ſit maior proportio a b <lb></lb><arrow.to.target n="marg645"></arrow.to.target><lb></lb>ad a g, quam k l ad k g, erit maior a b ad k l quàm a g ad k g.<lb></lb><arrow.to.target n="marg646"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003463"><margin.target id="marg645"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003464"><margin.target id="marg646"></margin.target>R<emph type="italics"></emph>up.<emph.end type="italics"></emph.end> 16. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003465">Sequitur rurſus, quod in eadem conſtitutione cuiuſcunque ma­</s>
        </p>
        <p type="main">
          <s id="id003466"><arrow.to.target n="marg647"></arrow.to.target><lb></lb>ior pars abſumetur, ea quantitas minori numero, uel numeri parte <lb></lb>abſumetur.</s>
        </p>
        <p type="margin">
          <s id="id003467"><margin.target id="marg647"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 7.</s>
        </p>
        <p type="main">
          <s id="id003468">Nam ſi minor erit continuo proportio a b ad a e, quàm k l ad k <lb></lb><arrow.to.target n="marg648"></arrow.to.target><lb></lb>o, &amp; a e ad e g, quàm k o ad o g, erit longe minor a b ad b g quàm k l <lb></lb>ad l g, igitur longe maior a b ad a g quam k l ad k g. </s>
          <s id="id003469">Igitur a g citius <lb></lb>abſumetur quam k g.</s>
        </p>
        <p type="margin">
          <s id="id003470"><margin.target id="marg648"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003471">Propoſitio centeſima octuageſima.</s>
        </p>
        <p type="main">
          <s id="id003472">Si aliqua quantitas in duas partes diuidatur, fueritque alicuius, <lb></lb>quantitatis ad partes illas compoſita proportio eiuſdem quan­<lb></lb>titatis ad partes alias quantitatis diuiſa aliter proportio eadem <lb></lb>componi.<lb></lb><arrow.to.target n="marg649"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003473"><margin.target id="marg649"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003474">Sit a b proportio ad partes c d quæ ſint c e, &amp; c d componens f, <lb></lb>dico quod non poterit c d aliàs diuidi, ut proportio a b ad illas <lb></lb>componat eandem proportionem f. </s>
          <s id="id003475">Aliter ſit diuiſa in g, &amp; erit mi­


<pb pagenum="203" xlink:href="015/01/222.jpg"></pb>nor c g, minor aut maior c d minore, capiam ergo c d minorem, erit <lb></lb>igitur proportio a b ad c d maioris exceſſus ad proportionem a b <lb></lb>ad c g, quàm ſit proportio a b ad g d, ma­<lb></lb><figure id="id.015.01.222.1.jpg" xlink:href="015/01/222/1.jpg"></figure><lb></lb>ior proportione a b ad c e, propterea quod <lb></lb>g e communis differentia maiorem habet <lb></lb>proportionem ad e d quam g c, igitur ma­<lb></lb>ius eſt aggregatum proportionum a b ad <lb></lb>c e, &amp; e d, <expan abbr="quã">quam</expan> eiuſdem a b ad c g &amp; g d, quod erat demonſtrandum.</s>
        </p>
        <p type="main">
          <s id="id003476">Propoſitio centeſima octuageſima prima.</s>
        </p>
        <p type="main">
          <s id="id003477">Cum fuerit aliqua proportio compoſita ex proportionibus pri­<lb></lb>mæ ad ſecundam &amp; tertiam, &amp; rurſus quartæ ad quintam &amp; ſex­<lb></lb>tam, ita ſe habebit proportio ſecundæ ad tertiam proportionem <lb></lb>quintæ ad ſextam, uelut producti ex proportione in ſecundam de­<lb></lb>tracta prima ad primam ad productum ex proportione in quin­<lb></lb>tam, detracta quarta ad quartam.</s>
        </p>
        <p type="main">
          <s id="id003478">Sit pro portio g compoſita ex proportionibus a <lb></lb><figure id="id.015.01.222.2.jpg" xlink:href="015/01/222/2.jpg"></figure><lb></lb>ad b &amp; c, &amp; proportionibus d ad e &amp; f, dico quod <lb></lb>quemadmodum b ad c, ad proportionem e ad f, ita <lb></lb>producti ex g in b, detracto a ad a ad productum ex <lb></lb>g in e, detracto d ad d. </s>
          <s id="id003479">Eſt enim, ut demonſtratum <lb></lb>eſt b ad c, ut productum ex g in b, detracto a ab a &amp; e ad f, ut pro­<lb></lb>ducti ex g in e, detracto d ad d, igitur cum æqualium ſint eædem <lb></lb>comparationes, erit ut proportionis b ad c ad proportionem e ad <lb></lb>f, ita producti ex g in b, detracto a ad a, ad productum eſt g in e, de­<lb></lb>tracto d ad d.</s>
        </p>
        <p type="main">
          <s id="id003480">Quare erit proportio b ad c ad proportionem e ad f, uelut reſi­<lb></lb>dui b detracto quod prouenit, diuiſo a per proportionem a ad pro <lb></lb>portionem reſidui e detracto quod prouenit diuiſo d per propor­<lb></lb>tionem ad ipſum d.</s>
        </p>
        <p type="main">
          <s id="id003481">Propoſitio centeſima octuageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id003482">Propoſita differentia proportionum partium ſimilium ad par­<lb></lb>tes aſſumptas propoſitaque proportione totius ad reſidua eandem <lb></lb>differentiam proportionum totius ad reliquum reſidui inuenire.</s>
        </p>
        <figure id="id.015.01.222.3.jpg" xlink:href="015/01/222/3.jpg"></figure>
        <p type="main">
          <s id="id003483">Sint datæ partes b c &amp; e f, ſimiles in compa­<lb></lb>ratione ad a b &amp; d e, &amp; data reſidua a g &amp; d h <lb></lb>in <expan abbr="cõparatione">comparatione</expan> a b &amp; d e, ſimilia in differentia <lb></lb>proportionis f e ad c l, ad proportionem <lb></lb>c b ad b k, dico quod data eſt differentia proportionis a b ad g k <lb></lb>ad proportionem d e &amp; f h. </s>
          <s id="id003484">Nam quia proportio f e ad c l, ad pro­


<pb pagenum="204" xlink:href="015/01/223.jpg"></pb>portionem b e ad c k data eſt, &amp; c f ad e d, ut b c ad b a, erit ut a c ad <lb></lb>l e contineat a b ad b k, ut f e ad e l, c b ad b k, ſed a b ad a d, ut d c ad <lb></lb>d h, igitur a b ad b d, ut d e ad c h. </s>
          <s id="id003485">Sunt ergo duæ quantitates a b &amp; <lb></lb>d c, quæ eandem habent compoſitam proportionem ad g k &amp; k b, <lb></lb>&amp; h l &amp; l e, quare per præcedentem proportionis h l ad l e, ad pro<lb></lb>portionem g k ad k b, ut h l detracto prouentu d e, diuiſi per propor<lb></lb>tionem ad d e ad proportionem g k, detracto prouentu a b, diuiſi <lb></lb>per eandem proportionem ad ipſum a b. </s>
          <s id="id003486">Si igitur nota eſt l e &amp; h l, <lb></lb>erit nota proportio reſidui h l detracto prouentu d e diuiſi per pro­<lb></lb>portionem, quare nota detractio g k detracto prouentu a b diuiſi <lb></lb>per eandem proportionem ad a b. </s>
          <s id="id003487">Eſt autem a b nota, &amp; propor­<lb></lb>tio nota, &amp; ideo prouentus, &amp; cum ſit proportio nota, erit ergo <lb></lb>reſiduum notum, cui addito prouentu fit tota g k nota, quod fuit <lb></lb>demonſtrandum.</s>
        </p>
        <p type="main">
          <s id="id003488">Propoſitio centeſima octuageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id003489">Spatium uitæ naturalis per ſpatium uitæ fortuitum declarare.</s>
        </p>
        <p type="main">
          <s id="id003490">Cum conſtet homines caſu uiuere ægrotantes primum ſæpe: </s>
        </p>
        <p type="main">
          <s id="id003491"><arrow.to.target n="marg650"></arrow.to.target><lb></lb>deinde uiuentes in aëre malo, &amp; ipſum intempeſtiuis horis ſub­<lb></lb>euntes triſtitijs, curis, uigilia, uenere, laboribus perperam ſe excru­<lb></lb>ciantes, <expan abbr="tũ">tum</expan> uerò immodico cibo &amp; potu, &amp; prauo, &amp; ſæpius, quàm <lb></lb>oporteat, &amp; intempeſtiuè, &amp; malè præparato, &amp; uario ſe replentes, <lb></lb>atque ſic alij ad ſexageſimum, alij ad ſeptuageſimum, rari octuage­<lb></lb>ſimo, rariores nonageſimo uel centeſimo anno ita <expan abbr="moriunt̃">moriuntur</expan>, ut non <lb></lb>caſu, neque ui aut morbo, ſed potius quaſi naturali quadam morte <lb></lb>abſumpti intereant: de quibus tantum eſt ſermo. </s>
          <s id="id003492">Atque ut exem­<lb></lb>plo commodiore utamur, capiamus annum octogeſimum, qui eſt <lb></lb>terminus communis uitæ humanæ, non ſolum noſtra ætate, ſed an­<lb></lb>tiquo tempore etiam fuit, ut Dauid teſtatur in Pſalmis, in Cantico <lb></lb>Moyſis: antea autem ſi quis moriatur, non naturali morte, ſed ui <lb></lb>morbi abſumptus exiſtimatur. </s>
          <s id="id003493">Certum eſt, quod ſi homo recta ra­<lb></lb>tione uiueret, quod aliquanto diutius uitam extenderet, neque enim <lb></lb>negare poſſumus, cum in magnis exceſsibus maximè ſectionis ue­<lb></lb>næ &amp; curarum, quin homo euidentur uitam breuiorem efficiat: <lb></lb>quod ergo euidentiſsimum eſt in magnis exceſsibus, in paruis ean­<lb></lb>dem habet uim licet occultiorem. </s>
          <s id="id003494">Errorem autem in uita hunc adeſ­<lb></lb>ſe perpetuum, quiſque intelligit qui noſtras actiones penſitare uelit, <lb></lb>cum ſaltem malam ſequamur conſuetudinem: iam ergo proponan­<lb></lb>tur iuxta dicta duę lineę a b uitę naturalis exquiſitę recte longior &amp; 


<pb pagenum="205" xlink:href="015/01/224.jpg"></pb>c d uitæ quam is uicturus eſt, id eſt, annorum octuaginta, quam <expan abbr="cõ­">con­<lb></lb></expan><arrow.to.target n="marg651"></arrow.to.target><lb></lb>ſtat eſſe breuiorem aliquanto. </s>
          <s id="id003495">Et proponatur error quadrageſimæ <lb></lb>partis in ipſa uita, quamuis ſit longe maior: quotuſquiſque enim eſt <lb></lb>qui non ſaltem edat bibatque quadrageſima parte, pluſquàm opor­<lb></lb>teat in comparatione ad naturam, id eſt, ut natura fatigatur quadra<lb></lb>geſima illa parte amplius quàm debeat: idem dico de laboribus, cu<lb></lb>ris, uigilijs, uenere. </s>
          <s id="id003496">Sed hoc non eſt generale: habetque multas exce­<lb></lb>ptiones inuicem pugnantes, ut tandem concludam non concoqui <lb></lb>plenè poſſe, &amp; ob id impurum manere, unde citò diſſoluitur, &amp; ca­<lb></lb>lorem etiam naturalem extinguit: atque etiam ob id, tum quia debi­<lb></lb>tos labores, &amp; multo minus ad perfectam ætatem perferre <expan abbr="nõ">non</expan> poſ­<lb></lb>ſunt, denſari nequit &amp; pingueſcere, ut duplici cauſa multo celerius <lb></lb>reſoluatur, una etiam calorem extinguat. </s>
          <s id="id003497">Sit ergo a e talis pars a b, <lb></lb>qualis c f, c d. </s>
          <s id="id003498">Cum ergo a b conſumi­<lb></lb><figure id="id.015.01.224.1.jpg" xlink:href="015/01/224/1.jpg"></figure><lb></lb>tur in octuaginta annis, ſemper ſeruat <lb></lb><expan abbr="proportionẽ">proportionem</expan> cum uita contracta, quæ <lb></lb>æqualiter abſumitur: quia portiones <lb></lb>illæ æquales ſunt in minore inuicem ſicut in maiore, &amp; inæquales <lb></lb>ſeruant eandem proportionem, ſumatur ergo a b annorum cclvij. <lb></lb></s>
          <s id="id003499">menſium v. &amp; abſumatur ſemper quantitas æqualis octuageſima <lb></lb>a e, &amp; quadrageſima a b &amp; reſiduorum.<lb></lb><figure id="id.015.01.224.2.jpg" xlink:href="015/01/224/2.jpg"></figure><arrow.to.target n="table27"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003500"><margin.target id="marg650"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003501"><margin.target id="marg651"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 179. <lb></lb>E<emph type="italics"></emph>t in cor.<emph.end type="italics"></emph.end> 1. <lb></lb><emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end> 2.</s>
        </p>
        <table>
          <table.target id="table27"></table.target>
          <row>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
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            <cell>14</cell>
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            <cell>106</cell>
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            <cell>68</cell>
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            <cell>250</cell>
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            <cell>42</cell>
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            <cell>34</cell>
            <cell>10</cell>
            <cell>69</cell>
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            <cell>242</cell>
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            <cell>30</cell>
            <cell>99</cell>
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            <cell>60</cell>
            <cell>19</cell>
            <cell>56</cell>
            <cell>32</cell>
            <cell>16</cell>
            <cell>70</cell>
            <cell>10</cell>
            <cell>38</cell>
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            <cell>3</cell>
            <cell>235</cell>
            <cell>28</cell>
            <cell>17</cell>
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            <cell>23</cell>
            <cell>31</cell>
            <cell>95</cell>
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            <cell>44</cell>
            <cell>58</cell>
            <cell>0</cell>
            <cell>57</cell>
            <cell>30</cell>
            <cell>24</cell>
            <cell>71</cell>
            <cell>9</cell>
            <cell>28</cell>
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          <row>
            <cell>4</cell>
            <cell>228</cell>
            <cell>33</cell>
            <cell>18</cell>
            <cell>148</cell>
            <cell>30</cell>
            <cell>32</cell>
            <cell>92</cell>
            <cell>23</cell>
            <cell>45</cell>
            <cell>55</cell>
            <cell>22</cell>
            <cell>58</cell>
            <cell>28</cell>
            <cell>34</cell>
            <cell>72</cell>
            <cell>8</cell>
            <cell>19</cell>
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            <cell>5</cell>
            <cell>222</cell>
            <cell>5</cell>
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            <cell>144</cell>
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            <cell>33</cell>
            <cell>89</cell>
            <cell>11</cell>
            <cell>46</cell>
            <cell>53</cell>
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            <cell>27</cell>
            <cell>6</cell>
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            <cell>139</cell>
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            <cell>86</cell>
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            <cell>74</cell>
            <cell>6</cell>
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            <cell>7</cell>
            <cell>209</cell>
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            <cell>0</cell>
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            <cell>22</cell>
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            <cell>3</cell>
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            <cell>9</cell>
            <cell>196</cell>
            <cell>37</cell>
            <cell>23</cell>
            <cell>126</cell>
            <cell>15</cell>
            <cell>37</cell>
            <cell>76</cell>
            <cell>35</cell>
            <cell>50</cell>
            <cell>44</cell>
            <cell>10</cell>
            <cell>63</cell>
            <cell>20</cell>
            <cell>29</cell>
            <cell>77</cell>
            <cell>2</cell>
            <cell>31</cell>
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          <row>
            <cell>10</cell>
            <cell>191</cell>
            <cell>1</cell>
            <cell>24</cell>
            <cell>122</cell>
            <cell>9</cell>
            <cell>38</cell>
            <cell>74</cell>
            <cell>0</cell>
            <cell>51</cell>
            <cell>42</cell>
            <cell>6</cell>
            <cell>64</cell>
            <cell>19</cell>
            <cell>9</cell>
            <cell>78</cell>
            <cell>1</cell>
            <cell>29</cell>
          </row>
          <row>
            <cell>11</cell>
            <cell>185</cell>
            <cell>10</cell>
            <cell>25</cell>
            <cell>118</cell>
            <cell>7</cell>
            <cell>39</cell>
            <cell>71</cell>
            <cell>6</cell>
            <cell>52</cell>
            <cell>40</cell>
            <cell>4</cell>
            <cell>65</cell>
            <cell>17</cell>
            <cell>30</cell>
            <cell>79</cell>
            <cell>0</cell>
            <cell>28</cell>
          </row>
          <row>
            <cell>12</cell>
            <cell>179</cell>
            <cell>25</cell>
            <cell>26</cell>
            <cell>114</cell>
            <cell>9</cell>
            <cell>40</cell>
            <cell>68</cell>
            <cell>15</cell>
            <cell>53</cell>
            <cell>38</cell>
            <cell>4</cell>
            <cell>66</cell>
            <cell>16</cell>
            <cell>13</cell>
            <cell>80</cell>
            <cell>0</cell>
            <cell>0</cell>
          </row>
          <row>
            <cell>13</cell>
            <cell>174</cell>
            <cell>6</cell>
            <cell>27</cell>
            <cell>110</cell>
            <cell>15</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>67</cell>
            <cell>14</cell>
            <cell>37</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="id003502">Vt corrigas tabulam, ſcito quod numerus quadrageſimæ cum <lb></lb>ſuperiore annorum numero à leua componit numerum quadrage<lb></lb>ſimæ ſuperioris ſimpliciter, aut abiectis quadragenarijs. </s>
          <s id="id003503">Velut è <lb></lb>regione trigeſimi anni, ſunt anni nonaginta nouem, quad. </s>
          <s id="id003504">17 è <lb></lb>directo anni 29, ſunt anni 103, quad. </s>
          <s id="id003505">0. ad de 17 quad. </s>
          <s id="id003506">ad 103 fit 120, <lb></lb>abijce 40 ter, nil ſupereſt, &amp; ita nulla eſt quadragenaria è regione <lb></lb>29 &amp; 103.</s>
        </p>
        <pb pagenum="106 [=206]" xlink:href="015/01/225.jpg"></pb>
        <p type="main">
          <s id="id003507">Rurſus cum deuenimus ad annos 79, ſuperſunt ſolum 28 qua­<lb></lb>dragenariæ, &amp; eſt minus anno, ſed hoc fieri ob fractiones &amp; nume­<lb></lb>rorum partes, &amp; etiam ſi eſſet aliquis error, eſſet magis ad augen­<lb></lb>dum numerum annorum 257, menſium ſex quàm ad diminutio­<lb></lb>nem, ideo non curaui de exacta ueritate.</s>
        </p>
        <p type="main">
          <s id="id003508">Præterea ex hac tabella dignoſcis, quod in ultimis annis parum <lb></lb>poteſt produci uita in comparatione ad primos, ueluti in 60 anno <lb></lb>ſuperſunt anni 20, ex uita ordinaria, ex exacta paulo plures quàm <lb></lb>25, ſcilicet 25 cum dimidio. </s>
          <s id="id003509">Ergo à 60 anno non poterit per quam­<lb></lb>uis cuſtodiam homo producere uitam plus annis quinque cum di­<lb></lb>midio. </s>
          <s id="id003510">Et ſi dicas tunc cuſtodia maximè opus eſt, &amp; magis quàm <lb></lb>unquam, reſpondeo quod uerum eſt, ſed non ad producendum ui­<lb></lb>tam, ſed ne in morbum incidas: nam ex quocunque morbo homo ab <lb></lb>ea ætate perit, cum habeat adeò imbecilles uires. </s>
          <s id="id003511">Ex hoc patet, <lb></lb>quod Alexius Cornarius, patritius Venetus, cum incœpiſſet cuſto<lb></lb>diam anno 36, cum poſſet uiuere 44 annis, iuxta rationem uitę com<lb></lb>munis, potuit producere eam annis 79, igitur annis 25 pluſquàm ui<lb></lb>xiſſet uita communi etiam quòd fuiſſet ſanus.</s>
        </p>
        <p type="main">
          <s id="id003512">Si ergo aliquis ſit uicturus centum annis uita communi adde­<lb></lb>mus eodem modo trigeſimam nonam partem, id eſt quadrageſi­<lb></lb>mam partem, &amp; quadrageſimam quadrageſimæ huic numero, &amp; <lb></lb>unum amplius, &amp; habebimus numerum ut infrà.<lb></lb><figure id="id.015.01.225.1.jpg" xlink:href="015/01/225/1.jpg"></figure><arrow.to.target n="table28"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table28"></table.target>
          <row>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>A<emph type="italics"></emph>n.<emph.end type="italics"></emph.end></cell>
            <cell>Q<emph type="italics"></emph>uad.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell></cell>
            <cell>257</cell>
            <cell>20</cell>
            <cell>87</cell>
            <cell>314</cell>
            <cell>33</cell>
            <cell>94</cell>
            <cell>383</cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>81</cell>
            <cell>265</cell>
            <cell>3</cell>
            <cell>88</cell>
            <cell>323</cell>
            <cell>34</cell>
            <cell>95</cell>
            <cell>394</cell>
            <cell>3</cell>
          </row>
          <row>
            <cell>82</cell>
            <cell>272</cell>
            <cell>34</cell>
            <cell>89</cell>
            <cell>333</cell>
            <cell>5</cell>
            <cell>96</cell>
            <cell>405</cell>
            <cell>6</cell>
          </row>
          <row>
            <cell>83</cell>
            <cell>280</cell>
            <cell>32</cell>
            <cell>90</cell>
            <cell>342</cell>
            <cell>26</cell>
            <cell>97</cell>
            <cell>416</cell>
            <cell>27</cell>
          </row>
          <row>
            <cell>84</cell>
            <cell>289</cell>
            <cell>0</cell>
            <cell>91</cell>
            <cell>352</cell>
            <cell>16</cell>
            <cell>98</cell>
            <cell>428</cell>
            <cell>13</cell>
          </row>
          <row>
            <cell>85</cell>
            <cell>297</cell>
            <cell>16</cell>
            <cell>92</cell>
            <cell>362</cell>
            <cell>16</cell>
            <cell>99</cell>
            <cell>440</cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>86</cell>
            <cell>306</cell>
            <cell>0</cell>
            <cell>93</cell>
            <cell>372</cell>
            <cell>27</cell>
            <cell>100</cell>
            <cell>452</cell>
            <cell>22</cell>
          </row>
        </table>
        <p type="main">
          <s id="id003513">Et ex hac tabula dignoſcemus quantum quiſque poſsit uiuere, <lb></lb>quouis tempore ætatis ſuæ, illud intelligendo quod non eſt eadem <lb></lb>menſura omnibus, ut neque uitæ ordinariæ, nec magnitudinis cor <lb></lb>porum, nec ingeniorum, nec eiuſmodi in aliquibus uita decreſcit <lb></lb>per uigeſimam partem, hic ſcilicet qui inordinatè uiuunt, alijs uix ſe<lb></lb>xageſima, quan&lt;08&gt; pauciſsimis. </s>
          <s id="id003514">Hic ergo numerus maximè concor­<lb></lb>dat cum experimentis duobus, <expan abbr="q̃">quae</expan> apparuerunt <expan abbr="parũ">parum</expan> ante <expan abbr="tẽpora">tempora</expan> no<lb></lb>ſtra, ſcilicet Ioannis de <expan abbr="tẽporibus">temporibus</expan>, qui uixit annis 361, &amp; Richardus <lb></lb>de temporibus, annis 400. Et ambo fuerunt milites Caroli Ma­<lb></lb>gni, nam non potuerunt omnino proſpicere uitæ rationi exquiſi­<lb></lb>tiſsimæ. </s>
          <s id="id003515"><expan abbr="Referũt">Referunt</expan> etiam in India noſtris <expan abbr="tẽporibus">temporibus</expan> uiuere ad centum 


<pb pagenum="207" xlink:href="015/01/226.jpg"></pb>quinquaginta annos, cuius cauſam transferunt in aërem: ego po­<lb></lb>tius in uitæ genus, abſtinent enim carnibus, ouis, caſeo &amp; uino, u­<lb></lb>tunturque fructibus tantum, &amp; uiuebant ſine ſolicitudine ulla &amp; cu­<lb></lb>ris. </s>
          <s id="id003516">Vnde rectè inſinuatum eſt etiam ultra hiſtoriam, quod Adam <lb></lb>eſſet perpetuò uicturus, ſi non deguſtaſſet fructum arboris boni &amp; <lb></lb>mali, id eſt, quod mors nobis obrepit ob, ſolicitudines &amp; curas. </s>
          <s id="id003517">A­<lb></lb>uenzoar autem cum uixerit multis cum curis, &amp; fuerit in carcere <lb></lb>Hali, &amp; ab eo per iniuriam uexatus, &amp; natus in malo aëre, ſola ratio­<lb></lb>ne uictus produxit uitam ad annos 135, ut teſtatur Auerroes, quid <lb></lb>euenturum erat, ſi in bono aëre educatus nihil graue, &amp; adeò diu­<lb></lb>turnum expertus fuiſſet:</s>
        </p>
        <p type="main">
          <s id="id003518">Pro uſu autem huius &amp; ſuperioris tabulæ, ſi quis proponat iu­<lb></lb>uenem ex ſtirpe eorum, qui uiuunt ſexaginta annis, iam natum de­<lb></lb>cem &amp; ſeptem annos, uelimusque ſcire quantum uiuere poſsit, uide è <lb></lb>regione 20 annorum in primo ordine, &amp; habes annos 139. Quad. <lb></lb>18. &amp; ab hoc numera 17 annos, &amp; habebis annos 37 è regione, <lb></lb>quorum ſunt anni 76. Quad. 35, id eſt, menſes 10, dies 15. uel iunge <lb></lb>17, numerum annorum exactorum, &amp; 20 numerum annorum defi­<lb></lb>cientium ab 80, fiunt anni 33, ut prius, è quorum regione habet an­<lb></lb>nos 76. quad. </s>
          <s id="id003519">35.</s>
        </p>
        <p type="main">
          <s id="id003520">At ſcio multos qui parum conſyderatè hæc legunt, obiecturos, <lb></lb>primum quod neque mihi, neque ulli alij potui, uel ad centum uel ad <lb></lb>nonaginta annos <expan abbr="uitã">uitam</expan> producere. </s>
          <s id="id003521"><expan abbr="Secundũ">Secundum</expan>, q̊d ſi uita humana eſſet <lb></lb>eiuſmodi, naturaliter eſſet ut in pluribus: at uix inuenire licet <expan abbr="aliquẽ">aliquem</expan> <lb></lb>qui exceſſerit centeſimum uigeſimum annum. </s>
          <s id="id003522">Et maximè cum ſcri­<lb></lb>ptum ſit: Non ſpiritum meum in carne ultra centum uiginti annos, <lb></lb>&amp; loquitur Deus. </s>
          <s id="id003523">Videtur etiam neceſſe hoc uolenti, cupere totam <lb></lb>uitam ſub incerto fine, &amp; non uacare, nec negotijs nec uoluptati, <lb></lb>quæ ſunt duo illa præcipua, quibus uita noſtra conſtat, &amp; maximè <lb></lb>amittere bona, adeò ſecura ob tam leuem &amp; inanem ſpem. </s>
          <s id="id003524">Abſur­<lb></lb>dum etiam eſſe hoc quod latuerit tot præclaros medicos atque phi­<lb></lb>loſophos, quorum nullus de hoc ſermonem fecit. </s>
          <s id="id003525">Hæc &amp; huiuſmo<lb></lb>di ſunt quę mihi obij ci poſſe ſentio. </s>
          <s id="id003526">At rogo quid admirabilius eſt, <lb></lb>an ſolem eſſe plus centies et ſexagies terra ac mari, an homines tam­<lb></lb>diu poſſe producere uitam? </s>
          <s id="id003527">Et plures imperito hoc quam illud cre<lb></lb>dituri ſunt: &amp; tamen res illa ita ſe habet, nec apud ſapientes dubia <lb></lb>eſt: nedum incredibilis. </s>
          <s id="id003528">Similiter quòd corpus adeò tenue, debeat <lb></lb>adeò celeriter circumferri, ut in uno ictu pulſus debeat peragere <lb></lb>ſpatium bis mille quingentorum millium paſſuum, &amp; tamen &amp; il­<lb></lb>lud demonſtrari poteſt euidentiſsimè. </s>
          <s id="id003529">Ergo ut ad obiecta reſpon­<lb></lb>deam ſerò mihi hoc inuenire <expan abbr="cõtigit">contigit</expan>, infeliciter natus, peius educa­


<pb pagenum="208" xlink:href="015/01/227.jpg"></pb>tus &amp; imbecilli corpore ac natura, quod aliâs dixi, nec forſan in <lb></lb>quibuſdam ſufficiat educatio ab initio, ſed requiritur ſucceſsio, <lb></lb>qualis fuit olim per multas ætates, ſic progenerantur gigantes &amp; <lb></lb>homines ad miraculum uſque, docui etiam exacta media ætate, hoc <lb></lb>uix fieri poſſe. </s>
          <s id="id003530">Contingunt præterea multa impedimenta. </s>
          <s id="id003531">Sufficit <lb></lb>nobis ſcire quid ſit in natura hominis, non quæro modò quomo­<lb></lb>do faciendum: nec eſt præſentis inſtituti, quin etiam ueriſimile eſt <lb></lb>ad hoc eſſe uiam quandam compendioſiorem, quæ minimè la­<lb></lb>tuerit antiquos, maximè Hebræos. </s>
          <s id="id003532">Et forſan etiam hoc noſtro tem­<lb></lb>pore haberi poſſet quamuis lateat. </s>
          <s id="id003533">Vnum eſt certum, oportere ab <lb></lb>initio uitæ (qui uiam hanc exquiſitam, quam hic trado, ſequi uo­<lb></lb>luerit) conſtituere formam uictus, &amp; tum maximè contractam, <lb></lb>quoniam (ut uiſum eſt in tabula) ex minimo errore, &amp; breui tempo<lb></lb>re plurimum temporis uitæ perit. </s>
          <s id="id003534">Oportet autem multa adeſſe, cor<lb></lb>pus moderatè ſanum, &amp; mediocriter ſaltem conſtitutum, inſtituto­<lb></lb>rem ſapientem, obedientiam pueri, &amp; per omnes ætates cum pati­<lb></lb>entia ſumma commoda diuitiarum, &amp; bonum aërem &amp; fortunam <lb></lb>blandientem noſtro propoſito, ne quis caſus in tanto tempore ad­<lb></lb>uerſus nos impediat, ob tot &amp; tanta quæ neceſſaria ſunt, &amp; aſsiduè, <lb></lb>ideo res hæc fabuloſa uiſa eſt ad hanc uſque diem, tum maximè quod <lb></lb>nemo eam docuerat. </s>
          <s id="id003535">De dicto Moyſis non laboro, cum ſimus me­<lb></lb>dici ac philoſophi non theologi. </s>
          <s id="id003536">Quin etiam poſt hæc uixit Abra­<lb></lb><arrow.to.target n="marg652"></arrow.to.target><lb></lb>hamus annis clxxv, Iſaacus autem clxxx, Iacobus cxlvij, ſed non la­<lb></lb><arrow.to.target n="marg653"></arrow.to.target><lb></lb>boro de his, uerùm relinquo illa ſapientibus: melius eſt ergo ut de­<lb></lb><arrow.to.target n="marg654"></arrow.to.target><lb></lb>monſtrationem adducam huius, cum experimento etiam coniun­<lb></lb>ctam. </s>
          <s id="id003537">Conſtat enim quod humidum pingue euaneſcit per ætates, <lb></lb>ſeu à calore innato, ſeu ab aëre conſumatur, &amp; quod humidum pin­<lb></lb>gue purum, ac denſum tardè abſumitur, ſicut apparet experimen­<lb></lb>to de oleo &amp; ſepo ſalitis, quæ durant longiori tempore, quam ſi nil <lb></lb>tale admiſtum habeant hæc pinguia, ſimiliter aqua quadruplo ce­<lb></lb>lerius, imo longe uelocius abſumitur oleo in uaſe feruente. </s>
          <s id="id003538">Et ita <lb></lb>de pinguedinibus uariorum animalium de ligno iunipero, quod <lb></lb>referunt durare in annum, cur alia non poſsint ad ſex dies. </s>
          <s id="id003539">Cer­<lb></lb>tum etiam eſt, quod coctio condenſet, &amp; eſt Philoſophi in quar­<lb></lb>to Metheororum. </s>
          <s id="id003540">Si ergo coctio perfecta fiat, &amp; puriſsimum hu­<lb></lb>midum reſtauretur, dubium non eſt, quin homo poſsit uiuere ſex­<lb></lb>cuplo plus aut <expan abbr="etiã">etiam</expan> octuplo: quia cùm res peruenit ad <expan abbr="quendã">quendam</expan> ter­<lb></lb>minum, tunc acquiritur perfectio <expan abbr="quędã">quędam</expan> ultra <expan abbr="omnẽ">omnem</expan> fidem, ſicut ui­<lb></lb>demus de auro, q̊d prorſus <expan abbr="etiã">etiam</expan> longo tempore ab ignibus <expan abbr="nõ">non</expan> abſu<lb></lb>mitur: adeò ut liceat dicere, forſan non eſſe contra rationem, quod <lb></lb>detur humidum, quod nunquàm à calore naturali abſumitur, quia 


<pb pagenum="209" xlink:href="015/01/228.jpg"></pb>non eſt par ratio de auro &amp; humido humano, nam in auro <expan abbr="nõ">non</expan> eſt ca<lb></lb>lor niſi ab exteriore igne, ſed in humido noſtro eſt calor intus, &amp; ſe­<lb></lb>cundum ſubſtantiam, ut ſaltem habeamus experimentum longiſ­<lb></lb>ſimæ uitæ &amp; humidi quod uix à calore, &amp; non niſi multis in ſeculis <lb></lb>abſumatur. </s>
          <s id="id003541">Atque hæc (ne incurramus irriſionem Galeni) de Phi­<lb></lb>loſopho qui pollicebatur perpetuitatem uitæ, quanquam non ob <lb></lb>id refugiam hoc, ut negem poſſe hominis uitam eſſe perpetuam, <lb></lb>quod Galenus <expan abbr="Philoſophũ">Philoſophum</expan> hoc dicentem irriſerit, ſed quòd uidea­<lb></lb>mus omnia ſublunaria interire, quòd ſciamus omne compoſitum <lb></lb>debere diſſolui, quoniam compoſitio ſit accidens, &amp; accidens eſt <lb></lb>medium inter ea quæ ſunt &amp; non ſunt: loquor de huiuſmodi acci­<lb></lb>dentibus quæ adueniunt. </s>
          <s id="id003542">Demum, quoniam calor ille ſit in ipſo hu <lb></lb>mido: ideo cum hęc non animaduerterit Galenus, potius fuit uates <lb></lb>in irridendo, quàm ſapiens, ut authoritate eius moueri debeamus. <lb></lb></s>
          <s id="id003543">Hanc coctionem non animaduerterunt medici, ſed ſolam illam bo­<lb></lb>nam quę eſt cauſa ſanitatis, quæ ſtat cum uigilia, labore &amp; ciborum <lb></lb>multitudine, cùm illa exacta non ſtet niſi cum optimis &amp; paucis <lb></lb>ualde cibis, quiete ac ſomno. </s>
          <s id="id003544">Et ideo ſunt ſex genera coctionum, di­<lb></lb>co quod ad perfectionem attinet corrupta, imperfecta, imperfecta <lb></lb>morboſa, imperfecta quæ emendari poteſt, has omnes uitare do­<lb></lb>cent medici: bona quæ eſt cum longa ſanitate, cui medici ſtudent: <lb></lb>ualde bona quam per umbram quaſi <expan abbr="cognouerũt">cognouerunt</expan>, &amp; exacta quam <lb></lb>nec per ſomnium quidem uiderunt, quæ ſola eſt cauſa tantæ lon­<lb></lb>gitudinis uitæ, cum tamen nunquam fuerit uel admodum parum <lb></lb>interrupta. </s>
          <s id="id003545">Hoc autem inter cætera oſtendit experimentum de ele­<lb></lb>phantis, quos Ariſtoteles ducentis annis uiuere conſtanter affir­<lb></lb>mat, alius dixit eſſe trecentis. </s>
          <s id="id003546">Vt conſtet iam in natura animalium <lb></lb>&amp; in genere caloris habentis magnum motum, &amp; ſubſtantiam te­<lb></lb>nuem hoc inueniri poſſe, ut excludamus plantas de <expan abbr="quarũ">quarum</expan> uita lon­<lb></lb>giſsima ſatis conſtat, ſed quia caret motu euidenti calor in illis, &amp; <lb></lb>ſubſtantia eſt craſſa animalium comparatione, non laboro. </s>
          <s id="id003547">At de <lb></lb>elephanto omnes confitentur quòd ſit omnium ingenioſiſsimum, <lb></lb>adeò ut multi homines illo induſtria &amp; cognitione inferiores eſſe <lb></lb>uideantur. </s>
          <s id="id003548">Neque etiam ueriſimile eſt quod natura hominem fecerit <lb></lb>hac in parte illo inferiorem, præſertim cum de nullo alio animali <lb></lb>apud Ariſtotelem dubium ſit, &amp; ubi modo aliquod dubium eſſet <lb></lb>propter querelam Theophraſti, &amp; illud quod ſolet prædicari de <lb></lb>ceruis, tanto magis ueriſimile eſt indignum fuiſſe hominem conce­<lb></lb>dere tot animalibus in diuturnitate uitæ. </s>
          <s id="id003549">Quam ob rem cum hæc <lb></lb>tractatio ad libros de tuenda Sanitate ſpectaret, homines ad eos re­<lb></lb>lego, nam ob id illos conſcripſi quòd uiderem Galenum nec hoc <pb pagenum="210" xlink:href="015/01/229.jpg"></pb>uidiſſe nec multa alia, ſed eorum loco longas &amp; inutiles diſputatio­<lb></lb>nes interſeruiſſe. </s>
          <s id="id003550">Verùm etiam, quoniam eam tractationem diuul­<lb></lb>ſit, ut alia cogamus quærere in libris de Alimentis, alia, de cibis bo­<lb></lb>ni &amp; mali ſucci: tum uerò &amp; tractatio ipſa eduliorum eſt imperfe­<lb></lb>cta, &amp; multa etiam deficiunt circa genera: in quo eſt ex cuſandus ob <lb></lb>uarietatem regionis &amp; ætatis. </s>
          <s id="id003551">Deeſt præterea maxima pars, quæ <lb></lb>nec ibi nec alibi habetur, ſcilicet, de ciborum præparatione. </s>
          <s id="id003552">Quod <lb></lb>etiam hæc latuerint tot præclaros uiros, quid mirum? </s>
          <s id="id003553">cum Hippo­<lb></lb>crates uixerit ſeculo illo agreſti, in quo non eſt mirandum, quod ali <lb></lb>quid, pauca quædam &amp; abſtruſa omiſerit, ſed quod tam multa tam <lb></lb>bene inuenerit, ut fuerit, ſicut de Pindaro dicitur, imò longè uerius <lb></lb>quam de Pindaro inimitabilis. </s>
          <s id="id003554">De Galeno quid mirum, qui non <lb></lb>niſi ueterum ſcripta collegit, atque utinam <expan abbr="ſaltẽ">ſaltem</expan> bene. </s>
          <s id="id003555">De Ariſtotele <lb></lb>is multa inuenit ſuo Marte, &amp; Theophraſtus longè plura. </s>
          <s id="id003556">De alijs, <lb></lb>dico tam medicis quàm philoſophis, hoc eſt, quod queror, quod <lb></lb>in ſpatio pene duorum millium annorum, non hoc quod ualde re­<lb></lb>conditum erat, ſed nec leue ullum experimentum, uel naturæ arca­<lb></lb>num, uel uitæ ſalutare auxilium inuenerit. </s>
          <s id="id003557">Sed litigant de nugis &amp; <lb></lb>rebus inutilibus, &amp; etiam quę ſciri <expan abbr="nõ">non</expan> poſſunt, ac plerunque non ſine <lb></lb>magna impietate. </s>
          <s id="id003558">Quod uerò neceſſe ſit amittere uoluptatem, &amp; <lb></lb>negocia prætermittere uolenti hanc uitam longam adipiſci, quæ <lb></lb>poſtmodum etiam ualde in certa eſt: dico quod quantum ad uolu­<lb></lb>ptates &amp; negocia, non eſſe neceſſe, ſed ſolum ſuperfluas res, &amp; dam<lb></lb>noſas &amp; irritas, quas etiam philoſophi &amp; ciuitatum inſtitutores, &amp; <lb></lb>morum cenſores docent debere uitari, etiam nullo propoſito emo­<lb></lb>lumento, at reliqua <expan abbr="cõſuetudo">conſuetudo</expan> efficit <expan abbr="nõ">non</expan> ſolum grata &amp; tolerabilia, <lb></lb>ſed etiam iucunda. </s>
          <s id="id003559">De incerto fine, quid eſt certum apud homines, <lb></lb>niſi hoc nihil certum eſſe? </s>
          <s id="id003560">Verum tamen ſi quis reſpiciat ad præ­<lb></lb>mium tam ſingulare eſt, &amp; nobile atque utile, ut non luſerit operam <lb></lb>immeritò, quicunque cum ſpe tam illuſtris commodi, &amp; tam exigua <lb></lb>iactura rerum, ac minore periculo ſe huic aleæ experiundæ commi­<lb></lb>ſerit. </s>
          <s id="id003561">Cum, ſi quis hoc ipſum adipiſcatur, uerè dici poſsit ſummum <lb></lb>bonum adeptum eſſe: Non ſolum compos factus diuturnitatis ui­<lb></lb>tæ, ſed cum illa tot uoluptatum, quæ in longo tempore percipiun­<lb></lb>tur ſcientiæ tot rerum, quas non niſi temporis longitudo oſtende­<lb></lb>re poteſt, tot denique caſus uidere tum opum in crementum, quod <lb></lb>quaſi certiſsimum eſt in longa ætate &amp; uſu ſapientia &amp; authoritate <lb></lb>plena, adeò ut fermè neceſſe ſit ad principatus ſpeciem deuenire, <lb></lb>qui tamdiu uixerit, tum gloria ipſa in comparabili. </s>
          <s id="id003562">Hæc autem ma­<lb></lb>xime accidere neceſſe eſt, quod ut uiſum eſt, quanto longior fuerit <lb></lb>ætas eo firmiores <expan abbr="etiã">etiam</expan> ſunt illius partes quæ ad mortis tempus ap­


<pb pagenum="211" xlink:href="015/01/230.jpg"></pb>propinquant pari ratione, ut ex tabella prima deprehendere licet, <lb></lb>quòd ſi cum hoc ſobolis felicitas accedat, non obſcurum eſt huiuſ­<lb></lb>modi poſſe dici ultimam hominis felicitatem apud eos, qui huma­<lb></lb>nas res aliquid eſſe putant. </s>
          <s id="id003563">Accidunt autem hæc ſponte in ſeculo­<lb></lb>rum renouationibus, cum humanum genus conſumitur, ſeu qui ſu<lb></lb>perſunt ob robur, ſeu ex terra geniti, ut dubitat Ariſtoteles. </s>
          <s id="id003564">Haſen <lb></lb>credit, tum ob aëris puritatem, &amp; maximè quòd alterutro modo <lb></lb>ex calidis regionibus &amp; ſublimibus locis homines reparari neceſ­<lb></lb>ſe ſit, tamen etiam ob uictus ſimplicitatem, cum in altera ſuperſint <lb></lb>ſoli piſces, in altera ne hi quidem, ut in Arcanis demonſtratum eſt. <lb></lb></s>
          <s id="id003565">Atque etiam ob curarum abſentiam: ſiquidem homines illi gau­<lb></lb>dent, reges ex agricolis haud dubiè terrarum facti, ac quaſi ſecu­<lb></lb>ri moleſtiarum ad hanc ætatem perueniunt longa ſpatia tempo­<lb></lb>ris, &amp; propagandæ ſobolis habentes, ut feliciſsimè uiuant, reſtituti <lb></lb>ex optimis quibuſcunque aureæ illi ætati, non ſolum ob uitæ ſyn­<lb></lb>ceritatem atque ſplendorem, ſed etiam longitudinem ſic appella­<lb></lb>tæ. </s>
          <s id="id003566">Quæ finem habuit dum ſatis (uti cœperunt) à Saturno in uſum <lb></lb>traductis: unde etiam falcis inſigne accepit. </s>
          <s id="id003567">Eadem tamen ætate <lb></lb>pauciſsimi ex infinitis diutius quam noſtra uiuere cœperunt, cæte­<lb></lb>ri omnes minus quam nunc, quòd neque ueſtitus corporum ab in­<lb></lb>undatione parta, neque aëris puritas à ſqualoribus maneret, &amp; edu<lb></lb>lia multo pauciora eſſent hominibus &amp; incondita.</s>
        </p>
        <p type="margin">
          <s id="id003568"><margin.target id="marg652"></margin.target>G<emph type="italics"></emph>en. </s>
          <s id="id003569">ca.<emph.end type="italics"></emph.end> 25.</s>
        </p>
        <p type="margin">
          <s id="id003570"><margin.target id="marg653"></margin.target>C<emph type="italics"></emph>ap.<emph.end type="italics"></emph.end> 35.</s>
        </p>
        <p type="margin">
          <s id="id003571"><margin.target id="marg654"></margin.target>C<emph type="italics"></emph>ap.<emph.end type="italics"></emph.end> 47.</s>
        </p>
        <p type="main">
          <s id="id003572">Propoſitio centeſima octuageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id003573">Quæcunque grauia in uorticibus aquarum merguntur, in me­<lb></lb>dio uorticis primum uerſa mergantur.</s>
        </p>
        <p type="main">
          <s id="id003574">Hanc proponit Ariſtoteles, ſed non quantum neceſſarium eſt <lb></lb><arrow.to.target n="marg655"></arrow.to.target><lb></lb>explicauit, unius enim quæſiti, id eſt, primi multiplicem rationem <lb></lb>reddit. </s>
          <s id="id003575">Sed neque illam perfectè, quod amborum cauſa una ſit, ac <lb></lb>coniuncta, ſic ergo uortex, cuius extremus <lb></lb>circulus a b centrum in aquæ ſuperficie c <lb></lb><figure id="id.015.01.230.1.jpg" xlink:href="015/01/230/1.jpg"></figure><lb></lb>capacitas uorticis d e, ut aqua feratur per <lb></lb>ſpatium d e f g, h k in maiore circulo na­<lb></lb>uis, aut aliud graue, quod natura ſua non <lb></lb>eſſet deſcenſurum (ut falſò exponitur de <lb></lb>lapide, nam lapis, nec reuoluitur, nec fer­<lb></lb>tur ad d e circulum intimum, ſed præoccu­<lb></lb>pat ex grauitate ſua fertur in imum) dico <lb></lb>q̊d h k prius circumuoluetur, in de trahetur <lb></lb>ad d e, &amp; ubi fuerit ibi <expan abbr="deſcẽdet">deſcendet</expan>, ſed ſi leuius <lb></lb>ſit neceſſariò peruenet ad c antequam deſcendat. </s>
          <s id="id003576">Cum ergo aqua 


<pb pagenum="212" xlink:href="015/01/231.jpg"></pb>grauis ſit tota, fertur ad circulum d e, ut deſcendat. </s>
          <s id="id003577">Sed &amp; quia de­<lb></lb>ſcendit per d e f g, &amp; magis ex centro e, ideo omnes partes circumui<lb></lb>cinæ trahuntur ad d e, &amp; ad e centrum ſuperficiei uorticis, tanquàm <lb></lb>ad centrum, ut deſcendant, atque id primum. </s>
          <s id="id003578">Cunque <expan abbr="lignũ">lignum</expan> deſcendat <lb></lb>partim propria grauitate, partim <expan abbr="attractũ">attractum</expan>, ſi fuerit leue corpus, ut plu­<lb></lb>ma, quod natura ſua <expan abbr="nõ">non</expan> deſcendat, neceſſe eſt ut <expan abbr="deſcẽdat">deſcendat</expan> ſola ui at­<lb></lb>tractionis, quę <expan abbr="nõ">non</expan> eſt tanta in toto d e <expan abbr="quãta">quanta</expan> in e, <expan abbr="igit̃">igitur</expan> oportet ut pri­<lb></lb>us perueniat ad c quàm deſcendat, quia contra <expan abbr="naturã">naturam</expan> <expan abbr="propriã">propriam</expan> de­<lb></lb>ſcendit ui <expan abbr="attractũ">attractum</expan>. </s>
          <s id="id003579">Cum uerò pars quæ in directo c eſt, uelociſsimè <lb></lb>deſcendat, conantur omnes partes aquę, quę circa ſunt deſcendere, <lb></lb>et <expan abbr="cũ">cum</expan> <expan abbr="nõ">non</expan> poſsint ſimul peruenire, mouentur ad illud linea, dico quia <lb></lb>habent initium in e, circulus autem <expan abbr="nullũ">nullum</expan> habet <expan abbr="initiũ">initium</expan>, igitur uiden­<lb></lb>tur moueri circulariter. </s>
          <s id="id003580">Sed cum in circulo partes à <expan abbr="cẽtro">centro</expan> <expan abbr="moueant̃">moueantur</expan>, <lb></lb>uelocius mouebuntur, uelocius in elica a b quàm l m, &amp; l m quàm <lb></lb>n o. </s>
          <s id="id003581">Et ob has duas cauſas mouebuntur uelocius partes quæ ſunt <lb></lb>circa c, quàm diſtantes ab <expan abbr="eodẽ">eodem</expan>, tum quia in medio, <expan abbr="tũ">tum</expan> quia tardius <lb></lb><expan abbr="mouent̃">mouentur</expan> motu elice. </s>
          <s id="id003582"><expan abbr="Declaratũ">Declaratum</expan> eſt. </s>
          <s id="id003583">n. </s>
          <s id="id003584">ſuperius quod unus motus in <lb></lb><expan abbr="eodẽ">eodem</expan> mobili <expan abbr="aliũ">alium</expan> impedit &amp; retardat. </s>
          <s id="id003585">Cum ergo h k ſit in ſpatio a b <lb></lb>l m &amp; aqua <expan abbr="rapiat̃">rapiatur</expan> motu, dico ad d e mouebit ad d e, &amp; motu dico <lb></lb>qui uidetur circularis, nam mouetur motu eius à quo <expan abbr="ſuſtinet̃">ſuſtinetur</expan>. </s>
          <s id="id003586">Mo­<lb></lb>uetur etiam ad d e, quoniam pars illa eſt humilior, nam ſemper de­<lb></lb>ſcendit, omne <expan abbr="aũt">aut</expan> quod mouetur partim eſt in termino, à quo, par­<lb></lb>tim ad quem, ideo partim iam aqua illa cum deſcendat humilior eſt <lb></lb>locus, igitur nauis ad <expan abbr="illũ">illum</expan> locum feretur. </s>
          <s id="id003587">Tertio, quia latus k impelli<lb></lb>tur, in maiore circulo, ideo maiore impetu <lb></lb><figure id="id.015.01.231.1.jpg" xlink:href="015/01/231/1.jpg"></figure><lb></lb>quàm h, quare <expan abbr="deſcẽdet">deſcendet</expan> &amp; circulo mouebi­<lb></lb>tur, <expan abbr="nã">nam</expan> ſi h quieſceret <expan abbr="palã">palam</expan> eſt, q̊d nauis circu<lb></lb>lariter <expan abbr="moueret̃">moueretur</expan>, ſed h fungitur uice <expan abbr="quieſcẽ­tis">quieſcen­<lb></lb>tis</expan>, quia tardius <expan abbr="mouet̃">mouetur</expan> <expan abbr="quã">quam</expan> k, <expan abbr="igit̃">igitur</expan> k moue­<lb></lb>bitur ad d e &amp; motu circulari aut participe <lb></lb>eius. </s>
          <s id="id003588">Quarta cauſa eſt, quoniam h cupit <expan abbr="de­ſcẽdere">de­<lb></lb>ſcendere</expan>, ut graue. </s>
          <s id="id003589">ergo ferri, ubi minus impe<lb></lb>diatur à motu <expan abbr="uiolẽto">uiolento</expan>, at minus <expan abbr="impedit̃">impeditur</expan> in <lb></lb>circulo, de qua a b, qua a b <expan abbr="cũ">cum</expan> maioris ſit ambitus a qua in co ulterius <lb></lb><expan abbr="fert̃">fertur</expan> <expan abbr="quã">quam</expan> in d e, ob hæc <expan abbr="oĩa">oina</expan> &amp; in mari &amp; fluminibus ac lacubus <expan abbr="cũ">cum</expan> na<lb></lb>ues fuerint in ambitu uorticis <expan abbr="iã">iam</expan> <expan abbr="rapiunt̃">rapiuntur</expan> ad <expan abbr="illũ">illum</expan>, &amp; circulari motu: <lb></lb>isque motus eſt <expan abbr="indiciũ">indicium</expan> ſubmerſionis, <expan abbr="quoniã">quoniam</expan> indicat <expan abbr="aquã">aquam</expan>, ibi propè <lb></lb><expan abbr="deſcẽdere">deſcendere</expan> rectà uerſus <expan abbr="cẽtrũ">centrum</expan>, &amp; ob id <expan abbr="prudẽtes">prudentes</expan> nautę magna ui uen<lb></lb>toru &amp; <expan abbr="remorũ">remorum</expan> ſępe <expan abbr="ſeruãt">ſeruant</expan> ſe, pręo<expan abbr="ccupãtes">ccupantes</expan> <expan abbr="motũ">motum</expan> <expan abbr="elicũ">elicum</expan> recto motu. <lb></lb></s>
          <s id="id003590">Cur <expan abbr="aũt">aut</expan> aqua <expan abbr="q̃">quae</expan> eſt in a, non potius <expan abbr="ferat̃">feratur</expan> per obliquam lineam ad d <lb></lb>uel g, &lt;08&gt; ad e uel c inde ex illis ad d uel g, præſertim <expan abbr="cũ">cum</expan> adſit breuior 


<pb pagenum="213" xlink:href="015/01/232.jpg"></pb>a e &amp; e d et a g breuior a e et c (ut docet Euclides) cauſa eſt quia aqua <lb></lb>quæ deſcendit per e d &amp; c g maiore impetu deſcendit quàm per ad <lb></lb>uel a g ut demonſtratum eſt, ergo non poterit quæ eſt in e d uel e g <lb></lb>loco dimoueri, nec cedere aquæ per obliquam lineam deſcendenti.</s>
        </p>
        <p type="margin">
          <s id="id003591"><margin.target id="marg655"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id003592">Propoſitio centeſima octuageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id003593">Cur homo ſedens quanto altius ſedet, &amp; quanto magis crura ad <lb></lb>femora &amp; femora ad pectus reclinata habet, facilius conſurgat, cum <lb></lb>tamen hæc oppoſito modo inuicem ſe habeant, declarare.</s>
        </p>
        <p type="main">
          <s id="id003594">Huius ſecundam partem Ariſtoteles in Mechanicis propoſuit, <lb></lb><arrow.to.target n="marg656"></arrow.to.target><lb></lb>ſed neque ſub adiecta dubitatione, ſedens n <lb></lb><figure id="id.015.01.232.1.jpg" xlink:href="015/01/232/1.jpg"></figure><lb></lb>altius a b pectus, b c femur, c d crus eiuſ­<lb></lb>dem uel æqualis, pectus g h, femur h k, crus <lb></lb>k l longior b f quam h n facit, ut facilius ſur­<lb></lb>gat a b c d quàm g h k l, &amp; tamen anguli <lb></lb>a b c &amp; b c d ſunt maiores g h k &amp; h k l, qui­<lb></lb>nimo cum uolumus ſurgere, contrahimus c d &amp; k l propè &amp; è re­<lb></lb>gione a b, igitur patetratio ſecundi, propior n eſt c d ipſi a b quanto <lb></lb>angulus a b c minor eſt, cui æqualis eſt b c d. </s>
          <s id="id003595">Cum ergo quanto pro <lb></lb>pior eſt c d ipſi a b eo facilius ſurgat, quoniam particeps magis di­<lb></lb>ſpoſitionis per quam ſurgit, propior autem quo anguli ſunt acuti­<lb></lb>ores, ideo facilius exurgit homo, quo contractiora ſunt crura, &amp; an<lb></lb>guli femorum ad crura &amp; pectus minora. </s>
          <s id="id003596">Huc usque Ariſtoteles &amp; <lb></lb>bene.</s>
        </p>
        <p type="margin">
          <s id="id003597"><margin.target id="marg656"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003598">Sed cur rurſus contractiora dum ſunt crura, homo facilius exur­<lb></lb>git? </s>
          <s id="id003599">Proponantur c f contracta ad perpendiculum, &amp; inclinetur b a <lb></lb>in o ut fiant b o &amp; f e aequidiſtantes, ita enim commodius ſurgimus: <lb></lb>nec aliter qui ſunt imbecilliores: quia ergo b eſt in directo f, ideo <lb></lb>muſculi femoris inferiores ob crus, &amp; ſuperiores ob pectus ſunt <lb></lb>magis tenſi &amp; anteriores cruris itidem, ideo maiore ui trahunt par<lb></lb>ticulam. </s>
          <s id="id003600">Vnde manente fixo f &amp; capite etiam &amp; pectore grauitate <lb></lb>ſua adiuuantibus, facilius homo exurgit quam ad latos angulos <lb></lb>cum contractio, ut dixi, muſculorum et inclinatio partium ſuperio­<lb></lb>rum fiat maior.</s>
        </p>
        <p type="main">
          <s id="id003601">Rurſus pro prima parte problematis, dico quòd quanto altior <lb></lb>eſt b f tanto facilius exurgit, nam ſupponatur angu­<lb></lb><figure id="id.015.01.232.2.jpg" xlink:href="015/01/232/2.jpg"></figure><lb></lb>lus reflexionis a h e æqualis a h c, &amp; b c k æqualis h k f, <lb></lb>igitur cum b f ſit breuior b f, erit h k breuior b c &amp; f k, <lb></lb>f c. quare b c femur, &amp; f c crus erunt uiolentius exten­<lb></lb>ſa quàm in ſitu h k, k f ergo, muſculi facilius erigent <lb></lb>ſedentem altiore loco quàm humiliore, quod erat de­<lb></lb>monſtrandum.</s>
        </p>
        <pb pagenum="214" xlink:href="015/01/233.jpg"></pb>
        <p type="main">
          <s id="id003602">Propoſitio centeſima octuageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id003603">Si fuerit proportio primæ &amp; ſecundæ quantitatis ad tertiam, ut <lb></lb>primæ &amp; quartæ ad quintam, fueritqúe quarta ſecunda maior, erit <lb></lb>proportio quartę ad quintam maior quàm ſecundæ ad tertiam. <lb></lb></s>
          <s id="id003604">Quod ſi fuerit maior quartę ad quintam, quàm ſecundę ad tertiam, <lb></lb>neceſſe eſt quartam ſecunda eſſe maiorem.</s>
        </p>
        <p type="main">
          <s id="id003605">Sit proportio a &amp; b ad c, ut a &amp; d ad e, ſitque d maior b, dico maio­<lb></lb><arrow.to.target n="marg657"></arrow.to.target><lb></lb>rem eſſe <expan abbr="proportionẽ">proportionem</expan> d ad e quàm b ad e, quod <lb></lb><figure id="id.015.01.233.1.jpg" xlink:href="015/01/233/1.jpg"></figure><lb></lb>ſi maior ſit proportio d ad c quàm b ad c, dico d <lb></lb>eſſe maiorem b. </s>
          <s id="id003606">Quoniam enim eſt d eſt maior <lb></lb>b ad d eſt maior a b per <expan abbr="communẽ">communem</expan> animi ſenten­<lb></lb>tiam, igitur cum ſit proportio a d ad e ut a b ad c, <lb></lb>erit e maior c, igitur minor proportio a ad e quam a ad c, at propor­<lb></lb><arrow.to.target n="marg658"></arrow.to.target><lb></lb>tio totius a d ad e eſt æqualis proportioni a b ad e, igitur ex com­<lb></lb><arrow.to.target n="marg659"></arrow.to.target><lb></lb>muni animi ſententia maior proportio d ad e, quam b ad c. </s>
          <s id="id003607">Rurſus, <lb></lb>ſi maior eſt proportio d ad e quàm b ad c, igitur per communem <lb></lb>animi ſententiam maior eſt a ad e quàm a ad c, igitur e maior quàm <lb></lb><arrow.to.target n="marg660"></arrow.to.target><lb></lb>c, ſed d maiorem habet proportionem ad e quàm b ad c, igitur d <lb></lb><arrow.to.target n="marg661"></arrow.to.target><lb></lb>maiorem quàm b.</s>
        </p>
        <p type="margin">
          <s id="id003608"><margin.target id="marg657"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003609"><margin.target id="marg658"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 14. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003610"><margin.target id="marg659"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003611"><margin.target id="marg660"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003612"><margin.target id="marg661"></margin.target>P<emph type="italics"></emph>er eadem <lb></lb>ſæpius repe­<lb></lb>titam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003613">Propoſitio centeſima octuageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id003614">Si eiſdem uiribus &amp; eadem proportione cum auxilio ponderis <lb></lb>tertij, quartum pondus moueatur quibus ſecundum auxilio primi, <lb></lb>neceſſe eſt quartum pondus tardiùs &amp; maiore cum difficultate <lb></lb>moueri quàm ſecundum.<lb></lb><arrow.to.target n="marg662"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003615"><margin.target id="marg662"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003616">Maneat prior figura, &amp; ſint uires a quæ cum pondere b moue­<lb></lb>ant c pondus, et cum d pondere eadem uires ſub eadem proportio­<lb></lb>ne moueant e, ſit autem pondus d maius quàm b, dico e tardius &amp; <lb></lb>difficilius moueri quàm c. </s>
          <s id="id003617">Nam ex præcedente e erit maius quàm <lb></lb>c, &amp; proportio d ad e maior quàm b ad c, &amp; proportio a ad e minor <lb></lb>quàm ad c, tum ergo propter uectem magis preſſum, tum quia d <lb></lb>non mouet e, niſi motum ab a, neceſſe eſt ut tardius &amp; maiore cum <lb></lb>difficultate admoueat e quo a b mouet c. </s>
          <s id="id003618">Et ideo eo perueniri po­<lb></lb>terit abſque dubio, ut a b moueat uelociter e &amp; a d, nullo mouente. <lb></lb></s>
          <s id="id003619">Quia hoc accidit cùm d non mouet c niſi quia motum ab a.</s>
        </p>
        <p type="main">
          <s id="id003620">Propoſitio centeſima octuageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id003621">Si uires aliquæ moueant cum ponderibus aliqua pondera, ut <lb></lb>compoſita proportio ſit eadem proportioni uirium &amp; duorum <lb></lb>ponderum mouentium aggregatum æquale duorum ponderum, <lb></lb>ubi maior fuerit partium inæqualitas, ibi erit maior difficultas.</s>
        </p>
        <p type="main">
          <s id="id003622">Sint uires a, &amp; aggregatum ponderum b c &amp; d e æqualia, &amp; a </s>
        </p>
        <p type="main">
          <s id="id003623"><arrow.to.target n="marg663"></arrow.to.target><lb></lb>cum f &amp; g moueat b &amp; c ſub proportionibus componentibus ean­


<pb pagenum="215" xlink:href="015/01/234.jpg"></pb>dem proportionem, quam componunt proportiones a &amp; h mo­<lb></lb>uendo d &amp; a, &amp; k mouendo e, &amp; ſit maior diffe­<lb></lb><figure id="id.015.01.234.1.jpg" xlink:href="015/01/234/1.jpg"></figure><lb></lb>rentia ponderis e ad d quàm c ad b, dico quod <lb></lb>maiore <expan abbr="cũ">cum</expan> difficultate mouebuntur d &amp; e quàm <lb></lb>b &amp; e. </s>
          <s id="id003624">Nam <expan abbr="cũ">cum</expan> differentia e &amp; d ſit maior quàm <lb></lb><arrow.to.target n="marg664"></arrow.to.target><lb></lb>c &amp; b, &amp; d e &amp; b c ſint æqualia, erit e maius c, igi­<lb></lb>tur e difficilius mouebitur ab a &amp; k quàm c ab a <lb></lb>&amp; g. </s>
          <s id="id003625">Itidem quia e tanto maius eſt c, quanto b <lb></lb>maius eſt d, &amp; proportio a k ad e &amp; a h ad d, conficiunt proportio­<lb></lb>nem a g ad c &amp; a f ad b, erit ut motus d e ſint tardiores &amp; difficilio­<lb></lb>res motibus b c, per regulam dialecticam, nam difficultas motus e <lb></lb>ſupra difficultatem motus c, eſt maior quam difficultas motus b <lb></lb>ſupra difficultatem motus d, igitur difficultas motus d &amp; e, maior <lb></lb>eſt difficultate motus b &amp; e, quod erat demonſtrandum.</s>
        </p>
        <p type="margin">
          <s id="id003626"><margin.target id="marg663"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003627"><margin.target id="marg664"></margin.target>P<emph type="italics"></emph>er præce­<lb></lb>dentem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003628">Propoſitio centeſima octuageſima nona.</s>
        </p>
        <p type="main">
          <s id="id003629">Si pondus minus ad longitudinem maiorem ſub æquali pro­<lb></lb>portione coaptetur, facilius deorſum trahetur quàm quod maius <lb></lb>eſt &amp; propius.</s>
        </p>
        <p type="main">
          <s id="id003630">Sit ſitula aquæ f annexa tigno <lb></lb><figure id="id.015.01.234.2.jpg" xlink:href="015/01/234/2.jpg"></figure><lb></lb><arrow.to.target n="marg665"></arrow.to.target><lb></lb>in e &amp; ad minuendum pondus <lb></lb>ad datur ex aduerſo elongius ſeu <lb></lb>uincatur pondus a, dico quod <lb></lb><expan abbr="cõmo">commo</expan> dius erit quàm ſi ęquale ad <lb></lb>grauitatem addatur b proprius <lb></lb>in e, nam quia b ęquiponderat in <lb></lb>d ut a in e, &amp; homo trahens ex e <lb></lb>plus poteſt quàm ex d, igitur fa­<lb></lb>cilius trahet ex e quam d. </s>
          <s id="id003631">Et <expan abbr="quo­niã">quo­<lb></lb>niam</expan> graue minus ponderat quan<lb></lb>to magis diſtat à medio, licet mo­<lb></lb>ueat magis, ergo inclinatum ad <lb></lb><arrow.to.target n="marg666"></arrow.to.target><lb></lb>medium, cum ergo moueatur <lb></lb><arrow.to.target n="marg667"></arrow.to.target><lb></lb>uelocius ex e quam d, &amp; ſemper <lb></lb><arrow.to.target n="marg668"></arrow.to.target><lb></lb>uelocius deſcendendo in com­<lb></lb>paratione a g h, igitur ſemper <lb></lb>magis &amp; magis uelociter ex e <lb></lb>quàm d ut ſit duplex incrementum &amp; comparatione c e ad c d &amp; <lb></lb>deſcenſus ad deſcenſum in utroque &amp; ſimiliter in reditu, quia facilius <lb></lb>impelletur ſurſum e quàm d per primam rationem.</s>
        </p>
        <p type="margin">
          <s id="id003632"><margin.target id="marg665"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003633"><margin.target id="marg666"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 45.</s>
        </p>
        <p type="margin">
          <s id="id003634"><margin.target id="marg667"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003635"><margin.target id="marg668"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 109.</s>
        </p>
        <p type="main">
          <s id="id003636">Propoſitio centeſima nonageſima.</s>
        </p>
        <p type="main">
          <s id="id003637">Si fuerit primum graue minus ſecundo, &amp; ſecundum minus ter­<lb></lb>tio, proportio autem primi ad ſecundum multo maior quàm ſecun 


<pb pagenum="216" xlink:href="015/01/235.jpg"></pb>di ad tertium, poſsibile erit propoſitis uiribus eiſdem addere pon­<lb></lb>dus ſecundo, ut ipſum &amp; <expan abbr="tertiũ">tertium</expan> moueantur facilius ab eiſdem uiri­<lb></lb>bus, &amp; primo uel ſecundo quam antea.</s>
        </p>
        <p type="main">
          <s id="id003638">Sit a <expan abbr="põdus">pondus</expan> minus, c maius, proportio a ad b multo maior quàm <lb></lb>b ad c, uires d, &amp; d cum a moueat b &amp; cum b mo<lb></lb><figure id="id.015.01.235.1.jpg" xlink:href="015/01/235/1.jpg"></figure><lb></lb>ueat c, dico quòd poterit addi pondus ad b ut d <lb></lb>cum a moueat b, &amp; d cum b moueat e maiore fa­<lb></lb>cilitate componendo proportiones quam antea: Cum enim fuerit <lb></lb>proportio d b ad c minima, <expan abbr="quãtumcunque">quantumcunque</expan> moueatur b facilè ab a d <lb></lb><arrow.to.target n="marg669"></arrow.to.target><lb></lb>plus refert difficultas c moti a b d: igitur cum addito pondere di­<lb></lb><arrow.to.target n="marg670"></arrow.to.target><lb></lb>midio quod a ſuperat b omnino uincat a d ipſum b, cum eo quod <lb></lb>additum eſt, &amp; tanto minor ſit difficultas motus c a b d cum ponde<lb></lb>re addito, ſequitur ut minor ſit difficultas motus b cum pondere <lb></lb>addito a b a d, &amp; motus c à b cum pondere addito &amp; d quàm b &amp; e <lb></lb>ab a &amp; b cum uiribus d.<lb></lb><arrow.to.target n="marg671"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003639"><margin.target id="marg669"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 188.</s>
        </p>
        <p type="margin">
          <s id="id003640"><margin.target id="marg670"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 187.</s>
        </p>
        <p type="margin">
          <s id="id003641"><margin.target id="marg671"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 28</s>
        </p>
        <p type="main">
          <s id="id003642">Ex hoc patet quod qui interpretati ſunt Ariſtotelem, cum non <lb></lb>poſsit nec intelligi nec demonſtrari, fucum fecerunt legentibus: ni­<lb></lb>hilominus hoc illis debemus, quod ſi Phrynis non fuiſſet, Timo­<lb></lb>theus non fuiſſet, nam niſi illi quod ſciuerunt protuliſſent in medi­<lb></lb>um, ego forſan aut illa non intellexiſſem aut neglexiſſem. </s>
          <s id="id003643">Itaque &amp; re­<lb></lb>liquas habes à nobis expoſitas licet non adeò diligenter, &amp; mo­<lb></lb>dum huiuſmodi exponendi. </s>
          <s id="id003644">Subij ciemus autem et hanc, ut obiectę <lb></lb>quæſtioni, quantum nerui ſit (ſi pœnitus quis res ſequi uelit, non <lb></lb>addictus nimis authoritati ueterum ut pedem figere uelit, ubi illi <lb></lb>res uix tactas reliquerunt) intelligamus.</s>
        </p>
        <p type="head">
          <s id="id003645">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id003646">Vocatur autem hæc proportio auxiliaris. </s>
          <s id="id003647">Cunque fuerit ęqualis d <lb></lb>&amp; a ad b ut d &amp; b ad e, dicetur auxiliaris æqualis.</s>
        </p>
        <p type="main">
          <s id="id003648">Propoſitio centeſima nonageſima prima.</s>
        </p>
        <p type="main">
          <s id="id003649">Cum fuerint duo pondera &amp; uires duxeriſque aggregatum ex ui­<lb></lb>ribus &amp; minore pondere in maius, addiderisque inſuper <expan abbr="quãtum">quantum</expan> eſt <lb></lb>productum dimidij uirium in ſe latus aggregati detracto dimidio <lb></lb>uirium, dicetur pondus auxiliare æqualis proportionis.</s>
        </p>
        <p type="main">
          <s id="id003650"><arrow.to.target n="marg672"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003651"><margin.target id="marg672"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003652">Sint pondera b minus, c maius, &amp; ducatur aggre­<lb></lb><figure id="id.015.01.235.2.jpg" xlink:href="015/01/235/2.jpg"></figure><lb></lb>gatum ex a uiribus &amp; b minore pondere in e, &amp; ei <lb></lb>addatur quadratum dimidij a, dico quod radix ſeu <lb></lb>latus huius detracto dimidio a eſt pondus auxiliare <lb></lb>æquale, ſit productum a b in e ſuperficies &amp; quadra­<lb></lb>tum dimidij a ſit e, ita quod tota d e ſit ſuperficies <lb></lb>quadrata, cuius latus ſit f g: f h autem dimidium a di­<lb></lb>co h g eſſe pondus auxiliare æquale. </s>
          <s id="id003653">Quia enim f g 


<pb pagenum="217" xlink:href="015/01/236.jpg"></pb>quadratum eſt æquale quadratis g h, h f &amp; duplo g h in h f, &amp; qua­</s>
        </p>
        <p type="main">
          <s id="id003654"><arrow.to.target n="marg673"></arrow.to.target><lb></lb>dratum fh eſt ęquale e ſuperficiei, erit quadratum h g minus ſuper­<lb></lb>ficie d in duplo g h in h f, quare productum a b in c erit ęquale qua­<lb></lb>drato g h in ſe &amp; a, nam duplo g h in h f &amp; iam duplum g h in h f eſt <lb></lb>ęquale producto g h in a, quia a eſt duplum h f, igitur qualis eſt pro <lb></lb><arrow.to.target n="marg674"></arrow.to.target><lb></lb>portio a b ad g h, talis g h &amp; a ad c, igitur per definitionem datam <lb></lb>g h &amp; quantitas grauitatis auxiliaris æquale.</s>
        </p>
        <p type="margin">
          <s id="id003655"><margin.target id="marg673"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003656"><margin.target id="marg674"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003657">Ex hoc manifeſtum eſt, quod ſi fuerit datum pondus tertium au­<lb></lb><arrow.to.target n="marg675"></arrow.to.target><lb></lb>xiliare, quod ſciemus quantum addendum uel detrahendum ut fi­<lb></lb>at pondus auxiliare æquale, nam inuenta g h ſi fuerit k maior adde­<lb></lb>mus quod deficit, &amp; ſi minor quàm k detrahemus ex k quod eſt <lb></lb>ſuperfluum.</s>
        </p>
        <p type="margin">
          <s id="id003658"><margin.target id="marg675"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003659">Et rurſus inuenta g h ut perficiamus pondus ęquale, augebimus <lb></lb><arrow.to.target n="marg676"></arrow.to.target><lb></lb>aliquantiſper, ut fiat æqualis ad unguem difficultas in motu: iuxta <lb></lb><arrow.to.target n="marg677"></arrow.to.target><lb></lb>doctrinam ſuperiùs datam.</s>
        </p>
        <p type="margin">
          <s id="id003660"><margin.target id="marg676"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="margin">
          <s id="id003661"><margin.target id="marg677"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 187.</s>
        </p>
        <p type="main">
          <s id="id003662">Propoſitio centeſima nonageſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id003663">Si ex medio diametri linea ad perpendiculum erigatur ad circu­<lb></lb>li peripheriam: ex eo puncto <expan abbr="autẽ">autem</expan> quotlibet lineæ ducantur ſeu in­<lb></lb>tus ad circumferentiam uſque, ſeu extra ad diametrum, erit proportio <lb></lb>totius lineæ ad totam, uelut mutuò partis ad partem.</s>
        </p>
        <p type="main">
          <s id="id003664">Ex media diametro a c. 1. <expan abbr="cẽtro">centro</expan> b, ducatur ad perpendiculum b d, <lb></lb><arrow.to.target n="marg678"></arrow.to.target><lb></lb>&amp; ex d lineæ d a d e d h, dico d e ad d a, ut d a ad d f, &amp; d h ad d a ut <lb></lb>d a ad d g, &amp; d e ad d h ut d g ad d f. </s>
          <s id="id003665">Quia n quod fit ex d em e f, æ­<lb></lb>quale eſt ei quod ex e c in e a, quod uerò ex e c in e a cum quadrato <lb></lb><arrow.to.target n="marg679"></arrow.to.target><lb></lb>b d ſeu b a ęquale eſt quadrato b e, igitur ex <lb></lb><figure id="id.015.01.236.1.jpg" xlink:href="015/01/236/1.jpg"></figure><lb></lb>e d in e f cum quadrato d b æquale qua­<lb></lb><arrow.to.target n="marg680"></arrow.to.target><lb></lb>drato b e, ex d e igitur in e f cum quadratis <lb></lb><arrow.to.target n="marg681"></arrow.to.target><lb></lb>d b &amp; b a æquale quadrato d e. </s>
          <s id="id003666">Quadratis <lb></lb><arrow.to.target n="marg682"></arrow.to.target><lb></lb>autem a b &amp; b d æquale quadratum d e: <lb></lb><arrow.to.target n="marg683"></arrow.to.target><lb></lb>igitur ex d e in e f cum quadrato d a æqua­<lb></lb><arrow.to.target n="marg684"></arrow.to.target><lb></lb>le quadrato d e. </s>
          <s id="id003667">At quadratum d e æquale <lb></lb>eſt his quæ ex d e in e f, &amp; f d igitur detra­<lb></lb><arrow.to.target n="marg685"></arrow.to.target><lb></lb>cto communi ex d e in e f, erit quadratum d <lb></lb>e æquale ei quod ex d e in d f, igitur d e ad <lb></lb><arrow.to.target n="marg686"></arrow.to.target><lb></lb>d a, ut d a ad d f. </s>
          <s id="id003668">Similiter quod fit ex h d in <lb></lb><arrow.to.target n="marg687"></arrow.to.target><lb></lb>d g, æquale eſt ei quod fit ex h g in g d cum <lb></lb>quadrato d g, at quod fit ex h g in g d eſt æquale ei quod fit ex c g in <lb></lb>g a, erit quod fit ex c g in g a cum quadrato d g ęquale ei quod fit ex <lb></lb>d h in d g. </s>
          <s id="id003669">Quadratum autem d g eſt æquale quadratis d b, b g igi­<lb></lb><arrow.to.target n="marg688"></arrow.to.target><lb></lb>tur d h in d g æquale eſt ei quod fit ex g a in c g cum quadratis b d <lb></lb>b g, at quod fit ex a g in g c cum quadrato b g eſt æquale quadrato 


<pb pagenum="218" xlink:href="015/01/237.jpg"></pb>b a igitur quod fit ex d h in d g eſt ęquale quadratis d b, b a quę ſunt <lb></lb>ęqualia quadrato a d, igitur quadratum a d eſt ęquale ei quod fit ex <lb></lb><arrow.to.target n="marg689"></arrow.to.target><lb></lb>h d in d g, quare proportio h d ad d a ut d a ad a g. </s>
          <s id="id003670">Quia ergo pro­<lb></lb><arrow.to.target n="marg690"></arrow.to.target><lb></lb>portio d e ad d a ut d a ad d f, &amp; d h ad d a ut d a ad d g, erit d e ad d h <lb></lb>ut d g ad d f.</s>
        </p>
        <p type="margin">
          <s id="id003671"><margin.target id="marg678"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003672"><margin.target id="marg679"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 36. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003673"><margin.target id="marg680"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003674"><margin.target id="marg681"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003675"><margin.target id="marg682"></margin.target>P<emph type="italics"></emph>er tandem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003676"><margin.target id="marg683"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003677"><margin.target id="marg684"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſex­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003678"><margin.target id="marg685"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003679"><margin.target id="marg686"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 35. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003680"><margin.target id="marg687"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003681"><margin.target id="marg688"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003682"><margin.target id="marg689"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003683"><margin.target id="marg690"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end><lb></lb>17. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003684">Vnde manifeſtum eſt omnes has lineas in ſuam interiorem par­<lb></lb><arrow.to.target n="marg691"></arrow.to.target><lb></lb>tem ductas rectangulum conſtituere ęquale quadrato quod circu­<lb></lb>lo eidem inſcribitur.</s>
        </p>
        <p type="margin">
          <s id="id003685"><margin.target id="marg691"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003686">Propoſitio centeſima nonageſima tertia.</s>
        </p>
        <p type="main">
          <s id="id003687">Rationem ponderis triplicem explicare.<lb></lb><arrow.to.target n="marg692"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003688"><margin.target id="marg692"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003689">Superius declaratum eſt quòd id quod quieſcit, habet motum </s>
        </p>
        <p type="main">
          <s id="id003690"><arrow.to.target n="marg693"></arrow.to.target><lb></lb>occultum. </s>
          <s id="id003691">Quærit autem Ariſtoteles cur ſecuris pondere preſſa <expan abbr="nõ">non</expan> <lb></lb>diuidit lignum, minore uerò ſed moto ſed modo diuidit? </s>
          <s id="id003692">Diximus <lb></lb><arrow.to.target n="marg694"></arrow.to.target><lb></lb>motum ineſſe qui perpetuo augetur, indicium eſt, quod ſi ex a de­<lb></lb>ſcendat, <expan abbr="maiorẽ">maiorem</expan> facit ictum, quoniam plurimus aër coadiuuat, ex d <lb></lb>autem occultum <expan abbr="ſolũ">ſolum</expan>, et eum qui fit ratione grauitatis, me­<lb></lb><figure id="id.015.01.237.1.jpg" xlink:href="015/01/237/1.jpg"></figure><lb></lb>dium ex medijs locis. </s>
          <s id="id003693">Omitto modo de motu aucto per <lb></lb>uim humanam, de quo uidetur quærere Ariſtoteles, quili­<lb></lb>bet enim aër addit ſuper motum iam acquiſitum &amp; fit hoc <lb></lb>argumentum centies ac millies maius, quoniam m eſt qui <lb></lb>diuidit, pondus autem non ponetrat. </s>
          <s id="id003694">Sicut ergo cuneus <lb></lb>magis diuidit lignum quam claua, ita quod mouetur ſine <lb></lb>proportione (ut ita dicam) non ſolum ob <expan abbr="impetũ">impetum</expan> neceſſe <lb></lb>eſt ut uehementer diuidat lignum aut lapidem ſubiectum, <lb></lb>&amp; non in proportione diſtantię. </s>
          <s id="id003695">Sicut ſi pondus in forma <lb></lb>ſecuris, &amp; ipſa ſecuris diuidit longe magis ligna quam cla­<lb></lb>uis maioris ponderis &amp; maiore ui deſcendens: ita pondus motum <lb></lb>quam immotum. </s>
          <s id="id003696">Hoc adeò perſpicuam habet cauſſam, ut quanto <lb></lb>plura uerba adderentur, eo redderetur res difficilior. </s>
          <s id="id003697">Habet ergo <lb></lb>propriam ſolum grauitatem &amp; motum occultum. </s>
          <s id="id003698">Cęterum eſt ter­<lb></lb>tium, genus <expan abbr="mediũ">medium</expan>, cum idem pondus appenſum eſt, ue­<lb></lb><figure id="id.015.01.237.2.jpg" xlink:href="015/01/237/2.jpg"></figure><lb></lb>lut f quod dico eſſe maius &amp; minus occultum quam ſi ia­<lb></lb>ceret in plano, quoniam ſicut tuber &amp; cauitas in qua iacet <lb></lb>ſimul tempore ſunt, natura tamen tuber eſt prius cauitate, <lb></lb>ita pondus appenſum prius eſt, contrà nixum uinculi na­<lb></lb>tura &amp; quodammodo tempore, ſemper enim grauat, &amp; illud ſem­<lb></lb>per reſiſtit ſupra illius grauitatem: Sed pondus quod eſt in plano <lb></lb>occultam omnino habet actionem bifariamque diſtinguitur a pon­<lb></lb>dere ſuſpenſo: Primum quòd pondus quod quieſcit &amp; contra in­<lb></lb>tendi principium ſimul non ſolum ſunt tempore ſed etiam natu­<lb></lb>ra. </s>
          <s id="id003699">Sed in appenſo, ut dixi, pondus prius grauat quam uincu­


<pb pagenum="219" xlink:href="015/01/238.jpg"></pb>lum contranitatur. </s>
          <s id="id003700">Secundò, quia pondus in plano non inchoat <lb></lb>motum ſed pendens inchoat, ideo quòd eſt in plano habet pror­<lb></lb>ſus occultum, quod pendet non: &amp; ſi ſit lignum eiuſdem molis &amp; <lb></lb>duritiei cui appenſum ſit f &amp; cui inſideat, magis atteretur id cui ap­<lb></lb><figure id="id.015.01.238.1.jpg" xlink:href="015/01/238/1.jpg"></figure><lb></lb>penditur, &amp; prius&lt;08&gt; cui inſidet. </s>
          <s id="id003701">Cæterúm quod <lb></lb>ad grauitatem attinet æqualia ſunt, nam aër in <lb></lb>utroque pellit deorſum, ac magis quod quieſcit <lb></lb>in plano: ſolum enim planum reſiſtit, in pendu­<lb></lb>lo onere etiam aer ſuppoſitus, quo fit ut quod <lb></lb>pendet, minus graue ſit. </s>
          <s id="id003702">Sed æqualia uidentur.</s>
        </p>
        <p type="margin">
          <s id="id003703"><margin.target id="marg693"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 26. <lb></lb><emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end> 38.</s>
        </p>
        <p type="margin">
          <s id="id003704"><margin.target id="marg694"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 19. <lb></lb>M<emph type="italics"></emph>echan.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003705">Propoſitio centeſima nonageſima quarta.</s>
        </p>
        <p type="main">
          <s id="id003706">Proportionem ponderis longioris in medio ſuſpenſi ad breuius. <lb></lb></s>
          <s id="id003707">illi æquale &amp; in medio ſuſpenſum, declarare.<lb></lb><arrow.to.target n="marg695"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003708"><margin.target id="marg695"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 27.</s>
        </p>
        <p type="main">
          <s id="id003709">Hanc generaliter propoſuit Ariſtoteles in Mechanicis, <expan abbr="oſtendit̃">oſtenditur</expan> <lb></lb><expan abbr="em̃">emm</expan> quod ſi a b in e, &amp; d e in f æqualia <lb></lb>pondera in medio <expan abbr="ſuſpendãtur">ſuſpendantur</expan>, quod <lb></lb><figure id="id.015.01.238.2.jpg" xlink:href="015/01/238/2.jpg"></figure><lb></lb>grauius erit a b quam d e. </s>
          <s id="id003710">Et hoc eſt <lb></lb>certum quia a &amp; b extrema plus di­<lb></lb>ſtant ab hypomochlio. </s>
          <s id="id003711">Sit igitur g h reſecta æqualis hic cinde d e, <lb></lb>pondus eſt æquale a b, erit g h minus pondere d e in k, igitur per <lb></lb>communem animi ſententiam k eſt æquale uerò ponderi a g &amp; h b, <lb></lb>igitur cum a g &amp; h b plus ponderent in ſitu ſuo quam in ſitu d e, <lb></lb>patet propoſitum quoad Ariſtotelem attinet, ſcilicet quod a b eſt <lb></lb>grauior d e.</s>
        </p>
        <p type="main">
          <s id="id003712">Vt modò oſtendam proportionem, erit proportio h b ad g h ut <lb></lb>ponderis h b ad totum <expan abbr="põdus">pondus</expan> g b, eadem ratione a g ad g h ut pon­</s>
        </p>
        <p type="main">
          <s id="id003713"><arrow.to.target n="marg696"></arrow.to.target><lb></lb>deris a g ad totum a h, a h autem eſt æqualis g b &amp; a g æqualis h b <lb></lb>ex communi animi <expan abbr="ſentẽtia">ſententia</expan>, &amp; pondus a h ęquale ponderi b g, quia <lb></lb>ſunt æquales &amp; in eodem ſitu: igitur a g, h b ad g h, ut ponderum <lb></lb>a g h b ad pondus g b. </s>
          <s id="id003714">Et ita patet quod quanto longior eſt a b in <lb></lb>comparatione ad d e, tanto a g &amp; h b in comparatione ad g h, igitur <lb></lb>tanto maior proportio ponderum a g h b ad pondus a h. </s>
          <s id="id003715">rurſus eſt <lb></lb>tanto maius quanto a b eſt longior per <expan abbr="demõſtrata">demonſtrata</expan> in prima parte, <lb></lb>igitur multo maius eſt pondus a g h b, quanto longior a b in com­<lb></lb>paratione ad d e.</s>
        </p>
        <p type="margin">
          <s id="id003716"><margin.target id="marg696"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 92. <emph type="italics"></emph>hu­<lb></lb>ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003717"><expan abbr="Exemplũ">Exemplum</expan> ſit ponderis a b 12 ponderis <expan abbr="lõgitudinis">longitudinis</expan> <expan abbr="pedũ">pedum</expan> quatuor, <lb></lb>d e pondus 12 longitudinis <expan abbr="duorũ">duorum</expan> pedum, <expan abbr="eruntigit̃">erunt igitur</expan> a g, g e, c h, h b <lb></lb>unius pedis ſingulę. </s>
          <s id="id003718">Et quia a g &amp; b h ſunt <expan abbr="dimidiũ">dimidium</expan> g h erunt ambæ <lb></lb>pariter æquales g h &amp; ideo pondus a g h b æqualia g b ponderi, <lb></lb>ſed pondus g b eſt librarum nouem, quia g b eſt dodratus a b, igi­<lb></lb>tur tota a b eſt ponderis quindecim, nam g h eſt ponderis ſex, eſt er­<lb></lb>go pondus a b quadrante maius d e.</s>
        </p>
        <pb pagenum="220" xlink:href="015/01/239.jpg"></pb>
        <p type="main">
          <s id="id003719">Propoſitio centeſima nonageſima quinta.</s>
        </p>
        <p type="main">
          <s id="id003720">Si lectus fiat dupla longitudine ad latitudinem melius ſuffulcie­<lb></lb>tur reſtibus ex medio ad angulos, &amp; eis æquidiſtantibus quam ſe­<lb></lb>cundum longitudinem &amp; latitudinem.<lb></lb><arrow.to.target n="marg697"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003721"><margin.target id="marg697"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003722">Hęc proponitur à Philoſopho in mechanicis, &amp; dico quod ſi a b </s>
        </p>
        <p type="main">
          <s id="id003723"><arrow.to.target n="marg698"></arrow.to.target><lb></lb>ſit dupla a c, &amp; <foreign lang="grc">α β α γ</foreign> dupla, &amp; diuidantur a b a c &amp; <foreign lang="grc">α β α γ</foreign> in quotuis <lb></lb>partes ęquales inuicem, nam ſupponitur a b ęqualis <foreign lang="grc">α β</foreign> &amp; a c æqua­<lb></lb>lis <foreign lang="grc">α γ</foreign>, &amp; ducantur rectæ lineæ decuſſatim &amp; ad rectos angulos, &amp; <lb></lb><expan abbr="ſecundũ">ſecundum</expan> id ſtatuantur reſtes, quod decuſſa­<lb></lb><figure id="id.015.01.239.1.jpg" xlink:href="015/01/239/1.jpg"></figure><lb></lb>tim poſitæ utiliores <expan abbr="erũt">erunt</expan>, omitto quod de­<lb></lb>centius ob ſpatiorum minorem differenti­<lb></lb>am. </s>
          <s id="id003724">Adducam ſolùm tres Philoſophi ratio­<lb></lb>nes: prima, quoniam ligna non adeò facilè <lb></lb>finduntur nec incuruantur tranſuerſim tra­<lb></lb>cta, ut recta &amp; ſecundum longitudinem, Et <lb></lb>ideò longè plus durabit <foreign lang="grc">α β γ δ</foreign> <expan abbr="quã">quam</expan> a b c d, <lb></lb>&amp; cum ſpondis rectoribus, &amp; ideò etiam <lb></lb>cum reſtibus magis intentis: &amp; erit firmior <lb></lb>&amp; pulchrior. </s>
          <s id="id003725">Secunda ratio eſt, quod cum <lb></lb>reſtes in ſecunda conſtitutione æquales inuicem ſint, in prima quæ <lb></lb>ſecundum latitudinem duplę, quę longiores erunt magis laxabun­<lb></lb>tur tranſuerſalibus, &amp; ita turpiores &amp; incommodæ breui redden­<lb></lb>tur, &amp; in ſecunda conſtitutione ęqualiter ſuſtinebunt pondus &amp; re­<lb></lb>uolutionem cubantis, tum ob æqualitatem longitudinis inter ſe, <lb></lb>tum ob ſitum ſimilem inter ſe, tum ad humanum decubitum <expan abbr="diſsi­milẽ">diſsi­<lb></lb>milem</expan>, nam (ut oſtenſum eſt) in præcedenti magis grauat pondus in <lb></lb>extremis quam in medio, &amp; magis laxantur ob id quæ ſunt ſecun­<lb></lb>dum eundem situm. </s>
          <s id="id003726">Et hanc cauſſam expoſitores non intellexe­<lb></lb>runt multi, multo minus tertiam, in qua faciunt demonſtrationem <lb></lb>Geometricam &amp; computantem numeris. </s>
          <s id="id003727">Deinde non animaduer<lb></lb>tunt quod in ſecunda figura aſſumunt quinque lineas, cum in prima <lb></lb>tantum aſſumpſiſſent quatuor. </s>
          <s id="id003728">Peius omnibus eſt quod demon­<lb></lb>ſtratio hæc cum de tranſuerſis ad magis tranſuerſas lineas ſit non <lb></lb>eſt ad propoſitum Ariſtotelis, qui in duabus primis rationibus <lb></lb>tranſuerſas comparauit his, quæ à latere ad latus &amp; à capite ad ca­<lb></lb>put deducuntur, ita ubi trifariam decepti ſunt, ibi maximè glori­<lb></lb>antur. </s>
          <s id="id003729">Miſerum nunc philoſophandi genus: uoluntque ſupercilium <lb></lb>eſſe loco doctrinæ. </s>
          <s id="id003730">Sint igitur lineæ ductæ ut uides, dico omnes <lb></lb>pariter acceptas in prima figura, eſſe longiores omnibus pariter ac­<lb></lb><arrow.to.target n="marg699"></arrow.to.target><lb></lb>ceptis in ſecunda figura, quod intendit <expan abbr="demõ">demon</expan>ſtrare Ariſtoteles. </s>
          <s id="id003731">O­<lb></lb>ſtenſo ergo de duabus, idem ſuppoſito numero equali de omnibus 


<pb pagenum="221" xlink:href="015/01/240.jpg"></pb>conſtat. </s>
          <s id="id003732">Demonſtrandum eſt ergo a b &amp; g q maiores eſſe <foreign lang="grc">αζ</foreign> &amp; <foreign lang="grc">ζβ</foreign>, <lb></lb>nam <foreign lang="grc">αγ</foreign> &amp; <foreign lang="grc">γζ</foreign> ſunt æquales &amp; <foreign lang="grc">ζδ</foreign> &amp; <foreign lang="grc">δβ</foreign> ex ſuppoſito, quare <foreign lang="grc">αζ</foreign> &amp; <foreign lang="grc">ζβ</foreign><lb></lb>æquales ſunt poteſtate quadrato, <foreign lang="grc">αβ</foreign> igitur ambæ iunctæ lineæ me­<lb></lb><arrow.to.target n="marg700"></arrow.to.target><lb></lb>diæ inter duplum <foreign lang="grc">αβ</foreign> &amp; ipſam <foreign lang="grc">αβ</foreign>, quadratum enim <foreign lang="grc">αζ</foreign> &amp; <foreign lang="grc">ζβ</foreign> coniun­<lb></lb>ctarum eſt duplum quadratis uniuſcuiusque earum pariter acceptis, <lb></lb><arrow.to.target n="marg701"></arrow.to.target><lb></lb>uelut &amp; quadratum mediæ inter duplum <foreign lang="grc">αβ</foreign> &amp; ipſam <foreign lang="grc">αβ</foreign>, at quadra­<lb></lb>tum coniunctæ ex a b &amp; a c eſt æquale duplo quadrati a b cum qua<lb></lb><arrow.to.target n="marg702"></arrow.to.target><lb></lb>drato a c, igitur ſuperat duplum quadrati <foreign lang="grc">α β</foreign> in quadrato a c, ſed <lb></lb><arrow.to.target n="marg703"></arrow.to.target><lb></lb>quod poteſt in duplum quadrati <foreign lang="grc">αβ</foreign> eſt aggregatum <foreign lang="grc">αζ</foreign> &amp; <foreign lang="grc">ζβ</foreign>, igitur <lb></lb>a b &amp; a d ſunt longiores iunctæ <foreign lang="grc">αζ</foreign> &amp; <foreign lang="grc">ζβ</foreign> quia poſſunt eo plus quan­<lb></lb><arrow.to.target n="marg704"></arrow.to.target><lb></lb>tum eſt quadratum a c.</s>
        </p>
        <p type="margin">
          <s id="id003733"><margin.target id="marg698"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 25.</s>
        </p>
        <p type="margin">
          <s id="id003734"><margin.target id="marg699"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 34. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003735"><margin.target id="marg700"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri­<lb></lb>mi &amp;<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſe­<lb></lb>cundi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003736"><margin.target id="marg701"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003737"><margin.target id="marg702"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſecun <lb></lb>di<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003738"><margin.target id="marg703"></margin.target>P<emph type="italics"></emph>er eandem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003739"><margin.target id="marg704"></margin.target>P<emph type="italics"></emph>er eandem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003740">Propoſitio centeſima nonageſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id003741">Si duo circuli ſuper eodem centro eodem motu transferuntur, <lb></lb>æquale ſpatium ſuperant.</s>
        </p>
        <p type="main">
          <s id="id003742">Sint duo circuli a b, c d ſuper eodem centro e qui transferantur <lb></lb><figure id="id.015.01.240.1.jpg" xlink:href="015/01/240/1.jpg"></figure><lb></lb><arrow.to.target n="marg705"></arrow.to.target><lb></lb>ſuper axe per <expan abbr="ſpatiũ">ſpatium</expan> c g dum reſoluitur c d, <lb></lb>tum ergo a erit in f, quia c d contingit pla­<lb></lb>num c g, igitur e c eſt ad <expan abbr="perpẽdiculum">perpendiculum</expan> c g, <lb></lb><arrow.to.target n="marg706"></arrow.to.target><lb></lb>ergo punctum a eſt in f &amp; a f æqualis c g, <lb></lb><arrow.to.target n="marg707"></arrow.to.target><lb></lb>igitur a b circulus ſolum reuolutus eſt ſe­<lb></lb>mel, &amp; tantum perambulauit ſpatij quan­<lb></lb>tum e d &amp; æquali uelo citate, cùm tamen ſeorſum ſit proportio ſpa­<lb></lb>tij ad <expan abbr="ſpatiũ">ſpatium</expan> ut circuli ad circulum. </s>
          <s id="id003743">Hæc eſt ſubtiliſsima <expan abbr="quæſtionũ">quæſtionum</expan> <lb></lb><arrow.to.target n="marg708"></arrow.to.target><lb></lb><expan abbr="propoſitarũ">propoſitarum</expan> ab Ariſtotele in mechanicis, quam ſic quidam ſoluunt. <lb></lb></s>
          <s id="id003744">Supponunt duo: <expan abbr="primũ">primum</expan> ſi quid ab aliquo mouetur nihil conferens <lb></lb><figure id="id.015.01.240.2.jpg" xlink:href="015/01/240/2.jpg"></figure><lb></lb>ad illum motum, <lb></lb>ex ſe ipſo per tan<lb></lb>tum mouebitur <lb></lb><expan abbr="ſpatiũ">ſpatium</expan>, per quan­<lb></lb>tum ab illo mo­<lb></lb>tore mouebitur: <lb></lb>Secundum, <expan abbr="eadẽ">eadem</expan> <lb></lb>potentia in <expan abbr="eodẽ">eodem</expan> <lb></lb>tempore diuerſo <lb></lb>modo duo mobi <lb></lb>lia mouebit ęqua <lb></lb>lia, cum <expan abbr="unũ">unum</expan> mo­<lb></lb>tui aſſentietur aliud <expan abbr="nõ">non</expan>. </s>
          <s id="id003745">quod ſi hæc mobilia ſeiuncta fuiſſent, quod <lb></lb>aptitudinem haberet <expan abbr="ſeiunctũ">ſeiunctum</expan> uelocius moueretur, quàm dum con<lb></lb>iunctum eſt. </s>
          <s id="id003746">Cum ergo inquiunt circulus c d moueatur ab a b cir­<lb></lb>culo, nec conferat quic&lt;08&gt; ad motum, ideo tantum tranſibit ſpatium <pb pagenum="222" xlink:href="015/01/241.jpg"></pb>c d quantum a b per primum ſuppoſitum. </s>
          <s id="id003747">Sed quoniam proposi­<lb></lb>to circulo alio non circa idem centrum, utpote k l reuoluetur &amp; <lb></lb>perueniet ad h ex demonſtratis. </s>
          <s id="id003748"><expan abbr="Reſpondet̃">Reſpondetur</expan> ad hoc, quod idem eſt, <lb></lb>quia unus circulus tantum per ſe mouetur circa centrum, reliqui <lb></lb>omnes non perſe circa centrum, ſed ab alio circulo primo mouen­<lb></lb>tur, ideò nihil refert ſeu ſint circa idem centrum ſeu circa aliud, hoc <lb></lb>enim fortuitum eſt. </s>
          <s id="id003749">Ideo ad argumentum reſpondent cauilloſam <lb></lb>eſſe <expan abbr="hãc">hanc</expan> diſputationem, cum ſupponat idem ambobus circulis per <lb></lb>ſe centrum eſſe. </s>
          <s id="id003750">Sed non eſt perſe, uerùm per <expan abbr="accidẽs">accidens</expan>. </s>
          <s id="id003751">At tamen de­<lb></lb>miror de huiuſmodi ſolutione. </s>
          <s id="id003752">Primum quod ipſemet. </s>
          <s id="id003753">Ariſtoteles <lb></lb>de hoc nos docuit in primo Poſteriorum dicens. </s>
          <s id="id003754">Non eſt igitur ex <lb></lb>uno in aliud genus <expan abbr="tranſcẽdentem">tranſcendentem</expan> demonſtrare, ut Geometricum <lb></lb>Arithmetica. </s>
          <s id="id003755">Et <expan abbr="Auerroẽs">Auerroens</expan> in Commento magno inquit, ea uerba <lb></lb>exponens. </s>
          <s id="id003756">Fieri non poteſt, ut demonſtratio transferatur de <lb></lb>arte in artem. </s>
          <s id="id003757">Et ibidem docet, quod neque ut ambæ præmiſ­<lb></lb>ſæ ſint communes, neque etiam maior tantum, ſicut exponebat Al­<lb></lb>pharabices. </s>
          <s id="id003758">Verùm dicit, ſolum licet in artibus, quæ ſunt in com­<lb></lb>paratione generis ad ſpeciem, ut ſit concluſio ueluti phyſica ma­<lb></lb>ior propoſitio, in ſubiecta ſcientia ueluti medicina. </s>
          <s id="id003759">Vnde <expan abbr="cõcludit">concludit</expan> <lb></lb>Philoſophus. </s>
          <s id="id003760">Propter hoc Geometrię non licet demonſtrare quod <lb></lb>contrariorum una eſt ſcientia: ſed neque quod duo cubi cubus, neque<lb></lb> alij ſcientiæ quod alterius: niſi in his quæ ita inter ſe habent ut alte­<lb></lb>ra ſub altera ſit, ueluti perſpectiua ad Geometricam, &amp; harmonica <lb></lb>ad <expan abbr="Arithmeticã">Arithmeticam</expan>. </s>
          <s id="id003761">Et poſt docet quod etiam non licet demonſtrare ex <lb></lb>communibus: hæc igitur ratio eſt ex alienis genere atque communi­<lb></lb>bus. </s>
          <s id="id003762">Quid, quòd non ſoluit difficultatem quę mathematica tota eſt <lb></lb>&amp; innititur manifeſtis principijs. </s>
          <s id="id003763">Debuit enim oſten dere quomo­<lb></lb>do tardius moueatur circulus maior ipſo minore: hoc enim eſt ne­<lb></lb>ceſſe ſi eodem tempore debent æqualia ſpatia pertranſire. </s>
          <s id="id003764">Accipia­<lb></lb>mus ergo quod manifeſtum eſt, ſcilicet uectionem eſſe hanc in qua <lb></lb>e centrum perpetuò per æquidiſtantem lineam fertur in m, nullum <lb></lb>autem circulum progreſſus centri eſſe cauſam niſi ut rota mouet <lb></lb>currum &amp; currus axem, reuolutio ergo notæ efficit ut ſpatium c g <lb></lb>pertranſeat nota, &amp; ideo motus ille circularis non eſt, quia circula­<lb></lb>ris motus fit manente centro, ſed eſt circulus progrediens uelut &amp; <lb></lb>punctum e: at in circulo, hoc eſt diſcrimen quòd puncta, uariantur <lb></lb>centrum autem non. </s>
          <s id="id003765">Dico ergo ut melius intelligas quòd talis mo­<lb></lb>tus eſt uelut famulorum fabrorum qui rotam circunducant <expan abbr="domũ">domum</expan> <lb></lb>impellentes, talis enim motus, eſt rectus, &amp; eſt impulſionis non au­<lb></lb>tem circularis. </s>
          <s id="id003766">Et ideò omnia puncta æqualiter mouentur, &amp; per <lb></lb>æquale ſpatium, accidit autem ut hic motus fiat circunuertendo, 


<pb pagenum="223" xlink:href="015/01/242.jpg"></pb>ſicut etiam ſi traheretur fune. </s>
          <s id="id003767">Et ſi quis obijciat quod hæc reſpon­<lb></lb>ſio eſt eadem cum illa quę tribuitur Ariſtoteli, dico quod non, quia <lb></lb>in illa ſupponuntur duo falſa, unum quod principium motus ali­<lb></lb>quando ſit in c d, aliquando in a b, quod pro ſecunda parte falſum <lb></lb>eſt: nam nunquàm principium poteſt eſſe in a b, nam ſi intelliga­<lb></lb>mus de modo motus, non mouetur nec a b nec c d motu circulari, <lb></lb>quoniam (ut dixi) motus eſt uectio, ſeu tractio, non circularis. </s>
          <s id="id003768">Sin <lb></lb>autem de cauſa motus rotæ illa eſt in circulo ſemper maximo, ſcili­<lb></lb>cet c d &amp; non a b. </s>
          <s id="id003769">Et cauſa erroris horum fuit duplex: cum enim ſci­<lb></lb>rent hanc rationem, dubitarunt an circulus c d motus eſſet potius <lb></lb>cauſa motus circuli a b, an contrà, ideò protulerunt ambos, ſicut illi <lb></lb>quibus ſublata eſt res aliqua, ut non errent, dicunt hic, uel hic ſubri­<lb></lb>puit rem meam. </s>
          <s id="id003770">Secunda fuit, quia neſciuerunt diſtinguere inter <lb></lb>motum per circulum &amp; motum circularem, cum ſit magnum diſcri<lb></lb>men: motus enim rotæ eſt per circulum, quia per circumferentiam <lb></lb>eius, quæ eſt circulus, non autem circularis. </s>
          <s id="id003771">Etſi ſuperius appella­<lb></lb>uerim circularem, cum diſtinxi in triplicem motum ſphęrę circum­<lb></lb>uolutionem, tunc non curaui de uerbis, quia uerba tum non erant <lb></lb>cauſa erroris.</s>
        </p>
        <p type="margin">
          <s id="id003772"><margin.target id="marg705"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id003773"><margin.target id="marg706"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003774"><margin.target id="marg707"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 34. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003775"><margin.target id="marg708"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 25.</s>
        </p>
        <p type="main">
          <s id="id003776">Ex hoc patet unum, quod eſt difficilius, ſcilicet quia certum eſt, <lb></lb><arrow.to.target n="marg709"></arrow.to.target><lb></lb>quòd tam c d quàm a b mouentur ſuper rectas, &amp; ita ut ſingula <lb></lb>puncta c d tangant ſingula puncta c g, &amp; a b ſingula puncta a f, &amp; <lb></lb>tamen c d circumferentia, aut non eſt æqualis rectæ c g, aut circum­<lb></lb>ferentia a b non eſt æqualis rectæ a f, aliter ſi ambæ circumferentiæ <lb></lb>ambabus rectis eſſent æquales, cum rectæ ſint æquales, ut demon­<lb></lb>ſtratum eſt, eſſent circumferentiæ etiam a b &amp; c d, æquales maior <lb></lb>minori, quod eſt impoſsibile. </s>
          <s id="id003777">Non ergo ualet argumentum, iſte cir<lb></lb>culus circumfertur ſuper rectam aliquam, ita ut cum redit ad idem <lb></lb>punctum rectam perambulauit ad unguem, ergo illius peripheria <lb></lb>eſt æqualis illi rectæ.</s>
        </p>
        <p type="margin">
          <s id="id003778"><margin.target id="marg709"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003779">Melius ergo fuiſſet huius reddere rationem, in quo eſt tota dif­<lb></lb><arrow.to.target n="marg710"></arrow.to.target><lb></lb>ficultas, nam illa (ut dixi) de motu circulari nulla eſt, ſi quis tam pe­<lb></lb>nitus introſpiciat. </s>
          <s id="id003780">Sit igitur ut rotæ axis c, tranſeat in f, &amp; quia e a &amp; <lb></lb>f g æquales ſunt a centro ad circumferentiam, &amp; a g æquidiſtans <lb></lb>b c, erit per demonſtrata punctum g in linea fh, &amp; ponamus quod <lb></lb>punctum fuerit m, quod translatum, &amp; retro reuolutum peruene­<lb></lb>rit ad h, &amp; ſecet e m a b circulum in n, dico quod n eſt punctum g, in <lb></lb>quo etiam eſt animaduertendum de ſtupore horum ſcribentium, <lb></lb>nec aduertentium quod puncta circulorum a b &amp; c d retro cedunt, <lb></lb>uerſus a &amp; c, &amp; non uerſus o &amp; p, &amp; hoc eſt quod decipit illos. 


<pb pagenum="224" xlink:href="015/01/243.jpg"></pb>Quia ergo m eſt h <lb></lb>&amp; e f, <expan abbr="igit̃">igitur</expan> cum n ſit <lb></lb>in linea e m, erit in <lb></lb>linea f h, ſed n eſt <lb></lb><expan abbr="etiã">etiam</expan> in circulo a b, <lb></lb>igitur <expan abbr="cũ">cum</expan> <expan abbr="nullũ">nullum</expan> ſit <lb></lb><expan abbr="punctũ">punctum</expan> aliud in li­<lb></lb>nea fh, et circulo g <lb></lb>q, &lt;08&gt; g eſt n <expan abbr="cõmu­nis">commu­<lb></lb>nis</expan> ſectio, igitur n <lb></lb>peruenit in g. </s>
          <s id="id003781">Vi­<lb></lb>des ergo quod m <lb></lb><figure id="id.015.01.243.1.jpg" xlink:href="015/01/243/1.jpg"></figure><lb></lb>retroceſsit per angulum m g h, n autem anteceſsit per angulum n <lb></lb>g f, qui eſt æqualis angulo m g h. </s>
          <s id="id003782">Ex quo liquet cauſa dictorum, &amp; <lb></lb>quod non intellexerunt quæſtionis fundamentum cum ferantur <lb></lb>ſingula puncta in una reuolutione æqualiter cum centro motu re­<lb></lb>cto: &amp; motu circumuolutionis ſunt immobilia, quia tantum retro­<lb></lb>cedunt in una medietate, quantum procedunt in alia.</s>
        </p>
        <p type="margin">
          <s id="id003783"><margin.target id="marg710"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003784">Propoſitio centeſima nonageſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id003785">Cur lances ad <expan abbr="locũ">locum</expan> <expan abbr="ſuũ">ſuum</expan> <expan abbr="ſuſpẽſi">ſuſpenſi</expan> <expan abbr="redeãt">redeant</expan> <expan abbr="impendẽtes">impendentes</expan> <expan abbr="nõ">non</expan>, <expan abbr="demõſtrare">demonſtrare</expan>.<lb></lb><arrow.to.target n="marg711"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003786"><margin.target id="marg711"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id003787">Aliâs cum uiderem apud Ariſtotelem &amp; eius expoſitores hoc </s>
        </p>
        <p type="main">
          <s id="id003788"><arrow.to.target n="marg712"></arrow.to.target><lb></lb>problema non ſum auſus, quia ex proprijs non mihi occurrebat <lb></lb>demonſtratio, rationem reddere, at confecta dialectica ſtatim appa <lb></lb>ruit modus. </s>
          <s id="id003789">Sit ergo libra a b appenſa ex trutina c d, &amp; ſit per pon­<lb></lb><figure id="id.015.01.243.2.jpg" xlink:href="015/01/243/2.jpg"></figure><lb></lb>dus educta loco e f, &amp; ſublato reuertitur <lb></lb>ad locum priorem: Et rurſus eadem ſi <lb></lb>immineat g d ſuſtentaculo <expan abbr="nõ">non</expan> mouetur: <lb></lb>igitur palam eſt quod in trutina d e gra­<lb></lb>uior eſt <expan abbr="quã">quam</expan> d f inſiſtens g d, <expan abbr="nõ">non</expan> eſt adeo <lb></lb>grauis, aut omnino non grauior. </s>
          <s id="id003790">Neque <lb></lb>poteſt id accidere quod in primo caſu <lb></lb>angulus e d c acutus, ſit in ſecundo obtu<lb></lb>ſus, nam ſi ob angulum e d c acutum <expan abbr="deſcẽdit">deſcendit</expan> in primo caſu e, in ſe­<lb></lb>cundo caſu deſcendet f, quia pariter f d g acutus eſt, &amp; æqualis e d c, <lb></lb>hoc autem non contingit. </s>
          <s id="id003791">Mira ne dicam ſtultitia an audacia <expan abbr="eorũ">eorum</expan>, <lb></lb>qui nihil intelligentes auſi ſunt, hæc pertractare, ſperantes in tot ſe­<lb></lb>culis nullum futurum, qui ignorantiam ſuam &amp; impoſtura depre­<lb></lb>hendat, dicunt enim quod in primo caſu producta quadam recta <lb></lb>ad perpendiculum, &amp; quæ ſit h k maiorem reddi d e quàm d f, neque <lb></lb>quomodo id fiat oſtendunt, &amp; ſi (ut dixi) maior ſit <expan abbr="quã">quam</expan> d fin primo <lb></lb>caſu maior d f quam d e in <expan abbr="ſecũdo">ſecundo</expan> caſu: ergo ſi in primo caſu d e de­<lb></lb>ſcendit, in ſecundo deſcendet magis d f, at hoc non accidit ſed ſtat. 


<pb pagenum="225" xlink:href="015/01/244.jpg"></pb>Oportet igitur hoc eſſe principium ex Dialectica, quod oſtendat e <lb></lb>grauiorem eſſe f in primo caſu, in ſecundo non eſſe grauiorem, aut <lb></lb>leuiorem, ut neque ad angulum refugere poſsimus. </s>
          <s id="id003792">Ergo ſupponere <lb></lb>oportet quæ manifeſta ſunt, e eſſe grauiorem f, aliter enim non de­<lb></lb>ſcenderet: non prohiberi autem in primo caſu motum prohiberi in <lb></lb>ſecundo, aliter uel grauior fieret f, uel maneret eadem grauitas: ſi­<lb></lb>quidem maneret grauitas, nec impediretur deſcendere e in ſe­<lb></lb>cundo caſu, ut in primo, at non deſcendit. </s>
          <s id="id003793">Si grauitas mutaretur, igi<lb></lb>tur f deſcenderet ſecundo caſu magis quam in primo. </s>
          <s id="id003794">Quod ſi di­<lb></lb>cas non tanto fieri grauiorem, igitur f magis depreſſa deſcendet <lb></lb>ſaltem, at nunquam deſcendit, igitur grauior eſt ſemper e quàm f, <lb></lb>ſed in ſecundo caſu impeditur motus non in primo. </s>
          <s id="id003795">Cauſa grauita­<lb></lb>tis eſt, quoniam d eſt centrum grauitatis, quia medium. </s>
          <s id="id003796">igitur cum <lb></lb><arrow.to.target n="marg713"></arrow.to.target><lb></lb>c &amp; d conſpirent contra f, neceſſe eſt e deſcendere per ſuperius de­<lb></lb>monſtrata, igitur e deſcendet in primo caſu, quia grauius eſt ut do­<lb></lb>cui nec impeditum. </s>
          <s id="id003797">At in ſecundo caſu e &amp; d ſunt grauiora, ſed d <lb></lb>eſt impeditum, quia non habet motum, niſi occultum inſidet enim <lb></lb><arrow.to.target n="marg714"></arrow.to.target><lb></lb>g d, igitur tantum ponderat e quam f, ergo prorſus non mouebun­<lb></lb>tur, facit &amp; ad hoc quòd quæuis latitudo d, ſuſtentaculi prohibet <lb></lb>motum, at deeſſe uix poteſt. </s>
          <s id="id003798">Vides ergo illos nugas palam agere. <lb></lb></s>
          <s id="id003799">Primum deeſt illis dialectica, deinde ingenium acre, deinde quod <lb></lb>maius eſt, uolunt confeſtim tranſire ex principijs ad remota theore­<lb></lb>mata, quod fieri non poteſt.</s>
        </p>
        <p type="margin">
          <s id="id003800"><margin.target id="marg712"></margin.target>Q<emph type="italics"></emph>ueſt.<emph.end type="italics"></emph.end> 7. <lb></lb>M<emph type="italics"></emph>echan.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003801"><margin.target id="marg713"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 45.</s>
        </p>
        <p type="margin">
          <s id="id003802"><margin.target id="marg714"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 193.</s>
        </p>
        <p type="main">
          <s id="id003803">Propoſitio centeſima nonageſima octaua.</s>
        </p>
        <p type="main">
          <s id="id003804">Cur ſolidum quod cubus <expan abbr="uocat̃">uocatur</expan>, pyramide ſtabilius ſit, oſtendere.</s>
        </p>
        <p type="head">
          <s id="id003805">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id003806">Si intra circulum triangulus æquilaterus deſcribatur, &amp; ab uno <lb></lb>angulorum per centrum rectà ducatur, angulum per æqualia diui­<lb></lb>det, &amp; trianguli latus, &amp; ad angulos rectos ei inſiſtet, ipſa uerò quæ <lb></lb>ex centro per æqualia uiciſsim à trianguli latere diuidetur.<lb></lb><figure id="id.015.01.244.1.jpg" xlink:href="015/01/244/1.jpg"></figure><lb></lb><arrow.to.target n="marg715"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003807"><margin.target id="marg715"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id003808">Sit a b c æquilaterus circulo inſcriptus, </s>
        </p>
        <p type="main">
          <s id="id003809"><arrow.to.target n="marg716"></arrow.to.target><lb></lb>cuius centrum d, ducaturque ad e f rectà per <lb></lb>centrum, &amp; ducantur d b &amp; d c, eritque ex hoc <lb></lb><arrow.to.target n="marg717"></arrow.to.target><lb></lb>triangulus a b d ęquilaterus triangulo a c d, <lb></lb><arrow.to.target n="marg718"></arrow.to.target><lb></lb>quare angulus b a d æqualis c a d, igitur ar­<lb></lb>cus b e æqualis c e, igitur arcus b e eſt ſexta <lb></lb><arrow.to.target n="marg719"></arrow.to.target><lb></lb>pars circuli, quare b e recta latus exagoni, <lb></lb>quare b e erit æqualis d e, igitur cum anguli <lb></lb><arrow.to.target n="marg720"></arrow.to.target><lb></lb>a d f ſint utrin que recti, erit d f æqualis f e, itaque <lb></lb><arrow.to.target n="marg721"></arrow.to.target><lb></lb>f d, tertia pars fa &amp; fb dimidium a b quia b c. </s>
        </p>
        <pb pagenum="226" xlink:href="015/01/245.jpg"></pb>
        <p type="margin">
          <s id="id003810"><margin.target id="marg716"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003811"><margin.target id="marg717"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 26. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003812"><margin.target id="marg718"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 28. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003813"><margin.target id="marg719"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>m. <lb></lb>15. <emph type="italics"></emph>quarti <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003814"><margin.target id="marg720"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003815"><margin.target id="marg721"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id003816">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id003817">Quadratum lateris trianguli æquilateri ſe habet ad illius ſuperfi<lb></lb>ciem, ut latus eius ad mediam lineam inter latus dodrantis, &amp; qua­<lb></lb>drantis proportione duplicata.<lb></lb><arrow.to.target n="marg722"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003818"><margin.target id="marg722"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003819">Quadratum a b eſt æquale quadratis a f, fb, &amp; quadruplum qua</s>
        </p>
        <p type="main">
          <s id="id003820"><arrow.to.target n="marg723"></arrow.to.target><lb></lb>drato b f, igitur quadratum a f eſt dodrans quadrati a b. </s>
          <s id="id003821">Quod ue­<lb></lb>rò fit ex a fin f b eſt medium proportione inter quadrata a f, f b, re­<lb></lb><arrow.to.target n="marg724"></arrow.to.target><lb></lb>ctangulum igitur ex a fin fb, eſt ex lateribus dodrantis a f, &amp; qua­<lb></lb><arrow.to.target n="marg725"></arrow.to.target><lb></lb>drantis b f quadrati a b, quare cum mediæ inter a f &amp; fb æquale fa­<lb></lb>ciat quadratum rectangulo a fin fb, erit proportio quadrati a b ad <lb></lb>quadratum mediæ inter a f, fb, ut lateris trianguli ad mediam inter <lb></lb><arrow.to.target n="marg726"></arrow.to.target><lb></lb>latera dodrantis, &amp; quadrantis quadrati lateris ipſius duplicata: re­<lb></lb><arrow.to.target n="marg727"></arrow.to.target><lb></lb>ctangulum autem a fin fb eſt æquale triangulo a b c, igitur propor<lb></lb>tio quadrati a b ad triangulum a b c eſt uelut lateris a b ad mediam <lb></lb>inter latera dodrantis &amp; quadrantis duplicata.</s>
        </p>
        <p type="margin">
          <s id="id003822"><margin.target id="marg723"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 27. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003823"><margin.target id="marg724"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>ſexti <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003824"><margin.target id="marg725"></margin.target>P<emph type="italics"></emph>er eandem <lb></lb>&amp;<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003825"><margin.target id="marg726"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end><lb></lb>20. <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>l.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003826"><margin.target id="marg727"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 41. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id003827">LEMMA TERTIVM.</s>
        </p>
        <p type="main">
          <s id="id003828">Propoſitio quadrati cubi ſphæræ incluſi ad triangulum pyrami<lb></lb>dis eidem ſphæræ incluſæ, eſt uelut lateris pyramidis ſeu trianguli <lb></lb>eius ad cathetum ſuum.<lb></lb><arrow.to.target n="marg728"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003829"><margin.target id="marg728"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003830">Proponatur enim ſphæræ diameter g, &amp; latus pyramidis b a, &amp; </s>
        </p>
        <p type="main">
          <s id="id003831"><arrow.to.target n="marg729"></arrow.to.target><lb></lb>latus cubi b h, quæ corpora illi ſphæræ includuntur: igitur g erit <lb></lb>poteſtate ſexquialtera ad a b, &amp; tripla ad b h, igitur b a eſt poteſtate <lb></lb><arrow.to.target n="marg730"></arrow.to.target><lb></lb>dupla ad b h, quod igitur fit ex b a in dimidium ſuum, eſt æquale <lb></lb>quadrato b h, igitur b h eſt media inter b a &amp; b f, b f enim eſt dimi­<lb></lb>dium b a, ut probatum eſt. </s>
          <s id="id003832">Quadratum igitur a b ſe habet ad trian­<lb></lb><arrow.to.target n="marg731"></arrow.to.target><lb></lb>gulum a b c, ut a b ad mediam inter a f &amp; fb duplicata: Quadratum <lb></lb>quoque a b ſe habet ad quadratum h b, ut a b ad mediam inter a b &amp; <lb></lb>b f, duplicata igitur proportio quadrati b h ad triangulum a b c, eſt <lb></lb><arrow.to.target n="marg732"></arrow.to.target><lb></lb>uelut lateris a b ad cathetum a f.</s>
        </p>
        <p type="margin">
          <s id="id003833"><margin.target id="marg729"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb>13. <emph type="italics"></emph>decimi­<lb></lb>tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003834"><margin.target id="marg730"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb>15. <emph type="italics"></emph>decimi­<lb></lb>tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003835"><margin.target id="marg731"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ſex <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end><lb></lb>L<emph type="italics"></emph>emmate<emph.end type="italics"></emph.end> 1.</s>
        </p>
        <p type="margin">
          <s id="id003836"><margin.target id="marg732"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 67.</s>
        </p>
        <p type="head">
          <s id="id003837">LEMMA QVARTVM.</s>
        </p>
        <p type="main">
          <s id="id003838">Proportio lateris pyramidis ad axem illius eſt poteſtate ſex­<lb></lb>quialtera.<lb></lb><arrow.to.target n="marg733"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003839"><margin.target id="marg733"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003840">Intelligatur baſis pyramidis triangulus a b c, &amp; conus pyrami­</s>
        </p>
        <p type="main">
          <s id="id003841"><arrow.to.target n="marg734"></arrow.to.target><lb></lb>dis k, &amp; quæ per centrum ſphæræ tranſit ex cono k d, cumque k d a <lb></lb>angulus rectus ſit, erit quadratum k a æquale quadratis k d, d a, at <lb></lb>d a eſt dupla d f, ut probatum eſt, igitur poteſtate ſexquitertia f b, <lb></lb>k a uerò eſt quadrupla poteſtate fb, quia fb eſt dimidium k a, igitur <lb></lb>k a eſt tripla poteſtate a d, igitur k a poteſtate ſexquialtera k d, quod <lb></lb>erat demonſtrandum.<lb></lb><arrow.to.target n="marg735"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003842"><margin.target id="marg734"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 47. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end><lb></lb>L<emph type="italics"></emph>emmate<emph.end type="italics"></emph.end> 1.</s>
        </p>
        <p type="margin">
          <s id="id003843"><margin.target id="marg735"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003844">Ex hoc patet quod proportio axis pyramidis ad latus cubi ea­<lb></lb>dem ſphæra circumſcriptorum eſt poteſtate ſexquitertia.</s>
        </p>
        <pb pagenum="227" xlink:href="015/01/246.jpg"></pb>
        <p type="main">
          <s id="id003845"><arrow.to.target n="marg736"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003846"><margin.target id="marg736"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003847">Quia enim k a eſt poteſtate dupla ad b b, &amp; ſeſquialtera poteſta<lb></lb>te ad k d, neceſſe eſt ut k d ſit ſexquitertia poteſtate ad b h.</s>
        </p>
        <p type="head">
          <s id="id003848">LEMMA QVINTVM.</s>
        </p>
        <p type="main">
          <s id="id003849">Priſma altitudinem habens pyramidis &amp; triangulum eiuſdem <lb></lb>baſim, æquale eſt cubo eidem ſphæræ inſcripto.</s>
        </p>
        <p type="main">
          <s id="id003850"><arrow.to.target n="marg737"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003851"><margin.target id="marg737"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id003852">Cum enim proportio quadrati b h ad triangulum a b c ſit uelut </s>
        </p>
        <p type="main">
          <s id="id003853"><arrow.to.target n="marg738"></arrow.to.target><lb></lb>a b ad a f, a b autem ad a f ſit ſexquitertia poteſtate ex demonſtratis, <lb></lb>erit quadratum b h ad triangulum a b c ſexquitertium poteſtate: at <lb></lb>cubi b h altitudo eſt ipſa b h, priſmatis autem a b c altitudo eſt k d, <lb></lb>k d autem potentia ſexquitertia ad b h, igitur priſma a b c eſt ęquale <lb></lb>cubo b h, quod fuit propoſitum.</s>
        </p>
        <p type="margin">
          <s id="id003854"><margin.target id="marg738"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>lem­<lb></lb>ma.<emph.end type="italics"></emph.end><lb></lb>L<emph type="italics"></emph>emmate<emph.end type="italics"></emph.end> 2.</s>
        </p>
        <p type="main">
          <s id="id003855">Ex hoc ſequitur, quod cum priſma ſit triplum ſuæ pyramidi, ut <lb></lb><arrow.to.target n="marg739"></arrow.to.target><lb></lb>ab Euclide habetur, quod cubus eſt triplus pyramidi, quam eadem <lb></lb><arrow.to.target n="marg740"></arrow.to.target><lb></lb>ſphæra circumſcribit.<lb></lb><arrow.to.target n="marg741"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003856"><margin.target id="marg739"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003857"><margin.target id="marg740"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb><emph type="italics"></emph>lemmatis<emph.end type="italics"></emph.end> 4.</s>
        </p>
        <p type="margin">
          <s id="id003858"><margin.target id="marg741"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 34. <emph type="italics"></emph>un­<lb></lb>decimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003859">Nunc uenio ad demonſtrationem propoſitionis, &amp; dico quod <lb></lb>corpus difficile eſt ad motum, uel ob magnitudinem baſis, cui inſi­</s>
        </p>
        <p type="main">
          <s id="id003860"><arrow.to.target n="marg742"></arrow.to.target><lb></lb>det, uel ob pondus, uel ob formam: nam corpus quod forma eſt <lb></lb><arrow.to.target n="marg743"></arrow.to.target><lb></lb>contracta, difficilè mouetur, ut pyramis, contrà, quod prominet à la<lb></lb>teribus, facile reuoluitur, ut corpus duodecim baſium pentagona­<lb></lb>rum, &amp; uiginti triangularum: ergo cubi ſedes eſt maior quàm ſua <lb></lb>pyramis, &amp; pondus triplo maius, &amp; etiam non prominet cubus, <lb></lb>ideò pro re ſtabili poſitum eſt corpus eiuſmodi. </s>
          <s id="id003861">Eo quod ob gra­<lb></lb>uitatem etiam, ut dixi, ſit ſtabilius pyramide eiuſdem ſphęrę. </s>
          <s id="id003862">Quod <lb></lb>ſi etiam aſſumeres pyramidem, cuius baſis eſſet æqualis quadrato <lb></lb>cubi, ipſa ſe haberet ad pyramidem ſphæræ in grauitate, uelut latus <lb></lb>trianguli ad ſuum cathetum, &amp; ideo proportio ponderis cubi ad <lb></lb>pyramidem eſſet, uelut tredecim ad quinque fermè: ergo ratione pon<lb></lb>deris eſſet longè ſtabilior cubus ipſa pyramide. </s>
          <s id="id003863">At in alijs corpori­<lb></lb>bus, quæ rationalia uocantur, non eſt tanta proportio ponderis, &amp; <lb></lb>baſis eſt minor &amp; forma prominet.</s>
        </p>
        <p type="margin">
          <s id="id003864"><margin.target id="marg742"></margin.target>E<emph type="italics"></emph>x<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>duode<lb></lb>cimi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003865"><margin.target id="marg743"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003866">Propoſitio centeſima nonageſima nona.</s>
        </p>
        <p type="main">
          <s id="id003867">Rationem remorum nauim impellentium inuenire.<lb></lb><arrow.to.target n="marg744"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003868"><margin.target id="marg744"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003869">Sit a remi extremum, quod manu apprehenditur, b ſcalmus cui <lb></lb>remus inſidet: c extremum aliud latius remi, quod uocant pal­<lb></lb>mam, transferatur nixu manus, &amp; motu corporis a in d, ut c per­</s>
        </p>
        <p type="main">
          <s id="id003870"><arrow.to.target n="marg745"></arrow.to.target><lb></lb>ueniat in e, ſunt enim æquales a b, d b, b c, b e etiam &amp; angu­<lb></lb>li a d b contrapoſiti, quare trianguli a b d &amp; c b e ſimiles, igitur <lb></lb>primum quanto maior propoſitio c b ad b a, tanto maior propor­<lb></lb><arrow.to.target n="marg746"></arrow.to.target><lb></lb>tio c ad a d, &amp; ita ex æquali motu longius transferetur remus, ſeu <lb></lb>palma. </s>
          <s id="id003871">Secundum, cum motus a d fiat nixu brachiorum &amp; corpo­<lb></lb>ris, quanto magis transfertur corpus eo minus opus erit brachio­


<pb pagenum="228" xlink:href="015/01/247.jpg"></pb>rum nixu, &amp; ita minus laborabunt. </s>
          <s id="id003872">Et <lb></lb><arrow.to.target n="marg747"></arrow.to.target><lb></lb>quo minus laborabunt brachia, plus <lb></lb>corpus laborabit. </s>
          <s id="id003873">Et ideò, ut declara­<lb></lb>tum eſt ſuprà, minor labor erit cum æ­<lb></lb>qualiter ambo laborabunt. </s>
          <s id="id003874">Tertium, <lb></lb>quo minor erit proportio c b ad b a, <lb></lb>eo maius ſpatium pertranſibit remex, <lb></lb>qui mouet ex a in d, ſed tanto facilius <lb></lb><arrow.to.target n="marg748"></arrow.to.target><lb></lb>mouebit, quia labor motus b c minue­<lb></lb><figure id="id.015.01.247.1.jpg" xlink:href="015/01/247/1.jpg"></figure><lb></lb>tur, ut ſuprà uiſum eſt per longitudinem a b &amp; d b, ut ſuprà demon <lb></lb>ſtrauimus. </s>
          <s id="id003875">Quartum, cùm remus tranſierit quoddam ſpatium <lb></lb>iuxta robur, puta ex c in e, neceſſe eſt ut eleuetur ſuper aquam, tum <lb></lb>quia impediret motum pro greſſus nauis, tum ut transferatur ante: <lb></lb>aliter ſi transferretur ante ſub aqua difficilius multo, quam per aë­<lb></lb>rem transferretur, &amp; retroageret tantundem nauim, quantum an­<lb></lb>tea retroactam impulit. </s>
          <s id="id003876">His per ſe notis dico, quòd translato remo <lb></lb>ex c in e, neceſſe eſt nauim contrà transferri ex f in g: nam quia impe<lb></lb>dimentum ex aqua tranſitur c in e, maius eſt quam nauis ſuper a­<lb></lb>quam, &amp; remus debet transferri ex a in d, &amp; non poteſt transferri <lb></lb>niſi uel ſtante naui, &amp; translato c in e, uel ſtante a b c remo, &amp; tranſ­<lb></lb>lata naui: &amp; tunc neceſſe eſt, ut e progrediatur ad h, ita deſſecabit a­<lb></lb>quam ch, ergo difficultas manet eadem fermè, ex his fit motus com<lb></lb>poſitus, ut palma non redeat uſque ad e, ſed maneat remus minus in­<lb></lb>clinatus, &amp; quaſi ad perpendiculum in h. </s>
          <s id="id003877">Et manifeſtum eſt, q̊d erit <lb></lb>motus compoſitus ex retro ceſſu remi &amp; pro ceſſu nauis. </s>
          <s id="id003878">Qui etiam <lb></lb>remiges circa medium ſunt minus laborarent, ſi remus æqualiter <lb></lb>promineret extra ſcalmum, ſed magis laborant, quia proportio eſt <lb></lb>eadem, &amp; a b eſt longior, &amp; craſsior remus, ut minus flectatur ob <lb></lb>longitudinem, aliter ſi eſſet æqualis craſsitudinis, &amp; multo longior <lb></lb>flecteretur aut frangeretur, ideò robuſtiores remiges ponuntur in <lb></lb>medio triremis. </s>
          <s id="id003879">Iuuatur præterea motus nauis prorſum ex percuſ­<lb></lb>ſione remi, &amp; impetu iam aquiſito cum nixu remi in aduerſum ſu­<lb></lb>perueniente. </s>
          <s id="id003880">Rurſus cum nauis transferatur eodem tempore antè <lb></lb>quò a progreditur ad d, manifeſtum eſt quòd magna pars eſt ex <lb></lb>motu nauis, non nixu corporis aut uirium: &amp; ita quod celerius mo<lb></lb>uetur ex c in h, ab initio dum nauis quieſcit, aut tardius mouetur, <lb></lb>tardius autem dum nauis progreditur.</s>
        </p>
        <p type="margin">
          <s id="id003881"><margin.target id="marg745"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003882"><margin.target id="marg746"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſexti <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003883"><margin.target id="marg747"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 188.</s>
        </p>
        <p type="margin">
          <s id="id003884"><margin.target id="marg748"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 71.</s>
        </p>
        <p type="main">
          <s id="id003885">Propoſitio ducenteſima.</s>
        </p>
        <p type="main">
          <s id="id003886">Cur temo <expan abbr="cũ">cum</expan> paruus ſit magnam nauim agere poteſt: &amp; cur cum <lb></lb>uarietas ſit in prora, ipſe conſtituatur in puppi. </s>
          <s id="id003887">Et cum tranſuerſim <lb></lb>ab aqua prematur, rectà nauim dirigat?</s>
        </p>
        <pb pagenum="229" xlink:href="015/01/248.jpg"></pb>
        <p type="main">
          <s id="id003888">Dixi quod in hipomochlio parua uarietas fit in motu: igitur à <lb></lb><arrow.to.target n="marg749"></arrow.to.target><lb></lb>leui cauſa magnum nauigium impellitur aut uariatur. </s>
          <s id="id003889">Cum enim a <lb></lb><expan abbr="trãsfertur">transfertur</expan> ad b, fit minima uarietas in e, igitur a parua poterit tranſ­<lb></lb><figure id="id.015.01.248.1.jpg" xlink:href="015/01/248/1.jpg"></figure><lb></lb>ferri, tum uero quod debuit <expan abbr="trãsferri">transferri</expan> ad c, transfertur ad <lb></lb>d, nam motus ipſe ab alia cauſa fit, uelut <expan abbr="uẽto">uento</expan> aut remis, <lb></lb>ita non eſt difficultas niſi propter motum aquæ, ſcilicet <lb></lb>ut tabula ſcindat illam. </s>
          <s id="id003890">Ad hoc autem contulit illud <lb></lb>quod intra nauim prominet ut uectis rationem habeat, <lb></lb>&amp; ob id facilius uerti.</s>
        </p>
        <p type="margin">
          <s id="id003891"><margin.target id="marg749"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003892">Similiter uarietas in puppi exigua eſt cauſa magnæ <lb></lb>uarietatis in prora, quod autem poteſt fieri paucioribus <lb></lb>&amp; faciliori modo id debet fieri, hac igitur cauſa in pup­<lb></lb>pi temonem conſtituere oportet ſeu guberna culum.</s>
        </p>
        <p type="main">
          <s id="id003893">Cum autem impellatur à mari, neceſſe eſt, ut à latere excipiat <lb></lb>aquam ita ut tantum pendeat in unam partem, quantum nauis in <lb></lb>aduerſam, nam ſi nauis non penderet, gubernaculum rectè dirige­<lb></lb><figure id="id.015.01.248.2.jpg" xlink:href="015/01/248/2.jpg"></figure><lb></lb>retur. </s>
          <s id="id003894">Vt ergo ex duobus obliquis <expan abbr="unũ">unum</expan> rectum conſtitui<lb></lb>tur, ita ex naui &amp; gubernaculo, nam ſint a b &amp; c b &amp; im­<lb></lb>pellatur ad d, impelletur per mediam lineam b e &amp; non <lb></lb>per a b neque c b, igitur oportet temonem pendere ex ad <lb></lb>uerſo inclinationis nauis. </s>
          <s id="id003895">Eſt etiam alia ratio, quoniam <lb></lb>nauis ſecurior redditur, nam quemadmodum quod in <lb></lb>medio eſt, facilius impellitur tranſuerſim, quàm quod pendet in <lb></lb>contrarium, ita &amp; in gubernaculo. </s>
          <s id="id003896">Eſt &amp; id ob neceſsitatem, quoni­<lb></lb>am motus aquæ plerumque eſt in partem, uelut &amp; uentus ad la­<lb></lb>tus eius ſitus, ſecundum quem moueri debet nauis. </s>
          <s id="id003897">Sicut igitur &amp; <lb></lb>uela &amp; malus inclinantur, ut motum directum efficiant, quia aliò <lb></lb>dirigitur nauis quam qui mouet uentus, ita de temone compara­<lb></lb>tione aquæ.</s>
        </p>
        <p type="main">
          <s id="id003898">Propoſitio ducenteſima prima.</s>
        </p>
        <p type="main">
          <s id="id003899">Si duæ lineæ non ſecantes circuli peripheriam in <expan abbr="unũ">unum</expan> <expan abbr="punctũ">punctum</expan>, ex <lb></lb>ea coëant, exterius neceſſe eſt illas peripheria <expan abbr="cõtenta">contenta</expan> eſſe maiores.</s>
        </p>
        <p type="head">
          <s id="id003900">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id003901">Si fuerit proportio primi ad ſecundum maior quàm tertij ad <lb></lb>quartum, erit primi ad tertium maior quàm ſecundi ad quartum.<lb></lb><arrow.to.target n="marg750"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003902"><margin.target id="marg750"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003903">Quamuis hoc demonſtretur à Campano, quia </s>
        </p>
        <p type="main">
          <s id="id003904"><arrow.to.target n="marg751"></arrow.to.target><lb></lb>tamen facile eſt hic adijcietur. </s>
          <s id="id003905">Sit igitur maior a <lb></lb>ad b quam c ad d, dico maiorem eſſe a ad c quam <lb></lb><figure id="id.015.01.248.3.jpg" xlink:href="015/01/248/3.jpg"></figure><lb></lb><arrow.to.target n="marg752"></arrow.to.target><lb></lb>b ad d, quia enim maior eſt a ad b quam c ad d fiat e ad b ut c ad e <lb></lb><arrow.to.target n="marg753"></arrow.to.target><lb></lb>eritque e minuſ quam a, e igitur ad c ut b ad d ſed maior a ad c quam <lb></lb>e ad e igitur maior a ad c quam b ad d.<lb></lb><arrow.to.target n="marg754"></arrow.to.target></s>
        </p>
        <pb pagenum="230" xlink:href="015/01/249.jpg"></pb>
        <p type="margin">
          <s id="id003906"><margin.target id="marg751"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>quin<lb></lb>ti <emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003907"><margin.target id="marg752"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003908"><margin.target id="marg753"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003909"><margin.target id="marg754"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id003910">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id003911">Si fuerint quatuor quanti­<lb></lb><arrow.to.target n="marg755"></arrow.to.target><lb></lb>tates, quarum exceſſus primæ <lb></lb>ſupra ſecundam, fit minor ex­<lb></lb><figure id="id.015.01.249.1.jpg" xlink:href="015/01/249/1.jpg"></figure><lb></lb>ceſſu tertię ſupra quartam, ſitque prima non minor tertia, erit propor<lb></lb><arrow.to.target n="marg756"></arrow.to.target><lb></lb>tio primæ ad ſecundam minor quàm tertiæ ad quartam.<lb></lb><arrow.to.target n="marg757"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003912"><margin.target id="marg755"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem. par <lb></lb>tes ambas.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003913"><margin.target id="marg756"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 10. <emph type="italics"></emph>quin<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003914"><margin.target id="marg757"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003915">Sit exceſſus a ſupra b c, g b minor exceſſu d ſupra e f qui ſit h e, di­</s>
        </p>
        <p type="main">
          <s id="id003916"><arrow.to.target n="marg758"></arrow.to.target><lb></lb>co quod proportio a ad b c eſt minor proportione d ad e f. </s>
          <s id="id003917">Quia <lb></lb>enim a eſt maior d, &amp; b g minor h e, erit maior proportio a ad b g <lb></lb><arrow.to.target n="marg759"></arrow.to.target><lb></lb>quàm d ad h e, igitur fiat a ad g k ut d ad h e, erit ergo g k maior g b <lb></lb><arrow.to.target n="marg760"></arrow.to.target><lb></lb>quare k e minor b c ex communi animi ſententia, eſt autem a ad k c <lb></lb>ut d ad e f, minor autem a ad c b quàm ad k c, igitur minor a ad b c <lb></lb>quam d ad e f.</s>
        </p>
        <p type="margin">
          <s id="id003918"><margin.target id="marg758"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 19. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003919"><margin.target id="marg759"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003920"><margin.target id="marg760"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>quin<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003921">Si intra circulum æquicurium, &amp; ſuper eandem baſim figura æ­<lb></lb>quilatera &amp; æquiangula <expan abbr="cõſtituatur">conſtituatur</expan>, <expan abbr="erũt">erunt</expan> omnia illius latera pariter <lb></lb>accepta minora duobus trianguli lateribus.<lb></lb><arrow.to.target n="marg761"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003922"><margin.target id="marg761"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003923">Sit ut proponitur, &amp; producantur b d &amp; <lb></lb>c e quæ concurrent intra triangulum, quia <lb></lb>anguli d b c &amp; e c b ſupponuntur ęquales, &amp; <lb></lb>ducta d e producantur d fl, &amp; e g l quæ <expan abbr="con­currẽt">con­<lb></lb>current</expan> intra triangulum k d e ut propter ean­<lb></lb>dem cauſam, igitur a b &amp; a c ſunt maiores k b <lb></lb>&amp; k c, ergo maiores k d, d b, &amp; k e, e c quia <lb></lb>ſunt eædem. </s>
          <s id="id003924">Ductę quo que de ſimili modo <lb></lb><figure id="id.015.01.249.2.jpg" xlink:href="015/01/249/2.jpg"></figure><lb></lb>k d &amp; d e, ſunt maiores l d &amp; l e, igitur l f, f d &amp; l g, g e, igitur a b &amp; a c <lb></lb>maiores ſunt b d, d f, f l c e e g g l pariter acceptis. </s>
          <s id="id003925">Rurſus ducta f g: <lb></lb>f l &amp; l g maiores ſunt m f &amp; m g, igitur a b &amp; a c ſunt maiores omni­<lb></lb>bus lateribus figuræ inſcriptæ.</s>
        </p>
        <p type="main">
          <s id="id003926"><arrow.to.target n="marg762"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003927"><margin.target id="marg762"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003928">Ex hoc patet quod latera polygoniæ fi­<lb></lb>guræ ęquilateræ &amp; æquiangulæ inſcriptę <lb></lb>portioni circuli ſunt minora lateribus tra­<lb></lb>pezij circunſcripti eidem peripheriæ.</s>
        </p>
        <figure id="id.015.01.249.3.jpg" xlink:href="015/01/249/3.jpg"></figure>
        <p type="main">
          <s id="id003929">Sit ergo trapezium a g h b circa periphe</s>
        </p>
        <p type="main">
          <s id="id003930"><arrow.to.target n="marg763"></arrow.to.target><lb></lb><expan abbr="riã">riam</expan> a b, &amp; in ea inſcripta figura polygonia <lb></lb>æquilatera &amp; æquiangula a c, d f b. </s>
          <s id="id003931">Et quia <lb></lb>trapezium eſt figura cuius oppoſita duo <lb></lb>latera ſunt ęqualia, &amp; duo anguli ſupra ba <lb></lb>ſim æquales: itemque duo in ſummitate inui<lb></lb>cem ęquales, <expan abbr="tãget">tanget</expan> in medio peripheriam <lb></lb><arrow.to.target n="marg764"></arrow.to.target><lb></lb>quod patet ductis lineis ex centro ad ex­<lb></lb><figure id="id.015.01.249.4.jpg" xlink:href="015/01/249/4.jpg"></figure><lb></lb>trema trapezij. </s>
          <s id="id003932">Et ideo etiam <expan abbr="punctũ">punctum</expan> medium polygoniæ, quare ex 


<pb pagenum="231" xlink:href="015/01/250.jpg"></pb>hoc leminate duo latera g d &amp; g a deducta ad æquicrurium, erunt <lb></lb>maiora lateribus polygonię, &amp; ſimiliter duo latera h d maiora late­<lb></lb>ribus polygoniæ incluſæ, ergo latera trapezij erunt maiora omni­<lb></lb>bus lateribus polygoniæ incluſæ.<lb></lb><arrow.to.target n="marg765"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003933"><margin.target id="marg763"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id003934"><margin.target id="marg764"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>pri­<lb></lb>mi, &amp;<emph.end type="italics"></emph.end> 16. <lb></lb><emph type="italics"></emph>tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003935"><margin.target id="marg765"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003936">Ex hoc habetur demonſtratio propoſitionis: ſint duæ lineæ a b <lb></lb>&amp; a c quæ comprehendant portionem cir­<lb></lb>culi b c, dico eas eſſe maiores b c portione, <lb></lb>ſi enim a b &amp; a c ſunt æquales diuiſo arcu <lb></lb>b c per æqualia in f, ducam contingentem </s>
        </p>
        <p type="main">
          <s id="id003937"><arrow.to.target n="marg766"></arrow.to.target><lb></lb>h f k, ſi non faciant triangulum æquicruri­<lb></lb>um b c d ſuper b c, &amp; cuius ambo latera pa<lb></lb>riter accepta ſint æqualia a b &amp; a c. </s>
          <s id="id003938">Et du­<lb></lb>cam contingentem &amp; habebo trapezium <lb></lb><arrow.to.target n="marg767"></arrow.to.target><lb></lb>h b, c k. </s>
          <s id="id003939">Quare ſi peripheria circuli b c eſt <lb></lb><figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg"></figure><lb></lb>minor d b &amp; d c pariter acceptis, habeo <expan abbr="intentũ">intentum</expan>, ſi non toties <expan abbr="diuidã">diuidam</expan> <lb></lb>peripheriam per æqualia ut fiat figura polygonia ſuper b c æquila­<lb></lb>tera &amp; æquiangula, cuius differentia a peripheria ſit minor differen<lb></lb>tia d b &amp; d c à trapezio b h, k c, id eſt, tribus eius lateribus, nam cum <lb></lb>d h &amp; d k ſint maiores h k, conſtat quod d b &amp; d e ſunt maiores h b, <lb></lb>&amp; k c &amp; h k igitur ſit differentia illa l, &amp; <expan abbr="differẽtia">differentia</expan> peripherię à lineis <lb></lb>polygoniæ minori: igitur cum peripheria ſit æqualis aut maior <lb></lb>d b &amp; d c, &amp; differentia a lateribus polygoniæ minor quàm d b &amp; <lb></lb>d c, a b, h b, h k, k c, erit minor proportio peripheriæ ad latera poly­<lb></lb><arrow.to.target n="marg768"></arrow.to.target><lb></lb>goniæ quàm d b &amp; d c ad tria latera trapezij, quare minor propor­<lb></lb><arrow.to.target n="marg769"></arrow.to.target><lb></lb>tio peripheriæ ad d b &amp; d c quàm laterum polygoniæ ad tria latera <lb></lb><arrow.to.target n="marg770"></arrow.to.target><lb></lb>trapezij, ſed latera polygoniæ ſunt minora tribus lateribus. </s>
          <s id="id003940">trapezij, <lb></lb><arrow.to.target n="marg771"></arrow.to.target><lb></lb>igitur peripheria b c eſt minor d b &amp; d e, quod erat <expan abbr="demonſtrandũ">demonſtrandum</expan>.</s>
        </p>
        <p type="margin">
          <s id="id003941"><margin.target id="marg766"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end> 1. <lb></lb><emph type="italics"></emph>primi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003942"><margin.target id="marg767"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>eiuſ­<lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003943"><margin.target id="marg768"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003944"><margin.target id="marg769"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003945"><margin.target id="marg770"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003946"><margin.target id="marg771"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. <lb></lb>3 <emph type="italics"></emph>lemmatis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id003947">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id003948">Hanc propoſitionem non ſcripſi quòd eſſet magni momenti, ſed <lb></lb>propter modum probandi, ſi enim reſpicis ex uno oppoſito ſcilicet <lb></lb>quod peripheria circuli ſit maior trianguli lateribus, oſtendo de­<lb></lb>monſtratione non ducente ad inconueniens, ſed ſimplici quod ipſa <lb></lb>peripheria eſt minor trianguli lateribus, &amp; hoc nunquam fuit <expan abbr="factũ">factum</expan> <lb></lb>ab aliquo, imò uidetur plane impoſsibile. </s>
          <s id="id003949">Et eſt res admirabilior <lb></lb>quæ inuenta ſit ab orbe condito, ſcilicet oſtendere aliquid ex ſuo <lb></lb>oppoſito, demonſtratione non ducente ad impoſsibile &amp; ita, ut <expan abbr="nõ">non</expan> <lb></lb>poſsit demonſtrari ea <expan abbr="demõ">demom</expan>ſtratione niſi per illud <expan abbr="ſuppoſitũ">ſuppoſitum</expan> quod <lb></lb>eſt contrarium concluſioni, uelut ſi quis demonſtraret quòd So­<lb></lb>crates eſt albus quia eſt niger, &amp; non poſſet demonſtrare aliter, &amp; <lb></lb>ideo eſt longè maius Chryſippeo Syllogiſmo.<lb></lb><arrow.to.target n="marg772"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003950"><margin.target id="marg772"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003951">Ex hoc patet quod pars lineæ exterioris quæ tangit circulum 


<pb pagenum="232" xlink:href="015/01/251.jpg"></pb>intercepta à linea ex centro longior eſt peripheria, ſimiliter in­<lb></lb>tercepta.</s>
        </p>
        <p type="main">
          <s id="id003952"><arrow.to.target n="marg773"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003953"><margin.target id="marg773"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003954">Sit portio circuli a e, &amp; linea a b intercepta à linea c b ex centro, <lb></lb><figure id="id.015.01.251.1.jpg" xlink:href="015/01/251/1.jpg"></figure><lb></lb>dico ab eſſe longiorem a e, ducatur b e æqualis a b, ad </s>
        </p>
        <p type="main">
          <s id="id003955"><arrow.to.target n="marg774"></arrow.to.target><lb></lb>circumferentiam, quæ illi obuiabit, ducanturque c a, c e <lb></lb><arrow.to.target n="marg775"></arrow.to.target><lb></lb>eritque angulus e c b æqualis a c b, igitur arcus a d, æ­<lb></lb>qualis d c, quare a d erit <expan abbr="dimidiũ">dimidium</expan> a e, &amp; a b dimidium <lb></lb><arrow.to.target n="marg776"></arrow.to.target><lb></lb>a b, b e, facta enim fuit b e æqualis a b, cum ergo per <lb></lb>præſentem duæ lineæ a b, b e, ſint maiores a e, igitur per commu­<lb></lb>nem animi ſententiam a b maior a d.</s>
        </p>
        <p type="margin">
          <s id="id003956"><margin.target id="marg774"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>tertij <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003957"><margin.target id="marg775"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id003958"><margin.target id="marg776"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end>|26. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id003959">Propoſitio ducenteſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id003960">Rationem ſtrepitus oſtendere.<lb></lb><arrow.to.target n="marg777"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003961"><margin.target id="marg777"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id003962">Fit ſtrepitus ob multitudinem aëris percuſsi, uelut cum tabulis <lb></lb>percutimus: &amp; cauitatum cauſa, unde ligna &amp; tabulæ leues magis <lb></lb>ſtrepunt, &amp; illud Virgilij:</s>
        </p>
        <p type="main">
          <s id="id003963">—Sonitumque dedere cauernæ.</s>
        </p>
        <p type="main">
          <s id="id003964">Tum uerò ob ictus impetum, impetus <expan abbr="autẽ">autem</expan> partim uelocitatis cau­<lb></lb>ſa, partim anguſtiæ loci. </s>
          <s id="id003965">Fulmen edit tonitru in quo &amp; caua nebula <lb></lb>excipit aërem, &amp; multum impetuque maximo delatum, <expan abbr="obſtrepũt">obſtrepunt</expan> au<lb></lb>tem metalla magis quam ligna eo quòd magis ob <expan abbr="continuitatẽ">continuitatem</expan> par <lb></lb>tes moueantur. </s>
          <s id="id003966">Indicio eſt, quod intenta ut æs &amp; tenuia <expan abbr="maiorẽ">maiorem</expan> ſtre<lb></lb>pitum edunt: &amp; dum ſonant tremunt, aurum autem parum ſonat, <lb></lb>quoniam denſiſsimum eſt, et minus intentum <expan abbr="argẽtum">argentum</expan>, minus den <lb></lb>ſum, &amp; magis intentum, quod autem intentum eſt totum ſimul mo<lb></lb>uetur, &amp; ob id ſtridet: lignum <expan abbr="autẽ">autem</expan> &amp; tabula ſonat, non quia ut me­<lb></lb>tallum percutiat aërem, ſed quia in eo aër percutitur. </s>
          <s id="id003967">Craſſum <expan abbr="autẽ">autem</expan> <lb></lb>metallum &amp; lignum non adeò ſonant: metallum quoniam non mo<lb></lb>uet aërem, non enim mouetur: lignum quoniam non mouetur, nec <lb></lb>in eo qui eſt incluſus aër, aër autem facilè mouetur, &amp; ob id in ligno <lb></lb>cauo, etiamſi craſſum ſit, ſtrepitus magnus editur. </s>
          <s id="id003968">Ergo etſi tenue <lb></lb>ſit metallum, quod infixum eſt tabulę, reſonat multum: <expan abbr="nõ">non</expan> quia mo<lb></lb>ueatur, ſed quoniam <expan abbr="aẽrem">aerrem</expan> in tabula <expan abbr="cõ">com</expan> cutit. </s>
          <s id="id003969">Neque enim tabula per <lb></lb>ſe ſola, quæ etiam nimis tunderetur ſonum edere magnum poteſt <lb></lb>quoniam cedit: Oportet <expan abbr="autẽ">autem</expan> non cedere quod reſonat, neque metal­<lb></lb>lum ſi craſſum, ſed hebetem <expan abbr="ſonũ">ſonum</expan> etiam tabulę infixum reddit, quo­<lb></lb>niam neque moueri poteſt infixum &amp; craſſum, nec cauernoſum eſt, &amp; <lb></lb>tamen excipit ictum, ne lignum reſonet. </s>
          <s id="id003970">Velox autem ictus <expan abbr="nõ">non</expan> acu­<lb></lb>tum <expan abbr="ſonũ">ſonum</expan> reddit, &amp; ſi cum impetu ſit: indicio eſt tonitru &amp; machinę <lb></lb>bellicæ igneę, contrà anguſta fiſtula <expan abbr="acutũ">acutum</expan> ſonum reddit, <expan abbr="etiã">etiam</expan> remiſ­<lb></lb>ſè inflata. </s>
          <s id="id003971">Igitur aër ſoni cauſa eſt ſecundum <expan abbr="motũ">motum</expan>, ubi ergo multus <lb></lb>aër &amp; magnus motus ibi ſonus magnus. </s>
          <s id="id003972">Multus quidem aut in ca­


<pb pagenum="233" xlink:href="015/01/252.jpg"></pb>uernoſo corpore, qui <expan abbr="grauiſsimũ">grauiſsimum</expan> edit <expan abbr="ſonũ">ſonum</expan> intercluſus, ut <expan abbr="etiã">etiam</expan> in uo <lb></lb>cibus, aut quia à magno corpore ſtridulus efficitur, aut inter duo <lb></lb>corpora, qui grauitate medius eſt. </s>
          <s id="id003973">Impetu uerò <expan abbr="efficit̃">efficitur</expan> intenſus non <lb></lb>magnus, nam tonitrus procul audimus non iſtum quamuis celerri­<lb></lb>mum, acutum uerò ob anguſtiam loci. </s>
          <s id="id003974">Atque hę cauſę ſunt ſonorum.</s>
        </p>
        <p type="main">
          <s id="id003975">Propoſitio ducenteſima tertia.</s>
        </p>
        <p type="main">
          <s id="id003976">Cur ſcytalis onera portentur facilius, explorare.</s>
        </p>
        <figure id="id.015.01.252.1.jpg" xlink:href="015/01/252/1.jpg"></figure>
        <p type="main">
          <s id="id003977">Demiror <expan abbr="nõ">non</expan> exactè cauſam <expan abbr="manifeſtiſsimã">manifeſtiſsimam</expan> </s>
        </p>
        <p type="main">
          <s id="id003978"><arrow.to.target n="marg778"></arrow.to.target><lb></lb>Ariſtotelem non <expan abbr="aſſecutũ">aſſecutum</expan> fuiſſe, aut potius ad <lb></lb><arrow.to.target n="marg779"></arrow.to.target><lb></lb>nos <expan abbr="corruptã">corruptam</expan> ſcripturam perueniſſe: nam qui <lb></lb>expo <expan abbr="nũt">nunt</expan> multo minus <expan abbr="intelligũt">intelligunt</expan>. </s>
          <s id="id003979">Sit ergo cur <lb></lb>rus humilis ſcytalis <expan abbr="iucumbẽs">iucumbens</expan> a b c. </s>
          <s id="id003980">Diximus <lb></lb><expan abbr="autẽ">autem</expan> ſuprà quid eſſet ſcytala &amp; currus rotis, <expan abbr="q̃">quae</expan> <lb></lb>ſuntlonge maiores ſcytalis e f g h, <expan abbr="demõſtran">demonſtran</expan> <lb></lb><expan abbr="dũ">dum</expan> eſt <expan abbr="ſcytalã">ſcytalam</expan>, quamuis minoris ambitus ma­<lb></lb>gis mouere &lt;08&gt; rotam, <expan abbr="cũ">cum</expan> ergo de una demon­<lb></lb>ſtrauerimus, de <expan abbr="oĩbus">oimbus</expan> erit <expan abbr="intelligendũ">intelligendum</expan>. </s>
          <s id="id003981">Quia <lb></lb>ergo ſcytala k l m habet hypomochlion in k et <lb></lb>m, &amp; <expan abbr="põdus">pondus</expan> premit in l, <expan abbr="igit̃">igitur</expan> rota uerſatilis mo<lb></lb><arrow.to.target n="marg780"></arrow.to.target><lb></lb><expan abbr="uebit̃">uebitur</expan> tanto facilius procedendo, quanta eſt <expan abbr="lõ">lom</expan> gitudo l m &amp; l k, ſed &amp; <lb></lb>rotulę illę <expan abbr="uerſabũt">uerſabunt</expan> hypomochlion, q̊d eſt l <expan abbr="cõparatione">comparatione</expan> k &amp; m col­<lb></lb>lopum, <expan abbr="igit̃">igitur</expan> facilius multo <expan abbr="uerſabit̃">uerſabitur</expan> currus à ſcytalis &lt;08&gt; rotis. </s>
          <s id="id003982">Et hoc <lb></lb>eſt quod dixit Philoſophus. </s>
          <s id="id003983">In utriſque. </s>
          <s id="id003984">n. </s>
          <s id="id003985">his <expan abbr="reuoluit̃">reuoluitur</expan> circulus et mo<lb></lb>tus <expan abbr="impellit̃">impellitur</expan>, intelligit <expan abbr="mutuã">mutuam</expan> <expan abbr="commutationẽ">commutationem</expan> hypomochlij cum col<lb></lb>lopibus, nam ut <expan abbr="trahãtur">trahantur</expan> rotulę <expan abbr="q̃">quae</expan> ſunt hypomochlij loco, collopes <lb></lb><expan abbr="terminant̃">terminantur</expan> in medio: ut <expan abbr="aũt">aunt</expan> <expan abbr="uertat̃">uertatur</expan> axis, qui &amp; hypomochlion in me­<lb></lb>dio <expan abbr="collopũ">collopum</expan> initium ſint rotulæ. </s>
          <s id="id003986">Ex quo <expan abbr="ſequit̃">ſequitur</expan>, q̊d quanto <expan abbr="lõgiores">longiores</expan> <lb></lb>erunt l k l t &amp; l m, tanto facilius <expan abbr="mouebunt̃">mouebuntur</expan> currus, at quanto humi­<lb></lb>liores, modò non obruantur in terra, quoniam tardius mouentur, <lb></lb>quæ minorem habent circuitum, quæ autem tardius mouentur, fa<lb></lb>cilius mouentur, ut ſuprà ſæpius demonſtratum eſt: Ob has ergo <lb></lb>duas cauſas pondera facilius feruntur curribus cum ſcytalis, quàm <lb></lb>cum rotis magnis modò terra non obruantur.</s>
        </p>
        <p type="margin">
          <s id="id003987"><margin.target id="marg778"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id003988"><margin.target id="marg779"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 114.</s>
        </p>
        <p type="margin">
          <s id="id003989"><margin.target id="marg780"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 71</s>
        </p>
        <p type="main">
          <s id="id003990">Propoſitio ducenteſima quarta.</s>
        </p>
        <p type="main">
          <s id="id003991">Cur pluribus trochleis pondera facilius eleuentur oſten dere.</s>
        </p>
        <p type="main">
          <s id="id003992">Dictum eſt ſatis de hoc in lib. 

de Subtilitate, at nunc quod ad de­<lb></lb><arrow.to.target n="marg781"></arrow.to.target><lb></lb>monſtrationem attinet <expan abbr="eorũ">eorum</expan> ſubijciam. </s>
          <s id="id003993">Quia. n. </s>
          <s id="id003994">ſingulę rotulę diffi <lb></lb>culter <expan abbr="mouent̃">mouentur</expan>, igitur neceſſe eſt ſingulas participes eſſe grauitatis, <lb></lb>igitur &amp; totam <expan abbr="grauitatẽ">grauitatem</expan> eſſe diuiſam: quare ut in <expan abbr="pręcedẽti">pręcedenti</expan> facilius <lb></lb>moueri. </s>
          <s id="id003995">Habent &amp; rotulę ipſę centrum ſeu axem hypomochlij, ſeu <lb></lb><arrow.to.target n="marg782"></arrow.to.target><lb></lb>fulcimenti loco, ambitum <expan abbr="aũt">aunt</expan> iuxta ſemidiametrum, uelut collopes <pb pagenum="234" xlink:href="015/01/253.jpg"></pb>ſeu uectes, quare tanto facilius mouebuntur quanto maiores <expan abbr="erũt">erunt</expan>, <lb></lb><figure id="id.015.01.253.1.jpg" xlink:href="015/01/253/1.jpg"></figure><lb></lb>&amp; ut plures. </s>
          <s id="id003996">Vna enim alterius loco fungitur uectis. </s>
          <s id="id003997">Trochlea qui­<lb></lb>dem eſt, ut uides, inſtrumentum longum ſuprà anguſtius, ſed non, <lb></lb>craſſum, in quo plures orbiculi ſolent collo cari, unde ſæpe numero <lb></lb>trochleæ nomine intelligimus orbiculos ei incluſos, circa quos fu­<lb></lb>nis uocatur, ut in trochleis &amp; orbiculi &amp; funes includuntur. </s>
          <s id="id003998">Succu­<lb></lb>lis etiam ſolent capita funium trahi: ut uectis auxilio imò nonnum­<lb></lb>quàm rotarum facilius pondera eleuantur.<lb></lb><arrow.to.target n="marg783"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id003999"><margin.target id="marg781"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id004000"><margin.target id="marg782"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 71.</s>
        </p>
        <p type="margin">
          <s id="id004001"><margin.target id="marg783"></margin.target>8. <emph type="italics"></emph>de<emph.end type="italics"></emph.end> R<emph type="italics"></emph>epub.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004002">Propoſitio ducenteſima quinta, ſuper uerbis Platonis, <lb></lb>de fine Reipub.</s>
        </p>
        <p type="main">
          <s id="id004003">“Eſt autem ei quod diuinitus generandum eſt circuitus, quem nu<lb></lb>merus <expan abbr="cõtinet">continet</expan> perfectus. </s>
          <s id="id004004">Humanæ uerò, in quo primum argumen<lb></lb>tationes ſuperantes, ut ſuperatæ tres diſtantiæ: quatuor autem ter­<lb></lb>minos accipientes, ſimilium &amp; diſsimilium, ab <expan abbr="undantiũ">undantium</expan> &amp; deficien<lb></lb>tium cuncta correſpondentia, &amp; rationem habentia inuicem effece<lb></lb>runt. </s>
          <s id="id004005">Quorum ſexquitertium fundamentum quinario <expan abbr="iunctũ">iunctum</expan> duas <lb></lb>efficit harmonias ter aucta quidem: æqualem æqualiter centum to<lb></lb>ties, quandam autem æqualem quidem, longitudine <expan abbr="aũt">aunt</expan> ſingulum <lb></lb>quidem numerorum à diametris <expan abbr="rationẽ">rationem</expan> habentibus quinarij indi <lb></lb>gentibus uno ſingulis: non habentibus rationem <expan abbr="aũt">aunt</expan> duobus, cen­<lb></lb>tum autem cuborum ternarij. </s>
          <s id="id004006">Totus autem hic numerus geometri <lb></lb>cus talem authoritatem habet ad potiorem deterioremque <expan abbr="genera­tionẽ">genera­<lb></lb>tionem</expan>. </s>
          <s id="id004007">Quem locum Ariſtoteles ita declarat. </s>
          <s id="id004008">Quorum ſexquiter­<lb></lb>tium fundamentum quinario coniunctum duas exhibet harmo­<lb></lb>nias, <expan abbr="inquiẽs">inquiens</expan>, <expan abbr="quãdo">quando</expan> numerus diagrammatis huius <expan abbr="efficiat̃">efficiatur</expan> ſolidus.”</s>
        </p>
        <p type="main">
          <s id="id004009"><arrow.to.target n="marg784"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004010"><margin.target id="marg784"></margin.target>Q<emph type="italics"></emph>uin<emph.end type="italics"></emph.end> P<emph type="italics"></emph>olyt.<emph.end type="italics"></emph.end><lb></lb>C<emph type="italics"></emph>ap.<emph.end type="italics"></emph.end> 12.</s>
        </p>
        <p type="main">
          <s id="id004011"><foreign lang="grc">Γυσθμὴν</foreign> <expan abbr="fundamẽtum">fundamentum</expan> interpretatus ſum, quod radix pro latere in <lb></lb>hac materia accipi poſſet. </s>
          <s id="id004012">Par eſt ut in diuina generatione numerus </s>
        </p>
        <p type="main">
          <s id="id004013"><arrow.to.target n="marg785"></arrow.to.target><lb></lb><expan abbr="acciperet̃">acciperetur</expan> perfectus: ut intelligat generationem confeſtim ſequi cor <lb></lb>ruptionem: nam ſermo eſt de corruptione, corrumpitur <expan abbr="aũt">aunt</expan> unum­<lb></lb>quodque ut aliud generetur, malum enim eſt ob bonum, non contrà. <lb></lb></s>
          <s id="id004014">Liquet autem ex Euclide talem numerum eſſe octies mille <expan abbr="centũ">centum</expan> ui­<lb></lb>ginti octo. </s>
          <s id="id004015">Et hic eſt finis <expan abbr="omniũ">omnium</expan> urbium diuinus, cuius <expan abbr="quadruplũ">quadruplum</expan> <lb></lb>uelut in cœli reſtitutionibus, ac continuato ordine ſolet obſeruari, <lb></lb>eſt propè annus magnus: ueriſimile eſt enim <expan abbr="tãto">tanto</expan> tempore <expan abbr="cõfundi">confundi</expan> <lb></lb>decima, ſcilicet totius circuitus parte. </s>
          <s id="id004016">Humanæ uerò intelligit qua­<lb></lb><figure id="id.015.01.253.2.jpg" xlink:href="015/01/253/2.jpg"></figure><arrow.to.target n="table29"></arrow.to.target><lb></lb>tuor à monade numeros, aut in quauis ratione principium li­<lb></lb>neam ſuperficiem corpus, ut <expan abbr="unũ">unum</expan>, duo, quatuor, octo pariter <lb></lb>octo: duodecim decem octo uiginti <expan abbr="ſeptẽ">ſeptem</expan>: inter hæc ſunt tria <lb></lb>ſpatia, &amp; octo cum uiginti ſeptem ſunt diſsimilia &amp; deficien­<lb></lb>tia: maiora <expan abbr="em̃">emm</expan> ſunt ſuis partibus à quibus numerantur. </s>
          <s id="id004017">Contrà de­<lb></lb>cem octo &amp; duodecim ſunt ſimilia atque ab <expan abbr="undãtia">undantia</expan>, &amp; correſponden 


<pb pagenum="235" xlink:href="015/01/254.jpg"></pb>tem habent rationem inuicem. </s>
          <s id="id004018">Hæc Ariſtoteles omittit, ut ad in­<lb></lb>troductionem, non rem pertinentia, uelut &amp; finem tanquàm ex <lb></lb>præcedentibus notum. </s>
          <s id="id004019">Vnde uerba Ariſtotelis ſunt ad unguem <lb></lb>eadem uerbis Platonis, ſcilicet: “Quorum ſexquitertium funda­<lb></lb>mentum quinario iunctum duas efficit harmonias: loco autem ter <lb></lb>aucta quidem, ſcribit Ariſtoteles: efficiatur ſolidus, id eſt cubus, ut <lb></lb>in quadratum ſuum ducatur: loco autem uerborum æqualem æ­<lb></lb>qualiter centum centies, uſque illuc à diametris rationem habenti­<lb></lb>bus quinarij ponit numerum diagrammatis.” Eſt autem diagram­<lb></lb>ma, quod Plato uocat diametrum, cum numerus poteſt fermè du­<lb></lb>plum numeri alterius, ut 3 duplum 2, &amp; 7 duplum 5, &amp; 17 duplum <lb></lb>12, &amp; ſemper numerus hic dimetiens, excedit duplum alterius uno, <lb></lb>quod ex his patet, quæ ab Euclide demonſtrata ſunt in decimo li­<lb></lb>bro. </s>
          <s id="id004020">Quare ſi debet eſſe quadratum eius monade maius duplo, al­<lb></lb>terius quadrati, &amp; duplum | alterius quadrati eſt par, igitur addi­<lb></lb>ta monade erit impar, ergo latus eius dimetiens impar ſemper: la­<lb></lb>tera autem ipſa quadratorum, quæ duplicantur aliquando pa­<lb></lb>ria ſunt ut 2, &amp; tunc quadratum dimetientis eſt unum plus duplo <lb></lb>ut 9 eſt maius 8 monade, ſi uerò latera imparia ſint, erit quadratum <lb></lb>dimetientis uno minus duplo, ut 49 quadratum 7 eſt minus uno <lb></lb>50, duplo 25, quadrati 5. Ex quo patet agnatio, ut ita dicam in­<lb></lb>ter 7 &amp; 5.</s>
        </p>
        <p type="margin">
          <s id="id004021"><margin.target id="marg785"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <table>
          <table.target id="table29"></table.target>
          <row>
            <cell>8</cell>
          </row>
          <row>
            <cell>12</cell>
          </row>
          <row>
            <cell>18</cell>
          </row>
          <row>
            <cell>27</cell>
          </row>
        </table>
        <p type="main">
          <s id="id004022">Cum ergo dicit, quorum ſexquitertia eſt, ac ſi diceret, ex horum <lb></lb>numerorum ſerie ſumemus ſeptenarium principium epitrite, &amp; di­<lb></lb>metientem 5, quos ſimul iungemus.</s>
        </p>
        <p type="main">
          <s id="id004023">Propoſitio ducenteſima ſexta.</s>
        </p>
        <figure id="id.015.01.254.1.jpg" xlink:href="015/01/254/1.jpg"></figure>
        <p type="main">
          <s id="id004024">Rhombi paſsiones quaſdam declarare.</s>
        </p>
        <p type="main">
          <s id="id004025">Sit a d recta diuiſa in k per æqualia, cui ſu­<lb></lb><arrow.to.target n="marg786"></arrow.to.target><lb></lb>perſtent k b &amp; k c ad perpendiculum inter ſe <lb></lb>æquales, &amp; ſingulæ <expan abbr="earũ">earum</expan> minores k a &amp; k d, <lb></lb><arrow.to.target n="marg787"></arrow.to.target><lb></lb>&amp; <expan abbr="perficiat̃">perficiatur</expan> figura quadrilatera a b d c, cuius <lb></lb>latera erunt omnia æqualia inuicem, &amp; angu<lb></lb>li a &amp; d oppoſiti, &amp; b &amp; c oppoſiti etiam inui<lb></lb>cem ęquales. </s>
          <s id="id004026">Sed b &amp; c maiores erunt a &amp; d: <lb></lb><arrow.to.target n="marg788"></arrow.to.target><lb></lb>&amp; ideo talem figuram appellauit Ariſtoteles rhombum à piſcis ſi­<lb></lb>militudine in medio latioris <expan abbr="quã">quam</expan> in extremis, cuius <expan abbr="tamẽ">tamen</expan> longitudo <lb></lb>latitudine maior eſt. </s>
          <s id="id004027">Dicit ergo Ariſtoteles, q̊d ſi rhombus ipſe cir­<lb></lb><arrow.to.target n="marg789"></arrow.to.target><lb></lb>cumuoluatur, ita ut b tranſiret per b a c, &amp; a per a c d, a maius ſpa­<lb></lb>tium tranſiret ex recta, ſcilicet a k d quàm b, quod tranſiret b k c. </s>
          <s id="id004028">Et <lb></lb>ad hoc aſſumit, quòd cum angulus c ſit maior a, igitur duæ lineæ <lb></lb>a c d ſunt minus curuæ quam duæ b a c, igitur b a c habent ratio­


<pb pagenum="236" xlink:href="015/01/255.jpg"></pb>nem currui, &amp; a c d recti. </s>
          <s id="id004029">Ergo ſi in æquali <expan abbr="tẽporis">temporis</expan> ſpatio b, ſuperet <lb></lb>b a c &amp; a, a c d, magis per rectam feretur a quàm b, ſed quod rectum <lb></lb>eſt maius occupat ſpatium: igitur uelocius fertur a in d compara­<lb></lb>tione habita ad a d quàm b in c, comparatione habita ad b c.</s>
        </p>
        <p type="margin">
          <s id="id004030"><margin.target id="marg786"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id004031"><margin.target id="marg787"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004032"><margin.target id="marg788"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 25. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004033"><margin.target id="marg789"></margin.target>Q<emph type="italics"></emph>uæſt.<emph.end type="italics"></emph.end> 23.<lb></lb>M<emph type="italics"></emph>ech.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004034">Pro intellectu reliquorum ab eo dictorum, &amp; quorundam mira­<lb></lb>bilium, proponatur alius rhombus illi ęqualis, in tabula pictus deli<lb></lb>neatis lateribus &amp; diametris, qui fit l m o n, &amp; diametri l p o &amp; m p <lb></lb>n, &amp; abſcindatur hic ex ſuperficie, &amp; ſuperponatur ita, ut puncta l m <lb></lb>o n ordinatim cadant, &amp; aptentur <expan abbr="pũctis">punctis</expan> a b d c, &amp; p aptetur ipſi k. <lb></lb></s>
          <s id="id004035">Et tunc ſi rhombus l o totus moueretur, neceſſe eſt, ut moueatur ſe­<lb></lb>cundum latus aliquod, ut pote l m, &amp; ęquidiſtans a b, igitur dicetur <lb></lb><figure id="id.015.01.255.1.jpg" xlink:href="015/01/255/1.jpg"></figure><lb></lb>moueri ſuper latus aliquod, ſcilicet a c: atque hic eſt mo<lb></lb>tus, quem Ariſtoteles uocat <expan abbr="motũ">motum</expan> a b ſuper latus a c. <lb></lb></s>
          <s id="id004036">Si <expan abbr="aũt">aunt</expan> fingamus quieſcere latus aliquod l o, uel pars <lb></lb>lateris, non poſſet omnino moueri in ſuperficie a d <lb></lb>rhombi: et ita <expan abbr="nõ">non</expan> perinde eſſet ac ſi a d rhombus mo<lb></lb>ueretur, quod tamen ſupponit Ariſtoteles. </s>
          <s id="id004037">Neque <expan abbr="etiã">etiam</expan> <lb></lb>ſi quieſceret punctum aliud quam p haberet ratio­<lb></lb>nem motus regularis, quod ab illo ſupponitur: reli­<lb></lb>quum eſt igitur, ut rhombus l o moueatur uice rhombi a d ſeruan­<lb></lb>do centrum, id eſt punctum p in puncto k. </s>
          <s id="id004038">Dicamus ergo primum <lb></lb>de motu compoſito Ariſtotelis, &amp; pòſt de noſtro.</s>
        </p>
        <p type="main">
          <s id="id004039">Moueatur l m ſuper a c, æquidiſtans ſemper a b, ut ſeruet ſitum <lb></lb>quem habebat ita, quod <expan abbr="extremũ">extremum</expan> lineæ l m ſit ſemper in linea a c, &amp; <lb></lb>l punctum quod gerit uicem a, deſcendat tantum in linea l m, quan­<lb></lb>tum l extremum in linea a c: dicit Philoſophus, quod a ſeu l ſemper <lb></lb>deſcendet in linea a d, &amp; erit in e a. </s>
          <s id="id004040">Supponatur quae latus l m fit f g, &amp; <lb></lb>erit l n, f t, ducatur <expan abbr="aũt">aunt</expan> ex r puncto ſectionis diametri, &amp; lateris l m li <lb></lb><arrow.to.target n="marg790"></arrow.to.target><lb></lb>near q, æquidiſtans a f, <expan abbr="igit̃">igitur</expan> rhombus a q r f eſt ſimilis rhombo toti <lb></lb>a b d c, &amp; proportio a f ad fr, ut a c ad c d, ſed a c eſt ęqualis c d, <expan abbr="igit̃">igitur</expan> a f <lb></lb>eſt æqualis f r, ſed l deſcendit in l m, <expan abbr="quantũ">quantum</expan> eſt a f ex ſuppoſito, <expan abbr="igit̃">igitur</expan> <lb></lb><expan abbr="punctũ">punctum</expan> l ſemper erit in linea a d. </s>
          <s id="id004041">Poſt deficiunt quædam uerba: ob <lb></lb>quæ nemo intellexit ſententiam Philoſophi, &amp; <expan abbr="tamẽ">tamen</expan> auſi ſunt impo<lb></lb>nere lectoribus, tan&lt;08&gt; intellexiſſent, tres ſimul errores admittendo, <lb></lb>ſcilicet Ariſtotelem ob propriam ignorantiam, ut ſtultum accuſan­<lb></lb>do, qui falſa dicat, &amp; demonſtrare nitatur: produnt ſe ipſos cum <lb></lb>ſua impudentia. </s>
          <s id="id004042">Et lectoribus imponere conantur, debet ergo ſic <lb></lb>legi (“b in ipſa b c diametro latum, ubi latus b d moueatur in late­<lb></lb>re b a, &amp; b æqualiter uerſus d in b d, æqualis enim eſt ipſa b e”) <lb></lb>Tunc enim conſtat ut hic dixi, m moueri per b c rectam ut l per a d: <lb></lb>Dicit ergo <expan abbr="cũ">cum</expan> b d <expan abbr="moueat̃">moueatur</expan> in b a, tranſit unico motu <expan abbr="totã">totam</expan> b a, &amp; pun<pb pagenum="237" xlink:href="015/01/256.jpg"></pb><expan abbr="ctũ">ctum</expan> tamen b, quod <expan abbr="mouet̃">mouetur</expan> duobus motibus, non pertranſit niſi b c, <lb></lb>quæ poteſt eſſe minor b a: nam <expan abbr="cõſtat">conſtat</expan> quod <expan abbr="quãdo">quando</expan> m erit in a, o erit <lb></lb>in e, &amp; quia m deſcendit in o, in eodem tempore, ergo o erit in c, &amp; <lb></lb><expan abbr="trãſiuit">tranſiuit</expan> ſemper per rectam b c: igitur m eſt minus <expan abbr="motũ">motum</expan> duobus mo<lb></lb>tibus quàm m l unico <expan abbr="tantũ">tantum</expan>. </s>
          <s id="id004043">Et quia aliquis dicere potuiſſet non eſt <lb></lb>mirum, quod m ſit minus motum duobus motibus quàm l m latus <lb></lb>unico tantum: quia m mouetur motu contrario motui lateris: nam <lb></lb>latus m o mouetur in latere b a aſcendendo, et punctum m uerſus o <lb></lb>in ipſo m o deſcendendo. </s>
          <s id="id004044">Dicit Philoſophus, hoc eſt mirum, quia <lb></lb>cum idem contingat in motu l, cuius latus mouetur per a c, &amp; l per l <lb></lb>m recedendo in partem contrariam, nihilominus uelocius motum <lb></lb>eſt l, quàm latus l m, quia a d eſt longior a c. </s>
          <s id="id004045">Ex quo patet, quae quęſtio <lb></lb>Philoſophi eſt una tantum, &amp; non duæ. </s>
          <s id="id004046">Et eſt cur motum duobus <lb></lb>motibus in rhombo, in uno mouetur uelocius latere tantum moto <lb></lb>uno motu, in alio tardius? </s>
          <s id="id004047">Et quia aliquis dicere poſſet, q̊d b c poſ­<lb></lb>ſet eſſe <expan abbr="lõgior">longior</expan> a c: Dicit Philoſophus, uerum eſt, ſed ego poſſum in­<lb></lb>uenire talem rhombum, qui etiam habeat a c longiorem, &amp; tunc ni­<lb></lb>hilominus <expan abbr="ſequit̃">ſequitur</expan> quod dico. </s>
          <s id="id004048">Aliud <expan abbr="aũt">aunt</expan>, quod docet ex hac demon­<lb></lb>ſtratione, eſt quae ex duobus motibus rectis diuerſis poteſt fieri unus <lb></lb>motus rectus diuerſus: igitur idem punctum, puta formica poterit <lb></lb>ſimul, &amp; ſemel moueri duobus motibus rectis diuerſis. </s>
          <s id="id004049">Et hoc eſt, <lb></lb>quia primus motus eſt rectus ſolum ſecundum formam, &amp; non ſe­<lb></lb>cundum materiam: &amp; alter ſecundus, ſcilicet miſtus eſt ſecundum <lb></lb>materiam &amp; non ſecundum formam per rectam.</s>
        </p>
        <p type="margin">
          <s id="id004050"><margin.target id="marg790"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>ſexti <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004051">Ex hoc <expan abbr="ſequit̃">ſequitur</expan> aliud magis <expan abbr="mirũ">mirum</expan>, et eſt iuxta <expan abbr="noſtrũ">noſtrum</expan> motum rhom<lb></lb>bi l o in rhombo a d, fixo centro p in centro k, &amp; <expan abbr="moueat̃">moueatur</expan> quomodo <lb></lb>libet, l, dico quod l f ſemper æqualis erit a f, quia <expan abbr="em̃">emm</expan> k l &amp; k a ſunt æ­<lb></lb><figure id="id.015.01.256.1.jpg" xlink:href="015/01/256/1.jpg"></figure><lb></lb>quales, <expan abbr="cũ">cum</expan> eſſent una linea ante motum ducta, l a erit <lb></lb>angulus k l a, æqualis angulo k a l, ſed angulus k a c <lb></lb><arrow.to.target n="marg791"></arrow.to.target><lb></lb>eſt æqualis angulo k l m, cum angulus k l m eſſet <expan abbr="idẽ">idem</expan> <lb></lb>angulo k a b, &amp; angulus k a b eſt <expan abbr="æq̃lis">æqualis</expan> angulo k a c, <lb></lb><arrow.to.target n="marg792"></arrow.to.target><lb></lb>igitur angulus k l m eſt æqualis angulo k a c, <expan abbr="igit̃">igitur</expan> reſi<lb></lb>duus fl a eſt æqualis reſiduo f a l, quare f a æqualis <lb></lb><arrow.to.target n="marg793"></arrow.to.target><lb></lb>fl. </s>
          <s id="id004052">Si igitur quantum procedit latus m l in a c, <expan abbr="tãtum">tantum</expan> <lb></lb>deſcendat punctum in linea l m punctum perpetuo, erit in linea a c, <lb></lb>&amp; per eam mouebitur. </s>
          <s id="id004053">Vnde ſequitur quod</s>
        </p>
        <p type="margin">
          <s id="id004054"><margin.target id="marg791"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004055"><margin.target id="marg792"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 34. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004056"><margin.target id="marg793"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004057">Quod <expan abbr="punctũ">punctum</expan> l <expan abbr="mouebit̃">mouebitur</expan> duobus </s>
          <s id="id004058">motibus. </s>
          <s id="id004059">uno recto in linea, ſcilicet <lb></lb><arrow.to.target n="marg794"></arrow.to.target><lb></lb>l m, &amp; altero circulari. </s>
          <s id="id004060">ſ. </s>
          <s id="id004061">circa <expan abbr="centrũ">centrum</expan> k, &amp; <expan abbr="tñ">tnm</expan> <expan abbr="mouebit̃">mouebitur</expan> uerè motu re­<lb></lb>cto <expan abbr="tm̃">tmm</expan> in alia linea, ſcilicet a c, &amp; hoc eſt <expan abbr="primũ">primum</expan> admirabile. </s>
          <s id="id004062">Aliud eſt</s>
        </p>
        <p type="margin">
          <s id="id004063"><margin.target id="marg794"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004064">Quod <expan abbr="punctũ">punctum</expan> l <expan abbr="mouebit̃">mouebitur</expan> duobus motibus, &amp; per ipſos <expan abbr="mouebit̃">mouebitur</expan> <lb></lb><arrow.to.target n="marg795"></arrow.to.target><lb></lb>ad <expan abbr="unguẽ">unguem</expan> uno motu ęquali uni <expan abbr="eorũ">eorum</expan>, ita q̊d alius motus nihil addet 


<pb pagenum="238" xlink:href="015/01/257.jpg"></pb>nec minuet. </s>
          <s id="id004065">Patet quia mouebitur, gratia exempli, primo motu ex l <lb></lb>in f, &amp; pòſt motu circulari, &amp; uerè erit motum ex a in f, qui motus <lb></lb>eſt æqualis motui priori propriò, &amp; ſolo ex l in f.</s>
        </p>
        <p type="margin">
          <s id="id004066"><margin.target id="marg795"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004067">Propoſitio ducenteſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id004068">Proportionem agentium naturalium in tranſmutatione con­<lb></lb>ſyderare.<lb></lb><arrow.to.target n="marg796"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004069"><margin.target id="marg796"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004070">Sit latitudo a b ad conuerſionem terræ in aurum me­<lb></lb>dium perfectionis a b ſit c, &amp; medium a c d b, cuius dimi­<lb></lb>dium ſit e b. </s>
          <s id="id004071">Et fiat commutatio a c in f g, tempore dimi­<lb></lb>dium f g, g h in g h deberet peruenire ad perfectionem d, <lb></lb>quoniam ratio a c ad c d, ut f g ad g h. </s>
          <s id="id004072">At uerò dum tranſi­<lb></lb>ret terra ad perfectionem c tota reſiſtebat, iam adepta per­<lb></lb>fectione a c non reſiſtit, niſi pro medietate, at proportio cu<lb></lb>iuslibet quantitatis ad dimidium alterius producitur ex <lb></lb>proportione eadem &amp; dupla, dupla igitur eſt proportio <lb></lb>agentis ad imperfectionem a c ei quæ eſt ad a b, igitur in di<lb></lb>midio temporis g h acquiret perfectionem c d, &amp; ſit g k di<lb></lb>midium g h, erit ergo tempus totum fk, in quo acquiret <lb></lb>a d. </s>
          <s id="id004073">At ratio hæc conſtare non poteſt, nam ſi diuidatur ſpa<lb></lb><figure id="id.015.01.257.1.jpg" xlink:href="015/01/257/1.jpg"></figure><lb></lb>tium a b in trientes fient trientes duo, &amp; quarta pars in perfectione <lb></lb>a d: ſed iam multo citius acquiret quam in fk tempore, quod eſt di­<lb></lb>midium &amp; octaua pars. </s>
          <s id="id004074">Sed hoc non cogit, quoniam partes primæ <lb></lb>ſunt ſemper contumaciores, &amp; ut diſponuntur fiunt magis obedi­<lb></lb>entes, non iuxta proportionem ſimpliciter, ſed ut ſunt in materia, <lb></lb>&amp; ideò hæc actio eſt ſimilior proportioni exceſſus, &amp; eſt Arithme­<lb></lb>tica quam capacitatis ſcilicet Geometricæ.</s>
        </p>
        <p type="main">
          <s id="id004075"><arrow.to.target n="marg797"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004076"><margin.target id="marg797"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004077">Ex hoc patet, quod res quæ ad ſummam maturitatem perueni­<lb></lb>unt, maximè <expan abbr="acquirũt">acquirunt</expan> perfectionem in exiguo tempore, ut gemmę, <lb></lb>aurum, infans. </s>
          <s id="id004078">Ergo oportet maximè iuxta finem cauere, ne detur <lb></lb>occaſio ulla accelerandi partum.</s>
        </p>
        <p type="main">
          <s id="id004079">Propoſitio ducenteſima octaua.</s>
        </p>
        <p type="main">
          <s id="id004080">Mota res à centro grauitatis per priorem motum in reditu uelo­<lb></lb>cius mouetur, quam ſi quieuerit.</s>
        </p>
        <p type="main">
          <s id="id004081"><arrow.to.target n="marg798"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004082"><margin.target id="marg798"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004083">Sit a b c lectus penſilis, in quo ho <lb></lb>mo aut patera, in qua aqua uel <expan abbr="ui­nũ">ui­<lb></lb>num</expan>, &amp; ſit <expan abbr="cẽtrum">centrum</expan> grauitatis d, quod <lb></lb>neceſſariò eſt in linea loci, cui anne<lb></lb>xus eſt lectus a g, &amp; in patera lo ci <lb></lb>medij manus continentis pateram <lb></lb><expan abbr="cũ">cum</expan> centro quæ ſit a g, quibus ſtan­<lb></lb>tibus oſtendendum eſt primo.</s>
        </p>
        <figure id="id.015.01.257.2.jpg" xlink:href="015/01/257/2.jpg"></figure>
        <pb pagenum="239" xlink:href="015/01/258.jpg"></pb>
        <p type="head">
          <s id="id004084">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id004085">Omne graue <expan abbr="motũ">motum</expan> à centro grauitatis, reſtituto ad eundem ſitum <lb></lb>pondere mobili aut inmobili, continente ultra centrum grauitatis <lb></lb>naturalis uiolenter fertur.</s>
        </p>
        <p type="main">
          <s id="id004086">Seu ſit pondus per ſe non fluctuans in penſili lecto, ſeu humor in </s>
        </p>
        <p type="main">
          <s id="id004087"><arrow.to.target n="marg799"></arrow.to.target><lb></lb>patera, quum <expan abbr="põdus">pondus</expan> moueatur ſolum ratione una, ſcilicet lecti pen­<lb></lb>ſilis homo uel plumbum, humor autem aqua uel uinum bifariam <lb></lb>&amp; ratione pateræ ſi mobilis ſit in a laxa manu, &amp; etiam per humo­<lb></lb>rem ipſum redeuntem ad locum <expan abbr="ſuũ">ſuum</expan>: adeò quòd ſi eſſet &amp; immobi­<lb></lb>lis patera, humor ſaltem reflueret propria inundatione ad locum <lb></lb>ſuum centri grauitatis, licet in patera eſſet immobilis locus grauita­<lb></lb>tis uelocius &amp; maiore cum impetu, adeò ut tranſeat uerſus e, <expan abbr="cũ">cum</expan> fu <lb></lb>erit motus primus ex e in f, et reſtitutio ex fin e: ſeu in immobili pon<lb></lb>dere mobilis continenti, ut in lecto penſili: ſeu in immobili conti­<lb></lb>nente, ſcilicet poſtquàm ad locum ſuum reſtitutum fuerit per uim <lb></lb>retenta patera à manu iuxta ſitum priorem in a, mobili autem con­<lb></lb>tento, id eſt, humore, multo autem magis contento, &amp; continente <lb></lb>mobilibus. </s>
          <s id="id004088">Vt ſi patera &amp; humor ipſe ſimul <expan abbr="moueãtur">moueantur</expan>, nam &amp; pate<lb></lb>ra tranſgredietur locum ſuum, &amp; humor duplici motu ſuperau­<lb></lb><arrow.to.target n="marg800"></arrow.to.target><lb></lb>ctus tranſgredietur motum naturalem. </s>
          <s id="id004089">Cum enim a d eſt remotum <lb></lb>a g, &amp; eſt in f, mouetur maiore impetu, quam ſit pro ratione pon­<lb></lb>deris, ut demonſtratum eſt, igitur tranſibit ad e, cum ergo redeat <lb></lb>ad g motu naturali, neceſſe eſt ut motus uiolentus ſit ualidior ea <lb></lb>parte naturalis, qua d reſiſtit, dum eſt in g, ne dimoueatur à g, ſi igi­<lb></lb>tur tractum ad c, ſuperauit uim qua manet in g, in eo quod moue­<lb></lb>tur ad f, igitur in reditu mouebitur tantum ultra g uerſus e, quan­<lb></lb>tum eſt acquiſitum ex ui tranſitus ultra g uerſus f, quanto ergo ma­<lb></lb>ior eſt arcus e d, tanto maior eſt d f, &amp; quanto maior eſt arcus d f, <lb></lb>tanto maior d h.</s>
        </p>
        <p type="margin">
          <s id="id004090"><margin.target id="marg799"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id004091"><margin.target id="marg800"></margin.target>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 30.</s>
        </p>
        <p type="main">
          <s id="id004092">Ex quo patet, quod quanto magis remouetur d à g, tanto maio­<lb></lb><arrow.to.target n="marg801"></arrow.to.target><lb></lb>re impetu fertur uerſus extremum aliud &amp; ultra medium.</s>
        </p>
        <p type="margin">
          <s id="id004093"><margin.target id="marg801"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="head">
          <s id="id004094">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id004095">Omne pondus appenſum eſt graue comparatione medij graui­<lb></lb>tatis, ad hoc ut ab eo remoueatur, quantum eſt pro ratione anguli <lb></lb>ex quo appenſum eſt.</s>
        </p>
        <p type="main">
          <s id="id004096">Sit d appenſum in a &amp; in b, &amp; ſit angulus c b d, triplus angu­<lb></lb><arrow.to.target n="marg802"></arrow.to.target><lb></lb>lo c a d, dico quod tripla eſt uis quæ transfert d in c ex b, ei quæ <lb></lb>transfert ex a, quoniam enim mixtus eſt in b &amp; a, igitur a d æqua­<lb></lb><arrow.to.target n="marg803"></arrow.to.target><lb></lb>lia ſpatia æquales uires exigentur: igitur uirium proportio ut <lb></lb>angulorum, at quanto maior eſt a d in proportione ab b d tanto <lb></lb>maior eſt proportio anguli c b d ad <expan abbr="angulũ">angulum</expan> c a d, igitur quanto ma­


<pb pagenum="240" xlink:href="015/01/259.jpg"></pb><figure id="id.015.01.259.1.jpg" xlink:href="015/01/259/1.jpg"></figure><lb></lb>ior eſt a d tanto facilius remouet ęquali ſpa<lb></lb><arrow.to.target n="marg804"></arrow.to.target><lb></lb>tio d uerſus e. </s>
          <s id="id004097">Et licet remoueantur ab ipſo <lb></lb>d, ſemper eadem proportio manebit, ma­<lb></lb><arrow.to.target n="marg805"></arrow.to.target><lb></lb>nente eadem longitudine b d &amp; a d, nam <lb></lb><arrow.to.target n="marg806"></arrow.to.target><lb></lb>proportio d f ad d c, eſt uelut f b d ad <lb></lb>c b d, &amp; ut d f ad d e, ita f a d ad c a d, quare <lb></lb>fb d ad c b d, uelut f a d ad c a d, quare fb d <lb></lb>ad f a d, ut c b d ad c a d, quod fuit pro­<lb></lb>poſitum.</s>
        </p>
        <p type="margin">
          <s id="id004098"><margin.target id="marg802"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id004099"><margin.target id="marg803"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004100"><margin.target id="marg804"></margin.target>P<emph type="italics"></emph>er ult.  &gt;ſex­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004101"><margin.target id="marg805"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>quin <lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004102"><margin.target id="marg806"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>eiuſ <lb></lb>dem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="id004103">LEMMA TERTIVM.</s>
        </p>
        <p type="main">
          <s id="id004104">Grauitatem ponderis appenſi aut fluidi <lb></lb>in comparatione ad remotionem à centro <lb></lb>grauitatis inuenire.<lb></lb><arrow.to.target n="marg807"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004105"><margin.target id="marg807"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004106">Nam cum d trahetur per planum ut ſuſpenſum, &amp; non tractum </s>
        </p>
        <p type="main">
          <s id="id004107"><arrow.to.target n="marg808"></arrow.to.target><lb></lb>a d, erit dimidium ponderis appenſi, igitur ex lemmate ſecundo, pa<lb></lb>tebit proportio laboris in remouendo d à loco proprio in quan­<lb></lb>cunque partem &amp; diſtantiam, &amp; in quouis loco ſit appenſum.</s>
        </p>
        <p type="margin">
          <s id="id004108"><margin.target id="marg808"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>hu­<lb></lb>ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004109">Ex hoc ſequitur, quod poterit annulus tam altè appendi, ut iuxta <lb></lb><arrow.to.target n="marg809"></arrow.to.target><lb></lb>proportionem angulí &amp; leuitatem propriam cum filo tenuiſsimo, <lb></lb>&amp; ut fuerit latus, &amp; poſitus è regione oris, ut ex ſermone circum­<lb></lb>agatur quaqua uerſus, &amp; percutiat labra uaſis aqua pleni fermè, ut <lb></lb>uideatur plane reſponſa dare.</s>
        </p>
        <p type="margin">
          <s id="id004110"><margin.target id="marg809"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="head">
          <s id="id004111">LEMMA QVARTVM.</s>
        </p>
        <p type="main">
          <s id="id004112">Quanto magis remotum fuerit pondus ex eodem centro à recta <lb></lb>linea, tanto maiore impetu agetur, ut ultra locum medium feratur <lb></lb>non æquali, ſed producta proportione.</s>
        </p>
        <p type="main">
          <s id="id004113">Sit a b, &amp; ut dictum eſt, non eſt ei pondus, niſi quatenus remoue­<lb></lb><arrow.to.target n="marg810"></arrow.to.target><lb></lb>tur a recta, &amp; in c ſummam habeat grauitatem, &amp; d ſit medium b c, <lb></lb><figure id="id.015.01.259.2.jpg" xlink:href="015/01/259/2.jpg"></figure><lb></lb>dico ergo quod multo maiore impetu feretur ex cin <lb></lb>b quam ex d, nam cum c ſit ſumma grauitas, erit ſal­<lb></lb>tem dupla grauitati d, ſed d grauitas eſt penè infinita, <lb></lb>ut demonſtratum eſt in comparatione ad b, ut iuxta <lb></lb>ſitum remotionis à linea b, cum ergo proportio ſin­<lb></lb><arrow.to.target n="marg811"></arrow.to.target><lb></lb>gularum partium c d ad ſingulas d b medietate b c diſtantes ſit ma­<lb></lb><figure id="id.015.01.259.3.jpg" xlink:href="015/01/259/3.jpg"></figure><lb></lb>ior dupla augendo, erit proportio c d ad d b, uelut pro­<lb></lb>poſita h k dupla g f, &amp; h e dupla e f, e k h ad e g f quadru­<lb></lb>pla, igitur &amp; eo maior quo acquiſitus eſt impetus ex de­<lb></lb>monſtratis, quare proportio motus &amp; impetus ex c in <lb></lb><arrow.to.target n="marg812"></arrow.to.target><lb></lb>b, eſt multo maior impetu ex d in b quadrupla pro­<lb></lb>portione.</s>
        </p>
        <pb pagenum="241" xlink:href="015/01/260.jpg"></pb>
        <p type="margin">
          <s id="id004114"><margin.target id="marg810"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id004115"><margin.target id="marg811"></margin.target>L<emph type="italics"></emph>emmate<emph.end type="italics"></emph.end> 2.</s>
        </p>
        <p type="margin">
          <s id="id004116"><margin.target id="marg812"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 30. <emph type="italics"></emph>hu<lb></lb>ius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004117">Ex his omnibus concluditur propoſitum in prima figura, &amp; eſt <lb></lb><arrow.to.target n="marg813"></arrow.to.target><lb></lb>quod ſi b c inclinetur uerſus e, mouebitur a d, certo impetu uerſus <lb></lb>e. </s>
          <s id="id004118">Et quia ſi prius b c inclinatum fuerit in f, redit a d, dum b c reuer­<lb></lb>titur ad proprium ſitum ultra lineam a d g uſque ad h per primum <lb></lb>lemma. </s>
          <s id="id004119">Et cum b c inclinatur ad b f peruenit, quantum b c inclina­<lb></lb>ta ad f, ſcilicet ad e, igitur ex motibus b c in f &amp; in e tanto plus mo­<lb></lb>uetur d ultra e, quantum eſt productum d e in d h, ‘ideo multo plus <lb></lb>quam ſi ſolum motum fuiſſet d ex recta a g, etiam quod non moue­<lb></lb>retur b c. </s>
          <s id="id004120">Multo plus ergo moto etiam b c, ut diximus.</s>
        </p>
        <p type="margin">
          <s id="id004121"><margin.target id="marg813"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004122">Propoſitio ducenteſima nona.</s>
        </p>
        <p type="main">
          <s id="id004123">Si ſuperficies rectangula in duas partes æquales diuiſa intelli­<lb></lb>gatur, quæ ambę quadratæ ſint, itemque in duas inæquales, erit pa­<lb></lb>rallelipedum ex latere mediæ partis in totum ſuperficiem maius ag<lb></lb><figure id="id.015.01.260.1.jpg" xlink:href="015/01/260/1.jpg"></figure><lb></lb>gregato parallelipedorum ex par­<lb></lb>tibus inæqualibus, in latera alte­<lb></lb>rius partis mutuo in eo, quod fit <lb></lb>ex differentia lateris minoris par­<lb></lb>tis a mediæ latere in differentiam <lb></lb>maioris partis ſuperficiei à media <lb></lb>ſuperficie bis, &amp; ex differentia am­<lb></lb>borum laterum inæqualium iun­<lb></lb>ctorum ad ambo latera æqualia <lb></lb>iuncta in minorem partem ſuperficiei.</s>
        </p>
        <p type="main">
          <s id="id004124">Proponatur a g diuiſa in duo quadrata æqualia a h, h b, &amp; late­<lb></lb><arrow.to.target n="marg814"></arrow.to.target><lb></lb>ra erunt a c, c b, &amp; in duo inæqualia a d d g, quarum latera ſint b c, <lb></lb>a f, dico quod parallelipeda a c in c g, &amp; c b in c k, &amp; ſunt æqualia pa<lb></lb>rallelipedo ex a c in a g, excedunt <lb></lb><figure id="id.015.01.260.2.jpg" xlink:href="015/01/260/2.jpg"></figure><arrow.to.target n="table30"></arrow.to.target><lb></lb>parallelipeda ex a f in d g, &amp; b c <lb></lb>in d k, in duplo f c in d h, cum eo <lb></lb>quod fit ex f e in d k ſemel. </s>
          <s id="id004125">Quia <lb></lb>ergo parallelipedum ex a e in a g <lb></lb>eſt æquale parallelipedis a f &amp; f c <lb></lb>in a h, h d, h k, quare parallelipe­<lb></lb>dis a f in a h, h d, d k, &amp; f c in d k, &amp; <lb></lb>c e in d k, &amp; f e in d k, &amp; f e in d h <lb></lb>bis. </s>
          <s id="id004126">Ad parallelipedum a fin d g, <lb></lb>eſt æquale parallelipedis a fin a h, h d. </s>
          <s id="id004127">Et parallelipedum b e in d k, <lb></lb>parallelipedis a f, f e, c e in d k. </s>
          <s id="id004128">Detractis ſimilibus relinquetur f c in <lb></lb>d l, l e, e h bis, quod eſt f c in d h bis, cum eo quod fit ex e f in d k ſi­<lb></lb>mul, quod eſt propoſitum.</s>
        </p>
        <pb pagenum="242" xlink:href="015/01/261.jpg"></pb>
        <p type="margin">
          <s id="id004129"><margin.target id="marg814"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <table>
          <table.target id="table30"></table.target>
          <row>
            <cell>1 a f in a h</cell>
            <cell>f c in a h bis</cell>
          </row>
          <row>
            <cell>2 a f in h d</cell>
            <cell>f e in d k</cell>
          </row>
          <row>
            <cell>3 a f in d k</cell>
            <cell></cell>
          </row>
          <row>
            <cell>4 f c in d k</cell>
            <cell></cell>
          </row>
          <row>
            <cell>5 c e in d k</cell>
            <cell></cell>
          </row>
          <row>
            <cell>1 a f in a h</cell>
            <cell>4 f c in d k</cell>
          </row>
          <row>
            <cell>2 a f in d h</cell>
            <cell>5 c e in d k</cell>
          </row>
          <row>
            <cell>3 a f in d k</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="id004130">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id004131">Dico etiam, quòd duæ lineæ b e &amp; af ſunt minores duabus a c, <lb></lb>c b ſimul iunctis, nam quia d b, e b, c b, ſunt in eadem proportione, <lb></lb>&amp; d b eſt maior e b, erit maior differentia d b ad e b, quam e b ad <lb></lb><arrow.to.target n="marg815"></arrow.to.target><lb></lb>c b, igitur maior d e quam e c, quare e c eſt minor medietate d c, &amp; <lb></lb>ideo multo minor medietate a c. </s>
          <s id="id004132">Et ſimiliter, quia a c eſt maior af, &amp; <lb></lb>a c, a f, a d ſunt in continua proportione, maior erit c f quam <lb></lb>fd, &amp; ideò conſtat quamuis longum eſſet, ſi quis uellet demon­<lb></lb>ſtrare perfectè, quod b e &amp; a f iunctæ ſunt minores tota a b ſeu du­<lb></lb>plo a c.</s>
        </p>
        <p type="margin">
          <s id="id004133"><margin.target id="marg815"></margin.target>P<emph type="italics"></emph>er conuer­<lb></lb>ſam quaſi<emph.end type="italics"></emph.end> 8. <lb></lb><emph type="italics"></emph>quinti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004134">Exemplum, ſint h b &amp; h a 25, &amp; a e, c b 5, producta mutua 250, <lb></lb>ſitqúe g d 49, &amp; erit b e 7, ſit autem d k 1, &amp; erit a f 1, quia ergo a f <lb></lb>eſt 1, a e 5, erit f c 4, &amp; quia e b eſt 7, &amp; b c 5, erit e c 2, quare etiam ef2, <lb></lb>productum ergo ex e b in d k eſt 7, &amp; ex a f in d g 49, totum ag­<lb></lb>gregatum 56, differentia a 250, eſt 194, qui ſit ex duplo fc, quod <lb></lb>eſt 8 in d h, quæ eſt 24, &amp; fit 192, &amp; ex fe, quæ eſt 2, in d k, quæ eſt 1, <lb></lb>&amp; fit: quod additum ad 192 facit 194. Similiter capio 450, cuius di­<lb></lb>midium eſt 225, c g &amp; c k 225, &amp; c a &amp; c b 15 ſingulæ. </s>
          <s id="id004135">Et ponatur <lb></lb>d g 441, eritqúe e b 21, &amp; d k 9, &amp; erit a f 3, igitur cum b e ſit 21, <lb></lb>&amp; b c 15, erit c e 6, a f uerò eſt 3, igitur f e eſt 6. Producta mu­<lb></lb>tua æqualia 6750, inæqualia 1521, differentia 5238, quia er­<lb></lb>go f c eſt 12, duplum eius eſt 24, ductum in d h, quæ eſt <lb></lb>216, nam d k ex ſuppoſito eſt 9, fiet ergo 5184, cui ſi addam, quod <lb></lb>fit ex f e, quæ eſt 6, in d k, quæ eſt 9, fitqúe 54, erit totum 5238, quod <lb></lb>erat propoſitum.<lb></lb><arrow.to.target n="marg816"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004136"><margin.target id="marg816"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004137">Ex hac demonſtratione liquet, quod ſi linea in duas partes æ­<lb></lb>quales diuidatur, &amp; duas inæquales, quòd parallelipeda æqua­<lb></lb>lium ſectionum pariter accepta excedent parallelipeda inæqua­<lb></lb>lium ſectionum, ſimul iuncta in eo quod fit ex tota linea in quadra­<lb></lb>tum differentiæ partium æqualium ab inæ qualibus.</s>
        </p>
        <p type="main">
          <s id="id004138">Propoſitio ducenteſima decima.</s>
        </p>
        <p type="main">
          <s id="id004139">Si duæ lineæ ad æquales angulos ab eodem puncto peripheriæ <lb></lb>circuli reflectantur, neceſſe eſt angulos cum dimetiente factos æ­<lb></lb>quales eſſe. </s>
          <s id="id004140">Vnde manifeſtum eſt protractam diametrum angu­<lb></lb>lum ſuppoſitum per æqualia diuidere.</s>
        </p>
        <p type="main">
          <s id="id004141"><arrow.to.target n="marg817"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004142"><margin.target id="marg817"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004143">Reſiliat radius d b c ad æquales angulos, ut fert natura rerum 


<pb pagenum="243" xlink:href="015/01/262.jpg"></pb>dum à plano reſilit (licet refragante Plutarcho) ita ut anguli c b e, &amp; <lb></lb>d b f ſint æquales, dico angulos ibidem d b a, &amp; c b a æquales eſſe: <lb></lb><figure id="id.015.01.262.1.jpg" xlink:href="015/01/262/1.jpg"></figure><lb></lb>&amp; quod ſi trahatur latus a b uſque ad g, quod anguli d b <lb></lb>g &amp; c b g etiam erunt ęquales. </s>
          <s id="id004144">Primum patet, quia an­<lb></lb>guli a b e &amp; a b c &amp; a b f æquales ſunt, ſunt enim reſi­<lb></lb>dui ad angulos contactus eiuſdem circuli &amp; rectæ, igi<lb></lb>tur additis æqualibus ex ſuppoſito c b e, d b f erunt </s>
        </p>
        <p type="main">
          <s id="id004145"><arrow.to.target n="marg818"></arrow.to.target><lb></lb>per communem animi ſententiam a b c &amp; a b d æqua­<lb></lb>les. </s>
          <s id="id004146">Secundum, cum ſint a b c &amp; a b d æquales, &amp; duo <lb></lb>anguli a b c, c b g æquales duobus rectis: itemque a b d, <lb></lb>d b g duobus rectis æquales: Et omnes recti inuicem æquales ex <lb></lb><arrow.to.target n="marg819"></arrow.to.target><lb></lb>petitione Euclidis erunt per communem animi ſententiam, æqua­<lb></lb>les reſidui quoque c b g &amp; d b g.</s>
        </p>
        <p type="margin">
          <s id="id004147"><margin.target id="marg818"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004148"><margin.target id="marg819"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004149">Ex hoc patet, eam quæ reſilit lineam ſemper ultra lineam à cen­<lb></lb><arrow.to.target n="marg820"></arrow.to.target><lb></lb>tro ad punctum, ex quo reſilit ductam ferri.</s>
        </p>
        <p type="margin">
          <s id="id004150"><margin.target id="marg820"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004151">Conſtat quia linea ex centro diuidit angulum per æqualia, ergo <lb></lb><arrow.to.target n="marg821"></arrow.to.target><lb></lb>cadit media inter illa quæ incidit, &amp; quæ reſilit.</s>
        </p>
        <p type="margin">
          <s id="id004152"><margin.target id="marg821"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004153">Ex hac etiam patet, quòd conſtituto angulo in cen­<lb></lb><arrow.to.target n="marg822"></arrow.to.target><lb></lb>tro a b c, &amp; ducta linea a d à puncto a, ſciemus quo reſi­<lb></lb>lit in linea b c: ducta enim c d, faciemus angulum c d e <lb></lb><arrow.to.target n="marg823"></arrow.to.target><lb></lb>æqualem a b c, &amp; erit angulus a d g æqualis angulo e d <lb></lb>h, igitur d e reſilit ex a b a d linea.</s>
        </p>
        <p type="margin">
          <s id="id004154"><margin.target id="marg822"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>m. </s>
          <s id="id004155">2.</s>
        </p>
        <p type="margin">
          <s id="id004156"><margin.target id="marg823"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <figure id="id.015.01.262.2.jpg" xlink:href="015/01/262/2.jpg"></figure>
        <p type="main">
          <s id="id004157">Propoſitio ducenteſima undecima.</s>
        </p>
        <p type="main">
          <s id="id004158">Si duæ lineæ ex duobus punctis peripheriam contingentes in <lb></lb>eandem partem protrahantur, ſemper magis diſtabunt inuicem ea <lb></lb>ex parte, &amp; nunquam concurrent.</s>
        </p>
        <figure id="id.015.01.262.3.jpg" xlink:href="015/01/262/3.jpg"></figure>
        <p type="main">
          <s id="id004159">Duæ ſemidiametri a b, a c ex terminis earum <lb></lb><arrow.to.target n="marg824"></arrow.to.target><lb></lb>duæ contingentes b f, c e, dico quod quanto <lb></lb>magis protrahentur in partem e f, tantò magis <lb></lb>diſtabunt, nunquàm concurrent: Nam angu­<lb></lb>lus a c g rectus eſt: angulus uerò c a d, ſi ſit re­<lb></lb><arrow.to.target n="marg825"></arrow.to.target><lb></lb>ctus e g, nun&lt;08&gt; concurret cum a d, æquidiſta­<lb></lb>bit enim ei: ſin aut ſit maior recto aut ex altera <lb></lb><arrow.to.target n="marg826"></arrow.to.target><lb></lb>parte erit minor, &amp; ita concurret, ergo in alte­<lb></lb><arrow.to.target n="marg827"></arrow.to.target><lb></lb>ram partem ductæ nunquàm concurrent, ſed perpetuo magis di­<lb></lb>ſtabunt. </s>
          <s id="id004160">Si ergo minor recto ſit angulus c a b, igitur e c ex eadem <lb></lb><arrow.to.target n="marg828"></arrow.to.target><lb></lb>parte concurret cum a d: concurrat ergo in g: &amp; quia e g cadit ex­<lb></lb><arrow.to.target n="marg829"></arrow.to.target><lb></lb>tra circulum, igitur diuidet b f, quæ tangit circulum. </s>
          <s id="id004161">Sit ergo ut di­


<pb pagenum="244" xlink:href="015/01/263.jpg"></pb>uidat in h, igitur h e &amp; h f cùm angulum conſtituant, quanto magis <lb></lb>protrahentur eo magis diſtabunt, nec unquam concurrent.</s>
        </p>
        <p type="margin">
          <s id="id004162"><margin.target id="marg824"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id004163"><margin.target id="marg825"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 29. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004164"><margin.target id="marg826"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004165"><margin.target id="marg827"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. &amp; 4. <lb></lb><emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004166"><margin.target id="marg828"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>petit.<emph.end type="italics"></emph.end><lb></lb> E<emph type="italics"></emph>uclid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004167"><margin.target id="marg829"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004168">Propoſitio ducenteſima duodecima.</s>
        </p>
        <p type="main">
          <s id="id004169">Si ab eodem puncto ad circuli peripheriam, lineæ quotuis du­<lb></lb>cantur, tres inuenire lineas, quæ <expan abbr="nõ">non</expan> in alium punctum reflectentur.<lb></lb><arrow.to.target n="marg830"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004170"><margin.target id="marg830"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004171">Quouis conſtituto puncto ueluti a extra circu<lb></lb>lum b c d, dico poſſe trahi tres lineas ad ipſam cir­<lb></lb>culi peripheriam, uelut a b, a c, a d, quæ ad alium <lb></lb>punctum non reflectentur. </s>
          <s id="id004172">Ducantur ergo a e ad </s>
        </p>
        <p type="main">
          <s id="id004173"><arrow.to.target n="marg831"></arrow.to.target><lb></lb>centrum, &amp; a b &amp; a d ad contingentes illius peri­<lb></lb>pheriam, quas conſtat non reflecti ſed progredi, <lb></lb><arrow.to.target n="marg832"></arrow.to.target><lb></lb>a c autem reflectitur in ſe ipſam per demonſtrata <lb></lb><arrow.to.target n="marg833"></arrow.to.target><lb></lb>ſuperius, igitur conſtat propoſitum.<lb></lb><figure id="id.015.01.263.1.jpg" xlink:href="015/01/263/1.jpg"></figure><lb></lb><arrow.to.target n="marg834"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004174"><margin.target id="marg831"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 17. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004175"><margin.target id="marg832"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 61. <emph type="italics"></emph>ter <lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004176"><margin.target id="marg833"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 210.</s>
        </p>
        <p type="margin">
          <s id="id004177"><margin.target id="marg834"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>m. </s>
          <s id="id004178">1.</s>
        </p>
        <p type="main">
          <s id="id004179">Ex hoc patet, quod omnia puncta ſub linea <lb></lb>contingente poſſunt reflecti ad ipſum per arcum <lb></lb>interceptum à contingente, &amp; ea quæ ad centrum.</s>
        </p>
        <p type="main">
          <s id="id004180"><arrow.to.target n="marg835"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004181"><margin.target id="marg835"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004182">Id eſt, quod omnia puncta infra lineam a b f ductam quantum­<lb></lb>libet poſſunt reflecti per arcum b c ad punctum a æqualibus an­<lb></lb>gulis. </s>
          <s id="id004183">Quoniam ex a per c b reflectuntur ad quælibet puncta infra <lb></lb>a b f, eo quòd termini ſunt punctum a, per ea quæ ſunt hic demon­<lb></lb>ſtrata, &amp; a b f, ipſa ergo ſi extrema in extremis, media in medijs con­<lb></lb>tinentur per regulam illam Dialecticam: igitur omnia puncta ſub <lb></lb>a b f etiam in infinitum producta continentur in reflexione à pun­<lb></lb>cto a per arcum b c.</s>
        </p>
        <p type="main">
          <s id="id004184"><arrow.to.target n="marg836"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004185"><margin.target id="marg836"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004186">Et rurſus, ſi à circulo ad circulum extremæ ducantur, nec illæ re­<lb></lb>flectentur, ſed tranſibunt: mediæ autem omnes reflecti poterunt à <lb></lb>quouis puncto.</s>
        </p>
        <figure id="id.015.01.263.2.jpg" xlink:href="015/01/263/2.jpg"></figure>
        <p type="main">
          <s id="id004187">Quia ſi a b ſit Sol, c d Luna, Sole <lb></lb>minor extremum in utroque lumina­<lb></lb>ri a c, b d quæ contingant utrunque <lb></lb>circulum, quod facile fiat, ductis a c <lb></lb>&amp; b d ex punctis non oppoſitis, æ­<lb></lb>quidiſtarent enim, ſed iuxta quan­<lb></lb>titatem dimetientis minoris. </s>
          <s id="id004188">Erit er­<lb></lb>go ut h e non reflectantur, aliæ o­<lb></lb>mnes mediæ reflectentur per demonſtrata à quolibet puncto, ergo <lb></lb>idem de totis circulis &amp; punctis.</s>
        </p>
        <p type="head">
          <s id="id004189">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id004190">Propoſitis duobus circulis lineam ambos <expan abbr="cõtingentem">contingentem</expan> ducere.</s>
        </p>
        <pb pagenum="245" xlink:href="015/01/264.jpg"></pb>
        <p type="main">
          <s id="id004191">Propoſitorum circulorum a &amp; b centra iungam recta a b, ſuper </s>
        </p>
        <p type="main">
          <s id="id004192"><arrow.to.target n="marg837"></arrow.to.target><lb></lb>quam ut ſemidiametrum deſcribo circulum b c, &amp; ex puncto a ad <lb></lb><arrow.to.target n="marg838"></arrow.to.target><lb></lb>perpendiculum a d, ex quo abſcindo æqualem ſemidiametro b e li­<lb></lb><arrow.to.target n="marg839"></arrow.to.target><lb></lb><figure id="id.015.01.264.1.jpg" xlink:href="015/01/264/1.jpg"></figure><lb></lb>neam d f, ex f duco a d perpendi­<lb></lb>culum f g, ex g in a duco a g, &amp; æ­<lb></lb>qualem angulo g a d, b a h abſcin <lb></lb>do h k <expan abbr="ęqualẽ">ęqualem</expan> d f ſeu b e, duco <expan abbr="aũt">aunt</expan> <lb></lb><arrow.to.target n="marg840"></arrow.to.target><lb></lb>b e, ut ſit <expan abbr="æquidiſtãs">æquidiſtans</expan> h k, duco h e, <lb></lb><arrow.to.target n="marg841"></arrow.to.target><lb></lb><expan abbr="quã">quam</expan> dico contangere utrunque <expan abbr="cir­culũ">cir­<lb></lb>culum</expan> b k: produco b k, &amp; quia duæ <lb></lb>lineæ b a &amp; a k ſunt ęquales duo­<lb></lb>bus lineis a g &amp; a f, duæ enim <lb></lb>prodeunt ab eodem centro, reli­<lb></lb>quæ ſunt reſidua æqualium d f &amp; h k, &amp; angulus b a k æqualis <lb></lb><arrow.to.target n="marg842"></arrow.to.target><lb></lb>g a f, ex ſuppoſito erit angulus g f a æqualis angulo b k a, g f a au­<lb></lb>tem rectus fuit, quia g f ad perpendiculum erecta fuit, itaque b k a <lb></lb>rectus eſt, &amp; ideo b k h rectus, quare <expan abbr="cũ">cum</expan> b e &amp; k h ſint æquales, &amp; æ­<lb></lb><arrow.to.target n="marg843"></arrow.to.target><lb></lb>quidiſtantes, erit angulus e oppoſitus b h k rectus, igitur duo angu<lb></lb>li e b k &amp; e h k duobus rectis æquales, quare cum ſint æquales inui<lb></lb><arrow.to.target n="marg844"></arrow.to.target><lb></lb>cem, quia oppoſiti in parallelogrammo uterque eorum rectus erit. <lb></lb><arrow.to.target n="marg845"></arrow.to.target><lb></lb>Recti ergo ſunt anguli e &amp; h, &amp; lineæ b e &amp; a h ex centris circulo­<lb></lb>rum, &amp; angulos Illos conſtituit lineæ e h, igitur e h contangit u­<lb></lb><arrow.to.target n="marg846"></arrow.to.target><lb></lb>trunque circulum.</s>
        </p>
        <p type="margin">
          <s id="id004193"><margin.target id="marg837"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="margin">
          <s id="id004194"><margin.target id="marg838"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>primi<emph.end type="italics"></emph.end><lb></lb> E<emph type="italics"></emph>lement.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004195"><margin.target id="marg839"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004196"><margin.target id="marg840"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004197"><margin.target id="marg841"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 31. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004198"><margin.target id="marg842"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004199"><margin.target id="marg843"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004200"><margin.target id="marg844"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 33. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004201"><margin.target id="marg845"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>pri <lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004202"><margin.target id="marg846"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004203">Propoſitio ducenteſima tertia decima.</s>
        </p>
        <p type="main">
          <s id="id004204">Propoſito circulo atque in eius peripheria puncto ſignato lineas <lb></lb>contingentes ultra citraque, &amp; etiam ab ipſomet deducere.</s>
        </p>
        <figure id="id.015.01.264.2.jpg" xlink:href="015/01/264/2.jpg"></figure>
        <p type="main">
          <s id="id004205">Sit circulus b c d, &amp; in eius peripheria c <lb></lb><arrow.to.target n="marg847"></arrow.to.target><lb></lb>punctum deſcriptum, &amp; ſumatur b d por­<lb></lb>tio minor quadrante, in qua punctum c, &amp; <lb></lb>ducantur a b, a c, &amp; ducantur b e, c f, d g, ad <lb></lb><arrow.to.target n="marg848"></arrow.to.target><lb></lb>perpendiculum, &amp; conſtat propoſitum, &amp; <lb></lb>quod nunquam ex eadem parte conuenient <lb></lb><arrow.to.target n="marg849"></arrow.to.target><lb></lb>ex eadem parte ex demonſtratis ſuprà.</s>
        </p>
        <p type="margin">
          <s id="id004206"><margin.target id="marg847"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="margin">
          <s id="id004207"><margin.target id="marg848"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004208"><margin.target id="marg849"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 221.</s>
        </p>
        <p type="main">
          <s id="id004209">Propoſitio ducenteſima quarta decima.</s>
        </p>
        <p type="main">
          <s id="id004210">Si extra circulum duo puncta ęqualiter à centro diſtantia ſignen<lb></lb>tur, erit punctum reflexionis æqualis, in medio arcus intercepti in­<lb></lb>ter lineas, quæ à centro ducuntur ad illa puncta. </s>
          <s id="id004211">Si uerò unum cen<lb></lb>tro proximius fuerit altero punctum æqualitatis in peripheria, tan<lb></lb>to longius uerſus breuiorem lineam, quanto punctum aliud à cen­<lb></lb>tro magis diſteterit.


<pb pagenum="246" xlink:href="015/01/265.jpg"></pb><arrow.to.target n="marg850"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004212"><margin.target id="marg850"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>_{m}.</s>
        </p>
        <p type="main">
          <s id="id004213">Sint puncta b c, æqualiter diſtantia à cen</s>
        </p>
        <p type="main">
          <s id="id004214"><arrow.to.target n="marg851"></arrow.to.target><lb></lb>tro a circuli d e, &amp; reflectantur c f, b f, dico f <lb></lb><arrow.to.target n="marg852"></arrow.to.target><lb></lb>eſſe in medio arcus d e: producta enim f a, <lb></lb>erunt anguli d a f &amp; e a f æquales: ſupponi­<lb></lb>tur enim <expan abbr="primũ">primum</expan> f eſſe in medio: igitur cum <lb></lb>a b &amp; a c ſint æquales, &amp; a f communis, erit <lb></lb>a f c æqualis a f b, igitur reflectentur æqua­<lb></lb>liter: ergo ſi ęqualiter reflectentur, ex f re­<lb></lb>flectentur, ut ex ſecunda parte: quare ex <lb></lb>medio.<lb></lb><figure id="id.015.01.265.1.jpg" xlink:href="015/01/265/1.jpg"></figure><lb></lb><arrow.to.target n="marg853"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004215"><margin.target id="marg851"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 21. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004216"><margin.target id="marg852"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>primi <emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004217"><margin.target id="marg853"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 210. <lb></lb>P<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004218">Sumatur rurſus punctum g, remotius ab <lb></lb>a quam b, dico quòd reflexio erit in arcu f e. <lb></lb></s>
          <s id="id004219">Nam non in e, quoniam fic g e d eſſet æqualis b e k, cui rurſus eſt æ­<lb></lb>qualis b e d, ergo g e d æqualis b e d, pars toti. </s>
          <s id="id004220">Sed neque ultra e, nam <lb></lb>multo magis pars æqualis eſſet toti aut maior etiam. </s>
          <s id="id004221">Sed neque ex f, <lb></lb>nam eadem ratione pars eſſet maior toto. </s>
          <s id="id004222">Neque in toto arcu f d: <lb></lb>nam ſit punctum l, &amp; ducantur al, g f, igitur g l a maior g f a, g f a au<lb></lb>tem maior e f a, igitur g l a maior c f a, ęqualis ex ſuppoſito b f a, b f a </s>
        </p>
        <p type="main">
          <s id="id004223"><arrow.to.target n="marg854"></arrow.to.target><lb></lb>rurſus maior b l a: multo igitur maior g l a quam b l a, non ergo re­<lb></lb>flexio æqualis eſſe poteſt. </s>
          <s id="id004224">Cum ergo reflexio fiat, &amp; non ex arcu d f, <lb></lb><arrow.to.target n="marg855"></arrow.to.target><lb></lb>nec puncto f, nec e, nec ultra e, nec extra d, erit neceſſarium, ut fiat ex <lb></lb>puncto in arcu e f.<lb></lb><arrow.to.target n="marg856"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004225"><margin.target id="marg854"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 21. <emph type="italics"></emph>pri<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004226"><margin.target id="marg855"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 1 C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. <lb></lb><emph type="italics"></emph>præcedentis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004227"><margin.target id="marg856"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004228">Ex hoc patet, quod linea a puncto ducta, quo <lb></lb>longius fertur, eo etiam longius reſilit.</s>
        </p>
        <figure id="id.015.01.265.2.jpg" xlink:href="015/01/265/2.jpg"></figure>
        <p type="main">
          <s id="id004229"><arrow.to.target n="marg857"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004230"><margin.target id="marg857"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004231">Cum enim a c b maior ſit a d b, &amp; angulus e c b </s>
        </p>
        <p type="main">
          <s id="id004232"><arrow.to.target n="marg858"></arrow.to.target><lb></lb>æqualis a c b &amp; f d b æqualis a d b, erunt duo an­<lb></lb>guli a c b &amp; e c b, maiores a d b &amp; f d b, quare <lb></lb>reliquus f d a maior a c e, igitur&#039;d f reſilit latius <lb></lb>quam c e.<lb></lb><arrow.to.target n="marg859"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004233"><margin.target id="marg858"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 21. <lb></lb><emph type="italics"></emph>tertij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004234"><margin.target id="marg859"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004235">Ex hoc patet, quod tales lineæ quæ reſiliunt <lb></lb>nunquam concurrent.</s>
        </p>
        <p type="main">
          <s id="id004236"><arrow.to.target n="marg860"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004237"><margin.target id="marg860"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004238">Scilicet c e &amp; d f nam conſtat ducta c d, angulos e c d f &amp; d e, ma­</s>
        </p>
        <p type="main">
          <s id="id004239"><arrow.to.target n="marg861"></arrow.to.target><lb></lb>iores eſſe duobus rectis, ergo non concurrentin partem e f.</s>
        </p>
        <p type="margin">
          <s id="id004240"><margin.target id="marg861"></margin.target>P<emph type="italics"></emph>er conuer­<lb></lb>ſam<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>petit.<emph.end type="italics"></emph.end><lb></lb>E<emph type="italics"></emph>uclid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004241">Propoſitio ducenteſima quinta decima.</s>
        </p>
        <p type="main">
          <s id="id004242">Punctum reflexionis punctorum inæqualiter diſtantium à cen­<lb></lb>tro, æqualiter diſtat à lineis ductis à centro ad puncta, æqualiter di <lb></lb>ſtantia alterutrinque.<lb></lb><arrow.to.target n="marg862"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004243"><margin.target id="marg862"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004244">Sint g h a &amp; b h a æquales, &amp; abſcindatur h f æqualis h b, &amp; pro­<lb></lb>ducatur h b uſque a d c, ut ſit h c æqualis h g, &amp; producantur f a &amp; 


<pb pagenum="247" xlink:href="015/01/266.jpg"></pb>c a, quæ ſecent peripheriam in d &amp; e, dico quod <lb></lb>punctum h eſt medium inter e &amp; l, item inter d &amp; </s>
        </p>
        <p type="main">
          <s id="id004245"><arrow.to.target n="marg863"></arrow.to.target><lb></lb>k. </s>
          <s id="id004246">Nam cum h f &amp; h b ſint æquales ex ſuppoſito, <lb></lb><arrow.to.target n="marg864"></arrow.to.target><lb></lb>&amp; anguli b h a &amp; g h a æquales, &amp; linea h a com­<lb></lb><arrow.to.target n="marg865"></arrow.to.target><lb></lb>munis, erit angulus b a h æqualis f a h, igitur ar­<lb></lb>cus h l æqualis arcui h e. </s>
          <s id="id004247">Similiter angulus g h a <lb></lb>eſt æqualis e h a &amp; c h æqualis h g exſuppoſito, &amp; <lb></lb>a h communis, igitur ut ſuprà angulus c a h æqua­<lb></lb>lis g a h, igitur per eandem arcus h k æqualis arcui <lb></lb>h d, quare h punctum in medio d &amp; k, &amp; in medio <lb></lb>etiam e &amp; l, quod eſt probandum.</s>
        </p>
        <p type="margin">
          <s id="id004248"><margin.target id="marg863"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 210.</s>
        </p>
        <p type="margin">
          <s id="id004249"><margin.target id="marg864"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004250"><margin.target id="marg865"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 26. <emph type="italics"></emph>ter­<lb></lb>tij<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <figure id="id.015.01.266.1.jpg" xlink:href="015/01/266/1.jpg"></figure>
        <p type="main">
          <s id="id004251">Propoſitio ducenteſima ſexta decima.</s>
        </p>
        <p type="main">
          <s id="id004252">Si fuerint circuli duo inæquales, &amp; extra utrunque punctum a d il­<lb></lb>lud ex minore reflexè per magnam partem minoris à maiore perue<lb></lb>nire poterunt.</s>
        </p>
        <figure id="id.015.01.266.2.jpg" xlink:href="015/01/266/2.jpg"></figure>
        <p type="main">
          <s id="id004253">Sint duo circuli, maior a b, mi­<lb></lb><arrow.to.target n="marg866"></arrow.to.target><lb></lb>nor c d, &amp; <expan abbr="punctũ">punctum</expan> g, extra utrun­<lb></lb>que, dico quod a d g ex c d <expan abbr="pote­rũt">pote­<lb></lb>runt</expan> reflexè produci a b in c d, quia <lb></lb>enim ex a b quibuſuis punctis <lb></lb>poſſunt duci lineæ reflexè ex c d, <lb></lb>&amp; ideo cum puncta in a b uarient <lb></lb>reflexionem ex c d, aliter pars eſ­<lb></lb>ſet æqualis toti, patet intentum.</s>
        </p>
        <p type="margin">
          <s id="id004254"><margin.target id="marg866"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004255">Ex hoc patet, quod oculus in <lb></lb><arrow.to.target n="marg867"></arrow.to.target><lb></lb>quauis parte terræ conſtitutus, in <lb></lb>qua Lunam uidere poſsit, poterit <lb></lb>eam uidere per radios reflexos à <lb></lb>Sole.</s>
        </p>
        <p type="margin">
          <s id="id004256"><margin.target id="marg867"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004257">Ex hoc rurſus patet, quod <expan abbr="eodẽ">eodem</expan> modo oculus poterit uidere ſu­<lb></lb><arrow.to.target n="marg868"></arrow.to.target><lb></lb>perficiei Lunę illuminatę <expan abbr="partẽ">partem</expan> p radios reflexos à Solis corpore.</s>
        </p>
        <p type="margin">
          <s id="id004258"><margin.target id="marg868"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>_{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004259">Hoc patet, quoniam ſi circuli Solis ſinguli, qui illuminant <expan abbr="Lunã">Lunam</expan> <lb></lb><arrow.to.target n="marg869"></arrow.to.target><lb></lb>oſtendunt per primum corrolarium huius <expan abbr="partẽ">partem</expan> circuli Lunæ per <lb></lb>radios Solis reflexos ab ipſa Luna, putà ſecundum portionem cir­<lb></lb>culi e f, igitur cum liceat in Sole accipere magnam partem ſuperfi­<lb></lb>ciei eius, quæ Lunam illuminat, in qua continentur infinitæ por­<lb></lb>tiones circulorum, &amp; hæ ſingulæ mittunt radios reflexos ex Luna <lb></lb>ad punctum g, igitur g uidebit portionem ſuperficiei Lunæ ſecun­<lb></lb>dum longitudinem e f per radios Solares à Luna reflexos: quod <lb></lb>eſt propoſitum.</s>
        </p>
        <pb pagenum="248" xlink:href="015/01/267.jpg"></pb>
        <p type="margin">
          <s id="id004260"><margin.target id="marg869"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004261">Propoſitio ducenteſima decima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id004262">Oculus uidet partem ſuperficiei Lunæ illuminatam à Sole per <lb></lb>radios reflexos à Solis corpore: nec tamen poteſt uidere imaginem <lb></lb>ipſius in Luna tanquam in ſpeculo.<lb></lb><arrow.to.target n="marg870"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004263"><margin.target id="marg870"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004264">Quoniam per illos, ut <expan abbr="demõſtratum">demonſtratum</expan> eſt, poteſt uidere, &amp; illi ſunt </s>
        </p>
        <p type="main">
          <s id="id004265"><arrow.to.target n="marg871"></arrow.to.target><lb></lb>robuſtiores, ergo per illos uidet, omnis enim operatio tribuitur di­<lb></lb>gniori cauſæ &amp; potentiori. </s>
          <s id="id004266">Item, quoniam uidemus Lunam in no­<lb></lb>cte immittere radios per feneſtram uelut Sol: irradiare autem non <lb></lb>eſt niſi habentis tantum lumen ex ſe, ut hoc poſsit facere, aut ut ſpar<lb></lb>gantur, aut ut reflectantur: ex ſe tantum non habet ut adparet hora <lb></lb>deliquij: neque ſpargit, ſic enim non impediret Solem hora deliquij, <lb></lb>Solis ergo reflectis. </s>
          <s id="id004267">Ergo uidemus per radios reflexos. </s>
          <s id="id004268">Non <expan abbr="tamẽ">tamen</expan> <lb></lb>per eam uidemus Solem, ut in ſpeculo obiecto, quoniam Luna pri<lb></lb><expan abbr="mũ">mum</expan> lucet proprio lumine, &amp; rubro ſicut pruna, quod autem debet <lb></lb>fungi uice ſpeculi, oportet ut careat colore, &amp; ſit uelut aqua, &amp; ut ſit <lb></lb>purum. </s>
          <s id="id004269">Deinde, quia Sol eſt maior Luna, ideò uidetur ut paries in <lb></lb>ſpeculo, uidetur enim non res reflexa, ſed quod ipſum ſpeculum ſit <lb></lb>paries, &amp; ita Sol uidetur, ut totum quoddam, &amp; non poteſt ob id <lb></lb>cognoſci. </s>
          <s id="id004270">Et etiam magnitudo luminis per quam oculus non po­<lb></lb>teſt diſtinguere Lunam ab imagine Solis: nam ea his quæ perſpe­<lb></lb>culum uidentur, oportet duo cognoſcere, ſpeculum, &amp; rem quæ ui <lb></lb>detur, ſed magnitudo luminis prohibet ſpeculum uideri, ergo non <lb></lb>poterit uideri aliud tanquam in ſpeculo, ſed ſolum ſpeculum cum <lb></lb>lumine tanquam res una. </s>
          <s id="id004271">Et ita de Luna. </s>
          <s id="id004272">Accedit magnitudo di­<lb></lb>ſtantiæ: nam in ſuperflua diſtantia non cognoſcitur ſuperficies ſpe­<lb></lb>culi, ſed ſolum rei obiectæ imago, &amp; illa habetur pro ſuperficie ſpe­<lb></lb>culi, ergo oculus non diſtinguit inter ſpeculum, &amp; rem uiſam, ideò <lb></lb>non uidet tanquam è ſpeculo. </s>
          <s id="id004273">Ex quo ſequitur, quod Luna iudica­<lb></lb>bitur longiùs abeſſe quàm abſit, quia quod uidemus ex ea eſt So­<lb></lb>lis imago, quæ longius multo abeſt à nobis ipſa Lunæ ſuperficie. <lb></lb></s>
          <s id="id004274">Cum ergo ſint quatuor cauſæ, quarum unaquæque impedire poſſet, <lb></lb>quominus Sol non uideatur in Luna tanquàm in ſpeculo, quanto <lb></lb>magis cùm omnes adſint in Luna, &amp; ſimul concurrant.</s>
        </p>
        <p type="margin">
          <s id="id004275"><margin.target id="marg871"></margin.target>I<emph type="italics"></emph>n præceden <lb></lb>ti.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004276">Propoſitio ducenteſima decima octaua.</s>
        </p>
        <p type="main">
          <s id="id004277">Rationem maculæ Lunæ indagare.<lb></lb><arrow.to.target n="marg872"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004278"><margin.target id="marg872"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004279">Supponamus primum quæ ſunt manifeſta, inde addamus quæ <lb></lb>ſunt ueriſimilia ualde, poſt ueriſimiliora ex dubijs, ubi ratio utrinque<lb></lb> pugnare uidetur, demum dicemus de quæſito. </s>
          <s id="id004280">Manifeſtum eſt igi­<lb></lb>tur, quod Luna diſtat à nobis circiter &lt;20&gt; X MP. dimetiens igitur or <lb></lb>bis Lunæ eſt circiter CCC&lt;18&gt;&lt;18&gt; MP. igitur ambitus &lt;21&gt;MP. igitur in hora 


<pb pagenum="249" xlink:href="015/01/268.jpg"></pb>circuit circiter XLII MP. </s>
          <s id="id004281">Ergo in ictu inſenſili penè, id eſt, tempore <lb></lb>ictus pulſus infantis laborantibus acutiſsima febre II MP. quoniam <lb></lb>quinque tales ictus continentur penè in ictu uno uiri temperatæ <lb></lb>naturæ, &amp; &lt;23&gt; ictus pulſus fermè uiri temperati complent ſpatium <lb></lb>horæ. </s>
          <s id="id004282">Igitur Luna mouetur rapidiſsimo motu &amp; ſimili motui ful­<lb></lb>guris. </s>
          <s id="id004283">Ex quo patet quod eſt corpus expers grauitatis &amp; perfe­<lb></lb>ctum, quare nec miſtum, nec uitiatum.</s>
        </p>
        <p type="main">
          <s id="id004284">Eſt etiam rotunda, tametſi enim ob diſtantiam maximam poſ­<lb></lb>ſet uideri rotunda, etiam quod non eſſet, ueriſimile tamen eſt, cum <lb></lb>umbram talem efficiat in deliquio Solis, &amp; cum exit è tenebris ter­<lb></lb>ræ, tum quia perfecta eſt quod ſit <expan abbr="rotũda">rotunda</expan>, aut prope rotunditatem, <lb></lb>ſed quod eſt perfectum &amp; diuinum (quia ſeruat æqualitatem, hoc <lb></lb>enim demonſtratum eſt, quod æquale ſolum reperitur in diuinis <lb></lb>quod ad motum attinet) exactè tale eſt, igitur Luna eſt exactè ro­<lb></lb>tunda in circuitu ſecundum ſuperficiem orbis. </s>
          <s id="id004285">Ergo etiam unde­<lb></lb>quaque &amp; ſecundum profunditatem: nam in commutatione <expan abbr="nõ">non</expan> poſ­<lb></lb>ſet latere inæqualitas. </s>
          <s id="id004286">Et etiam non eſt ueriſimile ullo modo, quod <lb></lb>corpus perfectum &amp; diuinum ſit informe. </s>
          <s id="id004287">Eſſet autem neceſſario <lb></lb>eiuſmodi, ſi eſſet exactè rotunda ſecundum longitudinem &amp; latitu­<lb></lb>dinem, &amp; ſecundum profunditatem alterius figuræ. </s>
          <s id="id004288">Veriſimilius <lb></lb>eſt ergo, Lunam eſſe ut ignem <expan abbr="quẽdam">quendam</expan> denſum per ſe lucidum, ſed <lb></lb>inæqualiter luminoſum, non ſolum ob ſubſtantiæ denſitatem, <lb></lb>ſed copiam luminis &amp; puritatem, quæ impuritas non illi accidit, <lb></lb>quia miſta, ſed quoniam eſt inæqualium partium partium rararum ac den­<lb></lb>ſarum &amp; mediarum. </s>
          <s id="id004289">Neque ſolum colluſtratur à lumine ex his quæ <lb></lb>diximus, tum etiam quia colluſtrata non lucent procul, ut neque <lb></lb>montes, qui plurimum abſunt, quamuis non tale procul ut Luna, <lb></lb>imò nec nix quę illis inſidet, ſed nix eſt multo <expan abbr="cãdidior">candidior</expan> per ſe quàm <lb></lb>Luna, quam conſtat lumine Solis deſtitutam eſſe <expan abbr="rubrã">rubram</expan>, ergo Luna <lb></lb>relucet radijs Solaribus eliſis uelut à ſpeculo. </s>
          <s id="id004290">Et ſi quis in orbe Lu­<lb></lb>næ eſſet media die ſerena, non uideret terram luminoſam, quæ mul<lb></lb>to maior eſt Luna, &amp; paulo plus à Sole diſtat, &amp; quando que illi pro­<lb></lb>pior eſt quàm Luna. </s>
          <s id="id004291">Macula autem Lunæ eſt qualis depingitur <lb></lb>cum ore, oculis &amp; naſo, ſed quod magis ſpectatur eſt os ipſum: <lb></lb><figure id="id.015.01.268.1.jpg" xlink:href="015/01/268/1.jpg"></figure><lb></lb>adeò ut Plutarchus non de macula Lunæ, ſed de ore Lu­<lb></lb>næ inſcripſerit. </s>
          <s id="id004292">Non uerti autem Lunam, ex hoc probat </s>
        </p>
        <p type="main">
          <s id="id004293"><arrow.to.target n="marg873"></arrow.to.target><lb></lb>Philoſophus ſecundo de Cœlo. </s>
          <s id="id004294"><expan abbr="Igit̃">Igitur</expan> ab Oriente in <expan abbr="Occi­dentẽ">Occi­<lb></lb>dentem</expan> uerti ſub, &amp; ſuprà neceſſe eſt. </s>
          <s id="id004295">Scilicet ut oculi infrà <lb></lb>os ſupra appareat. </s>
          <s id="id004296">Videtur autem magis in plenilunio <lb></lb>ob <expan abbr="differentiã">differentiam</expan> luminis, &amp; tota quoniam pars uerſus nos etiam tota <lb></lb>illuſtratur. </s>
          <s id="id004297">Et ex illo loco apparet, quod Auerroes neſciuit Geo­


<pb pagenum="250" xlink:href="015/01/269.jpg"></pb>metriam, sicut ſemper fuit mos Philoſophorum <expan abbr="cõtentioſorum">contentioſorum</expan>, ut <lb></lb>nil ſciant, ſed ſolum garrire. </s>
          <s id="id004298">audierat hoc ab aliquo malo Geome­<lb></lb>tra, &amp; repoſuit in ſuos libros: nam nos, ut ſuprà uidiſti, demonſtra­<lb></lb>uimus oppoſitum. </s>
          <s id="id004299">Quod uerò ſit macula illa ex umbra terræ, ue­<lb></lb>rum non eſt, quoniam una eſſet &amp; non diuiſa, &amp; occuparet totam il<lb></lb>lius faciem: nec eſt uerum quod mutaret ſitum, quia ſuperficies ter­<lb></lb>ræ eſt nonupla ſuperficiei Lunæ. </s>
          <s id="id004300">Sicut terræ ſuperficies eſt minor <lb></lb>trigeſima parte ſuperficiei Solis. </s>
          <s id="id004301">Nec ſpargitur lumen Solis in Lu­<lb></lb>na, nam ſic eſſet ambitus ut uia lactea: cum autem Luna delin­<lb></lb>quit in Oriente, eſt glauca &amp; purpurea, cum in cœli medio rubra, <lb></lb>cum in Occidente nigra uidetur, nam ab utraque parte tenebris ope­<lb></lb>ritur: ex Oriente ab umbra terræ, ab Occidente ab obſcuritate loci. <lb></lb></s>
          <s id="id004302">In medijs locis medijs coloribus, quos Aſtrologi terraticis tribu­<lb></lb>unt: hoc autem quandiu tota delituerit, quod tempus horam uix <lb></lb>implere poteſt. </s>
          <s id="id004303">Ergo partes peruiæ non remittunt lumen, ideò ob­<lb></lb>ſcuræ apparent, quod in uitreis ſpeculis à quorum partibus plum­<lb></lb>bum excidit: nam nigræ illæ apparent, reliquæ ſplendidæ, ob id ſy­<lb></lb>dera aliquando per illam relucent, &amp; aliquando non. </s>
          <s id="id004304">Et Solaris <lb></lb>eclypſis tempore, non lux tota Solis perit: atque ideo ut uidemus, &amp; <lb></lb>uariant colores eo tempore, non <expan abbr="tamẽ">tamen</expan> colluſtrat ſplendidè Sol ob <lb></lb><arrow.to.target n="marg874"></arrow.to.target><lb></lb>craſsitiem Lunaris corporis hæc inferiora, tum etiam ob diuerſita­<lb></lb>tem partium, &amp; ad ſitum. </s>
          <s id="id004305">Nam ſi Sol ſit ad ſitum a b, tranſibunt mul<lb></lb><figure id="id.015.01.269.1.jpg" xlink:href="015/01/269/1.jpg"></figure><lb></lb>ti radij, ſi c d pauciſsimi aut nulli, ſed ut ubi <lb></lb>tenuior eſt Luna in ambitu, &amp; Solis radij <lb></lb>denſiores tranſeunt, &amp; ſydera pellucent <lb></lb>contrarijs cauſis minus, ut iuxta medium <lb></lb>nequaquàm. </s>
          <s id="id004306">At Lunæ maculam radij effi­<lb></lb>ciunt, etiam ſi tota ſubtus opaca eſſet, cum <lb></lb>peruia uel tantillum fuerit in ſuperficie, ut <lb></lb>uenis opus non ſit. </s>
          <s id="id004307">Et iuxta hoc macula illa, ut liquet, ad perfectio­<lb></lb>nem corporis Lunæ pertinet magis quam pars ſplendida, quam­<lb></lb>uis prima cogitatione oppoſitum uideatur. </s>
          <s id="id004308">Eſt enim duplex perfe­<lb></lb>ctionis genus in cœleſtibus corporibus, &amp; ob denſitatem cum re­<lb></lb>mittit, &amp; ob perſpicuitatem cum à Sole, ut uniuerſali quo dam prin<lb></lb>ci pio illuminatur.</s>
        </p>
        <p type="margin">
          <s id="id004309"><margin.target id="marg873"></margin.target>T<emph type="italics"></emph>ex.<emph.end type="italics"></emph.end> 49.</s>
        </p>
        <p type="margin">
          <s id="id004310"><margin.target id="marg874"></margin.target>2. A<emph type="italics"></emph>poteles<emph.end type="italics"></emph.end><lb></lb>P<emph type="italics"></emph>tolem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004311">Propoſitio ducenteſima decima nona.</s>
        </p>
        <p type="main">
          <s id="id004312">Ratio nem eorum quæ apparent circa Solem ſpeculo in aqua po<lb></lb>ſito declarare.<lb></lb><arrow.to.target n="marg875"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004313"><margin.target id="marg875"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004314">Sit peluis a b aqua plena: ſpeculum in ea c d e f quadratum, aut <lb></lb>perfecte, aut oblongum ſub merſum in ea: Sol primum ſolus in g <pb pagenum="251" xlink:href="015/01/270.jpg"></pb>oculus ex aduerſo in <lb></lb>h, ita ut ad æquales <lb></lb>angulos poſsit uide­<lb></lb>re <expan abbr="Solẽ">Solem</expan> in k, dico q̊d <lb></lb>depreſſo oculo in m, <lb></lb>uidebit alium Solem <lb></lb>maiorem uerſus mar<lb></lb>ginem aduerſum in l, <lb></lb>&amp; longè ſplendidio­<lb></lb>rem: quia enim radij <lb></lb><expan abbr="reflectũtur">reflectuntur</expan> ex k, ut ro <lb></lb>buſti &amp; à medio den<lb></lb>ſiore ad rarius, qui <lb></lb>non <expan abbr="inflectent̃">inflectentur</expan>, erunt <lb></lb>pauci, &amp; ideò Sol in <lb></lb>k minor apparebit, et <lb></lb>languidior: maior au<lb></lb><figure id="id.015.01.270.1.jpg" xlink:href="015/01/270/1.jpg"></figure><lb></lb>tem pars deflectetur à <expan abbr="perpẽdiculari">perpendiculari</expan> ad m, igitur Sol apparebit ma­<lb></lb>ior &amp; ualidior longè ſplendentibus radijs, adeò ut uix ferri poſsit. <lb></lb></s>
          <s id="id004315">Sed quoniam angulus ex ſuppoſito m l ſ maior eſt h k e, igitur cum <lb></lb>oculus iudicet ſe uidere a d æquales angulos, uidebitur g depreſ­<lb></lb>ſior &amp; propior labro in t, ſicut n m eſt infra h, ita t infra g, quare <expan abbr="etiã">etiam</expan> <lb></lb>ut angulus m l ſ ſit æqualis angulo t l f, neceſſe eſt ut l ſit ultra k: ali­<lb></lb>ter t uideretur quaſi tangere aquam. </s>
          <s id="id004316">In hora autem deliquij Solis, <lb></lb>uelut hodie v. Idus Aprilis hora ſexta diei, <expan abbr="cũ">cum</expan> diligentiſsimi ſtatue­<lb></lb>rint medium eclipſis in quinta, &amp; ſuppoſita fuerit obſcuratio à Io­<lb></lb>anne Stadio partium nouem cum beſſe, &amp; tempus horæ unius &amp; <lb></lb>m: 26, fuit tamen maior &amp; longior: quoniam luminaria <expan abbr="fuerũt">fuerunt</expan> pro­<lb></lb>piora una parte caudæ Draconis, quam ipſe poſuerit in tabulis, &amp; <lb></lb>hoc quia ſupponit ęquinoctium tardius diebus duobus <expan abbr="quã">quam</expan> apud <lb></lb>Alphonſum: &amp; forſan ſufficiebat una dies, ſcilicet ut eſſet die deci­<lb></lb>ma Martij horis decem octo à meridie: nam tunc omnia reſpon­<lb></lb>dent obſeruationi: in qua apparuerunt quatuor Lunæ: &amp; quidem <lb></lb>ab initio fuerunt duæ orientaliores è regione, ſcilicet o p, &amp; una oc<lb></lb>cidentalior n, &amp; tantum diſtabat n a k quantum o: Et clarum erat <lb></lb>quòd p erat, ſicut ſecunda iris parua &amp; non candida, ſed rubra pur­<lb></lb>pureo miſta, quoniam ex reflexu o oriebatur: apparebat autem a la <lb></lb>tere illo, quoniam Luna dextram partem obtegebat, ideo illa erat <lb></lb>minus luminoſa, &amp; uerus Sol erat in k, modò Lunæ, modò Solis <lb></lb>imaginem referens ubi tranſiſſet eclipſis medium, non amplius <lb></lb>tres illæ Lunæ apparuerunt à dextra &amp; à ſiniſtra, ſed una ultra nos 


<pb pagenum="252" xlink:href="015/01/271.jpg"></pb>in q, &amp; duæ uerſus nos in r &amp; n <lb></lb>&amp; quæ erat in F, erat ſimiliter <lb></lb>parua &amp; purpurea rubraque, &amp; <lb></lb>mutato ſpeculo uariebatur ſi­<lb></lb>tus q &amp; r u, id eſt, ut modo eſ­<lb></lb>ſent quaſi in medio laterum e <lb></lb>&amp; f, quando que tranſuerſæ. </s>
          <s id="id004317">Et <lb></lb>hoc contigit ob <expan abbr="mutationẽ">mutationem</expan> lo­<lb></lb>ci k propter ſpeculi <expan abbr="uariationẽ">uariationem</expan>.</s>
        </p>
        <figure id="id.015.01.271.1.jpg" xlink:href="015/01/271/1.jpg"></figure>
        <p type="main">
          <s id="id004318">Cauſa eſt, quoniam Luna <expan abbr="cũ">cum</expan> <lb></lb>permeet Solem non è regione <lb></lb>recta lineæ oppoſitæ noſtro ui <lb></lb>ſui, &amp; ſolum <expan abbr="momẽto">momento</expan>, &amp; in lon<lb></lb>gis <expan abbr="temporũ">temporum</expan> interuallis poſsit <lb></lb>obtegere illum. </s>
          <s id="id004319">Sit ergo ut Sol <lb></lb>obtegatur à Luna medijs par­<lb></lb>tibus, &amp; ſint radij extremi in <lb></lb>ſpeculo: a c &amp; a d, igitur erunt <lb></lb>tanquam duo Soles, ſed uterque<lb></lb> illorum geminatur, ideò fiunt <lb></lb>tres: medius enim ob Lunæ <lb></lb>perſpicuitatem integer, appa­<lb></lb>ret, ideò modò ſub forma So­<lb></lb>lis, modò Lunæ laterones am­<lb></lb>bo ſub forma Lunæ: ideò <expan abbr="erũt">erunt</expan> <lb></lb>tres, quibus. </s>
          <s id="id004320">addita Luna p, quæ <lb></lb>eſt reflexa a ſecunda, fient qua­<lb></lb>tuor. </s>
          <s id="id004321">At dices cur non fit refle­<lb></lb>xus ſecundum directum oculi, <lb></lb>ut Lunæ appareant ultra citra­<lb></lb>que Solem? </s>
          <s id="id004322">Dico quod Luna <lb></lb>diuidente orbem reflexus fit ad latera, quia radij tranſuerſim ferun­<lb></lb>tur: cum autem non diuiditur fit prorſum &amp; retrorſum. </s>
          <s id="id004323">Sed cur di­<lb></lb>ces Lunari forma? </s>
          <s id="id004324">quoniam partes Solis quæ uigent, eiuſmodi for­<lb></lb>ma apparent, Iconem uides à latere.</s>
        </p>
        <p type="main">
          <s id="id004325">Propoſitio ducenteſima uigeſima.</s>
        </p>
        <p type="main">
          <s id="id004326">Cauſam cur Sol æſtiuis diebus exoriens umbram ad meridiem, <lb></lb>cum in meridie ad boream mittat, explorare.</s>
        </p>
        <p type="main">
          <s id="id004327"><arrow.to.target n="marg876"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004328"><margin.target id="marg876"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004329">Dico quod ubicunque terrarum in noſtro hemiſpherio, Sol ubi <lb></lb>fuerit in Oriente ſeu Occidente uidebitur, cum ſub circulo æquino<lb></lb>ctij fuerit è regione, nobis <expan abbr="etiã">etiam</expan> ſi homo ſub arctico circulo habitet, 


<pb pagenum="253" xlink:href="015/01/272.jpg"></pb>&amp; ita reſpicienti ad polum umbra erit à dextra in ſiniſtram, dum o­<lb></lb>ritur &amp; à ſiniſtra in dextram dum occidit. </s>
          <s id="id004330">Et quod dum erit in me­<lb></lb>ridie umbra uerget ad Septentrionem. </s>
          <s id="id004331">Tertiò dico, quòd in his <lb></lb>qui habitant uerſus Septentrionem à tropico cancri umbra in Me­<lb></lb>ridie, quo cunque tempore anni borealis erit. </s>
          <s id="id004332">Quarto, quòd ijſdem <lb></lb>toto dimidio anni ab æquinoctio uerno ad autumnale, umbræ o­<lb></lb>riente &amp; occidente Sole ſunt meridianæ tranſuerſæ: &amp; muri reſpi­<lb></lb>cientes boream illuminantur. </s>
          <s id="id004333">Sit finitor a b c d in regione boreali, <lb></lb>cuius uertex e &amp; f polus, eleuatio poli ſupra finitorem a f, æquino­<lb></lb>ctij circulus b q d, cui parallelus borealior Solis uia per cancri ini­<lb></lb>tium, g h l m n, circulus magnus per uerticem, &amp; interſectiones æ­<lb></lb>quinoctij, &amp; finitoris b h e m d, Meridiei ſemicirculus ſuperior a f e <lb></lb>l q c. </s>
          <s id="id004334">Cum ergo uertex regionis ſit in e, &amp; circulus magnus b h d <lb></lb>tranſiens per uerticem, tranſeat per centrum terræ ex diffinitione <lb></lb>circuli magni, &amp; linea à uertice grauium habitantium ſub uertice e, <lb></lb><figure id="id.015.01.272.1.jpg" xlink:href="015/01/272/1.jpg"></figure><lb></lb>tendat ad centrum terræ ex de­<lb></lb>monſtratis ab Ariſtotele, &amp; ſup<lb></lb>poſitis ab Aſtrologis, q̊d gra­<lb></lb>uia omnia tendunt ad centrum <lb></lb>terræ, erit quodlibet graue ſeu <lb></lb>murus ſeu homo, ſeu per ulti­<lb></lb>mam <expan abbr="petitionẽ">petitionem</expan>, ſeu per demon­</s>
        </p>
        <p type="main">
          <s id="id004335"><arrow.to.target n="marg877"></arrow.to.target><lb></lb>ſtrata in undecimo ab Euclide <lb></lb>murus, &amp; homo quiuis inco­<lb></lb>la regionis in ſuperficie circuli <lb></lb>uerticalis b e d. </s>
          <s id="id004336">Igitur dum Sol <lb></lb>eſt in b uel d, umbræ <expan abbr="erũt">erunt</expan> à dex<lb></lb>tro in ſiniſtrum, uel contrario <lb></lb>modo, &amp; ita Sol uidebitur eſſe è regione nobis: &amp; murus faciet um<lb></lb>bram <expan abbr="orientalẽ">orientalem</expan> uel occidentalem. </s>
          <s id="id004337">Et hoc eſt primum. </s>
          <s id="id004338">Et quoniam <lb></lb>cum Sol erit in Meridie, tum erit in q, igitur erit umbra ad Septen­<lb></lb>trionem, cum e ſit loco gnomonis &amp; murus. </s>
          <s id="id004339">Et hoc eſt ſecun dum. <lb></lb></s>
          <s id="id004340">Tertium etiam patet, quia Sol nun quam tranſibit <expan abbr="punctũ">punctum</expan> l in Me­<lb></lb>ridie uerſus boream, ſed regio ſupponitur borealior l, igitur tempo<lb></lb>re meridiei umbra ſemper hic borealis erit. </s>
          <s id="id004341">Et quoniam b h e m d <lb></lb>ſecat parallelos, qui ſunt in Septentrione ut puta tropicum in h <lb></lb>&amp; m, igitur oriente Sole, &amp; occidente rurſus per totum arcum g h <lb></lb>&amp; m n, uidebitur borealior quàm in b uel d parte arcus magni in­<lb></lb>tercepti inter arcum magnum tranſeuntem per uerticem &amp; locum <lb></lb>Solis, ubi ſecat finitorem &amp; puncta b, &amp; d: &amp; ita erunt umbræ Me­<lb></lb>ridionales toto hoc tempore, &amp; hoc eſt quartum.


<pb pagenum="254" xlink:href="015/01/273.jpg"></pb><arrow.to.target n="marg878"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004342"><margin.target id="marg877"></margin.target>F<emph type="italics"></emph>ropoſ.<emph.end type="italics"></emph.end> 1</s>
        </p>
        <p type="margin">
          <s id="id004343"><margin.target id="marg878"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004344">Ex quo ſequitur, quod in hoc toto tempore ueris &amp; æſtatis, cùm <lb></lb>Sol in Meridie uideatur eſſe poſt tergum, &amp; in Meridie, &amp; dum ori<lb></lb>tur à parte Septentrionis. </s>
          <s id="id004345">Ergo ab ortu Solis ad Meridiem uidebi­<lb></lb>tur ferri motu diurno, linea obliqua à <expan abbr="Septẽtrione">Septentrione</expan> in Meridiem: &amp; <lb></lb>à Meridie ad Occaſum, alia obliqua linea à Meridie in Septentrio­<lb></lb>nem: ut in figura, ut ſi Sol ſit in a in Oriente, b in Meridie, cin Occi­<lb></lb>dente, &amp; uertex nobis in e.</s>
        </p>
        <p type="main">
          <s id="id004346"><arrow.to.target n="marg879"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004347"><margin.target id="marg879"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004348">Sequitur etiam, quòd ſi tempore æſtatis <lb></lb><figure id="id.015.01.273.1.jpg" xlink:href="015/01/273/1.jpg"></figure><lb></lb>poſſemus in media nocte uidere Solem, in <lb></lb>cœli medio uideretur, tantundem uerſus bo<lb></lb>ream declinare, quantum in Meridie ad Me<lb></lb><expan abbr="ridiẽ">ridiem</expan>. </s>
          <s id="id004349">Et hoc quia circulus æquinoctij b q d, <lb></lb>tanto borealior eſt in parte inferiore circulo <lb></lb>per uerticem, quanto in ſuperiori eſt auſtra­<lb></lb>lior: quoniam circuli magni ſe ſecant per æ­<lb></lb>qualia. </s>
          <s id="id004350">Et ſi hoc eſt uerum de Sole ſub æqui­<lb></lb>noctij circulo, <expan abbr="quãto">quanto</expan> magis erit uerum de Sole ſub tropico æſtiuo?</s>
        </p>
        <p type="main">
          <s id="id004351"><arrow.to.target n="marg880"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004352"><margin.target id="marg880"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}. 3.</s>
        </p>
        <p type="main">
          <s id="id004353">Ex præcedenti patet, q̊d Sol in media nocte borealior uideretur <lb></lb>ſub æquinoctij circulo tanto, <expan abbr="quãto">quanto</expan> uidetur auſtralior ſe ipſo, dum <lb></lb>eſt ſub tropico cancri, quia circuli ſe ſecant ad angulos oppoſitos <lb></lb>æquales: igitur ſi uerticis circulus maiorem facit angulum ſuperio­<lb></lb><figure id="id.015.01.273.2.jpg" xlink:href="015/01/273/2.jpg"></figure><lb></lb>rem cum æquinoctij quam tro</s>
        </p>
        <p type="main">
          <s id="id004354"><arrow.to.target n="marg881"></arrow.to.target><lb></lb>pici borealis circulo, igitur &amp; <lb></lb>inferiorem: homo autem &amp; ui­<lb></lb>ſus iudicat auſtrale &amp; boreale <lb></lb>iuxta inclinationem circuli du<lb></lb>cti per <expan abbr="locũ">locum</expan> Solis ad circulum <lb></lb>ductum per locum uerticis.</s>
        </p>
        <p type="margin">
          <s id="id004355"><margin.target id="marg881"></margin.target>P<emph type="italics"></emph>er funilem<emph.end type="italics"></emph.end><lb></lb>15. <lb></lb>P<emph type="italics"></emph>ropoſ. pri­<lb></lb>mi<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004356">Propoſitio CCXXL</s>
        </p>
        <p type="main">
          <s id="id004357">Magnitudo Lunæ &amp; cæte­<lb></lb>rorum <expan abbr="aſtrorũ">aſtrorum</expan> dignoſcitur ex <lb></lb>proportione aliorum ad eam <lb></lb>iuxta diſtantiam: ipſius uerò <lb></lb>iuxta rationem pupillę ad Lu­<lb></lb>nam diſtantiæ ratione.<lb></lb><arrow.to.target n="marg882"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004358"><margin.target id="marg882"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004359">Sit pupilla a b, quæ in circu­<lb></lb>lo l m, poſita in eodem centro, <lb></lb>comprehendat portionem no <lb></lb>tam l m, ideo clauſo oculo alte­<lb></lb>ro eandem portionem uidebit <lb></lb>totius cœli, ut liquet ex demon 


<pb pagenum="255" xlink:href="015/01/274.jpg"></pb>ſtratis in Elementis Euclidis, igitur nota l m nota erit pupillæ, &amp; <lb></lb>ideo g h quanta ſit portio cœli, quia k eſt etiam quaſi centrum cœ­<lb></lb>li Lunæ, ſit ergo Luna c d, eritque tanta portio g h notæ, quanta e f <lb></lb>pars pupillæ, per quam uidetur ipſius a b: e f autem ſimiliter eſt no­<lb></lb>ta in n o, igitur &amp; c d in comparatione ad totum circulum. </s>
          <s id="id004360">Quia ue­<lb></lb>ro g h eſt nota, &amp; in Sole conſpicitur arcus notus æqualis, ergo erit <lb></lb>nota diuerſitas aſpectu ob diſtantiam noſtram à terræ centro, qua­<lb></lb>re altitudo Lunæ nota, &amp; eius magnitudo, eius enim ad ſemidiame<lb></lb>trum oculi, ut c d ad ef. </s>
          <s id="id004361">Hoc autem eſt craſſa Minerua additum, ut <lb></lb>quis intelligat difficiliora eſſe quæ craſſa uidentur, quàm quæ ela­<lb></lb>borata. </s>
          <s id="id004362">huiuſmodi autem diuina, de quibus mox dicendum erit.</s>
        </p>
        <p type="head">
          <s id="id004363">SECVNDA PARS DESVPER</s>
        </p>
        <p type="main">
          <s id="id004364">Principia.</s>
        </p>
        <p type="head">
          <s id="id004365">DIFFINITIO PRIMA.</s>
        </p>
        <p type="main">
          <s id="id004366">Proportio imperfecta ſeu poteſtate eſt duarum <expan abbr="quantitatũ">quantitatum</expan>, quæ <lb></lb>ſic ſe habent, ut nullæ duæ aliæ in eodem genere inueniri queant.</s>
        </p>
        <p type="head">
          <s id="id004367">DIFFINITIO SECVNDA.</s>
        </p>
        <p type="main">
          <s id="id004368">Proportio media eſt comparatio rei non habentis quantitatem, <lb></lb>quæ tamen mutari poſsit ad rem, quæ quantitatem habeat.</s>
        </p>
        <p type="head">
          <s id="id004369">DIFFINITIO TERTIA.</s>
        </p>
        <p type="main">
          <s id="id004370">Proportio ſublimis ſeu ordo dicitur duarum ſubſtantiarum, quę <lb></lb>quantitatem non habeant, comparatio.</s>
        </p>
        <p type="head">
          <s id="id004371">PETITIO PRIMA.</s>
        </p>
        <p type="main">
          <s id="id004372">Infinitum quod imaginem habet <expan abbr="quãtitatis">quantitatis</expan>, quantitatem autem <lb></lb>non habet, neque eſt quantitas.</s>
        </p>
        <p type="head">
          <s id="id004373">PETITIO SECVNDA.</s>
        </p>
        <p type="main">
          <s id="id004374">Repugnans eſt ſuper quod nulla eſt potentia.</s>
        </p>
        <p type="head">
          <s id="id004375">PETITIO TERTIA.</s>
        </p>
        <p type="main">
          <s id="id004376">Non poſſe ſuper ea quæ <expan abbr="repugnãt">repugnant</expan>, nullam declarat imperfectio­<lb></lb>nem, neque infinitum non eſſe negat.</s>
        </p>
        <p type="head">
          <s id="id004377">PETITIO QVARTA.</s>
        </p>
        <p type="main">
          <s id="id004378">Infinitum infinito maius eſſe non poteſt.</s>
        </p>
        <p type="main">
          <s id="id004379">Propoſitio ducenteſima uigeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id004380">Quantitates quæ æquales eſſe <expan abbr="nõ">non</expan> poſſunt in eodem genere, ma­<lb></lb>ius tamen &amp; minus recipiunt, ſunt in proportione poteſtatis.</s>
        </p>
        <p type="main">
          <s id="id004381">Sint propoſiti duo anguli, gratia exempli, a rectilineus, b uerò in </s>
        </p>
        <p type="main">
          <s id="id004382"><arrow.to.target n="marg883"></arrow.to.target><lb></lb><expan abbr="circumferẽtia">circumferentia</expan> circuli, qui poteſt eſſe maior, &amp; minor rectilineo pro­<lb></lb>poſito, &amp; nunquàm poteſt eſſe æqualis, ut declaratum eſt ſuprà, di­<lb></lb>co proportionem b ad a eſſe poteſtate, nam ut uiſum eſt, poteſt eſſe <lb></lb>maior &amp; minor, &amp; eſt maius &amp; minus uerè, &amp; ideò ſunt in eodem <lb></lb>genere, &amp; uterque eſt continua quantitas, igitur in tranſitu neceſſe <lb></lb>eſt, ut ſint æquales aliquando ſed non actu, hoc enim repugnat, igi­<lb></lb>tur poteſtate.</s>
        </p>
        <pb pagenum="256" xlink:href="015/01/275.jpg"></pb>
        <p type="margin">
          <s id="id004383"><margin.target id="marg883"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id004384">Propoſitio ducenteſima uigeſima tertia.</s>
        </p>
        <p type="main">
          <s id="id004385">Quantitates quæ actu æquales eſſe non poſſunt, in nulla pro­<lb></lb>portione actu eſſe poſſunt.<lb></lb><arrow.to.target n="marg884"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004386"><margin.target id="marg884"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004387">Sint duæ quantitates quæ æquales eſſe non poſsint, ut in priore <lb></lb>exemplo a &amp; b, dico quod non poſſunt eſſe in aliqua proportione <lb></lb>in actu, aliter ſint in proportione c, &amp; ducatur cin b, fiat d, erunt er­<lb></lb>go d &amp; a æquales, quod eſt contra ſuppoſitum, nam ſupponitur <lb></lb>quod nulla quantitas ex genere b ſit æqualis a, ſed d eſt ex genere </s>
        </p>
        <p type="main">
          <s id="id004388"><arrow.to.target n="marg885"></arrow.to.target><lb></lb>b &amp; æquale a, &amp; ideo ſuppoſitum non manet, igitur a &amp; b non ſunt <lb></lb>in aliqua proportione in actu.</s>
        </p>
        <p type="margin">
          <s id="id004389"><margin.target id="marg885"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004390">Propoſitio ducenteſima uigeſima quarta.</s>
        </p>
        <p type="main">
          <s id="id004391">Neque temporis totius ut imaginamur ipſum eſſe infinitum, neque<lb></lb> æui uitarum proportio ulla eſt ad tempus quod poteſtate eſt, ut po<lb></lb>tè diem uel menſem.<lb></lb><arrow.to.target n="marg886"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004392"><margin.target id="marg886"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id004393">Tempus ipſum ut <expan abbr="infinitũ">infinitum</expan> eſt, aut in actu eſt, aut refert quippiam <lb></lb>in actu, pars autem temporis ſolùm eſt poteſtate, quia nullum tem­<lb></lb>pus in actu eſt, neque annus, neque menſis, neque dies, neque hora aut mo­<lb></lb>mentum, ſed ſi totum tempus non eſſet actu, nihil eſſet actu, neque to<lb></lb>tum neque partes. </s>
          <s id="id004394">Igitur <expan abbr="totũ">totum</expan> tempus, uel aliquid loco eius eſt actu, <lb></lb>partes autem poteſtate, ſed ut uiſum proportio infiniti nulla eſt, &amp; <lb></lb>ad rem quæ actu non eſt, igitur tempus nullam habet proportio­<lb></lb>nem ad annos, neque menſes uel dies. </s>
          <s id="id004395">Quare qui dicunt, quod mille <lb></lb>anni ſunt unus dies, in philoſophia errant, ſecus apud Apoſtolum, <lb></lb>ubi de diuinitate agitur. </s>
          <s id="id004396">Ergo anni ſunt <expan abbr="longũ">longum</expan> tempus, &amp; dies bre­<lb></lb>ue, quia dicuntur in comparatione inter ſe, &amp; non ſecundum pro­<lb></lb>portionem ad infinitum. </s>
          <s id="id004397">Quia ſit infinitum a, &amp; d uæ quantitates b <lb></lb>maior, &amp; c minor, uel ergo proportio a ad b c, eſt una uel diuerſa, ſi </s>
        </p>
        <p type="main">
          <s id="id004398"><arrow.to.target n="marg887"></arrow.to.target><lb></lb>una, ergo b c erunt æquales, ſi maior eſt ad c quam ad b, ergo infi­<lb></lb>nitum eſt maius infinito, ergo non eſt infinitum, quod eſt con­<lb></lb><arrow.to.target n="marg888"></arrow.to.target><lb></lb>tra petita.</s>
        </p>
        <p type="margin">
          <s id="id004399"><margin.target id="marg887"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>quin­<lb></lb>ti<emph.end type="italics"></emph.end> E<emph type="italics"></emph>lem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="id004400"><margin.target id="marg888"></margin.target>4. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004401">Propoſitio ducenteſima uigeſima quinta.</s>
        </p>
        <p type="main">
          <s id="id004402">Proportio media non eſt ex ratione agentis ſed patientis.</s>
        </p>
        <figure id="id.015.01.275.1.jpg" xlink:href="015/01/275/1.jpg"></figure>
        <p type="main">
          <s id="id004403">Proponatur a quantitas, quę debeat mutari ab uir­<lb></lb><arrow.to.target n="marg889"></arrow.to.target><lb></lb>tute quæ non fit in materia, &amp; palam eſt quod non po<lb></lb>terit permutari in inſtanti, quia ſimul eſſet, &amp; non eſſet <lb></lb>ergo repugnaret, neque etiam poteſt non eſſe, ut demonſtratum eſt <lb></lb>in Hyperchen, quia repugnant neceſſario &amp; eſſentiæ Dei, neque mo­<lb></lb>uetur à certa proportione, quia b caret omni quantitate, ergo ni­<lb></lb><arrow.to.target n="marg890"></arrow.to.target><lb></lb>hil oſtendit uim ipſius b eſſe finitam, quod ergo moueatur tardè ce<pb pagenum="257" xlink:href="015/01/276.jpg"></pb>leriter paruum magnum, iſtud contingit totum ex conditionibus <lb></lb>a, id eſt, materiæ &amp; quantitatis: uelut, gratia exempli, ſi a eſſet in ua­<lb></lb>ſculo palmi, non poſſet implere iugerum, &amp; hoc <expan abbr="nõ">non</expan> oſtendit ullam <lb></lb>imperfectionem in b. </s>
          <s id="id004404">Et ſicut homines omnes ſunt in carcere huius <lb></lb>mundi, &amp; tamen uidentur eſſe ſibi liberi, &amp; appellant <expan abbr="ſolũ">ſolum</expan> illos eſſe <lb></lb>in carcere qui ſunt in ergaſtulo, ita omnis materia, &amp; omnis quan­<lb></lb>titas habet conditiones, per quas (ut ita <expan abbr="dicã">dicam</expan>) conſtringitur, &amp; repu<lb></lb>gnat eas mutari, &amp; ideò <expan abbr="uitã">uitam</expan> agunt ſine ulla proportione. </s>
          <s id="id004405">Quod ue <lb></lb>rò dictum eſt, ſupra dictum fuit, per exemplum dictum eſt, <expan abbr="nõ">non</expan> quia <lb></lb>ita ſit, finge ergo quod in aliquo pariete, non ſit albitudo, niſi unius <lb></lb>gradus, illa non operabitur niſi per unum <expan abbr="gradũ">gradum</expan>, etiam ſi calx eſſet <lb></lb>infinitè alba, &amp; ſimiliter de luce Solis, ergo omnes mentes mouent <lb></lb>ſine proportione, &amp; non poſſunt dici finitæ uel infinitæ, quia ipſæ <lb></lb>ſunt expertes omnis quantitatis, imò omnis relationis ad quantita<lb></lb>tem, &amp; hoc eſt quod latuit multos, &amp; maximè propter dictum Phi­<lb></lb>loſophi, eſt ergo omnis operatio iuxta id quod eſt in materia, &amp; <lb></lb>non quod una mens maiores habeat uires, alia cum non ſit in eis, <lb></lb>neque maius neque minus.</s>
        </p>
        <p type="margin">
          <s id="id004406"><margin.target id="marg889"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="margin">
          <s id="id004407"><margin.target id="marg890"></margin.target>P<emph type="italics"></emph>er<emph.end type="italics"></emph.end> 3. P<emph type="italics"></emph>etit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="id004408">Propoſitio ducenteſima uigeſima ſexta.</s>
        </p>
        <p type="main">
          <s id="id004409">Proportio ſublimis non conſiſtit in magnitudine, ſed ordine <lb></lb>iuxta quem differentia eſt eius quod eſt ante &amp; poſt.</s>
        </p>
        <p type="main">
          <s id="id004410">Non enim poteſt eſſe comparatio iuxta magnitudines motas, <lb></lb><arrow.to.target n="marg891"></arrow.to.target><lb></lb>quoniam uel ſunt corpora cœleſtia, uel elementaria, <expan abbr="elemẽtaria">elementaria</expan> eſſe <lb></lb>non poſſunt, quia illa cum ſint corruptioni obnoxia, id eſt, tranſmu<lb></lb>tationi, ſecundum qualitatem <expan abbr="nõ">non</expan> poſſunt eſſe ſubiecta in corpor ca­<lb></lb>rum ſubſtantiarum, neque à primis ſubſtantijs moueri, neque etiam ex­<lb></lb>cipere primò lumen ſuum, ſed mouentur per uim influxam à cœle­<lb></lb>ſtibus corporibus, neque etiam per motum corporum <expan abbr="cœleſtiũ">cœleſtium</expan>, nam <lb></lb>illa non mouentur ſecundum proportionem mentis ad corpus, ſed <lb></lb>iuxta rationem finis, à qua circumſcribuntur, &amp; ideo quod Satur­<lb></lb>nus moueatur uelociore motu, quàm Iuppiter ab Oriente in Occi­<lb></lb>dentem, hoc non eſt, quia uitæ quæ mouet Saturnum fit robuſtior <lb></lb>uita quę mouet Iouem, cum ſint una &amp; eadem: uel ſi dicas quod ſint <lb></lb>diuerſæ uita Saturni, non tamen eſt ualidior in comparatione ad <lb></lb>ſuum cœlum, uita Iouis non moueret celerius Saturnum ab Occi­<lb></lb>dente in Orientem, quàm uita Iouis Iouem, quod eſt falſum, ſed ta­<lb></lb>lis motus uelo citas eſt ratione finis, quia oportet ut pariter mouea­<lb></lb>tur eo motu, &amp; quia cœlum Saturni eſt maius, ideo celerius moue­<lb></lb>tur quam Iouis, &amp; hoc ratione corporis mobilis, &amp; <expan abbr="nõ">non</expan> ratione pro­<lb></lb>portionis ad corpus. </s>
          <s id="id004411">Dico etiam, quod non habent potestatem <lb></lb>aliam, per quam ſubeant proportionem, nam quęritur cuius com­


<pb pagenum="258" xlink:href="015/01/277.jpg"></pb>paratione illa proportio oriatur, nam non ad corpora, quia neque <lb></lb>ad cœleſtia, neque mortalia, ut dictum eſt, niſi fingamus alia corpora, <lb></lb>quod eſt abſurdum, neque etiam ratione incorporeorum, nam non <lb></lb>poſſunt deſtruere ſe inuicem, quia inferior non poteſt tollere ſupe­<lb></lb>riorem, neque multo minus poteſt uelle. </s>
          <s id="id004412">Hoc eſt enim nefas cogita­<lb></lb>re, neque ſuperior inferiorem, quam producit quam amat: &amp; ideo <lb></lb>dico, quod ſunt in proportione ſublimium, id eſt, ordine perfectio­<lb></lb>nis, qui conſiſtit in propinquitate ad primam cauſam. </s>
          <s id="id004413">exemplum, <lb></lb>Sol eſt longe perfectior ſua luce, quæ eſt ei propria, quia Sol eſt <lb></lb>ſubſtantia, &amp; lux eſt proprium, &amp; lux Solis eſt multo perfectior lu­<lb></lb>mine, cum ſit (ut dixi) lux proprium &amp; in Sole, tanquam in ſubie­<lb></lb>cto, lumen autem extra &amp; accidens. </s>
          <s id="id004414">Nec tamen dicendum eſt, quod <lb></lb>Sol ſit potentior luce, aut lux lumine, idem dico de anima &amp; facul­<lb></lb>tatibus eius, &amp; functionibus, inter quas nulla cadit proportio per­<lb></lb>fectionis, tamen differentia conſpicua eſt, &amp; ideo poterit impediri <lb></lb>functio, &amp; non facultas, et facultas tolli remanente anima. </s>
          <s id="id004415">Forſan di<lb></lb>ces, quod iſtę non ſunt ſubſtantiæ, &amp; ideò oporteret, ut omnia in­<lb></lb>corporea Deo ſolo excepto eſſent accidentia, dico quod in incor­<lb></lb>poreis non eſt ſicut in anima, quæ eſt iuncta corpori, neque ut in So­<lb></lb>le quod eſt corpus, ſed tanta eſt perfectio producti incorporei, <lb></lb>quod ipſum eſt ſubſtantia. </s>
          <s id="id004416">Et ratio eſt quia ſubſtantia differt ab ac­<lb></lb>cidente uel ratione corporis, ut aqua à frigiditate, &amp; hoc non eſt in <lb></lb>incorporeis, ut manifeſtum eſt, uel quia unum ſit ſubiectum alte­<lb></lb>rius, &amp; ideò ſubſtantia, ut eſt principium comparationis, &amp; in ſe <lb></lb>ipſa dicitur ſubſtantia, &amp; ut comparatur ad extra &amp; ad operatio­<lb></lb>nem ſuam, cuius eſt principium dicitur facultas: uelut uita cœle­<lb></lb>ſtis ſubſtantia eſt, ut uerò cœlum pulchritudine illius delectatum <lb></lb>mouetur ad obſequium, dicitur facultas in illa uita, &amp; non eſt niſi <lb></lb>ſubſtantia, tamen ipſius uitæ adeo ut ſola ratione differant. </s>
          <s id="id004417">Tertia <lb></lb>differentia eſt, quia ſubſtantia non eſt in ſubiecto, ſed facultas eſt in <lb></lb>ſubiecto, uerùm in incorporeis, ut dixi, non differunt niſi ſola ra­<lb></lb>tione, uelut pater &amp; homo, nam pater neceſſariò eſt homo, &amp; eſt <lb></lb>ſubſtantia, ut ad aliud comparatur. </s>
          <s id="id004418">Quarta differentia eſt ratione <lb></lb>propriæ naturæ quæ non dependet, nam ſubſtantia non pendet <lb></lb>ſicut accidens &amp; facultas, uerùm ubi genita fuit non amplius pen­<lb></lb>det: reſpondeo, quod in incorporeis producitur, &amp; non repugnet <lb></lb>productio ſubſtantiæ, quia ſi non repugnat generatio hominis, <lb></lb>quod ſit ſub ſtantia, multo minus etiam incorporeorum. </s>
          <s id="id004419">Relinqui­<lb></lb>tur ut obijcias, quoniam ſubſtantiæ incorporeæ ſemper fiunt, er­<lb></lb>go nunquam ſunt ueræ ſubſtantiæ: ad hoc reſpondendum eſt per <lb></lb>interemptionem, nam de uera reſponſione non eſt hic locus, quod 


<pb pagenum="259" xlink:href="015/01/278.jpg"></pb>cadem ratione qua producuntur uitæ, producuntur etiam cœli, at <lb></lb>cœlum nihilominus eſt uerè ſubſtantia, &amp; magis iſtis mortalibus, <lb></lb>ergo uel talis productio non eſt perpetua, uel, ut uerius dicam, eſt <lb></lb>ſimpliciter productio circumſcripta ab omni tempore præſenti, <lb></lb>præterito &amp; futuro. </s>
          <s id="id004420">Quare erit magis uera productio quam ſub­<lb></lb>ſtantiæ mortalis, ideo contingit hic error ex diſsimilitudine eo­<lb></lb>rum quæ maximè ſimilia eſſe uidentur, nam cùm accidentia pro­<lb></lb>ducantur in tribus temporibus, &amp; incorporea in nullo, ſubſtantia <lb></lb>autem mortales ſolum in uno tempore, ideò productio incorpo­<lb></lb>reorum uidetur eſſe ſimilis productioni accidentium, cum tamen <lb></lb>productio ſubſtantiæ mortalis ſit uerè media inter illas, nam ſub­<lb></lb>ſtantia mortalis producitur in uno tempore, accidens in omni <lb></lb>ſubſtantia immortalis in nullo, neceſſe eſt autem extrema magis <lb></lb>differre inter ſe quàm à media, igitur ſubſtantiæ in corporeæ ordi­<lb></lb>ne &amp; perfectione differunt, non tamen proportionem habent. </s>
          <s id="id004421">Et <lb></lb>ſi quis dicát, quod ultima ſubſtantia eſſet ęquè potens, ut Deus: re­<lb></lb>ſpondeo quod non eſt uerum, quia uel loqueris de perfectione, &amp; <lb></lb>ita demonſtratum eſt, quod Deus eſt ipſa perfectio, ultima ſub­<lb></lb>ſtantia eſt imperfectiſsima: uel loqueris de magnitudine, &amp; ita non <lb></lb>ſunt æquales prima &amp; ultima ſubſtantia, quia non poſſunt com­<lb></lb>parari, ſicut lumen non poteſt comparari lumini, quod ſit dul­<lb></lb>cius uel amarius, grauius uel leuius, maius enim &amp; minus, &amp; æ­<lb></lb>quales ſunt differentię quantitatum, uitæ autem non habent quan­<lb></lb>titatem operationis, quia, ut dixi, eſt abſolutiſsima ratione finis, ne­<lb></lb>que potentiam ad aliquid, quia ſunt in æterno actu, &amp; hoc ſecun­<lb></lb>dum philoſophos, &amp; iuxta rationem numinis naturalis, nam ſe­<lb></lb>cus religio &amp; fides tenent, quia ſupponunt mundum eſſe creatum, <lb></lb>&amp; ſic potentia differentiæ ab actu, quia Deus nunc creauit, &amp; antea <lb></lb>non creauerat, &amp; tamen poterat creare.</s>
        </p>
        <p type="margin">
          <s id="id004422"><margin.target id="marg891"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004423">Ex hoc patet, quod nulla ſubſtantia incorporea eſt finita nec infi<lb></lb><arrow.to.target n="marg892"></arrow.to.target><lb></lb>nita, nec extenſa nec contracta, quia omnia iſta pertinent ad quan­<lb></lb>titatem, quarum illę omnino ſunt expertes.</s>
        </p>
        <p type="margin">
          <s id="id004424"><margin.target id="marg892"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004425">Propoſitio ducenteſima uigeſima ſeptima.</s>
        </p>
        <p type="main">
          <s id="id004426">Vitæ iuxta numerum perfectionum in comparatione ad cogita­<lb></lb>tionem noſtram proportionem quandam habent.</s>
        </p>
        <p type="main">
          <s id="id004427">Velut Deus eſt per ſe primo abſolutum, &amp; cauſa omnium bo­<lb></lb><arrow.to.target n="marg893"></arrow.to.target><lb></lb>norum, &amp; eſſe, ſapientia uerò quæ generatur à primo bono, non eſt <lb></lb>cauſa omnium bonorum, quia ſic produceret primum bonum, <lb></lb>&amp; produceretur eſt tamen per ſe primo &amp; abſolutum bonum, 


<pb pagenum="260" xlink:href="015/01/279.jpg"></pb>amor autem eſt cauſa omnium bonorum poſteriorum, &amp; abſolu­<lb></lb>tum, &amp; per ſe ſed non primò, &amp; ita de uita quæ regit mundum, ipſa <lb></lb>non eſt abſoluta, neque per ſe primò, ſed ſolum cauſa omnium bono­<lb></lb>rum, eſt tamen abſoluta in ordine <expan abbr="bonorũ">bonorum</expan>, quæ retinuit, &amp; hoc mo­<lb></lb>do dicimus eſſe plures perſonas in diuinis plures mentes, &amp; ſub­<lb></lb>ſtantias incorporeas.</s>
        </p>
        <p type="margin">
          <s id="id004428"><margin.target id="marg893"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>m.</s>
        </p>
        <p type="main">
          <s id="id004429">Propoſitio ducenteſima uigeſima octaua.</s>
        </p>
        <p type="main">
          <s id="id004430">Proportionem ſcientiæ futurorum &amp; cæterorum occultorum <lb></lb>conſiderare.</s>
        </p>
        <p type="main">
          <s id="id004431">Septem licet ſint modi futura &amp; occulta prægnoſcendi, quędam <lb></lb><arrow.to.target n="marg894"></arrow.to.target><lb></lb>tamen ſunt communia omnibus, quædam multis: uaria quoque eſt <lb></lb>ratio horum, alia enim eſt proportio ſciendi, atque hæc duplex, uel ex <lb></lb>ratione intelligendi quæ ortum habet ex comparatione animæ ad <lb></lb>magnitudinem &amp; difficultatem eorum, quæ <expan abbr="cognoſcũtur">cognoſcuntur</expan>, quędam <lb></lb>ad modum quo <expan abbr="iudicãtur">iudicantur</expan>. </s>
          <s id="id004432">Alia rurſus eſt ratio proportionis modi <lb></lb>ad animam ipſam, ut quiſque propior fuerit ipſi aut remotior, alia <lb></lb>demum eſt differentiæ <expan abbr="ſignorũ">ſignorum</expan> aut cauſarum, ergo ut à propinqui­<lb></lb>tate initium ducam, ſeptem uidentur eſſe ordines, qui etiam ad per­<lb></lb>fectionem dijudicandi pertinent. </s>
          <s id="id004433">Primus eſt eorum quæ agimus <lb></lb>quibus prudentia dominatur, atque hic admodum certus eſt, ut in <lb></lb>negotijs publicis priuatis que uidemus, eſt <expan abbr="autẽ">autem</expan> duplex, ciuilis &amp; mili <lb></lb>taris. </s>
          <s id="id004434">Secundus eſt naturalium, eſt autem maximè euidens in tribus <lb></lb>medicina, agricultura &amp; nauigatione. </s>
          <s id="id004435">Tertius eſt eorum quæ ſunt <lb></lb>ſecundum naturam, ſed non per cauſas, uelut aſtrologia &amp; phyſio­<lb></lb>gnomia. </s>
          <s id="id004436">Eius <expan abbr="aũt">aunt</expan> tres ſunt partes phyſiognomia, metopoſcopia &amp; <lb></lb>chiromantia, namque aſtrologia etſi per cauſas ſit, magis tamen per <lb></lb>ſigna oſtendere uidetur, nam quod Iuppiter in aſcendente bonos <lb></lb>præbeat mores, cur magis hoc in loco uel illo, magna eſt quæſtio. <lb></lb></s>
          <s id="id004437">Quartus eſt conſenſus omnium nobiſcum atque fatale uin culum, in <lb></lb>quo genere ponuntur fulgrum caſus, exta, &amp; augurium &amp; hygro­<lb></lb>mantia. </s>
          <s id="id004438">In quinto modo ponuntur ea quæ cum anima noſtra con­<lb></lb>ſenſum habent, eiuſmodi ſunt uitæ aut genij aut eroes. </s>
          <s id="id004439">Sextus uerò <lb></lb>eſt ex origine, uelut ſunt Prophetæ &amp; uates Sybillæque, quorum uis <lb></lb>alia in ſe ipſis, ut prophetarum, alia uaporis ut Delphici oraculi, alia <lb></lb>aquę uelut in Colophonio oraculo. </s>
          <s id="id004440">Vltimum eſt præſtantiſsimum <lb></lb>idemque <expan abbr="remotiſsimũ">remotiſsimum</expan>, quod à Deo per preces <expan abbr="cõſequimur">conſequimur</expan>. </s>
          <s id="id004441">In omni­<lb></lb>bus ergo his iuuat præſtantia modi non auſpicium, &amp; exta paruam <lb></lb>habent ſignificationem, quæ uero à Deo maximam, alia enim eſt <lb></lb>proportio agentis, ut Dei alia modi agendi, uelut quæ per cauſas <lb></lb>fit melior quàm quæ per ſigna, alia impreſsionis lucis aut efficacis, <lb></lb>alia coniunctionis naturæ nobiſcum. </s>
          <s id="id004442">Quod uerò ad nos attinet, 


<pb pagenum="261" xlink:href="015/01/280.jpg"></pb>aliud eſt ex peritia artis, aliud ex iudicio acri, aliud ex diligentia. <lb></lb></s>
          <s id="id004443">Differentia autem cognoſcendi ſunt multorum aut paucorum ex­<lb></lb>actæ, uel non exactæ, ſecuræ aut dubiæ, atque horum omnium cauſa <lb></lb>eſt magnitudo proportionis, aut in origine ad <expan abbr="ſignificandũ">ſignificandum</expan>, aut in <lb></lb>anima ad <expan abbr="intelligẽdum">intelligendum</expan>. </s>
          <s id="id004444">Atque originis, ut dixi, multiplex eſt ratio, ſci <lb></lb>licet modi uel cauſæ uel efficaciæ, cùm uerò hæc omnia in unum <lb></lb>conuenerint, certiſsima &amp; exactiſsima fiet diuinatio, cum pauca &amp; <lb></lb>minus ualida, ut pote diſcurſus &amp; iudicium dubia, debilis &amp; pauco<lb></lb>rum. </s>
          <s id="id004445">Quæ uerò nugantur Porphyrius &amp; Iamblicus de his, omni­<lb></lb>no fabulis ſimilia ſunt, uideturque Iamblicus Porphyrio indixiſſe <lb></lb>bellum, ſed cum ignauo hoſte, ipſe longe deterior.</s>
        </p>
        <p type="margin">
          <s id="id004446"><margin.target id="marg894"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004447">Propoſitio ducenteſima uigeſima nona.</s>
        </p>
        <p type="main">
          <s id="id004448">Incorporea omnia unum ſunt, neque numerus eſt eorum.</s>
        </p>
        <p type="main">
          <s id="id004449">Videbitur ab initio paradoxum, ſed ubi &amp; modum &amp; demon­<lb></lb><arrow.to.target n="marg895"></arrow.to.target><lb></lb>ſtrationem ipſam deprehenderis, intelliges ita eſſe iuxta luminis na <lb></lb>turalis rationem, tum uerò maximè, cum id adiecero non prohibe­<lb></lb>re me, quin ut partes in homine numerentur. </s>
          <s id="id004450">Sed aliud eſt partes in <lb></lb>homine dinumerare, quæ numero ipſo non diſtinguuntur, ſed ſi <lb></lb>plures homines ſeorſum de earum numero interroges ſinguli di­<lb></lb>uerſa, nec <expan abbr="exiguõ">exiguom</expan> interuallo differentia reſpondebunt, ſed unus de­<lb></lb>cem puta, alius centum, alius innumerabiles pronuntiabit. </s>
          <s id="id004451">Quin <lb></lb>etiam quiſque qua ratione uelis illas diſtinguere interrogabit, at non <lb></lb>ſic de numero gregis pauidum, aut de pecunijs, in quibus nemo ab <lb></lb>altero diſſentiet, niſi cum in numerando errorem admiſerit. </s>
          <s id="id004452">Igitur <lb></lb>dico non eſſe numerum in incorporeis, nam finitus erit uel infini­<lb></lb>tus: ſi infinitus, numerus non erit, quoniam primum nullus Deus <lb></lb>erit nulla prima ſubſtantia: nam quomodo Deus erit aut Domi­<lb></lb>nus infinitorum, aut primus ubi non eſt ultimum? </s>
          <s id="id004453">Sed neque nume­<lb></lb>rus aliquis certus earum eſſe poteſt, cum primum non magis hic <lb></lb>quàm ille: neque enim definiuntur ullo termino, ſeu centum, ſeu mil­<lb></lb>le aut millies mille: nec cum ſubijciantur quantitati continuæ pote­<lb></lb>runt ſubijci numero, uel alteri cuipiam accidenti. </s>
          <s id="id004454">Sed omnia ſunt <lb></lb>unum, ita tamen quod perfectius eſt atque imperfectius diffuſum ab <lb></lb>ipſo infinito, cuius in extremo cohærent mentes noſtræ &amp; animæ, <lb></lb>&amp; cœlum, quæ communicatæ inferioribus atque corporibus illa <lb></lb>agunt, mutant &amp; ſeruant. </s>
          <s id="id004455">Ipſum quàm ultimum eſſe, eſt in mundo, <lb></lb>quod eſt corpus, &amp; eius pars præcipua cœlum deinde reliqua. <lb></lb></s>
          <s id="id004456">Omniaque mouentur &amp; transferuntur immobili primo principio, <lb></lb>quod cum illis <expan abbr="coniunctũ">coniunctum</expan> eſt: nam reliqua incorporea ab ipſo pro­<lb></lb>fluunt. </s>
          <s id="id004457">Eſt &amp; ratio Ariſtotelis in tertio decimo Theologicorum ſer<lb></lb><arrow.to.target n="marg896"></arrow.to.target><lb></lb>monum, Deus non eſt unus numeri ratione, ſed ita ut non ſit plura, 


<pb pagenum="262" xlink:href="015/01/281.jpg"></pb>igitur in mundo toto incorporeo non eſt numerus. </s>
          <s id="id004458">Si enim Deus <lb></lb>eſſet unus numero, non poſſet eſſe ens commune, &amp; uniuerſim am­<lb></lb>plectens cuncta, &amp; accidens contineret, quæ omnia ſunt falſa, abſur<lb></lb>da, nefaria &amp; impia, licet tamen (ut dixi) menti humanæ quæ omnia <lb></lb>reducit ad ſimilitudinem ſenſilium, à quibus originem traxit ſuæ <lb></lb>operationis fingere numeros, ſicut in partibus hominis, aut cœli, <lb></lb>aut aeris iuxta ſitum, aut magnitudinem. </s>
          <s id="id004459">Eſt etiam alius modus <lb></lb>iuxta quem Ariſtoteles numerauit mentes quæ mouent corpora <lb></lb>cœleſtia, quod abſurdum non eſt, uelut ſi quis numeret digitos, in <lb></lb>pulſante chelim, erunt quatuor aut ſex, non tamen eſt numerus ille <lb></lb>uerè plurium, cum ad unum hominem referuntur. </s>
          <s id="id004460">Et cum ſit mun­<lb></lb><arrow.to.target n="marg897"></arrow.to.target><lb></lb>dus hic imago ſuperioris, ut ille dicebat, &amp; inferior poteſtate conti­<lb></lb>neat infinitas partes, infinitas ordinis ratione ſuperior continebit. <lb></lb></s>
          <s id="id004461">Sed non infinitas numero. </s>
          <s id="id004462">Exempli gratia, proponamus quod So<lb></lb>lis uis dirigatur ad nos uſque impedita per nebulas, ut <expan abbr="nõnunquam">nonnunquam</expan> <lb></lb>contingit: erit ergo perfectio una, ſed ordinata omnium radiorum: <lb></lb>adeò quod ſi infinita uaſa applicarentur aqua plena infinitæ ratio­<lb></lb>nes iridis apparerent, quæ omnes continerentur poteſtate in radijs <lb></lb>illis ratione comparationis ad uaſa &amp; irides, per ſe autem, ut ſunt <lb></lb>perfectiones eſſent in actu.</s>
        </p>
        <p type="margin">
          <s id="id004463"><margin.target id="marg895"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id004464"><margin.target id="marg896"></margin.target>S<emph type="italics"></emph>up.<emph.end type="italics"></emph.end> 5.</s>
        </p>
        <p type="margin">
          <s id="id004465"><margin.target id="marg897"></margin.target>Lib. 7. <emph type="italics"></emph>cap.<emph.end type="italics"></emph.end><lb></lb>4.</s>
        </p>
        <p type="main">
          <s id="id004466">Propoſitio ducenteſima trigeſima.</s>
        </p>
        <p type="main">
          <s id="id004467">Proportio incorporeorum aſcendentium ſemper maior eſt.<lb></lb><arrow.to.target n="marg898"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="id004468"><margin.target id="marg898"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004469">Cum proportio illa ſit quaſi ſimilis decori, &amp; ideo muſicæ geo­<lb></lb><arrow.to.target n="marg899"></arrow.to.target><lb></lb>metrica maior eſt in maioribus ac magnitudinibus, ut ſuprà docui­<lb></lb>mus. </s>
          <s id="id004470">Sed non eſt neque geometrica, neque arithmetica, nec muſica, nec <lb></lb>per recenſum, eſſent enim quantitates quæ <expan abbr="compararent̃">compararentur</expan>: unaquęque<lb></lb> enim harum inter quantitates conſtituta: at illa eſt ut producentis <lb></lb>ad productum. </s>
          <s id="id004471">Et non comparantur quoad æternitatem, quia ut <lb></lb>aliâs declaraui, omnis ſubſtantia eſt æterna: quanto magis incor­<lb></lb>porea. </s>
          <s id="id004472">Quia ergo primum per <expan abbr="præcedẽtem">præcedentem</expan> habet rationem totius, <lb></lb>&amp; eſt infinitum, <expan abbr="ſecundũ">ſecundum</expan> ea parte qua recedit, quia primum non eſt, <lb></lb>plus diſtat a primo quam à tertio, igitur deſcendendo uſque ad pri­<lb></lb>ma elementa. </s>
          <s id="id004473">Sed obijcies de qualitatibus &amp; accidentibus: dico <lb></lb>quod habent <expan abbr="mediũ">medium</expan> eſſe, licet tempore infinito uincantur à ſubſtan<lb></lb>tijs, illę tamen etiam uincuntur &amp; abſque participatione perfectionis <lb></lb>illius cum <expan abbr="accidẽtia">accidentia</expan> participent eſſentia &amp; tempore, &amp; ſi quis dicat, <lb></lb>cur ergo Sol &amp; lupiter <expan abbr="nõ">non</expan> ſunt locati in ſupremis orbibus, cum ſint <lb></lb>nobiliores &amp; maiores &amp; potentiores cæteris erraticis: dico quòd <lb></lb>fuit ob mundum inferiorem, quoniam ſi fuiſſent altiores mundus <lb></lb>inferior frigore corrumperetur, quando quidem uel ſic frigore pre­<lb></lb>mantur, in hyeme etiam ſub torrida plaga, &amp; ſub polis ac iuxta eos 


<pb pagenum="263" xlink:href="015/01/282.jpg"></pb>ſemper. </s>
          <s id="id004474">Et orbes ſuperiores <expan abbr="nõ">non</expan> indigebant lumine Solis, quod ap­<lb></lb>paret in nocte ſerena, cum etiam adeò à nobis diſtent. </s>
          <s id="id004475">Vnde ſi cani­<lb></lb>cula eſſet in cœlo Lunæ, plus luminis afferret centuplo quàm Lu­<lb></lb>na, cùm diſtantia ſit quingentupla diſtantiæ Lunæ à terra. </s>
          <s id="id004476">Et ſi Sol <lb></lb>eſſet factus adeo maior, ut in orbe Saturni conſiſtens calefaceret ter<lb></lb>ram æqualiter, ut non exureretur in æſtate, hyeme neceſſe eſſet, ut ni <lb></lb>mium gelaſceret. </s>
          <s id="id004477">Sin autem æquale eſſet frigus in hyeme, exurere­<lb></lb>tur terra per æſtatem, quando quidem nec ſic illam pati poſsint, qui <lb></lb>in torrida plaga habitant. </s>
          <s id="id004478">Et ſi Sol eſſet ubi eſt Luna, &amp; eo minor <lb></lb>non illuminarentur orbes ſuperiores. </s>
          <s id="id004479">Ideo nobilitas non eſt in or­<lb></lb>bibus ob altitudinem, ſed ob ſubſtantiam incorpoream quæ illi do <lb></lb>minatur. </s>
          <s id="id004480">Et eſt in loco congruenti toti corpus, uita autem non eſt <lb></lb>in loco.</s>
        </p>
        <p type="margin">
          <s id="id004481"><margin.target id="marg899"></margin.target>P<emph type="italics"></emph>rop.<emph.end type="italics"></emph.end> 171.</s>
        </p>
        <p type="head">
          <s id="id004482">LEMMA.</s>
        </p>
        <figure id="id.015.01.282.1.jpg" xlink:href="015/01/282/1.jpg"></figure>
        <p type="main">
          <s id="id004483">Et proponantur a &amp; b in proportione dupla alti­<lb></lb>tudinum &amp; magnitudinum, &amp; <expan abbr="cõparentur">comparentur</expan> ad d, erit <lb></lb>ergo angulus a d c maior b d c, quare ſi ſunt æquales <lb></lb>uires in a b, refrigerabitur magis d ab a quam b, &amp; <lb></lb>ita patet utraque pars dicti in fine propoſitionis.</s>
        </p>
        <p type="main">
          <s id="id004484">Propoſitio ducenteſima trigeſima prima.</s>
        </p>
        <p type="main">
          <s id="id004485">Tres eſſe mundos, atque inter ipſos nullam eſſe proportionem: <lb></lb>nec numero eos definiri.</s>
        </p>
        <p type="main">
          <s id="id004486">Cum palam ſit eſſe corporeum mundum ut elementa, &amp; incor­<lb></lb><arrow.to.target n="marg900"></arrow.to.target><lb></lb>poreum ut Dei, &amp; medium eſſe neceſſe eſt uitarum &amp; hominum ac <lb></lb>cœleſtium, quòd primum ſenſu patet, ut cœli, hominum &amp; anima­<lb></lb>lium, atque plantarum, &amp; ratione etiam, quoniam extrema contraria <lb></lb><expan abbr="nõ">non</expan> propriè medio copulantur, ut incorporeum ac corporeum. </s>
          <s id="id004487">Di­<lb></lb>co igitur nullam eſſe inter hos proportionem atque numerum face­<lb></lb>re: nam de numero conſtat, quoniam non ſunt tres, quia ſint in ordi <lb></lb>ne numerorum, ſed ut principium, medium, finis, &amp; perfectum, per­<lb></lb>fectius, perfectiſsimum: ſcilicet poſitiuum, comparatiuum &amp; ſuper­<lb></lb>latiuum. </s>
          <s id="id004488">Et quoniam ſunt extrema cum medio, ideò ſunt in propor<lb></lb>tione ſublimi etiam &amp; non propria. </s>
          <s id="id004489">Quod ſi eſſent maximè mun­<lb></lb>di uitalis ad corpora, ſed corpora <expan abbr="nõ">non</expan> mouentur niſi iuxta finem ui­<lb></lb>tæ, &amp; non uim: ipſa enim ſi poſſet habere uoluntatem infinitam mo<lb></lb>ueret in inſtanti: quia corpora non reluctantur animabus ſuis, ſed <lb></lb>quantus eſt actus in animabus &amp; uitis, tanta eſt <expan abbr="potẽtia">potentia</expan> ad unguem <lb></lb>in corporibus, ergo non contingit proportio in mundo uitarum <lb></lb>uera niſi illa ſublimis. </s>
          <s id="id004490">Neque enim finita eſt quæ nullis circumſcribi­<lb></lb>tur terminis, neque infinita quo finitam pręſupponit, ſed neque inter <lb></lb>mundum &amp; incorporeum &amp; uitarum cùm mentes non moueant, 


<pb pagenum="264" xlink:href="015/01/283.jpg"></pb>uitæ moueant: &amp; quod mouet neceſſariò mouet, &amp; quod non po­<lb></lb>teſt mouere, quoniam omnia æterna ſunt: &amp; in ęternis idem eſt eſſe <lb></lb>ac poſſe: igitur inter mundum incorporeum &amp; uitarum nulla eſt <lb></lb>proportio uera, ſed ſolum ſublimis, nec numerus: niſi ut à nobis fin<lb></lb>gitur. </s>
          <s id="id004491">Velut ſi dicamus in tabula, &amp; in negocio eſt principium me­<lb></lb>dium finis, &amp; hæc poſſunt dici tria quatenus diſtinguuntur: ſed <expan abbr="nõ">non</expan> <lb></lb>ob hoc dicendum eſt tabulam, aut negotium habere tres partes, <lb></lb>multo minus eſſe tria negocia aut tres tabulas.</s>
        </p>
        <p type="margin">
          <s id="id004492"><margin.target id="marg900"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004493">Propoſitio ducenteſima trigeſima ſecunda.</s>
        </p>
        <p type="main">
          <s id="id004494">Omnis motus naturalis, quanto uelocior eſt, tanto propior eſt, <lb></lb>&amp; magis ſimillimus quieti.</s>
        </p>
        <p type="main">
          <s id="id004495">Hæc propoſitio primo intuitu uidetur eſſe falſa, quoniam cùm <lb></lb><arrow.to.target n="marg901"></arrow.to.target><lb></lb>motus ſit contrarius quieti, &amp; efficiat actiones quieti contrarias, <lb></lb>quantò uelocior erit tanto remotior à natura quietis &amp; magis diſsi<lb></lb>milis, propterea intelligere oportet primum, in quo ſenſu uerba <lb></lb>ſint accipienda, nam hæc propoſitio, &amp; authoritate, &amp; ſenſu &amp; du­<lb></lb>plici ratione euidenti manifeſta eſt. </s>
          <s id="id004496">Oportet igitur <expan abbr="primũ">primum</expan> ſcire quo <lb></lb>ad locum attinet tria eſſe diſcrimina: quietem in eodem: tranſitum <lb></lb>ad alium per medium: &amp; tranſitum ad alium ſine medio. </s>
          <s id="id004497">Duorum <lb></lb><expan abbr="primorũ">primorum</expan> exempla notiſsima ſunt, tertij eſt hoc, ſi urceus aqua ple­<lb></lb>nus exponatur Soli, &amp; efficiatur iridis imago in tabula: inde ſubla­<lb></lb>ta tabula eadem iris appareat in muro, erit tranſitus ſine media, quia <lb></lb>quod ſit eadem dubium non eſt, ijdem radij &amp; idem corpus ſpecu­<lb></lb>lare, quod uerò tranſeat ſine medio, <expan abbr="primũ">primum</expan> ſenſus docet, ſecundum <lb></lb>ratio, quia fit in inſtanti, ut Secundo de Anima. </s>
          <s id="id004498">Rurſus Sol illuſtret <lb></lb><arrow.to.target n="marg902"></arrow.to.target><lb></lb>urceum aqua plenum: appareat ex hoc iris in muro, interponatur <lb></lb>aliquid, &amp; transferatur urceus, apparebit iris alia loco, &amp; non tran­<lb></lb>ſiuit per medium, uidetur idem de intellectu, &amp; ui imaginandi, qui­<lb></lb>bus ex Germania tranſeo in Indiam ſubitò: &amp; eodem modo ex ani­<lb></lb>ma ſalicis, in hac planta fit tranſitus in proximam neque per medium, <lb></lb>quod etiam uidemus in igne &amp; ellychnio proximo, &amp; id ſæpe acci­<lb></lb>dit tum præſertim cum nuper extinctum fuerit.</s>
        </p>
        <p type="margin">
          <s id="id004499"><margin.target id="marg901"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="margin">
          <s id="id004500"><margin.target id="marg902"></margin.target>T<emph type="italics"></emph>ex.<emph.end type="italics"></emph.end> 121.</s>
        </p>
        <p type="main">
          <s id="id004501">Iam ergo id ſupponamus, quod etiam ad rem parum facit, ſed ad <lb></lb>intelligentiam ſatis, uideamus que, quare ſit quod motus opponatur <lb></lb>quieti, &amp; <expan abbr="manifeſtũ">manifeſtum</expan> eſt, quod differentia loci eſt cauſa, nam in quiete <lb></lb>res manet in eodem loco, in motu tranſit ad alium locum, &amp; quan­<lb></lb>tò medium eſt maius, tantò motus eſt manifeſtior, unde ſequitur, <lb></lb>quod in his quæ ualde lentè mouentur, illa uidentur quieſcere, &amp; <lb></lb>poſt aliquot tempus deprehendimus mota fuiſſe, nunquàm tamen <lb></lb>moueri, ſicut in Sole, Luna, ſtellis, unde illa opinio <expan abbr="Philoſophorũ">Philoſophorum</expan> <lb></lb>exiſtimantium omnia ſemper moueri, <expan abbr="nõ">non</expan> omnino poteſt tam bene 


<pb pagenum="265" xlink:href="015/01/284.jpg"></pb>reprobari, quia licet ſenſus <expan abbr="nõ">non</expan> cognoſcat moueri, cognoſcit tamen <lb></lb>mota eſſe, &amp; id ſufficit: multa ergo cognoſcuntur mota eſſe quæ <expan abbr="nõ">non</expan> <lb></lb>cognoſcuntur moueri, uelut lapis grauis ſuperſtans terræ, quem ui <lb></lb>demus poſt annum deſcendiſſe per duos digitos, &amp; tamen ſemper <lb></lb>uidetur quieſcere. </s>
          <s id="id004502">Igitur cum in pari tempore quę uelo citer mouen<lb></lb>tur plus ſpatij ſuperent, maius etiam relinquunt medium inter lo­<lb></lb>cum, &amp; locum, &amp; ob id magis remota ſunt à quiete, &amp; magis illi <expan abbr="cõ­traria">con­<lb></lb>traria</expan>: hæc igitur eſt ratio cur quæ uelocius moueantur, minus quie<lb></lb>ti ſimilia aut proxima exiſtimentur. </s>
          <s id="id004503">Dico ergo, quod illa quæ natu­<lb></lb>raliter uelociſsimè mouentur, ſunt magis ſimilia &amp; magis proxima <lb></lb>ipſis quieſcentibus quàm quæ tardè: cum enim omnis motus natu<lb></lb>ralis neceſſariò <expan abbr="etiã">etiam</expan> ſit regularis, ut qui à uirtute Dei fiat, erit uel per <lb></lb>lineam obliquam aut <expan abbr="rectã">rectam</expan>. </s>
          <s id="id004504">Quoniam uerò <expan abbr="multarũ">multarum</expan> recta eſt per­<lb></lb>fectiſsima, &amp; obliquarum circularis, erit omnis motus naturalis cir<lb></lb>cularis aut rectus: dico ergo quòd in utroque <expan abbr="uerũ">uerum</expan> eſt quod dicitur. <lb></lb></s>
          <s id="id004505">Et <expan abbr="primũ">primum</expan> in circulari ille motus eſt propinquior quieti, in quo par­<lb></lb>tes ſunt propinquiores ſuo loco, ſed ſi uelociſsimus ſit motus, nun­<lb></lb>quàm ita ſunt extra ſuum locum, qui enim in poteſtate ſint proxi­<lb></lb>mæ ei: ergo partes illę inde ſe habent ac ſi quiescerent. </s>
          <s id="id004506">Secunda ra­<lb></lb>tio, quia quod uelociſsimè <expan abbr="mouet̃">mouetur</expan>, abſque dubio tanto tempore quie <lb></lb>ſcit in ſuo loco quantò quod tardè: exemplum. </s>
          <s id="id004507">Luna in triginta an <lb></lb>nis quieſcit in principio arietis <expan abbr="quadringẽteis">quadringenteis</expan> per ſex horas, id eſt, <lb></lb>centum diebus in quadringentis uicibus, Saturnus <expan abbr="cẽtum">centum</expan> diebus <lb></lb>ſed ſemel tantum: ergo tantum Luna quieſcit, quantum Saturnus, <lb></lb><expan abbr="cõparatione">comparatione</expan> ad idem tempus addita pari ratione in alijs partibus, <lb></lb>ſed cum uelocius moueatur Luna quàm Saturnus minus quieſce­<lb></lb>re uidebitur Luna in alijs partibus quàm Saturnus, &amp; tantundem <lb></lb>in principio arietis Luna ut Saturnus, ergo cum Luna tantundem <lb></lb>in principio arietis quieſcat, quantum Saturnus in triginta annis, &amp; <lb></lb>in alijs partibus minus quàm Saturnus, igitur abſolutè Luna plus <lb></lb>quieſcit in principio arietis, quàm Saturnus dato tempore æquali <lb></lb>triginta <expan abbr="annorũ">annorum</expan>. </s>
          <s id="id004508">Et formatur demonſtratio hoc modo: Luna quan <lb></lb>do eſt in loco ipſo, puta in principio arietis, ibidem eſt actu, &amp; quie <lb></lb>ſcit per tantundem temporis <expan abbr="quantũ">quantum</expan> Saturnus, &amp; in omnibus alijs <lb></lb>locis data paritate, eſt ſemper propior ipſi principio arietis poteſta<lb></lb>te quam Saturnus, igitur Luna plus quieſcit in principio arietis <lb></lb>quam Saturnus, quia dum ibidem ſunt æqualiter <expan abbr="quieſcũt">quieſcunt</expan>, &amp; dum <lb></lb>ſunt extra, Luna ſemper eſt propior &amp; poteſtate magis in illo loco, <lb></lb>igitur Luna magis quieſcit in principio arietis quàm Saturnus. </s>
          <s id="id004509">Prę<lb></lb>terea, ſi Luna &amp; Saturnus mouerentur in æquali tempore, &amp; Luna <lb></lb>in paruo circulo, &amp; Saturnus in magno, dubium non eſſet, quin 


<pb pagenum="266" xlink:href="015/01/285.jpg"></pb>Luna non diceretur magis quieſcere in ſuo loco, &amp; diutius quàm <lb></lb>Saturnus, nam Luna ſemper eſſet prope locum ſuum, &amp; Saturnus <lb></lb>perſæpe uideretur procul. </s>
          <s id="id004510">Sed ſi moueantur in eodem circulo, &amp; <lb></lb>Luna moueatur uelociſsimè, Saturnus tardè: perinde erit, ac ſi Lu­<lb></lb>na moueatur in paruo circulo, &amp; Saturnus in magno, ergo quod <lb></lb>uelociſsimè mouetur eſt proximius quieti quàm quod tardè. </s>
          <s id="id004511">Illud <lb></lb>etiam idem manifeſtius erit in extremis, nam quod minimo ſpatio <lb></lb>mouetur propemodum non mouetur. </s>
          <s id="id004512">Sicut, ſi quid circa centrum <lb></lb>moueatur, adeò ut ipſum tangat, non dicetur moueri, ſed quieſcere <lb></lb>ibi, ſed quod uelociſsime mouetur, ſemper uerſatur circa idem, quia <lb></lb>nunquam multum abeſt, quia ibi non quieſcit, igitur quod uelociſ­<lb></lb>ſimè mouetur motu naturali circularſ eſt proximius quieti quam <lb></lb>quod tardè. </s>
          <s id="id004513">Demum, ſi aliquid moueretur in finita uelo citate motu <lb></lb>circulari, ſemper eſſet in eodem ſitu ſecundum partes &amp; immobile, <lb></lb>igitur quod infinita uelo citate mouetur, &amp; quieſcit. </s>
          <s id="id004514">Ergo quod ue­<lb></lb>lociſsimè mouetur cum magis diſtet ab oppoſito eius, quod infini­<lb></lb>ta tarditate mouetur, quàm quod tardè, magis etiam appropinqua<lb></lb>bit poteſtate in efficaci infinitæ uelo citati quàm quod tardè, igitur <lb></lb>quod uelociſsimè mouetur propius eſt quieſcenti quam quod tar­<lb></lb>dè. </s>
          <s id="id004515">Demonſtratum eſt enim in Dialecticis, argumentum oſtendere <lb></lb>ab eo quod eſt ſimpliciter tale ad id q̊d natura illi quo quo modo <lb></lb>tale eſt &amp; <expan abbr="cõuerſo">conuerſo</expan> modo. </s>
          <s id="id004516">Oſtendo modò quod ſimillimus: <expan abbr="quoniã">quoniam</expan> <lb></lb>illud eſt ſimilius quieti in quo quod fertur non poteſt dignoſci di­<lb></lb>ſtantia à priore loco, ſed in uelociſsimè motis hæc diſtantia non po<lb></lb>teſt dignoſci, igitur uelociſsimè mota uidentur planè quieſcere, <lb></lb>quod idem patet duobus experimentis manifeſtis. </s>
          <s id="id004517">Primum ſi quis <lb></lb>uideat rotas quibus acuuntur gladij moueri uſque ad certam ueloci­<lb></lb>tatem, augeri uidetur motus ille, uerùm cum adeo <expan abbr="cõcitatus">concitatus</expan> fuerit, <lb></lb>ut ſenſus non poſsit diſcernere, neque comprehendere illam ueloci­<lb></lb>tatem, &amp; rota non fuerit mota ab axe, ita ut titubet nec fuerit ulla in­<lb></lb>æqualitas, uidebitur omnino quieſcere, &amp; ita oculus dijudicat, &amp; <lb></lb>longè magis dijudicaret, ubi ad tantam motus perueniret uelocita<lb></lb>tem, ut nullo modo initium à fine diſtingui poſſet, ſicut eſt in motu <lb></lb>cœli, qui comparatus ad quemuis motum uelociſsimum artificio <lb></lb>factum, inſenſilem habet proportionem ob magnitudinem, &amp; ideo <lb></lb>talis motus cœleſtis eſt ſimillimus quieti. </s>
          <s id="id004518">Secundum <expan abbr="experimẽtum">experimentum</expan> <lb></lb>eſt, ſi eſſent duo homines habitantes Bononiæ, quorum unus iret <lb></lb>Mutinam, paulatim quieſcendo in quolibet loco per unam diem, <lb></lb>adeò ut in unoquoque anno maneret Mutinæ, &amp; prope per ſex men<lb></lb>ſes, &amp; prope Bononiam per ſex alios menſes in diuerſis locis, &amp; <lb></lb>una die tantum Bononiæ: alius uerò iret Mutinam ſingulo die, &amp; 


<pb pagenum="267" xlink:href="015/01/286.jpg"></pb>per omnia loca ſicut hirundo uolans quater &amp; quater rediret Bo­<lb></lb>noniam, nemini dubium eſt, quod hic ſecundus uideretur magis <lb></lb>quieſcere Bononiæ quàm primus, &amp; hoc quia in anno quilibet eo­<lb></lb>rum quieſceret per unam diem Bononiæ, &amp; in hoc eſſent æquales, <lb></lb>ſed ſecundus uideretur frequentius Bononiæ quàm primus, &amp; eti­<lb></lb>am eſſet poteſtate propior illi, adeò ut liceret cuilibet illum conue­<lb></lb>nire qualibet die magis quam primum: ergo duabus de cauſis ui­<lb></lb>deretur ſecundus magis quieſcere Bononiæ quam primus, &amp; in ter<lb></lb>tia æqualiter.</s>
        </p>
        <p type="main">
          <s id="id004519">Modò dico de recto motu, quoniam quanto celerius fertur per <lb></lb>medium ad ſuum locum, tanto minus temporis inſumit, ergo diu­<lb></lb>tius quieſcit in loco, minus eſt etiam tempus per quod mouetur in <lb></lb>comparatione ad quietem &amp; ſimpliciter, ergo in motu recto pro­<lb></lb>pius eſt quieti, quod uelociſsimè mouetur, pręterea inter duas quie <lb></lb>tes motus uelociſsimus eſt imperceptibilis. </s>
          <s id="id004520">Ergo motus uelociſsi­<lb></lb>mus eſt ſimilior quieti quàm minus uelox. </s>
          <s id="id004521">Accedit manifeſtiſsimè <lb></lb>illud quod ab initio diximus, ſcilicet, quia motus uelociſsimus eſt <lb></lb>medius inter motum tardum &amp; ſubitam mutationem, hoc enim eſt <lb></lb>manifeſtiſsimum, adeò ut dubitemus in motibus uelociſsimis, an <lb></lb>mobile tranſierit per medium, eſt enim primùm motus lentus, qui <lb></lb>fit ex tranſitu in longo tempore, &amp; uelociſsimus in paruo, &amp; muta­<lb></lb>tio ſine tempore. </s>
          <s id="id004522">Rurſus conſtituamus alium ordinem quietis mo­<lb></lb>tus, &amp; ſubitæ mutationis: &amp; ex dictis ſubita mutatio eſt propior <lb></lb>quieti <expan abbr="quã">quam</expan> motus: quo­<lb></lb><arrow.to.target n="marg903"></arrow.to.target><lb></lb>niam ſi motus eſſet me­<lb></lb>dius inter quietem &amp; <lb></lb>ſubitam mutationem, non eſſet, ut dictum eſt, ſubita mutatio quæ­<lb></lb>dam quies: nam in ſubita mutatione non pertranſitur medium: in <lb></lb>quiete non pertranſitur medium, in motu pertranſitur medium, igi<lb></lb>tur quies eſt propior ſubitæ mutationi quàm motui. </s>
          <s id="id004523">Sed ſubita mu<lb></lb>tatio eſt propior motui uelociſsimo quàm tardo, igitur quies eſt <lb></lb>propior motui uelociſsimo quam tardo.</s>
        </p>
        <p type="margin">
          <s id="id004524"><margin.target id="marg903"></margin.target>Subit. </s>
          <s id="id004525">Mut. </s>
          <s id="id004526">Motus uelo ciſ. </s>
          <s id="id004527">Motus Tar. <lb></lb></s>
          <s id="id004528">Quies ſubita Mut. </s>
          <s id="id004529">Motus</s>
        </p>
        <p type="main">
          <s id="id004530">Videtur &amp; hoc ſenſus manifeſtè oſtendere, quoniam cum lapis <lb></lb>deſcendit ſumma cum uelo citate, adeò ut non percipiatur, uidetur <lb></lb>quieſcere, &amp; non motus eſſe, &amp; hæc fuit ſententia multorum nobi­<lb></lb>liorum antiquorum, &amp; propterea oportet ut oſtendamus difficul­<lb></lb>tates, quæ contingunt in his.</s>
        </p>
        <p type="main">
          <s id="id004531">Dico igitur, quod motus naturales ſunt duorum generum, ut di<lb></lb><expan abbr="ctũ">ctum</expan> eſt, ſcilicet rectus &amp; circularis: &amp; motus differt à quiete in duo­<lb></lb>bus, in eo quod mutat locum, et in eo quod tranſit per medium mo<lb></lb>tus, ergo rectus uelociſsimus in eo quod tranſit per medium ma­


<pb pagenum="268" xlink:href="015/01/287.jpg"></pb>gis diſtat à quiete in eo quod plus de medio ſuperat quàm tardus, <lb></lb>&amp; eſt propinquior quieti in eo quod celerius quieſcit. </s>
          <s id="id004532">At motus cir<lb></lb>cularis uelociſsimus eſt propior quieti in tranſitu medij, &amp; in redi­<lb></lb>tu ad locum priorem: de reditu ad locum priorem clarum eſt per ſe: <lb></lb>de tranſitu medij, dico quod cum in prima medietate magis remo­<lb></lb>ueatur à medio quam motus tardus, &amp; in ſecunda medietate tan­<lb></lb>tundem, uelocius redeat. </s>
          <s id="id004533">Ergo in <expan abbr="ſecũda">ſecunda</expan> medietate eſt ſemper pro­<lb></lb>ximior motus uelociſsimus ipſi quieti, ſed in prima medietate q̊d <lb></lb>mouetur motu uelociſsimo propius eſt ſecundæ medietati ſemper <lb></lb>quam quod mouetur tardo motu, igitur quod mouetur uelociſsi­<lb></lb>mè circulariter eſt propius quieſcenti, quam quod mouetur tardè. <lb></lb></s>
          <s id="id004534">Et hoc eſt quia in ęternis motus eſt quies, &amp; ideo habent quandam <lb></lb>ſimilitudinem iuxta <expan abbr="perfectionẽ">perfectionem</expan> ſuam, ſicut ſi eſſent in circulo hoc <lb></lb><figure id="id.015.01.287.1.jpg" xlink:href="015/01/287/1.jpg"></figure><lb></lb>modo. </s>
          <s id="id004535">Mutatio ergo <expan abbr="cõue­nit">conue­<lb></lb>nit</expan> in corporeis quę <expan abbr="pendẽt">pendent</expan> <lb></lb>à corpore, ſicut lumini: qua­<lb></lb>tenus enim ſunt ex corpo­<lb></lb>reo, <expan abbr="occupãt">occupant</expan> diuerſum <expan abbr="locũ">locum</expan>, <lb></lb>quatenus eſt in corporei id, <lb></lb>agit ſine tranſitu per <expan abbr="mediũ">medium</expan> <lb></lb>&amp; in inſtanti, ergo in corpo­<lb></lb>rea ſimpliciter mutationem <lb></lb>recipiunt, non in tempore <lb></lb>neque in loco. </s>
          <s id="id004536">Videtur <expan abbr="autẽ">autem</expan> <lb></lb>uelo<expan abbr="ciſsimũ">ciſsimum</expan> dupliciter <expan abbr="etiã">etiam</expan> <lb></lb>nobis iuxta ſenſum, idque eſt <lb></lb>in quo ſenſus medij tranſitum non percipit, &amp; natura quod eſt pri­<lb></lb>mi mobilis. </s>
          <s id="id004537">At dubitare quis poteſt circa hoc, nam proprium mo­<lb></lb>tus eſt tangentia concutere, quietis autem minime: concutit autem <lb></lb>maximè quod uelociſsimè mouetur, ob hoc arbitrati ſunt homi­<lb></lb>nes quod uelociſsimus motus multò plus diſtaret à natura quietis <lb></lb>quam tardus, ſed hoc eſt quia non eadem eſt ratio uiolenti &amp; natu­<lb></lb>ralis: uiolenta enim non redeunt in ſe ipſa, nec habent rationem cir­<lb></lb>cularis, ſed potius recti &amp; infiniti, &amp; ideò in his quæ mouentur mo<lb></lb>tu recto naturali cadit uiolentia, non autem in his quæ mouentur <lb></lb>motu circulari naturali: <expan abbr="cõ">com</expan> cuſsio ergo eſt in motu uiolento, &amp; qua­<lb></lb>liſcunque motus uiolentus, quanto magis augetur tantò magis re­<lb></lb>cedit à contrario, tantò magis remouetur à natura contrarij, &amp; ha­<lb></lb>bet actiones contrarias ualidiores.</s>
        </p>
        <p type="main">
          <s id="id004538">Eſt etiam aliud penè ſimile argumentum in figuris ipſis, circulus <lb></lb>enim unica linea continetur, nulla tamen figura ab ea magis natura 


<pb pagenum="269" xlink:href="015/01/288.jpg"></pb>remota eſt triangulo: ſiquidem circulus capaciſsimus eſt, triangu­<lb></lb>lus omnium rectilin<expan abbr="earũ">earum</expan> minimè capax: ut contrà polygonię, quan<lb></lb>to plurium ſunt laterum eo capaciores ſunt, adeò ut octagona qua­<lb></lb>drangula, &amp; quæ eſt ſexdecim laterum æqualium, &amp; æquiangula­<lb></lb>rium plus contineat octagona, &amp; forma etiam ſit ſimilior circulo, <lb></lb>adeò ut cum excreuerit in multiplicem numerum rectangula figu­<lb></lb>ra huiuſmodi, ſcilicet æquilatera, &amp; æquiangula omnino ſenſum <lb></lb>fallat, uideaturque prorſus circulus. </s>
          <s id="id004539">Et <expan abbr="tamẽ">tamen</expan> figura plurium laterum, <lb></lb><expan abbr="quãto">quanto</expan> plurium laterum fuerit rem otior eſt à natura circuli, qui una <lb></lb>tantum linea continetur: plus enim diſtat centum ab uno quàm de­<lb></lb>cem, &amp; mille quàm centum. </s>
          <s id="id004540">Cauſa igitur eſt, quia (ut dixi) etiam in <lb></lb>naturalibus omnis natura rerum eſt, ut quaſi clanculum redeat in <lb></lb>ſe ipſam: nam circularis figura per triangulum ex rectis multum à <lb></lb>natura ſua recedit &amp; ambitu &amp; ſimilitudine: eadem per figuras quę <lb></lb>ex pluribus rectis conſtant ad ſui ſimilitudinem redit, nunquàm ta<lb></lb>men explet eandem naturam perfectè, cùm nulla poligonya figura <lb></lb>pro circulo exacto ſit: ita uidetur in naturalibus ad <expan abbr="idẽ">idem</expan> redire, quod <lb></lb>eſt poteſtate ſolum quadam generali diſsimile: actu uerò non idem <lb></lb>ad unguem. </s>
          <s id="id004541">Sed obijcies de motu quòd ſi tempus fiat breuius, ma­<lb></lb>gnitudo autem conſtet, erit (ut diximus) quod mouetur ſimile quie<lb></lb>ſcenti: at ubi tempus idem ſit, ſed magnitudo perpetuò augeatur, <lb></lb>non idem ut in cœlo: ueriſimile eſt enim quicquid eſt quod moue­<lb></lb>tur ulterius quam id quod cernitur nihilominus in uiginti quatu­<lb></lb>or horis, non autem celerius moueri: propterea cùm ſpatium tem­<lb></lb>poris prolixum ſit, non uidebitur quieſcere. </s>
          <s id="id004542">Nec obſtat quòd quiſ­<lb></lb>piam proportionem obijciat, ſi quidem multo minus uidebuntur <lb></lb>propiora centro quieſcere, namque illa tardius ex confeſſo mouen­<lb></lb>tur, at quod tardius mouetur, ut dictum eſt, moueri magis uidetur, <lb></lb>ideò proportionem illam ad aliud mobile referre oporteret, cum <lb></lb>nullum tale ſit. </s>
          <s id="id004543">Dicimus ergo quòd apud illas non uidetur motus <lb></lb>tardus, quia comprehendunt motum ante tempus, nobis <expan abbr="autẽ">autem</expan> hæc <lb></lb>accidunt, quia comprehendimus tempus ante motum. </s>
          <s id="id004544">Et <expan abbr="etiã">etiam</expan> quia <lb></lb>circa polos quaſi quieſcit, &amp; quod non poteſt aliquid comprehen­<lb></lb>di, ſimul moueri &amp; quieſcere, ut docebimus. </s>
          <s id="id004545">Et etiam quia motus <lb></lb>eſt ab illis, ſicut in nobis cum mouemur: <expan abbr="nõ">non</expan> enim ut mouemur nos <lb></lb>moueri deprehendimus, ſed ut moti ideò in his, non quod appa­<lb></lb>ret, ſed quod eſt ſpectare oportet: at ita eſt ut quæ uelociter ualde <lb></lb>mouentur, perinde ſunt quaſi ac ſi quieſcerent, adeò ut motus ſi in <lb></lb>inſtanti fieret eſſet quies, &amp; quies in incorporeis eſt motus, non in <lb></lb>tempore. </s>
          <s id="id004546">Videntur etiam aſtra quieſcere nobis, quoniam (ut dixi) <lb></lb>lineæ a e &amp; b e non poſſunt uideri moueri in e, oculus autem iudi­


<pb pagenum="270" xlink:href="015/01/289.jpg"></pb><figure id="id.015.01.289.1.jpg" xlink:href="015/01/289/1.jpg"></figure><lb></lb>cat moueri debere in e, non ex c <lb></lb>in d, ubi eſt amplum ſpatium <lb></lb>terræ comprehenſum, ergo a e <lb></lb>quieſcere uidetur in e, igitur &amp; <lb></lb>in a. </s>
          <s id="id004547">Quòd autem uideatur in e <lb></lb>quieſcere, patet, quia quod mo<lb></lb>tum uideri debet, oportet ut in <lb></lb>inſenſili tempore ſpatium ſen­<lb></lb>ſile pertranſierit: inſenſile au­<lb></lb>tem tempus eſt minus motu ue<lb></lb>lociſsimo pulſus, hic autem ma <lb></lb>ius exigit <expan abbr="tẽpus">tempus</expan> centeſima par­<lb></lb>te centeſimæ partis horę, igitur <lb></lb>diei ducenteſima quadrageſima milleſimæ partis, &amp; in hoc oportet <lb></lb>ut pertranſeat ſenſile ſpatium, quod eſt quinquageſima parte ulnæ <lb></lb>ſaltem maius. </s>
          <s id="id004548">Ergo ſi fiat inſtrumentum <expan abbr="quingẽtarum">quingentarum</expan> ulnarum am<lb></lb>bitus, q̊d in uiginti quatuor horis circumuoluatur, adeò lentè mo­<lb></lb>uebitur, ut quieſcere uideatur: tum uerò magis ob id quod dixi, <lb></lb>quoniam in centro quieſcere uidebitur, ergo in peripheria, ubi di­<lb></lb>ſtantia deprehendi poſsit. </s>
          <s id="id004549">Ergo nulla machina quæ uideatur mo­<lb></lb>ueri, conſtitui poteſt, quæ in horis XXIIII circumuertatur: quia non <lb></lb>tam magna fieri poteſt, ut ſpatium à centro ad circumferentiam ocu<lb></lb>lo non poſsit deprehendi.</s>
        </p>
        <p type="main">
          <s id="id004550">Et hoc uoluimus declarare ut intelligamus, quæ ſunt neceſſaria <lb></lb>ad mundum incorporeum.</s>
        </p>
        <p type="main">
          <s id="id004551">Propoſitio ducenteſima trigeſima tertia.</s>
        </p>
        <p type="main">
          <s id="id004552">Quod eſt in mundo incorporeo æternum, eſt beatum, ſecurum <lb></lb>immutabile ſecundum locum ſolum iuxta eſſentiam fit, iuxta quod <lb></lb>uelut à leui ſuſurro aquæ &amp; aura æſtiua demulcetur.</s>
        </p>
        <p type="main">
          <s id="id004553">Quod eſt ibi non eſt pars nec totum, eſſet enim quantum, aut nu<lb></lb><arrow.to.target n="marg904"></arrow.to.target><lb></lb>mero diſcretum, nec mutationem loci aut temporis habet, cum in <lb></lb>nullo eorum ſit, ideò nec habere poteſt, nec amittere, non eſt ibi infi<lb></lb>nitum, cuius nullus finis ſit, ſed dum emanat à priore ſecundum or­<lb></lb>dinem eſt ſumma uoluptas, qualis in his qui ad cognitionem &amp; feli<lb></lb>citatem <expan abbr="deueniũt">deueniunt</expan>. </s>
          <s id="id004554">Quę in illis cum æterna ſit &amp; ſecura, recipit quan <lb></lb>dam uariationem, in qua delectatur, uelut mortalia ex <expan abbr="cõtrarijs">contrarijs</expan> cau<lb></lb>ſis naturæ contrarijs affectibus: &amp; hoc eſt perpetuò nouum, quia <lb></lb>ſemper pendet &amp; recipit. </s>
          <s id="id004555">Et ob id eſt unum &amp; actu ſempiterno, <lb></lb>quod uerò eſt extra, eſt potentia, ideò infinitum, quod imaginatur <lb></lb>anima, quia in ordinatum priore ordine, qui eſt ante <expan abbr="limitẽ">limitem</expan> omnem, <lb></lb>neque enim dubium eſt, quin infinitum non ſit cauſa, ut non poſsit 


<pb pagenum="271" xlink:href="015/01/290.jpg"></pb>eſſe ordo ille ſecundus: ſed nos loquimur de primo. </s>
          <s id="id004556">Et ideò anima <lb></lb>noſtra ob materiæ coniunctionem appetit ordinem, &amp; lætatur in <lb></lb>eo ut inueniat finem in rebus, uelut in multis proprietatibus nume <lb></lb>rorum eſt manifeſtum. </s>
          <s id="id004557">Potentia enim eſt cauſa imaginandi infini­<lb></lb>tum, quia ſemper ultra aliquid eſſe poſſe putamus, eſt igitur poten­<lb></lb>tia actus imperfectus. </s>
          <s id="id004558">Anima ergo noſtra conuerſa eſt à Deo, res <lb></lb>poſt ſe in quibus inuenit potentia imperfectionem <foreign lang="grc">ἀταξιαν</foreign> pericu­<lb></lb>lum &amp; infinitum ad deſperationem tandem, quod quilibet uidere <lb></lb>poterit, qui ſe à diuinis auerterit: quantò enim plura habet, plura <lb></lb>deſunt. </s>
          <s id="id004559"><expan abbr="Multiplicẽtur">Multiplicentur</expan> filij, opes, honores, nil niſi laborem &amp; anxie­<lb></lb>tatem aucta inuenies. </s>
          <s id="id004560">Quomodo autem quod infinitum non eſt, <lb></lb>infinitam faciat potentiam? </s>
          <s id="id004561">uides in repræſentatione Solis quę infi<lb></lb>nita eſſet, ſi cœlum eſſet infinitum. </s>
          <s id="id004562">Dubitatione autem dignum eſ­<lb></lb>ſet, an ſi cœlum infinitum eſſet ubique Sol illuminaret: ſeu quia quæ­<lb></lb>ſitum nullum ſit, uiſit de eo quod non eſt, nihil autem non eſſe po­<lb></lb>teſt, aut quod non poſſet, quoniam uirtus corporea eſt. </s>
          <s id="id004563">Corporeo <lb></lb>autem omni finem ad eſſe neceſſe eſt. </s>
          <s id="id004564">Hanc nouitatem ergo alij tri­<lb></lb>pudium, alij muſicam &amp; ſonum cœleſtem interpretati ſunt.</s>
        </p>
        <p type="margin">
          <s id="id004565"><margin.target id="marg904"></margin.target>C<emph type="italics"></emph>o<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="main">
          <s id="id004566">Manifeſtum eſt igitur ſubſtantiam incorporei mundi, eſſe in <lb></lb><arrow.to.target n="marg905"></arrow.to.target><lb></lb>quadam mutatione perpetua ordinis, &amp; ſine motu, tempore &amp; lo­<lb></lb>co: unde amor &amp; uoluptas mutua, &amp; totum unum, ſicut anima cum <lb></lb>cognoſcit Deum, &amp; cum cognoſcit cœlum deſcendit, &amp; fit alia or­<lb></lb>dine. </s>
          <s id="id004567">Et hæc beatitudo in mundo illo eſt tanta, ut in com­<lb></lb>parabilis ſit noſtræ, quæ eſt umbra eius, etiam <lb></lb>quando eſt &amp; pura, etiam ſi eſſet per­<lb></lb>petua. </s>
          <s id="id004568">Igitur hic finis no­<lb></lb>ſter Diuinę naturæ <lb></lb>&amp; libri.</s>
        </p>
        <p type="margin">
          <s id="id004569"><margin.target id="marg905"></margin.target>C<emph type="italics"></emph>or<emph.end type="italics"></emph.end>^{m}.</s>
        </p>
        <p type="head">
          <s id="id004570">LIBRI DE PROPORTIONI­<lb></lb>BVS FINIS.</s>
        </p>
        <pb xlink:href="015/01/291.jpg"></pb>
      </chap>
    </body>
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  </text>
</archimedes>