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  <info>
    <author>Cardano, Girolamo</author>
    <title>Opus novum de proportionibus</title>
    <date>1570</date>
    <place>Basel</place>
    <translator/>
    <lang>la</lang>
    <cvs_file>carda_propo_015_la_1570.xml</cvs_file>
    <cvs_version/>
    <locator>015.xml</locator>
  </info>
  <text>
    <front>
      <section>
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        <p type="head">
          <s id="id000001">HIERONYMI <lb/>CARDANI MEDIO<lb/>LANENSIS, CIVISQVE BONO&shy;<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s>
        </p>
        <p type="head">
          <s id="id000002">OPVS NOVVM DE <lb/>PROPORTIONIBVS NVMERORVM, MO<lb/>TVVM, PONDERVM, SONORVM, ALIARVMQVE RERVM <lb/>men&longs;urandarum, non &longs;ol&ugrave;m Geometrico  more &longs;tabilitum, &longs;ed etiam <lb/>uarijs experimentis &amp; ob&longs;eruationibus rerum in natura, &longs;olerti <lb/>demon&longs;tratione illu&longs;tratum, ad multiplices u&longs;us ac&shy;<lb/>commodatum, &amp; in &lt;var&gt;V&lt;/var&gt; libros dige&longs;tum.</s>
        </p>
        <p type="head">
          <s id="id000003">PRAETEREA.</s>
        </p>
        <p type="head">
          <s id="id000004">ARTIS MAGN&AElig;, SIVE DE REGVLIS <lb/>ALGEBRAICIS, LIBER VNVS, ABSTRVSISSIMVS <lb/>&amp; inexhau&longs;tus plane totius Arithmetic&aelig; the&longs;aurus, ab <lb/>authore recens multis in locis recogni&shy;<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/>logi&longs;tic&aelig; &longs;u&aelig;, numeros recondita numerandi &longs;ubtilitate, &longs;ecundum Geo&shy;<lb/>metricas quantitates inquirentis, nece&longs;&longs;aria Coronis, <lb/>nunc demum in lucem edita.</s>
        </p>
        <p type="head">
          <s id="id000007">O<emph type="italics"/>pus<emph.end type="italics"/> P<emph type="italics"/>hy&longs;icis &amp;<emph.end type="italics"/> M<emph type="italics"/>athematicis imprimis <lb/>utile &amp; nece&longs;&longs;arium.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id000008">Cum C&aelig;&longs;. </s>
          <s id="id000009">Maie&longs;t. </s>
          <s id="id000010">Gratia &amp; Priuilegio.</s>
        </p>
        <p type="head">
          <s id="id000011">BASILE&AElig;.</s>
        </p>
      </section>
      <section>
        <pb xlink:href="015/01/005.jpg"/>
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        <p type="head">
          <s id="id000012">IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQVE <lb/>Bononien&longs;is, Medici, Pr&aelig;fatio ad M. A. </s>
          <s id="id000013">Amulium <lb/>Venetum Card. </s>
          <s id="id000014">Illu&longs;tri&longs;simum.</s>
        </p>
        <p type="main">
          <s id="id000015">Bene Dictum e&longs;t  meo iudicio &agrave; Platone M.  <lb/>A. </s>
          <s id="id000016">Amuli optime, beatas fore Re&longs;pub. </s>
          <s id="id000017">&longs;i uel <lb/>illarum domini &longs;apienti&aelig; amatores e&longs;&longs;ent, <lb/>aut qui &longs;apienti&aelig; e&longs;&longs;ent amatores domina&shy;<lb/>rentur, hoc ip&longs;um clar&egrave; intelligens, &longs;tudio &longs;a<lb/>pienti&aelig; nihil e&longs;&longs;e utilius humano generi: <lb/>quo &longs;imul &amp; pietas, &amp; iu&longs;titia, &amp; mutuus <lb/>amor hominum inter &longs;e &amp; eorum commo&shy;<lb/>da continerentur. </s>
          <s id="id000018">Nempe hi&longs;ce quatuor tota no&longs;tra felicitas com&shy;<lb/>prehenditur. </s>
          <s id="id000019">Si quidem pietate in Deos nihil  ni&longs;i &longs;anctum, &amp; pu&shy;<lb/>rum, &amp; illu&longs;tre &longs;apimus: hoc ip&longs;o primum quod &longs;upra nos e&longs;t, intel&shy;<lb/>ligimus, Deos ueneramur, gratias agimus, timor cum ueneratione <lb/>no&longs;tros animos &longs;ubit, &amp; de futura uita cogitamus, h&aelig;c ip&longs;a morta&shy;<lb/>lia &longs;i non negligentes &longs;altem paruifacientes. </s>
          <s id="id000020">Iu&longs;titiam autem ade&ograve; <lb/>nece&longs;&longs;ariam humano generi e&longs;&longs;e &longs;cimus, ut &longs;ine illa neque e&longs;&longs;e, nedum <lb/>ben&egrave; e&longs;&longs;e po&longs;s&iacute;mus, ut neque latronum c&oelig;tus ab&longs;que ea diu &longs;tare po&longs;&shy;<lb/>&longs;int. </s>
          <s id="id000021">Porr&ograve; quid dicam de concordia, &amp; mutua hominum beneuo&shy;<lb/>lentia, in quibus omnis uit&etail; human&etail; dulcedo repo&longs;ita e&longs;t: nec quis <lb/>&longs;u&longs;tineat uiuere, qui &longs;e omnibus odio&longs;um e&longs;&longs;e &longs;entiat. </s>
          <s id="id000022">His ip&longs;is fi&shy;<lb/>lios in &longs;pem alimus, parentes fouemus, fratres tuemur, &amp; adiuua&shy;<lb/>mus, amicis opitulamur, cum hominibus hilarem &amp; iucundam ui&shy;<lb/>tam ducimus. </s>
          <s id="id000023">Si quis &longs;erpentem in lecto haberet, nunquam &longs;om&shy;<lb/>num caperet: ita nihil mole&longs;tius e&longs;t in hac uita, quam e&longs;&longs;e cum quo <lb/>nolis, &amp; priuari con&longs;uetudine eorum cum quibus maxim&egrave; uiuere <lb/>cupias. </s>
          <s id="id000024">Quid enim habent Principes pr&aelig;cipuum cum tota illa po&shy;<lb/>tentia quam habent, ni&longs;i hoc unum, quod &longs;uis quos amant bene fa&shy;<lb/>cere po&longs;sint: nam reliqua omnia exerceri, uenari, edere, bibere, dor&shy;<lb/>mire, iter agere, loca am&aelig;na inui&longs;ere multis alijs conce&longs;&longs;um e&longs;t, ma&shy;<lb/>ioreque commodo qui in uita priuata degunt. </s>
          <s id="id000025">Si ergo principatum <lb/>cum tot laboribus, curis, periculis, &amp; merit&ograve; omnes appetunt: nec <lb/>e&longs;t in eo quicquam pr&aelig;cipuum pr&aelig;ter hoc, cui dubium e&longs;t quin <lb/>hoc non &longs;it &longs;ummum huius uit&aelig; hominibus bonum? </s>
          <s id="id000026">propter cu&shy;<lb/>ius uel dubiam &longs;pem eorum, qu&aelig; habent obliti mortales pericli&shy;<lb/>tantur. </s>
          <s id="id000027">Succedunt inde tot commoda, non &longs;olum utilia, &longs;ed pleraque<pb xlink:href="015/01/007.jpg"/>etiam nece&longs;&longs;aria, qu&aelig; nos &longs;apientia docet: huiu&longs;modi ergo omnia <lb/>c&ugrave;m libris contineantur, merit&ograve; optimus qui&longs;que librorum bono&shy;<lb/>rum perpetuitati atque in columitati fauere debet. </s>
          <s id="id000028">C. </s>
          <s id="id000029">Caligulam exe&shy;<lb/>cramur &longs;olum ob id quod Vergilij, &amp; T. </s>
          <s id="id000030">Liuij &longs;cripta delere cogi&shy;<lb/>tauerit. </s>
          <s id="id000031">Quid facturi e&longs;&longs;emus, &longs;i feci&longs;&longs;et quod cogitauerat? </s>
          <s id="id000032">E&longs;t in &longs;a&shy;<lb/>pientum monumentis bonum &longs;ine malo, mens &longs;ine corporea labe: <lb/>Virtutes ab&longs;que uitijs, grati&aelig; &amp; iucunditas &longs;ine &longs;orde, &amp; immundi&shy;<lb/>tia, uoluptas &longs;ine dolore, conuer&longs;atio ab&longs;que t&aelig;dio, deliti&aelig; ab&longs;que mi&longs;e<lb/>ria nuda, omnia bona pr&aelig;&longs;tant, atque laudabilia ab omnibus morta&shy;<lb/>litatis exuuijs libera, tantum commodi afferunt libri. </s>
          <s id="id000033">Sed &amp; in eo&shy;<lb/>rum electione ac &longs;tudijs modus, ac medio critas qu&aelig;dam &longs;eruanda <lb/>e&longs;t, qu&aelig; &longs;i quis neglexerit non leui incommodo afficietur: eam an&shy;<lb/>tiqui rationem alij proportionem appellarunt, non equidem etiam <lb/>in pertritis tam <expan abbr="facillim&atilde;">facillimam</expan>, ut rentur homines: nam in alijs rebus per&shy;<lb/>ob&longs;curam e&longs;&longs;e fatentur, ego difficillimam puto undique, &amp; magis for<lb/>&longs;an ubi non exi&longs;timamus. </s>
          <s id="id000034">Vnde plures decidere uidemus magnis <lb/>cum auxilijs, &amp; euidenti &longs;pe: quid aliud e&longs;t in cau&longs;a qu&agrave;m ignota <lb/>men&longs;ura rerum? </s>
          <s id="id000035">quam tamen plerique tenere &longs;e putant. </s>
          <s id="id000036">Ergo, c&ugrave;m <lb/>&longs;ummum bonum in hac men&longs;ura &longs;itum e&longs;&longs;e cernerem, ut clar&egrave; o&longs;ten<lb/>dunt mu&longs;ic&aelig; uoces, qu&aelig; non ni&longs;i indiuiduo &lpar;ut ita dicam&rpar; &longs;patio <lb/>&longs;eu loco &longs;tare po&longs;&longs;unt, ita &amp; in figuris picturarum &amp; &longs;tatuarum, &amp; <lb/>diebus decretorijs, &amp; negotijs ciuilibus oper&etail; pretium me factu&shy;<lb/>rum exi&longs;timaui, &longs;i omnia h&aelig;c qu&aelig; lat&egrave; patebant breuiter in unum <lb/>redegi&longs;&longs;em, <expan abbr="n&otilde;">non</expan> tantum ne lectorem t&aelig;dio afficerem, qu&agrave;m ut qu&ograve;d <lb/>ali&agrave;s do cui, breuibus tractationibus, &amp; plura continerentur, &amp; faci<lb/>lius docerentur. </s>
          <s id="id000037">Cum uer&ograve; bona fortuna qu&aelig;dam effeci&longs;&longs;et, ut tibi <lb/>libellum dedica&longs;&longs;em de Prouidentia ex con&longs;titutione temporum, <lb/>longe meliore occa&longs;ione nominis tui typographi obliti &longs;int, indi&shy;<lb/>gnum fore putaui, ut non &aelig;rea &lpar;quemadmodum cum Glauco Dio<lb/>medes&rpar; cum aureis commutarem. </s>
          <s id="id000038">Itaque infinitis licet circumuentus <lb/>negotijs totus huic oper&aelig; in cubui, atque ade&ograve; ut pr&aelig;ter &longs;pem unius <lb/>anni pen&egrave; &longs;patio liber ab&longs;olueretur. </s>
          <s id="id000039">Qui cum tibi &lpar;ut dixi&rpar; iam iur&egrave; <lb/>deberetur, e&ograve; tamen magis dedicandum putaui, quod non ego &longs;o&shy;<lb/>lum quanquam id maxim&egrave;, &longs;ed communis con&longs;en&longs;us ho&shy;<lb/>minum exi&longs;timet, te &longs;ingulari uirtute omnibus <lb/>&longs;tudio&longs;is plurimum fauere, <lb/>Vale.</s>
        </p>
      </section>
      <section>
        <pb xlink:href="015/01/008.jpg"/>
        <p type="head">
          <s id="id000040">TABVLA PRO&shy;<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"/></s>
        </p>
        <table>
          <table.target id="table1"/>
          <row>
            <cell>I.</cell>
            <cell>Proportionem <emph type="italics"/>in proportionem duci, e&longs;t &longs;uperiores numeros atque inferiores inuicem ducere.<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>pagina<emph.end type="italics"/> 6</cell>
          </row>
          <row>
            <cell>II.</cell>
            <cell>P<emph type="italics"/>roportio extremorum producitur ex intermedijs.<emph.end type="italics"/></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell>III.</cell>
            <cell>S<emph type="italics"/>i proportio ex duabus proportionibus in quatuor terminis producatur,   ip&longs;a uer&ograve; proportio inter duas alias quantitates fuerit con&longs;tituta: con&longs;urgent trecen&shy;ti &longs;exaginta modi productionis proportionis.<emph.end type="italics"/></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell>IIII.</cell>
            <cell>S<emph type="italics"/>i fuerit proportio primi ad &longs;ecundum, producta ex proportionibus tertij ad quartum,   &amp; quinti ad &longs;extum, producetur etiam ex proportione tertij ad &longs;extum, &amp; quinti ad   quartum.<emph.end type="italics"/></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell>V.</cell>
            <cell>S<emph type="italics"/>i fuerit proportio primi ad &longs;ecundum, producta ex proportione tertij ad quartum, &amp;   quinti ad &longs;extum: erit proportio tertij ad &longs;extum, producta ex proportionibus primi   ad &longs;ecundum, &amp; quarti ad quintum.<emph.end type="italics"/></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell>VI.</cell>
            <cell>E<emph type="italics"/>x trecentis &longs;exaginta modis producendarum proportionum triginta &longs;ex tantum e&longs;&longs;e   nece&longs;&longs;arios.<emph.end type="italics"/></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell>VII.</cell>
            <cell>I<emph type="italics"/>n modis qui nece&longs;&longs;ari&ograve; producuntur ex duabus proportionibus, cum du&aelig; quantitates ex   illis qu&aelig; modos conficiunt, &aelig;quales fuerint: proportio producta ad quatuor quanti&shy;tates omiologas reducetur.<emph.end type="italics"/></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell>VIII.</cell>
            <cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen&shy;tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in&shy;uicem proportio, ex primis proportionibus compo&longs;ita.<emph.end type="italics"/></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>IX.</cell>
            <cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen&shy;tur, minusque productum ex maiore detrahatur, erit re&longs;idui ad productum ex in&longs;e&shy;rioribus proportio uelut illa, qu&aelig; relinquitur detracta minore proportione ex ma&shy;iore.<emph.end type="italics"/></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>X.</cell>
            <cell>S<emph type="italics"/>i fuerit alicuius quantitatis ad unam partem proportio, uelut alterius partis ad &longs;ecun&shy;dam quantitatem, erit proportio cuiu&longs;uis quantitatis eiu&longs;dem generis ad &longs;ecundam   compo&longs;ita proportio, ex proportionibus eiu&longs;dem quantitatis, a&longs;&longs;umpt&aelig; ad utranque   partem prim&aelig; quantitatis &longs;eor&longs;um.<emph.end type="italics"/></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>XI.</cell>
            <cell>P<emph type="italics"/>roportio aggregati quarumlibet duarum quantitatum ad aggregatum duarum &aelig;qua&shy;lium <expan abbr="quantitat&utilde;">quantitatum</expan> e&longs;t, compo&longs;ita ex proportionibus primis, &amp; diui&longs;a per duplam.<emph.end type="italics"/></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell>XII.</cell>
            <cell>P<emph type="italics"/>ropo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que multiplicatione.<emph.end type="italics"/></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell>XIII.</cell>
            <cell>P<emph type="italics"/>roportio confu&longs;a aggregata prim&aelig; &amp; terti&aelig; quatuor quantitatum omiologarum ad   aggregatum &longs;ecund&aelig; &amp; quart&aelig;, e&longs;t uelut compo&longs;ita ex ei&longs;dem diui&longs;a per du&shy;plam.<emph.end type="italics"/></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell>XIIII.</cell>
            <cell>P<emph type="italics"/>roportiones confu&longs;&aelig; &amp; coniunct&aelig; in tribus quantitatibus inuicem commutantur.<emph.end type="italics"/></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell>XV.</cell>
            <cell>S<emph type="italics"/>i fuerint quatuor quantitates proportio confu&longs;a, aggregati prim&aelig; &amp; terti&aelig;, ad aggre&shy;gatum &longs;ecund&aelig; &amp; quart&aelig;, erit ut monadis addito prouentu, qui fit diui&longs;a differentia,   differentiarum prim&aelig; &amp; &longs;ecund&aelig;, atque quart&aelig; &amp; terti&aelig;, per aggregatum terti&aelig; &amp;   quart&aelig; ad ip&longs;am monadem.<emph.end type="italics"/></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell>XVI.</cell>
            <cell>O<emph type="italics"/>mnium quatuor quantitatum propo&longs;ita prima, qu&aelig; non minorem habet proportio&shy;nem ad &longs;uam corre&longs;pondentem qu&agrave;m alia ad aliam, erit proportio confu&longs;a illarum,<emph.end type="italics"/></cell>
            <cell/>
          </row>
          <pb xlink:href="015/01/009.jpg"/>
          <row>
            <cell/>
            <cell><emph type="italics"/>ut producti ex aggregato prim&aelig; &amp; terti&aelig;, in tertiam ad productum ex aggre gato terti&aelig; &amp; omiotat&aelig; ad &longs;ecundam in ip&longs;am quartam.<emph.end type="italics"/></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell>XVII.</cell>
            <cell>O<emph type="italics"/>mnes du&aelig; proportiones conuer&longs;&aelig; producunt &aelig;qualem proportionem.<emph.end type="italics"/></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell>XVIII.</cell>
            <cell>S<emph type="italics"/>i fuerint quotlibet quantitates in continua proportione multiplici pr&aelig;ter, <expan abbr="ultim&atilde;">ultimam</expan>   proportio uer&ograve; penultim&aelig; ad ultimam, qualis re&longs;idui prim&aelig; ad &longs;ecundam,   erit prim&aelig; ad aggregatum reliquarum, uelut penultim&aelig; ad ultimam.<emph.end type="italics"/></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell>XIX.</cell>
            <cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmetic&aelig; omiolog&aelig;, quarum exce&longs;&longs;us &longs;it &aelig;qualis   minim&egrave;, omnibus autem deficientibus &longs;upplementa ad &aelig;qualitatem maxim&egrave;   adiungantur, erunt quadrata omnium quantitatum &aelig;qualium, adiecto rur&longs;us   quadrato prim&aelig; cum eo quod fit ex minima primi ordinis in aggregatum o&shy;mnium quantitatum eiu&longs;dem, tripla aggregato quadratorum omnium quanti tatum primi ordinis pariter acceptis.<emph.end type="italics"/></cell>
            <cell>17</cell>
          </row>
          <row>
            <cell>XX.</cell>
            <cell>C<emph type="italics"/>um fuerint quatuor quantitates, fueritque <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> &aelig;qualis terti&aelig;, aut prima &aelig;qualis   quart&aelig;, erit proportio prim&aelig; ad quartam, aut terti&aelig; ad &longs;ecundam, producta   ex proportionibus prim&aelig; ad &longs;ecundam &amp; terti&aelig; ad quartam.<emph.end type="italics"/></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell>XXI.</cell>
            <cell>C<emph type="italics"/>um decu&longs;&longs;atim ducta fuerit prima in quartam, &amp; &longs;ecunda in tertiam, produ&shy;ctumque prim&aelig; in quartam, diui&longs;um fuerit per productum &longs;ecund&aelig; in tertiam,   erit proportio prim&aelig; ad &longs;ecundam, diui&longs;a per proport&iacute;onem terti&aelig; ad quar&shy;tam.<emph.end type="italics"/> E<emph type="italics"/>t &longs;imiliter interpo&longs;ita omiologa.<emph.end type="italics"/></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell>XXII.</cell>
            <cell>C<emph type="italics"/>um fuerit proportio prim&aelig; ad &longs;ecundam maior qu&agrave;m terti&aelig; ad quartam, erit   confu&longs;a ex his maior qu&agrave;m terti&aelig; ad quartam, minor autem qu&agrave;m prim&aelig; ad   &longs;ecundam.<emph.end type="italics"/></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>XXIII.</cell>
            <cell>O<emph type="italics"/>mnis motus naturalis ad locum &longs;uum e&longs;t: ide&ograve; per rectam lineam fit.<emph.end type="italics"/></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>XXIIII.</cell>
            <cell>O<emph type="italics"/>mnis motus circularis uoluntarius e&longs;t.<emph.end type="italics"/></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>XXV.</cell>
            <cell>T<emph type="italics"/>res &longs;unt motus omnino &longs;implices naturalis, uoluntarius, &amp; uiolentus.<emph.end type="italics"/></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell>XXVI.</cell>
            <cell>M<emph type="italics"/>otus ergo compo&longs;iti quatuor nece&longs;&longs;ari&ograve; &longs;unt &longs;pecies.<emph.end type="italics"/></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell>XXVII.</cell>
            <cell>M<emph type="italics"/>otus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus ex loco.<emph.end type="italics"/></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell>XXVIII.</cell>
            <cell>M<emph type="italics"/>otus quilibet uoluntarius aut uiolentus in aliquo medio fit.<emph.end type="italics"/></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell>XXIX.</cell>
            <cell>O<emph type="italics"/>mnis motus uoluntarius &aelig;qualis e&longs;t &longs;emper: &longs;impliciter etiam quilibet alius mo&shy;tus.<emph.end type="italics"/></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell>XXX.</cell>
            <cell>I<emph type="italics"/>n omni corpore mobili in medio partes medij re&longs;i&longs;tunt obui&aelig;, ali&aelig; impel&shy;lunt.<emph.end type="italics"/></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell>XXXI.</cell>
            <cell>O<emph type="italics"/>mnis motus naturalis in &aelig;quali medio ualidior e&longs;t in fine qu&agrave;m in principio.<emph.end type="italics"/>V<emph type="italics"/>iolentus contr&agrave;.<emph.end type="italics"/></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell>XXXII.</cell>
            <cell>O<emph type="italics"/>mne mobile naturaliter motum &longs;eu uiolenter uelocius mouetur in medio rariore   qu&agrave;m den&longs;iore.<emph.end type="italics"/> M<emph type="italics"/>aior quoque e&longs;t proportio finis motus in corpore rariore ad   finem motus in corpore den&longs;iore qu&agrave;m principij.<emph.end type="italics"/> I<emph type="italics"/>n uiolento autem celerius   perueniret ad finem motus in corpore den&longs;iore.<emph.end type="italics"/></cell>
            <cell>27</cell>
          </row>
          <row>
            <cell>XXXIII.</cell>
            <cell>O<emph type="italics"/>mnia duo mobilia &aelig;qualis undique magnitudinis qu&aelig; &aelig;quali in tempore &aelig;qualia   &longs;pacia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia medijs nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad   pondus, quem ad modum medij ad medium proportio duplicata.<emph.end type="italics"/></cell>
            <cell>27</cell>
          </row>
          <row>
            <cell>XXXIIII.</cell>
            <cell>P<emph type="italics"/>roportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t uelut eiu&longs;dem &longs;uperfi   ciei, ad latus eiu&longs;dem uer&ograve; ad monadem.<emph.end type="italics"/></cell>
            <cell>28</cell>
          </row>
          <row>
            <cell>XXXV.</cell>
            <cell>V<emph type="italics"/>ocum magnitudines excre&longs;cunt in acumine, non in grauitate, finis autem e&longs;t in   utroque extremo.<emph.end type="italics"/> P<emph type="italics"/>ropter hoc minima facta uariatione in hypate acut&aelig; uix   ferunt.<emph.end type="italics"/></cell>
            <cell>29</cell>
          </row>
          <row>
            <cell>XXXVI.</cell>
            <cell>S<emph type="italics"/>i proportio per proportionem minorem &aelig;quali ducatur, proportio minor pro&shy;<emph.end type="italics"/></cell>
            <cell/>
          </row>
          <pb xlink:href="015/01/010.jpg"/>
          <row>
            <cell/>
            <cell><emph type="italics"/>ducetur.<emph.end type="italics"/> V<emph type="italics"/>nde manife&longs;tum e&longs;t duas proportiones minores &aelig;qualitate <expan abbr="inuic&etilde;">inuicem</expan> du ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"/></cell>
            <cell>30</cell>
          </row>
          <row>
            <cell>XXXVII.</cell>
            <cell>S<emph type="italics"/>i plures homines, quorum per &longs;e nauim mouere po&szlig;int, aut pondus ferre &longs;imul   iuncti eam moueant, aut pondus ferant, erunt ill&aelig; proportiones coniunct&aelig; non product&aelig;.<emph.end type="italics"/></cell>
            <cell>30</cell>
          </row>
          <row>
            <cell>XXXVIII.</cell>
            <cell>O<emph type="italics"/>mne corpus tantum re&longs;i&longs;tit motui contrario &longs;uo nat&uacute;rali, quantum mouetur oc&shy;culto motu quie&longs;cendo.<emph.end type="italics"/></cell>
            <cell>31</cell>
          </row>
          <row>
            <cell>XXXIX.</cell>
            <cell>A<emph type="italics"/>b &aelig;quali aut minore ui qu&agrave;m &longs;it impedimentum non fit motus.<emph.end type="italics"/></cell>
            <cell>31</cell>
          </row>
          <row>
            <cell>XL.</cell>
            <cell>O<emph type="italics"/>mne corpus &longs;ph&aelig;ricum tangens planum in puncto mouetur ad latus per quam&shy;cunque uim, qu&aelig; medium diuidere pote&longs;t.<emph.end type="italics"/></cell>
            <cell>31</cell>
          </row>
          <row>
            <cell>XLI.</cell>
            <cell>S<emph type="italics"/>i fuerint du&aelig; quantitates &longs;umaturque toties <expan abbr="aggregat&utilde;">aggregatum</expan> maioris &amp; minoris, quo&shy;ties aggregatum minoris &amp; maioris, erit proportio confu&longs;a maioris aggregati   ad minus, minor quam multiplicis maioris ad multiplex minoris.<emph.end type="italics"/></cell>
            <cell>32</cell>
          </row>
          <row>
            <cell>XLII.</cell>
            <cell>T<emph type="italics"/>rahentium nauim, aut ferentium pondera proportiones in &longs;e inuicem, quomodo   ducere oporteat con&longs;iderare.<emph.end type="italics"/></cell>
            <cell>32</cell>
          </row>
          <row>
            <cell>XLIII.</cell>
            <cell>P<emph type="italics"/>roductionem ad additionem retrahere.<emph.end type="italics"/></cell>
            <cell>33</cell>
          </row>
          <row>
            <cell>XLIIII.</cell>
            <cell>S<emph type="italics"/>i fuerit proportio motoris ad id quod e&longs;t maximum non mouens, &amp; &longs;patium &amp;   tempus, nota erit etiam reliquorum nota.<emph.end type="italics"/></cell>
            <cell>33</cell>
          </row>
          <row>
            <cell>XLV.</cell>
            <cell>R<emph type="italics"/>ationem &longs;tater&aelig; o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>34</cell>
          </row>
          <row>
            <cell>XLVI.</cell>
            <cell>A<emph type="italics"/>n &longs;it aliqua proportio &amp; qualis inter animam &amp; uitas, &amp; &longs;ua corpora con&longs;ide&shy;rare.<emph.end type="italics"/></cell>
            <cell>35</cell>
          </row>
          <row>
            <cell>XLVII.</cell>
            <cell>S<emph type="italics"/>i duo mobilia &aelig;qualister in eodem circulo iuxta proprios motus moueantur, pro&shy;ductum temporis circuituum inuicem, erit &aelig;quale producto differenti&aelig; tempo   rum circuitus duct&aelig; in tempus coniunctionis prim&aelig;.<emph.end type="italics"/></cell>
            <cell>36</cell>
          </row>
          <row>
            <cell>XLVIII.</cell>
            <cell>S<emph type="italics"/>i tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum ac duorum coniun&shy;ctiones in temporibus commen&longs;is, illa tria mobilia denuo coniungentur in tem   pore producto ex denominatore diui&longs;ionis temporis maioris per minus in mi&shy;nus aut numeratore in maius.<emph.end type="italics"/></cell>
            <cell>37</cell>
          </row>
          <row>
            <cell>XLIX.</cell>
            <cell>P<emph type="italics"/>ropofitio mobilis in circulo circuitus tempore dataque ratione di&longs;tanti&aelig; ab illo mo   bilis circuitum inuenire, quod ex <expan abbr="eod&etilde;">eodem</expan> puncto di&longs;cedens <expan abbr="c&utilde;alio">cunalio</expan> mobili in dato   puncto <expan abbr="c&otilde;ueniat">conueniat</expan> &longs;ub <expan abbr="quoc&utilde;que">quocunque</expan> numero <expan abbr="circuitu&utilde;">circuituum</expan> <expan abbr="t&etilde;pus">tempus</expan> quoque <expan abbr="c&otilde;iunctionis">coniunctionis</expan>.<emph.end type="italics"/></cell>
            <cell>39</cell>
          </row>
          <row>
            <cell>L.</cell>
            <cell>O<emph type="italics"/>mnes circuituum portiones in ei&longs;dem temporibus repetuntur.<emph.end type="italics"/></cell>
            <cell>40</cell>
          </row>
          <row>
            <cell>LI.</cell>
            <cell>O<emph type="italics"/>perationes dictas exemplo declarare.<emph.end type="italics"/></cell>
            <cell>41</cell>
          </row>
          <row>
            <cell>LII.</cell>
            <cell>T<emph type="italics"/>ria mobilia coniuncta in <expan abbr="eod&etilde;">eodem</expan> puncto, quorum duo &amp; duo conueniant in partib.   incommen&longs;is inter &longs;e, in perpetuum in nullo unquam puncto conuenient.<emph.end type="italics"/></cell>
            <cell>42</cell>
          </row>
          <row>
            <cell>LIII.</cell>
            <cell>C<emph type="italics"/>irculorum &longs;e in aduer&longs;um mouentium proportionem declarare.<emph.end type="italics"/></cell>
            <cell>43</cell>
          </row>
          <row>
            <cell>LIIII.</cell>
            <cell>P<emph type="italics"/>roportio circuli ad &longs;uum diametrum per &longs;imilitudinem e&longs;t quarta pars periphe&shy;ri&aelig;.<emph.end type="italics"/> R<emph type="italics"/>ur&longs;usque eiu&longs;dem circuli ad peripheriam diametri quarta pars.<emph.end type="italics"/></cell>
            <cell>44</cell>
          </row>
          <row>
            <cell>LV.</cell>
            <cell>P<emph type="italics"/>roportionem medicamentorum per ordines &longs;up po&longs;ita &aelig;quali proportione in or&shy;dinibus per quantitates &amp; proportiones demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>44</cell>
          </row>
          <row>
            <cell>LVI.</cell>
            <cell>P<emph type="italics"/>roportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&longs;um e&longs;t duplicata ei   qu&aelig; ad numeri latus.<emph.end type="italics"/></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell>LVII.</cell>
            <cell>M<emph type="italics"/>otus rationem ad pondus inuenire.<emph.end type="italics"/></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell>LVIII.</cell>
            <cell>Q<emph type="italics"/>u&aelig; ex alto de&longs;cendunt, cur non eandem pro di&longs;tantia motus rationem in libero   a&euml;re &longs;eruent con&longs;iderare.<emph.end type="italics"/></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell>LIX.</cell>
            <cell>O<emph type="italics"/>mne mobile motum duobus motibus non ad idem tendentibus utroque &longs;eor&longs;um tar   dius mouetur &longs;imili motu.<emph.end type="italics"/></cell>
            <cell>50</cell>
          </row>
          <row>
            <cell>LX.</cell>
            <cell>O<emph type="italics"/>mne mobile motu naturali de&longs;cendentis parte, de&longs;cendit grauiore &longs;ecundum gra&shy;<emph.end type="italics"/></cell>
            <cell/>
          </row>
          <pb xlink:href="015/01/011.jpg"/>
          <row>
            <cell/>
            <cell><emph type="italics"/>uitatis centrum.<emph.end type="italics"/></cell>
            <cell>51</cell>
          </row>
          <row>
            <cell>LXI.</cell>
            <cell>P<emph type="italics"/>roportionum ictus ad pondus rei &amp; di&longs;tantiam generaliter con&longs;iderare.<emph.end type="italics"/></cell>
            <cell>52</cell>
          </row>
          <row>
            <cell>LXII.</cell>
            <cell>P<emph type="italics"/>roportionem motoris in plano ad motorem, qui eleuat pondus iuxta id quod   mouet, inuenire.<emph.end type="italics"/></cell>
            <cell>53</cell>
          </row>
          <row>
            <cell>LXIII.</cell>
            <cell>O<emph type="italics"/>mne graue quanto proximius alligatum plano, tant&ograve; facilius trabitur.<emph.end type="italics"/></cell>
            <cell>53</cell>
          </row>
          <row>
            <cell>LXIIII.</cell>
            <cell>O<emph type="italics"/>mne mobile quant&ograve; latius tanto tardius moustur in plano.<emph.end type="italics"/></cell>
            <cell>54</cell>
          </row>
          <row>
            <cell>LXV.</cell>
            <cell>P<emph type="italics"/>roportionem duorum mobilium inter &longs;e cum auxilio medij inuenire.<emph.end type="italics"/></cell>
            <cell>54</cell>
          </row>
          <row>
            <cell>LXVI.</cell>
            <cell>P<emph type="italics"/>roportionem laterum eptagoni, &amp; &longs;ubten&longs;arum con&longs;iderare, &amp; qu&aelig; &agrave; reflexa   proportione pendent.<emph.end type="italics"/></cell>
            <cell>55</cell>
          </row>
          <row>
            <cell>LXVII.</cell>
            <cell>S<emph type="italics"/>i fuerint aliquot quantitates ab una quantitate ali&aelig;que totidem ab eadem analo&shy;g&aelig;, erit proportio terti&aelig; unius ordinis ad tertiam alterius, ut &longs;ecund&aelig; ad &longs;e&shy;cundum duplicata, &amp; quart&aelig; ad quartam triplicata, quint&aelig; ad quintam   quadruplicata, atque &longs;ic de alijs.<emph.end type="italics"/></cell>
            <cell>57</cell>
          </row>
          <row>
            <cell>LXVIII.</cell>
            <cell>P<emph type="italics"/>ropo&longs;itio collectorum ab<emph.end type="italics"/> E<emph type="italics"/>uclide &amp;<emph.end type="italics"/> A<emph type="italics"/>rchimede.<emph.end type="italics"/></cell>
            <cell>57</cell>
          </row>
          <row>
            <cell>LXIX.</cell>
            <cell>P<emph type="italics"/>ropo&longs;itio collectorum ex quatuor libris<emph.end type="italics"/> A<emph type="italics"/>pollonij<emph.end type="italics"/> P<emph type="italics"/>ergei &amp;<emph.end type="italics"/> <expan abbr="q.">que</expan> S<emph type="italics"/>ereni.<emph.end type="italics"/></cell>
            <cell>59</cell>
          </row>
          <row>
            <cell>LXX.</cell>
            <cell>S<emph type="italics"/>i fuerint tres quantitates in continua proportione, ali&aelig;que totidem in continua   proportione poterunt con&longs;tituere tres quantitates in &aelig;quali differentia per&shy;uer&longs;im copulat&aelig;.<emph.end type="italics"/></cell>
            <cell>62</cell>
          </row>
          <row>
            <cell>LXXI.</cell>
            <cell>P<emph type="italics"/>roportionem leuitatis ponderis per uirgam torcularem attracti ad rectam &longs;u&shy;&longs;pen&longs;ionem inuenire.<emph.end type="italics"/></cell>
            <cell>63</cell>
          </row>
          <row>
            <cell>LXXII.</cell>
            <cell>P<emph type="italics"/>roportionem ponderis &longs;ph&aelig;r&aelig; pendentis ad a&longs;cendentem per accliue planum   inuenire.<emph.end type="italics"/></cell>
            <cell>63</cell>
          </row>
          <row>
            <cell>LXXIII.</cell>
            <cell>P<emph type="italics"/>roportionem ponderum attractorum penes figuram in plano inuenire.<emph.end type="italics"/></cell>
            <cell>64</cell>
          </row>
          <row>
            <cell>LXXIIII.</cell>
            <cell>P<emph type="italics"/>roportionem concutientis ad concu&longs;&longs;um in&longs;tabili inuenire.<emph.end type="italics"/></cell>
            <cell>64</cell>
          </row>
          <row>
            <cell>LXXV.</cell>
            <cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> immoti in aqua, ad <expan abbr="immot&utilde;">immotum</expan> in terra in excipiendo <expan abbr="ict&utilde;">ictum</expan> inuenire.<emph.end type="italics"/></cell>
            <cell>65</cell>
          </row>
          <row>
            <cell>LXXVI.</cell>
            <cell>P<emph type="italics"/>roportionem <expan abbr="duor&utilde;">duorum</expan> mobilium &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="concurrenti&utilde;">concurrentium</expan> per <expan abbr="rect&atilde;">rectam</expan> inuenire.<emph.end type="italics"/></cell>
            <cell>66</cell>
          </row>
          <row>
            <cell>LXXVII.</cell>
            <cell>P<emph type="italics"/>roportionem motus obliqui ad motum rectum in nauibus inuenire.<emph.end type="italics"/></cell>
            <cell>66</cell>
          </row>
          <row>
            <cell>LXXVIII.</cell>
            <cell>P<emph type="italics"/>roportionem nauis ad triremes quotuis concurrentes demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>67</cell>
          </row>
          <row>
            <cell>LXXIX.</cell>
            <cell>P<emph type="italics"/>roportionem medicamentorum purgantium inuicem declarare<emph.end type="italics"/></cell>
            <cell>68</cell>
          </row>
          <row>
            <cell>LXXX.</cell>
            <cell>P<emph type="italics"/>roportionem motus &longs;ecundum obliquum ad rectum in &longs;pacio declarare.<emph.end type="italics"/></cell>
            <cell>69</cell>
          </row>
          <row>
            <cell>LXXXI.</cell>
            <cell>Q<emph type="italics"/>ualis &longs;it angulus, per quem pote&longs;t moueri nauis ad rectum explorare.<emph.end type="italics"/></cell>
            <cell>70</cell>
          </row>
          <row>
            <cell>LXXXII.</cell>
            <cell>P<emph type="italics"/>roportionem uelorum indagare.<emph.end type="italics"/></cell>
            <cell>70</cell>
          </row>
          <row>
            <cell>LXXXIII.</cell>
            <cell>P<emph type="italics"/>roportionem rece&longs;&longs;us &agrave; recta uia ad obliquitatem inue&longs;tigare.<emph.end type="italics"/></cell>
            <cell>72</cell>
          </row>
          <row>
            <cell>LXXXIIII.</cell>
            <cell>D<emph type="italics"/><expan abbr="i&longs;tanti&atilde;">i&longs;tantiam</expan> centri terr&aelig; &agrave; centro mundi per motum lapidis<emph.end type="italics"/> H<emph type="italics"/>erculei declarare.<emph.end type="italics"/></cell>
            <cell>73</cell>
          </row>
          <row>
            <cell>LXXXV.</cell>
            <cell>P<emph type="italics"/>roportio ponderis unius grauis ad aliud &longs;ub eadem men&longs;ura e&longs;t ueluti eiu&longs;dem   ad differentiam ponderis ua&longs;is repleti ex altero graui, &amp; ex ambobus de&shy;tracto priore.<emph.end type="italics"/></cell>
            <cell>74</cell>
          </row>
          <row>
            <cell>LXXXVI.</cell>
            <cell>S<emph type="italics"/>i circuli in &aelig; quales &longs;eu in &longs;ph&aelig;ra &longs;eu in plano &longs;e &longs;ecuerint, nunqu&agrave;m oppo&longs;itos   angulos &aelig;quales habent.<emph.end type="italics"/></cell>
            <cell>77</cell>
          </row>
          <row>
            <cell>LXXXVII.</cell>
            <cell>P<emph type="italics"/>roportiones cra&szlig;itiei aqu&aelig; ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="c&otilde;paratione">comparatione</expan> ad radios demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>78</cell>
          </row>
          <row>
            <cell>LXXXVIII.</cell>
            <cell>I<emph type="italics"/><expan abbr="n&longs;trument&utilde;">n&longs;trumentum</expan><emph.end type="italics"/> A<emph type="italics"/>colingen, quo momenta temporum <expan abbr="deprehend&atilde;tur">deprehendantur</expan> fabricare.<emph.end type="italics"/></cell>
            <cell>79</cell>
          </row>
          <row>
            <cell>LXXXIX.</cell>
            <cell>P<emph type="italics"/>roportionem den&longs;itatis aqu&aelig; ad a&euml;rem per pondera inuenire.<emph.end type="italics"/></cell>
            <cell>82</cell>
          </row>
          <row>
            <cell>XC.</cell>
            <cell>R<emph type="italics"/>ationem impetus uiolenti extra mi&szlig;i ponderis ad &aelig;qualitatem reducere.<emph.end type="italics"/></cell>
            <cell>82</cell>
          </row>
          <row>
            <cell>XCI.</cell>
            <cell>P<emph type="italics"/>roportionem grauis cubi, &amp; &longs;ph&aelig;rici &aelig;qualium in accliui, &amp; de&longs;cen&longs;us eorum   demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>83</cell>
          </row>
          <row>
            <cell>XCII.</cell>
            <cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> ponderis &aelig;qualis iuxta longitudinis <expan abbr="c&otilde;paration&etilde;">comparationem</expan> demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>85</cell>
          </row>
          <row>
            <cell>XCIII.</cell>
            <cell>P<emph type="italics"/>ropter qd in <expan abbr="c&otilde;cu&szlig;ione">concu&szlig;ione</expan> <expan abbr="eti&atilde;">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manifi <expan abbr="&longs;i&utilde;">&longs;ium</expan>   e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, &amp; <expan abbr="qu&atilde;t&utilde;">quantum</expan> <expan abbr="retroced&atilde;t">retrocedant</expan> amb&aelig;.<emph.end type="italics"/></cell>
            <cell>86</cell>
          </row>
          <pb xlink:href="015/01/012.jpg"/>
          <row>
            <cell>XCIIII.</cell>
            <cell>S<emph type="italics"/>i <expan abbr="qu&atilde;titas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="qu&atilde;titas">quantitas</expan> nota &longs;imiliter.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i du&aelig;   proportiones not&aelig; fuerint, erit producta ex his atque diui&longs;a coniunctaque atque detra&shy;cta nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit totius ad partem proportio nota, erit et ad aliam partem nota:   &amp; alterius partis ad <expan abbr="alter&atilde;">alteram</expan> uno minor.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit partis ad partem, erit ad totum   monade minor atque nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit unius <expan abbr="qu&atilde;titatis">quantitatis</expan> ad duas <expan abbr="qu&atilde;titates">quantitates</expan> proportio   nota, erit &amp; <expan abbr="c&otilde;fu&longs;a">confu&longs;a</expan> ex eis nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerint trium quantitatum omiologarum, aut   quatuor analogarum omnes pr&aelig;ter unam cognit&aelig;, erunt &amp; illa alia cognita.<emph.end type="italics"/></cell>
            <cell>87</cell>
          </row>
          <row>
            <cell>XCV.</cell>
            <cell>C<emph type="italics"/>uiu&longs;uis trigoni rectanguli, aut cuius duo auguli &longs;int in dupla proportione, aut qui   circulo in&longs;criptus &longs;it cognita quantitate unius lateris in comparatione ad dimetien   <expan abbr="t&etilde;">tem</expan>, &longs;i proportio duorum laterum cognita fuerit, <expan abbr="er&utilde;t">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell>
            <cell>88</cell>
          </row>
          <row>
            <cell>XCVI.</cell>
            <cell>C<emph type="italics"/>um in <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> den&longs;um radij lumino&longs;i inciderint, quatuor fiunt luminis genera.<emph.end type="italics"/></cell>
            <cell>89</cell>
          </row>
          <row>
            <cell>XCVII.</cell>
            <cell>M<emph type="italics"/><expan abbr="ot&utilde;">otum</expan> inuer&longs;ionis in figuris in <expan abbr="c&otilde;paratione">comparatione</expan> ad <expan abbr="mot&utilde;">motum</expan> &longs;ph&aelig;r&aelig; in plano inue&longs;tigare.<emph.end type="italics"/></cell>
            <cell>91</cell>
          </row>
          <row>
            <cell>XCVIII.</cell>
            <cell>P<emph type="italics"/>roportionem ponderum &aelig;qualium per differentiam angulorum inuenire.<emph.end type="italics"/></cell>
            <cell>92</cell>
          </row>
          <row>
            <cell>XCIX.</cell>
            <cell>P<emph type="italics"/>roportionem grauitatum per multitudinem &longs;uppo&longs;itorum orbium o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>93</cell>
          </row>
          <row>
            <cell>C.</cell>
            <cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> grauitatis <expan abbr="ponder&utilde;">ponderum</expan> attractorum per <expan abbr="trochlear&utilde;">trochlearum</expan> <expan abbr="numer&utilde;">numerum</expan> inue&longs;tigare.<emph.end type="italics"/></cell>
            <cell>93</cell>
          </row>
          <row>
            <cell>CI.</cell>
            <cell>P<emph type="italics"/>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell>
            <cell>94</cell>
          </row>
          <row>
            <cell>CII.</cell>
            <cell>P<emph type="italics"/>roportionem motuum inuer&longs;ionis, &amp; attractionis in plano inuenire.<emph.end type="italics"/></cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>CIII.</cell>
            <cell>P<emph type="italics"/>roportionem eorundem in accliui demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>CIIII.</cell>
            <cell>P<emph type="italics"/>roportionem motus attractionis in decliui ad motum in plano determinare.<emph.end type="italics"/></cell>
            <cell>95</cell>
          </row>
          <row>
            <cell>CV.</cell>
            <cell>P<emph type="italics"/>roportionem ferentium pondus in pertica inuenire.<emph.end type="italics"/></cell>
            <cell>96</cell>
          </row>
          <row>
            <cell>CVI.</cell>
            <cell>Q<emph type="italics"/>uales proportiones angulorum doceant laterum proportiones.<emph.end type="italics"/> A<emph type="italics"/>tque uici&szlig;im deter&shy;minare.<emph.end type="italics"/></cell>
            <cell>97</cell>
          </row>
          <row>
            <cell>CVII.</cell>
            <cell>S<emph type="italics"/>i in circulo du&aelig; diametri ad rectum angulum &longs;e &longs;ecauerint: ali&aelig; uer&ograve; ad perpendicu&shy;lum ex diametro exicrint ad circum ferentiam, &longs;ingul&aelig; &longs;upra diametrum erunt ma   iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium   autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua   pr&aelig;terea portionis &longs;uperioris parte, qu&aelig; uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita   e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;p&otilde;dentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line &aelig; tran&longs;uer&longs;&aelig;.<emph.end type="italics"/></cell>
            <cell>100</cell>
          </row>
          <row>
            <cell>CVIII.</cell>
            <cell>P<emph type="italics"/>unctum &aelig;qualitatis differenti&aelig; de&longs;cen&longs;us &amp; remotionis &agrave; centro inuenire.<emph.end type="italics"/></cell>
            <cell>100</cell>
          </row>
          <row>
            <cell>CIX.</cell>
            <cell>R<emph type="italics"/>ationem libr&aelig; expendere.<emph.end type="italics"/></cell>
            <cell>101</cell>
          </row>
          <row>
            <cell>CX.</cell>
            <cell>S<emph type="italics"/>i du&aelig; &longs;ph&aelig;r&aelig; ex eadem materia de&longs;cendant in a&euml;re, eodem temporis momento ad   planum ueniunt.<emph.end type="italics"/></cell>
            <cell>104</cell>
          </row>
          <row>
            <cell>CXI.</cell>
            <cell>C<emph type="italics"/>ur ex medio tela ualidiorem ictum, &amp; naues in &longs;calmo &agrave; remo ac malo recipiant in&shy;de ex puppi explorare.<emph.end type="italics"/></cell>
            <cell>105</cell>
          </row>
          <row>
            <cell>CXII.</cell>
            <cell>C<emph type="italics"/>ur ex imo leuia longi&ugrave;s ferantur declarare,<emph.end type="italics"/></cell>
            <cell>106</cell>
          </row>
          <row>
            <cell>CXIII.</cell>
            <cell>C<emph type="italics"/>ur uirga longius mittatur &agrave; puero quam &agrave; uiro inueftigare.<emph.end type="italics"/></cell>
            <cell>107</cell>
          </row>
          <row>
            <cell>CXIIII.</cell>
            <cell>C<emph type="italics"/>ircularis motus differentias quatuor e&longs;&longs;e, earumque rationem contemplari.<emph.end type="italics"/></cell>
            <cell>108</cell>
          </row>
          <row>
            <cell>CXV.</cell>
            <cell>P<emph type="italics"/>roportionem motuum impul&longs;ionis, &amp; attractionis inter &longs;e, ab eadem ui decla&shy;rare.<emph.end type="italics"/></cell>
            <cell>110</cell>
          </row>
          <row>
            <cell>CXVI.</cell>
            <cell>C<emph type="italics"/>ur machin&aelig; oblong&aelig; igne&aelig; longius emittant &longs;ph&aelig;ram explorare.<emph.end type="italics"/></cell>
            <cell>111</cell>
          </row>
          <row>
            <cell>CXVII.</cell>
            <cell>I<emph type="italics"/>n curriculis maior e&longs;t uis pulueris copio&longs;ioris ampliore in &longs;pacio, qu&agrave;m paucioris in   minore iuxta proportionem eandem.<emph.end type="italics"/></cell>
            <cell>112</cell>
          </row>
          <row>
            <cell>CXVIII.</cell>
            <cell>Q<emph type="italics"/>uanta proportione decedat ictus in obliquum parietem ab eo qui e&longs;t ad perpendi&shy;culum declarare.<emph.end type="italics"/></cell>
            <cell>114</cell>
          </row>
          <row>
            <cell>CXIX.</cell>
            <cell>Q<emph type="italics"/>uantum ictus machin&aelig; procliuis ad angulum minuatur explorare.<emph.end type="italics"/></cell>
            <cell>115</cell>
          </row>
          <row>
            <cell>CXX</cell>
            <cell>P<emph type="italics"/>roportionem partium nauis ad eundem obliquum uentum explorare.<emph.end type="italics"/></cell>
            <cell>118</cell>
          </row>
          <row>
            <cell>CXXI.</cell>
            <cell>F<emph type="italics"/>labelli uires atque naturam declarare.<emph.end type="italics"/></cell>
            <cell>219</cell>
          </row>
          <row>
            <cell>CXXII.</cell>
            <cell>C<emph type="italics"/>ontemptus circa<emph.end type="italics"/> S<emph type="italics"/>olis rationem in umbris declarare.<emph.end type="italics"/></cell>
            <cell>120</cell>
          </row>
          <pb xlink:href="015/01/013.jpg"/>
          <row>
            <cell>CXXIII.</cell>
            <cell>C<emph type="italics"/>ognita ratione umbr&aelig; ad gnomonem &longs;inum, &amp; arcum altitudinis ab horizon&shy;te, quouis tempore digno&longs;cere.<emph.end type="italics"/></cell>
            <cell>121</cell>
          </row>
          <row>
            <cell>CXXIIII.</cell>
            <cell>P<emph type="italics"/>roportionem umbr&aelig; uer&longs;&aelig; e&longs;&longs;e ad gnomonem, uelut gnomonis ad umbram   uer&longs;am.<emph.end type="italics"/></cell>
            <cell>122</cell>
          </row>
          <row>
            <cell>CXXV.</cell>
            <cell>P<emph type="italics"/>roportionem dimetientis, &amp; peripheri&aelig; cuiuslibet circuli paralleli &aelig;quino&shy;ctiali per cognitam partem magni circuli demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>123</cell>
          </row>
          <row>
            <cell>CXXVI.</cell>
            <cell>C<emph type="italics"/>irculi horarij naturam declarare.<emph.end type="italics"/></cell>
            <cell>123</cell>
          </row>
          <row>
            <cell>CXXVII.</cell>
            <cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"/></cell>
            <cell>124</cell>
          </row>
          <row>
            <cell>CXXVIII.</cell>
            <cell>N<emph type="italics"/>ota amplitudine ortus, cuiu&longs;que puncti arcum &longs;emidiurnum inuenire.<emph.end type="italics"/></cell>
            <cell>124</cell>
          </row>
          <row>
            <cell>CXXIX.</cell>
            <cell>D<emph type="italics"/>ata altitudine<emph.end type="italics"/> S<emph type="italics"/>olis in quacunque regione, quacunque die di&longs;tantiam<emph.end type="italics"/> S<emph type="italics"/>olis &agrave; meri&shy;diano cogno&longs;cere.<emph.end type="italics"/></cell>
            <cell>124</cell>
          </row>
          <row>
            <cell>CXXX.</cell>
            <cell>D<emph type="italics"/>ata regionis altitudine, &amp; loco<emph.end type="italics"/> S<emph type="italics"/>olis proportionem gnomonis, tam ad um&shy;bram rectam qu&agrave;m uer&longs;am, uel etiam in cylindro determinare.<emph.end type="italics"/></cell>
            <cell>125</cell>
          </row>
          <row>
            <cell>CXXXI.</cell>
            <cell>S<emph type="italics"/>i line&aelig; alicui duplum alterius adiungatur, erit proportio duarum ad primam   maior qu&agrave;m dupli cum prima ad primam cum una adiecta.<emph.end type="italics"/></cell>
            <cell>126</cell>
          </row>
          <row>
            <cell>CXXXII.</cell>
            <cell>S<emph type="italics"/>i ad duas lineas quarum una alteri dupla &longs;it eadem linea addatur, erit aggrega&shy;ti ex minore, &amp; adiecta ad ip&longs;am minorem, minor proportio qu&agrave;m aggre&shy;gati ex maiore, &amp; adiecta ad ip&longs;am maiorem duplicata.<emph.end type="italics"/></cell>
            <cell>126</cell>
          </row>
          <row>
            <cell>CXXXIII.</cell>
            <cell>S<emph type="italics"/>i fuerint du&aelig; quantitates, <expan abbr="quar&utilde;">quarum</expan> una alteri dupla &longs;it: minuatur &agrave; minore qu&aelig;&shy;dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior pro&shy;portio, qu&agrave;m aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uer&ograve; minori addatur, &amp;   &agrave; maiore detrabatur, erit aggregati ad minorem minor proportio qu&agrave;m   maioris ad re&longs;iduum duplicata.<emph.end type="italics"/></cell>
            <cell>127</cell>
          </row>
          <row>
            <cell>CXXXIIII.</cell>
            <cell>S<emph type="italics"/>i rectangula &longs;uperficies &longs;it, cuius pars tertia quadrata &longs;it corpus, quod ex la&shy;tere quadrat&aelig; in re&longs;iduum &longs;uperficiei con&longs;tat, maius e&longs;t quouis corpore ex   eadem &longs;uperficies, aliter diui&longs;a con&longs;tituto.<emph.end type="italics"/></cell>
            <cell>127</cell>
          </row>
          <row>
            <cell>CXXXV.</cell>
            <cell>S<emph type="italics"/>i linea in duas partes, quarum una fit alteri dupla diuidatur, erit quod fit ex   tertia parte in quadratum re&longs;idui parallelipedum maius omni pararalleli&shy;pedo, quod ex diui&longs;ione eiu&longs;dem line&aelig; creari po&szlig;it.<emph.end type="italics"/></cell>
            <cell>128</cell>
          </row>
          <row>
            <cell>CXXXVI.</cell>
            <cell>D<emph type="italics"/>enominationes in infinitum extendere.<emph.end type="italics"/></cell>
            <cell>129</cell>
          </row>
          <row>
            <cell>CXXXVII.</cell>
            <cell>R<emph type="italics"/>ationem numerorum ex progre&szlig;ione declarare.<emph.end type="italics"/></cell>
            <cell>131</cell>
          </row>
          <row>
            <cell>CXXXVIII.</cell>
            <cell>M<emph type="italics"/>odos u&longs;us horum numerorum declarare.<emph.end type="italics"/></cell>
            <cell>131</cell>
          </row>
          <row>
            <cell>CXXXIX.</cell>
            <cell>R<emph type="italics"/>adices omnes &agrave; propo&longs;itis numeris extrahere.<emph.end type="italics"/></cell>
            <cell>132</cell>
          </row>
          <row>
            <cell>CXL.</cell>
            <cell>R<emph type="italics"/>adices per numeros fractos determinare.<emph.end type="italics"/></cell>
            <cell>133</cell>
          </row>
          <row>
            <cell>CXLI.</cell>
            <cell>N<emph type="italics"/>umeros fractos ad minores in ea <expan abbr="i&etilde;">iem</expan> proportione ualde propinqud deducere<emph.end type="italics"/></cell>
            <cell>136</cell>
          </row>
          <row>
            <cell>CXLII.</cell>
            <cell>D<emph type="italics"/><expan abbr="enomination&utilde;">enominationum</expan> in <expan abbr="crem&etilde;ta">crementa</expan> ex extrema cognita inuenire.<emph.end type="italics"/> E<emph type="italics"/>t <expan abbr="c&otilde;uer&longs;o">conuer&longs;o</expan> modo.<emph.end type="italics"/></cell>
            <cell>137</cell>
          </row>
          <row>
            <cell>CXLIII.</cell>
            <cell>S<emph type="italics"/>i linea in duas partes diuidatur, corpora qu&aelig; fiunt ex una parte in alterius   quadratum mutuo &aelig;qualia &longs;unt corpori, quod fit ex tota linea in &longs;uperfi&shy;ciem unius partis in alteram.<emph.end type="italics"/></cell>
            <cell>138</cell>
          </row>
          <row>
            <cell>CXLIIII.</cell>
            <cell>D<emph type="italics"/>uplum cubi medietatis maius e&longs;t aggregato corporum mutuorum, cuiuslibet   diui&longs;ionis quantum e&longs;t, quod fit ex tota in quadratum differenti&aelig;.<emph.end type="italics"/></cell>
            <cell>139</cell>
          </row>
          <row>
            <cell>CXLV.</cell>
            <cell>S<emph type="italics"/>i linea in duas partes diuidatur quadrata ambarum partium detracto eo, quod   fit ex una parte in alteram, &aelig;qualia &longs;unt producto unius in alteram cum   quadrato differenti&aelig;.<emph.end type="italics"/></cell>
            <cell>139</cell>
          </row>
          <row>
            <cell>CXLVI.</cell>
            <cell>C<emph type="italics"/>orpus quod fit ex linea diui&longs;a in &longs;uperficiem &aelig;qualem quadratis ambarum par   tium detracta &longs;uperficie unius partis in alteram, e&longs;t &aelig;quale aggregato cubo&shy;rum ambarum partium.<emph.end type="italics"/></cell>
            <cell>139</cell>
          </row>
          <row>
            <cell>CXLVII.</cell>
            <cell>P<emph type="italics"/>ropo&longs;ita linea diui&longs;a duas ei line as adijcere, ut proportio <expan abbr="additar&utilde;">additarum</expan> &longs;ingularium<emph.end type="italics"/></cell>
            <cell/>
          </row>
          <pb xlink:href="015/01/014.jpg"/>
          <row>
            <cell/>
            <cell><emph type="italics"/>&amp; partium &longs;imul iunctarum ad additas &longs;it mutua.<emph.end type="italics"/></cell>
            <cell>148</cell>
          </row>
          <row>
            <cell>CXLVIII.</cell>
            <cell>P<emph type="italics"/>ropo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus alijs lineis, &longs;ecun&shy;dum <expan abbr="ration&etilde;">rationem</expan> mutuam &longs;ingularum &longs;ingulis, <expan abbr="aggregat&utilde;">aggregatum</expan> ex una <expan abbr="adiectar&utilde;">adiectarum</expan>, &amp; par   te ad <expan abbr="aggregat&utilde;">aggregatum</expan> ex alia parte, &amp; adiecta &longs;e habeat, ut &longs;ecunda ad <expan abbr="terti&atilde;">tertiam</expan>.<emph.end type="italics"/></cell>
            <cell>140</cell>
          </row>
          <row>
            <cell>CXLIX.</cell>
            <cell>D<emph type="italics"/>atam lineam &longs;ic diuidere, ut proportio quadratorum ad dupium unius partis in   alteram &longs;it, ut line&aelig; dat&aelig; ad lineam datam.<emph.end type="italics"/></cell>
            <cell>141</cell>
          </row>
          <row>
            <cell>CL.</cell>
            <cell>P<emph type="italics"/>ropo&longs;itis duabus lineis, lineam communem utrique adiungere, ut &longs;it maioris ad ad&shy;ditam proportio, uelut quadratorum minoris, &amp; adiect&aelig; ad duplum unius in   alteram.<emph.end type="italics"/></cell>
            <cell>141</cell>
          </row>
          <row>
            <cell>CLI.</cell>
            <cell>P<emph type="italics"/>roportio differenti&aelig; quadratorum partium cuiu&longs;uis line&aelig;, ad quadratum diffe&shy;renti&aelig; illarum e&longs;t, uelut totius line&aelig; ad differentiam.<emph.end type="italics"/></cell>
            <cell>142</cell>
          </row>
          <row>
            <cell>CLII.</cell>
            <cell>S<emph type="italics"/>i linea in duas partes &aelig;quales, duasque in&aelig;quales diuidatur, fueritque proportio ag&shy;gregati ex maiore, &amp; dimidio ad ip&longs;am maiorem, uelut ex minore, &amp; ali&shy;qua linea ad ip&longs;am minorem, &amp; rur&longs;us aggregati ex minore, &amp; dimidio ad   ip&longs;am minorem, uelut aggregati ex maiore, &amp; alia addita ad ip&longs;am maiorem,   erit proportio dimidij ad partem unam in&aelig;qualem, uelut alterius partis in&aelig;&shy;qualis ad &longs;uam additam mutu&ograve;, &amp; etiam proportio additarum inuicem, uelut   proportio <expan abbr="parti&utilde;">partium</expan> <expan abbr="in&aelig;quali&utilde;">in&aelig;qualium</expan> duplicata, &amp; rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> line&aelig; a&longs;&longs;um&shy;pt&aelig; <expan abbr="medi&utilde;">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="em&utilde;">emum</expan> proportio dimidij <expan abbr="c&utilde;">cum</expan> addita   maiore ad <expan abbr="dimidi&utilde;">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minor&etilde;">minorem</expan>.<emph.end type="italics"/></cell>
            <cell>142</cell>
          </row>
          <row>
            <cell>CLIII.</cell>
            <cell>V<emph type="italics"/>im quamcunque manus multiplicare.<emph.end type="italics"/></cell>
            <cell>144</cell>
          </row>
          <row>
            <cell>CLIIII.</cell>
            <cell>S<emph type="italics"/>i line&aelig; dat&aelig; alia linea adiungatur, ab extremitatibus autem prioris line&aelig; du&aelig;   rect&aelig; in unum punctum concurrant proportionem habentes, quam mediam   inter tota m &amp; adiectam, &amp; adiectam erit punctus, concur&longs;us &agrave; puncto extre&shy;mo line&aelig; adiect&aelig; di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius li&shy;ne&aelig; &aelig;qua&apos;is medi&aelig;, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea   du&aelig; line&aelig; ad pr&aelig;dicta puncta producantur, ip&longs;&aelig; erunt in proportione medi&aelig;   ad adiectam.<emph.end type="italics"/></cell>
            <cell>145</cell>
          </row>
          <row>
            <cell>CLV.</cell>
            <cell>Q<emph type="italics"/>uadr atorum numerum proportionem &amp; inuentionem con&longs;iderare.<emph.end type="italics"/></cell>
            <cell>147</cell>
          </row>
          <row>
            <cell>CLVI.</cell>
            <cell>H<emph type="italics"/>orologiorum tempus multiplicare.<emph.end type="italics"/></cell>
            <cell>152</cell>
          </row>
          <row>
            <cell>CLVII.</cell>
            <cell>H<emph type="italics"/>orologiorum molarium rationem o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>154</cell>
          </row>
          <row>
            <cell>CLVIII.</cell>
            <cell>R<emph type="italics"/>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur ex&shy;plicare.<emph.end type="italics"/></cell>
            <cell>156</cell>
          </row>
          <row>
            <cell>CLIX.</cell>
            <cell>N<emph type="italics"/>ullus angulus rectilineus &aelig;qualis e&longs;&longs;e pote&longs;t alicui angulo contento recta, &amp; cir   culi portione.<emph.end type="italics"/></cell>
            <cell>158</cell>
          </row>
          <row>
            <cell>CLX.</cell>
            <cell>P<emph type="italics"/>ropo&longs;ita linea tribusque in ea &longs;ignis punctum inuenire, ex quo duct&aelig; tres line&aelig; ad   &longs;igna &longs;int in proportionibus datis.<emph.end type="italics"/></cell>
            <cell>162</cell>
          </row>
          <row>
            <cell>CLXI.</cell>
            <cell>S<emph type="italics"/>i fuerint duo trianguli, quorum ba&longs;es in eadem linea &longs;int con&longs;tituti, &amp; &aelig;quales   ad unum punctum terminati, &amp; latus unum commune inter reliqua quantita&shy;te medium nece&longs;&longs;e e&longs;t angulum &agrave; maioribus lineis <expan abbr="content&utilde;">contentum</expan> minorem e&longs;&longs;e.<emph.end type="italics"/></cell>
            <cell>162</cell>
          </row>
          <row>
            <cell>CLXII.</cell>
            <cell>P<emph type="italics"/>roportionem duorum orbium, quorum diametrorum conuex&aelig; partis, &amp; conca&shy;u&aelig; proportiones dat&aelig; &longs;int inue&longs;tigare.<emph.end type="italics"/></cell>
            <cell>164</cell>
          </row>
          <row>
            <cell>CLXIII.</cell>
            <cell>P<emph type="italics"/>roportionem uirium &longs;tellarum per motus &longs;uos indagare.<emph.end type="italics"/></cell>
            <cell>165</cell>
          </row>
          <row>
            <cell>CLXIIII.</cell>
            <cell>S<emph type="italics"/>yderum proportionem in magnitudine o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>166</cell>
          </row>
          <row>
            <cell>CLXV.</cell>
            <cell>P<emph type="italics"/>roportionem motuum omnium &longs;tellarum ad<emph.end type="italics"/> S<emph type="italics"/>olem con&longs;iderare.<emph.end type="italics"/></cell>
            <cell>167</cell>
          </row>
          <row>
            <cell>CLXVI.</cell>
            <cell>P<emph type="italics"/>roportiones mu&longs;icas &longs;uperpartientes in eas, qu&aelig; particul&aacute; una tantum abundant   reducere.<emph.end type="italics"/></cell>
            <cell>168</cell>
          </row>
          <pb xlink:href="015/01/015.jpg"/>
          <row>
            <cell>CLXVII.</cell>
            <cell>P<emph type="italics"/>roportionem mu&longs;icam ad &longs;apores &amp; odores coaptare.<emph.end type="italics"/></cell>
            <cell>176</cell>
          </row>
          <row>
            <cell>CLXVIII.</cell>
            <cell>P<emph type="italics"/>icturarum proportiones explicare.<emph.end type="italics"/></cell>
            <cell>179</cell>
          </row>
          <row>
            <cell>CLXIX.</cell>
            <cell>P<emph type="italics"/>roportionem mu&longs;icam in in&longs;trumentis declarare iuxta compo&longs;itionis ra&shy;tionem.<emph.end type="italics"/></cell>
            <cell>182</cell>
          </row>
          <row>
            <cell>CLXX.</cell>
            <cell>C<emph type="italics"/>oniugationes cuiu&longs;uis numeri breuiter inuenire.<emph.end type="italics"/></cell>
            <cell>185</cell>
          </row>
          <row>
            <cell>CLXXI.</cell>
            <cell>P<emph type="italics"/>ropo&longs;itis duobus quibuslibet numeris, quotuis alios &longs;eu in continuum &longs;eu   medios in continua proportione arithmetica, geometrica &amp; mu&longs;ica in&shy;uenire.<emph.end type="italics"/></cell>
            <cell>187</cell>
          </row>
          <row>
            <cell>CLXXII.</cell>
            <cell>P<emph type="italics"/>roportiones<emph.end type="italics"/> S<emph type="italics"/>tiphelij de&longs;cribere.<emph.end type="italics"/></cell>
            <cell>191</cell>
          </row>
          <row>
            <cell>CLXXIII.</cell>
            <cell>C<emph type="italics"/>irculum &longs;uper centro &longs;uo mouere &aelig;qualiter, ita quod omnia illius puncta   per rectam lineam moueantur ultro citroque.<emph.end type="italics"/></cell>
            <cell>192</cell>
          </row>
          <row>
            <cell>CLXXIIII.</cell>
            <cell>P<emph type="italics"/>rogre&longs;&longs;us &amp; regre&longs;&longs;us, tam &longs;ine latitudine qu&agrave;m cum latitudine in planetis   per &longs;olos concentricos circulos &aelig;qualiter motos demon&longs;trare.<emph.end type="italics"/></cell>
            <cell>194</cell>
          </row>
          <row>
            <cell>CLXXV.</cell>
            <cell>C<emph type="italics"/>au&longs;am uarietatis diametrorum ex &longs;uppo&longs;itis concentricis demon&longs;tra&shy;re.<emph.end type="italics"/></cell>
            <cell>195</cell>
          </row>
          <row>
            <cell>CLXXVI.</cell>
            <cell>R<emph type="italics"/>ationem centri grauitatis declarare.<emph.end type="italics"/></cell>
            <cell>197</cell>
          </row>
          <row>
            <cell>CLXXVII.</cell>
            <cell>S<emph type="italics"/>i proportio aliqua ex duabus proportionibus eiu&longs;dem quantitatis ad alias   duas componatur, erit proportio illarum duarum eadem proportioni   producti ex proportione in primam duarum quantitatum, detracta prio&shy;re illa quantitate, qu&aelig; ad duas comparatur, ad eandem priorem quanti&shy;tatem.<emph.end type="italics"/></cell>
            <cell>198</cell>
          </row>
          <row>
            <cell>CLXXVIII.</cell>
            <cell>P<emph type="italics"/>roportionem mi&longs;tionis metallorum, maxim&egrave; auri &amp; argenti declara&shy;re.<emph.end type="italics"/></cell>
            <cell>199</cell>
          </row>
          <row>
            <cell>CLXXIX.</cell>
            <cell>S<emph type="italics"/>i duobus totis du&aelig; portiones &longs;imiles ab&longs;cindantur ab ei&longs;dem denu&ograve;, &amp; ab&shy;&longs;ci&szlig;is portionibus partes e&aelig;dem auferantur, denuoque ac denu&ograve; quoties   libuerit &agrave; portionibus, &amp; &ugrave; re&longs;iduis ip&longs;arum quantitatum partes e&aelig;dem   auferantur, erit re&longs;idu&iacute; ad re&longs;iduum, ueluti totius ad totum.<emph.end type="italics"/></cell>
            <cell>200</cell>
          </row>
          <row>
            <cell>CLXXX.</cell>
            <cell>S<emph type="italics"/>i aliqua quantitas in duas partes diuidatur, fueritque alicuius quantitatis ad   partes illas compo&longs;ita proportio, non poterit eiu&longs;dem quantitatis ad par&shy;tes alias quantitatis diui&longs;a, aliter proportio eadem componi.<emph.end type="italics"/></cell>
            <cell>202</cell>
          </row>
          <row>
            <cell>CLXXXI.</cell>
            <cell>C<emph type="italics"/>um fuerit aliqua proportio, compo&longs;ita ex proportionibus prim&aelig; ad &longs;ecun&shy;dam &amp; tertiam, &amp; rur&longs;us quart&aelig; ad quintam &amp; &longs;extam: ita &longs;e habebit   proportio &longs;ecund&aelig; ad tertiam, ad proportionem quint&aelig; ad &longs;extam, uelut   producti ex proportione in &longs;ecundam detracta prima ad primam ad pro&shy;ductum ex proportione in quintam, detracta quarta ad quartam.<emph.end type="italics"/></cell>
            <cell>203</cell>
          </row>
          <row>
            <cell>CLXXXII.</cell>
            <cell>P<emph type="italics"/>ropo&longs;ita differentia proportionum partium &longs;imilium ad partes a&longs;&longs;umptas,   propo&longs;itaque proportione totius ad re&longs;idua eadem, differentiam propor&shy;tionum totius ad reliquum re&longs;idui inuenire.<emph.end type="italics"/></cell>
            <cell>203</cell>
          </row>
          <row>
            <cell>CLXXXIII.</cell>
            <cell>S<emph type="italics"/>pacium uit&aelig; naturalis per &longs;pacium uit&aelig; fortuitum declarare.<emph.end type="italics"/></cell>
            <cell>204</cell>
          </row>
          <row>
            <cell>CLXXXIIII.</cell>
            <cell>Q<emph type="italics"/>u&aelig;cunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri&shy;mum uer&longs;a mergantur.<emph.end type="italics"/></cell>
            <cell>211</cell>
          </row>
          <row>
            <cell>CLXXXV.</cell>
            <cell>C<emph type="italics"/>ur homo &longs;edens quanto altius &longs;edet, &amp; quanto magis crura ad f&oelig;mora, &amp;   f&oelig;mora ad pectus reclinata habet, facilius con&longs;urgat, cum tamen h&aelig;c op&shy;po&longs;ito modo inuicem &longs;e habeant, declarare.<emph.end type="italics"/></cell>
            <cell>213</cell>
          </row>
          <row>
            <cell>CLXXXVI.</cell>
            <cell>S<emph type="italics"/>i fuerit proportio prim&aelig; &amp; &longs;ecund&aelig; quantitatis ad tertiam, ut prim&aelig; &amp;   quart&aelig; ad quintam, fueritque quarta &longs;ecunda maior, erit proportio quar&shy;t&aelig; ad quintam maior qu&agrave;m &longs;ecund&aelig; ad tertiam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i fuerit maior<emph.end type="italics"/></cell>
            <cell/>
          </row>
          <pb xlink:href="015/01/016.jpg"/>
          <row>
            <cell/>
            <cell><emph type="italics"/>quart&aelig; ad quintam qu&agrave;m &longs;ecund&aelig; ad tertiam, nece&longs;&longs;e e&longs;t quartam &longs;ecunda e&longs;&longs;e   maiorem.<emph.end type="italics"/></cell>
            <cell>214</cell>
          </row>
          <row>
            <cell>CLXXXVII.</cell>
            <cell>S<emph type="italics"/>i ei&longs;dem uiribus &amp; &lsquo;eadem&rsquo; proportione cum auxilio ponderis tertij quar&shy;tum pondus moueatur quibus &longs;ecundum, auxilio primi nece&longs;&longs;e e&longs;t <expan abbr="quart&utilde;">quartum</expan> pon   dus tardius &amp; maiore cum difficultate moueri qu&agrave;m &longs;ecundum.<emph.end type="italics"/></cell>
            <cell>214</cell>
          </row>
          <row>
            <cell>CLXXXVIII.</cell>
            <cell>S<emph type="italics"/>i uires aliqu&aelig; moueant cum ponderibus aliqua pondera, ut compo&longs;ita pro&shy;portio &longs;it eadem proportioni uirium &amp; duorum ponderum mouentium ag&shy;gregatum &aelig;quale duorum ponderum, ubi maior fuerit partium in &aelig;qualitas,   ibi erit maior difficultas.<emph.end type="italics"/></cell>
            <cell>214</cell>
          </row>
          <row>
            <cell>CLXXXIX.</cell>
            <cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem &longs;ub &aelig;quali proportione coapte&shy;tar, facilius deor&longs;um trahetur qu&agrave;m quod maius e&longs;t &amp; propius.<emph.end type="italics"/></cell>
            <cell>215</cell>
          </row>
          <row>
            <cell>CXC.</cell>
            <cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, &amp; &longs;ecundum minus tertio, proportio   autem primi ad &longs;ecundum multo maior qu&agrave;m &longs;ecundi ad tertium, po&longs;ibile erit   propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um &amp; tertium mouea&shy;tur facili&ugrave;s ab ei&longs;dem uiribus, &amp; primo uel &longs;ecundo qu&agrave;m antea.<emph.end type="italics"/></cell>
            <cell>215</cell>
          </row>
          <row>
            <cell>CXCL.</cell>
            <cell>C<emph type="italics"/>um fuerint duo pondera &amp; uires, duxerisque aggregatum ex uiribus &amp; mi&shy;nore pondere in maius, addiderisque in&longs;uper quantum e&longs;t productum dimidij ui   rium in &longs;e latus aggregati detracto dimidio uirium, dicetur pondus auxiliare   &aelig;qualis proportionis.<emph.end type="italics"/></cell>
            <cell>215</cell>
          </row>
          <row>
            <cell>CXCII.</cell>
            <cell>S<emph type="italics"/>i ex medio diametri linea ad perpendiculum erigatur ad circuli peripheri&shy;am, ex eo puncto autem quotlibet line&aelig; ducantur &longs;eu intus ad circun ferentiam u&longs;que, &longs;eu extra ad diametrum, erit proportio totius line&aelig; ad totam uelut mu&shy;tuo partis ad partem.<emph.end type="italics"/></cell>
            <cell>217</cell>
          </row>
          <row>
            <cell>CXCIII.</cell>
            <cell>R<emph type="italics"/>ationem ponderis triplicem explicare.<emph.end type="italics"/></cell>
            <cell>218</cell>
          </row>
          <row>
            <cell>CXCIIII.</cell>
            <cell>P<emph type="italics"/>roportionem ponderis longioris in medio &longs;u&longs;pen&longs;i, ad breuius illi &aelig;quale &amp; in   medio &longs;u&longs;pen&longs;um declarare.<emph.end type="italics"/></cell>
            <cell>219</cell>
          </row>
          <row>
            <cell>CXCV.</cell>
            <cell>S<emph type="italics"/>i lectus fiat dupla longitudine ad latitudinem, melius &longs;uffulcietur re&longs;tibus   ex medio ad angulos &amp; eius &aelig;quidi&longs;tantibus qu&agrave;m &longs;ecundum longitudinem   &amp; latitudinem.<emph.end type="italics"/></cell>
            <cell>220</cell>
          </row>
          <row>
            <cell>CXCVI.</cell>
            <cell>S<emph type="italics"/>i duo circuli &longs;uper eodem centro eodem motu trans feruntur, &aelig;quale &longs;pacium   &longs;uperant.<emph.end type="italics"/></cell>
            <cell>221</cell>
          </row>
          <row>
            <cell>CXCVII.</cell>
            <cell>C<emph type="italics"/>ur lances ad locum &longs;uum &longs;u&longs;pen&longs;i redeant, impendentes <expan abbr="n&otilde;">non</expan>, <expan abbr="dem&otilde;&longs;trare">demon&longs;trare</expan>.<emph.end type="italics"/></cell>
            <cell>224</cell>
          </row>
          <row>
            <cell>CXCVIII.</cell>
            <cell>C<emph type="italics"/>ur &longs;olidum quod cubus uocatur<emph.end type="italics"/> P<emph type="italics"/>yramide &longs;tabilius &longs;it o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>225</cell>
          </row>
          <row>
            <cell>CXCIX.</cell>
            <cell>R<emph type="italics"/>ationem remorum nauim impellentium inuenire.<emph.end type="italics"/></cell>
            <cell>227</cell>
          </row>
          <row>
            <cell>CC.</cell>
            <cell>C<emph type="italics"/>ur temo cum paruus &longs;it, magnam nauim agere pote&longs;t, &amp; cur c&ugrave;m uarietas &longs;it   in prora, ip&longs;e con&longs;tituatur in puppi.<emph.end type="italics"/> E<emph type="italics"/>t cum transuer&longs;im ab aqua prematur   rect&agrave; nauim dirigat.<emph.end type="italics"/></cell>
            <cell>228</cell>
          </row>
          <row>
            <cell>CCI.</cell>
            <cell>S<emph type="italics"/>i du&aelig; line&aelig; non &longs;ecantes circuli peripheriam in unum punctum ex ea coe&shy;ant exterius, nece&longs;&longs;e e&longs;t illas peripheria contenta e&longs;&longs;e maiores.<emph.end type="italics"/></cell>
            <cell>229</cell>
          </row>
          <row>
            <cell>CCII.</cell>
            <cell>R<emph type="italics"/>ationem &longs;trepitus o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>232</cell>
          </row>
          <row>
            <cell>CCIII.</cell>
            <cell>C<emph type="italics"/>ur &longs;cytalis onera portentur facili&ugrave;s, explorare.<emph.end type="italics"/></cell>
            <cell>233</cell>
          </row>
          <row>
            <cell>CCIIII.</cell>
            <cell>C<emph type="italics"/>ur pluribus trochleis, pondera facilius eleuentur o&longs;tendere.<emph.end type="italics"/></cell>
            <cell>233</cell>
          </row>
          <row>
            <cell>CCV.</cell>
            <cell>S<emph type="italics"/>uper uerbis<emph.end type="italics"/> P<emph type="italics"/>latonis de fine<emph.end type="italics"/> R<emph type="italics"/>eipublic&aelig;.<emph.end type="italics"/></cell>
            <cell>234</cell>
          </row>
          <row>
            <cell>CCVI.</cell>
            <cell>R<emph type="italics"/>hombi pa&szlig;iones qua&longs;dam declarare.<emph.end type="italics"/></cell>
            <cell>235</cell>
          </row>
          <row>
            <cell>CCVII.</cell>
            <cell>P<emph type="italics"/>roportionem agentium naturalium in tran&longs;mutatione con&longs;iderare.<emph.end type="italics"/></cell>
            <cell>238</cell>
          </row>
          <row>
            <cell>CCVIII.</cell>
            <cell>M<emph type="italics"/>ota res &agrave; centro grauitatis per <expan abbr="prior&etilde;">priorem</expan> motum, in reditu uelocius mouetur   quam &longs;i quieuerit.<emph.end type="italics"/></cell>
            <cell>238</cell>
          </row>
          <pb xlink:href="015/01/017.jpg"/>
          <row>
            <cell>CCIX.</cell>
            <cell>S<emph type="italics"/>i &longs;uperficies rectangula in duas partes &aelig;quales diui&longs;a intelligatur, qu&aelig; am&shy;b&aelig; quadrat&aelig; &longs;int, itemque in duas in&aelig;quales, erit parallelipedum ex latere   medi&aelig; partis in totam &longs;uperficiem maius aggregato parallelipedorum ex   partibus in&aelig;qualibus in latera alterius partis mutuo, in eo, quod fit ex dif   ferentia lateris minoris partis &agrave; medi&aelig; latere in differentiam maioris par&shy;tis &longs;uperficiei &agrave; media &longs;uperficie bis, &amp; ex differentia amborum laterum   in&aelig;qualium iunctorum ad ambo latera, &aelig;qualia iuncta in minorem par&shy;tem &longs;uperficiei.<emph.end type="italics"/></cell>
            <cell>241</cell>
          </row>
          <row>
            <cell>CCX.</cell>
            <cell>S<emph type="italics"/>i du&aelig; line&aelig; ad &aelig;quales angulos ab eodem puncto peripheri&aelig; circuli refle&shy;ctantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos &aelig;quales e&longs;&longs;e.<emph.end type="italics"/> V<emph type="italics"/>nde ma&shy;nife&longs;tum e&longs;t, protractam diametrum angulum &longs;uppo&longs;itum per &aelig;qualia di&shy;uidere.<emph.end type="italics"/></cell>
            <cell>242</cell>
          </row>
          <row>
            <cell>CCXI.</cell>
            <cell>S<emph type="italics"/>i du&aelig; line&aelig; ex duobus punctis peripheriam contingentes, in eandem par&shy;tem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea ex parte, &amp; nun&shy;quam concurrent.<emph.end type="italics"/></cell>
            <cell>243</cell>
          </row>
          <row>
            <cell>CCXII.</cell>
            <cell>S<emph type="italics"/>i ab eodem puncto ad circuli peripheriam line&aelig; quotuis ducantur, tres inue&shy;nire lineas, qu&aelig; non in alium punctum reflectentur.<emph.end type="italics"/></cell>
            <cell>244</cell>
          </row>
          <row>
            <cell>CCXIII.</cell>
            <cell>P<emph type="italics"/>ropo&longs;ito circulo, atque in eius peripheria puncto &longs;ignato, lineas contingentes   ultra c&iacute;traque, &amp; eam ab ip&longs;omet deducere.<emph.end type="italics"/></cell>
            <cell>245</cell>
          </row>
          <row>
            <cell>CCXIIII.</cell>
            <cell>S<emph type="italics"/>i extra circulum duo puncta &aelig;qualiter &agrave; centro di&longs;tantia &longs;ignentur, erit pun&shy;ctum reflexionis &aelig;qualis in medio arcus intercepti inter lineas, qu&aelig; &agrave; cen   tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uer&ograve; unum centro proximius fuerit altero,   punctum &aelig;qualitatis in peripheria tant&ograve; longius, uer&longs;us breuiorem line&shy;am, quant&ograve; punctum aliud &agrave; centro magis di&longs;teterit.<emph.end type="italics"/></cell>
            <cell>245</cell>
          </row>
          <row>
            <cell>CCXV.</cell>
            <cell>P<emph type="italics"/>unctum reflexionis punctorum in&aelig;qualiter di&longs;tantium &agrave; centro, &aelig;qualiter   di&longs;tat &agrave; lineis, ductis &agrave; centro ad puncta &aelig;qualiter di&longs;tantia alterutrin&shy;que.<emph.end type="italics"/></cell>
            <cell>246</cell>
          </row>
          <row>
            <cell>CCXVI.</cell>
            <cell>S<emph type="italics"/>i fuerint circuli duo in&aelig;quales, &amp; extra utrunq&uacute;e punctum ad illud ex mi&shy;nore reflex&egrave; per magnam partem minoris &agrave; maiore perueuire pote&shy;runt.<emph.end type="italics"/></cell>
            <cell>247</cell>
          </row>
          <row>
            <cell>CCXVII.</cell>
            <cell>O<emph type="italics"/>culus uidet partem &longs;uperficiei<emph.end type="italics"/> L<emph type="italics"/>un&aelig; illuminatam &agrave;<emph.end type="italics"/> S<emph type="italics"/>ole per radios reflexos   &agrave;<emph.end type="italics"/> S<emph type="italics"/>olis corpore: nec tamen pote&longs;t uidere imaginem ip&longs;ius in<emph.end type="italics"/> L<emph type="italics"/>una tan   quam in &longs;peculo.<emph.end type="italics"/></cell>
            <cell>248</cell>
          </row>
          <row>
            <cell>CCXVIII.</cell>
            <cell>R<emph type="italics"/>ationem macul&aelig;<emph.end type="italics"/> L<emph type="italics"/>un&aelig; indagare.<emph.end type="italics"/></cell>
            <cell>248</cell>
          </row>
          <row>
            <cell>CCXIX.</cell>
            <cell>R<emph type="italics"/>ationem eorum qu&aelig; apparent circa<emph.end type="italics"/> S<emph type="italics"/>olem &longs;peculo in aqua po&longs;ito decla&shy;rare.<emph.end type="italics"/></cell>
            <cell>150</cell>
          </row>
          <row>
            <cell>CCXX.</cell>
            <cell>C<emph type="italics"/>au&longs;am cur<emph.end type="italics"/> S<emph type="italics"/>ol &aelig;&longs;tiuis diebus exoriens umbram ad meridiem, cum in meridie   ad boream mittat, explorare.<emph.end type="italics"/></cell>
            <cell>252</cell>
          </row>
          <row>
            <cell>CCXXI.</cell>
            <cell>M<emph type="italics"/>agnitudo<emph.end type="italics"/> L<emph type="italics"/>un&aelig; &amp; c&aelig;terorum a&longs;trorum digno&longs;citur ex proportione alio&shy;rum ad eam iuxta di&longs;tantiam: ip&longs;ius uer&ograve; iuxta rationem pupill&aelig; ad<emph.end type="italics"/> L<emph type="italics"/>u&shy;nam di&longs;tanti&aelig; ratione.<emph.end type="italics"/></cell>
            <cell>354</cell>
          </row>
          <row>
            <cell>CCXXII.</cell>
            <cell>Q<emph type="italics"/>uantitates qu&aelig; &aelig;quales e&longs;&longs;e non po&longs;&longs;unt in eodem genere, maius tamen &amp;   minus recipiunt, &longs;unt in proportione pote&longs;tatis.<emph.end type="italics"/></cell>
            <cell>255</cell>
          </row>
          <row>
            <cell>CCXXIII.</cell>
            <cell>Q<emph type="italics"/>uantitates qu&aelig; actu &aelig;quales e&longs;&longs;e non po&longs;&longs;unt, in nulla proportione actu   e&longs;&longs;e po&longs;&longs;unt.<emph.end type="italics"/></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>CCXXIIII.</cell>
            <cell>N<emph type="italics"/>eque temporis totius, ut imaginamur, ip&longs;um e&longs;&longs;e infinitum, neque &aelig;ui ui&shy;tarum proportio ulla e&longs;t ad tempus, quod pote&longs;tate e&longs;t, utpot&egrave; diem<emph.end type="italics"/></cell>
            <cell/>
          </row>
          <pb xlink:href="015/01/018.jpg"/>
          <row>
            <cell/>
            <cell><emph type="italics"/>uel men&longs;em.<emph.end type="italics"/></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>CCXXV.</cell>
            <cell>P<emph type="italics"/>roportio media non e&longs;t ex ratione agentis, &longs;ed patientis.<emph.end type="italics"/></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell>CCXXVI.</cell>
            <cell>P<emph type="italics"/>roportio &longs;ublimis non con&longs;i&longs;tit in magnitudine, &longs;ed ordine, iuxta quem diffe&shy;rentia e&longs;t eius quod e&longs;t ante &amp; po&longs;t.<emph.end type="italics"/></cell>
            <cell>257</cell>
          </row>
          <row>
            <cell>CCXXVII.</cell>
            <cell>V<emph type="italics"/>it&aelig; iuxta numerum perfectionum in comparatione ad cogitationem no&shy;&longs;tram proportionem quand am habent.<emph.end type="italics"/></cell>
            <cell>259</cell>
          </row>
          <row>
            <cell>CCXXVIII.</cell>
            <cell>P<emph type="italics"/>roportionem &longs;cienti&aelig; futurorum &amp; c&aelig;terorum occultorum con&longs;idera&shy;re.<emph.end type="italics"/></cell>
            <cell>260</cell>
          </row>
          <row>
            <cell>CCXXIX.</cell>
            <cell>I<emph type="italics"/>ncorporea omnia unum &longs;unt, neque numerus e&longs;t eorum.<emph.end type="italics"/></cell>
            <cell>261</cell>
          </row>
          <row>
            <cell>CCXXX.</cell>
            <cell>P<emph type="italics"/>roportio incorporeorum a&longs;cendentium &longs;emper maior e&longs;t.<emph.end type="italics"/></cell>
            <cell>262</cell>
          </row>
          <row>
            <cell>CCXXXI.</cell>
            <cell>T<emph type="italics"/>res e&longs;&longs;e mundos atque inter ip&longs;os nullam e&longs;&longs;e proportionem: nec numero cos   definiri.<emph.end type="italics"/></cell>
            <cell>263</cell>
          </row>
          <row>
            <cell>CCXXXII.</cell>
            <cell>O<emph type="italics"/>mnis motus naturalis quanto uelocior e&longs;t tanto propior e&longs;t &amp; magis &longs;imil   limus quieti.<emph.end type="italics"/></cell>
            <cell>264</cell>
          </row>
          <row>
            <cell>CCXXXIII.</cell>
            <cell>Q<emph type="italics"/>uod e&longs;t in mundo incorporeo &aelig;ternum e&longs;t, beatum, &longs;ecurum, immutabile   &longs;ecundum locum, &longs;olum iuxta e&longs;&longs;entiam fit: iuxta quod uelut &agrave; leui &longs;u&shy;&longs;urro aqu&aelig; &amp; aura &aelig;&longs;tiua demulcetur.<emph.end type="italics"/></cell>
            <cell>270</cell>
          </row>
        </table>
        <p type="head">
          <s id="id000041">FINIS.</s>
        </p>
        <pb xlink:href="015/01/019.jpg"/>
      </section>
    </front>
    <body>
      <chap>
        <pb pagenum="1" xlink:href="015/01/020.jpg"/>
        <p type="head">
          <s id="id000042">HIERONYMI CAR<lb/>DANI MEDIOLANENSIS, CI&shy;<lb/>VISQVE BONONIENSIS, MEDICI <lb/>de Proportionibus, &longs;eu Ope&shy;<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s>
        </p>
        <p type="main">
          <s id="id000043">Prima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000044">Proportio ab Euclide &longs;ic de&longs;cribitur, Qu&ograve;d <lb/>&longs;it duarum quantitatum eiu&longs;dem generis, <lb/>quod ad magnitudinem attinet, compara&shy;<lb/>tio certa.</s>
        </p>
        <p type="main">
          <s id="id000045">Secunda diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000046">Proportiones per &longs;imilitudinem <expan abbr="dic&utilde;tur">dicuntur</expan>, <lb/>c&ugrave;m quantitas quantitati <expan abbr="compara&ttilde;">comparatur</expan> alterius <lb/>generis, cui fingitur &aelig;qualis e&longs;&longs;e pote&longs;tate.</s>
        </p>
        <p type="main">
          <s id="id000047">Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c &aelig;quale line&aelig; b c.</s>
        </p>
        <figure id="id.015.01.020.1.jpg" xlink:href="015/01/020/1.jpg"/>
        <p type="main">
          <s id="id000048">Tertia diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000049">Proportio &aelig;qualis proportioni e&longs;t, c&ugrave;m eodem modo termini <lb/>&longs;e habent inuicem in utraque</s>
        </p>
        <p type="main">
          <s id="id000050">Quarta diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000051">Proportiones &longs;ecundum genus not&aelig; dicuntur, c&ugrave;m nouimus, <lb/>qu&ograve;d &longs;int maiores, aut minores. </s>
          <s id="id000052">Nam c&ugrave;m &aelig;quales &longs;unt, &longs;imul ne<lb/>ceffe e&longs;t, ut cogno&longs;camus genus, &amp; &longs;peciem.</s>
        </p>
        <p type="main">
          <s id="id000053">Quinta diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000054">Datum po&longs;itione e&longs;t: quod nece&longs;&longs;ari&ograve; ex po&longs;itis certam habet <lb/>quantitatem.</s>
        </p>
        <p type="main">
          <s id="id000055">Sexta diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000056">Datum &longs;impliciter dicitur, quod ex propo&longs;itis cogno&longs;ci pote&longs;t, <lb/>quantum &longs;it.</s>
        </p>
        <p type="main">
          <s id="id000057">Septima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000058">Proportiones pote&longs;tate <expan abbr="dicun&ttilde;">dicuntur</expan>, qu&aelig; &longs;ub comparatione aliarum <lb/><expan abbr="quantitat&utilde;">quantitatum</expan> nece&longs;&longs;ariam habentium <expan abbr="c&otilde;nexionem">connexionem</expan> <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cogno&longs;cuntur</expan>.</s>
        </p>
        <p type="main">
          <s id="id000059">H&aelig; autem &longs;unt aliquando eiu&longs;dem generis, cum primis ut nu&shy;<lb/>meri: aliquand&ograve; alterius, ut linearum &amp; &longs;uperficierum, angulorum, <lb/>&amp; arcuum: aliquando eiu&longs;dem generis, &amp; diuer&longs;arum &longs;pecierum, <lb/>ut arcuum per &longs;inus, qua utuntur A&longs;tronomi.</s>
        </p>
        <p type="main">
          <s id="id000060">Octaua diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000061">Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge</s>
        </p>
        <p type="main">
          <s id="id000062"><arrow.to.target n="marg1"/><lb/>neris, &longs;ed alterius a b altero dependentium, uelut motus ad tem&shy;





<pb pagenum="2" xlink:href="015/01/021.jpg"/>pus. </s>
          <s id="id000063">Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s>
        </p>
        <p type="margin">
          <s id="id000064"><margin.target id="marg1"/>C<emph type="italics"/>ar<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000065">Nona diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000066">Proportionum ali&aelig; dicuntur rhete, ali&aelig; alog&aelig;, rhet&aelig; qu&aelig; &longs;unt <lb/>ut numeri ad numerum, alog&aelig; qu&aelig; non &longs;unt numeri ad numerum.</s>
        </p>
        <p type="main">
          <s id="id000067">Decima diffinitio</s>
        </p>
        <p type="main">
          <s id="id000068">Proportio rhete alia &aelig;qualis, alia multiplex, uel &longs;ubmultiplex: <lb/>alia unius partis exce&longs;&longs;us, aut defectus, alia plurium, quam &longs;uper&shy;<lb/>partientem, aut &longs;upartientem uocant.</s>
        </p>
        <p type="main">
          <s id="id000069">Vndecima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000070">Cum diui&longs;o denominatore per numeratorem exit quantitas alo<lb/>ga, proportio dicitur aloga: &longs;i autem numerus integer, aut pars nu&shy;<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 e&longs;t, quoties recto ordine <lb/>tres quantitates in ei&longs;dem collo<expan abbr="can&ttilde;">cantur</expan>: ut &longs;int tres quan<lb/><figure id="id.015.01.021.1.jpg" xlink:href="015/01/021/1.jpg"/><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b &amp; b ad c, &amp; &longs;imiliter proportio c ad <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 e&longs;t, quoties ad eandem <lb/>quantitatem du&aelig; quantitates comparantur, tunc illarum propor&shy;<lb/>tio e&longs;t, qu&aelig; prodit una per alteram diui&longs;a.</s>
        </p>
        <p type="main">
          <s id="id000075">Sint proportiones a &amp; b ad c &amp; interponatur b inter a &amp; c, dico <lb/>proportionem a ad c diui&longs;am per proportionem a ad b, &amp; prodire <lb/>proportionem b ad c, con&longs;tat ex conuer&longs;a pr&aelig;cedentis.</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&shy;<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s>
        </p>
        <p type="main">
          <s id="id000078">Velut &longs;i comparentur a b &amp; b c ad d, inde tota <lb/><figure id="id.015.01.021.2.jpg" xlink:href="015/01/021/2.jpg"/><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e con<lb/><expan abbr="iunct&atilde;">iunctam</expan> ex duabus proportionibus a b ad d &amp; b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. </s>
          <s id="id000079">Hoc &amp; duo &longs;equentes &longs;icut &amp; du&etail; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon&shy;<lb/>&longs;trabitur e&longs;&longs;e. </s>
          <s id="id000080">nunc &longs;olum quomodo <expan abbr="intelligend&utilde;">intelligendum</expan> &longs;it proponimus.</s>
        </p>
        <p type="main">
          <s id="id000081">Quinta decima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000082">Detractionem proportionis &agrave; proportione intelligimus fieri <lb/>per <expan abbr="detraction&etilde;">detractionem</expan> minoris quantitatis &agrave; maiore, comparatam ad ean&shy;<lb/>dem quantitatem.</s>
        </p>
        <p type="main">
          <s id="id000083">Velut in exemplo &longs;uperiore detracta proportione b c ad d ex 


<pb pagenum="3" xlink:href="015/01/022.jpg"/>proportione a c ad d, relinquetur proportio a b ad d. </s>
          <s id="id000084">&amp; probatur <lb/>ex conuer&longs;ione pr&aelig;cedentis.</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/>radicum quantitatum illius iuxta unam, &amp; eandem rationem.</s>
        </p>
        <p type="main">
          <s id="id000087">Velut quadrat&aelig;, uel cub&aelig;, uel pronic&aelig;, uel uniner&longs;alis, uel alte&shy;<lb/>rius modi.</s>
        </p>
        <p type="main">
          <s id="id000088">Decima &longs;eptima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000089">C&ugrave;m fuerint du&aelig; proportiones &longs;imiles in tribus terminis con&shy;<lb/>tinuat&aelig;, dicetur proportio prim&aelig; quantitatis ad tertiam ueluti <lb/>prim&aelig; ad &longs;ecundam duplicata. </s>
          <s id="id000090">Et &longs;i &longs;int tres proportiones &longs;imiles <lb/>in quatuor terminis, dicetur proportio prim&aelig; quantitatis ad quar&shy;<lb/>tam triplicat&agrave; ei, qu&aelig; e&longs;t prim&aelig; ad &longs;ecundam,</s>
        </p>
        <p type="main">
          <s id="id000091">Decima octaua diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000092">Confu&longs;a proportio dicitur &longs;implicis, aut compo&longs;it&aelig; quantitatis <lb/>ad compo&longs;itam 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 qu&etail; in continua &longs;unt proportione Analog&aelig; <expan abbr="uocan&ttilde;">uocantur</expan>.</s>
        </p>
        <p type="main">
          <s id="id000095">Dictum e&longs;t hoc ad fugiendum nomen barbarum, etiam ut bre&shy;<lb/>uiter tamen po&longs;&longs;emus &longs;ententiam explicare.</s>
        </p>
        <p type="main">
          <s id="id000096">Vige&longs;ima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000097">Reflexa proportio dicitur cum trium quantitatum aggregatum <lb/>prim&aelig;, &amp; terti&aelig; &longs;e habet ad &longs;ecundam uelut &longs;ecunda ad tertiam,</s>
        </p>
        <p type="main">
          <s id="id000098">Vige&longs;ima prima diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000099">Trium quantitatum analogarum ali&aelig; quidem Geometric&aelig;, <lb/>c&ugrave;m proportio &longs;imilis e&longs;t: Ali&aelig; Arithmetic&aelig;, cum fuerit &aelig;qualis <lb/>exce&longs;&longs;us huc ind&egrave;: Ali&aelig; mu&longs;ic&aelig; cum fuerit proportio prim&aelig; ad ter<lb/>tiam multiplex, aut &longs;implex, aut compo&longs;ita exce&longs;&longs;us qu&aelig; &longs;implici <lb/>iuncta &longs;it ad multiplicis perfectionem: eadem autem &longs;it proportio <lb/>exce&longs;&longs;us prim&aelig;, &amp; &longs;ecund&aelig; ad exce&longs;&longs;um &longs;ecund&aelig; &longs;upra tertiam.</s>
        </p>
        <p type="main">
          <s id="id000100">Velut proportio 6. 4. 3. dupla e&longs;t utrinque, &amp; 6. 3. 2 tripla. </s>
          <s id="id000101">&amp; 28. 24. <lb/>21. &amp; 45. 40. 36. Geometrica uer&ograve; &amp; arithmetica facilius continuan&shy;<lb/>tur in quotquot quantitatibus, &longs;ed &amp; mu&longs;ica uelut 12. 8. 6. 4. 3. &amp; <lb/>proportio 8 ad 5 mu&longs;ica e&longs;t: quia proportio 5 ad 4 mu&longs;ica e&longs;t, &amp; <lb/>bene &longs;onans, igitur con&longs;titutis 8. 5. 4. cum 8 ad 4 ben&egrave; &longs;onet, &amp; 5 <lb/>ad 4, &amp; 4 &longs;it extrema non media inde 8. &amp; 5 ben&egrave; <expan abbr="&longs;on&atilde;t">&longs;onant</expan>. </s>
          <s id="id000102">nam in me&shy;<lb/>dijs <expan abbr="n&otilde;">non</expan> e&longs;t <expan abbr="uer&utilde;">uerum</expan>, ut in 9. 6. 4 bis diapente, &amp; 16. 12. 9 bis diate&longs;&longs;aron.</s>
        </p>
        <p type="main">
          <s id="id000103">Vige&longs;ima &longs;ecunda diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000104">Quantitates qu&aelig; &longs;imilem habent proportionem non continua&shy;<lb/>tam, omiolog&aelig; appellantur.</s>
        </p>
        <p type="main">
          <s id="id000105">Vige&longs;ima tertia diffinitio.</s>
        </p>
        <p type="main">
          <s id="id000106">Prima operatione con&longs;i&longs;tere dicuntur proportiones, c&ugrave;m inter <lb/>primo conflatas quantitates con&longs;titerint.</s>
        </p>
        <pb pagenum="4" xlink:href="015/01/023.jpg"/>
        <p type="main">
          <s id="id000107">PRIMA Animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000108">Omnis Proportio e&longs;t, aut &aelig;qualitatis, aut maior in&aelig;qualis, <lb/>aut minor.</s>
        </p>
        <p type="main">
          <s id="id000109">Secunda animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000110">Quilibet numerus tantus dicitur, quanta e&longs;t illius proportio ad <lb/>monadem.</s>
        </p>
        <p type="main">
          <s id="id000111">Dicimus enim quatuor, quod monadem quater contineat. </s>
          <s id="id000112">Et <lb/>duo cum dimidio c&ugrave;m monadem bis &amp; &longs;emis contineat.</s>
        </p>
        <p type="main">
          <s id="id000113">Tertia animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000114">Proportionem defectus, &longs;eu detract&aelig; quantitatis ad defectum <lb/>e&longs;&longs;e po&longs;&longs;e, ut quantitatis ad quantitatem dicuntur communes ani&shy;<lb/>mi &longs;ententi&aelig;, qu&aelig; ex intellectu &longs;olo terminorum, quod uer&aelig; &longs;int, <lb/>cogno&longs;cuntur. </s>
          <s id="id000115">Si ergo defectus e&longs;t quantitas, &amp; quantitas eiu&longs;dem <lb/>&longs;peciei, quia detrahitur, &amp; defectus non e&longs;t &longs;implicitur, &longs;ed detra&shy;<lb/>cto ergo per quartam petitionem: uel primam diffinitionem erit <lb/>proportio inter illas. </s>
          <s id="id000116">Sunt enim amb&aelig; detract&aelig;.</s>
        </p>
        <p type="main">
          <s id="id000117">Quarta animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000118">Inter quantitatem, &amp; defectum minorem quantitate, cuius e&longs;t de<lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. </s>
          <s id="id000119">Sit a b linea, &amp; detra&shy;<lb/>cta quantitas b c, non maior a b &amp; d &longs;it alia qu&aelig;uis quantitas eiu&longs;&shy;<lb/><figure id="id.015.01.023.1.jpg" xlink:href="015/01/023/1.jpg"/><lb/><expan abbr="d&etilde;">dem</expan> generis, dico qu&ograve;d inter d &amp; b c e&longs;t propor&shy;<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;&shy;<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. </s>
          <s id="id000120">Sed ut b c e&longs;t defectus, nulla e&longs;t propor&shy;<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, &amp; <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.</s>
        </p>
        <p type="main">
          <s id="id000121">Quinta animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000122">Cum proportio producitur ex proportionibus qu&aelig;libet illa&shy;<lb/>rum dicetur producta diui&longs;a per alteram.</s>
        </p>
        <p type="main">
          <s id="id000123">Sexta animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000124">&AElig;qualium quantitatum &longs;eu proportionum ad tertiam compa&shy;<lb/>rabilium eadem e&longs;t proportio atque uici&longs;sim. </s>
          <s id="id000125">H&aelig;c et&longs;i demon&longs;tre&shy;<lb/>tur ab Euclide, e&longs;t tamen hic generalior: &amp; &longs;atis per &longs;e nota. </s>
          <s id="id000126">Vt &longs;it <lb/>propior animi communi &longs;ententi&aelig;, qu&agrave;m rei demon&longs;trand&aelig;.</s>
        </p>
        <p type="main">
          <s id="id000127">Septima animi communis &longs;ententia.</s>
        </p>
        <p type="main">
          <s id="id000128">Ad quod quantitas proportionem habet infinitam, id in genere <lb/>illius quantitatis non comprehenditur.</s>
        </p>
        <p type="main">
          <s id="id000129">Nam proportio e&longs;t duarum quantitatum eiu&longs;dem generis com&shy;<lb/>paratio certa: at h&aelig;c comparatio certa non e&longs;t: non igitur quantita&shy;<lb/>tes amb&aelig; &longs;unt, aut non eiu&longs;dem generis.</s>
        </p>
        <pb pagenum="5" xlink:href="015/01/024.jpg"/>
        <p type="main">
          <s id="id000130">PRIMA Petitio.</s>
        </p>
        <p type="main">
          <s id="id000131">Si fuerit primi ad &longs;ecundum, ut tertij ad quartum, &amp; ex primo in <lb/>&longs;ecundum producatur &aelig;quale, aut maius, aut minus primo, uel <lb/>&longs;ecundo, producetur eodem modo ex tertio in quartum &etail;quale aut <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 po&longs;&longs;unt duci, diuidi, iungi, &amp; auferri, &amp; &longs;umi radix <lb/>in eis cuiu&longs;cunque generis, atque earum quantitatis, ut libet, po&longs;&longs;e <lb/>tran&longs;ponere.</s>
        </p>
        <p type="main">
          <s id="id000134">Tertia petitio.</s>
        </p>
        <p type="main">
          <s id="id000135">Proportionis cuiu&longs;uis nomen &agrave; denominatore &longs;upr&agrave; &longs;cripto, &amp; <lb/>numeratore infr&agrave; &longs;cripto &longs;umitur.</s>
        </p>
        <p type="main">
          <s id="id000136">Quarta petitio.</s>
        </p>
        <p type="main">
          <s id="id000137">Diui&longs;a quauis quantitate per aliam eiu&longs;dem generis, quod exit <lb/>proportio dicitur.</s>
        </p>
        <p type="main">
          <s id="id000138">Quinta petitio.</s>
        </p>
        <p type="main">
          <s id="id000139">Qu&etail;libet proportio e&longs;t uel inter duas quantitates, uel per unam <lb/>&longs;ignificatur.</s>
        </p>
        <p type="main">
          <s id="id000140">Nam per tertiam petitionem &longs;i &longs;int du&aelig; quantitates, qu&aelig; non ha<lb/>beant unius rationem, nomen &longs;umit proportio &agrave; duobus numeris, <lb/>&longs;in autem &longs;it altera monas, erit per &longs;ecundam animi communem &longs;en<lb/>tentiam, proportio numerus ip&longs;e Ide&ograve; patet, quod dicitur.</s>
        </p>
        <p type="main">
          <s id="id000141">Sexta petitio.</s>
        </p>
        <p type="main">
          <s id="id000142">Propo&longs;ita proportione quacunque, &amp; monade quantitatem inue<lb/>nire, qu&aelig; &longs;e habeat ad monadem in proportione propo&longs;ita.</s>
        </p>
        <p type="main">
          <s id="id000143">Nam c&ugrave;m per quartam petitionem diui&longs;a quantitate per quan&shy;<lb/>titatem exeat proportio, &amp; numerus ad <expan abbr="monad&etilde;">monadem</expan> &longs;e habeat, ut pro&shy;<lb/>portio, ideo &longs;umpta monade &longs;ecundum illum numerum, ille nume <lb/>rus e&longs;t quantitas qu&aelig;&longs;ita.</s>
        </p>
        <p type="main">
          <s id="id000144">Septima petitio.</s>
        </p>
        <p type="main">
          <s id="id000145">Quamlibet quantitatem per aliam eiu&longs;dem generis diuidere <lb/>po&longs;&longs;e.</s>
        </p>
        <p type="main">
          <s id="id000146">Octaua petitio.</s>
        </p>
        <p type="main">
          <s id="id000147">Proportionem in proportionem ducere po&longs;&longs;e: quamuis &longs;int in&shy;<lb/>ter quantitates diuer&longs;i generis.</s>
        </p>
        <p type="main">
          <s id="id000148">Quod dicitur de multiplicatione intelligendum e&longs;t de alijs ope&shy;<lb/>rationibus &longs;upr&agrave; enumeratis.</s>
        </p>
        <p type="main">
          <s id="id000149">Nona petitio.</s>
        </p>
        <p type="main">
          <s id="id000150">Monadem &longs;emper &longs;umere in quo cunque genere po&longs;&longs;e propo&longs;i&shy;<lb/>ta proportione.</s>
        </p>
        <pb pagenum="6" xlink:href="015/01/025.jpg"/>
        <p type="main">
          <s id="id000151">Nam licet diuidere per &longs;eptimam petitionem quantitatem per <lb/>quantitatem proportionis: &amp; quod exit, e&longs;t proportio per quar&shy;<lb/>tam petitionem, &amp; per &longs;ecundam animi communem &longs;ententiam <lb/>illa proportio e&longs;t numero &aelig;qualis: ergo diui&longs;a proportione, per &longs;i&shy;<lb/>milem numerum &longs;tatuetur monas.</s>
        </p>
        <p type="main">
          <s id="id000152">Decima petitio.</s>
        </p>
        <p type="main">
          <s id="id000153">In quouis genere quantitatum &longs;umere po&longs;&longs;e quantitatem, qu&aelig; <lb/><arrow.to.target n="marg2"/><lb/>&longs;e habeat ad monadem in proportione data. </s>
          <s id="id000154">Similem huic propo&shy;<lb/>nit Euclides in lineis generaliter: nos autem contr&agrave; generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s>
        </p>
        <p type="margin">
          <s id="id000155"><margin.target id="marg2"/>D<emph type="italics"/>uodecima <lb/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/>Vndecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000156">Monadem in quancunque quantitatem ductam &aelig;quale ip&longs;i pro&shy;<lb/>ducere. </s>
          <s id="id000157">Similiter &amp; proportionem &aelig;qualem.</s>
        </p>
        <p type="main">
          <s id="id000158">Nam cum aliqua quantitas augeat ducta aliqua minuat, nece&longs;&longs;e <lb/>e&longs;t aliquam e&longs;&longs;e, qu&aelig; nec augeat, nec minuat, &amp; h&aelig;c e&longs;t monas. <lb/></s>
          <s id="id000159">Idem dico de diui&longs;ione. </s>
          <s id="id000160">Aequalitas etiam ducta, uel diuidens non <lb/><arrow.to.target n="marg3"/><lb/>mutat proportionem: nec quantitatem ip&longs;am, igitur monas &aelig;qua&shy;<lb/>litatem refert. </s>
          <s id="id000161">Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s>
        </p>
        <p type="margin">
          <s id="id000162"><margin.target id="marg3"/>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="c&otilde;munis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></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 &aelig;qu&egrave; <lb/>multiplicibus a&longs;&longs;umptis, item que ad &longs;ecundam &amp; quartam, &amp; &longs;i mul&shy;<lb/>tiplex prim&aelig; maius e&longs;t multiplici &longs;ecund&aelig;, multiplex terti&aelig; &longs;it ma&shy;<lb/>ius multiplici quart&aelig;, &amp; &longs;i minus minus, &amp; &longs;i &aelig;quale &aelig;quale, idque<lb/> &longs;emper quouis modo a&longs;&longs;umptis his proportionibus ad primam &amp; <lb/>tertiam, &amp; ad &longs;ecundam &amp; quartam erit proportio prim&aelig; ad &longs;ecun<lb/>dam, ut terti&aelig; ad quartam. </s>
          <s id="id000165">H&aelig;c etiam a&longs;&longs;umitur ab Euclide. </s>
          <s id="id000166">Et per <lb/><arrow.to.target n="marg4"/><lb/>hanc intelligimus etiam conuer&longs;am.</s>
        </p>
        <p type="margin">
          <s id="id000167"><margin.target id="marg4"/>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/>diff.<emph.end type="italics"/> 6.</s>
        </p>
        <p type="main">
          <s id="id000168">Tertiadecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000169">Quantitates &aelig;quales, atque proportiones in qua&longs;uis quanti&shy;<lb/>tates duct&aelig; eandem &longs;eruant rationem. </s>
          <s id="id000170">Euclides hanc demon&longs;trat, <lb/>nos autem ad uitandum t&aelig;dium petimus concedi, &longs;ub qua in&shy;<lb/><arrow.to.target n="marg5"/><lb/>cluduntur diui&longs;io etiam additio, detractio, laterum omnium in&shy;<lb/>uentio.</s>
        </p>
        <p type="margin">
          <s id="id000171"><margin.target id="marg5"/>Q<emph type="italics"/>uarta quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000172">Quartadecima petitio.</s>
        </p>
        <p type="main">
          <s id="id000173">C&ugrave;m termini alicuius quantitatis eandem &longs;eruant rationem in <lb/>omnibus, &amp; firmi &longs;unt ac &longs;tabiles eiu&longs;dem rationis comparatione <lb/>content&aelig; partes &aelig;qualem &longs;eruant exce&longs;&longs;um, &longs;eu proportionem.</s>
        </p>
        <p type="main">
          <s id="id000174">PROPOSITIO prima.</s>
        </p>
        <p type="main">
          <s id="id000175">Proportionem in proportionem duci e&longs;t &longs;uperiores nume&shy;<lb/>ros atque inferiores inuicem ducere.</s>
        </p>
        <pb pagenum="7" xlink:href="015/01/026.jpg"/>
        <p type="main">
          <s id="id000176">Sit proportio line&aelig; a ad lineam b, ut anguli c ad angulum d, &longs;ta&shy;<lb/><arrow.to.target n="marg6"/><lb/>tuatur e monas in genere a <lb/><figure id="id.015.01.026.1.jpg" xlink:href="015/01/026/1.jpg"/><lb/>b, &amp; fiat f ad e, ut c ad d, &amp; du<lb/><arrow.to.target n="marg7"/><lb/>catur a in f &amp; b in e, &amp; pro&shy;<lb/>ducantur g &amp; h. </s>
          <s id="id000177">Quia ergo <lb/><arrow.to.target n="marg8"/><lb/>f e&longs;t proportio ip&longs;a, erit g ad <lb/><arrow.to.target n="marg9"/><lb/>a ut c ad d, &longs;ed h e&longs;t &aelig;qualis <lb/>b, igitur a ad h ut ad b. </s>
          <s id="id000178">Du&shy;<lb/>cta ergo dicetur proportio a <lb/><arrow.to.target n="marg10"/><lb/>ad b in proportionem c ad d <lb/>ducendo terminos proportionis, &longs;eu quantitatis recta &longs;cilicet &longs;u&shy;<lb/>periores cum &longs;uperioribus, &amp; inferiores cum inferioribus. </s>
          <s id="id000179">Nam &longs;i <lb/><arrow.to.target n="marg11"/><lb/>rur&longs;um con&longs;tituantur f ad e ut a ad b c&ugrave;m f &longs;it proportio, &amp; k ad f ut <lb/><arrow.to.target n="marg12"/><lb/>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb/>qu&aelig; e&longs;t fin proportionem c ad d, liquet igitur propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id000180"><margin.target id="marg6"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000181"><margin.target id="marg7"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000182"><margin.target id="marg8"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000183"><margin.target id="marg9"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000184"><margin.target id="marg10"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni&shy;<lb/>mi &longs;entent.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000185"><margin.target id="marg11"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000186"><margin.target id="marg12"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000187">Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunda</expan>.</s>
        </p>
        <p type="main">
          <s id="id000188">Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"/></s>
        </p>
        <p type="margin">
          <s id="id000189"><margin.target id="marg13"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000190">Sint a b c quantitates dico proportio&shy;<lb/><figure id="id.015.01.026.2.jpg" xlink:href="015/01/026/2.jpg"/><lb/>nem a ad c, produci ex proportione a ad b </s>
        </p>
        <p type="main">
          <s id="id000191"><arrow.to.target n="marg14"/><lb/>&amp; b ad c, &longs;tatuantur totidem &agrave; monade d e <lb/>f, er&uacute;ntque ex demon&longs;trantis ab Euclide in <lb/>quinto <expan abbr="Elem&etilde;torum">Elementorum</expan> in eadem proportio&shy;<lb/>ne, &longs;tatuatur ergo d prima quantitas e &longs;e&shy;<lb/>cunda &amp; tertia f quarta. </s>
          <s id="id000192">eritq&uacute;e per pr&aelig;ce&shy;<lb/><arrow.to.target n="marg15"/><lb/>dentem proportio productorum ex d in e <lb/>&amp; &longs;it g, &amp; in f &amp; &longs;it h, producta ex propor&shy;<lb/>tionibus d ad e &amp; e ad f, quare ex propor&shy;<lb/>tionibus a ad b &amp; b ad e, &longs;ed ex dictis cum <lb/>e &longs;it eadem, erit proportio d ad f, ut g ad h &amp; proportio, d ad f per <lb/>&aelig;quam proportionem ab Euclide demon&longs;tratam, ut a ad c, igitur <lb/><arrow.to.target n="marg16"/><lb/>proportio a ad c producitur ex proportionibus a ad b &amp; b ad c, &amp; <lb/>e&longs;t proportio ip&longs;a a ad c d numerus, ut o&longs;ten&longs;um e&longs;t.</s>
        </p>
        <p type="margin">
          <s id="id000193"><margin.target id="marg14"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&amp;<emph.end type="italics"/> 9. <lb/>P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000194"><margin.target id="marg15"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000195"><margin.target id="marg16"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000196">Ex hoc &longs;equitur, qu&ograve;d c&ugrave;m fuerit quantitas tertia monas ex pro&shy;<lb/><arrow.to.target n="marg17"/><lb/>portionibus inuicem ductis producetur prima quantitas.<lb/><arrow.to.target n="marg18"/></s>
        </p>
        <p type="margin">
          <s id="id000197"><margin.target id="marg17"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="margin">
          <s id="id000198"><margin.target id="marg18"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3</s>
        </p>
        <p type="main">
          <s id="id000199">Ex hoc &longs;equitur, qu&ograve;d conuer&longs;a proportio producitur ex con&shy;<lb/>uer&longs;is proportionibus.</s>
        </p>
        <p type="main">
          <s id="id000200">Propo&longs;itio tertia.</s>
        </p>
        <p type="main">
          <s id="id000201">Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uer&ograve; proportio inter duas alias quantitates fue&shy;


<pb pagenum="8" xlink:href="015/01/027.jpg"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s>
        </p>
        <p type="main">
          <s id="id000202"><arrow.to.target n="marg19"/></s>
        </p>
        <p type="margin">
          <s id="id000203"><margin.target id="marg19"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000204">H&etail;c propo&longs;itio ut pr&aelig;cedens &amp; <expan abbr="&longs;equ&etilde;tes">&longs;equentes</expan> tres ab Alchindo &longs;um&shy;<lb/>pt&aelig; &longs;unt, &amp; ab eo demon&longs;trantur. </s>
          <s id="id000205">Sit ergo proportio a ad b, pro&shy;<lb/><arrow.to.target n="table2"/><lb/><figure id="id.015.01.027.1.jpg" xlink:href="015/01/027/1.jpg"/>ducta ex proportione c ad d &amp; e ad f, con&longs;tat <lb/>qu&ograve;d cum &longs;int &longs;ex quantitates, qu&ograve;d fieri pote&shy;<lb/>runt quindecim coniugationes, quas po&longs;ui &agrave; la&shy;<lb/>tere facilitatis gratia, quibus re&longs;pondent totidem <lb/><arrow.to.target n="table3"/><lb/>conuer&longs;&aelig;: erunt ergo triginta. </s>
          <s id="id000206">Singul&aelig; autem ha <lb/>rum produci po&longs;&longs;unt duodecim modis: ductis <lb/><figure id="id.015.01.027.2.jpg" xlink:href="015/01/027/2.jpg"/>duodecim in triginta, fiunt trecenti &longs;exaginta mo <lb/>di. </s>
          <s id="id000207">Et hoc e&longs;t clarum per&longs;e, modo <expan abbr="dem&otilde;&longs;tremus">demon&longs;tremus</expan>, <lb/>quod &longs;inguli horum modorum po&longs;sint produ&shy;<lb/>ci duodecim modis, &amp; capiamus ab primam qu&etail; <lb/>pote&longs;t produci ex c d &amp; e f: Item ambabus con&shy;<lb/>uer&longs;is d c &amp; fe: &amp; rur&longs;us altera recta altera con&shy;<lb/>uer&longs;a: &amp; hoc bifariam c d &amp; f e, &amp; d c &amp; e f, &longs;unt er&shy;<lb/>go iam quatuor modi. </s>
          <s id="id000208">Totidem ex c e &amp; d f, toti&shy;<lb/>demque ex c f &amp; d e, igitur erunt duodecim mo&shy;<lb/>di, quibus produci po&longs;&longs;e intelligitur propor&shy;<lb/>tio a ad b.</s>
        </p>
        <table>
          <table.target id="table2"/>
          <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"/>
          <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">Propo&longs;itio quarta.</s>
        </p>
        <p type="main">
          <s id="id000210">Si fuerit proportio primi ad &longs;ecundum produ&shy;<lb/>cta ex proportionibus tertij ad quartum, &amp; quin <lb/>ti ad &longs;extum, producetur etiam ex proportione <lb/>tertij ad &longs;extum, &amp; quinti ad quartum.</s>
        </p>
        <p type="main">
          <s id="id000211">Sit proportio a b producta ex proportioni&shy;<lb/><arrow.to.target n="table4"/><lb/><figure id="id.015.01.027.3.jpg" xlink:href="015/01/027/3.jpg"/>bus c ad d, &amp; e ad f, dico quod etiam erit produ&shy;</s>
        </p>
        <table>
          <table.target id="table4"/>
          <row>
            <cell>a</cell>
            <cell>b</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"/><lb/>cta ex proportionibus c ad f, &amp; e ad d, di&longs;ponan&shy;<lb/>tur ut in figura &amp; fiat ex c in e g, &amp; ex d in fh, ergo <lb/><arrow.to.target n="marg21"/><lb/>per primam harum g ad h ut a ad b, &longs;ed per pr&aelig;&shy;<lb/>&longs;uppo&longs;ita in &longs;ecunda productione etiam prode&shy;<lb/>unt g &amp; h, igitur per primam propo&longs;itionem ha&shy;<lb/>rum a ad b proportio producitur ex proportionibus c ad f terti&aelig; <lb/>&longs;cilicet ad &longs;extam, &amp; e ad d quint&etail; ad quartam, quod fuit <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s>
        </p>
        <p type="margin">
          <s id="id000213"><margin.target id="marg20"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000214"><margin.target id="marg21"/>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000215">Propo&longs;itio quinta.</s>
        </p>
        <p type="main">
          <s id="id000216">Si fuerit proportio primi ad &longs;ecundum producta ex proportio&shy;<lb/>ne tertij ad quartum, &amp; quinta ad &longs;extum: erit proportio tertij ad <lb/>&longs;extum producta ex proportionibus primi ad &longs;ecundum, &amp; quar&shy;<lb/>ti ad quintum.</s>
        </p>
        <pb pagenum="9" xlink:href="015/01/028.jpg"/>
        <p type="main">
          <figure id="id.015.01.028.1.jpg" xlink:href="015/01/028/1.jpg"/>
          <s id="id000217">Sit proportio a ad b producta ex proportio&shy;<lb/><arrow.to.target n="marg22"/><lb/><arrow.to.target n="table5"/><lb/>nibus c ad d, &amp; e ad f, dico quod proportio c ad <lb/>f producitur ex proportione a ad b &amp; d ad e. </s>
          <s id="id000218">In&shy;<lb/>terponam d e inter c &amp; f, eritque ex &longs;ecunda pro&shy;<lb/>po&longs;itione repetita proportio c ad f producta ex <lb/>tribus proportionibus c ad d, d ad e, e ad f, &longs;ed <lb/>proportiones c ad d, &amp; e ad f producunt pro&shy;<lb/><figure id="id.015.01.028.2.jpg" xlink:href="015/01/028/2.jpg"/>portionem a ad b, igitur proportio c ad f produ<lb/>citur ex proportionibus a ad b, &amp; e ad f.<lb/><arrow.to.target n="table6"/></s>
        </p>
        <p type="margin">
          <s id="id000219"><margin.target id="marg22"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <table>
          <table.target id="table5"/>
          <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"/>
          <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">Propo&longs;itio &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id000221">Ex trecentis &longs;exaginta modis producenda&shy;<lb/>rum proportionum triginta &longs;ex tantum e&longs;&longs;e ne&shy;<lb/>ce&longs;&longs;arios.<lb/><arrow.to.target n="table7"/></s>
        </p>
        <table>
          <table.target id="table7"/>
          <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"/>
          <s id="id000222">Per quartam enim proportio a ad b produ&shy;<lb/><arrow.to.target n="marg23"/><lb/>citur bifariam, &amp; ex c ad d, &amp; e ad f, &amp; ex c ad f, &amp; <lb/>e ad d. </s>
          <s id="id000223">&amp; per pr&aelig;cedentem c ad f producitur ex <lb/>a ad b, &amp; d ad e, &amp; per quartam rur&longs;us ex a ad e, <lb/>&amp; d ad b. </s>
          <s id="id000224">Et per pr&aelig;cedentem rur&longs;us a ad e ex c <lb/>ad f &amp; b ad d, igitur per quartam eadem produ&shy;<lb/>cetur ex c ad d &amp; b ad f. </s>
          <s id="id000225">Quare per pr&aelig;ceden&shy;<lb/>tem c ad f ex a ad e, &amp; d ad b, &amp; ita di&longs;ponemus <lb/>hos modos in tabula. </s>
          <s id="id000226">Vides etiam <lb/><arrow.to.target n="table8"/><lb/><figure id="id.015.01.028.4.jpg" xlink:href="015/01/028/4.jpg"/>aliquos modos non produci, ut pri&shy;<lb/>mi ad quartum nec ad &longs;extum, &amp; li&shy;<lb/>quet, qu&ograve;d c&ugrave;m &longs;int quindecim o&shy;<lb/>mnes modi qui produci po&longs;&longs;e intelli&shy;<lb/>guntur, &amp; nouem tantum producan&shy;<lb/>tur &longs;ex e&longs;&longs;e, qui non producuntur, quos <lb/>&longs;eor&longs;um in tabula coniunxi. </s>
          <s id="id000227">Et con&shy;<lb/>&longs;tat etiam, quod totidem conuer&longs;i &longs;ci&shy;<lb/>licet decem octo <expan abbr="produc&utilde;tur">producuntur</expan>, de qui&shy;<lb/>bus diximus, ut &longs;int omnes triginta <lb/>&longs;ex, qui con&longs;tat ex duabus propo&longs;i&shy;<lb/>tionibus pr&aelig;mi&longs;sis, &amp; hac tertia, <expan abbr="qu&atilde;">quam</expan> <lb/>adiungemus &longs;cilicet, qu&ograve;d propor&shy;<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum, <lb/>&amp; quinti ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>. </s>
          <s id="id000228">Hoc enim ex pr&aelig;&shy;<lb/>cedentibus non liquet: ben&egrave; liquet <lb/>permutatis ordinibus, quod &longs;i pro&shy;<lb/>portio primi ad tertium producitur, 


<pb pagenum="9 &lsqb;=10&rsqb;" xlink:href="015/01/029.jpg"/>quod etiam propor&shy;<lb/><figure id="id.015.01.029.1.jpg" xlink:href="015/01/029/1.jpg"/><arrow.to.target n="marg24"/><lb/>tio primi ad <expan abbr="quint&utilde;">quintum</expan>. <lb/></s>
          <s id="id000229">Nam tertium, &amp; quin <lb/>tum, item que quartum, <lb/>&amp; &longs;extum non <expan abbr="diffe&shy;r&utilde;t">diffe&shy;<lb/>runt</expan> ni&longs;i ordine uolun<lb/>tario. </s>
          <s id="id000230">Ergo interpo&longs;i&shy;<lb/>to e inter a, &amp; c per &longs;e&shy;<lb/>cundam propo&longs;itio&shy;<lb/>nem proportio a ad c <lb/>producitur ex proportionibus a ad <lb/>e, &amp; e ad c, ut ex demon&longs;tratis in pr&aelig;&shy;<lb/>&longs;enti proportio a ad c producitur ex <lb/>c ad f &amp; b ad d. </s>
          <s id="id000231">Proportio ergo a ad <lb/>c producitur ex proportionibus e <lb/>ad c &amp; c ad f &amp; b ad d, at e ad c &amp; c ad <lb/>f producunt eam, qu&aelig; e&longs;t e ad f per <lb/><expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> propo&longs;itionem. </s>
          <s id="id000232">Igitur pro&shy;<lb/>portio a ad c producitur ex propor&shy;<lb/>tionibus b ad d &longs;ecundi ad quartum, <lb/>&amp; e ad f quinti ad &longs;extum. </s>
          <s id="id000233">H&aelig;c Al&shy;<lb/>chindus in &longs;uo libello: &longs;ed licet inge&shy;<lb/>nio &longs;a ualde: parum <expan abbr="tam&etilde;">tamen</expan> utilia olim <lb/><expan abbr="er&atilde;t">erant</expan> nece&longs;&longs;aria ad intelligendum ma&shy;<lb/>gnam <expan abbr="c&otilde;po&longs;itionem">compo&longs;itionem</expan> Ptolem&etail;i, nunc <lb/>po&longs;tquam Heber has &longs;ex quantita&shy;<lb/>tes traduxit ad quatuor, pror&longs;us h&aelig;c <lb/>&longs;cientia ulli u&longs;ui e&longs;&longs;e de&longs;ijt.<lb/><arrow.to.target n="table9"/></s>
        </p>
        <p type="margin">
          <s id="id000234"><margin.target id="marg23"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000235"><margin.target id="marg24"/>Modi qui <expan abbr="n&otilde;">non</expan> <lb/>producuntur <lb/>pri. ad quartu <lb/>pri. ad &longs;extum <lb/>&longs;ec. ad <expan abbr="terti&utilde;">tertium</expan> <lb/>&longs;ec. ad <expan abbr="quint&utilde;">quintum</expan> <lb/>tert. </s>
          <s id="id000236">ad quint. <lb/></s>
          <s id="id000237">quart. </s>
          <s id="id000238">ad &longs;ext.</s>
        </p>
        <table>
          <table.target id="table8"/>
          <row>
            <cell/>
            <cell>Primi ad &longs;ecundum.</cell>
          </row>
          <row>
            <cell>1</cell>
            <cell>tertij ad <expan abbr="quart&utilde;">quartum</expan>, &amp; quin</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad &longs;extum.</cell>
          </row>
          <row>
            <cell>2</cell>
            <cell>tertij ad &longs;extum, &amp; quin</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad quartum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Primi ad tertium.</cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>&longs;ecundi ad quartum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>quinti ad &longs;extum.</cell>
          </row>
          <row>
            <cell>4</cell>
            <cell>&longs;ecundi ad &longs;extum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>quinti ad quartum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Primi ad quintum.</cell>
          </row>
          <row>
            <cell>5</cell>
            <cell>&longs;ecundi ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>, &amp; ter&shy;</cell>
          </row>
          <row>
            <cell/>
            <cell>tij ad quartum.</cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>&longs;ecundi ad quartum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>tertij ad &longs;extum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Secundi ad quartum.</cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>primi ad tertium, &amp; &longs;ex</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad quintum.</cell>
          </row>
          <row>
            <cell>8</cell>
            <cell>primi ad quintum, et &longs;ex</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad tertium.</cell>
          </row>
          <row>
            <cell/>
            <cell>Secundi ad &longs;extum.</cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>primi ad <expan abbr="quint&utilde;">quintum</expan>, &amp; quar</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad tertium.</cell>
          </row>
          <row>
            <cell>10</cell>
            <cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, &amp; quar&shy;</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad quintum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Tertij ad quartum.</cell>
          </row>
          <row>
            <cell>11</cell>
            <cell>primi ad &longs;ecundum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>&longs;exti ad quintum.</cell>
          </row>
          <row>
            <cell>12</cell>
            <cell>primi ad quintum, &amp; &longs;ex</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad &longs;ecundum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Tertij ad &longs;extum.</cell>
          </row>
          <row>
            <cell>13</cell>
            <cell>primi ad &longs;ecundum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>quarti ad quintum.</cell>
          </row>
          <row>
            <cell>14</cell>
            <cell>primi ad quintum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>quarti ad &longs;ecundum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Quarti ad quintum.</cell>
          </row>
          <row>
            <cell>15</cell>
            <cell>&longs;ecundi ad primum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>tertij ad &longs;extum.</cell>
          </row>
          <row>
            <cell>16</cell>
            <cell>&longs;ecundi ad &longs;extum, &amp; ter</cell>
          </row>
          <row>
            <cell/>
            <cell>tij ad primum.</cell>
          </row>
          <row>
            <cell/>
            <cell>Quinti ad &longs;extum.</cell>
          </row>
          <row>
            <cell>17</cell>
            <cell>primi ad &longs;ecundum, &amp;</cell>
          </row>
          <row>
            <cell/>
            <cell>quarti ad tertium.</cell>
          </row>
          <row>
            <cell>18</cell>
            <cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, &amp; quar&shy;</cell>
          </row>
          <row>
            <cell/>
            <cell>ti ad &longs;ecundum.</cell>
          </row>
        </table>
        <table>
          <table.target id="table9"/>
          <row>
            <cell>a</cell>
            <cell>e c</cell>
            <cell>a e</cell>
            <cell>e c</cell>
          </row>
          <row>
            <cell/>
            <cell/>
            <cell>c b</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell/>
            <cell/>
            <cell>f d</cell>
            <cell>c</cell>
          </row>
          <row>
            <cell/>
            <cell/>
            <cell/>
            <cell>f</cell>
          </row>
        </table>
        <p type="main">
          <s id="id000239">Propo&longs;itio &longs;eptima.</s>
        </p>
        <figure id="id.015.01.029.2.jpg" xlink:href="015/01/029/2.jpg"/>
        <p type="main">
          <s id="id000240">In modis qui nece&longs;&longs;ari&ograve; produ&shy;<lb/>cuntur ex duabus proportionibus, <lb/>cum du&etail; quantitates ex illis, qu&etail; mo <lb/><figure id="id.015.01.029.3.jpg" xlink:href="015/01/029/3.jpg"/>dos conficiunt, &aelig;quales fuerint: pro&shy;<lb/><arrow.to.target n="table10"/><lb/>portio producta ad quatuor quanti&shy;<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"/></s>
        </p>
        <p type="margin">
          <s id="id000241"><margin.target id="marg25"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <table>
          <table.target id="table10"/>
          <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 &longs;ex quantitates a b c d e f, &amp; <lb/>producatur proportio a ad b ex pro&shy;<lb/>portione c ad d, &amp; e ad f, tu &longs;cis, qu&ograve;d <lb/>modi recepti &longs;unt prima cum &longs;ecunda, tertia uel quinta, &amp; &longs;ecunda <lb/>cum quarta, &amp; &longs;exta, &amp; tertia &longs;imiliter cum ei&longs;dem, &amp; quinta eodem <lb/>modo cum ei&longs;dem: &longs;i igitur du&etail; quantitates ex his, qu&etail; faciunt pro&shy;


<pb pagenum="11" xlink:href="015/01/030.jpg"/>portionem productam inter &longs;e fuerint &aelig;quales reducetur h&aelig;c pro&shy;<lb/>portio ad quatuor quantitates omologas, &longs;ciliter abiectis amba&shy;<lb/>bus &aelig;qualibus. </s>
          <s id="id000243">Sit gratia exempli prima &aelig;qualis quint&aelig;: &amp; quia <lb/>in octauo modo proportio <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum producitur ex pro&shy;<lb/>portione primi ad quintum, &amp; &longs;exti ad tertium, ergo per expo&longs;ita <lb/>proportio &longs;ecundi ad quartum, ut &longs;exti ad tertium, &amp; ita permutan&shy;<lb/>do, &amp; conuertendo &longs;ecundi ad &longs;extum, ut quarti ad tertium, &amp; tertij </s>
        </p>
        <p type="main">
          <s id="id000244"><arrow.to.target n="marg26"/><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s>
        </p>
        <p type="margin">
          <s id="id000245"><margin.target id="marg26"/>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000246">Propo&longs;itio octaua.</s>
        </p>
        <p type="main">
          <s id="id000247">Si duarum <expan abbr="proportion&utilde;">proportionum</expan> &longs;uperiores numeri alternatim cum infe<lb/>rioribus multiplicentur, atque coniungantur: erit proportio aggre&shy;<lb/>gati ad productum ex inferioribus inuicem proportio ex primis <lb/>proportionibus compo&longs;ita.</s>
        </p>
        <figure id="id.015.01.030.1.jpg" xlink:href="015/01/030/1.jpg"/>
        <p type="main">
          <s id="id000248">Sit proportio una a ad b, alia c ad d, ducatur b in <lb/><arrow.to.target n="marg27"/><lb/>c, fiatque e &amp; a in d, &amp; fiat f, iunganturque e &amp; f &amp; fiat h, <lb/>&amp; ducatur b in d et fiat g: dico <expan abbr="proportion&etilde;">proportionem</expan> h g com&shy;<lb/>po&longs;itam e&longs;&longs;e ex proportione a ad b, &amp; c ad d. </s>
          <s id="id000249">Quia <lb/><arrow.to.target n="marg28"/><lb/>enim ex b in c fit e, &amp; ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, &amp; &longs;imiliter, quia ex d in a fit f, &amp; ex d in b fit g, erit f ad <lb/>g ut a ad b. </s>
          <s id="id000250">Sed e &amp; f componunt h, igitur proportio h ad g e&longs;t com<lb/>po&longs;ita ex proportionibus e &amp; f ad g, igitur per communem animi <lb/>&longs;ententiam, &amp; diffinitionem compo&longs;it&aelig; proportionis, proportio h <lb/><arrow.to.target n="marg29"/><lb/>ad g compo&longs;ita e&longs;t ex proportionibus a ad b, &amp; c ad d, quod e&longs;t <lb/>propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id000251"><margin.target id="marg27"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000252"><margin.target id="marg28"/>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti&shy;<lb/>tione.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000253"><margin.target id="marg29"/>P<emph type="italics"/>er<emph.end type="italics"/> 14 <emph type="italics"/>diffi <lb/>nitionem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000254">Propo&longs;itio nona.</s>
        </p>
        <p type="main">
          <s id="id000255">Si duarum proportionum &longs;uperiores numeri alternatim cum <lb/>inferioribus multiplicentur, minusque productum ex maiore detra&shy;<lb/>hatur, erit re&longs;idui ad productum ex inferioribus proportio uelut <lb/>illa, qu&aelig; relinquitur detracta minore proportione ex maiore.</s>
        </p>
        <p type="main">
          <s id="id000256">H&aelig;c eodem modo probatur, ut pr&aelig;cedens, ni&longs;i quod h fit de&shy;<lb/><arrow.to.target n="marg30"/><lb/>tracto &egrave; minore: gratia exempli ex f, &amp; ita ex diffinitione patet pro&shy;<lb/>po&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id000257"><margin.target id="marg30"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. <lb/>152.</s>
        </p>
        <p type="main">
          <s id="id000258">Propo&longs;itio decima.</s>
        </p>
        <p type="main">
          <s id="id000259">Si fuerit alicuius quantitatis ad unam partem proportio uelut al<lb/>terius partis ad <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> quantitatem erit proportio cuiu&longs;uis quan<lb/>titatis eiu&longs;dem generis ad &longs;ecundam compo&longs;ita proportio ex pro&shy;<lb/>portionibus eiu&longs;dem quantitatis a&longs;&longs;umpt&aelig; ad utran que partem pri&shy;<lb/>m&aelig; quantitatis &longs;eor&longs;um.<lb/><arrow.to.target n="marg31"/></s>
        </p>
        <p type="margin">
          <s id="id000260"><margin.target id="marg31"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.030.2.jpg" xlink:href="015/01/030/2.jpg"/>
        <p type="main">
          <s id="id000261">Sit a b quantitas diui&longs;a in c, &amp; &longs;i c ut a b ad a c, <lb/>ita b c ad d: eritque iterum permutando a b ad b c, <lb/>ut a c ad d, &amp; &longs;umatur qu&aelig;dam quantitas e eiu&longs;&shy;


<pb pagenum="12" xlink:href="015/01/031.jpg"/>dem tamen generis, cum illis dico qu&ograve;d proportio e ad d e&longs;t com&shy;<lb/>po&longs;ita ex proportionibus e ad a c, &amp; e ad b c. </s>
          <s id="id000262">Po&longs;ita ergo e tan&lt;08&gt; &longs;u&shy;<lb/>periore numero, &amp; a c &amp; c b inferioribus, erit ex octaua propo&longs;itio&shy;<lb/>ne huius proportio productorum ex e in a c, &amp; coniunctorum, &amp; <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, &amp; e ad c b: at quod fit ex a c in c b, e&longs;t &aelig;quale ei quod fit ex a b in <lb/>d, eo qu&ograve;d a b, a c, c b &amp; d &longs;unt omiolog&aelig; per decimam &longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, &amp; e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. </s>
          <s id="id000263">per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor<lb/>tionibus e ad a c, &amp; e ad b c, quod fuit demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id000264"><margin.target id="marg32"/>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000265">Propo&longs;itio undecima.</s>
        </p>
        <p type="main">
          <s id="id000266">Proportio aggregati quarumlibet duarum quantitatum ad ag&shy;<lb/>gregatum duarum &aelig;qualium quantitatum e&longs;t compo&longs;ita ex pro&shy;<lb/>portionibus primis, &amp; diui&longs;a per duplam.<lb/><arrow.to.target n="marg33"/></s>
        </p>
        <p type="margin">
          <s id="id000267"><margin.target id="marg33"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000268">Sit proportio a ad c, &amp; b ad d, &amp; &longs;int c &amp; d <lb/><figure id="id.015.01.031.1.jpg" xlink:href="015/01/031/1.jpg"/><lb/>&aelig;quales, dico qu&ograve;d proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, &amp; b ad <lb/>d diui&longs;o compo&longs;ito per duplam. </s>
          <s id="id000269">Quia enim </s>
        </p>
        <p type="main">
          <s id="id000270"><arrow.to.target n="marg34"/><lb/>c &amp; d &longs;unt &aelig;quales, erit b ad c, ut b ad d, qua&shy;<lb/>re ex diffinitione c&ugrave;m proportio a b ad c d <lb/><arrow.to.target n="marg35"/><lb/>&longs;it compo&longs;ita ex proportionibus a ad c, &amp; b <lb/>ad c, erit etiam compo&longs;ita ex dictis ex propo&longs;itione a ad c, &amp; b ad d, <lb/><arrow.to.target n="marg36"/><lb/>&longs;tatuatur ergo e &aelig;qualis c d media inter a b &amp; c. </s>
          <s id="id000271">Et erit per &longs;ecun&shy;<lb/>dam propo&longs;itionem proportio aggregati a b ad c producta ex <lb/><arrow.to.target n="marg37"/><lb/>proportione aggregati a b ad c, &amp; e ad c, igitur proportio a b ad e <lb/>erit proportio a b ad c, diui&longs;a per proportionem e ad c, &longs;ed e ad c e&longs;t <lb/><arrow.to.target n="marg38"/><lb/>dupla: igitur proportio a b ad c d e&longs;t proportio a b ad c diui&longs;a per <lb/>duplam.</s>
        </p>
        <p type="margin">
          <s id="id000272"><margin.target id="marg34"/>E<emph type="italics"/>x &longs;exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/>com. </s>
          <s id="id000273">&longs;ententia.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000274"><margin.target id="marg35"/>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000275"><margin.target id="marg36"/>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000276"><margin.target id="marg37"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000277"><margin.target id="marg38"/>P<emph type="italics"/>er quintam <emph.end type="italics"/><lb/>A<emph type="italics"/>nim. </s>
          <s id="id000278">com. </s>
          <s id="id000279">&longs;en <lb/>tentiam.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000280">Propo&longs;itio duodecima.</s>
        </p>
        <p type="main">
          <s id="id000281">Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"/></s>
        </p>
        <p type="margin">
          <s id="id000282"><margin.target id="marg39"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/>10. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000283">Sint propo&longs;it&aelig; proportiones a ad c &amp; <lb/><figure id="id.015.01.031.2.jpg" xlink:href="015/01/031/2.jpg"/><lb/>b ad d, &amp; a&longs;&longs;umo e ad c, iuxta ea qu&aelig; Eu&shy;<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s>
        </p>
        <p type="main">
          <s id="id000284"><arrow.to.target n="marg40"/><lb/>proportio a e ad c, compo&longs;ita ex proportionibus a ad c, &amp; e ad c, <lb/>&longs;ed proportio e ad c e&longs;t, ut b ad d, igitur proportio a e ad c compo&shy;<lb/>&longs;ita e&longs;t ex proportionibus a ad c, &amp; b ad d.</s>
        </p>
        <p type="margin">
          <s id="id000285"><margin.target id="marg40"/>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. &longs;en <lb/>tentia.<emph.end type="italics"/></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"/>
        <figure id="id.015.01.032.1.jpg" xlink:href="015/01/032/1.jpg"/>
        <p type="main">
          <s id="id000287">Quia ergo ex c in b fit f, ex c in d h, erit f ad h, <lb/>ut b ad d, igitur ut e ad c, &longs;ed a ad c, ut g ad h igi<lb/><arrow.to.target n="marg41"/><lb/>tur a e ad c, ut k ad h, &longs;ed k ad h c&oacute;mponitur ex <lb/>proportionibus a ad c, &amp; b ad d. </s>
          <s id="id000288">Ex octaua ha <lb/>rum igitur proportio a c ad c compo&longs;ita e&longs;t ex <lb/>ei&longs;dem. </s>
          <s id="id000289">For&longs;an quis dicat hanc eandem e&longs;&longs;e <lb/>octau&aelig; &longs;ed <expan abbr="n&otilde;">non</expan> e&longs;t, in illa enim proportio com&shy;<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s>
        </p>
        <p type="margin">
          <s id="id000290"><margin.target id="marg41"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000291">Ex hoc &longs;equitur qu&ograve;d: Qu&aelig;libet du&aelig; quantitates quarum ag&shy;<lb/><arrow.to.target n="marg42"/><lb/>gregatum e&longs;t idem ad eam quantitatem, componunt eandem pro&shy;<lb/>portionem.</s>
        </p>
        <p type="margin">
          <s id="id000292"><margin.target id="marg42"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000293">Propo&longs;itio tertia decima.</s>
        </p>
        <p type="main">
          <s id="id000294">Proportio confu&longs;a aggregati prim&aelig; &amp; terti&aelig; quatuor quantita&shy;<lb/>tum omiologarum ad <expan abbr="aggregat&utilde;">aggregatum</expan> &longs;ecund&aelig; &amp; quart&aelig;, e&longs;t uelut com<lb/>po&longs;ita ex ei&longs;dem diui&longs;a per duplam.<lb/><arrow.to.target n="marg43"/></s>
        </p>
        <p type="margin">
          <s id="id000295"><margin.target id="marg43"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000296">Sint a ad b, ut c ad d, dico, qu&ograve;d erit confu&longs;a <lb/><figure id="id.015.01.032.2.jpg" xlink:href="015/01/032/2.jpg"/><arrow.to.target n="table11"/><lb/>proportio a c aggregati ad <expan abbr="aggregat&utilde;">aggregatum</expan> b d, com<lb/>po&longs;it&aelig; ex his proportionibus diui&longs;&aelig; per du&shy;<lb/>plam &aelig;qualis. </s>
          <s id="id000297">Erit enim aggregati ex a c ad aggregatum ex b d, ue&shy;<lb/>lut a ad b per 18 quinti Elementorum. </s>
          <s id="id000298">Sed proportiones a ad b, <lb/>&amp; c ad d componunt proportionem producti a in d, &amp; c in b per <lb/>octauam harum, ad <expan abbr="product&utilde;">productum</expan> ex b in d, productum uer&ograve; ex a in d <lb/>e&longs;t &aelig;quale producto ex b in c per decimam &longs;extam &longs;exti Elemento&shy;<lb/>rum, &amp; proportio producti ex b in c ad productum ex b in d e&longs;t ue <lb/>lut c ad d, quare ut aggregati a c ad aggregatum b d, igitur propor&shy;<lb/>tio compo&longs;ita ex a ad b, &amp; c ad d, e&longs;t uelut confu&longs;a bis &longs;umpta. </s>
          <s id="id000299">Igi&shy;<lb/>tur confu&longs;a e&longs;t uelut compo&longs;ita diui&longs;a per duplam per modum un&shy;<lb/>decim&aelig; huius.</s>
        </p>
        <table>
          <table.target id="table11"/>
          <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">Propo&longs;itio quarta decima.</s>
        </p>
        <p type="main">
          <s id="id000301">Proportiones confu&longs;&aelig;, &amp; coniunct&aelig; in tribus quantitatibus in&shy;<lb/>uicem commutantur.</s>
        </p>
        <figure id="id.015.01.032.3.jpg" xlink:href="015/01/032/3.jpg"/>
        <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"/><lb/>ad a b confu&longs;a e&longs;t, conuer&longs;a coniunct&aelig; a &amp; b ad <lb/><arrow.to.target n="marg45"/><lb/>c. </s>
          <s id="id000304">Nam per dicta proportio a b ad c efficit con&shy;<lb/>iunctam ex a b ad c: &longs;ed c ad a b conuer&longs;a e&longs;t eius, qu&aelig; e&longs;t a b ad c, &amp; <lb/>proportio c ad a b e&longs;t confu&longs;a eius, qu&aelig; e&longs;t c ad a &amp; b. </s>
          <s id="id000305">Igitur pro&shy;<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunct&aelig; in <lb/>ei&longs;dem.</s>
        </p>
        <p type="margin">
          <s id="id000306"><margin.target id="marg44"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000307"><margin.target id="marg45"/>14. <emph type="italics"/>diff.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000308">Ex quauis ergo illarum data, data erit &amp; reliqua.<lb/><arrow.to.target n="marg46"/></s>
        </p>
        <pb pagenum="14" xlink:href="015/01/033.jpg"/>
        <p type="margin">
          <s id="id000309"><margin.target id="marg46"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000310">Propo&longs;itio quinta decima.</s>
        </p>
        <p type="main">
          <s id="id000311">Si fuerint quatuor quantitas proportio confu&longs;a aggregati pri&shy;<lb/>m&aelig; &amp; terti&aelig; ad aggregatum &longs;ecund&aelig;, &amp; quart&aelig; erit ut monadis <lb/>addito prouentu, qui fit diui&longs;a differentia differentiarum prim&aelig; &amp; <lb/>&longs;ecund&aelig;, atque quart&aelig; &amp; terti&aelig; per aggregatum terti&aelig;, &amp; quart&aelig; ad <lb/>ip&longs;am monadem.</s>
        </p>
        <figure id="id.015.01.033.1.jpg" xlink:href="015/01/033/1.jpg"/>
        <p type="main">
          <s id="id000312">Sint quatuor quantitates a b, c, d, e f, &amp; <lb/><arrow.to.target n="marg47"/><lb/>&longs;it a b maior cin a h, &amp; e f maior d in f g, &amp; <lb/>differentia f g &amp; a h &longs;it a k: dico proportio&shy;<lb/>nem a b, &amp; d confu&longs;am ad c &amp; e f, e&longs;&longs;e ut mo<lb/>nadis addito prouentu, uel detracto a k diui&longs;&aelig; per aggregatum c. <lb/>&amp; e f ad ip&longs;am monadem, &amp; manife&longs;tum e&longs;t, qu&ograve;d pote&longs;t continge&shy;<lb/>re pluribus modis: Primus ut a b &longs;it maior c &amp; e f minor d, &amp; tunc <lb/>differenti&aelig; coniungentur, &amp; prouentus, addetur monadi. </s>
          <s id="id000313">Idem fa&shy;<lb/>ciendum erit &longs;i a b &longs;it maior c, &amp; e f &longs;it minor d, &longs;ed exce&longs;&longs;us &longs;uperet <lb/>defectum. </s>
          <s id="id000314">At &longs;i uel a b &longs;it minor c, &amp; e f maior d, uel ita minor, ut c <lb/>exce&longs;&longs;us &longs;upra b a &longs;it maior defectu, detrahemus prouentum &agrave; mo&shy;<lb/>nade. </s>
          <s id="id000315">Alia cautio e&longs;t qu&ograve;d &longs;i fuerint utrinque exce&longs;&longs;us, aut defectus, <lb/>minuemus minorem de maiore: &longs;i autem unus &longs;it exce&longs;&longs;us alter de&shy;<lb/>fectus, iungemus illos, &amp; po&longs;t diuidemus. </s>
          <s id="id000316">uno ergo demon&longs;trato <lb/>ut pote primo intelligentur reliqui. </s>
          <s id="id000317">Quia ergo b h e&longs;t &aelig;qualis c &amp; <lb/>e g &aelig;qualis d &amp; h k &aelig;qualis g f, erit ex communi animi &longs;ententia ag<lb/>gregatum ex d &amp; k b &aelig;quale aggregato ex c &amp; e f, igitur per dicta <lb/>proportio aggregati ad aggregatum e&longs;t unum. </s>
          <s id="id000318">at uer&ograve; diui&longs;a k a <lb/>per c &amp; e f fit quantum diui&longs;a eadem per b k, &amp; d, &longs;ed diui&longs;a k a per b <lb/>k, &amp; d iunctas, exit proportio a k ad aggregatum b k &amp; d: igitur di&shy;<lb/>ui&longs;a a k per aggregatum e f &amp; c, exibit eadem proportio, igitur a b <lb/>&amp; d ad aggregatum c &amp; e f e&longs;t coniuncta ex monade &amp; proportio&shy;<lb/>ne a k ad aggregatum c &amp; e f, quod erat demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id000319"><margin.target id="marg47"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.033.2.jpg" xlink:href="015/01/033/2.jpg"/>
        <p type="main">
          <s id="id000320">Ex hoc patet quod proportionum confu&longs;io <lb/><arrow.to.target n="marg48"/><lb/>fit iunctis denominatoribus numeratoris: mul&shy;<lb/>tiplicatio multiplicatis: additio multiplicatis <lb/>decu&longs;&longs;atim in numeratores ad productum ex <lb/>denominatoribus, ut in exemplis.</s>
        </p>
        <p type="margin">
          <s id="id000321"><margin.target id="marg48"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000322">Propo&longs;itio &longs;exta decima.</s>
        </p>
        <p type="main">
          <s id="id000323">Omnium quatuor quantitatum propo&longs;ita <lb/>prima, qu&aelig; non minorem habet proportionem <lb/>ad &longs;uam corre&longs;pondentem, qu&agrave;m alia ad aliam <lb/><figure id="id.015.01.033.3.jpg" xlink:href="015/01/033/3.jpg"/><lb/>erit proportio confu&longs;a illarum, ut pro&shy;<lb/>ducti ex aggregato prim&aelig; &amp; terti&aelig; in 


<pb pagenum="15" xlink:href="015/01/034.jpg"/>tertiam, ad productum ex aggregato terti&aelig; &amp; omiotat&aelig; ad &longs;ecun&shy;<lb/>dam in ip&longs;am quartam.</s>
        </p>
        <p type="main">
          <s id="id000324">H&aelig;c magis reducit confu&longs;am proportionem ad notitiam, qu&agrave;m, <lb/>pr&aelig;cedens, quia reducit ad proportionem <expan abbr="product&atilde;">productam</expan>, qu&etail; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, du&aelig; <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. </s>
          <s id="id000325">Summatur ergo a b, c, d &amp; e, &amp; non &longs;it maior propor&shy;<lb/>tio d ad e, qu&agrave;m a b ad c, &amp; &longs;tatuatur tunc prima a b, &longs;ecunda c, ter&shy;<lb/>tia d, quarta e, &amp; po&longs;tquam non e&longs;t minor ratio a b ad c, qu&agrave;m d ad <lb/>e, &longs;umatur a f ad c, ut d ad e. </s>
          <s id="id000326">licet enim hoc facere. </s>
          <s id="id000327">Dico quod pro&shy;<lb/>portio confu&longs;a a b &amp; d ad c &amp; e e&longs;t uelut producti ex aggregato a b <lb/>&amp; d in d ad productum ex aggregato a f &amp; d in e. </s>
          <s id="id000328">Statuatur aggre&shy;<lb/><arrow.to.target n="marg49"/><lb/>gatum a b &amp; d linea a d prima quantitas, &amp; aggregatum a f &amp; d, <lb/><figure id="id.015.01.034.1.jpg" xlink:href="015/01/034/1.jpg"/><lb/>a d &longs;ecunda quantitas, &amp; d tertia, <lb/>&amp; c quarta, &amp; ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri&shy;<lb/>mam propo&longs;itionem g ad h pro&shy;<lb/><arrow.to.target n="marg50"/><lb/>ducta ex proportionibus a b d ad <lb/>a f d, &amp; d ad c. </s>
          <s id="id000329">Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. </s>
          <s id="id000330">Proportio uer&ograve; a b d ad a f d, &amp; <lb/>a f d ad e c producunt proportio&shy;<lb/>nem a b d ad c &amp; e per &longs;ecundam propo&longs;itionem, harum igitur con&shy;<lb/>fu&longs;a a b ad c, &amp; d ad e, &amp; e&longs;t proportio a b d ad c &amp; e, producuntur <lb/>ex proportionibus a b d ad a f d, &amp; d ad e. </s>
          <s id="id000331">Ergo proportio g ad h <lb/>e&longs;t confu&longs;a ex a b ad e, &amp; d ad e, quod erat demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id000332"><margin.target id="marg49"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000333"><margin.target id="marg50"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000334">Propo&longs;itio decima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id000335">Omnes du&etail; proportiones conuer&longs;&aelig; producunt &aelig;qualem pro&shy;<lb/>portionem.<lb/><arrow.to.target n="table12"/></s>
        </p>
        <table>
          <table.target id="table12"/>
          <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 du&aelig; proportiones a ad b &amp; b ad a conuer&longs;a, <lb/><figure id="id.015.01.034.2.jpg" xlink:href="015/01/034/2.jpg"/><arrow.to.target n="marg51"/><lb/>dico, qu&ograve;d producunt proportionem &aelig;qualem. </s>
          <s id="id000337">fiat <lb/>enim b ad c, ut b ad a, erit igitur a &aelig;qualis c &amp; b c con<lb/><arrow.to.target n="marg52"/><lb/>uer&longs;a eius qu&aelig; e&longs;t a ad b, &longs;ed per &longs;ecundam harum <lb/>proportiones a ad b, &amp; b ad c producunt propor&shy;<lb/>tionem a ad c, igitur proportiones etiam a ad b &amp; b ad a produ&shy;<lb/>cunt eandem.</s>
        </p>
        <p type="margin">
          <s id="id000338"><margin.target id="marg51"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000339"><margin.target id="marg52"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni&shy;<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>&longs;ententiam.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000340">Propo&longs;itio decima octaua.</s>
        </p>
        <p type="main">
          <s id="id000341">Si fuerint quotlibet quantitates in continua proportione multi&shy;<lb/>plici pr&aelig;ter ultimam: proportio uer&ograve; penultim&aelig; ad ultimam qua&shy;<lb/>lis re&longs;idui prim&aelig; ad &longs;ecundam, erit prim&aelig; ad aggregatum reliqua&shy;<lb/>rum uelut penultim&aelig; ad ultimam.


<pb pagenum="16" xlink:href="015/01/035.jpg"/><arrow.to.target n="marg53"/></s>
        </p>
        <p type="margin">
          <s id="id000342"><margin.target id="marg53"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000343">Sint quantitates a b c d in continua proportione multiplici, &longs;ed <lb/>d ad e &longs;it uelut re&longs;idui a &amp; b ad b, dico proportionem a ad b c d e <lb/>e&longs;&longs;e ut d ad e. </s>
          <s id="id000344">Quia enim e&longs;t gnomonis e ad quadratum d, ut d ad e <lb/>ex &longs;uppo&longs;ito 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"/><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua&shy;<lb/><figure id="id.015.01.035.1.jpg" xlink:href="015/01/035/1.jpg"/><lb/>dratum e, igitur ut c quadrati ad d &amp; eiuncta, ita <lb/>d ad e. </s>
          <s id="id000346">Rur&longs;us, quia b quadrati ad c quadratum, <lb/><arrow.to.target n="marg55"/><lb/>ut c ad d erit gnomonis b ad quadratum c, ut <lb/>gnomonis c ad quadratum d, &amp; ita d ad e, igitur <lb/><arrow.to.target n="marg56"/><lb/>gnomonum b c cum quadrato d ad aggrega&shy;<lb/>tum c d e quadratorum, ut d ad e, &longs;ed c gno&shy;<lb/>mo cum d quadrato perficit c quadratum, <lb/>&amp; c quadratum cum gnomone b perficit <lb/>quadratum b, igitur proportio quadrati b <lb/>ad quadrata c d e, ut d quadrati a d e. </s>
          <s id="id000347">Et ita <lb/>repetendo de quotuis quantitatibus in infi<lb/>nitum u&longs;que. </s>
          <s id="id000348">H&aelig;c proponitur ab Archimede in libro de quadrato <lb/>&aelig;quali parabol&aelig;, &amp; minus generaliter &amp; pluribus demon&longs;tratur. <lb/></s>
          <s id="id000349">Ego tamen quia e&longs;t generalis, de&longs;cribam illam per corrolarium: ad&shy;<lb/>damque aliud quod ex hoc &longs;equitur.<lb/><arrow.to.target n="marg57"/></s>
        </p>
        <p type="margin">
          <s id="id000350"><margin.target id="marg54"/>13. P<emph type="italics"/>ropo&longs;. <lb/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000351"><margin.target id="marg55"/>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000352"><margin.target id="marg56"/>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000353"><margin.target id="marg57"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000354">Si fuerint quotlibet <expan abbr="qu&atilde;titates">quantitates</expan> omnes analog&aelig; pr&aelig;ter ultimam, <lb/>&longs;it autem penultima ad ultimam qualis re&longs;idui prim&aelig; &amp; &longs;ecund&aelig; <lb/>ad &longs;ecundam, erit proportio prim&aelig; ad aggregatum omnium alia&shy;<lb/>rum ueluti penultim&aelig; ad ultimam.</s>
        </p>
        <p type="main">
          <s id="id000355"><arrow.to.target n="marg58"/></s>
        </p>
        <p type="margin">
          <s id="id000356"><margin.target id="marg58"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000357">H&aelig;c enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/><figure id="id.015.01.035.2.jpg" xlink:href="015/01/035/2.jpg"/><lb/>exemplo autem in numeris &agrave; latere <lb/>po&longs;ito uides declarationem. </s>
          <s id="id000358">nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i<lb/>dui prim&aelig; &amp; &longs;ecund&aelig; ad ip&longs;am &longs;e&shy;<lb/>cundam &longs;cilicet ad 54.</s>
        </p>
        <p type="main">
          <s id="id000359"><arrow.to.target n="marg59"/></s>
        </p>
        <p type="margin">
          <s id="id000360"><margin.target id="marg59"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000361">Ex hoc patet etiam qu&ograve;d a&longs;&longs;umptis omnibus, &longs;ub multiplicibus <lb/>analogi&aelig; u&longs;que in infinitum prima quantitas e&longs;t multiplex aggre&shy;<lb/>gati omnium reliquarum numero 1 m: quo prima e&longs;t multiplex <lb/>&longs;ecund&aelig;.</s>
        </p>
        <p type="main">
          <s id="id000362"><arrow.to.target n="marg60"/></s>
        </p>
        <p type="margin">
          <s id="id000363"><margin.target id="marg60"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id000364">Si fuerint quotlibet quantitates in &longs;uper particulari proportio&shy;<lb/>ne analog&aelig;, erit proportio prim&aelig; ad aggregatum omnium in infi&shy;<lb/>nitum iuxta proportionem multiplicem conuer&longs;am illius partis.</s>
        </p>
        <p type="main">
          <s id="id000365"><arrow.to.target n="marg61"/></s>
        </p>
        <p type="margin">
          <s id="id000366"><margin.target id="marg61"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000367">Velut collect&aelig; in &longs;e&longs;quialtera dupl&aelig; in &longs;exquitertia tripl&aelig; in <lb/>&longs;exqui&longs;eptima &longs;eptupl&aelig;. </s>
          <s id="id000368">Vt capio 512 448 392 343, &amp; ita deinceps <lb/>u&longs;que in infinitum aggregatum omnium earum erit 3584. Septu&shy;


<pb pagenum="17" xlink:href="015/01/036.jpg"/>plum 512, &amp; aggregatum 18. 12. 8. 5 2/3, &amp; ita deinceps in &longs;exquialtera <lb/>erit 54 duplum 27 prim&aelig; in eo ordine.</s>
        </p>
        <p type="head">
          <s id="id000369">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id000370">Ex quo patet genus demon&longs;trandi nouun &amp; pulchrum: nam <lb/>&longs;upponatur 54, aggregatum duplum 27, prim&aelig; igitur addito 27 <lb/>ad 54, cum &longs;it dimidium, &amp; addito 13 1/2, dimidio 27 ad 27, nam ex <lb/>&longs;uppo&longs;ito quantitas &longs;equens e&longs;t &longs;exquialtera ad 27, igitur 81 e&longs;t du&shy;</s>
        </p>
        <p type="main">
          <s id="id000371"><arrow.to.target n="marg62"/><lb/>plum ad 40 1/2. Igitur conuertendo e&longs;t proportio aggregati prioris <lb/>ad 27 e&longs;t dupla, ergo aggregatum e&longs;t 54.<lb/><arrow.to.target n="marg63"/></s>
        </p>
        <p type="margin">
          <s id="id000372"><margin.target id="marg62"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000373"><margin.target id="marg63"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id000374">Ex hoc patet eandem generaliter quod proportio maioris quan<lb/>titatis ad aggregatum reliquarum analogarum e&longs;t, uelut eius quod <lb/>prouenit diui&longs;o quadrato maioris termini per differentiam eius, &amp; <lb/>&longs;equentis maioris in eadem proportione ad ip&longs;um maiorem.</s>
        </p>
        <p type="main">
          <s id="id000375"><arrow.to.target n="marg64"/></s>
        </p>
        <p type="margin">
          <s id="id000376"><margin.target id="marg64"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000377">Exemplum &longs;it proportio augens 25 &amp; 35 duarum quintarum, uo <lb/>lo &longs;cire quantum &longs;it aggregatum omnium citra 25, maximam acci&shy;<lb/>pio 35, ulteriorem ad 25, cuius differentia a 25 e&longs;t 10, cum quo diui&shy;<lb/>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. </s>
          <s id="id000378">Et facile po&shy;</s>
        </p>
        <p type="main">
          <s id="id000379"><arrow.to.target n="marg65"/><lb/>re&longs;t demon&longs;trari. </s>
          <s id="id000380">Si quis dicat in qua proportione &longs;unt infinit&aelig; <lb/>quantitates analog&aelig; cum 12, qu&aelig; iunct&aelig; efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, &amp; in ea pro&shy;<lb/>portione <expan abbr="er&utilde;t">erunt</expan> ill&aelig; quantitates, in qua &longs;unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, &amp; 132 diuide per 12, exeunt 11 &amp; 5, &amp; ita erunt in proportione 11 <lb/>ad 5 experiaris, &amp; inuenies, &amp; demon&longs;tratur ex prioribus.</s>
        </p>
        <p type="margin">
          <s id="id000381"><margin.target id="marg65"/>Q<emph type="italics"/>u&aelig;&longs;tio.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000382">Propo&longs;itio decimanona.</s>
        </p>
        <p type="main">
          <s id="id000383">Si fuerint aliquot quantitates arithmetic&aelig; omiolog&aelig;, quarum <lb/>exce&longs;&longs;us &longs;it &aelig;qualis minim&egrave;, omnibus autem deficientibus &longs;upple&shy;<lb/>menta ad &etail;qualitatem maxim&egrave; adiungantur, erunt quadrata omni&shy;<lb/>um quantitatum &aelig;qualium adiecto rur&longs;us quadrato prim&aelig; cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregat&utilde;">aggregatum</expan> omnium quan&shy;<lb/>titatum eiu&longs;dem tripla aggregato quadra&shy;<lb/><figure id="id.015.01.036.1.jpg" xlink:href="015/01/036/1.jpg"/><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"/><lb/>pariter acceptis.</s>
        </p>
        <p type="margin">
          <s id="id000384"><margin.target id="marg66"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000385">Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>
          <s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe&shy;<lb/>renti&etail; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po<lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a &amp; b, &amp; b &amp; c, &amp; c &amp; <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">aut</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;in <lb/>gulis harum, qu&aelig; &longs;int i k l m n o p, ita ut <expan abbr="o&etilde;s">oes</expan> <lb/>fiant &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>&agrave; maiori. </s>
          <s id="id000387">E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti<pb pagenum="18" xlink:href="015/01/037.jpg"/>tates, &amp; <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes &aelig;quales, &longs;i quinque in quinque, &longs;i decem in decem, ea ra<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis prim&aelig; partis, penultima ubi <lb/>e&longs;t finis &longs;ecund&aelig; partis, ante penultima ubi e&longs;t finis terti&aelig;, &amp; &longs;ic de <lb/>alijs. </s>
          <s id="id000388">Vocabo ergo primas <expan abbr="qu&atilde;titates">quantitates</expan> propo&longs;itas a b c d e f g h quan&shy;<lb/>titates primi ordinis, &longs;ed quantitates &aelig;quales qu&aelig; <expan abbr="con&longs;t&atilde;t">con&longs;tant</expan> ex quan <lb/>titatis. </s>
          <s id="id000389">primi ordinis, &amp; &longs;upplementis, appellabo quantitates &longs;ecun<lb/>di ordinis: ex quo patet qu&ograve;d prima <expan abbr="qu&atilde;titas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliqu&aelig; omnes differunt, quantitates uer&ograve; quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, &amp; &longs;unt una minus <expan abbr="qu&atilde;">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="qu&atilde;titates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e&shy;<lb/>ptem, &amp; &longs;i quantitates <expan abbr="ordin&utilde;">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="n&otilde;">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. </s>
          <s id="id000390">Erunt er<lb/>go &longs;upplementa i k l m n o p, qu&aelig; tanto erunt maiora quanto quan<lb/>titates primi ordinis &longs;unt minores, &amp; contr&agrave; tanto maiora, quanto <lb/><expan abbr="qu&atilde;titates">quantitates</expan> primi ordinis &longs;unt maiores. </s>
          <s id="id000391">quantitates <expan abbr="a&utilde;t">aut</expan> &longs;ecundi ordi<lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, &amp; hp. </s>
          <s id="id000392">H&aelig;c uolui pluribus <lb/>agere, ut dilucidior e&longs;&longs;et propo&longs;itio. </s>
          <s id="id000393">qu&aelig; licet <expan abbr="n&otilde;">non</expan> &longs;it difficilis, e&longs;t <expan abbr="tam&etilde;">tamen</expan> <lb/>confu&longs;a ualde propter multitudinem <expan abbr="quantitat&utilde;">quantitatum</expan> &amp; ordinum. </s>
          <s id="id000394">Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadrator&utilde;">quadratorum</expan> quantitatum &longs;ecundi ordinis pri<lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="c&utilde;">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="tripl&utilde;">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> primi ordinis, &amp; utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="qu&atilde;titates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. &amp; 9.4 &amp; 1. qu&aelig; iuncta <expan abbr="faci&utilde;t">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="qu&atilde;titatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, qu&aelig; &longs;unt octies 64, &amp; eis addiderimus unum <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>his, ut fiant nouies 64, &amp; erunt 556, &longs;imul iuncta &amp; eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti&shy;<lb/>tatum omnium primi ordinis, &amp; e&longs;t tale <expan abbr="product&utilde;">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadrator&utilde;">quadratorum</expan> primi or&shy;<lb/>dinis unius demon&longs;tratio h&etail;c e&longs;t. </s>
          <s id="id000395">Quia ex quarta &longs;ecundi Element. <lb/>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con<lb/>&longs;tant ex quatuor partibus quarum du&etail; &longs;unt quadrata partium, reli&shy;<lb/>qu&aelig; du&aelig; &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, &amp; quia h fuit &aelig;qua&shy;<lb/>lis 1, &amp; p &etail;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&etail;qualia">fuerunt &etail;qualia</expan> mutu&ograve; quanti<lb/>tatibus, &amp; ita c &aelig;qualis o &amp; k &aelig;qualis g &amp; d, &aelig;qualis n &amp; l, &aelig;qualis <lb/>f, e <expan abbr="a&utilde;t">aut</expan> &etail;qualis m. </s>
          <s id="id000396"><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod &longs;umptis duabus quantitatibus <lb/>&longs;ecundi ordinis habentibus <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> mutu&ograve; &aelig;qualia ip&longs;is quan<lb/>titatibus quod quadrata partium <expan abbr="er&utilde;t">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam &amp; h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua&shy;


<pb pagenum="19" xlink:href="015/01/038.jpg"/>drata partium &longs;unt quadrata b &amp; i, &amp; h &amp; p, &longs;ed b e&longs;t &aelig;qualis p, &amp; h <lb/>&aelig;qualis i. </s>
          <s id="id000397">Ergo quatuor quadrata b i &amp; h p &longs;unt dupla quadratis b <lb/>&amp; h, &amp; ita <expan abbr="conclud&atilde;">concludam</expan> de omnibus ubi du&aelig; quantitates duabus com<lb/>parantur: &longs;ed in e m quia e&longs;t &longs;ola una quantitas, i&longs;tud e&longs;t etiam cla&shy;<lb/>rius, quia quadrata e &amp; m &longs;unt dupla quadrato e &longs;oli eo, quod &amp; m <lb/><arrow.to.target n="marg67"/><lb/>&longs;unt &aelig;quales. </s>
          <s id="id000398">Igitur per demon&longs;trata ab Euclide erit proportio o&shy;<lb/>mnium quadratorum b i, c k, d l, e m, f n, g o, h p, ad quadrata b c d e <lb/>f g h, pariter accepta proportio dupla. </s>
          <s id="id000399">at uer&ograve; addito quadrato a <lb/>quadratis b c d e f g h, &amp; erunt quadrata omnium quantitatum, &amp; <lb/>quadratis b i, c k, d l, e m, f n, g o, h p, duplo quadrati a &longs;cilicet &longs;emel, <lb/>quia a e&longs;t ex &longs;ecundo ordine quantitatum, &amp; &longs;emel, quia hoc fuit a&longs;&shy;<lb/>&longs;umptum in Problemate. </s>
          <s id="id000400">Sequitur ut quadrata omnia <expan abbr="qu&atilde;titatum">quantitatum</expan> <lb/>&longs;ecundi ordinis, pro ut &longs;unt diui&longs;a in partes addito quadrato a, &longs;int <lb/>dupla quadratis primarum quant&iacute;tatum, &longs;imul pariter acceptis. </s>
          <s id="id000401">Re<lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illor&utilde;">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, &longs;cilicet h in aggregatum ip&longs;arum <lb/>quantitatum primi ordinis e&longs;&longs;e &aelig;quale quadratis, <expan abbr="quantitat&utilde;">quantitatum</expan> eiu&longs;&shy;<lb/>dem primi ordinis pariter acceptis. </s>
          <s id="id000402">Con&longs;tat igitur, quod duplum i<lb/>in b e&longs;t &aelig;quale duplo h in ip&longs;um b, quia h &amp; i &longs;unt &aelig;quales, &amp; du&shy;<lb/>plum k in ip&longs;um c, e&longs;t &aelig;quale quadruplo h in idem c, quia k e&longs;t du&shy;<lb/>pla h, &amp; &longs;imiliter duplum l in ip&longs;um d e&longs;t &aelig;quale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, &amp; ita procedendo erunt illa dupla producta &aelig;&shy;<lb/>qualia productis ex h in ip&longs;as quantitates toties &longs;umptis quantus <lb/>e&longs;t numerus, qui prouenit duplicato numero, &longs;ecundum <expan abbr="qu&etilde;">quem</expan> h con<lb/>tinetur in illo &longs;upplemento, exemplum uolo duplum producti lin <lb/>d bis, &longs;cio qu&ograve;d &longs;upplementum l continet h ter, duplicabo tria &amp; fi&shy;<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dupl&utilde;">duplum</expan> lin d &aelig;quale e&longs;t &longs;excuplo h in ip&longs;um d. </s>
          <s id="id000403">Quo con&shy;<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="qu&atilde;titatum">quantitatum</expan> e&longs;&longs;e &aelig;qualia quadratis ip&longs;a&shy;<lb/>rum quantitatum, igitur addemus <expan abbr="product&utilde;">productum</expan> ex h in &longs;ingulas quan&shy;<lb/>titates productis illis prioribus, &amp; fiet productum h in a &longs;emel, in b <lb/>ter, in c quinquies, in d &longs;epties, in e nouies, in f undecies, in g trede&shy;<lb/>cies, &amp; in h quindecies &aelig;quale duplo producti uniu&longs;cuiu&longs;que quan&shy;<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> ip&longs;a&shy;<lb/>rum quantitatum, at quadratum a e&longs;t &etail;quale producto ex h in eam, <lb/>qu&etail; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"/><lb/>h per demon&longs;trata ab Euclide, &amp; pariter de quadrato b, quod e&longs;t &etail;&shy;<lb/>quale ei quod fit ex h in eam qu&aelig; toties continet b, quotiens b con<lb/>tinet h, &amp; ita quadratum c &aelig;quale e&longs;t ei, quod continetur &longs;ub h, &amp; <lb/>habente proportionem ad b eandem, quam b ad h, &amp; &longs;imiliter de <lb/>quadrato c &amp; omnibus reliquis, u&longs;que ad h ip&longs;um. </s>
          <s id="id000404">Gratia ergo exem<pb pagenum="20" xlink:href="015/01/039.jpg"/>pli quadratum a, erit &aelig;quale producto ex h in omnes quantitates &longs;e&shy;<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, &amp; &longs;imiliter quotus e&longs;t numerus quan <lb/>t&iacute;tatum incipiendo &agrave; b, &amp; quotus e&longs;t numerus quantitatum incipi&shy;<lb/>endo &agrave; c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, &amp; ita de alijs, quadrata ergo om&shy;<lb/>nium quantitatum &longs;imul iuncta &longs;unt &aelig;qualia productis ex h in &longs;in&shy;<lb/>gulas illarum toties &longs;umptis, quoties ill&aelig; <expan abbr="c&otilde;tinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, &amp; <expan abbr="numer&atilde;do">numerando</expan> uer&longs;us a. <lb/></s>
          <s id="id000405">Rur&longs;us dico, quod productum multiplicis cuiuslibet <expan abbr="qu&atilde;titatis">quantitatis</expan> in <lb/>minimam, &longs;eu quadratum eiu&longs;dem quantitatis &etail;quale e&longs;t producto <lb/>eiu&longs;dem quantitatis, &amp; dupli omnium &longs;equentium primi ordinis in <lb/>ip&longs;am minimam quantitatem, uelut quadratum a e&longs;t &aelig;quale produ<lb/>cto ex h in a, &amp; in duplum b c d e f g h, hoc <expan abbr="aut&etilde;">autem</expan> facile e&longs;t probare in <lb/>his quantitatibus, quia &longs;i quadratum a e&longs;t &aelig;quale producto h in o&shy;<lb/>mnes quantitates &longs;ecundi ordinis, &amp; omnes quantitates &longs;ecundi or <lb/>dinis &longs;imul &longs;umpt&aelig; &longs;unt &etail;quales ip&longs;i a, &amp; duplo <expan abbr="reliquar&utilde;">reliquarum</expan> primi or <lb/>dinis, quia tales quantitates &longs;unt &aelig;quales &longs;uis &longs;upplementis uici&longs;&shy;<lb/>&longs;im, ut h cum i, k cum g, f cum l, e <expan abbr="c&utilde;">cum</expan> m, ergo tam &longs;upplementa, qu&agrave;m <lb/>quantitates primi ordinis &longs;unt dimidium quantitatum &longs;ecundi or&shy;<lb/>dinis, ergo duplum quantitatum primi ordinis e&longs;t dimidium quan<lb/>titatum &longs;ecundi ordinis, uer&ugrave;m de b dico idem accidere, quia qua&shy;<lb/>dratum b e&longs;t &etail;quale producto ex h in b, &amp; in duplum reliquarum &agrave; <lb/>b, &longs;cilicet duplum c d e f g h, &amp; hoc e&longs;t o&longs;tendere, quod i&longs;t&etail; quantita<lb/>tes &longs;unt dimidium totidem quantitatum &aelig;qualium b, nam c e&longs;t mi&shy;<lb/>nor b in h, &amp; &longs;upplementum p quod e&longs;t &aelig;quale ip&longs;i b, &longs;i tota h p fiat <lb/>&aelig;qualis ip&longs;i b, ut pote h q erit ip&longs;a q dempta h &aelig;qualis ip&longs;i c, ergo <lb/>quantitates primi ordinis &longs;emper &longs;unt &aelig;quales &longs;upplementis non <lb/>ueris, &longs;ed prioris quantitatis a&longs;&longs;umpt&aelig;, &longs;eu in comparatione ad il&shy;<lb/>lam, quadratum igitur b e&longs;t &aelig;quale producto ex h in b, &amp; in duplum <lb/>c d e f g h, &amp; &longs;imiliter per eadem, quadratum c e&longs;t &aelig;quale producto <lb/>ex h in c, &amp; in duplum d e f g h, &amp; &longs;ic de alijs. </s>
          <s id="id000406">Habemus ergo, quod <lb/>quadrata a b c d e f g h &longs;imul iuncta &longs;unt &aelig;qualia producto ex h in <lb/>a, &amp; in duplum reliquarum, &amp; ex h in b, &amp; in duplum reliquarum <lb/>&longs;equentium, &amp; producto ex h in c &longs;emel, &amp; in duplum &longs;equentium <lb/>u&longs;que ad h, &amp; ita de reliquis. </s>
          <s id="id000407">hoc enim e&longs;t, quod nuper demon&longs;traui&shy;<lb/>mus. </s>
          <s id="id000408">Antea quo que <expan abbr="dem&otilde;&longs;tratum">demon&longs;tratum</expan> e&longs;t, quod duplum b in i, c in k, d in <lb/>l, e in m, f in n, g in o, h in p, <expan abbr="c&utilde;">cum</expan> producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> a b c d e f g h <lb/>erat &etail;quale productis ex h in a &longs;emel, &amp; in b ter, &amp; in c quinquies, in <lb/>d &longs;epties, in e nouies, in fundecies, in g tredecies, in &longs;e ip&longs;am h quin&shy;<lb/>decies, detractis ergo p <expan abbr="ordin&etilde;">ordinem</expan>, &qring;d fit ex h in a ab utro que aggregato, <lb/>&amp; ex h in b c d e f g h bis <expan abbr="relinque&ttilde;">relinquetur</expan> ex una parte, quae fit ex h in b &longs;emel 


<pb pagenum="21" xlink:href="015/01/040.jpg"/>cum &longs;uis duplicatis &longs;equentibus, &amp; in c, &amp; in d, &amp; in reliquis pa&shy;<lb/>riter conduplicatis &longs;uis &longs;equentibus ex altera, quod fit ex h in b &longs;e&shy;<lb/>mel, in c ter, in d quinquies, in e &longs;epties, in f nouies, in g undecies, <lb/>in h tredecies, detractis ergo rur&longs;us quod fit ex h in b &longs;emel, &amp; ex <lb/>h in c d e f g h bis relinquetur, quod fit ex h in c, &amp; duplo &longs;equen&shy;<lb/>tium, &amp; d &amp; duplo &longs;equentium, &amp; e &amp; aliarum pariter: &amp; ex alia <lb/>parte, quod fit ex h in c &longs;emel, &amp; in d ter, &amp; in e quinquies, in f &longs;e&shy;<lb/>pties, in g nouies, in h undecies. </s>
          <s id="id000409">Ab his rur&longs;us detractis, qu&ograve;d fit <lb/>ex h in c &longs;emel, &amp; in &longs;equentes bis, relinquetur h in d &longs;emel cum &longs;uis <lb/>&longs;equentibus bis, &amp; in e &longs;emel cum &longs;uis &longs;equentibus &amp; in f, &amp; in g &amp; <lb/>in h pariter, &amp; ex alia parte, quod fit ex h in d &longs;emel, in e ter, f quin&shy;<lb/>quies, g &longs;epties, h nouies, ab his rur&longs;us detraho, quod fit ex h in d <lb/>&longs;emel, &amp; in &longs;equentes bis, relinquetur ex una parte, quod fit ex h <lb/>in e f g h cum duplo &longs;equentium ex alia, quod fit ex h in e &longs;e&shy;<lb/>mel, f ter, g quinquies, h &longs;epties, &amp; &longs;imiliter ab his detractis, quod <lb/>fit ex h in e &longs;emel, &amp; bis in &longs;equentes, relinquetur ex una par&shy;<lb/>te; quod fit ex h in f &longs;emel, &amp; in g h bis, &amp; in g &longs;emel, &amp; in h bis, <lb/>&amp; in h &longs;emel, &amp; ex alia, quod fit ex h in f &longs;emel, in g ter, in h quin&shy;<lb/>quies. </s>
          <s id="id000410">Iterum detractis, quod fit ex h in f &longs;emel, &amp; in g h bis com&shy;<lb/>muniter relinquetur, quod fit ex h in g &longs;emel, &amp; in h bis, &amp; in h &longs;e&shy;<lb/>mel, &amp; ex alia parte quod fit ex h in g &longs;emel, &amp; ex h in h ter. </s>
          <s id="id000411">Sed <lb/>i&longs;ta, qu&aelig; relicta &longs;unt iam, &longs;unt manife&longs;t&egrave; &aelig;qualia, ergo etiam pri&shy;<lb/>ma aggregata ab initio fuere &aelig;qualia, ergo &amp; &aelig;qualia illis qua&shy;<lb/>drata a b c d e f g h his, qu&aelig; fiunt, ex h in ea&longs;dem quantita&shy;<lb/>tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o, <lb/>h in p, &longs;ed iam his quadratis a b c d e f g h demon&longs;trata &longs;unt e&longs;&longs;e du&shy;<lb/>pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra&shy;<lb/>ti a, ergo quadrata omnium quantitatum &longs;ecundi ordinis cum <lb/>quadrato a rur&longs;us repetito, &amp; producto h in aggregatum quanti&shy;<lb/>tatum primi ordinis &longs;unt tripla quadratis quantitatum primi ordi&shy;<lb/>nis pariter acceptis, quod fuit propo&longs;itum, &amp; fuit Archimedis in li <lb/>bro de lineis &longs;piralibus, &amp; ego adieci hic propter modum demon<lb/>&longs;trandi, qui e&longs;t eleganti&longs;simus, &amp; procedit ex principijs arithmeti&shy;<lb/>cis, &amp; diuer&longs;is &agrave; communibus, &amp; ideo non reuoluitur, ut &longs;olent re&shy;<lb/>liqu&aelig; qu&aelig;&longs;tiones.</s>
        </p>
        <p type="margin">
          <s id="id000412"><margin.target id="marg67"/>I<emph type="italics"/>n<emph.end type="italics"/> 5. E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 12.</s>
        </p>
        <p type="margin">
          <s id="id000413"><margin.target id="marg68"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 6. E<emph type="italics"/>le.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 17.</s>
        </p>
        <p type="main">
          <s id="id000414">Propo&longs;itio uige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id000415">C&ugrave;m fuerint quatuor quantitates, fueritque &longs;ecunda &aelig;qualis ter&shy;<lb/>ti&aelig;, aut prim&aelig; &aelig;qualis quart&aelig;, erit proportio prim&aelig; ad quartam, <lb/>aut terti&aelig; ad &longs;ecundam producta ex proportionibus prim&aelig; ad &longs;e&shy;<lb/>cundam, &amp; terti&aelig; ad quartam.<lb/><arrow.to.target n="marg69"/></s>
        </p>
        <p type="margin">
          <s id="id000416"><margin.target id="marg69"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000417">C&ugrave;m enim quantitates h&aelig; non fuerint &etail;quales, <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> per &longs;ecun&shy;


<pb pagenum="22" xlink:href="015/01/041.jpg"/>dam harum, quod proportio prim&aelig; ad <expan abbr="quart&atilde;">quartam</expan> producitur ex pro&shy;<lb/>portione prim&aelig; ad &longs;ecundam, &longs;ecund&etail; ad tertiam, &amp; terti&etail; ad quar<lb/>tam: ergo non ex &longs;olis proportionibus prim&aelig; ad &longs;ecundam, &amp; ter&shy;<lb/>ti&aelig; ad quartam, &amp; &longs;imiliter ex prima harum proportio prim&etail; ad &longs;e&shy;<lb/>cundam, &amp; terti&aelig; ad quartam producunt proportionem producti <lb/>prim&aelig; in &longs;ecundam ad productum terti&aelig; in quartam. </s>
          <s id="id000418">Et in multi&shy;<lb/>plicatione proportio, qu&aelig; &longs;olet e&longs;&longs;e inter producta illa, &amp; e&longs;t qua&longs;i <lb/>duplicata e&longs;t inter ip&longs;as quantitates. </s>
          <s id="id000419">Sint igitur quantitates a b c d, <lb/>&amp; &longs;it b &aelig;qualis c, ponantur ergo recto ordine a b c d, eritque propor<lb/><figure id="id.015.01.041.1.jpg" xlink:href="015/01/041/1.jpg"/><lb/>tio a ad d producta ex proportioni&shy;<lb/>bus a ad b, b ad c, &amp; c ad d, producan&shy;<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. </s>
          <s id="id000420">proportio c ad f, erit igitur pro&shy;<lb/>portio e ad f, &longs;i multiplicetur per pro&shy;<lb/>portionem b ad c eadem qu&aelig; prius, &amp; </s>
        </p>
        <p type="main">
          <s id="id000421"><arrow.to.target n="marg70"/><lb/>producta iam e&longs;t eadem ei, qu&aelig; e&longs;t a <lb/>ad d, ergo proportio a ad d erit producta ex proportionibus a ad <lb/>b, c ad d per primam propo&longs;itionem. </s>
          <s id="id000422">Quod uer&ograve; diximus de pri&shy;<lb/>ma &amp; quarta &longs;i &longs;int &aelig;quales, manife&longs;tum e&longs;t, qu&ograve;d res redit ad idem <lb/>&longs;olum tran&longs;mutato ordine, ut tertia, &amp; quarta pr&aelig;mittantur prim&etail;, <lb/>&amp; &longs;ecund&aelig;. </s>
          <s id="id000423">H&aelig;c igitur propo&longs;itio nihil aliud innuit, qu&agrave;m quod <lb/>in hoc ca&longs;u productio, qu&aelig; &longs;olet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s>
        </p>
        <p type="margin">
          <s id="id000424"><margin.target id="marg70"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000425">Propo&longs;itio uige&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id000426">C&ugrave;m decu&longs;&longs;atim ducta fuerit prima in quartam, &amp; &longs;ecunda in ter<lb/>tiam; productumque prim&aelig; in quartam diui&longs;um fuerit per produ&shy;<lb/>ctum &longs;ecund&aelig; in tertiam erit proportio prim&aelig; ad &longs;ecundam diui&shy;<lb/>&longs;a per proportionem terti&aelig; ad quartam. </s>
          <s id="id000427">Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/><arrow.to.target n="marg71"/></s>
        </p>
        <p type="margin">
          <s id="id000428"><margin.target id="marg71"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.041.2.jpg" xlink:href="015/01/041/2.jpg"/>
        <p type="main">
          <s id="id000429">Primum exponamus &longs;ecundam partem, &longs;it <lb/>proportio a ad b, quam uolo diuidere per <lb/>proportionem c ad d, facio e ad b, ut c ad d, erit <lb/><arrow.to.target n="marg72"/><lb/>ergo per <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> harum proportio ad b pro&shy;<lb/>ducta ex proportione a ad e, &amp; e ad b, quare ex a ad e, &amp; c ad d, ergo <lb/>diui&longs;a proportione a ad b per proportionem c ad d exit proportio <lb/>a ad e, &amp; hic e&longs;t &longs;ecundus modus. </s>
          <s id="id000430">Primus autem modus ducatur a <lb/>in d &amp; fiat f, &amp; b in c &amp; fiat g, dico proportione f ad g e&longs;&longs;e prouen&shy;<lb/>tum proportionis a ad b, diuide per proportionem c ad d, ducatur <lb/>igitur c in f &amp; fiat h, &amp; d in g &amp; fiat k, quia igitur h producitur ex c <lb/>in f, &amp; f producitur ex a in d, ergo h producetur ex producto c in d, <lb/>in a, &amp; &longs;imiliter quia k producitur ex d in g, &amp; g producitur ex b in 


<pb pagenum="23" xlink:href="015/01/042.jpg"/>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/></s>
          <s id="id000431">erit a ad b ut h ad k, igitur ex prima harum cum ex c in f producatur <lb/>h, &amp; ex d in g k, &amp; dicatur produci proportio h ad k ex proportio&shy;<lb/>ne c ad d, &amp; f ad g, &amp; proportio h ad k &longs;it eadem, qu&aelig; a ad b, ergo <lb/>proportio a ad b producitur ex c ad d, &amp; f ad g, ergo diui&longs;a propor&shy;<lb/>tione a ad b prodibit proportio f ad g, quod fuit propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id000432"><margin.target id="marg72"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000433">Propo&longs;itio uige&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id000434">C&ugrave;m fuerit proportio prim&aelig; ad &longs;ecundam maior, qu&agrave;m terti&aelig; <lb/>ad quartam, erit confu&longs;a ex his maior qu&agrave;m terti&aelig; ad quartam, mi&shy;<lb/>nor autem qu&agrave;m prim&aelig; ad &longs;ecundam.</s>
        </p>
        <figure id="id.015.01.042.1.jpg" xlink:href="015/01/042/1.jpg"/>
        <p type="main">
          <s id="id000435">Sit proportio a ad b maior qu&agrave;m c <lb/><arrow.to.target n="marg73"/><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, qu&agrave;m c ad d, et minor qu&agrave;m <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiam decimam ha&shy;<lb/><arrow.to.target n="marg74"/><lb/>rum e c ad b d confu&longs;a minor qu&agrave;m a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo qu&ograve;d e ha&shy;<lb/>bet proportionem ad b, quam c ad d, qu&aelig; <expan abbr="aut&etilde;">autem</expan> c ad d minor, qu&aacute;m <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, qu&agrave;m a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi&shy;<lb/>nor, qu&agrave;m confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba&shy;<lb/>bitur, &amp; primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma&shy;<lb/>ior proportio a f ad b d, qu&agrave;m a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>eandem tertiam decimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, qu&agrave;m a ad b.</s>
        </p>
        <p type="margin">
          <s id="id000436"><margin.target id="marg73"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000437"><margin.target id="marg74"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000438">Propo&longs;itio uige&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id000439">Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li&shy;<lb/>neam fit.<lb/><arrow.to.target n="marg75"/></s>
        </p>
        <p type="margin">
          <s id="id000440"><margin.target id="marg75"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000441">Motus naturalis e&longs;t, ut con&longs;eruetur corpus, &amp; conueniat locus <lb/>corpori, igitur fit ad &longs;uum locum. </s>
          <s id="id000442">Locus autem dicitur in compara<lb/>tione ad uniuer&longs;um. </s>
          <s id="id000443">ideo omnis motus naturalis e&longs;t &agrave; centro mun&shy;<lb/>di &longs;ur&longs;um, uel ad centrum deor&longs;um. </s>
          <s id="id000444">Et quia quanto natura celerius <lb/>&longs;uum finem pote&longs;t a&longs;&longs;equi &lpar;quia finis bonus e&longs;t aliter non illum ap&shy;<lb/>peteret&rpar; eum qu&aelig;rit, c&ugrave;m &longs;it &longs;apienti&longs;sim&aelig; uit&aelig; mini&longs;tra: at linea re&shy;</s>
        </p>
        <p type="main">
          <s id="id000445"><arrow.to.target n="marg76"/><lb/>cta breui&longs;sima e&longs;t Euclide te&longs;te &agrave; puncto ad punctum, igitur omnis <lb/>motus naturalis e&longs;t &longs;ur&longs;um aut deor&longs;um per rectam lineam.</s>
        </p>
        <p type="margin">
          <s id="id000446"><margin.target id="marg76"/>D<emph type="italics"/>i&longs;t. tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000447">Propo&longs;itio uige&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id000448">Omnis motus circularis uoluntarius e&longs;t.</s>
        </p>
        <p type="main">
          <s id="id000449">Sit motus in circulo &longs;eu per circulum in orbe cuius &longs;it centrum, <lb/>&longs;it c mundi centrum: igitur ex diffinitione circuli tantum di&longs;tabit a, <lb/>quantum b ab ip&longs;o c: &longs;ed in motu naturali per pr&etail;cedentem nece&longs;&longs;e <lb/>e&longs;t, ut recta feratur ad c, uel recedat, igitur motus a e&longs;t uoluntarius, 


<pb pagenum="24" xlink:href="015/01/043.jpg"/><figure id="id.015.01.043.1.jpg" xlink:href="015/01/043/1.jpg"/><lb/>non naturalis. </s>
          <s id="id000450">nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. </s>
          <s id="id000451">Omnia ergo a&longs;tra feruntur <lb/>circa centrum mundi. </s>
          <s id="id000452">Sit modo rota e f g, di<lb/>co e non moueri motu circulari nam linea <lb/>e c <expan abbr="l&otilde;gior">longior</expan> e&longs;t g c, ergo recta mouetur ad cen<lb/>trum non circa centrum. </s>
          <s id="id000453">Indicio etiam id <lb/>e&longs;t: qu&ograve;d &longs;i in e ponatur fru&longs;tum aliquod <lb/>in&longs;igne plumbi in motu ad g per f de&longs;cen&shy;<lb/>det raptim: at dum ex g in e magna cum dif&shy;<lb/>ficultate, igitur motus hic non e&longs;t naturalis, <lb/>nec circularis. </s>
          <s id="id000454">nihil etiam hoc modo &longs;ponte mouetur. </s>
          <s id="id000455">Sed cum non <lb/>moueatur per rectam naturaliter, nec &aelig;quidi&longs;tans &agrave; centro per cir&shy;<lb/>culum relinquitur, ut moueatur motu uiolento, aut mi&longs;to, &longs;ed non <lb/>ex uoluntario, cum nullo modo moueatur &aelig;quidi&longs;tans &agrave; centro, <lb/>&longs;ed &longs;emper ab e line&aelig; ad centrum fiant breuiores, liquet e&longs;&longs;e mo&shy;<lb/>tum uiolentum: aut mi&longs;tum ex naturali, &amp; uiolento.</s>
        </p>
        <p type="main">
          <s id="id000456">Propo&longs;itio uige&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id000457">Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius &amp; <lb/>uiolentus.<lb/><arrow.to.target n="marg77"/></s>
        </p>
        <p type="margin">
          <s id="id000458"><margin.target id="marg77"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000459">Tres &longs;unt modi, quibus po&longs;&longs;unt moueri in comparatione ad cen<lb/>trum &longs;cilicet uel recta cum centro, uel &aelig;quidi&longs;tando &agrave; centro, uel <lb/>neutro modo, igitur tres motus. </s>
          <s id="id000460">Rur&longs;us uel &agrave; principio interiore <lb/>non intelligente, &amp; e&longs;t naturalis, uel intelligente &amp; e&longs;t uoluntarius: <lb/>uel exteriore &amp; e&longs;t uiolentus. </s>
          <s id="id000461">H&aelig;c autem diui&longs;io e&longs;t &longs;olum propria <lb/>non prima. </s>
          <s id="id000462">Nam e&longs;t uiolentus in recta ad centrum: ideo omnis, qui <lb/>non e&longs;t in recta ad centrum, nec &aelig;quidi&longs;tat, uiolentus e&longs;t: non ta&shy;<lb/>men omnis uiolentus e&longs;t extra rectam. </s>
          <s id="id000463">Attractio autem, qu&aelig; fit ob <lb/>raritatem corporum, &longs;eu, ut dicunt, &agrave; uacuo, uiolenta e&longs;t non natu&shy;<lb/>ralis ni&longs;i ratione finis, non agentis. </s>
          <s id="id000464">Sunt enim quatuor genera mo&shy;</s>
        </p>
        <p type="main">
          <s id="id000465"><arrow.to.target n="marg78"/><lb/>tus uiolenti ab Ari&longs;totele po&longs;ita, uectio, tractio, pul&longs;io, &amp; uolutio: <lb/>quanquam his non opus &longs;it in demon&longs;tratiua &longs;cientia. </s>
          <s id="id000466"><expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> enim <lb/>uolutionem ex tractione, &amp; pul&longs;ione apud illum con&longs;i&longs;tere.</s>
        </p>
        <p type="margin">
          <s id="id000467"><margin.target id="marg78"/>7. P<emph type="italics"/>hy&longs;. <lb/>cap.<emph.end type="italics"/> 2.</s>
        </p>
        <p type="main">
          <s id="id000468">Propo&longs;itio uige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id000469">Motus ergo compo&longs;iti quatuor nece&longs;&longs;ari&ograve; &longs;unt &longs;pecies.</s>
        </p>
        <p type="main">
          <s id="id000470">Si tantum &longs;unt tres &longs;pecies &longs;implicium, con&longs;tat ratione arithme&shy;<lb/>tica quatuor e&longs;&longs;e compo&longs;itorum. </s>
          <s id="id000471">Di&longs;quiramus ergo an &longs;int natura&shy;<lb/>liter tot &longs;pecies, for&longs;an enim repugnabit aliquis alicui. </s>
          <s id="id000472">Porr&ograve; uidea&shy;<lb/>mus prim&ograve;, quot &longs;int uiolentorum &longs;pecies: Prima erit cum non &longs;e&shy;<lb/>cundum rectam lineam fuerit: nec &agrave; centro &aelig;quidi&longs;tantem. </s>
          <s id="id000473">Secun&shy;<lb/>da cum fuerit &longs;ecundum rectam, &longs;ed non ad centrum. </s>
          <s id="id000474">Tertia cum <lb/>fuerit in recta ad centrum, &longs;ed contrario modo, uelut terr&aelig; &longs;ur&longs;um. 


<pb pagenum="25" xlink:href="015/01/044.jpg"/>Quarta c&ugrave;m in recta ad centrum, &longs;ecundum naturam, &longs;ed <expan abbr="n&otilde;">non</expan> &agrave; prin<lb/>cipio naturali. </s>
          <s id="id000475">Velut cum quis proijcit lapidem rect&agrave; in terram &egrave; <lb/>turri uiolentius, qu&agrave;m ille &longs;ua grauitate de&longs;cen&longs;urus e&longs;&longs;et. </s>
          <s id="id000476">Hic igi&shy;<lb/>tur motus e&longs;t compo&longs;itus ex naturali, &amp; uiolento. </s>
          <s id="id000477">Animalium au&shy;<lb/>tem motus uoluntarius e&longs;t, cum &longs;it &agrave; principio interiore cogno&longs;cen <lb/>te: &amp; &longs;it quatenus &agrave; principio in linea circulari &aelig;qualiter di&longs;tante &agrave; <lb/>centro: &longs;ed quia ob&longs;tat grauitas, ide&ograve; mi&longs;tus e&longs;t ex naturali, &amp; uo&shy;<lb/>luntario. </s>
          <s id="id000478">Sed circularis, &amp; uiolentus &longs;oli e&longs;&longs;e non po&longs;&longs;unt: nam uio <lb/>lentus e&longs;t nece&longs;&longs;ari&ograve; in corpore graui aut leui: &longs;ed omne corpus gra<lb/>ue aut leue, c&ugrave;m mouetur, naturaliter mouetur &longs;altem in fine: &amp; per <lb/>totum motum, motu &oacute;cculto, qui maxim&egrave; in hoc libro dignus e&longs;t <lb/>con&longs;ideratione, igitur motus uoluntarius, &amp; uiolentus non po&longs;&shy;<lb/>&longs;unt e&longs;&longs;e &longs;imul &longs;oli. </s>
          <s id="id000479">Erunt ergo &longs;ecundum naturam tant&ugrave;m tres &longs;pe&shy;<lb/>cies. </s>
          <s id="id000480">Velut c&ugrave;m quis &longs;candit, aut&longs; alit: E&longs;t enim motus naturalis &longs;al&shy;<lb/>tem in fine, &amp; uoluntarius, &amp; uiolentus. </s>
          <s id="id000481">Si quis autem uelit uiolen&shy;<lb/>tum cum uoluntario copulare dicemus con&longs;tare eam compo&longs;itio&shy;<lb/>nem in initio &longs;aliendi. </s>
          <s id="id000482">Motum autem occultum uocamus grauita&shy;<lb/>tem aut leuitatem.</s>
        </p>
        <p type="main">
          <s id="id000483">Propo&longs;itio uige&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id000484">Motus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus <lb/>exloco.</s>
        </p>
        <p type="main">
          <s id="id000485">H&aelig;c e&longs;t tertia differentia primarum &longs;pecierum motuum uolun&shy;<lb/>tarius fit manente corpore toto in eodem loco, ideo proprius e&longs;t <lb/>c&oelig;lo, corpora autem animalium in eodem loco feruntur: quia in <lb/>eodem orbe nata redire ad proprium locum. </s>
          <s id="id000486">Et ide&ograve;, ut dixi, e&longs;t mo<lb/>tus mi&longs;tus ex naturali, &amp; uoluntario, qui &longs;i per &longs;e fieret, non fatiga&shy;<lb/>ret mobile, c&ugrave;m ex utroque principio ab interiore ui procedat. </s>
          <s id="id000487">Sed <lb/>quia fit per mu&longs;culos, qui trahuntur: hic autem motus e&longs;t uiolen&shy;<lb/>tus, ide&ograve; per con&longs;equentiam fatigat. </s>
          <s id="id000488">Qui uer&ograve; naturalis, e&longs;t ut re&shy;<lb/>deat corpus ad &longs;uum locum, igitur naturalis e&longs;t ad locum. </s>
          <s id="id000489">Sed <lb/>uiolenti finis e&longs;t, ut protrudatur ex loco in quo e&longs;t, non habens cer&shy;<lb/>tum finem. </s>
          <s id="id000490">licet enim qui trahit, ad &longs;uum locum trahat, non tamen <lb/>ad locum mobilis.</s>
        </p>
        <p type="main">
          <s id="id000491">Propo&longs;itio uige&longs;imaoctaua.</s>
        </p>
        <p type="main">
          <s id="id000492">Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"/></s>
        </p>
        <p type="margin">
          <s id="id000493"><margin.target id="marg79"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000494">C&ugrave;m uacuum non detur, &amp; omnis motus naturalis &longs;it ad locum, <lb/>et uiolentus ex loco per pr&aelig;cedentem, igitur c&ugrave;m non &longs;it in medio, <lb/>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s>
        </p>
        <p type="main">
          <s id="id000495">Propo&longs;itio uige&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id000496">Omnis motus uoluntarius &aelig;qualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s>
        </p>
        <pb pagenum="26" xlink:href="015/01/045.jpg"/>
        <p type="main">
          <s id="id000497"><arrow.to.target n="marg80"/></s>
        </p>
        <p type="margin">
          <s id="id000498"><margin.target id="marg80"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id000499">Motus uoluntarius non habet, qu&ograve;d fatiget, &amp; &longs;umma perfectio <lb/>e&longs;t &aelig;qualitas, &amp; natura qu&aelig; mouet non debilitatur, igitur perpe&shy;<lb/>tuo per&longs;euerat &aelig;qualis. </s>
          <s id="id000500">neque enim e&longs;t, ut dixi, per medium corpus. <lb/></s>
          <s id="id000501">Naturalis quoque, &amp; uiolentus cum ratione proportionis mouentis <lb/>&longs;upra mobile per&longs;e non uarientur, &amp; ab &etail;quali proportione &etail;qua&shy;<lb/>lis uelo citas proueniat, igitur natura tales motus &longs;unt &etail;quales, nam <lb/>in utroque mouens, mouet &longs;ecundum ultimam &longs;uam uim.</s>
        </p>
        <p type="main">
          <s id="id000502">Propo&longs;itio trige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id000503">In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obui&aelig;, <lb/>ali&aelig; impellunt.</s>
        </p>
        <p type="main">
          <s id="id000504"><arrow.to.target n="marg81"/></s>
        </p>
        <p type="margin">
          <s id="id000505"><margin.target id="marg81"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000506">Sit mobile a cui partes &longs;ubiaceant direct&aelig; b, &amp; &longs;it graue. </s>
          <s id="id000507">Et pa&shy;<lb/>tet ne diuidatur b re&longs;i&longs;tere, cum autem &longs;uperauerit, partes b de&longs;cen&shy;<lb/>dunt ante a, &amp; trahunt partes c &amp; d adh&etail;rentes &longs;ecum, atque ita e c d f <lb/><figure id="id.015.01.045.1.jpg" xlink:href="015/01/045/1.jpg"/><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g &amp; h cum a tran&longs;it in b, ne detur uacuum, tran&shy;<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio<lb/>re impetu inferius.</s>
        </p>
        <p type="main">
          <s id="id000508"><arrow.to.target n="marg82"/></s>
        </p>
        <p type="margin">
          <s id="id000509"><margin.target id="marg82"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000510">Ex quo patet, quod in omni motu naturali, <lb/>uel uiolento fit augumentum uelocitatis ab initio &longs;altem u&longs;que <lb/>ad aliquid.</s>
        </p>
        <p type="main">
          <s id="id000511"><arrow.to.target n="marg83"/></s>
        </p>
        <p type="margin">
          <s id="id000512"><margin.target id="marg83"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000513">Et ide&ograve; etiam bellic&aelig; machin&aelig; cuiu&longs;cunque generis certam exi&shy;<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s>
        </p>
        <p type="main">
          <s id="id000514">Propo&longs;itio trige&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id000515">Omnis motus naturalis in &aelig;quali medio ualidior e&longs;t in fine, <lb/>qu&agrave;m in principio: uiolentus contr&agrave;.</s>
        </p>
        <p type="main">
          <s id="id000516"><arrow.to.target n="marg84"/></s>
        </p>
        <p type="margin">
          <s id="id000517"><margin.target id="marg84"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000518">C&ugrave;m enim ex pr&aelig;cedenti augeantur &longs;emper ob medium, &amp; cau&shy;<lb/>&longs;a, qu&aelig; mouet, &longs;it perpetua, &amp; &agrave; principio &aelig;terno, quod per dict&aelig; <lb/>&aelig;qualiter mouet, igitur motus ille fiet uelocior in fine qu&agrave;m in alia <lb/>parte temporis. </s>
          <s id="id000519">In uiolento autem, c&ugrave;m perueniat ad finem de&longs;init </s>
        </p>
        <p type="main">
          <s id="id000520"><arrow.to.target n="marg85"/><lb/>uis illa nece&longs;&longs;ari&ograve;, qu&aelig; mouet, &amp; &longs;uperatur &agrave; ui naturali, qu&aelig; mo&shy;<lb/>uet in contrarium, igitur antequam ce&longs;&longs;et motus fiet tardi&longs;simus <lb/>in fine.</s>
        </p>
        <p type="margin">
          <s id="id000521"><margin.target id="marg85"/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000522">Ex quo patet, qu&ograve;d motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe&shy;<lb/><arrow.to.target n="marg86"/><lb/>cie, ut c&ugrave;m quis iacit lapidem &egrave; turri: uel ex occulto naturali, &amp; uio&shy;<lb/>lento manife&longs;to: uelut c&ugrave;m quis iacit lapidem, &amp; de&longs;cendit po&longs;t mo<lb/><figure id="id.015.01.045.2.jpg" xlink:href="015/01/045/2.jpg"/><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, &amp; naturali oc&shy;<lb/>culto: uel ratione medij, &amp; hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me&shy;<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis, 


<pb pagenum="27" xlink:href="015/01/046.jpg"/>&amp; medij ad mobile, ac medium, quod re&longs;i&longs;tit. </s>
          <s id="id000523">Quarto ex motibus <lb/>imperfectis natura &longs;ua, &amp; non e&longs;t uera mi&longs;tio, &amp; hoc apparet in mo&shy;<lb/>tibus uoluntarijs animalium, qui non &longs;unt neque &aelig;quales, neque perfe <lb/>ct&egrave; circa medium: &longs;ed &longs;unt potius &longs;imiles uoluntarijs. </s>
          <s id="id000524">Et ideo de&shy;<lb/>mon&longs;trationes ill&aelig; Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s>
        </p>
        <p type="margin">
          <s id="id000525"><margin.target id="marg86"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000526">Propo&longs;itio trige&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id000527">Omne mobile naturaliter motum, &longs;eu uiolenter uelocius moue&shy;<lb/>tur in medio rariore, qu&agrave;m den&longs;iore. </s>
          <s id="id000528">Maior quoque e&longs;t proportio fi&shy;<lb/>nis motus in corpore rariore ad finem motus in corpore den&longs;iore, <lb/>qu&agrave;m principij. </s>
          <s id="id000529">In uiolento autem celeri&ugrave;s perueniet ad finem mo<lb/>tus in corpore den&longs;iore.</s>
        </p>
        <figure id="id.015.01.046.1.jpg" xlink:href="015/01/046/1.jpg"/>
        <p type="main">
          <s id="id000530">A mobile moueatur in b medio rariore, &amp; in c den&longs;io&shy;<lb/><arrow.to.target n="marg87"/><lb/>re, igitur b minus re&longs;i&longs;tit, qu&agrave;m c &amp; magis adiuuat, quia <lb/>ueloci&ugrave;s mouetur: igitur duplici de cau&longs;a a mouebitur <lb/>ueloci&ugrave;s in b qu&agrave;m in c: &amp; quia per corrolarium trige&longs;i&shy;<lb/>m&aelig;, &amp; pr&aelig;cedentis proportio finis &lpar;ubi &aelig;qualiter moueantur&rpar; ad <lb/>&longs;ua principia maior erit in d, qu&agrave;m in e: ergo per <expan abbr="dem&otilde;&longs;trata">demon&longs;trata</expan> &agrave; Cam <lb/>pano po&longs;ita d prima, b &longs;ecunda, e tertia, c quarta, maior erit propor&shy;<lb/>tio d ad e, qu&agrave;m b ad c quod fuit propo&longs;itum in naturali.</s>
        </p>
        <p type="margin">
          <s id="id000531"><margin.target id="marg87"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000532">Propo&longs;itio trige&longs;ima ertia.</s>
        </p>
        <p type="main">
          <s id="id000533">Omnia duo mobilia &aelig;qualis undique magnitudinis, qu&aelig; &aelig;quali <lb/>in tempore &aelig;qualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me&shy;<lb/>dijs, nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"/></s>
        </p>
        <p type="margin">
          <s id="id000534"><margin.target id="marg88"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000535">Sint duo mobilia a &amp; b magnitudine, &amp; forma omnino paria, <lb/>&amp; &longs;int media c &amp; d, exempli gratia: &amp; pertran&longs;eant &aelig;quale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei qu&aelig; e&longs;t raritatis c ad raritatem d. </s>
          <s id="id000536">Quia <lb/>enim feruntur &aelig;qualiter, nam in &aelig;quali tem&shy;<lb/><figure id="id.015.01.046.2.jpg" xlink:href="015/01/046/2.jpg"/><lb/>pore, &longs;eu eodem &aelig;qualia &longs;patia pertran&longs;e&shy;<lb/>unt, erit proportio potenti&aelig; a cum &longs;uo auxi&shy;<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au&shy;<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi <lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari&shy;<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten&shy;<lb/>do, tum adiuuando motum a, igitur proportio differenti&aelig; motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t &aelig;qua&shy;<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam &longs;extam &longs;exti Ele&shy;<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>qu&aelig; e&longs;t raritatis c ad raritatem d.</s>
        </p>
        <pb pagenum="28" xlink:href="015/01/047.jpg"/>
        <p type="head">
          <s id="id000537">SCHOLIVM PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id000538">Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur c raritas quatuor, d unum, a pondus duodecim libra&shy;<lb/><figure id="id.015.01.047.1.jpg" xlink:href="015/01/047/1.jpg"/><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, &amp; effi&shy;<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua&shy;<lb/>draginta octo, tota igitur proportio, qua mo&shy;<lb/>uebitur a in c, erit centum nonaginta duorum, &amp; hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="p&otilde;dus">pondus</expan> b centum nonaginta duo. </s>
          <s id="id000539">Pro&shy;<lb/>portio igitur b ad a e&longs;t &longs;ex de cupla, &amp; h&aelig;c e&longs;t duplicata quadrupl&aelig; <lb/>raritatis c ad raritatem d.</s>
        </p>
        <p type="main">
          <s id="id000540">Qu&ograve;d &longs;i quis neget tantundem augere c actionem a, quanto mi&shy;<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, &amp; &longs;it proportio b ad a dupli&shy;<lb/>cata ip&longs;i f, dico fe&longs;&longs;e proportionem c ad d, nam proportio b ad a <lb/>e&longs;t uelut actionis c ad d per decimam &longs;extam &longs;exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, &longs;ed ex <lb/>fin &longs;e fit proportio b ad a ex diffinitione proportionis duplicat&aelig;. <lb/></s>
          <s id="id000541">Sed ex duabus proportionibus a ad c, &amp; actionis ex c ad a produ&shy;<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptim&atilde;">decimam &longs;eptimam</expan> &longs;exti Elemento&shy;<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, &amp; actio&shy;<lb/>nis a in c, quare &aelig;qualis f, igitur proportio b ad a duplicata ei, qu&aelig; <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s>
        </p>
        <p type="head">
          <s id="id000542">SCHOLIVM SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id000543">Si autem media fuerint diuer&longs;arum rationum, ut aqua, &amp; a&euml;r non <lb/>demon&longs;trat argumentum, quia pondera inter &longs;e non &longs;eruant ratio&shy;<lb/>nem. </s>
          <s id="id000544">Nam lignum centum librarum ex &longs;alicis arbore, non magis <lb/>de&longs;cendit, qu&agrave;m lignum libr&aelig; unius. </s>
          <s id="id000545">Ide&ograve; nec in comparatione ad <lb/>medium a&euml;ris.</s>
        </p>
        <p type="main">
          <s id="id000546">Propo&longs;itio trige&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id000547">Proportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t ue&shy;<lb/>lut eiu&longs;dem &longs;uperficiei ad latus, eiu&longs;dem uer&ograve; ad monadem.</s>
        </p>
        <p type="main">
          <s id="id000548"><arrow.to.target n="marg89"/></s>
        </p>
        <p type="margin">
          <s id="id000549"><margin.target id="marg89"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000550">Sit cubus a b c eius quadrata, &longs;uperficies a <lb/><figure id="id.015.01.047.2.jpg" xlink:href="015/01/047/2.jpg"/><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. </s>
          <s id="id000551">Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, &amp; toties <lb/>etiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s>
        </p>
        <p type="main">
          <s id="id000552"><arrow.to.target n="marg90"/><lb/>dis &longs;ecundo Elementorum, &longs;i ergo monas e&longs;t <lb/>in continua proportione, habeo intentum: &longs;i <lb/>non ponatur e media inter a e &amp; d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo &aelig;qualis a b, igitur cum a c, e &amp; d &longs;int analog&aelig;, erunt &amp; a b c, <lb/>a b, &amp; d analog&aelig;, quod fuit demon&longs;trandum.</s>
        </p>
        <pb pagenum="39 &lsqb;=29&rsqb;" xlink:href="015/01/048.jpg"/>
        <p type="margin">
          <s id="id000553"><margin.target id="marg90"/>P<emph type="italics"/>rima ex<emph.end type="italics"/><lb/>C<emph type="italics"/>ampano.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000554">Propo&longs;itio trige&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id000555">Vocum magnitudines excre&longs;cunt in acumine non in grauitate, <lb/>finis autem e&longs;t in utroque extremo, propter hoc minima facta uaria&shy;<lb/>tione in hypate acut&aelig; uix ferunt.<lb/><arrow.to.target n="marg91"/></s>
        </p>
        <p type="margin">
          <s id="id000556"><margin.target id="marg91"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id000557">Quoniam facta uariatione in hypate, qu&aelig; e&longs;t <lb/>in Diapa&longs;on, uel bis D&iacute;apa&longs;on maiore interual&shy;<lb/><figure id="id.015.01.048.1.jpg" xlink:href="015/01/048/1.jpg"/><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in&shy;<lb/>teruallum ex c in d, igitur maior e&longs;t b d, qu&agrave;m a c <lb/>ergo &longs;ingul&aelig; uoces inter b &amp; d magis di&longs;tant, <lb/>qu&agrave;m inter a &amp; c, &amp; quanto magis appropin&shy;<lb/>quant ad d, igitur d maius e&longs;t qu&agrave;m b. </s>
          <s id="id000558">Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b &amp; <lb/>c ip&longs;o a. </s>
          <s id="id000559">O&longs;tenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus &longs;icut acuta ex uehementia. </s>
          <s id="id000560">Motus autem e&longs;t res, quies, <lb/>priuatio.</s>
        </p>
        <p type="main">
          <s id="id000561">Secundum &longs;ic: nam remi&longs;sio mota non feriet aurem, ide&ograve; &longs;onum <lb/>non pariet ob nimiam tarditatem. </s>
          <s id="id000562">At in ueloci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, &amp; non contrahitur ni&longs;i per mu&longs;cu&shy;<lb/>los, igitur contentio illa finem habet. </s>
          <s id="id000563">Si autem non &longs;it nece&longs;&longs;arium <lb/>habere, uel ualde procul po&longs;sit extendi contentio, ut in machinis <lb/>igneis &longs;trepitus fit maximus, nam motus, ut motus e&longs;t etiam in a&euml;re <lb/>nullum finem per &longs;e habet ni&longs;i ratione in&longs;trumenti, ergo &longs;trepitus <lb/>tantus e&longs;&longs;e pote&longs;t, ut ferm&egrave; ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s>
        </p>
        <figure id="id.015.01.048.2.jpg" xlink:href="015/01/048/2.jpg"/>
        <p type="main">
          <s id="id000564">Tertium &longs;ic &longs;it a b humi&shy;<lb/>lior uox, qu&aelig; excre&longs;cat &longs;e&shy;<lb/>mitonio minore &longs;olum in <lb/>c, &amp; &longs;it d e dupla ad ab &longs;e&shy;<lb/>cundum naturam, ut in uo&shy;<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de&shy;<lb/>cimam nonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> f e dupla c b, &amp; in acutis ubi ex&shy;<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, qu&aelig; iam diapa&shy;<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, &amp; ide&ograve; quadrupla b c, &longs;ed in acu&shy;<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fract&aelig; uocis, qui&longs;u&shy;<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo&shy;<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cendunt ad bis diapa<lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor&shy;<lb/>tionem habentis e&longs;t &aelig;quale ferm&egrave; toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, &amp; duo comata, manife&longs;tum e&longs;t, quod ex&shy;<lb/>ten&longs;io illa erit maxima in <expan abbr="c&otilde;parat&iacute;one">comparat&iacute;one</expan> grauioris uo cis a b. </s>
          <s id="id000565">Et ide&ograve; <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga&shy;


<pb pagenum="40 &lsqb;=30&rsqb;" xlink:href="015/01/049.jpg"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, ade&ograve; ut &aelig;gr&egrave; &agrave; pluribus fera&shy;<lb/>tur, &agrave; quibu&longs;dam 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/>breuitate intenderentur, &longs;ed ex con&longs;trictione ligul&aelig;, ut dicunt, &longs;u&shy;<lb/>per a&longs;peram arteriam uox ad diapa&longs;on acueretur addito impetu <lb/>proportione, ut ex con&longs;trictione, &amp; impetu <expan abbr="c&otilde;&longs;urgeret">con&longs;urgeret</expan> dupla pro&shy;<lb/>portio. </s>
          <s id="id000568">Hoc autem manife&longs;t&egrave; experimur in elymis in quibus null&aelig; <lb/>pror&longs;us facta mutatione in&longs;trumenti con&longs;tantibus digitis omni&shy;<lb/>bus pr&aelig;ter pollicem &longs;ini&longs;tr&aelig; uocem exacuimus ad diapa&longs;on, inde <lb/>etiam ad bis diapa&longs;on: &longs;icut declarauimus in commentarijs Epi&shy;<lb/>demiorum.</s>
        </p>
        <p type="main">
          <s id="id000569">Propo&longs;itio trige&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id000570">Si proportio per proportionem minorem &aelig;quali ducatur, pro&shy;<lb/>portio minor producetur. </s>
          <s id="id000571">Vnde manife&longs;tum e&longs;t duas proportio&shy;<lb/>nes minores &aelig;qualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s>
        </p>
        <p type="main">
          <s id="id000572"><arrow.to.target n="marg92"/></s>
        </p>
        <p type="margin">
          <s id="id000573"><margin.target id="marg92"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.049.1.jpg" xlink:href="015/01/049/1.jpg"/>
        <p type="main">
          <s id="id000574">Proportio a b ad c, quali&longs;cunque &longs;it, duca&shy;<lb/>tur in proportionem minorem &aelig;qualitate <lb/>f ad g, dico quod producta proportio erit <lb/>minor ea, qu&aelig; e&longs;t a b ad c fiat d ad a b, ut f <lb/>ad g, et erit per &longs;ecundam huius d ad c pro&shy;<lb/>ducta ex proportionibus a b ad c, &amp; f g. </s>
          <s id="id000575">Itemque per decimam quar&shy;<lb/><arrow.to.target n="marg93"/><lb/>tam quinti <expan abbr="Elementor&utilde;">Elementorum</expan> erit d minor a b, igitur maior a b ad c, qu&agrave;m <lb/>d ad c. igitur qu&agrave;m proportio a b ad c in proportionem f ad g. </s>
          <s id="id000576">Sit <lb/>autem utraque minor &aelig;qualitate ea, qu&aelig; a b ad c, &amp; ea qu&aelig; f ad g, di&shy;<lb/>co productam unaquaque earum e&longs;&longs;e minorem. </s>
          <s id="id000577">Quod enim &lpar;manen<lb/>tibus his, qu&aelig; dicta &longs;unt&rpar; minor &longs;it d ad c, quam a b ad c ex prima <lb/>parte o&longs;ten&longs;um e&longs;t. </s>
          <s id="id000578">Qu&ograve;d uer&ograve; etiam minor &longs;it d ad c, qu&agrave;m d ad <lb/>a b, &amp; ex con&longs;equenti qu&agrave;m f ad g demon&longs;tratur &longs;ic. </s>
          <s id="id000579">Quia enim mi&shy;<lb/>nor e&longs;t a b ad c, &aelig;qualitate erit a b minor c, fiat ergo h &aelig;qualis a b, <lb/>erit ergo d ad h, ut d ad a b per &longs;eptimam quinti Elementorum, at d <lb/>ad c minor qu&agrave;m d ad h per octauam eiu&longs;dem, igitur minor d ad c, <lb/>qu&agrave;m d ad a b, igitur patet propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id000580"><margin.target id="marg93"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000581">Propo&longs;itio trige&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id000582">Si plures homines, quorum nulli per &longs;e nauim mouere po&longs;sint, <lb/>aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, <lb/>erunt ill&aelig; proportiones coniunct&aelig; non product&aelig;.<lb/><arrow.to.target n="marg94"/></s>
        </p>
        <p type="margin">
          <s id="id000583"><margin.target id="marg94"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000584">C&ugrave;m enim primus non po&longs;sit mouere nec &longs;ecundus, erunt pro&shy;<lb/>portiones minores &aelig;qualitate, Ide&ograve; per &longs;ecundam partem pr&aelig;ce&shy;<lb/>dentis multo minus mouerent duo, qu&agrave;m unus. </s>
          <s id="id000585">Et &longs;i quatuor mo&shy;


<pb pagenum="41 &lsqb;=31&rsqb;" xlink:href="015/01/050.jpg"/>uerent unusque per &longs;e mouere non po&longs;&longs;et, adderetur &longs;i proportio <lb/>produceretur, fieret minor, ergo minus mouerent quinque qu&agrave;m <lb/>quatuor ex ij&longs;dem, quod e&longs;t ab&longs;urdum.</s>
        </p>
        <p type="main">
          <s id="id000586">Propo&longs;itio trige&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id000587">Omne corpus tant&ugrave;m re&longs;i&longs;tit motui contrario &longs;uo naturali quan <lb/>cum mouetur occulto motu quie&longs;cendo.</s>
        </p>
        <p type="main">
          <s id="id000588"><arrow.to.target n="marg95"/></s>
        </p>
        <p type="margin">
          <s id="id000589"><margin.target id="marg95"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id000590">Sit a corpus quie&longs;cens in pauimento b, &amp; mouetur in eo occul&shy;</s>
        </p>
        <p type="main">
          <s id="id000591"><arrow.to.target n="marg96"/><lb/>to motu uer&longs;us centrum, ut &longs;upr&agrave; ui&longs;um e&longs;t, contra&shy;<lb/><figure id="id.015.01.050.1.jpg" xlink:href="015/01/050/1.jpg"/><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue&shy;<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. </s>
          <s id="id000592">Manife&longs;tum e&longs;t autem, quod hic <lb/><arrow.to.target n="marg97"/><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/><arrow.to.target n="marg98"/></s>
        </p>
        <p type="margin">
          <s id="id000593"><margin.target id="marg96"/>I<emph type="italics"/>n commen.<emph.end type="italics"/><lb/>26. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000594"><margin.target id="marg97"/>P<emph type="italics"/>er<emph.end type="italics"/> 30. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000595"><margin.target id="marg98"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000596">Ex hoc patet cur naues &amp; currus ab initio tard&egrave; &amp; difficulter mo<lb/>ueantur, ubi moueri c&oelig;perint motus augetur: quoniam re&longs;i&longs;tunt </s>
        </p>
        <p type="main">
          <s id="id000597"><arrow.to.target n="marg99"/><lb/>per motum occultum naturalem qui maximus e&longs;t dum quie&longs;cunt, <lb/>ut etiam docebat philo&longs;ophus in mechanicis, nam motus ille natu&shy;<lb/>ralis e&longs;t, &amp; ide&ograve; contrarius uiolento: Ergo cum iam mouetur uio&shy;<lb/>lenter minus, mouetur naturaliter, igitur minus re&longs;i&longs;tit. </s>
          <s id="id000598">Declarabi&shy;<lb/>tur enim infr&agrave; qu&ograve;d omne quod mouetur duobus motibus tanto <lb/><arrow.to.target n="marg100"/><lb/>minus uno mouetur quanto magis altero.</s>
        </p>
        <p type="margin">
          <s id="id000599"><margin.target id="marg99"/>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 31.</s>
        </p>
        <p type="margin">
          <s id="id000600"><margin.target id="marg100"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>
        </p>
        <p type="main">
          <s id="id000601">Propo&longs;itio trige&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id000602">Ab &aelig;quali aut minore ui, qu&agrave;m &longs;it <expan abbr="impediment&utilde;">impedimentum</expan>, non fit motus.</s>
        </p>
        <p type="main">
          <s id="id000603">Sit a quod re&longs;i&longs;tat, ne &longs;ur&longs;um trahatur per decem, dico, quod <expan abbr="n&otilde;">non</expan> <lb/><arrow.to.target n="marg101"/><lb/>&longs;ur&longs;um trahetur neque &agrave; decem, neque minore: nam &longs;i impedimen&shy;<lb/>tum non e&longs;&longs;et, moueretur infra ut decem, ergo &longs;i traheretur &longs;ur&longs;um <lb/>per decem tantum moueretur &longs;ur&longs;um, <expan abbr="quant&utilde;">quantum</expan> deor&longs;um, ergo quie&shy;<lb/>&longs;ceret. </s>
          <s id="id000604">Si uer&ograve; &agrave; minore moueretur &agrave; maiore ui deor&longs;um, quam &longs;ur&shy;<lb/>&longs;um, ergo deor&longs;um &longs;impliciter non &longs;ur&longs;um.</s>
        </p>
        <p type="margin">
          <s id="id000605"><margin.target id="marg101"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000606">Propo&longs;itio quadrage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id000607">Omne corpus &longs;ph&aelig;ricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, qu&aelig; medium diuidere pote&longs;t.</s>
        </p>
        <figure id="id.015.01.050.2.jpg" xlink:href="015/01/050/2.jpg"/>
        <p type="main">
          <s id="id000608">Sit corpus ad unguem &longs;ph&aelig;ricum a tan&shy;<lb/><arrow.to.target n="marg102"/><lb/>gens planum b in puncto c &lpar;e&longs;t enim hoc <lb/>nece&longs;&longs;arium ex demon&longs;tratis ab Euclide in <lb/>decima&longs;exta Propo&longs;itione tertij Elemento&shy;<lb/>rum&rpar; dico, quod mouebitur &agrave; ui, qu&aelig; pote&longs;t <lb/>&longs;cindere a&euml;rem. </s>
          <s id="id000609">Nam cum non a&longs;cendat, nec <lb/>de&longs;cendat, &longs;ed qua&longs;i in circulo ad centrum <lb/>mundi moueatur, pondus non affert. </s>
          <s id="id000610">Neque<lb/> ratione magnitudinis contactus, cum &longs;it in <lb/>puncto &longs;olo, igitur remanet &longs;olum a&euml;ris impedimentum.


<pb pagenum="42 &lsqb;=32&rsqb;" xlink:href="015/01/051.jpg"/><arrow.to.target n="marg103"/></s>
        </p>
        <p type="margin">
          <s id="id000611"><margin.target id="marg102"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000612"><margin.target id="marg103"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000613">Ex hoc liquet, quod oportet b planum e&longs;&longs;e ex duri&longs;sima mate&shy;<lb/>ria, qu&aelig; nullo modo cedat, aliter tanget plu&longs;qu&agrave;m in puncto.</s>
        </p>
        <p type="main">
          <s id="id000614"><arrow.to.target n="marg104"/></s>
        </p>
        <p type="margin">
          <s id="id000615"><margin.target id="marg104"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000616">Vix fieri pote&longs;t, utin elementaribus &longs;ph&aelig;ra tangat planum in <lb/>puncto. </s>
          <s id="id000617">Vel quia planum non erit exact&egrave; rectum, uel non durum, <lb/>ut pror&longs;us non cedat, uel non ad &aelig;quilibrium po&longs;itum, uel &longs;ph&aelig;ra <lb/>non erit exact&egrave; rotunda.</s>
        </p>
        <p type="main">
          <s id="id000618">Propo&longs;itio quadrage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id000619">Si fuerint du&aelig; quantitates &longs;umaturque totius aggregatum maio&shy;<lb/>ris &amp; minoris, quoties aggregatum minoris, &amp; maioris, erit pro&shy;<lb/>portio confu&longs;a maioris aggregati ad minus, minor qu&agrave;m multipli&shy;<lb/>cis maioris ad multiplex minoris.</s>
        </p>
        <p type="main">
          <s id="id000620"><arrow.to.target n="marg105"/></s>
        </p>
        <p type="margin">
          <s id="id000621"><margin.target id="marg105"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000622">Sint du&aelig; magnitudines a &amp; b, &amp; &longs;it a maior <lb/><figure id="id.015.01.051.1.jpg" xlink:href="015/01/051/1.jpg"/><lb/>b, &amp; &longs;umatur exempli gratia a quater cum b &longs;e&shy;<lb/>mel, &amp; b quater cum a &longs;emel, dico, quod propor<lb/>tio &lpar;quam confu&longs;am e&longs;&longs;e liquet&rpar; aggregati primi ad &longs;ecundum, e&longs;t </s>
        </p>
        <p type="main">
          <s id="id000623"><arrow.to.target n="marg106"/><lb/>minor qu&agrave;m quadrupla. </s>
          <s id="id000624">Con&longs;tat enim quod proportio quadru&shy;<lb/>pli a ad a e&longs;t maior, quam b ad quadruplum b, cum una &longs;it quadru&shy;<lb/>pla, alia &longs;ub quadrupla, igitur per uige&longs;imam &longs;ecundam huius ag&shy;<lb/>gregati quadrupli a cum b &longs;emel, ad quadruplum b cum a &longs;emel mi <lb/><arrow.to.target n="marg107"/><lb/>nor, qu&agrave;m quadrupli a ad a, &amp; maior qu&agrave;m b ad quadruplum b, &amp; <lb/>e&longs;t pro intellectu Archimedis.</s>
        </p>
        <p type="margin">
          <s id="id000625"><margin.target id="marg106"/>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000626"><margin.target id="marg107"/>I<emph type="italics"/>n<emph.end type="italics"/> 2. <emph type="italics"/>lib. 

de<emph.end type="italics"/><lb/>A<emph type="italics"/>tqui pon&shy;<lb/>deran.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 10.</s>
        </p>
        <p type="main">
          <s id="id000627">Propo&longs;itio quadrage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id000628">Trahentium nauim, ut ferentium pondera proportiones in &longs;e in&shy;<lb/>uicem, quomodo ducere oporteat con&longs;iderare.<lb/><arrow.to.target n="marg108"/></s>
        </p>
        <p type="margin">
          <s id="id000629"><margin.target id="marg108"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000630">Hoc quomodo non po&longs;sit fieri &longs;upr&agrave; docuimus, nunc etiam ge&shy;</s>
        </p>
        <p type="main">
          <s id="id000631"><arrow.to.target n="marg109"/><lb/>neraliter dicam, cum con&longs;i&longs;tant h&aelig;c in duobus terminis, productio <lb/>uer&ograve; pr&aelig;&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, &amp; moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="n&otilde;">non</expan> &longs;int quatuor termini, nec tres, &egrave; quibus <lb/>unus &longs;it mouens, &amp; motum proportio non poterit produci. </s>
          <s id="id000632">Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, &amp; <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo&shy;<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo&shy;<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume&shy;<lb/>rus po&longs;&longs;et mouere in duplicata proportione ad unguem &longs;cilicet <lb/>5 1/16 ut e&longs;&longs;et uix hominum collectorum 30 3/8 at nullus e&longs;t numerus ho<lb/>minum qui collectus faciat hunc numerum, nam &longs;ex homines ex&shy;<lb/>plent numerum 27, &amp; &longs;eptem 31 1/2, &amp; ide&ograve; non pote&longs;t duci propor&shy;<lb/>tio. </s>
          <s id="id000633">Et ide&ograve; maximus e&longs;t error dicendo decem homines mouent na <lb/>uim proportione tripla, ergo triginta alij additis illis &longs;imiles robo&shy;<lb/>re mouebunt &agrave; proportione uiginti &longs;eptupla &longs;cilicet ducta nonu&shy;


<pb pagenum="33" xlink:href="015/01/052.jpg"/>pla in triplam. </s>
          <s id="id000634">Sed &longs;umpta proportione alio modo producitur. </s>
          <s id="id000635">Ve<lb/>lut &longs;i dicam, homines decem mouent nauim, aut <expan abbr="fer&utilde;t">ferunt</expan> pondus pro&shy;<lb/>portione tripla, igitur quadraginta homines idem facient propor&shy;<lb/>tione duodecupla &longs;cilicet quadrupla in triplam ducta. </s>
          <s id="id000636">Cum ergo <lb/>addo triginta homines, qui mouent in proportione nonupla, non <lb/>oportet ducere nonuplam in triplam, &longs;ed totum numerum accipe&shy;<lb/>re, &amp; quam proportionem habet ad partem, tandem habet uis mo&shy;<lb/>uens ad uim <expan abbr="mou&etilde;tem">mouentem</expan>. </s>
          <s id="id000637">Vnde &longs;i duo moueant in proportione &longs;ex&shy;<lb/>quialtera, &amp; &longs;ex in proportione quadrupla cum dimidia, &amp; iungan <lb/>tur, ut fiant octo, non oportebit ducere &longs;exquialteram, in quadru&shy;<lb/>plam &longs;exquialteram, &longs;ed cum octo ad duo &longs;it in proportione qua&shy;<lb/>drupla, &longs;umemus quadruplam ad &longs;exquialteram, qu&etail; erit &longs;excupla, <lb/>&amp; octo mouebunt, aut pondus gerentin proportione &longs;excupla.</s>
        </p>
        <p type="margin">
          <s id="id000638"><margin.target id="marg109"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s>
        </p>
        <p type="main">
          <s id="id000639">Propo&longs;itio quadrage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id000640">Productionem ad additionem retrahere.<lb/><arrow.to.target n="marg110"/></s>
        </p>
        <p type="margin">
          <s id="id000641"><margin.target id="marg110"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.052.1.jpg" xlink:href="015/01/052/1.jpg"/>
        <p type="main">
          <s id="id000642">Sit proportio a ad b dupla pote&longs;tate li&shy;<lb/>cet &longs;int quinque homines, &amp; &longs;int quindecim <lb/>homines c, &amp; habebunt ad b &longs;excuplam <lb/>proportionem per pr&aelig;cedentem. </s>
          <s id="id000643">Iuncta <lb/>ergo a, &amp; c per octauam huius <expan abbr="moueb&utilde;t">mouebunt</expan> <lb/>b proportione octupla, dico, quod &longs;i du&shy;<lb/>xeris <expan abbr="proportion&etilde;">proportionem</expan> c ad a plus uno. </s>
          <s id="id000644">i. </s>
          <s id="id000645">qua&shy;<lb/>druplam in proportionem a ad b, qu&aelig; e&longs;t dupla, proueniet eadem <lb/>octupla. </s>
          <s id="id000646">Nam quia in coniunctione &longs;ufficit iungere c cum a, &amp; &longs;u&shy;<lb/>mitur &longs;ecundum proportionem a ad b, igitur cum proportio a ad <lb/>b comparata ad proportionem c &amp; a ad b &longs;it, &longs;icut proportio c &amp; a <lb/>ad a, &amp; proportio c &amp; a ad a &longs;it, &longs;icut proportio c ad a, &amp; a ad a, &amp; <lb/>proportio a ad a habet rationem unius, igitur proportio aggregati <lb/>c a ad b e&longs;t producta ex proportione c ad a plus monade in propor<lb/>tionem a ad b, quod erat demon&longs;trandum.</s>
        </p>
        <p type="main">
          <s id="id000647">Propo&longs;itio quadrage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id000648">Si fuerit proportio motoris ad id, quod e&longs;t maximum non mo&shy;<lb/>uens &amp; &longs;patium, &amp; tempus, nota erit etiam reliquorum nota.</s>
        </p>
        <p type="main">
          <s id="id000649">S&aelig;pe contingit, ut quinque homines moueant nauim, &amp; &longs;patium <lb/>ad tempus notum, &amp; etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. </s>
          <s id="id000650">Sit ergo a numerus hominum, b na&shy;<lb/><figure id="id.015.01.052.2.jpg" xlink:href="015/01/052/2.jpg"/><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, &amp; g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, &amp; h &longs;patium, dico, quod erit f motor, &longs;eu numerus 


<pb pagenum="34" xlink:href="015/01/053.jpg"/>hominum notus. </s>
          <s id="id000651">Quoniam ergo notum e&longs;t a &amp; c, quia e&longs;t &aelig;quale <lb/>b, igitur proportio a ad b nota e&longs;t: &longs;ed iuxta illam a mouet b in d <lb/>tempore per e &longs;patium, igitur per pr&aelig;cedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. </s>
          <s id="id000652">Sed per eadem ut temporis d ad &longs;patium illud, <lb/>ita g ad h, ergo cum nota &longs;int d e f g erit etiam h, &amp; ita conuertendo.</s>
        </p>
        <p type="main">
          <s id="id000653">Propo&longs;itio quadrage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id000654">Rationem &longs;tater&aelig; o&longs;tendere.<lb/><arrow.to.target n="marg111"/></s>
        </p>
        <p type="margin">
          <s id="id000655"><margin.target id="marg111"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000656">Archimedes nititur huic fundamento, quod pondera, qu&aelig; pro&shy;<lb/>portionem mutuam habent, ut di&longs;tanti&aelig; &agrave; libella a, qu&aelig; &longs;u&longs;pen&shy;<lb/>duntur, &aelig;qualiter ponderant, &longs;it ergo libella a b, &amp; &longs;u&longs;pen&longs;a in a cen<lb/>trum mundi c, ad quod dirigitur pondus, &amp; liquet, quod ip&longs;um <lb/>non &longs;e inclinabit ex uige&longs;ima tertia propo&longs;itione. </s>
          <s id="id000657">Si ergo ponantur <lb/>lo co line&aelig; b d in e &amp; f, &amp; &longs;it proportio e b <lb/><figure id="id.015.01.053.1.jpg" xlink:href="015/01/053/1.jpg"/><lb/>ad b f, ut g ad h, dico, qu&ograve;d erit &aelig;quili&shy;<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et &verbar; &longs;u&longs;pen&longs;um h, moueretur in re&shy;<lb/>cta e h per eandem, quia ergo retinetur, mo&shy;<lb/>uetur per obliquam h k, &amp; &longs;umatur in pro&shy;<lb/>pin quum punctum in b e, &amp; n in &aelig;quali di&shy;<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon&shy;<lb/>dere h, &amp; in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, &amp; ita in b erit pen&egrave; infinita: quia quanta ui ex&shy;<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor&shy;<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o &aelig;qualis m p, &amp; n b, &amp; b m &aelig;quales, ut uer&ograve; g ad <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/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/></s>
          <s id="id000658">nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s>
        </p>
        <p type="main">
          <s id="id000659"><arrow.to.target n="marg112"/><lb/>ad n o, ita h ad m p, &longs;ed m p &amp; n o &longs;unt &aelig;quales, ergo tanta e&longs;t uis g <lb/>in f, quanta h in e.<lb/><arrow.to.target n="marg113"/></s>
        </p>
        <p type="margin">
          <s id="id000660"><margin.target id="marg112"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000661"><margin.target id="marg113"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000662">Ex quo patet, quod hypomochlion moueretur infinita ui, &longs;i po&longs;&shy;<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ide&ograve; pote&longs;t <lb/>aliqua ui retineri.</s>
        </p>
        <p type="main">
          <s id="id000663"><arrow.to.target n="marg114"/></s>
        </p>
        <p type="margin">
          <s id="id000664"><margin.target id="marg114"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000665">Et &longs;i quis po&longs;&longs;et capere ha&longs;tam in extremo puncto, non po&longs;&longs;et <lb/>eam mouere, etiam quod haberet robur infinitum, quia ab &aelig;quali <lb/>non fit motus per trige&longs;imamnonam propo&longs;itionem.</s>
        </p>
        <p type="main">
          <s id="id000666"><arrow.to.target n="marg115"/></s>
        </p>
        <p type="margin">
          <s id="id000667"><margin.target id="marg115"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id000668">Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu&shy;


<pb pagenum="35" xlink:href="015/01/054.jpg"/>pit ad centrum peruenire, &amp; pondus ei appen&longs;um non prohi&shy;<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede&shy;<lb/>re ex directo centri mundi: &amp; ut grauat hypomochlion faciens im&shy;<lb/>pre&longs;sionem.</s>
        </p>
        <p type="main">
          <s id="id000669"><arrow.to.target n="marg116"/></s>
        </p>
        <p type="margin">
          <s id="id000670"><margin.target id="marg116"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id000671">Et &longs;i terra tota e&longs;&longs;et appen&longs;a polo, moueretur magna ui: quoni&shy;<lb/>am uis eadem e&longs;t in polo, qu&aelig; in circulo toto &aelig;quinoctij.</s>
        </p>
        <p type="main">
          <s id="id000672"><arrow.to.target n="marg117"/></s>
        </p>
        <p type="margin">
          <s id="id000673"><margin.target id="marg117"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 5.</s>
        </p>
        <p type="main">
          <s id="id000674">Et rota, quanto uelocius mouetur in ambitu, tanto mi<lb/>norem habet uim: &longs;ed propter a&euml;rem, qui &longs;ecum circum&shy;<lb/><figure id="id.015.01.054.1.jpg" xlink:href="015/01/054/1.jpg"/><lb/>fertur, mouetur magno impetu, &amp; magnas facit l&aelig;&longs;iones. <lb/></s>
          <s id="id000675">Ide&ograve; hoc in cono non accidit.</s>
        </p>
        <p type="main">
          <s id="id000676"><arrow.to.target n="marg118"/></s>
        </p>
        <p type="margin">
          <s id="id000677"><margin.target id="marg118"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 6.</s>
        </p>
        <p type="main">
          <s id="id000678">Ex quo patet ratio eleuandi pondera magna per tra&shy;<lb/>bem, ut &agrave; latere uides.</s>
        </p>
        <p type="main">
          <s id="id000679">Propo&longs;itio quadrage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id000680">An &longs;it aliqua proportio, &amp; qualis inter animam, &amp; ui&shy;<lb/>tas, &amp; &longs;ua corpora con&longs;iderare.</s>
        </p>
        <p type="main">
          <s id="id000681"><arrow.to.target n="marg119"/></s>
        </p>
        <p type="margin">
          <s id="id000682"><margin.target id="marg119"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000683">Declarauimus motum c&oelig;li e&longs;&longs;e uoluntarium, ob&longs;equente c&oelig;&shy;<lb/>lo per uirtutem in eo infu&longs;am. </s>
          <s id="id000684">In animalibus autem, &amp; pr&aelig;cipu&egrave; <lb/>in homine notius e&longs;t hoc experientibus nobis in ip&longs;is: &longs;ed motus <lb/>hic, ut dixi &longs;upra, mi&longs;tus e&longs;t, ille uer&ograve; c&oelig;le&longs;tis ignotior e&longs;t. </s>
          <s id="id000685">Certum </s>
        </p>
        <p type="main">
          <s id="id000686"><arrow.to.target n="marg120"/><lb/>tamen e&longs;t plen&egrave; ob&longs;equi c&oelig;lum uit&aelig;, nec pror&longs;us repugnare. </s>
          <s id="id000687">So&shy;<lb/>let Ari&longs;toteli imponi, qu&ograve;d &longs;i adderetur a&longs;trum c&oelig;lo, qu&ograve;d c&oelig;lum <lb/>aut quie&longs;ceret, aut tardius moueretur: quod e&longs;t, ac &longs;i diceremus, <lb/>qu&ograve;d homo paruus &longs;i fieret maior, non e&longs;&longs;et ade&ograve; agilis, tanquam <lb/>motus ille e&longs;&longs;et ab externa cau&longs;a. </s>
          <s id="id000688">Im&ograve; perinde e&longs;&longs;et, ac &longs;i quis dice&shy;<lb/>ret, quod lapides magni minus uelociter de&longs;cenderent, quam par&shy;<lb/>ui. </s>
          <s id="id000689">Quin potius ut lapis magnus ueloci&ugrave;s mouetur: qu&agrave;m par&shy;<lb/>uus naturali motu, &amp; tardius pr&aelig;ternaturali, ita c&oelig;lum motu uo&shy;<lb/>luntario, &longs;i ita dici po&longs;&longs;et &aelig;qualius &amp; maiore cum efficacia, quan&shy;<lb/>to den&longs;ius. </s>
          <s id="id000690">Et ita &longs;i Ari&longs;toteles illud dixi&longs;&longs;et, o&longs;tendi&longs;&longs;et magnam <lb/>imperitiam. </s>
          <s id="id000691">Ide&ograve; quale iudicium debemus facere de Alexandro, &amp; <lb/><arrow.to.target n="marg121"/><lb/>Aueroe, qui hoc ei tribuunt. </s>
          <s id="id000692"><expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip&shy;<lb/>piam. </s>
          <s id="id000693">De Animalibus for&longs;an po&longs;&longs;et hoc dici, <expan abbr="quoni&atilde;">quoniam</expan>, ut &longs;upr&agrave; dixi&shy;<lb/>mus, motus ille mi&longs;tus e&longs;t. </s>
          <s id="id000694">Remanet ergo difficultas, <expan abbr="quoni&atilde;">quoniam</expan> &longs;i mo&shy;<lb/>tus i&longs;te non &agrave; proportione fit, quare non e&longs;t infinitus? </s>
          <s id="id000695">&amp; dico quae in <lb/>animalibus tres &longs;unt cau&longs;&aelig;, una, quia e&longs;t mi&longs;tus, &amp; habet repugnan<lb/>tiam: &longs;ecunda, quia e&longs;t de loco ad locum, motus autem c&oelig;li e&longs;t in lo<lb/>co: tertia e&longs;t communis etiam c&oelig;lo, et e&longs;t, <expan abbr="quoni&atilde;">quoniam</expan> non e&longs;t ratio finis. <lb/></s>
          <s id="id000696">Natura enim diuina non appetit mouere <expan abbr="t&atilde;">tam</expan> celeriter. </s>
          <s id="id000697">Quid e&longs;t ergo <lb/>proportio, <expan abbr="c&utilde;">cum</expan> &longs;it <expan abbr="ultim&utilde;">ultimum</expan> uoluntatis uit&etail;, ut obtemperet prim&aelig; cau&longs;&aelig;, <lb/>ideo illud e&longs;t <expan abbr="ultim&utilde;">ultimum</expan>, &qring; mouet. </s>
          <s id="id000698">E&longs;t <expan abbr="a&utilde;t">aut</expan> idem uelle, &amp; po&longs;&longs;e. </s>
          <s id="id000699">In natura 


<pb pagenum="46 &lsqb;=36&rsqb;" xlink:href="015/01/055.jpg"/>enim c&oelig;li e&longs;t ille appetitus, cuius principium e&longs;t uita: &amp; e&iacute;us uolun<lb/>tatis bonum ip&longs;um. </s>
          <s id="id000700">Et ideo h&aelig;c proportio <expan abbr="n&otilde;">non</expan> diuiditur. </s>
          <s id="id000701">In anima&shy;<lb/>libus autem non e&longs;t uis illa ni&longs;i, cum proportione, quia primum in&shy;<lb/>&longs;trumentum, quod recipit, &amp; e&longs;t &longs;piritus uim habet determinatam, <lb/>cum &longs;it uirtus in materia: ideo <expan abbr="n&otilde;">non</expan> mouet ni&longs;i cum certa proportio&shy;<lb/>ne, uelut lumen in medio in &longs;e non habet proportionem ni&longs;i ad lu&shy;<lb/>cem, &longs;ed ut e&longs;t in illo, pote&longs;t e&longs;&longs;e remi&longs;&longs;um, <expan abbr="ob&longs;cur&utilde;">ob&longs;curum</expan> &amp; hebes. </s>
          <s id="id000702">Qu&aelig;&shy;<lb/>ritur ergo quantitas illius? </s>
          <s id="id000703">&longs;i dicas, qu&ograve;d e&longs;t &agrave; luce: qu&aelig;ro quanti&shy;<lb/>tas lucis, unde &longs;it? </s>
          <s id="id000704">for&longs;an dicendum, qu&ograve;d uelutin motibus, quanto <lb/>den&longs;iora &longs;unt corpora tanto <expan abbr="mouen&ttilde;">mouentur</expan> maiore nixu, &amp; robore. </s>
          <s id="id000705">Nam <lb/>calor in materia augetur iuxta illius quantitatem: idem in luce, &amp; <lb/>reliquis. </s>
          <s id="id000706">Dico ergo proportionem e&longs;&longs;e infinitam: nam &longs;i corpus e&longs;&shy;<lb/>&longs;et infinitum &amp; optim&egrave; di&longs;po&longs;itum infinita ui moueretur &amp; agili&shy;<lb/>tate, ut enim maius e&longs;t eo maiores uires habet.</s>
        </p>
        <p type="margin">
          <s id="id000707"><margin.target id="marg120"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 27.</s>
        </p>
        <p type="margin">
          <s id="id000708"><margin.target id="marg121"/>T<emph type="italics"/>ex.<emph.end type="italics"/> 71. <lb/>2. <emph type="italics"/>de<emph.end type="italics"/> C<emph type="italics"/>&oelig;lo.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000709">Propo&longs;itio quadrage&longs;ima&longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id000710">Si duo mobilia &aelig;qualiter in eodem circulo iuxta proprios mo&shy;<lb/>tus moueantur, productum temporis circuituum inuicem erit &aelig;&shy;<lb/>quale producto differenti&aelig; temporum circuitus duct&aelig; in tempus <lb/>coniunctionis prim&aelig;.<lb/><arrow.to.target n="marg122"/></s>
        </p>
        <p type="margin">
          <s id="id000711"><margin.target id="marg122"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000712">Sint duo mobilia a &amp; b in eodem pun&shy;<lb/><figure id="id.015.01.055.1.jpg" xlink:href="015/01/055/1.jpg"/><lb/>cto, qu&aelig; &aelig;qualiter uer&longs;us eandem partem <lb/>moueantur &aelig;qualibus in temporibus, inui<lb/>cem tamen in &aelig;qualiter, ita quod a in f &amp; b <lb/>in g temporibus ab&longs;oluant circulum, &amp; ho <lb/>rum differentia &longs;it h. </s>
          <s id="id000713">Dum itaque a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif<lb/>ferentia, qu&aelig; &longs;uperanda e&longs;t, &amp; proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli&shy;<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu&shy;<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/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem &amp; cum circulo &amp; b c &amp; c d b diffe&shy;<lb/>rentia, &amp; &longs;it &longs; productum ex f in g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis prim&aelig;, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a &amp; a, b in reditu ex con<lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;&shy;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, qu&ograve;d a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a &amp; b &longs;int in d ultra perfectas circulationes uterque eorum pertran<lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir&shy;<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un&shy;


<pb pagenum="37" xlink:href="015/01/056.jpg"/>guem. </s>
          <s id="id000714">Hoch declarato ponatur m spatium compositum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, &amp; erit ex de&shy;<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, &amp; &longs;it p &longs;pa&shy;<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, &amp; <lb/>&longs;it q differentia o, &amp; p qu&etail; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m &amp; b, n, erit m ad n, ut a ad b, &amp; eadem ratio&shy;<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o &longs;it differentia m &amp; n, &amp; q, differentia o &amp; p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, &amp; ita circulus e&longs;t ana<lb/>logus inter &longs;patium pertran&longs;itum &agrave; motore uelociori, &amp; inter diffe&shy;<lb/>rentiam &longs;patij qu&aelig; accidit, dum uelocior motor pertran&longs;it circu&shy;<lb/>lum, id e&longs;t qu&ograve;d circulus a c d e&longs;t analogus inter c d l b, &amp; circulos <lb/>pertran&longs;itos a b a cum portione b d. </s>
          <s id="id000715">Reuertor igitur ad propo&longs;i&shy;<lb/>tum, cum &longs;it m ad o, ut o ad q, &amp; m ad o, ut n ad p, ex &longs;exta decima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;exta decima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, &amp; diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon&shy;<lb/>&longs;trandum.</s>
        </p>
        <p type="main">
          <s id="id000716"><arrow.to.target n="marg123"/></s>
        </p>
        <p type="margin">
          <s id="id000717"><margin.target id="marg123"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000718">Ex hoc patet, quod proportio temporis coniunctionis ad tem&shy;<lb/>pus tardioris motus circuitionis e&longs;t ueluti temporis circuitus uelo<lb/>cioris motoris ad differentiam temporis motus tardioris, &amp; uelo&shy;<lb/>cioris motoris in uno circuitu.</s>
        </p>
        <p type="main">
          <s id="id000719">Propo&longs;itio quadrage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id000720">Si tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commen&longs;is illa tria mobi&shy;<lb/>lia denu&ograve; coniungentur in tempore producto ex denominatore di <lb/>ui&longs;ionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s>
        </p>
        <p type="main">
          <s id="id000721"><arrow.to.target n="marg124"/></s>
        </p>
        <p type="margin">
          <s id="id000722"><margin.target id="marg124"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000723">Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. </s>
          <s id="id000724">Dico quod primum redibunt in numero producto ex <lb/>&longs;eptem quinque &amp; duobus, qui &longs;unt numeri primi, &amp; erit ille nume&shy;<lb/>rus &longs;eptuaginta annorum. </s>
          <s id="id000725">Nam in &longs;eptuaginta annis a perficiet tri&shy;<lb/>ginta quinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redib&utilde;t">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. </s>
          <s id="id000726">O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;it, ut in triginta quinque annis igitur b &amp; c per&shy;<lb/>ficient perfectos circuitus, ergo <expan abbr="redib&utilde;t">redibunt</expan> ad idem punctum, a autem <lb/>non redibit, quoniam eius circuitus non numerat trigintaquinque <lb/>aliter non fui&longs;&longs;et &longs;eptuaginta minimus numeratus ab a b c, cum 


<pb pagenum="38" xlink:href="015/01/057.jpg"/>ergo iam &longs;upponatur numerari a b &amp; c non numerabitur a b a, er&shy;<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b &amp; c. </s>
          <s id="id000727">Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/><figure id="id.015.01.057.1.jpg" xlink:href="015/01/057/1.jpg"/><lb/>quo illorum temporum, auferantur perfe&shy;<lb/>ct&aelig; circulationes, &amp; <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, du&aelig; quint&aelig; ex b, tres &longs;eptim&aelig; ex c, igi&shy;<lb/>tur oportebit ut h&aelig; portiones &longs;int &aelig;qua&shy;<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="c&otilde;ueniant">conueniant</expan>, ergo 1/2 &amp; 2/5 &amp; 3/7 &aelig;qui&shy;<lb/>ualebunt, quare proportio 7 ad 3 &amp; 5 ad 2 <lb/>&amp; 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 &amp; 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, &amp; duo non &longs;unt <lb/>in eadem proportione. </s>
          <s id="id000728">Rur&longs;us dicantur conuenire in annis qua&shy;</s>
        </p>
        <p type="main">
          <s id="id000729"><arrow.to.target n="marg125"/><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite&shy;<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t &aelig;quale uni, ergo duo erunt &aelig;qualia &longs;eptem. </s>
          <s id="id000730">Rur&longs;us dica&shy;<lb/>mus, quod in tempore annorum &lt;02&gt; quadrata decem &longs;imiliter aufe&shy;<lb/>ram integras reuolutiones, quas potero, &amp; erunt &lt;02&gt; 2 1/2 m: 1, &amp; &lt;02&gt; 2/5 &amp; <lb/>&lt;02&gt; 10/49 &aelig;qualia. </s>
          <s id="id000731">Hic uides infinita &longs;equi in conuenientia, qu&aelig; longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et &aelig;quale quinque, &amp; proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. </s>
          <s id="id000732">Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/><arrow.to.target n="marg126"/></s>
        </p>
        <p type="margin">
          <s id="id000733"><margin.target id="marg125"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 23</s>
        </p>
        <p type="margin">
          <s id="id000734"><margin.target id="marg126"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id000735">Ex hoc &longs;equitur, qu&ograve;d nullibi conuenient pr&aelig;terqu&agrave;m in eo&shy;<lb/>dem puncto, &longs;cilicet in quo ab initio coniuncti fuerunt.</s>
        </p>
        <p type="main">
          <s id="id000736"><arrow.to.target n="marg127"/></s>
        </p>
        <p type="margin">
          <s id="id000737"><margin.target id="marg127"/>C<emph type="italics"/>or<emph.end type="italics"/>m. </s>
          <s id="id000738">2.</s>
        </p>
        <p type="main">
          <s id="id000739">Sequitur denuo ex propo&longs;itione ip&longs;a repetita, &amp; primo corrola&shy;<lb/>rio, quod nullibi alibi conuenient qu&agrave;m in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam u&longs;que in &aelig;ternum.</s>
        </p>
        <p type="main">
          <s id="id000740">Sit rur&longs;us ut a circuat in annis duobus cum dimidio, b in tribus <lb/>cum tertia parte, cin quatuor cum quarta parte ducam per &longs;uos <lb/>denominatores, &amp; erit ut a in quinque annis. </s>
          <s id="id000741">b in decem, c in decem&shy;<lb/>&longs;eptem circuant, &amp; redeant ad idem punctum, &amp; quia quin que nu&shy;<lb/>merat decem, &amp; decem, &amp; decem&longs;eptem &longs;unt numeri inuicem pri&shy;<lb/>mi, ducam decem in decem&longs;eptem fiunt centum &longs;eptuaginta. </s>
          <s id="id000742">Con&shy;<lb/>&longs;tat igitur c quadrag&iacute;es, b quinquagies &longs;emel, a &longs;exagies octies cir&shy;<lb/>cumuerti, &amp; redire ad idem punctum: ergo rur&longs;us coibunt po&longs;t tot <lb/>annos in eo, dico modo, quod non ante: nam &longs;i non &longs;it, ut in trigin&shy;<lb/>ta tribus annis. </s>
          <s id="id000743">gratia exempli, aufero <expan abbr="decem&longs;ept&etilde;">decem&longs;eptem</expan>, decem, &amp; quin&shy;<lb/>que, &amp; relinquentur &longs;exdecim tria &amp; tria, &amp; rur&longs;us ex &longs;exdecim tres <pb pagenum="39" xlink:href="015/01/058.jpg"/>circuitus c, &amp; relinquentur 3 3/4 &longs;equetur igitur, ut &longs;it proportio 17 ad <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 &longs;i iam &longs;upponi<lb/>mus 17 &amp; 10 e&longs;&longs;e primos inuicem, ut in &longs;ecunda demon&longs;tratione./&gt;<lb/></s>
          <s id="id000744">Igitur &longs;equuntur eadem corrolaria, qu&aelig; dicta &longs;unt.</s>
        </p>
        <p type="main">
          <s id="id000745">Propo&longs;itio quadrage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id000746">Propo&longs;ito mobilis in circulo circuitus tempore, dataque ratione <lb/>di&longs;tanti&aelig; ab illo mobilis circuitum inuenire, quod ex eodem pun&shy;<lb/>cto di&longs;cedens cum alio mobili in dato puncto conueniat &longs;ub quo&shy;<lb/>cunque numero circuituum tempus quoque coniunctionis.</s>
        </p>
        <p type="main">
          <s id="id000747"><arrow.to.target n="marg128"/></s>
        </p>
        <p type="margin">
          <s id="id000748"><margin.target id="marg128"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.058.1.jpg" xlink:href="015/01/058/1.jpg"/>
        <p type="main">
          <s id="id000749">Sit in circuli peripheria a <expan abbr="p&utilde;ctus">punctus</expan>, qui cir <lb/>cuat &aelig;quali motu &lpar;hoc enim &longs;emper intel&shy;<lb/>ligitur&rpar; in b tempore: &amp; &longs;it datus punctus c <lb/>in quo di&longs;cedens e mobile ex coniunctio&shy;<lb/>ne cum a po&longs;t certos circuitus proprios, <lb/>aut etiam. </s>
          <s id="id000750">&longs;ine ulla circuitione perfecta de&shy;<lb/>beat conuenire. </s>
          <s id="id000751">Volo &longs;cire tempus circui&shy;<lb/>tionis e: &amp; etiam tempus coniunctionis. <lb/></s>
          <s id="id000752">Sit ergo primum ut ab&longs;que circuitione ulla e, a debeat comprehen&shy;<lb/>dere e in c po&longs;t numerum circuitionum ip&longs;ius a, qui &longs;it f. </s>
          <s id="id000753">nam &longs;i a o c <lb/>currit e in prima circuitione ip&longs;ius e, igitur a mouetur uelocius <lb/>qu&agrave;m e, cum ergo debeat attingere ip&longs;um e, nece&longs;&longs;e e&longs;t ut a pertran&shy;<lb/>&longs;eat prius per punctum ex quo di&longs;ce&longs;sit antequam redeat ad con&shy;<lb/>iunctionem e: ergo perficiet &longs;altem unam circuitionem. </s>
          <s id="id000754">Ducemus <lb/>ergo f in b, &amp; fiet g tempus circuitus aut circuituum a, &amp; quia &longs;pa&shy;<lb/>tium a c datum e&longs;t, &longs;it b temporis circuitus a ad h, uelut circuli to&shy;<lb/><arrow.to.target n="marg129"/><lb/>tius ad a c, &amp; iungatur g cum h &amp; fiat k. </s>
          <s id="id000755">Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, &amp; ducatur l in k, &amp; fiat m: dico m e&longs;&longs;e tem&shy;<lb/>pus circuitus e. </s>
          <s id="id000756">Con&longs;tat enim ex &longs;uppo&longs;ito, quod k e&longs;t tempus to&shy;<lb/>tum in quo a peruenit po&longs;t b circuitiones in c, &longs;i ergo e moueretur <lb/>per m tempus totum ex &longs;uppo&longs;ito perficeret circuitum, at quia cir&shy;<lb/>cuitus ad a c, ut monadis ad h, igitur etiam ut l ad monadem, ergo <lb/>proportio circuitus ad a c, ut m ad monadem: ergo &longs;i in m tran&longs;it to <lb/>tum circuitum in monade tran&longs;it a c: &longs;ed monas ducta in k facit k, <lb/>igitur e in tempore k perueniet in c, quod erat demon&longs;trandum. <lb/></s>
          <s id="id000757">Proponatur modo tempus reuolutionum e ip&longs;um d: eodem mo&shy;<lb/><arrow.to.target n="marg130"/><lb/>do agemus ducendo fin b fit g, addatur h &amp; fiat k, diuidatur k per <lb/>aggregatum d &amp; a e, &amp; exeat m, &lpar;idem enim e&longs;t diuidere per aggre&shy;<lb/>gatum d &amp; h, &amp; multiplicare per l&rpar; dico ergo ut in demon&longs;tratione <lb/>priore, quod m e&longs;t tempus circuitus e. </s>
          <s id="id000758">Nam cum k &longs;it tempus, in <lb/>quo a po&longs;t circuitus f peruenit ad c, ergo diui&longs;o ip&longs;o toto tempore 


<pb pagenum="40" xlink:href="015/01/059.jpg"/>per numerum reuolutionum d, &amp; partem reuolutionis exibit tem&shy;<lb/>pus unius reuolutionis.</s>
        </p>
        <p type="margin">
          <s id="id000759"><margin.target id="marg129"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000760"><margin.target id="marg130"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000761">Exemplum primi in re paul&ograve; ob&longs;curiore: &longs;it f 4 &amp; b 2 1/2 &amp; a c 4/5, du<lb/>cemus 4 in 2 1/2 fit 10, adde 4/5 6 quod e&longs;t 2 fit 12, diuide per 4/5 &longs;eu mul&shy;<lb/>tiplica per 5/4 quod idem e&longs;t, fit 15 circuitus e, in quatuor ergo circui&shy;<lb/>tibus, &amp; 4/5 qui &longs;unt duodecim anni perueniet a ad c, &amp; in duodecim <lb/>annis e perueniet ad c, nam 12 &longs;unt 4/5 ip&longs;ius 15. Similiter in &longs;ecundo <lb/>ca&longs;u &longs;it 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/>portionem b qualis a c e&longs;t totius circuitus, id e&longs;t 1/7, e&longs;t autem 1/7 2 1/3, 1/3 <lb/>fient 9 1/3, &longs;imiliter ponatur d 5, &amp; quia a c e&longs;t 1/7 erunt 36/7, diuide ergo <lb/>9 2/3 id e&longs;t 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s>
          <s id="id000762">Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita &longs;eptima parte, qu&aelig; e&longs;t 29/108 fient 2044/108 <lb/>&longs;eu 261/27, &amp; &longs;unt anni 9 18/27 &longs;eu 9 2/3, ergo in tanto tempore a faciet qua&shy;<lb/>tuor circuitus, &amp; &longs;eptimam partem, &amp; e quinque circuitus, &amp; &longs;e&shy;<lb/>ptimam.<lb/><arrow.to.target n="marg131"/></s>
        </p>
        <p type="margin">
          <s id="id000763"><margin.target id="marg131"/>C<emph type="italics"/>om./&gt;<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000764">Ex hoc patet, quod non coniungentur in alio loco, neque alio tem<lb/>pore ante pr&aelig;dictum tempus.</s>
        </p>
        <p type="main">
          <s id="id000765">Propo&longs;itio quinquage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id000766">Omnes circuituum portiones in eiu&longs;dem temporibus <expan abbr="repetun&ttilde;">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 congre&longs;&longs;u <lb/>iungantur in c, in &longs;ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in &longs;exto in h, in &longs;eptimo in k, in octauo in l. </s>
          <s id="id000768">Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cuique<lb/>tempora</expan> &longs;int &aelig;qualia, erunt &amp; circuitus totidem numero, &amp; exce&longs;&shy;<lb/>&longs;us &aelig;quales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"/><lb/>k l. </s>
          <s id="id000769">Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>&amp; portionis fuerit commen&longs;um circulo, &amp; <lb/>ita de b erunt omnia <expan abbr="c&otilde;men&longs;a">commen&longs;a</expan> ad circulum, </s>
        </p>
        <p type="main">
          <s id="id000770"><arrow.to.target n="marg132"/><lb/>&amp; etiam inter &longs;e. </s>
          <s id="id000771">Et &longs;i inter &longs;e aggregata, uel <lb/>portiones erunt, &amp; eodem modo reliqua. <lb/></s>
          <s id="id000772">Et quoniam circuli circulis commen&longs;i &longs;unt: <lb/>&longs;i portiones erunt inuicem commen&longs;&aelig; <expan abbr="er&utilde;t">erunt</expan>, <lb/>&amp; toti circuitus cum partibus commen&longs;i, &amp; <lb/>&longs;i non commen&longs;i, neque erunt inter &longs;e, neque ad circulum. </s>
          <s id="id000773">Et &longs;i totum <lb/>&longs;patium cum circuitibus erit unius generis, erunt duplicata, &amp; tri&shy;<lb/>plicata, &amp; quadruplicata eiu&longs;dem generis: quare cum &longs;patia ip&longs;a <lb/>detractis circuitibus uelut rhete habeant naturam reci&longs;i, &amp; &longs;patia <lb/>ip&longs;a tota &longs;int eiu&longs;dem generis, erunt &longs;patia, qu&aelig; relinquuntur eiu&longs;&shy;<lb/>dem generis. </s>
          <s id="id000774">Erunt tamen incommen&longs;a nece&longs;&longs;ari&ograve;, &longs;i partes fuerint <lb/>incommen&longs;&aelig; toti. </s>
          <s id="id000775">Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="eti&atilde;">etiam</expan> e&longs;t incommen&longs;a toti circulo: &amp; <expan abbr="eti&atilde;">etiam</expan> a k, &amp; k c. </s>
          <s id="id000776">Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, &amp; k a cum toto circulo &longs;emel e&longs;t commen&shy;


<pb pagenum="41" xlink:href="015/01/060.jpg"/>&longs;a a c, quia multiplex ei. </s>
          <s id="id000777">igitur cum circulus, &amp; a k diuidantur in cir&shy;<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, &amp; circulus &longs;it incommen&longs;us circulo, cum a k erit aggre<lb/></s>
          <s id="id000778">gatum ex circulo, &amp; a k incommen&longs;um ip&longs;i a k, &amp; a k pariter incom<lb/><arrow.to.target n="marg134"/><lb/>men&longs;a circulo. </s>
          <s id="id000779">Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, &amp; <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"/><lb/>erit c k incommen&longs;a a k &amp; a c, &amp; circulo ad dita a k. </s>
          <s id="id000780">Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, &amp; &longs;i <lb/><arrow.to.target n="marg136"/><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra&shy;<lb/>ctis, ut a k &amp; a l reci&longs;a: &amp; a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu<lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi&shy;<lb/>c&aelig; numeri, &longs;eu latera corporum rhete, a k uero &amp; a l, &amp; huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"/></s>
        </p>
        <p type="margin">
          <s id="id000781"><margin.target id="marg132"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/><emph type="italics"/>pr&aelig;cedentis.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000782"><margin.target id="marg133"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <emph type="italics"/>deci <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000783"><margin.target id="marg134"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000784"><margin.target id="marg135"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000785"><margin.target id="marg136"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000786"><margin.target id="marg137"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000787">Ex hoc patet, quod cum circulus po&longs;sit diuidi in infinita gene&shy;</s>
        </p>
        <p type="main">
          <s id="id000788"><arrow.to.target n="marg138"/><lb/>ra quantitatum, qu&aelig; non &longs;unt inuicem commen&longs;&aelig; cumque coniun&shy;<lb/>ctiones h&aelig; &longs;emper in eodem genere maneant, quod infinita pun&shy;<lb/>cta, &amp; infinitis in &longs;peciebus quantitatum remanebunt in quibus a <lb/>&amp; b in perpetuum nunquam conuenient. </s>
          <s id="id000789">Velut &longs;i coniunctio pri&shy;<lb/>ma fiat in &lt;02&gt; cu. </s>
          <s id="id000790">1/2 alicuius circuli, nunquam conuenient, neque in me&shy;<lb/>dietate, neque in quarta parte, nec octaua, nec tertia, nec &longs;exta, nec no&shy;<lb/>na, nec quinta, nec decima, &amp; &longs;ic de &longs;ingulis in genere commen&longs;a&shy;<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/>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&shy;<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; &longs;ic <lb/>de alijs.</s>
        </p>
        <p type="margin">
          <s id="id000794"><margin.target id="marg138"/>P<emph type="italics"/>er penulti&shy;<lb/>mam uige&longs;i&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000795">Propo&longs;itio quinquage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id000796">Operationes dictas exemplo declarare.<lb/><arrow.to.target n="marg139"/></s>
        </p>
        <p type="margin">
          <s id="id000797"><margin.target id="marg139"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000798">Supponamus in circulo pr&aelig;dicto a c &lt;02&gt; 7 con&longs;tat, quod e&longs;&longs;e non <lb/>pote&longs;t, quia &lt;02&gt; 7 e&longs;t maior monade, ideo toto circulo, quare non po<lb/>terit e&longs;&longs;e pars circuli, &longs;ed referetur ad <expan abbr="quantitat&etilde;">quantitatem</expan> certam, uelut quod <lb/>circulus &longs;it 10. &longs;emper ergo diuidemus &lt;02&gt; 7, &longs;eu eam portionem per <lb/>10 quantitatem circuli &amp; exibit &lt;02&gt; 7/100, &amp; h&aelig;c erit portio circuli, &amp; ita <lb/>&longs;i portio &longs;it &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/>ita de alijs.</s>
        </p>
        <p type="main">
          <s id="id000801">Sed cum ex repetitione cre&longs;cat portio illa, donec exuperet mo&shy;<lb/>nadem, aut aliquem quemuis numerum detracta monade aut nu&shy;<lb/>mero circuituum habebit rationem reci&longs;i. </s>
          <s id="id000802">Velut &lt;02&gt; 7/100 quater &longs;um&shy;<lb/>pta efficit &lt;02&gt; 112/100. Et hoc e&longs;t potentia rhete, &longs;ed &longs;i quis auferat mona&shy;<lb/>dem fiet &lt;02&gt; 112/100 m: 1, &amp; hoc e&longs;t reci&longs;um 1, &longs;cilicet 1 p: &lt;02&gt; v: 23/25 m: &lt;02&gt; 28/25, &longs;ed ta<lb/>men uer&egrave; e&longs;t linea media.</s>
        </p>
        <p type="main">
          <s id="id000803">Quod uer&ograve; non contingat coniungi in alio loco, neque tem&shy;<lb/>pore &longs;it, ut a b iungantur in c, &amp; &longs;it reuolutio a triplex integra, &amp; b 


<pb pagenum="42" xlink:href="015/01/061.jpg"/>&longs;excuplex, &amp; tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, &amp; a circuitus tres anni, &amp; quia circuitus b &longs;unt &longs;ex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"/><lb/>puncto. </s>
          <s id="id000804">Si enim prim&ugrave;m in eodem pun&shy;<lb/>cto, &amp;, gratia exempli, in quatuor annis <lb/>congruit enim, &amp; b dicamus quod per&shy;<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni&shy;<lb/>re ad c, &amp; erunt anni tres, &amp; 23/19, non ergo <lb/>anni quatuor. </s>
          <s id="id000805">Cum enim tempora di&shy;<lb/>uer&longs;a diuiduntur per numeros haben&shy;<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"/><lb/>numeri in eadem ratione. </s>
          <s id="id000806">Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, &amp; diui&longs;o 4 per 1 11/19 exit <lb/>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb/>non pote&longs;t e&longs;&longs;e &aelig;quale 1/3. Si enim per <lb/>pr&aelig;cedentem repetuntur, ergo non po&longs;&shy;<lb/>&longs;unt redire, donec iterum coniungantur in ip&longs;o a. </s>
          <s id="id000807">Si enim aliter &longs;it <lb/>ut ex e, igitur e c e&longs;t &aelig;qualis a c pars toti, quod contingere non po&shy;<lb/>te&longs;t. </s>
          <s id="id000808">Sin uer&ograve; coniunctio fiat in d, igitur per pr&aelig;cedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um&shy;<lb/>pti primi numeri. </s>
          <s id="id000809">Veluti in exemplo con&longs;tituimus, quod a, &amp; b <lb/>conueniunt in c in decem annis, &amp; a c e&longs;t tertia pars circuitus: er&shy;<lb/>go in triginta annis conueniunt in a, &amp; in quadraginta rur&longs;us in c. <lb/>&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con&shy;<lb/>gre&longs;&longs;u, &amp; diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, &amp; &longs;i per 3 exiret 13 1/3, &amp; mani&shy;<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu&shy;<lb/>merum, utpote 4 &amp; exit numerus cum eadem parte &longs;cilicet 6 1/3 &amp; <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro&shy;<lb/>portione. </s>
          <s id="id000810">Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo&shy;<lb/>dem modo.</s>
        </p>
        <table>
          <table.target id="table13"/>
          <row>
            <cell>Decem</cell>
            <cell/>
            <cell>Quatuor</cell>
            <cell/>
          </row>
          <row>
            <cell>3</cell>
            <cell>3 1/3</cell>
            <cell>1 11/19</cell>
            <cell>2 8/15&rpar;</cell>
          </row>
          <row>
            <cell>1 11/19</cell>
            <cell>6 1/3</cell>
            <cell/>
            <cell/>
          </row>
        </table>
        <p type="main">
          <s id="id000811">Propo&longs;itio quinquage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id000812">Tria mobilia coniuncta in eodem puncto, quorum duo, &amp; duo <lb/>conueniant in partibus in commen&longs;is inter &longs;e, in perpetuum in nul&shy;<lb/>lo unquam puncto conuenient.</s>
        </p>
        <p type="main">
          <s id="id000813"><arrow.to.target n="marg140"/></s>
        </p>
        <p type="margin">
          <s id="id000814"><margin.target id="marg140"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;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/>c in e, &amp; &longs;int a d, a e incommen&longs;&aelig;, dico qu&ograve;d a b c nunquam con&shy;<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio &agrave; primo: &longs;i non con&shy;<pb pagenum="43" xlink:href="015/01/062.jpg"/><figure id="id.015.01.062.1.jpg" xlink:href="015/01/062/1.jpg"/><lb/>ueniant in f, erunt ergo in g tempore re&shy;<lb/>uolutiones integr&aelig;, &amp; portio a f in&longs;uper. <lb/></s>
          <s id="id000816">Et quia h&aelig; con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, &amp; &longs;unt &longs;patia a d, &amp; b cum c, &amp; <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge&shy;<lb/>nere quantitatis a d, &amp; a e per quinqua&shy;<lb/>ge&longs;imam, harum ergo erunt commen&longs;&aelig;: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. </s>
          <s id="id000817">Et harum <lb/>propo&longs;itionum principium e&longs;t traditum <lb/>&agrave; Campano Nouarien&longs;i Euclidis expo&longs;itore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s>
        </p>
        <p type="main">
          <s id="id000818">Propo&longs;itio quinquage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id000819"><expan abbr="Circulor&utilde;">Circulorum</expan> &longs;e in aduer&longs;um mouentium proportionem declarare.</s>
        </p>
        <p type="main">
          <s id="id000820"><arrow.to.target n="marg141"/></s>
        </p>
        <p type="margin">
          <s id="id000821"><margin.target id="marg141"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000822">Sit orbis a b cuius cen&shy;<lb/><figure id="id.015.01.062.2.jpg" xlink:href="015/01/062/2.jpg"/><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu<lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al&shy;<lb/>teri orbi funiculo a l b, &amp; <lb/>&longs;it in uertice axis k m or&shy;<lb/>biculus &longs;olidus aut &longs;emi&shy;<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ<lb/>cta ex duabus proportio&shy;<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>&amp; &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo&shy;<lb/>ties <expan abbr="peripheri&atilde;">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con&shy;<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo&shy;<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit producta proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>&amp; &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>
        </p>
        <p type="main">
          <s id="id000823">Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo&shy;</s>
        </p>
        <p type="main">
          <s id="id000824"><arrow.to.target n="marg142"/>


<pb pagenum="44" xlink:href="015/01/063.jpg"/>uetur &longs;ur&longs;um &agrave; dextro in &longs;ini&longs;trum in inferiore parte, mouebitur &agrave; <lb/>&longs;ini&longs;tro in dextrum, &amp; uterque circulorum g &amp; k in &longs;uperiore parte, <lb/>&amp; in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore &agrave; &longs;ini<lb/>&longs;tro in dextrum, &amp; inferiore &agrave; dextro in &longs;ini&longs;trum, illi uer&ograve; duo or&shy;<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, qu&agrave;m inferio&shy;<lb/>re, &amp; proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>eorundem.<lb/><arrow.to.target n="marg143"/></s>
        </p>
        <p type="margin">
          <s id="id000825"><margin.target id="marg142"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="margin">
          <s id="id000826"><margin.target id="marg143"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id000827">Rur&longs;us cum a b circumuertatur cum manubrio c d f e, tanto uelo<lb/>cius circumuertetur, &amp; in ea proportione, qua d f continetur in c n, <lb/>&amp; in eodem tempore, in quo manubrium circumuertitur in eodem <lb/>axis circumuertitur, &amp; orbis, ut dictum e&longs;t, ergo in eodem tempo&shy;<lb/>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb/>uidebitur moueri axis ip&longs;o orbe, quanta e&longs;t proportio minoris in <lb/>&aelig;qualitatis ip&longs;ius axis, &longs;eu ambitus, &longs;eu &longs;emidimetientis ad ambi&shy;<lb/>tum, &longs;eu &longs;emidimetientem orbis.</s>
        </p>
        <p type="main">
          <s id="id000828">Propo&longs;itio quinquage&longs;imaquarta.</s>
        </p>
        <p type="main">
          <s id="id000829">Proportio circuli ad &longs;uum diametrum per <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> e&longs;t quar&shy;<lb/>ta pars peripheri&aelig;. </s>
          <s id="id000830">Rur&longs;usque eiu&longs;dem circuli ad peripheriam diame<lb/>tri quarta pars.</s>
        </p>
        <p type="main">
          <s id="id000831"><arrow.to.target n="marg144"/></s>
        </p>
        <p type="margin">
          <s id="id000832"><margin.target id="marg144"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000833">Quoniam enim &longs;uperficies circuli, ut ab <lb/><figure id="id.015.01.063.1.jpg" xlink:href="015/01/063/1.jpg"/><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi&shy;</s>
        </p>
        <p type="main">
          <s id="id000834"><arrow.to.target n="marg145"/><lb/>dio diametri in <expan abbr="dimidi&utilde;">dimidium</expan> peripheri&aelig; erit, ut <lb/>eadem fiat ex tota peripheria in <expan abbr="quart&atilde;">quartam</expan> par<lb/>tem diametri, &amp; ex tota diametro in quar&shy;<lb/>tam <expan abbr="part&etilde;">partem</expan> peripheri&etail;. </s>
          <s id="id000835">ergo proportio are&etail; <lb/>circuli ad diametrum per &longs;imilitudinem <lb/><arrow.to.target n="marg146"/><lb/>e&longs;t quarta pars peripheri&etail;, &amp; proportio are&etail; <lb/>ad <expan abbr="peripheri&atilde;">peripheriam</expan> e&longs;t quarta pars dimetientis, quod erat probandum.</s>
        </p>
        <p type="margin">
          <s id="id000836"><margin.target id="marg145"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000837"><margin.target id="marg146"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>diff.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000838">Propo&longs;itio quinquage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id000839">Proportionem medicamentorum per ordines &longs;uppo&longs;ita &aelig;quali <lb/>proportione in ordinibus per quantitates, &amp; proportiones de&shy;<lb/>mon&longs;trare.<lb/><arrow.to.target n="marg147"/></s>
        </p>
        <p type="margin">
          <s id="id000840"><margin.target id="marg147"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000841">Galenus libro quinto de Simplicibus medicamentis, quem &longs;e&shy;</s>
        </p>
        <p type="main">
          <s id="id000842"><arrow.to.target n="marg148"/><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentor&utilde;">medicamentorum</expan> iux&shy;<lb/>ta qualitates calidi, frigidi, &longs;icci, &amp; humidi, &amp; primus e&longs;t cum <expan abbr="medi&shy;cament&utilde;">medi&shy;<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&etail;melon, ab&shy;<lb/>&longs;ynthium, &amp; oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non l&aelig;dit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, &amp; l&aelig;dit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta&shy;<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue&shy;<lb/>lut pyretrum, piper, euphorbium c&aelig;pe aggre&longs;te, &amp; &longs;inapis, cina&shy;


<pb pagenum="45" xlink:href="015/01/064.jpg"/>momum autem, &amp; gingiber numerantur inter medicinas cal&iacute;das <lb/>tertij gradus, &amp; hoc opus comparatur ad corpus &longs;icut dicit Gale&shy;<lb/>nus, &amp; Serapio non ad linguam, ut medici no&longs;tri temporis interpre<lb/>tantur. </s>
          <s id="id000843">Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordi<lb/>nis, &amp; non l&aelig;dere linguam in gu&longs;tu, &amp; alia tertij ordinis, qu&aelig; non <lb/>&longs;olum l&aelig;det linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;&aelig; mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>
          <s id="id000844">Sed non oportet h&etail;c nunc tractar, enon &longs;olum quia <lb/>non &longs;it locus, &longs;ed etiam qu&ograve;d confu&longs;a &longs;it per &longs;e ip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem pr&aelig;&longs;en<lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. </s>
          <s id="id000845">Nam de &longs;icco, <lb/>&amp; humido, cum &longs;int long&egrave; minoris actionis, qu&agrave;m calidum, &amp; fri&shy;<lb/>gidum, &amp; pr&aelig;cipu&egrave; humidum, non uideo quomodo po&longs;sit Gale&shy;<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, qu&aelig; de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. </s>
          <s id="id000846">Et licet Serapio po&longs;uerit gingiber <lb/><arrow.to.target n="marg149"/><lb/>&amp; enulam &amp; zelim in tertio ordine calidorum &amp; humidorum: &amp; <lb/>inter frigidas, &amp; humidas in tertio portulacam, aizoum, &amp; uirgam <lb/>pa&longs;toris, &amp; fungos. </s>
          <s id="id000847">Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&etail; &longs;int humid&aelig;. </s>
          <s id="id000848">&longs;ecundum, <lb/>quando dicit medicinas cal&iacute;das, aut frigidas, atque hum&iacute;das in ter&shy;<lb/>tio ordine, intelligit &longs;olum de qualitate actiua &longs;cilicet caliditate, uel <lb/>frigiditate, &amp; non de humida qualitate, quod o&longs;tendit de gingibe&shy;<lb/>re, &amp; enula, dicens, quod &longs;unt calid&aelig; in tertio ordine, &amp; humid&aelig; <lb/>humido crudo, non au&longs;us addere ordinem, quia non u&iacute;dit ratio&shy;<lb/>nem, qua po&longs;&longs;ent dici humid&aelig; in tertio. </s>
          <s id="id000849">Et clarius in capite de zei&shy;<lb/>len, quem &longs;tatuerat inter medicinas calidas, &amp; humidas in tertio, di<lb/>cit quod e&longs;t calida in tertio, &amp; humida in primo, ergo non intelligit <lb/>per medicinas calidas &amp; humidas in tertio ordine, quod &longs;int humi&shy;<lb/>d&aelig; in tertio ordine. </s>
          <s id="id000850">Clarius etiam de frigidis &amp; humidis, nam por&shy;<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, &amp; quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, &amp; hu&shy;<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad&shy;<lb/>ijcit, quod e&longs;t &longs;iccum parum, &amp; de uirga pa&longs;toris nihil dicit de hu&shy;<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun&shy;<lb/>dum mentem Serapionis nulla e&longs;t medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, &longs;atis e&longs;t quod non excedunt &longs;ecun<lb/>dum ordinem in humido neque calida neque frigida, &longs;ed frigida &longs;unt <lb/>humidiora, ut fungi, &amp; portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, qu&agrave;m caliditas, &amp; calida magis hu<pb pagenum="46" xlink:href="015/01/065.jpg"/>mectant, quia magis penetrat uis medicamenti, &amp; h&aelig;c regula de <lb/>humido, &amp; &longs;icco e&longs;t generalis apud Serapionem, quod non intelli&shy;<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicin&aelig; &longs;icc&aelig; in tertio, &amp; for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, qu&aelig; in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con<lb/>&longs;iderandum relinquo.</s>
        </p>
        <p type="margin">
          <s id="id000851"><margin.target id="marg148"/>C<emph type="italics"/>ap. </s>
          <s id="id000852">ult.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000853"><margin.target id="marg149"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 336. <lb/>337. &amp; <lb/>338.</s>
        </p>
        <p type="main">
          <s id="id000854">Secunda difficultas e&longs;t maior, &amp; magis pertinet ad nos, &amp; e&longs;t, <lb/>qu&ograve;d non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliqu&atilde;">aliquam</expan> proportionem <lb/>&longs;eruarent, an omnino nullam, &longs;i enim nulla proportio &longs;eruatur, fieri <lb/>nullo modo pote&longs;t, ut per cognitionem temperatur&aelig; &longs;implicium <lb/>medicamentorum cogno&longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. </s>
          <s id="id000855">Sed &longs;i ordines &longs;er&shy;<lb/>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb/>&longs;it Arithmetica, uel Geometrica, uel Mu&longs;ica, &amp; nihil mirum e&longs;&longs;et, <lb/>quod e&longs;&longs;et Mu&longs;ica, ut ali&acirc;s docuimus, ubi tractauimus de differen&shy;<lb/>tia inter &longs;en&longs;um auditus, et ui&longs;us. </s>
          <s id="id000856">Sed quia de hac nullus medicus ui <lb/>detur intellexi&longs;&longs;e, omittam hanc tractationem. </s>
          <s id="id000857">Et quanqu&agrave;m Gale&shy;<lb/>nus po&longs;sit uideri non exi&longs;tima&longs;&longs;e, qu&ograve;d hi ordines non &longs;eruent <lb/>proportionem ullam, quia non au&longs;us e&longs;t tractare de temperamen&shy;<lb/>to medicamentorum compo&longs;itorum per rationem temperamen&shy;<lb/>ti &longs;implicium, nihilominus &longs;uppo&longs;ito quod ita e&longs;&longs;et, quod &longs;eruetur <lb/>altera proportionum, uolo o&longs;tendere rationem componendi in <lb/>utraque proportione &amp; Arithmetica, &amp; Geometrica. </s>
          <s id="id000858">Ex quo &longs;e&shy;<lb/>quitur, quod Aueroes qu&agrave;m o&longs;citanter tractauerit in quinto &longs;uo&shy;<lb/>rum collectaneorum de hoc, &amp; non di&longs;tinguit, neque docet pri&shy;<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, &amp; <lb/>cum in re tam clara pugnet pror&longs;us, ut c&oelig;cus ictus maximos eden&shy;<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, qu&agrave;m mal&egrave; agant qui ei in arduis tan&shy;<lb/>tum tribuunt fidei, &amp; authoritatis, &longs;ed h&aelig;c e&longs;t infelicitas no&longs;tra, &amp; <lb/>ira Deorum. </s>
          <s id="id000859">Suppo&longs;ito ergo quod prim&ograve; ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/><figure id="id.015.01.065.1.jpg" xlink:href="015/01/065/1.jpg"/><lb/>&amp; a &longs;it calida in primo gradu, &amp; b in ter&shy;<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu&shy;<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u&shy;<lb/>perficie rectangula b, hoc igitur erit to&shy;<lb/>tum mi&longs;tum, &amp; quia quantitas medicamenti non mutatur qu&aelig; e&longs;t <lb/>a, b, ergo talia corpora &aelig;quantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, &amp; b in tria, ergo 


<pb pagenum="47" xlink:href="015/01/066.jpg"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to&shy;<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paru&aelig; huiu&longs;modi, &amp; reliqu&aelig;, traduxi autem illas ad hunc <lb/>locum, &ldquo;quia pendent ex demon&longs;tratione hac: &ldquo;duc numerum ordi&shy;<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti&shy;<lb/>tatum, exibit numerus ordinis compo&longs;iti. </s>
          <s id="id000860">Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. </s>
          <s id="id000861">Secunda &longs;i ex pluribus diuer&longs;arum, qualitatum, &amp; ordi&shy;<lb/>num temperatum efficere uelis, duc qu&aelig; &longs;unt eiu&longs;dem qualitatis in <lb/>&longs;uas quantitates, &amp; iunge, quod fit, diuide per numerum ordinis <lb/>medicamenti contrarij, exibit quantitas illius, &longs;ub qua &longs;i iungatur, <lb/>fiet medicamentum temperatum. </s>
          <s id="id000862">Tertia cum nolueris ex tempera&shy;<lb/>to, &amp; alio cuiu&longs;cunque ordinis medicamen conficere ordinis re&shy;<lb/>mi&longs;sionis, detrahe numerum ordinis eius, quod conficere uis ex nu<lb/>mero ordinis eius, quod habes, &amp; cum re&longs;iduo diuide numerum <lb/>medicaminis, quod conficere uis, quod exit e&longs;t numerus quantita&shy;<lb/>tis medicamenti non temperati in comparatione ad temperatum.&rdquo; <lb/>Ex his potes propo&longs;itis quibu&longs;cunque medicamentis conficere <lb/>antidotum &longs;ub quo cunque ordine remi&longs;siore potenti&longs;simo ex il&shy;<lb/>lis. </s>
          <s id="id000863">Quarta in compo&longs;itione, qu&aelig; non fermente&longs;cit calida, calidis <lb/>iuncta &longs;emper opus augent, ut mel cum pipere. </s>
          <s id="id000864">Qu&aelig; autem &longs;ub mi<lb/>nore quantitate exhibentur non &longs;ub remi&longs;siore ordine agant, &longs;ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s>
        </p>
        <p type="main">
          <s id="id000865">Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, &amp; quo ad numerum etiam in primo, &amp; &longs;ecun<lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat &agrave; primo, quantum primus ab &aelig;qualitate, quia <lb/>unum &amp; duo &longs;eruant proportionem, &amp; &aelig;qualem di&longs;tantiam, &longs;ed in <lb/>c&aelig;teris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, &amp; quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/><figure id="id.015.01.066.1.jpg" xlink:href="015/01/066/1.jpg"/><lb/>&longs;cribemus ordines hoc modo, &amp; operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia&shy;<lb/>rum, &amp; medicina frigida in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo&shy;<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi&shy;<lb/>nuit relinquuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag&shy;<pb pagenum="48" xlink:href="015/01/067.jpg"/>gregatum unciarum exit unum, &amp; tertia, ergo erit calida in princi&shy;<lb/>pio &longs;ecundi ordinis. </s>
          <s id="id000866">Secundum exemplum &longs;int e&aelig;dem medicin&aelig;, <lb/>&amp; &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, &amp; <lb/>fiunt &longs;exdecim, &amp; duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex<lb/>decim, &amp; remanent duodecim, diuide per &longs;ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine &longs;ecundi gradus uides ergo di&longs;crimen. </s>
          <s id="id000867">rur&longs;us <lb/>&longs;int amb&aelig; medicin&aelig; calid&aelig;, &amp; ducemus, ut prius in tertio exem&shy;<lb/>plo, ubi proportio &longs;it Arithmetica iungendo duodecim cum qua&shy;<lb/>tuor, &amp; fient &longs;exdecim, diuide per &longs;ex, exeunt duo, &amp; du&aelig; terti&aelig;, er&shy;<lb/>go erit calida in medio tertij gradus, rur&longs;us in quarto exemplo iun<lb/>gemus &longs;edecim cum quatuor, &amp; fient uiginti, diuide per &longs;ex exi&shy;<lb/>bunt tria &amp; tertia, &amp; ita erit in medio tertij gradus, ut prius, &longs;ed &longs;i <lb/>ille quatuor unci&aelig; e&longs;&longs;ent calid&aelig; in quarto gradu, &amp; ill&aelig; du&aelig; unci&aelig; <lb/>in &longs;ecundo gradu, ut prius ducendo quatuor in quatuor fiunt &longs;ex&shy;<lb/>decim, &amp; duo in duo fiunt quatuor, iunge, &amp; fient uiginti, diuide <lb/>per &longs;ex exeunt tria cum tertia, ergo erit calida in principio quarti <lb/>gradus &longs;ecundum proportionem Arithmeticam, &longs;ed &longs;ecundum <lb/>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua&shy;<lb/>tuor ut prius, &longs;cilicet productum duorum in duo fiunt triginta &longs;ex, <lb/>diuide per &longs;ex, exeunt &longs;ex, &amp; quia &longs;ex ad quatuor maiorem habent <lb/>proportionem, qu&agrave;m octo ad &longs;ex ideo h&aelig;c medicina erit calida ul&shy;<lb/>tra medium quarti gradus, iam ergo uides rationem, &amp; differen&shy;<lb/>tiam horum.</s>
        </p>
        <p type="main">
          <s id="id000868">Quod &longs;i quis dicat, an debeat attendi Geometrica proportio in <lb/>medicamentis, an Arithmetica, re&longs;pondeo, qu&ograve;d ueri&longs;imilius e&longs;t de <lb/>Arithmetica, quia illa proportio etiam quod &longs;it minor quatuor ad <lb/>trium, qu&agrave;m trium ad duo, &amp; mult&ograve; minor qu&agrave;m duo ad unum ni&shy;<lb/>hilominus long&egrave; plus operatur, quia tertius ordo iam incipit e&longs;&longs;e <lb/>pr&aelig;ter naturam, &amp; uidemus, quod l&aelig;&longs;io facta in uulnerato, etiam <lb/>qu&ograve;d &longs;it quadruplo minor, plus nocet long&egrave;, qu&agrave;m in &longs;ano qua&shy;<lb/>druplo maior: quia termini pr&aelig;ter naturam &longs;unt uald&egrave; angu&longs;ti in <lb/>comparatione ad latitudinem naturalem, &longs;icut etiam uidemus in&shy;<lb/>tendendis chordis &longs;corpionum, quod ultima pars e&longs;t breuis &amp; ta&shy;<lb/>men homini tantam difficultatem adijcit. </s>
          <s id="id000869">Notandum e&longs;t etiam, <lb/>qu&ograve;d ob hoc diui&longs;erunt ordines in tres partes, uelut gingiber e&longs;t <lb/>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb/>principio, &amp; ita euphorbium e&longs;t calidum in principio quarti gra&shy;<lb/>dus, &longs;ed in fine principij piper, in principio principij aqua &longs;epara&shy;<lb/>tionis in medio quarti ordinis, &longs;ed oleum chalcanthi factum ea ar&shy;<lb/>te, ut exurat paleas, &longs;icut ignis e&longs;t calidum in fine quarti ordinis, &amp; <lb/>ita &longs;ufficiet diuidere propter eandem cau&longs;am primum, &amp; &longs;ecun&shy;


<pb pagenum="49" xlink:href="015/01/068.jpg"/>dum ordinem in duas tantum partes non ratione latitudinis, qu&aelig; <lb/>e&longs;t &aelig;qualis, uel etiam for&longs;an maior, &longs;ed ratione uarietatis operatio&shy;<lb/>nis qu&aelig; minus &longs;entitur, &amp; maxim&egrave; in primo ordine.</s>
        </p>
        <p type="main">
          <s id="id000870">Propo&longs;itio quinquage&longs;ima&longs;exta.</s>
        </p>
        <p type="main">
          <s id="id000871">Proportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&shy;<lb/>&longs;um e&longs;t duplicata ei, qu&aelig; ad numeri latus.<lb/><arrow.to.target n="marg150"/></s>
        </p>
        <p type="margin">
          <s id="id000872"><margin.target id="marg150"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id000873">Cum enim proportionis medium &longs;it latus numeri eo quod ex bi<lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, qu&aelig; demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmetic&aelig; de omnibus binomijs cum &longs;uis </s>
        </p>
        <p type="main">
          <s id="id000874"><arrow.to.target n="marg151"/><lb/>reci&longs;is, uel in quadratis lateribus erit &lt;02&gt; numeri media proportione <lb/>inter binomium, &amp; &longs;uum reci&longs;um, igitur cum proportio producto&shy;<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci&shy;<lb/><arrow.to.target n="marg152"/><lb/>&longs;a erunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo &lt;02&gt; nu<lb/><arrow.to.target n="marg153"/><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, &amp; omnia com&shy;<lb/>men&longs;a illi, e&longs;t duplicata ei qu&aelig; ad &lt;02&gt; numeri.<lb/><arrow.to.target n="marg154"/></s>
        </p>
        <p type="margin">
          <s id="id000875"><margin.target id="marg151"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro&shy;<lb/>po&longs;. </s>
          <s id="id000876">lib. 

de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000877"><margin.target id="marg152"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000878"><margin.target id="marg153"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>&longs;eptimi <lb/>eiu&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000879"><margin.target id="marg154"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>deci&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement:<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000880">Propo&longs;itio quinquage&longs;ima &longs;eptima.</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"/></s>
        </p>
        <p type="margin">
          <s id="id000883"><margin.target id="marg155"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000884">O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob a&euml;ris motum, ubi uer&ograve; h&aelig;ret e&longs;t ac &longs;i quie&longs;cat. <lb/></s>
          <s id="id000885">Eadem autem e&longs;t ratio in motis uiolenter, &amp; naturaliter dum &etail;qua&shy;<lb/>li impetu feruntur. </s>
          <s id="id000886">Sed &longs;ubit&ograve; po&longs;t etiam, quod motus &aelig;qualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im&shy;<lb/><figure id="id.015.01.068.1.jpg" xlink:href="015/01/068/1.jpg"/><lb/>pedimentum naturale. </s>
          <s id="id000887">Sed &longs;i uis mouens fuerit <lb/>ade&ograve; ualida ut proportio incrementi ex a&euml;re &longs;it <lb/>maior, qu&agrave;m impedimentum, &amp; in crementum al<lb/>terius mobilis naturaliter moti, motus ille uelo&shy;<lb/>cior fiet naturali, ut in &longs;ph&aelig;ris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad pr&aelig;&longs;entem <lb/>motum ratio e&longs;t eadem. </s>
          <s id="id000888">Quicunque ergo motus <lb/>minoris grauis cogit de&longs;cendere lancem ex ad&shy;<lb/>uer&longs;o proportionem habet eandem ad &longs;uum mo <lb/>bile quam habet graue &aelig;quiponderans. </s>
          <s id="id000889">Sit ergo <lb/>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb/>g, h, erit ergo ponderum h, g, f, e, ad &longs;e inuicem, &amp; ad a qualis mo&shy;<lb/>tuum ob di&longs;tantiam intentorum. </s>
          <s id="id000890">Experimentum ergo docet, qu&ograve;d <lb/>dimidium ponderis &aelig;quilibrium facit ex palmo minoris dimidio <lb/>motum manife&longs;tum, &amp; ex palmo quarta pars ponderis, ergo &longs;e ha&shy;<lb/>bent prope portionem.</s>
        </p>
        <p type="main">
          <s id="id000891">Propo&longs;itio quinquage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id000892">Qu&etail; ex alto de&longs;cendunt cur non eandem pro di&longs;tantia motus ra<lb/>tionem in libero a&euml;re &longs;eruent con&longs;iderare.</s>
        </p>
        <pb pagenum="50" xlink:href="015/01/069.jpg"/>
        <p type="main">
          <s id="id000893">A&euml;r in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed &amp; infra uerum minus manife&longs;t&egrave;. </s>
          <s id="id000894">At ca&shy;<lb/>&longs;u plerun que contingit, ut moueatur long&egrave; uehementius, &longs;eu ad ean&shy;<lb/>dem partem, &longs;eu aliam. </s>
          <s id="id000895">Qui uer&ograve; naturalis e&longs;t, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="c&otilde;tinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aqu&aelig; maris fluit ac refluit: aliter ne&shy;<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede&shy;<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quandoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latent nos. </s>
          <s id="id000896">Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictu&utilde;">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis a&euml;ris <lb/>e&longs;&longs;et continuus, in hora a&euml;r procederet ob ambitum terr&aelig; millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, im&ograve; ne po&longs;sit quidem, itaque <lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i ade&ograve; uelox &amp; in ea&shy;<lb/>dem parte <expan abbr="a&etilde;ris">aeris</expan> e&longs;&longs;et at que continuus. </s>
          <s id="id000897">Pr&aelig;terea tantus impetus nun&shy;<lb/>quam &agrave; minore motu, aut cau&longs;a &longs;uperaretur, ade&ograve; ut &longs;emper flatum <lb/>a&euml;ris orientalem &longs;entiremus. </s>
          <s id="id000898">Quotidie etiam aduenire ad nos a&euml;&shy;<lb/>rem ex Illyrico, Macedonia, My&longs;ia, Ponto, Byth&iacute;nia, Capadocia, Sy <lb/>ria, Babylonia, Hyrcanomar&iacute;, Bactrianis, Sac&iacute;s, Scythis, ac Seris, to&shy;<lb/>to pr&aelig;terea Oceano orientali tam ua&longs;to, &amp; Gallica noua, terraque flo<lb/>rida non &longs;olum res e&longs;t admirabilis&apos;, &amp; incredibilis, &longs;ed etiam aliena <lb/>&agrave; &longs;en&longs;u, &amp; ab his, qu&aelig; eueniunt. </s>
          <s id="id000899">A&apos;&longs;en&longs;u quidem, quoniam nebul&etail;, <lb/>qu&aelig; in a&euml;re mouentur, prim&ugrave;m non in eandem partem &longs;emper mo<lb/>uentur: nun quam autem ade&ograve; celeriter: at &longs;i a&euml;r &longs;ic circumuoluere&shy;<lb/>tur, mouerentur &amp; illa, qu&etail; in eo continentur, quotidieque a&euml;rem ex&shy;<lb/>periremur &amp; nubilo&longs;um, &amp; madidum propter mare. </s>
          <s id="id000900">Nechis, qu&aelig; <lb/>eueniunt hoc &longs;atis re&longs;pondet, nec nobis id contingeret, ut &longs;i pe&longs;ti <lb/>aliqua in regione no&longs;tra directa &longs;&aelig;uiret, ut a&euml;r &longs;ingulis diebus la&shy;<lb/>be ea infectus ad nos deferretur. </s>
          <s id="id000901">Moueri uer&ograve; a&euml;rem &longs;emper mani&shy;<lb/>fe&longs;ti&longs;simum e&longs;t tum experimento, tum ratione: ratione &longs;iquidem, <lb/>quod aqua &amp; c&oelig;lum naturaliter perpetu&ograve; mouentur, quare etiam <lb/>a&euml;r. </s>
          <s id="id000902">Experimento, qu&ograve;d ubi hiant o&longs;tia, &amp; ianu&aelig;, ibi perpetuus &longs;en&shy;<lb/>titur flatus. </s>
          <s id="id000903">Ergo &longs;i a pondus de&longs;cendat in c, ex alto fertur rect&agrave;, &longs;ed <lb/>&longs;i ex &longs;ublimi transferetur in b, &amp; indirecta, &amp; ad latus, unde ex <lb/>hoc &longs;equitur.</s>
        </p>
        <p type="main">
          <s id="id000904">Propo&longs;itio quin quage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id000905"><arrow.to.target n="marg156"/></s>
        </p>
        <p type="margin">
          <s id="id000906"><margin.target id="marg156"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id000907">Omne mobile motum duobus motibus non ad idem tendenti&shy;<lb/>bus, utro que &longs;eor&longs;um tardius mouetur &longs;imili motu.</s>
        </p>
        <pb pagenum="51" xlink:href="015/01/070.jpg"/>
        <p type="main">
          <s id="id000908">Sit a mobile, quod moueatur per a b c impul&longs;u uenti aut uiolen&shy;</s>
        </p>
        <p type="main">
          <s id="id000909"><arrow.to.target n="marg157"/><lb/><figure id="id.015.01.070.1.jpg" xlink:href="015/01/070/1.jpg"/><lb/>to cum naturali coniuncto: &amp; &longs;it terminus naturalis e, <lb/><arrow.to.target n="marg158"/><lb/>&amp; uiolenti d: uter que in directo c, dico, quod tardius per&shy;<lb/>ueniet ad c quam d, uel e. </s>
          <s id="id000910">De e manife&longs;tum e&longs;t, quoniam <lb/>motus a&euml;ris, qui intendit motum a, diu&iacute;ditur in partem, <lb/>qu&aelig; iuuat motum ad d, &amp; partem, qu&aelig; mouetur ad e, <lb/>igitur fit minor adiectio. </s>
          <s id="id000911">Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rect&aelig;: quare tardius perueniet ad c qu&agrave;m ad e du<lb/>plici ratione. </s>
          <s id="id000912">Dico etiam, quod tardius ad c qu&agrave;m d. </s>
          <s id="id000913">Quia enim <lb/>uis, qu&aelig; fert ad d repugnat ei, qu&aelig; fert ad e, &amp; uis, qu&aelig; fert ad e, re&shy;<lb/>pugnat ei qu&aelig; fert ad d, igitur tardius perueniet ad c, qu&agrave;m d. </s>
          <s id="id000914">Nec <lb/>potes dicere, qu&ograve;d uis, qu&aelig; fert ad c adiuuet ad motum &egrave; regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan&shy;<lb/>tum motus ad e retardabit motum ad d, tanto motus a c erit tard&iacute;&shy;<lb/>or ab&longs;olut&egrave; motu ad d. </s>
          <s id="id000915">Verum etiam e&longs;t, quod c e breuior erit a d, <lb/>quia motus ad e &longs;emper contrahit motum ad d naturalis uiolen&shy;<lb/>tum ob cau&longs;am dictam. </s>
          <s id="id000916">Vtr&ugrave;m uer&ograve; motus ad c ab&longs;olut&egrave; &longs;it tardi&shy;<lb/>or, qu&agrave;m ad d, non &longs;uppo&longs;ito, quod c e &longs;it &aelig;qualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s>
        </p>
        <p type="margin">
          <s id="id000917"><margin.target id="marg157"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000918"><margin.target id="marg158"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>bu&shy;ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000919">Ex hoc patet, quod motus &aelig;quidi&longs;tantis mobilis, finis e&longs;t mini&shy;<lb/><arrow.to.target n="marg159"/><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. </s>
          <s id="id000920">Velut &longs;i a mo<lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debili&longs;simus, quia non <lb/><figure id="id.015.01.070.2.jpg" xlink:href="015/01/070/2.jpg"/><lb/>e&longs;t motus actu: uiolentus autem &aelig;qualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/><arrow.to.target n="marg160"/><lb/>e&longs;&longs;et minor dimidio palmi. </s>
          <s id="id000921">Et etiam qu&ograve;d e&longs;&longs;et minor, quia ut di&shy;<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t &aelig;qualis uni eorum non impedito <lb/>uel minor.</s>
        </p>
        <p type="margin">
          <s id="id000922"><margin.target id="marg159"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000923"><margin.target id="marg160"/>P<emph type="italics"/>er<emph.end type="italics"/> 57. <emph type="italics"/>bu&shy;ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000924">Propo&longs;itio &longs;exage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id000925">Omne mobile motu naturali de&longs;cendens parte, de&longs;cendit gra&shy;<lb/>uiore &longs;ecundum grauitatis centrum.</s>
        </p>
        <p type="main">
          <s id="id000926">Sit a mobile, grauitatis centrum b, cuius pars ei pro&shy;<lb/><arrow.to.target n="marg161"/><lb/><figure id="id.015.01.070.3.jpg" xlink:href="015/01/070/3.jpg"/><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na&shy;<lb/>tura partis, &amp; totius: totius autem terr&aelig; natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"/><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. </s>
          <s id="id000927">Sed pars pro&shy;<lb/>ximior nece&longs;&longs;ari&ograve; e&longs;t grauior, quia centrum e&longs;t in medio grauita&shy;


<pb pagenum="52" xlink:href="015/01/071.jpg"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio&shy;<lb/>rem partem.<lb/><arrow.to.target n="marg163"/></s>
        </p>
        <p type="margin">
          <s id="id000928"><margin.target id="marg161"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id000929"><margin.target id="marg162"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>bu&shy;ius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id000930"><margin.target id="marg163"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000931">Ex hoc &longs;equitur, qu&ograve;d graue habens partes in&aelig;quales, &longs;eu &longs;ub&shy;<lb/>&longs;tantia, &longs;eu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="n&otilde;">non</expan> &longs;it, infr&agrave; opor&shy;<lb/>tet, ut circumuoluatur.</s>
        </p>
        <p type="main">
          <s id="id000932">Propo&longs;itio &longs;exage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id000933">Proportionem ictus ad pondus rei, &amp; di&longs;tantiam generaliter <lb/>con&longs;iderare.</s>
        </p>
        <p type="main">
          <s id="id000934"><arrow.to.target n="marg164"/></s>
        </p>
        <p type="margin">
          <s id="id000935"><margin.target id="marg164"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000936">Dictum e&longs;t &longs;uperius de proportione de&longs;cen&longs;us ad grauitatem: </s>
        </p>
        <p type="main">
          <s id="id000937"><arrow.to.target n="marg165"/><lb/>&amp; qu&ograve;d &longs;i graue de&longs;cendat ex alto impeditur &agrave; motu a&euml;ris: &amp; qu&ograve;d <lb/><arrow.to.target n="marg166"/><lb/>res, qu&aelig; mouetur duobus motibus non ad idem tendentibus tar&shy;<lb/><arrow.to.target n="marg167"/><lb/>dius mouetur, quam motus &longs;it unu&longs;qui&longs;que. </s>
          <s id="id000938">Dem&ugrave;m qu&ograve;d graue <lb/><arrow.to.target n="marg168"/><lb/>de&longs;cendens circumuoluitur, &longs;i pars grauior non &longs;it, deor&longs;um: &amp; an&shy;<lb/>tea ubi egimus de proportione motus ad grauitatem, quod h&etail;c in&shy;<lb/>telligenda &longs;unt pro ut po&longs;&longs;unt intelligi de motu etiam uiolento. <lb/></s>
          <s id="id000939">Cum ergo uideamus duo h&aelig;c, quod res acuta frangit caput, &longs;i ex <lb/>alto incidat, &longs;ed non concutit, lata concutit, &longs;ed non diuidit, premit <lb/>tamen carnem &longs;ubiectam: nec hoc accidit merito ponderis: nam ut <lb/>ui&longs;um e&longs;t &longs;emilibra lapidis, uel ferri cadens ex alto contundit caput, <lb/>&amp; uulnerat, &amp; non eleuat in &aelig;quilibrio, ut pot&egrave; ex alto cadens loco <lb/>per &longs;patium octo palmorum pondus &longs;exdecim librarum, &amp; a pon&shy;<lb/>dere &longs;exdecim librarum homo non l&aelig;ditur, nec uulneratur, ergo id <lb/>accidit ex alia cau&longs;a, &amp; e&longs;t, quod a&euml;r interceptus inter graue, &amp; cor&shy;<lb/>pus no&longs;trum non pote&longs;t dilabi tam cit&ograve;, ergo ne corpus penetret, <lb/>cogitur ingredi locum, cui e&longs;t obuius, at que ita concutere, &amp; diuide&shy;<lb/>re. </s>
          <s id="id000940">Ex quibus &longs;equuntur omnia h&aelig;c.<lb/><arrow.to.target n="marg169"/></s>
        </p>
        <p type="margin">
          <s id="id000941"><margin.target id="marg165"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 57.</s>
        </p>
        <p type="margin">
          <s id="id000942"><margin.target id="marg166"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 58.</s>
        </p>
        <p type="margin">
          <s id="id000943"><margin.target id="marg167"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>
        </p>
        <p type="margin">
          <s id="id000944"><margin.target id="marg168"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 60.</s>
        </p>
        <p type="margin">
          <s id="id000945"><margin.target id="marg169"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000946">Prim&ugrave;m &longs;i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars &longs;ubiecta, quia re&longs;ilit in corpus molle: nec &agrave; molli, quia retundi&shy;<lb/>tur, pote&longs;t uulnerari: ergo nullo modo. </s>
          <s id="id000947">Sed neque ade&ograve; concutit, <lb/>quia a&euml;r rediens, &amp; receptus in molli corpore pro parte, non uer&shy;<lb/>berat locum.</s>
        </p>
        <p type="main">
          <s id="id000948"><arrow.to.target n="marg170"/></s>
        </p>
        <p type="margin">
          <s id="id000949"><margin.target id="marg170"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000950">Secundum in omni colli&longs;ione &longs;eu duri, &longs;eu mollis, &longs;ed magis du&shy;<lb/>ri, dilabuntur partes a&euml;ris ad latera, ideo quod partes medi&aelig; pre&shy;<lb/>muntur. </s>
          <s id="id000951">Et quanto motus e&longs;t tardior.</s>
        </p>
        <p type="main">
          <s id="id000952"><arrow.to.target n="marg171"/></s>
        </p>
        <p type="margin">
          <s id="id000953"><margin.target id="marg171"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000954">Tertium in motu uelo ci fit maior ictus &amp; l&aelig;&longs;io, &amp; maiora omnia <lb/>quam pro proportione motus: quoniam ob uelo<expan abbr="citat&etilde;">citatem</expan> minus diffu<lb/>git a&euml;ris. </s>
          <s id="id000955">Et ide&ograve; fiunt grauia uulnera ex modico incremento uelo&shy;<lb/>citatis motus.</s>
        </p>
        <p type="main">
          <s id="id000956"><arrow.to.target n="marg172"/></s>
        </p>
        <p type="margin">
          <s id="id000957"><margin.target id="marg172"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000958">Quartum res lat&aelig;, dur&aelig; concutiunt, &amp; non uulnerant ni&longs;i &longs;int <lb/>cum magno impetu, aut ualde graues: acut&aelig; autem uulnerant, &longs;ed <lb/>non concutiunt, ni&longs;i parti acut&aelig; lata &longs;uccedat.</s>
        </p>
        <pb pagenum="53" xlink:href="015/01/072.jpg"/>
        <p type="main">
          <s id="id000959">Quintum, corpora dura magis l&aelig;duntur &agrave; latis, quia &longs;cindun&shy;</s>
        </p>
        <p type="main">
          <s id="id000960"><arrow.to.target n="marg173"/><lb/>tur, mollia autem &agrave; tenuibus, quia diuiduntur: nam mollitie excipi&shy;<lb/>unt a&euml;rem, &amp; ita &agrave; latis non ade&ograve; patiuntur, &amp; etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s>
        </p>
        <p type="margin">
          <s id="id000961"><margin.target id="marg173"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000962">Sextum, etiam in duris penetrat aliquid a&euml;ris, aliter tota frange&shy;<lb/><arrow.to.target n="marg174"/><lb/>rentur. </s>
          <s id="id000963">Con&longs;tat etiam omnem lapidem marmoreum, aut &longs;iliceum <lb/>e&longs;&longs;e poro&longs;um, ut dicunt. </s>
          <s id="id000964">Et etiam quia recipitur in mollioribus, er&shy;<lb/>go etiam in durioribus &amp; in duri&longs;simis: quod &longs;i non recipiant ut ui<lb/>trum, &amp; gemm&aelig; tota franguntur. </s>
          <s id="id000965">Hoc etiam uidetur &longs;en&longs;i&longs;&longs;e Philo<lb/>&longs;ophus, qui uult, qu&ograve;d res franguntur ob poros.</s>
        </p>
        <p type="margin">
          <s id="id000966"><margin.target id="marg174"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000967">Propo&longs;itio &longs;exage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id000968">Proportionem motoris in plano ad motorem, qui eleuat pon&shy;<lb/>dus iuxta id, quod mouet inuenire.</s>
        </p>
        <p type="main">
          <s id="id000969">Con&longs;titutum e&longs;t inuenire proportionem uirium, qu&aelig; eleuant <lb/><arrow.to.target n="marg175"/><lb/>pondus ad uires, qu&aelig; ip&longs;um in plano leui trahere po&longs;&shy;<lb/><figure id="id.015.01.072.1.jpg" xlink:href="015/01/072/1.jpg"/><lb/>&longs;unt. </s>
          <s id="id000970">Vires enim, qu&aelig; eleuant pondus a &longs;unt e&aelig;dem <lb/>puta b, qu&aelig; uero trahunt c, &longs;ed h&aelig; po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>&amp; maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/></s>
          <s id="id000971">Du&aelig; uer&ograve; po&longs;trem&aelig; cau&longs;&aelig; etiam per &longs;e per&longs;picu&aelig; &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si&shy;<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla&shy;<lb/>num tangit. </s>
          <s id="id000972">Nos pr&aelig;terea &longs;upponimus planum &aelig;quale undique <lb/>leue durum, &amp; corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe&shy;<lb/>rens, &amp; uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>qu&ograve;d in hoc ca&longs;u b e&longs;t duplum ad c. </s>
          <s id="id000973">Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, &amp; &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci&shy;<lb/>tur circa medium, nam plana &longs;uperficies parum differt &agrave; rotunda <lb/>terr&aelig; ob terr&aelig; magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="n&otilde;">non</expan> remouetur nul&shy;<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s>
        </p>
        <p type="margin">
          <s id="id000974"><margin.target id="marg175"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000975">Propo&longs;itio &longs;exage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id000976">Omne graue quanto proximius alligatum plano, tanto faci&shy;<lb/>lius trahitur.


<pb pagenum="54" xlink:href="015/01/073.jpg"/><arrow.to.target n="marg176"/></s>
        </p>
        <p type="margin">
          <s id="id000977"><margin.target id="marg176"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000978">Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"/><lb/>qu&ograve;d facilius trahetur per fe qu&agrave;m c b &amp; e b, qu&agrave;m <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a &amp; b communicabitur c, igitur erit minor qu&agrave;m in <lb/>c, &amp; hoc naturaliter. </s>
          <s id="id000979">Mathematica autem ratione quoniam ex a tra&shy;<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua <lb/>igitur attractio c per d e&longs;t debilior, qu&agrave;m per f. </s>
          <s id="id000980">Rur&longs;us &longs;i e trahitur <lb/>per d c&ugrave;m a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li&shy;<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s>
        </p>
        <p type="main">
          <s id="id000981"><arrow.to.target n="marg177"/><lb/>tero ad latus, ergo lentius ad f per d c qu&agrave;m f c, quod erat demon&shy;<lb/>&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id000982"><margin.target id="marg177"/>P<emph type="italics"/>er<emph.end type="italics"/> 59. <emph type="italics"/>bu&shy;ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id000983">Propo&longs;itio &longs;exage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id000984">Omne mobile quanto latius tanto tardius mouetur in plano.<lb/><arrow.to.target n="marg178"/></s>
        </p>
        <p type="margin">
          <s id="id000985"><margin.target id="marg178"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000986">Demon&longs;tratum e&longs;t &longs;uperius qu&ograve;d &longs;i mobile &longs;it &longs;ph&etail;ricum, &amp; tan</s>
        </p>
        <p type="main">
          <s id="id000987"><arrow.to.target n="marg179"/><lb/>gat planum in puncto, qu&ograve;d mouetur per quancunque uim aptam <lb/>diuidere medium. </s>
          <s id="id000988">Quia ergo &longs;i tangat in puncto facillime moue&shy;<lb/>tur, &longs;i in linea paul&ograve; difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici&shy;<lb/>lius mouetur. </s>
          <s id="id000989">Sit ergo mobile a b, quod moueatur uer&longs;us c, &amp; quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me&shy;<lb/><figure id="id.015.01.073.2.jpg" xlink:href="015/01/073/2.jpg"/><lb/>dietatis, &amp; pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, &amp; longius a b, tanto difficilius <lb/><arrow.to.target n="marg180"/><lb/>mouetur. </s>
          <s id="id000990">Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s>
        </p>
        <p type="margin">
          <s id="id000991"><margin.target id="marg179"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s>
        </p>
        <p type="margin">
          <s id="id000992"><margin.target id="marg180"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62</s>
        </p>
        <p type="main">
          <s id="id000993">Propo&longs;itio &longs;exage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id000994">Proportionem duorum mobilium inter &longs;e cum auxilio medij <lb/>inuenire.<lb/><arrow.to.target n="marg181"/></s>
        </p>
        <p type="margin">
          <s id="id000995"><margin.target id="marg181"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id000996">Graue de&longs;cendit naturaliter quatuor cau&longs;is: prima e&longs;t ponderis <lb/>magnitudo, unde quod grauius e&longs;t celerius de&longs;cendit. </s>
          <s id="id000997">Secund&ograve; ob <lb/>paruam medij repugnantiam, ideo quanto medium e&longs;t rarius &amp; <lb/>mobile tenuius, tanto celerius de&longs;cendit: contr&agrave; uer&ograve; tardius. </s>
          <s id="id000998">Ter&shy;<lb/>ti&ograve; ob impetum a&euml;ris &longs;ub &longs;equentis: &amp; ideo mobile qu&ograve;d ex eadem </s>
        </p>
        <p type="main">
          <s id="id000999"><arrow.to.target n="marg182"/><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne a&euml;r <lb/>cogatur celerius ferri: &amp; quanto diutius de&longs;cendit, tanto magis in&shy;<lb/>tenditur motus, at que augetur, ut &longs;upr&agrave; de claratum e&longs;t. </s>
          <s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab a&euml;re tran&longs;uer&longs;im moto, et &agrave; latere: ideo <lb/>leuia mobilia &amp; magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, &amp; magnam repugnantiam, &longs;ed quia tran&longs;uer<lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>
          <s id="id001001">Por&shy;


<pb pagenum="55" xlink:href="015/01/074.jpg"/>r&ograve; proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo ant&egrave;, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, &amp; figura non <lb/>eiu&longs;modi, nec leuitas, ut pror&longs;us non impellat, nedum ut moueat la<lb/>tus: figura quo que eadem ambobus relinquetur proportio motus <lb/>ad motum producta ex proportionibus incrementi in proportio&shy;<lb/><arrow.to.target n="marg185"/><lb/>nem ponderum, &amp; iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"/><lb/>motu a&euml;ris, ergo proportio unius motus producti ad alteram no&shy;<lb/>ta erit.</s>
        </p>
        <p type="margin">
          <s id="id001002"><margin.target id="marg182"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s>
        </p>
        <p type="margin">
          <s id="id001003"><margin.target id="marg183"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>
        </p>
        <p type="margin">
          <s id="id001004"><margin.target id="marg184"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s>
        </p>
        <p type="margin">
          <s id="id001005"><margin.target id="marg185"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>ha&shy;rum.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001006"><margin.target id="marg186"/>I<emph type="italics"/>n<emph.end type="italics"/> 61. <emph type="italics"/>ha&shy;rum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001007">Propo&longs;itio &longs;exage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id001008">Proportionem laterum eptagoni, &amp; &longs;ubten&longs;arum con&longs;iderare, <lb/>&amp; qu&aelig; &agrave; reflexa proportione pendent.<lb/><arrow.to.target n="marg187"/></s>
        </p>
        <p type="margin">
          <s id="id001009"><margin.target id="marg187"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id001010">Sit eptagonus a b d f g e c, &amp; &longs;ubten&longs;&aelig; b <lb/><figure id="id.015.01.074.1.jpg" xlink:href="015/01/074/1.jpg"/><lb/>c, &amp; f e duobus lateribus, tribus autem d c <lb/>d e, &amp; erunt &lpar;quia intelligitur eptagono &aelig;&shy;<lb/>quilatero, &amp; &aelig;quiangulo&rpar; b c &amp; e f inuicem <lb/>&aelig;quales: &amp; item d c, &amp; d e &aelig;quales: &amp; &longs;i du&shy;<lb/>cerentur b e &amp; c f inuicem &aelig;quales: &amp; ad a c <lb/>&amp; d g: quare cum angulus cb d con&longs;i&longs;tatin </s>
        </p>
        <p type="main">
          <s id="id001011"><arrow.to.target n="marg188"/><lb/>arcu c e g f d, &amp; angulus b d c in arcu b a c, <lb/>&amp; angulus b c d in arcu b d; &amp; &longs;it arcus c e g <lb/>f d duplus arcus b a c, quia c e g f d &longs;ubtendit quatuor latera epta&shy;<lb/>goni, &amp; arcus b a c duo, &amp; ita arcus etiam b a c duplus arcui b d <lb/>erit angulus d b e duplus angulo c d b, &amp; angulus c d b duplus an&shy;<lb/><arrow.to.target n="marg189"/><lb/>gulo b c d, quare per demon&longs;trata &agrave; nobis proportio laterum b d, <lb/>b c, c d, e&longs;t reflexa, igitur proportio d b &amp; b c, ad d c, ut d e ad b c, &amp; <lb/><arrow.to.target n="marg190"/><lb/>rur&longs;us proportio b d &amp; d e ad b e, ut b e ad b d. </s>
          <s id="id001012">Quare &longs;uppo&longs;ita <lb/>d b 1, b c 1 po&longs;itione, erit d c latus 1 quad. </s>
          <s id="id001013">p: 1 po&longs;itione. </s>
          <s id="id001014">Proportio <lb/><arrow.to.target n="marg191"/><lb/>uer&ograve;, ut dictum e&longs;t b d &amp; d c ad b c, id e&longs;t p: &lt;02&gt; 1 quad. </s>
          <s id="id001015">p: 1 pos, ad 1 <lb/>pos e&longs;t, ut b c ad b d, id e&longs;t 1 pos ad 1, igitur 1 p: &lt;02&gt; v: 1 quad. </s>
          <s id="id001016">p: 1 pos <lb/>&aelig;quatur quadrato b c, quod e&longs;t 1 quad. </s>
          <s id="id001017">igitur 1 quad. </s>
          <s id="id001018">m: 1 &aelig;quatur <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 &aelig;quatur 1 <lb/>quad. </s>
          <s id="id001023">p: 1 pos. </s>
          <s id="id001024">Additis igitur communiter quatuor quadratis fient <lb/>1 quad. </s>
          <s id="id001025">quad. </s>
          <s id="id001026">p: 2 quad. </s>
          <s id="id001027">p: 1 &aelig;qualia 5 quad. </s>
          <s id="id001028">p: 1 pos. </s>
          <s id="id001029">Et reducitur ad <lb/>1 cu. </s>
          <s id="id001030">&aelig;qualem 1 3/4 pos p: 7/8.</s>
        </p>
        <p type="margin">
          <s id="id001031"><margin.target id="marg188"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. &amp; 29. <emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001032"><margin.target id="marg189"/>P<emph type="italics"/>er ult. </s>
          <s id="id001033">&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001034"><margin.target id="marg190"/>D<emph type="italics"/>e<emph.end type="italics"/> S<emph type="italics"/>uh. lib.<emph.end type="italics"/> 16.</s>
        </p>
        <p type="margin">
          <s id="id001035"><margin.target id="marg191"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>diff.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001036">Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon&shy;<lb/>&longs;tratis &agrave; Ptolem&aelig;o quadratum b c, &amp; e&longs;t 1 quad e&longs;t &aelig;quale produ&shy;<lb/>cto ex b d in c e, quod e&longs;t 1, &amp; a b in d c, igitur detracto 1, produ&shy;<lb/>cto b d in c e ex 1 quad. </s>
          <s id="id001037">quadrato c b, relinquitur productum ex <lb/>a b in c d 1 quad. </s>
          <s id="id001038">m: 1, ergo diui&longs;o co per a b, qu&aelig; e&longs;t 1, relinquitur <lb/>c d 1 quad. </s>
          <s id="id001039">m: 1 huius uer&ograve; quadratum per <expan abbr="ead&etilde;">eadem</expan> demon&longs;trata &agrave; Pto&shy;


<pb pagenum="56" xlink:href="015/01/075.jpg"/>lem&aelig;o, &etail;quale e&longs;t rectangulis ex b c in de, &amp; b d in c e, igitur 1 quad. <lb/></s>
          <s id="id001040">quad. </s>
          <s id="id001041">m: 2 quad. </s>
          <s id="id001042">p: 1 e&longs;t &aelig;quale 1 producto b d in c e, &amp; producto b <lb/>cin d e detracto 1 communi, relinquetur productum ex b c in d e 1 <lb/>quad. </s>
          <s id="id001043">quad. </s>
          <s id="id001044">m: 2 quad. </s>
          <s id="id001045">igitur diui&longs;o 1 quad. </s>
          <s id="id001046">quad. </s>
          <s id="id001047">m: 2 quad. </s>
          <s id="id001048">per 1 <lb/>pos, exit 1 cu. </s>
          <s id="id001049">m: 2 pos &aelig;qualia d e, &amp; d e e&longs;t &aelig;qualis d c, ut ab initio <lb/>demon&longs;trauimus, &amp; d c fuit 1 quad. </s>
          <s id="id001050">m: 1, igitur 1 cu. </s>
          <s id="id001051">m: 2 &aelig;quantur 1 <lb/>quad. </s>
          <s id="id001052">m: 1, igitur 1 cu. </s>
          <s id="id001053">p: 1 &aelig;quantur 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/>cantur perpendiculares a f, d g &amp; c d, &amp; &longs;it c e i ca 1 pos, &amp; quia ut <lb/><arrow.to.target n="marg192"/><lb/>a e ad a c, ita d e ad b c, erit ergo b c &lpar;1 posp: 1&rpar;/&lpar;1 pos&rpar; quare b f &lpar;1/2 pos 1/2,&rpar;/&lpar;2 pos&rpar; &amp; <lb/>quia d h e&longs;t dimidium d e, erit d h, &amp; g f <lb/><figure id="id.015.01.075.1.jpg" xlink:href="015/01/075/1.jpg"/><lb/>1/2, cum ergo b f &longs;it &lpar;1/2 pos p: 1/2&rpar;/pos erit ergo di&shy;<lb/>ui&longs;a 1/2 pos per 1 pos, &amp; exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/&lpar;1 pos&rpar;. <lb/>&amp; eius quadratum 1/4/&lpar;1 quad&rpar;. igitur cum qua&shy;<lb/>dratum b d &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/&lpar;2 quad&rpar;g c autem e&longs;t compo&longs;ita ex e f, qu&aelig; <lb/>e&longs;t 1/2p: 1/2/&lpar;1 pos&rpar; &amp; f g qu&aelig; e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/&lpar;1 pos&rpar;, &amp; <expan abbr="quadrat&utilde;">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/&lpar;1 quad.&rpar; quare <expan abbr="&qtilde;drat&utilde;">quadratum</expan> e d &qring;d e&longs;t <lb/><arrow.to.target n="marg193"/><lb/>compo&longs;itum ex quadratis c g &amp; g d erit 2 p: 1/pos c a uer&ograve; e&longs;t &aelig;qua&shy;<lb/>lis c d, quia, ut demon&longs;tratum e&longs;t angulus d c e e&longs;t &longs;eptima pars <lb/>duorum rectorum, &amp; angulus b c e ei duplus, quare cum c f a &longs;it re&shy;<lb/>ctus erit ex trige&longs;ima &longs;ecunda primi Elementorum f a c tres &longs;epti&shy;<lb/>m&aelig; unius recti, ergo d a c 6/7 unius recti, d c a uer&ograve; 2/7 unius recti, quia <lb/><arrow.to.target n="marg194"/><lb/>e&longs;t &longs;eptima pars duorum rectorum, &iacute;gitur a d c e&longs;t 6/7 unius recti: igi&shy;<lb/>tur c d e&longs;t &aelig;qualis c a, ergo quadratum quadrato: igitur 1 quad. </s>
          <s id="id001056">p: 2 <lb/>pos p: 1, &aelig;quatur 2 p: 1/&lpar;1 pos&rpar; igitur 1 quad. </s>
          <s id="id001057">p: 2 pos, &aelig;quantur 1 p: 1/&lpar;1 pos&rpar;. <lb/>Quare 1 cub. </s>
          <s id="id001058">p: 2 quad. </s>
          <s id="id001059">&aelig;quatur 1 pos p: 1. <lb/><figure id="id.015.01.075.2.jpg" xlink:href="015/01/075/2.jpg"/><lb/>Sit etiam angulus a duplus b, &amp; b c dupla <lb/>b a: &amp; erit per eadem proportio a c, &amp; a b <lb/>ad c b, ut c b ad c a. </s>
          <s id="id001060">Ponamus ergo ab 1, erit <lb/>b c 2, &amp; a c 1 pos, &amp; a c, a b 1 pos p: 1, &amp; du&shy;<lb/>cta in a c fit 1 quad. </s>
          <s id="id001061">p: 1 pos, &amp; hoc e&longs;t &aelig;quale 4 quadrato b c per re&shy;<lb/>flex&aelig; proportionis diffinitionem. </s>
          <s id="id001062">Igitur a c e&longs;t &lt;02&gt; 4 1/4 m: 1/2, &amp; ita <lb/>de alijs.</s>
        </p>
        <p type="margin">
          <s id="id001063"><margin.target id="marg192"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001064"><margin.target id="marg193"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001065"><margin.target id="marg194"/>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001066">Propo&longs;itio &longs;exage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id001067">Si fuerint aliquot quantitates ab una quantitate, ali&aelig;que totidem 


<pb pagenum="57" xlink:href="015/01/076.jpg"/>ab eadem analog&aelig;, erit proportio terti&aelig; unius ordinis ad tertiam <lb/>alterius, ut &longs;ecund&aelig; ad &longs;ecundam duplicata, &amp; quart&aelig; ad quartam <lb/>triplicata, quint&aelig; ad quintam quadruplicata, at que &longs;ic de alijs.<lb/><arrow.to.target n="marg195"/></s>
        </p>
        <p type="margin">
          <s id="id001068"><margin.target id="marg195"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001069">Sint quantitates b c d e f, ab a in continua proportio&shy;<lb/><figure id="id.015.01.076.1.jpg" xlink:href="015/01/076/1.jpg"/><arrow.to.target n="table14"/><lb/>ne, &amp; ali&aelig; totidem g h k l m, dico quod proportio h c e&longs;t <lb/>duplicata ei, qu&aelig; e&longs;t g ad b, &amp; k ad d triplicata, &amp; l ad e <lb/>quadruplicata, &amp; &longs;ic deinceps, &longs;umatur enim unum, &amp; ab </s>
        </p>
        <table>
          <table.target id="table14"/>
          <row>
            <cell/>
            <cell>a</cell>
            <cell/>
          </row>
          <row>
            <cell>b</cell>
            <cell/>
            <cell>g</cell>
          </row>
          <row>
            <cell>c</cell>
            <cell/>
            <cell>h</cell>
          </row>
          <row>
            <cell>d</cell>
            <cell/>
            <cell>k</cell>
          </row>
          <row>
            <cell>e</cell>
            <cell/>
            <cell>l</cell>
          </row>
          <row>
            <cell>f</cell>
            <cell/>
            <cell>m</cell>
          </row>
          <row>
            <cell/>
            <cell>n</cell>
            <cell/>
          </row>
          <row>
            <cell>o</cell>
            <cell/>
            <cell>t</cell>
          </row>
          <row>
            <cell>p</cell>
            <cell><foreign lang="greek">a</foreign></cell>
            <cell>u</cell>
          </row>
          <row>
            <cell>q</cell>
            <cell><foreign lang="greek">b g</foreign></cell>
            <cell>x</cell>
          </row>
          <row>
            <cell>z</cell>
            <cell/>
            <cell>y</cell>
          </row>
          <row>
            <cell>s</cell>
            <cell/>
            <cell>z</cell>
          </row>
        </table>
        <p type="main">
          <s id="id001070"><arrow.to.target n="marg196"/><lb/>eo o p q r s in proportione b ad a, &amp; t u x y z in propor&shy;<lb/>tione g ad a, erit igitur p quadratum o, &amp; u quadratum t, <lb/>&amp; q cubus o, &amp; x cubus t, &amp; ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, qu&aelig; t ad o, &amp; x ad q triplicata ei, qu&aelig; t <lb/>ad o, &amp; pote&longs;t etiam demon&longs;trari generaliter ultra qua&shy;<lb/><arrow.to.target n="marg198"/><lb/>dratum, &amp; cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro&shy;<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem qu&aelig; t ad o, &amp; proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicat&aelig; <lb/><arrow.to.target n="marg199"/><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>qu&aelig; t ad o, &amp; &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>
          <s id="id001071">Quia ergo propor&shy;<lb/>tio q ad <foreign lang="greek">b</foreign> e&longs;t ut o ad t, patet, quod x ad q e&longs;t triplicata ei, qu&aelig; e&longs;t t ad <lb/>o, &amp; ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, &amp; o ad <lb/><arrow.to.target n="marg201"/><lb/>n, ut b ad a, &amp; n ad t, ut a ad g, &amp; t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, &amp; u ad p, ut h ad c, igitur cum &longs;it ut u ad p duplicata ei, qu&etail; <lb/>e&longs;t t ad o erit h ad e, duplicata ei qu&aelig; e&longs;t g ad b, &amp; ita de reliquis, &amp; <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, qu&aelig; e&longs;t t ad o, &longs;eu dicas p <lb/><arrow.to.target n="marg202"/><lb/>ad u duplicatam ei, qu&aelig; e&longs;t o ad t. </s>
          <s id="id001072">Aliter &amp; euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e &amp; h duplicata ei qu&aelig; e&longs;t b &amp; g <lb/>ad a, ut &longs;upra, &amp; quadrati b ad quadratum a, &amp; quadrati g ad qua&shy;<lb/><arrow.to.target n="marg203"/><lb/>dratum a duplicata his qu&aelig; b &amp; g ad a erunt b &amp; g quadratorum <lb/>ad quadratum a, uelut c &amp; h ad a. </s>
          <s id="id001073">Et conuertendo qua&shy;<lb/><arrow.to.target n="table15"/><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/><figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg"/>hic &amp; erit quadrati b ad <expan abbr="quadrat&utilde;">quadratum</expan> g, ita c ad h: &longs;ed qua&shy;<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s>
        </p>
        <p type="margin">
          <s id="id001074"><margin.target id="marg196"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>noni<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> &amp; 22. &amp; 23. <emph type="italics"/>octaui.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001075"><margin.target id="marg197"/>V<emph type="italics"/>ide per<emph.end type="italics"/> 23. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001076"><margin.target id="marg198"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/> &amp; 33. <emph type="italics"/>undeci&shy;mi.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001077"><margin.target id="marg199"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;e&shy;ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001078"><margin.target id="marg200"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 10.</s>
        </p>
        <p type="margin">
          <s id="id001079"><margin.target id="marg201"/>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001080"><margin.target id="marg202"/>P<emph type="italics"/>er<emph.end type="italics"/> 10 <emph type="italics"/>diff. </s>
          <s id="id001081">quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001082"><margin.target id="marg203"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <table>
          <table.target id="table15"/>
          <row>
            <cell><expan abbr="&qtilde;d">quad</expan>.</cell>
            <cell>b</cell>
            <cell>e</cell>
          </row>
          <row>
            <cell><expan abbr="&qtilde;d">quad</expan>.</cell>
            <cell>a</cell>
            <cell>a</cell>
          </row>
          <row>
            <cell><expan abbr="&qtilde;d">quad</expan>.</cell>
            <cell>g</cell>
            <cell>h</cell>
          </row>
        </table>
        <p type="main">
          <s id="id001083">Propo&longs;itio &longs;exage&longs;imaoctaua, collectorum ab Euclide <lb/>&amp; Archimede.</s>
        </p>
        <p type="main">
          <s id="id001084">Omnis cylindrus cono habenti ba&longs;im, &amp; altitudinem eandem <lb/><arrow.to.target n="marg204"/><lb/>triplus e&longs;t. </s>
          <s id="id001085">Omnis cylindrus &longs;ph&aelig;r&aelig; habenti eundem magnum <lb/><arrow.to.target n="marg205"/><lb/>circulum, &amp; altitudinem &longs;exquialter e&longs;t. </s>
          <s id="id001086">Omnis &longs;ph&aelig;ra dupla e&longs;t <lb/><arrow.to.target n="marg206"/><lb/>cono, cuius ba&longs;is e&longs;t eius circulus magnus, &amp; altitudo eadem, qu&aelig; <lb/>&longs;ph&aelig;r&aelig; ip&longs;ius. </s>
          <s id="id001087">Omnis &longs;uperficies &longs;ph&aelig;r&aelig; quadrupla e&longs;t maiori <lb/><arrow.to.target n="marg207"/><lb/>&longs;uo circulo. </s>
          <s id="id001088">Superficies portionis &longs;ph&aelig;r&aelig; e&longs;t &aelig;qualis circulo, cu <lb/><arrow.to.target n="marg208"/>


<pb pagenum="58" xlink:href="015/01/077.jpg"/>ius &longs;emidiameter e&longs;t linea ducta &agrave; uertice portionis ad finem illius.</s>
        </p>
        <p type="margin">
          <s id="id001089"><margin.target id="marg204"/>1</s>
        </p>
        <p type="margin">
          <s id="id001090"><margin.target id="marg205"/>2</s>
        </p>
        <p type="margin">
          <s id="id001091"><margin.target id="marg206"/>3</s>
        </p>
        <p type="margin">
          <s id="id001092"><margin.target id="marg207"/>4</s>
        </p>
        <p type="margin">
          <s id="id001093"><margin.target id="marg208"/>5</s>
        </p>
        <p type="main">
          <s id="id001094">Quilibet &longs;ector &longs;ph&aelig;r&aelig; &aelig;qualis e&longs;t cono, cuius ba&longs;is e&longs;t circu&shy;<lb/>lus &aelig;qualis &longs;uperficiei eiu&longs;dem portionis, altitudo uer&ograve; &longs;ph&aelig;r&aelig; &longs;e&shy;<lb/>midiameter. </s>
          <s id="id001095">Proportio &longs;ph&aelig;r&aelig; ad &longs;ectorem datum, e&longs;t duplica&shy;<lb/>ta ei, qu&etail; e&longs;t dimetientis ad lineam, qu&aelig; &agrave; uertice portionis ad lim&shy;<lb/>bum. </s>
          <s id="id001096">Cum enim &longs;ph&aelig;ra &longs;it &aelig;qualis cono, cuius ba&longs;is e&longs;t maior cir&shy;<lb/>culus, altitudo uer&ograve; dupla dimetienti per tertiam harum, qu&aelig; hic <lb/><arrow.to.target n="marg209"/><lb/>proponuntur: erit &longs;ph&aelig;ra &aelig;qualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it &aelig;qualis diametro &longs;ph&aelig;r&aelig;, altitudo uer&ograve; &longs;e&shy;<lb/>midiameter &longs;ph&aelig;r&aelig;. </s>
          <s id="id001097">At per &longs;extam harum &longs;ector &longs;ph&aelig;r&aelig; e&longs;t &aelig;qua&shy;<lb/>lis cono habenti altitudinem &longs;emidiametrum &longs;ph&aelig;r&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;ph&aelig;r&aelig; ad &longs;ecto&shy;<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;ph&aelig;r&aelig; ad <lb/>c&iacute;rculum &aelig;qualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t &aelig;qualis circulo, cuius &longs;emidiameter e&longs;t li&shy;<lb/>nea &agrave; uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;ph&aelig;&shy;<lb/>r&aelig; ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;ph&aelig;r&aelig;, aut &longs;emidimetiens e&longs;t &aelig;qualis dimetienti &longs;ph&aelig;r&aelig; <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea &agrave; uertice portionis ad <lb/>limbum. </s>
          <s id="id001098">Sed proportio talium circulorum e&longs;t duplicata propor&shy;<lb/><arrow.to.target n="marg211"/><lb/>tioni &longs;emidimetientium, igitur proportio &longs;ph&aelig;r&aelig; ad &longs;uum &longs;ecto&shy;<lb/>rem e&longs;t ueluti dimetientis &longs;ph&aelig;r&aelig; ad lineam, qu&aelig; &aacute; uertice portio&shy;<lb/><arrow.to.target n="marg212"/><lb/>nis ad limbum duplicata. </s>
          <s id="id001099">Cuicunque portioni &longs;ph&aelig;r&aelig; conus ille <lb/>habetur &aelig;qualis, qui ba&longs;im habeat eandem cum portione, altitudi&shy;<lb/>nem uer&ograve; lineam rectam, qu&aelig; ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;ph&aelig;r&aelig; un&agrave; cum alti&shy;<lb/>tudine reliqu&aelig; portionis habet ad eandem reliqu&aelig; portionis alti&shy;<lb/><arrow.to.target n="marg213"/><lb/>tudinem. </s>
          <s id="id001100">Earum &longs;ph&aelig;r&aelig; portionum, qu&aelig; &aelig;qualibus &longs;uperfi&shy;<lb/><arrow.to.target n="marg214"/><lb/>ciebus continentur medietas &longs;ph&aelig;r&aelig; maxima exi&longs;tit. </s>
          <s id="id001101">Proportio <lb/>&longs;uperficiei &longs;ph&aelig;r&aelig; plano diui&longs;&aelig; ad reliqu&aelig; portionis &longs;uperficiem, <lb/>&amp; re&longs;idui &longs;ectoris ad &longs;ectorem, e&longs;t uelut quadratorum duarum li&shy;<lb/>nearum qu&aelig; &agrave; uerticulis &longs;ectionum ad communem &longs;uperficiem <lb/>plani portiones &longs;ecantis de&longs;cendunt: nam &longs;ectorem &longs;ph&aelig;r&aelig;, dico <lb/><arrow.to.target n="marg215"/><lb/>corpus compo&longs;itum ex portione, &amp; cono illo. </s>
          <s id="id001102">Ille idem etiam defi&shy;<lb/>nit Ellip&longs;im coni a cuti anguli &longs;ectionem, quam dicit etiam fieri &longs;e&shy;<lb/><arrow.to.target n="marg216"/><lb/>cto cylindro per planum non ad angulos rectos &longs;tante &longs;uper cylin&shy;<lb/>dri axem. </s>
          <s id="id001103">Ab hac igitur coni acuti anguli &longs;ectione &longs;eu ellip&longs;i cir&shy;<lb/><arrow.to.target n="marg217"/><lb/>cumacta figura &longs;ph&aelig;roides corpus quod ba&longs;im rotundam habet, <lb/>uocat: id que duplex ob longum, quod fit diametro longiore quie&shy;<lb/>&longs;cente, &amp; prolatum quod fit quie&longs;cente breuiore: &longs;icut reliquam &longs;ci <lb/>licet parabolen aut hyperbolen, quia inferius non e&longs;t terminata, 


<pb pagenum="59" xlink:href="015/01/078.jpg"/>in cono rectangulo uocat rectanguli coni &longs;ectionem: ex qua cir&shy;<lb/>cumacta fit conoidale, quia planam habet ba&longs;im. </s>
          <s id="id001104">Si ergo in ea&shy;<lb/><arrow.to.target n="marg218"/><lb/>dem rectanguli coni &longs;ectione &agrave; plano portiones &aelig;quales habentes <lb/>diametros ab&longs;cindantur, ill&aelig; portiones erunt &aelig;quales. </s>
          <s id="id001105">Et triangu&shy;<lb/>li in ei&longs;dem portionibus in&longs;cripti &aelig;quales erunt. </s>
          <s id="id001106">Diametrum uo&shy;<lb/>cat in <expan abbr="quacunq&utilde;e">quacunqune</expan> portione lineam, qu&aelig; omnes lineas ba&longs;i &aelig;quidi&shy;<lb/>&longs;tantes per &aelig;qualia diuidit. </s>
          <s id="id001107">Omnis circuli cuius diameter e&longs;t ma<lb/><arrow.to.target n="marg219"/><lb/>ior diameter ellip&longs;is proportio ad ellip&longs;im e&longs;t uelut direct&egrave; diame&shy;<lb/>tri ellip&longs;is ad diametrum tran&longs;uer&longs;am. </s>
          <s id="id001108">Ex quo patet quod pro&shy;<lb/><arrow.to.target n="marg220"/><lb/>portio cuiuslibet circuli ad ellip&longs;im e&longs;t uelut quadrati &longs;u&aelig; diame&shy;<lb/>tri ad rectangulum recta, &amp; tran&longs;uer&longs;a diametro ellip&longs;is compre&shy;<lb/>hen&longs;um. </s>
          <s id="id001109">Ex hoc rur&longs;us &longs;equitur quod ellip&longs;is ad ellip&longs;im, ut re&shy;<lb/><arrow.to.target n="marg221"/><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s>
        </p>
        <p type="margin">
          <s id="id001110"><margin.target id="marg209"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. &amp; 15. <emph type="italics"/>duodeci mi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> E<emph type="italics"/>ucl.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001111"><margin.target id="marg210"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>duodecimi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001112"><margin.target id="marg211"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duodecimi<emph.end type="italics"/>, &amp; 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001113"><margin.target id="marg212"/>8</s>
        </p>
        <p type="margin">
          <s id="id001114"><margin.target id="marg213"/>9</s>
        </p>
        <p type="margin">
          <s id="id001115"><margin.target id="marg214"/>10</s>
        </p>
        <p type="margin">
          <s id="id001116"><margin.target id="marg215"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001117"><margin.target id="marg216"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001118"><margin.target id="marg217"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001119"><margin.target id="marg218"/>11</s>
        </p>
        <p type="margin">
          <s id="id001120"><margin.target id="marg219"/>12</s>
        </p>
        <p type="margin">
          <s id="id001121"><margin.target id="marg220"/>13</s>
        </p>
        <p type="margin">
          <s id="id001122"><margin.target id="marg221"/>14</s>
        </p>
        <p type="main">
          <s id="id001123">Si conoides &amp; &longs;ph&aelig;roides &longs;ecet plano &aelig;quidi&longs;tanti axi fiet &longs;e&shy;<lb/><arrow.to.target n="marg222"/><lb/>ctio conoidalis &longs;imilis ei &agrave; qua conoides &longs;eu &longs;ph&aelig;roides de&longs;cri&shy;<lb/>ptum e&longs;t. </s>
          <s id="id001124">Sin autem &longs;upra axem plano ad perpendiculum erecto <lb/>&longs;ectio circulus erit. </s>
          <s id="id001125">Et &longs;i &longs;ecentur obliqu&egrave; fiet ellip&longs;is, modo omnia <lb/>latera comprehendat. </s>
          <s id="id001126">Omnis portio conoidalis rectanguli, quam <lb/><arrow.to.target n="marg223"/><lb/>planum &longs;ecat, &longs;exquialtera e&longs;t, cono qui ba&longs;im &amp; axem eandem ha&shy;<lb/>bet. </s>
          <s id="id001127">Ex quo patet, quod &longs;i portio conoidalis rectanguli &amp; &longs;ph&aelig;&shy;<lb/><arrow.to.target n="marg224"/><lb/>r&aelig; medietas eandem ba&longs;im habeant &amp; axem eundem, medietas <lb/>&longs;ph&aelig;r&aelig; &longs;exquitertia erit conoidali portioni. </s>
          <s id="id001128">Et &longs;i eiu&longs;dem rectan<lb/><arrow.to.target n="marg225"/><lb/>guli conoidalis portiones ab&longs;cin dantur erit portionum propor&shy;<lb/>tio uelut quadratorum axium. </s>
          <s id="id001129">Cuiuslibet &longs;ph&aelig;roidis pars pla&shy;<lb/><arrow.to.target n="marg226"/><lb/>no per centrum ab&longs;ci&longs;&longs;a dupla e&longs;t cono ba&longs;im &amp; axem eadem ha&shy;<lb/>benti. </s>
          <s id="id001130">Si autem non &longs;uper centrum erit proportio earum ad co&shy;<lb/><arrow.to.target n="marg227"/><lb/>num ba&longs;im, &amp; axem eandem habentem uelut coniunct&aelig; ex axe al&shy;<lb/>terius partis &amp; dimidio axis &longs;ph&aelig;roidis ad axem alterius partis.</s>
        </p>
        <p type="margin">
          <s id="id001131"><margin.target id="marg222"/>15</s>
        </p>
        <p type="margin">
          <s id="id001132"><margin.target id="marg223"/>16</s>
        </p>
        <p type="margin">
          <s id="id001133"><margin.target id="marg224"/>17</s>
        </p>
        <p type="margin">
          <s id="id001134"><margin.target id="marg225"/>18</s>
        </p>
        <p type="margin">
          <s id="id001135"><margin.target id="marg226"/>19</s>
        </p>
        <p type="margin">
          <s id="id001136"><margin.target id="marg227"/>20</s>
        </p>
        <p type="main">
          <s id="id001137">Demum proportio partis conoidis obtu&longs;i anguli plano ab&longs;ci&longs;&shy;<lb/><arrow.to.target n="marg228"/><lb/>&longs;&aelig; ad conum, ba&longs;im &amp; axem eadem habentem e&longs;t ueluti line&aelig;, com<lb/>po&longs;it&aelig; ex axe portionis &amp; triplo adiect&aelig; ad compo&longs;itum ex axe <lb/>portionis &amp; duplo eiu&longs;dem adiect&aelig;. </s>
          <s id="id001138">Adiectam uocat hyperbolis <lb/>tran&longs;uer&longs;am. </s>
          <s id="id001139">Omnis cylindrus cono triplus e&longs;t habenti eandem <lb/><arrow.to.target n="marg229"/><lb/>ba&longs;im &amp; altitudinem. </s>
          <s id="id001140">Omnes cylindri coni &longs;ph&aelig;r&aelig; &longs;unt in pro&shy;<lb/><arrow.to.target n="marg230"/><lb/>portione corporum &longs;imilium planis &longs;uperficiebus contentarum.</s>
        </p>
        <p type="margin">
          <s id="id001141"><margin.target id="marg228"/>21</s>
        </p>
        <p type="margin">
          <s id="id001142"><margin.target id="marg229"/>22</s>
        </p>
        <p type="margin">
          <s id="id001143"><margin.target id="marg230"/>23</s>
        </p>
        <p type="main">
          <s id="id001144">Propo&longs;itio &longs;exage&longs;ima nona, collectorum ex quatuor libris <lb/>Apollonij Pergei &amp; <expan abbr="q.">que</expan> Sereni.</s>
        </p>
        <p type="main">
          <s id="id001145">Si fuerit linea bifariam diui&longs;a, eique in longum alia addita, &amp; rur&shy;<lb/><arrow.to.target n="marg231"/><lb/>&longs;us alia detracta, fueritque totius cum addita ad eam, qu&aelig; addita e&longs;t <lb/>ueluti re&longs;idui ad detractam erit line&aelig; com&shy;<lb/><figure id="id.015.01.078.1.jpg" xlink:href="015/01/078/1.jpg"/><lb/>po&longs;it&aelig; ex addita, &amp; dimidia ad dimidiam 


<pb pagenum="60" xlink:href="015/01/079.jpg"/>ip&longs;am uelut dimidi&aelig; ad differentiam eius, &amp; detract&aelig;. </s>
          <s id="id001146">Rur&longs;usque li&shy;<lb/>ne&aelig; compo&longs;it&aelig; ex dimidio &amp; re&longs;iduo dimidi&aelig; ac detract&aelig; ad li&shy;<lb/>neam compo&longs;itam ex addita &amp; detracta ut re&longs;idui dimidi&aelig;, &amp; de&shy;<lb/>tract&aelig; ad partem detractam. </s>
          <s id="id001147">Et rur&longs;us totius compo&longs;it&aelig; ad com&shy;<lb/>po&longs;itam ex dimidia &amp; addita, uelut compo&longs;it&aelig; ex addita, &amp; diffe&shy;<lb/>rentia ad ip&longs;am additam. </s>
          <s id="id001148">Velut &longs;it propo&longs;ita a b per &aelig;qualia diui&longs;a <lb/>in c, addita b d, &amp; detracta b e, &longs;it proportio a d ad d b, ut a e ad e b, <lb/>dico e&longs;&longs;e, 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&shy;<lb/><arrow.to.target n="marg232"/><lb/>rum ut a d ad c d uelut e d ad d b. </s>
          <s id="id001151">In parabole proportio partium <lb/>diametri ad uerticem terminantium duplicata e&longs;t proportioni li&shy;<lb/>nearum ab ei&longs;dem punctis ordinatim ductarum ad ip&longs;am &longs;ectio&shy;<lb/><arrow.to.target n="marg233"/><lb/>nem. </s>
          <s id="id001152">In hyperbole autem &amp; ellip&longs;i &amp; circuli circumferentia erit <lb/>quadratorum linearum ordinatim ductarum inter &longs;e uelut rectan&shy;<lb/><arrow.to.target n="marg234"/><lb/>gulorum partium diametri ad eadem puncta terminantium. </s>
          <s id="id001153">Et in <lb/>ei&longs;dem &longs;i &agrave; puncto peripheri&aelig; contingens ad diametrum ducatur, <lb/>&amp; ab eodem ordinata, erit ut partis diametri intercept&etail; inter extre&shy;<lb/>mum, &amp; ordinatam ad partem inter ordinatam &amp; peripheriam, ue&shy;<lb/>lut intercept&aelig; inter extremum &amp; contingentem ad interceptam <lb/><arrow.to.target n="marg235"/><lb/>exterius inter finem contingentis &amp; peripheriam. </s>
          <s id="id001154">Et in ei&longs;dem <lb/>quadratum &longs;emidiametri &aelig;quale e&longs;&longs;e rectangulo ex intercepta in&shy;<lb/>ter centrum &amp; ca&longs;um contingentis in interceptam inter centrum &amp; <lb/><arrow.to.target n="marg236"/><lb/>ca&longs;um ordinat&aelig; &agrave; loco contactus product&aelig;. </s>
          <s id="id001155">Si parabolen recta <lb/>linea contingens ad diametrum perueniat, &longs;umptoque puncto alio <lb/>in &longs;ectione &aelig;quidi&longs;tans ab eo ducatur contingenti: &amp; ab utroque <lb/>etiam ad diametrum ordinat&aelig;, demum &agrave; uertice &aelig;quidi&longs;tans illis, <lb/>&amp; &agrave; priore puncto diametro &aelig;quidi&longs;tans donec concurrant, erit <lb/>triangulus ex ordinata, &amp; &aelig;quidi&longs;tante &agrave; &longs;ecundo puncto, &amp; dia&shy;<lb/>metri parte contentus rectangulo ex prima ordinata &amp; parte dia&shy;<lb/>metri inter uerticem &amp; &longs;ecundam ordinatam contento &aelig;qualis.<lb/><arrow.to.target n="marg237"/></s>
        </p>
        <p type="margin">
          <s id="id001156"><margin.target id="marg231"/>1</s>
        </p>
        <p type="margin">
          <s id="id001157"><margin.target id="marg232"/>2</s>
        </p>
        <p type="margin">
          <s id="id001158"><margin.target id="marg233"/>3</s>
        </p>
        <p type="margin">
          <s id="id001159"><margin.target id="marg234"/>4</s>
        </p>
        <p type="margin">
          <s id="id001160"><margin.target id="marg235"/>5</s>
        </p>
        <p type="margin">
          <s id="id001161"><margin.target id="marg236"/>6</s>
        </p>
        <p type="margin">
          <s id="id001162"><margin.target id="marg237"/>7</s>
        </p>
        <p type="main">
          <s id="id001163">Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei &aelig;quidi&longs;tans ducatur ex ip&longs;a &longs;ectione, ubi iterum &longs;ecat &longs;ectionem <lb/>intercepta per &aelig;qualia diuidetur linea &agrave; puncto contingentis dia&shy;</s>
        </p>
        <p type="main">
          <s id="id001164"><arrow.to.target n="marg238"/><lb/>metro &aelig;quidi&longs;tanti ducta. </s>
          <s id="id001165">Idem uer&ograve; ferm&egrave; continget ducta li&shy;<lb/>nea &agrave; centro in locum contactus, &longs;ecabit enim omnes contingenti <lb/><arrow.to.target n="marg239"/><lb/>&aelig;quidi&longs;tantes in hyperbole, ellip&longs;i at que circulo. </s>
          <s id="id001166">E&longs;t autem omne <lb/>centrum in medio diametri: diameter autem in circulo &amp; ellip&longs;i il&shy;<lb/>las per &aelig;qualia diuidit intus enim e&longs;t: in contrapo&longs;itis inter uerti&shy;<lb/>cem, &amp; uerticem po&longs;ita e&longs;t exterius utriu&longs;que contingenti ad per&shy;<lb/>pendiculum in&longs;i&longs;tens. </s>
          <s id="id001167">In hyperbole autem exterius etiam adiacet, <lb/>ut in contrapo&longs;itis eadem &amp; tran&longs;uer&longs;a uocatur: cuius terminus e&longs;t <lb/>punctus concur&longs;us cum latere trianguli, qui conum per axem diui&shy;


<pb pagenum="61" xlink:href="015/01/080.jpg"/>dit: linea uer&ograve; tangens uerticem hyperbolis ad quam ordinat&aelig; <lb/><arrow.to.target n="marg240"/><lb/>po&longs;&longs;unt, Recta appellabitur. </s>
          <s id="id001168">Data recta linea po&longs;itione, aliaque ma<lb/>gnitudine data &amp; ang&uuml;lo parabolen, &amp; hyperbolen, &amp; ellip&longs;im, <lb/>&amp; contra po&longs;itas circa datam po&longs;itione tanqu&agrave;m diametrum de&shy;<lb/>&longs;cribere tanqu&agrave;m cono erecto, ut angulus ad uerticem &longs;ectionis <lb/>comprehen&longs;us &longs;it, &amp; per rectam rectangulum &aelig;quale comprehen&shy;<lb/>datur quadrato dat&aelig; line&aelig; magnitudine. </s>
          <s id="id001169">Si linea in duas partes <lb/><arrow.to.target n="marg241"/><lb/>diuidatur, eique utrinque &aelig;quales line&aelig; adiun&shy;<lb/><figure id="id.015.01.080.1.jpg" xlink:href="015/01/080/1.jpg"/><lb/>gantur erit rectangulum ex partibus totius &aelig;&shy;<lb/>quale rectangulis partium prioris line&aelig;, &amp; ex <lb/>priore linea cum una adiecta in eam, qu&aelig; adiecta e&longs;t. </s>
          <s id="id001170">Si hyperbo<lb/><arrow.to.target n="marg242"/><lb/>len recta linea in uertice contingat, &amp; utrinque ab&longs;cindatur, quan&shy;<lb/>tum e&longs;t, quod pote&longs;t in quartam partem rectanguli ex diametro <lb/>tran&longs;uer&longs;a hyperbolis, qu&aelig; exterius adiacetin eam, qu&aelig; recta dici&shy;<lb/>tur, ad quam, qu&aelig; ordinatim ducuntur, &longs;unt &aelig;quidi&longs;tantes line&aelig;, <lb/>qu&aelig; &agrave; &longs;ectionis centro ad terminos contingentis ducuntur &longs;emper <lb/>ip&longs;i &longs;ectioni magis appropinquabunt, nec unquam conuenient: &amp; <lb/>ob id a&longs;ymptoton appellantur. </s>
          <s id="id001171">Nec ull&aelig; ali&aelig; intra <expan abbr="angul&utilde;">angulum</expan> illum <lb/><arrow.to.target n="marg243"/><lb/>inueniri poterunt. </s>
          <s id="id001172">Vnde etiam intra <expan abbr="dat&utilde;">datum</expan> angulum de&longs;cribere do&shy;<lb/>cemur hyperbolen cuius anguli latera &longs;int a&longs;ymptota. </s>
          <s id="id001173">A&longs;ymptotis <lb/><arrow.to.target n="marg244"/><lb/>duabus propo&longs;itis uni hyperboli, in finitas al&iacute;as eidem a&longs;ymptotas <lb/>inuenire. </s>
          <s id="id001174">Duabus rectis a&longs;ymptotis infinitas &longs;ubijci po&longs;&longs;e hyperbo<lb/>les illis rectis, &amp; inter &longs;e a&longs;ymptotas. </s>
          <s id="id001175">Cum in duabus &longs;uperficie&shy;<lb/><arrow.to.target n="marg245"/><lb/>bus &aelig;quidi&longs;tantibus duo circuli &aelig;quales, quorum linea per cen&shy;<lb/>tra non e&longs;t ad perpendiculum earum infinitis planis &longs;ecantur, fiunt <lb/>in ip&longs;is line&aelig; &agrave; peripheria in peripheriam rect&aelig; qu&aelig; corpus cylin&shy;<lb/>dricum claudunt quod &longs;calenus cylindrus appellatur: long&egrave; alius <lb/>ab eo, qui fit recto cylindro per duo plana &aelig;quidi&longs;tantia, &longs;ed non <lb/>ad perpendiculum po&longs;ita di&longs;&longs;ecto. </s>
          <s id="id001176">nam eius extrem&aelig; &longs;uperficies <lb/>non circuli, &longs;ed ellip&longs;es &longs;unt. </s>
          <s id="id001177">Si &longs;calenus cylindrus plano non &aelig;&shy;<lb/><arrow.to.target n="marg246"/><lb/>quidi&longs;tanti ba&longs;i, &longs;ed ita ut angulos interiores &aelig;quales faciat angu&shy;<lb/>lis ba&longs;is &longs;ectio circulus erit: uocaturque h&aelig;c &longs;ectio &longs;ub contraria: nec <lb/>ulla pr&aelig;ter hanc &amp; ba&longs;i &aelig;quidi&longs;tantem &longs;ectio circulus e&longs;&longs;e pote&longs;t: <lb/>&longs;ed &longs;unt ellip&longs;es. </s>
          <s id="id001178">Super eundem circulum, &amp; &longs;ub eadem altitudi&shy;<lb/><arrow.to.target n="marg247"/><lb/>ne ellip&longs;es &longs;imiles in cono &amp; cylindro e&longs;&longs;e po&longs;&longs;unt, qu&aelig; ab eodem <lb/>plano fiant, docetque uel ba&longs;i uel cono uel cylindro, aut cono pro&shy;<lb/>po&longs;ito reliqua facere, quod e&longs;t ualde admirabile: cum ellip&longs;is cylin&shy;<lb/>drica &longs;emper &aelig;qualis &longs;it in utraque parte &agrave; diametro tran&longs;uer&longs;a <lb/>utrinque &aelig;qualiter di&longs;tante, conica uer&ograve; minor nece&longs;&longs;ari&ograve; &longs;it in &longs;u&shy;<lb/>periore parte uer&longs;us coni uerticem latior in inferiore, ubi partes a <lb/>diametro tran&longs;uer&longs;a &aelig;qualiter di&longs;teterint: ip&longs;&etail; autem non &longs;olum &longs;i&shy;


<pb pagenum="62" xlink:href="015/01/081.jpg"/><arrow.to.target n="marg248"/><lb/>miles, &longs;ed unam per&longs;&aelig;pe in utri&longs; que e&longs;&longs;e uult. </s>
          <s id="id001179">Sed &amp; hoc Archime&shy;<lb/>des dicere uidetur: line&aelig; duct&aelig; &agrave; uertice coni&longs;caleni ad perpendi&shy;<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axem coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>
        </p>
        <p type="margin">
          <s id="id001180"><margin.target id="marg238"/>8</s>
        </p>
        <p type="margin">
          <s id="id001181"><margin.target id="marg239"/>9</s>
        </p>
        <p type="margin">
          <s id="id001182"><margin.target id="marg240"/>10</s>
        </p>
        <p type="margin">
          <s id="id001183"><margin.target id="marg241"/>11</s>
        </p>
        <p type="margin">
          <s id="id001184"><margin.target id="marg242"/>12</s>
        </p>
        <p type="margin">
          <s id="id001185"><margin.target id="marg243"/>13</s>
        </p>
        <p type="margin">
          <s id="id001186"><margin.target id="marg244"/>14</s>
        </p>
        <p type="margin">
          <s id="id001187"><margin.target id="marg245"/>15</s>
        </p>
        <p type="margin">
          <s id="id001188"><margin.target id="marg246"/>16</s>
        </p>
        <p type="margin">
          <s id="id001189"><margin.target id="marg247"/>17</s>
        </p>
        <p type="margin">
          <s id="id001190"><margin.target id="marg248"/>18</s>
        </p>
        <p type="main">
          <s id="id001191">Propo&longs;itio &longs;eptuage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id001192">Si fuerint tres quantitates in continua proportione, ali&aelig;que toti&shy;<lb/>dem in continua proportione, poterunt con&longs;tituere tres quantita&shy;<lb/>tes in &aelig;quali differentia peruer&longs;im copulat&aelig;.<lb/><arrow.to.target n="marg249"/></s>
        </p>
        <p type="margin">
          <s id="id001193"><margin.target id="marg249"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id001194">Velut &longs;int a b c primi ordi&shy;<lb/><figure id="id.015.01.081.1.jpg" xlink:href="015/01/081/1.jpg"/><lb/>nis, &amp; d ef &longs;ecundi, &amp; &longs;it 28, </s>
        </p>
        <p type="main">
          <s id="id001195"><arrow.to.target n="marg250"/><lb/>b 4, c 2, &amp; d 2 1/4, e 1 1/2, f 1, tunc <lb/>iunctis a &amp; e fit 9 1/2, &amp; b &amp; d b <lb/>1/4, &amp; e cum f 3, at 3 &amp; 6 1/4 &amp; 9 1/2 <lb/>&aelig;qualiter di&longs;tant, nam diffe&shy;<lb/>rentia e&longs;t 3 1/4. At &longs;i iungatur <lb/>cum e, &amp; b cum f, &amp; c cum d <lb/>idem poterit contingere: ut in <lb/>figura uides, nam a e e&longs;t 8 1/2, <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&shy;<lb/>po&longs;ito, e&longs;t 1 1/2 p: &lt;02&gt; 1 1/4, qua excedit &amp; exceditur. </s>
          <s id="id001196">Dico modo, qua&longs;i <lb/>ex ordine coniungantur quale&longs;cunque proportiones fuerint, modo <lb/>non &longs;int amb&aelig; &aelig;qualitatis 1, ut b iungatur cum c, &amp; reliqu&aelig; ut li&shy;<lb/>bet, uelut a cum d, &amp; c cum f, uel a cum f, &amp; e cum d, nunquam fient <lb/><arrow.to.target n="marg251"/><lb/>&aelig;quales exce&longs;&longs;us, nam de primo e&longs;t clarum: nam &longs;i a cum d iun&shy;<lb/>gatur, &amp; amb&aelig; fuerint maxim&aelig;, maior e&longs;t differentia a ad b, qu&agrave;m <lb/>b ad c, &amp; maior etiam d ad e qu&agrave;m e ad f, ideo maior erit differentia <lb/>a &amp; d ad b e qu&agrave;m b e ad c f, quod erat probandum. </s>
          <s id="id001197">Eodem modo <lb/>&longs;ed laborio&longs;ius demon&longs;tratur reliquus modus &longs;cilicet, quod con&shy;<lb/>iunctio a f ad b e e&longs;t maior aut minor qu&agrave;m b e ad c d, ex hoc &longs;e&shy;<lb/>quuntur corrolaria.</s>
        </p>
        <p type="margin">
          <s id="id001198"><margin.target id="marg250"/>16</s>
        </p>
        <p type="margin">
          <s id="id001199"><margin.target id="marg251"/>17</s>
        </p>
        <p type="main">
          <s id="id001200">Primum, tres &aelig;quales quantitates non po&longs;&longs;unt diuidi in tres, &amp; <lb/>tres quantitates in continua proportione ordinat&egrave;, ut dixi, ni&longs;i u&shy;<lb/>triu&longs;que ordinis tres, ac tres inuicem &longs;int &aelig;quales.</s>
        </p>
        <p type="main">
          <s id="id001201">Secundum, tres quantitates in &aelig;quali exce&longs;&longs;u ordinate, ut dixi, <lb/>non po&longs;&longs;unt diuidi in tres, &amp; tres quantitates, qu&aelig; &longs;int in eadem <lb/>proportione quantumcunque proportiones ill&aelig; duorum ordinum <lb/>fint diuer&longs;&aelig;.</s>
        </p>
        <p type="main">
          <s id="id001202">Tertium, tres quantitates, qu&aelig; &longs;int in eadem proportione non <lb/>po&longs;&longs;unt diuidi ordinate in tres ac tres, qu&aelig; &longs;int in continua propor<lb/>tione ni&longs;i &longs;int amb&aelig; proportiones e&aelig;dem cum proportione ip&longs;a&shy;<lb/>rum quantitatum.</s>
        </p>
        <pb pagenum="63" xlink:href="015/01/082.jpg"/>
        <p type="main">
          <s id="id001203">Propo&longs;itio &longs;eptuage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra&shy;<lb/>cti ad rectam &longs;u&longs;pen&longs;ionem inuenire.</s>
        </p>
        <figure id="id.015.01.082.1.jpg" xlink:href="015/01/082/1.jpg"/>
        <p type="main">
          <s id="id001205">Sit torcularis uirga, cuius &longs;pir&aelig; a b per circui&shy;<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centupl&aelig; ad altitudinem a b, &amp; axis d c <lb/><arrow.to.target n="marg253"/><lb/>&longs;emidiametro b c centupla, &amp; quoniam per &longs;upe&shy;<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa&shy;<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen<lb/>dens per a b leuius quam per b <expan abbr="crect&atilde;">c rectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, &amp; d c &longs;int in eodem tem<lb/>pore, &amp; circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d qu&agrave;m b, &longs;ed per b circumductum cen&shy;<lb/>tuplo leuius e&longs;t, qu&agrave;m per rectam, igitur e ponderat &longs;olum particu&shy;<lb/>lam ex decem millibus recti ponderis.</s>
        </p>
        <p type="margin">
          <s id="id001206"><margin.target id="marg252"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="margin">
          <s id="id001207"><margin.target id="marg253"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s>
        </p>
        <p type="main">
          <s id="id001208">Propo&longs;itio &longs;eptuage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id001209">Proportionem ponderis &longs;ph&etail;r&aelig; pendentis ad a&longs;cendentem per <lb/>accliue planum inuenire</s>
        </p>
        <figure id="id.015.01.082.2.jpg" xlink:href="015/01/082/2.jpg"/>
        <p type="main">
          <s id="id001210">Sit &longs;ph&aelig;ra &aelig;qualis ponderi g in pun&shy;<lb/><arrow.to.target n="marg254"/><lb/>cto b, qu&aelig; debeat trahi &longs;uper b c accli&shy;<lb/>ue planum b e ad perpendiculum pla&shy;<lb/><arrow.to.target n="marg255"/><lb/>ni b f. </s>
          <s id="id001211">Quia ergo in b e mouetur a, qua&shy;<lb/>uis modica ui per dicta &longs;uperius, erit per <lb/>communem animi &longs;ententiam uis, qu&aelig; <lb/>mouebit a per e b nulla: per dicta uer&ograve; <lb/>a mouebitur ad f &longs;emper, a con&longs;tanti ui <lb/>&aelig;quali g, &amp; per b c a con&longs;tanti ui &aelig;qua&shy;<lb/>li k, &longs;icut per b d a con&longs;tanti &aelig;quali h, ergo per ultimam petitio&shy;<lb/>nem, cum termini &longs;eruent, quo ad partes eandem rationem &longs;in&shy;<lb/>guli per &longs;e, &amp; motus per b e &longs;it a nulla ui, erit proportio g ad k, ue&shy;<lb/>lut proportio uis, qu&aelig; mouet per b f ad uim, qu&aelig; mouet per <lb/>b c, &amp; uelut anguli per e b f recti ad angulum e b c, &amp; ita uis, <lb/>qu&aelig; mouet a per b f, &amp; e&longs;t, ut dictum e&longs;t, g ad uim, qu&aelig; mouet <lb/>per b d, &amp; e&longs;t h ex &longs;uppo&longs;ito, ut c b f ad e b d, igitur proportio dif&shy;<lb/>ficultatis motus a per b d ad idem a per b c, e&longs;t uelut h ad k, quod <lb/>erat demon&longs;trandum.</s>
        </p>
        <pb pagenum="64" xlink:href="015/01/083.jpg"/>
        <p type="margin">
          <s id="id001212"><margin.target id="marg254"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="margin">
          <s id="id001213"><margin.target id="marg255"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40. 7</s>
        </p>
        <p type="main">
          <s id="id001214">Propo&longs;itio &longs;eptuage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id001215">Proportionem ponderum attractorum penes figuram in pla&shy;<lb/>no inuenire.<lb/><arrow.to.target n="marg256"/></s>
        </p>
        <p type="margin">
          <s id="id001216"><margin.target id="marg256"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001217">Sint duo pondera &aelig;qualia in plano a &amp; b, &amp; &longs;it <lb/><figure id="id.015.01.083.1.jpg" xlink:href="015/01/083/1.jpg"/><lb/>a &longs;uperficies qua planum tangit dupla b &longs;uperfi&shy;<lb/>ciei, qua planum tangit: dico quod &longs;i trahantur ab <lb/>imo, quod erunt &aelig;qualia: &longs;u&longs;pendantur, &amp; erunt <lb/>&aelig;qualia ex &longs;uppo&longs;ito, &longs;ed a quie&longs;cens in plano e&longs;t <lb/>dimidium a &longs;u&longs;pen&longs;i, &amp; b quie&longs;cens in plano e&longs;t di<lb/>midium b &longs;u&longs;pen&longs;i ex demon&longs;tratis &longs;uperius, igi&shy;<lb/>tur per communem animi &longs;ententiam a &amp; b in pla&shy;<lb/>no &longs;unt &aelig;qualia.</s>
        </p>
        <p type="main">
          <s id="id001218"><arrow.to.target n="marg257"/></s>
        </p>
        <p type="margin">
          <s id="id001219"><margin.target id="marg257"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001220">Ex hoc manife&longs;tum e&longs;t, quod proportio uirium trahentium pon<lb/>dera in plano eadem e&longs;t, qu&aelig; ip&longs;orum ponderum dum &longs;u&longs;pendun&shy;<lb/>tur. </s>
          <s id="id001221">Vbi planum &aelig;quale &longs;it, &amp; &longs;olidum.</s>
        </p>
        <p type="main">
          <s id="id001222"><arrow.to.target n="marg258"/></s>
        </p>
        <p type="margin">
          <s id="id001223"><margin.target id="marg258"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s>
        </p>
        <p type="main">
          <s id="id001224">Propo&longs;itio &longs;eptuage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id001225">Proportionem concutientis ad concu&longs;&longs;um &longs;tabili inuenire.</s>
        </p>
        <p type="main">
          <s id="id001226"><arrow.to.target n="marg259"/></s>
        </p>
        <p type="margin">
          <s id="id001227"><margin.target id="marg259"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001228">Intelligo concutiens e&longs;&longs;e &longs;olidum, quod non frangitur, idque gra&shy;<lb/>uitate, &amp; impetu concutere, nam de duritie &longs;upponitur, &amp; grauitas, <lb/>ut demon&longs;trabitur in corrolario e&longs;t iuxta &longs;uperficiem inferiorem <lb/>ponderi comparatam. </s>
          <s id="id001229">Cum ergo motus concu&longs;sionis magnitudo <lb/>con&longs;tet ex grauitate, impetu &amp; figura, concu&longs;si autem ex pondere <lb/>&amp; connexione: multiplicatis inuicem partibus productorum pro&shy;<lb/>portio, erit proportio concu&longs;sionis: ut &longs;it grauitas decem, impetus <lb/>quadraginta: pondus icti centum connexio ut duo, ducemus qua&shy;<lb/>draginta in decem, &amp; fient quadringenta, et duo in centum, fient du<lb/>centa, igitur concu&longs;sio erit dupla.</s>
        </p>
        <p type="main">
          <s id="id001230"><arrow.to.target n="marg260"/></s>
        </p>
        <p type="margin">
          <s id="id001231"><margin.target id="marg260"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001232">Cum fuerit figura rotunda, concu&longs;sio erit integra in puncto: <lb/>quia &longs;ph&aelig;ra iacens in plano totum pondus in punctum cogit.</s>
        </p>
        <p type="main">
          <s id="id001233"><arrow.to.target n="marg261"/></s>
        </p>
        <p type="margin">
          <s id="id001234"><margin.target id="marg261"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001235">Si autem planum e&longs;t, quod ijcitur, proportio totius ad totum e&longs;t <lb/>minor, qu&agrave;m partis ad partem pro ratione quantitatis latitudinis. </s>
        </p>
        <p type="main">
          <s id="id001236"><arrow.to.target n="marg262"/><lb/>&longs;ed maior ratione a&euml;ris comprehen&longs;i, de quo infr&agrave;.<lb/><arrow.to.target n="marg263"/></s>
        </p>
        <p type="margin">
          <s id="id001237"><margin.target id="marg262"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 84.</s>
        </p>
        <p type="margin">
          <s id="id001238"><margin.target id="marg263"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001239">Cum proportio minor fuerit &longs;tabile, non poterit in &longs;olido plano <lb/>moueri: aliter fieret motus &agrave; debiliore, &amp; per pr&aelig;cedentem etiam <lb/>po&longs;&longs;et pari ratione eleuari.</s>
        </p>
        <p type="main">
          <s id="id001240"><arrow.to.target n="marg264"/></s>
        </p>
        <p type="margin">
          <s id="id001241"><margin.target id="marg264"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id001242">Cumque &longs;tabile non mouetur, &amp; omne agens agat aliquid nece&longs;&longs;e <lb/>e&longs;t, ut &longs;tabilis partes cedant, aut di&longs;&longs;oluantur. </s>
          <s id="id001243">Quanto ergo magis <lb/>cedit, tanto minus di&longs;&longs;oluitur.</s>
        </p>
        <pb pagenum="65" xlink:href="015/01/084.jpg"/>
        <p type="main">
          <s id="id001244">Cau&longs;&aelig; igitur qu&aelig; alleuiant ictum, ne di&longs;&longs;oluatur, &longs;unt &longs;eptem le&shy;</s>
        </p>
        <p type="main">
          <s id="id001245"><arrow.to.target n="marg265"/><lb/>uitas ictus, ponderis, fractura, mollities eius, quod icitur, mollities <lb/>eius, quod excipit ictum, motus eiu&longs;dem, &amp; figura lata, &amp; in&aelig;qua&shy;<lb/>lis. </s>
          <s id="id001246">Durities ergo, quatenus fractur&aelig; opponitur, aliud e&longs;t, quam ut <lb/>mollitiei: &amp; utra que e&longs;t cau&longs;a, qu&aelig; auget ictum, ut reliqu&aelig; <lb/> oppo&longs;it&aelig; minuunt, dicemus autem de his inferius.</s>
        </p>
        <p type="margin">
          <s id="id001247"><margin.target id="marg265"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 9.</s>
        </p>
        <p type="main">
          <s id="id001248">Propo&longs;itio &longs;eptuage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id001249">Proportionem immoti in aqua ad immotum in terra in excipien <lb/>do ictum inuenire.</s>
        </p>
        <p type="main">
          <s id="id001250">Sit pondus a in terra &aelig;quale b eiu&longs;dem natur&aelig; magnitudinis fi&shy;<lb/><arrow.to.target n="marg266"/><lb/>gur&aelig;, &amp; eodem in &longs;itu, quod &longs;it in aqua porr&ograve; a, &longs;i e&longs;&longs;et affixum ter&shy;<lb/>r&aelig; oportet, ut conuellatur, aut di&longs;&longs;oluatur aut frangatur. </s>
          <s id="id001251">Et clarum <lb/><figure id="id.015.01.084.1.jpg" xlink:href="015/01/084/1.jpg"/><lb/>e&longs;t, quod totum ictum excipit. </s>
          <s id="id001252">Si uer&ograve; <lb/>affixum non &longs;it, euertitur, &amp; tanto mino&shy;<lb/>rem partem excipit ictus, quanto faci&shy;<lb/>lior e&longs;t ad euer&longs;ionem. </s>
          <s id="id001253">Vnde nata fabu&shy;<lb/>la de quercu, qu&aelig; cum immobilis e&longs;&longs;et, <lb/>&amp; &longs;taret uento euer&longs;a e&longs;t, arundo flecten&shy;<lb/>do &longs;e, cecidit quidem, &longs;ed non e&longs;t eradi&shy;<lb/>cata. </s>
          <s id="id001254">Sermo igitur e&longs;t de b in&longs;identi aqu&etail; <lb/>in comparatione ad a, quando excipit <lb/>plenum ictum. </s>
          <s id="id001255">Cum ergo b tangitur, ex&shy;<lb/>cipit plenum ictum illo in&longs;tanti, &longs;ed quia <lb/>non excipitur ictus cedente materia, &amp; <lb/>antequam materia cedat b mouetur loco, quia in&longs;idet aqu&aelig;, ergo <lb/>non excipit ictum. </s>
          <s id="id001256">Proponatur ergo, quod moueatur b per c &longs;pa&shy;<lb/>tium in d tempore, &amp; &longs;it, ut idem b ab e ui trahatur per idem &longs;pa&shy;<lb/>tium in eodem tempore ex loco directo ad eandem partem: qua&shy;<lb/>lis ergo proportio e ad b, &amp; a&euml;rem, qui cum eo re&longs;i&longs;tit, talis propor&shy;<lb/>tio ictus f grauis puta in a ad ictum Y in b. </s>
          <s id="id001257">Quia per demon&longs;tra&shy;<lb/><arrow.to.target n="marg267"/><lb/>ta &longs;uperius proportio f ad a producitur ex proportionibus e ad b, <lb/><arrow.to.target n="marg268"/><lb/>&amp; a ad e, ergo diui&longs;a proportione f ad a per proportionem c ad b <lb/>exibit proportio ictus Y in a ad ictum Y in b quod erat demon&shy;<lb/>&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id001258"><margin.target id="marg266"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001259"><margin.target id="marg267"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 2.</s>
        </p>
        <p type="margin">
          <s id="id001260"><margin.target id="marg268"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. &amp; 43. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001261">Ex hoc patet, quod b quanto mollius, leuius, &amp; &longs;trictius in imo, <lb/><arrow.to.target n="marg269"/><lb/>&amp; in tenuiore aqua, eo minus l&aelig;detur. </s>
          <s id="id001262">Et quanto ictus lentior fue&shy;<lb/>rit etiam quod &longs;it grauius Y.</s>
        </p>
        <pb pagenum="66" xlink:href="015/01/085.jpg"/>
        <p type="margin">
          <s id="id001263"><margin.target id="marg269"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001264">Propo&longs;itio &longs;eptuage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id001265">Proportionem duorum mobilium &longs;ibi inuicem concurrentium <lb/>per rectam inuenire.<lb/><arrow.to.target n="marg270"/></s>
        </p>
        <p type="margin">
          <s id="id001266"><margin.target id="marg270"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001267">Iam cognito, quod mobilia, qu&aelig; loco mouentur per pr&aelig;ceden&shy;<lb/>tes, &longs;ed omnino quie&longs;cunt integros excipiunt ictus: alia quidem, <lb/>qu&aelig; concurrunt, non omnino re&longs;iliunt, alia uero re&longs;iliunt, &amp; qu&aelig; <lb/>re&longs;iliunt minores excipiunt ictus, &longs;equitur ut diuer&longs;a &longs;it compara&shy;<lb/>tio: nam erunt, qu&aelig; &longs;tando excipient ictus, &amp; h&aelig;c integros ut mu&shy;<lb/>ri, &amp; qu&aelig; concurrendo, nec re&longs;iliendo, ut equi cur&longs;u incitati: &amp; qu&aelig; <lb/>&longs;tando, &longs;ed re&longs;iliendo, ut naues &longs;tantes: &amp; qu&aelig; concurrendo, re&longs;i&shy;<lb/>liendo q&uacute;e ut naues uentis, &amp; triremes ab impul&longs;u: bifariam ergo <lb/>contingit intelligi, quod proponitur. </s>
          <s id="id001268">Sed in utroque etiam &longs;en&longs;u <lb/>uarietas e&longs;t: nam ut concurrit pars altera celerius, ita etiam magis <lb/>concutitur. </s>
          <s id="id001269">Et ideo &longs;it, ut proportio ict&ugrave;s &longs;it in comparatione ad <lb/>grauitatem dupl&aacute;, &amp; concurrant &aelig;qualiter, &amp; &longs;int &aelig;qu&egrave; grauia, &amp; <lb/>neutrum re&longs;iliat, erunt in proportione quadrupla, &amp; eodem mo&shy;<lb/>do &longs;i utrunque re&longs;iliat. </s>
          <s id="id001270">At &longs;i diuer&longs;o impetu ferantur, ut dixi, tria <lb/>erunt pr&aelig;cipu&egrave; con&longs;ideranda grauitas &longs;eu pondus, impetus, &amp; an <lb/>re&longs;iliat. </s>
          <s id="id001271">Quanto enim grauiora fuerint, &amp; maiore impetu agen&shy;<lb/>tur, &amp; non re&longs;ilierint eo maiorem ictum recipient: quanto leuio&shy;<lb/>ra, &amp; minore impetu, &amp; magis re&longs;ilierint, minus l&aelig;dentur. </s>
          <s id="id001272">Sed &amp; <lb/>in debilitando ictum con&longs;iderare oportet tria, quod re&longs;iliat, quod <lb/>diffugiat, quod circumuertatur: re&longs;iliunt naues, &longs;i ro&longs;tris concur&shy;<lb/>rant pleno ictu: &longs;i uer&ograve; non pleno ictu concurrant, &longs;ed diffugiant <lb/>hoc experimento compertum e&longs;t minimum e&longs;&longs;e ictum: &longs;i ro&longs;tro <lb/>tran&longs;uer&longs;um nauis feriatur medium, e&longs;t hoc.</s>
        </p>
        <figure id="id.015.01.085.1.jpg" xlink:href="015/01/085/1.jpg"/>
        <p type="main">
          <s id="id001273">Sit ergo ut a b nauis tangat ro&longs;tro b c &longs;ic ut <lb/>diffugiat, erit hypomochlium c, &amp; &longs;i tangat <lb/>e f hypomochlium e&longs;t in d dupla, ergo e&longs;t c b <lb/>ip&longs;i d e, igitur ictus duplo minor excipitur &agrave; <lb/>c b qu&agrave;m ef. </s>
          <s id="id001274">E&longs;t etiam tempus long&egrave; maius, <lb/>quo excipit ictum ef, qu&agrave;m b c: &longs;tatim enim di&longs;cedit b c occurritque<lb/>aliis partibus, in c f autem impingit, &amp; angulus a d c e&longs;t long&egrave; ma&shy;<lb/>ior recto, qu&agrave;m a b f: ob h&aelig;c igitur long&egrave; maior e&longs;t ictus c f qu&agrave;m <lb/>b c: uocant autem hoc declinationem.</s>
        </p>
        <p type="main">
          <s id="id001275">Propo&longs;itio &longs;eptuage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id001276">Proportionem motus obliqui ad motum rectum in nauibus <lb/>inuenire.</s>
        </p>
        <p type="main">
          <s id="id001277"><arrow.to.target n="marg271"/></s>
        </p>
        <p type="margin">
          <s id="id001278"><margin.target id="marg271"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001279">C&ugrave;m uentus fertur ad puppim rect&agrave;, naui&longs;q&uacute;e gubernaculum di<pb pagenum="67" xlink:href="015/01/086.jpg"/>rigitur, tendunturq&uacute;e uela ac expanduntur &longs;umma in parte mali, <lb/>tunc motus e&longs;t ueloci&longs;simus: fingamus autem, quod omnia ad <lb/>idem tendant pr&aelig;ter uentum, qui non directus &longs;it ad puppim, &longs;ed <lb/>&agrave; latere, ut uides, &amp; temo &longs;itin contrarium tantundem directus, &amp; <lb/>&longs;upponamus pro nunc, quod uelum &longs;it &longs;olum in anteriore parte <lb/>nauis, nam &longs;ecus e&longs;&longs;et nimis magna differentia, <lb/><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"/><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Qu&aelig;ritur igitur proportio motus b c ad mo&shy;<lb/>tum d e: fiat ergo c f &aelig;qualis e g, ita ut f angulus <lb/>rectus &longs;it, &amp; manife&longs;tum e&longs;t, quod h c maior e&longs;t <lb/>c f, cum ergo angulus f rectus &longs;it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod e&longs;t primum a, i&nacute;de noto angulo h c f <lb/>per ea, qu&aelig; tradita &longs;unt ab A&longs;trologis de &longs;inu &amp; <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k &aelig;qualis c h, igitur c k erit maior e g, &longs;i ergo perambula&shy;<lb/>bit &aelig;qualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota &longs;it c k, e&longs;t enim &aelig;qualis c h, erit temporis ad <lb/>tempus proportio nota. </s>
          <s id="id001280">Quod autem in &aelig;quali tempore mouebi&shy;<lb/>tur nauis per c k &amp; h c patet ex a&longs;&longs;umpto inferius declarando.</s>
        </p>
        <p type="main">
          <s id="id001281"><arrow.to.target n="marg272"/></s>
        </p>
        <p type="margin">
          <s id="id001282"><margin.target id="marg272"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 99.</s>
        </p>
        <p type="main">
          <s id="id001283">Propo&longs;itio &longs;eptuage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id001284">Propo&longs;itionem nauis ad triremes quotuis concurrentes de&shy;<lb/>mon&longs;trare.</s>
        </p>
        <p type="main">
          <s id="id001285">Sit nauis deferens pondus decuplo maius triremi, &amp; con&longs;tat, </s>
        </p>
        <p type="main">
          <s id="id001286"><arrow.to.target n="marg273"/><lb/>quod impul&longs;u &aelig;quabitur decem triremibus, ubi flante uento e <lb/>puppi &aelig;qualiter feratur in aduer&longs;um, quantum triremes ui homi&shy;<lb/>num. </s>
          <s id="id001287">Sed quoniam triremes impediuntur &agrave; uento licet &longs;ine uelis <lb/>&longs;int, habent enim &amp; ip&longs;&etail; malum, &amp; uelum, &longs;ed exigua comparatio&shy;<lb/><arrow.to.target n="marg274"/><lb/>ne nauium, ideo ictus ille multo ualidior e&longs;t ex demon&longs;tratis. </s>
          <s id="id001288">Cum <lb/>uero uis illa &longs;imul &longs;it, liquet, &apos;qu&ograve;d hoc in ca&longs;u ni&longs;i machin&aelig; ob&longs;ta&shy;<lb/>rent una nauis mille po&longs;&longs;et obruere triremes di&longs;iunctas per tantum <lb/>&longs;patium inter &longs;e, quantum e&longs;t id, in quo nauis pote&longs;t uenti impul&shy;<lb/>&longs;um recipere. </s>
          <s id="id001289">At impedimentorum maximum &longs;unt machin&aelig;, qu&aelig; <lb/>in nauim collimant &agrave; lateribus, cum triremes quaqu&acirc; uer&longs;um &longs;e a&shy;<lb/>gant, &amp; ob id proram &longs;olam exponunt ictibus, in quam difficile <lb/>e&longs;t collimare, &amp; &longs;i tangatur pars ea robu&longs;tior e&longs;t, nec periculum <lb/>euer&longs;ionis ade&ograve; in currit, ut &agrave; lateribus: nec enim ade&ograve; angu&longs;ta e&longs;t a <lb/>prora ad puppim nauis, quam &agrave; latere ad latus: his tot cau&longs;is mi&shy;<lb/>nus e&longs;t obnoxia machinis triremis, qu&aacute;m nauis. </s>
          <s id="id001290">Sed &amp; alia cau&longs;a <lb/>e&longs;t, quoniam nece&longs;&longs;e e&longs;t ut ob angulum laterum ad proram 


<pb pagenum="68" xlink:href="015/01/087.jpg"/>ictus dilabatur &longs;&etail;pius &longs;olum traiecta &longs;uperficie. </s>
          <s id="id001291">Secundum impe&shy;<lb/>dimentum e&longs;t &agrave; uento, &longs;i ualde obliquus &longs;it, nam ad rectum impul&shy;<lb/>&longs;um, multum debilitatur: aut &longs;i incon&longs;tans &longs;it, uiribusque remittatur. <lb/></s>
          <s id="id001292">Tertium uer&ograve; &longs;i triremes inuicem connex&aelig; &longs;int, ac &longs;e tangant, in <lb/>quas nauis dirigitur. </s>
          <s id="id001293">Sed &amp; hoc infr&agrave; demon&longs;trabitur nauim, ut le&shy;<lb/><arrow.to.target n="marg275"/><lb/>uior fuerit facilius elabi, &longs;ed ut pondere magis onerata grauiores <lb/>ictus inferre: ob hoc triremem inuenerunt mediam maximi u&longs;us <lb/><foreign lang="greek">a&rpar;mfh/rhn. </foreign></s>
          <s id="id001294">Galeonum uulg&ograve; uocant.</s>
        </p>
        <p type="margin">
          <s id="id001295"><margin.target id="marg273"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001296"><margin.target id="marg274"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 74.</s>
        </p>
        <p type="margin">
          <s id="id001297"><margin.target id="marg275"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 109.</s>
        </p>
        <p type="main">
          <s id="id001298">Propo&longs;itio &longs;eptuage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id001299">Proportionem medicamentorum purgantium inuicem de&shy;<lb/>clarare.<lb/><arrow.to.target n="marg276"/></s>
        </p>
        <p type="margin">
          <s id="id001300"><margin.target id="marg276"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001301">Scio, qu&agrave;m multa concurrant, etiam per &longs;e ad purgationem mul <lb/>titudo humorum pr&aelig;paratio locus propinquus, &longs;ed nobis &longs;er&shy;<lb/>mo e&longs;t pari &longs;ub conditione, ut &longs;it dimidia uncia Ca&longs;si&aelig; nigr&aelig; in tri&shy;<lb/>bus uicibus expurget libram humorum, &amp; uelim &longs;cire ab una un&shy;<lb/>cia, quoties expurgabitur, &amp; quantum. </s>
          <s id="id001302">Dico, quod in &longs;camonio, &amp; <lb/>agarico h&aelig;c ratio deprehendi pote&longs;t: in his autem medicamentis, <lb/>qu&aelig; magis leniunt, qu&agrave;m &agrave; proprietate educant, ut e&longs;t ca&longs;sia nigra, <lb/>ratio h&aelig;c non ualet, quoniam feces quando que pro maiore par&shy;<lb/>te educuntur, ita ut etiam multiplicato medicamento de&longs;it, quod <lb/>educatur. </s>
          <s id="id001303">Et quamuis humores iuxta proportionem trahat, cum <lb/>tamen feces proportionem non &longs;eruent, &longs;equitur: ut aggregati ad </s>
        </p>
        <p type="main">
          <s id="id001304"><arrow.to.target n="marg277"/><lb/>aggregatum proportio non &longs;eruetur. </s>
          <s id="id001305">At non e&longs;t facile po&longs;tmo&shy;<lb/>dum interno&longs;cere feces ab humoribus, quocirca uidetur propor&shy;<lb/>tio illa confundi. </s>
          <s id="id001306">Quod &longs;i medicamentum leniens, fiat ob quanti&shy;<lb/>tatem purgans humores, ut de multa ca&longs;sia nigra, tunc non pote&longs;t <lb/>a&longs;signari illa comparatio ni&longs;i ut e&longs;t medicamentum purgans. </s>
          <s id="id001307">Et &longs;it <lb/>gratia exempli, primum ut grana &longs;ex &longs;camonij purgent aliquem <lb/>ter, &amp; uncias decem bilis, dico iuxta rationem &longs;upra po&longs;itam, quod <lb/><arrow.to.target n="marg278"/><lb/>grana duodecim purgabunt iuxta proportionem duplam &longs;exqui&shy;<lb/>alteram, &longs;i duo grana nil purgant, &longs;ed commouent. </s>
          <s id="id001308">&aelig;qualia enim <lb/><arrow.to.target n="marg279"/><lb/>&longs;unt: ut quatuor &longs;int dupla, &amp; &longs;ex tripla, &amp; mouent ter, quia &longs;exqui&shy;<lb/>alteram habent proportionem ad exce&longs;&longs;um, igitur duodecim du&shy;<lb/>plam, &amp; &longs;exquialteram ad quatuor, nam decem ad quatuor e&longs;t du&shy;<lb/>pla &longs;exquialtera, &amp; purgabit &longs;epties cum nixu libras duas fer&shy;<lb/>me bilis. </s>
          <s id="id001309">Vt comparatio fiat exce&longs;&longs;us ad uim, qu&aelig; re&longs;i&longs;tit eodem <lb/>modo. </s>
          <s id="id001310">In ca&longs;sia ergo nigra &longs;i uncia <expan abbr="unan&otilde;">una non</expan> purga, &longs;ed lenit tantum, <lb/>&amp; du&aelig; unci&aelig; purgant ter, &amp; libram unam bilis, tres unci&aelig; duplam 


<pb pagenum="69" xlink:href="015/01/088.jpg"/>habent proportionem iuxta exce&longs;&longs;um ad unam, exce&longs;&longs;us igitur <lb/>duplum purgabunt, &amp; duplo magis, id e&longs;t pr&aelig;ter feces libras <lb/>duas bilis in &longs;ex uicibus.</s>
        </p>
        <p type="margin">
          <s id="id001311"><margin.target id="marg277"/>E<emph type="italics"/>x conuer&longs;a<emph.end type="italics"/> 18. <emph type="italics"/>quint.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001312"><margin.target id="marg278"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s>
        </p>
        <p type="margin">
          <s id="id001313"><margin.target id="marg279"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 42.</s>
        </p>
        <p type="main">
          <s id="id001314">Propo&longs;itio octuage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id001315">Proportionem motus &longs;ecundum obliquum ad rectum in &longs;pa&shy;<lb/>tio declarare.</s>
        </p>
        <p type="main">
          <s id="id001316">H&aelig;c u&iacute;detur &longs;imilis &longs;uperiori cuidam propo&longs;itioni, &longs;ed tamen in <lb/><arrow.to.target n="marg280"/><lb/>hoc differt, quoniam in c a &longs;upponimus nauim moueri, ut concu&shy;<lb/>tiat, hic autem iuxta motum &longs;olum: ut proponamus b nauim ferri <lb/><figure id="id.015.01.088.1.jpg" xlink:href="015/01/088/1.jpg"/><lb/>uer&longs;us a uento recto ex b in a: &longs;it autem uentus ex <lb/>cin a mouens nauim ex b in a: n&ograve;n enim moue&shy;<lb/>bit ut quidam putant in ratione c a ad b a: ut &longs;i ca <lb/>&longs;it &longs;exquiquarta ad b a, ut &aelig;quali impetu ex b &amp; <lb/>c flante uento moueretur tardius per c a, quam <lb/>per b a, quia &aelig;qualiter ex &longs;uppo&longs;ito: ergo tanto <lb/>tardius c fertur in a, quam b in idem quanto lon&shy;<lb/>gior e&longs;t c a, b a igitur &longs;i b perueniet in a in qua&shy;<lb/>tuor diebus c perueniet in idem a in quinque <lb/>diebus. </s>
          <s id="id001317">Hoc enim e&longs;t per &longs;e manife&longs;tum: &longs;ed non qu&aelig;rimus id, &longs;ed <lb/>ut uento c a &aelig;quali per c a ei, qui e&longs;t b a per b a, ubi b moueatur uen <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/>a per b a, quam per c a, at per c a tardius, quam ex b in a per &aelig;qua&shy;<lb/>lem uim, ergo multo tardius ex b in a per c a uentum, quam per uen <lb/>tum ex b in a. </s>
          <s id="id001321">Qu&aelig;rimus ergo compo&longs;itionem horum, ut &longs;it c <lb/>nauis, qu&aelig; debeat transferri ad a per uentum ex b, &amp; &longs;equitur, <lb/>quod tardius, quam ex c per uentum ex c in a, &amp; tardius ex b per <lb/>uentum ex cin a. </s>
          <s id="id001322">Ergo malus, qui in prora e&longs;t conuoluto eo, qui <lb/>e&longs;t in puppi, ut etiam Ari&longs;toteles docet tantundem nititur ad re&shy;<lb/><arrow.to.target n="marg281"/><lb/>ctum ex cin &aelig;quidi&longs;tantem locum ab a quantum c di&longs;tat ab con&shy;<lb/>tra temo, qui in puppi e&longs;t dirigitur ad h, &amp; &longs;i ualidius &longs;it uentus e&shy;<lb/>tiam adiuuante temonem, &longs;eu contra nitente, quantum licet mo&shy;<lb/>bili pondere nauis ad id latus, premitur enim nauis, qua&longs;i &longs;ubmer&shy;<lb/>gi debeat, uento in aduer&longs;um premente, ut &longs;i uentus repente huic <lb/>contrarius exoriatur, <expan abbr="pericul&utilde;">periculum</expan> &longs;ubeat, ne obruatur. </s>
          <s id="id001323">Cum ergo uen&shy;<lb/>tus ex b feratur, &aelig;quidi&longs;tans c h, &amp; c feratur per temonem in k, &amp; ab <lb/>oppo&longs;itis &aelig;qualis actio &longs;equatur, im&ograve; tota impeditur, ex c in h fere&shy;<lb/>tur iuxta proportionem anguli, quem con&longs;tituit h c cum a c ad to&shy;<lb/>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob 


<pb pagenum="70" xlink:href="015/01/089.jpg"/>uim uenti, &amp; ui&aelig; longitudinem, angulus uer&ograve; h c a &longs;it &longs;exta re&shy;<lb/>cti pars, feretur ex c uer&longs;us a ad quantitatem b a in quatuorde&shy;<lb/>cim horis: igitur rur&longs;us quanta e&longs;t proportio c a ad b a tan&shy;<lb/>tum e&longs;t temporis, in quo fertur ex c ad a ad quatuor decim horas <lb/>per uentum b a.</s>
        </p>
        <p type="margin">
          <s id="id001324"><margin.target id="marg280"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001325"><margin.target id="marg281"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 7. M<emph type="italics"/>echanica.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001326">Propo&longs;itio octuage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id001327">Qualis &longs;it angulus, per quem pote&longs;t moueri nauis ad rectum <lb/>explorare.<lb/><arrow.to.target n="marg282"/></s>
        </p>
        <p type="margin">
          <s id="id001328"><margin.target id="marg282"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001329">Cum in pr&aelig;cedenti propo&longs;itione o&longs;ten&longs;um &longs;it angulum k c a <lb/>oportere e&longs;&longs;e &aelig;qualem angulo h c a, ut feratur, c in a uento c h, nec <lb/>tamen pror&longs;us, &longs;ed temo magis inflectit uer&longs;us k quam uentus co&shy;<lb/>git uer&longs;us h: &longs;icut contra maiori ui uentus dirigit ad h, qu&agrave;m temo <lb/>ad k, ut nece&longs;&longs;e &longs;it nauim flecti ad k pondere, ideo &longs;i uentus e&longs;&longs;et <lb/>tran&longs;uer&longs;us periclitaretur, nece&longs;&longs;e e&longs;t, ut per omnes uentos, qui fe&shy;<lb/>runt ab ea, qu&aelig; ad perpendiculum &longs;uper c a, &amp; &longs;unt quatuor decim: <lb/>&longs;ed quoniam, ut dixi, pondere adiuuante uis uenti minor fit, nece&longs;&shy;<lb/>&longs;e e&longs;t, ut per uentos debiliores feratur magis ab extremis, qui pro&shy;<lb/>pe perpendiculum &longs;unt: ita ut numerus omnium &longs;it, cum leui&longs;simi <lb/>fuerint, quatuor decim, cum uiolenti&longs;simi, tres tantum proprius, &amp; <lb/>qui di&longs;tant trige&longs;ima &longs;ecunda parte totius circuli, id e&longs;t partibus un<lb/>decimi, cum quarta reliqui undecim, medij &longs;unt: ut tanto plures a&longs;&shy;<lb/>&longs;umi po&longs;sint &agrave; Nauclero, quanto molliores &longs;unt uenti, tanto pau&shy;<lb/>ciores, quo uiolentiores. </s>
          <s id="id001330">Tutius autem fuerit in ualidis uentis diri&shy;<lb/>gere nauim per uentum proximiorem, quam per ip&longs;ummet, qui re&shy;</s>
        </p>
        <p type="main">
          <s id="id001331"><arrow.to.target n="marg283"/><lb/>ct&egrave; tendit ad locum. </s>
          <s id="id001332">Veluti tendat nauis ex a in b, uentus tendat in <lb/>c ualidior, cumque magnus fuerit angulus c a b, ut pot&egrave; dodrans to&shy;<lb/>tius recti, ut e&longs;&longs;et temo dirigendus ad &longs;extum uentum altrin&longs;ecus di <lb/>rigemus &longs;olum ad quintum, ut feratur in d, &amp; hoc erit tanto cele&shy;<lb/>rius, &amp; celerius feratur per a d &amp; d b, qu&agrave;m &longs;i nauis recta lata e&longs;&longs;et <lb/>ex a in b. </s>
          <s id="id001333">in&longs;uper tutius.</s>
        </p>
        <p type="margin">
          <s id="id001334"><margin.target id="marg283"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 83</s>
        </p>
        <p type="main">
          <s id="id001335">Propo&longs;itio octuage&longs;ima&longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id001336">Proportionem uelorum indagare.<lb/><arrow.to.target n="marg284"/></s>
        </p>
        <p type="margin">
          <s id="id001337"><margin.target id="marg284"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001338">Vela tribus in locis di&longs;poni &longs;olent dolo b, quod in prora con&shy;<lb/>&longs;tituitur, &amp; in malo, qui ponitur in medio ratione, qu&aelig; inferius <lb/>o&longs;tendetur, &longs;ed non ad unguem, quia cum malus in anteriorem <lb/>partem &agrave; uento impellatur, &longs;i e&longs;&longs;et in medio, &longs;emper pr&aelig;meretur <lb/>nauis in anteriorem partem, ex quo duo magna incommoda &longs;eque <lb/>rentur: prim&ugrave;m ut periculum &longs;ubiret, ne inuer&longs;a in anteriorem par&shy;


<pb pagenum="71" xlink:href="015/01/090.jpg"/>tem &longs;ubmergeretur. </s>
          <s id="id001339">Secundum ne pre&longs;&longs;a in parte anteriore dif&shy;<lb/>ficilius aquas di&longs;&longs;ecaret, &amp; ob id longe tardius, moueretur. </s>
          <s id="id001340">Pro&shy;<lb/>pter h&aelig;c duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>&lpar;quod uulgati&longs;simum maioribus no&longs;tris &verbar;fuit&rpar; in parte magis <lb/>pror&aelig; proxima locabatur &agrave; gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien<lb/>te &agrave; ro&longs;tro in be&longs;&longs;e &agrave; puppi: Rarum fuit, &amp; memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou&amp;</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue&shy;<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>
          <s id="id001341">Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, &amp; omnium mini&shy;<lb/>mum, ut nauis facile ab eo inuerteretur. </s>
          <s id="id001342">Vnde etiam nunc minus <lb/>minime habent tam quantitate, quam etiam altitudine, quod uo&shy;<lb/>cant Trinehetum, &longs;olum enim &longs;u&longs;tinet nauim, qu&aelig; &agrave; uentis, uel un&shy;<lb/>dis mergi &longs;olet: ab undis ubi humilior e&longs;t, &agrave; uentis &agrave; lateribus, et an&shy;<lb/>teriore parte. </s>
          <s id="id001343">Vnde humile, &amp; exiguum uelum efficit, ut nauis ante&shy;<lb/>riore parte leuis, nec mergatur prona &agrave; uentis, nec aquas ea exci&shy;<lb/>piat, nec tamen impelli pote&longs;t nauis in &longs;copulos, nec euerti ob cau&shy;<lb/>&longs;as dictas: ob qu&aelig; in magnis tempe&longs;tatibus hoc ip&longs;o duntaxat uti <lb/>&longs;olent. </s>
          <s id="id001344">Quod et&longs;i nimium &longs;&aelig;uierint, etiam illud demittunt, &amp; &longs;i <lb/>fieri pote&longs;t, etiam malum ip&longs;am quamuis &longs;ine uelo &longs;it. </s>
          <s id="id001345">Sed plerun&shy;<lb/>que circumuolutam, &amp; implicatam &longs;olet antennam annexam, at&shy;<lb/>que &longs;u&longs;pen&longs;am habere. </s>
          <s id="id001346">Sed &amp; ne nauis pror&longs;um obruatur, quo&shy;<lb/>niam ea pars omnem uentorum uim excipere &longs;olet, &amp; ut leui&longs;sima <lb/>&longs;it ijdem Gubernatores puppim multa arena, lapillis q&uacute;e onerant. <lb/></s>
          <s id="id001347">Ergo uelocitas nauis &agrave; uentorum impetu, eorumq&uacute;e rectitudi&shy;<lb/>ne &agrave; uelorum magnitudine, &amp; loco humiliore, aut &longs;ublimiore ha&shy;<lb/>betur: tum nauis leuitate, &amp; forma. </s>
          <s id="id001348">Qu&aelig; enim non merguntur ut <lb/><foreign lang="greek">droma/des</foreign> &lpar;&longs;ic enim uocat Ari&longs;tophanes&rpar; eas, quas nunc uulgus fre&shy;<lb/>gatas appellat&rpar; qua&longs;i aquas innatantes cur&longs;u &longs;unt ueloci&longs;sim&aelig;. </s>
          <s id="id001349">Et <lb/>longiores latis. </s>
          <s id="id001350">Po&longs;t has &longs;unt, qu&aelig; carinam habent tenuem, ut fa&shy;<lb/>cile aquas diuidant. </s>
          <s id="id001351">Vltimo loco, qu&aelig; qua&longs;i medi&aelig;, ante quidem <lb/>tenues, p&ograve;&longs;t latiores ad uelocem cur&longs;um, &amp; ferendum onera apt&aelig;, <lb/>&amp; humiles altis: &amp; leui ex ligno. </s>
          <s id="id001352">Sed nos de uelorum uarieta&shy;<lb/>te loquimur, non ea&apos;, qu&aelig; ad malos pertinet. </s>
          <s id="id001353">Con&longs;tat enim me&shy;<lb/>dio loco plus mouere, quam in extremis, ut infr&agrave; docebi&shy;<lb/>mus. </s>
          <s id="id001354">Antiquo enim tempore opus non fuit malorum mul&shy;<lb/>titudine, quoniam &longs;ijderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maxim&egrave; ob mo&shy;<lb/>tus obliquitatem in circulo ui&longs;us: ide&ograve; mali multi confu&shy;<lb/>&longs;ionem in cur&longs;u, &amp; impedimentum in naui, maiu&longs;q&uacute;e pericu&shy;<lb/>lum attuli&longs;&longs;ent. </s>
          <s id="id001355">At nunc inuenta pyxide, &amp; lapidis Her&shy;


<pb pagenum="72" xlink:href="015/01/091.jpg"/>culei auxilio pluribus locis uela di&longs;po&longs;ita melius dirigunt iter, ut <lb/>qua&longs;i cra&longs;&longs;a minerua depictum, &amp; pote&longs;tate deformatum, ad amu&longs;&shy;<lb/>&longs;im contrahant. </s>
          <s id="id001356">Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uperficiei ueli ad uelum longitudine quidem, </s>
        </p>
        <p type="main">
          <s id="id001357"><arrow.to.target n="marg285"/><lb/>ac latitudine conflata per multiplicationem. </s>
          <s id="id001358">Altitudinis quo que ut <lb/><arrow.to.target n="marg286"/><lb/>infr&agrave; exponetur. </s>
          <s id="id001359">Ex quorum omnium ductu, qua&longs;i cubica, uel tri&shy;<lb/>plicata ratione, ut &longs;uperius o&longs;ten&longs;um e&longs;t, ratio uelocitatis motus na <lb/>uium conflatur.</s>
        </p>
        <p type="margin">
          <s id="id001360"><margin.target id="marg285"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 86.</s>
        </p>
        <p type="margin">
          <s id="id001361"><margin.target id="marg286"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 42.</s>
        </p>
        <p type="main">
          <s id="id001362">Propo&longs;itio octuage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id001363">Proportionem rece&longs;&longs;us &agrave; recta uia ad obliquitatem inue&longs;tigare.<lb/><arrow.to.target n="marg287"/></s>
        </p>
        <p type="margin">
          <s id="id001364"><margin.target id="marg287"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001365">Sit nauis in a itura in b &lpar;uentus rectus ad c, medius ad e&rpar; per <expan abbr="ob&shy;liqu&utilde;">ob&shy;<lb/>liquum</expan>, cum ergo tardius moueatur per a e qu&agrave;m a c &amp; per a b, quam <lb/>per a d, &amp; &longs;int ad perpendiculum b e, b d quas con&longs;tat e&longs;&longs;e breui&longs;si&shy;<lb/>mas earum, qu&aelig; ad a c &amp; ad a d. </s>
          <s id="id001366">Queritur igitur quando uelocius <lb/><figure id="id.015.01.091.1.jpg" xlink:href="015/01/091/1.jpg"/><lb/>ferretur ad b, an cum per a c, c b, an cum per a d, d b, <lb/>an cum per a b &longs;impliciter. </s>
          <s id="id001367">Et con&longs;tat quod a d &amp; d b <lb/>longiores &longs;unt a b, i&longs;tud enim demon&longs;tratum e&longs;t ab <lb/>Euclide in primo Elementorum, dico modo a c, &amp; </s>
        </p>
        <p type="main">
          <s id="id001368"><arrow.to.target n="marg288"/><lb/>c b e&longs;&longs;e longiores a d &amp; d b, nam quadrata a d &amp; d b <lb/>&amp; a c &amp; c b &longs;unt &aelig;qualia quadrato a b per dicta ibi&shy;<lb/><arrow.to.target n="marg289"/><lb/>dem, &amp; ideo quadrata a c &amp; c b &etail;qualia quadratis a d <lb/>&amp; d b, &longs;ed a d e&longs;t longior a c, quia ducta c d angulus <lb/>d c a e&longs;t obtu&longs;us, igitur ad maiorem a c per decimam <lb/>nonam primi Elementorum: quare per communem <lb/>animi &longs;ententiam quadratum a d maius e&longs;t quadrato a c, quare rur&shy;<lb/>&longs;us per communem animi &longs;ententiam quadratum c b maius e&longs;t <lb/>quadrato d b. </s>
          <s id="id001369">Cum ergo quadrata a d &amp; d b &aelig;qualia &longs;int quadra&shy;<lb/>tis a c &amp; c b, &amp; a d &longs;it maior a c &amp; c b maior d b, &longs;equitur per nonam <lb/>&longs;ecundi Elementorum, quod a c &amp; c d &longs;int maiores a d &amp; d b pari&shy;<lb/>ter acceptis. </s>
          <s id="id001370">Si ergo maior fuerit exce&longs;&longs;us qu&agrave;m proportio motus <lb/>per temonem cohibiti, ut &longs;upra ui&longs;um e&longs;t, tardius mouebitur per <lb/>a d, d b qu&agrave;m a b per a c, c b qu&agrave;m per a d, d b, &longs;ed &longs;i contr&agrave; maior &longs;it <lb/>proportio motus cohibiti &agrave; temone ad motum liberum qu&agrave;m ex&shy;<lb/><arrow.to.target n="marg290"/><lb/>ce&longs;&longs;us ad exce&longs;&longs;um uelocius mouebitur per a d d b, qu&agrave;m per a b, <lb/>&amp; per a c qu&agrave;m per a b. </s>
          <s id="id001371">Accedit huc e incommodo longioris ui&aelig;, <lb/>quod uento a c non poterit ferri nauis ex c d in b, quoniam antea <lb/>&aelig;gre ferebatur: &amp; nunc &aelig;grius per c b qu&agrave;m a b, plus enim di&longs;tat <lb/>uentus a c ab itinere c a qu&agrave;m &agrave; uento a b, ut ui&longs;um e&longs;t &longs;uperius, igi&shy;<lb/>tur multo melius e&longs;t &lpar;ni quid ob&longs;tet&rpar; ire per a b qu&agrave;m per <expan abbr="ull&atilde;">ullam</expan> aliam <lb/><arrow.to.target n="marg291"/><lb/>uiam: ni&longs;i &longs;tationes &longs;int in c d, uel periculum immineat in a b. </s>
          <s id="id001372">Vbi ta<lb/>men uenti &longs;ecundarent, tantum e&longs;t uirium in recto cur&longs;u, &amp; &aelig;quali 


<pb pagenum="73" xlink:href="015/01/092.jpg"/>uelocitate ferretur citius ex a in b per a d d b, &amp; etiam citius per a c, <lb/>c b in b quam per ip&longs;am a b, quod fuit propo&longs;itum declarare.</s>
        </p>
        <p type="margin">
          <s id="id001373"><margin.target id="marg288"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 20.</s>
        </p>
        <p type="margin">
          <s id="id001374"><margin.target id="marg289"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 47.</s>
        </p>
        <p type="margin">
          <s id="id001375"><margin.target id="marg290"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 80.</s>
        </p>
        <p type="margin">
          <s id="id001376"><margin.target id="marg291"/>P<emph type="italics"/>er<emph.end type="italics"/> 81. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001377">Propo&longs;itio octuage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id001378">Di&longs;tantiam centri terr&aelig; &agrave; centro mundi per motum lapidis Her<lb/>culei declarare.<lb/><arrow.to.target n="marg292"/></s>
        </p>
        <p type="margin">
          <s id="id001379"><margin.target id="marg292"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001380">Non me later Ari&longs;totelem exi&longs;timare centrum mundi e&longs;&longs;e cen&shy;<lb/>trum terr&aelig; illudque proba&longs;&longs;e, quod tamen ex demon&longs;tratione no&longs;tra <lb/>mathematica apparet nunc &longs;ubijciam, &amp; quid ad illius rationes di&shy;<lb/>cendum &longs;it, ali&acirc;s etiam dicendum erit: nam liber hic, ut mathemati&shy;<lb/>ca decet, e&longs;&longs;e debet ab omnibus contentionibus ab&longs;olutus. </s>
          <s id="id001381">Con&shy;<lb/>&longs;tat &longs;an&egrave; non e&longs;&longs;e propriam uim lapidis illius, ut qui non &longs;it circum&shy;<lb/>&longs;criptus &longs;ed fru&longs;tulum quoduis id pote&longs;t, neque per &longs;e, &longs;ed in ferro &amp; <lb/>pendulo, nec fieri pote&longs;t, ut &longs;it illius <expan abbr="t&atilde;quam">tanquam</expan> &longs;peciei unius lapidum, <lb/>&longs;ed qua&longs;i perfect&aelig; portionis cuiu&longs;dam generis terr&aelig;, qu&aelig; ab&longs;olu&shy;<lb/>ta &longs;it, cuius indicium e&longs;t illius copia, neque enim ullibi non inuenitur, <lb/>&amp; ubi ferrum effoditur, ut in Ilua In&longs;ula Tyrrheno mari, e&longs;t ergo fer <lb/><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"/><lb/>ri uis terr&aelig; marit&aelig;, qu&aelig; perfecta in &longs;uo ge&shy;<lb/>nere, ubi uim fecundam acceperit &agrave; ma&longs;cu&shy;<lb/>lo &longs;cilicet Herculeo lapide, qu&aelig;rit primum <lb/>ut de&longs;cendat, ubi hoc non po&longs;sit <expan abbr="&longs;alt&etilde;">&longs;altem</expan> qu&aelig;&shy;<lb/>rit, ut quie&longs;cere po&longs;sit. </s>
          <s id="id001382">Vt ergo quie&longs;cat &agrave; <lb/>motu c&oelig;li qui e&longs;t ab Oriente in Occiden&shy;<lb/>tem iuxta axis c&oelig;li &longs;itum &longs;e dirigit, quod <lb/>ille &longs;olus quie&longs;cat in &longs;uo motu, uel &longs;altem <lb/>tardi&longs;sim&egrave; moueatur: indicio e&longs;t quod &longs;i <lb/>extra &longs;itum illum acus ferrea imbuta eo lapide ponatur, &longs;tatim tre&shy;<lb/>mit uehementer, ade&ograve; ut nec momento ullo con&longs;i&longs;tat, &longs;ed mi&longs;er&egrave; &amp; <lb/>grauiter torqueri uideatur, non ergo quod &longs;entiat polorum locum <lb/>qui tantum abe&longs;t ab illa, ut nec ab homine perito mathematicarum, <lb/>&longs;ed quod uix illa c&oelig;li &longs;entiatur circa centrum mundi. </s>
          <s id="id001383">Cuius indi&shy;<lb/>cio e&longs;t Oceani maris, aquarum fluxus &amp; refluxus. </s>
          <s id="id001384">Duos ergo ha&shy;<lb/>bet motus terra perfecta, &longs;eu ferrum lapide Herculeo <expan abbr="imbut&utilde;">imbutum</expan> &longs;ub&shy;<lb/>ordinatos imperfectum perfecto: perfectus e&longs;t, ut de&longs;cendat ad cen<lb/>trum terr&aelig;, ut ibi quie&longs;cat: imperfectum, cum &agrave; perfecto prohibe&shy;<lb/>tur, ut quie&longs;cat &longs;altem extra centrum cum in clinatione ad centrum, <lb/>et hoc fiet &longs;i &longs;ecundum longitudinem acus dirigatur per axem mun <lb/>di, cum &longs;itu tamen de&longs;cen&longs;ui ad terr&aelig; centrum proximiore, ut &longs;&aelig;pi&shy;<lb/>us &longs;uperius declarauimus, dum de motu grauium &amp; pr&aelig;cipu&egrave; li&shy;<lb/>br&aelig;, &amp; centro grauitatis loqueremur. </s>
          <s id="id001385">Quibus demon&longs;tratis tum <lb/>experimento tum ratione &agrave; Fortunio Affaytato Cremonen&longs;i Me&shy;<lb/>dico, cum per h&aelig;c po&longs;tmodum cogeretur fateri acum ad polum 


<pb pagenum="74" xlink:href="015/01/093.jpg"/>tendere, cum tamen tendat &agrave; dextro latere &longs;cilicet ab Oriente no&shy;<lb/>uem partibus, &longs;eu decima parte unius recti in centro terr&aelig;, qu&aelig; e&longs;t <lb/>quadrage&longs;ima totius ambitus c&oelig;li. </s>
          <s id="id001386">Statuatur centrum mundi a, &amp; <lb/>b a c axis, &longs;ecundum quam mouetur motu diurno, ita l a dextra exit <lb/>oriens, k a &longs;ini&longs;tra occidens, &amp; &longs;tatuatur d centrum terr&aelig;, &longs;eu &longs;upr&agrave; <lb/>&longs;eu infr&agrave;, non tamen in linea b c, &longs;ed uel &longs;upr&agrave; in dextra parte, uel in&shy;<lb/>fr&agrave; in &longs;ini&longs;tra, ita ut ducta linea per illud punctum arcus b g &longs;it no&shy;<lb/>uem partium. </s>
          <s id="id001387">Con&longs;tituta ergo acu in e puncto, ubi linea h ad g &longs;ecat <lb/>peripheriam terr&etail; dico, quod acus dirigetur per h g, &amp; non per b c, <lb/>nam acus mouetur ad centrum per eam, &amp; in eo &longs;itu tota dirigitur, <lb/>quia omnes partes grauis con&longs;entiunt in motu principij grauitatis <lb/>ad centrum, hoc enim demon&longs;tratum: nixus ergo e&longs;t ut moueatur <lb/>per c d, &amp; in eo nixu qui e&longs;t quies cu&longs;todit lineam axis, qu&aelig; e&longs;t a b, <lb/>ut quie&longs;cat, ergo non quie&longs;cet, ni&longs;i in linea d g, quod erat demon&shy;<lb/>&longs;trandum. </s>
          <s id="id001388">Qu&aelig; autem &longs;equuntur ex his corrolaria omnia concor&shy;<lb/>dant cum experimentis. </s>
          <s id="id001389">Ergo hic &longs;ermo e&longs;t demon&longs;tratiuus, ut e&shy;<lb/>nim bene dixit Auerroes: Sermo demon&longs;tratiuus &longs;atisfacit omni&shy;<lb/>bus problematibus qu&aelig; <expan abbr="c&otilde;tingunt">contingunt</expan> circa principale qu&aelig;&longs;itum. </s>
          <s id="id001390">Ex <lb/>hoc ergo patet, quod angulus di&longs;tantia d ab a in latitudine e&longs;t deci&shy;<lb/>ma pars recti, et quod quanto magis di&longs;tatin longitudine centrum <lb/>terr&aelig; &agrave; centro mundi, tanto etiam minus di&longs;tatin latitudine. </s>
          <s id="id001391">H&aelig;c <lb/>enim &longs;unt demon&longs;trata clar&egrave; in mathematicis. </s>
          <s id="id001392">Vnde fieri po&longs;&longs;et <lb/>quod h&aelig;c quantitas di&longs;tanti&aelig; e&longs;&longs;et res, per quam exigua etiam &longs;i <lb/>non e&longs;&longs;et maior quatuor digitis &longs;ufficeret, modo etiam per ualde <lb/>paruum &longs;patium di&longs;taret ab eodem in longitudine. </s>
          <s id="id001393">De cau&longs;a au&shy;<lb/>tem huius differenti&aelig; ali&acirc;s dicendum erit, hic locus non e&longs;t, &longs;ed &longs;uf&shy;<lb/>ficit &longs;cire quod ita &longs;it, quod &longs;i mobilis &longs;it punctus d, clarum e&longs;t ali&shy;<lb/>quando futurum ut minus di&longs;tet g &agrave; b, aliquando ut &longs;it idem. </s>
          <s id="id001394">Et <lb/>quali&longs;cunque motus &longs;it, nece&longs;&longs;e e&longs;t eam di&longs;tantiam uariari.</s>
        </p>
        <p type="main">
          <s id="id001395">Propo&longs;itio octuage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id001396">Proportio ponderis unius grauis ad aliud &longs;ub eadem men&longs;ura <lb/>e&longs;t, ueluti eiu&longs;dem ad differentiam ponderis ua&longs;is repleti ex altero <lb/>graui, &amp; ex ambobus detracto priore.</s>
        </p>
        <p type="main">
          <s id="id001397"><arrow.to.target n="marg293"/></s>
        </p>
        <p type="margin">
          <s id="id001398"><margin.target id="marg293"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001399">Sit aurum a, &amp; liquor b, qu&aelig; repleant uas c, &amp; <lb/>pondus amborum &longs;it librarum quadraginta, &amp; <lb/><figure id="id.015.01.093.1.jpg" xlink:href="015/01/093/1.jpg"/><lb/>uas repletum liquore &longs;olo &longs;it librarum xxix, au&shy;<lb/>rum autem &longs;it ponderis librarum xij, igitur reli&shy;<lb/>quum erit ponderis xxviij, differentia ergo ua&shy;<lb/>&longs;is pleni, &amp; non pleni liquore e&longs;t libra una, pon&shy;<lb/>dus auri e&longs;t librarum duodecim: dico quod au&shy;<lb/>ri pondus e&longs;t duodecuplum ponderi liquoris, &amp; <pb pagenum="75" xlink:href="015/01/094.jpg"/>&longs;i fui&longs;&longs;et pondus amborum libr&aelig; xxxix, manentibus reliquis, &longs;eque <lb/>retur quod pondus liquoris e&longs;&longs;et xxvij, &amp; quia plenum uas &longs;uppo&shy;<lb/>nitur e&longs;&longs;e librarum xxix, e&longs;&longs;et differentia libr&aelig; ij, at auri pondus e&longs;t <lb/>libr&aelig; xij, igitur proportio ponderis auri ad liquorem e&longs;&longs;et &longs;excu&shy;<lb/>pla. </s>
          <s id="id001400">Nam &longs;i uas plenum liquore ex &longs;uppo&longs;ito e&longs;t librarum xxix, &amp; <lb/>cum auro xl, gratia exempli, &amp; auri pondus e&longs;t xij, igitur liquoris <lb/>pondus e&longs;t xxviij librarum: &longs;ed cum liquor &longs;it corpus &longs;imilium par&shy;<lb/>tium, igitur loci ad lo cum, ut ponderis ad pondus, ergo dum ade&longs;t <lb/>aurum, liquor occupat xxviij partes cxxxix, totius ua&longs;is igitur au&shy;<lb/>rum continet unam partem tantum, &amp; cum aurum pondus habeat <lb/>librarum xij, &amp; liquor unius: quia totum uas cxxxix librarum dum <lb/>e&longs;t plenum, &amp; e&longs;t diui&longs;um in xxix partes, igitur pondus unius par&shy;<lb/>tis liquoris e&longs;t una libra, igitur pondus auri e&longs;t duodecuplum ad <lb/>pondus liquoris quod fuit propo&longs;itum.</s>
        </p>
        <p type="main">
          <s id="id001401"><arrow.to.target n="marg294"/></s>
        </p>
        <p type="margin">
          <s id="id001402"><margin.target id="marg294"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001403">Ex quo &longs;equitur qu&ograve;d &longs;i ducatur pondus illud partis per pon&shy;<lb/>dus repleti ua&longs;is ex alio graui, &amp; productum diuidatur per differen<lb/>tiam illam, prodibit pondus ua&longs;is repleti liquore graui.</s>
        </p>
        <p type="main">
          <s id="id001404"><arrow.to.target n="marg295"/></s>
        </p>
        <p type="margin">
          <s id="id001405"><margin.target id="marg295"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001406">Exemplum, &longs;i pondus auri fuerit librarum xij, pondus ua&longs;is re&shy;<lb/>pleti liquore xxix librarum, pondus auri &amp; liquoris replentium <lb/>uas xxxix librarum, ducemus xij in xxix fit cccxlviij, diuido perij <lb/>differentiam xxvij ponderis ua&longs;is, repleti ex ambobus detracto au&shy;<lb/>ri pondere, &amp; xxix ponderis ua&longs;is repleti liquore exit clxxiiij, &amp; tan <lb/>tum auri uas illud continebit, nam cum du&aelig; partes quas occupa&shy;<lb/>bat aurum e&longs;&longs;ent ponderis librarum xij, totum quod erat partium <lb/>xxix, continebit decies &amp; quater cum dimidio illud aurum xij, aut <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 &longs;uperiore propo&longs;itione docui, quod ferrum e&longs;t ue&shy;<lb/>ra terra: uolui &longs;cire qualis e&longs;&longs;et proportio ferri ad aquam. </s>
          <s id="id001409">Accepi ur<lb/>ceum cuius aqua dum plenus e&longs;&longs;et ponderis, fuit unciarum &longs;ex, &amp; <lb/>&longs;eptuncis unci&aelig;, &amp; &longs;eptuncis duodecim&aelig; partis unci&aelig; &amp; pondus <lb/>ferri unci&aelig; &longs;eptem, &amp; triens unci&aelig; &amp; triens duodecim&aelig; partis un&shy;<lb/>ci&aelig;: &amp; ua&longs;is aqu&etail; &amp; ferro eodem repleti unci&aelig; tredecim, &amp; duode&shy;<lb/>cima &amp; &longs;eptunx duodecim&aelig; partis unci&aelig;. </s>
          <s id="id001410">Detrahemus ergo vij &amp; <lb/>trientem &amp; trientem duodecim&aelig;. </s>
          <s id="id001411">i. </s>
          <s id="id001412">7 &amp; 64/144 pondus ferri ex 13 19/144, &amp; <lb/>relinquentur 5 99/144, detrahe ex 6 81/144, pondere aqu&aelig; totius ua&longs;is relin<lb/>quuntur 17/18, diuide 7 64/144 per 17/18 exit proportio ponderis ferri ad pon<lb/>dus aqu&aelig; 7 15/17. Et hoc e&longs;t proximum ei quod dixit Philo&longs;ophus de <lb/>proportione ponderis terr&aelig; &amp; aqu&aelig;.</s>
        </p>
        <p type="main">
          <s id="id001413"><arrow.to.target n="marg296"/></s>
        </p>
        <p type="margin">
          <s id="id001414"><margin.target id="marg296"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001415">Ex hoc patet &longs;olutio problematis cuiu&longs;dam propo&longs;iti aliasque mi <lb/>nus bene &longs;oluti c&ugrave;m cau&longs;am habeat manife&longs;ti&longs;simam, &longs;cilicet quod 


<pb pagenum="76" xlink:href="015/01/095.jpg"/>ua&longs;e aqua pleno impo&longs;itis &longs;en&longs;im centum aureis coronatis nihil ef&shy;<lb/>funditur, non quod quicquam ab&longs;umatur in metallo, &longs;ed cau&longs;a e&longs;t <lb/>quod cum aurum &longs;it duplum pondere ferro, erit ex demon&longs;tratis <lb/>&longs;ex decuplum ad pondus aqu&aelig;. </s>
          <s id="id001416">Igitur cum &longs;it proportio ponderis <lb/>auri ad differentiam &longs;patij eadem, &longs;i &longs;it uas aqu&aelig; ponderis libr&aelig; <lb/>unius &amp; medi&aelig;, erit pondus totum xxiij unciarum, igitur aqua de&shy;<lb/>ficiet &longs;olum ex decimaoctaua parte &longs;eu cre&longs;cet ex impo&longs;itione auri, <lb/>&longs;ed illa pars in tumore aqu&aelig; ab&longs;umitur, <expan abbr="n&otilde;">non</expan> &longs;olum, quia <lb/><figure id="id.015.01.095.1.jpg" xlink:href="015/01/095/1.jpg"/><lb/>dum aureos imponimus plana &longs;olum &longs;it, &longs;ed quia non ex <lb/>quauis rotunditate defluit, aliter in urceo tam exiguo <lb/>non po&longs;&longs;et apparere rotunda: quod enim rotunditas to&shy;<lb/>tius terr&aelig;, qu&aelig; etiam planam o&longs;tendit totam unam re&shy;<lb/>gionem ad rotunditatem qu&aelig; apparet in exiguo urceo <lb/>aqu&aelig;. </s>
          <s id="id001417">E&longs;t igitur rotunditas illa potius ob lentorem aqu&etail; qui auge&shy;<lb/>tur &agrave; lentore argenti, &amp; etiam magis auri, cum &longs;en&longs;u digitorum per&shy;<lb/>cipiatur.</s>
        </p>
        <p type="main">
          <s id="id001418"><arrow.to.target n="marg297"/></s>
        </p>
        <p type="margin">
          <s id="id001419"><margin.target id="marg297"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001420">Ex hoc apparet ratio quomodo Archimedes potuerit deprehen<lb/>dere coronam &agrave; Hierone propo&longs;itam quantum auri &amp; argenti con<lb/>tineret. </s>
          <s id="id001421">Sit ergo uas a b aqua <expan abbr="plen&utilde;">plenum</expan> ponderis unciarum triginta, &amp; <lb/>cum libra auri &longs;it ponderis unciarum quadraginta unius, &amp; cum li&shy;<lb/>bra argenti ponderis unciarum quadraginta cum dimidio, igitur <lb/>erit auri pondus ad aqu&aelig; pondus duodecuplum, argenti autem <lb/>ad idem octuplum, quare auri ad <expan abbr="arg&etilde;tum">argentum</expan> pondus &longs;exquialterum. <lb/></s>
          <s id="id001422">Ponamus ergo quod corona impo&longs;ita ex auro &amp; argento &longs;olo fa&shy;<lb/>bricata &lpar;hoc enim &longs;upponere oportet&rpar; fuerit unciarum &longs;exaginta, <lb/>pondus autem aqu&aelig; content&etail; cum corona in ua&longs;e unciarum uigin<lb/>ti quatuor cum dimidio, &longs;cilicet totum octuaginta quatuor cum di&shy;<lb/>midia, erit ergo proportio ponderis coron&aelig; ad pondus aqu&aelig;, ut <lb/>cxx ad xi, aurum igitur e&longs;t proportione duodecuplum, argentum <lb/>autem octuplum, corona ut cxx ad xi. </s>
          <s id="id001423">Con&longs;tituantur &longs;ub ei&longs;dem ra&shy;<lb/>tionibus ducen do lxxxviij. </s>
          <s id="id001424">cxx. </s>
          <s id="id001425">cxxxij. </s>
          <s id="id001426">hoc e&longs;t ac &longs;i dicamus, accipe <lb/>partes ex cxxxij &amp; lxxxviij, tot ut faciant integrum &amp; componant <lb/>cxx. </s>
          <s id="id001427">Et ide&ograve; reduces ad minores numeros, &longs;cilicet 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"/><lb/>&amp; operaberis per regulam de con&longs;olatione monetarum, quas po&shy;<lb/>nemus infr&agrave;, &amp; fient auri partes octo &amp; argen<lb/><figure id="id.015.01.095.2.jpg" xlink:href="015/01/095/2.jpg"/><lb/>ti partes iij, nam cum duxeris iij in octo pon&shy;<lb/>dus argenti fiet xxiiij, &amp; cum duxeris viij in <lb/>xij, pondus auri fiet xcvi, igitur totum pon&shy;<lb/>dus erit cxx, diuidendum per xi, aggregatum <lb/>partium auri &amp; argenti, ita uero uncia ad unciam, ut tota corona mi <lb/>&longs;ta ad coronam puram auri &amp; argenti.</s>
        </p>
        <pb pagenum="77" xlink:href="015/01/096.jpg"/>
        <p type="margin">
          <s id="id001431"><margin.target id="marg298"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 178.</s>
        </p>
        <p type="main">
          <s id="id001432">Ex hoc etiam patet modus <expan abbr="cogno&longs;c&etilde;di">cogno&longs;cendi</expan> proportionem grauium <lb/><arrow.to.target n="marg299"/><lb/>inuicem per &longs;olam aquam, uelut auri ad plumbum, ad lapides uel <lb/>&aelig;s, aut &aelig;ris ad lapidem &amp; &longs;imilia, ut in pr&aelig;cedenti operatione de&shy;<lb/>prehendi&longs;ti: nam cum &longs;it nota proportio auri ad aquam &amp; &aelig;ris uel <lb/>lapidis ad eandem, erit auri ad &aelig;s uel lapidem nota.</s>
        </p>
        <p type="margin">
          <s id="id001433"><margin.target id="marg299"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id001434">Et &longs;imiliter &longs;ciemus per hoc accipere partes diuer&longs;orum, qu&etail; iun<lb/><arrow.to.target n="marg300"/><lb/>ct&aelig; faciant con&longs;titutum pondus. </s>
          <s id="id001435">Velut uolo facere ma&longs;&longs;am ex mel&shy;<lb/><figure id="id.015.01.096.1.jpg" xlink:href="015/01/096/1.jpg"/><lb/>le &amp; aqua, qu&aelig; impleat uas, quod mellis contineat <lb/>quindecim, aqu&aelig; duodecim, uolo ut contentum &longs;it <lb/>ponderis quatuordecim, operabor, ut in <expan abbr="c&otilde;&longs;olatio&shy;nibus">con&longs;olatio&shy;<lb/>nibus</expan>, ponam duas partes mellis &amp; unam aqu&aelig;, ut <lb/>uides in operatione &agrave; latere.</s>
        </p>
        <p type="margin">
          <s id="id001436"><margin.target id="marg300"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 5.</s>
        </p>
        <p type="main">
          <s id="id001437">Propo&longs;itio octuage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id001438">Si circuli in &aelig;quales, &longs;eu in &longs;ph&aelig;ra, &longs;eu in plano &longs;e &longs;ecuerint nun&shy;<lb/>quam oppo&longs;itos angulos &aelig;quales habent.</s>
        </p>
        <p type="main">
          <s id="id001439">Capiantur tres quart&aelig; circulorum magnorum a b, a c, b c, &amp; alia <lb/><arrow.to.target n="marg301"/><lb/>b d ad rectos angulos <expan abbr="er&utilde;tque">eruntque</expan> uici&longs;sim poli, &amp; ducatur per medium <lb/>parallelus, erit ergo e f &aelig;qualis e g, &amp; f e &aelig;qualis f g, &longs;ed ba&longs;is c g e&longs;t <lb/><figure id="id.015.01.096.2.jpg" xlink:href="015/01/096/2.jpg"/><lb/>quarta circuli, &amp; ba&longs;is c b dimidium quart&aelig; <lb/>circuli eo quod tota b a e&longs;t quarta circuli, igi&shy;<lb/>tur per modum 25 primi Elementorum qu&aelig; <lb/>tenet, erit angulus c f g maior oppo&longs;ito c f b. <lb/></s>
          <s id="id001440">Hoc autem tenet in eiu&longs;dem rationis &longs;uperfi&shy;<lb/>ciebus, quales &longs;unt h&aelig;, qu&aelig; &longs;unt &longs;uperficies eiu&longs;dem &longs;ph&etail;r&aelig;. </s>
          <s id="id001441">po&longs;&longs;et <lb/>etiam demon&longs;trari per modum quart&aelig; primi Elementorum. </s>
          <s id="id001442">Et eti&shy;<lb/>am con&longs;tituta &longs;ph&aelig;ra e f g, cuius hic circulus e&longs;&longs;et maior circulus, &amp; <lb/>non tangeret ni&longs;i in illa linea &longs;ph&aelig;ra maiorem, &amp; utrin que &longs;ecaret eo&shy;<lb/>dem circulo. </s>
          <s id="id001443">Et etiam per cordas &amp; trigonos rectilineos, auxilio <lb/><expan abbr="tam&etilde;">tamen</expan> regul&aelig; dialectic&aelig;. </s>
          <s id="id001444">Ex hoc &longs;equitur auxilio regul&aelig; dialectic&aelig;, <lb/><figure id="id.015.01.096.3.jpg" xlink:href="015/01/096/3.jpg"/><lb/>quod in omnibus parallelis a c d &amp; e f g cum b c circulo <lb/>maiore, &amp; per aliam regulam dialecticam in omnibus cira<lb/>culis in&aelig;qualibus inter &longs;e ad &aelig;quales angulos &longs;ecanti&shy;<lb/>bus &amp; ex tertia demum regula dialectica, &longs;equitur in o&shy;<lb/>mnibus circulis in &aelig;qualibus &longs;e &longs;ecantibus ad quemuis <lb/>angulum in &longs;ph&aelig;r&aelig; &longs;uperficie. </s>
          <s id="id001445">Sunt autem h&aelig; regul&aelig; medi&aelig; inter <lb/>axiomata &amp; demon&longs;trata. </s>
          <s id="id001446">Et ex logica propria illi arti. </s>
          <s id="id001447">In plano au&shy;<lb/><arrow.to.target n="marg302"/><lb/>tem &longs;patium d b c minus e&longs;t a b c, &longs;ed &longs;patium c b d e&longs;t unum, ergo <lb/>per communem animi &longs;ententiam &longs;patium a b d, maius e&longs;t &longs;patio <lb/>c b c, quod fuit probandum.</s>
        </p>
        <pb pagenum="78" xlink:href="015/01/097.jpg"/>
        <p type="margin">
          <s id="id001448"><margin.target id="marg301"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001449"><margin.target id="marg302"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>terd <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001450">Propo&longs;itio octuage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id001451">Proportionem cra&longs;sitiei aqu&aelig; ad a&euml;rem in comparatione ad ra&shy;<lb/>dios demon&longs;trare.<lb/><arrow.to.target n="marg303"/></s>
        </p>
        <p type="margin">
          <s id="id001452"><margin.target id="marg303"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001453">Sit in aheno a b c d in imo e dena<lb/><figure id="id.015.01.097.1.jpg" xlink:href="015/01/097/1.jpg"/><lb/>rius argenteus cera affixus uel cla&shy;<lb/>uo, quem uideat ex h impo&longs;ita aqua <lb/>clara u&longs;que ad f, uideat ex k, igitur per <lb/>aquam deflectitur &agrave; perpendiculo <lb/>per angulum k f n, &amp; in l, per angu&shy;<lb/>lum l g o cre&longs;cente aqua demum in <lb/>labro m a p, &amp; &longs;it e annexus, &amp; tabu<lb/>la h k l m &longs;it affixa &longs;olo uel pondere <lb/>firma foraminibus obliquis infr&agrave; <lb/>&longs;pectantibus, &amp; per a a&longs;picientibus extremitatem e. </s>
          <s id="id001454">Po&longs;&longs;umus ergo <lb/>imaginari primum, qu&ograve;d omnes inclinationes &longs;int &agrave; perpendicu&shy;<lb/>lari, dum exit aqua, &amp; ita denarius uideretur, uel in &longs;uperficie aqu&aelig; <lb/>in directo e, uel in recta ex oculo in imo, quorum neutrum uerum <lb/>e&longs;t. </s>
          <s id="id001455">Secundus modus e&longs;t, ut radius delatus e a flectatur ad k uel l, &amp; <lb/>hoc non quia in a non e&longs;t mutatio medij. </s>
          <s id="id001456">Tertius e&longs;t, ut linea ex ocu<lb/>lo ducta perueniat per punctum a ad &longs;uperficiem aqu&aelig;, &amp; ex ea <lb/>per directum ad denarium, &amp; tunc quia oculus iudicat &longs;e uidere <lb/>per rectam, ideo iudicabit &longs;e uidere per l a g in q, eo quod &longs;emper in <lb/>directo loci in quo e&longs;t e. </s>
          <s id="id001457">At quoniam non ex qua cunque di&longs;tantia ui&shy;<lb/>detur e, &longs;ed ex longinquiore loco, ubi uas fuerit humilius quod li&shy;<lb/>ne&aelig; ad a ex oculo, quanto a fuerit humilius, tanto propius ip&longs;i e <lb/>procedunt. </s>
          <s id="id001458">Et uer&longs;a uice line&aelig; ex e ad a, quanto e e&longs;t humilius ad <lb/>quencunque locum inflectuntur, tanto inferius <expan abbr="cad&utilde;t">cadunt</expan>. </s>
          <s id="id001459">Ergo cum fue<lb/>rint ad &aelig;quilibrium h, magis di&longs;tabunt ab e, &amp; ita e magis procul <lb/>uidebitur. </s>
          <s id="id001460">Cau&longs;a ergo triplex e&longs;t humilitas, uel altitudo ua&longs;is: humi <lb/>litas uel altitudo aqu&aelig;: &amp; labri ua&longs;is altitudo. </s>
          <s id="id001461">Sed hanc relinquere <lb/>po&longs;&longs;umus. </s>
          <s id="id001462">Difficultas ergo experimenti etiam rect&egrave; facti e&longs;t, quo&shy;<lb/>niam po&longs;ito ua&longs;e n c d &longs;olum, ut altitudo &longs;it tantum n e, procul ma&shy;<lb/>gis uidebitur e, qu&agrave;m &longs;i uas &longs;it a b c d, &amp; totum plenum. </s>
          <s id="id001463">Vbi autem <lb/>uas fit a b c d, magis procul uidebitur e cum &longs;uerit totum plenum, <lb/>quam cum fuerit plena &longs;ola pars n c d. </s>
          <s id="id001464">Sic difficile e&longs;t con&longs;iderare <lb/>an altitudo aqu&aelig; faciat ad ui&longs;ionem procul, cum in humiliore, &longs;ed <lb/>di&longs;sipari ua&longs;e longius uideatur in pauca, quia labrum non ob&longs;tat: <lb/>in eodem autem longius in pluri aqua, quia labrum etiam non ob&shy;<lb/>&longs;tat, &longs;ed alia ratione. </s>
          <s id="id001465">Vt ergo uideamus hoc experimentum, capie&shy;


<pb pagenum="79" xlink:href="015/01/098.jpg"/>mus duo ua&longs;a a b c d duplum h k l m &longs;ub eadem proportione alti&shy;<lb/>tudinis &amp; latitudinis, &amp; collocabimus ita ut p n radius &aelig;quidi&longs;tet <lb/>f e, &amp; collocabimus tabulas cum foraminibus, ut prius, &amp; g f p q <lb/><figure id="id.015.01.098.1.jpg" xlink:href="015/01/098/1.jpg"/><lb/>in &aelig;quilibrio, in de uidebimus, an q p &longs;it &aelig;qualis aut breuior, nam <lb/>longior e&longs;&longs;e non pote&longs;t, quoniam inflectitur a minore aqua, ideo <lb/>angulus p h q non pote&longs;t e&longs;&longs;e maior f a g, &longs;uppo&longs;ita p h &aelig;quali a f: <lb/>quod &longs;i non e&longs;&longs;et, &longs;ufficeret, ut q &amp; p e&longs;&longs;ent in &aelig;quilibrio uno, &amp; f g <lb/>alio. </s>
          <s id="id001466">Sed ueritas e&longs;t quod &agrave; maiore aqua maior fit reflexio: tum <lb/>quia in his, qu&aelig; &longs;unt &longs;ecundum naturam corpoream, &amp; &longs;ub&longs;tan&shy;<lb/>tiam den&longs;am, aut tenuem uarietas quantitatis uariat uires: tum <lb/>quia uidemus, quod in altiore aqua denarius uidetur magis cum <lb/>fundo elatus. </s>
          <s id="id001467">Igitur his cognitis experimentum fiat cum ua&longs;e ple&shy;<lb/>no. </s>
          <s id="id001468">Et &lpar;ut dixi&rpar; con&longs;iderabimus proportionem anguli f a g ad far, <lb/>&longs;eu f e c qu&aelig; &longs;an&egrave; e&longs;t notabilis: ade&ograve; ut &longs;it maior proportio aqu&aelig; ad <lb/>a&euml;rem comparatione grauium qu&agrave;m lucis.</s>
        </p>
        <p type="main">
          <s id="id001469"><arrow.to.target n="marg304"/></s>
        </p>
        <p type="margin">
          <s id="id001470"><margin.target id="marg304"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001471">Ex his cogno&longs;cemus comparatione eiu&longs;dem aqu&aelig; tenuitatem <lb/>a&euml;ris unius regionis in comparatione ad a&euml;rem alterius: nam ubi <lb/>remotius uidebitur denarius, ibi a&euml;r erit tenuior.</s>
        </p>
        <p type="main">
          <s id="id001472"><arrow.to.target n="marg305"/></s>
        </p>
        <p type="margin">
          <s id="id001473"><margin.target id="marg305"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001474">Et per idem in eadem regione comparationem aquarum. </s>
          <s id="id001475">Nam <lb/>cum &longs;it idem a&euml;r, &amp; uas, ac reliqua paria, ubi magis procul uidebi&shy;<lb/>tur denarius, aqua erit cra&longs;sior ide&ograve; deterior.</s>
        </p>
        <p type="main">
          <s id="id001476"><arrow.to.target n="marg306"/></s>
        </p>
        <p type="margin">
          <s id="id001477"><margin.target id="marg306"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001478">Sequitur etiam qu&ograve;d omnes res propiores in aqua uidentur, <lb/>quam &longs;int, &amp; ide&ograve; maiores: &amp; ob id etiam omnis aqua profundior <lb/>e&longs;t, quam uideatur. </s>
          <s id="id001479">Vt ingredi per&longs;&aelig;p&egrave; &longs;it periculo&longs;um.</s>
        </p>
        <p type="main">
          <s id="id001480">Propo&longs;itio octuage&longs;ima octaua. </s>
          <s id="id001481">De in&longs;trumento <lb/>momentorum.</s>
        </p>
        <p type="main">
          <s id="id001482">In&longs;trumentum Acolingen, quo momenta temporum deprehen<lb/>dantur fabricare.</s>
        </p>
        <pb pagenum="80" xlink:href="015/01/099.jpg"/>
        <p type="main">
          <s id="id001483"><arrow.to.target n="marg307"/></s>
        </p>
        <p type="margin">
          <s id="id001484"><margin.target id="marg307"/>C<emph type="italics"/>om.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001485">Et quoniam motus naturales fiunt in tempore: &amp; dicuntur ue&shy;<lb/>lociores, uel ob &longs;patium loci magnum, quod &longs;uperatur, uel ob tem<lb/>poris breuitatem in ueloci&longs;simis motibus, quod ad &longs;patia attinet, <lb/>facilius digno&longs;cuntur uelociores, quoniam &longs;patium maius &amp; ma&shy;<lb/>net, ut men&longs;urari commod&egrave; po&longs;sit: &longs;ed qu&ograve;d ad tempus, quanto tar<lb/>diores, quoniam in uelo cibus quantitas temporis e&longs;t exigua: &amp; e&shy;<lb/>tiam tempus ip&longs;um perpetu&ograve; diffluit: ide&ograve; difficillim&egrave; deprehendi <lb/>pote&longs;t. </s>
          <s id="id001486">Huius cau&longs;a excogitauimus in&longs;trumentum, quod uo caui&shy;<lb/>mus Acolingen: quod con&longs;tat tribus rotis: prima e&longs;t pedum duo&shy;<lb/>decim diametri, in ambitu autem habet denticulos ccclx &aelig;qua&shy;<lb/>les, &amp; &aelig;qualiter inter &longs;e di&longs;tantes, huius peripheri&aelig; funis cum pon&shy;<lb/>deribus in&longs;eritur, ita ut cum alijs duabus rotis renitentibus in una <lb/>hora circumagatur &aelig;qualiter. </s>
          <s id="id001487">Duodecim ex his denticulis curru&shy;<lb/>lis duodecim denticulorum axis &longs;ecund&aelig; rot&aelig; in&longs;eritur: &longs;ic ut cum <lb/>rota magna duodecim conuer&longs;a fuerit partibus, &longs;ecunda rota cu&shy;<lb/>ius axis &longs;it pedum duorum, &longs;cilicet &longs;excuplo maior circumuerta&shy;<lb/>tur. </s>
          <s id="id001488">Huius minoris ambitus diui&longs;us &longs;it in cxx partes &aelig;quales, &amp; <lb/>unicuique parti denticulus in&longs;ertus &longs;it: ita h&aelig;c rota tricies in una <lb/>hora conuertetur. </s>
          <s id="id001489">Singulis uer&ograve; denticulis currulis axis rot&aelig; ha&shy;<lb/>bentis denticulos quatuor in&longs;eratur, ita ut dum &longs;ecunda rota uer&shy;<lb/>titur &longs;emel minima circumuertatur tricies: nam pro &longs;ingulis qua&shy;<lb/>tuor denticulis, quibus media rota circumagetur, minima tota cir&shy;<lb/>cumuertetur, ideoq&uacute;e nongenties in una hora. </s>
          <s id="id001490">H&aelig;c minima ro&shy;<lb/>tula be&longs;&longs;em pedis in dimetiente habebit, ut &longs;it &longs;exta pars illius, in <lb/>ambitu autem diui&longs;a erit in xl partes, ut cum circumuer&longs;a fue&shy;<lb/>rit nongenties in una hora pertran&longs;ierit partes xxxvi. </s>
          <s id="id001491">Et cum <lb/>pul&longs;us hominis communis &longs;int in hora &lt;23&gt;, uel circa nouem partes <lb/>ex his rot&etail; minoris perficient circiter unam pul&longs;ationem ex dia&longs;to&shy;<lb/>le &amp; &longs;i&longs;tole, &longs;eu ex di&longs;tentione &amp; contractione perfectam: ut partis <lb/>unius conuer&longs;io fiat in nona parte, uel circa unius pul&longs;ationis pul&shy;<lb/>&longs;us humani: &amp; hoc e&longs;t minimum ferm&egrave;, quod ab humano &longs;en&shy;<lb/>&longs;u percipi po&longs;sit. </s>
          <s id="id001492">Erit etiam proportio rotarum eadem tam in dia&shy;<lb/>metris, qu&agrave;m circuitibus &longs;cilicet &longs;excupla, neque motus diffor&shy;<lb/>mis, quoniam maior tanto tardius mouebitur, quanto quod ue&shy;<lb/>locius mouetur etiam minus erit, tamen proportio uelo citatis ma&shy;<lb/>ioris ad minorem in &aelig;qualibus &longs;patijs uigintiquincupla, ut ma&shy;<lb/>ioris ad mediam quintupla, nam cum &longs;it &longs;excupla in ambitu, <lb/>&amp; tricies moueatur uelocius comparatione totius, &longs;equitur, ut <lb/>proportio &longs;patij, quod &longs;uperabit media ad &longs;patium, quod &longs;u&shy;<lb/>perabit maior in ei&longs;dem temporibus, erit quintupla, &longs;emper ad un&shy;<lb/>guem. </s>
          <s id="id001493">Et ita medi&aelig; ad minorem quintupla, &amp; ide&ograve; maioris ad 


<pb pagenum="81" xlink:href="015/01/100.jpg"/>minorem uelo citas uiginti quincupla, ut non &longs;it difformis, neque <lb/>periculo&longs;a, ut in rotis moletrinis, &amp; &longs;it diui&longs;a per medium iuxta <lb/>proportionem, cum &longs;it tanto uelocior minor media, quanto media <lb/>maiore. </s>
          <s id="id001494">Rur&longs;us proportio partium maioris ad medi&aelig; partes tripla <lb/>e&longs;t &longs;cilicet ccclx ad cxx, &amp; medi&aelig; ad <expan abbr="minor&etilde;">minorem</expan> tripla cxx ad xl, &amp; pro&shy;<lb/>portio e&longs;t &longs;excupla, iterum igitur partes maioris ad mediam, &amp; me&shy;<lb/>di&aelig; ad minorem erunt in dupla proportione, utrobique, &amp; e&longs;t pul&shy;<lb/>chrum. </s>
          <s id="id001495">Ide&ograve; partes etiam minim&aelig; rot&aelig; erunt &longs;atis magn&aelig;: nam <lb/>cum diameter &longs;it bes pedis, ambitus peripheri&aelig; erit duorum pe&shy;<lb/>dum. </s>
          <s id="id001496">1. unciarum uiginti quatuor: igitur diui&longs;a peripheria in xl par&shy;<lb/>te r, unaqu&aelig;que 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 in&longs;trumentum, utemur men&longs;ura expul&longs;u homi&shy;<lb/>nis de&longs;umpta, &longs;ed non e&longs;t ade&ograve; exacta. </s>
          <s id="id001499">Accedit aliud commodum, <lb/>qu&ograve;d cum in una hora circumuertantur partes xxxvi, id e&longs;t triginta <lb/>&longs;ex mille: &amp; octauus orbis circumuertatur in totidem annis, tot <lb/>erunt momenta ex his in una hora, quot anni in uno circuitu &longs;tella&shy;<lb/>rum fixarum. </s>
          <s id="id001500">Vt intelligamus, qu&agrave;m breui tran&longs;it una hora apud <lb/>nos, ita apud Deum, ut ita dicam &lpar;nam nulla in infinito proportio&rpar; <lb/>unus annus magnus, &amp; reditus rerum omnium. </s>
          <s id="id001501">Comparata etiam <lb/>rota minima ad rotam moletrini &longs;ic &longs;e habet, qu&ograve;d c&ugrave;m modica ad&shy;<lb/>e&longs;t, uer&longs;atur rota in una pul&longs;atione: cum &longs;atis abundans quinquies, <lb/>aut &longs;exies cum immodica duo decies.</s>
        </p>
        <figure id="id.015.01.100.1.jpg" xlink:href="015/01/100/1.jpg"/>
        <pb pagenum="82" xlink:href="015/01/101.jpg"/>
        <p type="main">
          <s id="id001502"><arrow.to.target n="marg308"/></s>
        </p>
        <p type="margin">
          <s id="id001503"><margin.target id="marg308"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001504">Ex hoc &longs;equitur, quod homo &longs;i moueretur uelo citate motus ro&shy;<lb/>t&aelig; moletrin&aelig; in &longs;ex ebdomadibus perueniret ad &longs;ydus Lun&aelig;, nam <lb/>rotarum earum, quibus ferrum acuitur &longs;emidimetiens communi&shy;<lb/>ter e&longs;t bes unius pa&longs;&longs;us, ide&ograve; dimetiens pa&longs;&longs;us cum triente: ambi&shy;<lb/>tus ergo quatuor pa&longs;&longs;us, &amp; xxi pars, colligamus nunc integra, in <lb/>uno ictu pul&longs;us circumagitur decies, id e&longs;t pa&longs;&longs;us xl, in hora &longs;unt <lb/>&lt;23&gt; pul&longs;ationes: in hora igitur &longs;patium pertran&longs;itum e&longs;t cxl pa&longs;&longs;uum <lb/>in M. horis, ergo erunt clx M. pa&longs;&longs;uum addita parte xxi, erunt clxviij <lb/>M. pa&longs;&longs;uum, &amp; tantum di&longs;tat luna &agrave; terra: &amp; M. hor&aelig; &longs;unt dies pen&egrave; <lb/>xlij, ebdomad&aelig; &longs;cilicet &longs;ex.</s>
        </p>
        <p type="main">
          <s id="id001505">Propo&longs;itio octuage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id001506">Proportionem den&longs;itatis aqu&aelig; ad a&euml;rem per pondera inuenire.</s>
        </p>
        <p type="main">
          <s id="id001507"><arrow.to.target n="marg309"/></s>
        </p>
        <p type="margin">
          <s id="id001508"><margin.target id="marg309"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001509">Contingit hoc multis modis: primum acceptis duabus &longs;ph&aelig;ru&shy;<lb/>lis &aelig;qualibus ex cry&longs;tali &longs;ub&longs;tantia unaque demi&longs;&longs;a ab alti&longs;sima turri, <lb/>&amp; men&longs;urato ictu per in&longs;trumentum pr&aelig;cedens, &amp; &longs;ub totidem <lb/>momentis alia demi&longs;&longs;a in aquam, in de &longs;ub eodem tempore dimen&shy;<lb/>&longs;a altitudine, erit proportio &longs;patij ad &longs;patium, ut den&longs;itatis aqu&aelig;, ad <lb/>den&longs;itatem a&euml;ris. </s>
          <s id="id001510">Item emi&longs;&longs;a &longs;ph&aelig;rula per in&longs;trumentum in a&euml;rem, <lb/>in de in aquam: &amp; &longs;umpta proportione. </s>
          <s id="id001511">Et uidimus &longs;corpionem, <lb/>qui <expan abbr="&longs;ph&aelig;rul&atilde;">&longs;ph&aelig;rulam</expan> creteam emittebat pedibus lxx, &amp; in aqua per unum <lb/>&amp; dimidium ade&ograve;, ut proportio fuerit, ut quinquaginta ad unum: <lb/>ide&ograve; e&longs;t fallax experimentum in uiolento motu: nam cum emitte&shy;<lb/>batur in aquam erat prop&egrave;, &amp; ob id in &longs;ummo robore: c&ugrave;m in a&euml;&shy;<lb/>rem, emittitur &longs;en&longs;im uis. </s>
          <s id="id001512">De hoc ergo loquar.</s>
        </p>
        <p type="main">
          <s id="id001513"><arrow.to.target n="marg310"/></s>
        </p>
        <p type="margin">
          <s id="id001514"><margin.target id="marg310"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001515">Et erumpentia ob id magis qu&agrave;m &egrave; terra, et minus qu&agrave;m ex a&euml;re: <lb/>diuiditur enim aqua cum graue petit fundum, &amp; aqua feruet: &amp; e&longs;t <lb/>mirabilius, qu&agrave;m utile.</s>
        </p>
        <p type="main">
          <s id="id001516">Propo&longs;itio nonage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id001517">Rationem impetus uiolenti extra mi&longs;si ponderis ad &aelig;qualita&shy;<lb/>tem reducere.</s>
        </p>
        <p type="main">
          <s id="id001518"><arrow.to.target n="marg311"/></s>
        </p>
        <p type="margin">
          <s id="id001519"><margin.target id="marg311"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001520">Sit uiolentum a quod moueatur per b c d e, e &longs;patium, &amp; quia <lb/>uiolentum contr&agrave; nititur naturali, cadat ergo in planum in e: &longs;unt <lb/>ergo tria con&longs;ideran da, primum quod, ut dixi ali&acirc;s, motus uiolen&shy;<lb/>tus pro certa di&longs;tantia augetur, &amp; cau&longs;am ibi reddidi, ut pot&egrave; u&longs;que <lb/>ad c, &longs;ed hoc e&longs;&longs;et difficile cognitu. </s>
          <s id="id001521">Secundum, quod ubi incipit de&shy;<lb/>cre&longs;cere, &longs;emper magis ac magis decre&longs;cit propter naturalem ni&shy;<lb/>xum contra operantem. </s>
          <s id="id001522">Tertium quod ubi de&longs;cendere incipit, ibi <lb/>e&longs;t &aelig;qualis uis uiolentum motum agens cum naturali. </s>
          <s id="id001523">Certum e&longs;t <lb/>etiam quod motus &aelig;qualis intelligitur erecta ad perpendiculum <lb/>e f, donec occurrat a d: &amp; diui&longs;a tota b f per tempus, locus ergo, in <lb/>quo mouetur per tantum &longs;patium, dicitur locus motus &aelig;qualis: 


<pb pagenum="83" xlink:href="015/01/102.jpg"/>qui &longs;it gratia exempli g h, cuius medium proportione &longs;it k, di&shy;<lb/>co k con&longs;i&longs;tere propiorem f, qu&agrave;m b, etiam&longs;i &aelig;qualiter mouere&shy;<lb/>tur. </s>
          <s id="id001524">Primum qu&ograve;d in tota g f declinat, &amp; totus motus e&longs;t lentior, <lb/>qu&agrave;m in tota b g, &amp; tamen tardatur tantundem, ergo per commu&shy;<lb/>nem animi &longs;ententiam, k e&longs;t propior f, qu&agrave;m b. </s>
          <s id="id001525">Secund&ograve;, quia per <lb/>&longs;ecundum &longs;uppo &longs;itum motus a uer&longs;us f, continu&egrave; fit lentior, igitur <lb/>per communem animi &longs;ententiam mult&ograve; longius e&longs;t tempus mo&shy;<lb/>tus a k, quam f, &amp; tanto maius &longs;patium. </s>
          <s id="id001526">Terti&ograve;, quia motus ex b uer<lb/>&longs;us c augetur, &amp; &longs;i e&longs;&longs;et &aelig;qualis adhuc mult&ograve; e&longs;&longs;et breuior k f quam <lb/>a k, igitur mult&ograve; magis hoc modo, &amp; triplicata ratione. </s>
          <s id="id001527">Si ergo b k <lb/><figure id="id.015.01.102.1.jpg" xlink:href="015/01/102/1.jpg"/><lb/>e&longs;&longs;et &longs;exquiquarta &longs;olum ip&longs;i k f, <lb/>erit b k dupla: ferm&egrave; ex triplicata <lb/>ratione ip&longs;i k f, &amp; iuxta eundem <lb/>modum ponemus mediam uim <lb/>xlvi pa&longs;sibus &agrave; &longs;corpione a quam <lb/>&amp; hoc modo erit prop&egrave; id quod e&longs;t.</s>
        </p>
        <p type="head">
          <s id="id001528">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001529">Dubitat autem Philo&longs;ophus in mechanicis qu&aelig; nam uis &longs;it, qu&etail; <lb/>moueat lapidem iam excu&longs;&longs;um? </s>
          <s id="id001530">&amp; dubium non e&longs;t quin ex parte &longs;it <lb/>a&euml;r motus tum ratione, quia mouetur ergo mouet, tum experimen <lb/>to, ut in fulminibus, &amp; his qu&aelig; uento impelluntur, ut hypophy&longs;is, <lb/>&longs;ed in &longs;corpionibus &amp; arcubus &amp; pilis id non &longs;ufficere uidetur. </s>
          <s id="id001531">Ita&shy;<lb/>que uelut &amp; caliditas &amp; frigiditas in corporibus natura contrarijs <lb/>aliquandiu manent, &amp; agunt ita &amp; uiolentos motus, idque Alexan&shy;<lb/>der &amp; Simplicius uolunt. </s>
          <s id="id001532">Inditio &longs;unt qu&ograve;d mota &amp; emi&longs;&longs;a ex lon&shy;<lb/>gioribus machinis quan quam non a&euml;rem continentibus, nec in&shy;<lb/>anibus tamen, longius eijciunt &longs;agittas &amp; mi&longs;silia, quoniam uis <lb/>illa firmius imprimitur, uelut etiam de lapidibus &amp; ferro, quod di&shy;<lb/>utius in igne moram traxit, aut continu&egrave; follibus ignitum e&longs;t, nam <lb/>etiam tanto tardius refrigeratur unum quod que horum, &amp; alia urit <lb/>&amp; accendit calore illo externo, quanquam natura frigidum &longs;it: di&shy;<lb/>cemus autem &amp; de hoc &longs;uo loco.</s>
        </p>
        <p type="main">
          <s id="id001533">Propo&longs;itio nonage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id001534">Proportionem grauis cubi, &amp; &longs;ph&aelig;rici &aelig;qualium in accliui, &amp; <lb/>de&longs;cen&longs;us eorum demon&longs;trare.</s>
        </p>
        <p type="main">
          <s id="id001535">Hic non pauca &longs;unt <expan abbr="c&otilde;&longs;ideranda">con&longs;ideranda</expan>: Primum <lb/><figure id="id.015.01.102.2.jpg" xlink:href="015/01/102/2.jpg"/><lb/>qu&ograve;d hoc intelligi pote&longs;t, uel de motibus at&shy;<lb/>tractionis, uel impul&longs;ionis, uel inuer&longs;ionis. <lb/></s>
          <s id="id001536">Secundum quod omne, quod impellitur &longs;uperi&ugrave;s, tantundem gra&shy;<lb/>uat attractum, quantum ad de&longs;cen&longs;um, &longs;i &longs;it rotundum, nam qua&shy;<lb/>drata, <expan abbr="eti&atilde;">etiam</expan> alia non de&longs;cendunt &longs;ponte in decliui, &amp; &longs;i &longs;it locus uald&egrave; 


<pb pagenum="84" xlink:href="015/01/103.jpg"/>decliuis, tanto minus de&longs;cendunt, quanto &longs;unt latiora. </s>
          <s id="id001537">Quia tamen <lb/>omnia difficili&ugrave;s de&longs;cendunt &longs;ph&aelig;ricis, &amp; facilius qu&agrave;m in plano, <lb/>ubi ponderant ni&longs;i per dimidium grauitatis, ide&ograve; proportio h&aelig;c <lb/>con&longs;tat ex proportione anguli de&longs;cen&longs;us ad totum rectum, &amp; ma&shy;<lb/>gnitudine &longs;uperficiei, qua incumbit ad pondus comparata. </s>
          <s id="id001538">Omne <lb/>enim graue, quanto grauius tam ad quietem, qu&agrave;m ad motum na&shy;<lb/>turalem potentius e&longs;t: hoc enim per&longs;picuum e&longs;t, quia quieti natu&shy;<lb/>rali motus uiolentus, &amp; motui naturali quies uiolenta opponitur: <lb/>quia ergo maiore ui opus e&longs;t ad motum pr&aelig;ter naturam, ergo &longs;e&shy;<lb/>cundum naturam etiam maiore ui quie&longs;cit. </s>
          <s id="id001539">A&longs;&longs;ump&longs;imus ergo cu&shy;<lb/>bum, ut magis notum. </s>
          <s id="id001540">Sph&aelig;ra igitur in omni decliui de&longs;cendit, <lb/>quia ut dictum e&longs;t, nil habet quod re&longs;i&longs;tat ad motum: &amp; ip&longs;a gra&shy;<lb/>uior e&longs;t in decliui, qu&agrave;m in plano, quia c pun&shy;<lb/>ctus cadit ultra e, ergo punctus contactus, &amp; <lb/><figure id="id.015.01.103.1.jpg" xlink:href="015/01/103/1.jpg"/><lb/>centrum grauitatis, &amp; centrum mundi, non &longs;unt <lb/>in una linea. </s>
          <s id="id001541">Si enim b c contangeretur, e&longs;&longs;et b c <lb/>plana. </s>
          <s id="id001542">Si uer&ograve; tangit, angulus e&longs;t maior angulo <lb/>contactus, ergo cum nece&longs;&longs;arium &longs;it, &aelig;quidi&longs;ta&shy;<lb/>re aliter non e&longs;&longs;et &longs;ph&aelig;ricum, oportet, ut eleue&shy;<lb/>tur ex parte c, &amp; de&longs;cendat uer&longs;us b, &amp; ide&ograve; ut <lb/>continuetur motus. </s>
          <s id="id001543">Si uer&ograve; &longs;it in linea conta&shy;<lb/>ctus b c f, &amp; &aelig;quidi&longs;tet non erit, ut dixi punctus <lb/>contactus in linea centrorum, &longs;ed in a c, cum &longs;uppo&longs;itum &longs;it lineam <lb/>a d e&longs;&longs;e lineam centrorum: maior e&longs;t ergo portio g c e, qu&agrave;m re&longs;i&shy;<lb/>duum, ergo de&longs;cendet in b. </s>
          <s id="id001544">Cubus uer&ograve; non de&longs;cendet, ni&longs;i cum di&shy;<lb/>midium d addito, quod intercipitur inter lineam mediam, &amp; qu&aelig; &agrave; <lb/>centro mundi ad punctum medium contactus u&longs;que qu&ograve; perueniat <lb/>ad oppo&longs;itam partem, eam habuerit proportionem ad idem me&shy;<lb/>dium eadem portione detracta, quem iuncta proportioni anguli <lb/>declinationis ad re&longs;iduum recti dimidiam proportionem efficiat. <lb/></s>
          <s id="id001545">Eademque ratio aliorum planorum. </s>
          <s id="id001546">Dico pr&aelig;terea qu&ograve;d motus <lb/>&longs;ph&aelig;r&aelig;, &amp; etiam corporum rectarum &longs;uperficierum in de&longs;cen&longs;u <lb/>alius e&longs;t &aelig;qualis, &amp; alius in&aelig;qualis, &amp; qua&longs;i &agrave; latere, uelut &longs;i angu&shy;<lb/>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb/>hoc, &amp; maxim&egrave; &longs;i non retineatur &aelig;qualiter, &amp; difficile &longs;it in medio <lb/>retinere, propterea prolap&longs;us hi melius <expan abbr="retin&etilde;tur">retinentur</expan> duobus uinculis, <lb/>qu&agrave;m in medio, non &longs;olum ob hanc &aelig;qualitatem, &amp; complexum <lb/>meliorem, &longs;ed <expan abbr="eti&atilde;">etiam</expan>, quod omnes motus, omnes ponderum nixus fa<lb/>cili&ugrave;s cohibentur, &amp; <expan abbr="deducun&ttilde;">deducuntur</expan> diui&longs;i in partes, &lt;08&gt; &longs;i toti contin <expan abbr="ean&ttilde;">eantur</expan>, <lb/>aut ui <expan abbr="trah&atilde;tur">trahantur</expan>. </s>
          <s id="id001547">Et ideo uincula in rami cibus duplicia dextra, &amp; &longs;ini<lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tam&euml;n longe &longs;unt meliora etiam ferreis, qu&aelig; <lb/>&longs;olum in medio nectantur.</s>
        </p>
        <pb pagenum="85" xlink:href="015/01/104.jpg"/>
        <p type="main">
          <s id="id001548"><arrow.to.target n="marg312"/></s>
        </p>
        <p type="margin">
          <s id="id001549"><margin.target id="marg312"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001550">Ex hoc etiam &longs;equitur, <lb/><figure id="id.015.01.104.1.jpg" xlink:href="015/01/104/1.jpg"/><lb/>quod c&ugrave;m omne graue <lb/>&longs;pont&egrave; &longs;emper appropin&shy;<lb/>quet centro mundi, &amp; a &longs;i <lb/>moueretur per planum e, <lb/>magis remoueretur &agrave; cen&shy;<lb/>tro mundi, ut per e c per ea <lb/>qu&aelig; diximus, &amp; quoniam <lb/>linea ex centro mundi ad <lb/>c longior e&longs;t, qu&agrave;m ad e, <lb/>mult&ograve; pote&longs;t enim e&longs;&longs;e, ut <lb/>in proportione diametri <lb/>quadrati ad latus eius, &amp; <lb/>etiam maior. </s>
          <s id="id001551">ergo poterit <lb/>e&longs;&longs;e ade&ograve; parum decliuis <lb/>linea c d, ut c punctus ma&shy;<lb/>gis di&longs;ter &agrave; centro mundi, <lb/>qu&agrave;m d, &amp; tamen feretur <lb/>ex d in c motu naturali, ut demon&longs;tratum e&longs;t, ergo per purum mo&shy;<lb/>tum naturalem poterit a remoueri &agrave; centro mundi. </s>
          <s id="id001552">Hoc uolui pro&shy;<lb/>ponere, ut intelligeres in plano uero c e non moueri a &longs;ponte, quia <lb/>c nece&longs;&longs;ari&ograve; altior e&longs;t d: &longs;i ergo mouebitur, non erit c e recta, &longs;ed <lb/>pars proportionis circuli &longs;uperficiei terr&aelig;, qu&aelig; &longs;en&longs;u &agrave; recta di&longs;tin&shy;<lb/>gui non poterit. </s>
          <s id="id001553">Hoc ergo e&longs;t primum, ex quo &longs;equitur.</s>
        </p>
        <p type="main">
          <s id="id001554"><arrow.to.target n="marg313"/></s>
        </p>
        <p type="margin">
          <s id="id001555"><margin.target id="marg313"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001556">Quod aliquid poterit uideri decliue, in quo non de&longs;cendet im&ograve; <lb/>erit, ut pot&egrave; &longs;i aliqua linea obliqua e&longs;&longs;et inter c e, &amp; f e, illa e&longs;&longs;et decli&shy;<lb/>uis &longs;pecie, &amp; re, &amp; tamen graue in illa non de&longs;cenderet, quia &agrave; cen&shy;<lb/>tro mundi magis remoueretur: hoc tamen e&longs;t perdifficile factu, &amp; <lb/>maxim&egrave; in parua di&longs;tantia, uel etiam unius miliaris. </s>
          <s id="id001557">Atque h&aelig;c <lb/>in leuigatis.</s>
        </p>
        <p type="main">
          <s id="id001558">Propo&longs;itio nonage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id001559">Proportionem ponderis &aelig;qualis iuxta longitudinis compara&shy;<lb/>tionem demon&longs;trare.</s>
        </p>
        <figure id="id.015.01.104.2.jpg" xlink:href="015/01/104/2.jpg"/>
        <p type="main">
          <s id="id001560">Hoc e&longs;t, quod Archimedes reliquit </s>
        </p>
        <p type="main">
          <s id="id001561"><arrow.to.target n="marg314"/><lb/>intactum, cum e&longs;&longs;et maxim&egrave; nece&longs;&longs;a&shy;<lb/>rium, &amp; o&longs;tendit magis ab&longs;tru&longs;a, &longs;ed <lb/>pace illius dixerim minus utilia. </s>
          <s id="id001562">Cum <lb/>ergo &longs;ump&longs;i&longs;&longs;em uirgam b f ponderis <lb/>unciarum xxiij, fui&longs;&longs;et b a uige&longs;ima quarta pars, b f fuit pondus &aelig;&shy;<lb/>quilibrij in b appen&longs;um librarum uiginti &longs;ex cum dimidia: fuit igi&shy;<lb/>tur proportio ponderis e f ad pondus f b, ut tredecim ferme ad 


<pb pagenum="86" xlink:href="015/01/105.jpg"/>unum. </s>
          <s id="id001563">Et rur&longs;us feci a b quintam partem a f, &amp; fuit a b unciarum <lb/>quatuor, &amp; pondus quod &aelig;quauit librarum quatuor, ide&ograve; du&shy;<lb/>plum ad pondus b f, &longs;icut c f ad c b: con&longs;tat enim qu&ograve;d pondus ap&shy;<lb/>pen&longs;um e&longs;t &aelig;quale ponderi cf. </s>
          <s id="id001564">Et rur&longs;us po&longs;ui b a quartam partem <lb/>b f, &amp; fuit pondus, quod &aelig;quauit in b du&aelig; libr&aelig;: ex quo manife&shy;<lb/>&longs;tum e&longs;t, qu&ograve;d proportio c f ad c b e&longs;t &longs;emper uelut ponderis c f ad <lb/>totam b f. </s>
          <s id="id001565">Et hoc e&longs;t, ac &longs;i dicamus, qu&ograve;d proportio ponderis c f ad <lb/>totam e&longs;t confu&longs;a ex proportione e f ad c b, &amp; c f, quod e&longs;t 1 p. </s>
          <s id="id001566">Id <lb/><arrow.to.target n="marg315"/><lb/>etiam declaratum e&longs;t in primo de Subtilitate. </s>
          <s id="id001567">Proponatur ergo <lb/>lemma, iam &longs;ic proportio ponderis cf ad pondus b c, e&longs;t primum <lb/>ut longitudinis cf, &longs;i e&longs;&longs;et &longs;u&longs;pen&longs;a in medio ad longitudinem b c, <lb/>quia &longs;upponuntur proportione &longs;imiles &longs;uis longitudinibus ma&shy;<lb/>gnitudines, &amp; pondera. </s>
          <s id="id001568">At c f &longs;u&longs;pen&longs;a in c, tanto e&longs;t grauior pon&shy;<lb/>dere proprio, quanto proportionis longitudinis cf ad cb quadra&shy;<lb/>tum, quia in &longs;e ducitur proportio: igitur proportio ponderis c f in <lb/>loco &longs;uo ad b c pondus e&longs;t confu&longs;a ex proportione longitudinis <lb/>cf ad c b, &amp; quadratis eiu&longs;dem proportionis longitudinis cf ad c <lb/>b. </s>
          <s id="id001569">Sed quadratum proportionis longitudinis cf ad cb e&longs;t &aelig;quale <lb/>producto proportionis longitudinis c f in ip&longs;am c f, propterea <lb/>qu&ograve;d ex proportione longitudinis cf ad cb in ip&longs;am c b fit c f, igi&shy;<lb/>tur proportio ponderis c f ad pondus c b e&longs;t confu&longs;a ex propor&shy;<lb/>tione ponderis c f ad pondus c b, &amp; proportione ponderis cf alicu<lb/>ius &longs;e habentis ad pondus cf, ut cf longitudo ad longitudinem <lb/>c b, igitur proportio ponderis cf ad pondus b f, ut cf ad c b in lon&shy;<lb/>gitudine, quod erat probandum.</s>
        </p>
        <p type="margin">
          <s id="id001570"><margin.target id="marg314"/>C<emph type="italics"/>om.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001571"><margin.target id="marg315"/>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001572">Propo&longs;itio nonage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id001573">Propter quid in concu&longs;sione etiam leui nauis loco moueatur <lb/>o&longs;tendere. </s>
          <s id="id001574">Vnde manife&longs;tum e&longs;t, duas naues &longs;ibi inuicem occur&longs;an <lb/>tes retrocedere, &amp; quantum retrocedant amb&aelig;.<lb/><arrow.to.target n="marg316"/></s>
        </p>
        <p type="margin">
          <s id="id001575"><margin.target id="marg316"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001576">Proponatur, quod proportio motus grauis in a d graue in aqua <lb/>&longs;it, uelut proportio ponderis attracti in terra ad den&longs;itatem aqu&aelig; <lb/>cum profunditate, nam ubi pondus &longs;upernataret aqu&aelig;, quia aqua <lb/>e&longs;t rotunda, e&longs;t ac &longs;i tangeret in puncto. </s>
          <s id="id001577">Quare per demon&longs;trata &longs;u&shy;<lb/>peri&ugrave;s mouebitur &agrave; quacunque ui, ergo nixus contrarius aduenit ob </s>
        </p>
        <p type="main">
          <s id="id001578"><arrow.to.target n="marg317"/><lb/>profunditatem, &amp; aqu&aelig; den&longs;itatem, &longs;ed quanto aqua den&longs;ior e&longs;t, <lb/>tanto minus nauis de&longs;cendit, &amp; quanto minus den&longs;a, tanto magis: <lb/>ergo pari modo ferm&egrave; redduntur mobiles, &amp; in aqua dulci &amp; &longs;al&longs;a, <lb/>ubi naues &longs;int &longs;imiles forma, pondere, magnitudine. </s>
          <s id="id001579">Quia ergo ne&shy;<lb/>ce&longs;&longs;e e&longs;t tabulam nauis e&longs;&longs;e duriorem, quam aqua ad re&longs;i&longs;tendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe&shy;<lb/>netret, cum ergo quod facilius e&longs;t, pr&aelig;cedat, difficilius ergo naues 


<pb pagenum="87" xlink:href="015/01/106.jpg"/>utrinque mouebuntur, &amp; quia inter duos quo&longs;cunque motus contra&shy;<lb/>rios <expan abbr="n&otilde;">non</expan> e&longs;&longs;e eos, ut utar uocabulo Auerrois quinto Phy&longs;icorum, ne&shy;<lb/>ce&longs;&longs;e e&longs;t, ut intercedat quies media, &amp; in quiete ab ictu, ut ui&longs;um e&longs;t <lb/>&longs;uperius, oportet, ut quod excipit ictum uel loco moueatur, uel ce&shy;<lb/><arrow.to.target n="marg318"/><lb/>dat, &amp; ictus penetret, uel a&euml;r non conden&longs;etur ob tarditatem ultra <lb/>metam, nec retro cedere pote&longs;t ex &longs;uppo&longs;ito, &amp; ictus e&longs;t magnus, <lb/>clarum e&longs;t, quod oportet, ut cedat, &amp; &longs;i durum &longs;it confringatur. <lb/></s>
          <s id="id001580">Proportio ergo rece&longs;&longs;us ad ictum e&longs;t ut temporis, &amp; magnitudinis <lb/>partis, qu&aelig; cedit, &amp; retro ce&longs;&longs;us po&longs;ito ictu tanquam monade.</s>
        </p>
        <p type="margin">
          <s id="id001581"><margin.target id="marg317"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s>
        </p>
        <p type="margin">
          <s id="id001582"><margin.target id="marg318"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 74.</s>
        </p>
        <p type="main">
          <s id="id001583">Propo&longs;itio nonage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id001584">Si quantitas aliqua nota atque proportio erit producta quantitas <lb/>nota &longs;imiliter. </s>
          <s id="id001585">Et &longs;i du&aelig; proportiones not&aelig; fuerint, erit producta <lb/>ex his atque diui&longs;a, coniunctaque, atque detracta nota. </s>
          <s id="id001586">Et &longs;i fuerit totius <lb/>ad partem proportio nota erit, &amp; ad aliam partem nota, &amp; alterius <lb/>partis ad alteram uno minor. </s>
          <s id="id001587">Et &longs;i fuerit partis ad partem, erit ad to<lb/>tum monade minor atque nota. </s>
          <s id="id001588">Et &longs;i fuerit unius quantitatis ad duas <lb/>quantitates proportio nota, erit &amp; confu&longs;a ex eis nota. </s>
          <s id="id001589">Et &longs;i fuerint <lb/>trium quantitatum omiologarum, aut quatuor analogarum, o&shy;<lb/>mnes pr&aelig;ter unam cognit&aelig; erunt, &amp; illa alia cognita.</s>
        </p>
        <figure id="id.015.01.106.1.jpg" xlink:href="015/01/106/1.jpg"/>
        <p type="main">
          <s id="id001590">Sit quantitas a b &amp; ducta in d proportionem, <lb/><arrow.to.target n="marg319"/><lb/>producat b c: dico quod duobus quibuslibet ex <lb/>his cognitis, erit cognitum tertium: nam cogni&shy;<lb/>tum quodlibet dicitur in comparatione ad &longs;impliciter cognitum, <lb/>quod e&longs;t unum per &longs;e omnibus cognitum. </s>
          <s id="id001591">Ob id Arithmetica e&longs;t <lb/>prima omnium di&longs;ciplinarum, quia habet principium cognitum, <lb/>&amp; id, quod e&longs;t, ad principium comparatum cognitum in illius com<lb/>paratione: neque aliter cognitum dici pote&longs;t. </s>
          <s id="id001592">Quia ergo d cognita <lb/>e&longs;t, erunt monades, &amp; partes cognit&aelig; in ea: aliter non e&longs;&longs;et cognita <lb/>b a, igitur cum cognita &longs;it, erit cognita per &longs;ingulas monades, quan<lb/>ta &longs;it. </s>
          <s id="id001593">Et &longs;i diceres qu&ograve;d b a non e&longs;t cognita per partem monadis: <lb/>dico quod pars monadis non e&longs;t incognita, quia cum monades <lb/>&longs;unt cognit&aelig;, e&longs;&longs;et d incognita. </s>
          <s id="id001594">Omnes enim, quod componitur ex <lb/>cognito &amp; incognito, e&longs;t incognitum, quia cognitum &longs;olum ratio&shy;<lb/>ne partis cognit&aelig;. </s>
          <s id="id001595">Si ergo pars monadis e&longs;t cognita, erit pars a b <lb/>qu&aelig;libet prout ex monade componitur &longs;impliciter cognita. </s>
          <s id="id001596">Su&shy;<lb/><arrow.to.target n="marg320"/><lb/>pere&longs;t, ut &longs;olum pars partis: &amp; dico quod illa etiam e&longs;t cognita: <lb/>quia &longs;i pars ab e&longs;&longs;et, monas e&longs;&longs;et cognita: e&longs;&longs;et enim pars ip&longs;a.</s>
        </p>
        <p type="margin">
          <s id="id001597"><margin.target id="marg319"/>C<emph type="italics"/>om.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001598"><margin.target id="marg320"/>E<emph type="italics"/>x &longs;ecunda <lb/>animi com&shy;<lb/>muni &longs;enten<lb/>tia.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001599">Sed &longs;i &longs;it pars, erit &longs;umpta &longs;ecundum partem monadis ip&longs;ius, <lb/>ide&ograve; erit cognita iuxta nomen, uelut dimidium e&longs;t dimidium mo&shy;<lb/>nadis, dimidium terti&aelig; partis monadis e&longs;t cognitum, quia tertia <lb/>pars e&longs;t cognita, &amp; &longs;cimus, quanta pars a&longs;&longs;umatur illius. </s>
          <s id="id001600">Ergo &longs;i a b, 


<pb pagenum="88" xlink:href="015/01/107.jpg"/>&amp; d cognit&aelig; &longs;unt erit &amp; b c, quod e&longs;t primum. </s>
          <s id="id001601">Per h&aelig;c eadem pro&shy;<lb/>bantur quatuor &longs;equentes partes eodem modo. </s>
          <s id="id001602">Sexta &longs;ic: &longs;it pro&shy;<lb/>portio a c ad c b, nota igitur in comparatione ad monadem, &longs;ed pro <lb/>portio a c ad c b b a e&longs;t monas, igitur proportio a c ad a b nota e&longs;t, <lb/>quoniam aliter non po&longs;&longs;et dici proportio a c ad b c nota. </s>
          <s id="id001603">Aliter, &longs;it <lb/>proportio a c ad c b e nota, ex &longs;uppo&longs;ito igitur conuer&longs;a nota qu&aelig; <lb/>&longs;it 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/>tur f g e&longs;t monas, f autem nota e&longs;t, igitur in comparatione ad mona&shy;<lb/><arrow.to.target n="marg321"/><lb/>dem, ergo re&longs;iduum g notum. </s>
          <s id="id001604">Cum uer&ograve; proportio a c ad c b com&shy;<lb/>ponatur ex proportione a b b c ad b c, &amp; proportio b c ad b c &longs;it <lb/>monas, &amp; proportio a c ad b c nota, erit proportio a b ad b c cogni<lb/><arrow.to.target n="marg322"/><lb/>ta, &amp; monade minor proportione a c ad b c. </s>
          <s id="id001605">Per idem octaua pars <lb/><figure id="id.015.01.107.1.jpg" xlink:href="015/01/107/1.jpg"/><lb/>demon&longs;trabitur. </s>
          <s id="id001606">Inde &longs;it proportio a ad b, &amp; ad c no&shy;<lb/>ta, erit ergo b, &amp; c ad a nota, quare b c ad a nota, &longs;ed <lb/><arrow.to.target n="marg323"/><lb/>h&aelig;c e&longs;t conuer&longs;a ad b c confu&longs;a, igitur proportio a <lb/>ad b confu&longs;a nota e&longs;t. </s>
          <s id="id001607">Vltimum &longs;it, &longs;int a b c omiolog&aelig;, &amp; &longs;int a &amp; b <lb/><arrow.to.target n="marg324"/><lb/>not&aelig; duo, quod c nota e&longs;t, nam a b, &longs;i not&aelig; &longs;unt, nota e&longs;t proportio <lb/>earum. </s>
          <s id="id001608">Ergo &amp; proportio b ad c ergo per primam partem huius <lb/><arrow.to.target n="marg325"/><lb/>cum &longs;it b nota, exit &amp; c. </s>
          <s id="id001609">Et &longs;i ponantur a c not&aelig;, dico, qu&ograve;d b nota <lb/>erit: nam proportio a c ad c nota e&longs;t, qu&aelig; &longs;it d, igitur d ad monadem <lb/>ut a ad c, ergo latus notum erit, quod ductum in c producit b, b igi&shy;<lb/><arrow.to.target n="marg326"/><lb/>tur nota. </s>
          <s id="id001610">Et &longs;imiliter in analogis &longs;int a b c not&aelig;: &amp; ide&ograve; erit propor&shy;<lb/>tio a ad b nota ergo c ad d. </s>
          <s id="id001611">cumque c nota &longs;it, ergo per primam par&shy;<lb/>tem huius erit d nota, quod fuit demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id001612"><margin.target id="marg321"/>P<emph type="italics"/>er demon&shy;<lb/>&longs;trat.<emph.end type="italics"/> 12. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001613"><margin.target id="marg322"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001614"><margin.target id="marg323"/>E<emph type="italics"/>x demon&longs;t.<emph.end type="italics"/><lb/>12. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001615"><margin.target id="marg324"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001616"><margin.target id="marg325"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001617"><margin.target id="marg326"/>E<emph type="italics"/>x<emph.end type="italics"/> 2. A<emph type="italics"/>nimi <lb/>&longs;ententia.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001618">Propo&longs;itio nonage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id001619">Cuiu&longs;uis trigoni rectanguli, aut cuius duo anguli &longs;int in dupla <lb/>proportione, aut qui circulo in&longs;criptus &longs;it cognita quantitate uni&shy;<lb/>us lateris in comparatione ad dimetientem &longs;i proportio <expan abbr="duor&utilde;">duorum</expan> la&shy;<lb/>terum cognita fuerit, erunt omnia eius latera cognita.<lb/><arrow.to.target n="marg327"/></s>
        </p>
        <p type="margin">
          <s id="id001620"><margin.target id="marg327"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001621">Non de cognitione propinqua <expan abbr="a&longs;tronomor&utilde;">a&longs;tronomorum</expan>, de qua abund&egrave; ab <lb/>Heber tractatum e&longs;t, &longs;ed de exacta, de qua &longs;uperius egi nunc &longs;ermo </s>
        </p>
        <p type="main">
          <s id="id001622"><arrow.to.target n="marg328"/><lb/>e&longs;t: &longs;it igitur primum a b c trigonus orthogonius: &amp; &longs;it a rectus, &amp; <lb/>proportio <expan abbr="duor&utilde;">duorum</expan> laterum cognita, dico, quod omnia latera cognita <lb/><arrow.to.target n="marg329"/><lb/><figure id="id.015.01.107.2.jpg" xlink:href="015/01/107/2.jpg"/><lb/>erunt: nam &longs;it proportio, gratia exempli, <lb/>a b ad b c, erit ergo quadrati a b ad qua&shy;<lb/>dratum b c cognita, quia duplicata: at <lb/>quadrata a b, &amp; a c perficiunt quadratum <lb/>b c, igitur proportio quadrati a b ad a c et <lb/>e&longs;t 1 p: cognita erit, quare &amp; a b ad a c, &amp; <expan abbr="eod&etilde;">eodem</expan> modo a c ad b c: quod <lb/>e&longs;t primum. </s>
          <s id="id001623">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a c igi&shy;<pb pagenum="89" xlink:href="015/01/108.jpg"/>tur &longs;i a b ponatur 1 b c erit 2, &amp; a c &lt;02&gt; 3. Rur&longs;us ponatur angulus b <lb/>duplus angulo c quali&longs;cunque &longs;it, erit per demon&longs;trata &longs;uperius pro&shy;<lb/>portio a b b c ad a c, ut a c ad a b, &longs;i igitur nota &longs;it proportio a c ad <lb/>a b, erit nota proportio a b b c ad a b per pr&aelig;cedentem. </s>
          <s id="id001624">Ergo per <lb/>eandem omnia nota &longs;cilicet b c ad b a, &amp; b c ad c a. </s>
          <s id="id001625">Et &longs;i e&longs;&longs;et nota <lb/>proportio a b ad b c, dico, quod e&longs;&longs;ent nota omnia, nam nota e&longs;&longs;et <lb/>a b, &amp; b c, &amp; quod fit ex a b in ip&longs;um aggregatum. </s>
          <s id="id001626">Sed hoc e&longs;t &aelig;&shy;<lb/><arrow.to.target n="marg330"/><lb/>quale quadrato a c, igitur notum e&longs;t quadratum a c ergo a c: igitur <lb/>proportio a b b c ad a c, &amp; a c ad a b. </s>
          <s id="id001627">Vt &longs;i a b e&longs;&longs;et 4 b c 5, e&longs;&longs;et a b b c <lb/>9 ducta in a b, qu&aelig; e&longs;t, fit 36, cuius latus e&longs;t b a c &longs;cilicet. </s>
          <s id="id001628">Et &longs;i e&longs;&longs;et <lb/>trigonus aliquis in circulo, cuius proportio duorum laterum &longs;it co<lb/>gnita ad dimetientem relata, &longs;equitur per demon&longs;trata &longs;upe&shy;<lb/>rius, quod etiam tertium latus erit cognitum in comparatione ad <lb/>eadem, &amp; ideo etiam proportio illorum laterum ad unguem co&shy;<lb/>gnita erit.</s>
        </p>
        <p type="margin">
          <s id="id001629"><margin.target id="marg328"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 97.</s>
        </p>
        <p type="margin">
          <s id="id001630"><margin.target id="marg329"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001631"><margin.target id="marg330"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 17.</s>
        </p>
        <p type="main">
          <s id="id001632">Multa pr&aelig;terea cognita e&longs;&longs;ent in hoc genere, qu&aelig; nunc pr&aelig;ter&shy;<lb/><arrow.to.target n="marg331"/><lb/>mitto, quia non &longs;unt ad finem nece&longs;&longs;aria. </s>
          <s id="id001633">Alia pr&aelig;terea per diligen&shy;<lb/>tem inqui&longs;itionem maioris artis qu&agrave;m alias edidimus. </s>
          <s id="id001634">tum uer&ograve; <lb/>etiam per nouas demon&longs;trationes.</s>
        </p>
        <p type="margin">
          <s id="id001635"><margin.target id="marg331"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001636">Propo&longs;itio nonage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id001637">Cum in per&longs;picuum den&longs;um radij lumino&longs;i inciderint, quatuor <lb/>fiunt luminis genera.<lb/><arrow.to.target n="marg332"/></s>
        </p>
        <p type="margin">
          <s id="id001638"><margin.target id="marg332"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001639">Sit &longs;ol a, &amp; per&longs;picuum den&longs;um, exempli gratia, ut ampula <lb/>magna aqua plena b c d, &amp; &longs;i &longs;it rotunda accendit ignem ex ad&shy;<lb/>uer&longs;o ut in e. </s>
          <s id="id001640">Dico ergo in b c d e&longs;&longs;e quatuor genera luminis. </s>
          <s id="id001641">Pri&shy;<lb/>mum quod e&longs;t ualidius, &amp; rect&agrave; tran&longs;it, ualidius enim e&longs;t, quod <lb/>tran&longs;it qu&agrave;m quod tran&longs;ire non pote&longs;t, &amp; etiam quia, ut dixi, <lb/>ignem accendit. </s>
          <s id="id001642">Secundum e&longs;t quod colligitur in ampula, &amp; dein&shy;<lb/>de &longs;pargitur <expan abbr="circ&utilde;circ&agrave;">circuncirc&agrave;</expan>, nam id ualidius e&longs;t, quia penetrat, &amp; re&longs;ilit <lb/>qu&agrave;m quod non penetrat, aut &longs;i penetrat, non &longs;pargitur, &amp; hoc dif&shy;<lb/>funditur circa uas, nec reflectitur rect&egrave;, &longs;ed qua&longs;i intro colligitur, &amp; <lb/>diuer&longs;a ratione diffunditur, e&longs;t tamen imbecillius primo, ut dictum <lb/>e&longs;t. </s>
          <s id="id001643">Tertium genus e&longs;t, quod illuminat intus ingrediendo, &longs;ed non <lb/>&longs;pargitur, &amp; hoc e&longs;t debilius &longs;ecundo, quia <expan abbr="n&otilde;">non</expan> pote&longs;t &longs;pargi. </s>
          <s id="id001644">Quar&shy;<lb/><figure id="id.015.01.108.1.jpg" xlink:href="015/01/108/1.jpg"/><lb/>tum e&longs;t, quod non ingreditur omnino, &longs;ed refle&shy;<lb/>ctitur, i&longs;tud e&longs;t ab&longs;que dubio imbecillimum, quo&shy;<lb/>niam penetrare non pote&longs;t. </s>
          <s id="id001645">Et licet in &longs;peculis <lb/>concauis radius reflexus uideatur e&longs;&longs;e ualidior, <lb/>&longs;tatim enim accendit ignem, hoc non contin&shy;<lb/>git, ni&longs;i quia in &longs;peculo cauo radij omnes col&shy;


<pb pagenum="90" xlink:href="015/01/109.jpg"/><expan abbr="ligun&ttilde;">liguntur</expan> ob <expan abbr="opac&utilde;">opacum</expan>, quod &agrave; tergo e&longs;t, neque <expan abbr="&longs;pargun&ttilde;">&longs;parguntur</expan>, neque <expan abbr="tran&longs;e&utilde;t">tran&longs;eunt</expan>, neque<lb/> combibuntur, ut ita dicam &longs;ed omnes <expan abbr="reflect&utilde;tur">reflectuntur</expan>. </s>
          <s id="id001646">Ex quo colligitur <lb/>quincuplex ordo radiorum iuxta rationem uirium, primus e&longs;t refle<lb/><expan abbr="xor&utilde;">xorum</expan> &agrave; &longs;peculo <expan abbr="c&otilde;cauo">concauo</expan>, &amp; hi &longs;unt <expan abbr="pot&etilde;ti&longs;simi">potenti&longs;simi</expan> ob <expan abbr="ration&etilde;">rationem</expan> <expan abbr="dict&atilde;">dictam</expan>, po&longs;t <lb/>quos &longs;unt radij, qui tran&longs;eunt per per&longs;picuum maxim&egrave; rotundum, <lb/>qui &amp; ip&longs;i generant ignem, &amp; debiliorem primo, deinde reliqui <lb/>tres &longs;equentes &longs;upra dicti. </s>
          <s id="id001647">Sextus e&longs;t radiorum, qui reflectuntur &agrave; <lb/>rebus non nitidis, ut &agrave; muris, &amp; tabulis, nam omnia dura reflectunt <lb/>&amp; etiam mollium pleraque, &amp; h&aelig;c reflexio e&longs;t ferm&egrave; infinita, &amp; ob id <lb/>cubicula etiam in angulis illuminantur.</s>
        </p>
        <p type="main">
          <s id="id001648"><arrow.to.target n="marg333"/></s>
        </p>
        <p type="margin">
          <s id="id001649"><margin.target id="marg333"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id001650">Ex hoc &longs;equitur, qu&ograve;d Luna remittit lumen, non reflectit, nam <lb/>&longs;ecus non illuminaret to tum orbem, &longs;ed &longs;olum portionem oppo&shy;<lb/>&longs;itam Soli, &amp; hoc etiam rar&ograve;, ergo combibitur, &amp; illu&longs;trat circun&shy;<lb/>circa ubique.</s>
        </p>
        <p type="main">
          <s id="id001651"><arrow.to.target n="marg334"/></s>
        </p>
        <p type="margin">
          <s id="id001652"><margin.target id="marg334"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id001653">In &longs;tellis lumen Solis pertran&longs;it aliter, &longs;i reflecteretur, non illumi&shy;<lb/>naret nos, aut apparerent, uelut comet&aelig;, quia pars una e&longs;&longs;et clarior <lb/>reliqua, &amp; &longs;i conbiberent lumen, non uiderentur &aelig;qu&egrave; clar&aelig;, cum <lb/>Sol e&longs;&longs;et propinquus, aut remotus.</s>
        </p>
        <p type="main">
          <s id="id001654"><arrow.to.target n="marg335"/></s>
        </p>
        <p type="margin">
          <s id="id001655"><margin.target id="marg335"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id001656">Luna tota intus illuminatur &agrave; Sole, quoniam &longs;i ante coniun&shy;<lb/>ctionem illuminatur &agrave; &longs;ini&longs;tra parte, &amp; combibit lumen per cor&shy;<lb/>rolarium primum, &amp; po&longs;t coniunctionem illuminatur &agrave; dex&shy;<lb/>tra, &amp; combibit pariter lumen, ergo e&longs;t tota natur&aelig; per&longs;picu&aelig;, &longs;ed <lb/>uidetur ob&longs;cura ex aduer&longs;o, propterea qu&ograve;d radij ualidiores refle&shy;<lb/>xi illu&longs;trant illam ex parte Solis, diffugiunt &agrave; contraria, quod ma&shy;<lb/>nife&longs;t&egrave; apparet in ampula expo&longs;ita Soli. </s>
          <s id="id001657">Pars enim clarior uer&longs;us <lb/>Solem uidetur, quam ex aduer&longs;o, hoc autem long&egrave; magis in Luna <lb/>ob di&longs;tantiam.</s>
        </p>
        <p type="main">
          <s id="id001658"><arrow.to.target n="marg336"/></s>
        </p>
        <p type="margin">
          <s id="id001659"><margin.target id="marg336"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id001660">In omni Solis eclip&longs;i fit colectio radiorum ad a&longs;pectum, &amp; <lb/>ideo in regione illa, in qua centrum Solis integitur &agrave; centro Lun&aelig;, <lb/>&amp; ubicunque fit, fit incendium per tertium corrolarium. </s>
          <s id="id001661">Hoc autem <lb/>fit &longs;emper in quauis coniunctione, &amp; dum Luna &longs;ilet in regione ae&shy;<lb/>ris, &longs;ed terris non &longs;ecund&ugrave;m centrum, uer&ugrave;m ad latitudinem, &amp; ad <lb/>Orientem ante coniunctionem cum Sole, &amp; ad Occidentem po&longs;t: <lb/>&longs;ed centra non &longs;unt in linea ui&longs;us.</s>
        </p>
        <p type="main">
          <s id="id001662"><arrow.to.target n="marg337"/></s>
        </p>
        <p type="margin">
          <s id="id001663"><margin.target id="marg337"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 5.</s>
        </p>
        <p type="main">
          <s id="id001664">Ex hoc &longs;equitur, quod oportet &longs;ub&longs;tantiam Lun&aelig; e&longs;&longs;e ualde cla&shy;<lb/>ram, cum uideamus ab ampula tam paruum lumen diffundi, &amp; ra&shy;<lb/>rum, &agrave; Luna uer&ograve; in uniuer&longs;um orbem, &amp; tam copio&longs;um, ut nece&longs;&shy;<lb/>&longs;arium &longs;it &longs;ub&longs;tantiam Lun&aelig; e&longs;&longs;e den&longs;am, &amp; lucidam ualde.</s>
        </p>
        <p type="head">
          <s id="id001665">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001666">Et &longs;i quis dicat, qu&ograve;d &longs;i incendium illud fieri po&longs;&longs;et in hora ecli&shy;<lb/>p&longs;is, &longs;equeretur, qu&ograve;d ut in ampula in medio Lun&aelig; uideretur ma&shy;


<pb pagenum="91" xlink:href="015/01/110.jpg"/>gnus &longs;plendor, referens corpus Solis. </s>
          <s id="id001667">Propterea dico, qu&ograve;d uel ac&shy;<lb/>cidit, quia homo non pote&longs;t ea hora intueri Solem, &amp; etiam e&longs;t im&shy;<lb/>peditus &agrave; radijs circum&longs;tantibus, cuius indicio e&longs;t, quod in &longs;pe&shy;<lb/>culo po&longs;ito in aqua, &longs;imile uidetur &longs;tellul&aelig; in centro Lun&etail;: &amp; hic e&longs;t <lb/>&longs;plendor Solis collectus in centro Lun&aelig;. </s>
          <s id="id001668">po&longs;&longs;et etiam dici, qu&ograve;d <lb/>Luna circa medium propter maculam non admitteret lumen, &amp; ita <lb/>e&longs;&longs;et in&aelig;qualium partium.</s>
        </p>
        <p type="main">
          <s id="id001669">Propo&longs;itio nonage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id001670">Motum inuer&longs;ionis in figuris in comparatione ad motum &longs;ph&aelig;<lb/>r&aelig; in plano inue&longs;tigare.</s>
        </p>
        <p type="main">
          <s id="id001671"><arrow.to.target n="marg338"/></s>
        </p>
        <p type="margin">
          <s id="id001672"><margin.target id="marg338"/>C<emph type="italics"/>om.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001673">Voco motum inuer&longs;ionis, qui &longs;imilis e&longs;t motui &longs;ph&aelig;r&aelig;, &longs;cili&shy;<lb/>cet circumuertendo graue &agrave; uertice, &amp; manife&longs;tum e&longs;t, qu&ograve;d in <lb/>quacunque figura, qua graue in&longs;idet plano per punctum ue&shy;</s>
        </p>
        <p type="main">
          <s id="id001674"><arrow.to.target n="marg339"/><lb/>lut ouata ip&longs;um mouetur &agrave; quauis ui, &longs;ed &longs;i in&longs;ideat per &longs;uperfi&shy;<lb/>ciem, quanto maior e&longs;t, &amp; humilior, tanto difficilius mouetur, <lb/>ide&ograve; in corpore uiginti ba&longs;ium, qu&ograve;d inter regularia uocata, plu&shy;<lb/>res habet, &longs;uperficies pro ratione &aelig;qualis ponderis, motus erit <lb/>longe facilior. </s>
          <s id="id001675">Alia cau&longs;a e&longs;t in&aelig;qualitas partium, unde qu&aelig; ro&shy;<lb/>tunda &longs;unt, quia prominent, facile mouentur, &amp; cum partes me&shy;<lb/>di&aelig; in&longs;i&longs;tant plano, quanto minores erunt tanto facilius moue&shy;<lb/>buntur ratione ponderis. </s>
          <s id="id001676">Vnde patet, qu&ograve;d corpora ouata faci&shy;<lb/>lius mouentur, etiam qu&agrave;m &longs;ph&aelig;rica, habent enim partem me&shy;<lb/>diam minorem, &amp; paria &longs;unt ratione ince&longs;&longs;us plani, &longs;ed a&euml;ris mul&shy;<lb/>titudine tardius, quoniam enim &longs;ph&aelig;ra &longs;ub &aelig;quali ambitu plus <lb/>continet corporis, ergo ouatum &aelig;quale &longs;ph&aelig;r&aelig; habet maio&shy;<lb/>rem ambitum ip&longs;a &longs;ph&aelig;ra. </s>
          <s id="id001677">H&aelig;c autem &agrave; Theone partim de&shy;<lb/>mon&longs;trata &longs;unt, partim ab Archimede, &amp; partim &agrave; nobis, ergo <lb/>motus ouati e&longs;t ferm&egrave; &aelig;qualis motui &longs;ph&aelig;r&aelig;, &amp; tardior e&longs;t con&shy;<lb/><figure id="id.015.01.110.1.jpg" xlink:href="015/01/110/1.jpg"/><lb/>citatus, qu&agrave;m &longs;ph&aelig;r&aelig;, quia &agrave; ma&shy;<lb/>iore excipitur a&euml;re, &amp; partes exte&shy;<lb/>riores non ita incumbunt in me&shy;<lb/>dium &longs;ecundum longitudinem. </s>
          <s id="id001678">Cu&shy;<lb/>bus uero tardior e&longs;t propter &aelig;qua&shy;<lb/>litatem, &amp; latitudinem &longs;uperficiei in&shy;<lb/>ferioris, omnium <expan abbr="aut&etilde;">autem</expan> minime pro&shy;<lb/>pter has cau&longs;as conus ambligonius, <lb/>&amp; quanto magis fuerit, ratio uero <lb/>eleuationis e&longs;t, ut &longs;it cubus b c, cuius <lb/>medium grauitatis &longs;it b &longs;uper pla&shy;


<pb pagenum="92" xlink:href="015/01/111.jpg"/>no de, &amp; eleuetur ex a, &amp; manife&longs;tum e&longs;t, quod in&longs;idebit per totam <lb/>lineam c f ip&longs;i plano, &amp; proportio grauitatis totius &longs;u&longs;pen&longs;i in com<lb/>paratione ad grauitatem eius, qui inuertit, e&longs;t, uelut proportio par&shy;<lb/>tis terminat&aelig; ad lineam c f uer&longs;us eum, qui eleuat ad partem, qu&aelig; <lb/>ultra e&longs;t, cum uer&ograve; h&aelig; partes not&aelig; &longs;int iuxta perpendiculum ex <lb/>centro grauitatis, manife&longs;tum e&longs;t, quod &longs;ciemus pondus corporis <lb/>a b cf, dum inuertitur in quo cunque &longs;itu ad pondus eius, dum &longs;u&shy;<lb/>&longs;penditur, &amp; clarum e&longs;t, qu&ograve;d c&ugrave;m centrum, &amp; medium grauitatis <lb/>fuerint in una linea per c f, tunc nulla erit grauitas.</s>
        </p>
        <p type="margin">
          <s id="id001679"><margin.target id="marg339"/>P<emph type="italics"/>er<emph.end type="italics"/> 40.</s>
        </p>
        <p type="main">
          <s id="id001680">Propo&longs;itio nonage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id001681">Proportionem ponderum &aelig;qualium per differentiam angulo&shy;<lb/>rum inuenire.<lb/><arrow.to.target n="marg340"/></s>
        </p>
        <p type="margin">
          <s id="id001682"><margin.target id="marg340"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001683">Sit a b, qu&aelig; &longs;i appen&longs;a e&longs;&longs;et ad &aelig;quidi&shy;<lb/><figure id="id.015.01.111.1.jpg" xlink:href="015/01/111/1.jpg"/><lb/>&longs;tantem terr&aelig; &longs;uperficiei, nulla ui po&longs;&longs;et ele</s>
        </p>
        <p type="main">
          <s id="id001684"><arrow.to.target n="marg341"/><lb/>uari, inflectatur ergo ad c punctum, omi&longs;&longs;a <lb/>c g, &amp; manife&longs;tum e&longs;t, quod &longs;i b c in&longs;i&longs;teret <lb/><arrow.to.target n="marg342"/><lb/>ad perpendiculum, ponderaret a c &longs;i e&longs;&longs;et in <lb/>&aelig;quilibrio, ponatur ergo accliuis in c d per <lb/>notum angulum. </s>
          <s id="id001685">Quia igitur b c ad c a no&shy;<lb/>ta e&longs;t, erit dicta &longs;uperi&ugrave;s notum pondus <lb/>b h, po&longs;ita h c &aelig;quali c a, quare totius a b, <lb/>&amp; iam fuit e k notum, &amp; punctus d notus: <lb/>hoc enim infr&agrave; demon&longs;trabitur, qualis igitur proportio line&aelig; <lb/><arrow.to.target n="marg343"/><lb/>tran&longs;uer&longs;&aelig; dl ad lineam de&longs;cendentem d m, talis differenti&aelig; pon&shy;<lb/>derum c m, &amp; c e, id e&longs;t partis ad partem. </s>
          <s id="id001686">h&aelig;c autem inferi&ugrave;s de&shy;<lb/>mon&longs;trabuntur. </s>
          <s id="id001687">Neque enim ab&longs;urdum e&longs;t in materijs mi&longs;tis, ali&shy;<lb/><arrow.to.target n="marg344"/><lb/>quando uti nondum demon&longs;tratis cum fuerint mathematica, quia <lb/>obtinent principij rationem, quod etiam facit Archimedes. </s>
          <s id="id001688">Ma&shy;<lb/>nife&longs;tum e&longs;t autem, quod in angulo m c d recti dimidio, propor&shy;<lb/>tio media erit. </s>
          <s id="id001689">Sed hoc bifariam contingere pote&longs;t &longs;cilicet, ut &longs;it <lb/>media, per quantitatem, &amp; per proportionem, e&longs;t autem media, ut <lb/><arrow.to.target n="marg345"/><lb/>demon&longs;trabitur infr&agrave; &longs;ecundum proportionem l d ad l e, propo&shy;<lb/>natur ergo c e b, erit latus quadrati &lt;02&gt; 72, igitur latus octogoni e&longs;t <lb/>&lt;02&gt; v: 72 m: &lt;02&gt; 2592, &amp; latus re&longs;idui &lt;02&gt; v: 72 p: &lt;02&gt; 2592. quadrata er&shy;<lb/>go partium ba&longs;is differunt in &lt;02&gt; 10368. Quare partes ba&longs;is &longs;unt <lb/>6 p: &lt;02&gt; 18, &amp; 6 m: &lt;02&gt; 18 &longs;cilicet l e, l d autem e&longs;t &lt;02&gt; 18, igitur differen&shy;<lb/>tia, &amp; proportio e&longs;t, qualis &lt;02&gt; 18 ad 6 m: &lt;02&gt; 18 ferm&ecirc;, ut 17 ad 7, &amp; ta&shy;<lb/>lis e&longs;t proportio ponderis c d ad pondus c e ratione in crementi, <lb/>&longs;eu differenti&aelig;. </s>
          <s id="id001690">Vt &longs;i pondus in c e e&longs;&longs;et decem librarum in c in 


<pb pagenum="93" xlink:href="015/01/112.jpg"/>quadraginta erit in c d triginta unius cum quarta, &longs;ed proportionis <lb/>ratione e&longs;&longs;et uiginti octo cum tertia.</s>
        </p>
        <p type="margin">
          <s id="id001691"><margin.target id="marg341"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2. <lb/>45. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001692"><margin.target id="marg342"/>P<emph type="italics"/>er<emph.end type="italics"/> 86. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001693"><margin.target id="marg343"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 99.</s>
        </p>
        <p type="margin">
          <s id="id001694"><margin.target id="marg344"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 97.</s>
        </p>
        <p type="margin">
          <s id="id001695"><margin.target id="marg345"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 98.</s>
        </p>
        <p type="main">
          <s id="id001696">Propo&longs;itio nonage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id001697">Proportionem grauitatum per multitudinem &longs;uppo&longs;itorum or <lb/>bium o&longs;tendere.<lb/><arrow.to.target n="marg346"/></s>
        </p>
        <p type="margin">
          <s id="id001698"><margin.target id="marg346"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001699">Omne, quod mouetur, mouetur &longs;ecundum naturam ponderis, <lb/>qu&aelig; in attractione, ut demon&longs;tratum e&longs;t, &aelig;qualis e&longs;t dimidio &longs;u&shy;<lb/>&longs;pen&longs;i, cum ergo diuidatur in multiplices partes motus uniu&longs;cuiu&longs;&shy;<lb/>que, e&longs;t &longs;ecundum dimidium illius partis, ut, &longs;i &longs;int &longs;ex rot&aelig; in cur&shy;<lb/>ru det, quod uehitur, &longs;it pondus &longs;exaginta librarum, unaqu&aelig;que </s>
        </p>
        <p type="main">
          <s id="id001700"><arrow.to.target n="marg347"/><lb/>rota habet pondus quinque librarum, &longs;cilicet diui&longs;o triginta per <lb/>&longs;ex, &amp; quia quod cunque mouetur &longs;ph&aelig;ric&egrave; non habet pondus, <lb/>ni&longs;i quantum premitur axis, ide&ograve; pondus &longs;exaginta librarum in <lb/>uehendo redditur l&aelig;&longs;us, quanto proportio producta minor e&longs;t <lb/>additione. </s>
          <s id="id001701">Exemplum, &longs;it deductum pondus &longs;exaginta librarum <lb/>per &longs;ex rotas ad uiginti quatuor, quia &longs;i rot&aelig; po&longs;&longs;ent circumduci, <lb/>ut in inuer&longs;ione dictum e&longs;t, &amp; e&longs;&longs;ent &aelig;quales, &amp; in &longs;olido &aelig;quali, <lb/>ac duro, nulla ui mouerentur, &longs;ed qua&longs;i per &longs;e, ergo &longs;uppo&longs;ito pon&shy;<lb/>dere uiginti quatuor librarum a&longs;&longs;umemus unamquamque partem, <lb/>&amp; ducemus eam in &longs;e ip&longs;am, &longs;cilicet detraham quintam partem ex <lb/>toto 30, fit 24, duc 30 in &longs;e, fit 900, duc 24 in &longs;e, fit 576, proportio ut <lb/>25 ad 16, at diui&longs;o 30 in &longs;ex partes, fit 5, detrahe quintam partem, fit <lb/>4, duc in &longs;e, fit 16, duc in &longs;ex, fit 96, igitur proportio 900 ad 96 e&longs;t ut <lb/>25 ad 2 2/3, quod ergo erat 16 factum e&longs;t 2 2/3, proportio ergo de&shy;<lb/>cre&longs;centis maior e&longs;t diui&longs;o per plura. </s>
          <s id="id001702">Sed plerunque additis ro&shy;<lb/>tis cre&longs;cit pondus nihilo &longs;ecius, redditur etiam leuius. </s>
          <s id="id001703">Sed &amp; de <lb/>hoc in &longs;equenti.</s>
        </p>
        <p type="margin">
          <s id="id001704"><margin.target id="marg347"/>P<emph type="italics"/>er<emph.end type="italics"/> 40.</s>
        </p>
        <p type="main">
          <s id="id001705">Propo&longs;itio cente&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id001706">Proportionem grauitatis ponderum attractorum per trochlea&shy;<lb/>rum numerum inue&longs;tigare.<lb/><arrow.to.target n="marg348"/></s>
        </p>
        <p type="margin">
          <s id="id001707"><margin.target id="marg348"/>C<emph type="italics"/>om.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001708">Ari&longs;toteles in Mechanicis cen&longs;et cau&longs;am leuitatis trochlearum </s>
        </p>
        <p type="main">
          <s id="id001709"><arrow.to.target n="marg349"/><lb/>e&longs;&longs;e in pondere eleuando, qu&ograve;d pondera auxilio uectium facilius <lb/>mouentur, qu&agrave;m manibus. </s>
          <s id="id001710">Rotul&aelig; uer&ograve; in trochleis uectes &longs;unt, <lb/>&amp; axis mi&longs;ta hypomochlij, ergo facilius pondus trahitur per u&shy;<lb/>nam rotulam, qu&agrave;m &longs;i manu traheretur, at uer&ograve; per duas tres, <lb/>unde tris pa&longs;&longs;us longe facilius, &amp; etiam facilius per quinque, unde <lb/>pentas pa&longs;&longs;us, nam quinque orbiculis, qua&longs;i totidem uectibus <lb/>diui&longs;um pondus manife&longs;t&egrave; fit leuius, &amp; ut dictum e&longs;t, tanquam <lb/>totidem uectibus pondus eleuatur, e&longs;tq&uacute;e proportio produ&shy;


<pb pagenum="94" xlink:href="015/01/113.jpg"/>cta, &longs;emperque prior hypomochlij locum habet, ueruntamen minus <lb/>a&longs;&longs;umit laboris, po&longs;terior uer&ograve; uectis maiorem partem &longs;ibi ponde&shy;<lb/>ris &longs;eruat, uelut in &longs;uccula etiam iugum traiectum per plures colo&shy;<lb/>pes facilius uertitur. </s>
          <s id="id001711">Et &longs;i quis dicat n&oacute;nne totum pondus in&longs;idet <lb/>prim&etail; trochle&aelig; per trochleam, intelligo nunc &longs;ol&ugrave;m rotulam cum <lb/>ip&longs;o axe, &longs;eu axiculo &lpar;ut dicunt&rpar; non autem in proprio &longs;ignificato, <lb/>in quo etiam funis traiectus, &amp; in&longs;idens rotul&aelig;, &longs;eu rotulis, nam <lb/>una trochlea plures continere&apos;pote&longs;t orbiculos, &amp; axes. </s>
          <s id="id001712">Licet ergo <lb/>pondus in&longs;ideat prim&aelig; trochle&aelig;, &longs;eu rotul&aelig;, in eo tamen, quod tra<lb/>hitur, diuiditur&apos;, licet non &aelig;qualiter dico, pr&aelig;ter id funis motum <lb/>intendi. </s>
          <s id="id001713">nam motus actionem auget, &amp; ide&ograve; quanto longior, eo fa&shy;<lb/>cilius mouet ob concu&longs;sionem, demum quia leuis e&longs;t rotula circa <lb/>axem, ut plus uecte po&longs;sit.</s>
        </p>
        <p type="margin">
          <s id="id001714"><margin.target id="marg349"/>I<emph type="italics"/>n<emph.end type="italics"/> M<emph type="italics"/>echan.<emph.end type="italics"/><lb/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 18.</s>
        </p>
        <p type="main">
          <s id="id001715">Propo&longs;itio cente&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id001716">Proportionem precij gemmarum ex tribus in eodem genere co<lb/>gnitis inuenire.<lb/><arrow.to.target n="marg350"/></s>
        </p>
        <p type="margin">
          <s id="id001717"><margin.target id="marg350"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001718">Solent gemmarij uendere adamantem ponderis unius grani <lb/>uno coronato, duorum autem granorum tribus coronatis, qua&shy;<lb/>tuor autem, gratia exempli, quadraginta coronatis, qu&etail;ritur quan&shy;<lb/>tum ualebit adamas octo granorum, quoniam ergo proportio <lb/>non &longs;eruatur. </s>
          <s id="id001719">E&longs;t enim in pondere utraque dupla, in precio autem <lb/>ex prima habetur tripla, ex &longs;ecunda habetur proportio maior, <lb/>qu&agrave;m tredecim ad unum, propterea utendum e&longs;t proportione <lb/>propinquiori, &longs;i &longs;atis faceret. </s>
          <s id="id001720">gratia exempli, in prima ad ditione fuit <lb/>unum granum, &amp; acqui&longs;iuit proportionem triplam, in &longs;ecunda fue<lb/>runt duo grana, &longs;i ergo acqui&longs;i&longs;&longs;et &longs;olum &longs;excuplam proportio&shy;<lb/>nem, haberemus intentum. </s>
          <s id="id001721">Propterea in i&longs;to ca&longs;u oportet demon&shy;<lb/>&longs;trare forma Geometrica, &longs;uppo&longs;ito, qu&ograve;d &longs;it figura recta ex uno la <lb/><figure id="id.015.01.113.1.jpg" xlink:href="015/01/113/1.jpg"/><lb/>tere a b, ita ut angulus, uel minimus capiat b c &aelig;qualem a b, &amp; ex <lb/>&aelig;quali b a c addito fiat b d tripla b c, &amp; ex angulo b a e duplo b a d, <lb/>fiat b c d e quadragintupla a b, &amp; iuxta rationem erit in infinitum. <lb/></s>
          <s id="id001722">Siue &longs;it parabole, &longs;iue hyperbole, &longs;eu &longs;it alia coincidentium.</s>
        </p>
        <pb pagenum="95" xlink:href="015/01/114.jpg"/>
        <p type="head">
          <s id="id001723">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id001724">Et nota, qu&ograve;d &longs;i res h&aelig;c e&longs;&longs;et naturalis, o&longs;tenderet infinitum in <lb/>rebus ex regula dialectica, &longs;ed quia ex <expan abbr="uol&utilde;taria">uoluntaria</expan>, nullas habet uires.</s>
        </p>
        <p type="main">
          <s id="id001725">Propo&longs;itio cente&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id001726">Proportionem motuum inuer&longs;ionis, &amp; attractionis in plano <lb/>inuenire.</s>
        </p>
        <p type="main">
          <s id="id001727">Et &longs;it, ut aliquid inuertatur, declaratum autem e&longs;t &longs;upr&agrave;, quid &longs;it </s>
        </p>
        <p type="main">
          <s id="id001728"><arrow.to.target n="marg351"/><lb/>inuer&longs;io, &amp; qu&agrave;m diuer&longs;a &longs;it rur&longs;us, &amp; qu&ograve;d attractio e&longs;t dimidium <lb/><arrow.to.target n="marg352"/><lb/>ponderis eleuati. </s>
          <s id="id001729">Cum ergo con&longs;tet in inuer&longs;ione, quanta &longs;it pro&shy;<lb/>portio ponderis &longs;u&longs;pen&longs;i ad pondus inuer&longs;um, &amp; pondus &longs;u&longs;pen&longs;i <lb/><arrow.to.target n="marg353"/><lb/>&longs;it duplum ponderi attracti, &longs;equitur, ut diui&longs;a proportione ponde<lb/>ris &longs;u&longs;pen&longs;i ad pondus inuer&longs;um per medium cogno&longs;catur propor<lb/>tio attractionis ad inuer&longs;ionem.</s>
        </p>
        <p type="margin">
          <s id="id001730"><margin.target id="marg351"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001731"><margin.target id="marg352"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 89.</s>
        </p>
        <p type="margin">
          <s id="id001732"><margin.target id="marg353"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s>
        </p>
        <p type="main">
          <s id="id001733">Ex hoc &longs;equitur, quod aliquod pondus trahi pote&longs;t, quod non <lb/><arrow.to.target n="marg354"/><lb/>pote&longs;t inuerti, hoc autem indiget longa declaratione, quam doce&shy;<lb/>bimus inferi&ugrave;s: &amp; tamen attigit hoc rar&ograve;.</s>
        </p>
        <p type="margin">
          <s id="id001734"><margin.target id="marg354"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001735">Propo&longs;itio cente&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id001736">Proportionem eorundem in accliui demon&longs;trare.</s>
        </p>
        <p type="main">
          <s id="id001737">Dupliciter pote&longs;t intelligi, uel de&longs;cendendo, uel a&longs;cendendo. <lb/><arrow.to.target n="marg355"/><lb/><arrow.to.target n="marg356"/><lb/>Sed ego nunc loquor de a&longs;cen&longs;u, contraria ratione intelliges de <lb/>de&longs;cen&longs;u, &amp; circa inuer&longs;ionem demon&longs;trata e&longs;t proportio eius <lb/>iuxta angulum a&longs;cen&longs;us, &amp; &longs;imiliter declarabitur de proportione <lb/><arrow.to.target n="marg357"/><lb/>attractionis iuxta eundem angulum a&longs;cen&longs;us, &amp; nuper declarata <lb/>e&longs;t proportio inuer&longs;ionis in plano ad attractionem, ex quibus &longs;e&shy;<lb/>quitur per ea, qu&aelig; dicam inferius, qu&ograve;d proportio cuiu&longs;uis mobi&shy;<lb/>lis inuer&longs;i ad attractum &longs;ub quibu&longs;cunque angulis nota erit.</s>
        </p>
        <p type="margin">
          <s id="id001738"><margin.target id="marg355"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001739"><margin.target id="marg356"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 72.</s>
        </p>
        <p type="margin">
          <s id="id001740"><margin.target id="marg357"/>I<emph type="italics"/>n &longs;equenti.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001741">Propo&longs;itio cente&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id001742">Proportionem motus attractionis in decliui ad motum in pla&shy;<lb/>no determinare.</s>
        </p>
        <p type="main">
          <s id="id001743">Si ab accliue, &longs;eu decliue in quo d ad attra&shy;<lb/><arrow.to.target n="marg358"/><lb/><arrow.to.target n="marg359"/><lb/><figure id="id.015.01.114.1.jpg" xlink:href="015/01/114/1.jpg"/><lb/>hendum, cuius nota e&longs;t ex &longs;uperioribus dif&shy;<lb/>ficultas in plano ratione figur&aelig; con&longs;tante, er&shy;<lb/>go ea qu&aelig;ritur proportio a&longs;cen&longs;us, &amp; quo&shy;<lb/>niam terminus ad perpendiculum e&longs;t dupla <lb/>proportio, &amp; iam grauitas in plano e&longs;t dimidium, ide&ograve; quicquid <lb/>acquiritur in eleuatione e&longs;t in comparatione ad illud dimidium, <lb/>cum ergo attractio &longs;ecundum eandem proportionem augeatur, er&shy;<lb/>go &longs;emper maior difficultas augebitur, ergo ab initio minimum 


<pb pagenum="96" xlink:href="015/01/115.jpg"/>erit di&longs;crimen ab attractione in plano. </s>
          <s id="id001744">Exempli gratia &longs;it, ut graue d <lb/>in plano &longs;it, ut quin que, &amp; &longs;u&longs;pen&longs;um decem, ergo in medio angulo <lb/>erit pen&egrave; &longs;eptem, &longs;ed &longs;eptem minus longe <expan abbr="di&longs;t&atilde;t">di&longs;tant</expan> &agrave; quin que, qu&agrave;m de&shy;<lb/>cem ad &longs;eptem, ergo in &longs;ecunda parte plus long&egrave; augebitur difficul<lb/>tas attractionis &longs;upra difficultatem in medio angulo accliui, quam <lb/>in prima parte &agrave; plano ad medium accliue, &amp; quoniam planum in <lb/>plano de&longs;cendit, tanto uehementius, quanto difficilius attrahitur, <lb/>ergo planum in decliui &longs;ublimi longe maiore impetu feretur infr&agrave; <lb/>quam &longs;it proportio anguli ad angulum. </s>
          <s id="id001745">Exempli gratia, planum in <lb/>medio angulo, &longs;i incipiat de&longs;cendere in dodrante multo lentius, <lb/>qu&agrave;m pro dimidio uirium de&longs;cen&longs;us totius anguli, im&ograve; initium de&shy;<lb/>&longs;cen&longs;us e&longs;t &agrave; medio recti ad unguem, ubi omnia plana &longs;int, &amp; duri&longs;&shy;<lb/>&longs;ima, &amp; cau&longs;a huius e&longs;t, quia omne graue tendit ad centrum, qu&ograve;d <lb/>maior pars ip&longs;ius grauis e&longs;t ultra medium grauitatis in decliui <lb/>humiliore.</s>
        </p>
        <p type="margin">
          <s id="id001746"><margin.target id="marg358"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001747"><margin.target id="marg359"/>E<emph type="italics"/>x<emph.end type="italics"/> 62. &amp; <lb/>64. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001748">Propo&longs;itio cente&longs;ima 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"/>
        <p type="main">
          <s id="id001750">H&aelig;c proponitur etiam &agrave; Philo&longs;o&shy;<lb/><arrow.to.target n="marg360"/><lb/>pho, &amp; ponatur ab, &amp; &longs;i pondus &longs;it in <lb/><arrow.to.target n="marg361"/><lb/>medio d grauat &aelig;qualiter utrunque, <lb/>nam in hoc con&longs;entit experimentum <lb/>cum ratione, at uer&ograve; &longs;i ponatur in cita, <lb/>ut b c &longs;it tripla b a uiderentur a &amp; b, tanquam hypomochlia, &amp; pon<lb/><arrow.to.target n="marg362"/><lb/>dus ip&longs;um b, ut grauior e&longs;&longs;et cb, quam c a. </s>
          <s id="id001751">Ari&longs;toteles, &longs;eu author <lb/>ille hoc uidens bifariam re&longs;pondet: primum qu&ograve;d hoc e&longs;t inuer&shy;<lb/><arrow.to.target n="marg363"/><lb/>&longs;um in&longs;trumentum, cum in c&aelig;teris motor &longs;it ex aduer&longs;o hypomo&shy;<lb/>chlij, hic in ip&longs;o, ge&longs;tans enim mouet &amp; hypomochlij in&longs;tar e&longs;t hu&shy;<lb/>merus. </s>
          <s id="id001752">At hoc uerum non e&longs;t: quod mouet enim e&longs;t pondus, &amp; e&longs;t <lb/>in c: nam a, &amp; contingit moueri: quia &longs;i &longs;tarent, idem &longs;equeretur. </s>
          <s id="id001753">Se&shy;<lb/>cunda re&longs;pon&longs;io e&longs;t, quod utrunque premit &longs;cilicet ferentes &amp; pon&shy;<lb/>dus, &amp; qu&ograve;d qui longior e&longs;t ab hypomochlio facilius mouet, &amp; <lb/>redit ad idem ferm&egrave;: nam in c con&longs;tituitur, quod moueri debet, ca&shy;<lb/>pita uectium &longs;unt a, &amp; b: motus autem e&longs;t ip&longs;um &longs;u&longs;tinere pondus. <lb/></s>
          <s id="id001754">At hoc non uidetur, quoniam ratio, qua uectis longior facilius mo<lb/>uet, e&longs;t ambitus magnitudo, ob quam motus redditur tardior, &amp; <lb/>ideo leuior: igitur non e&longs;t hoc uerum de motu occulto, &longs;icut e&longs;t gra<lb/>uis prementis, &longs;ed circumducente, cum in occulto uelut in &longs;tatera <lb/>contrarium accidere docuerimus ali&acirc;s. </s>
          <s id="id001755">Quidam dixere b premere <lb/>c uer&longs;us a, a contr&agrave; uer&longs;us b, &amp; ide&ograve; grauari magis a &agrave;b, qu&agrave;m b ab <lb/>a, quia maiorem uim habet b e, qu&agrave;m a c. </s>
          <s id="id001756">I&longs;tud fal&longs;um e&longs;t bifariam. <lb/></s>
          <s id="id001757">Primum, quia &amp; &longs;i a, &amp; b &longs;int in &aelig;quilibrio, ut nec unus in alterum 


<pb pagenum="97" xlink:href="015/01/116.jpg"/>incumbat, nec impellat, &longs;ed tantum &longs;u&longs;tineat nihilo &longs;ecius res uera <lb/>e&longs;t. </s>
          <s id="id001758">Et etiam quia non e&longs;t uerum, qu&ograve;d qui longius incumbit, ma&shy;<lb/>iorem uim inferat. </s>
          <s id="id001759">Propterea dicendum e&longs;t, qu&ograve;d qui ex commu&shy;<lb/>nibus propria nituntur demon&longs;trare, omnes corrumpunt di&longs;cipli&shy;<lb/>nas. </s>
          <s id="id001760">Nihil deterius e&longs;t his mon&longs;tris. </s>
          <s id="id001761">Nam et&longs;i h&aelig;c ratio uera e&longs;&longs;et: <lb/>non tamen reddit cau&longs;am, quia non e&longs;t ex proprijs principijs. </s>
          <s id="id001762">Dico <lb/>ergo, quod &longs;i c de&longs;cendat in e, per perpendiculum de&longs;cendet, igitur <lb/>d b e&longs;t longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus de&longs;cendit comparatione a, qu&agrave;m b, ergo plus grauat c ip&longs;um a <lb/>qu&agrave;m b, &longs;eu ex cau&longs;a, quod magis premat, &longs;eu ex effectu, qu&ograve;d ma&shy;<lb/>gis dece&longs;&longs;erit. </s>
          <s id="id001763">Cau&longs;a ergo erroris e&longs;t, quod &longs;i ponatur angulus f b a <lb/>&aelig;qualis angulo f a b, &amp; ponatur b f &etail;qualis b c, tun c in eodem tem&shy;<lb/>pore, in quo tran&longs;it dimidium c in e, tran&longs;ibit aliud dimidium c in f. <lb/></s>
          <s id="id001764">quia &longs;eparat&etail; partes grauiores &longs;unt in c b, qu&agrave;m c a, propter di&longs;tan&shy;<lb/>tiam ab hypomochlio, &longs;ed tunc uelocius mouentur, &amp; angulus fit <lb/>&etail;qualis. </s>
          <s id="id001765">Sed quando pondus e&longs;t unum, &amp; c de&longs;cendit ad e, cum de&shy;<lb/>&longs;cendat in&aelig;quali tempore, &amp; peragat maiorem angulum compa&shy;<lb/>ratione a, quam b, &longs;equitur, ut uelocius moueatur comparatione a <lb/>qu&agrave;m b. </s>
          <s id="id001766">Ergo &longs;i non mouetur, cum omnis potentia &longs;it &longs;imilis actui, <lb/>tum quia ab eo producitur, &amp; effectus e&longs;t &longs;imilis cau&longs;&aelig;: tum quia <lb/>e&longs;t initium actus, igitur etiam quod a b non inclinetur, nec de&longs;cen&shy;<lb/>dat, grauius erit pondus, comparatione a qu&agrave;m b, quod erat de&shy;<lb/>mon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id001767"><margin.target id="marg360"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001768"><margin.target id="marg361"/>Q<emph type="italics"/>us&longs;t.<emph.end type="italics"/> 59. <lb/>M<emph type="italics"/>echanic.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001769"><margin.target id="marg362"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s>
        </p>
        <p type="margin">
          <s id="id001770"><margin.target id="marg363"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 103.</s>
        </p>
        <p type="main">
          <s id="id001771">Ex hoc &longs;equitur, qu&ograve;d aliqua iuncta erunt grauiora re&longs;pectu u&shy;<lb/>nius, qu&aelig; erunt mutato ordine diui&longs;a leuiora. </s>
          <s id="id001772">Quoniam diui&longs;a, <lb/>qu&aelig; longius di&longs;tant &aelig;qualem, aut maiorem angulum faciunt, iun&shy;<lb/>cta minorem.</s>
        </p>
        <p type="main">
          <s id="id001773">Propo&longs;itio cente&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id001774">Quales proportiones angulorum doceant laterum proportio&shy;<lb/>nes. </s>
          <s id="id001775">At que uici&longs;sim determinare.</s>
        </p>
        <p type="main">
          <s id="id001776">Sit circulus a b c, cuius dimetiens, nota b d &longs;it b, erit ergo latus <lb/><arrow.to.target n="marg364"/><lb/><figure id="id.015.01.116.1.jpg" xlink:href="015/01/116/1.jpg"/><lb/>exagoni a b dimidium b d, id e&longs;t 3. igitur <lb/>cum angulus a &longs;it rectus, erit a d &lt;02&gt; 27 latus <lb/>trianguli. </s>
          <s id="id001777">Et latus quadrati per eandem &lt;02&gt;<lb/>18. Vt latus exagoni &longs;it &lt;02&gt; 9. Quadrati &lt;02&gt; 18 <lb/>Trianguli &lt;02&gt; 27, &amp; ita pote&longs;tate &longs;e habent <lb/>h&aelig;c ut 1. 2. 3. Et &longs;unt nota. </s>
          <s id="id001778">Et quia latus d e c <lb/>agoni e&longs;t &lt;02&gt; 11 1/4 m, 1 1/2. &amp; ip&longs;um erit notum. <lb/></s>
          <s id="id001779">Quare latus pentagoni e&longs;t &lt;02&gt; v 22 1/2 m: &lt;02&gt;<lb/>101 1/4 notum. </s>
          <s id="id001780">Et iam notum fuit latus epta&shy;<lb/>goni. </s>
          <s id="id001781">Habebimus igitur latera Trianguli 


<pb pagenum="98" xlink:href="015/01/117.jpg"/>quadrati pentagoni, &amp; eptagoni &aelig;quilaterorum nota: &amp; etiam <lb/>&longs;ubten&longs;orum duobus ex his. </s>
          <s id="id001782">Sit, gratia exempli, a b 3 &amp; b c &lt;02&gt; 11 1/4m: <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/>&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&shy;<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&shy;<lb/>tum diuide per 66, qu&aelig; e&longs;t 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/>6 1701/5184. Nec credas te errare, quoniam latus pentagoni e&longs;&longs;et, ac &longs;i an&shy;<lb/>gulus b rectus e&longs;&longs;et: &longs;ed quia e&longs;t obtu&longs;us, ideo a c e&longs;t alia linea, &amp; <lb/>maior latere pentagoni. </s>
          <s id="id001783">Et &longs;imiliter &longs;i a b, &amp; a c not&aelig; e&longs;&longs;ent, utpo&shy;<lb/><arrow.to.target n="marg365"/><lb/>te a b 3, ut prius a c 5 dico, qu&ograve;d b c nota e&longs;t: nam a d erit &lt;02&gt; 27, &amp; <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/>m: 9 quad. </s>
          <s id="id001784">igitur 30 m: pos &lt;02&gt; 27 &aelig;quantur &lt;02&gt; 324 m: 9 quad. </s>
          <s id="id001785">quare <lb/>900 p: 27 quad. </s>
          <s id="id001786">m: pos &lt;02&gt; 97200 <expan abbr="&aelig;qu&atilde;tur">&aelig;quantur</expan> 324 m: 9 quad. </s>
          <s id="id001787">igitur 576 <lb/>p: 16 quad. </s>
          <s id="id001788">&etail;quantur pos &lt;02&gt; 97200. Quadratum igitur p: 36 &etail;quan&shy;<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; &longs;imiliter &longs;i a c <lb/>&longs;it nota, puta 4 erit a b &longs;ubten&longs;a dimidio arcus a c nota. </s>
          <s id="id001789">Erit enim a e <lb/>2 ergo d e 3 p: &lt;02&gt; 5 et b e 3 m: &lt;02&gt; 5, <expan abbr="igi&ttilde;">igitur</expan> a b &lt;02&gt; v: 18 m, &lt;02&gt; 180. Igitur hoc <lb/>modo diuidendo, iungendo, &amp; detrahendo habebimus ex quatu&shy;<lb/>or illis &longs;implicibus trianguli quadrati. </s>
          <s id="id001790">Pentagoni, &amp; eptagoni in <lb/>numeras linearum magnitudines in circulo. </s>
          <s id="id001791">Et &longs;imiliter quouis mo <lb/>do, ut dictum e&longs;t, in quauis figura &aelig;quilatera, utpote &longs;uppo&longs;ito <lb/><figure id="id.015.01.117.1.jpg" xlink:href="015/01/117/1.jpg"/><lb/>quod de&longs;criptum &longs;it non angulum in <lb/>circulo &aelig;quilaterum, quod etiam erit <lb/>&aelig;quiangulum, &amp; &longs;it arcus a b duplus <lb/>arcui a c, erit angulus a c b duplus an&shy;<lb/>gulo a b c, &amp; angulus b a c in portione <lb/>b d e c &longs;excuplus a b c, &amp; triplus a c b. <lb/></s>
          <s id="id001792">Erit ergo per demon&longs;trata proportio <lb/><arrow.to.target n="marg366"/><lb/>b a ad a c, uelut a c, &amp; c b, ad a b: pro&shy;<lb/>portio autem a b arcus ad a c, ex &longs;up&shy;<lb/>po&longs;ito maior e&longs;t proportione rect&aelig; a b ad a c, igitur etiam propor&shy;<lb/>tione a c &amp; c b ad a b, ergo duo latera trianguli ad tertium minorem <lb/>habent proportionem, quam arcus ad arcum, quanto rect&aelig; ad re&shy;<lb/>ctam minor e&longs;t. </s>
          <s id="id001793">Sit rur&longs;us in triangulo b e d quomodolibet modo <lb/>&longs;it angulus b d e quadruplus angulo b e d, &amp; diuidatur d per &etail;qua&shy;<lb/>lia ducta d f, erit igitur proportio f d, d e ad f e, ut e f ad f d, &longs;ed e f ad <lb/><arrow.to.target n="marg367"/><lb/>f b ut d e ad d b. </s>
          <s id="id001794">igitur proportio b d, d e ad f b <expan abbr="c&otilde;po&longs;ita">compo&longs;ita</expan> ex propor&shy;<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/>producti ex e f in e d ad productum ex d fin d b. </s>
          <s id="id001796">Rur&longs;us ponamus, <lb/><arrow.to.target n="marg368"/><lb/>quod in quadrangulo a b c d prim&aelig; figur&aelig; &longs;it a b 4 b c 3 c d 5 ad 6 <lb/>dico, qu&ograve;d &longs;patium contentum erit notum. </s>
          <s id="id001797">Ductis rectis a c &amp; b d 


<pb pagenum="99" xlink:href="015/01/118.jpg"/>quomodolibet, ut &longs;e &longs;ecent in e, erunt anguli d c a, &amp; d b a &aelig;quales, <lb/><arrow.to.target n="marg369"/><lb/>quia in ea&dacute;em portione circuli a d, &amp; anguli a d e &etail;quales, quia con<lb/>tra &longs;e po&longs;iti. </s>
          <s id="id001798">igitur trianguli a b e, &amp; c d e &longs;imiles, &amp; proportio d c ad <lb/><arrow.to.target n="marg370"/><lb/>a b, ut c e ad b e, c d autem fuit 5 a b 4, igitur &longs;i b e ponatur 4 pos c e <lb/>erit 5 pos. </s>
          <s id="id001799">Per ea&longs;dem, &amp; eodem modo a d ad b c ut d e ad e c. igitur <lb/>po&longs;ita c e 5 pos erit e d 10 pos, tota igitur d b 14 pos. </s>
          <s id="id001800">Et quoniam ea&shy;<lb/><arrow.to.target n="marg371"/><lb/>dem proportio a e ad e b per eadem, &amp; e b fuit 4 pos: igitur a e e&longs;t 8 <lb/>pos, quare a e 13. po&longs;t productum igitur ex a c in d b, e&longs;t 182 quad. <lb/></s>
          <s id="id001801">&amp; hoc &aelig;quatur productis a b in c d, quod e&longs;t 20, &amp; b c in a d quod <lb/>e&longs;t 18, totum igitur e&longs;t 38, igitur res e&longs;t &lt;02&gt; 19/91. Quare not&etail; erunt line&aelig; <lb/>b e, e d, a e, &amp; e c, &longs;ed &longs;ufficit, ut cognita &longs;it a c, uel b d. </s>
          <s id="id001802">Per regulam <lb/>enim triangulorum erunt not&aelig; are&aelig; a b c, &amp; a d e, quare tota &longs;uper&shy;<lb/>ficies a b c d. </s>
          <s id="id001803">Et e&longs;t inuentum Scipionis Ferri Bononien&longs;is de quo <lb/>ali&acirc;s. </s>
          <s id="id001804">Pote&longs;t etiam inuenta a c uel b d haberi &longs;uperficies facilius <lb/>per catheros.</s>
        </p>
        <p type="margin">
          <s id="id001805"><margin.target id="marg364"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001806"><margin.target id="marg365"/>P<emph type="italics"/>er<emph.end type="italics"/> 52. E<emph type="italics"/>le <lb/>ment.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001807"><margin.target id="marg366"/>I<emph type="italics"/>n<emph.end type="italics"/> 16. <emph type="italics"/>de<emph.end type="italics"/><lb/>S<emph type="italics"/>ubtil.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001808"><margin.target id="marg367"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001809"><margin.target id="marg368"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001810"><margin.target id="marg369"/>P<emph type="italics"/>er<emph.end type="italics"/> 21. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001811"><margin.target id="marg370"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001812"><margin.target id="marg371"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001813">Sit modo obtu&longs;i angulus a b c, &amp; nota latera &longs;ingula, &amp; angu&shy;<lb/>lus a b c, &amp; producantur latera ad perpendicu&shy;<lb/><figure id="id.015.01.118.1.jpg" xlink:href="015/01/118/1.jpg"/><lb/>lum, ut &longs;int d &amp; e recti, &amp; quia anguli ad a &longs;unt <lb/>&aelig;quales, erunt anguli e b a, &amp; d e a &longs;emper &aelig;&shy;<lb/><arrow.to.target n="marg372"/><lb/>quales. </s>
          <s id="id001814">Et hoc idem contingit in acuti angulis <lb/>triangulis intus, &amp; e&longs;t utile mechanicum: &amp; <lb/>quia a b c notus e&longs;t, &amp; d notus, erunt anguli tri<lb/>goni d b c noti: &amp; &longs;i fuerit angulus a notus, <expan abbr="er&utilde;t">erunt</expan> anguli d a c &amp; e a b <lb/>noti, &amp; ideo anguli e b a, &amp; d c a: &amp; &longs;emper notum, quod fit ex b a <lb/>in a d, uel c a in a e, &longs;unt enim &etail;qualia inter &longs;e: etiam not&aelig; ad &amp; a c, <lb/>quoniam duplum horum e&longs;t exce&longs;&longs;us quadrati b c &longs;uper quadrata <lb/>a b, &amp; a c. </s>
          <s id="id001815">Quod uer&ograve; propositur&agrave; Monteregio de cognitione an&shy;<lb/>gulorum in triangulis non e&longs;t intelligendum, ut uerba &longs;ignificant, <lb/><arrow.to.target n="marg373"/><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s>
        </p>
        <p type="margin">
          <s id="id001816"><margin.target id="marg372"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001817"><margin.target id="marg373"/>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>&longs;e&shy;<lb/>cundi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001818">Et iterum ponamus, qu&ograve;d proportio a c c b ad a b &longs;it qualis a b <lb/>ad a c, dico qu&ograve;d angulus c duplus e&longs;t angulo b. </s>
          <s id="id001819">Si non ducatur c d <lb/><figure id="id.015.01.118.2.jpg" xlink:href="015/01/118/2.jpg"/><lb/>faciens angulum d c b duplum b, erit igitur pro&shy;<lb/>portio d c c b ad d b, ut d b ad d c. </s>
          <s id="id001820">Maior e&longs;t <expan abbr="aut&etilde;">autem</expan> <lb/>d c, qu&agrave;m a c, aut &aelig;qualis, aut minor, &longs;i &aelig;qualis, <lb/>igitur maior proportio d c c b ad b d qu&agrave;m b a, <lb/>igitur maior proportio b d ad d c quam b a ad a c <lb/>ad a c &amp; &aelig;quales &longs;unt igitur b d maior d a pars toto, quod e&longs;&longs;e non <lb/>pote&longs;t. </s>
          <s id="id001821">Si uer&ograve; d c ponatur maior a c, magis ex hoc &longs;equitur b d ma&shy;<lb/>iorem e&longs;&longs;e b a. </s>
          <s id="id001822">Quod &longs;i minor &longs;it d c qu&agrave;m a c. </s>
          <s id="id001823">Ex demon&longs;tratio&shy;<lb/>ne ip&longs;ius reflex&aelig; proportionis patet hoc contingere non po&longs;&longs;e. <lb/></s>
          <s id="id001824">Et &longs;imiliter patet conuer&longs;as in reliquis etiam ueras e&longs;&longs;e, non &longs;olum 


<pb pagenum="100" xlink:href="015/01/119.jpg"/>in proportionibus noti&longs;simis angulorum &longs;ed etiam in coniuncti&shy;<lb/>one &amp; detractione. </s>
          <s id="id001825">Et e&longs;t ex &longs;ubtili&longs;simis operationibus, qu&aelig; ho&shy;<lb/>mini in hoc genere eueniant.</s>
        </p>
        <p type="main">
          <s id="id001826">Propo&longs;itio cente&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id001827">Si in circulo duo diametri ad rectum angulum &longs;e &longs;ecauer int: ali&etail; <lb/>uer&ograve; ad perpendiculum ex diametro exierint ad circumferentiam, <lb/>&longs;ingul&aelig; &longs;upra diametrum erunt maiores portionibus reliquis dia&shy;<lb/>metri &longs;uperioribus, infra autem minores. </s>
          <s id="id001828">Dimidium autem porti&shy;<lb/>onis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit. </s>
          <s id="id001829">In ali&shy;<lb/>qua pr&aelig;terea portionis &longs;uperioris parte, qu&aelig; uer&longs;us diametrum <lb/>tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differentia partis diametri ei cor&shy;<lb/>re&longs;pondentis, quam line&aelig; tran&longs;uer&longs;&aelig;.</s>
        </p>
        <figure id="id.015.01.119.1.jpg" xlink:href="015/01/119/1.jpg"/>
        <p type="main">
          <s id="id001830">Sint du&etail; diametri a b, c d ad perpendi <lb/>culum &longs;ecantes &longs;e in centro, &amp; <expan abbr="duc&utilde;tur">ducuntur</expan> <lb/>&longs;upr f g k h, &amp; infra m l ad perpendicu&shy;<lb/>lum &longs;upra a b: dico f g e&longs;&longs;e maiorem f a, <lb/>&amp; k h k a, &amp; contr&agrave; minorem m l, qu&agrave;m <lb/>m a. </s>
          <s id="id001831">Per octauam enim &longs;exti, quod fit ex <lb/><arrow.to.target n="marg374"/><lb/>b f in f a &aelig;quale e&longs;t <expan abbr="&qtilde;drato">quadrato</expan> f g, &longs;ed b f e&longs;t <lb/>maior f g, quia b f e&longs;t maior c b, &amp; ideo <lb/>e c g f, ergo f g maior e&longs;t f a, m l <expan abbr="a&utilde;t">aut</expan> minor e&longs;t per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod e&longs;t primum. </s>
          <s id="id001832">Suppo&longs;ito etiam, qu&ograve;d <lb/><arrow.to.target n="marg375"/><lb/>a g arcus &longs;it dimidium a c, dico a f <expan abbr="minor&etilde;">minorem</expan> e&longs;&longs;e f e, nam quadratum e <lb/><arrow.to.target n="marg376"/><lb/>g &aelig;quale e&longs;t quadratis f e, &amp; f g, &amp; <expan abbr="quadrat&utilde;">quadratum</expan> a g quadratis f g &amp; f a <lb/>&amp; e g e&longs;t &etail;qualis lateri exagoni, &amp; a g latus octogoni, igitur e g ma&shy;<lb/><arrow.to.target n="marg377"/><lb/>ior g a, &amp; duo quadrata e f &amp; f g maiora duobus quadratis f g &amp; <lb/>f a, detracto igitur communi f g quadrato, patet propo&longs;itum.<lb/><arrow.to.target n="marg378"/></s>
        </p>
        <p type="margin">
          <s id="id001833"><margin.target id="marg374"/>P<emph type="italics"/>er<emph.end type="italics"/> 31. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001834"><margin.target id="marg375"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>tertij<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001835"><margin.target id="marg376"/>1. <emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001836"><margin.target id="marg377"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001837"><margin.target id="marg378"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/>15. <emph type="italics"/>quarti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001838">Cum rur&longs;us ex prima parte huius line&etail; f g &amp; k h &longs;int maiores f a, <lb/>&amp; k a &amp; ea &longs;it &aelig;qualis e c, nece&longs;&longs;e e&longs;t ut iuxta punctum c augeatur </s>
        </p>
        <p type="main">
          <s id="id001839"><arrow.to.target n="marg379"/><lb/>magis linea in ea, quam &longs;it differentia line&aelig; tran&longs;uer&longs;&aelig; ad lineam <lb/>tran&longs;uer&longs;am per communem animi &longs;ententiam, quod e&longs;t tertium.</s>
        </p>
        <p type="margin">
          <s id="id001840"><margin.target id="marg379"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001841">Propo&longs;itio cente&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id001842">Punctum &etail;qualitatis differenti&etail; de&longs;cen&longs;us, &amp; remotionis &agrave; cen&shy;<lb/>tro inuenire.</s>
        </p>
        <p type="main">
          <s id="id001843">Per pr&aelig;cedentem moto puncto a uer&longs;us c &longs;emper u&longs; que ad e, c ma<lb/><arrow.to.target n="marg380"/><lb/>gis di&longs;tat <expan abbr="p&utilde;ctum">punctum</expan> a linea a e, qu&agrave;m &agrave; puncto a uer&longs;us, quia linea n h <lb/>maior e&longs;t n a, &amp; per eandem dum appropinquat ad c cum e c fiat <lb/>&etail;qualis ea, maius fit in crementum in a e, qu&agrave;m re&longs;pectu line&aelig; tran&longs;&shy;<lb/>uer&longs;alis. </s>
          <s id="id001844">Volo ergo inuenire punctum hoc in quo fit mutatio: &amp; <lb/>diuido arcum ac per &aelig;qualia in f, &amp; dico illum e&longs;&longs;e punctum qu&aelig;&shy;<lb/>&longs;itum: accepto quouis puncto in e f, puta k, duco g o h p &etail;quidi&longs;tan<pb pagenum="101" xlink:href="015/01/120.jpg"/><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"/><lb/>tes a b, &amp; c d: erunt que anguli q &amp; n recti <lb/><arrow.to.target n="marg381"/><lb/>&amp; anguli f e a, &amp; f e c &etail;quales, igitur uter <lb/><arrow.to.target n="marg382"/><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &etail;qua <lb/><arrow.to.target n="marg383"/><lb/>lis n k, igitur c q &aelig;qualis e n, quare h p <lb/>&aelig;qualis g o, &longs;ed quod fit ex o k in k g e&longs;t <lb/><arrow.to.target n="marg384"/><lb/>&aelig;quale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"/><lb/>k h e&longs;t &aelig;qualis k g ex eisdem o&longs;tendi&shy;<lb/>tur f l m k quadratum e&longs;&longs;e. </s>
          <s id="id001845">Quia ergo <lb/>k h e&longs;t &aelig;qualis k g, &amp; k l &aelig;qualis k m, erit l g &aelig;qualis m h. </s>
          <s id="id001846">Er&shy;<lb/>go de&longs;cendendo ex g in f, quantum f l &longs;uperat l g, tantum de&longs;cen&shy;<lb/>dendo ex f in h, f m &longs;uperat m h per communem animi &longs;ententi&shy;<lb/>am. </s>
          <s id="id001847">At f m e&longs;t de&longs;cen&longs;us f in linea a e, &amp; m h di&longs;tantia, qu&aelig; acqui&shy;<lb/>ritur in linea f r, n m enim e&longs;t &aelig;qualis f r, igitur n h excedit f r in <lb/>h m, &amp; ita a n excedit a r in n r &etail;quali f m. </s>
          <s id="id001848">Quantum ergo in g f, <lb/>l f excedit l g, tantum in de&longs;cen&longs;u ex f in h, f m, qu&aelig; refert g l, ex&shy;<lb/>cedit h m, qu&aelig; refert f l. </s>
          <s id="id001849">Arcus autem f g e&longs;t &aelig;qualis arcui f h, <lb/>quod <expan abbr="c&utilde;">cum</expan> po&longs;&longs;em o&longs;tendere pluribus modis &longs;atis con&longs;tat, quia chor<lb/><arrow.to.target n="marg386"/><lb/>darum illorum quadrata &longs;unt inuicem &aelig;qualia, quia line&aelig; f m, &amp; <lb/><arrow.to.target n="marg387"/><lb/>f l item que m h &amp; l g &longs;unt &aelig;quales, &amp; anguli m, &amp; l recti. </s>
          <s id="id001850">Igitur cum <lb/>ad quod uis punctum in linea e f &longs;emper linea de&longs;cen&longs;us in parte <lb/>inferiore e&longs;t maior linea di&longs;tanti&aelig; tanto, quanto per &aelig;qualem ar&shy;<lb/>cum in &longs;uperiore linea di&longs;tanti&aelig; e&longs;t maior linea, de&longs;cen&longs;us &longs;equitur <lb/>per regulam Dialecticam quod punctus f, e&longs;t punctus &etail;qualitatis. <lb/></s>
          <s id="id001851">Per idem diceremus in quarta parte inferiore.</s>
        </p>
        <p type="margin">
          <s id="id001852"><margin.target id="marg380"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001853"><margin.target id="marg381"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001854"><margin.target id="marg382"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001855"><margin.target id="marg383"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 32. <lb/>&amp; 6.</s>
        </p>
        <p type="margin">
          <s id="id001856"><margin.target id="marg384"/>P<emph type="italics"/>er<emph.end type="italics"/> 34. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001857"><margin.target id="marg385"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>tertij<emph.end type="italics"/><lb/>E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001858"><margin.target id="marg386"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001859"><margin.target id="marg387"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001860">Propo&longs;itio cente&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id001861">Rationem libr&aelig; expendere.</s>
        </p>
        <p type="main">
          <s id="id001862">Cum libra moueatur, uelut rota circa axem, quia trutina manet, <lb/>ide&ograve; &longs;i pondus ponatur, dum iugum fuerit in linea a b nihil mo&shy;<lb/>uebitur, quia appetitus de&longs;cen&longs;us ex puncto a maximus e&longs;t, &amp; ni&shy;<lb/>hil iuuat motum extra naturam, idem dico de graui po&longs;ito in uerti&shy;<lb/>ce b a. </s>
          <s id="id001863">Nam duo &longs;unt motus in rota, &amp; in libra unus, per quem <lb/>dum fertur per arcum a f, gratia exempli de&longs;cendit, quantum e&longs;t <lb/><arrow.to.target n="marg388"/><lb/>a r, qu&aelig; e&longs;t minor dimidio e r, &amp; ide&ograve; minor e r, qu&aelig; e&longs;t maior di&shy;<lb/>midio, ut demon&longs;tratum e&longs;t, &amp; etiam minor r f, qu&aelig; &aelig;qualis e&longs;t r e <lb/><arrow.to.target n="marg389"/><lb/>per demon&longs;trata rur&longs;us: &amp; hic e&longs;t naturalis ut palam e&longs;t: alter pr&aelig;&shy;<lb/>ter <expan abbr="natur&atilde;">naturam</expan>, &amp; e&longs;t ferri ad latus, quoniam hoc e&longs;t <expan abbr="propri&utilde;">proprium</expan> immortali&shy;<lb/>bus: cun que hic &longs;it ad latus e&longs;t etiam <expan abbr="c&otilde;tra">contra</expan> naturam, quia magis di&longs;tat <lb/>a centro, nam e f e&longs;t longior c r, &longs;i ergo r ferretur in f, moueretur &agrave; <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"/>fertur, qu&agrave;m ex f in c: uelocius autem ex c u&longs;que ad medium: nam <lb/>plurimum de&longs;cendit. </s>
          <s id="id001865">Ex h ad b autem celerrim&egrave;, quoniam de&longs;cen&shy;<lb/>dit, &amp; appropinquat line&aelig; a b, ut uter que motus &longs;it naturalis. </s>
          <s id="id001866">Non <lb/>ergo mouetur pr&etail;ter naturam ni&longs;i quatenus longius recedit &agrave; linea <lb/>a b, unde in inferiore parte mouetur ad eandem, ide&ograve; de parte c b <lb/>tota per&longs;picua e&longs;t ratio, cur facillim&egrave; de&longs;cendat, &longs;imiliter &amp; tota, <lb/>hoc enim e&longs;t demon&longs;tratum. </s>
          <s id="id001867">Similiter &amp; quare difficillim&egrave; feratur <lb/>ex b u&longs; que ad p, &amp; ultra p u&longs; que ad directum r f: at de motu ex a in f, <lb/>quod debeat ferri, quia plus remouetur, quam de&longs;cendat, nulla e&longs;t <lb/>ratio: ut nec cur ex oppo&longs;ito f ad a difficilem &longs;e pr&aelig;&longs;tet: &amp; hoc e&longs;t, <lb/>quia tertiam rationem etiam ip&longs;e Ari&longs;toteles, &amp; qui eum &longs;equuti <lb/>&longs;unt, pr&aelig;termi&longs;it. </s>
          <s id="id001868">Ea autem e&longs;t, quod dum fertur ad g, uel f etiam li&shy;<lb/>cet non de&longs;cendat magis, qu&agrave;m remoueatur, ex a <lb/><figure id="id.015.01.121.1.jpg" xlink:href="015/01/121/1.jpg"/><lb/>ad centrum terr&aelig; tamen magis appropinquat. <lb/></s>
          <s id="id001869">Quia enim e a e&longs;t &etail;qualis e c, quoniam prodeunt <lb/>&agrave; centro circuli eiu&longs;dem, &amp; b e, &amp; e c &longs;unt maio&shy;<lb/>res b c, ide&ograve; b a erit maior b c, e&longs;t autem b cen&shy;<lb/><arrow.to.target n="marg390"/><lb/>trum mundi, ergo a motum ad c, appropinqua&shy;<lb/>uit ip&longs;i b</s>
        </p>
        <p type="margin">
          <s id="id001870"><margin.target id="marg388"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 98.</s>
        </p>
        <p type="margin">
          <s id="id001871"><margin.target id="marg389"/>I<emph type="italics"/>n pr&aelig;ceden <lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001872"><margin.target id="marg390"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id001873">Dico etiam quod libra ex chalybe tenui&longs;simo, <lb/>&amp; quanto <expan abbr="leuior&utilde;">leuiorum</expan> concharum, &amp; longioris iugi <lb/>10 exactior, quoniam lances ill&aelig; minori exce&longs;&longs;u <lb/>mouentur, quia plus di&longs;tant ab hypomochlio. <lb/></s>
          <s id="id001874">Sit ergo libra, cuius iugum a b trutina c: lances d &amp; e, alia libra, <lb/>cuius lances h, &amp; k, &amp; l m longiores, iugum f g. </s>
          <s id="id001875">Con&longs;tat, quod <lb/>qualis proportio f g ad a b, talis ambitus, ad ambitum: motus er&shy;<lb/>go &longs;i &longs;it &aelig;qualis utrarumque, igitur a tanto minore proportione <lb/><figure id="id.015.01.121.2.jpg" xlink:href="015/01/121/2.jpg"/>


<pb pagenum="103" xlink:href="015/01/122.jpg"/>mouebitur in h, quam in d, uelut &longs;it proportio f g ad a b dupla, ut <lb/>ergo &aelig;qualiter moueantur, &longs;i &longs;it dupla &longs;exquiquarta in d cum lan&shy;<lb/>ce ad e uacuam, erit in h &longs;exquialtera, &amp; mouebit &aelig;quali tempore. <lb/></s>
          <s id="id001876">Ergo iuxta hoc fient libr&aelig;, qu&aelig; examinabunt decimam, &amp; uige&longs;i&shy;<lb/>mam partem grani, quod e&longs;t nece&longs;&longs;arium in pretio&longs;is rebus, &amp; me&shy;<lb/>dicamentis potentibus, &amp; long&egrave; magis in mechanicis experimen&shy;<lb/>tis, &amp; maxim&egrave; qu&aelig; ad demon&longs;trationem pertinent magnitudinis <lb/>&longs;uperficierum, &amp; con&longs;tat res in tribus, in longitudine, f g iungi, in le <lb/>uitate materi&aelig; illius, &amp; lancium, nam tanto maior redditur propor<lb/>tio ponderis exigui, &amp; in firmitate iugi ac rectitudine. </s>
          <s id="id001877">ide&ograve; debet <lb/>fieri ex chalybe purgato, durato ac tenui&longs;simo, natura que leui, &amp; ut c <lb/>&longs;it in medio, &amp; mobilis f g.</s>
        </p>
        <p type="main">
          <s id="id001878">Con&longs;iderandum e&longs;t demum an f l &amp; g m &longs;int grauiores f h, &amp; <lb/>g k. </s>
          <s id="id001879">Vt enim grauiores extiterint minus facil&egrave; mouentur. </s>
          <s id="id001880">Viden&shy;<lb/>tur autem mihi, qui de his con&longs;crip&longs;erunt perperam contemp&longs;i&longs;&longs;e <lb/>hoc, con&longs;tat enim, qu&ograve;d dum l de&longs;cendit, remouetur a b n c tru&shy;<lb/>tina, &amp; m, qu&aelig; a&longs;cendit contra appropinquat. </s>
          <s id="id001881">Videtur autem hoc <lb/>bifariam contra naturam: nam ut diximus pondus applicat &longs;e ad <lb/>rectam n c, quia uer&longs;us centrum, &amp; etiam quia facit angulum ob&shy;<lb/>tu&longs;um, cum deberet, ut ab initio &longs;altem con&longs;tituere cum iugo re&shy;<lb/>ctum. </s>
          <s id="id001882">Et de m nihil mirum e&longs;t, cum acutum, ut &longs;e ad lineam, qu&aelig; ad <lb/>centrum retrahat. </s>
          <s id="id001883">Huiu&longs;modi pr&aelig;terij&longs;&longs;e Ari&longs;totelem, demiror, <lb/>qu&aelig; nimis fuerunt in con&longs;picuo, ut dubitem ne non &longs;uus &longs;it ille li&shy;<lb/>ber, qui eius pen&egrave; nihil &longs;apiat pr&aelig;ter ob&longs;curitatem. </s>
          <s id="id001884">Tentan&shy;<lb/>dum e&longs;t igitur horum cau&longs;as a&longs;signare. </s>
          <s id="id001885">nam qu&aelig; huiu&longs;modi po&shy;<lb/>te&longs;t e&longs;&longs;e doctrina ni&longs;i perfecta fuerit, in omnibus etenim nece&longs;&longs;e e&longs;t <lb/>aut omnia &longs;cire, aut ignorare. </s>
          <s id="id001886">In hoc igitur dico, quod h f, &longs;eu l f, <lb/>&longs;emper &aelig;quidi&longs;tant n c trutin&aelig;, ergo cum angulus f c n in clina&shy;<lb/>to iugo fiat obtu&longs;us de&longs;cendente pondere, &amp; n c g a&longs;cendente pon&shy;<lb/>dere fiat acutus, ergo angulus l f c tantundem fiet obtu&longs;ior, &amp; m g c <lb/>acutior, quanto anguli ad c tales &longs;unt. </s>
          <s id="id001887">Et cau&longs;a e&longs;t quia n c ratio&shy;<lb/>ne ponderis e&longs;t directa ad centrum, ergo oportet, ut pondera l, uel <lb/>h, &amp; m, uel k, &longs;i debent tendere ad centrum, ut f l, &amp; g m &aelig;quidi&shy;<lb/>&longs;tent n c, ni&longs;i quantum e&longs;t pro di&longs;tantia f, &agrave; puncto c, &amp; g a b eodem, <lb/>qu&aelig; comparata ad <expan abbr="centr&utilde;">centrum</expan> terr&etail;, &longs;eu mundi, e&longs;t in&longs;en&longs;ibilis omnino. <lb/></s>
          <s id="id001888">Circa h&aelig;c <expan abbr="notand&utilde;">notandum</expan> i&longs;tud mirabile &longs;cilicet, quod ratio motus, quan&shy;<lb/>tumuis exigua &longs;ufficit ad motus <expan abbr="mod&utilde;">modum</expan>, licet uelo citas <expan abbr="p&etilde;deat">pendeat</expan> ex gra<lb/>uitate, &amp; alijs. </s>
          <s id="id001889">Et quae graue, quod expers e&longs;t &longs;en&longs;us, debeat &longs;equi ratio <lb/>nem Geometricam uix &longs;apientibus <expan abbr="cognit&atilde;">cognitam</expan>, cau&longs;a tamen una e&longs;t, &amp; <lb/>per&longs;picua: <expan abbr="n&atilde;">nam</expan> omne graue e&longs;t in linea &agrave; centro <expan abbr="m&utilde;di">mundi</expan>: &longs;i <expan abbr="a&utilde;t">aut</expan> medium <lb/>grauis &longs;it extra <expan abbr="line&atilde;">lineam</expan>, uertitur ad illam, qu&etail; e&longs;t in eo, nam <expan abbr="centr&utilde;">centrum</expan> &longs;em<pb pagenum="104" xlink:href="015/01/123.jpg"/>per e&longs;t in <expan abbr="ead&etilde;">eadem</expan>. </s>
          <s id="id001890">Ergo &longs;ola inclinatio ad hoc ut <expan abbr="medi&utilde;">medium</expan> grauis &longs;it in li&shy;<lb/>nea <expan abbr="centror&utilde;">centrorum</expan> grauitatis &amp; terr&aelig;, &longs;ufficit. </s>
          <s id="id001891">E&longs;t ergo principium in &longs;e i&shy;<lb/>p&longs;o. </s>
          <s id="id001892">In appen&longs;is &longs;imiliter. </s>
          <s id="id001893">Trutina enim, &amp; finis iugi, &amp; grauis <expan abbr="cen&shy;tr&utilde;">cen&shy;<lb/>trum</expan> mundi <expan abbr="centr&utilde;">centrum</expan> &longs;unt in <expan abbr="ead&etilde;">eadem</expan> linea, ut e&longs;&longs;e po&longs;&longs;unt, cum exigua illa <lb/>&amp; &longs;ola di&longs;tantia intercedat. </s>
          <s id="id001894">&amp; hoc e&longs;t primum. </s>
          <s id="id001895">Quia ergo <expan abbr="iug&utilde;">iugum</expan> e&longs;t <lb/>ex materia &longs;olida, mouetur ratione, qu&aelig; dicta e&longs;t, lances autem <lb/>oportet cum filis appen&longs;i &longs;int, ut puncta f &amp; h, uel l, &amp; g k, uel g m <lb/>&longs;int in una linea cum centro terr&aelig;. </s>
          <s id="id001896">Et quia l magis di&longs;tat a b f quam <lb/>h, &amp; m a g magis, quam k, &amp; oportet faciant eandem inclinatio&shy;<lb/>nem, quia anguli trutin&aelig; cum iug&oacute; &longs;unt ijdem, &amp; linea cl e&longs;t ma&shy;<lb/>ior c h, &amp; c m, qu&agrave;m c k in quouis &longs;itu, ergo &longs;patium, quod ambitur, <lb/>e&longs;t maius ergo per d e mon&longs;trata &longs;uperius l e&longs;t grauius h etiam <lb/>pr&aelig;ter uinculorum additionem, &amp; m grauius k. </s>
          <s id="id001897">Quanto igi&shy;<lb/>tur longiores &longs;unt funiculi &agrave; libr&aelig; extremitate &longs;eu iugi, tanto gra&shy;<lb/>uius redditur pondus, quod tamen multi putant e&longs;&longs;e fal&longs;um: nec <lb/>aliquid referre, qu&ograve;d &longs;it longum, aut breue &longs;u&longs;tentaculum.</s>
        </p>
        <p type="main">
          <s id="id001898">Propo&longs;itio cente&longs;ima decima.</s>
        </p>
        <p type="main">
          <s id="id001899">Si du&aelig; &longs;ph&aelig;r&aelig; ex eadem materia de&longs;cendant in <expan abbr="a&etilde;">ae</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"/><lb/><arrow.to.target n="marg391"/></s>
        </p>
        <p type="margin">
          <s id="id001900"><margin.target id="marg391"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001901">Supponitur quod ex eodem loco. </s>
          <s id="id001902">Sermo enim <lb/>ab&longs;urda &longs;ub interpretatione nunquam ni&longs;i ab inui&shy;<lb/>dio&longs;o, uel imperito intelligi debet. </s>
          <s id="id001903">Sit ergo a tripla <lb/>ad b, &longs;ph&aelig;rula ad &longs;ph&aelig;rulam ex plumbo amb&aelig; fer&shy;<lb/>ro uel lapide eiu&longs;dem generis, dico, qu&ograve;d in&aelig;quali <lb/>tempore peruenient ad planum c d. </s>
          <s id="id001904">Nam a propor&shy;<lb/>tionem habet ad b, ut uiginti &longs;eptem ad unum. </s>
          <s id="id001905">pro&shy;<lb/>portio autem &longs;patij a ad &longs;patium b nonupla e&longs;t, &amp; <lb/>proportio den&longs;itatis a&euml;ris ad a&euml;rem e&longs;t tripla, propterea quod den&shy;<lb/>&longs;itas illa multiplicatur propter impetus magnitudinem. </s>
          <s id="id001906">nam &longs;i ro&shy;<lb/>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du&shy;<lb/>plo, qu&agrave;m &longs;it robur, ut quinque percutiat baculo, ut duo: propter <lb/>den&longs;itatem ergo maiorem a&euml;ris in a, quam in b: &amp; quoniam &longs;i &longs;ub <lb/>maiore impetu mouetur <expan abbr="a&etilde;r">aer</expan> &longs;ub a, quam &longs;ub b, igitur proportio <lb/>erit comparanda longitudini &agrave; centro a ad longitudinem a centro <lb/>b, qu&aelig; e&longs;t tripla. </s>
          <s id="id001907">Si ergo &longs;ubtripla e&longs;t ratio motus b ad a, quod <lb/>ad medium attinet, tripla autem propter uelocitatem di&longs;ce&longs;&longs;us a&euml;&shy;<lb/>ris &agrave; medio grauitatis, quod e&longs;t in &longs;uperficie e regione centri graui&shy;<lb/>tatis in linea ad centrum mundi, ut dictum e&longs;t in pr&aelig;cedenti: mani&shy;<lb/>fe&longs;tum e&longs;t, quod a, &amp; b in&aelig;quali tempore peruenient ad &longs;ubie&shy;<lb/>ctum planum, &amp; &aelig;quidi&longs;tans centris eorum. </s>
          <s id="id001908">Similiter &amp; in aqua: 


<pb pagenum="105" xlink:href="015/01/124.jpg"/>cum uer&ograve; uideatur in illa tanto celerius a de&longs;cendere, qu&agrave;m b, <lb/>quanto e&longs;t &longs;emidiameter a longior &longs;emidiametro b, liquet ex hoc, <lb/>quod &aelig;quali uelo citate de&longs;cendunt, &longs;ed ob uelocitatem motus in <lb/>a&euml;re latet di&longs;crimen anticipationis contactus &longs;oli a ante b, qui di&shy;<lb/>gno&longs;citur in aqua, ex quo patet exactam e&longs;&longs;e &aelig;qualitatem. </s>
          <s id="id001909">Sed re&longs;i&shy;<lb/>liunt &longs;emel in aqua amb&aelig;, cum pluries in a&euml;re a &longs;olo, quare etiam in <lb/>aqua perturbatur cognitio in parum accuratis, at que &longs;en&longs;u pr&aelig;ditis, <lb/>&longs;icut etiam in ca&longs;u, ne altera alteram perueniat, utra que comprehen&longs;a <lb/>duobus digitis, altera alteram tangente, &amp; u&longs;que ad centrum in <lb/>aquam demi&longs;sis &longs;imul digitis dilatatis dimittend&aelig; &longs;unt.</s>
        </p>
        <p type="main">
          <s id="id001910">Propo&longs;itio cente&longs;ima undecima.</s>
        </p>
        <p type="main">
          <s id="id001911">Cur ex medio tela ualidiorem ictum, &amp; naues in &longs;calmo &agrave; remo, <lb/>ac malo recipiant inde ex puppi explorare.</s>
        </p>
        <p type="main">
          <s id="id001912">Ari&longs;toteles uidetur in Mechanicis, &amp; qui eum &longs;equuti &longs;unt, ui&shy;</s>
        </p>
        <p type="main">
          <s id="id001913"><arrow.to.target n="marg392"/><lb/>dentur rem nauticam qu&ograve;d ad remos attinet, referre in longitu&shy;<lb/>dinem partis, qu&aelig; &longs;calmum tanqu&agrave;m hypomochlium interiacet <lb/>&amp; manum: ea enim circa medium nauis cum illa ibi &longs;it latior ma&shy;<lb/>ior e&longs;t. </s>
          <s id="id001914">Sed &amp; qui lembos ducunt, &amp; in puppe magis di&longs;tant &agrave; <lb/>&longs;calmo &amp; in prora, qu&agrave;m in medio nauis, nec tamen uelocius il&shy;<lb/>lam agunt: non qu&ograve;d ratio illa fal&longs;a &longs;it, &longs;ed quia uelocius ferun&shy;<lb/>tur etiam ob aliam cau&longs;am, qu&agrave;m &longs;it h&aelig;c, &amp; magis uniuer&longs;alem. <lb/></s>
          <s id="id001915">Primum igitur &longs;umamus, quod &longs;uperi&ugrave;s demon&longs;tratum e&longs;t &longs;cili&shy;<lb/><arrow.to.target n="marg393"/><lb/>cet, qu&ograve;d ubi pondus aliquod &aelig;quale undique tanquam in li&shy;<lb/>bra &longs;u&longs;pen&longs;um fuerit, proportio ponderis partium in&aelig;qualium <lb/>ad duas partes &aelig;quales, e&longs;t confu&longs;a ex proportione longitudi&shy;<lb/>nis earundem, &amp; quadrato eiu&longs;dem proportionis. </s>
          <s id="id001916">Sit ergo diui&shy;<lb/>&longs;a a b in c, &amp; fiat c e &aelig;qualis c a: proportio igitur ponderis b e ad <lb/>pondus e a e&longs;t compo&longs;ita ex proportione b e ad e a, &amp; quadrato <lb/><figure id="id.015.01.124.1.jpg" xlink:href="015/01/124/1.jpg"/><lb/>eius <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> longitudinem. </s>
          <s id="id001917">at po&longs;ita agi <lb/>na d g in medio a b, proportio ponderis b e <lb/>ad pondus ea e&longs;t, ueluti longitudinis b e <lb/>ad e a, igitur proportio <expan abbr="p&otilde;deris">ponderis</expan> b e ad e a, <lb/>cum agina e&longs;t extra medium in c, e&longs;t tanto <lb/>maior proportione b c ad ea, quantum e&longs;t quadratum illius pro&shy;<lb/><arrow.to.target n="marg394"/><lb/>portionis, ergo b e pondus maius e&longs;t, cum agina e&longs;t in c, qu&agrave;m in d. <lb/></s>
          <s id="id001918">igitur per <expan abbr="commun&etilde;">communem</expan> animi <expan abbr="&longs;ententi&atilde;">&longs;ententiam</expan> addito communi pondere a e, <lb/>erit pondus a b minus &longs;emper cum agina e&longs;t in d, &lt;08&gt; in ullo alio lo&shy;<lb/>co a b. </s>
          <s id="id001919">Ergo pondus a b apprehen&longs;um in d <expan abbr="mouebi&ttilde;">mouebitur</expan> a b &aelig;quali ui <lb/><arrow.to.target n="marg395"/><lb/>maiore proportione, &lt;08&gt; in ullo alio loco. </s>
          <s id="id001920">Ha&longs;tile ergo in medio ap&shy;<lb/>prehen&longs;um maiore ui mouebitur, qu&agrave;m in ulla alia parte. </s>
          <s id="id001921">Et &longs;i gra&shy;


<pb pagenum="106" xlink:href="015/01/125.jpg"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, &amp; &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. </s>
          <s id="id001922">Semper igitur ob hanc cau&shy;<lb/>&longs;am mota ex medio grauitatis, &longs;eu uelo, &longs;eu ramo, &longs;eu manu uelo&shy;<lb/>cius mouentur, qu&agrave;m ex alijs partibus. </s>
          <s id="id001923">In remo etiam pote&longs;t acce&shy;<lb/>dere illud commodum, cuius meminit Ari&longs;toteles. </s>
          <s id="id001924">Propter hoc igi<lb/>tur, qui malum in naui collo carunt tant&ugrave;m unum, in medio ferm&egrave; <lb/>eum collocarunt, ut antiqui: &amp; qui duos aut tres, maiorem cra&longs;sio&shy;<lb/><arrow.to.target n="marg396"/><lb/>rem &longs;cilicet, &amp; altiorem in medio con&longs;tituerunt.</s>
        </p>
        <p type="margin">
          <s id="id001925"><margin.target id="marg392"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001926"><margin.target id="marg393"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 86.</s>
        </p>
        <p type="margin">
          <s id="id001927"><margin.target id="marg394"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001928"><margin.target id="marg395"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id001929"><margin.target id="marg396"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 82.</s>
        </p>
        <p type="main">
          <s id="id001930">Propo&longs;itio cente&longs;imaduodecima.</s>
        </p>
        <p type="main">
          <s id="id001931">Cur ex imo leuia longius ferantur declarare.</s>
        </p>
        <p type="main">
          <s id="id001932">Iam uer&ograve; <expan abbr="c&otilde;&longs;ideremus">con&longs;ideremus</expan>, qu&ograve;d propo&longs;itum e&longs;t, non &longs;olum in com&shy;<lb/><arrow.to.target n="marg397"/><lb/>paratione ad medium, &longs;ed extremorum inuicem, mi&longs;&longs;a enim ab imo <lb/>uelo cius feruntur, qu&agrave;m &agrave; medio non &longs;olum manu, &longs;ed &longs;corpioni&shy;<lb/>bus, &amp; arcubus. </s>
          <s id="id001933">Videmus &amp; hoc ob&longs;eruare pueros uirgam lon&shy;<lb/>gius iacentes non ex medio, &longs;ed imo apprehen&longs;am, quoniam pars <lb/>ip&longs;a anterior, &amp; qu&aelig; manu apprehen&longs;a e&longs;t, uehementi impetu emit&shy;<lb/>titur: &amp; ut recipit impetum magis &aelig;qualem, longius fertur, nam <lb/>quod emittitur proportionem habet ad &longs;patium. </s>
          <s id="id001934">Cum ergo appre<lb/>hen&longs;a in medio uirga &longs;olum medietate anteriore impetum recipiat <lb/>per &longs;e, ob id minus fertur: at impetus &longs;equitur proportionem, ut ui&shy;<lb/>&longs;um e&longs;t, qu&aelig; e&longs;t circa medium ob leuitatem ponderis. </s>
          <s id="id001935">In leuibus <lb/>ergo maius &longs;patium &longs;uperabunt emi&longs;&longs;a ex imo, quoniam propor&shy;<lb/>tio &longs;patij eadem e&longs;t ad duplum, &amp; ad dimidium. </s>
          <s id="id001936">igitur ex imo fer&shy;<lb/>me duplum etiam &longs;patij &longs;uperabit: non tamen omnino quia maio&shy;<lb/>rem, ut dixi proportionem habet ad id, quod ex medio comprehen<lb/>&longs;um e&longs;t. </s>
          <s id="id001937">At in leuibus non e&longs;t nece&longs;&longs;arium, ut ex medio apprehen&shy;<lb/>dantur, quoniam etiam cum incremento illo ponderis iam leuia <lb/>&longs;unt: plus ergo facit longitudo eius, quod eiaculatur, qu&agrave;m impe&shy;<lb/><figure id="id.015.01.125.1.jpg" xlink:href="015/01/125/1.jpg"/><lb/>tus, cuius demon&longs;tratio e&longs;t h&aelig;c. </s>
          <s id="id001938">Sit uirga <lb/>a b apprehen&longs;a in medio ponderis unci&aelig; <lb/>medi&aelig;, &amp; in a d, ut &longs;it d a palmus, &amp; uige&longs;i&shy;<lb/>ma pars totius a b, erit ergo re&longs;iduum ad duplum, a d nonuplum, <lb/><arrow.to.target n="marg398"/><lb/>&amp; a b tota unciarum quin que cum dimidia, &longs;i igitur grauetur, quia in <lb/>&longs;itu recto e&longs;t medi&aelig; unci&aelig;, in &aelig;quidi&longs;tanti terr&aelig;, quin que unciarum <lb/>cum dimidio, erit in &longs;itu dimidij recti unciarum trium. </s>
          <s id="id001939">E&longs;t igitur <lb/>proportio &longs;excupla, &longs;i apprehendatur in medio, &amp; ad &aelig;quidi&longs;tan&shy;<lb/>tem, ad apprehen&longs;am in imo, &amp; ad angulum medium: at emi&longs;&longs;a ex <lb/><arrow.to.target n="marg399"/><lb/>a d habet totum a&euml;rem a b circumdantem impul&longs;um ex c b &longs;olum <lb/>dimidium reliqua pars ui trahitur, ergo proportio &longs;patij a b, erit <lb/>&longs;exdecupla ferm&egrave; &longs;patio b c, quoniam e&longs;t triplicata corporis ad cor<lb/>pus eius, qu&aelig; e&longs;t longitudinis ad longitudinem, &amp; quadruplicata 


<pb pagenum="107" xlink:href="015/01/126.jpg"/>re&longs;pectu a&euml;ris a c, qui re&longs;i&longs;tit apprehen&longs;a a b in c. </s>
          <s id="id001940">Et iam minus fere&shy;<lb/>batur quinta parte, ideo longius eiaculabitur triplo ex a, qu&agrave;m ex <lb/>c. </s>
          <s id="id001941">Nec tamen maiore impetu, quia obliqu&egrave; fertur, &amp; qu&aelig; obliqu&egrave; <lb/><expan abbr="feri&utilde;t">feriunt</expan>, minore cum impetu feriunt: at que eo magis &longs;i leuia fuerint: ab <lb/>a&euml;re enim circumambiente perturbantur, &amp; in incertum trudun&shy;<lb/>tur. </s>
          <s id="id001942">Qu&aelig; ergo grauia &longs;unt ex medio emi&longs;&longs;a, &amp; ad &aelig;quidi&longs;tantem <lb/>longius feruntur, &amp; maiore cum impetu, quia magis direct&egrave;: leuia <lb/>autem longius ex imo, &longs;ed minore cum impetu, &longs;i aliqua cau&longs;a &agrave; re&shy;<lb/>cto, &amp; &aelig;quidi&longs;tante declinauerint. </s>
          <s id="id001943">At &longs;i &agrave; &longs;uprema parte, &amp; iuxta <lb/>cu&longs;pidem, neque procul feruntur, neque cum impetu ob cau&longs;as di&shy;<lb/>ctas. </s>
          <s id="id001944">Eadem quoque ratio e&longs;t omnium machinarum: ide&ograve; ob lon&shy;<lb/>g&etail; longius eiaculantur, quoniam proportionem &longs;eruant ad cana&shy;<lb/><arrow.to.target n="marg400"/><lb/>lem. </s>
          <s id="id001945">Sed de hoc inferius agetur.</s>
        </p>
        <p type="margin">
          <s id="id001946"><margin.target id="marg397"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001947"><margin.target id="marg398"/>P<emph type="italics"/>er<emph.end type="italics"/> 86.</s>
        </p>
        <p type="margin">
          <s id="id001948"><margin.target id="marg399"/>P<emph type="italics"/>er<emph.end type="italics"/> 89.</s>
        </p>
        <p type="margin">
          <s id="id001949"><margin.target id="marg400"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 107.</s>
        </p>
        <p type="main">
          <s id="id001950">Propo&longs;itio cente&longs;imatertia decima.</s>
        </p>
        <p type="main">
          <s id="id001951">Cur uirga longius mittatur &agrave; puero, qu&agrave;m &agrave; uiro inue&longs;tigare.<lb/><arrow.to.target n="marg401"/></s>
        </p>
        <p type="margin">
          <s id="id001952"><margin.target id="marg401"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001953">Diligentia, &amp; u&longs;us puerilis efficit, ut uirga feratur &longs;ecundum me&shy;<lb/>dium rectianguli: uir autem non con&longs;tanter iacit, &amp; &longs;ecundum re&shy;<lb/>ctum, at rectus ince&longs;&longs;us in leuibus, quia ab a&euml;re in obliquum defle&shy;<lb/>ctitur uirga ob longitudinem efficit, ut inflectatur infr&agrave; celerius, &amp; <lb/>de&longs;inat citius motus, ac finiatur. </s>
          <s id="id001954">Tertia cau&longs;a e&longs;t, qu&ograve;d leui&longs;sima <lb/>non ade&ograve; recipiunt impetum ut grauia: nam leui&longs;simam &amp; exigu&shy;<lb/>am ligni portionem maximo nixu uix excutiemus &egrave; manu. </s>
          <s id="id001955">Cau&longs;a <lb/>ergo e&longs;t: quoniam uim, oportet, ut habeat, quod contra naturam <lb/>mouetur, ut naturaliter moueri po&longs;sit, qu&aelig;cunque igitur naturaliter <lb/>exiguum habent motum, ut pluma, palea, fe&longs;tuc&aelig; nulla ratione ue&shy;<lb/>hementer contra naturam agi po&longs;&longs;unt. </s>
          <s id="id001956">Qu&aelig;dam ergo &agrave; pueris lon<lb/>gius <expan abbr="iaci&utilde;tur">iaciuntur</expan> ob &longs;olam peritiam, &amp; exercitationem, qu&aelig;dam quo&shy;<lb/>niam ad angulum latiorem magis feruntur, qu&agrave;m &longs;it rectus, qu&aelig;&shy;<lb/>dam quoniam leui&longs;sima &longs;unt. </s>
          <s id="id001957">Sed &longs;i leuiora non feruntur ualido <lb/>motu uiolento, cur tamen &agrave; pueris iacta longius <expan abbr="fer&utilde;tur">feruntur</expan>? </s>
          <s id="id001958">Ratio e&longs;t, <lb/>quoniam maior uis deficiente obiecto magis fatigatur, atque ide&ograve; <lb/>minus mouet. </s>
          <s id="id001959">Propter h&aelig;c igitur omnia non &longs;ol&ugrave;m in pueris, &longs;ed <lb/>in machinis, qu&aelig; accommodata &longs;unt, melius impelluntur, a c lon&shy;<lb/>gius feruntur, qu&agrave;m leui&longs;sima. </s>
          <s id="id001960">nam nec palea &longs;corpione iacta tam <lb/>procul, qu&agrave;m &longs;agitta fertur, cum proportio maior &longs;it, tamen ad pa&shy;<lb/>leam, qu&agrave;m ad &longs;agittam. </s>
          <s id="id001961">Inde fit, ut quemadmodum Turca ille lite&shy;<lb/>ras &longs;ui Principis, cum timeret ad no&longs;tros propius accedere, lapidi al<lb/>ligatas longius emi&longs;it. </s>
          <s id="id001962">Cau&longs;am autem huius docet Ari&longs;toteles in <lb/>Mechanicis dum qu&aelig;rit cur, &amp; grauia &amp; leuia ualde longe proijci <lb/>nequeunt: nam grauia nimis, moueri <expan abbr="n&otilde;">non</expan> facil&egrave; po&longs;&longs;unt: leuia etiam <lb/>ualde ad rem mouere non ualent. </s>
          <s id="id001963">Ob h&aelig;c utra que ex his paruo cum 


<pb pagenum="108" xlink:href="015/01/127.jpg"/>impetu emittuntur, tamet&longs;i uehementer nitaris. </s>
          <s id="id001964">Sed &amp; leuia ferun&shy;<lb/>tur hac illac, ut non po&longs;sint retinere impetum prioris uiolenti&aelig;: in&shy;<lb/>natum enim e&longs;t, ut duorum motuum &longs;imul in eadem re uigentium, <lb/>cum illa proprio impetu feratur, unus alterum impediat: nam &longs;i ro&shy;<lb/>ta uehatur circulariter acta, non tamen ce&longs;&longs;abit, aut iminuetur impe<lb/>tus circulationis. </s>
          <s id="id001965">Multa ergo in huiu&longs;modi anomalis motibus con<lb/>&longs;ideranda &longs;unt, ut illorum impetum robur, ac locum definiamus.</s>
        </p>
        <p type="main">
          <s id="id001966">Ex hoc liquet, cur plumbe&aelig; &longs;ph&aelig;rul&aelig; longius ferantur &agrave; tor&shy;</s>
        </p>
        <p type="main">
          <s id="id001967"><arrow.to.target n="marg402"/><lb/>mento emi&longs;&longs;&aelig;, qu&agrave;m ligne&aelig;, etiam &longs;i non frangantur.</s>
        </p>
        <p type="margin">
          <s id="id001968"><margin.target id="marg402"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id001969">Propo&longs;itio cente&longs;ima quarta decima.</s>
        </p>
        <p type="main">
          <s id="id001970">Circularis motus differentias quatuor e&longs;&longs;e, earum q&uacute;e rationem <lb/>contemplari.</s>
        </p>
        <p type="main">
          <s id="id001971">In motu circulari aut axis <expan abbr="progredi&ttilde;">progreditur</expan>, aut &longs;uo loco manet. </s>
          <s id="id001972">Vtroque <lb/><arrow.to.target n="marg403"/><lb/>autem modo uel mouetur ab axe, uel circumferentia, igitur con&longs;tat <lb/>quatuor e&longs;&longs;e motuum differentias: quas cum tres proponat author <lb/>libri Mechanicarum, aut Ari&longs;totelem illum e&longs;&longs;e, credendum non <lb/>e&longs;t, aut illum &longs;tupidum dicere nece&longs;&longs;e e&longs;t, nam modum diuidendi <lb/>eum latui&longs;&longs;e quis putet. </s>
          <s id="id001973">cum rota igitur aut &longs;ph&aelig;ra in plano cir&shy;<lb/>cumagitur, motus e&longs;t ex circumferentia pr&aelig;grediente axe: ut pa&shy;<lb/>lam e&longs;t: motis enim loco nobis mouentur omnia, qu&aelig; &longs;unt in no&shy;<lb/>bis. </s>
          <s id="id001974">Cum uer&ograve; rot&aelig; &longs;ub curru &longs;unt, progreditur axis earum, &amp; rota <lb/>ob id cum quie&longs;cere nequeat, quia facilius circumuertitur, qu&agrave;m <lb/>trahatur, procedit, &amp; hic e&longs;t &longs;ecundus modus, quo rota ex circumfe<lb/>rentia mouetur, &amp; ex axe initium e&longs;t motus. </s>
          <s id="id001975">At uer&ograve; in rota molari, <lb/>&amp; quibus gladij exacuuntur, cum loco non moueantur, motus e&longs;t <lb/>ex axe: axis enim rotam circumagit, non rota axem, quie&longs;cit tamen <lb/>in eodem loco rota, &amp; axis &longs;cilicet, quia non progreditur, &longs;ed in lo&shy;<lb/>co mouetur: atque hic e&longs;t tertius modus. </s>
          <s id="id001976">Demum &longs;uccula putei, &amp; <lb/>ip&longs;a mouetur circulari motu, &amp; trochle&aelig; etiam, neque enim progre&shy;<lb/>diuntur: &longs;ed non ex axe mouentur, uer&ugrave;m &longs;uccula per coloppes cir<lb/>cumducitur, &amp; trochlea per funes, axis que in &longs;uccula mouetur, in tro<lb/>chleis autem quie&longs;cit pror&longs;us: dico mouetur, id e&longs;t circumducitur, <lb/>non quod progrediatur: ut non &longs;olum &longs;int quatuor modi, &longs;ed po&shy;<lb/>tius quin que, nam &amp; demon&longs;tratione o&longs;tenduntur, &amp; experimento <lb/>docente deprehenduntur. </s>
          <s id="id001977">Horum omnium liberrimus e&longs;t, primus <lb/>ex circumferentia progrediente toto, &longs;eu attracto &longs;eu impul&longs;o &amp; ue<lb/>loci&longs;simus, cuius cau&longs;am &longs;upr&agrave; o&longs;tendimus. </s>
          <s id="id001978">Proximus huic e&longs;t mo&shy;<lb/><arrow.to.target n="marg404"/><lb/>tus rotarum per axem, quoniam axis premit rotam interius &longs;o&shy;<lb/>lam, &amp; labitur: ideo que quod &amp; axis, &amp; rota intus &longs;int leui&longs;sima, pro&shy;<lb/>de&longs;t plurimum: &amp; aurig&aelig; axungia inungunt, &amp; nomen ab eo traxit 


<pb pagenum="109" xlink:href="015/01/128.jpg"/>axungia. </s>
          <s id="id001979">Et quae rota magna &longs;it: quoniam cum <expan abbr="n&otilde;">non</expan> rota, &longs;ed axis traha&shy;<lb/>tur in &aelig;quali tempore &amp; magna, &amp; parua trahitur: utra que uer&ograve; una <lb/>conuer&longs;ione tantam <expan abbr="line&atilde;">lineam</expan> rectam &longs;uperat, quanta e&longs;t rot&aelig; periphe&shy;<lb/>ria. </s>
          <s id="id001980">Quod &longs;i plures &longs;int rot&aelig; celerius feruntur, quia axis minus tan&shy;<lb/>to <expan abbr="rot&atilde;">rotam</expan> premit. </s>
          <s id="id001981">Et &longs;i rectus &longs;it axis, &amp; bene rotundus, &amp; foramen ro<lb/>tundum, &amp; latius, &amp; &egrave; duri&longs;simo ligno, ut non po&longs;sit in clinari: &amp; <lb/>rota ip&longs;a in ambitu &aelig;qualis, omnia h&aelig;c faciunt ad motus uelo cita<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"/></s>
        </p>
        <p type="margin">
          <s id="id001982"><margin.target id="marg403"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id001983"><margin.target id="marg404"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s>
        </p>
        <p type="margin">
          <s id="id001984"><margin.target id="marg405"/>I<emph type="italics"/>liad.<emph.end type="italics"/> 23.</s>
        </p>
        <p type="main">
          <s id="id001985"><foreign lang="greek">i&rpar;/xnia tu/pte o/deosi a/r &amp; ko/nin a&rpar;mfixuqῡnai</foreign>.</s>
        </p>
        <p type="main">
          <s id="id001986">Id e&longs;t, ue&longs;tigia per cu&longs;sit pedibus, ante que illa puluis pedibus ex&shy;<lb/>cu&longs;&longs;us &lpar;ue&longs;tigia &longs;cilicet relinquentibus&rpar; ingrederetur. </s>
          <s id="id001987">Principalis <lb/>autem cau&longs;a uelo citatis e&longs;t agens, uelut equi. </s>
          <s id="id001988">Sed inter <expan abbr="h&utilde;c">hunc</expan> motum <lb/>&amp; priorem medius e&longs;t Scital&aelig; uocat&aelig;, nam ut in primo axis proci&shy;<lb/>dit &amp; rotundum &agrave; &longs;uperficie circumagitur, licet axis etiam circum&shy;<lb/>ducatur, ut axis, &amp; rota, aut &longs;ph&aelig;ra duplici motu moueantur, &longs;ci&shy;<lb/>licet antror&longs;um, &amp; circumcirca, in rota currus duo ijdem motus <lb/>&longs;int, axis quo que antror&longs;um moueatur, &longs;ed non circumagatur: unde <lb/>impeditior e&longs;t hic motus: ita in Scytala utrunque utro que motu mo&shy;<lb/>uetur, &amp; circumcirca, &amp; antror&longs;um, at que id commune e&longs;t, cum pri&shy;<lb/>mo ita axis mouet rotas, non rot&aelig; axem, qu&ograve;d &longs;ecundo motui ro&shy;<lb/>tarum in curru proprium e&longs;t, ut tantum degenerent &agrave; primo motu, <lb/>quanto leuius uertuntur, qu&agrave;m in &longs;ecundo motu. </s>
          <s id="id001989">Trahitur ergo <lb/><figure id="id.015.01.128.1.jpg" xlink:href="015/01/128/1.jpg"/><lb/>iugum in &longs;citala, uelut in rotis currus, <lb/>&longs;ed e&longs;t annexum rotis non in curri&shy;<lb/>bus. </s>
          <s id="id001990">Propterea in primo motu trahi&shy;<lb/>tur, uel impellitur &agrave; &longs;uperficie: in &longs;e&shy;<lb/>cundo a b axe, &longs;ed non affixo rotis, unde &aelig;gr&egrave; trahuntur in &longs;cyta&shy;<lb/>la ab axe affixo rot&etail;. </s>
          <s id="id001991">Quare leuius qu&agrave;m in curru, difficilius qu&agrave;m <lb/>in rota uel &longs;ph&aelig;ra &agrave; &longs;uperficie extima circumacta. </s>
          <s id="id001992">Quartus modus <lb/>e&longs;t, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, &amp; rotis, quibus ferrum exacuitur. </s>
          <s id="id001993">E&longs;t enim hic &longs;imilior <lb/>primo, quia contrarius, in primo enim procedit rota, &amp; uertitur &agrave; <lb/>circumferentia, hic quie&longs;cit rota, &amp; mouetur ab axe. </s>
          <s id="id001994">Proximus huic <lb/>e&longs;t, qui fit in &longs;ucculis ob firmitatem axis: nam axis e&longs;t coniunctus <lb/>rot&aelig;. </s>
          <s id="id001995">Vltimus e&longs;t trochlearum, qui &amp; difficillimus: &longs;it enim &agrave; cir&shy;<lb/>cumferentia, &amp; axis di&longs;iunctus e&longs;t &agrave; trochlea: quod ad dit difficulta&shy;<lb/>tem. </s>
          <s id="id001996">Sed &amp; trochlea caret colloppibus. </s>
          <s id="id001997">Ergo uerum e&longs;t, quod o&shy;<lb/>mnia rotunda facilius circumaguntur, &longs;ed uaria ratione: nam plus <lb/>mota &longs;uper aliquo plano, ut in plau&longs;tris &amp; &longs;cytalis: minus in &longs;uccu&shy;<lb/>lis, &amp; rotis acuentibus ferrum, &amp; molis: nam &amp; &longs;i rotunditatem iu&shy;<lb/>uet ob &aelig;qualitatem ad conuer&longs;ionem, non tamen in his e&longs;t ad e&ograve; 


<pb pagenum="110" xlink:href="015/01/129.jpg"/>utilis. </s>
          <s id="id001998">Vtilitas ergo prima e&longs;t, cum circumuertitur in plano, uelut <lb/>in rotis &longs;cytalis, &amp; &longs;ph&aelig;ris. </s>
          <s id="id001999">Secunda qu&aelig; minor e&longs;t, cum &agrave; &longs;uperfi&shy;<lb/>cie circumuertitur, ut in trochleis. </s>
          <s id="id002000">Tertia cum &agrave; coloppis, qu&aelig; mi&shy;<lb/>nima e&longs;t omnium, ut in &longs;ucculis. </s>
          <s id="id002001">Motus autem c&oelig;li non e&longs;t ex tri&shy;<lb/>plici primo genere, cum &longs;it in loco, &amp; non ad locum, neque ut rot&aelig; <lb/>molaris: nam ille e&longs;t ex axe: nec ut in trochlea: nam in ea axis quie&longs;&shy;<lb/>cit ip&longs;um autem c&oelig;lum circa axem non uertitur, &longs;ed cum axe, &longs;i ta&shy;<lb/>men in&longs;ecabilis linea circumagi pote&longs;t dici. </s>
          <s id="id002002">Relinquitur ergo, ut <lb/>C&oelig;li motus propior &longs;it motui &longs;uccul&aelig;, qu&agrave;m alij motui. </s>
          <s id="id002003">Differt <lb/>ab eo in hoc, quod in &longs;uccula mouetur axis ab orbe: at in c&oelig;lo <lb/>ut non mouetur ab axe, ita nec axis ab orbe: cun que &longs;it motus &longs;im&shy;<lb/>plici&longs;simus, in alio genere collocandus e&longs;t: quando quidem in illo <lb/>nulla pars po&longs;sit dici primo, quod <expan abbr="nece&longs;&longs;ari&utilde;">nece&longs;&longs;arium</expan> e&longs;t in uno quo que <expan abbr="hor&utilde;">horum</expan>.</s>
        </p>
        <p type="main">
          <s id="id002004">Propo&longs;itio cente&longs;ima quinta decima.</s>
        </p>
        <p type="main">
          <s id="id002005">Proportionem motuum impul&longs;ionis, &amp; attractionis inter&apos;&longs;e ab <lb/>eadem ui declarare.</s>
        </p>
        <p type="main">
          <s id="id002006">Con&longs;tat, qu&ograve;d attractio cum fune longiore ualidior e&longs;t, quam </s>
        </p>
        <p type="main">
          <s id="id002007"><arrow.to.target n="marg406"/><lb/>cum manibus, quoniam e&longs;t cum motu quodam: motus autem au&shy;<lb/>get actionem, ideo attractio ualidior e&longs;t hac de cau&longs;a, &longs;ed &amp; impul&shy;<lb/>&longs;io cum baculo ualidior e&longs;t, quam cum manibus, quoniam licet col<lb/>ligere omnes uires in illo baculo, &amp; ip&longs;um applicare loco, unde fa&shy;<lb/>cilius impelli pote&longs;t. </s>
          <s id="id002008">Velut &longs;ph&aelig;ra ex medio latere: nam ibi magis <lb/>colliguntur uires, &amp; ad impellendum facilius e&longs;t, quodcunque leui&shy;<lb/>us e&longs;t. </s>
          <s id="id002009">Pars autem magis remota &agrave; centro grauitatis e&longs;t leuior, his <lb/>duabus cau&longs;is, &longs;ph&aelig;ra ex medio latere facilius ac magis impellitur. <lb/></s>
          <s id="id002010">Sed nos &longs;upponimus nunc applicationem &aelig;qualem e&longs;&longs;e, nam &longs;e&shy;<lb/>cus ad impellendum facilius e&longs;t applicare totum corpus, qu&agrave;m at&shy;<lb/>tractionem. </s>
          <s id="id002011">Pectore enim magna ui impellimus, nihil e&longs;t compar, <lb/>quo trahere po&longs;simus. </s>
          <s id="id002012">Sed, ut dixi, &longs;it baculus applicatus alicui la&shy;<lb/>pidi ea parte, qua facilius pote&longs;t impelli &amp; trahi, &amp; qu&aelig;ritur, qu&aelig; <lb/>maior &longs;it uis, an attrahendi? </s>
          <s id="id002013">&amp; dico qu&ograve;d homo, uel conatur trahe&shy;<lb/>re toto corpore, &amp; impellere, at que hoc modo magis trahit, qu&agrave;m <lb/>impellet, quoniam corporis pondus melius adhibetur in tractione <lb/>qu&agrave;m impul&longs;u: uel citra corporis pondus, &longs;ed &longs;ola ui membrorum: <lb/>&amp; tunc magis impellit, quoniam impul&longs;us fit corpore prono in <expan abbr="an&shy;terior&etilde;">an&shy;<lb/>teriorem</expan> partem, qu&aelig; inclinatio, &amp; motus e&longs;t naturalis magis, qu&agrave;m <lb/>in attractione in partem po&longs;teriorem. </s>
          <s id="id002014">Sed ubi nulla &longs;it diuer&longs;itas <lb/>neque horum, neque figurarum &aelig;qualis uis &aelig;qualem efficit motum: <lb/>quia impul&longs;us impellentis comparatione e&longs;t attractio re&longs;pectu al&shy;<lb/>terius. </s>
          <s id="id002015">Ver&ugrave;m non e&longs;t eadem uis nec prop&egrave; par impellendi, at que <lb/>attrahendi hominibus, cum attractio fiat per mu&longs;culos ad origi&shy;


<pb pagenum="111" xlink:href="015/01/130.jpg"/>nem &longs;uam naturaliter &longs;e retrahentibus impul&longs;ui nullum in&longs;trumen<lb/>tum &agrave; natura delegatum inuenio, nam ad exten&longs;ionem mu&longs;culi &longs;a&shy;<lb/>n&egrave; ex aduer&longs;o &longs;unt fabricati: cum ergo duo &longs;int tantum motus mu&shy;<lb/>&longs;culorum ten&longs;io, dum <expan abbr="retrah&utilde;tur">retrahuntur</expan> ad principium &longs;uum, &amp; remi&longs;sio, <lb/>dum membrum quie&longs;cit in naturali nullus erit locus impul&longs;ioni, <lb/>ni&longs;i ex con&longs;equentia non per &longs;e, quamobrem multo infirmiorem il&shy;<lb/>lum attractione in brachijs e&longs;&longs;e, nece&longs;&longs;e e&longs;t.</s>
        </p>
        <p type="margin">
          <s id="id002016"><margin.target id="marg406"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002017">Propo&longs;itio cente&longs;ima &longs;exta decima.</s>
        </p>
        <p type="main">
          <s id="id002018">Cur machin&aelig; ablong&aelig; igne&aelig; longius emittant &longs;ph&aelig;ram ex&shy;<lb/>plorare.<lb/><arrow.to.target n="marg407"/></s>
        </p>
        <p type="margin">
          <s id="id002019"><margin.target id="marg407"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002020">Quoniam ratio &longs;uperius adducta, neque in his, neque in hypophy&shy;</s>
        </p>
        <p type="main">
          <s id="id002021"><arrow.to.target n="marg408"/><lb/>&longs;is &lpar;uocant cerbatanas&rpar; non pote&longs;t &longs;atisfacere, cum tamen idem &longs;e&shy;<lb/>quatur in his, ut in illis uidetur, qua&longs;i uis e&longs;&longs;e in &longs;ph&aelig;rula &longs;ic emi&longs;&shy;<lb/>&longs;a, &amp; non in a&euml;re, quemadmodum dicebamus, coniuncto e&longs;&longs;e. </s>
          <s id="id002022">Ex <lb/>quo nece&longs;&longs;e e&longs;&longs;et, ut quod longius ferretur, etiam ualidiores ictus <lb/><figure id="id.015.01.130.1.jpg" xlink:href="015/01/130/1.jpg"/><lb/>inferret, hoc autem <lb/>non ita &longs;e habet, &longs;ed <lb/>ictus magnitudo <lb/>ex robore machi&shy;<lb/>narum tam ignea&shy;<lb/>rum, quam &longs;corpio <lb/>num pendet, nam <lb/>&longs;it a &longs;corpio ma&shy;<lb/>gnus, &longs;ed tenuis, ex <lb/>h&ograve;c palam e&longs;t lon&shy;<lb/>gius mittere &longs;agit&shy;<lb/>tam, qu&ograve;d &agrave; parua, <lb/>&amp; breui, quantun&shy;<lb/>uis cra&longs;&longs;a non lon&shy;<lb/>ge mittitur: at uer&ograve; <lb/>quod b cra&longs;&longs;us &amp; paruus maiore cum impetu mittat o&longs;tenditur <lb/>nam ea pondera &longs;agitt&aelig; mouet, qu&aelig; non pote&longs;t mouere a, igitur b <lb/>ualidiore robore mouet, quam a. </s>
          <s id="id002023">Pr&aelig;terea illud o&longs;tendit iugum fu&shy;<lb/>nis arcus cra&longs;siora duriora, qu&aelig; maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui &agrave; puero tendi poterit. </s>
          <s id="id002024">Non e&longs;t ergo eadem ratio mittendi <lb/>longius, &amp; ualidiore cum robore. </s>
          <s id="id002025">Eadem ergo cum ratio &longs;it in <lb/>machinis igneis, cra&longs;siores enim, &amp; latiores ac breuiores magis <lb/>concutiunt, quam longiores tenuiores minoris &longs;ph&aelig;r&aelig; capaces: <lb/>non &longs;olum ob magnitudinem &longs;ph&aelig;r&aelig; magis ill&aelig; concutiunt, &longs;ed, <lb/>ut dixi, ob maiorem impetus uim: cau&longs;a ergo e&longs;t manife&longs;ta in his, <lb/>&longs;ed non cau&longs;a, qua longius ferantur in longiore canali. </s>
          <s id="id002026">Sed uide&shy;


<pb pagenum="112" xlink:href="015/01/131.jpg"/>tur una, eadem que e&longs;&longs;e ratio in utri&longs;que. </s>
          <s id="id002027">Con&longs;tituatur can alis a b <lb/>lo&nacute;gior, &amp; c d breuior, ut &longs;it &longs;exquialter a b ad c d, &amp; &longs;it rur&longs;us <lb/><figure id="id.015.01.131.1.jpg" xlink:href="015/01/131/1.jpg"/><lb/>&longs;ph&aelig;rul&aelig; locus e in longiore, <lb/>&longs;exquialter in di&longs;tantia a b, qua <lb/>lis e&longs;t in f a d, &amp; erit per dicta <lb/>ab Euclide in quinto, ac &longs;exqui<lb/>altera c f. </s>
          <s id="id002028">Po&longs;&longs;emus igitur di&shy;<lb/>cere, quod uelut ab hypomo&shy;<lb/>chlio longiore &longs;patio circuma&shy;<lb/>gitur pondus: ita &amp; a b c, &amp; f. <lb/></s>
          <s id="id002029">Sed rur&longs;us incidimus in id, ut <lb/>maiore impetu feratur e qu&agrave;m f. </s>
          <s id="id002030">Ideo &longs;i concedatur maiore ferri ex <lb/>e, quam ex f non &longs;equitur, ut celerius, aut maiore impetu. </s>
          <s id="id002031">Percutit <lb/>puer pugno quanta ui pote&longs;t ac celerrim&egrave;, uir robu&longs;tus lent&egrave;, &amp; mi&shy;<lb/>nore impetu, &longs;ed tamen ictus long&egrave; maior e&longs;t. </s>
          <s id="id002032">E&longs;t enim ictus robur <lb/>non &agrave; uelo citate &longs;olum, &longs;ed maiore ex ponderis grauitate, qu&aelig; &longs;ola <lb/>premit, urget, &amp; frangit etiam &longs;ine motu. </s>
          <s id="id002033">Solum ergo id re&longs;tat du&shy;<lb/>bium, cur &longs;i grauius e&longs;t, moueatur eodem ferm&eacute; impetu: nam quo <lb/>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con<lb/>cutit, aut qua&longs;&longs;at, &longs;ed grauitas ad hoc plus facit impetu. </s>
          <s id="id002034">Palea maxi&shy;<lb/>mo impetu demi&longs;&longs;a non ferit, non ledit, &amp; celerius de&longs;cendit, fer&shy;<lb/>rum &longs;ola grauitate actum, im&ograve; etiam temperato ictu l&aelig;dit graui&shy;<lb/>ter, qua&longs;&longs;at, &amp; frangit: itaque f maiore indiget quantitate pyrij pulue&shy;<lb/>ris, qu&agrave;m e: &longs;iquidem tertia parte ponderis &longs;u&aelig; &longs;ph&aelig;r&aelig;: at maius <lb/>e&longs;t pondus f quam e, ergo maius pondus pulueris f qu&agrave;m e, ergo <lb/>maior uehementia ictus, &longs;iquidem ea &longs;equitur, robur cau&longs;&aelig; mouen<lb/>tis &longs;impliciter: ut concludamus longitudinem ictus &longs;equi propor&shy;<lb/>tionem motoris ad motum, &longs;ed uehementia robur motoris: nam &longs;i <lb/>ex portione mouet &aelig;quale pondus maiore cum impetu mouet, <lb/>quoniam maior e&longs;t proportio: &longs;i minore igitur pondus maius e&longs;t, <lb/>&amp;, ut dixi plus facit magnitudo ponderis cum leui ictu, qu&agrave;m ma&shy;<lb/>gnitudo ictus cum leui pondere. </s>
          <s id="id002035">Qu&aelig; ergo feruntur per longio&shy;<lb/>res canales maiore impetu feruntur, &amp; &longs;ocietatem <expan abbr="hab&etilde;t">habent</expan> a&euml;ris moti <lb/>per longius <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="u&itilde;s">uis</expan><lb/>motus confirmata e&longs;t, &amp; proportio eius, qu&ograve;d mouet, maior e&longs;t ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uer&ograve; f <expan abbr="mot&utilde;">motum</expan> minore propor&shy;<lb/>tione <expan abbr="ict&utilde;">ictum</expan> facit <expan abbr="maior&etilde;">maiorem</expan>, quia, ut dixi, <expan abbr="t&atilde;to">tanto</expan> grauius, e&longs;t quod ferit. </s>
          <s id="id002036">Quod <lb/><expan abbr="aut&etilde;">autem</expan> minus <expan abbr="ext&etilde;datur">extendatur</expan> machina a b quam c d, <expan abbr="n&utilde;c">nunc</expan> <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan> oportet.</s>
        </p>
        <p type="margin">
          <s id="id002037"><margin.target id="marg408"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 103.</s>
        </p>
        <p type="main">
          <s id="id002038">Propo&longs;itio cente&longs;ima decima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id002039">In cuniculis maior e&longs;t uis pulueris copio&longs;ioris ampliore in &longs;pa&shy;<lb/>tio, qu&agrave;m paucioris in minore iuxta proportionem eandem.</s>
        </p>
        <pb pagenum="113" xlink:href="015/01/132.jpg"/>
        <p type="main">
          <s id="id002040">Sit &longs;patium f d &longs;exqui tertium b e, puluis quo que in f d &longs;patio &longs;i&shy;<lb/><arrow.to.target n="marg409"/><lb/>militer &longs;exqui tertius pulueri b e pondere, &amp; manife&longs;tum e&longs;t, quod <lb/>dum conuertitur in ignem quali&longs;cunque &longs;it proportio &lpar;modo eadem <lb/>ignis ad puluerem&rpar; erit ignis in f d pariter &longs;exqui tertius igni in b e, <lb/>dico qu&ograve;d &longs;i cra&longs;sities f d &longs;it etiam &longs;exqui tertia cra&longs;sitiei b e, quod <lb/>poterit frangi, &amp; moueri f d quie&longs;cente b e. </s>
          <s id="id002041">Vnde idem in cuniculis <lb/>ut magnus cuniculus cum multo puluere po&longs;sit mouere montem <lb/>paruus cum puluere proportione re&longs;pondente priori non po&longs;sit. <lb/></s>
          <s id="id002042">Nam c&ugrave;m &aelig;qualia &longs;int omnia iuxta que rationem eandem, nece&longs;&longs;e e&longs;t <lb/>ut pro ratione extendantur, at in paruo &longs;patio minor fit den&longs;itas c&etail;&shy;<lb/>tera paria &longs;unt, ergo &agrave; paruo &longs;patio non tantus fit impetus, quantus <lb/>&agrave; magno. </s>
          <s id="id002043">Impetus etiam proportionem habet ad <expan abbr="p&otilde;dus">pondus</expan>, &amp; ad con&shy;<lb/>iunctionem, &agrave; maiore igitur impetu plura, &amp; maiora mouentur, &amp; <lb/>conuelluntur, quam &agrave; minore, ob h&aelig;c igitur minores cuniculi &longs;uc&shy;<lb/>cutiunt, maiores euertunt, maximi exturbant, &amp; proijciunt. </s>
          <s id="id002044">Nam <lb/>qui &longs;uccutiunt, ubi pondus, aut coniunctio maior &longs;it, qu&agrave;m ut di&shy;<lb/>&longs;trahere po&longs;sint, conden&longs;ant partes proximiores, &amp; rimas faciunt, <lb/>per quas exhalat ignis aut omnino extinguitur, aut conden&longs;atur. <lb/></s>
          <s id="id002045">At ergo in bellicis machinis, minus dilatat puluis, cum fuerit in lon <lb/>go canali, ob id ergo maiore impetu feruntur per illas, qu&agrave;m per <lb/>breuiores, etiam qu&ograve;d minor &longs;it puluis, minor &longs;it ignis. </s>
          <s id="id002046">Experimen <lb/>tum facies in canali, ubi &longs;ambuci medulla pro globulo flatu impel&shy;<lb/>lente expellitur ab&longs; que periculo: nam quanto minor fuerit canalis <lb/>ambitu ac longior eo maiore impetu pellitur. </s>
          <s id="id002047">For&longs;an qui&longs;piam nos <lb/>merit&ograve; poterit uideri <expan abbr="repreh&etilde;di&longs;&longs;e">reprehendi&longs;&longs;e</expan>, qu&ograve;d inanis glori&aelig; &longs;tudio per&shy;<lb/>nicio&longs;a humano generi doceam. </s>
          <s id="id002048">Quibus re&longs;pondeo, me nihil do cu <lb/>i&longs;&longs;e, quod &iacute;n humani generis detrimentum cedat, huiu&longs;modi que pr&etail;&shy;<lb/>cepta iam ob&longs;cura&longs;&longs;e, ut ne quid mali accidere po&longs;&longs;et hominibus ex <lb/>his: <expan abbr="n&atilde;">nam</expan> qu&ograve;d ad ea, qu&aelig; declarata, &longs;unt, cau&longs;as &longs;ol&ugrave;m retuli, effectus <lb/>ip&longs;i modi artis <expan abbr="nimi&utilde;">nimium</expan> feruntur, ac nimio plu&longs;quam <expan abbr="uell&etilde;">uellem</expan> in telligun&shy;<lb/>tur. </s>
          <s id="id002049">Vt cum ad copiam, ad magnitudinem, ad coacta imperia mi&longs;e&shy;<lb/>rorum re&longs;picio, nihil plus po&longs;sit addi. </s>
          <s id="id002050">Omnia enim huiu&longs;que <expan abbr="&longs;pect&atilde;t">&longs;pectant</expan> <lb/>ad potentiorum in crementa. </s>
          <s id="id002051">An ergo &longs;uccurrere afflictis, ob&longs;e&longs;sis, <lb/>cinctis, &aelig;quare <expan abbr="condition&etilde;">conditionem</expan>, liberare &agrave; &longs;eruitute etiam rebelles <expan abbr="n&otilde;">non</expan> li&shy;<lb/>cebit? </s>
          <s id="id002052">Ab initio fuimus omnes liberi: excogitata fuit regni ratio ad <lb/>commodum hominum, ea uer&longs;a e&longs;t per uim in <expan abbr="Tyrannid&etilde;">Tyrannidem</expan>. </s>
          <s id="id002053">Subtili <lb/>ergo ratione <expan abbr="occurrend&utilde;">occurrendum</expan> e&longs;t imbecillioribus: <expan abbr="n&atilde;">nam</expan> reliqua omnia ni&shy;<lb/>mis, ut dixi, qu&etail; ad cuniculos ad <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="machinar&utilde;">machinarum</expan> ad rectos <lb/>ictus ad <expan abbr="libram&etilde;ta">libramenta</expan> ad longitudinem &longs;patij, per quos globus ille de&shy;<lb/>fertur, nota &longs;unt improbis illis artificibus, nec no&longs;trum e&longs;t &longs;pectare, <lb/>cur id licuerit, po&longs;tquam Deus hanc uiolentiam e&longs;&longs;e uoluit. </s>
          <s id="id002054">Multa <lb/>damnamus, <expan abbr="&qtilde;">quae</expan> Deus e&longs;&longs;e uult: boni uiri e&longs;t <expan abbr="n&otilde;">non</expan> ni&longs;i opitulari homini&shy;<lb/>bus, <expan abbr="eti&atilde;">etiam</expan> malis modo bonis futuri <expan abbr="n&otilde;">non</expan> &longs;int <expan abbr="impedim&etilde;to">impedimento</expan>: <expan abbr="quamobr&etilde;">quamobrem</expan> 


<pb pagenum="114" xlink:href="015/01/133.jpg"/>ea tradenda &longs;unt, qu&aelig; oppre&longs;sis &longs;int auxilio: ea &longs;unt, qu&etail; &longs;ubtilibus <lb/><expan abbr="con&longs;t&atilde;t">con&longs;tant</expan> rationibus, et multiplicata <expan abbr="amitt&utilde;t">amittunt</expan> uim ut qua&longs;i <expan abbr="pr&etail;&longs;t&etilde;t">pr&etail;&longs;tent</expan> pau<lb/>ca multis, &amp; exigua magnis. </s>
          <s id="id002055">In c&etail;teris ob&longs;curare ita decet cuncta, <expan abbr="&qtilde;">quae</expan> <lb/>obe&longs;&longs;e po&longs;&longs;unt, aut quouis modo puerti ad malos u&longs;us <expan abbr="que&atilde;t">queant</expan>, ut di&shy;<lb/>cta <expan abbr="n&otilde;">non</expan> dicta e&longs;&longs;e <expan abbr="put&etilde;t">putent</expan>, hoc e&longs;t <expan abbr="offici&utilde;">officium</expan> <expan abbr="n&otilde;">non</expan> &longs;olum probi, &longs;ed <expan abbr="eti&atilde;">etiam</expan> pruden<lb/>tis uiri.</s>
        </p>
        <p type="margin">
          <s id="id002056"><margin.target id="marg409"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002057">Propo&longs;itio cente&longs;ima decima octaua.</s>
        </p>
        <p type="main">
          <s id="id002058">Quanta proportione decedat ictus in obliquum parietem ab eo, <lb/>qui e&longs;t ad perpendiculum declarare.</s>
        </p>
        <figure id="id.015.01.133.1.jpg" xlink:href="015/01/133/1.jpg"/>
        <p type="main">
          <s id="id002059">Sit paries b d e, ex a <expan abbr="fera&ttilde;">feratur</expan> in dictus, qui &longs;i <lb/><arrow.to.target n="marg410"/><lb/>e&longs;&longs;et in c d <expan abbr="pariet&etilde;">parietem</expan> e&longs;&longs;e ad perpendiculum, &amp; <lb/>ualidi&longs;simus, &longs;in uero in f g abraderet, &amp; <expan abbr="n&otilde;">non</expan> <lb/><expan abbr="c&otilde;qua&longs;&longs;aret">conqua&longs;&longs;aret</expan>. </s>
          <s id="id002060">Qu&aelig;ritur ergo ex b d e muro <lb/>qualis excipietur? </s>
          <s id="id002061">erit ergo proportio anguli c d a ad <expan abbr="angul&utilde;">angulum</expan> b d a, <lb/>ueluti ictus a d in d c ad <expan abbr="ict&utilde;">ictum</expan> in b d, <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t <expan abbr="a&utilde;t">aut</expan> &longs;equi proportio&shy;<lb/>nem, <expan abbr="quoni&atilde;">quoniam</expan> maxima uarietate <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> dum ex angulo b d a acuto fit <lb/>acutior, <expan abbr="quoni&atilde;">quoniam</expan> &longs;i b d c &longs;it <expan abbr="&qtilde;druplus">quadruplus</expan> b d a erit re&longs;iduus ad <expan abbr="dimidi&utilde;">dimidium</expan> b <lb/>d a nonuplus ip&longs;i dimidio, &amp; ad <expan abbr="quart&atilde;">quartam</expan> <expan abbr="part&etilde;">partem</expan> habebit proportionem <lb/><expan abbr="decemnou&etilde;">decem nouem</expan> ad <expan abbr="un&utilde;">unum</expan>. </s>
          <s id="id002062">Si ergo <expan abbr="eti&atilde;">etiam</expan> in <expan abbr="id&etilde;">idem</expan> tenderent, <expan abbr="n&otilde;">non</expan> efficerent mille <lb/>ictus &qring;d tres, cuius demon&longs;tratio h&etail;c e&longs;t. </s>
          <s id="id002063">Supponamus <expan abbr="proportion&etilde;">proportionem</expan> <lb/>b d c ad <expan abbr="&qtilde;rtam">quartam</expan> <expan abbr="part&etilde;">partem</expan> a d b ad dito re&longs;iduo ad b d c e&longs;&longs;e <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="decupl&atilde;">decuplam</expan>: <lb/><expan abbr="t&utilde;c">tunc</expan> ex duob. </s>
          <s id="id002064">ictibus centupla erit in d c ad <expan abbr="e&atilde;">eam</expan>, qu&etail; in b e, <expan abbr="eti&atilde;">etiam</expan> tribus <lb/>millecupla: nam <expan abbr="c&otilde;qua&longs;&longs;ata">conqua&longs;&longs;ata</expan> turri in primo ictu, id d decuplo magis <lb/>ad perpendiculum &lt;08&gt; in b d e <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> decima pars in ambitu d, &amp; illa <lb/>erit ergo <expan abbr="t&atilde;">tam</expan> di&longs;&longs;oluta, &amp; infirma ex &longs;uppo&longs;ito, &lt;08&gt; e&longs;t tota b e: &longs;ed ex &longs;e<lb/>cundo ictu decuplo magis <expan abbr="c&otilde;qua&longs;&longs;abi&ttilde;">conqua&longs;&longs;abitur</expan> illa pars, &lt;08&gt; b e ergo tota d c <lb/>centuplo magis <expan abbr="qua&longs;&longs;abi&ttilde;">qua&longs;&longs;abitur</expan> ex duob. </s>
          <s id="id002065">ictibus c d turris, &lt;08&gt; b e, &amp; ita in <lb/>tribus: ex <expan abbr="dec&etilde;">decem</expan> millibus ergo ictibus <expan abbr="eti&atilde;">etiam</expan> ad amu&longs;sim directis, <expan abbr="c&utilde;">cum</expan> ta<lb/><expan abbr="m&etilde;id">men id</expan> uix fieri po&longs;sit in <expan abbr="t&atilde;ta">tanta</expan> multitudine <expan abbr="n&otilde;">non</expan> plus <expan abbr="c&otilde;minue&ttilde;">comminuetur</expan> b d e, &lt;08&gt;<lb/>ex dec&euml; c d <expan abbr="&ptilde;ter">pnter</expan> <expan abbr="qu&atilde;">quam</expan> <expan abbr="exigu&utilde;">exiguum</expan> <expan abbr="quippi&atilde;">quippiam</expan> in &longs;uperficie. </s>
          <s id="id002066">Imo ut <expan abbr="declarat&utilde;">declaratum</expan> <lb/>e&longs;t multo minus repetita ratione multiplicis. </s>
          <s id="id002067">Ob id in arce <expan abbr="Medio&shy;lan&etilde;&longs;i">Medio&shy;<lb/>lanen&longs;i</expan> exterius lapidibus uiuis in <expan abbr="rotund&utilde;">rotundum</expan> diducta &longs;uperficie inter&shy;<lb/><figure id="id.015.01.133.2.jpg" xlink:href="015/01/133/2.jpg"/><lb/>uallo que <expan abbr="&qtilde;">quae</expan>drato hunc in <expan abbr="mod&utilde;">modum</expan> munit&etail; &longs;unt altiores tur<lb/>res. </s>
          <s id="id002068">Fiat ergo murus cuius proportio a d c ad b d a &longs;it &longs;ex<lb/>quitertia, erit que angulus b d c <expan abbr="dodr&atilde;s">dodrans</expan> recti, &amp; <expan abbr="par&utilde;">parum</expan> incli <lb/>natis, <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> b d c erit quarta pars recti, &amp; &longs;it tant&etail; ma&shy;<lb/>gnitudinis, at que duritiei, ac ade&ograve; ben&egrave; coniunctus fer&shy;<lb/><arrow.to.target n="table16"/><lb/>reis cathenis, ac &longs;tolonibus, ut po&longs;sit re&longs;i&longs;tere <expan abbr="machinar&utilde;">machinarum</expan> <expan abbr="fe&shy;renti&utilde;">fe&shy;<lb/>rentium</expan> <expan abbr="&longs;ph&etail;r&atilde;">&longs;ph&etail;ram</expan> <expan abbr="librar&utilde;">librarum</expan> ducentarum &lpar;qu&aelig; &longs;an&egrave; maxim&aelig; &longs;unt&rpar; <lb/><figure id="id.015.01.133.3.jpg" xlink:href="015/01/133/3.jpg"/>quinquaginta: <expan abbr="t&utilde;c">tunc</expan> cum proportio &longs;exquitertia nouies repeti&shy;<lb/>ta, ut in numeris uides, efficiat quinquies replicatis nouem <lb/>ictibus, fiet proportio decupla quinquies producta, qu&etail; e&longs;t cen <lb/><expan abbr="t&utilde;">tum</expan> millium ad <expan abbr="un&utilde;">unum</expan> in quadraginta quin que ictibus. </s>
          <s id="id002069"><expan abbr="Antequ&atilde;">Antequam</expan> <lb/>ergo peruenit ad quinquaginta ictus rectos nece&longs;&longs;e erit, ut 


<pb pagenum="115" xlink:href="015/01/134.jpg"/>multo plures centum millibus ictus excipiat ante &lt;08&gt; euertatur, qu&aelig; <lb/>recta &longs;i e&longs;&longs;et quinquaginta &longs;ol&ugrave;m potui&longs;&longs;et &longs;u&longs;tinere. </s>
          <s id="id002070">Qu&aelig; ergo hu<lb/>mana potentia &longs;ufficeret. </s>
          <s id="id002071">In arce Medio<expan abbr="lan&etilde;&longs;i">lanen&longs;i</expan> uidimus uix attactas <lb/>in illis extuberationibus lapideis. </s>
          <s id="id002072">Sed quoniam hic occurritur per <lb/>inclinationem machinarum, ide&ograve; de hoc <expan abbr="&longs;ermon&etilde;">&longs;ermonem</expan> &longs;um habiturus.</s>
        </p>
        <p type="margin">
          <s id="id002073"><margin.target id="marg410"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <table>
          <table.target id="table16"/>
          <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">Propo&longs;itio cente&longs;ima decima nona.</s>
        </p>
        <p type="main">
          <s id="id002075">Quantum ictus machin&etail; procliuis ad <expan abbr="angul&utilde;">angulum</expan> <expan abbr="minua&ttilde;">minuatur</expan> explorare.</s>
        </p>
        <p type="main">
          <s id="id002076">Huiu&longs;ce cau&longs;a <expan abbr="excogitar&utilde;t">excogitarunt</expan>, ut ictus ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <expan abbr="dirigere&ttilde;">dirigeretur</expan>, &amp; <lb/><arrow.to.target n="marg411"/><lb/><expan abbr="quanqu&atilde;">quanquam</expan> angulus d e f &longs;it &etail;quali angulo a b c, long&egrave; <expan abbr="t&ntilde;">tamen</expan> maior e&longs;t uis <lb/>a b &lt;08&gt; d e duplici cau&longs;a, &amp; <expan abbr="quoni&atilde;">quoniam</expan> a b e&longs;t <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> nat uram impetus <lb/><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"/><lb/>ignis, &amp; <expan abbr="eti&atilde;">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&etail; <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: &amp; &qring;d pars <lb/>&longs;uperior in b retineat <expan abbr="ict&utilde;">ictum</expan>, in e non retineat. </s>
          <s id="id002077">Sed caui<lb/>tas fiat maior in inferiore parte: cuius <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>quiliber facere pote&longs;t <expan abbr="c&utilde;">cum</expan> ha&longs;ta. </s>
          <s id="id002078">Huic ergo &longs;olerti&aelig;, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare ob&longs;tat <expan abbr="prim&utilde;">primum</expan>, quod <lb/>ictus ex decliui &longs;itu periculo&longs;ior e&longs;t pro machina, &amp; ma<lb/>xim&egrave; &qring;d retro impellit, quae ex retro ce&longs;&longs;a, po&longs;t &lt;08&gt; exone<lb/>rata e&longs;t, <expan abbr="digno&longs;ci&ttilde;">digno&longs;citur</expan>, &amp; ad <expan abbr="collimand&utilde;">collimandum</expan> decedit parte <expan abbr="ui&shy;ri&utilde;">ui&shy;<lb/>rium</expan> &longs;uarum, &qring;d et&longs;i <expan abbr="paru&utilde;">paruum</expan> &longs;it in ductu <expan abbr="t&ntilde;">tamen</expan>, &amp; <expan abbr="ictu&utilde;">ictuum</expan> mul<lb/>tiplicatione <expan abbr="magn&utilde;">magnum</expan> affert di&longs;crimen. </s>
          <s id="id002079">Habet &amp; <expan abbr="c&otilde;mo">commo</expan> <lb/>dum &longs;itus muri accliuis <expan abbr="terr&atilde;">terram</expan> <expan abbr="&longs;uppo&longs;it&atilde;">&longs;uppo&longs;itam</expan> ad perpendiculum, <expan abbr="&qtilde;">quae</expan> ictum <lb/>&longs;u&longs;tinet: ade&ograve; ut omnib. </s>
          <s id="id002080"><expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde &longs;it ac &longs;i ex perpen&shy;<lb/>diculo, et &etail;quidi&longs;tanti ad <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. </s>
          <s id="id002081">Venetus. </s>
          <s id="id002082">S. aliter Patauij cauit, <lb/>uidetur que, quae &longs;apienti&longs;simus &longs;it, &amp; eandem &longs;equatur ubi que normam, <lb/>po&longs;t &lt;08&gt; in <expan abbr="rotund&atilde;">rotundam</expan> figuram <expan abbr="tot&utilde;">totum</expan> urbis ambitum formauit, &amp; fo&longs;&longs;a la<lb/>ta, ac pro fundi&longs;sima aqua que perenni muniuit, &amp; <expan abbr="&longs;umm&atilde;">&longs;ummam</expan> muri partem <lb/><expan abbr="rotund&atilde;">rotundam</expan> in hunc <expan abbr="mod&utilde;">modum</expan> effecit <expan abbr="cau&atilde;">cauam</expan> que interius undique, ne cuniculis <lb/><figure id="id.015.01.134.2.jpg" xlink:href="015/01/134/2.jpg"/><lb/>po&longs;&longs;et euerti, &agrave; lateribus uer&ograve; humiles, ac cra&longs;si&longs;simas turres, ut nul<lb/>la ui po&longs;&longs;ent dirui, eas que tormentis bellicis, undique latera lu&longs;trantib. <lb/></s>
          <s id="id002083">reple&longs;&longs;et, illud diligenti&longs;sime cauit, ne murus humilior e&longs;&longs;et aduer&longs;a <lb/>ripa, &longs;ed ad <expan abbr="libell&atilde;">libellam</expan> tamen depre&longs;&longs;us, ut <expan abbr="eti&atilde;">etiam</expan> machinis in terram exten <lb/>&longs;is &longs;ph&etail;rul&aelig; non tangerent <expan abbr="mur&utilde;">murum</expan>: nam <expan abbr="c&utilde;">cum</expan> fo&longs;&longs;a &longs;it quadraginta pa&longs;&shy;<lb/>&longs;uum, excedat <expan abbr="a&utilde;t">aut</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pa&longs;&longs;u, ut quicquid <lb/>in ambitu e&longs;t uno ictu oculi cogno&longs;ci po&longs;sit, &amp; aggeris angulus ma<lb/>ior &longs;it uno pa&longs;&longs;u, <expan abbr="t&utilde;">tum</expan> magis adiecta cra&longs;sitie machin&etail; fieri non pote&longs;t, <lb/>ut ictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. </s>
          <s id="id002084">Eam ob cau&longs;am <expan abbr="eti&atilde;">etiam</expan> cauit, ne <expan abbr="&etail;difici&utilde;">&etail;dificium</expan> ul&shy;<lb/><figure id="id.015.01.134.3.jpg" xlink:href="015/01/134/3.jpg"/><lb/>lum, aut planta, uel colliculus e&longs;&longs;et cir&shy;<lb/>cum circa <expan abbr="urb&etilde;">urbem</expan> ad tria M. P. laborat hoc <lb/>periculo h&etail;c urbs, ne tota &etail;dificijs euer&shy;<lb/>&longs;is concidat. </s>
          <s id="id002085"><expan abbr="Turcar&utilde;">Turcarum</expan> enim Princeps di&shy;<lb/>dicit, ut in Nouo ca&longs;tro in Melit&etail; In&longs;ul&etail; <lb/>arce S. </s>
          <s id="id002086">Elmi appellata plu&longs; &lt;08&gt; mille icti&shy;<lb/>bus in &longs;ingulos dies imo M D obtundere 


<pb pagenum="116" xlink:href="015/01/135.jpg"/>munitiones. </s>
          <s id="id002087">Eum que impetum producere ad quindecim dies, &amp; ui&shy;<lb/>ginti tum etiam longius, ut facil&egrave; domos omnes euertat, homines <lb/>occidat: &longs;i qui &longs;uper&longs;unt tot in commodis obruuntur uigilijs, fame, <lb/>&longs;iti, puluere, ut inutiles reddantur. </s>
          <s id="id002088">Ide&ograve; huic <expan abbr="inc&otilde;modo">incommodo</expan> occurrunt <lb/>aggeribus intra m&oelig;nia erectis, in quos uis <expan abbr="torm&etilde;torum">tormentorum</expan> igneorum <lb/>emoritur. </s>
          <s id="id002089">Sed dices, cur ergo non pro muris erigere eos pr&aelig;&longs;tat, &amp; <lb/>minore &longs;umptu &longs;atis? </s>
          <s id="id002090">quoniam &longs;ubruuntur &agrave; fo&longs;&longs;oribus facillim&egrave;, &longs;i <lb/>ad illos peruenire po&longs;sit ho&longs;tis. </s>
          <s id="id002091">Ide&ograve; intra m&oelig;nia utili&longs;simi &longs;unt, pro <lb/>m&oelig;nijs parum pro&longs;unt. </s>
          <s id="id002092">Quod uer&ograve; ad te&longs;tudines attinet, &longs;ub qui&shy;<lb/>bus <expan abbr="lat&etilde;t">latent</expan> fo&longs;&longs;ores machin&aelig; laterales, &amp; &agrave; fronte &amp; ignes, &amp; aqua al&shy;<lb/>tior prohibent omnino iniuriam, qu&etail; ab his imminet. </s>
          <s id="id002093">C&aelig;terum hu&shy;<lb/>iu&longs;modi cum in longum <expan abbr="differun&ttilde;">differuntur</expan> morbis, illuuie, <expan abbr="inc&otilde;modis">incommodis</expan>, plu&shy;<lb/>uijs, frigoribus omnino <expan abbr="di&longs;&longs;olu&utilde;tur">di&longs;&longs;oluuntur</expan>, ut nulla multitudo huic operi <lb/>&longs;ufficere po&longs;sit. </s>
          <s id="id002094">Rhodus, Alba regia, Melita, Ca&longs;trum <expan abbr="nou&utilde;">nouum</expan>, Byzan<lb/>tium, &longs;i diferri potui&longs;&longs;ent tempora, non ce&longs;si&longs;&longs;ent uictori quantum&shy;<lb/>uis &longs;uperbo. </s>
          <s id="id002095">Vicit pertinacia, audacia que &longs;umma, <expan abbr="Corcyr&atilde;">Corcyram</expan>, Viennam <lb/>capere <expan abbr="n&otilde;">non</expan> potuit, quoniam in <expan abbr="long&utilde;">longum</expan> trahebatur oppugnatio. </s>
          <s id="id002096">Mul<lb/>t&aelig; machin&aelig;, &amp; pauci homines pr&aelig;d&aelig; ob&longs;e&longs;&longs;orum expo&longs;it&aelig; &longs;unt: <lb/>pauc&etail;, &amp; pauci homines ob&longs;idebuntur potius, quam ob&longs;idebunt. <lb/></s>
          <s id="id002097">Exercitus magnus di&longs;&longs;oluitur, &amp; &longs;emet ip&longs;um con&longs;umit, &longs;i nulla fiat <lb/>acce&longs;sio aut exigua quomodo &longs;tabit: &longs;i magna auxilia omnia cor&shy;<lb/>rumpuntur. </s>
          <s id="id002098">Contr&agrave; ob&longs;e&longs;sis auxilia &longs;i ueniant lu&longs;trata, &amp; munita, et <lb/>omnibus nece&longs;&longs;arijs ornata uiri integri <expan abbr="c&otilde;tra">contra</expan> fatigatos, &amp; fe&longs;&longs;os cor <lb/>pore, armati contra inermes, alacres contra torpidos &longs;uperueniunt. <lb/></s>
          <s id="id002099">Ob id pr&aelig;cipuum e&longs;t auxilium pr&etail;ter h&etail;c his, qui oppugnantur co<lb/>pia militum, qui per initia nun &lt;08&gt; quie&longs;cant diu noctu que, <expan abbr="uer&utilde;">uerum</expan> noctu <lb/>duo tubicines per&longs;&aelig;pe <expan abbr="exercit&utilde;">exercitum</expan> <expan abbr="in&longs;omn&etilde;">in&longs;omnem</expan> in armis tota nocte <expan abbr="c&otilde;tine">contine</expan> <lb/><expan abbr="b&utilde;t">bunt</expan>. </s>
          <s id="id002100">Serio <expan abbr="a&utilde;t">aut</expan> die pugnare, &amp; noctu <expan abbr="c&utilde;">cum</expan> minim&egrave; id <expan abbr="&longs;per&atilde;t">&longs;perant</expan>, &amp; fatigati <lb/>&longs;unt: mira euenire &longs;olent in his in&longs;peratis, ac audacibus eruptionib. <lb/></s>
          <s id="id002101">per&longs;&etail;pe <expan abbr="eti&atilde;">etiam</expan> omnino &longs;upra <expan abbr="fid&etilde;">fidem</expan>. </s>
          <s id="id002102">Ita <expan abbr="n&otilde;">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino &agrave; capto de&longs;inat ho&longs;tis, aut <expan abbr="loc&utilde;">locum</expan> occupet &longs;ibi <expan abbr="relict&utilde;">relictum</expan> po&shy;<lb/>tius &lt;08&gt; <expan abbr="qu&etilde;">quem</expan> elegerit. </s>
          <s id="id002103">nam <expan abbr="experiment&utilde;">experimentum</expan> frequens docuit, ubi ill&aelig; ma<lb/>gn&etail; uires &longs;uo arbitrio <expan abbr="loc&utilde;">locum</expan>, <expan abbr="qu&etilde;">quem</expan> <expan abbr="eleger&utilde;t">elegerunt</expan> obtinere potuerint, <expan abbr="tand&etilde;">tandem</expan> <lb/>potiri locis <expan abbr="qu&atilde;tumuis">quantumuis</expan> munitis in hoc &qring;d diximus <expan abbr="c&otilde;tra">contra</expan> <expan abbr="oppona&ttilde;">opponatur</expan>. <lb/></s>
          <s id="id002104">Etenim <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> modis <expan abbr="c&utilde;">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quor&utilde;">quorum</expan> duo &longs;unt ex<lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pn&longs;entem</expan> <expan abbr="con&longs;ideration&etilde;">con&longs;iderationem</expan> ob&longs;idio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tol&shy;li&ttilde;">tol&shy;<lb/>litur</expan>, &amp; proditio, <expan abbr="&qtilde;">quae</expan> cu&longs;to<expan abbr="d&utilde;">dum</expan> <expan abbr="uigil&atilde;tia">uigilantia</expan>, cuniculi, euer&longs;io &longs;uperioris muri, <lb/>euer&longs;io ab imo per machinas, cuniculi, &longs;eu &longs;uffo&longs;sio, urbis euer&longs;io, &longs;eu <lb/><expan abbr="&etail;dificior&utilde;">&etail;dificiorum</expan>: &amp; <expan abbr="&qtilde;uo">qua uo</expan>cant aggre&longs;sio, &longs;eu oppugnatio per &longs;calas, &amp; crates <lb/><expan abbr="c&utilde;">cum</expan> &longs;agittarijs: his omnib. </s>
          <s id="id002105"><expan abbr="&longs;atisfact&utilde;">&longs;atisfactum</expan> puto, pr&etail;ter &lt;08&gt; oppugnationi pro&shy;<lb/>pter <expan abbr="humilitat&etilde;">humilitatem</expan> <expan abbr="muror&utilde;">murorum</expan>: <expan abbr="n&atilde;">nam</expan> lignis <expan abbr="opplen&ttilde;">opplentur</expan>, at que fa&longs;ciculis, terra que fo&longs;<lb/>&longs;&etail;: nihil. </s>
          <s id="id002106">n. </s>
          <s id="id002107">re&longs;i&longs;tit immen&longs;&etail; illi pote&longs;tati, &amp; crudelitati <expan abbr="&longs;&etail;ui&longs;simor&utilde;">&longs;&etail;ui&longs;simorum</expan> ty<lb/><expan abbr="r&atilde;nor&utilde;">rannorum</expan>. </s>
          <s id="id002108"><expan abbr="Ver&utilde;">Verum</expan>, ut dixi, terra noctu <expan abbr="effodi&ttilde;">effoditur</expan>, ligna artificio&longs;is ignib. </s>
          <s id="id002109">eru<pb pagenum="117" xlink:href="015/01/136.jpg"/>untur. </s>
          <s id="id002110">Et longum e&longs;t opus &longs;iue per paucos, &longs;iue per multos quis ef&shy;<lb/>ficere conetur: ut non minus exigat temporis, qu&agrave;m ob&longs;idio: nam <lb/>multitudine unus alterum impedit, &amp; mortui uiuos, ut omnino res <lb/>&longs;it non &longs;peranda ni&longs;i aduer&longs;us inerti&longs;simos. </s>
          <s id="id002111">Pontes euertunt machi<lb/>n&aelig;, ignes que. </s>
          <s id="id002112">Sed ubi etiam muros obtinuerint ob rotunditatem in <lb/>illis con&longs;i&longs;tere non po&longs;&longs;unt. </s>
          <s id="id002113">Inde &agrave; defen&longs;oribus propul&longs;antur &longs;ari&longs;&shy;<lb/>&longs;is, telis, ignibus, tran&longs;uer&longs;is trabibus, machinis: illudque accedit com<lb/>modi, ut quanto plures eo facilius excutiantur. </s>
          <s id="id002114">Dixi non debere <lb/>uereri maxima etiam pr&aelig;ter id, quoniam &amp; i&longs;t&etail; ip&longs;&etail; tanto &longs;anguine <lb/>acqui&longs;it&etail; tanto deorum &amp; hominum iniuria modica &longs;cintilla ignis <lb/>&longs;ine munitionibus, exercitibus, &longs;iue machinis, ab&longs;que terr&aelig; <expan abbr="c&otilde;cu&longs;sio&shy;ne">concu&longs;sio&shy;<lb/>ne</expan>, aut inundatione, uel pe&longs;te euertuntur. </s>
          <s id="id002115">In illam mi&longs;eram lachry&shy;<lb/>mam patris &longs;cintilla ignis inferni, c&ugrave;m Deo placuerit, <expan abbr="mitti&ttilde;">mittitur</expan>, ex qua, <lb/>quod <expan abbr="coalit&utilde;">coalitum</expan> e&longs;t, multis &longs;eculis imperium luxu, crudelitate, &longs;tultitia <lb/>unius filij, uix uno lu&longs;tro toto di&longs;&longs;oluitur. </s>
          <s id="id002116">Hanc <expan abbr="&longs;cintill&atilde;">&longs;cintillam</expan> cum felici <lb/>etiam genio &longs;ecum ex utero detulit Alexander Magnus. </s>
          <s id="id002117">In alijs alij <lb/>genium &longs;ortiti &longs;unt, alij <expan abbr="&longs;cintill&atilde;">&longs;cintillam</expan> detulere ab Orco. </s>
          <s id="id002118">Ex imperio A&longs;&longs;y<lb/>riorum per luxum Sardanapalus: ex Medorum per <expan abbr="&longs;cintill&atilde;">&longs;cintillam</expan> A&longs;tya&shy;<lb/>ges: ex <expan abbr="Per&longs;ar&utilde;">Per&longs;arum</expan> per &longs;tultitiam Darius: ex <expan abbr="Romanor&utilde;">Romanorum</expan> Honorius. </s>
          <s id="id002119">Di <lb/>ces, h&etail;c quid ad proportionem? </s>
          <s id="id002120">Im&ograve; uelut machina ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <lb/>librata pauculo illo puluere Pyrio <expan abbr="urb&etilde;">urbem</expan> euertit, ita &longs;cintilla illa infer <lb/>ni ignis &longs;emini magni tyranni indita euertit at que di&longs;&longs;oluit totum re&shy;<lb/>gnum &longs;ine machinis, ut dixi, uel exercitibus ullis, &amp; quod maius e&longs;t <lb/>remedio nullo. </s>
          <s id="id002121">Sed puerulo indito luxus, ignaui&aelig;, crudelitatis atque <lb/>&longs;tultiti&etail; fontibus, mirabile dictu &longs;an&egrave;, &amp; ad proportionem diuino&shy;<lb/>rum <expan abbr="in&longs;trumentor&utilde;">in&longs;trumentorum</expan> pertinens. </s>
          <s id="id002122">Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>&amp; &aelig;dificia illius po&longs;&longs;e ab&longs;que aggeris erectione, &amp; &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoni&atilde;">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit &agrave; machinis: nec ho&longs;tes id cu<lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, quod m&oelig;nia &longs;olo <expan abbr="&aelig;quen&ttilde;">&aelig;quentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, &amp; in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&etail; &amp; in Cremonen&longs;i arce. <lb/></s>
          <s id="id002123">Ver&ugrave;m ni fallor, ut paruis arcibus &agrave; tanta ui tormentorum nullum <lb/>e&longs;t <expan abbr="pr&aelig;&longs;idi&utilde;">pr&aelig;&longs;idium</expan>, aut &longs;alutis &longs;pes, ita neque <expan abbr="c&otilde;uenit">conuenit</expan>, ut muris humilibus ag<lb/>geri confidant, nam &amp; pauci homines tanto labori non &longs;ufficerent, <lb/>&amp; agger cum fo&longs;&longs;a effo&longs;&longs;a &longs;cilicet terra defen&longs;ores nimis in <expan abbr="angu&longs;t&utilde;">angu&longs;tum</expan> <lb/>cogeret. </s>
          <s id="id002124">At in urbibus contra eueniet: muris enim erectis altius ma<lb/>chin&aelig; lapidum fru&longs;tis hominem <expan abbr="occid&etilde;t">occident</expan>: an percu&longs;&longs;a &longs;uperiore par <lb/>te ob coniunctionem inferior concutitur, &amp; in de <expan abbr="tot&utilde;">totum</expan> &longs;imul cadit, <lb/>ut uidimus Papi&etail;, quo <expan abbr="cad&etilde;te">cadente</expan>, &amp; fo&longs;&longs;a impletur, &amp; <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad &longs;ubruendum reliquas partes <expan abbr="pr&etail;be&ttilde;">pr&etail;betur</expan>: im&ograve; percul&longs;i defen&shy;


<pb pagenum="118" xlink:href="015/01/137.jpg"/>&longs;ores &longs;&aelig;pe muneris &longs;ui obliui&longs;cuntur, de&longs;ertaque ea parte liberum <lb/>ingre&longs;&longs;um ho&longs;tibus exhibent. </s>
          <s id="id002125">Tum uer&ograve; magis, quod non confi&shy;<lb/>dunt animo <expan abbr="n&otilde;">non</expan> ad id parato, po&longs;&longs;e aggerem &longs;ufficientem, &amp; in tam <lb/>breui tempore ex&longs;truere, &amp; etiam intelligunt, antequam erigatur, <lb/>patere &agrave; lateribus introitum ho&longs;tibus.</s>
        </p>
        <p type="margin">
          <s id="id002126"><margin.target id="marg411"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002127">Propo&longs;itio cente&longs;imauige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id002128">Proportionem partium nauis ad eundem obliquum uentum <lb/>explorare.<lb/><arrow.to.target n="marg412"/></s>
        </p>
        <p type="margin">
          <s id="id002129"><margin.target id="marg412"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002130">Sint mali in naui a b c, ad b e, c f uentus &egrave; regione g h k etiam ad <lb/>perpendiculum feratur, ut anguli g d a, h e b, k f c &longs;int &aelig;quales, dico <lb/>tamen diuer&longs;o modo affici: nam cum premitur a uer&longs;us l, c premi&shy;<lb/>tur uer&longs;us f: at &longs;i prematur c uer&longs;us n a, premitur uer&longs;us d, at &longs;i pre&shy;<lb/><figure id="id.015.01.137.1.jpg" xlink:href="015/01/137/1.jpg"/><lb/>matur b uer&longs;us m, &amp; a uer&shy;<lb/>&longs;us l, &longs;ed non quantum ex <lb/>g d, &amp; c uer&longs;us n, &longs;ed non <lb/>quantum ex k f, ab eodem <lb/>ergo uento contrarij mo&shy;<lb/>tus efficiuntur ex uelorum <lb/>diuer&longs;itate, etenim per uen <lb/>tum d feretur ad meridiem <lb/>nauis, &amp; per uelum f ad Se<lb/>ptentrionem etiam didu&shy;<lb/>cto auxilio e l a ui, quanto <lb/>magis cum illo: &amp; &longs;i uen&shy;<lb/>tus excipiatur in f uelo, <lb/>non iuuabit clauus, &amp; &longs;i in <lb/>d dirigetur, &amp; temperabitur motus, &amp; &longs;i in e medio modo. </s>
          <s id="id002131">Ergo &longs;i <lb/>uentus feratur rect&egrave; iuuabit, ut dici &longs;olet omnibus, &amp; plenis uelis <lb/>excipere, &longs;i ex obliquo demittere antennam puppis, &longs;in autem ual&shy;<lb/>de obliquus &longs;it, &longs;olo pror&aelig; uelo utemur. </s>
          <s id="id002132">Si ualidior qu&agrave;m oportet <lb/>humiliore. </s>
          <s id="id002133">Atque h&aelig;c po&longs;tmodum &longs;unt diligenter numeranda, ac <lb/>metienda: nunc &longs;ufficiat cau&longs;am reddidi&longs;&longs;e, &amp; admonui&longs;&longs;e diuer&longs;i&shy;<lb/>tatis motuum, qu&aelig; ex uelis contingit: nam e&ograve; fertur nauis, qu&ograve; <lb/>prora dirigitur. </s>
          <s id="id002134">Ergo cum puppis tanto feratur uer&longs;us meridiem <lb/>a b, quanto prora uer&longs;us meridiem a d, &amp; quanto puppis fertur uer<lb/>&longs;us <expan abbr="meridi&etilde;">meridiem</expan>, tanto prora fertur uer&longs;us boream, igitur quanto prora <lb/>fertur uer&longs;us meridiem a d, tanto uer&longs;us boream a b f, &longs;ed &longs;itus claui <lb/>pote&longs;t multo plus in comparatione ueli d, quam f &longs;cilicet, quia di&shy;<lb/>&longs;tantia a b a e&longs;t o a, &amp; di&longs;tantia e c e&longs;t o c, tanto plus ergo pote&longs;t cla&shy;<lb/>ui &longs;itus in comparatione ad uelum d, quam f, quanta e&longs;t proportio 


<pb pagenum="119" xlink:href="015/01/138.jpg"/>o a, ad o c, igitur clauus e&longs;t long&egrave; potentior in comparatione ueli <lb/>d, quam f, ergo uelum d minus agit nauim, quam f. </s>
          <s id="id002135">Sed ut extrema <lb/>&longs;e habent, ita medium eorum comparatione, igitur malus b e uali&shy;<lb/>dior e&longs;t, multo d a, &amp; infirmior c f. </s>
          <s id="id002136">Ver&ugrave;m, ut dixi, ob &longs;itum &longs;impli&shy;<lb/>citer ualidius e&longs;t, uelum e quam f, &amp; etiam quia, ut dixi, altior &amp; <lb/>cra&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, qu&agrave;m e f: <lb/>adde quartam qu&ograve;d uelum habet maius, antiquo tempore uoca&shy;<lb/>tum acatius. </s>
          <s id="id002137">At ut etiam docui c b non e&longs;t in medio, nec &aelig;quidi&longs;tat <lb/>ab a d &amp; c f, &longs;ed inclinatur ad proram ideoque imbecillior: cum ergo <lb/>&longs;it &aelig;qualium, &amp; paulo maiorum uirium, qu&agrave;m c f, &amp; tutior, &amp; me&shy;<lb/>lius agatur per <expan abbr="clau&utilde;">clauum</expan> qu&agrave;m c f, &amp; &longs;it a d nimis iu&longs;to imbecillis, pro&shy;<lb/>pterea b e mali, &amp; ueli maximus e&longs;t u&longs;us: ade&ograve; mali nomen per an&shy;<lb/>tonoma&longs;iam de ip&longs;o &longs;impliciter intelligatur.</s>
        </p>
        <p type="main">
          <s id="id002138">Propo&longs;itio cente&longs;ima uige&longs;ima 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 appen&longs;um, ut &longs;olet, in a, &amp; moueatur motu </s>
        </p>
        <p type="main">
          <s id="id002141"><arrow.to.target n="marg413"/><lb/>qua&longs;i circa axem p a q in parte inferiore, &amp; a&euml;r comprehen&longs;us &longs;ub <lb/>b h k, &amp; &longs;patium &longs;it 1 m figur&aelig; nauicularis, qu&aelig; con&longs;tat e&longs;&longs;e par&shy;<lb/>tem cylindri inanis ex formatione ab Euclide &longs;cripta: nam &longs;i pro&shy;<lb/>poneretur p a q ad perpendiculum &longs;uper&longs;tans plano, fieret circum&shy;<lb/>ducta a b c &longs;uperficie, qu&aelig; e&longs;&longs;et lata &longs;uperius, &longs;icut etiam inferius <lb/><arrow.to.target n="marg414"/><lb/>cylindrus: at &longs;uperius a b tenuis e&longs;t, &amp; angu&longs;ta, ergo fiet pars cy&shy;<lb/>lindri inanis: quia non circumuoluitur, donec redeat. </s>
          <s id="id002142">Ergo per di&shy;<lb/>cta &longs;uperius &longs;ectio illius p r q s per axem e&longs;t pars cuiu&longs;dam elly&shy;<lb/><arrow.to.target n="marg415"/><lb/>p&longs;is. </s>
          <s id="id002143">Et &longs;ectio qu&aelig;uis plan&aelig; &longs;uperficiei &aelig;quidi&longs;tans a b c uelut tu, <lb/>item que &aelig;quidi&longs;tans axi p a q e&longs;t &longs;uperficies rectangula, quarum <lb/>una e&longs;t &longs;imilis, &amp; &aelig;qualis b h k, e&longs;t in una &longs;uperficie cum axe p a q <lb/>alia uer&ograve; e&longs;t &aelig;quidi&longs;tans eidem axi maior aut minor &aelig;quidi&longs;tanti&shy;<lb/>um, &amp; ip&longs;a laterum, at que rectangula ac &longs;i cylindrus &longs;tans axi plano <lb/>&aelig;quidi&longs;tanti &longs;ecaretur iuxta longitudinem &longs;eu altitudinem &longs;uam: <lb/>&amp; manife&longs;tum e&longs;t, quod i&longs;ta duo plana, &amp; eorum &longs;uperficies &longs;ecant <lb/>&longs;e mutu&ograve; ad rectos angulos.</s>
        </p>
        <p type="margin">
          <s id="id002144"><margin.target id="marg413"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="margin">
          <s id="id002145"><margin.target id="marg414"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 11. <lb/><emph type="italics"/>diff.<emph.end type="italics"/> 21.</s>
        </p>
        <p type="margin">
          <s id="id002146"><margin.target id="marg415"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 69.</s>
        </p>
        <p type="main">
          <s id="id002147">Quibus con&longs;titutis, qui &longs;tabunt iuxta l, &amp; m longitudines a&euml;ris <lb/>moti, &amp; loci, per quem tran&longs;it flabellum, &longs;entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus &longs;it elatius a b ex&shy;<lb/>tremis, &longs;tantes, &amp; alti tangentur &agrave; uento agitato. </s>
          <s id="id002148">Si uero &longs;edeant, aer <lb/>primum non attinget illos, ut etiam quia &longs;ur&longs;um pellitur non per&shy;<lb/>ueniet ad illos, im&ograve; diffugiet, ergo non refrigerabuntur. </s>
          <s id="id002149">Qui uer&ograve; <lb/>&agrave; lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="n&otilde;">non</expan> refrigera<lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="qu&atilde;do">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, &amp; mouebi&shy;


<pb pagenum="120" xlink:href="015/01/139.jpg"/>tur diuer&longs;a ratione, quippe ab f in h, &amp; non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per &aelig;quidi&longs;tantem f, <lb/>&amp; g non poterunt refrige&shy;<lb/>rari. </s>
          <s id="id002150">Sed &longs;i humili loco &longs;e&shy;<lb/>deant, quoniam a&euml;r de&longs;cen <lb/>dit, ex l &amp; m uer&longs;us x, &amp; <lb/>etiam, quia erunt proximi <lb/>h k, <expan abbr="qu&atilde;do">quando</expan> fuerit in x, <expan abbr="refri&shy;gerabun&ttilde;">refri&shy;<lb/>gerabuntur</expan> ualde. </s>
          <s id="id002151">Qui <expan abbr="aut&etilde;">autem</expan> <lb/><expan abbr="er&utilde;t">erunt</expan> iuxta h &amp; k minus <expan abbr="re&shy;frigerabun&ttilde;">re&shy;<lb/>frigerabuntur</expan> utri&longs;que, &longs;ed pau<lb/>lulum in reditibus propin <lb/>quis, &amp; neque &longs;tantes, <expan abbr="neque&longs;ed&etilde;tes">neque <lb/>&longs;edentes</expan>, &longs;ed &longs;i altius attolla&shy;<lb/>tur h k. </s>
          <s id="id002152">Rur&longs;us &longs;i b h k fue&shy;<lb/>rit grauior eodem, ut de&shy;<lb/>&longs;cendat tanto impetu, <expan abbr="qu&atilde;&shy;to">quan&shy;<lb/>to</expan> a&longs;cendit attractum, ut <lb/>pote ex ligno tenui nucis, <lb/>tunc multo magis refrige&shy;<lb/>rabit, &amp; procul, <expan abbr="n&otilde;">non</expan> ob uim <lb/>ualidiorem, &longs;ed quoniam <lb/>celerius occur&longs;antes &longs;ibi <lb/>contrarijs motibus, ac <expan abbr="ue&shy;hem&etilde;tibus">ue&shy;<lb/>hementibus</expan> fiet colli&longs;io par<lb/>tium a&euml;ris, &amp; ideo in ambitum impelletur, &amp; undique cubiculum <lb/>refrigerabit, quod non faciet maius long&egrave; flabellum lento motu <lb/>agitatum, aut ex materia leui. </s>
          <s id="id002153">Idem multo magis contingeret, ubi <lb/>duo e&longs;&longs;ent flabella laquearibus appen&longs;a, qu&aelig; ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerrem</expan> mouerent, &longs;eu quod &longs;uperficies eo modo &longs;e haberent: &amp; &longs;i <lb/>flabella rotunda e&longs;&longs;ent, tunc maiorem ambitum a&euml;ris occuparent, <lb/>&amp; uelocius deficientibus angulis mouebuntur.</s>
        </p>
        <p type="main">
          <s id="id002154">Propo&longs;itio cente&longs;ima uige&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id002155">Contemptus circa &longs;olis rationem in umbris declarare.</s>
        </p>
        <p type="main">
          <s id="id002156">Con&longs;tat prim&ugrave;m &longs;olem, &amp; ex centro, &amp; toto eius ambitu illumi&shy;<lb/>nare hanc prim&ugrave;m diuer&longs;itatem, qu&aelig; aliquando tota diametro <lb/>computata dimidium unius partis totius c&oelig;li excedit: &longs;cioterici <lb/>negligunt, ut exiguam. </s>
          <s id="id002157">Secund&ograve; etiam diuer&longs;itatis illius, qua mo&shy;<lb/>d&ograve; &agrave; terra uer&longs;us ab&longs;idem defertur, mod&ograve; ad terram de&longs;cendere to&shy;<lb/>tidem uariata altitudine, non parum nullam habent rationem, &longs;eu 


<pb pagenum="121" xlink:href="015/01/140.jpg"/>qu&ograve;d tanta ne &longs;it, ut euidentem in gnomonibus faciat uarietatem, <lb/>&longs;eu qu&ograve;d incertum adhuc &longs;it, an id uer&egrave; &longs;oli accidat. </s>
          <s id="id002158">Tertium e&longs;t fi&shy;<lb/>nis umbr&aelig; ip&longs;ius gnomonis, qui incertus e&longs;t, ut pars non contem&shy;<lb/>nenda in dubium uertatur, quoniam &longs;en&longs;im ex ob&longs;curo in illumi&shy;<lb/>natum feratur, at tamen contemnitur etiam. </s>
          <s id="id002159">Quartum qu&ograve;d cum <lb/>&longs;ol moueatur in &longs;pira, fingitur qua&longs;i in parallelo &aelig;quinoctiali circu<lb/>lo circumagatur ab his, qui horologia de&longs;cribunt. </s>
          <s id="id002160">Quintum qu&ograve;d <lb/>cum in&aelig;qualiter in orbe &longs;uo moueatur quanuis exigua &longs;it h&aelig;c dif&shy;<lb/>ferentia, &aelig;qualiter <expan abbr="tam&etilde;">tamen</expan> moueri pr&aelig;&longs;upponitur. </s>
          <s id="id002161">Sextum e&longs;t, qu&ograve;d <lb/>dies &aelig;quales &longs;upponuntur, qui tamen tum ex ratione partis pera&shy;<lb/>grat&aelig;, tum ratione a&longs;cen&longs;us <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> &longs;unt in&aelig;quales, &amp; <expan abbr="tam&etilde;">tamen</expan> h&aelig;c in&shy;<lb/>qualitas <expan abbr="eti&atilde;">etiam</expan> in <expan abbr="horar&utilde;">horarum</expan> computatione pr&aelig;termittitur. </s>
          <s id="id002162">Sed &amp; h&etail;c ut <lb/>prior ratione magis, <expan abbr="qu&atilde;">quam</expan> &longs;en&longs;u <expan abbr="deprehendi&ttilde;">deprehenditur</expan>. </s>
          <s id="id002163"><expan abbr="Septim&utilde;">Septimum</expan> e&longs;t di&longs;crimen, <lb/>&qring;d oritur ex ui&longs;us circulo &longs;eu horizonte, &amp; circulo tran&longs;eunte p cen<lb/><expan abbr="tr&utilde;">trum</expan> mundi, nam horizon uere <expan abbr="t&atilde;to">tanto</expan> minor e&longs;t circulo magno, quan&shy;<lb/>tum e&longs;t &longs;emidiameter terr&etail;, <expan abbr="c&otilde;paratus">comparatus</expan> ad <expan abbr="&longs;emidiametr&utilde;">&longs;emidiametrum</expan> orbis c&oelig;le<lb/>&longs;tis, &longs;ed e&longs;t in&longs;en&longs;ilis quantitatis. </s>
          <s id="id002164"><expan abbr="Octau&utilde;">Octauum</expan> e&longs;t, quod trianguli ex gno&shy;<lb/>mone umbra, &amp; radijs &longs;olis latera non mutant lineas, qu&aelig; &agrave; &longs;ole ad <lb/>centrum terr&aelig; deueniunt, nec qu&ograve;d maius e&longs;t, radius &longs;olis ad uerti&shy;<lb/>cem hominis breuior habetur femidimetiente. </s>
          <s id="id002165">H&aelig;c <expan abbr="igi&ttilde;">igitur</expan> omnia <expan abbr="&longs;ci&shy;otericor&utilde;">&longs;ci&shy;<lb/>otericorum</expan> opifices non ob&longs;eruant, &longs;ed negligunt. </s>
          <s id="id002166">Verum quatuor <lb/>tant&ugrave;m altitudinem poli regionis locum &longs;olis in eclyptica locum <lb/>&longs;olis in circulo &aelig;quinoctialis, uel &aelig;quinoctiali parallelo, ex qui&shy;<lb/>bus tribus fit altitudo &longs;olis, una in circulo &longs;cilicet uerticali ab hori&shy;<lb/>zonte, &amp; differentia line&aelig; meridian&aelig; &agrave; linea uer&longs;us polum, quam <lb/><arrow.to.target n="marg416"/><lb/>o&longs;tendit lapis Herculeus, de qua dictum e&longs;t &longs;uperius.</s>
        </p>
        <p type="margin">
          <s id="id002167"><margin.target id="marg416"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 84.</s>
        </p>
        <p type="main">
          <s id="id002168">Propo&longs;itio cente&longs;ima uige&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id002169">Cognita ratione umbr&etail; ad gno<lb/>monem &longs;inum, &amp; arcum altitudi&shy;<lb/>nis ab horizonte quouis tempo&shy;<lb/>re digno&longs;cere.</s>
        </p>
        <figure id="id.015.01.140.1.jpg" xlink:href="015/01/140/1.jpg"/>
        <p type="main">
          <s id="id002170">Sit circulus magnus, in quo &longs;ol <lb/><arrow.to.target n="marg417"/><lb/>a f g &longs;uper&longs;tans ad perpendicu&shy;<lb/>lum circulo ui&longs;us f e g, quos mani <lb/>fe&longs;tum e&longs;t tran&longs;ire per idem cen&shy;<lb/>trum mundi c, quia magni &longs;unt, &amp; <lb/>&longs;it c d erecta ad perpendiculum <lb/>&longs;uper f g, nam perinde e&longs;t per &longs;e&shy;<lb/>ptimum contemptum, ac &longs;i &longs;uper&shy;<lb/><arrow.to.target n="marg418"/><lb/>ficies horizontis tran&longs;eat per terr&etail; centrum, &amp; pedes per octauum, <lb/><arrow.to.target n="marg419"/><lb/>ideo proportio e c ad c d umbr&aelig; ad gnomonem, ut b e ad b a, ergo 


<pb pagenum="123 &lsqb;=122&rsqb;" xlink:href="015/01/141.jpg"/>per demon&longs;trata b a cognita in comparatione a d e a, e a autem per <lb/>octauum contemptum e&longs;t dimetiens circuli, ergo a b &longs;inus notus, <lb/>&amp; arcus f a, quod e&longs;t primum cognitum. </s>
          <s id="id002171">Et hic quidem circulus <lb/>uerticalis dicitur, quia per illum tran&longs;it, aliter non e&longs;&longs;et ad perpen&shy;<lb/>diculum horizonti.<lb/><arrow.to.target n="marg420"/></s>
        </p>
        <p type="margin">
          <s id="id002172"><margin.target id="marg417"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002173"><margin.target id="marg418"/>P<emph type="italics"/>r&aelig;ced.<emph.end type="italics"/> P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002174"><margin.target id="marg419"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 113.</s>
        </p>
        <p type="margin">
          <s id="id002175"><margin.target id="marg420"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002176">Ex hoc &longs;equitur, quod altitudines &longs;olis &aelig;quales omnes in uno <lb/>&longs;unt circulo horizonti parallelo. </s>
          <s id="id002177">Et &longs;i &longs;ol fuerit in uno circulo ho&shy;<lb/>rizonti parallelo, altitudines &longs;olis, &amp; umbr&aelig; magnitudines &aelig;qua&shy;<lb/>les erunt.</s>
        </p>
        <p type="main">
          <s id="id002178"><arrow.to.target n="marg421"/></s>
        </p>
        <p type="margin">
          <s id="id002179"><margin.target id="marg421"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002180">Sol ni&longs;i bis in una die pote&longs;t e&longs;&longs;e in circulo horizonti parallelo, <lb/>&longs;emel ante meridiem, &amp; &longs;emel po&longs;t, tantundem ab eodem di&longs;tans.</s>
        </p>
        <p type="main">
          <s id="id002181"><arrow.to.target n="marg422"/></s>
        </p>
        <p type="margin">
          <s id="id002182"><margin.target id="marg422"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id002183">Cum ergo ita &longs;it, nece&longs;&longs;e e&longs;t umbras &aelig;quales, &amp; circulum hori&shy;<lb/>zonti <expan abbr="parallel&utilde;">parallelum</expan> fieri &longs;ub in &aelig;qualibus horis in diuer&longs;is &longs;emper die&shy;<lb/>bus, pr&aelig;terquam cum in punctis fuerit &aelig;qualis ab &etail;quinoctiali, &amp; <lb/>in eandem partem declinationis, &amp; hoc bis <expan abbr="c&otilde;tingit">contingit</expan> &longs;olum in anno <lb/>pro quolibet circulo parallelo, &longs;icut in eodem die etiam bis <expan abbr="t&atilde;tum">tantum</expan>, <lb/>ut dictum e&longs;t.</s>
        </p>
        <p type="main">
          <s id="id002184"><arrow.to.target n="marg423"/></s>
        </p>
        <p type="margin">
          <s id="id002185"><margin.target id="marg423"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002186">Nam exempli gratia, cum &longs;ol e&longs;t in initio Capricorni, &amp; in C&oelig;li <lb/>medio, minima e&longs;t umbra eius diei, &amp; totius anni. </s>
          <s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um&shy;<lb/>bra meridiei: at ei &aelig;qualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbr&aelig; &aelig;quales diuer&longs;orum dierum fi&shy;<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo ad circulum meridiei, quod erat de&shy;<lb/>mon&longs;trandum.</s>
        </p>
        <p type="main">
          <s id="id002188"><arrow.to.target n="marg424"/></s>
        </p>
        <p type="margin">
          <s id="id002189"><margin.target id="marg424"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id002190">Ex hoc &longs;equitur, quod horarum determinatio fit &longs;ecundum line&shy;<lb/>am in &aelig;qualem obliquam, qu&aelig; toti anno &longs;eruiat, ut &aelig;qualium um&shy;<lb/>brarum determinatio hararum &amp; partium eius numerum.</s>
        </p>
        <p type="main">
          <s id="id002191"><arrow.to.target n="marg425"/></s>
        </p>
        <p type="margin">
          <s id="id002192"><margin.target id="marg425"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 5.</s>
        </p>
        <p type="main">
          <s id="id002193">Ex quo colligitur modus faciendi gnomonem, &longs;eu per umbras <lb/>rectas, &longs;eu per uer&longs;as, qui docebit toto anno non <expan abbr="&longs;ol&utilde;">&longs;olum</expan> horas, &longs;ed mo <lb/>menta <expan abbr="pul&longs;u&utilde;">pul&longs;uum</expan>, de quibus <expan abbr="dict&utilde;">dictum</expan> e&longs;t quod MMMDC horam <expan abbr="perfici&utilde;t">perficiunt</expan>.</s>
        </p>
        <p type="main">
          <s id="id002194">Propo&longs;itio cente&longs;ima uige&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id002195">Proportionem umbr&aelig; uer&longs;&aelig; e&longs;&longs;e ad gnomonem, uelut gnomo&shy;<lb/>nis ad umbram uer&longs;am.</s>
        </p>
        <p type="main">
          <s id="id002196"><arrow.to.target n="marg426"/></s>
        </p>
        <p type="margin">
          <s id="id002197"><margin.target id="marg426"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002198">Vmbra uer&longs;a dicitur, quoties gnomo in pariete ad perpendicu&shy;<lb/>lum figitur, &longs;ic ut gnomo &aelig;quidi&longs;tet circulo horizontis. </s>
          <s id="id002199">Sit ergo <lb/>paries c k ad perpendiculum f g, &amp; h k a d gnomo ad perpendicu&shy;<lb/>lum parietis &amp; &longs;ol, ut prius in a, &amp; &longs;it primo k h tant&aelig; longitudinis </s>
        </p>
        <p type="main">
          <s id="id002200"><arrow.to.target n="marg427"/><lb/>ut umbr&aelig; locus &longs;it <expan abbr="p&utilde;ctus">punctus</expan> d, ut &longs;it radius a h d e, eritque angulus d u&shy;<lb/>trin que &aelig;qualis, &amp; propterea triangulus k h d &longs;imilis d c e. </s>
          <s id="id002201">Sit modo <lb/><arrow.to.target n="marg428"/><lb/>gnomo maior m l ip&longs;o h k &amp; c l maior c k &longs;eu &aelig;qualis, &amp; quam an&shy;<lb/>guli k &amp; l recti &longs;unt, &amp; anguli l m n, &amp; k h d &aelig;qualis, quia a n, &amp; a c 


<pb pagenum="113 &lsqb;=123&rsqb;" xlink:href="015/01/142.jpg"/>&longs;unt &aelig;quidi&longs;tantes per octauum contemptum, erunt per dicta tri&shy;<lb/>anguli &longs;imiles, igitur proportio l m gnomonis ad l n umbram <lb/>ut k h gnomonis ad k d umbram, &longs;ed k h, ad k d, ut c e umbr&aelig; ad c d <lb/>gnomonem: igitur proportio l m gnomonis ad l n <expan abbr="umbr&atilde;">umbram</expan>, ut um&shy;<lb/>br&aelig; c e ad c d gnomonem, quod fuit demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id002202"><margin.target id="marg427"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002203"><margin.target id="marg428"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002204">Ex hoc prim&ugrave;m patet &amp; pr&etail;cedenti, quod cognita proportione <lb/><arrow.to.target n="marg429"/><lb/>umbr&etail; uer&longs;&etail; ad gnomonem cogno&longs;citur &longs;inus &longs;olis, &amp; arcus altitu&shy;<lb/>dinis in circulo magno, &amp; e&longs;t altitudo ab horizontis parte, qu&aelig; <lb/>proximior e&longs;t loco &longs;olis, ut demon&longs;tratum &agrave; nobis in Geometricis.</s>
        </p>
        <p type="margin">
          <s id="id002205"><margin.target id="marg429"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002206">Sequitur etiam, qu&ograve;d c&ugrave;m umbra fuerit &aelig;qualis gnomoni, &longs;eu <lb/><arrow.to.target n="marg430"/><lb/>recta, &longs;eu uer&longs;a &longs;olis, uel Lun&aelig;, uel &longs;tell&aelig;, altitudo erit partium qua&shy;<lb/>draginta quin que: nam anguli d &amp; e, uel d &amp; h erunt &aelig;quales: igitur <lb/>arcus f a medietas quart&aelig; ide&ograve; partium xlv. </s>
          <s id="id002207">Et &longs;i gnomo fuerit ma&shy;<lb/>ior umbra uer&longs;a, uel minor recta, erit arcus f a minor xlv partibus, &longs;i <lb/>contr&agrave; maior. </s>
          <s id="id002208">Et hoc ubique terrarum. </s>
          <s id="id002209">Et ubi non po&longs;sit tantundem <lb/>eleuari, ut quando &longs;ol e&longs;t &longs;ub circulo capricorni, nunquam nobis <lb/><arrow.to.target n="marg431"/><lb/>gnomo &aelig;quabitur umbr&aelig; rect&aelig; &longs;ed &longs;emper erit minor, &amp; &longs;emper <lb/><arrow.to.target n="marg432"/><lb/>maior umbra uer&longs;a pari ratione.</s>
        </p>
        <p type="margin">
          <s id="id002210"><margin.target id="marg430"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="margin">
          <s id="id002211"><margin.target id="marg431"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>primi<emph.end type="italics"/><lb/>E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002212"><margin.target id="marg432"/>P<emph type="italics"/>er ult. </s>
          <s id="id002213">&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002214">Propo&longs;itio cente&longs;ima uige&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id002215">Proportionem dimetientis, &amp; peripheri&etail; cuiuslibet circuli paral <lb/>leli &aelig;quinoctiali per cognitam partem magni circuli demon&longs;trare.</s>
        </p>
        <p type="main">
          <s id="id002216">H&aelig;c erat tam clara, ut hic locum non mereretur: tam nece&longs;&longs;aria <lb/><arrow.to.target n="marg433"/><lb/>huic propo&longs;ito, ut non potuerit omitti. </s>
          <s id="id002217">Sit ergo Aequinoctij circu&shy;<lb/>lus a b portio circuli magni nota, a c parallelus circulus, &etail;quinoctij <lb/>circulo c d, erit igitur &longs;inus c d notus. </s>
          <s id="id002218">Et ide&ograve; <expan abbr="quadrat&utilde;">quadratum</expan> c d notum, <lb/><arrow.to.target n="marg434"/><lb/>ergo &amp; pars utraque b d d a nota. </s>
          <s id="id002219">Quare detracta a d ex d b relinqui&shy;<lb/>tur d g &aelig;qualis f c diametro paralleli a&longs;signari. </s>
          <s id="id002220">Quare proportio <lb/><arrow.to.target n="marg435"/><lb/>a b ad e f nota ex obiter &longs;upr&agrave; demon&longs;tratis, &amp; pariter ambi&shy;<lb/>tus circuli a b ad ambitum circuli c d, e&longs;t enim ut dimetientis ad di&shy;<lb/><arrow.to.target n="marg436"/><lb/>metientem.</s>
        </p>
        <p type="margin">
          <s id="id002221"><margin.target id="marg433"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002222"><margin.target id="marg434"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>tertij,<emph.end type="italics"/><lb/>&amp; 8. &amp; 17. <lb/><emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002223"><margin.target id="marg435"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun&shy;<lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002224"><margin.target id="marg436"/>P<emph type="italics"/>er<emph.end type="italics"/> 113. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002225">Propo&longs;itio cente&longs;ima uige&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id002226">Circuli horarij naturam declarare.<lb/><arrow.to.target n="marg437"/></s>
        </p>
        <p type="margin">
          <s id="id002227"><margin.target id="marg437"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <figure id="id.015.01.142.1.jpg" xlink:href="015/01/142/1.jpg"/>
        <p type="main">
          <s id="id002228">Circulus horarius e&longs;t circulus magnus <lb/>tran&longs;iens per <expan abbr="&longs;ol&etilde;">&longs;olem</expan>, aut lunam, aut quoduis <lb/>&longs;ydus, de quo agitur, &amp; per polos mundi, <lb/>ide&ograve; differt &agrave; circulo priore altitudinis So&shy;<lb/>lis, quia ille &longs;tat ad perpendiculum &longs;uper <lb/>horizontem, ni&longs;i cum tangitur uice meridi&shy;<lb/>ani, uterque tamen tran&longs;it per <expan abbr="centr&utilde;">centrum</expan> mundi, <lb/>ac &longs;olis. </s>
          <s id="id002229">Hic etiam ad &longs;imiles partes &aelig;qui&shy;<lb/>noctij circulum, &amp; omnes parallelos &longs;ecat. 


<pb pagenum="124" xlink:href="015/01/143.jpg"/>Et principalis e&longs;t meridianus, ide&ograve; ab illo A&longs;trologi horas utrinque<lb/> ante, &amp; po&longs;t numerant. </s>
          <s id="id002230">Ide&ograve; <expan abbr="clar&utilde;">clarum</expan> e&longs;t, qu&ograve;d hor&aelig; &agrave; meridie com&shy;<lb/>putat&aelig; &longs;unt <expan abbr="c&otilde;munes">communes</expan>, habitantibus &longs;ub quauis altitudine poli, &amp; <lb/>ubiuis &longs;it, &longs;ol mod&ograve; regiones &aelig;qualiter di&longs;tent &agrave; fortunatis, &longs;eu &longs;int <lb/>in eadem longitudine.</s>
        </p>
        <p type="main">
          <s id="id002231">Propo&longs;itio cente&longs;ima uige&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id002232">Data Poli altitudine ortus amplitudinem demon&longs;trare.<lb/><arrow.to.target n="marg438"/></s>
        </p>
        <p type="margin">
          <s id="id002233"><margin.target id="marg438"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002234">Sit horizon a d b &aelig;quinoctij circulus <lb/><figure id="id.015.01.143.1.jpg" xlink:href="015/01/143/1.jpg"/><lb/>a k f eclyptica c g, &amp; punctus ortus in ea g. <lb/></s>
          <s id="id002235">&amp; c initium arietis, &amp; g b amplitudo ortiua <lb/>&amp; c e, c f quart&aelig; circulorum, ut &longs;it e f maxi&shy;<lb/>ma &longs;olis declinatio, &amp; polus mundi borea&shy;<lb/>lis l, quia igitur l d nota e&longs;t ex &longs;uppo&longs;ito, &amp; <lb/>l k quadrans erit k h <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> ad dimidium <lb/>circuli notum. </s>
          <s id="id002236">Quia uer&ograve; &aelig;quinoctium, &amp; <lb/>Meridianus &longs;ecant &longs;e ad angulos rectos, &amp; <lb/>b a &aelig;quidi&longs;tat ab utro que polo, erit b polus <lb/>h d, quare b k, quarta circuli, &amp; angulus k <lb/>rectus. </s>
          <s id="id002237">Igitur &longs;umus in di&longs;po&longs;itione tabula&shy;<lb/>rum primi mobilis, ergo etiam oppo&longs;itus <lb/>triangulus, qui ei e&longs;t &aelig;qualis, &amp; &etail;quiangu&shy;<lb/>lus in eadem di&longs;po&longs;itione b m d, quare cum <lb/>data &longs;it g n declinatio <expan abbr="p&utilde;cti">puncti</expan> g dati, datus erit, &amp; arcus g b qu&aelig;&longs;itus.</s>
        </p>
        <p type="main">
          <s id="id002238">Propo&longs;itio cente&longs;ima uige&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id002239">Nota amplitudine ortus cuiu&longs;que <expan abbr="p&utilde;cti">puncti</expan> <expan abbr="arc&utilde;">arcum</expan> <expan abbr="&longs;emidiurn&utilde;">&longs;emidiurnum</expan> inuenire.</s>
        </p>
        <p type="main">
          <s id="id002240"><arrow.to.target n="marg439"/></s>
        </p>
        <p type="margin">
          <s id="id002241"><margin.target id="marg439"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002242">Sit in eadem figura nota g b, uolo illius <expan abbr="arc&utilde;">arcum</expan> &longs;emidiurnum. </s>
          <s id="id002243">Cum <lb/>ergo g n &longs;it declinatio, erit pars arcus Meridiani horarij per polos <lb/>tran&longs;euntis, compleatur ergo l g n o, &amp; quia g n nota e&longs;t, quia de&shy;<lb/>clinatio puncti dati, &amp; g b nota ex &longs;uppo&longs;ito, &amp; f angulus rectus, <lb/>quia e f e&longs;t portio meridiani, erit b n nota differentia a&longs;cen&longs;ionis a <lb/>quarta circuli k b, <expan abbr="igi&ttilde;">igitur</expan> tota k n arcus &longs;emidiurnus. </s>
          <s id="id002244"><expan abbr="Quoni&atilde;">Quoniam</expan> g p paral <lb/>lelus &longs;imilis e&longs;t k n, &amp; in eo <expan abbr="reuolui&ttilde;">reuoluitur</expan> Sol: ergo quando enim perue&shy;<lb/>niet ad p. </s>
          <s id="id002245">Po&longs;&longs;umus etiam &longs;ine inuentione arcus ortus amplitudi&shy;<lb/>nis per triangulum k m d ex notitia g n cogno&longs;cere eandem n b.</s>
        </p>
        <p type="main">
          <s id="id002246"><arrow.to.target n="marg440"/></s>
        </p>
        <p type="margin">
          <s id="id002247"><margin.target id="marg440"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002248">Ex his duabus &longs;equitur <expan abbr="c&otilde;uer&longs;a">conuer&longs;a</expan> &longs;cilicet, quae data magnitudine diei <lb/><expan abbr="cuiu&longs;c&utilde;que">cuiu&longs;cunque</expan> in quauis regione nota erit poli altitudo <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> regionis.</s>
        </p>
        <p type="main">
          <s id="id002249">Propo&longs;itio cente&longs;ima uige&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id002250">Data altitudine &longs;olis in quacunque regione quacunque die di&longs;tan&shy;<lb/>tiam &longs;olis &agrave; Meridiano cogno&longs;cere.</s>
        </p>
        <p type="main">
          <s id="id002251"><arrow.to.target n="marg441"/></s>
        </p>
        <p type="margin">
          <s id="id002252"><margin.target id="marg441"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002253">Sit Horizon a b c d &aelig;quinoctij circulus b e d. </s>
          <s id="id002254">Meridianus a e c <lb/>Polus mundi Borealis f uertex, g, <expan abbr="p&utilde;ctus">punctus</expan> in eclyptica h ducatur ex 


<pb pagenum="125" xlink:href="015/01/144.jpg"/>polo mundi circulus horarius f h k ad &aelig;quinoctij circulum, &amp; uer&shy;<lb/>ticalis circulus p h l u&longs;que ad Horizontem, &amp; circulus parallelus &aelig;&shy;<lb/>quinoctij circulo h m, &longs;it ergo h l altitudo &longs;olis nota, igitur h g nota </s>
        </p>
        <p type="main">
          <s id="id002255"><arrow.to.target n="marg442"/><lb/>erit re&longs;iduum quart&etail; circuli, &amp; &longs;imiliter h k <lb/><figure id="id.015.01.144.1.jpg" xlink:href="015/01/144/1.jpg"/><lb/>nota, quia declinatio puncti dati in eclypti<lb/>ca e&longs;t n nota dies, &amp; locus &longs;olis ex &longs;uppo&longs;i&shy;<lb/>to ergo nota fh <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> quart&etail; circuli no&shy;<lb/>ta e&longs;t <expan abbr="eti&atilde;">etiam</expan> g e, qu&aelig; e&longs;t &etail;qualis altitudini po&shy;<lb/>li ex &longs;uppo&longs;ito, ergo re&longs;iduum quadrantis <lb/>f g, ergo triangulus f g h notorum laterum <lb/>ergo notus angulus f, ergo arcus k e di&longs;tan <lb/><arrow.to.target n="marg443"/><lb/>tia &longs;umpta in &aelig;quinoctij circulo puncti h, <lb/>cui &longs;imilis e&longs;t arcus h m ex parallelo h m, nam quando k perueniet <lb/><arrow.to.target n="marg444"/><lb/>in e h perueniet in m, &amp; in &aelig;quali tempore, qua diui&longs;a per quinde&shy;<lb/>cim gradus, habebimus horas <expan abbr="di&longs;t&atilde;ti&etail;">di&longs;tanti&etail;</expan> &longs;olis &agrave; Meridie ante, uel po&longs;t, <lb/>&amp; minuta horarum dando quibuslibet gradibus quatuor minuta <lb/>hor&aelig;, &amp; quibuslibet minutis graduum quatuor &longs;ecunda hor&aelig;, &amp; <lb/>ita habebimus tempus exacti&longs;simum &agrave; Meridie in quacunque regi&shy;<lb/>one, &amp; in quacunque hora diei.</s>
        </p>
        <p type="margin">
          <s id="id002256"><margin.target id="marg442"/>P<emph type="italics"/>er<emph.end type="italics"/> 123. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002257"><margin.target id="marg443"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 34. <lb/><emph type="italics"/>lib.<emph.end type="italics"/> 4.</s>
        </p>
        <p type="margin">
          <s id="id002258"><margin.target id="marg444"/>D<emph type="italics"/>e<emph.end type="italics"/> T<emph type="italics"/>riang.<emph.end type="italics"/><lb/>M<emph type="italics"/>onteregij.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002259">Propo&longs;itio cente&longs;ima trige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id002260">Data regionis altitudine, &amp; loco &longs;olis proportionem gnomo&shy;<lb/>nis tam ad umbram rectam, quam uer&longs;am, uel etiam in cylindro de&shy;<lb/>terminare.</s>
        </p>
        <p type="main">
          <s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi&shy;<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, &amp; &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenorum con&longs;tructio&shy;<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, &amp; cylindricas &longs;cire ad <lb/>unguem, qualis hora, &amp; minutum, &amp; &longs;ecundum diei e&longs;&longs;et quocun&shy;<lb/>que anni tempore. </s>
          <s id="id002262">Plerique autem tam laborio&longs;&egrave; id conati &longs;unt de&shy;<lb/>mon&longs;trare, ut &longs;tudio&longs;os deterruerint ab opere: res autem ip&longs;a facil&shy;<lb/>lima e&longs;t. </s>
          <s id="id002263">Propo&longs;ita ergo Poli exacta altitudine &longs;olis in Meridie <lb/>declinatione addita uel detracta, habebis re&longs;iduum eius ad qua&shy;<lb/>drantem f g, &amp; &longs;imiliter habebis ex declinatione nota loci &longs;olis de&shy;<lb/>tracta &agrave; quadrante f h &amp; iuxta horam tuam, &amp; minutum multi&shy;<lb/><arrow.to.target n="marg446"/><lb/>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb/>g h, quare re&longs;iduum h l, igitur punctum umbr&etail; rect&etail;, uel uer&longs;&etail; ip&longs;i&shy;<lb/>us gnomonis ad unguem, &amp; ita con&longs;titues horologium exacti&longs;si&shy;<lb/>mum &longs;ecundum ea, qu&aelig; dixi in Corrolarijs &longs;upradictis, &amp; quia ho&shy;<lb/><arrow.to.target n="marg447"/><lb/>rizon a b c d &longs;ecat &aelig;quinoctialem in <expan abbr="c&etilde;tro">centro</expan> terr&aelig; ducta g h k, erunt <lb/>anguli b h g, &amp; k h l &etail;quales. </s>
          <s id="id002264">Igitur po&longs;ito g ortu puncti eclypti&shy;<lb/>c&aelig;, erit g b ortus amplitudo nota, &amp; ide&ograve; angulus b h g, &amp; k h l 


<pb pagenum="126" xlink:href="015/01/145.jpg"/><arrow.to.target n="marg448"/><lb/>notus, &amp; ita extendemus per totum annum. </s>
          <s id="id002265">Cum uer&ograve; fuerit g ele&shy;<lb/>uatus erit, ut <expan abbr="dem&otilde;&longs;tratum">demon&longs;tratum</expan> e&longs;t, in circulo magno uerticali, ergo an&shy;<lb/>gulus fiet in eodem circulo, quia gnomo e&longs;t etiam in illius &longs;uperfi&shy;<lb/>cie. </s>
          <s id="id002266">Ergo angulus erit &aelig;qualis angulo, quem faceret &longs;ol, &longs;i oriretur <lb/><arrow.to.target n="marg449"/><lb/><figure id="id.015.01.145.1.jpg" xlink:href="015/01/145/1.jpg"/><lb/>in puncto horizontis, quem &longs;ecat circulus <lb/>uerticalis &longs;ub ea altitudine: &longs;ed his e&longs;t no&shy;<lb/>tus: nam in priore figura g h f e&longs;t notus ea&shy;<lb/><arrow.to.target n="marg450"/><lb/><expan abbr="d&etilde;">dem</expan> ratione, qua f, &amp; ide&ograve; ei oppo&longs;itus k h n, <lb/>&amp; k rectus, e&longs;t enim f polus b d, &amp; h k decli<lb/>natio nota ergo k n, &amp; h n not&aelig;. </s>
          <s id="id002267">At e k, &amp; <lb/>g h fuere not&aelig;. </s>
          <s id="id002268">Ergo e n, &amp; g n, quare re&longs;i&shy;<lb/>du&aelig; n l &amp; n b not&aelig;. </s>
          <s id="id002269">E&longs;t autem angulus l <lb/>rectus. </s>
          <s id="id002270">ergo ortus amplitudo puncti l nota <lb/>&longs;cilicet arcus l b, ergo in pr&aelig;&longs;enti figura angulus m h b, ergo k h l. <lb/></s>
          <s id="id002271">igitur poterimus &longs;tatuere angulos umbrarum, &amp; iam po&longs;&longs;umus <lb/>determinare magnitudinem: ergo punctum ad <expan abbr="ungu&etilde;">unguem</expan> umbr&etail; qua&shy;<lb/>libet hora, &amp; parte hor&aelig; &longs;ingulis diebus in quacunque regione dat&aelig; <lb/>altitudinis poli uer&longs;a, &amp; rects. </s>
          <s id="id002272">In cylindrica autem eodem modo &longs;i&shy;<lb/>cut in uer&longs;a, e&longs;t enim &longs;pecies umbr&etail; uer&longs;&etail;, ni&longs;i quod analema ob ob&shy;<lb/>liquitatem cylindri melius aptatur, rotundum &longs;cilicet cum <expan abbr="rot&utilde;do">rotundo</expan>.</s>
        </p>
        <p type="margin">
          <s id="id002273"><margin.target id="marg445"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002274"><margin.target id="marg446"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. <emph type="italics"/>li.<emph.end type="italics"/> 4. <lb/><emph type="italics"/>loan. </s>
          <s id="id002275">de<emph.end type="italics"/> M<emph type="italics"/>on <lb/>teregij de<emph.end type="italics"/><lb/>T<emph type="italics"/>riang.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002276"><margin.target id="marg447"/>P<emph type="italics"/>er<emph.end type="italics"/> 123. <lb/><emph type="italics"/>uel<emph.end type="italics"/> 124. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002277"><margin.target id="marg448"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 123. <lb/>C<emph type="italics"/>orol.<emph.end type="italics"/> 1.</s>
        </p>
        <p type="margin">
          <s id="id002278"><margin.target id="marg449"/>P<emph type="italics"/>er<emph.end type="italics"/> 127. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002279"><margin.target id="marg450"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002280">Propo&longs;itio cente&longs;ima trige&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id002281">Si line&aelig; alicui dupla alterius <expan abbr="adiunga&ttilde;">adiungatur</expan>, erit proportio duarum ad <lb/><expan abbr="prim&atilde;">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 &longs;it b c, &amp; rur&longs;us ad b c c d <expan abbr="&aelig;&qacute;ualis">&aelig;qualis</expan> b c <lb/>dico, quod proportio a c ad a b e&longs;t maior, qu&agrave;m a d ad a c. </s>
          <s id="id002283">Propor<lb/><arrow.to.target n="marg451"/><lb/>tio enim c d ad c a minor e&longs;t, qu&agrave;m ad a b per octauam quinti E&shy;<lb/>lementorum. </s>
          <s id="id002284">Ergo minor d c ad c a qu&agrave;m c b ad a b, quia b c &amp; c d <lb/>&longs;unt &aelig;quales, ide&ograve; <expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> habent <expan abbr="proportion&etilde;">proportionem</expan> <lb/>ad a b: <expan abbr="igi&ttilde;">igitur</expan> coniungendo per 28. Quinti propor<lb/><figure id="id.015.01.145.2.jpg" xlink:href="015/01/145/2.jpg"/><lb/>tio d a ad a c minor, quam c a ad a b, quod erat demon&longs;trandum.<lb/><arrow.to.target n="marg452"/></s>
        </p>
        <p type="margin">
          <s id="id002285"><margin.target id="marg451"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="margin">
          <s id="id002286"><margin.target id="marg452"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002287">Propo&longs;itio cente&longs;ima trige&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id002288">Si ad duas lineas, quarum una alteri dupla &longs;it eadem linea adda&shy;<lb/>tur erit aggregati ex minore, &amp; a d adiecta ad ip&longs;am <expan abbr="minor&etilde;">minorem</expan> minor <lb/>proportio quam aggregati ex maiore, &amp; adiecta ad ip&longs;am maio&shy;<lb/>rem duplicata.</s>
        </p>
        <p type="main">
          <s id="id002289"><arrow.to.target n="marg453"/></s>
        </p>
        <p type="margin">
          <s id="id002290"><margin.target id="marg453"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002291">Sint du&aelig; line&etail; a b, &amp; c d. </s>
          <s id="id002292">&amp; &longs;it c d dupla ad a b, ad datur <expan abbr="c&otilde;munis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, &amp; uocetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi&shy;<lb/>nor duplicata f c ad c d, adijcia&shy;<lb/>tur d f &aelig;qualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t dupla ad a b, tota igitur <pb pagenum="127" xlink:href="015/01/146.jpg"/>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="permut&atilde;do">permutando</expan>, &amp; per <lb/>euer&longs;am ut e a ad a b, ita g c ad c d, ut g c ad c d <expan abbr="c&otilde;ponitur">componitur</expan> ex g e ad <lb/>f e, &amp; f c ad c d, igitur e a ad c b componitur ex ei&longs;dem. </s>
          <s id="id002294">Proportio <lb/>autem g c ad f c e&longs;t minor, quam f c ad c d, igitur minor qu&agrave;m du&shy;<lb/>plicata f c ad c d. </s>
          <s id="id002295">con&longs;tat uer&ograve; ex ei&longs;dem, quod proportio c a ad a b <lb/>maior e&longs;t duplicata g c ad f c.</s>
        </p>
        <p type="main">
          <s id="id002296">Propo&longs;itio cente&longs;ima trige&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id002297">Si fuerint du&aelig; quantitates, quarum una alteri dupla &longs;it: minua&shy;<lb/>tur &agrave; minore qu&aelig;dam <expan abbr="qu&atilde;titas">quantitas</expan> eademque maiori addatur, erit mino&shy;<lb/>ris ad <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> maior proportio, <expan abbr="qu&atilde;">quam</expan> aggregati ad <expan abbr="maior&etilde;">maiorem</expan> duplicata. <lb/></s>
          <s id="id002298">Si uer&ograve; minori addatur et &agrave; maiore detrahatur, erit aggregati ad mi<lb/>nore m minor proportio qu&agrave;m maioris ad re&longs;iduum duplicata.</s>
        </p>
        <p type="main">
          <s id="id002299"><arrow.to.target n="marg454"/></s>
        </p>
        <p type="margin">
          <s id="id002300"><margin.target id="marg454"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <figure id="id.015.01.146.1.jpg" xlink:href="015/01/146/1.jpg"/>
        <p type="main">
          <s id="id002301">Sit a b dupla c d, &amp; addatur qu&aelig;&shy;<lb/>dam ad b a, qu&etail; &longs;it a g, eadem detraha&shy;<lb/>tur ex c d &amp; &longs;it c h, dico, quod propor&shy;<lb/>tio e d ad d h maior e&longs;t, quam duplica&shy;<lb/>ta g b ad a b, &amp; rur&longs;us &longs;i qu&aelig;dam ad c &amp; minuatur ex a b utpot&egrave; <lb/>c f addatur c d, &amp; a e minuatur ex a b, erit proportio f d ad c d mi&shy;<lb/>nor duplicata a b ad g e. </s>
          <s id="id002302"><expan abbr="Prim&utilde;">Primum</expan> &longs;ic re&longs;ecentur a n &amp; k l &aelig;quales &longs;in&shy;<lb/>gul&aelig; c h, igitur a l dupla e&longs;t e h &amp; a b fuit dupla a d, c d igitur ut in <lb/>priore con&longs;titution&eacute; pr&aelig;cedentis a b ad l b, ut c d ad h d &amp; a b ad <lb/>b l maior, quam duplicata a b ad b k ut minor qu&agrave;m k b ad b l. </s>
          <s id="id002303">hoc <lb/>enim demon&longs;tratum e&longs;t in fine, igitur c d ad h d maior, qu&agrave;m du&shy;<lb/>plicata a k ad k b, &longs;ed a k ad k b maior e&longs;t per uige&longs;imam tertiam, hu&shy;<lb/>ius &longs;cilicet per demon&longs;trationem illius, qu&agrave;m g b ad b a, igitur mul&shy;<lb/>to maior c d ad d h, qu&agrave;m duplicata g b ad b a, quod e&longs;t primum.</s>
        </p>
        <p type="main">
          <s id="id002304">Secundum &longs;ic per eadem, addito enim duplo f c ip&longs;i <lb/><figure id="id.015.01.146.2.jpg" xlink:href="015/01/146/2.jpg"/><lb/>a b ut in &longs;ecunda figura, &amp; &longs;int a m, &amp; m n erit f d ad c d, <lb/>ut n a ad a b, quare cum n a ad a b &longs;it minor duplicata per <lb/>pr&aelig;cedentem in b ad a b, &amp; a b ad e b &longs;it maior, ut demon <lb/>&longs;tratum e&longs;t in uige&longs;ima tertia huius, qu&agrave;m m b ad a b, erit <lb/>f d ad d c multo minor duplicata a b ad b e, quod e&longs;t &longs;e&shy;<lb/>cundum.</s>
        </p>
        <p type="main">
          <s id="id002305">Propo&longs;itio cente&longs;ima trige&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id002306">Si rectangula &longs;uperficies &longs;it cuius pars tertia quadrata &longs;it, corpus <lb/>quod ex latere quadrat&aelig; in re&longs;iduum &longs;uperficiei con&longs;tat maius e&longs;t <lb/>quouis corpore ex eadem &longs;uperficies aliter diui&longs;a con&longs;tituto.</s>
        </p>
        <p type="main">
          <s id="id002307">Sit rectangulum a c cuius tertia pars c e &longs;it quadrata, dico quod <lb/><arrow.to.target n="marg455"/><lb/>corpus, quod <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> ex e d in a b e&longs;t maius omni corpore, quod fue <lb/>rit ex latere partis &longs;uperficiei a b in reliquam <expan abbr="part&etilde;">partem</expan>. </s>
          <s id="id002308">Si non diuidatur <lb/>uel &longs;upra uel infra, &amp; primo in f erit <expan abbr="aut&etilde;">autem</expan> proportio e d ad d f, ut e c ad 


<pb pagenum="128" xlink:href="015/01/147.jpg"/>e k, &amp; f a ad a e, ut &longs;uperficierum ip&longs;a&shy;<lb/><figure id="id.015.01.147.1.jpg" xlink:href="015/01/147/1.jpg"/><lb/>rum per primam &longs;exti Elementorum: at <lb/>per pr&aelig;cedentem maior e&longs;t proportio <lb/>e d ad d f, qu&agrave;m a f ad a e, duplicata igi&shy;<lb/>tur maior e&longs;t proportio e d ad eam, qu&etail; <lb/>pote&longs;t &longs;uper f c &longs;uperficiem, quam f a ad <lb/>a e, igitur maior, qu&agrave;m a k ad a b ex pri&shy;<lb/>ma &longs;exti Elementorum: igitur per trige<lb/>&longs;imam quartam undecimi. </s>
          <s id="id002309">Parallelipe&shy;<lb/>dum ex e d in a b maius e&longs;t parallelipedo ex ea, qu&aelig; pote&longs;t in f c &longs;u&shy;<lb/>perficiem in ip&longs;am &longs;uperficiem a k. </s>
          <s id="id002310">Si uer&ograve; diui&longs;io facta fuerit in g, <lb/>con&longs;tat ex pr&aelig;cedenti, quod minor e&longs;t proportio g e ad e d, qu&agrave;m <lb/>&longs;it duplicata e a ad a d a g, eam igitur minor proportio eius line&aelig;, <lb/>qu&aelig; pote&longs;t in g e &longs;uperficiem ad e d quam a b ad a h, igitur paralle&shy;<lb/>lipedum ex e d in a b e&longs;t maius parallelipedo ex ea, qu&aelig; pote&longs;t g c <lb/>in a h cum &longs;it a b ad a h, ut dictum e&longs;t, uelut a e ad a g.<lb/><arrow.to.target n="marg456"/></s>
        </p>
        <p type="margin">
          <s id="id002311"><margin.target id="marg455"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002312"><margin.target id="marg456"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002313">Manife&longs;tum e&longs;t autem, qu&ograve;d tale corpus e&longs;t &aelig;quale duplo cubi <lb/>lateris partis terti&aelig; quadrat&aelig;.</s>
        </p>
        <p type="main">
          <s id="id002314">Propo&longs;itio cente&longs;ima trige&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id002315">Si linea in duas partes, quarum una &longs;it alteri dupla, diuidatur <lb/>erit, quod fit ex tertia parte in quadratum re&longs;idui parallelipedum <lb/>maius omni parallelipedo, quod ex diui&longs;ione eiu&longs;dem line&aelig; crea&shy;<lb/>ri po&longs;sit.</s>
        </p>
        <p type="main">
          <s id="id002316"><arrow.to.target n="marg457"/></s>
        </p>
        <p type="margin">
          <s id="id002317"><margin.target id="marg457"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002318">Sit a c dupla b c, &amp; &longs;it quadratum ad ip&longs;ius a c, dico parallelipe&shy;<lb/><figure id="id.015.01.147.2.jpg" xlink:href="015/01/147/2.jpg"/><lb/>dum ex b c in a d maius e&longs;&longs;e quouis alio ex <lb/>diui&longs;ione line&aelig; a b &longs;imiliter creato. </s>
          <s id="id002319">Secetur <lb/>primo in e, &amp; fiat quadratum a f, eritque per <lb/>uige&longs;imam quintam. </s>
          <s id="id002320">Huius proportio c b <lb/>ad b c maior duplicata a e ad a c, quare ma&shy;<lb/>ior, quam a f ad a d per uige&longs;imam &longs;exti Ele<lb/>mentorum, igitur per trige&longs;imam quartam <lb/>undecimi, Parallelipedum ex b c in a d maius e&longs;t parallelipedo e b <lb/>in a f, quod e&longs;t demon&longs;trandum. </s>
          <s id="id002321">Si uer&ograve; diui&longs;io cadat in g, fiat qua&shy;<lb/>dratum a h, et erit per uige&longs;imamtertiam huius proportio g c ad c b <lb/>minor, quam duplicata c a ad a g: igitur minor, qu&agrave;m a d ad a h, igi&shy;<lb/>tur per eandem parallelipedum ex c b in a d maius e&longs;t parallelipe&shy;<lb/>do ex g b in a h.</s>
        </p>
        <p type="main">
          <s id="id002322"><arrow.to.target n="marg458"/></s>
        </p>
        <p type="margin">
          <s id="id002323"><margin.target id="marg458"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002324">Ex hoc liquet qu&ograve;d parallelipedum illud erit quadruplum cu&shy;<lb/>bo minoris partis, &amp; dimidium cubi maioris.</s>
        </p>
        <pb pagenum="129" xlink:href="015/01/148.jpg"/>
        <p type="main">
          <s id="id002325">Propo&longs;itio cente&longs;ima trige&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id002326">Denominationes in infinitum extendere.</s>
        </p>
        <p type="main">
          <s id="id002327">Inquit Euclides, &longs;i fuerint quotlibet quantitates ab uno in conti&shy;</s>
        </p>
        <p type="main">
          <s id="id002328"><arrow.to.target n="marg459"/><lb/><arrow.to.target n="marg460"/><lb/>nua proportione, erit tertius numerus quadratus, &amp; omnes alij &longs;e&shy;<lb/>quentes uno intermi&longs;&longs;o. </s>
          <s id="id002329">Tertia igitur in comparatione ad &longs;ecun&shy;<lb/>dam etiam, quod non &longs;it numerus, e&longs;t quadratum: e&longs;t enim tertia <lb/>ab uno quadratum &longs;ecund&aelig;, qu&aelig; e&longs;t proportio. </s>
          <s id="id002330">Detracto igitur <lb/>uno omnes quantitates lo co pari &longs;unt quadrat&aelig;: ut &longs;cias ergo cu&shy;<lb/>ius &longs;unt quadrat&aelig; diuide per medium, &amp; erit quadratum illius, er&shy;<lb/>go quadrage&longs;ima erit quadratum uige&longs;im&aelig;, &amp; uige&longs;ima decim&aelig;, <lb/>&amp; decima quint&aelig;, &amp; uige&longs;ima &longs;exta terti&aelig; decim&aelig;, &amp; ita de alijs. <lb/></s>
          <s id="id002331">Iuxta hoc dicemus, quod &longs;ecunda erit <expan abbr="quadrat&utilde;">quadratum</expan>, &amp; quarta quadra&shy;<lb/>tum quadrati, &amp; octaua <expan abbr="quadrat&utilde;">quadratum</expan> quadrati quadrati. </s>
          <s id="id002332">Et &longs;extadeci&shy;<lb/>ma quad quad quad quad. </s>
          <s id="id002333">&amp; ita trige&longs;ima &longs;ecunda quad quad quad <lb/>quad quad. </s>
          <s id="id002334">Quod autem quad. </s>
          <s id="id002335">e&longs;t quarta in ordine, ideo &amp; octa&shy;<lb/>ua &amp; duodecima &amp; decima&longs;exta, &amp; &longs;ic de alijs &longs;unt quadrata qua&shy;<lb/>drati, &amp; &longs;icut quarta e&longs;t quadratum quadrati prim&aelig;, ita octaua &longs;e&shy;<lb/>cund&aelig;, &amp; duodecima terti&aelig;, &amp; &longs;exta decima quart&aelig;, &amp; uige&longs;ima <lb/>quint&aelig;, &amp; ita &longs;emper diuidendo per quatuor.</s>
        </p>
        <p type="margin">
          <s id="id002336"><margin.target id="marg459"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002337"><margin.target id="marg460"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/> 8.</s>
        </p>
        <p type="main">
          <s id="id002338">Secunda regula dicebat ibidem Euclides, &longs;i fuerint quotlibet <lb/><arrow.to.target n="marg461"/><lb/>quantitates ab uno in continua proportione quartus, ab uno erit <lb/>cubus &longs;upple &longs;ecund&aelig;, &amp; ita duobus &longs;emper intermi&longs;sis, uno igi&shy;<lb/>tur ip&longs;o relicto quolibet loco ternario, ut tertia, &longs;exta, nona, duode&shy;<lb/>cima &longs;unt cubi, &amp; cubi eius quantitatis, qu&etail; exit diui&longs;o numero per <lb/>tria, uelut tertia prim&aelig;, &longs;exta &longs;ecund&aelig;, nona terti&etail;, duo decima quar <lb/>t&aelig;: &amp; ita tertia erit cubus nona cubus cubi, &amp; uige&longs;ima &longs;eptima cu&shy;<lb/>bus cubi cubi &longs;cilicet prim&aelig;. </s>
          <s id="id002339">Et trige&longs;ima nona e&longs;t cubus ter&shy;<lb/>ti&aelig; decim&aelig;.</s>
        </p>
        <p type="margin">
          <s id="id002340"><margin.target id="marg461"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. P<emph type="italics"/>ro&shy;<lb/>po&longs;.<emph.end type="italics"/> 8.</s>
        </p>
        <p type="main">
          <s id="id002341">Tertia regula quarta quantitas, ut ui&longs;um e&longs;t: e&longs;t quad quad. </s>
          <s id="id002342">Et <lb/>quinta e&longs;t relatum primum, quia 5 e&longs;t numerus primus, &amp; 7 e&longs;t re&shy;<lb/>latum &longs;ecundum, quia e&longs;t &longs;ecundus numerus primus: &amp; undecima <lb/>tertium: &amp; tertiadecima quartum: &amp; decima&longs;eptima quintum: &amp; <lb/>decimanona &longs;extum: &amp; uige&longs;ima tertia &longs;eptimum &amp; uige&longs;ima quin&shy;<lb/>ta, quia e&longs;t primus numerus pr&aelig;ter quam ad quintam, ide&ograve; e&longs;t rela&shy;<lb/>tum quint&aelig;, qu&aelig; e&longs;t relatum primum prim&aelig;, omnes ergo numeri <lb/>primi &longs;unt relata, alij omnes &longs;unt ex natura cubi uel quadrati. </s>
          <s id="id002343">Sed <lb/>relata &longs;unt inter &longs;e omnia diuer&longs;orum generum ni&longs;i <expan abbr="uige&longs;im&utilde;">uige&longs;imum</expan> quin&shy;<lb/>tum, quod e&longs;t relatum primum primi relati, &amp; quadrage&longs;imum no&shy;<lb/>num e&longs;t relatum &longs;ecundum relati &longs;ecundi. </s>
          <s id="id002344">Et ita cente&longs;imum uige&longs;i&shy;<lb/>mum primum e&longs;t relatum tertium tertij relati, reliqua, ut dixi, me&shy;<lb/>dia inter h&aelig;c &longs;unt &longs;ui generis.</s>
        </p>
        <pb pagenum="130" xlink:href="015/01/149.jpg"/>
        <p type="main">
          <s id="id002345">Quarta regula propo&longs;ita quantitate ab uno in continua propor<lb/>tione, &longs;i uis &longs;cire cuius natur&aelig; &longs;it detracto uno con&longs;idera, an po&longs;sit <lb/>diuidi per duo, e&longs;t quadratum medietatis, &amp; ita procedes diuiden&shy;<lb/>do u&longs;que ad numerum primum, qui uel e&longs;t 2, &amp; erit ex genere quad <lb/>quad. </s>
          <s id="id002346">uel 3, &amp; erit ex genere quadratorum cuborum, &amp; &longs;imiliter &longs;i <lb/>&longs;it 9, erit ex genere quadratorum cubi cubi. </s>
          <s id="id002347">Et &longs;i proueniat alius nu<lb/>merus primus, ut 5. 7. 11. 13. erit quadratum relati illius ordinis. </s>
          <s id="id002348">Et &longs;i <lb/>non pote&longs;t diuidi numerus quantitatum per 2 uide, &longs;i po&longs;sit diuidi <lb/>per 3, tunc erit cubus illius quantitatis, &amp; &longs;i illa quantitas, qu&aelig; pro&shy;<lb/>uenit ex diui&longs;ione: fuerit 3, uel potuerit diuidi per 3, erit cubus, uel <lb/>cubus cubi, &amp; ita deinceps. </s>
          <s id="id002349">Si uer&ograve; &longs;it alius numerus primus, ut 5. <lb/>7. 11. erit cubus relati. </s>
          <s id="id002350">Et ita &longs;i <expan abbr="n&otilde;">non</expan> po&longs;sit diuidi per 2, nec per 3, erit ex <lb/>genere relati. </s>
          <s id="id002351">Et tunc &longs;i po&longs;sit diuidi per alium numerum, ut 35, erit <lb/>relatum ex eo genere. </s>
          <s id="id002352">Vtpot&egrave; trige&longs;ima quinta quantitas e&longs;t rela&shy;<lb/>tum &longs;ecundum relati primi, &longs;eu relatum primum relati &longs;ecundi. <lb/></s>
          <s id="id002353">Nam quoties quantitas pote&longs;t diuidi per duos numeros, dicetur <lb/>&longs;ub utro que uici&longs;sim, ut duodecima pote&longs;t diuidi per 4 &amp; 3, ide&ograve; di&shy;<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/>quadratum cubi quadrati, &amp; quadratum cubicum quadrati ip&longs;ius <lb/>proportionis, ad quam omnia referri debent.</s>
        </p>
        <p type="main">
          <s id="id002357">Quinta regula ex pr&aelig;cedenti pendet, &amp; e&longs;t, quod denomina&shy;<lb/>tiones, &amp; proportiones uici&longs;sim commutantur: uelut 256 e&longs;t quad <lb/>quad quad, &amp; inter quad quad quad, &amp; quad quad &longs;unt quatuor ter <lb/>mini ip&longs;o computato, &amp; inter quad quad, &amp; quod ui&longs;i duo, ergo <lb/>quad quad quad continet plures proportiones, &amp; proportiones <lb/>duplicat&aelig; non con&longs;tituunt quad: nam 64 continet duas duplas <lb/>ad 16, non tamen e&longs;t quadratum 16, ideo oportet diligenter ani&shy;<lb/>maduertere.</s>
        </p>
        <p type="main">
          <s id="id002358">Sexta regula &longs;imiliter ex dictis pendet, &amp; e&longs;t, qu&ograve;d gratia exem&shy;<lb/>pli relatum primum comparatum ad primum terminum e&longs;t &longs;exta <lb/>quantitas, cum autem comparatur ad rem, iam pr&aelig;&longs;upponit pro&shy;<lb/>portionem. </s>
          <s id="id002359">Exemplum relatum primum proportionis 21/20 e&longs;t 4084101/3200000 <lb/>&amp; e&longs;t aliquanto maior &longs;exquiquarta, &amp; &longs;i colligas terminos 100. <lb/>105. 110 1/4 115 61/80 121 861/1600 127 19681/32000. Tu uides qu&ograve;d &longs;unt &longs;ex termini in <lb/>utra que computando primum, &longs;ed in 21/20 &longs;unt duo termini, &amp; in qua&shy;<lb/>drato tres, &amp; in quadrato quadrati per pr&aelig;cedentem, adduntur <lb/>duo &amp; ultimus &longs;cilicet &longs;extus fit ex relato ip&longs;o. </s>
          <s id="id002360">Ergo ultra propor&shy;<lb/>tionem &longs;unt tantum quatuor termini.</s>
        </p>
        <p type="main">
          <s id="id002361">Septima regula ad effugiendum omnes errores tu &longs;cis, qu&ograve;d <lb/>4096 quadratum 64 e&longs;t &longs;extus a 64, ad quem habet proportionem <lb/>quadrati, &amp; 64 e&longs;t &longs;imiliter &longs;extus ab uno illo &longs;cilicet non compu&shy;


<pb pagenum="131" xlink:href="015/01/150.jpg"/>tato, &amp; ita 64 habet rationem unius, &amp; licet comparetur ad 2 rem, <lb/>&amp; &longs;it &longs;extus ab eo, eo computato 4096 autem &agrave; 64 &longs;it &longs;eptimus, ta&shy;<lb/>men non e&longs;t eadem ratio, quia 64 non e&longs;t quadratum 2.</s>
        </p>
        <p type="main">
          <s id="id002362">Propo&longs;itio cente&longs;ima trige&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id002363">Rationem numerorum ex progre&longs;sione declarare.</s>
        </p>
        <p type="main">
          <s id="id002364">Micha&euml;l Stifelius rationem pulcherrimam tradidit ad inuentio&shy;<lb/><arrow.to.target n="marg462"/><lb/><arrow.to.target n="marg463"/><lb/>nem numerorum, qui uocantur multiplicandi, &amp; componitur hoc <lb/>modo. </s>
          <s id="id002365">Ex prima componitur 1 &amp; 2, faciunt 3. 1. 2. 3 faciunt 6. 1. 2. 3. 4 <lb/>faciunt 10, &amp; ita prima tabula con&longs;tituit &longs;ecundam recta &longs;erie nu&shy;<lb/>merorum iunctis o&shy;<lb/>mnibus ab uno. </s>
          <s id="id002366">Ter<lb/><figure id="id.015.01.150.1.jpg" xlink:href="015/01/150/1.jpg"/><arrow.to.target n="table17"/><lb/>tia fit ex &longs;ecunda &amp; <lb/>tertia, prim&ograve; a&longs;&longs;umi<lb/>tur 10 in tertia, ut in <lb/>&longs;ecunda, &amp; ex 10 &longs;e&shy;<lb/>cund&aelig;, &amp; 10 terti&aelig; <lb/>fit 20, &amp; ex 15 &longs;ecun&shy;<lb/>d&aelig;, &amp; 20 terti&aelig; fit <lb/>35, &amp; ex 21 &longs;ecund&aelig;, <lb/>&amp; 35 terti&aelig; fit 56, &amp; <lb/>ex 28, &amp; 56 fit 84. Et <lb/>quanta fit ex tertia, <lb/>&amp; ex &longs;e ip&longs;a. </s>
          <s id="id002367">primum <lb/>a&longs;&longs;umendo 35 ex ter<lb/>tia, &amp; ponitur pro <lb/>primo numero quart&aelig;, &amp; ex 35 terti&aelig;, &amp; 35 quart&aelig; fit 70 numerus <lb/>&longs;ecund&aelig; quart&aelig;: &amp; ita ex 56 &amp; 70 fit 126, &amp; ex 84, &amp; 126. 210. &amp; ita <lb/>quinta ex quarta &amp; &longs;e ip&longs;a, &amp; &longs;ic in infinitum.</s>
        </p>
        <p type="margin">
          <s id="id002368"><margin.target id="marg462"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002369"><margin.target id="marg463"/>P<emph type="italics"/>rim&aelig; &longs;u&aelig;<emph.end type="italics"/><lb/>A<emph type="italics"/>rith.<emph.end type="italics"/></s>
        </p>
        <table>
          <table.target id="table17"/>
          <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/>
          </row>
          <row>
            <cell>2</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell>3</cell>
            <cell>3</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell>4</cell>
            <cell>6</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell>5</cell>
            <cell>10</cell>
            <cell>10</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell>6</cell>
            <cell>15</cell>
            <cell>20</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell>7</cell>
            <cell>21</cell>
            <cell>35</cell>
            <cell>35</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell>8</cell>
            <cell>28</cell>
            <cell>56</cell>
            <cell>70</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/>
          </row>
          <row>
            <cell>10</cell>
            <cell>45</cell>
            <cell>120</cell>
            <cell>210</cell>
            <cell>252</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/>
          </row>
          <row>
            <cell>12</cell>
            <cell>66</cell>
            <cell>220</cell>
            <cell>495</cell>
            <cell>792</cell>
            <cell>924</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/>
          </row>
          <row>
            <cell>14</cell>
            <cell>91</cell>
            <cell>364</cell>
            <cell>1001</cell>
            <cell>2002</cell>
            <cell>3003</cell>
            <cell>3432</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 e&longs;t, qu&ograve;d binarius &longs;eruit &lt;02&gt; quadrat&aelig;, &amp; quia nihil <lb/>e&longs;t in eius directo, &longs;olus ip&longs;e &longs;eruiet &lt;02&gt; quadrat&aelig;. </s>
          <s id="id002371">Ternarius autem <lb/>cubic&aelig;, &amp; quia in eius directo e&longs;t alter ternarius, ille etiam &longs;eruiet <lb/>&lt;02&gt; cubic&aelig;. </s>
          <s id="id002372">Quaternarius autem &longs;eruiet quadrato quadrati, &amp; &longs;ena&shy;<lb/>rius, qui e&longs;t in illius directo. </s>
          <s id="id002373">Ergo quinarius &longs;eruiet &lt;02&gt; relat&etail; prim&etail;, <lb/>&amp; duo &longs;equentes numeri &longs;cilicet 10 &amp; 10, &amp; eo dem modo &longs;enarius <lb/>numeri duo &longs;equentes 15 &amp; 20 &longs;eruient cubo quadrati, &amp; ita etiam <lb/>&longs;eptenarius cum tribus &longs;equentibus numeris 21. 35 &amp; 35 &longs;eruient <lb/>rel. </s>
          <s id="id002374">&longs;ecundi radici, &amp; ita deinceps in infinitum.</s>
        </p>
        <p type="main">
          <s id="id002375">Propo&longs;itio cente&longs;ima trige&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id002376">Modos u&longs;us horum numerorum declarare.</s>
        </p>
        <p type="main">
          <s id="id002377">In quouis numero denominationis oportet tot addere o, quo&shy;<lb/><arrow.to.target n="marg464"/>


<pb pagenum="132" xlink:href="015/01/151.jpg"/>tus e&longs;t ordo, &amp; facere tot numeros &longs;equentes; quotus e&longs;t ordo, &amp; <lb/>&longs;emper minuere unam o, uelut quia quadrata &lt;02&gt; e&longs;t prima ad 2 ad&shy;<lb/>demus o, &amp; fiet 20, nec alium qu&etail;remus numerum. </s>
          <s id="id002378">Sed quia cubi&shy;<lb/>ca e&longs;t &longs;ecundo loco, habebit prima nota 00, &amp; fiet 300, &amp; &longs;ecundum <lb/>3 unam 0, &amp; fiet 30, &amp; in quadrato quadrati addemus 000 primo, <lb/>&amp; 00 &longs;ecundo, &amp; o tertio, &amp; ita habebimus 4000. 600. 40. &longs;ed quia <lb/>in tabula non e&longs;t 4 ultimum, addemus &longs;imilem primo &longs;emper. </s>
          <s id="id002379">In <lb/>relato primo, ergo habebimus 50000. 1000. 1000. 50. &amp; in cubo <lb/>quadrati 600000. 150000. 20000. 1500. 60. Manife&longs;tum e&longs;t, qu&ograve;d <lb/>his uice uer&longs;a a&longs;&longs;ump&longs;imus 15 &amp; 6 &longs;imiles prioribus addendo &longs;em&shy;<lb/>per ut dixi o minus, donec ad unam peruenerit. </s>
          <s id="id002380">Et ita in relato &longs;e&shy;<lb/>cundo 7000000. 2100000. 350000. 35000. 2100. 70. &amp; ita deinceps.</s>
        </p>
        <p type="margin">
          <s id="id002381"><margin.target id="marg464"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002382">Propo&longs;itio cente&longs;ima trige&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id002383">Radices omnes &agrave; propo&longs;itis numeris extrahere.<lb/><arrow.to.target n="marg465"/></s>
        </p>
        <p type="margin">
          <s id="id002384"><margin.target id="marg465"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002385">Propo&longs;itis quibu&longs;uis numeris utpot&egrave; 916132832, uolo detrahere <lb/>&lt;02&gt; relatam primam, primum habebo in tabula de&longs;cripta relata pri&shy;<lb/>ma numerorum &longs;implicium u&longs;que ad 10 uelut in exemplo. </s>
          <s id="id002386">Dein de <lb/><figure id="id.015.01.151.1.jpg" xlink:href="015/01/151/1.jpg"/><lb/>&longs;ub&longs;cribam pun&shy;<lb/>ctum &longs;ub prima <lb/>nota &agrave; dextra, &amp; <lb/>quia e&longs;t quarta in <lb/><figure id="id.015.01.151.2.jpg" xlink:href="015/01/151/2.jpg"/>ordine hoc, &longs;eu quinta denominatio &longs;ecun&shy;<lb/>dum no&longs;trum, omittam quatuor notas in&shy;<lb/>ter medias, &amp; &longs;ub&longs;cribam punctum aliud, <lb/>&amp; ita facerem &longs;i e&longs;&longs;ent plures qu&agrave;m decem <lb/>not&aelig;: relinquitur ergo ad <expan abbr="p&utilde;ctum">punctum</expan> primum <lb/>&agrave; &longs;ini&longs;tra 9161, cuius qu&etail;ro &lt;02&gt; relatam pri&shy;<lb/>mam in tabula, quam inuenio e&longs;&longs;e 6, nam <lb/><figure id="id.015.01.151.3.jpg" xlink:href="015/01/151/3.jpg"/>7776 eius relatum primum e&longs;t <lb/>proximius ex minoribus ad 9161, <lb/>detraho igitur 7776, ex numero <lb/>propo&longs;itio relinquitur. </s>
          <s id="id002387">Dein de<lb/>p&oacute;no 6 &amp; quadratum eius, &amp; cub. </s>
          <s id="id002388">&amp; quadratum <lb/>quadrati, quia, ut dixi, e&longs;t quarta denominatio a&shy;<lb/><figure id="id.015.01.151.4.jpg" xlink:href="015/01/151/4.jpg"/>pud illum, &amp; &egrave; regione numeros pr&aelig;cedentes in&shy;<lb/>uentos relati primi ex pr&aelig;cedenti propo&longs;itione: &amp; duco &longs;ingulos <lb/>cum &longs;uis collateralibus, ut uides etiam in figura, et cum ultimo pro&shy;<lb/>ducto, &longs;cilicet 64800000 diuido 138532832 exit 2, huius accipio o&shy;<lb/>mnes numeros ad relatum primum u&longs;que ut uides, &amp; pono minores <lb/>&egrave; regione maiorum, utpot&egrave; 2 &egrave; regione 1296 &amp; 50000, &amp; 4 &egrave; regio&shy;


<pb pagenum="133" xlink:href="015/01/152.jpg"/>ne 216 &amp; 10000, &amp; 8 &egrave; regione 36 &amp; 10000, &amp; 16 &egrave; regione 6, &amp; 50, <lb/>&amp; duco 6 in 50 fit 300, duco in 16 fit 4800, duco 36 in 1000 fit <lb/>36000, duco 36 in 8 fit 288000, duco etiam 216 in 10000 &amp; fit <lb/>2160000, &amp; duco hos per 4 fit 86400000, duco rur&longs;us 1296 in <lb/>50000 fit 64800000, duco in 2 fit 129600000. Demum addo 32 re&shy;<lb/>latum primum 2, &amp; fit &longs;umma omnium 138532832, &amp; ita habemus <lb/>radicem relatam primam dicti numeri e&longs;&longs;e 62. Et &longs;i numerus produ<lb/>ctus fui&longs;&longs;et maior oportui&longs;&longs;et accipere proximo minorem. </s>
          <s id="id002389">Inde per <lb/>regulam &longs;equentem addere minutias.</s>
        </p>
        <p type="main">
          <s id="id002390">Propo&longs;itio cente&longs;ima quadrage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id002391">Radices per numeros fractos determinare.</s>
        </p>
        <p type="main">
          <s id="id002392">Duplex e&longs;t modus, ut etiam docui in arithmeticis, &longs;cilicet ut pro </s>
        </p>
        <p type="main">
          <s id="id002393"><arrow.to.target n="marg466"/><lb/>radice quadrata addatur duo o, &amp; pro cuba tria, &amp; pro quadrata <lb/>quadrata quatuor, &amp; pro relata prima quinque, &amp; ita deinceps, &amp; <lb/>pr&etail; decimis &longs;emel, pro cente&longs;imis bis, pro mille&longs;imis ter, pro millia&shy;<lb/>ribus &longs;eu partibus earum quater, pro cente&longs;imis mille&longs;imis quin&shy;<lb/>quies, pro mille&longs;imis mille&longs;imarum &longs;exies, &amp; ita deinceps deinde <lb/>per pr&aelig;cedentem detrahere radicem, &amp; erit ualde exacta. </s>
          <s id="id002394">Exemplo <lb/>non utar, ni&longs;i qu&ograve;d &longs;i uelles radicem relatam 16 ad mille&longs;imas, acci&shy;<lb/>cipies radicem relatam numeri &agrave; latere propo&longs;iti, &amp; ita de alijs <lb/>1600000, 00000, 00000, &amp; &longs;i uelles &lt;02&gt; cub. </s>
          <s id="id002395">5 1/5 per mille&longs;imas, pri<lb/>mo addes ter 000, &amp; fiet 3000000000, inde &longs;ume 1/5 1000000000, <lb/>qui e&longs;t 200000000, &amp; adde ad 5000000000, fit 2500000000, <lb/>&amp; hoc quia unum refert numerum 1000000000 ex &longs;uppo&longs;ito &amp; 1/5 <lb/>e&longs;t 1/5 unius.</s>
        </p>
        <p type="margin">
          <s id="id002396"><margin.target id="marg466"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002397">Secundus modus e&longs;t, ut accipias proxim&egrave; maiorem, &amp; multipli&shy;<lb/>ca in &longs;e, &amp; detrahe numerum propo&longs;itum, &amp; re&longs;iduum diuide per <lb/>duplum radicis primo inuent&aelig;, &longs;i fuerit quadrata, &amp; per triplum <lb/>quadrati eiu&longs;dem &longs;i fuerit cubica, &amp; per quadruplum cubi, &longs;i fuerit <lb/>quadrata quadrata, &amp; per quincuplum quadrati quadrati, &amp; quod <lb/>exit detrahes ex priore radice, &amp; rur&longs;us quod relinquitur, multipli&shy;<lb/>ca in &longs;e, &amp; eodem modo agendo quod &longs;upere&longs;t &agrave; numero propo&longs;i&shy;<lb/>to, diuide per duplum radicis prioris, &longs;i &longs;it radix quadrata, uel per <lb/>triplum quadrati &longs;i &longs;it cubica, &amp; quod exit rur&longs;us detrahe, &amp; ita a&shy;<lb/>gendo, peruenies ad exacti&longs;simam radicem, exemplum uolo radi&shy;<lb/>cem quadratam 5 proxima maior e&longs;t 3, quadratum 9, differentia 4, <lb/>diuide per 6 duplum 3 exit 2/3, detrahe ex 3 fit 2 1/3, quadratum e&longs;t 49/9 <lb/>quod e&longs;t 5 4/9, rur&longs;us diuido 4/9 differentiam 5 4/9 &amp; 5 per 4 2/3 duplum <lb/>radicis prim&aelig; exit 2/21, detrahe ex 2 1/3, relinquitur 2 5/21, radix &longs;atis pro&shy;<lb/>pinqua, nam eius quadratum e&longs;t 5 4/441, in cubica &longs;imiliter uolo &lt;02&gt;<lb/>cu. </s>
          <s id="id002398">5, proxima maior e&longs;t 2, cubus 8, differentia 3, diuide per triplum 


<pb pagenum="134" xlink:href="015/01/153.jpg"/>quadrati 2 quod e&longs;t 12 exit 1/4 detrahe ex 2 fit 1 3/4 cuius cubus e&longs;t 5 23/64 <lb/>differentia e&longs;t 23/64 diuide per triplum quadrati 1 3/4 qu&ograve;d e&longs;t 9 3/16 exit <lb/>23/588 detrahe ex 1 3/4 <expan abbr="relinqu&utilde;tur">relinquuntur</expan> 1 107/147 cuius cubus e&longs;t 5 504449/3176523 Ita diuides <lb/>hunc exce&longs;&longs;um &longs;i placet per triplum quadrati 1 107/147 &amp; e&longs;t ferm&egrave; 9 exit <lb/>56050/3176523 qua&longs;i detrahe ex 1 107/147 relinquuntur 323159/453789.</s>
        </p>
        <p type="main">
          <s id="id002399">Tertius modus e&longs;t &longs;ubtilior, tu &longs;cis, &qring;d duo decima denominatio <lb/>e&longs;t quadrata &longs;ext&etail;, &amp; quadrata quad, terti&aelig;, &amp; cuba quarti, quarta <lb/>autem e&longs;t inter <expan abbr="terti&atilde;">tertiam</expan> &amp; &longs;extam &longs;ecunda quantitas in continua pro&shy;<lb/>portione: ergo inuenta &lt;02&gt; numeri propo&longs;iti &amp; &lt;02&gt; radicis inuent&aelig; <lb/><expan abbr="reduc&atilde;">reducam</expan> ad unam denominationem, et inter numeratores collo cabo <lb/>duas quantitates, quod facile erit &longs;en&longs;im procedendo, &amp; habebo &lt;02&gt;<lb/>cu. </s>
          <s id="id002400">qu&aelig;&longs;itam, &longs;cilicet minorem ex duabus intermedijs. </s>
          <s id="id002401">Et &longs;imiliter <lb/>pro relata prima, capiam &longs;exaginta denominationes, &amp; &longs;cis, qu&ograve;d <lb/>quinta decima e&longs;t &lt;02&gt; &lt;02&gt; &longs;exage&longs;im&etail;, &amp; decima e&longs;t &lt;02&gt; cu. </s>
          <s id="id002402">&lt;02&gt; &longs;exage&longs;im&etail;, <lb/>&amp; duodecima &lt;02&gt; relata prima &longs;exage&longs;im&aelig; per eandem inuenta, er&shy;<lb/>go &lt;02&gt; numeri propo&longs;iti tanquam ille &longs;it &longs;exage&longs;ima denominatio, <lb/>inueniam illius radicis inuent&aelig; &lt;02&gt; quadratam, &amp; cubicam, &amp; <lb/>quia duodecima quantitas qu&aelig; e&longs;t &lt;02&gt; relata prima numeri e&longs;t <lb/>&longs;ecunda, quatuor intermediarum inter ponam inter &lt;02&gt; quadra&shy;<lb/>tum, quadratum, &amp; cubicam quadratam quatuor numeros in <lb/>continua proportione, &amp; &longs;ecundus ex minoribus erit &lt;02&gt; relata <lb/>prima numeri propo&longs;iti. </s>
          <s id="id002403">Exemplum cubic&aelig; uolo &lt;02&gt; cu: 5 habui &lt;02&gt;<lb/>quadratam eius 2 5/21 &longs;ed uolo proximiorem diuidendo 4/441 per 4, <lb/>quod e&longs;t ferm&egrave; duplum 2 5/21 exit 1/441 detraho ex 2 5/21 relinquitur ualde <lb/>proxima &lt;02&gt; 5. 2 104/441 huius igitur radix quadrata, primo inuenta e&longs;t 1 1/2 <lb/>&longs;ecunda proximior e&longs;t 1 41/84 reduco ad eandem denominationem fi&shy;<lb/>ent 284/9261 2 416/1764 &amp; 1 861/1764 inter 3944, &amp; 2625, inueniemus duos nume&shy;<lb/>ros in continua proportione, ut uides, &amp; erit &longs;ecunda quantitas <lb/><figure id="id.015.01.153.1.jpg" xlink:href="015/01/153/1.jpg"/><lb/>3006/7641, quod e&longs;t 167/98 proximum ad 1 5/7, &lt;02&gt; cubica. </s>
          <s id="id002404">5. <lb/><expan abbr="n&atilde;">nam</expan> eius cubus e&longs;t 5. 13/343 at exacti&longs;sima e&longs;t ergo 1 69/98. <lb/>ut liquet. </s>
          <s id="id002405">Pro relata prima ergo ponamus, ut ue&shy;<lb/>lim &lt;02&gt; relatam <expan abbr="prim&atilde;">primam</expan> 25, accipio 5 &lt;02&gt; 25 cuius &lt;02&gt; e&longs;t, ut ui&longs;um e&longs;t, 2 104/441 <lb/>&longs;imiliter &lt;02&gt; cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb/>&amp; inueniam quatuor numeros in <expan abbr="c&otilde;tinua">continua</expan> proportione inter illos, <lb/>&amp; &longs;ecundus po&longs;t minimum ex illis erit &lt;02&gt; relata prima propinqui&longs;&shy;<lb/>&longs;ima 25. Quomodo uer&ograve; inueniantur facillim&egrave; illi termini, do&shy;<lb/>cui in &longs;exto libro operis perfecti.</s>
        </p>
        <p type="main">
          <s id="id002406">Quarta regula e&longs;t utilior, licet minus uideatur nobilis, &amp; e&longs;t fun&shy;<lb/>data in hoc, quod &longs;i a b &longs;it maior c &amp; eis ad dantur b e, &amp; d f &aelig;qua&shy;<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, &amp; ex <lb/>con&longs;equenti per <expan abbr="ui&atilde;">uiam</expan> fracti maior pars unius erit c f ip&longs;ius a e, qu&agrave;m <pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>
          <s id="id002407">Dico ergo quod maior e&longs;t proportio a b <lb/><figure id="id.015.01.154.1.jpg" xlink:href="015/01/154/1.jpg"/><lb/>ad c d, qu&agrave;m a e ad e f, fiat d g ad quam &longs;it b c ut <lb/><arrow.to.target n="marg467"/><lb/>a b ad c d, eritque a e ad c g ut a b ad c d, minor au&shy;<lb/>tem e&longs;t a e ad c f, quam ad c g, igitur minor a e ad <lb/>c f qu&agrave;m a b ad c d quod fuit propo&longs;itum. </s>
          <s id="id002408">Simili <lb/>ter &longs;i fuerint du&aelig; quantitates, a b &amp; c d, quarum a b &longs;it maiore, c d <lb/>autem eadem e minor, dico, qu&ograve;d dimidium aggregati a b &amp; c d <lb/>maiorem habebit proportionem ad e, qu&agrave;m c d &amp; minor, nam iun&shy;<lb/>cta b f &aelig;quali d e ad a b, ita ut f g &longs;it dimidium totius a f, q&ugrave;ia ergo <lb/><figure id="id.015.01.154.2.jpg" xlink:href="015/01/154/2.jpg"/><lb/>f g e&longs;t dimidium f a &amp; fb e&longs;t minor dimidio <lb/><arrow.to.target n="marg468"/><lb/>f a cum &longs;it minor b a, &amp; &longs;imiliter f g e&longs;t mi&shy;<lb/>nor a b, quia a b e&longs;t maior dimidio a f, quia <lb/>e&longs;t maior b f, ergo proportio g f ad c e&longs;t ma<lb/>ior quam b f ad e, ita quam c d ad e, &amp; mi&shy;<lb/><arrow.to.target n="marg469"/><lb/>nor qu&agrave;m a b ad e, quod fuit propo&longs;itum. </s>
          <s id="id002409">Quo ui&longs;o uolo &lt;02&gt; 1000 <lb/>quadratam, &amp; qu&ograve;d de quadrata dico, dico etiam de alijs radici&shy;<lb/>bus &amp; erit ex &longs;ecunda regula harum 31 39/62 &amp; quadratum erit 1000 <lb/>1521/3844. Iuxta ergo primam partem regul&aelig; 31 38/61 erit minus, &amp; in ueritate <lb/>in eo, quod fit ducendo, ut uides, &amp; hoc e&longs;t pro&shy;<lb/><figure id="id.015.01.154.3.jpg" xlink:href="015/01/154/3.jpg"/><lb/>ximum ad 11/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t ferm&egrave; 63 1/4 in 1/160 fient 63/160 detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24   /40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 &longs;unt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, &amp; erit &lt;02&gt; exa&shy;<lb/>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita<lb/>tem, &amp; propinquitatem in producto differentia e&longs;t 1/100 aut parum <lb/>maius quod ad radicem comparatum cum debeat diuidi per du&shy;<lb/>plum eius erit paulo maius 1/6300. Vnde facilior e&longs;t, &amp; breuior h&aelig;c <lb/>uia qu&agrave;m per 00 additus. </s>
          <s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimere</expan> &amp; cum pro<lb/>pinquitate ita facio. </s>
          <s id="id002411">Con&longs;idero qu&ograve;d 31 38/61 e&longs;t maius 1/6300 radice, di&shy;<lb/>uido 6300 per 62 exit 103 ferm&egrave;, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 &amp; habeo 3914/6283 hic denominator e&longs;t proxi&shy;<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam &lt;02&gt; 1000, <lb/>31 3913/6283 cuius quadratum e&longs;t 1000 minus 1/1048 hoc ut dixi diui&longs;um <lb/>per duplum &lt;02&gt; quod e&longs;t 63 e&longs;t omnino in&longs;en&longs;ile in radice.</s>
        </p>
        <p type="margin">
          <s id="id002412"><margin.target id="marg467"/>8. P<emph type="italics"/>ropo&longs;. <lb/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002413"><margin.target id="marg468"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem. <lb/><expan abbr="amplificat&atilde;">amplificatam</expan>.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002414"><margin.target id="marg469"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002415">Quinta regula e&longs;t omnium pulcherrima, &amp; e&longs;t communis omni <lb/>bus &amp; fractis &amp; integris &amp; omnibus generibus radicum, &amp; &longs;it ex&shy;<lb/>emplum, uolo &lt;02&gt; radicis &longs;upra&longs;cript&aelig; &longs;cilicet 31 3913/6283 multiplico 31 <lb/>in 6283, &amp; fit 194793, cui addo 3913, fit 198686 manife&longs;tum e&longs;t igi&shy;<lb/>tur, quod 198686/6283 &aelig;quiualet 31 3913/6283 hoc facto, quod e&longs;t commune om&shy;


<pb pagenum="136" xlink:href="015/01/155.jpg"/>nibus radicibus extrahendis pro radice quadrata, multiplicabo n&ugrave;<lb/>meratorem, qui e&longs;t 194686 per denominatorem, qui e&longs;t 6283, &amp; &longs;i <lb/>uoluero radicem cubicam, multiplicabo eundem numeratorem <lb/>per quadratum denominatoris, &amp; &longs;i uoluero radicem radicis, mul&shy;<lb/>tiplicabo per cubum, multiplicabo per quadratum quadratum <lb/>6283, &amp; ita de alijs una diminutione minore, &amp; eius qui prouenit <lb/>numeri &lt;02&gt; &longs;upra po&longs;ita denominatori erit &lt;02&gt; eiu&longs;modi, quam &longs;u&longs;ce&shy;<lb/>pi&longs;ti, uelut in exemplo fuit numerus 198686/6283 quia ergo uolo &lt;02&gt; quad. <lb/></s>
          <s id="id002416">multiplico 198686 in 6283, &amp; fit 1248344138, huius accipio &lt;02&gt;<lb/>quad. </s>
          <s id="id002417">qu&aelig; e&longs;t 35332, h&aelig;c autem e&longs;t diuidenda per 6283, &amp; exeunt <lb/>5 3917/12566, ecce uides radicem exactam admodum, &amp; facilem. </s>
          <s id="id002418">Volo rur&shy;<lb/>&longs;us &lt;02&gt; quadrat. </s>
          <s id="id002419">5 3917/12566, multiplico 12566 per 5 &amp; fit 62830, cui addo <lb/>3917, &amp; fit 66747, cui &longs;uppono 12566 denominatorem, fient ergo <lb/>66747/12566, manife&longs;tum e&longs;t igitur qu&ograve;d hoc &aelig;quiualet 5 3917/12566, &longs;i igitur mul<lb/>tiplicarem denominatorem per denominatorem &amp; numeratorem, <lb/>quod proueniret, e&longs;&longs;et &aelig;quale eidem numero, ergo &lt;02&gt; eius e&longs;&longs;et ea&shy;<lb/>dem cum &lt;02&gt; prioris, &longs;ed &lt;02&gt; denominatoris e&longs;&longs;et prior numerus, er&shy;<lb/>go &longs;ufficiet extrahere &lt;02&gt; producti ex denominatore in numerato&shy;<lb/>rem, &amp; ita productum erit ex denominatore in numeratorem <lb/>838742802, cuius &lt;02&gt; e&longs;t 28961, h&aelig;c igitur diui&longs;a per 12566 o&longs;ten&shy;<lb/>dit &lt;02&gt; 2 3892/12566. In hac autem quadrata e&longs;t alius modus &longs;ine multiplica&shy;<lb/>tione, &longs;ed non e&longs;t communis alijs, ubi &longs;tatueris denominatorem <lb/>pro denominatore &lt;02&gt;, utpote 12566, &amp; numeratorem 66747, con&shy;<lb/>&longs;titues medium &longs;en&longs;im augendo.</s>
        </p>
        <p type="main">
          <s id="id002420">Rur&longs;us uolo &lt;02&gt; relatam 2 3829/12566 reduco ad denominatorem, &amp; fit <lb/>ut prius 28961/12566, duco igitur 12566 ad quad. </s>
          <s id="id002421">quad. </s>
          <s id="id002422">&longs;ed &longs;ufficiet in hoc <lb/>ca&longs;u deducere ad minores denominationes, utpot&egrave; diuide 28961 <lb/>per 12566 exit 2 3829/12566 multiplico per 566 fit 1104 5862/12566, hoc detrahe <lb/>ex 28961 habebis 27856/12000, diuide igitur per 1000 habebis 12 &amp; 27 107/125 <lb/>at 108/126 &longs;unt 6/7, igitur habes 12 pro denominatore, &amp; 27 6/7 pro nume&shy;<lb/>ratore, quare erunt numeri 195/84, erit ergo per hanc regulam, ut ducas <lb/>84 ad quad. </s>
          <s id="id002423">quadrati, &amp; fit 49787136, duc in 195 fit 9708491520, <lb/>cuius &lt;02&gt; relata prima e&longs;t 99, igitur &lt;02&gt; relata prima 2 3829/12566 e&longs;t 1 15/84 pau&shy;<lb/>lo maior, id e&longs;t 1 13/70. Et nota quod &longs;i denominator haberet &lt;02&gt; illius <lb/>generis, quam qu&aelig;ris, &longs;ufficeret inuenire radicem eiu&longs;dem generis <lb/>ab&longs;que alia numerorum multiplicatione.</s>
        </p>
        <p type="main">
          <s id="id002424">Propo&longs;itio cente&longs;ima quadrage&longs;ima prima. &lpar;deducere.</s>
        </p>
        <p type="main">
          <s id="id002425">Numeros fractos ad minores in <expan abbr="ead&etilde;">eadem</expan> proportione ualde propinqua</s>
        </p>
        <p type="main">
          <s id="id002426">Cum plerunque numeri fracti habeantur per radices, ut aliquan&shy;<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino&shy;<lb/>res numeros, ut melius intelligi po&longs;sint &amp; facilius tractari, &amp; 


<pb pagenum="137" xlink:href="015/01/156.jpg"/>cum hoc &longs;it exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb/>uolo certa ratione ad minores diui&longs;iones deducere. </s>
          <s id="id002427">Deduco pri&shy;<lb/>m&ograve; totum ad fractiones ducendo 2 in 12566, &amp; addendo 3829, &amp; <lb/>fit 26961/12566, multiplico 12566 per 9, quia proportio unius ad alterum <lb/>e&longs;t ferm&egrave;, ut 9 ad 4, &amp; fit 113094, multiplico 4 in 28961 fit 115844, <lb/>hoc igitur e&longs;t maius, igitur proportio 28961 ad 12566 e&longs;t maior <lb/>qu&agrave;m 9 ad 4, detraho igitur 12566 ex 28961, relinquitur 16395, de&shy;<lb/>traho 113094 ex 115844, relinquitur 2750, diuido 2750 per 16395 <lb/>exit 55/328 addo 2 denominatori fit 55/330, quod e&longs;t 1/6, nam i&longs;t&aelig; additiones <lb/>paru&aelig; pr&aelig;ter qu&ograve;d parum uariant quantitatem etiam dum ad ex&shy;<lb/>amen reducuntur, nihil impediunt, detrahe igitur 1/6 &agrave; 9/4, &amp; ducendo <lb/>per 6, &amp; detrahendo 53/23, duco igitur primos numeros &longs;cilicet 28961/12566 <lb/>mutuo in 53/23, fiunt 665998, &amp; 666107, ita uides, quod proportio <lb/>53 ad 23 e&longs;t paulo minor, qu&agrave;m 28961 ad 12566, &amp; &aelig;quiualent 27/23<lb/>&amp; 2 3829/12566.</s>
        </p>
        <p type="margin">
          <s id="id002428"><margin.target id="marg470"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002429">Propo&longs;itio cente&longs;ima quadrage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id002430">Denominationum incrementa ex extrema cognita inuenire, &amp; <lb/>conuer&longs;o modo.</s>
        </p>
        <p type="main">
          <s id="id002431"><expan abbr="Quid&atilde;">Quidam</expan> per u&longs;uram <expan abbr="rediuiu&atilde;">rediuiuam</expan> fecit 40000 coronatos ex 40 in 40 <lb/><arrow.to.target n="marg471"/><lb/>annis. </s>
          <s id="id002432">Qu&etail;ro <expan abbr="qut&atilde;a">qutana</expan> fuerit u&longs;ura, &amp; <expan abbr="qu&atilde;do">quando</expan> habuit 1000 coronatos, <lb/><expan abbr="quid&atilde;">quidam</expan> uellent &longs;oluere per regulam trium quantitatum, in qua com&shy;<lb/>mitterentur maximi errores. </s>
          <s id="id002433">Et in ea multi &longs;unt modi, &amp; omnes fal&shy;<lb/>&longs;i pr&aelig;ter hanc uiam nulla e&longs;t uera, adde qu&ograve;d uellent multi per &longs;or&shy;<lb/>tem inuentam &longs;oluere augendo per &longs;ingulos annos, quod ade&ograve; <lb/>difficile e&longs;&longs;et, &amp; pen&egrave; foret impo&longs;sibile. </s>
          <s id="id002434">Ide&ograve; diuides 40000 per 40 <lb/>numerum &longs;ortis exit 1000, igitur in 40 annis unum fit mille, &longs;unt <lb/>ergo 40 denominationes ab uno, quarum quadrage&longs;ima e&longs;t 1000, <lb/>igitur uige&longs;ima e&longs;t &lt;02&gt; 1000 &verbar;&longs;cilicet &verbar;31 3913/6283, igitur decima e&longs;t &lt;02&gt; eius <lb/><arrow.to.target n="marg472"/><lb/>5 3917/12566 huius radix, erit quinta quantitas 2 7/23, cuius &lt;02&gt; relata prima, <lb/><arrow.to.target n="table18"/><lb/>erit proportio 1 13/70, cuius quadratum e&longs;t 1 1889/4900 &longs;eu <lb/>1 67/165 pro &longs;ecunda quantitate, duces ergo primam, <lb/><figure id="id.015.01.156.1.jpg" xlink:href="015/01/156/1.jpg"/>qu&aelig; e&longs;t 83/70 in quintam, qu&aelig; e&longs;t reducta ad mino&shy;<lb/>res fractiones facilitatis cau&longs;a 53/23, &amp; habebis &longs;ex&shy;<lb/>tam quantitatem 2 118/161, duco etiam quintam quan&shy;<lb/>titatem &longs;cilicet 53/23 in &longs;ecundam qu&aelig; e&longs;t 232/165, &amp; fit &longs;e&shy;<lb/>ptimi anni quantitas, duco igitur &longs;eptem anno&shy;<lb/>rum numerum, qui e&longs;t 3 14/61 in 31 38/61 fit 102 992/6283. At in <lb/>&longs;ex annis additis ad uiginti, fit tanto minus, quan&shy;<lb/>to 31 38/61 ductum in differentiam &longs;eptem, &amp; &longs;ex an&shy;<lb/>norum qu&aelig; e&longs;t 60/121, fit ergo 15 35/492. Quia ergo an&shy;


<pb pagenum="138" xlink:href="015/01/157.jpg"/>nuatim &longs;olum u&longs;ura adijcitur &longs;orti, &longs;ufficiet diuidere 2 992/6283 per 15 35/492 <lb/>&longs;cilicet multiplicando per 12 numerum men&longs;ium 2 992/6283 fit 25 5621/6283 di&shy;<lb/>uide 25 5621/6283 per 15 35/492, exit men&longs;is unus, &amp; dies 21, detrahe ex 27 an&shy;<lb/>nis, remanent anni 26, men&longs;es 10, dies 9, in quo tempore habuit <lb/>4000 aureos coronatos. </s>
          <s id="id002435">V&longs;ura autem fuit ut ui&longs;um 13/70, igitur per re&shy;<lb/>gulam trium duc 13 in 100 fit 1300, diuide 1300 per 70 exit 18 4/7 &amp; <lb/>tanta fuit pro centum. </s>
          <s id="id002436">Et cum computaueris in tribus annis, acqui&shy;<lb/>rit modico plus be&longs;&longs;e eius, quod habet. </s>
          <s id="id002437">Et ita in 13 annis, &amp; parua <lb/>illa parte perueniet ad decuplum eius, quod habet, &longs;cilicet 4000 au <lb/>reorum, &amp; habebit aureos 40000, ut propo&longs;itum e&longs;t.</s>
        </p>
        <p type="margin">
          <s id="id002438"><margin.target id="marg471"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002439"><margin.target id="marg472"/>P<emph type="italics"/>er<emph.end type="italics"/> 136. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <table>
          <table.target id="table18"/>
          <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 propo&longs;ita proportione numero que terminorum rediuiuam u&shy;<lb/>&longs;uram inuenire.</s>
        </p>
        <p type="main">
          <s id="id002442">Sit gratia exempli, in &longs;ex annis u&longs;ura rediuiua uige&longs;im&aelig;, erit&shy;<lb/>q&uacute;e proportio 21/20, cuius numeratorem &longs;exies ducam in &longs;e primum <lb/>bis fit 441: ergo ducto 441 in &longs;e fit q&uacute;e 194481 ductum in 441 <lb/>fit 85766121 &longs;exies ductum 21, quinquies autem ducam 20 deno&shy;<lb/><figure id="id.015.01.157.1.jpg" xlink:href="015/01/157/1.jpg"/><lb/>minatorem in &longs;e fit bis 400, ter 8000, <lb/>quinquies ergo 3200000, diuide nume&shy;<lb/>ratorem per denominatorem abiectis <lb/>quinque notis erit 26 2566121/3200000. Qu&aelig; propor<lb/>tio e&longs;t proxima 26 4/5 ad 20, &amp; ita ut 134 ad <lb/>100. Et &longs;i pigeret t&aelig;dij aut laboris po&longs;&longs;es <lb/>pro xij annis, ducere 134 in &longs;e, &amp; fit 17956 <lb/>diuide per 100 eadem ratione, exit 179 14/25 <lb/>&amp; ita 100 in xij annis, fit tantundem. </s>
          <s id="id002443">Et <lb/>ita pro xviij &amp; xx annis.</s>
        </p>
        <p type="main">
          <s id="id002444">Propo&longs;itio cente&longs;ima quadrage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id002445">Si linea in duas partes diuidatur, corpora, qu&aelig; fiunt ex una par&shy;<lb/>te in alterius quadratum mutu&ograve; &aelig;qualia &longs;unt corpori, quod fit ex <lb/>tota linea in &longs;uperficiem unius partis in alteram.<lb/><arrow.to.target n="marg473"/></s>
        </p>
        <p type="margin">
          <s id="id002446"><margin.target id="marg473"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002447">Sit a c diui&longs;a in a b, b c quadratum a b &longs;it <lb/><figure id="id.015.01.157.2.jpg" xlink:href="015/01/157/2.jpg"/><lb/>a d, <expan abbr="quadrat&utilde;">quadratum</expan> b c, &longs;it b e <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>ex a b in b e, a f dico qu&ograve;d corpora ex a b in <lb/>b e, &amp; b c in a d &aelig;qualia &longs;unt corpori ex a c <lb/>in a f. </s>
          <s id="id002448">Quia enim corpus ex a c in a f con&longs;tat <lb/>ex a b in a f, &amp; b c in a f, per primam &longs;ecun&shy;</s>
        </p>
        <p type="main">
          <s id="id002449"><arrow.to.target n="marg474"/><lb/>di Elementorum. </s>
          <s id="id002450">corpus autem ex a b in a f <lb/>e&longs;t &aelig;quale corpori ex b c in a d, &amp; corpus <lb/>ex b c in a f e&longs;t &aelig;quale corpori ex a b in b c <lb/>igitur con&longs;tat propo&longs;itum.</s>
        </p>
        <pb pagenum="140 &lsqb;=139&rsqb;" xlink:href="015/01/158.jpg"/>
        <p type="margin">
          <s id="id002451"><margin.target id="marg474"/>I<emph type="italics"/>d e&longs;t per <lb/>eius demon&shy;<lb/>&longs;trationem.<emph.end type="italics"/><lb/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>un <lb/>decimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002452">Propo&longs;itio cente&longs;ima quadrage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id002453">Duplum cubi medietatis maius e&longs;t aggregato corporum mutu&shy;<lb/>orum cuiuslibet diui&longs;ionis, quantum e&longs;t, quod fit ex tota in quadra <lb/>tum differenti&aelig;.<lb/><arrow.to.target n="marg475"/></s>
        </p>
        <p type="margin">
          <s id="id002454"><margin.target id="marg475"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002455">Sit a b diui&longs;a per &aelig;qualia in c, &amp; per in&aelig;qua&shy;<lb/>lia in d, dico, qu&ograve;d duplum cubi a c e&longs;t maius ag<lb/><figure id="id.015.01.158.1.jpg" xlink:href="015/01/158/1.jpg"/><lb/>gregato corporum ex a d in quadratum b d, &amp; b d in quadratum <lb/>a cin eo quod fit ex a b in quadratum c d, nam per <expan abbr="pr&aelig;cedent&etilde;">pr&aelig;cedentem</expan> du&shy;<lb/>plum cubi a c e&longs;t &aelig;quale corpori ex a b in quadratum a c: aggrega&shy;<lb/>tum quo que corporum ex a d in quadratum b d, &amp; b d in quadra&shy;<lb/>tum a d e&longs;t &etail;quale ei, quod fit ex a b in <expan abbr="rectangul&utilde;">rectangulum</expan> ex a d in d b. </s>
          <s id="id002456"><expan abbr="qua&shy;drat&utilde;">qua&shy;<lb/>dratum</expan> <expan abbr="aut&etilde;">autem</expan> a c e&longs;t maius rectangulo a d in d b quadrato c d differen<lb/>ti&aelig;, igitur duplum cubi a c excedit aggregatum <expan abbr="corpor&utilde;">corporum</expan> <expan abbr="mutuor&utilde;">mutuorum</expan> <lb/>in corpore ex a b in quadratum c d differenti&etail;, quod e&longs;t <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s>
        </p>
        <p type="main">
          <s id="id002457"><arrow.to.target n="marg476"/></s>
        </p>
        <p type="margin">
          <s id="id002458"><margin.target id="marg476"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002459">Propo&longs;itio cente&longs;ima quadrage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id002460">Si line a in duas partes diuidatur quadrata ambarum partium <lb/>detracto eo quod fit ex una parte in alteram, &etail;qualia &longs;unt producto <lb/>unius in alteram cum quadrato differenti&aelig;.</s>
        </p>
        <p type="main">
          <s id="id002461"><arrow.to.target n="marg477"/></s>
        </p>
        <p type="margin">
          <s id="id002462"><margin.target id="marg477"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002463">Sit linea a c diui&longs;a in b, &amp; &longs;it differentia a b, <lb/>b c, b d, dico quod quadrata a b &amp; b c detracto <lb/><figure id="id.015.01.158.2.jpg" xlink:href="015/01/158/2.jpg"/><lb/>eo quod fit ex a b in b c, &aelig;qualia &longs;unt producto a b in b c cum qua&shy;<lb/>drato b d. </s>
          <s id="id002464">Quoniam. </s>
          <s id="id002465">n. </s>
          <s id="id002466">quadrata a b, b c &aelig;qualia quadratis a d d b <lb/>b c &amp; productis ex a d in d b bis &amp; quod fit ex a b in b c &aelig;quale e&longs;t <lb/>ei quod fit ex a d in &longs;e cum eo quod fit ex a d in d b, quia a d e&longs;t &etail;qua </s>
        </p>
        <p type="main">
          <s id="id002467"><arrow.to.target n="marg478"/><lb/>lis b c ideo quadrata a b &amp; b c detracto eo quod fit ex a b in b c &longs;unt <lb/>&aelig;qualia quadratis a d d b, &amp; producto a d in d b &longs;emel: a c quadra&shy;<lb/><arrow.to.target n="marg479"/><lb/>tum a d cum producto a d in d b e&longs;t &aelig;quale producto a b in a d, &amp; <lb/>ex con&longs;equenti in b c, igitur re&longs;iduum quadratorum a b &amp; b c de&shy;<lb/>tracto producti a b in b c e&longs;t &aelig;quale a b in b c cum quadrato b d <lb/>quod fuit propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002468"><margin.target id="marg478"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002469"><margin.target id="marg479"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002470">Propo&longs;itio cente&longs;ima quadrage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id002471">Corpus quod fit ex linea diui&longs;a in &longs;uperficiem &etail;qualem quadra&shy;<lb/>tis ambarum partium detracta &longs;uperficie unius partis in <expan abbr="alter&atilde;">alteram</expan>, e&longs;t <lb/>&aelig;quale aggregato cuborum <expan abbr="ambar&utilde;">ambarum</expan> <expan abbr="parti&utilde;">partium</expan>.</s>
        </p>
        <figure id="id.015.01.158.3.jpg" xlink:href="015/01/158/3.jpg"/>
        <p type="main">
          <s id="id002472">Sic a b diui&longs;a in e quadrata partium e f &amp; <lb/><arrow.to.target n="marg480"/><lb/>b d detrahatur ex e f, f g &aelig;qualis a d, dico cor<lb/>pus ex a b in &longs;uperficies b d, d g &aelig;quale e&longs;&shy;<lb/>&longs;e cubis a c &amp; c b pariter acceptis, quia. </s>
          <s id="id002473">n. <lb/></s>
          <s id="id002474">ex a b in b d fiunt duo corpora cubus <lb/>b d &amp; corpus ex a d in quadratum d b hoc <lb/>autem e&longs;t &aelig;quale corpori ex b cin a d quia 


<pb pagenum="140" xlink:href="015/01/159.jpg"/>f&iacute;unt ex &aelig;qualibus lineis: at corpus quod fit ex a b in d g &aelig;quale e&longs;t <lb/>corporibus qu&aelig; fiunt ex a c, c b in &longs;uperficiem d g at cubus a c con&shy;<lb/>tinet duo corpora qu&etail; fiunt &amp; a c in d g &amp; g f, igitur cubus a c &longs;upe&shy;<lb/>rat productum ex a b in d g in producto ex a c in f g &amp; &longs;uperatur ab <lb/>eo in producto ex b c in d g, &longs;uperabatur etiam, ut ui&longs;um e&longs;t, cubus <lb/>b c &agrave; producto b a in d b in producto b cin c f, igitur cubi a c c b &longs;u&shy;<lb/>perantur &agrave; producto a b in ad in producto b c in c f &amp; in d g, quare <lb/>in producto b c in f e: &longs;i quidem f e &amp; f g &longs;unt &aelig;qualia ex &longs;uppo&longs;ito <lb/>&longs;uperant autem in producto ex c b in e f, igitur tantum e&longs;t in in quo <lb/>&longs;uperantur quantum e&longs;t id in quo &longs;uperant: ergo &longs;unt &aelig;qualia.</s>
        </p>
        <p type="margin">
          <s id="id002475"><margin.target id="marg480"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002476">Propo&longs;itio cente&longs;ima quadrage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id002477">Propo&longs;ita linea diui&longs;a duas ei lineas adijcere, ut proportio addita&shy;<lb/>rum &longs;ingularum &amp; partium &longs;imul iunctarum ad additas &longs;it mutua.<lb/><arrow.to.target n="marg481"/></s>
        </p>
        <p type="margin">
          <s id="id002478"><margin.target id="marg481"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002479">Sit linea a b diui&longs;a in c uolo eius <lb/><figure id="id.015.01.159.1.jpg" xlink:href="015/01/159/1.jpg"/><lb/>partibus addere lineas, ut propo&longs;i&shy;</s>
        </p>
        <p type="main">
          <s id="id002480"><arrow.to.target n="marg482"/><lb/>tum e&longs;t, &longs;tatuo mediam c d inter a e &amp; <lb/><arrow.to.target n="marg483"/><lb/>c b qu&aelig; &longs;it c d, &amp; facio ut c d ad c a ita <lb/>c a ad a e, &amp; ut d c ad c b ita c b ad b f, quia ergo d e media e&longs;t inter <lb/><arrow.to.target n="marg484"/><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/><arrow.to.target n="marg485"/><lb/>proportione, quare proportio e c ad c a ut c f ad b f &amp; e c ad ea ut <lb/>c f ad c b quod e&longs;t propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002481"><margin.target id="marg482"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002482"><margin.target id="marg483"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002483"><margin.target id="marg484"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002484"><margin.target id="marg485"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002485">Propo&longs;itio cente&longs;ima quadrage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id002486">Propo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus <lb/>alijs lineis &longs;ecundum rationem mutuam &longs;ingularum &longs;ingulis ag&shy;<lb/>gregatum ex una adiectarum &amp; parte ad aggregatum ex alia parte <lb/>&amp; adiecta &longs;e habeat, ut &longs;ecunda ad tertiam.<lb/><arrow.to.target n="marg486"/></s>
        </p>
        <p type="margin">
          <s id="id002487"><margin.target id="marg486"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002488">Sit a, b, c, d, propo&longs;it&aelig; line&etail;, <lb/><figure id="id.015.01.159.2.jpg" xlink:href="015/01/159/2.jpg"/><lb/>uolo diuidere a b ita in e ut <lb/>&longs;umpta &longs;ecundum proportio&shy;<lb/>nem alicuius quantitatis, puta <lb/>g ad a e &longs;ic b f ad e b &amp; ut g ad <lb/>e b &longs;ic g a ad a e ut &longs;it propor&shy;<lb/>tio g e ad e f ut c ad d. </s>
          <s id="id002489">Sint ergo <lb/>omnia <expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan> &amp; &longs;it g rectan&shy;<lb/>gulum ex a e in e b, cum ergo <lb/>g a contineat a e ut g continet e b, g autem continet e b &longs;ecundum <lb/>a e, igitur g a continet a e &longs;ecundum a c, ergo ex diffinitione qua&shy;</s>
        </p>
        <p type="main">
          <s id="id002490"><arrow.to.target n="marg487"/><lb/>drati a g e&longs;t quadratum a e. </s>
          <s id="id002491">Pari ratione b f e&longs;t quadratum b e. </s>
          <s id="id002492">pro&shy;<lb/>portio igitur g e ad e f cum &longs;it ut c ad e ex &longs;uppo&longs;ito erit ut ip&longs;i pro&shy;<lb/>portioni addamus, &amp; detrahamus ex duplo a b &amp; dimidium re&longs;i&shy;<lb/>dui ducamus in &longs;e, &amp; addamus aggregato quadrati a b cum ip&longs;a 


<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, &amp; latus eius detracto dimidio re&longs;idui erit b c linea, quare diui&shy;<lb/>&longs;io nota, &amp; e&longs;t ut dicamus : uolo diuidere datam lineam, ut quantita&shy;<lb/>tes adiect&aelig; &longs;ub mutua proportione ad unam tertiam cum parti&shy;<lb/>bus obtineant inter &longs;e proportionem datam.</s>
        </p>
        <p type="margin">
          <s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecun<lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002494">Propo&longs;itio cente&longs;ima quadrage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id002495">Datam lineam &longs;ic diuidere, ut proportio quadratorum ad du&shy;<lb/>plum unius partis in alteram &longs;it, ut line&etail; dat&aelig; ad lineam datam.</s>
        </p>
        <p type="main">
          <s id="id002496">Sit data a b quam uolo diuidere, ut proponitur &longs;ub proportio&shy;<lb/><arrow.to.target n="marg488"/><lb/>ne c d ad e, diuido a b bifariam in f, &amp; ab&longs;cindo <lb/><figure id="id.015.01.160.1.jpg" xlink:href="015/01/160/1.jpg"/><lb/>g d &aelig;qualem d e, &amp; inter c g <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> &amp; c e inter&shy;<lb/>pono proportione, &amp; ut h ad c g ita a f medietatis a b ad fk. </s>
          <s id="id002497">Omnia <lb/>i&longs;ta &longs;unt noti&longs;sima ex primo &amp; &longs;exto Elemento&shy;<lb/><figure id="id.015.01.160.2.jpg" xlink:href="015/01/160/2.jpg"/><lb/><expan abbr="r&utilde;">rum</expan> Euclidis. </s>
          <s id="id002498">Si ergo ab&longs;cindantur fk ex fa, dico <lb/>quod proportio quadratorum l k &amp; k a ad du&shy;<lb/>plum rectanguli a k in k b e&longs;t ut c d ad d e. </s>
          <s id="id002499">Quia. n. </s>
          <s id="id002500">c e ad c g dupli&shy;<lb/>cata e&longs;t ei qu&etail; e&longs;t h ad c g, duplicata e&longs;t <expan abbr="eti&atilde;">etiam</expan> ei qu&aelig; e&longs;t f a ad fk, qua&shy;<lb/>re ut quadrati a f ad fk, ita c e ad c g, igitur di&longs;iungendo c g ad g e ut <lb/>re&longs;idui quadrati k f ad re&longs;iduum quadrati a f, quare c g ad g d ut <lb/>quadrati k f ad dimidium re&longs;idui quadrati a f, igitur coniunctim c d <lb/>ad d g ut quadrati k f &amp; dimidij re&longs;idui quadrati a f ad ip&longs;um dimi&shy;<lb/>dium re&longs;idui. </s>
          <s id="id002501">At uer&ograve; cum g d &longs;it &aelig;qualis d e, erit c d ad d e ut qua&shy;<lb/>drati k f cum dimidio re&longs;idui &longs;&aelig;pius dicti ad ip&longs;um dimidium re&longs;i&shy;<lb/>dui. </s>
          <s id="id002502">Igitur etiam ut dupli quadrati k f cum re&longs;iduo ad <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;unt <lb/>enim omnia duplicata. </s>
          <s id="id002503">At <expan abbr="dupl&utilde;">duplum</expan> quadrati k f <expan abbr="c&utilde;">cum</expan> re&longs;iduo e&longs;t &aelig;qua&shy;<lb/>le quadratis a f &amp; f k, igitur quadratorum a f &amp; f k ad differentiam <lb/>eo rum proportio e&longs;t ut c d ad d e, igitur dupli quadratorum a f &amp; <lb/>f k ad duplum differenti&aelig; quadratorum a f &amp; fk ut c d ad d e. </s>
          <s id="id002504">Ve&shy;<lb/><arrow.to.target n="marg489"/><lb/>rum duplum quadratorum a f &amp; f k &aelig;quatur quadratis b k &amp; k a. <lb/><arrow.to.target n="marg490"/><lb/>Et duplum differenti&aelig; quadratorum a f &amp; fk e&longs;t &etail;quale duplo pro <lb/>ducti b k in k a, igitur proportio quadratorum k b &amp; k a ad <expan abbr="dupl&utilde;">duplum</expan> <lb/>producti k b in k a e&longs;t ueluti c d ad d e, quod e&longs;t propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002505"><margin.target id="marg488"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002506"><margin.target id="marg489"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002507"><margin.target id="marg490"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002508">Propo&longs;itio cente&longs;ima quinquage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id002509">Propo&longs;itis duabus lineis <expan abbr="line&atilde;">lineam</expan> communem <lb/><figure id="id.015.01.160.3.jpg" xlink:href="015/01/160/3.jpg"/><lb/>utrique adiungere, ut &longs;it maioris ad additam pro&shy;<lb/>portio, uelut quadratorum minoris &amp; adiect&aelig; <lb/>ad duplum unius in alteram.</s>
        </p>
        <p type="main">
          <s id="id002510">H&aelig;c e&longs;t qua&longs;i conuer&longs;a <expan abbr="pr&aelig;ced&etilde;tis">pr&aelig;cedentis</expan>. </s>
          <s id="id002511">Sit a ma&shy;<lb/><arrow.to.target n="marg491"/><lb/>ior, &amp; b c minor, &amp; fiat b d dupla b c, &longs;uper <expan abbr="qu&atilde;">quam</expan> <lb/>erigatur b f &aelig;qualis a; &amp; &longs;it rectangulum d f &amp; <lb/>de&longs;cribatur quadratum b c quod &longs;it b g re&longs;idu&etail; <lb/>&longs;uperficiei ad d f latus &longs;it h, dico h e&longs;&longs;e lineam qu&aelig;&longs;itam. </s>
          <s id="id002512">Superficies 


<pb pagenum="142" xlink:href="015/01/161.jpg"/>enim d f cum fiat ex a in duplum b c, dupla erit &longs;uperficiei a in b c, &longs;u<lb/>perficies f d, tota &aelig;quatur quadratis h &amp; b c, igitur quadrata h &amp; b <lb/>c dupla &longs;unt &longs;uperficiei a in b c, quod uer&ograve; fit ex a in duplum b c &longs;e <lb/>habet ad id quod fit ex h in duplum b c, ut a ad h, cum per eandem <lb/>lineam ducantur, igitur quod fit ex a in duplum b c, &amp; &longs;unt quadra&shy;<lb/>ta h &amp; b c, &longs;e habent ad duplum h in b c, ut a ad h, quod fuit de&shy;<lb/>mon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id002513"><margin.target id="marg491"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002514">Propo&longs;itio cente&longs;ima quinquage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id002515">Proportio differenti&aelig; quadratorum partium, cuiu&longs;uis line&aelig; ad <lb/>quadratum differenti&aelig; <expan abbr="illar&utilde;">illarum</expan> e&longs;t uelut totius line&etail; ad differentiam.<lb/><arrow.to.target n="marg492"/></s>
        </p>
        <p type="margin">
          <s id="id002516"><margin.target id="marg492"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002517">Sit a b diui&longs;a in puncto c, &amp; fiat c d &aelig;qualis <lb/>c b, manife&longs;tum e&longs;t quod differentia partium <lb/><figure id="id.015.01.161.1.jpg" xlink:href="015/01/161/1.jpg"/><lb/>e&longs;t a d, dico proportionem differenti&aelig; quadra <lb/>torum a c &amp; c b ad quadratum a d differenti&aelig; partium e&longs;&longs;e ut a b ad </s>
        </p>
        <p type="main">
          <s id="id002518"><arrow.to.target n="marg493"/><lb/>a d. </s>
          <s id="id002519">Quoniam differentia quadratorum a c &amp; c b e&longs;t, quod fit ex a d <lb/>in d c bis cum quadrato a d, &amp; ide&ograve; quod fit ex a d in d b cum qua&shy;<lb/>drato a d, &amp; ide&ograve; quod fit ex tota a b in a d. </s>
          <s id="id002520">Igitur differentia qua&shy;<lb/><arrow.to.target n="marg494"/><lb/>drato a c &amp; c b e&longs;t quod fit ex a b in a d, quare cum quadratum a d <lb/>fiat ex a d in a d, erit proportio a b ad a d, uelut differenti&aelig; quadra&shy;<lb/><arrow.to.target n="marg495"/><lb/>torum a c &amp; b c ad quadratum a d differenti&aelig; partium. </s>
          <s id="id002521">Quod fuit <lb/>propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002522"><margin.target id="marg493"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002523"><margin.target id="marg494"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002524"><margin.target id="marg495"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002525">Propo&longs;itio cente&longs;ima quinquage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id002526">Si linea in duas partes &aelig;quales duas que in &aelig;quales diuidatur, fue&shy;<lb/>ritque proportio aggregati ex maiore &amp; dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, &amp; aliqua linea ad ip&longs;am minorem, &amp; rur&longs;us aggre&shy;<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma&shy;<lb/>iore &amp; alia addita ad ip&longs;am maiorem, erit proportio dimidij ad par<lb/>tem unam in&aelig;qualem, uelut alterius partis in&aelig;qualis ad &longs;uam ad&shy;<lb/>ditam mutu&ograve;, &amp; etiam proportio additarum inuicem, uelut pro&shy;<lb/>portio partium in&aelig;qualium duplicata, &amp; rur&longs;us ip&longs;um dimidium <lb/>line&aelig; a&longs;&longs;umpt&aelig; medium erit proportione inter additas. </s>
          <s id="id002527">Demum <lb/>proportio dimidij cum ad dita maiore ad dimidium cum addita mi<lb/>nore, uelut maioris partis ad minorem.<lb/><arrow.to.target n="marg496"/></s>
        </p>
        <p type="margin">
          <s id="id002528"><margin.target id="marg496"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002529">Sit propo&longs;ita a b diui&longs;a per <lb/><figure id="id.015.01.161.2.jpg" xlink:href="015/01/161/2.jpg"/><lb/>&aelig;qualia in c per in&aelig;qualia in <lb/>d, &amp; &longs;it ut addantur a g &amp; b f, <lb/>ita ut proportio c a, &amp; a d ad a d &longs;it ueluti f d ad d b, &amp; c b &amp; b d ad <lb/>b d, uelut g d ad d a, &amp; h&aelig;c e&longs;t quarta <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> Archimedis de &longs;ph&etail;ra, <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/>fb ad b d. </s>
          <s id="id002530">Et &longs;imiliter quia e&longs;t c b &amp; b d ad b d, uelut g d ad d a erit 


<pb pagenum="143" xlink:href="015/01/162.jpg"/>c b ad b d, uelut g a ad a d, &amp; hoc e&longs;t primum. </s>
          <s id="id002531">Quia ergo c a e&longs;t &aelig;&shy;<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/>ad f b, per conuer&longs;am igitur a d ad b d, ut g a ad a d, &amp; ut b d ad fb, <lb/>interpo&longs;itis ergo a d &amp; d b inter a g &amp; b f cum compo&longs;ita &longs;it pro&shy;<lb/>portio a g ad b f ex proportione a g ad a d, &amp; ad d b, &amp; d b <lb/>ad b f, &amp; proportio a d ad d b, &longs;it &aelig;qualis proportioni <lb/><figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg"/><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&shy;<lb/>mon&longs;trata ab Alchindo e&longs;t duplicata proportioni a d ad <lb/>d b quod e&longs;t &longs;ecundum. </s>
          <s id="id002533">Rur&longs;us quia ex primo demon&shy;<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, &amp; d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d &amp; d b ad a c componunt proportionem produ&shy;<lb/>cti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, &amp; &longs;imiliter proportio b d ad b f &amp; a d ad a g com&shy;<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;it l ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli&shy;<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h &amp; </s>
        </p>
        <p type="main">
          <s id="id002534"><arrow.to.target n="marg497"/><lb/>l &longs;unt &aelig;quales, quia producuntur ex ei&longs;dem, igitur per demon&longs;tra&shy;<lb/>ta in quinto Elementorum Euclidis, k e&longs;t &aelig;quale m, ergo a c e&longs;t me&shy;<lb/>dia pro portione inter b f &amp; g a, quod e&longs;t tertium. </s>
          <s id="id002535">Quia uer&ograve; ex pri&shy;<lb/>mo demon&longs;trato e&longs;t fb ad b d, ut a c ad a d, &amp; c b ad idem b d, ut g a <lb/>ad idem a d erit coniungendo fb &amp; b c ad b d, ut coniun&shy;<lb/><figure id="id.015.01.162.3.jpg" xlink:href="015/01/162/3.jpg"/><lb/>gendo g a &amp; a c ad a d, &longs;ed fb &amp; b c componunt f c &amp; g a, <lb/>&amp; a c componunt g c, igitur ut f c ad b d, ita g c ad a d, er&shy;<lb/>go permutando g c ad f c, ut a d ad b d, quod e&longs;t quartum.</s>
        </p>
        <p type="margin">
          <s id="id002536"><margin.target id="marg497"/>I<emph type="italics"/>n<emph.end type="italics"/> P<emph type="italics"/>rop.<emph.end type="italics"/> 23 <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 9.</s>
        </p>
        <p type="main">
          <s id="id002537">Cum ergo punctum d fuerit datum, licet inuenire a g &amp; b f, faci&shy;<lb/>l&egrave;, ut Archimedes pr&aelig;&longs;upponit proportionem g d ad d f datam &amp; <lb/>qu&aelig;rit eam, qu&aelig; e&longs;t a d ad d b, &amp; peruenitur ad res numero triplo <lb/>quadrati dimidij line&aelig; a&longs;&longs;umpt&aelig; &aelig;quales cubo &amp; numero, qui &longs;it <lb/>ex duplo cubi dimidij in 1 m: ip&longs;a proportione, &amp; quod produci&shy;<lb/>tur diui&longs;o per 1 p: ip&longs;a proportione. </s>
          <s id="id002538">Veluti po&longs;ita a b 10, &amp; propor&shy;<lb/>tione quam uolo g d ad d f &longs;excupla, duco 5 dimidium 10 in &longs;e fit 25, <lb/>&amp; triplico, fit 75 numerus rerum. </s>
          <s id="id002539">Inde duco 5 idem dimidium ad <lb/>cubum fit 125, duplico fit 250, duco in 5, qui e&longs;t 1 m: proportione fit <lb/>1250, diuido per 7, qui e&longs;t 1 p: proportione exit 178 4/7 numerus, qui <lb/>cum cubo &aelig;quatur 75 rebus. </s>
          <s id="id002540">Cum ergo con&longs;tituta fuerit diui&longs;io in <lb/>c, non recipit proportionem g d ad f d quam uolueris, &longs;ed &longs;equitur <lb/>una &longs;ola ad <expan abbr="ill&atilde;">illam</expan>, &amp; e&longs;t mirabile, quoniam line&etail; uidentur &longs;umi liber&egrave;. <lb/></s>
          <s id="id002541">Sed non e&longs;t ita. </s>
          <s id="id002542">Et <expan abbr="eti&atilde;">etiam</expan> quia Archimedes <expan abbr="uide&ttilde;">uidetur</expan> a&longs;&longs;umere <expan abbr="ali&atilde;">aliam</expan> lineam, <lb/>&longs;ed non inue&longs;tigat eam, im&ograve; o&longs;tendit eam ex a&longs;&longs;umptis. </s>
          <s id="id002543">At Eutoci&shy;<lb/>us o&longs;tendit ambas, <expan abbr="un&atilde;">unam</expan> ex propria inuentione, aliam ex Diocle, &longs;ed 


<pb pagenum="144" xlink:href="015/01/163.jpg"/>una e&longs;t &longs;uperflua, quia ut dixi, una &longs;equitur ad aliam. </s>
          <s id="id002544">Ex hoc pa&shy;<lb/>tet cur Diocles a&longs;&longs;ump&longs;erit lineam unam, qu&aelig; e&longs;t a c, qu&aelig; &longs;e ha&shy;<lb/>bet ad a d, &amp; d b, ut uici&longs;sim a d, &amp; d b ad additas, quod e&longs;t pri&shy;<lb/>mum demon&longs;tratum. </s>
          <s id="id002545">Sic enim omittit primum quod proponit Ar<lb/>chimedes, &amp; a&longs;&longs;umit quod proximum e&longs;t: &amp; ide&ograve; Archimedes non <lb/>probat, nec pr&aelig;&longs;upponit, quod &agrave; Diocle probatur, &longs;cilicet datum <lb/>e&longs;&longs;e punctum d in linea a b, &longs;ed &longs;olum in linea g f, ide&ograve; cogitur pro&shy;<lb/>bare &longs;ecundum quod demon&longs;tratur ab Eutocio, &amp; &agrave; nobis demon <lb/>&longs;tratum e&longs;t &longs;upr&agrave;. </s>
          <s id="id002546">Archimedes <expan abbr="a&utilde;t">aut</expan> a&longs;&longs;umit <expan abbr="line&atilde;">lineam</expan> extra circulum, <expan abbr="qu&atilde;">quam</expan> <lb/>uocat b f, qu&aelig; e&longs;t &aelig;qualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/></s>
          <s id="id002547">Con&longs;tat ergo quod proportio g d ad d f e&longs;t data. </s>
          <s id="id002548">Et &longs;imiliter f g ad <lb/>g d, &amp; e&longs;t 1 pr&aelig; proportione data. </s>
          <s id="id002549">Vnde notandum quod datum <lb/>dicitur, &longs;impliciter cognitum alio modo, dicitur datum po&longs;itione, <lb/>quod e&longs;t certum &amp; tale, uelut &longs;i quis dicat, diuide 10 in duos nume&shy;<lb/>ros quadratos: hoc non e&longs;t datum, pote&longs;t enim diuidi pluribus mo <lb/>dis. </s>
          <s id="id002550">At &longs;i dicas ut una pars &longs;it alterius <expan abbr="quadrat&utilde;">quadratum</expan>, i&longs;tud antequ&agrave;m &longs;ci<lb/>untur partes, dicitur datum po&longs;itione. </s>
          <s id="id002551">Ergo datum po&longs;itione e&longs;t du<lb/>plex, uel ut ratio nota &longs;it, non autem quantitas, ut &longs;i dicam a b e&longs;t du<lb/>pla ad b c, utra que dicitur nota po&longs;itione, quo&shy;<lb/>niam ne&longs;cio quanta &longs;it a b. </s>
          <s id="id002552">Vel &longs;i quantitas e&longs;t <lb/><figure id="id.015.01.163.1.jpg" xlink:href="015/01/163/1.jpg"/><lb/>nota proportio ignota &longs;it, ut &longs;i a c &longs;it 10, &amp; &longs;it, <lb/>ut b c &longs;it &lt;02&gt; relata, a b erit punctus b, &amp; proportio a b ad b c data po<lb/>&longs;itione, non tamen nota. </s>
          <s id="id002553">Et &longs;i dicas igitur omnia, qu&aelig; habent deter<lb/>minationem erunt data po&longs;itione? </s>
          <s id="id002554">Dico quod non, quia oportet, <lb/>ut illa determinatio comprehendatur &longs;ub una ratione, eaque &longs;altem <lb/>generaliter cognita.</s>
        </p>
        <p type="main">
          <s id="id002555">Propo&longs;itio cente&longs;ima quinquage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id002556">Vim quan cun que manus multiplicare.<lb/><arrow.to.target n="marg498"/></s>
        </p>
        <p type="margin">
          <s id="id002557"><margin.target id="marg498"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002558">Cum enim radimus aut trahimus manife&longs;tum e&longs;t, </s>
        </p>
        <p type="main">
          <s id="id002559"><arrow.to.target n="marg499"/><lb/>quod ambabus manibus uis conduplicatur, &amp; ma&shy;<lb/><figure id="id.015.01.163.2.jpg" xlink:href="015/01/163/2.jpg"/><lb/>ior redditur, quanta e&longs;t proportio totius ad exce&longs;&shy;<lb/>&longs;um: uelut &longs;it a quod mouetur ab una manu uiribus <lb/>ut b, qu&aelig; &longs;unt exce&longs;&longs;us b d &longs;upra a, cum ergo propor<lb/>tio c b d ad a &longs;it compo&longs;ita ex proportionibus c &amp; <lb/>b d ad a manife&longs;tum e&longs;t, quod erit producta ex pro&shy;<lb/>portione c b d ad b d, &amp; b d ad a, &longs;ed e b d e&longs;t dupla <lb/>ad b d, quia e e&longs;t &aelig;qualis, c igitur proportio c b d ad <lb/><arrow.to.target n="marg500"/><lb/>a e&longs;t maior multo qu&agrave;m duorum exce&longs;&longs;uum, qui mo<lb/>uerent in proportione dupla: uelut &longs;i adderemus f 


<pb pagenum="145" xlink:href="015/01/164.jpg"/>ad d b &aelig;qualem b, multo maior e&longs;t ex communi animi &longs;ententia e f <lb/>b d <expan abbr="qu&atilde;">quam</expan> f b d, quia e continet f, &amp; quantum e&longs;t d in&longs;uper: cum ergo <lb/>b cum d moueat a in proportione b d ad a &amp; f cum d mouebit a in <lb/>proportione eadem qua b d, ergo per uiam additionis duplo ue&shy;<lb/>locius, qu&agrave;m dupla proportione, uer&ugrave;m dupla comparatione ad <lb/>proportionem b d ad a, non autem duplicata &longs;ed dupla, ut dixi, qu&etail; <lb/>erit maior qu&agrave;m dupla per <expan abbr="addition&etilde;">additionem</expan> exce&longs;&longs;us. </s>
          <s id="id002560">Ergo &longs;i addatur al&shy;<lb/>ter homo, erit dupla ad illam duplam, ueluti addendo &aelig;qualem d b <lb/>f e, ade&ograve; ut &longs;i proportio d b f e e&longs;&longs;et quintupla, mouerent illi duo in <lb/>proportione decupla. </s>
          <s id="id002561">Sed annexo baculo aut lima aut &longs;erra annu&shy;<lb/>lo h, ita ut circunuolui po&longs;sit h &aelig;quabit uires non &longs;olum d b f e &longs;ed <lb/>multorum hominum. </s>
          <s id="id002562">igitur multo plus aget homo ambabus ma&shy;<lb/>nibus radendo aut &longs;ecando cum g, qu&agrave;m quadrupla proportione <lb/>unius manus, &amp; hoc incrementum e&longs;t non &longs;olum magn&aelig; <lb/>utilitatis, &longs;ed ualde <expan abbr="acc&otilde;modatum">accommodatum</expan> in actionibus artificum <lb/>operum grauiorum. </s>
          <s id="id002563">Et huiu&longs;modi conduplicatio e&longs;t ratio <lb/>lim&aelig; quam &longs;urdam uocamus.</s>
        </p>
        <p type="margin">
          <s id="id002564"><margin.target id="marg499"/>P<emph type="italics"/>er<emph.end type="italics"/> 37.</s>
        </p>
        <p type="margin">
          <s id="id002565"><margin.target id="marg500"/>P<emph type="italics"/>er<emph.end type="italics"/> 2.</s>
        </p>
        <figure id="id.015.01.164.1.jpg" xlink:href="015/01/164/1.jpg"/>
        <p type="main">
          <s id="id002566">Propo&longs;itio cente&longs;ima quadrage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id002567">Si line&etail; dat&etail; alia linea adiungatur, ab extremitatibus autem pri&shy;<lb/>oris line&etail; du&aelig; rect&aelig; in unum punctum concurrant proportionem <lb/>habentes quam media inter totam &amp; adiectam, ad adiectam erit <lb/>punctus concur&longs;us &agrave; puncto extremo line&aelig; adiect&aelig; di&longs;tans per li&shy;<lb/>neam mediam. </s>
          <s id="id002568">Qu&ograve;d &longs;i ab extremo alicuius line&aelig; &aelig;qualis medi&aelig; <lb/>&longs;eu peripheria circuli cuius &longs;emidiameter &longs;it media linea du&aelig; line&aelig; <lb/>ad pr&aelig;dicta puncta producantur, ip&longs;&etail; erunt in proportione medi&etail; <lb/>ad adiectam.<lb/><arrow.to.target n="marg501"/></s>
        </p>
        <p type="margin">
          <s id="id002569"><margin.target id="marg501"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id002570">H&etail;c propo&longs;itio e&longs;t admirabilis: &amp; etiam de&longs;crip&longs;i, ut multa &longs;ecre&shy;<lb/>ta Dialectic&aelig; potius <expan abbr="aperiren&ttilde;">aperirentur</expan> quam quod huic propo&longs;ito <expan abbr="mult&utilde;">multum</expan> <lb/>congrueret. </s>
          <s id="id002571">Ide&ograve; potius &longs;cholij cau&longs;a po&longs;ita e&longs;t quam ip&longs;ius tracta&shy;<lb/>tionis: ut <expan abbr="mod&utilde;">modum</expan> demon&longs;trandi magis quam id, &qring;d <expan abbr="demon&longs;tra&ttilde;">demon&longs;tratur</expan>, re&shy;<lb/>&longs;picere oporteat. </s>
          <s id="id002572"><expan abbr="Con&longs;titua&ttilde;">Con&longs;tituatur</expan> ergo &lpar;per uiam problematis&rpar; linea a b <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/>ut g ad c, eritque g media inter a f &amp; f b, quod licet &longs;olum &longs;upponatur <lb/>ab Appollonio, <expan abbr="tam&etilde;">tamen</expan> facil&egrave; demon&longs;tratur &amp; &agrave; Commandino adie&shy;<lb/>cta e&longs;t <expan abbr="dem&otilde;">demon</expan>&longs;tratio. </s>
          <s id="id002573">Concurrant ergo ex a &amp; b du&etail; line&etail; in aliquod </s>
        </p>
        <p type="main">
          <s id="id002574"><arrow.to.target n="marg502"/><lb/>punctum, putat h ut &longs;it a h ad h b uelut c ad d, dico quod &longs;i ducat <lb/>h f quod ip&longs;a erit &aelig;qualis g, ducatur b l &aelig;quidi&longs;tans a h, &amp; quia <lb/><arrow.to.target n="marg503"/><lb/>ex &longs;uppo&longs;ito a h ad h b, ut g ad b f, erit b h ad h a, ut b f ad g, &amp; quia <lb/>trianguli a h f &amp; b l f &longs;unt &longs;imiles erit proportio a h ad b l, ueluti a f <lb/><arrow.to.target n="marg504"/><lb/>ad fb, igitur per &etail;quam proportionem b e h ad b l, ut a f ad g, &longs;ed ut <lb/><arrow.to.target n="marg505"/><lb/>a f ad g ita g ad b f ex &longs;uppo&longs;ito: &amp; ut a f ad g, it a h a ad h b, ex &longs;uppo 


<pb pagenum="146" xlink:href="015/01/165.jpg"/>&longs;ito igitur ut a h ad h b ita h b ad b l, &longs;ed angulus a h b e&longs;t &aelig;qualis <lb/>angulo h b l, ergo triangulus a h b e&longs;t <lb/>&longs;imilis triangulo h b l, quare angulus <lb/>b h l e&longs;t &etail;qualis angulo h a f, igitur du <lb/>orum triangulorum f a h, &amp; fb h duo <lb/><arrow.to.target n="marg506"/><lb/>anguli unius a &amp; f &longs;unt &aelig;quales duo&shy;<lb/>bus angulis, alterius igitur propor&shy;<lb/><figure id="id.015.01.165.1.jpg" xlink:href="015/01/165/1.jpg"/><lb/>tio a f ad fh re&longs;picientium angulos &etail;&shy;<lb/><arrow.to.target n="marg507"/><lb/>quales ut a h ad h b re&longs;picientium an&shy;<lb/><arrow.to.target n="marg508"/><lb/>gulum f, &longs;ed a h ad h b ut c ad d, ex &longs;up <lb/>po&longs;ito igitur a f ad f h, ut c ad d, &longs;ed ut c ad d ita a f ad g, ex &longs;uppo&longs;ito <lb/>ergo h f e&longs;t &aelig;qualis g.<lb/><arrow.to.target n="marg509"/></s>
        </p>
        <p type="margin">
          <s id="id002575"><margin.target id="marg502"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>pri <lb/>mi, &amp;<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002576"><margin.target id="marg503"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002577"><margin.target id="marg504"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002578"><margin.target id="marg505"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002579"><margin.target id="marg506"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri<lb/>mi, &amp;<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002580"><margin.target id="marg507"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002581"><margin.target id="marg508"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002582"><margin.target id="marg509"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002583">Cum ergo h&etail;c demon&longs;tratio &longs;it ex &longs;en&longs;u in uno puncto h, ide&ograve; ad <lb/>qu&aelig;libet puncta traduci pote&longs;t, qu&aelig; potero imaginari, &amp; ita pri&shy;<lb/>ma uocabitur &longs;en&longs;us, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> imaginandi: Et <expan abbr="quoni&atilde;">quoniam</expan> in demon&longs;tran&shy;<lb/>do non a&longs;&longs;umimus aliquid, quod &longs;it proprium alicui puncto, ni&longs;i <lb/>proportionem h a ad h b &longs;imilem e&longs;&longs;e c ad d, ideo hoc pertinet ad <lb/>intellectum, &amp; e&longs;t tertium. </s>
          <s id="id002584">Et idem dico &longs;i k e&longs;&longs;et ultra h quod po&shy;<lb/>te&longs;t contingere. </s>
          <s id="id002585">mod&ograve; k a ad k b &longs;it ut c ad d &amp; k f &longs;it &etail;qualis g idem <lb/>&longs;equetur, &amp; comprehenditur &longs;ub tertio &amp; pertinet ad intellectum, <lb/>&amp; quoniam demon&longs;tratur quod punctum k ubicunque &longs;umatur, e&longs;t <lb/>in &etail;quali <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> &agrave; puncto f &longs;cilicet per g lineam, erit &longs;emper in peri&shy;<lb/>pheria circuli, &amp; hoc pote&longs;t e&longs;&longs;e in infinitis locis &longs;impliciter &amp; extra <lb/>infinitum nihil e&longs;t, igitur &longs;ub hoc continetur conuer&longs;um &longs;cilicet, <lb/>quod a quolibet puncto circuli ductis lineis ad a &amp; b ip&longs;&etail; erunt in <lb/>proportione c ad d. </s>
          <s id="id002586">Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica &amp; hoc e&longs;t <foreign lang="greek">perila/mpousin</foreign> &amp; ferm&egrave; &longs;ummum in&shy;<lb/>tellectus humani. </s>
          <s id="id002587">Et pote&longs;t demon&longs;trari Geometric&egrave; duobus uer&shy;<lb/>bis. </s>
          <s id="id002588">Quia. n. </s>
          <s id="id002589"><expan abbr="f&longs;upponi&ttilde;">f &longs;upponitur</expan> &aelig;qualis g eo qu&ograve;d h e&longs;t in peripheria circu&shy;<lb/>li erit media inter a f &amp; f b, quare cum angulus f &longs;it communis, erit <lb/>proportio a h ad h b, laterum re&longs;picientium angulum f in utroque </s>
        </p>
        <p type="main">
          <s id="id002590"><arrow.to.target n="marg510"/><lb/>triangulo, uelut h f lateris in maiori ad f b latus in minori, quare <lb/><arrow.to.target n="marg511"/><lb/>cum ex &longs;uppo&longs;ito h f ad fb &longs;it ut c ad d, erit a ad b, ut c ad d. </s>
          <s id="id002591">Et uides <lb/>Apollonium, &amp; Pappium quanta &longs;uperflua adijciant in hac &longs;ecun&shy;<lb/><arrow.to.target n="marg512"/><lb/>da parte demon&longs;trationis, qu&aelig; e&longs;t prima apud illos, &amp; ducunt <expan abbr="un&atilde;">unam</expan> <lb/>lineam non nece&longs;&longs;ariam ex puncto b ad latus fh. </s>
          <s id="id002592">Vt <expan abbr="antiquor&utilde;">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria &amp; ingenio, qu&aelig; ferunt excel<lb/>lenti&longs;sima in illis, quantum nos ex Dialectica <foreign lang="greek">pẹrila/mpousin</foreign> inducen <lb/>tes. </s>
          <s id="id002593">e&longs;t enim &longs;ingulare hoc exemplum.<lb/><arrow.to.target n="marg513"/></s>
        </p>
        <p type="margin">
          <s id="id002594"><margin.target id="marg510"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002595"><margin.target id="marg511"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/><expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan><emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002596"><margin.target id="marg512"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>I<emph type="italics"/>n primo<emph.end type="italics"/> C<emph type="italics"/>o <lb/>nicor.<emph.end type="italics"/> A<emph type="italics"/>pol. <lb/>in<emph.end type="italics"/> P<emph type="italics"/>r&aelig;fat.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002597"><margin.target id="marg513"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002598">Ex hoc <expan abbr="eti&atilde;">etiam</expan> patet quod &longs;i circulus duceretur &longs;ecundum f k tran&shy;<lb/>&longs;iretque per m &amp; n e&longs;&longs;et 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"/>
        <p type="head">
          <s id="id002599">SCHOLIVM</s>
        </p>
        <p type="main">
          <s id="id002600">Ex hoc pater qualiter ex uera demon&longs;tratione &longs;en&longs;u o&longs;ten&longs;a per&shy;<lb/>uenimus ad quotquot imaginando, inde intellectu abiectis condi&shy;<lb/>tionibus non nece&longs;&longs;arijs facimus infinitum &amp; uniuer&longs;ale. </s>
          <s id="id002601">Demum <lb/>&longs;ine artis &longs;pecialis auxilio o&longs;tendimus theorema uniuer&longs;ale &lpar;quod <lb/>etiam poterat o&longs;tendi Geometric&egrave;, &longs;ed long&egrave; pulchrius e&longs;t, ac &longs;ubli&shy;<lb/>mius per <foreign lang="greek">perilampousin</foreign>, qua hoc ip&longs;o infinita alia docemus generaliter <lb/>per &longs;implicem <expan abbr="compreh&etilde;&longs;ionem">comprehen&longs;ionem</expan> o&longs;tendere&rpar; &longs;cilicet quod &agrave; quouis <lb/>puncto peripheri&etail; circuli, cuius &longs;emidiameter e&longs;t media proportio&shy;<lb/>ne inter totam exten&longs;am &agrave; centro u&longs;que exterius, &amp; partem qu&aelig;&apos; e&longs;t &agrave; <lb/>centro ad punctum de&longs;criptum &longs;ub proportione continua <expan abbr="datar&utilde;">datarum</expan> <lb/>linearum line&aelig; duct&aelig; ex eo ad punctum exterius, &amp; punctum de&shy;<lb/>&longs;criptum &longs;unt in proportione datarum linearum.</s>
        </p>
        <p type="main">
          <s id="id002602">Propo&longs;itio cente&longs;ima quinquage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id002603"><expan abbr="Quadrator&utilde;">Quadratorum</expan> <expan abbr="numeror&utilde;">numerorum</expan> proportionem &amp; <expan abbr="inuention&etilde;">inuentionem</expan> <expan abbr="c&otilde;&longs;iderare">con&longs;iderare</expan>.</s>
        </p>
        <figure id="id.015.01.166.1.jpg" xlink:href="015/01/166/1.jpg"/>
        <p type="main">
          <s id="id002604">Prim&ugrave;m oportet &longs;cire e&longs;&longs;e tres naturales <lb/>numerorum &longs;eries, primam Euclidis iuxta </s>
        </p>
        <p type="main">
          <s id="id002605"><arrow.to.target n="marg514"/><lb/>quamuis <expan abbr="proportion&etilde;">proportionem</expan>, in qua unum &amp; ter&shy;<lb/>tius &amp; quintus, &amp; ita uno &longs;emper intermi&longs;&shy;<lb/>&longs;o &longs;unt quadrati. </s>
          <s id="id002606">Primus quo que. </s>
          <s id="id002607">1. unum &amp; <lb/>quartus &amp; &longs;eptimus &amp; ita duobus intermi&longs;sis &longs;unt cubi. </s>
          <s id="id002608">In &longs;ecun&shy;<lb/>do ordine e&longs;t naturalis &longs;eries numerorum, ex qua colligitur alia, &amp; <lb/>ex illa bini quilibet &longs;e &longs;equentes con&longs;tituunt numerum <expan abbr="quadrat&utilde;">quadratum</expan>. <lb/></s>
          <s id="id002609">In tertia numeri impares, qui &longs;emper collati efficiunt quadratum.</s>
        </p>
        <p type="margin">
          <s id="id002610"><margin.target id="marg514"/>E<emph type="italics"/><expan abbr="xempl&utilde;">xemplum</expan><emph.end type="italics"/> 1.</s>
        </p>
        <figure id="id.015.01.166.2.jpg" xlink:href="015/01/166/2.jpg"/>
        <p type="main">
          <s id="id002611">Sit ergo propo&longs;itus numerus cui uelim <lb/>addere quadratum numerum, ut fiat qua&shy;<lb/><arrow.to.target n="marg515"/><lb/>dratus totus, accipe numerum quadratum <lb/>minorem illo quem uis, &amp; detrahe &agrave; propo<lb/>&longs;ito numero &longs;eu quadrato &longs;eu non re&longs;idu&shy;<lb/><arrow.to.target n="marg516"/><lb/>um, diuide per duplum &lt;02&gt; quadrati quod <lb/>detraxi&longs;ti, &qring;d exit duc in &longs;e fiet quadratus numerus, idem que additus <lb/>numero propo&longs;ito, faciet quadratum. </s>
          <s id="id002612">Velut capio 16 qui e&longs;t qua&shy;<lb/>dratus, aufero 9 quadratum <expan abbr="minor&etilde;">minorem</expan> relinquitur 7, diuido per 6 du&shy;<lb/>plum &lt;02&gt; 9, exit 1 1/6 quadratum eius e&longs;t 1 13/36 qui additus ad 16 facit 17 13/36 <lb/><expan abbr="quadrat&utilde;">quadratum</expan> cuius &lt;02&gt; e&longs;t 4 1/6.</s>
        </p>
        <p type="margin">
          <s id="id002613"><margin.target id="marg515"/>E<emph type="italics"/><expan abbr="xempl&utilde;">xemplum</expan><emph.end type="italics"/> 2.</s>
        </p>
        <p type="margin">
          <s id="id002614"><margin.target id="marg516"/>E<emph type="italics"/><expan abbr="xempl&utilde;">xemplum</expan><emph.end type="italics"/> 3.</s>
        </p>
        <p type="main">
          <s id="id002615">Ex hoc patet propo&longs;ito quouis numero <expan abbr="&qtilde;drato">quadrato</expan> modus inuenien&shy;<lb/><arrow.to.target n="marg517"/><lb/>di infinitos numeros quadratos qui <expan abbr="c&utilde;">cum</expan> illo iuncti facient <expan abbr="quadrat&utilde;">quadratum</expan>.</s>
        </p>
        <p type="margin">
          <s id="id002616"><margin.target id="marg517"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="head">
          <s id="id002617">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002618">Po&longs;&longs;em adducere demon&longs;trationes omnium <expan abbr="hor&utilde;">horum</expan>, &longs;ed reddere&shy;<lb/>tur res longa <expan abbr="c&utilde;">cum</expan> &longs;int manife&longs;t&etail; ex &longs;eptimo octauo &amp; nono Euclidis. <lb/></s>
          <s id="id002619">Exemplum &longs;ecundum capio mod&ograve; 14 qui non e&longs;t quadratus, aufe&shy;<lb/>ro 9, remanet 5, diuido per 6 duplum &lt;02&gt; 9 exit 5/6 <expan abbr="quadrat&utilde;">quadratum</expan> eius e&longs;t 25/36 


<pb pagenum="148" xlink:href="015/01/167.jpg"/>hic additus ad 14 con&longs;tituit 14 25/36 quadratum 3 5/6. Et ita 14 e&longs;t diffe&shy;<lb/>rentia duorum quadratorum, &longs;cilicet 25/36 &amp; 14 25/36.<lb/><arrow.to.target n="marg518"/></s>
        </p>
        <p type="margin">
          <s id="id002620"><margin.target id="marg518"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002621">Ex hoc habebis duo quadrata in datis terminis qu&aelig; different <lb/>dato numero, &amp; e&longs;t pulchrum. </s>
          <s id="id002622">Velut uolo duo quadrata qu&aelig; dif&shy;<lb/>ferant in 2, &amp; &lt;02&gt; minoris &longs;it inter 1 &amp; 2, tunc capies per regulam i&shy;<lb/>p&longs;am 2, &amp; auferes <expan abbr="numer&utilde;">numerum</expan> quadratum ita qu&ograve;d re&longs;iduum diui&longs;um <lb/>per duplum radicis efficiat <expan abbr="numer&utilde;">numerum</expan> inter 1 &amp; 2. Veluti capio 4/9 qua&shy;<lb/>dratum, aufero ex 2, relinquitur 1 5/9 diuido per duplum 2/13 radicis 4/9 &amp; <lb/>e&longs;t 1 1/3 &amp; exit 1 1/6, &amp; hic e&longs;t minor numerus cuius quadratum e&longs;t 1 13/36 <lb/>cui &longs;i addantur 2, fient 3 13/36 numerus quadratus 1 5/6.</s>
        </p>
        <p type="main">
          <s id="id002623"><arrow.to.target n="marg519"/></s>
        </p>
        <p type="margin">
          <s id="id002624"><margin.target id="marg519"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id002625">Cum autem uolueris duo quadrata qu&aelig; differant in 100, tunc <lb/>per regulam datam &longs;i auferes 1, peruenires ad numeros magnos &amp; <lb/>fractos, &amp; ideo melius e&longs;t quia numerus e&longs;t par, ut detrahas nume&shy;<lb/>rum parem quadratum, ita quod re&longs;iduum po&longs;sit diuidi per <expan abbr="dupl&utilde;">duplum</expan> <lb/>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb/>84 <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> non <expan abbr="p&otilde;t">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb/><expan abbr="detrah&atilde;">detraham</expan> 4 &amp; <expan abbr="relinque&ttilde;">relinquetur</expan> 96, diuido per <expan abbr="dupl&utilde;">duplum</expan> radicis quod e&longs;t 4 exit <lb/>24, cuius quadratum qua e&longs;t 576 addito 100 facit 676 <expan abbr="quadrat&utilde;">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam &longs;ed 9, quia relinquetur 24 qui pote&longs;t diui&shy;<lb/>di per &longs;e, duplum &lt;02&gt; 9 &amp; exit 4 cuius <expan abbr="quadrat&utilde;">quadratum</expan> e&longs;t 16, addito 33 fit 49.</s>
        </p>
        <p type="main">
          <s id="id002626">Secunda regula, cum uolueris propo&longs;ito uno numero quadra&shy;<lb/>to illum diuidere infinitis modis in duos numeros quadratos, cape <lb/>quemuis numerum quadratum per primum exemplum regul&etail; pri<lb/>m&aelig;, &amp; cum eo diuide numerum propo&longs;itum, &amp; qui proueniet erit <lb/>quadratus, <expan abbr="h&utilde;c">hunc</expan> ergo duces in partes numeri quadrati qu&etail; &longs;unt nu&shy;<lb/>meri <expan abbr="&qtilde;drati">quadrati</expan>, &amp; fient duo quadrati numeri, &amp; illi <expan abbr="compon&etilde;t">component</expan> <expan abbr="numer&utilde;">numerum</expan> <lb/><expan abbr="quadrat&utilde;">quadratum</expan> <expan abbr="prior&etilde;">priorem</expan> quem diui&longs;i&longs;ti. </s>
          <s id="id002627">quia multiplicatio fit per <expan abbr="eo&longs;d&etilde;">eo&longs;dem</expan> nu&shy;<lb/>meros qui &longs;unt partes diui&longs;oris. </s>
          <s id="id002628">Velut uolo facere de 4 duas partes <lb/>qu&etail; &longs;int <expan abbr="&qtilde;drati">quadrati</expan> numeri, capio <expan abbr="numer&utilde;">numerum</expan> <expan abbr="&qtilde;drat&utilde;">quadratum</expan> qui <expan abbr="c&otilde;pona&ttilde;">componatur</expan> ex duo&shy;<lb/>bus <expan abbr="&qtilde;dratis">quadratis</expan>, uelut 25, diuido 4 per 25 exit 4/25 <expan abbr="h&utilde;c">hunc</expan> duco per 9 &amp; 16 <expan abbr="&qtilde;dra&shy;tos">quadra&shy;<lb/>tos</expan> numeros <expan abbr="c&otilde;ponentes">componentes</expan> 25 <expan abbr="fi&utilde;t">fiunt</expan> 1 11/25 &amp; 2 14/25 <expan abbr="&qtilde;drati">quadrati</expan> 1 2/5 &amp; 1 3/5 Et hi <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="c&otilde;ponunt">componunt</expan> 4. Et ita po&longs;&longs;es diuidere infinitis modis, puta per 17 13/36 &amp; <lb/>per 169. Tertia regula cum unus numerus additus <lb/><figure id="id.015.01.167.1.jpg" xlink:href="015/01/167/1.jpg"/><lb/>primo &amp; detractis &agrave; <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> facit ambo quadrata, <expan abbr="id&etilde;">idem</expan> <lb/>numerus coniunctus cum differentia illorum nume&shy;<lb/>rorum &amp; detractus &agrave; primo &amp; additus &longs;ecundo facit <lb/>eo&longs;dem numeros quadratos, ueluti capio 10 primum <lb/>3 &longs;ecundum 6 additus ad 10 &amp; detractus &agrave; 7 efficit 6 <lb/>&amp; 1 quadratos dico quod iunctus 16 cum 3 differen&shy;<lb/>tia 10 &amp; 7 fit 9, qui detractus &agrave; 10 &amp; additus ad 7 effi&shy;<lb/>cit 1 &amp; 16 numeros quadratos priores.</s>
        </p>
        <pb pagenum="149" xlink:href="015/01/168.jpg"/>
        <p type="head">
          <s id="id002629">SCHOLIVM</s>
        </p>
        <p type="main">
          <s id="id002630">Sunt &amp; alij modi plures faciendi huiu&longs;modi, &longs;ed <expan abbr="n&otilde;">non</expan> &longs;unt ad e&ograve; ge<lb/>nerales, &amp; nihilo minus &longs;unt magis confu&longs;i, &amp; non aliquid plus.</s>
        </p>
        <p type="main">
          <s id="id002631">Quarta regula, <expan abbr="c&utilde;">cum</expan> uolueris <expan abbr="numer&utilde;">numerum</expan> aliquem non quad. </s>
          <s id="id002632">qui bifa<lb/><expan abbr="ri&atilde;">riam</expan> <expan abbr="compona&ttilde;">componatur</expan> ex duob. </s>
          <s id="id002633"><expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002634">uelut 10 ex 25, &amp; 25 &amp; 49 &amp; 1, <lb/><figure id="id.015.01.168.1.jpg" xlink:href="015/01/168/1.jpg"/><lb/>&amp; <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> a b numerus quad. </s>
          <s id="id002635">diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="&aelig;&qtilde;li">&aelig;quali</expan> c d gnomo <lb/>cir<expan abbr="c&utilde;&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="c&utilde;">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, &longs;it <expan abbr="&etail;&qtilde;lis">&etail;qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e &amp; e d, <expan abbr="&aelig;&qtilde;libus">&aelig;qualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="c&utilde;f">cunf</expan> qua&shy;<lb/>drato &etail;qualia duob. </s>
          <s id="id002636"><expan abbr="&longs;upplem&etilde;tis">&longs;upplementis</expan> a b <expan abbr="c&utilde;">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. </s>
          <s id="id002637">Maio&shy;<lb/>ra <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="exced&utilde;t">excedunt</expan> minora in duplo quad. </s>
          <s id="id002638">c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus &longs;upplementis <expan abbr="c&otilde;munibus">communibus</expan>, erit <expan abbr="dupl&utilde;">duplum</expan> quad. </s>
          <s id="id002639">c d <expan abbr="c&utilde;">cum</expan> f qua&shy;<lb/>drato &etail;qualia h g <expan abbr="&qtilde;drato">quadrato</expan>. </s>
          <s id="id002640">Ergo propo&longs;ito numero, put&agrave; 3 ducam in &longs;e <lb/>fit 9, <expan abbr="duc&atilde;">ducam</expan> 2 <expan abbr="minor&etilde;">minorem</expan> in &longs;e fit 4, duplicabo fit 8, detraho ex 9, <expan abbr="relinqui&ttilde;">relinquitur</expan> <lb/>1 numerus <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dic&atilde;">dicam</expan> &qring;d 3 <expan abbr="c&utilde;">cum</expan> duplo 2, &amp; erit <expan abbr="tot&utilde;">totum</expan> 7, e&longs;t unus <lb/>numerus, alter &lt;02&gt; 1. 1. 1, &amp; <expan abbr="hor&utilde;">horum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002641"><expan abbr="c&otilde;ponunt">componunt</expan> 50, <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002642">5. Et &longs;imi <lb/>liter capio 6 <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002643">36 <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002644">4. 32 differentia 4, numerus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002645">2, ideo <lb/>6 <expan abbr="c&utilde;">cum</expan> duplo 4, &amp; e&longs;t 14, e&longs;t unus numerus, alter 2, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002646">&longs;unt 200, <lb/><expan abbr="dimidi&utilde;">dimidium</expan> e&longs;t 100 <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002647">10 <expan abbr="c&otilde;po&longs;iti">compo&longs;iti</expan> ex 6 &amp; 4. Et ita capio 9, <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002648">eius 81 du<lb/><expan abbr="pl&utilde;">plum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002649">6. 72 differentia 9 numerus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002650"><expan abbr="igi&ttilde;">igitur</expan> cum duplo 6, &amp; e&longs;t 21, e&longs;t <lb/>unus <expan abbr="illor&utilde;">illorum</expan>, alter 3 <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002651">450, <expan abbr="dupl&utilde;">duplum</expan> 225 <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002652">15, qui con&longs;tat ex 9 &amp; 6. Et <lb/>ita capio 11 <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002653">cuius e&longs;t 121, <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002654">6 e&longs;t 72 differentia, 72 &amp; 21 e&longs;t <lb/>49 numerus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002655">7, <expan abbr="igi&ttilde;">igitur</expan> 23 qui con&longs;tat ex 11, &amp; duplo 6 numeri mino<lb/>ris e&longs;t unus numerus, alter e&longs;t 7 <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002656"><expan abbr="quor&utilde;">quorum</expan> &longs;unt 578. <expan abbr="dupl&utilde;">duplum</expan> 289, <expan abbr="&qtilde;d">quad</expan>. <lb/></s>
          <s id="id002657">17, qui con&longs;tat ex 11 &amp; 6. Quinta regula, per hoc inueniemus infini<lb/>tos numeros <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002658"><expan abbr="c&otilde;ponentes">componentes</expan> 32, nam <expan abbr="c&utilde;">cum</expan> 32 &longs;it duplus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002659"><expan abbr="diuid&atilde;">diuidam</expan> per <lb/>unum <expan abbr="aggregat&utilde;">aggregatum</expan> ex inuentis puta 578, &amp; quia ambo ex &longs;uppo&longs;ito <lb/>&longs;unt dupli ad <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002660">qui proueniet erit <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002661">&longs;cilicet 16/289, duc in numeros <expan abbr="&qtilde;&shy;dratos">qua&shy;<lb/>dratos</expan> qui componunt 578, &amp; &longs;unt 529 &amp; 49, &amp; fient 2 206/289 &amp; 29 83/289, <lb/>&amp; hi iuncti <expan abbr="fi&utilde;t">fiunt</expan> 32, quia &longs;unt multiplicat&aelig; partes numeri, per quem <lb/>e&longs;t diui&longs;us numerus. </s>
          <s id="id002662">Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s>
        </p>
        <p type="main">
          <s id="id002663">Sexta regula, ponamus mod&ograve; quod uelim diuidere 10, <expan abbr="c&otilde;po&longs;it&utilde;">compo&longs;itum</expan> ex <lb/>duob. </s>
          <s id="id002664"><expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002665">9 &amp; 1, &amp; non <expan abbr="dupl&utilde;">duplum</expan> numero <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002666">ita quod &longs;it diui&longs;us in alios <lb/>duos: <expan abbr="duc&atilde;">ducam</expan> 10 in 25 <expan abbr="c&otilde;po&longs;it&utilde;">compo&longs;itum</expan> ex duob. </s>
          <s id="id002667"><expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002668">fit 250/25, at 250 <expan abbr="c&otilde;poni&ttilde;">componitur</expan> aliter <lb/>ex duob. </s>
          <s id="id002669">quad. </s>
          <s id="id002670">&lt;08&gt; 225/25 &amp; 25/25, &longs;cilicet 169/25 &amp; 81/25, id e&longs;t 6 19/25 &amp; 3 6/25, qui &longs;unt <expan abbr="&qtilde;d">quad</expan>. <lb/></s>
          <s id="id002671">2 3/5 &amp; 1 4/5, &amp; ita uolo diuidere 13 in duo alia <expan abbr="&qtilde;drata">quadrata</expan> &lt;08&gt; 9 &amp; 4, duco 13 in <lb/>25 &amp; fit 325/25, qui nece&longs;&longs;ario <expan abbr="c&otilde;poni&ttilde;">componitur</expan> ex 225/25 &amp; 100/25, &longs;ed ego uolo &qring;d <expan abbr="c&otilde;po">compo</expan> <lb/><expan abbr="na&ttilde;">natur</expan> aliter, uelut ex 289/25 &amp; 63/25, &amp; ita ex 11 14/25 &amp; 1 11/25, qui &longs;unt numeri <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002672">com<lb/>ponentes 13, &amp; &lt;02&gt; &longs;unt 3 2/5 &amp; 1 1/5, &amp; in his opus e&longs;t indu&longs;tria, &longs;cilicet ut <lb/><expan abbr="multiplice&ttilde;">multiplicetur</expan> per numeros <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002673">ut proueniant numeri illi <expan abbr="bifari&atilde;">bifariam</expan> comp <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>
          <s id="id002674">Vt uer&ograve; uideamus <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, proponamus quae uelim diui <lb/>dere 6 in duos numeros <expan abbr="&qtilde;d">quad</expan>, <expan abbr="prim&utilde;">primum</expan> &longs;cire debes &qring;d non po&longs;&longs;unt e&longs;&longs;e 


<pb pagenum="150" xlink:href="015/01/169.jpg"/>integri ex ratione dicta, quia oporteret ut e&longs;&longs;ent ambo impares aut <lb/>pares, &amp; &longs;ic <expan abbr="differr&etilde;t">differrent</expan> numero pari, ergo oporteret ut e&longs;&longs;et unus me&shy;<lb/>dius numerus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002675">&longs;unt &amp; ali&etail; rationes, &longs;ed neque unus po&longs;&longs;et e&longs;&longs;e inte<lb/>ger, &amp; alius fractus, <expan abbr="n&otilde;">non</expan> e&longs;&longs;et. </s>
          <s id="id002676">n. </s>
          <s id="id002677">6 numerus integer: <expan abbr="relinqui&ttilde;">relinquitur</expan> ergo ut <lb/>&longs;int duo fracti: &longs;ed in numeris fractis <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002678">deductis ad minimas deno <lb/>minationes <expan abbr="oper&utilde;">operum</expan>, ut tam denominator &lt;08&gt; numerator habeat radi&shy;<lb/>ces, ergo oportet &qring;d hoc &longs;it in illis, &amp; quia iuncti debent facere inte&shy;<lb/>gros 6, nece&longs;&longs;e e&longs;t ut denominator &longs;it unus, &amp; <expan abbr="id&etilde;">idem</expan> in utroque, et &qring;d nu<lb/>meratores &longs;imul iuncti &longs;int <expan abbr="&longs;excupl&utilde;">&longs;excuplum</expan> denominatoris, &longs;i fracti <expan abbr="deb&etilde;t">debent</expan> <lb/>&etail;quipollere 6, ergo ille denominator <expan abbr="c&utilde;">cum</expan> &longs;it <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002679">&amp; numeratores am&shy;<lb/>bo &longs;int <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002680">&amp; &longs;int <expan abbr="&longs;excupl&utilde;">&longs;excuplum</expan> denominatoris, oportebit inuenire <expan abbr="nu&shy;mer&utilde;">nu&shy;<lb/>merum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002681">qui ductus in 6, faciat <expan abbr="numer&utilde;">numerum</expan> qui <expan abbr="c&otilde;poni&ttilde;">componitur</expan> ex duob. </s>
          <s id="id002682"><expan abbr="&qtilde;d">quad</expan>. <lb/></s>
          <s id="id002683">aut <expan abbr="c&otilde;poni&ttilde;">componitur</expan> &etail;qualiter, ergo proportio medietatis ad <expan abbr="medietat&etilde;">medietatem</expan> 6, e&longs;t <lb/>ueluti totius ad 6, &longs;ed totu continet 6 in <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002684">quia ex 6 in <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002685">fit <expan abbr="tot&utilde;">totum</expan>, <lb/>ergo ex medietate in <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002686">idem fit medietas, &longs;ed medietas e&longs;t nume&shy;<lb/>rus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002687">ergo 3 e&longs;&longs;et numerus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002688">quod e&longs;t fal&longs;um, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi &longs;int in&aelig; quales, &amp; ut 6 diuidatur in duas partes in&etail;quales, hoc <lb/><expan abbr="a&utilde;t">aut</expan> fit diuidendo quemlibet <expan abbr="numer&utilde;">numerum</expan> parem, qui <expan abbr="c&otilde;poni&ttilde;">componitur</expan> ex duob. <lb/></s>
          <s id="id002689">numeris <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002690">nam &longs;i e&longs;&longs;et impar, <expan abbr="n&otilde;">non</expan> po&longs;&longs;et prodire numerus integer, &amp; <lb/><expan abbr="c&utilde;">cum</expan> prouenerit numerus <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002691">ille erit <expan abbr="qu&etilde;">quem</expan> qu&etail;rimus, <expan abbr="n&atilde;">nam</expan> diui&longs;o 6 per to&shy;<lb/>tum <expan abbr="ill&utilde;">illum</expan> numerum, inde &qring;d prouenit multiplicato per numeros <expan abbr="&qtilde;d">quad</expan>, <lb/><expan abbr="c&otilde;ponentes">componentes</expan> illum <expan abbr="numer&utilde;">numerum</expan> productum, <expan abbr="producun&ttilde;">producuntur</expan> partes 6, qu&aelig; <expan abbr="er&utilde;t">erunt</expan> <lb/>numeri <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002692">quia denominator utriu&longs;que partis ex &longs;uppo&longs;ito e&longs;t nume <lb/>rus <expan abbr="&qtilde;dratus">quadratus</expan>, qui multiplicatus e&longs;t per 6, &amp; numeratores &longs;unt nume <lb/>ri <expan abbr="&qtilde;drati">quadrati</expan>, qui <expan abbr="c&otilde;ponebant">componebant</expan> <expan abbr="numer&utilde;">numerum</expan> <expan abbr="product&utilde;">productum</expan>, et tales partes <expan abbr="&etail;quan&ttilde;">&etail;quantur</expan> <lb/>6, quia numerus productus <expan abbr="componi&ttilde;">componitur</expan> ex numeratoribus, &amp; <expan abbr="produ&shy;ci&ttilde;">produ&shy;<lb/>citur</expan> tale <expan abbr="c&otilde;po&longs;itum">compo&longs;itum</expan> ex 6 in <expan abbr="denominator&etilde;">denominatorem</expan>, &amp; hic e&longs;t diui&longs;us per deno <lb/><expan abbr="minator&etilde;">minatorem</expan>, ergo prouenit 6, &longs;i <expan abbr="e&mtilde;">emm</expan> multiplicato 3 in 4 fit 12, diui&longs;o 12 per <lb/>4, exit nece&longs;&longs;ario idem 3. Pro colligendo ergo numeros omnes, qui <lb/><expan abbr="c&otilde;ponuntur">componuntur</expan> ex <expan abbr="&qtilde;dratis">quadratis</expan>, propones tibi &longs;eriem <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002693"><expan abbr="omni&utilde;">omnium</expan>, &amp; inde iun&shy;<lb/>ges, &amp; diuides per 6, &amp; <expan abbr="c&utilde;">cum</expan> prodierit <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="inueni&ttilde;">inuenitur</expan> denominator, <lb/>&amp; numeri <expan abbr="c&otilde;ponentes">componentes</expan> ip&longs;um erunt numeratores, et &longs;uppo&longs;iti deno <lb/>minatoribus <expan abbr="c&otilde;&longs;tituent">con&longs;tituent</expan> partes. </s>
          <s id="id002694">Vt uer&ograve; cogno&longs;cas, ex quibus po&longs;&shy;<lb/>&longs;it componi primum ex imparibus, non oportet a&longs;&longs;umere ni&longs;i 135, <lb/>quia 7 diui&longs;um per 6 relinquit 1, &amp; 9 diui&longs;um per 6, relinquit 3, &amp; 35 <lb/>diui&longs;um per 6 relinquit 5. ergo non pote&longs;t componi numerus im&shy;<lb/>par, qui diuidatur per 6, ut &longs;uper&longs;it impar alius qu&agrave;m 1. 3. 5. &longs;ed 1 &amp; 3 <lb/>&amp; 5, &amp; 5 componunt 4 &amp; 1, &amp; 1 &amp; 3 &amp; 5 componunt 2, &longs;cilicet abie&shy;<lb/>cto 6, ergo tales numeri <expan abbr="&qtilde;drati">quadrati</expan> &longs;i &longs;int impares, uel ambo terminan&shy;<lb/>tur in 3, ut 9 &amp; 81, qui faciunt 90, uel in 1 &amp; 5, &longs;ed nullus numerus <lb/>quadratus diui&longs;us per 6 terminatur in 5, quia 1 ductum in &longs;e produ&shy;<lb/>cit 1, &amp; 3 pro ducit 3, &amp; 5 pro ducit 1, ut 5 in 5 facit 25, &amp; 11 in 11 produ&shy;


<pb pagenum="151" xlink:href="015/01/170.jpg"/>cit 121, quibus diui&longs;is per 6 &longs;upere&longs;t 1. Quod etiam &longs;ic demon&longs;tratur <lb/>de 5, &amp; compo&longs;itis &agrave; 5, nam diui&longs;o 5 in 3 &amp; 2, quadratum eius <expan abbr="c&otilde;po&shy;nitur">compo&shy;<lb/>nitur</expan> ex duplo 3 in 2, in quo nihil &longs;upere&longs;t, &longs;i diuidatur per 6, &amp; ex <lb/>quadrato 3, qu&ograve;d e&longs;t 9, in quo &longs;upere&longs;t 3, &amp; ex quadrato 2 quod e&longs;t </s>
        </p>
        <p type="main">
          <s id="id002695"><arrow.to.target n="marg520"/><lb/>4, &longs;ed iunctis 4 &amp; 3, &amp; abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="duct&utilde;">ductum</expan>, &amp; diui<lb/>&longs;o producto relinquitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 &amp; <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u&shy;<lb/>pere&longs;t, &amp; duplo 5 in 12, in quo <expan abbr="eti&atilde;">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>&amp; ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="c&otilde;po&longs;ito">compo&longs;ito</expan> <lb/>ex 5 &amp; 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relinquat 5, igitur neque talis numerus pot&eacute;rit <expan abbr="c&otilde;poni">componi</expan> ex duo&shy;<lb/>bus quadratis, in quib. </s>
          <s id="id002696">&longs;uper&longs;it 5 &amp; 1, quia nullus e&longs;t, in quo &longs;uper&shy;<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de&shy;<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, &amp; relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, &amp; ita &longs;i ducerem 28 in &longs;e, &amp; diuide&shy;<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, &amp; tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e &amp; fit 784, &amp; diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;&aelig; per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dra&shy;tum">quadra&shy;<lb/>tum</expan> 2 e&longs;t 4, &amp; <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, &amp; iuncta faciunt 8, in quod ̊&longs;upere&longs;t 2, <expan abbr="c&otilde;">con</expan>fla&shy;<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="qu&aelig;&longs;it&utilde;">qu&aelig;&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>
          <s id="id002697"><expan abbr="duo&shy;r&utilde;">duo&shy;<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui&shy;<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari&shy;<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>
          <s id="id002698">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u&shy;<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus&apos;">quadratus&apos;</expan>.</s>
        </p>
        <p type="margin">
          <s id="id002699"><margin.target id="marg520"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E <emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002700">His ui&longs;is dico, quod con&longs;tat radices talium numerorum opor&shy;<lb/>tere e&longs;&longs;e in imparibus per additionem 6 incipiendo &agrave; 3, ut &longs;int <lb/>3. 9. 15. 21. 27. 33. 39. 45. 51. &amp; &longs;ic deinceps: in paribus au&shy;<lb/>tem per additionem eiu&longs;dem 6 incipiendo &agrave; 6, uelut 6. 12. <lb/>18. 24. 30. 36. 42. 48. 54. 60. Dico ergo quod diui&shy;<lb/>&longs;o numero illo compo&longs;ito per 6 in imparibus exibit numerus, 


<pb pagenum="152" xlink:href="015/01/171.jpg"/>qui diui&longs;us per 6 &longs;upererit 3, &amp; in paribus qui poterit diuidi per 6. <lb/>Quia <expan abbr="componun&ttilde;">componuntur</expan> ex huiu&longs;modi: uelut 3 in &longs;e facit 9, &amp; 25 in &longs;e facit <lb/>225, qui <expan abbr="i&utilde;cti">iuncti</expan> <expan abbr="faci&utilde;t">faciunt</expan> 234, diui&longs;o 235 per 6 exit 39, qui <expan abbr="iter&utilde;">iterum</expan> diui&longs;us per 6 <lb/>&longs;upere&longs;t 3, &amp; &longs;imiliter capio 6 &amp; 12, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="&qtilde;drata">quadrata</expan> &longs;unt 36 &amp; 144, &amp; <lb/><expan abbr="aggregat&utilde;">aggregatum</expan> 180, qui diui&longs;us per 6 exit 30, qui <expan abbr="iter&utilde;">iterum</expan> pote&longs;t diuidi per <lb/>6. Et hoc quia <expan abbr="quilibetillor&utilde;">quilibet illorum</expan> pote&longs;t diuidi per <expan abbr="&qtilde;drat&utilde;">quadratum</expan> 6 in paribus, <lb/>ergo aggregato diui&longs;o per 6 &qring;d prodit, <expan abbr="iter&utilde;">iterum</expan> poterit diuidi per 6. <lb/>Et in imparibus quodlibet <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> exuperat &longs;upra &longs;enarios in 3, <lb/><expan abbr="igi&ttilde;">igitur</expan> <expan abbr="aggregat&utilde;">aggregatum</expan> diui&longs;um in 2 pariet <expan abbr="numer&utilde;">numerum</expan> qui diui&longs;us per 3, exibit <lb/>numerus impar <expan abbr="c&otilde;po&longs;itus">compo&longs;itus</expan> ex &longs;enarijs &amp; 3. Illud ergo <expan abbr="quadrat&utilde;">quadratum</expan>, &qring;d <lb/>prodibit, uel erit <expan abbr="c&otilde;po&longs;itum">compo&longs;itum</expan> ex &longs;enarijs, uel &longs;upererit 3. Sed <expan abbr="c&utilde;">cum</expan> 3 nume <lb/>ret 6, ergo tres <expan abbr="&qtilde;drati">quadrati</expan> numeri &longs;cilicet duo, qui <expan abbr="c&otilde;ponunt">componunt</expan> <expan abbr="numer&utilde;">numerum</expan>, <lb/><arrow.to.target n="marg521"/><lb/>&amp; qui prodit per <expan abbr="diui&longs;ion&etilde;">diui&longs;ionem</expan> 6, erunt <expan abbr="c&otilde;po&longs;iti">compo&longs;iti</expan> inter &longs;e, ergo &amp; radices il<lb/>lorum. </s>
          <s id="id002701"><expan abbr="Igi&ttilde;">Igitur</expan> radix numeri <expan abbr="&qtilde;drati">quadrati</expan>, qui prouenit diui&longs;o aggregato <expan abbr="qua&shy;drator&utilde;">qua&shy;<lb/>dratorum</expan> per 6 e&longs;t ex <expan abbr="eod&etilde;">eodem</expan> ordine <expan abbr="impari&utilde;">imparium</expan>, &longs;i impares numeri <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="fuer&utilde;t">fuerunt</expan>, aut <expan abbr="pari&utilde;">parium</expan> &longs;i pares. </s>
          <s id="id002702">At hoc e&longs;&longs;e <expan abbr="n&otilde;">non</expan> pote&longs;t, <expan abbr="n&atilde;">nam</expan> fracti illi numeri, <lb/>qui <expan abbr="er&utilde;t">erunt</expan> radices, <expan abbr="n&otilde;">non</expan> <expan abbr="er&utilde;t">erunt</expan> minimi, &longs;ed diui&longs;i per 3 o&longs;tendent minores, <lb/>quod e&longs;t contra &longs;uppo&longs;itum, quare nullo modo 6 pote&longs;t diuidi in <lb/>duos numeros quadratos, neque integros, neque fractos, quod erat <lb/>demon&longs;trandum. </s>
          <s id="id002703">Habes igitur ex hoc demon&longs;trationem quando <lb/><expan abbr="n&otilde;">non</expan> po&longs;sit diuidi, &amp; quando po&longs;sit, quod po&longs;sit, &amp; quomodo &longs;imul.</s>
        </p>
        <p type="margin">
          <s id="id002704"><margin.target id="marg521"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>&longs;e&shy;<lb/>ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002705">Propo&longs;itio cente&longs;ima quinquage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id002706">Horologiorum tempus multiplicare.<lb/><arrow.to.target n="marg522"/></s>
        </p>
        <p type="margin">
          <s id="id002707"><margin.target id="marg522"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002708">Contingit quandoque &qring;d <expan abbr="horologior&utilde;">horologiorum</expan> tem<lb/><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"/><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aut</expan> maius efficere: id <lb/>duob. </s>
          <s id="id002709">modis po&longs;&longs;umus, <expan abbr="quor&utilde;">quorum</expan> unus diffici&shy;<lb/>lior e&longs;t &longs;ed perpetuus, &amp; long&egrave; nobilior, nam <lb/>grauitas ponderis uer&longs;atilis efficit <expan abbr="quid&etilde;">quidem</expan> <expan abbr="tar&shy;dior&etilde;">tar&shy;<lb/>diorem</expan>, &longs;ed difficilius <expan abbr="mobil&etilde;">mobilem</expan>, &amp; ob id grauio&shy;<lb/>re <expan abbr="p&otilde;dere">pondere</expan> in<expan abbr="digent&etilde;">digentem</expan>. </s>
          <s id="id002710">Sit ergo rota a b uer&longs;ati&shy;<lb/>lis, qu&aelig; certam men&longs;uram exigit pro quacunque funis parte corre&longs;peron<lb/>dentis uni denti ex centum, in quos di&longs;tincta &longs;it, curriculum <expan abbr="a&utilde;t">aut</expan> c d <lb/>quinque <expan abbr="denti&utilde;">dentium</expan>, per &qring;drota &longs;exaginta dentes <expan abbr="hab&etilde;s">habens</expan> <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> in <lb/><expan abbr="c&otilde;uer&longs;ione">conuer&longs;ione</expan>, <expan abbr="igi&ttilde;">igitur</expan> prim&etail; rot&etail; uities <expan abbr="circumfere&ttilde;">circumferetur</expan>, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> <expan abbr="d&etilde;tesque">dentesque</expan> M. CC. <lb/>rur&longs;us ad <expan abbr="h&atilde;c">hanc</expan> <expan abbr="&longs;ecund&atilde;">&longs;ecundam</expan> tertia <expan abbr="necta&ttilde;">nectatur</expan> cum curriculo &longs;ex <expan abbr="denti&utilde;">dentium</expan>, atque in <lb/>ea <expan abbr="d&etilde;tes">dentes</expan> &longs;eptuaginta duo, ut in una <expan abbr="c&otilde;uer&longs;ione">conuer&longs;ione</expan> &longs;int xiiij cccc, dentes <lb/><expan abbr="igi&ttilde;">igitur</expan> tot dentes in una <expan abbr="c&otilde;uer&longs;ione">conuer&longs;ione</expan> prim&etail; rot&etail; circumuoluentur. </s>
          <s id="id002711">Iam <lb/>uer&ograve; tempus illud poterit duplicari ac triplicari iuxta <expan abbr="tarditat&etilde;">tarditatem</expan> tem<lb/>poris uer&longs;atilis: <expan abbr="qu&atilde;to">quanto</expan> <expan abbr="igi&ttilde;">igitur</expan> pondero&longs;ius fuerit illud <expan abbr="t&etilde;pus">tempus</expan>, tanto tar&shy;<lb/>dius <expan abbr="mouebi&ttilde;">mouebitur</expan>, pauciores que circumuolutiones nece&longs;&longs;ari&etail; <expan abbr="er&utilde;t">erunt</expan> ad <expan abbr="ex&shy;pl&etilde;dam">ex&shy;<lb/>plendam</expan> unam <expan abbr="di&etilde;">diem</expan>: id e&longs;t horas 24, &longs;ed hoc in <expan abbr="c&otilde;modi">commodi</expan> accedet, qu&ograve;d <lb/>reuolutio indicis tanto tardior erit, ut <expan abbr="n&otilde;">non</expan> iu&longs;t&egrave; o&longs;ten dat horas: pro&shy;


<pb pagenum="153" xlink:href="015/01/172.jpg"/>po&longs;itum igitur e&longs;t, ut pondera tardius ferantur, index <expan abbr="a&utilde;t">aut</expan>, &amp; qu&etail; ad <lb/>indicem &longs;equuntur horarum demon&longs;trationes celerius aut eodem <lb/>modo ferantur. </s>
          <s id="id002712">Ponamus ergo po&longs;t&lt;08&gt; eadem e&longs;t ratio celerioris &amp; <lb/>&aelig;qu&eacute; uelocis, ponderis <expan abbr="a&utilde;t">aut</expan> tardius de&longs;cendentis, aut <expan abbr="c&otilde;tr&agrave;">contr&agrave;</expan> tardio&shy;<lb/>ris, aut &aelig;qualiter circumducti in dicis, celerioris <expan abbr="a&utilde;t">aut</expan> de&longs;cen&longs;us pon&shy;<lb/>deris, quod ad nullam <expan abbr="utilitat&etilde;">utilitatem</expan> profuturum uideo. </s>
          <s id="id002713">Sit ergo ut pon<lb/>dus uelim tardius de&longs;cendere, rotam <expan abbr="a&utilde;t">aut</expan> &etail;qualiter circumferri, dico <lb/>quod ex tempore mobili &longs;eu uer&longs;atili &lpar;&amp; e&longs;t ferrum, quod in &longs;um&shy;<lb/>mo horologij citra ultraque <expan abbr="fer&ttilde;">fertur</expan> tam in horologijs ponderum &lt;08&gt; mo <lb/>l&aelig;&rpar; id fieri non pote&longs;t: nam quantum tardabitur rota tertia &longs;ecunda <lb/>&amp; prima, atque ob id de&longs;cen&longs;us ponderum, tantum remorabitur rota <lb/>prima qu&aelig; indicem o&longs;tendit, ergo tantum index tardabitur quan&shy;<lb/>tum <expan abbr="p&otilde;dera">pondera</expan>, &amp; ut uno uerbo dicam, c&ugrave;m <expan abbr="ead&etilde;">eadem</expan> rota index circumfe&shy;<lb/>ratur, &amp; <expan abbr="p&otilde;dus">pondus</expan> de&longs;cendat, <expan abbr="quant&utilde;">quantum</expan> unum tardatur tantum &amp; aliud.</s>
        </p>
        <p type="main">
          <s id="id002714">Secundus modus e&longs;t, ut rota una totum tempus cum indice in ui<lb/>gintiquatuor horis circumuoluatur, &amp; currulis in quo funis minor <lb/>fiat: nece&longs;&longs;e e&longs;t <expan abbr="igi&ttilde;">igitur</expan>, ut circumuoluta rota aut &longs;emel aut bis, <expan abbr="&ttilde;er">tur</expan>, qua&shy;<lb/>ter decies, &amp; <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> pleno circuitu index, et &longs;ine errore: quo&shy;<lb/>niam tempus &amp; dentes men&longs;ur&aelig; re&longs;pondent: igitur &longs;ub ei&longs;dem cir&shy;<lb/>cuitibus numero eodemque tempore minus ex fune <expan abbr="de&longs;cend&etilde;t">de&longs;cendent</expan> in cur<lb/>ruli paruo &lt;08&gt; magno: quare mutatione indiget currulis, aut ut funis <lb/>circumuoluens rotam curriculum habeat <expan abbr="annex&utilde;">annexum</expan> rot&aelig; o&longs;ten denti <lb/>horas, in qua pauciores &longs;int dentes: nam in eodem tempore, &amp; cir&shy;<lb/>cuitu paucioribus uicibus circumuoluitur rota funis qu&aelig; grauita&shy;<lb/>te temporis, &amp; multitudine <expan abbr="denti&utilde;">dentium</expan> certam <lb/><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"/><lb/>&longs;eruabit <expan abbr="men&longs;ur&atilde;">men&longs;uram</expan>. </s>
          <s id="id002715">Sed in hoc nece&longs;&longs;e e&longs;t gra<lb/>uius efficere pondus, aut leuius <expan abbr="t&etilde;pus">tempus</expan> <expan abbr="quo&shy;ni&atilde;">quo&shy;<lb/>niam</expan> funis debilius circumuertit <expan abbr="rot&atilde;">rotam</expan>: minus <lb/><expan abbr="t&ntilde;">tn</expan> tard&egrave; quod &longs;it pro paruitatis circuitus ratione.</s>
        </p>
        <p type="main">
          <s id="id002716">Tertius modus facilior e&longs;t, &amp; magis com<lb/><expan abbr="p&etilde;dio&longs;us">pendio&longs;us</expan>: Sit horologium a b c, in quo rota <lb/>d qu&aelig; funem <expan abbr="c&otilde;tinet">continet</expan> ba&longs;is horologij e f, cui <lb/>firmiter &longs;int <expan abbr="app&etilde;&longs;&etail;">appen&longs;&etail;</expan> du&etail; trochle&etail; g &amp; h, &amp; fu <lb/>nis una parte trochle&etail; appen&longs;us in k, <expan abbr="duca&ttilde;">ducatur</expan> <lb/>ad inferiorem aliam trochleam l in&longs;eraturque <lb/>ibi orbiculo &longs;uo, &amp; redeat &agrave; dextra &longs;uperius <lb/><expan abbr="in&longs;era&ttilde;que">in&longs;eraturque</expan> orbiculo &longs;uperioris trochle&etail;, dedu<lb/><expan abbr="ca&ttilde;que">caturque</expan> uer&longs;us <expan abbr="&longs;ini&longs;tr&atilde;">&longs;ini&longs;tram</expan>: atque ibi <expan abbr="de&longs;cend&etilde;s">de&longs;cendens</expan> habe <lb/>at <expan abbr="p&otilde;dus">pondus</expan> tractorium in m, <expan abbr="deduca&ttilde;que">deducaturque</expan> &longs;upra <lb/>ad <expan abbr="rot&atilde;">rotam</expan> horologij d, et circumuolutus exeat <lb/>ip&longs;um, &amp; <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> ad tro<expan abbr="chle&atilde;n">chlean</expan>, &longs;ub que ea circumuolutus <expan abbr="iter&utilde;">iterum</expan> a&longs;cen<pb pagenum="154" xlink:href="015/01/173.jpg"/>dat &agrave; dextra parte, et circumuoluatur h cochle&etail; rediens ad &longs;ini&longs;tram <lb/>ibique de&longs;cendens connectatur trochle&aelig; in inferiori in o, cuius im&aelig; <lb/>parti annectatur pondus remorans in imo annexum parte troch&shy;<lb/>le&aelig; p. </s>
          <s id="id002717">Cum ergo trahitur n trochlea, trahitur funis ade&ograve; ut pon&shy;<lb/>dus m, tandem a&longs;cendat cum trochlea l prope k: quia ergo in duo&shy;<lb/>decim horis pondus m de&longs;cenderet per k l funem reuolutionibus <lb/>circa d rotam dicamus uiginti, ergo &longs;i debet de&longs;cendere &agrave; k ad l, per <lb/>funem duplicatam k l cum ip&longs;am nece&longs;&longs;e &longs;it obequitantem d reuo&shy;<lb/>lutionibus quadraginta circumuolui d, nam tota o h n d m g l k lon<lb/>g&egrave; maior e&longs;t duplo k l, nece&longs;&longs;e e&longs;t m de&longs;cendere tardius qu&agrave;m in du<lb/>plo temporis, quo de&longs;cenderet per rectum funem k l, quod erat de&shy;<lb/>mon&longs;trandum. </s>
          <s id="id002718">Et hanc appendicem uidi apud C&aelig;&longs;arem Odonum <lb/>Apulum medicum, uirum elegantem lepidique ingenij. </s>
          <s id="id002719">Memento <lb/>uer&ograve; quod ubi orbiculi non cederent funi, uel quia duriores in cir<lb/>cumuolutione, uel quia latius exciperent illum reduplicato fune <lb/>circa illos omnin o circumducuntur, &longs;ed difficilius ide&ograve; egent gra&shy;<lb/>uiori pondere.</s>
        </p>
        <p type="main">
          <s id="id002720">Propo&longs;itio cente&longs;ima quinquage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id002721">Horologiorum molarium rationem o&longs;tendere.</s>
        </p>
        <p type="main">
          <s id="id002722"><arrow.to.target n="marg523"/></s>
        </p>
        <p type="margin">
          <s id="id002723"><margin.target id="marg523"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002724">Sunt horum duo genera primum, &amp; anti<lb/><figure id="id.015.01.173.1.jpg" xlink:href="015/01/173/1.jpg"/><lb/>quius licet multo po&longs;terius eo quod pon&shy;<lb/>deribus ducitur, quod funiculo ex inte&longs;ti&shy;<lb/>nis ouium &longs;eu fidibus lir&aelig; agitur. </s>
          <s id="id002725">Sit igitur <lb/>axis f k erectus &longs;uper plano, cui per longum <lb/>coniuncta mola multiplicis &longs;pir&aelig; in fine, cu<lb/>ius c annectatur ferreo circulo, qui habeatur loco cap&longs;ul&aelig; b c, qu&aelig; <lb/>circumuolui po&longs;sit: huic <expan abbr="circ&utilde;ductus">circunductus</expan> funis d e multipliciter in pun<lb/>cto g, &longs;it autem e h in modum pyramidis &longs;en&longs;im in acutum, &longs;ed non <lb/>ualde per <expan abbr="&longs;pir&atilde;">&longs;piram</expan> exculptam de&longs;inentis, cui rota in uertice in&longs;erta den<lb/>&longs;iculo, &amp; uertatur h e, colligens funiculum tractum in &longs;pira uer&longs;us <lb/>apicem: unde funiculus circumuoluet b g d, <expan abbr="cap&longs;ul&atilde;">cap&longs;ulam</expan> uer&longs;us c, trahet <lb/>ergo molam, &amp; con&longs;tringet uiolenter <expan abbr="qu&atilde;tum">quantum</expan> fert longitudo funis <lb/>qu&aelig; circumuolui pote&longs;t a b e ad h: &amp; cum trahitur in d eremittitur, <lb/>non pote&longs;t mola &longs;tatim retrahere reluctantibus denticulis h l rot&aelig;, <lb/>&amp; alijs qu&aelig; implicantur curriculo m, a igitur mola con&longs;tructa uio&shy;<lb/>lenter mouet b g d, cap&longs;ulam motu contrario &agrave; c in d &amp; in g &amp; in b, <lb/>quare funis d e trahitur, &amp; trahit e h illum circumuoluendo contra&shy;<lb/>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb/>aliam <expan abbr="rot&atilde;">rotam</expan>, &amp; &longs;ic deinceps donec tempus moueatur, &amp; rota indicis. <lb/></s>
          <s id="id002726">Hic ade&longs;t cap&longs;ula, &amp; quod circumuertitur &agrave; claue non e&longs;t axis mol&etail; <lb/>&longs;ed extra molam, &longs;cilicet e h. </s>
          <s id="id002727">Et quoniam hac ratione quanto mola a 


<pb pagenum="155" xlink:href="015/01/174.jpg"/>magis <expan abbr="explicabi&ttilde;">explicabitur</expan>, tanto lentius trahet, &amp; uertet e h, ide&ograve; hoc ex &longs;tru<lb/>ctura auxilium pr&aelig;&longs;tatur, ut funis in inferiore parte <expan abbr="c&otilde;plexus">complexus</expan> latio&shy;<lb/>res orbes, &amp; &egrave; regione tanto uehementius uertat e h: &amp; ita uis qu&aelig; <lb/>remittitur ob mol&aelig; laxitatem, augetur tantundem ob &longs;itum &amp; ma&shy;<lb/>gnitudinem &longs;pirarum ut di&longs;tantiorum &longs;ua extremitate ab hypomo<lb/>chlio, quod e&longs;t axis coni e h, &longs;eu in&longs;tar axis.</s>
        </p>
        <p type="main">
          <s id="id002728">Alterum genus horologiorum cum mola &longs;ine fune loco cap&longs;ul&etail; <lb/>habet <expan abbr="rot&atilde;">rotam</expan> plano &longs;ub &longs;tratam, plenam denticulis axis, quo circum&shy;<lb/>agitur uiolenter, non e&longs;t extra molam, &longs;ed ei annexa e&longs;t mola intus, <lb/>exterius <expan abbr="a&utilde;t">aut</expan> rot&etail;; ergo circumducto axe mol&etail; uim patitur circulus <lb/>exterior, &longs;ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. </s>
          <s id="id002729">Vbi mola quan&shy;<lb/>tum decet con&longs;tricta e&longs;t &longs;ublato clauo &longs;tatim &longs;ecum trahit rotam, &amp; <lb/>illa <expan abbr="curricul&utilde;">curriculum</expan> rotas que alias, &amp; tempus agitur, &amp; index uertitur. </s>
          <s id="id002730">Sed <lb/>in hoc idem e&longs;t in commodum &longs;ine remedio <lb/><figure id="id.015.01.174.1.jpg" xlink:href="015/01/174/1.jpg"/><lb/>quod fuit in priore. </s>
          <s id="id002731">Vbi enim c&oelig;perit laxa&shy;<lb/>ri mola tanto tardius progrediuntur rot&aelig; <lb/>atque index. </s>
          <s id="id002732">Veluti axis a b cui &longs;ecun dum lon<lb/>gitudinem mol&aelig; caput interius annexum <lb/>e&longs;t altero circulo rot&aelig; in c d curriculum rot&aelig; e, implexum rot&aelig; f <lb/>clauus rotam retinens, donec circumducto a b mola con&longs;tringa&shy;<lb/>tur, &amp; latus eius trahat rotam ex c. </s>
          <s id="id002733">Inde &longs;ublato clauo circulus, &longs;eu <lb/>rota trahitur ex c in g, &amp; in famola, qu&aelig; etiam &longs;ecundum eandem <lb/>partem circumuoluta e&longs;t: igitur d circumagetur &agrave; rota &amp; reliqua. <lb/></s>
          <s id="id002734">Sed ut dixi con&longs;tructio h&aelig;c non &longs;atisfacit.</s>
        </p>
        <p type="main">
          <s id="id002735">Aliam ergo oportuit excogitare qu&etail; huiu&longs;modi e&longs;t. </s>
          <s id="id002736">Sub axe a b, <lb/>qui circumuertitur ad molam contrahendam rotam, collocant par <lb/>uam qu&aelig; e&longs;t, ut ita dicam, pars axis ima cui in&longs;eruntur dentes in am<lb/>bitu ea ratione, ut dum mola tenditur, premant denticulos interio&shy;<lb/>res, atque ita elabitur, totiesque circumducitur manente g f, donec <lb/>colligatur mola, qu&aelig; non ut in priore reliquo extremo ulli rot&aelig; <lb/>affixa e&longs;t, &longs;ed column&aelig; in continenti <lb/>opercula horologij. </s>
          <s id="id002737">Cum ergo mola <lb/>tenta retrahat axem a b contrario mo&shy;<lb/><figure id="id.015.01.174.2.jpg" xlink:href="015/01/174/2.jpg"/><lb/>tu, &amp; ille rotam mobilem, qu&aelig; cum <lb/>non po&longs;sit regredi propter auer&longs;os <lb/>dentes, mouet rotam f g contrario mo<lb/>tu, qu&aelig; circumacta per denticulos &longs;u&shy;<lb/>os curriculum agit, &amp; reliqua omnia <lb/>nece&longs;&longs;aria. </s>
          <s id="id002738">Cur autem cum laxatur mo <lb/>la, &amp; uertit lentius c e rotam coniun&shy;<lb/>ctam, ideoque g f, &amp; reliqua omnia <expan abbr="n&otilde;">non</expan> tardetur tempus, &amp; circumuo&shy;


<pb pagenum="156" xlink:href="015/01/175.jpg"/>lutio indicis cau&longs;a e&longs;t alia long&egrave; qu&agrave;m in priore, nam mola longior <lb/>fit cra&longs;sior, &amp; durior adeoque robu&longs;ta, &amp; rot&aelig; leues, ac tempus dum <lb/>laxata fuerit munus &longs;uum iu&longs;to in tempore obeant: quare nece&longs;&longs;e <lb/>e&longs;t, ut ab initio uehementius agat, &amp; celerius rotam cum axe qui tra<lb/>hitur &agrave; mola. </s>
          <s id="id002739">Ergo excogitarunt aliud genus retinaculi forma co&shy;<lb/>chle&aelig; quod ab initio moratur <expan abbr="uehem&etilde;ter">uehementer</expan> axem ne circumagatur, et <lb/>quanto magis mola explicatur eo minus retinet <expan abbr="impet&utilde;">impetum</expan> illius, adeo <lb/>ut uehementer retineat uehementem concitationem mediocriter <lb/>moderatam, &longs;egniter lentam, nullo modo iu&longs;tam: ita fit, ut &longs;emper <lb/>ferm&egrave; &aelig;qualiter moueatur. </s>
          <s id="id002740">Difficile e&longs;t tamen ad unguem &longs;eruare <lb/>moderationem, &amp; &aelig;qualitatem, &amp; magis etiam in his horologijs, <lb/>qu&aelig; uno circuitu mol&aelig; tempus <expan abbr="l&otilde;gius">longius</expan> exigunt: at difficilius etiam <lb/>efficere molam, qu&aelig; longo tempore duret, cum intenta ualde cele&shy;<lb/>rius moueat rotas, &amp; ob id breui ab&longs;oluat circuitum, mollior au&shy;<lb/>tem cit&ograve; remittatur. </s>
          <s id="id002741">Et ob id longior &amp; non ade&ograve; <lb/>dura melior e&longs;t. </s>
          <s id="id002742">Ratio autem cochle&aelig; ita &longs;e habet. <lb/><figure id="id.015.01.175.1.jpg" xlink:href="015/01/175/1.jpg"/><lb/>Circa axem mol&aelig; d deducitur cochlea a b c, qu&aelig; <lb/>dum laxatur mola cochlea mouetur ex b in c, atque<lb/> ita pariter laxatur uis cochle&aelig; retinentis axem.</s>
        </p>
        <p type="main">
          <s id="id002743">Propo&longs;itio cente&longs;ima quinquage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id002744">Rationem indicis mobilis cum rota horarum numerus per ictus <lb/>indicatur explicare.</s>
        </p>
        <p type="main">
          <s id="id002745"><arrow.to.target n="marg524"/></s>
        </p>
        <p type="margin">
          <s id="id002746"><margin.target id="marg524"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002747">Hoc fieri pote&longs;t in &longs;ingulo genere horologij trium <expan abbr="de&longs;criptor&utilde;">de&longs;criptorum</expan>. <lb/></s>
          <s id="id002748">Propterea &longs;ufficiat de uno o&longs;tendi&longs;&longs;e. </s>
          <s id="id002749">Sed &amp; in &longs;ingulo genere &longs;unt <lb/>multi modi, unius tamen reddidi&longs;&longs;e <expan abbr="ration&etilde;">rationem</expan> &longs;ufficiat. </s>
          <s id="id002750">Hoc <expan abbr="a&utilde;t">aut</expan> qua&shy;<lb/>tuor habet difficultates: prima ut horarum ictus conueniant cum <lb/>indice: &longs;ecunda ut conuer&longs;o indice conuertatur, &amp; rota ictuum: ter<lb/>tia ut ictuum numerus cum numero indicis conueniat. </s>
          <s id="id002751">Vnde mul&shy;<lb/>ta &longs;unt horologia, in quibus ictus unus &longs;olum auditur &longs;ingulis ho&shy;<lb/>ris, atque hic modus facilis e&longs;t: quarta cur in horum pleri&longs; que &longs;i non <lb/>pul&longs;ata &longs;tatim hora <expan abbr="transfera&ttilde;ur">transferatur</expan> index, non ce&longs;&longs;at pul&longs;atio: im&ograve; nec <lb/>retineri pote&longs;t, donec pondus illud de&longs;cenderit. </s>
          <s id="id002752">Ergo primi &amp; ter&shy;<lb/>tij ratio h&aelig;c habeatur, cum rota qu&etail; indicis rotam circumagit, per&shy;<lb/>uenerit ad hor&aelig; finem, denticulo &longs;oluit aliam, eleuans obicem, illa <lb/>mouetur &agrave; pondere proprio alio, &longs;cilicet ab illo quod tempus agit: <lb/>aut &longs;i &longs;it horologium mol&aelig; &agrave; mola alia propria, qu&aelig; malleos cir&shy;<lb/>cumacta perpetu&ograve; mouet, atque motura e&longs;&longs;et &longs;emper, donec pondus <lb/>ad terram de&longs;cenderet: uerum dum mouetur de&longs;cendit ferrum pro <lb/>quouis ictu quod in rot&aelig; limbum incidit, &amp; donec inciderit in eam <lb/>partem qu&aelig; lenis e&longs;t dilabitur, nec retinetur, &amp; ita eleuatur rur&longs;us, 


<pb pagenum="157" xlink:href="015/01/176.jpg"/>at uero cum in concauam partem incidit retineri nece&longs;&longs;e e&longs;t: atque ita <lb/>pondus non amplius de&longs;cendit, rota &longs;i&longs;titur, malleus manet immo&shy;<lb/>bilis: &longs;patia ergo qu&aelig; &longs;unt inter cauitates &longs;unt &longs;ecundum magnitu&shy;<lb/>dinem proportionis numer&oacute;rum <expan abbr="horar&utilde;">horarum</expan>, uel ad &longs;ex, uel ad duode&shy;<lb/>cim, uel ad uiginti &shy;<lb/><figure id="id.015.01.176.1.jpg" xlink:href="015/01/176/1.jpg"/><lb/> quatuor terminan&shy;<lb/>tium. </s>
          <s id="id002753">Ita quod, gra&shy;<lb/>tia exempli, &longs;it iam <lb/>in cauitate a duode&shy;<lb/>cim&etail; hor&aelig; uncus, di<lb/>uidam circulum to&shy;<lb/>tum in duas partes <lb/>&aelig;quales, quia in &longs;in <lb/>gulis medietatibus <lb/>propo&longs;itum e&longs;t, duo<lb/>decim facere cauita&shy;<lb/>tes pro unco retinen&shy;<lb/>do. </s>
          <s id="id002754">Et quia in una&shy;<lb/>quaque medietate o&shy;<lb/>portet, ut pul&longs;ent ho<lb/>r&aelig; lxxviij, &amp; pr&aelig;terea &longs;int ibi &longs;ex &longs;patia cauitatum, quarum &longs;ingul&aelig; <lb/>contineant, gratia exempli, duo &longs;patia unius ictus, ut certius retinea <lb/>tur uncus, <expan abbr="er&utilde;t">erunt</expan> igitur &longs;patia omnia nonaginta: diuidemus ergo me&shy;<lb/>dietatem circuli utranque in nonaginta partes &aelig;quales incipiendo <lb/>ab a, &amp; dabimus b prim&aelig; hor&etail; quod &longs;patium e&longs;t unius tantum par<lb/>tis ex nonaginta, po&longs;t de&longs;cribemus c cauitatem duarum partium, <lb/>ita ubi ictum unum dederit uncus, retinebitur in c, p&ograve;&longs;t accipiemus <lb/>duo &longs;patia, &amp; &longs;int &longs;ignificata d litera, po&longs;t qu&etail; faciemus cauitatem e: <lb/>&amp; ita uncus bis cadet in d, &amp; pul&longs;abunt duo ictus, &amp; p&ograve;&longs;t retinebi&shy;<lb/>tur uncus in e. </s>
          <s id="id002755">Et po&longs;t accipiam &longs;patium trium partium, quod &longs;it f, <lb/>&amp; po&longs;t de&longs;cribam cauitatem g duarum partium, atque ita procedam <lb/>u&longs;que ad duodecim.</s>
        </p>
        <p type="main">
          <s id="id002756">Ex quo manife&longs;tum e&longs;t pondus quod agit rotam uol&aelig; non de&shy;</s>
        </p>
        <p type="main">
          <s id="id002757"><arrow.to.target n="marg525"/><lb/>&longs;cendere, ni&longs;i dum hor&aelig; pul&longs;ant, &longs;ecus quie&longs;cere.</s>
        </p>
        <p type="margin">
          <s id="id002758"><margin.target id="marg525"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002759">Secundum, qu&ograve;d de&longs;cendit illud pondus plus &amp; minus, iuxta <lb/><arrow.to.target n="marg526"/><lb/>proportionem numeri horarum, ita quod quando pul&longs;abit una ho <lb/>ra parum ualde de&longs;cendet, cum &longs;ex hor&aelig; &longs;excuplo magis, cum duo&shy;<lb/>decim adhuc long&egrave; magis, id e&longs;t duplo plus qu&agrave;m cum pul&longs;ant <lb/>&longs;ex hor&aelig;.</s>
        </p>
        <p type="margin">
          <s id="id002760"><margin.target id="marg526"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002761">Secunda con&longs;tructio hanc habet rationem: Cum n rota indicis <lb/>coniuncta fuerit rot&aelig;, qu&aelig; transfert malleum, nece&longs;&longs;e e&longs;t ut un&agrave; fe&shy;


<pb pagenum="158" xlink:href="015/01/177.jpg"/>rantur: qui nim&ograve; illud magis mirum de quo illi non mirantur quia <lb/>frequens e&longs;t, &longs;cilicet cur aut quomodo &longs;i diui&longs;&aelig; &longs;unt ut cir<expan abbr="&ccedil;&utilde;ducto">&ccedil;unducto</expan> <lb/>indice non transferatur rota mallei, <expan abbr="p&otilde;dere">pondere</expan> tamen uer&longs;ata rota in&shy;<lb/>dicis in idem incidat, ut hor&aelig; qu&aelig; pul&longs;u declarantur ad unguem <lb/>&amp; in ei&longs;dem &longs;ectionibus <expan abbr="c&otilde;ueniant">conueniant</expan> cum horis quas index o&longs;tendit.</s>
        </p>
        <p type="main">
          <s id="id002762">Verum quia multis modis contingit ordinem horologiorum <lb/>peruerti: in &longs;imilibus quidem &longs;i hora indicis &longs;imul &amp; pul&longs;us un&agrave; <lb/>circumferuntur, &longs;ed tardius ambo index traducitur ad locum debi&shy;<lb/>tum, inde ponderi aliquid additur. </s>
          <s id="id002763">Si uer&ograve; ant&egrave; proce&longs;&longs;erit quam. <lb/></s>
          <s id="id002764">Sol in dicet ablato pondere, &longs;ines tempus fluere u&longs;que ad indicis lo&shy;<lb/>cum &longs;ine motu horologij, pondus quoque ip&longs;um minues. </s>
          <s id="id002765">At &longs;i pon&shy;<lb/>dus pul&longs;us in terram deuenerit uel prop&egrave;, expecta donec &longs;uper li&shy;<lb/>nea index fuerit, inde trahe, neque. </s>
          <s id="id002766">n. </s>
          <s id="id002767">excurret: nam &longs;i dum index e&longs;t in <lb/>medio hor&aelig; aut prop&egrave;, traxeris pondus pul&longs;us, non de&longs;inet de&longs;cen <lb/>dere, pul&longs;abuntq&uacute;e hor&aelig; donec ad terram pondus deuenerit, <lb/>qu&ograve;d &longs;i iam in errorem incideris pul&longs;entque hor&etail; &amp; de&longs;cendat, pon&shy;<lb/>dus, &longs;en&longs;im deducito indicem, cum. </s>
          <s id="id002768">n. </s>
          <s id="id002769">ad finem hor&etail; peruenerit ini&shy;<lb/>tiumque &longs;equentis, quoniam ferrum in interuallum deuenerit rota &amp; <lb/>pondus firmabitur. </s>
          <s id="id002770">Inde &longs;ublato <expan abbr="p&otilde;dere">pondere</expan> donec Sol ad <expan abbr="hor&atilde;">horam</expan> quam <lb/>index mon&longs;trat peruenerit, reddes pondus horologio. </s>
          <s id="id002771">Si ergo ho&shy;<lb/>ram pul&longs;u <expan abbr="eand&etilde;">eandem</expan> declarat quam index, bene e&longs;t, &longs;i non, <expan abbr="paulul&utilde;">paululum</expan> <expan abbr="uir&shy;gul&atilde;">uir&shy;<lb/>gulam</expan> eleua qu&etail; e&longs;t iuxta fores horologij pul&longs;abitque &longs;equens hora, id <lb/>uero toties repetes immoto in dies &amp; &longs;ublato, &longs;i uereris ne extra <expan abbr="in&shy;teruall&utilde;">in&shy;<lb/>teruallum</expan> ferrum feratur, &amp; ob id excurrat rota pul&longs;us <expan abbr="horar&utilde;">horarum</expan>, donec <lb/>hora pul&longs;et qu&aelig; cum indice conuenit, &longs;tatimque pondus quo hor&aelig; <lb/>pul&longs;ant &longs;ur&longs;um retrahes. </s>
          <s id="id002772">His quinque regulis u&longs;um di&longs;ces &longs;imilium <lb/>horologiorum, unumquodque autem proprias habet: &longs;ed du&aelig; pri&shy;<lb/>m&aelig; omni horologi&aelig; &longs;atisfaciunt. </s>
          <s id="id002773">Qu&ograve;d &longs;i h&aelig; non &longs;atisfaciunt iam <lb/>horologium laborat: tum uer&ograve; illud di&longs;&longs;oluere oportet &amp; deterge&shy;<lb/>re &amp; inungere, iuuat autem uel cap&longs;ula uel linteo perpetuo pul&shy;<lb/>uerem ab illo arcere. </s>
          <s id="id002774">Qu&ograve;d &longs;i nec &longs;ic re&longs;tituitur nece&longs;&longs;e e&longs;t di&longs;&longs;ol&shy;<lb/>uere &amp; antea con&longs;iderare impedimentum, p&ograve;&longs;t denticulum qui la&shy;<lb/>borat, plerunque. </s>
          <s id="id002775">n. </s>
          <s id="id002776">aliquem inuenies huius modi, quem lima aut alia <lb/>ratione re&longs;titues, &longs;emper aut&eacute;m hi ferm&egrave; re&longs;tituuntur: at qui mola <lb/>aguntur pr&aelig;ter rotarum &amp; axium &amp; indicum labores, mol&aelig; etiam <lb/>in&aelig;qualitati &amp; defectibus &longs;ubiciuntur, qui &longs;i nimis uelo citer agunt <lb/>rotas cum difficultate re&longs;tituuntur moderationi, &longs;i lentius rar&ograve; uel <lb/>nunquam emendantur, uix etiam noua inducta mola.</s>
        </p>
        <p type="main">
          <s id="id002777">Propo&longs;itio cente&longs;ima quinquage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id002778">Nullus angulus rectilineus &aelig;qualis e&longs;&longs;e pote&longs;t alicui angulo con<lb/>tento recta &amp; circuli portione.</s>
        </p>
        <pb pagenum="159" xlink:href="015/01/178.jpg"/>
        <p type="main">
          <s id="id002779">Sit angulus a &amp; circulus b c, dico non po&longs;&longs;e aliquem angulum <lb/><arrow.to.target n="marg527"/><lb/>contentum recta &amp; circuli portione e&longs;&longs;e illi <lb/><figure id="id.015.01.178.1.jpg" xlink:href="015/01/178/1.jpg"/><lb/>&aelig;qualem. </s>
          <s id="id002780">&longs;i enim e&longs;&longs;e po&longs;sit, &longs;it c b e. </s>
          <s id="id002781">duca&shy;<lb/>tur recta b d faciens rectilineum d b c &etail;qua<lb/><arrow.to.target n="marg528"/><lb/>lem a, erit igitur d b c &etail;qualis e b c per com&shy;<lb/>munem animi &longs;ententiam, &longs;eu ergo b d ca&shy;<lb/>dat intra circulum &longs;eu extra, erit pars &etail;qua&shy;<lb/>lis toti quod e&longs;&longs;e non pote&longs;t. </s>
          <s id="id002782">Sed neque po&shy;<lb/>te&longs;t cadere recta &longs;uper b e. </s>
          <s id="id002783">nam id e&longs;t contra demon&longs;trata ab Eucli&shy;<lb/><arrow.to.target n="marg529"/><lb/>de. </s>
          <s id="id002784">At &longs;i &longs;it angulus c b e exterior &longs;imiliter producta b d, &longs;eu intus, <lb/>&longs;eu extr&agrave; cadat, pars erit &aelig;qualis toti quod e&longs;&longs;e non pote&longs;t.</s>
        </p>
        <p type="margin">
          <s id="id002785"><margin.target id="marg527"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002786"><margin.target id="marg528"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002787"><margin.target id="marg529"/>23. E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002788">Ex hoc patet quod nullus angulus peripheria circuli &amp; recta <expan abbr="c&otilde;&shy;">con&shy;<lb/></expan><arrow.to.target n="marg530"/><lb/>tentus pote&longs;t e&longs;&longs;e &aelig;qualis recto, quia rectus etiam rectilineus e&longs;t.</s>
        </p>
        <p type="margin">
          <s id="id002789"><margin.target id="marg530"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id002790">Et rur&longs;us nullus angulus peripheria &amp; <lb/><arrow.to.target n="marg531"/><lb/><figure id="id.015.01.178.2.jpg" xlink:href="015/01/178/2.jpg"/><lb/>recta contentus &agrave; recta linea per &aelig;qualia <lb/>diuidi pote&longs;t, patet quia una pars e&longs;&longs;et an&shy;<lb/>gulus rectilineus, alia contentus recta &amp; pe<lb/>ripheria: i&longs;ti <expan abbr="aut&etilde;">autem</expan> non po&longs;&longs;unt e&longs;&longs;e &aelig;quales, <lb/>quare nec prior potuit per &aelig;qualia diuidi.</s>
        </p>
        <p type="margin">
          <s id="id002791"><margin.target id="marg531"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id002792">Ex hoc etiam patet quod &longs;patium con&shy;<lb/><arrow.to.target n="marg532"/><lb/><expan abbr="tent&utilde;">tentum</expan> &agrave; peripheria circuli nulli angulo rectilineo &etail;quale e&longs;&longs;e pote&longs;t. <lb/></s>
          <s id="id002793">nam dimidium e&longs;&longs;et &aelig;quale dimidio, quod e&longs;t contra demon&longs;trata.</s>
        </p>
        <p type="margin">
          <s id="id002794"><margin.target id="marg532"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="head">
          <s id="id002795">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id002796">Inter duos circulos qui &longs;e diuidant infinit&aelig; line&aelig; duci po&longs;&longs;unt. <lb/></s>
          <s id="id002797">Inter circulos autem qui &longs;e tangant, recta linea duci non pote&longs;t.<lb/><arrow.to.target n="marg533"/></s>
        </p>
        <p type="margin">
          <s id="id002798"><margin.target id="marg533"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002799">Sint duo circuli a b &amp; a c, qui &longs;e diuidant </s>
        </p>
        <p type="main">
          <s id="id002800"><arrow.to.target n="marg534"/><lb/>in a, &amp; ducatur ex centro inferioris d a &amp; <lb/><figure id="id.015.01.178.3.jpg" xlink:href="015/01/178/3.jpg"/><lb/>a d, &amp; ad d a cathetus a e, dico qu&ograve;d a e di&shy;<lb/>uidet angulum b a c ducatur ex centro &longs;u&shy;<lb/><arrow.to.target n="marg535"/><lb/>perioris a c b quod &longs;it f, fa cui cathetus a g, <lb/>quia ergo e a cadit infra a g, &amp; inter a g &amp; <lb/><arrow.to.target n="marg536"/><lb/>a b non pote&longs;t duci recta, igitur e a cadit in&shy;<lb/><figure id="id.015.01.178.4.jpg" xlink:href="015/01/178/4.jpg"/><lb/>tra a c b circulum. </s>
          <s id="id002801">Rur&longs;us tangant &longs;e circuli <lb/>c d &amp; c e, &amp; ducatur a b per centra <expan abbr="eor&utilde;">eorum</expan> qu&etail; <lb/>applicabit ad c, ex c ducatur cathetus c f &amp; <lb/><expan abbr="quoni&atilde;">quoniam</expan> c f contangit <expan abbr="circul&utilde;">circulum</expan> c e, l igitur, du&shy;<lb/>cta quauis linea infra c f, cadet intra <expan abbr="circul&utilde;">circulum</expan> <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"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002804"><margin.target id="marg535"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002805"><margin.target id="marg536"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></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="&aelig;quali&utilde;">&aelig;qualium</expan> circulorum <lb/>&longs;e &longs;ecantium &aelig;qualem rectilineum illi fabricare.</s>
        </p>
        <pb pagenum="160" xlink:href="015/01/179.jpg"/>
        <p type="main">
          <s id="id002808">Sit angulus a b c duabus peripherijs &aelig;qualium circulorum con<lb/><arrow.to.target n="marg537"/><lb/>tentus, uolo ei &aelig;qualem rectilineum fabricare, ducantur b d &amp; b e <lb/><arrow.to.target n="marg538"/><lb/>&aelig;quales, ut pote facto b centro eritque angulus d b a &aelig;qualis angu&shy;<lb/>lo e b c, addito utrique communi d b e ex peri<lb/><figure id="id.015.01.179.1.jpg" xlink:href="015/01/179/1.jpg"/><lb/>pheria &amp; recta, fiet angulus d b e ex rectis <lb/>&aelig;qualis a b c ex peripherijs, quod crat de&shy;<lb/>mon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id002809"><margin.target id="marg537"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002810"><margin.target id="marg538"/>P<emph type="italics"/>er modum<emph.end type="italics"/><lb/>8. <emph type="italics"/>primi<emph.end type="italics"/> E<emph type="italics"/>l.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002811">Ex hoc patet quod reliqua duo &longs;patia <lb/><arrow.to.target n="marg539"/><lb/>non po&longs;&longs;unt e&longs;&longs;e &aelig;qualia rectilineo. </s>
          <s id="id002812">Nam <lb/>&longs;patium b a c demon&longs;tratum e&longs;t &aelig;quale e&longs;&shy;<lb/>&longs;e rectilineo, &amp; b ad non e&longs;t &aelig;quale rectili&shy;<lb/>neo, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c a d non pote&longs;t e&longs;&longs;e &aelig;quale <lb/>angulo rectilineo, nam &longs;i &longs;ic &longs;it b a c &etail;quale <lb/>f g h &amp; c a d h g k, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="tot&utilde;">totum</expan>, b a d erit &etail;quale <lb/><arrow.to.target n="marg540"/><lb/>toti f g k quod e&longs;t contra <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan>, ide&ograve; neque<lb/>b a e quia b a c &amp; d a e &longs;unt <expan abbr="&aelig;&qtilde;lia">&aelig;qualia</expan> rectilineis <lb/>per &longs;e, &amp; <expan abbr="eti&atilde;">etiam</expan> pariter accepta. </s>
          <s id="id002813">Totum <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> a e&longs;t <expan abbr="&etail;&qtilde;le">&etail;quale</expan> quatuor, re&shy;<lb/>ctis ergo <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;cilicet &longs;patia c a d &amp; b a c pariter accepta &longs;unt <expan abbr="&etail;&qtilde;&shy;lia">&etail;qua&shy;<lb/>lia</expan> rectilineis &longs;patijs, &longs;ed <expan abbr="&longs;pati&utilde;">&longs;patium</expan> e a d non e&longs;t <expan abbr="&aelig;&qtilde;le">&aelig;quale</expan> rectilineo, ergo per<lb/>demon&longs;trata hic, nec b a e, <expan abbr="n&atilde;">nam</expan> &longs;i &longs;it, &longs;it ergo b a e &aelig;quale h g k &amp; quia <lb/>ambo &longs;patia b a e &amp; c a d &longs;unt <expan abbr="&aelig;&qtilde;lia">&aelig;qualia</expan> rectilineo ex demon&longs;tratis, &longs;it <lb/>ergo &aelig;qualia f g k, erit ergo ex communi animi &longs;ententia &longs;patium f <lb/>g h &aelig;quale &longs;patio c a d, quod e&longs;t contra primam partem corrolarij.</s>
        </p>
        <p type="margin">
          <s id="id002814"><margin.target id="marg539"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="margin">
          <s id="id002815"><margin.target id="marg540"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/><emph type="italics"/>pr&aelig;&longs;entis.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id002816">LEMMA TERTIVM.<lb/><arrow.to.target n="marg541"/></s>
        </p>
        <p type="margin">
          <s id="id002817"><margin.target id="marg541"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002818">Inter duas rectas lineas &longs;e tangentes circuli dati peripheriam </s>
        </p>
        <p type="main">
          <s id="id002819"><arrow.to.target n="marg542"/><lb/>ducere. </s>
          <s id="id002820">Sit circulus datus a b rectilineus <lb/><figure id="id.015.01.179.2.jpg" xlink:href="015/01/179/2.jpg"/><lb/>angulus c d e, uolo illum diuidere circuli <lb/> periferia data b f, duco perpendicularem <lb/>d g ex, d &longs;uper d c, &amp; facio g d &aelig;qualem a b <lb/><arrow.to.target n="marg543"/><lb/>&amp; duco circulum per d qui &longs;it d h qui cadet <lb/>infra d c &amp; ob id etiam &longs;upra d e, igitur di&shy;<lb/>uidet angulum c d e, quare cum circulus d h &longs;it &aelig;qualis circulo b f <lb/><arrow.to.target n="marg544"/><lb/>patet propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002821"><margin.target id="marg542"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/><expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan><emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002822"><margin.target id="marg543"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002823"><margin.target id="marg544"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 6.</s>
        </p>
        <p type="main">
          <s id="id002824">Ex hoc patet quod infinitis modis pote&longs;t diuidi angulus c d e <lb/><arrow.to.target n="marg545"/><lb/>peripheria b f, nam diui&longs;o per rectam c d e linea d k per &etail;qualia &amp; di <lb/><arrow.to.target n="marg546"/><lb/>ui&longs;o k d e per pr&aelig;&longs;entem peripheria b f, patet propo&longs;itum quoniam <lb/>angulus c d e pote&longs;tin infinitum recta diuidi, &amp; ita &longs;emper per peri&shy;<lb/>pheriam, unde patet propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002825"><margin.target id="marg545"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>diff. <lb/></s>
          <s id="id002826">tertij <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan>.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002827"><margin.target id="marg546"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id002828">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002829">Atque h&aelig;c omnia &longs;equuntur de mente Euclidis, qu&aelig; tamen ui&shy;<lb/>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe<pb pagenum="161" xlink:href="015/01/180.jpg"/>ria &amp; recta &longs;unt ex genere quantitatis continu&aelig;, &amp; qu&ograve;d detur ma&shy;<lb/>ius &amp; minus &amp; nunquam detur &etail;quale, uidetur ab&longs;urdum ne dum <lb/>admirabile. </s>
          <s id="id002830">Et maxim&egrave; quod etiam anguli ex peripheria &amp; recta <lb/>&longs;unt diuer&longs;orum generum inter &longs;e &amp; infinitorum. </s>
          <s id="id002831">Pr&etail;terea i&longs;tud re&shy;<lb/>pugnare uidetur ip&longs;imet Euclidi, dicenti duabus magnitudinibus <lb/><arrow.to.target n="marg547"/><lb/><arrow.to.target n="marg548"/><lb/>propo&longs;itis in&aelig;qualibus, &longs;i de maiore earum plus dimidio detraha&shy;<lb/>tur, atque iterum de re&longs;iduo maius dimidio, &amp; rur&longs;us de eo quod re&shy;<lb/>linquitur plus dimidio, nece&longs;&longs;e erit ut tandem minor minore quan&shy;<lb/>titas relinquatur. </s>
          <s id="id002832">Neque illud argumentum uidetur concludere an&shy;<lb/>gulus contactus, ex recta, &amp; circuli circumferentia non pote&longs;t recta <lb/>diuidi, &amp; rectilineus pote&longs;t diuidi, ergo rectilineus &longs;emper e&longs;t ma&shy;<lb/>ior angulo contactus, quia hoc contingit in angulo contactus pro<lb/>pter modum anguli, non paruitatem: &longs;icut etiam non ualet de figu&shy;<lb/><figure id="id.015.01.180.1.jpg" xlink:href="015/01/180/1.jpg"/><lb/>ra a lunari, &amp; quadrangulo b. </s>
          <s id="id002833">nam pote&longs;t b diuidi <lb/>ab angulo ad angulum recta &amp; a non pote&longs;t, &amp; <lb/>tamen a maius e&longs;t quam b, cum contineat ip&longs;am. <lb/></s>
          <s id="id002834">Proponantur ergo duo circuli a d e &amp; a f g qui &longs;e contingant in a, &amp; <lb/>eorum centra &longs;int b &amp; c &amp; ducantur rect&aelig; a f d &amp; a g e &amp; con&longs;tat <lb/>qui portiones a d &amp; a f &longs;imiles &longs;unt, <lb/><figure id="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"/><lb/>itemque a e &amp; a g, ducta enim a b c <lb/><arrow.to.target n="marg549"/><lb/>per centra circulorum ex contactu <lb/>tran&longs;ibit per illa: quare anguli h a g <lb/>&amp; h a e &longs;unt ijdem &amp; &longs;imiliter h a f <lb/>&amp; h a d ijdem, portiones ergo af &amp; <lb/>a d itemque a g &amp; a e &longs;imiles &longs;unt: an&shy;<lb/>gulus igitur g a e ex peripherijs &amp; <lb/><arrow.to.target n="marg550"/><lb/>e a d ex rectis &longs;unt ijdem in puncto <lb/>a: &longs;ed quod ad ba&longs;sim maior e&longs;t ba&shy;<lb/>&longs;is g e quam e d: hoc enim &longs;uppono <lb/>quod per &longs;e e&longs;t manife&longs;tum toties <lb/><expan abbr="diuid&etilde;do">diuidendo</expan> arcum d e ut fiat minor recta g e. </s>
          <s id="id002835">Quia ergo &longs;unt du&etail; ma&shy;<lb/>gnitudines, quarum ter mini &longs;unt ijdem ex una parte, &longs;cilicet pun&shy;<lb/>ctum a, ex alia autem unus e&longs;t maior altero, &longs;cilicet g e quam e f &amp; <lb/><arrow.to.target n="marg551"/><lb/>a d e peripheria e&longs;t maior recta a g e. </s>
          <s id="id002836">Ergo per regulam dialecti&shy;<lb/>cam &longs;i &longs;ub eadem proportione procederent, maius e&longs;&longs;et &longs;patium <lb/>&longs;emper inter peripherias qu&agrave;m rectas. </s>
          <s id="id002837">igitur angulus peripheria&shy;<lb/>rum e&longs;t maior angulo &agrave; rectis contento. </s>
          <s id="id002838">Cum angulus non &longs;it <lb/>ni&longs;i quidam habitus propinquitatis linearum, &longs;ed angulus con&shy;<lb/>tactus ex recta &amp; peripheria maior e&longs;t contento ex peripherijs cum <lb/>habeat rationem totius ad partem, igitur angulus contactus e&longs;t <lb/>maior dato angulo rectilineo.</s>
        </p>
        <pb pagenum="162" xlink:href="015/01/181.jpg"/>
        <p type="margin">
          <s id="id002839"><margin.target id="marg547"/>1. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002840"><margin.target id="marg548"/>10. E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002841"><margin.target id="marg549"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002842"><margin.target id="marg550"/>E<emph type="italics"/>x<emph.end type="italics"/> 10. <emph type="italics"/>diff. <lb/></s>
          <s id="id002843">tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002844"><margin.target id="marg551"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>deci&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002845">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id002846">Propo&longs;ita linea tribus que in ea &longs;ignis punctum inuenire, ex que <lb/>duct&aelig; tres line&aelig; ad &longs;igna &longs;int in proportionibus datis.<lb/><arrow.to.target n="marg552"/></s>
        </p>
        <p type="margin">
          <s id="id002847"><margin.target id="marg552"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002848">Sit data linea a b c in qua puncta dicta &amp; dat&aelig; tres line&etail; d e f, uo&shy;<lb/>lo inuenire punctum, puta g ex quo duct&aelig; tres <lb/>line&aelig; ad a b c puncta &longs;int in proportione a g ad </s>
        </p>
        <p type="main">
          <s id="id002849"><arrow.to.target n="marg553"/><lb/>g b, ut d ad e &amp; g b ad g c, ut e ad f. </s>
          <s id="id002850">Per pr&etail;ceden<lb/><figure id="id.015.01.181.1.jpg" xlink:href="015/01/181/1.jpg"/><lb/>tia inuenio circulum ex cuius peripheria omni&shy;<lb/>bus ex punctis duct&aelig; line&aelig; ad a b &longs;int in pro&shy;<lb/>portione d ad e, &amp; per idem circulum ex cuius <lb/>peripheria qu&aelig;libet line&aelig; duct&aelig; ad b c puncta <lb/>&longs;int in proportione c ad f, &longs;i igitur i&longs;ti duo circu&shy;<lb/>li &longs;e &longs;ecabunt in aliquo puncto puta g: liquet <lb/>quod line&aelig; duct&aelig; ex g ad a b c, erunt in propor<lb/>tione d e f.<lb/><arrow.to.target n="marg554"/></s>
        </p>
        <p type="margin">
          <s id="id002851"><margin.target id="marg553"/>P<emph type="italics"/>er<emph.end type="italics"/> 154.</s>
        </p>
        <p type="margin">
          <s id="id002852"><margin.target id="marg554"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002853">Ex quo liquet quod &longs;i uoluero ducere ad tria puncta data, tres <lb/>lineas in continua proportione data d ad e, &longs;ubijciam tertiam uel in<lb/>terponam, &longs;i uoluero mediam. </s>
          <s id="id002854">Et &longs;i uellem, ut e&longs;&longs;et a g ad g b dupli&shy;<lb/>cata ei qu&aelig; e&longs;t g b ad b c, &amp; uellem qu&ograve;d proportio d ad a d f data <lb/>e&longs;&longs;et, oporteret inuenire duas medias proportione inter d &amp; f, in de <lb/>operari cum una earum per modum propo&longs;itum. </s>
          <s id="id002855">Differt corrola&shy;<lb/>rium hoc &agrave; propo&longs;itione in hoc, quod in propo&longs;itione non qu&aelig;ri&shy;<lb/>mus ni&longs;i proportionem g a ad g b &amp; g b ad b c, non g a ad g c, neque <lb/>comparationem proportionum: at in corrolario qu&aelig;rimus tres <lb/>proportiones g a g b &amp; g c, &amp; comparationem proportionum in&shy;<lb/>ter &longs;e, &longs;cilicet &aelig;qualitatem.</s>
        </p>
        <p type="main">
          <s id="id002856">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id002857">Si fuerint duo trianguli quorum ba&longs;es in eadem linea &longs;int con&shy;<lb/>&longs;tituti &amp; &aelig;quales &amp; ad unum punctum terminati, &amp; latus unum <lb/>commune inter reliqua quantita&shy;<lb/><figure id="id.015.01.181.2.jpg" xlink:href="015/01/181/2.jpg"/><lb/>te medium, nece&longs;&longs;e e&longs;t angulum &agrave; <lb/>maioribus lineis contentum mi&shy;<lb/>norem e&longs;&longs;e.</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"/><lb/>quales proponuntur, &amp; &longs;it a d ma&shy;<lb/><arrow.to.target n="marg556"/><lb/>ior a b dico angulum d a c e&longs;&longs;e mi&shy;<lb/>norem. </s>
          <s id="id002860">Si non fiat angulus d a c &aelig;&shy;<lb/>qualis ex alia parte, &amp; oportet &longs;i non &longs;it minor ut uel cadat a d &longs;u&shy;<lb/><arrow.to.target n="marg557"/><lb/>per a b &amp; ducta a d ad &etail;qualitatem cadet infra b, ducta ergo d c erit <lb/>trigonus a d c maior a b c, quod e&longs;&longs;e non pote&longs;t cum &longs;int &aelig;quales. 


<pb pagenum="163" xlink:href="015/01/182.jpg"/>Si autem a d cadat extra a b ducatur d e: qu&aelig; &longs;i cadat &longs;upra b c uel <lb/>infra, cum totum &longs;it maius parte erit a d e, ut prius maior a b c quod <lb/><arrow.to.target n="marg558"/><lb/>e&longs;t contra Euclidem. </s>
          <s id="id002861">Reliquum e&longs;t ut d c cadat &longs;upra b c: hoc au&shy;<lb/><arrow.to.target n="marg559"/><lb/>tem e&longs;&longs;e non pote&longs;t, nam cum &longs;uppo&longs;uerimus a b e&longs;&longs;e minorem a c <lb/>erit angulus a c b minor angulo a b c, quare a c b e&longs;t minor recto, &amp; <lb/><arrow.to.target n="marg560"/><lb/>ide&ograve; a c d maior recto, at a c d &aelig;qualis e&longs;t a c d, alteri igitur a c d e&longs;t <lb/><arrow.to.target n="marg561"/><lb/>maior recto a c b minor, erit ergo pars maior toto.</s>
        </p>
        <p type="margin">
          <s id="id002862"><margin.target id="marg555"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002863"><margin.target id="marg556"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002864"><margin.target id="marg557"/>P<emph type="italics"/>er<emph.end type="italics"/> 38. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002865"><margin.target id="marg558"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002866"><margin.target id="marg559"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002867"><margin.target id="marg560"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002868"><margin.target id="marg561"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id002869">LEMMA.</s>
        </p>
        <p type="main">
          <s id="id002870">His demon&longs;tratis quis dicere po&longs;&longs;et ex &longs;uperius expo&longs;itis quod <lb/><arrow.to.target n="marg562"/><lb/>angulus rectilineus &longs;emper e&longs;&longs;et maior angulo contactus? </s>
          <s id="id002871">quia an&shy;<lb/>gulus contactus non pote&longs;t diuidi ni&longs;i obliqua linea, recti lineus <lb/>autem tam obliqua quam recta. </s>
          <s id="id002872">Propter hoc exponantur circuli <lb/><figure id="id.015.01.182.1.jpg" xlink:href="015/01/182/1.jpg"/><lb/>tres &longs;e tangentes a b, a c, a d hac rati&shy;<lb/>one ut a b, b c, c d &longs;int &aelig;quales, erunt <lb/><arrow.to.target n="marg563"/><lb/>enim centra omnia in linea conta&shy;<lb/>ctus, &amp; ducatur a e f g recta quomo<lb/><arrow.to.target n="marg564"/><lb/>dolibet: &amp; erunt ductis lineis b c, <lb/><arrow.to.target n="marg565"/><lb/>c f, d g anguli e f g recti, quare om&shy;<lb/>nes trigoni a b e, a c f, a d g, &longs;imiles <lb/><arrow.to.target n="marg566"/><lb/>&amp; ideo a e, e f, f g &aelig;quales, atque por&shy;<lb/>tiones a g, a f, a e, iuxta proportio&shy;<lb/>nem circulorum, quare a g, erit &longs;ex&shy;<lb/>quialtera a f &amp; a f dupla a e, igitur <lb/><arrow.to.target n="marg567"/><lb/>per pr&aelig;cedentem maior erit angu&shy;<lb/>lus e a f, quam f a g, &amp; a d a ex recta <lb/><arrow.to.target n="marg568"/><lb/>&amp; peripheria quam e a f, igitur augendo eadem ratione cum perue&shy;<lb/>niamus ad angulum b a g qui ferm&egrave; e&longs;t recto &aelig;qualis cum deficiat <lb/>&longs;olo angulo contactus, liquet angulum e a g e&longs;&longs;e long&egrave; maiorem <lb/>multis rectilineis. </s>
          <s id="id002873">I&longs;tud po&longs;&longs;et etiam demon&longs;trari uia Archimedis <lb/>diuidendo arcus g a in h &amp; f a in k bifariam ducendo que lineas re&shy;<lb/>ctas g h &amp; fk &amp; ita diuidendo h a in 1, &amp; k a in m bifariam, &amp; ducen&shy;<lb/>do rectas atque ita &longs;emper appropinquando puncto a. </s>
          <s id="id002874">Concludo er&shy;<lb/>go quod angulus <expan abbr="c&otilde;tactus">contactus</expan> ex recta &amp; peripheria e&longs;t maior multis <lb/>rectilineis. </s>
          <s id="id002875">Cau&longs;a autem erroris e&longs;t quod multi exi&longs;timarunt corro&shy;<lb/>larium illud e&longs;&longs;e Euclidis cum non &longs;it. </s>
          <s id="id002876">Nam Euclidi &longs;ufficit hoc <lb/>qu&ograve;d angulus contactus <expan abbr="n&otilde;">non</expan> po&longs;sit recta diuidi, nam eo utitur po&longs;t <lb/><expan abbr="mod&utilde;">modum</expan> in demon&longs;trationibus. </s>
          <s id="id002877">Eo uer&ograve; quod &longs;it minor omnibus re&shy;<lb/>ctilineis angulis non utitur, ide&ograve; etiam &longs;i <expan abbr="uer&utilde;">uerum</expan> fui&longs;&longs;et <expan abbr="n&otilde;">non</expan> addidi&longs;&longs;et: <lb/>quanto minus: cum uerum non &longs;it, ide&ograve; fuit <expan abbr="adiect&utilde;">adiectum</expan> ab aliquo qui <lb/><expan abbr="id&etilde;">idem</expan> fore credidit <expan abbr="n&otilde;">non</expan> po&longs;&longs;e diuidi recta linea &amp; e&longs;&longs;e minus quocunque<lb/> quod recta linea diuidi po&longs;&longs;et, quod apert&egrave; ut dixi fal&longs;um e&longs;t.</s>
        </p>
        <pb pagenum="164" xlink:href="015/01/183.jpg"/>
        <p type="margin">
          <s id="id002878"><margin.target id="marg562"/>L<emph type="italics"/>emmate<emph.end type="italics"/> 3. <lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 159.</s>
        </p>
        <p type="margin">
          <s id="id002879"><margin.target id="marg563"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002880"><margin.target id="marg564"/>P<emph type="italics"/>er<emph.end type="italics"/> 31. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002881"><margin.target id="marg565"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002882"><margin.target id="marg566"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002883"><margin.target id="marg567"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>diff <lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002884"><margin.target id="marg568"/>P<emph type="italics"/>er pr&aelig;ce&shy;<lb/>dentem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id002885">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id002886">Ratio autem qu&ograve;d omnis angulus contactus indiuiduus &longs;it, &longs;eu <lb/>duorum circulorum, &longs;eu circuli cum recta e&longs;t, quoniam cum fuerint <lb/>du&aelig; rationes contrari&aelig;, &amp; una perpetu&ograve; minuitur, alia manet ne&shy;<lb/>ce&longs;&longs;e e&longs;t, ut tandem, qu&aelig; minuitur, &longs;uperetur ab ea qu&aelig; manet: cum <lb/>ergo circuli curuitas maneat, &amp; angulus tendat in punctum perpe&shy;<lb/>tua diminutione nece&longs;&longs;e e&longs;t, ut curuitas circuli impediat diui&longs;io&shy;<lb/>nem rect&egrave;: &longs;ed hoc habet duplicem obicem. </s>
          <s id="id002887">Primum, quia nullus <lb/>angulus ex circumferentia &amp; recta po&longs;&longs;et diuidi: hoc autem fal&longs;um <lb/>e&longs;t manife&longs;t&egrave;, cum &longs;olus ille qui fit ex contactu line&aelig;, qu&aelig; non di&shy;<lb/>uidit circulum, diuidi non po&longs;sit. </s>
          <s id="id002888">Secund&ograve;, quod angulus conta&shy;<lb/>ctus duorum circulorum &longs;e exterius tangentium multo minus <lb/>po&longs;&longs;et diuidi angulo contactus interioris duorum circulorum, <lb/>quod tamen fal&longs;um e&longs;t: &amp; hoc animaduertit Campanus no&longs;ter, uir <lb/>acutus. </s>
          <s id="id002889">Dico ergo qu&ograve;d in his qui &longs;e tangunt exterius, non fit diui&shy;<lb/>&longs;io ni&longs;i &longs;emel: &amp; quamuis inclinentur mutu&ograve;, tamen in concur&longs;u <lb/>non aptantur, ut cum obuiat rect&aelig; aut cau&aelig; parti circuli quia ne&shy;<lb/>ce&longs;&longs;e e&longs;t, ut accedat, in alio autem di&longs;cedat: indicio e&longs;t quod circu&shy;<lb/>los &longs;e exterius tangentes, in puncto facil&egrave; de&longs;cribes, interius uix fie&shy;<lb/>ri pote&longs;t, &longs;ed uidentur coniuncti <lb/><figure id="id.015.01.183.1.jpg" xlink:href="015/01/183/1.jpg"/><lb/>per longum interuallum. </s>
          <s id="id002890">Ad aliud <lb/>dico, qu&ograve;d ille angulus ex recta &amp; <lb/>peripheria conuexa circuli propter <lb/>di&longs;ce&longs;&longs;um &longs;eruat maiorem inclina&shy;<lb/>tionem in quocunque puncto, qu&agrave;m <lb/>&longs;it acce&longs;&longs;us conuex&aelig; partis exterio&shy;<lb/>ris circuli.</s>
        </p>
        <p type="main">
          <s id="id002891">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima <lb/>&longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id002892">Proportionem duorum orbium <lb/>quorum diametrorum <expan abbr="c&otilde;uex&aelig;">conuex&aelig;</expan> par<lb/>tis, &amp; concau&aelig; proportiones dat&aelig; <lb/>&longs;int, inue&longs;tigare.</s>
        </p>
        <p type="main">
          <s id="id002893">Sint duo orbes a b c d &amp; e f g h, <lb/><arrow.to.target n="marg569"/><lb/>&amp; &longs;it proportio a d ad b c, data &amp; e <lb/>h ad f g, data &amp; rur&longs;us a d ad e h, di&shy;<lb/>co orbis proportionem a b c d ad <lb/><expan abbr="orb&etilde;">orbem</expan> e f g h e&longs;&longs;e <expan abbr="dat&atilde;">datam</expan>. </s>
          <s id="id002894">Quia. n. </s>
          <s id="id002895">propor<lb/>tio a d &longs;ph&aelig;r&etail; ad b c e&longs;t ueluti ad di <lb/>metientis ad b c <expan abbr="dimetient&etilde;">dimetientem</expan> triplicata, ide&ograve; <expan abbr="c&utilde;">cum</expan> nota &longs;it a d ad b c di <lb/><arrow.to.target n="marg570"/><lb/><expan abbr="metienti&utilde;">metientium</expan>, erit nota <expan abbr="eti&atilde;">etiam</expan> a d &longs;ph&aelig;r&aelig; ad b c <expan abbr="&longs;ph&etail;r&atilde;">&longs;ph&etail;ram</expan>. </s>
          <s id="id002896">quare orbis ad ad <lb/><expan abbr="&longs;ph&etail;r&atilde;">&longs;ph&etail;ram</expan> b c. nota e&longs;t <expan abbr="eti&atilde;">etiam</expan> proportio b c <expan abbr="dimeti&etilde;tis">dimetientis</expan> ad a d &amp; ad a d e h &amp; 


<pb pagenum="165" xlink:href="015/01/184.jpg"/>e h ad f g, igitur b c proportio dimetientis ad f g dimetientem nota. <lb/><arrow.to.target n="marg571"/><lb/>Quare &longs;ph&aelig;r&aelig; b c ad f g &longs;ph&aelig;ram. </s>
          <s id="id002897">at nota e&longs;t proportio f g ad e h <lb/>dimetientium igitur &amp; &longs;ph&aelig;rarum: igitur nota e&longs;t f g &longs;ph&aelig;r&aelig; ad or<lb/>bem e h, igitur cum nota &longs;it proportio orbis ad a d &longs;ph&aelig;ram b c, &amp; <lb/>b c &longs;ph&aelig;r&aelig; ad f g &longs;ph&aelig;ram, &amp; f g &longs;ph&aelig;r&aelig; ad orbem e h, erit propor<lb/>tio orbis a d ad orbem e h nota, quod e&longs;t propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id002898"><margin.target id="marg569"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002899"><margin.target id="marg570"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>duo <lb/>decimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id002900"><margin.target id="marg571"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem. <lb/></s>
          <s id="id002901">&amp;<emph.end type="italics"/> A<emph type="italics"/>lizam.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id002902">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id002903">Proportionem uirium &longs;tellarum per motus &longs;uos indagare.</s>
        </p>
        <p type="main">
          <s id="id002904">Mouentur &longs;tell&aelig; omnes ab Oriente in Occidentem die una, qui <lb/><arrow.to.target n="marg572"/><lb/>motus fit &agrave; prima mente, qu&aelig; mouet: ide&ograve; quod ad hoc attinet non <lb/>e&longs;t diuer&longs;itas: uer&ugrave;m in motibus ab Occidente in Orientem <expan abbr="c&utilde;">cum</expan> &longs;int <lb/>proprij, oportet con&longs;iderare tempus, in quo <expan abbr="circumuert&utilde;tur">circumuertuntur</expan>, &amp; ma<lb/>gnitudinem ambitus, &amp; inde magnitudinem orbis, qui circumagi&shy;<lb/>tur, &amp; horum trium facta comparatione digno&longs;citur robur uirium <lb/>&longs;tellarum &amp; uitarum qu&aelig; mouent eas. </s>
          <s id="id002905">Ponatur ergo, ut uelim pro&shy;<lb/>portionem uit&etail; Saturni ad uitam Lun&aelig;: erit ergo &lpar;ut docet Alphra<lb/><arrow.to.target n="marg573"/><lb/>ganus&rpar; Luna, cum e&longs;t in longitudine propiore, altitudinem habens <lb/>109000 M.P. &amp; cum e&longs;t in longitudine longiore 208500, tota igitur <lb/>dimetiens 417000 M.P. mane 218000 M.P. </s>
          <s id="id002906">Igitur proportio &longs;olida&shy;<lb/>rum &longs;ph&aelig;rarum e&longs;t uelut 72511713 ad 10360232, remanebit ergo <lb/>proportio orbis ad &longs;ph&aelig;ram elementorum, ut 62151481 ad <lb/>10360232, &amp; e&longs;t &longs;excuplum ferm&egrave;. </s>
          <s id="id002907">Rur&longs;us proportio dimetientis al&shy;<lb/>titudinis Saturni ad contentum e&longs;t uelut 2011 ad 1440, &amp; e&longs;t prop&egrave; <lb/>201 ad 114, quare 67 ad 38, quare &longs;ph&aelig;rarum ut 300000 ad 55000 <lb/>ferme. </s>
          <s id="id002908">Igitur fer&egrave; ut 60 ad 11. Rur&longs;us proportio dimetientis &longs;ph&aelig;&shy;<lb/>r&aelig; Saturni ad dimetientem &longs;ph&aelig;r&aelig; Lun&aelig; e&longs;t prop&egrave; 313, &amp; &longs;ph&aelig;ra&shy;<lb/>rum &longs;olidarum 306 317 10. Perinde e&longs;t. </s>
          <s id="id002909">Quia ergo proportio &longs;ph&aelig;&shy;<lb/>r&aelig; Saturni ad &longs;ph&aelig;ram Lun&aelig; e&longs;t 30631710, &amp; orbis Lun&aelig; e&longs;t 5/6 <lb/>&longs;olum &longs;ph&aelig;r&aelig; &longs;u&aelig; diuidemus 30631710 per 5/6, &amp; exibit proportio <lb/>&longs;ph&aelig;r&aelig; Saturni ad orbem Lun&aelig; 36758052, at quia proportio &longs;o&shy;<lb/>lid&aelig; &longs;ph&aelig;r&aelig; Saturni ad contentum e&longs;t ut 60 ad 11, erit &longs;ph&aelig;r&aelig; ad <lb/>orbem, ut 60 ad 49 re&longs;iduum, diuidam ergo 36758052 per 60, exe&shy;<lb/>unt 612634, &amp; ducam per 49, id e&longs;t per 100, fit 61263400, &amp; diuiden <lb/>do per 2, exit 30631700, detraho 612634, relinquitur proportio or&shy;<lb/>bis Saturni ad orbem Lun&aelig; 30019066.</s>
        </p>
        <p type="margin">
          <s id="id002910"><margin.target id="marg572"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002911"><margin.target id="marg573"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 21.</s>
        </p>
        <p type="main">
          <s id="id002912">Iam uer&ograve; circuitus Saturni ad circulum Lun&aelig;, proportio e&longs;t 313, <lb/>ut ui&longs;um e&longs;t, Lun&aelig; autem tempus per &longs;ex ductum e&longs;t 164 dies, Sa&shy;<lb/>turni 177 anni propemodum, qui &longs;unt dies 64649 diuide, duc <lb/>ergo 313 in 164, fiunt 51332. Idem ergo peragrat Luna in <lb/>51332 diebus, quod Saturnus in 64649, &amp; e&longs;t quo ad hoc agi&shy;


<pb pagenum="166" xlink:href="015/01/185.jpg"/>lior, ut ita dicam, quarta parte: at Saturnus, ut dictum e&longs;t, mouet or&shy;<lb/>bem 30019066, &longs;ed lenti&ugrave;s quinta parte, detrahe illam fiet robur Sa<lb/>turni in comparatione ad Lunam 24015253.</s>
        </p>
        <p type="main">
          <s id="id002913">E&longs;t tamen Luna multo agilior ob propinquitatem, &amp; ob uarie&shy;<lb/>tatem luminis, &amp; magnitudinem &longs;uperficiei. </s>
          <s id="id002914">Et etiam quod maius <lb/>e&longs;t ob id quod defert ad nos uires omnium &longs;yderum, nihilominus <lb/>quo ad uires uix e&longs;t comparatio.</s>
        </p>
        <p type="head">
          <s id="id002915">SCHOLIVM.<lb/><arrow.to.target n="marg574"/></s>
        </p>
        <p type="margin">
          <s id="id002916"><margin.target id="marg574"/>46</s>
        </p>
        <p type="main">
          <s id="id002917">Multum autem differt h&aelig;c propo&longs;itio &agrave; &longs;uperiore, nam in illa <lb/>qu&aelig;&longs;iuimus uim uitarum ex proportione ad &longs;ua corpora, qu&aelig; <lb/>quodammodo e&longs;t quodammodo, non hic autem exponimus uim <lb/>uitarum ex earum operatione. </s>
          <s id="id002918">Propterea &longs;ubijciemus breuiter alti&shy;<lb/>tudinem proportiones in minore longitudine &amp; maiori<lb/><arrow.to.target n="table19"/></s>
        </p>
        <table>
          <table.target id="table19"/>
          <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 h&aelig; <lb/>men&longs;ur&aelig; &longs;unt in comparatione ad &longs;emidiametrum terr&aelig;. </s>
          <s id="id002921">Et iuxta <lb/>id quod potuit &longs;ecundum rationem haberi: nam demon&longs;tratio &longs;ola <lb/>e&longs;t de altitudinibus Solis &amp; Lun&aelig;, &amp; eorum magnitudinibus &agrave; </s>
        </p>
        <p type="main">
          <s id="id002922"><arrow.to.target n="marg575"/><lb/>Ptolem&aelig;o in magna compo&longs;itione.</s>
        </p>
        <p type="margin">
          <s id="id002923"><margin.target id="marg575"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 5. <emph type="italics"/>cap.<emph.end type="italics"/><lb/>14. 15. <emph type="italics"/>&amp;<emph.end type="italics"/><lb/>16.</s>
        </p>
        <p type="main">
          <s id="id002924">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id002925">Syderum proportionem in magnitudine o&longs;tendere.<lb/><arrow.to.target n="table20"/></s>
        </p>
        <table>
          <table.target id="table20"/>
          <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 in&longs;ignium unaqu&aelig;que etiam minima, &longs;i <lb/><arrow.to.target n="marg576"/><lb/>credendum e&longs;t Alphragano, e&longs;t centies maior tota terra, unde ca&shy;<lb/>nem nece&longs;&longs;e e&longs;t centies mille maiorem e&longs;&longs;e, e&longs;t enim in eadem altitu<lb/>dine, &amp; dimetiens decuplus dimetienti &longs;tellarum &longs;ecund&aelig; magni&shy;<lb/>tudinis, quas ille in&longs;ignes uocat: aliter Saturnus non tantus e&longs;&longs;e <lb/>po&longs;&longs;et, cum &longs;it minimus a&longs;pectu.</s>
        </p>
        <pb pagenum="167" xlink:href="015/01/186.jpg"/>
        <p type="margin">
          <s id="id002927"><margin.target id="marg576"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 22.</s>
        </p>
        <p type="main">
          <s id="id002928">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id002929">Propo&longs;itionem motuum omnium <expan abbr="&longs;tellar&utilde;">&longs;tellarum</expan> ad &longs;olem con&longs;iderare.</s>
        </p>
        <p type="main">
          <s id="id002930">Videtur Sol qua&longs;i Rex in C&oelig;lo, nam omnes orbes cum illius <lb/><arrow.to.target n="marg577"/><lb/>motu conueniunt, &amp; uidetur es admiratione digna his, qui non <lb/>nouerunt, quanta &longs;it concordia omnium rerum, de qua infr&agrave; dice&shy;<lb/>mus. </s>
          <s id="id002931">Ergo Luna primum hoc habet, ut linea &aelig;qualis motu Solis <lb/>&longs;emper media &longs;it inter lineam &aelig;qualis motus Lun&etail; &amp; loci maxim&egrave; <lb/>in&aelig;qualitatis motus eius, ubi &longs;cilicet tardi&longs;sim&egrave; mouetur, Veneris <lb/>autem &amp; Mercurij ut motus &aelig;quales idem &longs;emper &longs;int cum motu <lb/>&aelig;quali, &amp; locus cum loco ip&longs;ius Solis ad unguem pr&aelig;ter id quod <lb/>infr&agrave; dicemus. </s>
          <s id="id002932">Trium uer&ograve; <expan abbr="&longs;uperior&utilde;">&longs;uperiorum</expan> ratio &longs;ic <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> ad Solem ut &agrave; <lb/>Ptolem&etail;o <expan abbr="ob&longs;eruat&utilde;">ob&longs;eruatum</expan> e&longs;t ex Hipparcho. </s>
          <s id="id002933">In omni re&longs;titutione cuiu&longs;&shy;<lb/>libet planet&etail; &longs;uperioris numerus <expan abbr="reuolution&utilde;">reuolutionum</expan> Solis &etail;qualis e&longs;t nu&shy;<lb/>mero <expan abbr="re&longs;titution&utilde;">re&longs;titutionum</expan> planet&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> <expan abbr="mot&utilde;">motum</expan> &aelig;qualitatis &amp; in&etail;qualita<lb/>tis pariter acceptis. </s>
          <s id="id002934">Velut Saturnus in annis quinquaginta nouem <lb/>die una &amp; horis decem octo quinquage&longs;ies &longs;epties per motum in&etail;&shy;<lb/>qualem ad <expan abbr="ungu&etilde;">unguem</expan>, per &aelig;qualem autem duabus reuolutionibus par <lb/>te in&longs;uper una &amp; quadraginta quin que minutijs, qu&aelig; re&longs;pondent di&shy;<lb/>ei uni, &amp; horis decem octo ex motu Solis, &amp; ita bis Saturnus reuol<lb/>uitur &longs;ecundum motum &aelig;qualitatis &amp; quinquage&longs;ies &longs;epties per <lb/>motum in&aelig;qualem &amp; &longs;imiliter. </s>
          <s id="id002935">Iupiter in annis 70, diebus trecen&shy;<lb/>tis &longs;exaginta, horis quatuor, &longs;exaginta quinque reuolutiones in&etail;qua<lb/>les perficiet &amp; &longs;ex &etail;quales, deficientibus ex &etail;qualibus quatuor par&shy;<lb/>tibus &amp; dextante quod e&longs;t <expan abbr="qu&atilde;tum">quantum</expan> peragraret Solin quatuor die&shy;<lb/>bus, &amp; dextante diei ad perfectionem &longs;cilicet annorum &longs;eptuaginta <lb/>atque unius. </s>
          <s id="id002936">Martis quo que &longs;tella in annis &longs;eptuaginta nouem, &amp; die&shy;<lb/>bus tribus &amp; horis ferm&egrave; quatuor triginta nouem facit in&aelig;quali&shy;<lb/>tatis reuolutiones: &aelig;qualitatis autem quadraginta duas, &amp; in&longs;uper <lb/>partes tres cum &longs;extante, quas manife&longs;tum e&longs;t peragrari &agrave; Sole in <lb/>diebus tribus atque horis quatuor. </s>
          <s id="id002937">Veneris quo que &longs;ydus in octo an&shy;<lb/>nis deficientibus diebus duobus &amp; quadrante, in&aelig;qualitatis quin&shy;<lb/>que perficit reuolutiones, &aelig;qualitatis autem tantundem ad un<expan abbr="gu&etilde;">guem</expan> <lb/>quantum Sol deficiente eadem parte &longs;eu diebus duobus &amp; qua&shy;<lb/>drante. </s>
          <s id="id002938">Mercurij quo que &longs;tella in quadraginta &longs;ex annis &amp; una die <lb/>&amp; hora una ferm&egrave; quadraginta &longs;ex ferm&egrave; perficit reuolutiones &aelig;&shy;<lb/>qualis motus &amp; in&longs;uper gradum unum cum portione re&longs;pondenti <lb/>portioni temporis, id e&longs;t, hor&aelig; ferm&egrave; uni: in &aelig;qualitatis autem cen&shy;<lb/>&longs;um quadraginta quin que. </s>
          <s id="id002939">Atque h&etail;c &longs;unt manife&longs;ti&longs;sima et ut dixi ad&shy;<lb/>miranda &longs;unt, pr&aelig;terea alia minus generalia, aut minus manife&longs;ta <lb/>aut non tanti momenti qu&aelig; con&longs;ult&ograve; pr&aelig;termitto, non e&longs;t. </s>
          <s id="id002940">n. </s>
          <s id="id002941">locus <lb/>hic docendi artes &longs;ingulas &longs;ed &longs;olum ea tractandi qu&aelig; ad argumen<pb pagenum="168" xlink:href="015/01/187.jpg"/>tum pertinent. </s>
          <s id="id002942">Igitur ut ad rem redeam. </s>
          <s id="id002943">Solis cum octauo Orbe ea <lb/>ratio e&longs;t, ut linea quam ille permeat eadem &longs;it quam qu&etail; fix&etail; &longs;tell&aelig;, <lb/>non. </s>
          <s id="id002944">n. </s>
          <s id="id002945">ad eandem di&longs;tantiam &amp; mente conceptam ab &aelig;quinoctijs <lb/>de&longs;cendentem ac &aelig;quidi&longs;tantem mouetur, &longs;ed ad eam &longs;ecundum <lb/>quam &longs;tell&etail; fix&etail; in octauo orbe mouentur in comparatione ad ecli&shy;<lb/>pticam &longs;uperioris orbis. </s>
          <s id="id002946">Porr&ograve; de his atque huiu&longs;modi in Paralipo&shy;<lb/>menis diximus, ubi etiam docuimus quomodo &longs;ecundum duos cir<lb/><arrow.to.target n="marg578"/><lb/>culos, qui &longs;olum circa &longs;uum centrum mouentur, punctus datus per<lb/>petu&ograve; in recta linea feratur.</s>
        </p>
        <p type="margin">
          <s id="id002947"><margin.target id="marg577"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id002948"><margin.target id="marg578"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 14. <lb/><emph type="italics"/>cap.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="main">
          <s id="id002949">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id002950">Proportiones mu&longs;icas &longs;uperpartientes in eas qu&aelig; particula una <lb/>tantum abundant reducere.<lb/><arrow.to.target n="marg579"/></s>
        </p>
        <p type="margin">
          <s id="id002951"><margin.target id="marg579"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id002952">Ptolem&etail;i hoc inuentum fuit, ut &amp; multa alia pr&etail;clara: itaque &longs;ta&shy;<lb/>tuendum e&longs;t, primum uoces &etail;quales non concentum efficere, quia <lb/>diuer&longs;&aelig; non &longs;unt, qu&etail; autem diuer&longs;&etail; &longs;unt, nihilominus proportio&shy;<lb/>ne con&longs;tant &longs;implici&longs;sima &amp; multiplici, tales optimam efficiunt ar&shy;<lb/>moniam. </s>
          <s id="id002953">Eiu&longs;modi &longs;unt qu&aelig; in dupla &longs;unt proportione, uocatur <lb/>autem diapa&longs;on. </s>
          <s id="id002954">1. qua&longs;i omnia comprehendens non &agrave; numero uo&shy;<lb/>cum uelut diapente &amp; diate&longs;&longs;aron &agrave; quatuor &amp; quin que uo cibus. </s>
          <s id="id002955">In <lb/>diapa&longs;o. </s>
          <s id="id002956">n. </s>
          <s id="id002957">omnia <expan abbr="c&otilde;prehendi">comprehendi</expan> uidentur. </s>
          <s id="id002958">1. omnes uo <expan abbr="c&utilde;">cum</expan> differenti&aelig;, <lb/><expan abbr="quanqu&atilde;">quanquam</expan> ex octo <expan abbr="tant&utilde;">tantum</expan> uo cibus con&longs;tet. </s>
          <s id="id002959">P&ograve;&longs;t &longs;unt qu&aelig; in <expan abbr="&qtilde;drupla">quadrupla</expan>, <lb/>unde bis diapa&longs;on, po&longs;t qu&aelig; in tripla, nam propior e&longs;t monadi &longs;eu &etail;&shy;<lb/>qualitati: &longs;ed non ade&ograve; &longs;implex ut bis diapa&longs;on. </s>
          <s id="id002960">Vocant <expan abbr="a&utilde;t">aut</expan> hanc <lb/>diapa&longs;on diapente: inde <expan abbr="&longs;ub&longs;equi&ttilde;">&longs;ub&longs;equitur</expan> octupla qu&etail; uix in uocibus </s>
          <s id="id002961">huma&shy;<lb/>nis habetur: <expan abbr="frequ&etilde;s">frequens</expan> in in&longs;trumentis, uo<expan abbr="ca&ttilde;que">caturque</expan> tris diapa&longs;on inde &longs;ex&shy;<lb/>cupla, &longs;eu bis diapa&longs;on diapente. </s>
          <s id="id002962">Quintupla <expan abbr="a&utilde;t">aut</expan> minus <expan abbr="c&otilde;cors">concors</expan> e&longs;t: <lb/>&longs;ed de hac inferius dicemus, atque de multiplicibus </s>
          <s id="id002963">dicta &longs;unto. </s>
          <s id="id002964">Sed de <lb/><expan abbr="c&otilde;">com</expan>centu ex particula &longs;uperaddita &longs;exquialtera &longs;exquitertia atque alijs <lb/>nunc agendum. </s>
          <s id="id002965">Clarum e&longs;t. </s>
          <s id="id002966">n. </s>
          <s id="id002967">has e&longs;&longs;e &longs;implici&longs;simas. </s>
          <s id="id002968">Cum ergo du<lb/>pla proportio non magis po&longs;sit diuidi &aelig;qualibus interuallis atque<lb/> &longs;implicibus proportionibus qu&agrave;m in &longs;exquialteram &amp; &longs;exquiter&shy;<lb/>tiam, uelut inter 4 &amp; 2 interpo&longs;ito 3. nam proportio 3 ad 2 e&longs;t &longs;ex&shy;<lb/>quialtera, &amp; 4 ad 3 &longs;exquitertia: nec melius pote&longs;t diuidi, at &longs;exqui&shy;<lb/>alteram &amp; &longs;exquitertiam quantumuis magnis numeris diuidere <lb/>non licebat melius aut commodius quam per &longs;exquioctauas: uelu&shy;<lb/>ti &longs;umpto numero 64 cui duplus e&longs;t 128, inter medius 96 qui cum <lb/>64 &longs;exquialteram facit proportionem, qu&aelig; &longs;uaui&longs;sima e&longs;t omni&shy;<lb/>um deductis multiplicibus, uocaturque diapente. </s>
          <s id="id002969">At qu&aelig; e&longs;t 128 ad <lb/>96 &longs;exquitertia e&longs;t minu&longs;que ben&egrave; &longs;onat per &longs;e, &longs;ed in acutioribus uo&shy;<lb/>cibus &longs;olum cum alijs ben&egrave; &longs;onat, uelut cum diapente, perficiens <lb/>diapa&longs;on, interuallum, ergo inter 96 &amp; 64 diui&longs;um per &longs;exquiocta&shy;<pb pagenum="169" xlink:href="015/01/188.jpg"/>uas producit 72 et 81, <expan abbr="n&atilde;">nam</expan> 72 ad 64 e&longs;t &longs;exquio<expan abbr="ctau&utilde;">ctauum</expan>, &longs;icut 81 ad 72. uer&ugrave;m <lb/>id accidebat in <expan abbr="c&otilde;modi">commodi</expan> quae 81 ad 64 <expan abbr="null&atilde;">nullam</expan> habet <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="commod&atilde;">commodam</expan>, <lb/>&amp; multo minus 96 ad 81, quare ui&longs;um e&longs;t Ptolem&etail;o ut &longs;ubtracta mona<lb/>de <expan abbr="fier&etilde;t">fierent</expan> termini 64, 72, 80, &amp; 96, proportio <expan abbr="a&utilde;t">aut</expan> 80 ad 64 <expan abbr="c&otilde;&longs;tituit">con&longs;tituit</expan> &longs;exqui<lb/><expan abbr="quart&atilde;">quartam</expan> atque <expan abbr="diton&utilde;">ditonum</expan>, proportio quoque 96 ad 72 <expan abbr="&longs;exquiterti&atilde;">&longs;exquitertiam</expan> <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> que. <lb/></s>
          <s id="id002970">Rur&longs;us proportio 128 ad 64 <expan abbr="c&otilde;poni&ttilde;">componitur</expan> ex proportionibus </s>
          <s id="id002971">80 ad 64, <expan abbr="&qtilde;">quae</expan> <expan abbr="habe&ttilde;">habetur</expan> <lb/>pro ditono ut <expan abbr="dict&utilde;">dictum</expan> e&longs;t, &amp; e&longs;t &longs;exquiquarta proportio. </s>
          <s id="id002972">At 128 cum 80 e&longs;t in <lb/>proportione &longs;uperpartiente tres quintas, <expan abbr="&qtilde;">quae</expan> <expan abbr="iter&utilde;">iterum</expan> e&longs;t con&longs;ona. </s>
          <s id="id002973">Regula <expan abbr="e&mtilde;">emm</expan> <lb/>e&longs;t quae ubi con&longs;onantia uo <expan abbr="c&utilde;">cum</expan> <expan abbr="diuida&ttilde;">diuidatur</expan> in duas partes, <expan abbr="quar&utilde;">quarum</expan> una &longs;it con&longs;o<lb/>nans, <expan abbr="reliqu&atilde;">reliquam</expan> <expan abbr="eti&atilde;">etiam</expan> e&longs;&longs;e <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan>, at <expan abbr="n&otilde;">non</expan> <expan abbr="c&otilde;uerti&ttilde;">conuertitur</expan>. </s>
          <s id="id002974">S&etail;pe. </s>
          <s id="id002975">n. </s>
          <s id="id002976">fit ut ex duabus <lb/></s>
          <s id="id002977">con&longs;onantibus di&longs;&longs;onans <expan abbr="c&otilde;po&longs;itio">compo&longs;itio</expan> <expan abbr="oria&ttilde;">oriatur</expan>, uelut ex duplici <expan abbr="diap&etilde;te">diapente</expan>, aut <lb/><expan abbr="diap&etilde;te">diapente</expan> <expan abbr="c&utilde;">cum</expan> ditono, &longs;ed ut ad <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> reuertar, alia diapa&longs;on e&longs;t inter 80 <lb/>&amp; 40, at inter 48 &amp; 40 e&longs;t &longs;emiditonus ut <expan abbr="o&longs;t&etilde;&longs;um">o&longs;ten&longs;um</expan> e&longs;t, uelut inter 96 &amp; <lb/>80, nam inter 45 &amp; 40 e&longs;t proportio &longs;exquioctaua, inter 48 <expan abbr="a&utilde;t">aut</expan> &amp; 45 &longs;ex&shy;<lb/>quiquinta decima, <expan abbr="igi&ttilde;">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="&qtilde;">quae</expan> e&longs;t &longs;uperbi&shy;<lb/>partiens tertias &longs;eu &longs;olida <expan abbr="c&utilde;">cum</expan> be&longs;&longs;e &longs;eu &longs;exta maior erit <expan abbr="c&otilde;&longs;onans">con&longs;onans</expan>. </s>
          <s id="id002978">Iam er<lb/>go uidemus detractione aut additione &longs;exquioctuage&longs;im&aelig;, concinnas <lb/>reddi uulgatiores armonias: <expan abbr="terti&atilde;">tertiam</expan> utran que <expan abbr="maior&etilde;">maiorem</expan> &longs;cilicet &amp; <expan abbr="minor&etilde;">minorem</expan>, ac <lb/>rur&longs;us <expan abbr="&longs;ext&atilde;">&longs;extam</expan> <expan abbr="maior&etilde;">maiorem</expan> atque minore <expan abbr="&qtilde;">quae</expan> in minoribus numeris &longs;cilicet &agrave; mo&shy;<lb/>nade ad octo po&longs;it&aelig; &longs;unt. </s>
          <s id="id002979">Vides pr&aelig;terea <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> in &longs;exquiquinta <lb/><arrow.to.target n="table21"/><lb/><expan abbr="c&otilde;&longs;tare">con&longs;tare</expan>: ade&ograve; ut &agrave; &longs;enario infra nihil inutile <lb/><figure id="id.015.01.188.1.jpg" xlink:href="015/01/188/1.jpg"/>reddatur. </s>
          <s id="id002980">Diate&longs;&longs;aron <expan abbr="a&utilde;t">aut</expan> cum primum di <lb/>uidi pote&longs;t, &longs;i &longs;ecus diuidatur &lt;08&gt; in <expan abbr="diton&utilde;">ditonum</expan> <lb/>&amp; <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, aut in &longs;emiditonum &amp; <expan abbr="ton&utilde;">tonum</expan>, <lb/>&longs;cilicet in duo <expan abbr="tant&utilde;">tantum</expan> interualla, non <expan abbr="c&otilde;mo&shy;dius">commo&shy;<lb/>dius</expan> <expan abbr="qu&atilde;">quam</expan> inter octo &amp; &longs;eptem &amp; &longs;ex diuidi <lb/>pote&longs;t. </s>
          <s id="id002981">Cum ergo octo ad <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> di&longs;&longs;ona &longs;it, <lb/>quippe nimis remota e&longs;t h&etail;c proportio &agrave; &longs;en<lb/>&longs;u humano: <expan abbr="quamobr&etilde;">quamobrem</expan> ex regula data, ne&shy;<lb/>que proportio <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> ad &longs;ex. </s>
          <s id="id002982">Sed dubitabis <lb/>merit&ograve;, quia <expan abbr="c&utilde;">cum</expan> diate&longs;&longs;aron diuidatur <expan abbr="bifa&shy;ri&atilde;">bifa&shy;<lb/>riam</expan>, in <expan abbr="diton&utilde;">ditonum</expan> &amp; <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, ac rur&longs;us in <expan abbr="&longs;e&shy;miditon&utilde;">&longs;e&shy;<lb/>miditonum</expan> &amp; <expan abbr="ton&utilde;">tonum</expan>, quarum altera <expan abbr="c&otilde;&longs;onans">con&longs;onans</expan> e&longs;t, reliqua <expan abbr="n&otilde;">non</expan>. </s>
          <s id="id002983"><expan abbr="Vide&ttilde;">Videtur</expan> ergo <lb/>infirmari regula illa, quae con&longs;onantia diui&longs;a &longs;i una pars <expan abbr="c&otilde;&longs;onet">con&longs;onet</expan>, alia non <lb/>po&longs;sit e&longs;&longs;e di&longs;&longs;onans, <expan abbr="n&atilde;">nam</expan> con&longs;tat <expan abbr="coni&utilde;">conium</expan> &amp; <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan> tam per &longs;e quam in <lb/><expan abbr="c&otilde;po&longs;itione">compo&longs;itione</expan> di&longs;&longs;onare: &amp; <expan abbr="n&otilde;">non</expan> <expan abbr="par&utilde;">parum</expan> &longs;ed acerb&egrave;. </s>
          <s id="id002984"><expan abbr="Ver&utilde;">Verum</expan> re&longs;pondeo diate&longs;&longs;a<lb/>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="e&mtilde;">emm</expan> per <lb/>&longs;e e&longs;t, di&longs;&longs;onans e&longs;t: at que &longs;ic in <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan> &amp; di&longs;&longs;onantem diuidi pote&longs;t: <lb/>quatenus <expan abbr="a&utilde;t">aut</expan> pars e&longs;t diapa&longs;on <expan abbr="c&otilde;&longs;onans">con&longs;onans</expan> in acutis: quan &lt;08&gt; <expan abbr="eti&atilde;">etiam</expan> adiecta <lb/>ditono aut &longs;emiditono &longs;upr&agrave; efficiat <expan abbr="&longs;ext&atilde;">&longs;extam</expan> maiorem aut <expan abbr="minor&etilde;">minorem</expan> parum <lb/>ben&egrave; &longs;onantes. </s>
          <s id="id002985">At quintupla proportio ut ab initio propo&longs;itum e&longs;t, <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> <lb/>bis diapa&longs;on, &amp; &longs;exquiquarta, ut plan&egrave; <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t: &longs;exquiquarta <expan abbr="a&utilde;t">aut</expan> 


<pb pagenum="170" xlink:href="015/01/189.jpg"/>ditonus: bis diapa&longs;on <expan abbr="a&utilde;t">aut</expan> quindecim uo cibus. </s>
          <s id="id002986">Omnes igitur decem, &amp; <lb/><expan abbr="&longs;ept&etilde;">&longs;eptem</expan> uoces, <expan abbr="&qtilde;">quae</expan> &longs;exdecim interuallis <expan abbr="di&longs;tinguun&ttilde;">di&longs;tinguuntur</expan>, con&longs;onantes &longs;unt: &amp; ex <lb/>genere ditoni, &amp; &longs;exquiquart&aelig;, &longs;ed paulo minus ben&egrave; <expan abbr="&longs;on&atilde;t">&longs;onant</expan> quod ditonus <lb/>ip&longs;e. </s>
          <s id="id002987">Igitur <expan abbr="quintupl&atilde;">quintuplam</expan> multiplicem ad &longs;ex <expan abbr="quiquart&atilde;">quiquartam</expan> reduximus. </s>
          <s id="id002988">Verum <lb/>ut o&longs;ten&longs;um e&longs;t &amp; decima&longs;eptima, <expan abbr="&qtilde;">quae</expan> bis diapa&longs;on <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan>, &amp; &longs;emiditono <lb/>ben&egrave; &longs;onat, h&etail;c <expan abbr="a&utilde;t">aut</expan> inter nonaginta &longs;ex &amp; uiginti: quadrupla <expan abbr="igi&ttilde;">igitur</expan> e&longs;t &amp; <lb/>&longs;uperquadripartiens quintas. </s>
          <s id="id002989">Diapa&longs;on quo que cum &longs;exta maiore &amp; mi<lb/>nore eandem habent rationem quam 16 ad 5, &amp; 10 ad 3, triplam utranque, <lb/>&longs;ed altera &longs;exquiquinta, altera &longs;exquitertia: bis diapa&longs;on uer&ograve; <expan abbr="c&utilde;">cum</expan> ei&longs;dem <lb/>ut uiginti ad tria, &amp; 32 ad quin que &longs;excupla utraque: &longs;ed altera &longs;uperbipar&shy;<lb/>tiens tertias, altera quintas. </s>
          <s id="id002990"><expan abbr="Manife&longs;t&utilde;">Manife&longs;tum</expan> e&longs;t igitur hanc diui&longs;ionem <expan abbr="n&otilde;">non</expan> &longs;o&shy;<lb/>lum concinnam magis e&longs;&longs;e &amp; &longs;uauem &longs;ed omnem <expan abbr="tonor&utilde;">tonorum</expan> &amp; &longs;emitonio&shy;<lb/>rum <expan abbr="nece&longs;sitat&etilde;">nece&longs;sitatem</expan> effugere. </s>
          <s id="id002991">Qu&ograve;d uer&ograve; in cau&longs;a fuit ut toni &amp; &longs;emitonia <lb/>in u&longs;u e&longs;&longs;ent, id e&longs;t, quoniam in <expan abbr="di&longs;c&etilde;do">di&longs;cendo</expan> nece&longs;&longs;e e&longs;t eandem &longs;eruari ratio&shy;<lb/>nem in <expan abbr="crementor&utilde;">crementorum</expan>, ne que arithmeticam &longs;ed <expan abbr="geometric&atilde;">geometricam</expan>. </s>
          <s id="id002992">Ide&ograve; <expan abbr="a&longs;c&etilde;&longs;us">a&longs;cen&longs;us</expan> per <lb/>tonos &amp; &longs;emitonia <expan abbr="c&otilde;modus">commodus</expan> fuit, nam duplicem <expan abbr="&longs;ol&utilde;">&longs;olum</expan> differentiam pue<lb/>ri u&longs;u a&longs;&longs;equi coguntur. </s>
          <s id="id002993">At uer&ograve; poterat &amp; per &longs;exqui&longs;extam diuidi dia<lb/>te&longs;&longs;aron, ut inter triginta &longs;ex &amp; quadraginta nouem interpo&longs;itis 42, ue&shy;<lb/>r&ugrave;m triplex <expan abbr="&longs;equeba&ttilde;">&longs;equebatur</expan> in <expan abbr="c&otilde;ueniens">conueniens</expan>: primum ut diate&longs;&longs;aron ad amu&longs;sim <lb/>non &longs;eruaretur, &longs;ed incidebat in cacophoniam, addita quadrage&longs;ima o&shy;<lb/>ctaua parte: deficiente <expan abbr="a&utilde;t">aut</expan> in duabus &longs;exqui&longs;eptimis numeris &longs;eu propor<lb/>tione &longs;exquitertia: ut inter 49 &amp; 64 loco 48 &amp; 64, uelut <expan abbr="eti&atilde;">etiam</expan> inter 48 ad <lb/>36, addita igitur monade in termino medio utrin que fit di&longs;&longs;onantia. </s>
          <s id="id002994">Se&shy;<lb/>cundum inconueniens, e&longs;t quae &longs;ic diuidente non &longs;eruabatur ratio &longs;exqui&shy;<lb/>quart&aelig; &amp; &longs;exquiquint&aelig; &longs;eu ditoni &amp; &longs;emiditoni, qu&aelig; uoces ben&egrave; &longs;o&shy;<lb/>nant. </s>
          <s id="id002995">Tertium inconueniens erat, qu&ograve;d h&aelig;c ratio diuidendi diapentes <lb/>minim&egrave; &longs;atisfaciebat, uelut inter 324 &amp; 216. Interponere enim nece&longs;&longs;e <lb/>erat 252 &amp; 294, unde incongrua rur&longs;us erat diui&longs;io. </s>
          <s id="id002996">His tot cau&longs;is cum <lb/>proportiones maiores non fatisfacerent ut &longs;exqui quinta qu&aelig; diate&longs;&longs;a&shy;<lb/>ron nullo modo &aelig;qualiter diuidere pote&longs;t, &amp; in diapente deficit &longs;exqui<lb/>uige&longs;ima quarta, ut inter 25 &amp; 36, coacti &longs;unt cum nec &longs;exqui&longs;exta nec <lb/>&longs;exqui&longs;eptima idone&aelig; e&longs;&longs;ent ad &longs;exquioctauam confugere.</s>
        </p>
        <table>
          <table.target id="table21"/>
          <row>
            <cell>Diapa&longs;on</cell>
            <cell>2</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Bis diapa&longs;on</cell>
            <cell>4</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Diapa&longs;on diapente</cell>
            <cell>3</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Tris diapa&longs;on</cell>
            <cell>8</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>Bis diapa&longs;on <expan abbr="diap&etilde;te">diapente</expan></cell>
            <cell>6</cell>
            <cell>1</cell>
          </row>
          <row>
            <cell>H&aelig;miolia</cell>
            <cell>3</cell>
            <cell>2</cell>
          </row>
          <row>
            <cell>H&aelig;mitrit&aelig;a</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 diapa&longs;on ditonus</cell>
            <cell>5</cell>
            <cell>1</cell>
          </row>
        </table>
        <p type="main">
          <s id="id002997">E&longs;t &amp; alia diui&longs;io toni in &longs;emitonia, <expan abbr="&qtilde;">quae</expan> e&longs;t uaria <expan abbr="pon&etilde;do">ponendo</expan> <expan abbr="ton&utilde;">tonum</expan> inter 18 <lb/>&amp; 16, media uox e&longs;t 17 &longs;emitonium maius inter 17 &amp; 16, &longs;ed minus inter <lb/>18 &amp; 17, <expan abbr="quor&utilde;">quorum</expan> differentia e&longs;t 1/288. Hic &longs;ubit admiratio quomodo <expan abbr="&longs;emi&shy;toni&utilde;">&longs;emi&shy;<lb/>tonium</expan> minus <expan abbr="apte&ttilde;">aptetur</expan> tam grat&egrave; in &longs;ymphonijs, maius <expan abbr="a&utilde;t">aunt</expan> <expan abbr="nequaqu&atilde;">nequaquam</expan>. </s>
          <s id="id002998">Ptole<lb/>m&etail;us hoc negaret, quia &longs;exquiquinta &longs;eu &longs;emiditonus <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> tono inte&shy;<lb/>gro, qui e&longs;t inter 90 &amp; 80, &amp; &longs;emitonio <expan abbr="plu&longs;qu&atilde;">plu&longs;quam</expan> maiore quod e&longs;t inter <lb/>96 &amp; 90, &amp; e&longs;t &longs;exquiquinta decima: <expan abbr="&qtilde;">quae</expan> maior e&longs;t tono maiore 1/255. Pro&shy;<lb/>pterea dicemus cau&longs;am e&longs;&longs;e quae po&longs;ito &longs;emiditono inter 81 &amp; 96, id e&longs;t, <lb/>27 &amp; 32 &longs;ublato tono, id e&longs;t, 234 &amp; 216, remanebit 13 differentia 256 ad <lb/>243, &longs;eu qualis e&longs;t 96 ad 91 &amp; 1/8 qu&aelig; e&longs;t ut 768 ad 729 et redit ad <expan abbr="id&etilde;">idem</expan>, &longs;cili<pb pagenum="171" xlink:href="015/01/190.jpg"/>cet, ut 256 ad 243, 13 autem e&longs;t paulo plus decimanona, ergo multo mi&shy;<lb/>nus &longs;emitonio minore. </s>
          <s id="id002999">&longs;ecundum <expan abbr="m&etilde;tem">mentem</expan> ergo Ptolem&aelig;i, po&longs;ito tono <lb/>inter 135, &amp; 120, &amp; &longs;emitonio maiore inter 128 &amp; 120 remanebit &longs;emito&shy;<lb/>nium minus ferm&egrave; inter 19 &amp; 18, id e&longs;t, 133 &amp; 126, qu&etail; proportio differt <lb/>&agrave; 135 &amp; 138. Si quis autem bene animaduertat, &longs;exquioctuage&longs;ima illa <lb/>adimitur, ex tono &amp; additur &longs;emitonio minori, &amp; h&aelig;c e&longs;t cau&longs;a qu&ograve;d <lb/>&longs;emitonium maius Ptolem&aelig;i &longs;it concinnum, quia additur tonis imper<lb/>fectis. </s>
          <s id="id003000">Dimidium autem &longs;emitonij minoris e&longs;t inter 36 &amp; 35, &amp; uocatur <lb/><expan abbr="c&otilde;ma">comma</expan>: &amp; e&longs;t minus &amp; maius: maius e&longs;t inter 35 &amp; 34, rur&longs;us <expan abbr="c&otilde;ma">comma</expan> mi&shy;<lb/>nus diuiditur in duas die&longs;es, minorem, qu&aelig; e&longs;t inter 72 &amp; 71, &amp; maio&shy;<lb/>rem, qu&etail; e&longs;t inter 71 &amp; 70, &amp; ide&ograve; manet difficultas quomodo intenta <lb/>uoce per die&longs;im fiat melior con&longs;onantia? </s>
          <s id="id003001">nam de remi&longs;sione po&longs;&longs;emus <lb/>dicere qu&ograve;d accipitur loco &longs;exquioctuage&longs;im&aelig;: &longs;ed in &longs;exquioctuage&shy;<lb/>&longs;ima remittitur de tono &longs;ecundum mentem Ptolem&aelig;i, in die&longs;i intendi&shy;<lb/>tur &longs;emitonium minus, &longs;icut o&longs;tendit experimentum, &longs;ed for&longs;an conue<lb/>niunt quia intentio &longs;emitonij minoris deducit &longs;emiditonum ad &longs;exqui<lb/>quintam: e&longs;t enim differentia &longs;emitonij minoris intenti hoc modo ad <lb/>&longs;emitonium minus, ut 136 ad 135: &longs;ed hoc e&longs;t long&egrave; minus &longs;exquioctua<lb/>ge&longs;ima, unum &longs;at e&longs;t, hanc e&longs;&longs;e ultimam diui&longs;ionem toni in octo par&shy;<lb/>tes, &amp; ut in diatonico toni dominantur, ita in chromatico &longs;emitonia in <lb/>enarmonico die&longs;es, &longs;ed die&longs;es fugitando &lpar;ut ita dicam&rpar; ac aures uelli&shy;<lb/>cando, mirum in modum oblectant audientes: uelut toni &longs;tando, un&shy;<lb/>de etiam nomen, &longs;emitonia medium modum obtinent.</s>
        </p>
        <p type="main">
          <s id="id003002">Tertium genus proportionis &lpar;omitto mod&ograve; <expan abbr="diui&longs;ion&etilde;">diui&longs;ionem</expan> temporum <lb/>binarij, ternarij, quinarij, qui ultimus e&longs;t eorum quos &longs;en&longs;us recipiat, <lb/>nam &longs;eptenarius propinquior e&longs;t binarij diui&longs;ioni ob octonarium, &amp; <lb/>modos illos &longs;atis notos Doricum, Lydium &amp; Phrigium, ac eiu&longs;modi&rpar; <lb/>e&longs;t Ptolem&aelig;i: rur&longs;us qui cum uideret de&longs;pectam futuram mu&longs;ic&aelig; con&shy;<lb/>templationem, conatus e&longs;t illius aliquod &longs;ingulare emolumentum <lb/>o&longs;tendere, quemadmodum fecit &amp; in libro de Pr&aelig;dictionibus, exi&longs;ti&shy;<lb/>mans ni illos compo&longs;ui&longs;&longs;et ueluti pr&etail;mium o&longs;tendentes tanti laboris <lb/>quantus nece&longs;&longs;arius uideretur ad intellectum librorum Magn&aelig; com&shy;<lb/>po&longs;itionis, futurum e&longs;&longs;e, ut hi negligerentur, ergo &amp; hoc in mu&longs;ic&aelig; li&shy;<lb/>bris o&longs;tendere molitus e&longs;t, &longs;cilicet, pr&aelig;clarum e&longs;&longs;e <expan abbr="aliqu&etilde;">aliquem</expan> huius <expan abbr="c&otilde;tem&shy;plationis">contem&shy;<lb/>plationis</expan> finem, quod <expan abbr="utin&atilde;">utinam</expan> non feci&longs;&longs;et, ne illud uer&egrave; de eo dici po&longs;&longs;et:</s>
        </p>
        <p type="main">
          <s id="id003003">&mdash;Non omnia po&longs;&longs;umus omnes.</s>
        </p>
        <p type="main">
          <s id="id003004">Virum enim hunc &longs;upra omnem humani ingenij <expan abbr="met&atilde;">metam</expan> fui&longs;&longs;e <expan abbr="n&otilde;">non</expan> nega&shy;<lb/>mus: &longs;ed hanc partem quam hic agit, ade&ograve; infeliciter tractat, ut malim <lb/>credere <expan abbr="tot&utilde;">totum</expan> illum tertium <expan abbr="libr&utilde;">librum</expan> fui&longs;&longs;e ab aliquo alio <expan abbr="adiect&utilde;">adiectum</expan>. </s>
          <s id="id003005">Etenim <lb/>quid turpius &longs;apienti homini &lt;08&gt; imitari uulgares illos? </s>
          <s id="id003006"><expan abbr="&longs;ept&etilde;">&longs;eptem</expan> planet&aelig;, <lb/>&longs;eptem mundi miracula, <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> artes liberales: quid enim &longs;imilitudo nu<pb pagenum="172" xlink:href="015/01/191.jpg"/>meri iuuare pote&longs;t, aut qu&agrave;m afferre utilitatem? </s>
          <s id="id003007">nimis cert&egrave; in <expan abbr="dign&utilde;">dignum</expan> e&longs;t <lb/>uti <expan abbr="argum&etilde;to">argumento</expan> &agrave; &longs;imilitudine &longs;umpto: tum maxim&egrave; ade&ograve; leui. </s>
          <s id="id003008">Sed quo&shy;<lb/>niam con&longs;tat omnia qu&aelig; in mundo &longs;unt ordine coniuncta e&longs;&longs;e, &amp; ne&shy;<lb/>ce&longs;sitate uinciri, ide&ograve; c&ugrave;m finis ip&longs;e uerus &longs;it, non tam debemus Ptole&shy;<lb/>m&aelig;um damnare, quae non probauerit, qu&agrave;m laudare, quod <expan abbr="ueritat&etilde;">ueritatem</expan> &longs;ine <lb/>ratione &longs;it a&longs;&longs;ectus. </s>
          <s id="id003009">S&aelig;pe enim accidit huiu&longs;modi uiris ade&ograve; pr&etail;&longs;tan&shy;<lb/>tibus ut ueritas detegatur, quam c&ugrave;m illi, ut mos e&longs;t <expan abbr="homin&utilde;">hominum</expan>, rationi&shy;<lb/>bus adornare nituntur, tran&longs;gredientes metam muneris, in ab&longs;urda &amp; <lb/>ineptias <expan abbr="incid&utilde;t">incidunt</expan>. </s>
          <s id="id003010">Ergo id mod&ograve; declarare aggrediar, &longs;upponens quae ue&shy;<lb/>rum e&longs;t, &longs;cilicet hanc mu&longs;icam <expan abbr="concinnitat&etilde;">concinnitatem</expan> cum diuinis <expan abbr="connex&atilde;">connexam</expan> e&longs;&longs;e, <lb/>&amp; ab illis originem ducere. </s>
          <s id="id003011">Ver&ugrave;m dubium e&longs;t, an &longs;oni propter nume <lb/>ros iucundi &longs;int, an propter aliud? </s>
          <s id="id003012">&amp; &longs;i propter aliud, cur ergo numeri <lb/>ad hoc &longs;unt nece&longs;&longs;arij? </s>
          <s id="id003013">&amp; cur ob&longs;eruare eos oportet ne ab illorum ordi <lb/>ne di&longs;iungi po&longs;sint? </s>
          <s id="id003014">Hoc <expan abbr="a&utilde;t">aut</expan> perfacil&egrave; <expan abbr="intelligi&ttilde;">intelligitur</expan>, &amp; &agrave; nobis ali&acirc;s decla&shy;<lb/>ratum e&longs;t, &longs;cilicet delectare nos, qu&aelig; percipiuntur qu&aelig;que ratione facta <lb/>uidentur, <expan abbr="quoni&atilde;">quoniam</expan> in his natur&aelig; uis relucet &amp; imago uniuer&longs;i, ergo dele<lb/>ctant nos, quoniam natur&etail; ordine nos con&longs;tamus. </s>
          <s id="id003015">Illud difficilius lon<lb/>g&egrave; &qring;d <expan abbr="tam&etilde;">tamen</expan> diligenti ob&longs;eruatione <expan abbr="dign&utilde;">dignum</expan> uidetur, &longs;cilicet, quonam pa<lb/>cto harmonia cum rebus c&oelig;le&longs;tibus aut humanis <expan abbr="c&otilde;iuncta">coniuncta</expan> &longs;it. </s>
          <s id="id003016">For&longs;an <lb/>&amp; illud ab re non e&longs;&longs;et intelligere, cur nullum animal pr&etail;ter hominem <lb/>capax &longs;it harmoni&aelig;? </s>
          <s id="id003017">an for&longs;an <expan abbr="quoni&atilde;">quoniam</expan> &longs;olus homo ratione participet, <lb/>&amp; ob id &longs;olus gaudet ratione? </s>
          <s id="id003018">ordinata <expan abbr="a&utilde;t">aut</expan> ratione <expan abbr="c&otilde;&longs;tant">con&longs;tant</expan> aut &longs;ola aut <lb/>maxim&egrave;, numerus autem quid aliud e&longs;t qu&agrave;m ordinis <expan abbr="&longs;eparator&utilde;">&longs;eparatorum</expan> ima&shy;<lb/>go. </s>
          <s id="id003019">Porr&ograve; h&aelig;c accipienda &longs;unt ex his qu&aelig; &longs;en&longs;ibus deprehenduntur, <lb/>qualia &longs;unt quae animus mouetur &amp; uarios affectus induit iuxta harmo&shy;<lb/>ni&aelig; diuer&longs;itatem l&aelig;titi&etail;, tri&longs;titi&etail;, impetus, remi&longs;sionis, timoris, &longs;pei, ira&shy;<lb/>cundi&aelig;, &amp; commi&longs;erationis. </s>
          <s id="id003020">Nos enim maxim&egrave; octo affectus mouent <lb/>mu&longs;ic&aelig; modulationes. </s>
          <s id="id003021">Secundum quid autem mouent? </s>
          <s id="id003022">uel quia con&shy;<lb/>&longs;on&aelig; aut di&longs;&longs;on&aelig;, uel quia concitat&etail; aut tard&aelig;, uel quod maius e&longs;t quae<lb/> tendant in acutum ad alacritatem, uel in grauem de&longs;inant &amp; remi&longs;&longs;um <lb/>&longs;onum ad <expan abbr="c&otilde;mi&longs;erationem">commi&longs;erationem</expan>, &amp; lachrymas, aut etiam ex modo tetrachor<lb/>dorum. </s>
          <s id="id003023">Illud &longs;an&egrave; non ob&longs;curum e&longs;t, <expan abbr="anim&atilde;">animam</expan> cum &longs;ono maxim&egrave; e&longs;&longs;e con<lb/><expan abbr="iunct&atilde;">iunctam</expan>, nam neque odoribus ut odores &longs;unt, neque &longs;aporibus, aut his qu&aelig; <lb/>tanguntur licet plurimum delectent, aut etiam l&aelig;dant, anima mouetur <lb/>ad affectus, licet, ut dixi, magis homo delectetur, aut tri&longs;titia afficiatur <lb/>quemadmodum ex &longs;onorum uaria natura, quod etiam in mor&longs;is &agrave; Ta<lb/>rantula &lpar;arane&etail; genus e&longs;t&rpar; deprehenditur. </s>
          <s id="id003024">Quinim&ograve; nec &agrave; luce nec &agrave; co<lb/>loribus aut pictura, ni&longs;i ut h&aelig;c ad memoriam <expan abbr="reuoc&atilde;t">reuocant</expan> ea, propter qu&aelig; <lb/>ad hilaritatem aut tri&longs;titiam uel iram, uel commi&longs;erationem mouemur. <lb/></s>
          <s id="id003025">Vnde <expan abbr="quo&longs;d&atilde;">quo&longs;dam</expan> reges ferunt iniurias acceptas iu&longs;si&longs;&longs;e depingi in aula ne <lb/>po&longs;&longs;ent obliui&longs;ci, at long&egrave; plures <expan abbr="curar&utilde;t">curarunt</expan>, ut potius <expan abbr="eor&utilde;">eorum</expan> facta egregia 


<pb pagenum="173" xlink:href="015/01/192.jpg"/>pingerentur continuata per memoriam uoluptate, quam dum illa &agrave;ge <lb/>rent, <expan abbr="c&otilde;ceperant">conceperant</expan>: nihilominus, neque color ip&longs;e, nec lux aut &longs;pectaculum <lb/>uel imagines po&longs;&longs;unt ade&ograve; mouere animi affectus, uel &longs;onus. </s>
          <s id="id003026">Nam <lb/>duo in uniuer&longs;um ex ui&longs;u ad animi affectus mouendos habentur, tene<lb/>br&aelig; ad tri&longs;titiam &amp; metum, pictura regionum <expan abbr="am&oelig;nar&utilde;">am&oelig;narum</expan> ad iucundita<lb/>tem, &longs;ed <expan abbr="ir&atilde;">iram</expan> qu&aelig; moueant pictur&aelig; alacritatem&uacute;e aut <expan abbr="c&otilde;mi&longs;erationem">commi&longs;erationem</expan>, <lb/>non habemus. </s>
          <s id="id003027">Videtur ergo ob h&aelig;c &longs;onus ip&longs;e magis anim&aelig; intimus <lb/>&lt;08&gt; ullum aliud &longs;en&longs;ile. </s>
          <s id="id003028">Quod &longs;i odoratus e&longs;t in <expan abbr="app&etilde;dicibus">appendicibus</expan> cerebri, ui <lb/>&longs;us in pupilla oculi, gu&longs;tus in lingu&aelig; neruis, ueri &longs;imile e&longs;t magis inti&shy;<lb/>mum e&longs;&longs;e auditum, &longs;cilicet in cerebro ip&longs;o, atque ob id magis ab illo mo&shy;<lb/>ueri animam. </s>
          <s id="id003029">Neque <expan abbr="e&mtilde;">emm</expan> in <expan abbr="a&etilde;re">aere</expan> concepto &agrave; concauitatibus auris, qui no<lb/>&longs;tri pars non e&longs;t: neque &agrave; tympano, c&ugrave;m &longs;uperflua fui&longs;&longs;et cauitas interior <lb/>omnis: neque enim inter pupillam &amp; cerebrum pars ulla cernitur ad ui&shy;<lb/>&longs;um adiuuandum idonea: &longs;ed &longs;olus &longs;ufficit con&longs;en&longs;us pupill&etail; cum cere<lb/>bro: nam ad nos per &longs;piritus differtur imago, non <expan abbr="e&mtilde;">emm</expan> ui&longs;us e&longs;&longs;et unus, <lb/>nec in uno tempore fieret, &longs;ed ueluti &egrave; <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> &longs;peculo &amp; decimo &longs;imul, <lb/>&amp; eodem tempore reflectitur imago, ut &agrave; primo ita &longs;en&longs;us ui&longs;us ex pu&shy;<lb/>pilla in cerebro &amp; in corde &amp; anima &longs;imul relucet. </s>
          <s id="id003030">At ergo non potuit <lb/>in tympano uel neruo den&longs;iore fieri auditus, &longs;ed in cerebro ip&longs;o, ob &qring;d <lb/>magis moueret affectus. </s>
          <s id="id003031">Sed &amp; magis incorporeus e&longs;t &longs;onus, ut qui <lb/>in&longs;trumentum proprium non afficiat, ni&longs;i cum immoderatus fuerit, at <lb/>omnis color, omnis lux oculum afficit, ac, ut ita dicam, tingit, neque &longs;uc&shy;<lb/>ce&longs;siones illas ob id ade&ograve; minutas oculus percipere pote&longs;t ut auris, <lb/>&longs;ed coinquinatur, ut ita dicam, priorum obiectorum reliquijs atque ima<lb/>ginibus. </s>
          <s id="id003032">Vt in uniuer&longs;um con&longs;tet puriorem e&longs;&longs;e auditus &longs;en&longs;um etiam <lb/>anim&aelig; no&longs;tr&aelig; propiorem qu&agrave;m ui&longs;um.</s>
        </p>
        <p type="main">
          <s id="id003033">Quibus con&longs;titutis uidendum e&longs;t, quomodo &longs;onus permutet affe&shy;<lb/>ctus: hoc autem <expan abbr="n&otilde;">non</expan> quia animam, qu&aelig; immortalis e&longs;t &amp; immateriaria, <lb/>&longs;ed quoniam aut corporis eam partem, qu&aelig; e&longs;t anim&aelig; in&longs;trumentum, <lb/>id e&longs;t, &longs;piritum, aut anim&aelig; <expan abbr="principal&etilde;">principalem</expan> coniunctionem qua corpori an&shy;<lb/>nexa e&longs;t. </s>
          <s id="id003034">Vt enim corpus de&longs;erit aut impeditur &agrave; corporis commercio <lb/>corpus immoritur: hoc pr&aelig;&longs;entiens animus, fiunt illa duo pr&aelig;uia ad <lb/>mortem timor &amp; tri&longs;titia. </s>
          <s id="id003035">Vt contr&agrave;, l&aelig;titia non e&longs;t ni&longs;i communicatio <lb/>anim&aelig; corpori, &amp; quatenus communicatur &longs;olum de uita cogitat, atque<lb/> ob id qua&longs;i immortalis, qui l&aelig;tatur obliui&longs;citur mortis. </s>
          <s id="id003036">Ergo anim&etail; ra<lb/>tio illa erit, qu&aelig; ut cogno&longs;cit perfect&egrave; exhilaratur dulcedine uo cum, &amp; <lb/>hoc fit in diapa&longs;on. </s>
          <s id="id003037">Vt uer&ograve; imperfect&egrave; diapente, ut imperfectius dia&shy;<lb/>te&longs;&longs;aron, at cum ex diate&longs;&longs;aro &amp; diapente perficitur diapa&longs;on, accidit ei <lb/><expan abbr="id&etilde;">idem</expan>, quod <expan abbr="qu&aelig;r&etilde;ti">qu&aelig;renti</expan> gemmas in matrice dum inuenit, &amp; ei qui ex tabulis <lb/>arcam <expan abbr="c&otilde;ficit">conficit</expan>, &amp; puero <expan abbr="c&utilde;">cum</expan> adole&longs;cit, &amp; generaliter ei qui ex imperfectis <lb/>perfecta colligit: ex quint&aelig; enim &amp; quart&aelig; &longs;en&longs;u <expan abbr="imperfectar&utilde;">imperfectarum</expan> con&longs;o&shy;


<pb pagenum="174" xlink:href="015/01/193.jpg"/>nantiarum percipit perfectam diapa&longs;on. </s>
          <s id="id003038">Videamus ergo an aliquid &longs;it <lb/>&longs;imile in anim&aelig; facultatibus, nec <expan abbr="dubi&utilde;">dubium</expan> e&longs;t quin ex &longs;en&longs;ibus. </s>
          <s id="id003039">exterioribus <lb/>atque interioribus fiat intelligentia. </s>
          <s id="id003040">Et &longs;en&longs;us <expan abbr="quid&etilde;">quidem</expan> exteriores &longs;exquiter <lb/>tia <expan abbr="c&otilde;&longs;tant">con&longs;tant</expan>: e&longs;t enim <expan abbr="illor&utilde;">illorum</expan> imperfecta cognitio: maior long&egrave; memori&etail; <lb/>unius &amp; rationis reliquarumque <expan abbr="facultat&utilde;">facultatum</expan>, ex quibus <expan abbr="intellig&etilde;tia">intelligentia</expan> oritur. <lb/></s>
          <s id="id003041">Iam uer&ograve; habemus exactam <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> facultatum anim&etail; human&etail;, <expan abbr="&qtilde;">quae</expan> <lb/>cogno&longs;cit. </s>
          <s id="id003042">Nunc ulterius procedamus et uideamus, an &longs;it aliqua <expan abbr="eti&atilde;">etiam</expan> con<lb/>iunctio inter illas, nam &longs;imilitudo et&longs;i &longs;it una originis cau&longs;a, non tamen <lb/>&longs;ola digna e&longs;t ut &agrave; Philo&longs;opho <expan abbr="numere&ttilde;">numeretur</expan> inter cau&longs;as ordinis &amp; natura&shy;<lb/>lis uinculi. </s>
          <s id="id003043">Non e&longs;t ut <expan abbr="tetrachordor&utilde;">tetrachordorum</expan> genera ad partes anim&etail; <expan abbr="c&otilde;paren&shy;tur">comparen&shy;<lb/>tur</expan>, <expan abbr="c&utilde;">cum</expan> &longs;int uoluntaria diui&longs;ione, non natura con&longs;tituta. </s>
          <s id="id003044">Sed &longs;i quis hoc <lb/>uelit, magis ad rationem proprietatis re&longs;piciat, &longs;uauitas in chromatico, <lb/>&longs;ubtilitas in Enarmonico, &longs;tabilitas in diatonico: Vt <expan abbr="Enarmonic&utilde;">Enarmonicum</expan> ad <lb/>mentem uer&egrave; referri po&longs;sit, <expan abbr="chromatic&utilde;">chromaticum</expan> ad &longs;en&longs;us: <expan abbr="diatonic&utilde;">diatonicum</expan> ad <expan abbr="uit&atilde;">uitam</expan> na <lb/>turalemque facultatem. </s>
          <s id="id003045">Sed, ut dixi, iam propius accedamus, <expan abbr="c&otilde;citatior">concitatior</expan> &longs;o<lb/>nus, ut Doricus ad alacritatem pertinet, ad pugnam, ad uim anim&etail; ira&shy;<lb/>&longs;cibilis: Phrygius ad <expan abbr="uoluptat&etilde;">uoluptatem</expan>, Lydius ad intelligentiam remi&longs;sione <lb/><expan abbr="corporeor&utilde;">corporeorum</expan> affectuum. </s>
          <s id="id003046">Sed <expan abbr="n&otilde;">non</expan> qu&etail;rere decet aut laborare, ut mal&egrave; in&shy;<lb/>uenta aut di&longs;tributa aptemus ordini natur&etail;, &longs;ed ut res rebus. </s>
          <s id="id003047">Diximus <lb/>quatuor e&longs;&longs;e <expan abbr="differ&etilde;tias">differentias</expan> <expan abbr="nobilior&utilde;">nobiliorum</expan> <expan abbr="affectu&utilde;">affectuum</expan> animi, &longs;cilicet, timoris, &longs;pei, <lb/><expan abbr="irac&utilde;di&etail;">iracundi&etail;</expan> &longs;eu &longs;&etail;uiti&etail; &amp; <expan abbr="c&otilde;mi&longs;erationis">commi&longs;erationis</expan>, l&etail;titi&etail;, tri&longs;titi&etail;, impetus ac remi&longs;&shy;<lb/>&longs;ionis. </s>
          <s id="id003048">Et <expan abbr="uide&ttilde;">uidetur</expan> mu&longs;ica nec hoc &etail;qualiter monere, &longs;ed <expan abbr="prim&utilde;">primum</expan> uideamus <lb/>an hi &longs;oli affectus &longs;int maximi, quippe dee&longs;&longs;e <expan abbr="uiden&ttilde;">uidentur</expan> amor atque odium. <lb/></s>
          <s id="id003049">Et mihi dubium non e&longs;t quin hi potenti&longs;simi &longs;int <expan abbr="omni&utilde;">omnium</expan> pr&aelig;ter <expan abbr="met&utilde;">metum</expan>. <lb/></s>
          <s id="id003050">Sed metus <expan abbr="c&utilde;">cum</expan> cau&longs;a, affectus propri&egrave; <expan abbr="n&otilde;">non</expan> e&longs;t, &longs;ed potius &longs;cientia <expan abbr="qu&aelig;d&atilde;">qu&aelig;dam</expan>. <lb/></s>
          <s id="id003051">Proprium enim perturbationum e&longs;t excedere rationem: at metus mor<lb/>tis, propri&etail; aut de filio, non e&longs;t &agrave; ratione alien&uacute;s, nec excedit metas, mod&ograve; <lb/>inanis non &longs;it aut fal&longs;us, ob hoc metum excludemus ab hoc negocio: <lb/>tum maxim&egrave; ob id quod nulla mu&longs;ica e&longs;t qu&aelig; <expan abbr="met&utilde;">metum</expan> excitet c&ugrave;m ea, <expan abbr="n&otilde;">non</expan> <lb/>opus &longs;it in eo, qui &longs;it cum ratione coniunctus. </s>
          <s id="id003052">Indicio e&longs;t quae potius <expan abbr="ill&utilde;">illum</expan> <lb/>excudit abrupta mu&longs;ica, &longs;icut &amp; omnia alia qu&aelig; perturbant rationem, <lb/>ueluti <expan abbr="&longs;olan&utilde;">&longs;olanum</expan> &amp; madrangora atque cicuta. </s>
          <s id="id003053">Amorem igitur &amp; odium <expan abbr="n&otilde;">non</expan> <lb/>excitat mu&longs;ica, quia amor &amp; odium alicuius &longs;unt amor &amp; odium, mu&longs;i <lb/>ca <expan abbr="a&utilde;t">aunt</expan> generales &longs;olum mouet animi affectus. </s>
          <s id="id003054">Et commi&longs;eratio, licet &longs;it <lb/>Didonis aut Phillidis, tamen e&longs;t generaliter mi&longs;erentis. </s>
          <s id="id003055">Qu&etail;ramus er&shy;<lb/>go rur&longs;us qui &longs;int affectus generales animi. </s>
          <s id="id003056">Et &longs;an&egrave; <expan abbr="uiden&ttilde;">uidentur</expan> e&longs;&longs;e l&aelig;titia <lb/>atque tri&longs;titia: impetus &amp; remi&longs;sio: &longs;&etail;uitia ac mi&longs;ericordia &amp; audacia. </s>
          <s id="id003057"><expan abbr="S&utilde;t">Sunt</expan> <lb/>tria ferme <expan abbr="c&otilde;i&utilde;cta">coniuncta</expan> &longs;imul impetus &amp; &longs;&aelig;uitia atque audacia, <expan abbr="quoni&atilde;">quoniam</expan> <expan abbr="c&utilde;">cum</expan> mo<lb/>tu perturbato animi &longs;unt eiecta ratione. </s>
          <s id="id003058">Ob id <expan abbr="un&utilde;quod">ununquod</expan>que <expan abbr="hor&utilde;">horum</expan> ab ira&shy;<lb/>cundia <expan abbr="deriua&ttilde;">deriuatur</expan>. </s>
          <s id="id003059">Quapropter &amp; ita <expan abbr="ration&etilde;">rationem</expan> expellit aut &longs;uppeditat. </s>
          <s id="id003060">at ra<lb/>tio <expan abbr="perturba&ttilde;">perturbatur</expan>, aut ab immodicis &longs;onis, aut in <expan abbr="c&otilde;ptis">comptis</expan> et magnas mutatio 


<pb pagenum="175" xlink:href="015/01/194.jpg"/>nes habentibus atque a&longs;peris. </s>
          <s id="id003061">H&aelig;c autem, ut ita dicam, nulla e&longs;t mu&longs;ica. <lb/></s>
          <s id="id003062">Sed neque mu&longs;ica ulla tri&longs;titiam gignit, cum ut dixi, tri&longs;titia nil aliud &longs;it &lt;08&gt;<lb/>mortis imago, mu&longs;ica <expan abbr="a&utilde;t">aut</expan> uitam fouet. </s>
          <s id="id003063">Vnde <expan abbr="n&otilde;">non</expan> immerit&ograve; fertur Xeno <lb/>philus mu&longs;icus <expan abbr="cent&utilde;">centum</expan> quinque annis &longs;ine aliquo <expan abbr="inc&otilde;modo">incommodo</expan> uixi&longs;&longs;e, quod <lb/>&longs;ingulare e&longs;&longs;e exemplum in humana uita refert Plinius. </s>
          <s id="id003064">Relinquitur igi<lb/>tur tandem, ut mu&longs;ica maxim&egrave; moueat tres affectus l&aelig;titiam, remi&longs;sio&shy;<lb/>nem &amp; mi&longs;ericordiam. </s>
          <s id="id003065">Et quod ex his po&longs;tmodum ad labores in&longs;urga&shy;<lb/>mus intentius, hoc non e&longs;t ex mu&longs;ic&etail; ui aut facultate, &longs;ed <expan abbr="c&otilde;&longs;equentibus">con&longs;equentibus</expan> <lb/>ad illa alia cau&longs;is. </s>
          <s id="id003066">Neque ergo <expan abbr="hor&utilde;">horum</expan> cau&longs;as ex diui&longs;ionibus atque di&longs;tribu&shy;<lb/>tionibus uoluntarijs mu&longs;ic&aelig; <expan abbr="c&otilde;&longs;iderare">con&longs;iderare</expan> oportet, &longs;ed ex ip&longs;a <expan abbr="rer&utilde;">rerum</expan> natura <lb/>atque e&longs;&longs;entia. </s>
          <s id="id003067">Veluti intentionis et remi&longs;sionis, a&longs;peritatis atque &longs;uauitatis <lb/>celeritatis ac tarditatis; <expan abbr="c&otilde;&longs;onantium">con&longs;onantium</expan> aut di&longs;&longs;onantium uo <expan abbr="c&utilde;">cum</expan> at que muta&shy;<lb/>tionis: h&aelig; enim differenti&etail; pr&aelig;cipu&etail; &longs;unt uo cum, uel etiam te&longs;te Ari&longs;to <lb/>tele. </s>
          <s id="id003068">Ver&ugrave;m <expan abbr="n&otilde;">non</expan> ob&longs;curum e&longs;t: quemadmodum remi&longs;siones fiant animi </s>
        </p>
        <p type="main">
          <s id="id003069"><arrow.to.target n="marg580"/><lb/>affectuum, <expan abbr="c&utilde;">cum</expan> remittuntur uoces aut intendantur ad <expan abbr="ear&utilde;">earum</expan> intentionem. <lb/></s>
          <s id="id003070">Sed non e&longs;t &aelig;qualis ratio, quoniam natura no&longs;tra ad <expan abbr="remi&longs;sion&etilde;">remi&longs;sionem</expan> natu&shy;<lb/>raliter inclinata e&longs;t, ad intentionem non ita, &longs;ed per uim <expan abbr="quand&atilde;">quandam</expan> aut me&shy;<lb/>dio uoluptatis, aut cum anima purior e&longs;t &agrave; corporis impedimentis. </s>
          <s id="id003071">Et <lb/>ob id ad &longs;tudia nil aptius e&longs;t pura &longs;obrietate: nihil ineptius crapula atque<lb/> temulentia. </s>
          <s id="id003072">At l&etail;titi&etail; cau&longs;&etail; &longs;unt, &amp; <expan abbr="c&otilde;">con</expan>cordia uo <expan abbr="c&utilde;">cum</expan>, &amp; mutatio ex a&longs;pera <lb/>in &longs;uauem, <expan abbr="n&otilde;">non</expan> &longs;ecus ac eius qui euadit &egrave; paupertate uel &egrave; mole&longs;tia aliqua <lb/>aut dolore aut alio <expan abbr="inc&otilde;modo">incommodo</expan>, tum inten&longs;io uo <expan abbr="c&utilde;">cum</expan> ac liber &longs;onus. </s>
          <s id="id003073">Vnde <lb/>in l&etail;titia &longs;olent homines exclamare. </s>
          <s id="id003074">At ad <expan abbr="c&otilde;mi&longs;erationem">commi&longs;erationem</expan> mouendam <lb/>omnia remitti oportet ex magna in parua, adeoque deficientem ex a&longs;pera <lb/>in leuem, ex ueloci in tardam, ex di&longs;&longs;ona in con&longs;onantem. </s>
          <s id="id003075">Antiqui ergo <lb/>&lpar;ut author e&longs;t C&aelig;lius Rhodiginius&rpar; Dorico ad temperantiam &amp; mode <lb/><arrow.to.target n="marg581"/><lb/>rationem utebantur, &longs;cilicet qu&ograve;d non haberet pr&aelig;cipites lap&longs;us, neque<lb/> arduas intentiones: Phrygio ad impetum &amp; bellicum ardorem, &longs;cilicet <lb/>per a&longs;peras intentiones: Lydio ad fletus &amp; lamentationes per ca&longs;us &amp; <lb/>remi&longs;siones longas ac &longs;uaues: ideo funeribus peculiaris: Mixolydio ad <lb/>commi&longs;erationem, ut defectiones interponantur &amp; breues abrupt&aelig;que<lb/> remi&longs;siones, iuuantque in hoc plurimum &amp; &longs;en&longs;us uerborum, familiaris <lb/>hic trag&aelig;dijs: Aeolicus qui &amp; Ionicus tranquillitatis animi author e&longs;t <lb/>&longs;omnumque conciliat: Dorico non ab&longs;imilis &longs;ed &longs;uauior &amp; mollior: ide&ograve; <lb/>chromatici generis. </s>
          <s id="id003076">Qu&etail; uer&ograve; ad c&oelig;li motus referuntur, diapa&longs;on qui&shy;<lb/>dem refertur ad motum diurnum, nam maximo con&longs;tat, &amp; exacti&longs;simo <lb/>interuallo, unusque e&longs;t in omnibus &amp; iucundi&longs;simus &amp; omnia continet, <lb/>uelut &amp; diurnus motus. </s>
          <s id="id003077">Proprius autem t&agrave;m erraticis qu&agrave;m fixis, qui <lb/>etiam &aelig;qualitati propinquior e&longs;t, &amp; ad maiorem di&longs;tantiam &longs;cilicet de&shy;<lb/>clinationis &longs;igniferi ab &aelig;quinoctij circulo ad diapente refertur. </s>
          <s id="id003078">Rur&longs;us <lb/>diate&longs;&longs;aron qu&ograve;d minimo <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> interuallo ac maxim&egrave; in&aelig;quali, &amp; per <lb/>&longs;e quidem qua&longs;i non nece&longs;&longs;ario ad motum in latitudinem <expan abbr="refer&ttilde;">refertur</expan>, is enim 


<pb pagenum="176" xlink:href="015/01/195.jpg"/>exiguus e&longs;t &amp; in&aelig;qualis. </s>
          <s id="id003079">Ex horum itaque duorum <expan abbr="c&otilde;po&longs;itione">compo&longs;itione</expan> quem&shy;<lb/>admodum et ex diate&longs;&longs;aro &amp; diapente conformatur diapa&longs;on, pulchra <lb/>con&longs;truitur exortus &amp; occa&longs;us &longs;yderum ratio, qu&aelig; primo motu <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan>.</s>
        </p>
        <p type="margin">
          <s id="id003080"><margin.target id="marg580"/>I<emph type="italics"/>n lib. 

de<emph.end type="italics"/> A<emph type="italics"/>u<lb/>dibilibus.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003081"><margin.target id="marg581"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. <emph type="italics"/>ca.<emph.end type="italics"/> 3.</s>
        </p>
        <p type="main">
          <s id="id003082">Porr&ograve; de participatione diapente, quam non <expan abbr="&longs;ol&utilde;">&longs;olum</expan> u&longs;urpamus in <expan abbr="in&shy;&longs;trum&etilde;tis">in&shy;<lb/>&longs;trumentis</expan> fi&longs;tularum organis dictis: &longs;ed <expan abbr="eti&atilde;">etiam</expan> in fidibus <expan abbr="monachordor&utilde;">monachordorum</expan> <lb/>&longs;eu <expan abbr="clauichordor&utilde;">clauichordorum</expan> &lpar;ita. </s>
          <s id="id003083">n. </s>
          <s id="id003084">nunc uo<expan abbr="can&ttilde;">cantur</expan> <expan abbr="in&longs;trum&etilde;ta">in&longs;trumenta</expan> quib. </s>
          <s id="id003085">caruerunt anti&shy;<lb/>qui&rpar; <expan abbr="n&otilde;">non</expan> alia e&longs;t ratio, qu&agrave;m <expan abbr="&qtilde;">quae</expan> dicta e&longs;t <expan abbr="con&longs;tituendar&utilde;">con&longs;tituendarum</expan> con&longs;onantiarum <lb/>in ditonis &amp; &longs;emiditonis &longs;extaque utraque. </s>
          <s id="id003086">Vt <expan abbr="e&mtilde;">emm</expan> quatuor con&longs;onanti&aelig; <lb/>&longs;uauiores <expan abbr="efficeren&ttilde;">efficerentur</expan>, nece&longs;&longs;e fuit <expan abbr="un&atilde;">unam</expan>, &longs;cilicet <expan abbr="diapent&etilde;">diapentem</expan> uariari. </s>
          <s id="id003087">Exempli <lb/>gratia, &longs;int fides expo&longs;it&etail; octo, &amp; ut <expan abbr="con&longs;titua&ttilde;">con&longs;tituatur</expan> proportio h ad c, ut 128 <lb/><figure id="id.015.01.195.1.jpg" xlink:href="015/01/195/1.jpg"/><arrow.to.target n="table22"/><lb/>ad 80, id e&longs;t ut 8 ad 5, c facta e&longs;t remi&longs;sior octoge&longs;ima, quare <expan abbr="c&utilde;">cum</expan> <lb/>81 diapente habeat ad 121 <expan abbr="c&utilde;">cum</expan> dimidio, erit ad 80 maior 1 1/2, id e&longs;t <lb/>octuage&longs;ima parte 120, quare intentior diapente. </s>
          <s id="id003088">At in diapa&longs;o <lb/>omnia ad <expan abbr="id&etilde;">idem</expan> redeunt: <expan abbr="hor&utilde;">horum</expan> etiam cau&longs;a &longs;emitonia nigra illa ad&shy;<lb/>dita &longs;unt. </s>
          <s id="id003089">Sed h&etail;c tractatio proprium <expan abbr="loc&utilde;">locum</expan> exigeret, &longs;ecus e&longs;&longs;et ni&shy;<lb/>mis curio&longs;i illa huc traducere. </s>
          <s id="id003090">quemadmodum, &amp; ut uellemus <lb/>Philo&longs;ophiam naturalem, <expan abbr="moral&etilde;">moralem</expan>, &amp; <expan abbr="mathematic&atilde;">mathematicam</expan> ad <expan abbr="mu&longs;ic&atilde;">mu&longs;icam</expan> tra<lb/>ducere <expan abbr="proportion&etilde;">proportionem</expan>. </s>
          <s id="id003091">Melius &longs;an&egrave; fui&longs;&longs;et &longs;ubtilioribus rationibus <lb/><expan abbr="h&atilde;c">hanc</expan> <expan abbr="m&etilde;&longs;uris">men&longs;uris</expan> <expan abbr="motu&utilde;">motuum</expan> <expan abbr="a&longs;tror&utilde;">a&longs;trorum</expan> pro ut <expan abbr="c&otilde;ueni&utilde;t">conueniunt</expan> &lpar;<expan abbr="quant&utilde;">quantum</expan> fieri potuit&rpar; apta&longs;&longs;e.</s>
        </p>
        <table>
          <table.target id="table22"/>
          <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>&longs;ol</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">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id003093">Proportionem mu&longs;icam ad &longs;apores &amp; odores coaptare.<lb/><arrow.to.target n="marg582"/></s>
        </p>
        <p type="margin">
          <s id="id003094"><margin.target id="marg582"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003095">Melius feci&longs;&longs;et Ptolem&etail;us, &longs;i <expan abbr="h&atilde;c">hanc</expan> proportionem ad &longs;apores &amp; odores <lb/>et picturas, <expan abbr="quemadmod&utilde;">quemadmodum</expan> inuenimus nos, applica&longs;&longs;et, uel ut Vitruuius <lb/>ad machinas, poterat <expan abbr="e&mtilde;">emm</expan> hoc &longs;cire, cum Vitruuius plu&longs;&lt;08&gt; centum quin&shy;<lb/>quaginta annis <expan abbr="Ptolem&etail;&utilde;">Ptolem&etail;um</expan> antece&longs;&longs;erit. </s>
          <s id="id003096">Et quan&lt;08&gt; Latin&egrave; &longs;crip&longs;erit, non <lb/>tam turp&egrave; erat latina legi&longs;&longs;e, aut <expan abbr="c&otilde;uer&longs;a">conuer&longs;a</expan> ab alio quopiam intellexi&longs;&longs;e, &lt;08&gt;<lb/>ne&longs;ciui&longs;&longs;e nece&longs;&longs;aria pulchraque inuenta aliorum clarorum uirorum, &amp; <lb/>quod deterius erat, <expan abbr="rer&utilde;">rerum</expan> memorabilium loco fabulas &longs;ubtexui&longs;&longs;e. </s>
          <s id="id003097">Ergo <lb/>ut ad rem ueniam: mu&longs;ica proportio bifariam <expan abbr="inueni&ttilde;">inuenitur</expan> in &longs;aporibus: &longs;im&shy;<lb/>pliciter, &amp; ex comparatione, &amp; &longs;impliciter quidem &longs;umma &longs;uauitas ad <lb/>diapa&longs;on refertur: e&longs;t enim &longs;uaui&longs;simus concen&longs;us in &longs;aporibus, ergo <lb/>dulce ei <expan abbr="re&longs;p&otilde;det">re&longs;pondet</expan>, ut &longs;implex, quid enim &longs;uauius e&longs;&longs;e pote&longs;t in utro que ge<lb/>nere. </s>
          <s id="id003098">At pinguis, qualis in carnibus &amp; ouis ben&egrave; pr&etail;paratis ad <expan abbr="diap&etilde;te">diapente</expan> <lb/>refertur, e&longs;t enim &amp; ip&longs;e &longs;uaui&longs;simus po&longs;t dulce, at que in &longs;uo genere perfe<lb/>ctus, diate&longs;&longs;aron uer&ograve; optim&egrave; &longs;al&longs;o <expan abbr="c&otilde;uenit">conuenit</expan>. </s>
          <s id="id003099">Hic enim per &longs;e improbus <lb/>e&longs;t &amp; in&longs;uauis, &longs;icut etiam &longs;apor &longs;al&longs;us e&longs;t, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> cum diapente <lb/>perficit diapa&longs;on, &amp; cum diapa&longs;o inutile e&longs;t, et di&longs;cordat, ita &longs;apor &longs;al&longs;us <lb/>cum pingui &longs;ummam delectationem affert: cum dulci ade&ograve; parum con<lb/>gruit, ut melius &longs;ocietur <expan abbr="c&utilde;">cum</expan> amaro, uelut in oliuis ben&egrave; &longs;al&longs;is. </s>
          <s id="id003100">Ergo &longs;al&shy;<lb/>&longs;us &longs;apor cum diate&longs;&longs;aro ad <expan abbr="ungu&etilde;">unguem</expan> congruit rur&longs;us &longs;emiditonus <expan abbr="c&utilde;">cum</expan> in&longs;i<lb/>pido, &amp; a&longs;tringens cum ditono conueniunt ad unguem, nam uterque <expan abbr="n&otilde;">non</expan> <lb/>illepidus, &amp; cum dulci conuenit, ita &longs;emiditonus &amp; ditonus cum diapa<pb pagenum="171 &lsqb;=177&rsqb;" xlink:href="015/01/196.jpg"/>&longs;o conueniunt, uterque etiam horum &longs;aporum parum mouet &longs;en&shy;<lb/>&longs;um, &amp; inter &longs;e &longs;unt qua&longs;i &longs;imiles quod ditono accidit &amp; &longs;emidito&shy;<lb/>no, &longs;ed &amp; neuter horum cum pingui conuenit, neque ditonus aut &longs;e&shy;<lb/>miditonus cum diapente congruit, di&longs;cordat enim h&etail;c compo&longs;itio <lb/>non parum. </s>
          <s id="id003101">Rur&longs;us &amp; in hoc &longs;imiles &longs;unt quod diate&longs;&longs;aron cum di&shy;<lb/>tono &amp; &longs;emiditono plurimum conuenit, ita &amp; in&longs;ipidum, &amp; a&longs;trin&shy;<lb/>gens cum &longs;al&longs;o bell&egrave; <expan abbr="c&otilde;ueniunt">conueniunt</expan>. </s>
          <s id="id003102">Diate&longs;&longs;aron enim cum ditono &longs;ex&shy;<lb/>tam efficit maiorem, &amp; cum &longs;emiditono minorem qu&etail; utrique con&longs;o<lb/>nant, non tamen plus &longs;uaues per &longs;e &longs;unt, qu&ograve;d dulci &amp; pingui care&shy;<lb/>ant, ut nec &longs;exta maior aut minor, &qring;d neque diapa&longs;on perficiant neque <lb/>diapente: Acris <expan abbr="aut&etilde;">autem</expan> &longs;apor &longs;exta maiori &longs;imilis e&longs;t, acidus minori: <lb/>mutuo conueniunt cum in&longs;ipido acris, &amp; cum a&longs;tringente acidus, <lb/>quemadmodum &amp; &longs;exta maior cum &longs;emiditono, &amp; minor cum di&shy;<lb/>tono copulatur perficientes diapa&longs;on: &longs;ed minus &longs;uauem, quia ab&shy;<lb/>e&longs;t diapente ibi, quia abe&longs;t pingue: au&longs;terum uero cum acri mode&shy;<lb/>rato conuenit, propterea bene uterque cum in&longs;ipido iungitur, unde <lb/>illud Epigrammatici:</s>
        </p>
        <p type="main">
          <s id="id003103">Vt &longs;apiant fatu&aelig; fabrorum prandia bet&aelig;, <lb/>O quam &longs;&aelig;pe petet uina piperque coquus.</s>
        </p>
        <p type="main">
          <s id="id003104">Piper enim acre e&longs;t, &amp; uinum au&longs;terum e&longs;t. </s>
          <s id="id003105">Et iu&longs;ta querela Cicero&shy;<lb/>nis in Epi&longs;tolis familiaribus, qui &agrave; maluis fatetur &longs;e uictum, ut deci&shy;<lb/>derit in lienteriam: conueniunt ambo hi &longs;apores <expan abbr="c&utilde;">cum</expan> dulci &amp; pingui, <lb/>uelut &amp; utraque &longs;exta maior &amp; minor cum diapa&longs;on &amp; diapente, at <lb/>neuter cum &longs;al&longs;o, nam neque diate&longs;&longs;aron cum &longs;extamaiore uel mino&shy;<lb/>re iungi pote&longs;t. </s>
          <s id="id003106">Amarus autem &longs;apor tono per&longs;imilis e&longs;t, di&longs;&longs;onus <lb/>enim per &longs;e e&longs;t &longs;emper, &amp; amarus per&longs;e odio&longs;us tonus origo e&longs;t o&shy;<lb/>mnium <expan abbr="con&longs;onantiar&utilde;">con&longs;onantiarum</expan>, ita omnes fructus, &longs;eu dulces &longs;eu a&longs;tringen&shy;<lb/>tes, &longs;eu acidi, &longs;eu acres prius amari &longs;unt: tonus pr&aelig;terea nulla cum <lb/>con&longs;onantia peius coit qu&agrave;m cum diapa&longs;o, ita neque amarus &longs;apor <lb/>infelicius iungitur qu&agrave;m cum dulci, amarus quo que &longs;apor cum nul&shy;<lb/>lo magis conuenit <expan abbr="qu&atilde;">quam</expan> cum &longs;al&longs;o, ita tonus additus diate&longs;&longs;aro, perfi<lb/>cit diapente dulci&longs;simam con&longs;onantiam, ut multi oliuas ben&egrave;&longs;al&longs;as <lb/>pr&aelig;tulerint fa&longs;ianis: tantum conuenit &longs;al&longs;o cum amaro, amarus, <lb/>quo que &longs;apor leuis non abhorret &agrave; pingui, deteriorem <expan abbr="tam&etilde;">tamen</expan> aliquan<lb/>to efficit, ut intortis ex ab&longs;ynthio ouis &amp; ca&longs;eo, atque in uitibus in <lb/>quibus coma ab&longs;ynthij in cocta fuit parum, degenerat tamen &longs;apor <lb/>ille &agrave; pingui: ita tono addito ad diapente fit &longs;exta maior, non ade&ograve; <lb/>&longs;uauis ut diapente, at tamen <expan abbr="n&otilde;">non</expan> pror&longs;us in&longs;uauis. </s>
          <s id="id003107">Similiter &longs;i tonus <lb/>addatur ad &longs;emiditonum aut ad ditonum ex altero fit diate&longs;&longs;aron, <lb/>qui non concordat ex reliquo tritonus omnium a&longs;perrimus. </s>
          <s id="id003108">Ergo <lb/>cum idem fiat coniuncto amaro cum in&longs;ipido, ac deterius <expan abbr="c&utilde;">cum</expan> a&longs;trin&shy;


<pb pagenum="172 &lsqb;=178&rsqb;" xlink:href="015/01/197.jpg"/>gente, uelut in acerbis glandibus, quibus nihil tri&longs;tius gu&longs;tari po&shy;<lb/>te&longs;t. </s>
          <s id="id003109">Manife&longs;tum e&longs;t igitur optim&egrave; conuenire hanc &longs;aporum diui&shy;<lb/>&longs;ionem cum mu&longs;ica proportione.</s>
        </p>
        <p type="main">
          <s id="id003110">Cumque &longs;apores ex &longs;eptem planetis pendent manife&longs;t&egrave;, Saturnus <lb/><expan abbr="e&mtilde;">emm</expan> habet a&longs;tringens, quoniam frigidus e&longs;t &amp; &longs;iccus. </s>
          <s id="id003111">Iupiter pingue <lb/><expan abbr="c&otilde;traria">contraria</expan> ratione, &amp; <expan abbr="quoni&atilde;">quoniam</expan> hic &longs;uauis e&longs;t, ille tri&longs;tis, acre &amp; au&longs;terum <lb/><expan abbr="c&otilde;ueniunt&longs;oli">conueniunt &longs;oli</expan>, apparetque in eis uis maxima ad <expan abbr="&longs;pirit&utilde;">&longs;piritum</expan> uitalem <expan abbr="c&otilde;fir">confir</expan> <lb/>mandum, uiresque <expan abbr="o&etilde;s">oens</expan> adauget, uelut &amp; Sol. </s>
          <s id="id003112">Venus habet dulce: de&shy;<lb/>mon&longs;tratione hoc non indiget. </s>
          <s id="id003113">Mars &longs;al&longs;um &amp; <expan abbr="c&utilde;">cum</expan> peruer&longs;&egrave; di&longs;po&longs;i&shy;<lb/>tus e&longs;t, <expan abbr="amar&utilde;">amarum</expan>. </s>
          <s id="id003114">Luna in&longs;ipidum. </s>
          <s id="id003115">Mercurius <expan abbr="acid&utilde;">acidum</expan>, etenim frigida e&longs;t <lb/>&amp; humida Luna, &amp; Mercurius <expan abbr="tenuitat&etilde;">tenuitatem</expan> quan dam habet <expan abbr="c&utilde;">cum</expan> tempe<lb/><expan abbr="ram&etilde;to">ramento</expan> moderato, cuiu&longs;modi ferm&egrave; e&longs;t acidus &longs;apor, quan&lt;08&gt; ad fri&shy;<lb/>giditatem declinet, <expan abbr="par&utilde;">parum</expan> enim habet <expan abbr="uiri&utilde;">uirium</expan> Mercurius &qring;d minima &longs;it <lb/>&longs;tellarum, ut &longs;upr&agrave; docuimus. </s>
          <s id="id003116">Huiu&longs;modi ergo ratione con&longs;iderata <lb/>Luna ad <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> pertinebit Mercurius ad <expan abbr="&longs;ext&atilde;">&longs;extam</expan> minorem, Sol <lb/>ad &longs;extam maiorem, Mars ad <expan abbr="tetrachord&utilde;">tetrachordum</expan>, Saturnus ad ditonum, <lb/>Iupiter ad diapente, Venus ad diapa&longs;on, unde plena illius dona uul<lb/>garis felicitatis opum honoris amoris &amp; uoluptatis, po&longs;t quem e&longs;t <lb/>Iupiter, ut &longs;ine his duobus omnino nulla po&longs;sit e&longs;&longs;e felicitas.</s>
        </p>
        <p type="main">
          <s id="id003117">Sed &amp; in circulo &longs;igniferi aliquam mu&longs;ica proportio habebit ra&shy;<lb/>tionem: diapa&longs;on <expan abbr="e&mtilde;">emm</expan> erit &amp; totius ad dimidium, &amp; be&longs;sis ad trien&shy;<lb/>tem, &amp; dimidij ad quadrantem, &amp; trientis ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diapente <expan abbr="a&utilde;t">aut</expan> <lb/>totius circuli ad be&longs;&longs;em, &amp; dodrantis ad <expan abbr="dimidi&utilde;">dimidium</expan>, &amp; dimidij ad tri&shy;<lb/>entem, &amp; <expan abbr="quadr&atilde;tis">quadrantis</expan> ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> totius circuli ad do<lb/>drantem, &amp; be&longs;sis ad <expan abbr="dimidi&utilde;">dimidium</expan>, &amp; trientis ad <expan abbr="quadr&atilde;tem">quadrantem</expan>: itaque in hoc <lb/>&longs;olo <expan abbr="c&utilde;">cum</expan> Ptolem&etail;o concordamus, in reliquis duobus ne&longs;cio qua ra&shy;<lb/>tione Ptolem&etail;us omi&longs;erit unam <expan abbr="c&otilde;iugationem">coniugationem</expan>, nam <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent qua&shy;<lb/>tuor in diapa&longs;on &amp; diapente, tres tantum numerauit. </s>
          <s id="id003118">Reliquas <expan abbr="a&utilde;t">aunt</expan> <lb/>quatuor per integra &longs;igna numerare licebit, ad <expan abbr="ration&etilde;">rationem</expan>, tamen a&longs;pe&shy;<lb/>ctuum deducere non po&longs;&longs;umus, propterea efficaciam quandam ha<lb/>bent etiam &longs;ignorum mutationes, &longs;ed harmoniam non perficiunt, <lb/>nam &amp; &longs;i &longs;umamus &longs;exquiquartam &amp; &longs;exquiquintam, ut in his &longs;ex&shy;<lb/>quialteram, &longs;eu diapente con&longs;tituamus, aut tria aut &longs;ex &longs;igna acci&shy;<lb/>pere oportebit: utrunque fuerit, reliqua pars ad diate&longs;&longs;aron pertinere <lb/>minim&egrave; pote&longs;t: quamobrem conuenientius e&longs;&longs;et meo iudicio, ut to<lb/>tus circulus non ad diapa&longs;on, uelut Ptolem&aelig;us, referretur, &longs;ed po&shy;<lb/>tius ad diapa&longs;on diapente: ita enim con&longs;titutis quatuor, quinque, <lb/>&longs;ex, duodecimque numeris, con&longs;taret tota ratio harmonica, diui&longs;o e&shy;<lb/>tiam diapente in ditonum &amp; &longs;emiditonum. </s>
          <s id="id003119">&longs;ed de hoc &longs;atis.</s>
        </p>
        <p type="main">
          <s id="id003120">Reuertamur ad &longs;apores, in quibus diximus aliam e&longs;&longs;e rationem <lb/>mu&longs;icam iuxta <expan abbr="c&otilde;po&longs;itionem">compo&longs;itionem</expan>: cum enim inter &longs;apores qui quoui&longs;&shy;


<pb pagenum="173 &lsqb;=179&rsqb;" xlink:href="015/01/198.jpg"/>modo conueniunt, dupla fuerit optimi &longs;aporis proport&iacute;o ad dete&shy;<lb/>riorem, medius uer&ograve; ad deteriorem &longs;exquitertia, optimus ad me&shy;<lb/>dium &longs;exquialtera, &longs;apor ille optimus erit. </s>
          <s id="id003121">Et primum quidem id <lb/>in pingui tanqu&agrave;m medio dulcique &amp; &longs;al&longs;o experiamur, &longs;imiliter in <lb/>&longs;al&longs;o, acri, atque in&longs;ipido. </s>
          <s id="id003122"><expan abbr="Manife&longs;t&utilde;">Manife&longs;tum</expan> e&longs;t enim quod horum optimus <lb/>e&longs;t in&longs;ipidus, quia per &longs;e ferri pote&longs;t, &longs;al&longs;us autem medius, acris de&shy;<lb/>terrimus, &longs;uperabit ergo in&longs;ipidus &longs;al&longs;um &longs;exquialtera, acrem du&shy;<lb/>pla proportione, &longs;al&longs;us acrem &longs;exquitertia. </s>
          <s id="id003123">Rur&longs;us dulcem copule&shy;<lb/>mus cum acri, &amp; cum in&longs;ipido aut cum acido, &amp; in&longs;ipido pr&aelig;&longs;tabit, <lb/>ut dulcis dupla, aut quadrupla, aut octupla proportione in&longs;ipi&shy;<lb/>dum &longs;uperet, id e&longs;t, per diapa&longs;on, uel bis diapa&longs;on, aut ter diapa&shy;<lb/>&longs;on: acidum uero in&longs;ipidum &longs;exquitertia &longs;uperabit. </s>
          <s id="id003124">Alia rur&longs;us ra&shy;<lb/>tio in coniunctionibus &longs;aporum ad &longs;en&longs;um uniu&longs;cuiu&longs;que referenda <lb/>e&longs;t, in quo enim e&longs;t &longs;umma uoluptas comparatione ad illum, hic &longs;ta <lb/>tuemus diapa&longs;on, optimumque con&longs;tituemus &longs;aporem, dimidium il<lb/>lius quod ad uires attinet ex minus iucundo &longs;exquitertium, ad il&shy;<lb/>lum minus iucundum ex medio. </s>
          <s id="id003125">Exempli gratia, proponamus ut <lb/>alicui au&longs;tera maxim&egrave; iucunda &longs;int &lpar;nam &longs;al&longs;a nemini, qu&ograve;d nullum <lb/>animal pr&aelig;ter hominem, im&ograve; ne plant&aelig; quidem ni&longs;i admodum <lb/>pauc&aelig;, &amp; &longs;ui generis &longs;al&longs;o alantur, iucunda e&longs;&longs;e po&longs;&longs;unt: cum &longs;al&longs;um <lb/>amari pars &longs;it, eoque deterius quod acutum &longs;it &longs;al&longs;um, unde in &longs;ale <lb/>nullum animal na&longs;citur: in ab&longs;ynthio, quanqu&agrave;m ualde amaro, exi&shy;<lb/>guum mu&longs;carum genus, nigrum tota &aelig;&longs;tate oritur, &amp; in ruta uer&shy;<lb/>miculi&rpar; is ergo au&longs;teri, quantum &longs;atis erit &longs;umet, dulcis <expan abbr="t&atilde;qu&agrave;m">tanqu&agrave;m</expan> me&shy;<lb/>dij. </s>
          <s id="id003126">gratia exempli &lpar;nam optima ad extremum oppo&longs;itum uix tran&shy;<lb/>&longs;ire queunt&rpar; be&longs;&longs;em accipito huius, gratia exempli, tanqu&agrave;m deter&shy;<lb/>rimi a&longs;tringentis dodrantem, ut &longs;it dulcis ad a&longs;tringentem dupla <lb/>proportio. </s>
          <s id="id003127">Sic ergo con&longs;tituetur iuxta naturam propriam mu&longs;ica <lb/>proportione &longs;apor iucundi&longs;simus.</s>
        </p>
        <p type="main">
          <s id="id003128">Idem quo que in odoribus &amp; eadem ratione, &longs;ed ex &longs;aporibus hoc <lb/>cum intellectum &longs;it, fru&longs;tra fuerit con&longs;umere tempus, eadem enim <lb/>in omnibus ad &longs;ciendum proportionem intelligenda erunt.</s>
        </p>
        <p type="main">
          <s id="id003129">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id003130">Picturarum proportiones explicare.</s>
        </p>
        <p type="main">
          <s id="id003131">E&longs;t pictura imago rei corpore&aelig; quanqu&agrave;m, &amp; per illam, &amp; acti&shy;</s>
        </p>
        <p type="main">
          <s id="id003132"><arrow.to.target n="marg583"/><lb/>ones, &amp; cogitationes, &longs;ed non ni&longs;i ut per corpora &longs;ignificantur: ut <lb/>ergo corpora ip&longs;a referamus. </s>
          <s id="id003133">coloribus opus e&longs;t, nam corpora, co&shy;<lb/>lorata &longs;unt, &longs;ecund&ograve; ip&longs;a rerum natura &longs;cientiaque illarum, unde pi&shy;<lb/>ctorem multi&longs;cium e&longs;&longs;e nece&longs;&longs;e e&longs;t. </s>
          <s id="id003134">tertium e&longs;t, ut minimas earum <lb/>differentias explicare norit. </s>
          <s id="id003135">quartum, ut affectiones, uelut in ira&shy;


<pb pagenum="174 &lsqb;=180&rsqb;" xlink:href="015/01/199.jpg"/>to ruborem, ciliorum <expan abbr="c&otilde;tractionem">contractionem</expan>, tumorem faciei in ambulante <lb/>inclinationem quandam, flexionem cruris atque &longs;imilia. </s>
          <s id="id003136">quintum e&longs;t <lb/>lux coloribus <expan abbr="exhib&etilde;da">exhibenda</expan>, &longs;ed de horum nullo propo&longs;itum e&longs;t hic lo&shy;<lb/>qui, quando quidem h&aelig;c u&longs;u magis &amp; con&longs;ideratione, qu&agrave;m ratio&shy;<lb/>ne con&longs;tent proportione&uacute;e, nec &longs;int ade&ograve; admiranda ut neque &longs;im&shy;<lb/>plex magnitudo <expan abbr="qu&atilde;&longs;exto">quan&longs;exto</expan> loco reponere po&longs;&longs;umus. </s>
          <s id="id003137">Tria ergo ui&shy;<lb/>dentur e&longs;&longs;e pr&aelig;cipua quorum nunc ratio habenda e&longs;&longs;et, ut &longs;int in <lb/>totum nouem, &longs;ed unum ex his relinquemus, tum quia alienum ab <lb/>hac con&longs;ideratione, tum quia alibi pertractatum atque etiam ab alijs, <lb/>neque ade&ograve; admiratione dignum &longs;cilicet magnitudo picturarum re&shy;<lb/>&longs;pondens magnitudini corporum iuxta &longs;itus differentiam, nam <lb/>qu&etail; altiores &longs;unt paulo latiores atque in &longs;uperiori magis parte quam <lb/>in inferiore, mult&ograve; autem longiores e&longs;&longs;e oportet, &longs;ic &amp; qu&aelig; &agrave; latere <lb/>erunt eadem ratione iuxta a&longs;pectus ingredientium rationem. </s>
          <s id="id003138">Ve&shy;<lb/>rum hoc ut dixi omittamus, &amp; de duplici miraculo in pictura lo&shy;<lb/>quamur, &longs;cilicet di&longs;tantia magna quam in parua tabella referimus, <lb/>et corporeitate quam in plano repr&etail;&longs;entamus. </s>
          <s id="id003139">Horum autem duo&shy;<lb/>rum aliqua communia &longs;unt aliqua propria. </s>
          <s id="id003140">Dicemus ergo <expan abbr="prim&utilde;">primum</expan> <lb/>de corpore ita pingendo, ut pal&agrave;m extra tabulam prominere uide<lb/>atur. </s>
          <s id="id003141">Hoc autem primum ex forma &longs;umitur, nam &longs;i corpus in plano <lb/>&longs;it nece&longs;&longs;e e&longs;t, ut partes illius qu&aelig;dam pror&longs;us ab&longs;condantur, par&shy;<lb/>tes ali&aelig; non pror&longs;us, ali&aelig; pror&longs;us &longs;int in con&longs;picuo. </s>
          <s id="id003142">Ergo pictu&shy;<lb/>ram talem fingere oportebit, qu&aelig; partes &longs;ingulas pro ratione o&longs;ten <lb/>dat aut occultet. </s>
          <s id="id003143"><expan abbr="Sec&utilde;da">Secunda</expan> ratio e&longs;t quod ima corporis ob&longs;cura &longs;unt, <lb/>&longs;umm&etail; partes lucid&etail; &amp; clar&aelig; ac lumine qua&longs;i dealbat&aelig;: media, me&shy;<lb/>dia quadam ratione ut in columnis, tantumque pote&longs;t h&aelig;c ratio, ut <lb/>uel &longs;ola picturas fallere nos faciat corpora eas e&longs;&longs;e putantes. </s>
          <s id="id003144">Opor&shy;<lb/>tet autem imum e&longs;&longs;e ad unguem &longs;imile in colore colori anguli loci <lb/>&amp; &longs;ummum parti qu&aelig; &longs;e oculis maxim&egrave; &longs;ubiectam pr&aelig;bet &amp; cla&shy;<lb/>ram: media uer&ograve; qualia ex umbris ob&longs;curari &longs;olent. </s>
          <s id="id003145">Tertia ratio e&longs;t <lb/>pro modo partium iuxta <expan abbr="obliquitat&etilde;">obliquitatem</expan> a&longs;pectus: nam in&longs;picienti a b <lb/>in c d ex e oculo: depingemus in c d iuxta obli&shy;<lb/><figure id="id.015.01.199.1.jpg" xlink:href="015/01/199/1.jpg"/><lb/>quitatem &longs;uam, quia cum c d uideatur per line&shy;<lb/>as e a c &amp; e b d, &amp; eleuatum in &longs;itu a b, nece&longs;&longs;e e&longs;t <lb/>ut uideatur in &longs;itu a b, ergo eleuatum &agrave; c d. </s>
          <s id="id003146">E&longs;t <lb/>&amp; alia con&longs;ideratio proportionis ad proxima <lb/>remotaque, grati a exempli, &longs;i homo e&longs;&longs;et po&longs;t co&shy;<lb/>lumnam a b, lateret eius pars, qu&aelig; e&longs;t propinquior parieti c d, ergo <lb/>&longs;i depinxerimus hominis partes tantum dextram, reliquum &longs;ub um<lb/>bra, cogitur oculus iudicare columnam eleuatam a pariete. </s>
          <s id="id003147">De&shy;<lb/>mum omnia h&aelig;c ita &longs;unt &longs;ubijcienda oculis, &amp; per minimas diffe&shy;


<pb pagenum="175 &lsqb;=181&rsqb;" xlink:href="015/01/200.jpg"/>rentias &amp; animaduer&longs;iones ita dijudicanda, atque experimento &longs;ub&shy;<lb/>ijcienda, tum proprio, tum aliorum non artis in expertium, ut res <lb/>pror&longs;us ab&longs;oluta uideatur, atque in hoc multum refert multiplices <lb/>partes &longs;ecundum longitudinem coloribus di&longs;tinguere ad hoc a&shy;<lb/>ptis, qui &longs;unt ob&longs;curus, &longs;ub ob&longs;curus, cinereus, qualis &longs;ilicis candi&shy;<lb/>dus &longs;ine luce, demum etiam aliquid nigri adijciendum, nam diui&longs;io <lb/>&longs;ecundum longitudinem multum impedit, hanc repr&aelig;&longs;entationem <lb/>iuuant, &amp; extrema ben&egrave; coaptata, uelut &longs;capi imi, &amp; capitula &amp; &longs;u&shy;<lb/>premi, <expan abbr="t&utilde;">tum</expan> trabeationes ex materia coron&aelig;, zofoni, t&oelig;nia, epi&longs;tylia, <lb/>plinthi, echini, hypotrachelia, a&longs;tagali, apophyges. </s>
          <s id="id003148">Qu&aelig; etiam in <lb/>parte inferiore <expan abbr="c&utilde;">cum</expan> &longs;pira &longs;eu ba&longs;i &amp; limbo &amp; toro &amp; plintho inferio&shy;<lb/>re, &amp; &longs;tylobata, et alia t&oelig;nia &longs;umma diligentia, &amp; cum eleuatione ac <lb/>magnitudine ultra column&aelig; limites extendantur. </s>
          <s id="id003149">Sicin &longs;tylobata <lb/>ratio diapente con&longs;tat, cui &longs;olet addi utrinque &longs;exta pars pro coro&shy;<lb/>nice, manife&longs;tum e&longs;t autem, quod in ea con&longs;tat mu&longs;ica ratio diapa&shy;<lb/>&longs;on ex diapente &amp; diate&longs;&longs;aro, compo&longs;iti nam du&aelig; &longs;ext&aelig; partes, alte<lb/>ra utrinque adiecta tertiam conficiunt ut &longs;it diate&longs;&longs;aron &longs;upr&agrave; diapen<lb/>te. </s>
          <s id="id003150">In regionibus autem &amp; &longs;patijs depingendis eadem ferm&egrave; &longs;eruan <lb/>da &longs;unt duobus tamen adiectis, <expan abbr="quor&utilde;">quorum</expan> unum e&longs;t ut longinqui&longs;sima <lb/>pars, <expan abbr="n&otilde;">non</expan> per nigrum aut ob&longs;curum, &longs;ed c&oelig;ruleum <expan abbr="color&etilde;">colorem</expan>, qualis in <lb/>c&oelig;lo determinanda e&longs;t &lpar;ni&longs;i nox fingatur&rpar; nam c&oelig;lum longi&longs;sim&egrave; <lb/>&agrave; nobis di&longs;tat, ita nubes coloribus proprijs, &amp; montes cum niui&shy;<lb/>bus, &amp; &longs;patia uelut fluminis alueus, mare, lacus, atque h&aelig;c omnia <lb/>per colores di&longs;tanti&aelig; finguntur, uelut fluminis pars propior clara <lb/>&amp; lympida, &amp; colore aqueo cernitur remota ob&longs;cura, qu&aelig; maxi&shy;<lb/>m&egrave; procul abe&longs;t nigra. </s>
          <s id="id003151">Sed maxima e&longs;t confirmatio in compara&shy;<lb/>tionibus: ut &longs;i arbores prop&egrave; magn&aelig; &longs;int, &amp; homines &amp; animalia, <lb/>in remotiore autem parte minimi, ac qua&longs;i puncti magnitudinem <lb/>referentes, atque ut in his mu&longs;ica non geometrica aut arithmeti&shy;<lb/>ca proportio &longs;eruetur. </s>
          <s id="id003152">Equidem &longs;i quis iudicio h&aelig;c con&longs;equa&shy;<lb/>tur, ac diligentia qu&aelig; &longs;cribi non po&longs;&longs;unt, &longs;ed contemplatione ha&shy;<lb/>bentur, &longs;en&longs;u quoque, quem experimentum docet, nec ip&longs;um man&shy;<lb/>dare literis, licet ex rationibus tamen, quas hic docemus intelli&shy;<lb/>get parum differre repr&aelig;&longs;entationem &agrave; re ip&longs;a corporea. </s>
          <s id="id003153">Sed de <lb/>his hactenus, qu&aelig; &longs;i diligentius quis per&longs;equi uelit &longs;ine <lb/>artis experientia, plus adimet perfectioni rei, <lb/>quam adijciet. </s>
          <s id="id003154">Hoc enim ali&acirc;s <lb/><arrow.to.target n="marg584"/><lb/>declarauimus.</s>
        </p>
        <pb pagenum="176 &lsqb;=182&rsqb;" xlink:href="015/01/201.jpg"/>
        <p type="margin">
          <s id="id003155"><margin.target id="marg583"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003156"><margin.target id="marg584"/>I<emph type="italics"/>n prima <emph.end type="italics"/><lb/>D<emph type="italics"/>islcffic&aelig;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003157">Propo&longs;itio cente&longs;ima &longs;exage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id003158">Proportionem mu&longs;icam in in&longs;trumentis declarare iuxta compo<lb/>&longs;itionis rationem.<lb/><arrow.to.target n="marg585"/></s>
        </p>
        <p type="margin">
          <s id="id003159"><margin.target id="marg585"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003160">Tria &longs;unt in&longs;trumentorum genera, in quibus maxim&egrave; relucet ra&shy;<lb/>tio compo&longs;itionis mu&longs;ic&aelig; qu&aelig; &agrave; nobis nunc &longs;unt demon&longs;tranda, <lb/>&longs;cilicet machin&aelig; bellic&etail;, ut catapult&aelig; &amp; bali&longs;t&etail; &amp; &longs;corpiones, &amp; hy<lb/>draulica in&longs;trumenta ad modulationes parata, qu&aelig; antiquo tem&shy;<lb/>pore maxim&egrave; in u&longs;u fuerunt nunc de&longs;ita, de quibus Vitruuius agit </s>
        </p>
        <p type="main">
          <s id="id003161"><arrow.to.target n="marg586"/><lb/>in decimo libro. </s>
          <s id="id003162">Tertium e&longs;t &aelig;neorum in&longs;trumentorum, quorum <lb/>etiam u&longs;us de&longs;ijt in &longs;c&oelig;nicis theatris, ad intendendam uocem cum <lb/>modulatione, ut etiam clamor audientium &amp; uulgi cum uoluptate <lb/><arrow.to.target n="marg587"/><lb/>excipiatur, de quo idem in quinto libro egit. </s>
          <s id="id003163">Sed nil melius qu&agrave;m <lb/>uerba ip&longs;ius explicare de hoc tractantis, &longs;unt autem h&aelig;c. </s>
          <s id="id003164">&ldquo;Mu&longs;icen <lb/>autem &longs;ciat oportet, uti canonicam rationem &amp; mathematicam no&shy;<lb/>tam habeat: pr&aelig;terea bali&longs;tarum, catapultarum, &longs;corpionum tem&shy;<lb/>peraturas po&longs;sit rect&egrave; facere. </s>
          <s id="id003165">In capitulis enim dextra ac &longs;ini&longs;tra <lb/>&longs;unt foramina homotonorum, per qu&etail; tenduntur ergatis aut &longs;ucu&shy;<lb/>lis &amp; uectibus &egrave; neruo torti funes, qui non pr&aelig;cluduntur, nec pr&aelig;&shy;<lb/>ligantur ni&longs;i &longs;onitus ad artificis aures certos &amp; &etail;quales fuerint. </s>
          <s id="id003166">Bra&shy;<lb/>chia enim qu&aelig; in eas tentiones includuntur cum extenduntur &aelig;&shy;<lb/>qualiter &amp; parter utraque plagam emittere debent. </s>
          <s id="id003167">Quod &longs;i non ho&shy;<lb/>motona fuerint, impedient directam telorum mi&longs;sionem. </s>
          <s id="id003168">Item the&shy;<lb/>atris ua&longs;a &aelig;rea, qu&etail; in cellis &longs;ub gradibus. </s>
          <s id="id003169">mathematica ratione <expan abbr="collo&shy;can&ttilde;">collo&shy;<lb/>cantur</expan>, &amp; <expan abbr="&longs;onit&utilde;">&longs;onitum</expan> di&longs;crimina, qu&etail; Gr&etail;ci <foreign lang="greek">&rpar;hx=eia</foreign> <expan abbr="uoc&atilde;t">uocant</expan>, ad &longs;ymphonias mu <lb/>&longs;icas &longs;iue concentus <expan abbr="componun&ttilde;">componuntur</expan>, diui&longs;a in circinatione diate&longs;&longs;aron <lb/>&amp; diapente &amp; diapa&longs;on, uti uox &longs;c&oelig;nici &longs;onitus <expan abbr="c&otilde;ueniens">conueniens</expan> in di&longs;po <lb/>&longs;itionibus, tactu <expan abbr="c&utilde;">cum</expan> o&longs;tenderit aucta <expan abbr="c&utilde;">cum</expan> <expan abbr="increm&etilde;to">incremento</expan> clarior et &longs;uauior <lb/>ad <expan abbr="&longs;pectator&utilde;">&longs;pectatorum</expan> perueniat aures. </s>
          <s id="id003170">Hydraulicas quo que machinas &amp; c&aelig;&shy;<lb/>tera <expan abbr="&qtilde;">quae</expan> &longs;unt &longs;imilia his organis &longs;ine mu&longs;icis rationibus. </s>
          <s id="id003171">efficere nemo <lb/>poterit. </s>
          <s id="id003172">Capiamus ergo primum illud &qring;d e&longs;t manife&longs;tius, &longs;cilicet de <lb/>hydraulicis organis quorum meminit Suetonius in Nerone: Reli&shy;<lb/>quam diei partem per organa hydraulica noui &amp; ignoti generis cir<lb/>cunduxit, o&longs;tenden&longs;que &longs;ingula de ratione ac difficultate cuiu&longs;que di&longs;&shy;<lb/>&longs;erens iam &longs;e prolaturum, ut con&longs;tet illa fui&longs;&longs;e magni opificij qu&aelig; <lb/>no&longs;tra &etail;tate de&longs;iere.&rdquo; Re&longs;tat unicum &amp; ualde leue <expan abbr="exempl&utilde;">exemplum</expan> auicul&aelig; <lb/>&aelig;ne&aelig; uelligne&aelig; re&longs;onantis. </s>
          <s id="id003173">Certum e&longs;t <expan abbr="a&etilde;re">aere</expan> effici &longs;onum, &longs;ed ita mi<lb/>&longs;ceri aqu&aelig;, ut dulcior &amp; mollior non &longs;olum euadat, &longs;ed etiam acuti&shy;<lb/>or ac modulatior. </s>
          <s id="id003174">Eadem autem ratio maris: &longs;ed cum aqu&aelig; corpus <lb/>moueatur, uidetur difficile &longs;eruare proportionem. </s>
          <s id="id003175">ea prima diffi&shy;<lb/>cultas. </s>
          <s id="id003176">&longs;ecunda e&longs;t, quod c&ugrave;m aqua moueatur, uix fieri po&longs;&longs;e uide&shy;<lb/>tur ut totum &longs;eruet uocis integrum tenorem. </s>
          <s id="id003177">tertia ob illius con&shy;


<pb pagenum="179 &lsqb;=183&rsqb;" xlink:href="015/01/202.jpg"/>&longs;umptionem. </s>
          <s id="id003178">Propterea nil mirum e&longs;t &longs;i Nexo de his &longs;ubtiliter di&shy;<lb/>&longs;putauit, mirum fuit quod in tanta animi perturbatione ni&longs;i ad <lb/>amentia, ut illi putant, referatur. </s>
          <s id="id003179">Sed quid iam amplius uagor, extat <lb/><arrow.to.target n="marg588"/><lb/>compendio&longs;a ratio con&longs;tructionis illius apud eundem Vitruuium <lb/>ubi Philander ex Atheneo &longs;onus hydradis &longs;uauis admodum atque <lb/><arrow.to.target n="marg589"/><lb/>iucundus auditu e&longs;t: ita ut omnes concinnitate capti conuerterent, <lb/>fuitque Alexendrin&etail; urbis inuentum authore Cte&longs;ibio ton&longs;ore, e&longs;t <lb/>autem magn&aelig; Clep&longs;ydr&aelig; in&longs;trumentum non ab&longs;imile, &longs;unt enim <lb/>fi&longs;tul&aelig; in aquam contort&aelig;, qu&aelig;, c&ugrave;m aqua &agrave; iuuene quopiam per&shy;<lb/>cutitur, axinis per organum tran&longs;euntibus inflantur, <expan abbr="periuc&utilde;dum&shy;q&uacute;e">periucundum&shy;<lb/>q&uacute;e</expan> &longs;onum emittunt. </s>
          <s id="id003180">E&longs;t autem ar&aelig; rotund&aelig; hoc in&longs;trumentum <lb/>per&longs;imile inuentumque Ptolem&aelig;i &longs;ecundi Euergit&etail; temporibus, de <lb/>quo eundem Cte&longs;ibium &longs;crip&longs;i&longs;&longs;e ferunt. </s>
          <s id="id003181">Fiebant autem ex &aelig;re &amp; <lb/>ba&longs;is e ligno cum regulis dextra ac &longs;ini&longs;tra &longs;calari regula compactis, <lb/>aqua autem in &etail;rea arca continebatur. </s>
          <s id="id003182">Facil&egrave; autem e&longs;t per h&aelig;c reli<lb/>qua inuenire: nam epi&longs;tomijs includebatur a&euml;r atque re&longs;erabatur, &amp; <lb/>modus erat per uectes: non tamen octo <expan abbr="fi&longs;tular&utilde;">fi&longs;tularum</expan> &amp; exin de uocum <lb/>numerum in&longs;trumentum id &longs;uperabat organa no&longs;tra ut locupleti&shy;<lb/>ora ita a&longs;periora. </s>
          <s id="id003183">Liquet ergo &longs;i fabrilis omnis ars ad Architectum <lb/>pertinet, illum etiam hac ratione oportere e&longs;&longs;e peritum mu&longs;ic&aelig;.<lb/><arrow.to.target n="marg590"/></s>
        </p>
        <p type="margin">
          <s id="id003184"><margin.target id="marg586"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 15. <emph type="italics"/>ad<emph.end type="italics"/><lb/>18. <emph type="italics"/>&amp; in <lb/>cap.<emph.end type="italics"/> 13.</s>
        </p>
        <p type="margin">
          <s id="id003185"><margin.target id="marg587"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 5.</s>
        </p>
        <p type="margin">
          <s id="id003186"><margin.target id="marg588"/>L<emph type="italics"/>ib,<emph.end type="italics"/> 10. <emph type="italics"/>cd,<emph.end type="italics"/><lb/>16.</s>
        </p>
        <p type="margin">
          <s id="id003187"><margin.target id="marg589"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 4. <emph type="italics"/>cap.<emph.end type="italics"/><lb/>24.</s>
        </p>
        <p type="margin">
          <s id="id003188"><margin.target id="marg590"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 5. <emph type="italics"/>ca.<emph.end type="italics"/> 5.</s>
        </p>
        <p type="main">
          <s id="id003189">&ldquo;De Va&longs;is uer&ograve; &aelig;neis theatri quod melius e&longs;t qu&agrave;m ut eundem <lb/>authorem con&longs;ulamus, dicentem ua&longs;a &etail;rea pro ratione magnitudi&shy;<lb/>nis theatri ita fabricentur, ut cum <expan abbr="tang&utilde;tur">tanguntur</expan>, &longs;onitum facere po&longs;sint <lb/>inter &longs;e diate&longs;&longs;aron diapente, ex ordine addit diapa&longs;on, po&longs;tea inter <lb/>&longs;edes theatri con&longs;titutis cellis ratione mu&longs;ica ibi collocentur: ita uti <lb/>nullum parietem tangant circaque habeant locum <expan abbr="uacu&utilde;">uacuum</expan> et &agrave; &longs;ummo <lb/>capite &longs;patium, ponantque inuer&longs;a &amp; habeant in parte qu&etail; &longs;pectat ad <lb/>&longs;cenam &longs;uppo&longs;itos cuneos ne minus alios &longs;emipede, contraque eas <lb/>cellas relinquantur apertur&etail; inferiorum graduum cubilibus lon&shy;<lb/>g&etail; pedes duos alt&aelig; &longs;emipedem. </s>
          <s id="id003190">Et &longs;i non erit ampla magnitudine <lb/>theatrum, media altitudinis tran&longs;uer&longs;aregio de&longs;ignetur, &amp; in ea tre<lb/>decim cell&aelig; duodecim &aelig;qualibus. 

interuallis di&longs;tantes <expan abbr="confornicen&ttilde;">confornicentur</expan> <lb/>uti ea echea qu&aelig; &longs;upra &longs;cripta &longs;unt, ad neten hyperboleon &longs;onan&shy;<lb/>tia in cellis qu&aelig; &longs;untin cornibus extremis utraque parte prima col&shy;<lb/>locentur, &longs;ecunda ab extremis diate&longs;&longs;aron ad <expan abbr="net&etilde;">netem</expan> diezeugmenon, <lb/>tertia diate&longs;&longs;aron ad neten parame&longs;on, quarta ad neten &longs;ynemme&shy;<lb/>non, quinta diate&longs;&longs;aron ad me&longs;en, &longs;exta diate&longs;&longs;aron ad hypaten me&shy;<lb/>&longs;en in medio unum diate&longs;&longs;aron ad hypaten hypaton. </s>
          <s id="id003191">Qu&aelig; sequun&shy;<lb/>tur &amp; ad intelligentiam pr&aelig;dictorum melius ex Gulielmo Philan&shy;<lb/>dro emendata &longs;ic tran&longs;cribemus: Eas regiones in tredecim cellas <lb/>diuidit &aelig;qualibus interuallis: id e&longs;t, cellas paribus uici&longs;sim inter&shy;


<pb pagenum="178 &lsqb;=184&rsqb;" xlink:href="015/01/203.jpg"/>&longs;ticijs di&longs;po&longs;itas di&longs;tribuit &longs;ex hinc atque hinc &amp; unam mediam, qu&aelig; <lb/>tamen non u&longs;us, &longs;ed partitionis &amp; re&longs;pon&longs;us cau&longs;a fit in media pr&etail;&shy;<lb/>cinctione. </s>
          <s id="id003192">In ima pr&aelig;cinctione ponuntur ua&longs;a qu&etail; habent harmo&shy;<lb/>ni&etail; <expan abbr="ration&etilde;">rationem</expan>, hoc modo. </s>
          <s id="id003193">In <expan abbr="cornu&utilde;">cornuum</expan> cellis collocantur qu&aelig; <expan abbr="&longs;onit&utilde;">&longs;onitum</expan> ha&shy;<lb/>bent netes hyperboleon. </s>
          <s id="id003194">Sub&longs;equuntur utrinque qu&aelig; &longs;unt ad neten <lb/>diezeugmenon interuallo con&longs;onantia diate&longs;&longs;aron. </s>
          <s id="id003195">In tertijs cel&shy;<lb/>lis &longs;unt qu&aelig; ad neten parame&longs;en interuallo item diate&longs;&longs;aron, qu&aelig; <lb/>&longs;unt in quartis tono &longs;olummodo di&longs;tant &amp; &longs;unt netes &longs;ynemenon. <lb/></s>
          <s id="id003196">In quintis cellis &longs;unt ad me&longs;en interuallo diate&longs;&longs;aron. </s>
          <s id="id003197">In &longs;extis cellis <lb/>ad hypaten me&longs;on, <expan abbr="it&etilde;">item</expan> diate&longs;&longs;aron &longs;patio. </s>
          <s id="id003198">In media cella &longs;unt ad hy<lb/>paten hypaton interuallo diate&longs;&longs;aron. </s>
          <s id="id003199">In media pr&aelig;cinctione &longs;unt <lb/>ua&longs;a chromatos, collocantur autem in cornibus ua&longs;a qu&aelig; &longs;unt ad <lb/>paraneten hyperbolem. </s>
          <s id="id003200">In &longs;ecundis cellis ad paraneten diezeugme <lb/><expan abbr="n&otilde;">non</expan> &longs;patio diate&longs;&longs;aron, in tertijs ad paraneten hynemenon &longs;patio dia <lb/>pente. </s>
          <s id="id003201">In quartis ad lichanon me&longs;on interuallo diate&longs;&longs;aron. </s>
          <s id="id003202">In quin<lb/>tis ad lichanon hypaton, <expan abbr="it&etilde;">item</expan> diate&longs;&longs;aron. </s>
          <s id="id003203">In &longs;extis ad parame&longs;en &qring;d <lb/>&longs;patium ad paraneten hyperboleon e&longs;t diapente ad paraneten hy&shy;<lb/>nemenon diate&longs;&longs;aron. </s>
          <s id="id003204">In chromatis media cella nulla &longs;unt ua&longs;a, <lb/>quod &agrave; lichano hypaton ad proslambanomenon, aut ad aliam o&shy;<lb/>mnino decem &amp; octo uocum nulla &longs;it con&longs;onantia, &longs;unt enim h&aelig;&shy;<lb/>mitonia tantum duo &amp; tonus. </s>
          <s id="id003205">In tertia pr&aelig;cinctione collocantur <lb/>ua&longs;a diatoni. </s>
          <s id="id003206">Etin cornibus quidem ea qu&aelig; &longs;unt ad paraneten, hy&shy;<lb/>perboleon. </s>
          <s id="id003207">In &longs;ecundis cellis ad paraneten diezeugmenon. </s>
          <s id="id003208">&longs;patio <lb/>diate&longs;&longs;aron. </s>
          <s id="id003209">In tertijs ad paraneten hynemenon diapente. </s>
          <s id="id003210">In quar&shy;<lb/>tis ad lichanon me&longs;on diate&longs;&longs;aron. </s>
          <s id="id003211">In quintis ad lichanon hypaton <lb/>diate&longs;&longs;aron. </s>
          <s id="id003212">In &longs;extis qu&aelig; ad proslambanomenon diate&longs;&longs;aron &longs;pa&shy;<lb/>tio. </s>
          <s id="id003213">In media qu&aelig; &longs;unt ad me&longs;en, quod ea ad proslambanomenon <lb/>habet con&longs;onantiam diapa&longs;on, &amp; ad lychanon hypaton diapente.&rdquo; </s>
        </p>
        <p type="main">
          <s id="id003214"><arrow.to.target n="marg591"/><lb/>H&aelig;c autem ex figura patent in opere de Subtilitate de&longs;cripta.</s>
        </p>
        <p type="margin">
          <s id="id003215"><margin.target id="marg591"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 16.</s>
        </p>
        <p type="main">
          <s id="id003216">Porr&ograve; quod ad machinas attinet. </s>
          <s id="id003217">Sit catapulta, cuius rudens a b <lb/>quam oportet trahere, &longs;i emittere debeat lapi&shy;<lb/><figure id="id.015.01.203.1.jpg" xlink:href="015/01/203/1.jpg"/><lb/>dem, aut &longs;corpio &longs;agittam ad aliquod &longs;ignum <lb/>puta c, cum ergo &longs;onus c a &amp; c b homotenus fue<lb/>rit, non &longs;olum &aelig;qualiter pertract&aelig; erunt c a &amp; <lb/>c b, &longs;ed etiam &aelig;quales: nam &longs;i &aelig;quales e&longs;&longs;ent, &amp; <lb/>in&etail;qualiter tract&aelig;, aut in&etail;quales &amp; in&aelig;qualiter <lb/>tract&etail; <expan abbr="&longs;on&utilde;">&longs;onum</expan> diuer&longs;um <expan abbr="redd&etilde;t">reddent</expan> euidenter. </s>
          <s id="id003218">At &longs;i in&shy;<lb/>&etail;quales &amp; <expan abbr="&etail;qual&etilde;">&etail;qualem</expan> &longs;onum reddant, erit <expan abbr="t&ntilde;">tnm</expan> ut fidis <lb/>not&aelig; qu&aelig; &longs;trepitum edit duplicem, &amp; effigiem <lb/>oculis <expan abbr="multiplic&etilde;">multiplicem</expan>, unde &longs;agitta in partem aduer&shy;<lb/>&longs;am dirigitur <expan abbr="rud&etilde;tis">rudentis</expan> intentioris, atque h&aelig;c ex Vitruuio eodem dum <lb/>de his agit.</s>
        </p>
        <pb pagenum="185" xlink:href="015/01/204.jpg"/>
        <p type="main">
          <s id="id003219">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id003220">Coniugationes cuiu&longs;uis numeri breuiter inuenire.</s>
        </p>
        <p type="main">
          <s id="id003221">Sint gratia exempli <expan abbr="dec&etilde;">decem</expan> homines, &amp; patet quod po&longs;&longs;ent e&longs;&longs;e &longs;in<lb/><arrow.to.target n="marg592"/><lb/>guli, &amp; hoc <expan abbr="dec&etilde;">decem</expan> modis, quia &longs;unt <expan abbr="dec&etilde;">decem</expan>, ut Petrus &amp; Ioannes: item, <lb/>po&longs;&longs;unt e&longs;&longs;e omnes &longs;imul, &amp; hoc uno modo tantum, &amp; po&longs;&longs;unt e&longs;&longs;e <lb/>duo, &amp; hoc pote&longs;t uariari <expan abbr="&qtilde;">qua</expan>draginta quinque modis: &amp; po&longs;&longs;unt e&longs;&longs;e <lb/>octo, &amp; manife&longs;tum e&longs;t, quod <expan abbr="totid&etilde;">totidem</expan> modis uariantur, &longs;cilicet qua&shy;<lb/>draginta quinque, nam cum erunt octo, duo <expan abbr="quirelinqu&utilde;tur">qui relinquuntur</expan>, uariari <lb/>po&longs;&longs;unt 45 modis, ergo &amp; illi octo ad <expan abbr="ungu&etilde;">unguem</expan> totidem modis. </s>
          <s id="id003222">Et &longs;i&shy;<lb/>militer tres quot modis uariantur tot modis <expan abbr="&longs;ept&etilde;">&longs;eptem</expan>, &amp; quot modis <lb/>quatuor tot &longs;ex: quinque autem quia &longs;unt dimidium decem, pluribus <lb/>modis uariantur. </s>
          <s id="id003223">Et ide&ograve; pro ordine huius detrahes <expan abbr="un&utilde;">unum</expan>, ut &longs;i &longs;int <lb/>undecim uiri pones decem, &longs;i decem pones <expan abbr="nou&etilde;">nouem</expan>, &amp; colliges natu&shy;<lb/>ralem seriem numerorum, ut infr&agrave; uides uno &longs;emper termino defi&shy;<lb/>ciente: &amp; ex priore ordine, ubi uidebis &longs;emper <expan abbr="eti&atilde;">etiam</expan> duplicari nume&shy;<lb/>ros: ut 3. 6. in de &longs;ub 6. 10. &amp; 20 &agrave;latere, &amp; &longs;ub 20 35. &amp; &agrave; latere 70 du&shy;<lb/>plum 35, &amp; &longs;ub <lb/><arrow.to.target n="table23"/><lb/><figure id="id.015.01.204.1.jpg" xlink:href="015/01/204/1.jpg"/>70 126, &amp; &agrave; late&shy;<lb/>re 252, &amp; hoc pro <lb/>cognitione &qring;d <lb/>rect&egrave; &longs;is opera&shy;<lb/>tus. </s>
          <s id="id003224">Secund&ograve; a&shy;<lb/>nimaduertes <expan abbr="&longs;e&shy;qu&etilde;tes">&longs;e&shy;<lb/>quentes</expan> ordines <lb/>fieri ex recta li&shy;<lb/>nea priorum, ue<lb/>lut &longs;extus ordo e&longs;t 7. 28. 84. 210. 462. ita incipiendo in primo ordi&shy;<lb/>ne &agrave; 7, &amp; tendendo ad dextram, inuenies illos eo&longs;dem numeros ad <lb/>unguem, &amp; ita in &longs;eptimo ordine 8. 36. 120. 330. &agrave; &longs;ini&longs;tra inuento 8 <lb/>in primo ordine, &amp; procedendo ad dextram, inuenies 36. 120. &amp; <lb/>330. Tertium e&longs;t quod numeri ultimi &agrave; medio &longs;unt ijdem, ut 462 &amp; <lb/>462. 330 &amp; 330. 165 &amp; 165. 55 &amp; 55. 11 &amp; 11. Et &longs;eor&longs;um, ut dixi, rema&shy;<lb/>net 1. Oportet igitur colligere numeros angulares, ut &agrave; latere ui&shy;<lb/>des, &amp; fit 2047 numerus coniugationum, tot enim modis po&longs;&longs;unt <lb/>uariari. </s>
          <s id="id003225">Et &longs;i e&longs;&longs;ent decem tantum, ut ab initio propo&longs;ui, primus or&shy;<lb/>do finitur ad 10, &longs;ecundus ad 45, tertius ad 120, quartus ad 210, quin<lb/>tus ad 252, &longs;extus redit ad 210, &longs;eptimus ad 120, octauus ad 45, no&shy;<lb/>nus ad 10, decimus ad 1. Et ita colligeretur &longs;umma ex extremis nu&shy;<lb/>meris angularibus 1023. Et tot erunt coniugationes. </s>
          <s id="id003226">Hic uides quia <lb/>numerus 10 e&longs;t par, et quod adempta monade, relinquitur 9, qui e&longs;t <lb/>impar qu&ograve;d medius qui pertinet ad quintum ordinem e&longs;t maxi&shy;


<pb pagenum="186" xlink:href="015/01/205.jpg"/>mus, &amp; e&longs;t 252, &amp; e&longs;t coniugatio quinarij: hoc uolui dixi&longs;&longs;e, <lb/><figure id="id.015.01.205.1.jpg" xlink:href="015/01/205/1.jpg"/><arrow.to.target n="table24"/><lb/>ut intelligeres rationes colligendi &longs;ingulos ordines &longs;eor&shy;<lb/>&longs;um. </s>
          <s id="id003227">Quod ergo attinet ad collectionem maximi numeri, <lb/>primus ordo &longs;eruit &longs;emper ultimo <expan abbr="relinqu&etilde;do">relinquendo</expan> monadem, <lb/>&amp; &longs;ecundus penultimo, &amp; tertius antepenultimo, &amp; ita de <lb/><figure id="id.015.01.205.2.jpg" xlink:href="015/01/205/2.jpg"/>alijs, nam &longs;i &longs;ecundus uariatur 55 modis, &amp;&apos;pen&shy;<lb/>ultimus uariabitur 55 modis. </s>
          <s id="id003228">Et &longs;i tertius uaria&shy;<lb/>tur 165 modis, antepenultimus uariatur 165 mo <lb/>dis. </s>
          <s id="id003229">Et ita de alijs.<lb/><arrow.to.target n="table25"/><lb/><arrow.to.target n="marg593"/></s>
        </p>
        <p type="margin">
          <s id="id003230"><margin.target id="marg592"/>C<emph type="italics"/>o.<emph.end type="italics"/> ^&lcub;m}</s>
        </p>
        <p type="margin">
          <s id="id003231"><margin.target id="marg593"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <table>
          <table.target id="table23"/>
          <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/>
          </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/>
          </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/>
          </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/>
          </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/>
          </row>
          <row>
            <cell>7</cell>
            <cell>28</cell>
            <cell>84</cell>
            <cell>210</cell>
            <cell>462</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/>
          </row>
          <row>
            <cell>9</cell>
            <cell>45</cell>
            <cell>165</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/>
          </row>
          <row>
            <cell>11</cell>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
            <cell/>
          </row>
        </table>
        <table>
          <table.target id="table24"/>
          <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"/>
          <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">H&aelig;c autem ratio &longs;atisfacit multum, &amp; e&longs;t ne&shy;<lb/>ce&longs;&longs;aria in temperiebus corporis humani. </s>
          <s id="id003233">Vt in <lb/>&longs;ecundo, De dentibus. </s>
          <s id="id003234">Et etiam ut qu&aelig;libet di&shy;<lb/>&longs;ciplina qu&agrave;m breui&longs;sim&egrave; tradi po&longs;sit, ut gratia <lb/>exempli, medicina tota in una pagina, dico me&shy;<lb/>dicina <expan abbr="n&otilde;">non</expan> &longs;olum Gr&aelig;corum, &longs;ed etiam Arabum <lb/>&amp; Latinorum, &amp; etiam long&egrave; plus: nam &longs;i tradatur uiginti quatuor <lb/>regulis simplicibus, &amp; ex illis fiant coniugationes 16777215, mani <lb/>fe&longs;tum e&longs;t quod erunt regul&aelig; omnes h&aelig; multo plures, qu&agrave;m con&shy;<lb/>tineantur in omnibus libris Gr&aelig;corum, &amp; Arabum, &amp; Latino&shy;<lb/>rum, qui extant. </s>
          <s id="id003235">Et tamen per&longs;picuum e&longs;t, uiginti quatuor regulas <lb/>una pagina commodi&longs;sim&egrave; contineri. </s>
          <s id="id003236">Et hoc ali&acirc;s docui, quan&shy;<lb/>qu&agrave;m credam me erra&longs;&longs;e in &longs;upputatione, nam locum inuenire non <lb/>potui. </s>
          <s id="id003237">Vnum e&longs;t id certum, qu&ograve;d h&aelig;c ratio qu&agrave;m nunc explicabo, <lb/>e&longs;t uera &amp; demon&longs;tratiua, &amp; facillima.</s>
        </p>
        <p type="main">
          <s id="id003238">Cum enim &longs;uperior &longs;it uera &amp; demon&longs;tratiua, non e&longs;t tamen fa&shy;<lb/>cilis, &amp; pr&aelig;cipu&egrave; in magnis numeris. </s>
          <s id="id003239">Et ide&ograve; inueni hanc, qu&aelig; &lpar;ut <lb/>dixi&rpar; facillima e&longs;t: adde numero propo&longs;ito monadem, in de confla&shy;<lb/>ri inuenias numerum &agrave; monade in eodem ordine, &amp; ab eo detra&shy;<lb/>cta monade habes numerum coniugationum. </s>
          <s id="id003240">Exemplum, &longs;i &longs;int <lb/>10 adde 1 fit 11. Vndecimus ergo numerus in proportione dupla <lb/>e&longs;t 1024, detrahe 1 &amp; relinquantur 1023 numerus coniugationum, <lb/>ut in priore &longs;upputatione. </s>
          <s id="id003241">Item &longs;i &longs;int 11 numeri adde 1 fit 12, duo de&shy;<lb/>cimus ergo numerus in proportione dupla e&longs;t 2048, detrahe 1 re&shy;<lb/>linquuntur 2047, coniugationes 11, ut prius in &longs;upr&agrave; &longs;cripto exem&shy;<lb/>plo. </s>
          <s id="id003242">Et ita pro uiginti quatuor regulis adde 1 fit 25, uige&longs;imus quin&shy;<lb/>tus igitur numerus in ordine dupl&aelig; proportionis &agrave; monade e&longs;t <lb/>16777216, ergo detracta monade relinquitur numerus &lpar;ut dixi&rpar; re&shy;<lb/>gularum &amp; coniugationum uiginti quatuor regularum, qu&aelig; ta&shy;<lb/>men non &longs;int contrari&aelig; inuicem: nam tunc e&longs;&longs;ent pauciores. </s>
          <s id="id003243">Et <lb/>quia in i&longs;tis numeris duplicandis po&longs;&longs;es facile incidere in errorem, <lb/>diuide ultimum per 16, &amp; &longs;i nihil &longs;upere&longs;t, rect&egrave; proce&longs;sit opus: &longs;in 


<pb pagenum="187" xlink:href="015/01/206.jpg"/>autem aliquid &longs;uper&longs;it, aberra&longs;ti. </s>
          <s id="id003244">Vt au&shy;<lb/><figure id="id.015.01.206.1.jpg" xlink:href="015/01/206/1.jpg"/><arrow.to.target n="table26"/><lb/>tem habeas numeros &longs;ingulorum or&shy;<lb/>dinum, in quauis multitudine, deduci&shy;<lb/>to numerum ordinis &agrave; primo, &amp; diui&shy;<lb/>de per numerum ordinis ip&longs;ius reli&shy;<lb/>quum, &amp; illud quod prouenit, duci&shy;<lb/>to in numerum maximum pr&aelig;ceden&shy;<lb/>tis ordinis, &amp; habebis numerum qu&aelig;&shy;<lb/>&longs;itum. </s>
          <s id="id003245">Velut &longs;i &longs;int undecim, uolo &longs;ci&shy;<lb/>re breuiter numeros, qui fiunt ex ua&shy;<lb/>riatione trium. </s>
          <s id="id003246">Primum deduco pro <lb/>&longs;ecundo ordine 1 ex 11 fit 10, diuido per <lb/>2 numerum ordinis, exit 5, duco in 11 fit <lb/>55 numerus &longs;ecundi ordinis. </s>
          <s id="id003247">Inde detra<lb/>ho 2, qui e&longs;t numerus differenti&aelig; ordi&shy;<lb/>nis tertij &agrave; primo ex 11, relinquitur 9, di&shy;<lb/>uido 9 per 3 <expan abbr="numer&utilde;">numerum</expan> ordinis exit 3, du&shy;<lb/>co 3 in 55 numerum &longs;ecundi fit 165, nu&shy;<lb/>merus tertij ordinis. </s>
          <s id="id003248">Similiter uolo nu<lb/>merum uariationum quatuor, deduco <lb/>3 differentiam 4 &agrave; primo ordine ab 11, <lb/>relinquitur 8. diuido 8 per 4 numerum ordinis, exit 2, duc 2 in 195 <lb/>fit 330. numerus quarti ordinis. </s>
          <s id="id003249">Similiter pro quinto detraho 4 dif&shy;<lb/>ferentiam &agrave; primo ordine, relinquitur 7, diuido per 5 numerum or&shy;<lb/>dinis exit 1 2/5, duco in 330 numerum pr&aelig;cedentis ordinis, fit 462 <lb/>numerus quinti ordinis.</s>
        </p>
        <table>
          <table.target id="table26"/>
          <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 manife&longs;t&egrave; modus conuertendi proportionem </s>
        </p>
        <p type="main">
          <s id="id003251"><arrow.to.target n="marg594"/><lb/>arithmeticam in proportionem mi&longs;tam: dico mi&longs;tam, quia opor&shy;<lb/>tet addere monadem in priore numero: dein de quia numerum <lb/>terminorum oportet &longs;umere iuxta numerum a&longs;signatum, &longs;cilicet <lb/>addita monade: demum, quia oportet detrahere monadem ip&longs;am. <lb/></s>
          <s id="id003252">E&longs;t tamen &longs;umpta &agrave; proportione Geometrica ut liquet, &longs;cilicet con&shy;<lb/>tinua dupla.</s>
        </p>
        <p type="margin">
          <s id="id003253"><margin.target id="marg594"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003254">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id003255">Propo&longs;itis duobus quibuslibet numeris, quotuis alios, &longs;eu in <lb/>continuum, &longs;eu medios in continua proportione arithmetica, geo&shy;<lb/>metrica &amp; mu&longs;ica inuenire.</s>
        </p>
        <p type="main">
          <s id="id003256">H&aelig;c tota propo&longs;itio pendet ex intellectu diffinitionis earum. <lb/><arrow.to.target n="marg595"/><lb/>Sint ergo propo&longs;iti duo numeri 2 &amp; 3, &amp; uelim tertium in conti&shy;<lb/><arrow.to.target n="marg596"/><lb/>nua proportione arithmetica, duplico quemuis, ut pote 3 fit 6, de&shy;


<pb pagenum="188" xlink:href="015/01/207.jpg"/>traho 2, reliquum remanet 4 tertius numerus. </s>
          <s id="id003257">Item uolo quar&shy;<lb/>tum, duplico 4 fit 8, detraho 3 remanet 5 quartus numerus: item <lb/>uolo minorem 3 &amp; 2, duplico 2 fit 4, detraho 3 remanet 1, &longs;i autem <lb/>uellem minorem uno, non po&longs;&longs;et, quia e&longs;&longs;et nihil, &longs;ed cre&longs;cendo <lb/>pote&longs;t extendi in infinitum, ita capio 2, &amp; &lt;02&gt; 10, duplico &lt;02&gt; 10, fit &lt;02&gt;<lb/>40, detraho 2, remanet &lt;02&gt; 40 m: 2, &amp; ita &longs;i uolo quartum numerum, <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&shy;<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, &longs;unt in con&shy;<lb/>tinua proportione arithmetica, &amp; ita pote&longs;t extendi in infini&shy;<lb/>tum. </s>
          <s id="id003258">Sed &longs;i uellem unum, aut duos, aut tres terminos, uel quouis <lb/>medio 5 arithmetic&aelig;, diuido differentiam per 1 p:numero termi&shy;<lb/>norum, &amp; partes addo minori numero. </s>
          <s id="id003259">Exemplum, uolo tres nu&shy;<lb/>meros medios inter 2 &amp; 7 in continua proportione arithmeti&shy;<lb/>ca, detraho 2 &agrave; 7 remanet 5, diuido 5 per 1 p: quam 3, id e&longs;t per 4, <lb/>exit 1 1/4, adde ergo 1 1/4 ad 2 fit 3 1/4 primus terminus, cui adde iterum <lb/>1 1/4 fit 4 1/2 &longs;ecundus terminus, cui adde iterum 1 1/4 fit 5 3/4 tertius <lb/>numerus: fient ergo quinque termini, hoc modo in continua pro&shy;<lb/>portione arithmetica 23 1/4 4 1/2 5 3/4 &amp; 7. Rur&longs;us uolo totidem, uolo <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/>qui e&longs;t 1 p: numero terminorum, exit &lt;02&gt; 2 m: 1/2, addo ergo &lt;02&gt; 2 m: <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/>&lt;02&gt; 8 p:1, &longs;ecundus terminus, cui etiam addo &lt;02&gt; 2 m: 1/2 fit &lt;02&gt; 18 m: <lb/>1/2, &amp; ita habes tres terminos medios in continua proportione <lb/>arithmetica inter 2 &amp; &lt;02&gt; 32, &amp; ita &longs;i uelles quatuor terminos, diui&shy;<lb/>deres differentiam per 5, &amp; &longs;i uelles quinque, diuideres per &longs;ex. </s>
          <s id="id003260">&amp; <lb/>ita de alijs quibu&longs;cunque.</s>
        </p>
        <p type="margin">
          <s id="id003261"><margin.target id="marg595"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003262"><margin.target id="marg596"/>D<emph type="italics"/>iff,<emph.end type="italics"/> 20.</s>
        </p>
        <p type="main">
          <s id="id003263">Pro Geometrica proponantur, gratia exempli, 2 &amp; 4, &longs;i uelim in <lb/>continua proportione tertium, duco 4 in &longs;emet fit 16, diuido per 2 <lb/>exit 8. &amp; &longs;i uelles quartum duc 8 in &longs;e fit 64, diuide per 4 exit 16 <lb/>quartus terminus, &amp; ita in infinitum, &amp; &longs;i uelles minorem 2, duc 2 <lb/>in &longs;e fit 4, diuide 4 per 4 exit 1 tertius terminus, &amp; ita &longs;i uelles mino&shy;<lb/>rem. </s>
          <s id="id003264">duc 1 in &longs;e fit 1, diuide per 2 exit 1/2 quartus terminus, &amp; ita ha&shy;<lb/>bes quo&longs;uis terminos, &amp; e&longs;t &longs;imilis arithmetic&aelig; h&aelig;c operatio, &longs;ed <lb/>in arithmetica duplicamus unum terminum, &amp; detrahimus alium: <lb/>in geometrica multiplicamus unum terminum ad productum, &amp; <lb/>diuidimus per alium. </s>
          <s id="id003265">Et &longs;i uelim terminum in continua proportio&shy;<lb/>ne 2 &amp; &lt;02&gt; 10, duco eodem modo &lt;02&gt; 10 in &longs;e fit 10, diuido per 2 fit 5 <lb/>tertius terminus, &amp; uelim quartum, duco 5 in &longs;e fit 25, diuido per &lt;02&gt;<lb/>10 exit &lt;02&gt; 62 1/2 quartus terminus.</s>
        </p>
        <p type="main">
          <s id="id003266">Et &longs;i uelles plures terminos medios in proportione geometrica, de <lb/>ducito maius extremum in &longs;e <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> <expan abbr="denomination&etilde;">denominationem</expan> <expan abbr="inferior&etilde;">inferiorem</expan>, id 


<pb pagenum="189" xlink:href="015/01/208.jpg"/>e&longs;t, &longs;i uolo duos terminos &longs;emel, &amp; dein de in minorem, &amp; &lt;02&gt;<lb/>cubica producti e&longs;t &longs;ecundus terminus, idem facio de minore in <lb/>&longs;e in de in maiorem, &amp; accipio &lt;02&gt; cu. </s>
          <s id="id003267">Exemplum, uolo duos termi&shy;<lb/>nos inter 2 &amp; 3, duco 3 in &longs;e fit 9, duco 2 in 9 fit 18, capio &lt;02&gt; cu. </s>
          <s id="id003268">18. hic <lb/>e&longs;t unus terminus, &amp; ita duco 2 in &longs;e fit 4, duco in 3 fit 12, capio &lt;02&gt; cu. <lb/></s>
          <s id="id003269">12 pro &longs;ecundo termino. </s>
          <s id="id003270">Et &longs;i uolo tres terminos, duco 3 in 3 fit 9, du<lb/>co 3 in 9 fit 27, duco 2 in 27 fit 54, &amp; &lt;02&gt; &lt;02&gt; 54 e&longs;t primus terminus. <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/>e&longs;t, &lt;02&gt; 36 e&longs;t &longs;ecundus terminus, &longs;imiliter duco 2 ad &longs;uum cubum fit <lb/>8, duco 3 in 8 fit 24, &amp; &lt;02&gt; &lt;02&gt; 24, e&longs;t tertius terminus. </s>
          <s id="id003272">Similiter uolo <lb/>quatuor terminos medios, duco 3 in 3 fit 9, duco 9 in 9 fit 81, duco 2 <lb/>in 81 fit 162, &amp; &lt;02&gt; relata prima 162, e&longs;t primus terminus, item duco 2 <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/>quartus terminus, item ducendo 3 ad cubum fit 27, &amp; 2 ad quadra&shy;<lb/>tum, &amp; fit 4, &amp; 4 in 27 fit 108, &amp; &lt;02&gt; relata prima 108, erit &longs;ecundus <lb/>terminus, &amp; &longs;imiliter ducendo 2 ad cubum fit 8, &amp; 3 ad quadratum <lb/>fit 9, &amp; 9 in 8 fit 72, &amp; &lt;02&gt; relata prima 72 e&longs;t tertius terminus. </s>
          <s id="id003273">Habe&shy;<lb/>bis ergo terminos in continua proportione 2, id e&longs;t, &lt;02&gt; relata pri&shy;<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/>relata prima 172, &amp; &lt;02&gt; relata prima 243, quod e&longs;t 3, &amp; ita de alijs in <lb/>infinitum.</s>
        </p>
        <p type="main">
          <s id="id003274">At pro mu&longs;ica, &longs;i &longs;int exhibiti duo numeri minores utpot&egrave; 2 &amp; 3, <lb/>uelim tertium terminum, diuido 2 per 1 differentiam exit 2, detraho <lb/>1 pro regula remanet 1, diuido 3 maiorem terminum per 1 exit 3, ad&shy;<lb/>de 3 ad 3, fit 6 maior terminus. </s>
          <s id="id003275">Similiter capio 3 &amp; 4, diuide 3 mino&shy;<lb/>rem terminum per 1 differentiam exit 3, detrahe 1 pro regula, relin&shy;<lb/>quitur 2, diuide 4 terminum medium per 2 exit 2, adde ad 4 fit 6 ma<lb/>ior terminus. </s>
          <s id="id003276">Stiphelius autem erat in &longs;ua regula, nam &longs;ic 12 4 &amp; 3 <lb/>e&longs;&longs;ent in continua proportione mu&longs;ica ex &longs;ua regula. </s>
          <s id="id003277">Dico ergo, <lb/>quod &longs;i proponantur 5 &amp; 7, &amp; uelim mu&longs;icam proportionem con&shy;<lb/>tinuare, detraho 5 de 7 relinquitur 2, diuido 5 per 2 exit 2 1/2, detra&shy;<lb/>he 1 pro regula remanet 1 1/2, diuide 7 per 1 1/2 exit 4 &amp; 2/3, adde ad 7 <lb/>fit 11 2/3, reduc ad integra multiplicando omnia per 3, habebis <lb/>35, 21, &amp; 15, in continua proportione mu&longs;ica, nam 35 ad 15 e&longs;t ut 7 <lb/>ad 3, &amp; 14 ad 6, e&longs;t ut 7 ad 3, e&longs;t autem 14 differentia 21 &amp; 35, &amp; 6 dif&shy;<lb/>ferentia 21 &amp; 15, &amp; ita po&longs;&longs;es continuare inueniendo quartum, <lb/>quintum, &longs;extum, in infinitum. </s>
          <s id="id003278">Rur&longs;us &longs;int propo&longs;iti duo termini <lb/>maiores, uelut 6 &amp; 4, detrahe 4 &agrave; 6 exit 2, diuide 6 per 2 exit 3, ad&shy;<lb/>de 1 pro regula fit 4, diuide 4 minorem terminum per 4 exit 1, de&shy;<lb/>trahe 1 ex 4, relinquitur 3 minor terminus, &amp; ita propo&longs;itis 6 &amp; 3 


<pb pagenum="190" xlink:href="015/01/209.jpg"/>differentia e&longs;t 3, diuide 6 per 3 differentiam exit 2, adde 1 pro re&shy;<lb/>gula fit 3, diuide 3 per 3 exit 1, detrahe ex 3 relinquitur 2 minor ter&shy;<lb/>minus, &amp; ita potes inuenire quotuis. </s>
          <s id="id003279">Gratia exempli, habeo 3 &amp; 2 <lb/>maiores, capio 1 differentiam, per quam diuido 3 exit 3, addo 1 <lb/>fit 4, diuido 2 minorem terminum per 4 exit 1/2, detrahe 1/2 ex <lb/>2, relinquuntur 1 1/2, erunt ergo 32 &amp; 1 1/2, 1. 6. 4. 3. duplican&shy;<lb/>do 2, ut prius in continua proportione mu&longs;ica, quia ergo 632 <lb/>&longs;unt in continua proportione mu&longs;ica, &amp; 32, &amp; 1 1/2 &longs;unt in con&shy;<lb/>tinua proportione mu&longs;ica, erunt duplicando 3. 4. 6. 12. in con&shy;<lb/>tinua proportione mu&longs;ica. </s>
          <s id="id003280">Rur&longs;us &longs;int propo&longs;iti maior, &amp; mi&shy;<lb/>nor terminus, ut 6 &amp; 2, diuides maiorem per minorem exit 3, <lb/>cui addes 1 fit 4, diuide 4 differentiam 6 &agrave; 2 per 4 iam inuentum <lb/>exiti, adde ad 2 fit 3 medius terminus, &longs;imiliter inter 6 &amp; 3, uolo me&shy;<lb/>dium terminum in proportione mu&longs;ica, detraho 3 &agrave; 6, relinquitur <lb/>3, &longs;imiliter diuido 6 maiorem terminum per 3 minorem terminum, <lb/>exit 2, addo 1 pro regula fit 3, diuido 3 differentiam iam &longs;eruatam <lb/>per hoc 3 iam inuentum exit 1, addo ad 3 minorem terminum fit 4, <lb/>medius terminus, &longs;ic uolo inter 4 &amp; 6 medium terminum in con&shy;<lb/>tinua proportione mu&longs;ica, diuido 6 per 4: exit 1 1/2, addo ei pro re&shy;<lb/>gula fit 2 1/2, diuide 2 differentiam 4 &amp; 6 per 2 1/2 exit 4/5, adde ad 4 <lb/>fit 4 4/5 terminus medius, duc omnes in 5, habebis integros nume&shy;<lb/>ros 30, 24 &amp; 20, &amp; &longs;unt pulcherrim&aelig; regul&aelig;, quia po&longs;&longs;es diui&shy;<lb/>dere 24 &amp; 20 interponendo medium, id e&longs;t capiendo 6 &amp; 5, diui&shy;<lb/>de 6 per 5 exit 1 1/5, adde 1 pro regula fit 2 1/5, diuide 1 differentiam <lb/>per 2 1/5 exit 5/11, adde ad 5 fient termini 5 5/11 &amp; 6, reduc ad integra fi&shy;<lb/>ent 55. 60. 66. &amp; quia 30. 24. &amp; 20, etiam erant in continua propor&shy;<lb/>tione, &amp; 30 ad 20, erat &longs;exquialter, ide&ograve; capiam &longs;exquialterum ad <lb/>55, &amp; e&longs;t 82 1/2, erunt ergo 82 1/2 66. 60. &amp; 55. in continua proportio&shy;<lb/>ne mu&longs;ica, ergo duplicando 165 132 120 &amp; 110, erunt in continua <lb/>proportione.</s>
        </p>
        <p type="main">
          <s id="id003281">Adnotat Stiphelius, quod cum fuerint tres termini in continua <lb/>proportione geometrica, &amp; inter primum &amp; tertium interpo&longs;itus <lb/>fuerit terminus in continua proportione arithmetica, quod ibi <lb/>erit proportio mu&longs;ica, &amp; dat exemplum de 12. 9. 8 &amp; 6, &longs;ed ita e&longs;t in&shy;<lb/>telligendum, ut a&longs;&longs;umpta proportione arithmetica, ut pot&egrave; 12 9 &amp; <lb/>6, in de ut e&longs;t 9 ad 6, ita fiat 12 ad 8, tunc i&longs;ti tres termini 128 &amp; 6 e&shy;<lb/>runt in continua proportione mu&longs;ica. </s>
          <s id="id003282">Et hoc e&longs;t pulchrum, &longs;i ita in&shy;<lb/>telligatur, &longs;cilicet ex proportione Geometrica &amp; Arithmetica con&shy;<lb/>&longs;tituere proportionem mu&longs;icam.</s>
        </p>
        <pb pagenum="185 &lsqb;=191&rsqb;" xlink:href="015/01/210.jpg"/>
        <p type="main">
          <s id="id003283">Ex hoc patet &qring;d in <expan abbr="proportion&etilde;">proportionem</expan> Arithmetica &amp; mu&longs;ica &longs;emper, &longs;i <lb/><arrow.to.target n="marg597"/><lb/>duo termini fuerint numeri, tertius erit numerus, &amp; in Geometrica <lb/>idem erit, &longs;i medius &amp; extremus fuerint numeri, erit alter extremus <lb/>numerus, &longs;ed tamen &longs;i unus euariet, omnes poterunt e&longs;&longs;e diuer&longs;i.</s>
        </p>
        <p type="margin">
          <s id="id003284"><margin.target id="marg597"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003285">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id003286">Proportiones Stiphelij de&longs;cribere.</s>
        </p>
        <p type="main">
          <s id="id003287">Con&longs;iderauit Michael Stiphelius quod &longs;ump&longs;it &agrave; <expan abbr="Bo&etilde;tio">Boentio</expan>, qua&longs;&shy;<lb/><arrow.to.target n="marg598"/><lb/>dam inueniri proportiones tribus numeris con&longs;titutis, qu&aelig; in nul&shy;<lb/>lo trium primorum generum continerentur, &longs;ed qu&aelig;dam tamen <lb/>geometricis ali&aelig; mu&longs;icis a&longs;similarentur, prima ergo Geometrica&shy;<lb/>rum e&longs;t, quoties proportio &longs;ecund&aelig; ad primam fuerit, uelut diffe&shy;<lb/>renti&aelig; &longs;ecund&aelig; &amp; prim&aelig; ad differentiam &longs;ecund&aelig; &amp; terti&aelig;. </s>
          <s id="id003288">Velut <lb/><arrow.to.target n="marg599"/><lb/>capio 2, 4, 5, proportio 4 ad 2 e&longs;t dupla talis e&longs;t 2 differenti&aelig; 4 &amp; 2 <lb/><arrow.to.target n="marg600"/><lb/>ad 1 differentiam 5 &amp; 4, nam in uera proportione Geometrica fit <lb/>conuer&longs;o modo, quia proportio &longs;ecund&aelig; ad primam e&longs;t, uelut dif&shy;<lb/>ferenti&etail; terti&aelig; &amp; &longs;ecund&aelig; ad differentiam &longs;ecund&aelig; &agrave; prima ut in 4. <lb/>6. &amp; 9 proportio 6 ad 4 e&longs;t uelut 3 differenti&aelig; 9 ad 6 ad 2 differen&shy;<lb/>tiam 6 &amp; 4.</s>
        </p>
        <p type="margin">
          <s id="id003289"><margin.target id="marg598"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003290"><margin.target id="marg599"/>2 1</s>
        </p>
        <p type="margin">
          <s id="id003291"><margin.target id="marg600"/>2 4 5</s>
        </p>
        <p type="main">
          <s id="id003292"><expan abbr="Sec&utilde;da">Secunda</expan> proportio quam ille appellat po&longs;teriorem, e&longs;t in qua pro <lb/>portio tertij ad &longs;ecundum e&longs;t uelut differenti&aelig; primi &amp; &longs;ecundi ad <lb/>differentiam &longs;ecundi &amp; tertij: Velut capio 1, 4, 6, proportio 6 ad 4 <lb/><arrow.to.target n="marg601"/><lb/>tertij &longs;cilicet, &amp; &longs;ecundum e&longs;t uelut 3 differenti&aelig; 4 &amp; 1, ad 2, differen&shy;<lb/><arrow.to.target n="marg602"/><lb/>tiam 6 &amp; 4, &amp; h&aelig;c &longs;imiliter differt &agrave; Geometrica uera in eo quo in <lb/>Geometrica uera oporteret, ut proportio tertij ad &longs;ecundum e&longs;&longs;et <lb/>ut differentia tertij &amp; &longs;ecundi ad differentiam &longs;ecundi &amp; primi. </s>
          <s id="id003293">Dif&shy;<lb/>fert &agrave; priore, quoniam in illa differenti&aelig; &longs;eruant eundem ordinem, <lb/>quanuis transferantur in hac uer&ograve; fit conuer&longs;us modus.</s>
        </p>
        <p type="margin">
          <s id="id003294"><margin.target id="marg601"/>3 2</s>
        </p>
        <p type="margin">
          <s id="id003295"><margin.target id="marg602"/>1 4 6</s>
        </p>
        <p type="main">
          <s id="id003296">Tertia e&longs;t ut &longs;it proportio differenti&aelig; prim&aelig; &amp; terti&aelig; ad diffe&shy;<lb/>rentiam prim&aelig; &amp; &longs;ecund&aelig;, uelut &longs;ecund&aelig; ad primam, in Geometri <lb/>ca autem e&longs;&longs;et &longs;icut aggregati &longs;ecund&aelig; &amp; prim&aelig; ad ip&longs;am primam, <lb/>tales ergo quantitates erunt uelut 4, 6, 7, nam proportio 6 ad 4 e&longs;t <lb/><arrow.to.target n="marg603"/><lb/>uelut 3 differenti&aelig; 4 &amp; 7 ad 2 differentiam 4 &amp; 6.<lb/><arrow.to.target n="marg604"/><lb/><arrow.to.target n="marg605"/></s>
        </p>
        <p type="margin">
          <s id="id003297"><margin.target id="marg603"/>3</s>
        </p>
        <p type="margin">
          <s id="id003298"><margin.target id="marg604"/>4 6 7</s>
        </p>
        <p type="margin">
          <s id="id003299"><margin.target id="marg605"/>2</s>
        </p>
        <p type="main">
          <s id="id003300">Quarta proportio &longs;imilis Geometric&aelig; e&longs;t cum fuerit proportio <lb/>differenti&aelig; prim&aelig; &amp; terti&aelig; ad differentiam terti&aelig; &amp; &longs;ecund&etail;, uelut <lb/>&longs;ecund&aelig; ad primam, uelut in 2, 3, 5 proportio differenti&aelig; 5 &amp; 2 qu&aelig; </s>
        </p>
        <p type="main">
          <s id="id003301"><arrow.to.target n="marg606"/><lb/><arrow.to.target n="marg607"/><lb/>e&longs;t 3 ad differentiam &longs;ecund&aelig; &amp; terti&aelig;, qu&aelig; e&longs;t 2 e&longs;t uelut 3 quantita<lb/><arrow.to.target n="marg608"/><lb/>tis &longs;ecund&aelig; ad 2 quantitatem primam.</s>
        </p>
        <p type="margin">
          <s id="id003302"><margin.target id="marg606"/>3</s>
        </p>
        <p type="margin">
          <s id="id003303"><margin.target id="marg607"/>2 3 5</s>
        </p>
        <p type="margin">
          <s id="id003304"><margin.target id="marg608"/>2</s>
        </p>
        <p type="main">
          <s id="id003305">Prima <expan abbr="aut&etilde;">autem</expan> <expan abbr="harmonicar&utilde;">harmonicarum</expan> qu&aelig; notha e&longs;t nec legitima, hoc modo <lb/>&longs;umitur: Vt &longs;it proportio prim&aelig; ad tertiam uelut differenti&aelig; &longs;ecun <lb/><arrow.to.target n="marg609"/><lb/>d&aelig; &amp; terti&aelig; ad differentiam &longs;ecund&aelig; &amp; prim&aelig;, ueluti capio 6 pri&shy;<lb/><arrow.to.target n="marg610"/><lb/>mam 5 &longs;ecundum 3 tertiam proportio 6 ad 3 e&longs;t dupla &longs;icut 2 diffe&shy;


<pb pagenum="186 &lsqb;=192&rsqb;" xlink:href="015/01/211.jpg"/>renti&aelig; &longs;ecund&aelig; &agrave; tertia ad 1 differentiam &longs;ecund&aelig; &agrave; prima. </s>
          <s id="id003306">Manife&shy;<lb/>&longs;tum e&longs;t autem quod in uera harmonica proportio differentiarum <lb/>e&longs;t prim&aelig; &amp; &longs;ecund&aelig; ad illam qu&aelig; &longs;ecund&aelig; &amp; terti&aelig;.</s>
        </p>
        <p type="margin">
          <s id="id003307"><margin.target id="marg609"/>1 2</s>
        </p>
        <p type="margin">
          <s id="id003308"><margin.target id="marg610"/>6 5 3</s>
        </p>
        <p type="main">
          <s id="id003309">Secunda notha harmonica e&longs;t, ut &longs;it propor&shy;<lb/><figure id="id.015.01.211.1.jpg" xlink:href="015/01/211/1.jpg"/><lb/>tio prim&aelig; ad tertiam, uelut differenti&aelig; prim&aelig; &agrave; <lb/>tertia ad differentiam &longs;ecund&aelig; &agrave; tertia, ponatur <lb/>25, prima 21, &longs;ecunda 15, tertia proportio 25 ad 15 <lb/>e&longs;t uelut 10 differenti&aelig; prim&etail; &agrave; tertia ad b differen<lb/>tiam &longs;ecund&aelig; &agrave; tertia.</s>
        </p>
        <p type="main">
          <s id="id003310">Tertia e&longs;t &longs;imilis priori, ni&longs;i quod &longs;umitur dif&shy;<lb/><figure id="id.015.01.211.2.jpg" xlink:href="015/01/211/2.jpg"/><lb/>ferentia prim&aelig; &agrave; &longs;ecunda pro ultimo termino. </s>
          <s id="id003311">Ex&shy;<lb/>emplum, 25 primus terminus, 19 &longs;ecundus, 15 ter&shy;<lb/>tius, proportio 25 ad 15 e&longs;t uelut 10 differenti&aelig; pri&shy;<lb/>m&aelig; a tertia ad b, differentiam prim&aelig; &agrave; &longs;ecunda. <lb/></s>
          <s id="id003312">Has proportiones quanqu&agrave;m exigu&aelig; utilitatis, proponere uo&shy;<lb/>lui, ut excogitatis aliquibus demon&longs;trationibus, uelut &longs;uperius <lb/>diximus, pulchra theoremata &amp; problemata tradi po&longs;&longs;ent.</s>
        </p>
        <p type="main">
          <s id="id003313">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id003314">Circulum &longs;uper centro &longs;uo mouere &aelig;qualiter, ita qu&ograve;d omnia <lb/>illius puncta per rectam lineam moueantur ultro citro que.<lb/><arrow.to.target n="marg611"/></s>
        </p>
        <p type="margin">
          <s id="id003315"><margin.target id="marg611"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003316">Sit a centrum circuli b c, &amp; &aelig;qualis ei <lb/><figure id="id.015.01.211.3.jpg" xlink:href="015/01/211/3.jpg"/><lb/>circulus d e, centrum eius b in circumfe&shy;<lb/>rentia circuli b c, fixum ita ut ibi mouea&shy;<lb/>tur ad motum circuli b c: &amp; moueatur b <lb/>uer&longs;us c &aelig;qualiter, &amp; e contrario motu <lb/>etiam regulariter, &amp; duplo uelocius ex e <lb/>uer&longs;us d, dico omnia puncta d e moue&shy;<lb/>ri in linea recta, &amp; primum capio pun&shy;<lb/>ctum d, quod &longs;it in linea recta centro&shy;<lb/>rum: &amp; moueatur b ad c, &amp; &longs;i circulus d e <lb/>e&longs;&longs;et immobilis, palam e&longs;t qu&ograve;d pun&shy;<lb/>ctum d cum &longs;it in una linea a b, cum b <lb/>perueniret in c, d e&longs;&longs;et in linea a c, put&agrave; in <lb/>h &longs;ecundum quantitatem, ergo b d ex </s>
        </p>
        <p type="main">
          <s id="id003317"><arrow.to.target n="marg612"/><lb/>centro c, de&longs;cribo circuli portionem h k, <lb/>duco etiam c k, erit ergo angulus h c k <lb/>duplus a, quare arcus h k duplus b c, <lb/>nam con&longs;i&longs;tunt in centris circulorum &aelig;&shy;<lb/>qualium: igitur cum ex h motu conuer&longs;o, &amp; duplo ueloci in codem <lb/>tempore feratur d perueniet in k, &amp; ita &longs;ecundum rectam lineam <lb/>erit motum eadem ratione ex d in k, quod erat demon&longs;trandum.</s>
        </p>
        <pb pagenum="187 &lsqb;=193&rsqb;" xlink:href="015/01/212.jpg"/>
        <p type="margin">
          <s id="id003318"><margin.target id="marg612"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003319">Ex hoc patet qu&ograve;d quando b <lb/><figure id="id.015.01.212.1.jpg" xlink:href="015/01/212/1.jpg"/><lb/><arrow.to.target n="marg613"/><lb/>erit in c peracta quarta circuli, ut in <lb/>&longs;ecunda figura erit per motum l e <lb/>in a: nam cum d a &longs;it dupla c b, igi&shy;<lb/>tur in eodem tempore l perueniet <lb/>ad a, in quo b perueniet ad c.</s>
        </p>
        <p type="margin">
          <s id="id003320"><margin.target id="marg613"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003321">Dico etiam, quod <expan abbr="qu&atilde;do">quando</expan> b per&shy;<lb/><arrow.to.target n="marg614"/><lb/>ueniet ad fin prima figura, d perue&shy;<lb/>niet ad g, quia permeabit totum cir<lb/>culum, &amp; a b d &longs;unt in una recta li&shy;<lb/>nea. </s>
          <s id="id003322">Et cum b perueniet ad m in &longs;e&shy;<lb/>cunda figura, d rur&longs;us perueniet ad a centrum.</s>
        </p>
        <p type="margin">
          <s id="id003323"><margin.target id="marg614"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003324">Ex hoc patet, qu&ograve;d punctum d permeabit lineam rectam &aelig;qua&shy;<lb/><arrow.to.target n="marg615"/><lb/>lem duplo diametri unius circuli, id e&longs;t, quantum e&longs;t linea a g in pri<lb/>ma figura.</s>
        </p>
        <p type="margin">
          <s id="id003325"><margin.target id="marg615"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id003326">Sequitur etiam, qu&ograve;d d punctum meabit et remeabit per rectam <lb/><arrow.to.target n="marg616"/><lb/>lineam ag, peragendo bis eam in uno circuitu circuli b c, &longs;eu duo&shy;<lb/>bus circuitibus d e.</s>
        </p>
        <p type="margin">
          <s id="id003327"><margin.target id="marg616"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id003328">O&longs;ten damus modo, quod pun<lb/><figure id="id.015.01.212.2.jpg" xlink:href="015/01/212/2.jpg"/><lb/>ctum d extra lineam centrorum, &longs;ci <lb/>licet in linea d c a f tran&longs;ibit per <expan abbr="re&shy;ct&atilde;">re&shy;<lb/>ctam</expan> eandem, ut in tertia figura pro&shy;<lb/>ducatur c d u&longs;que ad k, ita ut c k &longs;it <lb/>&aelig;qualis c a, erit ergo punctus d pri<lb/>m&aelig; figur&aelig; m &egrave; regione k terti&aelig;, &amp; <lb/>dum c mouetur ad e, d perueniat <lb/>ad g, erit ergo e g &aelig;qualis ea, &amp; &longs;e&shy;<lb/>cet circulus g h rectam a d in h, &amp; <lb/>ducatur c h. </s>
          <s id="id003329">Et erit ut prius angu&shy;<lb/>lus h e g duplus h a g, ergo arcus <lb/>g h duplus e c, ergo g remeauit in <lb/>h in tempore quo c feretur in e, <lb/>quare d de&longs;cendit per rectam in h.</s>
        </p>
        <p type="main">
          <s id="id003330">Dico rur&longs;us, qu&ograve;d quanto ma&shy;<lb/>gis d erit propinquum line&aelig; d g, <lb/>tanto minus de&longs;cendet in recta, <lb/>quanto magis propinquum longi<lb/>tudinibus medijs, <expan abbr="t&atilde;to">tanto</expan> celerius mo<lb/>uebitur, ade&ograve; ut in &longs;ecunda figura <lb/>apparet motum ex d in g, non de&longs;cendit ni&longs;i per d n, &amp; motum ex g <lb/>in l de&longs;cendit ex n in a centrum fixum. </s>
          <s id="id003331">De&longs;cendat ergo ex e in h &amp; h 


<pb pagenum="188 &lsqb;=194&rsqb;" xlink:href="015/01/213.jpg"/>in k per arcus &aelig;quales, &amp; ducantur arcus h l &amp; k m. </s>
          <s id="id003332">Quia n m &amp; n l <lb/>&longs;unt minores quarta circuli, &amp; maiores &longs;unt f e &amp; fl, &amp; angulus an&shy;<lb/>gulo non minor, patet propo&longs;itum. </s>
          <s id="id003333">Ita ergo motus, ut appropin&shy;<lb/>quant <expan abbr="p&utilde;ctis">punctis</expan> medijs &longs;unt uelociores, &amp; in &aelig;quali <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> &aelig;quales.</s>
        </p>
        <p type="main">
          <s id="id003334">Et hoc inuentum fuit Ludouici Ferrarij, cuius meminimus in Ar<lb/>te magna, &amp; nos ei &longs;ubtexuimus ex no&longs;tra inuentione, cuius ille de&shy;<lb/>mon&longs;trationem inuenire nequiuit.</s>
        </p>
        <p type="main">
          <s id="id003335">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id003336">Progre&longs;&longs;us &amp; regre&longs;&longs;us tam &longs;ine latitudine, qu&agrave;m cum latitudi&shy;<lb/>ne in planetis per &longs;olos concentricos circulos &aelig;qualiter motos de&shy;<lb/>mon&longs;trare.<lb/><arrow.to.target n="marg617"/></s>
        </p>
        <p type="margin">
          <s id="id003337"><margin.target id="marg617"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003338">Sit eclyptica a b c d, &amp; arcus regre&longs;&longs;us b c in partes <lb/><figure id="id.015.01.213.1.jpg" xlink:href="015/01/213/1.jpg"/><lb/>quatuor &aelig;quales diui&longs;us, &amp; de&longs;cribantur circuli duo b <lb/>h &amp; e k &longs;uper e &amp; f, &amp; &longs;upponatur orbis &longs;uperior &longs;ub <lb/>eclyptica tamen, cuius polus in f, qui circumagatur in du<lb/>plo temporis retroce&longs;&longs;us planet&aelig;, &amp; in di&longs;tantia circuli <lb/>e k &longs;ub puncto e eclyptic&aelig;, polus alterius orbis concen&shy;<lb/>trici inferioris, qui circumagatur in tempore retro ce&longs;&longs;us <lb/>planet&aelig;, &amp; planeta &longs;it in puncto 6, liquet ergo qu&ograve;d pla<lb/>neta ille in uno circuitu e k circuli permeabit b c &amp; re&shy;<lb/>meabit, &amp; &longs;emper erit &longs;ub ip&longs;a eclyptica. </s>
          <s id="id003339">Sed enim eclyptica habet <lb/>rationem rect&aelig; line&aelig;, ut quiuis circulus maximus. </s>
          <s id="id003340">Et &longs;i quis relu&shy;<lb/>ctetur fingamus rectam &longs;ubten&longs;am arcui b c, &amp; aliam po&longs;tmodum <lb/>&aelig;quidi&longs;tantem in eadem &longs;uperficie, &amp; in orbe inferiore, &amp; tunc pa&shy;<lb/>tebit liquid&ograve; propo&longs;itum. </s>
          <s id="id003341">Sed &longs;i uelim latitudinem de&longs;cribam, ma&shy;<lb/>ximam latitudinem &agrave; puncto b, &amp; ducam circulum magnum per <lb/>punctum illud: reliqua ut prius, ad unguem: nihil enim refert quod <lb/>ad demon&longs;trationem pr&aelig;cedentis attinet, &longs;eu a d ponatur eclypti&shy;<lb/>ca, &longs;eu alius circulus magnus.</s>
        </p>
        <p type="main">
          <s id="id003342"><arrow.to.target n="marg618"/></s>
        </p>
        <p type="margin">
          <s id="id003343"><margin.target id="marg618"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003344">Ex hoc patet cau&longs;a cur retroce&longs;&longs;us in initio, &amp; in fine &longs;int exigui, <lb/>in medio &longs;int magni im&ograve; maximi, &amp; quomodo perpetu&ograve; uarietur <lb/>latitudo in tempore retro ce&longs;&longs;us, &amp; ratio omnium, &amp; &longs;imiliter de in&shy;<lb/>crementis &amp; uelocitate motus.</s>
        </p>
        <p type="main">
          <s id="id003345"><arrow.to.target n="marg619"/></s>
        </p>
        <p type="margin">
          <s id="id003346"><margin.target id="marg619"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003347">Ex hoc &longs;equitur, quod cum erratica fuerit in centro &longs;eu polo f, &amp; <lb/>tunc mouetur ueloci&longs;s&iacute;m&egrave;, qu&ograve;d tamen erit in oppo&longs;ito &longs;olis, &amp; <lb/>tunc etiam ibi erit ip&longs;e polus, quare alter erit cum ip&longs;o &longs;ole.</s>
        </p>
        <p type="main">
          <s id="id003348"><arrow.to.target n="marg620"/></s>
        </p>
        <p type="margin">
          <s id="id003349"><margin.target id="marg620"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id003350">Et quia dum motus e&longs;t ueloci&longs;simi &longs;ecundum ordinem &longs;igno&shy;<lb/>rum, tunc erratica &longs;uperior e&longs;t &longs;oli iuncta, e&longs;tque in polo, oportet ut <lb/>polus f moueatur &longs;ecundum ordinem &longs;ignorum, ade&ograve; ut cum &longs;ol <lb/>peruenerit ad illius oppo&longs;itum, orbis &longs;uperior dimidium perfecerit 


<pb pagenum="195" xlink:href="015/01/214.jpg"/>circuitus, inferior autem integrum. </s>
          <s id="id003351">Ergo orbis &longs;uperior tanto tar&shy;<lb/>di&ugrave;s mouetur &longs;ole, quantum e&longs;t id quod peragit polus &longs;ine &aelig;quali <lb/>motu in orbe &longs;ignorum, per motum circunducentis orbis &longs;uperio&shy;<lb/>ris in tempore dimidij circuitus. </s>
          <s id="id003352">Inferior ergo cum moueatur du&shy;<lb/>plo ueloci&ugrave;s &longs;uperiore, ut dictum e&longs;t, igitur duplo uelocius &longs;ole, ni&shy;<lb/>&longs;i quantum e&longs;t duplum motus poli &longs;uperioris per motum orbis <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 ce&longs;&longs;us non &longs;olum illum quo pla&shy;<lb/>neta retrocedit, nam hic e&longs;t long&egrave; minor arcu proce&longs;&longs;us, &longs;ed in quo <lb/>motus in &aelig;qualis e&longs;t minor &aelig;quali, palam autem e&longs;t hunc fore &aelig;&shy;<lb/>qualem arcui uelocioris motus qu&agrave;m &longs;it motus &aelig;qualis.</s>
        </p>
        <p type="head">
          <s id="id003355">SCHOLIVM II.</s>
        </p>
        <p type="main">
          <s id="id003356">Cum ergo, dum erratica e&longs;t in polo orbis &longs;uperioris, ibi quie&longs;cat <lb/>motu eius, motu autem inferioris orbis ueloci&longs;sim&egrave; moueatur &longs;eu <lb/>progrediendo &longs;eu regrediendo motuque circulari, &amp; tamen per re&shy;<lb/>ctam lineam, igitur uideretur qu&ograve;d motus circularis partes po&longs;&longs;et <lb/>tran&longs;ire in rectum. </s>
          <s id="id003357">Re&longs;pondeo qu&ograve;d &longs;ufficit &longs;ola inclinatio ob ma&shy;<lb/>gnitudinem anguli: nam dum &longs;ydus transfertur extra centrum mo&shy;<lb/>tu orbis inferioris, mouetur uelociter quo ad angulum motu orbis <lb/>&longs;uperioris.</s>
        </p>
        <p type="main">
          <s id="id003358">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id003359">Cau&longs;am uarietatis diametrorum ex &longs;uppo&longs;itis concentricis de&shy;<lb/>mon&longs;trare.</s>
        </p>
        <p type="main">
          <s id="id003360">In tribus &longs;uperioribus planetis &amp; quibu&longs;cunque &longs;tellis octaui or&shy;</s>
        </p>
        <p type="main">
          <s id="id003361"><arrow.to.target n="marg621"/><lb/>bis manife&longs;tum e&longs;t, qu&ograve;d pars qu&aelig; re&longs;picit nos quant&ograve; remotior <lb/>fuerit &agrave; Sole, <expan abbr="t&atilde;to">tanto</expan> magis illuminatur. </s>
          <s id="id003362">Manife&longs;tum e&longs;t etiam &amp; expe&shy;<lb/>rimento &amp; ratione, qu&ograve;d illud quod magis lucet, &amp; e&longs;t <expan abbr="illuminat&utilde;">illuminatum</expan> <lb/>&agrave; Sole in nocte, maius uidetur, &longs;icut etiam de facibus nocturnis. </s>
          <s id="id003363">Et <lb/>rur&longs;us, quod &longs;ub &longs;tantia orbium circa loca qu&aelig; habentur pro polis <lb/>e&longs;t den&longs;ior, &amp; quod res in medio den&longs;o apparent maiores, &longs;icut de <lb/>pi&longs;cibus in aqua, denarijs &amp; baculis. </s>
          <s id="id003364">Demon&longs;tratum <expan abbr="a&utilde;t">aunt</expan> e&longs;t in pr&aelig;&shy;<lb/>cedenti, quod quando &longs;tella fuerit in polo orbis &longs;uperioris, qu&ograve;d <lb/>tunc maxim&egrave; retrocedit, &amp; ide&ograve; cum in tempore maximi retro ce&longs;&shy;<lb/>&longs;us &longs;it in oppo&longs;ito Solis <expan abbr="d&utilde;">dum</expan> tres &longs;uperiores &longs;unt in oppo&longs;itu Solis, <lb/>multo maiores duabus ex cau&longs;is e&longs;&longs;e uidentur, &amp; iuxta proportio&shy;<lb/>nem propinquitatis ad Solem commutant quantitatem &amp; tanto <lb/>minores apparent, quia non po&longs;&longs;unt, commutare <expan abbr="form&atilde;">formam</expan>, uelut Lu&shy;<lb/>na propter &aelig;qualitatem &longs;ub&longs;tanti&etail; &amp; luminis proprij copiam, qu&etail; <lb/>non &longs;init di&longs;cerni uarietatem figur&etail;. </s>
          <s id="id003365">In Luna autem &longs;ecus e&longs;t, nam in 


<pb pagenum="196" xlink:href="015/01/215.jpg"/>ip&longs;a di&longs;cernitur ob paucitatem luminis proprij figur&aelig; uarietas, &amp; <lb/>ob id non apparet maior, im&ograve; minor aut medi&aelig; quantitatis in op&shy;<lb/>po&longs;ito Solis, &longs;ed maxima in longitudinibus medijs, quoniam ibi <lb/>&longs;unt poli motus uarietatis ut dictum e&longs;t, qu&etail; habet locum retro ce&longs;&shy;<lb/>&longs;us, &longs;ed ob motus paruitatem Luna non pote&longs;t retrocedere, uer&ugrave;m <lb/>&longs;ol&ugrave;m motus tardatur. </s>
          <s id="id003366">Nam licet den&longs;itas &longs;it in c&oelig;lo &longs;uperiore &amp; <lb/>motus uelox nihilominus efficit imaginem maiorem, &longs;icut apparet <lb/>de pi&longs;ce in magna aqua in medio, &amp; in parua in imo, nam in parua <lb/>uidetur long&egrave; maior qu&agrave;m in magna, licet &longs;it in &aelig;quali di&longs;tantia. </s>
          <s id="id003367">In <lb/>Venere autem &amp; Mercurio eadem e&longs;t ratio di&longs;tanti&aelig; &agrave; Sole ut di&shy;<lb/>ctum e&longs;t in pr&aelig;cedenti. </s>
          <s id="id003368">Cum ergo &longs;ub Sole multum moueantur <lb/>motu differenti&aelig; uel &longs;ecundum &longs;ucce&longs;sionem, uel contra &longs;ucce&longs;&shy;<lb/>&longs;ionem in medijs longitudinibus, parum tunc uidentur e&longs;&longs;e mino&shy;<lb/>res, quia &longs;unt remotiores &agrave; polo orbis &longs;uperioris. </s>
          <s id="id003369">Quod autem pro<lb/>pinqui coniunctioni Solis, &amp; ueloces uideantur minores, i&longs;tud <lb/>contingit ob primam cau&longs;am, quia minus illuminantur, ea parte <lb/>qu&aelig; ad nos uergit. </s>
          <s id="id003370">Re&longs;tat ergo &longs;olum o&longs;tendere cur propinqui <lb/>Soli &amp; in retroce&longs;&longs;u <expan abbr="uide&atilde;tur">uideantur</expan> maiores, c&ugrave;m utraque ratio ob&longs;tet, &longs;unt <lb/>enim remoti &agrave; polo orbis &longs;uperioris &amp; propinqui Soli, cau&longs;a e&longs;t <lb/>quoniam apparent &longs;ol&ugrave;m in crepu&longs;culis quando &longs;unt &longs;ic di&longs;po&longs;iti, <lb/>&amp; tunc a&euml;r e&longs;t cra&longs;sior. </s>
          <s id="id003371">Qu&aelig; cau&longs;a facit, ut neque dum ueloci&longs;simi <lb/>&longs;unt &longs;emper parui uideantur, ide&ograve; non pote&longs;t con&longs;titui certa ratio. <lb/></s>
          <s id="id003372">im&ograve; i&longs;ta deducta &longs;unt potius ex fundamento fal&longs;o illius figmen&shy;<lb/>ti, quam ex &longs;en&longs;u &lpar;ita enim argumentantur&rpar; retro cedunt, ergo &longs;unt <lb/>propinquiores terr&aelig;, ergo uidentur maiores, &amp; ita fingunt &longs;en&shy;<lb/>&longs;u &longs;ehabere quod fal&longs;a ratione o&longs;tendere uidentur. </s>
          <s id="id003373">quodque i&longs;tud <lb/>&longs;it uerum, patet quia nullum <expan abbr="in&longs;trum&etilde;tum">in&longs;trumentum</expan> etiam in a&euml;re clari&longs;simo <lb/>Aegypti pote&longs;t o&longs;tendere differentiam minorem &longs;ex minutis, &amp; <lb/>hic e&longs;t ferm&egrave; diameter Mercurij, nec tanta e&longs;t differentia in Venere. <lb/></s>
          <s id="id003374">Reliquum e&longs;t ut &longs;atisfaciamus obiectioni quam faciunt de diuer&shy;<lb/>&longs;itate magnitudinis Lun&aelig; propter eclip&longs;im, nam uidetur e&longs;&longs;e ali&shy;<lb/>quando maior, &amp; aliquando minor in &aelig;quali di&longs;tantia &agrave; &longs;ectione <lb/>capitis &amp; caud&aelig; draconis, ade&ograve; ut non uideatur po&longs;&longs;e a&longs;signari. </s>
          <s id="id003375">di<lb/>co ergo huius cau&longs;am e&longs;&longs;e umbram ip&longs;ius Lun&aelig; dubiam, &longs;icut eti&shy;<lb/>am in crepu&longs;culis, quoniam Sol in diuer&longs;o &longs;itu facit diuer&longs;am um&shy;<lb/>bram comparatione oculi no&longs;tri, maior e&longs;t enim in hyeme qu&agrave;m <lb/>in &aelig;&longs;tate, &amp; qu&aelig; e&longs;t propior nobis qu&agrave;m qu&aelig; procul, &amp; qu&aelig; e&longs;t in <lb/>meridie qu&agrave;m iuxta Ortum uel Occa&longs;um, &amp; ide&ograve; tam parua diffe&shy;<lb/>rentia &amp; incerta, &amp; qu&aelig; aliquando uariat, nullo modo uitiare po&shy;<lb/>te&longs;t rationem motuum &aelig;ternorum.</s>
        </p>
        <pb pagenum="197" xlink:href="015/01/216.jpg"/>
        <p type="margin">
          <s id="id003376"><margin.target id="marg621"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003377">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id003378">Rationem centri grauitatis declarare.</s>
        </p>
        <p type="main">
          <s id="id003379">Duplicem rationem <expan abbr="c&etilde;tri">centri</expan> grauitatis inuenit Archimedes, unam <lb/><arrow.to.target n="marg622"/><lb/>&longs;u&longs;pen&longs;orum ponderum: alteram &longs;upernatantium aqu&aelig;, in qua&shy;<lb/>rum utraque &longs;ubtilitatis cert&egrave; e&longs;t quantum dignum e&longs;t authore illo <lb/>ingenio&longs;i&longs;simo, &longs;icut etiam in elica linea, fructus autem non pro ra&shy;<lb/>tione laboris, neque enim ab &aelig;tate illa u&longs;que nunc inuentus e&longs;t qui&longs;&shy;<lb/>quam, qui potuerit docere, nec ille idem qu&aelig; nam utilitas ex huiu&longs;&shy;<lb/>modi contemplatione haberetur, propterea totum hoc una propo<lb/>&longs;itione conclu&longs;imus.</s>
        </p>
        <p type="margin">
          <s id="id003380"><margin.target id="marg622"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003381">Dico igitur qu&ograve;d <expan abbr="c&etilde;trum">centrum</expan> grauitatis in appen&longs;is &aelig;qualibus qua&shy;<lb/>dratis aut quadrilateris parallelis e&longs;t, ubi &longs;e inter&longs;ecant du&aelig; diame&shy;<lb/>tri. </s>
          <s id="id003382">Et quod in triangulis e&longs;t punctus in quo concurrant tres line&aelig;, <lb/>duct&etail; ab angulis ad latera illa per &aelig;qualia &longs;ecando. </s>
          <s id="id003383">In quadrilatero <lb/>autem trapezio centrum grauitatis e&longs;t in puncto line&aelig;, qu&aelig; &longs;ecat <lb/>ambo latera oppo&longs;ita per &aelig;qualia, ita ut proportio partis eius li&shy;<lb/>ne&aelig;, qu&aelig; intercipitur &agrave; minore &aelig;quidi&longs;tantium, ad partem qu&aelig; in&shy;<lb/>tercipitur &agrave; maiore &aelig;quidi&longs;tantium, &longs;it ueluti dupli maioris &aelig;qui&shy;<lb/>di&longs;tantium cum minore ad duplum minoris &aelig;quidi&longs;tantium cum <lb/>maiore. </s>
          <s id="id003384">Cuiu&longs;cunque portionis &agrave; recta linea, &amp; rectanguli coni &longs;ecti&shy;<lb/>one comprehen&longs;&aelig;, centrum grauitatis diuidit diametrum portio&shy;<lb/>nis, ita ut pars eius ad uerticem terminata, &longs;it ad partem eam &longs;exqui&shy;<lb/>altera, qu&aelig; ad ba&longs;im portionis terminatur. </s>
          <s id="id003385">Cuiuslibet fru&longs;ti &agrave; &longs;ecti&shy;<lb/>one rectanguli coni ablati, centrum grauitatis e&longs;t in linea recta, qu&etail; <lb/>fru&longs;ti exi&longs;tit diametros: qua in quinque partes &aelig;quas diui&longs;a, cen&shy;<lb/>trum in quinta eius media exi&longs;tit, atque in eo eius puncto quo ip&longs;a <lb/>quinta &longs;ic diuiditur, ut portio eius propinquior minori ba&longs;i fru&shy;<lb/>&longs;ti ad reliquam eius portionem eam habeat proportionem, quam <lb/>habet &longs;olidum, cuius ba&longs;is &longs;it quadratum line&aelig; illius qu&aelig; fru&longs;ti ba&shy;<lb/>&longs;is maior extiterit.. Altitudo uer&oacute; i&longs;tis utri&longs;que &longs;imul &aelig;qualis line&aelig; <lb/>qu&aelig; dupla &longs;it minoris ba&longs;is fru&longs;ti, &amp; ba&longs;i maiori eiu&longs;dem, ad &longs;oli&shy;<lb/>dum quod ba&longs;im habeat quadratum ba&longs;is minoris fru&longs;ti, altitudi&shy;<lb/>nem uero i&longs;tis utri&longs;que &longs;imul &aelig;qualem line&aelig; qu&aelig; dupla &longs;it maioris <lb/>ba&longs;is, &amp; ba&longs;i minori. </s>
          <s id="id003386">Et h&aelig;c de prima, multa q&uacute;e alia pulchra de&shy;<lb/>clarat Federicus Comandinus, in &longs;uo libro de Centro grauitatis, ut <lb/>pote. </s>
          <s id="id003387">Quod cuiuslibet portionis conoidis rectanguli axis &agrave; cen&shy;<lb/>tro grauitatis ita diuiditur ut pars, qu&aelig; determinatur ad uerticem <lb/>reliqu&aelig;, qu&aelig; ad ba&longs;im terminatur dupla &longs;it, &amp; long&egrave; &longs;ubtiliora qu&etail; <lb/>quilibet uidere poterit apud illum.</s>
        </p>
        <pb pagenum="198" xlink:href="015/01/217.jpg"/>
        <p type="head">
          <s id="id003388">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id003389">Partes omnes con&longs;entiunt in grauitatem medij, quoniam una <lb/>aliam non uult centro mundi fieri propiorem.</s>
        </p>
        <p type="main">
          <s id="id003390">De &longs;ecunda pr&aelig;cipua &longs;unt, quod &longs;i magnitudo aliqua humido <lb/>leuior ea in grauitate proportionem habebit ad humidum &etail;qualis <lb/>molis, quam pars magnitudinis demer&longs;a ad totam magnitudinem, <lb/>&amp; hoc intelligitur quando magnitudo illa fuerit &egrave; genere &longs;olido&shy;<lb/>rum rectorum &amp; rectangulorum. </s>
          <s id="id003391">Secunda e&longs;t, qu&ograve;d qu&aelig; &longs;imilia <lb/>&longs;unt &longs;uperficiebus, ita ut axem habeant in medio, &longs;ecundum &longs;itum <lb/>axis merguntur &amp; prominent, &amp; &longs;i aliter mergantur, redeunt. </s>
          <s id="id003392">Ter&shy;<lb/>tia, quod qu&etail; angu&longs;tiora &longs;unt, ab oppo&longs;ita parte uer&ograve; latiora, incli&shy;<lb/>nantur ad partem acutiorem, quia &longs;ic facilius de&longs;cendunt. </s>
          <s id="id003393">Quarta <lb/>e&longs;t, de corporibus non &aelig;qualibus, ip&longs;a enim nece&longs;&longs;e e&longs;t, ut ab hac &longs;e <lb/>inflectant, &amp; ratio horum diuer&longs;a e&longs;t iuxta rationem proportionis <lb/>partium qu&aelig; merguntur adinuicem. </s>
          <s id="id003394">Quinta e&longs;t, qu&ograve;d mer&longs;a in hu&shy;<lb/>mido, quanto minus mer&longs;a fuerint, tanto facilius &amp; eo frequenti&shy;<lb/>us commutantur.</s>
        </p>
        <p type="main">
          <s id="id003395">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id003396">Si proportio aliqua ex duabus proportionibus eiu&longs;dem quanti <lb/>tatis ad alias duas componatur: erit proportio illarum duarum ea&shy;<lb/>dem proportioni producti ex proportione in primam duarum <lb/>quantitatum detracta priore illa quantitate, qu&aelig; ad duas compara<lb/>tur, ad eandem priorem quantitatem.</s>
        </p>
        <p type="main">
          <s id="id003397">Sit proportio a ad compo&longs;ita ex proportionibus c <lb/><arrow.to.target n="marg623"/><lb/><figure id="id.015.01.217.1.jpg" xlink:href="015/01/217/1.jpg"/><lb/>ad d &amp; c ad e, dico qu&ograve;d proportio d ad e e&longs;t, ut produ&shy;<lb/>cti ex proportione in d detracto c ad ip&longs;um c. </s>
          <s id="id003398">Et nos <lb/>&longs;uperius expo&longs;uimus conuer&longs;am huius. </s>
          <s id="id003399">Erit enim per <lb/><expan abbr="&longs;ecund&atilde;">&longs;ecundam</expan> demon&longs;trationem illius proportio a ad b, uelut producti <lb/>ex c in d, &amp; e ad productum d in e: at productum d in e &amp; in propor<lb/>tionem, e&longs;t idem quod productum proportionis in d in ip&longs;um e: igi<lb/>tur cum in uno &longs;it productum e in c, &amp; d in c, in alio productum a b <lb/>in d in de in e, qu&aelig; &longs;unt &aelig;qualia, detracto producto e in c ex produ&shy;<lb/>cto proportionis in d &amp; inde in e, relinquetur, productum c in d &aelig;&shy;<lb/>quale producto a b .i. </s>
          <s id="id003400">proportionis in productum d in e, detracto <lb/>numero c in e: igitur ducto c in d, &amp; diui&longs;o per productum a b in d <lb/>numero c, exibit e, igitur cum illud productum fiat ex d, &longs;cilicet in c, <lb/>&amp; ex e in productum proportionis in d dempto numero c, erit pro <lb/>portio d ad e, uelut producti ex d in proportionem, detracto e ad <lb/>ip&longs;um c, uelut c &longs;it 12, d 4, e 6, a b erit 5 proportio d ad e, uelut d in a b, <lb/>id e&longs;t 20, detracto c, &amp; e&longs;t 8 ad c 12.</s>
        </p>
        <pb pagenum="199" xlink:href="015/01/218.jpg"/>
        <p type="margin">
          <s id="id003401"><margin.target id="marg623"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003402">Ex demon&longs;tratione &longs;equitur, quod qualis e&longs;t proportio e ad a b, <lb/><arrow.to.target n="marg624"/><lb/>talis e&longs;t producti d in e, ad aggregatum eorum. </s>
          <s id="id003403">Si quis ergo dicat, <lb/>habeo 10, &amp; uolo inuenire duas quantitates, quarum differentia &longs;it <lb/>1, &amp; proportio 10, ad eas componat quintuplam, dices quintupla <lb/>e&longs;t dimidium 10, igitur in uenias duas quantitates, quarum differen<lb/>tia &longs;it 1, &amp; proportio producti unius in alteram ad aggregatum &longs;it <lb/>dupla. </s>
          <s id="id003404">Et hoc e&longs;t manife&longs;tum.</s>
        </p>
        <p type="margin">
          <s id="id003405"><margin.target id="marg624"/>C<emph type="italics"/>or<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id003406">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id003407">Proportionem mi&longs;tionis metallorum, maxim&egrave; auri &amp; argenti <lb/>declarare.</s>
        </p>
        <p type="main">
          <s id="id003408">Dubium non e&longs;t, quod mi&longs;tio non cogno&longs;catur ducto ponde&shy;<lb/><arrow.to.target n="marg625"/><lb/>re totius in partem auri uel argenti, &amp; productis collectis diui&longs;o <lb/>aggregato per aggregatum ponderis, idq&uacute;e e&longs;t per &longs;e manife&shy;<lb/>&longs;tum, nam qualis e&longs;t proportio partis ad partem, talis e&longs;t totius ad <lb/>totum.</s>
        </p>
        <p type="margin">
          <s id="id003409"><margin.target id="marg625"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003410">Sed e&longs;t genus mi&longs;tionis, quod uocant con&longs;olationem. </s>
          <s id="id003411">Veluti, <lb/>uolo ex argento perfectionis decem &amp; &longs;eptem, &amp; quinque, confla&shy;<lb/>re argenti ma&longs;&longs;am centum librarum perfectionis nouem, ita agen&shy;<lb/>dum e&longs;t. </s>
          <s id="id003412">Detrahe 9 &agrave; 10, &amp; omni maiori 10, relinqui&shy;<lb/>tur 1, hoc &longs;uppone 7 &amp; 5, item detrahe 7 &amp; 5, &amp; omne <lb/><figure id="id.015.01.218.1.jpg" xlink:href="015/01/218/1.jpg"/><lb/>minus 9 &agrave; 9, relinquitur 2 &amp; 4, iunge omnia re&longs;idua <lb/>fient 8, nam 4. 2. 11. Dicemus ergo quod 8 unci&etail; per&shy;<lb/>fectionis nouem componentur ex 6 uncijs perfe&shy;<lb/>ctionis decem &amp; una &longs;eptem alia quinque. </s>
          <s id="id003413">Po&longs;t di&shy;<lb/>ces, &longs;i unci&aelig; octo fiant 100, &longs;ex &amp; una, &amp; una, quot fient, eruntque un&shy;<lb/>ci&aelig; aut libr&aelig;, aut ut uocant march&aelig; perfectionis decem, &amp; duo de&shy;<lb/>cim cum dimidia, ac duodecim cum dimidia perfectionis, ut &longs;e&shy;<lb/>ptem &amp; ut quinque: licebit etiam propo&longs;itis terminis pluribus ex <lb/>repetita operatione idem facere, ueluti &longs;int ma&longs;&longs;&aelig; perfectionis 10. <lb/>7. 5. &amp; 2. uolo ma&longs;&longs;am perfectionis ut 8. Tu &longs;cis quod ex 10. 7 &amp; 5. <lb/>fit ma&longs;&longs;a perfectionis nouem data lege &longs;ub 6. 1 &amp; 1. nunc habeo iam <lb/>perfectam ut 9, aliam ut 2, detraho 2 ex 8, relinquitur 6 &amp; 8, x 9 re&shy;<lb/>linquitur 1, iunge fient 7, erunt ergo &longs;eptem unci&aelig;, in <lb/><figure id="id.015.01.218.2.jpg" xlink:href="015/01/218/2.jpg"/><lb/>quibus &longs;ex erunt perfectionis, ut 9 &amp; 1 perfectionis ut <lb/>2, &amp; totum erit perfectionis ut octo. </s>
          <s id="id003414">Duc ergo, ut ex&shy;<lb/>plores ueritatem, 6 in 9 fit 54, duc 2 in 1 fit 2, iunge fit 56 <lb/>diuide per 7 exit 8 perfectio qu&aelig;&longs;ita.</s>
        </p>
        <p type="main">
          <s id="id003415">Per idem intelliges detractionem ex ma&longs;&longs;a argenti perfectionis <lb/>7, detraxi quartam partem perfectionis 10, uolo &longs;cire dodrantem <pb pagenum="200" xlink:href="015/01/219.jpg"/>

qualis relinquatur perfectionis, duc quadrantem in 10 fit 30, duc 12 <lb/>in 7 fit 84, detrahe 30 ex 84, relinquitur 54, divide 54 per 9, re&longs;idu&shy;<lb/>um 12 &amp; 3, exit 6 perfectio re&longs;idui.</s>
          <lb/>
        </p>
        <p type="main">
          <s id="id003416">Si quis dicat propo&longs;itis argenti pondo 50 &amp; dodrante perfe&shy;<lb/>ctionis 11/18, uolo partem a&longs;&longs;umere, &amp; igne perficere, ita purum ar&shy;<lb/>gentum, quod relinquitur additum re&longs;iduo, efficiat ip&longs;um perfe&shy;<lb/>ctionis dextantis &amp; be&longs;sis unci&aelig; pro libra, &longs;eu 8/9, divide 11/18 per 8/9 exit <lb/>11/16, duc in pondo 50 cum dodrante, fiunt pondo 34, unci&aelig; 7 1/8, hoc <lb/>igitur erit aggregatum conflatum ex argento puro &amp; re&longs;iduo. </s>
          <s id="id003417">De&shy;<lb/>trahe igitur 11/18 ex integro, relinquitur 7/18, detrahe pondo 34, uncia <lb/>7 1/8 ex pondo 50 cum dodrante, relinquuntur pondo 15 unci&aelig; 6 7/8 <lb/>&lpar;pondo enim uncias continet &longs;ub hoc &longs;en&longs;u, quia u&longs;ui &longs;eruimus o&shy;<lb/>cto&rpar; divide per 7/8, exeunt pondo 40 unci&aelig; 6 1/4, &amp; tanta pars debuit <lb/>igne purgari. </s>
          <s id="id003418">In ea enim erunt puri argenti pondo 24, unci&aelig; 7 /78, <lb/>qu&aelig; addita re&longs;iduo, &longs;cilicet pondo 9, uncijs 7 3/4 conficiunt pondo <lb/>34 uncias 7 1/8 perfectionis dict&aelig;.</s>
          <lb/>
        </p>
        <p type="main">
          <s id="id003419">Quidam mi&longs;cuit uncias decem auri perfectionis dextantis, &amp; <lb/>partem perfectionis dextantis cum dimidio, &amp; aliud perfectionis <lb/>be&longs;sis concreuit ma&longs;&longs;a perfectionis dodrantis unciarum octuagin<lb/>ta, qu&aelig;runtur pondera reliquarum partium, &longs;ubtrahe 10 pondus <lb/>ex 80 pondere, relinquuntur 70 perfectionis 17 5/7, inde detrahe per <lb/>modum &longs;uperiorem, &amp; relinquuntur 3 2/7 &amp; 1 5/7, <lb/><figure id="id.015.01.219.1.jpg" xlink:href="015/01/219/1.jpg"/>iunge &longs;imul fiunt 5, dico ergo, &longs;i 6 producit <lb/>70, quid producet 3 2/7 &amp; 1 5/7, &amp; inuenies quod 1 5/7 <lb/>producet 24 &amp; 3 5/7 producet 46, qui iuncti faci&shy;<lb/>unt 70. </s>
          <s id="id003420">Igitur aurum perfectionis dextantis <lb/>cum dimidio fuit unciarum 46 aurum perfe&shy;<lb/>ctionis, be&longs;sis unciarum 24. </s>
          <s id="id003421">Reliqua interro&shy;<lb/>gata di&longs;&longs;olues per regulas Algebr&aelig;, horum <lb/>modo.</s>
        </p>
        <p type="main">
          <s id="id003422">Propo&longs;itio cente&longs;ima &longs;eptuage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id003423">Si duobus totis du&aelig; portiones &longs;imiles ab&longs;cindantur, ab ei&longs;dem <lb/>denuo, &amp; ab&longs;ci&longs;sis proportionibus partes e&aelig;dem auferantur, denuo<expan abbr="&qgrave;">que</expan><lb/> ac denuo, quoties libuerit &agrave; portionibus, &amp; &agrave; re&longs;iduis ip&longs;arum <lb/>quantitatum partes e&aelig;dem auferantur, erit re&longs;idui ad re&longs;iduum, ue <lb/>luti totius ad totum.</s>
        </p>
        <p type="margin">
          <s id="id003424"><margin.target id="marg906"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003425">Sint du&aelig; quanitates a b &amp; k l, &amp; ab&longs;ci&longs;&longs;&aelig; du&aelig; partes &longs;imiles ex <lb/>utraque b c &amp; l m, &amp; &longs;it propo&longs;ita aliqua proportio, qu&aelig; &longs;it h, &amp; <lb/>&longs;umatur portio b d ip&longs;ius b c, &longs;ecundum proportionem h, &amp; &longs;i&shy;<lb/>


<pb pagenum="201" xlink:href="015/01/220.jpg"/>militer l n ip&longs;ius l m, iuxta pro&shy;<lb/><figure id="id.015.01.220.1.jpg" xlink:href="015/01/220/1.jpg"/><lb/>portionem h, &longs;umatur rur&longs;us <lb/><arrow.to.target n="marg626"/><lb/>de ip&longs;ius a b pars &longs;ecundum h, <lb/><arrow.to.target n="marg627"/><lb/>&amp; n o ip&longs;ius k l, &longs;ecundum ean <lb/>dem proportionem. </s>
          <s id="id003426">Et rur&longs;us <lb/><arrow.to.target n="marg628"/><lb/>&longs;umatur e f &aelig;qualis d b, &amp; o p <lb/><arrow.to.target n="marg629"/><lb/>&aelig;qualis n l, ut &longs;int portiones <lb/>b c &amp; l m &longs;ecundum proportionem h, &amp; &longs;umatur f g ip&longs;ius a c, &longs;ecun <lb/><arrow.to.target n="marg630"/><lb/>dum proportionem h, &amp; p q ip&longs;ius k o, &longs;ecundum eandum propor&shy;<lb/><arrow.to.target n="marg631"/><lb/>tionem, &amp; ita procedendo &longs;emper, dico quod erit a g re&longs;idui ad k q <lb/><arrow.to.target n="marg632"/><lb/>re&longs;iduum, ut a b ad k l. </s>
          <s id="id003427">Quia enim a b ad b c, ut k l ad l m ex &longs;uppo&longs;i&shy;<lb/><arrow.to.target n="marg633"/><lb/>to, erit a b ad b d, ut k l ad l n: e&longs;t etiam a b ad d e, ut k l ad n o ex &longs;up&shy;<lb/>po&longs;ito, 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/><arrow.to.target n="marg634"/><lb/>&longs;us 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/>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&apos; g f, ut k l ad <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/>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/>erat demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id003432"><margin.target id="marg626"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003433"><margin.target id="marg627"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003434"><margin.target id="marg628"/>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>&amp;<emph.end type="italics"/><lb/>22. <emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003435"><margin.target id="marg629"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003436"><margin.target id="marg630"/>P<emph type="italics"/>er eandem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003437"><margin.target id="marg631"/>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>&amp;<emph.end type="italics"/><lb/>22 <emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003438"><margin.target id="marg632"/>P<emph type="italics"/>er ea&longs;dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003439"><margin.target id="marg633"/>P<emph type="italics"/>er<emph.end type="italics"/> 19 <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003440"><margin.target id="marg634"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003441">Ex hoc patet, quod et&longs;i proportio non maneat eadem in parti&shy;<lb/><arrow.to.target n="marg635"/><lb/>bus totius, &amp; partis modo &longs;it eadem in totis ad partes a&longs;&longs;umptas, et <lb/>in partibus ad partes a&longs;&longs;umptas, nihilominus &longs;equitur idem.</s>
        </p>
        <p type="margin">
          <s id="id003442"><margin.target id="marg635"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003443">Sequitur rur&longs;us, quod et&longs;i proportio eadem non maneat quan&shy;<lb/><arrow.to.target n="marg636"/><lb/>titatum a&longs;&longs;umptarum ad partes qu&aelig; &longs;umuntur, nec etiam partium <lb/>modo &longs;emper pars, qu&aelig; a&longs;&longs;umitur &longs;it totius pars, &amp; alia partis idem <lb/>ueratur.</s>
        </p>
        <p type="margin">
          <s id="id003444"><margin.target id="marg636"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003445">Velut &longs;i prima uice capiam b d partem b c, ut l n partem l m &longs;e&shy;<lb/><arrow.to.target n="marg637"/><lb/>cundum h proportionem, &amp; deinde capiam d e partem a b &amp; n o <lb/>partem k l &longs;ecundum proportionem r, qu&aelig; &longs;it alia ab h, &amp; &longs;ecunda <lb/>uice capiam e f partem b c, &amp; o p partem l m &longs;ecundum proportio&shy;<lb/>nem h, qu&aelig; &longs;it alia ab h &amp; r. </s>
          <s id="id003446">Et capiam f g partem a e &amp; p q partem <lb/>k o, &longs;ecundum eandem proportionem, &longs;ed tamen qu&aelig; non &longs;it ali&shy;<lb/>qua pr&aelig;dictarum, &longs;cilicet h r s, &longs;ed diuer&longs;a ab eis, &amp; uocetur t, dico <lb/>quod nihilominus erit proportio a g ad k q, ut a b ad k l, qu&aelig; pa&shy;<lb/>tent ex ui demon&longs;trationum, in quibus nil plus a&longs;&longs;umitur ad de&shy;<lb/>mon&longs;trandum, qu&agrave;m id quod proponitur in corrolarijs.</s>
        </p>
        <p type="margin">
          <s id="id003447"><margin.target id="marg637"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id003448">Ex hoc etiam &longs;equitur, quod &longs;ecundum quem numerum prima <lb/><arrow.to.target n="marg638"/><lb/>quantitas ab&longs;umetur, &longs;ecundum eundem ab&longs;umetur &amp; &longs;ecunda.</s>
        </p>
        <p type="margin">
          <s id="id003449"><margin.target id="marg638"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. .3.</s>
        </p>
        <p type="main">
          <s id="id003450">Velut &longs;i prima quantitas ab&longs;umatur ad unguem in quinta detra&shy;<lb/><arrow.to.target n="marg639"/><lb/>ctione, etiam &longs;ecunda k l in quinta detractione ad unguem ab&longs;ume<lb/>tur, quod patet per demon&longs;trata, nam re&longs;idua &longs;emper &longs;unt e&aelig;dem <lb/>partes ip&longs;arum quantitatum.</s>
        </p>
        <pb pagenum="202" xlink:href="015/01/221.jpg"/>
        <p type="margin">
          <s id="id003451"><margin.target id="marg639"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003452">Quarto &longs;equitur, quod &longs;i detractio fuerit facta eodem modo, &amp; <lb/><arrow.to.target n="marg640"/><lb/>fuerit proportio totius ad totum, ut re&longs;idui ad re&longs;iduum, erunt par <lb/>tes a&longs;&longs;umpt&aelig; &longs;imiles.</s>
        </p>
        <p type="margin">
          <s id="id003453"><margin.target id="marg640"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 4.</s>
        </p>
        <p type="main">
          <s id="id003454">Velut &longs;i fuerit facta detractio iuxta propo&longs;itionem, aut primum <lb/><arrow.to.target n="marg641"/><lb/>uel &longs;ecundum corrolarium, &amp; fuerit proportio a g ad k g, ut a b ad <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"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003456">Sequitur etiam, quod &longs;i fuerit a&longs;&longs;umpta proportio <expan abbr="primar&utilde;">primarum</expan> par&shy;<lb/><arrow.to.target n="marg642"/><lb/>tium eadem, &amp; facta fuerit detractio in omnibus pr&aelig;ter unam iux&shy;<lb/>ta dicta, &amp; fuerit totius ad totum, ut re&longs;idui ad re&longs;iduum, erit ut illa <lb/>etiam reliqua detractio, &longs;eu ad tota, &longs;eu ad partes &longs;it facta, &longs;ecundum <lb/>eandem proportionem.</s>
        </p>
        <p type="margin">
          <s id="id003457"><margin.target id="marg642"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 5.</s>
        </p>
        <p type="main">
          <s id="id003458">Velut &longs;i &longs;it proportio a b ad k l, ut a g ad k g, &amp; rur&longs;us ut b c ad <lb/><arrow.to.target n="marg643"/><lb/>l m, &amp; a&longs;&longs;umpt&aelig; &longs;int proportiones e&aelig;dem &longs;emper totius, &amp; totius <lb/>ad partes, &amp; re&longs;iduorum ad partes, etiam &amp; b c &amp; l m ad partes, eti&shy;<lb/>am excepta una &longs;eu quantitatum a b &amp; k l, &longs;eu re&longs;iduorum ut a c &amp; <lb/>k o, &longs;eu partium ut b c &amp; l m ad partes, dico quod h&aelig; partes etiam <lb/>erunt a&longs;&longs;umpt&aelig; &longs;ecundum eandem proportionem ad ip&longs;as magni&shy;<lb/>tudines, uel partes primas uel re&longs;idua.</s>
        </p>
        <p type="margin">
          <s id="id003459"><margin.target id="marg643"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003460">Sed &amp; id &longs;equitur ex his, quod cuiu&longs;cunque &longs;eu totius &longs;eu partis <lb/><arrow.to.target n="marg644"/><lb/>&longs;eu utriu&longs;que pars maior a&longs;&longs;umetur, erit maior proportio totius ad <lb/>totum qu&agrave;m re&longs;idui ad re&longs;iduum.</s>
        </p>
        <p type="margin">
          <s id="id003461"><margin.target id="marg644"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 6.</s>
        </p>
        <p type="main">
          <s id="id003462">H&aelig;c demon&longs;trantur &agrave; Campano, nam &longs;i &longs;it maior proportio a b <lb/><arrow.to.target n="marg645"/><lb/>ad a g, quam k l ad k g, erit maior a b ad k l qu&agrave;m a g ad k g.<lb/><arrow.to.target n="marg646"/></s>
        </p>
        <p type="margin">
          <s id="id003463"><margin.target id="marg645"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003464"><margin.target id="marg646"/>R<emph type="italics"/>up.<emph.end type="italics"/> 16. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003465">Sequitur rur&longs;us, quod in eadem con&longs;titutione cuiu&longs;cunque ma&shy;</s>
        </p>
        <p type="main">
          <s id="id003466"><arrow.to.target n="marg647"/><lb/>ior pars ab&longs;umetur, ea quantitas minori numero, uel numeri parte <lb/>ab&longs;umetur.</s>
        </p>
        <p type="margin">
          <s id="id003467"><margin.target id="marg647"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 7.</s>
        </p>
        <p type="main">
          <s id="id003468">Nam &longs;i minor erit continuo proportio a b ad a e, qu&agrave;m k l ad k <lb/><arrow.to.target n="marg648"/><lb/>o, &amp; a e ad e g, qu&agrave;m k o ad o g, erit longe minor a b ad b g qu&agrave;m k l <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/>ab&longs;umetur quam k g.</s>
        </p>
        <p type="margin">
          <s id="id003470"><margin.target id="marg648"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003471">Propo&longs;itio cente&longs;ima octuage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id003472">Si aliqua quantitas in duas partes diuidatur, fueritque alicuius, <lb/>quantitatis ad partes illas compo&longs;ita proportio eiu&longs;dem quan&shy;<lb/>titatis ad partes alias quantitatis diui&longs;a aliter proportio eadem <lb/>componi.<lb/><arrow.to.target n="marg649"/></s>
        </p>
        <p type="margin">
          <s id="id003473"><margin.target id="marg649"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003474">Sit a b proportio ad partes c d qu&aelig; &longs;int c e, &amp; c d componens f, <lb/>dico quod non poterit c d ali&agrave;s diuidi, ut proportio a b ad illas <lb/>componat eandem proportionem f. </s>
          <s id="id003475">Aliter &longs;it diui&longs;a in g, &amp; erit mi&shy;


<pb pagenum="203" xlink:href="015/01/222.jpg"/>nor c g, minor aut maior c d minore, capiam ergo c d minorem, erit <lb/>igitur proportio a b ad c d maioris exce&longs;&longs;us ad proportionem a b <lb/>ad c g, qu&agrave;m &longs;it proportio a b ad g d, ma&shy;<lb/><figure id="id.015.01.222.1.jpg" xlink:href="015/01/222/1.jpg"/><lb/>ior proportione a b ad c e, propterea quod <lb/>g e communis differentia maiorem habet <lb/>proportionem ad e d quam g c, igitur ma&shy;<lb/>ius e&longs;t aggregatum proportionum a b ad <lb/>c e, &amp; e d, <expan abbr="qu&atilde;">quam</expan> eiu&longs;dem a b ad c g &amp; g d, quod erat demon&longs;trandum.</s>
        </p>
        <p type="main">
          <s id="id003476">Propo&longs;itio cente&longs;ima octuage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id003477">Cum fuerit aliqua proportio compo&longs;ita ex proportionibus pri&shy;<lb/>m&aelig; ad &longs;ecundam &amp; tertiam, &amp; rur&longs;us quart&aelig; ad quintam &amp; &longs;ex&shy;<lb/>tam, ita &longs;e habebit proportio &longs;ecund&aelig; ad tertiam proportionem <lb/>quint&aelig; ad &longs;extam, uelut producti ex proportione in &longs;ecundam de&shy;<lb/>tracta prima ad primam ad productum ex proportione in quin&shy;<lb/>tam, detracta quarta ad quartam.</s>
        </p>
        <p type="main">
          <s id="id003478">Sit pro portio g compo&longs;ita ex proportionibus a <lb/><figure id="id.015.01.222.2.jpg" xlink:href="015/01/222/2.jpg"/><lb/>ad b &amp; c, &amp; proportionibus d ad e &amp; f, dico quod <lb/>quemadmodum b ad c, ad proportionem e ad f, ita <lb/>producti ex g in b, detracto a ad a ad productum ex <lb/>g in e, detracto d ad d. </s>
          <s id="id003479">E&longs;t enim, ut demon&longs;tratum <lb/>e&longs;t b ad c, ut productum ex g in b, detracto a ab a &amp; e ad f, ut pro&shy;<lb/>ducti ex g in e, detracto d ad d, igitur cum &aelig;qualium &longs;int e&aelig;dem <lb/>comparationes, erit ut proportionis b ad c ad proportionem e ad <lb/>f, ita producti ex g in b, detracto a ad a, ad productum e&longs;t g in e, de&shy;<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 re&longs;i&shy;<lb/>dui b detracto quod prouenit, diui&longs;o a per proportionem a ad pro <lb/>portionem re&longs;idui e detracto quod prouenit diui&longs;o d per propor&shy;<lb/>tionem ad ip&longs;um d.</s>
        </p>
        <p type="main">
          <s id="id003481">Propo&longs;itio cente&longs;ima octuage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id003482">Propo&longs;ita differentia proportionum partium &longs;imilium ad par&shy;<lb/>tes a&longs;&longs;umptas propo&longs;itaque proportione totius ad re&longs;idua eandem <lb/>differentiam proportionum totius ad reliquum re&longs;idui inuenire.</s>
        </p>
        <figure id="id.015.01.222.3.jpg" xlink:href="015/01/222/3.jpg"/>
        <p type="main">
          <s id="id003483">Sint dat&aelig; partes b c &amp; e f, &longs;imiles in compa&shy;<lb/>ratione ad a b &amp; d e, &amp; data re&longs;idua a g &amp; d h <lb/>in <expan abbr="c&otilde;paratione">comparatione</expan> a b &amp; d e, &longs;imilia in differentia <lb/>proportionis f e ad c l, ad proportionem <lb/>c b ad b k, dico quod data e&longs;t differentia proportionis a b ad g k <lb/>ad proportionem d e &amp; f h. </s>
          <s id="id003484">Nam quia proportio f e ad c l, ad pro&shy;


<pb pagenum="204" xlink:href="015/01/223.jpg"/>portionem b e ad c k data e&longs;t, &amp; c f ad e d, ut b c ad b a, erit ut a c ad <lb/>l e contineat a b ad b k, ut f e ad e l, c b ad b k, &longs;ed a b ad a d, ut d c ad <lb/>d h, igitur a b ad b d, ut d e ad c h. </s>
          <s id="id003485">Sunt ergo du&aelig; quantitates a b &amp; <lb/>d c, qu&aelig; eandem habent compo&longs;itam proportionem ad g k &amp; k b, <lb/>&amp; h l &amp; l e, quare per pr&aelig;cedentem proportionis h l ad l e, ad pro<lb/>portionem g k ad k b, ut h l detracto prouentu d e, diui&longs;i per propor<lb/>tionem ad d e ad proportionem g k, detracto prouentu a b, diui&longs;i <lb/>per eandem proportionem ad ip&longs;um a b. </s>
          <s id="id003486">Si igitur nota e&longs;t l e &amp; h l, <lb/>erit nota proportio re&longs;idui h l detracto prouentu d e diui&longs;i per pro&shy;<lb/>portionem, quare nota detractio g k detracto prouentu a b diui&longs;i <lb/>per eandem proportionem ad a b. </s>
          <s id="id003487">E&longs;t autem a b nota, &amp; propor&shy;<lb/>tio nota, &amp; ideo prouentus, &amp; cum &longs;it proportio nota, erit ergo <lb/>re&longs;iduum notum, cui addito prouentu fit tota g k nota, quod fuit <lb/>demon&longs;trandum.</s>
        </p>
        <p type="main">
          <s id="id003488">Propo&longs;itio cente&longs;ima octuage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id003489">Spatium uit&aelig; naturalis per &longs;patium uit&aelig; fortuitum declarare.</s>
        </p>
        <p type="main">
          <s id="id003490">Cum con&longs;tet homines ca&longs;u uiuere &aelig;grotantes primum &longs;&aelig;pe: </s>
        </p>
        <p type="main">
          <s id="id003491"><arrow.to.target n="marg650"/><lb/>deinde uiuentes in a&euml;re malo, &amp; ip&longs;um intempe&longs;tiuis horis &longs;ub&shy;<lb/>euntes tri&longs;titijs, curis, uigilia, uenere, laboribus perperam &longs;e excru&shy;<lb/>ciantes, <expan abbr="t&utilde;">tum</expan> uer&ograve; immodico cibo &amp; potu, &amp; prauo, &amp; &longs;&aelig;pius, qu&agrave;m <lb/>oporteat, &amp; intempe&longs;tiu&egrave;, &amp; mal&egrave; pr&aelig;parato, &amp; uario &longs;e replentes, <lb/>atque &longs;ic alij ad &longs;exage&longs;imum, alij ad &longs;eptuage&longs;imum, rari octuage&shy;<lb/>&longs;imo, rariores nonage&longs;imo uel cente&longs;imo anno ita <expan abbr="moriun&ttilde;">moriuntur</expan>, ut non <lb/>ca&longs;u, neque ui aut morbo, &longs;ed potius qua&longs;i naturali quadam morte <lb/>ab&longs;umpti intereant: de quibus tantum e&longs;t &longs;ermo. </s>
          <s id="id003492">Atque ut exem&shy;<lb/>plo commodiore utamur, capiamus annum octoge&longs;imum, qui e&longs;t <lb/>terminus communis uit&aelig; human&aelig;, non &longs;olum no&longs;tra &aelig;tate, &longs;ed an&shy;<lb/>tiquo tempore etiam fuit, ut Dauid te&longs;tatur in P&longs;almis, in Cantico <lb/>Moy&longs;is: antea autem &longs;i quis moriatur, non naturali morte, &longs;ed ui <lb/>morbi ab&longs;umptus exi&longs;timatur. </s>
          <s id="id003493">Certum e&longs;t, quod &longs;i homo recta ra&shy;<lb/>tione uiueret, quod aliquanto diutius uitam extenderet, neque enim <lb/>negare po&longs;&longs;umus, cum in magnis exce&longs;sibus maxim&egrave; &longs;ectionis ue&shy;<lb/>n&aelig; &amp; curarum, quin homo euidentur uitam breuiorem efficiat: <lb/>quod ergo euidenti&longs;simum e&longs;t in magnis exce&longs;sibus, in paruis ean&shy;<lb/>dem habet uim licet occultiorem. </s>
          <s id="id003494">Errorem autem in uita hunc ade&longs;&shy;<lb/>&longs;e perpetuum, qui&longs;que intelligit qui no&longs;tras actiones pen&longs;itare uelit, <lb/>cum &longs;altem malam &longs;equamur con&longs;uetudinem: iam ergo proponan&shy;<lb/>tur iuxta dicta du&etail; line&etail; a b uit&etail; naturalis exqui&longs;it&etail; recte longior &amp; 


<pb pagenum="205" xlink:href="015/01/224.jpg"/>c d uit&aelig; quam is uicturus e&longs;t, id e&longs;t, annorum octuaginta, quam <expan abbr="c&otilde;&shy;">con&shy;<lb/></expan><arrow.to.target n="marg651"/><lb/>&longs;tat e&longs;&longs;e breuiorem aliquanto. </s>
          <s id="id003495">Et proponatur error quadrage&longs;im&aelig; <lb/>partis in ip&longs;a uita, quamuis &longs;it longe maior: quotu&longs;qui&longs;que enim e&longs;t <lb/>qui non &longs;altem edat bibatque quadrage&longs;ima parte, plu&longs;qu&agrave;m opor&shy;<lb/>teat in comparatione ad naturam, id e&longs;t, ut natura fatigatur quadra<lb/>ge&longs;ima illa parte amplius qu&agrave;m debeat: idem dico de laboribus, cu<lb/>ris, uigilijs, uenere. </s>
          <s id="id003496">Sed hoc non e&longs;t generale: habetque multas exce&shy;<lb/>ptiones inuicem pugnantes, ut tandem concludam non concoqui <lb/>plen&egrave; po&longs;&longs;e, &amp; ob id impurum manere, unde cit&ograve; di&longs;&longs;oluitur, &amp; ca&shy;<lb/>lorem etiam naturalem extinguit: atque etiam ob id, tum quia debi&shy;<lb/>tos labores, &amp; multo minus ad perfectam &aelig;tatem perferre <expan abbr="n&otilde;">non</expan> po&longs;&shy;<lb/>&longs;unt, den&longs;ari nequit &amp; pingue&longs;cere, ut duplici cau&longs;a multo celerius <lb/>re&longs;oluatur, una etiam calorem extinguat. </s>
          <s id="id003497">Sit ergo a e talis pars a b, <lb/>qualis c f, c d. </s>
          <s id="id003498">Cum ergo a b con&longs;umi&shy;<lb/><figure id="id.015.01.224.1.jpg" xlink:href="015/01/224/1.jpg"/><lb/>tur in octuaginta annis, &longs;emper &longs;eruat <lb/><expan abbr="proportion&etilde;">proportionem</expan> cum uita contracta, qu&aelig; <lb/>&aelig;qualiter ab&longs;umitur: quia portiones <lb/>ill&aelig; &aelig;quales &longs;unt in minore inuicem &longs;icut in maiore, &amp; in&aelig;quales <lb/>&longs;eruant eandem proportionem, &longs;umatur ergo a b annorum cclvij. <lb/></s>
          <s id="id003499">men&longs;ium v. &amp; ab&longs;umatur &longs;emper quantitas &aelig;qualis octuage&longs;ima <lb/>a e, &amp; quadrage&longs;ima a b &amp; re&longs;iduorum.<lb/><figure id="id.015.01.224.2.jpg" xlink:href="015/01/224/2.jpg"/><arrow.to.target n="table27"/></s>
        </p>
        <p type="margin">
          <s id="id003500"><margin.target id="marg650"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003501"><margin.target id="marg651"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 179. <lb/>E<emph type="italics"/>t in cor.<emph.end type="italics"/> 1. <lb/><emph type="italics"/>&amp;<emph.end type="italics"/> 2.</s>
        </p>
        <table>
          <table.target id="table27"/>
          <row>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
          </row>
          <row>
            <cell/>
            <cell>257</cell>
            <cell>20</cell>
            <cell>14</cell>
            <cell>168</cell>
            <cell>32</cell>
            <cell>28</cell>
            <cell>106</cell>
            <cell>25</cell>
            <cell>41</cell>
            <cell>65</cell>
            <cell>27</cell>
            <cell>54</cell>
            <cell>36</cell>
            <cell>6</cell>
            <cell>68</cell>
            <cell>13</cell>
            <cell>23</cell>
          </row>
          <row>
            <cell>1</cell>
            <cell>250</cell>
            <cell>0</cell>
            <cell>15</cell>
            <cell>163</cell>
            <cell>24</cell>
            <cell>29</cell>
            <cell>103</cell>
            <cell>0</cell>
            <cell>42</cell>
            <cell>63</cell>
            <cell>2</cell>
            <cell>55</cell>
            <cell>34</cell>
            <cell>10</cell>
            <cell>69</cell>
            <cell>12</cell>
            <cell>10</cell>
          </row>
          <row>
            <cell>2</cell>
            <cell>242</cell>
            <cell>30</cell>
            <cell>16</cell>
            <cell>158</cell>
            <cell>21</cell>
            <cell>30</cell>
            <cell>99</cell>
            <cell>17</cell>
            <cell>43</cell>
            <cell>60</cell>
            <cell>19</cell>
            <cell>56</cell>
            <cell>32</cell>
            <cell>16</cell>
            <cell>70</cell>
            <cell>10</cell>
            <cell>38</cell>
          </row>
          <row>
            <cell>3</cell>
            <cell>235</cell>
            <cell>28</cell>
            <cell>17</cell>
            <cell>153</cell>
            <cell>23</cell>
            <cell>31</cell>
            <cell>95</cell>
            <cell>38</cell>
            <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>
          </row>
          <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>
          </row>
          <row>
            <cell>5</cell>
            <cell>222</cell>
            <cell>5</cell>
            <cell>19</cell>
            <cell>144</cell>
            <cell>2</cell>
            <cell>33</cell>
            <cell>89</cell>
            <cell>11</cell>
            <cell>46</cell>
            <cell>53</cell>
            <cell>7</cell>
            <cell>59</cell>
            <cell>27</cell>
            <cell>6</cell>
            <cell>73</cell>
            <cell>7</cell>
            <cell>11</cell>
          </row>
          <row>
            <cell>6</cell>
            <cell>215</cell>
            <cell>23</cell>
            <cell>20</cell>
            <cell>139</cell>
            <cell>18</cell>
            <cell>34</cell>
            <cell>86</cell>
            <cell>2</cell>
            <cell>47</cell>
            <cell>50</cell>
            <cell>34</cell>
            <cell>60</cell>
            <cell>25</cell>
            <cell>19</cell>
            <cell>74</cell>
            <cell>6</cell>
            <cell>4</cell>
          </row>
          <row>
            <cell>7</cell>
            <cell>209</cell>
            <cell>8</cell>
            <cell>21</cell>
            <cell>135</cell>
            <cell>0</cell>
            <cell>35</cell>
            <cell>82</cell>
            <cell>36</cell>
            <cell>48</cell>
            <cell>48</cell>
            <cell>24</cell>
            <cell>61</cell>
            <cell>23</cell>
            <cell>34</cell>
            <cell>75</cell>
            <cell>4</cell>
            <cell>38</cell>
          </row>
          <row>
            <cell>8</cell>
            <cell>203</cell>
            <cell>0</cell>
            <cell>22</cell>
            <cell>130</cell>
            <cell>25</cell>
            <cell>36</cell>
            <cell>79</cell>
            <cell>34</cell>
            <cell>49</cell>
            <cell>46</cell>
            <cell>16</cell>
            <cell>62</cell>
            <cell>22</cell>
            <cell>11</cell>
            <cell>76</cell>
            <cell>3</cell>
            <cell>34</cell>
          </row>
          <row>
            <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>
          </row>
          <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>67</cell>
            <cell>14</cell>
            <cell>37</cell>
            <cell/>
            <cell/>
            <cell/>
          </row>
        </table>
        <p type="main">
          <s id="id003502">Vt corrigas tabulam, &longs;cito quod numerus quadrage&longs;im&aelig; cum <lb/>&longs;uperiore annorum numero &agrave; leua componit numerum quadrage<lb/>&longs;im&aelig; &longs;uperioris &longs;impliciter, aut abiectis quadragenarijs. </s>
          <s id="id003503">Velut &egrave; <lb/>regione trige&longs;imi anni, &longs;unt anni nonaginta nouem, quad. </s>
          <s id="id003504">17 &egrave; <lb/>directo anni 29, &longs;unt anni 103, quad. </s>
          <s id="id003505">0. ad de 17 quad. </s>
          <s id="id003506">ad 103 fit 120, <lb/>abijce 40 ter, nil &longs;upere&longs;t, &amp; ita nulla e&longs;t quadragenaria &egrave; regione <lb/>29 &amp; 103.</s>
        </p>
        <pb pagenum="106 &lsqb;=206&rsqb;" xlink:href="015/01/225.jpg"/>
        <p type="main">
          <s id="id003507">Rur&longs;us cum deuenimus ad annos 79, &longs;uper&longs;unt &longs;olum 28 qua&shy;<lb/>dragenari&aelig;, &amp; e&longs;t minus anno, &longs;ed hoc fieri ob fractiones &amp; nume&shy;<lb/>rorum partes, &amp; etiam &longs;i e&longs;&longs;et aliquis error, e&longs;&longs;et magis ad augen&shy;<lb/>dum numerum annorum 257, men&longs;ium &longs;ex qu&agrave;m ad diminutio&shy;<lb/>nem, ideo non curaui de exacta ueritate.</s>
        </p>
        <p type="main">
          <s id="id003508">Pr&aelig;terea ex hac tabella digno&longs;cis, quod in ultimis annis parum <lb/>pote&longs;t produci uita in comparatione ad primos, ueluti in 60 anno <lb/>&longs;uper&longs;unt anni 20, ex uita ordinaria, ex exacta paulo plures qu&agrave;m <lb/>25, &longs;cilicet 25 cum dimidio. </s>
          <s id="id003509">Ergo &agrave; 60 anno non poterit per quam&shy;<lb/>uis cu&longs;todiam homo producere uitam plus annis quinque cum di&shy;<lb/>midio. </s>
          <s id="id003510">Et &longs;i dicas tunc cu&longs;todia maxim&egrave; opus e&longs;t, &amp; magis qu&agrave;m <lb/>unquam, re&longs;pondeo quod uerum e&longs;t, &longs;ed non ad producendum ui&shy;<lb/>tam, &longs;ed ne in morbum incidas: nam ex quocunque morbo homo ab <lb/>ea &aelig;tate perit, cum habeat ade&ograve; imbecilles uires. </s>
          <s id="id003511">Ex hoc patet, <lb/>quod Alexius Cornarius, patritius Venetus, cum inc&oelig;pi&longs;&longs;et cu&longs;to<lb/>diam anno 36, cum po&longs;&longs;et uiuere 44 annis, iuxta rationem uit&etail; com<lb/>munis, potuit producere eam annis 79, igitur annis 25 plu&longs;qu&agrave;m ui<lb/>xi&longs;&longs;et uita communi etiam qu&ograve;d fui&longs;&longs;et &longs;anus.</s>
        </p>
        <p type="main">
          <s id="id003512">Si ergo aliquis &longs;it uicturus centum annis uita communi adde&shy;<lb/>mus eodem modo trige&longs;imam nonam partem, id e&longs;t quadrage&longs;i&shy;<lb/>mam partem, &amp; quadrage&longs;imam quadrage&longs;im&aelig; huic numero, &amp; <lb/>unum amplius, &amp; habebimus numerum ut infr&agrave;.<lb/><figure id="id.015.01.225.1.jpg" xlink:href="015/01/225/1.jpg"/><arrow.to.target n="table28"/></s>
        </p>
        <table>
          <table.target id="table28"/>
          <row>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>A<emph type="italics"/>n.<emph.end type="italics"/></cell>
            <cell>Q<emph type="italics"/>uad.<emph.end type="italics"/></cell>
          </row>
          <row>
            <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 digno&longs;cemus quantum qui&longs;que po&longs;sit uiuere, <lb/>quouis tempore &aelig;tatis &longs;u&aelig;, illud intelligendo quod non e&longs;t eadem <lb/>men&longs;ura omnibus, ut neque uit&aelig; ordinari&aelig;, nec magnitudinis cor <lb/>porum, nec ingeniorum, nec eiu&longs;modi in aliquibus uita decre&longs;cit <lb/>per uige&longs;imam partem, hic &longs;cilicet qui inordinat&egrave; uiuunt, alijs uix &longs;e<lb/>xage&longs;ima, quan&lt;08&gt; pauci&longs;simis. </s>
          <s id="id003514">Hic ergo numerus maxim&egrave; concor&shy;<lb/>dat cum experimentis duobus, <expan abbr="&qtilde;">quae</expan> apparuerunt <expan abbr="par&utilde;">parum</expan> ante <expan abbr="t&etilde;pora">tempora</expan> no<lb/>&longs;tra, &longs;cilicet Ioannis de <expan abbr="t&etilde;poribus">temporibus</expan>, qui uixit annis 361, &amp; Richardus <lb/>de temporibus, annis 400. Et ambo fuerunt milites Caroli Ma&shy;<lb/>gni, nam non potuerunt omnino pro&longs;picere uit&aelig; rationi exqui&longs;i&shy;<lb/>ti&longs;sim&aelig;. </s>
          <s id="id003515"><expan abbr="Refer&utilde;t">Referunt</expan> etiam in India no&longs;tris <expan abbr="t&etilde;poribus">temporibus</expan> uiuere ad centum 


<pb pagenum="207" xlink:href="015/01/226.jpg"/>quinquaginta annos, cuius cau&longs;am transferunt in a&euml;rem: ego po&shy;<lb/>tius in uit&aelig; genus, ab&longs;tinent enim carnibus, ouis, ca&longs;eo &amp; uino, u&shy;<lb/>tunturque fructibus tantum, &amp; uiuebant &longs;ine &longs;olicitudine ulla &amp; cu&shy;<lb/>ris. </s>
          <s id="id003516">Vnde rect&egrave; in&longs;inuatum e&longs;t etiam ultra hi&longs;toriam, quod Adam <lb/>e&longs;&longs;et perpetu&ograve; uicturus, &longs;i non degu&longs;ta&longs;&longs;et fructum arboris boni &amp; <lb/>mali, id e&longs;t, quod mors nobis obrepit ob, &longs;olicitudines &amp; curas. </s>
          <s id="id003517">A&shy;<lb/>uenzoar autem cum uixerit multis cum curis, &amp; fuerit in carcere <lb/>Hali, &amp; ab eo per iniuriam uexatus, &amp; natus in malo a&euml;re, &longs;ola ratio&shy;<lb/>ne uictus produxit uitam ad annos 135, ut te&longs;tatur Auerroes, quid <lb/>euenturum erat, &longs;i in bono a&euml;re educatus nihil graue, &amp; ade&ograve; diu&shy;<lb/>turnum expertus fui&longs;&longs;et:</s>
        </p>
        <p type="main">
          <s id="id003518">Pro u&longs;u autem huius &amp; &longs;uperioris tabul&aelig;, &longs;i quis proponat iu&shy;<lb/>uenem ex &longs;tirpe eorum, qui uiuunt &longs;exaginta annis, iam natum de&shy;<lb/>cem &amp; &longs;eptem annos, uelimusque &longs;cire quantum uiuere po&longs;sit, uide &egrave; <lb/>regione 20 annorum in primo ordine, &amp; habes annos 139. Quad. <lb/>18. &amp; ab hoc numera 17 annos, &amp; habebis annos 37 &egrave; regione, <lb/>quorum &longs;unt anni 76. Quad. 35, id e&longs;t, men&longs;es 10, dies 15. uel iunge <lb/>17, numerum annorum exactorum, &amp; 20 numerum annorum defi&shy;<lb/>cientium ab 80, fiunt anni 33, ut prius, &egrave; quorum regione habet an&shy;<lb/>nos 76. quad. </s>
          <s id="id003519">35.</s>
        </p>
        <p type="main">
          <s id="id003520">At &longs;cio multos qui parum con&longs;yderat&egrave; h&aelig;c legunt, obiecturos, <lb/>primum quod neque mihi, neque ulli alij potui, uel ad centum uel ad <lb/>nonaginta annos <expan abbr="uit&atilde;">uitam</expan> producere. </s>
          <s id="id003521"><expan abbr="Secund&utilde;">Secundum</expan>, &qring;d &longs;i uita humana e&longs;&longs;et <lb/>eiu&longs;modi, naturaliter e&longs;&longs;et ut in pluribus: at uix inuenire licet <expan abbr="aliqu&etilde;">aliquem</expan> <lb/>qui exce&longs;&longs;erit cente&longs;imum uige&longs;imum annum. </s>
          <s id="id003522">Et maxim&egrave; cum &longs;cri&shy;<lb/>ptum &longs;it: Non &longs;piritum meum in carne ultra centum uiginti annos, <lb/>&amp; loquitur Deus. </s>
          <s id="id003523">Videtur etiam nece&longs;&longs;e hoc uolenti, cupere totam <lb/>uitam &longs;ub incerto fine, &amp; non uacare, nec negotijs nec uoluptati, <lb/>qu&aelig; &longs;unt duo illa pr&aelig;cipua, quibus uita no&longs;tra con&longs;tat, &amp; maxim&egrave; <lb/>amittere bona, ade&ograve; &longs;ecura ob tam leuem &amp; inanem &longs;pem. </s>
          <s id="id003524">Ab&longs;ur&shy;<lb/>dum etiam e&longs;&longs;e hoc quod latuerit tot pr&aelig;claros medicos atque phi&shy;<lb/>lo&longs;ophos, quorum nullus de hoc &longs;ermonem fecit. </s>
          <s id="id003525">H&aelig;c &amp; huiu&longs;mo<lb/>di &longs;unt qu&etail; mihi obij ci po&longs;&longs;e &longs;entio. </s>
          <s id="id003526">At rogo quid admirabilius e&longs;t, <lb/>an &longs;olem e&longs;&longs;e plus centies et &longs;exagies terra ac mari, an homines tam&shy;<lb/>diu po&longs;&longs;e producere uitam? </s>
          <s id="id003527">Et plures imperito hoc quam illud cre<lb/>dituri &longs;unt: &amp; tamen res illa ita &longs;e habet, nec apud &longs;apientes dubia <lb/>e&longs;t: nedum incredibilis. </s>
          <s id="id003528">Similiter qu&ograve;d corpus ade&ograve; tenue, debeat <lb/>ade&ograve; celeriter circumferri, ut in uno ictu pul&longs;us debeat peragere <lb/>&longs;patium bis mille quingentorum millium pa&longs;&longs;uum, &amp; tamen &amp; il&shy;<lb/>lud demon&longs;trari pote&longs;t euidenti&longs;sim&egrave;. </s>
          <s id="id003529">Ergo ut ad obiecta re&longs;pon&shy;<lb/>deam &longs;er&ograve; mihi hoc inuenire <expan abbr="c&otilde;tigit">contigit</expan>, infeliciter natus, peius educa&shy;


<pb pagenum="208" xlink:href="015/01/227.jpg"/>tus &amp; imbecilli corpore ac natura, quod ali&acirc;s dixi, nec for&longs;an in <lb/>quibu&longs;dam &longs;ufficiat educatio ab initio, &longs;ed requiritur &longs;ucce&longs;sio, <lb/>qualis fuit olim per multas &aelig;tates, &longs;ic progenerantur gigantes &amp; <lb/>homines ad miraculum u&longs;que, docui etiam exacta media &aelig;tate, hoc <lb/>uix fieri po&longs;&longs;e. </s>
          <s id="id003530">Contingunt pr&aelig;terea multa impedimenta. </s>
          <s id="id003531">Sufficit <lb/>nobis &longs;cire quid &longs;it in natura hominis, non qu&aelig;ro mod&ograve; quomo&shy;<lb/>do faciendum: nec e&longs;t pr&aelig;&longs;entis in&longs;tituti, quin etiam ueri&longs;imile e&longs;t <lb/>ad hoc e&longs;&longs;e uiam quandam compendio&longs;iorem, qu&aelig; minim&egrave; la&shy;<lb/>tuerit antiquos, maxim&egrave; Hebr&aelig;os. </s>
          <s id="id003532">Et for&longs;an etiam hoc no&longs;tro tem&shy;<lb/>pore haberi po&longs;&longs;et quamuis lateat. </s>
          <s id="id003533">Vnum e&longs;t certum, oportere ab <lb/>initio uit&aelig; &lpar;qui uiam hanc exqui&longs;itam, quam hic trado, &longs;equi uo&shy;<lb/>luerit&rpar; con&longs;tituere formam uictus, &amp; tum maxim&egrave; contractam, <lb/>quoniam &lpar;ut ui&longs;um e&longs;t in tabula&rpar; ex minimo errore, &amp; breui tempo<lb/>re plurimum temporis uit&aelig; perit. </s>
          <s id="id003534">Oportet autem multa ade&longs;&longs;e, cor<lb/>pus moderat&egrave; &longs;anum, &amp; mediocriter &longs;altem con&longs;titutum, in&longs;tituto&shy;<lb/>rem &longs;apientem, obedientiam pueri, &amp; per omnes &aelig;tates cum pati&shy;<lb/>entia &longs;umma commoda diuitiarum, &amp; bonum a&euml;rem &amp; fortunam <lb/>blandientem no&longs;tro propo&longs;ito, ne quis ca&longs;us in tanto tempore ad&shy;<lb/>uer&longs;us nos impediat, ob tot &amp; tanta qu&aelig; nece&longs;&longs;aria &longs;unt, &amp; a&longs;sidu&egrave;, <lb/>ideo res h&aelig;c fabulo&longs;a ui&longs;a e&longs;t ad hanc u&longs;que diem, tum maxim&egrave; quod <lb/>nemo eam docuerat. </s>
          <s id="id003535">De dicto Moy&longs;is non laboro, cum &longs;imus me&shy;<lb/>dici ac philo&longs;ophi non theologi. </s>
          <s id="id003536">Quin etiam po&longs;t h&aelig;c uixit Abra&shy;<lb/><arrow.to.target n="marg652"/><lb/>hamus annis clxxv, I&longs;aacus autem clxxx, Iacobus cxlvij, &longs;ed non la&shy;<lb/><arrow.to.target n="marg653"/><lb/>boro de his, uer&ugrave;m relinquo illa &longs;apientibus: melius e&longs;t ergo ut de&shy;<lb/><arrow.to.target n="marg654"/><lb/>mon&longs;trationem adducam huius, cum experimento etiam coniun&shy;<lb/>ctam. </s>
          <s id="id003537">Con&longs;tat enim quod humidum pingue euane&longs;cit per &aelig;tates, <lb/>&longs;eu &agrave; calore innato, &longs;eu ab a&euml;re con&longs;umatur, &amp; quod humidum pin&shy;<lb/>gue purum, ac den&longs;um tard&egrave; ab&longs;umitur, &longs;icut apparet experimen&shy;<lb/>to de oleo &amp; &longs;epo &longs;alitis, qu&aelig; durant longiori tempore, quam &longs;i nil <lb/>tale admi&longs;tum habeant h&aelig;c pinguia, &longs;imiliter aqua quadruplo ce&shy;<lb/>lerius, imo longe uelocius ab&longs;umitur oleo in ua&longs;e feruente. </s>
          <s id="id003538">Et ita <lb/>de pinguedinibus uariorum animalium de ligno iunipero, quod <lb/>referunt durare in annum, cur alia non po&longs;sint ad &longs;ex dies. </s>
          <s id="id003539">Cer&shy;<lb/>tum etiam e&longs;t, quod coctio conden&longs;et, &amp; e&longs;t Philo&longs;ophi in quar&shy;<lb/>to Metheororum. </s>
          <s id="id003540">Si ergo coctio perfecta fiat, &amp; puri&longs;simum hu&shy;<lb/>midum re&longs;tauretur, dubium non e&longs;t, quin homo po&longs;sit uiuere &longs;ex&shy;<lb/>cuplo plus aut <expan abbr="eti&atilde;">etiam</expan> octuplo: quia c&ugrave;m res peruenit ad <expan abbr="quend&atilde;">quendam</expan> ter&shy;<lb/>minum, tunc acquiritur perfectio <expan abbr="qu&etail;d&atilde;">qu&etail;dam</expan> ultra <expan abbr="omn&etilde;">omnem</expan> fidem, &longs;icut ui&shy;<lb/>demus de auro, &qring;d pror&longs;us <expan abbr="eti&atilde;">etiam</expan> longo tempore ab ignibus <expan abbr="n&otilde;">non</expan> ab&longs;u<lb/>mitur: ade&ograve; ut liceat dicere, for&longs;an non e&longs;&longs;e contra rationem, quod <lb/>detur humidum, quod nunqu&agrave;m &agrave; calore naturali ab&longs;umitur, quia 


<pb pagenum="209" xlink:href="015/01/228.jpg"/>non e&longs;t par ratio de auro &amp; humido humano, nam in auro <expan abbr="n&otilde;">non</expan> e&longs;t ca<lb/>lor ni&longs;i ab exteriore igne, &longs;ed in humido no&longs;tro e&longs;t calor intus, &amp; &longs;e&shy;<lb/>cundum &longs;ub&longs;tantiam, ut &longs;altem habeamus experimentum longi&longs;&shy;<lb/>&longs;im&aelig; uit&aelig; &amp; humidi quod uix &agrave; calore, &amp; non ni&longs;i multis in &longs;eculis <lb/>ab&longs;umatur. </s>
          <s id="id003541">Atque h&aelig;c &lpar;ne incurramus irri&longs;ionem Galeni&rpar; de Phi&shy;<lb/>lo&longs;opho qui pollicebatur perpetuitatem uit&aelig;, quanquam non ob <lb/>id refugiam hoc, ut negem po&longs;&longs;e hominis uitam e&longs;&longs;e perpetuam, <lb/>quod Galenus <expan abbr="Philo&longs;oph&utilde;">Philo&longs;ophum</expan> hoc dicentem irri&longs;erit, &longs;ed qu&ograve;d uidea&shy;<lb/>mus omnia &longs;ublunaria interire, qu&ograve;d &longs;ciamus omne compo&longs;itum <lb/>debere di&longs;&longs;olui, quoniam compo&longs;itio &longs;it accidens, &amp; accidens e&longs;t <lb/>medium inter ea qu&aelig; &longs;unt &amp; non &longs;unt: loquor de huiu&longs;modi acci&shy;<lb/>dentibus qu&aelig; adueniunt. </s>
          <s id="id003542">Demum, quoniam calor ille &longs;it in ip&longs;o hu <lb/>mido: ideo cum h&etail;c non animaduerterit Galenus, potius fuit uates <lb/>in irridendo, qu&agrave;m &longs;apiens, ut authoritate eius moueri debeamus. <lb/></s>
          <s id="id003543">Hanc coctionem non animaduerterunt medici, &longs;ed &longs;olam illam bo&shy;<lb/>nam qu&etail; e&longs;t cau&longs;a &longs;anitatis, qu&aelig; &longs;tat cum uigilia, labore &amp; ciborum <lb/>multitudine, c&ugrave;m illa exacta non &longs;tet ni&longs;i cum optimis &amp; paucis <lb/>ualde cibis, quiete ac &longs;omno. </s>
          <s id="id003544">Et ideo &longs;unt &longs;ex genera coctionum, di&shy;<lb/>co quod ad perfectionem attinet corrupta, imperfecta, imperfecta <lb/>morbo&longs;a, imperfecta qu&aelig; emendari pote&longs;t, has omnes uitare do&shy;<lb/>cent medici: bona qu&aelig; e&longs;t cum longa &longs;anitate, cui medici &longs;tudent: <lb/>ualde bona quam per umbram qua&longs;i <expan abbr="cognouer&utilde;t">cognouerunt</expan>, &amp; exacta quam <lb/>nec per &longs;omnium quidem uiderunt, qu&aelig; &longs;ola e&longs;t cau&longs;a tant&aelig; lon&shy;<lb/>gitudinis uit&aelig;, cum tamen nunquam fuerit uel admodum parum <lb/>interrupta. </s>
          <s id="id003545">Hoc autem inter c&aelig;tera o&longs;tendit experimentum de ele&shy;<lb/>phantis, quos Ari&longs;toteles ducentis annis uiuere con&longs;tanter affir&shy;<lb/>mat, alius dixit e&longs;&longs;e trecentis. </s>
          <s id="id003546">Vt con&longs;tet iam in natura animalium <lb/>&amp; in genere caloris habentis magnum motum, &amp; &longs;ub&longs;tantiam te&shy;<lb/>nuem hoc inueniri po&longs;&longs;e, ut excludamus plantas de <expan abbr="quar&utilde;">quarum</expan> uita lon&shy;<lb/>gi&longs;sima &longs;atis con&longs;tat, &longs;ed quia caret motu euidenti calor in illis, &amp; <lb/>&longs;ub&longs;tantia e&longs;t cra&longs;&longs;a animalium comparatione, non laboro. </s>
          <s id="id003547">At de <lb/>elephanto omnes confitentur qu&ograve;d &longs;it omnium ingenio&longs;i&longs;simum, <lb/>ade&ograve; ut multi homines illo indu&longs;tria &amp; cognitione inferiores e&longs;&longs;e <lb/>uideantur. </s>
          <s id="id003548">Neque etiam ueri&longs;imile e&longs;t quod natura hominem fecerit <lb/>hac in parte illo inferiorem, pr&aelig;&longs;ertim cum de nullo alio animali <lb/>apud Ari&longs;totelem dubium &longs;it, &amp; ubi modo aliquod dubium e&longs;&longs;et <lb/>propter querelam Theophra&longs;ti, &amp; illud quod &longs;olet pr&aelig;dicari de <lb/>ceruis, tanto magis ueri&longs;imile e&longs;t indignum fui&longs;&longs;e hominem conce&shy;<lb/>dere tot animalibus in diuturnitate uit&aelig;. </s>
          <s id="id003549">Quam ob rem cum h&aelig;c <lb/>tractatio ad libros de tuenda Sanitate &longs;pectaret, homines ad eos re&shy;<lb/>lego, nam ob id illos con&longs;crip&longs;i qu&ograve;d uiderem Galenum nec hoc <pb pagenum="210" xlink:href="015/01/229.jpg"/>uidi&longs;&longs;e nec multa alia, &longs;ed eorum loco longas &amp; inutiles di&longs;putatio&shy;<lb/>nes inter&longs;erui&longs;&longs;e. </s>
          <s id="id003550">Ver&ugrave;m etiam, quoniam eam tractationem diuul&shy;<lb/>&longs;it, ut alia cogamus qu&aelig;rere in libris de Alimentis, alia, de cibis bo&shy;<lb/>ni &amp; mali &longs;ucci: tum uer&ograve; &amp; tractatio ip&longs;a eduliorum e&longs;t imperfe&shy;<lb/>cta, &amp; multa etiam deficiunt circa genera: in quo e&longs;t ex cu&longs;andus ob <lb/>uarietatem regionis &amp; &aelig;tatis. </s>
          <s id="id003551">Dee&longs;t pr&aelig;terea maxima pars, qu&aelig; <lb/>nec ibi nec alibi habetur, &longs;cilicet, de ciborum pr&aelig;paratione. </s>
          <s id="id003552">Quod <lb/>etiam h&aelig;c latuerint tot pr&aelig;claros uiros, quid mirum? </s>
          <s id="id003553">cum Hippo&shy;<lb/>crates uixerit &longs;eculo illo agre&longs;ti, in quo non e&longs;t mirandum, quod ali <lb/>quid, pauca qu&aelig;dam &amp; ab&longs;tru&longs;a omi&longs;erit, &longs;ed quod tam multa tam <lb/>bene inuenerit, ut fuerit, &longs;icut de Pindaro dicitur, im&ograve; long&egrave; uerius <lb/>quam de Pindaro inimitabilis. </s>
          <s id="id003554">De Galeno quid mirum, qui non <lb/>ni&longs;i ueterum &longs;cripta collegit, atque utinam <expan abbr="&longs;alt&etilde;">&longs;altem</expan> bene. </s>
          <s id="id003555">De Ari&longs;totele <lb/>is multa inuenit &longs;uo Marte, &amp; Theophra&longs;tus long&egrave; plura. </s>
          <s id="id003556">De alijs, <lb/>dico tam medicis qu&agrave;m philo&longs;ophis, hoc e&longs;t, quod queror, quod <lb/>in &longs;patio pene duorum millium annorum, non hoc quod ualde re&shy;<lb/>conditum erat, &longs;ed nec leue ullum experimentum, uel natur&aelig; arca&shy;<lb/>num, uel uit&aelig; &longs;alutare auxilium inuenerit. </s>
          <s id="id003557">Sed litigant de nugis &amp; <lb/>rebus inutilibus, &amp; etiam qu&etail; &longs;ciri <expan abbr="n&otilde;">non</expan> po&longs;&longs;unt, ac plerunque non &longs;ine <lb/>magna impietate. </s>
          <s id="id003558">Quod uer&ograve; nece&longs;&longs;e &longs;it amittere uoluptatem, &amp; <lb/>negocia pr&aelig;termittere uolenti hanc uitam longam adipi&longs;ci, qu&aelig; <lb/>po&longs;tmodum etiam ualde in certa e&longs;t: dico quod quantum ad uolu&shy;<lb/>ptates &amp; negocia, non e&longs;&longs;e nece&longs;&longs;e, &longs;ed &longs;olum &longs;uperfluas res, &amp; dam<lb/>no&longs;as &amp; irritas, quas etiam philo&longs;ophi &amp; ciuitatum in&longs;titutores, &amp; <lb/>morum cen&longs;ores docent debere uitari, etiam nullo propo&longs;ito emo&shy;<lb/>lumento, at reliqua <expan abbr="c&otilde;&longs;uetudo">con&longs;uetudo</expan> efficit <expan abbr="n&otilde;">non</expan> &longs;olum grata &amp; tolerabilia, <lb/>&longs;ed etiam iucunda. </s>
          <s id="id003559">De incerto fine, quid e&longs;t certum apud homines, <lb/>ni&longs;i hoc nihil certum e&longs;&longs;e? </s>
          <s id="id003560">Verum tamen &longs;i quis re&longs;piciat ad pr&aelig;&shy;<lb/>mium tam &longs;ingulare e&longs;t, &amp; nobile atque utile, ut non lu&longs;erit operam <lb/>immerit&ograve;, quicunque cum &longs;pe tam illu&longs;tris commodi, &amp; tam exigua <lb/>iactura rerum, ac minore periculo &longs;e huic ale&aelig; experiund&aelig; commi&shy;<lb/>&longs;erit. </s>
          <s id="id003561">Cum, &longs;i quis hoc ip&longs;um adipi&longs;catur, uer&egrave; dici po&longs;sit &longs;ummum <lb/>bonum adeptum e&longs;&longs;e: Non &longs;olum compos factus diuturnitatis ui&shy;<lb/>t&aelig;, &longs;ed cum illa tot uoluptatum, qu&aelig; in longo tempore percipiun&shy;<lb/>tur &longs;cienti&aelig; tot rerum, quas non ni&longs;i temporis longitudo o&longs;tende&shy;<lb/>re pote&longs;t, tot denique ca&longs;us uidere tum opum in crementum, quod <lb/>qua&longs;i certi&longs;simum e&longs;t in longa &aelig;tate &amp; u&longs;u &longs;apientia &amp; authoritate <lb/>plena, ade&ograve; ut ferm&egrave; nece&longs;&longs;e &longs;it ad principatus &longs;peciem deuenire, <lb/>qui tamdiu uixerit, tum gloria ip&longs;a in comparabili. </s>
          <s id="id003562">H&aelig;c autem ma&shy;<lb/>xime accidere nece&longs;&longs;e e&longs;t, quod ut ui&longs;um e&longs;t, quanto longior fuerit <lb/>&aelig;tas eo firmiores <expan abbr="eti&atilde;">etiam</expan> &longs;unt illius partes qu&aelig; ad mortis tempus ap&shy;


<pb pagenum="211" xlink:href="015/01/230.jpg"/>propinquant pari ratione, ut ex tabella prima deprehendere licet, <lb/>qu&ograve;d &longs;i cum hoc &longs;obolis felicitas accedat, non ob&longs;curum e&longs;t huiu&longs;&shy;<lb/>modi po&longs;&longs;e dici ultimam hominis felicitatem apud eos, qui huma&shy;<lb/>nas res aliquid e&longs;&longs;e putant. </s>
          <s id="id003563">Accidunt autem h&aelig;c &longs;ponte in &longs;eculo&shy;<lb/>rum renouationibus, cum humanum genus con&longs;umitur, &longs;eu qui &longs;u<lb/>per&longs;unt ob robur, &longs;eu ex terra geniti, ut dubitat Ari&longs;toteles. </s>
          <s id="id003564">Ha&longs;en <lb/>credit, tum ob a&euml;ris puritatem, &amp; maxim&egrave; qu&ograve;d alterutro modo <lb/>ex calidis regionibus &amp; &longs;ublimibus locis homines reparari nece&longs;&shy;<lb/>&longs;e &longs;it, tamen etiam ob uictus &longs;implicitatem, cum in altera &longs;uper&longs;int <lb/>&longs;oli pi&longs;ces, in altera ne hi quidem, ut in Arcanis demon&longs;tratum e&longs;t. <lb/></s>
          <s id="id003565">Atque etiam ob curarum ab&longs;entiam: &longs;iquidem homines illi gau&shy;<lb/>dent, reges ex agricolis haud dubi&egrave; terrarum facti, ac qua&longs;i &longs;ecu&shy;<lb/>ri mole&longs;tiarum ad hanc &aelig;tatem perueniunt longa &longs;patia tempo&shy;<lb/>ris, &amp; propagand&aelig; &longs;obolis habentes, ut felici&longs;sim&egrave; uiuant, re&longs;tituti <lb/>ex optimis quibu&longs;cunque aure&aelig; illi &aelig;tati, non &longs;olum ob uit&aelig; &longs;yn&shy;<lb/>ceritatem atque &longs;plendorem, &longs;ed etiam longitudinem &longs;ic appella&shy;<lb/>t&aelig;. </s>
          <s id="id003566">Qu&aelig; finem habuit dum &longs;atis &lpar;uti c&oelig;perunt&rpar; &agrave; Saturno in u&longs;um <lb/>traductis: unde etiam falcis in&longs;igne accepit. </s>
          <s id="id003567">Eadem tamen &aelig;tate <lb/>pauci&longs;simi ex infinitis diutius quam no&longs;tra uiuere c&oelig;perunt, c&aelig;te&shy;<lb/>ri omnes minus quam nunc, qu&ograve;d neque ue&longs;titus corporum ab in&shy;<lb/>undatione parta, neque a&euml;ris puritas &agrave; &longs;qualoribus maneret, &amp; edu<lb/>lia multo pauciora e&longs;&longs;ent hominibus &amp; incondita.</s>
        </p>
        <p type="margin">
          <s id="id003568"><margin.target id="marg652"/>G<emph type="italics"/>en. </s>
          <s id="id003569">ca.<emph.end type="italics"/> 25.</s>
        </p>
        <p type="margin">
          <s id="id003570"><margin.target id="marg653"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 35.</s>
        </p>
        <p type="margin">
          <s id="id003571"><margin.target id="marg654"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 47.</s>
        </p>
        <p type="main">
          <s id="id003572">Propo&longs;itio cente&longs;ima octuage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id003573">Qu&aelig;cunque grauia in uorticibus aquarum merguntur, in me&shy;<lb/>dio uorticis primum uer&longs;a mergantur.</s>
        </p>
        <p type="main">
          <s id="id003574">Hanc proponit Ari&longs;toteles, &longs;ed non quantum nece&longs;&longs;arium e&longs;t <lb/><arrow.to.target n="marg655"/><lb/>explicauit, unius enim qu&aelig;&longs;iti, id e&longs;t, primi multiplicem rationem <lb/>reddit. </s>
          <s id="id003575">Sed neque illam perfect&egrave;, quod amborum cau&longs;a una &longs;it, ac <lb/>coniuncta, &longs;ic ergo uortex, cuius extremus <lb/>circulus a b centrum in aqu&aelig; &longs;uperficie c <lb/><figure id="id.015.01.230.1.jpg" xlink:href="015/01/230/1.jpg"/><lb/>capacitas uorticis d e, ut aqua feratur per <lb/>&longs;patium d e f g, h k in maiore circulo na&shy;<lb/>uis, aut aliud graue, quod natura &longs;ua non <lb/>e&longs;&longs;et de&longs;cen&longs;urum &lpar;ut fal&longs;&ograve; exponitur de <lb/>lapide, nam lapis, nec reuoluitur, nec fer&shy;<lb/>tur ad d e circulum intimum, &longs;ed pr&aelig;occu&shy;<lb/>pat ex grauitate &longs;ua fertur in imum&rpar; dico <lb/>&qring;d h k prius circumuoluetur, in de trahetur <lb/>ad d e, &amp; ubi fuerit ibi <expan abbr="de&longs;c&etilde;det">de&longs;cendet</expan>, &longs;ed &longs;i leuius <lb/>&longs;it nece&longs;&longs;ari&ograve; peruenet ad c antequam de&longs;cendat. </s>
          <s id="id003576">Cum ergo aqua 


<pb pagenum="212" xlink:href="015/01/231.jpg"/>grauis &longs;it tota, fertur ad circulum d e, ut de&longs;cendat. </s>
          <s id="id003577">Sed &amp; quia de&shy;<lb/>&longs;cendit per d e f g, &amp; magis ex centro e, ideo omnes partes circumui<lb/>cin&aelig; trahuntur ad d e, &amp; ad e centrum &longs;uperficiei uorticis, tanqu&agrave;m <lb/>ad centrum, ut de&longs;cendant, atque id primum. </s>
          <s id="id003578">Cunque <expan abbr="lign&utilde;">lignum</expan> de&longs;cendat <lb/>partim propria grauitate, partim <expan abbr="attract&utilde;">attractum</expan>, &longs;i fuerit leue corpus, ut plu&shy;<lb/>ma, quod natura &longs;ua <expan abbr="n&otilde;">non</expan> de&longs;cendat, nece&longs;&longs;e e&longs;t ut <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> &longs;ola ui at&shy;<lb/>tractionis, qu&etail; <expan abbr="n&otilde;">non</expan> e&longs;t tanta in toto d e <expan abbr="qu&atilde;ta">quanta</expan> in e, <expan abbr="igi&ttilde;">igitur</expan> oportet ut pri&shy;<lb/>us perueniat ad c qu&agrave;m de&longs;cendat, quia contra <expan abbr="natur&atilde;">naturam</expan> <expan abbr="propri&atilde;">propriam</expan> de&shy;<lb/>&longs;cendit ui <expan abbr="attract&utilde;">attractum</expan>. </s>
          <s id="id003579">Cum uer&ograve; pars qu&aelig; in directo c e&longs;t, ueloci&longs;sim&egrave; <lb/>de&longs;cendat, conantur omnes partes aqu&etail;, qu&etail; circa &longs;unt de&longs;cendere, <lb/>et <expan abbr="c&utilde;">cum</expan> <expan abbr="n&otilde;">non</expan> po&longs;sint &longs;imul peruenire, mouentur ad illud linea, dico quia <lb/>habent initium in e, circulus autem <expan abbr="null&utilde;">nullum</expan> habet <expan abbr="initi&utilde;">initium</expan>, igitur uiden&shy;<lb/>tur moueri circulariter. </s>
          <s id="id003580">Sed cum in circulo partes &agrave; <expan abbr="c&etilde;tro">centro</expan> <expan abbr="mouean&ttilde;">moueantur</expan>, <lb/>uelocius mouebuntur, uelocius in elica a b qu&agrave;m l m, &amp; l m qu&agrave;m <lb/>n o. </s>
          <s id="id003581">Et ob has duas cau&longs;as mouebuntur uelocius partes qu&aelig; &longs;unt <lb/>circa c, qu&agrave;m di&longs;tantes ab <expan abbr="eod&etilde;">eodem</expan>, tum quia in medio, <expan abbr="t&utilde;">tum</expan> quia tardius <lb/><expan abbr="mouen&ttilde;">mouentur</expan> motu elice. </s>
          <s id="id003582"><expan abbr="Declarat&utilde;">Declaratum</expan> e&longs;t. </s>
          <s id="id003583">n. </s>
          <s id="id003584">&longs;uperius quod unus motus in <lb/><expan abbr="eod&etilde;">eodem</expan> mobili <expan abbr="ali&utilde;">alium</expan> impedit &amp; retardat. </s>
          <s id="id003585">Cum ergo h k &longs;it in &longs;patio a b <lb/>l m &amp; aqua <expan abbr="rapia&ttilde;">rapiatur</expan> motu, dico ad d e mouebit ad d e, &amp; motu dico <lb/>qui uidetur circularis, nam mouetur motu eius &agrave; quo <expan abbr="&longs;u&longs;tine&ttilde;">&longs;u&longs;tinetur</expan>. </s>
          <s id="id003586">Mo&shy;<lb/>uetur etiam ad d e, quoniam pars illa e&longs;t humilior, nam &longs;emper de&shy;<lb/>&longs;cendit, omne <expan abbr="a&utilde;t">aut</expan> quod mouetur partim e&longs;t in termino, &agrave; quo, par&shy;<lb/>tim ad quem, ideo partim iam aqua illa cum de&longs;cendat humilior e&longs;t <lb/>locus, igitur nauis ad <expan abbr="ill&utilde;">illum</expan> locum feretur. </s>
          <s id="id003587">Tertio, quia latus k impelli<lb/>tur, in maiore circulo, ideo maiore impetu <lb/><figure id="id.015.01.231.1.jpg" xlink:href="015/01/231/1.jpg"/><lb/>qu&agrave;m h, quare <expan abbr="de&longs;c&etilde;det">de&longs;cendet</expan> &amp; circulo mouebi&shy;<lb/>tur, <expan abbr="n&atilde;">nam</expan> &longs;i h quie&longs;ceret <expan abbr="pal&atilde;">palam</expan> e&longs;t, &qring;d nauis circu<lb/>lariter <expan abbr="mouere&ttilde;">moueretur</expan>, &longs;ed h fungitur uice <expan abbr="quie&longs;c&etilde;&shy;tis">quie&longs;cen&shy;<lb/>tis</expan>, quia tardius <expan abbr="moue&ttilde;">mouetur</expan> <expan abbr="qu&atilde;">quam</expan> k, <expan abbr="igi&ttilde;">igitur</expan> k moue&shy;<lb/>bitur ad d e &amp; motu circulari aut participe <lb/>eius. </s>
          <s id="id003588">Quarta cau&longs;a e&longs;t, quoniam h cupit <expan abbr="de&shy;&longs;c&etilde;dere">de&shy;<lb/>&longs;cendere</expan>, ut graue. </s>
          <s id="id003589">ergo ferri, ubi minus impe<lb/>diatur &agrave; motu <expan abbr="uiol&etilde;to">uiolento</expan>, at minus <expan abbr="impedi&ttilde;">impeditur</expan> in <lb/>circulo, de qua a b, qua a b <expan abbr="c&utilde;">cum</expan> maioris &longs;it ambitus a qua in co ulterius <lb/><expan abbr="fer&ttilde;">fertur</expan> <expan abbr="qu&atilde;">quam</expan> in d e, ob h&aelig;c <expan abbr="o&itilde;a">oina</expan> &amp; in mari &amp; fluminibus ac lacubus <expan abbr="c&utilde;">cum</expan> na<lb/>ues fuerint in ambitu uorticis <expan abbr="i&atilde;">iam</expan> <expan abbr="rapiun&ttilde;">rapiuntur</expan> ad <expan abbr="ill&utilde;">illum</expan>, &amp; circulari motu: <lb/>isque motus e&longs;t <expan abbr="indici&utilde;">indicium</expan> &longs;ubmer&longs;ionis, <expan abbr="quoni&atilde;">quoniam</expan> indicat <expan abbr="aqu&atilde;">aquam</expan>, ibi prop&egrave; <lb/><expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> rect&agrave; uer&longs;us <expan abbr="c&etilde;tr&utilde;">centrum</expan>, &amp; ob id <expan abbr="prud&etilde;tes">prudentes</expan> naut&etail; magna ui uen<lb/>toru &amp; <expan abbr="remor&utilde;">remorum</expan> &longs;&etail;pe <expan abbr="&longs;eru&atilde;t">&longs;eruant</expan> &longs;e, pr&etail;o<expan abbr="ccup&atilde;tes">ccupantes</expan> <expan abbr="mot&utilde;">motum</expan> <expan abbr="elic&utilde;">elicum</expan> recto motu. <lb/></s>
          <s id="id003590">Cur <expan abbr="a&utilde;t">aut</expan> aqua <expan abbr="&qtilde;">quae</expan> e&longs;t in a, non potius <expan abbr="fera&ttilde;">feratur</expan> per obliquam lineam ad d <lb/>uel g, &lt;08&gt; ad e uel c inde ex illis ad d uel g, pr&aelig;&longs;ertim <expan abbr="c&utilde;">cum</expan> ad&longs;it breuior 


<pb pagenum="213" xlink:href="015/01/232.jpg"/>a e &amp; e d et a g breuior a e et c &lpar;ut docet Euclides&rpar; cau&longs;a e&longs;t quia aqua <lb/>qu&aelig; de&longs;cendit per e d &amp; c g maiore impetu de&longs;cendit qu&agrave;m per ad <lb/>uel a g ut demon&longs;tratum e&longs;t, ergo non poterit qu&aelig; e&longs;t in e d uel e g <lb/>loco dimoueri, nec cedere aqu&aelig; per obliquam lineam de&longs;cendenti.</s>
        </p>
        <p type="margin">
          <s id="id003591"><margin.target id="marg655"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id003592">Propo&longs;itio cente&longs;ima octuage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id003593">Cur homo &longs;edens quanto altius &longs;edet, &amp; quanto magis crura ad <lb/>femora &amp; femora ad pectus reclinata habet, facilius con&longs;urgat, cum <lb/>tamen h&aelig;c oppo&longs;ito modo inuicem &longs;e habeant, declarare.</s>
        </p>
        <p type="main">
          <s id="id003594">Huius &longs;ecundam partem Ari&longs;toteles in Mechanicis propo&longs;uit, <lb/><arrow.to.target n="marg656"/><lb/>&longs;ed neque &longs;ub adiecta dubitatione, &longs;edens n <lb/><figure id="id.015.01.232.1.jpg" xlink:href="015/01/232/1.jpg"/><lb/>altius a b pectus, b c femur, c d crus eiu&longs;&shy;<lb/>dem uel &aelig;qualis, pectus g h, femur h k, crus <lb/>k l longior b f quam h n facit, ut facilius &longs;ur&shy;<lb/>gat a b c d qu&agrave;m g h k l, &amp; tamen anguli <lb/>a b c &amp; b c d &longs;unt maiores g h k &amp; h k l, qui&shy;<lb/>nimo cum uolumus &longs;urgere, contrahimus c d &amp; k l prop&egrave; &amp; &egrave; re&shy;<lb/>gione a b, igitur patetratio &longs;ecundi, propior n e&longs;t c d ip&longs;i a b quanto <lb/>angulus a b c minor e&longs;t, cui &aelig;qualis e&longs;t b c d. </s>
          <s id="id003595">Cum ergo quanto pro <lb/>pior e&longs;t c d ip&longs;i a b eo facilius &longs;urgat, quoniam particeps magis di&shy;<lb/>&longs;po&longs;itionis per quam &longs;urgit, propior autem quo anguli &longs;unt acuti&shy;<lb/>ores, ideo facilius exurgit homo, quo contractiora &longs;unt crura, &amp; an<lb/>guli femorum ad crura &amp; pectus minora. </s>
          <s id="id003596">Huc usque Ari&longs;toteles &amp; <lb/>bene.</s>
        </p>
        <p type="margin">
          <s id="id003597"><margin.target id="marg656"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003598">Sed cur rur&longs;us contractiora dum &longs;unt crura, homo facilius exur&shy;<lb/>git? </s>
          <s id="id003599">Proponantur c f contracta ad perpendiculum, &amp; inclinetur b a <lb/>in o ut fiant b o &amp; f e aequidi&longs;tantes, ita enim commodius &longs;urgimus: <lb/>nec aliter qui &longs;unt imbecilliores: quia ergo b e&longs;t in directo f, ideo <lb/>mu&longs;culi femoris inferiores ob crus, &amp; &longs;uperiores ob pectus &longs;unt <lb/>magis ten&longs;i &amp; anteriores cruris itidem, ideo maiore ui trahunt par<lb/>ticulam. </s>
          <s id="id003600">Vnde manente fixo f &amp; capite etiam &amp; pectore grauitate <lb/>&longs;ua adiuuantibus, facilius homo exurgit quam ad latos angulos <lb/>cum contractio, ut dixi, mu&longs;culorum et inclinatio partium &longs;uperio&shy;<lb/>rum fiat maior.</s>
        </p>
        <p type="main">
          <s id="id003601">Rur&longs;us pro prima parte problematis, dico qu&ograve;d quanto altior <lb/>e&longs;t b f tanto facilius exurgit, nam &longs;upponatur angu&shy;<lb/><figure id="id.015.01.232.2.jpg" xlink:href="015/01/232/2.jpg"/><lb/>lus reflexionis a h e &aelig;qualis a h c, &amp; b c k &aelig;qualis h k f, <lb/>igitur cum b f &longs;it breuior b f, erit h k breuior b c &amp; f k, <lb/>f c. quare b c femur, &amp; f c crus erunt uiolentius exten&shy;<lb/>&longs;a qu&agrave;m in &longs;itu h k, k f ergo, mu&longs;culi facilius erigent <lb/>&longs;edentem altiore loco qu&agrave;m humiliore, quod erat de&shy;<lb/>mon&longs;trandum.</s>
        </p>
        <pb pagenum="214" xlink:href="015/01/233.jpg"/>
        <p type="main">
          <s id="id003602">Propo&longs;itio cente&longs;ima octuage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id003603">Si fuerit proportio prim&aelig; &amp; &longs;ecund&aelig; quantitatis ad tertiam, ut <lb/>prim&aelig; &amp; quart&aelig; ad quintam, fueritq&uacute;e quarta &longs;ecunda maior, erit <lb/>proportio quart&etail; ad quintam maior qu&agrave;m &longs;ecund&aelig; ad tertiam. <lb/></s>
          <s id="id003604">Quod &longs;i fuerit maior quart&etail; ad quintam, qu&agrave;m &longs;ecund&etail; ad tertiam, <lb/>nece&longs;&longs;e e&longs;t quartam &longs;ecunda e&longs;&longs;e maiorem.</s>
        </p>
        <p type="main">
          <s id="id003605">Sit proportio a &amp; b ad c, ut a &amp; d ad e, &longs;itque d maior b, dico maio&shy;<lb/><arrow.to.target n="marg657"/><lb/>rem e&longs;&longs;e <expan abbr="proportion&etilde;">proportionem</expan> d ad e qu&agrave;m b ad e, quod <lb/><figure id="id.015.01.233.1.jpg" xlink:href="015/01/233/1.jpg"/><lb/>&longs;i maior &longs;it proportio d ad c qu&agrave;m b ad c, dico d <lb/>e&longs;&longs;e maiorem b. </s>
          <s id="id003606">Quoniam enim e&longs;t d e&longs;t maior <lb/>b ad d e&longs;t maior a b per <expan abbr="commun&etilde;">communem</expan> animi &longs;enten&shy;<lb/>tiam, igitur cum &longs;it proportio a d ad e ut a b ad c, <lb/>erit e maior c, igitur minor proportio a ad e quam a ad c, at propor&shy;<lb/><arrow.to.target n="marg658"/><lb/>tio totius a d ad e e&longs;t &aelig;qualis proportioni a b ad e, igitur ex com&shy;<lb/><arrow.to.target n="marg659"/><lb/>muni animi &longs;ententia maior proportio d ad e, quam b ad c. </s>
          <s id="id003607">Rur&longs;us, <lb/>&longs;i maior e&longs;t proportio d ad e qu&agrave;m b ad c, igitur per communem <lb/>animi &longs;ententiam maior e&longs;t a ad e qu&agrave;m a ad c, igitur e maior qu&agrave;m <lb/><arrow.to.target n="marg660"/><lb/>c, &longs;ed d maiorem habet proportionem ad e qu&agrave;m b ad c, igitur d <lb/><arrow.to.target n="marg661"/><lb/>maiorem qu&agrave;m b.</s>
        </p>
        <p type="margin">
          <s id="id003608"><margin.target id="marg657"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003609"><margin.target id="marg658"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003610"><margin.target id="marg659"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003611"><margin.target id="marg660"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003612"><margin.target id="marg661"/>P<emph type="italics"/>er eadem <lb/>&longs;&aelig;pius repe&shy;<lb/>titam.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003613">Propo&longs;itio cente&longs;ima octuage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id003614">Si ei&longs;dem uiribus &amp; eadem proportione cum auxilio ponderis <lb/>tertij, quartum pondus moueatur quibus &longs;ecundum auxilio primi, <lb/>nece&longs;&longs;e e&longs;t quartum pondus tardi&ugrave;s &amp; maiore cum difficultate <lb/>moueri qu&agrave;m &longs;ecundum.<lb/><arrow.to.target n="marg662"/></s>
        </p>
        <p type="margin">
          <s id="id003615"><margin.target id="marg662"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003616">Maneat prior figura, &amp; &longs;int uires a qu&aelig; cum pondere b moue&shy;<lb/>ant c pondus, et cum d pondere eadem uires &longs;ub eadem proportio&shy;<lb/>ne moueant e, &longs;it autem pondus d maius qu&agrave;m b, dico e tardius &amp; <lb/>difficilius moueri qu&agrave;m c. </s>
          <s id="id003617">Nam ex pr&aelig;cedente e erit maius qu&agrave;m <lb/>c, &amp; proportio d ad e maior qu&agrave;m b ad c, &amp; proportio a ad e minor <lb/>qu&agrave;m ad c, tum ergo propter uectem magis pre&longs;&longs;um, tum quia d <lb/>non mouet e, ni&longs;i motum ab a, nece&longs;&longs;e e&longs;t ut tardius &amp; maiore cum <lb/>difficultate admoueat e quo a b mouet c. </s>
          <s id="id003618">Et ideo eo perueniri po&shy;<lb/>terit ab&longs;que dubio, ut a b moueat uelociter e &amp; a d, nullo mouente. <lb/></s>
          <s id="id003619">Quia hoc accidit c&ugrave;m d non mouet c ni&longs;i quia motum ab a.</s>
        </p>
        <p type="main">
          <s id="id003620">Propo&longs;itio cente&longs;ima octuage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id003621">Si uires aliqu&aelig; moueant cum ponderibus aliqua pondera, ut <lb/>compo&longs;ita proportio &longs;it eadem proportioni uirium &amp; duorum <lb/>ponderum mouentium aggregatum &aelig;quale duorum ponderum, <lb/>ubi maior fuerit partium in&aelig;qualitas, ibi erit maior difficultas.</s>
        </p>
        <p type="main">
          <s id="id003622">Sint uires a, &amp; aggregatum ponderum b c &amp; d e &aelig;qualia, &amp; a </s>
        </p>
        <p type="main">
          <s id="id003623"><arrow.to.target n="marg663"/><lb/>cum f &amp; g moueat b &amp; c &longs;ub proportionibus componentibus ean&shy;


<pb pagenum="215" xlink:href="015/01/234.jpg"/>dem proportionem, quam componunt proportiones a &amp; h mo&shy;<lb/>uendo d &amp; a, &amp; k mouendo e, &amp; &longs;it maior diffe&shy;<lb/><figure id="id.015.01.234.1.jpg" xlink:href="015/01/234/1.jpg"/><lb/>rentia ponderis e ad d qu&agrave;m c ad b, dico quod <lb/>maiore <expan abbr="c&utilde;">cum</expan> difficultate mouebuntur d &amp; e qu&agrave;m <lb/>b &amp; e. </s>
          <s id="id003624">Nam <expan abbr="c&utilde;">cum</expan> differentia e &amp; d &longs;it maior qu&agrave;m <lb/><arrow.to.target n="marg664"/><lb/>c &amp; b, &amp; d e &amp; b c &longs;int &aelig;qualia, erit e maius c, igi&shy;<lb/>tur e difficilius mouebitur ab a &amp; k qu&agrave;m c ab a <lb/>&amp; g. </s>
          <s id="id003625">Itidem quia e tanto maius e&longs;t c, quanto b <lb/>maius e&longs;t d, &amp; proportio a k ad e &amp; a h ad d, conficiunt proportio&shy;<lb/>nem a g ad c &amp; a f ad b, erit ut motus d e &longs;int tardiores &amp; difficilio&shy;<lb/>res motibus b c, per regulam dialecticam, nam difficultas motus e <lb/>&longs;upra difficultatem motus c, e&longs;t maior quam difficultas motus b <lb/>&longs;upra difficultatem motus d, igitur difficultas motus d &amp; e, maior <lb/>e&longs;t difficultate motus b &amp; e, quod erat demon&longs;trandum.</s>
        </p>
        <p type="margin">
          <s id="id003626"><margin.target id="marg663"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003627"><margin.target id="marg664"/>P<emph type="italics"/>er pr&aelig;ce&shy;<lb/>dentem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003628">Propo&longs;itio cente&longs;ima octuage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id003629">Si pondus minus ad longitudinem maiorem &longs;ub &aelig;quali pro&shy;<lb/>portione coaptetur, facilius deor&longs;um trahetur qu&agrave;m quod maius <lb/>e&longs;t &amp; propius.</s>
        </p>
        <p type="main">
          <s id="id003630">Sit &longs;itula aqu&aelig; f annexa tigno <lb/><figure id="id.015.01.234.2.jpg" xlink:href="015/01/234/2.jpg"/><lb/><arrow.to.target n="marg665"/><lb/>in e &amp; ad minuendum pondus <lb/>ad datur ex aduer&longs;o elongius &longs;eu <lb/>uincatur pondus a, dico quod <lb/><expan abbr="c&otilde;mo">commo</expan> dius erit qu&agrave;m &longs;i &etail;quale ad <lb/>grauitatem addatur b proprius <lb/>in e, nam quia b &etail;quiponderat in <lb/>d ut a in e, &amp; homo trahens ex e <lb/>plus pote&longs;t qu&agrave;m ex d, igitur fa&shy;<lb/>cilius trahet ex e quam d. </s>
          <s id="id003631">Et <expan abbr="quo&shy;ni&atilde;">quo&shy;<lb/>niam</expan> graue minus ponderat quan<lb/>to magis di&longs;tat &agrave; medio, licet mo&shy;<lb/>ueat magis, ergo inclinatum ad <lb/><arrow.to.target n="marg666"/><lb/>medium, cum ergo moueatur <lb/><arrow.to.target n="marg667"/><lb/>uelocius ex e quam d, &amp; &longs;emper <lb/><arrow.to.target n="marg668"/><lb/>uelocius de&longs;cendendo in com&shy;<lb/>paratione a g h, igitur &longs;emper <lb/>magis &amp; magis uelociter ex e <lb/>qu&agrave;m d ut &longs;it duplex incrementum &amp; comparatione c e ad c d &amp; <lb/>de&longs;cen&longs;us ad de&longs;cen&longs;um in utroque &amp; &longs;imiliter in reditu, quia facilius <lb/>impelletur &longs;ur&longs;um e qu&agrave;m d per primam rationem.</s>
        </p>
        <p type="margin">
          <s id="id003632"><margin.target id="marg665"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003633"><margin.target id="marg666"/>P<emph type="italics"/>er<emph.end type="italics"/> 45.</s>
        </p>
        <p type="margin">
          <s id="id003634"><margin.target id="marg667"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003635"><margin.target id="marg668"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 109.</s>
        </p>
        <p type="main">
          <s id="id003636">Propo&longs;itio cente&longs;ima nonage&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id003637">Si fuerit primum graue minus &longs;ecundo, &amp; &longs;ecundum minus ter&shy;<lb/>tio, proportio autem primi ad &longs;ecundum multo maior qu&agrave;m &longs;ecun 


<pb pagenum="216" xlink:href="015/01/235.jpg"/>di ad tertium, po&longs;sibile erit propo&longs;itis uiribus ei&longs;dem addere pon&shy;<lb/>dus &longs;ecundo, ut ip&longs;um &amp; <expan abbr="terti&utilde;">tertium</expan> moueantur facilius ab ei&longs;dem uiri&shy;<lb/>bus, &amp; primo uel &longs;ecundo quam antea.</s>
        </p>
        <p type="main">
          <s id="id003638">Sit a <expan abbr="p&otilde;dus">pondus</expan> minus, c maius, proportio a ad b multo maior qu&agrave;m <lb/>b ad c, uires d, &amp; d cum a moueat b &amp; cum b mo<lb/><figure id="id.015.01.235.1.jpg" xlink:href="015/01/235/1.jpg"/><lb/>ueat c, dico qu&ograve;d poterit addi pondus ad b ut d <lb/>cum a moueat b, &amp; d cum b moueat e maiore fa&shy;<lb/>cilitate componendo proportiones quam antea: Cum enim fuerit <lb/>proportio d b ad c minima, <expan abbr="qu&atilde;tumcunque">quantumcunque</expan> moueatur b facil&egrave; ab a d <lb/><arrow.to.target n="marg669"/><lb/>plus refert difficultas c moti a b d: igitur cum addito pondere di&shy;<lb/><arrow.to.target n="marg670"/><lb/>midio quod a &longs;uperat b omnino uincat a d ip&longs;um b, cum eo quod <lb/>additum e&longs;t, &amp; tanto minor &longs;it difficultas motus c a b d cum ponde<lb/>re addito, &longs;equitur ut minor &longs;it difficultas motus b cum pondere <lb/>addito a b a d, &amp; motus c &agrave; b cum pondere addito &amp; d qu&agrave;m b &amp; e <lb/>ab a &amp; b cum uiribus d.<lb/><arrow.to.target n="marg671"/></s>
        </p>
        <p type="margin">
          <s id="id003639"><margin.target id="marg669"/>P<emph type="italics"/>er<emph.end type="italics"/> 188.</s>
        </p>
        <p type="margin">
          <s id="id003640"><margin.target id="marg670"/>P<emph type="italics"/>er<emph.end type="italics"/> 187.</s>
        </p>
        <p type="margin">
          <s id="id003641"><margin.target id="marg671"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 28</s>
        </p>
        <p type="main">
          <s id="id003642">Ex hoc patet quod qui interpretati &longs;unt Ari&longs;totelem, cum non <lb/>po&longs;sit nec intelligi nec demon&longs;trari, fucum fecerunt legentibus: ni&shy;<lb/>hilominus hoc illis debemus, quod &longs;i Phrynis non fui&longs;&longs;et, Timo&shy;<lb/>theus non fui&longs;&longs;et, nam ni&longs;i illi quod &longs;ciuerunt protuli&longs;&longs;ent in medi&shy;<lb/>um, ego for&longs;an aut illa non intellexi&longs;&longs;em aut neglexi&longs;&longs;em. </s>
          <s id="id003643">Itaque &amp; re&shy;<lb/>liquas habes &agrave; nobis expo&longs;itas licet non ade&ograve; diligenter, &amp; mo&shy;<lb/>dum huiu&longs;modi exponendi. </s>
          <s id="id003644">Subij ciemus autem et hanc, ut obiect&etail; <lb/>qu&aelig;&longs;tioni, quantum nerui &longs;it &lpar;&longs;i p&oelig;nitus quis res &longs;equi uelit, non <lb/>addictus nimis authoritati ueterum ut pedem figere uelit, ubi illi <lb/>res uix tactas reliquerunt&rpar; intelligamus.</s>
        </p>
        <p type="head">
          <s id="id003645">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id003646">Vocatur autem h&aelig;c proportio auxiliaris. </s>
          <s id="id003647">Cunque fuerit &etail;qualis d <lb/>&amp; a ad b ut d &amp; b ad e, dicetur auxiliaris &aelig;qualis.</s>
        </p>
        <p type="main">
          <s id="id003648">Propo&longs;itio cente&longs;ima nonage&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id003649">Cum fuerint duo pondera &amp; uires duxeri&longs;que aggregatum ex ui&shy;<lb/>ribus &amp; minore pondere in maius, addiderisque in&longs;uper <expan abbr="qu&atilde;tum">quantum</expan> e&longs;t <lb/>productum dimidij uirium in &longs;e latus aggregati detracto dimidio <lb/>uirium, dicetur pondus auxiliare &aelig;qualis proportionis.</s>
        </p>
        <p type="main">
          <s id="id003650"><arrow.to.target n="marg672"/></s>
        </p>
        <p type="margin">
          <s id="id003651"><margin.target id="marg672"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003652">Sint pondera b minus, c maius, &amp; ducatur aggre&shy;<lb/><figure id="id.015.01.235.2.jpg" xlink:href="015/01/235/2.jpg"/><lb/>gatum ex a uiribus &amp; b minore pondere in e, &amp; ei <lb/>addatur quadratum dimidij a, dico quod radix &longs;eu <lb/>latus huius detracto dimidio a e&longs;t pondus auxiliare <lb/>&aelig;quale, &longs;it productum a b in e &longs;uperficies &amp; quadra&shy;<lb/>tum dimidij a &longs;it e, ita quod tota d e &longs;it &longs;uperficies <lb/>quadrata, cuius latus &longs;it f g: f h autem dimidium a di&shy;<lb/>co h g e&longs;&longs;e pondus auxiliare &aelig;quale. </s>
          <s id="id003653">Quia enim f g 


<pb pagenum="217" xlink:href="015/01/236.jpg"/>quadratum e&longs;t &aelig;quale quadratis g h, h f &amp; duplo g h in h f, &amp; qua&shy;</s>
        </p>
        <p type="main">
          <s id="id003654"><arrow.to.target n="marg673"/><lb/>dratum fh e&longs;t &etail;quale e &longs;uperficiei, erit quadratum h g minus &longs;uper&shy;<lb/>ficie d in duplo g h in h f, quare productum a b in c erit &etail;quale qua&shy;<lb/>drato g h in &longs;e &amp; a, nam duplo g h in h f &amp; iam duplum g h in h f e&longs;t <lb/>&etail;quale producto g h in a, quia a e&longs;t duplum h f, igitur qualis e&longs;t pro <lb/><arrow.to.target n="marg674"/><lb/>portio a b ad g h, talis g h &amp; a ad c, igitur per definitionem datam <lb/>g h &amp; quantitas grauitatis auxiliaris &aelig;quale.</s>
        </p>
        <p type="margin">
          <s id="id003655"><margin.target id="marg673"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>primi.<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003656"><margin.target id="marg674"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003657">Ex hoc manife&longs;tum e&longs;t, quod &longs;i fuerit datum pondus tertium au&shy;<lb/><arrow.to.target n="marg675"/><lb/>xiliare, quod &longs;ciemus quantum addendum uel detrahendum ut fi&shy;<lb/>at pondus auxiliare &aelig;quale, nam inuenta g h &longs;i fuerit k maior adde&shy;<lb/>mus quod deficit, &amp; &longs;i minor qu&agrave;m k detrahemus ex k quod e&longs;t <lb/>&longs;uperfluum.</s>
        </p>
        <p type="margin">
          <s id="id003658"><margin.target id="marg675"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003659">Et rur&longs;us inuenta g h ut perficiamus pondus &etail;quale, augebimus <lb/><arrow.to.target n="marg676"/><lb/>aliquanti&longs;per, ut fiat &aelig;qualis ad unguem difficultas in motu: iuxta <lb/><arrow.to.target n="marg677"/><lb/>doctrinam &longs;uperi&ugrave;s datam.</s>
        </p>
        <p type="margin">
          <s id="id003660"><margin.target id="marg676"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="margin">
          <s id="id003661"><margin.target id="marg677"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 187.</s>
        </p>
        <p type="main">
          <s id="id003662">Propo&longs;itio cente&longs;ima nonage&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id003663">Si ex medio diametri linea ad perpendiculum erigatur ad circu&shy;<lb/>li peripheriam: ex eo puncto <expan abbr="aut&etilde;">autem</expan> quotlibet line&aelig; ducantur &longs;eu in&shy;<lb/>tus ad circumferentiam u&longs;que, &longs;eu extra ad diametrum, erit proportio <lb/>totius line&aelig; ad totam, uelut mutu&ograve; partis ad partem.</s>
        </p>
        <p type="main">
          <s id="id003664">Ex media diametro a c. 1. <expan abbr="c&etilde;tro">centro</expan> b, ducatur ad perpendiculum b d, <lb/><arrow.to.target n="marg678"/><lb/>&amp; ex d line&aelig; 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/>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, &aelig;&shy;<lb/>quale e&longs;t ei quod ex e c in e a, quod uer&ograve; ex e c in e a cum quadrato <lb/><arrow.to.target n="marg679"/><lb/>b d &longs;eu b a &etail;quale e&longs;t quadrato b e, igitur ex <lb/><figure id="id.015.01.236.1.jpg" xlink:href="015/01/236/1.jpg"/><lb/>e d in e f cum quadrato d b &aelig;quale qua&shy;<lb/><arrow.to.target n="marg680"/><lb/>drato b e, ex d e igitur in e f cum quadratis <lb/><arrow.to.target n="marg681"/><lb/>d b &amp; b a &aelig;quale quadrato d e. </s>
          <s id="id003666">Quadratis <lb/><arrow.to.target n="marg682"/><lb/>autem a b &amp; b d &aelig;quale quadratum d e: <lb/><arrow.to.target n="marg683"/><lb/>igitur ex d e in e f cum quadrato d a &aelig;qua&shy;<lb/><arrow.to.target n="marg684"/><lb/>le quadrato d e. </s>
          <s id="id003667">At quadratum d e &aelig;quale <lb/>e&longs;t his qu&aelig; ex d e in e f, &amp; f d igitur detra&shy;<lb/><arrow.to.target n="marg685"/><lb/>cto communi ex d e in e f, erit quadratum d <lb/>e &aelig;quale ei quod ex d e in d f, igitur d e ad <lb/><arrow.to.target n="marg686"/><lb/>d a, ut d a ad d f. </s>
          <s id="id003668">Similiter quod fit ex h d in <lb/><arrow.to.target n="marg687"/><lb/>d g, &aelig;quale e&longs;t ei quod fit ex h g in g d cum <lb/>quadrato d g, at quod fit ex h g in g d e&longs;t &aelig;quale ei quod fit ex c g in <lb/>g a, erit quod fit ex c g in g a cum quadrato d g &etail;quale ei quod fit ex <lb/>d h in d g. </s>
          <s id="id003669">Quadratum autem d g e&longs;t &aelig;quale quadratis d b, b g igi&shy;<lb/><arrow.to.target n="marg688"/><lb/>tur d h in d g &aelig;quale e&longs;t ei quod fit ex g a in c g cum quadratis b d <lb/>b g, at quod fit ex a g in g c cum quadrato b g e&longs;t &aelig;quale quadrato 


<pb pagenum="218" xlink:href="015/01/237.jpg"/>b a igitur quod fit ex d h in d g e&longs;t &etail;quale quadratis d b, b a qu&etail; &longs;unt <lb/>&etail;qualia quadrato a d, igitur quadratum a d e&longs;t &etail;quale ei quod fit ex <lb/><arrow.to.target n="marg689"/><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&shy;<lb/><arrow.to.target n="marg690"/><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/>ut d g ad d f.</s>
        </p>
        <p type="margin">
          <s id="id003671"><margin.target id="marg678"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003672"><margin.target id="marg679"/>P<emph type="italics"/>er<emph.end type="italics"/> 36. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003673"><margin.target id="marg680"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003674"><margin.target id="marg681"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003675"><margin.target id="marg682"/>P<emph type="italics"/>er tandem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003676"><margin.target id="marg683"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003677"><margin.target id="marg684"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003678"><margin.target id="marg685"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003679"><margin.target id="marg686"/>P<emph type="italics"/>er<emph.end type="italics"/> 35. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003680"><margin.target id="marg687"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003681"><margin.target id="marg688"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003682"><margin.target id="marg689"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003683"><margin.target id="marg690"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&amp;<emph.end type="italics"/><lb/>17. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003684">Vnde manife&longs;tum e&longs;t omnes has lineas in &longs;uam interiorem par&shy;<lb/><arrow.to.target n="marg691"/><lb/>tem ductas rectangulum con&longs;tituere &etail;quale quadrato quod circu&shy;<lb/>lo eidem in&longs;cribitur.</s>
        </p>
        <p type="margin">
          <s id="id003685"><margin.target id="marg691"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003686">Propo&longs;itio cente&longs;ima nonage&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id003687">Rationem ponderis triplicem explicare.<lb/><arrow.to.target n="marg692"/></s>
        </p>
        <p type="margin">
          <s id="id003688"><margin.target id="marg692"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003689">Superius declaratum e&longs;t qu&ograve;d id quod quie&longs;cit, habet motum </s>
        </p>
        <p type="main">
          <s id="id003690"><arrow.to.target n="marg693"/><lb/>occultum. </s>
          <s id="id003691">Qu&aelig;rit autem Ari&longs;toteles cur &longs;ecuris pondere pre&longs;&longs;a <expan abbr="n&otilde;">non</expan> <lb/>diuidit lignum, minore uer&ograve; &longs;ed moto &longs;ed modo diuidit? </s>
          <s id="id003692">Diximus <lb/><arrow.to.target n="marg694"/><lb/>motum ine&longs;&longs;e qui perpetuo augetur, indicium e&longs;t, quod &longs;i ex a de&shy;<lb/>&longs;cendat, <expan abbr="maior&etilde;">maiorem</expan> facit ictum, quoniam plurimus a&euml;r coadiuuat, ex d <lb/>autem occultum <expan abbr="&longs;ol&utilde;">&longs;olum</expan>, et eum qui fit ratione grauitatis, me&shy;<lb/><figure id="id.015.01.237.1.jpg" xlink:href="015/01/237/1.jpg"/><lb/>dium ex medijs locis. </s>
          <s id="id003693">Omitto modo de motu aucto per <lb/>uim humanam, de quo uidetur qu&aelig;rere Ari&longs;toteles, quili&shy;<lb/>bet enim a&euml;r addit &longs;uper motum iam acqui&longs;itum &amp; fit hoc <lb/>argumentum centies ac millies maius, quoniam m e&longs;t qui <lb/>diuidit, pondus autem non ponetrat. </s>
          <s id="id003694">Sicut ergo cuneus <lb/>magis diuidit lignum quam claua, ita quod mouetur &longs;ine <lb/>proportione &lpar;ut ita dicam&rpar; non &longs;olum ob <expan abbr="impet&utilde;">impetum</expan> nece&longs;&longs;e <lb/>e&longs;t ut uehementer diuidat lignum aut lapidem &longs;ubiectum, <lb/>&amp; non in proportione di&longs;tanti&etail;. </s>
          <s id="id003695">Sicut &longs;i pondus in forma <lb/>&longs;ecuris, &amp; ip&longs;a &longs;ecuris diuidit longe magis ligna quam cla&shy;<lb/>uis maioris ponderis &amp; maiore ui de&longs;cendens: ita pondus motum <lb/>quam immotum. </s>
          <s id="id003696">Hoc ade&ograve; per&longs;picuam habet cau&longs;&longs;am, ut quanto <lb/>plura uerba adderentur, eo redderetur res difficilior. </s>
          <s id="id003697">Habet ergo <lb/>propriam &longs;olum grauitatem &amp; motum occultum. </s>
          <s id="id003698">C&etail;terum e&longs;t ter&shy;<lb/>tium, genus <expan abbr="medi&utilde;">medium</expan>, cum idem pondus appen&longs;um e&longs;t, ue&shy;<lb/><figure id="id.015.01.237.2.jpg" xlink:href="015/01/237/2.jpg"/><lb/>lut f quod dico e&longs;&longs;e maius &amp; minus occultum quam &longs;i ia&shy;<lb/>ceret in plano, quoniam &longs;icut tuber &amp; cauitas in qua iacet <lb/>&longs;imul tempore &longs;unt, natura tamen tuber e&longs;t prius cauitate, <lb/>ita pondus appen&longs;um prius e&longs;t, contr&agrave; nixum uinculi na&shy;<lb/>tura &amp; quodammodo tempore, &longs;emper enim grauat, &amp; illud &longs;em&shy;<lb/>per re&longs;i&longs;tit &longs;upra illius grauitatem: Sed pondus quod e&longs;t in plano <lb/>occultam omnino habet actionem bifariamque di&longs;tinguitur a pon&shy;<lb/>dere &longs;u&longs;pen&longs;o: Primum qu&ograve;d pondus quod quie&longs;cit &amp; contra in&shy;<lb/>tendi principium &longs;imul non &longs;olum &longs;unt tempore &longs;ed etiam natu&shy;<lb/>ra. </s>
          <s id="id003699">Sed in appen&longs;o, ut dixi, pondus prius grauat quam uincu&shy;


<pb pagenum="219" xlink:href="015/01/238.jpg"/>lum contranitatur. </s>
          <s id="id003700">Secund&ograve;, quia pondus in plano non inchoat <lb/>motum &longs;ed pendens inchoat, ideo qu&ograve;d e&longs;t in plano habet pror&shy;<lb/>&longs;us occultum, quod pendet non: &amp; &longs;i &longs;it lignum eiu&longs;dem molis &amp; <lb/>duritiei cui appen&longs;um &longs;it f &amp; cui in&longs;ideat, magis atteretur id cui ap&shy;<lb/><figure id="id.015.01.238.1.jpg" xlink:href="015/01/238/1.jpg"/><lb/>penditur, &amp; prius&lt;08&gt; cui in&longs;idet. </s>
          <s id="id003701">C&aelig;ter&uacute;m quod <lb/>ad grauitatem attinet &aelig;qualia &longs;unt, nam a&euml;r in <lb/>utroque pellit deor&longs;um, ac magis quod quie&longs;cit <lb/>in plano: &longs;olum enim planum re&longs;i&longs;tit, in pendu&shy;<lb/>lo onere etiam aer &longs;uppo&longs;itus, quo fit ut quod <lb/>pendet, minus graue &longs;it. </s>
          <s id="id003702">Sed &aelig;qualia uidentur.</s>
        </p>
        <p type="margin">
          <s id="id003703"><margin.target id="marg693"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 26. <lb/><emph type="italics"/>&amp;<emph.end type="italics"/> 38.</s>
        </p>
        <p type="margin">
          <s id="id003704"><margin.target id="marg694"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 19. <lb/>M<emph type="italics"/>echan.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003705">Propo&longs;itio cente&longs;ima nonage&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id003706">Proportionem ponderis longioris in medio &longs;u&longs;pen&longs;i ad breuius. <lb/></s>
          <s id="id003707">illi &aelig;quale &amp; in medio &longs;u&longs;pen&longs;um, declarare.<lb/><arrow.to.target n="marg695"/></s>
        </p>
        <p type="margin">
          <s id="id003708"><margin.target id="marg695"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 27.</s>
        </p>
        <p type="main">
          <s id="id003709">Hanc generaliter propo&longs;uit Ari&longs;toteles in Mechanicis, <expan abbr="o&longs;tendi&ttilde;">o&longs;tenditur</expan> <lb/><expan abbr="e&mtilde;">emm</expan> quod &longs;i a b in e, &amp; d e in f &aelig;qualia <lb/>pondera in medio <expan abbr="&longs;u&longs;pend&atilde;tur">&longs;u&longs;pendantur</expan>, quod <lb/><figure id="id.015.01.238.2.jpg" xlink:href="015/01/238/2.jpg"/><lb/>grauius erit a b quam d e. </s>
          <s id="id003710">Et hoc e&longs;t <lb/>certum quia a &amp; b extrema plus di&shy;<lb/>&longs;tant ab hypomochlio. </s>
          <s id="id003711">Sit igitur g h re&longs;ecta &aelig;qualis hic cinde d e, <lb/>pondus e&longs;t &aelig;quale a b, erit g h minus pondere d e in k, igitur per <lb/>communem animi &longs;ententiam k e&longs;t &aelig;quale uer&ograve; ponderi a g &amp; h b, <lb/>igitur cum a g &amp; h b plus ponderent in &longs;itu &longs;uo quam in &longs;itu d e, <lb/>patet propo&longs;itum quoad Ari&longs;totelem attinet, &longs;cilicet quod a b e&longs;t <lb/>grauior d e.</s>
        </p>
        <p type="main">
          <s id="id003712">Vt mod&ograve; o&longs;tendam proportionem, erit proportio h b ad g h ut <lb/>ponderis h b ad totum <expan abbr="p&otilde;dus">pondus</expan> g b, eadem ratione a g ad g h ut pon&shy;</s>
        </p>
        <p type="main">
          <s id="id003713"><arrow.to.target n="marg696"/><lb/>deris a g ad totum a h, a h autem e&longs;t &aelig;qualis g b &amp; a g &aelig;qualis h b <lb/>ex communi animi <expan abbr="&longs;ent&etilde;tia">&longs;ententia</expan>, &amp; pondus a h &etail;quale ponderi b g, quia <lb/>&longs;unt &aelig;quales &amp; in eodem &longs;itu: igitur a g, h b ad g h, ut ponderum <lb/>a g h b ad pondus g b. </s>
          <s id="id003714">Et ita patet quod quanto longior e&longs;t a b in <lb/>comparatione ad d e, tanto a g &amp; h b in comparatione ad g h, igitur <lb/>tanto maior proportio ponderum a g h b ad pondus a h. </s>
          <s id="id003715">rur&longs;us e&longs;t <lb/>tanto maius quanto a b e&longs;t longior per <expan abbr="dem&otilde;&longs;trata">demon&longs;trata</expan> in prima parte, <lb/>igitur multo maius e&longs;t pondus a g h b, quanto longior a b in com&shy;<lb/>paratione ad d e.</s>
        </p>
        <p type="margin">
          <s id="id003716"><margin.target id="marg696"/>P<emph type="italics"/>er<emph.end type="italics"/> 92. <emph type="italics"/>hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003717"><expan abbr="Exempl&utilde;">Exemplum</expan> &longs;it ponderis a b 12 ponderis <expan abbr="l&otilde;gitudinis">longitudinis</expan> <expan abbr="ped&utilde;">pedum</expan> quatuor, <lb/>d e pondus 12 longitudinis <expan abbr="duor&utilde;">duorum</expan> pedum, <expan abbr="eruntigi&ttilde;">erunt igitur</expan> a g, g e, c h, h b <lb/>unius pedis &longs;ingul&etail;. </s>
          <s id="id003718">Et quia a g &amp; b h &longs;unt <expan abbr="dimidi&utilde;">dimidium</expan> g h erunt amb&aelig; <lb/>pariter &aelig;quales g h &amp; ideo pondus a g h b &aelig;qualia g b ponderi, <lb/>&longs;ed pondus g b e&longs;t librarum nouem, quia g b e&longs;t dodratus a b, igi&shy;<lb/>tur tota a b e&longs;t ponderis quindecim, nam g h e&longs;t ponderis &longs;ex, e&longs;t er&shy;<lb/>go pondus a b quadrante maius d e.</s>
        </p>
        <pb pagenum="220" xlink:href="015/01/239.jpg"/>
        <p type="main">
          <s id="id003719">Propo&longs;itio cente&longs;ima nonage&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id003720">Si lectus fiat dupla longitudine ad latitudinem melius &longs;uffulcie&shy;<lb/>tur re&longs;tibus ex medio ad angulos, &amp; eis &aelig;quidi&longs;tantibus quam &longs;e&shy;<lb/>cundum longitudinem &amp; latitudinem.<lb/><arrow.to.target n="marg697"/></s>
        </p>
        <p type="margin">
          <s id="id003721"><margin.target id="marg697"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003722">H&etail;c proponitur &agrave; Philo&longs;opho in mechanicis, &amp; dico quod &longs;i a b </s>
        </p>
        <p type="main">
          <s id="id003723"><arrow.to.target n="marg698"/><lb/>&longs;it dupla a c, &amp; <foreign lang="greek">a b a g</foreign> dupla, &amp; diuidantur a b a c &amp; <foreign lang="greek">a b a g</foreign> in quotuis <lb/>partes &etail;quales inuicem, nam &longs;upponitur a b &etail;qualis <foreign lang="greek">a b</foreign> &amp; a c &aelig;qua&shy;<lb/>lis <foreign lang="greek">a g</foreign>, &amp; ducantur rect&aelig; line&aelig; decu&longs;&longs;atim &amp; ad rectos angulos, &amp; <lb/><expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> id &longs;tatuantur re&longs;tes, quod decu&longs;&longs;a&shy;<lb/><figure id="id.015.01.239.1.jpg" xlink:href="015/01/239/1.jpg"/><lb/>tim po&longs;it&aelig; utiliores <expan abbr="er&utilde;t">erunt</expan>, omitto quod de&shy;<lb/>centius ob &longs;patiorum minorem differenti&shy;<lb/>am. </s>
          <s id="id003724">Adducam &longs;ol&ugrave;m tres Philo&longs;ophi ratio&shy;<lb/>nes: prima, quoniam ligna non ade&ograve; facil&egrave; <lb/>finduntur nec incuruantur tran&longs;uer&longs;im tra&shy;<lb/>cta, ut recta &amp; &longs;ecundum longitudinem, Et <lb/>ide&ograve; long&egrave; plus durabit <foreign lang="greek">a b g d</foreign> <expan abbr="qu&atilde;">quam</expan> a b c d, <lb/>&amp; cum &longs;pondis rectoribus, &amp; ide&ograve; etiam <lb/>cum re&longs;tibus magis intentis: &amp; erit firmior <lb/>&amp; pulchrior. </s>
          <s id="id003725">Secunda ratio e&longs;t, quod cum <lb/>re&longs;tes in &longs;ecunda con&longs;titutione &aelig;quales inuicem &longs;int, in prima qu&aelig; <lb/>&longs;ecundum latitudinem dupl&etail;, qu&etail; longiores erunt magis laxabun&shy;<lb/>tur tran&longs;uer&longs;alibus, &amp; ita turpiores &amp; incommod&aelig; breui redden&shy;<lb/>tur, &amp; in &longs;ecunda con&longs;titutione &etail;qualiter &longs;u&longs;tinebunt pondus &amp; re&shy;<lb/>uolutionem cubantis, tum ob &aelig;qualitatem longitudinis inter &longs;e, <lb/>tum ob &longs;itum &longs;imilem inter &longs;e, tum ad humanum decubitum <expan abbr="di&longs;si&shy;mil&etilde;">di&longs;si&shy;<lb/>milem</expan>, nam &lpar;ut o&longs;ten&longs;um e&longs;t&rpar; in pr&aelig;cedenti magis grauat pondus in <lb/>extremis quam in medio, &amp; magis laxantur ob id qu&aelig; &longs;unt &longs;ecun&shy;<lb/>dum eundem situm. </s>
          <s id="id003726">Et hanc cau&longs;&longs;am expo&longs;itores non intellexe&shy;<lb/>runt multi, multo minus tertiam, in qua faciunt demon&longs;trationem <lb/>Geometricam &amp; computantem numeris. </s>
          <s id="id003727">Deinde non animaduer<lb/>tunt quod in &longs;ecunda figura a&longs;&longs;umunt quinque lineas, cum in prima <lb/>tantum a&longs;&longs;ump&longs;i&longs;&longs;ent quatuor. </s>
          <s id="id003728">Peius omnibus e&longs;t quod demon&shy;<lb/>&longs;tratio h&aelig;c cum de tran&longs;uer&longs;is ad magis tran&longs;uer&longs;as lineas &longs;it non <lb/>e&longs;t ad propo&longs;itum Ari&longs;totelis, qui in duabus primis rationibus <lb/>tran&longs;uer&longs;as comparauit his, qu&aelig; &agrave; latere ad latus &amp; &agrave; capite ad ca&shy;<lb/>put deducuntur, ita ubi trifariam decepti &longs;unt, ibi maxim&egrave; glori&shy;<lb/>antur. </s>
          <s id="id003729">Mi&longs;erum nunc philo&longs;ophandi genus: uoluntque &longs;upercilium <lb/>e&longs;&longs;e loco doctrin&aelig;. </s>
          <s id="id003730">Sint igitur line&aelig; duct&aelig; ut uides, dico omnes <lb/>pariter acceptas in prima figura, e&longs;&longs;e longiores omnibus pariter ac&shy;<lb/><arrow.to.target n="marg699"/><lb/>ceptis in &longs;ecunda figura, quod intendit <expan abbr="dem&otilde;">demon</expan>&longs;trare Ari&longs;toteles. </s>
          <s id="id003731">O&shy;<lb/>&longs;ten&longs;o ergo de duabus, idem &longs;uppo&longs;ito numero equali de omnibus 


<pb pagenum="221" xlink:href="015/01/240.jpg"/>con&longs;tat. </s>
          <s id="id003732">Demon&longs;trandum e&longs;t ergo a b &amp; g q maiores e&longs;&longs;e <foreign lang="greek">az</foreign> &amp; <foreign lang="greek">zb</foreign>, <lb/>nam <foreign lang="greek">ag</foreign> &amp; <foreign lang="greek">gz</foreign> &longs;unt &aelig;quales &amp; <foreign lang="greek">zd</foreign> &amp; <foreign lang="greek">db</foreign> ex &longs;uppo&longs;ito, quare <foreign lang="greek">az</foreign> &amp; <foreign lang="greek">zb</foreign><lb/>&aelig;quales &longs;unt pote&longs;tate quadrato, <foreign lang="greek">ab</foreign> igitur amb&aelig; iunct&aelig; line&aelig; me&shy;<lb/><arrow.to.target n="marg700"/><lb/>di&aelig; inter duplum <foreign lang="greek">ab</foreign> &amp; ip&longs;am <foreign lang="greek">ab</foreign>, quadratum enim <foreign lang="greek">az</foreign> &amp; <foreign lang="greek">zb</foreign> coniun&shy;<lb/>ctarum e&longs;t duplum quadratis uniu&longs;cuiusque earum pariter acceptis, <lb/><arrow.to.target n="marg701"/><lb/>uelut &amp; quadratum medi&aelig; inter duplum <foreign lang="greek">ab</foreign> &amp; ip&longs;am <foreign lang="greek">ab</foreign>, at quadra&shy;<lb/>tum coniunct&aelig; ex a b &amp; a c e&longs;t &aelig;quale duplo quadrati a b cum qua<lb/><arrow.to.target n="marg702"/><lb/>drato a c, igitur &longs;uperat duplum quadrati <foreign lang="greek">a b</foreign> in quadrato a c, &longs;ed <lb/><arrow.to.target n="marg703"/><lb/>quod pote&longs;t in duplum quadrati <foreign lang="greek">ab</foreign> e&longs;t aggregatum <foreign lang="greek">az</foreign> &amp; <foreign lang="greek">zb</foreign>, igitur <lb/>a b &amp; a d &longs;unt longiores iunct&aelig; <foreign lang="greek">az</foreign> &amp; <foreign lang="greek">zb</foreign> quia po&longs;&longs;unt eo plus quan&shy;<lb/><arrow.to.target n="marg704"/><lb/>tum e&longs;t quadratum a c.</s>
        </p>
        <p type="margin">
          <s id="id003733"><margin.target id="marg698"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 25.</s>
        </p>
        <p type="margin">
          <s id="id003734"><margin.target id="marg699"/>P<emph type="italics"/>er<emph.end type="italics"/> 34. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003735"><margin.target id="marg700"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri&shy;<lb/>mi &amp;<emph.end type="italics"/> 4. <emph type="italics"/>&longs;e&shy;<lb/>cundi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003736"><margin.target id="marg701"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003737"><margin.target id="marg702"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003738"><margin.target id="marg703"/>P<emph type="italics"/>er eandem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003739"><margin.target id="marg704"/>P<emph type="italics"/>er eandem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003740">Propo&longs;itio cente&longs;ima nonage&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id003741">Si duo circuli &longs;uper eodem centro eodem motu transferuntur, <lb/>&aelig;quale &longs;patium &longs;uperant.</s>
        </p>
        <p type="main">
          <s id="id003742">Sint duo circuli a b, c d &longs;uper eodem centro e qui transferantur <lb/><figure id="id.015.01.240.1.jpg" xlink:href="015/01/240/1.jpg"/><lb/><arrow.to.target n="marg705"/><lb/>&longs;uper axe per <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c g dum re&longs;oluitur c d, <lb/>tum ergo a erit in f, quia c d contingit pla&shy;<lb/>num c g, igitur e c e&longs;t ad <expan abbr="perp&etilde;diculum">perpendiculum</expan> c g, <lb/><arrow.to.target n="marg706"/><lb/>ergo punctum a e&longs;t in f &amp; a f &aelig;qualis c g, <lb/><arrow.to.target n="marg707"/><lb/>igitur a b circulus &longs;olum reuolutus e&longs;t &longs;e&shy;<lb/>mel, &amp; tantum perambulauit &longs;patij quan&shy;<lb/>tum e d &amp; &aelig;quali uelo citate, c&ugrave;m tamen &longs;eor&longs;um &longs;it proportio &longs;pa&shy;<lb/>tij ad <expan abbr="&longs;pati&utilde;">&longs;patium</expan> ut circuli ad circulum. </s>
          <s id="id003743">H&aelig;c e&longs;t &longs;ubtili&longs;sima <expan abbr="qu&aelig;&longs;tion&utilde;">qu&aelig;&longs;tionum</expan> <lb/><arrow.to.target n="marg708"/><lb/><expan abbr="propo&longs;itar&utilde;">propo&longs;itarum</expan> ab Ari&longs;totele in mechanicis, quam &longs;ic quidam &longs;oluunt. <lb/></s>
          <s id="id003744">Supponunt duo: <expan abbr="prim&utilde;">primum</expan> &longs;i quid ab aliquo mouetur nihil conferens <lb/><figure id="id.015.01.240.2.jpg" xlink:href="015/01/240/2.jpg"/><lb/>ad illum motum, <lb/>ex &longs;e ip&longs;o per tan<lb/>tum mouebitur <lb/><expan abbr="&longs;pati&utilde;">&longs;patium</expan>, per quan&shy;<lb/>tum ab illo mo&shy;<lb/>tore mouebitur: <lb/>Secundum, <expan abbr="ead&etilde;">eadem</expan> <lb/>potentia in <expan abbr="eod&etilde;">eodem</expan> <lb/>tempore diuer&longs;o <lb/>modo duo mobi <lb/>lia mouebit &etail;qua <lb/>lia, cum <expan abbr="un&utilde;">unum</expan> mo&shy;<lb/>tui a&longs;&longs;entietur aliud <expan abbr="n&otilde;">non</expan>. </s>
          <s id="id003745">quod &longs;i h&aelig;c mobilia &longs;eiuncta fui&longs;&longs;ent, quod <lb/>aptitudinem haberet <expan abbr="&longs;eiunct&utilde;">&longs;eiunctum</expan> uelocius moueretur, qu&agrave;m dum con<lb/>iunctum e&longs;t. </s>
          <s id="id003746">Cum ergo inquiunt circulus c d moueatur ab a b cir&shy;<lb/>culo, nec conferat quic&lt;08&gt; ad motum, ideo tantum tran&longs;ibit &longs;patium <pb pagenum="222" xlink:href="015/01/241.jpg"/>c d quantum a b per primum &longs;uppo&longs;itum. </s>
          <s id="id003747">Sed quoniam proposi&shy;<lb/>to circulo alio non circa idem centrum, utpote k l reuoluetur &amp; <lb/>perueniet ad h ex demon&longs;tratis. </s>
          <s id="id003748"><expan abbr="Re&longs;ponde&ttilde;">Re&longs;pondetur</expan> ad hoc, quod idem e&longs;t, <lb/>quia unus circulus tantum per &longs;e mouetur circa centrum, reliqui <lb/>omnes non per&longs;e circa centrum, &longs;ed ab alio circulo primo mouen&shy;<lb/>tur, ide&ograve; nihil refert &longs;eu &longs;int circa idem centrum &longs;eu circa aliud, hoc <lb/>enim fortuitum e&longs;t. </s>
          <s id="id003749">Ideo ad argumentum re&longs;pondent cauillo&longs;am <lb/>e&longs;&longs;e <expan abbr="h&atilde;c">hanc</expan> di&longs;putationem, cum &longs;upponat idem ambobus circulis per <lb/>&longs;e centrum e&longs;&longs;e. </s>
          <s id="id003750">Sed non e&longs;t per&longs;e, uer&ugrave;m per <expan abbr="accid&etilde;s">accidens</expan>. </s>
          <s id="id003751">At tamen de&shy;<lb/>miror de huiu&longs;modi &longs;olutione. </s>
          <s id="id003752">Primum quod ip&longs;emet. </s>
          <s id="id003753">Ari&longs;toteles <lb/>de hoc nos docuit in primo Po&longs;teriorum dicens. </s>
          <s id="id003754">Non e&longs;t igitur ex <lb/>uno in aliud genus <expan abbr="tran&longs;c&etilde;dentem">tran&longs;cendentem</expan> demon&longs;trare, ut Geometricum <lb/>Arithmetica. </s>
          <s id="id003755">Et <expan abbr="Auerro&etilde;s">Auerroens</expan> in Commento magno inquit, ea uerba <lb/>exponens. </s>
          <s id="id003756">Fieri non pote&longs;t, ut demon&longs;tratio transferatur de <lb/>arte in artem. </s>
          <s id="id003757">Et ibidem docet, quod neque ut amb&aelig; pr&aelig;mi&longs;&shy;<lb/>&longs;&aelig; &longs;int communes, neque etiam maior tantum, &longs;icut exponebat Al&shy;<lb/>pharabices. </s>
          <s id="id003758">Ver&ugrave;m dicit, &longs;olum licet in artibus, qu&aelig; &longs;unt in com&shy;<lb/>paratione generis ad &longs;peciem, ut &longs;it conclu&longs;io ueluti phy&longs;ica ma&shy;<lb/>ior propo&longs;itio, in &longs;ubiecta &longs;cientia ueluti medicina. </s>
          <s id="id003759">Vnde <expan abbr="c&otilde;cludit">concludit</expan> <lb/>Philo&longs;ophus. </s>
          <s id="id003760">Propter hoc Geometri&etail; non licet demon&longs;trare quod <lb/>contrariorum una e&longs;t &longs;cientia: &longs;ed neque quod duo cubi cubus, neque<lb/> alij &longs;cienti&aelig; quod alterius: ni&longs;i in his qu&aelig; ita inter &longs;e habent ut alte&shy;<lb/>ra &longs;ub altera &longs;it, ueluti per&longs;pectiua ad Geometricam, &amp; harmonica <lb/>ad <expan abbr="Arithmetic&atilde;">Arithmeticam</expan>. </s>
          <s id="id003761">Et po&longs;t docet quod etiam non licet demon&longs;trare ex <lb/>communibus: h&aelig;c igitur ratio e&longs;t ex alienis genere atque communi&shy;<lb/>bus. </s>
          <s id="id003762">Quid, qu&ograve;d non &longs;oluit difficultatem qu&etail; mathematica tota e&longs;t <lb/>&amp; innititur manife&longs;tis principijs. </s>
          <s id="id003763">Debuit enim o&longs;ten dere quomo&shy;<lb/>do tardius moueatur circulus maior ip&longs;o minore: hoc enim e&longs;t ne&shy;<lb/>ce&longs;&longs;e &longs;i eodem tempore debent &aelig;qualia &longs;patia pertran&longs;ire. </s>
          <s id="id003764">Accipia&shy;<lb/>mus ergo quod manife&longs;tum e&longs;t, &longs;cilicet uectionem e&longs;&longs;e hanc in qua <lb/>e centrum perpetu&ograve; per &aelig;quidi&longs;tantem lineam fertur in m, nullum <lb/>autem circulum progre&longs;&longs;us centri e&longs;&longs;e cau&longs;am ni&longs;i ut rota mouet <lb/>currum &amp; currus axem, reuolutio ergo not&aelig; efficit ut &longs;patium c g <lb/>pertran&longs;eat nota, &amp; ideo motus ille circularis non e&longs;t, quia circula&shy;<lb/>ris motus fit manente centro, &longs;ed e&longs;t circulus progrediens uelut &amp; <lb/>punctum e: at in circulo, hoc e&longs;t di&longs;crimen qu&ograve;d puncta, uariantur <lb/>centrum autem non. </s>
          <s id="id003765">Dico ergo ut melius intelligas qu&ograve;d talis mo&shy;<lb/>tus e&longs;t uelut famulorum fabrorum qui rotam circunducant <expan abbr="dom&utilde;">domum</expan> <lb/>impellentes, talis enim motus, e&longs;t rectus, &amp; e&longs;t impul&longs;ionis non au&shy;<lb/>tem circularis. </s>
          <s id="id003766">Et ide&ograve; omnia puncta &aelig;qualiter mouentur, &amp; per <lb/>&aelig;quale &longs;patium, accidit autem ut hic motus fiat circunuertendo, 


<pb pagenum="223" xlink:href="015/01/242.jpg"/>&longs;icut etiam &longs;i traheretur fune. </s>
          <s id="id003767">Et &longs;i quis obijciat quod h&aelig;c re&longs;pon&shy;<lb/>&longs;io e&longs;t eadem cum illa qu&etail; tribuitur Ari&longs;toteli, dico quod non, quia <lb/>in illa &longs;upponuntur duo fal&longs;a, unum quod principium motus ali&shy;<lb/>quando &longs;it in c d, aliquando in a b, quod pro &longs;ecunda parte fal&longs;um <lb/>e&longs;t: nam nunqu&agrave;m principium pote&longs;t e&longs;&longs;e in a b, nam &longs;i intelliga&shy;<lb/>mus de modo motus, non mouetur nec a b nec c d motu circulari, <lb/>quoniam &lpar;ut dixi&rpar; motus e&longs;t uectio, &longs;eu tractio, non circularis. </s>
          <s id="id003768">Sin <lb/>autem de cau&longs;a motus rot&aelig; illa e&longs;t in circulo &longs;emper maximo, &longs;cili&shy;<lb/>cet c d &amp; non a b. </s>
          <s id="id003769">Et cau&longs;a erroris horum fuit duplex: cum enim &longs;ci&shy;<lb/>rent hanc rationem, dubitarunt an circulus c d motus e&longs;&longs;et potius <lb/>cau&longs;a motus circuli a b, an contr&agrave;, ide&ograve; protulerunt ambos, &longs;icut illi <lb/>quibus &longs;ublata e&longs;t res aliqua, ut non errent, dicunt hic, uel hic &longs;ubri&shy;<lb/>puit rem meam. </s>
          <s id="id003770">Secunda fuit, quia ne&longs;ciuerunt di&longs;tinguere inter <lb/>motum per circulum &amp; motum circularem, cum &longs;it magnum di&longs;cri<lb/>men: motus enim rot&aelig; e&longs;t per circulum, quia per circumferentiam <lb/>eius, qu&aelig; e&longs;t circulus, non autem circularis. </s>
          <s id="id003771">Et&longs;i &longs;uperius appella&shy;<lb/>uerim circularem, cum di&longs;tinxi in triplicem motum &longs;ph&etail;r&etail; circum&shy;<lb/>uolutionem, tunc non curaui de uerbis, quia uerba tum non erant <lb/>cau&longs;a erroris.</s>
        </p>
        <p type="margin">
          <s id="id003772"><margin.target id="marg705"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003773"><margin.target id="marg706"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003774"><margin.target id="marg707"/>P<emph type="italics"/>er<emph.end type="italics"/> 34. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003775"><margin.target id="marg708"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 25.</s>
        </p>
        <p type="main">
          <s id="id003776">Ex hoc patet unum, quod e&longs;t difficilius, &longs;cilicet quia certum e&longs;t, <lb/><arrow.to.target n="marg709"/><lb/>qu&ograve;d tam c d qu&agrave;m a b mouentur &longs;uper rectas, &amp; ita ut &longs;ingula <lb/>puncta c d tangant &longs;ingula puncta c g, &amp; a b &longs;ingula puncta a f, &amp; <lb/>tamen c d circumferentia, aut non e&longs;t &aelig;qualis rect&aelig; c g, aut circum&shy;<lb/>ferentia a b non e&longs;t &aelig;qualis rect&aelig; a f, aliter &longs;i amb&aelig; circumferenti&aelig; <lb/>ambabus rectis e&longs;&longs;ent &aelig;quales, cum rect&aelig; &longs;int &aelig;quales, ut demon&shy;<lb/>&longs;tratum e&longs;t, e&longs;&longs;ent circumferenti&aelig; etiam a b &amp; c d, &aelig;quales maior <lb/>minori, quod e&longs;t impo&longs;sibile. </s>
          <s id="id003777">Non ergo ualet argumentum, i&longs;te cir<lb/>culus circumfertur &longs;uper rectam aliquam, ita ut cum redit ad idem <lb/>punctum rectam perambulauit ad unguem, ergo illius peripheria <lb/>e&longs;t &aelig;qualis illi rect&aelig;.</s>
        </p>
        <p type="margin">
          <s id="id003778"><margin.target id="marg709"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003779">Melius ergo fui&longs;&longs;et huius reddere rationem, in quo e&longs;t tota dif&shy;<lb/><arrow.to.target n="marg710"/><lb/>ficultas, nam illa &lpar;ut dixi&rpar; de motu circulari nulla e&longs;t, &longs;i quis tam pe&shy;<lb/>nitus intro&longs;piciat. </s>
          <s id="id003780">Sit igitur ut rot&aelig; axis c, tran&longs;eat in f, &amp; quia e a &amp; <lb/>f g &aelig;quales &longs;unt a centro ad circumferentiam, &amp; a g &aelig;quidi&longs;tans <lb/>b c, erit per demon&longs;trata punctum g in linea fh, &amp; ponamus quod <lb/>punctum fuerit m, quod translatum, &amp; retro reuolutum peruene&shy;<lb/>rit ad h, &amp; &longs;ecet e m a b circulum in n, dico quod n e&longs;t punctum g, in <lb/>quo etiam e&longs;t animaduertendum de &longs;tupore horum &longs;cribentium, <lb/>nec aduertentium quod puncta circulorum a b &amp; c d retro cedunt, <lb/>uer&longs;us a &amp; c, &amp; non uer&longs;us o &amp; p, &amp; hoc e&longs;t quod decipit illos. 


<pb pagenum="224" xlink:href="015/01/243.jpg"/>Quia ergo m e&longs;t h <lb/>&amp; e f, <expan abbr="igi&ttilde;">igitur</expan> cum n &longs;it <lb/>in linea e m, erit in <lb/>linea f h, &longs;ed n e&longs;t <lb/><expan abbr="eti&atilde;">etiam</expan> in circulo a b, <lb/>igitur <expan abbr="c&utilde;">cum</expan> <expan abbr="null&utilde;">nullum</expan> &longs;it <lb/><expan abbr="punct&utilde;">punctum</expan> aliud in li&shy;<lb/>nea fh, et circulo g <lb/>q, &lt;08&gt; g e&longs;t n <expan abbr="c&otilde;mu&shy;nis">commu&shy;<lb/>nis</expan> &longs;ectio, igitur n <lb/>peruenit in g. </s>
          <s id="id003781">Vi&shy;<lb/>des ergo quod m <lb/><figure id="id.015.01.243.1.jpg" xlink:href="015/01/243/1.jpg"/><lb/>retroce&longs;sit per angulum m g h, n autem antece&longs;sit per angulum n <lb/>g f, qui e&longs;t &aelig;qualis angulo m g h. </s>
          <s id="id003782">Ex quo liquet cau&longs;a dictorum, &amp; <lb/>quod non intellexerunt qu&aelig;&longs;tionis fundamentum cum ferantur <lb/>&longs;ingula puncta in una reuolutione &aelig;qualiter cum centro motu re&shy;<lb/>cto: &amp; motu circumuolutionis &longs;unt immobilia, quia tantum retro&shy;<lb/>cedunt in una medietate, quantum procedunt in alia.</s>
        </p>
        <p type="margin">
          <s id="id003783"><margin.target id="marg710"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003784">Propo&longs;itio cente&longs;ima nonage&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id003785">Cur lances ad <expan abbr="loc&utilde;">locum</expan> <expan abbr="&longs;u&utilde;">&longs;uum</expan> <expan abbr="&longs;u&longs;p&etilde;&longs;i">&longs;u&longs;pen&longs;i</expan> <expan abbr="rede&atilde;t">redeant</expan> <expan abbr="impend&etilde;tes">impendentes</expan> <expan abbr="n&otilde;">non</expan>, <expan abbr="dem&otilde;&longs;trare">demon&longs;trare</expan>.<lb/><arrow.to.target n="marg711"/></s>
        </p>
        <p type="margin">
          <s id="id003786"><margin.target id="marg711"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003787">Ali&acirc;s cum uiderem apud Ari&longs;totelem &amp; eius expo&longs;itores hoc </s>
        </p>
        <p type="main">
          <s id="id003788"><arrow.to.target n="marg712"/><lb/>problema non &longs;um au&longs;us, quia ex proprijs non mihi occurrebat <lb/>demon&longs;tratio, rationem reddere, at confecta dialectica &longs;tatim appa <lb/>ruit modus. </s>
          <s id="id003789">Sit ergo libra a b appen&longs;a ex trutina c d, &amp; &longs;it per pon&shy;<lb/><figure id="id.015.01.243.2.jpg" xlink:href="015/01/243/2.jpg"/><lb/>dus educta loco e f, &amp; &longs;ublato reuertitur <lb/>ad locum priorem: Et rur&longs;us eadem &longs;i <lb/>immineat g d &longs;u&longs;tentaculo <expan abbr="n&otilde;">non</expan> mouetur: <lb/>igitur palam e&longs;t quod in trutina d e gra&shy;<lb/>uior e&longs;t <expan abbr="qu&atilde;">quam</expan> d f in&longs;i&longs;tens g d, <expan abbr="n&otilde;">non</expan> e&longs;t adeo <lb/>grauis, aut omnino non grauior. </s>
          <s id="id003790">Neque <lb/>pote&longs;t id accidere quod in primo ca&longs;u <lb/>angulus e d c acutus, &longs;it in &longs;ecundo obtu<lb/>&longs;us, nam &longs;i ob angulum e d c acutum <expan abbr="de&longs;c&etilde;dit">de&longs;cendit</expan> in primo ca&longs;u e, in &longs;e&shy;<lb/>cundo ca&longs;u de&longs;cendet f, quia pariter f d g acutus e&longs;t, &amp; &aelig;qualis e d c, <lb/>hoc autem non contingit. </s>
          <s id="id003791">Mira ne dicam &longs;tultitia an audacia <expan abbr="eor&utilde;">eorum</expan>, <lb/>qui nihil intelligentes au&longs;i &longs;unt, h&aelig;c pertractare, &longs;perantes in tot &longs;e&shy;<lb/>culis nullum futurum, qui ignorantiam &longs;uam &amp; impo&longs;tura depre&shy;<lb/>hendat, dicunt enim quod in primo ca&longs;u producta quadam recta <lb/>ad perpendiculum, &amp; qu&aelig; &longs;it h k maiorem reddi d e qu&agrave;m d f, neque <lb/>quomodo id fiat o&longs;tendunt, &amp; &longs;i &lpar;ut dixi&rpar; maior &longs;it <expan abbr="qu&atilde;">quam</expan> d fin primo <lb/>ca&longs;u maior d f quam d e in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ca&longs;u: ergo &longs;i in primo ca&longs;u d e de&shy;<lb/>&longs;cendit, in &longs;ecundo de&longs;cendet magis d f, at hoc non accidit &longs;ed &longs;tat. 


<pb pagenum="225" xlink:href="015/01/244.jpg"/>Oportet igitur hoc e&longs;&longs;e principium ex Dialectica, quod o&longs;tendat e <lb/>grauiorem e&longs;&longs;e f in primo ca&longs;u, in &longs;ecundo non e&longs;&longs;e grauiorem, aut <lb/>leuiorem, ut neque ad angulum refugere po&longs;simus. </s>
          <s id="id003792">Ergo &longs;upponere <lb/>oportet qu&aelig; manife&longs;ta &longs;unt, e e&longs;&longs;e grauiorem f, aliter enim non de&shy;<lb/>&longs;cenderet: non prohiberi autem in primo ca&longs;u motum prohiberi in <lb/>&longs;ecundo, aliter uel grauior fieret f, uel maneret eadem grauitas: &longs;i&shy;<lb/>quidem maneret grauitas, nec impediretur de&longs;cendere e in &longs;e&shy;<lb/>cundo ca&longs;u, ut in primo, at non de&longs;cendit. </s>
          <s id="id003793">Si grauitas mutaretur, igi<lb/>tur f de&longs;cenderet &longs;ecundo ca&longs;u magis quam in primo. </s>
          <s id="id003794">Quod &longs;i di&shy;<lb/>cas non tanto fieri grauiorem, igitur f magis depre&longs;&longs;a de&longs;cendet <lb/>&longs;altem, at nunquam de&longs;cendit, igitur grauior e&longs;t &longs;emper e qu&agrave;m f, <lb/>&longs;ed in &longs;ecundo ca&longs;u impeditur motus non in primo. </s>
          <s id="id003795">Cau&longs;a grauita&shy;<lb/>tis e&longs;t, quoniam d e&longs;t centrum grauitatis, quia medium. </s>
          <s id="id003796">igitur cum <lb/><arrow.to.target n="marg713"/><lb/>c &amp; d con&longs;pirent contra f, nece&longs;&longs;e e&longs;t e de&longs;cendere per &longs;uperius de&shy;<lb/>mon&longs;trata, igitur e de&longs;cendet in primo ca&longs;u, quia grauius e&longs;t ut do&shy;<lb/>cui nec impeditum. </s>
          <s id="id003797">At in &longs;ecundo ca&longs;u e &amp; d &longs;unt grauiora, &longs;ed d <lb/>e&longs;t impeditum, quia non habet motum, ni&longs;i occultum in&longs;idet enim <lb/><arrow.to.target n="marg714"/><lb/>g d, igitur tantum ponderat e quam f, ergo pror&longs;us non mouebun&shy;<lb/>tur, facit &amp; ad hoc qu&ograve;d qu&aelig;uis latitudo d, &longs;u&longs;tentaculi prohibet <lb/>motum, at dee&longs;&longs;e uix pote&longs;t. </s>
          <s id="id003798">Vides ergo illos nugas palam agere. <lb/></s>
          <s id="id003799">Primum dee&longs;t illis dialectica, deinde ingenium acre, deinde quod <lb/>maius e&longs;t, uolunt confe&longs;tim tran&longs;ire ex principijs ad remota theore&shy;<lb/>mata, quod fieri non pote&longs;t.</s>
        </p>
        <p type="margin">
          <s id="id003800"><margin.target id="marg712"/>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 7. <lb/>M<emph type="italics"/>echan.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003801"><margin.target id="marg713"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s>
        </p>
        <p type="margin">
          <s id="id003802"><margin.target id="marg714"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 193.</s>
        </p>
        <p type="main">
          <s id="id003803">Propo&longs;itio cente&longs;ima nonage&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id003804">Cur &longs;olidum quod cubus <expan abbr="uoca&ttilde;">uocatur</expan>, pyramide &longs;tabilius &longs;it, o&longs;tendere.</s>
        </p>
        <p type="head">
          <s id="id003805">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id003806">Si intra circulum triangulus &aelig;quilaterus de&longs;cribatur, &amp; ab uno <lb/>angulorum per centrum rect&agrave; ducatur, angulum per &aelig;qualia diui&shy;<lb/>det, &amp; trianguli latus, &amp; ad angulos rectos ei in&longs;i&longs;tet, ip&longs;a uer&ograve; qu&aelig; <lb/>ex centro per &aelig;qualia uici&longs;sim &agrave; trianguli latere diuidetur.<lb/><figure id="id.015.01.244.1.jpg" xlink:href="015/01/244/1.jpg"/><lb/><arrow.to.target n="marg715"/></s>
        </p>
        <p type="margin">
          <s id="id003807"><margin.target id="marg715"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id003808">Sit a b c &aelig;quilaterus circulo in&longs;criptus, </s>
        </p>
        <p type="main">
          <s id="id003809"><arrow.to.target n="marg716"/><lb/>cuius centrum d, ducaturque ad e f rect&agrave; per <lb/>centrum, &amp; ducantur d b &amp; d c, eritque ex hoc <lb/><arrow.to.target n="marg717"/><lb/>triangulus a b d &etail;quilaterus triangulo a c d, <lb/><arrow.to.target n="marg718"/><lb/>quare angulus b a d &aelig;qualis c a d, igitur ar&shy;<lb/>cus b e &aelig;qualis c e, igitur arcus b e e&longs;t &longs;exta <lb/><arrow.to.target n="marg719"/><lb/>pars circuli, quare b e recta latus exagoni, <lb/>quare b e erit &aelig;qualis d e, igitur cum anguli <lb/><arrow.to.target n="marg720"/><lb/>a d f &longs;int utrin que recti, erit d f &aelig;qualis f e, itaque <lb/><arrow.to.target n="marg721"/><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"/>
        <p type="margin">
          <s id="id003810"><margin.target id="marg716"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003811"><margin.target id="marg717"/>P<emph type="italics"/>er<emph.end type="italics"/> 26. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003812"><margin.target id="marg718"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003813"><margin.target id="marg719"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>m. <lb/>15. <emph type="italics"/>quarti <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003814"><margin.target id="marg720"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003815"><margin.target id="marg721"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id003816">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id003817">Quadratum lateris trianguli &aelig;quilateri &longs;e habet ad illius &longs;uperfi<lb/>ciem, ut latus eius ad mediam lineam inter latus dodrantis, &amp; qua&shy;<lb/>drantis proportione duplicata.<lb/><arrow.to.target n="marg722"/></s>
        </p>
        <p type="margin">
          <s id="id003818"><margin.target id="marg722"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003819">Quadratum a b e&longs;t &aelig;quale quadratis a f, fb, &amp; quadruplum qua</s>
        </p>
        <p type="main">
          <s id="id003820"><arrow.to.target n="marg723"/><lb/>drato b f, igitur quadratum a f e&longs;t dodrans quadrati a b. </s>
          <s id="id003821">Quod ue&shy;<lb/>r&ograve; fit ex a fin f b e&longs;t medium proportione inter quadrata a f, f b, re&shy;<lb/><arrow.to.target n="marg724"/><lb/>ctangulum igitur ex a fin fb, e&longs;t ex lateribus dodrantis a f, &amp; qua&shy;<lb/><arrow.to.target n="marg725"/><lb/>drantis b f quadrati a b, quare cum medi&aelig; inter a f &amp; fb &aelig;quale fa&shy;<lb/>ciat quadratum rectangulo a fin fb, erit proportio quadrati a b ad <lb/>quadratum medi&aelig; inter a f, fb, ut lateris trianguli ad mediam inter <lb/><arrow.to.target n="marg726"/><lb/>latera dodrantis, &amp; quadrantis quadrati lateris ip&longs;ius duplicata: re&shy;<lb/><arrow.to.target n="marg727"/><lb/>ctangulum autem a fin fb e&longs;t &aelig;quale triangulo a b c, igitur propor<lb/>tio quadrati a b ad triangulum a b c e&longs;t uelut lateris a b ad mediam <lb/>inter latera dodrantis &amp; quadrantis duplicata.</s>
        </p>
        <p type="margin">
          <s id="id003822"><margin.target id="marg723"/>P<emph type="italics"/>er<emph.end type="italics"/> 27. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003823"><margin.target id="marg724"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;exti <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003824"><margin.target id="marg725"/>P<emph type="italics"/>er eandem <lb/>&amp;<emph.end type="italics"/> 11. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003825"><margin.target id="marg726"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&amp;<emph.end type="italics"/><lb/>20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>l.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003826"><margin.target id="marg727"/>P<emph type="italics"/>er<emph.end type="italics"/> 41. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id003827">LEMMA TERTIVM.</s>
        </p>
        <p type="main">
          <s id="id003828">Propo&longs;itio quadrati cubi &longs;ph&aelig;r&aelig; inclu&longs;i ad triangulum pyrami<lb/>dis eidem &longs;ph&aelig;r&aelig; inclu&longs;&aelig;, e&longs;t uelut lateris pyramidis &longs;eu trianguli <lb/>eius ad cathetum &longs;uum.<lb/><arrow.to.target n="marg728"/></s>
        </p>
        <p type="margin">
          <s id="id003829"><margin.target id="marg728"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003830">Proponatur enim &longs;ph&aelig;r&aelig; diameter g, &amp; latus pyramidis b a, &amp; </s>
        </p>
        <p type="main">
          <s id="id003831"><arrow.to.target n="marg729"/><lb/>latus cubi b h, qu&aelig; corpora illi &longs;ph&aelig;r&aelig; includuntur: igitur g erit <lb/>pote&longs;tate &longs;exquialtera ad a b, &amp; tripla ad b h, igitur b a e&longs;t pote&longs;tate <lb/><arrow.to.target n="marg730"/><lb/>dupla ad b h, quod igitur fit ex b a in dimidium &longs;uum, e&longs;t &aelig;quale <lb/>quadrato b h, igitur b h e&longs;t media inter b a &amp; b f, b f enim e&longs;t dimi&shy;<lb/>dium b a, ut probatum e&longs;t. </s>
          <s id="id003832">Quadratum igitur a b &longs;e habet ad trian&shy;<lb/><arrow.to.target n="marg731"/><lb/>gulum a b c, ut a b ad mediam inter a f &amp; fb duplicata: Quadratum <lb/>quoque a b &longs;e habet ad quadratum h b, ut a b ad mediam inter a b &amp; <lb/>b f, duplicata igitur proportio quadrati b h ad triangulum a b c, e&longs;t <lb/><arrow.to.target n="marg732"/><lb/>uelut lateris a b ad cathetum a f.</s>
        </p>
        <p type="margin">
          <s id="id003833"><margin.target id="marg729"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/>13. <emph type="italics"/>decimi&shy;<lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003834"><margin.target id="marg730"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/>15. <emph type="italics"/>decimi&shy;<lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003835"><margin.target id="marg731"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>L<emph type="italics"/>emmate<emph.end type="italics"/> 1.</s>
        </p>
        <p type="margin">
          <s id="id003836"><margin.target id="marg732"/>P<emph type="italics"/>er<emph.end type="italics"/> 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 e&longs;t pote&longs;tate &longs;ex&shy;<lb/>quialtera.<lb/><arrow.to.target n="marg733"/></s>
        </p>
        <p type="margin">
          <s id="id003839"><margin.target id="marg733"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003840">Intelligatur ba&longs;is pyramidis triangulus a b c, &amp; conus pyrami&shy;</s>
        </p>
        <p type="main">
          <s id="id003841"><arrow.to.target n="marg734"/><lb/>dis k, &amp; qu&aelig; per centrum &longs;ph&aelig;r&aelig; tran&longs;it ex cono k d, cumque k d a <lb/>angulus rectus &longs;it, erit quadratum k a &aelig;quale quadratis k d, d a, at <lb/>d a e&longs;t dupla d f, ut probatum e&longs;t, igitur pote&longs;tate &longs;exquitertia f b, <lb/>k a uer&ograve; e&longs;t quadrupla pote&longs;tate fb, quia fb e&longs;t dimidium k a, igitur <lb/>k a e&longs;t tripla pote&longs;tate a d, igitur k a pote&longs;tate &longs;exquialtera k d, quod <lb/>erat demon&longs;trandum.<lb/><arrow.to.target n="marg735"/></s>
        </p>
        <p type="margin">
          <s id="id003842"><margin.target id="marg734"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>L<emph type="italics"/>emmate<emph.end type="italics"/> 1.</s>
        </p>
        <p type="margin">
          <s id="id003843"><margin.target id="marg735"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003844">Ex hoc patet quod proportio axis pyramidis ad latus cubi ea&shy;<lb/>dem &longs;ph&aelig;ra circum&longs;criptorum e&longs;t pote&longs;tate &longs;exquitertia.</s>
        </p>
        <pb pagenum="227" xlink:href="015/01/246.jpg"/>
        <p type="main">
          <s id="id003845"><arrow.to.target n="marg736"/></s>
        </p>
        <p type="margin">
          <s id="id003846"><margin.target id="marg736"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003847">Quia enim k a e&longs;t pote&longs;tate dupla ad b b, &amp; &longs;e&longs;quialtera pote&longs;ta<lb/>te ad k d, nece&longs;&longs;e e&longs;t ut k d &longs;it &longs;exquitertia pote&longs;tate ad b h.</s>
        </p>
        <p type="head">
          <s id="id003848">LEMMA QVINTVM.</s>
        </p>
        <p type="main">
          <s id="id003849">Pri&longs;ma altitudinem habens pyramidis &amp; triangulum eiu&longs;dem <lb/>ba&longs;im, &aelig;quale e&longs;t cubo eidem &longs;ph&aelig;r&aelig; in&longs;cripto.</s>
        </p>
        <p type="main">
          <s id="id003850"><arrow.to.target n="marg737"/></s>
        </p>
        <p type="margin">
          <s id="id003851"><margin.target id="marg737"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003852">Cum enim proportio quadrati b h ad triangulum a b c &longs;it uelut </s>
        </p>
        <p type="main">
          <s id="id003853"><arrow.to.target n="marg738"/><lb/>a b ad a f, a b autem ad a f &longs;it &longs;exquitertia pote&longs;tate ex demon&longs;tratis, <lb/>erit quadratum b h ad triangulum a b c &longs;exquitertium pote&longs;tate: at <lb/>cubi b h altitudo e&longs;t ip&longs;a b h, pri&longs;matis autem a b c altitudo e&longs;t k d, <lb/>k d autem potentia &longs;exquitertia ad b h, igitur pri&longs;ma a b c e&longs;t &etail;quale <lb/>cubo b h, quod fuit propo&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id003854"><margin.target id="marg738"/>P<emph type="italics"/>er<emph.end type="italics"/> 3 <emph type="italics"/>lem&shy;<lb/>ma.<emph.end type="italics"/><lb/>L<emph type="italics"/>emmate<emph.end type="italics"/> 2.</s>
        </p>
        <p type="main">
          <s id="id003855">Ex hoc &longs;equitur, quod cum pri&longs;ma &longs;it triplum &longs;u&aelig; pyramidi, ut <lb/><arrow.to.target n="marg739"/><lb/>ab Euclide habetur, quod cubus e&longs;t triplus pyramidi, quam eadem <lb/><arrow.to.target n="marg740"/><lb/>&longs;ph&aelig;ra circum&longs;cribit.<lb/><arrow.to.target n="marg741"/></s>
        </p>
        <p type="margin">
          <s id="id003856"><margin.target id="marg739"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003857"><margin.target id="marg740"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/><emph type="italics"/>lemmatis<emph.end type="italics"/> 4.</s>
        </p>
        <p type="margin">
          <s id="id003858"><margin.target id="marg741"/>P<emph type="italics"/>er<emph.end type="italics"/> 34. <emph type="italics"/>un&shy;<lb/>decimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003859">Nunc uenio ad demon&longs;trationem propo&longs;itionis, &amp; dico quod <lb/>corpus difficile e&longs;t ad motum, uel ob magnitudinem ba&longs;is, cui in&longs;i&shy;</s>
        </p>
        <p type="main">
          <s id="id003860"><arrow.to.target n="marg742"/><lb/>det, uel ob pondus, uel ob formam: nam corpus quod forma e&longs;t <lb/><arrow.to.target n="marg743"/><lb/>contracta, difficil&egrave; mouetur, ut pyramis, contr&agrave;, quod prominet &agrave; la<lb/>teribus, facile reuoluitur, ut corpus duodecim ba&longs;ium pentagona&shy;<lb/>rum, &amp; uiginti triangularum: ergo cubi &longs;edes e&longs;t maior qu&agrave;m &longs;ua <lb/>pyramis, &amp; pondus triplo maius, &amp; etiam non prominet cubus, <lb/>ide&ograve; pro re &longs;tabili po&longs;itum e&longs;t corpus eiu&longs;modi. </s>
          <s id="id003861">Eo quod ob gra&shy;<lb/>uitatem etiam, ut dixi, &longs;it &longs;tabilius pyramide eiu&longs;dem &longs;ph&etail;r&etail;. </s>
          <s id="id003862">Quod <lb/>&longs;i etiam a&longs;&longs;umeres pyramidem, cuius ba&longs;is e&longs;&longs;et &aelig;qualis quadrato <lb/>cubi, ip&longs;a &longs;e haberet ad pyramidem &longs;ph&aelig;r&aelig; in grauitate, uelut latus <lb/>trianguli ad &longs;uum cathetum, &amp; ideo proportio ponderis cubi ad <lb/>pyramidem e&longs;&longs;et, uelut tredecim ad quinque ferm&egrave;: ergo ratione pon<lb/>deris e&longs;&longs;et long&egrave; &longs;tabilior cubus ip&longs;a pyramide. </s>
          <s id="id003863">At in alijs corpori&shy;<lb/>bus, qu&aelig; rationalia uocantur, non e&longs;t tanta proportio ponderis, &amp; <lb/>ba&longs;is e&longs;t minor &amp; forma prominet.</s>
        </p>
        <p type="margin">
          <s id="id003864"><margin.target id="marg742"/>E<emph type="italics"/>x<emph.end type="italics"/> 7. <emph type="italics"/>duode<lb/>cimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003865"><margin.target id="marg743"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003866">Propo&longs;itio cente&longs;ima nonage&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id003867">Rationem remorum nauim impellentium inuenire.<lb/><arrow.to.target n="marg744"/></s>
        </p>
        <p type="margin">
          <s id="id003868"><margin.target id="marg744"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003869">Sit a remi extremum, quod manu apprehenditur, b &longs;calmus cui <lb/>remus in&longs;idet: c extremum aliud latius remi, quod uocant pal&shy;<lb/>mam, transferatur nixu manus, &amp; motu corporis a in d, ut c per&shy;</s>
        </p>
        <p type="main">
          <s id="id003870"><arrow.to.target n="marg745"/><lb/>ueniat in e, &longs;unt enim &aelig;quales a b, d b, b c, b e etiam &amp; angu&shy;<lb/>li a d b contrapo&longs;iti, quare trianguli a b d &amp; c b e &longs;imiles, igitur <lb/>primum quanto maior propo&longs;itio c b ad b a, tanto maior propor&shy;<lb/><arrow.to.target n="marg746"/><lb/>tio c ad a d, &amp; ita ex &aelig;quali motu longius transferetur remus, &longs;eu <lb/>palma. </s>
          <s id="id003871">Secundum, cum motus a d fiat nixu brachiorum &amp; corpo&shy;<lb/>ris, quanto magis transfertur corpus eo minus opus erit brachio&shy;


<pb pagenum="228" xlink:href="015/01/247.jpg"/>rum nixu, &amp; ita minus laborabunt. </s>
          <s id="id003872">Et <lb/><arrow.to.target n="marg747"/><lb/>quo minus laborabunt brachia, plus <lb/>corpus laborabit. </s>
          <s id="id003873">Et ide&ograve;, ut declara&shy;<lb/>tum e&longs;t &longs;upr&agrave;, minor labor erit cum &aelig;&shy;<lb/>qualiter ambo laborabunt. </s>
          <s id="id003874">Tertium, <lb/>quo minor erit proportio c b ad b a, <lb/>eo maius &longs;patium pertran&longs;ibit remex, <lb/>qui mouet ex a in d, &longs;ed tanto facilius <lb/><arrow.to.target n="marg748"/><lb/>mouebit, quia labor motus b c minue&shy;<lb/><figure id="id.015.01.247.1.jpg" xlink:href="015/01/247/1.jpg"/><lb/>tur, ut &longs;upr&agrave; ui&longs;um e&longs;t per longitudinem a b &amp; d b, ut &longs;upr&agrave; demon <lb/>&longs;trauimus. </s>
          <s id="id003875">Quartum, c&ugrave;m remus tran&longs;ierit quoddam &longs;patium <lb/>iuxta robur, puta ex c in e, nece&longs;&longs;e e&longs;t ut eleuetur &longs;uper aquam, tum <lb/>quia impediret motum pro gre&longs;&longs;us nauis, tum ut transferatur ante: <lb/>aliter &longs;i transferretur ante &longs;ub aqua difficilius multo, quam per a&euml;&shy;<lb/>rem transferretur, &amp; retroageret tantundem nauim, quantum an&shy;<lb/>tea retroactam impulit. </s>
          <s id="id003876">His per &longs;e notis dico, qu&ograve;d translato remo <lb/>ex c in e, nece&longs;&longs;e e&longs;t nauim contr&agrave; transferri ex f in g: nam quia impe<lb/>dimentum ex aqua tran&longs;itur c in e, maius e&longs;t quam nauis &longs;uper a&shy;<lb/>quam, &amp; remus debet transferri ex a in d, &amp; non pote&longs;t transferri <lb/>ni&longs;i uel &longs;tante naui, &amp; translato c in e, uel &longs;tante a b c remo, &amp; tran&longs;&shy;<lb/>lata naui: &amp; tunc nece&longs;&longs;e e&longs;t, ut e progrediatur ad h, ita de&longs;&longs;ecabit a&shy;<lb/>quam ch, ergo difficultas manet eadem ferm&egrave;, ex his fit motus com<lb/>po&longs;itus, ut palma non redeat u&longs;que ad e, &longs;ed maneat remus minus in&shy;<lb/>clinatus, &amp; qua&longs;i ad perpendiculum in h. </s>
          <s id="id003877">Et manife&longs;tum e&longs;t, &qring;d erit <lb/>motus compo&longs;itus ex retro ce&longs;&longs;u remi &amp; pro ce&longs;&longs;u nauis. </s>
          <s id="id003878">Qui etiam <lb/>remiges circa medium &longs;unt minus laborarent, &longs;i remus &aelig;qualiter <lb/>promineret extra &longs;calmum, &longs;ed magis laborant, quia proportio e&longs;t <lb/>eadem, &amp; a b e&longs;t longior, &amp; cra&longs;sior remus, ut minus flectatur ob <lb/>longitudinem, aliter &longs;i e&longs;&longs;et &aelig;qualis cra&longs;situdinis, &amp; multo longior <lb/>flecteretur aut frangeretur, ide&ograve; robu&longs;tiores remiges ponuntur in <lb/>medio triremis. </s>
          <s id="id003879">Iuuatur pr&aelig;terea motus nauis pror&longs;um ex percu&longs;&shy;<lb/>&longs;ione remi, &amp; impetu iam aqui&longs;ito cum nixu remi in aduer&longs;um &longs;u&shy;<lb/>perueniente. </s>
          <s id="id003880">Rur&longs;us cum nauis transferatur eodem tempore ant&egrave; <lb/>qu&ograve; a progreditur ad d, manife&longs;tum e&longs;t qu&ograve;d magna pars e&longs;t ex <lb/>motu nauis, non nixu corporis aut uirium: &amp; ita quod celerius mo<lb/>uetur ex c in h, ab initio dum nauis quie&longs;cit, aut tardius mouetur, <lb/>tardius autem dum nauis progreditur.</s>
        </p>
        <p type="margin">
          <s id="id003881"><margin.target id="marg745"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003882"><margin.target id="marg746"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003883"><margin.target id="marg747"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 188.</s>
        </p>
        <p type="margin">
          <s id="id003884"><margin.target id="marg748"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 71.</s>
        </p>
        <p type="main">
          <s id="id003885">Propo&longs;itio ducente&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id003886">Cur temo <expan abbr="c&utilde;">cum</expan> paruus &longs;it magnam nauim agere pote&longs;t: &amp; cur cum <lb/>uarietas &longs;it in prora, ip&longs;e con&longs;tituatur in puppi. </s>
          <s id="id003887">Et cum tran&longs;uer&longs;im <lb/>ab aqua prematur, rect&agrave; nauim dirigat?</s>
        </p>
        <pb pagenum="229" xlink:href="015/01/248.jpg"/>
        <p type="main">
          <s id="id003888">Dixi quod in hipomochlio parua uarietas fit in motu: igitur &agrave; <lb/><arrow.to.target n="marg749"/><lb/>leui cau&longs;a magnum nauigium impellitur aut uariatur. </s>
          <s id="id003889">Cum enim a <lb/><expan abbr="tr&atilde;sfertur">transfertur</expan> ad b, fit minima uarietas in e, igitur a parua poterit tran&longs;&shy;<lb/><figure id="id.015.01.248.1.jpg" xlink:href="015/01/248/1.jpg"/><lb/>ferri, tum uero quod debuit <expan abbr="tr&atilde;sferri">transferri</expan> ad c, transfertur ad <lb/>d, nam motus ip&longs;e ab alia cau&longs;a fit, uelut <expan abbr="u&etilde;to">uento</expan> aut remis, <lb/>ita non e&longs;t difficultas ni&longs;i propter motum aqu&aelig;, &longs;cilicet <lb/>ut tabula &longs;cindat illam. </s>
          <s id="id003890">Ad hoc autem contulit illud <lb/>quod intra nauim prominet ut uectis rationem habeat, <lb/>&amp; ob id facilius uerti.</s>
        </p>
        <p type="margin">
          <s id="id003891"><margin.target id="marg749"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003892">Similiter uarietas in puppi exigua e&longs;t cau&longs;a magn&aelig; <lb/>uarietatis in prora, quod autem pote&longs;t fieri paucioribus <lb/>&amp; faciliori modo id debet fieri, hac igitur cau&longs;a in pup&shy;<lb/>pi temonem con&longs;tituere oportet &longs;eu guberna culum.</s>
        </p>
        <p type="main">
          <s id="id003893">Cum autem impellatur &agrave; mari, nece&longs;&longs;e e&longs;t, ut &agrave; latere excipiat <lb/>aquam ita ut tantum pendeat in unam partem, quantum nauis in <lb/>aduer&longs;am, nam &longs;i nauis non penderet, gubernaculum rect&egrave; dirige&shy;<lb/><figure id="id.015.01.248.2.jpg" xlink:href="015/01/248/2.jpg"/><lb/>retur. </s>
          <s id="id003894">Vt ergo ex duobus obliquis <expan abbr="un&utilde;">unum</expan> rectum con&longs;titui<lb/>tur, ita ex naui &amp; gubernaculo, nam &longs;int a b &amp; c b &amp; im&shy;<lb/>pellatur ad d, impelletur per mediam lineam b e &amp; non <lb/>per a b neque c b, igitur oportet temonem pendere ex ad <lb/>uer&longs;o inclinationis nauis. </s>
          <s id="id003895">E&longs;t etiam alia ratio, quoniam <lb/>nauis &longs;ecurior redditur, nam quemadmodum quod in <lb/>medio e&longs;t, facilius impellitur tran&longs;uer&longs;im, qu&agrave;m quod pendet in <lb/>contrarium, ita &amp; in gubernaculo. </s>
          <s id="id003896">E&longs;t &amp; id ob nece&longs;sitatem, quoni&shy;<lb/>am motus aqu&aelig; plerumque e&longs;t in partem, uelut &amp; uentus ad la&shy;<lb/>tus eius &longs;itus, &longs;ecundum quem moueri debet nauis. </s>
          <s id="id003897">Sicut igitur &amp; <lb/>uela &amp; malus inclinantur, ut motum directum efficiant, quia ali&ograve; <lb/>dirigitur nauis quam qui mouet uentus, ita de temone compara&shy;<lb/>tione aqu&aelig;.</s>
        </p>
        <p type="main">
          <s id="id003898">Propo&longs;itio ducente&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id003899">Si du&aelig; line&aelig; non &longs;ecantes circuli peripheriam in <expan abbr="un&utilde;">unum</expan> <expan abbr="punct&utilde;">punctum</expan>, ex <lb/>ea co&euml;ant, exterius nece&longs;&longs;e e&longs;t illas peripheria <expan abbr="c&otilde;tenta">contenta</expan> e&longs;&longs;e maiores.</s>
        </p>
        <p type="head">
          <s id="id003900">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id003901">Si fuerit proportio primi ad &longs;ecundum maior qu&agrave;m tertij ad <lb/>quartum, erit primi ad tertium maior qu&agrave;m &longs;ecundi ad quartum.<lb/><arrow.to.target n="marg750"/></s>
        </p>
        <p type="margin">
          <s id="id003902"><margin.target id="marg750"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003903">Quamuis hoc demon&longs;tretur &agrave; Campano, quia </s>
        </p>
        <p type="main">
          <s id="id003904"><arrow.to.target n="marg751"/><lb/>tamen facile e&longs;t hic adijcietur. </s>
          <s id="id003905">Sit igitur maior a <lb/>ad b quam c ad d, dico maiorem e&longs;&longs;e a ad c quam <lb/><figure id="id.015.01.248.3.jpg" xlink:href="015/01/248/3.jpg"/><lb/><arrow.to.target n="marg752"/><lb/>b ad d, quia enim maior e&longs;t a ad b quam c ad d fiat e ad b ut c ad e <lb/><arrow.to.target n="marg753"/><lb/>eritque e minu&longs; quam a, e igitur ad c ut b ad d &longs;ed maior a ad c quam <lb/>e ad e igitur maior a ad c quam b ad d.<lb/><arrow.to.target n="marg754"/></s>
        </p>
        <pb pagenum="230" xlink:href="015/01/249.jpg"/>
        <p type="margin">
          <s id="id003906"><margin.target id="marg751"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>quin<lb/>ti <emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003907"><margin.target id="marg752"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003908"><margin.target id="marg753"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003909"><margin.target id="marg754"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id003910">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id003911">Si fuerint quatuor quanti&shy;<lb/><arrow.to.target n="marg755"/><lb/>tates, quarum exce&longs;&longs;us prim&aelig; <lb/>&longs;upra &longs;ecundam, fit minor ex&shy;<lb/><figure id="id.015.01.249.1.jpg" xlink:href="015/01/249/1.jpg"/><lb/>ce&longs;&longs;u terti&etail; &longs;upra quartam, &longs;itque prima non minor tertia, erit propor<lb/><arrow.to.target n="marg756"/><lb/>tio prim&aelig; ad &longs;ecundam minor qu&agrave;m terti&aelig; ad quartam.<lb/><arrow.to.target n="marg757"/></s>
        </p>
        <p type="margin">
          <s id="id003912"><margin.target id="marg755"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem. par <lb/>tes ambas.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003913"><margin.target id="marg756"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003914"><margin.target id="marg757"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003915">Sit exce&longs;&longs;us a &longs;upra b c, g b minor exce&longs;&longs;u d &longs;upra e f qui &longs;it h e, di&shy;</s>
        </p>
        <p type="main">
          <s id="id003916"><arrow.to.target n="marg758"/><lb/>co quod proportio a ad b c e&longs;t minor proportione d ad e f. </s>
          <s id="id003917">Quia <lb/>enim a e&longs;t maior d, &amp; b g minor h e, erit maior proportio a ad b g <lb/><arrow.to.target n="marg759"/><lb/>qu&agrave;m d ad h e, igitur fiat a ad g k ut d ad h e, erit ergo g k maior g b <lb/><arrow.to.target n="marg760"/><lb/>quare k e minor b c ex communi animi &longs;ententia, e&longs;t autem a ad k c <lb/>ut d ad e f, minor autem a ad c b qu&agrave;m ad k c, igitur minor a ad b c <lb/>quam d ad e f.</s>
        </p>
        <p type="margin">
          <s id="id003918"><margin.target id="marg758"/>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003919"><margin.target id="marg759"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003920"><margin.target id="marg760"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003921">Si intra circulum &aelig;quicurium, &amp; &longs;uper eandem ba&longs;im figura &aelig;&shy;<lb/>quilatera &amp; &aelig;quiangula <expan abbr="c&otilde;&longs;tituatur">con&longs;tituatur</expan>, <expan abbr="er&utilde;t">erunt</expan> omnia illius latera pariter <lb/>accepta minora duobus trianguli lateribus.<lb/><arrow.to.target n="marg761"/></s>
        </p>
        <p type="margin">
          <s id="id003922"><margin.target id="marg761"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003923">Sit ut proponitur, &amp; producantur b d &amp; <lb/>c e qu&aelig; concurrent intra triangulum, quia <lb/>anguli d b c &amp; e c b &longs;upponuntur &etail;quales, &amp; <lb/>ducta d e producantur d fl, &amp; e g l qu&aelig; <expan abbr="con&shy;curr&etilde;t">con&shy;<lb/>current</expan> intra triangulum k d e ut propter ean&shy;<lb/>dem cau&longs;am, igitur a b &amp; a c &longs;unt maiores k b <lb/>&amp; k c, ergo maiores k d, d b, &amp; k e, e c quia <lb/>&longs;unt e&aelig;dem. </s>
          <s id="id003924">Duct&etail; quo que de &longs;imili modo <lb/><figure id="id.015.01.249.2.jpg" xlink:href="015/01/249/2.jpg"/><lb/>k d &amp; d e, &longs;unt maiores l d &amp; l e, igitur l f, f d &amp; l g, g e, igitur a b &amp; a c <lb/>maiores &longs;unt b d, d f, f l c e e g g l pariter acceptis. </s>
          <s id="id003925">Rur&longs;us ducta f g: <lb/>f l &amp; l g maiores &longs;unt m f &amp; m g, igitur a b &amp; a c &longs;unt maiores omni&shy;<lb/>bus lateribus figur&aelig; in&longs;cript&aelig;.</s>
        </p>
        <p type="main">
          <s id="id003926"><arrow.to.target n="marg762"/></s>
        </p>
        <p type="margin">
          <s id="id003927"><margin.target id="marg762"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id003928">Ex hoc patet quod latera polygoni&aelig; fi&shy;<lb/>gur&aelig; &etail;quilater&aelig; &amp; &aelig;quiangul&aelig; in&longs;cript&etail; <lb/>portioni circuli &longs;unt minora lateribus tra&shy;<lb/>pezij circun&longs;cripti eidem peripheri&aelig;.</s>
        </p>
        <figure id="id.015.01.249.3.jpg" xlink:href="015/01/249/3.jpg"/>
        <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"/><lb/><expan abbr="ri&atilde;">riam</expan> a b, &amp; in ea in&longs;cripta figura polygonia <lb/>&aelig;quilatera &amp; &aelig;quiangula a c, d f b. </s>
          <s id="id003931">Et quia <lb/>trapezium e&longs;t figura cuius oppo&longs;ita duo <lb/>latera &longs;unt &etail;qualia, &amp; duo anguli &longs;upra ba <lb/>&longs;im &aelig;quales: itemque duo in &longs;ummitate inui<lb/>cem &etail;quales, <expan abbr="t&atilde;get">tanget</expan> in medio peripheriam <lb/><arrow.to.target n="marg764"/><lb/>quod patet ductis lineis ex centro ad ex&shy;<lb/><figure id="id.015.01.249.4.jpg" xlink:href="015/01/249/4.jpg"/><lb/>trema trapezij. </s>
          <s id="id003932">Et ideo etiam <expan abbr="punct&utilde;">punctum</expan> medium polygoni&aelig;, quare ex 


<pb pagenum="231" xlink:href="015/01/250.jpg"/>hoc leminate duo latera g d &amp; g a deducta ad &aelig;quicrurium, erunt <lb/>maiora lateribus polygoni&etail;, &amp; &longs;imiliter duo latera h d maiora late&shy;<lb/>ribus polygoni&aelig; inclu&longs;&aelig;, ergo latera trapezij erunt maiora omni&shy;<lb/>bus lateribus polygoni&aelig; inclu&longs;&aelig;.<lb/><arrow.to.target n="marg765"/></s>
        </p>
        <p type="margin">
          <s id="id003933"><margin.target id="marg763"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003934"><margin.target id="marg764"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>pri&shy;<lb/>mi, &amp;<emph.end type="italics"/> 16. <lb/><emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003935"><margin.target id="marg765"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003936">Ex hoc habetur demon&longs;tratio propo&longs;itionis: &longs;int du&aelig; line&aelig; a b <lb/>&amp; a c qu&aelig; comprehendant portionem cir&shy;<lb/>culi b c, dico eas e&longs;&longs;e maiores b c portione, <lb/>&longs;i enim a b &amp; a c &longs;unt &aelig;quales diui&longs;o arcu <lb/>b c per &aelig;qualia in f, ducam contingentem </s>
        </p>
        <p type="main">
          <s id="id003937"><arrow.to.target n="marg766"/><lb/>h f k, &longs;i non faciant triangulum &aelig;quicruri&shy;<lb/>um b c d &longs;uper b c, &amp; cuius ambo latera pa<lb/>riter accepta &longs;int &aelig;qualia a b &amp; a c. </s>
          <s id="id003938">Et du&shy;<lb/>cam contingentem &amp; habebo trapezium <lb/><arrow.to.target n="marg767"/><lb/>h b, c k. </s>
          <s id="id003939">Quare &longs;i peripheria circuli b c e&longs;t <lb/><figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg"/><lb/>minor d b &amp; d c pariter acceptis, habeo <expan abbr="intent&utilde;">intentum</expan>, &longs;i non toties <expan abbr="diuid&atilde;">diuidam</expan> <lb/>peripheriam per &aelig;qualia ut fiat figura polygonia &longs;uper b c &aelig;quila&shy;<lb/>tera &amp; &aelig;quiangula, cuius differentia a peripheria &longs;it minor differen<lb/>tia d b &amp; d c &agrave; trapezio b h, k c, id e&longs;t, tribus eius lateribus, nam cum <lb/>d h &amp; d k &longs;int maiores h k, con&longs;tat quod d b &amp; d e &longs;unt maiores h b, <lb/>&amp; k c &amp; h k igitur &longs;it differentia illa l, &amp; <expan abbr="differ&etilde;tia">differentia</expan> peripheri&etail; &agrave; lineis <lb/>polygoni&aelig; minori: igitur cum peripheria &longs;it &aelig;qualis aut maior <lb/>d b &amp; d c, &amp; differentia a lateribus polygoni&aelig; minor qu&agrave;m d b &amp; <lb/>d c, a b, h b, h k, k c, erit minor proportio peripheri&aelig; ad latera poly&shy;<lb/><arrow.to.target n="marg768"/><lb/>goni&aelig; qu&agrave;m d b &amp; d c ad tria latera trapezij, quare minor propor&shy;<lb/><arrow.to.target n="marg769"/><lb/>tio peripheri&aelig; ad d b &amp; d c qu&agrave;m laterum polygoni&aelig; ad tria latera <lb/><arrow.to.target n="marg770"/><lb/>trapezij, &longs;ed latera polygoni&aelig; &longs;unt minora tribus lateribus. </s>
          <s id="id003940">trapezij, <lb/><arrow.to.target n="marg771"/><lb/>igitur peripheria b c e&longs;t minor d b &amp; d e, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>
        </p>
        <p type="margin">
          <s id="id003941"><margin.target id="marg766"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>&amp;<emph.end type="italics"/> 1. <lb/><emph type="italics"/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003942"><margin.target id="marg767"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>eiu&longs;&shy;<lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003943"><margin.target id="marg768"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003944"><margin.target id="marg769"/>P<emph type="italics"/>er<emph.end type="italics"/> 2 <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003945"><margin.target id="marg770"/>P<emph type="italics"/>er<emph.end type="italics"/> 1 <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003946"><margin.target id="marg771"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. <lb/>3 <emph type="italics"/>lemmatis.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id003947">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id003948">Hanc propo&longs;itionem non &longs;crip&longs;i qu&ograve;d e&longs;&longs;et magni momenti, &longs;ed <lb/>propter modum probandi, &longs;i enim re&longs;picis ex uno oppo&longs;ito &longs;cilicet <lb/>quod peripheria circuli &longs;it maior trianguli lateribus, o&longs;tendo de&shy;<lb/>mon&longs;tratione non ducente ad inconueniens, &longs;ed &longs;implici quod ip&longs;a <lb/>peripheria e&longs;t minor trianguli lateribus, &amp; hoc nunquam fuit <expan abbr="fact&utilde;">factum</expan> <lb/>ab aliquo, im&ograve; uidetur plane impo&longs;sibile. </s>
          <s id="id003949">Et e&longs;t res admirabilior <lb/>qu&aelig; inuenta &longs;it ab orbe condito, &longs;cilicet o&longs;tendere aliquid ex &longs;uo <lb/>oppo&longs;ito, demon&longs;tratione non ducente ad impo&longs;sibile &amp; ita, ut <expan abbr="n&otilde;">non</expan> <lb/>po&longs;sit demon&longs;trari ea <expan abbr="dem&otilde;">demom</expan>&longs;tratione ni&longs;i per illud <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan> quod <lb/>e&longs;t contrarium conclu&longs;ioni, uelut &longs;i quis demon&longs;traret qu&ograve;d So&shy;<lb/>crates e&longs;t albus quia e&longs;t niger, &amp; non po&longs;&longs;et demon&longs;trare aliter, &amp; <lb/>ideo e&longs;t long&egrave; maius Chry&longs;ippeo Syllogi&longs;mo.<lb/><arrow.to.target n="marg772"/></s>
        </p>
        <p type="margin">
          <s id="id003950"><margin.target id="marg772"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id003951">Ex hoc patet quod pars line&aelig; exterioris qu&aelig; tangit circulum 


<pb pagenum="232" xlink:href="015/01/251.jpg"/>intercepta &agrave; linea ex centro longior e&longs;t peripheria, &longs;imiliter in&shy;<lb/>tercepta.</s>
        </p>
        <p type="main">
          <s id="id003952"><arrow.to.target n="marg773"/></s>
        </p>
        <p type="margin">
          <s id="id003953"><margin.target id="marg773"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003954">Sit portio circuli a e, &amp; linea a b intercepta &agrave; linea c b ex centro, <lb/><figure id="id.015.01.251.1.jpg" xlink:href="015/01/251/1.jpg"/><lb/>dico ab e&longs;&longs;e longiorem a e, ducatur b e &aelig;qualis a b, ad </s>
        </p>
        <p type="main">
          <s id="id003955"><arrow.to.target n="marg774"/><lb/>circumferentiam, qu&aelig; illi obuiabit, ducanturque c a, c e <lb/><arrow.to.target n="marg775"/><lb/>eritque angulus e c b &aelig;qualis a c b, igitur arcus a d, &aelig;&shy;<lb/>qualis d c, quare a d erit <expan abbr="dimidi&utilde;">dimidium</expan> a e, &amp; a b dimidium <lb/><arrow.to.target n="marg776"/><lb/>a b, b e, facta enim fuit b e &aelig;qualis a b, cum ergo per <lb/>pr&aelig;&longs;entem du&aelig; line&aelig; a b, b e, &longs;int maiores a e, igitur per commu&shy;<lb/>nem animi &longs;ententiam a b maior a d.</s>
        </p>
        <p type="margin">
          <s id="id003956"><margin.target id="marg774"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>tertij <emph.end type="italics"/><lb/>E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003957"><margin.target id="marg775"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id003958"><margin.target id="marg776"/>P<emph type="italics"/>er<emph.end type="italics"/>&verbar;26. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id003959">Propo&longs;itio ducente&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id003960">Rationem &longs;trepitus o&longs;tendere.<lb/><arrow.to.target n="marg777"/></s>
        </p>
        <p type="margin">
          <s id="id003961"><margin.target id="marg777"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id003962">Fit &longs;trepitus ob multitudinem a&euml;ris percu&longs;si, uelut cum tabulis <lb/>percutimus: &amp; cauitatum cau&longs;a, unde ligna &amp; tabul&aelig; leues magis <lb/>&longs;trepunt, &amp; illud Virgilij:</s>
        </p>
        <p type="main">
          <s id="id003963">&mdash;Sonitumque dedere cauern&aelig;.</s>
        </p>
        <p type="main">
          <s id="id003964">Tum uer&ograve; ob ictus impetum, impetus <expan abbr="aut&etilde;">autem</expan> partim uelocitatis cau&shy;<lb/>&longs;a, partim angu&longs;ti&aelig; loci. </s>
          <s id="id003965">Fulmen edit tonitru in quo &amp; caua nebula <lb/>excipit a&euml;rem, &amp; multum impetuque maximo delatum, <expan abbr="ob&longs;trep&utilde;t">ob&longs;trepunt</expan> au<lb/>tem metalla magis quam ligna eo qu&ograve;d magis ob <expan abbr="continuitat&etilde;">continuitatem</expan> par <lb/>tes moueantur. </s>
          <s id="id003966">Indicio e&longs;t, quod intenta ut &aelig;s &amp; tenuia <expan abbr="maior&etilde;">maiorem</expan> &longs;tre<lb/>pitum edunt: &amp; dum &longs;onant tremunt, aurum autem parum &longs;onat, <lb/>quoniam den&longs;i&longs;simum e&longs;t, et minus intentum <expan abbr="arg&etilde;tum">argentum</expan>, minus den <lb/>&longs;um, &amp; magis intentum, quod autem intentum e&longs;t totum &longs;imul mo<lb/>uetur, &amp; ob id &longs;tridet: lignum <expan abbr="aut&etilde;">autem</expan> &amp; tabula &longs;onat, non quia ut me&shy;<lb/>tallum percutiat a&euml;rem, &longs;ed quia in eo a&euml;r percutitur. </s>
          <s id="id003967">Cra&longs;&longs;um <expan abbr="aut&etilde;">autem</expan> <lb/>metallum &amp; lignum non ade&ograve; &longs;onant: metallum quoniam non mo<lb/>uet a&euml;rem, non enim mouetur: lignum quoniam non mouetur, nec <lb/>in eo qui e&longs;t inclu&longs;us a&euml;r, a&euml;r autem facil&egrave; mouetur, &amp; ob id in ligno <lb/>cauo, etiam&longs;i cra&longs;&longs;um &longs;it, &longs;trepitus magnus editur. </s>
          <s id="id003968">Ergo et&longs;i tenue <lb/>&longs;it metallum, quod infixum e&longs;t tabul&etail;, re&longs;onat multum: <expan abbr="n&otilde;">non</expan> quia mo<lb/>ueatur, &longs;ed quoniam <expan abbr="a&etilde;rem">aerrem</expan> in tabula <expan abbr="c&otilde;">com</expan> cutit. </s>
          <s id="id003969">Neque enim tabula per <lb/>&longs;e &longs;ola, qu&aelig; etiam nimis tunderetur &longs;onum edere magnum pote&longs;t <lb/>quoniam cedit: Oportet <expan abbr="aut&etilde;">autem</expan> non cedere quod re&longs;onat, neque metal&shy;<lb/>lum &longs;i cra&longs;&longs;um, &longs;ed hebetem <expan abbr="&longs;on&utilde;">&longs;onum</expan> etiam tabul&etail; infixum reddit, quo&shy;<lb/>niam neque moueri pote&longs;t infixum &amp; cra&longs;&longs;um, nec cauerno&longs;um e&longs;t, &amp; <lb/>tamen excipit ictum, ne lignum re&longs;onet. </s>
          <s id="id003970">Velox autem ictus <expan abbr="n&otilde;">non</expan> acu&shy;<lb/>tum <expan abbr="&longs;on&utilde;">&longs;onum</expan> reddit, &amp; &longs;i cum impetu &longs;it: indicio e&longs;t tonitru &amp; machin&etail; <lb/>bellic&aelig; igne&etail;, contr&agrave; angu&longs;ta fi&longs;tula <expan abbr="acut&utilde;">acutum</expan> &longs;onum reddit, <expan abbr="eti&atilde;">etiam</expan> remi&longs;&shy;<lb/>&longs;&egrave; inflata. </s>
          <s id="id003971">Igitur a&euml;r &longs;oni cau&longs;a e&longs;t &longs;ecundum <expan abbr="mot&utilde;">motum</expan>, ubi ergo multus <lb/>a&euml;r &amp; magnus motus ibi &longs;onus magnus. </s>
          <s id="id003972">Multus quidem aut in ca&shy;


<pb pagenum="233" xlink:href="015/01/252.jpg"/>uerno&longs;o corpore, qui <expan abbr="graui&longs;sim&utilde;">graui&longs;simum</expan> edit <expan abbr="&longs;on&utilde;">&longs;onum</expan> interclu&longs;us, ut <expan abbr="eti&atilde;">etiam</expan> in uo <lb/>cibus, aut quia &agrave; magno corpore &longs;tridulus efficitur, aut inter duo <lb/>corpora, qui grauitate medius e&longs;t. </s>
          <s id="id003973">Impetu uer&ograve; <expan abbr="effici&ttilde;">efficitur</expan> inten&longs;us non <lb/>magnus, nam tonitrus procul audimus non i&longs;tum quamuis celerri&shy;<lb/>mum, acutum uer&ograve; ob angu&longs;tiam loci. </s>
          <s id="id003974">Atque h&etail; cau&longs;&etail; &longs;unt &longs;onorum.</s>
        </p>
        <p type="main">
          <s id="id003975">Propo&longs;itio ducente&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id003976">Cur &longs;cytalis onera portentur facilius, explorare.</s>
        </p>
        <figure id="id.015.01.252.1.jpg" xlink:href="015/01/252/1.jpg"/>
        <p type="main">
          <s id="id003977">Demiror <expan abbr="n&otilde;">non</expan> exact&egrave; cau&longs;am <expan abbr="manife&longs;ti&longs;sim&atilde;">manife&longs;ti&longs;simam</expan> </s>
        </p>
        <p type="main">
          <s id="id003978"><arrow.to.target n="marg778"/><lb/>Ari&longs;totelem non <expan abbr="a&longs;&longs;ecut&utilde;">a&longs;&longs;ecutum</expan> fui&longs;&longs;e, aut potius ad <lb/><arrow.to.target n="marg779"/><lb/>nos <expan abbr="corrupt&atilde;">corruptam</expan> &longs;cripturam perueni&longs;&longs;e: nam qui <lb/>expo <expan abbr="n&utilde;t">nunt</expan> multo minus <expan abbr="intellig&utilde;t">intelligunt</expan>. </s>
          <s id="id003979">Sit ergo cur <lb/>rus humilis &longs;cytalis <expan abbr="iucumb&etilde;s">iucumbens</expan> a b c. </s>
          <s id="id003980">Diximus <lb/><expan abbr="aut&etilde;">autem</expan> &longs;upr&agrave; quid e&longs;&longs;et &longs;cytala &amp; currus rotis, <expan abbr="&qtilde;">quae</expan> <lb/>&longs;untlonge maiores &longs;cytalis e f g h, <expan abbr="dem&otilde;&longs;tran">demon&longs;tran</expan> <lb/><expan abbr="d&utilde;">dum</expan> e&longs;t <expan abbr="&longs;cytal&atilde;">&longs;cytalam</expan>, quamuis minoris ambitus ma&shy;<lb/>gis mouere &lt;08&gt; rotam, <expan abbr="c&utilde;">cum</expan> ergo de una demon&shy;<lb/>&longs;trauerimus, de <expan abbr="o&itilde;bus">oimbus</expan> erit <expan abbr="intelligend&utilde;">intelligendum</expan>. </s>
          <s id="id003981">Quia <lb/>ergo &longs;cytala k l m habet hypomochlion in k et <lb/>m, &amp; <expan abbr="p&otilde;dus">pondus</expan> premit in l, <expan abbr="igi&ttilde;">igitur</expan> rota uer&longs;atilis mo<lb/><arrow.to.target n="marg780"/><lb/><expan abbr="uebi&ttilde;">uebitur</expan> tanto facilius procedendo, quanta e&longs;t <expan abbr="l&otilde;">lom</expan> gitudo l m &amp; l k, &longs;ed &amp; <lb/>rotul&etail; ill&etail; <expan abbr="uer&longs;ab&utilde;t">uer&longs;abunt</expan> hypomochlion, &qring;d e&longs;t l <expan abbr="c&otilde;paratione">comparatione</expan> k &amp; m col&shy;<lb/>lopum, <expan abbr="igi&ttilde;">igitur</expan> facilius multo <expan abbr="uer&longs;abi&ttilde;">uer&longs;abitur</expan> currus &agrave; &longs;cytalis &lt;08&gt; rotis. </s>
          <s id="id003982">Et hoc <lb/>e&longs;t quod dixit Philo&longs;ophus. </s>
          <s id="id003983">In utri&longs;que. </s>
          <s id="id003984">n. </s>
          <s id="id003985">his <expan abbr="reuolui&ttilde;">reuoluitur</expan> circulus et mo<lb/>tus <expan abbr="impelli&ttilde;">impellitur</expan>, intelligit <expan abbr="mutu&atilde;">mutuam</expan> <expan abbr="commutation&etilde;">commutationem</expan> hypomochlij cum col<lb/>lopibus, nam ut <expan abbr="trah&atilde;tur">trahantur</expan> rotul&etail; <expan abbr="&qtilde;">quae</expan> &longs;unt hypomochlij loco, collopes <lb/><expan abbr="terminan&ttilde;">terminantur</expan> in medio: ut <expan abbr="a&utilde;t">aunt</expan> <expan abbr="uerta&ttilde;">uertatur</expan> axis, qui &amp; hypomochlion in me&shy;<lb/>dio <expan abbr="collop&utilde;">collopum</expan> initium &longs;int rotul&aelig;. </s>
          <s id="id003986">Ex quo <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan>, &qring;d quanto <expan abbr="l&otilde;giores">longiores</expan> <lb/>erunt l k l t &amp; l m, tanto facilius <expan abbr="mouebun&ttilde;">mouebuntur</expan> currus, at quanto humi&shy;<lb/>liores, mod&ograve; non obruantur in terra, quoniam tardius mouentur, <lb/>qu&aelig; minorem habent circuitum, qu&aelig; autem tardius mouentur, fa<lb/>cilius mouentur, ut &longs;upr&agrave; &longs;&aelig;pius demon&longs;tratum e&longs;t: Ob has ergo <lb/>duas cau&longs;as pondera facilius feruntur curribus cum &longs;cytalis, qu&agrave;m <lb/>cum rotis magnis mod&ograve; terra non obruantur.</s>
        </p>
        <p type="margin">
          <s id="id003987"><margin.target id="marg778"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id003988"><margin.target id="marg779"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 114.</s>
        </p>
        <p type="margin">
          <s id="id003989"><margin.target id="marg780"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 71</s>
        </p>
        <p type="main">
          <s id="id003990">Propo&longs;itio ducente&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id003991">Cur pluribus trochleis pondera facilius eleuentur o&longs;ten dere.</s>
        </p>
        <p type="main">
          <s id="id003992">Dictum e&longs;t &longs;atis de hoc in lib. 

de Subtilitate, at nunc quod ad de&shy;<lb/><arrow.to.target n="marg781"/><lb/>mon&longs;trationem attinet <expan abbr="eor&utilde;">eorum</expan> &longs;ubijciam. </s>
          <s id="id003993">Quia. n. </s>
          <s id="id003994">&longs;ingul&etail; rotul&etail; diffi <lb/>culter <expan abbr="mouen&ttilde;">mouentur</expan>, igitur nece&longs;&longs;e e&longs;t &longs;ingulas participes e&longs;&longs;e grauitatis, <lb/>igitur &amp; totam <expan abbr="grauitat&etilde;">grauitatem</expan> e&longs;&longs;e diui&longs;am: quare ut in <expan abbr="pr&etail;ced&etilde;ti">pr&etail;cedenti</expan> facilius <lb/>moueri. </s>
          <s id="id003995">Habent &amp; rotul&etail; ip&longs;&etail; centrum &longs;eu axem hypomochlij, &longs;eu <lb/><arrow.to.target n="marg782"/><lb/>fulcimenti loco, ambitum <expan abbr="a&utilde;t">aunt</expan> iuxta &longs;emidiametrum, uelut collopes <pb pagenum="234" xlink:href="015/01/253.jpg"/>&longs;eu uectes, quare tanto facilius mouebuntur quanto maiores <expan abbr="er&utilde;t">erunt</expan>, <lb/><figure id="id.015.01.253.1.jpg" xlink:href="015/01/253/1.jpg"/><lb/>&amp; ut plures. </s>
          <s id="id003996">Vna enim alterius loco fungitur uectis. </s>
          <s id="id003997">Trochlea qui&shy;<lb/>dem e&longs;t, ut uides, in&longs;trumentum longum &longs;upr&agrave; angu&longs;tius, &longs;ed non, <lb/>cra&longs;&longs;um, in quo plures orbiculi &longs;olent collo cari, unde &longs;&aelig;pe numero <lb/>trochle&aelig; nomine intelligimus orbiculos ei inclu&longs;os, circa quos fu&shy;<lb/>nis uocatur, ut in trochleis &amp; orbiculi &amp; funes includuntur. </s>
          <s id="id003998">Succu&shy;<lb/>lis etiam &longs;olent capita funium trahi: ut uectis auxilio im&ograve; nonnum&shy;<lb/>qu&agrave;m rotarum facilius pondera eleuantur.<lb/><arrow.to.target n="marg783"/></s>
        </p>
        <p type="margin">
          <s id="id003999"><margin.target id="marg781"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004000"><margin.target id="marg782"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 71.</s>
        </p>
        <p type="margin">
          <s id="id004001"><margin.target id="marg783"/>8. <emph type="italics"/>de<emph.end type="italics"/> R<emph type="italics"/>epub.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004002">Propo&longs;itio ducente&longs;ima quinta, &longs;uper uerbis Platonis, <lb/>de fine Reipub.</s>
        </p>
        <p type="main">
          <s id="id004003">&ldquo;E&longs;t autem ei quod diuinitus generandum e&longs;t circuitus, quem nu<lb/>merus <expan abbr="c&otilde;tinet">continet</expan> perfectus. </s>
          <s id="id004004">Human&aelig; uer&ograve;, in quo primum argumen<lb/>tationes &longs;uperantes, ut &longs;uperat&aelig; tres di&longs;tanti&aelig;: quatuor autem ter&shy;<lb/>minos accipientes, &longs;imilium &amp; di&longs;similium, ab <expan abbr="undanti&utilde;">undantium</expan> &amp; deficien<lb/>tium cuncta corre&longs;pondentia, &amp; rationem habentia inuicem effece<lb/>runt. </s>
          <s id="id004005">Quorum &longs;exquitertium fundamentum quinario <expan abbr="iunct&utilde;">iunctum</expan> duas <lb/>efficit harmonias ter aucta quidem: &aelig;qualem &aelig;qualiter centum to<lb/>ties, quandam autem &aelig;qualem quidem, longitudine <expan abbr="a&utilde;t">aunt</expan> &longs;ingulum <lb/>quidem numerorum &agrave; diametris <expan abbr="ration&etilde;">rationem</expan> habentibus quinarij indi <lb/>gentibus uno &longs;ingulis: non habentibus rationem <expan abbr="a&utilde;t">aunt</expan> duobus, cen&shy;<lb/>tum autem cuborum ternarij. </s>
          <s id="id004006">Totus autem hic numerus geometri <lb/>cus talem authoritatem habet ad potiorem deterioremque <expan abbr="genera&shy;tion&etilde;">genera&shy;<lb/>tionem</expan>. </s>
          <s id="id004007">Quem locum Ari&longs;toteles ita declarat. </s>
          <s id="id004008">Quorum &longs;exquiter&shy;<lb/>tium fundamentum quinario coniunctum duas exhibet harmo&shy;<lb/>nias, <expan abbr="inqui&etilde;s">inquiens</expan>, <expan abbr="qu&atilde;do">quando</expan> numerus diagrammatis huius <expan abbr="efficia&ttilde;">efficiatur</expan> &longs;olidus.&rdquo;</s>
        </p>
        <p type="main">
          <s id="id004009"><arrow.to.target n="marg784"/></s>
        </p>
        <p type="margin">
          <s id="id004010"><margin.target id="marg784"/>Q<emph type="italics"/>uin<emph.end type="italics"/> P<emph type="italics"/>olyt.<emph.end type="italics"/><lb/>C<emph type="italics"/>ap.<emph.end type="italics"/> 12.</s>
        </p>
        <p type="main">
          <s id="id004011"><foreign lang="greek">*gusqmh\n</foreign> <expan abbr="fundam&etilde;tum">fundamentum</expan> interpretatus &longs;um, quod radix pro latere in <lb/>hac materia accipi po&longs;&longs;et. </s>
          <s id="id004012">Par e&longs;t ut in diuina generatione numerus </s>
        </p>
        <p type="main">
          <s id="id004013"><arrow.to.target n="marg785"/><lb/><expan abbr="accipere&ttilde;">acciperetur</expan> perfectus: ut intelligat generationem confe&longs;tim &longs;equi cor <lb/>ruptionem: nam &longs;ermo e&longs;t de corruptione, corrumpitur <expan abbr="a&utilde;t">aunt</expan> unum&shy;<lb/>quodque ut aliud generetur, malum enim e&longs;t ob bonum, non contr&agrave;. <lb/></s>
          <s id="id004014">Liquet autem ex Euclide talem numerum e&longs;&longs;e octies mille <expan abbr="cent&utilde;">centum</expan> ui&shy;<lb/>ginti octo. </s>
          <s id="id004015">Et hic e&longs;t finis <expan abbr="omni&utilde;">omnium</expan> urbium diuinus, cuius <expan abbr="quadrupl&utilde;">quadruplum</expan> <lb/>uelut in c&oelig;li re&longs;titutionibus, ac continuato ordine &longs;olet ob&longs;eruari, <lb/>e&longs;t prop&egrave; annus magnus: ueri&longs;imile e&longs;t enim <expan abbr="t&atilde;to">tanto</expan> tempore <expan abbr="c&otilde;fundi">confundi</expan> <lb/>decima, &longs;cilicet totius circuitus parte. </s>
          <s id="id004016">Human&aelig; uer&ograve; intelligit qua&shy;<lb/><figure id="id.015.01.253.2.jpg" xlink:href="015/01/253/2.jpg"/><arrow.to.target n="table29"/><lb/>tuor &agrave; monade numeros, aut in quauis ratione principium li&shy;<lb/>neam &longs;uperficiem corpus, ut <expan abbr="un&utilde;">unum</expan>, duo, quatuor, octo pariter <lb/>octo: duodecim decem octo uiginti <expan abbr="&longs;ept&etilde;">&longs;eptem</expan>: inter h&aelig;c &longs;unt tria <lb/>&longs;patia, &amp; octo cum uiginti &longs;eptem &longs;unt di&longs;similia &amp; deficien&shy;<lb/>tia: maiora <expan abbr="e&mtilde;">emm</expan> &longs;unt &longs;uis partibus &agrave; quibus numerantur. </s>
          <s id="id004017">Contr&agrave; de&shy;<lb/>cem octo &amp; duodecim &longs;unt &longs;imilia atque ab <expan abbr="und&atilde;tia">undantia</expan>, &amp; corre&longs;ponden 


<pb pagenum="235" xlink:href="015/01/254.jpg"/>tem habent rationem inuicem. </s>
          <s id="id004018">H&aelig;c Ari&longs;toteles omittit, ut ad in&shy;<lb/>troductionem, non rem pertinentia, uelut &amp; finem tanqu&agrave;m ex <lb/>pr&aelig;cedentibus notum. </s>
          <s id="id004019">Vnde uerba Ari&longs;totelis &longs;unt ad unguem <lb/>eadem uerbis Platonis, &longs;cilicet: &ldquo;Quorum &longs;exquitertium funda&shy;<lb/>mentum quinario iunctum duas efficit harmonias: loco autem ter <lb/>aucta quidem, &longs;cribit Ari&longs;toteles: efficiatur &longs;olidus, id e&longs;t cubus, ut <lb/>in quadratum &longs;uum ducatur: loco autem uerborum &aelig;qualem &aelig;&shy;<lb/>qualiter centum centies, u&longs;que illuc &agrave; diametris rationem habenti&shy;<lb/>bus quinarij ponit numerum diagrammatis.&rdquo; E&longs;t autem diagram&shy;<lb/>ma, quod Plato uocat diametrum, cum numerus pote&longs;t ferm&egrave; du&shy;<lb/>plum numeri alterius, ut 3 duplum 2, &amp; 7 duplum 5, &amp; 17 duplum <lb/>12, &amp; &longs;emper numerus hic dimetiens, excedit duplum alterius uno, <lb/>quod ex his patet, qu&aelig; ab Euclide demon&longs;trata &longs;unt in decimo li&shy;<lb/>bro. </s>
          <s id="id004020">Quare &longs;i debet e&longs;&longs;e quadratum eius monade maius duplo, al&shy;<lb/>terius quadrati, &amp; duplum &verbar; alterius quadrati e&longs;t par, igitur addi&shy;<lb/>ta monade erit impar, ergo latus eius dimetiens impar &longs;emper: la&shy;<lb/>tera autem ip&longs;a quadratorum, qu&aelig; duplicantur aliquando pa&shy;<lb/>ria &longs;unt ut 2, &amp; tunc quadratum dimetientis e&longs;t unum plus duplo <lb/>ut 9 e&longs;t maius 8 monade, &longs;i uer&ograve; latera imparia &longs;int, erit quadratum <lb/>dimetientis uno minus duplo, ut 49 quadratum 7 e&longs;t minus uno <lb/>50, duplo 25, quadrati 5. Ex quo patet agnatio, ut ita dicam in&shy;<lb/>ter 7 &amp; 5.</s>
        </p>
        <p type="margin">
          <s id="id004021"><margin.target id="marg785"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <table>
          <table.target id="table29"/>
          <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 &longs;exquitertia e&longs;t, ac &longs;i diceret, ex horum <lb/>numerorum &longs;erie &longs;umemus &longs;eptenarium principium epitrite, &amp; di&shy;<lb/>metientem 5, quos &longs;imul iungemus.</s>
        </p>
        <p type="main">
          <s id="id004023">Propo&longs;itio ducente&longs;ima &longs;exta.</s>
        </p>
        <figure id="id.015.01.254.1.jpg" xlink:href="015/01/254/1.jpg"/>
        <p type="main">
          <s id="id004024">Rhombi pa&longs;siones qua&longs;dam declarare.</s>
        </p>
        <p type="main">
          <s id="id004025">Sit a d recta diui&longs;a in k per &aelig;qualia, cui &longs;u&shy;<lb/><arrow.to.target n="marg786"/><lb/>per&longs;tent k b &amp; k c ad perpendiculum inter &longs;e <lb/>&aelig;quales, &amp; &longs;ingul&aelig; <expan abbr="ear&utilde;">earum</expan> minores k a &amp; k d, <lb/><arrow.to.target n="marg787"/><lb/>&amp; <expan abbr="perficia&ttilde;">perficiatur</expan> figura quadrilatera a b d c, cuius <lb/>latera erunt omnia &aelig;qualia inuicem, &amp; angu<lb/>li a &amp; d oppo&longs;iti, &amp; b &amp; c oppo&longs;iti etiam inui<lb/>cem &etail;quales. </s>
          <s id="id004026">Sed b &amp; c maiores erunt a &amp; d: <lb/><arrow.to.target n="marg788"/><lb/>&amp; ideo talem figuram appellauit Ari&longs;toteles rhombum &agrave; pi&longs;cis &longs;i&shy;<lb/>militudine in medio latioris <expan abbr="qu&atilde;">quam</expan> in extremis, cuius <expan abbr="tam&etilde;">tamen</expan> longitudo <lb/>latitudine maior e&longs;t. </s>
          <s id="id004027">Dicit ergo Ari&longs;toteles, &qring;d &longs;i rhombus ip&longs;e cir&shy;<lb/><arrow.to.target n="marg789"/><lb/>cumuoluatur, ita ut b tran&longs;iret per b a c, &amp; a per a c d, a maius &longs;pa&shy;<lb/>tium tran&longs;iret ex recta, &longs;cilicet a k d qu&agrave;m b, quod tran&longs;iret b k c. </s>
          <s id="id004028">Et <lb/>ad hoc a&longs;&longs;umit, qu&ograve;d cum angulus c &longs;it maior a, igitur du&aelig; line&aelig; <lb/>a c d &longs;unt minus curu&aelig; quam du&aelig; b a c, igitur b a c habent ratio&shy;


<pb pagenum="236" xlink:href="015/01/255.jpg"/>nem currui, &amp; a c d recti. </s>
          <s id="id004029">Ergo &longs;i in &aelig;quali <expan abbr="t&etilde;poris">temporis</expan> &longs;patio b, &longs;uperet <lb/>b a c &amp; a, a c d, magis per rectam feretur a qu&agrave;m b, &longs;ed quod rectum <lb/>e&longs;t maius occupat &longs;patium: igitur uelocius fertur a in d compara&shy;<lb/>tione habita ad a d qu&agrave;m b in c, comparatione habita ad b c.</s>
        </p>
        <p type="margin">
          <s id="id004030"><margin.target id="marg786"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004031"><margin.target id="marg787"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004032"><margin.target id="marg788"/>P<emph type="italics"/>er<emph.end type="italics"/> 25. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004033"><margin.target id="marg789"/>Q<emph type="italics"/>u&aelig;&longs;t.<emph.end type="italics"/> 23.<lb/>M<emph type="italics"/>ech.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004034">Pro intellectu reliquorum ab eo dictorum, &amp; quorundam mira&shy;<lb/>bilium, proponatur alius rhombus illi &etail;qualis, in tabula pictus deli<lb/>neatis lateribus &amp; diametris, qui fit l m o n, &amp; diametri l p o &amp; m p <lb/>n, &amp; ab&longs;cindatur hic ex &longs;uperficie, &amp; &longs;uperponatur ita, ut puncta l m <lb/>o n ordinatim cadant, &amp; aptentur <expan abbr="p&utilde;ctis">punctis</expan> a b d c, &amp; p aptetur ip&longs;i k. <lb/></s>
          <s id="id004035">Et tunc &longs;i rhombus l o totus moueretur, nece&longs;&longs;e e&longs;t, ut moueatur &longs;e&shy;<lb/>cundum latus aliquod, ut pote l m, &amp; &etail;quidi&longs;tans a b, igitur dicetur <lb/><figure id="id.015.01.255.1.jpg" xlink:href="015/01/255/1.jpg"/><lb/>moueri &longs;uper latus aliquod, &longs;cilicet a c: atque hic e&longs;t mo<lb/>tus, quem Ari&longs;toteles uocat <expan abbr="mot&utilde;">motum</expan> a b &longs;uper latus a c. <lb/></s>
          <s id="id004036">Si <expan abbr="a&utilde;t">aunt</expan> fingamus quie&longs;cere latus aliquod l o, uel pars <lb/>lateris, non po&longs;&longs;et omnino moueri in &longs;uperficie a d <lb/>rhombi: et ita <expan abbr="n&otilde;">non</expan> perinde e&longs;&longs;et ac &longs;i a d rhombus mo<lb/>ueretur, quod tamen &longs;upponit Ari&longs;toteles. </s>
          <s id="id004037">Neque <expan abbr="eti&atilde;">etiam</expan> <lb/>&longs;i quie&longs;ceret punctum aliud quam p haberet ratio&shy;<lb/>nem motus regularis, quod ab illo &longs;upponitur: reli&shy;<lb/>quum e&longs;t igitur, ut rhombus l o moueatur uice rhombi a d &longs;eruan&shy;<lb/>do centrum, id e&longs;t punctum p in puncto k. </s>
          <s id="id004038">Dicamus ergo primum <lb/>de motu compo&longs;ito Ari&longs;totelis, &amp; p&ograve;&longs;t de no&longs;tro.</s>
        </p>
        <p type="main">
          <s id="id004039">Moueatur l m &longs;uper a c, &aelig;quidi&longs;tans &longs;emper a b, ut &longs;eruet &longs;itum <lb/>quem habebat ita, quod <expan abbr="extrem&utilde;">extremum</expan> line&aelig; l m &longs;it &longs;emper in linea a c, &amp; <lb/>l punctum quod gerit uicem a, de&longs;cendat tantum in linea l m, quan&shy;<lb/>tum l extremum in linea a c: dicit Philo&longs;ophus, quod a &longs;eu l &longs;emper <lb/>de&longs;cendet in linea a d, &amp; erit in e a. </s>
          <s id="id004040">Supponatur quae latus l m fit f g, &amp; <lb/>erit l n, f t, ducatur <expan abbr="a&utilde;t">aunt</expan> ex r puncto &longs;ectionis diametri, &amp; lateris l m li <lb/><arrow.to.target n="marg790"/><lb/>near q, &aelig;quidi&longs;tans a f, <expan abbr="igi&ttilde;">igitur</expan> rhombus a q r f e&longs;t &longs;imilis rhombo toti <lb/>a b d c, &amp; proportio a f ad fr, ut a c ad c d, &longs;ed a c e&longs;t &etail;qualis c d, <expan abbr="igi&ttilde;">igitur</expan> a f <lb/>e&longs;t &aelig;qualis f r, &longs;ed l de&longs;cendit in l m, <expan abbr="quant&utilde;">quantum</expan> e&longs;t a f ex &longs;uppo&longs;ito, <expan abbr="igi&ttilde;">igitur</expan> <lb/><expan abbr="punct&utilde;">punctum</expan> l &longs;emper erit in linea a d. </s>
          <s id="id004041">Po&longs;t deficiunt qu&aelig;dam uerba: ob <lb/>qu&aelig; nemo intellexit &longs;ententiam Philo&longs;ophi, &amp; <expan abbr="tam&etilde;">tamen</expan> au&longs;i &longs;unt impo<lb/>nere lectoribus, tan&lt;08&gt; intellexi&longs;&longs;ent, tres &longs;imul errores admittendo, <lb/>&longs;cilicet Ari&longs;totelem ob propriam ignorantiam, ut &longs;tultum accu&longs;an&shy;<lb/>do, qui fal&longs;a dicat, &amp; demon&longs;trare nitatur: produnt &longs;e ip&longs;os cum <lb/>&longs;ua impudentia. </s>
          <s id="id004042">Et lectoribus imponere conantur, debet ergo &longs;ic <lb/>legi &lpar;&ldquo;b in ip&longs;a b c diametro latum, ubi latus b d moueatur in late&shy;<lb/>re b a, &amp; b &aelig;qualiter uer&longs;us d in b d, &aelig;qualis enim e&longs;t ip&longs;a b e&rdquo;&rpar; <lb/>Tunc enim con&longs;tat ut hic dixi, m moueri per b c rectam ut l per a d: <lb/>Dicit ergo <expan abbr="c&utilde;">cum</expan> b d <expan abbr="mouea&ttilde;">moueatur</expan> in b a, tran&longs;it unico motu <expan abbr="tot&atilde;">totam</expan> b a, &amp; pun<pb pagenum="237" xlink:href="015/01/256.jpg"/><expan abbr="ct&utilde;">ctum</expan> tamen b, quod <expan abbr="moue&ttilde;">mouetur</expan> duobus motibus, non pertran&longs;it ni&longs;i b c, <lb/>qu&aelig; pote&longs;t e&longs;&longs;e minor b a: nam <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> quod <expan abbr="qu&atilde;do">quando</expan> m erit in a, o erit <lb/>in e, &amp; quia m de&longs;cendit in o, in eodem tempore, ergo o erit in c, &amp; <lb/><expan abbr="tr&atilde;&longs;iuit">tran&longs;iuit</expan> &longs;emper per rectam b c: igitur m e&longs;t minus <expan abbr="mot&utilde;">motum</expan> duobus mo<lb/>tibus qu&agrave;m m l unico <expan abbr="tant&utilde;">tantum</expan>. </s>
          <s id="id004043">Et quia aliquis dicere potui&longs;&longs;et non e&longs;t <lb/>mirum, quod m &longs;it minus motum duobus motibus qu&agrave;m l m latus <lb/>unico tantum: quia m mouetur motu contrario motui lateris: nam <lb/>latus m o mouetur in latere b a a&longs;cendendo, et punctum m uer&longs;us o <lb/>in ip&longs;o m o de&longs;cendendo. </s>
          <s id="id004044">Dicit Philo&longs;ophus, hoc e&longs;t mirum, quia <lb/>cum idem contingat in motu l, cuius latus mouetur per a c, &amp; l per l <lb/>m recedendo in partem contrariam, nihilominus uelocius motum <lb/>e&longs;t l, qu&agrave;m latus l m, quia a d e&longs;t longior a c. </s>
          <s id="id004045">Ex quo patet, quae qu&etail;&longs;tio <lb/>Philo&longs;ophi e&longs;t una tantum, &amp; non du&aelig;. </s>
          <s id="id004046">Et e&longs;t cur motum duobus <lb/>motibus in rhombo, in uno mouetur uelocius latere tantum moto <lb/>uno motu, in alio tardius? </s>
          <s id="id004047">Et quia aliquis dicere po&longs;&longs;et, &qring;d b c po&longs;&shy;<lb/>&longs;et e&longs;&longs;e <expan abbr="l&otilde;gior">longior</expan> a c: Dicit Philo&longs;ophus, uerum e&longs;t, &longs;ed ego po&longs;&longs;um in&shy;<lb/>uenire talem rhombum, qui etiam habeat a c longiorem, &amp; tunc ni&shy;<lb/>hilominus <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan> quod dico. </s>
          <s id="id004048">Aliud <expan abbr="a&utilde;t">aunt</expan>, quod docet ex hac demon&shy;<lb/>&longs;tratione, e&longs;t quae ex duobus motibus rectis diuer&longs;is pote&longs;t fieri unus <lb/>motus rectus diuer&longs;us: igitur idem punctum, puta formica poterit <lb/>&longs;imul, &amp; &longs;emel moueri duobus motibus rectis diuer&longs;is. </s>
          <s id="id004049">Et hoc e&longs;t, <lb/>quia primus motus e&longs;t rectus &longs;olum &longs;ecundum formam, &amp; non &longs;e&shy;<lb/>cundum materiam: &amp; alter &longs;ecundus, &longs;cilicet mi&longs;tus e&longs;t &longs;ecundum <lb/>materiam &amp; non &longs;ecundum formam per rectam.</s>
        </p>
        <p type="margin">
          <s id="id004050"><margin.target id="marg790"/>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>&longs;exti <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004051">Ex hoc <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan> aliud magis <expan abbr="mir&utilde;">mirum</expan>, et e&longs;t iuxta <expan abbr="no&longs;tr&utilde;">no&longs;trum</expan> motum rhom<lb/>bi l o in rhombo a d, fixo centro p in centro k, &amp; <expan abbr="mouea&ttilde;">moueatur</expan> quomodo <lb/>libet, l, dico quod l f &longs;emper &aelig;qualis erit a f, quia <expan abbr="e&mtilde;">emm</expan> k l &amp; k a &longs;unt &aelig;&shy;<lb/><figure id="id.015.01.256.1.jpg" xlink:href="015/01/256/1.jpg"/><lb/>quales, <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent una linea ante motum ducta, l a erit <lb/>angulus k l a, &aelig;qualis angulo k a l, &longs;ed angulus k a c <lb/><arrow.to.target n="marg791"/><lb/>e&longs;t &aelig;qualis angulo k l m, cum angulus k l m e&longs;&longs;et <expan abbr="id&etilde;">idem</expan> <lb/>angulo k a b, &amp; angulus k a b e&longs;t <expan abbr="&aelig;&qtilde;lis">&aelig;qualis</expan> angulo k a c, <lb/><arrow.to.target n="marg792"/><lb/>igitur angulus k l m e&longs;t &aelig;qualis angulo k a c, <expan abbr="igi&ttilde;">igitur</expan> re&longs;i<lb/>duus fl a e&longs;t &aelig;qualis re&longs;iduo f a l, quare f a &aelig;qualis <lb/><arrow.to.target n="marg793"/><lb/>fl. </s>
          <s id="id004052">Si igitur quantum procedit latus m l in a c, <expan abbr="t&atilde;tum">tantum</expan> <lb/>de&longs;cendat punctum in linea l m punctum perpetuo, erit in linea a c, <lb/>&amp; per eam mouebitur. </s>
          <s id="id004053">Vnde &longs;equitur quod</s>
        </p>
        <p type="margin">
          <s id="id004054"><margin.target id="marg791"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004055"><margin.target id="marg792"/>P<emph type="italics"/>er<emph.end type="italics"/> 34. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004056"><margin.target id="marg793"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004057">Quod <expan abbr="punct&utilde;">punctum</expan> l <expan abbr="mouebi&ttilde;">mouebitur</expan> duobus </s>
          <s id="id004058">motibus. </s>
          <s id="id004059">uno recto in linea, &longs;cilicet <lb/><arrow.to.target n="marg794"/><lb/>l m, &amp; altero circulari. </s>
          <s id="id004060">&longs;. </s>
          <s id="id004061">circa <expan abbr="centr&utilde;">centrum</expan> k, &amp; <expan abbr="t&ntilde;">tnm</expan> <expan abbr="mouebi&ttilde;">mouebitur</expan> uer&egrave; motu re&shy;<lb/>cto <expan abbr="t&mtilde;">tmm</expan> in alia linea, &longs;cilicet a c, &amp; hoc e&longs;t <expan abbr="prim&utilde;">primum</expan> admirabile. </s>
          <s id="id004062">Aliud e&longs;t</s>
        </p>
        <p type="margin">
          <s id="id004063"><margin.target id="marg794"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004064">Quod <expan abbr="punct&utilde;">punctum</expan> l <expan abbr="mouebi&ttilde;">mouebitur</expan> duobus motibus, &amp; per ip&longs;os <expan abbr="mouebi&ttilde;">mouebitur</expan> <lb/><arrow.to.target n="marg795"/><lb/>ad <expan abbr="ungu&etilde;">unguem</expan> uno motu &etail;quali uni <expan abbr="eor&utilde;">eorum</expan>, ita &qring;d alius motus nihil addet 


<pb pagenum="238" xlink:href="015/01/257.jpg"/>nec minuet. </s>
          <s id="id004065">Patet quia mouebitur, gratia exempli, primo motu ex l <lb/>in f, &amp; p&ograve;&longs;t motu circulari, &amp; uer&egrave; erit motum ex a in f, qui motus <lb/>e&longs;t &aelig;qualis motui priori propri&ograve;, &amp; &longs;olo ex l in f.</s>
        </p>
        <p type="margin">
          <s id="id004066"><margin.target id="marg795"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004067">Propo&longs;itio ducente&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id004068">Proportionem agentium naturalium in tran&longs;mutatione con&shy;<lb/>&longs;yderare.<lb/><arrow.to.target n="marg796"/></s>
        </p>
        <p type="margin">
          <s id="id004069"><margin.target id="marg796"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004070">Sit latitudo a b ad conuer&longs;ionem terr&aelig; in aurum me&shy;<lb/>dium perfectionis a b &longs;it c, &amp; medium a c d b, cuius dimi&shy;<lb/>dium &longs;it e b. </s>
          <s id="id004071">Et fiat commutatio a c in f g, tempore dimi&shy;<lb/>dium f g, g h in g h deberet peruenire ad perfectionem d, <lb/>quoniam ratio a c ad c d, ut f g ad g h. </s>
          <s id="id004072">At uer&ograve; dum tran&longs;i&shy;<lb/>ret terra ad perfectionem c tota re&longs;i&longs;tebat, iam adepta per&shy;<lb/>fectione a c non re&longs;i&longs;tit, ni&longs;i pro medietate, at proportio cu<lb/>iuslibet quantitatis ad dimidium alterius producitur ex <lb/>proportione eadem &amp; dupla, dupla igitur e&longs;t proportio <lb/>agentis ad imperfectionem a c ei qu&aelig; e&longs;t ad a b, igitur in di<lb/>midio temporis g h acquiret perfectionem c d, &amp; &longs;it g k di<lb/>midium g h, erit ergo tempus totum fk, in quo acquiret <lb/>a d. </s>
          <s id="id004073">At ratio h&aelig;c con&longs;tare non pote&longs;t, nam &longs;i diuidatur &longs;pa<lb/><figure id="id.015.01.257.1.jpg" xlink:href="015/01/257/1.jpg"/><lb/>tium a b in trientes fient trientes duo, &amp; quarta pars in perfectione <lb/>a d: &longs;ed iam multo citius acquiret quam in fk tempore, quod e&longs;t di&shy;<lb/>midium &amp; octaua pars. </s>
          <s id="id004074">Sed hoc non cogit, quoniam partes prim&aelig; <lb/>&longs;unt &longs;emper contumaciores, &amp; ut di&longs;ponuntur fiunt magis obedi&shy;<lb/>entes, non iuxta proportionem &longs;impliciter, &longs;ed ut &longs;unt in materia, <lb/>&amp; ide&ograve; h&aelig;c actio e&longs;t &longs;imilior proportioni exce&longs;&longs;us, &amp; e&longs;t Arithme&shy;<lb/>tica quam capacitatis &longs;cilicet Geometric&aelig;.</s>
        </p>
        <p type="main">
          <s id="id004075"><arrow.to.target n="marg797"/></s>
        </p>
        <p type="margin">
          <s id="id004076"><margin.target id="marg797"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004077">Ex hoc patet, quod res qu&aelig; ad &longs;ummam maturitatem perueni&shy;<lb/>unt, maxim&egrave; <expan abbr="acquir&utilde;t">acquirunt</expan> perfectionem in exiguo tempore, ut gemm&etail;, <lb/>aurum, infans. </s>
          <s id="id004078">Ergo oportet maxim&egrave; iuxta finem cauere, ne detur <lb/>occa&longs;io ulla accelerandi partum.</s>
        </p>
        <p type="main">
          <s id="id004079">Propo&longs;itio ducente&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id004080">Mota res &agrave; centro grauitatis per priorem motum in reditu uelo&shy;<lb/>cius mouetur, quam &longs;i quieuerit.</s>
        </p>
        <p type="main">
          <s id="id004081"><arrow.to.target n="marg798"/></s>
        </p>
        <p type="margin">
          <s id="id004082"><margin.target id="marg798"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004083">Sit a b c lectus pen&longs;ilis, in quo ho <lb/>mo aut patera, in qua aqua uel <expan abbr="ui&shy;n&utilde;">ui&shy;<lb/>num</expan>, &amp; &longs;it <expan abbr="c&etilde;trum">centrum</expan> grauitatis d, quod <lb/>nece&longs;&longs;ari&ograve; e&longs;t in linea loci, cui anne<lb/>xus e&longs;t lectus a g, &amp; in patera lo ci <lb/>medij manus continentis pateram <lb/><expan abbr="c&utilde;">cum</expan> centro qu&aelig; &longs;it a g, quibus &longs;tan&shy;<lb/>tibus o&longs;tendendum e&longs;t primo.</s>
        </p>
        <figure id="id.015.01.257.2.jpg" xlink:href="015/01/257/2.jpg"/>
        <pb pagenum="239" xlink:href="015/01/258.jpg"/>
        <p type="head">
          <s id="id004084">LEMMA PRIMVM.</s>
        </p>
        <p type="main">
          <s id="id004085">Omne graue <expan abbr="mot&utilde;">motum</expan> &agrave; centro grauitatis, re&longs;tituto ad eundem &longs;itum <lb/>pondere mobili aut inmobili, continente ultra centrum grauitatis <lb/>naturalis uiolenter fertur.</s>
        </p>
        <p type="main">
          <s id="id004086">Seu &longs;it pondus per &longs;e non fluctuans in pen&longs;ili lecto, &longs;eu humor in </s>
        </p>
        <p type="main">
          <s id="id004087"><arrow.to.target n="marg799"/><lb/>patera, quum <expan abbr="p&otilde;dus">pondus</expan> moueatur &longs;olum ratione una, &longs;cilicet lecti pen&shy;<lb/>&longs;ilis homo uel plumbum, humor autem aqua uel uinum bifariam <lb/>&amp; ratione pater&aelig; &longs;i mobilis &longs;it in a laxa manu, &amp; etiam per humo&shy;<lb/>rem ip&longs;um redeuntem ad locum <expan abbr="&longs;u&utilde;">&longs;uum</expan>: ade&ograve; qu&ograve;d &longs;i e&longs;&longs;et &amp; immobi&shy;<lb/>lis patera, humor &longs;altem reflueret propria inundatione ad locum <lb/>&longs;uum centri grauitatis, licet in patera e&longs;&longs;et immobilis locus grauita&shy;<lb/>tis uelocius &amp; maiore cum impetu, ade&ograve; ut tran&longs;eat uer&longs;us e, <expan abbr="c&utilde;">cum</expan> fu <lb/>erit motus primus ex e in f, et re&longs;titutio ex fin e: &longs;eu in immobili pon<lb/>dere mobilis continenti, ut in lecto pen&longs;ili: &longs;eu in immobili conti&shy;<lb/>nente, &longs;cilicet po&longs;tqu&agrave;m ad locum &longs;uum re&longs;titutum fuerit per uim <lb/>retenta patera &agrave; manu iuxta &longs;itum priorem in a, mobili autem con&shy;<lb/>tento, id e&longs;t, humore, multo autem magis contento, &amp; continente <lb/>mobilibus. </s>
          <s id="id004088">Vt &longs;i patera &amp; humor ip&longs;e &longs;imul <expan abbr="moue&atilde;tur">moueantur</expan>, nam &amp; pate<lb/>ra tran&longs;gredietur locum &longs;uum, &amp; humor duplici motu &longs;uperau&shy;<lb/><arrow.to.target n="marg800"/><lb/>ctus tran&longs;gredietur motum naturalem. </s>
          <s id="id004089">Cum enim a d e&longs;t remotum <lb/>a g, &amp; e&longs;t in f, mouetur maiore impetu, quam &longs;it pro ratione pon&shy;<lb/>deris, ut demon&longs;tratum e&longs;t, igitur tran&longs;ibit ad e, cum ergo redeat <lb/>ad g motu naturali, nece&longs;&longs;e e&longs;t ut motus uiolentus &longs;it ualidior ea <lb/>parte naturalis, qua d re&longs;i&longs;tit, dum e&longs;t in g, ne dimoueatur &agrave; g, &longs;i igi&shy;<lb/>tur tractum ad c, &longs;uperauit uim qua manet in g, in eo quod moue&shy;<lb/>tur ad f, igitur in reditu mouebitur tantum ultra g uer&longs;us e, quan&shy;<lb/>tum e&longs;t acqui&longs;itum ex ui tran&longs;itus ultra g uer&longs;us f, quanto ergo ma&shy;<lb/>ior e&longs;t arcus e d, tanto maior e&longs;t d f, &amp; quanto maior e&longs;t arcus d f, <lb/>tanto maior d h.</s>
        </p>
        <p type="margin">
          <s id="id004090"><margin.target id="marg799"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004091"><margin.target id="marg800"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s>
        </p>
        <p type="main">
          <s id="id004092">Ex quo patet, quod quanto magis remouetur d &agrave; g, tanto maio&shy;<lb/><arrow.to.target n="marg801"/><lb/>re impetu fertur uer&longs;us extremum aliud &amp; ultra medium.</s>
        </p>
        <p type="margin">
          <s id="id004093"><margin.target id="marg801"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="head">
          <s id="id004094">LEMMA SECVNDVM.</s>
        </p>
        <p type="main">
          <s id="id004095">Omne pondus appen&longs;um e&longs;t graue comparatione medij graui&shy;<lb/>tatis, ad hoc ut ab eo remoueatur, quantum e&longs;t pro ratione anguli <lb/>ex quo appen&longs;um e&longs;t.</s>
        </p>
        <p type="main">
          <s id="id004096">Sit d appen&longs;um in a &amp; in b, &amp; &longs;it angulus c b d, triplus angu&shy;<lb/><arrow.to.target n="marg802"/><lb/>lo c a d, dico quod tripla e&longs;t uis qu&aelig; transfert d in c ex b, ei qu&aelig; <lb/>transfert ex a, quoniam enim mixtus e&longs;t in b &amp; a, igitur a d &aelig;qua&shy;<lb/><arrow.to.target n="marg803"/><lb/>lia &longs;patia &aelig;quales uires exigentur: igitur uirium proportio ut <lb/>angulorum, at quanto maior e&longs;t a d in proportione ab b d tanto <lb/>maior e&longs;t proportio anguli c b d ad <expan abbr="angul&utilde;">angulum</expan> c a d, igitur quanto ma&shy;


<pb pagenum="240" xlink:href="015/01/259.jpg"/><figure id="id.015.01.259.1.jpg" xlink:href="015/01/259/1.jpg"/><lb/>ior e&longs;t a d tanto facilius remouet &etail;quali &longs;pa<lb/><arrow.to.target n="marg804"/><lb/>tio d uer&longs;us e. </s>
          <s id="id004097">Et licet remoueantur ab ip&longs;o <lb/>d, &longs;emper eadem proportio manebit, ma&shy;<lb/><arrow.to.target n="marg805"/><lb/>nente eadem longitudine b d &amp; a d, nam <lb/><arrow.to.target n="marg806"/><lb/>proportio d f ad d c, e&longs;t uelut f b d ad <lb/>c b d, &amp; ut d f ad d e, ita f a d ad c a d, quare <lb/>fb d ad c b d, uelut f a d ad c a d, quare fb d <lb/>ad f a d, ut c b d ad c a d, quod fuit pro&shy;<lb/>po&longs;itum.</s>
        </p>
        <p type="margin">
          <s id="id004098"><margin.target id="marg802"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004099"><margin.target id="marg803"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004100"><margin.target id="marg804"/>P<emph type="italics"/>er ult.  &gt;&longs;ex&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004101"><margin.target id="marg805"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004102"><margin.target id="marg806"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>eiu&longs; <lb/>dem.<emph.end type="italics"/></s>
        </p>
        <p type="head">
          <s id="id004103">LEMMA TERTIVM.</s>
        </p>
        <p type="main">
          <s id="id004104">Grauitatem ponderis appen&longs;i aut fluidi <lb/>in comparatione ad remotionem &agrave; centro <lb/>grauitatis inuenire.<lb/><arrow.to.target n="marg807"/></s>
        </p>
        <p type="margin">
          <s id="id004105"><margin.target id="marg807"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004106">Nam cum d trahetur per planum ut &longs;u&longs;pen&longs;um, &amp; non tractum </s>
        </p>
        <p type="main">
          <s id="id004107"><arrow.to.target n="marg808"/><lb/>a d, erit dimidium ponderis appen&longs;i, igitur ex lemmate &longs;ecundo, pa<lb/>tebit proportio laboris in remouendo d &agrave; loco proprio in quan&shy;<lb/>cunque partem &amp; di&longs;tantiam, &amp; in quouis loco &longs;it appen&longs;um.</s>
        </p>
        <p type="margin">
          <s id="id004108"><margin.target id="marg808"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004109">Ex hoc &longs;equitur, quod poterit annulus tam alt&egrave; appendi, ut iuxta <lb/><arrow.to.target n="marg809"/><lb/>proportionem angul&iacute; &amp; leuitatem propriam cum filo tenui&longs;simo, <lb/>&amp; ut fuerit latus, &amp; po&longs;itus &egrave; regione oris, ut ex &longs;ermone circum&shy;<lb/>agatur quaqua uer&longs;us, &amp; percutiat labra ua&longs;is aqua pleni ferm&egrave;, ut <lb/>uideatur plane re&longs;pon&longs;a dare.</s>
        </p>
        <p type="margin">
          <s id="id004110"><margin.target id="marg809"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;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 &agrave; recta <lb/>linea, tanto maiore impetu agetur, ut ultra locum medium feratur <lb/>non &aelig;quali, &longs;ed producta proportione.</s>
        </p>
        <p type="main">
          <s id="id004113">Sit a b, &amp; ut dictum e&longs;t, non e&longs;t ei pondus, ni&longs;i quatenus remoue&shy;<lb/><arrow.to.target n="marg810"/><lb/>tur a recta, &amp; in c &longs;ummam habeat grauitatem, &amp; d &longs;it medium b c, <lb/><figure id="id.015.01.259.2.jpg" xlink:href="015/01/259/2.jpg"/><lb/>dico ergo quod multo maiore impetu feretur ex cin <lb/>b quam ex d, nam cum c &longs;it &longs;umma grauitas, erit &longs;al&shy;<lb/>tem dupla grauitati d, &longs;ed d grauitas e&longs;t pen&egrave; infinita, <lb/>ut demon&longs;tratum e&longs;t in comparatione ad b, ut iuxta <lb/>&longs;itum remotionis &agrave; linea b, cum ergo proportio &longs;in&shy;<lb/><arrow.to.target n="marg811"/><lb/>gularum partium c d ad &longs;ingulas d b medietate b c di&longs;tantes &longs;it ma&shy;<lb/><figure id="id.015.01.259.3.jpg" xlink:href="015/01/259/3.jpg"/><lb/>ior dupla augendo, erit proportio c d ad d b, uelut pro&shy;<lb/>po&longs;ita h k dupla g f, &amp; h e dupla e f, e k h ad e g f quadru&shy;<lb/>pla, igitur &amp; eo maior quo acqui&longs;itus e&longs;t impetus ex de&shy;<lb/>mon&longs;tratis, quare proportio motus &amp; impetus ex c in <lb/><arrow.to.target n="marg812"/><lb/>b, e&longs;t multo maior impetu ex d in b quadrupla pro&shy;<lb/>portione.</s>
        </p>
        <pb pagenum="241" xlink:href="015/01/260.jpg"/>
        <p type="margin">
          <s id="id004114"><margin.target id="marg810"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004115"><margin.target id="marg811"/>L<emph type="italics"/>emmate<emph.end type="italics"/> 2.</s>
        </p>
        <p type="margin">
          <s id="id004116"><margin.target id="marg812"/>P<emph type="italics"/>er<emph.end type="italics"/> 30. <emph type="italics"/>hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004117">Ex his omnibus concluditur propo&longs;itum in prima figura, &amp; e&longs;t <lb/><arrow.to.target n="marg813"/><lb/>quod &longs;i b c inclinetur uer&longs;us e, mouebitur a d, certo impetu uer&longs;us <lb/>e. </s>
          <s id="id004118">Et quia &longs;i prius b c inclinatum fuerit in f, redit a d, dum b c reuer&shy;<lb/>titur ad proprium &longs;itum ultra lineam a d g u&longs;que ad h per primum <lb/>lemma. </s>
          <s id="id004119">Et cum b c inclinatur ad b f peruenit, quantum b c inclina&shy;<lb/>ta ad f, &longs;cilicet ad e, igitur ex motibus b c in f &amp; in e tanto plus mo&shy;<lb/>uetur d ultra e, quantum e&longs;t productum d e in d h, &lsquo;ideo multo plus <lb/>quam &longs;i &longs;olum motum fui&longs;&longs;et d ex recta a g, etiam quod non moue&shy;<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"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004122">Propo&longs;itio ducente&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id004123">Si &longs;uperficies rectangula in duas partes &aelig;quales diui&longs;a intelli&shy;<lb/>gatur, qu&aelig; amb&etail; quadrat&aelig; &longs;int, itemque in duas in&aelig;quales, erit pa&shy;<lb/>rallelipedum ex latere medi&aelig; partis in totum &longs;uperficiem maius ag<lb/><figure id="id.015.01.260.1.jpg" xlink:href="015/01/260/1.jpg"/><lb/>gregato parallelipedorum ex par&shy;<lb/>tibus in&aelig;qualibus, in latera alte&shy;<lb/>rius partis mutuo in eo, quod fit <lb/>ex differentia lateris minoris par&shy;<lb/>tis a medi&aelig; latere in differentiam <lb/>maioris partis &longs;uperficiei &agrave; media <lb/>&longs;uperficie bis, &amp; ex differentia am&shy;<lb/>borum laterum in&aelig;qualium iun&shy;<lb/>ctorum ad ambo latera &aelig;qualia <lb/>iuncta in minorem partem &longs;uperficiei.</s>
        </p>
        <p type="main">
          <s id="id004124">Proponatur a g diui&longs;a in duo quadrata &aelig;qualia a h, h b, &amp; late&shy;<lb/><arrow.to.target n="marg814"/><lb/>ra erunt a c, c b, &amp; in duo in&aelig;qualia a d d g, quarum latera &longs;int b c, <lb/>a f, dico quod parallelipeda a c in c g, &amp; c b in c k, &amp; &longs;unt &aelig;qualia pa<lb/>rallelipedo ex a c in a g, excedunt <lb/><figure id="id.015.01.260.2.jpg" xlink:href="015/01/260/2.jpg"/><arrow.to.target n="table30"/><lb/>parallelipeda ex a f in d g, &amp; b c <lb/>in d k, in duplo f c in d h, cum eo <lb/>quod fit ex f e in d k &longs;emel. </s>
          <s id="id004125">Quia <lb/>ergo parallelipedum ex a e in a g <lb/>e&longs;t &aelig;quale parallelipedis a f &amp; f c <lb/>in a h, h d, h k, quare parallelipe&shy;<lb/>dis a f in a h, h d, d k, &amp; f c in d k, &amp; <lb/>c e in d k, &amp; f e in d k, &amp; f e in d h <lb/>bis. </s>
          <s id="id004126">Ad parallelipedum a fin d g, <lb/>e&longs;t &aelig;quale parallelipedis a fin a h, h d. </s>
          <s id="id004127">Et parallelipedum b e in d k, <lb/>parallelipedis a f, f e, c e in d k. </s>
          <s id="id004128">Detractis &longs;imilibus relinquetur f c in <lb/>d l, l e, e h bis, quod e&longs;t f c in d h bis, cum eo quod fit ex e f in d k &longs;i&shy;<lb/>mul, quod e&longs;t propo&longs;itum.</s>
        </p>
        <pb pagenum="242" xlink:href="015/01/261.jpg"/>
        <p type="margin">
          <s id="id004129"><margin.target id="marg814"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <table>
          <table.target id="table30"/>
          <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/>
          </row>
          <row>
            <cell>4 f c in d k</cell>
            <cell/>
          </row>
          <row>
            <cell>5 c e in d k</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/>
          </row>
        </table>
        <p type="head">
          <s id="id004130">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id004131">Dico etiam, qu&ograve;d du&aelig; line&aelig; b e &amp; af &longs;unt minores duabus a c, <lb/>c b &longs;imul iunctis, nam quia d b, e b, c b, &longs;unt in eadem proportione, <lb/>&amp; d b e&longs;t maior e b, erit maior differentia d b ad e b, quam e b ad <lb/><arrow.to.target n="marg815"/><lb/>c b, igitur maior d e quam e c, quare e c e&longs;t minor medietate d c, &amp; <lb/>ideo multo minor medietate a c. </s>
          <s id="id004132">Et &longs;imiliter, quia a c e&longs;t maior af, &amp; <lb/>a c, a f, a d &longs;unt in continua proportione, maior erit c f quam <lb/>fd, &amp; ide&ograve; con&longs;tat quamuis longum e&longs;&longs;et, &longs;i quis uellet demon&shy;<lb/>&longs;trare perfect&egrave;, quod b e &amp; a f iunct&aelig; &longs;unt minores tota a b &longs;eu du&shy;<lb/>plo a c.</s>
        </p>
        <p type="margin">
          <s id="id004133"><margin.target id="marg815"/>P<emph type="italics"/>er conuer&shy;<lb/>&longs;am qua&longs;i<emph.end type="italics"/> 8. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004134">Exemplum, &longs;int h b &amp; h a 25, &amp; a e, c b 5, producta mutua 250, <lb/>&longs;itq&uacute;e g d 49, &amp; erit b e 7, &longs;it autem d k 1, &amp; erit a f 1, quia ergo a f <lb/>e&longs;t 1, a e 5, erit f c 4, &amp; quia e b e&longs;t 7, &amp; b c 5, erit e c 2, quare etiam ef2, <lb/>productum ergo ex e b in d k e&longs;t 7, &amp; ex a f in d g 49, totum ag&shy;<lb/>gregatum 56, differentia a 250, e&longs;t 194, qui &longs;it ex duplo fc, quod <lb/>e&longs;t 8 in d h, qu&aelig; e&longs;t 24, &amp; fit 192, &amp; ex fe, qu&aelig; e&longs;t 2, in d k, qu&aelig; e&longs;t 1, <lb/>&amp; fit: quod additum ad 192 facit 194. Similiter capio 450, cuius di&shy;<lb/>midium e&longs;t 225, c g &amp; c k 225, &amp; c a &amp; c b 15 &longs;ingul&aelig;. </s>
          <s id="id004135">Et ponatur <lb/>d g 441, eritq&uacute;e e b 21, &amp; d k 9, &amp; erit a f 3, igitur cum b e &longs;it 21, <lb/>&amp; b c 15, erit c e 6, a f uer&ograve; e&longs;t 3, igitur f e e&longs;t 6. Producta mu&shy;<lb/>tua &aelig;qualia 6750, in&aelig;qualia 1521, differentia 5238, quia er&shy;<lb/>go f c e&longs;t 12, duplum eius e&longs;t 24, ductum in d h, qu&aelig; e&longs;t <lb/>216, nam d k ex &longs;uppo&longs;ito e&longs;t 9, fiet ergo 5184, cui &longs;i addam, quod <lb/>fit ex f e, qu&aelig; e&longs;t 6, in d k, qu&aelig; e&longs;t 9, fitq&uacute;e 54, erit totum 5238, quod <lb/>erat propo&longs;itum.<lb/><arrow.to.target n="marg816"/></s>
        </p>
        <p type="margin">
          <s id="id004136"><margin.target id="marg816"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004137">Ex hac demon&longs;tratione liquet, quod &longs;i linea in duas partes &aelig;&shy;<lb/>quales diuidatur, &amp; duas in&aelig;quales, qu&ograve;d parallelipeda &aelig;qua&shy;<lb/>lium &longs;ectionum pariter accepta excedent parallelipeda in&aelig;qua&shy;<lb/>lium &longs;ectionum, &longs;imul iuncta in eo quod fit ex tota linea in quadra&shy;<lb/>tum differenti&aelig; partium &aelig;qualium ab in&aelig; qualibus.</s>
        </p>
        <p type="main">
          <s id="id004138">Propo&longs;itio ducente&longs;ima decima.</s>
        </p>
        <p type="main">
          <s id="id004139">Si du&aelig; line&aelig; ad &aelig;quales angulos ab eodem puncto peripheri&aelig; <lb/>circuli reflectantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos &aelig;&shy;<lb/>quales e&longs;&longs;e. </s>
          <s id="id004140">Vnde manife&longs;tum e&longs;t protractam diametrum angu&shy;<lb/>lum &longs;uppo&longs;itum per &aelig;qualia diuidere.</s>
        </p>
        <p type="main">
          <s id="id004141"><arrow.to.target n="marg817"/></s>
        </p>
        <p type="margin">
          <s id="id004142"><margin.target id="marg817"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004143">Re&longs;iliat radius d b c ad &aelig;quales angulos, ut fert natura rerum 


<pb pagenum="243" xlink:href="015/01/262.jpg"/>dum &agrave; plano re&longs;ilit &lpar;licet refragante Plutarcho&rpar; ita ut anguli c b e, &amp; <lb/>d b f &longs;int &aelig;quales, dico angulos ibidem d b a, &amp; c b a &aelig;quales e&longs;&longs;e: <lb/><figure id="id.015.01.262.1.jpg" xlink:href="015/01/262/1.jpg"/><lb/>&amp; quod &longs;i trahatur latus a b u&longs;que ad g, quod anguli d b <lb/>g &amp; c b g etiam erunt &etail;quales. </s>
          <s id="id004144">Primum patet, quia an&shy;<lb/>guli a b e &amp; a b c &amp; a b f &aelig;quales &longs;unt, &longs;unt enim re&longs;i&shy;<lb/>dui ad angulos contactus eiu&longs;dem circuli &amp; rect&aelig;, igi<lb/>tur additis &aelig;qualibus ex &longs;uppo&longs;ito c b e, d b f erunt </s>
        </p>
        <p type="main">
          <s id="id004145"><arrow.to.target n="marg818"/><lb/>per communem animi &longs;ententiam a b c &amp; a b d &aelig;qua&shy;<lb/>les. </s>
          <s id="id004146">Secundum, cum &longs;int a b c &amp; a b d &aelig;quales, &amp; duo <lb/>anguli a b c, c b g &aelig;quales duobus rectis: itemque a b d, <lb/>d b g duobus rectis &aelig;quales: Et omnes recti inuicem &aelig;quales ex <lb/><arrow.to.target n="marg819"/><lb/>petitione Euclidis erunt per communem animi &longs;ententiam, &aelig;qua&shy;<lb/>les re&longs;idui quoque c b g &amp; d b g.</s>
        </p>
        <p type="margin">
          <s id="id004147"><margin.target id="marg818"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004148"><margin.target id="marg819"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004149">Ex hoc patet, eam qu&aelig; re&longs;ilit lineam &longs;emper ultra lineam &agrave; cen&shy;<lb/><arrow.to.target n="marg820"/><lb/>tro ad punctum, ex quo re&longs;ilit ductam ferri.</s>
        </p>
        <p type="margin">
          <s id="id004150"><margin.target id="marg820"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004151">Con&longs;tat quia linea ex centro diuidit angulum per &aelig;qualia, ergo <lb/><arrow.to.target n="marg821"/><lb/>cadit media inter illa qu&aelig; incidit, &amp; qu&aelig; re&longs;ilit.</s>
        </p>
        <p type="margin">
          <s id="id004152"><margin.target id="marg821"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004153">Ex hac etiam patet, qu&ograve;d con&longs;tituto angulo in cen&shy;<lb/><arrow.to.target n="marg822"/><lb/>tro a b c, &amp; ducta linea a d &agrave; puncto a, &longs;ciemus quo re&longs;i&shy;<lb/>lit in linea b c: ducta enim c d, faciemus angulum c d e <lb/><arrow.to.target n="marg823"/><lb/>&aelig;qualem a b c, &amp; erit angulus a d g &aelig;qualis angulo e d <lb/>h, igitur d e re&longs;ilit ex a b a d linea.</s>
        </p>
        <p type="margin">
          <s id="id004154"><margin.target id="marg822"/>C<emph type="italics"/>or<emph.end type="italics"/>m. </s>
          <s id="id004155">2.</s>
        </p>
        <p type="margin">
          <s id="id004156"><margin.target id="marg823"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <figure id="id.015.01.262.2.jpg" xlink:href="015/01/262/2.jpg"/>
        <p type="main">
          <s id="id004157">Propo&longs;itio ducente&longs;ima undecima.</s>
        </p>
        <p type="main">
          <s id="id004158">Si du&aelig; line&aelig; ex duobus punctis peripheriam contingentes in <lb/>eandem partem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea <lb/>ex parte, &amp; nunquam concurrent.</s>
        </p>
        <figure id="id.015.01.262.3.jpg" xlink:href="015/01/262/3.jpg"/>
        <p type="main">
          <s id="id004159">Du&aelig; &longs;emidiametri a b, a c ex terminis earum <lb/><arrow.to.target n="marg824"/><lb/>du&aelig; contingentes b f, c e, dico quod quanto <lb/>magis protrahentur in partem e f, tant&ograve; magis <lb/>di&longs;tabunt, nunqu&agrave;m concurrent: Nam angu&shy;<lb/>lus a c g rectus e&longs;t: angulus uer&ograve; c a d, &longs;i &longs;it re&shy;<lb/><arrow.to.target n="marg825"/><lb/>ctus e g, nun&lt;08&gt; concurret cum a d, &aelig;quidi&longs;ta&shy;<lb/>bit enim ei: &longs;in aut &longs;it maior recto aut ex altera <lb/><arrow.to.target n="marg826"/><lb/>parte erit minor, &amp; ita concurret, ergo in alte&shy;<lb/><arrow.to.target n="marg827"/><lb/>ram partem duct&aelig; nunqu&agrave;m concurrent, &longs;ed perpetuo magis di&shy;<lb/>&longs;tabunt. </s>
          <s id="id004160">Si ergo minor recto &longs;it angulus c a b, igitur e c ex eadem <lb/><arrow.to.target n="marg828"/><lb/>parte concurret cum a d: concurrat ergo in g: &amp; quia e g cadit ex&shy;<lb/><arrow.to.target n="marg829"/><lb/>tra circulum, igitur diuidet b f, qu&aelig; tangit circulum. </s>
          <s id="id004161">Sit ergo ut di&shy;


<pb pagenum="244" xlink:href="015/01/263.jpg"/>uidat in h, igitur h e &amp; h f c&ugrave;m angulum con&longs;tituant, quanto magis <lb/>protrahentur eo magis di&longs;tabunt, nec unquam concurrent.</s>
        </p>
        <p type="margin">
          <s id="id004162"><margin.target id="marg824"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004163"><margin.target id="marg825"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004164"><margin.target id="marg826"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004165"><margin.target id="marg827"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. &amp; 4. <lb/><emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004166"><margin.target id="marg828"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>petit.<emph.end type="italics"/><lb/> E<emph type="italics"/>uclid.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004167"><margin.target id="marg829"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004168">Propo&longs;itio ducente&longs;ima duodecima.</s>
        </p>
        <p type="main">
          <s id="id004169">Si ab eodem puncto ad circuli peripheriam, line&aelig; quotuis du&shy;<lb/>cantur, tres inuenire lineas, qu&aelig; <expan abbr="n&otilde;">non</expan> in alium punctum reflectentur.<lb/><arrow.to.target n="marg830"/></s>
        </p>
        <p type="margin">
          <s id="id004170"><margin.target id="marg830"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004171">Quouis con&longs;tituto puncto ueluti a extra circu<lb/>lum b c d, dico po&longs;&longs;e trahi tres lineas ad ip&longs;am cir&shy;<lb/>culi peripheriam, uelut a b, a c, a d, qu&aelig; ad alium <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"/><lb/>centrum, &amp; a b &amp; a d ad contingentes illius peri&shy;<lb/>pheriam, quas con&longs;tat non reflecti &longs;ed progredi, <lb/><arrow.to.target n="marg832"/><lb/>a c autem reflectitur in &longs;e ip&longs;am per demon&longs;trata <lb/><arrow.to.target n="marg833"/><lb/>&longs;uperius, igitur con&longs;tat propo&longs;itum.<lb/><figure id="id.015.01.263.1.jpg" xlink:href="015/01/263/1.jpg"/><lb/><arrow.to.target n="marg834"/></s>
        </p>
        <p type="margin">
          <s id="id004174"><margin.target id="marg831"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004175"><margin.target id="marg832"/>P<emph type="italics"/>er<emph.end type="italics"/> 61. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004176"><margin.target id="marg833"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 210.</s>
        </p>
        <p type="margin">
          <s id="id004177"><margin.target id="marg834"/>C<emph type="italics"/>or<emph.end type="italics"/>m. </s>
          <s id="id004178">1.</s>
        </p>
        <p type="main">
          <s id="id004179">Ex hoc patet, quod omnia puncta &longs;ub linea <lb/>contingente po&longs;&longs;unt reflecti ad ip&longs;um per arcum <lb/>interceptum &agrave; contingente, &amp; ea qu&aelig; ad centrum.</s>
        </p>
        <p type="main">
          <s id="id004180"><arrow.to.target n="marg835"/></s>
        </p>
        <p type="margin">
          <s id="id004181"><margin.target id="marg835"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004182">Id e&longs;t, quod omnia puncta infra lineam a b f ductam quantum&shy;<lb/>libet po&longs;&longs;unt reflecti per arcum b c ad punctum a &aelig;qualibus an&shy;<lb/>gulis. </s>
          <s id="id004183">Quoniam ex a per c b reflectuntur ad qu&aelig;libet puncta infra <lb/>a b f, eo qu&ograve;d termini &longs;unt punctum a, per ea qu&aelig; &longs;unt hic demon&shy;<lb/>&longs;trata, &amp; a b f, ip&longs;a ergo &longs;i extrema in extremis, media in medijs con&shy;<lb/>tinentur per regulam illam Dialecticam: igitur omnia puncta &longs;ub <lb/>a b f etiam in infinitum producta continentur in reflexione &agrave; pun&shy;<lb/>cto a per arcum b c.</s>
        </p>
        <p type="main">
          <s id="id004184"><arrow.to.target n="marg836"/></s>
        </p>
        <p type="margin">
          <s id="id004185"><margin.target id="marg836"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004186">Et rur&longs;us, &longs;i &agrave; circulo ad circulum extrem&aelig; ducantur, nec ill&aelig; re&shy;<lb/>flectentur, &longs;ed tran&longs;ibunt: medi&aelig; autem omnes reflecti poterunt &agrave; <lb/>quouis puncto.</s>
        </p>
        <figure id="id.015.01.263.2.jpg" xlink:href="015/01/263/2.jpg"/>
        <p type="main">
          <s id="id004187">Quia &longs;i a b &longs;it Sol, c d Luna, Sole <lb/>minor extremum in utroque lumina&shy;<lb/>ri a c, b d qu&aelig; contingant utrunque <lb/>circulum, quod facile fiat, ductis a c <lb/>&amp; b d ex punctis non oppo&longs;itis, &aelig;&shy;<lb/>quidi&longs;tarent enim, &longs;ed iuxta quan&shy;<lb/>titatem dimetientis minoris. </s>
          <s id="id004188">Erit er&shy;<lb/>go ut h e non reflectantur, ali&aelig; o&shy;<lb/>mnes medi&aelig; reflectentur per demon&longs;trata &agrave; quolibet puncto, ergo <lb/>idem de totis circulis &amp; punctis.</s>
        </p>
        <p type="head">
          <s id="id004189">SCHOLIVM.</s>
        </p>
        <p type="main">
          <s id="id004190">Propo&longs;itis duobus circulis lineam ambos <expan abbr="c&otilde;tingentem">contingentem</expan> ducere.</s>
        </p>
        <pb pagenum="245" xlink:href="015/01/264.jpg"/>
        <p type="main">
          <s id="id004191">Propo&longs;itorum circulorum a &amp; b centra iungam recta a b, &longs;uper </s>
        </p>
        <p type="main">
          <s id="id004192"><arrow.to.target n="marg837"/><lb/>quam ut &longs;emidiametrum de&longs;cribo circulum b c, &amp; ex puncto a ad <lb/><arrow.to.target n="marg838"/><lb/>perpendiculum a d, ex quo ab&longs;cindo &aelig;qualem &longs;emidiametro b e li&shy;<lb/><arrow.to.target n="marg839"/><lb/><figure id="id.015.01.264.1.jpg" xlink:href="015/01/264/1.jpg"/><lb/>neam d f, ex f duco a d perpendi&shy;<lb/>culum f g, ex g in a duco a g, &amp; &aelig;&shy;<lb/>qualem angulo g a d, b a h ab&longs;cin <lb/>do h k <expan abbr="&etail;qual&etilde;">&etail;qualem</expan> d f &longs;eu b e, duco <expan abbr="a&utilde;t">aunt</expan> <lb/><arrow.to.target n="marg840"/><lb/>b e, ut &longs;it <expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan> h k, duco h e, <lb/><arrow.to.target n="marg841"/><lb/><expan abbr="qu&atilde;">quam</expan> dico contangere utrunque <expan abbr="cir&shy;cul&utilde;">cir&shy;<lb/>culum</expan> b k: produco b k, &amp; quia du&aelig; <lb/>line&aelig; b a &amp; a k &longs;unt &etail;quales duo&shy;<lb/>bus lineis a g &amp; a f, du&aelig; enim <lb/>prodeunt ab eodem centro, reli&shy;<lb/>qu&aelig; &longs;unt re&longs;idua &aelig;qualium d f &amp; h k, &amp; angulus b a k &aelig;qualis <lb/><arrow.to.target n="marg842"/><lb/>g a f, ex &longs;uppo&longs;ito erit angulus g f a &aelig;qualis angulo b k a, g f a au&shy;<lb/>tem rectus fuit, quia g f ad perpendiculum erecta fuit, itaque b k a <lb/>rectus e&longs;t, &amp; ideo b k h rectus, quare <expan abbr="c&utilde;">cum</expan> b e &amp; k h &longs;int &aelig;quales, &amp; &aelig;&shy;<lb/><arrow.to.target n="marg843"/><lb/>quidi&longs;tantes, erit angulus e oppo&longs;itus b h k rectus, igitur duo angu<lb/>li e b k &amp; e h k duobus rectis &aelig;quales, quare cum &longs;int &aelig;quales inui<lb/><arrow.to.target n="marg844"/><lb/>cem, quia oppo&longs;iti in parallelogrammo uterque eorum rectus erit. <lb/><arrow.to.target n="marg845"/><lb/>Recti ergo &longs;unt anguli e &amp; h, &amp; line&aelig; b e &amp; a h ex centris circulo&shy;<lb/>rum, &amp; angulos Illos con&longs;tituit line&aelig; e h, igitur e h contangit u&shy;<lb/><arrow.to.target n="marg846"/><lb/>trunque circulum.</s>
        </p>
        <p type="margin">
          <s id="id004193"><margin.target id="marg837"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004194"><margin.target id="marg838"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>primi<emph.end type="italics"/><lb/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004195"><margin.target id="marg839"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004196"><margin.target id="marg840"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004197"><margin.target id="marg841"/>P<emph type="italics"/>er<emph.end type="italics"/> 31. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004198"><margin.target id="marg842"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004199"><margin.target id="marg843"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004200"><margin.target id="marg844"/>P<emph type="italics"/>er<emph.end type="italics"/> 33. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004201"><margin.target id="marg845"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004202"><margin.target id="marg846"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004203">Propo&longs;itio ducente&longs;ima tertia decima.</s>
        </p>
        <p type="main">
          <s id="id004204">Propo&longs;ito circulo atque in eius peripheria puncto &longs;ignato lineas <lb/>contingentes ultra citraque, &amp; etiam ab ip&longs;omet deducere.</s>
        </p>
        <figure id="id.015.01.264.2.jpg" xlink:href="015/01/264/2.jpg"/>
        <p type="main">
          <s id="id004205">Sit circulus b c d, &amp; in eius peripheria c <lb/><arrow.to.target n="marg847"/><lb/>punctum de&longs;criptum, &amp; &longs;umatur b d por&shy;<lb/>tio minor quadrante, in qua punctum c, &amp; <lb/>ducantur a b, a c, &amp; ducantur b e, c f, d g, ad <lb/><arrow.to.target n="marg848"/><lb/>perpendiculum, &amp; con&longs;tat propo&longs;itum, &amp; <lb/>quod nunquam ex eadem parte conuenient <lb/><arrow.to.target n="marg849"/><lb/>ex eadem parte ex demon&longs;tratis &longs;upr&agrave;.</s>
        </p>
        <p type="margin">
          <s id="id004206"><margin.target id="marg847"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="margin">
          <s id="id004207"><margin.target id="marg848"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004208"><margin.target id="marg849"/>P<emph type="italics"/>er<emph.end type="italics"/> 221.</s>
        </p>
        <p type="main">
          <s id="id004209">Propo&longs;itio ducente&longs;ima quarta decima.</s>
        </p>
        <p type="main">
          <s id="id004210">Si extra circulum duo puncta &etail;qualiter &agrave; centro di&longs;tantia &longs;ignen<lb/>tur, erit punctum reflexionis &aelig;qualis, in medio arcus intercepti in&shy;<lb/>ter lineas, qu&aelig; &agrave; centro ducuntur ad illa puncta. </s>
          <s id="id004211">Si uer&ograve; unum cen<lb/>tro proximius fuerit altero punctum &aelig;qualitatis in peripheria, tan<lb/>to longius uer&longs;us breuiorem lineam, quanto punctum aliud &agrave; cen&shy;<lb/>tro magis di&longs;teterit.


<pb pagenum="246" xlink:href="015/01/265.jpg"/><arrow.to.target n="marg850"/></s>
        </p>
        <p type="margin">
          <s id="id004212"><margin.target id="marg850"/>C<emph type="italics"/>o<emph.end type="italics"/>_&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004213">Sint puncta b c, &aelig;qualiter di&longs;tantia &agrave; cen</s>
        </p>
        <p type="main">
          <s id="id004214"><arrow.to.target n="marg851"/><lb/>tro a circuli d e, &amp; reflectantur c f, b f, dico f <lb/><arrow.to.target n="marg852"/><lb/>e&longs;&longs;e in medio arcus d e: producta enim f a, <lb/>erunt anguli d a f &amp; e a f &aelig;quales: &longs;upponi&shy;<lb/>tur enim <expan abbr="prim&utilde;">primum</expan> f e&longs;&longs;e in medio: igitur cum <lb/>a b &amp; a c &longs;int &aelig;quales, &amp; a f communis, erit <lb/>a f c &aelig;qualis a f b, igitur reflectentur &aelig;qua&shy;<lb/>liter: ergo &longs;i &etail;qualiter reflectentur, ex f re&shy;<lb/>flectentur, ut ex &longs;ecunda parte: quare ex <lb/>medio.<lb/><figure id="id.015.01.265.1.jpg" xlink:href="015/01/265/1.jpg"/><lb/><arrow.to.target n="marg853"/></s>
        </p>
        <p type="margin">
          <s id="id004215"><margin.target id="marg851"/>P<emph type="italics"/>er<emph.end type="italics"/> 21. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004216"><margin.target id="marg852"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>primi <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004217"><margin.target id="marg853"/>P<emph type="italics"/>er<emph.end type="italics"/> 210. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004218">Sumatur rur&longs;us punctum g, remotius ab <lb/>a quam b, dico qu&ograve;d reflexio erit in arcu f e. <lb/></s>
          <s id="id004219">Nam non in e, quoniam fic g e d e&longs;&longs;et &aelig;qualis b e k, cui rur&longs;us e&longs;t &aelig;&shy;<lb/>qualis b e d, ergo g e d &aelig;qualis b e d, pars toti. </s>
          <s id="id004220">Sed neque ultra e, nam <lb/>multo magis pars &aelig;qualis e&longs;&longs;et toti aut maior etiam. </s>
          <s id="id004221">Sed neque ex f, <lb/>nam eadem ratione pars e&longs;&longs;et maior toto. </s>
          <s id="id004222">Neque in toto arcu f d: <lb/>nam &longs;it punctum l, &amp; ducantur al, g f, igitur g l a maior g f a, g f a au<lb/>tem maior e f a, igitur g l a maior c f a, &etail;qualis ex &longs;uppo&longs;ito b f a, b f a </s>
        </p>
        <p type="main">
          <s id="id004223"><arrow.to.target n="marg854"/><lb/>rur&longs;us maior b l a: multo igitur maior g l a quam b l a, non ergo re&shy;<lb/>flexio &aelig;qualis e&longs;&longs;e pote&longs;t. </s>
          <s id="id004224">Cum ergo reflexio fiat, &amp; non ex arcu d f, <lb/><arrow.to.target n="marg855"/><lb/>nec puncto f, nec e, nec ultra e, nec extra d, erit nece&longs;&longs;arium, ut fiat ex <lb/>puncto in arcu e f.<lb/><arrow.to.target n="marg856"/></s>
        </p>
        <p type="margin">
          <s id="id004225"><margin.target id="marg854"/>P<emph type="italics"/>er<emph.end type="italics"/> 21. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004226"><margin.target id="marg855"/>P<emph type="italics"/>er<emph.end type="italics"/> 1 C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. <lb/><emph type="italics"/>pr&aelig;cedentis.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004227"><margin.target id="marg856"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004228">Ex hoc patet, quod linea a puncto ducta, quo <lb/>longius fertur, eo etiam longius re&longs;ilit.</s>
        </p>
        <figure id="id.015.01.265.2.jpg" xlink:href="015/01/265/2.jpg"/>
        <p type="main">
          <s id="id004229"><arrow.to.target n="marg857"/></s>
        </p>
        <p type="margin">
          <s id="id004230"><margin.target id="marg857"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004231">Cum enim a c b maior &longs;it a d b, &amp; angulus e c b </s>
        </p>
        <p type="main">
          <s id="id004232"><arrow.to.target n="marg858"/><lb/>&aelig;qualis a c b &amp; f d b &aelig;qualis a d b, erunt duo an&shy;<lb/>guli a c b &amp; e c b, maiores a d b &amp; f d b, quare <lb/>reliquus f d a maior a c e, igitur&apos;d f re&longs;ilit latius <lb/>quam c e.<lb/><arrow.to.target n="marg859"/></s>
        </p>
        <p type="margin">
          <s id="id004233"><margin.target id="marg858"/>P<emph type="italics"/>er<emph.end type="italics"/> 21. <lb/><emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004234"><margin.target id="marg859"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004235">Ex hoc patet, quod tales line&aelig; qu&aelig; re&longs;iliunt <lb/>nunquam concurrent.</s>
        </p>
        <p type="main">
          <s id="id004236"><arrow.to.target n="marg860"/></s>
        </p>
        <p type="margin">
          <s id="id004237"><margin.target id="marg860"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004238">Scilicet c e &amp; d f nam con&longs;tat ducta c d, angulos e c d f &amp; d e, ma&shy;</s>
        </p>
        <p type="main">
          <s id="id004239"><arrow.to.target n="marg861"/><lb/>iores e&longs;&longs;e duobus rectis, ergo non concurrentin partem e f.</s>
        </p>
        <p type="margin">
          <s id="id004240"><margin.target id="marg861"/>P<emph type="italics"/>er conuer&shy;<lb/>&longs;am<emph.end type="italics"/> 5. <emph type="italics"/>petit.<emph.end type="italics"/><lb/>E<emph type="italics"/>uclid.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004241">Propo&longs;itio ducente&longs;ima quinta decima.</s>
        </p>
        <p type="main">
          <s id="id004242">Punctum reflexionis punctorum in&aelig;qualiter di&longs;tantium &agrave; cen&shy;<lb/>tro, &aelig;qualiter di&longs;tat &agrave; lineis ductis &agrave; centro ad puncta, &aelig;qualiter di <lb/>&longs;tantia alterutrinque.<lb/><arrow.to.target n="marg862"/></s>
        </p>
        <p type="margin">
          <s id="id004243"><margin.target id="marg862"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004244">Sint g h a &amp; b h a &aelig;quales, &amp; ab&longs;cindatur h f &aelig;qualis h b, &amp; pro&shy;<lb/>ducatur h b u&longs;que a d c, ut &longs;it h c &aelig;qualis h g, &amp; producantur f a &amp; 


<pb pagenum="247" xlink:href="015/01/266.jpg"/>c a, qu&aelig; &longs;ecent peripheriam in d &amp; e, dico quod <lb/>punctum h e&longs;t medium inter e &amp; l, item inter d &amp; </s>
        </p>
        <p type="main">
          <s id="id004245"><arrow.to.target n="marg863"/><lb/>k. </s>
          <s id="id004246">Nam cum h f &amp; h b &longs;int &aelig;quales ex &longs;uppo&longs;ito, <lb/><arrow.to.target n="marg864"/><lb/>&amp; anguli b h a &amp; g h a &aelig;quales, &amp; linea h a com&shy;<lb/><arrow.to.target n="marg865"/><lb/>munis, erit angulus b a h &aelig;qualis f a h, igitur ar&shy;<lb/>cus h l &aelig;qualis arcui h e. </s>
          <s id="id004247">Similiter angulus g h a <lb/>e&longs;t &aelig;qualis e h a &amp; c h &aelig;qualis h g ex&longs;uppo&longs;ito, &amp; <lb/>a h communis, igitur ut &longs;upr&agrave; angulus c a h &aelig;qua&shy;<lb/>lis g a h, igitur per eandem arcus h k &aelig;qualis arcui <lb/>h d, quare h punctum in medio d &amp; k, &amp; in medio <lb/>etiam e &amp; l, quod e&longs;t probandum.</s>
        </p>
        <p type="margin">
          <s id="id004248"><margin.target id="marg863"/>P<emph type="italics"/>er<emph.end type="italics"/> 210.</s>
        </p>
        <p type="margin">
          <s id="id004249"><margin.target id="marg864"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004250"><margin.target id="marg865"/>P<emph type="italics"/>er<emph.end type="italics"/> 26. <emph type="italics"/>ter&shy;<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <figure id="id.015.01.266.1.jpg" xlink:href="015/01/266/1.jpg"/>
        <p type="main">
          <s id="id004251">Propo&longs;itio ducente&longs;ima &longs;exta decima.</s>
        </p>
        <p type="main">
          <s id="id004252">Si fuerint circuli duo in&aelig;quales, &amp; extra utrunque punctum a d il&shy;<lb/>lud ex minore reflex&egrave; per magnam partem minoris &agrave; maiore perue<lb/>nire poterunt.</s>
        </p>
        <figure id="id.015.01.266.2.jpg" xlink:href="015/01/266/2.jpg"/>
        <p type="main">
          <s id="id004253">Sint duo circuli, maior a b, mi&shy;<lb/><arrow.to.target n="marg866"/><lb/>nor c d, &amp; <expan abbr="punct&utilde;">punctum</expan> g, extra utrun&shy;<lb/>que, dico quod a d g ex c d <expan abbr="pote&shy;r&utilde;t">pote&shy;<lb/>runt</expan> reflex&egrave; produci a b in c d, quia <lb/>enim ex a b quibu&longs;uis punctis <lb/>po&longs;&longs;unt duci line&aelig; reflex&egrave; ex c d, <lb/>&amp; ideo cum puncta in a b uarient <lb/>reflexionem ex c d, aliter pars e&longs;&shy;<lb/>&longs;et &aelig;qualis toti, patet intentum.</s>
        </p>
        <p type="margin">
          <s id="id004254"><margin.target id="marg866"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004255">Ex hoc patet, quod oculus in <lb/><arrow.to.target n="marg867"/><lb/>quauis parte terr&aelig; con&longs;titutus, in <lb/>qua Lunam uidere po&longs;sit, poterit <lb/>eam uidere per radios reflexos &agrave; <lb/>Sole.</s>
        </p>
        <p type="margin">
          <s id="id004256"><margin.target id="marg867"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004257">Ex hoc rur&longs;us patet, quod <expan abbr="eod&etilde;">eodem</expan> modo oculus poterit uidere &longs;u&shy;<lb/><arrow.to.target n="marg868"/><lb/>perficiei Lun&etail; illuminat&etail; <expan abbr="part&etilde;">partem</expan> p radios reflexos &agrave; Solis corpore.</s>
        </p>
        <p type="margin">
          <s id="id004258"><margin.target id="marg868"/>C<emph type="italics"/>or<emph.end type="italics"/>_&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004259">Hoc patet, quoniam &longs;i circuli Solis &longs;inguli, qui illuminant <expan abbr="Lun&atilde;">Lunam</expan> <lb/><arrow.to.target n="marg869"/><lb/>o&longs;tendunt per primum corrolarium huius <expan abbr="part&etilde;">partem</expan> circuli Lun&aelig; per <lb/>radios Solis reflexos ab ip&longs;a Luna, put&agrave; &longs;ecundum portionem cir&shy;<lb/>culi e f, igitur cum liceat in Sole accipere magnam partem &longs;uperfi&shy;<lb/>ciei eius, qu&aelig; Lunam illuminat, in qua continentur infinit&aelig; por&shy;<lb/>tiones circulorum, &amp; h&aelig; &longs;ingul&aelig; mittunt radios reflexos ex Luna <lb/>ad punctum g, igitur g uidebit portionem &longs;uperficiei Lun&aelig; &longs;ecun&shy;<lb/>dum longitudinem e f per radios Solares &agrave; Luna reflexos: quod <lb/>e&longs;t propo&longs;itum.</s>
        </p>
        <pb pagenum="248" xlink:href="015/01/267.jpg"/>
        <p type="margin">
          <s id="id004260"><margin.target id="marg869"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004261">Propo&longs;itio ducente&longs;ima decima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id004262">Oculus uidet partem &longs;uperficiei Lun&aelig; illuminatam &agrave; Sole per <lb/>radios reflexos &agrave; Solis corpore: nec tamen pote&longs;t uidere imaginem <lb/>ip&longs;ius in Luna tanquam in &longs;peculo.<lb/><arrow.to.target n="marg870"/></s>
        </p>
        <p type="margin">
          <s id="id004263"><margin.target id="marg870"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004264">Quoniam per illos, ut <expan abbr="dem&otilde;&longs;tratum">demon&longs;tratum</expan> e&longs;t, pote&longs;t uidere, &amp; illi &longs;unt </s>
        </p>
        <p type="main">
          <s id="id004265"><arrow.to.target n="marg871"/><lb/>robu&longs;tiores, ergo per illos uidet, omnis enim operatio tribuitur di&shy;<lb/>gniori cau&longs;&aelig; &amp; potentiori. </s>
          <s id="id004266">Item, quoniam uidemus Lunam in no&shy;<lb/>cte immittere radios per fene&longs;tram uelut Sol: irradiare autem non <lb/>e&longs;t ni&longs;i habentis tantum lumen ex &longs;e, ut hoc po&longs;sit facere, aut ut &longs;par<lb/>gantur, aut ut reflectantur: ex &longs;e tantum non habet ut adparet hora <lb/>deliquij: neque &longs;pargit, &longs;ic enim non impediret Solem hora deliquij, <lb/>Solis ergo reflectis. </s>
          <s id="id004267">Ergo uidemus per radios reflexos. </s>
          <s id="id004268">Non <expan abbr="tam&etilde;">tamen</expan> <lb/>per eam uidemus Solem, ut in &longs;peculo obiecto, quoniam Luna pri<lb/><expan abbr="m&utilde;">mum</expan> lucet proprio lumine, &amp; rubro &longs;icut pruna, quod autem debet <lb/>fungi uice &longs;peculi, oportet ut careat colore, &amp; &longs;it uelut aqua, &amp; ut &longs;it <lb/>purum. </s>
          <s id="id004269">Deinde, quia Sol e&longs;t maior Luna, ide&ograve; uidetur ut paries in <lb/>&longs;peculo, uidetur enim non res reflexa, &longs;ed quod ip&longs;um &longs;peculum &longs;it <lb/>paries, &amp; ita Sol uidetur, ut totum quoddam, &amp; non pote&longs;t ob id <lb/>cogno&longs;ci. </s>
          <s id="id004270">Et etiam magnitudo luminis per quam oculus non po&shy;<lb/>te&longs;t di&longs;tinguere Lunam ab imagine Solis: nam ea his qu&aelig; per&longs;pe&shy;<lb/>culum uidentur, oportet duo cogno&longs;cere, &longs;peculum, &amp; rem qu&aelig; ui <lb/>detur, &longs;ed magnitudo luminis prohibet &longs;peculum uideri, ergo non <lb/>poterit uideri aliud tanquam in &longs;peculo, &longs;ed &longs;olum &longs;peculum cum <lb/>lumine tanquam res una. </s>
          <s id="id004271">Et ita de Luna. </s>
          <s id="id004272">Accedit magnitudo di&shy;<lb/>&longs;tanti&aelig;: nam in &longs;uperflua di&longs;tantia non cogno&longs;citur &longs;uperficies &longs;pe&shy;<lb/>culi, &longs;ed &longs;olum rei obiect&aelig; imago, &amp; illa habetur pro &longs;uperficie &longs;pe&shy;<lb/>culi, ergo oculus non di&longs;tinguit inter &longs;peculum, &amp; rem ui&longs;am, ide&ograve; <lb/>non uidet tanquam &egrave; &longs;peculo. </s>
          <s id="id004273">Ex quo &longs;equitur, quod Luna iudica&shy;<lb/>bitur longi&ugrave;s abe&longs;&longs;e qu&agrave;m ab&longs;it, quia quod uidemus ex ea e&longs;t So&shy;<lb/>lis imago, qu&aelig; longius multo abe&longs;t &agrave; nobis ip&longs;a Lun&aelig; &longs;uperficie. <lb/></s>
          <s id="id004274">Cum ergo &longs;int quatuor cau&longs;&aelig;, quarum unaqu&aelig;que impedire po&longs;&longs;et, <lb/>quominus Sol non uideatur in Luna tanqu&agrave;m in &longs;peculo, quanto <lb/>magis c&ugrave;m omnes ad&longs;int in Luna, &amp; &longs;imul concurrant.</s>
        </p>
        <p type="margin">
          <s id="id004275"><margin.target id="marg871"/>I<emph type="italics"/>n pr&aelig;ceden <lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004276">Propo&longs;itio ducente&longs;ima decima octaua.</s>
        </p>
        <p type="main">
          <s id="id004277">Rationem macul&aelig; Lun&aelig; indagare.<lb/><arrow.to.target n="marg872"/></s>
        </p>
        <p type="margin">
          <s id="id004278"><margin.target id="marg872"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004279">Supponamus primum qu&aelig; &longs;unt manife&longs;ta, inde addamus qu&aelig; <lb/>&longs;unt ueri&longs;imilia ualde, po&longs;t ueri&longs;imiliora ex dubijs, ubi ratio utrinque<lb/> pugnare uidetur, demum dicemus de qu&aelig;&longs;ito. </s>
          <s id="id004280">Manife&longs;tum e&longs;t igi&shy;<lb/>tur, quod Luna di&longs;tat &agrave; nobis circiter &lt;20&gt; X MP. dimetiens igitur or <lb/>bis Lun&aelig; e&longs;t 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"/>circuit circiter XLII MP. </s>
          <s id="id004281">Ergo in ictu in&longs;en&longs;ili pen&egrave;, id e&longs;t, tempore <lb/>ictus pul&longs;us infantis laborantibus acuti&longs;sima febre II MP. quoniam <lb/>quinque tales ictus continentur pen&egrave; in ictu uno uiri temperat&aelig; <lb/>natur&aelig;, &amp; &lt;23&gt; ictus pul&longs;us ferm&egrave; uiri temperati complent &longs;patium <lb/>hor&aelig;. </s>
          <s id="id004282">Igitur Luna mouetur rapidi&longs;simo motu &amp; &longs;imili motui ful&shy;<lb/>guris. </s>
          <s id="id004283">Ex quo patet quod e&longs;t corpus expers grauitatis &amp; perfe&shy;<lb/>ctum, quare nec mi&longs;tum, nec uitiatum.</s>
        </p>
        <p type="main">
          <s id="id004284">E&longs;t etiam rotunda, tamet&longs;i enim ob di&longs;tantiam maximam po&longs;&shy;<lb/>&longs;et uideri rotunda, etiam quod non e&longs;&longs;et, ueri&longs;imile tamen e&longs;t, cum <lb/>umbram talem efficiat in deliquio Solis, &amp; cum exit &egrave; tenebris ter&shy;<lb/>r&aelig;, tum quia perfecta e&longs;t quod &longs;it <expan abbr="rot&utilde;da">rotunda</expan>, aut prope rotunditatem, <lb/>&longs;ed quod e&longs;t perfectum &amp; diuinum &lpar;quia &longs;eruat &aelig;qualitatem, hoc <lb/>enim demon&longs;tratum e&longs;t, quod &aelig;quale &longs;olum reperitur in diuinis <lb/>quod ad motum attinet&rpar; exact&egrave; tale e&longs;t, igitur Luna e&longs;t exact&egrave; ro&shy;<lb/>tunda in circuitu &longs;ecundum &longs;uperficiem orbis. </s>
          <s id="id004285">Ergo etiam unde&shy;<lb/>quaque &amp; &longs;ecundum profunditatem: nam in commutatione <expan abbr="n&otilde;">non</expan> po&longs;&shy;<lb/>&longs;et latere in&aelig;qualitas. </s>
          <s id="id004286">Et etiam non e&longs;t ueri&longs;imile ullo modo, quod <lb/>corpus perfectum &amp; diuinum &longs;it informe. </s>
          <s id="id004287">E&longs;&longs;et autem nece&longs;&longs;ario <lb/>eiu&longs;modi, &longs;i e&longs;&longs;et exact&egrave; rotunda &longs;ecundum longitudinem &amp; latitu&shy;<lb/>dinem, &amp; &longs;ecundum profunditatem alterius figur&aelig;. </s>
          <s id="id004288">Veri&longs;imilius <lb/>e&longs;t ergo, Lunam e&longs;&longs;e ut ignem <expan abbr="qu&etilde;dam">quendam</expan> den&longs;um per &longs;e lucidum, &longs;ed <lb/>in&aelig;qualiter lumino&longs;um, non &longs;olum ob &longs;ub&longs;tanti&aelig; den&longs;itatem, <lb/>&longs;ed copiam luminis &amp; puritatem, qu&aelig; impuritas non illi accidit, <lb/>quia mi&longs;ta, &longs;ed quoniam e&longs;t in&aelig;qualium partium partium rararum ac den&shy;<lb/>&longs;arum &amp; mediarum. </s>
          <s id="id004289">Neque &longs;olum collu&longs;tratur &agrave; lumine ex his qu&aelig; <lb/>diximus, tum etiam quia collu&longs;trata non lucent procul, ut neque <lb/>montes, qui plurimum ab&longs;unt, quamuis non tale procul ut Luna, <lb/>im&ograve; nec nix qu&etail; illis in&longs;idet, &longs;ed nix e&longs;t multo <expan abbr="c&atilde;didior">candidior</expan> per &longs;e qu&agrave;m <lb/>Luna, quam con&longs;tat lumine Solis de&longs;titutam e&longs;&longs;e <expan abbr="rubr&atilde;">rubram</expan>, ergo Luna <lb/>relucet radijs Solaribus eli&longs;is uelut &agrave; &longs;peculo. </s>
          <s id="id004290">Et &longs;i quis in orbe Lu&shy;<lb/>n&aelig; e&longs;&longs;et media die &longs;erena, non uideret terram lumino&longs;am, qu&aelig; mul<lb/>to maior e&longs;t Luna, &amp; paulo plus &agrave; Sole di&longs;tat, &amp; quando que illi pro&shy;<lb/>pior e&longs;t qu&agrave;m Luna. </s>
          <s id="id004291">Macula autem Lun&aelig; e&longs;t qualis depingitur <lb/>cum ore, oculis &amp; na&longs;o, &longs;ed quod magis &longs;pectatur e&longs;t os ip&longs;um: <lb/><figure id="id.015.01.268.1.jpg" xlink:href="015/01/268/1.jpg"/><lb/>ade&ograve; ut Plutarchus non de macula Lun&aelig;, &longs;ed de ore Lu&shy;<lb/>n&aelig; in&longs;crip&longs;erit. </s>
          <s id="id004292">Non uerti autem Lunam, ex hoc probat </s>
        </p>
        <p type="main">
          <s id="id004293"><arrow.to.target n="marg873"/><lb/>Philo&longs;ophus &longs;ecundo de C&oelig;lo. </s>
          <s id="id004294"><expan abbr="Igi&ttilde;">Igitur</expan> ab Oriente in <expan abbr="Occi&shy;dent&etilde;">Occi&shy;<lb/>dentem</expan> uerti &longs;ub, &amp; &longs;upr&agrave; nece&longs;&longs;e e&longs;t. </s>
          <s id="id004295">Scilicet ut oculi infr&agrave; <lb/>os &longs;upra appareat. </s>
          <s id="id004296">Videtur autem magis in plenilunio <lb/>ob <expan abbr="differenti&atilde;">differentiam</expan> luminis, &amp; tota quoniam pars uer&longs;us nos etiam tota <lb/>illu&longs;tratur. </s>
          <s id="id004297">Et ex illo loco apparet, quod Auerroes ne&longs;ciuit Geo&shy;


<pb pagenum="250" xlink:href="015/01/269.jpg"/>metriam, sicut &longs;emper fuit mos Philo&longs;ophorum <expan abbr="c&otilde;tentio&longs;orum">contentio&longs;orum</expan>, ut <lb/>nil &longs;ciant, &longs;ed &longs;olum garrire. </s>
          <s id="id004298">audierat hoc ab aliquo malo Geome&shy;<lb/>tra, &amp; repo&longs;uit in &longs;uos libros: nam nos, ut &longs;upr&agrave; uidi&longs;ti, demon&longs;tra&shy;<lb/>uimus oppo&longs;itum. </s>
          <s id="id004299">Quod uer&ograve; &longs;it macula illa ex umbra terr&aelig;, ue&shy;<lb/>rum non e&longs;t, quoniam una e&longs;&longs;et &amp; non diui&longs;a, &amp; occuparet totam il<lb/>lius faciem: nec e&longs;t uerum quod mutaret &longs;itum, quia &longs;uperficies ter&shy;<lb/>r&aelig; e&longs;t nonupla &longs;uperficiei Lun&aelig;. </s>
          <s id="id004300">Sicut terr&aelig; &longs;uperficies e&longs;t minor <lb/>trige&longs;ima parte &longs;uperficiei Solis. </s>
          <s id="id004301">Nec &longs;pargitur lumen Solis in Lu&shy;<lb/>na, nam &longs;ic e&longs;&longs;et ambitus ut uia lactea: cum autem Luna delin&shy;<lb/>quit in Oriente, e&longs;t glauca &amp; purpurea, cum in c&oelig;li medio rubra, <lb/>cum in Occidente nigra uidetur, nam ab utraque parte tenebris ope&shy;<lb/>ritur: ex Oriente ab umbra terr&aelig;, ab Occidente ab ob&longs;curitate loci. <lb/></s>
          <s id="id004302">In medijs locis medijs coloribus, quos A&longs;trologi terraticis tribu&shy;<lb/>unt: hoc autem quandiu tota delituerit, quod tempus horam uix <lb/>implere pote&longs;t. </s>
          <s id="id004303">Ergo partes perui&aelig; non remittunt lumen, ide&ograve; ob&shy;<lb/>&longs;cur&aelig; apparent, quod in uitreis &longs;peculis &agrave; quorum partibus plum&shy;<lb/>bum excidit: nam nigr&aelig; ill&aelig; apparent, reliqu&aelig; &longs;plendid&aelig;, ob id &longs;y&shy;<lb/>dera aliquando per illam relucent, &amp; aliquando non. </s>
          <s id="id004304">Et Solaris <lb/>eclyp&longs;is tempore, non lux tota Solis perit: atque ideo ut uidemus, &amp; <lb/>uariant colores eo tempore, non <expan abbr="tam&etilde;">tamen</expan> collu&longs;trat &longs;plendid&egrave; Sol ob <lb/><arrow.to.target n="marg874"/><lb/>cra&longs;sitiem Lunaris corporis h&aelig;c inferiora, tum etiam ob diuer&longs;ita&shy;<lb/>tem partium, &amp; ad &longs;itum. </s>
          <s id="id004305">Nam &longs;i Sol &longs;it ad &longs;itum a b, tran&longs;ibunt mul<lb/><figure id="id.015.01.269.1.jpg" xlink:href="015/01/269/1.jpg"/><lb/>ti radij, &longs;i c d pauci&longs;simi aut nulli, &longs;ed ut ubi <lb/>tenuior e&longs;t Luna in ambitu, &amp; Solis radij <lb/>den&longs;iores tran&longs;eunt, &amp; &longs;ydera pellucent <lb/>contrarijs cau&longs;is minus, ut iuxta medium <lb/>nequaqu&agrave;m. </s>
          <s id="id004306">At Lun&aelig; maculam radij effi&shy;<lb/>ciunt, etiam &longs;i tota &longs;ubtus opaca e&longs;&longs;et, cum <lb/>peruia uel tantillum fuerit in &longs;uperficie, ut <lb/>uenis opus non &longs;it. </s>
          <s id="id004307">Et iuxta hoc macula illa, ut liquet, ad perfectio&shy;<lb/>nem corporis Lun&aelig; pertinet magis quam pars &longs;plendida, quam&shy;<lb/>uis prima cogitatione oppo&longs;itum uideatur. </s>
          <s id="id004308">E&longs;t enim duplex perfe&shy;<lb/>ctionis genus in c&oelig;le&longs;tibus corporibus, &amp; ob den&longs;itatem cum re&shy;<lb/>mittit, &amp; ob per&longs;picuitatem cum &agrave; Sole, ut uniuer&longs;ali quo dam prin<lb/>ci pio illuminatur.</s>
        </p>
        <p type="margin">
          <s id="id004309"><margin.target id="marg873"/>T<emph type="italics"/>ex.<emph.end type="italics"/> 49.</s>
        </p>
        <p type="margin">
          <s id="id004310"><margin.target id="marg874"/>2. A<emph type="italics"/>poteles<emph.end type="italics"/><lb/>P<emph type="italics"/>tolem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004311">Propo&longs;itio ducente&longs;ima decima nona.</s>
        </p>
        <p type="main">
          <s id="id004312">Ratio nem eorum qu&aelig; apparent circa Solem &longs;peculo in aqua po<lb/>&longs;ito declarare.<lb/><arrow.to.target n="marg875"/></s>
        </p>
        <p type="margin">
          <s id="id004313"><margin.target id="marg875"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004314">Sit peluis a b aqua plena: &longs;peculum in ea c d e f quadratum, aut <lb/>perfecte, aut oblongum &longs;ub mer&longs;um in ea: Sol primum &longs;olus in g <pb pagenum="251" xlink:href="015/01/270.jpg"/>oculus ex aduer&longs;o in <lb/>h, ita ut ad &aelig;quales <lb/>angulos po&longs;sit uide&shy;<lb/>re <expan abbr="Sol&etilde;">Solem</expan> in k, dico &qring;d <lb/>depre&longs;&longs;o oculo in m, <lb/>uidebit alium Solem <lb/>maiorem uer&longs;us mar<lb/>ginem aduer&longs;um in l, <lb/>&amp; long&egrave; &longs;plendidio&shy;<lb/>rem: quia enim radij <lb/><expan abbr="reflect&utilde;tur">reflectuntur</expan> ex k, ut ro <lb/>bu&longs;ti &amp; &agrave; medio den<lb/>&longs;iore ad rarius, qui <lb/>non <expan abbr="inflecten&ttilde;">inflectentur</expan>, erunt <lb/>pauci, &amp; ide&ograve; Sol in <lb/>k minor apparebit, et <lb/>languidior: maior au<lb/><figure id="id.015.01.270.1.jpg" xlink:href="015/01/270/1.jpg"/><lb/>tem pars deflectetur &agrave; <expan abbr="perp&etilde;diculari">perpendiculari</expan> ad m, igitur Sol apparebit ma&shy;<lb/>ior &amp; ualidior long&egrave; &longs;plendentibus radijs, ade&ograve; ut uix ferri po&longs;sit. <lb/></s>
          <s id="id004315">Sed quoniam angulus ex &longs;uppo&longs;ito m l &longs; maior e&longs;t h k e, igitur cum <lb/>oculus iudicet &longs;e uidere a d &aelig;quales angulos, uidebitur g depre&longs;&shy;<lb/>&longs;ior &amp; propior labro in t, &longs;icut n m e&longs;t infra h, ita t infra g, quare <expan abbr="eti&atilde;">etiam</expan> <lb/>ut angulus m l &longs; &longs;it &aelig;qualis angulo t l f, nece&longs;&longs;e e&longs;t ut l &longs;it ultra k: ali&shy;<lb/>ter t uideretur qua&longs;i tangere aquam. </s>
          <s id="id004316">In hora autem deliquij Solis, <lb/>uelut hodie v. Idus Aprilis hora &longs;exta diei, <expan abbr="c&utilde;">cum</expan> diligenti&longs;simi &longs;tatue&shy;<lb/>rint medium eclip&longs;is in quinta, &amp; &longs;uppo&longs;ita fuerit ob&longs;curatio &agrave; Io&shy;<lb/>anne Stadio partium nouem cum be&longs;&longs;e, &amp; tempus hor&aelig; unius &amp; <lb/>m: 26, fuit tamen maior &amp; longior: quoniam luminaria <expan abbr="fuer&utilde;t">fuerunt</expan> pro&shy;<lb/>piora una parte caud&aelig; Draconis, quam ip&longs;e po&longs;uerit in tabulis, &amp; <lb/>hoc quia &longs;upponit &etail;quinoctium tardius diebus duobus <expan abbr="qu&atilde;">quam</expan> apud <lb/>Alphon&longs;um: &amp; for&longs;an &longs;ufficiebat una dies, &longs;cilicet ut e&longs;&longs;et die deci&shy;<lb/>ma Martij horis decem octo &agrave; meridie: nam tunc omnia re&longs;pon&shy;<lb/>dent ob&longs;eruationi: in qua apparuerunt quatuor Lun&aelig;: &amp; quidem <lb/>ab initio fuerunt du&aelig; orientaliores &egrave; regione, &longs;cilicet o p, &amp; una oc<lb/>cidentalior n, &amp; tantum di&longs;tabat n a k quantum o: Et clarum erat <lb/>qu&ograve;d p erat, &longs;icut &longs;ecunda iris parua &amp; non candida, &longs;ed rubra pur&shy;<lb/>pureo mi&longs;ta, quoniam ex reflexu o oriebatur: apparebat autem a la <lb/>tere illo, quoniam Luna dextram partem obtegebat, ideo illa erat <lb/>minus lumino&longs;a, &amp; uerus Sol erat in k, mod&ograve; Lun&aelig;, mod&ograve; Solis <lb/>imaginem referens ubi tran&longs;i&longs;&longs;et eclip&longs;is medium, non amplius <lb/>tres ill&aelig; Lun&aelig; apparuerunt &agrave; dextra &amp; &agrave; &longs;ini&longs;tra, &longs;ed una ultra nos 


<pb pagenum="252" xlink:href="015/01/271.jpg"/>in q, &amp; du&aelig; uer&longs;us nos in r &amp; n <lb/>&amp; qu&aelig; erat in F, erat &longs;imiliter <lb/>parua &amp; purpurea rubraque, &amp; <lb/>mutato &longs;peculo uariebatur &longs;i&shy;<lb/>tus q &amp; r u, id e&longs;t, ut modo e&longs;&shy;<lb/>&longs;ent qua&longs;i in medio laterum e <lb/>&amp; f, quando que tran&longs;uer&longs;&aelig;. </s>
          <s id="id004317">Et <lb/>hoc contigit ob <expan abbr="mutation&etilde;">mutationem</expan> lo&shy;<lb/>ci k propter &longs;peculi <expan abbr="uariation&etilde;">uariationem</expan>.</s>
        </p>
        <figure id="id.015.01.271.1.jpg" xlink:href="015/01/271/1.jpg"/>
        <p type="main">
          <s id="id004318">Cau&longs;a e&longs;t, quoniam Luna <expan abbr="c&utilde;">cum</expan> <lb/>permeet Solem non &egrave; regione <lb/>recta line&aelig; oppo&longs;it&aelig; no&longs;tro ui <lb/>&longs;ui, &amp; &longs;olum <expan abbr="mom&etilde;to">momento</expan>, &amp; in lon<lb/>gis <expan abbr="tempor&utilde;">temporum</expan> interuallis po&longs;sit <lb/>obtegere illum. </s>
          <s id="id004319">Sit ergo ut Sol <lb/>obtegatur &agrave; Luna medijs par&shy;<lb/>tibus, &amp; &longs;int radij extremi in <lb/>&longs;peculo: a c &amp; a d, igitur erunt <lb/>tanquam duo Soles, &longs;ed uterque<lb/> illorum geminatur, ide&ograve; fiunt <lb/>tres: medius enim ob Lun&aelig; <lb/>per&longs;picuitatem integer, appa&shy;<lb/>ret, ide&ograve; mod&ograve; &longs;ub forma So&shy;<lb/>lis, mod&ograve; Lun&aelig; laterones am&shy;<lb/>bo &longs;ub forma Lun&aelig;: ide&ograve; <expan abbr="er&utilde;t">erunt</expan> <lb/>tres, quibus. </s>
          <s id="id004320">addita Luna p, qu&aelig; <lb/>e&longs;t reflexa a &longs;ecunda, fient qua&shy;<lb/>tuor. </s>
          <s id="id004321">At dices cur non fit refle&shy;<lb/>xus &longs;ecundum directum oculi, <lb/>ut Lun&aelig; appareant ultra citra&shy;<lb/>que Solem? </s>
          <s id="id004322">Dico quod Luna <lb/>diuidente orbem reflexus fit ad latera, quia radij tran&longs;uer&longs;im ferun&shy;<lb/>tur: cum autem non diuiditur fit pror&longs;um &amp; retror&longs;um. </s>
          <s id="id004323">Sed cur di&shy;<lb/>ces Lunari forma? </s>
          <s id="id004324">quoniam partes Solis qu&aelig; uigent, eiu&longs;modi for&shy;<lb/>ma apparent, Iconem uides &agrave; latere.</s>
        </p>
        <p type="main">
          <s id="id004325">Propo&longs;itio ducente&longs;ima uige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id004326">Cau&longs;am cur Sol &aelig;&longs;tiuis diebus exoriens umbram ad meridiem, <lb/>cum in meridie ad boream mittat, explorare.</s>
        </p>
        <p type="main">
          <s id="id004327"><arrow.to.target n="marg876"/></s>
        </p>
        <p type="margin">
          <s id="id004328"><margin.target id="marg876"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004329">Dico quod ubicunque terrarum in no&longs;tro hemi&longs;pherio, Sol ubi <lb/>fuerit in Oriente &longs;eu Occidente uidebitur, cum &longs;ub circulo &aelig;quino<lb/>ctij fuerit &egrave; regione, nobis <expan abbr="eti&atilde;">etiam</expan> &longs;i homo &longs;ub arctico circulo habitet, 


<pb pagenum="253" xlink:href="015/01/272.jpg"/>&amp; ita re&longs;picienti ad polum umbra erit &agrave; dextra in &longs;ini&longs;tram, dum o&shy;<lb/>ritur &amp; &agrave; &longs;ini&longs;tra in dextram dum occidit. </s>
          <s id="id004330">Et quod dum erit in me&shy;<lb/>ridie umbra uerget ad Septentrionem. </s>
          <s id="id004331">Terti&ograve; dico, qu&ograve;d in his <lb/>qui habitant uer&longs;us Septentrionem &agrave; tropico cancri umbra in Me&shy;<lb/>ridie, quo cunque tempore anni borealis erit. </s>
          <s id="id004332">Quarto, qu&ograve;d ij&longs;dem <lb/>toto dimidio anni ab &aelig;quinoctio uerno ad autumnale, umbr&aelig; o&shy;<lb/>riente &amp; occidente Sole &longs;unt meridian&aelig; tran&longs;uer&longs;&aelig;: &amp; muri re&longs;pi&shy;<lb/>cientes boream illuminantur. </s>
          <s id="id004333">Sit finitor a b c d in regione boreali, <lb/>cuius uertex e &amp; f polus, eleuatio poli &longs;upra finitorem a f, &aelig;quino&shy;<lb/>ctij circulus b q d, cui parallelus borealior Solis uia per cancri ini&shy;<lb/>tium, g h l m n, circulus magnus per uerticem, &amp; inter&longs;ectiones &aelig;&shy;<lb/>quinoctij, &amp; finitoris b h e m d, Meridiei &longs;emicirculus &longs;uperior a f e <lb/>l q c. </s>
          <s id="id004334">Cum ergo uertex regionis &longs;it in e, &amp; circulus magnus b h d <lb/>tran&longs;iens per uerticem, tran&longs;eat per centrum terr&aelig; ex diffinitione <lb/>circuli magni, &amp; linea &agrave; uertice grauium habitantium &longs;ub uertice e, <lb/><figure id="id.015.01.272.1.jpg" xlink:href="015/01/272/1.jpg"/><lb/>tendat ad centrum terr&aelig; ex de&shy;<lb/>mon&longs;tratis ab Ari&longs;totele, &amp; &longs;up<lb/>po&longs;itis ab A&longs;trologis, &qring;d gra&shy;<lb/>uia omnia tendunt ad centrum <lb/>terr&aelig;, erit quodlibet graue &longs;eu <lb/>murus &longs;eu homo, &longs;eu per ulti&shy;<lb/>mam <expan abbr="petition&etilde;">petitionem</expan>, &longs;eu per demon&shy;</s>
        </p>
        <p type="main">
          <s id="id004335"><arrow.to.target n="marg877"/><lb/>&longs;trata in undecimo ab Euclide <lb/>murus, &amp; homo quiuis inco&shy;<lb/>la regionis in &longs;uperficie circuli <lb/>uerticalis b e d. </s>
          <s id="id004336">Igitur dum Sol <lb/>e&longs;t in b uel d, umbr&aelig; <expan abbr="er&utilde;t">erunt</expan> &agrave; dex<lb/>tro in &longs;ini&longs;trum, uel contrario <lb/>modo, &amp; ita Sol uidebitur e&longs;&longs;e &egrave; regione nobis: &amp; murus faciet um<lb/>bram <expan abbr="oriental&etilde;">orientalem</expan> uel occidentalem. </s>
          <s id="id004337">Et hoc e&longs;t primum. </s>
          <s id="id004338">Et quoniam <lb/>cum Sol erit in Meridie, tum erit in q, igitur erit umbra ad Septen&shy;<lb/>trionem, cum e &longs;it loco gnomonis &amp; murus. </s>
          <s id="id004339">Et hoc e&longs;t &longs;ecun dum. <lb/></s>
          <s id="id004340">Tertium etiam patet, quia Sol nun quam tran&longs;ibit <expan abbr="punct&utilde;">punctum</expan> l in Me&shy;<lb/>ridie uer&longs;us boream, &longs;ed regio &longs;upponitur borealior l, igitur tempo<lb/>re meridiei umbra &longs;emper hic borealis erit. </s>
          <s id="id004341">Et quoniam b h e m d <lb/>&longs;ecat parallelos, qui &longs;unt in Septentrione ut puta tropicum in h <lb/>&amp; m, igitur oriente Sole, &amp; occidente rur&longs;us per totum arcum g h <lb/>&amp; m n, uidebitur borealior qu&agrave;m in b uel d parte arcus magni in&shy;<lb/>tercepti inter arcum magnum tran&longs;euntem per uerticem &amp; locum <lb/>Solis, ubi &longs;ecat finitorem &amp; puncta b, &amp; d: &amp; ita erunt umbr&aelig; Me&shy;<lb/>ridionales toto hoc tempore, &amp; hoc e&longs;t quartum.


<pb pagenum="254" xlink:href="015/01/273.jpg"/><arrow.to.target n="marg878"/></s>
        </p>
        <p type="margin">
          <s id="id004342"><margin.target id="marg877"/>F<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 1</s>
        </p>
        <p type="margin">
          <s id="id004343"><margin.target id="marg878"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 1.</s>
        </p>
        <p type="main">
          <s id="id004344">Ex quo &longs;equitur, quod in hoc toto tempore ueris &amp; &aelig;&longs;tatis, c&ugrave;m <lb/>Sol in Meridie uideatur e&longs;&longs;e po&longs;t tergum, &amp; in Meridie, &amp; dum ori<lb/>tur &agrave; parte Septentrionis. </s>
          <s id="id004345">Ergo ab ortu Solis ad Meridiem uidebi&shy;<lb/>tur ferri motu diurno, linea obliqua &agrave; <expan abbr="Sept&etilde;trione">Septentrione</expan> in Meridiem: &amp; <lb/>&agrave; Meridie ad Occa&longs;um, alia obliqua linea &agrave; Meridie in Septentrio&shy;<lb/>nem: ut in figura, ut &longs;i Sol &longs;it in a in Oriente, b in Meridie, cin Occi&shy;<lb/>dente, &amp; uertex nobis in e.</s>
        </p>
        <p type="main">
          <s id="id004346"><arrow.to.target n="marg879"/></s>
        </p>
        <p type="margin">
          <s id="id004347"><margin.target id="marg879"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 2.</s>
        </p>
        <p type="main">
          <s id="id004348">Sequitur etiam, qu&ograve;d &longs;i tempore &aelig;&longs;tatis <lb/><figure id="id.015.01.273.1.jpg" xlink:href="015/01/273/1.jpg"/><lb/>po&longs;&longs;emus in media nocte uidere Solem, in <lb/>c&oelig;li medio uideretur, tantundem uer&longs;us bo<lb/>ream declinare, quantum in Meridie ad Me<lb/><expan abbr="ridi&etilde;">ridiem</expan>. </s>
          <s id="id004349">Et hoc quia circulus &aelig;quinoctij b q d, <lb/>tanto borealior e&longs;t in parte inferiore circulo <lb/>per uerticem, quanto in &longs;uperiori e&longs;t au&longs;tra&shy;<lb/>lior: quoniam circuli magni &longs;e &longs;ecant per &aelig;&shy;<lb/>qualia. </s>
          <s id="id004350">Et &longs;i hoc e&longs;t uerum de Sole &longs;ub &aelig;qui&shy;<lb/>noctij circulo, <expan abbr="qu&atilde;to">quanto</expan> magis erit uerum de Sole &longs;ub tropico &aelig;&longs;tiuo?</s>
        </p>
        <p type="main">
          <s id="id004351"><arrow.to.target n="marg880"/></s>
        </p>
        <p type="margin">
          <s id="id004352"><margin.target id="marg880"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}. 3.</s>
        </p>
        <p type="main">
          <s id="id004353">Ex pr&aelig;cedenti patet, &qring;d Sol in media nocte borealior uideretur <lb/>&longs;ub &aelig;quinoctij circulo tanto, <expan abbr="qu&atilde;to">quanto</expan> uidetur au&longs;tralior &longs;e ip&longs;o, dum <lb/>e&longs;t &longs;ub tropico cancri, quia circuli &longs;e &longs;ecant ad angulos oppo&longs;itos <lb/>&aelig;quales: igitur &longs;i uerticis circulus maiorem facit angulum &longs;uperio&shy;<lb/><figure id="id.015.01.273.2.jpg" xlink:href="015/01/273/2.jpg"/><lb/>rem cum &aelig;quinoctij quam tro</s>
        </p>
        <p type="main">
          <s id="id004354"><arrow.to.target n="marg881"/><lb/>pici borealis circulo, igitur &amp; <lb/>inferiorem: homo autem &amp; ui&shy;<lb/>&longs;us iudicat au&longs;trale &amp; boreale <lb/>iuxta inclinationem circuli du<lb/>cti per <expan abbr="loc&utilde;">locum</expan> Solis ad circulum <lb/>ductum per locum uerticis.</s>
        </p>
        <p type="margin">
          <s id="id004355"><margin.target id="marg881"/>P<emph type="italics"/>er funilem<emph.end type="italics"/><lb/>15. <lb/>P<emph type="italics"/>ropo&longs;. pri&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004356">Propo&longs;itio CCXXL</s>
        </p>
        <p type="main">
          <s id="id004357">Magnitudo Lun&aelig; &amp; c&aelig;te&shy;<lb/>rorum <expan abbr="a&longs;tror&utilde;">a&longs;trorum</expan> digno&longs;citur ex <lb/>proportione aliorum ad eam <lb/>iuxta di&longs;tantiam: ip&longs;ius uer&ograve; <lb/>iuxta rationem pupill&etail; ad Lu&shy;<lb/>nam di&longs;tanti&aelig; ratione.<lb/><arrow.to.target n="marg882"/></s>
        </p>
        <p type="margin">
          <s id="id004358"><margin.target id="marg882"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004359">Sit pupilla a b, qu&aelig; in circu&shy;<lb/>lo l m, po&longs;ita in eodem centro, <lb/>comprehendat portionem no <lb/>tam l m, ideo clau&longs;o oculo alte&shy;<lb/>ro eandem portionem uidebit <lb/>totius c&oelig;li, ut liquet ex demon 


<pb pagenum="255" xlink:href="015/01/274.jpg"/>&longs;tratis in Elementis Euclidis, igitur nota l m nota erit pupill&aelig;, &amp; <lb/>ideo g h quanta &longs;it portio c&oelig;li, quia k e&longs;t etiam qua&longs;i centrum c&oelig;&shy;<lb/>li Lun&aelig;, &longs;it ergo Luna c d, eritque tanta portio g h not&aelig;, quanta e f <lb/>pars pupill&aelig;, per quam uidetur ip&longs;ius a b: e f autem &longs;imiliter e&longs;t no&shy;<lb/>ta in n o, igitur &amp; c d in comparatione ad totum circulum. </s>
          <s id="id004360">Quia ue&shy;<lb/>ro g h e&longs;t nota, &amp; in Sole con&longs;picitur arcus notus &aelig;qualis, ergo erit <lb/>nota diuer&longs;itas a&longs;pectu ob di&longs;tantiam no&longs;tram &agrave; terr&aelig; centro, qua&shy;<lb/>re altitudo Lun&aelig; nota, &amp; eius magnitudo, eius enim ad &longs;emidiame<lb/>trum oculi, ut c d ad ef. </s>
          <s id="id004361">Hoc autem e&longs;t cra&longs;&longs;a Minerua additum, ut <lb/>quis intelligat difficiliora e&longs;&longs;e qu&aelig; cra&longs;&longs;a uidentur, qu&agrave;m qu&aelig; ela&shy;<lb/>borata. </s>
          <s id="id004362">huiu&longs;modi 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 &longs;eu pote&longs;tate e&longs;t duarum <expan abbr="quantitat&utilde;">quantitatum</expan>, qu&aelig; <lb/>&longs;ic &longs;e habent, ut null&aelig; du&aelig; ali&aelig; 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 e&longs;t comparatio rei non habentis quantitatem, <lb/>qu&aelig; tamen mutari po&longs;sit ad rem, qu&aelig; quantitatem habeat.</s>
        </p>
        <p type="head">
          <s id="id004369">DIFFINITIO TERTIA.</s>
        </p>
        <p type="main">
          <s id="id004370">Proportio &longs;ublimis &longs;eu ordo dicitur duarum &longs;ub&longs;tantiarum, qu&etail; <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="qu&atilde;titatis">quantitatis</expan>, quantitatem autem <lb/>non habet, neque e&longs;t quantitas.</s>
        </p>
        <p type="head">
          <s id="id004373">PETITIO SECVNDA.</s>
        </p>
        <p type="main">
          <s id="id004374">Repugnans e&longs;t &longs;uper quod nulla e&longs;t potentia.</s>
        </p>
        <p type="head">
          <s id="id004375">PETITIO TERTIA.</s>
        </p>
        <p type="main">
          <s id="id004376">Non po&longs;&longs;e &longs;uper ea qu&aelig; <expan abbr="repugn&atilde;t">repugnant</expan>, nullam declarat imperfectio&shy;<lb/>nem, neque infinitum non e&longs;&longs;e negat.</s>
        </p>
        <p type="head">
          <s id="id004377">PETITIO QVARTA.</s>
        </p>
        <p type="main">
          <s id="id004378">Infinitum infinito maius e&longs;&longs;e non pote&longs;t.</s>
        </p>
        <p type="main">
          <s id="id004379">Propo&longs;itio ducente&longs;ima uige&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id004380">Quantitates qu&aelig; &aelig;quales e&longs;&longs;e <expan abbr="n&otilde;">non</expan> po&longs;&longs;unt in eodem genere, ma&shy;<lb/>ius tamen &amp; minus recipiunt, &longs;unt in proportione pote&longs;tatis.</s>
        </p>
        <p type="main">
          <s id="id004381">Sint propo&longs;iti duo anguli, gratia exempli, a rectilineus, b uer&ograve; in </s>
        </p>
        <p type="main">
          <s id="id004382"><arrow.to.target n="marg883"/><lb/><expan abbr="circumfer&etilde;tia">circumferentia</expan> circuli, qui pote&longs;t e&longs;&longs;e maior, &amp; minor rectilineo pro&shy;<lb/>po&longs;ito, &amp; nunqu&agrave;m pote&longs;t e&longs;&longs;e &aelig;qualis, ut declaratum e&longs;t &longs;upr&agrave;, di&shy;<lb/>co proportionem b ad a e&longs;&longs;e pote&longs;tate, nam ut ui&longs;um e&longs;t, pote&longs;t e&longs;&longs;e <lb/>maior &amp; minor, &amp; e&longs;t maius &amp; minus uer&egrave;, &amp; ide&ograve; &longs;unt in eodem <lb/>genere, &amp; uterque e&longs;t continua quantitas, igitur in tran&longs;itu nece&longs;&longs;e <lb/>e&longs;t, ut &longs;int &aelig;quales aliquando &longs;ed non actu, hoc enim repugnat, igi&shy;<lb/>tur pote&longs;tate.</s>
        </p>
        <pb pagenum="256" xlink:href="015/01/275.jpg"/>
        <p type="margin">
          <s id="id004383"><margin.target id="marg883"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id004384">Propo&longs;itio ducente&longs;ima uige&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id004385">Quantitates qu&aelig; actu &aelig;quales e&longs;&longs;e non po&longs;&longs;unt, in nulla pro&shy;<lb/>portione actu e&longs;&longs;e po&longs;&longs;unt.<lb/><arrow.to.target n="marg884"/></s>
        </p>
        <p type="margin">
          <s id="id004386"><margin.target id="marg884"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004387">Sint du&aelig; quantitates qu&aelig; &aelig;quales e&longs;&longs;e non po&longs;sint, ut in priore <lb/>exemplo a &amp; b, dico quod non po&longs;&longs;unt e&longs;&longs;e in aliqua proportione <lb/>in actu, aliter &longs;int in proportione c, &amp; ducatur cin b, fiat d, erunt er&shy;<lb/>go d &amp; a &aelig;quales, quod e&longs;t contra &longs;uppo&longs;itum, nam &longs;upponitur <lb/>quod nulla quantitas ex genere b &longs;it &aelig;qualis a, &longs;ed d e&longs;t ex genere </s>
        </p>
        <p type="main">
          <s id="id004388"><arrow.to.target n="marg885"/><lb/>b &amp; &aelig;quale a, &amp; ideo &longs;uppo&longs;itum non manet, igitur a &amp; b non &longs;unt <lb/>in aliqua proportione in actu.</s>
        </p>
        <p type="margin">
          <s id="id004389"><margin.target id="marg885"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004390">Propo&longs;itio ducente&longs;ima uige&longs;ima quarta.</s>
        </p>
        <p type="main">
          <s id="id004391">Neque temporis totius ut imaginamur ip&longs;um e&longs;&longs;e infinitum, neque<lb/> &aelig;ui uitarum proportio ulla e&longs;t ad tempus quod pote&longs;tate e&longs;t, ut po<lb/>t&egrave; diem uel men&longs;em.<lb/><arrow.to.target n="marg886"/></s>
        </p>
        <p type="margin">
          <s id="id004392"><margin.target id="marg886"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id004393">Tempus ip&longs;um ut <expan abbr="infinit&utilde;">infinitum</expan> e&longs;t, aut in actu e&longs;t, aut refert quippiam <lb/>in actu, pars autem temporis &longs;ol&ugrave;m e&longs;t pote&longs;tate, quia nullum tem&shy;<lb/>pus in actu e&longs;t, neque annus, neque men&longs;is, neque dies, neque hora aut mo&shy;<lb/>mentum, &longs;ed &longs;i totum tempus non e&longs;&longs;et actu, nihil e&longs;&longs;et actu, neque to<lb/>tum neque partes. </s>
          <s id="id004394">Igitur <expan abbr="tot&utilde;">totum</expan> tempus, uel aliquid loco eius e&longs;t actu, <lb/>partes autem pote&longs;tate, &longs;ed ut ui&longs;um proportio infiniti nulla e&longs;t, &amp; <lb/>ad rem qu&aelig; actu non e&longs;t, igitur tempus nullam habet proportio&shy;<lb/>nem ad annos, neque men&longs;es uel dies. </s>
          <s id="id004395">Quare qui dicunt, quod mille <lb/>anni &longs;unt unus dies, in philo&longs;ophia errant, &longs;ecus apud Apo&longs;tolum, <lb/>ubi de diuinitate agitur. </s>
          <s id="id004396">Ergo anni &longs;unt <expan abbr="long&utilde;">longum</expan> tempus, &amp; dies bre&shy;<lb/>ue, quia dicuntur in comparatione inter &longs;e, &amp; non &longs;ecundum pro&shy;<lb/>portionem ad infinitum. </s>
          <s id="id004397">Quia &longs;it infinitum a, &amp; d u&aelig; quantitates b <lb/>maior, &amp; c minor, uel ergo proportio a ad b c, e&longs;t una uel diuer&longs;a, &longs;i </s>
        </p>
        <p type="main">
          <s id="id004398"><arrow.to.target n="marg887"/><lb/>una, ergo b c erunt &aelig;quales, &longs;i maior e&longs;t ad c quam ad b, ergo infi&shy;<lb/>nitum e&longs;t maius infinito, ergo non e&longs;t infinitum, quod e&longs;t con&shy;<lb/><arrow.to.target n="marg888"/><lb/>tra petita.</s>
        </p>
        <p type="margin">
          <s id="id004399"><margin.target id="marg887"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="id004400"><margin.target id="marg888"/>4. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004401">Propo&longs;itio ducente&longs;ima uige&longs;ima quinta.</s>
        </p>
        <p type="main">
          <s id="id004402">Proportio media non e&longs;t ex ratione agentis &longs;ed patientis.</s>
        </p>
        <figure id="id.015.01.275.1.jpg" xlink:href="015/01/275/1.jpg"/>
        <p type="main">
          <s id="id004403">Proponatur a quantitas, qu&etail; debeat mutari ab uir&shy;<lb/><arrow.to.target n="marg889"/><lb/>tute qu&aelig; non fit in materia, &amp; palam e&longs;t quod non po<lb/>terit permutari in in&longs;tanti, quia &longs;imul e&longs;&longs;et, &amp; non e&longs;&longs;et <lb/>ergo repugnaret, neque etiam pote&longs;t non e&longs;&longs;e, ut demon&longs;tratum e&longs;t <lb/>in Hyperchen, quia repugnant nece&longs;&longs;ario &amp; e&longs;&longs;enti&aelig; Dei, neque mo&shy;<lb/>uetur &agrave; certa proportione, quia b caret omni quantitate, ergo ni&shy;<lb/><arrow.to.target n="marg890"/><lb/>hil o&longs;tendit uim ip&longs;ius b e&longs;&longs;e finitam, quod ergo moueatur tard&egrave; ce<pb pagenum="257" xlink:href="015/01/276.jpg"/>leriter paruum magnum, i&longs;tud contingit totum ex conditionibus <lb/>a, id e&longs;t, materi&aelig; &amp; quantitatis: uelut, gratia exempli, &longs;i a e&longs;&longs;et in ua&shy;<lb/>&longs;culo palmi, non po&longs;&longs;et implere iugerum, &amp; hoc <expan abbr="n&otilde;">non</expan> o&longs;tendit ullam <lb/>imperfectionem in b. </s>
          <s id="id004404">Et &longs;icut homines omnes &longs;unt in carcere huius <lb/>mundi, &amp; tamen uidentur e&longs;&longs;e &longs;ibi liberi, &amp; appellant <expan abbr="&longs;ol&utilde;">&longs;olum</expan> illos e&longs;&longs;e <lb/>in carcere qui &longs;unt in erga&longs;tulo, ita omnis materia, &amp; omnis quan&shy;<lb/>titas habet conditiones, per quas &lpar;ut ita <expan abbr="dic&atilde;">dicam</expan>&rpar; con&longs;tringitur, &amp; repu<lb/>gnat eas mutari, &amp; ide&ograve; <expan abbr="uit&atilde;">uitam</expan> agunt &longs;ine ulla proportione. </s>
          <s id="id004405">Quod ue <lb/>r&ograve; dictum e&longs;t, &longs;upra dictum fuit, per exemplum dictum e&longs;t, <expan abbr="n&otilde;">non</expan> quia <lb/>ita &longs;it, finge ergo quod in aliquo pariete, non &longs;it albitudo, ni&longs;i unius <lb/>gradus, illa non operabitur ni&longs;i per unum <expan abbr="grad&utilde;">gradum</expan>, etiam &longs;i calx e&longs;&longs;et <lb/>infinit&egrave; alba, &amp; &longs;imiliter de luce Solis, ergo omnes mentes mouent <lb/>&longs;ine proportione, &amp; non po&longs;&longs;unt dici finit&aelig; uel infinit&aelig;, quia ip&longs;&aelig; <lb/>&longs;unt expertes omnis quantitatis, im&ograve; omnis relationis ad quantita<lb/>tem, &amp; hoc e&longs;t quod latuit multos, &amp; maxim&egrave; propter dictum Phi&shy;<lb/>lo&longs;ophi, e&longs;t ergo omnis operatio iuxta id quod e&longs;t in materia, &amp; <lb/>non quod una mens maiores habeat uires, alia cum non &longs;it in eis, <lb/>neque maius neque minus.</s>
        </p>
        <p type="margin">
          <s id="id004406"><margin.target id="marg889"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="margin">
          <s id="id004407"><margin.target id="marg890"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. P<emph type="italics"/>etit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="id004408">Propo&longs;itio ducente&longs;ima uige&longs;ima &longs;exta.</s>
        </p>
        <p type="main">
          <s id="id004409">Proportio &longs;ublimis non con&longs;i&longs;tit in magnitudine, &longs;ed ordine <lb/>iuxta quem differentia e&longs;t eius quod e&longs;t ante &amp; po&longs;t.</s>
        </p>
        <p type="main">
          <s id="id004410">Non enim pote&longs;t e&longs;&longs;e comparatio iuxta magnitudines motas, <lb/><arrow.to.target n="marg891"/><lb/>quoniam uel &longs;unt corpora c&oelig;le&longs;tia, uel elementaria, <expan abbr="elem&etilde;taria">elementaria</expan> e&longs;&longs;e <lb/>non po&longs;&longs;unt, quia illa cum &longs;int corruptioni obnoxia, id e&longs;t, tran&longs;mu<lb/>tationi, &longs;ecundum qualitatem <expan abbr="n&otilde;">non</expan> po&longs;&longs;unt e&longs;&longs;e &longs;ubiecta in corpor ca&shy;<lb/>rum &longs;ub&longs;tantiarum, neque &agrave; primis &longs;ub&longs;tantijs moueri, neque etiam ex&shy;<lb/>cipere prim&ograve; lumen &longs;uum, &longs;ed mouentur per uim influxam &agrave; c&oelig;le&shy;<lb/>&longs;tibus corporibus, neque etiam per motum corporum <expan abbr="c&oelig;le&longs;ti&utilde;">c&oelig;le&longs;tium</expan>, nam <lb/>illa non mouentur &longs;ecundum proportionem mentis ad corpus, &longs;ed <lb/>iuxta rationem finis, &agrave; qua circum&longs;cribuntur, &amp; ideo quod Satur&shy;<lb/>nus moueatur uelociore motu, qu&agrave;m Iuppiter ab Oriente in Occi&shy;<lb/>dentem, hoc non e&longs;t, quia uit&aelig; qu&aelig; mouet Saturnum fit robu&longs;tior <lb/>uita qu&etail; mouet Iouem, cum &longs;int una &amp; eadem: uel &longs;i dicas quod &longs;int <lb/>diuer&longs;&aelig; uita Saturni, non tamen e&longs;t ualidior in comparatione ad <lb/>&longs;uum c&oelig;lum, uita Iouis non moueret celerius Saturnum ab Occi&shy;<lb/>dente in Orientem, qu&agrave;m uita Iouis Iouem, quod e&longs;t fal&longs;um, &longs;ed ta&shy;<lb/>lis motus uelo citas e&longs;t ratione finis, quia oportet ut pariter mouea&shy;<lb/>tur eo motu, &amp; quia c&oelig;lum Saturni e&longs;t maius, ideo celerius moue&shy;<lb/>tur quam Iouis, &amp; hoc ratione corporis mobilis, &amp; <expan abbr="n&otilde;">non</expan> ratione pro&shy;<lb/>portionis ad corpus. </s>
          <s id="id004411">Dico etiam, quod non habent potestatem <lb/>aliam, per quam &longs;ubeant proportionem, nam qu&etail;ritur cuius com&shy;


<pb pagenum="258" xlink:href="015/01/277.jpg"/>paratione illa proportio oriatur, nam non ad corpora, quia neque <lb/>ad c&oelig;le&longs;tia, neque mortalia, ut dictum e&longs;t, ni&longs;i fingamus alia corpora, <lb/>quod e&longs;t ab&longs;urdum, neque etiam ratione incorporeorum, nam non <lb/>po&longs;&longs;unt de&longs;truere &longs;e inuicem, quia inferior non pote&longs;t tollere &longs;upe&shy;<lb/>riorem, neque multo minus pote&longs;t uelle. </s>
          <s id="id004412">Hoc e&longs;t enim nefas cogita&shy;<lb/>re, neque &longs;uperior inferiorem, quam producit quam amat: &amp; ideo <lb/>dico, quod &longs;unt in proportione &longs;ublimium, id e&longs;t, ordine perfectio&shy;<lb/>nis, qui con&longs;i&longs;tit in propinquitate ad primam cau&longs;am. </s>
          <s id="id004413">exemplum, <lb/>Sol e&longs;t longe perfectior &longs;ua luce, qu&aelig; e&longs;t ei propria, quia Sol e&longs;t <lb/>&longs;ub&longs;tantia, &amp; lux e&longs;t proprium, &amp; lux Solis e&longs;t multo perfectior lu&shy;<lb/>mine, cum &longs;it &lpar;ut dixi&rpar; lux proprium &amp; in Sole, tanquam in &longs;ubie&shy;<lb/>cto, lumen autem extra &amp; accidens. </s>
          <s id="id004414">Nec tamen dicendum e&longs;t, quod <lb/>Sol &longs;it potentior luce, aut lux lumine, idem dico de anima &amp; facul&shy;<lb/>tatibus eius, &amp; functionibus, inter quas nulla cadit proportio per&shy;<lb/>fectionis, tamen differentia con&longs;picua e&longs;t, &amp; ideo poterit impediri <lb/>functio, &amp; non facultas, et facultas tolli remanente anima. </s>
          <s id="id004415">For&longs;an di<lb/>ces, quod i&longs;t&etail; non &longs;unt &longs;ub&longs;tanti&aelig;, &amp; ide&ograve; oporteret, ut omnia in&shy;<lb/>corporea Deo &longs;olo excepto e&longs;&longs;ent accidentia, dico quod in incor&shy;<lb/>poreis non e&longs;t &longs;icut in anima, qu&aelig; e&longs;t iuncta corpori, neque ut in So&shy;<lb/>le quod e&longs;t corpus, &longs;ed tanta e&longs;t perfectio producti incorporei, <lb/>quod ip&longs;um e&longs;t &longs;ub&longs;tantia. </s>
          <s id="id004416">Et ratio e&longs;t quia &longs;ub&longs;tantia differt ab ac&shy;<lb/>cidente uel ratione corporis, ut aqua &agrave; frigiditate, &amp; hoc non e&longs;t in <lb/>incorporeis, ut manife&longs;tum e&longs;t, uel quia unum &longs;it &longs;ubiectum alte&shy;<lb/>rius, &amp; ide&ograve; &longs;ub&longs;tantia, ut e&longs;t principium comparationis, &amp; in &longs;e <lb/>ip&longs;a dicitur &longs;ub&longs;tantia, &amp; ut comparatur ad extra &amp; ad operatio&shy;<lb/>nem &longs;uam, cuius e&longs;t principium dicitur facultas: uelut uita c&oelig;le&shy;<lb/>&longs;tis &longs;ub&longs;tantia e&longs;t, ut uer&ograve; c&oelig;lum pulchritudine illius delectatum <lb/>mouetur ad ob&longs;equium, dicitur facultas in illa uita, &amp; non e&longs;t ni&longs;i <lb/>&longs;ub&longs;tantia, tamen ip&longs;ius uit&aelig; adeo ut &longs;ola ratione differant. </s>
          <s id="id004417">Tertia <lb/>differentia e&longs;t, quia &longs;ub&longs;tantia non e&longs;t in &longs;ubiecto, &longs;ed facultas e&longs;t in <lb/>&longs;ubiecto, uer&ugrave;m in incorporeis, ut dixi, non differunt ni&longs;i &longs;ola ra&shy;<lb/>tione, uelut pater &amp; homo, nam pater nece&longs;&longs;ari&ograve; e&longs;t homo, &amp; e&longs;t <lb/>&longs;ub&longs;tantia, ut ad aliud comparatur. </s>
          <s id="id004418">Quarta differentia e&longs;t ratione <lb/>propri&aelig; natur&aelig; qu&aelig; non dependet, nam &longs;ub&longs;tantia non pendet <lb/>&longs;icut accidens &amp; facultas, uer&ugrave;m ubi genita fuit non amplius pen&shy;<lb/>det: re&longs;pondeo, quod in incorporeis producitur, &amp; non repugnet <lb/>productio &longs;ub&longs;tanti&aelig;, quia &longs;i non repugnat generatio hominis, <lb/>quod &longs;it &longs;ub &longs;tantia, multo minus etiam incorporeorum. </s>
          <s id="id004419">Relinqui&shy;<lb/>tur ut obijcias, quoniam &longs;ub&longs;tanti&aelig; incorpore&aelig; &longs;emper fiunt, er&shy;<lb/>go nunquam &longs;unt uer&aelig; &longs;ub&longs;tanti&aelig;: ad hoc re&longs;pondendum e&longs;t per <lb/>interemptionem, nam de uera re&longs;pon&longs;ione non e&longs;t hic locus, quod 


<pb pagenum="259" xlink:href="015/01/278.jpg"/>cadem ratione qua producuntur uit&aelig;, producuntur etiam c&oelig;li, at <lb/>c&oelig;lum nihilominus e&longs;t uer&egrave; &longs;ub&longs;tantia, &amp; magis i&longs;tis mortalibus, <lb/>ergo uel talis productio non e&longs;t perpetua, uel, ut uerius dicam, e&longs;t <lb/>&longs;impliciter productio circum&longs;cripta ab omni tempore pr&aelig;&longs;enti, <lb/>pr&aelig;terito &amp; futuro. </s>
          <s id="id004420">Quare erit magis uera productio quam &longs;ub&shy;<lb/>&longs;tanti&aelig; mortalis, ideo contingit hic error ex di&longs;similitudine eo&shy;<lb/>rum qu&aelig; maxim&egrave; &longs;imilia e&longs;&longs;e uidentur, nam c&ugrave;m accidentia pro&shy;<lb/>ducantur in tribus temporibus, &amp; incorporea in nullo, &longs;ub&longs;tantia <lb/>autem mortales &longs;olum in uno tempore, ide&ograve; productio incorpo&shy;<lb/>reorum uidetur e&longs;&longs;e &longs;imilis productioni accidentium, cum tamen <lb/>productio &longs;ub&longs;tanti&aelig; mortalis &longs;it uer&egrave; media inter illas, nam &longs;ub&shy;<lb/>&longs;tantia mortalis producitur in uno tempore, accidens in omni <lb/>&longs;ub&longs;tantia immortalis in nullo, nece&longs;&longs;e e&longs;t autem extrema magis <lb/>differre inter &longs;e qu&agrave;m &agrave; media, igitur &longs;ub&longs;tanti&aelig; in corpore&aelig; ordi&shy;<lb/>ne &amp; perfectione differunt, non tamen proportionem habent. </s>
          <s id="id004421">Et <lb/>&longs;i quis dic&aacute;t, quod ultima &longs;ub&longs;tantia e&longs;&longs;et &etail;qu&egrave; potens, ut Deus: re&shy;<lb/>&longs;pondeo quod non e&longs;t uerum, quia uel loqueris de perfectione, &amp; <lb/>ita demon&longs;tratum e&longs;t, quod Deus e&longs;t ip&longs;a perfectio, ultima &longs;ub&shy;<lb/>&longs;tantia e&longs;t imperfecti&longs;sima: uel loqueris de magnitudine, &amp; ita non <lb/>&longs;unt &aelig;quales prima &amp; ultima &longs;ub&longs;tantia, quia non po&longs;&longs;unt com&shy;<lb/>parari, &longs;icut lumen non pote&longs;t comparari lumini, quod &longs;it dul&shy;<lb/>cius uel amarius, grauius uel leuius, maius enim &amp; minus, &amp; &aelig;&shy;<lb/>quales &longs;unt differenti&etail; quantitatum, uit&aelig; autem non habent quan&shy;<lb/>titatem operationis, quia, ut dixi, e&longs;t ab&longs;oluti&longs;sima ratione finis, ne&shy;<lb/>que potentiam ad aliquid, quia &longs;unt in &aelig;terno actu, &amp; hoc &longs;ecun&shy;<lb/>dum philo&longs;ophos, &amp; iuxta rationem numinis naturalis, nam &longs;e&shy;<lb/>cus religio &amp; fides tenent, quia &longs;upponunt mundum e&longs;&longs;e creatum, <lb/>&amp; &longs;ic potentia differenti&aelig; ab actu, quia Deus nunc creauit, &amp; antea <lb/>non creauerat, &amp; tamen poterat creare.</s>
        </p>
        <p type="margin">
          <s id="id004422"><margin.target id="marg891"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004423">Ex hoc patet, quod nulla &longs;ub&longs;tantia incorporea e&longs;t finita nec infi<lb/><arrow.to.target n="marg892"/><lb/>nita, nec exten&longs;a nec contracta, quia omnia i&longs;ta pertinent ad quan&shy;<lb/>titatem, quarum ill&etail; omnino &longs;unt expertes.</s>
        </p>
        <p type="margin">
          <s id="id004424"><margin.target id="marg892"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004425">Propo&longs;itio ducente&longs;ima uige&longs;ima &longs;eptima.</s>
        </p>
        <p type="main">
          <s id="id004426">Vit&aelig; iuxta numerum perfectionum in comparatione ad cogita&shy;<lb/>tionem no&longs;tram proportionem quandam habent.</s>
        </p>
        <p type="main">
          <s id="id004427">Velut Deus e&longs;t per &longs;e primo ab&longs;olutum, &amp; cau&longs;a omnium bo&shy;<lb/><arrow.to.target n="marg893"/><lb/>norum, &amp; e&longs;&longs;e, &longs;apientia uer&ograve; qu&aelig; generatur &agrave; primo bono, non e&longs;t <lb/>cau&longs;a omnium bonorum, quia &longs;ic produceret primum bonum, <lb/>&amp; produceretur e&longs;t tamen per &longs;e primo &amp; ab&longs;olutum bonum, 


<pb pagenum="260" xlink:href="015/01/279.jpg"/>amor autem e&longs;t cau&longs;a omnium bonorum po&longs;teriorum, &amp; ab&longs;olu&shy;<lb/>tum, &amp; per &longs;e &longs;ed non prim&ograve;, &amp; ita de uita qu&aelig; regit mundum, ip&longs;a <lb/>non e&longs;t ab&longs;oluta, neque per &longs;e prim&ograve;, &longs;ed &longs;olum cau&longs;a omnium bono&shy;<lb/>rum, e&longs;t tamen ab&longs;oluta in ordine <expan abbr="bonor&utilde;">bonorum</expan>, qu&aelig; retinuit, &amp; hoc mo&shy;<lb/>do dicimus e&longs;&longs;e plures per&longs;onas in diuinis plures mentes, &amp; &longs;ub&shy;<lb/>&longs;tantias incorporeas.</s>
        </p>
        <p type="margin">
          <s id="id004428"><margin.target id="marg893"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s>
        </p>
        <p type="main">
          <s id="id004429">Propo&longs;itio ducente&longs;ima uige&longs;ima octaua.</s>
        </p>
        <p type="main">
          <s id="id004430">Proportionem &longs;cienti&aelig; futurorum &amp; c&aelig;terorum occultorum <lb/>con&longs;iderare.</s>
        </p>
        <p type="main">
          <s id="id004431">Septem licet &longs;int modi futura &amp; occulta pr&aelig;gno&longs;cendi, qu&etail;dam <lb/><arrow.to.target n="marg894"/><lb/>tamen &longs;unt communia omnibus, qu&aelig;dam multis: uaria quoque e&longs;t <lb/>ratio horum, alia enim e&longs;t proportio &longs;ciendi, atque h&aelig;c duplex, uel ex <lb/>ratione intelligendi qu&aelig; ortum habet ex comparatione anim&aelig; ad <lb/>magnitudinem &amp; difficultatem eorum, qu&aelig; <expan abbr="cogno&longs;c&utilde;tur">cogno&longs;cuntur</expan>, qu&etail;dam <lb/>ad modum quo <expan abbr="iudic&atilde;tur">iudicantur</expan>. </s>
          <s id="id004432">Alia rur&longs;us e&longs;t ratio proportionis modi <lb/>ad animam ip&longs;am, ut qui&longs;que propior fuerit ip&longs;i aut remotior, alia <lb/>demum e&longs;t differenti&aelig; <expan abbr="&longs;ignor&utilde;">&longs;ignorum</expan> aut cau&longs;arum, ergo ut &agrave; propinqui&shy;<lb/>tate initium ducam, &longs;eptem uidentur e&longs;&longs;e ordines, qui etiam ad per&shy;<lb/>fectionem dijudicandi pertinent. </s>
          <s id="id004433">Primus e&longs;t eorum qu&aelig; agimus <lb/>quibus prudentia dominatur, atque hic admodum certus e&longs;t, ut in <lb/>negotijs publicis priuatis que uidemus, e&longs;t <expan abbr="aut&etilde;">autem</expan> duplex, ciuilis &amp; mili <lb/>taris. </s>
          <s id="id004434">Secundus e&longs;t naturalium, e&longs;t autem maxim&egrave; euidens in tribus <lb/>medicina, agricultura &amp; nauigatione. </s>
          <s id="id004435">Tertius e&longs;t eorum qu&aelig; &longs;unt <lb/>&longs;ecundum naturam, &longs;ed non per cau&longs;as, uelut a&longs;trologia &amp; phy&longs;io&shy;<lb/>gnomia. </s>
          <s id="id004436">Eius <expan abbr="a&utilde;t">aunt</expan> tres &longs;unt partes phy&longs;iognomia, metopo&longs;copia &amp; <lb/>chiromantia, namque a&longs;trologia et&longs;i per cau&longs;as &longs;it, magis tamen per <lb/>&longs;igna o&longs;tendere uidetur, nam quod Iuppiter in a&longs;cendente bonos <lb/>pr&aelig;beat mores, cur magis hoc in loco uel illo, magna e&longs;t qu&aelig;&longs;tio. <lb/></s>
          <s id="id004437">Quartus e&longs;t con&longs;en&longs;us omnium nobi&longs;cum atque fatale uin culum, in <lb/>quo genere ponuntur fulgrum ca&longs;us, exta, &amp; augurium &amp; hygro&shy;<lb/>mantia. </s>
          <s id="id004438">In quinto modo ponuntur ea qu&aelig; cum anima no&longs;tra con&shy;<lb/>&longs;en&longs;um habent, eiu&longs;modi &longs;unt uit&aelig; aut genij aut eroes. </s>
          <s id="id004439">Sextus uer&ograve; <lb/>e&longs;t ex origine, uelut &longs;unt Prophet&aelig; &amp; uates Sybill&aelig;que, quorum uis <lb/>alia in &longs;e ip&longs;is, ut prophetarum, alia uaporis ut Delphici oraculi, alia <lb/>aqu&etail; uelut in Colophonio oraculo. </s>
          <s id="id004440">Vltimum e&longs;t pr&aelig;&longs;tanti&longs;simum <lb/>idemque <expan abbr="remoti&longs;sim&utilde;">remoti&longs;simum</expan>, quod &agrave; Deo per preces <expan abbr="c&otilde;&longs;equimur">con&longs;equimur</expan>. </s>
          <s id="id004441">In omni&shy;<lb/>bus ergo his iuuat pr&aelig;&longs;tantia modi non au&longs;picium, &amp; exta paruam <lb/>habent &longs;ignificationem, qu&aelig; uero &agrave; Deo maximam, alia enim e&longs;t <lb/>proportio agentis, ut Dei alia modi agendi, uelut qu&aelig; per cau&longs;as <lb/>fit melior qu&agrave;m qu&aelig; per &longs;igna, alia impre&longs;sionis lucis aut efficacis, <lb/>alia coniunctionis natur&aelig; nobi&longs;cum. </s>
          <s id="id004442">Quod uer&ograve; ad nos attinet, 


<pb pagenum="261" xlink:href="015/01/280.jpg"/>aliud e&longs;t ex peritia artis, aliud ex iudicio acri, aliud ex diligentia. <lb/></s>
          <s id="id004443">Differentia autem cogno&longs;cendi &longs;unt multorum aut paucorum ex&shy;<lb/>act&aelig;, uel non exact&aelig;, &longs;ecur&aelig; aut dubi&aelig;, atque horum omnium cau&longs;a <lb/>e&longs;t magnitudo proportionis, aut in origine ad <expan abbr="&longs;ignificand&utilde;">&longs;ignificandum</expan>, aut in <lb/>anima ad <expan abbr="intellig&etilde;dum">intelligendum</expan>. </s>
          <s id="id004444">Atque originis, ut dixi, multiplex e&longs;t ratio, &longs;ci <lb/>licet modi uel cau&longs;&aelig; uel efficaci&aelig;, c&ugrave;m uer&ograve; h&aelig;c omnia in unum <lb/>conuenerint, certi&longs;sima &amp; exacti&longs;sima fiet diuinatio, cum pauca &amp; <lb/>minus ualida, ut pote di&longs;cur&longs;us &amp; iudicium dubia, debilis &amp; pauco<lb/>rum. </s>
          <s id="id004445">Qu&aelig; uer&ograve; nugantur Porphyrius &amp; Iamblicus de his, omni&shy;<lb/>no fabulis &longs;imilia &longs;unt, uideturque Iamblicus Porphyrio indixi&longs;&longs;e <lb/>bellum, &longs;ed cum ignauo ho&longs;te, ip&longs;e longe deterior.</s>
        </p>
        <p type="margin">
          <s id="id004446"><margin.target id="marg894"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004447">Propo&longs;itio ducente&longs;ima uige&longs;ima nona.</s>
        </p>
        <p type="main">
          <s id="id004448">Incorporea omnia unum &longs;unt, neque numerus e&longs;t eorum.</s>
        </p>
        <p type="main">
          <s id="id004449">Videbitur ab initio paradoxum, &longs;ed ubi &amp; modum &amp; demon&shy;<lb/><arrow.to.target n="marg895"/><lb/>&longs;trationem ip&longs;am deprehenderis, intelliges ita e&longs;&longs;e iuxta luminis na <lb/>turalis rationem, tum uer&ograve; maxim&egrave;, cum id adiecero non prohibe&shy;<lb/>re me, quin ut partes in homine numerentur. </s>
          <s id="id004450">Sed aliud e&longs;t partes in <lb/>homine dinumerare, qu&aelig; numero ip&longs;o non di&longs;tinguuntur, &longs;ed &longs;i <lb/>plures homines &longs;eor&longs;um de earum numero interroges &longs;inguli di&shy;<lb/>uer&longs;a, nec <expan abbr="exigu&otilde;">exiguom</expan> interuallo differentia re&longs;pondebunt, &longs;ed unus de&shy;<lb/>cem puta, alius centum, alius innumerabiles pronuntiabit. </s>
          <s id="id004451">Quin <lb/>etiam qui&longs;que qua ratione uelis illas di&longs;tinguere interrogabit, at non <lb/>&longs;ic de numero gregis pauidum, aut de pecunijs, in quibus nemo ab <lb/>altero di&longs;&longs;entiet, ni&longs;i cum in numerando errorem admi&longs;erit. </s>
          <s id="id004452">Igitur <lb/>dico non e&longs;&longs;e numerum in incorporeis, nam finitus erit uel infini&shy;<lb/>tus: &longs;i infinitus, numerus non erit, quoniam primum nullus Deus <lb/>erit nulla prima &longs;ub&longs;tantia: nam quomodo Deus erit aut Domi&shy;<lb/>nus infinitorum, aut primus ubi non e&longs;t ultimum? </s>
          <s id="id004453">Sed neque nume&shy;<lb/>rus aliquis certus earum e&longs;&longs;e pote&longs;t, cum primum non magis hic <lb/>qu&agrave;m ille: neque enim definiuntur ullo termino, &longs;eu centum, &longs;eu mil&shy;<lb/>le aut millies mille: nec cum &longs;ubijciantur quantitati continu&aelig; pote&shy;<lb/>runt &longs;ubijci numero, uel alteri cuipiam accidenti. </s>
          <s id="id004454">Sed omnia &longs;unt <lb/>unum, ita tamen quod perfectius e&longs;t atque imperfectius diffu&longs;um ab <lb/>ip&longs;o infinito, cuius in extremo coh&aelig;rent mentes no&longs;tr&aelig; &amp; anim&aelig;, <lb/>&amp; c&oelig;lum, qu&aelig; communicat&aelig; inferioribus atque corporibus illa <lb/>agunt, mutant &amp; &longs;eruant. </s>
          <s id="id004455">Ip&longs;um qu&agrave;m ultimum e&longs;&longs;e, e&longs;t in mundo, <lb/>quod e&longs;t corpus, &amp; eius pars pr&aelig;cipua c&oelig;lum deinde reliqua. <lb/></s>
          <s id="id004456">Omniaque mouentur &amp; transferuntur immobili primo principio, <lb/>quod cum illis <expan abbr="coniunct&utilde;">coniunctum</expan> e&longs;t: nam reliqua incorporea ab ip&longs;o pro&shy;<lb/>fluunt. </s>
          <s id="id004457">E&longs;t &amp; ratio Ari&longs;totelis in tertio decimo Theologicorum &longs;er<lb/><arrow.to.target n="marg896"/><lb/>monum, Deus non e&longs;t unus numeri ratione, &longs;ed ita ut non &longs;it plura, 


<pb pagenum="262" xlink:href="015/01/281.jpg"/>igitur in mundo toto incorporeo non e&longs;t numerus. </s>
          <s id="id004458">Si enim Deus <lb/>e&longs;&longs;et unus numero, non po&longs;&longs;et e&longs;&longs;e ens commune, &amp; uniuer&longs;im am&shy;<lb/>plectens cuncta, &amp; accidens contineret, qu&aelig; omnia &longs;unt fal&longs;a, ab&longs;ur<lb/>da, nefaria &amp; impia, licet tamen &lpar;ut dixi&rpar; menti human&aelig; qu&aelig; omnia <lb/>reducit ad &longs;imilitudinem &longs;en&longs;ilium, &agrave; quibus originem traxit &longs;u&aelig; <lb/>operationis fingere numeros, &longs;icut in partibus hominis, aut c&oelig;li, <lb/>aut aeris iuxta &longs;itum, aut magnitudinem. </s>
          <s id="id004459">E&longs;t etiam alius modus <lb/>iuxta quem Ari&longs;toteles numerauit mentes qu&aelig; mouent corpora <lb/>c&oelig;le&longs;tia, quod ab&longs;urdum non e&longs;t, uelut &longs;i quis numeret digitos, in <lb/>pul&longs;ante chelim, erunt quatuor aut &longs;ex, non tamen e&longs;t numerus ille <lb/>uer&egrave; plurium, cum ad unum hominem referuntur. </s>
          <s id="id004460">Et cum &longs;it mun&shy;<lb/><arrow.to.target n="marg897"/><lb/>dus hic imago &longs;uperioris, ut ille dicebat, &amp; inferior pote&longs;tate conti&shy;<lb/>neat infinitas partes, infinitas ordinis ratione &longs;uperior continebit. <lb/></s>
          <s id="id004461">Sed non infinitas numero. </s>
          <s id="id004462">Exempli gratia, proponamus quod So<lb/>lis uis dirigatur ad nos u&longs;que impedita per nebulas, ut <expan abbr="n&otilde;nunquam">nonnunquam</expan> <lb/>contingit: erit ergo perfectio una, &longs;ed ordinata omnium radiorum: <lb/>ade&ograve; quod &longs;i infinita ua&longs;a applicarentur aqua plena infinit&aelig; ratio&shy;<lb/>nes iridis apparerent, qu&aelig; omnes continerentur pote&longs;tate in radijs <lb/>illis ratione comparationis ad ua&longs;a &amp; irides, per &longs;e autem, ut &longs;unt <lb/>perfectiones e&longs;&longs;ent in actu.</s>
        </p>
        <p type="margin">
          <s id="id004463"><margin.target id="marg895"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004464"><margin.target id="marg896"/>S<emph type="italics"/>up.<emph.end type="italics"/> 5.</s>
        </p>
        <p type="margin">
          <s id="id004465"><margin.target id="marg897"/>Lib. 7. <emph type="italics"/>cap.<emph.end type="italics"/><lb/>4.</s>
        </p>
        <p type="main">
          <s id="id004466">Propo&longs;itio ducente&longs;ima trige&longs;ima.</s>
        </p>
        <p type="main">
          <s id="id004467">Proportio incorporeorum a&longs;cendentium &longs;emper maior e&longs;t.<lb/><arrow.to.target n="marg898"/></s>
        </p>
        <p type="margin">
          <s id="id004468"><margin.target id="marg898"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004469">Cum proportio illa &longs;it qua&longs;i &longs;imilis decori, &amp; ideo mu&longs;ic&aelig; geo&shy;<lb/><arrow.to.target n="marg899"/><lb/>metrica maior e&longs;t in maioribus ac magnitudinibus, ut &longs;upr&agrave; docui&shy;<lb/>mus. </s>
          <s id="id004470">Sed non e&longs;t neque geometrica, neque arithmetica, nec mu&longs;ica, nec <lb/>per recen&longs;um, e&longs;&longs;ent enim quantitates qu&aelig; <expan abbr="compararen&ttilde;">compararentur</expan>: unaqu&etail;que<lb/> enim harum inter quantitates con&longs;tituta: at illa e&longs;t ut producentis <lb/>ad productum. </s>
          <s id="id004471">Et non comparantur quoad &aelig;ternitatem, quia ut <lb/>ali&acirc;s declaraui, omnis &longs;ub&longs;tantia e&longs;t &aelig;terna: quanto magis incor&shy;<lb/>porea. </s>
          <s id="id004472">Quia ergo primum per <expan abbr="pr&aelig;ced&etilde;tem">pr&aelig;cedentem</expan> habet rationem totius, <lb/>&amp; e&longs;t infinitum, <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ea parte qua recedit, quia primum non e&longs;t, <lb/>plus di&longs;tat a primo quam &agrave; tertio, igitur de&longs;cendendo u&longs;que ad pri&shy;<lb/>ma elementa. </s>
          <s id="id004473">Sed obijcies de qualitatibus &amp; accidentibus: dico <lb/>quod habent <expan abbr="medi&utilde;">medium</expan> e&longs;&longs;e, licet tempore infinito uincantur &agrave; &longs;ub&longs;tan<lb/>tijs, ill&etail; tamen etiam uincuntur &amp; ab&longs;que participatione perfectionis <lb/>illius cum <expan abbr="accid&etilde;tia">accidentia</expan> participent e&longs;&longs;entia &amp; tempore, &amp; &longs;i quis dicat, <lb/>cur ergo Sol &amp; lupiter <expan abbr="n&otilde;">non</expan> &longs;unt locati in &longs;upremis orbibus, cum &longs;int <lb/>nobiliores &amp; maiores &amp; potentiores c&aelig;teris erraticis: dico qu&ograve;d <lb/>fuit ob mundum inferiorem, quoniam &longs;i fui&longs;&longs;ent altiores mundus <lb/>inferior frigore corrumperetur, quando quidem uel &longs;ic frigore pre&shy;<lb/>mantur, in hyeme etiam &longs;ub torrida plaga, &amp; &longs;ub polis ac iuxta eos 


<pb pagenum="263" xlink:href="015/01/282.jpg"/>&longs;emper. </s>
          <s id="id004474">Et orbes &longs;uperiores <expan abbr="n&otilde;">non</expan> indigebant lumine Solis, quod ap&shy;<lb/>paret in nocte &longs;erena, cum etiam ade&ograve; &agrave; nobis di&longs;tent. </s>
          <s id="id004475">Vnde &longs;i cani&shy;<lb/>cula e&longs;&longs;et in c&oelig;lo Lun&aelig;, plus luminis afferret centuplo qu&agrave;m Lu&shy;<lb/>na, c&ugrave;m di&longs;tantia &longs;it quingentupla di&longs;tanti&aelig; Lun&aelig; &agrave; terra. </s>
          <s id="id004476">Et &longs;i Sol <lb/>e&longs;&longs;et factus adeo maior, ut in orbe Saturni con&longs;i&longs;tens calefaceret ter<lb/>ram &aelig;qualiter, ut non exureretur in &aelig;&longs;tate, hyeme nece&longs;&longs;e e&longs;&longs;et, ut ni <lb/>mium gela&longs;ceret. </s>
          <s id="id004477">Sin autem &aelig;quale e&longs;&longs;et frigus in hyeme, exurere&shy;<lb/>tur terra per &aelig;&longs;tatem, quando quidem nec &longs;ic illam pati po&longs;sint, qui <lb/>in torrida plaga habitant. </s>
          <s id="id004478">Et &longs;i Sol e&longs;&longs;et ubi e&longs;t Luna, &amp; eo minor <lb/>non illuminarentur orbes &longs;uperiores. </s>
          <s id="id004479">Ideo nobilitas non e&longs;t in or&shy;<lb/>bibus ob altitudinem, &longs;ed ob &longs;ub&longs;tantiam incorpoream qu&aelig; illi do <lb/>minatur. </s>
          <s id="id004480">Et e&longs;t in loco congruenti toti corpus, uita autem non e&longs;t <lb/>in loco.</s>
        </p>
        <p type="margin">
          <s id="id004481"><margin.target id="marg899"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 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"/>
        <p type="main">
          <s id="id004483">Et proponantur a &amp; b in proportione dupla alti&shy;<lb/>tudinum &amp; magnitudinum, &amp; <expan abbr="c&otilde;parentur">comparentur</expan> ad d, erit <lb/>ergo angulus a d c maior b d c, quare &longs;i &longs;unt &aelig;quales <lb/>uires in a b, refrigerabitur magis d ab a quam b, &amp; <lb/>ita patet utraque pars dicti in fine propo&longs;itionis.</s>
        </p>
        <p type="main">
          <s id="id004484">Propo&longs;itio ducente&longs;ima trige&longs;ima prima.</s>
        </p>
        <p type="main">
          <s id="id004485">Tres e&longs;&longs;e mundos, atque inter ip&longs;os nullam e&longs;&longs;e proportionem: <lb/>nec numero eos definiri.</s>
        </p>
        <p type="main">
          <s id="id004486">Cum palam &longs;it e&longs;&longs;e corporeum mundum ut elementa, &amp; incor&shy;<lb/><arrow.to.target n="marg900"/><lb/>poreum ut Dei, &amp; medium e&longs;&longs;e nece&longs;&longs;e e&longs;t uitarum &amp; hominum ac <lb/>c&oelig;le&longs;tium, qu&ograve;d primum &longs;en&longs;u patet, ut c&oelig;li, hominum &amp; anima&shy;<lb/>lium, atque plantarum, &amp; ratione etiam, quoniam extrema contraria <lb/><expan abbr="n&otilde;">non</expan> propri&egrave; medio copulantur, ut incorporeum ac corporeum. </s>
          <s id="id004487">Di&shy;<lb/>co igitur nullam e&longs;&longs;e inter hos proportionem atque numerum face&shy;<lb/>re: nam de numero con&longs;tat, quoniam non &longs;unt tres, quia &longs;int in ordi <lb/>ne numerorum, &longs;ed ut principium, medium, finis, &amp; perfectum, per&shy;<lb/>fectius, perfecti&longs;simum: &longs;cilicet po&longs;itiuum, comparatiuum &amp; &longs;uper&shy;<lb/>latiuum. </s>
          <s id="id004488">Et quoniam &longs;unt extrema cum medio, ide&ograve; &longs;unt in propor<lb/>tione &longs;ublimi etiam &amp; non propria. </s>
          <s id="id004489">Quod &longs;i e&longs;&longs;ent maxim&egrave; mun&shy;<lb/>di uitalis ad corpora, &longs;ed corpora <expan abbr="n&otilde;">non</expan> mouentur ni&longs;i iuxta finem ui&shy;<lb/>t&aelig;, &amp; non uim: ip&longs;a enim &longs;i po&longs;&longs;et habere uoluntatem infinitam mo<lb/>ueret in in&longs;tanti: quia corpora non reluctantur animabus &longs;uis, &longs;ed <lb/>quantus e&longs;t actus in animabus &amp; uitis, tanta e&longs;t <expan abbr="pot&etilde;tia">potentia</expan> ad unguem <lb/>in corporibus, ergo non contingit proportio in mundo uitarum <lb/>uera ni&longs;i illa &longs;ublimis. </s>
          <s id="id004490">Neque enim finita e&longs;t qu&aelig; nullis circum&longs;cribi&shy;<lb/>tur terminis, neque infinita quo finitam pr&etail;&longs;upponit, &longs;ed neque inter <lb/>mundum &amp; incorporeum &amp; uitarum c&ugrave;m mentes non moueant, 


<pb pagenum="264" xlink:href="015/01/283.jpg"/>uit&aelig; moueant: &amp; quod mouet nece&longs;&longs;ari&ograve; mouet, &amp; quod non po&shy;<lb/>te&longs;t mouere, quoniam omnia &aelig;terna &longs;unt: &amp; in &etail;ternis idem e&longs;t e&longs;&longs;e <lb/>ac po&longs;&longs;e: igitur inter mundum incorporeum &amp; uitarum nulla e&longs;t <lb/>proportio uera, &longs;ed &longs;olum &longs;ublimis, nec numerus: ni&longs;i ut &agrave; nobis fin<lb/>gitur. </s>
          <s id="id004491">Velut &longs;i dicamus in tabula, &amp; in negocio e&longs;t principium me&shy;<lb/>dium finis, &amp; h&aelig;c po&longs;&longs;unt dici tria quatenus di&longs;tinguuntur: &longs;ed <expan abbr="n&otilde;">non</expan> <lb/>ob hoc dicendum e&longs;t tabulam, aut negotium habere tres partes, <lb/>multo minus e&longs;&longs;e tria negocia aut tres tabulas.</s>
        </p>
        <p type="margin">
          <s id="id004492"><margin.target id="marg900"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004493">Propo&longs;itio ducente&longs;ima trige&longs;ima &longs;ecunda.</s>
        </p>
        <p type="main">
          <s id="id004494">Omnis motus naturalis, quanto uelocior e&longs;t, tanto propior e&longs;t, <lb/>&amp; magis &longs;imillimus quieti.</s>
        </p>
        <p type="main">
          <s id="id004495">H&aelig;c propo&longs;itio primo intuitu uidetur e&longs;&longs;e fal&longs;a, quoniam c&ugrave;m <lb/><arrow.to.target n="marg901"/><lb/>motus &longs;it contrarius quieti, &amp; efficiat actiones quieti contrarias, <lb/>quant&ograve; uelocior erit tanto remotior &agrave; natura quietis &amp; magis di&longs;si<lb/>milis, propterea intelligere oportet primum, in quo &longs;en&longs;u uerba <lb/>&longs;int accipienda, nam h&aelig;c propo&longs;itio, &amp; authoritate, &amp; &longs;en&longs;u &amp; du&shy;<lb/>plici ratione euidenti manife&longs;ta e&longs;t. </s>
          <s id="id004496">Oportet igitur <expan abbr="prim&utilde;">primum</expan> &longs;cire quo <lb/>ad locum attinet tria e&longs;&longs;e di&longs;crimina: quietem in eodem: tran&longs;itum <lb/>ad alium per medium: &amp; tran&longs;itum ad alium &longs;ine medio. </s>
          <s id="id004497">Duorum <lb/><expan abbr="primor&utilde;">primorum</expan> exempla noti&longs;sima &longs;unt, tertij e&longs;t hoc, &longs;i urceus aqua ple&shy;<lb/>nus exponatur Soli, &amp; efficiatur iridis imago in tabula: inde &longs;ubla&shy;<lb/>ta tabula eadem iris appareat in muro, erit tran&longs;itus &longs;ine media, quia <lb/>quod &longs;it eadem dubium non e&longs;t, ijdem radij &amp; idem corpus &longs;pecu&shy;<lb/>lare, quod uer&ograve; tran&longs;eat &longs;ine medio, <expan abbr="prim&utilde;">primum</expan> &longs;en&longs;us docet, &longs;ecundum <lb/>ratio, quia fit in in&longs;tanti, ut Secundo de Anima. </s>
          <s id="id004498">Rur&longs;us Sol illu&longs;tret <lb/><arrow.to.target n="marg902"/><lb/>urceum aqua plenum: appareat ex hoc iris in muro, interponatur <lb/>aliquid, &amp; transferatur urceus, apparebit iris alia loco, &amp; non tran&shy;<lb/>&longs;iuit per medium, uidetur idem de intellectu, &amp; ui imaginandi, qui&shy;<lb/>bus ex Germania tran&longs;eo in Indiam &longs;ubit&ograve;: &amp; eodem modo ex ani&shy;<lb/>ma &longs;alicis, in hac planta fit tran&longs;itus in proximam neque per medium, <lb/>quod etiam uidemus in igne &amp; ellychnio proximo, &amp; id &longs;&aelig;pe acci&shy;<lb/>dit tum pr&aelig;&longs;ertim cum nuper extinctum fuerit.</s>
        </p>
        <p type="margin">
          <s id="id004499"><margin.target id="marg901"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="margin">
          <s id="id004500"><margin.target id="marg902"/>T<emph type="italics"/>ex.<emph.end type="italics"/> 121.</s>
        </p>
        <p type="main">
          <s id="id004501">Iam ergo id &longs;upponamus, quod etiam ad rem parum facit, &longs;ed ad <lb/>intelligentiam &longs;atis, uideamus que, quare &longs;it quod motus opponatur <lb/>quieti, &amp; <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t, quod differentia loci e&longs;t cau&longs;a, nam in quiete <lb/>res manet in eodem loco, in motu tran&longs;it ad alium locum, &amp; quan&shy;<lb/>t&ograve; medium e&longs;t maius, tant&ograve; motus e&longs;t manife&longs;tior, unde &longs;equitur, <lb/>quod in his qu&aelig; ualde lent&egrave; mouentur, illa uidentur quie&longs;cere, &amp; <lb/>po&longs;t aliquot tempus deprehendimus mota fui&longs;&longs;e, nunqu&agrave;m tamen <lb/>moueri, &longs;icut in Sole, Luna, &longs;tellis, unde illa opinio <expan abbr="Philo&longs;ophor&utilde;">Philo&longs;ophorum</expan> <lb/>exi&longs;timantium omnia &longs;emper moueri, <expan abbr="n&otilde;">non</expan> omnino pote&longs;t tam bene 


<pb pagenum="265" xlink:href="015/01/284.jpg"/>reprobari, quia licet &longs;en&longs;us <expan abbr="n&otilde;">non</expan> cogno&longs;cat moueri, cogno&longs;cit tamen <lb/>mota e&longs;&longs;e, &amp; id &longs;ufficit: multa ergo cogno&longs;cuntur mota e&longs;&longs;e qu&aelig; <expan abbr="n&otilde;">non</expan> <lb/>cogno&longs;cuntur moueri, uelut lapis grauis &longs;uper&longs;tans terr&aelig;, quem ui <lb/>demus po&longs;t annum de&longs;cendi&longs;&longs;e per duos digitos, &amp; tamen &longs;emper <lb/>uidetur quie&longs;cere. </s>
          <s id="id004502">Igitur cum in pari tempore qu&etail; uelo citer mouen<lb/>tur plus &longs;patij &longs;uperent, maius etiam relinquunt medium inter lo&shy;<lb/>cum, &amp; locum, &amp; ob id magis remota &longs;unt &agrave; quiete, &amp; magis illi <expan abbr="c&otilde;&shy;traria">con&shy;<lb/>traria</expan>: h&aelig;c igitur e&longs;t ratio cur qu&aelig; uelocius moueantur, minus quie<lb/>ti &longs;imilia aut proxima exi&longs;timentur. </s>
          <s id="id004503">Dico ergo, quod illa qu&aelig; natu&shy;<lb/>raliter ueloci&longs;sim&egrave; mouentur, &longs;unt magis &longs;imilia &amp; magis proxima <lb/>ip&longs;is quie&longs;centibus qu&agrave;m qu&aelig; tard&egrave;: cum enim omnis motus natu<lb/>ralis nece&longs;&longs;ari&ograve; <expan abbr="eti&atilde;">etiam</expan> &longs;it regularis, ut qui &agrave; uirtute Dei fiat, erit uel per <lb/>lineam obliquam aut <expan abbr="rect&atilde;">rectam</expan>. </s>
          <s id="id004504">Quoniam uer&ograve; <expan abbr="multar&utilde;">multarum</expan> recta e&longs;t per&shy;<lb/>fecti&longs;sima, &amp; obliquarum circularis, erit omnis motus naturalis cir<lb/>cularis aut rectus: dico ergo qu&ograve;d in utroque <expan abbr="uer&utilde;">uerum</expan> e&longs;t quod dicitur. <lb/></s>
          <s id="id004505">Et <expan abbr="prim&utilde;">primum</expan> in circulari ille motus e&longs;t propinquior quieti, in quo par&shy;<lb/>tes &longs;unt propinquiores &longs;uo loco, &longs;ed &longs;i ueloci&longs;simus &longs;it motus, nun&shy;<lb/>qu&agrave;m ita &longs;unt extra &longs;uum locum, qui enim in pote&longs;tate &longs;int proxi&shy;<lb/>m&aelig; ei: ergo partes ill&etail; inde &longs;e habent ac &longs;i quiescerent. </s>
          <s id="id004506">Secunda ra&shy;<lb/>tio, quia quod ueloci&longs;sim&egrave; <expan abbr="moue&ttilde;">mouetur</expan>, ab&longs;que dubio tanto tempore quie <lb/>&longs;cit in &longs;uo loco quant&ograve; quod tard&egrave;: exemplum. </s>
          <s id="id004507">Luna in triginta an <lb/>nis quie&longs;cit in principio arietis <expan abbr="quadring&etilde;teis">quadringenteis</expan> per &longs;ex horas, id e&longs;t, <lb/>centum diebus in quadringentis uicibus, Saturnus <expan abbr="c&etilde;tum">centum</expan> diebus <lb/>&longs;ed &longs;emel tantum: ergo tantum Luna quie&longs;cit, quantum Saturnus, <lb/><expan abbr="c&otilde;paratione">comparatione</expan> ad idem tempus addita pari ratione in alijs partibus, <lb/>&longs;ed cum uelocius moueatur Luna qu&agrave;m Saturnus minus quie&longs;ce&shy;<lb/>re uidebitur Luna in alijs partibus qu&agrave;m Saturnus, &amp; tantundem <lb/>in principio arietis Luna ut Saturnus, ergo cum Luna tantundem <lb/>in principio arietis quie&longs;cat, quantum Saturnus in triginta annis, &amp; <lb/>in alijs partibus minus qu&agrave;m Saturnus, igitur ab&longs;olut&egrave; Luna plus <lb/>quie&longs;cit in principio arietis, qu&agrave;m Saturnus dato tempore &aelig;quali <lb/>triginta <expan abbr="annor&utilde;">annorum</expan>. </s>
          <s id="id004508">Et formatur demon&longs;tratio hoc modo: Luna quan <lb/>do e&longs;t in loco ip&longs;o, puta in principio arietis, ibidem e&longs;t actu, &amp; quie <lb/>&longs;cit per tantundem temporis <expan abbr="quant&utilde;">quantum</expan> Saturnus, &amp; in omnibus alijs <lb/>locis data paritate, e&longs;t &longs;emper propior ip&longs;i principio arietis pote&longs;ta<lb/>te quam Saturnus, igitur Luna plus quie&longs;cit in principio arietis <lb/>quam Saturnus, quia dum ibidem &longs;unt &aelig;qualiter <expan abbr="quie&longs;c&utilde;t">quie&longs;cunt</expan>, &amp; dum <lb/>&longs;unt extra, Luna &longs;emper e&longs;t propior &amp; pote&longs;tate magis in illo loco, <lb/>igitur Luna magis quie&longs;cit in principio arietis qu&agrave;m Saturnus. </s>
          <s id="id004509">Pr&etail;<lb/>terea, &longs;i Luna &amp; Saturnus mouerentur in &aelig;quali tempore, &amp; Luna <lb/>in paruo circulo, &amp; Saturnus in magno, dubium non e&longs;&longs;et, quin 


<pb pagenum="266" xlink:href="015/01/285.jpg"/>Luna non diceretur magis quie&longs;cere in &longs;uo loco, &amp; diutius qu&agrave;m <lb/>Saturnus, nam Luna &longs;emper e&longs;&longs;et prope locum &longs;uum, &amp; Saturnus <lb/>per&longs;&aelig;pe uideretur procul. </s>
          <s id="id004510">Sed &longs;i moueantur in eodem circulo, &amp; <lb/>Luna moueatur ueloci&longs;sim&egrave;, Saturnus tard&egrave;: perinde erit, ac &longs;i Lu&shy;<lb/>na moueatur in paruo circulo, &amp; Saturnus in magno, ergo quod <lb/>ueloci&longs;sim&egrave; mouetur e&longs;t proximius quieti qu&agrave;m quod tard&egrave;. </s>
          <s id="id004511">Illud <lb/>etiam idem manife&longs;tius erit in extremis, nam quod minimo &longs;patio <lb/>mouetur propemodum non mouetur. </s>
          <s id="id004512">Sicut, &longs;i quid circa centrum <lb/>moueatur, ade&ograve; ut ip&longs;um tangat, non dicetur moueri, &longs;ed quie&longs;cere <lb/>ibi, &longs;ed quod ueloci&longs;sime mouetur, &longs;emper uer&longs;atur circa idem, quia <lb/>nunquam multum abe&longs;t, quia ibi non quie&longs;cit, igitur quod ueloci&longs;&shy;<lb/>&longs;im&egrave; mouetur motu naturali circular&longs; e&longs;t proximius quieti quam <lb/>quod tard&egrave;. </s>
          <s id="id004513">Demum, &longs;i aliquid moueretur in finita uelo citate motu <lb/>circulari, &longs;emper e&longs;&longs;et in eodem &longs;itu &longs;ecundum partes &amp; immobile, <lb/>igitur quod infinita uelo citate mouetur, &amp; quie&longs;cit. </s>
          <s id="id004514">Ergo quod ue&shy;<lb/>loci&longs;sim&egrave; mouetur cum magis di&longs;tet ab oppo&longs;ito eius, quod infini&shy;<lb/>ta tarditate mouetur, qu&agrave;m quod tard&egrave;, magis etiam appropinqua<lb/>bit pote&longs;tate in efficaci infinit&aelig; uelo citati qu&agrave;m quod tard&egrave;, igitur <lb/>quod ueloci&longs;sim&egrave; mouetur propius e&longs;t quie&longs;centi quam quod tar&shy;<lb/>d&egrave;. </s>
          <s id="id004515">Demon&longs;tratum e&longs;t enim in Dialecticis, argumentum o&longs;tendere <lb/>ab eo quod e&longs;t &longs;impliciter tale ad id &qring;d natura illi quo quo modo <lb/>tale e&longs;t &amp; <expan abbr="c&otilde;uer&longs;o">conuer&longs;o</expan> modo. </s>
          <s id="id004516">O&longs;tendo mod&ograve; quod &longs;imillimus: <expan abbr="quoni&atilde;">quoniam</expan> <lb/>illud e&longs;t &longs;imilius quieti in quo quod fertur non pote&longs;t digno&longs;ci di&shy;<lb/>&longs;tantia &agrave; priore loco, &longs;ed in ueloci&longs;sim&egrave; motis h&aelig;c di&longs;tantia non po<lb/>te&longs;t digno&longs;ci, igitur ueloci&longs;sim&egrave; mota uidentur plan&egrave; quie&longs;cere, <lb/>quod idem patet duobus experimentis manife&longs;tis. </s>
          <s id="id004517">Primum &longs;i quis <lb/>uideat rotas quibus acuuntur gladij moueri u&longs;que ad certam ueloci&shy;<lb/>tatem, augeri uidetur motus ille, uer&ugrave;m cum adeo <expan abbr="c&otilde;citatus">concitatus</expan> fuerit, <lb/>ut &longs;en&longs;us non po&longs;sit di&longs;cernere, neque comprehendere illam ueloci&shy;<lb/>tatem, &amp; rota non fuerit mota ab axe, ita ut titubet nec fuerit ulla in&shy;<lb/>&aelig;qualitas, uidebitur omnino quie&longs;cere, &amp; ita oculus dijudicat, &amp; <lb/>long&egrave; magis dijudicaret, ubi ad tantam motus perueniret uelocita<lb/>tem, ut nullo modo initium &agrave; fine di&longs;tingui po&longs;&longs;et, &longs;icut e&longs;t in motu <lb/>c&oelig;li, qui comparatus ad quemuis motum ueloci&longs;simum artificio <lb/>factum, in&longs;en&longs;ilem habet proportionem ob magnitudinem, &amp; ideo <lb/>talis motus c&oelig;le&longs;tis e&longs;t &longs;imillimus quieti. </s>
          <s id="id004518">Secundum <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>e&longs;t, &longs;i e&longs;&longs;ent duo homines habitantes Bononi&aelig;, quorum unus iret <lb/>Mutinam, paulatim quie&longs;cendo in quolibet loco per unam diem, <lb/>ade&ograve; ut in unoquoque anno maneret Mutin&aelig;, &amp; prope per &longs;ex men<lb/>&longs;es, &amp; prope Bononiam per &longs;ex alios men&longs;es in diuer&longs;is locis, &amp; <lb/>una die tantum Bononi&aelig;: alius uer&ograve; iret Mutinam &longs;ingulo die, &amp; 


<pb pagenum="267" xlink:href="015/01/286.jpg"/>per omnia loca &longs;icut hirundo uolans quater &amp; quater rediret Bo&shy;<lb/>noniam, nemini dubium e&longs;t, quod hic &longs;ecundus uideretur magis <lb/>quie&longs;cere Bononi&aelig; qu&agrave;m primus, &amp; hoc quia in anno quilibet eo&shy;<lb/>rum quie&longs;ceret per unam diem Bononi&aelig;, &amp; in hoc e&longs;&longs;ent &aelig;quales, <lb/>&longs;ed &longs;ecundus uideretur frequentius Bononi&aelig; qu&agrave;m primus, &amp; eti&shy;<lb/>am e&longs;&longs;et pote&longs;tate propior illi, ade&ograve; ut liceret cuilibet illum conue&shy;<lb/>nire qualibet die magis quam primum: ergo duabus de cau&longs;is ui&shy;<lb/>deretur &longs;ecundus magis quie&longs;cere Bononi&aelig; quam primus, &amp; in ter<lb/>tia &aelig;qualiter.</s>
        </p>
        <p type="main">
          <s id="id004519">Mod&ograve; dico de recto motu, quoniam quanto celerius fertur per <lb/>medium ad &longs;uum locum, tanto minus temporis in&longs;umit, ergo diu&shy;<lb/>tius quie&longs;cit in loco, minus e&longs;t etiam tempus per quod mouetur in <lb/>comparatione ad quietem &amp; &longs;impliciter, ergo in motu recto pro&shy;<lb/>pius e&longs;t quieti, quod ueloci&longs;sim&egrave; mouetur, pr&etail;terea inter duas quie <lb/>tes motus ueloci&longs;simus e&longs;t imperceptibilis. </s>
          <s id="id004520">Ergo motus ueloci&longs;si&shy;<lb/>mus e&longs;t &longs;imilior quieti qu&agrave;m minus uelox. </s>
          <s id="id004521">Accedit manife&longs;ti&longs;sim&egrave; <lb/>illud quod ab initio diximus, &longs;cilicet, quia motus ueloci&longs;simus e&longs;t <lb/>medius inter motum tardum &amp; &longs;ubitam mutationem, hoc enim e&longs;t <lb/>manife&longs;ti&longs;simum, ade&ograve; ut dubitemus in motibus ueloci&longs;simis, an <lb/>mobile tran&longs;ierit per medium, e&longs;t enim prim&ugrave;m motus lentus, qui <lb/>fit ex tran&longs;itu in longo tempore, &amp; ueloci&longs;simus in paruo, &amp; muta&shy;<lb/>tio &longs;ine tempore. </s>
          <s id="id004522">Rur&longs;us con&longs;tituamus alium ordinem quietis mo&shy;<lb/>tus, &amp; &longs;ubit&aelig; mutationis: &amp; ex dictis &longs;ubita mutatio e&longs;t propior <lb/>quieti <expan abbr="qu&atilde;">quam</expan> motus: quo&shy;<lb/><arrow.to.target n="marg903"/><lb/>niam &longs;i motus e&longs;&longs;et me&shy;<lb/>dius inter quietem &amp; <lb/>&longs;ubitam mutationem, non e&longs;&longs;et, ut dictum e&longs;t, &longs;ubita mutatio qu&aelig;&shy;<lb/>dam quies: nam in &longs;ubita mutatione non pertran&longs;itur medium: in <lb/>quiete non pertran&longs;itur medium, in motu pertran&longs;itur medium, igi<lb/>tur quies e&longs;t propior &longs;ubit&aelig; mutationi qu&agrave;m motui. </s>
          <s id="id004523">Sed &longs;ubita mu<lb/>tatio e&longs;t propior motui ueloci&longs;simo qu&agrave;m tardo, igitur quies e&longs;t <lb/>propior motui ueloci&longs;simo quam tardo.</s>
        </p>
        <p type="margin">
          <s id="id004524"><margin.target id="marg903"/>Subit. </s>
          <s id="id004525">Mut. </s>
          <s id="id004526">Motus uelo ci&longs;. </s>
          <s id="id004527">Motus Tar. <lb/></s>
          <s id="id004528">Quies &longs;ubita Mut. </s>
          <s id="id004529">Motus</s>
        </p>
        <p type="main">
          <s id="id004530">Videtur &amp; hoc &longs;en&longs;us manife&longs;t&egrave; o&longs;tendere, quoniam cum lapis <lb/>de&longs;cendit &longs;umma cum uelo citate, ade&ograve; ut non percipiatur, uidetur <lb/>quie&longs;cere, &amp; non motus e&longs;&longs;e, &amp; h&aelig;c fuit &longs;ententia multorum nobi&shy;<lb/>liorum antiquorum, &amp; propterea oportet ut o&longs;tendamus difficul&shy;<lb/>tates, qu&aelig; contingunt in his.</s>
        </p>
        <p type="main">
          <s id="id004531">Dico igitur, quod motus naturales &longs;unt duorum generum, ut di<lb/><expan abbr="ct&utilde;">ctum</expan> e&longs;t, &longs;cilicet rectus &amp; circularis: &amp; motus differt &agrave; quiete in duo&shy;<lb/>bus, in eo quod mutat locum, et in eo quod tran&longs;it per medium mo<lb/>tus, ergo rectus ueloci&longs;simus in eo quod tran&longs;it per medium ma&shy;


<pb pagenum="268" xlink:href="015/01/287.jpg"/>gis di&longs;tat &agrave; quiete in eo quod plus de medio &longs;uperat qu&agrave;m tardus, <lb/>&amp; e&longs;t propinquior quieti in eo quod celerius quie&longs;cit. </s>
          <s id="id004532">At motus cir<lb/>cularis ueloci&longs;simus e&longs;t propior quieti in tran&longs;itu medij, &amp; in redi&shy;<lb/>tu ad locum priorem: de reditu ad locum priorem clarum e&longs;t per &longs;e: <lb/>de tran&longs;itu medij, dico quod cum in prima medietate magis remo&shy;<lb/>ueatur &agrave; medio quam motus tardus, &amp; in &longs;ecunda medietate tan&shy;<lb/>tundem, uelocius redeat. </s>
          <s id="id004533">Ergo in <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> medietate e&longs;t &longs;emper pro&shy;<lb/>ximior motus ueloci&longs;simus ip&longs;i quieti, &longs;ed in prima medietate &qring;d <lb/>mouetur motu ueloci&longs;simo propius e&longs;t &longs;ecund&aelig; medietati &longs;emper <lb/>quam quod mouetur tardo motu, igitur quod mouetur ueloci&longs;si&shy;<lb/>m&egrave; circulariter e&longs;t propius quie&longs;centi, quam quod mouetur tard&egrave;. <lb/></s>
          <s id="id004534">Et hoc e&longs;t quia in &etail;ternis motus e&longs;t quies, &amp; ideo habent quandam <lb/>&longs;imilitudinem iuxta <expan abbr="perfection&etilde;">perfectionem</expan> &longs;uam, &longs;icut &longs;i e&longs;&longs;ent in circulo hoc <lb/><figure id="id.015.01.287.1.jpg" xlink:href="015/01/287/1.jpg"/><lb/>modo. </s>
          <s id="id004535">Mutatio ergo <expan abbr="c&otilde;ue&shy;nit">conue&shy;<lb/>nit</expan> in corporeis qu&etail; <expan abbr="pend&etilde;t">pendent</expan> <lb/>&agrave; corpore, &longs;icut lumini: qua&shy;<lb/>tenus enim &longs;unt ex corpo&shy;<lb/>reo, <expan abbr="occup&atilde;t">occupant</expan> diuer&longs;um <expan abbr="loc&utilde;">locum</expan>, <lb/>quatenus e&longs;t in corporei id, <lb/>agit &longs;ine tran&longs;itu per <expan abbr="medi&utilde;">medium</expan> <lb/>&amp; in in&longs;tanti, ergo in corpo&shy;<lb/>rea &longs;impliciter mutationem <lb/>recipiunt, non in tempore <lb/>neque in loco. </s>
          <s id="id004536">Videtur <expan abbr="aut&etilde;">autem</expan> <lb/>uelo<expan abbr="ci&longs;sim&utilde;">ci&longs;simum</expan> dupliciter <expan abbr="eti&atilde;">etiam</expan> <lb/>nobis iuxta &longs;en&longs;um, idque e&longs;t <lb/>in quo &longs;en&longs;us medij tran&longs;itum non percipit, &amp; natura quod e&longs;t pri&shy;<lb/>mi mobilis. </s>
          <s id="id004537">At dubitare quis pote&longs;t circa hoc, nam proprium mo&shy;<lb/>tus e&longs;t tangentia concutere, quietis autem minime: concutit autem <lb/>maxim&egrave; quod ueloci&longs;sim&egrave; mouetur, ob hoc arbitrati &longs;unt homi&shy;<lb/>nes quod ueloci&longs;simus motus mult&ograve; plus di&longs;taret &agrave; natura quietis <lb/>quam tardus, &longs;ed hoc e&longs;t quia non eadem e&longs;t ratio uiolenti &amp; natu&shy;<lb/>ralis: uiolenta enim non redeunt in &longs;e ip&longs;a, nec habent rationem cir&shy;<lb/>cularis, &longs;ed potius recti &amp; infiniti, &amp; ide&ograve; in his qu&aelig; mouentur mo<lb/>tu recto naturali cadit uiolentia, non autem in his qu&aelig; mouentur <lb/>motu circulari naturali: <expan abbr="c&otilde;">com</expan> cu&longs;sio ergo e&longs;t in motu uiolento, &amp; qua&shy;<lb/>li&longs;cunque motus uiolentus, quanto magis augetur tant&ograve; magis re&shy;<lb/>cedit &agrave; contrario, tant&ograve; magis remouetur &agrave; natura contrarij, &amp; ha&shy;<lb/>bet actiones contrarias ualidiores.</s>
        </p>
        <p type="main">
          <s id="id004538">E&longs;t etiam aliud pen&egrave; &longs;imile argumentum in figuris ip&longs;is, circulus <lb/>enim unica linea continetur, nulla tamen figura ab ea magis natura 


<pb pagenum="269" xlink:href="015/01/288.jpg"/>remota e&longs;t triangulo: &longs;iquidem circulus capaci&longs;simus e&longs;t, triangu&shy;<lb/>lus omnium rectilin<expan abbr="ear&utilde;">earum</expan> minim&egrave; capax: ut contr&agrave; polygoni&etail;, quan<lb/>to plurium &longs;unt laterum eo capaciores &longs;unt, ade&ograve; ut octagona qua&shy;<lb/>drangula, &amp; qu&aelig; e&longs;t &longs;exdecim laterum &aelig;qualium, &amp; &aelig;quiangula&shy;<lb/>rium plus contineat octagona, &amp; forma etiam &longs;it &longs;imilior circulo, <lb/>ade&ograve; ut cum excreuerit in multiplicem numerum rectangula figu&shy;<lb/>ra huiu&longs;modi, &longs;cilicet &aelig;quilatera, &amp; &aelig;quiangula omnino &longs;en&longs;um <lb/>fallat, uideaturque pror&longs;us circulus. </s>
          <s id="id004539">Et <expan abbr="tam&etilde;">tamen</expan> figura plurium laterum, <lb/><expan abbr="qu&atilde;to">quanto</expan> plurium laterum fuerit rem otior e&longs;t &agrave; natura circuli, qui una <lb/>tantum linea continetur: plus enim di&longs;tat centum ab uno qu&agrave;m de&shy;<lb/>cem, &amp; mille qu&agrave;m centum. </s>
          <s id="id004540">Cau&longs;a igitur e&longs;t, quia &lpar;ut dixi&rpar; etiam in <lb/>naturalibus omnis natura rerum e&longs;t, ut qua&longs;i clanculum redeat in <lb/>&longs;e ip&longs;am: nam circularis figura per triangulum ex rectis multum &agrave; <lb/>natura &longs;ua recedit &amp; ambitu &amp; &longs;imilitudine: eadem per figuras qu&etail; <lb/>ex pluribus rectis con&longs;tant ad &longs;ui &longs;imilitudinem redit, nunqu&agrave;m ta<lb/>men explet eandem naturam perfect&egrave;, c&ugrave;m nulla poligonya figura <lb/>pro circulo exacto &longs;it: ita uidetur in naturalibus ad <expan abbr="id&etilde;">idem</expan> redire, quod <lb/>e&longs;t pote&longs;tate &longs;olum quadam generali di&longs;simile: actu uer&ograve; non idem <lb/>ad unguem. </s>
          <s id="id004541">Sed obijcies de motu qu&ograve;d &longs;i tempus fiat breuius, ma&shy;<lb/>gnitudo autem con&longs;tet, erit &lpar;ut diximus&rpar; quod mouetur &longs;imile quie<lb/>&longs;centi: at ubi tempus idem &longs;it, &longs;ed magnitudo perpetu&ograve; augeatur, <lb/>non idem ut in c&oelig;lo: ueri&longs;imile e&longs;t enim quicquid e&longs;t quod moue&shy;<lb/>tur ulterius quam id quod cernitur nihilominus in uiginti quatu&shy;<lb/>or horis, non autem celerius moueri: propterea c&ugrave;m &longs;patium tem&shy;<lb/>poris prolixum &longs;it, non uidebitur quie&longs;cere. </s>
          <s id="id004542">Nec ob&longs;tat qu&ograve;d qui&longs;&shy;<lb/>piam proportionem obijciat, &longs;i quidem multo minus uidebuntur <lb/>propiora centro quie&longs;cere, namque illa tardius ex confe&longs;&longs;o mouen&shy;<lb/>tur, at quod tardius mouetur, ut dictum e&longs;t, moueri magis uidetur, <lb/>ide&ograve; proportionem illam ad aliud mobile referre oporteret, cum <lb/>nullum tale &longs;it. </s>
          <s id="id004543">Dicimus ergo qu&ograve;d apud illas non uidetur motus <lb/>tardus, quia comprehendunt motum ante tempus, nobis <expan abbr="aut&etilde;">autem</expan> h&aelig;c <lb/>accidunt, quia comprehendimus tempus ante motum. </s>
          <s id="id004544">Et <expan abbr="eti&atilde;">etiam</expan> quia <lb/>circa polos qua&longs;i quie&longs;cit, &amp; quod non pote&longs;t aliquid comprehen&shy;<lb/>di, &longs;imul moueri &amp; quie&longs;cere, ut docebimus. </s>
          <s id="id004545">Et etiam quia motus <lb/>e&longs;t ab illis, &longs;icut in nobis cum mouemur: <expan abbr="n&otilde;">non</expan> enim ut mouemur nos <lb/>moueri deprehendimus, &longs;ed ut moti ide&ograve; in his, non quod appa&shy;<lb/>ret, &longs;ed quod e&longs;t &longs;pectare oportet: at ita e&longs;t ut qu&aelig; uelociter ualde <lb/>mouentur, perinde &longs;unt qua&longs;i ac &longs;i quie&longs;cerent, ade&ograve; ut motus &longs;i in <lb/>in&longs;tanti fieret e&longs;&longs;et quies, &amp; quies in incorporeis e&longs;t motus, non in <lb/>tempore. </s>
          <s id="id004546">Videntur etiam a&longs;tra quie&longs;cere nobis, quoniam &lpar;ut dixi&rpar; <lb/>line&aelig; a e &amp; b e non po&longs;&longs;unt uideri moueri in e, oculus autem iudi&shy;


<pb pagenum="270" xlink:href="015/01/289.jpg"/><figure id="id.015.01.289.1.jpg" xlink:href="015/01/289/1.jpg"/><lb/>cat moueri debere in e, non ex c <lb/>in d, ubi e&longs;t amplum &longs;patium <lb/>terr&aelig; comprehen&longs;um, ergo a e <lb/>quie&longs;cere uidetur in e, igitur &amp; <lb/>in a. </s>
          <s id="id004547">Qu&ograve;d autem uideatur in e <lb/>quie&longs;cere, patet, quia quod mo<lb/>tum uideri debet, oportet ut in <lb/>in&longs;en&longs;ili tempore &longs;patium &longs;en&shy;<lb/>&longs;ile pertran&longs;ierit: in&longs;en&longs;ile au&shy;<lb/>tem tempus e&longs;t minus motu ue<lb/>loci&longs;simo pul&longs;us, hic autem ma <lb/>ius exigit <expan abbr="t&etilde;pus">tempus</expan> cente&longs;ima par&shy;<lb/>te cente&longs;im&aelig; partis hor&etail;, igitur <lb/>diei ducente&longs;ima quadrage&longs;ima mille&longs;im&aelig; partis, &amp; in hoc oportet <lb/>ut pertran&longs;eat &longs;en&longs;ile &longs;patium, quod e&longs;t quinquage&longs;ima parte uln&aelig; <lb/>&longs;altem maius. </s>
          <s id="id004548">Ergo &longs;i fiat in&longs;trumentum <expan abbr="quing&etilde;tarum">quingentarum</expan> ulnarum am<lb/>bitus, &qring;d in uiginti quatuor horis circumuoluatur, ade&ograve; lent&egrave; mo&shy;<lb/>uebitur, ut quie&longs;cere uideatur: tum uer&ograve; magis ob id quod dixi, <lb/>quoniam in centro quie&longs;cere uidebitur, ergo in peripheria, ubi di&shy;<lb/>&longs;tantia deprehendi po&longs;sit. </s>
          <s id="id004549">Ergo nulla machina qu&aelig; uideatur mo&shy;<lb/>ueri, con&longs;titui pote&longs;t, qu&aelig; in horis XXIIII circumuertatur: quia non <lb/>tam magna fieri pote&longs;t, ut &longs;patium &agrave; centro ad circumferentiam ocu<lb/>lo non po&longs;sit deprehendi.</s>
        </p>
        <p type="main">
          <s id="id004550">Et hoc uoluimus declarare ut intelligamus, qu&aelig; &longs;unt nece&longs;&longs;aria <lb/>ad mundum incorporeum.</s>
        </p>
        <p type="main">
          <s id="id004551">Propo&longs;itio ducente&longs;ima trige&longs;ima tertia.</s>
        </p>
        <p type="main">
          <s id="id004552">Quod e&longs;t in mundo incorporeo &aelig;ternum, e&longs;t beatum, &longs;ecurum <lb/>immutabile &longs;ecundum locum &longs;olum iuxta e&longs;&longs;entiam fit, iuxta quod <lb/>uelut &agrave; leui &longs;u&longs;urro aqu&aelig; &amp; aura &aelig;&longs;tiua demulcetur.</s>
        </p>
        <p type="main">
          <s id="id004553">Quod e&longs;t ibi non e&longs;t pars nec totum, e&longs;&longs;et enim quantum, aut nu<lb/><arrow.to.target n="marg904"/><lb/>mero di&longs;cretum, nec mutationem loci aut temporis habet, cum in <lb/>nullo eorum &longs;it, ide&ograve; nec habere pote&longs;t, nec amittere, non e&longs;t ibi infi<lb/>nitum, cuius nullus finis &longs;it, &longs;ed dum emanat &agrave; priore &longs;ecundum or&shy;<lb/>dinem e&longs;t &longs;umma uoluptas, qualis in his qui ad cognitionem &amp; feli<lb/>citatem <expan abbr="deueni&utilde;t">deueniunt</expan>. </s>
          <s id="id004554">Qu&etail; in illis cum &aelig;terna &longs;it &amp; &longs;ecura, recipit quan <lb/>dam uariationem, in qua delectatur, uelut mortalia ex <expan abbr="c&otilde;trarijs">contrarijs</expan> cau<lb/>&longs;is natur&aelig; contrarijs affectibus: &amp; hoc e&longs;t perpetu&ograve; nouum, quia <lb/>&longs;emper pendet &amp; recipit. </s>
          <s id="id004555">Et ob id e&longs;t unum &amp; actu &longs;empiterno, <lb/>quod uer&ograve; e&longs;t extra, e&longs;t potentia, ide&ograve; infinitum, quod imaginatur <lb/>anima, quia in ordinatum priore ordine, qui e&longs;t ante <expan abbr="limit&etilde;">limitem</expan> omnem, <lb/>neque enim dubium e&longs;t, quin infinitum non &longs;it cau&longs;a, ut non po&longs;sit 


<pb pagenum="271" xlink:href="015/01/290.jpg"/>e&longs;&longs;e ordo ille &longs;ecundus: &longs;ed nos loquimur de primo. </s>
          <s id="id004556">Et ide&ograve; anima <lb/>no&longs;tra ob materi&aelig; coniunctionem appetit ordinem, &amp; l&aelig;tatur in <lb/>eo ut inueniat finem in rebus, uelut in multis proprietatibus nume <lb/>rorum e&longs;t manife&longs;tum. </s>
          <s id="id004557">Potentia enim e&longs;t cau&longs;a imaginandi infini&shy;<lb/>tum, quia &longs;emper ultra aliquid e&longs;&longs;e po&longs;&longs;e putamus, e&longs;t igitur poten&shy;<lb/>tia actus imperfectus. </s>
          <s id="id004558">Anima ergo no&longs;tra conuer&longs;a e&longs;t &agrave; Deo, res <lb/>po&longs;t &longs;e in quibus inuenit potentia imperfectionem <foreign lang="greek">a&rpar;tacian</foreign> pericu&shy;<lb/>lum &amp; infinitum ad de&longs;perationem tandem, quod quilibet uidere <lb/>poterit, qui &longs;e &agrave; diuinis auerterit: quant&ograve; enim plura habet, plura <lb/>de&longs;unt. </s>
          <s id="id004559"><expan abbr="Multiplic&etilde;tur">Multiplicentur</expan> filij, opes, honores, nil ni&longs;i laborem &amp; anxie&shy;<lb/>tatem aucta inuenies. </s>
          <s id="id004560">Quomodo autem quod infinitum non e&longs;t, <lb/>infinitam faciat potentiam? </s>
          <s id="id004561">uides in repr&aelig;&longs;entatione Solis qu&etail; infi<lb/>nita e&longs;&longs;et, &longs;i c&oelig;lum e&longs;&longs;et infinitum. </s>
          <s id="id004562">Dubitatione autem dignum e&longs;&shy;<lb/>&longs;et, an &longs;i c&oelig;lum infinitum e&longs;&longs;et ubique Sol illuminaret: &longs;eu quia qu&aelig;&shy;<lb/>&longs;itum nullum &longs;it, ui&longs;it de eo quod non e&longs;t, nihil autem non e&longs;&longs;e po&shy;<lb/>te&longs;t, aut quod non po&longs;&longs;et, quoniam uirtus corporea e&longs;t. </s>
          <s id="id004563">Corporeo <lb/>autem omni finem ad e&longs;&longs;e nece&longs;&longs;e e&longs;t. </s>
          <s id="id004564">Hanc nouitatem ergo alij tri&shy;<lb/>pudium, alij mu&longs;icam &amp; &longs;onum c&oelig;le&longs;tem interpretati &longs;unt.</s>
        </p>
        <p type="margin">
          <s id="id004565"><margin.target id="marg904"/>C<emph type="italics"/>o<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="main">
          <s id="id004566">Manife&longs;tum e&longs;t igitur &longs;ub&longs;tantiam incorporei mundi, e&longs;&longs;e in <lb/><arrow.to.target n="marg905"/><lb/>quadam mutatione perpetua ordinis, &amp; &longs;ine motu, tempore &amp; lo&shy;<lb/>co: unde amor &amp; uoluptas mutua, &amp; totum unum, &longs;icut anima cum <lb/>cogno&longs;cit Deum, &amp; cum cogno&longs;cit c&oelig;lum de&longs;cendit, &amp; fit alia or&shy;<lb/>dine. </s>
          <s id="id004567">Et h&aelig;c beatitudo in mundo illo e&longs;t tanta, ut in com&shy;<lb/>parabilis &longs;it no&longs;tr&aelig;, qu&aelig; e&longs;t umbra eius, etiam <lb/>quando e&longs;t &amp; pura, etiam &longs;i e&longs;&longs;et per&shy;<lb/>petua. </s>
          <s id="id004568">Igitur hic finis no&shy;<lb/>&longs;ter Diuin&etail; natur&aelig; <lb/>&amp; libri.</s>
        </p>
        <p type="margin">
          <s id="id004569"><margin.target id="marg905"/>C<emph type="italics"/>or<emph.end type="italics"/>^&lcub;m}.</s>
        </p>
        <p type="head">
          <s id="id004570">LIBRI DE PROPORTIONI&shy;<lb/>BVS FINIS.</s>
        </p>
        <pb xlink:href="015/01/291.jpg"/>
      </chap>
    </body>
    <back/>
  </text>
</archimedes>