Vitruvius, De architectura libri decem, 1567

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          <div xml:id="echoid-div718" type="section" level="2" n="135">
            <p style="it">
              <s xml:id="echoid-s20383" xml:space="preserve">
                <pb o="273" file="514.01.305" n="305" rhead="NONVS."/>
              illa h i. </s>
              <s xml:id="echoid-s20384" xml:space="preserve">quæ lineam a c, tangat, in ſigno, i, a quo ſigno cadat æquidiſtans ipſi a f, & </s>
              <s xml:id="echoid-s20385" xml:space="preserve">in ſignum d,
                <lb/>
              terminet, & </s>
              <s xml:id="echoid-s20386" xml:space="preserve">ſit illa i d. </s>
              <s xml:id="echoid-s20387" xml:space="preserve">His poſitis, ut facilius res explicetur, uocabimus lineas illas a b. </s>
              <s xml:id="echoid-s20388" xml:space="preserve">f &</s>
              <s xml:id="echoid-s20389" xml:space="preserve">, h i, d
                <lb/>
              c, primas æquidiſtantes, ſed lincas a f. </s>
              <s xml:id="echoid-s20390" xml:space="preserve">g h. </s>
              <s xml:id="echoid-s20391" xml:space="preserve">i d. </s>
              <s xml:id="echoid-s20392" xml:space="preserve">ſecundas æquidiſtantes, ſimiliter duo erunt maiora
                <lb/>
              trigona, quorum primum erit a b c, cuius angulus b, rectus est. </s>
              <s xml:id="echoid-s20393" xml:space="preserve">aliud uero trigonum est a f c, & </s>
              <s xml:id="echoid-s20394" xml:space="preserve">il-
                <lb/>
              lud primum, hoc ſecundum trigonum appellabimus. </s>
              <s xml:id="echoid-s20395" xml:space="preserve">in primo inerant trigona, quæ a primis æquidiſtantibus
                <lb/>
              fiunt, & </s>
              <s xml:id="echoid-s20396" xml:space="preserve">ſunt g f c. </s>
              <s xml:id="echoid-s20397" xml:space="preserve">i h e. </s>
              <s xml:id="echoid-s20398" xml:space="preserve">c d e. </s>
              <s xml:id="echoid-s20399" xml:space="preserve">& </s>
              <s xml:id="echoid-s20400" xml:space="preserve">hæc quoniam æquis angulis conſtant,ut patet ex vigeſimanona
                <lb/>
              primi, proportione reſpondentia latera, habent, ut ex quarta ſexti elicitur. </s>
              <s xml:id="echoid-s20401" xml:space="preserve">Similiter quoniam ſecunda trigona,
                <lb/>
              & </s>
              <s xml:id="echoid-s20402" xml:space="preserve">ſecundis æquidistantibus lineis facta, æqualibus angulis conſtant, ſine dubio latera quoque habebunt mu
                <lb/>
              tua ratione comparabilia. </s>
              <s xml:id="echoid-s20403" xml:space="preserve">Igitur quemadmodum in primis trigonis æquidistantibus proportione ſibi reſpon-
                <lb/>
              dent eæ lineæ, ita ut ſicuti ſe habet a e, ad e g, itaſe habent b e, ad e f. </s>
              <s xml:id="echoid-s20404" xml:space="preserve">ita in ſecundis æquidistan-
                <lb/>
                <note position="left" xlink:label="note-514.01.305-01" xlink:href="note-514.01.305-01a" xml:space="preserve">10</note>
              tibus,ut ſe habent a e, ad e g, ſic ſe habet f e, ad e h, & </s>
              <s xml:id="echoid-s20405" xml:space="preserve">iterum quemadmodum in primis, ut ſe
                <lb/>
              habet f e. </s>
              <s xml:id="echoid-s20406" xml:space="preserve">ad e h, ita g e, ad e i, ita in ſecundis parallelis, ut ſe habebit g e, ad e i, ita ſeha-
                <lb/>
              bebit b e, ad e d, ſunt igitur continuo proportionales a b, f g, h i, c d, quoniam ſicut eſt b c,
                <lb/>
              ad e f, ita eſt a b. </s>
              <s xml:id="echoid-s20407" xml:space="preserve">ad f g, &</s>
              <s xml:id="echoid-s20408" xml:space="preserve">ſicuti ſe habet f c, ad c h, ita ſe habet f g, ad h i, & </s>
              <s xml:id="echoid-s20409" xml:space="preserve">uti ſe ha-
                <lb/>
              bet g e, ad e d, ita h i, ad c d. </s>
              <s xml:id="echoid-s20410" xml:space="preserve">Propoſitis igitur duabus rectis a b, & </s>
              <s xml:id="echoid-s20411" xml:space="preserve">c d, duas medias in-
                <lb/>
              uenimus ſcilicet f g, & </s>
              <s xml:id="echoid-s20412" xml:space="preserve">h i, quod facere propoſitum fuit. </s>
              <s xml:id="echoid-s20413" xml:space="preserve">Hæc Eratoſthenis demonstratio est, & </s>
              <s xml:id="echoid-s20414" xml:space="preserve">licet
                <lb/>
              uelit ipſe lineas a b, & </s>
              <s xml:id="echoid-s20415" xml:space="preserve">c d, rectas eſſe ſupra lineam b d, non tamen eſt dubitandum, quin eadem con-
                <lb/>
              cluſio colligatur quocunque modo utraque linea cadat in b d, modo ſimiles utraque faciat angulos, & </s>
              <s xml:id="echoid-s20416" xml:space="preserve">ſi-
                <lb/>
              mili ſint ratione æquidis̃tantes. </s>
              <s xml:id="echoid-s20417" xml:space="preserve">Quoniam res omnis in eo ſita est, quod ex his trigonorum angulos æquales ha
                <lb/>
              bentium,latera proportione ſibi reſpondent. </s>
              <s xml:id="echoid-s20418" xml:space="preserve">Quareſiplures duabus medias inter duas datas a b, & </s>
              <s xml:id="echoid-s20419" xml:space="preserve">c d,
                <lb/>
                <note position="left" xlink:label="note-514.01.305-02" xlink:href="note-514.01.305-02a" xml:space="preserve">20</note>
              inuenire mens fuerit, eodem modo plures parallelas, ſiue ex primis, ſiue ex ſecundis formabimus. </s>
              <s xml:id="echoid-s20420" xml:space="preserve">Organum
                <lb/>
              uerò quo demonſtrationis præhabitæ uſus haberi poſſit huiuſmodi fabrica struitur. </s>
              <s xml:id="echoid-s20421" xml:space="preserve">Lamnam æream; </s>
              <s xml:id="echoid-s20422" xml:space="preserve">uel li-
                <lb/>
              gneam tabellam ſume quadrangularem oblongam rectis angulis quadratam, ſitq; </s>
              <s xml:id="echoid-s20423" xml:space="preserve">ea in exemplum a b d
                <lb/>
              c, ſi deformare duas medias ratione comparabiles duabus datis, uelimus,aliquot lamnellas ſuper eam opor-
                <lb/>
              tebit aptare hoc modo. </s>
              <s xml:id="echoid-s20424" xml:space="preserve">Sume lamnellas tenues ex aliqua materia ſolida quadrangulares, unam ex ijs in media
                <lb/>
              tabula obfirmato ita, ut mouerinequeat, ſitq́;</s>
              <s xml:id="echoid-s20425" xml:space="preserve">ea e f g h, & </s>
              <s xml:id="echoid-s20426" xml:space="preserve">in ſignis e, & </s>
              <s xml:id="echoid-s20427" xml:space="preserve">f, affixis habeatregulas,
                <lb/>
              ut circumuolui poſſin in ſuis axiculis. </s>
              <s xml:id="echoid-s20428" xml:space="preserve">Eſto regula una e m, altera f n. </s>
              <s xml:id="echoid-s20429" xml:space="preserve">Sed lamnella alia ſit k d e,
                <lb/>
              ita in maiori lamna collocata ut admoueri poſſit uerſus obfirmatam lamnellam e f g h, atque etiam ab ea
                <lb/>
              amoueri, ita ut latera ſemper habeat æquidiſtantia lateri f h, ſitq́ in ea etiam regula in ſigno k,quæitem
                <lb/>
              conuolui poſſit, & </s>
              <s xml:id="echoid-s20430" xml:space="preserve">ſit illa k o, quæ cum alijsregulis ſcilicet e m, & </s>
              <s xml:id="echoid-s20431" xml:space="preserve">f n, ita ponatur, ut omnes ſint pa-
                <lb/>
                <note position="left" xlink:label="note-514.01.305-03" xlink:href="note-514.01.305-03a" xml:space="preserve">30</note>
              rallelæ, & </s>
              <s xml:id="echoid-s20432" xml:space="preserve">earum communes ſectiones in lineis a g, f h, & </s>
              <s xml:id="echoid-s20433" xml:space="preserve">l, ſigno, ſint in eademrecta m n l o, ſi-
                <lb/>
              militer a m, ſitæqualis d k. </s>
              <s xml:id="echoid-s20434" xml:space="preserve">quoniam a m, uel minimo diſcrimineaccedit d k. </s>
              <s xml:id="echoid-s20435" xml:space="preserve">Hæc, cum ita ſint ex-
                <lb/>
              pedita inter duas rectas a b, & </s>
              <s xml:id="echoid-s20436" xml:space="preserve">c d, dantur duæ mediæ comparabiles ratione continua, quæſunt e n,
                <lb/>
              & </s>
              <s xml:id="echoid-s20437" xml:space="preserve">f o, quod manifeſtum eſt ob ſupradictas rationes. </s>
              <s xml:id="echoid-s20438" xml:space="preserve">Cæterum ſi ſorte duæ propoſitælineæ, quales ſunt ſ,
                <lb/>
              & </s>
              <s xml:id="echoid-s20439" xml:space="preserve">t, quibus opus ſit medias comparabiles inuenire, non erunt æquales lineis in organo poſitis, quæſunt a
                <lb/>
              b, & </s>
              <s xml:id="echoid-s20440" xml:space="preserve">e r. </s>
              <s xml:id="echoid-s20441" xml:space="preserve">efficiatur ut quemadmodum ſe habet ſ, ad t, itaſe habeat a b, ad r d, efficietur autem
                <lb/>
              ſi lamella k d e, prore proprius lamellam obfirmatam admouebitur, uel retrocedet in pari ſemper, & </s>
              <s xml:id="echoid-s20442" xml:space="preserve">
                <lb/>
              æquidis̃tanti laterum ratione. </s>
              <s xml:id="echoid-s20443" xml:space="preserve">& </s>
              <s xml:id="echoid-s20444" xml:space="preserve">quoniam ipſi a b. </s>
              <s xml:id="echoid-s20445" xml:space="preserve">& </s>
              <s xml:id="echoid-s20446" xml:space="preserve">r d, quæ ſunt in organo duæ mediæ compara-
                <lb/>
              biles inuentæ ſunt, conſequenter ipſi s. </s>
              <s xml:id="echoid-s20447" xml:space="preserve">ipſi t. </s>
              <s xml:id="echoid-s20448" xml:space="preserve">propoſitis, duæ erunt mediæ mediæ comparabiles inuentæ. </s>
              <s xml:id="echoid-s20449" xml:space="preserve">quo
                <lb/>
              igitur maiori artificio elaboratum organum feceris, eo facilius duas inuenies medias proportione conti-
                <lb/>
                <note position="left" xlink:label="note-514.01.305-04" xlink:href="note-514.01.305-04a" xml:space="preserve">40</note>
              nua reſpondentes, ideo capita mobilium lamnellarum in ſuos utrinque canales immittentur, ita ut per
                <lb/>
              lamnam decurrant leniter, & </s>
              <s xml:id="echoid-s20450" xml:space="preserve">æquabiliter. </s>
              <s xml:id="echoid-s20451" xml:space="preserve">& </s>
              <s xml:id="echoid-s20452" xml:space="preserve">Quod ſi plures quàm duas medias inuenire quis uelit,
                <lb/>
              lamnellarum, & </s>
              <s xml:id="echoid-s20453" xml:space="preserve">regularum additamento id commode faciet. </s>
              <s xml:id="echoid-s20454" xml:space="preserve">animaduertendum autem oportere lon-
                <lb/>
              gas eſſe regulas, ut cum opus fuerit lamnellas abſcedere a media, regulæ ad ſectiones linearum prædi-
                <lb/>
              ctas pertingere poſſint, tangantq́;</s>
              <s xml:id="echoid-s20455" xml:space="preserve">ſuperius organi latus regulæ omnes, e m, f x, r u, immo ( ut me-
                <lb/>
              lius dicam ) tam longæ ſint, quantum eſſet diagonia lamnellæ obfirmatæ e f g h. </s>
              <s xml:id="echoid-s20456" xml:space="preserve">Atque hæc innentio fuit
                <lb/>
              Eratoſthenis.</s>
              <s xml:id="echoid-s20457" xml:space="preserve">Sed ut facilior uſus inſrumenti appareat,nec ſolum inter duas duas,ſed inter duas plures compa-
                <lb/>
              rabiles inuenire poſſimus.</s>
              <s xml:id="echoid-s20458" xml:space="preserve">Sint duærectæ a b. </s>
              <s xml:id="echoid-s20459" xml:space="preserve">c d. </s>
              <s xml:id="echoid-s20460" xml:space="preserve">incidant ambæ in rectam, & </s>
              <s xml:id="echoid-s20461" xml:space="preserve">æquidiſtent,tantumq́
                <lb/>
              addatur lineæ c d. </s>
              <s xml:id="echoid-s20462" xml:space="preserve">ut æqua ſit lineæ a b. </s>
              <s xml:id="echoid-s20463" xml:space="preserve">cuius ſummum ſit e. </s>
              <s xml:id="echoid-s20464" xml:space="preserve">Ab a, uſque ad e, ducatur linea,& </s>
              <s xml:id="echoid-s20465" xml:space="preserve">
                <lb/>
              ita claudatur ſuperficies a b c e. </s>
              <s xml:id="echoid-s20466" xml:space="preserve">Partiatur inde b c. </s>
              <s xml:id="echoid-s20467" xml:space="preserve">in partes tres, quarum una ſit ubi f. </s>
              <s xml:id="echoid-s20468" xml:space="preserve">ultra f ali-
                <lb/>
                <note position="left" xlink:label="note-514.01.305-05" xlink:href="note-514.01.305-05a" xml:space="preserve">50</note>
              quanto amplius notetur g. </s>
              <s xml:id="echoid-s20469" xml:space="preserve">ut a puncto b. </s>
              <s xml:id="echoid-s20470" xml:space="preserve">ad g. </s>
              <s xml:id="echoid-s20471" xml:space="preserve">ſit pluſquam tertia pars lineæ b c. </s>
              <s xml:id="echoid-s20472" xml:space="preserve">ſimili ratione in li-
                <lb/>
              nea a c. </s>
              <s xml:id="echoid-s20473" xml:space="preserve">notetur ſignum, quod ab a distet quantum g. </s>
              <s xml:id="echoid-s20474" xml:space="preserve">ab ipſo b. </s>
              <s xml:id="echoid-s20475" xml:space="preserve">et ſit illud h. </s>
              <s xml:id="echoid-s20476" xml:space="preserve">nectatur g, cum
                <lb/>
              a, & </s>
              <s xml:id="echoid-s20477" xml:space="preserve">h. </s>
              <s xml:id="echoid-s20478" xml:space="preserve">& </s>
              <s xml:id="echoid-s20479" xml:space="preserve">a. </s>
              <s xml:id="echoid-s20480" xml:space="preserve">cum d. </s>
              <s xml:id="echoid-s20481" xml:space="preserve">& </s>
              <s xml:id="echoid-s20482" xml:space="preserve">g h. </s>
              <s xml:id="echoid-s20483" xml:space="preserve">ſecet a d in i. </s>
              <s xml:id="echoid-s20484" xml:space="preserve">ſimiliter ſecetur tantum de a b, quantum est à g.
                <lb/>
              </s>
              <s xml:id="echoid-s20485" xml:space="preserve">ad i. </s>
              <s xml:id="echoid-s20486" xml:space="preserve">& </s>
              <s xml:id="echoid-s20487" xml:space="preserve">ſpatium illud ſit b k. </s>
              <s xml:id="echoid-s20488" xml:space="preserve">ducaturq́ linea ab i ad k. </s>
              <s xml:id="echoid-s20489" xml:space="preserve">donec tangat lineam g a. </s>
              <s xml:id="echoid-s20490" xml:space="preserve">in l. </s>
              <s xml:id="echoid-s20491" xml:space="preserve">Quo-
                <lb/>
              niam ergo ex trigeſiſma tertia primi elementoruma b. </s>
              <s xml:id="echoid-s20492" xml:space="preserve">æquidiſtat g i h, & </s>
              <s xml:id="echoid-s20493" xml:space="preserve">ex poſitione g i, & </s>
              <s xml:id="echoid-s20494" xml:space="preserve">b k. </s>
              <s xml:id="echoid-s20495" xml:space="preserve">
                <lb/>
              ſunt æquales, ſequitur b g. </s>
              <s xml:id="echoid-s20496" xml:space="preserve">æquidiſtare i l. </s>
              <s xml:id="echoid-s20497" xml:space="preserve">Præterea de lineis g c. </s>
              <s xml:id="echoid-s20498" xml:space="preserve">& </s>
              <s xml:id="echoid-s20499" xml:space="preserve">h c. </s>
              <s xml:id="echoid-s20500" xml:space="preserve">auferantur æquales par-
                <lb/>
              tes duæ uerſus i l. </s>
              <s xml:id="echoid-s20501" xml:space="preserve">& </s>
              <s xml:id="echoid-s20502" xml:space="preserve">ſint illæ g m. </s>
              <s xml:id="echoid-s20503" xml:space="preserve">& </s>
              <s xml:id="echoid-s20504" xml:space="preserve">h n. </s>
              <s xml:id="echoid-s20505" xml:space="preserve">iunganturq́; </s>
              <s xml:id="echoid-s20506" xml:space="preserve">i m, & </s>
              <s xml:id="echoid-s20507" xml:space="preserve">m n, ex præallegata propoſitione
                <lb/>
              g l. </s>
              <s xml:id="echoid-s20508" xml:space="preserve">& </s>
              <s xml:id="echoid-s20509" xml:space="preserve">m i. </s>
              <s xml:id="echoid-s20510" xml:space="preserve">erunt æquidiſtantes, & </s>
              <s xml:id="echoid-s20511" xml:space="preserve">eadem ratione g h, & </s>
              <s xml:id="echoid-s20512" xml:space="preserve">m n. </s>
              <s xml:id="echoid-s20513" xml:space="preserve">ſecet quoq; </s>
              <s xml:id="echoid-s20514" xml:space="preserve">linea m n. </s>
              <s xml:id="echoid-s20515" xml:space="preserve">lineam a
                <lb/>
              d. </s>
              <s xml:id="echoid-s20516" xml:space="preserve">in o. </s>
              <s xml:id="echoid-s20517" xml:space="preserve">& </s>
              <s xml:id="echoid-s20518" xml:space="preserve">ſumatur de b k tantum,quantum eſt m o. </s>
              <s xml:id="echoid-s20519" xml:space="preserve">& </s>
              <s xml:id="echoid-s20520" xml:space="preserve">ſit pars illa b p. </s>
              <s xml:id="echoid-s20521" xml:space="preserve">& </s>
              <s xml:id="echoid-s20522" xml:space="preserve">ab o. </s>
              <s xml:id="echoid-s20523" xml:space="preserve">uerſus p. </s>
              <s xml:id="echoid-s20524" xml:space="preserve">du-
                <lb/>
              catur linea, donec tangat i m. </s>
              <s xml:id="echoid-s20525" xml:space="preserve">in q. </s>
              <s xml:id="echoid-s20526" xml:space="preserve">Si ergo linea m e, erit æqualis lineæ o q. </s>
              <s xml:id="echoid-s20527" xml:space="preserve">bene erit. </s>
              <s xml:id="echoid-s20528" xml:space="preserve">Sedſi m c. </s>
              <s xml:id="echoid-s20529" xml:space="preserve">
                <lb/>
                <note position="left" xlink:label="note-514.01.305-06" xlink:href="note-514.01.305-06a" xml:space="preserve">60</note>
              </s>
            </p>
          </div>
        </div>
      </text>
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