Clavius, Christoph, Geometria practica

Table of contents

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[131.] PROBLEMA XIV.
Page: 153 (123)
[132.] PROBLEMA XV.
Page: 153 (123)
[133.] PROBLEMA XVI.
Page: 156 (126)
[134.] PROBLEMA XVII.
Page: 158 (128)
[135.] PROBLEMA XVIII.
Page: 158 (128)
[136.] PROBLEMA XIX.
Page: 159 (129)
[137.] PROBLEMA XX.
Page: 159 (129)
[138.] PROBLEMA XXI.
Page: 160 (130)
[139.] PROBLEMA XXII.
Page: 160 (130)
[140.] PROBLEMA XXIII.
Page: 161 (131)
[141.] PROBLEMA XXIV.
Page: 161 (131)
[142.] PROBLEMA XXV.
Page: 161 (131)
[143.] SCHOLIVM.
Page: 162 (132)
[144.] PROBLEMA XXVI.
Page: 162 (132)
[145.] PROBLEMA XXVII.
Page: 164 (134)
[146.] PROBLEMA XXVIII.
Page: 166 (163)
[147.] PROBLEMA XXIX.
Page: 167 (137)
[148.] PROBLEMA XXX.
Page: 167 (137)
[149.] PROBLEMA XXXI.
Page: 168 (138)
[150.] PROBLEMA XXXII.
Page: 169 (139)
[151.] PROBLEMA XXXIII.
Page: 169 (139)
[152.] PROBLEMA XXXIV.
Page: 170 (140)
[153.] PROBLEMA XXXV.
Page: 171 (141)
[154.] PROBLEMA XXXVI.
Page: 172 (142)
[155.] PROBLEMA XXXVII.
Page: 172 (142)
[156.] PROBLEMA XXXVIII.
Page: 173 (143)
[157.] PROBLEMA XXXIX.
Page: 174 (144)
[158.] ALITER.
Page: 174 (144)
[159.] ALITER.
Page: 175 (145)
[160.] PROBLEMA XL.
Page: 175 (145)
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          <pb o="108" file="138" n="138" rhead="GEOMETR. PRACT."/>
          <note style="it" position="right" xml:space="preserve">
            <lb/>
          Vt A a, differentia Quotientum. diuiſo \\ latere Quadrati per vtramque vm- \\ bram verſam, # ad A a, dif- \\ ferentiam \\ ſtationum. # Ita F G, vt 1. # ad FG,
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s4178" xml:space="preserve">producetur altitudo F G, inpartibus differentiæ ſtationum A a, nota, & </s>
            <s xml:id="echoid-s4179" xml:space="preserve">adiecta
              <lb/>
            menſoris ſtatura F L, tota altitudo GL, nota fiet.</s>
            <s xml:id="echoid-s4180" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div270" type="section" level="1" n="117">
          <head xml:id="echoid-head120" xml:space="preserve">ALITER.</head>
          <p>
            <s xml:id="echoid-s4181" xml:space="preserve">EX differentia H I, vmbrarum verſarum in latus quadrati gignatur nume-
              <lb/>
            rus N: </s>
            <s xml:id="echoid-s4182" xml:space="preserve">& </s>
            <s xml:id="echoid-s4183" xml:space="preserve">ex vmbra verſa D E, in verſam d H, fiat O. </s>
            <s xml:id="echoid-s4184" xml:space="preserve">Productis autem lateribus
              <lb/>
            BC, bc, vſque ad R, P, in radijs, abſcindatur B Q, ipſi b P, æqualis. </s>
            <s xml:id="echoid-s4185" xml:space="preserve">Et quia, vt pro-
              <lb/>
            blemate 3. </s>
            <s xml:id="echoid-s4186" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s4187" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4188" xml:space="preserve">demonſtratum eſt, ita eſt OR, differentia vmbrarumrectarum
              <lb/>
              <note symbol="a" position="left" xlink:label="note-138-02" xlink:href="note-138-02a" xml:space="preserve">4. ſexti.</note>
            ad A a, differentiam ſtationum, vt vmbra recta maior BR, ad AF, diſtantiam: </s>
            <s xml:id="echoid-s4189" xml:space="preserve"> Vt autem B R, ad B A, ita eſt A F, ad F G: </s>
            <s xml:id="echoid-s4190" xml:space="preserve">& </s>
            <s xml:id="echoid-s4191" xml:space="preserve">permutando, vt B R, ad A F, ita B A, ad
              <lb/>
            F G; </s>
            <s xml:id="echoid-s4192" xml:space="preserve">erit quoque QR, ad A a, vt B A, ad F G, & </s>
            <s xml:id="echoid-s4193" xml:space="preserve">permutando Q R, ad B A, vt A a,
              <lb/>
              <note symbol="b" position="left" xlink:label="note-138-03" xlink:href="note-138-03a" xml:space="preserve">4. ſexti.</note>
            ad F G. </s>
            <s xml:id="echoid-s4194" xml:space="preserve">Dico iam, vt eſt QR, ad AB, ita eſſe numerum N, ad numerum O.</s>
            <s xml:id="echoid-s4195" xml:space="preserve">
              <note symbol="c" position="left" xlink:label="note-138-04" xlink:href="note-138-04a" xml:space="preserve">16. ſexti.</note>
            enim ſit, vt DE, ad DA, ita AB, ad BR;</s>
            <s xml:id="echoid-s4196" xml:space="preserve"> erit rectangulum ſub DE, BR, rectangulo ſub D A, A B, hoc eſt quadrato lateris æquale. </s>
            <s xml:id="echoid-s4197" xml:space="preserve">Eademque ratione erit rectangu-
              <lb/>
            lum ſub d H, b P, eidem quadrato laterisæquale; </s>
            <s xml:id="echoid-s4198" xml:space="preserve">ideo que rectangulum ſub DE,
              <lb/>
            BR, rectangulo ſub d H, b P, æquale erit. </s>
            <s xml:id="echoid-s4199" xml:space="preserve"> Igitur erit, vt D E, ad D H, ita b P,
              <note symbol="d" position="left" xlink:label="note-138-05" xlink:href="note-138-05a" xml:space="preserve">16. ſexti.</note>
            B R: </s>
            <s xml:id="echoid-s4200" xml:space="preserve">& </s>
            <s xml:id="echoid-s4201" xml:space="preserve">conuertendo, vt d H, ad D E, ita B R, ad b P, vel ad B Q: </s>
            <s xml:id="echoid-s4202" xml:space="preserve">& </s>
            <s xml:id="echoid-s4203" xml:space="preserve">permutando
              <lb/>
              <note symbol="e" position="left" xlink:label="note-138-06" xlink:href="note-138-06a" xml:space="preserve">19. quinti.</note>
            vttota d H, ad totam BR, ita DE, vel d I, ablata ad b P, vel ad BQ@ ablatam. </s>
            <s xml:id="echoid-s4204" xml:space="preserve"> Igi- tur & </s>
            <s xml:id="echoid-s4205" xml:space="preserve">reliqua HI, ad reliquam QR, erit, vt tota dH, adtotam BR, vel vtablata dI,
              <lb/>
            ad ablatam BQ;</s>
            <s xml:id="echoid-s4206" xml:space="preserve">
              <unsure/>
            & </s>
            <s xml:id="echoid-s4207" xml:space="preserve">permutando, vt H I, ad d I, ita QR, ad B Q, vel ad b P. </s>
            <s xml:id="echoid-s4208" xml:space="preserve">
              <note symbol="f" position="left" xlink:label="note-138-07" xlink:href="note-138-07a" xml:space="preserve">23. ſexti.</note>
            portio autem numeri N, ad numerum O, componitur ex proportionibus HI, ad
              <lb/>
            d I, vel ad DE, & </s>
            <s xml:id="echoid-s4209" xml:space="preserve">lateris a d, ad d H: </s>
            <s xml:id="echoid-s4210" xml:space="preserve">propterea quod N, factus eſt ex HI, differen-
              <lb/>
            tia vmbrarum verſarum in latus quadratia d; </s>
            <s xml:id="echoid-s4211" xml:space="preserve">Atvero O, ex vmbra verſa D E, in
              <lb/>
            verſam d H, ex conſtructione. </s>
            <s xml:id="echoid-s4212" xml:space="preserve">Cum ergo ſit, vt paulo ante oſtendimus, quem-
              <lb/>
              <note symbol="g" position="left" xlink:label="note-138-08" xlink:href="note-138-08a" xml:space="preserve">4. ſexti.</note>
            admodum HI, ad d I, ita QR. </s>
            <s xml:id="echoid-s4213" xml:space="preserve">ad b P, Itẽ vtlatus a d, ad DH, ita b P, ad b a, com- ponetur quoque proportio N, ad O, ex proportionibus QR, ad b P, & </s>
            <s xml:id="echoid-s4214" xml:space="preserve">b P, ad
              <lb/>
            b a. </s>
            <s xml:id="echoid-s4215" xml:space="preserve">Sed ex his eiſdem componitur proportio QR, ad ba. </s>
            <s xml:id="echoid-s4216" xml:space="preserve">Igitur eadem eſt pro-
              <lb/>
            portio N, ad O, quæ QR, ad b a, vel BA. </s>
            <s xml:id="echoid-s4217" xml:space="preserve">quod oſtendere volebamus eſt autem
              <lb/>
            vt QR, ad BA, ita A a, differentiaſtationum ad altitudinem F G, vt ſupra oſtendi-
              <lb/>
            mus prope initium huius demonſtrationis. </s>
            <s xml:id="echoid-s4218" xml:space="preserve">Igitur ſi fiat,</s>
          </p>
          <note style="it" position="right" xml:space="preserve">
            <lb/>
          Vt N, numer{us}, qui fit ex HI, \\ differentia vmbrarum verſa- \\ rum in lat{us} quadrati 1000. # ad numerum O, fa- \\ ctum ex vmbra ver- \\ ſa D E, in verſam dH. # Ita A a, dif- \\ ferentiaſta- \\ tionum # ad F G, \\ altitudi- \\ dinem,
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s4219" xml:space="preserve">deprehenſa erit altitudo FG, in partibus differentiæ ſtationum A a, &</s>
            <s xml:id="echoid-s4220" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4221" xml:space="preserve"/>
          </p>
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          <head xml:id="echoid-head121" xml:space="preserve">ALITER</head>
          <p>
            <s xml:id="echoid-s4222" xml:space="preserve">
              <emph style="sc">Redvcatvr</emph>
            vtraque vmbra verſa ad rectam, vt adinitium huius libri in
              <lb/>
            quadrati conſtructione Num. </s>
            <s xml:id="echoid-s4223" xml:space="preserve">7. </s>
            <s xml:id="echoid-s4224" xml:space="preserve">docuimus. </s>
            <s xml:id="echoid-s4225" xml:space="preserve">Nam per has vmbras rectas altitudi-
              <lb/>
            nem FG, nanciſcemur, vti Num. </s>
            <s xml:id="echoid-s4226" xml:space="preserve">2. </s>
            <s xml:id="echoid-s4227" xml:space="preserve">in ſequenti tradetur.</s>
            <s xml:id="echoid-s4228" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4229" xml:space="preserve">2. </s>
            <s xml:id="echoid-s4230" xml:space="preserve">
              <emph style="sc">Secetvr</emph>
            deinde vtraque vmbra recta in E, H, vt in 2. </s>
            <s xml:id="echoid-s4231" xml:space="preserve">figura problema-
              <lb/>
            tis 3. </s>
            <s xml:id="echoid-s4232" xml:space="preserve">quæ hic repetatur. </s>
            <s xml:id="echoid-s4233" xml:space="preserve">Et quoniam ob ſimilem triangulorum A B E, A F </s>
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