Clavius, Christoph, Geometria practica

Table of handwritten notes

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        <div xml:id="echoid-div54" type="section" level="1" n="19">
          <pb o="41" file="071" n="71" rhead="LIBER PRIMVS."/>
          <p>
            <s xml:id="echoid-s1586" xml:space="preserve">
              <emph style="sc">Qvando</emph>
            particula minor cum vno gradu repetita incidit in gradum ali-
              <lb/>
            quem præcisè, auferendi erunt ex gradibus percurſis tot gradus, quoties parti-
              <lb/>
            cula illa cum vno gradurepetita fuit. </s>
            <s xml:id="echoid-s1587" xml:space="preserve">Reliquus enim numerus erit Numerator
              <lb/>
            fractionis: </s>
            <s xml:id="echoid-s1588" xml:space="preserve">Denominator autem erit, qui prius. </s>
            <s xml:id="echoid-s1589" xml:space="preserve">Vtſi particula illa minor cum
              <lb/>
            1. </s>
            <s xml:id="echoid-s1590" xml:space="preserve">gradu repetita ſepties inciditin 10. </s>
            <s xml:id="echoid-s1591" xml:space="preserve">gradum; </s>
            <s xml:id="echoid-s1592" xml:space="preserve">demendi erunt 7. </s>
            <s xml:id="echoid-s1593" xml:space="preserve">gradus repetiti.
              <lb/>
            </s>
            <s xml:id="echoid-s1594" xml:space="preserve">Atq; </s>
            <s xml:id="echoid-s1595" xml:space="preserve">ita habebuntur iterum {3/7}. </s>
            <s xml:id="echoid-s1596" xml:space="preserve">vnius gradus in data particula.</s>
            <s xml:id="echoid-s1597" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1598" xml:space="preserve">13. </s>
            <s xml:id="echoid-s1599" xml:space="preserve">
              <emph style="sc">Si</emph>
            viciſsim ex quouis gradu auferre velimus particulam quotlibet mi-
              <lb/>
              <note position="right" xlink:label="note-071-01" xlink:href="note-071-01a" xml:space="preserve">Quo pacto ex
                <lb/>
              quouis gradu
                <lb/>
              particula quot
                <lb/>
              libet Minuto-
                <lb/>
              rum abſcin-
                <lb/>
              datur.</note>
            nutorum, ita agendum erit. </s>
            <s xml:id="echoid-s1600" xml:space="preserve">In quadrante ſuperiore B C, accipiatur circino arcus
              <lb/>
            tot graduum, quot minuta deſiderantur; </s>
            <s xml:id="echoid-s1601" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s1602" xml:space="preserve">(vt confuſionem vitemus) in ar-
              <lb/>
            cuminteruallo ſemidiametri quadrantis B C, deſcriptum transferatur. </s>
            <s xml:id="echoid-s1603" xml:space="preserve">Si enim
              <lb/>
            hic arcus in 60. </s>
            <s xml:id="echoid-s1604" xml:space="preserve">partes æquales ſecetur, (primum, videlicet in duas: </s>
            <s xml:id="echoid-s1605" xml:space="preserve">deinde vna
              <lb/>
            harũ iterum in duas: </s>
            <s xml:id="echoid-s1606" xml:space="preserve">Tertio vna harũ in tres; </s>
            <s xml:id="echoid-s1607" xml:space="preserve">ac poſtremo vna harũ in quinq;)
              <lb/>
            </s>
            <s xml:id="echoid-s1608" xml:space="preserve">cõtinebit vna particula ſexageſima numerum minutorum propoſitũ. </s>
            <s xml:id="echoid-s1609" xml:space="preserve">Verbigra-
              <lb/>
            tia, ſi particula quæratur continens 50. </s>
            <s xml:id="echoid-s1610" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1611" xml:space="preserve">accipiemus in arcu F G, adinterual-
              <lb/>
            lum ſemidiametri A C, deſcripto arcum F G, arcui C Z, graduum 50. </s>
            <s xml:id="echoid-s1612" xml:space="preserve">æqualem,
              <lb/>
            eumque in 60. </s>
            <s xml:id="echoid-s1613" xml:space="preserve">partes æquales ſecabimus, primum in duas in puncto K: </s>
            <s xml:id="echoid-s1614" xml:space="preserve">Deinde
              <lb/>
            arcum F K, iterum in duas in puncto L; </s>
            <s xml:id="echoid-s1615" xml:space="preserve">tertio arcum F L, in tres in punctis T, V. </s>
            <s xml:id="echoid-s1616" xml:space="preserve">
              <lb/>
            ac tandem arcum F V, in quinque. </s>
            <s xml:id="echoid-s1617" xml:space="preserve">Vna namque harum 5. </s>
            <s xml:id="echoid-s1618" xml:space="preserve">particularum compre-
              <lb/>
            hendet 50. </s>
            <s xml:id="echoid-s1619" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1620" xml:space="preserve">ac proinde ſi transferatur circino in quemlibet gradum Quadrã-
              <lb/>
            tis B C, abſciſſa erunt 50. </s>
            <s xml:id="echoid-s1621" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s1622" xml:space="preserve">ex eo gradu. </s>
            <s xml:id="echoid-s1623" xml:space="preserve">Hoc ita demonſtro. </s>
            <s xml:id="echoid-s1624" xml:space="preserve">Quoniam eſt,
              <lb/>
            vt arcus 60. </s>
            <s xml:id="echoid-s1625" xml:space="preserve">graduum ad 1. </s>
            <s xml:id="echoid-s1626" xml:space="preserve">gradum, ita arcus C Z, vel F G, graduum 50. </s>
            <s xml:id="echoid-s1627" xml:space="preserve">ad ſexa-
              <lb/>
            geſimam partem eiuſdem arcus F G; </s>
            <s xml:id="echoid-s1628" xml:space="preserve">erit permutando, vt arcus 60. </s>
            <s xml:id="echoid-s1629" xml:space="preserve">graduum ad
              <lb/>
            arcum F G, graduum 50; </s>
            <s xml:id="echoid-s1630" xml:space="preserve">ita 1. </s>
            <s xml:id="echoid-s1631" xml:space="preserve">gradus ad ſexageſimam particulam arcus F G: </s>
            <s xml:id="echoid-s1632" xml:space="preserve">& </s>
            <s xml:id="echoid-s1633" xml:space="preserve">
              <lb/>
            conuertendo, vt arcus F G, graduum 50. </s>
            <s xml:id="echoid-s1634" xml:space="preserve">ad arcum graduum 60. </s>
            <s xml:id="echoid-s1635" xml:space="preserve">ita particula ſe-
              <lb/>
            xageſima arcus F G, ad 1. </s>
            <s xml:id="echoid-s1636" xml:space="preserve">gradum. </s>
            <s xml:id="echoid-s1637" xml:space="preserve">Cum ergo arcus F G, contineat {50/60}, arcus 60. </s>
            <s xml:id="echoid-s1638" xml:space="preserve">
              <lb/>
            graduum, continebit quoq; </s>
            <s xml:id="echoid-s1639" xml:space="preserve">particula ſexageſima arcus F G, {50/60}. </s>
            <s xml:id="echoid-s1640" xml:space="preserve">vnius gradus,
              <lb/>
            hoc eſt, 50. </s>
            <s xml:id="echoid-s1641" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s1642" xml:space="preserve">Quod eſt propoſitum.</s>
            <s xml:id="echoid-s1643" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1644" xml:space="preserve">
              <emph style="sc">Sed</emph>
            hæcres incommodiſsima eſt in paruis Quadrantibus, præſertim ſi pau-
              <lb/>
            ca Minuta, vtpote 1. </s>
            <s xml:id="echoid-s1645" xml:space="preserve">2. </s>
            <s xml:id="echoid-s1646" xml:space="preserve">vel 3. </s>
            <s xml:id="echoid-s1647" xml:space="preserve">abſcindenda ſint. </s>
            <s xml:id="echoid-s1648" xml:space="preserve">Quis enimin Quadrante exiguo
              <lb/>
            arcum 1. </s>
            <s xml:id="echoid-s1649" xml:space="preserve">gradus, vel 2. </s>
            <s xml:id="echoid-s1650" xml:space="preserve">vel 3. </s>
            <s xml:id="echoid-s1651" xml:space="preserve">in 60. </s>
            <s xml:id="echoid-s1652" xml:space="preserve">particulas diſtribuat? </s>
            <s xml:id="echoid-s1653" xml:space="preserve">Quamobrem commo-
              <lb/>
            diusid, quod proponitur, efficiemus hacratione. </s>
            <s xml:id="echoid-s1654" xml:space="preserve">Ex eo dem quadranteſupe-
              <lb/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="34"/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="35"/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="36"/>
            riore B C, arcus grad. </s>
            <s xml:id="echoid-s1655" xml:space="preserve">61. </s>
            <s xml:id="echoid-s1656" xml:space="preserve">transferatur in arcum X Y, ad interuallum ſemidiame-
              <lb/>
            tri A C. </s>
            <s xml:id="echoid-s1657" xml:space="preserve">deſcriptum, ab X, vſque ad Y. </s>
            <s xml:id="echoid-s1658" xml:space="preserve">Atque hic arcus X Y, in 6
              <emph style="sub">0</emph>
            . </s>
            <s xml:id="echoid-s1659" xml:space="preserve">partes æquales
              <lb/>
            ſecetur; </s>
            <s xml:id="echoid-s1660" xml:space="preserve">Primum ſcilicet in 2. </s>
            <s xml:id="echoid-s1661" xml:space="preserve">deinde vtraq; </s>
            <s xml:id="echoid-s1662" xml:space="preserve">ſemiſsis in 3. </s>
            <s xml:id="echoid-s1663" xml:space="preserve">Deinde prima pars in
              <lb/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="34"/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="35"/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="36"/>
            10. </s>
            <s xml:id="echoid-s1664" xml:space="preserve">particulas æquales diuidatur, ita vt quælibet harum particularũ ſit {1/60}. </s>
            <s xml:id="echoid-s1665" xml:space="preserve">arcus
              <lb/>
            XY. </s>
            <s xml:id="echoid-s1666" xml:space="preserve">Et quoniam vna harum particularum eſt ad arcum XY, vt 1. </s>
            <s xml:id="echoid-s1667" xml:space="preserve">gradus Qua-
              <lb/>
            drantis B C, ad arcum 60. </s>
            <s xml:id="echoid-s1668" xml:space="preserve">graduum; </s>
            <s xml:id="echoid-s1669" xml:space="preserve">cum vtrobique proportio ſit ſubſexagecu-
              <lb/>
            pla; </s>
            <s xml:id="echoid-s1670" xml:space="preserve">erit permutando vna illarum particularum ad 1.</s>
            <s xml:id="echoid-s1671" xml:space="preserve">
              <unsure/>
            grad. </s>
            <s xml:id="echoid-s1672" xml:space="preserve">vt arcus XY, ad arcum
              <lb/>
            60. </s>
            <s xml:id="echoid-s1673" xml:space="preserve">graduum. </s>
            <s xml:id="echoid-s1674" xml:space="preserve">Quocirca quemadmodum arcus X Y, arcum grad. </s>
            <s xml:id="echoid-s1675" xml:space="preserve">60. </s>
            <s xml:id="echoid-s1676" xml:space="preserve">continet
              <lb/>
            ſemel, & </s>
            <s xml:id="echoid-s1677" xml:space="preserve">inſuper vnam eius partem ſexageſimam, id eſt, 1. </s>
            <s xml:id="echoid-s1678" xml:space="preserve">ad grad. </s>
            <s xml:id="echoid-s1679" xml:space="preserve">ex conſtru-
              <lb/>
            ctione; </s>
            <s xml:id="echoid-s1680" xml:space="preserve">Ita quoq; </s>
            <s xml:id="echoid-s1681" xml:space="preserve">vna illarum particularum comprehendet 1. </s>
            <s xml:id="echoid-s1682" xml:space="preserve">gradum ſemel, & </s>
            <s xml:id="echoid-s1683" xml:space="preserve">
              <lb/>
            inſuper vnam partem ſexageſimam vnius gradus, hoc eſt, 1. </s>
            <s xml:id="echoid-s1684" xml:space="preserve">Minutum. </s>
            <s xml:id="echoid-s1685" xml:space="preserve">Ex quo
              <lb/>
            fit, vt duæ particulæ complectantur 2. </s>
            <s xml:id="echoid-s1686" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1687" xml:space="preserve">& </s>
            <s xml:id="echoid-s1688" xml:space="preserve">inſuper 2. </s>
            <s xml:id="echoid-s1689" xml:space="preserve">Minuta. </s>
            <s xml:id="echoid-s1690" xml:space="preserve">Atvero 3. </s>
            <s xml:id="echoid-s1691" xml:space="preserve">par-
              <lb/>
            ticulæ contineant 3. </s>
            <s xml:id="echoid-s1692" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1693" xml:space="preserve">& </s>
            <s xml:id="echoid-s1694" xml:space="preserve">3. </s>
            <s xml:id="echoid-s1695" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s1696" xml:space="preserve">& </s>
            <s xml:id="echoid-s1697" xml:space="preserve">ſic deinceps.</s>
            <s xml:id="echoid-s1698" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1699" xml:space="preserve">
              <emph style="sc">Itaqve</emph>
            ſi in Quadrantem BC, transferatur vna particula ſexageſima arcus
              <lb/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="34"/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="35"/>
              <handwritten xlink:label="hd-071-1" xlink:href="hd-071-1a" number="36"/>
            XY, à puncto C, vel à quouis gradu, habebitur 1. </s>
            <s xml:id="echoid-s1700" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s1701" xml:space="preserve">in 2. </s>
            <s xml:id="echoid-s1702" xml:space="preserve">gradu, vel quouis a-
              <lb/>
            lio. </s>
            <s xml:id="echoid-s1703" xml:space="preserve">Et ſi duæ particulæ transferantur, habebuntur in tertio gradu duo Minuta:</s>
            <s xml:id="echoid-s1704" xml:space="preserve"/>
          </p>
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