Clavius, Christoph, Geometria practica
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              <pb o="40" file="070" n="70" rhead="GEOMETR. PRACT."/>
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            particulam ſemiſſe gradus minorem non ſuperare min. </s>
            <s xml:id="echoid-s1478" xml:space="preserve">24. </s>
            <s xml:id="echoid-s1479" xml:space="preserve">(Nam ſi ſuperaret
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            24. </s>
            <s xml:id="echoid-s1480" xml:space="preserve">minuta, non poſſemus ratione iam explicanda in ouirere minuta; </s>
            <s xml:id="echoid-s1481" xml:space="preserve">propterea
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            quod circumducendo circinum, to tum Quadrantem excederemus, vt patebit.)
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            </s>
            <s xml:id="echoid-s1482" xml:space="preserve">commodius minuta datæ particulæ cognoſcemus hoc modo. </s>
            <s xml:id="echoid-s1483" xml:space="preserve">Datam particu-
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            lam cum gradu præcedenti primo loco quadruplicabimus: </s>
            <s xml:id="echoid-s1484" xml:space="preserve">deinde huncarcum
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            quadruplum duplabimus, vt hab eamus octuplum arcus ex particula, & </s>
            <s xml:id="echoid-s1485" xml:space="preserve">vno
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            gradu compoſiti; </s>
            <s xml:id="echoid-s1486" xml:space="preserve">tertio arcum hunc octuplum iterum duplabimus, vt fiat ar-
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            c
              <emph style="sub">9</emph>
            ſedecupl
              <emph style="sub">9</emph>
            arcus ex data particula, & </s>
            <s xml:id="echoid-s1487" xml:space="preserve">vno gradu cõpoſiti; </s>
            <s xml:id="echoid-s1488" xml:space="preserve">quarto hunc arcũ
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            rurſus duplabimus; </s>
            <s xml:id="echoid-s1489" xml:space="preserve">& </s>
            <s xml:id="echoid-s1490" xml:space="preserve">quinto tandem hunc duplum iterum duplabimus; </s>
            <s xml:id="echoid-s1491" xml:space="preserve">vt
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            habeatur arcus continens arcum ex particula pro poſita, & </s>
            <s xml:id="echoid-s1492" xml:space="preserve">vno gradu ſexagies
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            & </s>
            <s xml:id="echoid-s1493" xml:space="preserve">quater, cuius extremum punctũ
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            m diligenter notetur. </s>
            <s xml:id="echoid-s1494" xml:space="preserve">Nam ſi ex toto arcu
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            ab eo puncto incipiendo, auferatur arcus quadruplus particulæ datæ vnâ cum
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            vno gradu, ſupererit arcus ſexagecuplus eiuſdem particulæ vnâ cum vno gra-
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            du; </s>
            <s xml:id="echoid-s1495" xml:space="preserve">ex quo deniqueſi demantur 60. </s>
            <s xml:id="echoid-s1496" xml:space="preserve">gradus, reliqui gradus numerum minu-
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            torum indicabunt. </s>
            <s xml:id="echoid-s1497" xml:space="preserve">Quòd ſi data particula ſemiſſe gradus fuerit maior, explo-
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            tabimus eodem modo minuta inreliqua particula minore comprehenſa. </s>
            <s xml:id="echoid-s1498" xml:space="preserve">Hæc
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            namque minuta ex 60. </s>
            <s xml:id="echoid-s1499" xml:space="preserve">detracta relinquent minutain particula illa maiore com-
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            prehenſa. </s>
            <s xml:id="echoid-s1500" xml:space="preserve">Benè autem vides, quando data particula minor parum à ſemiſſe
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            gradus differt, tutius eſſe quintuplare illam vnâ cum vno gradu; </s>
            <s xml:id="echoid-s1501" xml:space="preserve">deinde hunc
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            arcum duplare, tertio hunc arcum triplare, & </s>
            <s xml:id="echoid-s1502" xml:space="preserve">poſtremò hunc iterum duplare.
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            </s>
            <s xml:id="echoid-s1503" xml:space="preserve">Hac enim ratione ex vltimo arcu duplato auferenditantum ſunt 60. </s>
            <s xml:id="echoid-s1504" xml:space="preserve">gradus, & </s>
            <s xml:id="echoid-s1505" xml:space="preserve">
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            nunquam totus quadrans exhauritur, vt patet.</s>
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            <s xml:id="echoid-s1507" xml:space="preserve">12. </s>
            <s xml:id="echoid-s1508" xml:space="preserve">
              <emph style="sc">Neqve</emph>
            verò ſemper opus eſt, vt particula illa gradus, vel particula mi-
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            nor vnâ cum vno gradu ſexagies repetatur, ſed ſatis eſt, vt ea aliquoties repeti-
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            ta incidat præciſè in aliquem gradum; </s>
            <s xml:id="echoid-s1509" xml:space="preserve">quod non raro accidere ſolet. </s>
            <s xml:id="echoid-s1510" xml:space="preserve">Nam
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            tunc conſtituetur fractio, cuius numerator eſt numerus graduum percurſorũ:
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            </s>
            <s xml:id="echoid-s1511" xml:space="preserve">Denominator autem numerus tot vnitatum, quoties particula circino repetita
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            fuerit. </s>
            <s xml:id="echoid-s1512" xml:space="preserve">Verbi gratia, ſi aliqua grad
              <emph style="sub">9</emph>
            particula vicies repetita incidat in 6. </s>
            <s xml:id="echoid-s1513" xml:space="preserve">gradũ, cõ
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            plectetur particula illa {6/20}. </s>
            <s xml:id="echoid-s1514" xml:space="preserve">vnius gradus. </s>
            <s xml:id="echoid-s1515" xml:space="preserve">Quare ſi numerator 6. </s>
            <s xml:id="echoid-s1516" xml:space="preserve">per 60. </s>
            <s xml:id="echoid-s1517" xml:space="preserve">multi-
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            plicetur, & </s>
            <s xml:id="echoid-s1518" xml:space="preserve">pro ductus numerus 360. </s>
            <s xml:id="echoid-s1519" xml:space="preserve">per denominatorem 20. </s>
            <s xml:id="echoid-s1520" xml:space="preserve">diuidatur, in dica-
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            bit Quotiens 18. </s>
            <s xml:id="echoid-s1521" xml:space="preserve">particulam illam continere 18. </s>
            <s xml:id="echoid-s1522" xml:space="preserve">minuta. </s>
            <s xml:id="echoid-s1523" xml:space="preserve">Sic ſi alia particula ſe-
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            pties repetita incidat in tertium gradum, comprehendet ea {3/7}. </s>
            <s xml:id="echoid-s1524" xml:space="preserve">vnius gradus. </s>
            <s xml:id="echoid-s1525" xml:space="preserve">Si
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            igitur numerator 3. </s>
            <s xml:id="echoid-s1526" xml:space="preserve">per 60. </s>
            <s xml:id="echoid-s1527" xml:space="preserve">multiplicetur, & </s>
            <s xml:id="echoid-s1528" xml:space="preserve">numerus productus 180. </s>
            <s xml:id="echoid-s1529" xml:space="preserve">per de-
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            nominatorem 7. </s>
            <s xml:id="echoid-s1530" xml:space="preserve">diuidatur, reperientur 25. </s>
            <s xml:id="echoid-s1531" xml:space="preserve">min. </s>
            <s xml:id="echoid-s1532" xml:space="preserve">Et quia in diuiſione ſuperſunt
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            5. </s>
            <s xml:id="echoid-s1533" xml:space="preserve">ſi ea rurſum multiplicentur per 60. </s>
            <s xml:id="echoid-s1534" xml:space="preserve">productuſque numerus 300. </s>
            <s xml:id="echoid-s1535" xml:space="preserve">per eundem
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            denominatorem 7. </s>
            <s xml:id="echoid-s1536" xml:space="preserve">diuidatur, dabit quotiens adhuc 42 {6/7}. </s>
            <s xml:id="echoid-s1537" xml:space="preserve">ſecunda.</s>
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            <s xml:id="echoid-s1539" xml:space="preserve">
              <emph style="sc">Demonstratio</emph>
            huius praxis hæc eſt. </s>
            <s xml:id="echoid-s1540" xml:space="preserve">Quoniam in priori exemplo, ita
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            ſe habent 20. </s>
            <s xml:id="echoid-s1541" xml:space="preserve">gradus ad 1. </s>
            <s xml:id="echoid-s1542" xml:space="preserve">gradum, vt particula vicies repetita ad vnam parti-
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            culam; </s>
            <s xml:id="echoid-s1543" xml:space="preserve">erit permutando arcus 20. </s>
            <s xml:id="echoid-s1544" xml:space="preserve">graduum ad arcum continentem particulam
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            vicies, hoc eſt, ad arcum 6. </s>
            <s xml:id="echoid-s1545" xml:space="preserve">graduum, vt 1. </s>
            <s xml:id="echoid-s1546" xml:space="preserve">gradus ad vnam particulam: </s>
            <s xml:id="echoid-s1547" xml:space="preserve">& </s>
            <s xml:id="echoid-s1548" xml:space="preserve">con-
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            uertendo 6. </s>
            <s xml:id="echoid-s1549" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1550" xml:space="preserve">ad 20. </s>
            <s xml:id="echoid-s1551" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1552" xml:space="preserve">vt 1 particula ad 1. </s>
            <s xml:id="echoid-s1553" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1554" xml:space="preserve">Cum ergo 6. </s>
            <s xml:id="echoid-s1555" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1556" xml:space="preserve">ſint
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            {6/20}. </s>
            <s xml:id="echoid-s1557" xml:space="preserve">graduum 20. </s>
            <s xml:id="echoid-s1558" xml:space="preserve">continebit quo que vna particula {6/20}. </s>
            <s xml:id="echoid-s1559" xml:space="preserve">vnius gradus, quod eſt
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            propoſitum. </s>
            <s xml:id="echoid-s1560" xml:space="preserve">In poſteriori verò exemplo, quia ita ſe habent 7. </s>
            <s xml:id="echoid-s1561" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1562" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s1563" xml:space="preserve">gra-
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            dum, vt particula ſepties repetita, hoc eſt, vt 3. </s>
            <s xml:id="echoid-s1564" xml:space="preserve">gradus ad 1. </s>
            <s xml:id="echoid-s1565" xml:space="preserve">particuiam; </s>
            <s xml:id="echoid-s1566" xml:space="preserve">erit per-
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            mutando arcus 7. </s>
            <s xml:id="echoid-s1567" xml:space="preserve">graduum ad 3 grad. </s>
            <s xml:id="echoid-s1568" xml:space="preserve">vt 1. </s>
            <s xml:id="echoid-s1569" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1570" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s1571" xml:space="preserve">particulam: </s>
            <s xml:id="echoid-s1572" xml:space="preserve">& </s>
            <s xml:id="echoid-s1573" xml:space="preserve">conuerten-
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            do 3. </s>
            <s xml:id="echoid-s1574" xml:space="preserve">gradus ad 7. </s>
            <s xml:id="echoid-s1575" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1576" xml:space="preserve">vt vna particula ad 1. </s>
            <s xml:id="echoid-s1577" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s1578" xml:space="preserve">Cum ergo 3. </s>
            <s xml:id="echoid-s1579" xml:space="preserve">gradus ſint {3/7}.
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            </s>
            <s xml:id="echoid-s1580" xml:space="preserve">ſeptem graduum, complectetur quoque 1. </s>
            <s xml:id="echoid-s1581" xml:space="preserve">particula {5/7}. </s>
            <s xml:id="echoid-s1582" xml:space="preserve">vnius gradus; </s>
            <s xml:id="echoid-s1583" xml:space="preserve">quod eſt
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            propoſitum. </s>
            <s xml:id="echoid-s1584" xml:space="preserve">eadem que in cęteris ratio eſt.</s>
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