Clavius, Christoph
,
Geometria practica
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GEOMETR. PRACT.
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particulam ſemiſſe gradus minorem non ſuperare min. </
s
>
<
s
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">24. </
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<
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xml:space
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">(Nam ſi ſuperaret
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24. </
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>
<
s
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="
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xml:space
="
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">minuta, non poſſemus ratione iam explicanda in ouirere minuta; </
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>
<
s
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="
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"
xml:space
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">propterea
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<
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<
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<
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quod circumducendo circinum, to tum Quadrantem excederemus, vt patebit.)
<
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/>
</
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>
<
s
xml:id
="
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"
xml:space
="
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">commodius minuta datæ particulæ cognoſcemus hoc modo. </
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>
<
s
xml:id
="
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"
xml:space
="
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">Datam particu-
<
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lam cum gradu præcedenti primo loco quadruplicabimus: </
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>
<
s
xml:id
="
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"
xml:space
="
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">deinde huncarcum
<
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quadruplum duplabimus, vt hab eamus octuplum arcus ex particula, & </
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>
<
s
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="
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"
xml:space
="
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">vno
<
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gradu compoſiti; </
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>
<
s
xml:id
="
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"
xml:space
="
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">tertio arcum hunc octuplum iterum duplabimus, vt fiat ar-
<
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c
<
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style
="
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">9</
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>
ſedecupl
<
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style
="
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">9</
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>
arcus ex data particula, & </
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>
<
s
xml:id
="
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"
xml:space
="
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">vno gradu cõpoſiti; </
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>
<
s
xml:id
="
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"
xml:space
="
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">quarto hunc arcũ
<
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rurſus duplabimus; </
s
>
<
s
xml:id
="
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"
xml:space
="
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">& </
s
>
<
s
xml:id
="
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"
xml:space
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">quinto tandem hunc duplum iterum duplabimus; </
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>
<
s
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="
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xml:space
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">vt
<
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habeatur arcus continens arcum ex particula pro poſita, & </
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>
<
s
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="
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"
xml:space
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">vno gradu ſexagies
<
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& </
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>
<
s
xml:id
="
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xml:space
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">quater, cuius extremum punctũ
<
unsure
/>
m diligenter notetur. </
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>
<
s
xml:id
="
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"
xml:space
="
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">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; </
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>
<
s
xml:id
="
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="
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">ex quo deniqueſi demantur 60. </
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>
<
s
xml:id
="
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"
xml:space
="
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">gradus, reliqui gradus numerum minu-
<
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torum indicabunt. </
s
>
<
s
xml:id
="
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xml:space
="
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">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
="
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xml:space
="
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">Hæc
<
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/>
namque minuta ex 60. </
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>
<
s
xml:id
="
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xml:space
="
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">detracta relinquent minutain particula illa maiore com-
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<
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<
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="
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prehenſa. </
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>
<
s
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="
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xml:space
="
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">Benè autem vides, quando data particula minor parum à ſemiſſe
<
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/>
gradus differt, tutius eſſe quintuplare illam vnâ cum vno gradu; </
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>
<
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="
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">deinde hunc
<
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arcum duplare, tertio hunc arcum triplare, & </
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>
<
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">poſtremò hunc iterum duplare.
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</
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<
s
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">Hac enim ratione ex vltimo arcu duplato auferenditantum ſunt 60. </
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>
<
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">gradus, & </
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<
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<
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nunquam totus quadrans exhauritur, vt patet.</
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</
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<
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<
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="
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>
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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<
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<
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ta incidat præciſè in aliquem gradum; </
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>
<
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">quod non raro accidere ſolet. </
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>
<
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">Nam
<
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tunc conſtituetur fractio, cuius numerator eſt numerus graduum percurſorũ:
<
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</
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>
<
s
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xml:space
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">Denominator autem numerus tot vnitatum, quoties particula circino repetita
<
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fuerit. </
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>
<
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">Verbi gratia, ſi aliqua grad
<
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="
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">9</
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>
particula vicies repetita incidat in 6. </
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>
<
s
xml:id
="
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xml:space
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">gradũ, cõ
<
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plectetur particula illa {6/20}. </
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<
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">vnius gradus. </
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<
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<
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<
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plicetur, & </
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<
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">pro ductus numerus 360. </
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<
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<
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bit Quotiens 18. </
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<
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<
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<
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xml:id
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xml:space
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">Sic ſi alia particula ſe-
<
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pties repetita incidat in tertium gradum, comprehendet ea {3/7}. </
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<
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>
<
s
xml:id
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xml:space
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">Si
<
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igitur numerator 3. </
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<
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<
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">multiplicetur, & </
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>
<
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<
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nominatorem 7. </
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<
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xml:id
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">diuidatur, reperientur 25. </
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<
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">min. </
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<
s
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xml:space
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5. </
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<
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>
<
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">productuſque numerus 300. </
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>
<
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">per eundem
<
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denominatorem 7. </
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<
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">diuidatur, dabit quotiens adhuc 42 {6/7}. </
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<
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</
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<
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<
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<
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style
="
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">Demonstratio</
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>
huius praxis hæc eſt. </
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<
s
xml:id
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xml:space
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ſe habent 20. </
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<
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xml:id
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">gradus ad 1. </
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>
<
s
xml:id
="
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xml:space
="
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">gradum, vt particula vicies repetita ad vnam parti-
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culam; </
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>
<
s
xml:id
="
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"
xml:space
="
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">erit permutando arcus 20. </
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>
<
s
xml:id
="
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xml:space
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">graduum ad arcum continentem particulam
<
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vicies, hoc eſt, ad arcum 6. </
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>
<
s
xml:id
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xml:space
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">graduum, vt 1. </
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<
s
xml:id
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xml:space
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">gradus ad vnam particulam: </
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>
<
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="
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xml:space
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">& </
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>
<
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xml:id
="
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uertendo 6. </
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<
s
xml:id
="
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">grad. </
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>
<
s
xml:id
="
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">ad 20. </
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>
<
s
xml:id
="
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"
xml:space
="
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">grad. </
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>
<
s
xml:id
="
echoid-s1552
"
xml:space
="
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">vt 1 particula ad 1. </
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>
<
s
xml:id
="
echoid-s1553
"
xml:space
="
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">grad. </
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>
<
s
xml:id
="
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="
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">Cum ergo 6. </
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>
<
s
xml:id
="
echoid-s1555
"
xml:space
="
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">grad. </
s
>
<
s
xml:id
="
echoid-s1556
"
xml:space
="
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">ſint
<
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/>
{6/20}. </
s
>
<
s
xml:id
="
echoid-s1557
"
xml:space
="
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">graduum 20. </
s
>
<
s
xml:id
="
echoid-s1558
"
xml:space
="
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">continebit quo que vna particula {6/20}. </
s
>
<
s
xml:id
="
echoid-s1559
"
xml:space
="
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">vnius gradus, quod eſt
<
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/>
propoſitum. </
s
>
<
s
xml:id
="
echoid-s1560
"
xml:space
="
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">In poſteriori verò exemplo, quia ita ſe habent 7. </
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>
<
s
xml:id
="
echoid-s1561
"
xml:space
="
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">grad. </
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>
<
s
xml:id
="
echoid-s1562
"
xml:space
="
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">ad 1. </
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>
<
s
xml:id
="
echoid-s1563
"
xml:space
="
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">gra-
<
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/>
dum, vt particula ſepties repetita, hoc eſt, vt 3. </
s
>
<
s
xml:id
="
echoid-s1564
"
xml:space
="
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">gradus ad 1. </
s
>
<
s
xml:id
="
echoid-s1565
"
xml:space
="
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">particuiam; </
s
>
<
s
xml:id
="
echoid-s1566
"
xml:space
="
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">erit per-
<
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/>
mutando arcus 7. </
s
>
<
s
xml:id
="
echoid-s1567
"
xml:space
="
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">graduum ad 3 grad. </
s
>
<
s
xml:id
="
echoid-s1568
"
xml:space
="
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">vt 1. </
s
>
<
s
xml:id
="
echoid-s1569
"
xml:space
="
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">grad. </
s
>
<
s
xml:id
="
echoid-s1570
"
xml:space
="
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">ad 1. </
s
>
<
s
xml:id
="
echoid-s1571
"
xml:space
="
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">particulam: </
s
>
<
s
xml:id
="
echoid-s1572
"
xml:space
="
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">& </
s
>
<
s
xml:id
="
echoid-s1573
"
xml:space
="
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">conuerten-
<
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do 3. </
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>
<
s
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="
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">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}.
<
lb
/>
</
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
<
lb
/>
propoſitum. </
s
>
<
s
xml:id
="
echoid-s1584
"
xml:space
="
preserve
">eadem que in cęteris ratio eſt.</
s
>
<
s
xml:id
="
echoid-s1585
"
xml:space
="
preserve
"/>
</
p
>
</
div
>
</
text
>
</
echo
>
