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[Commentary:
There is a reference on this and the four following folios to page 212 of Commandino's edition of Mathematicae collectiones
(Pappus . Page 212 contains Proposition 85, also denoted Lemma XI.
Problema V. Propos. LXXXV.
Semicirculo positione dato , & dato puncto , describere per semicirculum, qualis est , ita vt ducatur contingens , fiat ipsi æqualis.
Given a semicircle and a point , draw through a semicircle , so that when the tangent is drawn, is equal to . ]
Problema V. Propos. LXXXV.
Semicirculo positione dato , & dato puncto , describere per semicirculum, qualis est , ita vt ducatur contingens , fiat ipsi æqualis.
Given a semicircle and a point , draw through a semicircle , so that when the tangent is drawn, is equal to . ]
Pappus. 212.
Hic habetur usus determinatæ
[Translation: Here is found the use of a determinate ]
[Translation: Here is found the use of a determinate ]
Semicirculo positione dato , & dato puncto : Describere per semicirculum,
qualis est , ita ut ducatur contingens , fiat ipsi
[Translation: Given a semicircle and a point , draw through a semicircle , so that when the tangent is drawn, is equal to .
qualis est , ita ut ducatur contingens , fiat ipsi
[Translation: Given a semicircle and a point , draw through a semicircle , so that when the tangent is drawn, is equal to .
Constructio:
Centro , intervallo , describitur periferia ; Et inscribatur
æqualis . A puncto ducatur perpendicularis ad .
Dividatur bisariam in puncto . fiat æqualis . et æqualis .
Dividatur bisariam in . et fiat æqualis vel . Sit
perpendicularis ad . et ducatur , cui fiat æqualis . Dico quod
est centrum semicirculo quæsiti. qui describatur et sit . a puncto
ducatur contingens et producatur ad . Dico quod æqualis
est
[Translation: Construction
With centre and radius , there isdrawn the circumference . And there is inscribed equal to . From the point there is drawn perpendicular to . The line is bisected at the point . Make equal to and equal to . The line is bisected at , and make euqal to or . Let be perpendicular to and is drawn, to which make equal. I say that is the centre of the semicircle sought, which is drawn, and is . From the point there is drawn the tangent extended to . I say that is equal to .
Centro , intervallo , describitur periferia ; Et inscribatur
æqualis . A puncto ducatur perpendicularis ad .
Dividatur bisariam in puncto . fiat æqualis . et æqualis .
Dividatur bisariam in . et fiat æqualis vel . Sit
perpendicularis ad . et ducatur , cui fiat æqualis . Dico quod
est centrum semicirculo quæsiti. qui describatur et sit . a puncto
ducatur contingens et producatur ad . Dico quod æqualis
est
[Translation: Construction
With centre and radius , there isdrawn the circumference . And there is inscribed equal to . From the point there is drawn perpendicular to . The line is bisected at the point . Make equal to and equal to . The line is bisected at , and make euqal to or . Let be perpendicular to and is drawn, to which make equal. I say that is the centre of the semicircle sought, which is drawn, and is . From the point there is drawn the tangent extended to . I say that is equal to .
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