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[Commentary:
At the end of Chapter XIX of Variorum responsorum liber VIII (1593) there is a lengthy section entitled 'Eis procheiron scholia'
(Viete 1593d, Chapter 19, . Section IV is on spherical geometry. The 18th and final proposition contains four statements, which Harriot here translated into symbolic notation
(Viete 1593d, Chapter 19, PROCHEIRON, Section IV, Prop .
18 Sit triangulum sphaericum ABD, & in peripheria BD cadat segmentum orthogonii AC.
Primo dico esse transsinuosa anguli BAC ad transsinuosa anguli DAC, sicut prosinum peripheria AB ad prosinum peripheri AD.
Secundo dico esse transsinuosam peripheriæ CB ad transsinuosam peripheriæ CD, sicut transsinuosam peripheriae AB ad transsinuosam peripheriæ AD.
Tertio dico esse sinum CD ad sinum CB, sicut prosinum anguli B ad prosinum anguli D.
Denique & quarto dico esse sinum anguli BAC ad sinum anguli DAC, sicut transsinuosam anguli D ad transsinuosam anguli B.
18. Let there be a spherical triangle ABD, and to the arc BD there falls an orthogonal line AC.
First I say that the secant of angle BAC to the secant of angle DAC is as the tangent of the arc AB to the tangent of the arc AD.
Second I say that the secant of the arc CB to the secant of the arc CD is as the secant of the arc AB to the secant of the arc AD.
Third I say that the sine of CD to the sine of DB is as the tangent of angle B to the tangent of angle D.
Fourth and last I say that the sine of angle BAC to the sine of angle DAC is as the secant of angle D to the secant of angle B.]
18 Sit triangulum sphaericum ABD, & in peripheria BD cadat segmentum orthogonii AC.
Primo dico esse transsinuosa anguli BAC ad transsinuosa anguli DAC, sicut prosinum peripheria AB ad prosinum peripheri AD.
Secundo dico esse transsinuosam peripheriæ CB ad transsinuosam peripheriæ CD, sicut transsinuosam peripheriae AB ad transsinuosam peripheriæ AD.
Tertio dico esse sinum CD ad sinum CB, sicut prosinum anguli B ad prosinum anguli D.
Denique & quarto dico esse sinum anguli BAC ad sinum anguli DAC, sicut transsinuosam anguli D ad transsinuosam anguli B.
18. Let there be a spherical triangle ABD, and to the arc BD there falls an orthogonal line AC.
First I say that the secant of angle BAC to the secant of angle DAC is as the tangent of the arc AB to the tangent of the arc AD.
Second I say that the secant of the arc CB to the secant of the arc CD is as the secant of the arc AB to the secant of the arc AD.
Third I say that the sine of CD to the sine of DB is as the tangent of angle B to the tangent of angle D.
Fourth and last I say that the sine of angle BAC to the sine of angle DAC is as the secant of angle D to the secant of angle B.]
Demonstratio eorum quæ desiderant
in Vieta. lib. 8. respons. pag. 41.b.
sectione
[Translation: Demonstration of what is missing in Viète, Responsorum liber VIII, page 41v, section ]
in Vieta. lib. 8. respons. pag. 41.b.
sectione
[Translation: Demonstration of what is missing in Viète, Responsorum liber VIII, page 41v, section ]
Hinc apparet mendam
esse apud Vietam. nam
[Translation: Here is is clear that it is wrong in Viète, for ]
esse apud Vietam. nam
[Translation: Here is is clear that it is wrong in Viète, for ]
Istæ quatuor conclusiones
sexeis variari possunt ut
ex analogijs rectangulorm est
[Translation: These four conclusions have six variations as is clear from the ratios for ]
sexeis variari possunt ut
ex analogijs rectangulorm est
[Translation: These four conclusions have six variations as is clear from the ratios for ]
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