709355
[Commentary:
On this page Harriot examines a particular case arising from Proposition VII of Supplementum geometriæ
(Viète 1593c, Prop , when the fourth proportional is twice the first. The same proposition is the subject of Chapter V of Viète's Variorum responsorum libri VIII
(Viete 1593d, Chapter .
Caput V
Propositio
Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione
Construct four lines in continued proportion, whose extremes are in double
The Variorum refers to the Supplementum, indicating that the Supplementum was written first. ]
Caput V
Propositio
Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione
Construct four lines in continued proportion, whose extremes are in double
The Variorum refers to the Supplementum, indicating that the Supplementum was written first. ]
Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag.
[Translation: On a corollary to Proposition 7 of the Supplement. Also Chapter 5, Variorum liber responsorum, page 4.
[Translation: On a corollary to Proposition 7 of the Supplement. Also Chapter 5, Variorum liber responsorum, page 4.
Sit prima proportionalium, et ea
cuius quadratum est triplum quadrati .
Tum est dupla ad ; et per assumptum
ex poristicis in alia charta demonstratum
erit quarta proportionalis. Per propositione est secunda et tertia.
Sed est æqualis propter similitudine triangulorum et , et
analogiam precedentam ut sequitur.
Analogia precedens.
[…]
Et per similitudi-
num Δorum[…]
Ergo. continue
[Translation: Let be the first proportional, and that whose square is three times the square of .
Then is twice ; and by taking it from the proof demonstrated in the other sheet, will be the fourth proportional. By the proposition is the second and the third.
But is equal to because of similar triangles and , and
the precding ratio, as follows.
preceding ratio.
And by similar triangles.
Therefore are continued proportionals.
[Commentary: The other sheet mentioned in this paragraph appears to be Add MS f. .
cuius quadratum est triplum quadrati .
Tum est dupla ad ; et per assumptum
ex poristicis in alia charta demonstratum
erit quarta proportionalis. Per propositione est secunda et tertia.
Sed est æqualis propter similitudine triangulorum et , et
analogiam precedentam ut sequitur.
Analogia precedens.
[…]
Et per similitudi-
num Δorum[…]
Ergo. continue
[Translation: Let be the first proportional, and that whose square is three times the square of .
Then is twice ; and by taking it from the proof demonstrated in the other sheet, will be the fourth proportional. By the proposition is the second and the third.
But is equal to because of similar triangles and , and
the precding ratio, as follows.
preceding ratio.
And by similar triangles.
Therefore are continued proportionals.
[Commentary: The other sheet mentioned in this paragraph appears to be Add MS f. .
Datis igitur extremis in ratione dupla, mediæ ita compendiosæ
[Translation: Therefore given the extremes in double ratio, the mean is briefly ]
Sit maxima bisariam divisa in puncto et intervallo describatur
circulus. Et sit prima minima inscripta et producta ad partes .
Ducatur ita ut sit æqualis . et acta fit linea .
Quatuor igitur continue proportionales ex supra demonstratis
[Translation: Let the maximum be cut in half at the point and with radius there is described a circle. And let the minimum be inscribed and produced to the point . Construct so that is equal to , and let the line be joined.
Therefore there are the four continued proportionals that were demonstrated ]
[Translation: Therefore given the extremes in double ratio, the mean is briefly ]
Sit maxima bisariam divisa in puncto et intervallo describatur
circulus. Et sit prima minima inscripta et producta ad partes .
Ducatur ita ut sit æqualis . et acta fit linea .
Quatuor igitur continue proportionales ex supra demonstratis
[Translation: Let the maximum be cut in half at the point and with radius there is described a circle. And let the minimum be inscribed and produced to the point . Construct so that is equal to , and let the line be joined.
Therefore there are the four continued proportionals that were demonstrated ]