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[Commentary:
On this page Harriot investigates Propositions 20 and 21 from Supplementum geometriæ
(Viète 1593c, Props 20, .
Proposition XX.
Constituere triangulum aæquicrurum, ut differntia inter basin & alterum e cruribus fit ad basin, sicut quadratum cruris ad quadratum compositae ex crure &
To construct an isosceles triangle so that the difference between the base and either of the legs is to the base as the square of a leg is to the square of the sum of a leg and the base.
Proposition XXI.
Si fuerit triangulum aequicrurum, fit autem differentia inter basin & alterum e cruribus ad basin, sicut quadratum cruris ad quadratum compositæ ex crure & base: quae a termino basis ducetur ad crus linea recta ipsi cruri æquale, secabit bisariam angulum ad basin.
If there is an isosceles triangle, and moreover the [ratio of the] difference between the base and either of the legs, to the base, is equal to the square of the leg to the square of the sum of a leg and the base, then a line drawn from the [end of the] base to the leg, equal [in length] to that leg, will bisect the angle at the
There is a reference to Propostion 19 of the Supplementum (see Add MS 6785 f. ). There are also references to Euclid's Propositions , , , and .
The angle at the centre of a circle is double the angle at the circumference, when they have the same part of the circumference for a base.
If from a point without a circle two straight lines be drawn to it, one of which is a tangent to the circle, and the other cuts it; the rectangle under the whole cutting line and the external segment is equal to the square of the tangent.
To find a fourth proportional to three given lines.
Equal parallograms which have one angle each equal have the sides about the equal angles reciprocally proportional. ]
Proposition XX.
Constituere triangulum aæquicrurum, ut differntia inter basin & alterum e cruribus fit ad basin, sicut quadratum cruris ad quadratum compositae ex crure &
To construct an isosceles triangle so that the difference between the base and either of the legs is to the base as the square of a leg is to the square of the sum of a leg and the base.
Proposition XXI.
Si fuerit triangulum aequicrurum, fit autem differentia inter basin & alterum e cruribus ad basin, sicut quadratum cruris ad quadratum compositæ ex crure & base: quae a termino basis ducetur ad crus linea recta ipsi cruri æquale, secabit bisariam angulum ad basin.
If there is an isosceles triangle, and moreover the [ratio of the] difference between the base and either of the legs, to the base, is equal to the square of the leg to the square of the sum of a leg and the base, then a line drawn from the [end of the] base to the leg, equal [in length] to that leg, will bisect the angle at the
There is a reference to Propostion 19 of the Supplementum (see Add MS 6785 f. ). There are also references to Euclid's Propositions , , , and .
The angle at the centre of a circle is double the angle at the circumference, when they have the same part of the circumference for a base.
If from a point without a circle two straight lines be drawn to it, one of which is a tangent to the circle, and the other cuts it; the rectangle under the whole cutting line and the external segment is equal to the square of the tangent.
To find a fourth proportional to three given lines.
Equal parallograms which have one angle each equal have the sides about the equal angles reciprocally proportional. ]
prop. 20.
[Translation: Proposition 20 from the ]
[Translation: Proposition 20 from the ]
Constituere triangulum aæquicrurum; ut differntia inter basin et alterum
e cruribus fit ad basin, sicut quadratum cruris ad quadratum compositæ
ex crure et
[Translation: To construct an isosceles triangle so that the difference between the base and either of the legs is to the base, as the square of a leg is to the square of the sum of a leg and the base.
e cruribus fit ad basin, sicut quadratum cruris ad quadratum compositæ
ex crure et
[Translation: To construct an isosceles triangle so that the difference between the base and either of the legs is to the base, as the square of a leg is to the square of the sum of a leg and the base.
per 19,p fiat
Et ponatur in circumferentia, vel .
Et ducantur recta
Triangulum est quod
[Translation: By Proposition 19,
And in the circumference, put or . and constructing the line , the triangle is as required.
Et ponatur in circumferentia, vel .
Et ducantur recta
Triangulum est quod
[Translation: By Proposition 19,
And in the circumference, put or . and constructing the line , the triangle is as required.
prop.
[Translation: Proposition ]
[Translation: Proposition ]
Si fuerit triangulum aequicrurum: fit autem differentia inter basin et alterum
e cruribus ad basin; sicut quadratum cruris ad quadratum compositæ ex crure et base.
Quae a termino basis ducetur ad crus linea recta ipsi cruri æquale: secabit
bisariam angulum ad
[Translation: If there is an isosceles triangle, and moreover the [ratio of the] difference between the base and either of the legs, to the base, is equal to the square of the leg to the square of the sum of a leg and the base, then a line drawn from the [end of the] base to the leg, equal [in length] to that leg, will bisect the angle at the ]
e cruribus ad basin; sicut quadratum cruris ad quadratum compositæ ex crure et base.
Quae a termino basis ducetur ad crus linea recta ipsi cruri æquale: secabit
bisariam angulum ad
[Translation: If there is an isosceles triangle, and moreover the [ratio of the] difference between the base and either of the legs, to the base, is equal to the square of the leg to the square of the sum of a leg and the base, then a line drawn from the [end of the] base to the leg, equal [in length] to that leg, will bisect the angle at the ]
secat bisariam angulum .
Nam ex hypothesi.
[…]
Et per 36,3 el
Ergo. per 14, 6 el
[…]
Consequenter:
Et subducendo
Ergo: per 2,6: el: et sunt parallelæ
Et: Angulus æqualis angulo .
Sed. per 20,3: Angulus est duplus anguli Hoc est:
Ergo angulus sectis est bisariam a recta .
Quod erat
[Translation: bisects the angle .
For from the hypothesis
And by Elements III.36
Therefore by Elements VI.14
Consequently:
And subtracting
Therefore, by Elements VI.2, and are parallel.
And angle is equal to angle .
But by Elements III.20, angle is twice angle , that is .
Therefore angle is cut in two by the line .
Which was to be ]
Nam ex hypothesi.
[…]
Et per 36,3 el
Ergo. per 14, 6 el
[…]
Consequenter:
Et subducendo
Ergo: per 2,6: el: et sunt parallelæ
Et: Angulus æqualis angulo .
Sed. per 20,3: Angulus est duplus anguli Hoc est:
Ergo angulus sectis est bisariam a recta .
Quod erat
[Translation: bisects the angle .
For from the hypothesis
And by Elements III.36
Therefore by Elements VI.14
Consequently:
And subtracting
Therefore, by Elements VI.2, and are parallel.
And angle is equal to angle .
But by Elements III.20, angle is twice angle , that is .
Therefore angle is cut in two by the line .
Which was to be ]