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[Commentary:
On this page Harriot investigates Proposition 15 from Supplementum geometriæ
(Viète 1593c, Prop .
Proposition XV.
Si e circumferential circuli cadant in diametrum perpendiculares duæ, una in centro, altera extra & ad perpendicularem in centro agatur ex puncto incidentiæ perpendicularis alterius, linea recta faciens cum diametro angulum æqualem trienti recti; a puncto autem quo acta illa secat perpendiculare in centro, ducatur alia linea recta ad angulum semicirculi: triplum quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est
If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean
The working contains a reference to Euclid's Elements, Proposition .
If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts, together with twice the rectangle contained by the parts. ]
Proposition XV.
Si e circumferential circuli cadant in diametrum perpendiculares duæ, una in centro, altera extra & ad perpendicularem in centro agatur ex puncto incidentiæ perpendicularis alterius, linea recta faciens cum diametro angulum æqualem trienti recti; a puncto autem quo acta illa secat perpendiculare in centro, ducatur alia linea recta ad angulum semicirculi: triplum quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est
If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean
The working contains a reference to Euclid's Elements, Proposition .
If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts, together with twice the rectangle contained by the parts. ]
prop. 15.
[Translation: Proposition 15 from the ]
[Translation: Proposition 15 from the ]
Si e circumferential circuli cadant in
diametrum perpendiculares duæ; una in
centro; altera extra centrum: et ad per-
pendicularem in centro agatur ex puncto
incidentiæ perpendicularis alterius, linea
recta faciens cum diametro angulum æqualem
trienti recti, a puncto autem quo acta illa secat
perpendiculare in centro, ducatur alia
linea recta ad angulum semicirculi; Triplum
quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum,
quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est
[Translation: If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean ]
Sit diameter circuli , a cuius circumferentia cadat perpendiculariter et fit
minus segmentum, maius, verum centro. Sed et cadat quoque e circumferentia
perpendiculariter , et ex ducatur recta ita ut angulus sit æqualis trienti
recti, unde fiat dupla ipsius ; et iungatur . Dico triplum quadratum ex
æquari quadrato ex , una cum quadrato ex et quadrato ex
[Translation: Let be the diameter of a circle, from whose circumference there falls perpendicularly , and let be the lesser segment, the greater, and the centre. But there also falls perpendicularly from the circumference , and from there is drawn a line so that the angle is equal to a third of a right angle, whence is twice ; and is joined. I say that three times the square on is equal to the square on together with the square on and the squareon .
Etiam
per 4,2 El.
[…] Addatur utrovisque
[…] Ergo
[Translation: Also by Elements II.4
Hence the ]
diametrum perpendiculares duæ; una in
centro; altera extra centrum: et ad per-
pendicularem in centro agatur ex puncto
incidentiæ perpendicularis alterius, linea
recta faciens cum diametro angulum æqualem
trienti recti, a puncto autem quo acta illa secat
perpendiculare in centro, ducatur alia
linea recta ad angulum semicirculi; Triplum
quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum,
quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est
[Translation: If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean ]
Sit diameter circuli , a cuius circumferentia cadat perpendiculariter et fit
minus segmentum, maius, verum centro. Sed et cadat quoque e circumferentia
perpendiculariter , et ex ducatur recta ita ut angulus sit æqualis trienti
recti, unde fiat dupla ipsius ; et iungatur . Dico triplum quadratum ex
æquari quadrato ex , una cum quadrato ex et quadrato ex
[Translation: Let be the diameter of a circle, from whose circumference there falls perpendicularly , and let be the lesser segment, the greater, and the centre. But there also falls perpendicularly from the circumference , and from there is drawn a line so that the angle is equal to a third of a right angle, whence is twice ; and is joined. I say that three times the square on is equal to the square on together with the square on and the squareon .
Etiam
per 4,2 El.
[…] Addatur utrovisque
[…] Ergo
[Translation: Also by Elements II.4
Hence the ]
Hinc tale Consectarium potest
[Translation: Here a Consequence of this kind may be ]
[Translation: Here a Consequence of this kind may be ]
Datis tribus continue proportionalibus: invenire lineam cuius
quadratum sit tertia pars adgregati quadratorum e tribus
[Translation: Given three continued proportionals, find a line whose square is a third of the sum of the squares of all three proportionals.
quadratum sit tertia pars adgregati quadratorum e tribus
[Translation: Given three continued proportionals, find a line whose square is a third of the sum of the squares of all three proportionals.