601301
[Translation: Most ]
[Commentary:
The first reference on this page is to a lemma proved by Commandino on page 198v of his edition of Mathematicae collectiones
(Pappus . Harriot's diagram imitates Commandino's except that he has changed some of the lettering.
Towards the end there are references to Euclid's .
In a given triangle to inscribe a circle.
About a given triangle to circumscribe a circle.
There is also a reference to In duos Archimedis aequiponderantium libros (Guidobaldi . ]
Towards the end there are references to Euclid's .
In a given triangle to inscribe a circle.
About a given triangle to circumscribe a circle.
There is also a reference to In duos Archimedis aequiponderantium libros (Guidobaldi . ]
Pappus. pag.
Quod hic construitur est in altera
[Translation: What is here constructed is in another ]
[Translation: What is here constructed is in another ]
Sit triangulum . Ducantur
perpendicularis , , quæ sese
intersecant in puncto . et ab
ad agatur recta usque ad .
Dico quod est linea perpendicu-
laris lineæ .
[Translation: Let there be a triangle . Draw the perpendiculars , , which intersect each other at the point . And from ad construct a line as far as .
I say that is a line perpendicular to the line . *
perpendicularis , , quæ sese
intersecant in puncto . et ab
ad agatur recta usque ad .
Dico quod est linea perpendicu-
laris lineæ .
[Translation: Let there be a triangle . Draw the perpendiculars , , which intersect each other at the point . And from ad construct a line as far as .
I say that is a line perpendicular to the line . *
Demonstratio in
altera charta.
In huius demonstratione
Erravit
[Translation: The demonstration is in another sheet.
In this demonstration Commndino has ]
altera charta.
In huius demonstratione
Erravit
[Translation: The demonstration is in another sheet.
In this demonstration Commndino has ]
Demonstratio
* puncta , , , , sunt in
peripheria quoniam anguli ad et
sunt recti. agatur igitur circulus, .
et puncta sunt etiam in perpheria
semicirculi , [???] causa [???]
[Translation: Demonstration
* The points , , , are on a circumference because the angles at et are right angles. Therefore construct the circle . The points et are also on the circumference of the semicircle , [???] because [???].
* puncta , , , , sunt in
peripheria quoniam anguli ad et
sunt recti. agatur igitur circulus, .
et puncta sunt etiam in perpheria
semicirculi , [???] causa [???]
[Translation: Demonstration
* The points , , , are on a circumference because the angles at et are right angles. Therefore construct the circle . The points et are also on the circumference of the semicircle , [???] because [???].
Anguli et sunt æqualis quia
sunt in peripheria [???] super eandem
basis . Tum et æqualis
quia insistunt super . æqualis igitur
anguli et . et
sunt æqualis quia [???] ad verticem.
Ergo, tertius angulus æquatur
tertio recto. quod demonstrantur
[Translation: The agnles et are equal because the are on a circumference [???] on the same base . Then and are equal because they stand on . Therefore angles and are equal. et are euqal because [???] to the vertex.
Therefore, the third angle is equal to the third , a right angle. Which was to be proved.
sunt in peripheria [???] super eandem
basis . Tum et æqualis
quia insistunt super . æqualis igitur
anguli et . et
sunt æqualis quia [???] ad verticem.
Ergo, tertius angulus æquatur
tertio recto. quod demonstrantur
[Translation: The agnles et are equal because the are on a circumference [???] on the same base . Then and are equal because they stand on . Therefore angles and are equal. et are euqal because [???] to the vertex.
Therefore, the third angle is equal to the third , a right angle. Which was to be proved.
Nota.
Tres lineæ ductæ
per tres angulos et
Biscantes angulos: Euclid. el. 4,4.
Bisecantes latera: Archimed. de æqui:
perpenduclares ad latera Hic demonstratur.
Tres, perpendiculares in medijs laterum, El. 5,4. Sese
interse-
cant in
uno
[Translation: Note.
Three lines drawn through three angles.
Bisection of angules: Euclid, Elements, IV,4.
Bisection of sides: Archimedes, De æqui:
Perpendiculars to the sides: demonstrated here.
Three perpendiculars in the middle of the sides, Elements, IV.5. Intersecting each other in a single point.
Tres lineæ ductæ
per tres angulos et
Biscantes angulos: Euclid. el. 4,4.
Bisecantes latera: Archimed. de æqui:
perpenduclares ad latera Hic demonstratur.
Tres, perpendiculares in medijs laterum, El. 5,4. Sese
interse-
cant in
uno
[Translation: Note.
Three lines drawn through three angles.
Bisection of angules: Euclid, Elements, IV,4.
Bisection of sides: Archimedes, De æqui:
Perpendiculars to the sides: demonstrated here.
Three perpendiculars in the middle of the sides, Elements, IV.5. Intersecting each other in a single point.
Alia propositio: seu conclusio
Superioribus constructis et demonstratis; fiat præterea triangulum . Dico quod
eius anguli bisecantur per lineas , , et
[Translation: Another proposition: or conclusion
By the above construction and demonstration; make also the triangle . I say that its angles are bisected by the lines , , and .
Superioribus constructis et demonstratis; fiat præterea triangulum . Dico quod
eius anguli bisecantur per lineas , , et
[Translation: Another proposition: or conclusion
By the above construction and demonstration; make also the triangle . I say that its angles are bisected by the lines , , and .
Nam: Anguli et æquantur, quia in perpheria et insistunt super .
Angulus etiam , æqualis quia trainguli æquianguli;
duo enim recti et duo ad verticem; ergo tertius tertio æqualis.
Sed angulus æqualis quia in peripheria super . Ergo
anguli et sunt æqualis. Simili modo probatur bisectio
duorum
[Translation: For the angles and are equal because they are on a circumference and stand on .
Also the angle is equal to because in an equiangular triangle, for two are right angles and two are at the vertex. Therefore the third is equal to the third.
But the angle is equal because on the circumference standing on . Therefore the angles and are equal. In a similar way there can be proved the bisection of the others.
Angulus etiam , æqualis quia trainguli æquianguli;
duo enim recti et duo ad verticem; ergo tertius tertio æqualis.
Sed angulus æqualis quia in peripheria super . Ergo
anguli et sunt æqualis. Simili modo probatur bisectio
duorum
[Translation: For the angles and are equal because they are on a circumference and stand on .
Also the angle is equal to because in an equiangular triangle, for two are right angles and two are at the vertex. Therefore the third is equal to the third.
But the angle is equal because on the circumference standing on . Therefore the angles and are equal. In a similar way there can be proved the bisection of the others.
Etiam triangula: sunt similia.
Tertio est una linea reflexa ad
angulis æqualis intra triangluum
[Translation: Also the triangles , , , are similar.
Third, is a line of reflection at equal angles inside triangle .
Tertio est una linea reflexa ad
angulis æqualis intra triangluum
[Translation: Also the triangles , , , are similar.
Third, is a line of reflection at equal angles inside triangle .
[Translation: Most ]
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