605303
[Commentary:
The reference on this page is to a lemma proved by Commandino on page 198v of his edition of Mathematicae collectiones
(Pappus . ]
pappus. pag. 198.
[Translation: Pappus, page ]
[Translation: Pappus, page ]
vide digraphum in altera
[Translation: see the diagram on the other ]
[Commentary: The other sheet is Add MS f. .
[Translation: see the diagram on the other ]
[Commentary: The other sheet is Add MS f. .
Sit triangulum . Ducantur perpendiculares , , quæ sese
intersecant in puncto . et ab ad agatur recta usque ad .
Dico quod linea est perpendicularis lineæ .
Quoniam anguli et sunt recta: Quadrilaterum est
in circulo, qui desribatur. Deinde circa triangulum describatur etiam
circulus et ducatur usque ad in perpheria,
, usque ad , et ad . agatur etiam lineæ
, , , , ,
[Translation: Let there be a triangle . Draw perpendiculars , which intersect each other at the point . And from to construct a line as far as .
I say that the line is perpendicular to the line .
Because the angles and are right angles, the quadrilateral is in a circle, which may be drawn. Then around the triangle there is also drawn a circle and is drawn as far as on the circumference, as far as , and to . ALso there are constructed the lines , , , , , .
intersecant in puncto . et ab ad agatur recta usque ad .
Dico quod linea est perpendicularis lineæ .
Quoniam anguli et sunt recta: Quadrilaterum est
in circulo, qui desribatur. Deinde circa triangulum describatur etiam
circulus et ducatur usque ad in perpheria,
, usque ad , et ad . agatur etiam lineæ
, , , , ,
[Translation: Let there be a triangle . Draw perpendiculars , which intersect each other at the point . And from to construct a line as far as .
I say that the line is perpendicular to the line .
Because the angles and are right angles, the quadrilateral is in a circle, which may be drawn. Then around the triangle there is also drawn a circle and is drawn as far as on the circumference, as far as , and to . ALso there are constructed the lines , , , , , .
Demonstratio:
Quoniam est quadrilaterum in circulo:
ergo: Duobus rectis.
Sed: . quia insistunt super .
et: . quia anguli ad verticem.
et: Duobus rectis.
[Translation: Demonstration
Because is a quadrilateral in a circle, therefore two right angles.
But , because they stand on .
and , because they are angles at the vertex.
and two right angles.
Quoniam est quadrilaterum in circulo:
ergo: Duobus rectis.
Sed: . quia insistunt super .
et: . quia anguli ad verticem.
et: Duobus rectis.
[Translation: Demonstration
Because is a quadrilateral in a circle, therefore two right angles.
But , because they stand on .
and , because they are angles at the vertex.
and two right angles.
* Atque inde: . cum angulus æqualibus
Ergo etiam: = . cum angulis et lateribus.
Ergo: et recti.
Ergo: perpendicularis est ad . Quæsitum.
Ex eadem diagrammate nullis alijs modis conlusio infertur ut
inspicienti facile
[Translation: * And thence triangle , because of equal angles.
Therefore also triangle = triangle because of angles and sides.
Therefore and are right angles.
Therefore is perpendicular to . Sought.
From the same diagram, the conclusion is inferred in no other way, as is easily clear from ]
Ergo etiam: = . cum angulis et lateribus.
Ergo: et recti.
Ergo: perpendicularis est ad . Quæsitum.
Ex eadem diagrammate nullis alijs modis conlusio infertur ut
inspicienti facile
[Translation: * And thence triangle , because of equal angles.
Therefore also triangle = triangle because of angles and sides.
Therefore and are right angles.
Therefore is perpendicular to . Sought.
From the same diagram, the conclusion is inferred in no other way, as is easily clear from ]
Triangula igitur et sunt æquiangula.
[…]
Sed est rectus. ergo: .
[Translation: Therefore the triangles and are equiangular.
But is a right angle, so therefore is . Sought.
[…]
Sed est rectus. ergo: .
[Translation: Therefore the triangles and are equiangular.
But is a right angle, so therefore is . Sought.