1910
Iisdem positis: In linea sumatur quovis punctum
et erigatur perpendicularis quæ erit parallela lineaæ
et secabit lineam in puncto . agatur etiam recta
, quæ secabit in puncto . Manifestum est
[Translation: The same things being supposed, in the line there is taken any point and there is constructed the perpendicular which will be parallel to the line and will cut the line in the point . There is also constructed the line , which will cut in the point .
et erigatur perpendicularis quæ erit parallela lineaæ
et secabit lineam in puncto . agatur etiam recta
, quæ secabit in puncto . Manifestum est
[Translation: The same things being supposed, in the line there is taken any point and there is constructed the perpendicular which will be parallel to the line and will cut the line in the point . There is also constructed the line , which will cut in the point .
Dico primo quod est maior quam .
connectantur puncta , ,
et fiat æqualis lineæ
sitque acta recta quæ
necessario secabit lineam in puncto .
Quoniam et in puncto . Quoniam
In triangulo , latera
et sunt æqualia,
anguli etiam ad et
sunt etiam
[Translation: I say first that is greater than .
Connect the points and , and make equal to the line ; let there also be constructed the line which will necessarily cut the line in the point and in the point . Because in the triangle , the sides and are equal, and also the angles at and are also equal.
connectantur puncta , ,
et fiat æqualis lineæ
sitque acta recta quæ
necessario secabit lineam in puncto .
Quoniam et in puncto . Quoniam
In triangulo , latera
et sunt æqualia,
anguli etiam ad et
sunt etiam
[Translation: I say first that is greater than .
Connect the points and , and make equal to the line ; let there also be constructed the line which will necessarily cut the line in the point and in the point . Because in the triangle , the sides and are equal, and also the angles at and are also equal.
Sed angulus ad nempe hoc est est
æqualis duobus angulis interioris et . Duobus videlicet
interioris et oppositis triangulis . Ergo maior est anguli,
maior etiam angulo qui est ad verticem
anguli . Ergo, angulos hoc est est
maior quam angulus . Ergo lineam est maior quam .
multo igitur maior quam . Sed linea , æqualis est
ergo est maior quam . Sed etiam est æqualis , cum
sunt latera subtendentia æqualis angulos æqualium triangulum
nempe, et . Ergo est maior quam , quod primo
demonstrandum
[Translation: But the angle at , namely , that is is equal to two interior angles et , clearly the two interior and opposite angles in triangle . Therefore is the greater angle.
The angle is also greater, which is at the vertex of angle . Therefore, the angle , that is , is greater than the angle . Therefore the line is greater than , and therefore is much greater than . But the line is equal to , therefore is greater than . But also is equal to , since they are sides subtending equal angles of equal triangles, namely, et . Therefore is greater than , which was to be demonstrated first.
æqualis duobus angulis interioris et . Duobus videlicet
interioris et oppositis triangulis . Ergo maior est anguli,
maior etiam angulo qui est ad verticem
anguli . Ergo, angulos hoc est est
maior quam angulus . Ergo lineam est maior quam .
multo igitur maior quam . Sed linea , æqualis est
ergo est maior quam . Sed etiam est æqualis , cum
sunt latera subtendentia æqualis angulos æqualium triangulum
nempe, et . Ergo est maior quam , quod primo
demonstrandum
[Translation: But the angle at , namely , that is is equal to two interior angles et , clearly the two interior and opposite angles in triangle . Therefore is the greater angle.
The angle is also greater, which is at the vertex of angle . Therefore, the angle , that is , is greater than the angle . Therefore the line is greater than , and therefore is much greater than . But the line is equal to , therefore is greater than . But also is equal to , since they are sides subtending equal angles of equal triangles, namely, et . Therefore is greater than , which was to be demonstrated first.
Dico secundo, ut ad , seu ut ad , ita ad .
Hoc enim manifestum ex similitudine triangulorum et ,
seu et
[Translation: I say, second, that as to , or as to , so is to .
For this is clear from the similarity of triangles and , or and .
Hoc enim manifestum ex similitudine triangulorum et ,
seu et
[Translation: I say, second, that as to , or as to , so is to .
For this is clear from the similarity of triangles and , or and .
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