467234
De
[Translation: On ]
[Translation: On ]
A puncto extra circulum
ducere lineam secantem:
ut partes, exterior et
interior, sint in ratione
[Translation: From a point outside a circle draw a line cutting it, so that the external and internal parts are in a given ]
ducere lineam secantem:
ut partes, exterior et
interior, sint in ratione
[Translation: From a point outside a circle draw a line cutting it, so that the external and internal parts are in a given ]
Sit () punctum datum, extra
circulum cuius centrum .
et per centrum agatur recta
infinita quæ secabit
periferium circuli dati in punctis
, . et fiat æqualis
[Translation: Let be the given point outside the circle whose centre is , and through the centre there is constructed an infinite line which cuts the circumference of the given circle at the points and ; and make equal to .
circulum cuius centrum .
et per centrum agatur recta
infinita quæ secabit
periferium circuli dati in punctis
, . et fiat æqualis
[Translation: Let be the given point outside the circle whose centre is , and through the centre there is constructed an infinite line which cuts the circumference of the given circle at the points and ; and make equal to .
Sit ratio data ut ad .
oportet a puncto ducere
rectum secantem, nempe
ut pars exterior ad partem
interiorem , sit ut ad
[Translation: Let the given ratio be as to .
It is required to draw a cutting line from the point , namely so that the external part to the internal part is as to .
oportet a puncto ducere
rectum secantem, nempe
ut pars exterior ad partem
interiorem , sit ut ad
[Translation: Let the given ratio be as to .
It is required to draw a cutting line from the point , namely so that the external part to the internal part is as to .
producatur , et fiat æqualis .
Deinde secetur in puncto ut
ad sit ut ad . et fiat æqualis . Inde fit
ut partes , , ; sint in ratione , ,
[Translation: Extend and make equal to .
Then is cut at point so that to is as to , and make equal to . Thence the parts , , will be in the ratio , , .
Deinde secetur in puncto ut
ad sit ut ad . et fiat æqualis . Inde fit
ut partes , , ; sint in ratione , ,
[Translation: Extend and make equal to .
Then is cut at point so that to is as to , and make equal to . Thence the parts , , will be in the ratio , , .
Porro super fiat circulus super diametrum . quæ
secabit circulum datum in puncto . Et agatur recta , quæ
secabitur a circulo dato in puncto .
Dico quod ratio ad , est ut ad
[Translation: Then make the circle with diameter , whcih will cut the given circle at the point . And construct the line , which will be cut by the given circle at the point .
I say that the ratio to is as to .
secabit circulum datum in puncto . Et agatur recta , quæ
secabitur a circulo dato in puncto .
Dico quod ratio ad , est ut ad
[Translation: Then make the circle with diameter , whcih will cut the given circle at the point . And construct the line , which will be cut by the given circle at the point .
I say that the ratio to is as to .
Centro intervallo , fiat circulus . Continuetur ad .
et connectantur , , puncta. [???] , , et , . Quoniam anguli
ad et sunt recti (sunt enim in semicirculos.) et sunt
parallelæ. Et inde triangula et sunt similia. Quare ut
ad , sic ad etiam ad . Et quoniamm et sunt æqualis
per constructionem, ob parallelismum circulorum , et , erunt
et etiam æquales. Sed etiam æqualis est . Igitur ut ad
ita ad , etiam ad . Quare etiam alterne ut ad
ita ad . Sed ut ad ita ad supra per constructionem.
Ergo ad est ut ad . Factum est igitur quod
[Translation: With centre and radius make the circle . Extend to and connect the points , . [???] , , and , . Since the angles at and are straight lines (for they are in a semicircle) and are paralle. And hence the triangles and are similar. Whence as to , so are to and also to . And since and are equal by construction, on account of the parallelism of the circles and , the lines and are also equal. But is also equal to . Therefore as is to so is to and to . Whence also, alternatively, as is to , so is to . But as is to so is to above by construction. Therefore to is as to . Therefore it is done as was required.
et connectantur , , puncta. [???] , , et , . Quoniam anguli
ad et sunt recti (sunt enim in semicirculos.) et sunt
parallelæ. Et inde triangula et sunt similia. Quare ut
ad , sic ad etiam ad . Et quoniamm et sunt æqualis
per constructionem, ob parallelismum circulorum , et , erunt
et etiam æquales. Sed etiam æqualis est . Igitur ut ad
ita ad , etiam ad . Quare etiam alterne ut ad
ita ad . Sed ut ad ita ad supra per constructionem.
Ergo ad est ut ad . Factum est igitur quod
[Translation: With centre and radius make the circle . Extend to and connect the points , . [???] , , and , . Since the angles at and are straight lines (for they are in a semicircle) and are paralle. And hence the triangles and are similar. Whence as to , so are to and also to . And since and are equal by construction, on account of the parallelism of the circles and , the lines and are also equal. But is also equal to . Therefore as is to so is to and to . Whence also, alternatively, as is to , so is to . But as is to so is to above by construction. Therefore to is as to . Therefore it is done as was required.
Corollarium.
Patet quod angulus est rectus et in semicirculo cuius
diameter
[Translation: Corollary
It is clear that the angle is a right angle in a semicircle whose diameter is .
Patet quod angulus est rectus et in semicirculo cuius
diameter
[Translation: Corollary
It is clear that the angle is a right angle in a semicircle whose diameter is .