659330
Sit data linea .
Dividatur in duas partes
secundum datam rationem
in puncto, .
Sit maior, minor.
producatur, , ad , ita
ut
[Translation: Let there be a given line . Divide it into two parts according to the given ratio at the point . Let be greater, smaller. Extend to so that:
Dividatur in duas partes
secundum datam rationem
in puncto, .
Sit maior, minor.
producatur, , ad , ita
ut
[Translation: Let there be a given line . Divide it into two parts according to the given ratio at the point . Let be greater, smaller. Extend to so that:
Centro, , intervallo
fiat periferia .
Dico quod:
ubicunque signatur in periferia
punctum, ,
[Translation: With centre and radius , draw the circumference .
I say that, wherever the point is placed on the circumference:
fiat periferia .
Dico quod:
ubicunque signatur in periferia
punctum, ,
[Translation: With centre and radius , draw the circumference .
I say that, wherever the point is placed on the circumference:
Nam per construcitonem
[…]
In duobus triangulis et
angulus ad, , est
[Translation: For by the construction
In the two triangles and the agnle at is shared.
[…]
In duobus triangulis et
angulus ad, , est
[Translation: For by the construction
In the two triangles and the agnle at is shared.
Nullum punctum extra periferiam vel intra,
habet eandem conditionem.
Sit punctum , extra periferiam.
agatur , et .
secabit periferiam in puncto .
agatur ,
[Translation: No point outside the circumference or inside has the same conditions.
Let the point be outside the circumference.
Construct and .
will cut the circumference at the point .
Construct and .
habet eandem conditionem.
Sit punctum , extra periferiam.
agatur , et .
secabit periferiam in puncto .
agatur ,
[Translation: No point outside the circumference or inside has the same conditions.
Let the point be outside the circumference.
Construct and .
will cut the circumference at the point .
Construct and .
per supra demonstrata:
[…]
Dico quod: , et non sunt in eadem
[Translation: By what was demosntrated above:
I say that and are not in the same ratio.
[…]
Dico quod: , et non sunt in eadem
[Translation: By what was demosntrated above:
I say that and are not in the same ratio.
Hoc est unum latus trianglui maius est
duobus reliquis quod est
[Translation: That is, one side of the triangle is greater than the two others, which is ]
duobus reliquis quod est
[Translation: That is, one side of the triangle is greater than the two others, which is ]
Simili ratione probabitur quod punctum,
intra circulum idem feret absurdum.
videlicet quod sit maior quam .
Quare nullum punctum
extra vel intra &
[Translation: By similar reasoning it may be proved that the point inside the circle leades to the same absurdity, namely that is greater than .
Whence no point outside or inside ]
intra circulum idem feret absurdum.
videlicet quod sit maior quam .
Quare nullum punctum
extra vel intra &
[Translation: By similar reasoning it may be proved that the point inside the circle leades to the same absurdity, namely that is greater than .
Whence no point outside or inside ]