7940
[Commentary:
The problem pursued in this and many other folios in Add MS 6784 is 'the cutting-off of a ratio',
as set out in Pappus, Mathematicae collectiones
(Pappus , Book 7. For a statement of the problem see Add MS 6784
f. . ]
De resectione
[Translation: The cutting off of a ]
[Translation: The cutting off of a ]
Sint dati linea AB AD C
oportet resecare et in ratione data
ad .
Sit iam factum, ut:
tum in datam
[Translation: Let there be given lines and and a point .
It is required to cut off and in the fiven ratio to . Let it be already done, so that:
then in the given ]
oportet resecare et in ratione data
ad .
Sit iam factum, ut:
tum in datam
[Translation: Let there be given lines and and a point .
It is required to cut off and in the fiven ratio to . Let it be already done, so that:
then in the given ]
Synthesis.
A puncto agatur parallela
et fiat æqualis . et parallela
æqualis . et per puncta et ducatur
usque ad . quæ secabit in .
Demonstratio
[Translation: Synthesis
From the point there is constructed parallel to , and make equal to , and parallel to equal to . And through the points and there is drawn as far as , which will cut in .
The demonstration is ]
A puncto agatur parallela
et fiat æqualis . et parallela
æqualis . et per puncta et ducatur
usque ad . quæ secabit in .
Demonstratio
[Translation: Synthesis
From the point there is constructed parallel to , and make equal to , and parallel to equal to . And through the points and there is drawn as far as , which will cut in .
The demonstration is ]
Quatuor possunt esse resectiones ab uno
puncto dato et duabus solummodo lineis
datis, secundum quantitatem rationes datæ.
Sed numquam punctibus duabus, nisi in uno casu
rationis æqualitatis, scilicet, cum sit æqualis
[Translation: There are four possible sections from one given point and just two given lines, according to the size of the given ratio.
But never from two points, unless in the case of equal ratio, namely, if is equal to .
puncto dato et duabus solummodo lineis
datis, secundum quantitatem rationes datæ.
Sed numquam punctibus duabus, nisi in uno casu
rationis æqualitatis, scilicet, cum sit æqualis
[Translation: There are four possible sections from one given point and just two given lines, according to the size of the given ratio.
But never from two points, unless in the case of equal ratio, namely, if is equal to .
Et sic de
cæteris.
Nullum vide in
hac propositione
difficultatem.
De Infinitis
speculatio ut
altera pagina
[Translation: And so for the rest.
There is nothing of difficulty in this proposition.
On consideration of infinity as in the other page, ]
[Commentary: The other page mentioned here is the reverse of this one, Add MS f. .
cæteris.
Nullum vide in
hac propositione
difficultatem.
De Infinitis
speculatio ut
altera pagina
[Translation: And so for the rest.
There is nothing of difficulty in this proposition.
On consideration of infinity as in the other page, ]
[Commentary: The other page mentioned here is the reverse of this one, Add MS f. .