375188
[Translation: turn ]
[Commentary:
The problem pursued in this and the seven following folios is 'on tangencies',
as set out in Pappus, Mathematicae collectiones, Book 7. In Commandino's edition of 1588
(Pappus , the problem is stated on page 159–159v.
Punctis, & rectis lineis, & circulis quibuscumque duobus datis circulum describere magnitudine datum, qui per datum punctum, vel data puncta transeat; contingat autem vnamquamque datarum
Given any two points, lines, or circles in position, to draw a circle of a given size, which passes through any given point or points and touches any of the given lines.]
Punctis, & rectis lineis, & circulis quibuscumque duobus datis circulum describere magnitudine datum, qui per datum punctum, vel data puncta transeat; contingat autem vnamquamque datarum
Given any two points, lines, or circles in position, to draw a circle of a given size, which passes through any given point or points and touches any of the given lines.]
1.) Ad locum de tactibus (.
[Translation: On the place of ]
[Translation: On the place of ]
De parabola.
[Translation: On the parabola. ]
[Translation: On the parabola. ]
Sit curvum parabola.
cuius axis , producta.
recta, .
fiat, .
et sit quælibet linea ordinatim
applicata .
agatur .
Dico quod:
[Translation: Let the curve be a parabola, with axis extended, a straight line.
Make and let be any ordinate. Construct . I say that .
cuius axis , producta.
recta, .
fiat, .
et sit quælibet linea ordinatim
applicata .
agatur .
Dico quod:
[Translation: Let the curve be a parabola, with axis extended, a straight line.
Make and let be any ordinate. Construct . I say that .
Corollarium. 1m.
Hinc patet quod curva parabolæ
, est locus centrorum omnium
circulorum qui possunt contingere
lineam , et punctum .
Huiusmodi punctum in variis
scriptis nostris, usurpandimus
appellate, parabolæ
[Translation: Corollary 1.
Here it is clear that the parabolic curve is the locus of the centres of all the circles that can touch the line and the point .
In this way, the point , in my various writings, I may call the centroid of the parabola.
Hinc patet quod curva parabolæ
, est locus centrorum omnium
circulorum qui possunt contingere
lineam , et punctum .
Huiusmodi punctum in variis
scriptis nostris, usurpandimus
appellate, parabolæ
[Translation: Corollary 1.
Here it is clear that the parabolic curve is the locus of the centres of all the circles that can touch the line and the point .
In this way, the point , in my various writings, I may call the centroid of the parabola.
Corollarium. 2m.
Patet etiam pro descriptione
parabolæ: si dividatur in partes
æquales, et continuentur in producta
erit prima centrali æqualis numero
. 2a centralis .
3a centralis . &c.
ut si dividatur in 3 partes
æquales, erit ad prima centralis
3 + 1, hoc est 4. , 5. , 6. , 7
, 8. &
[Translation: Corollary 2.
The description of the parabola is also clear. If is divided into equal parts and they are continued in extension, the first from the centre will be equal in numbers to , the second from the centre to , the third to , etc. so if is divided into 3 equal parts, the first from the centre will be 3 + 1, that is, 4, , , , , etc.
Patet etiam pro descriptione
parabolæ: si dividatur in partes
æquales, et continuentur in producta
erit prima centrali æqualis numero
. 2a centralis .
3a centralis . &c.
ut si dividatur in 3 partes
æquales, erit ad prima centralis
3 + 1, hoc est 4. , 5. , 6. , 7
, 8. &
[Translation: Corollary 2.
The description of the parabola is also clear. If is divided into equal parts and they are continued in extension, the first from the centre will be equal in numbers to , the second from the centre to , the third to , etc. so if is divided into 3 equal parts, the first from the centre will be 3 + 1, that is, 4, , , , , etc.
Corollarium. 3m.
Sit angulus rectus et
et sit parallela . fiat .
centro , et intervallo,
agatur periferia, et secabit
in puncto
agatur perpendic: .
Dico quod:
Et punctum est in
parabola. Ut patet
ex
[Translation: Corollary 3.
Let be a right angle and , and let be parallel to . Make .
With centre and radius construct a circumference, and it willl cut at the point . Construct perpendicular to .
I say that . And the point is on the parabola. As is clear from the suppositions.
Sit angulus rectus et
et sit parallela . fiat .
centro , et intervallo,
agatur periferia, et secabit
in puncto
agatur perpendic: .
Dico quod:
Et punctum est in
parabola. Ut patet
ex
[Translation: Corollary 3.
Let be a right angle and , and let be parallel to . Make .
With centre and radius construct a circumference, and it willl cut at the point . Construct perpendicular to .
I say that . And the point is on the parabola. As is clear from the suppositions.
[Translation: turn ]