4724
[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, Book 7. In Commandino's edition of 1588, the problem is stated on page 158v
(Pappus .
Per datum punctum rectam lineam ducere secantem a duabus rectis lineis positione datis ad data in ipsis puncta lineas, quæ proportionem habeant eandem datæ proportioni.
Through a given point, draw a line cutting off line segments from two lines given in position, to points given in them, which have a ratio the same as a given ratio.
Using Harriot's lettering, the problem may be stated as follows. Given the lines , , the point , and the ratio , construct the line so that . There are many variations of the problem according to the relative positions of , , and , which Harriot explored in this and other folios.
This example, in Add MS f. f. , demonstrates the use of Viète's concepts of zetetic, poristic, analysis, and synthesis. ]
Per datum punctum rectam lineam ducere secantem a duabus rectis lineis positione datis ad data in ipsis puncta lineas, quæ proportionem habeant eandem datæ proportioni.
Through a given point, draw a line cutting off line segments from two lines given in position, to points given in them, which have a ratio the same as a given ratio.
Using Harriot's lettering, the problem may be stated as follows. Given the lines , , the point , and the ratio , construct the line so that . There are many variations of the problem according to the relative positions of , , and , which Harriot explored in this and other folios.
This example, in Add MS f. f. , demonstrates the use of Viète's concepts of zetetic, poristic, analysis, and synthesis. ]
b.1) De sectione
[Translation: On the cutting off of a ]
[Translation: On the cutting off of a ]
Data:
. . lineaæ infinitæ
. . termini.
. punctum.
. .
[Translation: Given:
, , infinite lines.
, , endpoints.
, a point.
, a ratio.
. . lineaæ infinitæ
. . termini.
. punctum.
. .
[Translation: Given:
, , infinite lines.
, , endpoints.
, a point.
, a ratio.
Quæsitum:
Ducere lineam
Ut fiat: .
Hoc fit si ducetur , vel , vel .
Quæritur igitur una, nempe
[Translation: Sought:
To draw the ine so that .
This is done if there are drawn , or , or .
Therefore, one is sought, namely .
Ducere lineam
Ut fiat: .
Hoc fit si ducetur , vel , vel .
Quæritur igitur una, nempe
[Translation: Sought:
To draw the ine so that .
This is done if there are drawn , or , or .
Therefore, one is sought, namely .
Interpretatio
[Translation: Interpretation of the ]
[Translation: Interpretation of the ]
Zetetice 1a
Sit iam factum.
Et agatur .
fiat: parall: .
. parall. .
agatur recta quæ secabit in .
[Translation: Zetetic 1.
Let it be already done, and the line constructed.
Then make parallel to and parallel to .
The line is constructed which cuts in .
]
Sit iam factum.
Et agatur .
fiat: parall: .
. parall. .
agatur recta quæ secabit in .
[Translation: Zetetic 1.
Let it be already done, and the line constructed.
Then make parallel to and parallel to .
The line is constructed which cuts in .
]
Ergo ad determinatum sectione
eductam est. datur igitur
[Translation: Therefore the determination of the section is brought out. Therefore is given.
eductam est. datur igitur
[Translation: Therefore the determination of the section is brought out. Therefore is given.
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