837419
[Commentary:
This page contains a fully worked example of Viète's method of zetesis, porismus, and exegesis.
There are several references to propositions from Euclid's Elements; in order of appearance: VI.20 , VI.12 , VI.13 , IX.19 , and VI.22 .
VI.20 Similara polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.
VI.12 To three given straight lines to find a fourth proportional.
VI.13 To two given straight lines to find a mean proportional.
V.9 Magnitudes which have the same ratio to the same are equal to one another: and magnitudes to which the same has the same ratio are equal.
If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
IX.19 Given three nubers, to investigate when it is possible to find a fourth proportional to them.
VI.22 If four straight lines be proportional, the rectilinear figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and similarly described upon them be proportional, the straight lines will themselves also be proportional. ]
There are several references to propositions from Euclid's Elements; in order of appearance: VI.20 , VI.12 , VI.13 , IX.19 , and VI.22 .
VI.20 Similara polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.
VI.12 To three given straight lines to find a fourth proportional.
VI.13 To two given straight lines to find a mean proportional.
V.9 Magnitudes which have the same ratio to the same are equal to one another: and magnitudes to which the same has the same ratio are equal.
If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
IX.19 Given three nubers, to investigate when it is possible to find a fourth proportional to them.
VI.22 If four straight lines be proportional, the rectilinear figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and similarly described upon them be proportional, the straight lines will themselves also be proportional. ]
Problema. usus habetur in aciebus
Data media trium media proportionalium, et ratione extremarum: invenire
[Translation: Given the mean of three mean proportionals and the ratio of the extremes, find the extremes. ]
[Translation: Given the mean of three mean proportionals and the ratio of the extremes, find the extremes. ]
Sit media proportionalium data . et ratio extremarum ut ad .
et disponatur ita. . .
[Translation: Let the mean of the given proportionals be and the ratio of the extremes as to , and display them thus.
et disponatur ita. . .
[Translation: Let the mean of the given proportionals be and the ratio of the extremes as to , and display them thus.
Ponatur for prima termina
Ergo per canone proportionalium.
consequenter, et de hypothesi.
vel etiam per 12 symbolum
per resolutionem
per reductione a communi
multiplicatione
vel etiam per parabolismum
Ergo Analogia per constitutione
Et sua munia implevit
[Translation: Put for the first term
Therefore by the rules of proportionals.
Consequently, and by hypothesis
Or also in symbols
By resolution
By reduction by ocmmon multiplication
Or also by parabolismus
Therefore the ratio by constitution.
And its role satisfies the zetesis. ]
Ergo per canone proportionalium.
consequenter, et de hypothesi.
vel etiam per 12 symbolum
per resolutionem
per reductione a communi
multiplicatione
vel etiam per parabolismum
Ergo Analogia per constitutione
Et sua munia implevit
[Translation: Put for the first term
Therefore by the rules of proportionals.
Consequently, and by hypothesis
Or also in symbols
By resolution
By reduction by ocmmon multiplication
Or also by parabolismus
Therefore the ratio by constitution.
And its role satisfies the zetesis. ]
Conclusio per zetetin illata, per interpretatione, ita enunciatur:
ut tertia trium propotionalium ad primam, ita quadratum secunda ad quadratum primam
cuius propositionis veritas manifeste patet per collorarium 20 a prop 6 ti elementorum.
si elementum non occurret, argumentatio recederet per vestigia zetetos, usque
ad quæsiti
[Translation: Conclusion from that zetesis, by interpretation, thus stated:
As the third of three proportinals to the first, so is the square of the second to the square of the first, the truth of which proposition is clearly manifest by the corollary to proposition 20 of the 6th book of the Elements.
If the element does not occur, the argument goes back to the vestige of the zetetic, as far as the supposed thing sought. ]
ut tertia trium propotionalium ad primam, ita quadratum secunda ad quadratum primam
cuius propositionis veritas manifeste patet per collorarium 20 a prop 6 ti elementorum.
si elementum non occurret, argumentatio recederet per vestigia zetetos, usque
ad quæsiti
[Translation: Conclusion from that zetesis, by interpretation, thus stated:
As the third of three proportinals to the first, so is the square of the second to the square of the first, the truth of which proposition is clearly manifest by the corollary to proposition 20 of the 6th book of the Elements.
If the element does not occur, the argument goes back to the vestige of the zetetic, as far as the supposed thing sought. ]
Exegesis: ad
Exegesis: ad
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