697349
[Commentary:
On this page Harriot investigates Proposition 18 from Supplementum geometriæ
(Viète 1593c, Prop .
Proposition XVIII.
Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris
If two triangles are each isosceles, equal to one another in their legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg.
This page refers to several previous propositions from the Supplementum, namely Proposition 12 and 14b (Add MS 6784 f. ), Proposition 16 (add MS 6784 f. ) and Proposition 17 (add MS 6784 f. ). ]
Proposition XVIII.
Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris
If two triangles are each isosceles, equal to one another in their legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg.
This page refers to several previous propositions from the Supplementum, namely Proposition 12 and 14b (Add MS 6784 f. ), Proposition 16 (add MS 6784 f. ) and Proposition 17 (add MS 6784 f. ). ]
prop. 18.
[Translation: Proposition 18 from the ]
[Translation: Proposition 18 from the ]
Si duo triangula fuerint aequicrura singula, et ipsa alterum alteri cruribus aequalia; angulus
autem qui est ad basin secundi sit triplus anguli qui est ad basin primi. Triplum solidum
sub quadrato cruris communis, et dimidia base primi multata continuatave longitudine
ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem
dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi et ejusdem
cruris
[Translation: If two triangles are each isosceles, equal to one another in their legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg.
autem qui est ad basin secundi sit triplus anguli qui est ad basin primi. Triplum solidum
sub quadrato cruris communis, et dimidia base primi multata continuatave longitudine
ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem
dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi et ejusdem
cruris
[Translation: If two triangles are each isosceles, equal to one another in their legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg.
Sit triangulum primum , secundum
. quorum crura et anguli sint
ut exigit propositio. Et sit dupla
. Tum quadratum erit triplum quadrati
Â
[Translation: Let the first triangle be and the second , whose sides and angles are as specified in the proposition. And let be twice . Then the square of is three times the square of .
. quorum crura et anguli sint
ut exigit propositio. Et sit dupla
. Tum quadratum erit triplum quadrati
Â
[Translation: Let the first triangle be and the second , whose sides and angles are as specified in the proposition. And let be twice . Then the square of is three times the square of .
Nam:
per 15,p […] Hoc est, in notis proportionalium quas notum 12,p
1o. Ducantur omnia per
[…]
Hoc est in notis
[Translation: For by Proposition 15 that is, in the notation for proportionals noted in Proposition 12,
1. Multiply everything by .
That is, in the notation of Proposition ]
per 15,p […] Hoc est, in notis proportionalium quas notum 12,p
1o. Ducantur omnia per
[…]
Hoc est in notis
[Translation: For by Proposition 15 that is, in the notation for proportionals noted in Proposition 12,
1. Multiply everything by .
That is, in the notation of Proposition ]
2o. Ducantur omnia per
[…]
Hoc est in notis
[Translation: 2. Multiply everything by .
That is, in the notation of Proposition ]
[…]
Hoc est in notis
[Translation: 2. Multiply everything by .
That is, in the notation of Proposition ]
Deinde per 16.p
Hoc est in notis 12,p.
Sed: per consect: 14.p
Ergo patet
[Translation: Thence by Proposition 16,
That is, in the notation of Proposition 12
But by the consequence of Proposition 14,
Thus the propostion is ]
Hoc est in notis 12,p.
Sed: per consect: 14.p
Ergo patet
[Translation: Thence by Proposition 16,
That is, in the notation of Proposition 12
But by the consequence of Proposition 14,
Thus the propostion is ]
Cum 16a et 17a prop. basis notabatur ()
ideo eius partes
Scilicet et alijs vocalibus notandæ sunt. pro nota ()
et pro , (). et servent easdem notas quas ibi
habuerunt. Videlicet , () et , ().
Propositum igitur simplicibus notis ita
[Translation: Since in Propositions 16 adn 17, the base is denoted by , therefore its parts, namely and may be denoted by other names; for put the letter and for the letter . For and use the same notation as they had there, namely and .
In simple notation the proposition may therefore be ]
Scilicet et alijs vocalibus notandæ sunt. pro nota ()
et pro , (). et servent easdem notas quas ibi
habuerunt. Videlicet , () et , ().
Propositum igitur simplicibus notis ita
[Translation: Since in Propositions 16 adn 17, the base is denoted by , therefore its parts, namely and may be denoted by other names; for put the letter and for the letter . For and use the same notation as they had there, namely and .
In simple notation the proposition may therefore be ]
igitur:
Quando æquatio est sub ista
forma:
erit duplex vel. . vel.
[Translation: When the equation is in this form, is twofold, either or .
Quando æquatio est sub ista
forma:
erit duplex vel. . vel.
[Translation: When the equation is in this form, is twofold, either or .
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