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[Commentary:
This is the first of a set of twenty-one pages exploring propositions from Supplementum geometriæ
(Viète . In the Effectionum geometricarum
(Viète , which preceded it, Viète gave Euclidean constructions to demonstrate relationships between proportional lines, and showed that they corresponded to quadratic, or sometimes quartic, equations. This, however, gave him only a limited range of constructions, or equations, insufficient for the requirements of the analytic art by which he meant to leave no problem unsolved (nulla non problema solvere, In artem analyticen isagoge,
(Viète , final sentence). The Supplementum geometricarum was intended to remedy this shortcoming (defecta Geometriæ) by offering constructions that went beyond the limitations of ruler and compass. Thus the first statement of the book is:
A quovis puncto ad duas quavis lineas rectam ducere, interceptam ab iis præfinito possibili quocumque intersegmento.
To draw a straight from any point to any two straight lines, the intercept between them being any possible predefined distance.
Such constructions are sometimes knownn as neusis constructions.
On this page, Harriot examines Propositions 3 and (Viète 1593c, Props 3, .
Propositio III.
Si duae lineae rectae a puncto extra circulum eductae ipsum secent, pars autem exterior primae fit proportionalis inter partem exteriorem secundae & partem interiorem ejusdem: erit quoque pars exterior secundae proportionalis inter partem exteriorem primae & partem interiorem
If two straight lines drawn from a point outside a circle cut it in such a way that the external part of the first is a proportional between the external and internal parts of the second, the external part of the second will be a proportional between the external and internal parts of the
Propositio IV.
Si duae lineae rectae a puncto extra circulum eductae ipsum secent quod autem fit sub partibus exterioribus eductarum, aequale fit ei quod fit sub intertioribus: exteriores partes permutatim sumptae, erunt continue proportionales inter partes
If two straight lines drawn from a point outside a circle cut it, and moreover the product of the external parts is equal to that of the internal parts, the external parts taken in turn will be continued proportionals between the internal
There is also a reference to Euclid, .
If a whole be to a whole as a part subtracted be to a part subtracted, then the remainder is also to the remainder as the whole is to the whole ]
A quovis puncto ad duas quavis lineas rectam ducere, interceptam ab iis præfinito possibili quocumque intersegmento.
To draw a straight from any point to any two straight lines, the intercept between them being any possible predefined distance.
Such constructions are sometimes knownn as neusis constructions.
On this page, Harriot examines Propositions 3 and (Viète 1593c, Props 3, .
Propositio III.
Si duae lineae rectae a puncto extra circulum eductae ipsum secent, pars autem exterior primae fit proportionalis inter partem exteriorem secundae & partem interiorem ejusdem: erit quoque pars exterior secundae proportionalis inter partem exteriorem primae & partem interiorem
If two straight lines drawn from a point outside a circle cut it in such a way that the external part of the first is a proportional between the external and internal parts of the second, the external part of the second will be a proportional between the external and internal parts of the
Propositio IV.
Si duae lineae rectae a puncto extra circulum eductae ipsum secent quod autem fit sub partibus exterioribus eductarum, aequale fit ei quod fit sub intertioribus: exteriores partes permutatim sumptae, erunt continue proportionales inter partes
If two straight lines drawn from a point outside a circle cut it, and moreover the product of the external parts is equal to that of the internal parts, the external parts taken in turn will be continued proportionals between the internal
There is also a reference to Euclid, .
If a whole be to a whole as a part subtracted be to a part subtracted, then the remainder is also to the remainder as the whole is to the whole ]
In prop: 3am
[Translation: From the 3rd proposition of the ]
[Translation: From the 3rd proposition of the ]
sunt partes ablatæ a et et in eadem ratione
partes reliquæ sunt quæ per 19,5 sunt etiam in eadem ratione.
Ergo ut supra.
Ergo si sit media proportionalis inter et
erit inter et
[Translation: the parts are taken from and in the same ratio.
the remaining parts are , , which by [Euclid's Elements] V.19 are also in the same ratio.
Therefore as above.
Therefore if is the mean proportional between and , then will be between and .
partes reliquæ sunt quæ per 19,5 sunt etiam in eadem ratione.
Ergo ut supra.
Ergo si sit media proportionalis inter et
erit inter et
[Translation: the parts are taken from and in the same ratio.
the remaining parts are , , which by [Euclid's Elements] V.19 are also in the same ratio.
Therefore as above.
Therefore if is the mean proportional between and , then will be between and .
In
[Translation: From the ]
[Translation: From the ]
continue proportionales ut supra
[…]
Et per synæresin
Ex ratione constructionis
[…]
Ergo . . . . continuæ
[Translation: continued proportionals as above
And by synæresis
By reason of the construction
Therefore , , , are continued proportionals.
[…]
Et per synæresin
Ex ratione constructionis
[…]
Ergo . . . . continuæ
[Translation: continued proportionals as above
And by synæresis
By reason of the construction
Therefore , , , are continued proportionals.
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