TRACT 21.

LOGARITHMS.

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the vertex of the hyperbola in this case also being still l,thesame as the side of the square in the right-angled hyperbola.But the areas of the square and rhombus, and consequentlythe logarithms of any one and the same number or ratio, dif-fering according to the sine of the angle of the asymptotes.And the area of the square or rhombus, or any inscribed pa-rallelogram, is also the same thing as what was by Cotescalled the modulus of the system of logarithms ; which mo-dulus will therefore be expressed by the numerical measureof the sine of the angle formed by the asymptotes, to theradius 1 ; as that is the same with the number expressing thearea of the said square or rhombus, the side being 1: whichis another definition of the modulus to be added to those weremarked above, in treating of the logarithmic curve. Andthe evident reason of this is, that in the beginning of thegeneration of these areas, from the vertex of the hyperbola,the nascent increment of the abscisse drawn into the altitude1, is to the increment of the area, as radius is to the sine ofthe angle of the ordinate and abscisse, or of the asymptotes ;and at the beginning of the logarithms, the nascent incrementof the natural numbers is to the increment of the logarithms,as 1 is to the modulus of the system. Hence we easily dis-cover that the angle formed by the asymptotes of the hyper-bola exhibiting Briggs’s system of logarithms, will be 25 deg.44 minutes, 25 l seconds, this being the angle whose sine is0 , 4342944819 &c, the modulus of this system.

Or indeed any one hyperbola will express all possible sys-tems of logarithms whatever, namely, if the square or rhom-bus inscribed at the vertex, or, which is the same thing, anyparallelogram inscribed between the asymptotes and the curveat any other point, be expounded by the modulus of thesystem ; or, which is the same, by expounding the area, in-tercepted between two ordinates which are to each other inthe ratio of 10 to 1, by the logarithm of that ratio in theproposed system.

As to the first remarks on the analogy between logarithmsand the hyperbolic spaces; it having been shown by Gregory

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