100 REPORT — 1856. 



Tte foregoing theory decides a controversy long Carried on between 

 Leibnitz and J. Bernoulli on the subject of the logarithms of negative num- 

 bers. Leibnitz insisted they were imaginary, while Bernoulli argued they 

 were real, and the same as the logarithms of equal positive numbers. Euler 

 espoused the side of the former, while D'Alembert coincided with the views 

 of Bernoulli. Indeed, if we derive the theory of logarithms from the pro- 

 perties of the hyperbola (as geometers always have done), it will not be easy 

 satisfactorily to answer the argument of Bernoulli — that as an hyperbolic 

 area represents the logarithm of a positive number, denoted by the positive 

 abscissa+.r, so a negative number, according to conventional usage, being 

 represented by the negative abscissa— .r, the corresponding hyperbolic area 

 should denote its logarithm also. And this is the more remarkable, because 

 by Van Huraet's method the quadrature of the hyperbola itself depends on 

 the rectification of the parabola, as shown in XXXV. All this obscurity is 

 cleared up by the theory developed in the text, which completely establishes 

 the correctness of the views of Leibnitz and Euler. 



It is somewhat remarkable in the history of mathematical science, that 

 although the arithmetical properties of logarithms have been familiarly known 

 to every geometer since the time of Napier, their inventor, or rather dis- 

 coverer, no mathematician has hitherto divined their true geometrical origin. 

 And this is the more singular, because the properties of the logarithms of 

 imaginary numbers are intimately connected with those of the circle. No 

 satisfactory reason has been shown why this should be so. The logarithmic 

 curve which has been devised to represent one well-known property of loga- 

 rithms, is a transcendental curve, and has no connexion with the circle. 

 Neither has any attempt been made to show how the Napierian base e, an 

 abstract isolated incommensurable number, may be connected with our 

 known geometrical knowledge. Had the circle never been made a geome- 

 trical conception, the same obscurity might probably have hung over the 

 signification of tt, which has hitherto concealed from us the real interpreta- 

 tion of the Napierian base e. 



This affords another instance, were any needed, to show how thin the veil 

 may be which is suflicient to conceal from us the knowledge of apparently 

 the simplest truths, the clue to whose discovery is even already in our hands. 

 The geometrical origin of logarithms and the trigonometry of the parabola 

 ought, in logical sequence, to have been developed by Napier, or by one of 

 his immediate successors. They had many indications to direct them aright 

 in their investigations. So true it is that men, in the contemplation of remote 

 truths, often overlook those that are lying before their feet ! 



I have shown in this memoir that the theory of logarithms is a result of the 

 solution of the geometrical problem to find and compare the lengths of arcs 

 of a parabola, just as plane trigonometry is nothing but the development of 

 the same problem for the circle. I have shown, too, elsewhere*, that elliptic 

 integrals of the three orders do in all cases representthe lengths of curves which 

 are the symmetrical intersections of the surfaces of a sphere or a paraboloid 

 by ruled surfaces. These functions divide themselves into two distinct groups, 

 representing spherical and paraboloidal curves, and by no rational trans- 

 formation can we pass from the one group to the other. The transition is 

 always made by the help of imaginary transformations, as when we pass from 

 the real logaritiims of the parabola to the imaginary logarithms of the circle. 

 When we take plane sections of those surfaces, that is to say, a circle and a 



* " Researches on the Geometrical Properties of Elliptic Integrals," Philosophical 

 Transactions for 1852, p. 316. 



