Article 40
Appendix.

40.1 Historical and Bibliographical.

What is probably the earliest suggestion of the analogy between the sector of the circle and that of the hyperbola is found in Newton’s Principia (Bk. 2, prop. 8 et seq.) in connection with the solution of a dynamical problem. On the analytical side, the first hint of the modified sine and cosine is seen in Roger Cotes’ Harmonica Mensurarum (1722), where he suggests the possibility of modifying the expression for the area of the prolate spheroid so as to give that of the oblate one, by a certain use of the operator $\sqrt{-1}$. The actual inventor of the hyperbolic trigonometry was Vincenzo Riccati, S.J. (Opuscula ad res Phys. et Math. pertinens, Bononiæ, 1757). He adopted the notation $Sh.\phi$, $Ch.\phi$ for the hyperbolic functions, and $Sc.\phi$, $Cc.\phi$ for the circular ones. He proved the addition theorem geometrically and derived a construction for the solution of a cubic equation. Soon after, Daviet de Foncenex showed how to interchange circular and hyperbolic functions by the use of $\sqrt{-1}$, and gave the analogue of De Moivre’s theorem, the work resting more on analogy, however, than on clear definition (Reflex. sur les quant. imag., Miscel. Turin Soc., Tom. 1). Johann Heinrich Lambert systematized the subject, and gave the serial developments and the exponential expressions. He adopted the notation $sinhu$, etc., and introduced the transcendent angle, now called the gudermanian, using it in computation and in the construction of tables (l. c. page 30). The important place occupied by Gudermann in the history of the subject is indicated on page §.

The analogy of the circular and hyperbolic trigonometry naturally played a considerable part in the controversy regarding the doctrine of imaginaries, which occupied so much attention in the eighteenth century, and which gave birth to the modern theory of functions of the complex variable. In the growth of the general complex theory, the importance of the “singly periodic functions” became still clearer, and was gradually developed by such writers as Ferroni (Magnit. expon. log. et trig., Florence, 1782); Dirksen (Organon der tran. Anal., Berlin, 1845); Schellbach (Die einfach. period. funkt., Crelle, 1854); Ohm (Versuch eines volk. conseq. Syst. der Math., Nürnberg, 1855); Hoüel (Theor. des quant. complex, Paris, 1870). Many other writers have helped in systematizing and tabulating these functions, and in adapting them to a variety of applications. The following works may be especially mentioned: Gronau (Tafeln, 1862, Theor. und Anwend., 1865); Forti (Tavoli e teoria, 1870); Laisant (Essai, 1874); Gunther (Die Lehre ..., 1881). The last-named work contains a very full history and bibliography with numerous applications. Professor A. G. Greenhill, in various places in his writings, has shown the importance of both the direct and inverse hyperbolic functions, and has done much to popularize their use (see Diff. and Int. Calc., 1891). The following articles on fundamental conceptions should be noticed: Macfarlane, On the definitions of the trigonometric functions (Papers on Space Analysis, N. Y., 1894); Haskell, On the introduction of the notion of hyperbolic functions (Bull. N. Y. M. Soc., 1895). Attention has been called in Arts. 30 and 37 to the work of Arthur E. Kennelly in applying the hyperbolic complex theory to the plane vectors which present themselves in the theory of alternating currents; and his chart has been described on page § as a useful substitute for a numerical complex table (Proc. A. I. E. E., 1895). It may be worth mentioning in this connection that the present writer’s complex table in Art. 39 is believed to be the only one of its kind for any function of the general argument $x+iy$.

40.2 Exponential Expressions as Definitions.

For those who wish to start with the exponential expressions as the definitions of $sinhu$ and $coshu$, as indicated on page §, it is here proposed to show how these definitions can be easily brought into direct geometrical relation with the hyperbolic sector in the form $\frac{x}{a}=cosh\frac{S}{K}$, $\frac{y}{b}=sinh\frac{S}{K}$, by making use of the identity ${cosh}^{2}u-{sinh}^{2}u=1$, and the differential relations $dcoshu=sinhudu$, $dsinhu=coshudu$, which are themselves immediate consequences of those exponential definitions. Let $OA$, the initial radius of the hyperbolic sector, be taken as axis of $x$, and its conjugate radius $OB$ as axis of $y$; let $OA=a$, $OB=b$, angle $AOB=\omega$, and area of triangle $AOB=K$, then $K=\frac{1}{2}absin\omega$. Let the coordinates of a point $P$ on the hyperbola be $x$ and $y$, then $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Comparison of this equation with the identity ${cosh}^{2}u-{sinh}^{2}u=1$ permits the two assumptions $\frac{x}{a}=coshu$ and $\frac{y}{b}=sinhu$, wherein $u$ is a single auxiliary variable; and it now remains to give a geometrical interpretation to $u$, and to prove that $u=\frac{S}{K}$, wherein $S$ is the area of the sector $OAP$. Let the coordinates of a second point $Q$ be $x+\Delta x$ and $y+\Delta y$, then the area of the triangle $POQ$ is, by analytic geometry, . Now the sector $POQ$ bears to the triangle $POQ$ a ratio whose limit is unity, hence the differential of the sector $S$ may be written . By integration $S=Ku$, hence $u=\frac{S}{K}$, the sectorial measure (p. §); this establishes the fundamental geometrical relations $\frac{x}{a}=cosh\frac{S}{K},\frac{y}{b}=sinh\frac{S}{K}$.