This compendium of hyperbolic trigonometry was ﬁrst published as a chapter in Merriman and Woodward’s Higher Mathematics. There is reason to believe that it supplies a need, being adapted to two or three diﬀerent types of readers. College students who have had elementary courses in trigonometry, analytic geometry, and diﬀerential and integral calculus, and who wish to know something of the hyperbolic trigonometry on account of its important and historic relations to each of those branches, will, it is hoped, ﬁnd these relations presented in a simple and comprehensive way in the ﬁrst half of the work. Readers who have some interest in imaginaries are then introduced to the more general trigonometry of the complex plane, where the circular and hyperbolic functions merge into one class of transcendents, the singly periodic functions, having either a real or a pure imaginary period. For those who also wish to view the subject in some of its practical relations, numerous applications have been selected so as to illustrate the various parts of the theory, and to show its use to the physicist and engineer, appropriate numerical tables being supplied for these purposes.
With all these things in mind, much thought has been given to the mode of approaching the subject, and to the presentation of fundamental notions, and it is hoped that some improvements are discernible. For instance, it has been customary to deﬁne the hyperbolic functions in relation to a sector of the rectangular hyperbola, and to take the initial radius of the sector coincident with the principal radius of the curve; in the present work, these and similar restrictions are discarded in the interest of analogy and generality, with a gain in symmetry and simplicity, and the functions are deﬁned as certain characteristic ratios belonging to any sector of any hyperbola. Such deﬁnitions, in connection with the fruitful notion of correspondence of points on conics, lead to simple and general proofs of the addition-theorems, from which easily follow the conversion-formulas, the derivatives, the Maclaurin expansions, and the exponential expressions. The proofs are so arranged as to apply equally to the circular functions, regarded as the characteristic ratios belonging to any elliptic sector. For those, however, who may wish to start with the exponential expressions as the deﬁnitions of the hyperbolic functions, the appropriate order of procedure is indicated on page §, and a direct mode of bringing such exponential deﬁnitions into geometrical relation with the hyperbolic sector is shown in the Appendix.