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What, however, has been so successfully effected in other departments of mathematics has not yet been attempted with regard to Trigonometry. We have some excellent works on this subject, whose value it would ill become me to depreciate. But such of them as go extensively into the business of Trigonometry and its applications are too large and expensive for general circulation; while others, being confined almost entirely to the elements, exclusive of the applications, must of necessity be restricted, both in point of circulation and utility. There is one treatise, that of Emerson, which is a most copious store-house of curious and elegant theorems: but they are so obscured by a defective notation, that the perusal of greater part of the book must, to a mathematical student, be as perplexing as the solution of a perpetual string of enigmas.
It has been my aim to steer into a middle course, between that in which is presented a mere commonplace book of principles and theorems, and that which, by leadingfar into the detail of multifarious processes and methods, precludes the study of the science, except by the sacrifice of much time and expense. In order to this I have endeavoured to be select in my materials, and have, for the most part, observed unity of method. A book of three times the size and price might have been drawn up with far less intellectual labour (for the fatigue of selecting from a fund of valuable materials, and casting the result into one mould, is not slight); but such a work would not have tended to accomplish the purpose I have in view. By adopting a small type and a full page, and confining myself chiefly to the analytical mode of investigation, I have been able to introduce, and I shall rejoice if it be thought I have treated perspicuously, a greater variety of the applications of plane and spherical Trigonometry than are to be found in . other work on the subject with which I am acquainted. In the first three chapters I have exhibited the theory of Plane Trigonometry geometrically, and have shown the application of that theory to the logarithmic solutions of the usual cases into which this portion of the subject is conveniently distributed. I have endeavoured to conduct these introductory inquiries with the utmost perspicuity; that the student, by obtaining a thorough comprehension of the principal topics of research, and by seeing a little of their utility, may enter with the greater relish upon the subsequent investigations; and by tracing the correspondence of these results with such as will afterwards appear in the analytical theory of Plane Trigonometry, may be prepared to lean with full confidence upon the analytical formulae that are in other places to be laid before him. The deductions from theory in the succeeding chapters are usually obtained by analytical processes; and their utility is shown in the logarithmic and trigonometric solution of a great number of problems, classified under the particular heads of the several applications, as specified in the table of contents. The whole, except what relates to the minute variations of the sides and angles of triangles, and the differential analogies which apply to them, may, I am persuaded, be readily comprehended by any person who is tolerably conversant with the elements of Geometry and Algebra. In the composition of the work I have freely availed myself of all such matter as was likely to answer my purpose, especially in the productions of foreign mathematicians. The plan and method are of course my own: the materials have been collected, almost of necessity, from all quarters. In addition to the acknowledgments which will occur in different parts of this little volume, it would be unjust not to say here, that the theory of Projections, the general problem in reference to Dialling, and the comprehensive table of Differential Equations for the variations of triangles, are taken, simply with such alterations as fitted them better for general usefulness, from the Chevalier Delambre's admirable treatise on Astronomy, in three volumes quarto. The transferring such curious and valuable matter from an expensive treatise in a foreign language, into a cheap volume in our own, will not I hope be regarded as performing a trifling service to the English student. I am aware that there are some persons, into whose hands this work may fall, who will not approve it as they would have done had the demonstrations been exclusively geometrical. This is in consequence of a prejudice against the analytical processes, most singularly cherished in a country where the modern analysis has received some of its most valuable improvements: a prejudice which, though it is rapidly weakening, still retains its hold upon the minds of several respectable mathematicians; and on account of which it may be expedient to assign some of the reasons that have induced me to appropriate so large a portion of the following volume as I have done to the analytical or algebraical mode of deducing properties and theorems. 1. It is more concise, and therefore allows of the introduction of a much greater quantity and variety of matter, in any proposed space, than could possibly be exhibited and demonstrated according to the geometrical method of the antients. 2. This method is more uniform than the other, as well as more general and comprehensive. In the geometrical method as it is usually conducted, however convincing and elegant, the demonstration of one property or theorem may not have the remotest analogy to that which will serve to establish the truth of another. The demonstrations of a series of propositions such as are obviously connected in the logical arrangement of a treatise, may probably have nothing common in their appearance, except that they are all geometrical; nor shall the manner of demonstrating one proposition suggest necessarily a single hint that may apply to the de
monstration of the very next. The separate chains of demonstration of the two propositions may be as distinct (if I may be pardoned so familiar an allusion) as the processes by which a sword and a needle are manufactured. In the one case both are geometrical, in the other both are mechanical; but neither of the two, whether geometrical or mechanical, although beautifully adapted to their purpose, need be at all alike. It is not thus with regard to the analytical method: the processes have all more or less of resemblance, they are all conducted by the same general rules; and they commonly lead to universal results, from which particular corollaries are deducible at pleasure. The analytical method is at the same time much the most comprehensive. There are several curious and useful theorems to be found in the analytical treatises on trigonometry, which have not yet, to my knowledge, been demonstrated in any other way; and not a few which I am persuaded do not admit of any other kind of proof. 3. This method is also much the easiest. The processes themselves are, in the main, conducted with the greatest possible simplicity; the substitutions and transformations are generally natural and obvious: in truth, so much so, that a student no sooner attains a competent acquaintance with the manner of conducting his investigation, than he will be enabled to develope practical theorems nearly as fast as he can write them down. Nor is the mode of inquiry such as need encumber the memory; the operations being general, the requisite first principles few. This is a great recommendation; because every unnecessary load upon the , memory tends more or less to weaken our mental elasticity, and impede the intellectual operations. I am happy to fortify my opinion on this point by an observation of the most profound mathematician and natural philosopher now living, LAPLAce. “Préférez (says he) dans l’en“seignement les méthodes générales, * 3. “les présenter de la manière la plus simple, et vous “verrez en même tems qu’elles sont presque toujours “les plus faciles.” 4. The analytical method of establishing the principles, and deducing the formulae of trigonometry, has this farther advantage, that it connects it more intimately with the principal topics of mixed mathematics, and causes it to become a portal to the higher mechanics and the celestial physics. Any person who has looked, however cursorily, into the best treatises on statics, dynamics, and physical astronomy, especially those which have been published on the continent, must have observed that they abound with trigonometrical formulae. And they who have gone a little below the surface, know that several of the most striking results of physical astronomy turn upon some obvious trigonometrical truth. Thus, to select only one class of instances, our countryman Simpson, in his researches into that part of the celestial physics which relates to the moon (Miscellaneous Tracts, p. 179), having shown that no terms enter the equation of the orbit but what are expressible by the cosine of an arc, or the cosines of its multiples, and, therefore, that no terms enter that equation but what by a regular increase and decrease return to their former values; immediately infers that the moon's “mean motion, and the greatest quantities of the several equations, undergo no change from gravity.” Frisi advanced still farther in the same line of induction. And farther yet Lagrange and Laplace; who have demonstrated that no term of the form Ax nt,
A or A tan nT, or A cosec nT, or
(T denoting the
sin not time) can enter the analytical expression for any of the inequalities of the planetary motions, or those of their satellites: and have thus proved that the system is stable, all its irregularities being confined within certain limits; just as all the modifications in the magnitude and posi