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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 seme 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 demonstration 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, attachez-vous à
“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 gra‘vity.” - 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 A × nor,
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 position of the sines and cosines of arcs in the same circle are confined within limits, such as the theory of trigonometry assigns them. This consideration stamps a value upon the researches in this department of science which they would not otherwise possess; and in order that the mathematical student may fully avail himself of it, it is requisite that he understand the analytical method. Lastly, this method is preferable to the geometrical, because it tends to communicate to the student the habit of investigation, which that does not. It is one thing to be able to demonstrate, or to be able to understand by means of a demonstration, that a proposition is true or false: it is a totally distinct one to be able to investigate propositions which shall inevitably be true. In this point of view I have often been struck with what I cannot but regard as a singular defect in the manner of teaching geometry, which prevails in most mathematical seminaries. If a student so apply himself to the admirable ELEMENTs of Euclid, or to those of Legendre, or others which need not be specified, as to understand and feel the force of each demonstration, and trace the exquisite concatenation and mutual dependence of the parts, his logical habits of arrangement, and the classification of his thoughts in reasoning, must be improved; and superadded to this there may be a fondness for geometrical pursuits. But this latter consequence does not necessarily follow; for which the principal reason is, that he has not been taught the use of his instruments. That method of teaching geometry is essentially defective which does not include geometrical analysis; and yet, axiomatic as this would seem to be, Euclid's Elements, or books for a similar purpose, are almost universally studied; while Euclid's Data are almost as universally neglected. For like reasons, every department of mathematics should be so taught as to enable the student, nay to stimulate him, to pursue his researches; and this, as in every other region of abstract science, so in trigonometry, must, if the pursuit is intended to be at all extensive, be conducted, for the most part, (I de not say,” exclusively) according to the principles of analysis. . . . . * . .