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tion 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 do not say, exclusively) according to the principles of analysis.


Royal Military Academy,
Woolwich, Feb, 1816,

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29, line 8, for answer read answers
37, line 2, for and is read it is

41, line s, for read:

156, Er. 1. Add, In the Connaissance des Tems, pour l'an 1817, there is given a table by M. Burckhardt to shorten the computation of this problem.

167, line 9 from bottom, for limited, read limits

212, last line but one, for +2 V op, read F 2 V #.


Preliminary Definitions and Principles.

1.THE word Trigonometry signifies the measure of triangles. But in an enlarged sense we comprehend under this name, the science by which we are enabled to determine the positions and dimensions of the different parts of space, by means of the previous knowledge of some of those parts. -2. If we conceive any different points whatever, posited in space, to be joined one to another by right lines, there will be presented to our consideration three things: 1st. The lengths of those lines. 2dly. The angles they respectively form. 3dly. The angles formed respectively, by planes in which those lines are, or may be imagined to be, comprehended. On the comparison of these three objects depends the solution of the various questions which can be proposed, as to the measure of extension, and of its parts.

3. The intersections of three or more lines in one and the same plane, constitute angles limited by right lines, and plane or rectilinear triangles, or polygons susceptible of being resolved into triangles. And the inter-sections of three or more planes, form plane angles and triedral or polyedral surfaces, the determination of the magnitudes and relations of which is facilitated by re

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