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TRIGONOMETRY is a department of mathematical
science which in its nature is essentially analytical.
It borrows from geometry no principle except the
proportionality of the sides of similar triangles,
and even this property may perhaps be more simply
and clearly expressed by the language of analysis
than by the phraseology of geometry. All the
results of this science are matters of computation,
the quantities engaged in it are all numerical, and
the operations to which these quantities are sub-
mitted are arithmetical. Some of the processes and
quantities no doubt admit of geometrical expression;
but in many instances this is not the case. Rela-
tions are contemplated and combinations are formed
wholly foreign to geometry; problems are solved
and investigations instituted under which the feeble,



The system has however been changed, and analytical science has in these countries at length obtained that attention as an elementary part of mathematical education, to which its importance so justly entitles it. Students who are about to commence trigonometry have now generally obtained a competent knowledge of algebra, and the reasons which hitherto rendered it expedient to treat the subject geometrically no longer exist. In the following treatise I have accordingly brought to my aid the powerful resources of analysis. On the property of similar triangles, already mentioned, as a basis, I have attempted to raise the whole superstructure of trigonometrical science by reasoning purely analytical. Nor have I found it necessary to resort to any principles beyond what must be considered the rudiments of algebra, except in those higher departments of trigonometry which are only read by students who have made considerable progress in mathematics. Those who are conversant with the first principles of elementary algebra are competent to study all those parts of the present work which are necessary for the elements of natural philosophy, and which are distinguished by an asterisk in the table of contents.

The power and facility of investigation which the student obtains by the use of the analytical method are not its only advantages. The great generality of the theorems, the beautiful symmetry which reigns among the groups of results, the order with which they are developed one from another, offering themselves as unavoidable consequences of the method, and almost independent of the will or the skill of the author, the singular fitness with which the symbolical language of analysis adapts itself so as to represent, even to the eye, all this order and harmony, are effects too conspicuous not to be immediately noticed. Nor is the elegant form which the science thus receives from the hand of analysis a mere object pleasurable to contemplate, but barren of utility. All this order and symmetry, which is given as well to the matter as the form, as well to the things expressed as to the characters which express them, not only serves to impress the knowledge indelibly on the memory, but is the fruitful source of further improvement and dis


The table of contents presents so complete an analysis of the work, that any further account of its arrangement would be superfluous. The first

three sections of the second part might, perhaps, more properly come under the title of spherical geometry. However, as the formulæ and theorems of spherical trigonometry have an intimate and necessary connexion with the subject of these sections, and as they are not contained in other works commonly used in the universities, to have omitted them would be going too far in the sacrifice of utility to system. I have devoted considerable attention to the section on the solution of spherical triangles, and hope that I have succeeded in rendering the discussion of it more full and satisfactory than is usual.

Geodesy, a subject of peculiar interest and much neglected in trigonometrical works, occupies the tenth section of the second part. This would form an interesting subject for a separate treatise; but as we have no such work in our language, I conceived that it might be useful to introduce in the present work the rapid sketch contained in that section. For the materials of it I am indebted for the most part to Base du Système Metrique, &c. of Delambre, Traité de Géodésie of Puissant, The Survey of England and Wales by General

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