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THE TREATISE on TRIGONOMETRY by Mr. AIRY was written expressly for the ENCYCLOPEDIA METROPOLITANA. The author's engagements, as Astronomer Royal, leaving him little leisure, he was unable to make those amendments in the work that were required by the plan of the Revised Edition of the ENCYCLOPÆDIA. The Treatise was therefore placed in the competent hands of Professor BLACKBURN; who, in superintending its passing through the press, has added questions, and made such other improvements as, it is hoped, will render it more suitable both for a class book and for private study.


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TRIGONOMETRY (Τριγωνομετρία, from τριγωνός, a triangle, and μετρέω, I measure), the science of triangles, the branch of mathematics which treats of the application of arithmetic to geometry. The term was originally restricted to signify the science which gives the relation of the parts of triangles described on a plane or spherical surface; but it is now understood to comprehend all theorems respecting the properties of angles and circular arcs, and the lines belonging to them. This latter department is frequently called the arithmetic of sines.

In the application of mathematics to physics, no branch is more important than trigonometry. It is the connecting link by which we are enabled to combine, in their fullest extent, the practical exactness of arithmetical calculations with the hypothetical accuracy of geometrical constructions. Without it, the former could never have been applied to physics, and the limit of the errors of the latter would have depended on the skill of the practical geometer. By the substitution of numerical calculations for graphical constructions, we are enabled to obtain results to any desired degree of accuracy. With trigonometry, in fact, astronomy first received such a degree of exactness as justly to merit the name of science; and every improvement that has been made in trigonometry to the present time, has been attended with corresponding improvements in all parts of physical science.

The following will be the arrangement of the present treatise :The first section will contain the definitions of the terms most frequently in use; in the second will be given the principal theorems relating to trigonometrical lines; the third will explain the use of subsidiary angles; the fourth will contain all the most important propositions of plane trigonometry; the fifth, those of spherical geometry; and the sixth, those of spherical trigonometry. In the seventh will be given formulæ for small corresponding variations of the parts of triangles; and the eighth will contain some theorems which require for their investigation a more refined analysis. The ninth will treat of some expressions peculiar to geodetic operations; and the tenth will explain the construction of trigonometrical tables.


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