both because it is the strictest and the easiest, and because I think the Mathematical Student should be early introduced to the method.

The succeeding Chapters are devoted to an exposition of the nature of the Trigonometrical ratios, and to the demonstration by geometrical constructions of the principal propositions required for the Solution of Triangles. To these I have added a general explanation of the applications of these propositions in Trigonometrical Surveying : and I have concluded with a proof of the formulæ for the sine and cosine of the sum of two angles treated (as it seems to me they should be) as examples of the Elementary Theory of Projection. Having learned thus much the Student has gained a knowledge of Trigonometry as originally understood, and may apply his knowledge in Surveying; and he has also reached a point from which he may advance into Analytical Trigonometry and its use in Natural Philosophy.

Thinking that others may have felt the same want as myself, I have published the Tract instead of merely printing it for the use of my Class.

H. B.

ELEMENTS

OF

PLANE TRIGONOMETRY.

TRIGONOMETRY (from Tpíywvov, triangle, and μerpéw, I measure) is the science of the numerical relations between the sides and angles of triangles.

This Treatise is intended to demonstrate, to those who have learned the principal propositions in the first six books of Euclid, so much of Trigonometry as was originally implied in the term, that is, how from given values of some of the sides and angles of a triangle to calculate, in the most convenient way, all the others.

A few propositions supplementary to Euclid are premised as introductory to the propositions of Trigonometry as usually understood.

CHAPTER I.

OF THE MENSURATION OF THE CIRCLE.

DEF. I. A magnitude or ratio, which is fixed in value. by the conditions of the question, is called a Constant. A magnitude or ratio, which is not fixed in

DEF. 2.

B. T.

I

point to the next, then if the number of points be conceived to increase and the distance between each two to diminish continually, the extremities remaining fixed, the limit of the sum of the straight lines is called the LENGTH OF THE CURVE.

PROP. I. The lengths of similar arcs are proportional to their chords.

For let any number of points be taken in the one and the points be joined by straight lines so as to inscribe a polygon in it, and let a similar polygon be inscribed in the other, the perimeters of the two polygons are proportional to the chords, or the ratio of the perimeter of the one to its chord is equal to the ratio of the perimeter of the other to its chord. Then if the number of sides of the polygons increase these two ratios vary but remain always equal to each other, therefore (Lemma) their limits are equal. But the limit of the ratio of the perimeter of the polygon to the chord is (Def. 5) the ratio of the length of the curve to its chord, therefore the ratio of the length of the one curve to its chord is equal to the ratio of the length of the other curve to its chord, or the lengths of similar finite curve lines are proportional to their chords.

COR. I. Since semicircles are similar curves and the diameters are their chords, the ratio of the semi-circumference to the diameter is the same for all circles.

If this ratio be denoted, as is customary, by

umerically

the circumference ÷ the diameter = π,

and the circumference = 2πR.

π

then

COR. 2. The angle subtended at the centre of a circle y an arc equal to the radius is the same for all circles. For

value by the conditions of the question and which is conceived to change its value by lapse of time, or otherwise, is called a VARIABLE.

DEF. 3. If a variable shall be always less than a given constant, but shall in time become greater than any less constant, the given constant is the SUPERIOR LIMIT of the variable and if the variable shall be always greater than a given constant but in time shall become less than any greater constant, the given constant is the INFERIOR LIMIT of the variable.

:

LEMMA. If two variables are at every instant equal their limits are equal.

For if the limits be not equal, the one variable shall necessarily in time become greater than the one limit and less than the other, while at the same instant the other variable shall be greater than both limits or less than both limits, which is impossible, since the variables are always equal.

DEF. 4.

Curvilinear segments are similar when, if on the chord of the one as base any triangle be described with its vertex in the arc, a similar triangle with its vertex in the other arc can always be described on its chord as base; and the arcs are SIMILAR CURVES.

COR. I. Arcs of circles subtending equal angles at the centres are similar curves.

COR. 2. If a polygon of any number of sides be inscribed in one of two similar curves, a similar polygon can be inscribed in the other.

DEF. 5. Let a number of points be taken in a terminated curve line, and let straight lines be drawn from each