Euler's and Legendre's verification formulas, may be used to test the accuracy of the tables. The latter formulas are (see Exs. 7– 10, Ch. XII.),

sin(36°+A) — sin (36° — A) — sin (72°+A)+sin(72°—À)=sin A, (4) cos(36°+4)+cos(36° — A) — cos(72°+A)—cos(72° — A)=cos A. (5)

EXERCISES.

1. Test the tables of natural sines and cosines by means of formulas (4), (5), taking A equal to 4°, 10°, 15°, and other values. 2. Assuming the functions of 1° as known, calculate the sines of 2o, 3o, 4o, 5o, 6o, by formula (1). 3. By means of formulas (2), (3), calculate the sines and cosines of 33°, 37°, 41°, 47°, 53°, 67°, and other angles.

98. Trigonometry defined. Branches of trigonometry. Before concluding this text-book it may be well to indicate to the student the relation of the part of trigonometry treated in the preceding pages to the subject as a whole, and also to try to give him a little idea of another branch of trigonometry; namely, analytical trigonometry.

In Chapters II.-IX., plane angles, the solution of plane triangles, and applications connected therewith were discussed. This is what is usually known as plane trigonometry. The study of solid angles, the solution of spherical triangles, and the associated practical applications, constitute spherical trigonometry. These branches of mathematics are founded on geometrical considerations, and may be looked upon as applications of algebra to geometry. Pure mathematics is sometimes regarded as consisting of two great branches; namely, geometry and analysis. Analysis includes algebra, infinitesimal calculus, and other subjects which employ the symbols, rules, and methods of algebra, and do not rest upon conceptions of space. (Geometrical ideas may be used in analysis, however, for the sake of exposition and illustration, and, on the other hand, algebra may be employed in expounding the principles of geometry.) Since the eighteenth century, trigonometry has also been treated as a branch of analysis.*

* The meaning of the word "analysis" thus used in mathematics, should not be confounded with the ordinary meaning of the word, or with the meaning attached to the term "analysis" in logic.

98.]

ANALYTICAL TRIGONOMETRY.

163

Analytical (or algebraical) trigonometry treats of the general relations of angles and their trigonometric functions without any reference to measurement. It discusses, among other things, the development of exponential and logarithmic series, the connections between trigonometric and exponential functions, the expansions of an angle and its trigonometric functions into infinite series, the calculation of π, the summation of series, and the factorization of certain algebraic expressions. The properties stated in formulas, (1)-(3) Art. 44, (1)-(8) Art. 50, (1)-(8) Art. 52, (1)-(3) Art. 93, are analytical properties, and can be derived without the aid of geometry. Analytical trigonometry includes hyperbolic trigonometry; that is, the treatment of what are called the hyperbolic functions. While the trigonometric functions may be defined and discussed on a geometrical basis, as done in this book (and this is the easiest way for beginners), it may be stated that they can also be defined and their properties deduced on a purely algebraic basis. It is beyond the scope of this work to show this, but the student may obtain a little light on the subject by reading Notes A and D. It may be stated further, that, under certain restrictions, some of the most important theorems and properties found in analytical trigonometry can be derived easily in an elementary course in the infinitesimal calculus. It has been pointed out that the trigonometric functions can be defined in a purely geometrical manner, and in a purely algebraic manner; they can also be given definitions depending on the infinitesimal calculus, and their properties deduced therefrom. Finally, it may be said that trigonometry is merely a brief chapter in the modern Theory of Functions, and may be defined as the science of singly periodic functions (see Art. 78). For a treatment of trigonometry, either as a part of algebra, or, as "an elementary illustration of the application of the Theory of Functions," see Lock, Higher Trigonometry; Loney (Part II.), Analytical Trigonometry; W. E. Johnson, Treatise on Trigonometry, Chaps. XII.-XXII.; Casey, A Treatise on Plane Trigonometry; Levett and Davison, Elements of Plane Trigonometry (Parts II., III., Real Algebraical Quantity, Complex Quantity); Hayward, Vector Algebra and Trigonometry; Hobson, A Treatise on Plane Trigonometry; Chrystal, Algebra, Part I., Chap. XII.; Part II., Preface, and Chaps. XXIX., XXX.

3. The sine and cosine of the sum of two angles. [Supplementary to

Art. 46.]

Let the construction be made as indicated in Art. 46. Then

sin (A + B) = proj. OP on OY_ proj. OQ on OY proj. QP on OY

ОР

OP

+

OP [Theorem in (2).]

proj. OP on OX _ proj. OQ on OX proj. QP on OX

=

cos (A+B) =

OP

+

OP

OQ proj. QP on OX QP

+

In the projection proof of the addition formulas for the sine and cosine, A and B can have any magnitudes, positive or negative. The formulas for sin (A – B), cos (A – B), can also be derived by substituting - B for + B in the addition formulas.

NOTE C.

[Supplementary to Arts. 9, 72.]

ON THE LENGTH AND AREA OF A CIRCLE.

1. The main purpose of this note is to outline a method of approximating to the value of ; that is, to the ratio of the length of a circle to its diameter. This method depends only on elementary geometry.* There are simpler and more expeditious methods of finding, but they require a greater knowledge of mathematics than beginners in trigonometry generally possess.

By the methods of elementary geometry, as shown in the texts of Euclid and others, regular polygons of 3, 4, 5, 6, 15 sides can be inscribed in, and circumscribed about a given circle. Moreover, inscribed and circumscribing regular polygons of 2, 4, 8, 16, ......., times each of those numbers of sides can also be constructed by successively bisecting the arcs subtended by the sides, and joining the consecutive points of division. This process can evidently be

* A section on the mensuration of the circle is given in many geometries. Reference may be made to the geometries of Beman and Smith (Ginn & Co.), Gore (Longmans, Green, & Co.), Phillips and Fisher (Harpers), and others.

carried on until the inscribed and circumscribing polygons have an infinitely great number of sides; that is, regular polygons of 3.2", 4.2", 5.2", 15.2" sides, n being any positive integer, can be inscribed in, or circumscribed about, a given circle.

2. Outline of a proof of the theorem that the lengths of circles are proportional to their diameters.

(a) The length of a circle is greater than the perimeter of an inscribed polygon, and is less than the perimeter of a circumscribing polygon of any finite number of sides.

(b) As the number of sides of a regular polygon inscribed in, or circumscribed about, a circle is increased, the length of the perimeter of the polygon approaches nearer and nearer to the length of the circle. In other words, by increasing the number of sides, the difference between the length of the perimeter of the polygon and the length of the circle may be made as small as one please, and this difference approaches zero when the number of sides approaches infinity.

(c) Let any two circles be taken, and let the radii be R, r. Let AB be a side of a regular polygon of n sides inscribed in the circle having centre O

B

FIG. 92.

and radius R, and let ab be the side of a regular polygon of n sides inscribed in the circle having centre o and radius r. Let P denote the perimeter of the first polygon, p that of the second; let C denote the length of the first circle, and c that of the second. Then

where D, d may each be made smaller than any assignable quantity by making the number of sides, n, infinitely great.

The polygons are similar, since they are regular and have the same number of sides. Hence, by geometry,