Euler's and Legendre's verification formulas, may be used to test the accuracy of the tables. The latter formulas are (see Exs. 710, Ch. XII.),

sin(36°+A) — sin (36° — A) — sin (72°+A)+sin(72° — A)=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 2°, 3°, 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.

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.

APPENDIX.

NOTE A.

HISTORICAL SKETCH.

The most ancient mathematical writing known at the present time is an Egyptian papyrus preserved in the British Museum. It is the work of Ahmes, an Egyptian priest who lived at least seventeen hundred years B.C., and is believed to have been founded on older works dating as far back as 3400 B.C. The treatise is concerned with practical mathematics, and merely gives rules for making geometrical constructions and determining areas. The area of an isosceles triangle is obtained by taking the product of half the base and one of the sides. The area of a circle is found by deducting from the diameter one-ninth of its length, and squaring the remainder-a proceeding which is equivalent to taking = = 3.1604.. ....

The ancient Greeks brought geometry to a high state of perfection, but showed little aptitude for algebra and trigonometry. They were not inclined to be satisfied with approximate results, and regarded the practical application of mathematics as degrading to the science. Trigonometry was invented to supply practical needs, and its development, in the earlier stages, was due to men of the Egyptian, the Hindoo, and the Semitic races.

Astronomy was one of the studies most cultivated by the ancients, but astronomy could not advance, or even become a science, without the aid of trigonometry. Hipparchus of Nicea in Bithynia, the greatest astronomer of antiquity, who flourished about 160-120 B.C., is regarded as the founder of trigonometry, which he developed solely as a necessary part of astronomy. Moreover, trigonometry continued to exist, for the most part, merely as a handmaid of astronomy for over eighteen hundred years. On this account, the theorems of spherical trigonometry were developed earlier than those of plane trigonometry. Of the writings of Hipparchus, all but one have been lost; but it is known that he constructed a table of chords, which serves the same purpose as a table of natural sines. Hero of Alexandria, who flourished some time between 155 and 100 B.C., and is supposed to have been a native Egyptian, found the area of a triangle in terms of its sides, and placed

engineering and land-surveying on a scientific basis. Ptolemy, a native of Egypt, the records of whose observations cover the period 127-151 A.D., wrote the Syntaxis Mathematica (called the Almagest by the Arabs), a work founded on the investigations of Hipparchus. This was regarded as a kind of astronomical Bible for thirteen hundred years, until the Ptolemaic theory, namely, that the sun, planets, and stars revolve around the earth, was shown to be erroneous by Copernicus and Galileo. The Almagest is divided into thirteen books. Book I. treats of plane and spherical trigonometry, contains a very accurate table of chords, probably derived from Hipparchus, and shows the method of forming the table. It develops spherical before plane trigonometry, and does not give the solution of plane triangles. "Whereas the Ptolemaic system (of astronomy) was ... . . overthrown, the theorems of Hipparchus and Ptolemy, on the other hand, will be, as Delambre * says, forever the basis of trigonometry." †

Whatever advance was made in trigonometry during the thousand years after Ptolemy, was due to the Hindoos and Arabs. The Hindoos had tables of the half-chords, or sines, and found that the arc equal in length to the radius contained 3438'. Aryabhatta (476–530 A.D.?) wrote a work containing sections on astronomy, spherical and plane trigonometry. This contained tables of natural sines of the angles in the first quadrant at intervals of 340, the sine being defined as the semi-chord of twice the angle. He gave 3.1416 as the value of T. Other writers were Brahmagupta, born 598, and Bhaskara, about 1150, who gave some trigonometric formulas. The Hindoos knew how to solve plane and spherical right triangles.

During the period of the Dark Ages in Europe, the sciences of the Greeks and Hindoos were preserved, and, to some slight extent, improved by the Arabs. The latter studied trigonometry only for the sake of astronomy. The term sine is due to the celebrated Arabian astronomer Al Battani (Albatagnius), a native of Syria, who died about 930 A.D. Another Arabian astronomer, Abú 'l Wafá (940–998), a native of Persia, was the first to introduce the tangent of the arc into the science; he calculated a table of tangents. Among the Western Arabs, to whom the development of the subject is indebted, were Ibn Yunos of Cairo (died 1008), and Gabir ben Aflah, who was born at Seville and who died at Cordova in the latter part of the eleventh century. The latter wrote an astronomy in nine books, the first of which is devoted to trigonometry; he also contributed to the advancement of spherical trigonometry.

The next stage in the history of trigonometry is marked by the introduction of the Arabian works into Europe, and the development of the arithmetical part of the subject, especially the calculation of tables. This was largely

* Jean Baptiste Delambre (1749–1822), a French mathematician who derived important formulas in spherical trigonometry.

† Ency. Brit., Art. Ptolemy.