VII. AREAS 21. According to the definition already given (sec. 16) two polygons are equivalent, or equal in area, if they can be divided into the same number of triangles which are congruent in pairs. We have proved (sec. 16) that a triangle is equivalent to an isosceles birectangular quadrilateral having its summit equal to one side of the triangle, and each summit angle equal to half the sum of the angles of the triangle. Now, in either the Lobachevskian or the Riemannian geometry an isosceles birectangular quadrilateral is fully determined by its summit and summit angles, for if ABCD (Fig. 20) and ABEF are two isosceles birectangular quadrilaterals with the same summit EF and the same summit angles E and F, their bases CD and AB must coincide. Otherwise, the quadrilateral ABCD would have four right angles, which is impossible (sec. 18). Hence follows the theorem: In the Lobachevskian and Riemannian geometries, two triangles are equivalent if a side and the sum of the angles of one are equal to a side and the sum of the angles of another. 22. A triangle may be constructed having the same area and the same angle sum as a given triangle, and having one side arbitrarily assumed within certain wide limits. Let ABC (Fig. 21) be a given triangle and BCKL the isosceles birectangular quadrilateral constructed as in sec. 16. Let be a given length. With as a radius and B as a center described an arc of a circle cutting KL in M. Connect B and M and prolong BM to A' so that MA'- BM. Connect A' and C. Then A'BC is the required triangle, as is readily shown. That the construction may be possible, it is necessary, on the one hand, that BM >BK, a condition which is certainly met if l>AB. On the other hand, it is necessary, in the Riemannian geometry, that I should be less than the constant 24 (sec. 15). If, now, we have two triangles with the same angle sum, we may take greater than one side of each, and replace each by an equivalent one with the same angle sum and a side equal tol. The two new triangles are equivalent (sec. 21). Hence: Any two triangles with the same angle sum are equivalent. 23. Consider any triangle ABC (Fig. 22) and draw from A a straight line to any point D of the base. We shall call this line a transversal and shall say that the triangle is divided transversally. Now if s is the sum of the angles of the triangle ABC, and s1 and s the sum of the angles of the triangles ABD and ADC respectively, we have A B D FIG. 22. S=81+82−2 rt. 4s. If we adopt such a unit of angle measure that a right angle In the Lobachevskian geometry, -s is positive (sec. 20) and is called the defect of the triangle. In the Riemannian geometry 8- is positive and is called the excess of the triangle. Hence we may state the theorem: If a triangle is divided transversally, the sum of the defects, or excesses, of the parts, is equal to the defect, or excess, of the triangle. The theorem evidently remains true if the triangle is further subdivided by successive transversals of the parts, as shown, for example, in Fig. 23. Further Hilbert * has shown that any division of a triangle may be reduced to transverse divisions. We have accordingly the more general theorem: FIG. 23. In the Lobachevskian and Riemannian geometries the defect, or excess, of any triangle is equal to the sum of the defects, or excesses, of triangles which are formed from it by any system of division. 24. Since equivalent triangles may be divided into the same number of triangles congruent in pairs (sec. 21), and since obviously congruent triangles have the same defect, or excess, it follows that any two equivalent triangles have the same defect, or excess. The converse theorem has been proved in sec. 22. We are now enabled to take the defect, or excess, of a triangle as the measure of its area, since the essential properties of a measure of area are that two triangles with the same area have the same measure, that two triangles with the same measure have the same area, and that the measure of a whole is the sum of the measure of its parts. Hence we may say: In the Lobachevskian geometry the area of a triangle is equal to a constant times its defect. In the Riemannian geometry, the area of a triangle is equal to a constant times its excess. The value of the constant depends evidently upon the unit of area employed. The area of a polygon is found by dividing it into triangles. * Grundlagen der Geometrie, Vol. VII of Wissenschaft und Hypothese, Leipzig, 1909. VIII. NON-EUCLIDEAN TRIGONOMETRY 25. The definitions of the trigonometric functions as given in the elementary trigonometry are evidently not available in non-Euclidean geometries, since these definitions are based upon properties of similar triangles which are true only in the Euclidean geometry. Lobachevsky met this difficulty by the construction of a "limit-surface," or horisphere, on which the Euclidean geometry and trigonometry are valid at the same time that the Lobachevskian geometry is valid on the plane. By the aid of this surface and the sphere he obtained the formulas which will be found in sec. 34. This method, however, cannot be applied to the Riemannian geometry. We shall therefore follow a more general method which has also the advantage of operating entirely in the plane. The method, however, is not as elementary as the other, and we shall be obliged to state some results without proof and to give a mere outline of other proofs.* We start with the purely analytic definitions of the trigonometric functions. That is, er being defined by the series where 1. These functions obey all the formulas of trigometry and if x is a real number they are real. * For complete proofs and historical notes consult Coolidge, The Elements of Non-Euclidean Geometry, Oxford, 1909, expecially Chapter IV, where all the requirements of rigor are met. If x is pure imaginary, the above equations lead to the hyperbolic functions, which are defined by the following equations: If x is real, the hyperbolic functions are real, and formulas for this use are readily obtained, if needed, from the trigonometric functions. The following properties of cos x are important for us: If cos x1, x is real; if cos x>1, x is pure imaginary, except perhaps for multiples of the period 27 which may always be added. If we place cos mx=f(x), f(x) satisfies the functional equation f(x+y)+f(x− y) = 2ƒ (x)f(y). Conversely, if f(x) is a continuous function of x satisfying the above equation, then f(x) = cos mx, m being a constant, real or complex. 26. The sine and cosine of an acute angle may be defined as follows. The extension to angles of any size is then made as in the ordinary trigonometry. Let A (Fig. 24) be any acute angle and MP the perpendicular from any point P of one side to the other side, then it may be AM shown that approaches a limit as AP approaches zero, and AM that lim is a continuous function of A, which satisfies the functional equation of sec. 25. Hence lim AM cos mA Since AM< AP, the coefficient m is real, and if we adopt a sys |