and consequently if N denote any number, we shall have 1 log. 10N= log. 10log N; that is, com. log N-nap. log.N__nap. log.N = nap.log.10 2.3025851 =;43429448×nap.log.N; and the modulus of the common system is, therefore, M= 1 2.3025851 =.43429448 .. 2 M=.86858896 Hence, to construct a table of common logarithms, we have log(P+1)=logP+.86858896 5.0 2P+1+3(2P+1)**5(2P+1)° Expounding P successively by 1,2,3, &c., we get log. 4-2 log. 2 +... )=.4771213 log. 5=log. 12=log. 10—log 2=1—log. 2 log. 6=log. 2+log. 3 Since log.1=2M (n+n3 + !n3 +‡n2 +... ...) &c. and thus we have a series for computing the logs. of all numbers, without knowing the log. of the previous number. EXAMPLES IN LOGARITHMS. (1.) Given the log. of 2=0.3010300, to find the logs. of 25 and .0125. therefore log. 25-2 log. 10-2 log. 2=1.3979406 125 1 = 1 Again .0125 = 10000 80 10×22 .. log. .0125 log. 1-log. 10-3 log. 2=-1-3 log. 2 (2.) Calculate the common logarithm of 17. =2.0969100 Ans. 1.2304489. (3.) Given the logs. of 2 and 3 to find the logarithm of 12.5. Ans. 1+2 log. 3-2 log. 2. (4.) Having given the logs. of 3 and .21, to find the logarithm of 83349. Ans. 6+2 log. 3+3 log. .21. ON EXPONENTIAL EQUATIONS. An exponential equation is an equation in which the unknown quantity appears in the form of an exponent or index; thus, the following are exponential equations: = X When the equation is of the form a b, or abc, the value of x is readily obtained by logarithms, as we have already seen above. But if the equation be of the form xa, the value of x may be obtained by a rule of approximation, as in the following example : Ex. Given a 100, to find an approximate value of x. = The value of x is evidently between 3 and 4, since 33 = 27 and 4 256; hence, taking the logs. of both sides of the equation, we have = Then, as the difference of the results is to the difference of the assumed numbers, so is the least error to a correction of the assumed number corresponding to the least error; that is, .098451 .1: : .002689 : .00273; hence x 3.6-.00273-3.59727, nearly Again, by forming the value of x* for x=3·5972, we find the error to be-.0000841, and for x=3.5973, the error is + .0000149; hence, as .000099 .0001 :: .0001 .0000151; therefore x=3.5973-.0000151-3.5972849, the value nearly. EXAMPLES FOR PRACTICE. (1.) Find a from the equation ax = 5 Ans. 2.129372. (2.) Solve the equation x = 123456789 Ans. 8.6400268. (3) Find x from the equation a = 2000. Ans. 4.8278226. Since the properties of logarithms afford great facilities in performing complicated arithmetical operations upon large numbers, it becomes desirable to have the logarithms of sines, cosines, tangents, &c. computed and arranged in tables; but most of these numbers being less than unity, their logarithms would, of course, be negative. To avoid this inconvenience. all the trigonometrical functions calculated in the manner explained in Chap. IV, are multiplied by a large number, and, the operation being performed upon all, their relative value is not altered. This number may, of course, be any whatever, provided it be so large, that, when the numerical values of trigonometrical quantities are multiplied by it, their logarithms may be positive numbers. The number employed for this purpose in the common tables is 10000000000 or 10, which is usually represented by the symbol R. The sine of 1', as computed above, ist Sin. 1'=.0002908882 .... a number much smaller than unity, and whose logarithm would consequently be negative. When multiplied by 10" it becomes =2908882.. a number whose logarithm is 6.4637261, and consequently we find in our tables, log. sin. 1' 6.4637261. A table constituted upon this principle is called a Table of Logarithmic Sines, Cosines, Tangents, &c. and by this nearly all the practical operations of trigonometry are usually performed. It is manifest, from these remarks, that before we can apply formulæ deduced in the preceding chapters to practical purposes, we must transform them in such a manner as to render the several trigonometrical quantities identical with those registered in our tables. The sines, cosines, &c. we have hitherto employed, are called Trigonometrical quantities calculated to a radius unity; those registered in the tables, Trigonometrical quantities calculated to radius R. The problem to be solved therefore is To transform an expression calculated to a radius unity, to another calculated to a radius R Let us represent sin. to radius unity by m. R by n. and so for all the other trigonometrical quantities. Hence, in order to transform an expression calculated to radius unity, to another calculated to radius R, we must divide each of the trigonometrical quantities by R. If any of the trigonometrical quantities enter in the square, cube, &c. these must of course be divided by R2 R3, &c. As observed above, R may be any given number whatever, the number usually employed in the ordinary tables being 10", and therefore log. R = 10 Take as an example such an expression as a sin. 8 = b tan. ❤ in order to reduce this to an expression which we can compute by our tables we must, according to the above rule, divide each of the trigonometrical quantities by the proper power of R the expression then becomes. Or, clearing of fractions, Or, a R sin. = b tan.' a+ 8 log. a + log. R+ log. sin. = log. b + 2 log. tan. an expression which may be calculated by the tables. If the expression calculated to radius unity be of the form it requires no modification, for if we divide both terms of the fraction sin. sin. ❤ by R, we shall not alter its value. We need not prosecute this subject farther, as numerous examples of these transformations will occur at every step in the succeeding chapters. CHAPTER VI. ON THE SOLUTION OF RIGHT-ANGLED TRIANGLES. Every plane triangle being considered to consist of 6 parts, the three sides, and the three angles, if any of these three parts be given, we can, in general, determine the remaining parts by trigonometry. In right-angled triangles, the right angle is always known, and therefore any two other parts being given, we can, in general, determine the rest. We shall thus have five different cases. 1. When one of the acute angles and the hypothenuse is given. 2. When one of the acute angles and a side is given. 3. When the hypothenuse and one side is given. 4. When the two sides are given. 5. When the two acute angles are given. Let ABC be a right-angled triangle, C the right ngle. Let the sides opposite to the angles A and B be denoted respectively by a, b, and let the hypothenuse be called c. Case 1, Given A, c, required B, a. b. A+B=90° A b a B B=90°-A whence B is known -- -(1) By Chap. III. prop. 1, a=c sin. A |