Principles of geometry, mensuration, trigonometry, land-surveying, and levelling, with their application to the solution of practical problems in estimation, surveying and railway engineering

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1848 Excerpt: ...of 100 is 2; 1000 = 103, the logarithm of 1000 is 3; or if N be any number such that N--10, then x is the logarithm of N, or log Ts=x. Here the 10 is called the base of the system. In the hyperbolic system the base is 2-71828, which number is usually represented by e. 15. Since 1 = 10,.. log 1=0; 10= 10i,... log 10= 1; 100 = 102,..log 100 = 2; 1000 = 103,.. log 1000=3; and so on. Hence it follows that the logarithm of a number between 1 and 10 will be some decimal quantity; the log. of any number between 10 and 100 will be between 1 and 2, or 1 + a fraction; the log. of any number between 100 and 1000 will be between 2 and 3, or 2 + a fraction; and so on generally, the log. of any number having n integral figures will contain n--1 units + a fraction. The whole number in the logarithm is called the index or characteristic, and the decimal part is called the mantissa; thus it has been found that 406'5 = lO2,60906, or log 406.5 = 2.60906, where 2 is the characteristic and.60906 is the mantissa. As the characteristic is always 1 less than the number of figures in the integral part of the number, the logarithm given in the tables is only the decimal part of the required logarithm, and the integral part or characteristic must therefore be supplied by the calculator. 16. The logarithm of the product of any numbers is found by adding together the logarithms of the factors. Let A, B, c, &c. be any numbers, and x, y, z, &c. their logarithms, taking a as the base; then, A = ax, B = a", c = a", &c.; whence by multiplication, A x B x c X &c. = a X a" x az x &c. = a31 c..'. log (a X B X C X &c) = X + y + IS + &c = log a + log B + log c + &c. Ex. Find by logarithms the value of 36 x 2.45. Here log (36 x 2.45) = log 36 + log 2...