= s 9", the number 9 being opposite 15.8, which, to the nearest unit, may be called 16. 105 60 105 60 It will be remembered that the proportional parts opposite 10, 20, 30, 40, 50, when divided by 10 (that is, when the period is moved one place to the left), give the products of 60 by 1, 2, 3, 4, and 5. From those parts we may, therefore, find by inspection the products of 5 by all the digits from 1 to 9; and, in what follows, we shall proceed as if the products 1.75, 3.50, 5.25, 7.00, 8.75 of 15 by 1, 2, 3, 4, and 5 were actually printed in the table opposite those digits; that is, it will be assumed that the proportional parts run in this order: 1.75, 3.50, 5.25, 7.00, 8.75, 10.5, 12.3, 14.0, etc., up to 87.5, the corresponding numbers on the left being, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50. The proportional parts 1.75, 3.50, 5.25, 7.00, 8.75 will be referred to as proportional parts found in the table, corresponding to 1, 2, 3, 4, and 5 seconds, respectively. This being understood, the number s of odd seconds in the angle is determined as follows: Rule.-Find 1, l', L1, and (= tabular difference, l' 1 or D), as before. Look for the tabular difference D in the column of proportional parts. Look for L-1 in the column of proportional parts directly under D. If L I is found there, the number horizontally opposite it on the left of the vertical line is the required number of seconds s. If Ll is not found under D, take the proportional part next lower, which call p. Find the difference between L I and p, and look among the proportional parts under D for this difference, or the part nearest to it, whether higher or lower. Call this part p'. Add the numbers horizontally opposite p and p' on the left of the vertical line. The result will be the required number of seconds s. EXAMPLE 1.-Find the angle whose logarithmic tangent is 1.42822(= L). Look 7) is SOLUTION. 1 = 1.42805, A = 15° 0′, L − 1 = 17, D = 51. ing in the column marked p. p. for 51, the number 17 (= L found under it, horizontally opposite the number 20 on the left of the vertical column. Therefore, s = 20", and 1.42822 log tan 15° 0′ 20′′. Ans. EXAMPLE 2.- Find the angle whose logarithmic cosine is I.52783(= L). = = SOLUTION.I.52811, A = 70° 17′, L 28, D = 36. The proportional part under 36 next lower than 28 is 24; 28 - 24 = = 4; the proportional part nearest 4 is 4.2; the number horizontally opposite 24 is 40; and the number horizontally opposite 4.2 is 7; hence, s = 40+ 7 47", and therefore = = EXAMPLE 3.-Find the angle whose logarithmic sine is 1.66191(Z). SOLUTION.- 1 = Ï.66173; A 27° 19′; L - 7 = 18; D = 24. Looking in the p. p. column for 24, the proportional part next lower than 18 is 16(= p), horizontally opposite which is 40. 18-p 1816 2. This difference is found among the proportional parts in the table (since it is the same as 20 with the decimal point moved one place to = = I.66191 = log sin 27° 19′ 45′′. Ans. EXAMPLE 4.-Find the angle whose logarithmic cotangent is 1.00375(= L). SOLUTION.- = 1.00427; A 84° 14'; / L = 52; D = = 126. The proportional part under 126 next lower than 52 is 42(= p), which corresponds to 20"; 52 10. The proportional part nearest to which corresponds to 5" (= 50) · Therefore, s = 20"+5" = 25", and 1.00375 = log cot 84° 14' 25". Ans. 10 1. 2. EXAMPLES FOR PRACTICE Find the angle whose logarithmic sine is 1.78988. Ans. 38° 3′ 20′′ 5. Find the angle whose logarithmic cotangent is .31789. Ans. 25° 41′ 9′′ 6. Find the angle whose logarithmic cosine is 1.34567. Ans. 77° 11′ 38′′ 7. Find the angle whose logarithmic cotangent is 1.00381. Ans. 84° 14′ 22′′ 8. Find the angle whose logarithmic tangent is 1.00300. 9. Find the angle whose logarithmic sine is 2.99001. Ans. 84° 19′ 42′′ Ans. 5° 36′ 30′′ 32. Tabular Logarithms Increased by 10.-In printing a table of logarithms of the trigonometric functions, the characteristics cannot be omitted, since they cannot be ascertained by inspection. To avoid printing the bars over the negative characteristics, the latter are in many tables increased by 10. The logarithms as then printed are tabular logarithms, and are denoted by L sin, L cos, etc. Thus, GENERAL PRINCIPLE OF INTERPOLATION 33. It has been explained in some of the preceding articles how to determine the natural or the logarithmic functions of any angle containing an odd number of seconds, and therefore, not found in the table; also, how to find the angle corresponding to a given function, when that function is not in the table but lies between two values given in the table. The operation by which such intermediate values are determined from a table is called interpolation. The values that are actually given in the table are called tabular values. For example, in the table of logarithmic functions already described are found all angles that lie between 0° and 90° and contain no odd seconds, and also the logarithmic sines, cosines, etc. of such angles; those are all tabular values. Angles containing odd seconds are not in the table, nor are their logarithmic functions. Both these angles and their functions are intermediate values, and it is in connection with them that interpolation is used. 34. The general principle of interpolation, to be explained presently, is of the utmost importance, and of great value to the engineer, whose work requires the frequent use of tables of various kinds. That principle, although only approximately true, applies to nearly all tables with which the engineer has to deal, and the student should endeavor to make himself thoroughly familiar with it. X F Let a table be constructed on the general type shown on the margin, the left-hand column containing values of a quantity X, and the right-hand column corresponding values of some quantity whose values depend on the values of X. Thus, the values of X may be the natural numbers 1, 2, 3, 4, etc., and the corresponding values of F may be the logarithms or the square roots of those numbers; or the values of X may be angles, and those of F may be sines, cosines, etc., either natural or logarithmic. So far x f. f. as the principle of interpolation is concerned, it is immaterial what kind of quantity is represented by X, and what kind of quantity is tabulated under F. It should be stated, however, that the principle applies only to tables in which the differences between consecutive values of X and the differences between the corresponding values of F do not vary very rapidly. Let x, and x,, as shown in the above general form, be two consecutive values of X given in the table, and f, and f, the corresponding values of F. Let x be a value of X lying between x and x,, and f the corresponding value of F Neither x nor f is in the table, but one of them is given, and the problem is to find the other by interpolation. For instance, if the table is one of natural tangents in which the angles increase by whole minutes, x, and x, may be, respectively, 31° 42′ and 31° 43', and f, and f, their corresponding tangents; while x may be any angle between 31° 42′ and 31° 43', and f its tangent. Either x may be given to find f; or f may be given to find x. The quantity by which the tabular value x, must be algebraically increased in order to obtain x will be called the increment of x,, and denoted by i(x,), read increment of x、 (mathematicians use the notation 4x,, read delta x,). We have, then, x = x1 + i(x1) Using a similar notation for f1, f = f1 + i(f1) (1) (2) If x is given, i(x,) may be assumed as given, since i(x,) =x- X1. Then i(f) is determined by interpolation, as explained below, and f is found from formula 2. Similarly, if f is given, i(f) is likewise given, and x is found by interpolation. The difference, as x, of two consecutive values of X, will be called the interval of X; and that between two consecutive values of F, the interval of F. The notation I(x,), read interval of x,, will be used to denote the interval X2X1. Similarly, (f) will denote the interval f, — f1. The principle of interpolation is this: The increments i(x,) and i(f) are to each other as the corresponding intervals I(x,) and I(f1); or, algebraically, This formula is very easily remembered on account of its symmetry. The following, derived from it, serve, respectively, to find i (f) when x is given, and i (x,) when f is given: The last two formulas may be stated in the form of a general principle, as follows: Either increment is equal to the corresponding interval multiplied by the ratio of the other increment to the other interval. It is easy to remember what the numerator of this ratio is, by noticing that the ratio is |