ERRATA. 137 P. 16.-Seventh line of Art. 19, for 2 sin 20°, read 2 sin 30°. P. 77.-Second line of Ex. 3, for east of A, read north of A; and for east from C, read west from C. Seventh line of Art. 72, for % of a second, read % of a minute. P. 165. Second line from bottom, for Fig. 109, read Fig. 111. P. 170.-Fifteenth line, for Fig. 109, read Fig. 111. P. 172.-For Section VII, read Section VI. P. 196.-For Section VIII, read Section VII. P. 215.-Second line of Remarks, for Secs. 13 and 14, read Secs. 11 P. 216.-Tenth line from bottom, for sections, read stations. A MANUAL ANUAL OF LAND SURVEYING. A brief treatment of the subjects of Logarithms and 1. The Logarithm of any given number is the expo- nent with which a fixed number must be affected to pro- Thus, the logarithms of all powers of the base are in- 2. The logarithms of numbers intervening between exact powers of the base are composed of an integral and a fractional part. The integral part of a logarithm is called the Index or Characteristic; and the fractional 3. Properties of Common Logarithms.- From the nature of exponents considered in connection with the decimal notation result the following principles: I. The logarithm of any exact power of 10 is a positive integer one less than the number of places of figures in the expression of the number. For, by definition, log 102 = p. But 10 is expressed decimally by 1 with p ciphers annexed, and hence contains p+1 places of figures. II. In general, the characteristic of the common logarithm of any integral number is one less than the number of figures in the expression of the number. For, denoting the characteristic of the logarithm by c, and the decimal part by d, we may write log 10+α = c+d. But, as d < 1, 10+d, expressed decimally, contains c+1 figures. III. The characteristic of the common logarithm of any fractional number less than 1, expressed decimally, is negative and numerically one more than the number of zeros immediately following the decimal point. The numerator 10+d, under the decimal notation, has c+1 figures, as already observed, while the expression of the fraction by the same notation requires p places of figures. Since the decimal under consideration is, by hypothesis, not a mixed one, the number c+1 of figures in the numerator cannot exceed the number p of figures at the right of the decimal point, but may be less, the deficiency being supplied by the zeros prefixed to the c+1 figures of the numerator. In either case p-(c+1) is the number of zeros immediately following the decimal point. Let p−(c+1)=n, log 10c-p+d= c-p+d, of which c-p is the index, Substituting the value of c-p found above, we have SCHOLIUM.-In the notation of the negative characteristic, the minus sign is usually written above the characteristic. Thus, log 0.00045 = 4.653213. IV. The characteristic of the common logarithm of a mixed decimal is the same as the characteristic of the logarithm of the integral part. For, using the same notation as above, c-p+1 is the number of places of figures in the integral part; and c—p is the characteristic of the logarithm of the given decimal. V. The decimal part of the common logarithm of a number is not changed by multiplying or dividing the number by any power of 10. For the logarithm of 10+ is c+d, and the logarithm of 10c+d±p is c±p+d in which, since c and p are integers, cp is the characteristic and d is the decimal part, the same after the multiplication or division of 10+d by 10 as it was before. 4. Table of Logarithms.-In Table I at the end of the book, are given the logarithms of numbers from 1 to 10000, true to six decimal places. The characteristics, however, are usually not entered in the Table, but are supplied by inspection (Prin. II). The manner of using the Table is illustrated in the solution of the following problems: Prob. 1.-Given a number, to find its logarithm. CASE 1. When the number is between 1 and 1000. Solution. We find the given number in the left hand column of the Table, and opposite to it in the next column, the decimal part of the logarithm sought, the two left hand figures of the next preceding full set of six figures being understood where only four figures occur. To the decimal part of the logarithm as thus obtained, we prefix the proper characteristic as found by inspection. Thus, log 87 = 1.939519; log 237 = 2.374748. CASE 2.-When the number is between 1000 and 10000. Solution. We find the number expressed by the left hand three figures of the given number in the left hand column, and then pass horizontally across the page to the column headed by the right hand figure of the given number. Here we find either the six figures of the mantissa or the right hand four figures of it. In the latter case, the left hand two figures of the next preceding number of six figures are to be prefixed, together with the proper characteristic. Thus, log 4712 205 == 3.970997. CASE 3. When the number exceeds 10000. Solution. We find as in Case 2 the mantissa of the logarithm of the number expressed by the left hand four figures of the given number. We then take the number opposite in the column headed "Diff" and multiply it by the remaining figures of the given number, rejecting from the right of the product as many figures as there are in the multiplier. We then add the part of the product retained, to the mantissa as before found, and prefix the proper characteristic. SCH.-When the left hand figure of those rejected is 5 or more, the part retained is to be increased by 1. Thus, log 34256 = 4.534661+127X.6=4.534737; and log 283745 = 5.452859+153X.45=5.452928. CASE 4.- When the number is a decimal. Solution. We find the mantissa, regarding the number as integral, and prefix the proper characteristic, (Prin. III or IV). Thus, log 0.472 1.673942; log 37.25=1.571126. Prob 2.-Given any logarithm, to find the corresponding number. CASE 1.- When the mantissa is found in the Table. Solution. We take out the corresponding number and |