FIVE-PLACE LOGARITHMIC AND TRIGONOMETRIC TABLES ARRANGED BY G. A. WENTWORTH, A.M. AND G. A. HILL, A.M. BOSTON, U.S.A., AND LONDON 1896 HARVARD COLLEGE LIBRARY GIFT OF MISS ELLEN L. WENTWORTH MAY 8 1939 Entered according to Act of Congress, in the year 1882, by G. A. WENTWORTH AND G. A. HILL in the office of the Librarian of Congress at Washington INTRODUCTION. 1. If the natural numbers are regarded as powers of ten, the exponents of the powers are the Common or Briggs Logarithms of the numbers. If A and B denote natural numbers, a and b their logarithms, then 10" A, 10" B; or, written in logarithmic form, = = 2. The logarithm of a product is found by adding the logarithms of its factors. 3. The logarithm of a quotient is found by subtracting the logarithm of the divisor from that of the dividend. 4. The logarithm of a power of a number is found by multiplying the logarithm of the number by the exponent of the power. 5. The logarithm of the root of a number is found by dividing the logarithm of the number by the index of the root. 6. The logarithms of 1, 10, 100, etc., and of 0.1, 0.01, 0.001, etc., are integral numbers. The logarithms of all other numbers are fractions. 1 and 10, the logarithm is between O and 1. 1 and 2. If the number is between 7. If the number is less than 1, the logarithm is negative (§ 6), but is written in such a form that the fractional part is always positive. For the number may be regarded as the product of two factors, one of which lies between 1 and 10, and the other is a negative power of 10; the logarithm will then take the form of a difference whose minuend is a positive proper fraction, and whose subtrahend is a positive integral number. Therefore (§ 2), log 0.48 = log 4.8+ log 0.1 0.68124-1. (Page 1.) = Therefore, log 0.0007 log 7 + log 0.0001 = 0.84510 — 4. The logarithm 0.84510-4 is often written 4.84510. 8. Every logarithm, therefore, consists of two parts: a positive or negative integral number, which is called the Characteristic, and a positive proper fraction, which is called the Mantissa. Thus, in the logarithm 3.52179, the integral number 3 is the characteristic, and the fraction .52179 the mantissa. In the logarithm 0.78254-2, the integral number - 2 is the characteristic, and the fraction 0.78254 is the mantissa. 9. If the logarithm is negative, it is customary to change the form of the difference so that the subtrahend shall be 10 or a multiple of 10. This is done by adding to both minuend and subtrahend a number which will increase the subtrahend to 10 or a multiple of 10. Thus, the logarithm 0.78254 -2 is changed to 8.78254-10 by adding 8 to both minuend and subtrahend. The logarithm 0.92737-13 is changed to 7.92737-20 by adding 7 to both minuend and subtrahend. 10. The following rules are derived from § 6: If the number is greater than 1, make the characteristic of the logarithm one unit less than the number of figures on the left of the decimal point. If the number is less than 1, make the characteristic of the logarithm negative, and one unit more than the number of zeros between the decimal point and the first significant figure of the given number. If the characteristic of a given logarithm is positive, make the number of figures in the integral part of the corresponding number one more than the number of units in the characteristic. If the characteristic is negative, make the number of zeros between the decimal point and the first significant figure of the corresponding number one less than the number of units in the characteristic. Thus, the characteristic of log 7849.27 = 3; the characteristic of log 0.037 =- 28.00000— 10. If the characteristic is 4, the corresponding number has five figures in its integral part. If the characteristic is 3, that is, 7.00000 10, the corresponding fraction has two zeros between the decimal point and the first significant figure. 11. The logarithms of numbers that can be derived one from another by multiplication or division by an integral power of 10 have the same mantissa. For, multiplying or dividing a number by an integral power of 10 will increase or diminish its logarithm by the exponent of that power of 10; and since this exponent is an integer, the mantissa of the logarithm will be unaffected. 12. In this table (pp. 1-19) the vertical columns headed N contain the numbers, and the other columns the logarithms. On page 1 both the characteristic and the mantissa are printed. On pages 2-19 the mantissa only is printed. The fractional part of a logarithm can be expressed only approximately, and in a five-place table all figures that follow the fifth are rejected. Whenever the sixth figure is 5, or more, the fifth figure is increased by 1. The figure 5 is written when the value of the figure in the place in which it stands, together with the succeeding figures, is more than 44, but less than 5. Thus, if the mantissa of a logarithm written to seven places is 5328732, it is written in this table (a five-place table) 53287. If it is 5328751, it is written If it is 5328461 or 5328499, it is written in this table 53285. 53288. Again, if the mantissa is 5324981, it is written 53250; and if it is 4999967, it is written 50000. |