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Page 64 - C' (89) (90) (91) (92) (93) 112. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.
Page 128 - ... confusion need arise from this method in finding a number from its logarithm; for although the logarithm 6.141136 represents either the number 1,384,000, or the decimal .0001384, yet these are so diverse in their values that we can never be uncertain in a given problem which to adopt. The table XXIV. contains the mantissas of logarithms, carried to six places of decimals, for numbers between 1 and 9999, inclusive. The first three figures of a number are given in the first column, the fourth at...
Page 131 - ... the second, the log sin in the third, and in the fourth are the last three figures of a logarithm which is the difference between the log sin and the logarithm of the number of seconds in the first column. The first three figures and the characteristic of this logarithm are placed, once for all, at the head of the column. To find the log sin of an arc less than 2░ given to seconds. Reduce the given arc to seconds, and take...
Page 128 - ... with the two. If a number (after cutting off the ciphers at either end) consists of not more than four figures, the mantissa may be taken direct from the table ; but by interpolation the logarithm of a number having six figures may be obtained. The last column contains the average difference of consecutive logarithms on the same line, but for a given case the difference needs to be verified by actual subtraction, at least so far as the last figure is concerned. The lower part of the page contains...
Page 130 - Take out from the proper column of the table the logarithm corresponding to the given number of degrees and minutes. If there be any seconds multiply them by the adjoining tabular difference, and apply their product as a correction to the logarithm already taken out. The correction is to be added If the logarithms of the table are increasing with the angle, or subtracted if they are decreasing as the angle increases. In the first quadrant the log sines and tangents increase, and the log. cosines...
Page 127 - All numbers which consist of the same figures standing in the same order have the same mantissa, regardless of the position of the decimal point in the number, or of the number of ciphers which precede or follow the significant figures of the number. The value of the characteristic depends entirely on the position of the decimal point in the number. It is always one less than the number of figures in the number to the left of the decimal point. The value is therefore diminished by one every time...
Page 130 - The logarithmic sine, tangent, etc., of an arc is the logarithm of the natural sine, tangent, etc., of the same arc, but with 10 added to the characteristic to avoid negatives. This table gives log sines, tangents, cosines, and cotangents for every minute of the quadrant. With the number of degrees at the left side of the page are to be read the minutes in the left-hand column; with the degrees on the right-hand side are to be read the minutes in the right-hand column. When the degrees appear at...
Page 130 - The number derived from a six-place logarithm is not reliable beyond the sixth figure. At the end of table XXIV. is a small table of logarithms of numbers from 1 to 100, with the characteristic prefixed, for easy reference when the given number does not exceed two digits. But the same mantissas may be found in the larger table. TABLE XXV.— The logarithmic sine, tangent, etc.
Page 133 - Find in the proper column two consecutive logarithms between which the given logarithm falls. If the title of the given function is found at the top of that column read the degrees from the top of the page ; if at the bottom read from the bottom. Find the value of (7 — Il or (q -\- Tl, as the case may require, corresponding to the given log (interpolating for the last figure if necessary).
Page 132 - These two pages may be used in the same way when the given angle lies between 88░ and 92░, or between 178░ and 180░; but if the number of degrees be found at the bottom of the page, the title of each column will be found there also; and if the number of degrees be found on the...