ed at the end of the operations, in which the logarithm was used. 575 1000 The same rule applies also to decimal fractions. Thus to obtain the logarithm of 0,575, which is no other than we add to the logarithm of 575, the arithmetical complement of the logarithm of 1000. And by employing the arithmetical complements, instead of the negative logarithms of fractions, the values of the fractions, expressed in decimals, may be found without difficulty in the Table. And when we know, that the given logarithm is, or contains, one or more arithmetical complements, we know that its characteristic is too large, by as many tens as there are arithmetical complements in the logarithm. Thus, if it exceed such a number of tens, it will be easy to diminish it, and find the number, that belongs to the logarithm; which will be a whole number, or a whole number joined to a fraction. But if the characteristic be below the number of tens, of which it is supposed to contain too many, it belongs to a fraction, which may be found in the following manner. seek, according to what was explained in article (242 and following), to what number the given logarithm corresponds; and when found, we separate upon the right of it, as many tens of figures, as is the excess of tens în the characteristic. For example, if we have 8,732235, for the logarithm resulting from an operation, in which there was an arithmetical complement; we perceive, as its characteristic is less than one ten, that it belongs to a fraction. We first seek (243) to what number 8,732235, considered as the logarithm of a whole number, corresponds; and find, that it corresponds to 539802600; separating 10 figures from this number, we have 0,0539802600, for the approximate value of the fraction, to which the given logarithm corresponds. We But, as it is very seldom necessary to obtain the fractions to such a degree of precision as in the last example, we abridge the operation by diminishing the characteristic of the given logarithm, as many units as will bring it within the Table; and then taking the corresponding number, we separate from it as many figures less than was required by the preceding rule, as units were taken from the characteristic. Thus, using the last ex ample, we diminish the characteristic 5 units, and finding that the corresponding logarithm is 5398, we separate 5 figures only, and have 0,05398. In the elevation of powers, we must observe, that by multiplying (229) the logarithm by a number, which expresses the degree of the power, we shall multiply at logarithm whose characteristic is too large. Thus, for example, in raising a number to its cube, if there be an arithmetical complement in the given logarithm; that is, if the characteristic be too great by 10 units, the characteristic of the logarithm of the cube will be too great by 30 units; and so of the other powers. It will then be easy to reduce it to its just value. In the extraction of roots, to prevent mistake when there are arithmetical complements in the logarithms used, care must be taken to add to, or subtract from, the characteristic, such a number of tens as will make it too large by as many tens as there are units in the number, that expresses the degree of the root. And having, according to the common rule, divided by the number, that expresses the degree of the root, the characteristic will be too great by precisely 10 units. For example, if the cube root be required of 1; we add to the logarithm of 276, the arithmetical complement of the logarithm of 547. Logarithm of 276 276 2,440909 Arith. complt. of the log of 547 7,262013 To the characteristic of this logarithm we add 20; it then becomes 29,702922, and is too great by 3 tens. The third part of this logarithm, which is, 9,900974, is the logarithm of the cube root required, but with a characteristic too large by 10 units. Thus, according to what has been said above, we find that the cube root is 0,7961, to four places of decimals. Arithmetical complements are chiefly used in Trigonometrical calculations. U TABLE OF LOGARITHMS 59 1,770852 981.991226| 1372,136721 60 1,778151 61 1,785330 100 2,000000; 139 2,143015 20 1,3010301 29 1,462398 361,556303 37 1,568202 381,579784 1412,149219 142 2,152288 143 2,155336 144 2,158362 1452,161368 105 2,021189 751,875061114 2,056905 153 2,184691 bers. Num- Loga- Num- 178 2,250420 Loga rit hms. 189 2,276462 Num- Loga- Num- Logarithras. bers.. rithms., 156 2,193125 167 2,222716| 157 2,195900 168|2,225309|| 1792,252853|| 1902, 2,278754 1582,198657 169 2,227887 180 2,255273 191 2,281033 1592,201397 170|2,230449|| 181 2,257679|| 192 2,283301 160/2,204120 1712,232996 1822,260071|| 193 2,285557 1612,206826 1722,235528 1832,262451|| 1942,287802 162 2,209513 1732,238046 184 2,264818 195/2,290035 163 2,212188 1742,240549 1852,267172|| 1962,292256161 2,214844 1752,243038 186 2,269513 1652,217484 1762,243551|| 187 2,271842 166 2,220108|| 177 2,247973 188 2,274158 197 2,294466 1982,296665 1992,298853 200 2,301030 This Table has not been inserted so much with a view to practical use, as to illustrate the formation, and manner of using Logarithms. Since going to press this work has undergone a thorough 5, 8th line from bottom, for hundred th, read thousandth. 14, 13th line from top, for ,316, read 3160. 23, 16th line from top, for 3743, &c. read 650867379- 32, 11th line from bottom, for 54, read 54. 9th line from top, for ,33, read 0,33. 16th line from top, for 175, read 168,33. 65, 18th line from top, for $1496,22cts. r. $1946,22cts. 76, 10th line from top, for 11, read 104. 79, 2nd line from top, for, read. 36 17th line from top, for, read so 22nd from top, for, read 1539. 38433་ 5th line from top, for the answer, read 240, 256, 153. do. for 17500, read 18500 136 1 136 97, 3d line from bottom, for rds. read lin. 123, 11th line from bottom, for 17,28, read 17,93 &c, |