the logarithm of the quantity into the index of the power; (Art. 45.) that is, by repeating the logarithms as many times as there are units in the index. To involve a quantity on the scale, then, take in the compasses the linear logarithm, and double it, treble it, &c., according to the index of the proposed power. Ex. 1. Required the square of 9. Extend the compasses from 1 to 9. Twice this extent will reach to 81, the square. 2. Required the cube of 4. The extent from 1 to 4 repeated three times, will reach to 64 the cube of 4. 183. On the other hand, to perform evolution on the scale; take half, one third, &c., of the logarithm of the quantity, according to the index of the proposed root. Ex. 1. Required the square root of 49. Half the extent from 1 to 49, will reach from 1 to 7, the root. 2. Required the cube root of 27. One third the distance from 1 to 27, will extend from 1 to 3, the root. 184. The Rule of Three may be performed on the scale, in the same manner as in logarithms, by adding the two middle terms, and from the sum, subtracting the first term. (Art. 52.) But it is more convenient in practice to begin by subtracting the first term from one of the others. If four quantities are proportional, the quotient of the first divided by the second, is equal to the quotient of the third divided by the fourth. (Alg. 364.) Thus, if a b c d, then α C and b d' α C = b d. (Alg. 380.) But in logarithms, subtraction takes the place of division; so that, log.a-log.blog.c-log.d. Or, log. a-log.c-log.blog.d, Hence, 185. On the scale, the difference between the first and second terms of a proportion, is equal to the difference between the third and fourth. Or, the difference between the first and third terms, is equal to the difference between the second and fourth. The difference between the two terms is taken, by extending the compasses from one to the other. If the second term be greater than the first; the fourth must be greater than the third; if less, less. (Alg. 395.*) Therefore if the compasses * Euc. 14. 5. extend forward from left to right, that is, from a less number to a greater, from the first term to the second; they must also extend forward from the third to the fourth. But if they extend backward, from the first term to the second; they must extend the same way, from the third to the fourth. Ex. 1. In the proportion 3: 8: : 12: 32, the extent from 3 to 8, will reach from 12 to 32; Or, the extent from 3 to 12, will reach from 8 to 32. 2. If 54 yards of cloth cost 48 dollars, what will 18 yards cost? 54 48 18: 16 The extent from 54 to 48, will reach backwards from 18 to 16. 3. If 63 gallons of wine cost 81 dollars, what will 35 gallons cost? 63 81: 35: 45 The extent from 63 to 81, will reach from 35 to 45. The Line of Sines. 186. The line on Gunter's scale marked SIN. is a line of logarithmic sines, made to correspond with the line of numbers. The whole extent of the line of numbers, (Art. 179.) is from 1 to 100, whose logs. are 0.00000 and 2.00000, or from 10 to 1000, whose logs. are 1.00000 and 3.00000, or from 100 to 10000, whose logs. are 2.00000 4.00000, the difference of the indices of the two extreme logarithms being in each case 2. Now the logarithmic sine of 0° 34' 22" 41" is 8.00000 And the sine of 90° (Art. 95.) is 10.00000 Here also the difference of the indices is 2. If then the point directly beneath one extremity of the line of numbers, be marked for the sine of 0° 34′ 22′′ 41""; and the point beneath the other extremity, for the sine of 90°; the interval may furnish the intermediate sine; the divisions on it being made to correspond with the decimal part of the logarithmic sines in the tables.* To represent the sines less than 34′ 22′′ 41"", the scale must be extended on the left indefinitely. For, as the sine of an arc approaches to 0, its logarithm, which is negative, increases without limit. (Art. 15.) * The first dividing stroke in the line of Sines is generally at 0° 40', a little farther to the right than the beginning of the line of numbers. The next division is at 0° 50′; then begins the numbering of the degrees, 1, 2, 3, 4, &c., from left to right. The Line of Tangents, 187. The first 45 degrees on this line are numbered from left to right, nearly in the same manner as on the line of Sines. The logarithmic tangent of 0° 34' 22" 35" is 8.00000 And the tangent of 45°, (Art. 95.) 10.00000 The difference of the indices being 2, 45 degrees will reach to the end of the line. For those above 45° the scale ought to be continued much farther to the right. But as this would be inconvenient, the numbering of the degrees, after reaching 45, is carried back from right to left. The same dividing stroke answers for an arc and its complement, one above and the other below 45°, For, (Art. 93. Propor. 9.) tan R R cot. In logarithms, therefore, (Art. 184.) tan-R-R-cot. That is, the difference between the tangent and radius, is equal to the difference between radius and the cotangent: in other words, one is as much greater than the tangent of 45°, as the other is less. In taking, then, the tangent of an arc greater than 45°, we are to suppose the distance between 45 and the division marked with a given number of degrees, to be added to the whole line, in the same manner as if the line were continued out. In working proportions, extending the compasses back, from a less number to a greater, must be considered the same as carrying them forward in other cases. See art. 185. Trigonometrical Proportions on the Scale. 188. In working proportions in trigonometry by the scale; the extent from the first term to the middle term of the same name, will reach from the other middle term to the fourth term. (Art. 185.) In a trigonometrical proportion, two of the terms are the lengths of sides of the given triangle; and the other two are tabular sines, tangents, &c. The former are to be taken from the line of numbers; the latter, from the lines of logarithmic sines and tangents. If one of the terms is a secant, the calculation cannot be made on the scale, which has commonly no line of secants. It must be kept in mind that radius is equal to the sine of 90°, or to the tangent of 45°. (Art. 95.) Therefore, whenever radius is a term in the proportion, one foot of the compasses must be set on the end of the line of sines or of tangents. 189. The following examples are taken from the proportions which have already been solved by numerical calculation. Ex. 1. In Case I, of right angled triangles, (Art. 134. ex. 1.) R 45 sin 32° 20′ ; 24 Here the third term is a sine; the first term radius is, therefore, to be considered as the sine of 90°. Then the extent from 90° to 32° 20' on the line of sines, will reach from 45 to 24 on the line of numbers. As the compasses are set back from 90° to 32° 20'; they must also be set back from 45. (Art. 185.) 2. In the same case, if the base be made radius, (page 60.) R 38 tan 32° 20' : 24 Here, as the third term is a tangent, the first term radius is to be considered the tangent of 45°. Then the extent from 45° to 32° 20' on the line of tangents, will reach from 38 to 24 on the line of numbers. 3. If the perpendicular be made radius, (page 60.) R 24 tan 57° 40': 38 The extent from 45° to 57° 40' on the line of tangents, will reach from 24 to 38 on the line of numbers. For the tangent of 57° 40' on the scale, look for its complement 32° 20'. (Art. 187.) In this example, although the compasses extend back from 45° to 57° 40'; yet, as this is from a less number to a greater, they must extend forward on the line of numbers. (Arts. 185, 187.) 4. In art. 135, 35 R26: sin 48° The extent from 35 to 26 will reach from 90° to 48°. R 48 tan 271° : 243 4 5. In art. 136, The extent from 45° to 2710, will reach from 48 to 243. 6. In art. 150, ex. 1. Sin 74° 30': 32:: sin 56° 20′ 27. For other examples, see the several cases in Sections III. and IV. 190. Though the solutions in trigonometry may be effected by the logarithmic scale, or by geometrical construction, as well as by arithmetical computation; yet the latter method is by far the most accurate. The first is valuable principally for the expedition with which the calculations are made by it. The second is of use, in presenting the form of the triangle to the eye. But the accuracy which attends arithmetical operations, is not to be expected, in taking lines from a scale with a pair of compasses.* *See note G. |