177. It will be observed, that the divisions and subdivisions decrease, from left to right; as in the tables of logarithms, the differences decrease. The difference between the logarithms of 10 and 100, is no greater, than the difference between the logarithms of 1 and 10. 178. The line of numbers, as it has been here explained, furnishes the logarithms of all numbers between 1 and 100. And if the indices of the logarithms be neglected, the same scale may answer for all numbers whatever. For the decimal part of the logarithm of any number is the same, as that of the number multiplied or divided by 10, 100, &c. (Art. 14.) In logarithmic calculations, the use of the indices is to determine the distance of the several figures of the natural numbers from the place of units. (Art. 11.) But in those cases in which the logarithmic line is commonly used, it will not generally be difficult to determine the local value of the figures in the result. 179. We may, therefore, consider the point under 1 at the left hand, as representing the logarithm of 1, or 10, or 100; or to, or Too, &c., for the decimal part of the logarithm of each of these is 0. But if the first 1 is reckoned 10, all the succeeding numbers must also be increased in a tenfold ratio; so as to read, on the first half of the line, 20, 30, 40, &c., and on the other half, 200, 300, &c. The whole extent of the logarithmic line, is from 1 to 100, or from 10 to 1000, or from 100 to 10000, &c. or from 0.1 to 10, or from 0.01 to 1, or from 0.001 to 0.1, &c. Different. values may, on different occasions, be assigned to the several numbers and subdivisions marked on this line. But for any one calculation, the value must remain the same. Ex. Take from the scale 365. As this number is between 10 and 1000, let the 1 at the beginning of the scale, be reckoned 10. Then, from this point to the second 3 is 300; to the 6th dividing stroke is 60; and half way from this to the next stroke is 5. 180. Multiplication, division, &c., are performed by the line of numbers, on the same principle, as by common logarithms. Thus, To multiply by this line, add the logarithms of the two factors; (Art. 37.) that is, take off, with the dividers, that length of line which represents the logarithm of one of the factors, and apply this so as to extend forward from the end of that which represents the logarithm of the other factor. The sum of the two will reach to the end of the line representing the logarithm of the product. Ex. Multiply 9 into 8. The extent from 1 to 8, added to that from 1 to 9, will be equal to the extent from 1 to 72, the product. 181. To divide by the logarithmic line, subtract the logarithm of the divisor from that of the dividend; (Art. 41.) that is, take off the logarithm of the divisor, and this extent set back from the end of the logarithm of the dividend, will reach to the logarithm of the quotient. Ex. Divide 42 by 7. The extent from 1 to 7, set back from 42, will reach to 6, the quotient. 182. Involution is performed in logarithms, by multiplying 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 dividers 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 dividers 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. 315.) But in logarithms, subtraction takes the place of division; so that, log. a-log. b-log. c-log. d. Or, log. a-log. c-log. b— log. 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 dividers 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.* Therefore, if the dividers extend forward from left to right, that is, from a less number to a greater, from the first term to the second; * Euclid, 14, 5. 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 yds. 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? 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 whose logs. are 0.00000 and 2.00000, whose logs. are 1.00000 and 3.00000, 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.* 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.) is 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 * 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.) |